Vector Form of Symmetry Degree

We study the symmetries of periodic solutions obtained from Hopf bifurcation in systems with finite abelian symmetries. The H mod K theorem gives necessary and sufficient conditions for the existence of periodic solutions with spatial symmetries K and spatio-temporal symmetries H in systems with finite symmetry group $\Gamma$. Our main result, the abelian Hopf H mod K theorem, gives necessary and sufficient conditions for when these H mod K periodic solutions can occur by Hopf bifurcation when $\Gamma$ is a finite abelian group. We give examples of our results in the case when the symmetry group $\Gamma=\mathbf{Z}_l\times\mathbf{Z}_k$ acts on $\mathbf{R}^l\times\mathbf{R}^k$ by permutation of coordinates. In this case, we classify the H mod K periodic solutions that are obtainable by a generic Hopf bifurcation and show that there exist families of H mod K periodic solutions that cannot be obtained by Hopf bifurcation.

When one attempts to generalize the results of Bost and Connes [BC] to algebraic number fields, one has to face sooner or later the fact that in a number field there is no unique factorization in terms of primes. As is well known, this failure is twofold: the ring of integers has nontrivial units, and even if one considers integers modulo units, (equivalently the principal integral ideals), it turns out that factorization in terms of these can fail too, essentially because 'irreducible' does not mean 'prime'. The first difficulty, with the units, already arises in the situation of [BC], Remark 33.b], but is easily dealt with by considering elements fixed by a symmetry corresponding to complex conjugation. In the existing generalizations the lack of unique factorization has been dealt with in various ways. It has been eliminated, through replacing the integers by a principal ring that generates the same field [HLe], it has been sidestepped, by basing the construction of the dynamical system on the additive integral adeles with multiplication by (a section of) the integral ideles [Coh], and it has been ignored, by considering an almost normal subgroup that makes no reference to multiplication [ALR]. These simplifications make the construction and analysis of interesting dynamical systems possible, but they come at a price. Indeed, the noncanonical choices introduced in [HLe] and [Coh] lead to phase transitions with groups of symmetries that are not obviously isomorphic to actual Galois groups of maximal abelian extensions, and have slightly perturbed zeta functions in the case of [HLe], while the units not included in the almost normal subgroup in [ALR], reappear as a (possibly infinite) group of symmetries under which KMS states have to be invariant, which causes severe difficulties in their computation.

We investigate the phenomenology of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of a system of coupled nonlinear Schrödinger equations(NLSEs) with a combination of linear and nonlinear double-well potentials (DWPs), including self- and cross-phase-modulation (SPM and XPM) nonlinear terms. The DWP structure is represented by a symmetric pair of delta functions. This setup can be realized in optics and for matter waves in Bose-Einstein condensates. It is commonly known that SSB states are completely stable in the single one-dimensional NLSE with a combination of DWP and attractive SPM nonlinearity. The objective of the present work is to elaborate on the SSB and SASB phenomenology in the two-component system. One factor that affects the phenomenology (actually, the stability of stationary states) in this context is the relative sign of the two components, the states with identical/opposite signs defined as ones with in-phase/out-of-phase components. We demonstrate that the SSB/SASB, with both components originating from spatially symmetric or antisymmetric modes, of the out-of-phase type, in the coupled NLSEs including the SPM and XPM terms with opposite signs, are similar to their counterparts in the single NLSE. On the other hand, the SSB/SASB of in-phase states in the coupled NLSEs, including the SPM and XPM terms with opposite signs, are quite different from the ones in the single NLSE. In particular, we find that the in-phase SSB/SASB states are unstable in a part of the intervals of their propagation constantif the system includes SPM and XPM terms with opposite signs. As another feature that is possible only in the two-component system, we address mixed-parity states, originally built of spatially symmetric and antisymmetric components, and find that such states, featuring SSB or SASB, are stable in a part of their interval of the propagation constant. We clarify that the spatially antisymmetric component is a dominant one in determining the asymmetry of the established state if it is coupled to the symmetric one, in the case of the repulsive SPM. On the other hand, the symmetric component dominates if it is coupled to the antisymmetric one in the case of the attractive SPM.

Spontaneous chiral symmetry breaking in QCD with massless quarks at infinite volume can be seen in a finite box by studying, for instance, the dependence of the chiral condensate from the volume and the quark mass. We perform a feasibility study of this program by computing the quark condensate on the lattice in the quenched approximation of QCD at small quark masses. We carry out simulations in various topological sectors of the theory at several volumes, quark masses and lattice spacings by employing fermions with an exact chiral symmetry, and we focus on observables which are infrared stable and free from mass-dependent ultraviolet divergences. The numerical calculation is carried out with an exact variance-reduction technique, which is designed to be particularly efficient when spontaneous symmetry breaking is at work in generating a few very small low-lying eigenvalues of the Dirac operator. The finite-size scaling behaviour of the condensate in the topological sectors considered agrees, within our statistical accuracy, with the expectations of the chiral effective theory. Close to the chiral limit we observe a detailed agreement with the first Leutwyler-Smilga sum rule. By comparing the mass, the volume and the topology dependence of our results with the predictions of the chiral effective theory, we extract the corresponding low-energy constant.

We study physical implications of general CP symmetry including CP-like symmetry. Various scattering amplitudes of CP asymmetry are calculated in CP-like symmetric models. We explicitly show that the CP-like transformation leads to a specific relation between different CP asymmetries. The resultant relation is similar to the one obtained in GUT baryogenesis and sphaleron processes, where we also obtain a required condition for generating particle number asymmetry in CP-like symmetric models. In addition, we propose a generalization of a CP-like transformation for continuous symmetry groups. Since the CP transformation is an outer automorphism, which depends on the internal symmetry group, it turns out that the physical CP and CP-like symmetries can be mutually converted through the spontaneous symmetry breaking (SSB) of the internal symmetry. We investigate properties of physical CP asymmetry in both CP and CP-like symmetric phases, and find that the spontaneous CP violation and restoration can be observed even in models with continuous groups. We demonstrate that CP-like symmetric models with continuous Lie groups can be naturally realized in physical CP symmetric models through the SSB.

Floquet systems often exhibit dynamical symmetries (DS) that govern the time-dependent dynamics and result in selection rules. When a DS is broken, selection rule deviations are expected. Typically, information about the symmetry-breaking perturbation/phase and the time-dependent dynamics can be extracted from these deviations, hence they are regarded as a background free gauge of symmetry breaking. However, to date, DS breaking & selection rule deviations are not described by a general approach, thus there is no universal insight about the interplay between selection rule deviations, the symmetry breaking perturbation, and the broken DS. Here we consider DS breaking in Floquet systems from a general standpoint, formulating a general theory that analytically connects the symmetry-broken and fully symmetric systems. Using an external laser (of arbitrary frequency and polarization), as a model DS breaking perturbation, we discover that the broken symmetry systematically imposes selection rules on the symmetry-broken system, which physically manifest as scaling laws of selection rule deviations. We term these rules ‘selection rules for breaking selection rules’—a new concept in physics. We numerically validate the analytical theory in the context of high harmonic generation. Our discovery is a general feature of nonlinear wave-mixing phenomena, and we expect it to apply to any Floquet system (classical & quantum) and to any DS breaking mechanism (either by intrinsic or extrinsic elements of the system).

Spontaneous chiral symmetry breaking is a ubiquitous property in nature and diverse fields of modern physics. However, such symmetry breaking has been elusive experimentally in the optical systems, which usually demands multiple identical subsystems [1]. As a prominent photonic device, the ultrahigh-Q whispering-gallery mode (WGM) microresonator supports clockwise (CW) and counterclockwise (CCW) propagating waves coupled to each other, leading to symmetric and antisymmetric standing-wave modes with equal CW and CCW amplitudes. The demonstrations of such overall chirality have to rely on external perturbations to a resonator, either by breaking the parity or time-reversal symmetry [2,3]. The chirality with unbalanced CW and CCW components not only attracts general interest in physics, but also is of importance in novel devices [2,3]. Here, we experimentally demonstrate the spontaneous chirality in a single WGM microresonator (Fig. 1) without any explicit breaking of parity or time-reversal symmetry.

Multi-Higgs models equipped with global symmetry groups, either exact or softly broken, offer a rich framework for constructions beyond the Standard Model and lead to remarkable phenomenological consequences. Knowing all the symmetry options within each class of models can guide its phenomenological exploration, as confirmed by the vast literature on the two- and three-Higgs-doublet models. Here, we begin a systematic study of finite non-abelian symmetry groups which can be imposed on the scalar sector of the four-Higgs-doublet model (4HDM) without leading to accidental symmetries. In this work, we derive the full list of such non-abelian groups available in the 4HDM that can be constructed as extensions of cyclic groups by their automorphism groups. This list is remarkably restricted but it contains cases which have not been previously studied. Since the methods we develop may prove useful for other classes of models, we present them in a pedagogical manner.

The paradigm of spontaneous symmetry breaking encompasses the breaking of the rotational symmetries O(3) of isotropic space to a discrete subgroup, i.e., a three-dimensional point group. The subgroups form a rich hierarchy and allow for many different phases of matter with orientational order. Such spontaneous symmetry breaking occurs in nematic liquid crystals, and a highlight of such anisotropic liquids is the uniaxial and biaxial nematics. Generalizing the familiar uniaxial and biaxial nematics to phases characterized by an arbitrary point-group symmetry, referred to as generalized nematics, leads to a large hierarchy of phases and possible orientational phase transitions. We discuss how a particular class of nematic phase transitions related to axial point groups can be efficiently captured within a recently proposed gauge theoretical formulation of generalized nematics [K. Liu, J. Nissinen, R.-J. Slager, K. Wu, and J. Zaanen, Phys. Rev. X 6, 041025 (2016)2160-330810.1103/PhysRevX.6.041025]. These transitions can be introduced in the model by considering anisotropic couplings that do not break any additional symmetries. By and large this generalizes the well-known uniaxial-biaxial nematic phase transition to any arbitrary axial point group in three dimensions. We find in particular that the generalized axial transitions are distinguished by two types of phase diagrams with intermediate vestigial orientational phases and that the window of the vestigial phase is intimately related to the amount of symmetry of the defining point group due to inherently growing fluctuations of the order parameter. This might explain the stability of the observed uniaxial-biaxial phases as compared to the yet to be observed other possible forms of generalized nematic order with higher point-group symmetries.

Systems of ordinary differential equations having a finite symmetry group are considered. One-parameter local bifurcations of symmetric equilibria corresponding to a double pair of purely imaginary eigenvalues are studied. It is shown that in one case a two-dimensional torus is generated from the equilibrium. The torus contains limit cycles; their number does not depend on the values of the parameter. The trajectories of the system that do not leave a certain fixed domain may only tend to the equilibrium under study or to the 2-dimensional torus or to one of two (disjoint) limit cycles. In all the other cases an invariant surface is generated from the equilibrium which is diffeomorphic to the three-dimensional sphere. The behaviour of the trajectories on this surface depends on the symmetry group and is not studied in this paper. In the appendix we provide information on codimension 1 bifurcations corresponding to double zero eigenvalues.

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

Spontaneous discrete symmetry breaking during inflation and the NMSSM domain wall problem

The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Molmer [1]).

The Standard Model of field and particles is the theory that provides the best description of the known phenomenology of the particle physics up to now. Data collected in the last years, mainly by the experiments at the big particle accelerators (SPS, LEP, TEVATRON, HERA, SLAC), allowed to test the agreement between measurements and theoretical calculations with a precision of 10-3 / 10-4. The Standard Model is a Quantum Field Theory based on the gauge symmetry group SU(3)C x SU(2)L x U(1)Y , with spontaneous symmetry breaking. This gauge group includes the color symmetry group of the strong interaction, SU(3)C, and the symmetry group of the electroweak interactions, SU(2)L x U(1)Y. The formulation of the Standard Model as a gauge theory guarantees its renormalizability, but forbids explicit mass terms for fermions and gauge bosons. The masses of the particles are generated in a gauge-invariant way by the Higgs Mechanism via a spontaneous breaking of the electroweak symmetry. This mechanism also implies the presence of a massive scalar particle in the mass spectrum of the theory, the Higgs boson. This particle is the only one, among the basic elements for the minimal formulation of the Standard Model, to have not been confirmed by the experiments yet. For this reason in the last years the scientific community has been focusing an increasing fraction of its efforts on the search of the Higgs boson. The mass of the Higgs boson is a free parameter of the Standard Model, but the unitarity of the theory requires values not higher than 1 TeV and the LEP experiments excluded values smaller than 115 GeV. To explore this range of masses is under construction at CERN the Large Hadron Collider (LHC), a proton-proton collider with a center of mass energy of 14 TeV and a 1034 cm-2 s-1 peak luminosity. According to the present schedule, this machine will start to provide collisions for the experiments at the end of 2008. In the meanwhile the only running accelerator able to provide collisions suitable for the search of the Higgs boson is the Tevatron at Fermilab, a proton-antiproton collider with a center of mass energy of 1.96 TeV working at 3 • 1032cm-2s-1 peak luminosity. These features make the Tevatron able for the direct search of the Higgs boson in the 115-200 GeV mass range. Since the coupling of the Higgs boson is proportional to the masses of the particles involved, the decay in b{bar b} has the largest branching ratio for Higgs mass < 135 GeV and thus the events Z/W + $b\bar{b}$ are the main background to the Higgs signal in the most range favored by Standard Model fits. In this thesis a new technique to identify Heavy Flavour quarks inside high - PT jets is applied to events with a reconstructed Z boson to provide a measurement of the Z+b and Z+c inclusive cross sections. The study of these channels represent also a test of QCD in high transferred momentum regime, and can provide information on proton pdf. This new Heavy Flavour identication technique (tagger) provides an increased statistical separation between b, c and light flavours, using a new vertexing algorithm and a chain of artificial Neural Networks to exploit as much information as possible in each event. For this work I collaborated with the Universita di Roma 'La Sapienza' group working in the CDF II experiment at Tevatron, that has at first developed this tagger. After a brief theoretical introduction (chapter 1) and a description of the experimental apparatus (chapter 2), the tagger itself and its calibration procedure are described in chapter 3 and 4. The chapter 5 is dedicated to the event selection and the chapter 6 contains the results of the measurement and the study of the systematic errors.

We show how trinification models based on the gauge group SU(3)C× SU(3)L× SU(3)R realized near the TeV scale can provide naturally a variety of dark matter (DM) candidates. These models contain a discrete T parity which may remain unbroken even after spontaneous symmetry breaking. The lightest T-odd particle, which could be a fermion, a scalar, or a gauge boson, can constitute the dark matter of the universe. This framework naturally admits a doublet-singlet fermionic DM, a singlet scalar DM, or a vector boson DM. Here we develop the vector boson DM scenario wherein the DM couples off-diagonally with the usual fermions and vector-like fermions present in the theory. We show consistency of this framework with dark matter relic abundance and direct detection limits as well as LHC constraints. We derive upper limits of 900 GeV on the vector gauge boson DM mass and 4.5 TeV on the vector-like quark masses. We also show the consistency of spontaneous gauge symmetry breaking down to Standard Model times an extra U(1) while preserving the T-parity.