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Quark deconfinement and gluon condensate in a weak magnetic field

We study QCD finite energy sum rules (FESR) for the axial-vector current correlator in the presence of a magnetic field, in the weak field limit and at zero temperature. We find that the perturbative QCD as well as the hadronic contribution to the sum rules get explicit magnetic field-dependent corrections and that these in turn induce a magnetic field dependence on the deconfinement phenomenological parameter s_0 and on the gluon condensate. The leading corrections turn out to be quadratic in the field strength. We find from the dimension d=2 first FESR that the magnetic field dependence of s_0 is proportional to the absolute value of the light-quark condensate. Hence, it increases with increasing field strength. This implies that the parameters describing chiral symmetry restoration and deconfinement behave similarly as functions of the magnetic filed. Thus, at zero temperature the magnetic field is a catalysing agent of both chiral symmetry breaking and confinement. From the dimension d=4 second FESR we obtain the behavior of the gluon condensate in the presence of the external magnetic field. This condensate also increases with increasing field strength.


I. INTRODUCTION
Lately, the properties of strongly interacting matter in the presence of external magnetic fields has become a very active research field. One of the driving motivations behind this interest is the possibility to study experimentally such properties in peripheral collisions of heavy nuclei at high energy. In addition, recent lattice QCD (LQCD) results show that the critical temperature for deconfinement/chiral symmetry restoration decreases with increasing field strength [1]. This behavior is dubbed inverse magnetic catalysis, and it reveals an unexpected, non-trivial phenomenon: in a thermal environment, near the transition temperature, the presence of a magnetic field acting on strongly interacting matter hinders the formation of a quark-anti-quark condensate. LQCD calculations [2] show that the quark condensate does increase with increasing magnetic field at low temperatures. This behavior corresponds to magnetic catalysis. However, as the temperature increases approaching the crossover region T ≃ 150 MeV, the quark condensate reaches a maximum value smaller than at T = 0 (for the same field strength). Subsequently, the condensate decreases as a function of the field strength. Finally, for temperatures above the cross-over values the condensate decreases monotonically as a function of the magnetic field. Some of the possible scenarios aiming to understand this behavior include (i) invoking a fermion paramagnetic contribution to the pressure with a sufficiently large magnetization [3], (ii) the competition between the valence and sea contributions at the phase transition [4] produced by a back reaction of the Polyakov loop, which depends on the magnetic field [5], (iii) magnetic inhibition due to neutral meson fluctuations in a strong magnetic field [6], (iv) accounting for non-perturbative effects by means of Schwinger-Dyson Equations and renormalization group analyses [7][8][9], (v) a decreasing magnetic field and temperature dependent coupling with [10][11][12] and without [13,14] plasma screening effects, (vi) the proper account of the gluon contribution during the phase transition [15], and (vii) quark anti-screening due to the anomalous magnetic moment of quarks in strong [16] and weak [17] fields. On the other hand, this behavior is not obtained in mean field approaches describing the thermal environment [18][19][20][21][22][23], nor when calculations beyond mean field do not include magnetic effects on the coupling constants [24]. For recent reviews see Refs. [25,26].
Given the dual nature of the QCD phase transition, a pertinent question is to what extent inverse magnetic catalysis is due to the mechanisms of either chiral symmetry restoration and/or of deconfinement. One way to address this question is to find a relation between deconfinement and chiral symmetry restoration parameters as a function of the magnetic field. Since the transition happens for temperatures in the realm of non-perturbative phenomena, the relation searched for needs to carry nonperturbative information. An extensively used tool in the context of effective models at finite temperature and zero [27] and finite [22] magnetic field is the Polyakov loop [28]. When coupled to quark degrees of freedom this loop sheds light on how chiral symmetry and deconfinement behave during the QCD transition as a function of the field intensity. Another non-perturbative tool that does not rely on effective models is that of QCD Finite Energy Sum Rules (FESR). This approach has been successfully applied both at zero [29] and at finite temperature [30] to understand hadronic properties. Of particular mention are (i) the prediction, in this framework, of the survival of charmonium and bottonium above the critical temperature [31], (ii) the temperature behavior of the hadronic width of charmonium and bottonium [31] in qualitative agreement with bottonium results from LQCD [32], and (iii) the description of the di-muon spectrum in heavy-ion collisions [33], in excellent agreement with data [34] in the region of the rho-meson peak. A key parameter that emerges from this analysis signalling quark-gluon deconfinement is the squared energy threshold, s 0 , above which the hadronic spectral function is well approximated by perturbative QCD (pQCD). An interesting relation between s 0 and the quark condensate qq , whereby the former is proportional to the latter has been found, in the absence of a magnetic field and at finite temperature, in [35], and at finite temperature and density in [36].
In this paper we use FESR in the axial-vector channel, and in the presence of an external magnetic field, to explore the relation between (i) the deconfinement and chiral symmetry restoration parameters, s 0 and qq , and (ii) obtain the behavior of the gluon condensate as a function of the magnetic field intensity at zero temperature. Other formulations of the QCD sum rules, also at T = 0, have been used to explore the dependence of heavy-quark meson-masses and mixing as a function of the magnetic field strength [37]. The paper is organized as follows: In Sec. II we set up the FESR. We show that since the pQCD contribution to the current correlator receives corrections that can be expressed as powers of the magnetic field strength divided by powers of the squared energy s, there appear additional terms that contribute to higher order FESR. In Sec. III we explicitly compute the pQCD corrections to the current correlator and solve the first two FESR to find the dependence of s 0 and the gluon condensate on the magnetic field strength. We show that for magnetic field strengths eB smaller than s 0 , the former follows the magnetic field dependence of the quark condensate, which we parametrize from LQCD results [2]. We also show that the magnetic field dependence of the gluon condensate receives non-trivial corrections from the pQCD sector and that overall it is a monotonically increasing function of the field strength. We summarize and conclude in Sec. IV, leaving for the appendices the explicit computation of the imaginary part of the hadronic contribution and the vanishing of the first order correction in the field strength of the pQCD contribution to the axial-vector current correlator.

II. FINITE ENERGY QCD SUM RULES IN THE PRESENCE OF A MAGNETIC FIELD
The charged axial-vector current correlator in the absence of a magnetic field and at T = 0 can be written as where : is the (charged) axialvector current, q µ is the four-momentum carried by the current, with s ≡ q 2 > 0 the squared energy. The functions Π A,0 (q 2 ) are free of kinematical singularities, an important property needed in writing dispersion relations and sum rules. Concentrating on e.g. Π 0 (q 2 ) and invoking the Operator Product Expansion (OPE) of current correlators at short distances beyond perturbation theory, one of the two pillars of the QCD sum rule method, one has , and µ 2 is a renormalization scale. The Wilson coefficients C N depend on the Lorentz indexes and quantum numbers of the currents, and on the local gauge invariant operatorŝ O 2N built from the quark and gluon fields in the QCD Lagrangian. These operators are ordered by increasing dimensionality and the Wilson coefficients are calculable in pQCD. The unit operator above has dimension d = 0 and C 0Î stands for the purely perturbative contribution normalized according to Since there are no dimension d = 2 operators built from the QCD fields, it is generally assumed that the OPE starts at dimension d = 4. This is fully confirmed by determinations of condensates from experimental data [38]. The dimension d = 4 in the chiral limit is proportional to the renormalization group invariant gluon condensate The second pillar of the QCD sum rule method is to consider an integration contour in the complex square energy plane, as in Fig. 1, and invoke Cauchy's theorem assuming that QCD can be used on the circle of radius |s 0 |, provided |s 0 | is large enough (quark-hadron duality). On the real axis there is a discontinuity associated with the hadronic states entering the spectral function. Since there are no further singularities this leads to the FESR (5) with N ≥ 1, and Π QCD 0 (s) given by the OPE, Eq. (2). It will be shown later that in the presence of a magnetic field, and in the weak field limit eB < s 0 , the Wilson coefficients acquire themselves a B-field dependence. In this work we shall compute the corrections to the FESR due to a weak magnetic field, which can be expressed as a series in powers of eB. Since the magnetic field carries dimension of energy squared, on dimensional grounds one finds the replacements 2N are dimensionless quantities that can be computed in pQCD at a given order in eB. Substituting Eqs. (6) and (2) Note that in general the presence of the magnetic field mixes operators of different dimension in the FESR. For instance, the first two sum rules (N = 1, 2) become In order to set up explicitly the relevant FESR we start by computing the hadronic contribution. The axialvector current in the presence of a magnetic field can be interpolated by the charged pion current where f π = 130.28 (14) MeV [39] is the pion decay constant, π + the pion field, and A µ = (B/2)(0, −y, x, 0) the vector potential in the symmetric gauge, which gives rise to a constant magnetic field along theẑ direction. Therefore, the axial-vector correlator in the hadronic sector can be written as where we have used the fact that the charged pion propagator G π (x, y) in the presence of a magnetic field can be written as a product of a transnationally invariant piece G π (x − y) and a phase factor e ieΦ(x,y) . The phase factor does not depend on the integration path so that choosing a straight line trajectory it can be written as It is easy to show that the above phase factor can be gauged away by a suitable gauge transformation of the vector potential. Hence, we keep only the translational invariant part of the hadronic correlator whose Fourier transform is where G π (q 2 ) stands for the Fourier transform of G π (x − y). Using Schwinger's proper time method this quantity can be written as where m π = 0 as we consider the chiral limit. Hereafter we shall use the notation Consequently Extending the integration in Eq. (14) to the lower complex plane, it can be shown [40] that the charged pion propagator is expressed in terms of a sum over Landau levels Hereafter we study the behavior of the correlator setting q 2 ⊥ = 0. In this limit, Eq. (17) becomes Therefore the hadronic contribution to the correlator becomes explicitly As we show in Appendix A, the imaginary part of Eq. (19) in the weak field limit, eB < s 0 , is given by It is important to mention that Eq. (20) holds provided s 0 satisfies also the condition s 0 < 3eB, as explicitly shown in Appendix A. This means that the weak field condition (eB < s 0 ) has to be supplemented with a further restriction for Eq. (20) to remain valid. Recall that s 0 represents the onset for the pQCD description for the axial-vector spectral density and that this quantity is a decreasing function of temperature [33]. For cold nuclear matter, as in the case of a neutron star, the condition eB < s 0 < 3eB may be difficult to meet, specially for a weak field where not only the situation eB < s 0 , but even neB < s 0 , (n ≥ 1) can happen. However, for a heavyion collision, around the deconfinement/chiral symmetry restoration transition, when s 0 has dropped off to small values, the weak field condition can also be made compatible with eB < s 0 < 3eB. Hereafter we keep in mind this last observation as the working scenario, aiming to eventually incorporate thermal effects to describe the behavior of a magnetized medium near the phase transition.
Using Eq. (20), the hadronic line integral in the FESR is given explicitly by Substituting Eq. (21) into the QCD sum rules Eqs. (8)- (9) gives In order to solve these equations, we now proceed to compute explicitly the coefficients C 0 .

III. SOLUTIONS FOR s0 AND C4 O4
To perform the perturbative calculation of the coefficients C of the weak field expansion of the quark propagator in the presence of a constant magnetic field [41], and in the chiral limit, up to order O(B 2 ) where e q is the absolute value of the quark's charge.
The pQCD contribution to the axial-vector current correlator in the presence of a magnetic field is depicted in Fig. 2, where we also define the kinematics. The thick internal lines represent the full quark propagators in the magnetic field background. To first order in e q B only one of the two quark propagators carries the magnetic effects. This is depicted in Fig. 3 where the wavy line starting from a cross represents the external magnetic field. The two diagrams in Fig. 3 that determine the coefficient C (1) 0 , vanish identically when contracted with the momenta carried by the axial-vector currents. This is due to a straightforward application of Furry's theorem, and to the fact that the vector and axial-vector correlators are chiral symmetric. However, we show explicitly in Appendix B that each of the contributions in the diagrams, Fig. 3, vanishes.
The first non-trivial magnetic contribution to the pQCD axial-vector current correlator is of order (e q B) 2 . The relevant diagrams are shown in Fig. 4. First, we compute the diagram where one magnetic field line is attached to each one of the two quark propagators. For these, we use Eq. (23) to first order in e q B. We call this contribution Π (11) µν (q 2 ), and its explicit expression is Since according to Eq. (5), we are interested in the magnetic corrections to the coefficient of the longitudinal structure, Π 0 (q 2 ) we project Π (11) µν (q 2 ) with q µ q ν and define Π (11) 0 (q 2 ) = q µ q ν Π (11) µν (q 2 ). Using together with gives The evaluation of the trace yields We now use the Feynman parametrization and the change of variable to obtain where we have discarded terms with odd powers of l and defined ∆ = x(x−1)q 2 . The integrals over l are computed by means of with n = 4 and d = 4. Using Eq. (33) in Eq. (32), and after integrating over x, we find In the limit q 2 ⊥ → 0 Eq. (34) becomes In a similar fashion we compute the diagrams in Fig. 4 to second order in eB in the u-quark and in the dquark propagator. Calling the longitudinal projections Π , respectively, the result is Adding all three contributions, and using the absolute values q u = 2/3 e, q d = 1/3 e, and N c = 3, we obtain the coefficient of the longitudinal structure of the axialvector current correlator to second order in the magnetic field Using this result together with the first equation in Eq. (6), we obtain the Wilson coefficient of the pQCD contribution to second order in the magnetic field The last ingredient needed to find s 0 and C 4 O 4 is the magnetic field dependence of f π . Invoking the Gell-Mann-Oakes-Renner (GMOR) relation f π is related to the light quark condensate qq where the constant B is given by and m π , m u , and m d stand for the vacuum values of these masses. We assume that in the presence of a magnetic field the GMOR relation still holds and that the above masses do not change significantly. The light-quark condensate in the presence of the magnetic field has been computed in Ref. [2]. We make use of this result, and parametrize the magnetic field dependence of the lightquark condensate with a quadratic fit where a = 0.85 GeV −2 , b = 0.34 GeV −4 and (eB) is given in GeV 2 . The data from Ref. [2] together with the fit are shown in Fig. 5. Using this information we finally write The solutions for s 0 and for C 4 O 4 as functions of eB are plotted in Figs. 6 and 7, respectively. Note that s 0 is proportional to the absolute value of the light-quark condensate, and that together with C 4 O 4 it increases with increasing magnetic field.

IV. SUMMARY AND CONCLUSIONS
In this paper we studied QCD FESR for the axialvector current correlator in the presence of a magnetic field in the weak field limit eB < s 0 , where s 0 is the squared energy threshold for the onset of pQCD. We have shown that the presence of the field modifies both the pQCD as well as the hadronic sectors of the FESR. The direct modification to the hadronic contribution for the equation that governs the behavior of s 0 vanishes nontrivially. The only change in this sector comes from the dependence of the pion decay constant on the magnetic field. In turn, this pion decay constant is proportional to the quark condensate through the GMOR relation. The magnetic field dependence of s 0 is thus proportional to the magnetic field dependence of the absolute value of the light-quark condensate. This means that the behavior of these two parameters as a function of the magnetic field is similar. Therefore the magnetic field both helps the formation of the condensate and acts against deconfinement. Next, we computed explicitly the magnetic corrections to the pQCD contribution which affect the behavior of the gluon condensate as a function of field strength. In the weak field limit the first correction is quadratic in the field. The gluon condensate also grows as a function of the field strength which goes hand in hand with the behavior of the magnetic field, both as a catalyst of chiral symmetry breaking and confinement.
The results obtained here should serve as a basis for studies at finite temperature in an external magnetic field. Thus, providing clues on the relation between chiral symmetry restoration and deconfinement at the transition temperature. This research is in progress and will be reported elsewhere.
Appendix B: First order magnetic correction to the pQCD contribution to the axial-vector current correlator The contribution to the pQCD axial-vector current correlator of order (eB) is given by Π B µν (q 2 ) = Π (10) µν (q 2 ) + Π (01) µν (q 2 ) where Π (10) µν (q 2 ) = N c (q u B) Note that the traces in Eqs. (52) are equal except for an overall sign, due to the ordering of elements inside the trace. Therefore, we evaluate one of the traces, where u ρ and b ρ are four-vectors describing the particle's rest frame and the direction of the magnetic field, respectively. In the rest frame, there are given by u = (1, 0, 0, 0) b = (0, 0, 0, 1).
Using that we can write Eq. (53) as Notice that when Eq. (56) is contracted with q µ q ν the coefficient of the longitudinal structure vanishes.