Novel Master Formula for Twist-3 Soft-Gluon-Pole Mechanism to Single Transverse-Spin Asymmetry
aa r X i v : . [ h e p - ph ] J un Novel Master Formula for Twist-3 Soft-Gluon-PoleMechanism to Single Transverse-Spin Asymmetry
Yuji Koike and Kazuhiro Tanaka ∗
1- Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan2- Department of Physics, Juntendo University, Inba-gun, Chiba 270-1695, JapanWe prove that twist-3 soft-gluon-pole (SGP) cross section for single spin asymmetriesis determined by a certain “primordial” twist-2 cross section up to kinematic and colorfactors in the leading order perturbative QCD. This unveils universal structure behindthe SGP cross sections in a variety of hard processes, and also the special role of thescale invariance in the corresponding primordial cross section, which leads to remarkablesimplification of the SGP cross sections for the production of massless particle, such asthose for pion production p ↑ p → πX and direct-photon production p ↑ p → γX . The single transverse-spin asymmetry (SSA) is observed as “T-odd” effect proportionalto ~S ⊥ · ( ~p × ~q ) in the cross section for the scattering of transversely polarized proton withmomentum p and spin S ⊥ , off unpolarized particle with momentum p ′ , producing a particlewith momentum q which is observed in the final state. Famous examples [2] are p ↑ p → πX with the large asymmetry A N ∼ . ep ↑ → eπX , by HERMES and COMPASS Collaborations.The Drell-Yan (DY) process, p ↑ p → ℓ + ℓ − X , and the direct γ production, p ↑ p → γX , atRHIC, J-PARC, etc. are also expected to play important roles for the study of SSA.The SSA requires, (i) nonzero q ⊥ originating from transverse motion of quark or gluon;(ii) proton helicity flip; and (iii) interaction beyond Born level to produce the interferingphase for the cross section. For processes with q ⊥ ∼ Λ QCD , all (i)-(iii) may be gener-ated nonperturbatively from the T-odd, transverse-momentum-dependent parton distribu-tion/fragmentation functions [3]. For large q ⊥ ≫ Λ QCD , (i) should come from perturbativemechanism, while the nonperturbative effects can participate in the other two, (ii) and (iii),allowing us to obtain large SSA. This is realized as the twist-3 mechanism in QCD for theSSA. Recently we have thoroughly discussed the collinear-factorization property and gaugeinvariance in the twist-3 mechanism in the context of the SSA in SIDIS [4]. We have alsorevealed universal structure behind the twist-3 mechanism [5, 6], which we discuss here.Figure 1: Typical diagrams for DY processwith q ⊥ ≫ Λ QCD . The cross × denotes thepole contribution of the parton propagator.As an example, consider the DY produc-tion of the dilepton with q ⊥ ≫ Λ QCD : Thelarge q ⊥ of (i) is provided by hard interac-tion as the recoil from a hard final-state par-ton, as illustrated in Fig. 1. Proton helic-ity flip (ii) is provided by the participationof the coherent, nonperturbative gluon fromthe transversely polarized proton, the lowerblob in Fig. 1, associated with the twist-3 quark-gluon correlation functions such as G F ( x , x ) with x ( x ) the lightcone mo-mentum fraction of the quark leaving from ∗ Supported by the Grant-in-Aid for Scientific Research No. B-19340063.
DIS 2007 / ( k + iε ) = P (1 /k ) − iπδ ( k ). Depending on thevalue of the coherent-gluon’s momentum k g at the corresponding poles, these are soft-gluonpole (SGP) for k g = 0, and soft-fermion pole (SFP) and hard pole (HP) for k g = 0.Among these three-types of poles, the SGP deserves special attention; indeed, the SGP isconsidered to give dominant twist-3 mechanism in many phenomenological applications (seee.g. [4, 5]). We have derived the master formula for the SGP cross section, which embodiesthe remarkable structure that the SGP contributions from many Feynman diagrams of thetype of Fig. 1 are united into a derivative of the 2-to-2 partonic Born diagrams without thecoherent-gluon line: The SSA for the DY process can be expressed as [5] dσ DYtw-3,SGP [ dω ] = πM N C F ǫ σpnS ⊥ X j =¯ q,g B j Z dx ′ x ′ Z dxx f j ( x ′ ) ∂H jq ( x ′ , x ) ∂ ( x ′ p ′ σ ) G qF ( x, x ) , (1)where j = ¯ q and g represent the “ q ¯ q -annihilation” and “ qg -scattering” channels, respectively,corresponding to the left and right diagrams in Fig. 1. f j ( x ′ ) denotes the twist-2 partondistribution functions for the unpolarized proton, and M N is the nucleon mass representingnonperturbative scale associated with the twist-3 correlation function G qF ( x, x ) for the flavor q . The sum over quark and antiquark flavors is implicit for the index q as q = u, ¯ u, d, ¯ d, · · · .[ dω ] = dQ dyd q ⊥ denotes the relevant differential elements with Q = q and y the rapidityof the virtual photon. The color factors are introduced as B q = 1 / (2 N c ) and − / (2 N c ) forquark and antiquark flavors, respectively, B g = N c /
2, and C F = ( N c − / (2 N c ). Thederivative with respect to p ′ is taken under p ′ = 0, and H jq ( x ′ , x ) denote the partonichard-scattering functions which participate in the unpolarized twist-2 cross section for DYprocess as dσ unpol , DYtw-2 [ dω ] = X j =¯ q,g Z dx ′ x ′ Z dxx f j ( x ′ ) H jq ( x ′ , x ) f q ( x ) . (2)Namely, in order to obtain the explicit formula for the twist-3 SGP contributions to theSSA, knowledge of the twist-2 unpolarized cross section is sufficient.A proof of (1) is described in detail in [5]: Only the initial-state interaction (ISI) diagramslike Fig. 1, where the coherent gluon couples to an “external parton” associated with aninitial-state hadron, survive as the SGP contributions for DY process, while the SGPs fromthe other diagrams cancel out combined with the corresponding “mirror” diagrams [7]. Forsuch ISI diagrams, the coherent-gluon insertion and the associated SGP structure can bedisentangled from the partonic subprocess, keeping the remaining factors mostly intact. Forthe scalar-polarized coherent-gluon, this is performed using Ward identity; moreover, alsofor the transversely-polarized coherent-gluons that are relevant at the twist-3 level, this canbe performed exactly through the logic different from the scalar-polarized case [5]. Using thebackground field gauge, the three-gluon coupling relevant to the qg -scattering channel canbe disentangled similarly as the quark-gluon coupling case. After disentangling ISI with thecoherent gluons, the collinear expansion in terms of the parton transverse momenta givesthe twist-3 cross section (1) at the SGP, as the response of 2-to-2 partonic subprocess to thechange of the transverse momentum carried by the external parton, to which the coherentgluon had coupled. Note that B ¯ q and B g in (1) come from the insertion of the color matrix t a in the fundamental and adjoint representations, respectively, into the 2-to-2 subprocess,as the remnant of the coherent-gluon insertion via the quark-gluon and three-gluon vertices.2 DIS 2007 he hard-scattering functions in the twist-2 factorization formula (2) are expressed as H jq ( x ′ , x ) = ( α em α s e q / πN c sQ ) b σ jq (ˆ s, ˆ t, ˆ u ) δ (cid:0) ˆ s + ˆ t + ˆ u − Q (cid:1) , where s = ( p + p ′ ) , andexplicit form of b σ jq (ˆ s, ˆ t, ˆ u ) in terms of the partonic Mandelstam variables ˆ s, ˆ t and ˆ u can befound in Eq. (28) in [5]. The derivative in (1) can be performed through that for the ˆ u , andthis may act on either b σ jq or the delta function in H jq ( x ′ , x ), where the latter case producesthe derivative of the twist-3 correlation function, dG qF ( x, x ) /dx , as well as the non-derivativeterm ∝ G qF ( x, x ), by partial integration with respect to x . Our master formula (1) yields [5] dσ DYtw-3,SGP dQ dyd q ⊥ = α em α s e q πN c sQ πM N C F ǫ pnS ⊥ q ⊥ X j =¯ q,g B j Z dx ′ x ′ Z dxx δ (cid:0) ˆ s + ˆ t + ˆ u − Q (cid:1) f j ( x ′ ) × (cid:26) b σ jq − ˆ u x dG qF ( x, x ) dx + (cid:20) b σ jq ˆ u − ∂ b σ jq ∂ ˆ u − ˆ s ˆ u ∂ b σ jq ∂ ˆ s − ˆ t − Q ˆ u ∂ b σ jq ∂ ˆ t (cid:21) G qF ( x, x ) (cid:27) . (3)This novel expression not only reproduces the known pattern [7] that the partonic hardscattering functions associated with the derivative term are directly proportional to thoseparticipating in the twist-2 unpolarized process, b σ jq , but also reveals the structure thatwas hidden in the corresponding formula in the literature [7]: the partonic hard-scatteringfunctions associated with the non-derivative term are also completely determined by b σ jq .We obtain the SSA for the direct γ production in the real-photon limit, Q →
0; here onlythe massless particles participate in the 2-to-2 Born subprocess, so that the correspondingpartonic cross sections b σ jq are scale-invariant as (ˆ u∂/∂ ˆ u + ˆ s∂/∂ ˆ s + ˆ t∂/∂ ˆ t ) b σ jq = 0. Conse-quently, (3) reduces to the compact structure where − b σ jq / ˆ u appears both for derivative andnon-derivative terms, as the coefficient for the combination, xdG qF ( x, x ) /dx − G qF ( x, x ) [5].The DY process can be formally transformed into SIDIS, ep ↑ → eπX , crossing theinitial unpolarized proton into the final-state pion with momentum P h , and the virtualphoton into the initial-state one. The proof of (1) discussed above is unaffected by theanalytic continuation corresponding to this “crossing transformation”: p ′ → − P h , x ′ → /z , f ¯ q ( x ′ ) → D q ( z ), f g ( x ′ ) → D g ( z ), and q µ → − q µ , where D j ( z ) denote the twist-2 partonfragmentation functions for the final-state pion, and the new q µ gives Q = − q . Our masterformula (1) now gives the SGP contribution to the SSA in SIDIS, which is actually associatedwith the corresponding final-state interaction (FSI) diagrams, as ( C q ≡ B ¯ q , C g ≡ B g ) [5] dσ SIDIStw-3,SGP [ dω ] = πM N C F z f ǫ pnS ⊥ P h ⊥ ∂∂q T dσ unpol , SIDIStw-2 [ dω ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f q ( x ) → G qF ( x,x ) , D j ( z ) →C j zD j ( z ) , (4)in a frame where the 3-momenta ~q and ~p of the virtual photon and the transversely polarizednucleon are collinear along the z axis. [ dω ] = dx bj dQ dz f dq T dφ , where, as usual, x bj = Q / (2 p · q ), z f = p · P h /p · q , q T = P h ⊥ /z f , and φ is the azimuthal angle between the leptonand hadron planes. The twist-2 unpolarized cross section in the RHS of (4) is known to haveseveral terms with different φ -dependence, proportional to 1 , cos φ , and cos 2 φ , respectively(see [4]). Performing the derivative in (4) explicitly, the result obeys the similar pattern as(3) with derivative and non-derivative terms, for each azimuthal dependence, and unveils thestructure behind the complicated formula obtained by direct evaluation of the diagrams [4].Our master formula can be extended to “QCD-induced” pp collisions, like p ↑ p → πX [6].For example, the qq → qq scattering subprocess induce the twist-2 unpolarized cross section, E h d σ pp → πX tw-2 d P h = α s s Z dzz dx ′ x ′ dxx δ (ˆ s + ˆ t + ˆ u ) D q ( z ) f q ( x ′ ) f q ( x ) b σ U (ˆ s, ˆ t, ˆ u ) , (5) DIS 2007 p ( p ) + p ( p ′ ) → π ( P h ) + X , where E h ≡ P h , and b σ U (ˆ s, ˆ t, ˆ u ) = ( C F /N c )(ˆ s + ˆ u ) / ˆ t +( C F /N c )(ˆ s + ˆ t ) / ˆ u + ( − C F /N c )ˆ s / (ˆ t ˆ u ) is the qq → qq unpolarized cross section [8]. TheSGP contribution from the interference between qqg → qq and qq → qq is generated by FSIand ISI with the coherent gluon as in Fig. 2 (a) and (b), which can be treated similarly asthe SIDIS and DY cases, respectively, and yields [6] the twist-3 cross section for p ↑ p → πX : E h d σ pp → πX tw-3,SGP d P h = πM N α s s Z dzz dx ′ x ′ dxx δ (ˆ s + ˆ t + ˆ u ) D q ( z ) f q ( x ′ ) (cid:20) x dG qF ( x, x ) dx − G qF ( x, x ) (cid:21) × (cid:20) z ǫ S ⊥ P h pn + x ′ ˆ t ˆ s ǫ S ⊥ p ′ pn (cid:21) (cid:18) ˆ s b σ F (ˆ s, ˆ t, ˆ u )ˆ t ˆ u − b σ I (ˆ s, ˆ t, ˆ u )ˆ u (cid:19) , (6)Figure 2: (a) and (b) as FSI and ISI diagramsfor SGP mechanism, respectively. White cir-cles denote hard scattering between quarks.where the hard cross sections from the FSIand ISI diagrams, b σ W = A W, (ˆ s + ˆ u ) / ˆ t + A W, (ˆ s + ˆ t ) / ˆ u + A W, ˆ s / (ˆ t ˆ u ) for W = F and I , are the same as the above b σ U , ex-cept for the associated color factors A W,i that come from the color insertion of t a ,similarly as B j of (1). Note that the com-bination, xdG qF ( x, x ) /dx − G qF ( x, x ), in (6)is a consequence of the scale invariance, b σ U (ˆ s, ˆ t, ˆ u ) = b σ U ( λ ˆ s, λ ˆ t, λ ˆ u ), similarly as in p ↑ p → γX discussed below (3). These re-markable structures arise universally for allthe other relevant channels associated withthe “primordial” 2-to-2 partonic subprocesses, q ¯ q → q ¯ q , q ¯ q → gg , qg → qg , etc. (see also [8]).We have derived the novel master formula which gives the twist-3 SGP contributionsto the SSA entirely in terms of the knowledge of the “primordial” twist-2 cross section.This is based on the new approach, which allows us to disentangle ISI as well as FSI withthe soft coherent-gluon, irrespective of the details of the corresponding partonic subprocess.Thus our single master formula is applicable universally to all processes relevant to SSA,including QED-induced processes like DY process, direct γ production, SIDIS, etc., and alsoQCD-induced processes like p ↑ p → πX , p ↑ p → X , pp → Λ ↑ X , etc. For SSA associatedwith the chiral-even twist-3 function G qF ( x, x ), the primordial twist-2 process is unpolarizedas discussed above, while for SSA associated with the chiral-odd functions, the primordialprocess is the polarized one [6]. The primordial twist-2 structure behind the SGP mechanismmanifests gauge invariance of the results, and unveil the remarkable role of scale invariance. References [1] Slides: http://indico.cern.ch/contributionDisplay.py?contribId=160&sessionId=4&confId=9499 [2] See the articles on the data at RHIC, and those from HERMES, COMPASS, etc. in these proceedings.[3] D. Boer, Phys. Rev.
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