Physics Letters B 666 (2008) 44–47

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Physics Letters B

www.elsevier.com/locate/physletb

Single spin asymmetry in inclusive hadron production in pp scattering fromCollins mechanism

Feng Yuan a,b,∗a Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAb RIKEN BNL Research Center, Building 510A, Brookhaven National Laboratory, Upton, NY 11973, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 April 2008Received in revised form 6 June 2008Accepted 25 June 2008Available online 4 July 2008Editor: B. Grinstein

PACS:12.38.Bx13.88.+e12.39.St

We study the Collins mechanism contribution to the single transverse spin asymmetry in inclusive hadronproduction in pp scattering p↑ p → π X from the leading jet fragmentation. The azimuthal asymmetricdistribution of hadron in the jet leads to a single spin asymmetry for the produced hadron in the Labframe. The effect is evaluated in a transverse momentum dependent model that takes into account thetransverse momentum dependence in the fragmentation process. We find the asymmetry is comparablein size to the experimental observation at RHIC at

√s = 200 GeV.

© 2008 Elsevier B.V. All rights reserved.

Single-transverse spin asymmetries (SSA) in hadronic processeshave a long history [1,2]. They are defined as the spin asymmetrieswhen we flip the transverse spin of one of the hadrons involvedin the scattering: A = (dσ(S⊥) − dσ(−S⊥))/(dσ(S⊥) + dσ(−S⊥)).Recent experimental studies of SSAs in polarized semi-inclusivelepton–nucleon deep inelastic scattering (SIDIS) [3,4], in hadroniccollisions [5–9], and in the relevant e+e− annihilation process [10],have renewed the theoretical interest in SSAs and in understand-ing their roles in hadron structure and Quantum Chromodynamics(QCD). Among the theoretical approaches proposed in the QCDframework, the transverse momentum dependent (TMD) partondistribution approach [11–18] and the twist-3 quark–gluon correla-tion approach [19,20] are the most discussed in the last few years,and it has been demonstrated that these two approaches are actu-ally consistent with each other in the overlap regions where bothapply [21].

For the SSAs in hadron production, two important contribu-tions have been identified in the literature: one is associated withthe so-called Sivers effect [11,20] from the incoming polarized nu-cleon; and one with the Collins effect [12] in the fragmentationprocess for the final state hadron. A third contribution associ-ated with the twist-three effects from the unpolarized nucleonwas found very small [22]. Both effects shall contribute to theSSA in inclusive hadron production in nucleon–nucleon scatter-ing, for example, in pion production in p↑ p → π X . However, how

* Corresponding author at: Nuclear Science Division, Lawrence Berkeley NationalLaboratory, Berkeley, CA 94720, USA.

E-mail address: fyuan@lbl.gov.

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.06.066

the transverse momentum dependent Sivers and Collins functionscontribute to the inclusive hadron production in p↑ p → π X isnot clear, because the large p⊥ of the final state hadron has nodirect connection with the intrinsic transverse momentum of par-tons in nucleon or the transverse momentum in the fragmentationprocess. Therefore, these effects can only be evaluated in a model-dependent way [13,23]. Meanwhile, for the Sivers effects, the initialand final state interactions are crucial to the nonzero SSA in thehadronic processes [15], which have not yet been implementedin the model calculations [13,23]. Thus, it is more appropriate toadopt the twist-3 quark–gluon correlation approach for the Siverscontribution in p↑ p → π X , which takes into account the initialand final state interaction effects in the formalism [20].

For the Collins effect, a twist-3 extension to the fragmentationprocess has been formulated in [24]. However, a universality argu-ment for the Collins function [25] would indicate the contributionscalculated in [24] vanishes. This universality has also been recentlyextended to pp collisions [26]. Therefore, to establish a consistentframework for the twist-three quark–gluon correlation contribu-tion in the fragmentation process, we need further theoretical de-velopments. Before that, it is worthwhile to investigate the Collinseffects contribution to the inclusive hadron’s SSA p↑ p → π X byextending the results of [26] and using a transverse momentumdependent model in the quark fragmentation. This is what we willexplore in this paper. Earlier works on the Collins contribution canbe found in [27–29].

In our model, we assume a transversely polarized quark is pro-duced in hard partonic processes with transverse momentum P⊥and rapidity y1. This transversely polarized quark then fragmentsinto a final state hadron with azimuthal asymmetric distribution

F. Yuan / Physics Letters B 666 (2008) 44–47 45

(relative to the jet) according to the Collins effect [26]. The finalstate hadron’s momentum will naturally be the jet’s momentum ina fraction of zh plus the fragmentation momentum of hadron rela-tive to the jet: PhT . Thus, the azimuthal asymmetry found in [26]will lead to an azimuthal asymmetry of final state hadron in theLab frame, which is exactly the experimental measurement of theleft–right asymmetry AN . Our approach is a semi-classic picture,in the sense that the quark jet production comes from the hardpartonic processes and is calculated from a collinear factorizationapproach, whereas the fragmentation process takes the TMD ef-fects explicitly. This assumption, of course, will introduce sometheoretical uncertainties. However, we argue that our results shallprovide a good estimate on how large the Collins effects contributeto the inclusive hadron’s SSA in p↑ p → π X . It is important to notethat, to make reliable predictions for the inclusive process in ppscattering at the transverse momentum region of our interest, wehave to take into account the high order perturbative resumma-tion corrections, and the power corrections as well [30]. It will beinterested to study how this affect the spin asymmetries we areexploring in this Letter, as well as other spin-dependent observ-ables.

There have been calculations for the Collins effects contribu-tions to inclusive hadron’s SSA p↑ p → π X in the transverse mo-mentum dependent approach similar to our model, where theCollins contribution was found strongly suppressed [29].1 However,from the following calculations, we will find that the contribu-tions are as large as the SSAs observed by the RHIC experimentsat

√s = 200 GeV. In our model, we will follow the Collins effects

in a jet fragmentation in single transversely polarized pp scatter-ing [26]. As illustrated in Fig. 1, we studied the process,

p(P A, S⊥) + p(P B) → jet(P J ) + X → H(Ph) + X, (1)

where a transversely polarized proton with momentum P A scat-ters on another proton with momentum P B , and produces a jetwith momentum P J (transverse momentum P⊥ and rapidity y1in the Lab frame). The three momenta of P A , P B and P J formthe so-called reaction plane. Inside the produced jet, the hadronsare distributed around the jet axes. A particular hadron H willcarry certain longitudinal momentum fraction zh of the jet, andits transverse momentum PhT relative to the jet axis will definean azimuthal angle with the reaction plane: φh , shown in Fig. 1.Thus, the hadron’s momentum is defined as �Ph = zh �P J + �PhT . Therelative transverse momentum PhT is orthogonal to the jet’s mo-mentum P J : �PhT · �P J = 0. Similarly, we can define the azimuthalangle of the transverse polarization vector of the incident polar-ized proton: φS . The Collins effect will contribute to an azimuthalasymmetry for hadron production in term of sin(φh −φS). The dif-ferential cross section can be written as [26]

dσ

dy1 dy2 dP 2⊥ dzh d2 PhT

= dσ

dP .S.= dσU U

dP .S.+ |S⊥| |PhT |

Mhsin(φh − φs)

dσT U

dP .S., (2)

where dP .S. = dy1 dy2 dP 2⊥ dz d2 PhT represents the phase spacefor this process, y1 and y2 are rapidities for the jet P J and thebalancing jet, respectively, P⊥ is the jet transverse momentum,and the final observed hadron’s kinematic variables zh and PhT aredefined above. dσU U and dσT U are the spin-averaged and single-transverse-spin-dependent cross section terms, respectively. Theyare defined as [26]

1 After some corrections, this approach also predicts sizable contribution from theCollins effects [31].

Fig. 1. Illustration of the kinematics for the azimuthal distribution of hadrons in aleading jet fragmentation in single transversely polarized pp scattering.

dσU U

dP .S.=

∑a,b,c

x′ fb(x′)xfa(x)Dhc (zh, PhT )Huu

ab→cd,

dσT U

dP .S.=

∑b,q

x′ fb(x′)xδqT (x)δq(zh, PhT )HCollinsqb→qb. (3)

Here, x and x′ are the momentum fractions carried by the par-ton “a” and “b” from the incident hadrons, respectively. In theabove equation, fa and fb are the associated parton distributions,Dq(zh, PhT ) is the TMD quark fragmentation function, δqT (x) is thequark transversity distribution, and δq(zh, PhT ) the Collins frag-mentation function. The hard factors for the spin-averaged crosssections are identical to the differential partonic cross sections:Huu

ab→cd = dσ uuab→cd/dt , and the spin-dependent hard factors have

been calculated in [26].In the following, we will study how the above azimuthal asym-

metry contributes to the SSA in inclusive hadron production in ppscattering p↑ p → π X , especially at RHIC energy. In order to esti-mate this contribution, we assume that the hadron production isdominated by the leading jet fragmentation, and the Collins effectsdiscussed above shall lead to a nonzero azimuthal asymmetry inthe Lab frame, for example, in term of sin(Φh − ΦS ) where Φhand ΦS are the azimuthal angles of the final state hadron andthe polarization vector in the Lab frame. Following this assump-tion, the hadron production follows two steps: jet production andhadron fragmentation. In the fragmentation process, as we men-tioned above, the hadron’s momentum �Ph will be

�Ph = zh �P J + �PhT . (4)

If we choose the jet transverse momentum direction as x direc-tion as we plotted in Fig. 1, the final hadron’s momentum can beparameterized as follows:

Phx = zh P⊥ + PhT cosφh cos θ,

Phy = PhT sin φh,

Phz = zh P J z − PhT cosφh sin θ, (5)

where P⊥ is the transverse momentum of the jet in the Lab frame,θ the polar angle between the jet and plus z direction (the polar-ized nucleon momentum direction): y1 ≈ η = − ln tan(θ/2), andy1 and η are the rapidity and pseudorapidity of the hadron, re-spectively. We can also work out the general results for any az-imuthal angle (Φ J ) of the jet in the Lab frame. At RHIC experiment,a sizable single spin asymmetry has been observed in the forwarddirection, which means θ ≈ 0. We further assume that PhT P⊥ ,so that the rapidity of the hadron will approximately equal to thejet’s rapidity. The uncertainties coming from this approximationcan be further studied by taking into account the full kinematicsin the fragmentation process. With the above kinematics of Phx ,Phy , and Phz , we will be able to derive the transverse momentumPh⊥ and azimuthal angle Φh for the final state hadron in the Labframe.

By integrating the fragmentation functions over zh and PhT , wewill obtain the differential cross sections and the spin asymme-tries depending on the final state hadron’s kinematics, y1 and �Ph⊥ ,

46 F. Yuan / Physics Letters B 666 (2008) 44–47

where �Ph⊥ is hadron’s transverse momentum in the Lab frame. Letus first estimate roughly how the above effect contributes to theSSA for pion production in pp scattering p↑ p → π X , especiallyfor the sign. Suppose the incident nucleon A is polarized alongthe y direction, and we assume that π+ is dominated by the va-lence u-quark fragmentation in the forward rapidity region. TheHERMES data show that the Collins function for u-quark fragmen-tation into π+ is negative if the u-quark transversity distributionis positive in the valence region [32]. From the differential crosssection Eq. (2), we will find that the π+ will prefer to be pro-duced with φh around 0, which will lead to an increase of π+production in +x direction. That means this contribution will re-sult into a positive left–right (AN ) asymmetry for π+ . Similarly, wefind that the contribution to π− left–right asymmetry is negative,and that for π0 will be determined by the contributions from bothu and d quarks. These estimates are consistent with the experi-mental trends for the SSAs in pion productions at RHIC [5,6,8,9].

Quantitatively, we can perform our calculations for the spinasymmetries at RHIC energy. With the above kinematics, we canwrite down the differential cross section for inclusive hadron pro-duction pp → π X coming from the leading jet fragmentation, de-pending on the final state hadron’s kinematics,

dσ uu

dy1 d2 Ph⊥=

∫dy2 dP 2⊥

1

πdΦ J dzh Θ(P⊥ − k0)Θ(Λ − PhT )

× xfa(x)x′ fb(x′)Dc(zh, PhT )Huu, (6)

where the jet’s transverse momentum �P⊥ is integrated out, andalso the associated azimuthal angle Φ J . From the rotation invari-ance of the above expression, the differential cross section will beazimuthal symmetric for the final state hadron. Thus, it will notdepend on the azimuthal angle Φh . In the above equation, we haveimposed two cuts for the momenta P⊥ and PhT . The minimumvalue for P⊥ is necessary to guarantee that the fragmentation iscoming from the leading jet production. A cutoff on PhT is neededbecause in our model we have assumed that PhT P⊥ . The theo-retical uncertainties coming from these cutoffs can be further stud-ied by varying these two parameters. Similarly the spin-dependentcross section can be written as

d�σ U T (S⊥)

dy1 d2 Ph⊥=

∫dy2 k⊥ dP⊥

1

πdΦ J dzh Θ(P⊥ − k0)Θ(Λ − PhT )

× |PhT |Mh

sin(φh − ΦS + Φ J )xδqT (x)x′ fb(x′)

× δq(zh, PhT )HCollinsqb→qb, (7)

where ΦS is the azimuthal angle of the transverse polarizationvector S⊥ in the Lab frame, and its relative angle to the jet definedin Fig. 1 φS can be written as φs = ΦS − Φ J . Following above, wefurther define Φh as the azimuthal angle of the produced hadronin the Lab frame, which is different from the above φh . From thesedifferential cross sections, the left-right asymmetry AN is calcu-lated as

AN = 〈2 sin(ΦS − Φh)d�σ U T 〉〈dσ uu〉 . (8)

In the numerical simulations, we use simple Gaussian parameteri-zations for the TMD fragmentation functions,

Dc(zh, PhT ) = 1

π〈p2⊥〉e−P 2hT /〈p2⊥〉Dc(zh),

δq(zh, PhT ) = 2Mh

(π〈p2⊥〉)3/2e−P 2

hT /〈p2⊥〉δq(1/2)(zh), (9)

where Dc(zh) is the integrated fragmentation function, and δq(1/2)

the so-called half-moment of the Collins function. The above pa-rameterizations have been chosen to give the right normaliza-tion for the two fragmentation functions. In the following nu-merical calculations, we choose the parameters 〈p2⊥〉 = 0.2 GeV2,Λ = 1 GeV, and k0 = 1 GeV.

The half-moment of the Collins functions δq(1/2)(zh) have beendetermined from the HERMES data by assuming some functionalform dependence on zh [32–34]. In [32], they are parameter-ized as δq(1/2) = C zh(1 − zh)Dc(zh) for the favored and unfavoredones. These parameterizations have to be updated, because the di-hadron production in e+e− annihilation from BELLE experimentsshowed a strong increase of the asymmetry with zh [10]. To beconsistent with this observation, we re-parameterize the Collinsfunctions as follows [33]:

δqπ(1/2)

fav.(zh) = C ′

f zh Dπ+u (zh),

δqπ(1/2)

unfav.(zh) = C ′

u zh Dπ+d (zh). (10)

With the new parameters modified from [32]: C ′f = 0.61C f and

C ′u = 0.65Cu , we will be able to reproduce the Collins asymmetries

for π± from HERMES, assuming the quark transversity distribu-tions follow the parameterizations in [35].

In Fig. 2, we show the numerical estimates from our model cal-culations of the Collins mechanism contributions to the SSA in π0

production in pp scattering p↑ p → π0 X at RHIC at√

s = 200 GeV:

Fig. 2. Model predictions for the Collins contribution to the SSA in π0 production at RHIC at√

s = 200 GeV: left panel as function of Ph⊥ for xF > 0.4 and integrate overall rapidity; right panel as function of xF for two different rapidity bins, y = 3.3,3.7, respectively. The dashed curve in the left panel is the prediction with the cutoff onthe transverse momentum PhT in the fragmentation process: PhT < Λ = 0.7 GeV, whereas the solid curve in the left panel and the two curves in the right panel are all forΛ = 1 GeV.

F. Yuan / Physics Letters B 666 (2008) 44–47 47

left panel as function of Ph⊥ for xF > 0.4 and integrate over allrapidity; the right panel as functions of xF for two different ra-pidities: y = 3.3,3.7, respectively. Similar results are also obtainedfor the charged pions. The decrease of the SSA as Ph⊥ decreasesis due to our modeling the Collins effects in the fragmentationprocess. Numerically, this decrease comes from a lower cut forthe jet transverse momentum P⊥ > k0 in our formalism Eqs. (6),(7), which limits the Collins contribution to the asymmetries atPh⊥ below the cut-off k0. At this kinematic region, however, asoft mechanism may be responsible for the cross section and theasymmetries, and its contribution may further change the Ph⊥ de-pendence.

In order to study the model dependence of the above results,we also calculated the Collins contribution to the SSA with a differ-ent cutoff for the transverse momentum PhT in the fragmentationprocess: PhT < Λ = 0.7 GeV. We showed this prediction in theleft panel of Fig. 2. The variation between different cutoffs can beviewed as the theoretical uncertainties in our model. How to im-prove this part deserves further investigations.

These plots show that the Collins contributions to the SSA ininclusive hadron production in pp scattering p↑ p → π X are notnegligible, rather comparable in size to what we observed at RHICfor charged and neutral pions [5,6,8,9]. However, we will not in-tend to compare them with the real data on these SSAs, for whichwe have to take into account the Sivers contributions as well [20].

In conclusion, in this Letter, we have studied the Collins mech-anism contribution to the inclusive hadron’s SSA in pp scatteringp↑ p → π X at RHIC in a model where the hadron productioncomes from the leading jet fragmentation with transverse momen-tum dependence. We found that their contributions to the SSA forinclusive π0 production at RHIC are the same size as the experi-mental measurements. Our results shall stimulate more theoreticalinvestigations toward a fully understanding for the longstandingSSA phenomena in hadronic processes.

Acknowledgements

We thank Gerry Bunce, Les Bland, Matthias Grosse-Perdekamp,Jianwei Qiu, Werner Vogelsang for discussions and comments. Wealso thank Mauro Anselmino, Umberto D’Alesio, and Elliot Leaderfor the communications concerning the calculations of [29]. Thiswork was supported in part by the US Department of Energyunder contract DE-AC02-05CH11231. We are grateful to RIKEN,Brookhaven National Laboratory and the US Department of Energy(contract number DE-AC02-98CH10886) for providing the facilitiesessential for the completion of this work.

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