ELRADGEN 2.0: Monte Carlo generator for simulation of radiative events in polarized elastic electron-proton scattering
aa r X i v : . [ h e p - ph ] D ec EPJ manuscript No. (will be inserted by the editor)
ELRADGEN 2.0: Monte Carlo generator forsimulation of radiative events in polarized elasticelectron-proton scattering
I. Akushevich , A. Ilyichev , a and N. Shumeiko Duke University, Durham, USA National Center of Particle and High Energy Physics, 220040 Minsk, Belarus
Abstract.
A new version of Monte Carlo generator ELRADGEN for simulation ofreal photon emission in elastic electron-proton scattering is presented. The exten-sions in the new version include opportunity to deal with polarized particles: lon-gitudinally polarized electron and arbitrary polarized proton. Simulation strategy,specifications of used kinematics, structure of the contributions to the observedcross section, cross-checks, and numerical results for BLAST experimental setupare presented and briefly discussed.
Exclusive real photon production in lepton-nucleon scattering plays a rather important rolein the investigation of the nucleon structure. The measurement of this process in differentkinematical regions allows researchers to obtain the information about the generalized partondistributions [1,2] and the generalized polarizabilities [3,4]. In some cases this process appearsas background to lepton-nucleon scattering in elastic [5] and inelastic [6] channels.Here we present a new version of the Monte Carlo generator ELRADGEN. The previous one[7] was developed for simulation of real hard photon emission from the lepton legs as backgroundto unpolarized elastic electron-proton scattering. In the new version we extend the generatorto deal with initial polarized particles: longitudinally polarized lepton and arbitrary polarizedproton. Both new and previous versions of this generator are based on the results of ref. [8]which were obtained using the Bardin-Shumeiko covariant approach for the extraction andcancellation of the infrared divergence [9].
For the simulation of exclusive radiative events in polarized electron-proton scattering e ( k , ξ L ) + p ( p , η ) −→ e ′ ( k ) + p ′ ( p ) + γ ( k ) (1)( k = 0, k = k = m , p = p = M ) we choose three kinematic variables: a transfermomentum squared t = − ( k − k − k ) , the inelasticity v = ( p + k ) − M , and the azimuthalangle φ k between the planes ( q , k ) and ( k , k ) depicted in Fig. 1 (a). Together with thekinematic variables characterizing Born contribution to elastic scattering Q = − q = − ( k − k ) , S = 2 k p , φ, (2) a e-mail: [email protected] I. Akushevich, A. Ilyichev, N. Shumeiko: ELRADGEN 2.0 qk k k z x y φ k π/2 − φ (a) zx y θ k η φ ηη ξθ η k φ L η t η (b) Fig. 1.
Three-vectors decomposition in Lab system: a) momenta leptons and photon; b) polarizedvector for longitudinally polarized lepton ξ L and arbitrary polarized nucleon η . they represent a full set variables for reconstruction of the four–vectors of all final particles inany frame.The four–vectors ξ L and η in (1) describe the longitudinally polarized electron and thearbitrary polarized proton, respectively. The proton polarization is characterized by two angles θ η and φ η as presented in Fig. 1 (b). The explicit expressions for polarized vectors can be foundin [6].Simulation of exclusive radiative events requires a separation of the total (or the radiativelycorrected) cross section σ obs into the contribution of real hard photon emission σ rad ( v min )and remaining part containing Born, soft photon, and additional virtual particle contributions σ BSV ( v min ). It can be performed by introducing a separation parameter, namely, the minimuminelasticity value v min that can be associated with missing mass square resolution of the detector[10] when the final proton are not detected. The sum of these two positive parts σ obs = σ rad ( v min ) + σ BSV ( v min ) (3)does not depend on v min while σ BSV ( v min ) and σ rad ( v min ) do. The numerical details of thesedependencies are presented in the next section.The explicit expressions for these two contributions are similar to those for σ rad ( v min ) and σ non − rad ( v min ) from [7] with two exceptions: i) abbreviation BV S is used instead of non − rad for the remaining part of the cross section and ii) two structure functions representing thecontributions of the polarized parts of the cross sections are additionally used.The strategy for simulation of an event can be outlined as follows. Two contributions σ rad ( v min ) and σ BSV ( v min ) to σ obs are calculated using a predetermined value of the v min .Then the channel of scattering (i.e., the process with or without real hard photon emission) issimulated according to the partial contributions of these two parts to the radiatively correctedcross section. If the channel with the real hard photon emission is chosen, three photonic vari-ables t , v and φ k are simulated according to their calculated distributions as they contributeto σ rad ( v min ) (see Fig.2 and ref. [7] for details). Finally using these variables together withBorn ones (2) which can be simulated according to the Born cross section or be externallypredetermined, the four-vectors of all final particles in any frame are reconstructed.The cross section of the process (1) when the real hard photon emitted from the lepton legsis expressed through the nucleon form factors which depend only on one integration variable,namely, on t . Therefore to have an convenient opportunity to apply this generator to different Here and later we define σ ≡ dσ/dQ dφ . Akushevich, A. Ilyichev, N. Shumeiko: ELRADGEN 2.0 3 -4 -3 -2 -1 t (GeV ) ρ (t) (GeV -2 ) -3 -2 -1 v (GeV ) ρ (v) (GeV -2 ) t=0.2 GeV t=0.35 GeV t=0.8 GeV -5 -4 -3 -2 -1
110 -150 -100 -50 0 50 100 150 φ k ( degree ) ρ ( φ k ) ( degree -1 ) t=0.2 GeV , v=0.1 GeV t=0.35 GeV , v=0.68 GeV t=0.8 GeV , v=1.2 GeV Fig. 2.
Histograms (points) and corresponding probability densities (solid lines) for variables describingthe exclusive real hard photon production in polarized electron proton scattering at BLAST kinematicconditions [11] ( E beam = 850 MeV, Q = 0 . , θ η = 48 ), with φ = φ η , P L P N = − v min = 10 − GeV . fits or models of nucleon form factors, the integration over t has to be used numerical only, whilethe analytical integration over the other photonic variables v and φ k is possible and allows tospeed up the process of event generation. The analytical integration over these two variableswas used in previous version of this generator for unpolarized scattering [7]. However when wedeal with the arbitrary polarized proton the analytical integration over v is a rather difficultdue to non-trivial dependence of transverse component of the proton polarized vector on thisvariable. Therefore, in present version of the generator the analytical integration over v is usedfor unpolarized particle scattering only. One cross-check which has to be done first is to investigate how well the simulated distributionsof photonic variables t , v and φ k reproduce those theoretically calculated. Fig. 2 provides suchan illustration for the polarized electron proton scattering at BLAST kinematic conditions [11]:the longitudinally polarized electrons with energy 850 MeV scatter off the proton that polarizedunder angle θ η = 48 to the lepton beam. Plots for the photonic variable distributions contribut-ing to σ rad ( v min ) demonstrate excellent quality of the simulation procedure. Here P L ( P N ) isthe degree of electron (proton) polarization, the solid lines corresponds to probability densitiescalculated numerically and the points show the event distributions simulated by ELRADGEN.Nucleon form factors are taken from [12]. In the left plot the theoretical t -distribution is pre-sented when the integration over two other photonic variables has been performed. The sharppeak corresponds the situation when t = Q . On the second plot one can see v -distributions atthe fixed t and integrated φ k variables. Here the peak at t = 0 . near small v correspondsthe infrared divergence (that cut off by v min ). The peaks on other two distributions correspondthe situation when the real photon emitted along the momentum of scattered lepton. The rightplot presents φ k -distribution for fully differential contribution of real hard photon emission.As one can see, the momenta of the most of emitted photons are concentrated near scatteringplane i.e. when φ k = 0.Two studies were illustrated in Tab. 1 : i) the investigation of the v min -dependence of σ rad ( v min ), σ BSV ( v min ) and their sum; and ii) the comparison of the ratio of the radiativelycorrected cross section σ obs to the Born contribution σ obtained by our generator and Fortrancode MASCARAD [8] with the same input parameters as a simplest comparison of these twocodes. Specifically, Tab. 1 demonstrates that the observable cross section does not almost changewith decreasing v min from 1 to 10 − GeV while its components σ rad ( v min ) and σ BSV ( v min )change essentially: σ rad ( v min ) increases and σ BSV ( v min ) decreases. The comparison with MAS-CARAD results in a good agreement as well. I. Akushevich, A. Ilyichev, N. Shumeiko: ELRADGEN 2.0
Table 1.
The v min -dependence of the radiative, BSV and observable contributions to electron-protonscattering with polarized target for different spin orientation in the Born units and results of comparisonwith MASCARAD [8] at BLAST kinematic conditions [11] ( E beam = 850 MeV, Q = 0 . , θ η = 48 ), with φ = φ η . v min σ rad /σ σ BSV /σ σ obs /σ GeV ELRADGEN ELRADGEN ELRADGEN MASCARAD P L P N − − − − In the present report the new version of the Monte Carlo generator ELRADGEN for simulationof real photon events within elastic electron-proton scattering generalized for longitudinallypolarized lepton and arbitrary polarized target is presented.Numerical test of new version of this code shows a good agreement with the Fortran codeMASCARAD [8] and reveals lack of dependence on minimum inelasticity value v min withaccuracy up to 1%. Besides we found that the distributions of generated radiative events arein coincidence with corresponding probability density.The present approach is rather general and can be extended in many other different waysincluding i) the development of this generator for transferred polarization from lepton beam torecoil proton [13] for measurement of electromagnetic form-factors of the proton in polarizedscattering [14,15]; ii) its further generalization for the investigation of electroweak effects suchas axial form factors of the nucleon [16] and parity violation elastic scattering [17]; and iii) itsgeneralization for practical involvement in the experiments with the measurement of generalizedparton distribution [1,2] as well as generalized polarizabilities [3,4]. We would like to acknowledge useful discussion with E. A. Kuraev. The one of us (A. I.) would liketo thank the staff of MIT Bates Center as well as O. F. Filoti for their generous hospitality during hisvisit.
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