Chiral symmetry,the 1/N expansion and the SU(N) thirring model

In the two-dimensional SU(N) Thirring model, the 1/N expansion seems to predict spontaneous breaking of the continuous chiral symmetry. This is impossible in two-dimensions. Reasoning along the lines of Berezinski, Kosterlitz and Thouless for the two-dimensional XY model, we argue that, in fact, rather than showing long-range order, $\text{〈}\text{ψ}\text{ψ(x)}\text{ψ}\text{ψ(0)〉}$ vanishes in this model as |x|^?1/N at large |x|. The 1/N expansion is, in fact, a rather good guide to the properties of this model.     

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.     

Instantons and condensates in supersymmetric CPN−1 models

The partition function in the multi-instanton background for the supersymmetric CP^N?1 model is calculated and is shown to be supersymmetry invariant. This is used to calculate Green's function involving N pairs of $\text{ψ}\text{ψ(x}{}_{\mathrm{i}}\text{)}$, and this turns out to be independent of the positions. A possible interpretation is that the composite field $\text{ψ}\text{ψ}$ acquires a vacuum expectation value. The results are in agreement with those obtained in the 1/N expansion.     

On the thermodynamics and scaling behaviour of SU(2) gauge theory with fermion feedback

We study SU(2) lattice gauge theory with four species (N_f = 4) of light dynamical fermions by microcanonical simulation methods. Relatively large lattices, long runs and small quark masses are considered. On a 8³ × 16 lattice $\text{〈}\text{ψ}\text{ψ〉}$ is measured and good evidence for asymptotic freedom with fermion feedback is found. The scaling window begins at β = 4/g² ≈ 1.85. On a 12³ × 6 lattice SU(2) thermodynamics is studied systematically. The chiral symmetry restoration transition is found at β = 1.925 ± .025. The crossover from hadronic matter to the quark-gluon plasma is abrupt.     

Spontaneous chiral symmetry breaking for a U(N) gauge theory on a lattice

We study the breakdown of chiral invariance by calculating, in the infinite coupling, large-N limit, the generating functional of a U(N) gauge theory with one fermion, expressed on a lattice with the naive, chiral symmetric action. We compute the link integral over the gauge fields and the expression obtained after the integration over the fermions is recast under the form of a generating functional for bosonic fields. Then, a saddle-point method allows the calculation of the order parameter $\text{〈}\text{ψ}\text{ψ〉}$ for which a non-zero value signals the spontaneous breakdown of chiral symmetry. The analysis of the fluctuations around the saddle point allows one to exhibit the Goldstone modes corresponding to those global symmetries of the fermionic lattice action which are simultaneously broken.     

Monte Carlo simulation of SU(2) gauge theory with fermions on a four-dimensional lattice

After integration over the fermions in an SU(2) lattice gauge theory, the effective fermionic action may be expressed as a sum over all possible closed gauge field loops with corresponding weight factors. We approximate this sum and perform a Monte Carlo simulation of a coupled fermion-gauge system on a 4⁴ lattice. We compare our results for 〈S_eff〉 and $\text{〈}\text{ψ}\text{ψ〉}$ for different values of the gauge field coupling β and fermion coupling κ with the free fermion theory on a lattice. 〈S_eff〉 turns out to be quite small for $\text{κ?}\text{1}\text{8}$.     

Chiral phase transition in lattice QCD with dynamical quarks

The effect of virtual light quark loops on the chiral phase transition in lattice QCD with staggered fermions is investigated by employing the pseudo-fermion Monte Carlo method on a 6³ × 2 lattice. The variation in the order parameter $\text{〈}\text{ψ}\text{ψ〉}$ is found to become less sharp than the quenched case, indicating a second order chiral phase transitioon in the full theory.     

Chiral symmetry breakdown,anomaly, and the PCAC relation on a lattice

Lattice fermion formulation is investigated using a solvable model which resembles quantum chromodynamics. CP₂^N?1 models with quarks are formulated on a lattice. For dynamical quarks, a generalized formulation of the Wilson and the Osterwalder-Seiler lattice fermion is used. In the $\text{1}\text{N}$ expansion, the spontaneous breakdown of chiral symmetry (which is softly broken by the quark mass) apparently occurs in this model, and the “pion” mass is calculated. From the above results, it is shown that the above lattice fermion formulations have the desired continuum limit. The axial-vector current is investigated and it is proved that the usual anomaly appears in the continuum limit and the PCAC relation is satisfied.     

Towards the phase structure of euclidean lattice gauge theories with fermions

We propose the phase structure of abelian and non-abelian lattice gauge theories with fermions. We especially analyse Wilson's lattice action with euclidean discrete space-time. We mainly analyse $\text{〈}\text{ψ}{}_{\mathrm{n}}\text{ψ}{}_{\mathrm{n}}\text{〉}$ as an order parameter for the fermion-gauge coupled system. The Wilson loop integral and plaquette-plaquette two-point function are also useful in working out abelian phase diagrams. We will discuss physical implications of the phase diagrams, especially for the mass spectrum in the lattice continuum limit and chiral symmetry breaking. The 1/N expansion and a random walk idea are used in the formulation and play an important role in computing meson and baryon propagators in the strong coupling limit.     

Calculation of baryon masses in quantum chromodynamics

The polarization operator of quark currents with baryon quantum numbers is considered in quantum chromodynamics. The non-zero mean vacuum values of the field operator products are taken into account. The sum rules are obtained assuming that in the virtuality region ～1 GeV, among the mean vacuum values violating the chiral invariance, the most important is $\text{〈0|}\text{ψ}\text{ψ|0〉}$. Saturating these sum rules by the lowest baryonic states one is able to calculate the masses of the isobar Δ and nucleon N, M_Δ = 1.4 GeV, M_N = 1 GeV, up to 15% through the known value $\text{〈0|}\text{ψ}\text{ψ|0〉}$. The mass splitting in the baryonic decuplet $\text{M}{}_{\mathrm{\Sigma 1}}\text{?M}{}_{\mathrm{\Delta }}\text{= 125}\text{MeV}$ is calculated in first order in the current strange quark mass m_s = 150 MeV. Certain results for other baryonic resonances have also been obtained.     

Monte Carlo calculations with fermions: The Schwinger model

An algorithm for Monte Carlo simulation of lattice gauge theories with fermions is presented. The method is applied to the Schwinger model with two flavors of massless fermions, formulated on a two-dimensional euclidean lattice. Preliminary results of the Monte Carlo iteration of this system are presented, with special emphasis on the behavior of the Wilson loop and bilocal chiral correlation functions such as $\text{〈}\text{ψ}\text{(1 + γ}{}_5\text{)ψ(x)}\text{ψ}\text{(1 ? γ}{}_5\text{)ψ(y)〉}$.     

Charged pion production in e+e− annihilation at 14, 22 and 34 GeV CM energy

The inclusive production of π^± mesons in e⁺e^? annihilation has been measured at c.m. energies of 14, 22 and 34 GeV for pion momenta between 0.3 ans 10 GeV/c. The fraction of pions among the charged hadrons is above 90% at 0.4 GeV/c and decreases to about 50% at high momenta. The scaled cross sections ($\text{s}\text{β}$) $\text{d}\text{σ}\text{d}\text{x}$ at 14, 22 and 34 GeV as well as the 5.2 GeV data from DASP have a rather similar x dependence. After integration over the x range from 0.2 to 0.6 the cross sections indicate a monotonic decrease with increasing centre-of-mass energy.     

Pions as spin waves: Chiral symmetry breaking in lattice gauge theory

In hamiltonian lattice gauge theory, the fermion vacuum at lowest order in 1/g² can be determined from degenerate perturbation theory plus mean field-spin wave techniques. Using compact QED as an example, we show that: (i) chiral symmetry is spontaneously broken; and (ii) $\text{m}{}_{\mathrm{<mtext>pseudoGoldstone</mtext>}}{}^2\text{∝ m}{}_{\mathrm{<mtext>fermion</mtext>}}\text{〈}\text{ψ}\text{ψ〉}$. The pseudoscalar pseudoGoldstone particles—the “pions” of this abelian theory—correspond to antiferromagnetic spin wave excitations of the fermion vacuum.     

Chiral symmetry on the lattice with Wilson fermions

The chiral properties of the continuum limit of lattice QCD with Wilson fermions are studied. We show that a partially conserved axial current can be defined, satisfying the usual current algebra requirements.A proper definition of the chiral symmetry order parameter, $\text{〈0|}\text{ψ}\text{ψ|0〉}$, is given, and the chiral properties of composite operators are investigated. The implications of our analysis to the lattice determination of non-leptonic weak amplitudes are also discussed.     

Pion form factor from 480 MeV to 1100 MeV

The pion form factor is measured in the reaction e⁺e^? → π⁺π^? for center of mass energies in the range 480–1100 MeV. Our results are first analysed in terms of the conventional Vector Meson Dominance formalism, and then taking into account the ωπ inelastic channel. The result of this later formalism is a pion form factor (F_π) which fits quite well all the existing data on F_π both in the timelike and spacelike regions, and pion mean square radius of 〈r_π²〉 = 0.460 ± 0.011 fm² or $\text{〈r}{}_{\mathrm{\pi }}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 0.678 ± 0.008}\text{fm}$.     

Beyond leading order QCD perturbative corrections to the pion form factor

The order α_s²(Q²) corrections to the pion form factor, F_π(Q²), are calculated using perturbative quantum chromodynamics and dimensional regularization. In the $\text{MS}$ renormalization scheme these corrections are large. This means that reliable perturbative predictions cannot be made until momentum transfers, Q, of about 300–400 GeV are reached or unless one can sum the large perturbative terms to all orders. Attempts to reorganize the perturbation series so that the first term gives a better approximation of the complete sum indicate that at Q = 10 GeV the pion form factor may be about a factor of two larger than the leading order result.     

The S-matrix of the kinks of the (gyψ)2 model

We consider the particle-kink and kink-kink S-matrix elements of the two-dimensional $\text{(}\text{ψ}\text{ψ)}{}^2$ model, where the Majorana spinor ψ is an O(N) isovector. Our results confirm many qualitative ideas about the model, including the mass spectrum, the decoupling at N = 4, and the isospinor nature of the kinks.     

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.     

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%