Table Of ContentPredicting the sinφ Transverse Single-spin Asymmetry of Pion Production at an
S
Electron Ion Collider
Xiaoyu Wang1 and Zhun Lu1,∗
1Department of Physics, Southeast University, Nanjing 211189, China
We study the transverse single-spin asymmetry with a sinφ modulation in semi-inclusive deep
S
inelastic scattering. Particularly, we consider the case in which the transverse momentum of the
finalstate hadron is integrated out. Thus, theasymmetry is merely contributed by thecoupling of
the transversity distribution function h1(x) and the twist-3 collinear fragmentation function H˜(z).
Using the available parametrization of h1(x) from SIDIS data and the recent extracted result for
H˜(z), we predict the sinφ asymmetry for charged and neutral pion production at an Electron
S
6 Ion Collider. We find that the asymmetry is sizable and could be measured. We also study the
1 impact of the leading-order QCD evolution effect and find that it affects the sinφS asymmetry at
0 EIC considerably.
2
PACSnumbers: 13.60.-r,13.60.Le,13.88.+e
r
p
A
I. INTRODUCTION hadron fragmentation function, which will give rise to a
2 sin(φ +φ )asymmetrybasedonthecollinearfactoriza-
R S
1 tionformalism. Inthisapproachtwohadronsfragmented
Transverse single-spin asymmetry (SSA) in semi-
from a projectile quark should be detected. The idea
] inclusive processes has been recognized as a useful tool
h was used to extract the transversity in Ref. [9]. In the
to probe the spin structure of nucleon. One of the
p Drell-Yan process, the transversity may be accessed by
fundamental observables encoding the nucleon structure
- measuring the double transverse spin asymmetry, which
p is the transversity distribution of quarks, denoted by
iscontributedbythe convolutionofthe quarktransvere-
e hq(x), which describes the transverse polarization of the
h 1 sity and the antiquark transversity.
quark inside a transversely polarized nucleon: hq(x) =
[ 1 Inthiswork,weproposeanalternativeapproachtoac-
f f . As a chiral-odd distribution, transver-
3 siqt↑y/pi⇑s−ratqh↓e/rp⇑difficult to be probed, compared to the c3escshitrraaln-osdvedrsfirtaygmineSnItDatIiSo.nWfuenwctiilolns,hodwentohtaedt tbhye tH˜w(izst)-,
v
other two lead-twist collinear distribution functions: the
4 can serve as a “spin analyzer” to probe the transversity
unpolarized distribution and the helicity distribution,
7 distribution. As demonstrated in Ref. [10], within the
which have been extensively studied and measured. To
5 collinearpicture (or equivalently, the transversemomen-
1 manifest the effect of transversity in a process, another tum of the outgoing hadron is integrated out), only the
0 chiral-odd function is needed to couple with transver- coupling between the function H˜(z) and the transver-
. sity to ensure chirality conservation. It has been shown
1 sity remains as a contribution to the sinφ azimuthal
that twopromising processesmay be appliedto measure S
0 modulation in the leptoproduction of single hadron off
6 the transversity distributions. One is the semi-inclusive
a transversely polarized nucleon. The advantage of this
1 deeply inelastic scattering (SIDIS) [1], and the other is
approach is that the transverse momentum of the final-
: the Drell-Yan process [2].
v state hadron is not necessarily to be measured, in con-
Xi Under theframeworkofthetransversemomentumde- trasttotheCollinseffect. However,thefeasibilityofthis
pendent(TMD)factorization[3],inSIDISthechiral-odd approach has never been tested so far, partly because of
ar probe canbe the Collins function H1⊥(z,p2T) [1], a TMD the very limited knowledge of the almost unknown func-
fragmentationfunctionthatdescribesthefragmentingof tion H˜(z). This situation may be changed, since recent
atransverselypolarizedquarktoanunpolarizedhadron. phenomenological studies [11, 12] of the transverse SSA
The corresponding observable is the Collins asymmetry in pp↑ π+X provide veryuseful constraintsonH˜(z).
with a sin(φ +φ ) modulation. Here φ and φ are →
h S S h Particularly, the authors in Ref. [12] have adopted the
the azimuthal angles of the transverse spin of the nu-
SSAdataatSTAR[13–16]andBRAHMS[17]ofRHICto
cleonandthe outgoinghadron,respectively. The Collins extractthe function Hˆℑ (z,z ) within the frameworkof
asymmetry has been measured by the HERMES Collab- FU 1
collinear twist-3 factorization. The function Hˆℑ (z,z )
oration [4], the COMPASS Collaboration [5], and the FU 1
is the imaginary part of the twist-3 fragmentation func-
Jefferson Lab Hall A Collaboration [6]. The SIDIS data
tion H (z,z ), which involves the F-type quark-quark-
combined with the e+e− annihilation data were applied FU 1
gluoncorrelation[11, 18, 19]; and it can be connectedto
to extract [7, 8] the transversity of the up and down
H˜(z) by the integral
quarks. Another probe in SIDIS is the chiral-odd di-
∞ dz 1
H˜h/q(z)=2z3 1 Hˆh/q,ℑ(z,z ). (1)
z2 1 1 FU 1
Zz 1 z − z1
∗Electronicaddress: [email protected] A calculation using the spectator model also shows that
2
the size of H˜(z) is substantial [20]. Therefore, it is target has the general form [10]:
intriguing to study the consequence of sizable H˜(z) in
the sinφ asymmetry in SIDIS. More over, the study of d6σ α2 y2 γ2
S = (1+ )
the asymmetry in SIDIS may provide a check whether dxdydzdφ dφ dP2 xyQ22(1 ε) 2x
H˜(z) can cause the target SSAs in proton-proton colli- h S hT −
sions. Since the SSA data used to extract HˆFℑU(z,z1) × 2ε(1+ε) sinφSFUsiTnφS(x,z,PT)
are collected in the rather high energy region (mainly +psin(2φ φn)Fsin(2φh−φS)(x,z,P )
√s = 200 GeV) for which typically P > 1GeV, h− S UT T
h⊥
+ leading twist terms , (3)
where the collinear twist-3 factorization is applicable, }
we present the prediction of the sinφ asymmetry in
S
where ε is the ratio of the longitudinal and transverse
SIDIS at a future Electron Ion Collider (EIC) [21]. We
photon flux
calculate the sinφ asymmetry of charged and neutral
S
pion production at the energy √s = 45 GeV, which
1 y 1γ2y2
is comparable to the RHIC energy. To do this, we ε= − − 4 . (4)
adopt the available parametrization of the transversity 1 y+ 1y2+ 1γ2y2
− 2 4
for up and down quarks [8] and the recent extraction
for Hˆπ+/u,d¯,ℑ(z,z ) [12]. Furthermore, we include the In Eq. (3), FsinφS and Fsin(2φh−φS) are the twist-3
FU 1 UT UT
leading-order (LO) QCD evolution of the transversity structure functions which contribute to the sinφS and
distribution and H˜(z), and compare the corresponding the sin(2φh φS) azimuthal asymmetries, respectively.
−
results to those without evolution. Particularly, FUsiTnφS(x,z,PhT) can be expressed as [10]
The remained contentof the paper is organizedas fol-
aloswyms.mInetSreycitnioSnI.DIII,Swinestehteucpoltlhineefaorrmpiacltiusmre.ofInthSeescitnioφnS. FUsiTnφS(x,z,PhT)= 2QMC( xfTD1− MMhh1Hz˜!
III, we presentthe numericalcalculationofthe asymme-
triesinthe leptoproductionofchargedandneutralpions k p M G˜⊥
T · T xh H⊥+ hg
at EIC. Some conclusion is addressed in Sec. IV. − 2MMh " T 1 M 1T z !
M D˜⊥
xh⊥H⊥ hf⊥ , (5)
II. FORMALISM OF THE sinφS ASYMMETRY − T 1 − M 1T z !#)
IN SIDIS
where k and p are the transverse momenta of the in-
T T
The processwe study is the pion electroproductionoff
coming and outgoing quarks, M is the mass of the out-
h
a transversely polarized proton target:
going hadron, and the notation [ωfD] defines the con-
C
volution:
e(ℓ)+p↑(P) e(ℓ′)+π(P )+X(P ), (2)
h X
−→
where l and l′ stand for the momenta of the incoming C[ωfD]=x e2q d2pTd2kTδ(2)(pT −kT −PhT/z)
andoutgoingleptons,andP andP denotethemomenta q Z
h X
of the target nucleon and the final-state hadron (in our ω(p ,k )fq(x,k2)Dq(z,p2). (6)
T T T T
case the hadron is the pion meson), respectively. The
reference frame of the process under study is shown in Eq.(5) containsconvolutionsofthe twist-3distributions
Fig.1, inwhichthe momentumofvirtualphotondefines and the twist-2 fragmentation functions, as well as con-
the axis of z, φ denotes the the azimuthal angle of the volutions of the twist-3 fragmentation functions and the
h
final hadron around the virtual photon, and φ stands twist-2 distribution functions.
S
for the angle between the lepton scattering plane and In Ref. [22], the role of the twist-3 TMD distributions
thedirectionofthetransversespinofthenucleontarget. h (x,k2) and h⊥(x,k2) in the sinφ asymmetries as
T T T T S
The invariants used to express the differential cross functions of x, z and P = P was studied. In this
hT hT
| |
section are defined as work, however, we will consider the particular case in
whichthetransversemomentumoftheoutgoingpionme-
Q2 P q P P
x= , y = · , z = · h , son is integrated out, or equivalently, the case in which
2P q P l P q only the longitudinal momentum fraction z of pion is
· · ·
2Mx measured. Thus, after d2P is performed, the differ-
γ = , Q2 = q2, s=(P +l)2. hT
Q − ential cross section in Eq. (3) turns to the form
R
As usual, q = ℓ ℓ′ is defined as the momentum of the d4σ 2α2 y2 γ2
− = 1+
virtual photon, and M denotes the proton mass. Up to dxdydzdφ xyQ2 2(1 ε) 2x
twist-3level,thesix-fold(x, y,z,φ ,φ andP )differ- S − (cid:18) (cid:19)
h S hT 2ε(1+ε) sinφ FsinφS(x,z) . (7)
entialcrosssectioninSIDISwithatransverselypolarized × S UT
p
3
y
x
z
l′ S
⊥
l φs
PhT
φh
Ph
hadronplane
lepton plane
FIG. 1: The definition of the azimuthal angles in SIDIS. Ph stands for the momentum of the produced hadron, S⊥ is the
transverse component of thespin vectorS with respect to thevirtual photon momentum.
Here, the structure function FsinφS(x,z) is the where F is the unpolarized structure function:
UT UU
collinear counterpart of the original structure function
FsinφS(x,z,P ) [10] F (x,z)=x e2fq(x)Dq(z), (12)
UT hT UU q 1 1
q
X
FsinφS(x,z)= d2P FsinφS(x,z,P )
UT hT UT hT with fq(x) and Dq(z) are the unpolarized distribution
Z 1 1
2M H˜ function and fragmentation function, respectively. In a
=−Z d2PhT QhC"h1 z # sbiemwilarirttweanya,sthe sinφS asymmetry as a function of z can
2M H˜q(z,p2)
=−x Qh e2q d2kT z2 d2pThq1(x,kT2) z T AsUinTφs(z)
Xq Z Z dx dy α2 y2 (1+ γ2) 2ǫ(1+ǫ)Fsinφs(x,z)
=−x e2q2MQhhq1(x)H˜qz(z), (8) = R R dxxyQ2d2y(x1−yαQ2ǫ)22(1y−2ǫ2)x(1p+ 2γx2)FUU(xU,Tz) .
q
X (13)
R R
where only the convolution of the transversity and the
twist-3 collinear fragmentation function H˜q(z) remains. The function H˜(z) appearing in Eq. (8) is related to
In Eq. (8) we have used the integration the imaginarypartofthe twist-3 fragmentationfunction
Hˆq(z,z ) via Eq. (1). Useful information for Hˆq,ℑ(z,z )
1 FU 1
d2P δ(2)(p k P /z)=z2, (9) may be obtained from Ref. [12], in which the authors
hT T T hT
Z − − have extracted Hˆ(π/u,d¯),ℑ(z,z ) from the data [13–17] of
FU 1
andthe Collinearfragmentationfunctionisconnectedto pp↑ πX using the ansatz (at the scale Q2 =1GeV2):
the TMD function by →
Hˆπ+/(u,d¯),ℑ(z,z ) N
H˜q(z)=z2Z d2pTH˜q(z,p2T). (10) D1π+/(u,Fd¯U)(z)D1π+/(u,d¯1)(z/z1) = 2IfavfaJvfav zαfav(z/z1)α′fav
Therefore, the experimental investigation of the sinφS (1 z)βfav(1 z/z1)βf′av,(14)
× − −
azimuthal asymmetry without detecting P may pro-
hT
vide analternativeopportunity to measurethe transver- with N , α , α′ , β , β′ the parameters in the
fav fav fav fav fav
sity. To test the feasibility of this approach, we define model, and I and J , the forms of which are given
fav fav
the x-dependent sinφ asymmetry in Ref. [12], are the functions of the above parameters.
S
The disfavored fragmentation functions Hˆπ+/(d,u¯),ℑ are
AsinφS(x) FU
UT parameterizedin full analogyto (14) withthe additional
dy dzxyαQ222(1y−2ǫ)(1+ 2γx2) 2ǫ(1+ǫ)FUsiTnφs(x,z) parameters Ndis, αdis, α′dis, βdis, βd′is. The analysis in
=R R dy dzxyαQ222(1y−2ǫ)(1p+ γ2x2)FUU(x,z) , Rineft.h[e12t]rasnhsovwesrstehaStSAHˆFqi,nUℑ(tzh,ez1p)↑pplaysπaXn imprpoocretsasn.tTrohlee
R R (11) π− fragmentationfunctionsmaybe→fixedthroughcharge
4
zH~ +/d(z) 0.0 zH~ +/u(z)
0.4
0.2 2 2
-0.2 Q2=1GeV 2 ~z
Q=100GeV with H ( ) evolving as D1(z)
2 2 ~z
Q=100GeV with H ( ) evolving as h1(x)
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
z z
FIG.2: ResultofzH˜π+/d(z)(left panel)andzH˜π+/u(z)(rightpanel)attheinitialscaleQ2 =1GeV2 (solid lines,takenfrom
Ref. [12]) and the evolved results at Q2 =100GeV2 (dotted lines: evolvingas Dq(z), dashed lines: evolving as hq(x);)
1 1
conjugation: the corresponding impact on the sinφ asymmetry. To
S
do this, we apply the HOPPET package [26] to perform
Hˆπ−/(d,u¯),ℑ(z,z )=Hˆπ+/(u,d¯),ℑ(z,z ), (15) theevolution,andwemodifytheHOPPETtoolkittoin-
FU 1 FU 1
clude the LO DGLAP evolution kernel for the transver-
HˆFπ−U/(u,d¯),ℑ(z,z1)=HˆFπ+U/(d,u¯),ℑ(z,z1), (16) sity distribution. Since the QCD evolution of H˜(z) is
still unknown, in this calculation we adopt two different
andtheπ0fragmentationfunctionsaregivenbytheaver- choicestoevolveH˜(z). InthefirstchoiceweevolveH˜(z)
ageofthefragmentationfunctionsforπ+ andπ−. There- by employing the same evolution as the fragmentation
fore, we will apply the parametrization for Hˆh/q,ℑ in function D (z), following the scheme used in Ref. [12]
Ref. [12] and use Eq. (1) to obtain H˜q(z) needed in the for consiste1ncy. As a comparison, in the second choice
calculation. we assume that its evolution is the same as that of the
For the transversity distribution function hq1(x), we transversity. This is motivated by the fact that H˜(z)
adopt the standard parameterization from Ref. [8] (at is also a chiral-odd fragmentation function. In the left
the initial scale Q2 =2.41GeV2) and right panels of Fig. 2, we plot the z-dependence of
the disfavored and favored twist-3 fragmentation func-
1
hq(x)= T(x)[fq(x)+gq(x)], (17) tionszH˜π+/d(z)andzH˜π+/u atthe scalesQ2 =1GeV2
1 2Nq 1 1 (shown by the solid lines) and Q2 = 100GeV2 (dotted
with lines: evolving the same as D1q(z); dashed lines: evolv-
ing the same as hq(x)), respectively. We find that in the
1
(α+β)α+β region z < 0.4, the evolution from low Q to higher Q
T(x)=NT xα(1 β)β . (18)
Nq q − ααββ increases the sizes of both H˜π+/u and H˜π+/d; while it
decreases their sizes in the region z > 0.4. In addition,
The values of the parameters NqT,α, and β in Eq. (18) thereisslightlydifferencebetweenthetwoevolvedresults
are taken from Table. II in Ref. [8]. In order to of H˜ at the small-z region. However,if the uncertainties
be in consistence with the choices in Ref. [8], we ap- for the parametrization of H˜ were taken into account,
ply the parametrization for the unpolarized distribution the two curves calculated from two different evolution
f1q(x) from Ref. [23] and that for the helicity distribu- approaches would be statistically equivalent.
tion g1q(x) from Ref. [24], respectively. For the unpolar- Forthe kinematicalregionthatis availableatEIC,we
ized integrated fragmentation function D1q(z), we adopt adopt the following choice [21]
theleading-order(LO)settheDSSparametrization[25],
which is also chosen in Ref. [12]. Q2 >1GeV2, 0.001<x<0.4, 0.01<y <0.95,
(19)
0.2<z <0.8, √s=45 GeV, W >5 GeV,
III. NUMERICAL RESULTS AT AN EIC
where W is invariantmass of the virtualphoton-nucleon
In this section, we apply the formalism introduced system and W2 = (P +q)2 1−xQ2. Using the above
≈ x
above to estimate the sinφ asymmetry at EIC. As the kinematical configuration and applying Eqs. (11) and
S
kinematics at EIC covers a wide range of Q, it is neces- (13), we predict the sinφ asymmetry of pion produc-
S
sary to consider the QCD evolution of the distribution tion in SIDIS at EIC after the transverse momentum of
and fragmentation functions, particularly, the transver- pionis integrated. The correspondingresultsareplotted
sity and the twist-3 function H˜(z). In the present calcu- in Fig. 3, in which the upper, middle and lower pan-
lation we implement their LO QCD evolution to study els show the sinφ asymmetries at EIC for π+, π0 and
S
5
0.04 0.04
+
AsUinT s AUsinT s with H~ ( z ) evolving as D1(z)
EIC
0.03 energy s =45 GeV 0.03 AUsinT s with H~ ( z ) evolving as h1(x)
AUsinT s without evolution
0.02 0 .02
0.01 0.01
0.00 0.00
1E-3 0.01 0.1 0.2 0.4 0.6 0.8
x z
sin s
AUT
0.00 0.00
-0.02 -0.02
0 AsUinT s with H~ ( z ) evolving as D1(z)
EIC AsUinT s with H~ ( z ) evolving as h1(x)
energy s =45 GeV AsUinT s without evolution
-0.04 -0.04
1E-3 0.01 0.1 0.2 0.4 0.6 0.8
x z
0.00 0.00
sin s
AUT
-0.03 -0.03
-0.06 - -0.06 AsUinT s with H~ ( z ) evolving as D1(z)
EIC AsUinT s with H~ ( z ) evolving as h1(x)
-0.09 energy s =45 GeV -0.09 AUsinT s without evolution
1E-3 0.01 0.1 0.2 0.4 0.6 0.8
x z
FIG. 3: Transverse SSA AsinφS of π+, π0 and π− production in SIDIS at EIC for √s = 45 GeV. The left panels show the
UT
x-dependentasymmetry, while the right ones for thez-dependentasymmetry.
π−, respectively. Ineachpanel, we plot the asymmetries the asymmetry for π− is somewhat larger than those for
as functions of x (left figure) and z (right figure). The π+ and π0. This is because hu and H˜π+/d are posi-
1
solid lines denote the asymmetries without considering tive, while hd and H˜π+/u are negative. The numer-
1
theevolutionofthedistributionandfragmentationfunc- ical results show that the asymmetry for π0 is close
tions in Eqs. (11) and (13) (calculated at a fixed scale to (AsinφS,π+ +AsinφS,π−)/2, which is consistent with
Q2 = 1GeV2). The dotted and dashed lines correspond UT UT
tothetwodifferenttreatmentsontheevolutionofH˜q(z). the assumption that H˜π0/q is the average of H˜π+/q and
H˜π−/q. We also find that the x-dependent asymme-
We find that the sinφ asymmetries for the charged try has a peak at the intermediate x region, around
S
and neutral pion production are sizable, about several 0.1 < x < 0.2. For both charged and neutral pion pro-
percent. Therefore, there is a great opportunity to mea- ductions, the asymmetry reaches the maximum magni-
sure the sinφ asymmetry in SIDIS at a future EIC. tude at the region z 0.6.
S ∼
In addition, the asymmetry for π+ is positive, whereas An important observation is that the evolution effect
the asymmetries for π− and π0 are both negative; and for the sinφ asymmetry is substantial in certain kine-
S
6
maticalregionatEIC.Firstofall,asshownbythedotted verse SSA at EIC through the coupling h (x) H˜(z), in
1
lines(H˜(z)evolvesthesameasD (z))inFig.3,themag- the particularcasethatthe transversemomen⊗tumofthe
1
nitudesofthex-dependentandz-dependentasymmetries final-state hadron is integrated out. In our estimate we
forπ+,π0andπ−areallreducedbyQCDevolution. Sec- applied the standard parametrization for the transver-
ondly,atsmall-xregion(x<0.02)theevolutiondoesnot sity and the available extraction for the fragmentation
affect the x-dependent asymmetries, while the evolution function Hˆℑ (z,z ). In addition, the LO evolution ef-
FU 1
effect is sizable in the valence-x region, especially in the fects for the distribution and fragmentation functions
case of π− production. For the z-dependence asymme- were included. The numerical prediction shows that the
tries, the evolution effect may be observed in the region asymmetriesforthechargedandneutralpionsareallsiz-
z >0.4. Thirdly,theevolutioneffectforπ−productionis able,aboutseveralpercent. Therefore,itisquitepromis-
strongerthanthatforπ+andπ0production. Finally,the ing that the sinφ asymmetries of meson production in
S
uncertainties of the fragmentation function H˜(z), which SIDIS could be measured at the kinematics of EIC. We
will result in the uncertainties of the asymmetries, are alsofoundtheinclusionofevolutioneffectmaybeimpor-
notconsideredinourcalculation. Ifthe uncertaintiesfor tant for the interpretation of future experimental data.
the asymmetrieswereincluded,mostlikelytheevolution In conclusion, our study demonstrates that it is feasi-
effects fromthe statisticalviewpointare notsodramatic ble to access the transversity via transverse SSA of sin-
in the kinematical region of EIC. Nevertheless, the evo- gle meson production in SIDIS within the framework of
lution almost does not change the signs and the shapes collinear factorization.
of the asymmetries. As a comparison, we also show the
asymmetries(thedashedlines)withH˜ evolvingthesame
as h . It is found that the asymmetries in this case are
1
similar to the results calculated from the D evolution
1
for H˜, although slightly difference is observed in the x-
dependent asymmetries for π0 and π− production.
Acknowledgements
IV. CONCLUSION
This work is partially supported by the National Nat-
Inthiswork,wehaveimplementedthetwist-3collinear uralScience FoundationofChina (Grants No.11575043,
fragmentation function H˜(z) to study the sinφ trans- and No. 11120101004),and by the Qing Lan Project.
S
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7
120 (2009).
0.04 0.04
+
sin ~
s
sin s AUT with H ( z ) evolving as h1(x)
A
UT
sin ~
EIC s
A with H ( z ) evolving as D (z)
UT 1
0.03 0.03
energy s =45 GeV
sin
s
A without evolution
UT
0.02 0 .02
0.01 0.01
0.00 0.00
1E-3 0.01 0.1 0.2 0.4 0.6 0.8
z
x
~ /u
zH (z)
~ /d
zH (z)
0.0
0.4
2 2
0.2
Q =1GeV
~
H
-0.2
2 2
Q =100GeV with evolution as h (x)
1
~
H
2 2
Q =100GeV with evolution as Ref.[12]
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
z z
