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****** OUTSIDE FRONT COVER ******
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****** INSIDE FRONT COVER ******
For copies of this Booklet and of the full Review to be sent to addressesin the Americas, Australasia, or the Far East, visit
http://pdg.lbl.gov/pdgmailor write to
Particle Data GroupMS 50R6008Lawrence Berkeley National LaboratoryBerkeley, CA 94720-8166, USA
From all other areas, visithttp://www.cern.ch/library
or write toCERN Scientific Information ServiceCH-1211 Geneva 23Switzerland
To make comments or corrections, send e-mail to [email protected]. Weacknowledge all e-mail via e-mail. No reply indicates nonreceipt. Pleasetry again.
Visit our WWW site: http://pdg.lbl.gov/
The publication of the Review of Particle Physics is supported bythe Director, Office of Science, Office of High Energy and NuclearPhysics, the Division of High Energy Physics of the U.S. Department ofEnergy under Contract No. DE–AC02–05CH11231; by the U.S. NationalScience Foundation under Agreement No. PHY-0652989; by the EuropeanLaboratory for Particle Physics (CERN); by an implementing arrangementbetween the governments of Japan (MEXT: Ministry of Education,Culture, Sports, Science and Technology) and the United States (DOE)on cooperative research and development; and by the Italian NationalInstitute of Nuclear Physics (INFN).
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1
PARTICLE PHYSICS BOOKLET
Extracted from the Review of Particle Physics∗
K. Nakamura et al. (Particle Data Group), JP G 37, 075021 (2010)(next edition: July 2012)
Particle Data GroupK. Nakamura, K. Hagiwara, K. Hikasa, H. Murayama, M. Tanabashi,T. Watari, C. Amsler, M. Antonelli, D.M. Asner, H. Baer, H.R. Band,R.M. Barnett, T. Basaglia, E. Bergren, J. Beringer, G. Bernardi, W. Bertl,H. Bichsel, O. Biebel, E. Blucher, S. Blusk, R.N. Cahn, M. Carena,A. Ceccucci, D. Chakraborty, M.-C. Chen, R.S. Chivukula, G. Cowan,O. Dahl, G. D’Ambrosio, T. Damour, D. de Florian, A. de Gouvea,T. DeGrand, G. Dissertori, B. Dobrescu, M. Doser, M. Drees, D.A. Edwards,S. Eidelman, J. Erler, V.V. Ezhela, W. Fetscher, B.D. Fields, B. Foster,T.K. Gaisser, L. Garren, H.-J. Gerber, G. Gerbier, T. Gherghetta,G.F. Giudice, S. Golwala, M. Goodman, C. Grab, A.V. Gritsan,J.-F. Grivaz, D.E. Groom, M. Grunewald, A. Gurtu, T. Gutsche,H.E. Haber, C. Hagmann, K.G. Hayes, M. Heffner, B. Heltsley, J.J. Hernandez-Rey,A. Hocker, J. Holder, J. Huston, J.D. Jackson, K.F. Johnson, T. Junk,A. Karle, D. Karlen, B. Kayser, D. Kirkby, S.R. Klein, C. Kolda,R.V. Kowalewski, B. Krusche, Yu.V. Kuyanov, Y. Kwon, O. Lahav,P. Langacker, A. Liddle, Z. Ligeti, C.-J. Lin, T.M. Liss, L. Littenberg,K.S. Lugovsky, S.B. Lugovsky, J. Lys, H. Mahlke, T. Mannel, A.V. Manohar,W.J. Marciano, A.D. Martin, A. Masoni, D. Milstead, R. Miquel,K. Monig, M. Narain, P. Nason, S. Navas, P. Nevski, Y. Nir, K.A. Olive,L. Pape, C. Patrignani, J.A. Peacock, S.T. Petcov, A. Piepke, G. Punzi,A. Quadt, S. Raby, G. Raffelt, B.N. Ratcliff, P. Richardson, S. Roesler,S. Rolli, A. Romaniouk, L.J. Rosenberg, J.L. Rosner, C.T. Sachrajda,Y. Sakai, G.P. Salam, S. Sarkar, F. Sauli, O. Schneider, K. Scholberg,D. Scott, W.G. Seligman, M.H. Shaevitz, M. Silari, T. Sjostrand,J.G. Smith, G.F. Smoot, S. Spanier, H. Spieler, A. Stahl, T. Stanev,S.L. Stone, T. Sumiyoshi, M.J. Syphers, J. Terning, M. Titov, N.P. Tkachenko,N.A. Tornqvist, D. Tovey, T.G. Trippe, G. Valencia, K. van Bibber,G. Venanzoni, M.G. Vincter, P. Vogel, A. Vogt, W. Walkowiak, C.W. Walter,D.R. Ward, B.R. Webber, G. Weiglein, E.J. Weinberg, J.D. Wells,A. Wheeler, L.R. Wiencke, C.G. Wohl, L. Wolfenstein, J. Womersley,C.L. Woody, R.L. Workman, A. Yamamoto, W.-M. Yao, O.V. Zenin,J. Zhang, R.-Y. Zhu, P.A. Zyla
Technical Associates:G. Harper, V.S. Lugovsky, P. Schaffner
∗The full Review lists all the data, with references, used in obtainingthe values given in the Particle Summary Tables. It also containsmuch additional information. Some of the material that does appearin this Booklet is only an abbreviated version of what appears in thefull Review.
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2
PARTICLE PHYSICS BOOKLET TABLE OF CONTENTS
1. Physical constants (rev.) . . . . . . . . . . . . . . . 42. Astrophysical constants (rev.) . . . . . . . . . . . . . 6
Summary Tables of Particle PhysicsGauge and Higgs bosons . . . . . . . . . . . . . . . . 8Leptons . . . . . . . . . . . . . . . . . . . . . . 13Quarks . . . . . . . . . . . . . . . . . . . . . . 21Mesons . . . . . . . . . . . . . . . . . . . . . . 23Baryons∗ . . . . . . . . . . . . . . . . . . . . . 135Searches . . . . . . . . . . . . . . . . . . . . . . 161Tests of conservation laws∗ . . . . . . . . . . . . . . 165
Reviews, Tables, and Plots9. Quantum chromodynamics (new)∗ . . . . . . . . . . 169
10. Electroweak model and constraints on new physics (rev.)∗ 17211. Cabibbo-Kobayashi-Maskawa quark mixing matrix (rev.)∗ 18112. CP violation (rev.)∗ . . . . . . . . . . . . . . . . 18813. Neutrino Mass, Mixing and Oscillations (new)∗ . . . . . 19314. Quark model (rev.)∗ . . . . . . . . . . . . . . . . 20016. Structure functions (rev.)∗ . . . . . . . . . . . . . . 20319. Big-bang cosmology (rev.)∗ . . . . . . . . . . . . . 20821. The Cosmological Parameters (rev.)∗ . . . . . . . . . 21422. Dark matter (rev.)∗ . . . . . . . . . . . . . . . . . 21823. Cosmic Microwave Background (rev.)∗ . . . . . . . . . 22124. Cosmic rays (new)∗ . . . . . . . . . . . . . . . . . 22425. Accelerator physics of colliders (rev.)∗ . . . . . . . . . 22526. High-energy collider parameters (rev.)∗ . . . . . . . . 22627. Passage of particles through matter (rev.)∗ . . . . . . . 22928. Particle detectors at accelerators (rev.)∗ . . . . . . . . 24429. Particle detectors for non-accelerator physics (new)∗ . . 25630. Radioactivity and radiation protection (rev.)∗ . . . . . 26331. Commonly used radioactive sources . . . . . . . . . . 26532. Probability (rev.)∗ . . . . . . . . . . . . . . . . . 26733. Statistics (rev.)∗ . . . . . . . . . . . . . . . . . . 27136. Clebsch-Gordan coefficients, spherical harmonics,
and d functions . . . . . . . . . . . . . . . . . . 28539. Kinematics (rev.)∗ . . . . . . . . . . . . . . . . . 28740. Cross-section formulae for specific processes (rev.)∗ . . . 29641. Plots of cross sections and related quantities (rev.)∗ . . . 3016. Atomic and nuclear properties of materials ∗ . . . . . . 3024. Periodic table of the elements (rev.) . . . inside back cover
∗Abridged from the full Review of Particle Physics.
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3
The following are found only in the full Review and on the Web:http://pdg.lbl.gov
3. International System of Units (SI)5. Electronic structure of the elements7. Electromagnetic relations8. Naming scheme for hadrons
15. Grand Unified Theories17. Fragmentation functions in e+e− annihilation
and lepton-nucleon DIS (rev.)18. Experimental tests of gravitational theory (rev.)20. Big-bang nucleosynthesis (rev.)34. Monte Carlo techniques (rev.)35. Monte Carlo particle numbering scheme (rev.)37. SU(3) isoscalar factors and representation matrices38. SU(n) multiplets and Young diagrams
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4 1. Physical constants
Tab
le1.
1.R
evie
wed
2010
byP.
J.M
ohr
(NIS
T).
The
set
ofco
nsta
nts
excl
udin
gth
ela
stgr
oup
(whi
chco
me
from
the
Par
ticl
eD
ata
Gro
up)
isre
com
men
ded
byC
OD
ATA
for
inte
rnat
iona
lus
e.T
he1-
σun
cert
aint
ies
inth
ela
stdi
gits
are
give
nin
pare
nthe
ses
afte
rth
eva
lues
.Se
eth
efu
lled
itio
nof
this
Rev
iew
for
refe
renc
esan
dfu
rthe
rex
plan
atio
n.
Quan
tity
Sym
bol
,eq
uat
ion
Val
ue
Unce
rtai
nty
(ppb)
spee
dof
light
inva
cuum
c29
979
245
8m
s−1
exac
t∗P
lanc
kco
nsta
nth
6.62
606
896
(33)×1
0−34
Js
50P
lanc
kco
nsta
nt,r
educ
ed�≡
h/2π
1.05
457
162
8(53
)×10
−34
Js
50=
6.58
211
899
(16)×1
0−22
MeV
s25
elec
tron
char
gem
agni
tude
e1.
602
176
487(
40)×
10−1
9C
=4.
803
204
27(1
2)×1
0−10
esu
25,2
5co
nver
sion
cons
tant
�c
197.
326
9631
(49)
MeV
fm25
conv
ersi
onco
nsta
nt(�
c)2
0.38
937
930
4(19
)G
eV2
mba
rn50
elec
tron
mas
sm
e0.
510
998
910(
13)M
eV/c2
=9.
109
382
15(4
5)×1
0−31
kg25
,50
prot
onm
ass
mp
938.
272
013(
23)M
eV/c2
=1.
672
621
637(
83)×
10−2
7kg
25,5
0=
1.00
727
646
677
(10)
u=
1836
.152
672
47(8
0)m
e0.
10,0
.43
deut
eron
mas
sm
d18
75.6
1279
3(47
)M
eV/c2
25un
ified
atom
icm
ass
unit
(u)
(mas
s12C
atom
)/12
=(1
g)/(
NA
mol
)93
1.49
402
8(23
)M
eV/c2
=1.
660
538
782(
83)×
10−2
7kg
25,5
0
perm
ittivi
tyof
free
spac
eε 0
=1/
μ0c2
8.85
418
781
7..
.×1
0−12
Fm
−1ex
act
perm
eabi
lity
offr
eesp
ace
μ0
4π×
10−7
NA−2
=12
.566
370
614
...×1
0−7
NA−2
exac
t
fine-
stru
ctur
eco
nsta
ntα
=e2
/4πε 0
�c
7.29
735
253
76(5
0)×1
0−3
=1/
137.
035
999
679(
94)†
0.68
,0.6
8cl
assi
cale
lect
ron
radi
usr e
=e2
/4π
ε 0m
ec2
2.81
794
028
94(5
8)×1
0−15
m2.
1(e
−C
ompt
onw
avel
engt
h)/2
π− λ e
=�/m
ec
=r e
α−1
3.86
159
264
59(5
3)×1
0−13
m1.
4B
ohr
radi
us(m
nucl
eus
=∞
)a∞
=4π
ε 0�2/m
ee2
=r e
α−2
0.52
917
720
859
(36)×1
0−10
m0.
68w
avel
engt
hof
1eV
/cpa
rtic
lehc/
(1eV
)1.
239
841
875(
31)×
10−6
m25
Ryd
berg
ener
gyhcR
∞=
mee4
/2(
4πε 0
)2�2
=m
ec2
α2/2
13.6
0569
193
(34)
eV25
Tho
mso
ncr
oss
sect
ion
σT
=8π
r2 e/3
0.66
524
585
58(2
7)ba
rn4.
1
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1. Physical constants 5
Boh
rm
agne
ton
μB
=e�
/2m
e5.
788
381
7555
(79)×1
0−11
MeV
T−1
1.4
nucl
ear
mag
neto
nμ
N=
e�/2m
p3.
152
451
2326
(45)×1
0−14
MeV
T−1
1.4
elec
tron
cycl
otro
nfr
eq./
field
ωe cy
cl/B
=e/
me
1.75
882
015
0(44
)×10
11
rad
s−1
T−1
25pr
oton
cycl
otro
nfr
eq./
field
ωp cy
cl/B
=e/
mp
9.57
883
392
(24)×1
07ra
ds−
1T−1
25
grav
itat
iona
lcon
stan
t‡G
N6.
674
28(6
7)×1
0−11
m3
kg−1
s−2
1.0×
105
=6.
708
81(6
7)×1
0−39
�c
(GeV
/c2
)−2
1.0×
105
stan
dard
grav
itat
iona
lacc
el.
g N9.
806
65m
s−2
exac
t
Avo
gadr
oco
nsta
ntN
A6.
022
141
79(3
0)×1
023
mol−1
50B
oltz
man
nco
nsta
ntk
1.38
065
04(2
4)×1
0−23
JK−1
1700
=8.
617
343(
15)×
10−5
eVK−1
1700
mol
arvo
lum
e,id
ealg
asat
STP
NA
k(2
73.1
5K
)/(1
0132
5Pa)
22.4
1399
6(39
)×10
−3m
3m
ol−1
1700
Wie
ndi
spla
cem
ent
law
cons
tant
b=
λm
axT
2.89
776
85(5
1)×1
0−3
mK
1700
Stef
an-B
oltz
man
nco
nsta
ntσ
=π
2k4/60
�3c2
5.67
040
0(40
)×10
−8W
m−2
K−4
7000
Ferm
ico
uplin
gco
nsta
nt∗∗
GF
/(�
c)3
1.16
637
(1)×
10−5
GeV
−290
00
wea
k-m
ixin
gan
gle
sin2
θ(M
Z)
(MS)
0.23
116
(13)
††5.
6×
105
W±
boso
nm
ass
mW
80.3
99(2
3)G
eV/c2
2.9×
105
Z0
boso
nm
ass
mZ
91.1
876(
21)
GeV
/c2
2.3×
104
stro
ngco
uplin
gco
nsta
ntα
s(m
Z)
0.11
84(7
)5.
9×
106
π=
3.14
159
265
358
979
323
8e
=2.
718
281
828
459
045
235
γ=
0.57
721
566
490
153
286
1
1in
≡0.
0254
m
1A
≡0.
1nm
1ba
rn≡
10−2
8m
2
1G
≡10
−4T
1dy
ne≡
10−5
N
1er
g≡
10−7
J
1eV
=1.
602
176
487(
40)×
10−1
9J
1eV
/c2
=1.
782
661
758(
44)×
10−3
6kg
2.99
792
458
×10
9es
u=
1C
kT
at30
0K
=[3
8.68
168
5(68
)]−1
eV0
◦ C≡
273.
15K
1at
mos
pher
e≡
760
Tor
r≡
101
325
Pa
∗T
hem
eter
isth
ele
ngth
ofth
epa
thtr
avel
edby
light
inva
cuum
duri
nga
tim
ein
terv
alof
1/29
979
245
8of
ase
cond
.†
At
Q2
=0.
At
Q2≈
m2 W
the
valu
eis∼
1/12
8.•‡
Abs
olut
ela
bm
easu
rem
ents
ofG
Nha
vebe
enm
ade
only
onsc
ales
ofab
out
1cm
to1
m.
∗∗Se
eth
edi
scus
sion
inSe
c.10
,“E
lect
row
eak
mod
elan
dco
nstr
aint
son
new
phys
ics.
”††
The
corr
espo
ndin
gsi
n2θ
for
the
effec
tive
angl
eis
0.23
146(
12).
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6 2. Astrophysical constants2.A
ST
RO
PH
YSIC
AL
CO
NSTA
NT
SA
ND
PA
RA
MET
ER
STab
le2.
1.R
evis
edM
ay20
10by
E.B
ergr
enan
dD
.E.G
room
(LB
NL).
Fig
ures
inpa
rent
hese
sgi
ve1-
σun
cert
aint
ies
inla
stpl
ace(
s).
Thi
sta
ble
repr
esen
tsne
ithe
ra
crit
ical
revi
ewno
ran
adju
stm
ent
ofth
eco
nsta
nts,
and
isno
tin
tend
edas
apr
imar
yre
fere
nce.
See
the
full
edit
ion
ofth
isRev
iew
for
refe
renc
esan
dde
taile
dex
plan
atio
ns.
Quan
tity
Sym
bol
,eq
uat
ion
Val
ue
Ref
eren
ce,fo
otnot
e
spee
dof
light
c29
979
245
8m
s−1
exac
t[4]
New
toni
angr
avitat
iona
lcon
stan
tG
N6.
674
3(7)
×10
−11
m3kg
−1s−
2[1
]P
lanc
km
ass
√
�c/
GN
1.22
089
(6)×
1019
GeV
/c2
[1]
=2.
176
44(1
1)×
10−8
kgP
lanc
kle
ngth
√
�G
N/c3
1.61
625
(8)×
10−3
5m
[1]
stan
dard
grav
itat
iona
lacc
eler
atio
ng N
9.80
665
ms−
2≈
π2
exac
t[1]
jans
ky(fl
uxde
nsity)
Jy10
−26
Wm
−2H
z−1
defin
itio
n
trop
ical
year
(equ
inox
toeq
uino
x)(2
011)
yr31
556
925.
2s≈
π×
107
s[5
]si
dere
alye
ar(fi
xed
star
tofix
edst
ar)
(201
1)31
558
149.
8s≈
π×
107
s[5
]m
ean
side
real
day
(201
1)(t
ime
betw
een
vern
aleq
uino
xtr
ansi
ts)
23h
56m
04.s 0
9053
[5]
astr
onom
ical
unit
au,A
149
597
870
700(
3)m
[6]
pars
ec(1
au/1
arc
sec)
pc3.
085
677
6×
1016
m=
3.26
2..
.ly
[7]
light
year
(dep
reca
ted
unit
)ly
0.30
66
...pc
=0.
946
053
...×
1016
mSc
hwar
zsch
ildra
dius
ofth
eSu
n2G
NM
�/c2
2.95
325
007
70(
2)km
[8]
Sola
rm
ass
M�
1.98
84(
2)×
1030
kg[9
]So
lar
equa
tori
alra
dius
R�
6.95
51(4
)×10
8m
[10]
Sola
rlu
min
osity
L�
3.84
27(
14)
×10
26
W[1
1]Sc
hwar
zsch
ildra
dius
ofth
eE
arth
2GN
M⊕
/c2
8.87
005
594
(2)m
m[1
2]E
arth
mas
sM
⊕5.
972
2(6)
×10
24
kg[1
3]E
arth
mea
neq
uato
rial
radi
usR⊕
6.37
813
7×
106
m[5
]
lum
inos
ity
conv
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:55
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2. Astrophysical constants 7pr
esen
tda
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MB
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pera
ture
T0
2.72
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1)km
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3)◦ )
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[25]
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:55
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SUMMARY TABLES OF PARTICLE PROPERTIES
Extracted from the Particle Listings of theReview of Particle Physics
K. Nakamura et al. (Particle Data Group), JP G 37, 075021 (2010)Available at http://pdg.lbl.gov
c©Regents of the University of California
(Approximate closing date for data: January 15, 2010)
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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< �.7 × ��−8 *&9��5 �477
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
�������� ����� ����� � ��
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π� π� π� π−π−π− < 6.� × ��−6 *&9��5 �7��
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&) ������− < �.8 × ��−3 *&9��5 �686
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:16
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����� �� ���������� ��� ������������
TESTS OF CONSERVATION LAWS
Updated May 2010 by L. Wolfenstein (Carnegie-Mellon University),T.G. Trippe (LBNL), and C.-J. Lin (LBNL).
In keeping with the current interest in tests of conservation laws,
we collect together a Table of experimental limits on all weak and
electromagnetic decays, mass differences, and moments, and on a
few reactions, whose observation would violate conservation laws.
The Table is given only in the full Review of Particle Physics, not in
the Particle Physics Booklet. For the benefit of Booklet readers, we
include the best limits from the Table in the following text. Limits
in this text are for CL=90% unless otherwise specified. The Table is
in two parts: “Discrete Space-Time Symmetries,” i.e., C, P , T , CP ,
and CPT ; and “Number Conservation Laws,” i.e., lepton, baryon,
hadronic flavor, and charge conservation. The references for these
data can be found in the the Particle Listings in the Review. A
discussion of these tests follows.
CPT INVARIANCE
General principles of relativistic field theory require invariance un-
der the combined transformation CPT . The simplest tests of CPT
invariance are the equality of the masses and lifetimes of a particle
and its antiparticle. The best test comes from the limit on the mass
difference between K0 and K0. Any such difference contributes to
the CP -violating parameter ε. Assuming CPT invariance, φε, the
phase of ε should be very close to 44◦. (See the review “CP Viola-
tion in KL decay” in this edition.) In contrast, if the entire source
of CP violation in K0 decays were a K0 − K0
mass difference, φε
would be 44◦ + 90◦.Assuming that there is no other source of CPT violation than this
mass difference, it is possible to deduce that[1]
mK
0 − mK0 ≈2(mK0
L− mK0
S) |η| ( 2
3φ+− + 1
3φ00 − φSW)
sin φSW,
where φSW = (43.51 ± 0.05)◦, the superweak angle. Using our
best values of the CP -violation parameters, we get |(mK
0 −mK0)/mK0 | ≤ 0.8 × 10−18 at CL=90%. Limits can also be placed
on specific CPT -violating decay amplitudes. Given the small value
of (1 − |η00/η+−|), the value of φ00 − φ+− provides a measure of
CPT violation in K0L → 2π decay. Results from CERN [1] and
Fermilab [2] indicate no CPT -violating effect.
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������������ ����� �� ���������� ���
CP AND T INVARIANCE
Given CPT invariance, CP violation and T violation are equiv-
alent. The original evidence for CP violation came from the
measurement of |η+−| = |A(K0L → π+π−)/A(K0
S → π+π−)| =
(2.232±0.011)×10−3. This could be explained in terms of K0–K0
mixing, which also leads to the asymmetry [Γ(K0L → π−e+ν) −
Γ(K0L → π+e−ν)]/[sum] = (0.334± 0.007)%. Evidence for CP vio-
lation in the kaon decay amplitude comes from the measurement of
(1 − |η00/η+−|)/3 = Re(ε′/ε) = (1.65 ± 0.26) × 10−3. In the Stan-
dard Model much larger CP -violating effects are expected. The
first of these, which is associated with B–B mixing, is the param-
eter sin(2β) now measured quite accurately to be 0.671 ± 0.023.
A number of other CP -violating observables are being measured
in B decays; direct evidence for CP violation in the B decay am-
plitude comes from the asymmetry [Γ(B0 → K−π+) − Γ(B0 →
K+π−)]/[sum] = −0.098 ± 0.013. Direct tests of T violation are
much more difficult; a measurement by CPLEAR of the difference
between the oscillation probabilities of K0 to K0 and K0 to K0 is
related to T violation [3]. Other searches for CP or T violation
involve effects that are expected to be unobservable in the Standard
Model. The most sensitive are probably the searches for an electric
dipole moment of the neutron, measured to be < 2.9× 10−26 e cm,
and the electron (0.07 ± 0.07) × 10−26 e cm. A nonzero value re-
quires both P and T violation.
CONSERVATION OF LEPTON NUMBERS
Present experimental evidence and the standard electroweak theory
are consistent with the absolute conservation of three separate lep-
ton numbers: electron number Le, muon number Lμ, and tau num-
ber Lτ , except for the effect of neutrino mixing associated with
neutrino masses. Searches for violations are of the following types:
a) ΔL = 2 for one type of charged lepton. The best limit
comes from the search for neutrinoless double beta decay (Z, A)
→ (Z + 2, A)+e−+e−. The best laboratory limit is t1/2 > 1.9×1025
yr (CL=90%) for 76Ge.
b) Conversion of one charged-lepton type to another. For
purely leptonic processes, the best limits are on μ → eγ and
μ → 3e, measured as Γ(μ → eγ)/Γ(μ →all) < 1.2 × 10−11 and
Γ(μ → 3e)/Γ(μ → all) < 1.0 × 10−12. For semileptonic pro-
cesses, the best limit comes from the coherent conversion pro-
cess in a muonic atom, μ−+ (Z, A) → e− + (Z, A), measured as
Γ(μ−Ti → e−Ti)/Γ(μ−Ti → all) < 4.3 × 10−12. Of special interest
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����� �� ���������� ��� ������������
is the case in which the hadronic flavor also changes, as in KL → eμ
and K+ → π+e−μ+, measured as Γ(KL → eμ)/Γ(KL → all) <
4.7 × 10−12 and Γ(K+ → π+e−μ+)/Γ(K+ → all) < 1.3 × 10−11.
Limits on the conversion of τ into e or μ are found in τ decay
and are much less stringent than those for μ → e conversion, e.g.,
Γ(τ → μγ)/Γ(τ → all) < 4.4 × 10−8 and Γ(τ → eγ)/Γ(τ → all) <
3.3 × 10−8.
c) Conversion of one type of charged lepton into an-
other type of charged antilepton. The case most studied is
μ−+(Z, A) → e+ + (Z − 2, A), the strongest limit being Γ(μ−Ti →e+Ca)/Γ(μ−Ti → all) < 3.6 × 10−11.
d) Neutrino oscillations. It is expected even in the standard
electroweak theory that the lepton numbers are not separately con-
served, as a consequence of lepton mixing analogous to Cabibbo-
Kobayashi-Maskawa quark mixing. However, if the only source of
lepton-number violation is the mixing of low-mass neutrinos then
processes such as μ → eγ are expected to have extremely small
unobservable probabilities. For small neutrino masses, the lepton-
number violation would be observed first in neutrino oscillations,
which have been the subject of extensive experimental searches.
Strong evidence for neutrino mixing has come from atmospheric
and solar neutrinos. The SNO experiment has detected the total
flux of neutrinos from the sun measured via neutral current interac-
tions and found it greater than the flux of νe. This confirms previous
indications of a deficit of νe. Furthermore, evidence for such oscilla-
tions for reactor ν has been found by the KAMLAND detector. A
global analysis combining all solar neutrino data (SNO, Borexino,
Super-Kamiokande, Chlorine, Gallium) and the KamLAND data
yields Δ(m2) = (7.59 ± 0.20) × 10−5 eV2[4].
Underground detectors observing neutrinos produced by cosmic
rays in the atmosphere have found a factor of 2 deficiency of up-
ward going νμ compared to downward. This provides compelling
evidence for νμ disappearance, for which the most probable ex-
planation is νμ → ντ oscillations with nearly maximal mixing.
This mixing space can also be explored by accelerator-based long-
baseline experiments. The most recent result from MINOS gives
Δ(m2) = (2.43 ± 0.13) × 10−3 eV2[5].
CONSERVATION OF HADRONIC FLAVORS
In strong and electromagnetic interactions, hadronic flavor
is conserved, i.e. the conversion of a quark of one flavor
(d, u, s, c, b, t) into a quark of another flavor is forbidden. In the
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������������ ����� �� ���������� ���
Standard Model, the weak interactions violate these conservation
laws in a manner described by the Cabibbo-Kobayashi-Maskawa
mixing (see the section “CKM Mixing Matrix”). The way in which
these conservation laws are violated is tested as follows:(a) ΔS = ΔQ rule. In the strangeness-changing semileptonic decay
of strange particles, the strangeness change equals the change in
charge of the hadrons. Tests come from limits on decay rates such
as Γ(Σ+ → ne+ν)/Γ(Σ+ → all) < 5 × 10−6, and from a detailed
analysis of KL → πeν, which yields the parameter x, measured to
be (Re x, Im x) = (−0.002±0.006, 0.0012±0.0021). Corresponding
rules are ΔC = ΔQ and ΔB = ΔQ.
(b) Change of flavor by two units. In the Standard Model this
occurs only in second-order weak interactions. The classic example
is ΔS = 2 via K0 − K0
mixing, which is directly measured by
m(KL) − m(KS) = (0.5292 ± 0.0009) × 1010 hs−1. The ΔB =
2 transitions in the B0 and B0s systems via mixing are also well
established. The measured mass differences between the eigenstates
are (mB0H− mB0
L) = (0.507±0.005)×1012 hs−1 and (mB0
sH− mB0
sL)
= (17.77 ± 0.12) × 1012 hs−1. There is now strong evidence of
ΔC = 2 transition in the charm sector with the mass difference
mD0H− mD0
L= (2.39+0.59
−0.63) × 1010 hs−1. All results are consistent
with the second-order calculations in the Standard Model.
(c) Flavor-changing neutral currents. In the Standard Model
the neutral-current interactions do not change flavor. The low
rate Γ(KL → μ+μ−)/Γ(KL → all) = (6.84 ± 0.11) × 10−9 puts
limits on such interactions; the nonzero value for this rate is at-
tributed to a combination of the weak and electromagnetic in-
teractions. The best test should come from K+ → π+νν,
which occurs in the Standard Model only as a second-order weak
process with a branching fraction of (0.4 to 1.2)×10−10. Com-
bining results from BNL-E787 and BNL-E949 experiments yield
Γ(K+ → π+νν)/Γ(K+ → all) = (1.7 ± 1.1) × 10−10[6]. Limits
for charm-changing or bottom-changing neutral currents are much
less stringent: Γ(D0 → μ+μ−)/Γ(D0 → all) < 1.3 × 10−6 and
Γ(B0 → μ+μ−)/Γ(B0 → all) < 1.5 × 10−8. One cannot isolate
flavor-changing neutral current (FCNC) effects in non leptonic de-
cays. For example, the FCNC transition s → d+(u+u) is equivalent
to the charged-current transition s → u + (u + d). Tests for FCNC
are therefore limited to hadron decays into lepton pairs. Such de-
cays are expected only in second-order in the electroweak coupling
in the Standard Model.
See the full Review of Particle Physics for references and Summary Tables.
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9. Quantum chromodynamics 169
9. QUANTUM CHROMODYNAMICS
Written October 2009 by G. Dissertori (ETH, Zurich) and G.P. Salam(LPTHE, Paris).
9.1. Basics
Quantum Chromodynamics (QCD), the gauge field theory thatdescribes the strong interactions of colored quarks and gluons, is the SU(3)component of the SU(3)×SU(2)×U(1) Standard Model of Particle Physics.
The Lagrangian of QCD is given by
L =∑
q
ψq,a(iγμ∂μδab − gsγμtCabAC
μ − mqδab)ψq,b −14FA
μνFA μν , (9.1)
where repeated indices are summed over. The γμ are the Dirac γ-matrices.The ψq,a are quark-field spinors for a quark of flavor q and mass mq, witha color-index a that runs from a = 1 to Nc = 3, i.e. quarks come in three“colors.” Quarks are said to be in the fundamental representation of theSU(3) color group.
The ACμ correspond to the gluon fields, with C running from 1 to
N2c − 1 = 8, i.e. there are eight kinds of gluon. Gluons are said to be in
the adjoint representation of the SU(3) color group. The tCab correspondto eight 3 × 3 matrices and are the generators of the SU(3) group. Theyencode the fact that a gluon’s interaction with a quark rotates the quark’scolor in SU(3) space. The quantity gs is the QCD coupling constant.Finally, the field tensor FA
μν is given by
FAμν = ∂μAA
ν − ∂νAAμ − gs fABCAB
μ ACν [tA, tB] = ifABCtC , (9.2)
where the fABC are the structure constants of the SU(3) group.Neither quarks nor gluons are observed as free particles. Hadrons are
color-singlet (i.e. color-neutral) combinations of quarks, anti-quarks, andgluons.
Ab-initio predictive methods for QCD include lattice gauge theoryand perturbative expansions in the coupling. The Feynman rules of QCDinvolve a quark-antiquark-gluon (qqg) vertex, a 3-gluon vertex (bothproportional to gs), and a 4-gluon vertex (proportional to g2
s).Useful color-algebra relations include: tAabt
Abc = CF δac, where CF ≡
(N2c −1)/(2Nc) = 4/3 is the color-factor (“Casimir”) associated with gluon
emission from a quark; fACDfBCD = CAδAB where CA ≡ Nc = 3 is thecolor-factor associated with gluon emission from a gluon; tAabt
Bab = TRδAB,
where TR = 1/2 is the color-factor for a gluon to split to a qq pair.
The fundamental parameters of QCD are the coupling gs (or αs =g2s
4π)
and the quark masses mq.
9.1.1. Running coupling : In the framework of perturbative QCD(pQCD), predictions for observables are expressed in terms of the renor-malized coupling αs(μ2
R), a function of an (unphysical) renormalizationscale μR. When one takes μR close to the scale of the momentum transferQ in a given process, then αs(μ2
R � Q2) is indicative of the effectivestrength of the strong interaction in that process.
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170 9. Quantum chromodynamics
The coupling satisfies the following renormalization group equation(RGE):
μ2R
dαs
dμ2R
= β(αs) = −(b0α2s + b1α
3s + b2α
4s + · · ·) (9.3)
where b0 = (11CA − 4nfTR)/(12π) = (33 − 2nf )/(12π) is the 1-loopbeta-function coefficient, and the 2, 3 and 4-loop coefficients areb1 = (17C2
A − nfTR(10CA + 6CF ))/(24π2) = (153 − 19nf )/(24π2),b2 = (2857 − 5033
9 nf + 32527 n2
f )/(128π3), and b3 = ((1497536 + 3564ζ3) −
(1078361162 + 6508
27 ζ3)nf + (50065162 + 6472
81 ζ3)n2f + 1093
729 n3f )/(4π)4 [9,10], the last
two specifically in the MS scheme (here ζ3 � 1.2020569). The minus signin Eq. (9.3) is the origin of asymptotic freedom, i.e. the fact that thestrong coupling becomes weak for processes involving large momentumtransfers (“hard processes”), αs ∼ 0.1 for momentum transfers in the100 GeV –TeV range.
The β-function coefficients, the bi, are given for the coupling of aneffective theory in which nf of the quark flavors are considered light(mq � μR), and in which the remaining heavier quark flavors decouplefrom the theory. One may relate the coupling for the theory with nf + 1light flavors to that with nf flavors through an equation of the form
α(nf +1)s (μ2
R) = α(nf )s (μ2
R)
(
1 +∞∑
n=1
n∑
�=0
cn� [α(nf )s (μ2
R)]n ln� μ2R
m2h
)
, (9.4)
where mh is the mass of the (nf +1)th flavor, and the first few cn�
coefficients are c11 = 16π , c10 = 0, c22 = c211, c21 = 19
24π2 , and c20 = − 1172π2
when mh is the MS mass at scale mh (c20 = 724π2 when mh is the pole
mass — see the review on “Quark Masses”). Terms up to c4� are tobe found in Refs. 11, 12. Numerically, when one chooses μR = mh, thematching is a small effect, owing to the zero value for the c10 coefficient.
Working in an energy range where the number of flavors is constant, asimple exact analytic solution exists for Eq. (9.3) only if one neglects allbut the b0 term, giving αs(μ2
R) = (b0 ln(μ2R/Λ2))−1. Here Λ is a constant
of integration, which corresponds to the scale where the perturbatively-defined coupling would diverge, i.e. it is the non-perturbative scale ofQCD. A convenient approximate analytic solution to the RGE thatincludes also the b1, b2, and b3 terms is given by (e.g. Ref. 13),
αs(μ2R) � 1
b0t
(
1 − b1
b20
ln t
t+
b21(ln2 t − ln t − 1) + b0b2
b40t2
−b31(ln
3 t − 52
ln2 t − 2 ln t +12) + 3b0b1b2 ln t − 1
2b20b3
b60t3
)
, t ≡ lnμ2
R
Λ2,
(9.5)again parametrized in terms of a constant Λ. Note that Eq. (9.5) is one ofseveral possible approximate 4-loop solutions for αs(μ2
R), and that a valuefor Λ only defines αs(μ2
R) once one knows which particular approximationis being used. An alternative to the use of formulas such as Eq. (9.5)is to solve the RGE exactly, numerically (including the discontinuities,Eq. (9.4), at flavor thresholds). In such cases the quantity Λ is not defined
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9. Quantum chromodynamics 171
at all. For these reasons, in determinations of the coupling, it has becomestandard practice to quote the value of αs at a given scale (typically MZ)rather than to quote a value for Λ.
The value of the coupling, as well as the exact forms of the b2, c10 (andhigher order) coefficients, depend on the renormalization scheme in whichthe coupling is defined, i.e. the convention used to subtract infinities inthe context of renormalization. The coefficients given above hold for acoupling defined in the modified minimal subtraction (MS) scheme [14],by far the most widely used scheme.
9.3.4. Measurements of the strong coupling constant : For thisreview it was decided to quote a recent analysis by Bethke [172], whichincorporates results with recently improved theoretical predictions and/orexperimental precision. The central value is determined as the weightedaverage of the individual measurements. For the error an overall, a-prioriunknown, correlation coefficient is introduced and determined by requiringthat the total χ2 of the combination equals the number of degrees offreedom. The world average quoted in Ref. 172 is
αs(M2Z) = 0.1184± 0.0007 .
It is worth noting that a cross check performed in Ref. 172, consisting inexcluding each of the single measurements from the combination, resultedin variations of the central value well below the quoted uncertainty,and in a maximal increase of the combined error up to 0.0012. Mostnotably, excluding the most precise determination from lattice QCD givesonly a marginally different average value. Nevertheless, there remains anapparent and long-standing systematic difference between the results fromstructure functions and other determinations of similar accuracy. This isevidenced in Fig. 9.2 (left), where the various inputs to this combination,evolved to the Z mass scale, are shown. Fig. 9.2 (right) provides strongestevidence for the correct prediction by QCD of the scale dependence of thestrong coupling.
0.11 0.12 0.13α (Μ )s Z
Quarkonia (lattice)
DIS F2 (N3LO)
τ-decays (N3LO)
DIS jets (NLO)
e+e– jets & shps (NNLO)
electroweak fits (N3LO)
e+e– jets & shapes (NNLO)
Υ decays (NLO)
QCD α (Μ ) = 0.1184 ± 0.0007s Z
0.1
0.2
0.3
0.4
0.5
αs (Q)
1 10 100Q [GeV]
Heavy Quarkoniae+e– AnnihilationDeep Inelastic Scattering
July 2009
Figure 9.1: Left: Summary of measurements of αs(M2Z), used as
input for the world average value; Right: Summary of measurementsof αs as a function of the respective energy scale Q. Both plots aretaken from Ref. 172.
Further discussion and all references may be found in the full Review.
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172 10. Electroweak model and constraints on new physics
10. ELECTROWEAK MODEL AND
CONSTRAINTS ON NEW PHYSICS
Revised November 2009 by J. Erler (U. Mexico) and P. Langacker(Institute for Advanced Study).
10.1. Introduction
The standard electroweak model (SM) is based on the gaugegroup [1] SU(2) × U(1), with gauge bosons W i
μ, i = 1, 2, 3, andBμ for the SU(2) and U(1) factors, respectively, and the cor-responding gauge coupling constants g and g′. The left-handedfermion fields of the ith fermion family transform as doublets
Ψi =(
νi
�−i
)
and(
uid′i
)
under SU(2), where d′i ≡ ∑
j Vij dj , and V
is the Cabibbo-Kobayashi-Maskawa mixing matrix. (Constraints on Vand tests of universality are discussed in Ref. 2 and in the Section on“The CKM Quark-Mixing Matrix”. The extension of the formalism toallow an analogous leptonic mixing matrix is discussed in the Section on“Neutrino Mass, Mixing, and Flavor Change”.) The right-handed fieldsare SU(2) singlets. In the minimal model there are three fermion families
and a single complex Higgs doublet φ ≡(
φ+
φ0
)
which is introduced formass generation.
After spontaneous symmetry breaking the Lagrangian for the fermionfields, ψi, is
LF =∑
i
ψi
(
i �∂ − mi − gmiH
2MW
)
ψi
− g
2√
2
∑
i
Ψi γμ (1 − γ5)(T+ W+μ + T− W−
μ )Ψi
− e∑
i
qi ψi γμ ψi Aμ
− g
2 cos θW
∑
i
ψi γμ(giV − gi
Aγ5)ψi Zμ . (10.1)
θW ≡ tan−1(g′/g) is the weak angle; e = g sin θW is the positron electriccharge; and A ≡ B cos θW + W 3 sin θW is the (massless) photon field.W± ≡ (W 1 ∓ iW 2)/
√2 and Z ≡ −B sin θW + W 3 cos θW are the massive
charged and neutral weak boson fields, respectively. T+ and T− are theweak isospin raising and lowering operators. The vector and axial-vectorcouplings are
giV ≡t3L(i) − 2qi sin2 θW , (10.2a)
giA ≡t3L(i), (10.2b)
where t3L(i) is the weak isospin of fermion i (+1/2 for ui and νi; −1/2for di and ei) and qi is the charge of ψi in units of e.
The second term in LF represents the charged-current weak interac-tion [3,4]. For example, the coupling of a W to an electron and a neutrinois
− e
2√
2 sin θW
[
W−μ e γμ(1 − γ5)ν + W+
μ ν γμ (1 − γ5)e]
. (10.3)
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10. Electroweak model and constraints on new physics 173
For momenta small compared to MW , this term gives rise to the effectivefour-fermion interaction with the Fermi constant given (at tree level, i.e.,lowest order in perturbation theory) by GF /
√2 = g2/8M2
W . CP violationis incorporated in the SM by a single observable phase in Vij . The thirdterm in LF describes electromagnetic interactions (QED), and the last isthe weak neutral-current interaction.
In Eq. (10.1), mi is the mass of the ith fermion ψi. In the presenceof right-handed neutrinos, Eq. (10.1) gives rise also to Dirac neutrinomasses. The possibility of Majorana masses is discussed in the Section on“Neutrino Mass, Mixing, and Flavor Change”.
H is the physical neutral Higgs scalar which is the only remaining partof φ after spontaneous symmetry breaking. The Yukawa coupling of Hto ψi, which is flavor diagonal in the minimal model, is gmi/2MW . Innon-minimal models there are additional charged and neutral scalar Higgsparticles [10].
10.2. Renormalization and radiative corrections
The SM has three parameters (not counting the Higgs boson mass,MH , and the fermion masses and mixings). A particularly useful setcontains the Z-boson mass, MZ = 91.1876 ± 0.0021 GeV, which hasbeen determined from the Z lineshape scan at LEP 1; the fine structureconstant, α = 1/137.035999084(51), which is currently best determinedfrom the e± anomalous magnetic moment; and the Fermi constant, GF =1.166364(5) × 10−5 GeV−2, which is derived from the muon lifetimeformula. See the full Review for more details.
With these inputs, sin2 θW and the W boson mass, MW , can becalculated when values for mt and MH are given; conversely (as is done atpresent), MH can be constrained by sin2 θW and MW . The value of sin2 θWis extracted from Z pole observables and neutral-current processes [11,60],and depends on the renormalization prescription. There are a number ofpopular schemes [61–68] leading to values which differ by small factorsdepending on mt and MH , including the MS definition s 2
Z and the on-shelldefinition s2
W ≡ 1 − M2W /M2
Z .
Experiments are at such level of precision that complete O(α) radiativecorrections must be applied. These are discussed in the full edition of thisReview. A variety of related cross-section and asymmetry formulae arealso discussed there.
10.3.1. W and Z decays :
The partial decay width for gauge bosons to decay into masslessfermions f1f2 (the numerical values include the small electroweakradiative corrections and final state mass effects) is
Γ(W+ → e+νe) =GF M3
W
6√
2π≈ 226.31± 0.07 MeV , (10.33a)
Γ(W+ → uidj) =CGF M3
W
6√
2π|Vij |2 ≈ 706.18± 0.22 MeV |Vij |2(10.33b)
Γ(Z → ψiψi) =CGF M3
Z
6√
2π
[
gi2V + gi2
A
]
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174 10. Electroweak model and constraints on new physics
≈
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
167.21± 0.02 MeV (νν),83.99 ± 0.01 MeV (e+e−),
300.20± 0.06 MeV (uu),382.98± 0.06 MeV (dd),375.94∓ 0.04 MeV (bb).
(10.33c)
For leptons C = 1, while for quarks
C = 3[
1 +αs(MV )
π+ 1.409
α2s
π2 − 12.77α3
s
π3 − 80.0α4
s
π4
]
, (10.34)
where the 3 is due to color and the factor in brackets represents theuniversal part of the QCD corrections [164] for massless quarks [165].The O(α4
s) contribution in Eq. (10.34) is new [166]. The Z → f f widthscontain a number of additional corrections: universal (non-singlet) topquark mass contributions [167]; fermion mass effects and further QCDcorrections proportional to m2
q(M2Z) [168] which are different for vector
and axial-vector partial widths; and singlet contributions starting fromtwo-loop order which are large, strongly top quark mass dependent, familyuniversal, and flavor non-universal [169]. The QED factor 1 + 3αq2
f/4π,as well as two-loop order ααs and α2 self-energy corrections [170] are alsoincluded. Working in the on-shell scheme, i.e., expressing the widths interms of GF M3
W,Z , incorporates the largest radiative corrections from therunning QED coupling [61,171]. Electroweak corrections to the Z widthsare then incorporated by replacing g i2
V,A by the effective couplings g i2V,A
defined in Eq. (10.31) of the full Review. Hence, in the on-shell scheme theZ widths are proportional to 1 + ρt, where ρt = 3GF m2
t /8√
2π2. The MS
normalization accounts also for the leading electroweak corrections [66].There is additional (negative) quadratic mt dependence in the Z → bbvertex corrections [172] which causes Γ(bb) to decrease with mt. Thedominant effect is to multiply Γ(bb) by the vertex correction 1 + δρbb,
where δρbb ∼ 10−2(− 12
m2t
M2Z
+ 15). In practice, the corrections are included
in g bV,A, as discussed before.
For 3 fermion families the total widths are predicted to be
ΓZ ≈ 2.4957±0.0003 GeV , ΓW ≈ 2.0910±0.0007 GeV . (10.35)
We have assumed αs(MZ) = 0.1200. An uncertainty in αs of ±0.0016introduces an additional uncertainty of 0.05% in the hadronic widths,corresponding to ±0.8 MeV in ΓZ . These predictions are to be comparedwith the experimental results ΓZ = 2.4952 ± 0.0023 GeV [11] andΓW = 2.085 ± 0.042 GeV [173] (see the Gauge & Higgs Boson ParticleListings for more details).
10.4. Precision flavor physics
In addition to cross-sections, asymmetries, parity violation, W andZ decays, there is a large number of experiments and observables testingthe flavor structure of the SM. These are addressed elsewhere in thisReview, and generally not included in this Section. However, we identifythree precision observables with sensitivity to similar types of new physicsas the other processes discussed here. The branching fraction of the flavorchanging transition b → sγ is of comparatively low precision, but since
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10. Electroweak model and constraints on new physics 175
it is a loop-level process (in the SM) its sensitivity to new physics (andSM parameters, such as heavy quark masses) is enhanced. The τ -leptonlifetime and leptonic branching ratios are primarily sensitive to αs andnot affected significantly by many types of new physics. However, havingan independent and reliable low energy measurement of αs in a globalanalysis allows the comparison with the Z lineshape determination of αs
which shifts easily in the presence of new physics contributions. By far themost precise observable discussed here is the anomalous magnetic momentof the muon (the electron magnetic moment is measured to even greaterprecision, but its new physics sensitivity is suppressed by an additionalfactor of m2
e/m2μ). Its combined experimental and theoretical uncertainty
is comparable to typical new physics contributions. See the full Review formore details.
10.5. Experimental results
The values for mt [6], MW [160,207], neutrino scattering [98,114–116],the weak charges of the electron [125], cesium [132,133] and thallium [134],the b → sγ observable [174–175], the muon anomalous magneticmoment [189], and the τ lifetime are listed in Table 10.5. Likewise, theprincipal Z pole observables can be found in Table 10.6 where the LEP 1averages of the ALEPH, DELPHI, L3, and OPAL results include commonsystematic errors and correlations [11]. The heavy flavor results of LEP 1and SLD are based on common inputs and correlated, as well [11]. Notethat the values of Γ( + −), Γ(had), and Γ(inv) are not independent ofΓZ , the R�, and σhad and that the SM errors in those latter are largelydominated by the uncertainty in αs. Also shown in both Tables are theSM predictions for the values of MZ , MH , αs(MZ), Δα
(3)had and the heavy
quark masses shown in Table 10.4 in the full Review. The predictionsresult from a global least-square (χ2) fit to all data using the minimizationpackage MINUIT [208] and the electroweak library GAPP [90]. Inmost cases, we treat all input errors (the uncertainties of the values)as Gaussian. The reason is not that we assume that theoretical andsystematic errors are intrinsically bell-shaped (which they are not) butbecause in most cases the input errors are combinations of many different(including statistical) error sources, which should yield approximatelyGaussian combined errors by the large number theorem. Thus, it suffices ifeither the statistical components dominate or there are many componentsof similar size. An exception is the theory dominated error on the τlifetime, which we recalculate in each χ2-function call since it dependsitself on αs. Sizes and shapes of the output errors (the uncertainties ofthe predictions and the SM fit parameters) are fully determined by the fit,and 1σ errors are defined to correspond to Δχ2 = χ2 − χ2
min = 1, and donot necessarily correspond to the 68.3% probability range or the 39.3%probability contour (for 2 parameters).
The agreement is generally very good. Despite the few discrepanciesdiscussed in the following, the fit describes well the data with aχ2/d.o.f. = 43.0/44. Only the final result for gμ − 2 from BNL and A
(0,b)FB
from LEP 1 are currently showing large (2.5 σ and 2.7 σ) deviations.In addition, A0
LR (SLD) from hadronic final states differs by 1.8 σ. TheSM prediction of σhad (LEP 1) moved closer to the measurement valuewhich is slightly higher. Rb, whose measured value deviated in the pastby as much as 3.7 σ from the SM prediction, is now in agreement, anda 2 σ discrepancy in QW (Cs) has also been resolved. g2
L from NuTeV is
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176 10. Electroweak model and constraints on new physics
Table 10.3: Principal non-Z pole observables, compared withthe SM best fit predictions. The first MW value is from theTevatron [207] and the second one from LEP 2 [160]. The valuesof MW and mt differ from those in the Particle Listings when theyinclude recent preliminary results. g2
L, which has been adjustedas discussed in Sec. 10.3, and g2
R are from NuTeV [98] and havea very small (−1.7%) residual anti-correlation. e-DIS [123] andthe older ν-DIS constraints from CDHS [92], CHARM [93], andCCFR [94] are included, as well, but not shown in the Table. Theworld averages for gνe
V,A are dominated by the CHARM II [116]results, gνe
V = −0.035± 0.017 and gνeA = −0.503 ± 0.017. The errors
are the total (experimental plus theoretical) uncertainties. The ττ
value is the τ lifetime world average computed by combining thedirect measurements with values derived from the leptonic branchingratios [5]; in this case, the theory uncertainty is included in theSM prediction. In all other SM predictions, the uncertainty is fromMZ , MH , mt, mb, mc, α(MZ), and αs, and their correlations havebeen accounted for. The column denoted Pull gives the standarddeviations for the principal fit with MH free, while the columndenoted Dev. (Deviation) is for MH = 117 GeV fixed.
Quantity Value Standard Model Pull Dev.
mt [GeV] 173.1± 1.3 173.2 ± 1.3 −0.1 −0.5MW [GeV] 80.420± 0.031 80.384± 0.014 1.2 1.5
80.376± 0.033 −0.2 0.1g2L 0.3027± 0.0018 0.30399± 0.00017 −0.7 −0.6
g2R 0.0308± 0.0011 0.03001± 0.00002 0.7 0.7
gνeV −0.040± 0.015 −0.0398± 0.0003 0.0 0.0
gνeA −0.507± 0.014 −0.5064± 0.0001 0.0 0.0
QW (e) −0.0403± 0.0053 −0.0473± 0.0005 1.3 1.2QW (Cs) −73.20± 0.35 −73.15± 0.02 −0.1 −0.1QW (Tl) −116.4± 3.6 −116.76± 0.04 0.1 0.1ττ [fs] 291.09± 0.48 290.02± 2.09 0.5 0.5Γ(b→sγ)
Γ(b→Xeν)
(
3.38+0.51−0.44
)
× 10−3 (3.11 ± 0.07)× 10−3 0.6 0.6
12 (gμ − 2 − α
π ) (4511.07± 0.77)× 10−9 (4509.13± 0.08)× 10−9 2.5 2.5
currently in agreement with the SM but this statement is preliminary (seeSec. 10.3).
Ab can be extracted from A(0,b)FB when Ae = 0.1501 ± 0.0016 is taken
from a fit to leptonic asymmetries (using lepton universality). The result,Ab = 0.881± 0.017, is 3.2 σ below the SM prediction† and also 1.6 σ belowAb = 0.923 ± 0.020 obtained from AFB
LR (b) at SLD. Thus, it appears that
† Alternatively, one can use A� = 0.1481± 0.0027, which is from LEP 1alone and in excellent agreement with the SM, and obtain Ab = 0.893±0.022which is 1.9 σ low. This illustrates that some of the discrepancy is relatedto the one in ALR.
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10. Electroweak model and constraints on new physics 177
Table 10.4: Principal Z pole observables and their SM predictions(cf. Table 10.3). The first s2
� (A(0,q)FB ) is the effective angle extracted
from the hadronic charge asymmetry while the second is thecombined lepton asymmetry from CDF [157] and DØ [158]. Thethree values of Ae are (i) from ALR for hadronic final states [152];(ii) from ALR for leptonic final states and from polarized Bhabbascattering [154]; and (iii) from the angular distribution of the τpolarization at LEP 1. The two Aτ values are from SLD and thetotal τ polarization, respectively.
Quantity Value Standard Model Pull Dev.
MZ [GeV] 91.1876± 0.0021 91.1874± 0.0021 0.1 0.0ΓZ [GeV] 2.4952± 0.0023 2.4954± 0.0009 −0.1 0.1Γ(had) [GeV] 1.7444± 0.0020 1.7418± 0.0009 — —Γ(inv) [MeV] 499.0± 1.5 501.69± 0.07 — —Γ( + −) [MeV] 83.984± 0.086 84.005± 0.015 — —σhad[nb] 41.541± 0.037 41.484± 0.008 1.5 1.5Re 20.804± 0.050 20.735± 0.010 1.4 1.4Rμ 20.785± 0.033 20.735± 0.010 1.5 1.6Rτ 20.764± 0.045 20.780± 0.010 −0.4 −0.3Rb 0.21629± 0.00066 0.21578± 0.00005 0.8 0.8Rc 0.1721± 0.0030 0.17224± 0.00003 0.0 0.0A
(0,e)FB 0.0145± 0.0025 0.01633± 0.00021 −0.7 −0.7
A(0,μ)FB 0.0169± 0.0013 0.4 0.6
A(0,τ)FB 0.0188± 0.0017 1.5 1.6
A(0,b)FB 0.0992± 0.0016 0.1034± 0.0007 −2.7 −2.3
A(0,c)FB 0.0707± 0.0035 0.0739± 0.0005 −0.9 −0.8
A(0,s)FB 0.0976± 0.0114 0.1035± 0.0007 −0.6 −0.4
s2� (A
(0,q)FB ) 0.2324± 0.0012 0.23146± 0.00012 0.8 0.7
0.2316± 0.0018 0.1 0.0Ae 0.15138± 0.00216 0.1475± 0.0010 1.8 2.2
0.1544± 0.0060 1.1 1.30.1498± 0.0049 0.5 0.6
Aμ 0.142 ± 0.015 −0.4 −0.3Aτ 0.136 ± 0.015 −0.8 −0.7
0.1439± 0.0043 −0.8 −0.7Ab 0.923 ± 0.020 0.9348± 0.0001 −0.6 −0.6Ac 0.670 ± 0.027 0.6680± 0.0004 0.1 0.1As 0.895 ± 0.091 0.9357± 0.0001 −0.4 −0.4
at least some of the problem in A(0,b)FB is experimental. Note, however, that
the uncertainty in A(0,b)FB is strongly statistics dominated. The combined
value, Ab = 0.899±0.013 deviates by 2.8 σ. It would be difficult to accountfor this 4.0% deviation by new physics that enters only at the level ofradiative corrections since about a 20% correction to κb would be necessaryto account for the central value of Ab [211]. If this deviation is due tonew physics, it is most likely of tree-level type affecting preferentiallythe third generation. Examples include the decay of a scalar neutrinoresonance [209], mixing of the b quark with heavy exotics [210], and
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178 10. Electroweak model and constraints on new physics
a heavy Z ′ with family-nonuniversal couplings [212,213]. It is difficult,however, to simultaneously account for Rb, which has been measured onthe Z peak and off-peak [214] at LEP 1. An average of Rb measurementsat LEP 2 at energies between 133 and 207 GeV is 2.1 σ below the SMprediction, while A
(b)FB (LEP 2) is 1.6 σ low [160].
The left-right asymmetry, A0LR = 0.15138± 0.00216 [152], based on all
hadronic data from 1992–1998 differs 1.8 σ from the SM expectation of0.1475 ± 0.0010. The combined value of A� = 0.1513 ± 0.0021 from SLD(using lepton-family universality and including correlations) is also 1.8 σabove the SM prediction; but there is now experimental agreement betweenthis SLD value and the LEP 1 value, A� = 0.1481± 0.0027, obtained froma fit to A
(0,�)FB , Ae(Pτ ), and Aτ (Pτ ), again assuming universality.
The observables in Table 10.3 and Table 10.4, as well as some otherless precise observables, are used in the global fits described below. Inall fits, the errors include full statistical, systematic, and theoreticaluncertainties. The correlations on the LEP 1 lineshape and τ polarization,the LEP/SLD heavy flavor observables, the SLD lepton asymmetries,and the deep inelastic and ν-e scattering observables, are included. Thetheoretical correlations between Δα
(5)had and gμ−2, and between the charm
and bottom quark masses, are also accounted for.The data allow a simultaneous determination of MZ , MH , mt, and
the strong coupling αs(MZ). (mc, mb, and Δα(3)had are also allowed to
float in the fits, subject to the theoretical constraints [5,17] describedin Sec. 10.1–Sec. 10.2. These are correlated with αs.) αs is determinedmainly from R�, ΓZ , σhad, and ττ and is only weakly correlated with theother variables. The global fit to all data, including the CDF/DØ averagemt = 173.1 ± 1.3 GeV, yields the result in Table 10.4 (the MS top quarkmass given there corresponds to mt = 173.2 ± 1.3 GeV). The weak mixingangle is determined to
s 2Z = 0.23116± 0.00013, s2
W = 0.22292± 0.00028,
where the larger error in the on-shell scheme is due to the strongersensitivity to mt, while the corresponding effective angle is related byEq. (10.32), i.e., s2
� = 0.23146± 0.00012.The weak mixing angle can be determined from Z pole observables,
MW , and from a variety of neutral-current processes spanning a verywide Q2 range. The results (for the older low energy neutral-currentdata see Refs. 58 and 59) shown in Table 10.7 of the full Review are inreasonable agreement with each other, indicating the quantitative successof the SM. The largest discrepancy is the value s 2
Z = 0.23193 ± 0.00028from the forward-backward asymmetries into bottom and charm quarks,which is 2.7 σ above the value 0.23116 ± 0.00013 from the global fit toall data. Similarly, s 2
Z = 0.23067 ± 0.00029 from the SLD asymmetries(in both cases when combined with MZ) is 1.7 σ low. The SLD resulthas the additional difficulty (within the SM) of implying very low andexcluded [161] Higgs masses. This is also true for s 2
Z = 0.23100± 0.00023from MW and MZ and — as a consequence — for the global fit. Wehave therefore included in Table 10.3 and Table 10.4 an additional column(denoted Deviation) indicating the deviations if MH = 117 GeV is fixed.
The extracted Z pole value of αs(MZ) is based on a formula withnegligible theoretical uncertainty if one assumes the exact validityof the SM. One should keep in mind, however, that this value,
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10. Electroweak model and constraints on new physics 179
Table 10.5: Values of s 2Z , s2
W , αs, and MH [in GeV] for various(combinations of) observables. Unless indicated otherwise, the topquark mass, mt = 170.9±1.9 GeV, is used as an additional constraintin the fits. The (†) symbol indicates a fixed parameter.
Data s 2Z s2
W αs(MZ) MH
All data 0.23116(13) 0.22292(28) 0.1183(15) 90+27−22
All indirect (no mt) 0.23118(14) 0.22283(34) 0.1183(16) 112+110− 52
Z pole (no mt) 0.23121(17) 0.22311(59) 0.1198(28) 90+114− 44
LEP 1 (no mt) 0.23152(21) 0.22376(67) 0.1213(30) 170+234− 93
SLD + MZ 0.23067(29) 0.22201(54) 0.1183 (†) 33+27−17
A(b,c)FB + MZ 0.23193(28) 0.22484(76) 0.1183 (†) 389+264
−158
MW + MZ 0.23100(23) 0.22262(48) 0.1183 (†) 67+38−28
MZ 0.23128(6) 0.22321(17) 0.1183 (†) 117 (†)QW (APV) 0.2314(14) 0.2233(14) 0.1183 (†) 117 (†)QW (e) 0.2332(15) 0.2251(15) 0.1183 (†) 117 (†)νμ-N DIS (isoscalar) 0.2335(18) 0.2254(18) 0.1183 (†) 117 (†)Elastic νμ(νμ)-e 0.2311(77) 0.2230(77) 0.1183 (†) 117 (†)e-D DIS (SLAC) 0.222(18) 0.213(19) 0.1183 (†) 117 (†)Elastic νμ(νμ)-p 0.211(33) 0.203(33) 0.1183 (†) 117 (†)
αs(MZ) = 0.1198 ± 0.0028, is very sensitive to such types of new physicsas non-universal vertex corrections. In contrast, the value derived fromτ decays, αs(MZ) = 0.1174+0.0018
−0.0016, is theory dominated but less sensitiveto new physics. The two values are in remarkable agreement with eachother. They are also in agreement with other recent values, such asfrom jet-event shapes at LEP [215] (0.1202 ± 0.0050) the averagefrom HERA [216] (0.1198 ± 0.0032), and the most recent unquenchedlattice calculation [217] (0.1183 ± 0.0008). For more details and otherdeterminations, see our Section 9 on “Quantum Chromodynamics” in thisReview.
Using α(MZ) and s 2Z as inputs, one can predict αs(MZ) assuming
grand unification. One predicts [218] αs(MZ) = 0.130 ± 0.001 ± 0.01 forthe simplest theories based on the minimal supersymmetric extensionof the SM, where the first (second) uncertainty is from the inputs(thresholds). This is slightly larger, but consistent with the experimentalαs(MZ) = 0.1183 ± 0.0015 from the Z lineshape and the τ lifetime, aswell as with other determinations. Non-supersymmetric unified theoriespredict the low value αs(MZ) = 0.073 ± 0.001 ± 0.001. See also the noteon “Supersymmetry” in the Searches Particle Listings.
The data indicate a preference for a small Higgs mass. There isa strong correlation between the quadratic mt and logarithmic MHterms in ρ in all of the indirect data except for the Z → bb vertex.
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180 10. Electroweak model and constraints on new physics
Therefore, observables (other than Rb) which favor mt values higherthan the Tevatron range favor lower values of MH . MW has additionalMH dependence through ΔrW which is not coupled to m2
t effects. Thestrongest individual pulls toward smaller MH are from MW and A0
LR,
while A(0,b)FB favors higher values. The difference in χ2 for the global fit is
Δχ2 = χ2(MH = 300 GeV) − χ2min ≈ 25. Hence, the data favor a small
value of MH , as in supersymmetric extensions of the SM. The centralvalue of the global fit result, MH = 90+27
−22 GeV, is below the direct lowerbound, MH ≥ 114.4 GeV (95% CL) [161].
The 90% central confidence range from all precision data is
55 GeV ≤ MH ≤ 135 GeV. (10.36)
Including the results of the direct searches at LEP 2 [161] and theTevatron [219] as extra contributions to the likelihood function drivesthe 95% upper limit to MH ≤ 147 GeV. As two further refinements, weaccount for (i) theoretical uncertainties from uncalculated higher ordercontributions by allowing the T parameter (see next subsection) subjectto the constraint T = 0 ± 0.02, (ii) the MH dependence of the correlationmatrix which gives slightly more weight to lower Higgs masses [220]. Theresulting limits at 95 (90, 99)% CL are, respectively,
MH ≤ 149 (145, 194) GeV. (10.37)
10.7. Constraints on new physics
A number of authors [226–231] have considered the general effects onneutral-current and Z and W boson observables of various types of heavy(i.e., Mnew � MZ) physics which contribute to the W and Z self-energiesbut which do not have any direct coupling to the ordinary fermions. Inaddition to non-degenerate multiplets, which break the vector part of weakSU(2), these include heavy degenerate multiplets of chiral fermions whichbreak the axial generators. These effects can be described, for example,by the so-called T and S parameters, respectively, where the latter istypically rather large in Technicolor theories.
-1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
S
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
T
all: MH = 117 GeV
all: MH = 340 GeV
all: MH = 1000 GeV
ΓZ, σ
had, R
l, R
q
asymmetries
MW
ν scattering
e scattering
APV
Figure 10.1: 1 σ constraints (39.35%) on S and T from variousinputs combined with MZ . S and T represent the contributions ofnew physics only.
Further discussion and all references may be found in the full Review.
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11. The CKM quark-mixing matrix 181
11. THE CKM QUARK-MIXING MATRIX
Revised February 2010 by A. Ceccucci (CERN), Z. Ligeti (LBNL), andY. Sakai (KEK).
11.1. Introduction
The masses and mixings of quarks have a common origin in theStandard Model (SM). They arise from the Yukawa interactions of thequarks with the Higgs condensate. When the Higgs field acquires a vacuumexpectation value, quark mass terms are generated. The physical states areobtained by diagonalizing the up and down quark mass matrices by fourunitary matrices, V
u,dL,R. As a result, the charged current W± interactions
couple to the physical up and down-type quarks with couplings given by
VCKM ≡ V uL V
d†L =
⎛
⎝
Vud Vus VubVcd Vcs VcbVtd Vts Vtb
⎞
⎠ . (11.2)
This Cabibbo-Kobayashi-Maskawa (CKM) matrix [1,2] is a 3 × 3unitary matrix. It can be parameterized by three mixing angles and aCP -violating phase,
V =
⎛
⎝
c12c13 s12c13 s13e−iδ
−s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13
s12s23−c12c23s13eiδ −c12s23−s12c23s13eiδ c23c13
⎞
⎠ , (11.3)
where sij = sin θij , cij = cos θij , and δ is the phase responsible for allCP -violating phenomena in flavor changing processes in the SM. Theangles θij can be chosen to lie in the first quadrant.
It is known experimentally that s13 � s23 � s12 � 1, and it isconvenient to exhibit this hierarchy using the Wolfenstein parameterization.We define [4–6]
s12 = λ =|Vus|
√
|Vud|2 + |Vus|2, s23 = Aλ2 = λ
∣
∣
∣
∣
Vcb
Vus
∣
∣
∣
∣
,
s13eiδ = V ∗
ub = Aλ3(ρ + iη) =Aλ3(ρ + iη)
√1 − A2λ4
√1 − λ2[1 − A2λ4(ρ + iη)]
. (11.4)
These ensure that ρ + iη = −(VudV∗ub)/(VcdV ∗
cb) is phase-conventionindependent and the CKM matrix written in terms of λ, A, ρ and η isunitary to all orders in λ. To O(λ4),
V =
⎛
⎝
1 − λ2/2 λ Aλ3(ρ − iη)−λ 1 − λ2/2 Aλ2
Aλ3(1 − ρ − iη) −Aλ2 1
⎞
⎠ + O(λ4) . (11.5)
Unitarity implies∑
i VijV∗ik = δjk and
∑
j VijV∗kj = δik. The six
vanishing combinations can be represented as triangles in a complex plane.The most commonly used unitarity triangle arises from
Vud V ∗ub + Vcd V ∗
cb + Vtd V ∗tb = 0 , (11.6)
by dividing each side by VcdV∗cb (see Fig. 1). The vertices are exactly (0, 0),
(1, 0) and, due to the definition in Eq. (11.4), (ρ, η). An important goal offlavor physics is to overconstrain the CKM elements, many of which canbe displayed and compared in the ρ, η plane.
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182 11. The CKM quark-mixing matrix
Figure 11.1: Sketch of the unitarity triangle.
11.2. Magnitudes of CKM elements
11.2.1. |Vud| :The most precise determination of |Vud| comes from the study of
superallowed 0+ → 0+ nuclear beta decays, which are pure vectortransitions. Taking the average of the nine most precise determinations [8]yields [9]
|Vud| = 0.97425± 0.00022. (11.7)
11.2.2. |Vus| :The magnitude of Vus is extracted from semileptonic kaon decays or
leptonic kaon decays. Combining the data on K0L → πeν, K0
L → πμν,K± → π0e±ν, K± → π0μ±ν and K0
S → πeν gives |Vus| = 0.2246± 0.0012with the unquenched lattice QCD calculation value, f+(0) = 0.9644 ±0.0049 [12]. The KLOE measurement of the K+ → μ+ν(γ) branchingratio [16] with the lattice QCD value, fK/fπ = 1.189 ± 0.007 [17] leadsto |Vus| = 0.2259 ± 0.0014. The average of these two determinations isquoted by Ref. [9] as
|Vus| = 0.2252± 0.0009. (11.8)
11.2.3. |Vcd| :The most precise determination of |Vcd| is based on neutrino and
antineutrino interactions. The difference of the ratio of double-muonto single-muon production by neutrino and antineutrino beams isproportional to the charm cross section off valence d-quarks. Combiningthe results [26–29], we obtain
|Vcd| = 0.230± 0.011. (11.9)
11.2.4. |Vcs| :The determination of |Vcs| is possible from semileptonic D or leptonic
Ds decays. Using the recent D+s → μ+ν [37,38,40] and D+
s → τ+ν [38,41]data gives |Vcs| = 1.030 ± 0.038 with fDs = (242.8 ± 3.2 ± 5.3)MeV [42].The recent D → K�ν measurements [24,25,43] combined with the latticeQCD calculation of the form factor [23] gives |Vcs| = 0.98 ± 0.01 ± 0.10.Averaging these two determinations, we obtain
|Vcs| = 1.023 ± 0.036. (11.10)
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11. The CKM quark-mixing matrix 183
11.2.5. |Vcb| :The determination of |Vcb| from inclusive semileptonic B decays use
the semileptonic rate measurement together with the leptonic energyand the hadronic invariant-mass spectra. Determinations from exclusiveB → D(∗)�ν decays are based on the fact that in the mb,c � ΛQCD limitall form factors are given by a single Isgur-Wise function [49], which isnormalized at zero recoil. The Vcb and Vub minireview [48] quotes thecombination with a scaled error as
|Vcb| = (40.6 ± 1.3) × 10−3. (11.11)
11.2.6. |Vub| :The determination of |Vub| from inclusive B → Xu�ν decay suffers from
large B → Xc�ν backgrounds. In most regions of phase space where thecharm background is kinematically forbidden the rate is determined bynonperturbative shape functions. At leading order in ΛQCD/mb there isonly one such function, which is related to the photon energy spectrumin B → Xsγ [50,51]. The large and pure BB samples at the B factoriespermit the selection of B → Xu�ν decays in events where the other B isfully reconstructed [56]. With this full-reconstruction tag method, one canmeasure the four-momenta of both the leptonic and hadronic systems, andaccess wider kinematic regions because of improved signal purity.
To extract |Vub| from exclusive channels, the form factors have to beknown. Unquenched lattice QCD calculations of the B → π�ν form factorfor q2 > 16 GeV2 are available [57,58]. The theoretical uncertainties inthe inclusive and exclusive determinations are different. The Vcb and Vubminireview [48] quotes the combination
|Vub| = (3.89 ± 0.44)× 10−3. (11.12)
11.2.7. |Vtd| and |Vts| :These CKM elements cannot be measured from tree-level top quark
decays, so one has to use B–B oscillations or loop-mediated rare Kand B decays. The mass difference of the two neutral B meson masseigenstates is well measured, Δmd = (0.507± 0.005) ps−1 [60]. For the B0
ssystem, CDF measured Δms = (17.77 ± 0.10 ± 0.07) ps−1 [61] with morethan 5σ significance. Using unquenched lattice QCD calculations [63] andassuming |Vtb| = 1, we find
|Vtd| = (8.4 ± 0.6) × 10−3, |Vts| = (38.7 ± 2.1) × 10−3. (11.13)
Several uncertainties are reduced in the lattice QCD calculation of theratio Δmd/Δms, which gives a new and significantly improved constraint,
∣
∣Vtd/Vts∣
∣ = 0.211± 0.001 ± 0.005. (11.14)
11.2.8. |Vtb| :The determination of |Vtb| from top decays uses the ratio of branching
fractions B(t → Wb)/B(t → Wq) = |Vtb|2/(∑
q |Vtq |2) = |Vtb|2, whereq = b, s, d [69,70]. The direct determination of |Vtb| without assumingunitarity has become possible from the single top quark production crosssection. The (2.76+0.58
−0.47) pb [71] average cross section measured by DØ [72]and CDF [73] implies
|Vtb| = 0.88 ± 0.07 . (11.15)
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184 11. The CKM quark-mixing matrix
11.3. Phases of CKM elements
The angles of the unitarity triangle are
β = φ1 = arg(
−VcdV∗cb
VtdV ∗tb
)
, α = φ2 = arg(
− VtdV ∗tb
VudV∗ub
)
,
γ = φ3 = arg(
−VudV ∗ub
VcdV ∗cb
)
. (11.16)
Since CP violation involves phases of CKM elements, many measurementsof CP -violating observables can be used to constrain these angles and theρ, η parameters.
11.3.1. ε and ε′ :The measurement of CP violation in K0–K0 mixing, |ε| = (2.233 ±
0.015) × 10−3 [75], provides constraints in the ρ, η plane bounded byhyperbolas approximately. The dominant uncertainties are due to the bagparameter and the parametric uncertainty proportional to σ(A4) [i.e.,σ(|Vcb|4)].
The measurement of ε′ provides a qualitative test of the CKMmechanism because its nonzero experimental average, Re(ε′/ε) =(1.67 ± 0.23) × 10−3 [75], demonstrated the existence of direct CPviolation, a prediction of the KM ansatz. While Re(ε′/ε) ∝ Im(VtdV ∗
ts),this quantity cannot easily be used to extract CKM parameters, becauseof large hadronic uncertainties.
11.3.2. β / φ1 :The time-dependent CP asymmetry of neutral B decays to a final state
f common to B0 and B0 is given by [85,86]
Af =Γ(B0(t) → f) − Γ(B0(t) → f)Γ(B0(t) → f) + Γ(B0(t) → f)
= Sf sin(Δm t) − Cf cos(Δm t),
(11.18)where Sf = 2 Imλf/(1 + |λf |2), Cf = (1 − |λf |2)/(1 + |λf |2), andλf = (q/p)(Af/Af ). Here q/p describes B0–B0 mixing and, to a good
approximation in the SM, q/p = V ∗tbVtd/VtbV
∗td = e−2iβ+O(λ4) in the usual
phase convention. Af (Af ) is the amplitude of B0 → f (B0 → f) decay.If f is a CP eigenstate and amplitudes with one CKM phase dominate,then |Af | = |Af |, Cf = 0 and Sf = sin(arg λf ) = ηf sin 2φ, where ηf isthe CP eigenvalue of f and 2φ is the phase difference between the B0 → fand B0 → B0 → f decay paths.
The b → ccs decays to CP eigenstates (B0 → charmonium K0S,L)
are the theoretically cleanest examples, measuring Sf = −ηf sin 2β. Theworld average is [64]
sin 2β = 0.673 ± 0.023 . (11.20)
This measurement of β has a four-fold ambiguity. Of these, β → π/2 − β(but not β → π + β) has been resolved by a time-dependent angularanalysis of B0 → J/ψK∗0 [90,91] and a time-dependent Dalitz plotanalysis of B0 → D0h0 (h0 = π0, η, ω) [92,93].
The b → sqq penguin dominated decays have the same CKM phaseas the b → ccs tree dominated decays, up to corrections suppressedby λ2. Therefore, decays such as B0 → φK0 and η′K0 provide sin 2βmeasurements in the SM. If new physics contributes to the b → s
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11. The CKM quark-mixing matrix 185
loop diagrams and has a different weak phase, it would give rise toSf �= −ηf sin 2β and possibly Cf �= 0. The results and their uncertaintiesare summarized in Fig. 12.3 and Table 12.1 of Ref. [86].
11.3.3. α / φ2 :Since α is the phase between V ∗
tbVtd and V ∗ubVud, only time-dependent
CP asymmetries in b → uud dominated modes can directly measure it.In such decays the penguin contribution can be sizable. Then Sπ+π− nolonger measures sin 2α, but α can still be extracted using the isospinrelations among the B0 → π+π−, B0 → π0π0, and B+ → π+π0
amplitudes and their CP conjugates [94]. Because the isospin analysisgives 16 mirror solutions, and the sizable experimental error of B0 → π0π0,only a loose constraint is obtained at present.
The B0 → ρ+ρ− decay can in general have a mixture of CP -even andCP -odd components. However, the longitudinal polarization fractions inB+ → ρ+ρ0 and B0 → ρ+ρ− are measured to be close to unity [96], whichimplies that the final states are almost purely CP -even. Furthermore,B(B0 → ρ0ρ0) = (0.73+0.27
−0.28) × 10−6 implies that the effect of the penguindiagrams is small. The isospin analysis gives α = (89.9 ± 5.4)◦ [95] with amirror solution at 3π/2 − α.
The final state in B0 → ρ+π− decay is not a CP eigenstate, butmixing induced CP violations can still occur in the four decay amplitudes,B0, B0 → ρ±π∓. Because of the more complicated isospin relations, thetime-dependent Dalitz plot analysis of B0 → π+π−π0 gives the bestmodel independent extraction of α [99]. The Belle [100] and BABAR [101]measurements yield α = (120+11
−7 )◦ [95].Combining these three decay modes [95], α is constrained as
α = (89.0+4.4−4.2)
◦. (11.23)
11.3.4. γ / φ3 :The angle γ does not depend on CKM elements involving the top
quark, so it can be measured in tree level B decays. This is an importantdistinction from α and β, implying that the measurements of γ are unlikelyto be affected by physics beyond the SM.
The interference of B− → D0K− (b → cus) and B− → D0K−(b → ucs) transitions can be studied in final states accessible in bothD0 and D0 decays [85]. It is possible to extract from the data the Band D decay amplitudes, the relative strong phases, and γ. Analyses intwo-body D decays using the GLW [103,104] and ADS methods [105] havebeen made by the B factories [106], as well as in a Dalitz plot analysisof D0, D0 → KSπ+π− [109,110]. Combining these analyses [95], γ isconstrained as
γ = (73+22−25)
◦. (11.25)
11.4. Global fit in the Standard Model
Using the independently measured CKM elements mentioned inthe previous sections, the unitarity of the CKM matrix can bechecked. We obtain |Vud|2 + |Vus|2 + |Vub|2 = 0.9999 ± 0.0006 (1st row),|Vcd|2 + |Vcs|2 + |Vcb|2 = 1.101± 0.074 (2nd row), |Vud|2 + |Vcd|2 + |Vtd|2 =1.002 ± 0.005 (1st column), and |Vus|2 + |Vcs|2 + |Vts|2 = 1.098 ± 0.074(2nd column), respectively. For the second row, a more stringent check
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186 11. The CKM quark-mixing matrix
γ
γ
α
α
dmΔKε
Kε
smΔ & dmΔ
ubV
βsin 2
(excl. at CL > 0.95) < 0βsol. w/ cos 2
excluded at CL > 0.95
α
βγ
ρ-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
η
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5excluded area has CL > 0.95
Figure 11.2: 95% CL constraints on the ρ, η plane.
is obtained subtracting the sum of the first row from the measurementof
∑
u,c,d,s,b |Vij |2 from the W leptonic branching ratio [35], yielding|Vcd|2 + |Vcs|2 + |Vcb|2 = 1.002 ± 0.027. The sum of the three angles,α + β + γ = (183+22
−25)◦, is also consistent with the SM expectation.
The CKM matrix elements can be most precisely determined by aglobal fit that uses all available measurements and imposes the SMconstraints. There are several approaches to combining the experimentaldata [6,95,102,118], which provide similar results. The results for theWolfenstein parameters are
λ = 0.2253± 0.0007 , A = 0.808+0.022−0.015 ,
ρ = 0.132+0.022−0.014 , η = 0.341± 0.013 . (11.26)
The allowed ranges of the magnitudes of all nine CKM elements are
VCKM =
⎛
⎝
0.97428± 0.00015 0.2253± 0.0007 0.00347+0.00016−0.00012
0.2252± 0.0007 0.97345+0.00015−0.00016 0.0410+0.0011
−0.0007
0.00862+0.00026−0.00020 0.0403+0.0011
−0.0007 0.999152+0.000030−0.000045
⎞
⎠,
(11.27)and the Jarlskog invariant is J = (2.91+0.19
−0.11) × 10−5. Fig. 11.2 illustratesthe constraints on the ρ, η plane from various measurements and the globalfit result. The shaded 95% CL regions all overlap consistently around theglobal fit region.
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11. The CKM quark-mixing matrix 187
11.5. Implications beyond the SM
The effects in B, K, and D decays and mixings due to high-scale physics(W , Z, t, h in the SM, or new physics particles) can be parameterized byoperators composed of SM fields, obeying the SU(3)×SU(2)×U(1) gaugesymmetry. The observable effects of non-SM interactions are encoded inthe coefficients of these operators, and are suppressed by powers of thenew physics scale. In the SM, these coefficients are determined by just thefour CKM parameters, and the W , Z, and quark masses. For example,Δmd, Γ(B → ργ), and Γ(B → Xd�
+�−) are all proportional to |VtdV ∗tb|2
in the SM, however, they may receive unrelated new physics contributions.Similar to measurements of sin 2β in tree- and loop-dominated decays,overconstraining the magnitudes and phases of flavor-changing neutralcurrent amplitudes give good sensitivity to new physics.
To illustrate the level of suppression required for non-SM contributions,consider a class of models in which the dominant effect of new physics is tomodify the neutral meson mixing amplitudes [121] by (zij/Λ2)(qiγ
μPLqj)2.New physics with a generic weak phase may still contribute to mesonmixings at a significant fraction of the SM [125,118]. The data imply thatΛ/|zij|1/2 has to exceed about 104 TeV for K0 –K0 mixing, 103 TeV forD0 – D0 mixing, 500TeV for B0 –B0 mixing, and 100TeV for B0
s –B0s
mixing [118,123]. Thus, if there is new physics at the TeV scale, |zij | � 1is required. Even if |zij | are suppressed by a loop factor and |V ∗
tiVtj |2 (inthe down quark sector), as in the SM, one expects percent-level effects,which may be observable in forthcoming experiments.
The CKM elements are fundamental parameters, so they should bemeasured as precisely as possible. The overconstraining measurements ofCP asymmetries, mixing, semileptonic, and rare decays severely constrainthe magnitudes and phases of possible new physics contributions toflavor-changing interactions. When new particles are seen at the LHC, itwill be important to know the flavor parameters as precisely as possible tounderstand the underlying physics.
Further discussion and all references may be found in the full Review ofParticle Physics. The numbering of references and equations used herecorresponds to that version.
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188 12. CP violation in meson decays
12. CP VIOLATION IN MESON DECAYS
Revised September 2009 by D. Kirkby (UC Irvine) and Y. Nir (WeizmannInstitute).
The CP transformation combines charge conjugation C with parityP . Under C, particles and antiparticles are interchanged, by conjugatingall internal quantum numbers, e.g., Q → −Q for electromagnetic charge.Under P , the handedness of space is reversed, �x → −�x. Thus, for example,a left-handed electron e−L is transformed under CP into a right-handedpositron, e+
R.We observe that most phenomena are C- and P -symmetric, and
therefore, also CP -symmetric. In particular, these symmetries arerespected by the gravitational, electromagnetic, and strong interactions.The weak interactions, on the other hand, violate C and P in the strongestpossible way. For example, the charged W bosons couple to left-handedelectrons, e−L , and to their CP -conjugate right-handed positrons, e+
R,but to neither their C-conjugate left-handed positrons, e+
L , nor theirP -conjugate right-handed electrons, e−R. While weak interactions violate Cand P separately, CP is still preserved in most weak interaction processes.The CP symmetry is, however, violated in certain rare processes, asdiscovered in neutral K decays in 1964 [1], and observed in recent yearsin B decays.
The present measurements of CP asymmetries provide some ofthe strongest constraints on the weak couplings of quarks. Futuremeasurements of CP violation in K, D, B, and Bs meson decayswill provide additional constraints on the flavor parameters of theStandard Model, and can probe new physics. In this review, we give theformalism and basic physics that are relevant to present and near futuremeasurements of CP violation in meson decays.
12.1. Formalism
In this section, we present a general formalism for, and classificationof, CP violation in the decay of a pseudoscalar meson M that mightbe a charged or neutral K, D, B, or Bs meson. Subsequent sectionsdescribe the CP -violating phenomenology, approximations, and alternativeformalisms that are specific to each system.
12.1.1. Charged- and neutral-meson decays : We define decayamplitudes of M (which could be charged or neutral) and its CPconjugate M to a multi-particle final state f and its CP conjugate f as
Af = 〈f |H|M〉 , Af = 〈f |H|M〉 ,
Af = 〈f |H|M〉 , Af = 〈f |H|M〉 , (12.13)where H is the Hamiltonian governing weak interactions.
12.1.2. Neutral-meson mixing : A state that is initially a superposi-tion of M0 and M
0, say
|ψ(0)〉 = a(0)|M0〉 + b(0)|M0〉 , (12.17)will evolve in time acquiring components that describe all possible decayfinal states {f1, f2, . . .}, that is,
|ψ(t)〉 = a(t)|M0〉 + b(t)|M0〉 + c1(t)|f1〉 + c2(t)|f2〉 + · · · . (12.18)If we are interested in computing only a(t) and b(t), we can use a simplifiedformalism. The simplified time evolution is determined by a 2× 2 effective
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12. CP violation in meson decays 189
Hamiltonian H that is not Hermitian, since otherwise the mesons wouldonly oscillate and not decay. Any complex matrix, such as H, can bewritten in terms of Hermitian matrices M and Γ as
H = M − i
2Γ . (12.19)
The eigenvectors of H have well-defined masses and decay widths. Tospecify the components of the strong interaction eigenstates, M0 and M
0,in the light (ML) and heavy (MH) mass eigenstates, we introduce twocomplex parameters, p and q:
|ML〉 ∝ p |M0〉 + q |M0〉|MH〉 ∝ p |M0〉 − q |M0〉 , (12.20)
with the normalization |q|2 + |p|2 = 1.Solving the eigenvalue problem for H yields
(
q
p
)2
=M∗
12 − (i/2)Γ∗12
M12 − (i/2)Γ12. (12.22)
12.1.3. CP -violating observables : All CP -violating observables in Mand M decays to final states f and f can be expressed in terms of phase-convention-independent combinations of Af , Af , Af , and Af , togetherwith, for neutral-meson decays only, q/p. CP violation in charged-mesondecays depends only on the combination |Af/Af |, while CP violation
in neutral-meson decays is complicated by M0 ↔ M0 oscillations, and
depends, additionally, on |q/p| and on λf ≡ (q/p)(Af/Af ).Defining x ≡ Δm/Γ and y ≡ ΔΓ/(2Γ), one obtains the following
time-dependent decay rates:dΓ
[
M0phys(t) → f
]
/dt
e−ΓtNf=
(
|Af |2 + |(q/p)Af |2)
cosh(yΓt) +(
|Af |2 − |(q/p)Af |2)
cos(xΓt)
+ 2Re((q/p)A∗fAf ) sinh(yΓt) − 2 Im((q/p)A∗
fAf ) sin(xΓt) ,
(12.30)
dΓ[
M0phys(t) → f
]
/dt
e−ΓtNf=
(
|(p/q)Af |2 + |Af |2)
cosh(yΓt) −(
|(p/q)Af |2 − |Af |2)
cos(xΓt)
+ 2Re((p/q)AfA∗f ) sinh(yΓt) − 2 Im((p/q)AfA
∗f ) sin(xΓt) ,
(12.31)where Nf is a common, time-independent, normalization factor.
12.1.4. Classification of CP -violating effects : We distinguish threetypes of CP -violating effects in meson decays:
I. CP violation in decay is defined by|Af/Af | �= 1 . (12.34)
In charged meson decays, where mixing effects are absent, this isthe only possible source of CP asymmetries:
Af± ≡ Γ(M− → f−) − Γ(M+ → f+)Γ(M− → f−) + Γ(M+ → f+)
=|Af−/Af+ |2 − 1
|Af−/Af+ |2 + 1. (12.35)
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190 12. CP violation in meson decays
II. CP (and T ) violation in mixing is defined by|q/p| �= 1 . (12.36)
In charged-current semileptonic neutral meson decays M, M →�±X (taking |A�+X | = |A�−X | and A�−X = A�+X = 0, as is thecase in the Standard Model, to lowest order in GF , and in most ofits reasonable extensions), this is the only source of CP violation,and can be measured via the asymmetry of “wrong-sign” decaysinduced by oscillations:
ASL(t) ≡dΓ/dt
[
M0phys(t) → �+X
] − dΓ/dt[
M0phys(t) → �−X
]
dΓ/dt[
M0phys(t) → �+X
]
+ dΓ/dt[
M0phys(t) → �−X
]
=1 − |q/p|41 + |q/p|4 . (12.37)
Note that this asymmetry of time-dependent decay rates isactually time-independent.
III. CP violation in interference between a decay without mixing,M0 → f , and a decay with mixing, M0 → M
0 → f (such aneffect occurs only in decays to final states that are common to M0
and M0, including all CP eigenstates), is defined by
Im(λf ) �= 0 , (12.38)with
λf ≡ q
p
Af
Af. (12.39)
This form of CP violation can be observed, for example, using theasymmetry of neutral meson decays into final CP eigenstates fCP
AfCP(t) ≡
dΓ/dt[
M0phys(t) → fCP
] − dΓ/dt[
M0phys(t) → fCP
]
dΓ/dt[
M0phys(t) → fCP
]
+ dΓ/dt[
M0phys(t) → fCP
]. (12.40)
If ΔΓ = 0 and |q/p| = 1, as expected to a good approximation forB mesons, but not for K mesons, then AfCP
has a particularlysimple form (see Eq. (12.74), below). If, in addition, the decayamplitudes fulfill |AfCP
| = |AfCP|, the interference between
decays with and without mixing is the only source of theasymmetry and AfCP
(t) = Im(λfCP) sin(xΓt).
12.2. Theoretical Interpretation: The KM Mechanism
Of all the Standard Model quark parameters, only the Kobayashi-Maskawa (KM) phase is CP -violating. The Cabibbo-Kobayashi-Maskawa(CKM) mixing matrix for quarks is described in the preceding CKMreview and in the full edition.
12.3. K Decays
The decay amplitudes actually measured in neutral K decays refer tothe mass eigenstates KL and KS , rather than to the K and K statesreferred to in Eq. (12.13). The final π+π− and π0π0 states are CP -even.In the CP limit, KS(KL) would be CP -even (odd), and therefore would(would not) decay to two pions. We define CP -violating amplitude ratiosfor two-pion final states,
η00 ≡ 〈π0π0|H|KL〉〈π0π0|H|KS〉
, η+− ≡ 〈π+π−|H|KL〉〈π+π−|H|KS〉
. (12.50)
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12. CP violation in meson decays 191
Another important observable is the asymmetry of time-integratedsemileptonic decay rates:
δL ≡ Γ(KL → �+ν�π−) − Γ(KL → �−ν�π
+)Γ(KL → �+ν�π
−) + Γ(KL → �−ν�π+)
. (12.51)
CP violation has been observed as an appearance of KL decays totwo-pion final states [19],|η00| = (2.222± 0.010)× 10−3 |η+−| = (2.233± 0.010)× 10−3 (12.52)
|η00/η+−| = 0.9950± 0.0008 , (12.53)
where the phase φij of the amplitude ratio ηij has been determined:φ00 = (43.7 ± 0.08)◦, φ+− = (43.4 ± 0.07)◦, (12.54)
CP violation has also been observed in semileptonic KL decays [19]δL = (3.32 ± 0.06) × 10−3 , (12.56)
where δL is a weighted average of muon and electron measurements, aswell as in KL decays to π+π−γ and π+π−e+e− [19]. CP violation inK → 3π decays has not yet been observed [19,29].
Historically, CP violation in neutral K decays has been described interms of parameters ε and ε′. The observables η00, η+−, and δL are relatedto these parameters, and to those of Section 12.1, by
η00 =1 − λπ0π0
1 + λπ0π0= ε − 2ε′ , η+− =
1 − λπ+π−1 + λπ+π−
= ε + ε′ ,
δL =1 − |q/p|21 + |q/p|2 =
2Re(ε)1 + |ε|2 . (12.57)
12.4. D Decays
First evidence for D0–D0 mixing has been recently obtained [34–36].Long-distance contributions make it difficult to calculate the StandardModel prediction for the D0–D0 mixing parameters. Therefore, the goal ofthe search for D0–D0 mixing is not to constrain the CKM parameters, butrather to probe new physics. Here CP violation plays an important role.Within the Standard Model, the CP -violating effects are predicted to benegligibly small, since the mixing and the relevant decays are described,to an excellent approximation, by physics of the first two generations.Observation of CP violation in D0–D0 mixing (at a level much higherthan O(10−3)) will constitute an unambiguous signal of new physics.At present, the most sensitive searches involve the D → K+K− andD → K±π∓ modes.12.5. B and Bs Decays
The upper bound on the CP asymmetry in semileptonic B decays [20]implies that CP violation in B0 − B
0 mixing is a small effect (we useASL/2 ≈ 1 − |q/p|, see Eq. (12.37)):
ASL = (−0.4 ± 5.6) × 10−3 =⇒ |q/p| = 1.0002± 0.0028. (12.70)Thus, for the purpose of analyzing CP asymmetries in hadronic B
decays, we can useλf = e
−iφM(B)(Af/Af ) , (12.72)
where φM(B) refers to the phase of M12 appearing in Eq. (12.42) that
is appropriate for B0 − B0 oscillations. Within the Standard Model, the
corresponding phase factor is given by
e−iφM(B) = (V ∗
tbVtd)/(VtbV∗td) . (12.73)
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192 12. CP violation in meson decays
Some of the most interesting decays involve final states that arecommon to B0 and B
0 [40,41]. It is convenient to rewrite Eq. (12.40) forB decays as [42–44]
Af (t) = Sf sin(Δmt) − Cf cos(Δmt),
Sf ≡ 2 Im(λf )
1 +∣
∣λf
∣
∣
2 , Cf ≡ 1 − ∣
∣λf
∣
∣
2
1 +∣
∣λf
∣
∣
2 , (12.74)
where we assume that ΔΓ = 0 and |q/p| = 1. An alternative notation inuse is Af ≡ −Cf , but this Af should not be confused with the Af ofEq. (12.13).
A large class of interesting processes proceed via quark transitions ofthe form b → qqq′ with q′ = s or d. For q = c or u, there are contributionsfrom both tree (t) and penguin (pqu , where qu = u, c, t is the quark in theloop) diagrams (see Fig. 12.2) which carry different weak phases:
Af =(
V ∗qbVqq′
)
tf +∑
qu=u,c,t
(
V ∗qubVquq′
)
pquf . (12.75)
For B → J/ψKS and other b → ccs processes, we can neglect the pqu
terms. Consequently, within the Standard Model, we have:
SψKS= Im
(
−V ∗tbVtdVcbV
∗cd
VtbV∗tdV ∗
cbVcd
)
, CψKS= 0.
12.6. Summary and OutlookCP violation has been experimentally established in neutral K and B
meson decays:
1. All three types of CP violation have been observed in K → ππdecays:
Re(ε′) =16
(∣
∣
∣
∣
∣
Aπ0π0
Aπ0π0
∣
∣
∣
∣
∣
−∣
∣
∣
∣
∣
Aπ+π−Aπ+π−
∣
∣
∣
∣
∣
)
= (2.5 ± 0.4) × 10−6(I)
Re(ε) =12
(
1 −∣
∣
∣
∣
q
p
∣
∣
∣
∣
)
= (1.66 ± 0.02)× 10−3 (II)
Im(ε) = − 12Im(λ(ππ)I=0
) = (1.57 ± 0.02)× 10−3 . (III)
(12.86)2. Direct CP violation has been observed, first in B0 → K+π− decays
(and more recently also in B → π+π−, B0 → ηK∗0, and B+ → ρ0K+
decays), and CP violation in interference of decays with and withoutmixing has been observed, first in B → J/ψKS decays and relatedmodes (as well as other final CP eigenstates: η′KS , K+K−KS ,J/ψπ0 and π+π−):
AK+π− =|AK−π+/AK+π− |2 − 1|AK−π+/AK+π− |2 + 1
= −0.098± 0.013 (I)
SψK = Im(λψK) = 0.673 ± 0.023 . (III)(12.87)
Searches for additional CP asymmetries are ongoing in B, D, andK decays, and current limits are consistent with Standard Modelexpectations.
Further discussion and all references may be found in the full Review ofParticle Physics.
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13. Neutrino mixing 193
13. NEUTRINO MASS, MIXING, AND OSCILLATIONS
Written May 2010 by K. Nakamura (IPMU, U. Tokyo and KEK) andS.T. Petcov (SISSA/INFN, Trieste and IPMU, U. Tokyo).
I. Massive neutrinos and neutrino mixing. It is a well-establishedexperimental fact that the neutrinos and antineutrinos which take partin the standard charged current (CC) and neutral current (NC) weakinteraction are of three varieties (types) or flavours: electron, νe and νe,muon, νμ and νμ, and tauon, ντ and ντ . The notion of neutrino type orflavour is dynamical: νe is the neutrino which is produced with e+, orproduces an e−, in CC weak interaction processes, etc. The flavour ofa given neutrino is Lorentz invariant. Among the three different flavourneutrinos and antineutrinos, no two are identical.
The experiments with solar, atmospheric, reactor and acceleratorneutrinos [4–16,20,21,22] have provided compelling evidences for theexistence of neutrino oscillations [17,18], transitions in flight betweenthe different flavour neutrinos νe, νμ, ντ (antineutrinos νe, νμ, ντ ),caused by nonzero neutrino masses and neutrino mixing. The existence ofoscillations implies that if a given flavour neutrino, say νμ, with energyE is produced in some weak interaction process, the probability that itwill change into a different flavour neutrino, say ντ , after traveling asufficiently large distance L, P (νμ → ντ ; E, L), is different from zero. Ifthe νμ → ντ oscillation or transition probability P (νμ → ντ ; E, L) �= 0, theprobability that νμ will not change into a neutrino of a different flavour,i.e., the “νμ survival probability”, P (νμ → νμ; E, L), will be smaller thanone. One would observe a “disappearance” of muon neutrinos on the wayfrom the νμ source to the detector if only νμ are detected and they takepart in oscillations.
Oscillations of neutrinos are a consequence of the presence of neutrinomixing, or lepton mixing, in vacuum. In the formalism used to constructthe Standard Model, this means that the LH flavour neutrino fields νlL(x),which enter into the expression for the lepton current in the CC weakinteraction Lagrangian, are given by:
νlL(x) =∑
j
Ulj νjL(x), l = e, μ, τ, (13.1)
where νjL(x) is the LH component of the field of a neutrino νj possessing amass mj and U is a unitary matrix - the neutrino mixing matrix [1,17,18].Eq. (13.1) implies that the individual lepton charges Ll, l = e, μ, τ , are notconserved.
All neutrino oscillation data, except for the LSND result [23], can bedescribed assuming 3-neutrino mixing in vacuum. The number of massiveneutrinos νj , n, can, in general, be bigger than 3 if, e.g., there exist sterileneutrinos [1] and they mix with the flavour neutrinos. It follows from thecurrent data that at least 3 of the neutrinos νj , say ν1, ν2, ν3, must belight, m1,2,3 � 1 eV, and must have different masses, m1 �= m2 �= m3. Atpresent there are no compelling experimental evidences for the existenceof more than 3 light neutrinos.
Being electrically neutral, the massive neutrinos νj can be Dirac orMajorana particles [27,28]. The first possibility is realized when thereexists a lepton charge L carried by νj (e.g., L = Le + Lμ + Lτ , L(νj) = 1),which is conserved by the particle interactions. The neutrino νj has adistinctive antiparticle νj : νj differs from νj by the value of L it carries
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194 13. Neutrino mixing
(L(νj) = − 1). The neutrinos νj can be Majorana particles if no leptoncharge is conserved (see, e.g., Ref. 29). A massive Majorana particle χj isidentical with its antiparticle χj : χj ≡ χj . On the basis of the existingneutrino data it is impossible to determine whether the massive neutrinosare Dirac or Majorana fermions.
In the case of n neutrino flavours and n massive neutrinos, the n × nunitary neutrino mixing matrix U can be parametrised by n(n − 1)/2Euler angles and n(n + 1)/2 phases. If the massive neutrinos νj are Diracparticles, only (n− 1)(n− 2)/2 phases are physical and can be responsiblefor CP violation in the lepton sector. In this respect the neutrino mixingwith Dirac massive neutrinos is similar to the quark mixing. For n = 3there is one CP violating phase in U , “the Dirac CP violating phase.” CPinvariance holds if U is real, U∗ = U .
If, however, the massive neutrinos are Majorana fermions, νj ≡ χj , theneutrino mixing matrix U contains n(n−1)/2 CP violation phases [30,31],i.e., by (n− 1) phases more than in the Dirac neutrino case: in contrast toDirac fields, the massive Majorana neutrino fields cannot “absorb” phases.In this case U can be cast in the form [30]
U = V P (13.2)
where the matrix V contains the (n − 1)(n − 2)/2 Dirac CP violationphases, while P is a diagonal matrix with the additional (n− 1) MajoranaCP violation phases α21, α31,..., αn1,
P = diag(
1, eiα212 , ei
α312 , ..., ei
αn12
)
. (13.3)
The Majorana phases will conserve CP if [32] αj1 = πqj , qj = 0, 1, 2,j = 2, 3, ..., n. In this case exp[i(αj1 −αk1)] = ±1 is the relative CP-parityof Majorana neutrinos χj and χk. The condition of CP invariance of theleptonic CC weak interaction reads [29]:
U∗lj = Ulj ρj , ρj = − i ηCP (χj) = ±1 , (13.4)
where ηCP (χj) is the CP parity of the Majorana neutrino χj [32].In the case of n = 3 there are 3 CP violation phases - one Dirac and
two Majorana. Even in the mixing involving only 2 massive Majorananeutrinos there is one physical CP violation Majorana phase.II. Neutrino oscillations in vacuum. Neutrino oscillations are aquantum mechanical consequence of the existence of nonzero neutrinomasses and neutrino (lepton) mixing, Eq. (13.1), and of the relativelysmall splitting between the neutrino masses.
Suppose the flavour neutrino νl is produced in a CC weak interactionprocess and is observed by a neutrino detector capable of detecting alsoneutrinos νl′ , l′ �= l. If lepton mixing, Eq. (13.1), takes place and themasses mj of all neutrinos νj are sufficiently small, the state of theneutrino νl, |νl〉, will be a coherent superposition of the states |νj〉 ofneutrinos νj :
|νl〉 =∑
j
U∗lj |νj ; pj〉, l = e, μ, τ , (13.5)
pj being the 4-momentum of νj . For the state vector of the flavourantineutrino νl, produced in a weak interaction process, we get:
|νl〉 =∑
j
Ulj |νj ; pj〉 , l = e, μ, τ . (13.7)
Note the presence of U in Eq. (13.5) and U∗ in Eq. (13.7).
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13. Neutrino mixing 195
We will consider the case of relativistic neutrinos νj , which correspondsto the conditions in both past and currently planned future neutrinooscillation experiments [36]. If the spectrum of neutrino masses isnot degenerate, the states |νj ; pj〉 in the r.h.s. of Eq. (13.5) will havedifferent energies and different momenta, independently of the process inwhich they were produced: pj �= pk, or Ej �= Ek, pj �= pk, j �= k, where
Ej =√
p2j + m2
j , pj ≡ |pj |. The deviations of Ej and pj from the values
for a massless neutrino E and p = E, are proportional to m2j/E0, E0 being
a characteristic energy of the process, and are extremely small.Suppose that the neutrinos are observed via a CC weak interaction
process and that in the detector’s rest frame they are detected after timeT after emission, after traveling a distance L. Then the amplitude of theprobability that neutrino νl′ will be observed if neutrino νl was producedby the neutrino source can be written as [33,35,37]:
A(νl → νl′) =∑
j
Ul′j Dj U†jl , l, l′ = e, μ, τ , (13.8)
Dj ≡ Dj(pj ; L, T ) = e−ipj(xf−x0) = e−i(EjT−pjL) , (13.9)
where Dj describes the propagation of νj between the source and thedetector, U†
jl and Ul′j are the amplitudes to find νj in the initial and inthe final flavour neutrino state, respectively, x0 and xf are the space-timecoordinates of the points of neutrino production and detection, T = (tf−t0)and L = k(xf − x0), pj = kpj. What is relevant for the calculation of theprobability P (νl → νl′) = |A(νl → νl′)|2 is the interference factor DjD
∗k.
The latter depends on the phase δϕjk = (Ej − Ek)T − (pj − pk)L, whichcan be cast in the form:
δϕjk = (Ej − Ek)[
T − Ej + Ek
pj + pkL
]
+m2
j − m2k
pj + pkL . (13.10)
The first term in the r.h.s. of Eq. (13.10) vanishes i) if [39]T = (Ej +Ek)L/(pj+pk) = L/v, v = (Ej/(Ej +Ek))vj +(Ek/(Ej +Ek))vkbeing the “average” velocity of νj and νk, where vj,k = pj,k/Ej,k, or ii)if one assumes [40] that Ej = Ek = E0. The relation T = L/v has notemerged so far from any dynamical (wave packet) calculations. If thestates of νj and νj in Eq. (13.5) - Eq. (13.7) have the same 3-momentum[37,41], pj = pk = p, the first term in the r.h.s. of Eq. (13.10) isnegligible, being suppressed by the additional factor (m2
j + m2k)/p2 since
for relativistic neutrinos L = T up to terms ∼ m2j,k/p2. One arrives at the
same conclusion if Ej �= Ek, pj �= pk, j �= k [33].Although the cases considered above are physically quite different, they
lead to the same result for the phase difference δϕjk:
δϕjk∼=
m2j − m2
k
2pL = 2π
L
Lvjk
sgn(m2j − m2
k) , p =pj + pk
2, (13.11)
where Lvjk is the neutrino oscillation length associated with Δm2
jk,
Lvjk = 4π
p
|Δm2jk|
∼= 2.48 mp[MeV ]
|Δm2jk|[eV 2]
(13.12)
We can consider p to be the zero neutrino mass momentum, p = E. Thephase difference δϕjk, Eq. (13.11), is Lorentz-invariant.
Eq. (13.9) corresponds to a plane-wave description of the propagation ofneutrinos νj . In the wave packet treatment of the problem, the interference
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196 13. Neutrino mixing
between the states of νj and νk is subject to a number of conditions, thelocalization condition and the condition of overlapping of the wave packetsof νj and νk at the detection point being the most important (see, e.g.,[33,35,37]) .
For the νl → νl′ and νl → νl′ oscillation probabilities we get fromEq. (13.8), Eq. (13.9), and Eq. (13.11):
P (νl → νl′) =∑
j
Rl′ljj + 2
∑
j>k
|Rl′ljk| cos(
Δm2jk
2pL − φl′l
jk), (13.13)
P (νl → νl′) =∑
j
Rl′ljj + 2
∑
j>k
|Rl′ljk| cos(
Δm2jk
2pL + φl′l
jk), (13.14)
where l, l′ = e, μ, τ , Rl′ljk = Ul′j U∗
lj Ulk U∗l′k and φl′l
jk = arg(
Rl′ljk
)
.
It follows from Eq. (13.8) - Eq. (13.10) that in order for neutrinooscillations to occur, at least two neutrinos νj should not be degeneratein mass and lepton mixing should take place, U �= 1. The oscillationseffects can be large if at least for one Δm2
jk we have |Δm2jk|L/(2p) =
2πL/Lvjk � 1, i.e. the oscillation length Lv
jk is of the order of, or smaller,than source-detector distance L (otherwise the oscillations will not havetime to develop before neutrinos reach the detector).
We see from Eq. (13.13) and Eq. (13.14) that P (νl → νl′) = P (νl′ → νl).This is a consequence of CPT invariance. The conditions of CP invarianceread [30,42,43]: P (νl → νl′) = P (νl → νl′), l, l′ = e, μ, τ . In the case ofCPT invariance, which we will assume to hold, we get for the survivalprobabilities: P (νl → νl) = P (νl → νl), l, l′ = e, μ, τ . Thus, the studyof the “disappearance” of νl and νl, caused by oscillations in vacuum,cannot be used to test the CP invariance in the lepton sector. It followsfrom Eq. (13.13) - Eq. (13.14) that we can have CP violation effects inneutrino oscillations only if U is not real. Eq. (13.2) and Eq. (13.13) -Eq. (13.14) imply that P (νl → νl′) and P (νl → νl′) do not depend onthe Majorana phases in the neutrino mixing matrix U [30]. Thus, i) inthe case of oscillations in vacuum, only the Dirac phase(s) in U can causeCP violating effects leading to P (νl → νl′) �= P (νl → νl′), l �= l′, andii) the experiments investigating the νl → νl′ and νl → νl′ oscillationscannot provide information on the nature - Dirac or Majorana, of massiveneutrinos [30,44].
As a measure of CP violation in neutrino oscillations we can considerthe asymmetry: A
(l′l)CP = P (νl → νl′) − P (νl → νl′) = −A
(ll′)CP . In the case
of 3-neutrino mixing one has [45]: A(μe)CP = −A
(τe)CP = A
(τμ)CP ,
A(μe)CP = 4 JCP
(
sinΔm2
32
2pL + sin
Δm221
2pL + sin
Δm213
2pL
)
, (13.18)
where JCP = Im(
Uμ3U∗e3Ue2U
∗μ2
)
is analogous to the rephasing invariantassociated with the CP violation in the quark mixing [46]. Thus, JCPcontrols the magnitude of CP violation effects in neutrino oscillations inthe case of 3-neutrino mixing. Even if JCP �= 0, we will have A
(l′l)CP = 0
unless all three sin(Δm2ij/(2p))L �= 0 in Eq. (13.18).
Consider next neutrino oscillations in the case of one neutrinomass squared difference “dominance”: suppose that |Δm2
j1| � |Δm2n1|,
j = 2, ..., (n − 1), |Δm2n1|L/(2p)�1 and |Δm2
j1|L/(2p) � 1, so that
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13. Neutrino mixing 197
exp[i(Δm2j1 L/(2p)] ∼= 1, j = 2, ..., (n − 1). Under these conditions we
obtain from Eq. (13.13) and Eq. (13.14), keeping only the oscillating termsinvolving Δm2
n1: P (νl(l′) → νl′(l)) ∼= P (νl(l′) → νl′(l)),
P (νl(l′) → νl′(l)) ∼= δll′ − 4|Uln|2[
δll′ − |Ul′n|2]
sin2 Δm2n1
4pL . (13.20)
It follows from the neutrino oscillation data that in the case of 3-neutrinomixing, one of the two independent neutrino mass squared differences,say Δm2
21, is much smaller in absolute value than the second one, Δm231:
|Δm221|/|Δm2
31| ∼= 0.032, |Δm231| ∼= 2.4 × 10−3 eV2. Eq. (13.20) with
n = 3, describes with a relatively good precision the oscillations of i)reactor νe ( l, l′ = e) on a distance L ∼ 1 km, corresponding to theCHOOZ and the Double Chooz, Daya Bay and RENO experiments, and ofii) the accelerator νμ (l, l′ = μ), seen in the K2K and MINOS experiments.The νμ → ντ oscillations, which the OPERA experiment is aiming todetect, can be described in the case of 3-neutrino mixing by Eq. (13.20)with n = 3 and l = μ, l′ = τ .
In certain cases the dimensions of the neutrino source, ΔL, and/orthe energy resolution of the detector, ΔE, have to be included inthe analysis of the neutrino oscillation data. If [29] 2πΔL/Lv
jk � 1,and/or 2π(L/Lv
jk)(ΔE/E) � 1, the interference terms in P (νl → νl′) andP (νl′ → νl) will be strongly suppressed and the neutrino flavour conversionwill be determined by the average probabilities: P (νl → νl′) = P (νl →νl′) ∼=
∑
j |Ul′j |2 |Ulj |2. Suppose next that in the case of 3-neutrino mixing,|Δm2
21|L/(2p) ∼ 1, while |Δm231(32)|L/(2p) � 1, and the oscillations due
to Δm231(32) are strongly suppressed (averaged out) due to integration
over the region of neutrino production, etc. In this case we get for the νe
and νe survival probabilities: P (νe → νe) = P (νe → νe) ≡ Pee,
Pee∼= |Ue3|4 +
(
1 − |Ue3|2)2
[
1 − sin2 2θ12 sin2 Δm221
4pL
]
(13.26)
with θ12 determined by cos2 θ12 = |Ue1|2/(1 − |Ue3|2), sin2 θ12 =|Ue2|2/(1 − |Ue3|2). Eq. (13.26) describes the effects of reactor νe
oscillations observed by the KamLAND experiment (L ∼ 180 km).The data of ν-oscillations experiments is often analyzed assuming
2-neutrino mixing: |νl〉 = |ν1〉 cos θ + |ν2〉 sin θ, |νx〉 = −|ν1〉 sin θ +|ν2〉 cos θ, where θ is the neutrino mixing angle in vacuum and νx isanother flavour neutrino or sterile (anti-) neutrino, x = l′ �= l or νx ≡ νs.In this case we have [41]: Δm2 = m2
2 − m21 > 0,
P 2ν(νl → νl) = 1 − sin2 2θ sin2 πL
Lv , Lv = 4π p/Δm2 , (13.30)
P 2ν(νl → νx) = 1−P 2ν(νl → νl). Eq. (13.30) with l = μ, x = τ was used,e.g., in the atmospheric neutrino data analysis [13], in which the firstcompelling evidence for neutrino oscillations was obtained.III. Matter effects in neutrino oscillations. When neutrinospropagate in matter (e.g., in the Earth, Sun or a supernova), theircoherent forward-scattering from the particles present in matter canchange drastically the pattern of neutrino oscillations [25,26,52]. Thus,the probabilities of neutrino transitions in matter can differ significantlyfrom the corresponding vacuum oscillation probabilities.
In the case of, e.g., solar νe transitions in the Sun and 3-neutrinomixing, the oscillations due to Δm2
31 are strongly suppressed by the
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198 13. Neutrino mixing
Figure 13.10: The regions of squared-mass splitting and mixingangle favored or excluded by various experiments. The figure wascontributed by H. Murayama (University of California, Berkeley, andIPMU, University of Tokyo).
averaging over the region of neutrino production in the Sun. The νe
undergo transitions into (νμ + ντ )/√
2. As a consequence of the effectsof the solar matter, the solar νe transitions observed by the Super-Kamiokande and SNO experiments exhibit a characteristic dependenceon sin2 θ12: P 3ν� (νe → νe) ∼= |Ue3|4 + (1 − |Ue3|2)2 sin2 θ12. The datashow that P 3ν� ∼= 0.3, which is a strong evidence for solar mattereffects in the transitions [85] since in the case of transitions in vacuumP 3ν� ∼= |Ue3|4 + (1 − |Ue3|2)2(1 − 0.5 sin2 2θ12)� 0.48, where we used theexperimental limits |Ue3|2 < 0.056 and sin2 2θ12 � 0.93.IV. The evidence for flavour neutrino oscillations. The evidencefor flavour neutrino oscillations/transitions is compelling. We discuss therelevant data in the full edition. The regions of the oscillation parameterspace favored or excluded by various neutrino oscillation experiments areshown in Fig. 13.10.
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13. Neutrino mixing 199
V. Three neutrino mixing. All compelling data on neutrino oscillationscan be described assuming 3-flavour neutrino mixing in vacuum. In thiscase U can be parametrised as
U =
⎡
⎣
c12c13 s12c13 s13e−iδ
−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e
iδ s23c13s12s23 − c12c23s13e
iδ −c12s23 − s12c23s13eiδ c23c13
⎤
⎦
× diag(1, eiα212 , ei
α312 ) . (13.77)
where cij = cos θij , sij = sin θij , the angles θij = [0, π/2], δ = [0, 2π] is theDirac CP violation phase and α21, α31 are the two Majorana phases. Theexisting neutrino oscillation data allow us to determine the parameterswhich drive the solar νe and the dominant atmospheric νμ → ντ
oscillations, Δm221 > 0, sin2 θ12, and |Δm2
31|, sin2 2θ23, with a relativelygood precision, and to obtain rather stringent limit on the angle θ13:Δm2
21∼= 7.65 × 10−5 eV 2, sin2 θ12
∼= 0.304, |Δm231| ∼= 2.40 × 10−3 eV 2,
sin2 2θ23∼= 1, sin2 θ13 < 0.056 (at 3σ). These results imply that
Δm221 � |Δm2
31| and that θ23∼= π/4, θ12
∼= π/5.4 and θ13 < π/13. Thus,the pattern of neutrino mixing is drastically different from the pattern ofquark mixing.
The existing neutrino oscillation data do not allow to determine thesign of Δm2
31(32). Correspondingly, two types of neutrino mass spectrum
are possible: i) with normal ordering: m1 < m2 < m3, Δm231 > 0,
Δm221 > 0, m2(3)=(m2
1 + Δm221(31))
12 ; ii) with inverted ordering (IO):
m3 < m1 < m2, Δm232(31) < 0, Δm2
21 > 0, m2=(m23 + Δm2
23)12 , m1=
(m23 + Δm2
23 − Δm221)
12 . Depending on the values of the lightest neutrino
mass [113], min(mj), the neutrino mass spectrum can also be: a) normalhierarchical (NH): m1 � m2 < m3; or b) inverted hierarchical (IH):m3 � m1 < m2; or c) quasi-degenerate (QD): m1
∼= m2∼= m3 ≡ m0,
m2j � |Δm2
21,31|, m0 � 0.10 eV. All three types of spectrum are compatiblewith the existing data.
At present no experimental information on the CP violation phases inthe mixing matrix U is available. The JCP factor (see Eq. (13.18)) inthe “standard” parametrisation (Eq. (13.77)) of U has the form: JCP =(1/8) cos θ13 sin 2θ12 sin 2θ23 sin 2θ13 sin δ. Thus, the size of CP violationeffects in neutrino oscillations depends on the currently unknown values ofthe “small” angle θ13 and the Dirac phase δ.
The Majorana nature of massive neutrinos νj manifests itself in theexistence of |ΔL| = 2 processes: μ− + (A, Z) → μ+ + (A, Z − 2), etc.The latter can be triggered by the exchange of the neutrino νj if itis identical with its antiparticle. The only feasible experiments havingthe potential of establishing that the massive neutrinos are Majoranaparticles are at present the experiments searching for (ββ)0ν -decay:(A, Z) → (A, Z + 2) + e− + e− (see e.g., Ref. 124). One may expect thatthe dominant contribution in the (ββ)0ν -decay amplitude is due to theexchange of the light Majorana neutrinos νj . In this case the (ββ)0ν -decayrate is proportional to | < m > |2, where < m > is the “(ββ)0ν -decayeffective Majorana mass”, | <m> | = |∑j U2
ej mj |.Determining the type of neutrino mass spectrum, the status of the CP
symmetry in the lepton sector and the nature of massive neutrinos areamong the major goals of the future studies in neutrino physics.
For details and references, see the full Review.
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200 14. Quark model
14. QUARK MODEL
Revised September 2009 by C. Amsler (University of Zurich), T. DeGrand(University of Colorado, Boulder), and B. Krusche (University of Basel).
14.1. Quantum numbers of the quarks
Quarks are strongly interacting fermions with spin 1/2 and, byconvention, positive parity. Antiquarks have negative parity. Quarks havethe additive baryon number 1/3, antiquarks -1/3. Table 14.1 gives theother additive quantum numbers (flavors) for the three generations ofquarks. They are related to the charge Q (in units of the elementarycharge e) through the generalized Gell-Mann-Nishijima formula
Q = Iz +B + S + C + B + T
2, (14.1)
where B is the baryon number. The convention is that the flavor of aquark (Iz , S, C, B, or T) has the same sign as its charge Q. With thisconvention, any flavor carried by a charged meson has the same sign as itscharge, e.g., the strangeness of the K+ is +1, the bottomness of the B+
is +1, and the charm and strangeness of the D−s are each −1. Antiquarks
have the opposite flavor signs.
14.2. Mesons
Mesons have baryon number B = 0. In the quark model, they are qq ′bound states of quarks q and antiquarks q ′ (the flavors of q and q′ maybe different). If the orbital angular momentum of the qq ′ state is �, thenthe parity P is (−1)�+1. The meson spin J is given by the usual relation|� − s| ≤ J ≤ |� + s|, where s is 0 (antiparallel quark spins) or 1 (parallelquark spins). The charge conjugation, or C-parity C = (−1)�+s, is definedonly for the qq states made of quarks and their own antiquarks. TheC-parity can be generalized to the G-parity G = (−1)I+�+s for mesonsmade of quarks and their own antiquarks (isospin Iz = 0), and for thecharged ud and du states (isospin I = 1).
The mesons are classified in JPC multiplets. The � = 0 states are thepseudoscalars (0−+) and the vectors (1−−). The orbital excitations � = 1are the scalars (0++), the axial vectors (1++) and (1+−), and the tensors(2++). Assignments for many of the known mesons are given in Tables14.2 and 14.3. Radial excitations are denoted by the principal quantumnumber n. The very short lifetime of the t quark makes it likely thatbound-state hadrons containing t quarks and/or antiquarks do not exist.
States in the natural spin-parity series P = (−1)J must, according tothe above, have s = 1 and hence, CP = +1. Thus, mesons with naturalspin-parity and CP = −1 (0+−, 1−+, 2+−, 3−+, etc.) are forbidden inthe qq ′ model. The JPC = 0−− state is forbidden as well. Mesons withsuch exotic quantum numbers may exist, but would lie outside the qq ′model (see section below on exotic mesons).
Following SU(3), the nine possible qq ′ combinations containing thelight u, d, and s quarks are grouped into an octet and a singlet of lightquark mesons:
3 ⊗ 3 = 8⊕ 1 . (14.2)
A fourth quark such as charm c can be included by extending SU(3)to SU(4). However, SU(4) is badly broken owing to the much heavier c
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14. Quark model 201
quark. Nevertheless, in an SU(4) classification, the sixteen mesons aregrouped into a 15-plet and a singlet:
4⊗ 4 = 15⊕ 1 . (14.3)
The weight diagrams for the ground-state pseudoscalar (0−+) andvector (1−−) mesons are depicted in Fig. 14.1. The light quark mesonsare members of nonets building the middle plane in Fig. 14.1(a) and (b).
sD0D
sD
–D
0K–π π
������������������������������������������
������������������������������������������������������������������������������������
���������������������������������������������������������������������������������������������������������������������������������������������
���������������������������������������������������������������
+K
–K
(a)
sD
DD
sD
−ρ +ρ�����������������������������������
����������������������������������������������������������������������
���������������������������������������������������������������������������������������������������������������������������������������������
���������������������������������������������������������������
K
(b)
*0
K*−
*+K*0
D 0*D*−
*−
*+
−
*+
−cdcu−cs−
us−ds−
su− sd− ud
−
uc−sc−
dc−
0ρ ωφψJ/
uc−sc−
dc−
−cdcu−cs−
+
D+
+
K 0
us−ds−
su− sd−
du−
du−
0D
ηη′ηc
π 0
ud−
K 0*
C
I
Y
Figure 14.1: SU(4) weight diagram showing the 16-plets for thepseudoscalar (a) and vector mesons (b) made of the u, d, s, and cquarks as a function of isospin I, charm C, and hypercharge Y =S+B − C
3. The nonets of light mesons occupy the central planes to
which the cc states have been added.
Isoscalar states with the same JPC will mix, but mixing between thetwo light quark isoscalar mesons, and the much heavier charmonium orbottomonium states, are generally assumed to be negligible.14.3. Baryons: qqq states
Baryons are fermions with baryon number B = 1, i.e., in the mostgeneral case, they are composed of three quarks plus any number ofquark - antiquark pairs. Although recently some experimental evidencefor (qqqqq) pentaquark states has been claimed (see review on PossibleExotic Baryon Resonance), so far all established baryons are 3-quark (qqq)configurations. The color part of their state functions is an SU(3) singlet,a completely antisymmetric state of the three colors. Since the quarks arefermions, the state function must be antisymmetric under interchange ofany two equal-mass quarks (up and down quarks in the limit of isospinsymmetry). Thus it can be written as
| qqq 〉A = | color 〉A × | space, spin, flavor 〉S , (14.21)
where the subscripts S and A indicate symmetry or antisymmetry underinterchange of any two equal-mass quarks. Note the contrast with thestate function for the three nucleons in 3H or 3He:
|NNN 〉A = | space, spin, isospin 〉A . (14.22)
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202 14. Quark model
This difference has major implications for internal structure, magneticmoments, etc.
The “ordinary” baryons are made up of u, d, and s quarks. The threeflavors imply an approximate flavor SU(3), which requires that baryonsmade of these quarks belong to the multiplets on the right side of
3⊗ 3⊗ 3 = 10S ⊕ 8M ⊕ 8M ⊕ 1A . (14.23)
Ω++ccc
Ξ ++ccΞ +
cc
Ω+cc
Σ ++c
Ξ +cΞ 0
c
Ω −Ξ 0
Σ +
Δ+Δ0Δ−
Σ −
Ξ −
Δ++
(b)
Ξ +c
Σ ++c
Ξ 0
n pΞ c
0
(a)
ddcdsc
udc
uscuuc
uuduus
ussdss
udddds
ddd
dssdds
ussuus
uududduds
ssc
uscdscuuc
uccscc
dcc
Ω+cc
Ξ ++ccΞ +
cc
Σ 0c
uuu
Σ 0
Ξ −
Σ −Σ +Λ,Σ 0
udc
Σ +cΛ+
c,
cΣ +
Ω 0c
Σ 0c
dcc ucc
ddc
uds
ssc
scc
sss
Ω 0c
Figure 14.2: SU(4) multiplets of baryons made of u, d, s, and cquarks. (a) The 20-plet with an SU(3) octet. (b) The 20-plet withan SU(3) decuplet.
Here the subscripts indicate symmetric, mixed-symmetry, or anti-symmetric states under interchange of any two quarks. The 1 is a udsstate (Λ1), and the octet contains a similar state (Λ8). If these have thesame spin and parity, they can mix. The mechanism is the same as forthe mesons (see above). In the ground state multiplet, the SU(3) flavorsinglet Λ1 is forbidden by Fermi statistics. Section 37, on “SU(3) IsoscalarFactors and Representation Matrices,” shows how relative decay rates in,say, 10 → 8⊗ 8 decays may be calculated. The addition of the c quark tothe light quarks extends the flavor symmetry to SU(4). However, due tothe large mass of the c quark, this symmetry is much more strongly brokenthan the SU(3) of the three light quarks. Figures 14.2(a) and 14.2(b) showthe SU(4) baryon multiplets that have as their bottom levels an SU(3)octet, such as the octet that includes the nucleon, or an SU(3) decuplet,such as the decuplet that includes the Δ(1232). All particles in a givenSU(4) multiplet have the same spin and parity. The charmed baryons arediscussed in more detail in the “Note on Charmed Baryons” in the ParticleListings. The addition of a b quark extends the flavor symmetry to SU(5);the existence of baryons with t-quarks is very unlikely due to the shortlifetime of the top.
For further details, see the full Review of Particle Physics.
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16. Structure functions 203
16. STRUCTURE FUNCTIONS
Updated September 2007 by B. Foster (University of Oxford), A.D. Martin(University of Durham), and M.G. Vincter (Carleton University).
16.1. Deep inelastic scatteringHigh-energy lepton-nucleon scattering (deep inelastic scattering) plays
a key role in determining the partonic structure of the proton. Theprocess �N → �′X is illustrated in Fig. 16.1. The filled circle in this figurerepresents the internal structure of the proton which can be expressed interms of structure functions.
k
k
q
P, M W
Figure 16.1: Kinematic quantities for the description of deepinelastic scattering. The quantities k and k′ are the four-momentaof the incoming and outgoing leptons, P is the four-momentum of anucleon with mass M , and W is the mass of the recoiling system X .The exchanged particle is a γ, W±, or Z; it transfers four-momentumq = k − k′ to the nucleon.
Invariant quantities:
ν =q · PM
= E − E′ is the lepton’s energy loss in the nucleon rest frame(in earlier literature sometimes ν = q · P ). Here, E and E′are the initial and final lepton energies in the nucleon restframe.
Q2 = −q2 = 2(EE′ − −→k · −→k ′) − m2
� − m2�′ where m�(m�′) is the initial
(final) lepton mass. If EE′ sin2(θ/2) � m2� , m2
�′ , then
≈ 4EE′ sin2(θ/2), where θ is the lepton’s scattering angle with respectto the lepton beam direction.
x =Q2
2Mνwhere, in the parton model, x is the fraction of the nucleon’s
momentum carried by the struck quark.
y =q · Pk · P =
ν
Eis the fraction of the lepton’s energy lost in the nucleon
rest frame.
W 2 = (P + q)2 = M2 + 2Mν − Q2 is the mass squared of the system Xrecoiling against the scattered lepton.
s = (k + P )2 =Q2
xy+ M2 + m2
� is the center-of-mass energy squared of the
lepton-nucleon system.
The process in Fig. 16.1 is called deep (Q2 � M2) inelastic (W 2 � M2)scattering (DIS). In what follows, the masses of the initial and scatteredleptons, m� and m�′ , are neglected.
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204 16. Structure functions
16.1.1. DIS cross sections :d2σ
dx dy= x (s − M2)
d2σ
dx dQ2 =2π Mν
E′d2σ
dΩNrest dE′ . (16.1)
In lowest-order perturbation theory, the cross section for the scattering ofpolarized leptons on polarized nucleons can be expressed in terms of theproducts of leptonic and hadronic tensors associated with the coupling ofthe exchanged bosons at the upper and lower vertices in Fig. 16.1 (seeRefs. 1–4)
d2σ
dxdy=
2πyα2
Q4
∑
j
ηj Lμνj W j
μν . (16.2)
For neutral-current processes, the summation is over j = γ, Z and γZrepresenting photon and Z exchange and the interference between them,whereas for charged-current interactions there is only W exchange,j = W . (For transverse nucleon polarization, there is a dependence onthe azimuthal angle of the scattered lepton.) Lμν is the lepton tensorassociated with the coupling of the exchange boson to the leptons. Forincoming leptons of charge e = ±1 and helicity λ = ±1,
Lγμν = 2
(
kμk′ν + k′μkν − k · k′gμν − iλεμναβkαk′β)
,
LγZμν =(ge
V + eλgeA) Lγ
μν , LZμν = (ge
V + eλgeA)2 Lγ
μν ,
LWμν =(1 + eλ)2 Lγ
μν , (16.3)
where geV = − 1
2+ 2 sin2 θW , ge
A = − 12
. Although here the helicityformalism is adopted, an alternative approach is to express the tensors inEq. (16.3) in terms of the polarization of the lepton.
The factors ηj in Eq. (16.2) denote the ratios of the correspondingpropagators and couplings to the photon propagator and coupling squared
ηγ = 1 ; ηγZ =
(
GF M2Z
2√
2πα
) (
Q2
Q2 + M2Z
)
;
ηZ = η2γZ ; ηW = 1
2
(
GF M2W
4πα
Q2
Q2 + M2W
)2
. (16.4)
The hadronic tensor, which describes the interaction of the appropriateelectroweak currents with the target nucleon, is given by
Wμν =14π
∫
d4z eiq·z ⟨
P, S∣
∣
∣
[
J†μ(z), Jν(0)
]∣
∣
∣ P, S⟩
, (16.5)
where S denotes the nucleon-spin 4-vector, with S2 = −M2 and S ·P = 0.16.2. Structure functions of the proton
The structure functions are defined in terms of the hadronic tensor (seeRefs. 1–3)
Wμν =(
−gμν +qμqν
q2
)
F1(x, Q2) +PμPν
P · q F2(x, Q2)
− iεμναβqαPβ
2P · q F3(x, Q2)
+ iεμναβqα
P · q[
Sβg1(x, Q2) +(
Sβ − S · qP · q Pβ
)
g2(x, Q2)]
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16. Structure functions 205
+1
P · q[
12
(
PμSν + SμPν
)
− S · qP · q PμPν
]
g3(x, Q2)
+S · qP · q
[
PμPν
P · q g4(x, Q2) +(
−gμν +qμqν
q2
)
g5(x, Q2)
]
(16.6)
where
Pμ = Pμ − P · qq2 qμ, Sμ = Sμ − S · q
q2 qμ . (16.7)
The cross sections for neutral- and charged-current deep inelasticscattering on unpolarized nucleons can be written in terms of the structurefunctions in the generic form
d2σi
dxdy=
4πα2
xyQ2ηi
{(
1 − y − x2y2M2
Q2
)
F i2
+ y2xF i1 ∓
(
y − y2
2
)
xF i3
}
, (16.8)
where i = NC, CC corresponds to neutral-current (eN → eX) or charged-current (eN → νX or νN → eX) processes, respectively. For incomingneutrinos, LW
μν of Eq. (16.3) is still true, but with e, λ corresponding tothe outgoing charged lepton. In the last term of Eq. (16.8), the − sign istaken for an incoming e+ or ν and the + sign for an incoming e− or ν.The factor ηNC = 1 for unpolarized e± beams, whereas
ηCC = (1 ± λ)2ηW (16.9)
with ± for �±; and where λ is the helicity of the incoming lepton andηW is defined in Eq. (16.4); for incoming neutrinos ηCC = 4ηW . The CCstructure functions, which derive exclusively from W exchange, are
FCC1 = FW
1 , FCC2 = FW
2 , xFCC3 = xFW
3 . (16.10)
The NC structure functions Fγ2 , F
γZ2 , FZ
2 are, for e±N → e±X , given byRef. [5],
FNC2 = F
γ2 − (ge
V ±λgeA)ηγZF
γZ2 + (ge 2
V +ge 2A ±2λge
V geA) ηZFZ
2 (16.11)
and similarly for FNC1 , whereas
xFNC3 = −(ge
A±λgeV )ηγZxF
γZ3 +[2ge
V geA±λ(ge 2
V +ge 2A )]ηZxFZ
3 . (16.12)
The polarized cross-section differenceΔσ = σ(λn = −1, λ�) − σ(λn = 1, λ�) , (16.13)
where λ�, λn are the helicities (±1) of the incoming lepton and nucleon,respectively, may be expressed in terms of the five structure functionsg1,...5(x, Q2) of Eq. (16.6). Thus,
d2Δσi
dxdy=
8πα2
xyQ2ηi
{
−λ�y
(
2 − y − 2x2y2 M2
Q2
)
xgi1 + λ�4x3y2 M2
Q2gi2
+ 2x2yM2
Q2
(
1 − y − x2y2 M2
Q2
)
gi3
−(
1 + 2x2yM2
Q2
) [(
1 − y − x2y2 M2
Q2
)
gi4 + xy2gi
5
]}
(16.14)
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206 16. Structure functions
with i = NC or CC as before. In the M2/Q2 → 0 limit, Eq. (16.8) andEq. (16.14) may be written in the form
d2σi
dxdy=
2πα2
xyQ2 ηi[
Y+F i2 ∓ Y−xF i
3 − y2F iL
]
,
d2Δσi
dxdy=
4πα2
xyQ2ηi
[
−Y+gi4 ∓ Y−2xgi
1 + y2giL
]
, (16.16)
with i = NC or CC, where Y± = 1 ± (1 − y)2 and
F iL = F i
2 − 2xF i1 , gi
L = gi4 − 2xgi
5 . (16.17)
In the naive quark-parton model, the analogy with the Callan-Grossrelations [6] F i
L = 0, are the Dicus relations [7] giL = 0. Therefore,
there are only two independent polarized structure functions: g1 (parityconserving) and g5 (parity violating), in analogy with the unpolarizedstructure functions F1 and F3.16.2.1. Structure functions in the quark-parton model :
In the quark-parton model [8,9], contributions to the structurefunctions F i and gi can be expressed in terms of the quark distributionfunctions q(x, Q2) of the proton, where q = u, u, d, d etc.The quantityq(x, Q2)dx is the number of quarks (or antiquarks) of designated flavorthat carry a momentum fraction between x and x + dx of the proton’smomentum in a frame in which the proton momentum is large.
For the neutral-current processes ep → eX ,[
F γ2 , F γZ
2 , FZ2
]
= x∑
q
[
e2q , 2eqg
qV , gq 2
V + gq 2A
]
(q + q) ,
[
Fγ3 , F
γZ3 , FZ
3
]
=∑
q
[
0, 2eqgqA, 2g
qV g
qA
]
(q − q) ,
[
gγ1 , gγZ
1 , gZ1
]
= 12
∑
q
[
e2q , 2eqg
qV , gq 2
V + gq 2A
]
(Δq + Δq) ,
[
gγ5 , g
γZ5 , gZ
5
]
=∑
q
[
0, eqgqA, g
qV g
qA
]
(Δq − Δq) , (16.18)
where gqV = ± 1
2− 2eq sin2 θW and g
qA = ± 1
2, with ± according to whether
q is a u− or d−type quark respectively. The quantity Δq is the differenceq ↑ −q ↓ of the distributions with the quark spin parallel and antiparallelto the proton spin.
For the charged-current processes e−p → νX and νp → e+X , thestructure functions are:FW−
2 = 2x(u + d + s + c . . .) , FW−3 = 2(u − d − s + c . . .) , (16.19)
gW−1 = (Δu + Δd + Δs + Δc . . .) , gW−
5 = (−Δu + Δd + Δs − Δc . . .) ,
where only the active flavors are to be kept and where CKM mixing hasbeen neglected. For e+p → νX and νp → e−X , the structure functionsFW+
, gW+are obtained by the flavor interchanges d ↔ u, s ↔ c in the
expressions for FW−, gW−
. The structure functions for scattering on aneutron are obtained from those of the proton by the interchange u ↔ d.For both the neutral- and charged-current processes, the quark-partonmodel predicts 2xF i
1 = F i2 and gi
4 = 2xgi5.
Further discussion may be found in the full Review of Particle Physics.
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16. Structure functions 207
Q2 (GeV2)
F2(
x,Q
2 ) *
2i x
H1ZEUSBCDMSE665NMCSLAC
10-3
10-2
10-1
1
10
10 2
10 3
10 4
10 5
10 6
10 7
10 8
10 9
10-1
1 10 102
103
104
105
106
Figure 16.7: The proton structure function Fp2 measured
in electromagnetic scattering of positrons on protons (colliderexperiments ZEUS and H1), in the kinematic domain of the HERAdata, for x > 0.00006 (cf. Fig. 16.10 for data at smaller x and Q2),and for electrons (SLAC) and muons (BCDMS, E665, NMC) on afixed target. Statistical and systematic errors added in quadratureare shown. The data are plotted as a function of Q2 in bins of fixedx. Some points have been slightly offset in Q2 for clarity. The ZEUSbinning in x is used in this plot; all other data are rebinned to the xvalues of the ZEUS data. For the purpose of plotting, F
p2 has been
multiplied by 2ix , where ix is the number of the x bin, ranging fromix = 1 (x = 0.85) to ix = 28 (x = 0.000063).
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208 19. Big-Bang cosmology
19. BIG-BANG COSMOLOGY
Revised September 2009 by K.A. Olive (University of Minnesota) andJ.A. Peacock (University of Edinburgh).
19.1. Introduction to Standard Big-Bang ModelThe observed expansion of the Universe [1,2,3] is a natural (almost
inevitable) result of any homogeneous and isotropic cosmological modelbased on general relativity. In order to account for the possibility that theabundances of the elements had a cosmological origin, Alpher and Hermanproposed that the early Universe which was once very hot and dense(enough so as to allow for the nucleosynthetic processing of hydrogen),and has expanded and cooled to its present state [4,5]. In 1948, Alpherand Herman predicted that a direct consequence of this model is thepresence of a relic background radiation with a temperature of order afew K [6,7]. It was the observation of the 3 K background radiation thatsingled out the Big-Bang model as the prime candidate to describe ourUniverse. Subsequent work on Big-Bang nucleosynthesis further confirmedthe necessity of our hot and dense past. These relativistic cosmologicalmodels face severe problems with their initial conditions, to which the bestmodern solution is inflationary cosmology.19.1.1. The Robertson-Walker Universe :
The observed homogeneity and isotropy enable us to describe the overallgeometry and evolution of the Universe in terms of two cosmologicalparameters accounting for the spatial curvature and the overall expansion(or contraction) of the Universe. These two quantities appear in the mostgeneral expression for a space-time metric which has a (3D) maximallysymmetric subspace of a 4D space-time, known as the Robertson-Walkermetric:
ds2 = dt2 − R2(t)[
dr2
1 − kr2+ r2 (dθ2 + sin2 θ dφ2)
]
. (19.1)
Note that we adopt c = 1 throughout. By rescaling the radial coordinate,we can choose the curvature constant k to take only the discrete values+1, −1, or 0 corresponding to closed, open, or spatially flat geometries.
19.1.2. The redshift :The cosmological redshift is a direct consequence of the Hubble
expansion, determined by R(t). A local observer detecting light from adistant emitter sees a redshift in frequency. We can define the redshift as
z ≡ ν1 − ν2
ν2� v12
c, (19.3)
where ν1 is the frequency of the emitted light, ν2 is the observed frequencyand v12 is the relative velocity between the emitter and the observer.While the definition, z = (ν1 − ν2)/ν2 is valid on all distance scales,relating the redshift to the relative velocity in this simple way is only trueon small scales (i.e., less than cosmological scales) such that the expansionvelocity is non-relativistic. For light signals, we can use the metric givenby Eq. (19.1) and ds2 = 0 to write
1 + z =ν1
ν2=
R2
R1. (19.5)
This result does not depend on the non-relativistic approximation.19.1.3. The Friedmann-Lemaıtre equations of motion :
The cosmological equations of motion are derived from Einstein’sequations
Rμν − 12gμνR = 8πGNTμν + Λgμν . (19.6)
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19. Big-Bang cosmology 209
Gliner [17] and Zeldovich [18] have pioneered the modern view, inwhich the Λ term is taken to the rhs and interpreted as an effectiveenergy–momentum tensor Tμν for the vacuum of Λgμν/8πGN. It iscommon to assume that the matter content of the Universe is a perfectfluid, for which
Tμν = −pgμν + (p + ρ)uμuν , (19.7)
where gμν is the space-time metric described by Eq. (19.1), p is theisotropic pressure, ρ is the energy density and u = (1, 0, 0, 0) is thevelocity vector for the isotropic fluid in co-moving coordinates. With theperfect fluid source, Einstein’s equations lead to the Friedmann-Lemaıtreequations
H2 ≡(
R
R
)2
=8π GN ρ
3− k
R2 +Λ3
, (19.8)
andR
R=
Λ3− 4πGN
3(ρ + 3p) , (19.9)
where H(t) is the Hubble parameter and Λ is the cosmological constant.The first of these is sometimes called the Friedmann equation. Energyconservation via T
μν;μ = 0, leads to a third useful equation
ρ = −3H (ρ + p) . (19.10)Eq. (19.10) can also be simply derived as a consequence of the first lawof thermodynamics. For Λ = 0, it is clear that the Universe must beexpanding or contracting.
19.1.4. Definition of cosmological parameters :The Friedmann equation can be used to define a critical density such
that k = 0 when Λ = 0,
ρc ≡ 3H2
8π GN= 1.88 × 10−26 h2 kg m−3
= 1.05 × 10−5 h2 GeV cm−3 ,
(19.11)
where the scaled Hubble parameter, h, is defined byH ≡ 100 h km s−1 Mpc−1
⇒ H−1 = 9.78 h−1 Gyr
= 2998 h−1 Mpc .
(19.12)
The cosmological density parameter Ωtot is defined as the energy densityrelative to the critical density,
Ωtot = ρ/ρc . (19.13)Note that one can now rewrite the Friedmann equation as
k/R2 = H2(Ωtot − 1) . (19.14)From Eq. (19.14), one can see that when Ωtot > 1, k = +1 and theUniverse is closed, when Ωtot < 1, k = −1 and the Universe is open, andwhen Ωtot = 1, k = 0, and the Universe is spatially flat.
It is often necessary to distinguish different contributions to thedensity. It is therefore convenient to define present-day density parametersfor pressureless matter (Ωm) and relativistic particles (Ωr), plus thequantity ΩΛ = Λ/3H2. In more general models, we may wish to drop theassumption that the vacuum energy density is constant, and we thereforedenote the present-day density parameter of the vacuum by Ωv. TheFriedmann equation then becomes
k/R20 = H2
0 (Ωm + Ωr + Ωv − 1) , (19.15)
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210 19. Big-Bang cosmology
where the subscript 0 indicates present-day values. Thus, it is the sum ofthe densities in matter, relativistic particles, and vacuum that determinesthe overall sign of the curvature. Note that the quantity −k/R2
0H20 is
sometimes (unfortunately) referred to as Ωk.
19.1.5. Standard Model solutions :During inflation and again today the expansion rate for the Universe
is accelerating, and domination by a cosmological constant or some otherform of dark energy should be considered.
Let us first assume a general equation of state parameter for a singlecomponent, w = p/ρ which is constant. In this case, Eq. (19.10) can bewritten as ρ = −3(1 + w)ρR/R and is easily integrated to yield
ρ ∝ R−3(1+w) . (19.16)Note that at early times when R is small, k/R2 in the Friedmann equationcan be neglected so long as w > −1/3. Curvature domination occurs atrather late times (if a cosmological constant term does not dominatesooner). For w �= −1,
R(t) ∝ t2/[3(1+w)] . (19.17)
19.1.5.2. A Radiation-dominated Universe:In the early hot and dense Universe, it is appropriate to assume an
equation of state corresponding to a gas of radiation (or relativisticparticles) for which w = 1/3. In this case, Eq. (19.16) becomes ρ ∝ R−4.Similarly, one can substitute w = 1/3 into Eq. (19.17) to obtain
R(t) ∝ t1/2 ; H = 1/2t . (19.18)
19.1.5.3. A Matter-dominated Universe:Non-relativistic matter eventually dominates the energy density over
radiation. A pressureless gas (w = 0) leads to the expected dependenceρ ∝ R−3, and, if k = 0, we get
R(t) ∝ t2/3 ; H = 2/3t . (19.19)If there is a dominant source of vacuum energy, acting as a cosmological
constant with equation of state w = −1. This leads to an exponentialexpansion of the Universe
R(t) ∝ e√
Λ/3t . (19.20)The equation of state of the vacuum need not be the w = −1 of Λ,and may not even be constant [19,20,21]. There is now much interestin the more general possibility of a dynamically evolving vacuum energy,for which the name ‘dark energy’ has become commonly used. A varietyof techniques exist whereby the vacuum density as a function of timemay be measured, usually expressed as the value of w as a function ofepoch [22,23]. The best current measurement for the equation of state(assumed constant) is w = −1.006+0.067
−0.068 [24]. Unless stated otherwise,we will assume that the vacuum energy is a cosmological constant withw = −1 exactly.
The presence of vacuum energy can dramatically alter the fate of theUniverse. For example, if Λ < 0, the Universe will eventually recollapseindependent of the sign of k. For large values of Λ > 0 (larger thanthe Einstein static value needed to halt any cosmological expansion orcontraction), even a closed Universe will expand forever. One way toquantify this is the deceleration parameter, q0, defined as
q0 = − RR
R2
∣
∣
∣
∣
∣
0
=12Ωm + Ωr +
(1 + 3w)2
Ωv . (19.21)
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19. Big-Bang cosmology 211
This equation shows us that w < −1/3 for the vacuum may lead toan accelerating expansion. Current data indicate that vacuum energy isindeed the largest contributor to the cosmological density budget, withΩv = 0.74 ± 0.03 and Ωm = 0.26 ± 0.03 if k = 0 is assumed (5-year meanWMAP) [24].
19.2. Introduction to Observational Cosmology
19.2.1. Fluxes, luminosities, and distances :The key quantities for observational cosmology can be deduced quite
directly from the metric.(1) The proper transverse size of an object seen by us to subtend an
angle dψ is its comoving size dψ r times the scale factor at the time ofemission:
d� = dψ R0r/(1 + z) . (19.22)
(2) The apparent flux density of an object is deduced by allowingits photons to flow through a sphere of current radius R0r; but photonenergies and arrival rates are redshifted, and the bandwidth dν is reduced.These relations lead to the following common definitions:
angular-diameter distance: DA = (1 + z)−1R0r
luminosity distance: DL = (1 + z) R0r .(19.24)
These distance-redshift relations are expressed in terms of observablesby using the equation of a null radial geodesic plus the Friedmannequation:
R0
R(t)dt =
1H(z)
dz =1
H0
[
(1 − Ωm − Ωv − Ωr)(1 + z)2
+ Ωv(1 + z)3+3w + Ωm(1 + z)3 + Ωr(1 + z)4]−1/2
dz .
(19.25)
The main scale for the distance here is the Hubble length, 1/H0.In combination with Cepheid data from the HST and a direct
geometrical distance to the maser galaxy NGC4258, SNe resultsextend the distance ladder to the point where deviations from uniformexpansion are negligible, leading to the best existing direct value for H0:74.2 ± 3.6 km s−1Mpc−1 [25]. Better still, the analysis of high-z SNe hasallowed the first meaningful test of cosmological geometry to be carriedout.
19.2.3. Age of the Universe :
The dynamical result for the age of the Universe may be written as
H0t0 =∫ ∞
0
dz
(1 + z) [(1 + z)2(1 + Ωmz) − z(2 + z)Ωv]1/2, (19.28)
where we have neglected Ωr and chosen w = −1. Over the range of interest(0.1 <∼ Ωm <∼ 1, |Ωv| <∼ 1), this exact answer may be approximated to a
few % accuracy byH0t0 � 2
3 (0.7Ωm + 0.3 − 0.3Ωv)−0.3 . (19.29)
For the special case that Ωm + Ωv = 1, the integral in Eq. (19.28) can beexpressed analytically as
H0t0 =2
3√
Ωvln
1 +√
Ωv√1 − Ωv
(Ωm < 1) . (19.30)
The present consensus favors ages for the oldest clusters of about12 Gyr [36,37].
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212 19. Big-Bang cosmology
These methods are all consistent with the age deduced from studiesof structure formation, using the microwave background and large-scalestructure: t0 = 13.69± 0.13 Gyr [24], where the extra accuracy comes atthe price of assuming the Cold Dark Matter model to be true.
19.3. The Hot Thermal Universe
19.3.1. Thermodynamics of the early Universe :Through much of the radiation-dominated period, thermal equilibrium
is established by the rapid rate of particle interactions relative to theexpansion rate of the Universe. In equilibrium, it is straightforward tocompute the thermodynamic quantities, ρ, p, and the entropy density, s.
In the Standard Model, a chemical potential is often associated withbaryon number, and since the net baryon density relative to the photondensity is known to be very small (of order 10−10), we can neglect anysuch chemical potential when computing total thermodynamic quantities.
For photons, we have (in units where � = kB = 1)
ργ =π2
15T 4 ; pγ =
13ργ ; sγ =
4ργ
3T; nγ =
2ζ(3)π2
T 3 . (19.39)
Eq. (19.10) can be converted into an equation for entropy conservation,d(sR3)/dt = 0 . (19.40)
For radiation, this corresponds to the relationship between expansion andcooling, T ∝ R−1 in an adiabatically expanding universe. Note also thatboth s and nγ scale as T 3.
19.3.2. Radiation content of the Early Universe :At the very high temperatures associated with the early Universe,
massive particles are pair produced, and are part of the thermal bath.If for a given particle species i we have T mi, then we can neglectthe mass and the thermodynamic quantities are easily computed. Ingeneral, we can approximate the energy density (at high temperatures) byincluding only those particles with mi T . In this case, we have
ρ =
(
∑
B
gB +78
∑
F
gF
)
π2
30T 4 ≡ π2
30N(T )T 4 , (19.41)
where gB(F ) is the number of degrees of freedom of each boson (fermion)and the sum runs over all boson and fermion states with m T .Eq. (19.41) defines the effective number of degrees of freedom, N(T ), bytaking into account new particle degrees of freedom as the temperature israised.
The value of N(T ) at any given temperature depends on the particlephysics model. In the standard SU(3)×SU(2)×U(1) model, we can specifyN(T ) up to temperatures of O(100) GeV. The change in N (ignoring masseffects) can be seen in the table below. At higher temperatures, N(T ) willbe model-dependent.
In the radiation-dominated epoch, Eq. (19.10) can be integrated(neglecting the T -dependence of N) giving us a relationship between theage of the Universe and its temperature
t =(
9032π3GNN(T )
)1/2
T−2 . (19.42)
Put into a more convenient formt T 2
MeV = 2.4[N(T )]−1/2 , (19.43)where t is measured in seconds and TMeV in units of MeV.
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19. Big-Bang cosmology 213
Temperature New Particles 4N(T )T < me γ’s + ν’s 29me < T < mμ e± 43mμ < T < mπ μ± 57mπ < T < T †
c π’s 69Tc < T < mstrange π’s + u, u, d, d + gluons 205ms < T < mcharm s, s 247mc < T < mτ c, c 289mτ < T < mbottom τ± 303mb < T < mW,Z b, b 345mW,Z < T < mHiggs W±, Z 381mH < T < mtop H0 385mt < T t, t 427
†Tc corresponds to the confinement-deconfinement transition betweenquarks and hadrons.
19.3.7. Nucleosynthesis :An essential element of the standard cosmological model is Big-Bang
nucleosynthesis (BBN), the theory which predicts the abundances of thelight element isotopes D, 3He, 4He, and 7Li. Nucleosynthesis takes place ata temperature scale of order 1 MeV. The nuclear processes lead primarilyto 4He, with a primordial mass fraction of about 25%. Lesser amounts ofthe other light elements are produced: about 10−5 of D and 3He and about10−10 of 7Li by number relative to H. The abundances of the light elementsdepend almost solely on one key parameter, the baryon-to-photon ratio,η. The nucleosynthesis predictions can be compared with observationaldeterminations of the abundances of the light elements. Consistencybetween theory and observations leads to a conservative range of
5.1 × 10−10 < η < 6.5 × 10−10 . (19.54)η is related to the fraction of Ω contained in baryons, Ωb
Ωb = 3.66 × 107η h−2 , (19.55)or 1010η = 274Ωbh2.
19.4. The Universe at late timesWe are beginning to inventory the composition of the Universe:
total: Ω = 1.005± 0.0.006 (from CMB anisotropy)
matter: Ωm = 0.26 ± 0.03baryons: Ωb = 0.044 ± 0.004
CDM: ΩCDM = Ωm − Ωb
neutrinos: 0.001 <∼ Ων <∼ 0.05
dark energy: Ωv = 0.74 ± 0.03photons: Ωγ = 4.6 × 10−5
Further discussion and all references may be found in the full Review ofParticle Physics. The numbering of references and equations used herecorresponds to that version.
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214 21. The Cosmological Parameters
21. THE COSMOLOGICAL PARAMETERS
Updated September 2009, by O. Lahav (University College London) andA.R. Liddle (University of Sussex).
21.1. Parametrizing the Universe
The term ‘cosmological parameters’ is forever increasing in its scope,and nowadays includes the parametrization of some functions, as wellas simple numbers describing properties of the Universe. The originalusage referred to the parameters describing the global dynamics of theUniverse, such as its expansion rate and curvature. Also now of greatinterest is how the matter budget of the Universe is built up from itsconstituents: baryons, photons, neutrinos, dark matter, and dark energy.We need to describe the nature of perturbations in the Universe, throughglobal statistical descriptors such as the matter and radiation powerspectra. There may also be parameters describing the physical state of theUniverse, such as the ionization fraction as a function of time during theera since recombination. Typical comparisons of cosmological models withobservational data now feature between five and ten parameters.
21.1.1. The global description of the Universe : The completepresent state of the homogeneous Universe can be described by giving thecurrent values of all the density parameters and of the Hubble parameterh. These also allow us to track the history of the Universe back in time,at least until an epoch where interactions allow interchanges between thedensities of the different species, which is believed to have last happenedat neutrino decoupling, shortly before Big Bang Nucleosynthesis (BBN).To probe further back into the Universe’s history requires assumptionsabout particle interactions, and perhaps about the nature of physical lawsthemselves.
21.1.2. The standard cosmological model : The basic set of cos-mological parameters is therefore as shown in Table 21.1. The spatialcurvature does not appear in the list, because it can be determined fromthe other parameters using Eq. 21.1. The total present matter densityΩm = Ωcdm + Ωb is usually used in place of the dark matter density.
As described in Sec. 21.3, models based on these eleven parameters areable to give a good fit to the complete set of high-quality data availableat present, and indeed some simplification is possible. Observations areconsistent with spatial flatness, and indeed the inflation models so fardescribed automatically generate negligible spatial curvature, so we canset k = 0; the density parameters then must sum to unity, and so onecan be eliminated. The neutrino energy density is often not taken as anindependent parameter. Provided the neutrino sector has the standardinteractions, the neutrino energy density, while relativistic, can be relatedto the photon density using thermal physics arguments, and it is currentlydifficult to see the effect of the neutrino mass, although observations oflarge-scale structure have already placed interesting upper limits. Thisreduces the standard parameter set to nine. In addition, there is noobservational evidence for the existence of tensor perturbations (thoughthe upper limits are quite weak), and so r could be set to zero. Presentlyn is in a somewhat controversial position regarding whether it needs tobe varied in a fit, or can be set to the Harrison–Zel’dovich value n = 1.Parameter estimation [2] suggests n = 1 is ruled out at some significance,
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21. The Cosmological Parameters 215
Table 21.1: The basic set of cosmological parameters. We givevalues (with some additional rounding) as obtained using a fit ofa ΛCDM cosmology with a power-law initial spectrum to WMAP5data alone [2]. Tensors are assumed zero except in quoting alimit on them. The exact values and uncertainties depend on boththe precise data-sets used and the choice of parameters allowedto vary (see Table 21.2 for the former). Limits on ΩΛ and hweaken if the Universe is not assumed flat. The density perturbationamplitude is specified by the derived parameter σ8. Uncertaintiesare one-sigma/68% confidence unless otherwise stated.
Parameter Symbol Value
Hubble parameter h 0.72 ± 0.03Total matter density Ωm Ωmh2 = 0.133 ± 0.006Baryon density Ωb Ωbh2 = 0.0227± 0.0006Cosmological constant ΩΛ ΩΛ = 0.74 ± 0.03Radiation density Ωr Ωrh
2 = 2.47 × 10−5
Neutrino density Ων See Sec. 21.1.2Density perturb. amplitude at k=2Kpc Δ2
R (2.41 ± 0.11)× 10−9
Density perturb. spectral index n n = 0.963+0.014−0.015
Tensor to scalar ratio r r < 0.43 (95% conf.)Ionization optical depth τ τ = 0.087± 0.017Bias parameter b See Sec. 21.3.4
but Bayesian model selection techniques [9] suggest the data is notconclusive. With n set to one, this leaves seven parameters, which is thesmallest set that can usefully be compared to the present cosmologicaldata set. This model (usually with n kept as a parameter) is referred toby various names, including ΛCDM, the concordance cosmology, and thestandard cosmological model.
21.2. Extensions to the standard model
21.2.1. More general perturbations : The standard cosmology as-sumes adiabatic, Gaussian perturbations. Adiabaticity means that alltypes of material in the Universe share a common perturbation, so thatif the space-time is foliated by constant-density hypersurfaces, then allfluids and fields are homogeneous on those slices, with the perturbationscompletely described by the variation of the spatial curvature of the slices.Gaussianity means that the initial perturbations obey Gaussian statistics,with the amplitudes of waves of different wavenumbers being randomlydrawn from a Gaussian distribution of width given by the power spectrum.Note that gravitational instability generates non-Gaussianity; in thiscontext, Gaussianity refers to a property of the initial perturbations,before they evolve significantly.
The simplest inflation models, based on one dynamical field, predictadiabatic fluctuations and a level of non-Gaussianity which is too smallto be detected by any experiment so far conceived. For present data, theprimordial spectra are usually assumed to be power laws.
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216 21. The Cosmological Parameters
21.2.1.2. Isocurvature perturbations: An isocurvature perturbation isone which leaves the total density unperturbed, while perturbing therelative amounts of different materials. If the Universe contains N fluids,there is one growing adiabatic mode and N − 1 growing isocurvaturemodes (for reviews see Ref. 12 and Ref. 13). These can be excited, forexample, in inflationary models where there are two or more fields whichacquire dynamically-important perturbations. If one field decays to formnormal matter, while the second survives to become the dark matter, thiswill generate a cold dark matter isocurvature perturbation.
In general, there are also correlations between the different modes,and so the full set of perturbations is described by a matrix giving thespectra and their correlations. Constraining such a general construct ischallenging, though constraints on individual modes are beginning tobecome meaningful, with no evidence that any other than the adiabaticmode must be non-zero.
21.2.2. Dark energy : While the standard cosmological model givenabove features a cosmological constant, in order to explain observationsindicating that the Universe is presently accelerating, further possibilitiesexist under the general heading ‘dark energy’.† A particularly attractivepossibility (usually called quintessence, though that word is usedwith various different meanings in the literature) is that a scalar fieldis responsible, with the mechanism mimicking that of early Universeinflation [15]. As described by Olive and Peacock, a fairly model-independent description of dark energy can be given just using theequation of state parameter w, with w = −1 corresponding to acosmological constant. In general, the function w could itself vary withredshift, though practical experiments devised so far would be sensitiveprimarily to some average value weighted over recent epochs. For high-precision predictions of CMB anisotropies, it is better to use a scalar-fielddescription in order to have a self-consistent evolution of the ‘sound speed’associated with the dark energy perturbations.
Present observations are consistent with a cosmological constant, butoften w is kept as a free parameter to be added to the set describedin the previous section. Most, but not all, researchers assume the weakenergy condition w ≥ −1. In the future, it may be necessary to use a moresophisticated parametrization of the dark energy.
21.3. Probes
21.3.1. Direct measures of the Hubble constant : One of the mostreliable results on the Hubble constant comes from the Hubble SpaceTelescope Key Project [18]. This study used the empirical period–luminosity relations for Cepheid variable stars to obtain distances to 31galaxies, and calibrated a number of secondary distance indicators (TypeIa Supernovae, Tully–Fisher relation, surface brightness fluctuations, andType II Supernovae) measured over distances of 400 to 600 Mpc. Theyestimated H0 = 72 ± 3 (statistical) ± 7 (systematic) km s−1 Mpc−1. A
† Unfortunately this is rather a misnomer, as it is the negative pressureof this material, rather than its energy, that is responsible for giving theacceleration. Furthermore, while generally in physics matter and energyare interchangeable terms, dark matter and dark energy are quite distinctconcepts.
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21. The Cosmological Parameters 217
recent study [19] of 240 Cepheids observed with an improved cameraonboard the Hubble Space Telescope has yielded an even more accuratefigure, H0 = 74 ± 4 km s−1 Mpc−1 (including both statistical andsystematic errors). The major sources of uncertainty in these results aredue to the heavy element abundance of the Cepheids and the distance tothe fiducial nearby galaxy (called the Large Magellanic Cloud) relative towhich all Cepheid distances are measured.
21.3.4.3. Limits on neutrino mass from galaxy surveys and other probes:Large-scale structure data can put an upper limit on the ratio Ων/Ωm dueto the neutrino ‘free streaming’ effect [37,38]. For example, by comparingthe 2dF galaxy power spectrum with a four-component model (baryons,cold dark matter, a cosmological constant, and massive neutrinos), it isestimated that Ων/Ωm < 0.13 (95% confidence limit), giving Ων < 0.04 ifa concordance prior of Ωm = 0.3 is imposed. The latter corresponds to anupper limit of about 2 eV on the total neutrino mass, assuming a priorof h ≈ 0.7 [39]. Potential systematic effects include biasing of the galaxydistribution and non-linearities of the power spectrum. A similar upperlimit of 2 eV has been derived from CMB anisotropies alone [40–42]. Theabove analyses assume that the primordial power spectrum is adiabatic,scale-invariant and Gaussian. Additional cosmological data sets haveimproved the results [43,44]. An upper limit on the total neutrino massof 0.17 eV was reported by combining a large number of cosmologicalprobes [45].
21.4. Bringing observations together
Although it contains two ingredients—dark matter and dark energy—which have not yet been verified by laboratory experiments, the ΛCDMmodel is almost universally accepted by cosmologists as the bestdescription of the present data. The basic ingredients are given by theparameters listed in Sec. 21.1.1, with approximate values of some of thekey parameters being Ωb ≈ 0.04, Ωcdm ≈ 0.21, ΩΛ ≈ 0.74, and a Hubbleconstant h ≈ 0.72. The spatial geometry is very close to flat (and usuallyassumed to be precisely flat), and the initial perturbations Gaussian,adiabatic, and nearly scale-invariant.
The baryon density Ωb is now measured with quite high accuracyfrom the CMB and large-scale structure, and is consistent with thedetermination from BBN; Fields and Sarkar in this volume quote therange 0.019 ≤ Ωbh2 ≤ 0.024 (95% confidence).
While ΩΛ is measured to be non-zero with very high confidence, thereis no evidence of evolution of the dark energy density. The WMAP teamfind the limit w < −0.86 at 95% confidence from a compilation of dataincluding SNe Ia, with the cosmological constant case w = −1 giving anexcellent fit to the data.
One parameter which is very robust is the age of the Universe, as thereis a useful coincidence that for a flat Universe the position of the first peakis strongly correlated with the age. The WMAP5 result is 13.69 ± 0.13Gyr (assuming flatness). This is in good agreement with the ages of theoldest globular clusters [61] and radioactive dating [62].
For further details and all references, see the full Review of ParticlePhysics. See also “Astrophysical Constants,” table 2.1 in this Booklet.
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218 22. Dark matter
22. DARK MATTER
Revised September 2009 by M. Drees (Bonn University) and G. Gerbier(Saclay, CEA).
22.1. Theory
22.1.1. Evidence for Dark Matter : The existence of Dark (i.e.,non-luminous and non-absorbing) Matter (DM) is by now well established.An important example is the measurement of galactic rotation curves. Therotational velocity v of an object on a stable Keplerian orbit with radiusr around a galaxy scales like v(r) ∝ √
M(r)/r, where M(r) is the massinside the orbit. If r lies outside the visible part of the galaxy and masstracks light, one would expect v(r) ∝ 1/
√r. Instead, in most galaxies one
finds that v becomes approximately constant out to the largest values of rwhere the rotation curve can be measured. This implies the existence ofa dark halo, with mass density ρ(r) ∝ 1/r2, i.e., M(r) ∝ r, and a lowerbound on the DM mass density, ΩDM
>∼ 0.1.The observation of clusters of galaxies tends to give somewhat larger
values, ΩDM � 0.2. These observations include measurements of thepeculiar velocities of galaxies in the cluster, which are a measure of theirpotential energy if the cluster is virialized; measurements of the X-raytemperature of hot gas in the cluster, which again correlates with thegravitational potential felt by the gas; and—most directly—studies of(weak) gravitational lensing of background galaxies on the cluster.
The currently most accurate, if somewhat indirect, determination ofΩDM comes from global fits of cosmological parameters to a variety ofobservations; see the Section on Cosmological Parameters for details. Forexample, using measurements of the anisotropy of the cosmic microwavebackground (CMB) and of the spatial distribution of galaxies, Ref. 3 findsa density of cold, non–baryonic matter
Ωnbmh2 = 0.110± 0.006 , (22.1)
where h is the Hubble constant in units of 100 km/(s·Mpc). Some part ofthe baryonic matter density [3],
Ωbh2 = 0.0227± 0.0006 , (22.2)
may well contribute to (baryonic) DM, e.g., MACHOs [4] or cold moleculargas clouds [5].
The most recent estimate of the DM density in the “neighborhood” ofour solar system is 0.3 GeV cm−3.
22.1.2. Candidates for Dark Matter : Candidates for non-baryonicDM in Eq. (22.1) must satisfy several conditions: they must be stableon cosmological time scales (otherwise they would have decayed by now),they must interact very weakly with electromagnetic radiation (otherwisethey wouldn’t qualify as dark matter), and they must have the right relicdensity. Candidates include primordial black holes, axions, and weaklyinteracting massive particles (WIMPs).
The existence of axions [9] was first postulated to solve the strongCP problem of QCD; they also occur naturally in superstring theories.They are pseudo Nambu-Goldstone bosons associated with the (mostly)spontaneous breaking of a new global “Peccei-Quinn” (PQ) U(1) symmetryat scale fa; see the Section on Axions in this Review for further details.Although very light, axions would constitute cold DM, since they were
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22. Dark matter 219
produced non-thermally. At temperatures well above the QCD phasetransition, the axion is massless, and the axion field can take any value,parameterized by the “misalignment angle” θi. At T <∼ 1 GeV, the axiondevelops a mass ma due to instanton effects. Unless the axion fieldhappens to find itself at the minimum of its potential (θi = 0), it will beginto oscillate once ma becomes comparable to the Hubble parameter H .These coherent oscillations transform the energy originally stored in theaxion field into physical axion quanta. The contribution of this mechanismto the present axion relic density is [9]
Ωah2 = κa
(
fa/1012 GeV)1.175
θ2i , (22.5)
where the numerical factor κa lies roughly between 0.5 and a few.If θi ∼ O(1), Eq. (22.5) will saturate Eq. (22.1) for fa ∼ 1011 GeV,comfortably above laboratory and astrophysical constraints [9]; thiswould correspond to an axion mass around 0.1 meV. However, if the post-inflationary reheat temperature TR > fa, cosmic strings will form duringthe PQ phase transition at T � fa. Their decay will give an additionalcontribution to Ωa, which is often bigger than that in Eq. (22.5) [10],leading to a smaller preferred value of fa, i.e., larger ma. On the otherhand, values of fa near the Planck scale become possible if θi is for somereason very small.
Weakly interacting massive particles (WIMPs) χ are particles withmass roughly between 10 GeV and a few TeV, and with cross sections ofapproximately weak strength. Their present relic density can be calculatedreliably if the WIMPs were in thermal and chemical equilibrium withthe hot “soup” of Standard Model (SM) particles after inflation. Theirpresent relic density is then approximately given by (ignoring logarithmiccorrections) [11]
Ωχh2 � const. · T 30
M3Pl〈σAv〉 � 0.1 pb · c
〈σAv〉 . (22.6)
Here T0 is the current CMB temperature, MPl is the Planck mass, c isthe speed of light, σA is the total annihilation cross section of a pairof WIMPs into SM particles, v is the relative velocity between the twoWIMPs in their cms system, and 〈. . .〉 denotes thermal averaging. Freezeout happens at temperature TF � mχ/20 almost independently of theproperties of the WIMP. Notice that the 0.1 pb in Eq. (22.6) containsfactors of T0 and MPl; it is, therefore, quite intriguing that it “happens”to come out near the typical size of weak interaction cross sections.
The currently best motivated WIMP candidate is, therefore, the lightestsuperparticle (LSP) in supersymmetric models [12] with exact R-parity(which guarantees the stability of the LSP). Detailed calculations [15]show that the lightest neutralino will have the desired thermal relic densityEq. (22.1) in at least four distinct regions of parameter space. χ could be(mostly) a bino or photino (the superpartner of the U(1)Y gauge bosonand photon, respectively), if both χ and some sleptons have mass below∼ 150 GeV, or if mχ is close to the mass of some sfermion (so that its relicdensity is reduced through co-annihilation with this sfermion), or if 2mχis close to the mass of the CP-odd Higgs boson present in supersymmetricmodels. Finally, Eq. (22.1) can also be satisfied if χ has a large higgsinoor wino component.
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220 22. Dark matter
22.2. Experimental detection of Dark Matter
22.2.2. Axion searches : Axions can be detected by looking fora → γ conversion in a strong magnetic field [26]. Such a conversionproceeds through the loop-induced aγγ coupling, whose strength gaγγis an important parameter of axion models. There currently are twoexperiments searching for axionic DM. They both employ high qualitycavities. The cavity “Q factor” enhances the conversion rate on resonance,i.e., for mac2 = �ωres. One then needs to scan the resonance frequency inorder to cover a significant range in ma or, equivalently, fa.
22.2.3. Basics of direct WIMP search : The WIMP mean velocityinside our galaxy relative to its center is expected to be similar to thatof stars, i.e., a few hundred kilometers per second at the location of oursolar system. For these velocities, WIMPs interact with ordinary matterthrough elastic scattering on nuclei. With expected WIMP masses in therange 10 GeV to 10 TeV, typical nuclear recoil energies are of order of 1to 100 keV.
Expected interaction rates depend on the product of the local WIMPflux and the interaction cross section. The first term is fixed by the localdensity of dark matter, taken as 0.3 GeV/cm3 (see above), the meanWIMP velocity, typically 220 km/s, and the mass of the WIMP. Theexpected interaction rate then mainly depends on two unknowns, the massand cross section of the WIMP (with some uncertainty [6] due to the halomodel). This is why the experimental observable, which is basically thescattering rate as a function of energy, is usually expressed as a contour inthe WIMP mass–cross section plane.
The cross section depends on the nature of the couplings. For non-relativistic WIMPs, one in general has to distinguish spin-independentand spin-dependent couplings. The former can involve scalar and vectorWIMP and nucleon currents (vector currents are absent for MajoranaWIMPs, e.g., the neutralino), while the latter involve axial vector currents(and obviously only exist if χ carries spin). Due to coherence effects, thespin-independent cross section scales approximately as the square of themass of the nucleus, so higher mass nuclei, from Ge to Xe, are preferredfor this search. For spin-dependent coupling, the cross section depends onthe nuclear spin factor; used target nuclei include 19F, 23Na, 73Ge, 127I,129Xe, 131Xe, and 133Cs.
Cross sections calculated in MSSM models induce rates of at most1 evt day−1 kg−1 of detector, much lower than the usual radioactivebackgrounds. This indicates the need for underground laboratories toprotect against cosmic ray induced backgrounds, and for the selection ofextremely radio-pure materials.
The typical shape of exclusion contours can be anticipated from thisdiscussion: at low WIMP mass, the sensitivity drops because of thedetector energy threshold, whereas at high masses, the sensitivity alsodecreases because, for a fixed mass density, the WIMP flux decreases∝ 1/mχ. The sensitivity is best for WIMP masses near the mass of therecoiling nucleus.
Further discussion and all references may be found in the full Review.The numbering of references and equations used here corresponds to thatversion.
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23. Cosmic microwave background 221
23. COSMIC MICROWAVE BACKGROUND
Revised August 2009 by D. Scott (University of British Columbia) andG.F. Smoot (UCB/LBNL).
23.2. Description of CMB Anisotropies
Observations show that the CMB contains anisotropies at the 10−5
level, over a wide range of angular scales. These anisotropies are usuallyexpressed by using a spherical harmonic expansion of the CMB sky:
T (θ, φ) =∑
�m
a�mY�m(θ, φ).
The vast majority of the cosmological information is contained in thetemperature 2-point function, i.e., the variance as a function only ofangular separation, since we notice no preferred direction. Equivalently,the power per unit ln � is �
∑
m |a�m|2 /4π.23.2.1. The Monopole :
The CMB has a mean temperature of Tγ = 2.725 ± 0.001 K (1σ) [7],which can be considered as the monopole component of CMB maps,a00. Since all mapping experiments involve difference measurements,they are insensitive to this average level. Monopole measurementscan only be made with absolute temperature devices, such as theFIRAS instrument on the COBE satellite [7]. Such measurementsof the spectrum are consistent with a blackbody distribution overmore than three decades in frequency (with some recent evidencefor deviation at low frequencies [8]) . A blackbody of the measuredtemperature corresponds to nγ = (2ζ(3)/π2)T 3
γ � 411 cm−3 andργ = (π2/15)T 4
γ � 4.64 × 10−34 g cm−3 � 0.260 eVcm−3.
23.2.2. The Dipole :The largest anisotropy is in the � = 1 (dipole) first spherical harmonic,
with amplitude 3.355 ± 0.008 mK [6]. The dipole is interpreted to be theresult of the Doppler shift caused by the solar system motion relative tothe nearly isotropic blackbody field, as confirmed by measurements of theradial velocities of local galaxies [9].
The dipole is a frame-dependent quantity, and one can thus determinethe ‘absolute rest frame’ as that in which the CMB dipole would be zero.
23.2.3. Higher-Order Multipoles :The variations in the CMB temperature maps at higher multipoles
(� ≥ 2) are interpreted as being mostly the result of perturbations in thedensity of the early Universe, manifesting themselves at the epoch of thelast scattering of the CMB photons. In the hot Big Bang picture, theexpansion of the Universe cools the plasma so that by a redshift z � 1100(with little dependence on the details of the model), the hydrogen andhelium nuclei can bind electrons into neutral atoms, a process usuallyreferred to as recombination [12]. Before this epoch, the CMB photonsare tightly coupled to the baryons, while afterwards they can freely streamtowards us.
Theoretical models generally predict that the a�m modes are Gaussianrandom fields to high precision, e.g., standard slow-roll inflation’s non-Gaussian contribution is expected to be one or two orders of magnitudebelow current observational limits [13]. Although non-Gaussianity ofvarious forms is possible in early Universe models, tests show thatGaussianity is an extremely good simplifying approximation [14], with
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222 23. Cosmic microwave background
only some relatively weak indications of non-Gaussianity or statisticalanisotropy at large scales. Such signatures found in existing WMAPdata are generally considered to be subtle foreground or instrumentalartefacts [15,16].
A statistically isotropic sky means that all ms are equivalent, i.e., thereis no preferred axis. Together with the assumption of Gaussian statistics,the variance of the temperature field (or equivalently the power spectrumin �) then fully characterizes the anisotropies. The power summed overall ms at each � is (2� + 1)C�/(4π), where C� ≡ ⟨|a�m|2⟩. Thus averagesof a�ms over m can be used as estimators of the C�s to constrain theirexpectation values, which are the quantities predicted by a theoreticalmodel. For an idealized full-sky observation, the variance of each measuredC� (i.e., the variance of the variance) is [2/(2� + 1)]C2
� . This samplinguncertainty (known as ‘cosmic variance’) comes about because each C�is χ2 distributed with (2� + 1) degrees of freedom for our observablevolume of the Universe. For fractional sky coverage, fsky, this variance isincreased by 1/fsky and the modes become partially correlated.
It is important to understand that theories predict the expectationvalue of the power spectrum, whereas our sky is a single realization.Hence the cosmic variance is an unavoidable source of uncertainty whenconstraining models; it dominates the scatter at lower �s, while the effectsof instrumental noise and resolution dominate at higher �s [17].
23.2.4. Angular Resolution and Binning :There is no one-to-one conversion between multipole � and the angle
subtended by a particular spatial scale projected onto the sky. However,a single spherical harmonic Y�m corresponds to angular variations ofθ ∼ π/�. CMB maps contain anisotropy information from the size of themap (or in practice some fraction of that size) down to the beam-size ofthe instrument, σ. One can think of the effect of a Gaussian beam asrolling off the power spectrum with the function e−�(�+1)σ2
.
23.5. Current Anisotropy Data
There has been a steady improvement in the quality of CMB datathat has led to the development of the present-day cosmological model.Probably the most robust constraints currently available come fromthe combination of the WMAP five year data [34] with smaller scaleresults from the ACBAR [35] and QUAD [36] experiments (together withconstraints from other cosmological data-sets). We plot power spectrumestimates from these experiments, as well as BOOMERANG [38] andCBI [39] in Fig. 23.2. Other recent experiments also give powerfulconstraints, which are quite consistent with what we describe below.
23.6. CMB Polarization
Since Thomson scattering of an anisotropic radiation field also generateslinear polarization, the CMB is predicted to be polarized at the roughly5% level of the temperature anisotropies [42]. Polarization is a spin-2 fieldon the sky, and the algebra of the modes in �-space is strongly analogousto spin-orbit coupling in quantum mechanics [43]. The linear polarizationpattern can be decomposed in a number of ways, with two quantitiesrequired for each pixel in a map, often given as the Q and U Stokesparameters. However, the most intuitive and physical decomposition is ageometrical one, splitting the polarization pattern into a part that comesfrom a divergence (often referred to as the ‘E-mode’) and a part with acurl (called the ‘B-mode’) [44]. More explicitly, the modes are defined
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23. Cosmic microwave background 223
Figure 23.2: Band-power estimates from the WMAP, BOOMERANG,QUAD, CBI, and ACBAR experiments.
in terms of second derivatives of the polarization amplitude, with theHessian for the E-modes having principle axes in the same sense as thepolarization, while the B-mode pattern can be thought of simply as a45◦ rotation of the E-mode pattern. Globally one sees that the E-modeshave (−1)� parity (like the spherical harmonics), while the B-modes have(−1)�+1 parity.
Since inflationary scalar perturbations give only E-modes, while tensorsgenerate roughly equal amounts of E- and B-modes, then the determinationof a non-zero B-mode signal is a way to measure the gravitational wavecontribution (and thus potentially derive the energy scale of inflation),even if it is rather weak. However, one must first eliminate the foregroundcontributions and other systematic effects down to very low levels.23.8. Constraints on Cosmologies
The 5-year WMAP data alone, together with constraints from Hubbleconstant determination [18], supernovae [64] and baryon acousticoscillations [29], within the context of a 6 parameter family of models(which fixes Ωtot = 1 and r = 0), yield the following results [14]:A = (2.44 ± 0.10) × 10−9, n = 0.960 ± 0.013, h = 0.705 ± 0.013,Ωbh2 = 0.0227 ± 0.0006, Ωmh2 = 0.136 ± 0.004 and τ = 0.084 ± 0.016.Note that for h, the CMB data alone provide only a very weak constraint,unless spatial flatness or some other cosmological data are used. For Ωbh2,the precise value depends sensitively on how much freedom is allowed inthe shape of the primordial power spectrum.
For Ωtot, perhaps the best WMAP constraint is 1.006± 0.006, from thecombination with supernova and baryon acoustic oscillation constraints(and setting w = −1). The 95% confidence upper limit on r is 0.43using WMAP alone, tightening to r < 0.22 with the addition of otherdata [14]. This limit depends on how the slope n is restricted andwhether dn/d lnk �= 0 is allowed.
Further discussion and all references may be found in the full Review.
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224 24. Cosmic rays
24. COSMIC RAYS
Written August 2008 by D.E. Groom (LBNL).
At ∼10 GeV/nucleon the primary cosmic rays are mostly protons (79%)and alpha particles (15% of the cosmic ray nucleons). The abundanceratios of heavier nuclei follow the solar abundance ratios. The exceptionis the Li-Be-B group, which is overabundant by a large factor relative tosolar abundance, probably due to the spallation of heavier nuclei in theinterstellar medium. The intensity from a few GeV to beyond 100 TeV isproportional to E−2.7. The spectrum is evidently truncated at a few times1019 eV by inelastic collisions with the CMB (GZK mechanism).
The primary cosmic rays initiate hadronic cascades in the atmosphere,whose thickness is 11.5 λI or 28 X0. Most of the energy is converted togamma rays via π0 → γγ and deposited by ionization in EM showers; fewhadrons reach the ground. For charged pions, π± → μ±+ν competes withfurther nuclear interaction. The muon energy spectrum at the surfaceis almost flat below 1 GeV, gradually steepens to reflect the primaryspectrum in the 10–100 GeV range, and steepens further at higher energiesbecause pions with Eπ >∼ 100 GeV tend to interact before they decay.Above ∼ 10 TeV, and below ∼ 100TeV, when prompt muon productionbecomes important, the energy spectrum of atmospheric muons is onepower steeper than the primary spectrum.
The average muon flux at the surface goes as cos2 θ, characteristicof ∼3 GeV muons. The rate in a thin horizontal detector is roughly1 cm−2min−1; it is half this in a vertical detector.
The vertical muon flux for E > 1 GeV goes through a broad maximumat an atmospheric depth h≈ 170 g cm−2, but for h >∼ 300 g cm−2 itcan be fairly well represented by I/Isurface = exp[(h − 1033 gcm−2)/(630 g cm−2)]. Calculators to convert h to altitude in a ”standardatmosphere” can be found on the web. At the summit of Mauna Kea(4205 m = 600 g cm−2) the vertical flux is about twice that at sea level.
The vertical p + n flux at sea level is about 2% of the muon flux, butscales with depth as exp(−h/λI). The pion flux is 50 times smaller. Thee+/e− flux for E > 1GeV averages about 0.004 of the muon flux, but theseEM shower remnants are much more abundant at lower energies.
The energy loss rate for muons is usually written as a(E) + b(E)E,where a(E) (ionization) and b(E) (radiative loss rate/E) are slowly-varying functions of E. Ionization and radiative loss rates are equal atthe muon critical energy Eμc, which is typically several hundred GeV.If a(Eμc) and b(Eμc) are assumed constant, then the range is given byln(1 + E/Eμc). (Straggling is extremely important, but it is also neglectedhere.) Furthermore, since the differential muon flux at very high energiesis proportional to E−α (one power steeper than the primary spectrum),the total vertical flux at depth X is proportional to (ebX − 1)−α+1.This function, with b = 4.0 × 10−6 cm2g−1 and α = 3.6, normalizedto 0.013m−2s−1sr−1 at X = 1 km.w.e. (km water equivalent), gives areasonable fit to the depth-intensity data shown in Fig. 24.5 of the fullReview. The flux at depths >∼ 10–20 km.w.e., entirely due to neutrinointeractions, is about 2 × 10−9 m−2s−1sr−1.
For details and references, see the full Review.
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25. Accelerator physics of colliders 225
25. ACCELERATOR PHYSICS OF COLLIDERS
Revised August 2005 by K. Desler and D. A. Edwards (DESY).
25.2. LuminosityThe event rate R in a collider is proportional to the interaction cross
section σint and the factor of proportionality is called the luminosity:R = L σint . (25.1)
If two bunches containing n1 and n2 particles collide with frequency f ,the luminosity is
L = fn1n2
4πσxσy(25.2)
where σx and σy characterize the Gaussian transverse beam profiles in thehorizontal (bend) and vertical directions and to simplify the expressionit is assumed that the bunches are identical in transverse profile, thatthe profiles are independent of position along the bunch, and the particledistributions are not altered during collision. Whatever the distributionat the source, by the time the beam reaches high energy, the normal formis a good approximation thanks to the central limit theorem of probabilityand the diminished importance of space charge effects.
The beam size can be expressed in terms of two quantities, one termedthe transverse emittance, ε, and the other, the amplitude function, β. Thetransverse emittance is a beam quality concept reflecting the process ofbunch preparation, extending all the way back to the source for hadronsand, in the case of electrons, mostly dependent on synchrotron radiation.The amplitude function is a beam optics quantity and is determined bythe accelerator magnet configuration. When expressed in terms of σ andβ the transverse emittance becomes
ε = πσ2/β . (25.3)Of particular significance is the value of the amplitude function at theinteraction point, β∗. Clearly one wants β∗ to be as small as possible; howsmall depends on the capability of the hardware to make a near-focus atthe interaction point.
Eq. (25.2) can now be recast in terms of emittances and amplitudefunctions as
L = fn1n2
4√
εx β∗x εy β∗
y. (25.4)
Thus, to achieve high luminosity, all one has to do is make high populationbunches of low emittance to collide at high frequency at locations wherethe beam optics provides as low values of the amplitude functions aspossible.
Further discussion and references may be found in the full Review ofParticle Physics.
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226 26. High-energy collider parameters26.H
IGH
-EN
ER
GY
CO
LLID
ER
PA
RA
MET
ER
S:e+
e−
Collid
ers
(I)
Upd
ated
inea
rly
2010
wit
hnu
mbe
rsre
ceiv
edfr
omre
pres
enta
tive
sof
the
colli
ders
(con
tact
J.B
erin
ger,
LB
NL).
For
exis
ting
(fut
ure)
colli
ders
the
late
stac
hiev
ed(d
esig
n)va
lues
are
give
n.qu
antiti
esar
e,w
here
appr
opri
ate,
r.m
.s.;
Han
dV
indi
cate
hori
zont
alan
dve
rtic
aldi
rect
ions
;s.
c.st
ands
for
supe
rcon
duct
ing.
Par
amet
ers
for
the
defu
nct
SPE
AR
,D
OR
IS,
PE
TR
A,
PE
P,SL
C,T
RIS
TA
N,an
dV
EP
P-2
Mco
llide
rsm
aybe
foun
din
our
1996
edit
ion
(Phy
s.R
ev.D
54,1
July
1996
,Par
tI)
.
VE
PP
-200
0(N
ovos
ibir
sk)
VE
PP
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(Nov
osib
irsk
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orne
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Phy
sics
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0819
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0819
99
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sics
end
date
——
2008
—20
13
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imum
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eV)
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ity
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330
450
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ebe
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Inte
ract
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1
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26. High-energy collider parameters 227
HIG
H-E
NER
GY
CO
LLID
ER
PA
RA
MET
ER
S:e+
e−
Collid
ers
(II)
Upd
ated
inea
rly
2010
(con
tact
J.B
erin
ger,
LB
NL).
For
exis
ting
(fut
ure)
colli
ders
the
late
stac
hiev
ed(d
esig
n)va
lues
are
give
n.Q
uant
itie
sar
e,w
here
appr
opri
ate,
r.m
.s.;
Han
dV
indi
cate
hori
zont
alan
dve
rtic
aldi
rect
ions
;s.c
.st
ands
for
supe
rcon
duct
ing.
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KB
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date
—20
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inal
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Inte
ract
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ons
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ime
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ains
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nam
icbe
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eam
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t.
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228 26. High-energy collider parametersH
IGH
-EN
ER
GY
CO
LLID
ER
PA
RA
MET
ER
S:ep,pp,pp,and
Heavy
Ion
Collid
ers
Upd
ated
inea
rly
2010
(con
tact
J.B
erin
ger,
LB
NL).
For
exis
ting
(fut
ure)
colli
ders
the
late
stac
hiev
ed(d
esig
n)va
lues
are
give
n.Q
uant
itie
sar
e,w
here
appr
opri
ate,
r.m
.s.;
Han
dV
indi
cate
hori
zont
alan
dve
rtic
aldi
rect
ions
;s.c
.st
ands
for
supe
rcon
duct
ing;
pkan
dav
ede
note
peak
and
aver
age
valu
es.
HE
RA
(DE
SY)
TE
VAT
RO
N(F
erm
ilab)
RH
IC(B
rook
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n)LH
C†
(CE
RN
)
Phy
sics
star
tda
te19
9219
8720
0120
0020
0420
0220
0920
10
Phy
sics
end
date
2007
——
—
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ticl
esco
llide
dep
pp
pp
(pol
.)A
uA
uC
uC
ud
Au
pp
Pb
Pb
Max
imum
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eV)
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ity
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ract
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ons
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umbe
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10.
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27. Passage of particles through matter 229
27. PASSAGE OF PARTICLES THROUGH MATTER
Revised January 2010 by H. Bichsel (University of Washington), D.E.Groom (LBNL), and S.R. Klein (LBNL).
27.1. Notation
Table 27.1: Summary of variables used in this section. Thekinematic variables β and γ have their usual meanings.
Symbol Definition Units or Value
α Fine structure constant 1/137.035 999 11(46)(e2/4πε0�c)
M Incident particle mass MeV/c2
E Incident part. energy γMc2 MeVT Kinetic energy MeV
mec2 Electron mass × c2 0.510 998 918(44) MeV
re Classical electron radius 2.817 940 325(28) fme2/4πε0mec
2
NA Avogadro’s number 6.022 1415(10)× 1023 mol−1
ze Charge of incident particleZ Atomic number of absorberA Atomic mass of absorber g mol−1
K/A 4πNAr2emec
2/A 0.307 075 MeV g−1 cm2
for A = 1 g mol−1
I Mean excitation energy eV (Nota bene!)δ(βγ) Density effect correction to ionization energy loss�ωp Plasma energy
√
ρ 〈Z/A〉 × 28.816 eV(√
4πNer3e mec
2/α) (ρ in g cm−3)Ne Electron density (units of re)−3
wj Weight fraction of the jth element in a compound or mixturenj ∝ number of jth kind of atoms in a compound or mixture— 4αr2
eNA/A (716.408 g cm−2)−1 for A = 1 g mol−1
X0 Radiation length g cm−2
Ec Critical energy for electrons MeVEμc Critical energy for muons GeVEs Scale energy
√
4π/α mec2 21.2052 MeV
RM Moliere radius g cm−2
27.2. Electronic energy loss by heavy particles [1–32]
27.2.1. Moments and cross sections :The electronic interactions of fast charged particles with speed v = βc
occur in single collisions with energy losses E [1], leading to ionization,atomic, or collective excitation. Most frequently the energy losses aresmall (for 90% of all collisions the energy losses are less than 100 eV). Inthin absorbers few collisions will take place and the total energy loss willshow a large variance [1]; also see Sec. 27.2.7 below. For particles with
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230 27. Passage of particles through matter
charge ze more massive than electrons (“heavy” particles), scattering fromfree electrons is adequately described by the Rutherford differential crosssection [2],
dσR(E; β)dE
=2πr2
emec2z2
β2
(1 − β2E/Tmax)E2
, (27.1)
where Tmax is the maximum energy transfer possible in a single collision.But in matter electrons are not free. For electrons bound in atomsBethe [3] used “Born Theorie” to obtain the differential cross section
dσB(E; β)dE
=dσR(E, β)
dEB(E) . (27.2)
At high energies σB is further modified by polarization of the medium,and this “density effect,” discussed in Sec. 27.2.4, must also be included.Less important corrections are discussed below.
The mean number of collisions with energy loss between E and E + dEoccuring in a distance δx is Neδx (dσ/dE)dE, where dσ(E; β)/dE containsall contributions. It is convenient to define the moments
Mj(β) = Ne δx
∫
Ej dσ(E; β)dE
dE ,
so that M0 is the mean number of collisions in δx, M1 is the mean energyloss in δx, M2 − M2
1 is the variance, etc. The number of collisions isPoisson-distributed with mean M0. Ne is either measured in electrons/g(Ne = NAZ/A) or electrons/cm3 (Ne = NA ρZ/A).
Muon momentum
1
10
100
Sto
ppin
g po
wer
[MeV
cm
2 /g]
Lin
dhar
d-S
char
ff
Bethe Radiative
Radiativeeffects
reach 1%
μ+ on Cu
Without δ
Radiativelosses
βγ0.001 0.01 0.1 1 10 100 1000 104 105 106
[MeV/c] [GeV/c]
1001010.1 100101 100101
[TeV/c]
Anderson-Ziegler
Nuclearlosses
Minimumionization
Eμc
μ−
Fig. 27.1: Stopping power (= 〈−dE/dx〉) for positive muons in copper as afunction of βγ = p/Mc over nine orders of magnitude in momentum (12 ordersof magnitude in kinetic energy). Solid curves indicate the total stoppingpower. Data below the break at βγ ≈ 0.1 are taken from ICRU 49 [4], anddata at higher energies are from Ref. 5. Vertical bands indicate boundariesbetween different approximations discussed in the text. The short dottedlines labeled “μ− ” illustrate the “Barkas effect,” the dependence of stoppingpower on projectile charge at very low energies [6].
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27. Passage of particles through matter 231
27.2.2. Stopping power at intermediate energies : The mean rateof energy loss by moderately relativistic charged heavy particles, M1/δx,is well-described by the “Bethe” equation,
−⟨
dE
dx
⟩
= Kz2Z
A
1β2
[
12
ln2mec
2β2γ2Tmax
I2 − β2 − δ(βγ)2
]
. (27.3)
It describes the mean loss rate in the region 0.1 <∼ βγ <∼ 1000 forintermediate-Z materials with an accuracy of a few %. At the lower limitthe projectile velocity becomes comparable to atomic electron “velocities”(Sec. 27.2.3), and at the upper limit radiative effects begin to be important(Sec. 27.6). Both limits are Z dependent. Here Tmax is the maximumkinetic energy which can be imparted to a free electron in a single collision,and the other variables are defined in Table 27.1. A minor dependenceon M at the highest energies is introduced through Tmax, but for allpractical purposes 〈dE/dx〉 in a given material is a function of β alone.With the symbol definitions and values given in Table 27.1, the units areMeV g−1cm2.
Few concepts in high-energy physics are as misused as 〈dE/dx〉. Themain problem is that the mean is weighted by very rare events with largesingle-collision energy deposits. Even with samples of hundreds of events adependable value for the mean energy loss cannot be obtained. Far betterand more easily measured is the most probable energy loss, discussed inSec. 27.2.7. It is considerably below the mean given by the Bethe equation.
In a TPC (Sec. 28.6.5), the mean of 50%–70% of the samples with thesmallest signals is often used as an estimator.
Although it must be used with cautions and caveats, 〈dE/dx〉 asdescribed in Eq. (27.3) still forms the basis of much of our understandingof energy loss by charged particles. Extensive tables are available[5,4,pdg.lbl.gov/AtomicNuclearProperties/].
The function as computed for muons on copper is shown as the“Bethe” region of Fig. 27.1. Mean energy loss behavior below this regionis discussed in Sec. 27.2.3, and the radiative effects at high energyare discussed in Sec. 27.6. Only in the Bethe region is it a functionof β alone; the mass dependence is more complicated elsewhere. Thestopping power in several other materials is shown in Fig. 27.2. Exceptin hydrogen, particles with the same velocity have similar rates of energyloss in different materials, although there is a slow decrease in the rateof energy loss with increasing Z. The qualitative behavior difference athigh energies between a gas (He in the figure) and the other materialsshown in the figure is due to the density-effect correction, δ(βγ), discussedin Sec. 27.2.4. The stopping power functions are characterized by broadminima whose position drops from βγ = 3.5 to 3.0 as Z goes from 7 to100. The values of minimum ionization go roughly as 0.235 − 0.28 ln(Z),in MeVg−1cm−2, for Z > 6.
Eq. (27.3) may be integrated to find the total (or partial) “continuousslowing-down approximation” (CSDA) range R for a particle which losesenergy only through ionization and atomic excitation. Since dE/dx inthe “Bethe region” depends only on β, R/M is a function of E/M orpc/M . In practice, range is a useful concept only for low-energy hadrons(R <∼ λI , where λI is the nuclear interaction length), and for muons belowa few hundred GeV (above which radiative effects dominate). R/M as afunction of βγ = p/Mc is shown for a variety of materials in Fig. 27.4.
The mass scaling of dE/dx and range is valid for the electronic lossesdescribed by the Bethe equation, but not for radiative losses, relevant onlyfor muons and pions.
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232 27. Passage of particles through matter
1
2
3
4
5 6
8
10
1.0 10 100 1000 10 0000.1
Pion momentum (GeV/c)
Proton momentum (GeV/c)
1.0 10 100 10000.1
1.0 10 100 10000.1
1.0 10 100 1000 10 0000.1
−d
E/d
x (M
eV g
−1cm
2 )
βγ = p/Mc
Muon momentum (GeV/c)
H2 liquid
He gas
CAl
FeSn
Pb
Figure 27.2: Mean energy loss rate in liquid (bubble chamber)hydrogen, gaseous helium, carbon, aluminum, iron, tin, and lead.Radiative effects, relevant for muons and pions, are not included.These become significant for muons in iron for βγ >∼ 1000, and atlower momenta in higher-Z absorbers. See Fig. 27.21.For a particle with mass M and momentum Mβγc, Tmax is given by
Tmax =2mec
2 β2γ2
1 + 2γme/M + (me/M)2. (27.4)
In older references [2,7] the “low-energy” approximation Tmax =2mec
2 β2γ2, valid for 2γme/M � 1, is often implicit. For hadrons withE � 100 GeV, it is limited by structure effects.
Estimates of the mean excitation energy I based on experimentalstopping-power measurements for protons, deuterons, and alpha particlesare given in ICRU 37 [10]; see also pdg.lbl.gov.AtomicNuclearProperties.
27.2.4. Density effect : As the particle energy increases, its electricfield flattens and extends, so that the distant-collision contribution toEq. (27.3) increases as lnβγ. However, real media become polarized,limiting the field extension and effectively truncating this part of thelogarithmic rise [2–7,18–20]. At very high energies,
δ/2 → ln(�ωp/I) + lnβγ − 1/2 , (27.5)where δ(βγ)/2 is the density effect correction introduced in Eq. (27.3)and �ωp is the plasma energy defined in Table 27.1. A comparison withEq. (27.3) shows that |dE/dx| then grows as lnβγ rather than lnβ2γ2,and that the mean excitation energy I is replaced by the plasma energy
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27. Passage of particles through matter 233
0.05 0.10.02 0.50.2 1.0 5.02.0 10.0Pion momentum (GeV/c)
0.1 0.50.2 1.0 5.02.0 10.0 50.020.0
Proton momentum (GeV/c)
0.050.02 0.1 0.50.2 1.0 5.02.0 10.0Muon momentum (GeV/c)
βγ = p/Mc
1
2
5
10
20
50
100
200
500
1000
2000
5000
10000
20000
50000
R/M
(g c
m−2
GeV
−1)
0.1 2 5 1.0 2 5 10.0 2 5 100.0
H2 liquidHe gas
PbFe
C
Figure 27.4: Range of heavy charged particles in liquid (bubblechamber) hydrogen, helium gas, carbon, iron, and lead. For example:For a K+ whose momentum is 700 MeV/c, βγ = 1.42. For lead weread R/M ≈ 396, and so the range is 195 g cm−2.
�ωp. Since the plasma frequency scales as the square root of the electrondensity, the correction is much larger for a liquid or solid than for a gas,as is illustrated by the examples in Fig. 27.2.
The remaining relativistic rise comes from the β2γ2 growth of Tmax,which in turn is due to (rare) large energy transfers to a few electrons.When these events are excluded, the energy deposit in an absorbinglayer approaches a constant value, the Fermi plateau (see Sec. 27.2.6below). At extreme energies (e.g., > 332 GeV for muons in iron, andat a considerably higher energy for protons in iron), radiative effects aremore important than ionization losses. These are especially relevant forhigh-energy muons, as discussed in Sec. 27.6.
27.2.5. Energetic knock-on electrons (δ rays) : The distribution ofsecondary electrons with kinetic energies T I is [2]
d2N
dTdx=
12
Kz2Z
A
1β2
F (T )T 2
(27.7)
for I � T ≤ Tmax, where Tmax is given by Eq. (27.4). Here β is thevelocity of the primary particle. The factor F is spin-dependent, but isabout unity for T � Tmax. For spin-0 particles F (T ) = (1 − β2T/Tmax);forms for spins 1/2 and 1 are also given by Rossi [2]. For incidentelectrons, the indistinguishability of projectile and target means that the
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234 27. Passage of particles through matter
range of T extends only to half the kinetic energy of the incident particle.Additional formulae are given in Ref. 22. Equation (27.7) is inaccurate forT close to I. The cosine of the production angle is essentially unity inpractical cases.
δ rays of even modest energy are rare. For β ≈ 1 particle, for example,on average only one collision with Te > 1 keV will occur along a pathlength of 90 cm of Ar gas [1].
27.2.6. Restricted energy loss rates for relativistic ionizing par-ticles : Further insight can be obtained by examining the mean energydeposit by an ionizing particle when energy transfers are restricted toT ≤ Tcut ≤ Tmax. The restricted energy loss rate is
−dE
dx
∣
∣
∣
∣
T<Tcut
= Kz2Z
A
1β2
[
12
ln2mec
2β2γ2Tcut
I2 − β2
2
(
1 +Tcut
Tmax
)
− δ
2
]
.
(27.9)This form approaches the normal Bethe function (Eq. (27.3)) asTcut → Tmax. It can be verified that the difference between Eq. (27.3) andEq. (27.9) is equal to
∫ TmaxTcut
T (d2N/dTdx)dT , where d2N/dTdx is givenby Eq. (27.7).
Since Tcut replaces Tmax in the argument of the logarithmic term ofEq. (27.3), the βγ term producing the relativistic rise in the close-collisionpart of dE/dx is replaced by a constant, and |dE/dx|T<Tcut
approachesthe constant “Fermi plateau.” (The density effect correction δ eliminatesthe explicit βγ dependence produced by the distant-collision contribution.)This behavior is illustrated in Fig. 27.6, where restricted loss rates fortwo examples of Tcut are shown in comparison with the full Bethe dE/dxand the Landau-Vavilov most probable energy loss (to be discussed inSec. 27.2.7 below).
27.2.7. Fluctuations in energy loss : For detectors of moderatethickness x (e.g. scintillators or LAr cells),* the energy loss probabilitydistribution f(Δ; βγ, x) is adequately described by the highly-skewedLandau (or Landau-Vavilov) distribution [24,25]. The most probableenergy loss is [26]
Δp = ξ
[
ln2mc2β2γ2
I+ ln
ξ
I+ j − β2 − δ(βγ)
]
, (27.10)
where ξ = (K/2) 〈Z/A〉 (x/β2) MeV for a detector with a thickness x ing cm−2, and j = 0.200 [26]. † While dE/dx is independent of thickness,Δp/x scales as a ln x + b. The density correction δ(βγ) was not includedin Landau’s or Vavilov’s work, but it was later included by Bichsel [26].The high-energy behavior of δ(βγ) (Eq. (27.5)), is such that
Δp −→βγ>∼100
ξ
[
ln2mc2ξ
(�ωp)2+ j
]
. (27.11)
Thus the Landau-Vavilov most probable energy loss, like the restrictedenergy loss, reaches a Fermi plateau. The Bethe dE/dx and Landau-Vavilov-Bichsel Δp/x in silicon are shown as a function of muon energyin Fig. 27.6. The case x/ρ = 1600 μm was chosen since it has about
* G <∼ 0.05–0.1, where G is given by Rossi [Ref. 2, Eq. 2.7.10]. It isVavilov’s κ [25].
† Rossi [2], Talman [27], and others give somewhat different values forj. The most probable loss is not sensitive to its value.
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27. Passage of particles through matter 235
the same stopping power as does 3 mm of plastic scintillator. Foldingin experimental resolution displaces the peak of the distribution, usuallytoward a higher value.
Landau/Vavilov/Bichsel Δp/x for :
Bethe
Tcut = 10 dE/dx|minTcut = 2 dE/dx|min
Restricted energy loss for :
0.1 1.0 10.0 100.0 1000.0
1.0
1.5
0.5
2.0
2.5
3.0M
eV g
−1 c
m2
(Ele
cton
ic lo
ses
only
)
Muon kinetic energy (GeV)
Silicon
x/ρ = 1600 μm320 μm80 μm
Figure 27.6: Bethe dE/dx, two examples of restricted energyloss, and the Landau most probable energy per unit thickness insilicon. The change of Δp/x with thickness x illustrates its a ln x + bdependence. Minimum ionization (dE/dx|min) is 1.664 MeV g−1 cm2.Radiative losses are excluded. The incident particles are muons.
The mean of the energy-loss given by the Bethe equation, Eq. (27.3),is ill-defined experimentally and is not useful for describing energy lossby single particles. (It finds its application in dosimetry, where only bulkdeposit is of relevance.) It rises as lnβγ because Tmax increases as β2γ2.The large single-collision energy transfers that increasingly extend the longtail are rare, making the mean of an experimental distribution consistingof a few hundred events subject to large fluctuations and sensitive to cutsas well as to background. The most probable energy loss should be used.
For very thick absorbers the distribution is less skewed but neverapproaches a Gaussian. In the case of Si illustrated in Fig. 27.6, the mostprobable energy loss per unit thickness for x ≈ 35 g cm−2 is very close tothe restricted energy loss with Tcut = 2 dE/dx|min.
The Landau distribution fails to describe energy loss in thin absorberssuch as gas TPC cells [1] and Si detectors [26], as shown clearly inFig. 1 of Ref. 1 for an argon-filled TPC cell. Also see Talman [27]. WhileΔp/x may be calculated adequately with Eq. (27.10), the distributionsare significantly wider than the Landau width w = 4ξ [Ref. 26, Fig. 15].Examples for thin silicon detectors are shown in Fig. 27.7.
27.2.8. Energy loss in mixtures and compounds : A mixture orcompound can be thought of as made up of thin layers of pure elementsin the right proportion (Bragg additivity). In this case,
dE
dx=
∑
wjdE
dx
∣
∣
∣
∣
j, (27.12)
where dE/dx|j is the mean rate of energy loss (in MeV g cm−2)in the jth element. Eq. (27.3) can be inserted into Eq. (27.12) tofind expressions for 〈Z/A〉, 〈I 〉, and 〈δ〉; for example, 〈Z/A〉 =∑
wjZj/Aj =∑
njZj/∑
njAj . However, 〈I 〉 as defined this way is
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236 27. Passage of particles through matter
an underestimate, because in a compound electrons are more tightlybound than in the free elements, and 〈δ〉 as calculated this way has littlerelevance, because it is the electron density that matters. If possible,one uses the tables given in Refs. 20 and 28, or the recipes given inRef. 21 (repeated in Ref. 5), which include effective excitation energiesand interpolation coefficients for calculating the density effect correction.27.3. Multiple scattering through small anglesA charged particle traversing a medium is deflected by many small-anglescatters. Most of this deflection is due to Coulomb scattering from nuclei,and hence the effect is called multiple Coulomb scattering. (However, forhadronic projectiles, the strong interactions also contribute to multiplescattering.) The Coulomb scattering distribution is well represented by thetheory of Moliere [33]. It is roughly Gaussian for small deflection angles,but at larger angles (greater than a few θ0, defined below) it behaves likeRutherford scattering, with larger tails than does a Gaussian distribution.
If we define
θ0 = θ rmsplane =
1√2
θrmsspace . (27.13)
then it is usually sufficient to use a Gaussian approximation for the central98% of the projected angular distribution, with a width given by [34,35]
θ0 =13.6 MeV
βcpz
√
x/X0
[
1 + 0.038 ln(x/X0)]
. (27.14)
Here p, βc, and z are the momentum, velocity, and charge number of theincident particle, and x/X0 is the thickness of the scattering medium inradiation lengths (defined below). This value of θ0 is from a fit to Molieredistribution [33] for singly charged particles with β = 1 for all Z, and isaccurate to 11% or better for 10−3 < x/X0 < 100.27.4. Photon and electron interactions in matter
27.4.1. Radiation length : High-energy electrons predominantly loseenergy in matter by bremsstrahlung, and high-energy photons by e+e−pair production. The characteristic amount of matter traversed for theserelated interactions is called the radiation length X0, usually measured ing cm−2. It is both (a) the mean distance over which a high-energy electronloses all but 1/e of its energy by bremsstrahlung, and (b) 7
9 of the meanfree path for pair production by a high-energy photon [37]. It is alsothe appropriate scale length for describing high-energy electromagneticcascades. X0 has been calculated and tabulated by Y.S. Tsai [38]:
1X0
= 4αr2eNA
A
{
Z2[Lrad − f(Z)]
+ Z L′rad
}
. (27.22)
For A = 1 g mol−1, 4αr2eNA/A = (716.408 g cm−2)−1. Lrad and L′
radare given in Table 27.2. The function f(Z) is an infinite sum, but forelements up to uranium can be represented to 4-place accuracy byf(Z) = a2[(1+a2)−1+0.20206−0.0369 a2+0.0083 a4−0.002 a6 , (27.23)
where a = αZ [39].
27.4.2. Energy loss by electrons : At low energies electrons andpositrons primarily lose energy by ionization, although other processes(Møller scattering, Bhabha scattering, e+ annihilation) contribute, asshown in Fig. 27.10. While ionization loss rates rise logarithmically withenergy, bremsstrahlung losses rise nearly linearly (fractional loss is nearlyindependent of energy), and dominates above a few tens of MeV in mostmaterials.
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27. Passage of particles through matter 237
Table 27.2: Tsai’s Lrad and L′rad, for use in calculating the
radiation length in an element using Eq. (27.22).
Element Z Lrad L′rad
H 1 5.31 6.144He 2 4.79 5.621Li 3 4.74 5.805Be 4 4.71 5.924
Others > 4 ln(184.15 Z−1/3) ln(1194 Z−2/3)
Ionization loss by electrons and positrons differs from loss by heavyparticles because of the kinematics, spin, and the identity of the incidentelectron with the electrons which it ionizes. Complete discussions andtables can be found in Refs. 9, 10, and 28.
At very high energies and except at the high-energy tip of thebremsstrahlung spectrum, the cross section can be approximated in the“complete screening case” as [38]
dσ/dk = (1/k)4αr2e
{
(43 − 4
3y + y2)[Z2(Lrad − f(Z)) + Z L′rad]
+ 19 (1 − y)(Z2 + Z)
}
,(27.26)
where y = k/E is the fraction of the electron’s energy transfered to theradiated photon. At small y (the “infrared limit”) the term on the secondline ranges from 1.7% (low Z) to 2.5% (high Z) of the total. If it is ignoredand the first line simplified with the definition of X0 given in Eq. (27.22),we have
dσ
dk=
A
X0NAk
(
43 − 4
3y + y2)
. (27.27)
This formula is accurate except in near y = 1, where screening maybecome incomplete, and near y = 0, where the infrared divergenceis removed by the interference of bremsstrahlung amplitudes fromnearby scattering centers (the LPM effect) [41,42] and dielectricsuppression [43,44]. These and other suppression effects in bulk mediaare discussed in Sec. 27.4.5.
Except at these extremes, and still in the complete-screeningapproximation, the number of photons with energies between kmin andkmax emitted by an electron travelling a distance d � X0 is
Nγ =d
X0
[
43
ln(
kmax
kmin
)
− 4(kmax − kmin)3E
+k2max − k2
min
2E2
]
. (27.28)
27.4.3. Critical energy : An electron loses energy by bremsstrahlungat a rate nearly proportional to its energy, while the ionization loss ratevaries only logarithmically with the electron energy. The critical energyEc is sometimes defined as the energy at which the two loss rates areequal [46]. Among alternate definitions is that of Rossi [2], who definesthe critical energy as the energy at which the ionization loss per radiationlength is equal to the electron energy. Equivalently, it is the same as thefirst definition with the approximation |dE/dx|brems ≈ E/X0. This formhas been found to describe transverse electromagnetic shower developmentmore accurately (see below).
The accuracy of approximate forms for Ec has been limited by thefailure to distinguish between gases and solid or liquids, where there isa substantial difference in ionization at the relevant energy because of
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238 27. Passage of particles through matter
the density effect. Separate fits to Ec(Z), using the Rossi definition, havebeen made with functions of the form a/(Z + b)α, but α was found to beessentially unity. For Z > 6 we obtain
Ec ≈ 610 MeVZ + 1.24
(solids and liquids) , ≈ 710 MeVZ + 0.92
(gases) .
Since Ec also depends on A, I, and other factors, such forms are at bestapproximate.
Photon energy
100
10
10–4
10–5
10–6
1
0.1
0.01
0.001
10 eV 100 eV 1 keV 10 keV 100 keV 1 MeV 10 MeV 100 MeV 1 GeV 10 GeV 100 GeV
Abs
orpt
ion
len
gth
λ (
g/cm
2)
Si
C
Fe Pb
H
Sn
Fig. 27.16: The photon mass attenuation length (or meanfree path) λ = 1/(μ/ρ) for various elemental absorbers as afunction of photon energy. The mass attenuation coefficient isμ/ρ, where ρ is the density. The intensity I remaining aftertraversal of thickness t (in mass/unit area) is given by I = I0exp(−t/λ). The accuraccy is a few percent. For a chemicalcompound or mixture, 1/λeff ≈ ∑
elements wZ/λZ , where wZ isthe proportion by weight of the element with atomic numberZ. The processes responsible for attenuation are given inFig. 27.10. Since coherent processes are included, not all theseprocesses result in energy deposition.
27.4.4. Energy loss by photons : Contributions to the photon crosssection in a light element (carbon) and a heavy element (lead) areshown in Fig. 27.14. At low energies it is seen that the photoelectriceffect dominates, although Compton scattering, Rayleigh scattering, andphotonuclear absorption also contribute. The photoelectric cross sectionis characterized by discontinuities (absorption edges) as thresholds forphotoionization of various atomic levels are reached. Photon attenuationlengths for a variety of elements are shown in Fig. 27.15, and data for30 eV< k <100 GeV for all elements is available from the web pages givenin the caption. Here k is the photon energy.
The increasing domination of pair production as the energy increases isshown in Fig. 27.17 of the full Review. Using approximations similar tothose used to obtain Eq. (27.27), Tsai’s formula for the differential crosssection [38] reduces to
dσ
dx=
A
X0NA
[
1 − 43x(1 − x)
]
(27.29)
in the complete-screening limit valid at high energies. Here x = E/k is thefractional energy transfer to the pair-produced electron (or positron), and
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27. Passage of particles through matter 239
k is the incident photon energy. The cross section is very closely relatedto that for bremsstrahlung, since the Feynman diagrams are variants ofone another. The cross section is of necessity symmetric between x and1 − x, as can be seen by the solid curve in See the review by Motz, Olsen,& Koch for a more detailed treatment [51].
Eq. (27.29) may be integrated to find the high-energy limit for the totale+e− pair-production cross section:
σ = 79 (A/X0NA) . (27.30)
Equation Eq. (27.30) is accurate to within a few percent down to energiesas low as 1 GeV, particularly for high-Z materials.
27.4.5. Bremsstrahlung and pair production at very high ener-gies : At ultrahigh energies, Eqns. 27.26–27.30 will fail because ofquantum mechanical interference between amplitudes from different scat-tering centers. Since the longitudinal momentum transfer to a given centeris small (∝ k/E(E − k), in the case of bremsstrahlung), the interactionis spread over a comparatively long distance called the formation length(∝ E(E − k)/k) via the uncertainty principle. In alternate language, theformation length is the distance over which the highly relativistic electronand the photon “split apart.” The interference is usually destructive.Calculations of the “Landau-Pomeranchuk-Migdal” (LPM) effect may bemade semi-classically based on the average multiple scattering, or morerigorously using a quantum transport approach [41,42].
In amorphous media, bremsstrahlung is suppressed if the photon energyk is less than E2/(E + ELPM ) [42], where*
ELPM =(mec
2)2αX0
4π�cρ= (7.7 TeV/cm) × X0
ρ. (27.31)
Since physical distances are involved, X0/ρ, in cm, appears. The energy-weighted bremsstrahlung spectrum for lead, k dσLPM/dk, is shown inFig. 27.11 of the full Review. With appropriate scaling by X0/ρ, othermaterials behave similarly.
For photons, pair production is reduced for E(k − E) > k ELPM . Thepair-production cross sections for different photon energies are shown inFig. 27.15 of the full Review.
If k � E, several additional mechanisms can also produce suppression.When the formation length is long, even weak factors can perturbthe interaction. For example, the emitted photon can coherentlyforward scatter off of the electrons in the media. Because of this,for k < ωpE/me ∼ 10−4, bremsstrahlung is suppressed by a factor(kme/ωpE)2 [44]. Magnetic fields can also suppress bremsstrahlung.In crystalline media, the situation is more complicated, with coherentenhancement or suppression possible [42].27.5. Electromagnetic cascades
When a high-energy electron or photon is incident on a thick absorber, itinitiates an electromagnetic cascade as pair production and bremsstrahlunggenerate more electrons and photons with lower energy. The longitudinaldevelopment is governed by the high-energy part of the cascade, andtherefore scales as the radiation length in the material. Electron energieseventually fall below the critical energy, and then dissipate their energyby ionization and excitation rather than by the generation of more shower
* This definition differs from that of Ref. 52 by a factor of two. ELPMscales as the 4th power of the mass of the incident particle, so that ELPM =(1.4 × 1010 TeV/cm) × X0/ρ for a muon.
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240 27. Passage of particles through matter
particles. In describing shower behavior, it is therefore convenient tointroduce the scale variables t = x/X0 and y = E/Ec, so that distanceis measured in units of radiation length and energy in units of criticalenergy.
The mean longitudinal profile of the energy deposition in anelectromagnetic cascade is reasonably well described by a gammadistribution [58]:
dE
dt= E0 b
(bt)a−1e−bt
Γ(a)(27.33)
The maximum tmax occurs at (a − 1)/b. We have made fits to showerprofiles in elements ranging from carbon to uranium, at energies from 1GeV to 100 GeV. The energy deposition profiles are well described byEq. (27.33) with
tmax = (a − 1)/b = 1.0 × (ln y + Cj) , j = e, γ , (27.34)where Ce = −0.5 for electron-induced cascades and Cγ = +0.5 forphoton-induced cascades. To use Eq. (27.33), one finds (a − 1)/b fromEq. (27.34), then finds a either by assuming b ≈ 0.5 or by finding amore accurate value from Fig. 27.19. The results are very similar for theelectron number profiles, but there is some dependence on the atomicnumber of the medium. A similar form for the electron number maximumwas obtained by Rossi in the context of his “Approximation B,” [2] butwith Ce = −1.0 and Cγ = −0.5; we regard this as superseded by the EGS4result.
The “shower length” Xs = X0/b is less conveniently parameterized,since b depends upon both Z and incident energy, as shown in Fig. 27.19.As a corollary of this Z dependence, the number of electrons crossing aplane near shower maximum is underestimated using Rossi’s approxima-tion for carbon and seriously overestimated for uranium. Essentially thesame b values are obtained for incident electrons and photons. For manypurposes it is sufficient to take b ≈ 0.5.
The gamma function distribution is very flat near the origin, whilethe EGS4 cascade (or a real cascade) increases more rapidly. As a resultEq. (27.33) fails badly for about the first two radiation lengths, which areexcluded from fits. Because fluctuations are important, Eq. (27.33) shouldbe used only in applications where average behavior is adequate.
The transverse development of electromagnetic showers in differentmaterials scales fairly accurately with the Moliere radius RM , givenby [60,61]
RM = X0 Es/Ec , (27.35)where Es ≈ 21 MeV (Table 27.1), and the Rossi definition of Ec is used.
Measurements of the lateral distribution in electromagnetic cascadesare shown in Refs. 60 and 61. On the average, only 10% of the energylies outside the cylinder with radius RM . About 99% is containedinside of 3.5RM , but at this radius and beyond composition effectsbecome important and the scaling with RM fails. The distributions arecharacterized by a narrow core, and broaden as the shower develops. Theyare often represented as the sum of two Gaussians, and Grindhammer [59]describes them with the function
f(r) =2r R2
(r2 + R2)2, (27.37)
where R is a phenomenological function of x/X0 and lnE.At high enough energies, the LPM effect (Sec. 27.4.5) reduces the cross
sections for bremsstrahlung and pair production, and hence can causesignificant elongation of electromagnetic cascades [42].
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27. Passage of particles through matter 241
27.6. Muon energy loss at high energyAt sufficiently high energies, radiative processes become more importantthan ionization for all charged particles. For muons and pions in materialssuch as iron, this “critical energy” occurs at several hundred GeV. (Thereis no simple scaling with particle mass, but for protons the “criticalenergy” is much, much higher.) Radiative effects dominate the energyloss of energetic muons found in cosmic rays or produced at the newestaccelerators. These processes are characterized by small cross sections,hard spectra, large energy fluctuations, and the associated generation ofelectromagnetic and (in the case of photonuclear interactions) hadronicshowers [62–70]. At these energies the treatment of energy loss as auniform and continuous process is for many purposes inadequate.
It is convenient to write the average rate of muon energy loss as [71]−dE/dx = a(E) + b(E)E . (27.38)
Here a(E) is the ionization energy loss given by Eq. (27.3), and b(E)is the sum of e+e− pair production, bremsstrahlung, and photonuclearcontributions. To the approximation that these slowly-varying functionsare constant, the mean range x0 of a muon with initial energy E0 is givenby
x0 ≈ (1/b) ln(1 + E0/Eμc) , (27.39)
where Eμc = a/b.The “muon critical energy” Eμc can be defined more exactly as the
energy at which radiative and ionization losses are equal, and can befound by solving Eμc = a(Eμc)/b(Eμc). This definition corresponds to thesolid-line intersection in Fig. 27.12 of the full Review, and is different fromthe Rossi definition we used for electrons. It serves the same function:below Eμc ionization losses dominate, and above Eμc radiative effectsdominate. The dependence of Eμc on atomic number Z is shown in Fig.27.22 in the full Review.
The radiative cross sections are expressed as functions of the fractionalenergy loss ν. The bremsstrahlung cross section goes roughly as 1/ν overmost of the range, while for the pair production case the distributiongoes as ν−3 to ν−2 [72]. “Hard” losses are therefore more probablein bremsstrahlung, and in fact energy losses due to pair production mayvery nearly be treated as continuous. The simulated [70] momentumdistribution of an incident 1 TeV/c muon beam after it crosses 3 m ofiron is shown in Fig. 27.23 of the full Review. The hard bremsstrahlungphotons and hadronic debris from photonuclear interactions inducecascades which can obscure muon tracks in detector planes and reducetracking efficiency [74].27.7. Cherenkov and transition radiation [75,76,32]A charged particle radiates if its velocity is greater than the local phasevelocity of light (Cherenkov radiation) or if it crosses suddenly from onemedium to another with different optical properties (transition radiation).Neither process is important for energy loss, but both are used inhigh-energy physics detectors.
Cherenkov radiation. The cosine of the angle θc of Cherenkov radiation,relative to the particle’s direction, for a particle with velocity βc in amedium with index of refraction n, is 1/nβ, or
tan θc =√
β2n2 − 1 ≈√
2(1 − 1/nβ) (27.40)for small θc, e.g., in gases. The threshold velocity βt is 1/n, andγt = 1/(1 − β2
t )1/2. Therefore, βtγt = 1/(2δ + δ2)1/2, where δ = n − 1.Values of δ for various commonly used gases are given as a function of
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242 27. Passage of particles through matter
pressure and wavelength in Ref. 77. For values at atmospheric pressure,see Table 6.1. Data for other commonly used materials are given inRef. 78.
Practical Cherenkov radiator materials are dispersive. Let ω be thephoton’s frequency, and let k = 2π/λ be its wavenumber. The photonspropage at the group velocity vg = dω/dk = c/[n(ω) + ω(dn/dω)]. In anon-dispersive medium, this simplifies to vg = c/n.
In his classical paper, Tamm [79] showed that for dispersive media theradiation is concentrated in a thin conical shell whose vertex is at themoving charge, and whose opening half-angle η is given by
cot η =[
d
dω(ω tan θc)
]
ω0
=[
tan θc + β2ω n(ω)dn
dωcot θc
]
ω0
, (27.41)
where ω0 is the central value of the small frequency range underconsideration. (See Fig. 27.24.) This cone has a opening half-angle η,and, unless the medium is non-dispersive (dn/dω = 0), θc + η = 900. TheCherenkov wavefront ‘sideslips’ along with the particle [80]. This effectmay have timing implications for ring imaging Cherenkov counters [81],but it is probably unimportant for most applications.
The number of photons produced per unit path length of a particlewith charge ze and per unit energy interval of the photons is
d2N
dEdx=
αz2
�csin2 θc =
α2z2
re mec2
(
1 − 1β2n2(E)
)
≈ 370 sin2 θc(E) eV−1cm−1 (z = 1) , (27.42)or, equivalently,
d2N
dxdλ=
2παz2
λ2
(
1 − 1β2n2(λ)
)
. (27.43)
The index of refraction n is a function of photon energy E = �ω. Forpractical use, Eq. (27.42) must be multiplied by the photodetector responsefunction and integrated over the region for which β n(ω) > 1.
When two particles are within <∼ 1 wavelength, the electromagneticfields from the particles may add coherently, affecting the Cherenkovradiation. The radiation from an e+e− pair at close separation issuppressed compared to two independent leptons [82].
Coherent radio Cherenkov radiation from electromagnetic showers(containing a net excess of e− over e+) has been used to study cosmic rayair showers [84] and to search for νe induced showers.
Transition radiation. The energy I radiated when a particle with chargeze crosses the boundary between vacuum and a medium with plasmafrequency ωp is αz2γ�ωp/3, where
�ωp =√
4πNer3e mec
2/α =√
ρ (in g/cm3) 〈Z/A〉 × 28.81 eV . (27.45)
For styrene and similar materials, �ωp ≈ 20 eV; for air it is 0.7 eV.The number spectrum dNγ/d(�ω diverges logarithmically at low
energies and decreases rapidly for �ω/γ�ωp > 1. About half the energyis emitted in the range 0.1 ≤ �ω/γ�ωp ≤ 1. Inevitable absorption in apractical detector removes the divergence. For a particle with γ = 103,the radiated photons are in the soft x-ray range 2 to 40 keV. The γdependence of the emitted energy thus comes from the hardening of thespectrum rather than from an increased quantum yield.
The number of photons with energy �ω > �ω0 is given by the answer
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27. Passage of particles through matter 243
10−3
10−2
10−4
10−5101 100 1000
25 μm Mylar/1.5 mm airγ = 2 ×104
Without absorption
With absorption
200 foils
Single interface
X-ray energy ω (keV)
dS/d
( ω
)D
iffe
rent
ial y
ield
per
inte
rfac
e (k
eV/k
eV)
Figure 27.25: X-ray photon energy spectra for a radiator consistingof 200 25μm thick foils of Mylar with 1.5 mm spacing in air (solidlines) and for a single surface (dashed line). Curves are shown withand without absorption. Adapted from Ref. 85.
to problem 13.15 in Ref. 32,
Nγ(�ω > �ω0) =αz2
π
[
(
lnγ�ωp
�ω0− 1
)2
+π2
12
]
, (27.46)
within corrections of order (�ω0/γ�ωp)2. The number of photons above afixed energy �ω0 � γ�ωp thus grows as (ln γ)2, but the number above afixed fraction of γ�ωp (as in the example above) is constant. For example,for �ω > γ�ωp/10, Nγ = 2.519 αz2/π = 0.59%× z2.
The particle stays “in phase” with the x ray over a distance called theformation length, d(ω). Most of the radiation is produced in a distanced(ω) = (2c/ω)(1/γ2 + θ2 + ω2
p/ω2)−1. Here θ is the x-ray emission angle,characteristically 1/γ. For θ = 1/γ the formation length has a maximumat d(γωp/
√2) = γc/
√2 ωp. In practical situations it is tens of μm.
Since the useful x-ray yield from a single interface is low, in practicaldetectors it is enhanced by using a stack of N foil radiators—foilsL thick, where L is typically several formation lengths—separated bygas-filled gaps. The amplitudes at successive interfaces interfere to causeoscillations about the single-interface spectrum. At increasing frequenciesabove the position of the last interference maximum (L/d(w) = π/2), theformation zones, which have opposite phase, overlap more and more andthe spectrum saturates, dI/dω approaching zero as L/d(ω) → 0. This isillustrated in Fig. 27.25 for a realistic detector configuration.
For regular spacing of the layers fairly complicated analytic solutionsfor the intensity have been obtained [85]. (See also Ref. 86 and referencestherein.) Although one might expect the intensity of coherent radiationfrom the stack of foils to be proportional to N2, the angular dependenceof the formation length conspires to make the intensity ∝ N .
Further discussion and all references may be found in the full Review ofParticle Physics. The equation and reference numbering corresponds tothat version.
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244 28. Detectors at accelerators
28. PARTICLE DETECTORS AT ACCELERATORS
This is an abridgment of the discussion given in the full Review ofParticle Physics (the “full Review”); the equation and reference numberingcorresponds to that version. The quoted numbers are usually based ontypical devices, and should be regarded only as rough approximationsfor new designs. A more detailed discussion of detectors can be found inRefs. 1 and 56.
28.1. Summary of detector spatial resolution, temporalresolution, and deadtime
In this section we give various parameters for common detectorcomponents. The quoted numbers are usually based on typical devices,and should be regarded only as rough approximations for new designs.More detailed discussions of detectors and their underlying physics canbe found in books by Ferbel [1], Kleinknecht [2], Knoll [3], Green [4],Leroy & Rancoita [5], and Grupen [6]. In Table 28.1 are given typicalresolutions and deadtimes of common detectors.
Table 28.1: Typical resolutions and deadtimes of commondetectors. Revised September 2009.
Resolution DeadDetector Type Accuracy (rms) Time Time
Bubble chamber 10–150 μm 1 ms 50 msaStreamer chamber 300 μm 2 μs 100 msProportional chamber 50–100 μma 2 ns 200 nsDrift chamber 50–100 μm 2 nsd 100 nsScintillator — 100 ps/na 10 nsEmulsion 1 μm — —Liquid argon drift [7] ∼175–450 μm ∼ 200 ns ∼ 2 μsMicro-pattern gas detectors [8] 30–40 μm < 10 ns 20 nsResistive plate chamber [9] � 10 μm 1–2 ns —Silicon strip pitch/(3 to 7)a a a
Silicon pixel 2 μma a a
a See full Review for qualifications and assumptions.28.2. Photon detectorsUpdated September 2009 by D. Chakraborty (Northern Illinois U) andT. Sumiyoshi (Tokyo Metro U).
Most detectors in high-energy, nuclear, and astrophysics rely on thedetection of photons in or near the visible range, 100 nm � λ� 1000nm,or E ≈ a few eV. This range covers scintillation and Cherenkov radiationas well as the light detected in many astronomical observations.
Generally, photodetection involves generating a detectable electricalsignal proportional to the (usually very small) number of incident photons.28.2.1. Vacuum photodetectors : Vacuum photodetectors can bebroadly subdivided into three types: photomultiplier tubes, microchannelplates, and hybrid photodetectors.
28.2.1.1. Photomultiplier tubes: A versatile class of photon detectors,vacuum photomultiplier tubes (PMT) has been employed by a vast major-ity of all particle physics experiments to date [11]. Both “transmission-”and “reflection-type” PMT’s are widely used. In the former, the pho-tocathode material is deposited on the inside of a transparent window
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28. Detectors at accelerators 245
through which the photons enter, while in the latter, the photocathodematerial rests on a separate surface that the incident photons strike. Thecathode material has a low work function, chosen for the wavelength bandof interest. When a photon hits the cathode and liberates an electron (thephotoelectric effect), the latter is accelerated and guided by electric fieldsto impinge on a secondary-emission electrode, or dynode, which then emitsa few (∼ 5) secondary electrons. The multiplication process is repeatedtypically 10 times in series to generate a sufficient number of electrons,which are collected at the anode for delivery to the external circuit. Thetotal gain of a PMT depends on the applied high voltage V as G = AV kn,where k ≈ 0.7–0.8 (depending on the dynode material), n is the numberof dynodes in the chain, and A a constant (which also depends on n).Typically, G is in the range of 105–106.
28.2.2. Gaseous photon detectors : In gaseous photomultipliers(GPM) a photoelectron in a suitable gas mixture initiates an avalanchein a high-field region, producing a large number of secondary impact-ionization electrons. In principle the charge multiplication and collectionprocesses are identical to those employed in gaseous tracking detectorssuch as multiwire proportional chambers, micromesh gaseous detectors(Micromegas), or gas electron multipliers (GEM). These are discussed inSec. 28.6.4.
28.2.3. Solid-state photon detectors : In a phase of rapid develop-ment, solid-state photodetectors are competing with vacuum- or gas-baseddevices for many existing applications and making way for a multitude ofnew ones. Compared to traditional vacuum- and gaseous photodetectors,solid-state devices are more compact, lightweight, rugged, tolerant tomagnetic fields, and often cheaper. They also allow fine pixelization, areeasy to integrate into large systems, and can operate at low electricpotentials, while matching or exceeding most performance criteria. Theyare particularly well suited for detection of γ- and X-rays. Except forapplications where coverage of very large areas or dynamic range isrequired, solid-state detectors are proving to be the better choice.
Silicon photodiodes (PD) are widely used in high-energy physics asparticle detectors and in a great number of applications (including solarcells!) as light detectors. The structure is discussed in some detail inSec. 28.7.
Very large arrays containing O(107) of O(10 μm2)-sized photodioidespixelizing a plane are widely used to photograph all sorts of thingsfrom everyday subjects at visible wavelengths to crystal structureswith X-rays and astronomical objects from infrared to UV. To limitthe number of readout channels, these are made into charge-coupleddevices (CCD), where pixel-to-pixel signal transfer takes place overthousands of synchronous cycles with sequential output through shiftregisters [16]. Thus, high spatial resolution is achieved at the expense ofspeed and timing precision. Custom-made CCD’s have virtually replacedphotographic plates and other imagers for astronomy and in spacecraft.
In avalanche photodiodes (APD), an exponential cascade of impactionizations initiated by the initial photogenerated e-h pair under a largereverse-bias voltage leads to an avalanche breakdown [17]. As a result,detectable electrical response can be obtained from low-intensity opticalsignals down to single photons.
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246 28. Detectors at accelerators
28.3. Organic scintillatorsRevised September 2007 by K.F. Johnson (FSU).
Organic scintillators are broadly classed into three types, crystalline,liquid, and plastic, all of which utilize the ionization produced bycharged particles to generate optical photons, usually in the blue to greenwavelength regions [21]. Plastic scintillators are by far the most widelyused. Crystal organic scintillators are practically unused in high-energyphysics.
Densities range from 1.03 to 1.20 g cm−3. Typical photon yields areabout 1 photon per 100 eV of energy deposit [22]. A one-cm-thickscintillator traversed by a minimum-ionizing particle will therefore yield≈ 2 × 104 photons. The resulting photoelectron signal will depend on thecollection and transport efficiency of the optical package and the quantumefficiency of the photodetector.
Decay times are in the ns range; rise times are much faster. Ease offabrication into desired shapes and low cost has made plastic scintillatorsa common detector component. Recently, plastic scintillators in theform of scintillating fibers have found widespread use in tracking andcalorimetry [25].
28.3.3. Scintillating and wavelength-shifting fibers :The clad optical fiber is an incarnation of scintillator and wavelength
shifter (WLS) which is particularly useful [33]. Since the initialdemonstration of the scintillating fiber (SCIFI) calorimeter [34], SCIFItechniques have become mainstream [35].
SCIFI calorimeters are fast, dense, radiation hard, and can haveleadglass-like resolution. SCIFI trackers can handle high rates and areradiation tolerant, but the low photon yield at the end of a long fiber (seebelow) forces the use of sensitive photodetectors. WLS scintillator readoutof a calorimeter allows a very high level of hermeticity since the solid angleblocked by the fiber on its way to the photodetector is very small.28.4. Inorganic scintillators:Revised September 2009 by R.-Y. Zhu (California Institute of Technology)and C.L. Woody (BNL).
Inorganic crystals form a class of scintillating materials with muchhigher densities than organic plastic scintillators (typically ∼ 4–8 g/cm3)with a variety of different properties for use as scintillation detectors. Dueto their high density and high effective atomic number, they can be usedin applications where high stopping power or a high conversion efficiencyfor electrons or photons is required. These include total absorptionelectromagnetic calorimeters (see Sec. 28.9.1), which consist of a totallyactive absorber (as opposed to a sampling calorimeter), as well as servingas gamma ray detectors over a wide range of energies. Many of thesecrystals also have very high light output, and can therefore provideexcellent energy resolution down to very low energies (∼ few hundredkeV).
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28. Detectors at accelerators 247
28.5. Cherenkov detectorsRevised September 2009 by B.N. Ratcliff (SLAC).
Although devices using Cherenkov radiation are often thought of asonly particle identification (PID) detectors, in practice they are usedover a broader range of applications including; (1) fast particle counters;(2) hadronic PID; and (3) tracking detectors performing completeevent reconstruction. Cherenkov counters contain two main elements;(1) a radiator through which the charged particle passes, and (2) aphotodetector. As Cherenkov radiation is a weak source of photons, lightcollection and detection must be as efficient as possible. The refractiveindex n and the particle’s path length through the radiator L appearin the Cherenkov relations allowing the tuning of these quantities forparticular applications.
Cherenkov detectors utilize one or more of the properties of Cherenkovradiation discussed in the Passages of Particles through Matter section(Sec. 27 of this Review): the prompt emission of a light pulse; the existenceof a velocity threshold for radiation; and the dependence of the Cherenkovcone half-angle θc and the number of emitted photons on the velocity ofthe particle and the refractive index of the medium.28.6. Gaseous detectors
28.6.1. Energy loss and charge transport in gases : Revised March2010 by F. Sauli (CERN) and M. Titov (CEA Saclay).
Gas-filled detectors localize the ionization produced by chargedparticles, generally after charge multiplication. The statistics of ionizationprocesses having asymmetries in the ionization trails, affect the coordinatedetermination deduced from the measurement of drift time, or of the centerof gravity of the collected charge. For thin gas layers, the width of theenergy loss distribution can be larger than its average, requiring multiplesample or truncated mean analysis to achieve good particle identification.The energy loss of charged particles and photons in matter is discussedin Sec. 27. Table 28.5 provides values of relevant parameters in somecommonly used gases at NTP (normal temperature, 20◦ C, and pressure,1 atm) for unit-charge minimum-ionizing particles (MIPs) [57–63].
Table 28.5: Properties of noble and molecular gases at normaltemperature and pressure (NTP: 20◦ C, one atm). EX , EI : firstexcitation, ionization energy; WI : average energy per ion pair;dE/dx|min, NP , NT : differential energy loss, primary and totalnumber of electron-ion pairs per cm, for unit charge minimumionizing particles.
Gas Density, Ex EI WI dE/dx|min NP NTmgcm−3 eV eV eV keV cm−1 cm−1 cm−1
He 0.179 19.8 24.6 41.3 0.32 3.5 8Ne 0.839 16.7 21.6 37 1.45 13 40Ar 1.66 11.6 15.7 26 2.53 25 97Xe 5.495 8.4 12.1 22 6.87 41 312CH4 0.667 8.8 12.6 30 1.61 28 54C2H6 1.26 8.2 11.5 26 2.91 48 112iC4H10 2.49 6.5 10.6 26 5.67 90 220CO2 1.84 7.0 13.8 34 3.35 35 100CF4 3.78 10.0 16.0 54 6.38 63 120
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248 28. Detectors at accelerators
When an ionizing particle passes through the gas, it creates electron-ionpairs, but often the ejected electrons have sufficient energy to further ionizethe medium. As shown in Table 28.5, the total number of electron-ionpairs (NT ) is usually a few times larger than the number of primaries(NP ).
The probability for a released electron to have an energy E or largerfollows an approximate 1/E2 dependence (Rutherford law), taking intoaccount the electronic structure of the medium. The number of electron-ion pairs per primary ionization, or cluster size, has an exponentiallydecreasing probability; for argon, there is about 1% probability for primaryclusters to contain ten or more electron-ion pairs [59].
Once released in the gas, and under the influence of an applied electricfield, electrons and ions drift in opposite directions and diffuse towardsthe electrodes. The drift velocity and diffusion of electrons depend verystrongly on the nature of the gas. Large drift velocities are achieved byadding polyatomic gases (usually CH4, CO2, or CF4) having large inelasticcross sections at moderate energies, which results in “cooling” electronsinto the energy range of the Ramsauer-Townsend minimum (at ∼ 0.5 eV)of the elastic cross-section of argon. In a simple approximation, gas kinetictheory provides the drift velocity v as a function of the mean collisiontime τ and the electric field E: v = eEτ/me (Townsend’s expression). Inthe presence of an external magnetic field, the Lorentz force acting onelectrons between collisions deflects the drifting electrons and modifies thedrift properties.
If the electric field is increased sufficiently, electrons gain enough energybetween collisions to ionize molecules. Above a gas-dependent threshold,the mean free path for ionization, λi, decreases exponentially with the field;its inverse, α = 1/λi, is the first Townsend coefficient. In wire chambers,most of the increase of avalanche particle density occurs very close tothe anode wires, and a simple electrostatic consideration shows that thelargest fraction of the detected signal is due to the motion of positiveions receding from the wires. The electron component, although very fast,contributes very little to the signal. This determines the characteristicshape of the detected signals in the proportional mode: a fast rise followedby a gradual increase.
28.6.2. Multi-Wire Proportional and Drift Chambers : RevisedMarch 2010 by Fabio Sauli (CERN) and Maxim Titov (CEA Saclay).
Multiwire proportional chambers (MWPCs) [65,66], introduced in thelate ’60’s, detect, localize and measure energy deposit by charged particlesover large areas. A mesh of parallel anode wires at a suitable potential,inserted between two cathodes, acts almost as a set of independentproportional counters. Electrons released in the gas volume drift towardsthe anodes and produce avalanches in the increasing field.
Detection of charge on the wires over a predefined threshold providesthe transverse coordinate to the wire with an accuracy comparable to thatof the wire spacing. The coordinate along each wire can be obtained bymeasuring the ratio of collected charge at the two ends of resistive wires.Making use of the charge profile induced on segmented cathodes, theso-called center-of gravity (COG) method, permits localization of tracksto sub-mm accuracy.
Drift chambers, developed in the early ’70’s, can be used to estimate thelongitudinal position of a track by exploiting the arrival time of electronsat the anodes if the time of interaction is known [69]. The distancebetween anode wires is usually several cm, allowing coverage of large areasat reduced cost.
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28. Detectors at accelerators 249
28.6.4. Micro-Pattern Gas Detectors : Revised March 2010 byFabio Sauli (CERN) and Maxim Titov (CEA Saclay)
By using pitch size of a few hundred μm, an order of magnitudeimprovement in granularity over wire chambers, these detectors offerintrinsic high rate capability (> 106 Hz/mm2), excellent spatial res-olution (∼ 30 μm), multi-particle resolution (∼ 500 μm), and singlephoto-electron time resolution in the ns range.
The Gas Electron Multiplier (GEM) detector consists of a thin-foilcopper-insulator-copper sandwich chemically perforated to obtain a highdensity of holes in which avalanches occur [86]. The hole diameter istypically between 25 μm and 150 μm, while the corresponding distancebetween holes varies between 50 μm and 200 μm. The central insulatoris usually (in the original design) the polymer Kapton, with a thicknessof 50 μm. Application of a potential difference between the two sides ofthe GEM generates the electric fields. Each hole acts as an independentproportional counter. Electrons released by the primary ionization particlein the upper conversion region (above the GEM foil) drift into the holes,where charge multiplication occurs in the high electric field (50–70 kV/cm).Most of avalanche electrons are transferred into the gap below the GEM.Several GEM foils can be cascaded, allowing the multi-layer GEMdetectors to operate at overall gas gain above 104 in the presence of highlyionizing particles, while strongly reducing the risk of discharges.
The micro-mesh gaseous structure (Micromegas) is a thin parallel-plateavalanche counter. It consists of a drift region and a narrow multiplicationgap (25–150 μm) between a thin metal grid (micromesh) and the readoutelectrode (strips or pads of conductor printed on an insulator board).Electrons from the primary ionization drift through the holes of the meshinto the narrow multiplication gap, where they are amplified. The smallamplification gap produces a narrow avalanche, giving rise to excellentspatial resolution: 12 μm accuracy, limited by the micro-mesh pitch, hasbeen achieved for MIPs, as well as very good time resolution and energyresolution (∼ 12% FWHM with 6 keV x rays) [89].
The performance and robustness of GEM and Micromegas haveencouraged their use in high-energy and nuclear physics, UV and visiblephoton detection, astroparticle and neutrino physics, neutron detectionand medical physics.
28.6.5. Time-projection chambers : Written September 2007 by D.Karlen (U. of Victoria and TRIUMF, Canada)
The Time Projection Chamber (TPC) concept, invented by DavidNygren in the late 1970’s [74], is the basis for charged particle trackingin a large number of particle and nuclear physics experiments. A uniformelectric field drifts tracks of electrons produced by charged particlestraversing a medium, either gas or liquid, towards a surface segmented into2D readout pads. The signal amplitudes and arrival times are recorded toprovide full 3D measurements of the particle trajectories. The intrinsic 3Dsegmentation gives the TPC a distinct advantage over other large volumetracking detector designs which record information only in a 2D projectionwith less overall segmentation, particularly for pattern recognition inevents with large numbers of particles.
Gaseous TPC’s are often designed to operate within a strong magneticfield (typically parallel to the drift field) so that particle momenta canbe estimated from the track curvature. For this application, precisespatial measurements in the plane transverse to the magnetic field aremost important. Since the amount of ionization along the length of thetrack depends on the velocity of the particle, ionization and momentum
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250 28. Detectors at accelerators
measurements can be combined to identify the types of particles observedin the TPC.
Gas amplification of 103–104 at the readout endplate is usually requiredin order to provide signals with sufficient amplitude for conventionalelectronics to sense the drifted ionization. Until recently, the gasamplification system used in TPC’s have exclusively been planes of anodewires operated in proportional mode placed close to the readout pads.Performance has been recently improved by replacing these wire planeswith micro-pattern gas detectors, namely GEM [86] and Micromegas [88]devices.
Diffusion degrades the position information of ionization that drifts along distance. For a gaseous TPC, the effect can be alleviated by thechoice of a gas with low intrinsic diffusion or by operating in a strongmagnetic field parallel to the drift field with a gas which exhibits asignificant reduction in transverse diffusion with magnetic field.
28.6.6. Transition radiation detectors (TRD’s) : Written August2007 by P. Nevski (BNL) and A. Romaniouk (Moscow Eng. & Phys. Inst.)
Transition radiation (TR) x rays are produced when a highly relativisticparticle (γ >∼ 103) crosses a refractive index interface, as discussed inSec. 27.7. The x rays, ranging from a few keV to a few dozen keV, areemitted at a characteristic angle 1/γ from the particle trajectory. Sincethe TR yield is about 1% per boundary crossing, radiation from multiplesurface crossings is used in practical detectors. In the simplest concept, adetector module might consist of low-Z foils followed by a high-Z activelayer made of proportional counters filled with a Xe-rich gas mixture.The atomic number considerations follow from the dominant photoelectricabsorption cross section per atom going roughly as Z n/E3
x, where n variesbetween 4 and 5 over the region of interest, and the x-ray energy is Ex. Tominimize self-absorption, materials such as polypropylene, Mylar, carbon,and (rarely) lithium are used as radiators. The TR signal in the activeregions is in most cases superimposed upon the particle’s ionization losses.These drop a little faster than Z/A with increasing Z, providing anotherreason for active layers with high Z.
The TR intensity for a single boundary crossing always increases withγ, but for multiple boundary crossings interference leads to saturationnear a Lorentz factor γ sat = 0.6 ω1
√�1�2/c [101], where ω1 is the radiator
plasma frequency, �1 is its thickness, and �2 the spacing. In most ofthe detectors used in particle physics the radiator parameters are chosento provide γ sat ≈ 2000. Those detectors normally work as thresholddevices, ensuring the best electron/pion separation in the momentumrange 1 GeV/c <∼ p <∼ 150 GeV/c.
The discrimination between electrons and pions can be based on thecharge deposition measured in each detection module, on the number ofclusters—energy depositions observed above an optimal threshold (usuallyin the 5 to 7 keV region), or on more sophisticated methods analyzingthe pulse shape as a function of time. The total energy measurementtechnique is more suitable for thick gas volumes, which absorb most ofthe TR radiation and where the ionization loss fluctuations are small. Thecluster-counting method works better for detectors with thin gas layers,where the fluctuations of the ionization losses are big.
Recent TRDs for particle astrophysics are designed to directly measurethe Lorentz factor of high-energy nuclei by using the quadratic dependenceof the TR yield on nuclear charge; see Cherry and Muller papers inRef. 103.
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28. Detectors at accelerators 251
28.7. Semiconductor detectorsUpdated September 2009 by H. Spieler (LBNL).
28.7.1. Materials Requirements : Semiconductor detectors are essen-tially solid state ionization chambers. Absorbed energy forms electron-holepairs, i.e., negative and positive charge carriers, which under an appliedelectric field move towards their respective collection electrodes, wherethey induce a signal current. The energy required to form an electron-holepair is proportional to the bandgap. In tracking detectors the energy lossin the detector should be minimal, whereas for energy spectroscopy thestopping power should be maximized, so for gamma rays high-Z materialsare desirable.
Measurements on silicon photodiodes [112] show that for photonenergies below 4 eV one electron-hole (e-h) pair is formed per incidentphoton. The mean energy Ei required to produce an e-h pair peaks at4.4 eV for a photon energy around 6 eV. Above ∼1.5 keV it assumesa constant value, 3.67 eV at room temperature. It is larger than thebandgap energy because momentum conservation requires excitation oflattice vibrations (phonons). For minimum-ionizing particles, the mostprobable charge deposition in a 300 μm thick silicon detector is about3.5 fC (22000 electrons). Other typical ionization energies are 2.96 eV inGe, 4.2 eV in GaAs, and 4.43 eV in CdTe.
Since both electronic and lattice excitations are involved, the variancein the number of charge carriers N = E/Ei produced by an absorbedenergy E is reduced by the Fano factor F (about 0.1 in Si and Ge). Thus,σN =
√FN and the energy resolution σE/E =
√
FEi/E. However, themeasured signal fluctuations are usually dominated by electronic noise orenergy loss fluctuations in the detector.
A major effort is to find high-Z materials with a bandgap that issufficiently high to allow room-temperature operation while still providinggood energy resolution. Compund semiconductors, e.g., CdZnTe, can allowthis, but typically suffer from charge collection problems, characterizedby the product μτ of mobility and carrier lifetime. In Si and Geμτ > 1 cm2 V−1 for both electrons and holes, whereas in compoundsemiconductors it is in the range 10−3–10−8. Since for holes μτ is typicallyan order of magnitude smaller than for electrons, detector configurationswhere the electron contribution to the charge signal dominates—e.g., stripor pixel structures—can provide better performance.
28.7.2. Detector Configurations : A p-n junction operated at reversebias forms a sensitive region depleted of mobile charge and sets up anelectric field that sweeps charge liberated by radiation to the electrodes.Detectors typically use an asymmetric structure, e.g., a highly dopedp electrode and a lightly doped n region, so that the depletion regionextends predominantly into the lightly doped volume.In a planar device the thickness of the depleted region is
W =√
2ε (V + Vbi)/Ne =√
2ρμε(V + Vbi) , (28.1)where V = external bias voltage
Vbi = “built-in” voltage (≈ 0.5 V for resistivities typically used in Sidetectors)
N = doping concentratione = electronic chargeε = dielectric constant = 11.9 ε0 ≈ 1 pF/cm in Siρ = resistivity (typically 1–10 kΩ cm in Si)μ = charge carrier mobility
= 1350 cm2 V−1 s−1 for electrons in Si
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252 28. Detectors at accelerators
= 450 cm2 V−1 s−1 for holes in SiIn SiW = 0.5 [μm/
√Ω-cm · V] ×
√
ρ(V + Vbi) for n-type Si, and
W = 0.3 [μm/√
Ω-cm · V] ×√
ρ(V + Vbi) for p-type Si.
Large volume (∼ 102–103 cm3) Ge detectors are commonly configured ascoaxial detectors, e.g., a cylindrical n-type crystal with 5–10 cm diameterand 10 cm length with an inner 5–10mm diameter n+ electrode and anouter p+ layer forming the diode junction. Ge can be grown with verylow impurity levels, 109–1010 cm−3 (HPGe), so these large volumes can bedepleted with several kV.
28.7.3. Signal Formation : The signal pulse shape depends on theinstantaneous carrier velocity v(x) = μE(x) and the electrode geometry,which determines the distribution of induced charge (e.g., see Ref. 111,pp. 71–83). Charge collection time decreases with increasing bias voltage,and can be reduced further by operating the detector with “overbias,”i.e., a bias voltage exceeding the value required to fully deplete the device.The collection time is limited by velocity saturation at high fields (in Siapproaching 107 cm/s at E > 104 V/cm); at an average field of 104 V/cmthe collection time is about 15 ps/μm for electrons and 30 ps/μm for holes.In typical fully-depleted detectors 300 μm thick, electrons are collectedwithin about 10 ns, and holes within about 25 ns.
Position resolution is limited by transverse diffusion during chargecollection (typically 5 μm for 300 μm thickness) and by knock-on electrons.Resolutions of 2–4 μm (rms) have been obtained in beam tests. Inmagnetic fields, the Lorentz drift deflects the electron and hole trajectoriesand the detector must be tilted to reduce spatial spreading (see “Halleffect” in semiconductor textbooks).
Electrodes can be in the form of cm-scale pads, strips, or μm-scalepixels. Various readout structures have been developed for pixels, e.g.,CCDs, DEPFETs, monolithic pixel devices that integrate sensor andelectronics (MAPS), and hybrid pixel devices that utilize separate sensorsand readout ICs connected by two-dimensional arrays of solder bumps.For an overview and further discussion see Ref. 111.
28.7.4. Radiation Damage : Radiation damage occurs through twobasic mechanisms:
1. Bulk damage due to displacement of atoms from their latticesites. This leads to increased leakage current, carrier trapping, andbuild-up of space charge that changes the required operating voltage.Displacement damage depends on the nonionizing energy loss andthe energy imparted to the recoil atoms, which can initiate a chain ofsubsequent displacements, i.e., damage clusters. Hence, it is criticalto consider both particle type and energy.
2. Surface damage due to charge build-up in surface layers, which leadsto increased surface leakage currents. In strip detectors the inter-stripisolation is affected. The effects of charge build-up are stronglydependent on the device structure and on fabrication details. Sincethe damage is proportional to the absorbed energy (when ionizationdominates), the dose can be specified in rad (or Gray) independentof particle type.
Strip and pixel detectors have remained functional at fluences beyond1015 cm−2 for minimum ionizing protons. At this damage level, chargeloss due to recombination and trapping becomes significant and thehigh signal-to-noise ratio obtainable with low-capacitance pixel structures
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28. Detectors at accelerators 253
20 30 50 70 9040 60 80 100Z
10
0
20
30
40
50
1
0
2
3
4
5
ΛI/ρ
(cm
)
ΛI
X0/
ρ (
cm)
X0
CuFe Ru Pd W Au Pb U
Figure 28.21: Nuclear interaction length λI/ρ (circles) andradiation length X0/ρ (+’s) in cm for the chemical elements withZ > 20 and λI < 50 cm.
extends detector lifetime. The higher mobility of electrons makes themless sensitive to carrier lifetime than holes, so detector configurations thatemphasize the electron contribution to the charge signal are advantageous,e.g., n+ strips or pixels on a p-substrate. The occupancy of the defectcharge states is strongly temperature dependent; competing processescan increase or decrease the required operating voltage. It is criticalto choose the operating temperature judiciously (−10 to 0◦C in typicalcollider detectors) and limit warm-up periods during maintenance. Fora more detailed summary see Ref. 117 and and the web-sites of theROSE and RD50 collaborations at http://RD48.web.cern.ch/rd48 andhttp://RD50.web.cern.ch/rd50. Materials engineering, e.g., introducingoxygen interstitials, can improve certain aspects and is under investigation.At high fluences diamond is an alternative, but operates as an insulatorrather than a reverse-biased diode.
28.9. Calorimeters
A calorimeter is designed to measure the energy deposition and itsdirection for a contained electromagnetic (EM) or hadronic shower. Thecharacteristic interaction distance for an electromagnetic interaction isthe radiation length X0, which ranges from 13.8 g cm−2 in iron to 6.0 gcm−2 in uranium.* Similarly, the characteristic nuclear interaction lengthλI varies from 132.1 g cm−2 (Fe) to 209 g cm−2 (U).† In either case,a calorimeter must be many interaction lengths deep, where “many” isdetermined by physical size, cost, and other factors. EM calorimeters tendto be 15–30 X0 deep, while hadronic calorimeters are usually compromisedat 5–8 λI . In real experiments there is likely to be an EM calorimeterin front of the hadronic section, which in turn has less sampling densityin the back, so the hadronic cascade occurs in a succession of differentstructures.
* X0 = 120 g cm−2 Z−2/3 to better than 5% for Z > 23.† λI = 37.8 g cm−2 A0.312 to within 0.8% for Z > 15.
See pdg.lbl.gov/AtomicNuclearProperties for actual values.
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254 28. Detectors at accelerators
In all cases there is a premium on small λI/ρ and X0/ρ (both withunits of length). These quantities are shown for Z > 20 for the chemicalelements in Fig. 28.21.
These considerations are for sampling calorimeters consisting of metallicabsorber sandwiched or (threaded) with an active material which generatessignal. The active medium may be a scintillator, an ionizing noble liquid,a gas chamber, a semiconductor, or a Cherenkov radiator.
There are also homogeneous calorimeters, in which the entire volumeis sensitive, i.e., contributes signal. Homogeneous calorimeters (so farusually electromagnetic) may be built with inorganic heavy (high density,high 〈Z〉) scintillating crystals, or non-scintillating Cherenkov radiatorssuch as lead glass and lead fluoride. Scintillation light and/or ionizationin noble liquids can be detected. Nuclear interaction lengths in inorganiccrystals range from 17.8 cm (LuAlO3) to 42.2 cm (NaI).28.9.1. Electromagnetic calorimeters :Revised October 2009 by R.-Y. Zhu (California Inst. of Technology).
The development of electromagnetic showers is discussed in the sectionon “Passage of Particles Through Matter” (Sec. 27 of this Review).
The energy resolution σE/E of a calorimeter can be parametrized asa/
√E ⊕ b ⊕ c/E, where ⊕ represents addition in quadrature and E is in
GeV. The stochastic term a represents statistics-related fluctuations suchas intrinsic shower fluctuations, photoelectron statistics, dead material atthe front of the calorimeter, and sampling fluctuations. For a fixed numberof radiation lengths, the stochastic term a for a sampling calorimeter isexpected to be proportional to
√
t/f , where t is plate thickness and fis sampling fraction [123,124]. While a is at a few percent level for ahomogeneous calorimeter, it is typically 10% for sampling calorimeters.The main contributions to the systematic, or constant, term b aredetector non-uniformity and calibration uncertainty. In the case of thehadronic cascades discussed below, non-compensation also contributes tothe constant term. One additional contribution to the constant term forcalorimeters built for modern high-energy physics experiments, operatedin a high-beam intensity environment, is radiation damage of the activemedium. This can be minimized by developing radiation-hard activemedia [47] and by frequent in situ calibration and monitoring [46,124].
28.9.2. Hadronic calorimeters : [1–5,124]Written April 2008 by D. E. Groom (LBNL).
Most large hadron calorimeters are sampling calorimeters which areparts of complicated 4π detectors at colliding beam facilities. Typically,the basic structure is plates of absorber (Fe, Pb, Cu, or occasionally U orW) alternating with plastic scintillators (plates, tiles, bars), liquid argon(LAr), or gaseous detectors. The ionization is measured directly, as in LArcalorimeters, or via scintillation light observed by photodetectors (usuallyPMT’s). Waveshifting fibers are often used to solve difficult problems ofgeometry and light collection uniformity. Silicon sensors are being studiedfor ILC detectors; in this case e-h pairs are collected.
In an inelastic hadronic collision a significant fraction fem of theenergy is removed from further hadronic interaction by the productionof secondary π0’s and η’s, whose decay photons generate high-energyelectromagnetic (EM) cascades. Charged secondaries (π±, p, . . . ) depositenergy via ionization and excitation, but also interact with nuclei,producing spallation protons and neutrons, evaporation neutrons, andrecoiling nuclei in highly excited states. The charged collision productsproduce detectable ionization, as do the showering γ-rays from the promptde-excitation of highly excited nuclei. The recoiling nuclei generate little
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28. Detectors at accelerators 255
or no detectable signal. The neutrons lose kinetic energy in elasticcollisions over hundreds of ns, gradually thermalize and are captured,with the production of more γ-rays—usually outside the acceptance gateof the electronics. Between endothermic spallation losses, nuclear recoils,and late neutron capture, a significant fraction of the hadronic energy(20%–35%, depending on the absorber and energy of the incident particle)is invisible.
For h/e �= 1, when h and e are the hadronic and electromagneticcalorimeter responses, respectively, fluctuations in fem significantlycontribute to the resolution, in particular contributing a larger fraction ofthe variance at high energies. Since the fem distribution has a tail on thehigh side, the calorimeter response is non-Gaussian with a high-energy tailif h/e < 1. Noncompensation (h/e �= 1) thus seriously degrades resolutionas well as producing a nonlinear response.
It is clearly desirable to compensate the response, i.e., to designthe calorimeter such that h/e = 1. This is possible only in a samplingcalorimeter, where several variables can be chosen or tuned:1. Decrease the EM sensitivity. The absorber usually has higher 〈Z〉 than
does the sensor, the EM energy deposit rate, relative to minimumionization, is greater than this ratio in the sensor.
2. Increase the hadronic sensitivity. The abundant neutrons have alarge n-p scattering cross section, with the production of low-energyscattered protons in hydrogenous sampling materials such as butane-filled proportional counters or plastic scintillator. (When scattering offa nucleus with mass number A, a neutron can at most lose 4/(1 + A)2of its kinetic energy.)Motivated very much by the work of Brau, Gabriel, Bruckmann, and
Wigmans [134], several groups built calorimeters which were very nearlycompensating. The degree of compensation was sensitive to the acceptancegate width, and so could be somewhat tuned.
The average longitudinal distribution rises to a smooth peak, increasingslowly with energy but about one nuclear interaction length (λI ) intothe calorimeter. After several interaction lengths its fall is reasonablyexponential. It has been found that a gamma distribution fairly welldescribes the longitudinal development of an EM shower, as discussed inSec. 27.5.
The transverse energy deposit is characterized by a central coredominated by EM cascades, together with a wide “skirt” produced bywide-angle hadronic interactions [139].
Further discussion and all references may be found in the full Reviewof Particle Physics.The numbering of references and equations used herecorresponds to that version.
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256 29. Detectors for non-accelerator physics
29. PARTICLE DETECTORS FOR
NON-ACCELERATOR PHYSICS
Written 2009 (see the various sections for authors).29.1. Introduction
Non-accelerator experiments have become increasingly important inparticle physics. These include classical cosmic ray experiments, neutrinooscillation measurements, and searches for double-beta decay, darkmatter candidates, and magnetic monopoles. The experimental methodsare sometimes those familiar at accelerators (plastic scintillators, driftchambers, TRD’s, etc.) but there is also instrumentation either notfound at accelerators or applied in a radically different way. Examplesare atmospheric scintillation detectors (Fly’s Eye), massive Cherenkovdetectors (Super-Kamiokande, IceCube), ultracold solid state detectors(CDMS). And, except for the cosmic ray detectors, there is a demand forradiologically ultra-pure materials.
In this section, some more important detectors special to terrestrialnon-accelerator experiments are discussed. Techniques used in bothaccelerator and non-accelerator experiments are described in Sec. 28,Particle Detectors at Accelerators, some of which have been modified toaccommodate the non-accelerator nuances. Space-based detectors also usesome unique methods, but these are beyond the present scope of RPP.
29.2. High-energy cosmic-ray hadron and gamma-raydetectors
29.2.1. Atmospheric fluorescence detectors :Written September 2009 by L.R. Wiencke (Colorado School of Mines).
Cosmic-ray fluorescence detectors (FD) use the atmosphere as agiant calorimeter to measure isotropic scintillation light that traces thedevelopment profiles of extensive air showers (EAS). The EASs observedare produced by the interactions of high-energy (E > 1017eV) subatomicparticles in the stratosphere and upper troposphere. The amount ofscintillation light generated is proportional to energy deposited in theatmosphere and nearly independent of the primary species.
The scintillation light is emitted between 290 and 430 nm, whenrelativistic charged particles, primarily electrons and positrons, excitenitrogen molecules in air, resulting in transitions of the 1P and 2P systems.
An FD element (telescope) consists of a non-tracking spherical mirror(3.5–13 m2 and less than astronomical quality), a close-packed “camera”of PMTs near the focal plane, and flash ADC readout system with a pulseand track-finding trigger scheme [7]. Simple reflector optics (12◦ × 16◦degree field of view (FOV) on 256 PMTs) and Schmidt optics (30◦ × 30◦on 440 PMTs), including a correcting element, have been used.
The EAS generates a track consistent with a light source moving atv = c across the FOV. The number of photons (Nγ) as a function ofatmospheric depth (X) can be expressed as [6]
dNγ
dX=
dEtotdep
dX
∫
Y (λ, P, T, u) · τatm(λ, X) · εFD(λ)dλ , (29.1)
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29. Detectors for non-accelerator physics 257
where τatm(λ, X) is atmospheric transmission, including wavelength (λ)dependence, and εFD(λ) is FD efficiency. εFD(λ) includes geometricfactors and collection efficiency of the optics, quantum efficiency ofthe PMTs, and other throughput factors. The typical systematicuncertainties, Y (10–15%), τatm (10%) and εFD (photometric calibration10%), currently dominate the total reconstructed EAS energy uncertainty.ΔE/E of 20–25% is possible, provided the geometric fit of the EAS axis isconstrained by multi-eye stereo projection, or by timing from a colocatedsparse array of surface detectors.
29.2.2. Atmospheric Cherenkov telescopes for high-energy γ-ray astronomy :Written August 2009 by J. Holder (Bartol Research Inst., Univ. ofDelaware).
Atmospheric Cherenkov detectors achieve effective collection areas of∼ 105 m2 by employing the Earth’s atmosphere as an intrinsic part of thedetection technique. A hadronic cosmic ray or high energy γ-ray incidenton the Earth’s atmosphere triggers a particle cascade, or air shower.Relativistic charged particles in the cascade produce Cherenkov radiation,which is emitted along the shower direction, resulting in a light pool onthe ground with a radius of ∼ 130 m. Maximum emission occurs whenthe number of particles in the cascade is largest. The Cherenkov light atground level peaks at a wavelength, λ ≈ 300–350 nm. The photon densityis typically ∼ 100 photons/m2 at 1 TeV, arriving in a brief flash of a fewnanoseconds duration.
Modern atmospheric Cherenkov telescopes consist of large (> 100m2)segmented mirrors on steerable altitude-azimuth mounts. A camera, madefrom an array of up to 1000 photomultiplier tubes (PMTs) covering afield-of-view of up to 5.0◦ in diameter, is placed at the mirror focus andused to record a Cherenkov image of each air shower. Images are recordedat a rate of a few hundred Hz, the vast majority of which are due toshowers with hadronic cosmic-ray primaries. The shape and orientation ofthe Cherenkov images are used to discriminate γ-ray photon events fromthis cosmic-ray background, and to reconstruct the photon energy andarrival direction.
The total Cherenkov yield from the air shower is proportional to theenergy of the primary particle. The energy resolution of this technique,also energy-dependent, is typically 15–20 at energies above a few hundredGeV. Energy spectra of γ-ray sources can be measured over a wide range;potentially from ∼ 50 GeV to ∼ 100 TeV, depending upon the instrumentcharacteristics, source strength, and exposure time.
29.3. Large neutrino detectors
29.3.1. Deep liquid detectors for rare processes :Written September 2009 by K. Scholberg & C.W. Walter (Duke University)
Deep, large detectors for rare processes tend to be multi-purposewith physics reach that includes not only solar, reactor, supernova andatmospheric neutrinos, but also searches for baryon number violation,searches for exotic particles such as magnetic monopoles, and neutrinoand cosmic ray astrophysics in different energy regimes. The detectorsmay also serve as targets for long-baseline neutrino beams for neutrino
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258 29. Detectors for non-accelerator physics
oscillation physics studies. In general, detector design considerations canbe divided into high-and low-energy regimes, for which background andevent reconstruction issues differ. The high-energy regime, from about100 MeV to a few hundred GeV, is relevant for proton decay searches,atmospheric neutrinos and high-energy astrophysical neutrinos. Thelow-energy regime (a few tens of MeV or less) is relevant for supernova,solar, reactor and geological neutrinos.
Large water Cherenkov and scintillator detectors (see Table 29.1)usually consist of a volume of transparent liquid viewed by photomultipliertubes (PMTs). Because photosensors lining an inner surface represent adriving cost that scales as surface area, very large volumes can be usedfor comparatively reasonable cost. A common configuration is to have atleast one concentric outer layer of liquid material separated from the innerpart of the detector to serve as shielding against ambient background. Ifoptically separated and instrumented with PMTs, an outer layer may alsoserve as an active veto against entering cosmic rays and other background
Because in most cases one is searching for rare events, large detectorsare usually sited underground to reduce cosmic-ray related background(see Chapter 24). The minimum depth required varies according to thephysics goals [22].
29.3.1.1. Liquid scintillator detectors:Past and current large underground detectors based on hydrocarbon
scintillators include LVD, MACRO, Baksan, Borexino, KamLAND andSNO+. Experiments at nuclear reactors include Chooz, Double Chooz,Daya Bay, and RENO. Organic liquid scintillators for large detectorsare chosen for high light yield and attenuation length, good stability,compatibility with other detector materials, high flash point, low toxicity,appropriate density for mechanical stability, and low cost.
Scintillation detectors have an advantage over water Cherenkovdetectors in the lack of Cherenkov threshold and the high light yield.However, scintillation light emission is nearly isotropic, and thereforedirectional capabilities are relatively weak.
29.3.1.2. Water Cherenkov detectors:Very large-imaging water detectors reconstruct ten-meter-scale
Cherenkov rings produced by charged particles (see Sec. 28.5.0).The first such large detectors were IMB and Kamiokande. The onlycurrently existing instance of this class is Super-Kamiokande (Super-K).
Cherenkov detectors are excellent electromagnetic calorimeters, and thenumber of Cherenkov photons produced by an e/γ is nearly proportionalto its kinetic energy. The number of collected photoelectrons dependson the scattering and attenuation in the water along with the photo-cathode coverage, quantum efficiency and the optical parameters of anyexternal light collection systems or protective material surrounding them.Event-by-event corrections are made for geometry and attenuation.
High-energy (∼100 MeV or more) neutrinos from the atmosphere orbeams interact with nucleons; for the nucleons bound inside the 16Onucleus, the nuclear effects both at the interaction, and as the particlesleave the nucleus must be considered when reconstructing the interaction.Various event topologies can be distinguished by their timing and fitpatterns, and by presence or absence of light in a veto.
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29. Detectors for non-accelerator physics 259
Low-energy neutrino interactions of solar neutrinos in water arepredominantly elastic scattering off atomic electrons; single electron eventsare then reconstructed. At solar neutrino energies, the visible energyresolution (∼ 30%/
√
ξ Evis(MeV)) is about 20% worse than photoelectroncounting statistics would imply. At these energies, radioactive backgroundsbecome a dominant issue.
The Sudbury Neutrino Observatory (SNO) detector [27] is the onlyinstance of a large heavy water detector and deserves mention here. Inaddition to an outer 1.7 kton of light water, SNO contained 1 kton of D2O,giving it unique sensitivity to neutrino neutral current (νx+d → νx+p+n),and charged current (νe + d → p + p + e−) deuteron breakup reactions.
29.3.2. Neutrino telescopes :Written November 2009 by A. Karle (University of Wisconsin).
Neutrino telescopes are large water or ice Cherenkov detectors designedto do neutrino astronomy in the energy range 1011–1018 eV. The primaryphysics goal is the detection of high-energy extra-terrestrial neutrinos.Neutrinos offer a unique view of the high-energy Universe because they arenot deflected by magnetic fields like charged cosmic rays, and can travelvast distances without being absorbed like high-energy photons.
Neutrino telescopes detect neutrinos via the process νl(νl)+N → l±+Xof primarily upward-going or horizontal neutrinos interacting with anucleon N of the matter comprising or surrounding the detector volume.A significant fraction of the neutrino energy Eν will be carried awayby the produced lepton. For the important case of a lepton being amuon, the angle Δψ between the parent neutrino and the muon is≈ 0.7◦(Eν/TeV)−0.7 [28].
The small cross sections combined with small expected fluxes ofhigh-energy cosmic neutrinos necessitate very large, km-scale detectionvolumes. Neutrino telescopes make use of large natural transparent targetmedia such as deep sea water or the deep glacial ice in Antarctica. Thetarget volume is instrumented with large photomultipliers (PMTs). Thegoal is to maximize the number of detected Cherenkov photons producedby energetic charged particles, and thus to maximize the number of wellreconstructed events while keeping the density of PMTs (and thus thecost of the experiment) low. The sensors are deployed at depths of oneto several km to reduce backgrounds from the cosmic-ray-generated muonflux.
The IceCube Neutrino Observatory [37] is a km-scale detector which willbe completed in 2011. IceCube will consist of 5160 optical sensors deployedon 86 strings covering a volume of 1 km3. A surface detector component,IceTop, forms an air shower array with 320 standard photomultipliersdeployed in 160 IceTop tanks on the ice surface directly above the strings.The optical sensors are located at depths between 1450m and 2450m.The detector is constructed by drilling holes in the ice with a hot-waterdrill.
Figure 29.3 contains three different predictions for the flux of neutrinosassociated with the still-enigmatic sources of the highest energy cosmicrays. Kilometer-scale neutrino detectors have the sensitivity to revealgeneric cosmic-ray sources with an energy density in neutrinos comparableto their energy density in cosmic rays and in TeV γ rays.
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260 29. Detectors for non-accelerator physics
No detection of astrophysical neutrinos has been made yet. Two limitsfrom AMANDA to such E neutrino fluxes are shown: one is a diffusemuon neutrino flux limit [48], the other is a limit based on all flavoranalysis of non-contained events in the PeV to EeV energy region [49]
Eν [GeV] 106 107 108 109 1010 1011 1012102 103 104 105
10−10
10−9
10−8
10−7
10−4
10−3
10−6
10−5
Atmospheric neutrino modelGamma-ray burstsWaxman-Bahcall boundCosmogenic neutrino fluxAMANDA-II unfolded νμAMANDA II νμ × 3AMANDA II cascades prelim.IceCube projected 1 year νμ × 3
Eν
dNν/
dEν
[GeV
cm
−2 s
−1 s
r−1 ]
2
Figure 29.3: Measured atmospheric neutrino fluxes above 100 GeVare shown together with a few generic models for astrophysicalneutrinos and some limits. See full-color version on color pages atend of book.
29.4. Large time-projection chambers for rare eventdetection
Written August 2009 by M. Heffner (LLNL).TPCs in non-accelerator particle physics experiments are principally
focused on rare event detection (e.g., neutrino and dark matterexperiments) and the physics of these experiments can place dramaticallydifferent constraints on the TPC design (only extensions of the traditionalTPCs are discussed here). The drift gas or liquid is usually the target ormatter under observation and due to very low signal rates a TPC withthe largest possible active mass is desired. The large mass complicatesparticle tracking of short and sometimes very low-energy particles. Otherspecial design issues include efficient light collection, background rejection,internal triggering, and optimal energy resolution.
The liquid-phase TPC can have a high density at low pressure thatresults in very good self-shielding and compact installation with lightweightcontainment. The down sides are the need for cryogenics, slower chargedrift, tracks shorter than typical electron diffusion distances, lower-energyresolution (e.g., xenon) and limited charge readout options. Slower chargedrift requires long electron lifetimes, placing strict limits on the oxygenand other impurities with high electron affinity.
A high-pressure gas phase TPC has no cryogenics and density is easilyoptimized for the signal, but a large heavy-pressure vessel is required.
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29. Detectors for non-accelerator physics 261
Although self shielding is reduced, it can in some cases approach that ofthe liquid phase; in xenon at 50 atm the density is about half that of wateror about 1/6 of liquid xenon. A significant feature of high pressure xenongas is the energy resolution.
Rare-event TPCs can be designed to detect scintillation light as well ascharge to exploit the anti-correlation to improve energy resolution and/orsignal to noise [63]. Electroluminescence can be used to proportionallyamplify the drifted ionization, and it does not suffer the fluctuations ofan avalanche or the small signals of direct collection. It works by settingup at the positive end of the drift volume parallel meshes or wire arrayswith an electric field larger than the drift field, but less than the fieldneeded for avalanche. In xenon, this is 3–6 kV cm−1 bar−1 for good energyresolution.
Differentiation of nuclear and electron recoils at low-energy deposition isimportant as a means of background rejection. The nuclear recoil depositsa higher density of ionization than an electron recoil and this results ina higher geminate recombination resulting in a higher output of primaryscintillation and lower charge. The ratio of scintillation to charge can beused to distinguish the two. In the case of an electroluminescence readout,this is done simply with the ratio of primary light to secondary light.
29.5. Sub-Kelvin detectors
Written September 2009 by S. Golwala (Caltech).Detectors operating below 1 K, also known as “low-temperature”
or “cryogenic” detectors, use <∼meV quanta (phonons, superconductingquasiparticles) to provide better energy resolution than is typicallyavailable from conventional technologies. Such resolution can provideunique advantages to applications reliant on energy resolution, suchas beta-decay experiments seeking to measure the νe mass or searchesfor neutrinoless double-beta decay. In addition, the sub-Kelvin modeis combined with conventional (eV quanta) ionization or scintillationmeasurements to provide discrimination of nuclear recoils from electronrecoils, critical for searches for WIMP dark matter and for coherentneutrino-nucleus scattering.
29.5.1. Thermal Phonons :The most basic kind of low-temperature detector employs a dielectric
absorber coupled to a thermal bath via a weak link. A thermistor monitorsthe temperature of the absorber. The energy E deposited by a particleinteraction causes a calorimetric temperature change by increasing thepopulation of thermal phonons. The fundamental sensitivity is
σ2E = ξ2kT [T C(T ) + βE] , (29.5)
where C is the heat capacity of the detector, T is the temperature ofoperation, k is Boltzmann’s constant, and ξ is a dimensionless factor oforder unity that is precisely calculable from the nature of the thermal linkand the non-thermodynamic noises (e.g., Johnson and/or readout noise).The energy resolution typically acquires an additional energy dependencedue to deviations from an ideal calorimetric model that cause positionand/or energy dependence in the signal shape. The rise time of responseis limited by the internal thermal conductivity of the absorber.
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262 29. Detectors for non-accelerator physics
29.5.2. Athermal Phonons and Superconducting Quasiparticles :The advantage of thermal phonons is also a disadvantage: energy
resolution degrades as√
M where M is the detector mass. This motivatesthe use of athermal phonons. There are three steps in the development ofthe phonon signal. The recoiling particle deposits energy along its track,with the majority going directly into phonons. The recoil and bandgapenergy scales (keV and higher, and eV, respectively) are much largerthan phonon energies (meV), so the full energy spectrum of phonons ispopulated, with phase space favoring the most energetic phonons.
Another mode is detection of superconducting quasiparticles insuperconducting crystals. Energy absorption breaks superconductingCooper pairs and yields quasiparticles, electron-like excitations that candiffuse through the material and that recombine after the quasiparticlelifetime.
29.5.3. Ionization and Scintillation :While ionization and scintillation detectors usually operate at much
higher temperatures, ionization and scintillation can be measured atlow temperature and can be combined with a “sub-Kelvin” techniqueto discriminate nuclear recoils from background interactions producingelectron recoils, which is critical for WIMP searches and coherentneutrino-nucleus scattering. With ionization, such techniques are based onLindhard theory [55], which predicts substantially reduced ionization yieldfor nuclear recoils relative to electron recoils. For scintillation, applicationof Birks’ law Sec. 28.3.0) yields a similar prediction.
29.6. Low-radioactivity background techniques
Written August 2009 by A. Piepke (University of Alabama).
The physics reach of low-energy rare event searches e.g. for dark matter,neutrino oscillations, or double beta decay is often limited by backgroundcaused by radioactivity. Depending on the chosen detector design, theseparation of the physics signal from this unwanted interference can beachieved on an event-by-event basis by active event tagging, utilizingsome unique event feature, or by reducing the radiation backgroundby appropriate shielding and material selection. In both cases, thebackground rate is proportional to the flux of background-creatingradiation. Its reduction is thus essential for realizing the full physicspotential of the experiment. In this context, “low energy” may be definedas the regime of natural, anthropogenic, or cosmogenic radioactivity, all atenergies up to about 10 MeV. Following the classification of [64], sourcesof background may be categorized into the following classes:
1. environmental radioactivity,2. radioimpurities in detector or shielding components,3. radon and its progeny,4. cosmic rays,5. neutrons from natural fission, (α, n) reactions and from cosmic-ray
muon spallation and capture.
Further discussion and all references may be found in the full Reviewof Particle Physics.The numbering of references and equations used herecorresponds to that version.
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30. Radioactivity and radiation protection 263
30. RADIOACTIVITY AND RADIATION PROTECTION
Revised Sept. 2007 by S. Roesler (CERN) and J.C. Liu (SLAC).
30.1. Definitions
The International Commission on Radiation Units and Measurements(ICRU) recommends the use of SI units. Therefore we list SI units first,followed by cgs (or other common) units in parentheses, where they differ.
• Activity (unit: Becquerel):1 Bq = 1 disintegration per second (= 27 pCi).
• Absorbed dose (unit: Gray): The absorbed dose is the energyimparted by ionizing radiation in a volume element of a specified materialdivided by the mass of this volume element.
1 Gy = 1 J/kg (= 104 erg/g = 100 rad)= 6.24 × 1012 MeV/kg deposited energy.
• Kerma (unit: Gray): Kerma is the sum of the initial kinetic energies ofall charged particles liberated by indirectly ionizing particles in a volumeelement of the specified material divided by the mass of this volumeelement.• Exposure (unit: C/kg of air [= 3880 Roentgen†]): The exposure isa measure of photon fluence at a certain point in space integrated overtime, in terms of ion charge of either sign produced by secondary electronsin a small volume of air about the point. Implicit in the definition isthe assumption that the small test volume is embedded in a sufficientlylarge uniformly irradiated volume that the number of secondary electronsentering the volume equals the number leaving (so-called charged particleequilibrium).
Table 30.1: Radiation weighting factors, wR.
Radiation type wR
Photons 1Electrons and muons 1Neutrons, En < 1 MeV 2.5 + 18.2 × exp[−(ln En)2/6]
1 MeV ≤ En ≤ 50 MeV 5.0 + 17.0 × exp[−(ln(2En))2/6]En > 50 MeV 2.5 + 3.25 × exp[−(ln(0.04En))2/6]
Protons and charged pions 2Alpha particles, fissionfragments, heavy ions 20
• Equivalent dose (unit: Sievert [= 100 rem (roentgen equivalent inman)]): The equivalent dose HT in an organ or tissue T is equal to thesum of the absorbed doses DT,R in the organ or tissue caused by differentradiation types R weighted with so-called radiation weighting factors wR:
HT =∑
R
wR × DT,R . (30.1)
† This unit is somewhat historical, but appears on some measuring in-struments. One R is the amount of radiation required to liberate positiveand negative charges of one electrostatic unit of charge in 1 cm3 of air atstandard temperature and pressure (STP)
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264 30. Radioactivity and radiation protection
It expresses long-term risks (primarily cancer and leukemia) from low-levelchronic exposure. The values for wR recommended recently by ICRP [1]are given in Table 29.1. • Effective dose (unit: Sievert): The sum of theequivalent doses, weighted by the tissue weighting factors wT (
∑
T wT = 1)of several organs and tissues T of the body that are considered to be mostsensitive [2], is called “effective dose” E:
E =∑
T
wT × HT . (30.2)
30.2. Radiation levels [3]
• Natural annual background, all sources: Most world areas, whole-body equivalent dose rate ≈ (0.4–4) mSv (40–400 mrem). Can range upto 50 mSv (5 rem) in certain areas. U.S. average ≈ 3.6 mSv, including≈ 2 mSv (≈ 200 mrem) from inhaled natural radioactivity, mostly radonand radon daughters. (Average is for a typical house and varies by morethan an order of magnitude. It can be more than two orders of magnitudehigher in poorly ventilated mines. 0.1–0.2 mSv in open areas.)• Cosmic ray background (sea level, mostly muons):∼ 1 min−1 cm−2 sr−1. For more accurate estimates and details, see theCosmic Rays section (Sec. 24 of this Review).• Fluence (per cm2) to deposit one Gy, assuming uniform irradiation:≈ (charged particles) 6.24×109/(dE/dx), where dE/dx (MeV
g−1 cm2), the energy loss per unit length, may be obtained from Figs.27.2 and 27.4 in Sec. 27 of the Review, and pdg.lbl.gov/AtomicNuclearProperties.≈ 3.5 × 109 cm−2 minimum-ionizing singly-charged particles in carbon.≈ (photons) 6.24×109/[Ef/�], for photons of energy E (MeV),
attenuation length � (g cm−2), and fraction f � 1 expressing the fractionof the photon’s energy deposited in a small volume of thickness � � butlarge enough to contain the secondary electrons.≈ 2 × 1011 photons cm−2 for 1 MeV photons on carbon (f ≈ 1/2).
• Recommended limits of effective dose to radiation workers(whole-body dose):∗
EU/Switzerland: 20 mSv yr−1
U.S.: 50 mSv yr−1 (5 rem yr−1)†• Lethal dose: The whole-body dose from penetrating ionizing radiationresulting in 50% mortality in 30 days (assuming no medical treatment)is 2.5–4.5 Gy (250–450 rad), as measured internally on body longitudinalcenter line. Surface dose varies due to variable body attenuation and maybe a strong function of energy.• Cancer induction by low LET radiation: The cancer inductionprobability is about 5% per Sv on average for the entire population [2].
Footnotes:∗ The ICRP recommendation [2] is 20 mSv yr−1 averaged over 5 years,
with the dose in any one year ≤ 50 mSv.† Many laboratories in the U.S. and elsewhere set lower limits.
See full Review for references and further details.
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31. Commonly used radioactive sources 265
31. COMMONLY USED RADIOACTIVE SOURCES
Table 31.1. Revised November 1993 by E. Browne (LBNL).
Particle PhotonType of Energy Emission Energy Emission
Nuclide Half-life decay (MeV) prob. (MeV) prob.2211Na 2.603 y β+, EC 0.545 90% 0.511 Annih.
1.275 100%5425Mn 0.855 y EC 0.835 100%
Cr K x rays 26%5526Fe 2.73 y EC Mn K x rays:
0.00590 24.4%0.00649 2.86%
5727Co 0.744 y EC 0.014 9%
0.122 86%0.136 11%Fe K x rays 58%
6027Co 5.271 y β− 0.316 100% 1.173 100%
1.333 100%6832Ge 0.742 y EC Ga K x rays 44%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
→ 6831Ga β+, EC 1.899 90% 0.511 Annih.
1.077 3%9038Sr 28.5 y β− 0.546 100%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
→ 9039Y β− 2.283 100%
10644Ru 1.020 y β− 0.039 100%- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
→ 10645Rh β− 3.541 79% 0.512 21%
0.622 10%10948Cd 1.267 y EC 0.063 e− 41% 0.088 3.6%
0.084 e− 45% Ag K x rays 100%0.087 e− 9%
11350Sn 0.315 y EC 0.364 e− 29% 0.392 65%
0.388 e− 6% In K x rays 97%13755Cs 30.2 y β− 0.514 94% 0.662 85%
1.176 6%
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266 31. Commonly used radioactive sources
13356Ba 10.54 y EC 0.045 e− 50% 0.081 34%
0.075 e− 6% 0.356 62%Cs K x rays 121%
20783Bi 31.8 y EC 0.481 e− 2% 0.569 98%
0.975 e− 7% 1.063 75%1.047 e− 2% 1.770 7%
Pb K x rays 78%22890Th 1.912 y 6α: 5.341 to 8.785 0.239 44%
3β−: 0.334 to 2.246 0.583 31%2.614 36%
(→22488Ra → 220
86Rn → 21684Po → 212
82Pb → 21283Bi → 212
84Po)24195Am 432.7 y α 5.443 13% 0.060 36%
5.486 85% Np L x rays 38%24195Am/Be 432.2 y 6 × 10−5 neutrons (4–8 MeV) and
4 × 10−5γ’s (4.43 MeV) per Am decay24496Cm 18.11 y α 5.763 24% Pu L x rays ∼ 9%
5.805 76%25298Cf 2.645 y α (97%) 6.076 15%
6.118 82%Fission (3.1%)
≈ 20 γ’s/fission; 80% < 1 MeV≈ 4 neutrons/fission; 〈En〉 = 2.14 MeV
“Emission probability” is the probability per decay of a given emission;because of cascades these may total more than 100%. Only principalemissions are listed. EC means electron capture, and e− meansmonoenergetic internal conversion (Auger) electron. The intensity of 0.511MeV e+e− annihilation photons depends upon the number of stoppedpositrons. Endpoint β± energies are listed. In some cases when energiesare closely spaced, the γ-ray values are approximate weighted averages.Radiation from short-lived daughter isotopes is included where relevant.
Half-lives, energies, and intensities are from E. Browne and R.B. Firestone,Table of Radioactive Isotopes (John Wiley & Sons, New York, 1986), recentNuclear Data Sheets, and X-ray and Gamma-ray Standards for DetectorCalibration, IAEA-TECDOC-619 (1991).Neutron data are from Neutron Sources for Basic Physics and Applications(Pergamon Press, 1983).
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32. Probability 267
32. PROBABILITY
Revised September 2009 by G. Cowan (RHUL).
The following is a much-shortened version of Sec. 31 of the full Review.Equation, section, and figure numbers follow the Review.
32.2. Random variables
• Probability density function (p.d.f.): x is a random variable.Continuous: f(x; θ)dx = probability x is between x to x + dx, given
parameter(s) θ;Discrete: f(x; θ) = probability of x given θ.
• Cumulative distribution function:
F (a) =∫ a
−∞f(x) dx . (32.6)
Here and below, if x is discrete-valued, the integral is replaced by a sum.The endpoint a is induced in the integral or sum.
• Expectation values: Given a function u:
E [u(x)] =∫ ∞
−∞u(x) f(x) dx . (32.7)
• Moments:
nth moment of a random variable: αn = E[xn] , (32.8a)
nth central moment: mn = E[(x − α1)n] . (32.8b)
Mean: μ ≡ α1 . (32.9a)
Variance: σ2 ≡ V [x] ≡ m2 = α2 − μ2 . (32.9b)
Coefficient of skewness: γ1 ≡ m3/σ3.Kurtosis: γ2 = m4/σ4 − 3 .
Median: F (xmed) = 1/2.
• Marginal p.d.f.: Let x,y be two random variables with joint p.d.f.f(x, y).
f1(x) =∫ ∞
−∞f(x, y) dy ; f2(y) =
∫ ∞
−∞f(x, y) dx . (32.10)
• Conditional p.d.f.:
f4(x|y) = f(x, y)/f2(y) ; f3(y|x) = f(x, y)/f1(x) .
• Bayes’ theorem:
f4(x|y) =f3(y|x)f1(x)
f2(y)=
f3(y|x)f1(x)∫
f3(y|x′)f1(x′) dx′. (32.11)
• Correlation coefficient and covariance:
μx =∫ ∞
−∞
∫ ∞
−∞xf(x, y) dx dy , (32.12)
ρxy = E[
(x − μx)(y − μy)]
/σx σy ≡ cov[x, y]/σx σy ,
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268 32. Probability
σx =∫ ∞
−∞
∫ ∞
−∞(x − μx)2 f(x, y) dx dy . Note ρ2
xy ≤ 1.
• Independence: x,y are independent if and only if f(x, y) = f1(x) · f2(y);then ρxy = 0, E[u(x) v(y)] = E[u(x)] E[v(y)] and V [x+y] = V [x]+V [y].
• Change of variables: From x = (x1, . . . , xn) to y = (y1, . . . , yn):g(y) = f (x(y)) · |J | where |J | is the absolute value of the determinant ofthe Jacobian Jij = ∂xi/∂yj. For discrete variables, use |J | = 1.
32.3. Characteristic functions
Given a pdf f(x) for a continuous random variable x, the characteristicfunction φ(u) is given by (31.6). Its derivatives are related to the algebraicmoments of x by (31.7).
φ(u) = E[
eiux]
=∫ ∞
−∞eiuxf(x) dx . (32.17)
i−n dnφ
dun
∣
∣
∣
∣
u=0=
∫ ∞
−∞xnf(x) dx = αn . (32.18)
If the p.d.f.s f1(x) and f2(y) for independent random variables x andy have characteristic functions φ1(u) and φ2(u), then the characteristicfunction of the weighted sum ax+ by is φ1(au)φ2(bu). The additional rulesfor several important distributions (e.g., that the sum of two Gaussiandistributed variables also follows a Gaussian distribution) easily followfrom this observation.
32.4. Some probability distributions
See Table 32.1.
32.4.2. Poisson distribution :The Poisson distribution f(n; ν) gives the probability of finding exactlyn events in a given interval of x (e.g., space or time) when the eventsoccur independently of one another and of x at an average rate of ν perthe given interval. The variance σ2 equals ν. It is the limiting case p → 0,N → ∞, Np = ν of the binomial distribution. The Poisson distributionapproaches the Gaussian distribution for large ν.For example, a large number of radioactive nuclei of a given type willresult in a certain number of decays in a fixed time interval. If this intervalis small compared to the mean lifetime, then the probability for a givennucleus to decay is small, and thus the number of decays in the timeinterval is well modeled as a Poisson variable.
32.4.3. Normal or Gaussian distribution :Its cumulative distribution, for mean 0 and variance 1, is usually tabulatedas the error function
F (x; 0, 1) = 12
[
1 + erf(x/√
2)]
. (32.24)
For mean μ and variance σ2, replace x by (x− μ)/σ. The error function isacessible in libraries of computer routines such as CERNLIB.P (x in range μ ± σ) = 0.6827,P (x in range μ ± 0.6745σ) = 0.5,E[|x − μ|] =
√
2/πσ = 0.7979σ,
half-width at half maximum =√
2 ln 2 · σ = 1.177σ.
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32. Probability 269
Table
32.1
.Som
eco
mm
onpro
bab
ility
den
sity
funct
ions,
wit
hco
rres
pon
din
gch
arac
teri
stic
funct
ions
and
mea
ns
and
varian
ces.
Inth
eTab
le,Γ(k
)is
the
gam
ma
funct
ion,eq
ual
to(k
−1)
!w
hen
kis
anin
tege
r.
Pro
babi
lity
dens
ity
func
tion
Cha
ract
eris
tic
Dis
trib
utio
nf
(var
iabl
e;pa
ram
eter
s)fu
nction
φ(u
)M
ean
Var
ianc
eσ
2
Uni
form
f(x
;a,b
)=
{
1/(b
−a)
a≤
x≤
b
0ot
herw
ise
eibu
−ei
au
(b−
a)i
u
a+
b
2(b
−a)2
12
Bin
omia
lf(r
;N,p
)=
N!
r!(N
−r)
!prqN
−r(q
+pei
u)N
Np
Npq
r=
0,1,
2,..
.,N
;0≤
p≤
1;
q=
1−
p
Poi
sson
f(n
;ν)=
νne−
ν
n!
;n
=0,
1,2,
...
;ν
>0
exp[
ν(e
iu−
1)]
νν
Nor
mal
(Gau
ssia
n)f(x
;μ,σ
2)=
1σ√ 2π
exp(−(
x−
μ)2
/2σ
2)
exp(
iμu−
1 2σ
2u
2)
μσ
2
−∞<
x<
∞;
−∞<
μ<
∞;
σ>
0
Mul
tiva
riat
eG
auss
ian
f(x
;μ,V
)=
1(2
π)n
/2√
|V|
exp
[
iμ·u
−1 2u
TV
u]
μV
jk
×ex
p[
−1 2(x
−μ
)TV
−1(x
−μ
)]
−∞<
xj
<∞
;−
∞<
μj
<∞
;|V
|>0
χ2
f(z
;n)=
zn/2−1
e−z/2
2n/2Γ(n
/2)
;z≥
0(1
−2i
u)−
n/2
n2n
Stud
ent’s
tf(t
;n)=
1 √ nπ
Γ[(n
+1)
/2]
Γ(n
/2)
(
1+
t2 n
)−(
n+
1)/
2
—0
for
n≥
2n/(n
−2)
for
n≥
3−∞
<t<
∞;
nno
tre
quir
edto
bein
tege
r
Gam
ma
f(x
;λ,k
)=
xk−1
λke−
λx
Γ(k
);
0<
x<
∞;
(1−
iu/λ)−
kk/λ
k/λ
2
kno
tre
quir
edto
bein
tege
r
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270 32. Probability
For n Gaussian random variables xi, the joint p.d.f. is the multivariateGaussian:
f(x; μ, V ) =1
(2π)n/2√|V | exp
[
− 12 (x − μ)T V −1(x − μ)
]
, |V | > 0 .
(32.27)V is the n × n covariance matrix; Vij ≡ E[(xi − μi)(xj − μj)] ≡ ρij σi σj ,and Vii = V [xi]; |V | is the determinant of V . For n = 2, f(x; μ, V ) is
f(x1, x2; μ1, μ2, σ1, σ2, ρ) =1
2πσ1σ2
√
1 − ρ2× exp
{ −12(1 − ρ2)
[
(x1 − μ1)2
σ21
− 2ρ(x1 − μ1)(x2 − μ2)σ1σ2
+(x2 − μ2)2
σ22
]}
. (32.28)
The marginal distribution of any xi is a Gaussian with mean μi andvariance Vii. V is n × n, symmetric, and positive definite. Therefore forany vector X, the quadratic form XT V −1X = C, where C is any positivenumber, traces an n-dimensional ellipsoid as X varies. If Xi = xi − μi,then C is a random variable obeying the χ2 distribution with n degreesof freedom, discussed in the following section. The probability that Xcorresponding to a set of Gaussian random variables xi lies outsidethe ellipsoid characterized by a given value of C (= χ2) is given by1 − Fχ2(C; n), where Fχ2 is the cumulative χ2 distribution. This maybe read from Fig. 33.1. For example, the “s-standard-deviation ellipsoid”occurs at C = s2. For the two-variable case (n = 2), the point X liesoutside the one-standard-deviation ellipsoid with 61% probability. Theuse of these ellipsoids as indicators of probable error is described inSec. 33.3.2.4; the validity of those indicators assumes that μ and V arecorrect.
32.4.4. χ2 distribution :If x1, . . . , xn are independent Gaussian random variables, the sumz =
∑ni=1(xi − μi)2/σ2
i follows the χ2 p.d.f. with n degrees of freedom,which we denote by χ2(n). More generally, for n correlated Gaussianvariables as components of a vector X with covariance matrix V ,z = XT V −1X follows χ2(n) as in the previous section. For a set of zi,each of which follows χ2(ni),
∑
zi follows χ2(∑
ni). For large n, the χ2
p.d.f. approaches a Gaussian with mean μ = n and variance σ2 = 2n.The χ2 p.d.f. is often used in evaluating the level of compatibility betweenobserved data and a hypothesis for the p.d.f. that the data might follow.This is discussed further in Sec. 33.2.2 on tests of goodness-of-fit.
32.4.6. Gamma distribution :For a process that generates events as a function of x (e.g., space or time)according to a Poisson distribution, the distance in x from an arbitrarystarting point (which may be some particular event) to the kth eventfollows a gamma distribution, f(x; λ, k). The Poisson parameter μ is λper unit x. The special case k = 1 (i.e., f(x; λ, 1) = λe−λx) is called theexponential distribution. A sum of k′ exponential random variables xi isdistributed as f(
∑
xi; λ, k′).The parameter k is not required to be an integer. For λ = 1/2 andk = n/2, the gamma distribution reduces to the χ2(n) distribution.
See the full Review for further discussion and all references.
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33. Statistics 271
33. STATISTICS
Revised September 2009 by G. Cowan (RHUL).
There are two main approaches to statistical inference, which wemay call frequentist and Bayesian. In frequentist statistics, probability isinterpreted as the frequency of the outcome of a repeatable experiment.The most important tools in this framework are parameter estimation,covered in Section 33.1, and statistical tests, discussed in Section 33.2.Frequentist confidence intervals, which are constructed so as to coverthe true value of a parameter with a specified probability, are treated inSection 33.3.2. Note that in frequentist statistics one does not define aprobability for a hypothesis or for a parameter.
In Bayesian statistics, the interpretation of probability is more generaland includes degree of belief (called subjective probability). One can thenspeak of a probability density function (p.d.f.) for a parameter, whichexpresses one’s state of knowledge about where its true value lies. UsingBayes’ theorem Eq. (32.4), the prior degree of belief is updated by thedata from the experiment. Bayesian methods for interval estimation arediscussed in Sections 33.3.1 and 33.3.2.6
Following common usage in physics, the word “error” is often used inthis chapter to mean “uncertainty.” More specifically it can indicate thesize of an interval as in “the standard error” or “error propagation,” wherethe term refers to the standard deviation of an estimator.
33.1. Parameter estimationHere we review the frequentist approach to point estimationof
parameters. An estimator θ (written with a hat) is a function of the datawhose value, the estimate, is intended as a meaningful guess for the valueof the parameter θ.
33.1.1. Estimators for mean, variance and median : Suppose wehave a set of N independent measurements, xi, assumed to be unbiasedmeasurements of the same unknown quantity μ with a common, butunknown, variance σ2. Then
μ =1N
N∑
i=1
xi (33.4)
σ2 =1
N − 1
N∑
i=1
(xi − μ)2 (33.5)
are unbiased estimators of μ and σ2. The variance of μ is σ2/N and thevariance of σ2 is
V[
σ2]
=1N
(
m4 − N − 3N − 1
σ4)
, (33.6)
where m4 is the 4th central moment of x. For Gaussian distributed xi,this becomes 2σ4/(N − 1) for any N ≥ 2, and for large N , the standarddeviation of σ (the “error of the error”) is σ/
√2N . Again, if the xi are
Gaussian, μ is an efficient estimator for μ, and the estimators μ and σ2
are uncorrelated. Otherwise the arithmetic mean (33.4) is not necessarilythe most efficient estimator.
If the xi have different, known variances σ2i , then the weighted average
μ =1w
N∑
i=1
wixi (33.7)
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272 33. Statistics
is an unbiased estimator for μ with a smaller variance than an unweightedaverage; here wi = 1/σ2
i and w =∑
i wi. The standard deviation of μ is1/
√w.
33.1.2. The method of maximum likelihood : Suppose we have aset of N measured quantities x = (x1, . . . , xN ) described by a joint p.d.f.f(x; θ), where θ = (θ1, . . . , θn) is set of n parameters whose values areunknown. The likelihood function is given by the p.d.f. evaluated with thedata x, but viewed as a function of the parameters, i.e., L(θ) = f(x; θ).If the measurements xi are statistically independent and each follow thep.d.f. f(x; θ), then the joint p.d.f. for x factorizes and the likelihoodfunction is
L(θ) =N∏
i=1
f(xi; θ) . (33.8)
The method of maximum likelihood takes the estimators θ to be thosevalues of θ that maximize L(θ).
Note that the likelihood function is not a p.d.f. for the parameters θ;in frequentist statistics this is not defined. In Bayesian statistics, one canobtain from the likelihood the posterior p.d.f. for θ, but this requiresmultiplying by a prior p.d.f. (see Sec. 33.3.1).
It is usually easier to work with ln L, and since both are maximized forthe same parameter values θ, the maximum likelihood (ML) estimatorscan be found by solving the likelihood equations,
∂ ln L
∂θi= 0 , i = 1, . . . , n . (33.9)
In evaluating the likelihood function, it is important that anynormalization factors in the p.d.f. that involve θ be included.
The inverse V −1 of the covariance matrix Vij = cov[θi, θj ] for a set ofML estimators can be estimated by using
(V −1)ij = − ∂2 ln L
∂θi∂θj
∣
∣
∣
∣
θ
. (33.10)
For finite samples, however, Eq. (33.10) can result in an underestimateof the variances. In the large sample limit (or in a linear model withGaussian errors), L has a Gaussian form and lnL is (hyper)parabolic. Inthis case, it can be seen that a numerically equivalent way of determinings-standard-deviation errors is from the contour given by the θ′ such that
ln L(θ′) = lnLmax − s2/2 , (33.11)where lnLmax is the value of lnL at the solution point (comparewith Eq. (33.56)). The extreme limits of this contour on the θi axisgive an approximate s-standard-deviation confidence interval for θi (seeSection 33.3.2.4).
33.1.3. The method of least squares : The method of least squares(LS) coincides with the method of maximum likelihood in the followingspecial case. Consider a set of N independent measurements yi at knownpoints xi. The measurement yi is assumed to be Gaussian distributed withmean F (xi; θ) and known variance σ2
i . The goal is to construct estimatorsfor the unknown parameters θ. The likelihood function contains the sumof squares
χ2(θ) = −2 lnL(θ) + constant =N
∑
i=1
(yi − F (xi; θ))2
σ2i
. (33.13)
The set of parameters θ which maximize L is the same as those whichminimize χ2.
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33. Statistics 273
The minimum of Equation (33.13) defines the least-squares estimatorsθ for the more general case where the yi are not Gaussian distributedas long as they are independent. If they are not independent but ratherhave a covariance matrix Vij = cov[yi, yj ], then the LS estimators aredetermined by the minimum of
χ2(θ) = (y − F (θ))T V −1(y − F (θ)) , (33.14)where y = (y1, . . . , yN ) is the vector of measurements, F (θ) is thecorresponding vector of predicted values (understood as a column vectorin (33.14)), and the superscript T denotes transposed (i.e., row) vector.
In many practical cases, one further restricts the problem to thesituation where F (xi; θ) is a linear function of the parameters, i.e.,
F (xi; θ) =m
∑
j=1
θjhj(xi) . (33.15)
Here the hj(x) are m linearly independent functions, e.g., 1, x, x2, . . . , xm−1,or Legendre polynomials. We require m < N and at least m of the ximust be distinct.
Minimizing χ2 in this case with m parameters reduces to solving asystem of m linear equations. Defining Hij = hj(xi) and minimizing χ2
by setting its derivatives with respect to the θi equal to zero gives the LSestimators,
θ = (HT V −1H)−1HT V −1y ≡ Dy . (33.16)
The covariance matrix for the estimators Uij = cov[θi, θj ] is given byU = DV DT = (HT V −1H)−1 . (33.17)
Expanding χ2(θ) about θ, one finds that the contour in parameter spacedefined by
χ2(θ) = χ2(θ) + 1 = χ2min + 1 (33.23)
has tangent planes located at approximately plus-or-minus-one standarddeviation σ
θfrom the LS estimates θ.
As the minimum value of the χ2 represents the level of agreementbetween the measurements and the fitted function, it can be used forassessing the goodness-of-fit; this is discussed further in Section 33.2.2.
33.1.5. Propagation of errors : Consider a set of n quantitiesθ = (θ1, . . . , θn) and a set of m functions η(θ) = (η1(θ), . . . , ηm(θ)).Suppose we have estimated θ = (θ1, . . . , θn), using, say, maximum-likelihood or least-squares, and we also know or have estimated thecovariance matrix Vij = cov[θi, θj ]. The goal of error propagation is todetermine the covariance matrix for the functions, Uij = cov[ηi, ηj ], whereη = η(θ ). In particular, the diagonal elements Uii = V [ηi] give thevariances. The new covariance matrix can be found by expanding thefunctions η(θ) about the estimates θ to first order in a Taylor series.Using this one finds
Uij ≈∑
k,l
∂ηi
∂θk
∂ηj
∂θl
∣
∣
∣
∣
θVkl . (33.29)
This can be written in matrix notation as U ≈ AV AT where the matrix ofderivatives A is
Aij =∂ηi
∂θj
∣
∣
∣
∣
θ
, (33.30)
and AT is its transpose. The approximation is exact if η(θ) is linear.
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274 33. Statistics
33.2. Statistical tests
33.2.1. Hypothesis tests : Consider an experiment whose outcome ischaracterized by a vector of data x. A hypothesis is a statement aboutthe distribution of x. It could, for example, define completely the p.d.f.for the data (a simple hypothesis), or it could specify only the functionalform of the p.d.f., with the values of one or more parameters left open (acomposite hypothesis).
A statistical test is a rule that states for which values of x a givenhypothesis (often called the null hypothesis, H0) should be rejected infavor of its alternative H1. This is done by defining a region of x-spacecalled the critical region; if the outcome of the experiment lands in thisregion, H0 is rejected, otherwise it is accepted.
Rejecting H0 if it is true is called an error of the first kind. Theprobability for this to occur is called the size or significance level of thetest, α, which is chosen to be equal to some pre-specified value. It can alsohappen that H0 is false and the true hypothesis is the alternative, H1. IfH0 is accepted in such a case, this is called an error of the second kind,which will have some probability β. The quantity 1− β is called the powerof the test relative to H1.
Often one tries to construct a test to maximize power for a givensignificance level, i.e., to maximize the signal efficiency for a givensignificance level. The Neyman–Pearson lemma states that this is done bydefining the acceptance region such that, for x in that region, the ratio ofp.d.f.s for the hypotheses H1 (signal) and H0 (background),
λ(x) =f(x|H1)f(x|H0)
, (33.31)
is greater than a given constant, the value of which is chosen to givethe desired signal efficiency. Here H0 and H1 must be simple hypotheses,i.e., they should not contain undetermined parameters. The lemma isequivalent to the statement that (33.31) represents the test statistic withwhich one may obtain the highest signal efficiency for a given purity for theselected sample. It can be difficult in practice, however, to determine λ(x),since this requires knowledge of the joint p.d.f.s f(x|H0) and f(x|H1).
In the usual case where the likelihood ratio (33.31) cannot beused explicitly, there exist a variety of other multivariate classifiersthat effectively separate different types of events. Methods often usedin HEP include neural networks or Fisher discriminants (see [10]).Recently, further classification methods from machine-learning have beenapplied in HEP analyses; these include probability density estimation(PDE) techniques, kernel-based PDE (KDE or Parzen window), supportvector machines, and decision trees. Techniques such as “boosting” and“bagging” can be applied to combine a number of classifiers into a strongerone with greater stability with respect to fluctuations in the training data.
33.2.2. Significance tests : Often one wants to quantify the level ofagreement between the data and a hypothesis without explicit reference toalternative hypotheses. This can be done by defining a statistic t, whichis a function of the data whose value reflects in some way the level ofagreement between the data and the hypothesis.
The hypothesis in question, say, H0, will determine the p.d.f. g(t|H0)for the statistic. The significance of a discrepancy between the data andwhat one expects under the assumption of H0 is quantified by giving thep-value, defined as the probability to find t in the region of equal or lessercompatibility with H0 than the level of compatibility observed with the
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33. Statistics 275
actual data. For example, if t is defined such that large values correspondto poor agreement with the hypothesis, then the p-value would be
p =∫ ∞
tobs
g(t|H0) dt , (33.32)
where tobs is the value of the statistic obtained in the actual experiment.The p-value should not be confused with the size (significance level) of atest, or the confidence level of a confidence interval (Section 33.3), both ofwhich are pre-specified constants.
The p-value is a function of the data, and is therefore itself a randomvariable. If the hypothesis used to compute the p-value is true, thenfor continuous data, p will be uniformly distributed between zero andone. Note that the p-value is not the probability for the hypothesis;in frequentist statistics, this is not defined. Rather, the p-value is theprobability, under the assumption of a hypothesis H0, of obtaining data atleast as incompatible with H0 as the data actually observed.
When estimating parameters using the method of least squares, oneobtains the minimum value of the quantity χ2 (33.13). This statistic canbe used to test the goodness-of-fit, i.e., the test provides a measure ofthe significance of a discrepancy between the data and the hypothesizedfunctional form used in the fit. It may also happen that no parametersare estimated from the data, but that one simply wants to compare ahistogram, e.g., a vector of Poisson distributed numbers n = (n1, . . . , nN ),with a hypothesis for their expectation values νi = E[ni]. As thedistribution is Poisson with variances σ2
i = νi, the χ2 (33.13) becomesPearson’s χ2 statistic,
χ2 =N
∑
i=1
(ni − νi)2
νi. (33.34)
If the hypothesis ν = (ν1, . . . , νN ) is correct, and if the expected valuesνi in (33.34) are sufficiently large (in practice, this will be a goodapproximation if all νi > 5), then the χ2 statistic will follow the χ2
p.d.f. with the number of degrees of freedom equal to the number ofmeasurements N minus the number of fitted parameters. The minimizedχ2 from Eq. (33.13) also has this property if the measurements yi areGaussian.
Assuming the goodness-of-fit statistic follows a χ2 p.d.f., the p-value forthe hypothesis is then
p =∫ ∞
χ2f(z; nd) dz , (33.35)
where f(z; nd) is the χ2 p.d.f. and nd is the appropriate number of degreesof freedom. Values can be obtained from Fig. 33.1 or from the CERNLIBroutine PROB or the ROOT function TMath::Prob.
Since the mean of the χ2 distribution is equal to nd, one expects in a“reasonable” experiment to obtain χ2 ≈ nd. Hence the quantity χ2/nd issometimes reported. Since the p.d.f. of χ2/nd depends on nd, however,one must report nd as well if one wishes to determine the p-value. Thep-values obtained for different values of χ2/nd are shown in Fig. 33.2.
33.2.3. Bayesian model selection : In Bayesian statistics, all of one’sknowledge about a model is contained in its posterior probability, whichone obtains using Bayes’ theorem. Thus one could reject a hypothesis Hif its posterior probability P (H |x) is sufficiently small. The difficulty hereis that P (H |x) is proportional to the prior probability P (H), and there
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276 33. Statistics
1 2 3 4 5 7 10 20 30 40 50 70 1000.001
0.002
0.005
0.010
0.020
0.050
0.100
0.200
0.500
1.000
p-va
lue
for
test
α fo
r co
nfi
den
ce in
terv
als
3 42 6 8
10
15
20
25
30
40
50
n = 1
χ2
Figure 33.1: One minus the χ2 cumulative distribution, 1−F (χ2; n),for n degrees of freedom. This gives the p-value for the χ2 goodness-of-fit test as well as one minus the coverage probability for confidenceregions (see Sec. 33.3.2.4).
0 10 20 30 40 500.0
0.5
1.0
1.5
2.0
2.5
Degrees of freedom n
50%
10%
90%99%
95%
68%
32%
5%
1%
χ2/n
Figure 33.2: The ‘reduced’ χ2, equal to χ2/n, for n degreesof freedom. The curves show as a function of n the χ2/n thatcorresponds to a given p-value.
will not be a consensus about the prior probabilities for the existenceof new phenomena. Nevertheless one can construct a quantity called theBayes factor (described below), which can be used to quantify the degreeto which the data prefer one hypothesis over another, and is independentof their prior probabilities.
Consider two models (hypotheses), Hi and Hj , described by vectors
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33. Statistics 277
of parameters θi and θj , respectively. Some of the components willbe common to both models and others may be distinct. The full priorprobability for each model can be written in the form
π(Hi, θi) = P (Hi)π(θi|Hi) , (33.36)Here P (Hi) is the overall prior probability for Hi, and π(θi|Hi) isthe normalized p.d.f. of its parameters. For each model, the posteriorprobability is found using Bayes’ theorem,
P (Hi|x) =∫
L(x|θi, Hi)P (Hi)π(θi|Hi) dθi
P (x), (33.37)
where the integration is carried out over the internal parameters θi of themodel. The ratio of posterior probabilities for the models is therefore
P (Hi|x)P (Hj |x)
=∫
L(x|θi, Hi)π(θi|Hi) dθi∫
L(x|θj , Hj)π(θj |Hj) dθj
P (Hi)P (Hj)
. (33.38)
The Bayes factor is defined as
Bij =∫
L(x|θi, Hi)π(θi|Hi) dθi∫
L(x|θj , Hj)π(θj |Hj) dθj. (33.39)
This gives what the ratio of posterior probabilities for models i and jwould be if the overall prior probabilities for the two models were equal.If the models have no nuisance parameters i.e., no internal parametersdescribed by priors, then the Bayes factor is simply the likelihood ratio.The Bayes factor therefore shows by how much the probability ratio ofmodel i to model j changes in the light of the data, and thus can beviewed as a numerical measure of evidence supplied by the data in favourof one hypothesis over the other.
Although the Bayes factor is by construction independent of the overallprior probabilities P (Hi) and P (Hj), it does require priors for all internalparameters of a model, i.e., one needs the functions π(θi|Hi) and π(θj |Hj).In a Bayesian analysis where one is only interested in the posterior p.d.f.of a parameter, it may be acceptable to take an unnormalizable functionfor the prior (an improper prior) as long as the product of likelihoodand prior can be normalized. But improper priors are only defined up toan arbitrary multiplicative constant, which does not cancel in the ratio(33.39). Furthermore, although the range of a constant normalized prioris unimportant for parameter determination (provided it is wider thanthe likelihood), this is not so for the Bayes factor when such a prior isused for only one of the hypotheses. So to compute a Bayes factor, allinternal parameters must be described by normalized priors that representmeaningful probabilities over the entire range where they are defined.
An exception to this rule may be considered when the identicalparameter appears in the models for both numerator and denominator ofthe Bayes factor. In this case one can argue that the arbitrary constantswould cancel. One must exercise some caution, however, as parameterswith the same name and physical meaning may still play different roles inthe two models. Both integrals in equation (33.39) are of the form
m =∫
L(x|θ)π(θ) dθ , (33.40)
which is called the marginal likelihood (or in some fields called the evidence).A review of Bayes factors including a discussion of computational issues isRef. [26].
33.3. Intervals and limitsWhen the goal of an experiment is to determine a parameter θ,
the result is usually expressed by quoting, in addition to the pointestimate, some sort of interval which reflects the statistical precision of the
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278 33. Statistics
measurement. In the simplest case, this can be given by the parameter’sestimated value θ plus or minus an estimate of the standard deviation ofθ, σ
θ. If, however, the p.d.f. of the estimator is not Gaussian or if there
are physical boundaries on the possible values of the parameter, thenone usually quotes instead an interval according to one of the proceduresdescribed below.
33.3.1. Bayesian intervals : As described in Sec. 33.1.4, a Bayesianposterior probability may be used to determine regions that will have agiven probability of containing the true value of a parameter. In the singleparameter case, for example, an interval (called a Bayesian or credibleinterval) [θlo, θup] can be determined which contains a given fraction 1− αof the posterior probability, i.e.,
1 − α =∫ θup
θlo
p(θ|x) dθ . (33.41)
Sometimes an upper or lower limit is desired, i.e., θlo can be set to zeroor θup to infinity. In other cases, one might choose θlo and θup such thatp(θ|x) is higher everywhere inside the interval than outside; these arecalled highest posterior density (HPD) intervals. Note that HPD intervalsare not invariant under a nonlinear transformation of the parameter.
If a parameter is constrained to be non-negative, then the prior p.d.f.can simply be set to zero for negative values. An important example isthe case of a Poisson variable n, which counts signal events with unknownmean s, as well as background with mean b, assumed known. For thesignal mean s, one often uses the prior
π(s) ={
0 s < 01 s ≥ 0 . (33.42)
In the absence of a clear discovery, (e.g., if n = 0 or if in any case n iscompatible with the expected background), one usually wishes to place anupper limit on s (see, however, Sec. 33.3.2.6 on “flip-flopping” concerningfrequentist coverage). Using the likelihood function for Poisson distributedn,
L(n|s) =(s + b)n
n!e−(s+b) , (33.43)
along with the prior (33.42) in (33.24) gives the posterior density for s.An upper limit sup at confidence level (or here, rather, credibility level)1 − α can be obtained by requiring
1 − α =∫ sup
−∞p(s|n)ds =
∫ sup−∞ L(n|s)π(s) ds
∫ ∞−∞ L(n|s)π(s) ds
, (33.44)
where the lower limit of integration is effectively zero because of the cut-offin π(s). By relating the integrals in Eq. (33.44) to incomplete gammafunctions, the equation reduces to
α = e−sup
∑nm=0(sup + b)m/m!∑n
m=0 bm/m!. (33.45)
This must be solved numerically for the limit sup. For the special case ofb = 0, the sums can be related to the quantile F−1
χ2 of the χ2 distribution(inverse of the cumulative distribution) to give
sup = 12F−1
χ2 (1 − α; nd) , (33.46)
where the number of degrees of freedom is nd = 2(n + 1). The quantile ofthe χ2 distribution can be obtained using the CERNLIB routine CHISIN,or the ROOT function TMath::ChisquareQuantile. It so happens thatfor the case of b = 0, the upper limits from Eq. (33.46) coincide numerically
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33. Statistics 279
with the values of the frequentist upper limits discussed in Section 33.3.2.5.Values for 1 − α = 0.9 and 0.95 are given by the values νup in Table 33.3.
As in any Bayesian analysis, it is important to show how the resultwould change if one uses different prior probabilities. For example, onecould consider the Jeffreys prior as described in Sec. 33.1.4. For thisproblem one finds the Jeffreys prior π(s) ∝ 1/
√s + b for s ≥ 0 and zero
otherwise. As with the constant prior, one would not regard this asrepresenting one’s prior beliefs about s, both because it is improper andalso as it depends on b. Rather it is used with Bayes’ theorem to producean interval whose frequentist properties can be studied.
33.3.2. Frequentist confidence intervals :
33.3.2.1. The Neyman construction for confidence intervals: Considera p.d.f. f(x; θ) where x represents the outcome of the experiment and θis the unknown parameter for which we want to construct a confidenceinterval. The variable x could (and often does) represent an estimator forθ. Using f(x; θ), we can find for a pre-specified probability 1 − α, and forevery value of θ, a set of values x1(θ, α) and x2(θ, α) such that
P (x1 < x < x2; θ) = 1 − α =∫ x2
x1
f(x; θ) dx . (33.47)
This is illustrated in Fig. 33.3: a horizontal line segment [x1(θ, α),x2(θ, α)] is drawn for representative values of θ. The union of suchintervals for all values of θ, designated in the figure as D(α), is knownas the confidence belt. Typically the curves x1(θ, α) and x2(θ, α) aremonotonic functions of θ, which we assume for this discussion.
Possible experimental values x
para
met
er θ x2(θ), θ2(x)
x1(θ), θ1(x)
����������������
x1(θ0) x2(θ0)
D(α)
θ0
Figure 33.3: Construction of the confidence belt (see text).
Upon performing an experiment to measure x and obtaining a valuex0, one draws a vertical line through x0. The confidence interval for θis the set of all values of θ for which the corresponding line segment[x1(θ, α), x2(θ, α)] is intercepted by this vertical line. Such confidenceintervals are said to have a confidence level (CL) equal to 1 − α.
Now suppose that the true value of θ is θ0, indicated in the figure.We see from the figure that θ0 lies between θ1(x) and θ2(x) if and only
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280 33. Statistics
if x lies between x1(θ0) and x2(θ0). The two events thus have the sameprobability, and since this is true for any value θ0, we can drop thesubscript 0 and obtain
1 − α = P (x1(θ) < x < x2(θ)) = P (θ2(x) < θ < θ1(x)) . (33.48)In this probability statement, θ1(x) and θ2(x), i.e., the endpoints of theinterval, are the random variables and θ is an unknown constant. If theexperiment were to be repeated a large number of times, the interval[θ1, θ2] would vary, covering the fixed value θ in a fraction 1 − α of theexperiments.
The condition of coverage in Eq. (33.47) does not determine x1 and x2uniquely, and additional criteria are needed. The most common criterionis to choose central intervals such that the probabilities excluded belowx1 and above x2 are each α/2. In other cases, one may want to reportonly an upper or lower limit, in which case the probability excluded belowx1 or above x2 can be set to zero. Another principle based on likelihoodratio ordering for determining which values of x should be included in theconfidence belt is discussed in Sec. 33.3.2.2
When the observed random variable x is continuous, the coverageprobability obtained with the Neyman construction is 1 − α, regardless ofthe true value of the parameter. If x is discrete, however, it is not possibleto find segments [x1(θ, α), x2(θ, α)] that satisfy Eq. (33.47) exactly for allvalues of θ. By convention, one constructs the confidence belt requiringthe probability P (x1 < x < x2) to be greater than or equal to 1 − α.This gives confidence intervals that include the true parameter with aprobability greater than or equal to 1 − α.
33.3.2.4. Gaussian distributed measurements: An important exampleof constructing a confidence interval is when the data consists of a singlerandom variable x that follows a Gaussian distribution; this is oftenthe case when x represents an estimator for a parameter and one has asufficiently large data sample. If there is more than one parameter beingestimated, the multivariate Gaussian is used. For the univariate case withknown σ,
1 − α =1√2πσ
∫ μ+δ
μ−δe−(x−μ)2/2σ2
dx = erf(
δ√2 σ
)
(33.53)
is the probability that the measured value x will fall within ±δ of the truevalue μ. From the symmetry of the Gaussian with respect to x and μ, thisis also the probability for the interval x± δ to include μ. Fig. 33.4 shows aδ = 1.64σ confidence interval unshaded. The choice δ = σ gives an intervalcalled the standard error which has 1 − α = 68.27% if σ is known. Valuesof α for other frequently used choices of δ are given in Table 33.1.
Table 33.1: Area of the tails α outside ±δ from the mean of aGaussian distribution.
α δ α δ
0.3173 1σ 0.2 1.28σ
4.55 ×10−2 2σ 0.1 1.64σ
2.7 ×10−3 3σ 0.05 1.96σ
6.3×10−5 4σ 0.01 2.58σ
5.7×10−7 5σ 0.001 3.29σ
2.0×10−9 6σ 10−4 3.89σ
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33. Statistics 281
−3 −2 −1 0 1 2 3
f (x; μ,σ)
α /2α /2
(x−μ) /σ
1−α
Figure 33.4: Illustration of a symmetric 90% confidence interval(unshaded) for a measurement of a single quantity with Gaussianerrors. Integrated probabilities, defined by α, are as shown.
We can set a one-sided (upper or lower) limit by excluding above x + δ(or below x − δ). The values of α for such limits are half the values inTable 33.1.
The relation (33.53) can be re-expressed using the cumulativedistribution function for the χ2 distribution as
α = 1 − F (χ2; n) , (33.54)for χ2 = (δ/σ)2 and n = 1 degree of freedom. This can be obtained fromFig. 33.1 on the n = 1 curve or by using the CERNLIB routine PROB orthe ROOT function TMath::Prob.
For multivariate measurements of, say, n parameter estimatesθ = (θ1, . . . , θn), one requires the full covariance matrix Vij = cov[θi, θj ],which can be estimated as described in Sections 33.1.2 and 33.1.3. Underfairly general conditions with the methods of maximum-likelihood orleast-squares in the large sample limit, the estimators will be distributedaccording to a multivariate Gaussian centered about the true (unknown)values θ, and furthermore, the likelihood function itself takes on aGaussian shape.
The standard error ellipse for the pair (θi, θj) is shown in Fig. 33.5,corresponding to a contour χ2 = χ2
min + 1 or lnL = lnLmax − 1/2. Theellipse is centered about the estimated values θ, and the tangents to theellipse give the standard deviations of the estimators, σi and σj . Theangle of the major axis of the ellipse is given by
tan 2φ =2ρijσiσj
σ2j − σ2
i
, (33.55)
where ρij = cov[θi, θj ]/σiσj is the correlation coefficient.The correlation coefficient can be visualized as the fraction of the
distance σi from the ellipse’s horizontal centerline at which the ellipsebecomes tangent to vertical, i.e., at the distance ρijσi below the centerlineas shown. As ρij goes to +1 or −1, the ellipse thins to a diagonal line.
As in the single-variable case, because of the symmetry of the Gaussianfunction between θ and θ, one finds that contours of constant lnL orχ2 cover the true values with a certain, fixed probability. That is, the
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282 33. Statistics
θ i
φ
θ i
jσ
θj
iσ
jσ
iσ
^
θ j^
ij iρ σ
innerσ
Figure 33.5: Standard error ellipse for the estimators θi and θj . Inthis case the correlation is negative.
Table 33.2: Δχ2 or 2Δ ln L corresponding to a coverage probability1 − α in the large data sample limit, for joint estimation of mparameters.
(1 − α) (%) m = 1 m = 2 m = 368.27 1.00 2.30 3.5390. 2.71 4.61 6.2595. 3.84 5.99 7.8295.45 4.00 6.18 8.0399. 6.63 9.21 11.3499.73 9.00 11.83 14.16
confidence region is determined byln L(θ) ≥ ln Lmax − Δ ln L , (33.56)
or where a χ2 has been defined for use with the method of least-squares,χ2(θ) ≤ χ2
min + Δχ2 . (33.57)Values of Δχ2 or 2Δ ln L are given in Table 33.2 for several values of thecoverage probability and number of fitted parameters.
For finite data samples, the probability for the regions determined byequations (33.56) or (33.57) to cover the true value of θ will depend onθ, so these are not exact confidence regions according to our previousdefinition.
33.3.2.5. Poisson or binomial data: Another important class ofmeasurements consists of counting a certain number of events, n. In thissection, we will assume these are all events of the desired type, i.e., thereis no background. If n represents the number of events produced in areaction with cross section σ, say, in a fixed integrated luminosity L, thenit follows a Poisson distribution with mean ν = σL. If, on the other hand,one has selected a larger sample of N events and found n of them to havea particular property, then n follows a binomial distribution where theparameter p gives the probability for the event to possess the propertyin question. This is appropriate, e.g., for estimates of branching ratios orselection efficiencies based on a given total number of events.
For the case of Poisson distributed n, the upper and lower limits on themean value ν can be found from the Neyman procedure to be
νlo = 12F−1
χ2 (αlo; 2n) , (33.59a)
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33. Statistics 283
νup = 12F−1
χ2 (1 − αup; 2(n + 1)) , (33.59b)
where the upper and lower limits are at confidence levels of 1 − αlo and1 − αup, respectively, and F−1
χ2 is the quantile of the χ2 distribution
(inverse of the cumulative distribution). The quantiles F−1χ2 can be
obtained from standard tables or from the CERNLIB routine CHISIN. Forcentral confidence intervals at confidence level 1− α, set αlo = αup = α/2.
Table 33.3: Lower and upper (one-sided) limits for the mean νof a Poisson variable given n observed events in the absence ofbackground, for confidence levels of 90% and 95%.
1 − α =90% 1 − α =95%
n νlo νup νlo νup
0 – 2.30 – 3.001 0.105 3.89 0.051 4.742 0.532 5.32 0.355 6.303 1.10 6.68 0.818 7.754 1.74 7.99 1.37 9.155 2.43 9.27 1.97 10.516 3.15 10.53 2.61 11.847 3.89 11.77 3.29 13.158 4.66 12.99 3.98 14.439 5.43 14.21 4.70 15.71
10 6.22 15.41 5.43 16.96
It happens that the upper limit from Eq. (33.59a) coincides numericallywith the Bayesian upper limit for a Poisson parameter, using a uniformprior p.d.f. for ν. Values for confidence levels of 90% and 95% are shownin Table 33.3. For the case of binomially distributed n successes out of Ntrials with probability of success p, the upper and lower limits on p arefound to be
plo =nF−1
F [αlo; 2n, 2(N − n + 1)]
N − n + 1 + nF−1F [αlo; 2n, 2(N − n + 1)]
, (33.60a)
pup =(n + 1)F−1
F [1 − αup; 2(n + 1), 2(N − n)]
(N − n) + (n + 1)F−1F [1 − αup; 2(n + 1), 2(N − n)]
. (33.60b)
Here F−1F is the quantile of the F distribution (also called the Fisher–
Snedecor distribution; see [4]).
33.3.2.6. Difficulties with intervals near a boundary:A number of issues arise in the construction and interpretation of
confidence intervals when the parameter can only take on values in arestricted range. Important examples are where the mean of a Gaussianvariable is constrained on physical grounds to be non-negative and wherethe experiment finds a Poisson-distributed number of events, n, whichincludes both signal and background. Application of some standard recipescan lead to intervals that are partially or entirely in the unphysical region.Furthermore, if the decision whether to report a one- or two-sided interval
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284 33. Statistics
is based on the data, then the resulting intervals will not in general coverthe parameter with the stated probability 1 − α.
Several problems with such intervals are overcome by using the unifiedapproach of Feldman and Cousins [27]. Properties of these intervals aredescribed further in the Review. Table 33.4 gives the unified confidenceintervals [ν1, ν2] for the mean of a Poisson variable given n observed eventsin the absence of background, for confidence levels of 90% and 95%. Thevalues of 1 − α given here refer to the coverage of the true parameterby the whole interval [ν1, ν2]. In Table 33.3 for the one-sided upper andlower limits, however, 1−α referred to the probability to have individuallyνup ≥ ν or νlo ≤ ν.
Table 33.4: Unified confidence intervals [ν1, ν2] for a the meanof a Poisson variable given n observed events in the absence ofbackground, for confidence levels of 90% and 95%.
1 − α =90% 1 − α =95%
n ν1 ν2 ν1 ν2
0 0.00 2.44 0.00 3.091 0.11 4.36 0.05 5.142 0.53 5.91 0.36 6.723 1.10 7.42 0.82 8.254 1.47 8.60 1.37 9.765 1.84 9.99 1.84 11.266 2.21 11.47 2.21 12.757 3.56 12.53 2.58 13.818 3.96 13.99 2.94 15.299 4.36 15.30 4.36 16.77
10 5.50 16.50 4.75 17.82
Another possibility is to construct a Bayesian interval as described inSection 33.3.1. The presence of the boundary can be incorporated simplyby setting the prior density to zero in the unphysical region. Advantagesand pitfalls of this approach are discussed further in the Review.
Another alternative is presented by the intervals found from thelikelihood function or χ2 using the prescription of Equations (33.56) or(33.57). As in the case of the Bayesian intervals, the coverage probabilityis not, in general, independent of the true parameter. Furthermore, theseintervals can for some parameter values undercover.
In any case it is important to report sufficient information so that theresult can be combined with other measurements. Often this means givingan unbiased estimator and its standard deviation, even if the estimatedvalue is in the unphysical region. It is also useful to report the likelihoodfunction or an appropriate summary of it. Although this by itself is notsufficient to construct a frequentist confidence interval, it can be used tofind the Bayesian posterior probability density for any desired prior p.d.f.
Further discussion and all references may be found in the full Review ofParticle Physics; the equation and reference numbering corresponds tothat version.
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36. Clebsch-Gordan coefficients 285
36. CLEBSCH-GORDAN COEFFICIENTS,
SPHERICAL HARMONICS, AND d FUNCTIONS
+1
5/2
5/2
+3/2
3/2
+3/2
1/5
4/5
4/5
−1/5
5/2
5/2
−1/2 3/5
2/5 −1 −2
3/2
−1/2 2/5
5/2
3/2
−3/2
−3/2 4/5
1/5
−4/51/5
−1/2
−21
−5/25/2
−3/5
−1/2
+1/2
+1−1
/22/
53/
5−2
/5−1
/2
2 +2
+3/2
+3/2
5/2
+5/2
5/2
5/2
3/2
1/2
1/2
−1/3 −1 +10
1/6
+1/2
+1/2
−1/2
−3/2
+1/2 2/5
1/15
−8/1
5
+1/2
1/10
3/103/5
5/2
3/2
1/2
−1/2 1/6
−1/3
5/2
5/2
−5/2 1
3/2
−3/2
−3/52/5
−3/2 −3
/2
3/5
2/5
1/2 −1
−10
−1/2
8/15
−1/1
5−2
/5 −1/2
−3/2
−1/2
3/10 3/5
1/10
+3/2 +3
/2+1
/2−1
/2
+3/2
+1/2
+2+1 +2 +1
0 +12/
53/
5
3/2
3/5
−2/5 −1 +10
+3/2
1+1
+3 +11 0
3
1/3+2 2/3
2
3/2
3/2
1/3
2/3
+1/2 0 −1
1/2
+1/2 2/3
−1/3
−1/2
+1/2
1
+11 0
1/2
1/2
−1/2
0 01/
2
−1/2
1 1−1−1
/2
1 1−1
/2+1
/2
+1/2
+1/2
+1/2
−1/2
−1/2
+1/2
−1/2 −13/2
2/3
3/2
−3/2 1
1/3
−1/2
−1/21/2
1/3
−2/3
+1+1
/2 +1 0
+3/2
2/3
3
3
3
3
3 1−1
−2−3
2/3
1/3−2
2
1/3
−2/3−2
0 −1 −2
−1 0 +1
−1 2/5
8/15
1/15
2 −1
−1 −2−1 0
1/2
−1/6
−1/3
1 −11/
10−3
/10
3/5
02 0
1 03/
10−2
/53/
100
1/2
−1/2
1/5
1/5
3/5
+1
+1 −100−1 +1
1/15
8/15 2/5
2
+22 +1 1/2
1/2
1
1/2
2 0
1/6
1/6
2/3
1
1/2
−1/20 0
2
2 −2 1−1
−1
1 −1 1/2
−1/2
−1 1/2
1/2
0 0
0 −1
1/3
1/3
−1/3
−1/2+1 −1
−1 0
+10
0+1
−1
2 1 00
+1
+1+1
+1 1/3
1/6
−1/2
1 +1 3/5
−3/1
01/
10
−1/3 −1 0
+1 0+2
+1+2
3
+3/2
+1/2
+1 1/4
2
2 −11
2 −2 1
−1 1/4
−1/2
1/2
1/2
−1/2
−1/2
+1/2
−3/2
−3/2
1/21 0
03/
4+1
/2−1
/2−1
/2
2+1 3/4
3/4 −3
/41/
4
−1/2
+1/2
−1/41
+1/2
−1/2
+1/21
+1/2
3/5 0 −1
+1/2
0
+1/2
3/2
+1/2
+5/2
+2 −
1/2
+1/
2+2
+1 +
1/21
2×1
/2
3/2
×1/2
3/2
×12
×1
1×1
/2
1/2
×1/2
1×1
Not
atio
n:
JJ
MM
...
...
. . .
. . .m1
m2
m1
m2
Coe
ffic
ien
ts
Not
e:A
squa
re-r
oot
sign
isto
beun
ders
tood
over
ever
yco
effici
ent,
e.g.
,fo
r−8
/15
read
−√
8/15
.
Y0 1
=√
3 4πco
sθ
Y1 1
=−√
3 8πsi
nθei
φ
Y0 2
=√
5 4π
(
3 2co
s2θ−
1 2
)
Y1 2
=−√
15 8πsi
nθco
sθei
φ
Y2 2
=1 4
√
15 2πsi
n2θ
e2iφ
Y−m �
=(−
1)m
Ym∗
�
〈j 1j 2
m1m
2|j 1
j 2JM
〉=
(−1)
J−j
1−j
2〈j 2
j 1m
2m
1|j 2
j 1JM
〉
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286 36. Clebsch-Gordan coefficients
−1/5
2
2/7
2/7
−3/7
3
1/2
−1/2 −1 −2
−2 −1
04
1/2
1/2−3
3
1/2
−1/2 −2
1−44
−2
1/5
−27/
70
+1/2
7/2
+7/2
7/2
+5/2 3/7
4/7 +2 +1 0
1+2 +1
+4 14
4 +2 3/14
3/144/7
+2 1/2
−1/20
+2 −1 0 +1 +2
+2 +1 0 −132
4
1/14
1/143/7
3/7+1
3
1/5
−1/5
3/10
−3/1
0+12 +2 +1 0 −1 −2
−2 −1 0 +1 +2
3/7
3/7
−1/1
4−1
/14+1
1
43
2
2/7
2/7
−2/7
1/14
1/14
4
1/14
1/143/7
3/7
3
3/10
−3/1
0
1/5
−1/5−1
−2
−2 −1 0
0 −1 −2
−1 0 +1
+1 0 −1 −2
−12
4
3/14
3/144/7−2
−2−2
3/7
3/7
−1/1
4−1
/14−1
1
1/5
−3/1
03/
10−1
10
01/
70
1/70
8/35
18/3
58/
35
01/
10
−1/1
0
2/5
−2/50
00 0
2/5
−2/5
−1/1
0
1/10
01/
5
1/5
−1/5
−1/5
1/5
−1/5
−3/1
03/
10+1
2/7
2/7
−3/7
+3 1/2 +2 +1 0
1/2
+2+2 +2 +1
+2+1+3 1/2
−1/2 0 +1 +23
4
+1/2
+3/2
+3/2
+2+5
/2 4/7
7/2
+3/2 1/7
4/7
2/7
5/2
+3/2 +2 +1 −10
16/3
5
−18/
351/
35
1/35
12/3
518
/35
4/35
3/2
+3/2
+3/2
−3/2
−1/2
+1/22/5
−2/5
7/2
7/2
4/35
18/3
512
/35
1/35
−1/2
5/2
27/7
03/
35−5
/14
−6/3
5
−1/2
3/2
7/2
7/2
−5/2 4/7
3/7
5/2
−5/2 3/7
−4/7
−3/2
−2
2/7
4/7
1/7
5/2
−3/2 −1 −2
18/3
5−1
/35
−16/
35
−3/2 1/5
−2/5 2/5
−3/2
−1/23/2
−3/2
7/2 1
−7/2
−1/2 2/5
−1/5 0 0 −1 −22/5
1/2
−1/2
1/10
3/10
−1/5
−2/5
−3/2
−1/2
+1/2
5/2
3/2
1/2
+1/2 2/5
1/5
−3/2
−1/2
+1/2
+3/2
−1/1
0
−3/1
0
+1/2 2/5
2/5 +1 0 −1 −20
+333 +2
2 +21
+3/2
+3/2
+1/2
+1/2
1/2
−1/2
−1/2
+1/2
+3/2
1/2
32
3 01/
20
1/20
9/20
9/20
21
3 −1 1/5
1/5
3/5
2
3
3 1−3
−2 1/2
1/2
−3/2
2
1/2
−1/2
−3/2−2
−1 1/2
−1/2
−1/2
−3/20
1 −13/
10
3/10
−2/5
−3/2
−1/2
00
1/4
1/4
−1/4
−1/4
09/
20
9/20
+1/2
−1/2
−3/2
−1/2
0−1
/20
01/
4
1/4
−1/4
−1/4
−3/2
−1/2
+1/2
1/2
−1/20
1
3/10
3/10
−3/2
−1/2
+1/2
+3/2
+3/2
+1/2
−1/2
−3/2
−2/5+1
+1+1 1/5
3/5
1/5
1/2
+3/2
+1/2
−1/2
+3/2
+3/2
−1/5
+1/2
6/35
5/14
−3/3
5
1/5
−3/7
−1/2
+1/2
+3/25/2
2×3
/2
2×2
3/2
×3/2
−3
d� m
,0=
√
4π2�
+1
Ym �
e−im
φ
dj m
′ ,m=
(−1)
m−m
′ dj m
,m′=
dj −m
,−m
′d
1 0,0
=co
sθd
1/2
1/2,1
/2
=co
sθ 2
d1/2
1/2,−
1/2
=−
sin
θ 2
d1 1,1
=1
+co
sθ2
d1 1,0
=−
sin
θ√ 2
d1 1,−
1=
1−
cosθ
2
d3/2
3/2,3
/2
=1
+co
sθ2
cos
θ 2
d3/2
3/2,1
/2
=−√ 3
1+
cosθ
2si
nθ 2
d3/2
3/2,−
1/2
=√ 3
1−
cosθ
2co
sθ 2
d3/2
3/2,−
3/2
=−
1−
cosθ
2si
nθ 2
d3/2
1/2,1
/2
=3
cosθ
−1
2co
sθ 2
d3/2
1/2,−
1/2
=−
3co
sθ+
12
sin
θ 2
d2 2,2
=(
1+
cosθ
2
)
2
d2 2,1
=−
1+
cosθ
2si
nθ
d2 2,0
=√ 6 4
sin2
θ
d2 2,−
1=
−1−
cosθ
2si
nθ
d2 2,−
2=
(
1−
cosθ
2
)
2
d2 1,1
=1
+co
sθ2
(2co
sθ−
1)
d2 1,0
=−√
3 2si
nθ
cosθ
d2 1,−
1=
1−
cosθ
2(2
cosθ
+1)
d2 0,0
=(
3 2co
s2θ−
1 2
)
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40. Kinematics 287
40. KINEMATICS
Revised January 2000 by J.D. Jackson (LBNL) and June 2008 by D.R.Tovey (Sheffield).
Throughout this section units are used in which � = c = 1. The followingconversions are useful: �c = 197.3 MeV fm, (�c)2 = 0.3894 (GeV)2 mb.
40.1. Lorentz transformations
The energy E and 3-momentum p of a particle of mass m form a4-vector p = (E,p) whose square p2 ≡ E2 − |p|2 = m2. The velocity ofthe particle is β = p/E. The energy and momentum (E∗,p∗) viewed froma frame moving with velocity βf are given by
(
E∗p∗‖
)
=(
γf −γfβf−γfβf γf
) (
Ep‖
)
, p∗T
= pT , (40.1)
where γf = (1−β2f )−1/2 and p
T(p‖) are the components of p perpendicular
(parallel) to βf . Other 4-vectors, such as the space-time coordinates ofevents, of course transform in the same way. The scalar product of two4-momenta p1 · p2 = E1E2 − p1 · p2 is invariant (frame independent).
40.2. Center-of-mass energy and momentum
In the collision of two particles of masses m1 and m2 the totalcenter-of-mass energy can be expressed in the Lorentz-invariant form
Ecm =[
(E1 + E2)2 − (p1 + p2)2]1/2
,
=[
m21 + m2
2 + 2E1E2(1 − β1β2 cos θ)]1/2
, (40.2)
where θ is the angle between the particles. In the frame where one particle(of mass m2) is at rest (lab frame),
Ecm = (m21 + m2
2 + 2E1 lab m2)1/2 . (40.3)
The velocity of the center-of-mass in the lab frame isβcm = plab/(E1 lab + m2) , (40.4)
where plab ≡ p1 lab andγcm = (E1 lab + m2)/Ecm . (40.5)
The c.m. momenta of particles 1 and 2 are of magnitude
pcm = plabm2
Ecm. (40.6)
For example, if a 0.80 GeV/c kaon beam is incident on a proton target,the center of mass energy is 1.699 GeV and the center of mass momentumof either particle is 0.442 GeV/c. It is also useful to note that
Ecm dEcm = m2 dE1 lab = m2 β1 lab dplab . (40.7)
40.3. Lorentz-invariant amplitudes
The matrix elements for a scattering or decay process are written interms of an invariant amplitude −iM . As an example, the S-matrix for2 → 2 scattering is related to M by
〈p′1p′2 |S| p1p2〉 = I − i(2π)4 δ4(p1 + p2 − p′1 − p′2)
× M (p1, p2; p′1, p′2)(2E1)1/2 (2E2)1/2 (2E′
1)1/2 (2E′
2)1/2
. (40.8)
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288 40. Kinematics
The state normalization is such that〈p′|p〉 = (2π)3δ3(p− p′) . (40.9)
40.4. Particle decays
The partial decay rate of a particle of mass M into n bodies in its restframe is given in terms of the Lorentz-invariant matrix element M by
dΓ =(2π)4
2M|M |2 dΦn (P ; p1, . . . , pn), (40.10)
where dΦn is an element of n-body phase space given by
dΦn(P ; p1, . . . , pn) = δ4 (P −n
∑
i=1
pi)n
∏
i=1
d3pi
(2π)32Ei. (40.11)
This phase space can be generated recursively, viz.dΦn(P ; p1, . . . , pn) = dΦj(q; p1, . . . , pj)
× dΦn−j+1 (P ; q, pi+1, . . . , pn)(2π)3dq2 , (40.12)
where q2 = (∑j
i=1 Ei)2 −∣
∣
∣
∑ji=1 pi
∣
∣
∣
2. This form is particularly useful in
the case where a particle decays into another particle that subsequentlydecays.
40.4.1. Survival probability : If a particle of mass M has meanproper lifetime τ (= 1/Γ) and has momentum (E,p), then the probabilitythat it lives for a time t0 or greater before decaying is given by
P (t0) = e−t0 Γ/γ = e−Mt0 Γ/E , (40.13)
and the probability that it travels a distance x0 or greater is
P (x0) = e−Mx0 Γ/|p| . (40.14)
40.4.2. Two-body decays :
p1, m1
p2, m2
P, M
Figure 40.1: Definitions of variables for two-body decays.
In the rest frame of a particle of mass M , decaying into 2 particleslabeled 1 and 2,
E1 =M2 − m2
2 + m21
2M, (40.15)
|p1| = |p2|
=
[(
M2 − (m1 + m2)2) (
M2 − (m1 − m2)2)]1/2
2M, (40.16)
and
dΓ =1
32π2|M |2 |p1|
M2dΩ , (40.17)
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40. Kinematics 289
where dΩ = dφ1d(cos θ1) is the solid angle of particle 1. The invariant massM can be determined from the energies and momenta using Eq. (40.2)with M = Ecm.
40.4.3. Three-body decays :
p1, m1
p3, m3
P, M p2, m2
Figure 40.2: Definitions of variables for three-body decays.
Defining pij = pi + pj and m2ij = p2
ij , then m212 + m2
23 + m213 =
M2 + m21 + m2
2 + m23 and m2
12 = (P − p3)2 = M2 + m23 − 2ME3, where
E3 is the energy of particle 3 in the rest frame of M . In that frame,the momenta of the three decay particles lie in a plane. The relativeorientation of these three momenta is fixed if their energies are known.The momenta can therefore be specified in space by giving three Eulerangles (α, β, γ) that specify the orientation of the final system relative tothe initial particle [1]. Then
dΓ =1
(2π)51
16M|M |2 dE1 dE2 dα d(cos β) dγ . (40.18)
Alternatively
dΓ =1
(2π)51
16M2|M |2 |p∗1| |p3| dm12 dΩ∗
1 dΩ3 , (40.19)
where (|p∗1|, Ω∗1) is the momentum of particle 1 in the rest frame of 1
and 2, and Ω3 is the angle of particle 3 in the rest frame of the decayingparticle. |p∗1| and |p3| are given by
|p∗1| =
[(
m212 − (m1 + m2)2
) (
m212 − (m1 − m2)2
)]
2m12
1/2
, (40.20a)
and
|p3| =
[(
M2 − (m12 + m3)2) (
M2 − (m12 − m3)2)]1/2
2M. (40.20b)
[Compare with Eq. (40.16).]If the decaying particle is a scalar or we average over its spin states,
then integration over the angles in Eq. (40.18) gives
dΓ =1
(2π)31
8M|M |2 dE1 dE2
=1
(2π)31
32M3 |M |2 dm212 dm2
23 . (40.21)
This is the standard form for the Dalitz plot.
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290 40. Kinematics
40.4.3.1. Dalitz plot: For a given value of m212, the range of m2
23 isdetermined by its values when p2 is parallel or antiparallel to p3:
(m223)max =
(E∗2 + E∗
3 )2 −(
√
E∗22 − m2
2 −√
E∗23 − m2
3
)2
, (40.22a)
(m223)min =
(E∗2 + E∗
3 )2 −(
√
E∗22 − m2
2 +√
E∗23 − m2
3
)2
. (40.22b)
Here E∗2 = (m2
12 − m21 + m2
2)/2m12 and E∗3 = (M2 −m2
12 −m23)/2m12 are
the energies of particles 2 and 3 in the m12 rest frame. The scatter plotin m2
12 and m223 is called a Dalitz plot. If |M |2 is constant, the allowed
region of the plot will be uniformly populated with events [see Eq. (40.21)].A nonuniformity in the plot gives immediate information on |M |2. Forexample, in the case of D → Kππ, bands appear when m(Kπ) = mK∗(892),reflecting the appearance of the decay chain D → K∗(892)π → Kππ.
(m23)max
0 1 2 3 4 5 0
2
4
6
8
10
m12 (GeV2)
m23
(G
eV2 )
(m1+m2)2
(M−m3)2
(M−m1)2
(m2+m3)2
(m23)min2
2
2
2
Figure 40.3: Dalitz plot for a three-body final state. In thisexample, the state is π+K0p at 3 GeV. Four-momentum conservationrestricts events to the shaded region.
40.4.4. Kinematic limits :
40.4.4.1. Three-body decays: In a three-body decay (Fig. 40.2) themaximum of |p3|, [given by Eq. (40.20)], is achieved when m12 = m1 +m2,i.e., particles 1 and 2 have the same vector velocity in the rest frame of thedecaying particle. If, in addition, m3 > m1, m2, then |p
3|max > |p
1|max,
|p2|max. The distribution of m12 values possesses an end-point or
maximum value at m12 = M − m3. This can be used to constrain themass difference of a parent particle and one invisible decay product.
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40. Kinematics 291
40.4.5. Multibody decays : The above results may be generalized tofinal states containing any number of particles by combining some of theparticles into “effective particles” and treating the final states as 2 or 3“effective particle” states. Thus, if pijk... = pi + pj + pk + . . ., then
mijk... =√
p2ijk... , (40.25)
and mijk... may be used in place of e.g., m12 in the relations in Sec. 40.4.3or Sec. 40.4.4 above.
40.5. Cross sections
p3, m3
pn+2, mn+2
...
p1, m1
p2, m2
Figure 40.5: Definitions of variables for production of an n-bodyfinal state.
The differential cross section is given by
dσ =(2π)4|M |2
4√
(p1 · p2)2 − m21m
22
× dΦn(p1 + p2; p3, . . . , pn+2) . (40.26)
[See Eq. (40.11).] In the rest frame of m2(lab),√
(p1 · p2)2 − m21m
22 = m2p1 lab ; (40.27a)
while in the center-of-mass frame√
(p1 · p2)2 − m21m
22 = p1cm
√s . (40.27b)
40.5.1. Two-body reactions :
p1, m1
p2, m2
p3, m3
p4, m4
Figure 40.6: Definitions of variables for a two-body final state.
Two particles of momenta p1 and p2 and masses m1 and m2 scatterto particles of momenta p3 and p4 and masses m3 and m4; theLorentz-invariant Mandelstam variables are defined by
s = (p1 + p2)2 = (p3 + p4)2
= m21 + 2E1E2 − 2p1 · p2 + m2
2 , (40.28)t = (p1 − p3)2 = (p2 − p4)2
= m21 − 2E1E3 + 2p1 · p3 + m2
3 , (40.29)
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292 40. Kinematics
u = (p1 − p4)2 = (p2 − p3)2
= m21 − 2E1E4 + 2p1 · p4 + m2
4 , (40.30)
and they satisfys + t + u = m2
1 + m22 + m2
3 + m24 . (40.31)
The two-body cross section may be written asdσ
dt=
164πs
1|p1cm|2 |M |2 . (40.32)
In the center-of-mass framet = (E1cm − E3cm)2 − (p1cm − p3cm)2 − 4p1cm p3cm sin2(θcm/2)
= t0 − 4p1cm p3cm sin2(θcm/2) , (40.33)
where θcm is the angle between particle 1 and 3. The limiting valuest0 (θcm = 0) and t1 (θcm = π) for 2 → 2 scattering are
t0(t1) =[
m21 − m2
3 − m22 + m2
4
2√
s
]2
− (p1 cm ∓ p3 cm)2 . (40.34)
In the literature the notation tmin (tmax) for t0 (t1) is sometimes used,which should be discouraged since t0 > t1. The center-of-mass energiesand momenta of the incoming particles are
E1cm =s + m2
1 − m22
2√
s, E2cm =
s + m22 − m2
1
2√
s, (40.35)
For E3cm and E4cm, change m1 to m3 and m2 to m4. Then
pi cm =√
E2i cm − m2
i and p1cm =p1 lab m2√
s. (40.36)
Here the subscript lab refers to the frame where particle 2 is at rest. [Forother relations see Eqs. (40.2)–(40.4).]
40.5.2. Inclusive reactions : Choose some direction (usually the beamdirection) for the z-axis; then the energy and momentum of a particle canbe written as
E = mT
cosh y , px , py , pz = mT
sinh y , (40.37)
where mT , conventionally called the ‘transverse mass’, is given by
m2T
= m2 + p2x + p2
y . (40.38)
and the rapidity y is defined by
y =12
ln(
E + pz
E − pz
)
= ln(
E + pz
mT
)
= tanh−1(pz
E
)
. (40.39)
Note that the definition of the transverse mass in Eq. (40.38) differsfrom that used by experimentalists at hadron colliders (see Sec. 40.6.1below). Under a boost in the z-direction to a frame with velocity β,y → y − tanh−1 β. Hence the shape of the rapidity distribution dN/dy isinvariant, as are differences in rapidity. The invariant cross section may
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40. Kinematics 293
also be rewritten
Ed3σ
d3p=
d3σ
dφdy pT dpT
=⇒ d2σ
π dy d(p2T
). (40.40)
The second form is obtained using the identity dy/dpz = 1/E, and thethird form represents the average over φ.
Feynman’s x variable is given by
x =pz
pz max≈ E + pz
(E + pz)max(pT |pz|) . (40.41)
In the c.m. frame,
x ≈ 2pz cm√s
=2mT sinh ycm√
s(40.42)
and= (ycm)max = ln(
√s/m) . (40.43)
The invariant mass M of the two-particle system described in Sec. 40.4.2can be written in terms of these variables as
M2 = m21 + m2
2 + 2[ET (1)ET (2) coshΔy − pT (1) · pT (2)] , (40.44)
whereET (i) =
√
|pT (i)|2 + m2i , (40.45)
and pT (i) denotes the transverse momentum vector of particle i.For p � m, the rapidity [Eq. (40.39)] may be expanded to obtain
y =12
lncos2(θ/2) + m2/4p2 + . . .
sin2(θ/2) + m2/4p2 + . . .
≈ − ln tan(θ/2) ≡ η (40.46)
where cos θ = pz/p. The pseudorapidity η defined by the second line isapproximately equal to the rapidity y for p � m and θ � 1/γ, and in anycase can be measured when the mass and momentum of the particle areunknown. From the definition one can obtain the identities
sinh η = cot θ , cosh η = 1/ sin θ , tanh η = cos θ . (40.47)
40.5.3. Partial waves : The amplitude in the center of mass for elasticscattering of spinless particles may be expanded in Legendre polynomials
f(k, θ) =1k
∑
�
(2� + 1)a�P�(cos θ) , (40.48)
where k is the c.m. momentum, θ is the c.m. scattering angle, a� =(η�e
2iδ� − 1)/2i, 0 ≤ η� ≤ 1, and δ� is the phase shift of the �th partialwave. For purely elastic scattering, η� = 1. The differential cross sectionis
dσ
dΩ= |f(k, θ)|2 . (40.49)
The optical theorem states that
σtot =4π
kIm f(k, 0) , (40.50)
and the cross section in the �th partial wave is therefore bounded:
σ� =4π
k2(2� + 1)|a�|2 ≤ 4π(2� + 1)
k2. (40.51)
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294 40. Kinematics
40.5.3.1. Resonances: The Breit-Wigner (nonrelativistic) form for anelastic amplitude a� with a resonance at c.m. energy ER, elastic widthΓel, and total width Γtot is
a� =Γel/2
ER − E − iΓtot/2, (40.54)
where E is the c.m. energy.The spin-averaged Breit-Wigner cross section for a spin-J resonance
produced in the collision of particles of spin S1 and S2 is
σBW (E) =(2J + 1)
(2S1 + 1)(2S2 + 1)π
k2
BinBoutΓ2tot
(E − ER)2 + Γ2tot/4
, (40.55)
where k is the c.m. momentum, E is the c.m. energy, and B in and B outare the branching fractions of the resonance into the entrance and exitchannels. The 2S + 1 factors are the multiplicities of the incident spinstates, and are replaced by 2 for photons. This expression is valid only foran isolated state. If the width is not small, Γtot cannot be treated as aconstant independent of E. There are many other forms for σBW , all ofwhich are equivalent to the one given here in the narrow-width case. Someof these forms may be more appropriate if the resonance is broad.
The relativistic Breit-Wigner form corresponding to Eq. (40.54) is:
a� =−mΓel
s − m2 + imΓtot. (40.56)
A better form incorporates the known kinematic dependences, replacingmΓtot by
√s Γtot(s), where Γtot(s) is the width the resonance particle
would have if its mass were√
s, and correspondingly mΓel by√
s Γel(s)where Γel(s) is the partial width in the incident channel for a mass
√s:
a� =−√
s Γel(s)s − m2 + i
√s Γtot(s)
. (40.57)
For the Z boson, all the decays are to particles whose masses are smallenough to be ignored, so on dimensional grounds Γtot(s) =
√s Γ0/mZ ,
where Γ0 defines the width of the Z, and Γel(s)/Γtot(s) is constant. A fulltreatment of the line shape requires consideration of dynamics, not justkinematics. For the Z this is done by calculating the radiative correctionsin the Standard Model.
40.6. Transverse variables
At hadron colliders, a significant and unknown proportion of the energyof the incoming hadrons in each event escapes down the beam-pipe.Consequently if invisible particles are created in the final state, their netmomentum can only be constrained in the plane transverse to the beamdirection. Defining the z-axis as the beam direction, this net momentumis equal to the missing transverse energy vector
EmissT = −
∑
i
pT (i) , (40.58)
where the sum runs over the transverse momenta of all visible final stateparticles.
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40. Kinematics 295
40.6.1. Single production with semi-invisible final state :Consider a single heavy particle of mass M produced in association
with visible particles which decays as in Fig. 40.1 to two particles, ofwhich one (labeled particle 1) is invisible. The mass of the parent particlecan be constrained with the quantity MT defined by
M2T ≡ [ET (1) + ET (2)]2 − [pT (1) + pT (2)]2
= m21 + m2
2 + 2[ET (1)ET (2) − pT (1) · pT (2)] , (40.59)
wherepT (1) = Emiss
T . (40.60)
This quantity is called the ‘transverse mass’ by hadron colliderexperimentalists but it should be noted that it is quite different fromthat used in the description of inclusive reactions [Eq. (40.38)]. Thedistribution of event MT values possesses an end-point at Mmax
T = M . Ifm1 = m2 = 0 then
M2T = 2|pT (1)||pT (2)|(1 − cosφ12) , (40.61)
where φij is defined as the angle between particles i and j in the transverseplane.
40.6.2. Pair production with semi-invisible final states :
p11
, mp44
, mp
, mp
3 1
22
, m
M M
Figure 40.9: Definitions of variables for pair production of semi-invisible final states. Particles 1 and 3 are invisible while particles 2and 4 are visible.
Consider two identical heavy particles of mass M produced such thattheir combined center-of-mass is at rest in the transverse plane (Fig. 40.9).Each particle decays to a final state consisting of an invisible particle offixed mass m1 together with an additional visible particle. M and m1 canbe constrained with the variables MT2 and MCT which are defined inRefs. [4] and [5].
Further discussion and all references may be found in the full Review ofParticle Physics. The numbering of references and equations used herecorresponds to that version.
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296 41. Cross-section formulae for specific processes
41. CROSS-SECTION FORMULAE
FOR SPECIFIC PROCESSES
Revised October 2009 by H. Baer (Florida State University) and R.N.Cahn (LBNL).PART I: STANDARD MODEL PROCESSES
Setting aside leptoproduction (for which, see Sec. 16 of this Review),the cross sections of primary interest are those with light incident particles,e+e−, γγ, qq, gq , gg, etc., where g and q represent gluons and lightquarks. The produced particles include both light particles and heavyones - t, W , Z, and the Higgs boson H . We provide the production crosssections calculated within the Standard Model for several such processes.41.1. Resonance Formation
Resonant cross sections are generally described by the Breit-Wignerformula (Sec. 16 of this Review).
σ(E) =2J + 1
(2S1 + 1)(2S2 + 1)4π
k2
[
Γ2/4(E − E0)2 + Γ2/4
]
BinBout, (41.1)
where E is the c.m. energy, J is the spin of the resonance, and thenumber of polarization states of the two incident particles are 2S1 + 1and 2S2 + 1. The c.m. momentum in the initial state is k, E0 is thec.m. energy at the resonance, and Γ is the full width at half maximumheight of the resonance. The branching fraction for the resonance intothe initial-state channel is Bin and into the final-state channel is Bout.For a narrow resonance, the factor in square brackets may be replaced byπΓδ(E − E0)/2.41.2. Production of light particles
The production of point-like, spin-1/2 fermions in e+e− annihilationthrough a virtual photon, e+e− → γ∗ → ff , at c.m. energy squared s is
dσ
dΩ= Nc
α2
4sβ[1 + cos2 θ + (1 − β2) sin2 θ]Q2
f , (41.2)
where β is v/c for the produced fermions in the c.m., θ is the c.m.scattering angle, and Qf is the charge of the fermion. The factor Nc is 1for charged leptons and 3 for quarks. In the ultrarelativistic limit, β → 1,
σ = NcQ2f4πα2
3s= NcQ
2f
86.8 nbs (GeV2)
. (41.3)
The cross section for the annihilation of a qq pair into a distinct pairq′q′ through a gluon is completely analogous up to color factors, with thereplacement α → αs. Treating all quarks as massless, averaging over thecolors of the initial quarks and defining t = −s sin2(θ/2), u = −s cos2(θ/2),one finds
dσ
dΩ(qq → q′q′) =
α2s
9s
t2 + u2
s2 . (41.4)
Crossing symmetry gives
dσ
dΩ(qq′ → qq′) =
α2s
9s
s2 + u2
t2. (41.5)
If the quarks q and q′ are identical, we have
dσ
dΩ(qq → qq) =
α2s
9s
[
t2 + u2
s2 +s2 + u2
t2− 2u2
3st
]
, (41.6)
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41. Cross-section formulae for specific processes 297
and by crossing
dσ
dΩ(qq → qq) =
α2s
9s
[
t2 + s2
u2+
s2 + u2
t2− 2s2
3ut
]
. (41.7)
Annihilation of e+e− into γγ has the cross section
dσ
dΩ(e+e− → γγ) =
α2
2s
u2 + t2
tu. (41.8)
The related QCD process also has a triple-gluon coupling. The crosssection is
dσ
dΩ(qq → gg) =
8α2s
27s(t2 + u2)(
1tu
− 94s2 ) . (41.9)
The crossed reactions are
dσ
dΩ(qg → qg) =
α2s
9s(s2 + u2)(− 1
su+
94t2
) , (41.10)
dσ
dΩ(gg → qq) =
α2s
24s(t2 + u2)(
1tu
− 94s2
) , (41.11)
dσ
dΩ(gg → gg) =
9α2s
8s(3 − ut
s2− su
t2− st
u2) . (41.12)
Lepton-quark scattering is analogous (neglecting Z exchange)dσ
dΩ(eq → eq) =
α2
2se2qs2 + u2
t2, (41.13)
eq is the quark charge. For ν-scattering with the four-Fermi interactiondσ
dΩ(νd → −u) =
G2F s
4π2 , (41.14)where the Cabibbo angle suppression is ignored. Similarly
dσ
dΩ(νu → +d) =
G2F s
4π2
(1 + cos θ)2
4. (41.15)
For deep inelastic scattering (presented in more detail in Section 16)we consider quarks of type i carrying a fraction x = Q2/(2Mν) of thenucleon’s energy, where ν = E − E′ is the energy lost by the lepton in thenucleon rest frame. With y = ν/E we have the correspondences
1 + cos θ → 2(1 − y) , dΩcm → 4πfi(x)dx dy , (41.16)where the latter incorporates the quark distribution, fi(x). We find
dσ
dx dy(eN → eX) =
4πα2xs
Q4
12
[
1 + (1 − y)2]
×[49(u(x) + u(x) + . . .)+
19(d(x) + d(x) + . . .)
]
(41.17)where now s = 2ME is the cm energy squared for the electron-nucleoncollision and we have suppressed contributions from higher mass quarks.
Similarly,dσ
dx dy(νN → −X) =
G2F xs
π[(d(x) + . . .) + (1 − y)2(u(x) + . . .)] , (41.18)
dσ
dx dy(νN → +X) =
G2F xs
π[(d(x) + . . .) + (1 − y)2(u(x) + . . .)] . (41.19)
Quasi-elastic neutrino scattering (νµn → μ−p, νµp → μ+n) is directlyrelated to the crossed reaction, neutron decay.
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298 41. Cross-section formulae for specific processes
41.3. Hadroproduction of heavy quarksFor hadroproduction of heavy quarks Q = c, b, t, it is important to
include mass effects in the formulae. For qq → QQ, one hasdσ
dΩ(qq → QQ) =
α2s
9s3
[
(m2Q − t)2 + (m2
Q − u)2 + 2m2Qs
]
, (41.20)
while for gg → QQ one hasdσ
dΩ(gg → QQ) =
α2s
32s
[
6s2
(m2Q − t)(m2
Q − u)
−m2
Q(s − 4m2Q)
3(m2Q − t)(m2
Q − u)+
43
(m2Q − t)(m2
Q − u) − 2m2Q(m2
Q + t)
(m2Q − t)2
+43
(m2Q − t)(m2
Q − u) − 2m2Q(m2
Q + u)
(m2Q − u)2
−[
3(m2
Q − t)(m2Q − u) + m2
Q(u − t)
s(m2Q − t)
−3(m2
Q − t)(m2Q − u) + m2
Q(t − u)
s(m2Q − u)
]
.
(41.21)
41.4. Production of Weak Gauge Bosons
41.4.1. W and Z resonant production :
Resonant production of a single W or Z is governed by the partial widths
Γ(W → iνi) =
√2GF m3
W
12π(41.22)
Γ(W → qiqj) = 3
√2GF |Vij |2m3
W
12π(41.23)
Γ(Z → ff) = Nc
√2GF m3
Z
6π
×[
(T3 − Qf sin2 θW )2 + (Qf sin θW )2]
. (41.24)
The weak mixing angle is θW . The CKM matrix elements are Vij . Nc is 3for qq and 1 for leptonic final states. These widths along with associatedbranching fractions may be applied to the resonance production formulaof Sec. 41.1 to gain the total W or Z production cross section.
41.4.2. Production of pairs of weak gauge bosons :
The cross section for ff → W+W− is given in term of the couplings of theleft-handed and right-handed fermion f , = 2(T3 − QxW ), r = −2QxW ,where T3 is the third component of weak isospin for the left-handed f , Qis its electric charge (in units of the proton charge), and xW = sin2 θW :
dσ
dt=
2πα2
Ncs2
{
⎡
⎣
(
Q + + r
4xW
s
s − m2Z
)2
+
(
− r
4xW
s
s − m2Z
)2⎤
⎦A(s, t, u)
+1
2xW
(
Q +
2xW
s
s − m2Z
)
(Θ(−Q)I(s, t, u)− Θ(Q)I(s, u, t))
+1
8x2W
(Θ(−Q)E(s, t, u) + Θ(Q)E(s, u, t))
}
, (41.26)
where Θ(x) is 1 for x > 0 and 0 for x < 0, and where
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41. Cross-section formulae for specific processes 299
A(s, t, u) =
(
tu
m4W
− 1
)(
14− m2
W
s+ 3
m4W
s2
)
+s
m2W
− 4,
I(s, t, u) =
(
tu
m4W
− 1
)(
14− m2
W
2s− m4
W
st
)
+s
m2W
− 2 + 2m2
W
t,
E(s, t, u) =
(
tu
m4W
− 1
)(
14
+m4
W
t2
)
+s
m2W
, (41.27)
and s, t, u are the usual Mandelstam variables with s = (pf + pf )2, t =(pf − pW−)2, u = (pf − pW+)2. The factor Nc is 3 for quarks and 1 forleptons.
The analogous cross-section for qiqj → W±Z0 is
dσ
dt=
πα2|Vij |26s2x2
W
{(
1s − m2
W
)2 [(
9 − 8xW
4
)
(
ut − m2W m2
Z
)
+ (8xW − 6) s(
m2W + m2
Z
)]
+
[
ut − m2W m2
Z − s(m2W + m2
Z)s − m2
W
]
[
j
t− i
u
]
+ut − m2
W m2Z
4(1 − xW )
[
2jt2
+2iu2
]
+s(m2
W + m2Z)
2(1 − xW )ij
tu
}
, (41.28)
where i and j are the couplings of the left-handed qi and qj as definedabove. The CKM matrix element between qi and qj is Vij .
The cross section for qiqi → Z0Z0 is
dσ
dt=
πα2
964i + r4
i
x2W (1 − x2
W )2s2
[
t
u+
u
t+
4m2Zs
tu− m4
Z
(
1t2
+1u2
)
]
.
(41.29)
41.5. Production of Higgs Bosons
41.5.1. Resonant Production :
The Higgs boson of the Standard Model can be produced resonantlyin the collisions of quarks, leptons, W or Z bosons, gluons, or photons.The production cross section is thus controlled by the partial width of theHiggs boson into the entrance channel and its total width. The partialwidths are given by the relations
Γ(H → ff) =GF m2
fmHNc
4π√
2
(
1 − 4m2f/m2
H
)3/2, (41.30)
Γ(H → W+W−) =GF m3
HβW
32π√
2
(
4 − 4aW + 3a2W
)
, (41.31)
Γ(H → ZZ) =GF m3
HβZ
64π√
2
(
4 − 4aZ + 3a2Z
)
. (41.32)
where Nc is 3 for quarks and 1 for leptons and where aW = 1 − β2W =
4m2W /m2
H and aZ = 1 − β2Z = 4m2
Z/m2H . The decay to two gluons
proceeds through quark loops, with the t quark dominating. Explicitly,
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300 41. Cross-section formulae for specific processes
Γ(H → gg) =α2
sGF m3H
36π3√
2
∣
∣
∣
∣
∣
∑
q
I(m2q/m2
H)
∣
∣
∣
∣
∣
2
, (41.33)
where I(z) is complex for z < 1/4. For z < 2 × 10−3, |I(z)| is small so thelight quarks contribute negligibly. For mH < 2mt, z > 1/4 and
I(z) = 3
[
2z + 2z(1 − 4z)(
sin−1 12√
z
)2]
, (41.34)
which has the limit I(z) → 1 as z → ∞.
41.5.2. Higgs Boson Production in W ∗ and Z∗ decay :
The Standard Model Higgs boson can be produced in the decay ofa virtual W or Z (“Higgstrahlung”): In particular, if k is the c.m.momentum of the Higgs boson,
σ(qiqj → WH) =πα2|Vij |236 sin4 θW
2k√s
k2 + 3m2W
(s − m2W )2
(41.35)
σ(ff → ZH) =2πα2(2f + r2
f )
48Nc sin4 θW cos4 θW
2k√s
k2 + 3m2Z
(s − m2Z)2
. (41.36)
where and r are defined as above.
41.5.3. W and Z Fusion :Just as high-energy electrons can be regarded as sources of virtual photonbeams, at very high energies they are sources of virtual W and Z beams.For Higgs boson production, it is the longitudinal components of the W sand Zs that are important. The distribution of longitudinal W s carryinga fraction y of the electron’s energy is
f(y) =g2
16π2
1 − y
y, (41.37)
where g = e/ sin θW . In the limit s � mH � mW , the rate Γ(H →WLWL) = (g2/64π)(m3
H/m2W ) and in the equivalent W approximation
σ(e+e− → νeνeH) =1
16m2W
(
α
sin2 θW
)3
×[(
1 +m2
H
s
)
logs
m2H
− 2 + 2m2
H
s
]
. (41.38)
There are significant corrections to this relation when mH is not largecompared to mW . For mH = 150 GeV, the estimate is too high by 51%for
√s = 1000 GeV, 32% too high at
√s = 2000 GeV, and 22% too high
at√
s = 4000 GeV. Fusion of ZZ to make a Higgs boson can be treatedsimilarly. Identical formulae apply for Higgs production in the collisionsof quarks whose charges permit the emission of a W+ and a W−, exceptthat QCD corrections and CKM matrix elements are required. Even inthe absence of QCD corrections, the fine-structure constant ought to beevaluated at the scale of the collision, say mW . All quarks contribute tothe ZZ fusion process.
Further discussion and all references may be found in the full Review; theequation and reference numbering corresponds to that version.
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42. Plots of cross sections and related quantities 301
10-4
10-3
10-2
10-1
1
10
10 2
1 10 102
103
104
➚➘⇓
⇓
⇓
Tot
al c
ross
sec
tion
(mb)
⇓
104103102101.6
0.2
0.1
0.0
-0.1
-0.2
⇓
104103102101.6
⇓
104103102101.6
p−(p)p Σ−p
K∓p
π∓p
γp
γγ
p−p π−p
K−p
√sGeV
√sGeV
√sGeV
√sGeV
ppπ+p K+p
Re (T )Im (T )
Figure 40.10: Summary of hadronic, γp, and γγ total cross sections,and ratio of the real to imaginary parts of the forward hadronicamplitudes. Corresponding computer-readable data files may be foundat http://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPASgroup, IHEP, Protvino, August 2005)
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302 6. Atomic and nuclear properties of materials6.A
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6. Atomic and nuclear properties of materials 303
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ass
0.42
101
95.9
158.
07.
871.
255
6.22
0St
anda
rdro
ck0.
5000
066
.810
1.3
26.5
41.
688
2.65
0
Met
hane
(CH
4)
0.62
334
54.0
73.8
46.4
7(2
.417
)(0
.667
)90
.68
111.
7[4
44.]
Eth
ane
(C2H
6)
0.59
861
55.0
75.9
45.6
6(2
.304
)(1
.263
)90
.36
184.
5B
utan
e(C
4H
10)
0.59
497
55.5
77.1
45.2
3(2
.278
)(2
.489
)13
4.9
272.
6O
ctan
e(C
8H
18)
0.57
778
55.8
77.8
45.0
02.
123
0.70
321
4.4
398.
8Par
affin
(CH
3(C
H2) n
≈23C
H3)
0.57
275
56.0
78.3
44.8
52.
088
0.93
0N
ylon
(typ
e6,
6/6)
0.54
790
57.5
81.6
41.9
21.
973
1.18
Pol
ycar
bona
te(L
exan
)0.
5269
758
.383
.641
.50
1.88
61.
20Pol
yeth
ylen
e([
CH
2C
H2] n
)0.
5703
456
.178
.544
.77
2.07
90.
89Pol
yeth
ylen
ete
reph
thal
ate
(Myl
ar)
0.52
037
58.9
84.9
39.9
51.
848
1.40
Pol
ymet
hylm
etha
cryl
ate
(acr
ylic
)0.
5393
758
.182
.840
.55
1.92
91.
191.
49Pol
ypro
pyle
ne0.
5599
856
.178
.544
.77
2.04
10.
90Pol
ysty
rene
([C
6H
5C
HC
H2] n
)0.
5376
857
.581
.743
.79
1.93
61.
061.
59Pol
ytet
raflu
oroe
thyl
ene
(Tefl
on)
0.47
992
63.5
94.4
34.8
41.
671
2.20
Pol
yvin
ylto
luen
e0.
5414
157
.381
.343
.90
1.95
61.
031.
58
Alu
min
umox
ide
(sap
phir
e)0.
4903
865
.598
.427
.94
1.64
73.
970
2327
.32
73.
1.77
Bar
ium
flour
ide
(BaF
2)
0.42
207
90.8
149.
09.
911.
303
4.89
316
41.
2533
.1.
47C
arbo
ndi
oxid
ega
s(C
O2)
0.49
989
60.7
88.9
36.2
01.
819
(1.8
42)
[449
.]So
lidca
rbon
diox
ide
(dry
ice)
0.49
989
60.7
88.9
36.2
01.
787
1.56
3Su
blim
esat
194.
7K
Ces
ium
iodi
de(C
sI)
0.41
569
100.
617
1.5
8.39
1.24
34.
510
894.
215
53.
1.79
Lithi
umflu
orid
e(L
iF)
0.46
262
61.0
88.7
39.2
61.
614
2.63
511
21.
1946
.1.
39Lithi
umhy
drid
e(L
iH)
0.50
321
50.8
68.1
79.6
21.
897
0.82
096
5.Lea
dtu
ngst
ate
(PbW
O4)
0.41
315
100.
616
8.3
7.39
1.22
98.
300
1403
.2.
20Si
licon
diox
ide
(SiO
2,f
used
quar
tz)
0.49
930
65.2
97.8
27.0
51.
699
2.20
019
86.
3223
.1.
46So
dium
chlo
ride
(NaC
l)0.
5550
971
.211
0.1
21.9
11.
847
2.17
010
75.
1738
.1.
54So
dium
iodi
de(N
aI)
0.42
697
93.1
154.
69.
491.
305
3.66
793
3.2
1577
.1.
77W
ater
(H2O
)0.
5550
958
.583
.336
.08
1.99
21.
000(
0.75
6)27
3.1
373.
11.
33
Silic
aae
roge
l0.
5009
365
.097
.327
.25
1.74
00.
200
(0.0
3H
2O
,0.
97Si
O2)
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
304 NOTES
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
NOTES 305
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*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
306 NOTES
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****** INSIDE BACK COVER ******
Tab
le1.
1.R
evis
ed20
10by
D.E
.G
room
(LB
NL),
and
E.
Ber
gren
.A
tom
icw
eigh
tsof
stab
leel
emen
tsar
ead
apte
dfr
omth
eC
omm
issi
onon
Isot
opic
Abun-
dan
ces
and
Ato
mic
Wei
ghts
,“A
tom
icW
eigh
tsof
the
Ele
men
ts20
07,”
http://www.chem.qmul.ac.uk/iupac/AtWt/.
The
atom
icnum
ber
(top
left
)is
the
num
ber
ofpro
tons
inth
enucl
eus.
The
atom
icm
ass
(bot
tom
)of
ast
able
elem
ents
isw
eigh
ted
by
isot
opic
abundan
ces
inth
eE
arth
’ssu
rfac
e.If
the
elem
ent
has
no
stab
leis
otop
e,th
eat
omic
mas
s(i
npar
enth
eses
)of
the
mos
tst
able
isot
ope
curr
entl
yknow
nis
give
n.
Inth
isca
seth
em
ass
isfr
omhttp://www.nndc.bnl.gov/amdc/masstables/Ame2003/mass.mas03
and
the
longe
st-liv
edis
otop
eis
from
www.nndc.bnl.gov/ensdf/zaform.jsp.
The
exce
ption
sar
eT
h,
Pa,
and
U,w
hic
hdo
hav
ech
arac
teri
stic
terr
estr
ialco
mpos
itio
ns.
Ato
mic
mas
ses
are
rela
tive
toth
em
ass
of12C
,defi
ned
tobe
exac
tly
12unifi
edat
omic
mas
sunit
s(u
)(a
ppro
x.
g/m
ole)
.R
elat
ive
isot
opic
abundan
ces
ofte
nva
ryco
nsi
der
ably
,bot
hin
nat
ura
lan
dco
mm
erci
alsa
mple
s;th
isis
reflec
ted
inth
enum
ber
ofsi
gnifi
cant
figu
res
give
n.
Pre
vio
usl
yco
nfirm
edel
emen
t11
2w
asnam
edC
oper
nic
ium
(Cp).
The
dis
cove
ryof
elem
ent
114
was
confirm
edin
2009
.T
her
ear
eno
other
confirm
edel
emen
tsw
ith
Z>
112.
1 IA18
VIIIA
1H
Hydro
gen
1.0
0794
2 IIA
13 IIIA
14 IVA
15 VA
16 VIA
17 VIIA
2H
eH
eliu
m
4.0
02602
3Li
Lit
hiu
m
6.9
41
4Be
Ber
ylliu
m
9.0
12182
PER
IOD
ICTA
BLE
OF
TH
EELEM
EN
TS
5B
Boro
n
10.8
11
6C
Carb
on
12.0
107
7N
Nit
rogen
14.0
067
8O
Oxygen
15.9
994
9F
Flu
ori
ne
18.9
984032
10N
eN
eon
20.1
797
11N
aSodiu
m
22.9
8976928
12
Mg
Magnes
ium
24.3
050
3IIIB
4IV
B5 VB
6V
IB7
VIIB
89
VIII
1011 IB
12 IIB
13
Al
Alu
min
um
26.9
815386
14
Si
Silic
on
28.0
855
15
PP
hosp
h.
30.9
73762
16
SSulfur
32.0
65
17
Cl
Chlo
rine
35.4
53
18
Ar
Arg
on
39.9
48
19
KPota
ssiu
m
39.0
983
20
Ca
Calc
ium
40.0
78
21
Sc
Sca
ndiu
m
44.9
55912
22
Ti
Tit
aniu
m
47.8
67
23
VV
anadiu
m
50.9
415
24
Cr
Chro
miu
m
51.9
961
25
Mn
Manganes
e
54.9
38045
26
Fe
Iron
55.8
45
27
Co
Cobalt
58.9
33195
28N
iN
ickel
58.6
934
29
Cu
Copper
63.5
46
30
Zn
Zin
c
65.3
8
31
Ga
Galliu
m
69.7
23
32
Ge
Ger
man.
72.6
4
33
As
Ars
enic
74.9
2160
34
Se
Sel
eniu
m
78.9
6
35
Br
Bro
min
e
79.9
04
36
Kr
Kry
pto
n
83.7
98
37
Rb
Rubid
ium
85.4
678
38
Sr
Str
onti
um
87.6
2
39
YY
ttri
um
88.9
0585
40
Zr
Zir
coniu
m
91.2
24
41N
bN
iobiu
m
92.9
0638
42
Mo
Moly
bd.
95.9
6
43
Tc
Tec
hnet
.
(97.9
0722)
44
Ru
Ruth
en.
101.0
7
45
Rh
Rhodiu
m
102.9
0550
46
Pd
Palladiu
m
106.4
2
47
Ag
Silver
107.8
682
48
Cd
Cadm
ium
112.4
11
49
InIn
diu
m
114.8
18
50
Sn
Tin
118.7
10
51
Sb
Anti
mony
121.7
60
52
Te
Tel
lurium
127.6
0
53
IIo
din
e
126.9
0447
54
Xe
Xen
on
131.2
93
55
Cs
Ces
ium
132.9
054519
56
Ba
Barium
137.3
27
57–71
Lanth
a-
nid
es
72H
fH
afn
ium
178.4
9
73
Ta
Tanta
lum
180.9
4788
74
WTungst
en
183.8
4
75
Re
Rhen
ium
186.2
07
76
Os
Osm
ium
190.2
3
77
IrIr
idiu
m
192.2
17
78
Pt
Pla
tinum
195.0
84
79
Au
Gold
196.9
66569
80H
gM
ercu
ry
200.5
9
81
Tl
Thalliu
m
204.3
833
82
Pb
Lea
d
207.2
83
Bi
Bism
uth
208.9
8040
84
Po
Polo
niu
m
(208.9
8243)
85
At
Ast
ati
ne
(209.9
8715)
86
Rn
Radon
(222.0
1758)
87
Fr
Fra
nci
um
(223.0
1974)
88
Ra
Radiu
m
(226.0
2541)
89–103
Act
inid
es
104
Rf
Ruth
erfo
rd.
(267.1
22)
105
Db
Dubniu
m
(268.1
25)
106
Sg
Sea
borg
.
(271.1
33)
107
Bh
Bohrium
(270.1
34)
108
Hs
Hass
ium
(269.1
34)
109
Mt
Mei
tner
.
(276.1
51)
110
Ds
Darm
stadt.
(281.1
62)
111
Rg
Roen
tgen
.
(280.1
64)
112
Cp
Coper
nic
ium
(277)
114 (2
88)
Lan
than
ide
seri
es57
La
Lanth
an.
138.9
0547
58
Ce
Cer
ium
140.1
16
59
Pr
Pra
seodym
.
140.9
0765
60N
dN
eodym
.
144.2
42
61
Pm
Pro
met
h.
(144.9
1275)
62
Sm
Sam
ari
um
150.3
6
63
Eu
Euro
piu
m
151.9
64
64
Gd
Gadolin.
157.2
5
65
Tb
Ter
biu
m
158.9
2535
66
Dy
Dysp
ros.
162.5
00
67H
oH
olm
ium
164.9
3032
68
Er
Erb
ium
167.2
59
69
Tm
Thulium
168.9
3421
70
Yb
Ytt
erbiu
m
173.0
54
71
Lu
Lute
tium
174.9
668
Act
inid
ese
ries
89
Ac
Act
iniu
m
(227.0
2775)
90
Th
Thorium
232.0
3806
91
Pa
Pro
tact
in.
231.0
3588
92
UU
raniu
m
238.0
2891
93N
pN
eptu
niu
m
(237.0
4817)
94
Pu
Plu
toniu
m
(244.0
6420)
95
Am
Am
eric
.
(243.0
6138)
96
Cm
Curium
(247.0
7035)
97
Bk
Ber
kel
ium
(247.0
7031)
98
Cf
Califo
rn.
(251.0
7959)
99
Es
Ein
stei
n.
(252.0
830)
100
Fm
Fer
miu
m
(257.0
9510)
101
Md
Men
del
ev.
(258.0
9843)
102
No
Nobel
ium
(259.1
010)
103
Lr
Law
renc.
(262.1
10)
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2010 12:56
****** OUTSIDE BACK COVER ******
2010
JULY
AU
GU
ST
SEP
TEM
BER
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
OC
TO
BER
NO
VEM
BER
DEC
EM
BER
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
2011
JULY
AU
GU
ST
SEP
TEM
BER
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
OC
TO
BER
NO
VEM
BER
DEC
EM
BER
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
2011
JAN
UA
RY
FEB
RU
AR
YM
AR
CH
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
AP
RIL
MA
YJU
NE
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
30
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
2012
JAN
UA
RY
FEB
RU
AR
YM
AR
CH
SM
TW
TF
SS
MT
WT
FS
SM
TW
TF
S1
23
45
67
89
1011
1213
1415
1617
1819
2021
2223
2425
2627
2829
3031
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
29
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31A
PR
ILM
AY
JUN
ES
MT
WT
FS
SM
TW
TF
SS
MT
WT
FS
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
31
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
Prepared for the U.S. Department of Energyunder Contract DE-AC02-05CH11231
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2012 12:56
*** NOTE TO PUBLISHER OF Particle Physics Booklet ***Please use crop marks to align pages August 11, 2012 12:56