# Quark matter under strong magnetic fields in the su(3) Nambu–Jona-Lasinio Model

###### Abstract

In the present work we use the mean field approximation to investigate quark matter described by the su(3) Nambu–Jona-Lasinio model subject to a strong magnetic field. We consider two cases: pure quark matter and quark matter in -equilibrium possibly present in magnetars. The results are compared with the ones obtained with the su(2) version of the model. The energy per baryon of magnetized quark matter becomes more bound than nuclear matter made of iron nuclei, for around G. When the NJL model is applied to stellar matter, the maximum mass configurations are always above 1.45 and may be as high as 1.86 for a central magnetic field of G. These numbers are within the masses of observed neutron stars.

PACS number(s): 24.10.Jv,26.60+c,11.10.-z,11.30Qc

## I Introduction

In non-central heavy ion collisions such as the ones performed at RHIC and LHC-CERN, physicists have been looking for a possible signature of the presence of CP-odd domains in the presumably formed quark-gluon plasma phase qgp . The study of deconfined quark matter subject to strong external magnetic fields is then mandatory if one intends to understand the physics taking place in such colliders.

Neutron stars with very strong magnetic fields of the order of G are known as magnetars and they are believed to be the sources of the intense gamma and X rays detected in 1979 duncan ; kouve . The hypothesis that some neutron stars are constituted by unbound quark matter cannot be completely ruled out quarks since the Bodmer-Witten conjecture conjecture cannot be tested on earthly experiments. This conjecture implies that the true ground state of all matter is (unbound) quark matter because theoretical predictions show that its energy per baryon at zero pressure is lower than Fe binding energy.

In the present work our aim is to investigate quark matter described by the su(3) version of the Nambu-Jona-Lasinio njl model exposed to strong magnetic fields. In the case of pure quark matter, as predicted by the QCD phase transition possibly taking place in heavy ion collisions, the magnetic field is certainly external. In the case of neutron stars, the magnetic field can be generated by the alignment of charged particles that are spinning very rapidly. We next use an external field to mimic the real situation, which we do not know how to determine. Albeit in an approximate way, the effect of the magnetic field on the macroscopic quantities as radius and masses can be obtained.

Recently the su(2) version of the NJL model was used to treat both situations described above prc . We have shown that, for pure quark matter, the energy per baryon for magnetized quark matter has a minimum which is lower than the one determined for magnetic free quark matter. We have also obtained that a magnetic field of the order of G barely affects the effective mass as compared with the results for matter not subject to the magnetic field. For G matter is totally polarized for chemical potentials below 490 MeV. For small values of the magnetic fields the number of filled Landau levels (LL) is large and the quantisation effects are washed out, while for large magnetic fields the chiral symmetry restoration occurs for smaller values of the chemical potentials. When -equilibrium is enforced, the numerical results show that, for the the su(2) case, only very high magnetic fields (G) affect the equation of state (EOS) in a noticeable way.

The inclusion of the -quarks, necessary in the su(3) NJL model, poses some new numerical difficulties and some questions that need to be addressed. Those problems are tackled through out the paper. One of the questions was raised in buballa96 ; buballa99 and refers to the stability of quark matter described by the NJL model. The authors show that it is not absolute stable. As already mentioned, in prc we have seen that the inclusion of the magnetic field increases stability in the su(2) version and the same behavior is expected in the su(3) NJL, which is shown next.

The paper is organized is such a way that all calculations already shown explicitly in prc are not repeated but all important differences are outlined. In sections II and III the formalism (mean field theory) and the equations of state are shown and in section IV the final results are displayed and the conclusions are drawn.

## Ii General formalism

In order to consider (three flavor) quark stars in equilibrium with strong magnetic fields one may define the following lagrangian density

(1) |

where the quark sector is described by the su(3) version of the Nambu–Jona-Lasinio model

(2) |

where and are given by:

(3) |

(4) |

where represents a quark field with three flavors, is the corresponding (current) mass matrix while represents the quark electric charge and denotes the Gell-Mann matrices. Here, we consider . The term is the t’Hooft interaction which represents a determinant in flavor space which, for three flavor, gives a six-point interaction buballa

(5) |

where is the usual three-dimensional Levi-Civita symbol. The lagrangian also contains the term which is symmetric under global transformations and corresponds to a 4-point interaction in flavor space. In the appendix we discuss the steps to obtain in the mean-field approximation (MFA).

The leptonic sector is given by

(6) |

where . One recognizes this sector as being represented by the usual QED type of lagrangian density. As usual, and are used to account for the external magnetic field. Then, since we are interested in a static and constant magnetic field in the direction, .

## Iii The EOS

We need to evaluate the thermodynamical potential for the three flavor quark sector, , which as usual can be written as where represents the pressure, the energy density, the temperature, the entropy density, and the chemical potential.

For the present study, just the zero temperature case is important and, as a consequence, the term with the entropy vanishes. The total pressure for three flavor in equilibrium is given by

(7) |

where our notation means that is evaluated in terms of the quark effective mass, , which is determined in a (nonperturbative) self consistent way while is evaluated at the leptonic bare mass, . The term arises due to the electromagnetic term in the original lagrangian density. The subscript indicates normalized pressures. Here, our normalization choice is such that at () and at () implying that .

### iii.1 Quark Contribution to the EOS

In the mean field approximation the pressure can be written as

(8) |

where an irrelevant term has been discarded. The pressure due to the three quarks is diagrammatically represented in figure 1a.

For a given flavor, the term is given by

(9) |

and the condensates, are given by

(10) |

where all the traces are to be taken over color () and Dirac space, but not flavor. In order to obtain results valid at finite and in the presence of an external magnetic field one can use the following replacements

In the above relations, , with representing the Matsubara frequencies for fermions while represents the Landau levels (LL) and represents the spin states which, at , must be treated separately. The case in which we are interested can be easily obtained after the above substituions (see Ref. prc ).

The effective quark masses can be obtained self consistently from (see figure 1b)

(11) |

with being any permutation of . So, to determine the EOS for the su(3) NJL at finite density and in the presence of a magnetic field we need to know the condensates, , as well as the contribution from the gas of quasiparticles, . Both quantities, which are related by , have been evaluated with great detail in Ref. prc . Here, we just quote the results

(12) |

where the vacuum contribution reads

(13) |

where we have defined with representing a non covariant ultra violet cut off. The evaluations performed in Ref. prc also give the following finite magnetic contribution

(14) |

where while where is the Riemann-Hurwitz zeta function wolfram . Finally, after integration, the medium contribution can be written as

(15) | |||||

where , . The upper Landau level (or the nearest integer) is defined by

(16) |

Finally, the condensates entering the quark pressure at finite density and in the presence of an external magnetic field can also be written as

(17) |

where

(18) |

(19) | |||||

and

(20) | |||||

From the pressure one can obtain the density, , corresponding to each different flavor, which is given by

(21) |

where , since .

The quark contribution to the energy density is

(22) |

where .

Throughout this paper we consider the following set of parameters buballa : , , , and .

### iii.2 Lepton Contribution to the EOS

The leptonic contribution, has also been evaluated in detail in Ref. prc where the normalization requirement at has been adopted. The result shows that, at the one loop level, only the following (finite) medium contribution has to be considered

Then, the leptonic density is also easily evaluated yielding

(24) |

where . Finally, the leptonic energy density reads

(25) |

The lepton masses are and .

## Iv Results and conclusions

In the sequel we consider two different situations of quark matter under a strong magnetic field: a) pure quark matter with the same chemical potential for all quark flavors; b) -equilibrium quark stellar matter.

We first discuss the properties of pure quark matter with equal chemical potentials for all flavors, namely the behavior of the dynamical quark masses, the chiral symmetry restoration with density and the energy per baryon. In Fig. 2 we display the masses of quarks and as function of the chemical potential for different values of the magnetic field and the two versions of the NJL model. For the magnetic field intensities used, one can clearly identify the filling of different Landau levels causing the usual kinks in the curves. For the three intensities considered the chiral symmetry is approximately restored for MeV.

It is interesting to see that although the general behavior is the same, the effect of the LL is more pronounced in the su(2) version.

In Fig. 3 the mass of the quark is shown as a function of the chemical potential for different values of the magnetic field. One can see how drastically it falls around MeV. For magnetic free quark matter, this is the same behavior shown in Fig. 3 of schaffner . One can observe that the curve is no longer smooth when is turned on, but the values of the strange quark mass do not vary much. According to schaffner , the fact that the strange quark mass remains relatively high as compared with the masses of the other two quarks is the main reason why deconfined quark matter may not be likely to appear in the core of hybrid neutron stars. For a magnetic field larger than 10 the restoration of chiral symmetry for the -quark occurs in steps and starts at a smaller chemical potential than the B=0 case.

The phenomenon of magnetic catalysis, which enhances chiral symmetry breaking, has been well discussed within the version of the NJL model klimenko . Here, for reference, we show the vacuum effective mass of the three quarks as a function of the magnetic field in Fig. 4. For G the vacuum masses increase dramatically with the magnetic field as expected. A similar increase of the vacuum mass was also obtained for the version of NJL in klimenko ; prc and the effect is related to the fact that the B field facilitates the binding by antialigning the helicities of the quark and the antiquark, which are then bound by the NJL interaction. As shown in Fig. 4, an interesting result of the version is that, due to its larger electric charge, the quark has an effective mass that becomes larger than that of the quark for G.

In Fig.5 the baryonic density is shown as a function of the quark chemical potential for two values of the magnetic field and for both versions of the NJL model, and . As already noticed in prc , once again, for small values of the magnetic fields the number of filled LL is quite large and the effects of the quantization are less visible. Due to the Landau quantization, the increase of the strength of the magnetic field gives rise to a decrease of the number of the filled LL and the amplitude of the oscillations is more clear in the graphics. For each value of the magnetic field, the kink appearing at the smallest chemical potential corresponds to the case when only the first LL has been occupied.

In Fig. 6 one can see that the inclusion of the magnetic field makes matter more and more bound in both versions of the model. For the present set of parameters, the energy per baryon of magnetized quark matter becomes more bound than nuclear matter made of iron nuclei, MeV for around G.

We next consider stellar matter made out of quarks, electrons and muons in -equilibrium, as possibly occurring in the interior of magnetars. It is worth mentioning that, in this case, the three different quarks bear different chemical potentials, determined by the chemical equilibrium conditions

We start by plotting the quark effective masses for different values of the magnetic field in Fig. 7. It is seen that the results for non-magnetized matter () almost coincide with the ones obtained for G. A decrease of the quark mass starts only at fm. This behavior had already been discussed in dp04 . If the magnetic field is strong enough the mass of quark occurs in finite jumps which may give rise to an increase of the strangeness fraction as shown in Fig. 8.

The quark fractions , are shown in Fig. 8. Again the results for are similar to the ones for G. For strong enough fields the quark fractions increase with a reduction of the quark fraction. The quark fraction has a sudden increase for fm but above fm remains below the fraction.

In Fig. 9 the EOS for different values of the magnetic field is shown. For magnetic fields as large as G the differences are very small as compared with non-magnetized matter. For larger fields there is an overall net softening of the EOS.

It is well known that at the surface the magnetic field should not be larger than G. We have introduced a density dependent magnetic field as in chakra97 ; prc :

(26) |

where G is the magnetic field at the surface, is the magnetic field at the interior of the star for large densities and the parameters and were chosen in such a way that the field increases fast with density to its central value but still describes correctly the surface of the star where the pressure is zero. We show the equations of state for quark matter in -equilibrium and a density dependent magnetic field within both versions of the NJL model in Fig. 10. As implicit in eq. (26), the field at the surface is G. The magnetic field makes the EOS harder with consequences in the gravitational and baryonic masses of compact stars, whose properties are obtained from the integration of the Tolman-Oppenheimer-Volkoff equations, which use as input the EOS obtained with the density dependent magnetic field. The results are displayed both in Fig. 11 and in Table 1, from where it is seen that both the gravitational and the baryonic masses increase with the increase of the magnetic field for an intensity larger than G for the su(3) version and G for the su(2) NJL. However, the increase of the gravitational mass is larger than the increase of the baryonic mass because the contribution of the magnetic field becomes more and more important as the field increases. This explains the decrease of the central energy/baryonic density for the stronger fields considered.

Another important effect of the field on the properties of the stars is the increase of the radius of the star with the largest radius, which may be as high as 9.5 Km for the NJL. In general, the maximum mass star configurations for the version of the NJL model are smaller with smaller radius, Km, in average 2 Km smaller than the corresponding stars in the version of the NJL model.

Within the NJL the maximum mass configurations are always above 1.45 and may be as high as 1.86 for a central magnetic field of G. These numbers are within the masses of observed neutron stars. On the other hand the version of the NJL model forsees too small star masses except for very large magnetic fields.

(G) | () | () | (Km) | fm | fm | (G) |
---|---|---|---|---|---|---|

0 | 1.46 | 1.53 | 8.93 | 7.49 | 1.19 | 10 |

10 | 1.46 | 1.53 | 8.93 | 7.49 | 1.19 | 1.610 |

510 | 1.47 | 1.54 | 8.88 | 7.94 | 1.24 | 8.810 |

110 | 1.50 | 1.58 | 8.78 | 8.36 | 1.25 | 1.810 |

210 | 1.61 | 1.69 | 8.53 | 9.64 | 1.25 | 3.610 |

510 | 1.86 | 1.88 | 8.81 | 9.26 | 1.01 | 5.010 |

0 | 1.29 | 1.24 | 7.09 | 13.68 | 1.86 | 10 |

110 | 1.29 | 1.25 | 7.08 | 13.85 | 1.88 | 1.210 |

110 | 1.38 | 1.33 | 7.01 | 14.52 | 1.72 | 4.010 |

210 | 1.49 | 1.41 | 7.11 | 14.47 | 1.49 | 5.710 |

The effects of the anomalous magnetic moments has been shown to be relevant prakash ; wei ; aurora and we intend to take them into account in the next calculations.

The color superconductivity (CS) cfl , which allows the quarks near the Fermi surface to form Cooper pairs that condense and break the color gauge symmetry mga is known to be present in the QCD phase diagram at sufficiently high densities. The effect of strong magnetic fields on the CS properties of quark matter, which can be drastic for sufficiently high fields, has already been studied by several authors b-cs . It would be important to investigate how this SC phase could affect the properties of quark stars under strong magnetic fields. However, it could be that CS is only affected by magnetic fields stronger than the ones considered in the present paper, which, however, predicts already a very high maximum mass, M. The largest magnetic field we got in the center of a quark star is G, while in b-cs it is shown that a noticeable effect requires fields above G.

## Acknowledgments

This work was partially supported by the Capes/FCT n. 232/09 bilateral collaboration, by CNPq (Brazil), by FCT and FEDER (Portugal) under the project CERN/FP/83505/2008 and by Compstar, an ESF Research Networking Programme.

## Appendix A The su(3) NJL model in the MFA

In this appendix the main steps in order to obtain the Nambu-Jona-Lasinio lagrangian given eq.(2) in the mean field approximation are explicitly shown. Firstly, we consider the term given in eq.(3). For later convenience, we define the matrix elements of and its adjoint as hatsuda :

where i, j are flavor labels. From these definitions, one can easily show that:

(27) | |||||

where is the trace operator in flavor space. So, adding and subtracting these expressions, we can rewrite the NJL symmetric four-point interaction term as:

(28) | |||||

The summation involved in the latter equality can be performed noting that an arbitrary matrix in the =3 flavor space, can be expanded in terms of Gell-Mann matrices as follows:

(29) |

The expansion coefficients are obtained using the the Gell-Mann matrices property: . So, we can write:

(30) |

where in the latter term we have used that the Gell-Mann matrices are hermitian, i.e., =. We then evaluate in the mean field approximation linearizing the interaction terms. We follow refs.buballa ; hatsuda approximating the product of two operators and by:

(31) |

Therefore, calculating explicitly the trace involved in eq.(28) and taking into account the prescription above, , can be written in the MFA as:

(32) |

where we have used:

(33) |

The only 3 non-vanishing terms are the condensates which were defined in eq.(10). Finally, we consider the t’Hooft term, eq.(4), which is a six-point interaction in the su(3) flavor space. Notice that term involves the product of three operators which we linearize analogously to eq.(31):

So, in the MFA the determinants which appear in the t’Hooft term can be written as:

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