Timelike deeply virtual Compton scattering with a linearly polarized real (or quasi-real) photon beam
TTimelike deeply virtual Compton scattering with alinearly polarized real (or quasi-real) photon beam
A.T. Goritschnig
Institute of Physics - Theory Division, University of Graz, Graz, Austria
B. Pire
CPHT, École Polytechnique, CNRS, 91128 Palaiseau, France
Jakub Wagner ∗ National Center for Nuclear Research (NCBJ), Warsaw, PolandE-mail: [email protected]
We calculate timelike virtual Compton scattering amplitudes in the generalized Bjorken scal-ing regime and focus on a new polarization asymmetry in the scattering process with a linearlypolarized photon beam in the medium energy range, which will be studied intensely at JLab12experiments. We demonstrate that new observables help us to access the polarized quark andgluon generalized parton distributions ˜ H ( x , ξ , t ) and ˜ E ( x , ξ , t ) . XXII. International Workshop on Deep-Inelastic Scattering and Related Subjects,28 April - 2 May 2014Warsaw, Poland ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] J u l inearly polarized timelike DVCS Jakub Wagner
1. Introduction
The spin structure of the nucleon remains after decades of experimental efforts an open ques-tion of hadronic physics; in particular the contribution of gluons to the nucleon spin stays at a ratherunsatisfactory qualitative level, even for integrated parton distributions. A more refined version ofthis puzzle may be addressed through the polarized generalized parton distributions (GPDs) bothfor the parton helicity conserving ones ˜ H ( x , ξ , t ) and ˜ E ( x , ξ , t ) , and the parton helicity flip ones H T ( x , ξ , t ) , E T ( x , ξ , t ) , ˜ H T ( x , ξ , t ) and ˜ E T ( x , ξ , t ) . Accessing them through various deep exclusiveprocesses turns out to be more difficult than anticipated. Experimental techniques have recentlybeen developed to yield a linearly polarized, intense photon beam at JLab, using a forward taggerto detect electrons scattered at very small angles at CLAS12 [1] or using coherent bremsstrahlungtechnique at the GlueX experiment [2].We recently demonstrated [3] that such a set up will allow to perform new tests of the polarizedquark and gluon GPDs ˜ H ( x , ξ , t ) and ˜ E ( x , ξ , t ) , through the study of timelike Compton scattering(TCS) if proper observables are measured. We investigated, at leading twist and next to leading α S order, the photoproduction of lepton pairs off a proton where both a timelike deeply virtualCompton scattering and a Bethe-Heitler (BH) process contribute. When studying their interferenceone gets direct access to the amplitude of the Compton process. Through it one gets informationon the nucleon structure in terms of GPDs. We introduce new observables which are sensitive todifferent angular dependencies of the outgoing leptons to project out cross-section contributionsassociated with particular GPDs. We focus on the medium energy range which will be studiedintensely at JLab12 experiments [4] .The hard exclusive photoproduction of a lepton pair through TCS[5, 6] γ ( q in ) N ( p ) → γ ∗ ( q out ) N (cid:48) ( p (cid:48) ) → l − ( k ) l + ( k (cid:48) ) N (cid:48) ( p (cid:48) ) , (1.1)shares many features with deeply virtual Compton scattering (DVCS)[7], in the sense that it obeysa factorization property in the generalized Bjorken limit of large Q and fixed ratio Q / s with q in = q out = Q > s = ( p + q in ) . (1.2)These two reactions allow one to access the same knowledge of the nucleon structure encoded inuniversal GPDs[8]. Their amplitudes are related at Born order by a simple complex conjugation,but they significantly differ [9] at next to leading order (NLO) in the strong coupling constant α s .Moreover, the discovery [10] of the importance of NLO corrections driven by gluonic GPDs allowsto get an easy access to the gluonic content of nucleons in spacelike and timelike DVCS reactions,although they do not contribute at the Born order. These two processes are the most promising onesto access the three-dimensional partonic picture of the nucleon [11] with the help of the extractionof quark and gluon GPDs.We consider the process in which a linearly polarized real photon interacts with an unpolarizedproton and produces a lepton pair. To specify the variables we use in the description of this processwe introduce two coordinate systems. The first one, similar to the one introduced in [5], is the γ p center-of-momentum frame. In the case of a linearly polarized photon we now have a distinguishedtransverse direction given by the polarization vector, which we choose to point in the x -direction.2 inearly polarized timelike DVCS Jakub Wagner
We define the scattering angle θ c . m . through:sin θ c . m . = | (cid:126) p × (cid:126) p (cid:48) || (cid:126) p || (cid:126) p (cid:48) | = ∆ T | (cid:126) p (cid:48) | ∼ O (cid:32)(cid:114) − ts (cid:33) , (1.3)and the angle Φ h between the polarization vector and the hadronic plane as:sin Φ h = (cid:126) ε ( q in ) · (cid:126) n = (cid:126) ε ( q in ) · (cid:126) p (cid:48) × (cid:126) p | (cid:126) p (cid:48) × (cid:126) p | , (1.4)where (cid:126) n is the vector normal to the hadronic plane. Two additional angles θ and φ are defined, asin [5], as the polar and azimuthal angles of (cid:126) k in the l + l − center-of-momentum frame with the z -axisalong − (cid:126) p (cid:48) , and the other axes such that (cid:126) p lies in the x - z plane and has positive x -component.The full Compton scattering amplitude in its factorized form is conveniently expressed throughCompton Form Factors H , E , ˜ H , ˜ E : A µν ( η , t ) = − e ( p + p (cid:48) ) · n − ¯ u ( p (cid:48) ) (cid:20) g µν T (cid:16) H ( η , t ) / n − + E ( η , t ) i σ αρ n − α ∆ ρ M (cid:17) + i ε µν T (cid:16) ˜ H ( η , t ) / n − γ + ˜ E ( η , t ) ∆ + α n − α γ M (cid:17) (cid:21) u ( p ) , (1.5)where : { H , E } ( η , t ) = + (cid:90) − dx (cid:32) ∑ q T q ( x , η ) { H q , E q } ( x , η , t ) + T g ( x , η ) { H g , E g } ( x , η , t ) (cid:33)(cid:8) ˜ H , ˜ E (cid:9) ( η , t ) = − (cid:90) − dx (cid:32) ∑ q ˜ T q ( x , η ) (cid:8) ˜ H q , ˜ E q (cid:9) ( x , η , t ) + ˜ T g ( x , η ) (cid:8) ˜ H g , ˜ E g (cid:9) ( x , η , t ) (cid:33) . (1.6)The renormalized coefficient functions T q , g , ˜ T q , g calculated at LO and NLO in α S can be found in[10]. The scaling skewness variable η is defined as: η ≡ − ( q in − q out ) · ( q in + q out )( p + p (cid:48) ) · ( q in + q out ) ≈ Q s − Q . (1.7)
2. Cross sections and observables
The physical process where we study TCS is the exclusive photoproduction of a lepton pair.But to this process contribute both TCS and a BH mechanism, where the incoming photon fluc-tuates into the lepton pair, which then goes on-shell via the exchange of a virtual photon with theproton, cf. figure 1. It turns out that the BH contribution dominates over the TCS one in the wholekinematical range under consideration but the quantum mechanical interference of these two am-plitudes allows us to extract information on the Compton process itself. The interference part canbe accessed through the study of the angular distribution of the produced lepton pair.Up to the leading order accuracy the contribution of the TCS mechanism to the differential γ p → l + l − p photoproduction cross section for fixed polarization of the incoming photon reads (we3 inearly polarized timelike DVCS Jakub Wagner p p ! q in q out l − l + p p ! q in − ∆ l − l + Figure 1:
Two mechanisms contributing to the photoproduction of lepton pairs: timelike Compton scattering(Left) and Bethe-Heitler (Right). average over the initial proton and sum over all final particle polarizations): d σ ( TCS ) dQ dtd Ω l + l − d Φ h (2.1) = α π s Q (cid:0) − sin θ cos ( Φ h + φ ) (cid:1) (cid:104) ( − η ) | H | − ( η + t M ) | E | − η Re ( H ∗ E ) (cid:105) + α π s Q (cid:0) − sin θ sin ( Φ h + φ ) (cid:1) (cid:104) ( − η ) | ˜ H | − η t M | ˜ E | − η Re ( ˜ H ∗ ˜ E ) (cid:105) , where d Ω l + l − = d ( cos θ ) d φ .The contribution of the interference between the TCS and BH mechanisms to the differentialcross section for the γ p → l + l − p process reads: d σ ( INT ) dQ dtd Ω l + l − d Φ h ≡ d σ ( INT ) unpol dQ d ... + d σ ( INT ) linpol dQ d ... , (2.2)where: d σ ( INT ) unpol dQ d ... = α π s Q (cid:18) s | ∆ ⊥ | Qt (cid:19) + cos θ sin θ cos φ Re (cid:104) H F − t M E F − η ˜ H ( F + F )) (cid:105) , d σ ( INT ) linpol dQ d ... = − α π s Q (cid:18) s | ∆ ⊥ | Qt (cid:19) sin θ cos ( Φ h + φ ) Re (cid:104) H F − t M E F + η ˜ H ( F + F )) (cid:105) . The extraction of information from the interference part through the study of the angular distribu-tion of the lepton pair relies on the following observation: if one reverses the lepton charge, whichis equivalent to an exchange k ↔ k (cid:48) , the TCS and the BH amplitude transform with opposite signs.Thus, the cross sections associated with the TCS and the BH mechanism transform even under thisexchange, but the interference cross section is odd. If one now considers observables which alsochange sign when exchanging k and k (cid:48) , they give direct access to the interference term.We define two observables sensitive to the unpolarized and linearly polarized part of interfer-ence cross section. The first one is similar to the R ratio defined in [5] in the case of an unpolarized4 inearly polarized timelike DVCS Jakub Wagner ! ! ! ! ! ! ! Η C ! ! ! ! ! t ! C Figure 2: (Left) C as a function of η calculated for Q = and t = t ( η ) . (Right) C as a functionof t for Q = and η = .
1. Different curves correspond to different polarized GPD models ˜ H g = {− , , , , } · ˜ H GKg . Calculations where performed with NLO accuracy and α S = . photon beam : ˜ R = (cid:82) π d Φ h (cid:82) π d φ cos ( φ ) (cid:82) π / π / sin θ d θ d σ dtdQ d Ω d Φ h (cid:82) π d Φ h (cid:82) π d φ (cid:82) π / π / sin θ d θ d σ dtdQ d Ω d Φ h . (2.3)The second observable projects out the d σ ( INT ) linpol part of the interference cross section:˜ R = (cid:82) π d Φ h cos ( Φ h ) (cid:82) π d φ cos ( φ ) (cid:82) π / π / sin θ d θ d σ dtdQ d Ω d Φ h (cid:82) π d Φ h (cid:82) π d φ (cid:82) π / π / sin θ d θ d σ dtdQ d Ω d Φ h . (2.4)Making use of ˜ R and ˜ R we can define the following observable which is sensitive only to theinterference term and which provides us with information on ˜ H : C = ˜ R ˜ R = − π + π Re (cid:2) H F − t M E F − η ˜ H ( F + F ) (cid:3) Re (cid:2) H F − t M E F + η ˜ H ( F + F ) (cid:3) . (2.5)In order to illustrate the dependence of the proposed observable on ˜ H we make use of the GKGPDs model based on fits to Deeply Virtual Meson Production [12]. We present our numericalestimates with the GPD E set to zero, as it is largely unknown and also suppressed by the smallkinematic factor t M . On the left-hand side of figure 2 we present the ratio C as a function of η , calculated for Q = and a minimal value of t = t ( η ) = − η M / ( − η ) . To studythe sensitivity of C on the GPD ˜ H we have varied the gluonic contribution, taking it as ˜ H g = {− , , , , } · ˜ H gGK , where ˜ H gGK is given by the GK model. On the right-hand side of figure 2we show C as a function of t for Q = and η = .
1. In both cases significant effects of theorder of 30% are visible proving that C is a good observable sensitive to ˜ H .
3. Electroproduction with a small electron scattering angle.
In the unpolarized electron scattering process, the virtual photon polarization is given by [13]: ε = (cid:34) + Q γ + ν Q γ tan ( θ e (cid:48) / ) (cid:35) − (3.1)5 inearly polarized timelike DVCS Jakub Wagner where ν is the photon energy and θ e (cid:48) the electron scattering angle. The longitudinal polarization isgiven by ε L = Q γ ν ε and in the limit Q γ << ν the polarization density matrix ρ = ( + ε ) ∼ ε / L ( − ε ) ∼ ε / L ε L , (3.2)describes real transverse photons. The cross section d σ ep dQ γ d ν ∼ F ( Q γ , ν ) M ∗ λ ρ λλ (cid:48) M λ (cid:48) (3.3)is given by the incoherent sum of two polarizations: d σ ep dQ γ d ν = F ( Q γ , ν ) (cid:20) ( + ε ) σ xx γ p + ( − ε ) σ yy γ p (cid:21) = F ( Q γ , ν ) (cid:104) ( − ε ) σ unp γ p + ε σ linpol γ p (cid:105) where F ( Q γ , ν ) is the photon flux. In that case we can define the observable, analogous to the onedefined by the Eq.(2.5), in the following way: C ep = ˜ R ep ˜ R ep = − εε C . (3.4) Acknowledgements
This work is partly supported by the Polish NCN Grant No. DEC-2011/01/D/ST2/02069 andthe Austrian FWF Grant No. J 3163-N16.
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