CompAZ Parametrization of the Compton spectra for gamma-gamma collider at TESLA Prepared by A.F.Zarnecki First version, Nov. 21, 2001 Updated, March 26, 2001 Introduction ============ The study is based on the gamma-gamma luminosity spectra generated by Valery Telnov in October 2001 (for 100, 250 and 400 GeV electron beam. As expected, the high energy part of the spectra can be described by the convolution of two independent photon energy distributions. It turned out that such a convolution gives also a resonable description for collisions, where only one gamma has large energy (close to the maximum energy available in the Compton scattering) and the other one is softer (with energy down to about a quater of the maximum photon energy). However, photon energy spectra in this range is not described by simple Compton formula. Parametrization =============== CompAZ parametrization is based on the ideas presented in the paper by I.F.Ginzburg and G.L.Kotkin (hep-ph/9905462, Eur.Phys.J.C13,295-300,2000). The photon energy spectra is assumed to consists of three components: * "single" Compton scattering as described by Ginzburg & Kotkin with 'z' parameter corrected for non-linear effects (z=4*E_e*E_l/m_e^2) * scattering involving 2 photons, which corresponds to doubling 'z' parameter (this process produces high energy "step", extending above x_max) * scattering on secondary (already scattered once) electrons this contribution is calculated by numerical convolution of two Compton spectra (for primary and secondary scattering) Fit === The parametrization was fitted simultaneously to photon energy spectra for 100, 250 and 400 GeV electron beam energy. Uncorrelated part of the spectra was chosen by the requirement that the other photon has high energy (greater than 0.4, 0.6 and 0.65 of electron beam energy respectively). With only 10(!) free parameters (including normalization) a very good description of the energy spectra down to x_gamma of about 0.1 is obtained for all energies. The parametrization also gives resonable description of the beam polarisation, with only two additional assumptions (and no additional parameters): scattering involving 2 photons results in high polarisation weakly depending on energy; secondary electrons are unpolarized. Normalization ============= Normalization is defined by the requirement that the two photon distribution at high x (x=E_gamma/E_e) p(x_1,x_2) = f(x_1) * f(x_2) describes the spectra generated by V.Telnov p(x_1,x_2) = 1/N * dN/dx_1/dx_2 As low x part of the spectra (x< ~0.1) is not properly described and underestimated, the normalization of the single photon distribution varies from about 0.8 at 50 GeV to about 0.55 at 500 GeV. This means that the two photon spectra obtained from the convolution of the two distributions describes between 65 and 30% of all events (but all high x events). When calculating numbers of expected events product of the two single photon spectra should be multiplied by the total gamma-gamma luminosity. If needed, normalization of the spectra can be obtained from call to event generating routine CompAZGN (see last section below). Validity ======== Energy dependence was fitted using spectra at 100, 250 and 400 GeV. In most cases parameter were found to depend linearly on energy. Parametrisation should work properly for electron beam energy: 50 <= Ee <= 500 GeV To avoid possible problems (also numerical) the routine returns values of 0 outside this energy range (warning message is also printed). Proper description of the spectra is obtained for W > ~0.3*W_max where W_max = 2*E_e*x_max and x_max = z/(z+1) is the maximum energy fraction for photon produced in "single Compton" scattering This means that the energy spectrum for the photon is resonably described for x > 0.15 * x_max assuming that the other photon is in the high energy peak x_2 > 0.6 * x_max Routine ======= Fortran function 'compaz.f': Real Function Compaz(x,E,icase) Input parameters: ----------------- x (Real) Fraction of beam energy carried by photon ( 0 < x < 1 ) E (Real) Electron beam energy [GeV] ( 50 < E < 500 ) Icase (Integer) Choice of output distribution: Icase = 0 Full spectra - sum of three contributions 1 Main contribution from direct Compton process 2 Contribution from scattering with 2 photons 3 Contribution from scattering on secondary electrons 10 Polarisation of the full spectra 11 Polarisation of the 1st component (direct Compton) 12 Polarisation of the 2nd component (two photon) 13 Polarisation of the 3rd component (secondary electron) 20 Effective z parameter 21 maximum x value for direct Compton 22 maximum x value for scattering with 2 photons Output value: ------------- Value of the selected energy distribution (icase=0...3), polarization (icase=10...13) or parameter (icase=20...22) Zero value (for both density and polarization) is returned for x < 0 or x > x_max2 (maximum x for scattering with 2 photons) For wrong 'icase', too small or too large 'E' wwaring message is printed and zero value is returned. Two internal functions are also included: * Real Function Compazc(x,z,Pol) Real x,z,Pol Calculating 'ideal' spectrum of compton backscattered photons * Real Function Compazp(x,Pl,Pb,z) Real x,Pl,Pb,z Calculating polarization of the compton spectra Event Generation ================ Single photon spectra ===================== As CompAZ routine itself is quite time consuming (involves numerical integration), dedicated routine 'compazgn.f' was prepared for very fast photon energy generation: Subroutine CompAZGN(Ein,Eg,Pg) Input: ------ Ein (Real) - electron beam energy [GeV] 50.<=Abs(Ein)<=500. Output: ------- Eg (Real) - generated photon energy Pg (Real) - photon polarization for this energy At first call (or when input beam energy changes) an array of probabilities (N=1000) is calculated from CompAZ. Variable bin size is used to describe details of the distribution properly. Links between bins with low and high probabilities are set to speed up generation. Generation itself is very fast. To select the energy bin and to smear energy value inside the bin 3 uniform distributed random numbers are generated per event (using CERNLIB routine RANMAR). Normalization: -------------- when called with Ein<0, routine returns normalization of CompAZ spectra for E=ABS(Ein) as Eg (Pg is not used). Gamma-Gamma events ================== Assuming that photon energies are independent (which is true for large W) CompAZGN routine can be also used to generate gamma-gamma events. However, this method is not very effective when studing narrow resonance production (e.g. Higgs). Therefor, dedicated routine has been prepared for fast event generation: Subroutine CompAZG2(Ein,Wmin,Wmax,E1,P1,E2,P2) Input: ------ Ein - electron beam energy [GeV] 50.<=Abs(Ein)<=500. Wmin - minimum value of W_gg Wmax - maximum value og W_gg if Wmax<=0 -> up to the limit Output: ------- E1,E2 - generated photon energies P1,P2 - corresponding photon polarizations Method: ------- Depending on the W range given, routine chooses one of the following algorithms: * for Wmin=Wmax => one dimensional method rapidity y of the gamma-gamma system is generated * for Wmin two-dimensional generation in (W,y) space * for Wmin generates two independent photons using CompAZGN Normalization: -------------- when called with Ein<0, routine returns normalization of 2-D spectra (in selected W range) for E=ABS(Ein). Normalization is stored in E1: E1 = 1/L * \int\limits_{Wmin}^{Wmax} dL/dW , E2,P1 and P2 are not used (set to 0). For narrow W range (generation in W-y space) normalization is calculated by numerical integration and ia available already at the first call. However, if wide W range is selected, and generation is based on CompAZGN (see above), normalization is calculated from the fraction of accepted events and should be checked only after the generation is finished (!). If Wmin=Wmax, returned value is E1 = 1/L * dL/dW [1/GeV] Invariant mass spectra ====================== Invariant mass distribution of colliding photons can be calculated with function CompazW: real function compazw(W,E,ic) Input: ------ E (Real) - electron beam energy [GeV] 50.<=Abs(Ein)<=500. W (Real) - photon-photon center of mass energy Ic (Integer) - choice of output distribution: Ic = 0 full spectra - sum of J=0 and J=2 1 J=0 spectra only 2 J=2 spectra only Output value: ------------- Two photon invariant mass distribution, relative to the integrated photon-photon luminosity: compazw(W,...) = 1/L * dL/dW [1/GeV] Integral of CompazW corresponds to the normalization of CompAZ spectra (see comments above).