This example uses a custom QED model created with FeynRules. Please evaluate the file FeynCalc/Examples/FeynRules/QED/GenerateModelYukawa.m before running it for the first time.
description = "Renormalization, Yukawa, MS and MSbar, 1-loop";
If[ $FrontEnd === Null,
$FeynCalcStartupMessages = False;
Print[description];
];
If[ $Notebooks === False,
$FeynCalcStartupMessages = False
];
$LoadAddOns = {"FeynArts"};
<< FeynCalc`
$FAVerbose = 0;
FCCheckVersion[10, 0, 0];\text{FeynCalc }\;\text{10.0.0 (dev version, 2024-08-07 16:59:34 +02:00, 2f62a22c). For help, use the }\underline{\text{online} \;\text{documentation},}\;\text{ visit the }\underline{\text{forum}}\;\text{ and have a look at the supplied }\underline{\text{examples}.}\;\text{ The PDF-version of the manual can be downloaded }\underline{\text{here}.}
\text{If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.}
\text{Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!}
\text{FeynArts }\;\text{3.11 (3 Aug 2020) patched for use with FeynCalc, for documentation see the }\underline{\text{manual}}\;\text{ or visit }\underline{\text{www}.\text{feynarts}.\text{de}.}
\text{If you use FeynArts in your research, please cite}
\text{ $\bullet $ T. Hahn, Comput. Phys. Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260}
We keep scaleless B0 functions, since otherwise the UV part would not come out right.
$KeepLogDivergentScalelessIntegrals = True;FAPatch[PatchModelsOnly -> True];
(*Patched 4 FeynArts models.*)params = {InsertionLevel -> {Particles}, Model -> FileNameJoin[{"LY", "LY"}],
GenericModel -> FileNameJoin[{"LY", "LY"}], ExcludeParticles -> {}};
top[i_, j_] := CreateTopologies[1, i -> j,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}];
topCT[i_, j_] := CreateCTTopologies[1, i -> j,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}];
{diagFermionSE, diagFermionSECT} = InsertFields[#, {F[10]} -> {F[10]},
Sequence @@ params] & /@ {top[1, 1], topCT[1, 1]};
{diagScalarSE, diagScalarSECT} = InsertFields[#, {S[1]} -> {S[1]},
Sequence @@ params] & /@ {top[1, 1], topCT[1, 1]};
{diagVertexFFS, diagVertexFFSCT} = InsertFields[#, {F[10], S[1]} -> {F[10]},
Sequence @@ params] & /@ {top[2, 1], topCT[2, 1]};
{diagVertexSSSS, diagVertexSSSSCT} = InsertFields[#, {S[1], S[1]} -> {S[1], S[1]},
Sequence @@ params] & /@ {top[2, 2], topCT[2, 2]};diag1[0] = diagFermionSE[[0]][diagFermionSE[[1]], diagFermionSECT[[1]]];
diag2[0] = diagScalarSE[[0]][diagScalarSE[[1]], diagScalarSECT[[1]]];
diag3[0] = diagVertexFFS[[0]][diagVertexFFS[[1]], diagVertexFFSCT[[1]]];
diag4[0] = diagVertexSSSS[[0]][diagVertexSSSS[[1]], diagVertexSSSSCT[[1]]];Paint[diag1[0], ColumnsXRows -> {2, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 256 {2, 1}];
Paint[diag2[0], ColumnsXRows -> {2, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 256 {2, 1}];
Paint[diag3[0], ColumnsXRows -> {2, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 256 {2, 1}];
Paint[diag4[0], ColumnsXRows -> {3, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 256 {3, 1}];
The 1/(2Pi)^D prefactor is implicit.
Fermion self-energy including the counter-term
amp1[0] = FCFAConvert[CreateFeynAmp[diag1[0], Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p}, OutgoingMomenta -> {p},
LoopMomenta -> {l}, UndoChiralSplittings -> True,
ChangeDimension -> D, List -> False, SMP -> True,
FinalSubstitutions -> {}, Contract -> True]-\frac{(-i g).(\gamma \cdot l+\text{Mx}).(-i g)}{\left(l^2-\text{Mx}^2\right).\left((l-p)^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i \;\text{Mx} (\text{Zmx} \;\text{Zx}-1)+i (\text{Zx}-1) \gamma \cdot p
Scalar self-energy including the counter-term
amp2[0] = FCFAConvert[CreateFeynAmp[diag2[0], Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p}, OutgoingMomenta -> {p},
LoopMomenta -> {l}, UndoChiralSplittings -> True,
ChangeDimension -> D, List -> False, SMP -> True, Contract -> True]\frac{\text{la}}{2 \left(l^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i \;\text{Mphi}^2 (\text{Zmphi} \;\text{Zphi}-1)+i p^2 (\text{Zphi}-1)
Fermion-scalar vertex including the counter-term
amp3[0] = FCFAConvert[CreateFeynAmp[diag3[0], Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p1, k}, OutgoingMomenta -> {p2},
LoopMomenta -> {l}, UndoChiralSplittings -> True, ChangeDimension -> D,
List -> False, SMP -> True, Contract -> True]-\frac{i (-i g).(\gamma \cdot (k+l)+\text{Mx}).(-i g).(\gamma \cdot l+\text{Mx}).(-i g)}{\left(l^2-\text{Mx}^2\right).\left((k+l)^2-\text{Mx}^2\right).\left((k+l-\text{p2})^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i g \left(\text{Zg} \sqrt{\text{Zphi}} \;\text{Zx}-1\right)
Scalar self-interaction vertex including the counter-term
amp4[0] = FCFAConvert[CreateFeynAmp[diag4[0], Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p1, p2}, OutgoingMomenta -> {p3, p4},
LoopMomenta -> {l}, UndoChiralSplittings -> True, ChangeDimension -> D,
List -> False, SMP -> True, Contract -> True]-\frac{2 \;\text{tr}((\text{Mx}-\gamma \cdot l).(-i g).(\gamma \cdot (-l-\text{p2})+\text{Mx}).(-i g).(\gamma \cdot (-l-\text{p2}+\text{p4})+\text{Mx}).(-i g).(\gamma \cdot (-l-\text{p2}+\text{p3}+\text{p4})+\text{Mx}).(-i g))}{\left(l^2-\text{Mx}^2\right).\left((l+\text{p2})^2-\text{Mx}^2\right).\left((l+\text{p2}-\text{p4})^2-\text{Mx}^2\right).\left((l+\text{p2}-\text{p3}-\text{p4})^2-\text{Mx}^2\right)}-i \;\text{la} \left(\text{Zla} \;\text{Zphi}^2-1\right)
amp1[1] = amp1[0] // ReplaceAll[#, {Zx -> 1 + alpha dZx,
Zmx -> 1 + alpha dZmx}] & // Series[#, {alpha, 0, 1}] & //
Normal // ReplaceAll[#, alpha -> 1] &-\frac{(-i g).(\gamma \cdot l+\text{Mx}).(-i g)}{\left(l^2-\text{Mx}^2\right).\left((l-p)^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i \;\text{Mx} (\text{dZmx}+\text{dZx})+i \;\text{dZx} \gamma \cdot p
Tensor reduction allows us to express the electron self-energy in tems of the Passarino-Veltman coefficient functions.
amp1[2] = TID[amp1[1], l, ToPaVe -> True]\frac{i \pi ^2 g^2 \left(\gamma \cdot p \left(\text{Mphi}^2 \left(-\xi _{S(1)}\right)+\text{Mx}^2+p^2\right)+2 \;\text{Mx} p^2\right) \;\text{B}_0\left(p^2,\text{Mx}^2,\text{Mphi}^2 \xi _{S(1)}\right)}{2 p^2}+\frac{i \pi ^2 g^2 \gamma \cdot p \;\text{A}_0\left(\text{Mphi}^2 \xi _{S(1)}\right)}{2 p^2}-\frac{i \pi ^2 g^2 \;\text{A}_0\left(\text{Mx}^2\right) \gamma \cdot p}{2 p^2}-i (\text{dZmx} \;\text{Mx}+\text{dZx} \;\text{Mx}-\text{dZx} \gamma \cdot p)
Discard all the finite pieces of the 1-loop amplitude
amp1Div[0] = PaVeUVPart[amp1[2], Prefactor -> 1/(2 Pi)^D] //
FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // Normal //
FCHideEpsilon // SelectNotFree2[#, {SMP["Delta"], dZx,
dZmx}] & // Simplify // Collect2[#, DiracGamma] &\frac{i \left(32 \pi ^2 \;\text{dZx}+\Delta g^2\right) \gamma \cdot p}{32 \pi ^2}-\frac{i \;\text{Mx} \left(16 \pi ^2 \;\text{dZmx}+16 \pi ^2 \;\text{dZx}-\Delta g^2\right)}{16 \pi ^2}
Equating the result to zero and solving for dZx and dZmx we obtain the renormalization constants in the minimal subtraction schemes.
solMSbar1 = FCMatchSolve[amp1Div[0], {g, la, Mx, DiracGamma, SMP}];
solMS1 = solMSbar1 /. SMP["Delta"] -> 1/Epsilon\text{FCMatchSolve: Solving for: }\{\text{dZmx},\text{dZx}\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{dZmx}\to \frac{3 g^2}{32 \pi ^2 \varepsilon },\text{dZx}\to -\frac{g^2}{32 \pi ^2 \varepsilon }\right\}
amp2[0]\frac{\text{la}}{2 \left(l^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i \;\text{Mphi}^2 (\text{Zmphi} \;\text{Zphi}-1)+i p^2 (\text{Zphi}-1)
amp2[1] = amp2[0] // ReplaceRepeated[#, {Zphi -> 1 + alpha dZphi,
Zmphi -> 1 + alpha dZmphi}] & // Series[#, {alpha, 0, 1}] & //
Normal // ReplaceAll[#, alpha -> 1] &\frac{\text{la}}{2 \left(l^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i \;\text{Mphi}^2 (\text{dZmphi}+\text{dZphi})+i \;\text{dZphi} p^2
Tensor reduction allows us to express the scalar self-energy in tems of the Passarino-Veltman coefficient functions.
amp2[2] = TID[amp2[1], l, ToPaVe -> True]\frac{1}{2} i \pi ^2 \;\text{la} \;\text{A}_0\left(\text{Mphi}^2 \xi _{S(1)}\right)-i \left(\text{dZmphi} \;\text{Mphi}^2+\text{dZphi} \;\text{Mphi}^2-\text{dZphi} p^2\right)
Discard all the finite pieces of the 1-loop amplitude
amp2Div[0] = PaVeUVPart[amp2[2], Prefactor -> 1/(2 Pi)^D] //
FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // Normal //
FCHideEpsilon // SelectNotFree2[#, {SMP["Delta"], dZphi, dZmphi}] & // Simplify //
Collect2[#, p, Mphi] &i \;\text{dZphi} p^2-\frac{i \;\text{Mphi}^2 \left(32 \pi ^2 \;\text{dZmphi}+32 \pi ^2 \;\text{dZphi}-\Delta \;\text{la} \xi _{S(1)}\right)}{32 \pi ^2}
Equating this to zero and solving for dZphi and dZmphi obtain the renormalization constants in the minimal subtraction schemes.
solMSbar2 = FCMatchSolve[amp2Div[0], {g, la, Mphi, p, SMP, GaugeXi}]
solMS2 = solMSbar2 /. SMP["Delta"] -> 1/Epsilon;\text{FCMatchSolve: Following coefficients trivially vanish: }\{\text{dZphi}\to 0\}
\text{FCMatchSolve: Solving for: }\{\text{dZmphi}\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{dZphi}\to 0,\text{dZmphi}\to \frac{\Delta \;\text{la} \xi _{S(1)}}{32 \pi ^2}\right\}
amp3[1] = amp3[0] // ReplaceRepeated[#, {Zphi -> 1 + alpha dZphi,
Zx -> 1 + alpha dZx, Zg -> 1 + alpha dZg}] & //
Series[#, {alpha, 0, 1}] & // Normal // ReplaceAll[#, alpha -> 1] &-\frac{i (-i g).(\gamma \cdot (k+l)+\text{Mx}).(-i g).(\gamma \cdot l+\text{Mx}).(-i g)}{\left(l^2-\text{Mx}^2\right).\left((k+l)^2-\text{Mx}^2\right).\left((k+l-\text{p2})^2-\text{Mphi}^2 \xi _{S(1)}\right)}-i g \left(\text{dZg}+\frac{\text{dZphi}}{2}+\text{dZx}\right)
The result of the tensor reduction is quite large, since we keep the full gauge dependence and do not specify the kinematics
amp3[2] = TID[amp3[1], l, ToPaVe -> True, UsePaVeBasis -> True]i \pi ^2 g^3 \;\text{B}_0\left(\text{p2}^2,\text{Mx}^2,\text{Mphi}^2 \xi _{S(1)}\right)+i \pi ^2 g^3 \left(-(\gamma \cdot k).(\gamma \cdot k)-\text{Mx} \gamma \cdot k+2 \;\text{Mx}^2\right) \;\text{C}_0\left(k^2,\text{p2}^2,k^2-2 (k\cdot \;\text{p2})+\text{p2}^2,\text{Mx}^2,\text{Mx}^2,\text{Mphi}^2 \xi _{S(1)}\right)-i \pi ^2 g^3 ((\gamma \cdot k).(\gamma \cdot k)+2 \;\text{Mx} \gamma \cdot k) \;\text{C}_1\left(k^2,-2 (k\cdot \;\text{p2})+k^2+\text{p2}^2,\text{p2}^2,\text{Mx}^2,\text{Mx}^2,\text{Mphi}^2 \xi _{S(1)}\right)-i \pi ^2 g^3 ((\gamma \cdot k).(\gamma \cdot \;\text{p2})+2 \;\text{Mx} \gamma \cdot \;\text{p2}) \;\text{C}_1\left(\text{p2}^2,-2 (k\cdot \;\text{p2})+k^2+\text{p2}^2,k^2,\text{Mx}^2,\text{Mphi}^2 \xi _{S(1)},\text{Mx}^2\right)-\frac{1}{2} i g (2 \;\text{dZg}+\text{dZphi}+2 \;\text{dZx})
Discard all the finite pieces of the 1-loop amplitude
amp3Div[0] = PaVeUVPart[amp3[2], Prefactor -> 1/(2 Pi)^D] // DiracSimplify //
FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // Normal //
FCHideEpsilon // SelectNotFree2[#, {SMP["Delta"], dZphi,
dZx, dZg}] & // ReplaceAll[#, Join[solMSbar1, solMSbar2]] & //Simplify // FCFactorOut[#, g] &g \left(\frac{3 i \Delta g^2}{32 \pi ^2}-i \;\text{dZg}\right)
Equating this to zero and solving for dZg we obtain the renormalization constant in the minimal subtraction schemes.
solMSbar3 = FCMatchSolve[amp3Div[0], {g, SMP}]
solMS3 = solMSbar3 /. SMP["Delta"] -> 1/Epsilon;\text{FCMatchSolve: Solving for: }\{\text{dZg}\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{dZg}\to \frac{3 \Delta g^2}{32 \pi ^2}\right\}
amp4[1] = amp4[0] // ReplaceRepeated[#, {Zphi -> 1 + alpha dZphi,
Zla -> 1 + alpha dZla}] & //
Series[#, {alpha, 0, 1}] & // Normal // ReplaceAll[#, alpha -> 1] &-\frac{2 \;\text{tr}((\text{Mx}-\gamma \cdot l).(-i g).(\gamma \cdot (-l-\text{p2})+\text{Mx}).(-i g).(\gamma \cdot (-l-\text{p2}+\text{p4})+\text{Mx}).(-i g).(\gamma \cdot (-l-\text{p2}+\text{p3}+\text{p4})+\text{Mx}).(-i g))}{\left(l^2-\text{Mx}^2\right).\left((l+\text{p2})^2-\text{Mx}^2\right).\left((l+\text{p2}-\text{p4})^2-\text{Mx}^2\right).\left((l+\text{p2}-\text{p3}-\text{p4})^2-\text{Mx}^2\right)}-i \;\text{la} (\text{dZla}+2 \;\text{dZphi})
The result of the tensor reduction is quite large, since we keep the full gauge dependence and do not specify the kinematics
amp4[2] = TID[amp4[1], l, ToPaVe -> True, UsePaVeBasis -> True]4 i \pi ^2 \;\text{B}_0\left(\text{p2}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-4 i \pi ^2 \;\text{B}_0\left(\text{p2}^2-2 (\text{p2}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-4 i \pi ^2 \;\text{B}_0\left(\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2\right) g^4+4 i \pi ^2 \;\text{B}_0\left(\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-4 i \pi ^2 \;\text{C}_0\left(\text{p2}^2,\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) \left(4 \;\text{Mx}^2-\text{p2}\cdot \;\text{p4}\right) g^4-4 i \pi ^2 \;\text{C}_0\left(\text{p3}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) \left(4 \;\text{Mx}^2+\text{p2}\cdot \;\text{p3}-\text{p3}^2-\text{p3}\cdot \;\text{p4}\right) g^4-4 i \pi ^2 \;\text{C}_0\left(\text{p3}^2,\text{p4}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) \left(4 \;\text{Mx}^2+\text{p3}\cdot \;\text{p4}\right) g^4-4 i \pi ^2 \;\text{C}_0\left(\text{p2}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) \left(2 \;\text{Mx}^2-\text{p2}^2+\text{p2}\cdot \;\text{p3}+\text{p2}\cdot \;\text{p4}+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2\right) g^4-4 i \pi ^2 \;\text{D}_0\left(\text{p2}^2,\text{p4}^2,\text{p3}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p4})+\text{p4}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) \left(16 \;\text{Mx}^4-4 \;\text{p2}^2 \;\text{Mx}^2+4 (\text{p2}\cdot \;\text{p3}) \;\text{Mx}^2+4 (\text{p2}\cdot \;\text{p4}) \;\text{Mx}^2-4 \;\text{p3}^2 \;\text{Mx}^2-4 (\text{p3}\cdot \;\text{p4}) \;\text{Mx}^2-4 \;\text{p4}^2 \;\text{Mx}^2+(\text{p2}\cdot \;\text{p4}) \;\text{p3}^2-\text{p2}^2 (\text{p3}\cdot \;\text{p4})+2 (\text{p2}\cdot \;\text{p4}) (\text{p3}\cdot \;\text{p4})-(\text{p2}\cdot \;\text{p3}) \;\text{p4}^2\right) g^4-8 i \pi ^2 (\text{p2}\cdot \;\text{p3}+\text{p2}\cdot \;\text{p4}) \;\text{C}_1\left(\text{p2}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-8 i \pi ^2 \left(\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2\right) \;\text{C}_1\left(\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-8 i D \pi ^2 \;\text{C}_{00}\left(\text{p2}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-8 i \pi ^2 \;\text{p2}^2 \;\text{C}_{11}\left(\text{p2}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-8 i \pi ^2 \left(\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2\right) \;\text{C}_{11}\left(\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p2}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-16 i \pi ^2 (\text{p2}\cdot \;\text{p3}+\text{p2}\cdot \;\text{p4}) \;\text{C}_{12}\left(\text{p2}^2,\text{p2}^2-2 (\text{p2}\cdot \;\text{p3})-2 (\text{p2}\cdot \;\text{p4})+\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{p3}^2+2 (\text{p3}\cdot \;\text{p4})+\text{p4}^2,\text{Mx}^2,\text{Mx}^2,\text{Mx}^2\right) g^4-i (\text{dZla}+2 \;\text{dZphi}) \;\text{la}
Discard all the finite pieces of the 1-loop amplitude
amp4Div[0] = PaVeUVPart[amp4[2], Prefactor -> 1/(2 Pi)^D] // DiracSimplify //
FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // Normal //
FCHideEpsilon // SelectNotFree2[#, {SMP["Delta"], dZphi,
dZla}] & // ReplaceAll[#, Join[solMSbar1, solMSbar2]] & // Simplify-\frac{1}{2} i \left(2 \;\text{dZla} \;\text{la}+\frac{\Delta g^4}{\pi ^2}\right)
Equating this to zero and solving for dZg we obtain the renormalization constant in the minimal subtraction schemes.
solMSbar4 = FCMatchSolve[amp4Div[0], {g, SMP, la}]
solMS4 = solMSbar4 /. SMP["Delta"] -> 1/Epsilon;\text{FCMatchSolve: Solving for: }\{\text{dZla}\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{dZla}\to -\frac{\Delta g^4}{2 \pi ^2 \;\text{la}}\right\}
Join[solMSbar1, solMSbar2, solMSbar3, solMSbar4] // TableForm\begin{array}{l} \;\text{dZmx}\to \frac{3 \Delta g^2}{32 \pi ^2} \\ \;\text{dZx}\to -\frac{\Delta g^2}{32 \pi ^2} \\ \;\text{dZphi}\to 0 \\ \;\text{dZmphi}\to \frac{\Delta \;\text{la} \xi _{S(1)}}{32 \pi ^2} \\ \;\text{dZg}\to \frac{3 \Delta g^2}{32 \pi ^2} \\ \;\text{dZla}\to -\frac{\Delta g^4}{2 \pi ^2 \;\text{la}} \\ \end{array}