Eigenmath Demo

-- Verify Casimir trick for Bhabha scattering

p = sqrt(E^2 - m^2)

p1 = (E, 0, 0, p)
p2 = (E, 0, 0, -p)

p3 = (E,
      p expsin(theta) expcos(phi),
      p expsin(theta) expsin(phi),
      p expcos(theta))

p4 = (E,
      -p expsin(theta) expcos(phi),
      -p expsin(theta) expsin(phi),
      -p expcos(theta))

v11 = (p1[4], p1[2] + i p1[3], E + m, 0)
v12 = (p1[2] - i p1[3], -p1[4], 0, E + m)

u21 = (E + m, 0, p2[4], p2[2] + i p2[3])
u22 = (0, E + m, p2[2] - i p2[3], -p2[4])

v31 = (p3[4], p3[2] + i p3[3], E + m, 0)
v32 = (p3[2] - i p3[3], -p3[4], 0, E + m)

u41 = (E + m, 0, p4[4], p4[2] + i p4[3])
u42 = (0, E + m, p4[2] - i p4[3], -p4[4])

v1 = (v11,v12)
u2 = (u21,u22)
v3 = (v31,v32)
u4 = (u41,u42)

N = (E + m)^4

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)

gammaT = transpose(gamma)
gammaL = transpose(dot(gmunu,gamma))

v1bar = dot(conj(v1),gamma0) -- adjoint of v1
u4bar = dot(conj(u4),gamma0) -- adjoint of u4

"Sum over spin states"

S = 0

for(s1,1,2,for(s2,1,2,for(s3,1,2,for(s4,1,2,
  X13 = dot(v1bar[s1],gammaT,v3[s3]),
  X42 = dot(u4bar[s4],gammaL,u2[s2]),
  X12 = dot(v1bar[s1],gammaT,u2[s2]),
  X43 = dot(u4bar[s4],gammaL,v3[s3]),
  a = -1/t dot(X13,X42) + 1/s dot(X12,X43),  -- a is an amplitude
  f = a conj(a),                             -- f is a probability density function
  S = S + f
))))

"Casimir trick"

pslash1 = dot(p1,gmunu,gamma)
pslash2 = dot(p2,gmunu,gamma)
pslash3 = dot(p3,gmunu,gamma)
pslash4 = dot(p4,gmunu,gamma)

X1 = pslash1 - m I
X2 = pslash2 + m I
X3 = pslash3 - m I
X4 = pslash4 + m I

T1 = contract(dot(X1,gammaT,X3,gammaT),1,4)
T2 = contract(dot(X4,gammaL,X2,gammaL),1,4)
f11 = contract(dot(T1,transpose(T2)))

T = contract(dot(X1,gammaT,X2,gammaT,X4,gammaL,X3,gammaL),1,6)
f12 = contract(contract(T,1,3))

T1 = contract(dot(X1,gammaT,X2,gammaT),1,4)
T2 = contract(dot(X4,gammaL,X3,gammaL),1,4)
f22 = contract(dot(T1,transpose(T2)))

f = f11/t^2 - f12/(s t) - conj(f12)/(s t) + f22/s^2
f

"Verify Casimir trick (1=ok)"

S/N == f