# The phase diagram for the Nambu–Jona-Lasinio model with ’t Hooft and eight-quark interactions

###### Abstract

It is shown that the endpoint of the first order transition line which merges into a crossover regime in the phase diagram of the Nambu–Jona-Lasinio model, extended to include the six-quark ’t Hooft and eight-quark interaction Lagrangians, is pushed towards vanishing chemical potential and higher temperatures with increasing strength of the OZI-violating eight-quark interactions. We clarify the connection between the location of the endpoint in the phase diagram and the mechanism of chiral symmetry breaking at the quark level. We show how the interactions affect the number of effective quark degrees of freedom. We are able to obtain the correct asymptotics for this number at large temperatures by using the Pauli-Villars regularization.

###### pacs:

11.10.Wx, 11.30.Rd, 11.30.Qc## I Introduction

The last two decades have witnessed great efforts towards the understanding of the QCD phase diagram, in terms of effective low energy theories paralleled by QCD lattice calculations, see e.g. the recent reviews Wilczek:2001 -Fukushima:2008 , or the paper Weise:2006 . The domain of small to moderate baryonic chemical potential MeV and temperatures MeV, is of specific relevance for relativistic heavy ion collisions. Of fundamental importance in the study of the phase diagram are chiral symmetry and confinement, however the finite size () as well as the difference in the bare quark masses () pose major problems both from the calculational point of view and in the implications due to deviations from the ideal situations, where the chiral condensate and the Polyakov loop are known to be the appropriate order parameters to characterize the phase state of the quark-gluon system.

The present study focuses on the chiral symmetry breaking aspects related to non-zero current quark mass values. Our arguments will be based on the successful model of Nambu–Jona-Lasinio (NJL) Nambu:1961 , combined with the breaking flavor determinant of ’t Hooft Hooft:1976 -Reinhardt:1988 (NJLH). Moreover the most general chiral invariant non derivative eight-quark () interactions Osipov:2005b are included. These terms were proven to render the effective potential of the NJLH model globally stable, and their effect has been thoroughly studied in low energy characteristics of pseudoscalar and scalar mesons Osipov:2006a , at finite temperature Osipov:2007b ; Osipov:2008 and in presence of a constant magnetic field Osipov:2007c . These studies have lead at instances to sizeable and unforeseen effects. Of particular importance for the present work is that the strength of the coupling is strongly correlated with the temperature and slope at which the crossover occurs and that it can be regulated together with the four-quark coupling, leaving the meson spectra at unaffected (with exception of the scalar meson mass which decreases with increasing coupling) Osipov:2008 . As a result the symmetry breaking for large couplings is induced by the ’t Hooft coupling strength, as opposed to the case with small coupling, where the dynamical breaking of symmetry is controlled by the coupling strength Osipov:2006a . We would like to comment on a natural question which arises here, namely, can higher order many-quark interactions be also important? With regard to this, explicit argumets of A. A. Andrianov and V. A. Andrianov are known Andrianov:1993 , which show that the structure of the QCD-motivated models at low energies with effective multi-fermion interactions and a finite cut-off in the chiral symmetry-breaking regime should contain only the vertices with four, six and eight-fermion interactions in four dimensions. This result explains partly the approximation used in our work.

Thus, in this paper we give a quantitative account of local multi-fermion forces on the phase diagram in the plane, by comparing the results for two sets of parameters, in the small and large coupling regimes of the strengths, corresponding to the two above mentioned alternative mechanisms of chiral symmetry breaking.

Throughout the paper we work for simplicity in the isospin limit , breaking explicitly the chiral symmetry to the (isospin-hypercharge) subgroup, and take the same baryonic chemical potential for all quark species. Generalizations to take into account the nonzero isospin chemical potential can be implemented as for instace in Klimenko:2006 .

## Ii Effective Lagrangian

The explicit form of the multi-quark Lagrangian considered is presented in Osipov:2005b ; Osipov:2006a

(1) |

Quark fields have color and flavor indices which are suppressed, . Here

(2) | |||||

(3) | |||||

(4) | |||||

(5) |

The matrices acting in flavor space, are normalized such that ; , and are the standard Gell-Mann matrices; are chiral projectors and the determinant is over flavor indices. The large behaviour of the model is reflected in the dimensionful coupling constants, , which count as , , and or less. As a result the NJL interaction (2) dominates over at large , as one would expect, because Zweig’s rule is exact at . Let us note that the -interaction breaks Zweig’s rule as well.

Since the coupling constants are dimensionful, the model is not renormalizable. We use the cut-off to render quark loops finite. The global chiral symmetry of the Lagrangian (1) at is spontaneously broken to the group, showing the dynamical instability of the fully symmetric solutions of the theory. In addition, the current quark mass , being a diagonal matrix in flavor space with elements , explicitly breaks this symmetry down, retaining only the reduced symmetries of isospin and hypercharge conservation, if .

The model has been bosonized in the framework of functional integrals in the stationary phase approximation leading to the following effective mesonic Lagrangian at

(6) |

written in terms of the scalar, , and pseudoscalar, , nonet valued quantum fields. The result of the stationary phase integration at leading order, , is shown here as a series in growing powers of and . The result of the remaining Gaussian integration over the quark fields is given by . Here the Laplacian in euclidean space-time is associated with the euclidean Dirac operator (the are antihermitian and obey ); is the constituent quark mass matrix (to explore the properties of the spontaneously broken theory, we define quantum fields as having vanishing vacuum expectation values in the asymmetric phase).

The expression for the one-quark-loop action has been obtained by using a modified inverse mass expansion of the heat kernel associated to the given Laplacian Osipov:2001 . The procedure takes into account the differences in the nonstrange and strange constituent quark masses in a chiral invariant way at each order of the expansion, being the generalized Seeley–DeWitt coefficients of the new series. This modification distinguishes our calculation from the one made in Ebert:1986 . In fact we consider the series up to and including the order that corresponds to the first nontrivial step in the expansion of the induced effective hadron Lagrangian at long distances. At this stage meson fields obtain their kinetic terms, but are still considered to be elementary objects. The information about their quark-antiquark origin enters only through the coefficients such as the average

(7) |

over the 1-loop euclidean momentum integrals with vertices ()

(8) |

For the explicit evaluation of we use the Pauli–Villars regularization method with two subtractions in the integrand. The procedure is fully defined by the insertion of the particular operator

(9) |

Here the covariant cut-off is a free dimensionful parameter which characterizes the scale of the chiral symmetry breaking in the effective model considered. To the order of the heat kernel series truncated, only the integrals are needed. These are quadratic and logarithmic divergent respectively with , all other are finite. Note that the recurrence relation

(10) |

is fulfilled. If is known for one value of , then the function may be computed for other values of by successive applications of the relation.

In the are determined via the stationary phase conditions. These conditions and the pattern of explicit symmetry breaking show that in general can have only three non-zero components at most with indices , i.e. , which can be found from a system of three independent equations

(11) |

Here , . The matrix valued constants of higher order, like for instance in , are uniquely determined once the are known Osipov:2004a ; Osipov:2006a . The stability of the effective potential is guaranteed if the system (11) has only one real solution. For that the couplings must fulfill the inequalities Osipov:2005b : .

TABLE I. Parameters of the model: (MeV), (GeV), (MeV), (GeV), (GeV). We also show the corresponding values of constituent quark masses and (MeV).

a | 5.9 | 186 | 359 | 554 | 851 | 10.92 | 1001 | ||

b | 5.9 | 186 | 359 | 554 | 851 | 7.03 | 1001 |

TABLE II. The masses, weak decay constants of light pseudoscalars (in MeV), the singlet-octet mixing angle (in degrees), and the quark condensates expressed as usual by positive combinations in MeV.

a | |||||||||
---|---|---|---|---|---|---|---|---|---|

b |

TABLE III. The masses of the scalar nonet (in MeV), and the corresponding singlet-octet mixing angle (in degrees).

a | |||||
---|---|---|---|---|---|

b |

In this paper we use the two parameter sets of Table I, which differ only in the choice of the coupling and the strength . Set (b) is the same as in Osipov:2008 (there was a misprint in the value for the constituent strange quark mass, which we corrected). Tables II-III display the numerical fits at (input is denoted by a *). The only difference in the observables of the two sets occurs in the singlet-octet flavor mixing channel of the scalars, mainly in the -meson (i.e. ) mass. The model parameters are kept unchanged in the calculation of the and dependent solutions of the gap equations (see next sections).

It is worthwhile to stress that there is an essential difference between the two alternative ground states chosen here as the configurations on top of which the and effects are studied: Case (a) corresponds to the standard picture of the NJL hadronic vacuum. In this picture chiral symmetry is spontaneously broken at when . Case (b) corresponds to a new alternative, related to the pattern where . In this case chiral symmetry can be broken only due to the six-quark interactions, when exceeds some critical value (the -interactions could in principle also induce symmetry breaking, however the mass spectra are then not well reproduced). One can hardly distinguish between the two cases at , the spectra of and low-lying mesons do not show much difference: the model parameters are the same, except for the correlated and values. The larger value of in the case (b) is a signal of the increasing role played by the eight-quark OZI-violating interactions, but this does not affect the value of the mixing angle , and only slightly diminishes . Such insensitivity follows from the observation that the stationary phase equations (11) and mass formulae of the light and states Osipov:2006a only depend on the couplings and through the linear combination , except for the and states inside the scalar nonet. However as soon as or are finite, the start to change due to their intrinsic dependence, acquired through the coupling to the quark loop integrals in the gap equations, eqs. (12) below. The dependence of the combination above is steered by the strength . This is the main reason why the -interactions may strongly affect the thermodynamic observables, without changing the spectra at .

## Iii Thermodynamic potential

### iii.1 Case of vanishing temperarure and chemical potential

Before addressing the thermodynamical potential it is instructive to briefly discuss the effective potential of the model at . Using standard techniques Osipov:2004a , we obtain from the gap-equations

(12) |

the effective potential as a function of three independent variables . If the parameters of the model are fixed in such a way that eqs. (11) have only one real solution, the effective potential is

(13) | |||||

where , and we extend definition (7) for index with

(14) |

Here has the explicit form

(15) |

for the given choice of regulator.

The first integral in (13) accounts for the leading order stationary phase contribution. The second integral describes the quark one-loop part. Since both integrands in (13) are exact differentials, the line integrals depend only on the end points. The low limit of the integrals is adjusted so that (to understand this, it is enough to notice that the power-series expansion of at small starts from ; this term does not depend on and, therefore, does not affect the physical content of the theory; so we simply subtract it, calculating the potential energy of the system with regard to the energy of the symmetric vacuum in the imaginary world of massless quarks).

### iii.2 Case of finite temperature and chemical potential

The extension to finite and of the bosonized Lagrangian (II) is effected through the quark loop integrals (see eq.(8)). Due to the recurrence relation (10) it is sufficient to get it just for one of them, , by introducing the Matsubara frequencies, , and the chemical potential, , through the substitutions Kapusta

(16) |

Inserting (16) into we obtain

(17) |

The sum over is evaluated to give

(18) |

where we use the abbreviations . Thus we have and

(19) |

where the quark, anti-quark occupation numbers are given by

(20) |

Notice that , therefore the vacuum piece is well isolated from the matter part. The remaining integral containing the quark number occupation densities is strictly finite, the dependent terms being a remnant of the Pauli–Villars regularization scheme. At small and we have

(21) | |||||

The special feature of this integral is that it vanishes exponentially with .

Now we are ready to evaluate the thermodynamical potential. Indeed, the gap-equations at finite and are

(22) |

Consequently,

(23) | |||||

where the function does not depend on , and therefore cannot be determined from the gap equation; it will be found from other arguments in the end of this section, but obviously . The integral is the immediate generalization of the case (14)

(24) |

where the medium contribution to is

(25) | |||||

with and referring to the occupation numbers for massless and massive particles correspondingly, , . In spite of the fact that the integral is convergent, we still keep the regularization to be consistent. Note that the action of the operator (see eq. (9)) on any smooth function, depending on through the energy, , can also be expressed in terms of momentum as , where

(26) |

Then, noting that, for instance,

(27) | |||||

where we used the fact the surface term disappears, we get finally

(28) |

It is important to realize that the expansion of the integral (28) for small values of and at starts from the term

(29) |

This leading contribution in arises from the combination and does not depend on the cut-off, i.e. the Pauli-Villars regulator does not affect the leading order of the low-temperature asymptotics of the integral. To make this clear let us consider the typical integral in eq. (28)

(30) |

If , then the integral can be evaluated explicitly and is found to be , leading us to the result (29). If , we obtain at once the estimate at small

(31) |

We conclude that the integral vanishes exponentially for small and does not contribute to the leading order term in eq. (29). It is then clear that the action of the Pauli-Villars regulator on the integrand, which consists in subtracting the contribution of the massive Pauli-Villars states, will not affect the leading term as well.

One might worry that the low-temperature expansion of the integral (29) starts from the unphysical contribution which corresponds to the massless quark states. This fear would be valid if the potential had not the term . Let us fix the -independent function