Universality of transverse momentum dependent correlation
functionsPiet Mulders
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.