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TMDs at small x
Jian Zhou
ShanDong University
Parton Distributions in Modern Era, @TD Lee Institute, SJTU, 2016, Dec.
Outline:
TMD factorization at small x
Spin dependent TMDs at small x
1: unpolarized target
2: transversely polarized target
Elliptic gluon GTMD inside a large nucleus
Parton distributions at small x
Gluon radiation at small x
Derive the TMD factorization formula I
Starting from the full CGC result,
Balitsky 1996
Gelis, Jalilian-Marian 2003
Taylor expanding the impact factor (PT >>QS ),
Integrating out k1T,
F. Dominguez, B-W. Xiao, F. Yuan 2011
F. Dominguez, C. Marquet, B-W. Xiao, F. Yuan 2011
Derive the TMD factorization formula II
The cross section then reads,
One can identify,
F. Dominguez, C. Marquet, BW. Xiao, F. Yuan 2011
CGC
TMD
Derivative of impact factor in kT
Hard part
Derivative of Wilson lines in xT
Gluon TMDs
Physical picture behind the effective TMD factorization
Low jet PT ≤ KT ( only CGC applicable )
High jet PT >> KT ( both CGC and TMD applicable )
Gluons can not resolve the internal
structure of the color dipole system.
four Wilson lines
Collapse to two semi-finite Wilson lines
An additional hard scale is
essential to ensure a local
probe
NLO analysis
Gluon initiated Drell-Yan process
2
M
 M2 >>pT2, TMD factorization, ln 2 resummed by Collins-Soper equation
pT
1982-1983, Collins, Soper
S
2
 S>>M , Kt factorization, ln 2 resummed by BFKL equation
M
1991, Catani, Ciafaloni and Hautmann
1991, Collins and R. K. Ellis
The overlap region S  M 2  pT2
An explicit NLO cross section calculation shows that both the large logarithm appear.
2013, Mueller, Xiao, Yuan
Such joint resummation has been also disscussed in other literatures.
2015, Balitsky and Tarasov; 2015 Marzani
In collinear calculation ln
M2

2
absorbed into PDF.
One natural question:
Does it accommodate both type large logarithms?
S
ln 2
M
M2
ln 2
pT
The main strategy
In a simple quark model, at tree level:
 both large logarithms are absent at LO;
how is it dressed by quantum corrections at NLO?
Real graphs
Sample diagrams:
 Fig.a is the only diagram contributing to
both the CS and the BFKL evolution kernel
 Calculation formulated in the Ji-Ma-Yuan scheme.
One remark
In the infinite momentum frame P+ --> ∞
BFKL
rapidity divergence


0

l



dk
dk
dk

  


0 k
l k
k
Collins-Soper
rapidity divergence
Virtual graphs
Fig.a & Fig.b
Fig.c
BFKL kernel
CS kernel
Collecting all results
In the leading logarithm approximations,
Small x gluon TMD at NLO reads,
2016, ZJ
1
ln
x
S
ln 2
M
x 2 2
ln 2
kT
M2
ln 2
pT
The resulting gluon TMD indeed simultaneously satisfies the both
BFKL equation:
CS equation:
Small x TMDs in CGC at NLO
Sample diagrams
Collinear divergence is absent
Match small x TMDs onto multiple point functions instead of PDFs.
Two step evolution: S
M
2
2
kT
Xiao, Yuan and ZJ, in preparation.
NLO analysis further justifies the use of TMD factorization at small x.
Spin dependent TMDs at small x
1: Inside an unpolarized target
Spin dependent gluon distributions
Unpolarized
Circular polarization
(helicity distribution)
Linear polarization
19
Linearly polarized gluon TMD
Mulders, Rodrigues, 2001
nonperturabtive object
1: fit to experimental data
2: computed on lattice
3: or ...
Quasi-classical model--the MV model
At small x, a semi-hard scale(QS) emerges, gluon TMD in the MV model:
Kovchegov, 96
J. Marian, Kovner, Mclerran & Weigert, 97
21
Gluon TMDs in the MV model
The linearly polarized gluon TMDs in the MV model,
Metz & ZJ, 2011
Weizsäcker-Williams(WW) distribution:
Dipole distribution:
positivity bound saturated for any value of kt
CGC
PCGC(polarized CGC)
2: Inside a transversely polarized target
Three T-odd gluon TMDs
Identify 6 leading power gluon TMDs for a transversely polarized target
(8 in total). Among them, 3 gluon TMDs are T-odd distributions.
Mulders, Rodrigues, 2001
Are the T-odd gluon TMDs relevant at small x?
Quark target model
Meissner, Metz, Goeke; 2007
No 1/x enhancement for T-odd distributions.
Collinear twist-3 contributions
The WW case
 Leading 1/x terms cancel out between different contributions.
The Dipole case (in the small x limit)
Boer, Echevarria, Mulders, ZJ; 2016
Goes beyond the DGLAP treatment
 How to formulate them in the small x formalism?
T-odd gluon TMDs & the odderon
Starting point,
Using time reversal invariance and parity symmetry, at small x one obtains,
Schematically,
Nothing but an odderon operator in CGC
Kovchegov, Szymanowski & Wallon 2004
Hatta, Iancu, Itakura & McLerran 2005
How to identify the spin correlation in the
odderon operator?
Odderon in the MV model
Spin independent odderon has been studied in the MV/Dipole model
Kovchegov, Szymanowski & Wallon 2004
Jeon,Venugopalan 2005
Spin dependent odderon in the MV model
ZJ 2013
 Impact parameter dependent
valence quark distribution (GPDs)
M. Burkardt 2000, 2003
Valence quark
distribution
Spin dependent odderon
Caution: The application of the MV to a proton is less well justified.
In the momentum space:
ZJ 2013
Two different paramertrizations
Equaling two parmetrizations:
Simple algebra leads to
Boer, Echevarria, Mulders, ZJ; 2016
All of three dipole type T-odd gluon TMDs become identical at small x!
SSA in jet production in the backward region
of pp collisions
LO
Dumitru & J. Marian 2002
NLO
Chirilli, Xiao & Yuan 2012
ZJ 2013
Unpolarized gluon distribution
Spin dependent odderon
AnDY at RHIC, 2012
Elliptic gluon GTMD inside a nucleus
GTMD operator definition
See also Bowen's talk and Yoshitaka's talk
Motivations:
 tomography picture of nucleon
 encode information on parton OAM
In the small x limit:
Hatta, Xiao and Yuan, 2016
Dilute limit in the MV model
which is a trivial result,
Taylor expansion leads to
Saturation regime
less trivial to obtain,
• Uniform color source distribution leads to a vanishing eGTMD.
Parametrization of b dependence of Qs
ZJ, 2016
Observable
Quasi-elastic virtual photon-nucleus scattering,
azimuthal independent cross section
cos 2phi modulation
Buchmuller, Gehrmann and Hebecker, 99
Kovchegove and McLerran, 99
ZJ, 2016
Summary
At small x:
 TMD factorization is applicable
 Very rich polarization dependent phenomenology
 Gluon tomography induced by saturation effect
Back up slides
Cos2Ф asymmetries in eA and pA collisions
Di-jet in eA
Metz & ZJ 2011
γ*- jet production in pA
parameter free!
also verified by the calculation performed in the position space
Dominguez, Qiu, Xiao & Yuan 2011
Courtesy of C. Lorce
Physical picture behind the effective TMD factorization
Low jet PT ≤ KT ( only CGC applicable )
High jet PT >> KT ( both CGC and TMD applicable )
Gluons can not resolve the internal
structure of the color dipole system.
four Wilson lines
Collapse to two semi-finite Wilson lines
Time reversal invariance & parity symmetry
Starting point,
Due to time reversal invariance and parity symmetry,
Schematically,
Therefore,
Summary II
Small x physics:
Powerfull theoretical tools




Dipole approach ﹠MV model
BFKL and BK equations
CGC and JIMWLK equation
…
TMD/Spin physics:
Novel ideas/concepts
 T-odd effect
 Universality v.s. process dependent TMDs
 Color entanglement effect
 …
The study of hardon/nucleus structure is benefiting
from the joint force of small x physics and spin physics.