Double transverse spin asymmetry in the p ↑ p ¯ ↑ Drell-Yan process from Sivers functions
aa r X i v : . [ h e p - ph ] A p r Double transverse spin asymmetry in the p ↑ ¯ p ↑ Drell-Yan process from Sivers functions
Zhun Lu, Bo-Qiang Ma, ∗ and Ivan Schmidt † Center of subatomic studies and Departamento de F´ısica, UniversidadT´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile School of Physics and MOE Key Laboratory of Heavy Ion Physics, Peking University, Beijing 100871, China
We show that the transverse double spin asymmetry (DSA) in the Drell-Yan process contributedonly from the Sivers functions can be picked out by the weighting function Q T M (cos( φ − φ S ) cos( φ − φ S ) + 3 sin( φ − φ S ) sin( φ − φ S )). The asymmetry is proportional to the product of two Siversfunctions from each hadron f ⊥ (1)1 T × f ⊥ (1)1 T . Using two sets of Sivers functions extracted from thesemi-inclusive deeply elastic scattering data at HERMES, we estimate this asymmetry in the p ↑ ¯ p ↑ Drell-Yan process which is possible to be performed in HESR at GSI. The prediction of DSA in theDrell-Yan process contributed by the function g T ( x, k T ), which can be extracted by the weightingfunction Q T M (3 cos( φ − φ S ) cos( φ − φ S ) + sin( φ − φ S ) sin( φ − φ S )), is also given at GSI. PACS numbers: 13.88.+e, 13.85.Qk
I. INTRODUCTION
The Sivers effect [1] was proposed originally to explainthe large single spin asymmetries (SSA) observed in in-clusive pion hadro-production ( p ↑ p → πX ) at FNAL [2].The effect can be quantitatively described by a k T -dependent distribution named as Sivers function [3, 4] f ⊥ T ( x, k T ), which is the distribution of unpolarized par-tons in a transversely polarized proton. It arises froma non-trivial correlation between the nucleon transversespin and the intrinsic transverse momenta in the nucleon.Despite its (naively) T -odd property [5], Sivers functionhas been proven to be non-vanishing [6] due to its specialgauge-link property [7, 8, 9].Recently the SSA measured in semi-inclusive deeplyinelastic scattering (SIDIS) processes with transverselypolarized targets at HERMES [10, 11, 12] and COM-PASS [13, 14], has been shown to be interpreted bythe Sivers effect. The asymmetry is identified by theangular dependence sin( φ − φ S ), where φ and φ S de-note respectively the azimuthal angles of the producedhadron and of the nucleon spin polarization, with respectto the lepton scattering plane. The coexistent Collinsasymmetry [5], with a angular dependence sin( φ + φ S ),has also been measured in those experiments. Thedata on the Sivers SSA has been utilized by differentgroups [15, 16, 17, 18, 19] to extract the Sivers functionsof the proton, especially those for the u and d quarks, onthe basis of the generalized factorization [20, 21]. Thosesets of parametrization of the Sivers functions are qual-itatively in agreement [22] among themselves, and wereapplied to predict the Sivers SSA in various processes inthe established or planed facilities, such as the SIDIS atJLab, and the Drell-Yan process at COMPASS, RHIC ∗ Corresponding author. Electronic address:[email protected] † Corresponding author. Electronic address: [email protected] and GSI.In this paper, we will investigate the role of the Siversfunction on the transverse double spin asymmetry (DSA)in the Drell-Yan process. The transverse DSA has beeninvestigated [23] for many years, and is believed to haveadvantage to unravel the transverse spin property ofthe nucleon [24], especially the transversity distribution h ( x ) [25]. Various azimuthal asymmetries contributedby different k T -dependent distribution functions havebeen analyzed and given in Refs. [26] and [27]. Asshown in Ref. [27], the Sivers function contributes tothe DSA in the Drell-Yan process through the product f ⊥ T × f ⊥ T . However, this DSA is mixed with the contri-bution from another k T -dependent distribution function g T ( x, k T ). We will show that through the appropri-ate weighting function Q T M (cos( φ − φ S ) cos( φ − φ S ) +3 sin( φ − φ S ) sin( φ − φ S )), the asymmetry from theSivers function can be isolated without mixing with thecontribution from other functions. Using two sets of pa-rameterizations [16, 18] of the Sivers functions we cal-culate the double spin asymmetry from the Sivers func-tions in the p ↑ ¯ p ↑ Drell-Yan process at GSI. An asymme-try around 1 % is predicted. The asymmetries estimatedfrom these two sets of Sivers functions are quantitativelydifferent. Therefore measuring the DSA in the Drell-Yanprocess can provide new information on Sivers functions,especially their sizes. The transverse DSA contributedby g T ( x, k T ) through the product g T × g T can alsobe picked out by another weighting function. We esti-mate this asymmetry by adopting a g T coming from thecombination of a Lorentz invariance relation presentedin Refs. [28, 29] and the Wandura-Wilzeck approxima-tion [30]. II. EXTRACTING DSA CONTRIBUTED BYTHE SIVERS FUNCTIONS
The importance of the transverse-momentum distri-butions of quarks for a full understanding of the struc-ture of hadrons has been widely recognized in the lastdecade [4, 29, 31, 32]. A comprehensive leading-twist treelevel analysis of the (spin dependent) Drell-Yan processin terms of k T -dependent distributions has been givenin Ref. [26]. The role of the T -odd k T -dependent distri-butions in this process has been presented in Ref. [27].In the Collins-Soper frame [33] the leading order unpo-larized differential cross-section for the Drell-Yan process h ( P ) + h ( P ) → γ ∗ ( q ) + X → l + ( l ) + l − ( l ) + X hasthe form [27] dσ (0) ( h h → l ¯ lX ) d Ω dx dx d q T = α em Q X q e q ( A ( y ) F [ f q f ¯ q ]+ B ( y )cos2 φ F " (2ˆ h · p T ˆ h · k T − p T · k T ) h ⊥ q h ⊥ ¯ q M M (1)where q denotes the quark flavors, the notation F [ f f ] = Z d p ⊥ d k ⊥ δ ( p T + k T − q T ) × f ( x , p T ) f ( x , k T ) (2)shows the convolution of transverse momenta, Q = q is the invariance mass of the lepton pair, q T is the trans-verse momentum of the lepton pair, ˆ h = q T /Q T , φ is theangle between the hadron plane and the lepton plane,and A ( y ) = (cid:18) − y + y (cid:19) = 14 (1 + cos θ ) , (3) B ( y ) = y (1 − y ) = 14 sin θ, (4)in the c.m. frame of the lepton pair.The function h ⊥ in the second line of (1) is the Boer-Mulders function [4], the chiral-odd partner of the Siversfunction. This function has attracted a lot of interest [34,35, 36] recently because it can account for the anomalouscos 2 φ asymmetries [37, 38] observed in the unpolarizedDrell-Yan process, given by the second term of Eq. (1)has shown.The leading order differential cross-section for the dou-ble transversely polarized Drell-Yan process is [27] dσ (2) ( h ↑ h ↑ → l ¯ lX ) d Ω dx dx d q T = α em Q X q ( . . . + A ( y )2 | S T | | S T | cos(2 φ − φ S − φ S ) ×F " ˆh · p T ˆh · k T f ⊥ q T f ⊥ ¯ q T − g q T g ¯ q T M M − A ( y )2 | S T | | S T | cos( φ − φ S ) cos( φ − φ S ) ×F " p T · k T f ⊥ q T f ⊥ ¯ q T M M − A ( y )2 | S T | | S T | sin( φ − φ S ) sin( φ − φ S ) ×F (cid:20) p T · k T g q T g ¯ q T M M (cid:21) ) , (5)The . . . indicates the terms which will not contribute inour analysis below, φ S and φ S are the angles between S T , S T and the lepton plane, respectively.As shown in (5), the Sivers function can contributeto the transverse DSA through the product f ⊥ T × f ⊥ T .However this asymmetry is mixed with the asymmetry towhich it contributes another k T -dependent distribution g T ( x, k T ). The main goal of this paper is to isolatethe asymmetry contributed by the Sivers function. Thestarting point is the method introduced in Ref. [39], bywhich one integrates the differential cross section with aproper weighting function W ( Q T , φ, φ S , φ S ), as follows: h W ( Q T , φ, φ S , φ S ) i = Z dφdφ S d q T dσ ( h h → l ¯ lX ) d Ω dx dx d q T × W ( Q T , φ, φ S , φ S ) . (6)With the above weighting procedure, one can pick up theterms in which one is interested. Besides this, one cande-convolute the transverse momentum integration in amodel independent way.The unpolarized angular independent cross section canbe picked out by using the weighting function 1, fromEq. (1): (cid:18) A ( y ) α em Q (cid:19) − · h i UU = 4 π X q e q f q ( x ) f ¯ q ( x ) (7)We denote W C = cos( φ − φ S ) cos( φ − φ S ) and W S =sin( φ − φ S ) sin( φ − φ S ). Given the weighting function Q T M W C (assuming M = M = M , i.e. the colliding twohadrons are nucleons), we can obtain the following termfrom (5): (cid:18) A ( y ) α em Q (cid:19) − · (cid:28) Q T M p W C (cid:29) T T = π X q e q n h f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x ) − g (1) q T ( x ) g (1)¯ q T ( x ) i − f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x ) o = π X a e a h f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x ) − g (1) q T ( x ) g (1)¯ q T ( x ) i , (8)where f ⊥ (1)1 T ( x ) and g (1)1 T ( x ) are the first k T -moments, de-fined as: f ⊥ (1)1 T ( x ) = Z d k T k T M f ⊥ T ( x, k T ) , (9) g (1)1 T ( x ) = Z d k T k T M g T ( x, k T ) . (10)The factor Q T introduced in the weighting function en-sures that the transverse momentum integration in (8)can be de-convoluted (for details, refer to the Appendix).Again, applying the weighting function Q T M W S on (5), wearrive at (cid:18) A ( y ) α em Q (cid:19) − · (cid:28) Q T M p W S (cid:29) T T = − π X q e q n h f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x ) − g (1) q T ( x ) g (1)¯ q T ( x ) i − g (1) q T ( x ) g (1)¯ q T ( x ) o = π X a e a h − f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x )+ g (1) q T ( x ) g (1)¯ q T ( x ) i , (11)Therefore, combining (8) and (11), we can extract theterm contributing to the transverse DSA and coming onlyfrom the Sivers functions: (cid:18) A ( y ) α em Q (cid:19) − · (cid:28) Q T M p ( W C + 3 W S ) (cid:29) T T = − π X q e q f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x ) , (12)with the weighting function Q T M ( W C + 3 W S ).By taking the ratio between (13) and (7), we definethe weighted double spin asymmetry as follows A fT T = D Q T M ( W C + 3 W S ) E T T h i UU = − P q e q f ⊥ (1) q T ( x ) f ⊥ (1)¯ q T ( x ) P q e q f q ( x ) f ¯ q ( x ) . (13)The above equation thus provides a possibility to studythe Sivers function by measuring the transverse DSA inthe Drell-Yan process.Also, from (8), (11) and (7) we can get another typeof DSA: A gT T = D Q T M (3 W C + W S ) E T T h i UU = − P q e q g (1) q T ( x ) g (1)¯ q T ( x ) P q e q f q ( x ) f ¯ q ( x ) , (14)which is contributed only by g T . III. NUMERICAL RESULTS
In this section we will give numerical results on theDSA from the Sivers functions. We consider the trans-versely polarized proton antiproton Drell-Yan process, x A fTT FIG. 1: The DSA in the proton antiproton Drell-Yan processat GSI coming only from the Sivers functions, and calculatedfrom Eq. (13). The kinematics is s = 45 GeV and Q =2 . . The solid and dashed curve use the Sivers functionsin Ref. [16] and in Ref. [18], respectively. where the valence Sivers functions are involved, so thata larger asymmetry should be measured compared tothe p ↑ p ↑ Drell-Yan process. The p ↑ ¯ p ↑ Drell-Yan processis possible to be performed in the planned high energystorage ring (HESR) [40] at GSI. We study the trans-verse DSA at GSI from the Sivers functions, based onEq. (13). For this end we need to know the input forthe Sivers functions. Several groups [16, 18, 19] haveparameterized the Sivers functions based on the data ofSIDIS at HERMES [11, 12], and partially based on COM-PASS data [13]. The kinematics in GSI can be chosen asthe c.m. energy s = 45 GeV . For the invariance masssquare of the lepton pair we choose Q = 2 . , whichis close to the scale at HERMES. Therefore these sets ofSivers functions extracted from the data of HERMES canbe applied to predict the asymmetries at GSI in the kine-matics regime we give above. We will adopt two sets ofSivers functions, which are the sets in Refs. [16] and [18],respectively. The Sivers functions in Ref. [19] can not beapplied here since in that paper f ⊥ (1 / T ( x ) is given whilewe use f ⊥ (1)1 T ( x ) in our calculation.To use these Sivers functions one should notice that T -odd distribution functions in the DIS and in the Drell-Yan process have a minus sign difference [7]. However inthe p ↑ ¯ p ↑ Drell-Yan process two Sivers functions appear inthe product, therefore the sign difference doesn’t matterhere and the functions can be used directly.In Ref. [16] the Sivers functions are parameterized as − k T M f ⊥ ,q T ( x, k T ) = N q ( x ) f q ( x ) g ( k T ) h ( k T ) , (15) A gTT x FIG. 2: The DSA in the proton antiproton Drell-Yan processat GSI coming from the function g T , defined in Eq. (14).The kinematics is s = 45 GeV and Q = 2 . . with N q ( x ) = N q x a q (1 − x ) b q ( a q + b q ) ( a q + b q ) a a q q b b q q , (16) g ( k T ) = e − k T / h k T i π h k T i , (17)for q = u, d . For the function h ( k T ) two options areconsidered:( a ) h ( k T ) = 2 k T M k T + M , ( b ) √ e p T M ′ e − k T /M ′ . (18)In our calculation we will adopt option (b) in Eq.(18),and the central values of their fit. This parametriza-tion has taken advantage of the more precise data [12] atHERMES.In Ref. [18] the authors give the set of Sivers functionsfor the u and d quark as, xf ⊥ (1) ,u T ( x ) = − xf ⊥ (1) ,d T = − . x . (1 − x ) , (19)extracted from the published HERMES data [11], andwhose form is based on the limit of a large number ofcolours N c .For the unpolarized distribution we use theMRST2001(LO set) parametrization [41]. In Fig. 1we present the DSA from Sivers functions at GSI, as afunction of x . A sizable asymmetry is predicted. Theasymmetry (solid line) based on the Sivers functions fromRef. [16] is much larger than the asymmetry (dashedline) based on the Sivers functions from Ref. [18]. As ex-plained in Ref. [42], taking into account the more precisedata [12] of HERMES, larger Sivers functions can be extracted compared to the parametrization in Eq. (19),which will lead a larger asymmetry compared to thedashed curve in Fig. 1. Thus the difference between theasymmetries from the two sets of Sivers functions may bereduced. Depending on the accuracy of the experimentalmeasurements on the transverse DSA at GSI, usefulconstraints on the Sivers functions could be obtained,but it might be hard to distinguish between differentparameterizations without high precision measurements.Finally, we will predict the DSA contributed by thefunction g T ( x, k T ) at GSI. This function, describing lon-gitudinal polarization of quarks in the transversely po-larized target, also plays role in the double polarized(longitudinal-transverse) SIDIS process [28, 43]. A treat-ment on g T ( x, k T ) is the so-called Lorentz invariancerelation that connect the first k T moment of g T ( x, k T )with the twist-three distribution function g ( x ): g q ( x ) = ddx g q (1)1 T ( x ) . (20)Using the Wandzura and Wilczek approximation for g q g q ( x ) ≈ − g a ( x ) + Z x dy g q ( x ) y , (21)the following relation was derived in Ref. [28] g (1) q T ( x ) ≈ x Z x dy g q ( x ) y . (22)For the polarized parton distribution we apply theGRSV2001 (standard scenario) parametrization [44], andfor the unpolarized distribution we use GRV98 LOparametrization [45], following the choice in Ref. [43].In Fig. 2 we show the DSA contributed by g T ( x, k T ) inthe p ↑ ¯ p ↑ Drell-Yan process at GSI with s = 45 GeV and Q = 2 . . An asymmetry of 1% is predicted.We end this section with some comment. In our cal-culation, especially in the case of Siver DSA, we choose Q = 2 . . This value is consistent with the aver-aged scale h Q i in the HERMES experiment, from whichthe Sivers functions were extracted. Therefore, the pa-rameterizations for Sivers functions in Refs. [16, 18] canbe applied here without further assumptions. Experi-mental measurements at GSI can also cover the contin-uous Drell-Yan masses 2 − Q in the range 4 −
25 GeV . To estimate the asym-metries in this region one should use the fitted functionsevolved to the relevant scale, which is not trivial for the k T -dependent distributions [46]. Therefore we assumethat the ratios in Eqs.(13) and (14) scale with Q . Inthis region, The result is similar to the one which can beobtained at the fixed value of Q = 2 . . Also thereis the subtlety that the next to leading order correction ofthe hard process could lead the substantial K -factor onthe transversely polarized cross-section. Since we calcu-late an asymmetry, which is essentially a ratio where the Q dependences in the numerator and denominator tendto cancel each other, the effect of both the Q dependenceand K-factors do not introduce a strong influence on theresulting prediction coming from Eqs. (13) and (14). IV. SUMMARY
We have performed an analysis of the transverse DSAin the Drell-Yan process contributed by the Sivers func-tions through the term f ⊥ T × f ⊥ T . The asymmetry canbe isolated through the appropriate weighting function Q T M (cos( φ − φ S ) cos( φ − φ S )+3 sin( φ − φ S ) sin( φ − φ S )),without mixing with the contribution from other distri-bution functions. Using two sets of Sivers functions pa-rameterizing the SSA data in the SIDIS process, we cal-culate the double spin asymmetry in the p ↑ ¯ p ↑ Drell-Yanprocess from the Sivers functions at GSI. An asymmetryaround to 1 % is predicted. The asymmetries estimatedfrom these two sets of Sivers functions are quantitativelydifferent. Therefore measurements of the DSA in Drell-Yan process can provide new information on the Siversfunctions, especially their sizes. The transverse DSA con-tributed by g T ( x, k T ) through the product g T × g T inthe Drell-Yan process can also be picked out by a weight-ing function. We estimate this asymmetry at GSI byadopting g T from the combination of the Lorentz invari-ance relation and the Wandura-Wilzeck approximation.The investigation on the double transversely polarizedDrell-Yan process thus can shed light on the knowledge of k T -dependent distribution functions, including the Siversfunctions. Acknowledgments
This work is partially supported by National NaturalScience Foundation of China (Nos. 10421503, 10575003,10505011, 10528510), by the Key Grant Project of Chi-nese Ministry of Education (No. 305001), by the Research Fund for the Doctoral Program of Higher Education(China), by Fondecyt (Chile) under Project No. 3050047.
APPENDIX: MOMENTS
To derive (8) and (11) we have used the following trans-verse momentum integrations: Z d k T d p T δ ( q T − k T − p T ) Q T M ( k T · p T ) × f ( x , k T ) f ( x , p T )= 1 M Z d k T d p T ( k T + p T ) k T · p T × f ( x , k T ) f ( x , p T )= 2 M Z d k T d p T ( k T · p T ) f ( x , k T ) f ( x , p T )= 2 M Z d k T d p T (cid:16) k T p T + k T p T (cid:17) × f ( x , k T ) f ( x , p T )= 4 M f (1) ( x ) f (1) ( x ) . (A.1) Z d k T d p T δ ( q T − k T − p T ) Q T M ˆ h · k T ˆ h · p T × f ( x , k T ) f ( x , p T )= 1 M Z d k T d p T ( k T + p T ) · k T ( k T + p T ) · p T × f ( x , k T ) f ( x , p T )= 1 M Z d k T d p T ( p T k T + ( k T · p T ) ) × f ( x , k T ) f ( x , p T )= 6 M f (1) ( x ) f (1) ( x ) (A.2)In the above integrals, the terms containing odd numbersof k iT or p iT vanish after being integrated over k T or p T . [1] D. Sivers, Phys. Rev. D , 83 (1990); , 261 (1991).[2] D.L.Adams et.al , Phys. Lett. B , 201 (1991); ,462 (1991); Z. Phys. C , 181 (1992).[3] M. Anselmino, M. Boglione, and F. Murgia, Phys. Lett.B , 164 (1995).[4] D. Boer and P.J. Mulders, Phys. Rev. D , 5780 (1998).[5] J.C. Collins, Nucl. Phys. B396 , 161 (1993).[6] S.J. Brodsky, D.S. Hwang, and I. Schmidt, Phys. Lett. B , 99 (2002); Nucl. Phys.
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