Timelike Compton Scattering off the Proton: beam and/or target spin asymmetries
aa r X i v : . [ h e p - ph ] D ec Timelike Compton Scattering off the Proton:beam and/or target spin asymmetries.
Marie Bo¨er , Michel Guidal Institut de Physique Nucl´eaire, CNRS-IN2P3, Universit´e Paris-Sud F-91406 Orsay, France
DOI: h ttp://dx.doi.org/10.3204/DESY-PROC-2014-04/256 We present a sample of results of our work to be published soon on Timelike Comptonscattering off the proton, in the framework of the Generalized Parton Distributions for-malism.
N' (p')N (p) g* (q) g (q') e (k)e (k') hard soft x+ x x- x t - + Figure 1: Leading twist TCS diagram.More than 40 years after the discovery of point-like components within the proton, its quarks andgluons structure is still not well understood and isstill intensively studied. Hard exclusive processeson the proton provide access to the GeneralizedParton Distributions (GPDs) [1, 2, 3, 4] whichcontain informations about the longitudinal mo-mentum and the spatial transverse distributionsof partons inside the proton (in a frame where thenucleon has an “infinite” momentum along its lon-gitudinal direction). Such a hard exclusive processis the Deeply Virtual Compton scattering processwhich corresponds to the reaction γ ( ∗ ) P → γ ( ∗ ) P and to the scattering of a high-energy virtual photon off a quark inside the proton. There aretwo particular cases of deep Compton processes. “Spacelike” Deeply Virtual Compton Scatter-ing (DVCS) corresponds to the case where the incoming photon is emitted by a lepton beamand has a high spacelike virtuality and and where the final photon is real. The DVCS processhas been studied for the past ∼
15 years and is still intensively studied both theoretically andexperimentally. The second particular case of deep Compton scattering is the Timelike Comp-ton Scattering (TCS) process. It corresponds to the case where the incoming photon is realand the final photon has a high timelike virtuality and decays into a lepton pair (see Fig. 1).Contrary to DVCS, there is no published experimental data yet for TCS. Both DVCS and TCSgive access to the same proton GPDs in the QCD leading twist formalism. The study of TCSin parallel to DVCS is a very powerful way to check the universality of GPDs and/or to studyhigher twist effects. The reaction γP → e + e − P also involves the Bethe-Heitler process, wherethe incoming real photon creates a lepton pair, which then interacts with the proton. It is notsensitive to the GPDs but to the form factors. It can be calculated with a few percent accuracy. PANIC14 Amplitudes and observables
The four vectors involved are indicated in Fig. 1. According to QCD factorization theorems, atsufficiently large Q ′ = ( k + k ′ ) (photon’s virtuality), we can decompose the TCS amplitudeinto a soft part, parameterized by the GPDs, and a hard part, exactly calculable by Feynmandiagrams techniques. We work in a frame where the average protons and the average photonsmomenta, respectively P and ¯ q , are collinear along the z -axis and in opposite directions. Wedefine the lightlike vectors along the positive and negative z directions as ˜ p µ = P + / √ , , , n µ = 1 /P + · / √ , , , − P + ≡ ( P + P ) / √
2. We have the properties ˜ p = n = 0 and ˜ p · n = 1. In this frame, the TCS amplitude can be written in the asymptotic limit(mass terms are neglected with respect to Q ′ ) with the Ji convention for GPDs [5]: T T CS = − e q ′ ¯ u ( k ) γ ν υ ( k ′ ) ǫ µ ( q )
12 ( − g µν ) ⊥ Z − dx (cid:18) x − ξ − iǫ + 1 x + ξ + iǫ (cid:19) . (cid:18) H ( x, ξ, t )¯ u ( p ′ ) nu ( p ) + E ( x, ξ, t )¯ u ( p ′ ) iσ αβ n α ∆ β M u ( p ) (cid:19) − i ǫ νµ ) ⊥ Z − dx (cid:18) x − ξ − iǫ − x + ξ + iǫ (cid:19) . (cid:18) ˜ H ( x, ξ, t )¯ u ( p ′ ) nγ u ( p ) + ˜ E ( x, ξ, t )¯ u ( p ′ ) γ ∆ .n M u ( p ) (cid:19) , (1)where x is the quark longitudinal momentum fraction, ∆ = ( p ′ − p ) is the momentum transfer, t = ∆ and ξ is defined as ξ = − ( p − p ′ ) . ( q ′ + q )( p + p ′ ) . ( q ′ + q ) . (2)In Eq. 1, we used the metric( − g µν ) ⊥ = − g µν + ˜ p µ n ν + ˜ p ν n µ , ( ǫ νµ ) ⊥ = ǫ νµαβ n α ˜ p β . (3)The Bethe-Heitler amplitude reads: T BH = − e ∆ ¯ u ( p ′ ) (cid:18) γ ν F ( t ) + iσ νρ ∆ ρ M F ( t ) (cid:19) u ( p ) ǫ µ ( q )¯ u ( k ) (cid:18) γ µ k − 6 q ( k − q ) γ ν + γ ν q − 6 k ′ ( q − k ′ ) γ µ (cid:19) υ ( k ′ ) , (4)where F ( t ) and F ( t ) are the proton Dirac and Pauli form factors. At fixed beam energy, thecross section of the photoproduction process depends on four independant kinematic variables,which we choose as: Q ′ , t and the two angles θ and φ of the decay electron in the γ ∗ center ofmass. The 4-differential unpolarized cross section reads: d σdQ ′ dtd Ω ( γp → p ′ e + e − ) = 1(2 π )
164 1(2
M E γ ) | T BH + T T CS | , (5)where | T BH + T T CS | is averaged over the target proton and beam polarizations and summedover the final proton spins.We define the single and double spin asymmetries as: A ⊙ U ( A Ui ) = σ + − σ − σ + + σ − , A ⊙ i = ( σ ++ + σ −− ) − ( σ + − + σ − + ) σ ++ + σ −− + σ + − + σ − + , (6)2 PANIC14 here the first index of A corresponds to the polarization state of the beam and the second onecorresponds to the polarization state of the target. A ⊙ U is the circularly polarized beam spinasymmetry. The + and − superscripts in σ correspond to the two photon spin states, right andleft polarized. A Ui are the single target spin asymmetries where the + and − superscripts referto the target spin orientations along the axis i = x, y, z . The axis x and y are perpendicularto the incoming proton direction (along the z -axis) in the γP center of mass frame and arerespectivelly in the scattering plane and perpendicular to this plane. A ⊙ i are the double spinasymmetries with a circularly polarized beam and with a polarized target. We finally definethe single linearly polarized beam spin asymmetry as A ℓU (Ψ) = σ x (Ψ) − σ y (Ψ) σ x (Ψ) + σ y (Ψ) , (7)where Ψ is the angle between the photon polarization vector and the γP → γ ∗ P ′ plane andwhere σ x ( σ y ) indicate a photon polarized in the x -( y -)direction. Figure 2: Spin asymmetries as a function of − t . Top left: A ⊙ U for BH+TCS. Top right: A ℓU forBH and BH+TCS. Bottom left: A Uz for BH+TCS. Bottom right: A ⊙ x for BH and BH+TCS.All calculations are done at ξ = 0 . Q ′ = 7 GeV , φ = 90 ◦ and θ integrated over [45 ◦ , ◦ ]. A ℓU is also shown for θ = 45 ◦ and θ = 90 ◦ .We performed our calculations using the GPD parameterization of the VGG model [6, 7, 8].We focus here on the spin asymmetries. Figure 2 shows the circularly (top row left) and linearly PANIC14 t . One should note that A ⊙ U is particularly sensitive to the GPDs as it is exactly 0 for BH alone. It comes from the fact thatthis asymmetry is sensitive to the imaginary part of the amplitudes and the BH amplitude ispurely real. We also show A ⊙ U with a factorized- t ansatz instead of a Reggeized- t ansatz for the H GPD which illustrates the sensitivity to the GPD modeling. In contrast, the A ℓU asymmetry,which is strong, is dominated by the BH and the TCS makes up only small deviations. Indeed,this asymmetry is sensitive to the real part of the amplitudes.We display in Fig. 2 (bottom row) two examples of asymmetries with a polarized target: A Uz and A ⊙ x (double spin asymmetry). We present the results for TCS+BH with differentGPDs contributions and parameterizations. All single target spin asymmetries are zero for theBH alone as they are proportional to the imaginary part of the amplitudes. This makes the A Ui asymmetries privileged observables to study GPDs. On the contrary, it is more difficultto access GPDs with double spin asymmetries as the BH alone produces a strong double spinasymmetry. We have presented a sample of our results to be published soon, namely the t -dependence ofsingle and double spin asymmetries for the γP → e + e − P reaction which we analyzed in theframework of the GPD formalism. We didn’t discuss this here due to lack of space but we alsocompared our unpolarized cross sections and our single beam spin asymmetries with those ofthe earlier work of Refs [9, 10] and they are in agreement at the few percent level. We haveintroduced in our work the target polarization in order to define the single and double spinasymmetries with polarized targets. We have also introduced some higher twist corrections andgauge invariance restoration terms.As the BH contribution alone doesn’t contribute to single target spin asymmetries and tocircularly polarized beam spin asymmetries, these observables are good candidates to studyGPDs. Such measurements can be envisaged at the JLab 12 GeV facility. In particular, aproposal has been accepted for the CLAS12 experiment (JLab Hall B) to measure the unpolar-ized BH+TCS cross section [11]. The work that we presented here can open the way to newcomplementary experimental programs with polarized beams and/or targets. References [1] K. Goeke, M. V. Polyakov and M. Vanderhaeghen, Prog. Part. Nucl. Phys. , 401 (2001).[2] M. Diehl, Phys. Rept. , 41 (2003).[3] A.V. Belitsky, A.V. Radyushkin, Phys. Rept. , 1 (2005).[4] M. Guidal, H. Moutarde and M. Vanderhaeghen, Rept. Prog. Phys. , 066202 (2013).[5] X. Ji, Phys.Rev.Lett. , 610 (1997); Phys.Rev.D , 7114 (1997).[6] M. Vanderhaeghen, P.A.M. Guichon and M. Guidal, Phys. Rev. Lett. , 675 (2002).[10] A. T. Goritschnig, B. Pire and J. Wagner, Phys. Rev. D (2014) 094031[11] I. Albayrak et al. and the CLAS Collaboration, JLab PAC 39 Proposal, (2012).4