tmd universality

TMD Universality

Piet Mulders

[email protected]

Workshop on Opportunities for Drell Yan at RHICBNL, May 11, 2011

TMD UniversalityP.J. Mulders

Nikhef and VU University, Amsterdam

The basic idea of PDFs is achieving a factorized description with soft and hard parts, soft parts being portable and hard parts being calculable. In the leading contributions at high energies, the PDFs can be interpreted as probabilities. Beyond the collinear treatment one considers not only the dependence on partonic momentum fractions x, but also the dependence on the transverse momentum pT of the partons. Experimentally, transverse momentum dependent functions (TMDs) provide a rich phenomenology of azimuthal asymmetries for produced hadrons or jet-jet asymmetries. Furthermore inclusion of transverse momentum dependence provides an explanation for single spin asymmetries. An important issue is the universality of TMDs, which we study for some characteristic hard processes, where we focus on the pecularities coming from the color flow in the hard part. This color flow in the hard process gives rises to a variety of Wilson lines in the description of the cross section. These give rise to color entanglement, in particular in situations that the color flow is not just a simple transfer of color from initial or final state.

We argue that these Wilson lines can be combined into the appropriate gauge links for TMD correlators in cases where only the transverse momentum of partons in a single (incoming) hadron is relevant (1-parton un-integrated or 1PU processes). Such a situation occurs in single weighted cross sections, which consists of a sum of 1PU processes or if absence of any polarization makes all explicit transverse momentum effects vanish. For 1PU processes one finds TMDs with a complex gauge link structure depending on the color flow of the hard process. In the case of single weighted cross sections the results are the gluonic pole or Qiu-Sterman matrix elements appearing with calculable color factors.

WORKSHOP ON OPPORTUNITIES FOR DY AT RHIC

MAY 2011, BNL

Introduction

• Isolating hard process (factorization)– Study of quark and gluon structure of hadrons– Account for hadronic physics to study hard process

• Beyond collinear approach– Include mismatch of parton momentum p and xP

(fraction of hadron momentum)– TMDs with novel features

• Operator structure of TMDs– Color gauge invariance as guiding principle– Appearance of TMDs in hard processes– Gauge links in 1-particle un-integrated (1PU) processes

INTRODUCTION

Hard part: 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

External particles:

__

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

HARD PROCESS

( , ) ( , ) ( )i j iju p s u p s p m

Soft part: hadronic matrix elements

• For hard scattering process involving electrons and photons the link to external particles is, indeed, the ‘one-particle wave function’

• Confinement, however, leads to hadrons as ‘sources’ for quarks

• … and ‘source’ for quarks + gluons

• … and ….

.( )0 , ( , ) ipii p s u p s e

.( ) ii

pX P e

1 1( ). .( ) ( ) i pi

p ipAX P e

INTRODUCTION

Thus, the nonperturbative input for calculating hard processes involves [instead of ui(p)uj(p)] forward matrix elements of the form

Soft part: hadronic matrix elements

3

3( , ) | | | | ( )

(2) 0)

)(

2(0j i

Xij X

X

d Pp P P X X P P P p

E

4 .4

1( , ) | |

(2( (

)0) )i p

i ij jp P d e P P

quarkmomentum

INTRODUCTION

_

(0) ( )| |( )j iP PA

PDFs and PFFs

Basic idea of PDFs is to get a full factorized description of high energy scattering processes

21 2| ( , ,...) |H p p

1 2 1 1 1 2 2

, ... 1 2 1 1

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

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

a b

ab c c

P P dp p P p P

p p k K

calculable

defined (!) &portable

INTRODUCTION

Give a meaning to integration variables!

Example: Drell-Yan process

1 1

1 1 1 1

( , ) ( , )

( , ) ~ ( ) ( )su p s u p s

p P p m f p

• High energy limits number of soft matrix elements that contribute (twist expansion).

• Expand parton momenta (for DY take e.g. n= P2/P1.P2 )

• For meaningful separation of hard and soft, integrate over p.P and look at (x,pT). This shows that separation fails beyond ‘twist 3’.

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

. ~ 1x p p n

~ Q ~ M ~ M2/Q

Ralston and Soper 79, …

(NON-)COLLINEARITY

Rather than considering general correlator (p,P,…), one integrates over p.P = p (~MR

2, which is of order M2)

and/or pT (which is of order 1)

The integration over p = p.P makes time-ordering automatic. This works for (x) and (x,pT)

This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes (untruncated!), which satisfy particular analyticity and support properties, etc.

Integrated quark correlators: collinear and

TMD

2.

3 . 0(0) ( )

( . )( , ; )

(2 )q i pT

iij njT

d P dx p n e P P

.

. 0

( . )( (0); )

(2 )( )ij

T

q i pj i n

d Px n e P P

TMD

collinear

(NON-)COLLINEARITY

Jaffe (1984), Diehl & Gousset (1998), …

lightfront

lightcone

Relevance of transverse momenta?

At high energies fractional parton momenta fixed by kinematics (external momenta)

DY

Also possible for transverse momenta of partons

DY

2-particle inclusive hadron-hadron scattering

K2

K1

pp-scattering

1 11 1Tp Px p

2 22 2Tp Px p 1 2 21 1

1 2 1 2

. ..

. .

p P q Px p n

P P P P

1 1 2 2 1 2T T Tq q x P x P p 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

Care is needed: we need more than one hadron and knowledge of hard process(es)!

Boer & Vogelsang

Seco

nd s

cale

!

Oppertunities of TMDs

TMD quark correlators (leading part, unpolarized) including T-odd part

Interpretation: quark momentum distribution f1q(x,pT) and its

transverse spin polarization h1q(x,pT) both in an unpolarized

hadronThe function h1

q(x,pT) is T-odd (momentum-spin correlations!)

TMD gluon correlators (leading part, unpolarized)

Interpretation: gluon momentum distribution f1g(x,pT) and its

linear polarization h1g(x,pT) in an unpolarized hadron (both are

T-even)

122 2

1 12

1( , ) ( , )( , )

2

vT T T

g Tg g

T TT

p p gx f x p h x pp g

x M

2 21

]1

[ ( ,( , ) ( , ))2

q q TqT T T

p Px p i

Mf x p h x p

(NON-)COLLINEARITY

Twist expansion of (non-local) correlators

Dimensional analysis to determine importance of matrix elements(just as for local operators)maximize contractions with n to get leading contributions

‘Good’ fermion fields and ‘transverse’ gauge fieldsand in addition any number of n.A() = An(x) fields (dimension zero!)but in color gauge invariant combinations

Transverse momentum involves ‘twist 3’.

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

n n n ni iD i gA

T T T Ti iD i gA dim 0:

dim 1:

OPERATOR STRUCTURE

Matrix elements containing (gluon) fields produce gauge link

… essential for color gauge invariant definition

Soft parts: gauge invariant definitions

4[ ] . [ ]

[0, ]4( ; )

(2( )

)(0)C i p C

ij j i

dp P e P U P

[ ][0, ]

0

expCig ds AU

P

OPERATOR STRUCTURE

++ …

Any path yields a (different) definition

Gauge link results from leading gluons

Expand gluon fields and reshuffle a bit:

1 1 1 1 1 11

1( ) . ( ) ( ) ... ( ) ( ) ...

. .n n

T T

PA p n A p iA p A p p iG p

n P p n

OPERATOR STRUCTURE

Coupling only to final state partons, the collinear gluons add up to a U+ gauge link, (with

transverse connection from AT

Gn

reshuffling)

A.V. Belitsky, X.Ji, F. Yuan, NPB 656 (2003) 165D. Boer, PJM, F. Pijlman, NPB 667 (2003) 201

Even simplest links for TMD correlators non-trivial:

Gauge-invariant definition of TMDs: which gauge links?

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

[] []T

TMD

collinear

OPERATOR STRUCTURE

These merge into a ‘simple’ Wilson line in collinear (pT-integrated) case

Featuring: phases in gauge theories

'

( ) ( ')x

xig ds A

i iP Pe Px x

.

'ie ds A

Pe COLOR ENTANGLEMENT

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, which in general can have 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

[]

C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147F Dominguez, B-W Xiao, F Yuan, PRL 106 (2011) 022301

OPERATOR STRUCTURE

Gauge invariance for DY

COLOR ENTANGLEMENT

1[ , ]

2 2( ) (0 )

2[ , ] 2[ ,0 ]

1[0 , ]

1 1( ) (0 )

1 1 2 2

1 1 2 2

[0 , ] [ , ] [ , ] [ ,0 ]

[ ] [ ]† [ ] [ ]†[0 , ] [0 , ] 1 2[ ] [ ]n n n n

U U U U

W W W p W p

2 1[ ] [ ]† *11 1 1 2 2 2

[ ] [ †]1 1 2 2

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

ˆ( , ) ( , )

c

p pDY c q T c q T N

q T q T qq

d Tr W p x p Tr x p W p

x p x p

Strategy:transverse moments

Employing simple color flow possibilities, e.g. in gg J. Qiu, M. Schlegel, W. Vogelsang, ArXiv 1103.3861 (hep-ph)

Complications (example: qq

qq)

[ ](11) [ ](1) [ ](1)1( , ')... ... ( )... ( ') ( ), ( ') ... ... ( )... ( ')

2k k kU p p p p U p U p p p

T.C. Rogers, PJM, PR D81 (2010) 094006

U+[n] [p1,p2,k1]

modifies color flow, spoiling universality(and factorization)

COLOR ENTANGLEMENT

Color entanglement

[ ](21) [ ](2) [ ](1)

[ ](1) [ ](1) [ ](1)

[ ](1) [ ](2)

1( , ') ( ) ( ')

41

( ) ( ') ( )41

( ') ( )4

k k k

k k k

k k

U p p U p U p

U p U p U p

U p U p

COLOR ENTANGLEMENT

Color disentanglement for 1PU

COLOR ENTANGLEMENT

2 1 2 1 2 2 1 1 1 1

[ ][ ,0 ] [0 , ][0 , ] [ , ][ , ] [ , ] [0 , ] [ , ] [0 , ]

nU U U U U U W 2 1 2 1 1 1 2 2 1 1 1 1

[ ] [ ] [ ]† [ ] [ ][0 , ][0 , ] [ , ][ , ] [ , ] [ ,0 ] [0 , ] [0 , ] [0 , ] 2 1[ ] [ ]n n n n nU U U U W W W W p W p

[( ) ] *1 1 1

*2 2

~ [ ( , ) ( ) ]

[ ( ) ( ) ]

c T b a

b ac

Tr x p z

Tr x z

M.G.A. Buffing, PJM; in preparation

Collinear treatment for all-but-one parton (p1):

1 1( ) (0 )

1[ ,0 ]

2[ , ] 2[ ,0 ]

1[ , ]

1 2[ , ][ , ]

2 2( ) (0 )

1 2[0 , ][0 , ] 2 20

t T

1-parton unintegrated

• Resummation of all phases spoils universality • Transverse moments (pT-weighting) feels

entanglement• Special situations for only one transverse

momentum, as in single weighted asymmetries

• But: it does produces ‘complex’ gauge links• Applications of 1PU is looking for gluon h1

g (linear gluon polarization) using jet or heavy quark production in ep scattering (e.g. EIC), D. Boer, S.J. Brodsky, PJM, C. Pisano, PRL 106 (2011) 132001

2 2 2 21 2 1 2

2 2 2 21 1 2 1 2 2

... ... ( )

... ...

T T T T T T T

T T T T T T

d q q d p d p q p p

d p p d p d p d p p

COLOR ENTANGLEMENT

Full color disentanglement? NO!

COLOR ENTANGLEMENT

2 1 2 1 1 1 2 2 1 1 1 1

[ ] [ ] [ ]†[0 , ][0 , ] [ , ][ , ] [ , ] [ ,0 ] [0 , ] [0 , ] [0 , ]

[ ] [ ]2 1[ ] [ ]

n n n

n n

U U U U W W W

W p W p

1 1( ) (0 )

1[ ,0 ]

2[ , ] 2[ ,0 ]

1[ , ]

1 2[ , ][ , ]

†

†

[( ) ] * [( ) ]1 1

[( ) ] * [( ) ]2 2

~ [ ( ) ( ) ]

[ ( ) ( ) ]

c b a

b ac

Tr p k

Tr p k

2 2( ) (0 )

Loop 1:

X

a

b*

b*

a

M.G.A. Buffing, PJM; in preparation

1 2[0 , ][0 , ]

Result for integrated cross section

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

x x z

Collinear cross section

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

[ ]... ...ˆ ˆ Dab c ab c

D

(partonic cross section)

APPLICATIONS

Gauge link structure becomes irrelevant!

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

Result for single weighted cross section

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

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z

Single weighted cross section (azimuthal asymmetry)

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

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)


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

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z

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

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ( )] [ ]

1 1 2 ... 1,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z [ ( )] [ ]

1 1 2 ... 1,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

p x x z

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)


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

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

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

G Gx x C x x

universal matrix elements

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

T-even T-odd

(operator structure)

1( , )G x x x APPLICATIONS

Qiu, Sterman; Koike; Brodsky, Hwang, Schmidt, …

G(p,pp1)

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)


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

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

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

G Gx x C x x

universal matrix elements

Examples are:

CG[U+] = 1, CG

[U] = -1, CG[W U+] = 3, CG

[Tr(W)U+] = Nc

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)


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

,

ˆ~ ( ) ( ) ( )...C D DT a b ab c c

D abc

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

G Gx x C x x

1 1 2 ... 1

1 1 2 [ ] ... 1

ˆ~ ( ) ( ) ( )...

ˆ( , ) ( ) ( )...

T a b ab c cabc

G a b a b c cabc

p x x z

x x x z

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

).

)ˆ ˆU C D Da b c G ab c

D

C (gluonic pole cross section)

APPLICATIONS

11 1 2 1

[ ( )] [ ]...2

,1

( , ) ( ) ( )...ˆ~ T

C D Da b ab c c

D abcT

x p x zd

d p

(1PU)

T-odd part

Higher pT moments

• Higher transverse moments

• involve yet more functions

• Important application: there are no complications for fragmentation, since the ‘extra’ functions G, GG, … vanish. using the link to ‘amplitudes’; L. Gamberg, A. Mukherjee, PJM, PRD 83 (2011) 071503 (R)

• In general, by looking at higher transverse moments at tree-level, one concludes that transverse momentum effects from different initial state hadrons cannot simply factorize.

( ), ( , ), ( , , )G GGx x x x x x

SUMMARY

1 1 1[ ] ... 2( ) ( ... ) ( , )NNT T T Tx d p p p traces x p

Conclusions

• Color gauge invariance produces a jungle of Wilson lines attached to all parton legs, although the gauge connections themselves have a nicely symmetrized form

• Easy cases are collinear and 1-parton un-integrated (1PU) processes, with in the latter case for the TMD a complex gauge link, depending on the color flow in the tree-level hard process

• Example of 1PU processes are the terms in the sum of contributions to single weighted cross sections

• Single weighted cross sections involve T-even ‘normal weighting’ and T-odd gluonic pole matrix elements (SSA’s)

• Gluonic pole matrix elements in fragmentation correlators vanish, thus treatment of fragmentation TMDs is universal (physical picture: observation of jet direction)

• Furthermore, there is the issue of factorization! (next talk)SUMMARY

Thank you

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tmd universality

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