Integral from Lancaster's QFT In Chapter 32 of Lancaster's Quantum Field Theory for the Gifted Amateur, renormalization is discussed. The amplitudes of various one-loop Feynman diagrams which are corrections to the vertex in a $\phi^4$ scalar field theory are made finite by imposing a momentum cutoff. Lancaster writes
$$\int_0^{\Lambda} \frac{d^4q}{(2\pi)^4)} 
\frac{i}{q^2 - m^2 + i \epsilon} 
\frac{i}{(p-q)^2 - m^2 + i \epsilon} 
= - 4i a \ln\left(\frac{\Lambda}{p}\right),$$
where $a$ is "a numerical constant whose exact value isn't important to us." First of all, I would like to know the exact value anyway. Secondly, what exactly do the bounds of the integral mean here? Is each component of the four-momentum cut off at $\Lambda$, or is it just the magnitude? I would like to know how to actually get the result Lancaster gives.
 A: To answer your second question, $\Lambda$ is the Euclidean cut-off. That is, the Euclidean momentum-squared is constrained by
$$ q_0^2 + \vec{q}^2 \le \Lambda^2. $$
In actual computations you probably want to Wick-rotate the formal diverging integral first, and then impose the constraint on the Euclidean momentum.
Before you ask: this expression is not Lorentz-invariant. Momentum cutoff breaks Lorentz invariance, and we will have to explicitly check the results of the theory for Lorentz-invariance after renormalization (spoiler alert: Lorentz invariance turns out to be unbroken in $\phi^4$ theory at arbitrary level of loops).
To partially answer your first question. I don't know the exact value of $a$. It can be calculated explicitly:


*

*You Wick-rotate the $dq_0$ integral (deform the integration contour in the complex plane passing to the Euclidean momentum).

*You use Feynman's parameterization to account for $p$. Ideally you want to pass to another integration variable $k_{\mu}$ such that $p$ no longer appears in the integral. Of course it still appears in the result because the change of variables itself is dependent on $p$.

*Note that the second step is not straightforward, because the change of variables introduces nontrivial $p$-dependent regularization constraints on $k$ (unlike the constraint on $q$). But since you only want to compute this in $\Lambda \gg p$ limit, you can just ignore this issues and use the same constraint as you would for $q$:
$$ k_0^2 + \vec{k}^2 \le \Lambda^2. $$

*You pass to spherical coordinates in $\mathbb{R}^4$ and take the integral, which would give you the value of $a$.

