A self energy calculation

Problem

Task: calculate


with
S(k) = i/(kslash - m - bslash )

Solution

<< "HighEnergyPhysics`fc`"

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The denominator of the propagator S(k) is

sden = GS[k] - GS[b] . GA[5] - m

Ansatz for the inverse propagator (thanx to Rainer Scharf from Leipzig University)

invden = x[1]*GS[k] + x[2]*GS[b] + x[3]*GS[k] . GA[5] + x[4]*GS[b] . GA[5] + x[5]*GA[5] + 
   x[6]*GS[k] . GS[b] + x[7]*GS[k] . GS[b] . GA[5] - m

tmp = Select2[Collect2[DiracOrder[Calc[sden . invden]], DiracGamma], DiracGamma]

eqn = Union[(Select2[#1, x] & ) /@ List @@ Select2[tmp, DiracGamma]]

Length[eqn]

so = Map[Factor, Flatten[Solve[eqn == 0, Array[x, 7]]], {2}]

num = m^2 + ScalarProduct[b, b] - ScalarProduct[k, k]

NumProp = (num*#1 & ) /@ (invden /. so)

DenomProp = DiracOrder[Calc[sden . NumProp]]

Prop = NumProp/DenomProp

s[b_, k_] = I*FCE[Prop]

InputForm[s[b, k]]

(I*(2*m*GS[k] . GS[b] . GA[5] - 
   2*GS[k] . GA[5]*SP[b, k] - 
   2*m*GA[5]*SP[b, k] - 2*GS[b]*SP[b, k] - 
   m*(m^2 + SP[b, b] - SP[k, k]) + 
   GS[k]*(-m^2 + SP[b, b] + SP[k, k]) + 
   GS[b] . GA[5]*(m^2 + SP[b, b] + 
     SP[k, k])))/(m^4 + 2*m^2*SP[b, b] + 
  SP[b, b]^2 - 4*SP[b, k]^2 - 
  2*m^2*SP[k, k] + 2*SP[b, b]*SP[k, k] + 
  SP[k, k]^2)

selfenergystart = Trick[GA[mu] . s[b, k + p] . GA[nu] . s[b, k]]

selfden = Factor[Denominator[selfenergystart]]

selfnum = Numerator[selfenergystart]

selfenergy = Collect2[Tr[selfnum], {mu,nu}]

result = FCE[selfenergy/selfden]; 

InputForm[result]

(-8*FV[b, nu]*FV[p, mu]*(2*m^2*SP[b, k] - 
    2*SP[b, b]*SP[b, k] - m^2*SP[b, p] - 
    SP[b, b]*SP[b, p] - 2*SP[b, k]*
     SP[k, k] - SP[b, p]*SP[k, k] - 
    2*SP[b, k]*SP[k, p] - SP[b, k]*
     SP[p, p]) - 8*FV[b, mu]*FV[p, nu]*
   (2*m^2*SP[b, k] - 2*SP[b, b]*SP[b, k] - 
    m^2*SP[b, p] - SP[b, b]*SP[b, p] - 
    2*SP[b, k]*SP[k, k] - SP[b, p]*
     SP[k, k] - 2*SP[b, k]*SP[k, p] - 
    SP[b, k]*SP[p, p]) - 16*FV[b, nu]*
   FV[k, mu]*(2*m^2*SP[b, k] - 
    2*SP[b, b]*SP[b, k] + m^2*SP[b, p] - 
    SP[b, b]*SP[b, p] - 2*SP[b, k]*
     SP[k, k] - SP[b, p]*SP[k, k] - 
    2*SP[b, k]*SP[k, p] - SP[b, k]*
     SP[p, p]) - 16*FV[b, mu]*FV[k, nu]*
   (2*m^2*SP[b, k] - 2*SP[b, b]*SP[b, k] + 
    m^2*SP[b, p] - SP[b, b]*SP[b, p] - 
    2*SP[b, k]*SP[k, k] - SP[b, p]*
     SP[k, k] - 2*SP[b, k]*SP[k, p] - 
    SP[b, k]*SP[p, p]) - 8*FV[k, mu]*
   FV[k, nu]*(m^4 - 6*m^2*SP[b, b] + 
    SP[b, b]^2 + 4*SP[b, k]^2 + 
    4*SP[b, k]*SP[b, p] - 2*m^2*SP[k, k] + 
    2*SP[b, b]*SP[k, k] + SP[k, k]^2 - 
    2*m^2*SP[k, p] + 2*SP[b, b]*SP[k, p] + 
    2*SP[k, k]*SP[k, p] - m^2*SP[p, p] + 
    SP[b, b]*SP[p, p] + SP[k, k]*SP[p, p]) - 
  4*FV[k, nu]*FV[p, mu]*
   (m^4 - 6*m^2*SP[b, b] + SP[b, b]^2 + 
    4*SP[b, k]^2 + 4*SP[b, k]*SP[b, p] - 
    2*m^2*SP[k, k] + 2*SP[b, b]*SP[k, k] + 
    SP[k, k]^2 - 2*m^2*SP[k, p] + 
    2*SP[b, b]*SP[k, p] + 2*SP[k, k]*
     SP[k, p] - m^2*SP[p, p] + 
    SP[b, b]*SP[p, p] + SP[k, k]*SP[p, p]) - 
  4*FV[k, mu]*FV[p, nu]*
   (m^4 - 6*m^2*SP[b, b] + SP[b, b]^2 + 
    4*SP[b, k]^2 + 4*SP[b, k]*SP[b, p] - 
    2*m^2*SP[k, k] + 2*SP[b, b]*SP[k, k] + 
    SP[k, k]^2 - 2*m^2*SP[k, p] + 
    2*SP[b, b]*SP[k, p] + 2*SP[k, k]*
     SP[k, p] - m^2*SP[p, p] + 
    SP[b, b]*SP[p, p] + SP[k, k]*SP[p, p]) - 
  8*FV[b, mu]*FV[b, nu]*
   (m^4 + 2*m^2*SP[b, b] + SP[b, b]^2 + 
    4*SP[b, k]^2 + 4*SP[b, k]*SP[b, p] - 
    2*m^2*SP[k, k] + 2*SP[b, b]*SP[k, k] + 
    SP[k, k]^2 - 2*m^2*SP[k, p] + 
    2*SP[b, b]*SP[k, p] + 2*SP[k, k]*
     SP[k, p] + m^2*SP[p, p] + 
    SP[b, b]*SP[p, p] + SP[k, k]*SP[p, p]) - 
  4*MT[mu, nu]*(m^6 + m^4*SP[b, b] - 
    m^2*SP[b, b]^2 - SP[b, b]^3 - 
    4*m^2*SP[b, k]^2 + 4*SP[b, b]*
     SP[b, k]^2 - 4*m^2*SP[b, k]*SP[b, p] + 
    4*SP[b, b]*SP[b, k]*SP[b, p] + 
    2*m^2*SP[b, p]^2 + 2*SP[b, b]*
     SP[b, p]^2 - 3*m^4*SP[k, k] + 
    2*m^2*SP[b, b]*SP[k, k] - 
    3*SP[b, b]^2*SP[k, k] + 4*SP[b, k]^2*
     SP[k, k] + 4*SP[b, k]*SP[b, p]*
     SP[k, k] + 2*SP[b, p]^2*SP[k, k] + 
    3*m^2*SP[k, k]^2 - 3*SP[b, b]*
     SP[k, k]^2 - SP[k, k]^3 - 
    3*m^4*SP[k, p] + 2*m^2*SP[b, b]*
     SP[k, p] - 3*SP[b, b]^2*SP[k, p] + 
    4*SP[b, k]^2*SP[k, p] + 6*m^2*SP[k, k]*
     SP[k, p] - 6*SP[b, b]*SP[k, k]*
     SP[k, p] - 3*SP[k, k]^2*SP[k, p] + 
    2*m^2*SP[k, p]^2 - 2*SP[b, b]*
     SP[k, p]^2 - 2*SP[k, k]*SP[k, p]^2 - 
    m^4*SP[p, p] - 2*m^2*SP[b, b]*SP[p, p] - 
    SP[b, b]^2*SP[p, p] + 4*SP[b, k]^2*
     SP[p, p] + 2*SP[b, k]*SP[b, p]*
     SP[p, p] + 2*m^2*SP[k, k]*SP[p, p] - 
    2*SP[b, b]*SP[k, k]*SP[p, p] - 
    SP[k, k]^2*SP[p, p] + m^2*SP[k, p]*
     SP[p, p] - SP[b, b]*SP[k, p]*SP[p, p] - 
    SP[k, k]*SP[k, p]*SP[p, p]) - 
  4*I*(3*m^4 + 2*m^2*SP[b, b] - SP[b, b]^2 - 
    4*SP[b, k]^2 - 4*SP[b, k]*SP[b, p] - 
    2*m^2*SP[k, k] - 2*SP[b, b]*SP[k, k] - 
    SP[k, k]^2 - 2*m^2*SP[k, p] - 
    2*SP[b, b]*SP[k, p] - 2*SP[k, k]*
     SP[k, p] - m^2*SP[p, p] - 
    SP[b, b]*SP[p, p] - SP[k, k]*SP[p, p])*
   LC[mu, nu][b, p] + 
  8*I*(2*m^2*SP[b, k] - 2*SP[b, b]*
     SP[b, k] + m^2*SP[b, p] - 
    SP[b, b]*SP[b, p] - 2*SP[b, k]*
     SP[k, k] - SP[b, p]*SP[k, k] - 
    2*SP[b, k]*SP[k, p] - SP[b, k]*SP[p, p])*
   LC[mu, nu][k, p])/
 ((m^4 + 2*m^2*SP[b, b] + SP[b, b]^2 - 
   4*SP[b, k]^2 - 2*m^2*SP[k, k] + 
   2*SP[b, b]*SP[k, k] + SP[k, k]^2)*
  (m^4 + 2*m^2*SP[b, b] + SP[b, b]^2 - 
   4*SP[b, k]^2 - 8*SP[b, k]*SP[b, p] - 
   4*SP[b, p]^2 - 2*m^2*SP[k, k] + 
   2*SP[b, b]*SP[k, k] + SP[k, k]^2 - 
   4*m^2*SP[k, p] + 4*SP[b, b]*SP[k, p] + 
   4*SP[k, k]*SP[k, p] + 4*SP[k, p]^2 - 
   2*m^2*SP[p, p] + 2*SP[b, b]*SP[p, p] + 
   2*SP[k, k]*SP[p, p] + 4*SP[k, p]*
    SP[p, p] + SP[p, p]^2))

TimeUsed[]