Hadron Structure Theory II

Alexei Prokudin

l Lecture I:Structure of the nucleon

l Lecture IITransverse Momentum Dependent distributions (TMDs)Semi Inclusive Deep Inelastic Scattering (SIDIS)

l TutorialCalculations of SIDIS structure functions using Mathematica

• Lecture IIIAdvanced topics. Evolution of TMDs

The plan:

6

How do we study the structure of the nucleon?

Deep Inelastic Scattering (DIS)

In order to access distributions we could usedeep inelastic scattering

The energy is big enough totransform the proton ina lot of final states

Bjorken limit is

Distributions measured in deep inelastic scattering

This sum makes it sensitive to parton structure!

Deep Inelastic Scattering (DIS)

Distributions and parton model

Parton model is a logical step, partons are pointlike and dilute, so the photon interactswith them incoherently

Distributions and parton model

CONSTANT!

Parton model is a logical step, partons are pointlike and dilute, so photon interactswith them incoherently

Factorization

3

W

µ⌫

DY

=

X

f

|Hf

(Q;µ/Q)|µ⌫

⇥Z

d

2k1T

d

2k2T

F

f/P1(x

1

,k1T

;µ; ⇣

1

) F

f/P2(x

2

,k2T

;µ; ⇣

2

) �

(2)

(k1T

+ k2T

� qT

)

+ Y (q

T

, Q)

+O✓✓

⇤

Q

◆a

◆

˜

F

f/P1(x

1

,bT

;µ, ⇣

1

) =

Zd

2kT

e

�ikT ·bTF

f/P1(x

1

,kT

;µ, ⇣

F

)

˜

K(bT

;µ) =

Zd

2kT

e

�ikT ·bTK(k

T

;µ)

@

@ ln

p⇣

F

F

f/P1(x

1

,kT

;µ, ⇣

F

) =

Zd

2qT

K(qT

;µ)F

f/P1(x

1

,kT

� qT

;µ, ⇣

F

)

d

d lnµ

K(k

T

;µ) = ��

K

(g(µ)) �(kT

)

d

d lnµ

F

f/P1(x

1

,kT

;µ, ⇣

F

) = �

F

(g(µ); ⇣

F

/µ

2

)F

f/P1(x

1

,kT

;µ, ⇣

F

)

k⇤(kT

) ⌘ k̂T

qk

2

min

+ k

2

T

µ⇤(kT ) ⌘ C

1

k⇤

↵

s

(µ⇤(kT ))kT!0

= ↵

s

(C

1

k

min

)

b⇤(bT

) ⌘ bTp

1 + b

2

T

/b

2

max

µ⇤(bT ) = C

1

/b⇤

↵

s

(µ⇤(bT ))bT!1= ↵

s

(C

1

/b

max

)

d�

dqT

· · ·

P

1

P

2

k

1

⌘ k k

2

⌘ q � k

q + k (34)

3

W

µ⌫

DY

=

X

f

|Hf

(Q;µ/Q)|µ⌫

⇥Z

d

2k1T

d

2k2T

F

f/P1(x

1

,k1T

;µ; ⇣

1

) F

f/P2(x

2

,k2T

;µ; ⇣

2

) �

(2)

(k1T

+ k2T

� qT

)

+ Y (q

T

, Q)

+O✓✓

⇤

Q

◆a

◆

˜

F

f/P1(x

1

,bT

;µ, ⇣

1

) =

Zd

2kT

e

�ikT ·bTF

f/P1(x

1

,kT

;µ, ⇣

F

)

˜

K(bT

;µ) =

Zd

2kT

e

�ikT ·bTK(k

T

;µ)

@

@ ln

p⇣

F

F

f/P1(x

1

,kT

;µ, ⇣

F

) =

Zd

2qT

K(qT

;µ)F

f/P1(x

1

,kT

� qT

;µ, ⇣

F

)

d

d lnµ

K(k

T

;µ) = ��

K

(g(µ)) �(kT

)

d

d lnµ

F

f/P1(x

1

,kT

;µ, ⇣

F

) = �

F

(g(µ); ⇣

F

/µ

2

)F

f/P1(x

1

,kT

;µ, ⇣

F

)

k⇤(kT

) ⌘ k̂T

qk

2

min

+ k

2

T

µ⇤(kT ) ⌘ C

1

k⇤

↵

s

(µ⇤(kT ))kT!0

= ↵

s

(C

1

k

min

)

b⇤(bT

) ⌘ bTp

1 + b

2

T

/b

2

max

µ⇤(bT ) = C

1

/b⇤

↵

s

(µ⇤(bT ))bT!1= ↵

s

(C

1

/b

max

)

d�

dqT

· · ·

P

1

P

2

k

1

⌘ k k

2

⌘ q � k

q + k (34)

2

ps = 1.8 TeV / 1.97 TeV (13)

�

D

e↵

= 11mb (14)

ps = 7 TeV (15)

Zd

2b�

2n(s, b; pct) = �

inc

2n (s, pct) (16)

�

D

e↵

(17)

⇡ 34 mb (18)

p

ct = 3.5 GeV (19)

�

1 di↵

(s, b; p

ct) = �

diff(s, b; p

ct)� �

diff(s, b; p

ct)

1X

n=1

(�1)

n�1

�

2n(s, b; pct) (20)

�

1 di↵

(s, b; p

ct) = �

diff(s, b; p

ct) exp {��

2

(s, b; p

ct)} (21)

�

1 di↵

(s, b; p

ct) = �

diff(s, b; p

ct) exp {��

2

(s, b; p

ct)} (22)

�

n di↵

(s, b; p

ct) =

1

n!

�

diff(s, b; p

ct)

nexp {��

2

(s, b; p

ct)} (23)

�

diff(s, b; p

ct) =

�1� exp

���

diff(s, b; p

ct) �

exp {��

2

(s, b; p

ct)} (24)

�(s, b) = 1� exp [��h(s, b; pct)� �s(s, b; p

ct) + · · · ] (25)

�h(s, b; pct) (26)

d�

dq

2

T

(27)

qT (28)

q

2 ⇠ Q

2 � ⇤

2

QCD

(29)

qT ⇠ ⇤

QCD

(30)

⇤QCD ⌧ k

1T ⌧ Q (31)

qT ⌧ Q (32)

P (33)

II. DISCUSSION

...................

Acknowledgments

This work was supported by...

2

ps = 1.8 TeV / 1.97 TeV (13)

�

D

e↵

= 11mb (14)

ps = 7 TeV (15)

Zd

2b�

2n(s, b; pct) = �

inc

2n (s, pct) (16)

�

D

e↵

(17)

⇡ 34 mb (18)

p

ct = 3.5 GeV (19)

�

1 di↵

(s, b; p

ct) = �

diff(s, b; p

ct)� �

diff(s, b; p

ct)

1X

n=1

(�1)

n�1

�

2n(s, b; pct) (20)

�

1 di↵

(s, b; p

ct) = �

diff(s, b; p

ct) exp {��

2

(s, b; p

ct)} (21)

�

1 di↵

(s, b; p

ct) = �

diff(s, b; p

ct) exp {��

2

(s, b; p

ct)} (22)

�

n di↵

(s, b; p

ct) =

1

n!

�

diff(s, b; p

ct)

nexp {��

2

(s, b; p

ct)} (23)

�

diff(s, b; p

ct) =

�1� exp

���

diff(s, b; p

ct) �

exp {��

2

(s, b; p

ct)} (24)

�(s, b) = 1� exp [��h(s, b; pct)� �s(s, b; p

ct) + · · · ] (25)

�h(s, b; pct) (26)

d�

dq

2

T

(27)

qT (28)

q

2 ⇠ Q

2 � ⇤

2

QCD

(29)

qT ⇠ ⇤

QCD

(30)

⇤QCD ⌧ k

1T ⌧ Q (31)

qT ⌧ Q (32)

P (33)

k + q (34)

II. DISCUSSION

...................

Acknowledgments

This work was supported by...

3

P (33)

ˆ

k + q (34)

ˆ

k (35)

d�̂ (36)

electron quark ! electron quark (37)

d� (38)

electron proton ! electron proton (39)

d� =

Zd⇠f(⇠) d�̂ (40)

L

µ⌫W

µ⌫(41)

II. DISCUSSION

...................

Acknowledgments

This work was supported by...

3

P (33)

ˆ

k + q (34)

ˆ

k (35)

d�̂ (36)

electron quark ! electron quark (37)

d� (38)

electron proton ! electron proton (39)

d� =

Zd⇠f(⇠) d�̂ (40)

Lµ⌫ W

µ⌫(41)

II. DISCUSSION

...................

Acknowledgments

This work was supported by...

3

P (33)

ˆ

k + q (34)

ˆ

k (35)

d�̂ (36)

electron quark ! electron quark (37)

d� (38)

electron proton ! electron proton (39)

d� =

Zd⇠f(⇠) d�̂ (40)

Lµ⌫ W

µ⌫(41)

E

0 d�

d

3l0=

2↵em

sQ

4

Lµ⌫Wµ⌫

(42)

II. DISCUSSION

...................

Acknowledgments

This work was supported by...

p p

Leptonic tensor

Hadronic tensor

Distributions and parton model

This diagram is called “handbag diagram”

- parton distribution

Distributions and parton model

Why quarks are on mass-shell?

This one is virtual! However the main contribution comes from

Distributions and parton model

Definition of parton distribution

Distributions and parton model

Definition of parton distribution

Fourier transform from coordinate to momentum space

Distributions and parton model

Definition of parton distribution

Quark field operator

Definition of parton distribution

The proton state vector

Distributions and parton model

Definition of parton distribution

Position of the field in coordinate space

Distributions and parton model

Distributions and parton model Definition of parton distribution

This matrix element is called “bilocal”

Distributions and parton model

What do we know about quark momentum? Suppose that protonis moving along Z direction with a high momentum, then

“Big”component

is a new variable called lightcone momentumfraction

Distributions and parton model

What do we know about quark momentum?

“Big”component

“Small” component

“Small” component

Distributions and parton model

What do we know about quark momentum?

“Big”component

“Transverse” component

Distributions and parton model

What do we know about hadronic tensor?

Quarks are “probed” at value of

Gauge invariance

The quark and the remnant are colored thus they interactvia gluon exchanges! If “–” and perpendicular component of parton momentum areneclected, than in configuration space only “–” component survives,

This object is called Wilson line

For DIS:

ip · ⇠ = ixP

+⇠

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Factorization

Distribution

Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),

used in other processes

3

What Do We Know About Glue in Matter?

• Scaling violation: dF2/dlnQ2 and

linear DGLAP Evolution !

G(x,Q2)!

Deep Inelastic Scattering : d

2" ep#eX

dxdQ2

=4$%e.m.

2

xQ4

1& y +y

2

2

'

( )

*

+ , F2(x,Q2) &

y2

2FL (x,Q2)

-

. /

0

1 2

Gluons dominate low-x wave function

)20

1( !xG

)20

1( !xS

vxu

vxd

!

Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),

used in other process

Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),

used in other process

Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),

used in other process

3

What Do We Know About Glue in Matter?

• Scaling violation: dF2/dlnQ2 and

linear DGLAP Evolution !

G(x,Q2)!

Deep Inelastic Scattering : d

2" ep#eX

dxdQ2

=4$%e.m.

2

xQ4

1& y +y

2

2

'

( )

*

+ , F2(x,Q2) &

y2

2FL (x,Q2)

-

. /

0

1 2

Gluons dominate low-x wave function

)20

1( !xG

)20

1( !xS

vxu

vxd

!

Transverse structure: Momentum vs Position

Variables are related by 2 dimensional Fourier transform

At the level of squared amplitudes one has

The ‘average’ transverse momentum is Fourier conjugate to position difference (TMD)

¯̃ (k?, z�) =

Zd2z?e

�iz?k? ̄(z?, z�)

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z?k? � y?l? =1

2(z? � y?)(k? + l?) +

1

2(z? + y?)(k? � l?)

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The momentum transfer is Fourier conjugate to ‘average’ position (GPD)

¯̃ (k?) ̃(l?) =

Zd2z?d

2y?e�i(z?k?�y?l?) ̄(z?) (y?)

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32

GPDs TMDs

DVCS SIDIS

ensures hard scale, pointlike interactionmomentum transfer can be variedindependently

Connection to 3D structure Burkardt (2000)Burkardt (2003)

Drell-Yan frame Weiss (2009)

ensures hard scale, pointlike interactionfinal hadron transverse momentumcan be varied independently

Connection to 3D structure

is the transverse separation of parton fieldsin configuration space

Ji, Ma, Yuan (2004)Collins (2011)

AP (2012)

Kotzinian (1995),

Mulders,

Tangerman (1995),

Boer, Mulders (1998)

Ji (1997)

Radyushkin (1997)

Imaginary part, momentum transfer is zero

Transverse Momentum Dependent distributions

Gauge link

Ensures gauge invariance ofthe distribution, cannot be canceled by gauge choice

SIDIS in IMF:

Struckquark

Transverse Momentum Dependent distributionsIndividual TMDs can be projected out of the correlator

1

2Tr

�

+ �(x, k?)

�= f1 �

"

jkk

j?S

kT

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Unpolarized quarks

Longitudinally polarized quarks

Transversely polarized quarks

jk ⌘ (kj?kk? � 1

2k 2?�

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