Longitudinal Vector Bosons and the Equivalence Theorem

Longitudinal weak-vector amplitudes are well behaved at high energy because gauge, Goldstone, and Higgs contributions obey common Ward identities. For external energies $E\gg m_V$ at fixed scattering angle,

$$\mathcal M(V_L^{a_1}\cdots V_L^{a_n}\to X) =C\,\mathcal M(\phi^{a_1}\cdots\phi^{a_n}\to X) +O\!\left(\frac{m_V}{E}\right),$$

where $\phi^a$ are the corresponding Goldstone fields and $C=1$ at tree level in a standard normalization. At loop level $C$, the gauge choice, external-state convention, and renormalization scheme must be specified.

Required background. The Higgs doublet and electroweak symmetry breaking supplies the Goldstone directions and the radial mode. Partial-wave unitarity supplies the amplitude bounds used after the cancellation.

Helpful background. The Goldstone theorem supplies the global-symmetry pole argument whose gauge-theory descendant underlies the equivalence relation.

Longitudinal polarization and the Ward identity

For an on-shell massive vector of momentum $p$,

$$\epsilon_L^\mu(p)=\frac{p^\mu}{m_V} +O\!\left(\frac{m_V}{E}\right).$$

Inserting only the leading term into individual diagrams appears to generate powers of $E/m_V$. Those powers are not separately physical. The broken-gauge Ward identity relates contraction of an amputated vector amplitude with $p_\mu$ to the amplitude with the associated Goldstone insertion. After wave-function and mixing factors are included, it yields the equivalence theorem.

The useful hypotheses are:

• every replaced vector is external and on its physical mass shell;
• all hard invariants are large compared with $m_V^2$, at fixed nonexceptional angles;
• the Goldstone amplitude is computed in the same gauge and renormalization convention;
• infrared, collinear, threshold, and resonance regions receive their own treatment;
• at loop level the finite factor $C$ and external residues are retained.

Chanowitz and Gaillard give a systematic formulation and its domain for high-energy weak interactions Chanowitz and Gaillard 1985, §§2–3, pp. 383–397. The theorem does not say that a longitudinal vector literally becomes a physical scalar; it states an asymptotic relation between properly normalized amplitudes.

Representative cancellation in longitudinal W⁺W⁻ → ZZ scattering

At high energy the equivalent Goldstone process is $w^+w^-\to zz$. In a polar form of the one-doublet scalar sector, the derivative Goldstone interaction contributes

$$\mathcal A_{\rm deriv}(s)=\frac{s}{v^2},$$

while radial-Higgs exchange contributes, away from its pole,

$$\mathcal A_h(s) =-\frac{s^2}{v^2(s-m_h^2)}.$$

Their sum is

$$\mathcal A(s) =-\frac{m_h^2}{v^2}\frac{s}{s-m_h^2}.$$

Two limits expose the physics:

$$\mathcal A(s)\simeq \begin{cases} s/v^2, & |s|\ll m_h^2,\\[4pt] -m_h^2/v^2, & s\gg m_h^2. \end{cases}$$

Below the radial-mode scale, the nonlinear Goldstone theory has the expected energy-growing amplitude. Above it, Higgs exchange cancels that growth and leaves a constant. In the direct vector calculation, gauge diagrams first cancel the apparent $E^4/m_V^4$ terms, and the Higgs contribution cancels the remaining $E^2/m_V^2$ behavior. Schwartz gives both descriptions and their agreement in Schwartz 2014, §29.2, pp. 588–592.

The expression is a tree-level illustration, not a line shape. Near $s=m_h^2$, a width-resummed pole treatment and the nonresonant amplitude are required. At fixed high energy it also omits electroweak loop logarithms and channel mixing.

Partial-wave and scheme checks

For an amplitude independent of the scattering angle in this channel, the $J=0$ projection in the convention

$$a_0(s)=\frac{1}{32\pi} \int_{-1}^{1}d(\cos\theta)\,\mathcal A(s,\theta)$$

is $a_0=\mathcal A/(16\pi)$. Coupled charge and isospin channels must be diagonalized before applying the strongest unitarity condition; one channel alone does not reproduce the complete scalar-sector bound. Lee, Quigg, and Thacker perform that coupled-channel analysis Lee, Quigg, and Thacker 1977, §§II–III, pp. 1521–1527.

Useful independent checks are:

Energy-power check. Sum a gauge-invariant set of diagrams before taking the high-energy limit. Residual $E^4$ behavior indicates a broken Ward identity; residual $E^2$ behavior in the minimal one-doublet model usually indicates a missing Higgs or Goldstone contribution.

Equivalence check. Compare the full longitudinal amplitude with the Goldstone amplitude while increasing all hard invariants together. Their relative difference should scale as $m_V/E$, up to loop and convention factors.

Gauge check. Individual Goldstone graphs and the factor $C$ can depend on the gauge; the physical vector amplitude cannot. Testing only one diagram is not a gauge-independence test.

Low-energy check. Expanding the Higgs-exchange denominator for $|s|\ll m_h^2$ must recover $s/v^2$. This is the matching limit to the nonlinear theory.

Common pitfalls

Replacing internal vectors by Goldstones. The elementary theorem concerns external longitudinal states. Internal propagators require the complete gauge-fixed calculation and cannot be replaced line by line.

Using it at threshold. When $E$ is comparable to $m_V$, the nominal correction is not small. Compute the vector amplitude directly.

Keeping only the constant high-energy answer. The pole region and the low-energy $s/v^2$ behavior carry different physics. The unsimplified amplitude records both and displays where tree perturbation theory fails.

Handoff

A controlled equivalence-theorem result passes

$$\left\{\text{external channel and helicities};\ s,t,u;\ E/m_V;\ \text{gauge and renormalization scheme};\ C;\ \text{pole exclusions};\ \text{partial-wave convention}\right\}.$$

Use it with Higgs self-interactions for scalar-sector interpretations and electroweak renormalization for loop-consistent inputs. Broader perturbative-unitarity applications belong to Precision Standard Model.

References

• Chanowitz, Michael S., and Mary K. Gaillard. “The TeV Physics of Strongly Interacting W’s and Z’s.” Nuclear Physics B 261 (1985): 379–431. DOI.
• Lee, Benjamin W., C. Quigg, and H. B. Thacker. “Weak Interactions at Very High Energies: The Role of the Higgs-Boson Mass.” Physical Review D 16, no. 5 (1977): 1519–1531. DOI.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §29.2, pp. 588–592. DOI.