TMD Evolution

04/22/23 1

TMD Evolution

Feng Yuan Lawrence Berkeley National Laboratory

TMDs: center piece of nucleon structure

2

Long. Momentum distributions

NucleonSpin

3D imagingTransverse-momentum-dependent and Generalized PDFs

QCD:Factorization,Universality,Evolution,Lattice, …

kt-dependence crucial to the saturation

3

TMDs at small-x

TMDs in valence region

Quark Sivers function leads to an azimuthal asymmetric distribution of quark in the transverse plane

04/22/23 4

Alex Prokudin@EIC-Whitepaper

Evolution is crucial to strength the TMD probes

5

Two particle correlations from pp to dAu

Evolution?Saturation?

Sign change of Sivers asymmetry

04/22/23 6

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

04/22/23 7

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

04/22/23 8

DGLAP vs CSS

DGLAP for integrated parton distributionsOne hard scale

(Q)=H(Q/) f1()… Collins-Soper-Sterman for TMDs

Two scales, large double logs

04/22/23 9

Evolution vs resummation

Any evolution is to resum large logarithms DGLPA resum single large logarithms CSS evolution resum double logarithms

04/22/23 10

11

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

12

How Large of the Resummation effects

ResumResum

NLONLOKulesza, Sterman, Vogelsang, 02

13

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)

14

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)

15

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

04/22/23 17

Idilbi-Ji-Ma-Yuan, PRD04

Boer, NPB, 2002

Final resum form

Sudakov the same

04/22/23 18

Coefficients at one-loop order

04/22/23 19

Constraints from SIDIS

04/22/23 20

Sun, Yuan, 1308.5003

21

DIS and Drell-Yan

Initial state vs. final state interactions

“Universality”: QCD prediction

HERMES/COMPASS

* *

Drell-Yan DIS

Predictions for COMPASS

04/22/23 22

Drell-Yan, π- (190GeV)p

Q2~3-6GeV2 Q2~16-30GeV2

COMPASS

Fermilab Drell-Yan

120GeV proton beam

04/22/23 23

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

04/22/23 24

25

Drell-Yan at Fixed TargetQT spectrum from E288, PRD23,604(81)

Valence region

26

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.

04/22/23 27

√S = 500GeV

Drell-Yan Q=6GeV

Sun, Yuan, 1308.5003

04/22/23 28

√S = 510GeV

-0.06-0.06

Rapidity of W Rapidity of W

Pt(GeV)Pt(GeV)

y=0y=0

29

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!!

04/22/23 30

Kt-dependent observables

04/22/23 31

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

04/22/23 33

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

04/22/23 34

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

04/22/23 35

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

04/22/23 37

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

2

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

04/22/23 39