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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.

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  • $\begingroup$ Hi Klein Four. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Nov 12 '16 at 7:44
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    $\begingroup$ Thanks for letting me know. This question is in no way a homework problem. The textbook simply cites this result, and then uses it for the rest of the chapter. I'd just like to see how this result is found. This is purely for my own edification. I am a student, but I'm not even in a class which covers this material. $\endgroup$ – Klein Four Nov 12 '16 at 8:13
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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:

  1. You Wick-rotate the $dq_0$ integral (deform the integration contour in the complex plane passing to the Euclidean momentum).
  2. 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 $p$.
  3. 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 $q$: $$ k_0^2 + \vec{k}^2 \le \Lambda^2. $$
  4. You pass to spherical coordinates in $\mathbb{R}^4$ and take the integral, which would give you the value of $a$.
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