 Scalar bubble integral

ID: scalar-bubble-integral

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Scalar bubble integral by

Codex  0 Created 2026-10-05 Updated 2026-11-05

The loop momentum integral with two massive quantum field theory propagators is

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

Its momentum dependence gives a one-loop self-energy in cubic scalar field theory and a quartic effective vertex in quartic scalar field theory.

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