+ -

Simulation of e e -pair production by muons

+-------------+                                               +----------##
| Geant 3.10  |               GEANT User's Guide              | PHYS451  ##
+-------------+                                               +----------##
                                   

                                   

Author(s) : L.Urban                                    Submitted: 26.10.84
Origin    : Same                                         Revised: 19.12.92
                                  

Subroutines

                    +--------------------------------+
                    |CALL GPAIRM  |
                    +--------------------------------+
                                  

+ - GPAIRM generates the e e -pair radiated by a high energetic muon. It uses the following input and output:

input:

via common block /GCTRAK/

output:

via common block /GCKING/

GPAIRM is called automatically from the tracking routine GTMUON if, and when, the parent muon reaches its radiation point during the tracking.

Method

The double differential cross-section for the process can be written [bib-LOHM]:

       2                                4                    2                            2
    ((d sigma)/ (dnudrho))    =    alpha ((2)/(3pi))(Zlambda)  #((1-nu)/ (nu))[phi + (m/M) phi(1)
                                                                                  e           mu

All the quantities in the expression above are defined in PHYS 450. By computing this cross-section for different (nu,rho) points, it can be seen:

2
1. the shape of the functions ((d sigma)/(dnudrho)) and 2 ((dsigma)/(dnu))int drho((d sigma)/(dnudrho)) pratically does not depend on Z
2. the dominant contribution comes from the low nu region:

       nu    =(4m/ E)<=nu< =100*nu                                        (2)
         min                      min
   

   2
3. in this low region (d sigma/dnudrho) is flat as a function of rho

   Therefore, we propose the following sampling method as a rough approximation:

   1. In the low nu region the differential cross-section

                                        2
          ((dsigma)/ (dnu))= int drho((d sigmaK)/(dnudsigma))
      

      can be approximated as:

                                               1/2        a
          ((dsigma)/ (dnu))# [1-((nu   )/(nu))]    ((1)/(v ))    with:  a =2-((ln E)/(10))    (E in GeV)(3)
                                    min
      

      We can write:

          ((dsigma)/ (dnu))# f(nu)g(nu)                                      (4)
      

      where,

                                    a-1                   a-1           a
          f(nu) =(((a- 1))/(((1)/(nu   ))- (((1)/(nu   )))   ))((1)/ (nu ))  (5)
                                                    max
                                    c
      

      is the normalized distribution in the interval [nu ,nu ] and c max

                                    1/2
          g(nu) =[1- ((nu   )/(nu))]                                         (6)
                         min
      

      is the rejection function.
   2. r and r being two uniformly distributed random numbers in the 1 2 interval [0,1]:

              -    Sample nu from the distribution f(nu) as:
                                    a-1            a-1   (
                   nu =(((1- r )/(nu   ))((r )/ (nu   ))) (1)/(1- a))        (7)
                              1             1
                                    c              max
              -    Accept nu if r < =g(nu)                                   (8)
                                 2
      

   3. Then compute

                              2    2                          1/2
          rho   (nu) =[1- ((6M )/(E (1- nu)))][1-((4m)/(nuE))]               (9)
             max
      

      and generate rho uniformly in the range [-rho ,+rho ]. max max

      After the succesful sampling of (nu,rho), GPAIRM generates the polar + - angles of the radiated e e -pair with respect to an axis defined along the parent muon's momentum. Theta is assigned the approximate average value:

          Theta =((M)/ (E))                                                 (10)
      

      + - + phi is generated isotropically and phi = phi + pi