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Deep Inelastic Scattering and Global Fits of Parton Distributions
– lecture 1–
Alberto AccardiHampton U. and Jefferson Lab
HUGS 2011
HUGS 2011 – Lecture [email protected] 2
Plan of the lectures Lecture 1 – Motivation
– Quarks, gluons, hadrons– Deep Inelastic Scattering (DIS)
Lecture 2 – Parton model– DIS revisited– Parton Distribution Functions (PDFs)– QCD factorization, universality of PDFs
Lecture 3 – Global PDF fits– DIS, Drell-Yan (DY) lepton pairs, W and Z production, hadronic jets– Choices, choices, choices...– Errors: statistic, systematic, theory uncertainties
Lecture 4 – Examples and applications– Fits as community service (e.g., measure PDFs, apply to LHC)– Fits as a tool to study hadron and nuclear structure
HUGS 2011 – Lecture [email protected] 3
Resources Textbooks
– Povh et al., “Particles and Nuclei,” Springer, 1999
– Halzen, Martin, “Quarks and leptons,” John Wiley and sons, 1984
– Lenz et al. (Eds.), “Lectures on QCD. Applications,” Springer, 1997
• esp. lectures by Levy, Rith, Jaffe
– Devenish, Cooper-Sarkar, “Deep Inelastic Scattering,” Oxford U.P., 2004
– Feynman, “Photon-hadron interactions,” Addison Wesley, 1972
Articles and lectures
– J.Owens, “PDF and global fitting”, 2007 CTEQ summer school
– S.Forte, “Parton distributions at the dawn of the LHC”, arXiv:1011.5247
– W.K.Tung, “pQCD and parton structure of the nucleon”, CTEQ summer school website
– These lectures: http://www.jlab.org/~accardi/lectures/
Come and talk to me:
– Office: A223; phone: 757-269-6363
HUGS 2011 – Lecture [email protected] 4
Lecture 1 – Motivation
An illustrated introduction– Hadrons are made of quarks and gluons– How to probe the partonic structure of hadrons
Deep Inelastic Scattering (DIS)– A bit (!) of kinematics– Cross section– Structure functions
A taste of the parton model
HUGS 2011 – Lecture [email protected] 5
An illustrated introduction
HUGS 2011 – Lecture [email protected] 6
Motivation: quarks, gluon, hadrons...
The strong force is described in terms of colored quarks and gluons:
But only color neutral hadrons can be detected – color confinement
– How can one understand, say, proton and neutrons in terms of quark and gluons?
– And, for that matter, what's the evidence for quarks and gluons?
LQCD = ¹Ã (i°¹@¹ ¡m)à ¡ g¡¹Ã°¹TaÃ
¢Aa¹ ¡ 1
4Ga¹ºG
¹ºa
a quark a gluon“color”
HUGS 2011 – Lecture [email protected] 7
nucleons
Hadrons are made of quarks
6 flavors (and 3 colors):
up, down, strange – light
charm, bottom, top – heavy
spin ½isospin (u = ½, d = – ½)strangeness (s = 1)
confined in colorless hadrons– mesons – 2 quarks– baryons – 3 quarks– tetraquarks (?)– pentaquarks (???)
HUGS 2011 – Lecture [email protected] 8
x =p+parton
p+nucleonp§ =
1p2(p0 § p3)Fractional momentum:
Nucleons are made of 3 quarks…
HUGS 2011 – Lecture [email protected] 9
Nucleons are made of 3 quarks…
x =p+parton
p+nucleonp§ =
1p2(p0 § p3)Fractional momentum:
HUGS 2011 – Lecture [email protected] 10
… and gluons, and sea quarks …
x =p+parton
p+nucleonp§ =
1p2(p0 § p3)Fractional momentum:
HUGS 2011 – Lecture [email protected] 11
… and gluons, and sea quarks …
x =p+parton
p+nucleonp§ =
1p2(p0 § p3)Fractional momentum:
HUGS 2011 – Lecture [email protected] 12
... but this is another story …
[see A.Deshpande, and A.Prokudin]
… spinning and orbiting around !
HUGS 2011 – Lecture [email protected] 13
(and interacting inside the nucleus)
EMC effect discovered more than 30 years ago:– quarks / hadrons modified inside a nucleus– A ≠ p,n– still a theoretical mystery
?? ... yet another story …
[see V. Guzey]
(we will only discuss brieflythe deuteron)
HUGS 2011 – Lecture [email protected] 14
Evidence for quarks and gluons – a whirlwind tour
Baryon spectroscopy – light sector (u, d, s), ground state
– J=3/2+: |q1↑,q2↑,q3↑⟩
– J=1/2+: |q1↑,q2↑,q3↓⟩
Fig: [from Povh et al.]
totally symmetric w.fn.
HUGS 2011 – Lecture [email protected] 15
Evidence for quarks and gluons – a whirlwind tour
Baryon spectroscopy – light sector (u, d, s), ground state
– J=3/2+: |q1↑,q2↑,q3↑⟩ spin symmetric, color antisymmetric
– J=1/2+: |q1↑,q2↑,q3↓⟩ spin antisymmetrics, color symmetric
Fig: [from Povh et al.]
HUGS 2011 – Lecture [email protected] 16
Evidence for quarks and gluons – a whirlwind tour
e+ + e { annihilation into hadrons – quark-mediated process
R =¾(e+e¡ ! hadrons)
¾(e+e¡ ! ¹+¹¡)= Ncolors
X
q
e2q
e+ + e¡ ! q + ¹q ! hadrons
[from Halzen-Martin]
HUGS 2011 – Lecture [email protected] 17
Evidence for quarks and gluons – a whirlwind tour
Jets in high-energy e+ + e { collisions
– Hadron produced in 2, 3, … N, high-energy collimated “jets”– Evidence of common origin from a parton
Fig.: 2- and 3-jet events observed by the JADE detector at PETRA[from Povh et al.]
HUGS 2011 – Lecture [email protected] 18
Probing the nucleon parton structure
Need large momentum transfer Q2 = q¹ q¹ to resolve parton structure
Example 1: Deep Inelastic Scattering (DIS)
e§ + p ! e§ + X
e+(e¡) + p ! ¹º(º) + XQ2 = ¡p2°;Z
HUGS 2011 – Lecture [email protected] 19
Probing the nucleon parton structure
Need large momentum transfer Q2 = q¹ q¹ to resolve parton structure
Example 2: Drell-Yan lepton pair creation (DY)
Q2 = (p` + p¹̀)2
p + p (¹p) ! ` + ¹̀+ X
HUGS 2011 – Lecture [email protected] 20
Probing the nucleon parton structure
Need large momentum transfer Q2 = q¹ q¹ to resolve parton structure
Example 3: jet production in p+p collisions
Q2 = E2jet
p + p (¹p) ! jet + X
HUGS 2011 – Lecture [email protected] 21
Deep Inelastic Scattering
HUGS 2011 – Lecture [email protected] 22
Kinematics
Inclusive lepton-hadron scattering:
– Notation:– Masses:
Virtual photonmomentum
Hadronic final statemomentum
Neglect compared to Q2 (MeV vs. GeV) The photon is virtual: q2 < 0
m2¸ = 0 ) j~̀ j = E
q2 = (l ¡ l0)2 = m2¸ + m2
¸ ¡ 2` ¢ `0
= ¡2EE0 + j~̀ j j ~̀0 j cos µ = ¡2EE0(1 ¡ cos µ) · 0
p¹X =P
i2X p¹i
q¹ = `¹ ¡ `0¹
p2 = p¹p¹ p ¢ q = p¹q
¹
¸ = e; ¹; º; :::
N = p; n; :::
p2 = M2 `2 = m2¸
HUGS 2011 – Lecture [email protected] 23
Kinematics
Inclusive lepton-hadron scattering:
– Notation:– Masses:
Virtual photonmomentum
Hadronic final statemomentum
Measuring q(see [Levy])– Scattered lepton– Hadronic final state– Mixed methods
q = `¡ `0 = pX ¡ p
p¹X =P
i2X p¹i
q¹ = `¹ ¡ `0¹
p2 = p¹p¹ p ¢ q = p¹q
¹
p2 = M2 `2 = m2`
¸ = e; ¹; º; :::
N = p; n; :::
HUGS 2011 – Lecture [email protected] 24
Kinematics
Lorentz invariants
Q2 = ¡q2 xB =¡q2
2p ¢ q º =p ¢ qp
p2
inelasticity *
Bjorken x beam energy loss *virtuality
(final state)invariant mass
center-of-massenergy
y =p ¢ qp ¢ ` W 2 = (p + q)2 s = (p + `)2
– Ex.1 (easy) – Do these 6 invariants exhaust all possibilities?
– Ex.2 (easy) – Are all 6 independent of each other?
º =Q2
2MxBs¡M2 =
Q2
2xByW 2 = M2 + Q2
µ1
xB¡ 1
¶
* interpretation valid in hadron rest frame, see later
HUGS 2011 – Lecture [email protected] 25
Kinematics
Kinematic limits on invariants
– Q2 > 0
– 0 < xB < 1
• By baryon number conservation the final state must containat least 1 proton, if the target is a proton:
Then,
• All final state hadrons are on-shell:
Then,
so that
pX = pproton +P
i pi
p2X = M2 +X
i
pproton ¢ pi +X
i;j
pi ¢ pj
p2i = M2i ) Ei ¸ j~pi j
pj ¢ pj = EiEj ¡ j~pij j~pij cos µij ¸ EiEj ¡ j~pij j~pij ¸ 0
W 2 = p2X ¸ M2 ) Q2 1 ¡ xBxB
¸ 0 ) 0 · xB · 1
pi pj
HUGS 2011 – Lecture [email protected] 26
Kinematics
Kinematic limits on invariants, cont'd
– º ≥ 0
• Ex.3 (easy) Hint: use definition and
– 0 ≤ y ≤ 1
Consider the “target rest frame” such that
Then,
• Ex.4 (hard): find a Lorentz invariant proof
p = (M;~0T ; 0) ` = (E`;~0T ; `z) `0 = (E0`; ~̀0T ; `
0z)
y =M(E` ¡E0`)
ME`= 1 ¡ E0`
E`
~p = 0
W 2 ¡M 2 ¸ 0
HUGS 2011 – Lecture [email protected] 27
Kinematics
Kinematic plane (simplified)
– Pick xB , Q2, y as independent variables
experimental limitations,or theoretical prejudice
xB < 1
logxB
logQ2
y1 < 1
y2 < 1yi ¼
Q2
xsi
s2 > s1
HUGS 2011 – Lecture [email protected] 28
Kinematics
Kinematic plane – in practice
HERA
e+p collider fixed target experiments
HUGS 2011 – Lecture [email protected] 29
Cross section
DIS in the target rest frame (for collider, Breit frames see [Levy])
– invariants
Cross section
– Ex.5 (med) : prove this (hint: work out dQ2/d )– Ex.6 (hard) : directly show that r.h.s. Is Lorentz invariant
p = (M;~0T ; 0) ` = (E`;~0T ; `z)
q = (q0; ~qT ; qz) `0 = (E0`; ~̀0T ; `
0z)
d¾
dxBdQ2=
y
xB
¼
E0`
d¾
dE0`d
º = p ¢ q=M = q0 = E` ¡E0`y = p ¢ q=p ¢ ` = º=E`
Q2 = 2E`E0`(1 ¡ cos µ) = 4E`E
0` sin2(µ=2)
d = 2¼ sin µdµ
d¾
dxBdQ2=
º
xB
d¾
dºdQ2
=y
Q2
d¾
dxBdy=
y
xB
d¾
dydQ2
– and also
HUGS 2011 – Lecture [email protected] 30
Cross section
Deep inelastic(large W, more particles)
Inelastic(resonance region)
(W for only a few particles)
Elastic: W=M(no energy to produce
any other particle) Q2;W 2 ! 1 xB ¯xed
HUGS 2011 – Lecture [email protected] 31
Cross section
HUGS 2011 – Lecture [email protected] 32
Structure functions
Leptonic and hadronic tensors
– Electron is elementary: L can be calculated perturbatively
Lorentz decomposition + gauge invariance = structure functions
0 only for W, Z 0 boson exchanges
W¹º(p; q) =³¡ g¹º +
q¹qº
q2
´F1(xB; Q2)
+³p¹ ¡ p ¢ q
q2q¹´³
pº ¡ p ¢ qq2
qº´F2(xB ; Q2)
+ i"¹º®¯p®q¯
2p ¢ q F3(xB; Q2)
d¾ / L¹º(`; q)W¹º(p; q)
HUGS 2011 – Lecture [email protected] 33
Structure functions
Lorentz decomposition + gauge invariance = structure functions
W¹º(p; q) =³¡ g¹º +
q¹qº
q2
´F1(xB; Q2)
+³p¹ ¡ p ¢ q
q2q¹´³
pº ¡ p ¢ qq2
qº´F2(xB ; Q2)
+ i"¹º®¯p®q¯
2p ¢ q F3(xB; Q2)
Note:
– F1 , F2 , F3 are Lorentz invariants
(the tensor structure is explcitly factorized out)– Most general structure compatible with Lorentz and gauge
invariance, no missing functions– Ex.7 (med) : convince yourself of these 2 statements
(gauge invariance implies qW = 0)
HUGS 2011 – Lecture [email protected] 34
Structure functions
Same as if target was a free spin ½ particle:The photon is scattering on quasi-free quarks
HUGS 2011 – Lecture [email protected] 35
Structure functions
Note: evolution with Q 2 and rise at low-x (here come the gluons!)
10¡5 < x < 0:6
HUGS 2011 – Lecture [email protected] 36
Structure functions and cross section
The cross section (for a exchange) reads
Using
one obtains
L¹º = 2¡`¹`0º + `0¹`
0º ¡ ` ¢ `0g¹º
¢
d¾
dxdQ2=
2¼®2y2
Q4L¹º(`; q)W
¹º(p; q)
d¾
dxBdQ2=
4¼®2
xBQ4
n³1 ¡ y ¡ x2By2
M2
Q2
´F2(xB ; Q2)
+ y2xBF1(xB ; Q2) ¨³y ¡ y2
2
´xBF3(xB ; Q2)
o
HUGS 2011 – Lecture [email protected] 37
A taste of the parton model
HUGS 2011 – Lecture [email protected] 38
A taste of the parton model
The photon scatters on quasi-free quarks
– Empirical evidence: F2 = 2xBF1
– Photon wave-length in rest frame*, neglect proton mass M/Q ≪ 1:
– E.g., for x=0.1, Q 2 =4 GeV2 (and putting back c and hbar), = 10-17 m = 10-2 fm
to be compared with Rp ≈ 1 fm
¸ =1
j~q j =1p
º2 + Q2¼ 1
º=
2Mx
Q2
*we'll see a more general derivation
HUGS 2011 – Lecture [email protected] 39
A taste of the parton model
DIS ≈ photon-quark elastic scattering
Interpretation of xB
– Parton carries fraction x of proton's momentum: k = x p
– 4-momentum conservation: k' = k+q– Partons have zero mass: k 2 = k' 2 = 0
The virtual photon probes quarks with x = xB
x =Q2
2p ¢ q = xB
p
k'quark i p
protonproton
k
HUGS 2011 – Lecture [email protected] 40
A taste of the parton model
Beware: There is an inconsistency in the derivation:
– From k = xp follows that quarks are massive, Mq = xM !!
A heuristic way out is to work in the “infinite momentum frame”, where so that one can neglect the proton's mass:
– This frame is also important to better justify the parton model– But quarks should be massless in any frame
• The problem lies in the definition of x• We'll see a better solution in tomorrow's lecture
– Similarly, the vector 3-momentum is not a Lorentz-invariant scale• In fact, M can be neglected compared to Q, not ,
as we shall see
Ep =p
~p 2 + M2 ! j~pjj~p j ! 1
j~p j
HUGS 2011 – Lecture [email protected] 41
Lecture 1 – recap
Hadrons are made of quarks and gluons– Partonic structure probed in DIS, DY, jets, .... (we'll see more)
Deep Inelastic Scattering (DIS)– The master method to measure quarks and gluons– Invariant kinematics, target rest frame– Cross section, parametrized in terms of structure functions
A taste of the parton model– Phenomenological evidence for quasi-free quarks, gluons– Some trouble in heuristic, textbook arguments
Next lecture: Parton model , parton distributions (PDF)– (QCD improved) parton model– More kinematics– Factorization, universality: Drell-Yan, W and Z, jets