Chiral Invariance Research Articles

Introducing thermal excitations for color potentials.

A difficulty appears when introducing the temperature for a chiral-invariant color potential V(r) between quarks. The naive gap equation at finite temperature, obtained from a minimum principle and a Bogoliubov approximation applied to the free energy, has the following problems: (1) in the limit T\ensuremath{\rightarrow}0, it is inconsistent with the gap equation formulated directly at T=0; (2) it is not invariant under the transformation V(r)\ensuremath{\rightarrow}V(r)+const, as one would expect if the Hilbert space is restricted to color singlets, as required by confinement. These difficulties are solved if one requires the thermal excitations to be global color singlets, or, equivalently, if one restricts to the color singlets the trace in the calculation of the free energy. We obtain the corresponding gap equation in the infinite-volume thermodynamic limit. This equation has now a T\ensuremath{\rightarrow}0 limit that possesses the same ground-state solution as the zero-temperature gap equation. Moreover, in the case of a confining potential the same ground-state solution remains when we switch on the temperature, so that chiral invariance is not restored at any value of the temperature. Because translational invariance is assumed, particle-hole thermal excitations are constructed as color-singlet pairs of plane waves that, due to the confining interaction, possess infinite energy. The particle-hole pairs cannot be excited with a finite cost of energy, preventing chiral-symmetry restoration at any temperature.

Lowering of proton mass with respect to neutron state ing′-quark electromagnetic solition theory—I

The modification of electrodynamics for theg′-quark permitting chiral invariance, CA, and PCAC to hold affords to relate the nucleon electromagnetic mass splitting to the pionic case previously treated by Ward's identities. This gives correct sign and order of magnitude for this effect and finite positive proton charge.

Quantum theory of a massless spinning particle

The classical and quantum mechanics of a free massless point-like fermion is presented. The action is invariant under local 1-dimensional reparametrizations and supersymmetry, as well as a large set of rigid symmetries, including conformal and chiral symmetries. An infinite set of rigid symmetries is derived starting from the chiral invariance, but only a finite subset does not vanish on shell. The BRST-transformations corresponding to thed=1 local symmetries are constructed and the quantization is performed. Finally, the BRST-cohomology is investigated and the conditions producing the physical states are derived.

Phase structure of generalized Gross-Neveu models

A phase structure of models (1) withn spinor multiplets has been considered in the space-time of dimensionD=2,3. In the case whenn=2 andD=3 there may occur vacuums |+> violating chiral invariance, as well as |−> violatingP,T symmetry of the model. Atn, D=2 there may exist a phase, where a hierarchy of fermion multiplet masses arises in a dynamical way. The properties of Gross-Neveu model have been dealt with atD=3,n=1 and the temperature and chemical potential not equal to zero.

A New Electronic Pinning Mechanism for Sliding Charge Density Waves

AbstractIt is shown that the time component of the chiral Noether current is a constant of motion in a quasi‐one‐dimensional charge density wave (CDW) system in the absence of a disordered potential. Therefore this system possesses a current‐carrying equilibrium state which is characterized as the minimum of an accordingly generalized thermodynamic potential. A random impurity potential breaks the chiral invariance so that a current‐carrying equilibrium state in the strict sense does not exist in this case. It is assumed, however, that the sliding CDW state can approximately be described as a quasi‐equlibrium in the coordinate frame co‐moving with the velocity ν. It is shown that the density of states in the co‐moving frame exhibits tails in the Peierls gap which enhance the thermodynamic potential of the quasi‐equilibrium proportional to |ν| so that the current‐driving Lagrange parameter in this potential must exceed some threshold value before the sliding CDW state is thermodynamically favoured against the resting one. This is an electronic pinning mechanism which acts independently of the conventional phase pinning.

What is a realistic picture of hadrons?

• Chiral invariance has been a strong guide in constructing the meson presence in nuclei. Whereas low-energy theorems have pinned down isovector exchange currents, by gauging the baryon density anomaly, a good description at both low and high momenta of the deuteron (isoscalar) magnetic form factor has been achieved. • At low energies, both the nucleon and plun can be divided into an interior quark core and exterior meson cloud. At high energies one sees the quarks in both regions. The quark core in strange baryons tends to be somewhat larger than in the nucleon because there is little pion cloud to compress it. From the anomalously large tensor coupling of the p-meson to the nucleon, the nucleon quark core is found to have a radius R ∼ 0.5 fm> • A case is made that nonperturbative QCD should be used for low-energy descriptions, although perturbative QCD may be applicable to systems with strangeness. • It is argued that the exclusion principle operating at the quark level does not have very large effects in low-energy nuclear physics. • Two-phase models, with a Wigner (perturbative) phase at short distances should be used to properly incorporate the asymptotic freedom property of QCD. • Through nonperturbative QCD, elegant and deep connections can be made between nuclear physics phenomena and the QCD anomalies.

1/N expansion in the Gross-Neveu model with arbitrary number of fermion multiplets

The two-dimensional four-fermion theory with arbitrary number of spinor field multiplets is considered in the leading order of the 1/N expansion. This model is asymptotically free, dynamical breaking of chiral invariance occurs in it, and the fermions acquire masses. In the particular case of two multiplets, it is shown that there exist at least two different phases with massive fermions. The conditions under which the theory does not have ghosts are found. In the case of equal fermion masses, the masses of the bosonic states are calculated.

Particular solutions of the equations for the lowest Green's functions in the infrared region of quantum chromodynamics

The solutions of the Schwinger-Dyson equations for the ghost and quark propagators in the infrared region of quantum chromodynamics are studied. It is shown that there exists a gauge in which the equation for the ghost propagator has the free solution. This result is used to justify the possibility of omitting the contributions of the ghosts from the gauge identity for the quark-gluon vertex in this distinguished gauge. Under this condition, a new class of solutions to the equation for the quark propagator is obtained; the class includes solutions with broken chiral invariance that are manifestly nonanalytic with respect to the coupling constant.

Bifurcation of the quark self-energy: Infrared and ultraviolet cutoffs.

The quark self-energy in massless QCD is studied in the approximation that both the quark-gluon vertex and the gluon propagator remain bare. It is shown that chiral invariance is not spontaneously broken at a critical coupling ${\ensuremath{\lambda}}_{c}$g0, unless both infrared and ultraviolet cutoffs are introduced.

Particular solutions of the equation for the quark propagator in the infra-red region of QCD

A number of particular solutions of the Schwinger-Dyson equation for the quark propagator in the infra-red region of QCD is obtained taking account of the gauge identities. The solutions are of a nonperturbative character, some of them break the chiral invariance. The obtained solutions are analysed with the help of the effective potential method.

Phase transitions in the O(N) model with several coupling constants

A study is made of the phase transitions of the four-fermion two-dimensional model (1) with several coupling constants in the leading order of the 1/N expansion in the presence of a temperature T and chemical potential ..mu... It is shown that for sufficiently large T or ..mu.. the chiral invariance of the theory, which is spontaneously broken for T, ..mu.. = O, is restored irrespective of the values of the invariant constants. The critical values of the temperature and the chemical potential are found.

Evaluation of the propagator of Susskind fermions by summation of non-reversal random walks

We consider the Susskind action for quarks on the lattice which possesses a chiral invariance as the quark mass m goes to zero. The propagator G( y, x) in a gluon field U may be expanded in a series whose terms correspond to all random walks from x to y with hopping parameter κ = m −1. It is observed that a step followed by its reversal (a spike) is independent of U because of factors UU −1. It follows that the contribution from the insertion of all possible trees (made of spikes) at all sites of each walk may be evaluated by simple combinatorics. The propagator in a generic gluon field U is thereby expressed as a sum over all non-reversal walks with renormalized hopping parameter κ′ = 2κ/[1 + √(1 + 28κ 2)] < √ 1 7 (⪡ κ for small m) . The observation that non-reversal random walks define a Markov chain provides a new expression for the propagator as the inverse of a new matrix. The spectrum of the new matrix is found for generic U. It is shown that the new expression for the propagator may be evaluated by an iteration procedure with ratio of successive errors bounded by 1 − 1 8 √ 1 7 m + O(m 2) , whereas the corresponding bound for the original expression is 1− 1 128 m 2 + O(m 4) . The advantage of the new expression for small m is obvious.

Vacuum energy in the bag model

The vacuum energy of the Yang-Mills field is examined for the conditions of the bag model. The dominance of high-frequency effects results in a vacuum energy that decomposes naturally into a volume energy, a surface energy, and higher shape energies. These quantities are identified with the parameters of the bag model. The imposition of confining boundary conditions for all frequencies is shown to be inconsistent since this would result in the bag constant and certain of the shape tensions being infinite. The manner in which the boundary conditions should be relaxed at high frequency is discussed. The most naive procedure for relaxing the boundary conditions, which is to apply confining conditions only on modes of frequency less than some cutoff frequency, results in a negative bag constant and surface tension and would render the vacuum unstable against the spontaneous breaking of Poincaré invariance. Consideration of the manner by which the interacting electromagnetic field avoids a similar instability suggests that a more realistic way to relax the boundary conditions on the bag surface is to endow the vacuum exterior to the bag with a frequency-dependent dielectric constant and magnetic permeability. In this picture the stability of the vacuum is restored, the surface tension is finite and positive, and the bag constant is zero at least to lowest order in the coupling. It is pointed out that the fermion contributions to the bag constant and the surface tension may relate to the spontaneous breaking of chiral invariance. The aim throughout is to examine the bag model, as it relates to vacuum energy, strictly in its own terms with an emphasis on questions of principle. All too often is heard the alibi that since the theory itself is only approximate, the mathematics need be no better. In truth the opposite follows. Granted that the model represents but a part of nature, we are to find what such an ideal picture implies, a result strictly derived serves to test the model; a false result proves nothing but the failure of the theorist. To call an error by a sweeter name does not correct it. The oversimplification or extension afforded by the model is not error: the model, if well made, shows at least how the universe might behave, but logical errors bring us no closer to the reality of any universe. (Truesdell and Toupin 1960).

Generalization of the Gross-Neveu model to the case of several coupling constants

The two-dimensional massless four-fermion theory with several coupling constants is considered in the 1/N approximation. This model is asymptotically free. For all values of the coupling constants, there is dynamical breaking of the chiral invariance in it, and the fermions acquire masses. It is shown that there exist two different phases with massive fermions. Necessary and sufficient conditions under which the theory does not have ghosts are found. In the leading order in 1/N, the model contains bosons with two different masses.

Renormalization of the axial-vector current in QCD.

Following the method of Ioffe and Smilga, the propagation of the baryon current in an external constant axial-vector field is considered. The close similarity of the operator-product expansion with and without an external field is shown to arise from the chiral invariance of gauge interactions in perturbation theory. Several sum rules corresponding to various invariants both for the nucleon and the hyperons are derived. The analysis of the sum rules is carried out by two independent methods, one called the ratio method and the other called the continuum method, paying special attention to the nondiagonal transitions induced by the external field between the ground state and excited states. Up to operators of dimension six, two new external-field-induced vacuum expectation values enter the calculations. Previous work determining these expectation values from PCAC (partial conservation of axial-vector current) are utilized. Our determination from the sum rules of the nucleon axial-vector renormalization constant GA, as well as the Cabibbo coupling constants in the SU3-symmetric limit (ms=0), is in reasonable accord with the experimental values. Uncertainties in the analysis are pointed out. The case of broken flavor SU3 symmetry is also considered. While in the ratio method, the results are stable for variation of the fiducial interval of the Borel mass parameter over which the left-hand side and the right-hand side of the sum rules are matched, in the continuum method the results are less stable. Another set of sum rules determines the value of the linear combination 7F-5D to be ≊0, or D/(F+D)≊(7/12). .AE

A non-relativistic quark model for baryons with pion cloud

We incorporate chiral invariance into a non-relativistic scheme for baryon structure in an approximate way, by introducing additional mesonic degrees of freedom. We proceed to calculate static properties for the nucleon and isobar. Using standard few-body techniques we are able to eliminate center-of-mass spurious motion from the wave functions. In our scheme a D-state admixture arises in the nucleon and isobar wave functions. There are parametrizations compatible with sphericity leading to a strong quenching of the effective quark axial coupling constant. Others lead to deformed ground-state baryons with almost no quenching of the weak-coupling constant.

Supersymmetric unified compositeness and the quark-lepton generation structure.

We attempt to construct a realistic model, incorporating the ideas of supersymmetry, compositeness, and grand unification. Unification dictates the preon/spectator SU(3) x SU(2) x U(1) assignments, while supersymmetry tackles the hierarchy problem and partially protects chiral invariance. It is primarily the presence of scalar preons which allows for a 't Hooft--Appelquist--Carazzone consistent composite generation structure. The various quarks and leptons are internally distinguished, with U(1)/sub R/ serving as a broken horizontal symmetry. To demonstrate our idea, unified SU(10) is invoked to give birth to supersymmetric SU(5) hypercolor compositeness with four low-lying standard families.

Light-meson spectrum with a Nambu-Goldstone pion.

Chiral invariance is dynamically broken for a confining Lorentz vector interaction between massless quarks. We solve the Bethe-Salpeter equation in the broken vacuum for the case of the complete light-meson spectrum, with radial and orbital excitations, up to \ensuremath{\sim}2.5 GeV. A main phenomenological conclusion is that the hyperfine $\ensuremath{\rho}\ensuremath{-}\ensuremath{\pi}$ splitting comes essentially from the Nambu-Goldstone phenomenon, independently of any spin-spin interaction. The SU(6) \ensuremath{\bigotimes} O(3) classification remains, although badly broken. Degeneracy of chiral doublets is restored at high mass.

Quark model of light mesons with dynamically broken chiral symmetry.

We study the meson spectrum in a model with a confining Lorentz-vector: and hence chiral-invariant: interaction between massless quark fields. As shown in a previous work, chiral invariance is spontaneously broken. In the case of the harmonic oscillator, as the Fourier transform of the potential is the Laplacian of a delta function, the Bethe-Salpeter (BS) equation: a system of linear integral equations in general: splits into a system of differential equations that we solve in the broken vacuum. Without appealing to any spin-spin interaction, we find, besides the massless pseudoscalar, a vector meson at the right scale and an excited pion and two vectors in the 1--2-GeV region. Moreover, we find a large L-S splitting with the expected ordering for a vector interaction. We study in detail the BS wave function for the pion in motion, necessary to compute axial-vector-current matrix elements, and recover well known relations of current algebra. We compute f/sub ..pi../ and find on general grounds that f/sub piprime/ = 0 in the chiral limit, where ..pi..' is any radially excited pion. The pion satisfies the expected dispersion law for a Goldstone boson, ..omega..(p)..-->..cp (p..-->..0). .AE

Isovector electromagnetic exchange currents in the chiral approach

The problem studied in the paper concerns the structure of the isovector e.m. MEC operators of the A1−, ϱ- and π ranges in the interval of intermediate energies. The two main dynamic principles which we invoke in our considerations are the current conservation and the gauge chiral invariance. Respecting them consistently allows us to describe correctly the interaction of NA1 ϱπ system with the external e.m. field. Our main results are as follows: (i) We verified that our Lagrangian approach is consistent with the prediction of the low energy theorem at threshold. (ii) We showed explicitly the continuity equation which the longitudinal parts of our currents obey. (iii) We proved the equivalence relation for the MEC operator of the pion range and demonstrated the existence of the seagull current in the MEC operator built up using PS πN coupling. This new term influences strongly the exchange charge density.