Loyal Durand

We calculate the complete matrix of two-body scattering amplitudes for the scattering of longitudinally polarized gauge bosons $W_L^\pm$, $Z_L$ and Higgs bosons to two loops in the high-energy, heavy-Higgs limit $\sqrt{s}\gg M_H\gg M_W$. Use of the Goldstone boson equivalence theorem reduces the problem to one involving only the scalar fields $w^\pm$, $z$ (the Goldstone bosons of the original theory) and the Higgs boson. Renormalization of the scattering amplitudes requires the calculation of the self-energy functions $\Pi _i^0(M_i^2)$, the renormalization constants $Z_i$, and the bare quartic Higgs coupling $\lambda _0$ to two loops. The results will be useful in other calculations. To facilitate the calculations, we introduce a powerful new technique for evaluating integrals over Feynman parameters in dimensional regularization which is based on a Barnes' type representation of the binomial expansion. We also collect some useful integrals which extend the tables given by Devoto and Duke

We use the results of Maher et al. (preceding paper) to construct the matrix of $j=0\mathrm{}$ partial-wave two-body and 2 → 3 scattering amplitudes for the scattering of longitudinally polarized gauge bosons $W_L^{\pm }$, $Z_L$ and Higgs bosons $H$ correct to two loops in the high-energy, heavy-Higgs-boson limit $\sqrt{s}\gg M_H\gg M_W$. We show explicitly that the energy dependence of the 2 → 2 amplitudes can be completely absorbed into a running quartic Higgs boson coupling ${\lambda }_s={\lambda }_s(s, M_H^2)$ and factors which involve small anomalous dimensions and remain near unity. After diagonalizing the matrix of partial-wave amplitudes we use an Argand-diagram analysis to show that the elastic scattering amplitudes are approximately unitary and weakly interacting for ${\lambda }_s\lesssim 2.3$, but that three-loop corrections are necessary to restore unitarity for larger values of ${\lambda }_s$. That is, the interactions in the Higgs sector of the standard model are effectively strong with respect to the perturbative expansion for ${\lambda }_s\gtrsim 2.3$. The bound ${\lambda }_s\lesssim 2.3$ for a weakly interacting theory translates to a physical Higgs boson mass $M_H\lesssim 380$ GeV if the bound is to hold for energies up to a few TeV, or $M_H\le 155$ GeV in perturbatively unified theories with mass scales of order ${10}^{16}$ GeV.

We calculate the complete matrix of two-body scattering amplitudes for the scattering of longitudinally polarized gauge bosons $W_L^{\pm }$, $Z_L$ and Higgs bosons to two loops in the high-energy, heavy-Higgs-boson limit $\sqrt{s}\gg M_H\gg M_W$. Use of the Goldstone boson equivalence theorem reduces the problem to one involving only the scalar fields $w^{\pm }$, $z$ (the Goldstone bosons of the original theory) and the Higgs boson. Renormalization of the scattering amplitudes requires the calculation of the self-energy functions ${\Pi }_i^0\left(M_i^2\right)$, the renormalization constants $Z_i$, and the bare quartic Higgs boson coupling ${\lambda }_0$ to two loops. The results will be useful in other calculations. To facilitate the calculations we introduce a powerful new technique for evaluating integrals over Feynman parameters in dimensional regularization which is based on a Barnes-type representation of the binomial expansion. We also collect some useful integrals which extend the tables given by Devoto and Duke.

We reexamine the theory of hadronic photon-nucleon interactions at the quark-gluon level. The possibility of multiple parton collisions in a single photon-nucleon collision requires an eikonal treatment of the high-energy scattering process. We give a general formulation of the theory in which the $\gamma p$ cross section is expressed as a sum over properly eikonalized cross sections for the interaction of the virtual hadronic components of the photon with the proton, with each cross section weighted by the probability with which that component appears in the photon, and then develop a detailed model which includes contributions from light vector mesons and from excited virtual states described in a quark-gluon basis. The parton distribution functions which appear can be related approximately to those in the pion, while a weighted sum gives the distribution functions for the photon. We use the model to make improved QCD-based predictions for the total inelastic photon-nucleon and photon-nucleus cross sections at energies relevant for DESY HERA experiments and cosmic-ray observations. We emphasize the importance in this procedure of including a soft-scattering background such that the calculated cross sections join smoothly with low-energy data. Our results show clearly that high-energy measurements of the total inelastic $\gamma p$ cross section can impose strong constraints on the gluon and quark distributions in the photon, and indirectly on those in the pion.

We discuss the theory of jet events in high-energy photon-proton interactions using a model which gives a good description of the data available on total inelastic $\gamma p$ cross sections up to $\sqrt{s}=210\mathrm{}$ GeV. We show how to calculate the jet cross sections and jet multiplicities and give predictions for these quantities for energies appropriate for experiments at the DESY $\mathrm{ep}$ collider HERA and for very-high-energy cosmic ray observations.

We use the results of Maher {\em et al.\/} (preceding paper) to construct the matrix of $j=0$ partial-wave two-body and $2\rightarrow3$ scattering amplitudes for the scattering of longitudinally polarized gauge bosons $W_L^\pm$, $Z_L$ and Higgs bosons $H$ correct to two loops in the high-energy, heavy-Higgs limit $\sqrt{s}\gg M_H\gg M_W$. We show explicitly that the energy dependence of the $2\rightarrow2$ amplitudes can be completely absorbed into a running quartic Higgs coupling $\lambda_s= \lambda_s(s,M_H^2)$ and factors which involve small anomalous dimensions and remain near unity. After diagonalizing the matrix of partial-wave amplitudes, we use an Argand-diagram analysis to show that the elastic scattering amplitudes are approximately unitary and weakly interacting for $\lambda_s\alt2.3$, but that three-loop corrections are necessary to restore unitarity for larger values of $\lambda_s$. That is, the interactions in the Higgs sector of the standard model are effectively strong with respect to the perturbative expansion for $\lambda_s\agt2.3$. The bound $\lambda_s\alt2.3$ for a weakly interacting theory translates to a physical Higgs mass $M_H\alt380$ GeV if the bound is to hold for energies up to a few TeV, or $M_H\leq155$ GeV in perturbatively unified theories with mass scales of order $10^{16}$ GeV.

We calculate the complete matrix of two-body scattering amplitudes for the scattering of longitudinally polarized gauge bosons $W_L^\pm$, $Z_L$ and Higgs bosons to two loops in the high-energy, heavy-Higgs limit $\sqrt{s}\gg M_H\gg M_W$. Use of the Goldstone boson equivalence theorem reduces the problem to one involving only the scalar fields $w^\pm$, $z$ (the Goldstone bosons of the original theory) and the Higgs boson. Renormalization of the scattering amplitudes requires the calculation of the self-energy functions $\Pi _i^0(M_i^2)$, the renormalization constants $Z_i$, and the bare quartic Higgs coupling $\lambda _0$ to two loops. The results will be useful in other calculations. To facilitate the calculations, we introduce a powerful new technique for evaluating integrals over Feynman parameters in dimensional regularization which is based on a Barnes' type representation of the binomial expansion. We also collect some useful integrals which extend the tables given by Devoto and Duke.

We reexamine the unitarity constraints on the high-energy scattering of longitudinally polarized $W\text{'}\mathrm{s}$ and $Z\text{'}\mathrm{s}$ and Higgs bosons in the standard model including one-loop corrections. Using an Argand diagram analysis, we find that the $j=0\mathrm{}$ scattering amplitudes are approximately unitary and weakly interacting at order ${\lambda }^2$ for Higgs-boson couplings $\lambda (s,M_H^2)\lesssim 2$, but that corrections of order ${\lambda }^3$ or higher must be included to restore perturbative unitarity for larger values of $\lambda$. We show also that two-loop $\left[\mathrm{O}\right({\lambda }^3\left)\right]$ corrections cannot extend the range of validity of perturbation theory beyond $\lambda \approx 2.2$. An analysis of inelastic 2 → 4 scattering in the $W_L^{\pm }$, $Z_L$, $H$ system gives an independent but weaker limit $\lambda (s,M_H^2)\lesssim 5$. The limit $\lambda (s,M_H^2)<\mathrm{}2\mathrm{}$ translates to a physical-Higgsboson mass $M_H\lesssim 400$ GeV if the bound is to hold up to energies of a few TeV, or $M_H\lesssim 160$ GeV in perturbatively unified theories with a mass scale of order ${10}^{15}$ GeV. For masses much larger than these bounds, low-order perturbation theory fails and the Higgs sector of the standard model becomes effectively strongly interacting.

We investigate the partial-wave scattering of longitudinally polarized $W^{\pm }$ and $Z$ bosons for energies $M_W\ll \sqrt{s}\ll M_H$ using an Argand-diagram analysis. We find that, for large Higgs-boson masses $M_H$, partial-wave unitarity is badly violated at one loop at energies $\sqrt{s}\ll M_H$, and the standard model of the electroweak interactions becomes effectively strongly interacting in this sector. The implications of this and other similar results are summarized.