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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 1accardi@jlab.org 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

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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

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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

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An illustrated introduction

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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”

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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 (???)

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x =p+parton

p+nucleonp§ =

1p2(p0 § p3)Fractional momentum:

Nucleons are made of 3 quarks…

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Nucleons are made of 3 quarks…

x =p+parton

p+nucleonp§ =

1p2(p0 § p3)Fractional momentum:

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… and gluons, and sea quarks …

x =p+parton

p+nucleonp§ =

1p2(p0 § p3)Fractional momentum:

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… and gluons, and sea quarks …

x =p+parton

p+nucleonp§ =

1p2(p0 § p3)Fractional momentum:

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... but this is another story …

[see A.Deshpande, and A.Prokudin]

… spinning and orbiting around !

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(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)

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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.

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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.]

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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]

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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.]

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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

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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

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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

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Deep Inelastic Scattering

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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¸

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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; :::

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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

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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

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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

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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

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Kinematics

Kinematic plane – in practice

HERA

e+p collider fixed target experiments

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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

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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

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Cross section

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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)

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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)

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Structure functions

Same as if target was a free spin ½ particle:The photon is scattering on quasi-free quarks

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Structure functions

Note: evolution with Q 2 and rise at low-x (here come the gluons!)

10¡5 < x < 0:6

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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

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A taste of the parton model

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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

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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

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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

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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