Thermomagnetic Effects in an External Magnetic Field in the Logarithmic-Quark Sigma Model

Transcription

1 International Journal o Modern Physics and Application 017; (6): ISSN: hermomagnetic Eects in an External Magnetic Field in the Logarithmic-Quark Sigma Model Mohamed Abu-Shady Department o Mathematics, Faculty o Science, Menouia University, Shebin El-Koom, Egypt address Dr.abushady@gmail.com Keywords Chiral Lagrangian Density, Magnetic Catalysis, Mean-Field Approximation Received: October 16, 017 Accepted: November 1, 017 Published: November 16, 017 Citation Mohamed Abu-Shady. hermomagnetic Eects in an External Magnetic Field in the Logarithmic-Quark Sigma Model. International Journal o Modern Physics and Application. Vol., No. 6, 017, pp Abstract he phenomenon o magnetic catalysis o chiral symmetry breaking in the quantum chromodynamic theory in the ramework o logarithmic quark sigma model is studied. hermodynamic properties are calculated in the mean-ield approximation such as the pressure, the entropy density, the energy density, and the measure interaction. he pressure, the entropy density, and the energy density increase with increasing temperature and/or an external magnetic ield. he critical temperature increases with increasing an external magnetic ield. In addition, the chiral phase transition is crossover in the presence o an external magnetic ield at absent o baryonic chemical potential when explicit symmetry breaking is included. A comparison is presented with the original sigma model and other works. A conclusion indicates that the logarithmic quark model enhances the magnetic catalysis phenomenon. 1. Introduction he study o the inluence o an external magnetic ields on the undamental properties o quantum chromodynamic (QCD) theory such as the coninement o quark and gluons at low energy and asymptotic reedom at high energy is still a matter o great theoretical and experimental [ 1 ]. he transition rom composite objects to colored quarks and gluons has several characteristics o both theoretical and phenomenological relevance. One such characteristic is the equation o state which is the undamental relation encoding the thermodynamic properties o the system [ 1 ]. So ar, most estimates have been carried out at vanishing chemical potential with the aid o eective 5 and the Numbu-Jona-Lasinio (NJL) theories such as the linear sigma model (LSM [ ] model [ 6,7] in the mean ield approximation. he linear sigma model exhibits many o global symmetries o QCD theory. he model was originally introduced by Gell-Mann and Levy [ 8 ] with the purpose o describing pion-nucleon interactions. During the last years an impressive amount o work has been done with this model. he idea is to consider it as an eective low energy approach or QCD theory that the model has some aspects o QCD theory such as the chiral symmetry. hus, the model is successully to describe the most o hadron properties at low energy such as in Res. [ 5,9,10 ]. Some observables that are calculated in this model are conlict with experimental data. So researchers interest to modiy this model to provide a good description o hadron properties such as in Res. [ ]

2 50 Mohamed Abu-Shady: hermomagnetic Eects in an External Magnetic Field in the Logarithmic-Quark Sigma Model In Re. [ 16 ], the authors have modiied the linear sigma model by including the logarithmic mesonic potential and study its eect on the phase transition at inite temperature. In addition, the comparison with other models is done. On the same hand, the logarithmic sigma model successully describes nucleon properties at inite temperature and 17. chemical potential [ ] o continue the investigation that started in Re. [ 18 ]. In this paper, we investigate the eect o external magnetic ield on the thermodynamic properties in the ramework o logarithmic quark sigma model at inite temperature and chemical potential. So or no attempts have done to investigate the thermodynamic properties in the ramework o the logarithmic sigma model. his paper is organized as ollows: he original sigma model is briely presented in Sec.. Next, the eective logarithmic mesonic potential in the presence o external magnetic ield is presented in Sec. 3. he results are discussed and are compared with other works in Secs. and 5, respectively. Finally, the summary and conclusion are presented in Sec. 6.. he Chiral Sigma Model with Original Eective Potential he interactions o quarks via the exchange o σ and π- meson ields are given by the Lagrangian density [5] as ollows: µ 1 µ µ L( r ) = iψγ µ Ψ + ( µ σ σ + µ π. π) + gψ ( σ + iγ5τ. π ) Ψ U1( σ, π), (1) with ( ) π 1 (, ) λ U σ π = σ + π ν + m πσ, () where U ( σ, π ) is the meson-meson interaction potential and Ψ,σ and π are the quark, sigma, and pion ields, respectively. In the mean-ield approximation, the meson ields are treated as time-independent classical ields. his means that we replace the power and products o the meson ields by corresponding powers and the products o their expectation values. he meson-meson interactions in Eq. () lead to hidden chiral SU() SU() σ r taking on a vacuum expectation value symmetry with ( ) σ = π, (3) where π = 93MeV is the pion decay constant. he inal term in Eq. () included to break the chiral symmetry explicitly. It leads to the partial conservation o axial-vector isospin current (PCAC). he parameters λ and ν can be expressed in terms o π, sigma and pion masses as, σ π π m m λ =, () π. m ν = π (5) λ 3. he Eective Logarithmic Potential in the Presence o Magnetic Field In this section, the logarithmic mesonic potential U ( σ, π ) is applied. In Eq. (6), the logarithmic potential is included with the external magnetic ield at inite temperature and baryonic chemical potential [ ] as ollows, where U ( σ, π) = U ( σ, π) + U + U + U, (6) e Vaccum Matter Medium U σ, π λ σ π λ σ π log m πσ, (7) σ + π = π π ( ) ( ) ( ) In Eq. 7, the logarithmic potential satisies the chiral symmetry when m π 0 as well as in the original potential in Eq.. Spontaneous chiral symmetry breaking gives a nonzero vacuum expectation or σ and the explicit chiral symmetry breaking term in Eq. 7 gives the pion its mass. where σ = π (8) σ 7mπ m λ1 =, (9) 1 mσ mπ 1π λ =. (10) For details, see Res. [ 17,18 ]. o include the external magnetic ield in the present model, we ollow Re. [ ] by including the pure ermionic vacuum contribution in the ree

3 International Journal o Modern Physics and Application 017; (6): potential energy. Since this model is renormalizable the usual procedure is to regularize divergent integrals using dimensional regularization and to subtract the ultra violet divergences. his procedure gives the ollowing result U N cn g g σ + π Vacum = σ + π ( π) 3 ( ) ( ) ( ln( )), Λ (11) where N c = 3 and N = are color and lavor degrees o reedom, respectively and Λ is mass scale, d Nc (1,0) 1 Matter ζ π = u x U = ( q B) [ ( 1, x ) ( x x )ln x + ] (1) In Eq. 1, we have used x g ( σ + π ) = and ( q B) (1,0) d ζ( z, x ) ζ ( 1, x ) dz z = 1 = that represents the Riemann-Hurwitz unction, and also q is the absolute value o quark electric charge in the external magnetic ield with intense B. d [ EP, k( B) + µ ] [ EP, k( B) µ ] Nc Medium = α ( ) [ ln 1 ln 1 ], k z ( π ) (13) = uk = 0 U q B dp e e where P, k( ) z E B = P + k q B + M,M is the eective sel-consistent quark mass and µ is baryonic chemical potential.. Discussion o Results In this section, the eective potential o the logarithmic sigma model is studied. For this purpose, the eective potential is numerically calculated in Eq. (6). he parameters o the present model are the coupling constant g and the sigma mass m σ. he choice o ree parameters o g and m σ based on Re. [ ]. he parameters are usually chosen so that the chiral symmetry is spontaneously broken in the vacuum and the expectation values o the meson ields. In this work, two dierent sets o parameters are considered in order to get high and a low value or sigma mass. he irst set is given by Λ = 16.8 MeV which yields m π = 138 MeV and m σ = 600 MeV. he second set as the irst, yielding m σ = 00 MeV. he quantities such as the P dimensionless o pressure density,, the dimensionless energy E S n and the dimensionless entropy density 3 calculated. hese quantities can be readily obtained by the eective potential that deines in Eq. ( 6 ), gives the negative pressure U ( σ, π ) = P. hen the net quark number e density is obtained rom ρ = n dpn dµ are, and the entropy density dpn rom s = while the energy density is d En = Pn + S + µρ. hese quantities are displayed in Figures (1,, 3,, 5, and 6). First o all, one observes that the quantities are calculated at zero and inite strong magnetic ield in the two cases at vanishing o chemical potential and non-vanishing o chemical potential. Pn/^ Figure 1. he dimensionless pressure density is plotted as a unction o temperature at µ=0 and µ=0.1 GeV at vanishing magnetic ield. In Figure 1, the dimensionless pressure density is plotted as a unction o temperature at vanishing o magnetic ield ( B ). At vanishing o chemical potential ( µ = 0), the pressure increases with increasing temperature. he behavior is good qualitative agreement with recent lattice calculation 1, in which the pressure increases with increasing in Re. [ ] temperature. Also, the behavior o pressure is in qualitative agreement with the original sigma model and the NJL model. On the same hand, the behavior o pressure is qualitative [ ].0E- 1.6E- 1.E- 8.0E-5.0E-5 chemical potential = E (GeV) agreement work o awik et al. [ 3 ], in which the hadron gas and the Polyakov linear sigma models are applied in their calculations. At µ 0, we note that the curve shits to higher values, in particular, at higher-values o temperature.

4 5 Mohamed Abu-Shady: hermomagnetic Eects in an External Magnetic Field in the Logarithmic-Quark Sigma Model In Figure, the eect o strong magnetic ield is studied on the behavior o pressure, then one chooses magnetic ield 0. 1,,3. he qualitative eb = GeV as in Res. [ ] agreement is noted between Figure 1 represents the case o vanishing magnetic ield and Figure represents the presence o strong magnetic ield. he eect o magnetic ield appears on the pressure value that the pressure increases strongly at higher-values o temperature. his behavior is qualitative agreement with Res. [ 1,,3 ]. En/^ Pn/^ (GeV) Figure. he dimensionless pressure density is plotted as a unction o temperature or at µ=0 and µ=0.1 GeV at strong magnetic ield eb = 0. GeV²..5E-3 chemical potential = (GeV) Figure. he dimensionless energy density is plotted as a unction o temperature at µ=0 and µ=0. GeV at non-vanishing magnetic ield eb =0. GeV². In Figure 3, the dimensionless energy density is plotted as a unction o temperature. he energy density increases with increasing temperature at vanishing and nonvanishing chemical potential. At vanishing o chemical potential ( µ = 0), the energy density increases with increasing temperature up to 0.5 GeV and then slowly decreases at larger values o temperature above critical temperature due to E n taking ratio. By increasing magnetic ield as in Figure, the behavior o the energy density is a qualitative agreement with the behavior at vanishing magnetic ield as in Figure 3. he qualitative agreement with Res. [ 1,,3 ] is noted. 1.0E-3.0E-3 1.5E-3 8.0E- En/^ 1.0E-3 S/^3 6.0E- 5.0E-.0E- 0.0E (GeV) Figure 3. he dimensionless energy density is plotted as a unction o temperature or at µ=0 and µ=0. GeV at vanishing magnetic ield..0e (GeV) Figure 5. he dimensionless entropy density is plotted as a unction o temperature or at µ=0 and µ=0.1 GeV at vanishing magnetic ield.

5 International Journal o Modern Physics and Application 017; (6): S/^3 c (GeV) Figure 6. he dimensionless entropy density is plotted as a unction o temperature or at µ=0 and µ=0.1 GeV at magnetic ield eb = 0. GeV². In Figure 5, the dimensionless entropy density is plotted as a unction o temperature. One notes that the entropy density increases with increasing temperature at vanishing and nonvanishing chemical potential. he eect o temperature appears on entropy density at higher values o temperatures. By increasing magnetic ield as in Figure 6, the behavior o the entropy density is a qualitative agreement with the behavior at vanishing magnetic ield as in Figure 5. Also, the entropy density decreases at higher temperature above critical temperature c = 0. GeV. his due to taking ratio S 3 Delta/^ (GeV) (GeV) Figure 7. he dimensionless interaction measure is plotted as a unction o temperature or dierent values o magnetic ield eb = 0., 0.3, and 0. GeV² are arranged rom down to up. Figure 8. Critical o temperature is plotted as a unction o magnetic ield at vanishing o baryonic chemical potential. In Figure 7, the measure interaction is plotted as a unction o temperature. he measure interaction increases with increasing temperature up to the top curve and then gradually decreases with increasing temperature. By increasing magnetic ield, we note that the top o each curve shits to the direction o increasing temperature. his behavior is an agreement with original sigma model and the NJL model. his behavior is interpreted as crossover phase [ ] transition. In Figure 8, the critical temperature increases with increasing magnetic ield. hus, the logarithmic quark enhances magnetic catalysis as well as in the original sigma. By including Polyakov loop model and the NJL model [ ] to take coninement into account. he results show that c related to deconinment also increases with increasing 6. On other hand, the irst lattice magnetic ield as in [ ] attempt to solve the problem taken two quark lavors and high values o pion mass ( m π = 00 00MeV) conirming that c eb (GeV^) 7. increasing with increasing magnetic ield [ ] 5. Summary and Conclusion In this work, the eective logarithmic potential have been employed to study thermodynamic properties in the presence o magnetic ield. So, the novelty in this work, thermodynamic properties is investigated in the ramework o logarithmic quark sigma model. he present results are 7 and agreement with irst lattice calculations as in Re. [ ] other models as in Res. [,6 ]. In addition, the present calculations are carried out beyond the zero chemical potential which are not taken many recent works. he conclusion indicates that the logarithmic quark sigma model enhances magnetic catalysis at inite temperature and

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