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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
+
−
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
+
−
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
(↑↑)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
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
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)
transversely pol. quark
Chromodynamic lensing
Distortion in impact parameter(related to GPD ET)
Burkardt, PRD66 (02)
transversely pol. quark
Chromodynamic lensing
Distortion in transverse momentum(related to Boer Mulders function)
Final-state interaction(lensing function)
Burkardt, PRD66 (02)
transversely pol. quark
Chromodynamic lensing
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
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
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,u
t0
mV
][
/)=
(e
G!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
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
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
2⊥,i)⊗ΨSpin(xi,k⊥,iλi)
|!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
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
Recommended