TMD Evolution
Feng Yuan
Lawrence Berkeley National Laboratory
4/7/2015
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TMDs: center piece of nucleon
structure
QCD:
Factorization,
Universality,
Evolution,
Lattice, …
Long. Momentum
distributions
Nucleon
Spin
3D imaging
Transverse-momentum-dependent
and Generalized PDFs
2
TMDs at small-x

kt-dependence
crucial to the
saturation
3
TMDs in valence region
Alex Prokudin
@EIC-Whitepaper
Quark Sivers function leads to an azimuthal asymmetric
distribution of quark in the transverse plane
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Evolution is crucial to strength
the TMD probes

Two particle correlations from pp to dAu
Evolution?
Saturation?
5
Sign change of Sivers
asymmetry
Drell-Yan, π- (190GeV)p
COMPASS
Q2~3-6GeV2
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Q2~16-30GeV2
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Outlines
General theory background
 Applying to single spin asymmetries
 Consistent resummation in high enegy

 BFKL
vs Sudakov
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Collinear vs TMD factorization
TMD factorization is an extension and
simplification to the collinear factorization
 Extends to the region where collinear fails
 Simplifies the kinematics

 Power
counting, correction 1/Q neglected
(PT,Q)=H(Q) f1(k1T,Q) f2(k2T, Q) S(T)
 There is no x- and kt-dependence in the hard
factor
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DGLAP vs CSS

DGLAP for integrated parton distributions
 One
hard scale
(Q)=H(Q/) f1()…

Collins-Soper-Sterman for TMDs
 Two
scales, large double logs
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Evolution vs resummation
Any evolution is to resum large logarithms
 DGLPA resum single large logarithms
 CSS evolution resum double logarithms

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Sudakov Large Double Logarithms
Sudakov, 1956


Differential cross section depends on Q1,
where Q2>>Q12>>2QCD
We have to resum these large logs to make
reliable predictions
 QT:
Dokshitzer, Diakonov, Troian, 78; Parisi
Petronzio, 79; Collins, Soper, Sterman, 85
 Threshold: Sterman 87; Catani and Trentadue 89
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How Large of the Resummation
effects
Resum
NLO
Kulesza, Sterman, Vogelsang, 02
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Collins-Soper-Sterman Resummation
Introduce a new concept, the Transverse
Momentum Dependent PDF
 Prove the Factorization in terms of the
TMDs
(PT,Q)=H(Q) f1(k1T,Q) f2(k2T, Q) S(T)
 Large Logs are resummed by solving the
energy evolution equation of the TMDs

(Collins-Soper
81, Collins-Soper-Ste
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CSS Formalism (II)


K and G obey the renormalization group
eq.
The large logs will be resummed into the
exponential form factor
 A,B,C
functions are perturbative calculable.
(Collins-Soper-Sterman 85)
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Two Large Scales Processes

Very success in applications,
 DIS
and Drell-Yan at small PT (QT Resum)
 DIS and Drell-Yan at large x (Threshold
Resum)
 Higgs production at small PT or large x
 Thrust distribution
 Jet shape function
…
ResBos: Nadolsky, et al., PRD 2003
CSS resummation built in
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Single Transverse Spin
Asymmetry

Separate the singular and regular parts

TMD factorization in b-space
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Kang, Xiao, Yuan, PRL 11;
Rogers et al., PRD, 2012 16
Evolution equations
Idilbi-Ji-Ma-Yuan, PRD04
Boer, NPB, 2002
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Final resum form

Sudakov the same
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Coefficients at one-loop order
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Constraints from SIDIS
Sun, Yuan, 1308.5003
4/7/2015
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DIS and Drell-Yan

Initial state vs. final state interactions
*
*
Drell-Yan
DIS
HERMES/C
OMPASS

“Universality”: QCD prediction
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Predictions for COMPASS
Drell-Yan, π- (190GeV)p
COMPASS
Q2~3-6GeV2
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Q2~16-30GeV2
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Fermilab Drell-Yan
120GeV proton beam
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Few words on Drell-Yan at
RHIC

Never been measured before at a collider
 Fixed
target
 W/Z at Tevatron/LHC

Understand the x-evolution of the TMDs,
saturation?
 Compared
to that from HERA
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Drell-Yan at Fixed Target
QT spectrum from E288, PRD23,604(81)
Valence
region
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At very large Q2 (e.g., Z0 and W boson),
No longer a Gaussian
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Predictions at RHIC
√S = 500GeV
Drell-Yan Q=6GeV
Additional theory
uncertainties:
x-dependence of
the TMDs comes
from a fit to fixed
target drell-yan
and w/z production
at Tevatron
---Nadolsky et al.
Sun, Yuan, 1308.5003
4/7/2015
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y=0
Pt(GeV)
y=0
√S = 510GeV
Pt(GeV)
-0.06
-0.06
Rapidity of4/7/2015
W
Rapidity of W
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QCD evolution reduces the asymmetries about
a factor of 3 for W/Z as compared to Drell-Yan
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Uniqueness of forward RHIC
physics
Investigate the sign change of Sivers
asymmetries and the associated QCD
evolution effects in Drell-Yan and W SSAs
 Mapping out the saturation physics in dihadron and single-hadron production in
forward pA collisions

Complementary to the EIC Missions!!
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Kt-dependent observables
PJ>>KT
KT

CSS


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Hard processes probe
the kt-dependent gluon
distributions directly
Saturation phenomena
manifest in the
observables
Xiao,Yuan, et al,
PRL106, 022301 (2011)
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PRL105, 062001 (2010)
Resummation: Sudakov vs BFKL
Sudakov double logs can be re-summed
in the small-x saturation formalism
 Radiated gluon momentum

Soft gluon, α~β<<1
 Collinear gluon, α~1, β<<1
 Small-x collinear gluon, 1-β<<1, α0

 Rapidity
divergence
Mueller, Xiao, Yuan, PRL110,082301 (2013);
4/7/2015 arXiv:1308.2993
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Final result

Double logs at one-loop order

Collins-Soper-Sterman resummation
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Comments
Sudakov double logs can be re-summed
consistently in the small-x formalism
 Kinematics of double logs and small-x
evolution are well separated

 Soft
vs collinear gluons
If Qs is small, back to dilute region
 If Qs is large (~Q), we can safely
neglect the Sudakov effects

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Sudakov leading double logs:
general hard processes

Each incoming parton contributes to a half
of the associated color factor
 Initial

gluon radiation, aka, TMDs
Soft gluon radiation in collinear calculation
also demonstrates this rule
 Sterman,
et al
 Sub-leading logs will be much complicated,
usually a matrix form
Mueller, Xiao, Yuan, PRL110,082301 (2013);
4/7/2015 arXiv:1308.2993
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all order
factorization
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Similar calculations for pp collisions:
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Zhu HX, et al., PRL110 (2013) 082001
Dijet azimuthal correlation at colliders
preliminary
LO
NLLresummation
Peng Sun, et al.
will be extended to di-hadrons,
PRL 94, 221801 (2005)
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Two particle correlations in
Central dAu collisions


η1~η2~3.2
Q2sA~0.85A(1/3) Qsp2
Stasto,Xiao,Yuan,PLB716,430(2012)
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Conclusions
TMDs are important tool to investigate the
partonic structure of nucleon/nucleus, and
the associated QCD dynamics
 Although complicated, the evolution
effects have been well understood

 Provide
solid ground for phen. Applications
 Unique place to study QCD
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