Some new opportunities for spin physics
at small x
Jian Zhou
Regensburg University
Based on: Phys.Rev. D84 (2011) 051503. A. Metz and ZJ
Phys.Rev. D85 (2012) 114004. A. Schäfer and ZJ
Phys.Rev. D87 (2013) 054010. E. Akcakaya, A. Schäfer and ZJ
arXiv:1308.4961. A. Schäfer and ZJ
arXiv:1308.5912. ZJ
Outline:
 Quark Boer-Mulders distribution inside a
large nucleus
 Gluon Boer-Mulders distribution inside a
large nucleus
 Transverse single spin asymmetris at small
x
 Summary
Quark BM distribution inside a nucleus
and KT broadening
Operator definiation for quark TMDs in SIDIS
Unpolarized quark TMD
quark Boer-Mulders TMD
Gauge link and leading nuclear effect
Liang, Wang & ZJ 2008
Evaluating part of the gauge link in the MV model
Color source outside a nucleon is zero
 Neglect the correlation between different nucleons
 Describe the large nucleus as a homogenous system
Relation between nulceon matrix and nucleus one
with
Relations between nucleon TMDs and nucleus ones:
Schaefer, ZJ 2013
In the semi-hard region, where
with
Nulcear TMD containing more complicated gauge link structure
Photon-jet production in pA collisions:
Bomhof, Mulders & Pijlman 2004
 Gauge link structure
Evaluating multiple Wilson lines by means of recursion
Blaizot, Gelis & Venugopalan 2004
Relation between nucleon TMDs and nucleus ones in photo-jet production
Schaefer, ZJ 2013
In the semi-hard region:
Remark:
 Our method only applies for nuclear TMDs at intermediate and large x
Phenomenological applications
 Process dependent KT broadening
 KT dependence of the
asymmetris in SIDIS and DY
Gluon BM distribution inside a large
nucleus
Distribution of linearly polarized gluons: definition
Mulders, Rodrigues, 2001
Gluon TMD distributions receive the contribution from ISI/FSI→
make gauge link process dependent
in the adjoint representation
in the fundamental representation
or in the adjoint representation
Weizsäcker-Williams distribution
(probability interpretation)
Dominguez, Marquet, Xiao & Yuan, 2011
Dipole distribution
Distribution of linearly polarized gluons: dynamics I
At small x, due to the presence of semi-hard scale(QS), usual unp. gluon distributions
can be computed using Mclerran-Venugopalan model,
Kovchegov, 96
J. Marian, Kovner, Mclerran & Weigert, 97
Following the similar procedure, linearly polarized gluon distribution also can be
computed in the MV model.
Distribution of linearly polarized gluons: dynamics III
WW type:
Metz, ZJ, 2011
Large kt, dilute limit
same perturbative tail
Small kt, dense medium limit
Dipole type:
positivity bound saturated for any value of kt
Remarks: small x evolution.
Dominguez, Qiu, Xiao & Yuan 2011
How to probe the gluon BM?
 Di-jet production in eA scatterings
 Photo-jet production in pA
 Heavy quark pair production in pA
Cos 2 Ф azimuthal asymmetries
Ф: kt Λ Pt
Cos2Ф asymmetries in eA and pA collisions
Effective TMD factorization at small x
Di-jet in eA
Dominguez, Xiao & Yuan 2011
Dominguez, Marque, Xiao & Yuan 2011
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
Heavy quark pair production in pA collisions(Correlation limit):
Akcakaya, Schafer & ZJ 2012
New type of gluon TMDs
Dilute limit:
Boer, Mulders & Pisano 2009
Large Nc and forward limits:
Dominguez, Marquet, Xiao & Yuan 2011
 Extend the earlier work by taking into account
polarization effect, finite Nc effect, and nozero kT effect from proton side.
Transverse single spin asymmetris
p( ↑) + p →π + X
L
π
p
p
R
TMD factorization V.S. Collinear twist-3 approach
 TMD factorization approach:
quark Siverse distribution
gluon Siverse distribution
 Collinear twist-3 approach:
quark-gluon correlation (ETQS function)
tri-gluon correlations
We focuse on tri-gluon correlations, in particular, its asymptotic behavior at small x
Operator defination for tri-gluon correlations
Ji 1992
Kang & Qiu 2008
Receive contributions from:
Beppu, Koike, Tanaka & Yoshida 2010
In the light cone gauge with Mandelstam-Leibbrandt(ML) prescription
 Diagram a
 Diagram b
 Diagrams c, d, e
Differ from the existing results
Kang & Qiu 2008
Braun, Manashov & Pirnay 2009
More work will be needed to settle this issue !
The asymptotic behavior at small x
Within the leading logarithm lnx accuracy, we made approximations
Symmetry relations
Beppu, Koike, Tanaka & Yoshida 2010
Schafer, ZJ 2013
 Phenomenological implications
I.
Gluon Sivers function could change sign at small x
II. Gluon Sivers function dies out quickly with decreasing x
III. C-odd tri-gluon correlation dominates SSA at small x
AUT → Constant, when x →0
?
SSAs at small x in the CGC formalism
 The odderon origin of SSAs
Heavy fermion model
CGC calculation
Ahmedov, Akushevich, Kuraev & Ratcliffe 1999
Ahmedov, Antonov, Bartos, Kuraev & Zemlyanaya 2003
kovchegov & Sievert 2012
a
µ
c
ρ
d abc A A A
In PQCD:
 BKP equation
BLV Solution
b
ν
Bartels 1980, Kwiecinski & Praszalowicz 1980
Bartels, Lipatov & Vacca 2000
SSA in the forward region of pA collisions
kovchegov & Sievert 2012
The interaction between Spin depenent LC wave function & the odderon from
unpolarized nucleon/nucleus
 Spin dependent odderon (transversely polarized proton)
 Formulation in Mueller’s dipole model
Kovchegov, Szymanowski & Wallon 2004
Formulation in the CGC
Hatta, Iancu, Itakura & McLerran 2005
Wilson line
Odderon in the MV model
Mclerran, Venugopalan 1994
Jeon, Venugopalan 2004
The expectation value of the odderon operator
Jeon,Venugopalan 2005
Valence quark distribution
Odderon in the MV model
 Impact parameter dependent valence quark distribution
M. Burkardt 2000, 2003
ZJ 2013
 An earlier attempt to connect SSA phenomena to GPD E has been made.
M. Burkardt 2003
 Remark: The application of the MV model to proton may not be well justified.
SSA in jet production in the backward region
of pp collisions
Unpolarized gluon distribution
Polarized odderon
LO
Dumitru & J. Marian 2002
NLO
Chirilli, Xiao & Yuan 2012
 Remark 1
 Remark 2
 Remark 3
BLV solution
SSA for open charm production in SIDIS
LO
Mueller 1999; Mclerran & Venugopalan 1999
NLO
Balitsky & Chirilli 2011; Beuf 2012
Hard part ( transversely polarized photon)
 Collinear twist-3 approach
Kang, Qiu 2008
Beppu, Koike, Tanaka & Yoshida 2010
KT moment of the odderon & C-odd tri-gluon correlation
 SSA in open charm production and DY/direct photon
Twist-3 approach
Kang, Qiu, Vogelsang & Yuan
2008
Koike & Yoshida 2011, 2012
Summary
What can be done at RHIC
•

•

Observable I:
Deliverable I:
•
Observable II:

Deliverable II:
•
Observable III:

Deliverable III:
for photon-jet production in pA collisions
Dipole-type Boer-Mulders gluon distribution
in heavy quark production in pA collisions
process dependent BM gluon TMDs
in DY process in pA collisions with high
nuclear quark BM function
Observable IV: SSAs for jet and open charm production in pp collisions
Deliverable IV: C-odd tri-gluon correlation & the polarized odderon & BLV
 Golden
 Silver
Spin physics
& TMDs
Bronze measurements at EIC
•
Observable I:
for single hadron production in eA collisions ( high
•
Deliverable I:
nuclear quark BM function
•
Observable II:
for di-jet production in eA collisions.
•
Deliverable II:
WW-type gluon BM distribution.
•
Observable III:
SSAs for c and
•
production in ep collisions.
Deliverable III:
C-odd tri-gluon correlation & the polarized odderon & BLV solution
& Orbital angular momentum of valence quarks
)
Back up slides
KT broadening effect
Momentum broadening: sensitive to the medium properties
How is the quark Boer-Mulders distribution affected by cold nuclear matter ?
Distribution of linearly polarized gluons: phenomenology
Cos 2 Ф azimuthal asymmetries in the various processes:
Ф: kt Λ Pt
 dijet or heavy quark pair production in ep
Boer, Brodsky, Mulders & Pisano, 2011
 heavy quark pair production in pp
Boer, Brodsky, Mulders & Pisano, 2011
 photon pair production in pp
Nadolsky, Balazs, Berger & Yuan, 2007
Qiu, Schlegel & Vogelsang, 2011
 virtual photon-jet production in pA
Metz, ZJ, 2011
It also affects the transverse momentum spectrum of Higgs produced in pp/pA
Mantry & Petriello, 2010
Catani & Grazzini, 2011
Boer, Dunnen, Pisano, Schlegel & Vogelsang, 2011
Sun, Xiao & Yuan, 2011
Schafer, ZJ, 2012
Boer, Dunnen, Pisano & Schlegel, 2013
Matching between TMD and CGC at small x
Dijet production in eA scattering
 high gluon density
→higher twist contribution is not necessarily much smaller than
leading twist contribution (one can not take TMD factorization for granted)
ull CGC
esults
Extrapolating to
the Correlation limit
Hard part
independent kT
Multiple point function
two point function
gluon TMDs
Effective TMD factorization at small x
F. Dominguez, B-W. Xiao, F. Yuan 2011
F. Dominguez, C. Marquet, B-W. Xiao, F. Yuan 2011
 Our task: extend to the azimuthal dependent cross sections…
Cos2Ф asymmetry for dijet production in eA
Eikonal scattering
Where
I. Balitsky, 96
L. D. Mclerran, R. Venugopalan, 99
Scattering amplitude:
Integration over k1t is dominated by the kinematical region k1t ～ Qs
(Qs typical small x gluon transverse momentum)
Cos2Ф asymmetry for dijet production in eA I
In the correlation limit k1t ～ Qs<<lt, employ the power expansion
Neglected terms are suppressed by a power of Qs/lt
Taking a partial integration,
Cos2Ф asymmetry for dijet production in eA II
The cross section reads
with the help of the formula
One can identify,
Quark pair production in pA collisions
In hybrid approach (Lipatov approximation & CGC):
Akcakaya, Schafer & ZJ 2012
Feynman rules:
Catani, Ciafaloni & Hautmann 91
Collins & Ellis 91
Quark pair production in TMD factorization
Bomhof, Mulders, Pijlman, 2004
related to the derivative of four point function at small x
Differential cross section
Four point function
in agreement with the existing result
Blaizot, Gelis & Venugopalan 2004
Employing power expansion