Transcript Template

International School, Orsay
June 2012
Transverse momenta of partons in
high-energy scattering processes
Piet Mulders
1
[email protected]
Introduction
• What are we after?
– Structure of proton (quarks and gluons)
– Use of proton as a tool
• What are our means?
– QCD as part of the Standard Model
2
Valence structure of hadrons:
global properties of nucleons
•
•
•
•
•
•
mass
charge
spin
magnetic moment
isospin, strangeness
baryon number






Quarks as
constituents
Mp  Mn  940 MeV
Qp = 1, Qn = 0
s=½
gp  5.59, gn  -3.83
I = ½: (p,n) S = 0
B=1
u
u
d
proton
3
3 colors
A real look at the proton
g + N  ….
Nucleon excitation spectrum
E ~ 1/R ~ 200 MeV
R ~ 1 fm
4
A (weak) look at the nucleon
n

p + e- + n
 = 900 s
 Axial charge
GA(0) = 1.26
5
A virtual look at the proton
_
g*  N N
g* + N  N
6
Local – forward and off-forward m.e.
Local operators (coordinate space densities):
 P ' | O( x) | P   e
i . x


G1 (t ) - i  G2 (t ) 
P
t  2
P’
Form factors
Static properties:
G1 (0)   P | O( x) | P 
G2 (0)   P | x O( x) | P 
Examples:
(axial) charge
mass
spin
magnetic moment
angular momentum
7
Nucleon densities from virtual look
Gi (t )  i ( x)
neutron
proton
•
•
•
•
charge density  0
u more central than d?
role of antiquarks?
n = n0 + pp- + … ?
8
Quark and gluon operators
Given the QCD framework, the operators are known quarkic or
gluonic currents such as
(axial) vector currents
Vq ( x)  q ( x)g  q( x)
Aq ' q ( x)  q ( x)g  g 5 q '( x)
probed in specific combinations
by photons, Z- or W-bosons
J (g )  23 Vu - 13 Vd - 13 Vs  ...
J ( Z ) 
1
2
V
u

- Au  - 34 sin 2 W Vu  ...
J (W )  Vud ' - Aud '  ...
energy-momentum currents
Tnq ( x) ~ q ( x)g { Dn } q( x)
TnG ( x) ~ G ( x)Gn ( x)
probed by gravitons
9
Towards the quarks themselves
• The current provides the densities but
only in specific combinations, e.g. quarks
minus antiquarks and only flavor
weighted
• No information about their correlations,
(effectively) pions, or …
• Can we go beyond these global
observables (which correspond to local
operators)?
• Yes, in high energy (semi-)inclusive
measurements we will have access to
non-local operators!
• LQCD (quarks, gluons) known!
10
Non-local probing
Nonlocal forward operators (correlators):




 P | O x - 2y , x  2y | P   P | O - 2y ,  2y | P 
Specifically useful: ‘squares’
Selectivity
at high
energies:
q=p

O x - 2y , x 
y
2




  † x - 2y ... x 
y
2

Momentum space densities of -ons:
 dy e
ip. y
    | P  
 P| †
y
2
y
2
 P - p |   0  | P   f ( p)
2
11
A hard look at the proton

Hard virtual momenta ( q2 = Q2 ~ many GeV2) can couple to
(two) soft momenta
g* + N  jet
g*  jet + jet
12
Experiments!
13
QCD & Standard Model
•
•
•
QCD framework (including electroweak theory)
_ provides the
_ machinery
to calculate cross sections, e.g. g*q  q, qq  g*, g*  qq, qq  qq,
qg  qg, etc.
E.g.
qg  qg
Calculations work for plane waves
0  i( s ) ( ) p, s  ui ( p, s) e -ip.
14
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’
0  i ( ) p, s  ui ( p, s) e-ip.
• Confinement, however, implies hadrons as
‘sources’ for quarks
X  i ( ) P e ip.
• … and also as ‘source’ for quarks + gluons
X  i ( ) A ( ) P ei ( p - p1 ). ip1 .
• … and also ….
15
PARTON CORRELATORS
Soft part: hadronic matrix elements
Thus, the nonperturbative
_ input for calculating 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
(2p ) 2EX
quark ij ( p, P) 
momentum
1
4
i p.
d

e
 P | j (0)  i ( ) | P 
4 
(2p )
 P | j (0) A () i ( ) | P 
16
INTRODUCTION
PDFs and PFFs
Basic idea of PDFs is to get a full factorized description of high
energy scattering processes
  | H ( p1, p2 ,...) |2
calculable
defined (!)
&
portable
 ( P1 , P2 ,...)   ...dp1... a ( p1 , P1;  )  b ( p2 , P2 ;  )
Give a meaning to
integration variables!
  ab,c... ( p1 , p2 ,...;  )   c (k1 , K1;  )....
17
INTRODUCTION
Example: DIS
18
Principle for DIS
q
p
 u ( p, s ) u ( p, s )
s
 ( p, P) ~ ( p  m) f ( p)
P
• Instead of partons use correlators
• Expand parton momenta (for SIDIS take e.g. n= Ph/Ph.P)
p  x P  pT   n
~Q
~M
~ M2/Q
 2
Px2pM
 p.n ~ 1
  p.P - xM 2 ~ M 2
19
(calculation of) cross section in DIS
Full calculation
+
+
(x)
LEADING (in 1/Q)
x = xB = -q2/P.q
+
+…
20
Lightcone dominance in DIS
Result for DIS
q
p
P
n
n
2MW ( P, q)  - gT
1
2
 dx dp.P d
2
pT Tr[( p, P)g ] ( x - xB )

 - 12 gTn Tr[( xB )g  ]
22
Parametrization of lightcone correlator
leading part
• M/P+ parts appear as M/Q terms in cross section
• T-reversal applies to (x)  no T-odd functions
Jaffe & Ji
Jaffe & Ji
NP B 375 (1992) 527
PRL 71 (1993) 2547
23
Basis of
partons
 ‘Good part’ of Dirac
space is 2-dimensional
 Interpretation of DF’s
unpolarized quark
distribution
helicity or chirality
distribution
transverse spin distr.
or transversity
24
Matrix representation
for M = [(x)g+]T
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
Quark production
matrix, directly
related to the
helicity formalism
Anselmino et al.
 Off-diagonal elements (RL or LR) are chiral-odd functions
 Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY
25
Next example: SIDIS
26
(calculation of) cross section in SIDIS
Full calculation
+
LEADING
(in 1/Q)
+
+
+…
27
Lightfront dominance in SIDIS
Three external momenta
P Ph q
transverse directions relevant
qT = q + xB P – Ph/zh
or
qT = -Ph^/zh
Result for SIDIS
q
p
Ph
k
P
2MW n ( P, Ph , q)   d 2 pT  d 2 kT
Ph
qT  q  xB P zh
 Tr[( xB , pT )g  ( zh , kT )g  ] 2 ( pT  qT - kT )
- 12 gTn  d 2 pT  d 2 kT
 Tr[( xB , pT )g  ] Tr[( zh , kT )g - ] 2 ( pT  qT - kT )
29
Parametrization of (x,pT)
• Also T-odd functions
are allowed
• Functions h1^ (BM)
and f1T^ (Sivers)
nonzero!
• Similar functions (of
course) exist as
fragmentation
functions (no Tconstraints) H1^
(Collins) and D1T^
30
Interpretation
unpolarized quark
distribution
need pT
helicity or chirality
distribution
need pT
transverse spin distr.
or transversity
need pT
need pT
Matrix representation
for M = [[±](x,pT)g+]T
 pT-dependent
functions
T-odd: g1T  g1T – i f1T^ and h1L^  h1L^ + i h1^ (imaginary parts)
32
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
Jaffe (1984), Diehl & Gousset (1998), …
Integrated quark correlators:
collinear and TMD
Rather than considering general correlator (p,P,…), one thus
integrates over p.P = p- (~MR2, which is of order M2)
2
d
(

.
P
)
d
T i p.
q
ij ( x, pT ; n)  
e
P  j (0) i ( ) P
3
(2p )
and/or pT (which is of order 1)
 qij ( x ; n)  
d ( .P) i p.
e
P  j (0) i ( ) P
(2p )
 .n T  0
TMD
 .n 0
lightfront
collinear
lightcone
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.
33
(NON-)COLLINEARITY
Summarizing oppertunities of TMDs
TMD quark correlators (leading part, unpolarized) including T-odd part
[  ]q

pT  P
 q
2
^q
2
( x, pT )   f1 ( x, pT )  i h1 ( x, pT )

M

 2
Interpretation: quark momentum distribution f1q(x,pT) and its
transverse spin polarization h1^q(x,pT) both in an unpolarized hadron
The function h1^q(x,pT) is T-odd (momentum-spin correlations!)
TMD gluon correlators (leading part, unpolarized)
 v
n
1


p
p

g
1
n
g
2
T T
2 T
 n
(
x
,
p
)

g
f
(
x
,
p
)



g
T
T
1
T
2
2x 
M


 ^g
2
 h 1 ( x, pT ) 


Interpretation: gluon momentum distribution f1g(x,pT) and its linear
polarization h1^g(x,pT) in an unpolarized hadron (both are T-even)
34
(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
dim[ (0)n  ( )]  2
dim[ F n (0) F n ( )]  2
‘Good’ fermion fields and ‘transverse’ gauge fields
and in addition any number of n.A() = An(x) fields (dimension zero!)
but in color gauge invariant combinations
dim 0:
dim 1:
in  iDn  in  gAn
iT  iDT  iT  gAT
Transverse momentum involves ‘twist 3’.
35
OPERATOR STRUCTURE
Soft parts: gauge invariant definitions
+
+…
Matrix elements containing A (gluon) fields produce gauge link

[C ]
U[0,
]



 P exp  -ig  ds A 
 0

Any path yields a
(different) definition
… essential for color gauge invariant definition
4
d
 i p.
[C ]
[C ]
ij ( p; P)  
e
P

(0)
U
j
[0, ] i ( ) P
4
(2p )
36
OPERATOR STRUCTURE
A.V. Belitsky, X.Ji, F. Yuan, NPB 656 (2003) 165
D. Boer, PJM, F. Pijlman, NPB 667 (2003) 201
Gauge link results from leading gluons
Expand gluon fields and reshuffle a bit:
P
1

 An ( p1 ) p1  iGTn ( p1 )  ...
A ( p1 )  n. A( p1 )
 i AT ( p1 )  ... 
n.P
p1.n

Coupling only to final state
partons, the collinear gluons
add up to a U+ gauge link,
(with transverse connection
from AT  Gn reshuffling)
37
OPERATOR STRUCTURE
Gauge-invariant definition of TMDs:
which gauge links?
d ( .P)d 2T i p.
[C ]
 ( x, pT ; n)  
e
P

(0)
U
j
[0, ] i ( ) P
3
(2p )
d ( .P) i p.
q
[n]
 ij ( x; n)  
e
P  j (0)U[0,
 ] i ( ) P  .n T  0
(2p )
q[ C ]
ij
 .n  0
TMD
collinear
Even simplest links for TMD correlators non-trivial:
[-]
T
[]
These merge into a ‘simple’ Wilson line in collinear (pT-integrated) case 38
OPERATOR STRUCTURE
C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147
F Dominguez, B-W Xiao, F Yuan, PRL 106 (2011) 022301
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, ]
 (2p )3
 .n  0
The most general TMD gluon correlator contains two links, which in
general can have 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[-,]
39
OPERATOR STRUCTURE
M.G.A. Buffing, PJM, 1105.4804
Gauge invariance for SIDIS
U[0 , ]U[ , ]U[ ,- ]U[ -,0 ]
 W[ n[0,]  ]W-[ n[0,]† ]  W[ n ][ p1 ]W-[ n ]† [k1 ]
Strategy:
transverse moments
d SIDIS  Trc [W[ p2 ][ p1 ]  q ( x1 , p1T )] Trc [ q ( x2 , p2T )W-[ p1 ]†[ p2 ]] *
 [q ] ( x1 , p1T ) [q- †] (k1 , k1T ) ˆ g q q
Problems with double T-odd functions in DY
40
COLOR ENTANGLEMENT
T.C. Rogers, PJM, PR D81 (2010) 094006
Complications (example: qq

qq)
U+[n] [p1,p2,k1]
modifies color flow,
spoiling universality
(and factorization)
[ k ](11)
U 
( p, p ')...... ( p)... ( p ') 
1 [ k ](1)
[ k ](1)
U  ( p),U 
( p ') ...... ( p)... ( p ')

41
2
COLOR ENTANGLEMENT
Color
entanglement
1
U -[ k ](21) ( p, p ')  U -[ k ](2) ( p)U -[ k ](1) ( p ')
4
1
 U -[ k ](1) ( p)U -[ k ](1) ( p ')U -[ k ](1) ( p)
4
1
 U -[ k ](1) ( p ')U -[ k ](2) ( p)
4
42
COLOR ENTANGLEMENT
Featuring: phases in gauge theories
 '  Pe 

ie ds . A
 i ( x) P  Pe 
-ig
x'
x
ds A
 i ( x ') P
43
COLOR ENTANGLEMENT
Conclusions
• TMDs enter in processes with more than one hadron involved (e.g.
SIDIS and DY)
• Rich phenomenology (Alessandro Bacchetta)
• Relevance for JLab, Compass, RHIC, JParc, GSI, LHC, EIC and LHeC
• Role for models using light-cone wf (Barbara Pasquini) and lattice
gauge theories (Philipp Haegler)
• Link of TMD (non-collinear) and GPDs (off-forward)
• 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
• Finding gauge links is only first step, (dis)proving QCD factorization
is next (recent work of Ted Rogers and Mert Aybat)
44
SUMMARY
Jaffe (1984), Diehl & Gousset (1998), …
Thus we need integrated correlators
Rather than considering general correlator (p,P,…), one integrates
over p.P = p- (~MR2, which is of order M2)
2
d
(

.
P
)
d
T i p.
q
ij ( x, pT ; n)  
e
P  j (0) i ( ) P
3
(2p )
and/or pT (which is of order 1)
 qij ( x ; n)  
d ( .P) i p.
e
P  j (0) i ( ) P
(2p )
 .n T  0
TMD
 .n 0
lightfront
collinear
lightcone
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.
45
(NON-)COLLINEARITY
PARTON CORRELATORS
Large values of momenta
• Calculable!
pT2 - x p M 2
p2 
hard
1- xp
p.P 
M R2 
x
p0  P  p0T
xp
T
 - pT
M << pT < Q
x p ( pT2 - M 2 )
2 x (1 - x p )
0
( x - x p ) pT2  x p (1 - x)M 2
( x  x p  1)
p0T ~ M
0
 ( p, P ) 
x (1 - x p )
s
2
T
p
...
0
etc.
46
Bacchetta, Boer, Diehl, M
JHEP 0808:023, 2008 (arXiv:0803.0227)
T-odd  single spin asymmetry
 Wn(q;P,S;Ph,Sh) = -Wn(-q;P,S;Ph,Sh)
*
 Wn
(q;P,S;Ph,Sh) = Wn(q;P,S;Ph,Sh)
_
__ __
 Wn(q;P,S;Ph,Sh) = Wn(q;P, -S;Ph, -Sh)
___ _
_
*
* (q;P,S;P ,S ) = W (q;P,S;P ,S )
 Wn
h h
n
h h
_
_
symmetry
structure
hermiticity
parity
time
reversal
• with time reversal constraint only even-spin asymmetries
• the time reversal constraint cannot be applied in DY or in  1-particle
inclusive DIS or ee• In those cases single spin asymmetries can be used to measure T-odd
quantities (such as T-odd distribution or fragmentation functions)
47
Boer & Vogelsang
Relevance of transverse momenta in
hadron-hadron scattering
p1  x1 P1  p1T
p2  x2 P2  p2T
At high energies fractional parton momenta fixed by
kinematics (external momenta)
DY
p1.P2 q.P2
x1  p1.n 

P1.P2 P1.P2
Also possible for transverse momenta of partons
f2 - f1
K1^
DY
qT  q - x1P1 - x2 P2  p1T  p2T
2-particle inclusive hadron-hadron scattering
qT  z1-1K1  z2-1K2 - x1P1 - x2 P2
f K
2^
pp-scattering
 p1T  p2T - k1T - k2T
Care is needed: we need more than one
hadron and knowledge of hard process(es)!
48
NON-COLLINEARITY
Lepto-production of pions
H1^ is T-odd
and chiral-odd
49
Gauge link in DIS
• In limit of large Q2 the result
of ‘handbag diagram’ survives
• … + contributions from A+ gluons
ensuring color gauge invariance
A+
Ellis, Furmanski, Petronzio
Efremov, Radyushkin
A+ gluons
 gauge link
50
Distribution
including the gauge link (in SIDIS)
A+
One needs also AT
G+ =  +AT
AT()= AT(∞) +  d G+
Belitsky, Ji, Yuan, hep-ph/0208038
Boer, M, Pijlman, hep-ph/0303034
From 0AT() m.e.
Generic hard processes
• E.g. qq-scattering as hard
subprocess
• Matrix elements involving
parton 1 and additional
gluon(s) A+ = A.n appear
at same (leading) order in
‘twist’ expansion
• insertions of gluons
collinear with parton 1 are
possible at many places
• this leads for correlator
involving parton 1 to gauge
links to lightcone infinity
C. Bomhof, P.J. Mulders and F. Pijlman,
PLB 596 (2004) 277 [hep-ph/0406099];
EPJ C 47 (2006) 147 [hep-ph/0601171]
Link structure
for fields in
correlator 1
52
SIDIS
SIDIS  [U+] = [+]
DY  [U-] = [-]
53
A2

2 hard processes: qq
• E.g. qq-scattering as hard
subprocess
• The correlator (x,pT) enters
for each contributing term in
squared amplitude with
specific link

qq
U□ = U+U-†
□)U+]
[Tr(U
(x,pT)
[U□U+](x,pT)
54
Gluons
d ( .P)d 2T i p.
[C ]
n
[ C ']
n
 g ( x, pT ; C, C ')  
e
P
U
F
(0
)
U
F
( ) P
[ ,0]
[0, ]
3
(2p )

• Using 3x3 matrix representation for U, one finds in gluon correlator
appearance of two links, possibly with different paths.
• Note that standard field displacement involves C = C’
F ( )  U[[C,] ] F ( )U[[C,] ]
55
 .n  0
Integrating [±](x,pT)  [±](x)
d ( .P)d 2T i p.
†
n
T
n
 ( x, pT )  
e
P

(0)
U
U
U
[0,  ] [0T ,T ] [ , ] ( ) P
3
(2p )
[]
collinear
correlator
d ( .P)

 ( x)  
e
[]
(2p )
i p.
n
P  † (0)U[0,
 ] ( ) P
 .n T 0
56
 .n  0
Integrating [±](x,pT)  [±](x)
transverse
moment
 [  ] ( x)   d 2 pT pT [  ] ( x, pT )
d ( .P)d 2T i p.
†
n

T
n
 ( x)   d pT 
e
P

(0)
U
i

U
U
[0,  ] T
[0T ,T ] [  , ] ( ) P
3
(2p )
d ( .P) i p.
n

 [  ] ( x)  
e [ P  † (0)U[0,
iD
]
T  ( ) P
(2p )
 []
2
 .n  0

n
n
n
n
- P  (0)U[0,
d
(

.
P
)
U
gG
(

)
U
 ] 
[ , ]
[ , ]  ( ) P

 [  ] ( x)  D ( x)   dx1
i
G ( x, x - x1 )
x1 i
= FaD (x) + ò dx1 P
i
x
G(p,p-p1)
FaG (x, x - x1 ) ± p FaG (x, x)
1
Fa¶ (x)
T-even
T-odd
57
]
LC
Gluonic poles
• Thus:
[U](x) = (x)
[U](x)
~
=  (x) + CG[U] pG(x,x)
• Universal gluonic pole m.e. (T-odd for distributions)
• pG(x) contains the weighted T-odd functions h1^(1)(x)
[Boer-Mulders] and (for transversely polarized hadrons) the
function f1T^(1)(x) [Sivers]
~
• (x) contains the T-even functions h1L^(1)(x) and g1T^(1)(x)
• For SIDIS/DY links: CG[U±] = ±1
• In other hard processes one encounters different factors:
CG[U□ U+] = 3, CG[Tr(U□)U+] = Nc
58
Efremov and Teryaev 1982; Qiu and Sterman 1991
Boer, Mulders, Pijlman, NPB 667 (2003) 201
A2

2 hard processes: qq
• E.g. qq-scattering as hard
subprocess
• The correlator (x,pT) enters
for each contributing term in
squared amplitude with
specific link

qq
U□ = U+U-†
□)U+]
[Tr(U
(x,pT)
[U□U+](x,pT)
59
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268
examples: qqqq in pp
Tr (U )
U
U U
Nc
D1
q 
2
Nc  1
2
Nc
-1
[( )  ]

-
2
2
Nc
[ ]
-1

  
D4
2
Nc
-1
p 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
p G
CG [D3] = CG [D4]
60
Bacchetta, Bomhof, D’Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153
examples: qqqq in pp
D1
q 
1
2
Nc
†

[(
-1
)]

2
Nc - 2
2
Nc
-1
[ -]

  -
2
Nc - 3
2
Nc
-1
p G
For Nc:
CG [D1]  -1
(color flow as DY)
q 
2
Nc
2
Nc
-1

[(
†
)]
-
1
2
Nc
[-]
-1

  
2
Nc  1
2
Nc
-1
p G
61
Gluonic pole cross sections
• In order to absorb the factors CG[U], one can define specific hard
cross sections for gluonic poles (which will appear with the functions
in transverse moments)
• for pp:
ˆ[ q ]qqq  CG[U ( D)]ˆ [ D]
ˆ qqqq  ˆ [ D]

[ D]

[ D]
(gluonic pole cross section)
etc.
dˆ qqqq
• for SIDIS:
for DY:
ˆ [q] q   ˆ q q
dˆ(q)qqq
ˆ[q]q   - ˆqq 
y
62
Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]
examples: qgqg in pp
weigted
Transverse momentum dependent
q 
D1
2
Nc
2
Nc
-1
[( ) - ]

-
1
2
Nc
[]
-1

  -
Nc
2
1
2
Nc
-1
p G
Only one factor, but
more DY-like than
SIDIS
D2
D3
D4
Note: also
etc.
63
e.g. relevant in Bomhof, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019
examples: qgqg
weighted
Transverse momentum dependent
D1
q 
2
Nc
2
Nc - 1
[( ) - ]

-
1
2
Nc - 1
[]

  -
2
Nc
2
1
Nc - 1
p G
D2
D3
D4
D5
64
examples: qgqg
weighted
Transverse momentum dependent
q 
2
Nc
2
Nc - 1
[( ) - ]

-
1
2
Nc - 1
[]

  -
2
Nc
2
1
Nc - 1
p G
65
examples: qgqg
weighted
Transverse momentum dependent
q 
q 
q 
2
q

NNc2
c
2
2( N c - 1)
2
2( N c - 1)
4
Nc
2
( Nc
- 1)

[(
[( )(
)( † )  ]

2
[( )(
†

†
)]
2
2

-
Nc
2( N c - 1)
q 
[( ) - ]
Nc - 1
Nc
) N]c2
2
2
Nc
1
[( ) - ]
2
2

[(


[]
) -]
2
( Nc
2
Nc
2
Nc
[( ) - ]
- 1)
-1

2
[(
)(
-
[]

2
Nc - 1
1
1
  -
[  ]   -
  p G
2 N2 -1
c -1
2( N c -N1)
2
-
1
†
-
)]
N c -c 1
1
2
Nc
-
[ ]
-1

1
2
Nc
-1
[ ]
  

2
Nc
1
2
Nc - 1
1
2
Nc - 1
Nc
2
1
2
Nc
-1
p G
p G
p G
   p G
66
examples: qgqg
weighted
Transverse momentum dependent
It is also possible to group
the TMD
2
2
Nc
Nc  1
1
[(
)
]
[

]
functions in a smart
into
q way
 two!
- 2 
  - 2 p G
2
N
1
N
1
Nc - 1
c
c
(nontrivial for nine diagrams/four
color-flow possibilities)
2
2
N 2
Nc 1
1 1
[( )( † ) N]c2
[( ) - ]
[]
[( )( )  ]
q  N2Nc22c 

† 2  [( ) -]2-2 2 
[  ]  - p G
2
2
2( Ncc - 1)
[( ) - ]
c -1
2( N
- 1) [( )( 2() N]c - 1) 2( NN c -N1)
N 1N-c 1- 1 [  ]
q
 
†
c
2
2( N c
p G 
q 
- 1)


2
Nc
[( )(
42
N
Ncc - 1 [( )(

2
2
( N c - 1)
†
†
) ]
c
2
2( N c
- 1)

-
c
2
Nc
-1

  -
1
2
Nc - 1
p G
1
- 2 2 [ ]
N
)]
- N2c c- 12 [( ) -] ( N c - 1)
2
But still no factorization!
Nc
q  2 [(
Nc - 1
)(
†
)]
1
2
Nc
-
[ ]
-1

1
2
Nc
-1
[  ]
  

Nc
2
1
2
Nc
-1
p G
   p G
67
‘Residual’ TMDs
• We find that 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 )
no definite
T-behavior
definite T-behavior
• The residuals satisfies (x) = 0 and pG(x,x) = 0, i.e. cancelling kT
contributions; moreover they most likely disappear for large kT
68
pT  P
 q
2
^q
2
 ( x, pT )   f1 ( x, pT )  i h1 ( x, pT )

M  2

 q
q
^ q pT  P
 ( x, pT )   f1  i h1

M

 2
PnpT 
p

g 5g 
M q
T
^q
^q
q i[ P , n ] 
 e  f
g
h

2 
M
M
2 
q
1 q
M q
 ( x) 
f1 ( x) P  e ( x)
2
2
1 q
M
 q ( x) 
f1 ( x)  p  m  
2x
2
q
m q 
 q
e
(
x
)
f1 ( x) 

Mx


69
d ( .P)d 2T i p.
( x, pT )  
e
P  (0) ( ) P  .n0
3
(2p )
pT  P
 q
q
2
^q
2
 ( x, pT )   f1 ( x, pT )  i h 1 ( x, pT )

M

 2
d ( .P) i p.
( x)   d 2 pT ( x, pT )  
e
P  (0) ( ) P  .n 0
(2p )
P q
q
 ( x) 
f1 ( x)
2
d ( .P) i p.

2

 ( x)   d pT pT ( x, pT )  
e
P iT ( (0) ( )) P  .n 0
(2p )
2

 ^q
p
P
P
q
^ q (1)
2
2
T
  ( x) 
h 1 ( x)   d pT 
h
(
x
,
p
T)
2  1
70
2
2
 2M 
T
T
 q
2 2
^q
2 2 kT 
2 z  ( z, kT )   D1 ( z, z kT )  i H1 ( x, z kT )  K
M

 q
q
^ q kT 
2 z  ( z, kT )   D1  i H1
K
Mh 

KnkT 
 q

g 5g 
k
^q
^q
q i[ K , n ] 
T
 Mh  E  D
G
H

M
M
2
h
h


2 q
 q
q
q i[ K , n ] 
 ( z)  D1 ( z ) K  M h  E  H

z
2


1 q
q
 ( z) 
D1 ( z )  k  m   ...
71
2
q
pT  P
 q
2
^q
2
 ( x, pT )   f1 ( x, pT )  ih1 ( x, pT )

M  2


q
q
2
^q
2 g 5 pT  P
 L ( x, pT )   SL g1L ( x, pT ) g 5  SL h1L ( x, pT )

M  2

æ q
öP
q
2
q
2 pT .ST
FT (x, pT ) = ç h (x, pT ) g 5 ST - g1T (x, pT )
g5 ÷
1T
M
è
ø2
PnpT ST
æ
öP
e
q
2
^q
2 pT .ST g 5 pT
÷÷
+ çç f1T (x, pT )
- h (x, pT )
1T
M
M
M ø2
è
q
U
72
P
 ( x)  f ( x)
2
q
U
q
1
P
 ( x)  S L g ( x) g 5
2
q
L
q
1
P
 ( x)  h ( x) g 5 ST
2
q
T
q
1
73
74
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
2

2
d
q
q
...
d
 T T  p1T
2
2
d
p
...

 2T (qT - p1T - p2T )
  d 2 p1T p1T  d 2 p2T ... 
2
d
 p1T
2

d
p
p
2
T
2
T ...

• 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
75
COLOR ENTANGLEMENT
M.G.A. Buffing, PJM; in preparation
Full color disentanglement? NO!
 (2 )  (02 )
[ 1 , - ]
[ - ,0 1 ]
[  , 1 ][  ,2 ]
a
a
[0 1 ,  ][02 ,  ]
[2 ,-]
b*
[ - ,02 ]
 (1 )  (01 )
X
~ Trc [
[( )  ]
Trc [
Loop 1:
b*
( p1 ) 
[( )  ]
*
b
[( ) -† ]
( p2 ) 
b*
(k1 ) a ]
[( ) -† ]
( k 2 ) a ]
n]
[n]
[ n ]†
U[0 2 , ][01 , ]U[ , 2 ][ ,1 ]U[1 ,- ]U[ -,01 ]  W[ [0
W
W
 [01 ,1 ] - [01 , 1 ]
2 , 2 ]
 W[ n ][ p2 ]W [ n ][ p1 ]
76
COLOR ENTANGLEMENT
Result for integrated cross section
d
[ C1 ( D )]
[ D]
ˆ
~

(
x
,
p
)

(
x
)


a
1
1T
b
2
ab c...  c ( z1 )...
2
d p1T D,abc
(1PU)
Collinear cross section
[C ]

( x)   d 2 pT [ C ] ( x, pT )
Gauge link structure
becomes irrelevant!
 ~  a ( x1 )b ( x2 ) ˆ ab c... c ( z1 )...
abc
[ D]
ˆ abc...  ˆ ab
c...
(partonic cross section)
D
77
APPLICATIONS
Result for single weighted cross section
d
[ C1 ( D )]
[ D]
ˆ
~

(
x
,
p
)

(
x
)


a
1
1T
b
2
ab c...  c ( z1 )...
2
d p1T D,abc
(1PU)
Single weighted cross section (azimuthal asymmetry)
 [C ] ( x)   d 2 pT pT [ C ] ( x, pT )
p1T  ~
 [C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc... c ( z1 )...
D,abc
78
APPLICATIONS
Result for single weighted cross section
d
[ C1 ( D )]
[ D]
ˆ
~

(
x
,
p
)

(
x
)


a
1
1T
b
2
ab c...  c ( z1 )...
2
d p1T D,abc
p1T 
p1T 
p1T 
p1T 
(1PU)
[ 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
79
APPLICATIONS
Qiu, Sterman; Koike; Brodsky, Hwang, Schmidt, …
Result for single weighted cross section
d
[ C1 ( D )]
[ D]
ˆ
~

(
x
,
p
)

(
x
)


a
1
1T
b
2
ab c...  c ( z1 )...
2
d p1T D,abc
p1T  ~
(1PU)
 [C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc... c ( z1 )...
D,abc
 [C ]

[U (C )]
 [
C]

( x)   ( x)  CG p G ( x, x)
 [C ]
T-even
G(p,p-p1)
G ( x, x - x1 )
universal matrix T-odd
elements
(operator structure)
G(x,x) is gluonic pole
(x1 = 0) matrix element
(color entangled!)
80
APPLICATIONS
Result for single weighted cross section
d
[ C1 ( D )]
[ D]
ˆ
~

(
x
,
p
)

(
x
)


a
1
1T
b
2
ab c...  c ( z1 )...
2
d p1T D,abc
p1T  ~
(1PU)
 [C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc... c ( z1 )...
D,abc
 [C ]

[U (C )]
 [
C]

( x)   ( x)  CG p G ( x, x)
 [C ]
universal matrix
elements
Examples are:
CG[U+] = 1, CG [U- ] = -1, CG[W U+] = 3, CG [Tr(W)U+]81
= Nc
APPLICATIONS
Result for single weighted cross section
d
[ C1 ( D )]
[ D]
ˆ
~

(
x
,
p
)

(
x
)


a
1
1T
b
2
ab c...  c ( z1 )...
2
d p1T D,abc
p1T  ~
 [C ( D )]
[ D]
ˆ

(
x
)

(
x
)

 a
1
b
2
abc... c ( z1 )...
D,abc
 [C ]

p1T  ~
(1PU)
[U (C )]
 [
C]

( x)   ( x)  CG p G ( x, x)
 [C ]


 a ( x1 )b ( x2 ) ˆ abc... c ( z1 )...
abc
  p G a ( x1 , x1 ) b ( x2 ) ˆ[ a ]b c...  c ( z1 )...
T-odd part
abc
[ D]
ˆ[ a ]bc...   CG[U (C ( D ))]ˆ ab
c...
D
(gluonic pole
cross section)
82
APPLICATIONS
Higher pT moments
• Higher transverse moments
[ N ]1 ... N ( x)   d 2 pT ( pT1 ... pT1 - traces )  ( x, pT )
•
involve yet more functions



(
x
),

(
x
,
x
),


G
GG ( x, x, x)
• 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.
83
SUMMARY
DIS
q2  -Q2
P2  M 2
Q2
2 P.q 
xB
q
P2  M 2
n
q  xB P
P.q
P
84
DIS
Basis 1
n  P
q  xB P
n-  n 
P.q
Basis 2
q
Q
ˆt  q  2 xB P
Q
zˆ  -
q
P
85
Principle for DY
 u ( p , s) u ( p , s)
s
1
1
 ( p1 , P1 ) ~ ( p 1  m) f ( p1 )
• Instead of partons use correlators
• Expand parton momenta (for DY take e.g. n1= P2/P1.P2)
p  x P  pT   n
~Q
~M
~
M2/Q
x  p  p.n ~ 1
  p.P - xM 2 ~ M 2
86
(NON-)COLLINEARITY