Mesonic corrections to the quark propagator in the Nambu-Jona-Lasinio model

Corrections to the chiral condensate and quark mass in the Nambu-Jona-Lasinio model for the four-dimensional cutoff and dimensional-analytical regularizations are calculated within the mean-field expansion in the bilocal quark-source formalism. It is shown that pion corrections to quark masses are zero in both regularizations. This coincidence of the results suggests that the zero pion contribution to the quark mass is a fact of the Nambu-Jona-Lasinio model and is independent of regularization. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 31–36, July, 2006.     

Proof of chiral symmetry breaking in strongly coupled lattice gauge theory

We study chiral symmetry in the strong coupling limit of lattice gauge theory with staggered fermions and show rigorously that chiral symmetry is broken spontaneously in massless QED and the gauge-invariant Nambu-Jona-Lasinio model if the dimension of spacetime is at least four. The results for the chiral condensate as a function of the mass imply that the mean-field approximation is an upper bound for this observable which becomes exact as the dimension goes to infinity. For the model with gauge groupU(N),N=2,3,4, we prove that chiral long-range order exists at zero mass in four or more dimensions. Address after August 1991: Mathematics Department, University of British Columbia, Vancouver, Canada V6T1Y4     

Equations from non-linear chiral transformations

In comparison with theWT chiral identity which is indispensable for renormalization theory, relations deduced from the non-linear chiral transformation have a totally different physical significance. We wish to show that non-linear chiral transformations are powerful tools to deduce useful integral equations for propagators. In contrast to the case of linear chiral transformations, identities derived from non-linear ones contain more involved radiative effects and are rich in physical content. To demonstrate this fact we apply the simplest non-linear chiral transformation to the Nambu-Jona-Lasinio model, and show how our identity is related to the Dyson-Schwinger equation and Bethe-Salpeter amplitudes of the Higgs and π. Unlike equations obtained from the effective potential, our resultant equation is exact and can be used for events beyond the LEP energy.     

Two- and three-point functions in the extended NJL model

The two-point functions in generalized Nambu-Jona-Lasinio models are calculated to all orders in momenta and quark masses to leading order in 1/N _c . The use of Ward identities and the heat-kernel expansion allows for a large degree of regularization independence. We also show how this approach works to the same order for three-point functions on the example of the vectorpseudoscalar-pseudoscalar three-point function. The inclusion of the chiral anomaly effects at this level is shown by calculating the pseudoscalar-vector-vector three-point function to the same order. Finally we comment on how (vector-) meson-dominance comes out in the presence of explicit chiral symmetry breaking in both the anomalous and the non-anomalous sectors.     

Mean-field expansion and meson effects in chiral condensate of analytically regularized Nambu-Jona-Lasinio model

Scalar meson contributions in chiral quark condensate are calculated in the analytically regularized Nambu-Jona-Lasinio model using the framework of mean-field expansion in bilocal-source formalism. The sigma-meson contribution for physical values of the parameters is found to be small. Pion contribution is found to be significant and should be taken into account for the choice of the parameter values.     

Chiral Soliton Model with Dynamical Restoration of Chiral Symmetry from Quark Flavour Dynamics

The Nambu-Jona-Lasinio model as a particular model of quark flavour dynamics is bosonized in the low-energy regime using a generalized heat kernel expansion with local mass scale. The resulting effective meson Lagrangian defines a chiral soliton model, the solutions of which show a partial restoration of chiral symmetry inside the soliton.     

Meson-loop effects in the NJL model at zero and nonzero temperature

We compare two different possibilities of including meson-loop corrections in the Nambu-Jona-Lasinio model: a strict 1/N _c expansion in the next-to-leading order and a nonperturbative scheme corresponding to a one-meson-loop approximation to the effective action. Both schemes are consistent with chiral symmetry, in particular, with the Goldstone theorem and the Gell-Mann-Oakes-Renner relation. The numerical part at zero temperature focuses on the pion and the ρ-meson sector. For the latter, meson-loop corrections are crucial in order to include the dominant ρ → ππ-decay channel, while the standard Hartree + RPA approximation only contains unphysical $m_\pi ,f_\pi ,\left\langle {\bar \psi \psi } \right\rangle $ -decay channels. We find that $q\bar q$ , and quantities related to the ρ-meson self-energy can be described reasonably with one parameter set in the 1/N _c-expansion scheme, whereas we did not succeed in obtaining such a fit in the nonperturbative scheme. We also investigate the temperature dependence of the quark condensate. Here, we find consistency with the chiral perturbation theory to the lowest order. Similarities and differences of both schemes are discussed.     

On theO(1/N 2)β-function of the Nambu-Jona-Lasinio model with non-abelian chiral symmetry

We present the formalism for computing the critical exponent corresponding to the β-function of the Nambu-Jona-Lasinio model withSU(M)×SU(M) continuous chiral symmetry atO(1/N ²) in a largeN expansion, whereN is the number of fermions. We find that the equations can only be solved for the caseM=2 and subsequently an analytic expression is then derived. This contrasting behavior between theM=2 andM>2 cases, which appears first atO(1/N ²), is related to the fact that the anomalous dimensions of the bosonic fields are only equivalent forM=2.     

Massive Meson Fluctuation in NJL Model

Based on the self-consistent scheme beyond the mean-field approximation in the large N_c expansion, including current quark mass explicitly, a general scheme of SU(2) NJL model is developed. To ensure the quark self-energy expanded in the proper order of N_c,an approximate internal meson propagator is deduced, which is in order of O(l/N_c). In our scheme, adopting the method of external momentum expansion, all the Feynman diagrams are calculated in a unified way by only expanding the quark propagator. Our numerical results show that, different &om the mean-field approximation in which the explicitly chiral symmetry breaking is invisible, the effect of finite pion mass can be seen clearly when beyond the meanfield approximation.     

Density-Dependence of Nuclear Symmetry Energy: Role of QCD Chiral Phase Transition

The density-dependence of symmetry energy is of particular importance to many problems in nuclear physics and astrophysics. By using the functional path integral method, we show explicitly the relation between nuclear symmetry energy and isospin susceptibility. The latter one is found to be a probe to the QCD chiral phase transition. We further found in the Nambu-Jona-Lasinio model calculations that, the nuclear symmetry energy has an abrupt change at the critical nuclear density where the chiral symmetry restores partially.     

Diquark confinement in an extended NJL model

In a Nambu-Jona-Lasinio model supplemented with an infrared cutoff in addition to the ultraviolet cutoff we study the issue whether diquarks are confined when the model is extended beyond the rainbow-ladder approximation. The gap equation, obtained in a truncation scheme motivated via a non-trivial quark-gluon vertex function, is solved to determine the constituent quark mass if chiral symmetry is spontaneously broken. In a second step, the Bethe-Salpeter equations for mesons and diquarks beyond the ladder approximation are derived, taking care to preserve Goldstone's theorem in the pion channel. While the obtained masses of pseudo-scalar and vector mesons are only moderately shifted compared to the values in the ladder approximation, we observe that scalar diquarks disappear from the physical spectrum and therefore are confined. For axial-vector diquarks we observe indications that the same mechanism may also work, but the NJL model allows no conclusive answer in this channel.     

Nonlocal PNJL model beyond mean field

A nonlocal chiral quark model is consistently extended beyond mean field using a strict 1/N _c expansion scheme. It is found that the 1/N _c corrections lead to a lowering of the temperature of the chiral phase transition in comparison with the mean-field result. On the other hand, near the phase transition the 1/N _c expansion breaks down and a nonperturbative scheme for the inclusion of mesonic correlations is needed in order to describe the phase transition region.     

Density fluctuations in the presence of spinodal instabilities

Density fluctuations resulting from spinodal decomposition in a nonequilibrium first-order chiral phase transition are explored. We show that such instabilities generate divergent fluctuations of conserved charges along the isothermal spinodal lines appearing in the coexistence region. Thus, divergent density fluctuations could be a signal not only for the critical end point but also for the first-order phase transition expected in strongly interacting matter. We also compute the mean-field critical exponent at the spinodal lines. Our analysis is performed in the mean-field approximation to the Nambu-Jona-Lasinio model formulated at finite temperature and density. However, our main conclusions are expected to be generic and model independent.     

Meson Effects on the Chiral Condensate at Finite Density

Meson corrections on the chiral condensate up to next-to-leading order in a 1/N_c expansion at finite density are investigated in the NJL model with explicit chiral symmetry breaking. Compared with mean-field results, the chiral phase transition is still of the first order while the properties near the critical density for chiral phase transition are found to change significantly.     

Topologically non-trivial chiral transformations

The role of chiral transformations in effective theories modeling Quantum Chromo Dynamics is reviewed. In the context of the Nambu-Jona-Lasinio model the hidden gauge and massive Yang-Mills approaches to vector mesons are linked by a special chiral transformation which removes the chiral field from the scalar-pseudoscalar sector. The chirally rotated axial vector meson field (à _μ ) transforms homogeneously under flavor rotations and may thus be dropped without violating chiral symmetry. The fermion determinant for static meson field configurations is computed by summing the discretized eigenvalues of the Dirac Hamiltonian. It is discussed how the local chiral transformation loses its unitary character in a finite model space. This technical issue proves to be crucial for the construction of the soliton within the Nambu-Jona-Lasinio model when the chirally rotated axial vector field is neglected. In the background of this soliton the valence quark is strongly bound, and its eigenenergy turns out to be negative. This important feature, which usually is generated by non-vanishing axial vector profiles, is thus maintained by the simplificationà _μ = 0.     

Meson Effects on the Chiral Condensate at Finite Density

Meson corrections on the chiral condensate up to next-to-leading order in a 1/Nc expansion at finite densityare investigated in the NJL model with explicit chiral symmetry breaking. Compared with mean-field results, the chiralphase transition is still of the first order while the properties near the critical density for chiral phase transition are foundto change significantly.     

A Variational Approach for Hadronic Properties in NJL Model

We develop a variational approach for the Nambu-Jona-Lasinio (NJL) model with explicit chiral symmetry breaking and compute the constituent quark mass, the pion decay constant and the vacuum condensate. For a simple ansatz for the hadronic wavefunction, we also analyze the effects of the current quark mass and the ultraviolet cutoff on the static properties of hadrons.     

Mean-field theory of the three-state chiral clock model

The phase diagram of the three-state chiral clock model, which is known to exhibit commensurate and incommensurate ordered modulated structures, is investigated in the mean-field approximation. First a numerical analysis of the mean-field equations is presented. It is based in the main on the observation that these equations define a non-linear mapping in a four dimensional space. This method of analyzing the mean-field theory proves particularly useful in the determination of the pinning transition of the incommensurate structures. Next the phase diagram is investigated analytically by means of a Landau expansion modified such as to include domain walls. It is found that in the vicinity of the order-disorder transition most features of the phase diagram can be explained quantitatively by this expansion. Finally we present a systematic lowtemperature expansion of the mean-field theory, showing that the low-temperature phase diagram obtained in the mean-field approximation is different from that of the full model.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

On the chiral transition temperature in bilocal effective QCD

The breakdown of chiral symmetry in a many-quark system at finite temperature is studied by considering a bilocal quark interaction derived from a field theoretical treatment of QCD. We find that such an approach yields a chiral transition temperature which is about 30% lower compared to the standard NJL model prediction.