TMD Evolution
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Transcript of TMD Evolution
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TMD Evolution
Feng Yuan Lawrence Berkeley National Laboratory
TMDs: center piece of nucleon structure
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Long. Momentum distributions
NucleonSpin
3D imagingTransverse-momentum-dependent and Generalized PDFs
QCD:Factorization,Universality,Evolution,Lattice, …
kt-dependence crucial to the saturation
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TMDs at small-x
TMDs in valence region
Quark Sivers function leads to an azimuthal asymmetric distribution of quark in the transverse plane
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Alex Prokudin@EIC-Whitepaper
Evolution is crucial to strength the TMD probes
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Two particle correlations from pp to dAu
Evolution?Saturation?
Sign change of Sivers asymmetry
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Drell-Yan, π- (190GeV)p
Q2~3-6GeV2 Q2~16-30GeV2
COMPASS
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 distributionsOne 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 Differential cross section depends on Q1, where
Q2>>Q12>>2
QCD
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
Sudakov, 1956
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How Large of the Resummation effects
ResumResum
NLONLOKulesza, 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 xThrust distribution Jet shape function…
ResBos: Nadolsky, et al., PRD 2003 CSS resummation built in
Single Transverse Spin Asymmetry
Separate the singular and regular parts
TMD factorization in b-space
04/22/23 16Kang, Xiao, Yuan, PRL 11;Rogers et al., PRD, 2012
Evolution equations
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Idilbi-Ji-Ma-Yuan, PRD04
Boer, NPB, 2002
Final resum form
Sudakov the same
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Coefficients at one-loop order
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Constraints from SIDIS
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Sun, Yuan, 1308.5003
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DIS and Drell-Yan
Initial state vs. final state interactions
“Universality”: QCD prediction
HERMES/COMPASS
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Drell-Yan DIS
Predictions for COMPASS
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Drell-Yan, π- (190GeV)p
Q2~3-6GeV2 Q2~16-30GeV2
COMPASS
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 targetW/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 TargetQT 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
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.
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√S = 500GeV
Drell-Yan Q=6GeV
Sun, Yuan, 1308.5003
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√S = 510GeV
-0.06-0.06
Rapidity of W Rapidity of W
Pt(GeV)Pt(GeV)
y=0y=0
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QCD evolution reduces the asymmetries abouta factor of 3 for W/Z as compared to Drell-Yan
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
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CSS
PJ>>KT
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)
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, α0
Rapidity divergence04/22/23 32
Mueller, Xiao, Yuan, PRL110,082301 (2013);arXiv:1308.2993
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 separatedSoft 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 ruleSterman, et alSub-leading logs will be much complicated,
usually a matrix form
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Mueller, Xiao, Yuan, PRL110,082301 (2013);arXiv:1308.2993
all order
factorization
04/22/23 36Similar calculations for pp collisions:Zhu HX, et al., PRL110 (2013) 082001
Dijet azimuthal correlation at colliders
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PRL 94, 221801 (2005)
preliminary
Peng Sun, et al.
LO
NLL-resummation
will be extended to di-hadrons,
Two particle correlations in Central dAu collisions
η1~η2~3.2
Q2sA~0.85A(1/3) Qsp
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38Stasto,Xiao,Yuan,PLB716,430(2012)
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 understoodProvide solid ground for phen. ApplicationsUnique place to study QCD
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