Magnetic test of chiral dynamics in QCD

Strong magnetic fields in the range $ eB\gg m^2_\pi $ effectively probe internal quark structure of chiral mesons and test basic parameters of chiral theory, such as $\lan\bar q q\ran, f_\pi$. We argue on general grounds that $\lan\bar q q\ran$ should grow linearly with $eB$ when charged quark degrees of freedom come into play. To make explicit estimates we extend previously formulated chiral theory, including quark degrees of freedom, to the case of strong magnetic fields and show that the quark condensate $|\lan \bar q q\ran|_{u,d}$ grows quadratically with $eB$ for $eB\la 0.2$ GeV$^2$ and linearly for higher field values. These results agree quantitatively with recent lattice data and differ from $\chi PT$ predictions.


Introduction
Strong magnetic fields (MF) are expected to be present in the early universe [1,2,3], in neutron stars (magnetars) [4,5], and also in non-central heavy-ion collisions [6]. It is very interesting how MF can modify strong interactions, deduced from QCD. In particular, it was recently shown [7], how MF influence asymptotic freedom behavior of the QCD strong coupling constant and gluon exchange interaction. The interplay of confinement and MF in neutral mesons was studied in [8], showing a dramatic change of masses with growing MF.
Of a particular interest is the influence of MF on the chiral symmetry breaking (CSB) dynamics, and in a more general setting, on symmetry breaking pattern in field theory.
It was emphasized in [10], that MF has the tendency to strengthen the chiral condensate, the phenomenon called there the magnetic catalysis, see [?] for a review.
The behavior of chiral condensate in MF was studied in the chiral perturbation theory (χP T ) [11,12,13,14], in Polyakov-Nambu-Jona-Lasinio (PNJL) [15] and in other models [16]. On the lattice the corresponding analysis was done for quenched QCD in [17,18], in n f = 2 QCD in [19,20] and in n f = 4 SU(2) theory in [21]. In all cases both in models and on the lattice the chiral condensate grows with MF, however in different way.
Recently a comprehensive analysis of this problem on the lattice with physical quark masses was performed in [22] and has shown a quadratic growth of condensate up to eB ∼ 0.2 GeV 2 and approximately linear behavior above this value.
This behavior contradicts the (χP T ) results, see e.g. [11,14], where linear growth was found with the slope almost twice as small as in [22], and also contradicts PNJL quadratic growth at large eB [15].
As it was mentioned in [13], the reasonable region of χP T application is for eB < ∼ m 2 π , where the quark condensate grows quadratically, in qualitative agreement with lattice data however the linear behavior up to 1 GeV 2 is outside of χP T reliable region.
Indeed, the physical degrees of freedom in χP T are associated with the strucrureless Nambu-Goldstone mesons, and not with quarks and antiquarks, the role of the latter becomes more important with growing MF, and should be decisive for eB > ∼ σ, where σ = 0.18 GeV 2 is the string tension. As it was shown in [8] quark and antiquark are strongly attracted to each other in the plain perpendicular to MF, as it follows from the linear growth of the corresponding probability: |ψ qq (0)| 2 ∼ eB.
This phenomenon was called in [23] the magnetic focusing, and is the origin of the strong enhancement of the hyperfine interaction in MF [24]. Moreover, magnetic focusing also can produce a linear amplification with MF of atomic, nuclear and hadronic reaction yields, as shown in [25].
It seems reasonable, that some quantities in the chiral theory, like chiral condensate, are proportional to |ψ qq (0)| 2 , and therefore should grow linearly with eB. Indeed, one can write the quark condensate (in the Euclidean (1) = | S q (x, y)Sq(y, x)(m −D) x |, and one can visualize the probability amplitudes of the qq emission at the point x and absorption at the point y, combining into the factor |ψ qq (0)| 2 . This would bring us the linear behavior which is not connected to any chiral degrees of freedom. Therefore one can expect, that in any model, which takes into account the general structure (1) of the quark condensate, one would end up with the linear growth (2), and the main emphasis should be on the exact quantitative form, i.e. on the coefficient in front of eB in (2). here the recent accurate lattice analysis in [22] gives a good check of analytic results, which will be used below.
It is the purpose of the present paper to study quark condensate in MF starting from the basic QCD equations, derived earlier without MF in [26,27,28,29]. It was shown there, that GMOR relations and expressions for qq and f π can be derived from the basic QCD quantities: string tension, α s and current quark masses in good agreement with experiment and lattice data.
Recently in [30] these results were extended to account for growing current quark masses m q , and in particular the dependence of qq on m q was established to be in agreement with lattice data [31].
In the present paper we follow the same line of the formalism of [26,27,28,29], but now adding MF, and find the behavior of qq for u and d quarks and their average with growing MF.
As a result we observe in the resulting dependence of the average qq and ūu , d d , the same features and good quantitative agreement with the lattice data obtained in [22]. The physical reason for this dependence of qq on eB is clarified below in the paper.
The paper is organized as follows. In the next section a general derivation of GMOR relation and expressions for qq and f π are given, in section 3 the MF dependence of basic terms is established, in section 4 results are discussed and prospectives are given.

Effective chiral Lagrangian and the quark condensate
We start with the situation without MF. In this case in [28,29] the effective Lagrangian for Nambu-Goldstone (NG) mesons with qq degrees of freedom was obtained in the form and M(x) in the local limit is a quark confining interaction, M(x) ∼ σ|x| for |x| → ∞ see Fig. 1, whileφ = φ a t a , φ a (x) is the octet of NG mesons. It is important, that M s (x), appearing at the vertices of the Green's function in Fig.1, are finite, where λ is the vacuum correlation length, λ ∼ = (0.1 ÷ 0.15) fm [32]. Expanding (3) to the quadratic in φ terms, one obtains the GMOR relation, e.g. for the pion where the quark condensate ψ ψ is It was also found in [27], that trΛ can be expressed via the qq Green's function and where the spectral decomposition is [27] and c n = mn 2 ψ n (0). Finally, the quark condensate for i = u, d, s quarks is Eq. (10) expresses quark condensate in terms of M(0) and the reference spectrum in the PS channel, i.e. the spectrum, where chiral symmetry is not taken into account, but only confinement term M(x) is present, the corresponding masses m n , calculated in [27] are m 0 = 0.4 GeV, m 1 = 1.35 GeV, m 2 = 1.85 GeV. Taking λ = 0.1 fm and M(0) = 0.15 GeV, one obtains (cf appendix of the second paper in [27]) One can see, that our values are in the correct ballpark, close to average lattice and phenomenological values. Now we turn to the case of nonzero MF. In this case one should replace , so that the quark condensate now contains contribution of both quarks and antiquarks, Moreover, MF destroys both spin and isospin quantum numbers of the meson, and therefore one must distinguish in the Green's function of our neutral qq system not only (uū) and (dd) components separately, but also the qq spin projections (+−) and (−+), since they correspond to different mass eigenvalues.
As a result Eq. (10) in case of MF can be rewritten for each flavor separately, which can be rewritten in the form (i = u, d, s) whereω = ω 1 ω 2 ω 1 +ω 2 . The eigenvaluesm n,i of H i depend on ω 1 , ω 2 , and the final eigenvalues m n,i entering in (13) are obtained as stationary points in variation over ω 1 , ω 2 ∂m n,i ∂ω 1,2 ω (0) To avoid purely numerical calculations, one can simplify the hamiltonian (15), replacing linear confinement V conf (η) in (15) by a suitable quadratic form, with coefficient found from the stationary point condition To check the accuracy of this replacement, one can compare the lowest mass eigenvalues and ψ n (0) first without MF. In particular, m For ψ n (0) the corresponding results are This coincidence of |ψ 0 (0)| 2 will be of special importance in what follows, since the main effect of MF, as will be seen, is the increase of |ψ n (0)| 2 in (13) due to MF. It is clear, that with growing B the size of meson is decreasing, and the difference between V (lin) conf and V (quadr) conf will be even more suppressed, since these both interactions vanish simultaneously, V As a result of (17) one can immediately write the analytic expressions for ψ n (0) and m n of the following form, e.g.

The MF dependence of quark condensates
We are interested in eigenfunctions and eigenvalues of (+−) and (−+) states both for u and d quarks. For the (+−) state, one can see in (22), (21) that at large B, m (+−) 0,n 3 tends to a constant limit, together with ω 1 = ω 2 . In this case the parameter c in (20) and (22) which can be expressed in general as and β(B), a(B) are changing in finite limits and one obtains that c +− (B) ≈ 1, for all B, and m +− tends to a constant limit for B → ∞. Hence one can write for lowest levels n 3 = 0, 1, 2, ...
Note, that in the limit B → 0 this expression for |ψ(0)| 2 yields equal values for n 3 = 0, 1, 2 as it should be for pure linear confining interaction. For the (−+) case the situation is different, and at large B the stationary point value ω (−+) 0 ≈ 2|e q |B + σ 4 , and the parameter c −+ is increasing with B: As a result, the |ψ (−+) n (0)| 2 can be written as  Combining (22) and (26) we therefore can write where c −+ is given in (25). We shall be using notations of [22] for the increment of quark condensate as a function of MF (denoted τ i (B) in [10]) The resulting values of ∆ i (B) from (27), (28) are compared in Table 1 with the corresponding lattice calculations in [22].
One can see a reasonable agreement between our theory and lattice data. One can simplify the B dependence of Eq. (27), writing The form (29) also satisfactorily describes data of [22] and [20], taking M i as a fitting parameter. In the case of [20]the fitted values of M i are approximately i.e. are 2 and 1.5 times larger, than σ (the corresponding c = c +− = c −+ ∼ = 4 and 4.25 for u and d cases), however the agreement with lattice data is worse. Qualitatively the same situation (with even larger M 2 i )takes place in comparison with [17,18].
As a whole the behavior (27), (29) correctly reproduces the main qualitative features of the quark condensate as a function of eB: the quadratic behavior proportional to (e i B) 2 at small B and linear behavior ∼ |e i B| at large B. It differs from the results of others approaches. In particular, χP T [11] predicts linear behavior in eB with the slope much smaller, than in lattice data [22].
As it was mentioned in [20], for ∆ u and ∆ d there is a simple relation which is satisfied in lattice data, and, of course, is trivially satisfied in our definitions.

Discussion and conclusions
We have used our formalism for chiral dynamics, presented in [26,27,28,29,30], which is derived not from purely symmetry considerations, but from the QCD quark dynamics, where chiral symmetry appears approximately in the small m q limit of the effective QCD Lagrangian. In this way all basic degrees of freedom are connected to the confined quarks, and it is finally the confinement, which dictates properties of fundamental chiral quantities qq , f π etc., and gives them numerical values, expressed via σ. This is in contrast with standard chiral ideas, where chiral constants are postulated from the beginning and chiral symmetry breaking is not connected to the basic QCD quark dynamics.
Actually our reasoning for the calculation of quark condensate is very simple. After one derives scalar confining interaction M(x), which acts on each quark or antiquark, one can write quark condensate qq = −N c trΛ as since the term with∂ is odd and vanishes. The last quantity on the r.h.s. of (32) is the qq Green's function, proportional to ψ 2 n (0) for each n state. The only difference in presence of MF is the replacement∂ → (∂ − ieÂ (e) ), which immediately gives the proportionality qq ∼ ψ 2 (r = 0; eB). This latter quantity is linearly rising with |eB|, since MF is "focusing" quark-antiquark system at small distances. Note the similarity of (32) and (1).
This phenomenon of "magnetic focusing" is of a general character and in the case of the chiral condensate it actually explains dynamically its growth with eB, which was named before in [10, ?] "magnetic catalysis". Recently the effect of "magnetic focusing" in the hyperfine interaction in hydrogen was studied in [23], and in the case of molecular, nuclear and hadronic procession [25].
External magnetic field is here crucial for our understanding of chiral dynamics, and using MF one may decide, what is the role of quark dynamics in the chiral phenomena. In this respect the comparison of the behavior of quark condensate (or ∆ (B)) as a function of B in different models and lattice data is showing the following: 1. Standard chiral theory at large B ∼ σ gives linear behavior (qualitatively correct) but with the wrong slope.
2. Our approach, Eq. (13) predicts quadratic behavior at small B B < ∼ σ and linear at larger B with slopes different for u and d quarks, both in agreement with existing lattice data.
Physically, it is clear, that MF, acting on quark charges, discloses the internal quark structure of PS mesons, while the effective chiral Lagrangian, as in [11,12] describes only internal multipionic degrees of freedom. The latter can be important only for B ≪ 1/r 2 0 , where r 0 ≈ 0.6 fm is the pionic radius, i.e. for B ≪ 0.1 GeV 2 , while for large B the standard chiral picture is irrelevant, as it is confirmed by lattice data.
These considerations suggest the idea, that the true chiral dynamics can be derived e.g. from the effective Lagrangian (3), and should finally display coexisting quark and chiral symmetric degrees of freedom, demonstrating how the latter disappear (suppressed) for growing quark masses. How it happens with the pionic spectra, was explained in [30], demonstrating the unifying spectrum based on chiral and quark degrees of freedom at the same time.
The author is grateful for discussions to N.O.Agasian, M.A.Andreichikov and B.O.Kerbikov.