The Coleman-Weinberg (CW) renormalization scheme for renormalization-group improvement of the effective potential is particularly valuable for CW symmetry-breaking mechanisms (including the challenging case of models with multiple scalar fields). CW mechanism is typically studied using models with classical scale invariance which not only provide a possibility for an alternative symmetry breaking mechanism but also partially address the gauge hierarchies through dimensional transmutation. As outlined in our discussion section, when the couplings are not large, models with CW symmetry-breaking mechanisms have also been shown to naturally provide the strong first-order phase transition necessary for stochastic gravitational wave signals. A full understanding of the CW-MS scheme transformation of couplings thus becomes important in the era of gravitational wave detection and precision coupling measurements. A generalized Coleman-Weinberg (GCW) renormalization scheme is formulated and methods for transforming scalar self-couplings between the GCW and MS (minimal-subtraction) renormalization schemes are developed. Scalar $λΦ^4$ theory with global $O(4)$ symmetry is explicitly studied up to six-loop order to explore the magnitude of this scheme transformation effect on the couplings. The dynamical rescaling of renormalization scales between the GCW and MS schemes can lead to significant (order of 10\%) differences in the coupling at any order, and consequently GCW-MS scheme transformation effects must be considered within precision determinations of scalar couplings in extensions of the Standard Model.
In Coleman-Weinberg symmetry breaking, all dimensionful parameters vanish and the symmetry is broken by loop corrections. Before Coleman-Weinberg symmetry breaking in the Standard Model was experimentally ruled out, it had already been excluded on cosmological grounds. In this Brief Report, the cosmological analysis is carried out for Coleman-Weinberg models with extended Higgs sectors, which are not experimentally ruled out, and general constraints on such models are given.
We study the finite temperature effective potential of the Higgs scalar in GUTs with Coleman-Weinberg symmetry breaking. The critical temperature is derived without employing a high temperature approximation to the effective potential, and the limitations of such approximations are discussed.
We obtain the renormalization group improvement of the effective potential for the Coleman-Weinberg model by resumming the leading logarithms which have three different mass scales. Then, we investigate the effect of the multi-mass scale on the prediction of the magnitude of the Higgs boson mass by considering the two-loop effective potential.
We perform a non-perturbative study of the Coleman-Weinberg phase transition in scalar QED. Our method permits a consistent treatment of the effective potential near the origin, a region not accessible to perturbation theory. As a result, we establish reliably the first order character of the phase transition for arbitrary values of the scalar coupling.
The Coleman-Weinberg mechanism provides a procedure by which a scalar field, which initially has no mass parameters, acquires a mass due to the anomalous nature of scale symmetry. Loop corrections trigger a spontaneous symmetry breaking and the appearance of a non-trivial vacuum. We first review the basic example of the ColemanWeinberg mechanism, scalar Quantum Electrodynamics, in a perturbative regime where the scalar particle becomes massive through photon loops. We then present the main results of this article, what we name the gravitational Coleman-Weinberg mechanism: we analyse the same e…
In the standard model (SM) the condition that the Higgs mass parameter vanishes is stable under radiative corrections and yields a theory that can be renormalized using dimensional regularization. Thus, this model allows to predict the Higgs boson mass. However, it is phenomenologically ruled out in its minimal version. Here, we present a phenomenologically viable, minimal extension which only includes an additional SM singlet and a U(1)$_X$ gauge symmetry.
We present a realistic non-supersymmetric inflation model based on a gauged U(1)$_{B-L}$ symmetry and a tree-level Higgs potential. The inflaton is identified with the scalar field which spontaneously breaks U(1)$_{B-L}$, and we include radiative corrections à la Coleman-Weinberg in the inflaton potential. If the scalar spectral index $n_s$ lies close to 0.96, as indicated by the recent Planck and WMAP 9-yr measurements, the tensor-to-scalar ratio $r$, a canonical measure for gravity waves, exceeds 0.01. Thus, according to this model, gravity waves should be found in the near future. In this c…
We apply the idea of spiral inflation to Coleman-Weinberg potential, and show that inflation matching well observations is allowed for a symmetry-breaking scale ranging from an intermediate scale to GUT scale even if the quartic coupling $λ$ is of $\mathcal{O}(0.1)$. The tensor-to-scalar ratio can be of $\mathcal{O}(0.01)$ in case of GUT scale symmetry-breaking.
We explore the Coleman-Weinberg phase transition in regions outside the validity of perturbation theory. For this purpose we study a Euclidean field theory with two scalars and discrete symmetry in four dimensions. The phase diagram is established by a numerical solution of a suitable truncation of exact non-perturbative flow equations. We find regions in parameter space where the phase transition (in dependence on the mass term) is of the second or the first order, separated by a triple point. Our quantitative results for the first order phase transition compare well to the standard perturbat…
The tricritical point, which separates first and second order phase transitions in three-dimensional superconductors, is studied in the four-dimensional Coleman-Weinberg model, and the similarities as well as the differences with respect to the three-dimensional result are exhibited. The position of the tricritical point in the Coleman-Weinberg model is derived and found to be in agreement with the Thomas-Fermi approximation in the three-dimensional Ginzburg-Landau theory. From this we deduce a special role of the tricritical point for the Standard Model Higgs sector in the scope of the latest…
The Higgs mechanism may be a quantum phenomenon, i.e., a Coleman-Weinberg potential generated by the explicit breaking of scale symmetry in Feynman loops. We review the relationship of scale symmetry, trace anomalies, and emphasize the role of the renormalization group in determining Coleman- Weinberg potentials. We propose a simple phenomenological model with "maximal visibility" at the LHC containing a "dormant" Higgs doublet (no VEV, coupled to standard model gauge interactions $SU(2)\times U(1)$) with a mass of $\sim 380$ GeV. We discuss the LHC phenomenology and UV challenges of such a mo…
We discuss an extension of the standard model by fields not charged under standard model gauge symmetry in which the electroweak symmetry breaking is driven by the Higgs quartic coupling itself without the need for a negative mass term in the potential. This is achieved by a scalar field S with a large coupling to the Higgs field at the electroweak scale which is driven to very small values at high energies by the gauge coupling of a hidden symmetry under which S is charged. This model can remain perturbative all the way to the Planck scale. The Higgs boson is fully SM-like in its couplings to…
It is possible to delay the hierarchy problem, by replacing the standard Higgs-sector by the Coleman-Weinberg mechanism, and at the same time ensure perturbative naturalness through the so-called Veltman conditions. As we showed in a previous study, minimal models of this type require the introduction of an extra singlet scalar further coupled to new fermions. In this constrained setup the Higgs mass was close to the observed value and the new scalar mass was below a TeV scale. Here we first extend the previous analysis by taking into account the important difference between running mass and p…
We briefly summarize and update a class of inflationary models from the early eighties based on a quartic (Coleman-Weinberg) potential for a gauge singlet scalar (inflaton) field. For vacuum energy scales comparable to the grand unification scale, the scalar spectral index n_s=0.94-0.97, in very good agreement with the WMAP three year results. The tensor to scalar ratio r<~0.14, while alpha=dn/dlnk is =~-10^-3. An SO(10) version naturally explains the observed baryon asymmetry via non-thermal leptogenesis.
We study $SU(8)$ symmetry breaking induced by minimizing the Coleman-Weinberg effective potential for a third rank antisymmetric tensor scalar field in the 56 representation. Instead of breaking $SU(8) \supset SU(3) \times SU(5)$, we find that the stable minimum of the potential breaks the original symmetry according to $SU(8) \supset SU(3) \times Sp(4)$. Using both numerical and analytical methods, we present results for the potential minimum, the corresponding Goldstone boson structure and BEH mechanism, and the group-theoretic classification of the residual states after symmetry breaking.
We present the impact of non-minimal coupling $ξϕ^2 R$ on the inflationary parameters by taking into account the models of single-field inflation with the inflaton that has a non-zero vacuum expectation value ($v$) after the period of inflation in Palatini gravity. We discuss the well-known symmetry breaking type potentials, namely the Higgs potential and Coleman-Weinberg potential. We show the inflationary predictions of these potentials, for both $ϕ>v$ and $ϕ<v$ inflation, the regions in the $v-ξ$ plane for which the values of $n_s$ and $r$ are in agreement with the recent measurements. We a…
We apply the multiple point criticality principle to inflationary model building and study Coleman-Weinberg inflation when the scalar potential is quadratic in the logarithmic correction. We analyze also the impact of a non-minimal coupling to gravity under two possible gravity formulation: metric or Palatini. We compare the eventual compatibility of the results with the final data release of the Planck mission.
We revisit the problem of the gauge invariance in the Coleman-Weinberg model in which a $U(1)$ gauge symmetry is driven spontaneously broken by radiative corrections. It was noticed in previous work that masses in this model are not gauge invariant at one-loop order. In our analysis, we use the dressed propagators of scalars which include a resummation of the one-loop self-energy correction to the tree-level propagator. We calculate the one-loop self-energy correction to the vector meson using these dressed propagators. We find that the pole mass of the vector meson calculated using the dresse…
It has been previously shown that the linear inflation appears naturally as a solution of Coleman-Weinberg inflation, provided that the inflaton has a non-minimal coupling to gravity and the Planck scale is dynamically generated. We revisit the previous study by improving the discussion of reheating and by comparing the results of the metric and the Palatini formulations of non-minimal gravity. We find that both formulations predict linear inflation but a different number of $e$-folds. If the non-minimal coupling is larger than one, future experimental sensitivity can discriminate between the …
We revisit the small field Coleman-Weinberg (CW) inflation, which has the following two problems. First, the smallness of the slow roll parameter $ε$ requires the inflation scale to be very low. Second, the spectral index $n_s \approx1+2 η$ tends to become smaller compared to the observed value. In this letter, we consider two possible effects on the dynamics of inflation: radiatively generated non-minimal coupling to gravity $ξϕ^2 {\cal R}$ and condensation of fermions coupled to the inflaton as $ϕ\barψψ$. We show that the fermion condenate can solve the above problems.
Revised and reduced the size of the text. Removed figure 2 in previous text. So, old figure 3 and 4 become new figure 2 and 3, respectively. Added figure 4 for first order phase transition observation in the strong coupling region. Postscript files for the figures are available upon request. To be published in Phys. Lett. B.
We discuss how the non-minimal coupling $ξϕ^2R$ between the inflaton and the Ricci scalar affects the predictions of single field inflation models where the inflaton has a non-zero vacuum expectation value (VEV) $v$ after inflation. We show that, for inflaton values both above the VEV and below the VEV during inflation, under certain conditions the inflationary predictions become approximately the same as the predictions of the Starobinsky model. We then analyze inflation with double-well and Coleman-Weinberg potentials in detail, displaying the regions in the $v$-$ξ$ plane for which the spect…
We revisit a single field inflationary model based on Coleman-Weinberg potentials. We show that in small field Coleman-Weinberg inflation, the observed amplitude of perturbations needs an extremely small quartic coupling of the inflaton, which might be a signature of radiative origin. However, the spectral index obtained in a standard cosmological scenario turns out to be outside the 2 sigma region of the Planck data. When a non-standard cosmological framework is invoked, such as brane-world cosmology in the Randall-Sundrum model, the spectral index can be made consistent with Planck data with…
We extend the Coleman--Weinberg inflationary model where a scalar field $ϕ$ is non-minimally coupled to gravity with the addition of the $R^2$ term. We express the theory in terms of two scalar fields and going to the Einstein frame we employ the Gildener--Weinberg formalism, compute the one-loop effective potential and essentially reduce the problem to the case of single-field inflation. It turns out that there is only one free parameter, namely, the mixing angle between the scalars. For a wide range of this angle, we compute the inflationary observables which are in agreement with the latest…
We study viable small-field Coleman-Weinberg (CW) inflation models with the help of non-minimal coupling to gravity. The simplest small-field CW inflation model (with a low-scale potential minimum) is incompatible with the cosmological constraint on the scalar spectral index. However, there are possibilities to make the model realistic. First, we revisit the CW inflation model supplemented with a linear potential term. We next consider the CW inflation model with a logarithmic non-minimal coupling and illustrate that the model can open a new viable parameter space that includes the model with …
We study properties of Abrikosov-Nielsen-Olesen (ANO) strings with the Coleman-Weinberg (CW) potential, which we call CW-ANO strings. While the scale-invariant scalar potential has a topologically trivial vacuum admitting no strings at the classical level, quantum correction allows topologically nontrivial vacua and stable string solutions. We find that the system of the CW potential exhibits significant difference from that of the conventional Abelian-Higgs model with the quadratic-quartic potential. While a single-winding string is qualitatively similar in both systems, and the static interv…
Having discovered a candidate for the final piece of the Standard Model, the Higgs boson, the question remains why its vacuum expectation value and its mass are so much smaller than the Planck scale (or any other high scale of new physics). One elegant solution was provided by Coleman and Weinberg, where all mass scales are generated from dimensionless coupling constants via dimensional transmutation. However, the original Coleman-Weinberg scenario predicts a Higgs mass which is too light; it is parametrically suppressed compared to the mass of the vectors bosons, and hence is much lighter tha…
Keio University（庆应义塾大学）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）；Department of Physics, Keio University（庆应义塾大学物理系）；Waseda University（早稻田大学）；Kumamoto Health Science University（熊本县立大学）；Hokkaido University（北海道大学）
We discuss a classically conformal radiative neutrino model with gauged B$-$L symmetry, in which the B$-$L symmetry breaking can occur through the Coleman-Weinberg mechanism. As a result, Majorana mass term is generated and EW symmetry breaking also occurs. We show some allowed parameters to satisfy several theoretical and experimental constraints. Theoretical constraints are inert conditions and Coleman-Weinberg condition. Experimental bounds are lepton flavor violation(especially μ-> e γ), the current bound on the $Z'$ mass at LHC, in additions to the neutrino oscillations.
The five-loop effective potential and the associated summation of subleading logarithms for O(4) globally-symmetric massless $λϕ^4$ field theory in the Coleman-Weinberg renormalization scheme $\frac{d^4V}{dϕ^4}|_{ϕ= μ} = λ$ (where $μ$ is the renormalization scale) is calculated via renormalization-group methods. An important aspect of this analysis is conversion of the known five-loop renormalization-group functions in the minimal-subtraction (MS) scheme to the Coleman-Weinberg scheme.
We discuss how the non-minimal coupling $ξϕ^2 R$ between the inflaton and the Ricci scalar affects predictions of single field inflation models in Palatini formalism. To transition radiation dominated era, the inflaton field $ϕ$ must interact to matter fields at the end of inflation. Interactions of the inflaton with other fields lead to radiative corrections to the inflationary potential. These radiative corrections can be explained at leading order by Coleman-Weinberg (CW) one-loop corrections. In this work, using two different prescriptions debated in the literature, the effect of radiative…
The Standard Model with an added Higgs portal interaction and no explicit mass terms is a classically scale-invariant theory. In this case the scale of electroweak symmetry breaking can be induced radiatively by the Coleman-Weinberg mechanism operational in a hidden sector, and then transmitted to the Standard Model through the Higgs portal. The smallness of the generated values for the Higgs vev and mass, compared to the UV cutoff of our classically scale-invariant effective theory, is naturally explained by this mechanism. We show how these classically conformal models can generate the baryo…
Axionlike particle-assisted supercooling chiral phase transition in QCD: Identifying Coleman-Weinberg type-chiral phase transition in QCD-like scenarios
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心和物理学院）；Center for Fundamental Physics, School of Mechanics and Physics, Anhui University of Science and Technology（安徽理工大学力学与物理学院基础物理中心）；Department of Information and Mathematical Sciences, Tokyo Woman’s Christian University（东京女子基督教大学信息数学科学系）；RIKEN Center for Computational Science（理化学研究所计算科学中心）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）
We continue our study of Coleman-Weinberg symmetry breaking induced by a third rank antisymmetric tensor scalar, in the context of the $SU(8)$ model we proposed earlier. We focus in this paper on qualitative features that will determine whether the model can make contact with the observed particle spectrum. We discuss the mechanism for giving the spin $\frac{3}{2}$ field a mass by the BEH mechanism, and analyze the remaining massless spin $\frac{1}{2}$ fermions, the global chiral symmetries, and the running couplings after symmetry breaking. We note that the smallest gluon mass matrix eigenval…
The perturbative effective potential V in the massless $λϕ^4$ model with a global O(N) symmetry is uniquely determined to all orders by the renormalization group functions alone when the Coleman-Weinberg renormalization condition $\frac{d^4V}{dϕ^4}|_{ϕ= μ} = λ$ is used, where $μ$ represents the renormalization scale. Systematic methods are developed to express the n-loop effective potential in the Coleman-Weinberg scheme in terms of the known n-loop minimal subtraction (MS) renormalization-group functions. Moreover, it also proves possible to sum the leading- and subsequent-to-leading-logarith…
We show that both the Planck and electroweak mass scales can be generated from conformal gravity via the Coleman-Weinberg mechanism of dimensional transmutation. At the first step, the Planck scale is generated via the Coleman-Weinberg mechanism in the sector of conformal gravity, which means that radiative corrections associated with gravitons induce spontaneous symmetry breakdown of a local conformal symmetry. At the second step, the vacuum expectation value of a scalar field is transmitted to the sector of the standard model via a potential involving the conformally invariant part and the c…
In this paper, we derive the standard model with classical conformal invariance from the Yang--Mills--Higgs model in noncommutative geometry (NCG). In the ordinary context of the NCG, the {\it distance matrix} $M_{nm}$ which corresponds to the vacuum expectation value of Higgs fields is taken to be finite. However, since $M_{nm}$ is arbitrary in this formulation, we can take all $M_{nm}$ to be zero. In the original composite scheme, the Higgs field itself vanishes with the condition $M_{nm} = 0$. Then, we adopt the elemental scheme, in which the gauge and the Higgs bosons are regarded as eleme…
We investigate the impact of Coleman-Weinberg inflation on the stochastic gravity wave background spectrum emitted by intermediate scale cosmic strings. The string network is partially inflated and re-enters the horizon at later times after the end of inflation, such that the short string loops are not produced. This leads to a significant modification of the gravity wave spectrum that we explore in detail. We find that Coleman-Weinberg inflation can help to satisfy the Parkes Pulsar Timing Array (PPTA) bound for dimensionless string tension values in the range $Gμ> 1.1\times 10^{-10}$. We als…
We study non-minimal Coleman-Weinberg inflation in the Palatini formulation of gravity in the presence of an $R^2$ term. The Planck scale is dynamically generated by the vacuum expectation value of the inflaton via its non-minimal coupling to the curvature scalar $R$. We show that the addition of the $R^2$ term in Palatini gravity makes non-minimal Coleman-Weinberg inflation again compatible with observational data.
Under the assumption of classical conformal invariance, we study the Coleman-Weinberg symmetry breaking mechanism in the minimal left-right symmetric model. This model is attractive as it provides a natural framework for small neutrino masses and the restoration of parity as a good symmetry of nature. We find that, in a large fraction of the parameter space, the parity symmetry is maximally broken by quantum corrections in the Coleman-Weinberg potential, which are a consequence of the conformal anomaly. As the left-right symmetry breaking scale is connected to the Planck scale through the loga…
We show that the Planck mass scale can be generated from conformal gravity in the Weyl conformal geometry via the Coleman-Weinberg mechanism of dimensional transmutation where quantum corrections stemming from the gravitational field and the Weyl gauge field trigger the symmetry breakdown of a local Weyl symmetry. It is also shown that the vacuum expectation value of a scalar field is transmitted to a sector of the standard model through a potential involving the scale invariant part and the contribution from the Coleman-Weinberg mechanism, thereby generating the electroweak scale.
In the present work we study GUT Coleman-Weinberg inflation with a nonminimal coupling to gravity. In this kind of model one usually finds that either the nonminimal coupling is large or the inflaton self-coupling in unnaturally small. We have shown here that the model is in agreement with the recent results from Planck for natural values of the couplings.
We consider a coupling of dilaton gravity to the classically scale-invariant B-L extended standard model which has been recently proposed as a phenomenologically viable model realizing the Coleman-Weinberg mechanism of breakdown of the electroweak symmetry. It is shown in the present model that without recourse to the Coleman-Weinberg mechanism, the B-L gauge symmetry is broken in the process of spontaneous symmetry breakdown of scale invariance at the tree level and as a result the B-L gauge field becomes massive via the Higgs mechanism. Since the dimensionful parameter is only the Planck mas…
We study gravitational waves from the first-order electroweak phase transition in the $SU(N_c)$ gauge theory with $N_f/N_c\gg 1$ ("large $N_f$ QCD") as a candidate for the walking technicolor, which is modeled by the $U(N_f)\times U(N_f)$ linear sigma model with classical scale symmetry (without mass term), particularly for $N_f=8$ ("one-family model"). This model exhibits spontaneous breaking of the scale symmetry as well as the $U(N_f)\times U(N_f)$ radiatively through the Coleman-Weinberg mechanism $\grave{a}$ la Gildener-Weinberg, thus giving rise to a light pseudo dilaton (techni-dilaton)…
We apply the covariant derivative expansion of the Coleman-Weinberg potential to vector-like fermion models, matching the UV theory to the relevant dimension-6 operators in the standard model effective field theory. The $γ$ matrix induced complication in the fermionic covariant derivative expansion is studied in detail, and all the contributing combinations are enumerated. From this analytical result we also provide numerical constraints for a generation of vector-like quarks and vector-like leptons.
Gildener-Weinberg (GW) models of electroweak symmetry breaking are especially interesting because the low mass and nearly Standard Model couplings of the $125\,{\rm GeV}$ Higgs boson, $H$, are protected by approximate scale symmetry. Another important but so far under-appreciated feature of these models is that a sum rule bounds the masses of the new charged and neutral Higgs bosons appearing in {\em all} these models to be below about $500\,{\rm GeV}$. Therefore, they are within reach of LHC data currently or soon to be in hand. Also so far unnoticed of these models, certain cubic and quartic…
We obtain the renormalization group(RG) functions for the $O(N)$ scalar field theory and the Higgs-Yukawa field theory with the Coleman-Weinberg mechanism in which the symmetry breaking occurs radiatively by using the method proposed previously in case of the neutral scalar field theory.
There is a surprising connection between the top quark and Higgs alignment in Gildener-Weinberg multi-Higgs doublet models. Were it not for the top quark and its large mass, the coupling of the 125 GeV Higgs boson $H$ to gauge bosons and fermions would be indistinguishable from those of the Standard Model Higgs. The top quark's coupling to a single Higgs doublet breaks this perfect alignment in higher orders of the Coleman-Weinberg loop expansion of the effective potential. But the effect is still small, $< {\cal{O}}(1\%)$, and probably experimentally inaccessible.
We consider a coupling of conformal gravity to the classically scale-invariant B-L extended standard model which has been recently proposed as a phenomenologically viable model realizing the Coleman-Weinberg mechanism of breakdown of the electro-weak symmetry. As in a globally scale-invariant dilaton gravity, it is also shown in a locally scale-invariant conformal gravity that without recourse to the Coleman-Weinberg mechanism, the B-L gauge symmetry is broken in the process of spontaneous symmetry breakdown of the local scale invariance (Weyl invariance) at the tree level and as a result the …
The Coleman-Weinberg mechanism can realise different phases of dynamical symmetry breaking. In each phase a combination of scalars, corresponding to the pseudo-Goldstone boson of scale invariance, has a loop-suppressed mass. We show that additional scalars, beyond the pseudo-Goldstone bosons, can become light at critical points in the parameter space where two different phases co-exist. We present a minimal implementation of the mechanism in multi-scalar models, detailing how loop-suppressed masses and mixings can be computed. We discuss realisations of the resulting multi-phase criticality pr…
The phase transition "Coulomb-confinement" in the U(1) regularized gauge theory is considered in the framework of dual Abelian Higgs model of scalar monopoles (shortly: Higgs monopole model). The effective potential analogous to the Coleman-Weinberg's one, improved by the renormalization group equation, is constructed, using beta-functions in the two-loop approximation. The value of QED critical fine structure constant was calculated: $α_{crit}\approx 0.208$, in correspondence with lattice result: $α_{crit}^{lat}\approx 0.20\pm 0.015$.
Radiaitve mechanism of conformal symmetry breaking in a comformal-invariant version of the Standard Model is considered. The Coleman-Weinberg mechanism of dimensional transmutation in this system gives rise to finite vacuum expectation values and, consequently, masses of scalar and spinor fields. A natural bootstrap between the energy scales of the top quark and Higgs boson is suggested.
