One-loop four-point amplitudes

Define the scattering matrix through $S=1+iT$ and the connected matrix element of T by a momentum-conserving delta function times $\mathcal M$. The interaction $-\lambda\phi^4/4!$ gives tree amplitude $-\lambda$. At one loop, joining two quartic vertices by two internal lines produces a bubble in each of the s, t, and u channels. Each labeled channel has the same factor one half, while its scalar integral depends on the corresponding momentum invariant.

In $d=4-2\epsilon$, write the bare quartic coupling as $\lambda_B=\mu^{2\epsilon}(\lambda+\delta\lambda)$. We display the reduced amplitude after factoring one overall $\mu^{2\epsilon}$, and use the scalar loop measure $\mu^{2\epsilon}d^dk/(2\pi)^d$ inside its coefficient. This convention avoids multiplying the scale factor twice. After subtraction and the four-dimensional limit, the amplitude is dimensionless. For on-shell equal masses, the invariants satisfy $s+t+u=4m^2$, so setting all three to zero is not a physical scattering configuration.

The three bubble poles do not depend on the external invariants, so a single local quartic counterterm cancels their sum. In modified minimal subtraction, its value is three times the common channel pole coefficient. It cannot remove arbitrary momentum-dependent logarithms without changing the theory’s local operator content. A momentum-subtraction scheme instead fixes a coupling by a chosen off-shell vertex condition and generally differs by finite terms. Translating the coupling definition is essential when comparing finite amplitudes between schemes.

Define a finite channel function as minus the parameter integral of the causal logarithm. The renormalized amplitude is the tree term plus the sum of those three functions with their common coefficient. Permuting s, t, and u leaves this identical-scalar expression unchanged. Above the two-particle threshold a channel can become complex. A real Euclidean matching point is useful for defining parameters, but its vertex should not be reported as an on-shell cross section without analytic continuation and the remaining scattering factors.

At fixed external invariants and mass, each finite channel has explicit derivative two with respect to logarithm of mu. Three channels therefore generate a positive explicit scale derivative proportional to six times their common coefficient. The tree term changes because the renormalized coupling runs. Its one-loop beta function cancels that derivative. Changing lambda inside the loop term would contribute only at the next perturbative order in this check, so it is consistently omitted here.

Residual scale dependence after truncation is a higher-order effect, not a license to treat the scale as an independently measured parameter. Its variation can diagnose sensitivity but is not a rigorous complete uncertainty bound. Also, a one-loop correction to a probability involves interference between tree and loop amplitudes; squaring only the loop part starts at a different order. The wave lab illustrates coherent addition and cancellation, with two fixed-frequency classical components rather than scalar scattering channels or renormalization-group evolution.