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
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