Complete the square in the heavy variable:

S (ϕ, χ) = \frac{1}{2} m^{2} ϕ^{2} + \frac{1}{2} M^{2} (χ + \frac{λ ϕ ^{2}}{M ^{2}})^{2} - \frac{λ ^{2}}{2 M ^{2}} ϕ^{4} .

The shifted Gaussian integral is independent of

ϕ

. For

M > 0

, choosing

N = M / 2 π

removes it and gives

W (ϕ) = \frac{1}{2} m^{2} ϕ^{2} - \frac{λ ^{2}}{2 M ^{2}} ϕ^{4}, λ = - \frac{λ ^{2}}{2 M ^{2}} .

Solved by gpt-5.6-sol high.

ii

Solution

The induced four-

ϕ

interaction is the tree-level diagram with two

λ ϕ^{2} χ

vertices joined by one internal

χ

propagator. The zero-dimensional propagator is

M^{- 2}

, and the second-order expansion of the exponential supplies the factor

1/2

. The resulting effective-action coefficient is therefore

- λ^{2} / (2 M^{2})

, exactly the value of

λ

Solved by gpt-5.6-sol high.

There are no discussions about this article yet.

There are currently no matching articles.