Use two external slow-field legs and internal fast-mode propagators. Through the requested orders, the connected mass-correction topologies are:

order $g_{0}$ : one quartic vertex with one fast tadpole;
order $g_{0}^{2}$ : two quartic vertices joined either by three fast lines, or by two fast lines with a fast tadpole on the vertex carrying no external legs;
order $λ_{0}$ : one sextic vertex with two fast tadpole loops;
order $λ_{0} g_{0}$ : a sextic and a quartic vertex joined by two fast lines, with the remaining fast legs closed into tadpoles in the two possible external-leg placements; joined by four fast lines with two external legs on the sextic vertex; or joined by three fast lines with one sextic tadpole and one external leg on each vertex.

A nominal one-line bridge at order

λ_{0} g_{0}

vanishes in a sharp momentum-shell scheme at small external momentum because that line would have to carry momentum outside the fast shell. These descriptions specify the same diagrams without depending on a particular drawing convention for vertices and external legs.

ii

 0  0

Solution

 0  0

Write

I_{j} = \int_{Λ/ ζ < ∣ k ∣ < Λ} \frac{d ^{d} k}{( 2 π ) ^{d}} G_{0} (k)^{j}, G_{0} (k) = \frac{1}{k ^{2} + μ _{0}^{2}},

(1)

and, for the three-line topology,

J_{3} = \int_{+} \frac{d ^{d} k d ^{d} q}{( 2 π ) ^{2 d}} G_{0} (k) G_{0} (q) G_{0} (k + q),

(2)

and define the four-line integral

J_{4} = \int_{+} \frac{d ^{d} k d ^{d} q d ^{d} ℓ}{( 2 π ) ^{3 d}} G_{0} (k) G_{0} (q) G_{0} (ℓ) G_{0} (k + q + ℓ),

(3)

with every propagator momentum restricted to the fast shell.

The first cumulant contains

λ_{0} (2 6) ⟨(ϕ^{+})^{4} ⟩_{+} (ϕ^{-})^{2} = 45 λ_{0} I_{1}^{2} (ϕ^{-})^{2} .

(4)

Since the quadratic free energy is

\frac{1}{2} μ^{2} (ϕ^{-})^{2}

, this gives

δ μ^{2}_{λ_{0}} = 90 λ_{0} I_{1}^{2} .

(5)

The mixed term in the second cumulant is

- (⟨ V_{6} V_{4} ⟩_{+} - ⟨ V_{6} ⟩_{+} ⟨ V_{4} ⟩_{+}) .

(6)

Wick contraction with the printed normalization

V_{6} = λ_{0} \int ϕ^{6}

and

V_{4} = g_{0} \int ϕ^{4}

gives, at zero external momentum,

δ μ^{2}_{λ_{0} g_{0}} = - 2 λ_{0} g_{0} (1620 I_{1}^{2} I_{2} + 1440 I_{1} J_{3} + 360 J_{4}) .

(7)

The coefficients respectively combine the two placements of both external legs in the two-line topology, the three-line topology with one external leg on each vertex, and the four-line topology. Couplings normalized as