Longitudinal Vector Bosons and the Equivalence Theorem

At tree level in the one-doublet electroweak theory, gauge identities cancel the apparent fourth-power growth of longitudinal-vector scattering, and Higgs exchange cancels the remaining second-power growth above the Higgs mass. The equivalence theorem makes this mechanism easier to calculate: an amplitude with energetic external longitudinal vectors is related to one with Goldstone insertions,

\mathcal M_L=C\,\widehat{\mathcal M}_\phi+B.

Here $C$ contains external-state normalization factors, the hat includes conventional Goldstone phases, and $B$ is an additive remainder. Each replaced vector must have energy much larger than its mass in the chosen frame; approximating the vector amplitude by the Goldstone term also requires $|B|\ll|C\widehat{\mathcal M}_\phi|$ . We derive a tree-level scalar-sector cancellation below and distinguish this gaugeless calculation from the full electroweak amplitude.

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 distinction between physical Goldstone particles of a broken global symmetry and the gauge-fixed scalar insertions used here.

Longitudinal polarization and the Ward identity

For a positive-energy on-shell vector with $p^\mu=(E,\mathbf p)$ and mass $m_V$ , choose its longitudinal polarization along $\mathbf p$ . Writing $k=|\mathbf p|>0$ gives

\begin{aligned} \epsilon_L^\mu&=\frac1{m_V}(k,E\widehat{\mathbf p}),\\ r^\mu&\equiv\epsilon_L^\mu-\frac{p^\mu}{m_V} =\frac{m_V}{E+k}(-1,\widehat{\mathbf p}). \end{aligned}

Thus $p\cdot\epsilon_L=0$ , $\epsilon_L^2=-1$ , and every component of $r^\mu$ is $O(m_V/E)$ when $E\gg m_V$ . The statement is about the polarization vector; it does not yet bound the amplitude obtained by contracting it.

For one external vector, let $\mathcal T_\mu$ be the amputated amplitude with its external residue included and the other external states physical. The broken-gauge Ward identity, with the vector–Goldstone mixing factors included, relates $p^\mu\mathcal T_\mu/m_V$ to a Goldstone insertion. Consequently,

\mathcal M_L =\frac{p^\mu}{m_V}\mathcal T_\mu+r^\mu\mathcal T_\mu =C\widehat{\mathcal M}_\phi+B, \qquad B=r^\mu\mathcal T_\mu.

For several replaced legs, the corresponding identity expands into a Goldstone term and a sum of terms containing at least one $r^\mu$ insertion. Those terms, collectively $B$ , involve mixed vector–Goldstone amplitudes. Their size depends on those amplitudes as well as on $m_V/E$ . This is a proof sketch of the high-energy reduction; the renormalized identity and its mixing factors are given in He, Kuang, and Yuan 1995, § 2, equations (1)–(2b), manuscript pp. 4–5, PDF.

To make the phase convention explicit, the cited identity uses $-i\phi$ for each Goldstone insertion. With every leg treated as incoming, we write $\widehat{\mathcal M}_\phi=(-i)^n\mathcal M_\phi$ for $n$ replacements and $C=\prod_j C_j$ . Canonically normalized tree fields give $C_j=1$ ; beyond tree level their finite factors depend on the gauge, residue and renormalization conventions. Crossing to an incoming/outgoing process must preserve these external-state conventions. The four-Goldstone amplitude below has $(-i)^4=1$ .

The useful hypotheses are:

every replaced vector is external and on its physical mass shell;
every such vector has $E_j\gg m_j$ in the frame where its longitudinal polarization is defined, and the hard invariants are large compared with the vector masses squared at fixed nonexceptional angles;
the Goldstone insertions use the same external momenta, gauge and renormalization prescription, with the specified residue and phase factors;
infrared, collinear, threshold and resonance regions receive their own treatment; and
the additive remainder is smaller than the retained Goldstone contribution to the accuracy required.

The Goldstone quantities in this identity are amputated insertions at the vector external momenta; they are not a separate physical Goldstone S-matrix in a general gauge. The last condition is independent of a large total collision energy. In particular, a zero or a strong suppression of the Goldstone amplitude can make a relative-error estimate meaningless. Equations (9)–(10b) of He, Kuang, and Yuan 1995, § 2, manuscript p. 8, PDF state both the kinematic requirement and this amplitude comparison. The theorem concerns properly normalized amplitudes, not a longitudinal vector becoming a physical scalar.

For $W$ and $Z$ bosons, the external amplitudes used here are perturbative on-shell subprocesses. Connecting them to observables with decaying bosons needs the pole or narrow-width treatment of unstable particles, rather than treating an unstable boson as an exact asymptotic state.

Representative cancellation in longitudinal W⁺W⁻ → ZZ scattering

We first calculate $w^+w^-\to zz$ in the gaugeless scalar sector: set $g,g'\to0$ at fixed $v$ and scalar coupling $\lambda$ , work at tree level, and take the Goldstones massless. This isolates the scalar contribution; it is a separate approximation from replacing external longitudinal vectors by Goldstones. The perturbative scalar potential has $\lambda>0$ and $m_h^2=2\lambda v^2$ .

Write the same doublet as

H=\frac{v+h}{\sqrt2}\,U\begin{pmatrix}0\\1\end{pmatrix}, \qquad U=\exp\!\left(\frac{i\tau^a\pi^a}{v}\right),

where $\tau^a$ are Pauli matrices, $w^\pm=(\pi^1\mp i\pi^2)/\sqrt2$ , and $z=\pi^3$ . The scalar Lagrangian becomes

\begin{aligned} \mathcal L_{\rm scalar} &=\frac12(\partial h)^2 +\frac{(v+h)^2}{4}\operatorname{tr}(\partial_\mu U^\dagger\partial^\mu U) -V(h),\\ V(h)&=\frac12m_h^2h^2+\lambda vh^3+\frac{\lambda}{4}h^4, \end{aligned}

up to an irrelevant constant. The quadratic angular term is $\tfrac12(\partial\pi^a)^2$ , so these Goldstones are canonically normalized. Expanding the trace gives the interactions needed for this process:

\begin{aligned} \mathcal L_{h\pi\pi}&=\frac{h}{v}(\partial\pi^a)^2,\\ \mathcal L_{4\pi}&=\frac1{6v^2} \bigl[(\pi^a\partial_\mu\pi^a)(\pi^b\partial^\mu\pi^b)\\ &\hspace{3.3em}-(\pi^a\pi^a)\partial_\mu\pi^b\partial^\mu\pi^b\bigr]. \end{aligned}

For example, the terms of $\mathcal L_{4\pi}$ containing one $w^+$ , one $w^-$ and two $z$ fields are

\begin{aligned} \mathcal L_{w^+w^-zz}=\frac1{3v^2}\bigl[ &z\partial_\mu z\,\partial^\mu(w^+w^-)\\ &-z^2\partial_\mu w^+\partial^\mu w^- -w^+w^-(\partial z)^2\bigr]. \end{aligned}

Take $q_1,q_2$ to be the incoming $w^+,w^-$ momenta and $q_3,q_4$ the negatives of the outgoing $z$ momenta, so $\sum_iq_i=0$ , $q_i^2=0$ , and $s=(q_1+q_2)^2$ . Contracting the four fields, including the two identical $z$ contractions, gives

\mathcal A_{\rm deriv} =\frac{2q_1\cdot q_2+2q_3\cdot q_4 -(q_1+q_2)\cdot(q_3+q_4)}{3v^2} =\frac{s}{v^2}.

The polar expansion and the crossed $w^+z\to w^+z$ contact calculation are also given in Schwartz 2014, § 29.2.1, p. 591, equations (29.30)–(29.31).

The all-incoming $h\pi^a\pi^b$ vertex is $-2i(q_1\cdot q_2)\delta^{ab}/v$ : the two derivatives supply the minus sign and the identical fields supply the factor of two. Both $hw^+w^-$ and $hzz$ therefore give $-is/v$ in this channel. Multiplying by the Higgs propagator yields

\begin{aligned} i\mathcal A_h(s) &=\left(-\frac{is}{v}\right) \frac{i}{s-m_h^2+i0} \left(-\frac{is}{v}\right),\\ \mathcal A_h(s)&=-\frac{s^2}{v^2(s-m_h^2)} \qquad (s\ne m_h^2). \end{aligned}

Away from the pole, their sum is

\begin{aligned} \mathcal A(s) &=\frac{s}{v^2}-\frac{s^2}{v^2(s-m_h^2)}\\ &=-\frac{m_h^2}{v^2}\frac{s}{s-m_h^2}. \end{aligned}

The cancellation follows from the fixed relative coefficient of the contact and exchange terms in a single doublet kinetic term. Changing the Higgs coupling independently would leave energy growth. Two limits expose the result:

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

Below the radial-mode scale the nonlinear Goldstone theory has an energy-growing amplitude. Above it, Higgs exchange cancels that growth. All amplitudes displayed here are dimensionless; the two vertices and one propagator give the same dimensions and sign as the contact calculation.

Relating the scalar calculation to vector scattering

In the direct $W_L^+W_L^-\to Z_LZ_L$ calculation, the $t$ -channel and $u$ -channel vector graphs and the four-vector contact graph first cancel their apparent fourth-power growth. After factoring out their common prefactor, the leading momentum polynomials obey

(2tu+t^2)+(2tu+u^2)-(t^2+4tu+u^2)=0.

Their remaining energy-growing term is $s/v^2$ , and the leading Higgs exchange is $-s/v^2$ when $s\gg m_h^2$ . This is the crossed version of the explicit vector calculation in Schwartz 2014, § 29.2, pp. 589–590, equations (29.22)–(29.26): its $W_L^+Z_L\to W_L^+Z_L$ variables map as $(s,t,u)_{WZ}=(t,s,u)_{WW\to ZZ}$ . Only the energy-growing terms are being compared here.

Three qualifications are consequential:

Gauge contributions remain. At nonzero $g,g'$ , the complete Goldstone amplitude also has gauge-interaction contributions of order $g^2$ or $g'^2$ at fixed angle. Taking the energy large does not eliminate these constant terms relative to the scalar constant $-2\lambda$ . Using the scalar result as the dominant full answer requires a separate comparison, such as $g^2,g'^2\ll\lambda$ away from cancellations, with $\lambda$ itself still perturbative. The additive equivalence-theorem remainder must be controlled as well. The distinction between external-state equivalence and neglecting gauge interactions is explicit in He, Kuang, and Yuan 1995, § 3, equations (21a,b), manuscript p. 16, PDF.
The low-energy limit has two scales. The scalar expansion $|s|\ll m_h^2$ is valid within the scalar model. Interpreting it as high-energy longitudinal scattering additionally requires $m_V^2\ll s,|t|,|u|\ll m_h^2$ at fixed angle. Such a parametrically broad window requires $m_h\gg m_V$ ; it must not be assumed for arbitrary parameters.
The tree pole is excluded. The expression is not a resonance line shape. Near $s=m_h^2$ , a consistent width-resummed pole treatment and the nonresonant terms are needed. Electroweak loop logarithms and finite residue factors also lie beyond the tree scalar calculation.

Partial-wave and scheme checks

For the angle-independent scalar amplitude above, the labeled-amplitude projection

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

gives $a_0=\mathcal A/(16\pi)$ . This transition amplitude is not by itself a normalized elastic-channel eigenvalue: the $zz$ channel contains identical particles. Form normalized two-particle states, include the other coupled channels, and diagonalize the partial-wave matrix before imposing an elastic eigenvalue bound. The partial-wave treatment explains these normalization and coupled-channel conditions. A large tree eigenvalue diagnoses failure of a perturbative approximation, not failure of exact unitarity.

Useful independent checks are:

Energy-power check. Sum the complete gauge-related set of graphs before taking the high-energy limit. At energies above both vector and Higgs masses, uncanceled fourth- or second-power growth in this renormalizable one-doublet tree amplitude signals a missing contribution or an inconsistent coupling. Below the Higgs scale, the $s/v^2$ term is expected.

Equivalence check. At the same external momenta and with the phase and residue factors included, calculate $B=\mathcal M_L-C\widehat{\mathcal M}_\phi$ . Compare it to the retained contribution and the required absolute accuracy. Do not divide by an amplitude that vanishes, or infer a universal relative error solely from $m_V/E$ .

Gauge check. Goldstone amplitudes, the factors $C_j$ and $B$ can depend on the gauge; their combination giving the physical vector amplitude cannot. A test of only one graph or only the scalar-sector approximation does not establish gauge independence of the full prediction.

Low-energy check. Expanding the Higgs-exchange denominator must recover $s/v^2+O(s^2/(v^2m_h^2))$ . The application to longitudinal vectors still needs the two-scale window above.

Common pitfalls

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

Using total energy as the only test. A soft external vector, a forward singularity or a suppressed Goldstone amplitude can defeat an otherwise tempting approximation. Check each replaced leg and the actual remainder in the specified frame.

Keeping only the constant high-energy answer. The pole region and the low-energy $s/v^2$ behavior have different interpretations. Even far above the Higgs mass, the scalar constant is only one part of the full electroweak answer unless the omitted gauge terms are independently negligible.

Check your understanding

Repeat the scalar calculation in Cartesian fields, using

V=\frac{\lambda}{4} \left(2v\sigma+\sigma^2+\chi^a\chi^a\right)^2, \qquad m_h^2=2\lambda v^2.

Here $\sigma$ is the Cartesian radial fluctuation and $\chi^a$ are canonically normalized Cartesian Goldstones. Show that the constant four-Goldstone vertex and $\sigma$ exchange reproduce the polar-coordinate result, even though the separate diagrams differ.

If the leading Goldstone amplitude vanishes at a chosen kinematic point, what does $r^\mu=O(m_V/E)$ establish about the relative error? What must be estimated instead?

Solution

The terms needed in the potential are $\lambda v\sigma\chi^a\chi^a$ and $\lambda(\chi^a\chi^a)^2/4$ . With $w^\pm=(\chi^1\mp i\chi^2)/\sqrt2$ and $z=\chi^3$ , the all-incoming vertices are $-2i\lambda v$ for both $\sigma w^+w^-$ and $\sigma zz$ , and $-2i\lambda$ for $w^+w^-zz$ . Thus, away from the pole,

\begin{aligned} \mathcal A_{\rm Cartesian} &=-2\lambda-\frac{(2\lambda v)^2}{s-m_h^2}\\ &=-\frac{m_h^2}{v^2} -\frac{m_h^4}{v^2(s-m_h^2)}\\ &=-\frac{m_h^2s}{v^2(s-m_h^2)}. \end{aligned}

This agrees with the polar result. A nonsingular local field reparameterization changes the allocation among contact and exchange diagrams, while their on-shell sum agrees.

At a zero of $C\widehat{\mathcal M}_\phi$ , the ratio $B/(C\widehat{\mathcal M}_\phi)$ is undefined. The small polarization components alone do not bound $B$ without the amplitudes they multiply. Estimate the mixed amplitudes in $B$ , keep the first nonzero contribution, and judge an absolute error or the error relative to that nonzero result.

Using the result

A reusable calculation specifies the external channel and helicities, the frame and hard invariants, the gauge and residue conventions, the factors $C_j$ , the estimated remainder $B$ , any further gaugeless approximation, and the excluded pole or infrared regions. A partial-wave application also specifies normalized channel states.

Use Higgs self-interactions for the scalar potential and electroweak renormalization for loop-consistent inputs. Broader perturbative-unitarity applications belong to Precision Standard Model.

References

He, Hong-Jian, Yu-Ping Kuang, and C.-P. Yuan. “Equivalence Theorem and Probing the Electroweak Symmetry Breaking Sector.” Physical Review D 51 (1995): 6463–6473. DOI. Open PDF, arXiv:hep-ph/9410400v2, 3 March 1995. The locators above use the manuscript’s printed pagination.
Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.