TMD Evolution

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TMD Evolution

• Feng Yuan
• Lawrence Berkeley National Laboratory

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TMDs center piece of nucleon structure
QCD Factorization, Universality, Evolution, Latti
ce,
Nucleon Spin
Long. Momentum distributions
3D imaging Transverse-momentum-dependent and
Generalized PDFs
3
TMDs at small-x

• kt-dependence crucial to the saturation

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TMDs in valence region
Alex Prokudin _at_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?
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Sign change of Sivers asymmetry
Drell-Yan, p- (190GeV)p
COMPASS
Q216-30GeV2
Q23-6GeV2
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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
  Q2gtgtQ12gtgt?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
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-Sterman 85)
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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

Kang, Xiao, Yuan, PRL 11 Rogers et al., PRD, 2012
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Evolution equations
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
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DIS and Drell-Yan

• Initial state vs. final state interactions
• Universality QCD prediction

?
?
DIS
Drell-Yan
HERMES/COMPASS
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Predictions for COMPASS
Drell-Yan, p- (190GeV)p
COMPASS
Q216-30GeV2
Q23-6GeV2
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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

• 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.

vS 500GeV
Drell-Yan Q6GeV
Sun, Yuan, 1308.5003
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y0
y0
vS 510GeV
Pt(GeV)
Pt(GeV)
-0.06
-0.06
Rapidity of 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 di-hadron
  and single-hadron production in forward pA
  collisions
• Complementary to the EIC Missions!!

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Kt-dependent observables
PJgtgtKT
KT

• Hard processes probe the kt-dependent gluon
  distributions directly
• Saturation phenomena manifest in the observables
• Xiao,Yuan, et al,
• PRL106, 022301 (2011)
• PRL105, 062001 (2010)

CSS
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Resummation Sudakov vs BFKL

• Sudakov double logs can be re-summed in the
  small-x saturation formalism
• Radiated gluon momentum
• Soft gluon, aßltlt1
• Collinear gluon, a1, ßltlt1
• Small-x collinear gluon, 1-ßltlt1, a?0
• Rapidity divergence

Mueller, Xiao, Yuan, PRL110,082301
(2013) arXiv1308.2993
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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) arXiv1308.2993
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• all order
• factorization

Similar calculations for pp collisions Zhu HX,
et al., PRL110 (2013) 082001
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Dijet azimuthal correlation at colliders
LO
preliminary
NLL- resummation
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

• ?1?23.2
• Q2sA0.85A(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