Higgs scalar potential in the exponential parametrization in arbitrary gauge.
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We study the parametrization and gauge dependences in the Higgs field coupled to gravity in the context of asymptotic safety. We use the exponential parametrization to derive the fixed points for the cosmological constant, Planck mass, Higgs mass and its coupling, keeping arbitrary gauge parameters $\alpha$ and $\beta$. We find that the beta functions for the Higgs potential are expressed in terms of redefined Planck mass such that the apparent gauge dependence is absent. Only the trace mode of the gravity fluctuations couples to the Higgs potential and it tends to decouple in the large $\beta$ limit, but the anomalous dimension becomes large, invalidating the local potential approximation. There are also singularities for some values of the gauge parameters but well away from these, we find rather stable fixed points and critical exponents. We thus find that there are regions for the gauge parameters to give stable fixed points and critical exponents against the change of gauge parameters. The Higgs coupling is confirmed to be irrelevant for the reasonable choice of gauge parameters.Keywords:
Planck mass
Parametrization (atmospheric modeling)
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We consider both the Abelian Higgs model and the impact of a minimal length in the un-particle sector. It is shown that even if the Higgs field takes a non-vanishing v.e.v., gauge interaction keeps its long range character leading to an effective gauge symmetry restoration. The effect of a quantum gravity induced minimal length on a physical observable is then estimated by using a physically-based alternative to the usual Wilson loop approach. Interestingly, we obtain an ultraviolet finite interaction energy described by a confluent hyper-geometric function, which shows a remarkable richness of behavior.
Minimal models
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We show that, despite of the reparametrization symmetry of the Lagrangian describing the interaction between a scalar field and gauge vector bosons, the dynamics of the Higgs mechanism is really affected by the representation gauge chosen for the Higgs field. Actually, we find that, varying the parametrization for the two degrees of freedom of the complex scalar field, we obtain different expressions for the Higgs mass: in its turn this entails different expressions for the critical temperatures, ranging from zero to a maximum value, as well as different expressions for other basic thermodynamical quantities.
Parametrization (atmospheric modeling)
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BRST quantization
Quantum gauge theory
Gauge covariant derivative
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It has been known for some time that General Relativity can be regarded as a Yang-Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an SO(1,4) or SO(2,3) gauge field $A^{a}_{\ph{a}b\mu}$ and Higgs field $V^{a}$ which acts to break the symmetry down to that of the Lorentz group SO(1,3). This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field $\Phi$ of electroweak theory is taken as a genuine dynamical field satisfying a Klein-Gordon equation, the gauge independent component $V^2$ of the Higgs-type field $V^a$ is typically regarded as non-dynamical. Instead, in many treatments $V^a$ does not appear explicitly in the formalism or is required to satisfy $V^2\equiv \eta_{ab}V^{a}V^{b}=const.$ by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection $A^{a}_{\ph{a}b\mu}$ and Higgs field $V^a$ as genuine dynamical fields. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar-tensor model of gravity. A specific choice based on the symmetry group SO(1,4) yields a positive cosmological constant and an effective mass $M$ of the gravitational Higgs field ensuring the constancy of $V^2$ at low energies and agreement with empirical data if $M$ is sufficiently large. More general actions are discussed corresponding to variants of Chern-Simons modified gravity and scalar-Euler form gravity.
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We study four-dimensional gauge theories coupled to fermions in the fundamental and meson-like scalars. All requisite beta functions are provided for general gauge group and fermion representation. In the regime where asymptotic freedom is absent, we determine all interacting fixed points using perturbation theory up to three loop in the gauge and two loop in the Yukawa and quartic couplings. We find that the conformal window of ultraviolet fixed points is narrowed-down by finite-$N$ corrections beyond the Veneziano limit. We also find a new infrared fixed point whose main features such as scaling exponents, UV-IR connecting trajectories, and phase diagram are provided. Both fixed points collide upon varying the number of fermion flavours $N_{\rm f}$, and conformality is lost through a saddle-node bifurcation. We further revisit the prospect for ultraviolet fixed points in the large $N_{\rm f}$ limit where matter field fluctuations dominate. Unlike at weak coupling, we do not find clear evidence for new scaling solutions even in the presence of scalar and Yukawa couplings.
Asymptotic freedom
Yukawa potential
Infrared fixed point
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The one-loop effective action of the Abelian and non-Abelian Higgs models has been studied in various gauges, in the context of instanton and sphaleron transition, bubble nucleation, and most recently in nonequilibrium dynamics. Gauge invariance is expected on account of Nielsen's theorem if the classical background field is an extremum of the classical action, i.e., a solution of the classical equation of motion. We substantiate this general statement for the one-loop effective action, as computed using mode functions. We show that in the gauge-Higgs sector there are two types of modes that satisfy the same equation of motion as the Faddeev-Popov modes. We apply the general analysis to the computation of the fluctuation determinant for bubble nucleation in the SU(2) Higgs model in the 't Hooft gauge with general gauge parameter $\ensuremath{\xi}.$ We show that due to the cancellation of the modes mentioned above, the fluctuation determinant is independent of $\ensuremath{\xi}.$
Sphaleron
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We enumerate the micro-states in Higgs theories, addressing (i) the number of vacuum states and (ii) the appropriate measure in the quantum path integral. To address (i) we explicitly construct the set of ground state wave-functionals in the field basis focussing on scalar modes $\theta(x)$. Firstly, we show that in the limit in which the gauge coupling is zero, we obtain an infinite set of degenerate ground states at large volume distinguished by $\theta(x)\to\theta(x)+\theta_0$, spontaneously breaking the global symmetry, as is well known. We then show that at finite gauge coupling there is a unique ground state at large volume since the wave-functional only depends on $\nabla\theta$ in the IR, and we explain this at the level of the Lagrangian. Since gauge fields fall off exponentially from sources there are no conserved charges or symmetries in the Higgs phase; so the Higgs mechanism is the removal of symmetry from the theory. We show how physical features of defects, such as cosmic strings in the abelian Higgs model, are best understood in this context. Since there is a unique ground state, we address (ii) whether the path integral is a volume measure for the radial Higgs field $\mathcal{D}\rho\,\rho^{N-1}$ from the $N$ components of the Higgs multiplet, or a line measure $\mathcal{D}\rho$ as the $N-1$ would-be Goldstones can be removed in unitary gauge. We prove that the need to avoid quartic divergences demands a tower of counter terms that resum to exactly give the volume measure. So the size of the Hilbert space in the zero gauge coupling case and finite gauge coupling case are in one-to-one correspondence, despite the degeneracy of the ground state being lifted in the latter. As a cosmological application, we point out that the volume measure can make it exponentially more unlikely in $N(=4)$ for the Standard Model Higgs to relax to the electroweak vacuum in the early universe.
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Dilaton
Quintessence
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We construct asymptotically free renormalization group trajectories for the generic non-Abelian Higgs model in four-dimensional spacetime. These ultraviolet-complete trajectories become visible by generalizing the renormalization/boundary conditions in the definition of the correlation functions of the theory. Though they are accessible in a controlled weak-coupling analysis, these trajectories originate from threshold phenomena which are missed in a conventional perturbative analysis relying on the deep Euclidean region. We identify a candidate three-parameter family of renormalization group trajectories interconnecting the asymptotically free ultraviolet regime with a Higgs phase in the low-energy limit. We provide estimates of their low-energy properties in the light of a possible application to the standard model Higgs sector. Finally, we find a two-parameter subclass of asymptotically free Coleman-Weinberg-type trajectories that do not suffer from a naturalness problem.
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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 Fr\"ohlich-Morchio-Strocchi (FMS) framework. In particular a relation between the bound-state Higgs and the elementary Higgs field is obtained within ${R}_{\ensuremath{\xi}}$ 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.
Invariant mass
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