Electroweak unitarity in vector-boson scattering

In the leading nonlinear Goldstone theory with the Higgs omitted, the massless amplitude function is s divided by v squared. Custodial symmetry relates its crossed versions. Combining three times the direct function with the two crossed functions gives the isospin-zero amplitude. Since the massless Mandelstam variables sum to zero, this combination reduces to twice s divided by v squared and is independent of scattering angle at this order.

For the custodial isospin amplitudes, use the partial-wave convention with an angular integral divided by sixty-four pi. The zero-angular-momentum projection is then s divided by sixteen pi v squared. This normalization differs from the frequently quoted thirty-two-pi integral for a distinguishable single-channel amplitude; state normalization and channel structure must travel with the formula. Applying a bound to one transition amplitude without identifying its normalized eigenchannel can otherwise introduce an erroneous factor.

Write an elastic eigenchannel S-matrix as one plus twice i times its partial-wave amplitude. Exact elastic unitarity places that amplitude on an Argand circle, and inelasticity places it inside the corresponding disk. A real tree contribution is perturbatively consistent only while its real part has magnitude at most one half. Saturating this criterion identifies where neglected terms can no longer be presumed small, rather than proving that nature has a nonunitary S-matrix.

For the leading isospin-zero expression, this gives a characteristic center-of-mass energy square root of eight pi times v. The estimate is not an exact particle-production threshold, a theorem about a particular resonance mass, or an experimental exclusion. Coupled channels, higher orders and additional states can change how unitarity is maintained. A low-energy effective description is expected to be replaced or reorganized when its expansion becomes strongly interacting.

In the gaugeless linear model, the leading amplitude for charged Goldstones into two neutral Goldstones is the derivative contribution s over v squared minus the Higgs-exchange term s squared divided by v squared times s minus Higgs mass squared. Their sum tends to minus Higgs mass squared over v squared at energy much larger than that mass. The cancellation is controlled by the Higgs couplings dictated by the same vacuum scale.

Near the Higgs pole the zero-width expression is inappropriate, and elsewhere finite gauge couplings and vector masses add terms beyond this scalar approximation. Its constant high-energy limit does not eliminate every unitarity constraint: sufficiently large scalar self-coupling can still make perturbation theory unreliable, and full coupled-channel analyses constrain combinations of amplitudes. The useful conclusion is that the Higgs provides a definite mechanism for cancelling the dangerous energy growth, with its numerical success assessed in a complete specified regime.