Name: Lingxiao Xu Date: 12/21/14-05:01:05 PM Z
Hi, guys:
Recently I am trying to evaluate the process muon decay into electron
and photon in a theory, which contains just 2 kinds of leptons(muon and
electron),photon,another kind of fermion and one neutral scalar. One
lepton and fermion can couple with the neutral scalar via Yukawa-type
interaction, further normal QED vertex is allowed in this theory.
I just want to get the amplitude of this process at one loop level.
Because of Ward-Takahashi Identity, the divergence in vertex correction
type diagram should cancel the divergence in other two self energy type
diagrams.
Here comes my problem, when I write the amplitude with p1,p2,k in
different ways, in some ways the divergences cancel but some others
don’t. PS:muon has mass m2 and momentum p2, electron has mass m1 and
momentum p1 and photon has momentum k, where we have p2=p1+k.
Is there any bug ??? I am using FeynCalc 8.2.0 with Mathematica 9 at
Win7.
Thanks for the help!!
Regards,
Lingxiao
Here is code which might be helpful to solve my problem;
The divergences just don’t cancel,
In[2]:= (*some shorthands*)
dm[mu_] := DiracMatrix[mu, Dimension -> D]
ds[p_] := DiracSlash[p]
gA := I (AL dm[7] + AR dm[6])(*lepton scalar fermion
Yukawa vertex*)
gB := I (BL dm[7] + BR dm[6])(*fermion scalar lepton
Yukawa vertex*)
sp[p_, q_] := ScalarProduct[p, q]
In[7]:= onshell = {sp[p1, p1] -> m1^2, sp[p2,
p2] -> m2^2,
sp[k, k] -> 0, sp[k, p1] -> (m2^2 - m1^2)/2,
sp[k, p2] -> (m2^2 - m1^2)/2, sp[p1,
Polarization[k]] -> p2epk,
sp[p2, Polarization[k]] -> p2epk};
In[8]:= div = {B0[m1^2, mf^2, ms^2] -> Div,
B0[m2^2, mf^2, ms^2] -> Div, B0[0, mf^2, ms^2] ->
Div,
B0[0, mf^2, mf^2] -> Div, B0[0, ms^2, ms^2] ->
Div};
In[9]:= num1 =
SpinorUBar[p1, m1].gA.(ds[q + p1] + mf).ds[
Polarization[k]].(ds[q + p2] +
mf).gB.SpinorU[p2, m2] // FCI;
amp1 = num1 FeynAmpDenominator[PropagatorDenominator[q + p1,
mf],
PropagatorDenominator[q + p2, mf],
PropagatorDenominator[q, ms]];
num2 = SpinorUBar[p1, m1].gA.(ds[q + p1] +
mf).gB.(ds[p1] + m2).ds[
Polarization[k]].SpinorU[p2, m2] // FCI; amp2
=
num2 FeynAmpDenominator[PropagatorDenominator[q + p1,
mf],
PropagatorDenominator[p1, m2], PropagatorDenominator[q,
ms]];
num3 = SpinorUBar[p1, m1].ds[
Polarization[k]].(ds[p2] + m1).gA.(ds[q +
p2] + mf).gB.SpinorU[
p2, m2] // FCI;
amp3 = num3 FeynAmpDenominator[PropagatorDenominator[p2,
m1],
PropagatorDenominator[q + p2, mf],
PropagatorDenominator[q, ms]];
SetOptions[OneLoop, Dimension -> D];
ans = OneLoop[q, amp1 + amp2 + amp3] /. onshell /. div //
PaVeReduce //
Simplify;
test = Coefficient[ans, Div] // Simplify
Out[16]= -(1/2) I \[Pi]^2 (AL BR \[LeftDoubleBracketingBar]\CurlyPhi.(\[Gamma]\[CenterDot]\CurlyEpsilon).\[Gamma]^6.\CurlyPhi\[RightDoubleBracketingBar]+AR BL \[LeftDoubleBracketingBar]\CurlyPhi.(\[Gamma]\[CenterDot]\CurlyEpsilon).\[Gamma]^7.\CurlyPhi\[RightDoubleBracketingBar])