Goldstone equivalence theorem

Use the metric with positive time component and negative spatial components. For an on-shell massive vector travelling along positive z, the longitudinal polarization has time component momentum over mass and z component energy over mass. Its scalar product with the momentum vanishes and its norm is minus one. At high energy both displayed components are large, making a diagram-by-diagram calculation appear to grow with powers of energy over vector mass.

Subtract momentum divided by mass from this polarization. The remaining time and z components have magnitude mass divided by energy plus momentum, which scales as mass over energy when the particle is relativistic. The apparent leading term can be converted, using gauge-theory identities, into a Goldstone-field amplitude. Simply deleting it as though it were the polarization of a massless gauge boson would miss the massive theory's nonzero symmetry-breaking contribution.

After gauge fixing and reduction to external states, Ward or Slavnov–Taylor identities relate the momentum contraction to a Goldstone insertion. For several longitudinal external bosons, the leading term includes one conversion factor for each line and a phase depending on field conventions. At tree level a convenient convention makes the finite factors unity. Beyond tree level they depend on wavefunction normalization and the renormalization scheme, while the final physical prediction remains gauge independent.

The remainder consists of amplitudes with at least one small residual-polarization insertion. It is an additive contribution involving other tensor amplitudes, not universally a fixed percentage of the leading Goldstone term. The usual high-energy power counting assumes a controlled kinematic regime and comparison scales. Thresholds, a very forward enhancement, an accidental leading zero, or another large mass ratio can invalidate a naive numerical error estimate even if the external vector energy is large.

The theorem applies to longitudinal external states and does not replace arbitrary internal gauge propagators by scalars. In realistic collider analyses, vector bosons are unstable and reconstructed through decay products. Isolating a polarization channel and using a narrow-width approximation therefore introduces additional assumptions. A consistent complete amplitude is required near resonances and when separating diagrams would spoil gauge cancellations. The theorem is most useful as a carefully specified organizing approximation.

It also explains why a small vector mass does not always make the longitudinal contribution negligible: the momentum-like polarization compensates factors of mass in its interactions. Taking a gauge coupling to zero at fixed vacuum scale and taking the vacuum scale to zero at fixed coupling are different parameter limits, especially when Higgs and fermion masses depend on the same vacuum. One must say which ratios stay fixed instead of treating every formal massless limit as the same physical theory.