A model for chiral symmetry breaking in QCD

A recently proposed model for dynamical breaking of chiral symmetry in QCD is extended and developed for the calculation of pion and chiral symmetry breaking parameters. The pion is explicitly realized as a massless Goldstone boson and as a bound state of the constituent quarks. We compute, in the limit of exact chiral symmetry, M_Q, the constituent quark mass $\text{?}{}_{\mathrm{\pi }}$ the pion decay coupling, $\text{〈}\text{u}\text{u〉}$, the constituent quark loop density, μ_π²/m_q, the ratio of the Goldstone boson mass squared to the bare quark mass, and 〈r²〉_π, the pion electromagnetic charge radius squared.     

Dimensional reduction and flavor chirality

We show that phenomenologically realistic flavor-chiral Yang-Mills-Higgs theories in 4 dimensions can be derived by dimensional reduction of 10-dimensional vectorlike and gauge theories, where the extra 6 dimensions form a compact coset space with scale size r. The dimensional reduction often implies a symmetry breaking pattern like that of the electroweak theory, in which case it is natural to propose $\text{r ? G}{}_{\mathrm{<mtext>F</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Quantum effects then determine the short-distance behavior of the theory, including any additional symmetry breaking.     

From high energy supersymmetry breaking to low energy physics through decoupling

We reconsider a realistic model of electroweak and strong interactions with calculable mass spectrum at the tree level in which supersymmetry and an extra gauge group factor ?(1) beyond SU(3) × SU(2) × U(1) are both broken at very high energies: $\text{M}{}_{\mathrm{<mtext>SUSY</mtext>}}\text{?(M}{}_{\mathrm{<mtext>W</mtext>}}\text{M)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, $\text{M}{}_{\mathrm{<mtext>U</mtext><mtext>?</mtext><mtext>(1)</mtext>}}\text{?M}\text{with}\text{M?M}{}_{\mathrm{<mtext>W</mtext>}}$. In spite of these high-energy scales, especially the large scale of supersymmetry breaking, the low energy spectrum - including the relevant Higgs boson - is decoupled from the heavy degrees of freedom. Due to the “non-renormalization” theorems this decoupling persists to all orders in perturbation theory.     

Chiral symmetry breaking in the action formulation of lattice gauge theory

We show, in the euclidean path-integral formulation of strong-coupling lattice gauge theory, that continuous chiral symmetry is dynamically broken, and obtain the standard current algebra result that $\text{m}{}_{\mathrm{<mtext>pseudo-Goldstone</mtext>}}{}^2\text{～ m}{}_{\mathrm{<mtext>quark</mtext>}}\text{〈}\text{ψ}\text{ψ〉}$. We also remark that the center of the gauge group does not seem very relevant for this result; chiral symmetry breaking is a property of strong-coupling lattice theories both in the case where quark color is confined, and also in the case where it is screened by gauge field fluctuations.     

Spontaneous CP violation in a Z4 horizontal symmetry model for flavor mixing

The implications of a Z₄ horizontal symmetry model of flavor mixing for CP violation are studied in the framework of minimal SU(2)_L × SU(2)_R × U(1)_{B – L} gauge theory. We show that CP violation in this model arises purely from right-handed currents. We also note that spontaneous breaking of CP symmetry requires a fine tuning of coupling parameters to the level of , which can be avoided by the inclusion of one additional singlet Higgs field, of the kind recently introduced for other purposes.     

Quark mass matrices in left-right symmetric gauge theories

The most general left-right symmetry for SU(2)_L×SU(2)_R×U(1) gauge theories with any number of flavours and with at most two scalar multiplets transforming as $\text{q}\text{q}$ bilinears is analyzed. In order to get additional constraints on the structure of quark mass matrices, all possible horizontal groups (continuous or discrete) are investigated. We give a complete classification of physically inequivalent quark mass matrices for four and six flavours. It is argued that our methods and results are also applicable in the case of dynamical symmetry breaking. Parity invariance and horizontal symmetry are shown to imply CP conservation on the lagrangian level. For all non-trivial three-generation models there is spontaneous CP violation, which in most cases turns out to be naturally small. Several six-flavour models predict $\text{m}{}_{\mathrm{<mtext>t</mtext>}}\text{?m}{}_{\mathrm{<mtext>b</mtext>}}\text{(}\text{m}{}_{\mathrm{<mtext>u</mtext>}}\text{m}{}_{\mathrm{<mtext>c</mtext>}}\text{m}{}_{\mathrm{<mtext>d</mtext>}}\text{m}{}_{\mathrm{<mtext>s</mtext>}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and are, therefore, already ruled out experimentally. In the remaining few realistic models, predictions for the weak mixing angles are made.     

Higgs particle production by Z→Hγ

The rate for the decay of a Z-boson into a Higgs boson and monochromatic photon is computed to leading order in the standard SU(2) × U(1) gauge theory. The coupling has contributions from fermion and W-boson loops. The W-boson loop dominates unless the number of heavy fermion generations exceeds six. The branching ratio computed from the W-boson loop contribution, B(Z→Hγ), is approximately 2 × 10^?6$\text{(1?(}\text{M}{}_{\mathrm{<mtext>H</mtext>}}{}^2\text{M}{}_{\mathrm{<mtext>Z</mtext>}}{}^2\text{))}{}^3$.     

Quark masses

In quark gluon theory with very small bare masses, -$\text{ψ}$$\text{M}$ψ, spontaneous breakdown of chiral symmetry generates sizable masses M_u, M_d, M_s, … We find ($\text{M}{}_{\mathrm{<mtext>u</mtext>}}\text{+ M}{}_{\mathrm{<mtext>d</mtext>}}\text{) /2 ≈ m}\text{p}\text{/ √6 ≈ 312}\text{MeV}\text{,}\text{and}\text{M}{}_{\mathrm{<mtext>s</mtext>}}\text{≈ 432}\text{MeV}$. Scalar densities have well determined non-zero vaccum expectations $\text{〈0|u}{}^{\mathrm{a}}\text{|0〉 ≡ 〈0|ψ(x) (λ}{}^{\mathrm{a}}\text{/2)ψ(x)/0〉 ≈ ?}{}_{\mathrm{\pi }}{}^2\text{M}{}^{\mathrm{a}}\text{,}\text{i.e}\text{〈0? u}{}^{\mathrm{o}}\text{/vb0〉 ≈ 8 × 10}{}^{\mathrm{?3}}\text{(}\text{GeV}\text{)}{}^3$ at an SU(3) breaking of the vacuum c′ ≡ 〈0|u⁸|〉/〈0|u^o|0〉 ≈ ? 16%     

Coupling supersymmetric Yang-Mills theories to supergravity

We extend the technique of Cremmer et al. to couple arbitrary chiral multiplets with supersymmetric Yang-Mills interactions to N = 1 supergravity. We present the general form of the lagrangian and the detailed form of the scalar potential is spelled out. In the case of N chiral multiplets, “minimally” coupled to supergravity, we derive, in the absence of gauge interactions, a model-independent mass formula Supertrace $\text{M}{}^2\text{= Σ}{}_{\mathrm{J}}\text{(?)}{}^{\mathrm{2J}}\text{(2J + 1)m}{}_{\mathrm{J}}{}^2\text{= 2(N ? 1)m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}{}^2$, where $\text{m}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ is the gravitino mass. A concrete example of the super Higgs effect involving N chiral multiplets is exhibited.     

Electromagnetic field in the gauge SU(3) × SU(3) chiral group

The massless electromagnetic Yang-Mills field is explicitly constructed as a linear combination of 16 gauge fields of the chiral SU(3) × SU(3) group within the framework of the plasmon generating mechanism [1]. The remaining 15 gauge fields acquire a mass through the non-zero vacuum expectation values of the auxiliary scalar multiplet which transforms according to the (8,8) representation of the gauge group. The tadpoles with non-zero hypercharge which are required for the existence of the only massless electromagnetic potential A_μ are due to the natural mixing of charged weak currents with ΔS = 0 and ΔS = 1. The relevance of this phenomenon to the Cabibbo angle is briefly discussed. Also presented is a theorem concerning an admissible form of the zero-order mass term of gauge fields when the canonical number is unknown.     

Obtaining Mw = Mz cos θ in technicolor theories

We show that the successful relation M_w = M_z cos θ is preserved in the technicolor formulation of the dynamical Higgs mechanism provided only that the creation operators for Goldstone bosons associated with broken generators belong to the $\text{I}{}_{\mathrm{<mtext>w</mtext>}}\text{=}\text{1}\text{2}$ representation of the weak isospin group. We present a plausibility argument that this is indeed the case. No additional isospin or isospin-like global SU(2) symmetries are then required allowing isospin to be spontaneously broken. This may be of help in producing a large $\text{m}{}_{\mathrm{<mtext>c</mtext>}}\text{m}{}_{\mathrm{<mtext>s</mtext>}}$ splitting. It is also shown how the weak hyperchange interaction can produce substantial vacuum isospin breaking in a theory which is only marginally asymptotically free. This mechanism predicts , providing a natural explanation for small neutrino masses.     

Comparison of the lattice Λ parameter with the continuum Λ parameter in massless QCD

The ratio of the scale parameter Λ in massless QCD defined on a lattice to the one in the continuum theory is determined by performing one-loop renormalization of the coupling constant. Our calculation method on a lattice directly relates Λ^lattice to the continuum one in the minimal subtraction scheme. The effect of incorporation of massless quarks depends on a parameter λ which is introduced to avoid trouble with fermions on a lattice. For λ=1, which is Wilson's value, the ratio previously calculated by Hasenfratz in the pure gauge theory is changed as follows:

$\text{Δ}{}_{\mathrm{\alpha =1}}{}^{\mathrm{MOM}}\text{Δ}{}^{\mathrm{lattice}}\text{=83.5}\text{for pure SU(3) gauge theory;}$

$\text{Δ}{}_{\mathrm{\alpha =1}}{}^{\mathrm{MOM}}\text{Δ}{}^{\mathrm{lattice}}\text{=105.7}\text{for QCD with 3 flavors;}$

$\text{Δ}{}_{\mathrm{\alpha =1}}{}^{\mathrm{MOM}}\text{Δ}{}^{\mathrm{lattice}}\text{=105.7}\text{117.0 for QCD with 4 flavors.}$

Critical properties of the lattice QCD will also be discussed briefly.     

Theory of a condensed charged-bose,charged-fermi gas

Field-theoretic methods are used to derive the low-temperature properties of a dense, neutral gas of condensed charged (+Ze) bosons (mass M) and degenerate charged (?e) fermions (mass m). The density-density correlation function exhibits two collective modes for m ? M: a plasmon branch and a phonon branch with speed $\text{c = V}{}_{\mathrm{F}}\text{(}\text{Zm}\text{3M}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Boson single-particle excitations are gapless and coincide with the phonon mode at low k. Inclusion of the two interacting dynamical components leads to small shifts in the boson and fermion ground-state energy and boson depletion relative to the values obtained for models with rigid, inert, neutralizing backgrounds.     

Parity violation in atoms in SO(3) gauge models

We have evaluated the parity-violation contribution in atoms in the framework of SO(3) gauge theory. Various hadronic models have been used: first, for simplicity, the unrealistic five-quark one, next, others involving three ordinary SU(3) triplets for which all unwanted strangeness-changing processes are suppressed, up to order $\text{Gα}\text{or}\text{GαΔM}{}^2\text{M}{}_{\mathrm{<mtext>W</mtext>}}{}^2$. In the free quark approximation, we obtain quite similar parity-violation effects which are proportional to $\text{GαΔM}{}^2\text{M}{}_{\mathrm{<mtext>W</mtext>}}{}^2$ (ΔM² is the difference of squared masses of leptons (M_X0² ? M_ν² = M_X0²), or of quarks (ΔM_q²)). Namely, in large atoms (Z ? 1) the electronic contribution which is proportional to

gives the largest effect ($\text{σ}{}_{\mathrm{?}}\text{,}\text{p}{}_{\mathrm{?}}\text{and}\text{m}{}_{\mathrm{?}}$are the spin, momentum operators and mass of the lepton). Parity-violating effects in SO(3) gauge models are ?10^?4 smaller than those evaluated in the Weinberg theory with a neutral parity-violating current and will remain undetectable in the near future.     

The ′T Hooft-Polyakov monopole near the Prasad-Sommerfield limit

A perturbative classical monopole solution for the SO(3) gauge theory is constructed in the limit of small but non-vanishing Higgs potential. This corresponds to the limit $\text{μ}{}^2\text{2M}{}_{\mathrm{W}}{}^2\text{=}\text{λ}\text{? 1}$, where μ equals the mass of the scalar particle and M_W equals the mass of the intermediate vector particles. The monopole solution and mass are found to involve non-analytic functions of λ: $\text{γ}$ and λ ln λ. The monopole mass M_m is calculated to order $\text{μ}{}^2\text{M}{}_{\mathrm{W}}$ as

$\text{M}{}_{\mathrm{m}}\text{=}\text{4π}\text{e}{}^2\text{M}{}_{\mathrm{w}}\text{1+}\text{1}\text{2}\text{μ}\text{M}{}_{\mathrm{w}}\text{+}\text{1}\text{2}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}\text{ln}\text{μ}\text{M}{}_{\mathrm{w}}\text{+0.7071}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}$

.     

Properties of quark matter governed by quantum chromodynamics (II). Renormalization schemes in quark matter

Renormalization schemes are examined (in the Coulomb gauge) for quantum chromodynamics in the presence of quark matter. We demand that the effective coupling constant for all schemes become congruent with the vacuum QCD running coupling constant as the matter chemical potential, μ, goes to zero. Also, to enable us to standardize with the vacuum QCD running coupling constant at some asymptotic momentum transfer, $\text{|}\text{p}{}_0\text{|,}\text{we keep}\text{μ ? ¦}\text{p}{}_0\text{¦}$, to ensure that the matter contribution is negligible at this point. This means all schemes merge with vacuum QCD at $\text{|}\text{p}{}_0\text{|}$ and beyond. Two renormalization group invariants are shown to emerge: (i) the effective or invariant charge, g_inv², which is, however, scheme dependent and (ii) g²(M)/S(M), where S(M)^?1 is the Coulomb propagator, which is scheme independent. The only scheme in which g_inv² is scheme independent and identical to g²(M)/S(M) is the screened charged scheme (previous paper) characterised by the normalization of the entire Green function, S^?1, to unity. We conclude that this is the scheme to be used if one wants to identify with the experimental effective coupling in perturbation theory. However, if we do not restrict to perturbation theory all schemes should be allowed. Although we discuss matter QCD in the Coulomb gauge, the above considerations are quite general to gauge theories in the presence of matter.     

Poincaré gauge invariance and the dynamical role of spin in gravitational theory

We classify, according to the number of independent gauge fields, Poincaré gauge invariant theoretical frameworks of describing gravity into three categories. One of them may provide the dynamical definition of the spin tensor S and that of the energy-momentum tensor T, resulting in the response equation of matter to gravity with the gravitational field strengths, D′ and F, coupled to the former tensors

$\text{T}{}_{\mathrm{\nu }}{}^{\mathrm{\mu }}\text{;}{}_{\mathrm{\mu }}\text{=D′}{}_{\mathrm{\mu \lambda \nu }}\text{T}{}^{\mathrm{\mu \lambda }}\text{+F}{}_{\mathrm{\mu \lambda \nu \rho }}\text{S}{}^{\mathrm{\rho \mu \lambda }}$

, where the right-hand side represents spin force densities. In the absence of spin the response reduces to the conventional one of general relativity, i.e., without the spin forces. For the electromagnetic field the phase-gauge invariance requires the same conclusion as for a scalar field. For a spin $\text{1}\text{2}$ particle there is torsion, which deflects its trajectory from geodesic; an explicit expression for torsion takes a simple form of the axial vector current $\text{ψ}\text{γ}{}_5\text{γ}{}_{\mathrm{k}}\text{ψ}$.     

Magnetic monopoles in SU(N) gauge theories

The potential $\text{A}\text{(}\text{r}\text{) ≡}\text{M}\text{(}\text{r}\text{?}\text{×}\text{n}\text{?}\text{)(r?}\text{r}\text{·}\text{n}\text{?}\text{)}{}^{\mathrm{?1}}$ is a static solution to the classical theory of non-abelian gauge fields coupled to a point magnetic source, for any matrix $\text{M}$ in the Lie algebra of the gauge group G. This solution is rotationally invariant if the eigenvalues of $\text{M}$ in the adjoint representation of G are quantized in half-integer units, but is stable to small perturbations only if all non-vanishing eigenvalues are $\text{±}\text{1}\text{2}$. In this paper, for the gauge groups G = SU(N), it is shown which sets of eigenvalues of $\text{M}$ are consistent with the group structure, which consistent sets are gauge inequivalent, and which consistent gauge inequivalent sets correspond to stable monopoles. It is found that there are N inequivalent stable monopoles, including the trivial case $\text{M}\text{=}\text{0}$. Equivalence here is with respect to non-singular gauge transformations—the symmetry transformations of the classical theory. Singular gauge transformations are, in contrast, not symmetries but they are nevertheless useful for classifying solutions and for relating the above concept of local stability to the global, or topological, stability associated with the Dirac strings. In this context, it is shown that there are N distinct topological classes of monopoles, with the group structure of the center $\text{Z}{}_{\mathrm{N}}\text{?Π}{}_1\text{(}\text{SU}\text{(N)/}\text{Z}{}_{\mathrm{N}}\text{)}$ of SU(N), that each class contains exactly one stable monopole, and that any other monopole in the same class has a strictly larger value of the magnetic charge magnitude $\text{tr}\text{M}{}^2$. This leads to an interesting physical picture of local stability as a consequence of the minimization of magnetic energy. The paper concludes with some comments on related topics: the empirical absence of magnetic charge, `t Hooft's calculation of magnetic energy, magnetic confinement, and spontaneously broken theories.