Beam single spin asymmetry of neutral pion production in semi-inclusive deep inelastic scattering

0.3 MB·English

by  Wenjuan Mao

Beam single spin asymmetry of neutral pion production in semi-inclusive deep
inelastic scattering
Wenjuan Mao and Zhun Lu∗
Department of Physics, Southeast University, Nanjing 211189, China
We study the beam spin asymmetry Asinφh in semi-inclusive π0 electroproduction contributed
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by the T-odd twist-3 distribution function g⊥(x,k2). We calculate this transverse momentum
T
dependent distribution function for the u and d quarks inside the proton in a spectator model
includingthescalarandtheaxial-vectordiquarkcomponents. Usingthemodelresults,weestimate
the asymmetry Asinφh in the ep → e′π0X process in which the lepton beam is longitudinally
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polarized. The model prediction is compared with the data measured by the CLAS and HERMES
Collaborations, andit isfoundthatournumericalresultsagreewith experimentaldatareasonably.
2
Especially, our results can well describe the CLAS data at the region where the Bjorken x and the
1 piontransversemomentumisnotlarge. Wealso makeapredictionon theasymmetryAsinφh inπ0
0 LU
electroproduction at CLAS12 using thesame model calculation.
2

c PACSnumbers: 12.39.-x,13.60.-r,13.88.+e
e
D
I. INTRODUCTION odd chiral-even TMD, g⊥ can be regarded as an analog
1
of the Sivers function [19] at the twist-3 level, because
3
both of them require quark transverse motion as well as
Understanding the origins of the single spin asym-
] initial- or final-state interactions [22–24] via soft-gluon
h metries (SSAs) appearing in high-energy semi-inclusive
exchanges to receive nonzero contributions. Therefore,
p processes is one of the important goals of QCD spin
studying beam SSAs may provide a unique opportunity
- physics [1–4]. Substantial SSAs have been measured by
p tounraveltheroleofquarkspin-orbitcorrelationattwist
e theHERMESCollaboration[5–10],theCOMPASSCol- 3.
h laboration[11–13], and the Jefferson Lab (JLab) [14–18]
[ in semi-inclusive deep inelastic scattering(SIDIS). It is In this work,we present an analysis on the beam SSA

3 found that the T-odd transverse momentum dependent Asinφh in neutral pion production, based on the effect
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v (TMD) parton distribution functions (DFs) [19–24] or from g⊥. We calculate the function g⊥(x,k2) for the
T
0 fragmentation functions (FFs) [25], under the TMD fac- u and d quarks inside the proton, using the spectator
9 torization [26] framework, play central roles in the ob- model with scalar and axial-vector diquarks. Different
7 servedSSAs. TheT-oddTMDsdescribethecorrelations typesofspectatormodelhavebeenwidelyusedtocalcu-
4
betweenthe transversemotionofthe partonandits own late TMDs for the nucleon [22, 34–42] and the pion [43–
.
0 spinorthespinoftheinitial-statehadron,therebyencod- 46]. We will adopt a specific model given in Ref. [41], in
1 ingmuchricherinformationaboutthepartonicstructure whichtheauthorsconsidertheisospinofvectordiquarks
2 as well as the QCD dynamics inside hadrons than what to distinguish the isoscalar (ud-like) and isovector (uu-
1
can be learned from the collinear DFs. like)spectators. Furthermore,in thatmodel the free pa-
:
v In the particular case of a longitudinally polarized rametersarefixedbyreproducingtheparametrizationof
i
X beam colliding on an unpolarized target, an asymme- unpolarizedandlongitudinallypolarizedpartondistribu-
r try with sinφh modulation, the so-called beam SSA, tions. The above-mentioned feature of the model allows
a emerges. The CLAS Collaboration [14, 17, 18] at JLab us to perform the phenomenological analysis in a deep
and the HERMES Collaboration [8] have measured this sense. Using the calculated g⊥u and g⊥d, we estimate
asymmetryinpionelectroproductioninthemagnitudeof the beam SSA Asinφh in neutral pion production at the
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several percents that cannot be explained by perturba- kinematics of CLAS and compare our results with the
tiveQCD[27]. Differentmechanismshavebeenproposed CLAS data [17] measured recently with high precision.
to generate such asymmetry, such as the Boer-Mulders We also make a comparisonbetween our calculationand
effect [28] and the Collins effect [29, 30], involving the theπ0 datameasuredbytheHERMESCollaboration[8]
chiral-odd distribution or fragmentation functions. In for further testing. Finally, we present the prediction of
Refs. [31–33], a new source contributing to the beam Asinφh for π0 production at CLAS12.
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SSA has been identified, either from the model calcu-
lations [31, 32] or from the updated decomposition of Thepaperisorganizedasfollows. InSec.II,wepresent
the unpolarized quark-quark correlator [33], where the thedetailsonthecalculationofg⊥inthespectatormodel
twist-3 TMD g⊥(x,k2) plays a crucial role. As a T- with an axial-vector diquark, and discuss the flavor, x
T and k dependencies of g⊥. In Sec. III, we analyze the
T
beamSSAAsinφh inπ0productionnumericallyatCLAS,
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HERMES, and CLAS12 based on the calculated g⊥ for
∗Electronicaddress: [email protected] theuanddquarks. WeaddressourconclusioninSec.IV.
2
II. g⊥ OF THE PROTON IN THE SPECTATOR Since g⊥ is aT-oddDF,weneedto considerthe effect
MODEL WITH AN AXIAL-VECTOR DIQUARK ofthegaugelinktogenerateanonzeroresult,inanalogy
to the Sivers function and the Boer-Mulders function.
In this section, we present the detailed calculation on However,calculations on g⊥ basedon the scalardiquark
theT-oddtwist-3TMDg⊥(x,k2)oftheprotoninaspec- model(withpointlike proton-quark-diquarkcoupling)as
T
tatormodelwithaxial-vectordiquark. Thestartingpoint wellasthequark-targetmodel[48]showthatg⊥ andthe
ofthecalculationisthegauge-invariantquark-quarkcor- other twist-3 T-odd TMDs suffer light-cone divergence.
relator for the unpolarized nucleon This feature is in contrast to the case of leading-twist
T-odd TMDs and has been recognized as a theoretical
dξ−d2ξ challengeinderivingaTMDfactorizationproofinSIDIS
Φ[+](x,k )= Teik·ξ P ψ¯ (0) [0−, −]
T (2π)3 h | j L ∞ at twist 3 [48, 49]. In Refs. [48, 50], the authors pointed
Z out that light-cone divergence can be avoided by means
[0 ,ξ ] [ −,ξ−]ψ (ξ)P , (1)
×L T T L∞ i | i of a phenomenological approach in which form factors
are applied for the proton-quark-diquark coupling. In
where [+] denotes that the gauge link appearing in Φ
Ref. [51], all sixteen twist-3 T-odd TMDs were calcu-
is future-pointing, corresponding to the SIDIS process.
As an interaction-independent twist-3 distribution, g⊥ lated in the scalar diquark model by adopting the dipo-
lar form factor for the proton-quark-diquark coupling.
naturally appears in the decomposition of the quark-
quark correlator at the subleading order of 1/P+ ex- Inthis paper,weextendthe calculationsinRefs.[48,51]
using the spectator model with an axial-vector diquark
pansion, after including the light-cone vector n that
− to obtain g⊥ for both the u and d quarks. The model
defines the direction along which the path-order expo-
we apply in the calculation is the version developed in
nential runs [33, 47]
Ref.[41],whichwasoriginallyproposedinRef.[34]. The
samemodelhasbeenadoptedtocalculatevarioustwist-3
Φ[+](x,kT) = dk−Φ[+](P,k;n−) quark-gluon-quarkcorrelators in Ref. [52]. The common
(cid:12)(cid:12)O(PM+) Z feature of spectator models is that the protonwith mass
(cid:12) M ǫρσγ k M is supposed to be constituted by a quark with mass
(cid:12) = g⊥γ T ρ Tσ + ,
2P+ 5 M ··· m and a diquark with mass MX, and the diquark X can
(cid:26) (cid:27)
(2) be either a scalar one (denoted by s) or an axial-vector
one (denoted by v).
here, stands for the other twist-3 TMDs that arenot
···
taken into account in this paper. The distribution g⊥,
therefore,canbe deducedfromΦ[+](x,kT)bytakingthe We perform the calculation in the Feynman gauge
trace with proper Dirac matrices and expand the gauge link to the first order (one-gluon
exchange) to obtain the quark-quark correlators con-
ǫαρk
T Tρ g⊥(x,k2)=Tr[Φ[+]γαγ ]. (3) tributed by the scalar diquark and the axial-vector di-
− P+ T 5 quark components
1 d4q 1 (k/+m) 1
Φ (x,k ) ie dk− U¯(P,S)Υ (k2)
sij T ≡− q(2π)4 (2π)4 q++iǫ s k2 m2+iǫ (P k)2 M2+iǫ
Z Z (cid:20) − (cid:21)j(cid:20) − − s (4)
Γ+ 1 (k/ /q+m)
s − Υ ((k q)2)U(P,S) +H.c.,
× q2+iε(P k+q)2 M2+iǫ (k q)2 m2+iǫ s −
− − s (cid:21)(cid:20) − − (cid:21)i(cid:12)k+=xP+
Φvij(x,kT)≡−ieq(2π1)4 (2dπ4q)4 dk−q+1+iǫ U¯(P,S)Υρv(k2)k2(k/+m2m+)iǫ (P dρkα)(2(cid:12)(cid:12)(cid:12)P −Mk2)+iǫ
Z Z (cid:20) − (cid:21)j(cid:20) − − v (5)
Γ+,αβ d (P k+q) (k/ /q+m)
v βσ − − Υσ((k q)2)U(P,S) +H.c.,
× q2+iε(P k+q)2 M2+iǫ (k q)2 m2+iǫ v −
− − v (cid:21)(cid:20) − − (cid:21)i(cid:12)k+=xP+
(cid:12)
(cid:12)
(cid:12)
where e is the charge of the quark, Υ denotes the and Γµ or Γµ,αβ is the vertex between the gluon and the
q s/v s v
nucleon-quark-diquarkvertex with the form [34]
g (k2)
Υ (k2)=g (k2), Υµ(k2)= v γµγ5, (6)
s s v √2
3
scalar diquark or the axial-vector diquark the dipolar form factor for g (k2)
X
Γµ =ie (2P 2k+q)µ, (7)
Γµv,αβs =−siev[(−2P −2k+q)µgαβ −(P −k+q)αgµβ gX(k2)=NX|kk22−−Λm2X2|2
(P k)βgµα], (8) (k2 m2)(1 x)2
− − =N − − , (10)
X (k2 +L2 )2
with e denoting the charge of the scalar/axial-vector T X
s/v
diquark. In Eq.(5) we use d to denote the summa-
µν
forX =s, v,whereΛ isthecutoffparameter,N isthe
tionoverthepolarizationsoftheaxial-vectordiquarkfor X X
couplingconstant(whichalsoservesasthenormalization
which we choose the following form [53]
constant), and L2 has the form
X
(P k) n + (P k) n
µ −ν ν −µ
d (P k)= g + − −
µν − − µν (P k) n L2 =(1 x)Λ2 +xM2 x(1 x)M2. (11)
− · − X − X X − −
M2
v n n . (9)
− [(P k) n ]2 −µ −ν Performing the integrations in Eqs. (4) and (5) over
−
− · k−, q+ and q−, the quark-quark correlators for the un-
The advantage of the above choice has been argued in polarized nucleon in the spectator model are simplified
Ref. [41]. To obtain finite results for g⊥, we also choose as
(1 x)3 1 d2q [(k/ /q+m)(P/+M)(k/+m)]
Φ (x,k ) ie e N2 − T − , (12)
s T ≡− q s s 32π3P+ (L2s+kT2)2 Z (2π)2 qT2(L2s+(kT −qT)2)2 (cid:12)(cid:12)kq++==0xP+
Φ (x,k ) ie N2 (1−x)2 1 d2qT d (P k)( iΓ+,αβ)d (P (cid:12)(cid:12) k+q)
v T ≡− q v128π3(P+)2(L2+k2)2 (2π)2 ρα − − σβ −
v T Z
[(k/ /q+m)γσ(P/ M)γρ(k/+m)]
− − . (13)
× qT2(L2v+(kT −qT)2)2 (cid:12)(cid:12)kq++==0xP+
(cid:12)
(cid:12)
UsingEq.(3)andperformingtheintegrationoverq ,we scalar and the axial-vector diquark components 1
T
arriveatthe expressionsforthedistributiong⊥ fromthe
1 (T1h−erex)i(s1a−ty2pxo)Min2Rienf.t[h5e1]n,uthmeerfaacttoorrs(o1f−thxe)Λri2sgh+t-(h1a+ndx)sMides2o−f Eqs.(32)and(36)shouldbe(1−x)Λ2s+(1+x)Ms2−(1−x)M2.
N2(1 x)2e e (1 x)Λ2+(1+x)M2 (1 x)M2
g⊥s(x,k2)= s − s q − s s − − , (14)
T − (32π3) 4π L2(L2+k2)3
(cid:20) s s T (cid:21)
N2(1 x)2e e (1 x)(xM +m)2+(1 x)2M2 M2+xL2
g⊥v(x,k2)= v − v q − − − v v
T (32π3) 4π (1 x)L2(L2+k2)3
(cid:20) − v v T
x L2+k2
ln v T . (15)
−(1 x)k2(L2+k2)2 L2
− T v T (cid:18) v (cid:19)(cid:21)
The function g⊥ for the u and d quarks can be con- eter Λa′. Hence, by applying the relation between quark
structed by g⊥s and g⊥v. Here, we follow the approach flavorsanddiquarktypes,wecanobtainthedistributions
in Ref. [41] in which the two isospin states of the vector g⊥u and g⊥d by the following form [41]:
diquark are distinguished; that is, the vector isoscalar
diquark v(ud) is denoted by a with mass Ma and cutoff g⊥u =c2g⊥s+c2g⊥a, (16)
s a
parameterΛ ,whilethevectorisovectordiquarkv(uu)is
denotedbyaa′ withdifferentmassMa′ andcutoffparam- g⊥d =c′a2g⊥a′, (17)
4
Diquark MX(GeV) ΛX(GeV) cX NX
Scalar s (ud) 0.822 0.609 0.847 11.400 00..88 x gu vs. kT 00..44 x gu vs. x
Axial-vectora (ud) 1.492 0.716 1.061 28.277 00..66 x gd vs. kT 00..33 x gd vs. x
Axial-vector a′ (uu) 0.890 0.376 0.880 4.091 00..44 x=0.3 00..22 kT=0.3GeV
TABLE I: Values for the free parameters of the model 00..22 00..11
taken from Ref. [41], which are fixed by reproducing the
00..00 00..00
parametrization of unpolarized [54] and longitudinally polar-
ized [55] parton distributions. 0.0 0.2 0.4 kT 0.6 0.8GeV1.0 00..00 00..22 00..44 x 00 ..66 00..88 11..00
FIG. 1: Left panel: model results for xg⊥u (solid line) and
where cs, ca, and ca′ represent different couplings that mxgo⊥ddel(dreassuhletds floinrex)ga⊥sufu(nsocltiidonlisnoef)kaTndatxgx⊥=d 0(d.3a;shriegdhtlinpea)neals:
are the free parameters of the model. The same form
also holds for the other TMDs. functions of x at kT =0.3GeV .
For the parameters M , Λ and c (X = s,a,a′)
X X X
needed in the calculation, we also adopt the values from
an unpolarized hydrogen target. Earlier, the HERMES
Ref. [41], as shown in the first three columns of Table. I.
Collaboration measured the beam SSAs for neutral and
The fourth column shows the values for the correspond-
charged pions using a 27.6GeV beam. We compare the
ingnormalizationconstantsN (X =s,a,a′),whichare
X resultsfor Asinφh with the neutralpiondatafromCLAS
obtained from the normalizationcondition for the unpo- LU
and HERMES to test our model calculation. Then, we
larizedTMD.Thequarkmassischosenasm=0.3GeV.
will make new prediction on the asymmetry Asinφh for
To convert our calculation to real QCD, we use the fol- LU
π0 electroproduction at CLAS12 using the same model
lowingreplacementforthe combinationofthe chargesof
calculationto study the prospects to access g⊥ at CLAS
the quark q and the spectator diquark X
after the 12GeV upgrade is realized.
eqeX The semi-inclusive leptoproduction process that we
C α (18)
4π →− F s study can be expressed as
and choose αs 0.3 in our calculation. The minus sign e(ℓ) + p(P) e′(ℓ′) + h(P ) + X(P ), (19)
≈ h X
in the above equation comes from the fact that in QCD, →
a hadron is color-neutral; thus, the color charges eq and where ℓ and ℓ′ are the four-momentum of the incom-
eX should have the opposite signs. ing and scattered lepton, and P and Ph are the four-
In the left panel of Fig. 1, we plot the functions xg⊥u momentum of the target nucleon and the detected final-
(solid line) and xg⊥d (dashed line) vs kT at x = 0.3, state hadron h, respectively.
while in the right panel of the same figure, we display The variables to express the SIDIS cross section are
the x dependence of xg⊥u and xg⊥d at kT = 0.3GeV. defined as
The plots in Fig. 1 correspondto the results in the scale
µ20 =0.3GeV2, which is the scale used in Ref. [41] to fit x= Q2 , y = P ·q, z = P ·Ph, γ = 2Mx,
the parametrization of f1(x) [54]. As we can see from 2P q P l P q Q
· · ·
Fig. 1, the dominance of u quark contribution is evident Q2 = q2, s=(P +ℓ)2, W2 =(P +q)2, (20)
in the adopted spectator model; that is, g⊥u is several −
timeslargerthang⊥d bysize. Ourresultsshowthatg⊥u where q = ℓ ℓ′ is the four-momentum of the virtual
is positive for all x and k regions;while g⊥d is negative −
T photonandW is theinvariantmassofthe hadronicfinal
in the small x region and turns to be positive in the re-
state.
gionx>0.15,i.e.,thereisanodeinthexdependenceof
The reference frame we adopt here is that the virtual
g⊥d. Also,thek dependenciesoftheuanddquarkdis-
T photon and the target proton are collinear and along
tributions are different since g⊥d approaches zero faster
the z axis, with the photon moving toward the target
than g⊥u when k increases.
T in the positive z direction, as shown in Fig. 2. We use
k to denote the intrinsic transverse momentum of the
T
quark inside the proton for the DFs, with P to denote
T
III. NUMERICAL RESULTS FOR BEAM SPIN
the transverse momentum of the detected hadron. The
ASYMMETRY
transverse momentum of the hadron h with respect to
the directionofthe fragmentingquark is denoted by p ,
In this section, we will use our model resulting g⊥ which appears in the TMD FFs. The azimuthal angTle
to calculate the beam SSA Asinφh in π0 electroproduc- between the lepton and the hadron planes is defined as
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tion,asprecisemeasurementsonAsinφh ofaneutralpion φ , following the Trento convention [56].
LU h
for different x and P bins have been performed by the Up to subleading order of 1/Q, the differential cross
T
CLAS Collaboration [17] at JLab recently, in SIDIS by sectionofSIDISforalongitudinallypolarizedbeam(with
a 5.776GeV longitudinally polarized electron beam off helicity λ ) off an unpolarized hadron has the following
e
5
In the parton model, the two structure functions in
Eq.(21)canbeexpressedastheconvolutionofTMDDFs
andFFs,basedonthe tree-levelfactorizationadoptedin
Ref. [57]. With the help of the notation
[wfD]=x e2 d2k d2p δ2(zk P +p )
C q T T T − T T
q Z Z
X
w(k ,p )fq(x,k2)Dq(z,p2), (23)
FIG.2: Thekinematicalconfiguration fortheSIDISprocess. × T T T T
Theinitial andscattered leptonic momentadefinethelepton
plane (x−z plane), while the detected hadron momentum FUU and FLsiUnφh can be written as [57]
togetherwiththezaxisidentifythehadronproductionplane.
F = [f D ], (24)
UU 1 1
C
2M Pˆ k M E˜
general expression [57]: Fsinφh = T · T h h⊥ +xg⊥D
LU Q C " M M 1 z 1!
dσ =2πα2 y2 1+ γ2 F Pˆ p M G˜⊥
dxdydzhdPT2dφh xyQ22(1−ε)(cid:16) 2x(cid:17){ UU − TM·h T Mh f1 z +xeH1⊥!#, (25)
+λ 2ε(1 ε)sinφ Fsinφh ,
e − h LU
p o(21) where PˆT = PPTT with PT = |PT|, M and Mh are the
nucleon and hadron masses, respectively. The functions
where FUU is the helicity-averaged structure function, G˜⊥andE˜ aretheinteraction-dependenttwist-3FFsthat
while Fsinφh is helicity-dependent structure function re- comefromthequark-gluon-quarkcorrelatorforFFs. The
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sulting from the antisymmetric part of the unpolarized formerone is T-odd,while the later one is T-even. They
hadronic tensor. The first and second subscript of the can be connected to the interaction-independent twist-3
above structure functions indicate the polarization of FFs G⊥ [38] and E [58] by the following relations [57]:
beam and target, respectively. It is Fsinφh that gives
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rise to the sinφh beam SSA. The ratio of the longitudi- G˜⊥ = G⊥ m H⊥, E˜ = E m D . (26)
nal and transverse photon flux is given by z z − M 1 z z − M 1
h h
ε= 1−y−γ2y2/4 . (22) The beam-spin asymmetry AsLinUφ in single-pion pro-
1 y+y2/2+γ2y2/4 duction off an unpolarized target thus is expressed as
−
dx dy dz 1 y2 1+ γ2 2ε(1 ε)Fsinφ
Asinφ(P )= xyQ22(1−ε) × 2x − LU (27)
LU T R R dxR dy dz 1 y(cid:16)2 1(cid:17)p+ γ2 F
xyQ22(1−ε) × 2x UU
R R R (cid:16) (cid:17)
for the P -dependent asymmetry. The x-dependent and effect eH⊥, where H⊥ is the T-odd Collins FF [25], and
T 1 1
the z-dependent asymmetries can be defined in a similar e is the chiral-odd twist-3 DF [58, 60].
way.
In the following, we will calculate the beam SSA in
The structure function Fsinφh receives various contri- semi-inclusive pion electroproductioncontributed by the
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butions from the convolution of the twist-2 and twist-3 g⊥D term. To dothis, wefirstneglectthe quark-gluon-
1
TMD DFs and FFs, as shown in Eq. (25). The h⊥E quark correlators (often referred to as the Wandzura-
1
term, instead of the h⊥E˜ in Eq. (25), has been calcu- Wilczek approximation [61]) for FFs, which is equiva-
1
lated in Ref. [28]. The contribution from the g⊥D term lent to setting all the functions with a tilde to zero. It
1
has been studied in Refs. [31–33, 59]. These two terms is worthwhile to point out that a calculation from spec-
generatetheasymmetrythroughtheeffectsoftheT-odd tator model [50] as well as a model-independent anal-
distribution functions, namely, the twist-2 Boer-Mulders ysis [62] of the T-odd collinear quark-gluon-quark cor-
function h⊥ [21] and the twist-3 g⊥. Each distribution relators shows that the gluonic (partonic) pole contri-
1
represents a specific spin-orbit correlation of the initial butions for FFs vanish. The FF G˜⊥(x,p2) appears in
T
quark inside the nucleon. The beam SSA of the π+ me- the decompositionof the T-oddpart ofthe TMD quark-
son has also been analyzed [29, 30] based on the Collins gluon-quarkcorrelator[57,63],forwhichthegluonicpole
6
contribution should play an essential role. Whether the

vanishing gluonic pole matrix elements for collinear FFs h
canbe generalizedto the case ofTMD FFs deserves fur- sin A0LU.10 0.05<PT<0.2 0.2<PT<0.4
ther study [64]. Nevertheless, we ignore the G˜⊥ and E˜ 0.05
contributions based on the Wandzura-Wilczek approxi-
mation. Therefore, there are only two terms inside the 0.00
square brackets in the right-hand side of Eq. (25) re-
-0.05
maining. Moreover, we consider merely the beam SSA
of the π0 production, since the fascinating fact that the 0.10 0.4<PT<0.6 0.6<PT<0.8
favoredand the unfavoredCollins functions have similar
0.05
sizes but opposite signs [65, 66] suggests that the eH⊥
term leads vanishing beam SSA in π0 electroproduction1.
0.00
Specifically, the isospin symmetry determines that the
π0 FF should be the average of π+ and π− FFs, so that -0.05
iHng1⊥πc0a/lqcu=la(tHio1⊥n,fawve+cHan1⊥juunsft)/t2ak≈e 0in.tTohaucsc,oiunntthtehefoltleorwm- 0.2 0. 3 0.4 x 0.2 0. 3 0.4 x
g⊥D and obtain
1
2Mx Pˆ k FIG. 3: Beam SSA AsLinUφh in π0 electroproduction con-
FLsiUnφh ≈ Q e2q d2kT TM· T xg⊥q(x,k2T) tributedbyg⊥ asafunctionofxfordifferentPT ranges. The
q=u,d Z (cid:26) solid and the dashed lines correspond to the results without
X (cid:2) andwiththekinematicalconstraints(31)onkT,respectively.
Dq z,(P zk )2 . (28) Dataare from Ref. [17], theerror bars for thedata including
× 1 T − T thesystematic and statistical uncertainties.
(cid:27)
(cid:0) (cid:1)(cid:3)
For the unpolarized TMDs fq(x,k2), we adopt the
1 T
resultsfromthesamespectatormodelcalculation[41]for
consistency. For the unpolarized TMD FFs Dq z,p2 , h
we assume that their pT dependencies have a 1GaussTian sin 0A.LU10 0.1<x<0.2 0.2<x<0.3
(cid:0) (cid:1)
form 0.05
Dq z,p2 =Dq(z) 1 e−p2T/hp2Ti; (29) 0.00
1 T 1 π p2
h Ti
(cid:0) (cid:1) -0.05
here, p2 is the Gaussian width for p2, which is cho-
sen ash Tpi2 = 0.2 Gev2, following theTfitted result in 0.10 0.3<x<0.4 0.4<x<0.6
Ref. [67h].TFior the integrated FFs Dq(z), we adopt the
1 0.05
Kretzer parametrization [68].

To perform the numerical calculation on the asymme- 0.00
try Asinφh in π0 production at CLAS, we adopt the fol-
LU -0.05
lowing kinematical cuts [17]:
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
0.4<z <0.7, Q2 >1GeV2, W2 >4GeV2, PT(GeV) PT(GeV)
P >0.05GeV, M (eπ0)>1.5GeV, (30)
T x
FIG. 4: Beam SSA Asinφh in π0 electroproduction con-
LU
whereMx(eπ0)isthemissing-massvaluefortheeπ0 sys- tributedbyg⊥ asafunctionofPT fordifferentxranges. The
tem. Finally, in our calculation, we consider two differ- solid and the dashed lines correspond to the results without
ent cases concerning the kinematical constraints on the andwiththekinematicalconstraints(31)onkT,respectively.
intrinsic transverse momentum of the initial quarks [69]. Dataare from Ref. [17], theerror bars for thedata including
Thefirstcaseisthatwedonotimposeanyconstraintfor thesystematic and statistical uncertainties.
k inthecalculation;the secondcaseisthatweconsider
T
the following kinematical constraints that are derived in
Ref. [69]: analyzedinRef.[69]. InourcalculationofAsinφh,which
LU
is also a twist-3 observable, we find again that the kine-
k2 (2 x)(1 x)Q2, for 0<x<1; maticalconstraints(31)modify the sizesofthe asymme-
T ≤ − − (31)
(kT2 ≤ (x1(−12−xx))2 Q2, for x<0.5. tryFiingucreert3aisnhkoiwnsemthaeticraelsureltgsioonfs.the beam SSA Asinφh
LU
The aboveconstraintsfix the upper limit for the allowed for π0 production as a function of x, compared with the
k range;thus,theycanmodifycertainazimuthalasym- CLASdata(fullcircles)measuredusinga5.776GeVelec-
T
metries substantially,suchas the the twist-3 Cahneffect tronbeam[17]. Thefourpanelscorrespondtotheasym-
7
metryintegratedoverfourdifferentP ranges. Fromthe
T
comparisonbetweenthe theoreticalcurvesand the data,
one can see that our results qualitatively describe the x h0.10
dependence of the asymmetry. Especially, for the ranges sin ALU
0.05GeV<P <0.2GeV and 0.2GeV<P <0.4GeV,
T T
our model calculation predicts rather flat curves, well 0.05
agreeing with the data. We would like to point out that

the curve for the range 0.2GeV < P < 0.4GeV is
T
similar to the calculation on the g⊥D term (the solid 0.00
1
line in Fig. 4 of Ref. [18]) based on the models from
Refs. [39, 70]. For higher P ranges, deviation between
T -0.05
our calculation and the data is found in the larger x re-
gion.
0.3 0.6 0.9 0.1 0.3 0.6 0.9
In Fig. 4, we display the same asymmetry from our z x PT(GeV)
calculation, but as a function of the π0 transverse mo-
mentum PT. Here, the four panels in Fig. 4 correspond FIG.5: ThebeamSSAAsinφh forπ0 productioninSIDISat
totheasymmetryintegratedoverfourdifferentxranges. LU
HERMES vs z (left panel), x (central panel), and PT (right
Agreement between the theoretical calculation and the panel). The solid and the dashed lines correspond to the
data is found in the lower PT region (PT <0.5GeV) for results without and with the kinematical constraints (31) on
x<0.4. Morespecifically,ourtheoreticalcurvesincrease kT, respectively. The thin and thick lines in the central and
withincreasingP inthewholeP regionwithinthefig- right panels correspond to the results for the ranges 0.2 <
T T
ure,whiletheexperimentaldataincreasewithincreasing z < 0.5 and 0.5 < z < 0.8. Data are from Ref. [8], with
P then approach a maximum at around P 0.4GeV, open circles, full circles, and open squares for 0.2 <z < 0.5,
T T
≈ 0.5<z <0.8, and 0.8<z <1. The error bars represent the
indicating that our prediction overestimates the experi-
statistical uncertainty.
mental data in the region P >0.5GeV.
T
From the comparison between the theoretical calcu-
tlaetrimoncaanndactchoeunCtLfAorStdhaetba,eawme ScoSnAcliundπe0thpartodtuhcetigo⊥nDa1t sinh A0LU.10
CLASintheregionsxandPT arenotlarge(x<0.4and 0.08
P < 0.5 GeV). However, our model overestimates the
T
data in the higher P and x regions. This disagreement 0.06
T
mightbeexplainedbytheabsenceofothercontributions
0.04
inEq.(25)thatweneglectinthecalculation(suchasthe
h⊥E˜), or by the possibility that the tree-level factoriza- 0.02
1
tion is not suitable in this region.
0.00
Furthermore, we also compare our calculation of the 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8
Asinφh asymmetry for neutral pion production in SIDIS z x PT (GeV)
wiLtUh the data measured by the HERMES Collabora- FIG. 6: Beam SSA Asinφh of π0 at CLAS12 as a function of
LU
tion [8]. The HERMES measurement uses a longitudi- z (left panel), x (central panel), and PT (right panel). The
solid and the dashed lines correspond to the results without
nallypolarized27.6GeVpositronbeamoffthehydrogen
gas target. In addition, the following kinematics are ap- andwiththekinematicalconstraints(31)onkT,respectively.
plied in the calculation [8]
try Asinφh at CLAS12, which can be measured with a
0.023<x<0.4, 0<y <0.85, 1GeV2 <Q2 <15GeV2, LU
12GeV polarized electron beam. The kinematical cuts
W2 >4GeV2, 2GeV<Ph <15GeV, (32) for CLAS12 applied in the calculation are [71]
where P is the energy of the detected final-state π0 in 0.08<x<0.6, 0.2<y <0.9, 0.3<z <0.8,
h
the target rest frame. In the left, central, and right pan- Q2 >1GeV2, W2 >4GeV2, 0.05GeV<P <0.8GeV.
T
els of Fig. 5, we show the asymmetry vs z, x, and P
T (33)
and compare it with the HERMES data [8]. It is found
that the theoretical curves agree with the experimental In the left, central, and right panels of Fig. 6, we plot
data within the statistical uncertainty. The predicted the z, x, and P dependencies of the asymmetry, re-
T
z dependence of the asymmetry is rather flat, which is spectively. Our calculation shows that the beam SSA
consistent with the the data in the mid-z region. The at CLAS12 is smaller than that at CLAS, but is still
calculation without the constraints in Eq. (31) describes sizable. Our result indicates that there is no obvious
the data in the smaller x region better than that with z dependence of the asymmetry in π0 production; this
the constraints. is understandable since in our approach, the same FFs
Finally, we present the prediction of the asymme- Dq(z,p2)appearinboththe numeratorandthe denom-
1 T
8
inator of the expression for Asinφh. Therefore, the pre- thermore,we considerthe specific caseofπ0 production,
LU
cision measurement of the z dependence of AsLinUφh in π0 in which the eH1⊥ term should give vanishing contribu-
electroproductionatCLAS12canverifythe roleofg⊥ in tionbecause the favoredand the unfavoredCollins func-
beam SSA. tions have similar sizes but opposite signs. Thus, we
canperformthe phenomenologicalanalysisonthe asym-
metry Asinφh in π0 production at CLAS and HERMES
LU
IV. CONCLUSION justbasedonthe g⊥D termsafely. The comparisonbe-
1
tween our theoretical calculation and the data indicates
In this work,we have performedthe calculationof the thatthe g⊥D1 termcanaccountforthe beamSSAin π0
T-oddtwist-3TMD distributiong⊥(x,k2)for the u and production measured by the CLAS Collaboration in the
T
d quarks inside the proton in the spectator model with region x < 0.4 and PT < 0.5GeV, where our theoreti-
scalarand axial-vectordiquarks. The difference between cal curves describe the data fairly well. In addition, our
the isoscalar (ud-like) and isovector (uu-like) spectators calculatedasymmetryat the HERMESkinematic region
for the axial-vector diquark is considered in the calcula- is consistent with the HERMES measurements after the
tion. We make use of the single-gluonexchange between errorbarsofthedataareconsidered. Ourstudysuggests
thestruckquarkandthespectatortogeneratetheT-odd that the T-odd twist-3 distribution g⊥ plays an impor-
structure. Toobtainafiniteresult,wechoosethedipolar tantroleinthebeamSSAinSIDIS,especiallyinthecase
form factor for the nucleon-quark-diquark coupling. We of neutral pion production.
findthatg⊥u andg⊥d havedifferentxandk dependen-
T
cies. First, g⊥u is positive forall x andk regions,while
T
g⊥d is negative in the small x regionand turns out to be Acknowledgements
positive in the region x >0.15. Second, g⊥d approaches
to zero faster than g⊥u with increasing kT. This work is partially supported by National Natural
Using the model results, we analyze the beam SSA Science Foundation of China with Grant No. 11005018,
Asinφh in semi-inclusive pion electroproduction. We ap- by SRF for ROCS from SEM, and by the Teaching and
LU
ply the Wandzura-Wilczek approximation for the FFs; Research Foundation for Outstanding Young Faculty of
thatis,weignorethecontributionsfromG˜⊥ andE˜. Fur- Southeast University.
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