Solution
= Solution
Keep the mostly-plus <Minkowski metric> and the <Fourier transform> $e^{ip\cdot x}$. Write $\not p=\gamma^ap_a$, with $\{\gamma^a,\gamma^b\}=2\eta^{ab}$. To fix the otherwise unspecified phase of $\gamma^5$ and the <Dirac adjoint>, take
$$
\gamma^a=-i\Gamma^a,\qquad
\bar\psi=-i\psi^\dagger\gamma^0,\qquad
\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3,
$$
where $\Gamma^a$ are ordinary mostly-minus <gamma matrices>. Then $(\gamma^5)^2=-1$, $\{\gamma^5,\gamma^a\}=0$, and the fermionic time-derivative term is $i\psi^\dagger\partial_t\psi$. In these conventions the interaction with real $g$ is Hermitian: in mostly-minus notation it is $ig\bar\psi_{\rm std}\Gamma^5\psi\phi$. If one instead calls $i\gamma^5$ the square-one <chirality> matrix, its coefficient must be $-ig$ to represent the same interaction. These phase choices leave physical <relativistic scattering cross-sections> unchanged.
Expanding $e^{iS_{\rm int}}$ yields the following <Feynman rules> with relativistically normalized external states:
* A <real scalar field> line carrying <four-momentum> $r$ contributes $-i/(r^2+m^2-i0)$.
* An oriented <Dirac field> line contributes the <Dirac propagator>
$$
S_F(r)=\frac{i(M-i\not r)}{r^2+M^2-i0}.
$$
This follows from $(i\not r+M)(M-i\not r)=(r^2+M^2)I$.
* Each <pseudoscalar Yukawa interaction> vertex has one scalar leg, one incoming fermion arrow and one outgoing fermion arrow, and contributes $ig\gamma^5$. It also contributes $(2\pi)^4\delta^{(4)}(\sum r)$ with all vertex momenta counted incoming.
* External incoming particles contribute $u(p,s)$ and outgoing particles $\bar u(p,s)$. External incoming <antiparticles> contribute $\bar v(p,s)$ and outgoing <antiparticles> $v(p,s)$, with the corresponding fermion arrows. External scalar factors are one. Choose $u^\dagger u=v^\dagger v=2E$, $(i\not p+M)u=0$, and $(-i\not p+M)v=0$; all external momenta are on the appropriate <mass shell>.
* Integrate each independent loop <four-momentum> with $d^4r/(2\pi)^4$. Keep matrix factors in their order along a fermion line, take a trace around a closed fermion loop, and include a minus sign for each closed fermion loop. Permuting external identical <fermions> contributes the corresponding <fermionic sign>; the <Wick theorem> determines the <Feynman-diagram symmetry factors>.
Strip the overall <four-momentum conservation> delta function when defining the <scattering amplitude>. There are no further bare interaction vertices, no <gauge fixing> and no <Faddeev-Popov ghost fields> in this theory. Renormalized higher-order calculations add the required <counterterms>; these are additional to the rules of the displayed classical <Lagrangian density>.