Cutkosky cuts and perturbative unitarity

Write $S=1+iT$ and normalize one-particle states by $\langle\mathbf p|\mathbf q\rangle=2E_{\mathbf p}(2\pi)^3\delta^3(\mathbf p-\mathbf q)$. Removing the overall four-momentum delta function defines the invariant amplitude. Expanding $S^\dagger S=1$ gives a relation between the anti-Hermitian part of T and $T^\dagger T$. For a forward matrix element, it becomes twice the imaginary part of the forward amplitude equal to a sum of absolute squares over physical intermediate states.

At a given perturbative order, the right side uses amplitudes of lower orders whose product reaches that order. In quartic scalar theory, tree amplitudes are order lambda, so their squared contribution constrains the order-lambda-squared absorptive part. The intermediate-state sum includes species, polarizations, momentum integrals, and symmetry factors for indistinguishable particles. Counting two identical scalar momenta as ordered without division by two would double the result. This state-counting factor has the same numerical value as the bubble coefficient here, but its origin should still be tracked independently.

The distribution identity for a scalar denominator states that its discontinuity across the pole is minus two pi i times a delta function. For a physical cut, energy flow through the cut selects the positive-energy part. Multiplying by the propagator’s numerator i turns that replacement into a positive two-pi delta factor before the remaining diagram factors and conjugate subamplitudes are assembled. One must not apply a mnemonic replacement while dropping the prescription or reversing the cut orientation silently.

The Lorentz-invariant two-particle phase space integrates two positive-energy measures and one four-momentum delta function. For equal masses and timelike total momentum squared s, its integral is beta divided by eight pi when the two momenta are treated as distinguishable labels. An identical pair requires an additional factorial division in the intermediate-state sum. The phase-space square root vanishes at the same threshold as the bubble logarithm. This equality is a physical consistency test of signs and normalization, not a numerical coincidence.

The scalar s-channel loop contributes $\lambda^2F(s)/(32\pi^2)$ to the finite amplitude. Its imaginary part is therefore lambda squared times beta divided by thirty-two pi. Twice that quantity equals one half times the two-body phase space times the squared tree amplitude. In physical forward equal-mass scattering, the t and u channels do not acquire this same s-channel two-particle cut. A real ultraviolet counterterm also has no absorptive contribution to this check.

A cut constrains a discontinuity; it does not determine every real subtraction constant of an amplitude. Dispersion relations need appropriate subtraction information to reconstruct the real part. Gauge theories add the requirement that only physical states survive the complete unitarity sum, even when an intermediate gauge-fixed calculation uses additional fields. The Bloch lab demonstrates norm conservation for one closed pure two-state model. It neither sums multiparticle phase space nor implements Cutkosky rules or an interacting scattering matrix.