Higher-order radiative corrections, the heavy-Higgs decoupling ambiguity, and the high-energy regime of longitudinal WW scattering

In 1989 Veltman and Ynduráin computed the one-loop radiative corrections to the scattering of longitudinally polarized W bosons in an SU(2) electroweak gauge theory with a very heavy Higgs boson, in the kinematic window $M_W^2 \ll s,t,u \ll m_H^2$ (with $s,t,u$ the Mandelstam invariants, $M_W$ the W mass and $m_H$ the Higgs mass). To parametrize the approach to the $m_H \to \infty$ limit they introduced a gauge-singlet scalar, the U particle, coupled to the Higgs doublet through what is now called a Higgs-portal coupling, and found that the coefficients of the logarithms $\log(s/m_H^2)$ are U-independent while finite non-logarithmic remnants are not, so the heavy-Higgs limit of the amplitude is intrinsically ambiguous. The calculation left open the higher-order corrections and the resummation of the large logarithms in that window. Later literature resolved the full-structure one-loop question, computed two-loop amplitudes in the complementary high-energy window $\sqrt{s} \gg m_H \gg M_W$ and the local two-loop low-energy quartic couplings, and reframed the decoupling ambiguity as an effective-field-theory matching question. The open problem — the complete two-loop and resummed corrections in the original low-energy window, together with a well-defined decoupling/matching prescription that renders the heavy-Higgs limit of the amplitude unambiguous — was explicitly identified as an open "possible calculation" of "considerable interest" by Maher, Durand and Riesselmann (1993) and remains open. The genuinely nonperturbative regime $s \gtrsim m_H^2$, which lies outside the source window $s,t,u \ll m_H^2$, motivates the reliability question but is not part of this target.

Background

In the Standard Model of particle physics, electroweak symmetry is broken by a complex scalar doublet whose neutral component acquires a vacuum expectation value $v \approx 246$ GeV; this Higgs mechanism gives the W and Z bosons their masses and leaves one physical excitation, the Higgs boson. The scattering of longitudinally polarized vector bosons ($W_L$, $Z_L$: helicity-zero states whose polarization vectors grow with energy like $E/M_W$) is the classic probe of this symmetry-breaking sector. At tree level the $W_L W_L$ amplitude grows as $s/v^2$ (with $s$ the squared center-of-mass energy), and partial-wave unitarity — the requirement that each angular-momentum partial wave of a two-body amplitude respect the unitarity bound $|\mathrm{Re}\,a_J| \le 1/2$ — is maintained at high energy only because Higgs exchange cancels the growing terms. Lee, Quigg and Thacker used this argument to bound $m_H \lesssim 1$ TeV: the heavier the Higgs, the higher the energy at which the cancellation sets in and the stronger the Higgs self-coupling $\lambda \sim G_F m_H^2$ (with $G_F$ the Fermi constant), until perturbation theory fails. In the opposite limit, Veltman's screening theorem states that a very heavy Higgs affects low-energy observables at most logarithmically. Two tools make high-energy computations tractable: the Goldstone-boson equivalence theorem, which replaces longitudinal W amplitudes at $E \gg M_W$ by amplitudes of the corresponding massless Goldstone scalars up to corrections of order $M_W/E$, and renormalization-group (RG) resummation of the large logarithms generated by the scale hierarchy. In this setting Veltman and Ynduráin (1989) computed the one-loop corrections to four-longitudinal-W scattering in a simplified electroweak theory built on the weak-isospin gauge group SU(2) with a Higgs doublet, no fermions, and a vanishing weak mixing angle $\theta_W$ (the parameter that in the full electroweak theory mixes the W and Z gauge fields). Because the $m_H \to \infty$ limit of diagrams with internal Higgs lines is not unique — how a heavy sector decouples depends on how it is embedded — they introduced an auxiliary gauge-singlet scalar, the U particle, as a bookkeeping device parametrizing this ambiguity (one of the early appearances of the real-scalar-singlet, or Higgs-portal, extension, in which an extra singlet scalar couples to the Higgs doublet). Their main findings: the renormalized one-loop amplitude contains terms of order $s^2 \log(s/m_H^2)/M_W^4$ whose coefficients are U-independent when the U shares the Higgs mass, and finite non-logarithmic terms of order $s^2/M_W^4$ that do depend on the U couplings, so the infinite-Higgs-mass limit is defined only after specifying how it is taken; only the counterterm $\delta_2$ (the SU(2)-doublet wavefunction renormalization) feeds the leading $g^4 s^2$ structure (with $g$ the SU(2) gauge coupling); and the one-loop corrections remain at the ten-percent level at sub-TeV energies. The discovery of a 125 GeV Higgs boson at the LHC (the Large Hadron Collider at CERN) in 2012 settled nature's choice for the Standard Model, but the theoretical questions survive for the heavy-Higgs theory itself and for its modern descendants: strongly interacting electroweak symmetry breaking, the electroweak chiral Lagrangian (HEFT, the effective field theory of electroweak symmetry breaking in which the Higgs-like scalar is treated as a particle independent of the symmetry-breaking sector rather than as part of a doublet), Higgs-portal singlet extensions, and effective-field-theory (EFT) matching, in which heavy fields are integrated out and their effects encoded in the coefficients of higher-dimensional operators.

Problem Statement

The problem, posed by the source calculation and documented as open by Maher, Durand and Riesselmann (1993), is to obtain a complete, scheme-unambiguous, quantitatively controlled prediction for the scattering of longitudinally polarized W bosons in an electroweak theory whose Higgs boson is parametrically heavy, and to determine the energy and Higgs-mass range in which that prediction is trustworthy. In the kinematic window $M_W^2 \ll s,t,u \ll m_H^2$ — with $s,t,u$ the Mandelstam scattering invariants, $M_W$ the W-boson mass and $m_H$ the Higgs mass — where Veltman and Ynduráin computed the one-loop amplitude for an SU(2) model with vanishing weak mixing angle $\theta_W$ and no fermions, the components of the question are: (1) Higher-order corrections — compute the complete two-loop radiative corrections to the $W_L W_L \to W_L W_L$ amplitude and its crossed and inelastic channels in this window, including the renormalization-group resummation of the large logarithms $\log(s/m_H^2)$ and the summation of the powerlike non-decoupling terms of order $G_F m_H^2$ (terms that grow with the Higgs mass instead of vanishing as $m_H \to \infty$, with $G_F$ the Fermi constant); where feasible, extend the computation to higher orders; determine whether higher orders enhance, cancel, or qualitatively modify the one-loop result, and establish where the fixed-order expansion fails. (2) Decoupling prescription — determine which finite terms surviving the $m_H \to \infty$ limit are universal and which are fixed only by matching to a specified ultraviolet completion (a complete theory above the heavy scale whose low-energy limit is the model under study), replacing the U-particle bookkeeping (the auxiliary gauge-singlet scalar introduced in the source calculation to parametrize how the heavy Higgs sector decouples) by a well-defined scheme or matching condition. (3) Full electroweak structure — quantify how reinstating the weak mixing angle, fermion loops, and electromagnetic interactions modifies the results and their decoupling sensitivity, given that the W is unstable beyond leading order. (4) High-energy regime — extend the prediction to and beyond $s \sim m_H^2$, connecting the low-energy window to the regime where tree-level partial-wave unitarity binds and perturbation theory in the Higgs self-coupling fails. The conceptual core of (2) and component (3) have been addressed in the literature (see References 14, 18, 20 and 21); those references supply the concept of the matching prescription of (2), not the concrete, model-specific prescription, which remains part of the open core. Component (3) is supplied as settled context. The surviving open core is the component (1) computation — the complete two-loop and resummed amplitude in the source window — together with the decoupling/matching prescription of (2) that renders the target quantity well defined. Component (4) lies outside the source window $s,t,u \ll m_H^2$: it motivates the reliability question for the nonperturbative regime, but only model-dependent or expansion-based treatments exist there. The range in which the prediction is trustworthy is correspondingly confined to the source window.

References

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