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 Incoming links: Five-line banana integral

Past exam of the quantum field theory course of the University of Oxford / 2021 / 3 / c / ii / Solution 2026-10-25

To include all momentum-dependent contributions to the two-point function, put one external line at each interaction vertex. Let

r

be the number of lines connecting the vertices, and

t_{A}, t_{B}

their numbers of tadpole diagrams. For vertex valences

v_{A}, v_{B} \in {4, 6}

r + 2 t_{A} = v_{A} - 1, r + 2 t_{B} = v_{B} - 1.

(1)

The integer

r

is odd, and

r = 1

gives a one-particle-reducible Feynman diagram and is excluded. Thus the complete list is

$v_{A} = v_{B} = 4$ : $r = 3$ , $t_{A} = t_{B} = 0$ , the sunset diagram.
$v_{A} = 4, v_{B} = 6$ : $r = 3$ , $t_{A} = 0, t_{B} = 1$ , the sunset diagram with one tadpole diagram.
$v_{A} = v_{B} = 6$ : $r = 3$ , $t_{A} = t_{B} = 1$ , the sunset diagram with one tadpole diagram at each vertex.
$v_{A} = v_{B} = 6$ : $r = 5$ , $t_{A} = t_{B} = 0$ , the five-line banana integral topology.

The coefficient of the mixed graph includes the reflected external attachment as well as its reciprocal Feynman-diagram symmetry factor. If both external lines attach to the same interaction vertex, no external momentum enters the connecting internal lines, and that Feynman diagram cannot contribute to wave-function renormalization.

For incoming external momentum

p

, assign the connecting lines momenta

k_{1}, \dots, k_{r - 1}, p - \sum_{j} k_{j}

. Define

B_{r, M} (p) = \int \prod_{j = 1}^{r - 1} \frac{d ^{3} k _{j}}{( 2 π ) ^{3}} [\prod_{j = 1}^{r - 1} D (k_{j})] D (p - \sum_{j = 1}^{r - 1} k_{j}), r = 3, 5.

(2)

The momentum space Feynman rules give, in the order listed,

C_{44} (- i g_{1})^{2} B_{3, M} (p), C_{46} (- i g_{1}) (- i g_{2}) T_{M} B_{3, M} (p), C_{66, 3} (- i g_{2})^{2} T_{M}^{2} B_{3, M} (p), C_{66, 5} (- i g_{2})^{2} B_{5, M} (p) .

(3)

Each

C

is a reciprocal Feynman-diagram symmetry factor, including any equivalent external attachments. There are respectively

2, 3, 4, 4

independent loop momenta.

If “contribute to wavefunction renormalization” means requiring a divergent kinetic counterterm after subtraction of all subdivergences, only the last topology qualifies. The first three still have nontrivial external momentum dependence and give finite contributions to a normalization condition; their raw divergent factors arise from mass counterterms and interaction vertex counterterms. Keeping all four here makes this distinction explicit. The ultraviolet divergence analysis in d establishes it directly.

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Past exam of the quantum field theory course of the University of Oxford / 2021 / 3 / d / Solution 2026-10-25

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There are three independent nonfactorized momentum integrals, namely a tadpole integral, the three-line sunset diagram integral, and the five-line banana integral. The six displayed Feynman diagrams involve six complete integral expressions:

T, T^{2}, B_{3} (P), T B_{3} (P), T^{2} B_{3} (P), B_{5} (P) .

(1)

The distinction between independent building blocks and their products removes an ambiguity in counting “different momentum integrals”. For the strict divergent-wave-function renormalization interpretation of ii, the graphs in c reduce to

T, T^{2}, B_{5}

, comprising three full integral expressions and two independent nonfactorized building blocks.

Make a Wick rotation and use

m > 0

to separate ultraviolet divergences from infrared divergences. Write

D_{E} (k) = 1/ (k^{2} + m^{2})

, and let

T, B_{r} (P)

be the Euclidean versions of the integrals above, with their harmless overall factors of

i

removed. A spherical ultraviolet cutoff gives

T_{Λ} = \frac{1}{2 π ^{2}} \int_{0}^{Λ} \frac{k ^{2} d k}{k ^{2} + m ^{2}} = \frac{1}{2 π ^{2}} [Λ - m arctan (Λ/ m)] = \frac{Λ}{2 π ^{2}} - \frac{m}{4 π} + O (Λ^{- 1}) .

(2)

Thus the one-vertex quartic graph is linearly divergent, and the one-vertex sextic graph is quadratically divergent as

T_{Λ}^{2}

For the other two building blocks, the spatial Fourier transform is especially transparent. The three-dimensional Euclidean scalar propagator is

G (x) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{e ^{ik \cdot x}}{k ^{2} + m ^{2}} = \frac{e ^{- m ∣ x ∣}}{4 π ∣ x ∣} .

(3)

The convolution theorem converts the

r

parallel lines into

G (x)^{r}

. Angular integration gives, with a short-distance cutoff

a = Λ^{- 1}

B_{r}^{(a)} (P) = \frac{4 π}{( 4 π ) ^{r}} \int_{a}^{\infty} ρ^{2 - r} e^{- r m ρ} \frac{s i n ( Pρ )}{Pρ} d ρ, P = ∣ P_{E} ∣.

(4)

This is a different convenient cutoff regularization for the connected block; power-divergent coefficients depend on the cutoff convention, whereas the conclusions about powers and the presence of a kinetic logarithm do not. The sinc function is understood to be

1

P = 0

For

r = 3

, the integrand starts as

ρ^{- 1} + O (1)

, so

B_{3}^{(a)} (P) = \frac{1}{16 π ^{2}} lo g \frac{1}{ma} + O (1) .

(5)

Its divergent part is independent of external momentum. In particular,

B_{3} (P) - B_{3} (0) = \frac{1}{16 π ^{2}} \int_{0}^{\infty} \frac{e ^{- 3 m ρ}}{ρ} [\frac{s i n ( Pρ )}{Pρ} - 1] d ρ = - \frac{P ^{2}}{864 π ^{2} m ^{2}} + O (P^{4}) .

(6)

The finite difference and its finite derivative show why the pure three-line sunset diagram does not need a divergent wave-function renormalization in three dimensions.

For

r = 5

, the small-distance Taylor expansion is

ρ^{- 3} e^{- 5 m ρ} \frac{s i n ( Pρ )}{Pρ} = ρ^{- 3} - 5 m ρ^{- 2} + (\frac{25 m ^{2}}{2} - \frac{P ^{2}}{6}) ρ^{- 1} + O (1) .

(7)

Consequently

B_{5}^{(a)} (P)_{div} = \frac{1}{256 π ^{4}} [\frac{1}{2 a ^{2}} - \frac{5 m}{a} + (\frac{25 m ^{2}}{2} - \frac{P ^{2}}{6}) lo g \frac{1}{ma}] .

(8)

In particular, the nonzero

P^{2} lo g Λ

term requires a genuine divergent wave-function renormalization, in addition to the mass counterterm. The coefficients here refer to the displayed position-space cutoff and have not included Feynman-diagram symmetry factors or bare couplings.

The comparison with the superficial degree of divergence is

vertices and integral 4 : T 6 : T^{2} 4, 4 : B_{3} 4, 6 : T B_{3} 6, 6 : T^{2} B_{3} 6, 6 : B_{5} (n_{4}, n_{6}) (1, 0) (0, 1) (2, 0) (1, 1) (0, 2) (0, 2) ω 120122 leading unrenormalized growth Λ Λ^{2} lo g Λ Λ lo g Λ Λ^{2} lo g Λ Λ^{2}

(9)

Thus the actual leading power agrees with

ω

in every case. The additional logarithms in products come from ultraviolet subdivergences, which uniform power counting alone does not resolve.

For

T B_{3}

and

T^{2} B_{3}

, differentiation with respect to

P^{2}

removes the logarithmic divergence of

B_{3}

, but leaves divergent factors

T

T^{2}

multiplying a finite, nonpolynomial function of

P^{2}

. These are subdivergent contributions, not new nonlocal counterterms: subtracting each tadpole diagram subdivergence and the momentum-independent

B_{3}

subdivergence removes them. For example, in a factorized subtraction scheme

T B_{3} \mapsto (T - T_{div}) (B_{3} - B_{3, div})

, and likewise for

T^{2} B_{3}

. The remaining

B_{5}

kinetic logarithm explains the strict interpretation in ii. A dimensional regularization scheme can suppress power divergences; it does not invalidate this cutoff-based comparison with

ω

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