"Moller scattering 1"

"Verify formulas"

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

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

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

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

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

u31 = (E + m, 0, p3[4], p3[2] + i p3[3]) / sqrt(E + m)
u32 = (0, E + m, p3[2] - i p3[3], -p3[4]) / sqrt(E + m)

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

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))

u1 = (u11,u12)
u2 = (u21,u22)
u3 = (u31,u32)
u4 = (u41,u42)

u3bar = dot(conj(u3),gamma0)
u4bar = dot(conj(u4),gamma0)

M1(a,b,c,d) = dot(
 dot(u4bar[d],gammaT,u2[b]),
 dot(u3bar[c],gammaL,u1[a])
)

M2(a,b,c,d) = dot(
 dot(u3bar[c],gammaT,u2[b]),
 dot(u4bar[d],gammaL,u1[a])
)

MM11 = sum(a,1,2,sum(b,1,2,sum(c,1,2,sum(d,1,2,
 M1(a,b,c,d) conj(M1(a,b,c,d))
))))

MM12 = sum(a,1,2,sum(b,1,2,sum(c,1,2,sum(d,1,2,
 M1(a,b,c,d) conj(M2(a,b,c,d))
))))

MM21 = sum(a,1,2,sum(b,1,2,sum(c,1,2,sum(d,1,2,
 M2(a,b,c,d) conj(M1(a,b,c,d))
))))

MM22 = sum(a,1,2,sum(b,1,2,sum(c,1,2,sum(d,1,2,
 M2(a,b,c,d) conj(M2(a,b,c,d))
))))

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

f11 = 8 s^2 + 8 u^2 - 64 m^2 s - 64 m^2 u + 192 m^4
f12 = -8 s^2 + 64 m^2 s - 96 m^4
f22 = 8 s^2 + 8 t^2 - 64 m^2 s - 64 m^2 t + 192 m^4

check(MM11 == f11)
check(MM12 == f12)
check(MM22 == f22)

check(MM11 - MM12 - MM21 + MM22 == f11 - 2 f12 + f22)

MM = 1/4 e^4 (f11 / t^2 - 2 f12 / (t u) + f22 / u^2)

m = 0

check(f11 == 8 s^2 + 8 u^2)
check(f12 == -8 s^2)
check(f22 == 8 s^2 + 8 t^2)

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

a = (1 + cos(theta/2)^4) / sin(theta/2)^4
b = 2 / (sin(theta/2)^2 cos(theta/2)^2)
c = (1 + sin(theta/2)^4) / cos(theta/2)^4

check(MM == 2 e^4 (a + b + c))

check(MM == 4 e^4 (cos(theta)^2 + 3)^2 / sin(theta)^4)

m = quote(m)

"ok"