Magnetized vector boson gas at any temperature
G. Quintero Angulo, L. C. Suárez González, A. Pérez Martínez, H. Pérez Rojas
MMagnetized vector boson gas at any temperature
G. Quintero Angulo, ∗ L. C. Su´arez Gonz´alez, † A. P´erez Mart´ınez, ‡ and H. P´erez Rojas § Facultad de F´ısica, Universidad de la Habana,San L´azaro y L, Vedado, La Habana 10400, Cuba Instituto de Cibern´etica, Matem´atica y F´ısica (ICIMAF),Calle E esq a 15 Vedado 10400 La Habana Cuba
We study the thermodynamic properties of a relativistic magnetized neutral vectorboson gas at any temperature. By comparing the results with the low temperatureand the non relativistic descriptions of this gas, we found that the fully relativisticcase can be separated in two regimes according to temperature. For low tempera-tures, magnetic field effects dominate and the system shows a spontaneous magneti-zation, its pressure splits in two components and, eventually, a transversal magneticcollapse might occur. In the high temperature region, the gas behavior is led by pairproduction. The presence of antiparticles preserves the isotropy in the pressure, andincreases the magnetization and the total pressure of the system by several orders.Astrophysical implications of those behaviors are discussed.
Keywords: magnetized vector boson gas, magnetic BEC, antiparticles ∗ gquintero@fisica.uh.cu † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] J a n I. INTRODUCTION
The obtaining in the laboratory of the BEC for composite particles constituted one of themilestone of physics at the end of the last century, and it came along with the demonstrationthat Bose-Einstein condensation and superfluidity can also be considered as the limitingstates of another more general phenomenon: fermion pairing [1–6]. Since the experimentalachievement of Bose-Einstein condensation (BEC) [7], bosonic gases have attracted lot oftheoretical attention, not only for the condensate per se , but also for other interestingphenomena linked with it like the BEC collapse called ’Bose-Nova’ [8], the diffuse BEC ofmagnetized charged gases [9–15] and the Bose-Einstein ferromagnetism [16, 17].In Earth, BEC is still an exotic state restricted to lab, however astrophysical and cos-mological environments provide appropriate conditions for its natural existence. Hypothet-ically, several types of mesons condensates –pions and kaons– exist inside neutron stars[18–20]. Moreover, in the last few years new observational evidence has came to reinforcethe supposition of a NS superfluid interior [21–23]. This superfluid is composed by pairedprotons and neutrons that, given the star´s typical inner conditions, are expected to be inan intermediate situation between the BCS and the BEC limits [24]. Usually, the pairednucleons are described in the BCS limit, although some descriptions in the BEC limit havealso been developed giving birth to BEC stars models [24–27]. Boson stars are less popularthan fermion stars models to describe compact objects, nevertheless, self-gravitating bosonsystems have been studied since last century not only in connection to compact objects, butalso as sources of dark mater and black holes [25, 28–30].Besides dark matter stars, Bose-Einstein condensates are popular in cosmology as analternative to the standard CDM model in small scales ( ∼
10 kpc or less) [31]. Such modelsare based on the supposition that dark matter is composed of very light hypothetical bosons( m ∼ − − − eV), known as axions [31–35]. Large-scale predictions of axion modelsare the same as in CDM, but small-scale predictions seem to be more in accordance withobservations [31]. The BEC of axions has not only proven to be a viable dark mattercandidate [31, 33], but it also provides plausible explanations for dark energy and its relationwith dark matter [32, 33].A common approximation of all these theoretical studies is the assumption that theBose gas is at zero or low temperature. This is also the usual approximation in the caseof fermion gases in astrophysical environments. In these scenarios, fermion densities are sohigh that thermal fluctuations become negligible even at the billions of kelvins reached insideneutron stars. However, this limit does not work as well for bosons, because due to BECthey are very sensitive to environmental changes (variations in particle density, temperatureand magentic field) as we have already reported in some preliminary studies on magnetizedspin-one particles at finite temperature [36, 37].Spin-one boson gases are of great interest due to their unique magnetic properties inconnection with BEC [16] and with astrophysical magnetic field generation [27]. All thisbackground, makes us focus this article in the thermodynamic properties of a magnetizedneutral vector boson gas (NVBG) at any temperature. In Section II and III we reviewthe equations of motion of neutral vector bosons and the thermodynamic potential of thecorresponding gas. Section IV is devoted to condensation, while Section V encloses themagnetic properties. In Section VI the equations of state (EoS) are discussed. Concludingremarks are listed in Section VII, while mathematical details are given in the appendix.The numerical calculations and plots have been done for a composite spin-one bosonformed by two paired neutrons, with mass m = 2 m N and magnetic moment κ = 2 µ N , being m N and µ N the mass and the magnetic moment of the neutron. This kind of effective bosonsmight be created in the core of neutron stars (see [27] and references therein). Apart fromthe astrophysical inspiration, the discussions of our results are valid for any massive neutralvector boson gas and could be applied to phenomena in condensate matter [17] and heavy-ions colliders [38–40]. Along the paper, the results for the relativistic vector boson gas at anytemperature are compared with the ones coming for the low temperature [41] and the non-relativistic [36] treatment of this gas. As we shall see, this provide a better understandingof the underlaying physics, as well as a quick way to detect the high temperature effects. II. EQUATION OF MOTION OF A NEUTRAL VECTOR BOSON
The Lagrangian of neutral vector bosons under the action of an external magnetic field isan extension of the original Proca Lagrangian for spin-one particles that includes particle-field interactions [42, 43] L = − F ην F ην − ψ ην ψ ην + m ψ η ψ η + imκ ( ψ η ψ ν − ψ ν ψ η ) F ην , (1)where the index η and ν run from 1 to 4, F ην is the electromagnetic tensor and ψ ην , ψ η areindependent field variables that follow the equations of motion [42] ∂ η ψ ην − m ψ ν + 2 iκmψ η F ην = 0 , (2) ψ ην = ∂ η ψ ν − ∂ ν ψ η , (3)that in the momentum space read [41] (cid:0) ( p η p η + m ) δ νη − p ν p η − iκmF ην (cid:1) ρ η = 0 . (4)Thus, the vector boson propagator is D − ην = ( p η p η + m ) δ νη − p ν p η − iκmF ην . (5)Taking the magnetic field uniform, constant and in p direction B = B e , the generalizedSakata-Taketani Hamiltonian for the six component wave equation of the vector boson isobtained from Eq. (2) [42, 43] H = σ m + ( σ + iσ ) p m − iσ ( p · S ) m − ( σ − iσ ) κ S · B , (6)with p = ( p ⊥ , p ) and p ⊥ = p + p . The σ i are the 2 × S i are the3 × S is diagonal and S = { S , S , S } .The spectrum of the bosons described by Eq. (6) is ε ( p , p ⊥ , B, s ) = (cid:114) m + p + p ⊥ − κsB (cid:113) p ⊥ + m , (7)where s = 0 , ± B enters in the energy spectrum coupled with thetransverse momentum component p ⊥ (see the last term in the previous equation). This cou-pling reflects the breaking of the SO (3) symmetry of the free system and the axial symmetryimposed by the magnetic field. A difference with magnetized charged quantum particles ishere the absence of Landau quantization in the transversal momentum component, a directconsequence of the electric neutrality of the bosons we are studying [41]. S = √ , S = i √ , S = The ground state energy of the neutral vector bosons ( s = 1 and p = p ⊥ = 0) is ε (0 , B ) = √ m − κBm = m √ − b, (8)with b = BB c and B c = m κ . For the values of m and κ we are considering is B c = 7 . × G .From Eq. (8) follows that the rest energy of the magnetized vector bosons decreases withthe magnetic field and is zero for B = B c . At this point the system becomes unstable [41].However, in the present paper we will neither deal with this phenomenon nor going beyond B c , since the maximum magnetic field expected inside NSs is around (cid:39) × G [19].
III. THERMODYNAMIC POTENTIAL OF THE MAGNETIZED SPIN-ONEGAS
To obtain the thermodynamical potential of the magnetized NVBG we will follow theprocedure showed in [41]. We start from the spectrum Eq. (7) and the definitionΩ(
B, µ, T ) = Ω st ( B, µ, T ) + Ω vac ( B ) , (9)where Ω vac ( B ) = (cid:88) s = − , , ∞ (cid:90) p ⊥ dp ⊥ dp (2 π ) ε ( p , p ⊥ B, s ) , (10)is the zero-point energy or vacuum term and is only B-dependent. After regularization (seeAppendix A), the vacuum contribution readsΩ vac ( b ) = − m π (cid:0) b (66 − b ) − − b − b )(1 − b ) log(1 − b ) (11) − b − b )(1 + b ) log(1 + b ) (cid:1) . Ω st is the statistical contribution of particles/antiparticles. It depends on the magneticfield intensity B , the chemical potential µ and the absolute temperature T = 1 /β , and canbe written as Ω st ( B, µ, T ) = (cid:88) s = − , , ∞ (cid:90) p ⊥ dp ⊥ dp (2 π ) β ln (cid:0) f + BE f − BE (cid:1) , (12)where f ± BE = (cid:2) − e − ( ε ∓ µ ) β (cid:3) stands for particles/antiparticles.To compute Ω st we rewrite it asΩ st ( B, µ, T ) = (cid:88) s = − , , Ω st ( s ) , (13)being Ω st ( s ) the contribution of each spin state. Using the Taylor expansion of the logarithm,Ω st ( s ) is transformed intoΩ st ( s ) = − π β ∞ (cid:88) n =1 e nµβ + e − nµβ n ∞ (cid:90) p ⊥ dp ⊥ ∞ (cid:90) −∞ dp e − nβε ( p ,p ⊥ ,B,s ) , (14)where e nµβ stands for the particles and e − nµβ for the antiparticles.After integration over p , partial integration over p ⊥ and the change of variables x =( m + p ⊥ + α ) − α , Eq. (14) becomesΩ st ( s ) = − y π β ∞ (cid:88) n =1 e nµβ + e − nµβ n K ( nβy ) − α π β ∞ (cid:88) n =1 e nµβ + e − nµβ n ∞ (cid:90) y dx x K ( nβx ) √ x + α , (15)with K l ( x ) the McDonald function of order l , y = m √ − sb and α = smb/
2. NowΩ st ( b, µ, T ) readsΩ st ( b, µ, T ) = − (cid:88) s ∞ (cid:88) n =1 e nµβ + e − nµβ π nβ y nβ K ( nβy ) − α ∞ (cid:90) y dx x √ x + α K ( nβx ) . (16)We obtain the thermodynamic potential of the magnetized neutral vector boson gas atany temperature by adding Eqs. (11) and (16). The thermodynamic magnitudes derivedfrom Eqs. (9–11–16) will be study and compare with those that come from two importantcases: the relativistic low temperature limit (LT) [41] and the non–relativistic limit (NR)[36].The low temperature limit is obtained by assuming T << m in Eq. (16), and is equivalentto neglect the antiparticles contribution as well as those of the spin states with s = 0 , − p , p ⊥ , κB << m . These approximations are equivalentto neglect the vacuum and the antiparticles contributions, and lead to the NR spectrum ε ( p, s ) = m + (cid:126)p / m − sκB . Details of the computation of the NR thermodynamic quantitiesare shown in Appendix C.According to the assumptions of the LT and the NR limits, to consider the magnetizedNVBG at any temperature is equivalent to keep in Eqs. (9–16–11) the contributions of theantiparticles, as well as those of the vacuum and all the spin states.For a Bose gas, the particle desnsity is ρ = ρ gs − ∂ Ω ∂µ , (17)where ρ gs stands for the density in the ground state ε = m √ − b (the condensed ones),while the term − ∂ Ω ∂µ = − ∂ Ω st ∂µ accounts for the density in the excited states. In Eq. (17), ρ gs is such that ρ gs = 0 for T ≥ T c , while ρ gs > T < T c , being T c the critical temperatureof condensation. Deriving with respect to the chemical potential in Eq. (16), we obtain thefollowing expression for ρ at any temperature ρ = ρ gs + (cid:88) s ∞ (cid:88) n =1 e nµβ − e − nµβ π y nβ K ( nβy ) + α ∞ (cid:90) y dx x √ x + α K ( nβx ) . (18)In Eq. (18), the particle density in the excited states can be written as the differencebetween the particles ( ρ + ) and the antiparticles ( ρ − ) in the system: ρ + − ρ − = − ∂ Ω ∂µ . Theparticle/antiparticle density is thus obtained by taking only the terms with e nµβ or e − nµβ respectively.Fig. 1 shows the fraction of non-condensed particles/antiparticles ( ρ + /ρ and ρ − /ρ ) as afunction of the temperature and the magnetic field for ρ = 1 . × cm − . The antiparticledensity begins to be noticeable at T (cid:38) m/ B . However, to appreciatethis last effect ones requires magnetic fields close to B c . Note that the curves for b = 0 and b = 0 . B in pair production is not relevant, but it could be importantfor particles with weaker critical fields. IV. BOSE–EINSTEIN CONDENSATION
Bose-Einstein condensation occurs when µ = m √ − b and ρ gs = 0 [44]. Setting this inEq. (18) we get the following expression for the critical curve (i.e. for the implicit dependenceof ρ , T and B in the transition points): ρ c = (cid:88) s ∞ (cid:88) n =1 e nm √ − bβ − e − nm √ − bβ π y nβ K ( nβy ) + α ∞ (cid:90) y dx x √ x + α K ( nβx ) . (19) ρ + / ρ ,b = ρ - / ρ ,b = ρ + / ρ ,b = ρ - / ρ ,b = ρ + / ρ ,b = ρ - / ρ ,b = / m P a r t i c l e f r ac t i on Figure 1. Particle (solid lines) and antiparticle (dashed lines) fraction as function of temperaturefor several values of the magnetic field. The horizontal lines indicate the BEC critical temperature T c ( b ) (see Eq. (19) in next section). Bose-Einstein condensation of magnetized Bose gases depends on three parameters: tem-perature, density and magnetic field; so the gas can reach the condensate in several ways.For instance, it condenses for fixed ρ and b when the temperature decreases; for fixed T and b , when the density increases; and for fixed ρ and T if the magnetic field augments [41].We have illustrated these behaviors in Figs. 2 and 3, that correspond to the BEC phasediagrams in the ρ vs T and the T vs b planes respectively. RLTNR10 - - - - - / m ρ / m B = RLTNR10 - - - - - / m ρ / m B = G Figure 2. BEC phase diagram in the ρ vs T plane. The white region corresponds to the free gaswhile the colored one corresponds to the condensed state. The lines indicate the critical curves ρ c ( T, b ) for the different descriptions of the NVBG.
Fig. 2 shows the NVBG critical curves (Eq. (19)), denoted as R, along with the LT andthe NR limits. Note that ρ c >> m at T c >> m , which is the condition for a Bose gasto condense at relativistic temperatures [45]. In the low temperature region there is nodifference in the behavior of the R and NR critical curves; they separate around T (cid:39) m signaling the appearance of the antiparticles. On the other hand, from these plots is evidentthat the LT approximation is not valid in the non–magnetized case, something we will discussin–depth later. For B = 10 G, the LT critical curve coincides with the other two until T (cid:39) − m , indicating that this limit is not entirely correct above those temperatures. RLTNR0.0 0.2 0.4 0.6 0.8 1.00.050.100.501 B / Bc T / m Figure 3. BEC phase diagram in the T vs B plane for ρ = 1 . × cm − . The white regioncorresponds to the free gas, and the colored one to the condensate. In Fig. 3 we draw the BEC phase diagram in the T vs B plane. As it is shown, theincreasing of B augments T c in the relativistic cases, and, as B → B c ( b →
1) the criticaltemperature of the relativistic gases diverge, while in the NR limit it approaches the constantvalue T NRc ( ∞ ) = πm ( ρ/ζ (3 / / , where ζ ( x ) is the Riemann zeta function [36]. Thesaturation of T NRc ( b ) is caused by the absence of a critical magnetic field. For magnetic fieldsbefore saturation, we find that increasing B increases T NRc in an noticeable way, driving thesystem to condensation. But when the magnetic field reaches the saturated region, furtherchanges on it barely affects T NRc .The divergence of the critical temperature of the relativistic gases when b → ρ LTc ( T, b ) in the LT limit. The result is ρ LTc ( T, b ) = ζ (3 / √ − b ) (cid:18) m √ − bπβ (cid:19) / . (20)From the above expression it is easily seen that ρ LTc ( T, c ) = 0 for b = 1. Since the extensionof our calculations to b > b = 0 the R and NR critical temperaturescoincide, T c (0) = T NRc (0) = πm (cid:16) ρ ζ (3 / (cid:17) / , while T LTc (0) = πm (cid:16) ρζ (3 / (cid:17) / . This differencearises because in the LT limit, the spin states with s = 0 , − s = 1. Therefore, once this approximation is done, the B = 0 case can not be recovered. This is in agreement with Fig. 2 and highlights that theLT limit is not suitable for weak magnetic fields. Finally, let us note that T LTc (0) = T NRc ( ∞ ),since in the NR limit, b → ∞ drives the system to a state in which all the particles arealigned with the magnetic field, i.e., they all are in the s = 1 state. V. MAGNETIC PROPERTIES
In this section we focus on the dependence of the magnetization of the gas on the tem-perature and the magnetic field. The explicit analytical form of this dependence is derivedfrom the definition M = κ √ − b ρ gs − ∂ Ω st ∂B − ∂ Ω vac ∂B . (21)The first term in Eq. (21), M gs = κ √ − b ρ gs , stands for the magnetization of the condensedparticles. It has to be added because all of the condensed bosons are aligned to the field,but when the condensate is present Ω st only accounts for the particles in the excited states.The other two terms corresponds to the magnetization of the free particles M st = − ∂ Ω st ∂B and the vacuum M vac = − ∂ Ω vac ∂B . They read M st = (cid:88) s κsπ β ∞ (cid:88) n =1 e nµβ + e − nµβ n my (2 − bs ) K ( nβy ) + ∞ (cid:90) y dx x x + α ) / K ( nβx ) , (22)and M vac = − κm π (cid:8) b ( b − − b − b + 7) log(1 − b ) − b − b −
7) log(1 + b ) (cid:9) . (23)1Fig. 4 shows the total magnetization of the gas as a function of temperature for ρ =1 . × cm − and two values of B , 10 and 5 × G. The LT and NR limits weredrawn for comparison, as well as M vac . RLTNR M vac / m M / ( k ρ ) B = G RLTNR M vac / m M / ( k ρ ) B = × G Figure 4. The magnetization as a function of temperature for ρ = 1 . × cm − . For B = 10 G the vacuum magnetization is negligible, and the R, NR and LT curvescoincide for T → kρ . As the temperature increases, the NR magnetizationdecreases and goes to zero for T → ∞ [36]. The magnetization of the relativistic gas showsthe same behavior as the non-relativistic limit up to T ∼ . m , while after this value M ( µ, T, b ) begins to grow and increases in several orders with respect to the NR case. Thishas two causes: the increase in particle/antiparticle density due to pair creation (note inFig. 1 that the antiparticles presence starts to be noticeable around those temperatures),and the difference between the effective magnetic moment of particles in opposite spin states s = ±
1. Let us note that the magnetic energy per particle is ε (0 , , B, s ) = m (cid:112) − κBs/m ,( p = p ⊥ = 0), thus the effective magnetic moment of the particles in each spin state can beobtained as d ( B, s ) = ∂ε (0 , B, s ) ∂B = κs √ − bs . (24) d ( B, ±
1) = ± κ/ √ ∓ b , therefore | d ( B, | > | d ( B, − | and an excess of positive mag-netization appears in the system even if the temperature is high enough to overcome themagnetic field and keep the particles/antiparticles equally distributed in each spin state.This behavior is exclusive of the all temperature relativistic case due to the non–linear de-pendence of the effective magnetic moment on s and B . In the LT limit the spin state s = − s = ± B and are equal in modulus ( d NR ( B, ±
1) = ± κ ).Both panels of Fig. 4 show that the LT magnetization also decreases when T increases, butits behavior is quite different from the other two cases, becoming negative around T ∼ . m (the point where the curve ends). However, this negative magnetization does not imply thegas having a diamagnetic behavior; it is again a consequence of neglecting the states with s = 0 and s = − T (cid:46) − m , something that can be also appreciated in theright panel of Fig. 2.For B = 5 × G , M vac is higher than the maximum of M NR and comparable to M st .As a consequence, the magnetization of the relativistic cases differ from M NR at T = 0.Since the LT limit works better for strong magnetic field, for B = 5 × G the LT and theR magnetization curves coincide in a larger interval of temperature.Fig. 4 highlights the importance of considering the effects of antiparticles and the vacuum,which are usually neglected. In particular, in the case of antiparticles, they begin to berelevant for T ∼ . m , which for bosons formed by two neutrons is equivalent to T ∼ K, a relatively high temperature for astrophysical environments. But if we consider a lighterparticle, such as positronium, T ∼ . m equals T ∼ K, a temperature achievable inthe early stages of neutron star life.It is also interesting to analyze the limit b → M vac ( b = 0) = 0, whilesetting b = 0 in Eq. (22) gives M st = (cid:88) s m κs π β ∞ (cid:88) n =1 z n + z − n n (cid:26) K ( nm/T ) + (cid:90) ∞ m xK ( nx/T ) dx (cid:27) , (25)but if we sum by s = ± T c , ρ gs ( T ) (cid:54) = 0, and the magnetization is different from zero even if B = 0 M ± ( µ, T,
0) = κρ gs ( T ) . (26)Eq. (26) demonstrates that a spin-one BEC that was under the action of an externalmagnetic field, will remain magnetized even if the external magnetic field is somehow “dis-connected” [36, 41]. This phenomenon, known as Bose-Einstein ferromagnetism [16], hasbeen observed for condensates of diluted atomic gases [17]. For the ideal gas of bosons weare treating, this ferromagnetic behavior is not caused by a spin-spin interaction, but it3is a consequence of BEC, since all the bosons in the ground state have s = 1. Neverthe-less, in experimental situations the weak spin-spin interactions could also contribute to theoccurrence of Bose-Einstein ferromagnetism.To check out the connection between the magnetic behavior of the gas and the Bose–Einstein condensation, we will look at the specific heat and the magnetic susceptibility,whose maximum signal the corresponding phase transitions.To compute the specific heat C v = ∂E/∂T , we need the internal energy density E = Ω − T ∂ Ω ∂T − µ ∂ Ω ∂µ . (27)After derivation of the thermodynamical potential with respect to the temperature andsome simplifications, we find the entropy of the gas S = − ∂ Ω /∂T to be S = − µT ( ρ + − ρ − ) − T Ω st + (cid:88) s ∞ (cid:88) n =1 e nµβ + e − nµβ n (cid:26) y π [ K ( nβy ) + K ( nβy )]+ αn π T ∞ (cid:90) y dx x √ x + α K ( nβx ) , (28)and combining Eqs.(18), (28) and (27), the internal energy can be written as E = − Ω st + Ω vac + (cid:88) s ∞ (cid:88) n =1 e nµβ + e − nµβ n (cid:26) y T π [ K ( nβy ) + K ( nβy )]+ αn π ∞ (cid:90) y dx x √ x + α K ( nβx ) , while the specific heat is C v = S + (cid:88) s ∞ (cid:88) n =1 (cid:26) y ( z n + z − n )4 π n [ K ( nβy ) + K ( nβy )] (29)+ y ( z n + z − n )8 π T [ K ( nβy ) + 2 K ( nβy ) + K ( nβy )] − µ ( z n − z − n ) T (cid:20) y ( K ( nβy ) + K ( nβy ))4 π + αn π T (cid:90) ∞ y x K ( nβx ) dx √ x + α (cid:21) + αn ( z n + z − n )2 π T (cid:90) ∞ y x K ( nβx ) dx √ x + α (cid:27) . The magnetic susceptibility χ = − ∂ M /∂B turns out to be χ = χ T >T c + χ vac , Free gas χ T
3) log(1 − b ) (cid:9) . Fig. 5 shows the specific heat and the magnetic susceptibility as a function of temperaturefor ρ = 1 . × cm − and several values of the magnetic field. As in the non-relativisticcase [36], the peaks of both magnitudes occur at the condensation temperature (the solidvertical lines), indicating the appearance of the BEC and the passing of the system fromthe paramagnetic to the ferromagnetic behavior. This reinforces our conclusion that theferromagnetic behavior of the gas below T c is a consequence of condensation. From Eqs. (30-32) follows that at B = 0, χ diverges for all T < T c . The latter was also obtained in[16, 36] for the non-relativistic susceptibility and constitutes another evidence of the relationbetween BEC and Bose-Einstein ferromagnetism. Another interesting feature of Fig. 5 isthe high temperature magnetic susceptibility. When B = 0, χ decreases with T . But atfinite magnetic field, χ → χ vac when T increases. This is consistent with the fact thatthe magnetization augments with the temperature rather than canceling out. As we havealready seen, this is a consequence of the presence of a finite fraction of antiparticles in thesystem and a main difference with respect to the NR limit. The effect of antiparticles is alsopresent in the specific heat, that for high temperature increases instead of tending to theclassical value 3 / b = = = T / m C v / ρ χ ,b = χ ,b = χ vacío ,b = χ ,b = χ vacío ,b = - - / m χ Figure 5. The specific heat (left) and the magnetic susceptibility (right) as a function of B and T for ρ = 1 . × cm − . The vertical lines signal the temperature of condensation. VI. ANISOTROPIC PRESSURES
Now we analyze how antiparticles and magnetic field affects the parallel P (cid:107) = − Ω andperpendicular P ⊥ = − Ω − M B pressures of the gas. According to their definition, P (cid:107) and P ⊥ are the spatial components of the statistical average of the energy momentum tensor ofthe system of bosons under the action of an uniform and constant external magnetic field[46–48]. In this context “parallel” and “perpendicular” is said with respect to the magneticfield direction. This anisotropy in the pressure is important for the the gravitational stabilityof astronomical objects [49], and is also connected with an interesting phenomenon knownas quantum magnetic collapse [50]. P // ,b = P ⊥ ,b = P vac ,b = P // ,b = P ⊥ ,b = P vac ,b = T / m P r ess u r e ( M e V / f m ) Figure 6. The pressures as functions of the temperature and the magnetic field for ρ = 1 . × cm − . Fig. 6 shows the pressures vs the temperature for ρ = 1 . × cm − and various values6of the magnetic field. The vacuum pressure P vac = − Ω vac is also drawn for comparison.In this plot two regions can be clearly identified: in the first one, T > m and temperaturedominates; therefore, the difference between the pressures is negligible. On the contrary, inthe second one
T < m , the magnetic field dominates, and the presence of the magnetizedvacuum in P (cid:107) and the term −M B in P ⊥ are apparent. As T → P (cid:107) tends to the constant value − Ω vac ( b ), while P ⊥ becomes negativeat the point at which − Ω = M B . (Let us recall that for a Bose gas the statistical part ofthe pressure − Ω st goes to zero with temperature.)The differences between the pressures resulting from the relativistic calculation at alltemperature and their non-relativistic and low temperature counterparts are shown in Fig. 7.They are three: first, the presence of antiparticles in the region of high temperatures causea difference of several orders between the R and the NR pressures; second, in the relativisticcases the parallel pressure at low temperatures is dominated by − Ω vac ( b ), while in theNR limit P (cid:107) tends to zero with T ; and third, the value of temperature where P ⊥ = 0 isunderestimated in the NR limit and overestimated in the LT approximation. Note that RLTNR P vac T / m P a r a ll e l p r ess u r e ( M e V / f m ) RLTNR0.05 0.10 0.50 1 510 T / m P e r p e nd i c u l a r p r ess u r e ( M e V / f m ) Figure 7. The R, NR and LT parallel and perpendicular pressures as functions of temperaturefor ρ = 1 . × cm − and B = 10 G. below the temperature at which P ⊥ = 0, the perpendicular pressure becomes negative andthe gas is unstable. Such instability is known as quantum magnetic collapse [50]. It seems tosuggest that, for a fixed temperature, the magnetic field presence imposes an upper boundon the boson’s density needed to sustain it. Nevertheless, in a system composed by manykind of particles, such a limit will depend also on the other species and the interactionsamong them.7 VII. CONCLUDING REMARKS
We computed the exact analytic expressions for the thermodynamic quantities of a rel-ativistic magnetized neutral vector boson gas at any temperature, including magnetizationand the second derivatives of the thermodynamical potential ( C v and χ ). Our calculationswere inspired by astrophysics, however, they have a per se interest and can be useful inother scenarios like particle physics and condensed matter physics.The numerical study of the magnetized NVBG in astrophysical conditions allowed us toevaluate the relative influence of the particle density, the magnetic field and temperaturein the system. Depending on T there are two distinct regimes in the behavior of the gas.For T << m the effects of the magnetic field dominate the system and, in particular, wechecked its relevant role in Bose–Einstein condensation, pressure anisotropy and quantummagnetic collapse.When
T >> m the temperature effects dominate, being the most important, the existenceof a non-negligible fraction of antiparticles. In general, the density of antiparticles is nolonger negligible around T ∼ . m , although their effects are most strongly manifestedfor T (cid:38) m , being both values of temperature quite independent of the magnetic field. Therelevance of antiparticles is specially evident for the magnetization and the pressure of thegas, since they cause an increase in various orders in both magnitudes.When B → T in the relativistic case, the antiparticles plays the main role, contributing to allthe magnitudes and introducing non trivial differences between both situations. Somethingsimilar happens with the vacuum pressure and magnetization: they are usually neglected,however, they effects are important for high magnetic field. Therefore, the NR limit is valid8only for low temperatures and weak magnetic fields.The LT limit, on the contrary, works well for relatively strong magnetic fields B ≥ . B c .Looking at some fixed values we find that, for example, for B = 10 G, this approximationis valid for temperatures such that
T << − m . This implies that for paired neutrons thislimit cannot be used for T ∼ K, which is a possible temperature in early stages of NS.In addition, we found that increasing T in the LT limit leads to a negative magnetization,while our all temperature study shows that the magnetization of the gas is always positive.Hence the importance of using the exact expressions even for low temperatures. VIII. ACKNOWLEDGMENTS
We thank E. Mart´ınez Rom´an and C. Reigosa Soler for their review and useful commentson the manuscript. The authors have been supported by the grant No. 500.03401 of thePNCB-MES, Cuba, and the grant of the Office of External Activities of the Abdus SalamInternational Centre for Theoretical Physics (ICTP) through NT-09.
Appendix A: Vacuum thermodynamic potential
Here we compute the vacuum contribution to the thermodynamic potential (Eq.(11))following [41]. We start from Ω vac definitionΩ vac = (cid:88) s = − , , ∞ (cid:90) p ⊥ dp ⊥ dp (2 π ) ε ( p ⊥ , p , B, s )with ε ( p ⊥ , p , B, s ) = (cid:114) p + p ⊥ + m − κsB (cid:113) p ⊥ + m . We integrate over p and p ⊥ with the help of the equivalence √ a = − √ π ∞ (cid:90) dyy − / ( e − ya −
1) (A1)and the introduction of the small quantity δ as lower limit of the integral (cid:112) a ( δ ) = − √ π ∞ (cid:90) δ dyy − / e − ya . (A2)9The latter is done to regularize the divergence of the a dependent term and to eliminate theterm that does not depends on a .Now, let’s make a ( δ ) = ε = p + p ⊥ + m − κsB (cid:112) p ⊥ + m . As a consequence ε = − √ π ∞ (cid:90) δ dyy − / e − y ( p + p ⊥ + m − κsB √ p ⊥ + m ) , (A3)where we dropped out the explicit dependence of the spectrum on p ⊥ , p , B and s to simplifythe writing. Inserting Eq.(A3) in Eq.(A1), the vacuum thermodynamic potential reads asfollows Ω vac = − π / (cid:88) s = − , , ∞ (cid:90) δ dyy − / ∞ (cid:90) dp ⊥ p ⊥ ∞ (cid:90) −∞ dp e − y ( p + p ⊥ + m − κsB √ p ⊥ + m ) . (A4)After integration over p we obtainΩ vac = − π (cid:88) s = − , , ∞ (cid:90) δ dyy − ∞ (cid:90) dp ⊥ p ⊥ e − y ( p ⊥ + m − κsB √ p ⊥ + m ) . (A5)Eq.(A5) may be simplified by two successive changes of variables. The first one is z = (cid:112) m + p ⊥ − sκB , and Eq.(A5) becomesΩ vac = − π (cid:88) s = − , , ∞ (cid:90) δ dyy − e − y ( m − msκB ) + sκB ∞ (cid:90) δ dyy − ∞ (cid:90) z dze − y ( z − s κ B ) , (A6)where z = m − sκB . The second change of variables is w = z − z in the last term ofEq.(A6). If, in addition, we sum over the spin and recall that b = B/B c with B c = m/ κ ,Ω vac can be written asΩ vac = − π ∞ (cid:90) δ dyy − e − ym (1 + 2 cosh [ m by ]) (A7)+ mb ∞ (cid:90) δ dyy − ∞ (cid:90) dwe − y ( m − w ) sinh[ mb ( m − w ) y ] . Finally, to take the limit δ → m by ] and sinh[ mb ( m − w ) y ]the first terms in their series expansion and obtainΩ vac = − π ∞ (cid:90) dyy − e − ym { m by ] − − m b y } (A8) − mb π ∞ (cid:90) dyy − ∞ (cid:90) dwe − y ( m − w ) { sinh[ mb ( m − w ) y ] − mb ( m − w ) y − [ mb ( m − w ) y ] / (cid:9) , Appendix B: Thermodynamic potential of the NVBG in the low temperature limit
To obtain the low temperature limit of the statistical thermodynamic potential Eq. (16),we transform Eq. (15) by computing the integral on its second term [46] I = ∞ (cid:90) y dz x √ x + α K ( nβx ) . (B1)Let’s introduce the following form for K ( nβx ) K ( nβx ) = 1 nβx ∞ (cid:90) dte − t − n β x t . (B2)If we substitute (B2) in (B1), the integration over x can be carried out I = √ πn β ∞ (cid:90) dt √ te − t + n β α t erf c (cid:32) nβ (cid:112) y + α √ t (cid:33) . (B3)To integrate over t in (B3) we replace the complementary error function erf c ( x ) by itsseries expansion erf c ( x ) (cid:119) e − x √ πx (cid:32) − ∞ (cid:88) w =1 ( − w (2 w − x ) w (cid:33) . (B4)After integration Eq. (B3) becomes I = z nβ (cid:112) y + α K ( nβy ) − y nβ (cid:112) y + α ∞ (cid:88) w =1 ( − w (2 w − y + α ) w (cid:18) y nβ (cid:19) w K − ( w +2) ( nβy ) . (B5)By sustituting Eq. (B5) in Eq. (15), Ω st ( s ) readsΩ st ( s ) = − y π β (cid:32) α (cid:112) z + α (cid:33) ∞ (cid:88) n =1 e nµβ + e − nµβ n K ( nβy ) (B6) − αy π β (cid:112) y + α ∞ (cid:88) n =1 e nµβ + e − nµβ n ∞ (cid:88) w =1 ( − w (2 w − y + α ) w (cid:18) y nβ (cid:19) w K − ( w +2) ( nβy ) . Taking the low temperature limit T (cid:28) m in Eq. (B6) is equivalent to make β → ∞ . Inthis limit all the terms in Ω st ( s ) goes to zero except for the first one, thereforeΩ st ( s ) ∼ = − y π β (cid:32) α (cid:112) y + α (cid:33) ∞ (cid:88) n =1 e nµβ + e − nµβ n K ( nβy ) . (B7)1In addition K ( nβy ) ∼ = √ πe − nβy √ nβy = √ πe − nβy √ nβy , and Ω st ( s ) can be written asΩ st ( s ) ∼ = − y / / π / β / (cid:32) α (cid:112) y + α (cid:33) ∞ (cid:88) n =1 e nβ ( µ − y ) + e − nβ ( µ + y ) n / . For β (cid:29)
1, the antiparticles term e − nβ ( µ + z ) goes to zero for all spin eigenvalues s = 0 , ± µ + y ( y = m √ − sb ) is always a positive quantity. So, in the low temperaturelimit the antiparticles contribution can be neglected.Similarly, the particle term e nβ ( µ − z ) goes to zero for s = 0 , −
1, since in these cases µ − y is negative. However, when s = 1, e nβ ( µ − z ) goes to 1, because µ → m √ − b = y ( s = 1). Asa consequence, the contribution of the particles with spin states s = 0 , − st ∼ = Ω st (1). Finally, since Ω st (1) admits further simplifications, the statistical part ofthe thermodynamical potential in the low temperature limit is equal toΩ LTst ( b, µ, T ) = − (cid:0) m √ − b (cid:1) / / π / β / (2 − b ) Li / ( e βµ (cid:48) ) , (B8)where Li n ( x ) = (cid:80) ∞ l =1 x l /l n is the polylogarithmic function of order n and µ (cid:48) = µ − m √ − b .Using Eq. (B8) instead of Eq. (16) in Eq. (9) the thermodynamic magnitudes are com-puted in the relativistic low temperature limit [41]. They read ρ LT = ρ LTgs + ε / Li / ( e µ (cid:48) β ) √ π πβ / (2 − b ) (B9a) M LTst = κ √ − b ρ LT , (B9b) P LT (cid:107) = − Ω LTst − Ω vac , (B9c) P LT ⊥ = − Ω LTst − Ω vac − M LT B, (B9d) E LT = m √ − bρ LT + Ω vac −
32 Ω
LTst , (B9e)with ρ LTgs = ρ (cid:2) − ( T /T
LTc ) / (cid:3) the density of condensed particles and µ (cid:48) = − ζ (3 / T π (cid:34) − (cid:18) T LTc T (cid:19) / (cid:35) Θ( T − T LTc ) , (B10) T LTc = 1 m √ − b (cid:34) √ π π (2 − b ) ρ LT ζ (3 / (cid:35) / , (B11)2where T LTc is the LT critical temperature of condensation and ζ ( x ) is the Riemann zetafunction. Appendix C: Thermodynamic potential in the non relativistic limit
In this appendix we compute the thermodynamic potential of the magnetized vectorboson gas in the non–relativistic limit as done in [36]. We start from the non-relativisticspectrum ε ( p, s ) = (cid:126)p / m − sκB and consider the density of states of the gas, that is g ( (cid:15) ) = 4 πV (2 π (cid:126) ) (cid:88) s = − , , (cid:90) ∞ dp p δ (cid:18) (cid:15) − (cid:126)p m + sκB (cid:19) , where (cid:15) is the boson energy. Let us note that the rigorous no relativistic limit obtained fromthe spectrum Eq. (7) in the NR limit p , p ⊥ , κB << m is ε ( p, s ) = m + (cid:126)p / m − sκB. (C1)However, to simplify the calculations, we have done the rescaling ε → ε − m . This isequivalent to do the substitution µ → µ − m in the thermodynamic potential, and the onlymagnitude affected is the energy density, but it can be easily corrected by the addition of mρ .After doing the integration over p and the sum over the spin states s , g ( (cid:15) ) becomes g ( (cid:15) ) = 4 πmV (2 π (cid:126) ) (cid:104)(cid:112) m ( (cid:15) − κB ) + √ m(cid:15) + (cid:112) m ( (cid:15) + κB ) (cid:105) . (C2)Note that Eq. (C2) can be separated in three terms, each one corresponding to a specificspin state. Since the thermodynamical potential Ω NR ( µ, T, B ) isΩ NR ( µ, T, B ) = TV (cid:90) ∞ d(cid:15)g ( (cid:15) ) ln ( f BE ( (cid:15), µ )) , ∀ µ < (cid:15), (C3)with f BE ( (cid:15), µ ) = (cid:2) − e β ( µ − (cid:15) ) (cid:3) − , it can also be separated in three terms Ω NR ( µ, T, B ) =Ω − ( µ, T, B ) + Ω ( µ, T, B ) + Ω + ( µ, T, B ), corresponding to the states with s = − s = 0and s = 1 respectively. Integrating over the energy in Eq. (C3) one getsΩ − ( µ, T, B ) = − Tλ Li / ( z − ) , (C4)Ω ( µ, T, B ) = − Tλ Li / ( z ) , (C5)Ω + ( µ, T, B ) = − Tλ Li / ( z + ) , (C6)3where λ = (cid:112) π/mT is the thermal wavelength, z = e µ/T is the fugacity and z σ = ze σ κBT where σ = − , +.Eqs. (C4) and (C6) allow to compute all the thermodynamic magnitudes of the non–relativistic neutral vector boson gas. In particular, one has the following expressions for theparticle density ρ NR = ρ NRgs ( T, B ) + ρ − ( µ, T, B ) + ρ ( µ, T, B ) + ρ + ( µ, T, B ) , (C7)where ρ gs stands for the particles in the condensate and ρ − ( µ, T, B ) = Li / ( z − ) λ ,ρ ( µ, T, B ) = Li / ( z ) λ ,ρ + ( µ, T, B ) = Li / ( z + ) λ , correspond to the density of particles with spin state −
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