= Solution
Normalize the vacuum <path integral> by $Z[0]$. The scalar <two-point correlation function> is $Z[0]^{-1}\int\mathcal D\phi\,\phi(x)\phi(0)e^{iS_F}$, and the spinor <two-point correlation function> is $Z[0]^{-1}\int\mathcal D\psi\mathcal D\bar\psi\,\psi(x)\bar\psi(0)e^{iS_F}$. Vacuum boundary conditions make these time-ordered <Feynman propagators>. After <integration by parts>, the scalar quadratic <action> is $-\frac12\int\phi(-\partial^2+m^2)\phi$.
The <Schwinger-Dyson equation> follows by integrating a <functional derivative> of $\phi(0)e^{iS_F}$: the derivative of the insertion supplies $\delta^d(x)$, and the <action> derivative supplies the kinetic operator. The analogous <left Grassmann derivative> calculation, or differentiation of the <Grassmann Gaussian integral>, gives
$$
(-\partial^2+m^2)G_\phi(x)=-i\delta^d(x),\qquad(\gamma^\mu\partial_\mu+M)G_\psi(x)=-i\delta^d(x).
$$
Using $\partial_\mu\mapsto ip_\mu$ and the <Clifford algebra>, $(i\gamma\cdot p+M)(-i\gamma\cdot p+M)=p^2+M^2$. Consequently
$$
\boxed{G_\phi(p)=\frac{-i}{p^2+m^2-i0},\qquad G_\psi(p)=\frac{-i(-i\gamma\cdot p+M)}{p^2+M^2-i0}.}
$$
The free <quantum effective action> is quadratic, so all scalar <one-particle-irreducible vertices> with $n>2$ vanish. Its two-point vertex is the stated inverse kinetic form $-p^2-m^2$.
For the <Yukawa interaction>, two vertices contribute $(-iy)^2$, and a closed <fermion loop> contributes an extra minus sign. Tracing the two spinor numerators gives $4[M^2-k\cdot(k-p)]$, since the one-gamma traces vanish. Removing the overall $i$ from the amplitude gives the displayed loop integral. Define
$$
I(M)=\frac1{(2\pi)^di}\int\frac{d^dk}{k^2+M^2-i0},\quad
J(p;M)=\frac1{(2\pi)^di}\int\frac{d^dk}{(k^2+M^2-i0)((k-p)^2+M^2-i0)}.
$$
The numerator decomposition and translation invariance of <dimensional regularization> reduce it to
$$
\widehat\tau_2^{(1)}=-4y^2\left[\left(2M^2+\frac{p^2}2\right)J(p;M)-I(M)\right].
$$
The supplied tadpole <pole> is $I(M)\sim-2M^2/(\varepsilon16\pi^2)$. A <Feynman parameter> combines the two bubble denominators. Shifting its loop momentum gives mass squared $M^2+x(1-x)p^2$; differentiating the tadpole integral with respect to this squared mass gives the double-denominator <pole> $2/(\varepsilon16\pi^2)$, independent of $x$. Thus $J(p;M)\sim2/(\varepsilon16\pi^2)$ and
$$
\boxed{\widehat\tau_2^{(1)}\sim-\frac{y^2}{\varepsilon16\pi^2}(4p^2+24M^2),\qquad a=4,\quad b=24.}
$$
The <counterterms> contribute $-Ap^2-B$, so their minimal <pole> parts are
$$
A=-\frac{4y^2}{\varepsilon16\pi^2},\qquad B=-\frac{24y^2M^2}{\varepsilon16\pi^2}.
$$
Combining the kinetic terms gives <wavefunction renormalization> $Z_\phi=1+A$, and combining the mass terms gives $Z_\phi m_0^2=m^2+B$. Therefore
$$
\boxed{Z_\phi=1-\frac{4y^2}{\varepsilon16\pi^2},\qquad m_0^2=\frac{m^2+B}{Z_\phi}.}
$$
These are the <Yukawa scalar self-energy pole coefficients>; finite parts depend on the chosen <renormalization condition>.
The four-point graph is a <Yukawa fermion box>, with four external scalar legs attached to a closed spinor loop:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-44-yukawa-box.png]
{title=Fermion box with four external scalar legs in a Yukawa theory}
{height=350}
Each high-momentum <dirac propagator> is $O(k^{-1})$. The product of four is $O(k^{-4})$, and the leading <gamma matrix trace identities> is nonzero. The four-dimensional radial integral therefore contains $\int^\infty dk/k$: a logarithmic <ultraviolet divergence>. This local four-scalar divergence cannot be absorbed by scalar mass or field normalization. Add $-\lambda\phi^4/4!$, and, using the stipulated four-point <pole> normalization, take $\delta\lambda=-8y^4/(\varepsilon16\pi^2)$ so that $-\delta\lambda$ cancels it.
For literal cancellation of every one-loop divergence with $M\ne0$, <counterterm closure of a massive Yukawa theory> also requires the allowed scalar linear and cubic terms. A constant scalar background shifts the fermion mass to $M+y\phi$; the divergent local fermion contribution contains a polynomial proportional to $(M+y\phi)^4$. Its linear and cubic terms are not forbidden by a symmetry when the fermion mass is nonzero. A closed renormalizable family therefore has
$$
\mathcal L=-\frac12(\partial\phi)^2-\bar\psi(\gamma\cdot\partial+M)\psi-y\bar\psi\psi\phi-V(\phi),\qquad
V(\phi)=\Lambda+h\phi+\frac{m^2}2\phi^2+\frac{\kappa}{3!}\phi^3+\frac\lambda{4!}\phi^4,
$$
with field, mass and coupling redefinitions for both scalar and spinor fields. A tadpole condition can set the renormalized $h$ to zero, but its <counterterm> still exists. The vacuum constant is needed if vacuum energy is retained. If an exact discrete chiral symmetry is imposed with $M=0$, the scalar potential can be even and the odd terms are forbidden; the essential new interaction is then the quartic one.
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