g_{2 k}

has one

2 k

-valent interaction vertex. Attach the two external lines to it and pair its remaining

2 k - 2

legs into

k - 1

tadpole diagrams. The graph is connected, has

L = k - 1

independent loop momenta, and is momentum independent after amputation apart from the overall Dirac delta function. In particular, first order in the coupling need not mean one loop.

The number of contractions with fixed external labels is

(2 k) (2 k - 1) \frac{( 2 k - 2 )!}{2 ^{k - 1} ( k - 1 )!} .

Dividing by the interaction factorial

(2 k)!

1/ [2^{k - 1} (k - 1)!]

. Use

m > 0

for the mass in the lowest-order propagator and define the tadpole integral

T_{D} (m) = \int \frac{d ^{D} q}{( 2 π ) ^{D}} \frac{1}{q ^{2} + m ^{2}} .

The linear-order correction to the amputated function is consequently

Γ_{lin}^{(2)} (p_{1}, p_{2}) = - (2 π)^{D} δ^{(D)} (p_{1} + p_{2}) \frac{μ ^{d_{k}} g _{2 k}}{2 ^{k - 1} ( k - 1 )!} T_{D} (m)^{k - 1} .

This is the correction to the paper's amputated connected correlator, not a definition of an inverse-propagator self-energy with an unspecified sign. At this order using

m_{0}

instead of

m

gives the same lowest-order integral.

T_{D} (m) = \frac{1}{( 4 π ) ^{D /2}} \int_{0}^{\infty} d α α^{- D /2} e^{- α m^{2}} = \frac{( m ^{2} ) ^{D /2 - 1}}{( 4 π ) ^{D /2}} Γ (1 - D /2) .

This first converges for

0 < D < 2

D = 4 - 2 ϵ

, the Gamma function expansion yields

μ^{2 ϵ} T_{D} (m) = - \frac{m ^{2}}{16 π ^{2}} [\frac{1}{ϵ} + 1 - γ_{E} + lo g \frac{4 π μ ^{2}}{m ^{2}} + O (ϵ)] .

Thus its highest ultraviolet pole is

Γ_{lin}^{(2)} = (2 π)^{D} δ^{(D)} (p_{1} + p_{2}) \frac{( - 1 ) ^{k} g _{2 k} μ ^{4 - 2 k} ( m ^{2} ) ^{k - 1}}{2 ^{k - 1} ( k - 1 )! ( 16 π ^{2} ) ^{k - 1} ϵ ^{k - 1}} + O (ϵ^{- (k - 2)}) .

For

k = 2

the remainder means finite terms. For

k > 2

the unrenormalized graph has a higher-order pole from its independent ultraviolet subdivergences, rather than only a simple pole. A massless tadpole integral is scaleless and vanishes in dimensional regularization; that does not justify discarding the massive ultraviolet divergence by setting