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1
Transverse momenta of partons in high-energy scattering
processesPiet Mulders
International School, Orsay June 2012
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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
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3 colors
Valence structure of hadrons:
global properties of nucleons
duu
proton
• mass• charge• spin• magnetic moment• isospin, strangeness• baryon number
Mp Mn 940 MeV Qp = 1, Qn = 0 s = ½ gp 5.59, gn -3.83 I = ½: (p,n) S = 0 B = 1
Quarks as constituents
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A real look at the proton
+ N ….
Nucleon excitation spectrumE ~ 1/R ~ 200 MeVR ~ 1 fm
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A (weak) look at the nucleon
n p + e +
= 900 s Axial charge GA(0) = 1.26
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A virtual look at the proton
N N + N N_
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Local – forward and off-forward m.e.
1.
2( ) (' | ) | )( i x GP O tP e t Gx i
2t
1(0) | ( ) |G P O x P
2 (0) | ( ) |G P x O x P
Local operators (coordinate space densities):
P P’
Static properties:
Form factors
Examples:(axial) chargemassspinmagnetic momentangular momentum
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Nucleon densities from virtual look
proton
neutron
• charge density 0• u more central than d?• role of antiquarks?• n = n0 + p+ … ?
( ) ( )i iG t x
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Quark and gluon operators
Given the QCD framework, the operators are known quarkic or gluonic currents such as
probed in specific combinationsby photons, Z- or W-bosons
( ) 2 1 13 3 3 ...u d sJ V V V
( ) 21 42 3 sin ...Z u u u
WJ V A V
( ) ' ' ...W ud udJ V A
'5( ) ( ) '( )q qA x q x q x
( ) ( ) ( )qV x q x q x
(axial) vector currents
{ }( ) ~ ( ) ( )qT x q x D q x
( ) ~ ( ) ( )GT x G x G x
energy-momentum currents
probed by gravitons
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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!
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Non-local probing
†2 2 2 2, ...y y y y
x x x xO
2 2 2 2| , | | , |y y y yx xP O P P O P
2. †2 2| | | 0 | ( )y yip ydy e P P P p P f p
Nonlocal forward operators (correlators):
Specifically useful: ‘squares’
Momentum space densities of -ons:
Selectivity at high energies: q = p
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A hard look at the proton
Hard virtual momenta ( q2 = Q2 ~ many GeV2) can couple to (two) soft momenta
+ N jet jet + jet
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Experiments!
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QCD & Standard Model
• QCD framework (including electroweak theory) provides the machinery to calculate cross sections, e.g. *q q, qq *, * qq, qq qq, qg qg, etc.
• E.g. qg qg
• Calculations work for plane waves
__
) .( ( )0 , ( , )s ipi ip s u p s e
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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, implies hadrons as ‘sources’ for quarks
• … and also as ‘source’ for quarks + gluons
• … and also ….
.( )0 , ( , ) ipii p s u p s e
.( ) ii
pX P e
1 1( ). .( ) ( ) i pi
p ipAX P e
PARTON CORRELATORS
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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
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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!
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Example: DIS
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• Instead of partons use correlators• Expand parton momenta (for SIDIS take e.g. n=
Ph/Ph.P)
Principle for DIS
q
P
2 2P M
( , ) ( , )
( , ) ~ ( ) ( )su p s u p s
p P p m f p
Tx pp P n 2 2. ~p P xM M
. ~ 1x p p n
~ Q ~ M ~ M2/Q
p
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(calculation of) cross section in DIS
Full calculation
+ …
+ +
+LEADING (in 1/Q)
x = xB = q2/P.q
(x)
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Lightcone dominance in DIS
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Result for DIS
q
P
212
12
2 ( , ) . [ ( , ) ] ( )
[ ( ) ]
T T B
T B
MW P q g dxdp P d p Tr p P x x
g Tr x
p
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Parametrization of lightcone correlator
Jaffe & Ji NP B 375 (1992) 527Jaffe & Ji PRL 71 (1993) 2547
leading part
• M/P+ parts appear as M/Q terms in cross section• T-reversal applies to (x) no T-odd functions
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Basis of
partons
‘Good part’ of Dirac space is 2-dimensional
Interpretation of DF’s
unpolarized quarkdistribution
helicity or chiralitydistribution
transverse spin distr.or transversity
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Off-diagonal elements (RL or LR) are chiral-odd functions Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY
Matrix representation
for M = [(x)+]T
Quark production matrix, directly related to thehelicity formalism
Anselmino et al.
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
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Next example: SIDIS
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(calculation of) cross section in SIDIS
Full calculation
+
+ …
+
+LEADING (in 1/Q)
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Lightfront dominance in SIDIS
Three external momentaP Ph q
transverse directions relevant
qT = q + xB P – Ph/zh or
qT = -Ph/zh
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Result for SIDIS
2 2
2
2 212
2
2 ( , , )
[ ( , ) ( , ) ] ( )
[ ( , ) ] [ ( , ) ] ( )
h T T
B T h T T T T
T T T
B T h T T T T
MW P P q d p d k
Tr x p z k p q k
g d p d k
Tr x p Tr z k p q k
q
P
Ph
pk
hT B
h
Pq q x P
z
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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 T-constraints) H1
(Collins) and D1T
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Interpretation
unpolarized quarkdistribution
helicity or chiralitydistribution
transverse spin distr.or transversity
need pT
need pT
need pT
need pT
need pT
T-odd
T-odd
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pT-dependent functions
T-odd: g1T g1T – i f1T and h1L
h1L + i h1
(imaginary parts)
Matrix representationfor M = [±]
(x,pT)+]T
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
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Rather than considering general correlator (p,P,…), one thus 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
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Summarizing 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
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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
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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
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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
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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
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TMD correlators: gluons2
[ , '] . [ ] [ '][ ,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
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Gauge invariance for SIDIS
COLOR ENTANGLEMENT
[0 , ] [ , ] [ , ] [ ,0 ]
[ ] [ ]† [ ] [ ]†[0, ] [0, ] 1 1[ ] [ ]n n n n
U U U U
W W W p W k
2 1[ ] [ ]† *1 1 1 2 2 2
[ ] [ †]1 1 1 1
[ [ ] ( , )] [ ( , ) [ ]]
ˆ( , ) ( , )
p pSIDIS c q T c q T
q T q T q q
d Tr W p x p Tr x p W p
x p k k
Strategy:transverse moments
Problems with double T-odd functions in DY
M.G.A. Buffing, PJM, 1105.4804
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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
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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
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Featuring: phases in gauge theories
'
( ) ( ')x
xig ds A
i iP Pe Px x
.
'ie ds A
Pe COLOR ENTANGLEMENT
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44
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)
SUMMARY
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45
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.
Thus we need integrated correlators
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
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46
Large values of momenta
• Calculable!2 2
2 01T 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
p0T ~ M
M << pT < Q
hard
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47
T-odd single spin asymmetry
• with time reversal constraint only even-spin asymmetries• the time reversal constraint cannot be applied in DY or in 1-
particle inclusive DIS or ee
• In those cases single spin asymmetries can be used to measure T-odd quantities (such as T-odd distribution or fragmentation functions)
*
*
W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)
W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)
W(q;P,S;Ph,Sh) = W(q;P, S;Ph, Sh)
W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)
_
___
_ ____
__ _time
reversal
symmetrystructure
parity
hermiticity
*
*
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48
Relevance of transverse momenta in hadron-hadron
scatteringAt 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
Second scale!
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Lepto-production of pions
H1 is T-odd
and chiral-odd
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Gauge link in DIS
• In limit of large Q2 the resultof ‘handbag diagram’ survives
• … + contributions from A+ gluonsensuring color gauge invariance
A+ gluons gauge link
Ellis, Furmanski, PetronzioEfremov, Radyushkin
A+
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Distribution
From AT() m.e.
including the gauge link (in SIDIS)
A+
One needs also AT
G+ = +AT
AT()= AT
(∞) + d G+
Belitsky, Ji, Yuan, hep-ph/0208038Boer, M, Pijlman, hep-ph/0303034
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52
Generic hard processes
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
• 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
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SIDIS
SIDIS [U+] = [+]
DY [U-] = []
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A 2 2 hard processes: qq qq
• E.g. qq-scattering as hard subprocess
• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link
[Tr(U□)U+](x,pT)
U□ = U+U†
[U□U+](x,pT)
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55
Gluons
2. [ ] [ ']
[ ,0] [0, ]3 . 0
( . )( , ; , ')
((0
)) ( )
2i p C CT n
gn
T n
d P dx p C C e P U PF FU
• 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 ’
[ ] [ ][ , ] [ , ]( ) ( )C CF U F U
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Integrating [±](x,pT) [±]
(x)
[ ] . †[0, ] . 0
( . )( ) (0) ( )
(2 ) T
i p n
n
d Px e P U P
collinear correlator
2[ ] . †
[0, ] [0 , ] [ , ]3 . 0
( . )( , ) (0) ( )
(2 ) T T
i p n T nTT n
d P dx p e P U U U P
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2[ ] 2 . †
[0, ] [0 , ] [ , ]3 . 0
( . )( ) (0) ( )
(2 ) T T
i p n T nTT T n
d P dx d p e P U i U U P
Integrating [±](x,pT) [±]
(x)
[ ] . †[0, ]
( . )( ) (0) ( )
(2 )[i p n
T
d Px e P U iD P
[0, ] [ , ] [ , ](0) ( . ) ( ) ( ) ]n n n nLCP U d P U gG U P
[ ]
1 11
( ) ( ) ( , )D G
i
x ix x dx x x x
[ ] 2 [ ]( ) ( , )T T Tx d p p x p transve
rse moment
G(p,pp1)
T-even T-odd
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58
Gluonic poles
• Thus: [U](x) = (x)
[U](x) =
(x) + CG[U] G
(x,x)
• Universal gluonic pole m.e. (T-odd for distributions)
• 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
Efremov and Teryaev 1982; Qiu and Sterman 1991Boer, Mulders, Pijlman, NPB 667 (2003) 201
~
~
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59
A 2 2 hard processes: qq qq
• E.g. qq-scattering as hard subprocess
• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link
[Tr(U□)U+](x,pT)
U□ = U+U†
[U□U+](x,pT)
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60
examples: qqqq in pp
2 2[( ) ] [ ]
2 2 2
1 52
1 1 1
c cq G
c c c
N N
N N N
2 2 2[( ) ] [ ]
2 2 2
2 1 3
1 1 1
c c cq G
c c c
N N N
N N N
CG [D1]
D1
= CG [D2]
= CG [D4]CG
[D3]
D2
D3
D4
( )
c
Tr U
NU
U U
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268
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examples: qqqq in pp
†2 2
[( ) ] [ ]2 2 2
2 31
1 1 1
c cq G
c c c
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
D1
For Nc:
CG [D1] 1
(color flow as DY)
Bacchetta, Bomhof, D’Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153
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62
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:
etc.
• for SIDIS:
for DY:
Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206]
[ ]
[ ]
ˆ ˆ Dqq qq
D
[ ( )] [ ][ ]
[ ]
ˆ ˆU D Dq q qq G
D
C
[ ]ˆ ˆq q q q
[ ]ˆ ˆq q qq
(gluonic pole cross section)
y
( )ˆ q q qqd
ˆqq qqd
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63
examples: qgqin pp
2 2[( ) ] [ ]
2 2 2
11
1 1 1
c cq G
c c c
N N
N N N
D1
D2
D3
D4
Only one factor, but more DY-like than SIDIS
Note: also etc.
Transverse momentum dependent weigted
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64
examples: qgqg
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, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019
Transverse momentum dependent weighted
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examples: qgqg
2 2[( ) ] [ ]
2 2 2
11
1 1 1
c cq G
c c c
N N
N N N
Transverse momentum dependent weighted
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examples: qgqg
†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
Transverse momentum dependent weighted
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examples: qgqg
†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
†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 also possible to group the TMD functions in a smart way into two!(nontrivial for nine diagrams/four color-flow possibilities)
But still no factorization!
Transverse momentum dependent weighted
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68
‘Residual’ TMDs
• We find that we can work with basic TMD functions [±](x,pT) + ‘junk’
• The ‘junk’ constitutes process-dependent residual TMDs
• The residuals satisfies (x) = 0 and G(x,x) = 0, i.e. cancelling kT contributions; moreover they most likely disappear for large kT
† †
†[( )( ) ]
[( )( ) ] [ ] [( )( ) ] [ ]
( , )
( , ) ( , ) ( , )
T
T T T
x p
x p x p x p
[ ] [ ] [ ] [ ]( , ) 2 ( , ) ( , ) ( , )T T T Tx p x p x p x p
definite T-behavior
no definiteT-behavior
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69
2 21 1( , () , )( , )
2T
T Tq q q
Tf x p hp P
x p iM
x p
5
1 1( , )2
[ , ]
2 2
T
q q TqT
q q q qPnp
T
p Px p i
M
pM i P n
Me h
M
f h
f g
1 (1
( )2
) (2
)q q qf xM
x P e x
1 1
1( ) ( ) ( )
2 2( )q qq qM m
x ef x x f xx Mx
p m
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2 21 1( , () , )( , )
2T
T Tq q q
Tf x p hp P
x p iM
x p
1( )2
( )q qf xP
x
2.
3 . 0
( . )( , ()
(2 )0) ( )i pT
T n
d P dx p e P P
2 .
. 0
( . )( ) ( , ) (0) ( )
(2 ) T
i pT T n
d Px d p x p e P P
(1)1
22 2
12( ) ( , )
2 2 2( )q qT
T Tq pP P
x d p h x pM
h x
2 .
. 0
( . )( ) ( , ) ( (0) ( ))
(2 ) T
i pT nTT T
d Pix d p p x p e P P
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71
2 2 2 21 12 ( , ) ( , ) ( , )qq T
Tq
T T
kz z k iD z z k H x k K
Mz
1
5
12 ( , )
[ , ]
2
T
q TT
h
K
q
nkT
hq
h
q q
h
q
q
kz z k i K
M
k i
D H
E D GK n
MM M
H
1
2 [ , ]( ) )
2(qq
hq qD z E
i K nHz K M
z
1
1( ) .( .
2) .qq D z k mz
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2 21 1( , ) , ) , )
2( (q Tq
U Tq
T Tf x p hp P
x p iM
x p
1
525
21( , ) ( , ) , )
2(
L
Tqq qT L TLL T LS Sg x p
p Px px p
Mh
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1( )2
( )qqU f
Pxx
51( )2
( )qqLL g
PxSx
1 5(2
)) (q TqT
Pxx h S
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74
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75
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
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76
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 , ]
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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 D
ab c ab cD
(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)
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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)
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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
[ ( )] [ ]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)
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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 [ ] [ [ ] [ ])] (( ) ( ) ( , )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)
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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 [ ] [ [ ] [ ])] (( ) ( ) ( , )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)
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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 [ ] [ [ ] [ ])] (( ) ( ) ( , )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
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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
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DIS
q
P
2 2q Q
2 2P M2
2 .B
QP q
x 2 2P M
.Bq x P
nP q
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DIS
q
P
n P
.Bq x P
n nP q
Basis 1
ˆq
zQ
2ˆ Bq x Pt
Q
Basis 2
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Principle for DY
1 1
1 1 1 1
( , ) ( , )
( , ) ~ ( ) ( )su p s u p s
p P p m f p
• Instead of partons use correlators
• Expand parton momenta (for DY take e.g. n1= P2/P1.P2)
Tx pp P n 2 2. ~p P xM M
. ~ 1x p p n
~ Q ~ M ~ M2/Q
(NON-)COLLINEARITY