We investigate the phenomenological consequences of a strict gauge-invariant formulation of the Higgs particle. This requires a description of the observable scalar particle in terms of a bound state structure. Although this seems to be at odds with the common treatment of electroweak particle physics at first glance, the properties of the bound state can be described in a perturbative fashion due to the Fröhlich-Morchio-Strocchi (FMS) framework. In particular a relation between the bound-state Higgs and the elementary Higgs field is obtained within <math><msub><mi>R</mi><mi>ξ</mi></msub></math> gauges such that the main quantitative properties of the conventional description reappear in leading order of the FMS expansion. Going beyond leading order, we show that the pole structure of the elementary and the bound-state propagator coincide to all orders in a perturbative expansion. However, slight deviations of scattering amplitudes containing off-shell Higgs contributions can be caused by the internal bound state structure. We perform a consistent perturbative treatment to all orders in the FMS expansion to quantify such deviations and demonstrate how gauge-invariant expressions arrange in a natural way at the one-loop level. This also provides a gauge-invariant Higgs spectral function which is not plagued by positivity violations or unphysical thresholds. Furthermore, the mass extracted from the gauge-invariant bound state is only logarithmically sensitive to the scale of new physics at one-loop order in contrast to its elementary counterpart.

Austria

Axel Maas

René Sondenheimer

Gauge-invariant description of the Higgs resonance and its phenomenological implications

We apply the method proposed by Fröhlich, Morchio, and Strocchi to analyze the bound state spectrum of various gauge theories with a Brout-Englert-Higgs mechanism. These serve as building blocks for theories beyond the standard model but also stress the exceptional role of the standard model weak group. We will show how the Fröhlich-Morchio-Strocchi mechanism relates gauge-invariant bound state operators to invariant objects of the remaining unbroken gauge group after gauge fixing. In particular, this provides a strict gauge-invariant formulation of the latter in terms of the original gauge symmetry without using the terminology of spontaneous gauge symmetry breaking. We also demonstrate that the Fröhlich-Morchio-Strocchi approach allows us to put new constraints on theories beyond the standard model by pure field-theoretical considerations. Particularly the conventional construction of grand unified theories has to be rethought.

Analytical relations for the bound state spectrum of gauge theories with a Brout-Englert-Higgs mechanism

Recent studies have provided evidence for the existence of new asymptotically free trajectories in non-Abelian particle models without asymptotic symmetry in the high-energy limit. We extend these results to a general $$\mathrm {SU}(N_{\mathrm {L}})\times \mathrm {SU}(N_{\mathrm {c}})$$ <math><mrow><mi>SU</mi><mrow><mo>(</mo><msub><mi>N</mi><mi>L</mi></msub><mo>)</mo></mrow><mo>×</mo><mi>SU</mi><mrow><mo>(</mo><msub><mi>N</mi><mi>c</mi></msub><mo>)</mo></mrow></mrow></math>  Higgs-Yukawa model that includes the non-Abelian sector of the standard model, finding further confirmation for such scenarios for a wide class of regularizations that account for threshold behavior persisting to highest energies. We construct these asymptotically free trajectories within conventional $$\overline{\text {MS}}$$ <math><mover><mtext>MS</mtext><mo>¯</mo></mover></math>  schemes and systematic weak coupling expansions. The existence of these solutions is argued to be a scheme-independent phenomenon, as demonstrated for mass-dependent schemes based on general momentum-space infrared regularizations. A change of scheme induces a map of the theory’s coupling space onto itself, which in the present case also translates into a reparametrization of the space of asymptotically free solutions.

Germany

Holger Gies

Alessandro Ugolotti

Luca Zambelli

Scheme dependence of asymptotically free solutions

Higgs-portal interactions of fermionic dark matter — in contrast to fermions coupled via Yukawa interactions — can have a stabilizing effect on the standard-model Higgs potential. A non-perturbative renormalization-group analysis reveals that, similar to higher-order operators in the Higgs potential itself, the fermionic portal coupling can increase the metastability scale by only about one order of magnitude. Assuming a thermal freeze-out via the Higgs-portal coupling, this regime of very weakly coupled dark matter is in conflict with relic-density constraints. Conversely, fermionic dark matter with the right relic abundance requires either a low cutoff scale of the effective field theory or a strongly interacting scalar sector. This results in a triviality problem in the scalar sector which persists at the non-perturbative level. The corresponding breakdown of the effective field theory suggests a larger dark sector to be present not too far above the dark-fermion mass-scale.

Aaron Held

Higgs stability-bound and fermionic dark matter

$$\mathbb {Z}_2$$ <math><msub><mi>Z</mi><mn>2</mn></msub></math> -Yukawa-QCD models are a minimalistic model class with a Yukawa and a QCD-like gauge sector that exhibits a regime with asymptotic freedom in all its marginal couplings in standard perturbation theory. We discover the existence of further asymptotically free trajectories for these models by exploiting generalized boundary conditions. We construct such trajectories as quasi-fixed points for the Higgs potential within different approximation schemes. We substantiate our findings first in an effective-field-theory approach, and obtain a comprehensive picture using the functional renormalization group. We infer the existence of scaling solutions also by means of a weak-Yukawa-coupling expansion in the ultraviolet. In the same regime, we discuss the stability of the quasi-fixed point solutions for large field amplitudes. We provide further evidence for such asymptotically free theories by numerical studies using pseudo-spectral and shooting methods.

Asymptotic freedom in $$\mathbb {Z}_2$$ <math><msub><mi>Z</mi><mn>2</mn></msub></math> -Yukawa-QCD models

The possible violation of the conventional lower Higgs mass stability bound by the discovered Higgs boson has far reaching consequences within particle physics and cosmology. We discuss the possibility that nonpolynomial bare interactions seeded at some high-momentum scale can considerably diminish the lower Higgs mass bound without introducing a metastability in the Higgs effective potential. For this, we classify various deformations of the usual quartic bare potential regarding their impact on stable IR physics. We perform the analysis in a large $$N_{\mathrm {f}}$$ <math><msub><mi>N</mi><mi>f</mi></msub></math>  expansion, addressing the convergence of the obtained results by taking $$1/N_{\mathrm {f}}$$ <math><mrow><mn>1</mn><mo>/</mo><msub><mi>N</mi><mi>f</mi></msub></mrow></math>  corrections into account as well. In addition, we investigate the renormalization group flow of the scalar potential on a nonperturbative level. Within these approximations, we are able to identify bare potentials that lead to Higgs masses below stability mass bounds obtained from finite-order polynomial bare interactions without introducing a metastability in the effective potential.

Nonpolynomial Higgs interactions and vacuum stability

We investigate the impact of operators of higher canonical dimension on the lower Higgs-mass consistency bound by means of generalized Higgs–Yukawa interactions. Analogously to higher-order operators in the bare Higgs potential in an effective field theory approach, the inclusion of higher-order Yukawa interactions, e.g., ϕ3ψ¯ψ , leads to a diminishing of the lower Higgs-mass bound and thus to a shift of the scale of new physics towards larger scales by a few orders of magnitude without introducing a metastability in the effective Higgs potential. We observe that similar renormalization group mechanisms near the weak-coupling fixed point are at work in both generalizations of the microscopic action. Thus, a combination of higher-dimensional operators with generalized Higgs as well as Yukawa interactions does not lead to an additive shift of the lower mass bound, but it relaxes the consistency bounds found recently only slightly. On the method side, we clarify the convergence properties of different projection and expansion schemes for the Yukawa potential used in the functional renormalization group literature so far.

Japan

Matthias Warschinke

Impact of generalized Yukawa interactions on the lower Higgs-mass bound

We study the renormalization flow of the Higgs potential as a function of both field amplitude and energy scale. This overcomes limitations of conventional techniques that rely, e.g., on an identification of field amplitude and RG scale, or on local field expansions. Using a Higgs–Yukawa model with discrete chiral symmetry as an example, our global flows in field space clarify the origin of possible metastabilities, the fate of the pseudo-stable phase, and provide new information as regards the renormalization of the tunnel barrier. Our results confirm the relaxation of the lower bound for the Higgs mass in the presence of more general microscopic interactions (higher-dimensional operators) to a high quantitative accuracy.

Julia Borchardt

Global flow of the Higgs potential in a Yukawa model

In view of the measured Higgs mass of 125 GeV, the perturbative renormalization group evolution of the Standard Model suggests that our Higgs vacuum might not be stable. We connect the usual perturbative approach and the functional renormalization group which allows for a straightforward inclusion of higher-dimensional operators in the presence of an ultraviolet cutoff. In the latter framework we study vacuum stability in the presence of higher-dimensional operators. We find that their presence can have a sizable influence on the maximum ultraviolet scale of the Standard Model and the existence of instabilities. Finally, we discuss how such operators can be generated in specific models and study the relation between the instability scale of the potential and the scale of new physics required to avoid instabilities.

Canada

United Kingdom

Astrid Eichhorn

Joerg Jaeckel

Tilman Plehn

Michael Scherer

The Higgs mass and the scale of new physics

We study a chiral Yukawa model mimicking the Higgs–top–bottom sector of the standard model. We re-analyze the conventional arguments that relate a lower bound for the Higgs mass with vacuum stability in the light of exact results for the regularized fermion determinant as well as in the framework of the functional renormalization group. In both cases, we find no indication for vacuum instability nor meta-stability induced by top fluctuations if the cutoff is kept finite but arbitrary. A lower bound for the Higgs mass arises for the class of standard bare potentials of ϕ4 type from the requirement of a well-defined functional integral (i.e., stability of the bare potential). This consistency bound can, however, be relaxed considerably by more general forms of the bare potential without necessarily introducing new metastable minima.