Name: Lingxiao Xu (email_not_shown)
Date: 12/21/14-05:32:13 PM Z

In my previous message, I've made two small mistakes.
1)I should use PaVeReduce before "div" substitution, namely
ans = -I/Pi^2 (OneLoop[q, amp1 + amp2 + amp3] /. onshell //
PaVeReduce) /. div // Simplify;
2)further need to define sp[p1, p2] -> (m1^2 + m2^2)/2 in "onshell".
then the result is zero.

However, I just made some modification in the amplitude with the relation p2=p1+k, then the result is not zero.

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, p2] -> (m1^2 + m2^2)/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 + p2 - k] + mf).ds[
Polarization[k]].(ds[q + p2] + mf).gB.SpinorU[p2, m2] // FCI;
amp1 = num1 FeynAmpDenominator[PropagatorDenominator[q + p2 - k, mf],
PropagatorDenominator[q + p2, mf], PropagatorDenominator[q, ms]]

num2 = SpinorUBar[p1,
m1].gA.(ds[q + p2 - k] + mf).gB.(ds[p1] + m2).ds[
Polarization[k]].SpinorU[p2, m2] // FCI; amp2 =
num2 FeynAmpDenominator[PropagatorDenominator[q + p1, mf],
PropagatorDenominator[p2 - k, 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 = -I/Pi^2 (OneLoop[q, amp1 + amp2 + amp3] /. onshell //
PaVeReduce) /. div // Simplify;
test = Coefficient[ans, Div] // Simplify

Out[10]= \[CurlyPhi](p1,m1).(I (AL \[Gamma]^7+AR \[Gamma]^6)).(\[Gamma]\[CenterDot](-k+p2+q)+mf).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).(mf+\[Gamma]\[CenterDot](p2+q)).(I (BL \[Gamma]^7+BR \[Gamma]^6)).\[CurlyPhi](p2,m2)/((-k+p2+q)^2-mf^2).((p2+q)^2-mf^2).(q^2-ms^2)

Out[11]= \[CurlyPhi](p1,m1).(I (AL \[Gamma]^7+AR \[Gamma]^6)).(\[Gamma]\[CenterDot](-k+p2+q)+mf).(I (BL \[Gamma]^7+BR \[Gamma]^6)).(m2+\[Gamma]\[CenterDot]p1).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[CurlyPhi](p2,m2)/((p1+q)^2-mf^2).((p2-k)^2-m2^2).(q^2-ms^2)

Out[13]= \[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).(m1+\[Gamma]\[CenterDot]p2).(I (AL \[Gamma]^7+AR \[Gamma]^6)).(mf+\[Gamma]\[CenterDot](p2+q)).(I (BL \[Gamma]^7+BR \[Gamma]^6)).\[CurlyPhi](p2,m2)/(p2^2-m1^2).((p2+q)^2-mf^2).(q^2-ms^2)

Out[16]= (1/(m1^2-m2^2))(2 (AR BL m1-AL BR m2) \[LeftDoubleBracketingBar]p2epk \[CurlyPhi](p1,m1).\[Gamma]^7.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]+2 (AL BR m1-AR BL m2) \[LeftDoubleBracketingBar]p2epk \[CurlyPhi](p1,m1).\[Gamma]^6.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]-AL BR m1^2 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^6.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]+AL BR m2^2 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^6.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]+AL BR m2 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]k).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^7.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]-AL BR m1 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]k).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^6.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]-AR BL m1^2 \[LeftDoubleBracketingBar]\[CurlyP
hi](p1,m1).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^7.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]+AR BL m2^2 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^7.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]-AR BL m1 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]k).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^7.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar]+AR BL m2 \[LeftDoubleBracketingBar]\[CurlyPhi](p1,m1).(\[Gamma]\[CenterDot]k).(\[Gamma]\[CenterDot]\[CurlyEpsilon](k)).\[Gamma]^6.\[CurlyPhi](p2,m2)\[RightDoubleBracketingBar])

Regards,
Lingxiao

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