"Muon pair production"

clear

p = sqrt(E^2 - m^2)
rho = sqrt(E^2 - M^2)
p1 = (E,0,0,p)
p2 = (E,0,0,-p)
p3 = (E,
      rho*expsin(theta)*expcos(phi),
      rho*expsin(theta)*expsin(phi),
      rho*expcos(theta))
p4 = (E,
      -rho*expsin(theta)*expcos(phi),
      -rho*expsin(theta)*expsin(phi),
      -rho*expcos(theta))
I = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1))
gmunu = ((1,0,0,0),(0,-1,0,0),(0,0,-1,0),(0,0,0,-1))
gamma0 = ((1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,-1))
gamma1 = ((0,0,0,1),(0,0,1,0),(0,-1,0,0),(-1,0,0,0))
gamma2 = ((0,0,0,-i),(0,0,i,0),(0,i,0,0),(-i,0,0,0))
gamma3 = ((0,0,1,0),(0,0,0,-1),(-1,0,0,0),(0,1,0,0))
gamma = (gamma0,gamma1,gamma2,gamma3)
pslash1 = dot(p1,gmunu,gamma)
pslash2 = dot(p2,gmunu,gamma)
pslash3 = dot(p3,gmunu,gamma)
pslash4 = dot(p4,gmunu,gamma)
gammaT = transpose(gamma)
-- T1 is the first trace matrix
T1 = contract(dot(pslash1+m*I,gammaT,pslash2-m*I,gammaT),1,4)
-- T2 is the second trace matrix
T2 = contract(dot(pslash4-M*I,gammaT,pslash3+M*I,gammaT),1,4)
T2 = dot(gmunu,T2,gmunu)
-- A is the product of T1 and T2
A = contract(dot(T1,transpose(T2)))
-- B is the expression in theta
B = 1 + expcos(theta)^2 +
    (m^2+M^2)/E^2 * expsin(theta)^2 +
    m^2*M^2/E^4 * expcos(theta)^2

check(A/(64*E^4)-B == 0)
