Aspects of TMD evolution of azimuthal

asymmetries

Daniël Boer @ QCD evolution Workshop

Jefferson Lab, May 9, 2013

“Evolution” of TMD Factorization

• Collins & Soper, 1981: e+e- → h1 h2 X [NPB 193 (1981) 381]

• Ji, Ma & Yuan, 2004/5: SIDIS & DY [PRD 71 (2005) 034005 & PLB 597 (2004) 299]

• J. Collins, 2011: “Foundations of perturbative QCD” [Cambridge University Press]

• Sun, Xiao & Yuan, 2011: Higgs production (gluon TMDs) [PRD 84 (2011) 094005]

• Echevarria, Idilbi & Scimemi, 2012: DY (SCET) [JHEP 1207 (2012) 002] • Ma, Wang & Zhao, 2012: quarkonium production [arXiv:1211.7144]

Main differences among the various approached:- treatment of rapidity/LC divergences, in order to make each factor well-defined- redistribution of terms to avoid large logarithms

Application to azimuthal asymmetries

• DB, 2001: CS 81, Collins effect in e+e- and SIDIS [NPB 603 (2001) 195]

• Idilbi, Ji, Ma & Yuan, 2004: JMY, DLLA, SIDIS & DY [PRD70 (2004) 074021] • DB, 2009: CS 81 modified, Collins effect in e+e- [NPB 806 (2009) 23]

• Aybat & Rogers, 2011: JCC, Sivers function [PRD 83 (2011) 114042] Aybat, Collins, Qiu, Rogers, 2012 [PRD 85 (2012) 034043] Aybat, Prokudin & Rogers, 2012: Sivers effect in SIDIS [PRL 108 (2012) 242003]

• Anselmino, Boglione, Melis, 2012: Sivers effect in SIDIS [PRD 86 (2012) 014028] Godbole, Misra, Mukherjee, Rawoot: SSA J/ψ production [arXiv:1304.2584]

• Sun & Yuan, 2013: Sivers effect in SIDIS [arXiv:1304.5037]

• DB, 2013: update of older work using JCC, Sivers effect [arXiv:1304.5387]

Main differences among the various approached:- treatment of nonperturbative Sudakov factor- treatment of leading logarithms

Outline

I. TMD evolution of azimuthal asymmetries

II. Comparison of Q dependence of asymmetries at high and low QT

III. Weighted asymmetries

TMD evolution of azimuthal asymmetries

Part I

For a short summary cf. J.C. Collins, arXiv:1107.4123

2011: new TMD factorization

Form of TMD factorization proven:

dσ = H × convolution of AB + high-qT correction (Y ) + power-suppressed

A & B are TMD pdfs or FFs, the soft factor (U) has been absorbed in them, so as to have a TMD definition that is free from rapidity and Wilson-line self-energy divergences

A and B in transverse coordinate space are functions of:

x, bT, a rapidity ζ , and the renormalization scale μ

Factorization applies to e+e- → h1 h2 X, SIDIS and DY

Evolution of Sivers function

Aybat & Rogers, PRD 83 (2011) 114042Aybat, Collins, Qiu, Rogers, PRD 85 (2012) 034043

) S

h

sin

(UT

A

(GeV) hP0 0.2 0.4 0.6 0.8 1 1.2

0

0.05

0.1

0.15

HERMESCOMPASS

TMD evolution

) S

h

sin

(UT

A

)2 (GeV2Q20 40 60 80 100

0.01

0

0.01

0.02

0.03

0.04

HERMES, COMPASS

EIC

RHIC

Evolution of the Sivers Asymmetry

Aybat, Prokudin & Rogers,PRL 108 (2012) 242003

Integrated asymmetry falls off fast

Not like 1/Qα, but in the considered range between1/Q and 1/Q2

Sp(b,Q,Q0) =CF

π

Q2

Q20

dµ2

µ2αs(µ) ln

Q2

µ2− 16

33− 2nfln

Q2

Q20

ln

ln

µ2b/Λ

2

ln (Q20/Λ

2)

fa1 (x, b

2; ζF , µ) Db1(z, b

2; ζD, µ) = e−S(b,Q,Q0)fa1 (x, b

2;Q20, Q0) D

b1(z, b

2;Q20, Q0)

New TMD factorization expressions

Using only this last expression in the factorization expression is valid for Q2 very large, when the restriction b2 << 1/Λ2 is justified

If also larger b contributions are important, at small QT, then one needs to include a nonperturbative Sudakov factor

dσ

dΩd4q=

d2b e−ib·qT W (b, Q;x, y, z) +O

Q2

T /Q2

W (b, Q;x, y, z) =

a

fa1 (x, b

2; ζF , µ)Da1(z, b

2; ζD, µ)H (y,Q;µ)

b∗ = b/

1 + b2/b2max ≤ bmaxW (b) ≡ W (b∗) e−SNP (b)

SNP (b,Q) = ln(Q2/Q20)g1(b) + gA(xA, b) + gB(xB , b) Q0 =

1

bmax

W(b*) can be calculated within perturbation theoryIn general the nonperturbative Sudakov factor is of the form:

Collins, Soper & Sterman, NPB 250 (1985) 199

Nonperturbative Sudakov factor

The g.. functions are not calculable in perturbation theory and need to be fitted to data, in fact, they are necessary to describe available data

SNP (b,Q,Q0) =

0.184 ln

Q

2Q0+ 0.332

b2

Recently criticized by Sun & Yuan, arXiv:1304.5037

New SNP by Aybat & Rogers (x=0.1):

Aab(x, z,QT ) =f⊥ a1T (x;Q0)Da

1(z;Q0)

M2f b1(x;Q0)Db

1(z;Q0)A(QT )

A(QT ) ≡ M

db b2 J1(bQT ) exp (−Sp(b∗, Q,Q0)− SNP (b,Q/Q0))db b J0(bQT ) exp (−Sp(b∗, Q,Q0)− SNP (b,Q/Q0))

Assume that the TMDs of b* are slowly varying functions of b in the dominant b region:

DB, arXiv:1304.5387

Sivers asymmetry expression

This can also be directly compared to old expressions, allowing to estimate impact of new factorization on old results

Under this assumption, the same factor appears in e+e- → h1 h2 X, SIDIS and DY and in all asymmetries involving one b-odd TMD, such as the Collins asymmetry

Here the focus will be on Q and QT dependence, rather than on x and z dependence

Aab(x, z,QT ) ≡db b2 J1(bQT ) f⊥ a

1T (x, b2∗;Q20, Q0) Da

1(z, b2∗;Q

20, Q0) exp (−Sp(b∗, Q,Q0)− SNP (b,Q/Q0))

MQT

db b J0(bQT )f b

1(x, b2∗;Q

20, Q0)Db

1(z, b2∗;Q

20, Q0) exp (−Sp(b∗, Q,Q0)− SNP (b,Q/Q0))

TMD evolution of the Sivers asymmetry

DB, arXiv:1304.5387

) S

h

sin

(UT

A

)2 (GeV2Q20 40 60 80 100

0.01

0

0.01

0.02

0.03

0.04

HERMES, COMPASS

EIC

RHIC

0 20 40 60 80 1000

0.5

1

1.5

Q

1!Q0.68A"QT,max#

0 20 40 60 80 1001.61.82.02.22.42.62.83.0

Q

QT,max

0 1 2 3 4 50

0.5

1

1.5

QT

A!QT" 906030103.33

Q #GeV$

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5

A

QT [GeV]

Q=10 GeV

Q=90 GeV

Comparison old and new factorization

“CS 81”

SARNP (b,Q,Q0) =

0.184 ln

Q

2Q0+ 0.332

b2

SLYNP (b,Q,Q0) =

0.58 ln

Q

2Q0+ 0.11

b2

Ladinsky & Yuan, PRD 50 (1994) R4239

Very similar!

0 1 2 3 4 50

0.5

1

1.5

QT

A!QT" 906030103.33

Q #GeV$

AB09(QT ) ≡ M

db b2 J1(bQT ) U(b∗;Q0,αs(Q0)) exp (−SB09(b∗, Q,Q0)− SNP (b,Q/Q0))db b J0(bQT ) U(b∗;Q0,αs(Q0)) exp (−SB09(b∗, Q,Q0)− SNP (b,Q/Q0))

Use instead of Ladinsky-Yuan, the new Aybat-Rogers SNP:

Q behavior of azimuthal asymmetries not affect very much, but new TMD factorization preferred because of its small b (<1/Q) behavior

Old factorization, new SNP

Conclusion: the peak of the Sivers asymmetry decreases as 1/Q0.7±0.1

Test needs large Q2 range, requires an EIC Peak of the asymmetry shifts slowly towards higher QT, also offers a test

0 1 2 3 4 50

0.5

1

1.5

QT

AB09!QT" 906030103.33

Q #GeV$

0 20 40 60 80 1000

0.5

1

1.5

Q

1!Q0.65AB09"QT,max#

Double Collins effect gives rise to an azimuthal asymmetry cos 2φ in e+e- → h1 h2 XDB, Jakob Mulders, NPB 504 (1997) 345

- T

sT

sk kπTπ

H⊥1 =

T

Double Collins Effect

b = |b|

∆(z, b) =M

4

D1(z, b

2)PM

+

∂

∂b2H

⊥1 (z, b2)

2 b PM2

Collins effect (NPB 396 (1993) 161):

Clearly observed in experiment by BELLE (R. Seidl et al., PRL '06; PRD '08) and BaBar (I. Garzia at Transversity 2011 & this workshop)

DB, NPB 603 (2001) 195 & 806 (2009) 23

Considerable Sudakov suppression ~1/Q(effectively twist-3)

Double Collins Asymmetrydσ(e+e− → h1h2X)

dz1dz2dΩd2qT

∝ 1 + cos 2φ1A(qT )

A(QT ) =

a e

2a sin

2 θ H⊥(1)a1 (z1;Q0) H

⊥(1)a1 (z2;Q0)

b e2b(1 + cos2 θ) Db

1(z1;Q0) Db1(z2;Q0)

A(QT )

A(Q,QT , Q0) = M2

db b3 J2(bQT ) U(b∗;Q0,αs(Q0)) exp (−S(b∗, Q,Q0)−SNP (b,Q/Q0))db b J0(bQT ) U(b∗;Q0,αs(Q0)) exp (−S(b∗, Q,Q0)−SNP (b,Q/Q0))

Involves approximation Δ(z,b*) ≈ Δ(z,cst) i.o.w. perturbative tail of TMDs is dropped

Gives same result for cos2φ in SIDIS & DY 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5

A

QT [GeV]

Q=10 GeV

Q=90 GeV

“CS 81”

Comparison of high and low QT

Part II

Angular integrated cross section

The CSS expression matches CS factorization of the angular integrated cross section at low QT to fixed order pQCD collinear factorization calculations at high QT

Collins, Soper & Sterman, NPB 250 (1985) 199

If one considers the high QT limit of the low QT result, then the small b region becomes dominant and the perturbative tail of the TMDs must be included

This leads to integral over momentum fractions and mixing between q & g

For the unpolarized, angular independent part of the cross section, this matchesonto the leading logarithmic behavior of the high QT expressions obtained in collinear factorization at fixed order in pQCD

Angular independent part

The low QT limit of the high QT expression:

The high QT limit of the low QT expression:

From insertion of the high QT limit of the TMDs:

Works to all orders (CSS resummation)

Angular dependent part

The “right” logarithm:

The “wrong” splitting functions:

For cos φ and cos 2φ dependences of the cross section this does not work!DB, Vogelsang, PRD 74 (2006) 014004; Bacchetta, DB, Diehl, Mulders, JHEP 0808 (2008) 023

The low QT limit of the high QT expression for the cos φ dependence:

Not those of the tail of the unpolarized TMDs

Perturbative tails of cos φ

In the low QT TMD region a cos φ asymmetry is generated at twist-3: ~ f⊥ D1 + f1 D⊥ It so happens that the “wrong” splitting functions are of the perturbative tails of f⊥ & D⊥

Bacchetta, DB, Diehl, Mulders, JHEP 0808 (2008) 023

High QT limit of cos φ asymmetry at low QT

This “almost match” hints at modified TMD factorization at twist-3

(note: the tails of the chiral odd functions are suppressed at high QT)

Insert tails in “naive” twist-3 TMD factorized expression (which includes Cahn effect):

For details see: Bacchetta, DB, Diehl, Mulders, JHEP 0808 (2008) 023

In other words, assuming TMD factorization beyond leading twist, the f⊥ D1 cos φ term at low QT almost matched onto the fixed order collinear factorization result at high QT

cos 2φ

Bacchetta, DB, Diehl, Mulders, JHEP 0808 (2008) 0230

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8QT

ν = νh⊥1+ νpert +O(

Q2T

Q2or

M2

Q2T

)

Nontrivial since a ratio of sums becomes approximately a sum of ratios

The low QT part falls off approx. as 1/Q, which is slower than the 1/Q2 at high QT

This would be have been a problem if one had to match them

The cos 2φ asymmetry as function of QT has different high and low QT contributions

At low QT: ~ h1⊥ H1⊥, with M2/QT2 suppressed high-QT tail given by chiral-odd QS effect At high QT: ~ f1 D1, which is QT2/Q2 suppressed at low QT

The two contributions both need to be included, which is not double counting

The Qiu-Sterman effect determines the large pT behavior of the Sivers effect

This yields precisely the high QT result!Ji, Qiu, Vogelsang, Yuan, PRL 97 (2006) 082002; PLB 638 (2006) 178;Koike, Vogelsang, Yuan, PLB 659 (2008) 878

Matching Sivers and Qiu-Sterman effects

A CSS resummation expression for the Sivers asymmetry has been derived Kang, Xiao, Yuan, PRL 107 (2011) 152002

f1(x,p2T )

p2TM2

∼ αs1

p2T

(K ⊗ f1) (x)

f⊥1T (x,p

2T )

p2TM2

∼ αsM2

p4T

(K ⊗ TF ) (x)

TF (x, x)A+=0∝ F.T. P | ψ(0)

dη− F+α(η−) γ+ ψ(ξ−) |P

Qiu & Sterman, PRL 67 (1991) 2264

Evolution of TF is known; tricky because dTF(x,x)/dlnμ depends on TF(x,y) for x≠y Kang, Qiu, PRD 79 (2009) 016003; Zhou, Yuan, Liang, PRD 79 (2009) 114022Vogelsang, Yuan, PRD 79 (2009) 094010; Braun, Manashov, Pirnay, PRD 80 (2009) 114002

Evolution of the high-QT tail

Braun, Manashov, PirnayPRD 80 (2009) 114002

It evolves logarithmically with Q2, but considerably faster than f1

Simplification in the large x limit

Weighted Asymmetries

Part III

Weighted asymmetries were first considered, because azimuthal asymmetries in differential cross sections are convolution expressions of TMDs that appear in different processes in different ways; weighting projects out “portable” functions Kotzinian, Mulders, PLB 406 (1997) 373; DB, Mulders, PRD 57 (1998) 5780

Why weight?

This is only nonzero for b-odd TMDs, such as the Sivers function

d2qT qαT

dσ

d2qT−→

d2b δ(b)

∂

∂bαW (b)

−→ U(0) exp(−S(0)) f⊥1T (x) D1(z)

Specific weighted asymmetries are insensitive to Sudakov suppression (1/Qα) DB, NPB 603 (2001) 195

E.g. for single spin asymmetries the specific weighted integral is:

The only Q2 dependence that remains is through H(Q;αs(Q)), i.e. logarithmic

Works for Sivers because it “matches”, but not for cos 2φ because of the dominance of the Y term, which is unrelated to the leading twist TMD expression

“Works” is tricky, since integral may still diverge

To by-pass these tricky issues due to the perturbative tail of the asymmetries, one can consider a new type of weighting: Bessel weighting DB, Gamberg, Musch, Prokudin, JHEP 10 (2011) 021

Why Bessel weight?This weighting assumes that: - integral converges- integral over TMD expression (without Y term) is fine

Conventional weight for Sivers asymmetry: W ≡ |Ph⊥|/zM sin(φh − φS)

Bessel weighting: |Ph⊥|n → Jn(|Ph⊥|BT )n!

2

BT

n

In the limit BT → 0 conventional weights are retrieved

“TMD region”, “Y region”,

||res

1/

“good” -range

Fourier transfo rm

||

1/||res

If BT is not too small, the TMD region should dominateAllows to suppress Y term contribution & allows calculation of TMDs on the lattice!

Why Bessel weight?

In the limit BT → 0 of conventional weights, Y term becomes very important and divergences may arise

The cos 2φ asymmetry as function of QT has different high and low QT contributions

At low QT: ~ h1⊥ H1⊥ At high QT: ~ f1 D1, i.e. dominated by the perturbative contribution

Weighted cos 2φ asymmetry

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8QT

For cos 2φ asymmetry the appropriate weighting would be with QT2:

d2qT q2T

dσ

d2qT

Unfortunately sensitive mainly to the high QT part of the asymmetry ~QT2/Q2

Teaches us nothing about TMD part

Bessel weighting allows to emphasize the TMD region and suppress the Y contribution

py(x)TU =

d2pT py Φ(+)[γ+](x, pT , P, S, µ2, ζ, ρ)d2pT Φ(+)[γ+](x, pT , P, S, µ2, ζ, ρ)

S±=0, ST=(1,0)

= Mf⊥(1)1T (x;µ2, ζ, ρ)

f (0)1 (x;µ2, ζ, ρ)

py(x)BTTU =

d|pT | |pT |

dφp

2J1(|pT |BT )BT

sin(φp − φS)Φ(+)[γ+](x, pT , P, S, µ2, ζ, ρ)d|pT | |pT |

dφpJ0(|pT |BT ))Φ(+)[γ+](x, pT , P, S, µ2, ζ, ρ)

|ST |=1

= Mf⊥(1)1T (x,BT ;µ2, ζ, ρ)

f (0)1 (x,BT ;µ2, ζ, ρ)

Sivers shiftConsider the average transverse momentum shift orthogonal to a given transverse polarization:

And its Bessel-weighted analogue:

For BT in the TMD region, operators only multiplicatively renormalized (corresponding to TMD factorization without operator mixing and no x integral)

Therefore, scale dependence cancels in ratio [Musch et al., 2011]

Sivers function on the lattice

SIDIS DY

SiversShift, ud quarks

Ζ 0.39,bT 0.12 fm,

mΠ 518 MeV

10 5 0 5 10 0.6

0.4

0.2

0.0

0.2

0.4

0.6

Ηv lattice units

mNf 1T11

f 110 GeV

Sivers Shift SIDIS,ud quarks

Ζ 0.39,

mΠ 518 MeV

0.0 0.2 0.4 0.6 0.8

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

bT fmmNf 1T11

f 110 GeV

The first `first-principle’ demonstration in QCD that the Sivers function is nonzeroIt clearly corroborates the sign change relation!

ξ−

ξT

ξ−

ξTMusch, Hägler, Engelhardt, Negele & Schäfer, PRD 85 (2012) 094510

Brodsky, Hwang & Schmidt '02; Collins '02; Belitsky, Ji & Yuan '03 f⊥[SIDIS]1T = −f⊥[DY]

1T to be tested

After taking Mellin moments, the Bessel weighted Sivers shift yields a well-defined quantity <kT x ST>(n,BT), that can be evaluated on the lattice

Some model calculations show a node, but not for all flavorsd: Lu, Ma, NPA 741 (2004) 200; Courtoy, Fratini, Scopetta, Vento, PRD 78 (2008) 034002u: Bacchetta, Conti, Radici, PRD 78 (2008) 074010

f⊥[SIDIS]1T = −f⊥[DY]

1T to be tested

Overall sign relation test

f⊥u1T (x, k2T ) = −f⊥d

1T (x, k2T ) +O(N−1c )

For the experimental test the functions must be compared at the same x and kT2

The function may have a scale dependent node as a function x and/or kT

DB, PLB 702 (2011) 242; Kang, Qiu, Vogelsang, Yuan, PRD 83 (2011) 094001

The node can be at different places for different flavors, although one expects:

Pobylitsa, hep-ph/0301236; Drago, PRD 71 (2005) 057501

TMD evolution also important for the sign relation test

Conclusions

I. TMD evolution of azimuthal asymmetries: - New 2011 TMD factorization does not dramatically change partial power law fall-off of azimuthal asymmetries compared to CS 81

- Integrals of the asymmetries may fall off faster than peak

- Dependence on SNP and its correct form to be further investigated

II. Comparison of Q dependence of asymmetries at high and low QT:Matches and mismatches of high and low QT contributions occur: Sivers asymmetry matches; cos φ almost matches, but requires twist-3 TMD factorization; cos 2φ does not match, but is understood

Q dependence of the high and low QT contributions generally differs

III. Weighted Asymmetries:Bessel weighting allows to project out “portable” functions, to consider convergent integrals over all QT , to emphasize the TMD region and suppress Y region, allows for lattice evaluations of TMDs