Universality of transverse momentum dependent correlation

functionsPiet Mulders

mulders@few.vu.nl

INT SeattleSeptember 16, 2009

Yerevan - 2009

Outline

Introduction: correlation functions in QCD

PDFs (and FFs): from unintegrated to collinear

Transverse momenta: collinear and non-collinear parton correlatorsDistribution functions (collinear, TMD)Observables: azimuthal asymmetries

Gauge linksResumming multi-gluon interactions: Initial/final statesColor flow dependence

The master formula and some applications

Universality

Conclusions

OUTLINE

QCD & Standard Model

QCD framework (including electroweak theory) provides the machinery to calculate transition amplitudes, e.g. g*q q, qq g*, g* qq, qq qq, qg qg, etc.Example: qg qg

Calculations work for plane waves

__

) .( ( )0 , ( , )s ipi ip s u p s e

INTRODUCTION

From hadrons to partons

• For hard scattering process involving electrons and photons the link to external particles is

• Confinement leads to hadrons as ‘sources’ for quarks

• and ‘source’ for quarks + gluons

• and ….

) .( ( )0 , ( , )s ipi ip s u p s e

) .( ( ) ii

psX P e

1 1( ).) .( ( ) ( )s p ii

i p pX PA e

INTRODUCTION

Parton p belongs to hadron P: p.P ~ M2

For all other momenta K: p.K ~ P.K ~ s ~ Q2 Introduce a generic lightlike vector n satisfying P.n=1, then n ~ 1/Q The vector n gets its meaning in a particular hard process, n = K/K.P

e.g. in SIDIS: n = Ph/P.Ph (or n determined by P and q; doesn’t matter at leading order!)

Expand quark momentum

Scales in hard processes

PARTON CORRELATORS

Tx pp P n 2 2. ~p P xM M

. ~ 1x p n

~ Q ~ M ~ M2/Q

Ralston and Soper 79, …

The correlator (p,P;n) contains (in principle) information on hadron structure and can be expanded in a number of ‘Dirac structures’, as yet unintegrated, spectral functions/amplitudes depending on invariants: Si(p.n, p.P, p2) or Si(x, pT

2, MR2)

Which (universal) quantities can be measured/isolated (identifying specific observables/asymmetries) in which of (preferably several) processes and in which way (relevant variables) at unintegrated, TMD or collinear level? What about going beyond tree-level (factorization)?

Complication: color gauge invariance and colored remnants, higher order calculations in QCD

Spectral functions or unintegrated PDFs

PARTON CORRELATORS

More on this in talk of Ted Rogers (Friday)

‘Physical’ regions

• Support of (unintegrated) spectral functions

• ‘physical’ regions with pT

2 < 0 and MR2 > 0

• Distribution region: 0 < x < 1

• Fragmentation (z = 1/x): x > 1

PARTON CORRELATORS

PP

‘Physical’

Large values of momenta

• Calculable!2 2

2 01

T p

p

p x Mp

x

2 2( ). 0

2 (1 )p T

p

x p Mp P

x x

2 22 ( ) (1 )

0(1 )

p T pR

p

x x p x x MM

x x

0 0 ( 1)T pp

xp P p x x

x

T Tp 2( , ) ...s

T

p Pp

etc.

Bacchetta, Boer, Diehl, M JHEP 0808:023, 2008 (arXiv:0803.0227)

PARTON CORRELATORS

Leading partonic structure of hadrons

With P1.P2 ~ P1.K1 ~ P1.K2 ~ s (large) we get kinematic separation of soft and hard parts

Integration d… = d(p1.P1)… leaves (x,pT) and (z,kT) containing nonlocal operator combinations with or F

INTRODUCTION

1 2.

.

1 hT

k K z M

nk K k n

z K

2.

.T

p P xM

P np xP p n

2.

0

†3 .

( . )( , ) 0 , ,( 00

(2)

)( )i kT

T nX

d K dz k e K X K X

Fragmentation functions

Distribution functions

2.

. 0

†( . )( , )

(2 )(0) ( )i pT

T n

d P dx p e P P

Leading partonic structure of hadrons

With P1.P2 ~ P1.K1 ~ P1.K2 ~ s (large) we get kinematic separation of soft and hard parts

Integration d… = d(p1.P1)… leaves (x,pT) and (z,kT)

distributioncorrelator

fragmentationcorrelator

PP

(x, pT)

KK

(z, kT)

PARTON CORRELATORS

Integrate over p.P = p (which is of order M2) The integration over p = p.P makes time-ordering automatic (Jaffe, 1984). This works for (x) and (x,pT)

This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes, which satisfy particular support properties, etc.For collinear correlators (x), this can be extended to off-forward correlators

Integrating quark correlators

2.

3 . 0(0) ( )

( . )( , ; )

(2 )q i pT

iij njT

d P dx p n e P P

.

. 0

( . )( )

(2 )(0) ( )ij

T

q i pj i n

d Px e P P

TMD

collinear

NON-COLLINEARITY

Jaffe (1984), Diehl & Gousset, …

Relevance of transverse momenta (can they be measured?)

In a hard process one probes quarks and gluons

Parton momenta fixed by kinematics (external momenta)

DIS

SIDIS

Also possible for transverse momenta of partons

SIDIS

2-particle inclusive hadron-hadron scattering

K2

K1

pp-scattering

Tp x pP 1

Tz kk P 2. / . / 2 . Bx p n P n Q P q x

. / . . / .h h hz K n k n P K P q z

1T B h h

T T

q q x P z K

k p

1 11 1 2 2 1 1 2 2

1 2 1 2

T

T T T T

q z K z K x P x P

p p k k

NON-COLLINEARITY

We need more than one hadron and knowledge of hard process(es)!Boer &

Vogelsang

Twist expansion

Not all correlators are equally importantMatrix elements expressed in P, p, n allow twist expansion(just as for local operators) (maximize contractions with n to get dominant contributions)

Leading correlators involve ‘good fields’ in front form dynamics

Above arguments are valid and useful for both collinear and TMD fctsIn principle any number of gA.n = gA+ can be included in correlators

dim[ (0) ( )] 2n dim[ (0) ( )] 2n nF F

12 [ , ]i

T TgF D D i A

PARTON CORRELATORS

. na n a a

Barbara Pasquini, ...

dim[ (0) ( )] 3TnD

Gauge link essential for color gauge invariance

Arises from all ‘leading’ matrix elements containing Basic (simplest) gauge links for TMD correlators:

TMD correlators: quarks

2[ ] . [ ]

[0, ]3 . 0

( . )( , (0) ( ); )

(2 ) jq C i p CTij T i n

d P dx p n e P U P

. [ ][0, ] . 0

( . )( ; )

(2 )(0) ( )ij

T

q i p n

nj i

d Px n e P U P

[ ][0, ]

0

expCig ds AU

P

[] []

T

TMD

collinear

NON-COLLINEARITY

TMD correlators: gluons

2[ , '] . [ ] [ ']

[ ,0] [0, ]3 . 0

( . )( , ; )

(2 )(0) ( )C C i p C CT

g Tn

n

nd P dx p n e P U U F PF

The most general TMD gluon correlator contains two links, possibly with different paths.Note that standard field displacement involves C = C’

Basic (simplest) gauge links for gluon TMD correlators:

[ ] [ ][ , ] [ , ]( ) ( )C CF U F U

g[] g

[]

g[] g

[]

NON-COLLINEARITY

TMD master formula

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

APPLICATIONS

Note that the summation over D is over diagrams and color-flow,

e.g. for qq qq subprocess:

† †1

[ ]11

† †1

[ ]12

2

2

2

[ ][( ) ] [( ) ] [( ) ] [( ) ]2

[ ][ ] [ ] [ ] [ ]

1

1

2

1

(1') (2 ')

(1') (2 ') ....

ˆ~ (1) (2)

ˆ(1) (2)

D

D

c

c

c

Dq q qq qq q q

T

Dq q qq qq q q

N

N

N

d

d q

* + quark gluon + quark

2 2

2 21 1

1 1[ ... ... ]N N

N N N

† † † †2 2

2 2

[ ] [ ] [ , ] [ ] [ ] [ , ]1 12 1 1

ˆ ˆ[ ]N Nq q qg q g q q qg q gN N N

T

d

d q

†[ ] [ ]2

ˆq q q qT

d

d q

Compare this with *q q:

EXAMPLE

Four diagrams, each with two color flow possibilities

|M|2

quark + antiquark gluon + photon

2 2

2 21 1

1 1[ ... ... ]N N

N N N

† † † † †2 2

2 2

[ ( )] [ ( )] [ , ] [ ] [ ] [ , ]1 12 1 1

ˆ ˆ[ ]N Nq q qq g g q q qq g gN N N

T

d

d q

†[ ] [ ]2

ˆq q qqT

d

d q

Compare this with qqbar *:

EXAMPLE

Four diagrams, each with two similar color flow possibilities

|M|2

Result for integrated cross section

1 2 ... 1ˆ~ ( ) ( ) ( )...a b ab c cabc

x x z

Integrate into collinear cross section

[ ] 2 [ ]( ) ( , )C CT Tx d p x p

[ ]... ...ˆ ˆ D

ab c ab cD

(partonic cross section)

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

APPLICATIONS

Collinear parametrizations

• Gauge invariant correlators distribution functions• Collinear quark correlators (leading part, no n-dependence)

• i.e. massless fermions with momentum distribution f1q(x) = q(x), chiral

distribution g1q(x) = q(x) and transverse spin polarization h1

q(x) = q(x) in a spin ½ hadron

• Collinear gluon correlators (leading part)

• i.e. massless gauge bosons with momentum distribution f1g(x) = g(x)

and polarized distribution g1g(x) = g(x)

1 5151( ) ( (2

)( ) )Lqq q q

T

Px S Sf x g x h x

1 1( )( () )1

2g

TLg Tgx g if Sx x

xg

L T

P

MS SS

PARAMETRIZATION

COLLINEAR DISTRIBUTION AND FRAGMENTATION FUNCTIONS

1g

1h

1f

( )xeven

odd

U

T

L

1G

1H

1D

( )xeven

odd

U

T

L

Including flavor index one commonly writes

1 ( ) ( )qg x q x

1 ( ) ( )qf x q x

1 ( ) ( )qh x q x

1gg

1gf

( )g xflip

U

T

L

For gluons one commonly writes:

1 ( ) ( )gg x g x

1 ( ) ( )gf x g x

Result for weighted cross section

[ ( )] [ ]1 2 ... 1

,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z

Construct weighted cross section (azimuthal asymmetry)

[ ] 2 [ ]( ) ( , )C CT T Tx d p p x p

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

APPLICATIONS

New info on hadrons (cf models/lattice) Allows T-odd structure (exp. signal: SSA)

Result for weighted cross section

[ ( )] [ ]1 2 ... 1

,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z

[ ( )] [ ]1 2 ... 1

,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z [ ( )] [ ]

1 2 ... 1,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z [ ( )] [ ]

1 2 ... 1,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

APPLICATIONS

Drell-Yan and photon-jet production

• Drell-Yan

• Photon-jet production in hadron-hadron scattering

1 2 † † †2 2

2

† †

2

[ ( )] [ ( )] [ , ]12 1

[ ] [ ] [ , ]11

ˆ[

ˆ ]

H H JJN N

q q qq g gN NT

q q qq g gN

d

d q

1 2 †[ ] [ ]2

ˆH H

q q qqT

d

d q

APPLICATIONS

qT = p1T + p2T

Drell-Yan and photon-jet production

• Weighted Drell-Yan

• Photon-jet production in hadron-hadron scattering

1 2 † † †2 2

2

† †

2

[ ( )] [ ( )] [ , ]12 1

[ ] [ ] [ , ]11

ˆ[

ˆ ]

H H JJN N

q q qq g gN NT

q q qq g gN

d

d q

†1 2 [ ] [ ]ˆ ˆH H

T q q qq q q qqq

APPLICATIONS

Drell-Yan and photon-jet production

• Weighted Drell-Yan

• Photon-jet production in hadron-hadron scattering

1 2 † † †2 2

2

† †

2

[ ( )] [ ( )] [ , ]12 1

[ ] [ ] [ , ]11

ˆ[

ˆ ]

H H JJN N

q q qq g gN NT

q q qq g gN

d

d q

1 2 [ ] ˆ ...H HT q q qqq

APPLICATIONS

Consider only p1T contribution

Drell-Yan and photon-jet production

• Weighted Drell-Yan

• Weighted photon-jet production in hadron-hadron scattering

2 21 2

2

2

[ ( )]11

[ ]11

ˆ[

ˆ ] ...

H H JJ N NT q q qq g gN N

q q qq g gN

q

1 2 [ ] ˆ ...H HT q q qqq

APPLICATIONS

Result for weighted cross section

[ ( )] [ ]1 2 ... 1

,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z [ ] [ [ ] [ ])] (( ) ( ) ( , )C C U C C

G Gx x C x x

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

universal matrix elements

G(x,x) is gluonic pole (x1 = 0) matrix element

T-even T-odd

(operator structure)

1( , )G x x x

G(p,pp1)

APPLICATIONS

Qiu, Sterman; Koike; …

1H

TMD DISTRIBUTION AND FRAGMENTATION FUNCTIONS

1g

1h

1f

( )xeven

odd

U

T

L

1Tg1Lh

( )xeven

odd

U

T

L

1Tf

1h

( , )G x x even

odd

U

T

L

Todd

Teven

1TD

1LH

( )x

even odd

U

T

L

1TG

1G

1H

1D

( )xeven

odd

U

T

L

( , )G x x even

odd

U

T

L

Gamberg, Mukherjee, Mulders, 2008; Metz 2009, …

Gluonic poles (Spectral analysis)

G(x,x-x1)G(x,x-x1)

G(x,x) = 0G(x,x)

LimX1 0

Gamberg, Mukherjee, M, 2008; Metz 2009, …

Result for weighted cross section

[ ( )] [ ]1 2 ... 1

,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z [ ] [ [ ] [ ])] (( ) ( ) ( , )C C U C C

G Gx x C x x

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

universal matrix elements

Examples are:

CG[U+] = 1, CG

[U] = -1, CG[U□ U+] = 3, CG

[Tr(U□)U+] = Nc

APPLICATIONS

Result for weighted cross section

[ ( )] [ ]1 2 ... 1

,

ˆ~ ( ) ( ) ( )... .....C D DT a b ab c c

D abc

q x x z [ ] [ [ ] [ ])] (( ) ( ) ( , )C C U C C

G Gx x C x x

1 2 ... 1

1 1 2 [ ] ... 1

ˆ~ ( ) ( ) ( )... .....

ˆ( , ) ( ) ( )... .....

T a b ab c cabc

G a b a b c cabc

q x x z

x x x z

( ([ ] [ ][ ] ... . .

).

)ˆ ˆU C D Da b c G ab c

D

C (gluonic pole cross section)

'1 2 1

1 1 2 2 1 1

[ ( )] [ ( )] [ ( )][ ]...2

,

( , ) ( , ) ( , )ˆ~ ...T T T

C D C D C DDa b ab c c

D abcT

x p x p z kd

d q

APPLICATIONS

Drell-Yan and photon-jet production

• Weighted Drell-Yan

• Weighted photon-jet production in hadron-hadron scattering

2 21 2

2

2

[ ( )]11

[ ]11

ˆ[

ˆ ] ...

H H JJ N NT q q qq g gN N

q q qq g gN

q

1 2 [ ] ˆ ...H HT q q qqq

[ ]q q G q

[ ( )]q q G q

[ ]q q G q

APPLICATIONS

Drell-Yan and photon-jet production

• Weighted Drell-Yan

• Weighted photon-jet production in hadron-hadron scattering

1 2

2

211

ˆ[

ˆ ] ...

H H JJT q q qq g g

NG q q qq g gN

q

1 2 ˆ ˆ ...H HT q q qq G q q qqq

[ ]ˆ q q

[ ]ˆ q q g Note: color-flow in this case is more SIDIS like than DY

APPLICATIONS

Gluonic pole cross sections

• For quark distributions one needs normal hard cross sections

• For T-odd PDF (such as transversely polarized quarks in unpolarized proton) one gets modified hard cross sections

• for DIS:

• for DY:

Gluonic pole cross sections

[ ]

[ ]

ˆ ˆ Dqq qq

D

[ ( )] [ ]

[ ][ ]

ˆ ˆU D Dq q qq G

D

C

[ ]ˆ ˆq q q q

[ ]ˆ ˆq q qq

y

(gluonic pole cross section)

[ ]ˆ q q qqd

ˆqq qqd

APPLICATIONS

Bomhof, M, JHEP 0702 (2007) 029 [hep-ph/0609206]

TMD-universality : qg qg

D1

D2

D3

D4

D5

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

e.g. relevant in Bomhof, M, Vogelsang, Yuan, PRD 75 (2007) 074019

UNIVERSALITY

Transverse momentum dependent weighted

TMD-universality: qg qg

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

UNIVERSALITY

Transverse momentum dependent weighted

TMD-universality : qg qg

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

†2

[( )( ) ] [ ]2 2

1

1 1

cq G

c c

N

N N

†4 2 2

[( )( ) ] [( ) ] [ ]2 2 2 2 2 2

11

( 1) ( 1) 1 1

c c cq G

c c c c

N N N

N N N N

UNIVERSALITY

Transverse momentum dependent weighted

†2

[( )( ) ] [ ]2 2

1

1 1

cq G

c c

N

N N

TMD-universality : qg qg

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

2 2[( ) ] [ ]

2 2 2

11

1 1 1

c cq G

c c c

N N

N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2 2

1 1

2( 1) 2( 1) 1 1

c cq G

c c c c

N N

N N N N

†4 2 2

[( )( ) ] [( ) ] [ ]2 2 2 2 2 2

11

( 1) ( 1) 1 1

c c cq G

c c c c

N N N

N N N N

†2 2

[( )( ) ] [( ) ] [ ]2 2 2

1

2( 1) 2( 1) 1

c c

c c c

N N

N N N

†2

[( )( ) ] [ ]2 2

1

1 1

cG

c c

N

N N

It is possible to group the TMD functions in a smart way into two particular TMD functions(nontrivial for eight diagrams/four color-flow possibilities)

We get process dependent and (instead of diagram-dependent)But still no factorization!

[ ][ ]( , )q g qgTx p

][ ][ ( , )

G

q g qgTx p

Bomhof, M, NPB 795 (2008) 409 [arXiv: 0709.1390]

UNIVERSALITY

Transverse momentum dependent weighted

We can work with basic TMD functions [±](x,pT) + ‘junk’The ‘junk’ constitutes process-dependent residual TMDs

Thus:

The junk gives zero after integrating ((x) = 0) and after weighting ((x) = 0), i.e. cancelling kT contributions; moreover it most likely also disappears for large pT

Universality for TMD correlators?

† †

†[( )( ) ]

[( )( ) ] [ ] [( )( ) ] [ ]

( , )

( , ) ( , ) ( , ) ( , )

T

T T T T

x p

x p x p x p x p

[ ] [ ] [ ] [ ]( , ) 2 ( , ) ( , ) ( , )T T T Tx p x p x p x p

[[ ] [ ] [ ] [ ]1 12 2

]( , ) ( , ) ( , ) ( , )UU even odd UT T T TGCx p x p x p x p

[ ] [ ([ ] [ ] [ ] [ ]1 [ ] )12 2

] [ ]q g qg U qeven oddg qg q g qgGC

Bomhof, M, NPB 795 (2008) 409 [arXiv: 0709.1390]

UNIVERSALITY

(Limited) universality for TMD functions

[ ] [ ] [ ]1 12 2( , )even

Tx p

[ ] [ ] [ ]1 12 2( , )odd

Tx p

( )x( )x

( , )G x x

[ ] [ , ] [ , ]1 12 2( , )even

Tx p

[ ] [ , ] [ , ]1 12 2( , )odd

F Tx p

[ ] [ , ] [ , ] [ ]1 12 2' ( , ) ( , ) ( , )even even

T T Tx p x p x p

[ ] [ , ] [ , ]1 12 2( , )odd

D Tx p

( )x

( , )FG x x

( )x

( , )DG x x

QU

AR

KS

GLU

ON

S

Bomhof, M, NPB 795 (2008) 409 [arXiv: 0709.1390]

UNIVERSALITY

TMD DISTRIBUTION FUNCTIONS

1Lg

1f

[ ]( , )evenTx p

even

odd

U

T

L

1Tg1Lh

1Tf

1h

1Th

1Th

oddeven

[ ]( , )oddTx p

N

@ twist 2

2

1 ( , )q

T Th x k

2

1 ( , )q

L Th x k2

1 ( , )q

Th x k

2

1 ( , )q

T Th x k

2

1 ( , )q

T Tf x k

2

1 ( , )q

Tf x k

2

1 ( , )q

L Tg x k

2

1 ( , )q

T Tg x k

Courtesy: Aram Kotzinian

Conclusions

Transverse momentum dependence, experimentally important for single spin asymmetries, theoretically challenging (consistency, gauges and gauge links, universality, factorization)For leading integrated and weighted functions factorization is possible, but it requires besides the normal ‘partonic cross sections’ use of ‘gluonic pole cross sections’ and it is important to realize that qT-effects generally come from all partonsIf one realizes that the hard partonic part including its colorflow is an experimental handle, one recovers universality in terms of the (possibly large) number of possible gauge links, which can be separated into junk and a limited number of TMDs that give nonzero integrated and/or weighted results.

CONCLUSIONS

Thanks to experimentalists (HERMES, COMPASS, JLAB, RHIC, JPARC, GSI, …) for their continuous support.