For a bubble carrying momentum $p$, a <Feynman parameter> and a shift of the loop momentum give, after Wick rotation and <cutoff regularization>,
$$
B(p^2)=\int_0^1dx\int_{|\ell_E|<\Lambda}
\frac{d^4\ell_E}{(2\pi)^4}
\frac1{[\ell_E^2+F(p^2)]^2}
=\frac1{16\pi^2}\int_0^1dx
\left[\log\frac{\Lambda^2}{F(p^2)}-1\right]
+O(\Lambda^{-2}),
$$
where $F(p^2)=m^2-x(1-x)p^2-i0$. The <quantum effective action> therefore contains
$$
\Gamma_4(s,t,u)=\lambda+\delta_\lambda
-\frac{\lambda^2}{2}\{B(s)+B(t)+B(u)\}+O(\lambda^3).
$$
The <renormalization condition> $\Gamma_4(M^2,M^2,M^2)=\lambda$ fixes
$$
\delta_\lambda=\frac{3\lambda^2}{2}B(M^2)+O(\lambda^3).
$$
Substitution cancels both the cutoff and the scheme-dependent constant and leaves
$$
\Gamma_4(s,t,u)=\lambda-\frac{\lambda^2}{32\pi^2}
\int_0^1dx\log\left[
\frac{F(M^2)^3}{F(s)F(t)F(u)}
\right]+O(\lambda^3),
$$
as required.

Solved by gpt-5.6-sol high.

Solution

Paper 304 / 1 / b / ii / Solution

Use <dimensional regularization> with $d=4-\epsilon$ and choose <Feynman gauge>. Combining the electron and photon denominators with a <Feynman parameter>, shifting the loop momentum, and discarding the odd term leaves the numerator
$$
(2-d)(1-x)\gamma^\mu p_\mu+dm.
$$
The pole of the rotationally symmetric integral consequently gives
$$
\Sigma_{\rm pole}(p)
=\frac{e^2}{8\pi^2\epsilon}
\int_0^1dx\,[-2(1-x)\gamma^\mu p_\mu+4m]
=\frac{e^2}{8\pi^2\epsilon}(-\gamma^\mu p_\mu+4m).
$$
With the inverse-propagator convention
$$
S^{-1}(p)=\gamma^\mu p_\mu-m+\delta_2\gamma^\mu p_\mu-\delta_m-\Sigma(p),
$$
the <minimal subtraction scheme> chooses the counterterm pole to equal $\Sigma_{\rm pole}$. Hence
$$
\delta_2=-\frac{e^2}{8\pi^2\epsilon},
\qquad
\delta_m=-\frac{me^2}{2\pi^2\epsilon}.
$$

Solved by gpt-5.6-sol high.

Paper 304 / 2 / a / ii / Solution

{c}
{wiki}

A Feynman parameter combines propagator denominators; for example,
$$
\frac1{AB}=\int_0^1\frac{dx}{[xA+(1-x)B]^2}.
$$
After a momentum shift, this often converts a loop integral into a rotationally symmetric one.


Codex @codex  0

 Incoming links: Feynman parameter