Impact of the triple-gluon correlation functions on the single spin asymmetries in pp collisions
aa r X i v : . [ h e p - ph ] J u l Impact of the triple-gluon correlation functionson the single spin asymmetries in pp collisions
Yuji Koike ∗ and Shinsuke Yoshida † ∗ Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan † Graduate School of Science and Technology, Niigata University, Ikarashi, Niigata 950-2181,Japan
Abstract.
We calculate the single-spin-dependent cross section formula for the D -meson produc-tion and the direct-photon production in the pp collision induced by the twist-3 triple-gluon corre-lation functions in the transversely polarized nucleon. We also present a model calculation for theasymmetries in comparison with the preliminary data given by RHIC, showing the impact of thecorrelation functions on the asymmetries. Keywords: single spin asymmetry, twist-3, triple-gluon correlation function
PACS:
1. INTRODUCTION
Understanding the origin of the large single spin asymmetries (SSAs) observed in var-ious high-energy semi-inclusive processes have been a big challenge during the pastdecades. The SSA can be generated as a consequence of the multiparton correlationsinside the hadrons in the collinear factorization approach which is valid when the trans-verse momentum of final state hadron can be regarded as hard. Recently the mesure-ment of SSA for heavy meson production by the PHENIX collaboration [1] have moti-vated theoretical works for multigluon correlation inside the transversly polarized pro-ton which is represented by the triple-gluon correlation functions [2, 3] because heavyquarks fragmenting into final state meson are mainly produced by the gluon fusionmechanism.In this work, we study the contribution of the triple-gluon correlation functions toSSA for the D -meson and the direct photon productions in the pp collision [4, 5].We will derive the corresponding single-spin dependent cross sections by applyingthe formalism developed for the semi-inclusive deep inelastic scattering [3]. We willalso present a model estimate for the triple-gluon correlation functions by comparingour result with the RHIC preliminary data for the D -meson production [1]. Finally weperform numerical calculation of the asymmetry for the direct photon production byusing the models obtained from p ↑ p → DX to see its impact on the SSA for this process.
2. TRIPLE-GLUON CORRELATION FUNCTIONS
Triple-gluon correlation functions for the transversely polarized nucleon are defined asthe color-singlet nucleon matrix element composed of the three gluon’s field strengthensors F ab . Corresponding to the two structure constants for the color SU(3) group, d bca and f bca , one obtains two independent triple-gluon correlation functions O ( x , x ) and N ( x , x ) as [3] O abg ( x , x ) = − g ( i ) Z d l p Z d m p e i l x e i m ( x − x ) h pS | d bca F b nb ( ) F g nc ( m n ) F a na ( l n ) | pS i = iM N h O ( x , x ) g ab e g pnS + O ( x , x − x ) g bg e a pnS + O ( x , x − x ) g ga e b pnS i , (1) N abg ( x , x ) = − g ( i ) Z d l p Z d m p e i l x e i m ( x − x ) h pS | i f bca F b nb ( ) F g nc ( m n ) F a na ( l n ) | pS i = iM N h N ( x , x ) g ab e g pnS − N ( x , x − x ) g bg e a pnS − N ( x , x − x ) g ga e b pnS i , (2) where M N is the nucleon mass, S is the transverse-spin vector for the nucleon, n is thelight-like vector satisfying p · n = F b n ≡ F br n r etc . The gauge-link operators which restore gauge invariance of the correlation functionsare suppressed in (1) and (2) for simplicity. D -MESON PRODUCTION IN p p COLLISION
Applying the formalism for the contribution of the triple-gluon correlation functionsto SSA developed in [3], the twist-3 cross section for p ↑ ( p , S ⊥ ) + p ( p ′ ) → D ( P h ) + X (center-of-mass energy √ S ) can be obtained in the following form [4]: P h d D s d P h = a s M N p S e P h pnS ⊥ (cid:229) f = c ¯ c Z dx ′ x ′ G ( x ′ ) Z dzz D f ( z ) Z dxx d ( ˜ s + ˜ t + ˜ u ) z ˜ u × (cid:20) d f (cid:26)(cid:18) ddx O ( x , x ) − O ( x , x ) x (cid:19) ˆ s O + (cid:18) ddx O ( x , ) − O ( x , ) x (cid:19) ˆ s O + O ( x , x ) x ˆ s O + O ( x , ) x ˆ s O (cid:27) + (cid:26)(cid:18) ddx N ( x , x ) − N ( x , x ) x (cid:19) ˆ s N + (cid:18) ddx N ( x , ) − N ( x , ) x (cid:19) ˆ s N + N ( x , x ) x ˆ s N + N ( x , ) x ˆ s N (cid:27)(cid:21) , (3) where d c = d ¯ c = − D f ( z ) represents the c → D or ¯ c → ¯ D fragmentationfunctions, G ( x ′ ) is the unpolarized gluon density, p c is the four-momentum of the c (or ¯ c ) quark (mass m c ) fragmenting into the final D (or ¯ D ) meson and ˜ s , ˜ t , ˜ u aredefined as ˜ s = ( xp + x ′ p ′ ) , ˜ t = ( xp − p c ) − m c , ˜ u = ( x ′ p ′ − p c ) − m c . The hard crosssections ˆ s O , O , O , O and ˆ s N , N , N , N are listed in [4]. The cross section (3) receivescontributions from O ( x , x ) , O ( x , ) , N ( x , x ) and N ( x , ) separately, which differs fromthe previous result [2].We perform numerical estimate for A N based on (3). Since | ˆ s O , O , N , N | ≪| ˆ s O , O , N , N | and ˆ s O ≃ ˆ s O ∼ ˆ s N ≃ − ˆ s N , we assume the relation for the fourfunctions as O ( x , x ) = O ( x , ) = N ( x , x ) = − N ( x , ) for simplicity. For the functionalform of each functions, we employ the following two models:Model 1 : O ( x , x ) = K G x G ( x ) , (4)Model 2 : O ( x , x ) = K ′ G √ x G ( x ) , (5)where K G and K ′ G are the constants to be determined so that the calculated asymmetry isconsistent with the RHIC data [1].or the numerical calculation, we use GJR08 [6] for G ( x ) and KKKS08 [7] for D f ( z ) .We calculate A N for the D and ¯ D mesons at the RHIC energy at √ S =
200 GeV and thetransverse momentum of the D -meson P T = m = q P T + m c with the charm quark mass m c = . -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N (a) x F -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N (b) x F -0.4-0.3-0.2-0.1 0 0.1 0.2-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N (c) x F -0.4-0.3-0.2-0.1 0 0.1 0.2-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N (d) x F FIGURE 1.
Results of A DN for D (a) and ¯ D (b) for Model 1 in (4) with K G = . A DN for D (c)and ¯ D (d) for Model 2 in (5) with K ′ G = . Fig. 1 shows the result of A N for the D and ¯ D mesons together with the preliminarydata [1] denoted by the short bars. The sign of the contribution from { O ( x , x ) , O ( x , ) } changes between D and ¯ D as shown in (3), which causes the large difference between A N for the D and ¯ D . If one reverses the relative sign between O and N ,the result for the D and ¯ D mesons will be interchanged. The values K G = .
002 and K ′ G = . A N does not overshoot the RHIC data. By comparing the resultsfor the models 1 and 2 in Fig. 1, one sees that the behavior of A N at x F < x behavior of the triple-gluon correlation functions. Therefore A N at x F < x behavior of the three-gluon correlationfunctions.
4. DIRECT PHOTON PRODUCTION IN p p
COLLISION
Applying the same formalism, the twist-3 cross section for the direct photon production, p ↑ ( p , S ⊥ ) + p ( p ′ ) → g ( q ) + X , induced by the triple-gluon correlation functions can beobtained as [5] E g d s d q = a em a s M N p S (cid:229) a Z dx ′ x ′ f a ( x ′ ) Z dxx d ( ˆ s + ˆ t + ˆ u ) e qpnS ⊥ u × (cid:20) d a (cid:16) ddx O ( x , x ) − O ( x , x ) x + ddx O ( x , ) − O ( x , ) x (cid:17) − ddx N ( x , x ) + N ( x , x ) x + ddx N ( x , ) − N ( x , ) x (cid:21) (cid:18) N ˆ s + ˆ u ˆ s ˆ u (cid:19) , (6) where f a ( x ′ ) is the twist-2 unpolarized quark density, d a = ( − ) for quark (antiquark)and ˆ s , ˆ t , ˆ u are defined as ˆ s = ( xp + x ′ p ′ ) , ˆ t = ( xp − q ) , ˆ u = ( x ′ p ′ − q ) . As shown in(6), the combinations O ( x , x ) + O ( x , ) and N ( x , x ) − N ( x , ) appear in the cross sectionaccompanying the common partonic hard cross section which is the same as the twist-2ard cross section for the qg → q g scattering. This result differs from the previous studyin [8].We performed a numerical calculation for A g N for the following two cases: Case 1; O ( x , x ) = O ( x , ) = N ( x , x ) = − N ( x , ) and Case 2; O ( x , x ) = O ( x , ) = − N ( x , x ) = N ( x , ) . We use GJR08 [6] for f q ( x ′ ) and the models (4) and (5) with K G = .
002 and K ′ G = . A DN data. We calculate A g N at the RHICenergy at √ S =
200 GeV and the transverse momentum of the photon q T = m = q T . -0.1-0.08-0.06-0.04-0.02 0-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N (a) x F -0.1-0.08-0.06-0.04-0.02 0-0.6 -0.4 -0.2 0 0.2 0.4 0.6 A N (b) x F A N (c) x F A N (d) x F FIGURE 2. (a) A N for Case 1 with Model 1. (b) A N for Case 1 with Model 2. (c) A N for Case 2 withModel 1. (d) A N for Case 2 with Model 2. Fig. 2 shows the result for A g N for each case. One can see A N at x F > A N at x F < x behavior of the triple-gluon correlation functionsas in the case of p ↑ p → DX . At negative x F , large- x ′ region of the unpolarized quarkdistributions and the small- x region of the triple-gluon distributions are relevant. Forthe above case 1, only antiquarks in the unpolarized nucleon are active and thus lead tosmall A g N as shown in Figs. 2(a) and (b). On the other hand, for the case 2, quarks inthe unpolarized nucleon are active and thus lead to large A g N as shown in Figs. 2(c) and(d). Therefore A g N at x F < O and N . ACKNOWLEDGMENTS
This work is supported by the Grand-in-Aid for Scientific Research (No. 23540292 andNo. 22.6032) from the Japan Society for the Promotion of Science.
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