We continue our analysis of the general N-Higgs-doublet model and focus of the Higgs potential description in the space of gauge orbits. We develop a geometric technique that allows us to study the global minimum of the potential without explicitly finding its position. We discuss symmetry patterns of the NHDM potential, and illustrate the general discussion with various specific variants of the three-Higgs-doublet model.
We study the scalar sector of the general N-Higgs-doublet model via geometric constructions in the space of gauge orbits. We give a detailed description of the shape of the orbit space both for general N and, in more detail, for N=3. We also comment on remarkable analogies between NHDM and quantum information theory.
We prove that a generic multi-Higgs-doublet model (NHDM) generally must contain terms in the potential that violate the custodial symmetry. This is done by showing that the O(4) violating terms of the NHDM potential cannot be excluded by imposing a symmetry on the NHDM Lagrangian. Hence we expect higher-order corrections to necessarily introduce such terms. We also note, in the case of custodially symmetric Higgs-quark couplings, that vacuum alignment will lead to up-down mass degeneration; this is not true if the vacua are not aligned.
Conditions for CP violation in the scalar potential sector of general N-Higgs-doublet models (NHDMs) are analyzed from a group theoretical perspective. For the simplest two-Higgs-doublet model (2HDM) potential, a minimum set of conditions for explicit and spontaneous CP violation is presented. The conditions can be given a clear geometrical interpretation in terms of quantities in the adjoint representation of the basis transformation group for the two doublets. Such conditions depend on CP-odd pseudoscalar invariants. When the potential is CP invariant, the explicit procedure to reach the rea…
We derive basis-independent, necessary and sufficient conditions for the custodial symmetry in N-Higgs-doublet models (NHDM) for N > 2, and apply them on some 3HDM examples.
After elimination of the redundant variables, gauge theories may still exhibit symmetries associated with the gauge fields. The role of these residual gauge symmetries is discussed within the Abelian Higgs model and the Georgi-Glashow model. In the different phases of these models, these symmetries are realized differently. The characteristics of emergence and disappearance of the symmetries are studied in detail and the implications for the dynamics in Coulomb, Higgs, and confining phases are discussed.
We continue to explore the consequences of the recently discovered Minkowski space structure of the Higgs potential in the two-Higgs-doublet model. Here, we focus on the vacuum properties. The search for extrema of the Higgs potential is reformulated in terms of 3-quadrics in the 3+1-dimensional Minkowski space. We prove that 2HDM cannot have more than two local minima in the orbit space and that a twice-degenerate minimum can arise only via spontaneous violation of a discrete symmetry of the Higgs potential. Investigating topology of the 3-quadrics, we give concise criteria for existence of n…
Models addressing the naturalness of a light Higgs boson typically employ symmetries, either bosonic or fermionic, to stabilize the Higgs mass. We consider a setup with the minimal amount of symmetries: four shift symmetries acting on the four components of the Higgs doublet, subject to the constraints of linearly realized SU(2)xU(1) electroweak symmetry. Up to terms that explicitly violate the shift symmetries, the effective lagrangian can be derived, irrespective of the spontaneously broken group G in the ultraviolet, and is universal in all models where the Higgs arises as a pseudo-Nambu-Go…
B.W.Lee/K.Symanzik proved that Ward-Takahashi identities and tadpole renormalization force all ultra-violet quadratic divergences (UV-QD) to be absorbed into the physical renormalized pseudo-scalar pion mass in O(4)LSM (linear sigma models) across the Higgs-VEV vs. Pion-Mass-Squared half-plane. We show that all UV-QD vanish identically in the "Goldstone mode" Zero-Pion-Mass spontaneous symmetry breaking (SSB) limit. The Higgs mass is protected to all loop-orders and Goldstone-mode O(4)LSM has no Higgs fine-tuning problem (HFTP). We insist that self-consistent renormalization of the Standard Mo…
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等技术学院）；Instituto de Física Corpuscular (IFIC), CSIC‐Universitat de València（瓦伦西亚大学西班牙国家研究委员会粒子物理研究所）；University of Bergen（卑尔根大学）
Extra-dimensional components of gauge fields in higher-dimensional gauge theories will play a role of the Higgs field and become tachyonic after Kaluza-Klein compactifications on internal spaces with (topologically nontrivial) gauge field backgrounds. Its condensation is then expected to break gauge symmetries spontaneously. But, contrary to the expectation, some models exhibit restoration of gauge symmetries. In this paper, by considering all the massive Kaluza-Klein excitations of gauge fields, we explicitly show that some of them indeed become massless at the minimum of the Higgs potential …
We introduce and systematically study an expansive class of "orbifold Higgs" theories in which the weak scale is protected by accidental symmetries arising from the orbifold reduction of continuous symmetries. The protection mechanism eliminates quadratic sensitivity of the Higgs mass to higher scales at one loop (or more) and does not involve any new states charged under the Standard Model. The structures of the Higgs and top sectors are universal and determined exclusively by group theoretical considerations. The twin Higgs model fits within our framework as the simplest example of an orbifo…
We continue the study of the nonrelativistic short-distance completions of a naturally light Higgs, focusing on the interplay between the gauge symmetries and the polynomial shift symmetries. We investigate the naturalness of nonrelativistic scalar quantum electrodynamics with a dynamical critical exponent $z=3$ by computing leading power law divergences to the scalar propagator in this theory. We find that power law divergences exhibit a more refined structure in theories that lack boost symmetries. Finally, in this toy model, we show that it is possible to preserve a fairly large hierarchy b…
We consider six-dimensional supergravity with gauge group $SO(10)\times U(1)_A$, compactified on the orbifold $T^2 / \mathbb{Z}_2$. Three quark-lepton generations arise as zero modes of a bulk $\bf16$-plet due to magnetic flux of the anomalous $U(1)_A$. Boundary conditions at the four fixed points break $SO(10)$ to subgroups whose intersection is the Standard Model gauge group. The gauge and Higgs sector consist of "split" $SO(10)$ multiplets. As consequence of the $U(1)_A$ flux, squarks and sleptons are much heavier than gauge bosons, Higgs bosons, gauginos and higgsinos. We thus obtain a pic…
We study leptons in holographic composite Higgs models, namely in models possibly admitting a weakly coupled description in terms of five-dimensional (5D) theories. We introduce two scenarios leading to Majorana or Dirac neutrinos, based on the non-abelian discrete group $S_4\times \Z_3$ which is responsible for nearly tri-bimaximal lepton mixing. The smallness of neutrino masses is naturally explained and normal/inverted mass ordering can be accommodated. We analyze two specific 5D gauge-Higgs unification models in warped space as concrete examples of our framework. Both models pass the curre…
It has been long known that when spacetime symmetry is spontaneously broken, some of the broken generators may not give rise to independent gapless, Nambu-Goldstone excitations. We provide two complementary viewpoints of this phenomenon. On the one hand, we show that the corresponding field degrees of freedom have the same symmetry transformation properties as massive, matter fields. The "inverse Higgs constraints", sometimes employed to eliminate these modes from the theory, are reinterpreted as giving a field parametrization that makes these transformation properties manifest. On the other h…
We study F-Theory compactifications to four dimensions that exhibit discrete gauge symmetries. Geometrically these arise by deforming elliptic fibrations with two sections to a genus-one fibration with a bi-section. From a four-dimensional field-theory perspective they are remnant symmetries from a Higgsed U(1) gauge symmetry. We implement such symmetries in the presence of an additional SU(5) symmetry and associated matter fields, giving a geometric prescription for calculating the induced discrete charge for the matter curves and showing the absence of Yukawa couplings that are forbidden by …
We present a toy model that exhibits clash-of-symmetries style Higgs field kink configurations in a Randall-Sundrum-like spacetime. The model has two complex scalar fields Phi_{1,2}, with a sextic potential obeying global U(1)xU(1) and discrete Phi_1 <--> Phi_2 interchange symmetries. The scalar fields are coupled to 4+1 dimensional gravity endowed with a bulk cosmological constant. We show that the coupled Einstein-Higgs field equations have an interesting analytic solution provided the sextic potential adopts a particular form. The 4+1 metric is shown to be that of a smoothed-out Randall-Sun…
We perform a systematic study of generic accidental Higgs-family and CP symmetries that could occur in the two-Higgs-doublet-model potential, based on a Majorana scalar-field formalism which realizes a subgroup of GL(8,C). We derive the general conditions of convexity and stability of the scalar potential and present analytical solutions for two non-zero neutral vacuum expectation values of the Higgs doublets for a typical set of six symmetries, in terms of the gauge-invariant parameters of the theory. By means of a homotopy-group analysis, we identify the topological defects associated with t…
The fundamental Higgs doublet may be replaced in the Standard Model by certain non-perturbative four-quark interactions, whose effect is to induce a composite Higgs sector responsible for electroweak symmetry breaking. A simple composite two-Higgs-doublet model is presented. The four-quark interactions arise naturally if there are either extra spatial dimensions or larger gauge symmetries at a multi-TeV scale. Some theoretical and phenomenological implications of these scenarios are discussed.
Gauge symmetries are known to be respected by gravity because gauge charges carry flux lines, but global charges do not carry flux lines and are not conserved by gravitational interaction. For discrete symmetries, they are spontaneously broken in the Universe, forming domain walls. Since the realization of discrete symmetries in the Universe must involve the vacuum expectation values of Higgs fields, a string-like configuration (hair) at the intersection of domain walls in the Higgs vacua can be realized. Therefore, we argue that discrete charges are also respected by gravity.
We describe a model of quarks which identifies the large global symmetries of little Higgs models with the global flavor symmetries that arise in a deconstruction of the extra-dimensional 'topological insulator' model of flavor. The nonlinearly realized symmetries of little Higgs theories play a critical role in determining the flavor structure of fermion masses and mixing. All flavor physics occurs at the few TeV scale in this model, yet flavor changing neutral currents arising from the new physics are naturally smaller than those generated radiatively in the standard model, without having to…
An intriguing feature of the Standard Model is that the representations of the unbroken gauge symmetries are vector-like whereas those of the spontaneously broken gauge symmetries are chiral. Here we provide a toy model which shows that a natural explanation of this property could emerge in higher dimensional field theories and discuss the difficulties that arise in the attempt to construct a realistic theory. An interesting aspect of this type of models is that the 4D low energy effective theory is not generically gauge invariant. However, the non-invariant contributions to the observable qua…
Nonlinearly realized Abelian global symmetries can be reformulated as local shift symmetries gauged by three-form gauge fields. The anomalous symmetries of the Standard Model (such as Peccei-Quinn or $B+L$) can be dualized to local symmetries gauged by the Chern-Simons three-forms of the Standard Model gauge group. In this description the strong CP problem can be reformulated as the problem of a massless three-form field in QCD, which creates an arbitrary CP-violating constant four-form electric field in the vacuum. Both the axion as well as the massless quark solutions amount to simply Higgsi…
Outer automorphisms of symmetries ("symmetries of symmetries") in relativistic quantum field theories are studied, including charge conjugation (C), space-reflection (P) , and time-reversal (T) transformations. The group theory of outer automorphisms is pedagogically introduced and it is shown that CP transformations are special outer automorphisms of the global, local, and space-time symmetries of a theory. It is shown that certain discrete groups allow for a group theoretical prediction of parameter independent CP violating complex phases with fixed geometrical values. The remainder of this …
The embedding of a $p$-brane into higher dimensional spacetime breaks not only translational symmetries transverse to the worldvolume, but also Lorentz symmetries. There exist formulations for $p$-brane actions which associate Goldstone bosons with the generators of the broken Lorentz symmetries. These Goldstone bosons are unphysical, in that they can be eliminated in favour of other Goldstone bosons either via their equations of motion or via the imposition of an inverse Higgs constraint. In this paper, we examine the inter-relationship between the coset parameterization necessary to implemen…
We describe the theoretical ideas, developed between the 1950s-1970s, which led to the prediction of the Higgs boson, the particle that was discovered in 2012. The forces of nature are based on symmetry principles. We explain the nature of these symmetries through an economic analogy. We also discuss the Higgs mechanism, which is necessary to avoid some of the naive consequences of these symmetries, and to explain various features of elementary particles.
We investigate the asymptotic symmetry group of a scalar field minimally-coupled to an abelian gauge field using the Hamiltonian formulation. This extends previous work by Henneaux and Troessaert on the pure electromagnetic case. We deal with minimally coupled massive and massless scalar fields and find that they behave differently insofar as the latter do not allow for canonically implemented asymptotic boost symmetries. We also consider the abelian Higgs model and show that its asymptotic canonical symmetries reduce to the Poincaré group in an unproblematic fashion.
A retrospective historical overview of the phenomenon of spontaneous symmetry breaking (SSB) in quantum theory, the issue that has been implemented in particle physics in the form of the Higgs mechanism. The main items are: -- The Bogoliubov's microscopical theory of superfluidity (1946); -- The BCS-Bogoliubov theory of superconductivity (1957); -- Superconductivity as a superfluidity of Cooper pairs (Bogoliubov - 1958); -- Transfer of the SSB into the QFT models (early 60s); -- The Higgs model triumph in the electro-weak theory (early 80s). The role of the Higgs mechanism and its status in th…
We study four dimensional N=1 gauge theories that arise on the worldvolume of D3-branes probing complex cones over del Pezzo surfaces. Global symmetries of the gauge theories are made explicit by using a correspondence between bifundamental fields in the quivers and divisors in the underlying geometry. These global symmetries are hidden, being unbroken when all inverse gauge couplings of the quiver theory vanish. In the broken phase, for finite gauge couplings, only the Cartan subalgebra is manifest as a global symmetry. Superpotentials for these models are constructed using global symmetry in…
Higher-form symmetries have proved useful in constraining the dynamics of a number of quantum field theories. In the context of the Argyres-Douglas (AD) theories of the $(G,G')$ type, we find that the 1-form symmetries are invariant under the Higgs branch flow, and that they are captured by the non-Higgsable sector at a generic point on the Higgs branch of the AD theory in question. As a consequence, dimensional reduction of an AD theory with a non-trivial 1-form symmetry to 3d leads to a free sector. We utilize these observations, along with other results, to propose systematically the mirror…
Many models of beyond Standard Model physics connect flavor symmetry with a discrete group. Having this symmetry arise spontaneously from a gauge theory maintains compatibility with quantum gravity and can be used to systematically prevent anomalies. We minimize a number of Higgs potentials that break gauge groups to discrete symmetries of interest, and examine their scalar mass spectra.
We give a solution to the long-standing problem of constructing the generators of hidden symmetries of the quantum Higgs oscillator, a particle on a d-sphere moving in a central potential varying as the inverse cosine-squared of the polar angle. This superintegrable system is known to possess a rich algebraic structure, including a hidden SU(d) symmetry that can be deduced from classical conserved quantities and degeneracies of the quantum spectrum. The quantum generators of this SU(d) have not been constructed thus far, except at d=2, and naive quantization of classical conserved quantities l…
In contrast to internal symmetries, there is no general proof that the coset construction for spontaneously broken spacetime symmetries leads to universal dynamics. One key difference lies in the role of Goldstone bosons, which for spacetime symmetries includes a subset which are inessential for the non-linear realisation and hence can be eliminated. In this paper we address two important issues that arise when eliminating inessential Goldstones. The first concerns the elimination itself, which is often performed by imposing so-called inverse Higgs constraints. Contrary to claims in the litera…
It is shown that conformally invariant theories can be obtained within the framework of the coset space construction. The corresponding technique is applicable for the construction of representations of the unbroken conformal group, as well as of a spontaneously broken one. A special role of the "Nambu-Goldstone fields" for special conformal transformations is clarified - they ensure self-consistency of a theory by guaranteeing that discrete symmetries are indeed symmetries of the theory. A generalization of the developed construction to a special class of symmetry groups with a non-linear rea…
We apply different integrability analysis procedures to a reduced (spatially homogeneous) mechanical system derived from an off-shell non-minimally coupled N=2 Maxwell-Chern-Simons-Higgs model that presents BPS topological vortex excitations, numerically obtained with an ansatz adopted in a special - critical coupling - parametric regime. As a counterpart of the regularity associated to the static soliton-like solution, we investigate the possibility of chaotic dynamics in the evolution of the spatially homogeneous reduced system, descendant from the full N=2 model under consideration. The ori…
We describe a new kind of transition between topologically distinct $N=2$ type II Calabi--Yau vacua through points with enhanced non-abelian gauge symmetries together with fundamental charged matter hypermultiplets. We connect the appearance of matter to the local geometry of the singularity and discuss the relation between the instanton numbers of the Calabi--Yau manifolds taking part in the transition. In a dual heterotic string theory on $K3\times T^2$ the process corresponds to Higgsing a semi-classical gauge group or equivalently to a variation of the gauge bundle. In special cases the si…
We study spontaneous gauge symmetry breaking and the Higgs mechanism in nonlocal field theories. Motivated by the level truncated action of string field theory, we consider a class of nonlocal field theories with an exponential factor of the d'Alembertian attached to the kinetic and mass terms. Modifications of this kind are known to make mild the UV behavior of loop diagrams and thus have been studied not only in the context of string theory but also as an alternative approach to quantum gravity. In this paper we argue that such a nonlocal theory potentially includes a ghost mode near the non…
In gauge theory, Higgs fields are responsible for spontaneous symmetry breaking. In classical gauge theory on a principal bundle P, a symmetry breaking is defined as the reduction of a structure group of this principal bundle to a subgroup H of exact symmetries. This reduction takes place iff there exists a global section of the quotient bundle P/H. It is a classical Higgs field. A metric gravitational field exemplifies such a Higgs field. We summarize the basic facts on the reduction in principal bundles and geometry of Higgs fields. Our goal is the particular covariant differential in the pr…
We address emergent higher-form symmetry in Higgs phases with superfluidity. The emergent symmetry appears if a matter field is invariant under a transformation of a common subgroup of gauge and global $\mathrm{U}(1)$ symmetries. We explicitly construct the symmetry generator that is topological and gauge invariant in a low-energy effective theory. Such emergent symmetry is helpful to distinguish the Higgs and other phases with superfluidity, such as phases of QCD at finite density. We also discuss the possibility of phase transition between these phases.
Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.
A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode, is a collective excitation that arises in a system involving spontaneous breaking of a continuous symmetry, along with a gapless Nambu-Goldstone mode. It has been known in the previous studies that a pure amplitude MG mode emerges in superconductors if the dispersion of fermions exhibits the particle-hole (p-h) symmetry. However, clear understanding of the relation between the symmetry of the Hamiltonian and the MG modes has not been reached. Here we reveal the fundamental connection between the discrete symmetry of th…
