arXiv subjects
Publications and source records attributed to Kurt Riesselmann.
This contribution reviews the latest results of the perturbative calculations of heavy-Higgs amplitudes. A comparison of perturbative results with nonperturbative lattice calculations is made, and the theoretical uncertainties of the lower and upper bound on the Standard Model Higgs mass are presented.
The two-loop Higgs mass upper bounds are reanalyzed. Previous results for a cutoff scale $Λ\approx$ few TeV are found to be too stringent. For $Λ=10^{19}$ GeV we find $M_H < 180 \pm 4\pm 5$ GeV, the first error indicating the theoretical uncertainty, the second error reflecting the experimental uncertainty due to $ m_t = 175 \pm 6 $ GeV. We also summarize the lower bounds on $M_H$. We find that a SM Higgs mass in the range of 160 to 170 GeV will certainly allow for a perturbative and well-behaved SM up to the Planck-mass scale $Λ_{Pl}\simeq 10^{19}$ GeV, with no need for new physics to set in below this scale.
This contribution reviews the latest results of the perturbative calculations of heavy-Higgs two-loop amplitudes. A comparison of perturbative results with nonperturbative lattice calculations is made, and the theoretical uncertainties of the lower and upper bound on the Standard Model Higgs mass are presented.
This lecture reviews various Higgs-sector amplitudes which have been calculated to two loops in the Higgs quartic coupling. After explaining the framework of these calculatins, the perturbative behaviour of the amplitudes is discussed, and perturbative upper bounds on the Higgs boson mass are given.
Matching conditions relate couplings to particle masses. We discuss the importance of one-loop matching conditions in Higgs and top-quark sector as well as the choice of the matching scale. We argue for matching scales $μ_{0,t} \simeq m_t$ and $μ_{0,H} \simeq max[ m_t, M_H ]$. Using these results, the two-loop Higgs mass upper bounds are reanalyzed. Previous results for $Λ\approx$ few TeV are found to be too stringent. For $Λ=10^{19}$ GeV we find $M_H < 180 \pm 4\pm 5$ GeV, the first error indicating the theoretical uncertainty, the second error reflecting the experimental uncertainty due to $m_t=175\pm6$ GeV.
The evaluation of high-energy cross sections involving the SM Higgs boson requires the use of the Higgs running coupling $λ(μ)$. Taking $μ$ to be equal to the center-of-mass energy $\sqrt{s}$ of the scattering process, the perturbative approach fails for relatively small values of the Higgs mass and coupling, $λ(\sqrt{s})\approx 2.2$. Performing an approximate resummation of ``bubble'' Feynman diagrams, we find the scale $μ=\sqrt{s}/ e \approx \sqrt{s}/2.7$ to yield reliable perturbative results, even for large Higgs mass and coupling. The new perturbative upper limit on the Higgs running coupling is $λ(\sqrt{s}/ e)\approx 4$.
Previous work has suggested that perturbation theory is unreliable for Higgs- and Goldstone-boson scattering, at energies above the Higgs mass, for relatively small values of the Higgs quartic coupling $λ(μ)$. By performing a summation of nonlogarithmic terms, we show that perturbation theory is in fact reliable up to relatively large coupling. This eliminates the possibility of a strongly-interacting standard Higgs model at energies above the Higgs mass, complementing earlier studies which excluded strong interactions at energies near the Higgs mass. The summation can be formulated in terms of an appropriate scale in the running coupling, $μ=\sqrt{s}/e\approx\sqrt{s}/2.7$, so it can easily be incorporated in renormalization-group improved tree-level amplitudes as well as higher-order calculations.
The amplitudes for $2\rightarrow 2$ scattering processes involving longitudinally polarized gauge bosons $( W_L^\pm, Z_L )$ and the Higgs boson are analyzed up to two loops. Assuming $M_H >> M_W$, the trilinear Higgs coupling, $λv$, is dominant for energies of $\sqrt{s}$ < 1.5 -- 2 $M_H$. For larger values of $\sqrt{s}$, the quartic coupling, $λ$, becomes dominant, allowing for a simpler calculation of higher-order corrections. The high-energy amplitudes display a large logarithmic dependence on $\sqrt{s}$ which can be resummed using renormalization group techniques. The resummation of leading-log terms is sufficient for Higgs masses of less than 350 GeV. For 350 < $M_H$ < 450 GeV, a next-to-leading-log resummation is necessary. For even larger values of $M_H$, the perturbative approach fails completely since two-loop terms become in magnitude larger than one-loop terms. Choosing the $\overline{\rm MS}$ renormalization scheme instead of the OMS scheme, the coefficients of the perturbative series increase in magnitude, making the breakdown of perturbation theory even more apparent. In conclusion, the Standard Model cross sections presented here have very large uncertainties if $M_H\gtrsim 450$ GeV and $\sqrt{s} \gtrsim 2 M_H$, reducing the sensitivity to contributions from new physics significantly.
The calculation of the leading electroweak corrections to physical transition matrix elements in powers of $M_H^2/v^2$ can be greatly simplified in the limit $M_H^2\gg M_W^2,\, M_Z^2$ through the use of the Goldstone boson equivalence theorem. This theorem allows the vector bosons $W^\pm$ and $Z$ to be replaced by the associated scalar Goldstone bosons $w^\pm$, $z$ which appear in the symmetry breaking sector of the Standard Model in the limit of vanishing gauge couplings. In the present paper, we extend the equivalence theorem systematically to include the Yukawa interactions between the fermions and the Higgs and Goldstone bosons of the Standard Model. The corresponding Lagrangian ${\cal L}_{EQT}$ is given, and is formally renormalized to all orders. The renormalization conditions are formulated both to make connection with physical observables and to satisfy the requirements underlying the equivalence theorem. As an application of this framework, we calculate the dominant radiative corrections to fermionic Higgs decays at one loop including the virtual effects of a heavy top quark. We apply the result to the decays $H\rightarrow t\bar{t}$ and $H\rightarrow b\bar{b}$, and find that the equivalence theorem results including fermions are quite accurate numerically for Higgs-boson masses $M_H> 400\,(350)$ GeV, respectively, even for $m_t=175$ GeV.
We discuss different aspects of the Higgs self-interaction in the MS-bar and the on-mass-shell (OMS) scheme. The running coupling λ(μ) is investigated in great detail. The three-loop coefficient of the β-function in the OMS scheme is derived, and the three-loop running coupling is calculated. The breakdown of perturbation theory for large Higgs masses M_H is analyzed in three physical observables for which two-loop results are known. Requiring the dependence on the renormalization scale to diminish order-by-order in λ, we find that perturbation theory breaks down for M_H=O(700 GeV) in Higgs decays. Similarly, M_H must be smaller than O(400 GeV) for perturbatively calculated cross sections to be trustworthy up to cm energies of O(2 TeV). If the Higgs sector shall be perturbative up to the GUT scale, the Higgs must be lighter than O(150 GeV). For the two-loop observables examined, the apparent convergence of the perturbation series is better in the OMS scheme than in the MS-bar scheme.
The equivalence theorem is an extremely useful tool to calculate heavy Higgs {\it and} top-quark effects for processes that have center-of-mass-energies (much) larger than the $W$ boson mass. After an explanation of the renormalization procedures involved, the results for one- and two-loop radiative corrections to the fermionic Higgs decay, $H\rightarrow f\bar f$, are given and discussed. Finally, the renormalization scheme dependence is examined, and the reliability of the perturbative series is investigated.
We calculate the dominant ${\rm O}\left(G_F^2M_H^4\right)$ two-loop electroweak corrections to the fermi\-onic decay widths of a heavy Higgs boson in the Standard Model. Use of the Goldstone-boson equivalence theorem reduces the problem to one involving only the physical Higgs boson $H$ and the Goldstone bosons $w^\pm$ and $z$ of the unbroken theory. The two-loop corrections are opposite in sign to the one-loop electroweak corrections, exceed the one-loop corrections in magnitude for $M_H>1114\ {\rm GeV}$, and increase in relative magnitude as $M_H^2$ for larger values of $M_H$. We conclude that the perturbation expansion in powers of $G_FM_H^2$ breaks down for $M_H\approx 1100\ {\rm GeV}$. We discuss briefly the QCD and the complete one-loop electroweak corrections to $H\rightarrow b\bar{b}, \,t\bar{t}$, and comment on the validity of the equivalence theorem. Finally we note how a very heavy Higgs boson could be described in a phenomenological manner.
We present the dominant two-loop ${\rm O}\left(G_F^2M_H^4\right)$ electroweak corrections to the fermi\-onic decay widths of a high-mass Higgs boson in the Standard Model. The corrections are negative and quite significant, and are larger in magnitude than the one-loop electroweak corrections for $M_H\gsim 400\ {\rm GeV}$. This indicates the onset of a breakdown of perturbation theory in the Higgs sector of the Standard Model at this surprisingly low value of the Higgs-boson mass.
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 $Π_i^0(M_i^2)$, the renormalization constants $Z_i$, and the bare quartic Higgs coupling $λ_0$ 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 {\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 $λ_s= λ_s(s,M_H^2)$ 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 $λ_s\alt2.3$, but that three-loop corrections are necessary to restore unitarity for larger values of $λ_s$. That is, the interactions in the Higgs sector of the standard model are effectively strong with respect to the perturbative expansion for $λ_s\agt2.3$. The bound $λ_s\alt2.3$ 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.