The quark-hadron thermodynamics in magnetic field

# The quark-hadron thermodynamics in magnetic field

V.D.Orlovsky and Yu.A.Simonov
Institute of Theoretical and Experimental Physics
117218, Moscow, B.Cheremushkinskaya 25, Russia
###### Abstract

Nonperturbative treatment of quark-hadron transition at nonzero temperature and chemical potential in the framework of Field Correlator Method is generalized to the case of nonzero magnetic field B. A compact form of the quark pressure for arbitrary is derived. As a result the transition temperature is found as a function of B and , which depends on only parameters: vacuum gluonic condensate and the field correlator , which defines the Polyakov loops and it is known both analytically and on the lattice. A moderate (25%) decrease of for changing from zero to 1 GeV is found. A sequence of transition curves in the plane is obtained for in the same interval, monotonically decreasing in scale for growing .

## 1 Introduction

Strong magnetic fields (m.f.) are now a subject of numerous studies [1, 2, 3, 4, 5, 6, 7, 8], since they can be present in different physical systems. Namely, in cosmology m.f. of the order of Gauss or higher can occur during strong and electroweak phase transition [1, 2]. In noncentral heavy ion collisions one can expect m.f. Gauss [3, 4, 5, 6], while in some classes of neutron stars m.f. can reach the magnitude of Gauss, or even more in the cental regions [7]. All this makes it necessary to study the effects of strong m.f. in all possible physical situations and using different methods, for a recent review see [8].

One of most interesting aspects of strong m.f. is its influence on the QCD hadron-quark phase transition, which can occur both in astrophysics (neutron stars) and heavy ion experiments. On the theoretical side many model QCD calculations have predicted the increase of the critical temperature with growing B [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], and only few obtained an opposite result [36, 37], see [38, 39] for reviews and additional references. Recently the lattice data of [40] with physical pion mass and extrapolated to continuum have demonstrated the decreasing critical temperature as a function of . This phenomenon was called the inverse magnetic catalysis and the further study of the m.f. dependence of the quark condensate and of its magnetic susceptibility was done in [41] and [42] respectively. It is our purpose in this paper to exploit the formalism of Field Correlators (FC) developed earlier for the QCD phase transition at zero m.f. [43, 44, 45, 46, 47, 48, 49] to study the same problem in the case of arbitrary m.f.

The advantage of the FC method is that it is based only on the fundamental QCD input: gluonic condensate , string tension , and current quark masses. In contrast to [36], where the same basic principle [44] was used, but pions were elementary in CPTh, it treats all hadrons, including pions, as or systems, which allows to consider high m.f. with . As a result the critical temperature decreases with the growing as in lattice data of [40].

Recently the FC method was successfully applied to the study of phase transition in neutron stars [50, 51] without m.f. and in the case of strange quarks and strange matter in [52]. It is interesting to investigate the role of m.f. in these transitions and our results below may be a reasonable starting point for this analysis.

The paper is organized as follows. In section 2 the general FC formalism as applied to the hadron-quark transition is given, and in section 3 the contribution of magnetic field is explicitly taken into account. In section 4 the quark and hadron thermodynamic potentials are estimated at large m.f. and the corresponding transition temperature is found. In section 5 the case of nonzero chemical potential is treated and in section 6 a discussion of results and prospectives is presented.

## 2 General formalism

We shall follow the ideas of [43, 44, 45, 46, 47, 48, 49] (see [53] for a review) and consider the low-temperature hadron phase as the hadron gas in the confining background vacuum field, and the total free energy can be represented as

 F=εvacV3+Fh, (1)

where

 εvac=β(αs)16αs⟨GaμνGaμν⟩+∑qmq⟨¯qq⟩, (2)

and is the hadron free energy, which in absence of magnetic field and treating hadrons as elementary can be written as [54, 55]

 −Fh/V3=∑iP(i)h=∑igiT2π2∫∞0dpp2ηln(1+ηe−βEi), (3)

where or for bosons or fermions respectively and in the relativistic case , while is the spin-isospin multiplicity of hadron . Taking integral in (3), one can write as

 P(i)h=giT42π2∞∑n=1(−η)n+1(βmi)2n2K2(nβmi), (4)

where is the Mc Donald function. As an example of another starting point we present below the derivation of the quark pressure from the statistical sum in the form of the generating function with the proper time integration [45, 47, 48, 49].

 1TFq=12lndet(m2q−^D2)=−12tr∫∞0ξ(s)dsse−sm2q+s^D2 (5)

The latter expression can be written as a path integral with the background field containing both electromagnetic and color potential , [53, 56, 57, 58]

 1TFq(A,A(e))=−12tr∫∞0ξ(s)dssd4x¯¯¯¯¯¯¯¯¯¯¯(Dz)wxxe−K−sm2q⟨Wσ(Cn)⟩, (6)

where ,

 Wσ(Cn)=PFPAexp(ig∫CnAμdzμ+ie∫CnA(e)μdzμ)exp∫s0(gσμνFμν+eσμνF(e)μν)dτ, (7)

and

 ¯¯¯¯¯¯¯¯¯¯¯(Dz)wxy=n∏m=1d4Δzk(m)(4πε)2∑n=0,±1,±2(−)nd4p(2π)4eipμ(∑Δzμ(m)−(x−y)−nβδμ4). (8)

It was shown in [48, 49], that can be written as

 Pq=2Nc∫∞0dsse−m2qs∞∑n=1(−)n+1[S(n)(s)+S(−n)(s)] (9)

and

 S(n)(s)=∫(¯¯¯¯¯¯¯Dz)wone−K1NctrWσ(Cn) (10)

and in the case, when only one-particle contribution is retained,

 S(n)(s)=116π2s2e−n2β24s−JEn, (11)

and defines the Polyakov loop configuration expressed via the field correlator [48, 49, 59]

 JEn=nβ2∫nβ0dν(1−νnβ)∫∞0ξdξDE1(√ν2+ξ2), (12)

where is the colorelectric correlator, which stays nonzero above the deconfinement temperature.

The insertion of (11) into (9) yields

 1T4Pq=Ncnf4π2∞∑n=1(−)n+1n4∫∞0dss3e−m2qs−n2β24s−JEn=
 (13)

where

At this point it is convenient to give one more representation of , namely, using [58] one can extract in the fluctuating part

 z4(τ)=¯z4(τ)+z4(τ),  ¯z4(τ)=2ωτ=tE,  s=T42ω,  T4=nβ (14)
 Pq=2Ncnf∫∞0dωω√ω2π∞∑n=1(−)n+1√nβ∫D3ze−K(w)−JEn (15)
 K(ω)=∫nβ0dtE⎛⎝ω2+m22ω+ω2(d\boldmathzdtE)2⎞⎠,  mq≡m. (16)

In this way we obtain the form, equivalent to (13)

 Pq=Ncnfπ2β2∞∑n=1(−)n+1n2∫∞0ωdωe−(m22ω+ω2)nβ−JEn. (17)

Neglecting and for one obtains the standard result

 Pq=4NcnfT4π2∞∑n=1(−)n+1n4=7Ncnfπ2180T4, (18)

where we have used

 ∫∞0ωdωe−(m22ω+ω2)nβ=2m2K2(mnβ). (19)

For the following it will be useful to keep in (15) the integration, contained in , which yields

 Pq=nfNc√π∫d3p(2π)3∞∑n=1(−)n+1√2nβ∫∞0dω√ωe−⎛⎝m2+\boldmathp22ω+ω2⎞⎠nβ. (20)

One can see, that (20) coincides with (4), when . Finally, one obtains from (4) or (18) the pressure for gluons

 Pg=(N2c−1)2T4π2∞∑n=1⟨Ωn⟩+⟨Ω∗n⟩2n4, (21)

where ), and is the adjoint Polyakov loop.

## 3 Quark and hadron thermodynamics in magnetic field

We discuss here the one-particle thermodynamics in constant homogeneous m.f. along axis, in which case one should replace in (3) by the well-known expression [55], which in the relativistic case has the form

 Eσn⊥(B)=√p2z+(2n⊥+1−¯σ)|eq|B+m2q,    ¯σ≡eq|eq|σz,σz=±1. (22)

We also take into account, that the phase space of an isolated quark in m.f. is changed as follows [54]

 V3d3p(2π)3→dpz2π|eqB|2πV3, (23)

and hence (3) can be rewritten as

 Pq(B)=∑n⊥,σ2NcT|eqB|2π12(χ(μ)+χ(−μ)), (24)

where

 χ(μ)≡∫dpz2πln(1+exp(¯μ−Eσn⊥(B)T)). (25)

We have introduced in(24) the chemical potential with the averaged Polyakov loop factor , (see [53] for a corresponding treatment without m.f.)

 ¯μ=μ−¯J,  ¯Lμ=exp(μ−¯JT). (26)

Eq.(24) can be integrated over with the result

 Pq(B)=Nc|eqB|Tπ2∑n⊥,σ∞∑n=1(−)n+1n12(¯Lnμ+¯Ln−μ)εσn⊥K1(nεσn⊥T), (27)

where

 εσn⊥=√|eqB|(2n⊥+1−¯σ)+m2q. (28)

The same result for can be obtained, extending (20) to the case of nonzero , using (23) and replacing the exponent in (20) as

 (m2q+\boldmathp22ω+ω2)nβ→(m2q+p2z+(2n⊥+1−¯σ)|eqB|2ω+ω2)nβ, (29)
 Pq=2Nc|eqB|(2π)2∞∑n=1(−)n+1nβ∫∞0dωe−((εσ)22ω+ω2)nβ, (30)

and using the equation

 ∫∞0dωe−(λ22ω+ω2)τ=2λK1(λτ), (31)

we come to the Eq. (27).

We turn now to thermodynamics of hadrons in m.f. The difficulty here is that hadrons are not elementary objects, unlike quarks, and we cannot use for them the energy expressions like (22). Hadrons in m.f. were studied analytically in [60, 61, 62, 63] and on the lattice in [64, 65, 41]. We can use for them an expression of the type of Eq. (24) or Eq. (27), however we should write it in a more general way for the charged hadrons

 P(i)H(B)=gi|eHB|T2π∫dPz2π12(χ(μi)+χ(−μi)), (32)

where

 χ(μi)≡∑n⊥,siln(1+exp(μi−EsiN⊥(B)T)), (33)

and we take into account, that the total hadron energy depends on the set of 2d oscillator numbers for each of constituents and on the set of spin projections of all constituents. For very large m.f. is the string tension) one can approximate as an average of the sum of constituents (2 for mesons and 3 for baryons),

 (34)

where the average is taken with the functions , satisfying , and taking into account confining dynamics along axis (see explicit expressions in the Appendix). In the approximation used in [60, 61, 62, 63], when confinement is quadratic, the functions are the oscillator eigenfunctions. For neutral hadrons one should use instead of (32) the form (4), where m.f. acts on the multiplicity and the mass , which can strongly depend on m.f., as it is in the case of and mesons, see [61, 62].

At this point it is useful to compare the systematics of hadrons without m.f. with that of strong m.f. In the first case one classifies a hadron, using e.g. spin , partly isospin , orbital momentum and radial quantum number , or else total angular momentum . For strong m.f. both spin (or and isospin are not conserved and one has e.g. instead of 2 states linear combinations and similarly splits into 2 states: .

A similar situation occurs in baryons: neutron, splits into

A specific role is here played by the so-called “zero states”: those are states for which all constituents have factors in (34) equal to zero:

 2n⊥(k)+1−ek|ek|¯σk=0,  k=1,2,...ν. (35)

Masses of zero states decrease fast with m.f. and for can be lower than in absence of m.f. [61, 62]. Therefore the role of these states in the forming of exponentially grows, while energies of all other states according to (34) grow proportionally to . Thus in only the states are zero states, while for the neutron with the only zero state is . In this way the most part of all excited hadron states have energies growing with m.f. and their contribution is strongly suppressed for . However, the same situation occurs for the system of free quarks at large m.f., which can be clearly seen comparing (34) with energies of free quarks, therefore the main difference occurs for not large m.f., when and hadron energies change less rapidly than those of free quarks, and hence may grow faster with than , which finally results in the decreasing , as we show below.

Indeed, for each quark the zero states constitute one half of states, namely the states with , and the corresponding pressure is proportional to tending to at large m.f., thus growing linearly with .

For hadrons in the same limit the charged zero states contribute in (32) the amount , while neutral zero states, like , contribute . Therefore for the growth of quark pressure with is faster then that of hadrons, and one can assert, that for the inequality holds

 ΔPq(B,T)≡Pq(B,T)−Pq(0,T)>ΔPh(B,T)=Ph(B,T)−Ph(0,T). (36)

In the next section we shall show, that (36) leads to the decreasing of the deconfinement temperature with growing independently of the character of this transition. In particular, the above arguments were based on the single-line approximation for quarks [48, 49], when quarks are treated as independent and vacuum fields create only Polyakov line contributions ( in (27)). A more accurate treatment, taking into account the interaction due to the correlator (cf Eq. (12)), shows, that the pairs can form bound states in this interaction [59], and with increasing m.f. the binding energy grows, which lowers the mass, thus leading to the growth of the quark pressure. Moreover, the introduction of this “intermediate state of deconfinement”, existing in the narrow region near , consisting of bound and decaying pairs strongly affects the nature of the deconfining transition, making it softer. In addition, the colorelectric string tension, which disappears at , decreases gradually in the same region, lifting in this way the hadron pressure and making the transition continuous. However this remark does not change qualitatively the considerations of the present paper and will be treated in detail elsewhere.

## 4 The quark-antiquark contribution to the pressure at nonzero m.f.

As it was discussed above, the nonperturbative contribution to a single quark is given by Eqs. (11), (12), where can also be written as

 Lfund=exp(−JE1)=exp(−V1(∞)2T), (37)

where is the nonperturbative (np) colorelecric interaction generated by the field correlator [59]

 V1(r,T)=∫β0(1−νT)dν∫r0ξdξDE1(√ν2+ξ2). (38)

As it was argued in [66], the asymptotics of is expressed via the gluelump mass GeV [67] and can be written as

 V1(r,T)=V1(∞,T)−A1M20K1(M0r)M0r+O(TM0), (40)

and

 V1(∞,T)=A1M20[1−TM0(1−e−M0/T)],  A1M20≈6αs(M0)σfM0≈0.5GeV. (41)

At .

Above the value of is decreasing, as seen from (41), (40). This is in agreement with lattice data on Polyakov loops in [68]. Recently, the potential was studied on the lattice in [69], yielding a behavior similar for at .

We now consider the hadron and quark-gluon pressure in the single-line (the independent particle) approximations with the purpose to define the deconfinement temperature as a function of m.f.

One starts with the total pressure in the confined phase, phase I, which can be written in the form, generalizing the results of [43, 44, 45] for the case of nonzero m.f.

 PI=|εvac|+∑iP(i)H(B), (42)

where is given in (2), and we assume, that the gluonic condensate does not depend on m.f. in the first approximation, while the quark condensate grows with m.f., as shown analytically in [70] and on the lattice [41, 42], however we neglect this contribution in the first approximation and discuss its importance at large is the concluding section.

In the deconfined phase (phase II) the pressure can be written in the form (cf [43, 44, 45, 48])

 PII=12|ε(g)vac|+∑qPq(B)+Pg (43)

where is given in (24)-(26), and we assume, that vacuum colormagnetic fields, retained in the deconfined phase at , create one-half of vacuum condensate as it happens for .

 12|ε(g)vac|≅(11−23nf)32ΔG2,  ΔG2≈12G2. (44)

Taking into account the chemical potential , one can rewrite (27) as

 Pq(B)=∑q=u,d,...Nc|eqB|Tπ2∑n⊥,σ=±1∞∑n=1(−)n+1ncoshμnTLnfundεσn⊥K1(nεσn⊥T) (45)

Finally, for gluon pressure we are neglecting the influence of m.f., which appears in higher orders, and write

As a result we define the deconfinement temperature from the equality

 PI(T=Tc)=PII(T=Tc). (47)

The contribution of zero levels of light quarks clearly dominates in (45), when , so that keeping for simplicity only the , terms for small , one has

 ¯Pq(B)≈P(0)q(B)=Ncnf|¯eqB|T2π2LfundcoshμT, (48)

where,   for .

Neglecting as a first approximation the hadron pressure and in (42), one obtains an equation for :

 12|ε(g)vac|=P(0)gl+P(0)q(B), (49)

and finally, neglecting the term , and for large , one obtains the asymptotic expression

 T2c=(11−23nf)G2π264Ncnf|¯eq|BLfundcoshμT. (50)

For , we take GeV GeV [71], and we obtain

 Tc(eB=1GeV2)≅0.125 GeV. (51)

For the same parameters and in [71] one gets GeV. These values are in a reasonable agreement with the corresponding lattice data in [42], GeV GeV, GeV.

One can now take into account at large also the contribution of gluons, and mesons, since the mass of tends to zero for , while that of grows as and does not contribute appreciably to the pressure. One can check that solving equation

 12|ε(g)vac|+Pπ0=P(0)gl+P(0)q(B) (52)

one obtains GeV), which is 2% larger, than in (51).

At large and GeV, Eq.(50) yields

 T2c=0.00205 GeV2eBexp(0.25 GeVTc) (53)

and one obtains GeV and a slow decrease for larger ,

To investigate the behavior of at all values of and we take into account all Landau levels, as it was done in the Appendix, and write resulting expression for in the case (cf. Eq. (A 5)),

 Pq(B)=NceqBTπ2∞∑n=1(−)n+1n¯Ln{mqK1(nmqT)+
 +2TneqB+m2qeqBK2(nT√eqB+m2q)−neqB12TK0(nT√m2q+eqB)} (54)

In the limiting case of small quark mass, one can rewrite (54) as

 Pq(B)≈NceqBTπ2¯L{T+2TK2(√eqBT)−eqB12TK0(√eqBT)}. (55)

Note, that Eq. (55) for small tends to the limiting -independent form (13), (18). We shall use the forms (54), (55) at all values of , and hence recalculate (49) with , and from (54), (55).

As a result from Eq. (49) one obtains the curve shown in Fig. 1 together with the points obtained on the lattice in [40].

## 5 The case of nonzero chemical potential

We first consider the case of very large , when one can retain only the lowest Landau levels of quarks.

For nonzero and one can keep only zero Landau level and rewrite (24) in the form

 Pq(B)=Nc2π2eqB(ϕ(μ)+ϕ(−μ)), (56)

where is

 ϕ(μ)=∫∞0pzdpz1+epz−¯μT,  ¯μ=μ−¯J≡μ−V1(∞)2. (57)

At large one can use the expansion [55]

 ϕ(μ)≈(μ−¯J)22+π26T2; (58)

and one obtains in the lowest approximation (neglecting and gluon contribution at large ), which yields ( the critical value of the chemical potential ,

 ¯μ2c=1.386G2¯eqB;  μc=V1(∞)2+1.18√G2eB. (59)

For one has GeV, where we have assumed, that is independent of .

Near the critical point the critical curve is easily obtained from (56), (58).

 (μc−V1(∞)2)2+π23T2c=1.386G2eB. (60)

For small the lattice data of [72] reveal, that depends on and may become negative for large and .

However, for small the behavior of in [72] is compatible with our assumption, that is weakly dependent on and being around 0.5 GeV, which supports our form of the phase transition curve (60). Also the dependence of the color screening potential was studied in [69], and was found to be rather moderate for and and 1.35. Therefore we can assume, that the behavior (60) is qualitatively correct for large , GeV, and it should go over for into the form found earlier in [71].

Now we turn to the case of arbitrary m.f. As shown in the appendix, one can sum up in (24) over in the following way

 Pq(B)=NceqB2π2{ϕ(μ)+ϕ(−μ)+23(λ(μ)+λ(−μ))eqB−eqB24(τ(μ)+τ(−μ))}, (61)

where is given in (57), while are

 λ(μ)=∫∞0p4dp√p2+~m2q1exp(√p2+~m2q−¯μT)+1, (62)
 τ(μ)=∫∞0dpz√p2+~m2q1exp(√p2z+~m2q−¯μT)+1. (63)

Here . One can see, that decay exponentially for , and hence one returns to Eq.(56) in this limit. In the opposite case, when , one recovers the form (61) with only present, which exactly coincides with one, studied in [71].

One can now calculate the transition curve in the plane for different values of , using (61) for in the equation . The resulting sequence of curves and GeV is shown in Fig 2. One can see from (59), that asymptotically for large <