= Solution
Let the masses of $\Phi$ and $\phi$ be $M$ and $\mu$, respectively. Write their external <four-momenta> as $P=k_1+k_2$, with $P^2=M^2$ and $k_1^2=k_2^2=\mu^2$. The condition $M>2\mu$ opens the two-body decay <phase space>. Use mostly-minus <Minkowski spacetime>, the <Dirac propagator> $i(\not k+m)/(k^2-m^2+i0)$, and the convention that an amputated connected diagram contributes $i\mathcal M$.
There is no tree-level three-scalar <Feynman vertex> in the displayed interaction. The leading <decay amplitude> has one $\Phi$ <Yukawa interaction> and two $\phi$ <Yukawa interactions>, so is of order $Gg^2$. The <Yukawa fermion triangle> is:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-48-fermion-triangle.png]
{title=Fermion triangle for scalar decay with its loop-momentum routing and Yukawa vertex factors}
The two inequivalent cyclic orderings around the <fermion loop> are related by reversal and interchange of $k_1,k_2$. They give equal integrals: reversing all internal <four-momenta> leaves the masses and denominators unchanged, and the scalar <gamma matrix trace identities> below give the same numerator. Both must be included. The <closed fermion loop sign> supplies $-1$, and each ordering has the product
$$
(-1)(-iG)(-ig)^2i^3=-Gg^2.
$$
There is no extra $1/2!$ in the <decay amplitude>. Such a factor belongs to integration over the identical-particle final <phase space> when calculating a decay rate.
Route $a=\ell$, $b=\ell+k_1$, $c=\ell-k_2$ and define $D_a=a^2-m^2+i0$, and similarly for $D_b,D_c$. A regulated leading <decay amplitude> is
$$
\boxed{i\mathcal M_{\mathrm{loop}}=-2Gg^2\mu_R^{2\epsilon}
\int\frac{d^{4-2\epsilon}\ell}{(2\pi)^{4-2\epsilon}}
\frac{\operatorname{tr}[(\not a+m)(\not b+m)(\not c+m)]}{D_aD_bD_c}.}
$$
Here <dimensional regularization> preserves translation of the <loop momentum>, and $\mu_R$ is its reference mass scale. Factors $\mu_R^{2\epsilon}$ account for the loop's dimension; dimensionally continued couplings can equivalently carry these powers. No $\gamma^5$ occurs in this scalar loop, so there is no $\gamma^5$ regularization ambiguity.
Using <gamma matrix trace identities>, traces with an odd number of <gamma matrices> vanish, while $\operatorname{tr}1=4$ and $\operatorname{tr}(\gamma^\mu\gamma^\nu)=4g^{\mu\nu}$. Thus the numerator simplifies completely to
$$
\begin{aligned}
N&=4m\bigl[m^2+a\cdot b+a\cdot c+b\cdot c\bigr]\\
&=4m\bigl[m^2+3\ell^2+2\ell\cdot(k_1-k_2)-k_1\cdot k_2\bigr],
\qquad k_1\cdot k_2=\frac{M^2-2\mu^2}{2}.
\end{aligned}
$$
For a reduction to standard scalar integrals, disregard the vanishing infinitesimals only in the numerator. Since $D_a+D_b+D_c=3\ell^2+2\ell\cdot(k_1-k_2)+2\mu^2-3m^2$ up to those infinitesimals,
$$
N=4m\left[D_a+D_b+D_c+4m^2-\mu^2-\frac{M^2}{2}\right].
$$
Define the scalar bubble and <scalar triangle Feynman integral> by
$$
B(s)=\mu_R^{2\epsilon}\int\frac{d^{4-2\epsilon}\ell}{(2\pi)^{4-2\epsilon}}
\frac1{(\ell^2-m^2+i0)((\ell+q)^2-m^2+i0)},\quad q^2=s,
\qquad
C=\mu_R^{2\epsilon}\int\frac{d^{4-2\epsilon}\ell}{(2\pi)^{4-2\epsilon}}\frac1{D_aD_bD_c}.
$$
Cancelling each denominator in turn gives two bubbles with external invariant $\mu^2$ and one with invariant $M^2$. Therefore
$$
\boxed{i\mathcal M_{\mathrm{loop}}=-8mGg^2\left[2B(\mu^2)+B(M^2)+\left(4m^2-\mu^2-\frac{M^2}{2}\right)C\right].}
$$
This explicitly displays the complete <gamma matrix> trace simplification, the two loop orientations and the overall phase convention.
There is a <renormalization> qualification to this answer. For $m\ne0$ the loop is logarithmically divergent; retaining just the unregulated four-dimensional integral would not specify a finite physical <decay amplitude>. To see the divergence, a <Feynman parameter> and a shifted <loop momentum> give
$$
B(s)=\frac{i}{16\pi^2}\left[\frac1{\bar\epsilon}-\int_0^1dx\,\log\frac{m^2-x(1-x)s-i0}{\mu_R^2}\right]+O(\epsilon),
\qquad \frac1{\bar\epsilon}=\frac1\epsilon-\gamma_E+\log(4\pi).
$$
The <scalar triangle Feynman integral> is ultraviolet finite. Hence
$$
\mathcal M_{\mathrm{loop,div}}=-\frac{24mGg^2}{16\pi^2\bar\epsilon}.
$$
A <cubic counterterm from a Yukawa fermion triangle>, $\mathcal L_{\mathrm{ct}}=-\delta h\,\Phi\phi^2/2$, contributes $-\delta h$ to $\mathcal M$ and cancels the pole with $\delta h_{\mathrm{div}}=-24mGg^2/(16\pi^2\bar\epsilon)$. This is an instance of <counterterm closure of a massive Yukawa theory>. After subtraction the physical answer is $\mathcal M=-h_R+\mathcal M_{\mathrm{loop,ren}}$, where a <renormalization condition> fixes the finite cubic coupling $h_R$. The displayed interaction alone does not supply that condition. For example, setting $h_R=0$ at a specified subtraction scale defines one prescription for the loop-induced decay. The regulated expressions above remain the lowest-order loop answer without such extra data. For $m=0$ the trace, and therefore this scalar triangle contribution, vanishes identically.
Now take the <Hermitian> <pseudoscalar Yukawa interaction> with the sign printed in the PDF. Varying with respect to the <Dirac adjoint> gives the <Dirac equation>
$$
\boxed{(i\gamma^\mu\partial_\mu-m-g\phi+iG\Phi\gamma^5)\psi=0.}
$$
Varying with respect to $\psi$ and integrating by parts gives the <adjoint Dirac equation>
$$
i(\partial_\mu\bar\psi)\gamma^\mu+(m+g\phi)\bar\psi-iG\Phi\bar\psi\gamma^5=0.
$$
The factor $i$ in the coupling matters: $\bar\psi\gamma^5\psi$ is anti-Hermitian, so this factor makes the interaction <Hermitian> for real $G$.
For the <axial-current divergence for scalar and pseudoscalar backgrounds>, set $S=m+g\phi$ and $H=G\Phi$. The two equations are equivalently
$$
\gamma^\mu\partial_\mu\psi=-iS\psi-H\gamma^5\psi,\qquad
(\partial_\mu\bar\psi)\gamma^\mu=iS\bar\psi+H\bar\psi\gamma^5.
$$
In the divergence of the <axial current>, use $\{\gamma^5,\gamma^\mu\}=0$ and $(\gamma^5)^2=1$:
$$
\begin{aligned}
\partial_\mu(\bar\psi\gamma^\mu\gamma^5\psi)
&=(\partial_\mu\bar\psi)\gamma^\mu\gamma^5\psi
-\bar\psi\gamma^5\gamma^\mu\partial_\mu\psi\\
&=(iS\bar\psi+H\bar\psi\gamma^5)\gamma^5\psi
-\bar\psi\gamma^5(-iS\psi-H\gamma^5\psi).
\end{aligned}
$$
Thus
$$
\boxed{\partial_\mu j_5^\mu=2i(m+g\phi)\bar\psi\gamma^5\psi+2G\Phi\bar\psi\psi.}
$$
This is the classical field-equation identity requested here. It includes explicit breaking by both backgrounds; it is not a claim that a renormalized quantum <axial current> has no <chiral anomaly> when gauge interactions are present.
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