Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
HEAVY-QUARK FRAGMENTATION FUNCTIONS
IN e
�
e � COLLISIONS
Carlo OleariUniversita di Milano-Bicocca, Milan
DIS2006, Tzukuba
21 April 2006
� � Theoretical introduction
– collinear logarithms
– soft logarithms
� � Non-perturbative contributions
� � Results for c (and b) fragmentationfunctions
� � Conclusions
What is a fragmentation function?
pZ/γ
q
e-
e+Q
X
The differential cross section for the production of a massive quark Q in the process
e
�
e
� � Z
�
γ
�
q
� � Q
�
p� � X
is calculable in PQCD, because the mass acts as a cut-off for final-state collinear singularities
dσdx
�
x, q2 , m2 � �
�
∑n � 0
a
n
�
x, q2 , m2, µ2 �αn
s�
µ2 �
x � 2 p � qq2 E �
�
q2
2
with µ = renormalization scale and αs
αs
� �
2π
�
.
The heavy-quark fragmentation function (FF) in e
�
e � annihilation is defined as
D
�
x, q2 , m2 � 1σtot
dσdx
�
x, q2 , m2 �
Differential cross-section at order αs
pZ/γ
q
e-
e+Q
Q
pZ/γ
q
e-
e+Q
Q
x � EQ�
EQ
�
max
1σ0
dσdx
� δ
�
1 � x
� � αs
�
q2 �
2πCF
�
1 � lnq2
m2
�
1 � x2
1 � x�
��
�
ln
�
1 � x
�
1 � x
��
�
1 � x2 �
� 21 � x2
1 � xlog x � 1
2
�
11 � x
��
�
x2 � 6x � 2
� �
�
23π2 �
52
�
δ
�
1 � x
��
� ��
m2
q2
�
1
0
�
11 � x
��
f
�
x
�
1
0
�
f
�
x�
� f�
1
�
1 � x
�
� �
σ � σ0
�
1 � αs
π
� ��
m2
q2
�
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 2
Two main issues
� � Is this a “well-behaved” perturbation expansion?
� � We do NOT “see” free quarks, even if they are heavy, butbound states (mesons and baryons). How can we incorporatethese non-perturbative (hadronization) effects?
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 3
Collinear and soft logarithms
1σ0
dσdx
�
�
∑n � 0
a
n
�
x, q2 , m2, µ2 �
�
αs
�
µ2 �
2π
� n
� δ
�
1 � x
� � αs
�
q2 �
2πCF
�
1 � lnq2
m2
�
1 � x2
1 � x
��
�
�
ln
�
1 � x�
1 � x�
��
1 � x2 �
� 21 � x2
1 � xlog x � 1
2
�
11 � x
��
�
x2 � 6x � 2
� �
�
23π2 �
52
�
δ
�
1 � x
��
� ��
m2
q2
�
� � In the limit q2 � m2,
a
n
��
logq2
m2
� n
and if αs log q2
m2
� 1 we cannot truncate the series at some fixed order, because eachterm in the series is of the same order as the first one � � we have to resum these largecontributions
� � Same for soft logarithms, that arise when x � 1.
Quasi-collinear behavior
M
k
pp+k
� � M
k
pp/z
S
�
p � k
� 2 � m2 � 2p � k
� 2k0
�
� �
p
�
2 � m2 �� �
p
�
cosθQg�
t = (t0,0,tz) η = (1,0,-1)
k⊥ = (0,k⊥,0)
pν � ztν �
ξ
�
ην � kν �
kν � �
1 � z
�
tν �
ξ
� �
ην � kν �
p2 � t2 � m2 k2 � 0 η2 � 0
t � k � � 0 η � k � � 0 kν � k �
ν
� � k2 �
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 5
Factorization theorem
� � factorization of the squared amplitude
� � �
p, k; . . .
� � 2 � � � �
p
�
z; . . .
� � 2 8παs
2 p � kS � � � �
p
�
z; . . .
� � 2 8παsz
�1 � z
�
k2 �
� �
1 � z
�
2m2S
S z,m2
k2 �
� CF
�
1 � z2
1 � z
�
2z
�
1 � z
�
m2
k2 �
� �
1 � z�
2m2
�
� � factorization of the phase space
d
�
n � 1
� d
�
n � 1d3 p
�
2π
�
32p0
d3k
�
2π
�
32k0δ4 �
. . . � p � k
� � d�
n � 1d3t
�
2π
�
32t0δ4 �
. . . � t
�
� �� �
d
n
116π2
dzz
�
1 � z
� dk2 �
Then we can iterate to take into account multiple quasi-collinear emissions
dσn � 1
� � � �
p, k; . . .
� � 2d
�
n � 1
� � � �
t; . . .
� � 2d
�
nαs
2π
1
0dz
µ2f
0dk2 �
1k2 �
� �
1 � z
�
2m2S
� dσnαs
2πCF
1
0dz
1 � z2
1 � zlog
�
1 � µ2f
m21
�
1 � z
�
2
�� . . .
Factorization theorem
In the limit m2 � q2 we can neglect terms proportional to powers ofm2 �
q2, and the single inclusive heavy-quark cross section can be writtenas (factorization theorem)
dσdx
�
x, q2 , m2 � � ∑i
1
x
dzz
dσidz
�
z, q2 , µ2F
Di
� xz
, µ2F , m2
dσidz
MS-subtracted partonic cross section, for the production of the par-
ton i (process dependent)
Di MS fragmentation functions for the parton i to “fragment” into theheavy quark Q (process independent)
µF factorization scale
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 7
Example
dσdx
�
x, q2 , m2 � � ∑i
1
x
dzz
dσidz
�
z, q2 , µ2F
Di
� xz
, µ2F , m2
if i � g then
Z / γ
Q
Q
i=g
� �� �
m � 0 !
process dependent
dσidz
� Di
i=g
Q
Q� �� �
m
� � 0
process independent
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 8
Resummation of collinear logs
dσdx
�
x, q2 , m2 � � ∑i
1
x
dzz
dσidz
�
z, q2 , µ2F
Di
� xz
, µ2F , m2
How to choose µF?
If µ2F
� q2 then no large logarithms of q2 �
µ2F appear in dσi
�
dz and its perturbative expan-
sion is reliable.The large logarithms are moved in the fragmentation functions that obey the (µF
� µ)
DGLAP evolution equations
dDi
d log µ2
�
x, µ2 , m2 � � ∑j
1
x
dzz
Pi j� x
z, αs
�
µ2 �
D j
�
z, µ2 , m2 �
where the splitting functions Pi j have the expansion
Pi j
�
x, αs
�
µ
�
� αs
�
µ
�
P
0
i j
�x
� �
α2s
�
µ
�
P
1
i j
�
x
� �
α3s
�
µ
�
P
2
i j
�
x
� � � �
α4s
The DGLAP equations resum correctly all the large logarithms
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 9
Leading Log, Next-to-Leading Log. . .
Introducing the shorthand notation L � log
�
q2 �
m2
�
✓ with an expansion for Pi j up to order αs, we resum all terms of the form
dσdx
�
x, q2 , m2 ��������
LL
�
�
∑n � 0
β
n
�
x
� �
αsL
� n
� �� �
�
αsL
� n � � Leading Log
✓ with an expansion for Pi j up to order α2s , we resum all terms of the form
dσdx
�
x, q2 , m2 ��������
NLL
�
�
∑n � 0
β
n
�
x
� �
αsL� n �
�
∑n � 0
γ
n
�
x
�
αs
�
αsL
� n
� �� �
αs
�
αsL
� n � � Next-to-Leading Log
✓ . . . so on, so forth.
The time-like splitting functions are known up to the third order [Mitov, Moch and Vogt, hep-
ph/0604053]
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 10
Initial condition
How can we compute the initial condition for Di
�
x, µ2 , m2 �
?
Di
�
x, µ2 , m2 � � d
0
i δ
�
1 � x
� �
αs
�
µ2 �
d
1
i
�
x, µ2 , m2 � � � �
α2s
.
Use the factorization theorem
dσdx
�
x, q2 , m2 � � ∑i
1
x
dzz
dσidz
�
z, q2 , µ2
Di� x
z, µ2 , m2
and match the expansion up to order αs of the left- and right-hand side to obtain d
0
i and
d
1
i [Mele and Nason, Nucl. Phys. B361 (91) 626]
dσdx
�
x, q2 , m2 � � a
0
�
x, q2 , m2 � � a
1 �
x, q2 , m2, µ2 �
αs
�
µ2 � � � �
α2s
dσidx
�
x, q2 , µ2 � � a
0
i
�
x
� � a
1
i�
x, q2 , µ2 �
αs
�
µ2 � � � �
α2s
The d
2
i terms are known too, and have been computed by [Melnikov and Mitov, �� �� � � �� �� �� ��
;
Mitov, �� �� � � � � �� � � ], following a different strategy.
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 11
Final recipe for collinear-log resummation
✓ start with Di
�
x, µ20 , m2 �
, with µ20
� m2, so that no large loga-rithms of the ratio µ2
0
�
m2 appear in the initial conditions
✓ evolve Di
�
x, µ20 , m2 �
from the low to the high energy scale µ
with the DGLAP equation to obtain Di�
x, µ2 , m2 �
✓ use the factorization theorem to compute the resummedcross section.
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 12
Soft logarithms
In the region of the phase space of multiple soft-gluon emission (x � 1), the differentialcross section contains enhanced terms proportional to
a
n
� c
n
LLlog2n � 1 �
1 � x
�
1 � x �� c
n
NLLlog2n � 2 �
1 � x
�1 � x �
� . . .
These terms can be organized in towers of log N, where we introduce the Mellin transform
f
�
N
� �
1
0dx xN � 1 f
�
x
� � � 1
0dx xN � 1 logk �
1 � x
�
1 � x �� logk � 1 N
The large-N contributions come from the regions where x � 1, associated to thebremsstrahlung spectrum of soft and collinear emission.Up to now, it is known how to resum all the Leading Log and Next-to-Leading Log [Dok-
shitzer, Khoze and Troyan, �� �� � � �� � � � ; Cacciari and Catani, �� �� � � � � � � �� � �
]
�
∑n � 0
c
n
LL αns logn �
1 N � log N gLL�
αs log N
�
�
∑n � 0
c
n
NLL αns logn N � gNLL
�
αs log N
�
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 13
Non-perturbative effects
✗ The weak point of the factorization theorem comes from the initial condition for theevolution of the fragmentation function, which is computed as a power expansionin terms of αs
�
m
�
: irreducible, non-perturbative uncertainties of order
�
QCD
�
m arepresent.
✗ The soft-gluon resummation functions gLL and gNLL contain singularities at large Nwhich signal the eventual failure of perturbation theory and hence the onset of non-perturbative phenomena.
� � in the initial condition, the region
�
1 � x
�
m � �(m
�
N � �
in moment space) issensitive to the decay of excited states of the heavy-flavoured hadrons, where
�
isa typical hadronic scale of a few hundreds MeV.
� � in the coefficient functions, when
�
1 � x�
q2 � �2, the mass of the recoil systemapproaches typical hadronic scales.
The matching of perturbative results with non-perturbative physics is a delicate problem,which rests, first of all, on a proper definition of the perturbative series.
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 14
Non-perturbative fragmentation function
We assume that all these effects are described by a non-perturbative fragmen-tation function DH
NP, that takes into account all low-energy effects, includingthe process of the heavy quark turning into a heavy-flavoured hadron. Thefull resummed cross section, including non-perturbative corrections, is thenwritten as
dσH
dx
�
x, q2 � � ∑i
dσidx
�
x, q2 , µ2F
� Di
�
x, µ2F , m2
� DHNP
�
x
�
The non-perturbative part DHNP is
what is missing to go from the par-tonic cross section to the hadronicone � � very sensitive to the per-turbative part
It is expected to be universal � � exctract it from e
�
e � data and use it inhadronic heavy-quark production.
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 15
Non-perturbative fragmentation function
The Mellin transform of the full resummed cross section, including non-perturbative corrections, is
σH
�
N, q2 � � σQ
�
N, q2 , m2 �
DNP
�
N
�
We [Cacciari, Nason and C.O., �� � � � � �� � � � �
] have fitted CLEO and BELLE D
�
datausing the two-component form for DNP
DNP
�
x
� � Norm. �
�
δ
�
1 � x
� � c
�
1 � x
� axb
Na,b
�
, Na,b
�
1
0
�
1 � x
� axb
Simple phenomenological interpretation
� � the hard term (the delta function) corresponds, in some sense, to the directexclusive production of the D
�
� � the rest accounts for D
�
’s produced in the decay chain of higher reso-nances.
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 16
Non-perturbative fragmentation function
DNP
�
x
� � Norm. �
�
δ
�
1 � x
� � c
�
1 � x
� axb
Na,b
�
, Na,b
�
1
0
�
1 � x� axb
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 17
CLEO and BELLE D
�
fits
✓ D mesons data fits near the
� �
4S
�
mass (10.6 GeV)
✓ Very good fit in the whole x range
✓ More fits in [Cacciari, Nason and
C.O.], where D
� � D X have beenmodeled from decay chains andbranching ratios
✓ B mesons data fits near the Z0
mass (91.2 GeV)
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 18
ALEPH D
�
fits at LEP
✗ very few useful points
✗ large error bars
DNP
�
x
� � Norm. �
�δ
�1 � x
� � c
�
1 � x
� axb
Na,b
�
ALEPH a � 2.4
�
1.2 , b � 13.9
�
5.7 c � 5.9
�
1.7
CLEO
�
BELLE a � 1.8
�
0.2 , b � 11.3
�
0.6 c � 2.46
�
0.07
What about the supposed universality of the non-perturbative fragmentation function?
ALEPH with CLEO/BELLE parameters
✗ Discrepancy in the large-x (large-N)region
✗ The ratio data/theory well modeled by
11 � 0.044
�
N � 1�
Possible explanation
We can only speculate about the possible origin of the discrepancy.If this were due to a non-perturbative correction to the coefficient function ofthe form
� � 1 � C
�
N � 1
�
q2 this would lead to the extra factor
1 � C
N � 1
M2Z
1 � C
N � 1
M2�
� � � 11 � 0.044
�
N � 1
� if C � 5 GeV2 large value!!
� � 1 � C
�
N � 1
�
Ewith E �
�
q2
�
2 then C � 0.52 GeV, a much more accept-
able value.
Demonstrating the absence (or the existence) of 1
�
E corrections in fragmenta-tion functions would be a very interesting result, since it would help to vali-date or disprove renormalon-based predictions.
Carlo Oleari Heavy-Quark Fragmentation Functions in e
e
�
Collisions 21
Conclusions
σQ
�
N, M2Z , m2 �
σQ
�
N, M2� , m2
� � aq
�
N, M2Z , µ2
Z
�
1 �
αs
�
µ2Z
� �
πE
�
N, µ2Z , µ2�
� 1 �
αs
�
µ2�� �
π
aq
�
N, M2� , µ2��
� � low-scale effects, both at the heavy-quarkmass scale and at the non-perturbativelevel, cancel completely
� � its prediction is then entirely perturbative(the evolution function E is totally pertur-bative)
� � scale-variation effects do not explain thediscrepancy
� � mass effects in charm production onthe
� �
4S
�
where computed at order α2s ,
and found to be small [Nason and C.O.,
�� �� � � �� � � � ��
]
Unfortunately, the low precision of the available data does not allow, at the moment, to resolvethe issue of the absence (or the existence) of 1
�
E corrections in fragmentation functions.