Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 304 / 2

Table of contents
- Solution 2

Solution

For incoming momenta, the Euclidean momentum-space Feynman rules are

a $ϕ$ propagator $(p^{2} + m_{0}^{2})^{- 1}$ ;
a $χ$ propagator $(p^{2} + M_{0}^{2})^{- 1}$ ;
a vertex with two $ϕ$ legs and one $χ$ leg equal to $- λ_{0} (2 π)^{d} δ^{(d)} (p_{1} + p_{2} + p_{3})$ ;
one integration $\int d^{d} ℓ / (2 π)^{d}$ for each loop momentum, with division by the Feynman-diagram symmetry factor.

The one-loop one-particle-irreducible diagrams with at most three external legs can be classified by their external species. There is a one-point

χ

tadpole diagram made from a

ϕ

loop. There are two two-point bubble diagrams: the

ϕ

self-energy has one internal

ϕ

and one internal

χ

, while the

χ

self-energy has two internal

ϕ

lines and symmetry factor

1/2

. For three external legs, three vertices make a triangle: one triangle corrects the

ϕϕ χ

vertex and contains two internal

ϕ

lines and one internal

χ

line; another has three external

χ

legs and a

ϕ

loop, generating a

χ^{3}

interaction. The symmetry

ϕ \mapsto - ϕ

forbids amplitudes with an odd number of external

ϕ

legs. The classical

ϕϕ χ

vertex is the corresponding tree-level three-point diagram.

Adopt the self-energy convention

G^{(2)} (p) = \frac{1}{p ^{2} + m ^{2} - Π ( p ^{2} )} .

(1)

This follows by summing the geometric series of exact propagators separated by amputated one-particle-irreducible two-point insertions. At one loop, after writing

λ_{0} = μ^{ϵ /2} λ + O (λ^{3})

Π_{1} (p^{2}) = λ^{2} μ^{ϵ} \int \frac{d ^{6 - ϵ} ℓ}{( 2 π ) ^{6 - ϵ}} \frac{1}{( ℓ ^{2} + m ^{2} ) (( ℓ + p ) ^{2} + M ^{2} )} .

(2)

Introduce a Feynman parameter and shift the loop momentum. With

Δ (x) = x m^{2} + (1 - x) M^{2} + x (1 - x) p^{2},

(3)

dimensional regularization gives

Π_{1} (p^{2}) = \frac{λ ^{2} μ ^{ϵ}}{( 4 π ) ^{3 - ϵ /2}} Γ (- 1 + \frac{ϵ}{2}) \int_{0}^{1} d x Δ (x)^{1 - ϵ /2} .

(4)

Using

Γ (- 1 + ϵ /2) = - 2/ ϵ + γ - 1 + O (ϵ)

produces

Π_{1} (p^{2}) = \frac{1}{ϵ} (A + B p^{2}) + C (p^{2}, μ) + O (ϵ),

(5)

where

A = - \frac{λ ^{2}}{( 4 π ) ^{3}} (m^{2} + M^{2}), B = - \frac{λ ^{2}}{3 ( 4 π ) ^{3}}

(6)

and one convenient integral form of the finite part is

C (p^{2}, μ) = \frac{λ ^{2}}{( 4 π ) ^{3}} \int_{0}^{1} d x Δ (x) [lo g \frac{Δ ( x )}{4 π μ ^{2}} + γ - 1] .

(7)

Changing the definition of the dimensional-regularization scale only moves a finite constant between

C

and the counterterm.

To make the two-point function finite, write

ϕ_{0} = Z_{ϕ}^{1/2} ϕ

, express

m_{0}^{2}

in terms of a renormalized mass and a mass counterterm, and choose the pole parts of

δ Z_{ϕ}

and

δ m^{2}

to cancel

B p^{2} / ϵ

and

A / ϵ

. In the minimal subtraction scheme no additional finite pieces are removed. The physical mass is the pole mass, so with the self-energy convention above it obeys

m_{phys}^{2} = m^{2} (μ) - Π_{MS, fin} (- m_{phys}^{2}, μ) .

(8)

The explicit

μ

dependence of the finite self-energy cancels the running of

m (μ)

, leaving

m_{phys}

independent of the renormalization scale.

The superficial degree of divergence counts the ultraviolet power before subdivergences and symmetry cancellations are considered. At

d = 6

, a connected graph made from cubic vertices has

D = 6 L - 2 I = 6 - 2 E,

(9)

where

E

is its number of external legs. Hence one-, two-, and three-point functions can have quartic, quadratic, and logarithmic superficial divergences, whereas graphs with more external legs are superficially convergent.

Full renormalization also requires the