Azimuthal Spin Asymmetries in Light-Cone Quark model · Polarized Drell-Yan at COMPASS Azimuthal spin asymmetries in light-cone constituent quarkmodels S. Boﬃ∗,1,2 A. V. Efremov∗,3

Azimuthal Spin Asymmetries

in

Light-Cone Quark model

Barbara PasquiniPavia U. & INFN, Pavia (Italy)

In questo manuale sono illustrate le regole base per la corretta applicazione del marchio Università di Pavia. Il logo, i caratteri tipografici e i colori scelti sono infatti gli elementi che partecipano alla costruzione dell’identità visiva di qualsiasi attore che voglia presentarsi al mercato, sia esso un prodotto mass market, un’Istituzione o un ateneo. Sono il suo volto commerciale ma anche istituzionale, quello che permetterà all’Università di Pavia di essere riconoscibile nel tempo agli occhi del suo pubblico interno, ma anche esterno. Proprio per il ruolo centrale che rivestono, tali elementi devono essere rappresentati e utilizzati

secondo regole precise e inderogabili, al fine di garantire la coerenza e l’efficacia dell’intero sistema di identità visiva. Per questo è importante che il manuale, nella sua forma cartacea o digitale, venga trasmesso a tutti coloro che in futuro si occuperanno di progettare elementi di comunicazione per l’Università di Pavia.

INTRODUZIONE: L’IMMAGINE UNIVERSITÀ DI PAVIA

UNIVERSITÀ DI PAVIA

!TMDs from Light-Cone Quark Model for proton and pion

!Model results for Drell-Yan asymmetry

!Conclusions

Outline

!Drell-Yan at COMPASS with pion beam and transversely polarized target

‣asymmetry with convolution of proton pretzelosity and pion Boer-Mulders

‣physical content of proton pretzelosity

‣transverse-spin structure of quarks in the pion

Polarized Drell-Yan at COMPASS

Azimuthal spin asymmetries in light-cone constituent quark models

S. Bo!!,1, 2 A. V. Efremov!,3 B. Pasquini,1, 2 P. Schweitzer,4 and F. Yuan!5, 6

1Dipartimento di Fisica Nucleare e Teorica, Universita degli Studi di Pavia, Pavia, Italy2Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Pavia, Italy

3Joint Institute for Nuclear Research, Dubna, 141980 Russia4Department of Physics, University of Connecticut, Storrs, CT 06269, USA

5RIKEN BNL Research Center, Building 510A, BNL, Upton, NY 11973, U.S.A.6Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.

(Dated: Spring 2010)

First notes on azimuthal and/or spin asymmetries due to T-odd TMDs in the Drell-Yan process(DY) and possibly (at a later stage) also semi-inclusive deeply inelastic scattering (SIDIS) (?). Thepurpose is to prepare practical predictions, most practically for COMPASS meeting 26 April 2010,on the basis of results from light-cone quark models for TMDs in nucleon [1], and pion(!) [2].The “!” marks not-informed authors, who might be interested and could become prospectiveinformed-co-authors, upon agreement.

PACS numbers: 12.38.Bx, 12.39.St, 13.85.QkKeywords:

I. INTRODUCTION

We study DY, see Fig. 1. This Figure and its caption are shamelessly borrowed from [3]. We will work in theCollins-Soper frame (CS) which is a particular dilepton rest frame, see [3] for a nice and pedagogical review.

X

H

H

Xl

la

b

+

−

qq

a

b−

X

H

H

Xl

la

b

+

−

a

b

q−q

FIG. 1: Amplitude for dilepton production in parton model approximation. Both diagrams have to be taken into account. Thespectator systems Xa and Xb of the two hadrons are not detected.

The kinematical variables are s = (p1 + p2)2, the dilepton invariant mass Q2 = (k1 + k2)2 with p1/2 (and k1/2)indicating the momenta of the incoming proton and hadron h (and the outgoing lepton pair), and the rapidity

y =1

2ln

p1(k1 + k2)

p2(k1 + k2). (1)

The parton momenta x1/2 in the TMDs are fixed in terms of s, Q2 and y as follows

xa/b =

!

Q2

se±y , (2)

and typically the observables are presented as functions of y. Below we will have COMPASS in mind, where thekinematics is s ! 400 GeV2 and Q2 = 20 GeV2.

Alternatively, one can use xF = xa " xb with xaxb = Q2/s.

a

bµ+

µ-

dσ

d4qdΩ=

α2

Fq2σU

1 +D[sin2 θ]A

cos 2φU cos 2φ

+|ST |D[sin2 θ]

Asin(2φ+φS)

T sin(2φ+ φS) +Asin(2φ−φS)T sin(2φ− φS)

+ AsinφS

T sinφS

! DY cross section at leading-twist for transversely polarized target:

Ha : proton target

Hb : pion beam

Asin(2φ+φS)T

Acos 2φU Boer-Mulders functions of incoming hadrons

Boer-Mulders functions of pion beam and pretzelosity of nucleon target

AsinφS

T unpolarized TMD of pion beam and Sivers function of nucleon target

Asin(2φ−φS)T Boer-Mulders functions of pion beam and transversity of nucleon target

talk of C. Quintans

Polarized Drell-Yan at COMPASS

Azimuthal spin asymmetries in light-cone constituent quark models

S. Bo!!,1, 2 A. V. Efremov!,3 B. Pasquini,1, 2 P. Schweitzer,4 and F. Yuan!5, 6

1Dipartimento di Fisica Nucleare e Teorica, Universita degli Studi di Pavia, Pavia, Italy2Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Pavia, Italy

3Joint Institute for Nuclear Research, Dubna, 141980 Russia4Department of Physics, University of Connecticut, Storrs, CT 06269, USA

5RIKEN BNL Research Center, Building 510A, BNL, Upton, NY 11973, U.S.A.6Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.

(Dated: Spring 2010)

First notes on azimuthal and/or spin asymmetries due to T-odd TMDs in the Drell-Yan process(DY) and possibly (at a later stage) also semi-inclusive deeply inelastic scattering (SIDIS) (?). Thepurpose is to prepare practical predictions, most practically for COMPASS meeting 26 April 2010,on the basis of results from light-cone quark models for TMDs in nucleon [1], and pion(!) [2].The “!” marks not-informed authors, who might be interested and could become prospectiveinformed-co-authors, upon agreement.

PACS numbers: 12.38.Bx, 12.39.St, 13.85.QkKeywords:

I. INTRODUCTION

We study DY, see Fig. 1. This Figure and its caption are shamelessly borrowed from [3]. We will work in theCollins-Soper frame (CS) which is a particular dilepton rest frame, see [3] for a nice and pedagogical review.

X

H

H

Xl

la

b

+

−

qq

a

b−

X

H

H

Xl

la

b

+

−

a

b

q−q

FIG. 1: Amplitude for dilepton production in parton model approximation. Both diagrams have to be taken into account. Thespectator systems Xa and Xb of the two hadrons are not detected.

The kinematical variables are s = (p1 + p2)2, the dilepton invariant mass Q2 = (k1 + k2)2 with p1/2 (and k1/2)indicating the momenta of the incoming proton and hadron h (and the outgoing lepton pair), and the rapidity

y =1

2ln

p1(k1 + k2)

p2(k1 + k2). (1)

The parton momenta x1/2 in the TMDs are fixed in terms of s, Q2 and y as follows

xa/b =

!

Q2

se±y , (2)

and typically the observables are presented as functions of y. Below we will have COMPASS in mind, where thekinematics is s ! 400 GeV2 and Q2 = 20 GeV2.

Alternatively, one can use xF = xa " xb with xaxb = Q2/s.

a

bµ+

µ-

dσ

d4qdΩ=

α2

Fq2σU

1 +D[sin2 θ]A

cos 2φU cos 2φ

+|ST |D[sin2 θ]

Asin(2φ+φS)

T sin(2φ+ φS) +Asin(2φ−φS)T sin(2φ− φS)

+ AsinφS

T sinφS

! DY cross section at leading-twist for transversely polarized target:

Ha : proton target

Hb : pion beam

Asin(2φ+φS)T

Acos 2φU Boer-Mulders functions of incoming hadrons

Boer-Mulders functions of pion beam and pretzelosity of nucleon target

AsinφS

T unpolarized TMD of pion beam and Sivers function of nucleon target

Asin(2φ−φS)T Boer-Mulders functions of pion beam and transversity of nucleon target

talk of C. Quintans

Table 9: Expected statistical errors for various asymmetries assuming two years of datataking and a beam momentum of 190 GeV/c.

Asymmetry Dimuon mass (GeV/c2)2 < Mµµ < 2.5 J/ψ region 4 < Mµµ < 9

δ Acos 2φ

U0.0020 0.0013 0.0045

δ Asin φST

0.0062 0.0040 0.0142

δ Asin(2φ+φS)T

0.0123 0.008 0.0285

δ Asin(2φ−φS)T

0.0123 0.008 0.0285

!!"#"$#%#"&'( "&') "&'* "&'+ & &'+ &'* &') &'("&',

"&'&-

&

&'&-

&',

&',-

&'+

&'+-.#/012

34

!!"#"$#%#"&'( "&') "&'* "&'+ & &'+ &'* &') &'("&',

"&'&-

&

&'&-

&',

&',-

&'+

&'+-

#56/2+74

!!"#"$#%#"&'( "&') "&'* "&'+ & &'+ &'* &') &'("&',

"&'&-

&

&'&-

&',

&',-

&'+

&'+-

8.#92#/012:+34

!!"#"$#%#"&'( "&') "&'* "&'+ & &'+ &'* &') &'("&',

"&'&-

&

&'&-

&',

&',-

&'+

&'+-

8.#"2#/012:+34

Figure 28: Theoretical predictions and expected statistical errors on the Sivers (top-left),Boer–Mulders (top-right), sin(2φ + φS) (bottom-left) and sin(2φ − φS) (bottom-right)asymmetries for a DY measurement π

−p → µ

+µ−X with a 190 GeV/c π

− beam in thehigh-mass region 4 GeV/c

2< Mµµ < 9 GeV/c

2. In case of the Sivers asymmetry also thesystematic error is shown (smaller error bar).

59

AsinφS

T

Asin(2φ+φS)T

AsinφS

T

Asin(2φ−φS)T

f1⨂Sivers

BM⨂Pretzelosity BM⨂Transversity

BM⨂BMAnselmino et al., PRD79 (2009)

Zhang et al., PRD77 (2008)

Sissakian et al., Phys.Part.Nucl.41 (2010)

COMPASS-II Proposal

COMPASS collaboration, COMPASS II proposal, CERN-SPSC-2010-014, SPSC-P-340

no theoretical predictions!

DY: 4 < Mµµ < 9 GeV

Boer-Mulders function of the pion

(↑↑)LC (↑↓)LC (↓↓)LC

LCWF:

invariant under boost, independent of Pµ

internal variables:

[Brodsky, Pauli, Pinsky, ’98]

Light-Cone Wave Function of the Pion

ΨLCπβ (xi,k⊥,i)

Lzq =0Lz

q = -1

Jzq = -Lzq

2 independent wave function amplitudes isospin symmetry

parity time reversal

Lzq = 1

LCWF: eigenstate of total orbital angular momentum

|P,π =

β

d[1][2]ΨLC

πβ (xi,k⊥,i)δij√3q†iλi

(1)q†jλ2|0

ΨLCπβ (xi,k⊥,i,λi) = ΨMom(xi,k

2⊥,i)⊗ΨSpin(xi,k⊥,iλi)

Phenomenological Light-Cone Wave Function of the Pion

ΨMom(xi,k2⊥,i)! : momentum-space component gaussian shape

two parameters: mq and gaussian width fitted to exp. charge radius and pion decay constant

! : spin-dependent part eigenstate of the total angular momentum operator in light-front dynamics (Lz = 0, | Lz | =1)ΨSpin(xi,k⊥,i,λi)

ΨLCπβ (xi,k⊥,i,λi) = ΨMom(xi,k


Phenomenological Light-Cone Wave Function of the Pion

ΨMom(xi,k2⊥,i)! : momentum-space component gaussian shape

two parameters: mq and gaussian width fitted to exp. charge radius and pion decay constant

0

0.2

0.4

0.6

0.8

1

0 0.1 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4

Electromagnetic Form Factor

Q2 (GeV2) Q2 (GeV2)

F! F!Q2

! : spin-dependent part eigenstate of the total angular momentum operator in light-front dynamics (Lz = 0, | Lz | =1)ΨSpin(xi,k⊥,i,λi)

Model calculation of pion Boer-Mulders

h⊥1T

one-gluon exchange approximation

−

±

overlap of LCWFs in S and P waves

0

++ −

±0 00

+ h.c.+ h.c.+

Mismatch of helicity between initial and final state !Lz = ± 1

Model calculation of pion Boer-Mulders

h⊥1T

one-gluon exchange approximation

−

±

overlap of LCWFs in S and P waves

0

++ −

±0 00

+ h.c.+ h.c.+

Mismatch of helicity between initial and final state !Lz = ± 1

h⊥1T [SIDIS] = −h⊥

1T [DY ]

crossing

SIDISfinal state interactions

DYinitial state interactions

h⊥u1

h⊥u1 (π+) = h⊥d

1 (π+) = h⊥d1 (π−) = h⊥u

1 (π−)

Boer-Mulders functions in LCCQM

Pion Proton

xk2⊥ k2⊥ x

h⊥u1

h⊥u1

h⊥u1 (π+) = h⊥d

1 (π+) = h⊥d1 (π−) = h⊥u

1 (π−)

Boer-Mulders functions in LCCQM

Pion Proton

xk2⊥ k2⊥ x

h⊥u1

B.P., Schweitzer, Efremov, in preparation

P-D int.

S-P int.

S-P int.

P-D int.

TOTTOT

-3-2.5

-2-1.5

-1-0.5

0

0 0.2 0.4 0.6 0.8 1

h (1)u = h (1)d1 1

-

xB.P., F. Yuan, PRD81(2010)

TOT = S-P int.

downuph⊥(1)q1 =

d2k⊥

k2⊥

2M2ph⊥q1h⊥(1)u

1 =d2k⊥

k2⊥

2m2πh⊥u1

Burkardt, PRD66 (02)

unpolarized quarks

Chromodynamic lensing


transversely pol. quark


Distortion in impact parameter(related to GPD ET)




Distortion in transverse momentum(related to Boer Mulders function)

Final-state interaction(lensing function)




Distortion in transverse momentum(related to Boer Mulders function)

Final-state interaction(lensing function)

Boer-Mulders function lensing function FT of chiral-odd GPD

2m2πh

⊥(1)1 (x) ≈

d2bTbT · I(x,bT )

∂

∂b2TEπT (x,b

2T )

model-dependent relation

<by>=0.197 fm

Frederico, Pace, BP, Salme’,PRD80 (2009)

Light-Cone CQM

Broemmel et al., PRL101,2008

Lattice

<by>=( 0.151 ± 0.024) fm

Pion GPDs in impact parameter space

order terms. We take, however, the result of Ref. [12] as aguide to estimate the L dependence of our lattice data,fitting B!;u

T10!t " 0#=m! to the form c0 $ c1m2! $

c2m2! exp!%m!L#. This fit, represented by shaded bands

in Fig. 3, gives B!;uT10!t " 0# " 1:47!18# GeV%1 at L " 1

and m! & 440 MeV, compared to B!;uT10!t " 0# "

1:95!27# GeV%1 at L& 1:65 fm as represented by thediamond in the lowest panel of Fig. 3. The typical correc-tions for B!;u

T20!t " 0#=m! are similar. Within present sta-tistics, we do not see a clear volume dependence of thecorresponding p-pole masses for n " 1; 2.

The pion mass dependence of B!;uTn0!t " 0#=m! is shown

in Fig. 4. The darker shaded bands show fits based on theansatz we just described. Data points and error bands havebeen shifted to L " 1. For m! " 140 MeV, we obtainB!;uT10!t " 0#=m! " 1:54!24# GeV%1 with mp "

0:756!95# GeV and B!;uT20!t " 0#=m! " 0:277!71# GeV%1

with mp " 1:130!265# GeV, where in both cases we haveset p " 1:6. The errors of the forward values include theuncertainties from finite volume effects. The light shadedbands in Fig. 4 show fits restricted tom! < 650 MeV using1-loop ChPT [5] plus the volume-dependent termc2m

2! exp!%m!L#. We note that the ChPT extrapolation

gives larger values for B!;uT10!t " 0# at the physical point

than the linear extrapolation in m2!.

To compute the lowest two moments of the density inEq. (1), we further need the GFFs A!

n0!t# with n " 1; 2. ForA!;u10 !t# " F!!t#, we refer to our results in Ref. [6]. A

detailed analysis of A!;u20 !t# will be presented in Ref. [11],

and first results are given in Ref. [4]. We fit A!;un0 !t# to a

p-pole parametrization with p " 1, which provides anexcellent description of the lattice data and is consistentwith power counting for t ! %1. Fourier transforming theparametrizations of the momentum-space GFFs, we obtainthe densities "n!b?; s?#. In Fig. 5, we show "n"1!b?; s?#for up quarks in a !$ together with corresponding profileplots for fixed bx. Compared to the unpolarized case on theleft, the right-hand side of Fig. 5 shows strong distortionsfor transversely polarized quarks and thus a pronouncedspin structure. The difference between p " 1:6 and p " 2for B!;u

Tn0 is negligible within errors. The negative values of

11.5

22.5

11.5

22.5

3m 760...850 MeV

11.5

22.5

m 590...620 MeV

m 400...450 MeV

1.2 1.4 1.6 1.8 2 2.2L fm][

B01T

,ut

0m

V]

[/)

=(

eG

!1

="

="

="

""

5.40

5.29

5.25

5.20

#

FIG. 3. Study of finite size effects of B!;uT10!t " 0#=m!. Shaded

bands represent a combined fit (restricted tom!L > 3) inm! andL as described in the text. The dashed lines show the infinite-volume limit of the fit.

0 0.2 0.4 0.6 0.8 1m2 GeV2

00.5

11.5

22.5

33.5

n 1

n 2

5 .405 .295 .255 .20

][

B0nT,u

t0

mV

][

/)=

(e

G1 !

($2)

=

=

"

""

#

FIG. 4 (color online). Pion mass dependence of B!;uTn0!t "

0#=m!. The shaded bands represent fits as explained in the text.

0 0.002 0.004 0.006 0.008 0.01

a2 fm ][ 2

0

1

2

3

4

B01T,u

t0

mV

][

/)=

(e

G1 !

m

=

=760...850 MeV ($2)"

"

""

m 590...620 MeV5.40

5.29

5.25

5.20

#

FIG. 2. Study of discretization errors in B!;uT10!t " 0#=m!.

FIG. 5 (color online). The lowest moment of the densities ofunpolarized (left) and transversely polarized (right) up quarks ina !$ together with corresponding profile plots. The quark spin isoriented in the transverse plane as indicated by the arrow. Theerror bands in the profile plots show the uncertainties in B!;u

T10!t "0#=m! and the p-pole masses at mphys

! from a linear extrapola-tion. The dashed-dotted lines show the uncertainty from a ChPTextrapolation (light shaded band).

PRL 101, 122001 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending

19 SEPTEMBER 2008

122001-3










0:756!95# GeV and B!;uT20!t " 0#=m! " 0:277!71# GeV%1











11.5

22.5

11.5

22.5

3m 760...850 MeV

11.5

22.5

m 590...620 MeV

m 400...450 MeV

1.2 1.4 1.6 1.8 2 2.2L fm][

B01T

,ut

0m

V]

[/)

=(

eG

!1

="

="

="

""

5.40

5.29

5.25

5.20

#



0 0.2 0.4 0.6 0.8 1m2 GeV2

00.5

11.5

22.5

33.5

n 1

n 2

5 .405 .295 .255 .20

][

B0nT,u

t0

mV

][

/)=

(e

G1 !

($2)

=

=

"

""

#



0 0.002 0.004 0.006 0.008 0.01

a2 fm ][ 2

0

1

2

3

4

B01T,u

t0

mV

][

/)=

(e

G1 !

m

=

=760...850 MeV ($2)"

"

""

m 590...620 MeV5.40

5.29

5.25

5.20

#






19 SEPTEMBER 2008

122001-3










0:756!95# GeV and B!;uT20!t " 0#=m! " 0:277!71# GeV%1











11.5

22.5

11.5

22.5

3m 760...850 MeV

11.5

22.5

m 590...620 MeV

m 400...450 MeV

1.2 1.4 1.6 1.8 2 2.2L fm][

B01T,u

t0

mV

][

/)=

(e

G!1

="

="

="

""

5.40

5.29

5.25

5.20

#



0 0.2 0.4 0.6 0.8 1m2 GeV2

00.5

11.5

22.5

33.5

n 1

n 2

5 .405 .295 .255 .20

][

B0nT,u

t0

mV

][

/)=

(e

G1 !

($2)

=

=

"

""

#



0 0.002 0.004 0.006 0.008 0.01

a2 fm ][ 2

0

1

2

3

4

B01T,u

t0

mV

][

/)=

(e

G1 !

m

=

=760...850 MeV ($2)"

"

""

m 590...620 MeV5.40

5.29

5.25

5.20

#






19 SEPTEMBER 2008

122001-3

GPD for unpolarized quark

chiral-odd GPD for pol. quark in the x direction

Model calculations of pion Boer-Mulders function

Zhun Lu, B.Q. MaPRD70 (2011)

Diquark spectator model and LCCQM with one gluon exchange approximationBM funct. proportional to "s ⇒ overall normalization constant depending on the model scale

Spectator Quark model|k⊥h⊥

1π|/Mπf1π

mass and f1!x;k2?" is the unpolarized quark distribution

of the proton.

III. COS2! ASYMMETRIES IN UNPOLARIZEDDRELL-YAN PROCESS

The general form of the angular differential crosssection for unpolarized !#p Drell-Yan process is

1"

d"!

$ 34!

1#% 3

!

1% #cos2$%%sin2$ cos&

% '2sin2$cos2&

"

; (19)

where & is the angle between the lepton plane and theplane of the incident hadrons in the lepton pair center ofmass frame (see Fig. 4). The experimental data show largevalue of ' near to 30%, which can not be explained byperturbative QCD. Many theoretical approaches havebeen proposed to interpret the experimental data, suchas high-twist effect [21,22], and factorization breakingmechanism [23]. In Ref. [4] Boer demonstrated that un-suppressed cos2& asymmetries can arise from a productof two chiral-odd h?1 which depends on transverse mo-mentum. In Ref. [18] the cos2& asymmetry in unpolar-ized p "p ! l"lX Drell-Yan process has been estimatedfrom h?1 !x;k2

?" for the proton computed by quark-sca-lar-diquark model. The maximum of ' in that case is inthe order of 30%.

In this section we give a simple estimate of cos2&asymmetry in unpolarized !#p Drell-Yan process,

from h?1! computed by our model. The leading orderunpolarized Drell-Yan cross section expressed in theCollins-Soper frame [24] is

d"!h1h2 ! l"lX"d!dx1dx2d2q?

$ (2

3Q2

X

a; "a

#

A!y"F &f1 "f1' % B!y"

( cos2&F$

!2h ) p?h ) k?"

# !p? ) k?"h?1 "h?1M1M2

%&

; (20)

where Q2 $ q2 is the invariance mass square of the leptonpair, and the vector h $ q?=QT . We have used the nota-tion

F &f1 "f1' $Z

d2p?d2k?)2!p? % k? # q?"fa1 !x;p2?"

( "fa1! "x;k2?": (21)

From Eq. (20) one can give the expression for theasymmetry coefficient ' [4]:

' $ 2X

a;a

e2aF$

!2h ) p?h ) k?" # !p? ) k?"h?1 "h?1M1M2

%,

X

a; "ae2aF &f "f'

$ 2M1M2

P

a; "ae2aFa

P

a; "ae2aGa

: (22)

Our model calculation has shown "h?1! $ h?1!. Thus in!#p unpolarized Drell-Yan process we can assumeu-quark dominance, which means the main contributionto asymmetry comes from "h?; "u

1! ! "x;k2?" ( h?;u

1 !x;p2?",

since "u in !# and u in proton are both valence quarks.Then we have ' * 2Fu=!M!MGu". To evaluate ', we useour model result for "h?; "u

1! and "f1!, and we adopt h?;u1 and

f1 from Ref. [18]. Using the p? integration to eliminatethe delta function in the denominator and numerator inEq. (22) one arrives at

Fu $Z 1

0djk?j

Z 2!

0d$qk

A!Apjk?jk2?!k2

? % B!"ln!k2

? % B!

B!

"

( !k2? # 2k2

?cos2$qk % jk?jjq?jcos$qk"

!k2? % f"!k2

? % f% Bp"

( ln!k2

? % f% Bp

Bp

"

; (23)

l

P h

z

l’

P

FIG. 4. Angular definitions of unpolarized Drell-Yan processin the lepton pair center of mass frame.

0. . . . . .0.1

0.2

0.3

0.4

0. . . .0.0

0.1

0.2

0.3

0.4

X

a

|k!h! 1 "|/(M

"f1 ")

|k!|

b

GeV

20 0 4 0 6 00 8 1 0 0 0. 015 1 5 2 0

FIG. 3. Model prediction of jk?h?1!!x;k2?"j=&M!f1!!x;k2

?"'as a function of x and jk?j.

ZHUN LU; AND BO-QIANG MA PHYSICAL REVIEW D 70 094044

094044-4

fixed k⊥ = 0.3 GeV fixed x=0.15

|k⊥h⊥1π|/Mπf1π

x k⊥ [GeV]

mπ ↔ Mp

pion BM = proton BM

Model calculations of pion Boer-Mulders function

Zhun Lu, B.Q. MaPRD70 (2011)

Diquark spectator model and LCCQM with one gluon exchange approximationBM funct. proportional to "s ⇒ overall normalization constant depending on the model scale

Spectator Quark model|k⊥h⊥

1π|/Mπf1π

mass and f1!x;k2?" is the unpolarized quark distribution

of the proton.

III. COS2! ASYMMETRIES IN UNPOLARIZEDDRELL-YAN PROCESS

The general form of the angular differential crosssection for unpolarized !#p Drell-Yan process is

1"

d"!

$ 34!

1#% 3

!

1% #cos2$%%sin2$ cos&

% '2sin2$cos2&

"

; (19)

where & is the angle between the lepton plane and theplane of the incident hadrons in the lepton pair center ofmass frame (see Fig. 4). The experimental data show largevalue of ' near to 30%, which can not be explained byperturbative QCD. Many theoretical approaches havebeen proposed to interpret the experimental data, suchas high-twist effect [21,22], and factorization breakingmechanism [23]. In Ref. [4] Boer demonstrated that un-suppressed cos2& asymmetries can arise from a productof two chiral-odd h?1 which depends on transverse mo-mentum. In Ref. [18] the cos2& asymmetry in unpolar-ized p "p ! l"lX Drell-Yan process has been estimatedfrom h?1 !x;k2

?" for the proton computed by quark-sca-lar-diquark model. The maximum of ' in that case is inthe order of 30%.

In this section we give a simple estimate of cos2&asymmetry in unpolarized !#p Drell-Yan process,

from h?1! computed by our model. The leading orderunpolarized Drell-Yan cross section expressed in theCollins-Soper frame [24] is

d"!h1h2 ! l"lX"d!dx1dx2d2q?

$ (2

3Q2

X

a; "a

#

A!y"F &f1 "f1' % B!y"

( cos2&F$

!2h ) p?h ) k?"

# !p? ) k?"h?1 "h?1M1M2

%&

; (20)

where Q2 $ q2 is the invariance mass square of the leptonpair, and the vector h $ q?=QT . We have used the nota-tion

F &f1 "f1' $Z

d2p?d2k?)2!p? % k? # q?"fa1 !x;p2?"

( "fa1! "x;k2?": (21)

From Eq. (20) one can give the expression for theasymmetry coefficient ' [4]:

' $ 2X

a;a

e2aF$

!2h ) p?h ) k?" # !p? ) k?"h?1 "h?1M1M2

%,

X

a; "ae2aF &f "f'

$ 2M1M2

P

a; "ae2aFa

P

a; "ae2aGa

: (22)

Our model calculation has shown "h?1! $ h?1!. Thus in!#p unpolarized Drell-Yan process we can assumeu-quark dominance, which means the main contributionto asymmetry comes from "h?; "u

1! ! "x;k2?" ( h?;u

1 !x;p2?",

since "u in !# and u in proton are both valence quarks.Then we have ' * 2Fu=!M!MGu". To evaluate ', we useour model result for "h?; "u

1! and "f1!, and we adopt h?;u1 and

f1 from Ref. [18]. Using the p? integration to eliminatethe delta function in the denominator and numerator inEq. (22) one arrives at

Fu $Z 1

0djk?j

Z 2!

0d$qk

A!Apjk?jk2?!k2

? % B!"ln!k2

? % B!

B!

"

( !k2? # 2k2

?cos2$qk % jk?jjq?jcos$qk"

!k2? % f"!k2

? % f% Bp"

( ln!k2

? % f% Bp

Bp

"

; (23)

l

P h

z

l’

P

FIG. 4. Angular definitions of unpolarized Drell-Yan processin the lepton pair center of mass frame.

0. . . . . .0.1

0.2

0.3

0.4

0. . . .0.0

0.1

0.2

0.3

0.4

X

a

|k!h! 1 "|/(M

"f1 ")

|k!|

b

GeV

20 0 4 0 6 00 8 1 0 0 0. 015 1 5 2 0

FIG. 3. Model prediction of jk?h?1!!x;k2?"j=&M!f1!!x;k2

?"'as a function of x and jk?j.

ZHUN LU; AND BO-QIANG MA PHYSICAL REVIEW D 70 094044

094044-4

fixed k⊥ = 0.3 GeV fixed x=0.15

|k⊥h⊥1π|/Mπf1π

x

L. Gamberg, M. Schlegel / Physics Letters B 685 (2010) 95–103 101

and agree reasonably well with each other. Because the light conecomponents in Eq. (34) are already integrated out and the remain-ing integration range is over a 2-dimensional transverse Euclideanspace, and because the gauge dependent part of the gluon prop-agator does not contribute, it is natural to apply the Euclideanresults in Landau gauge of the Dyson–Schwinger framework. Oneunique feature of Dyson–Schwinger studies of the gluon propa-gator is that it rises like (k2)2!!1 in the infra-red limit with auniversal coe!cient ! " 0.595. This makes it infrared finite in con-trast to the perturbative propagator. A fit to the results for thenon-perturbative gluon propagator has been given in Refs. [82,87,88],

Z!p2,µ2" = p2D!1!p2,µ2"

=#

"s(p2)

"s(µ2)

$1+2## c( p2

$2 )! + d( p2

$2 )2!

1+ c( p2

$2 )! + d( p2

$2 )2!

$2

, (36)

with the parameters c = 1.269, d = 2.105, and # = ! 944 . These fits

for the running coupling and the gluon propagator merge withthe spirit of the eikonal methods described above since closedfermion loops (quenched approximation) were neglected. By usingthe non-perturbative propagator (36), we partly reintroduce gluonself-interactions that were originally neglected in the generalizedladder approximation. According to Ref. [88] the fitting functionsEqs. (35) and (36) were adjusted to Dyson–Schwinger results ob-tained at a very large renormalization scale, the mass of the topquark, µ2 = 170 GeV2, which defines the normalization in (36).Since the lensing function deals with soft physics, intuitively weprefer a much lower hadronic scale which sets the normalization,µ = $QCD # 0.2 GeV. In the spirit of Sudakov form factors we alsoassume that the scale at which the gluons are exchanged is givenby the transverse gluon momentum that we integrate over. In thisway the running coupling serves as a vertex form factor that addi-tional cuts off large gluon transverse momenta.

Our ansatz for the eikonal phase given by Dyson–Schwingerquantities then reads,

%DS!|$zT |"= 2

%%

0

dkT kT"s!k2T

"J0

!|$zT |kT

"Z!k2T ,$2

QCD"/k2T . (37)

The numerical result for this ansatz is shown in the center panelof Fig. 2. We plot this function for various scale $QCD = 0 GeV,0.2 GeV, 0.5 GeV, 0.7 GeV. Although the choice of this scale israther arbitrary we observe only a very mild dependence on thisscale as long as it remains soft. We further observe that the phasedoesn’t exceed a value of 4–4.5 & %max/4 # 1.15. Thus this featuremakes the application of the power series of the color function inSU(3) reliable since %/4 never exceeds 1.5 in the lensing function,Eq. (30) and in turn in the calculation of the Boer–Mulders func-tion in Eq. (5).

Finally, we insert our ansatz for the eikonal phase into the lens-ing functions (30) for a U (1), SU(2) and SU(3) color function. Weplot the results in Fig. 3 for a color function for U (1), SU(2), SU(3).While we observe that all lensing functions fall off at large trans-verse distances, they are quite different in size at small distances.However for each case, the all order calculation sums up to anexponential of the eikonal phase where one observes oscillationsfrom the Bessel function J0 of the first kind. Despite these oscilla-tions the lensing function remains negative.

6. The pion Boer–Mulders function

In this section we use the eikonal model for the lensing func-tion together with the spectator model for the GPD H&

1 to present

Fig. 3. The pion Boer–Mulders function, xm2&h',(1)

1 (x) vs. x calculated by means ofthe relation to the chirally-odd GPD H&

1 for an SU(3), SU(2), U (1) gauge theory.

predictions of the relation (5) for the first moment of the pionBoer–Mulders function h'(1)

1 .We start by fixing the model parameters in (13). We encounter

six free model parameters ms , mq , $, ', g& and n that we need todetermine by fitting to pion data. In order to do so we determinethe chiral-even GPD F&

1 (for definition and notation see Ref. [48])in the spectator model,

F&1

!x,0,! $(2

T"

= g2&2(2&)2

e2'2$2

#(1! x)$2

$D2T + $2(x)

$2n!2 2&%

0

d)

(1%

0

dzz2n!3[1 ! 2z ! z x(1!x)m2

&!2msmq$D2T +$2(x)

]e! 2'2( $D2T +$2(x))

(1!x)z

[1 ! 4z(1 ! z)$D2T

$D2T +$2(x)

cos2 )]n.

(38)

When integrated over x, the GPD reduces to the pion form factorF&+

(Q 2) = !F&!(Q 2). An experimental fit of the pion form fac-

tor to data is presented in Refs. [89,90], and up to Q 2 = 2.45 GeV2

is displayed by the monopole formula Ffit(Q 2) = (1 + 1.85Q 2)!1.This procedure is expected to predict the t-dependence of thechirally-odd GPD H&

1 reasonably well up to Q 2 = 2.45 GeV2. Inorder to fix the x-dependence of H&

1 we fit the collinear limitF&1 (x,0,0) to the valence quark distribution in a pion. A parame-

terization for F&1 (x) was given by GRV in Ref. [91] at a scale µ2 =

2 GeV2. Reasonable agreements of the form factor- and collinearlimit of Eq. (38) with the data fits are found for the parametersmq = 0.834 GeV, ms = 0.632 GeV, $ = 0.067 GeV, ' = 0.448 GeV,n = 0.971, g& = 3.604. Details of the fitting procedure for this andan analogous calculation for the Sivers function will be presentedin a future publication [80].

With the predicted GPD H&1 and the lensing function

I i(x, $bT ) ) biT /|bT |I(x, |bT |) as input we use (5) to give a pre-diction for the valence contribution to the first kT -moment of thepion Boer–Mulders function,

m2&h

'(1)1 (x) = 2&

%%

0

dbT b2T I(x,bT )*

*b2TH&

1!x,b2T

". (39)

Numerical results for m2&h

'(1)1 (x) are presented in Fig. 3 for a

U (1), SU(2) and SU(3) gauge theory. One observes that all re-sults are negative which reflects the sign of the lensing function. Itwas argued in Ref. [66] that a negative sign of the lensing func-tions indicates attractive FSIs. We find that this is valid in anAbelian perturbative model as well as our non-perturbative model

Gamberg, SchlegelPLB685 (2010)

Beyond one-gluon exchange approx.

TMD-GPD relation + lensing function from eikonal methods

k⊥ [GeV]

mπ ↔ Mp

pion BM = proton BM

Proton Pretzelosity

total quark helicity Jqz

! classification of LCWFs in orbital angular momentum components

Jz = Jzq + Lz

q

[Ji, J.P. Ma, Yuan, 03; Burkardt, Ji, Yuan, 02]

Lzq =2Lz

q =1Lzq =0Lz

q = -1

(↓↓↓)LC(↑↓↓)LC(↑↑↓)LC(↑↑↑)LCJzq

Light-Cone Wave Function of the Proton

MODEL

[B.P., Cazzaniga, Boffi, PRD78 (2008)]

! momentum-space component: spherically symmetric

two parameters fitted to anomalous magnetic moments of proton and neutron

!spin-dependent part eigenstate of the total angular momentum operator (S, P, D waves)

Ψfλβ(xi,k⊥,i,λi) = ΨMom(xi,k


|!Lz|=2from the initial to the final nucleon state

h⊥1T

h⊥(1)u1T h⊥(1)d

1T

Pretzelosity

!0.4 !0.2 0.0 0.2 0.4

!0.4

!0.2

0.0

0.2

0.4

kx !GeV"

ky!G

ev"

P-P int.

S-D int.

!0.4 !0.2 0.0 0.2 0.4

!0.4

!0.2

0.0

0.2

0.4

kx !GeV"

ky!G

ev"

P-P int.

S-D int.

up downh⊥1T

model-dependent relation

[She, Zhu, Ma, 2009; Avakian, Efremov, Schweitzer, Yuan, 2010] first derived in LC-diquark model and bag model

valid in all quark models with spherical symmetry in the rest frame [Lorce’, BP, PLB710 (2012)]

chiral even and charge even chiral odd and charge odd

no operator identityrelation at level of matrix

elements of operators

OAM and Pretzelosity

Application to Observables

xv = 1 xg = xsea = 0

Fixing the scale of the LCCQM

Evolve in energy until 2nd moment=1Find !02~0.1GeV2 +"!02

We use RGE and one first principle based assumption.Then we set scenarios ...

(uv + dv)

µ2

0

n=2

= 1

(uv + dv)

Q2 = 10GeV2

n=2

= 0.36

Say there exists a scale at which there is no sea and no gluon, then

QCD evolution introduces gluons and sea quarks: Roberts, “The structure of the proton”

! there exists a scale at which there are no sea and gluons

! the valence quarks carry the whole momentum of the hadron

Parisi & Petronzio, Phys. Lett. B 62 (1976)Traini et al, Nucl. Phys. A 614, 472 (1997)

xv = 1 xg = xsea = 0

Fixing the scale of the LCCQM

Evolve in energy until 2nd moment=1Find !02~0.1GeV2 +"!02

We use RGE and one first principle based assumption.Then we set scenarios ...

(uv + dv)

µ2

0

n=2

= 1

(uv + dv)

Q2 = 10GeV2

n=2

= 0.36

Say there exists a scale at which there is no sea and no gluon, then

QCD evolution introduces gluons and sea quarks: Roberts, “The structure of the proton”

! there exists a scale at which there are no sea and gluons

! the valence quarks carry the whole momentum of the hadron

Q2 = 4 GeV2

xv = 0.47

Q2 = 10 GeV2

xv = 0.36

xv = 1

Q20 ≈ 0.31 GeV2

Q20 ≈ 0.26 GeV2

Pion

Proton

NLO evolution

NLO evolution

xv = 1

hadronic scale of the modelfrom experiments

Parisi & Petronzio, Phys. Lett. B 62 (1976)Traini et al, Nucl. Phys. A 614, 472 (1997)

The BM-Pretzelosity Asymmetry in !p Drell Yan

[Arnold, Metz, Schlegel, PRD79, (2008)]

with

Numerator

Denominator

= proton spin angle in the CM frameφp = lepton angle in the CS frameφ

The BM-Pretzelosity Asymmetry in !p Drell Yan

[Arnold, Metz, Schlegel, PRD79, (2008)]

with

Numerator

Denominator

= proton spin angle in the CM frameφp = lepton angle in the CS frameφ

Fsin(2φ+φp)TU = BGausse2uh

⊥ (1/2)u/p1T (xp)h

⊥ (1)u/π−

1 (xπ) F 1UU = e2uf

u/p1 (xp)f

u/π−

1 (xπ)

h⊥(1/2)u/p1T =

dk⊥

k⊥2Mp

h⊥u/p1T (x, k2⊥)

h⊥(1)u/π−

1T =dk⊥

k2⊥

2m2πh⊥u/π1T (x, k2⊥)

Gaussian Ansatz

xF = xπ − xp

π−p

Q2 = 20 GeV2

x

Results from the LCCQM

-0.05

-0.04

-0.03

-0.02

-0.01

0

-0.75 -0.5 -0.25 0 0.25 0.5 0.75-0.05

-0.04

-0.03

-0.02

-0.01

0

-0.75 -0.5 -0.25 0 0.25 0.5 0.75xF = xπ − xp

π−p

hadronic scale

approximate evolution

pretzelosity and BM function evolved like transversity

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

-0

0 0.2 0.4 0.6 0.8 1

1/2 moment of pretzelosity first moment of pion BM

x

hadronic scale

evolution to Q2=20 GeV2

B.P., Schweitzer, Efremov, in preparation

xF

Q2 = 20 GeV2

Q2 = 20 GeV2

xfu/p1

x

xf u/π−

1

LCCQM

GRV

LCCQM

Aicher,Schaefer,Vogelsang,PRL105(2010)

Model dependence from f1p and f1#

-0.05

-0.04

-0.03

-0.02

-0.01

0

-0.75 -0.5 -0.25 0 0.25 0.5 0.750

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

SMRS, PRD45(1992)

-0.05

-0.04

-0.03

-0.02

-0.01

0

-0.75 -0.5 -0.25 0 0.25 0.5 0.750

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1B.P., Schweitzer, Efremov, in preparation

Z. Lu et al. / Physics Letters B 696 (2011) 513–517 515

Fig. 1. The sin(! + !S ) asymmetries for "±p! " µ+µ#X process at COMPASS. Solid and dashed curves are the results for "# and "+ productions, respectively.

Fig. 2. Similar to Fig. 1, but for the sin(3! # !S ) asymmetries. The thin curves are calculated with a cut 1.0 ! qT ! 2.0 GeV.

a number of processes [37] and transversity distributions relatedto the Collins asymmetry at HERMES [38]. This model is also suc-cessful in the prediction of the dihadron production asymmetryat COMPASS [39,40]. So it is worth trying to apply this modelto the Drell–Yan kinematics at COMPASS. Besides this, we needthe Boer–Mulders function for a pion [41,42], of which the knowl-edge is limited, and we will use the parametrization in Ref. [41],which was obtained in a quark spectator antiquark model. Thepion parton distributions we adopt were demonstrated [41] to givea good description on the cos2! asymmetries measured in the un-polarized "N Drell–Yan process [43], where a large and increasingasymmetry was observed in the qT region below 3 GeV, thus ourmodel has been checked to be reasonable in this region. Anotherimportant feature we should remember is that this T -odd functionhas a different sign in the Drell–Yan process with that in the SIDISprocess [18,20,44],

h$1 |DY = #h$

1 |SIDIS. (9)

In Ref. [41], the Boer–Mulders function is calculated for the SIDISprocess, so we will make a sign change for our parametrization inour Letter. However, we should be careful that due to the chiral-odd nature of Boer–Mulders function, it always couples with an-other chiral-odd function for being probed. This makes it very dif-ficult to obtain the information of this function, especially its sign.In the unpolarized Drell–Yan process, the Boer–Mulders functioncouples with itself, therefore it is impossible to determine its sign.In the SIDIS process, the Boer–Mulders function is combined withthe Collins function, the extraction [45] of which also relies on theazimuthal asymmetry of hadron production in e+e# annihilation

process. Also we will stress that unlike many other calculations,we do not make the ansatz that the transverse momentum depen-dence of the TMDs has a pure Gaussian form, but just deduce itfrom the model. That is, we evaluate the integration over the par-ton transverse momenta numerically. The experiment we consideris for COMPASS, where the kinematics we will use are [46]

%s = 18.9 GeV, 0.1 < x1 < 1, 0.05 < x2 < 0.5,

4 ! M ! 8.5 GeV, 0 ! qT ! 4 GeV (if qT is integrated).

We will investigate the xF ,M and qT dependence of the asym-metries. The integration range can be determined as follows.

• For the xF /M dependence, given a fixed xF /M , the range forM/xF is determined by Eq. (4) so that xmin

1,2 < x1,2(xF ,M) <

xmax1,2 .

• For the qT dependence, the range for M is 4 ! M ! 8.5 GeV,and the range for xF is determined by Eq. (4) so that xmin

1,2 <

x1,2(xF ,M) < xmax1,2 .

In Figs. 1 and 2 (thick curves), we plot the sin(! + !S ) asymmetryand the sin(3! # !S ) asymmetry in the " p! Drell–Yan at COM-PASS, respectively. We can clearly see from the two figures that theasymmetries for the "#p! process are much larger than those forthe "+p! process, because that the former process is dominatedby u quark while the latter is dominated by d quark. COMPASS willconduct a "#p! plan in the near future, however, we will also givethe prediction on the "+p! process as a supplement, and expect

Predictions from other models

Zhun Lu, B.-Q. Ma, PLB696(2011)

talk of Zhun Lu

Light-cone quark spectator model

cut in 1.0 GeV < qT < 2.0 GeV

cut in 1.0 GeV < qT < 2.0 GeV

π+

π−

π+

π− π−

π+

Asin(3φh−φS)UT ∝

q e

2qh

⊥ q1T ⊗H

⊥ q1

INT 12-49W, February 10th, 2012Gunar Schnell

( )Sh3sin

UTA!! "

chiral-odd ! needs Collins FF (or similar)

leads to sin(3φ-φs) modulation in AUT

proton and deuteron data consistent with zero

cancelations? pretzelosity=zero? or just the additional suppression by two powers of Ph!

quark pol.

U L T

nucl

eon

pol.

U f1 h⊥1

L g1L h⊥1L

T f⊥1T g1T h1, h⊥1T

Twist-2 TMDs

31

Pretzelosity

-0.02

0

0.02

0.04

2 !

sin

(3"-" S

)#U$

%+HERMES7.3% scale uncertainty

PRELIMINARY

-0.1

-0.05

0

0.05 %0

-0.04

-0.02

0

0.02

0.04

10-1

x

%-

0.4 0.6z

0.5 1Ph$ [GeV]

INT 12-49W, February 10th, 2012Gunar Schnell

( )Sh3sin

UTA!! "

chiral-odd ! needs Collins FF (or similar)

leads to sin(3φ-φs) modulation in AUT

proton and deuteron data consistent with zero

cancelations? pretzelosity=zero? or just the additional suppression by two powers of Ph!

quark pol.

U L T

nucl

eon

pol.

U f1 h⊥1

L g1L h⊥1L

T f⊥1T g1T h1, h⊥1T

Twist-2 TMDs

31

Pretzelosity

-0.02

0

0.02

0.04

2 !

sin

(3"-" S

)#U$

%+HERMES7.3% scale uncertainty

PRELIMINARY

-0.1

-0.05

0

0.05 %0

-0.04

-0.02

0

0.02

0.04

10-1

x

%-

0.4 0.6z

0.5 1Ph$ [GeV]

Pretzelosity from SIDIS

leads to sin (3$-$s) modulation in AUT

proton (HERMES) and deuteron (COMPASS) data consistent with zero

pretzelosity equal to zero? or just the suppression by third power of 1/Ph⊥?

Gunar Schnell, INT workshop, 2012

" experiment planned at CLAS12 (H. Avakian at al., LOI 12-06-108)

! Polarized Drell-Yan at COMPASS with pion beam: BM# pretzelosityproton

Summary

complementary to SIDIS to extract information on proton pretzelosity

new tool to learn about the transverse spin structure of quark in the pion

! Predictions in a Light-Cone Quark model

⊗

small asymmetry of the order of 1-2%, with larger contribution at large xF

small model dependence from f1 of protonlarger uncertainties from f1 of pion

effects of evolution are important and need further studies

! Future plans: extend the analysis within LCCQM to other spin asymmetries of (un)polarized DY

Documents

Azimuthal Spin Asymmetries in Light-Cone Quark model · Polarized Drell-Yan at COMPASS Azimuthal spin asymmetries in light-cone constituent quarkmodels S. Boﬃ∗,1,2 A. V. Efremov∗,3