Gauge invariance and fermion doubling on the lattice

(1) We consider a possible chiral invariant solution of the lattice fermion doubling problem. This makes the unwanted states decouple in the continuum limit, at least in the non-interacting theory. The introduction of gauge interactions restores doubling. We examine how local gauge invariance makes all the species in a doubled spectrum act alike. (2) We generalise earlier results to show how gauge invariance forces a doubled spectrum on us even when other considerations do not.     

Discrete symmetries in Kaluza-Klein theories

We investigate discrete symmetries in theories of higher-dimensional (d > 4) gravity and their consequences for the reduced four-dimensional theory, obtained for a ground state which is a direct product of four-dimensional Minkowski space and a compact d ? 4 dimensional internal space. If the action of pure d-dimensional gravity coupled to spinors is invariant under time reversal or reflection of an odd number of spacelike co-ordinates, the reduced four-dimensional theory has a non-trivial parity or CT symmetry not consistent with observation. A non-trivial d-dimensional charge conjugation results in an unwanted doubling of the four-dimensional fermion spectrum. As a consequence, realistic theories can only be obtained for Majorana-Weyl spinors in d = 2 mod 8 dimensions. The constraints are less stringent if supplementary fields are introduced in d dimensions. For d = 11 supergravity, for example, parity and CT invariance can be broken by a non-vanishing field strength of the totally antisymmetric three-index tensor.A ground state invariant under reflections of “internal” co-ordinates often gives rise to a non-trivial charge conjugation in four dimensions. We find that the ground state of a realistic Kaluza-Klein theory should not be invariant under any non-trivial internal co-ordinate reflection (which cannot be obtained by a gauge transformation). We finally comment on a possible solution of the strong-CP problem from Kaluza-Klein theories and discuss prospectives for finding internal spaces admitting chiral fermions.     

Confining but chirally symmetric dense and cold matter

The possibility for existence of cold, dense chirally symmetric matter with confinement is reviewed. The answer to this question crucially depends on the mechanism of mass generation in QCD and interconnection of confinement and chiral symmetry breaking. This question can be clarified from spectroscopy of hadrons and their axial properties. Almost systematical parity doubling of highly excited hadrons suggests that their mass is not related to chiral symmetry breaking in the vacuum and is approximately chirally symmetric. Then there is a possibility for existence of confining but chirally symmetric matter. We clarify a possible mechanism underlying such a phase at low temperatures and large density. Namely, at large density the Pauli blocking prevents the gap equation to generate a solution with broken chiral symmetry. However, the chirally symmetric part of the quark Green function as well as all color non-singlet quantities are still infrared divergent, meaning that the system is with confinement. A possible phase transition to such a matter is most probably of the first order. This is because there are no chiral partners to the lowest lying hadrons.     

Chiral dynamics and meson with non-commutative dipole field in gauge/gravity dual

Apply the T-duality and smeared twist to the D3-brane solution one can construct the supergravity backgrounds which may dual to supersymmetric or non-supersymmetric non-commutative dipole field theory. We introduce D7-brane probe into the dual supergravity background to study the chiral dynamics and meson spectrum therein. We first find that the non-commutative dipole field does not induce the chiral symmetry breaking even if the supersymmetry was completely broken, contrast to the conventional believing that the chiral symmetry will be broken in the non-supersymmetric theory. Next, we find that the dipole field does not modify the meson spectrum in the supersymmetric theory while it will reduce the meson bound-state energy in the non-supersymmetric theory. We also evaluate the static quark–anti-quark potential and see that the dipole field has an effect to produce attractive force between the quark and anti-quark.     

Introduction to Topological Phases and Electronic Interactions in (2+1) Dimensions

A brief introduction to topological phases is provided, considering several two-band Hamiltonians in one and two dimensions. Relevant concepts of the topological insulator theory, such as: Berry phase, Chern number, and the quantum adiabatic theorem, are reviewed in a basic framework, which is meant to be accessible to non-specialists. We discuss the Kitaev chain, SSH, and BHZ models. The role of the electromagnetic interaction in the topological insulator theory is addressed in the light of the pseudo-quantum electrodynamics (PQED). The well-known parity anomaly for massless Dirac particle is reviewed in terms of the Chern number. Within the continuum limit, a half-quantized Hall conductivity is obtained. Thereafter, by considering the lattice regularization of the Dirac theory, we show how one may obtain the well-known quantum Hall conductivity for a single Dirac cone. The renormalization of the electron energy spectrum, for both small and large coupling regime, is derived. In particular, it is shown that massless Dirac particles may, only in the strong correlated limit, break either chiral or parity symmetries. For graphene, this implies the generation of Landau-like energy levels and the quantum valley Hall effect.     

Reflection positivity and graviton doubling in Euclidean lattice gravity

Lattice Euclidean quantum gravity, formulated in analogy to the usual gauge theories, is considered. We prove that positivity is satisfied only for a special set of quantities which, however, in the continuum limit furnish the expectation values of the closed loops and correlation functions thereof. We work out the perturbative limit by expanding around a flat background in order to examine the particle content of the theory. A doubling phenomenon (analogous to the doubling of lattice chiral fermions) appears; such a phenomenon is shown to be of general nature for a class of lattice formulations of gravity.     

Parity doubling and SU(2)LxSU(2)R restoration in the hadron spectrum

We construct the most general nonlinear representation of chiral SU(2)LxSU(2)R broken down spontaneously to the isospin SU(2), on a pair of hadrons of same spin and isospin and opposite parity. We show that any such representation is equivalent, through a hadron field transformation, to two irreducible representations on two hadrons of opposite parity with different masses and axial-vector couplings. This implies that chiral symmetry realized in the Nambu-Goldstone mode does not predict the existence of degenerate multiplets of hadrons of opposite parity nor any relations between their couplings or masses.     

Chirality in rotational energy level clusters

Hougen’s role in developing our understanding of the symmetry classification of the energy levels of molecules is briefly reviewed. For chiral molecules the application of these symmetry ideas shows that chiral states have mixed parity. High angular momentum states of H₂X molecules, in which rotational energy level clusters form, do have mixed parity and are chiral. We show here that such states can have long lifetimes.     

Electroweak interactions in chiral molecules: two-component density functional theory study of vibrational frequency shifts in polyhalomethanes

In this paper, a two-component density functional theory study of parity violation-induced vibrational frequency shifts in chiral polyhalomethanes is reported and the prospects to detect in these compounds, for the first time, signals of parity violation in molecular systems are discussed. The recent synthesis of enantiomerically enriched CHClFI has renewed interest in examining electroweak corrections for this class of compounds. Utilizing a (quasi-relativistic) two-component zeroth-order regular approximation approach to molecular parity violation, together with density functional theory, the parity violation-induced relative vibrational frequency splittings Δ ν_pv/ν between the C–F stretching fundamental of polyhalomethane enantiomers are computed. The relative splitting in CHClFI is raised compared with CHBrClF, for which upper bounds were determined experimentally. To facilitate measurement, molecules possessing more pronounced relative splittings are desirable. Therefore, the chiral methane derivative CHAtFI is considered, which exhibits a significantly larger electroweak contribution that induces a vibrational frequency splitting on the order of the experimental resolution previously reported for CHBrClF. Employing compounds containing heavy nuclei such as astatine may thus be necessary with present detection methods.     

A supersymmetric gluon model

It is pointed out that parity doubling does not provide a satisfactory resolution of the conflict between parity and fermion-number conservation in supersymmetric gauge theories. A new generalized gauge principle is proposed which overcomes this difficulty for both Abelian and non-Abelian local symmetries.     

Parity doubling among the baryons

We study the evidence for and possible origins of parity doubling among the baryons. First we explore the experimental evidence, finding a significant signal for parity doubling in the non-strange baryons, but little evidence among strange baryons. Next we discuss potential explanations for this phenomenon. Possibilities include suppression of the violation of the flavor singlet axial symmetry (U(1_)A$U(1{)}_A$) of QCD, which is broken by the triangle anomaly and by quark masses. A conventional Wigner–Weyl realization of the SU(2_)L×SU(2_)R$\mathit{SU}(2{)}_L\times \mathit{SU}(2{)}_R$ chiral symmetry would also result in parity doubling. However this requires the suppression of families of chirally invariant   operators by some other dynamical mechanism. In this scenario the parity doubled states should decouple from pions. We discuss other explanations including connections to chiral invariant short distance physics motivated by large _Nc$N_c$ arguments as suggested by Shifman and others, and intrinsic deformation of relatively rigid highly excited hadrons, leading to parity doubling on the leading Regge trajectory. Finally we review the spectroscopic consequences of chiral symmetry using a formalism introduced by Weinberg, and use it to describe two baryons of opposite parity.     

QCD vacuum as a disordered medium: a simplified model for the QCD dirac operator

We model the QCD Dirac operator as a power-law random banded matrix (RBM) with the appropriate chiral symmetry. Our motivation is the form of the Dirac operator in a basis of instantonic zero modes with a corresponding gauge background of instantons. We compare the spectral correlations of this model to those of an instanton liquid model (ILM) and find agreement well beyond the Thouless energy. In the bulk of the spectrum the dimensionless Thouless energy of the RBM scales with the square root of system size in agreement with the ILM and chiral perturbation theory. Near the origin the scaling in the RBM remains the same as in the bulk which agrees with chiral perturbation theory but not with the ILM. Finally we discuss how this RBM should be modified in order to describe the spectral correlations of the QCD Dirac operator at the finite temperature chiral restoration transition.     

N and Δ parity doublets in the baryon spectrum

The N and Δ excitation spectrum exhibits parity doublets, i.e. states of equal total angular momentum but with opposite parity being almost degenerate in mass. Among others, it has been suggested by L.Ya. Glozman that the parity-doublet structure may be due to effective chiral restoration.     

Chiral-symmetry realization for even- and odd-parity baryon resonances

Baryon resonances with even and odd parity are collectively investigated from the viewpoint of chiral symmetry (ChS). We propose a quartet scheme where Delta's and N(*)'s with even and odd parity form a chiral multiplet. This scheme gives parameter-free constraints on the baryon masses in the quartet, which are consistent with observed masses with spin 1 / 2,3 / 2,5 / 2. The scheme also gives selection rules in the one-pion decay: The absence of the parity nonchanging decay N(1720)-->piDelta(1232) is a typical example which should be confirmed experimentally to unravel the role of ChS in baryon resonances.     

CN REALIZATION OF THE CHIRAL SYMMETRY IN THE LATTICE GAUGE THEORY CONTAINING THE U(I) CHIRAL GAUGE FIELD

We investigate the lattice gauge theory with a gauge group UxV(1) where U is a non-Abelian group and V(1) is a chiral Abelian group.In the strong coupling approximation for the U gauge field we find that the V(1) chiral symmetry is realized through parity doublets of composite fermions for strong V(1) coupling and that it is spontaneousely broken through a techni color-like mechanism for weak V(1) coupling.     

Spin,parity, and nature of the xi(1620) resonance

Using a unitary extension of chiral perturbation theory with a lowest-order s-wave SU(3) chiral Lagrangian we study low-energy meson-baryon scattering in the strangeness S=-2 sector. A scattering-matrix pole is found around 1605 MeV which corresponds to an s-wave Xi resonance with J(P)=1/2(-). We identify this resonance with the Xi(1620) state, quoted by the Particle Data Group with I=1/2 but with unknown spin and parity. The addition of the S=-2 state to the recently computed Lambda(1670), Sigma(1620), and N(1535) states completes the octet of J(P)=1/2(-) resonances dynamically generated in this chiral unitary approach.     

A Re-Examination of the Arguments for Baryon Parity Doublets

The arguments for baryon parity doubling are re-examined by use of a spin formalism developed by the author together with B. E. LAURENT . It is shown that the transformation between good parity amplitudes and helicity flip and non-flip amplitudes in πN scattering is ambiguous. It is further shown that the unitary condition for boson-fermion scattering gives no information of the parity of intermediate states contributing to a good parity amplitude. These ambiguities follow from the fact that for boson-fermion scattering the CPT transformation of an amplitude does not commute with the projection on a definite parity. It is finally shown how a Regge pole model with linear trajectories without parity doubling can be constructed without any specific dynamical assumptions or artifices. The constraints on the residues needed in this case do not violate any general principles.     

The U(1) anomaly in a chiral invariant lattice action

A chiral invariant lattice fermion action which violates reflection positivity and avoids the doubling of fermion species is studied in perturbation theory in two-dimensional spacetime. The absence of anomalies in the lattice regularization is shown to be compatible with the identification of current operators which reproduce the correct (axial) vector current Ward identities.     

Composite massless fermions in strongly coupled lattice gauge theories

We show that a class of strongly coupled lattice gauge theories with fermions in real representations of the gauge group do not have chiral symmetry breaking. The resulting spectrum of massless composite fermions satisfies 't Hooft's constraints if the model is naively extrapolated to the continuum limit. We argue that it is in fact the correct spectrum of the continuum gauge theory.     

Lattice QCD with exponentially small chirality breaking

A new multifermion formulation of lattice QCD is proposed. The model is free of spectrum doubling and preserves all non-anomalous chiral symmetries up to exponentially small corrections. It is argued that a small number of fermion fields may provide a good approximation making computer simulations feasible.