Ferromagnetic properties of quark matter -an origin of magnetic field in compact stars-
aa r X i v : . [ h e p - ph ] O c t Ferromagnetic properties of quark matter- an origin of magnetic field in compact stars -
Toshitaka TatsumiDepartment of PhysicsKyoto UniversityKyoto 606-8502Japan
Email: [email protected]
The phase diagram of QCD in the density ( ρ B ) - temperature ( T ) plane has been ex-plored by many authors; QGP in high- T region or color superconductivity in high- ρ B region is a typical phase in that plane. Here we are interested in low temperature andmoderate density region relevant to compact stars, where magnetic order is expected.Origin of the magnetic field in compact stars is one of the long-standing problemssince the first discovery of pulsars in early seventies. Recent discovery of magne-tars with huge magnetic field of O (10 − G)has revived this problem. Since nuclearmatter is developed inside compact stars, we are tempted to consider spontaneousspin polarization of nucleons as a microscopic origin of the magnetic field. Realisticcalculations have been performd for polarized nuclear matter, but they lead us to thenegative results [1]. In ref.[2] possibility of spin polarization of quark matter has beensuggested by a simple consideration in analogy with itinerant electrons, where theFock exchange interaction is responsible to ferromagnetism [3]. A weakly first-orderphase transition has been demonstrated around nuclear density. Using this resultwe can roughly estimate the magnitude of the magnetic field at the surface to be O (10 − G), which may explain the magnetic field of magnetars. The coexistence offerromagnetism with color superconductivity has been also discussed in ref. [4].Here we apply the Landau Fermi-liquid theory (FLT) to elucidate the criticalbehavior of the magnetic phase transition at finite density and temperature [5]. Weevaluate the magnetic susceptibility of quark matter. The divergence and sign changeof the magnetic susceptibility is a signal of the magnetic instability to the ferromag-netic phase, since its inverse measures the curvature of the free energy at the originwith respect to the magnetization. Thus quarks near the Fermi surface are respon-sible to the magnetic transition and the spin dependent quark-quark interaction andthe density of states near the Fermi surface are the key ingredients within FLT.1heoretically we find a non-Fermi-liquid behavior of the magnetic susceptibility.Itis well known that there appears a non-Fermi-liquid behavior in the expression of thespecific heat in QCD as well as QED, which is caused by the transverse gauge fieldbecause it is not statically screened [6].
Within the Landau Fermi-liquid theory (FLT) we assume a one-to-one correspondencebetween the states of the free Fermi gas and those of the interacting system [5]. Quarksare treated as quasi-particles carrying the same quantum numbers of the free quarks,and the quasi-particle distribution function is simply given by the Fermi-Dirac one, n ( k , ζ ) = [1 + exp( β ( ǫ k ,ζ − µ ))] − (1)with the quasi-particle energy ǫ k ,ζ specified by the momentum k and a spin quantumnumber ζ = ± In the following we consider the color-symmetric interaction among quasi-particlesthat can be written as the sum of two parts, the spin independent ( f s k , q ) and dependent( f a k , q ) terms; f k ζ, q ζ ′ = f s k , q + ζ ζ ′ f a k , q . (2)Since quark matter is color singlet as a whole, the Fock exchange interaction givesa leading contribution. We, hereafter, consider the one-gluon-exchange interaction(OGE). For a pair with color index ( a, b ), the Fock exchange interaction gives afactor ( λ α ) ab ( λ α ) ba = 1 / − / (2 N c ) δ ab , which is always positive for any pair. Hencethe situation is very similar to electron gas in QED. Since we are interested in theelectromagnetic properties of quark matter, only the color symmetric interaction isrelevant, which is written as f k ζ, q ζ ′ = 1 N c X a,b f k ζa, q ζ ′ b = mE k mE q M k ζ, q ζ ′ , (3)with the invariant matrix element, M k ζ, q ζ ′ = − g N c tr ( λ α / λ α / M µν ( k, ζ ; q, ζ ′ ) D µν ( k − q ) , (4)where M µν ( k, ζ ; q, ζ ′ ) = tr [ γ µ ρ ( k, ζ ) γ ν ρ ( q, ζ ′ )].Since the OGE interaction is a long-range force and we consider the small energy-momentum transfer between quasi-particles, we must treat the gluon propagator by2aking into account HDL resummation. Thus we take into account the screeningeffect, D µν ( k − q ) = P tµν D t ( p ) + P lµν D l ( p ) − ξ p µ p ν p (5)with p = k − q , where D t ( l ) ( p ) = ( p − Π t ( l ) ) − , and the last term represents the gaugedependence with a parameter ξ . P t ( l ) µν is the projection operator onto the transverse(longitudinal) mode, P tµν = (1 − g µ )(1 − g ν ) − g µν − p µ p ν | p | ! P lµν = − g µν + p µ p ν p − P tµν . (6)The self-energies for the transverse and longitudinal gluons are given asΠ l ( p , p ) = X f = u,d,s m D,f + i πm D,f u F,f p | p | ! Π t ( p , p ) = − i X f = u,d,s πu F,f m D,f p | p | , (7)in the limit p / | p | →
0, with u F,f ≡ k F,f /E F,f and the Debye mass for each flavor, m D,f ≡ g µ f k F,f / π . Thus the longitudinal gluons are statically screened to havethe Debye mass, while the transverse gluons are dynamically screened by the Landaudamping, in the limit p / | p | →
0. Accordingly, the screening effect for the transversegluons is ineffective at T = 0, where soft gluons ( p / | p | →
0) contribute. At finitetemperature, gluons with p ∼ O ( T ) can contribute due to the diffuseness of theFermi surface and the transverse gluons are effectively screened. We consider the linear response of the normal(unpolarized) quark matter by applyinga small magnetic field B . Using the Gordon identity, the coupling term with theuniform magnetic field ( A = B × r /
2) can be written as Z d x L int = e q Z d x ¯ q γ · A q = µ q Z d x ¯ q [ L + Σ ] · B q, (8)with µ q being the Dirac magneton. We discard the contribution of the orbital angularmomentum h L i by assuming the uniform distribution of quarks. Thus the magneti-zation h M i f for each flavor can be written as h M i f = V − h Z d x ¯ q f Σ z q f i , (9)3here we take B // ˆ z . Accordingly the magnetic susceptibility is defined as χ M = X f = u,d,s χ fM = X f = u,d,s ∂ h M i f ∂B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B =0 . (10)Hereafter, we shall concentrate on one flavor and omit the flavor indices because themagnetic susceptibility is given by the sum of the contribution from each flavors. Themagnetic susceptibility is proportional to the number difference between different spinstates ( ζ = ± δn k ζ =+1 − δn k ζ = − = ∂n k ∂ǫ k [ − g D µ q B + δǫ k ζ =+1 − δǫ k ζ = − ] (11)with the gyromagnetic ratio g D ∼
2, where δǫ k ζ = N c X ζ ′ = ± Z d q (2 π ) f k ζ ; q ζ ′ δn q ζ ′ . (12)Magnetic susceptibility is then written in terms of the quasi-particle interaction, χ M = (cid:18) ¯ g D µ q (cid:19) N ( T )1 + N ( T ) ¯ f a (13)where ¯ g D is an angle average of g D , and ¯ f a is the Landau-Migdal parameter averagedover the Fermi surface [11, 12]. T = 0 N ( T ) is the effective density of states at the Fermi surface, and is simply written as N − (0) = π N c k F v F (14)in the limit of zero temperature . Eq. (14) defines the Fermi velocity, which is givenby using the Lorentz transformation [5], v F ≡ ∂n k ∂ǫ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | k | = k F = k F µ − N c k F π f s , (15)where f s is a spin-averaged Landau-Migdal parameter.Finally the magnetic susceptibility at zero temperature can be written in termsof the Landau-Migdal parameters, χ M = χ Pauli " N c k F µπ (cid:18) − f s + ¯ f a (cid:19) − , (16)4here χ Pauli is the usual one for the Pauli paramagnetism, χ Pauli = ¯ g D µ q N c k F µ/ π . The quasiparticle interaction on the Fermi surface can be written as f k ζ, q ζ ′ | | k | = | q | = k F = − C g m E F h − M D L ( k − q ) + M ii D T ( k − q ) i , (17)with the effective coupling strength, C g = N c − N c g .We can see that the both Landau parameters f s , ¯ f a include the infrared singular-ities due to the absence of the static screening for the transverse gluons; D T ( k − q ) ∼− / ( k − q ) = − / k F (1 − cos θ ˆ kq ) in this case, so that the logarithmic diver-gences appear in the Landau parameters through the integral over the relative angle, R d Ω ˆ kq / (1 − cos θ ˆ kq ).Finally magnetic susceptibility is given as a sum of the contributions of the bareinteraction and the static screening effect. We can see that the logarithmic divergencesexactly cancel each other to give a finite result for susceptibility [8, 9].( χ M /χ Pauli ) − = 1 − C g N c µ π E F k F h m (2 E F + m ) −
12 ( E F + 4 E F m − m ) κ ln 2 κ i , (18)with κ = m D / k F . Obviously this expression is reduced to the simple OGE casewithout screening in the limit κ →
0; one can see that the interaction among masslessquarks gives a null contribution for the magnetic transition. The effect of the staticscreening for the longitudinal gluons gives the contribution of g ln(1 /g ). In thenonrelativistic limit, it recovers the corresponding term in the RPA calculation ofelectron gas [3, 8, 9]. -20-15-10-5 0 5 10 15 20 0 0.5 1 1.5 2 kF [1/fm] χ M / χ P au li α s =2.2 m s=300MeV m u= m d=0 Figure 1: Magnetic susceptibility at T = 0. The solid curve shows the result usingsimple OGE, while the dashed and dash-dotted ones show the screening effects with N f = 1(only s quarks) and with N f = 3(u, d, and s quarks) respectively.5n Fig. 1, we plot the magnetic susceptibility at T = 0[8, 9]. We take the QCDcoupling constant as α s ≡ g / π = 2 . m s = 300MeV in-ferred from the MIT bag model. We consider here the MIT bag model as an effectivemodel succeeded in reproducing the low-lying hadron spectra. The coupling constantlooks rather large, but this value is required for the color magnetic interaction to ex-plain the mass splitting of hadrons with different spins; e.g. for nucleon and ∆ isobar.We think this feature is relevant in our study, because the coupling constant is closelyrelated to the strength of the spin-spin interaction between quarks in this model.Moreover, the quark density in the MIT bag model is moderate, 0 . − , which isthe similar one we are interested in. Note that the perturbation method should bestill meaningful even for this rather large coupling, since the renormalization-groupanalysis has shown that the relevant expansion parameter is not the gauge couplingconstant g but the product of g with the Fermi velocity v F , which always goes tozero as one approaches to the Fermi surface [7].One can see that the magnetic susceptibility for the simple OGE without screeningdiverges around k F = 1 . − . This is consistent with the previous result for theenergy calculation. [8, 9]. One may expect that the screening effect weakens the Fockexchange interaction so that the critical density get lower once we take into accountthe screening effect. However, this is not necessarily the case in QCD. The screeningeffect behaves in different ways depending on the number of flavors. Compare theresults for the N f = 3 with the one for N f = 1. In the case of N f = 1 , κ ≤ N f = 3, κ > At finite temperature, the magnetic susceptibility is given by χ M = (cid:18) ¯ g D µ q (cid:19) h N − ( T ) + ¯ f al + ¯ f at i − (19)where ¯ f al and ¯ f at denote the longitudinal and transverse parts of ¯ f a respectively[11, 12]. First, we evaluate the effective density of states on the Fermi surface definedby N ( T ) = N c π Z ∞ ǫ dω dkdω k βe β ( ω − µ ) ( e β ( ω − µ ) + 1) , (20)6ith ǫ ≡ ǫ | k | =0 . The quasi-particle energy ω should be given as a solution of theequation, ω = E k ( ω ) + ReΣ + ( ω, k ( ω )) , (21)where we discard the imaginary part within the quasi-particle approximation.The one-loop self-energy is almost independent of the momentum, and can bewritten as [10]ReΣ + ( ω, k ) ∼ ReΣ + ( µ, k F ) − C f g u F π ( ω − µ ) ln Λ | ω − µ | + ∆ reg ( ω − µ ) (22)around ω ∼ µ with C f = ( N c − / (2 N c ) and u F = k F /E k F . Λ is a cut-off factorand should be an order of the Debye mass, Λ ∼ O ( m D ). Note that the anomalousterm in Eq. (22) appears from the dynamic screening of the transverse gluons, andthe contribution by the longitudinal gluons is summarized in the regular function∆ reg ( ω − µ ) of O ( g ). Within the approximation given by Eqs. (22) and ( ?? ), theself-energy is independent of spatial momentum k and thus we omit the argument k hereafter. The renormalization factor z + ( k ) is then given by the equation, z + ( k ) =(1 − ∂ ReΣ + ( ω ) /∂ω | ω = ǫ k ) − , and we have z + ( k ) − ∼ − C f g u F π ln | ǫ k − µ | . (23)It exhibits a logarithmic divergence as ǫ k → µ , which causes non-Fermi liquid behavior[7]. Eventually, N ( T ) is written as, N ( T ) ≃ N c π Z ∞ ǫ dω − ∂ ReΣ + ( ω ) ∂ω ! k ( ω ) E k ( ω ) βe β ( ω − µ ) ( e β ( ω − µ ) + 1) . (24)We can separate the contribution by the longitudinal gluons N l ( T ) from N ( T ). Sincethe longitudinal gluon exchange is short-ranged by the Debye screening mass, it be-comes almost temperature independent, N l ( T ) = N c k F E F π f sl ;1 , (25)with the Landau-Migdal parameter f sl ;1 , f sl ;1 = − N − c C f g E F k F h κk F + 2 E F i [(1 + κ ) I ( κ ) − , (26)where κ = P f m D,f / k F and I ( κ ) = 12 Z − du − u + κ ≃
12 ln (cid:18) κ (cid:19) ≃ ln( g − ) . (27)7o evaluate the transverse contribution, N t ( T ) = N ( T ) − N l ( T ), we only use thetransverse part in Eq. (22): substituting Eq. (22) into Eq. (24), we obtain the leadingorder contribution , N t ( T ) = N c k s µπ h π k F − m ) k F T + C f g u F
24 (2 k F − m ) k F T ln (cid:18) Λ T (cid:19) + C f g u F π ln (cid:18) Λ T (cid:19) i + O ( g T ) , (28)after some manipulation. N t ( T ) has a term proportional to ln T , which gives a sin-gularity at T = 0. This singularity corresponds to the logarithmic divergence of theLandau-Migdal parameter f s at T = 0. The chemical potential µ in Eq. (28) implic-itly includes the temperature dependence. To extract the proper temperature depen-dence in χ M we must carefully take into account the temperature dependence of µ .Using the thermodynamic relation µ = − ( ∂F/∂n ) | T with the free energy F = E − T s ,we have [12] µ ( T ) = µ − π k F + m k F E F T C f g u F π ln (cid:18) Λ T (cid:19)! + O ( g T ) . (29)We can see that µ includes T ln T term due to the dynamic screening effect for thetransverse gluons, besides the usual T term.As for the spin-dependent Landau-Migdal parameter, the leading-order contribu-tion at finite temperature comes from the transverse component ¯ f at ; it has a loga-rithmic singularity at T = 0 due to the dynamic screening effect. In this section, weshall see that the logarithmic divergences of N − ( T ) and ¯ f at at T = 0 cancel out eachother to give a finite contribution to the magnetic susceptibility. ¯ f at is given by¯ f at = − N c N − ( T ) Z d k (2 π ) ∂n ( ǫ k ) ∂ǫ k ¯ f at ; k,k s (30)with ¯ f at : k,k s = − Z d Ω k π Z d Ω q π m E s E k C f N − c g M iia D t ( k − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | q | = k s (31)where M iia is the spin-dependent component of M ii in Eq.(4), and k s = k F + O ( T )is defined by ǫ k s = µ .The real part of the transverse propagator isRe D t ( k − q ) (cid:12)(cid:12)(cid:12) | q | = k s = ( k − q ) { ( k − q ) } + (cid:16) P f πu F,f m D,f (cid:17) E k − E s ) ( k − q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | q | = k s (32) We discard here the temperature independent term of O ( g ), which cannot be given only by Eq.(22). However, we can recover it by taking the T → k in Eq. (30) can be performed as in Eq. (21). Finally we finda leading-order contribution at T = 0,¯ f at ∼ N − ( T ) C f g π k s E s " π k s − m ) k s T ln T − + O ( g T ) ∼ C f g N c E s µ ln T − . (33)Compare Eq. (33) with Eq. (27). Since E s = E F + O ( T ) and k s = k F + O ( T ) aswe shall see, the ln T terms cancel each other in the magnetic susceptibility (19).( χ M /χ Pauli ) − = 1 − C f g π E F k F h m (2 E F + m ) −
12 ( E F + 4 E F m − m ) κ ln 2 κ i + π k F E F − m + m E F ! T + C f g u F
72 (2 k F + k F m + m ) k F E F T ln (cid:18) Λ T (cid:19) + O ( g T ) . (34)In Fig.2, we plot the magnetic susceptibility given by Eq. (34). At T =0, themagnetic susceptibility is positive at higher densities and the quark matter is in theparamagnetic phase there. At the critical density where the magnetic susceptibilitydiverges( k cF ∼ . − ), there occurs a magnetic phase transition from the para-magnetic phase to the ferromagnetic phase and the quark matter remains in theferromagnetic phase below k cF .At T =30 MeV, there appear two critical densities at which the magnetic sus-ceptibility diverges. We denote these densities k cF and k cF ( k cF < k cF ). In thiscase, k cF ≃ . − and k cF ≃ . − . At densities below k cF and above k cF ,the magnetic susceptibility is positive, which corresponds to the paramagnetic phase,on the other hand, at densities between two critical densities, it becomes negativecorresponding to the ferromagnetic phase.At T =50 MeV, there are still two critical densities ( k cF ≃ . − and k cF ≃ . − ), but the range between these two densities becomes narrower than at T =30MeV.At T =60 MeV, there is no longer divergence in the magnetic susceptibility andquark matter is in the paramagnetic phase at any density.We show a magnetic phase diagram of QCD on the density-temperature planein Fig.3. The four curves corresponds to the critical curves given by Eq.(34) underfour different assumptions: below the curves the quark matter is in the ferromagneticphase, while it is in the paramagnetic phase above the critical curves. The magnetictransition occurs on the critical curves. 9 kF [1/fm] χ M / χ P au li Figure 2: Magnetic susceptibility at finite temperature. The dotted, dashed, dash-dotted, and solid curves show the results at T =0, 30, 50, and 60 MeV respectively.For the solid curve, we have used the full expression Eq.(34), on the other hand,for the dashed, dash-dotted, and dotted curves, we have ignored the dynamic screen-ing( i.e. the T ln T term), static screening( i.e. the κ ln κ term), and both of the twoscreenings in Eq. (34) respectively.Compare the result with the full expression (34) with the one without the non-Fermi-liquid effect i.e. T ln T dependence. In the case without the T ln T term, theferromagnetic phase can be sustained till over T = 60 MeV, while it can be at most T = 60MeV including T ln T dependence. It turns out that the dynamic screeningworks against the magnetic instability and can reduce the ferromagnetic region in thephase diagram up to a point, but this effect is not so large.The dash-dotted curve is the result without the static screening or κ ln κ termin Eq.(34). The static screening effect works in favor of the magnetic instability toenlarge the ferromagnetic region. As discussed in [9], it depends on the number offlavors whether the static screening works for the ferromagnetism or not, which ispeculiar to QCD.The maximum Curie temperature T max c is around 60MeV, which is achieved at k F ≃ . − . Note that this is still low temperature, since T max c /k F ≪
1. Thus ourlow-temperature expansion is legitimate over all points on the critical curve. One ofthe interesting phenomenological implications may be related to thermal evolutionof magnetars; during the supernova expansions temperature rises up to several tensMeV, which is so that ferromagnetic phase transition may occur in the initial coolingstage to produce huge magnetic field. 10 T [ M e V ] kF [1/fm] Figure 3: Magnetic phase diagram in the density-temperature plane. The solid,dashed, dash-dotted, dotted curves show the results for the full expression Eq. (34),the one without the T ln T term, without the κ ln κ term, and without the T ln T and κ ln κ terms in Eq. (34). The open (filled) circle indicates the Curie temperatureat k F = 1 . .
6) fm − while the squares show those when we disregard the T ln T dependence. We have discussed the critical behavior of the magnetic susceptibility in the density-temperature plane within the Fermi liquid theory. We have found a novel non-Fermi-liquid behavior and phase boundary by a perturbative calculations. Somenon-pertubative effects such as instanton effects should be taken into account atmoderate densities. This is important not theoretically but also phenomenologically;more realistic estimate of the critical density or the Curie temperature is needed whenwe face phenomena in compact stars.There are various ideas such as amplification of the fossil field for the origin of themagnetic field in compact stars. So it should be very interesting if we can distinguishthese ideas through observations. To this end we must consider not only magneticevolution but also thermal evolution; if ferromagnetic state is realized, spin wavesshould be excited which affect the thermal evolution of compact stars [8].This work was partially supported by the Grant-in-Aid for the Global COE Pro-gram “The Next Generation of Physics, Spun from Universality and Emergence”from the Ministry of Education, Culture, Sports, Science and Technology (MEXT)of Japan and the Grant-in-Aid for Scientific Research (C) (20540267).11 eferences [1] I. Bombaci et al.,Phys. Lett.
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