"Annihilation 1"

"Verify formulas"

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

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

p3 = (E,
      E sin(theta) cos(phi),
      E sin(theta) sin(phi),
      E cos(theta))

p4 = (E,
      -E sin(theta) cos(phi),
      -E sin(theta) sin(phi),
      -E cos(theta))

u11 = (E + m, 0, p1[4], p1[2] + i p1[3]) / sqrt(E + m)
u12 = (0, E + m, p1[2] - i p1[3], -p1[4]) / sqrt(E + m)

v21 = (p2[4], p2[2] + i p2[3], E + m, 0) / sqrt(E + m)
v22 = (p2[2] - i p2[3], -p2[4], 0, E + m) / sqrt(E + m)

u1 = (u11,u12)
v2 = (v21,v22)

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))

q1 = p1 - p3
q2 = p1 - p4

q1 = dot(q1,gmunu,gamma)
q2 = dot(q2,gmunu,gamma)

K1 = dot(-i e gammaT, q1 + m I, -i e gammaT)
K2 = dot(-i e gammaT, q2 + m I, -i e gammaT)

v2bar = dot(conj(v2),gamma0)

M1(a,b) = dot(v2bar[b], K1, u1[a])
M2(a,b) = dot(v2bar[b], K2, u1[a])

MM11 = sum(a,1,2,sum(b,1,2,
 contract(dot(M1(a,b), gmunu, transpose(conj(M1(a,b))), gmunu))
))

MM12 = sum(a,1,2,sum(b,1,2,
 contract(dot(M1(a,b), gmunu, conj(M2(a,b)), gmunu))
))

MM21 = sum(a,1,2,sum(b,1,2,
 contract(dot(M2(a,b), gmunu, conj(M1(a,b)), gmunu))
))

MM22 = sum(a,1,2,sum(b,1,2,
 contract(dot(M2(a,b), gmunu, transpose(conj(M2(a,b))), gmunu))
))

s = dot(p1 + p2, gmunu, p1 + p2)
t = dot(p1 - p3, gmunu, p1 - p3)
u = dot(p1 - p4, gmunu, p1 - p4)

f11 = 8 t u - 24 m^2 t - 8 m^2 u - 8 m^4
f12 = 8 m^2 s - 32 m^4
f22 = 8 t u - 8 m^2 t - 24 m^2 u - 8 m^4

check(MM11 == e^4 f11)
check(MM12 == e^4 f12)
check(MM22 == e^4 f22)

check(MM11 + MM12 + MM21 + MM22 == e^4 (f11 + 2 f12 + f22))

d11 = (t - m^2)^2
d12 = (t - m^2) (u - m^2)
d22 = (u - m^2)^2

MM = 1/4 e^4 (f11 / d11 + 2 f12 / d12 + f22 / d22)

m = 0

check(f11 == 8 t u)
check(f12 == 0)
check(f22 == 8 t u)

check(s == 4 E^2)
check(t == -2 E^2 (1 - cos(theta)))
check(u == -2 E^2 (1 + cos(theta)))

check(MM == 2 e^4 (u / t + t / u))

m = quote(m)

"ok"