Magnetic susceptibility at zero and nonzero chemical potential in QCD and QED
aa r X i v : . [ h e p - ph ] J un Magnetic susceptibility at zero and nonzerochemical potential in QCD and QED
V. D. Orlovsky and Yu. A. Simonov,Institute of Theoretical and Experimental PhysicsBolshaya Cheremushkinskaya 25, Moscow 117218, Russia
Abstract
Magnetic susceptibility of the quark matter in QCD is calculatedin a closed form for an arbitrary chemical potential µ . For small µ , µ ≪ T , √ eB ≪ T , a strong dependence on temperature T is found dueto Polyakov line factors. In the opposite case of small T , √ eB > ∼ T , theoscillations as functions of eB occur, characteristic of the de Haas-vanAlphen effect. Results are compared with available lattice data. The important role of magnetic fields (m.f) in nature has recently become atopic of a vivid interest. Strong m.f. are expected in cosmology [1] and inastrophysics (magnetars) [2], very large m.f. can occur in heavy-ion collisions[3], where a temperature transition to the quark-gluon matter is expected.For a modern review of these topics see [4].For a theory the m.f. effects play an additional role of a crucial test, whichproves or disproves the assumed intrinsic dynamics, or shows its boundaries.Of special importance strong m.f. are in QCD, since both hadrons andthe quark matter are possible parts of neutron stars and m.f. can occur inheavy ion collisions.Recently a new approach was suggested to treat QCD and QED in m.f.,based on the relativistic Hamiltonians, deduced from the QCD path integral[5, 6, 7]. 1 striking result of this approach is the strong reduction of the hadronmasses due to m.f. in mesons [8, 9, 10], and in baryons [11]. For the neutronthe mass is twice as small for eB = 0 . . These results for mesons aresupported by lattice data [12, 13, 14].Another feature of m.f. is the lowering of the temperature T c of thetransition from the hadronic to the quark-gluon matter, which was found inthe same path integral approach [15, 16] and supported by the lattice data[17]. It is a purpose of the present paper to develop the theory further and tofind in a simple closed form the magnetic susceptibility (m.s.) of the quarkmatter for an arbitrary chemical potential µ . Recently this type of analysiswas done for the zero µ [18] and the numerical results for m.s. have beencompared to lattice data [19, 20, 21, 22], showing a good agreement.The nonzero µ case is interesting from several points of view. First ofall it covers the regions of µ, T, eB which are present also in the case ofthe electron gas, and where the effects of the Pauli paramagnetism [23] andLandau diamagnetism [24] occur, moreover there also the de Haas-van Alphenphenomenon is possible, see the regular course [25] for a general discussion.As we shall see, in the case of the quark gas a simple modification occurs in allthese effects, and in addition another, and possibly more important region, µ ≪ T exists, where our method allows to obtain simple general results.The paper is organized as follows. In the next section general expressionsfor the thermodynamical potentials in m.f. are derived, in section 3 theexpression for the m.s. ˆ χ is deduced. Section 4 is devoted to the m.s. atnonzero µ , while in the section 5 the classical Pauli, Landau and de Haas-vanAlphen effects are demonstrated for the quark matter.In section 6 the main results are summarized and perspectives are given. We start with the case of the electron gas in m.f., where the thermodynamicalpotential Ω(
V, T, µ ) (or rather Ω( T ) − Ω(0) ≡ − P ) , which we use in whatfollows, can be written as [25] 2 e ( B, µ, T ) = X n ⊥ ,σ eBT π X ( µ ) , X ( µ ) = Z dp z π ln µ − E σn ⊥ ( B ) T !! , (1)where E σn ⊥ ( B ) = q p z + (2 n ⊥ + 1 − σ ) eB + m e , σ = ± . (2)Note, that (2) is the relativistic generalization of the standard expres-sion [25] in the theory of the electron (or electron-positron) gas in m.f. atnonzero temperature. It was a subject of an intensive study during the last50 years, see e.g. [26, 27, 28, 29, 30]. The QED relativistic thermodynamicalpotential in m.f. at finite T and density was obtained in [26], and using thegeneralized Fock-Schwinger method for µ = 0 , T = 0 in [27]. In the case ofthe zero temperature and nonzero µ, B the useful form of the effective actionwas obtained in [28], and finally the full expression for nonzero, µ, T, B waspresented in [29]. Simple forms of effective Lagrangians for T = 0 and oscil-lations as function of m.f. are obtained in [30]. For further developments anddiscussions and limiting cases see also [31]. These results have been exploitedand augmented by the study of the quark-antiquark gas also in magnetic field[32, 33, 34].In the latter case one can write for a given sort of quarks and antiquarkssimilarly to (1), if one neglects the effect of the vacuum QCD fields on quarks P q ( B, µ, T ) = X N c e q BT π ( X q ( µ ) + X q ( − µ )) , (3)and X q ( µ ) has the same form as in (1), (2) with e = e q ≡ | e q | , and m e → m q .However, the vacuum QCD fields, which are responsible for confinementat T < T c [35], also affect the quark gas. The theory of both confinedand deconfined matter was suggested in [36] and finally formulated, basingon the path integral formalism and the Field Correlator Method (FCM) in[37, 38, 39], for a review see [40].In this formalism, neglecting the q ¯ q weakly bound states around T c (the“Single Line Approximation” SLA [37]) one arrives at the simple modificationof the expression (3), where one should replace in X ( µ ) the chemical potential µ as follows exp µT → exp µ q T L ( T ) , (4)3here L ( T ) = exp (cid:16) − V ( ∞ ,T )2 T (cid:17) is the average value of the fundamental Polyakovline, which was studied analytically in [39, 40] and numerically on the latticein [41].As a result of integration over dp z in X ( µ ) one arrives at the expression[15], containing a sum over Matsubara numbers P q ( B, µ, T ) = N c e q BTπ X n ⊥ ,σ ∞ X n =1 ( − ) n +1 n L n e nµT + e − µnT T ε σn ⊥ K nε σn ⊥ T ! (5)with K ( z ) – the modified Bessel function and ε σn ⊥ = q e q B (2 n ⊥ + 1 − σ ) + m q . (6)Another form of (5) was obtained in [15] by direct summing P n ⊥ ,σ X q ( µ ) in(3), which gives the integral expressions P q ( B, µ, T ) = N c e q B π ( ψ ( µ )+ ψ ( − µ )) , ψ ( µ ) = φ ( µ )+ 23 λ ( µ ) e q B − e q Bτ ( µ )24 , (7)where φ ( µ ), λ ( µ ) and τ ( µ ) are integrals over momenta given in (35),(36),(37).In what follows we shall be mostly using the form (5), which was summedup over n ⊥ , σ for µ = 0 in [15] P q ( B, µ, T ) = N c e q BTπ ∞ X n =1 ( − ) n +1 n L n { m q K (cid:18) nm q T (cid:19) ++ 2 Tn e q B + m q e q B K (cid:18) nT q e q B + m q (cid:19) − ne q B T K (cid:18) nT q m q + e q B (cid:19) } . (8)It is easy to see, that the case of µ > L n → L n ch (cid:18) µnT (cid:19) = L nµ + L n − µ , L µ ≡ e µT L. (9)The form (8) or its nonzero µ equivalent (9) have a nice property ofyielding correct limiting values for 1) e q B →
0, 2) e q B → ∞ , 3) T ≫ m q , e q B .In the first case only the second term inside curly brackets in (8) con-tributes and one has P q (0 , µ, T ) = N c T m q π ∞ X n =1 ( − ) n n +1 n L nµ + L n − µ K (cid:18) nm q T (cid:19) . (10)4n the second case only the first term inside curly brackets survives andwe obtain P q ( B, µ, T ) | B →∞ = N c e q BT m q π ∞ X n =1 ( − ) n +1 n L nµ + L n − µ K (cid:18) nm q T (cid:19) . (11)At large T , T ≫ q m q + e q B , the leading term in (8) is again the secondin the curly brackets and one has P q ( B, µ, T → ∞ ) = 4 N c T π ∞ X n =1 ( − ) n +1 n L nµ + L n − µ , (12)which yields for µ = 0 , L = 1 the standard result P q ( B, µ = 0 , T → ∞ ) = 7 π N c T , ¯ P q = X q P q = 7 π N c T n f . (13)At this point one should stress the importance of the explicit summationover n , especially when µ = 0. This indeed can be done as in [15] with theresult given in (7).Finally we should comment on the accuracy of our representation (8),(9), which is obtained, when the summation over n ⊥ with σ = − ∞ X n ⊥ =0 F ( n ⊥ + 12 ) ∼ = Z ∞ F ( x ) dx + 124 F ′ (0) , (14)which yields the first term in the curly brackets in (8), (9). This substantiatesthe good accuracy of the total expression in the whole region of parametersexcept for a narrow region T ≪ m q , T < ∼ e q B m q ≪ µ − m q ≡ µ where anoscillating regime sets in, considered in the next sections. As an additionalcheck of this accuracy we show in the next section that in the expansion of P q in powers of ( e q B ) k the terms with k = 1 and 3 vanish identically.One finds that (14) is accurate within the terms O (cid:18) √ m q + e q BT (cid:19) , when q m q + e q B ≪ T , while in the opposite case the sum (14) is much smallerthan the term with σ = 1. 5 Magnetic susceptibility of the quark mat-ter
In this section we are specifically interested in the e q B dependence of P q ( B, µ, T )and first of all in the quadratic term of this expansion – the magnetic sus-ceptibility (m.s.). To this end we are exploiting the integral representationof K n K ν ( z ) = 12 (cid:18) z (cid:19) ν Z ∞ e − t − z t t ν − dt, K ν = K − ν , (15)which allows one to write expansions2 Tn ( e q B + m q ) K n q e q B + m q T = 2 T m q n ∞ X k =0 e q Bn T m q ! k ( − ) k k ! K k − (cid:18) nm q T (cid:19) , (16) K n q m q + e q BT = ∞ X k =0 ne q B T m q ! k ( − ) k k ! K k (cid:18) nm q T (cid:19) (17)and as a result Eqs. (8), (9) assume the form P q ( B, µ, T ) − P q (0 , µ, T ) = ( e q B ) N c π ∞ X n =1 ( − ) n +1 ( L nµ + L n − µ )2 f n , (18) f n = ∞ X k =0 ( − ) k k ! ne q B T m q ! k K k (cid:18) nm q T (cid:19) " k + 1)( k + 2) − . (19)Note, that the linear term in (16) exactly cancels the term m q K (cid:16) nm q T (cid:17) in (8), so that the sum in (18) starts with the quadratic term, also the cubicterm vanishes in (19). Hence one can define the m.s. ˆ χ q P q ( B, µ, T ) − P q (0 , µ, T ) = ˆ χ q e q B ) + O (( e q B ) ) . (20)As the result, one arrives at the following expression for ˆ χ q ˆ χ q ( T, µ ) = N c π ∞ X n =1 ( − ) n +1 L nµ + L n − µ K (cid:18) nm q T (cid:19) , (21)with L µ ≡ L exp µT . 6s the next step we are using for K ( z ) the relation K (cid:18) nm q T (cid:19) = 12 Z ∞ dxx e − n (cid:18) x + m qx T (cid:19) (22)and obtain the final expression, summing over n ,ˆ χ q ( T, µ ) = N c π J q ( µ ) + J q ( − µ )2 , J q ( µ ) = 12 Z ∞ dxx L µ e − (cid:18) x + m qx T (cid:19) L µ e − (cid:16) x + m qx T (cid:17) . (23)The total m.s. ˆ χ ( T, µ ) for a quark ensemble with n f species is defined asˆ χ ( T, µ ) = X q ˆ χ q ( T, µ ) (cid:18) e q e (cid:19) . (24)Note, that one can define a more general form, appropriate for the comparisonwith numerical simulations, when one simply extracts the quadratic term( e q B ) , leaving m.f. nonzero in the rest terms, namelyˆ χ q ( B, T, µ ) = 2 P q ( B, µ, T )( e q B ) = N c π ∞ X n =1 ( − ) n +1 L nµ + L n − µ ϕ n ( m q + e q B ) (25)with ϕ n ( m q + e q B ) = ln Tn q e q B + m q + 0 . . (26)One can see in (25), (26), that m q enters in ˆ χ q ( B, T, µ ) always in combinationwith e q B , so that one can define an effective mass( m q ) eff = m q + e q B, (27)and e q B is of the order of the minimal m.f. present in the lattice measurementof ˆ χ q , which is usually larger, than m u , m d .Note, however, that the series over n in Eq. (25) is not well convergent for L µ > n yielding(23).One can see in (21), (25), that at large T ≫ m q , q m q + e q B , each termin (21), (25) behaves as ∼ ln Tm q , implying that χ q > χ q , when m q > m q .7owever, summation over n yields in (23) the denominator which flattens thelogarithmic grows of the first term in the sum. This situation is especiallyinteresting for the free case, when L µ ≡ µ = 0), in which case ˆ χ (0) q ≈ N c π P n ( − n +1 K (cid:16) nm q T (cid:17) and summation over n leads to the Eq. (23) with L µ = 1. The numerical result for the first term ˆ χ (0) q ( n = 1) = N c π K (cid:16) m q T (cid:17) and for the whole sum is shown in Fig. 1. The corresponding expression ofˆ χ (0) q for the electron gas, which is twice as small, can be found in [21], and in[29]. m q /T q (n=1) q summed Figure 1: Magnetic susceptibility in SI units ( χ q = π ˆ χ q ) in free case, L µ = 1,Eq. (23) (solid line), in comparison with the first term in the sum (21).As the next step one must define the Polyakov line, which in the neigh-borhood of T c was found analytically in [39, 40] as L ≡ L ( V ) ( T ) = exp − V ( ∞ , T )2 T ! , V ( ∞ , T ) ≈ V ( ∞ , T c ) = 0 . . (28)Note, that by derivation in [37] the Polyakov line L ( V ) ( T ) takes into accountonly the single quark interaction with the vacuum, given by V ( ∞ , T ), hencethe superscript V , while on the lattice [41] one measures the full Polyakov8ine, which can be expressed via the free energy F ( ∞ , T ) ,L ( F ) ( T ) = exp − F ( ∞ , T )2 T ! . (29)As argued in [40], F < V and hence L ( F ) ( T ) > L ( V ) ( T ). In [18] bothforms of L ( T ) have been used for comparison with lattice data for m.s. with-out chemical potential.It is interesting to compare ˆ χ q ( T, µ ) for three different sorts of quarks, u, d, s . Using (23) with m q → m q ( ef f ), one can find three curves for m u ( ef f ) = 68 MeV, m d ( ef f ) = 49 MeV, m s ( ef f ) = 111 MeV and µ = 0,which are in good agreement with the lattice data from [21], see Fig. 2 (leftgraph), where we use L = L ( V ) from (28). The sum of different quarkscontributions is shown on Fig. 2, right graph. T, GeV q (T), Eq. (23), =0 u [21] d [21] s [21] T, GeV (T), Eq. (24), =0 lattice [21] lattice [20]
Figure 2: Magnetic susceptibility in SI units ( χ q = π ˆ χ q ) as a function oftemperature for different sorts of quarks (left graph) and the total magneticsusceptibility (right graph) for the case µ = 0 in comparison with lattice data[20, 21]. For µ > P q ( B ) = N c T e q B π ( ψ ( µ ) + ψ ( − µ )) , (30)9here ψ ( µ ) = X n ⊥ σ Z ∞−∞ dp z π ln ¯ µ − E σn ⊥ ( B ) T !! , (31)and ¯ µ = µ − ¯ J = µ − V ( ∞ , T )2 , ¯ L µ = exp ¯ µ − ¯ JT ! = ¯ L exp (cid:18) ¯ µT (cid:19) , (32) E σn ⊥ ( B ) = q p z + (2 n ⊥ + 1 − σ ) e q B + m q . (33)Separately out in (31) the term σ = 1 , n ⊥ = 0 one can rewrite ψ ( µ ) as(see Appendix of [15] for details) ψ ( µ ) = 1 πT ( φ ( µ ) + 23 λ ( µ ) e q B − e q B τ ( µ ) ) , (34)where φ ( µ ) does not depend on e q B , φ ( µ ) = Z ∞ p z dp z e pz − ¯ µT , (35) λ ( µ ) = Z ∞ p dp q p + ˜ m q
11 + exp (cid:18) √ p + ˜ m q − ¯ µT (cid:19) , (36) τ ( µ ) = Z ∞ dp q p + ˜ m q (cid:18) (cid:18) √ p + ˜ m q − ¯ µT (cid:19)(cid:19) , (37)and ˜ m q = m q + e q B .It is clear, that with 4 dimensionful parameters ¯ µ, m q , T, e q B one has morethan 6 limiting regions. Therefore in this section we shall confine ourselvesto only three situations, out of which two were treated in [25] for the electrongas and called there a) the case of weak fields, T ≪ ε F = µ , T ≪ m q , eB m q ≪ T, (38)and b) the case of strong fields, T < ∼ eB m q ≪ µ , T ≪ m q , µ . (39)10n addition, there is another interesting region, namely T ≫ µ, e q B < T ,leading some access to the numerical simulations, which will be considerednow, while the cases a) and b) are discussed in the next section.We consider here the case of small µ, µ ≪ T , and small m.f., √ eB ≪ T ,when the possible region of oscillations due to the sum over integrals n ⊥ in(2) is unimportant, and one can replace the sum by the integral, as it is donein (8), (9) using (14). In this case one can use (23) with L µ given by (9)and L ( T ) due to (28), (29). We note here, that the influence of µ on L ( T ) isexpected here to be negligible, see e.g. lattice data in [42].In Fig. 3 we show a typical behavior of ˆ χ q ( T, µ ), given by (23) for L ( T ) = L ( V ) ( T ) from (28). For m u = 68 MeV (as for Fig. 2) and µ ≡ µ u =(0 , , χ u as a function of T in theinterval (150-400) MeV. T, GeV u , u =0 u , u =0.1 GeV u , u =0.2 GeV Figure 3: Magnetic susceptibility χ u from (23) in SI units ( χ q = π ˆ χ q ) as afunction of temperature for nonzero values of chemical potential µ .Another possible characteristics of the small µ influence is the quarknumber susceptibility of ˆ χ q ( T, µ q ), given byˆ χ ( µ ) q ( T ) ≡ ∂ ˆ χ q ( T, µ ) ∂µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ =0 = N c T π ∂ J q ( µ ) ∂µ + ∂ J q ( − µ ) ∂µ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ =0 . (40)Differentiating (23) one obtains 11 ( µ ) q ( T ) = N c π Z ∞ dxx L µ e − (cid:18) x + m qx T (cid:19) − L µ e − (cid:18) x + m qx T (cid:19) L µ e − (cid:16) x + m qx T (cid:17) + ( µ → − µ ) . (41)This combined quark-number and magnetic susceptibility is a generaliza-tion of the powerful technic of the study of the chemical potential influence onthermodynamic potentials and phase transition on the lattice, (see a recentpaper [43] for a discussion and references).One should note, that the corresponding quark number susceptibility(q.n.s.) was calculated for zero m.f. in the framework of our approach in [38].To this end one can use (30), (36), since only λ ( µ ) survives for e q B ≪ m q ,and one has P q ( B = 0) = N c π λ (0) + ∞ X k =2 k ! ∂ k λ ( µ − q ) ∂ (cid:16) µ q T (cid:17) k (cid:18) µ q T (cid:19) k (42)and λ ( µ q ) is given by (36).The equivalent series for ˆ χ q ( T, µ ) is obtained by the replacement in (42), λ ( µ q ) → ( J q ( µ ) + J q ( − µ )). We consider now the cases of the weak and strong fields, Eqs. (38) and (39)respectively, essentially the material of §§ µ for quarks contains the quark mass, µ = m q + µ , while µ depends on density, µ = ε F for electron gas.We start with the case a) eB m q ≪ T ≪ m q , µ .Here one can use (30), (34) and take into account, that the quadratic in e q B terms come only from λ ( µ ) and τ ( µ ). We neglect the terms O (cid:18) e q Bm q (cid:19) (and hence ψ ( − µ ) in (30)) and omitting V ( ∞ , T ) write the exponent in the12ntegrand of (36) asexp q p + ˜ m q − ¯ µT = exp p m q T − ¯ µ T ! , ¯ µ = µ + m q , ¯ µ = µ − e q B m q ≡ µ − m q − e q B m q , (43)and the integral (36) can be rewritten, using the variable z = p m q T , λ ( µ ) = T (cid:18) m q T (cid:19) / Z ∞ z / dze z − ¯ µ /T + 1 . (44)The integral on the r.h.s. of (44) is exactly of the type considered in [25], §
58, where the asymptotic series was obtained in powers (cid:16) ¯ µ T (cid:17) k . Keeping theleading term, one has λ ( µ ) = T (cid:18) m q T (cid:19) / (cid:18) ¯ µ T (cid:19) / + O (cid:18) ¯ µ T (cid:19) / ! . (45)Expanding ¯ µ = ˜ µ − e q B m q , and keeping the term (cid:16) e q B m q (cid:17) , one obtains from(30), (34) the paramagnetic contribution to P q P (2) q = N c ( e q B ) π µ m q ! / , (46)and for τ ( µ ) one has τ ( µ ) = s m q T (cid:18) ¯ µ T (cid:19) / . (47)Inserting these values into (30), (34) one obtains for the τ ( µ ) the contri-bution to ˆ χ , which is (cid:16) − (cid:17) of the contribution of λ ( µ ) χ = N c π ( e q ) s µ m q , (48)which coincides with the total m.s. of the electron gas in the weak m.f., givenin [25], when m q = m e , N c = 1 , µ = ε F .The same result can obtained directly from (23) inserting there L µ =exp (cid:16) µ + m q T (cid:17) , and using instead of x the variable εT = x + m q T x − m q T . In the13imit T ≪ m q one obtains J q ≈ q m q R ∞ dε √ ε e ε − µ T , which using the sametechnic as in (45) yields q µ m q and one gets the result (48).We now turn to the case b), T < ∼ e q B m q ≪ µ , which is interesting for us,since it provides the oscillating behavior, which is not present in our form(31), see discussion in §
59 of [25].Indeed, the form (34) obtains, when one considers e q B outside of theinterval b), or else, when one averages the result over some interval of e q B ,comprising many values of n ⊥ in (31).To this end we rewrite ψ ( µ ) in (31), separating the first term σ = − F (0) + ∞ X n =1 F ( n ) = Z ∞ F ( x ) dx + 2 Re ∞ X k =1 Z ∞ dxF ( x ) e πikx , (49)where F ( x ) = Z ∞−∞ dp z π ln ¯ µ − q p z + m q + 2 e q BxT . (50)and ¯ µ = m q + µ − V ( ∞ ,T )2 for quarks and ¯ µ = m e + µ for the electron gas.For small T , e q Bm q as compared to m q (nonrelativistic situation), we writethe exponent as (cid:16) µ − p z m q − e q Bxm q (cid:17) T and we are in the exact correspondencewith the equations in §
60 of [25], when one replaces our µ , m q , e q by µ, m, e of the electron gas. The resulting expression for P q ( B ) (30) is P q ( B ) = N c e q B π ( φ ( µ ) + φ ( − µ )) − N c T ( e q B ) / π ∞ X k =1 cos (cid:16) πµ m q ke q B − π (cid:17) k / sh (cid:16) π kT m q e q B (cid:17) . (51)One can expect for the quark gas the same oscillations as in the de Haas-van Alphen effect, but the period of oscillations for quarks in e q B is µ m q and we assume T ≪ m q , hence this is improbable for a deconfined quark gas,where T c > m q , q = u, d, s. Therefore we shall try to proceed, assuming onlythat
T < ∼ e q B m q , but allowing for T ≫ m q . Then the exponent in (50) can berewritten as ¯ µ − m q − p z − e q BxT (¯ µ + q m q + p z + 2 e q Bx ) ≡ δµ T − ( p z + 2 e q Bx )2 M q T , (52)14here δµ = ¯ µ − m q M q and 2 M q ≡ ¯ µ + q m q + p z + 2 e q Bx .Approximating M q by some average value, not depending on p z , one canexploit the final Eq. (51), replacing there m q by M q and M by δµ . As aresult one expects the oscillations of P q ( B ) for growing e q B for e q BM q T > ∼ T ≪ δµ .Indeed, the oscillating term in the integral (31), using (49), (50) can bewritten as (cf. §
60 of [25]) I k = − e q B Z ∞−∞ Z ∞ ln " δµT − p z + 2 e q Bx M q T ! e πikx dp z dx. (53)Introducing new variable ε = p z +2 e q Bx M q instead of x , one obtains for theoscillating part of (53)¯ I k = − Z ∞−∞ Z ∞ ln " δµ − εT ! exp iπkεM q e q B ! exp − iπkp z e q B ! dεdp z . (54)In (54) the essential part of integration region is p z ∼ e q B , while for theoscillating regime δµ ∼ ε, δµ ≫ e q B M q , therefore one replaces the lower limitof the ε integration by zero. Moreover, 2 M q ≈ ¯ µ + q m q + 2 e q Bx > ∼ ¯ µ ≫ m q ,and δµ ≈ ¯ µ .Hence the final form of the oscillating part of the thermodynamic poten-tial can be written instead of (51) as∆ P q ( B ) = − N c ( e q B ) / T π ∞ X k =1 cos (cid:16) µ e q B k − π (cid:17) k / sh (cid:16) π kT ¯ µe q B (cid:17) . (55)One can see, that the relativistic quark gas potential (55) contains amuch larger denominator due to T ≫ m q , as compared to the electron gaspotential (51), leading to a relatively smaller amplitude of oscillations. We have developed above the theory of m.s. of the quark-antiquark matterin m.f., based on the explicit expressions for the thermodynamic potentialsobtained by us in [15]. The case of m.s. for zero chemical potential µ was15tudied in our previous paper [18], where it was shown, that m.s. ˆ χ ( T ) isa strong function of T , growing with T due to Polyakov line factors. Thisbehavior agrees well with recent lattice calculations [19, 20, 22], when onetakes into account a possible modification of the effective quark mass as in(27). In the present paper we further examined the zero µ m.s., calculatingm.s. for different quarks ( u, d, s ) and comparing with lattice data of [21]in our Fig. 2. As an additional topic we consider the free quark-antiquarkgas m.s., which obtains from (23) putting L µ ≡
1, and compare it withthe corresponding m.s. of the electron gas from [30]. We observe a strongmodification of the result due to the sum over Matsubara numbers. Themain part of our results belong to the case of nonzero chemical potential µ in sections 4 and 5. Here m.s. ˆ χ ( T, µ ) has different behavior in the regionsof small and large µ , µ ≪ T and µ ≫ T . In the first case, considered insection 4, one can define the double magnetic- quark number susceptibilities.In the case of large µ, µ ≫ T and eB m q ≪ T , one obtains the standard Pauliparamagnetism [23] and Landau diamagnetism [24] contributions to the m.s.given in (47).Finally, in the case of large µ and large m.f. one arrives at the Landautheory of the de Haas-Van Alphen effect, written for nonrelativistic quarksin (50). The generalization to the case of relativistic quarks for T ≫ m q is obtained in (54) and shows much milder amplitude of oscillations withgrowing e q B . As it is we have developed the full theory of m.s. of the quarkgas interacting with the QCD vacuum in the so-called Single Line Approxi-mation (SLA) [37], when the interaction enters in the form of Polyakov lines.This allows to obtain m.s. at zero or small µ , and a good agreement wasfound with lattice data at least in the first case. In this approximation theinterquark interactions are disregarded, however at larger µ (and hence largerquark densities) this effect can become important and this was discussed in[37, 40]. In SLA the QCD phase diagram in the µ − T plane was found in [15]and does not contain critical points. However for larger µ the interquark in-teraction becomes important and depends on µ both in the confined [44] anddeconfined [39, 45] states. As a result the problem of the quark-hadron (qh)matter transitions should be solved with the full account of the interquark(beyond SLA) interactions. One aspect of this transition – the formationof the multiquark states and the nucleon matter was considered in [46], andshown to be important for the quark cores of neutron stars.The authors are grateful to M. D’Elia and G. Endrodi for stimulating16orrespondence.The RFBR grant 1402-00395 is gratefully acknowledged. References [1] T. Vachaspati, Phys. Lett. B , 258 (1991); D. Grasso and H. R. Ru-binstein, Phys. Rept. , 163 (2001).[2] R. C. Duncan and C. Thompson, Astrophys. J.
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