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Transcript of 1 Transverse momenta of partons in high-energy scattering processes Piet Mulders International...
1
Transverse momenta of partons in high-energy scattering
processesPiet Mulders
International School, Orsay June 2012
2
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
3
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
4
A real look at the proton
+ N ….
Nucleon excitation spectrumE ~ 1/R ~ 200 MeVR ~ 1 fm
5
A (weak) look at the nucleon
n p + e +
= 900 s Axial charge GA(0) = 1.26
6
A virtual look at the proton
N N + N N_
7
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
8
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
9
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
10
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!
11
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
12
A hard look at the proton
Hard virtual momenta ( q2 = Q2 ~ many GeV2) can couple to (two) soft momenta
+ N jet jet + jet
13
Experiments!
14
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
15
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
16
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
17
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!
18
Example: DIS
19
• 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
20
(calculation of) cross section in DIS
Full calculation
+ …
+ +
+LEADING (in 1/Q)
x = xB = q2/P.q
(x)
Lightcone dominance in DIS
22
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
23
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
24
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
25
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
26
Next example: SIDIS
27
(calculation of) cross section in SIDIS
Full calculation
+
+ …
+
+LEADING (in 1/Q)
Lightfront dominance in SIDIS
Three external momentaP Ph q
transverse directions relevant
qT = q + xB P – Ph/zh or
qT = -Ph/zh
29
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
30
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
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
32
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
33
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
34
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
35
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
36
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
37
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
38
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
39
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
40
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
41
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
42
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
43
Featuring: phases in gauge theories
'
( ) ( ')x
xig ds A
i iP Pe Px x
.
'ie ds A
Pe COLOR ENTANGLEMENT
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
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
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
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
*
*
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!
49
Lepto-production of pions
H1 is T-odd
and chiral-odd
50
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+
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
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
53
SIDIS
SIDIS [U+] = [+]
DY [U-] = []
54
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)
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
56
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
57
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
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
~
~
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)
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
61
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
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
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
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
65
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
66
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
67
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
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
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
70
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
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
72
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
73
1( )2
( )qqU f
Pxx
51( )2
( )qqLL g
PxSx
1 5(2
)) (q TqT
Pxx h S
74
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
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 , ]
77
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)
78
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)
79
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)
80
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)
81
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)
82
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
83
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
84
DIS
q
P
2 2q Q
2 2P M2
2 .B
QP q
x 2 2P M
.Bq x P
nP q
85
DIS
q
P
n P
.Bq x P
n nP q
Basis 1
ˆq
zQ
2ˆ Bq x Pt
Q
Basis 2
86
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