Transcript Transverse momentum in hard processes Piet Mulders

Yerevan - 2009
TPSH 2009
Yerevan, Armenia
July 22-26, 2009
Transverse momentum
in hard processes
Piet Mulders
[email protected]
OUTLINE
Outline
Introduction: partons in high energy scattering processes
Transverse momenta: collinear and non-collinear parton correlators
Distribution functions (collinear, TMD)
Observables: azimuthal asymmetries
Gauge links
Resumming multi-gluon interactions: Initial/final states
Color flow dependence
The master formula and some applications
Universality
Conclusions
PARTONS
From partons to OPE
• Hard scattering process involves quarks and
gluons. Link to external particles:
0  i( s ) ( ) p, s  ui ( p, s) eip.
• Confinement leads to hadrons as ‘sources’ for
quarks
X  i( s ) ( ) P e ip.
• and ‘source’ for quarks + gluons
X  i( s ) ( ) A ( ) P e i ( p  p1 ). ip1 .
• and ….
PARTON CORRELATORS
From partons to OPE
Thus, the theoretical description/calculation
for hard processes
_
involves [instead of ui(p)uj(p)] forward matrix elements of the form
d 3 PX
ij ( p, P)  
 P |  j (0) | X  X |  i (0) | P   ( P  PX  p)
3
(2 ) 2 EX
1
4
i p .

(
p
,
P
)

d

e
 P | j (0)  i ( ) | P 
quark ij
4 
(2 )
momentum
G(p,pp1)
P
P
 P | j (0) A ( ) i ( ) | P 
PARTON CORRELATORS
Probing quark momenta
Correlator depends on quark momentum p
ij ( p, P) 
1
4
i p .
d

e
 P | j (0)  i ( ) | P 
4 
(2 )
Actually, it is not the unintegrated correlator that is measurable.
Hard processes provide a direction (lightlike vector n, satisfying P.n = 1)
the vector n gets its meaning in a particular hard process
e.g. in SIDIS: n = Ph/P.Ph (or n determined by P and q)
Expand quark momentum
p  xP  pT  ( p.P  xM 2 )n
Integrate over p.P = p (which is of order M2)
( with P.n  1)
NON-COLLINEARITY
Integrating quark correlators
d ( .P)d 2T i p.
 ( x, pT ; n)  
e
P  j (0) i ( ) P
3
(2 )
q
ij
 qij ( x)  
d ( .P) i p.
e
P  j (0) i ( ) P
(2 )
 .n T  0
 .n  0
TMD
collinear
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
Jaffe (1984), Diehl & Gousset, …
NON-COLLINEARITY
Relevance of transverse momenta
In a hard process one probes quarks and gluons
p  x P  pT
Parton momenta fixed by kinematics (external momenta)
k  z 1 P  kT
DIS
x  p.n / P.n  Q 2 / 2 P.q  xB
SIDIS
z  K h .n / k.n  P.K h / P.q  zh
Also possible for transverse momenta of partons
f2  f1
K1
SIDIS
qT  q  xB P  zh1K h
 kT  pT
2-particle inclusive hadron-hadron scattering
qT  z11 K1  z21 K 2  x1 P1  x2 P2
f K
2
pp-scattering
Boer & Vogelsang
 p1T  p2T  k1T  k2T
We need more than one hadron and
knowledge of hard process(es)!
PARTON CORRELATORS
Twist expansion
Not all correlators are equally important
Matrix elements expressed in P, p, n allow twist expansion
(just as for local operators)
(maximize contractions with n to get dominant contributions)
dim[ (0)n  ( )]  2
a.n  a n  a 
dim[ F n (0) F n ( )]  2

dim[ (0) nD
 T  ( )]  3
Leading correlators involve ‘good fields’ in front form dynamics
   12        
F   gi [ D , DT ]  i  AT
Pasquini (yesterday), …
Above arguments are valid and useful for both collinear and TMD fcts
In principle any number of gA.n = gA+ can be included in correlators
NON-COLLINEARITY
TMD correlators: quarks
d ( .P)d 2T i p.
[C ]
 ( x, pT ; n)  
e
P

(0)
U
j
[0, ] i ( ) P
3
(2 )
d ( .P) i p.
q
[n]
 ij ( x; n)  
e
P  j (0)U[0,
 ] i ( ) P  .n T  0
(2 )
q[ C ]
ij
 .n  0
TMD
collinear
Gauge link essential for color gauge invariance

[C ]
U [0,
]


 P exp  ig  ds  A 
 0

Arises from all ‘leading’ matrix elements containing  A...A 
Basic (simplest) gauge links for TMD correlators:
[]
[]
T
NON-COLLINEARITY
TMD correlators: gluons
2
d
(

.
P
)
d
T i p.
[ C ,C ']
[C ]
n
[ C ']
n

(
x
,
p
;
n
)

e
P
U
F
(0)
U
F
( ) P
g
T
[ ,0]
[0, ]
 (2 )3
The most general TMD gluon correlator contains two links, possibly
with different paths.
Note that standard field displacement involves C = C’
F  ( )  U[[C,] ] F  ( )U[[C,] ]
Basic (simplest) gauge links for gluon TMD correlators:
g[,]
g[,]
g[,]
g[,]
 .n  0
GAUGE LINKS
Process dependence of gauge link
In order to determine gauge
link for (x1,p1T) one must
consider all collinear gluon
insertions
For a given hard subprocess,
only the insertions attached to
other ‘external partons’ matter
They give links to lightcone
±infinity for parton fields in
(x1,p1T) and the connecting
pieces for the full gauge links
U[0,] in the correlator
C. Bomhof, P.J. Mulders and F. Pijlman,
PLB 596 (2004) 277 [hep-ph/0406099];
EPJ C 47 (2006) 147 [hep-ph/0601171]
Two color-flow possibilities
Link structure
for fields in
correlator 1
GAUGE LINKS
Process dependence of gauge link
E.g. qq-scattering as hard
subprocess
The correlator (x1,p1T)
enters each contributing
term in squared amplitude
with specific link
[Tr (U )U  ]


[U U  ]
U  U U  †
( x, pT )
( x, pT )
APPLICATIONS
TMD master formula
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
Note that the summation over D is over
diagrams and color-flow,
e.g. for qq qq subprocess:
d
[(
~

q
d 2 qT
)]
(1) 
[( )  ]
q
2
(2)
Nc  1
2
Nc
1
1
) † ]
[ D1 ]
qq  qq
[ † ]
(1')
q
[D ]
[(
ˆ qq

 qq
q
(1') [(q
) † ]
(2 ')
 [ D11 ]
 
[ ]
q
(1) 
[ ]
q
(2)
2
2
Nc
1
ˆ
 [ D12 ]


[ † ]
(2 ')
q
 ....
EXAMPLE
quark + antiquark  gluon + photon
|M|2
Four diagrams,
each with two
similar color flow
possibilities
d

2
d qT
N 2 1 N 2
N
N 2 1
N 2 1 N 2
N
N 2 1
[

[

[  ( )]
q

[ † (
q
 N 211 ...
...
†
ˆ qq  g 
)]
Compare this with qqbar  *:
[  , † ]
g

1
2
N 1
[ ]
q
 
]
[ † ]
q
ˆ qq  g 
d
[  ] [ † ]
  q  q ˆ qq 
2
d qT
[  , † ]
g
]
EXAMPLE
* + quark  gluon + quark
|M|2
Four diagrams,
each with two
color flow
possibilities
d

2
d qT

N 2 1 N 2
N
N 2 1
[
N 2 1 N 2
N
N 2 1
[
 ˆ  q qg 
[]
q
 N 211 ...
...
[ † ]
q
Compare this with *q  q:

[  , † ]
g

1
N 2 1
 ˆ  q qg 
[]
q
]
[ † ]
q
d
[]
[ † ]
  q ˆ  q q  q
2
d qT

[  , † ]
g
]
APPLICATIONS
Result for integrated cross section
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
Integrate into collinear cross section
[C ]

( x)   d 2 pT [ C ] ( x, pT )
 ~   a ( x1 )b ( x2 ) ˆ abc... c ( z1 )...
abc
[ D]
ˆ abc...   ˆ ab
c ...
D
(partonic cross section)
PARAMETRIZATION
Collinear parametrizations
• Gauge invariant correlators  distribution functions
• Collinear quark correlators (leading part, no n-dependence)
 ( x) 
q
f
q
1
( x)  S g ( x)  5  h ( x)  5 ST 
q
L 1
q
1
P
2
S  SL
P
M
• i.e. massless fermions with momentum distribution f1q(x) = q(x), chiral
distribution g1q(x) = q(x) and transverse spin polarization h1q(x) =
q(x) in a spin ½ hadron
• Collinear gluon correlators (leading part)
1
 g
 g
 g ( x) 

g
f
(
x
)

i
S


T
1
L T g1 ( x ) 
2x

• i.e. massless gauge bosons with momentum distribution f1g(x) = g(x)
and polarized distribution g1g(x) = g(x)
 ST
COLLINEAR DISTRIBUTION AND FRAGMENTATION FUNCTIONS
( x)
c
even
U
f1
L
 g ( x)
odd
f1q ( x)  q ( x)
g1
T
g1q ( x)  q ( x)
h1
( x)
c
even
odd
U D1
L G1
T
Including flavor index
one commonly writes
H1
h1q ( x)   q( x)
flip
U
f1 g
L
g1g
T
For gluons one
commonly writes:
f1g ( x)  g ( x)
g1g ( x)  g ( x)
APPLICATIONS
Result for weighted cross section
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
Construct weighted cross section (azimuthal asymmetry)
 [C ] ( x)   d 2 pT pT [C ] ( x, pT )
qT  ~
 [ C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
ab c...  c ( z1 )...  .....
D , abc
 New info on hadrons (cf models/lattice)
 Allows T-odd structure (exp. signal: SSA)
APPLICATIONS
Result for weighted cross section
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
qT 
qT 
qT 
qT 
[ D]
~   [aC ( D )] ( x1 )b ( x2 ) ˆ ab
c...  c ( z1 )...  .....
[ D]
, abc
~ D
 [aC ( D )] ( x1 )b ( x2 ) ˆ ab
c...  c ( z1 )...  .....
[ D]
, abc
~ D
 [aC ( D )] ( x1 )b ( x2 ) ˆ ab
c...  c ( z1 )...  .....
[ D]
, abc
~ D
 [aC ( D )] ( x1 )b ( x2 ) ˆ ab
c...  c ( z1 )...  .....
D , abc
APPLICATIONS
Drell-Yan and photon-jet production
qT = p1T + p2T
• Drell-Yan
d H1H 2 
[  ] [ † ]
  q  q ˆ qq 
2
d qT
• Photon-jet production in hadron-hadron scattering (including only
one of the contributions at the parton level, omitting qgq)
d H1H 2  JJ

2
d qT
N 2 1 N 2
N
N 2 1
[

1
2
N 1

[ † (
q
[  ( )]
q

[]
q
[ † ]
q
 
†
ˆ qq  g 
)]
ˆ qq  g 
[  , † ]
g
[  , † ]
g
]
APPLICATIONS
Drell-Yan and photon-jet production
• Weighted Drell-Yan
qT 

H1H 2 
 [ † ]
  q qˆ qq   q  q ˆ qq 
 [ ]
• Photon-jet production in hadron-hadron scattering (including only
one of the contributions at the parton level, omitting qgq)
d H1H 2  JJ

2
d qT
N 2 1 N 2
N
N 2 1
[

1
2
N 1

[ † (
q
[  ( )]
q

[]
q
[ † ]
q
 
†
ˆ qq  g 
)]
ˆ qq  g 
[  , † ]
g
[  , † ]
g
]
APPLICATIONS
Drell-Yan and photon-jet production
• Weighted Drell-Yan (contribution of first parton only)
qT  H1H 2    [q ] q ˆ qq   ...
• Photon-jet production in hadron-hadron scattering (including only
one of the contributions at the parton level, omitting qgq)
d H1H 2  JJ

2
d qT
N 2 1 N 2
N
N 2 1
[

1
2
N 1

[ † (
q
[  ( )]
q

[]
q
[ † ]
q
 
†
ˆ qq  g 
)]
ˆ qq  g 
[  , † ]
g
[  , † ]
g
]
APPLICATIONS
Drell-Yan and photon-jet production
• Weighted Drell-Yan (contribution of first parton only)
qT  H1H 2    [q ] qˆ qq   ...
• Weighted Photon-jet production in hadron-hadron scattering
(looking at contribution of first parton only and including only one of
the contributions at the parton level, omitting qgq)
qT 

H1H 2  JJ

N 2 1 N 2
N
N 2 1
[
 [  ( )]
 q
 qˆ qq  g  g
 N 211  [q ] qˆ qq  g  g ]  ...
APPLICATIONS
Result for weighted cross section
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
qT  ~
 [ C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc...  c ( z1 )...  .....
D , abc
C]
C]
 [C ] ( x)   [
( x)  CG[U (C )] G[
( x, x )
T-even
universal matrix T-odd
elements
(operator structure)
G(p,pp1)

 G ( x, x  x1 )
G(x,x) is gluonic pole
matrix element
TMD DISTRIBUTION AND FRAGMENTATION FUNCTIONS
( x)
c
even
U
f1
U
g1
L
L
T
odd
h1
c
T
( x)
c
even
  ( x)
  G ( x, x )
even
c
odd

1L
c
L
h
g1T
T
even
odd
c
U
L
U D1
U
L G1
L
H1L
H1
T G1T D1T
T
even
T
odd
f1T
 G ( x, x)
H 1
T
odd
h1
U
  ( x)
odd
even
even
odd
T
Gamberg, Mukherjee, Mulders, 2008; Metz 2009, …
APPLICATIONS
Result for weighted cross section
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
qT  ~
 [ C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc...  c ( z1 )...  .....
D , abc
C]
C]
 [C ] ( x)   [
( x)  CG[U (C )] G[
( x, x )
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
d
[ C1 ( D )]
[ C2 ( D )]
[ C1' ( D )]
[D]
~  a
( x1 , p1T )  b
( x2 , p2T ) ˆ abc...  c
( z1 , k1T )...
2
d qT D ,abc
qT  ~
 [ C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc...  c ( z1 )...  .....
D , abc
C]
C]
 [C ] ( x)   [
( x)  CG[U (C )] G[
( x, x )
qT  ~  a ( x1 )b ( x2 ) ˆ abc...  c ( z1 )...  .....
abc
   G a ( x1 , x1 )b ( x2 ) ˆ[ a ]bc...  c ( z1 )...  .....
abc
[ D]
ˆ[ a ]bc...   CG[U (C ( D ))]ˆ ab
c...
D
(gluonic pole
cross section)
APPLICATIONS
Drell-Yan and photon-jet production
 [q]   q   G q
• Weighted Drell-Yan
qT  H1H 2    [q ] qˆ qq   ...
• Weighted Photon-jet production in hadron-hadron scattering
(looking at only one of the contributions at the parton level,
omitting qgq)
 [  ( )]

 q
qT 

H1H 2  JJ

N 2 1 N 2
N
N 2 1
[
  q   G q
 [q ( )] qˆ qq  g  g
 N 211  [q ] qˆ qq  g  g ]  ...
 [q]   q   G q
APPLICATIONS
Drell-Yan and photon-jet production
ˆ[ q ]q 
• Weighted Drell-Yan
qT  H1H 2    q  qˆ qq    G q  q  ˆ qq    ...
• Weighted Photon-jet production in hadron-hadron scattering
(looking at only one of the contributions at the parton level,
omitting qgq)
qT  H1H 2  JJ  [ q  qˆ qq  g  g
  G q  q


N 2 1
N 2 1
ˆ qq  g   g ]  ...
Note: color-flow in this case
is more SIDIS like than DY
ˆ[ q ]q  g
APPLICATIONS
example: photon-jet production in pp
weighted
Transverse momentum dependent
D1
D2
q 
2
Nc
2
Nc  1

[( )  ]

1
2
Nc  1

[]
ˆ[ q ] g q   NN 2 11 ˆ[ q ] g q
2
D3
Here colorflow is more
DY-like than SIDIS
D4
ˆ[ q ]q  g  
Bacchetta, Bomhof, D’Alesio, M, Murgia
PRL 99 (2007) 212002
N 2 1
N 2 1
ˆ[ q ]q  g
  
2
Nc
2
1
Nc  1
 G
GAUGE LINKS
example: qq
Tr (U )
qq in pp

U
U U
Nc
 q ( x, pT ) 
2
Nc  1
2
Nc  1
[(
)]

2
2
Nc  1
[
]
color flow factors (summed to 1)
 q ( x, pT ) 
2
2 Nc
2
Nc  1

[( )  ]

2
Nc  1
2
Nc  1
[
]
Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005) 034030 [hep-ph/0505268]
APPLICATIONS
example: qq
Tr (U )

U
U U
Nc
D1
q 
2
Nc  1
2
Nc
1

[( )  ]
qq in pp

2
2
Nc
[ ]
1

  
D4
2
Nc
1
 G
CG [D1] = CG [D2]
D2
D3
2
Nc  5
q 
2
2 Nc
2
Nc
1

[( )  ]

2
Nc  1
2
Nc
1
[ ]

  
2
Nc  3
2
Nc
1
 G
CG [D3] = CG [D4]
Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005) 034030; hep-ph/0505268
APPLICATIONS
Gluonic pole cross sections
• For quark distributions one needs
normal hard cross sections
ˆ qq qq   ˆ [ D ]
• For T-odd PDF (such as transversely
polarized quarks in unpolarized proton)
one gets modified hard cross sections
ˆ[ q ]q qq   CG[U ( D )]ˆ [ D ]
[D]
[D]
(gluonic pole cross section)
dˆ qq qq
• for DIS:
ˆ [ q ] q   ˆ q  q
• for DY:
ˆ[ q ]q    ˆ qq 
dˆ[ q ]q qq
y
Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]
UNIVERSALITY
example: qg
qg

weighted
Transverse momentum dependent
D1
q 
2
Nc
2
Nc  1

[( )  ]

1
2
Nc  1

[]
  
2
Nc
2
1
Nc  1
 G
D2
D3
D4
D5
e.g. relevant in Bomhof, M, Vogelsang, Yuan, PRD 75 (2007) 074019
UNIVERSALITY
example: qg
qg

weighted
Transverse momentum dependent
q 
2
Nc
2
Nc  1

[( )  ]

1
2
Nc  1

[]
  
2
Nc
2
1
Nc  1
 G
UNIVERSALITY
example: qg
qg

weighted
Transverse momentum dependent
q 
 q 
q 
2
q

NNc 2
c
2
2( N c  1)
2
2( N c  1)
4
Nc
2
( Nc
 1)

[(
[( )(
)( † )  ]

2
†

[( )(
†
) N]c2
2

2( N c  1)
)]
q 

2
Nc
2
Nc  1
2
Nc
[( )  ]
2
2
[(


[]
c 1
2( N c N1)
2
Nc
2
( Nc
2
Nc
2
Nc
1

[( )  ]
1
 1)
[(
) ]

2
†
)]


2
Nc  1
1
1
[]
[  ]
   2 2   G
N 1
[( )  ]
)(

1


N c c 1
1
2
Nc
[ ]
1
1
2
Nc
1

[  ]
  
  
  

1
2
Nc
2
Nc  1
1
2
Nc  1
Nc
2
1
2
Nc
1
 G
 G
 G
   G
UNIVERSALITY
example: qg
qg

weighted
Transverse momentum dependent
It is possible to group the
2 TMD
2
Nc
Nc  1
1
[( )  ]
[]
 q way



   2  G
functions in a smart
2 into two 2
Nc  1
Nc  1
Nc  1
particular TMD functions
(nontrivial for eight diagrams/four
2
2
color-flow
possibilities)
NNc
Nc 1
1 1
1
[( )( ) N]
[( )  ]
[]
 q   2     2 

   2  G

  2

q
 
[( )(
2
2( N
Nc  1)
c
2
2( N c  1)
 G 
q 
2
c
2
2( N c  1)
2
N4c
N2c

†
)]
[( )(
[(

N
2 c 1
2
( N c  1)
†
[( )(
)(
†
2
c
2
2( N c  1)
)]
†
†

)]
)]
[( )  ]
[]

2
2( NNc 2  1)
Nc  1
c
2
2( N c  1)

12
N
2c

[ [(] ) ]
 N 2c  1 2 
( N c  1)
[( )  ]


G
2
N c 1Nc 1 1
2
Nc
1
1
2
Nc

[ ]
1
Nc  1
[ ]

  
We get process dependent [[ q ] g qg ] ( x, p )

T
[
[
q
]
g

qg
]
and  
( x,Np2T )
G
1
[( )( † )  ]
[ ]
c






(instead of diagram-dependent)
q
2
2
Nc  1
Nc  1
But still no factorization!
Nc
2
1
2
Nc
1
 G
   G
Bomhof, Mulders, NPB 795 (2008) 409 [arXiv: 0709.1390]
UNIVERSALITY
Universality for TMD correlators?
We can work with basic TMD functions [±](x,pT) + ‘junk’
The ‘junk’ constitutes process-dependent residual TMDs

[( )(
†
)]
( x, pT )   ( x, pT )  

[]
[( )(
†
)]
( x, pT )  [  ] ( x, pT ) 

 [(
[
]
( x, pT )  2[  ] ( x, pT )  [  ] ( x, pT )   [
]
)( † )  ]
( x , pT )
( x, pT )
[U ] ( x, pT )  12 [ even ] ( x, pT )  12 CG[U ] [ odd ] ( x, pT )   [U ] ( x, pT )
Thus: [[ q ] g qg ]  12 [ even ]  12 CG[U ([ q ] g qg )] [ odd ]   [[ q ] g qg ]
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
Bomhof, Mulders, NPB 795 (2008) 409 [arXiv: 0709.1390]
UNIVERSALITY
QUARKS
(Limited) universality for TMD functions
[ even ] ( x, pT )  12 [  ]  12 [  ]
  ( x)
  G ( x, x )
[ odd ] ( x, pT )  12 [  ]  12 [  ]
[ even ] ( x, pT )  12 [  , ]  12 [ ,  ]
GLUONS
( x)
( x)
  ( x)
[Fodd ] ( x, pT )  12 [  , ]  12 [ , ]
GF ( x, x)
[Dodd ] ( x, pT )  12 [  , ]  12 [ , ]
GD ( x, x)
 '[even] ( x, pT )  12 [ ,]  12 [ ,]  [ even] ( x, pT )   ( x, pT )
Bomhof, Mulders, NPB 795 (2008) 409 [arXiv: 0709.1390]
TMD DISTRIBUTION FUNCTIONS
[ odd ] ( x, pT )
[ even ] ( x, pT )
c
even
U
f1
L
g1L
T
odd
even
odd
h1

1L
h

1T
g1T h1T h
f1T
CONCLUSIONS
Conclusions
Transverse momentum dependence, experimentally
important for single spin asymmetries, theoretically
challenging (consistency, gauges, 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
partons
For TMDs there is no simple universality, but if 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, of which
most can be considered ‘junk’.