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3 MONTE CARLO SIMULATION OF VIRTUAL COMPTON SCATTERING AT MAMI L. Van Hoorebeke c, J. Berthot b,p.y. Bertin b, V. Breton b, W.U. Boeglin g,r. Bohm, N. Degrane c, N. D'Hose a, M. Distler, J.E. Ducret a, R. Eelho,H. Fonvieille b,j.frierich, J.M. Frierich, R. Geiges, Th. Gousset a,p.a.m. Guichon a, H.Holvoet c, Ch. Hye-Wright e,p. Jennewein, M. Kahrau,S. Kerhoas a, M. Korn, H. Kramer, K.W. Krygier, V. Kune, B.Lannoy c,d. Lhuillier a, A. Liesenfel, C. Marchan a, D. Marchan a, J. Martino a, H. Merkel, K. Merle,G.DeMeyer c, J. Mougey a, R. Neuhausen, E. Oermann f, Th. Pospischil, G. Quemener b,o.ravel b, Y. Roblin b,j.roche a, G. Rosner,D. Ryckbosch c,p. Sauer,H.Schmieen,S.Schart,G.Tamas,M. Vanerhaeghen a,r.van e Vyver c,j.van e Wiele h,p.vernin a,a.wagner, Th. Walcher,S.Wolf a CEA DAPNIA-SPhN, C.E. Saclay, France b LPC, Univ. Blaise Pascal, IN2P3 Aubiere, France IIKW, NFWO an Department of Subatomic an Raiation Physics, University of Gent, Belgium Institut fur Kernphysik, Universitat Mainz, Germany e Ol Dominium University, Virginia, U.S.A f C.E.B.A.F, Virginia, U.S.A g Floria International University, Miami,Floria, U.S.A h IPN, IN2P3 Orsay, France c The Monte Carlo simulation evelope specically for the VCS experiments taking place at MAMI is fully escribe. This simulation can generate events accoring to the Bethe-Heitler + Born cross section behaviour an takes into account resolution eteriorating eects. It is use to etermine soli angles for the various experimental settings. 1 Introuction Determining experimental ierential cross sections for the p(e; e p) reaction is one of the necessary steps in reaching the nal goal of the VCS experiments 1 at MAMI/Mainz: to measure for the rst time the generalize polarizabilities 2 of the proton. To erive ierential cross sections from the measure ata the soli angle for a given ata set has to be known. In general, if one performs an ieal experiment which is free of resolution eects one can write the ratio of the number of counts etecte in a given phase space bin N bin an the integrate luminosity L as: 1

4 N bin L = = = = Z R Z R = :( 1 + ) = : 1 (1) 1 A : 2 (2) Equation 1 shows that this ratio can be expresse as the mean ierential cross section over the bin, multiplie by a soli angle 1 which is a pure "geometrical" quantity. On the other han, equation 2 shows that one can also write this ratio as the prouct of the actual ierential cross section value (=) at a given point somewhere in the bin (eg. the centre point) multiplie by a soli angle 2, which eviates from 1 byavalue epening on the relative cross section eviation from the (=) value over the bin. However, in reality resolution eects always play a role. The target thickness introuces eects such as multiple scattering an energy loss of incoming an outgoing particles an the particle etectors have an intrinsic resolution. As a result, the number of particles actually being etecte in a phase space bin iers from the number of particles present in this bin at the point ofinteraction. To etermine 1 or 2 taking into acount also resolution eects an the actual etection geometry, a Monte Carlo simulation can be use: events are generate in a given phase space accoring to a given cross section behaviour an all resolution eects an the actual etector geometry are implemente. The number of counts obtaine in a bin combine with the integrate luminosity for the simulation an the use cross section yiels a correcte for the above eects. A simulation using a constant cross section behaviour yiels 1 ( becomes ) while the use of the actual cross section behaviour yiels 2. Determining 1 an especially 2 is the goal of the Monte Carlo simulation that has been written for the VCS experiments at MAMI. Below a esription of the present state of this simulation (which is still evolving) is given. 2

5 2 Cross section behaviour implementation The p(e; e p) reaction is escribe by a ve fol ierential cross section 5 =k e ;cm epening on the value of the incoming an outgoing lab electron momenta k an k, the lab electron scattering angle e an the cm outgoing real photon angles an '. In the present version of this simulation, the actual behaviour of the p(e; e p) cross section is approximate by the Bethe-Heitler + Born (BH+B) contribution. The BH+B contribution to the cross section can be perfectly calculate once the elastic form factors of the nucleon are known. The polarizability contribution is estimate at most to be of the orer of a 1% eect on the BH+B contribution an is neglecte. Internal raiative corrections are not taken into account at present. Events are generate in the phase space accoring to the BH+B cross section behaviour using the acceptance-rejection metho 3 with a constant value as an envelope. However, since the BH+B cross section increases very rapily when the outgoing real photon irection comes close to the incoming or outgoing electron irections, the eciency of the acceptance-rejection metho can be rather low (easily less than 1%), which means that one nees a fast metho to calculate cross section values in orer to generate events at an acceptable rate. The BH+B cross section values are calculate using a theoretical coe 4 yieling about 15 cross sections per secon on a SUN SPARC1 workstation. To obtain cross sections at a usable rate for the acceptance-rejection metho, a four imensional cross section matrix has been constructe having k, e, an ' as variables. The cross section value at each matrix point is calculate for k equal to the incoming beam momentum k i using the above mentione coe. The matrix covers the total necessary phase space in the four variables for a full simulation: k varies from the elastic peak value own to the minimum momentum acceptance of the electron spectrometer, e covers all possible electron scattering angles for a given spectrometer setting an the two real cm photon angles cover the half-sphere above the leptonic plane, which is sucient because of the symmetry of the cross section aroun the leptonic plane. The cross section value at a ranom point in the phase space is obtaine by logarithmic interpolation in the matrix, about 1 times faster than the theoretical coe can provie it. The accuracy of the interpolate cross section values is better than.5% in the region of interest. A cross section matrix contains about 41 cross section values, an is containe in a binary le with a size of 1.6 MB. To obtain cross section values for k-values lower than the incoming beam momentum k i (ue to energy loss of the incoming beam in the target before 3

6 the actual VCS-process takes place) without having to increase the matrixsize, at present the approximation (k; k ; e ;;') (k i ;kummy ; e ;;') when s(k; k ; e )=s(k i ;kummy ; e ) is use: from the actual set (k; ) a set k (k i ;kummy ) is etermine using the cm-energy relationship, the latter set being covere by the cross section matrix. The error mae by this approximation increases with increasing eviation of k from k i, but for 97% of the events it is smaller than 3% in the region of interest. 3 The simulation proceure The simulation proceure consists of three separate programs: VCSSIM, RES- OLUTION an ANALYSIS. The rst program, VCSSIM, generates events accoring to a given cross section behaviour, an simulates the resolution eteriorating processes happening in the target an in the foils the particles have to pass through before arriving in a spectrometer. The target cell (liqui hyrogen) an spectrometer geometry (spectrometers A an B of hall A1 in Mainz are use to etect the proton an electron, respectively) are fully implemente. The coe oes not track the particles through the magnetic spectrometers, it only checks the acceptance of the particles by the collimators at the spectrometers' entrance an their momentum acceptance. A constant cross section for the p(e; e p) reaction, or as mentione above, the BH+B contribution to the cross section can be use. Roughly speaking, in orer to obtain events the following steps are taken. First the beam position on the target is generate (to simulate the horizontal an vertical movement of the incoming electron beam neee to prevent local overheating) an uniformly istribute along the beam line an interaction point is chosen. Using the pathlength of the incoming electron through the target, the multiple scattering an energy loss of this particle is etermine. This yiels the incoming electron four vector k at the point of interaction. The electron energy loss contains a collision part an an (external) bremsstrahlung part. The latter contribution can be large, an therefore at this point acheck is mae in case this electron elastically scatters at the minimum scattering angle consiere in the simulation phase space, whether its momentum is larger than the minimum momentum acceptance of spectro B or not. If this is not the case, everything up to this point starts all over again. If this momentum is inee larger, the acceptance-rejection routine starts to generate an event. This yiels a scattere electron four vector k', an also a cm real outgoing photon four vector. For the outgoing photon, the coe actually generates the complete cm 4 phase space (except for the small regions aroun the electron irections 4

7 where the BH+B cross section is larger than the envelope value in case the BH+B cross section behaviour is chosen). The real photon is then transforme to the lab to obtain the four vector. The four vector of the recoil proton can now be calculate as p'=p+k-k'-. Using the path lengths of outgoing electron an proton through the target, their four vectors at the point ofinteraction k' an p' are subject to multiple scattering an energy loss, an afterwars the particles are tracke towars the spectrometers to etermine whether they are accepte or not. Then the generation of a new event starts. For the simulation of the energy loss by collision, one can choose to use the mean energy loss, the most probable energy loss, or a more realistic energy loss istribution. The output of the VCSSIM program is twofol. First, an Ntuple, containing the events accepte by both spectrometers (both with regar to angular acceptance an momentum acceptance). For each event the momentum components of the outgoing electron an proton in their respective spectrometer frames, an the coorinates of the point of origin of the event in the target are store. Also a proton spectrometer inex is store. Inee, since the coe generates the complete cm 4 phase space for the outgoing photon, the complete recoil proton cone is reprouce. To take avantage of this, up to 1 ierent proton etector settings can be ene for which the result is obtaine in one simulation run. The VCSSIM Ntuple contains what is physically presente at the spectrometers' entrance. Seconly, also a ata le is prouce containing the integrate luminosity, mean target thickness an some statistical information regaring the simulation run. The secon program, RESOLUTION, introuces the resolution eects of the spectrometers on the events in the Ntuple originating from VCSSIM. The momentum size an irection of the etecte particles are change using Gaussian istribute ranom numbers having the spectrometer resolution values at the target position as FWHM. Also the coorinate along the beam line of the point of origin is change in the same way. The output is an Ntuple similar to the rst one, but now the momentum components of the particles an the point of origin have been subject to the spectrometers' resolution eects. This Ntuple contains what one coul call the equivalent of reconstructe experimentally obtaine particle momenta at the reconstructe point of origin in the target. The eects of the spectrometer resolution on the ata can as such be stuie without the nee to rerun the complete simulation over an over again. The thir an nal part of the simulation, the ANALYSIS program, reconstructs using the particle momenta containe in the secon Ntuple, for each event all physical observables that are also reconstructe from the experimental ata. Before the reconstruction process, the particle momenta are correcte for the mean energy loss in the target, calculate using path lengths 5

8 erive from the coorinates of the origin. The simulation analysis proceure is as such equivalent to the analysis proceure of the experimental ata. The physical observables are store in an Ntuple, an the obtaine istributions can then be compare qualitatively an quantitatively with the experimental ones. 4 Calculation of soli angles Once the phase space bins for which ierential cross section have to be obtaine are well ene, the ata in the output Ntuple from the ANALYSIS program, in combination with the integrate luminosity L from the VCSSIM program can be use to calculate soli angles for these bins using equation 3: bin = L sim : N sim,bin (3) 5 k : e :;cm In this formula N sim,bin is the number of simulate events present in the ene bin, while 5 =k e ;cm stans for the ierential cross section use in the simulation proceure. If a constant value has been use, equation 3 will yiel 1;bin. Application of this 1;bin to the measure ata will give rise to the mean experimental cross section in the bin. On the other han, if the simulation has been performe using the BH+B cross section, the value of 5 =k e ;cm in equation 3 equals the BH+B cross section value at the centre of the bin. This proceure will give rise to 2;bin. Applying this 2;bin to the measure ata will give rise to a very goo approximation of the actual cross section value at the centre of the bin, provie of course that the BH+B cross section has a "shape" very close to the actual cross section "shape". The experimental mean ierential cross section (obtaine using 1;bin ) shoul be compare with the theoretical mean ierential cross section for a bin. To reaily obtain the mean value of the BH+B ierential cross section in each of the ene bins, the simulate BH+B ata can be use as shown in equation 4: 5 k : e : ;cm bin = N sim,bin,bh+b L: 1;bin (4) 6

9 Ω (MeV.sr 2 ) q =111.5 MeV (a) Ω (MeV.sr 2 ) q =111.5 MeV (b) cos(θ) cos(θ) Figure 1: 1 (full line) an 2 (ashe line) for two proton spectrometer settings (labele (a) an (b)) at q =111.5 MeV 5 Results As an example, gure 1 shows the obtaine 1;bin (full line) an 2;bin (ashe line) for two ierent proton spectrometer settings at an outgoing cm real photon energy q of MeV. The bins are ene as 96:5MeV <q < 126:5MeV, cos() = :1, 158 <'<22. The pure statistical error on the soli angles in the plateau region is about 1%. 2;bin is subject to an aitional systematic error estimate to be about 2%, ue to the approximations in the BH+B cross section behaviour. It turns out that for these examples the ierence between the two s is at most of the orer of 1%. 6 Summary an outlook The simulation escribe above is exible: all resolution eteriorating eects can inepenently be switche on or o an it is possible to use a constant cross section or the BH+B cross section to generate events. Due to the multiple proton spectrometer option several settings at the same real outgoing photon energy can be simulate in one run, while the moularity of the coe gives the possibility to stuy spectrometer resolution eects in an ecient way. The implementation of the cross section matrix for the BH+B cross section option allows to obtain events at a very acceptable rate. The simulation is still evolving. One of the features that certainly has to 7

10 be improve is the implementation of the k-epenence of the BH+B cross section. Also a very important improvement will be the implementation of internal raiative corrections 5. The virtual part of these corrections will most probably be implemente as a correcte BH+B cross section, while for the real part two options are open: the use of an equivalent raiator approximation or the introuction of the (missing mass) 2 as an aitional sampling variable. References 1. J.Roche et al, these proceeings 2. P.A.M. Guichon et al, Nucl. Phys. A 591, 66 (1995). 3. Review of particle properties, Phys. Lett. B 24, 1 (1988). 4. M.Vanerhaeghen, Phys. Lett. B 368, 13 (1996). 5. D.Marchan et al, these proceeings 8