Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 3 / b

Table of contents
- Solution b

Solution

Put

d = 6 - ϵ

. The one-loop two-point bubble has symmetry factor

1/2

and, after a Feynman parameter and momentum shift, its pole is proportional to

- \frac{2}{( 4 π ) ^{3} ϵ} \int_{0}^{1} d x [m^{2} + x (1 - x) p_{E}^{2}] = - \frac{2}{( 4 π ) ^{3} ϵ} (m^{2} + \frac{p _{E}^{2}}{6}) .

Cancelling the pole in the minimal subtraction scheme gives, with

ϵ = 6 - d

,

δ_{Z} = - \frac{λ ^{2}}{6 ( 4 π ) ^{3} ϵ}, δ_{m^{2}} = - \frac{λ ^{2} m ^{2}}{( 4 π ) ^{3} ϵ} .

If one defines dimensional regularization by

d = 6 - 2 ϵ

, these same poles are written with

2 ϵ

in place of

ϵ

.

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