Nambu-Goldstone Fields Research Articles

Complete CP eigen-bases of mesonic chiral Lagrangian up to p8-order

Chiral perturbation theory systematically describes the low energy dynamics of meson and baryons using nonlinear Nambu-Goldstone fields. Using the Young tensor technique, we construct the pure mesonic effective operators up to p8-order, one-to-one corresponding to contact amplitudes with the on-shell Adler zero condition. The off-shell external sources, non-vanishing under equation-of-motion conditions, are also added to the operator bases. We also show the invariant tensor bases using the Young tableau is equivalent to the trace bases with Cayley-Hamilton relations. Separated into different CP eigenstates, at Op8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({p}^8\\right) $$\\end{document} we obtain the operator lists of the 567 C+P+ operators, 483 C+P- operators, 376 C-P+ operators, and 408 C-P- operators for SU(2) case, while there are 1959 C+P+ operators, 1809 C+P- operators, 1520 C-P+ operators, and 1594 C-P- operators for SU(3) case, consistent with results using the Hilbert series.

Self-gravitating SU(5) Higgs domain walls as a braneworld

Five-dimensional domain walls in gauged $SU(5)$ generate a position-dependent symmetry breaking pattern along the additional dimension. We analyze the perturbative stability and the four-dimensional (4D) spectrum of these walls in the self-gravitating case, in terms of diffeomorphism-invariant and Lie algebra gauge-invariant field fluctuations. We show that tachyonic modes are absent, ensuring perturbative stability. As expected, gravitational tensor and vector fluctuations behave like their counterparts in the standard ${Z}_{2}$ domain walls. All the Lie algebra valued fluctuations exhibit towers of 4D massive modes, which propagate in the bulk, with a continuous spectrum starting from zero. All the would-be 4D Nambu-Goldstone fields, which are gravitationally trapped in the case of a global symmetry, are nontrivially absent. However, we find no localizable 4D gauge bosons, either massless or massive. Instead, quasilocalizable discrete 4D massive modes for the gauge field fluctuations are found, along the spontaneously broken directions.

P-Branes with AdS_{p+1} vacuum as models of R^2 gravity

Branes with constant mean curvature of their hyper-worldsheets of codimension 1 are treated as the Nambu-Goldstone fields of the broken Poincare symmetry. Mapping of their action into quadratic curvature gravity action with spontaneously generated gravity, is shown. Equation for the brane potential extremals and its solution describing hyper-ws of constant curvature are found. For membranes in {mathbf {R}}^{1,3} this extremum is shown to be a saddle 3-dim. hypersurface which defines classically instable vacuum.

Dynamical generation of three flavor mixing

In this paper we study patterns of dynamical symmetry breaking for a class of models characterized by SU(3)V × SU(3)A × U(1)V chiral-flavor symmetry. In particular, we show how dynamical mixing generation implies the existence of exotic condensates on vacuum, mixing fermion and antifermions with different flavors. By means of Ward–Takahashi identities we also derive Haag expansions for Nambu–Goldstone fields.

Solving puzzles of spontaneously broken spacetime symmetries

We establish a classical analog of the Nambu-Goldstone theorem for spontaneous breaking of spacetime symmetries. It provides a counting rule for independent Nambu-Goldstone fields and states which of them are gapped. We demonstrate that only those symmetry group generators give rise to independent Nambu-Goldstone fields that act non-trivially on a vacuum at the origin of coordinates. Other generators give rise to auxiliary fields that must be excluded from a theory by the means of inverse Higgs constraints. The physical meaning of the inverse Higgs phenomenon and an application of our results to theories of massive gravity are discussed.

Coset space construction for the conformal group

The technique for constructing conformally invariant theories within the coset space construction is developed. It reproduces all consequences of the conformal invariance and Lagrangians of widely-known conformal field theories. The method of induced representations, which plays the key role in the construction, allows to reveal a special role of the "Nambu-Goldstone fields" for special conformal transformations. Namely, their dependence on the coordinates turns out to be fixed by the symmetries. This results in the appearance of the constraints on possible forms of Lagrangians, which ensure that discrete symmetries are indeed symmetries of the theory.

On massive Nambu-Goldstone fields

We consider two theories undergoing the same spontaneous symmetry breaking pattern but with different representations of an order parameter under the Lorentz group. The effective description of the first theory includes a massive Nambu-Goldstone field, while in the second example it is absent. Based on this, we clarify the physical meaning of massive Nambu-Goldstone fields as non-radial modes needed to describe the fluctuations of an order parameter. The connection with the inverse Higgs phenomenon is also discussed.

Exploring dynamical gluon mass generation in three dimensions

In the d=3 gluon mass problem in pure-glue non-Abelian $SU(N)$ gauge theory we pay particular attention to the observed (in Landau gauge) violation of positivity for the spectral function of the gluon propagator. This causes a large bulge in the propagator at small momentum. Mass is defined through $m^{-2}=\Delta (p=0)$, where $\Delta(p)$ is the scalar function for the gluon propagator in some chosen gauge, it is not a pole mass and is generally gauge-dependent, except in the gauge-invariant Pinch Technique (PT). We truncate the PT equations with a new method called the vertex paradigm that automatically satisfies the QED-like Ward identity relating the 3-gluon PT vertex function with the PT propagator. The mass is determined by a homogeneous Bethe-Salpeter equation involving this vertex and propagator. This gap equation also encapsulates the Bethe-Salpeter equation for the massless scalar excitations, essentially Nambu-Goldstone fields, that necessarily accompany gauge-invariant gluon mass. The problem is to find a good approximate value for $m$ and at the same time explain the bulge, which by itself leads, in the gap equation for the gluon mass, to excessively large values for the mass. Our point is not to give a high-accuracy determination of $m$ but to clarify the way in which the propagator bulge and a fairly accurate estimate of $m$ can co-exist, and we use various approximations that illustrate the underlying mechanisms. The most critical point is to satisfy the Ward identity. In the PT we estimate a gauge-invariant dynamical gluon mass of $m \approx Ng^2/(2.48 \pi)$. We translate these results to the Landau gauge using a background-quantum identity involving a dynamical quantity $\kappa$ such that $m=\kappa m_L$, where $m_L^{-2}\equiv \Delta_L(p=0)$. Given our estimates for $m,\kappa$ the relation is fortuitously well-satisfied for $SU(2)$ lattice data.

Effective field theory for spacetime symmetry breaking

We discuss the effective field theory for spacetime symmetry breaking from the local symmetry point of view. By gauging spacetime symmetries, the identification of Nambu-Goldstone (NG) fields and the construction of the effective action are performed based on the breaking pattern of diffeomorphism, local Lorentz, and (an)isotropic Weyl symmetries as well as the internal symmetries including possible central extensions in nonrelativistic systems. Such a local picture distinguishes, e.g., whether the symmetry breaking condensations have spins and provides a correct identification of the physical NG fields, while the standard coset construction based on global symmetry breaking does not. We illustrate that the local picture becomes important in particular when we take into account massive modes associated with symmetry breaking, whose masses are not necessarily high. We also revisit the coset construction for spacetime symmetry breaking. Based on the relation between the Maurer-Cartan one form and connections for spacetime symmetries, we classify the physical meanings of the inverse Higgs constraints by the coordinate dimension of broken symmetries. Inverse Higgs constraints for spacetime symmetries with a higher dimension remove the redundant NG fields, whereas those for dimensionless symmetries can be further classified by the local symmetry breaking pattern.

Nambu–Goldstone dynamics and generalized coherent-state functional integrals

This paper gives a new method of attack on the Nambu–Goldstone dynamics in spontaneously broken theories. Since the target space of the Nambu–Goldstone fields is a group coset space, their effective quantum dynamics can be naturally phrased in terms of generalized coherent-state functional integrals. As an explicit example of this line of reasoning, we construct a low-energy effective Lagrangian for the Heisenberg ferromagnet in a broken phase. The leading field configuration in the WKB approximation leads to the Landau–Lifshitz equation for quantum ferromagnet. The corresponding linearized equations allow one to identify the Nambu–Goldstone boson with a ferromagnetic magnon.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Nonlinear realization of N=2 superconformal symmetry and brane effective actions

Since there is an incompatibility of simultaneously nonlinear breaking the superconformal symmetry and the dilatation symmetry with the dilaton taken as the compensator field, in the present paper is shown an alternative mechanism of spontaneous breaking the N=2 superconformal symmetry to the N=0 case. By using the approach of nonlinear transformations one finds that it leads to a space-filling brane theory with Weyl scale W(1,3) symmetry. The dynamics of the resulting Weyl scale invariant brane, along with that of other Nambu–Goldstone fields, is derived in terms of the building blocks of the vierbein and the covariant derivative from the Maurer–Cartan one-forms. A general coupling of the matter fields localized on the brane world volume to these NG fields is also constructed.

Inhomogeneity driven by Higgs instability in a gapless superconductor

The fluctuations of the Higgs and pseudo Nambu-Goldstone fields in the 2SC phase with mismatched pairing are described in the nonlinear realization framework of the gauged Nambu--Jona-Lasinio model. In the gapless 2SC phase, not only Nambu-Goldstone currents can be spontaneously generated, but the Higgs field also exhibits instablity. The Nambu-Goldstone currents generation indicates the formation of the single plane wave LOFF state and breaks rotation symmetry, while the Higgs instability favors spatial inhomogeneity and breaks translation invariance. In this paper, we focus on the Higgs instability which has not drawn much attention yet. The Higgs instability cannot be removed without a long range force, thus it persists in the gapless superfluidity and induces phase separation. In the case of g2SC state, the Higgs instability can only be partially removed by the electric Coulomb energy. However, it is not excluded that the Higgs instability might be completely removed in the charge neutral gCFL phase by the color Coulomb energy.

Nonlinear realization and Weyl scale invariantp=2brane

The action of Weyl scale invariant $p=2$ brane which breaks the target super-Weyl scale symmetry in the $N=1$, $D=4$ superspace down to the lower dimensional Weyl symmetry $W(1,2)$ is derived by the approach of nonlinear realization. The dual form action for the Weyl scale invariant supersymmetric D2 brane is also constructed. The interactions of localized matter fields on the brane with the Nambu-Goldstone fields associated with the breaking of the symmetries in the superspace and one spatial translation directions are obtained through the Cartan one-forms of the Coset structures. The covariant derivatives for the localized matter fields are also obtained by introducing Weyl gauge field as the compensating field corresponding to the local scale transformation on the brane world volume.

Tensor model and dynamical generation of commutative non-associative fuzzy spaces

The rank-three tensor model may be regarded as a theory of dynamical fuzzy spaces, because a fuzzy space is defined by a three-index coefficient of the product between functions on it, fa*fb = Cabcfc. In this paper, this proposal is applied to the dynamical generation of commutative non-associative fuzzy spaces. It is numerically shown that fuzzy flat tori and fuzzy spheres of various dimensions are classical solutions of the rank-three tensor model. Since these solutions are obtained for the same coupling constants of the tensor model, the cosmological constant and the dimensions are not fundamental but can be regarded as dynamical quantities. The symmetry of the model under the general linear transformation can be identified with a fuzzy analogue of the general coordinate transformation symmetry in general relativity. This symmetry of the tensor model is broken at the classical solutions. This feature may make the model a concrete finite setting for applying the old idea of obtaining gravity as Nambu–Goldstone fields of the spontaneous breaking of the local translational symmetry.

Chiral Lagrangian with confinement from the QCD Lagrangian

An effective Lagrangian for the light quark in the field of a static source is derived systematically using the exact field correlator expansion. The lowest Gaussian term is bosonized using nonlocal colorless bosonic fields and a general structure of effective chiral Lagrangian is obtained containing all set of fields. The new and crucial result is that the condensation of scalar isoscalar field which is a usual onset of chiral symmetry breaking and is constant in space-time, assumes here the form of the confining string and contributes to the confining potential, while the rest bosonic fields describe mesons with the q\bar q quark structure and pseudoscalars play the role of Nambu-Goldstone fields. Using derivative expansion the effective chiral Lagrangian is deduced containing both confinement and chiral effects for heavy-light mesons. The pseudovector quark coupling constant is computed to be exactly unity in the local limit,in agreement with earlier large N_c arguments.

Non-linear realizations of superconformal and W-algebras as embeddings of strings

We propose a simple method for constructing representations of (super) conformal and non-linear W-type algebras in terms of their subalgebras and corresponding Nambu-Goldstone fields. We apply it to N = 2 and N = 1 superconformal algebras and describe in this way various embeddings of strings and superstrings for which these algebras and their subalgebras define world-sheet symmetries. Besides reproducing the known examples, we present some new ones, in particular an embedding of the bosonic string with additional U(1) affine symmetry into N = 2 superstring. We also apply our method to the non-linear W 3 (2) algebra and demonstrate that the linearization procedure worked out for it some time ago gets a natural interpretation as a kind of string embedding. All these embeddings include the critical ones as particular cases.

Non-linear realizations of superconformal and W-algebras as embeddings of strings

We propose a simple method for constructing representations of (super)conformal and nonlinear W-type algebras in terms of their subalgebras and corresponding Nambu-Goldstone fields. We apply it to N=2 and N=1 superconformal algebras and describe in this way various embeddings of strings and superstrings for which these algebras and their subalgebras define world-sheet symmetries. Besides reproducing the known examples, we present some new ones, in particular an embedding of the bosonic string with additional U(1) affine symmetry into N=2 superstring. We also apply our method to the nonlinear $W_3^{(2)}$ algebra and demonstrate that the linearization procedure worked out for it some time ago gets a natural interpretation as a kind of string embedding. All these embeddings include the critical ones as particular cases.

Nambu-Goldstone fields, anomalies and WZ terms

We construct the Wess-Zumino terms of anomalous gauge theories in cases the anomalous transformations do not form a subgroup or in situations where the separation between anomalous and non-anomalous transformations cannot be done in a local way. The extra variables corresponding to all gauge degrees of freedom are introduced so that the WZ term possesses additional gauge symmetries. It effectively kills the degrees of freedom of non-anomalous transformations, leaving those of anomalous transformations as propagating fields. Our formalism can be seen as a generalization of the non-linear realization method of Lie groups and applicable also to effective gauge field theories of Nambu-Goldstone fields associated with spontaneously broken global symmetries. As an example we discuss the BRST invariant formulation of 21) gravity in the Weyl invariant regularization and reproduce the local form of the WZ term.

Regularization at the Next-to-Leading Order in the Top-Mode Standard Model without Gauge Bosons

We study Higgs condensation H∼ttin the top-mode standard model at the next-to-leading (NTL) order in 1/Ncby calculating the effective potential as a function of a hard mass termσ0of the top quark. We include the effects of the third generation quarks, the Higgs and the Nambu–Goldstone fields, and the leading quantum chromodynamic effects, but not the effects of the transverse components of the electroweak gauge bosons. The resulting effective theory contains finite energy cutoff parameters (Λf, Λb) for the fermionic and the bosonic degrees of freedom. Condensation is supposed to take place at energiesΛ∼Λf∼Λb. The paper describes how to regularize the integrals over the fermionic momenta in a way free of momentum branching ambiguities and how to treat the terms of 1/Ncexpansion in a mutually consistent manner. This is achieved by the proper time approach, employing specifically the proper time cutoff (PTC) or a Pauli–Villars (PV) regulator. For comparison, we use also the covariant spherical (S) cutoff. On the other hand, for the bosonic momenta we have to use the covariant spherical cutoff. We discuss how to ensure the validity of the Goldstone theorem. Demanding that the NTL contributions not “wash out” the leading ones, we obtain rather low bounds for the cutoffs:Λ=O (1 TeV). The results for the corresponding cases with PTC, PV, and S regularization differ only marginally. Thus, in the described framework, the 1/Ncexpansion approach has a predictive power only ifΛ=O (1 TeV), a result largely independent of the regularization procedure.

Spontaneous Breakdown of N=1 Superconformal Symmetry into Conformal Symmetry

Within a framework of the non-linear realization, we investigate the transformation law of the Nambu-Goldstone fields and their effective lagrangian when N =1 superconformal symmetry is spontaneously broken down to conformal symmetry. The effective lagrangian of the Nambu-Goldstone fields contains only non-linear terms and no quadratic kinetic terms.