Radiative corrections to semi-inclusive deep inelastic scattering induced by lepton and photon pair electroproduction

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1 arxiv: v1 [he-h] 30 Nov 2013 Radiative corrections to semi-inclusive dee inelastic scattering induced by leton and hoton air electroroduction A. Ilyichev National Scientific and Educational Center of Particle and High Energy Physics, Belarusian State University, Minsk, Belarus M. Osienko Istituto Nazionale di Fisica Nucleare, Sezione di Genova, Genoa, Italy Abstract The contribution of leton and hoton air roduction in escattering to the cross section of semi-inclusive dee inelastic scattering e e X has been calculated. The numerical results showed a large contribution of this rocesses at φ h = 180 o, in a good agreement with reliminary exerimental data. 1 Introduction Semi-inclusive dee inelastic scattering (SIDIS) allows to investigate not only arton distributions but also shed light on arton hadronization rocess. During the data analysis of this reaction it is necessary to account for the QED radiative corrections soiling exerimental observables. The lowest order radiative corrections to SIDIS were first calculated for the scattering of the olarized articles and three-fold cross section in [1]. These calculations allowed to develo the code SIRAD, which reresented a modification of DIS rogram POLRAD [2]. Then in [3] the ily@he.by 1

2 h }X Figure 1: Semi-inclusive reaction. calculation of radiative corrections was generalized for unolarized fivefold cross section and code HAPRAD was develoed. In all these calculations Bardin-Shumeiko covariant aroach [4] was used for cancellation of infrared divergences. At last in [5] the contribution of exclusive radiative tail was studied and a new version of HAPRAD (named 2.0) was written 1. In the resent work we consider another background rocess for a secific semi-inclusive reaction, in which the detected hadron is identical to the initial one. Preliminary exerimental data from JLab [6] showed that in the kinematic region near φ h = 180 o the measured cross section exceeded the calculated one-hoton exchange (Born) contribution and the excess could not be described by the radiative corrections calculated before. It was noticed that the main contribution in this region came from the exclusive leton air electroroduction. In site of its higher QED order (suressed by α 2 ) this rocess is enhanced by the square of roton electromagnetic form-factors at low Q 2. Therefore, in some secific kinematics the contribution of this rocess comete with SIDIS reaction cross section. The matrix elements of the leton air electroroduction and the hase sace of this rocess are considered in the next section. The contribution from two hoton emission are also considered there as the rocess of the same order. Numerical result and conclusion resented in the next sections. 2 Method of calculation Semi-inclusive reaction (see Fig. 1) e ( )+() e ( )+( h )+X (1) 1 All of these code can be found at htt:// 2

3 h h h h a) b) h h h h c) d) Figure 2: The four airs of gauge invariant grahs for the lowest order of the leton air electroroduction in e-scattering. can be describe by five variables: Q 2 = q 2 = ( ) 2, x = Q2 2q, x = 1 hq q, t = ( h) 2, φ h, (2) where φ h is the angle between (, ) and (, h ) lanes. The cross-section of electroroduction of the leton air in the escattering e ( )+() e ( )+( h )+e (l )+e (l + ) (3) at the lowest order reads: dσ = 1 2S M MdΓ, (4) with S = 2. We consider now the semi-inclusive rocess in which detected hadron is identical to the initial one. In order to use SIDIS variables the hase sace of this rocess can be transformed in the following way: dγ = 1 (2π) 8 d 3 20 d 3 h 2 h0 dγ 2l = S3 x dq2 dxdx dtdφ h 2 9 π 7 Q 2 S S 2 x +4M 2 Q 2dΓ 2l, (5) 3

4 h h h Figure 3: The grahs for the two hoton electroroduction in e-scattering. where S x = 2q and the hase sace of the leton air has a form: dγ 2l = d3 l + 2l +0 d 3 l 2l 0 δ 4 ( + h l l + ) = 1 8 dω R, (6) with dω R being the solid angle of leton air defined in its center mass system (l + +l = 0). The matrix element can be written as a sum of the four airs of gauge invariant diagrams M = M a +M b +M c +M d resented in Fig. 2. The exlicit exressions for each term read: M a = 4πα [ 2α ˆkγ α ū(k qh 2 2 ) γ µ 2 k + 2k ] 2α +γ αˆk +2 k γ µ u( ) ū(l )γ α u( l + )Jµ, h M c = 4πα ū(l q 2 qh 2 ) [ ū( )γ α u( )J h µ, γ µˆqγ α 2l +α q 2 2l + q + 2l α +γ αˆq q 2 2l q γ µ ] u( l + ) M b,d = M a,c ( l ), (7) where k = l + +l, q h = q k, Jµ h = ū( h ) [γ ν ] h µ F 1 +iσ ν µν 2M F 2 u() (8) and F 1,2 are the roton form factors. Additional contribution of the same order comes from two hoton emission which is shown in Fig. 3: e ( )+() e ( )+( h )+γ(κ 1,ε 1 )+γ(κ 2,ε 2 ), (9) where ε 1,2 are hoton olarization vectors. 4

5 d 5 σ/dxdq 2 dx P dt P dφ µb/gev 4 /rad x Q 2 =2 GeV 2 x=0.24 x =0.59 -t =1.95 GeV 2 Born rad. ex.tail e + e φ degrees d 5 σ/dxdq 2 dx P dt P dφ µb/gev 4 /rad x Q 2 =2 GeV 2 x=0.27 x =0.47 -t =1.19 GeV 2 Born rad. ex.tail e + e φ degrees Figure4: φ h -deendence ofthedifferent contributionstothesemi-inclusive rocess e e X. The matrix element of this rocess reads: M 2γ = 4πα [ ū(k qh 2 2 ) (2µ +γ µˆq h )ˆε 2 (2 ε 1 ˆκ 1ˆε 1 ) 2 κ 1 ( 2kκ 1 ) (2ε 2 + ˆε 2ˆκ 2 )γ µ (2 ε 1 ˆκ 1ˆε 1 ) 4 κ 1 κ 2 + (2ε 2 + ˆε 2ˆκ 2 )ˆε 1 γ µ (2µ ˆq h γ µ ) 2( +2k ) κ 2 ] +(κ 1,ε 1 κ 1,ε 1 ) u( ), (10) with k = κ 1 + κ 2, while the cross section and hase sace looks like (5) and (4), resectively. 3 Numerical results The φ h -deendence of the different contributions to the semi-inclusive reaction e+ e + +X is shown in Fig. 4. In the region near φ h = 180 o a narrow eak is observed. Such behavior cannot be described by the Born contribution far as it has φ h -deendence inconsistent with the usual unolarized SIDIS distribution A+Bcosφ h +Ccos2φ h. Radiative corrections calculatedin[3]aswell asexclusive radiativetail[5](radandex. taillines 5

6 resectively) cannot imrove this situation. The dominant contribution in this region comes from the exclusive electroroduction of leton airs in e-scattering discussed above. Instead the two hoton emission, shown in Fig. 3, is almost negligible desite being of the same α-order. Moreover, detailed numerical estimates showed that most of correction strength came from the square of matrix element M b (see Fig. 2 (b) and equation (7)). These calculations are in good agreement with reliminary exerimental data from [6]. 4 Conclusion Radiative corrections to semi-inclusive electroroduction of roton induced by the exclusive roduction of leton and hoton airs have been calculated for the first time. Numerical results showed imortant contribution (mainly from diagrams Fig. 2 (b)) in the region of φ h = 180 o, in good agreement with reliminary exerimental data. The resented aroach is rather general and can be extended on analysis of A -roduction [7]. Acknowledgments The one of authors (A.I.) would like to thanks INFN staff for generous hositality during his visits. References [1] A. V. Soroko and N. M. Shumeiko, Sov. J. Nucl. Phys. 53, 628 (1991) [Yad. Fiz. 53, 1015 (1991)]. [2] I. Akushevich, A. Ilyichev, N. Shumeiko, A. Soroko and A. Tolkachev, Comut. Phys. Commun. 104, 201 (1997) [3] I. Akushevich, N. Shumeiko and A. Soroko, Eur. Phys. J. C 10, 681 (1999) [4] D. Y. Bardin and N. M. Shumeiko, Nucl. Phys. B 127, 242 (1977). [5] I. Akushevich, A. Ilyichev and M. Osienko, Phys. Lett. B 672, 35 (2009) 6

7 [6] Study of Fracture Functions in Polarized and Unolarized Semiinclusive Electron Scattering off the Proton, M. Osienko et al., roosal for CLAS aroved analysis (2005). [7] J.D. Bjorken, R. Essig, P. Schuster, N. Toro, Phys. Rev. D80, (2009). 7