Magnetic field-dependence of the neutral pion mass in the linear sigma model coupled to quarks: The weak field case

We compute the neutral pion mass dependence on a magnetic field in the weak field approximation at one-loop order. The calculation is carried out within the linear sigma model coupled to quarks and using Schwinger's proper-time representation for the charged particle propagators. We find that the neutral pion mass decreases with the field strength provided the boson self-coupling magnetic field corrections are also included. The calculation should be regarded as the setting of the trend for the neutral pion mass as the magnetic field is turned on.


I. INTRODUCTION
Magnetic fields are involved in the properties of a large variety of physical systems including heavy-ion collisions [1,2], the interior of compact astrophysical objects [3][4][5] and even the early universe [6][7][8]. It has been estimated that the magnetic field strength |eB| in peripheral heavy-ion collisions reaches values equivalent to a few times the pion mass squared, both at RHIC and at the LHC [9]. The effects of such magnetic fields cannot be overlooked in a complete description of these systems and its understanding contributes, at a fundamental level, to a better characterization of the properties of QCD matter [10][11][12][13][14][15][16][17][18].
One of these properties is the behavior of meson masses as a function of |eB|. These properties have been the subject of intense study in several recent works. Within the framework of Lattice QCD (LQCD) calculations, it has been shown that the neutral pion mass decreases monotonically as |eB| grows [19]. The latter calculation solved an existing disagreement between the quenched Wilson and overlap fermions formulations of LQCD [20,21].
In effective QCD models, the neutral pion mass has also been found to decrease with the increase of the magnetic field intensity. Some of these calculations resort to use the Nambu-Jona-Lasino (NJL) model and its extensions [22][23][24][25]. In particular, Ref. [26] considers a twoflavor NJL model, employing a magnetic field dependent coupling, fitted to reproduce lattice QCD results [27] for the quark condensates [28]. The B-dependence of the pion mass has also been studied using chiral perturbation theory [29,30]. Also, Ref. [31] resorts to the use of phenomenological Lagrangians for the one-loop calculation of the pion effective mass in the weak field approximation (|eB| << m 2 π ), finding that for a pseudoscalar coupling, the pion mass decreaces, whereas for a pseudovector coupling the mass increases, as a function of the magnetic field.
In this work we explore the idea that magnetic fielddependent couplings, computed self-consistently, can account for the decrease of the neutral pion mass as a function of the magnetic field strength. For this purpose, we resort to use the Linear Sigma Model coupled to quarks (LSMq) to compute the neutral pion self-energy in the presence of a weak magnetic field, accounting for the oneloop corrections to the boson self-coupling. From the selfenergy, we then obtain the magnetic field dependence of the neutral pion mass. The work is organized as follows. For completeness, in Sec. II we summarize the properties of the LSMq Lagrangian after both, spontaneous and explicit breaking of symmetry are implemented. In Sec. III the one-loop contribution to the pion self-energy in a weak magnetic field is computed taking into account the quark and meson contributions. In Sec. IV we compute the magnetic field effects on the boson self-coupling. In Sec. V we compute the magnetic field effects on the vacuum expectation value. Finally, in Sec. VI we find the explicit expression for the magnetic field-dependence of the neutral pion mass. We also discuss our results and conclude. We leave for the appendices the explicit calculations of the various quantities involved.

II. LINEAR SIGMA MODEL COUPLED TO QUARKS WITH AN EXPLICIT SYMMETRY BREAKING TERM
The Lagrangian for the LSMq is given by where ψ is an SU (2) isospin doublet, π = (π 1 , π 2 , π 3 ) is an isospin triplet, and σ is an isospin singlet, with the covariant derivative. A µ is the vector potential corresponding to an external magnetic field directed along theẑ axis. In the symmetric gauge, it is given by The gauge field couples only to the charged pion combinations, namely, To allow for spontaneous symmetry breaking, we let the σ field to develop a vacuum expectation value v After this shift, the Lagrangian can be rewritten as The quarks, sigma and the three pions have masses given by respectively. Also, we notice from Eq. (6) that the treelevel potential is given by which has a minimum located at v 0 = a 2 λ .
Therefore, the masses evaluated at v 0 are In order to consider a non-vanishing pion mass, we add to the Lagrangian an explicit symmetry breaking term and thus where m π ≈ 140 MeV. As a consequence of the explicit symmetry breaking, the V tree (v) is modified and therefore the minimum of the tree-level potential becomes Consequently, although the expressions for the masses as functions of v, Eq. (7), remain the same, their values at the minimum of the potential with explicit symmetry breaking become Also, notice that from Eq. (7), we can fix the value of a from the relation For m π ≈ 140 MeV and when considering the sigma mass in the range m σ ≈ 400 − 600 MeV, we get a ≈ 225 − 390 MeV. In the spirit of LQCD calculations, we later on consider also different values for m π and accordingly, we also extend the ranges for m σ and a.

III. ONE-LOOP PION SELF-ENERGY
The self-energy for the neutral pion has four contributions The one-loop diagrams for the first two contributions are depicted in Figs. 1 and 2. Notice that for Π π 0 ,σ there are no magnetic corrections, given that the particles in the loop are neutral. We now proceed to compute the first two contributions to the self-energy.

A. Quark loop
The contribution from the one-loop Feynman diagram made out of fermions to the neutral pion self-energy iss depicted in Fig. 1. In the presence of a magnetic field, this corresponds to the expression is the Schwinger's proper time representation of the charged fermion propagator. In the weak magnetic field limit, this propagator can be written as a power series in |eB| which, up to order O(eB) 2 , is written as [32] iS where the transverse and parallel components of any vector are split, with respect to the magnetic field direction, and we use the definitions Therefore, the contribution from the quark-antiquark loop to the self-energy can in turn be split into tree terms The first term is the usual vacuum piece which contributes only to the pion mass renormalization. The term linear in |q f B| gives a vanishing contribution. The result for O(q f B) 2 , in the limit of vanishing four-momentum q → 0, is given by (see details in Appendix A) −iΠ

B. Meson loop
The meson loops consist of tadpole diagrams, like the one depicted in Fig. 2. The expression for the only diagram that contributes to the magnetic field correction to the pion mass is given by where D B (k) is the propagator for a charged scalar boson in a magnetic field. In the weak field limit, this can also be expressed as a power series in |eB| which, up to order O(eB) 2 , is given by [33] iD The first term in Eq. (22) corresponds to the vacuum contribution, which is magnetic field independent and ultraviolet divergent. As usual, this term contributes to the pion mass renormalization. The magnetic fielddependent terms are finite. Thus, the one-loop contribution to the magnetic field corrections to the neutral pion mass, up to O(eB) 2 due to charged mesons, is written as (see details in Appendix B) C. One-loop magnetic modification to neutral pion mass In order to compute the magnetic field-induced modification to the pion mass, M π (B), we need to find the solution to the equation in the limit q → 0 and q 0 = M π (B). For the time being, for simplicity, instead of working with the solution of Eq. (24), which requires a numerical treatment, let us first use the approximation which allows for an analytic treatment. Nevertheless, in Sec. VI, we show the results when explicitly solving Eq. (24). Using Eqs. (20) and (23) we get where we sum over the number of quark-flavors N f , and thus account for the absolute values of their electric charge. Using Eq. (13) and simplifying, we obtain Notice that the dependence of the coupling g cancels in the above expression. Therefore, the magnetic fielddependent neutral pion mass is given by where we recall that m π is the vacuum pion mass.
Since the parameters a and m π are of the same order, 5/9/(1 + a 2 /m 2 π ) > 1/96. Naively it would seem that the neutral pion mass increases with the magnetic field strength. However, before drawing this conclusion, we observe that the calculation is not yet complete given that we also need to incorporate the one-loop magneticfield correction to λ and v 0 . We now set up to find such corrections.

IV. MAGNETIC CORRECTION TO THE BOSON SELF-COUPLING
In order to compute the magnetic field correction to λ, we compute the Feynman diagrams depicted in Fig. 3. We write the effective coupling up to one-loop order as where ∆λ is given by [15], where is the contribution from neutral boson fields in the loop (i = σ, π 0 ), with and corresponds to the contribution from charged boson fields in the loop (i = π ± ). The charged-boson propagator is given in Eq. (22). After computing both contributions and considering the pure magnetic correction, we obtain in the limiting of vanishing external four-momentum (see details in Appendix C) and therefore, using Eqs. (29) and (13) we have Notice that the magnetic field-induced corrections to the boson self-coupling make the effective coupling to decrease with the increase of the magnetic field strength.

V. MAGNETIC CORRECTION TO THE VACUUM EXPECTATION VALUE
To include the magnetic correction to the vacuum expectation value, we start from the one-loop effective potential [15]. This includes the tree-level potential, the one-loop boson contribution, the one-loop fermion contribution and the counter-terms that come from requiring the vacuum stability conditions. These contributions are respectively. The vacuum stability conditions [34] are introduced to ensure that v 0 and the sigma-mass maintain their tree level values, even after including the vacuum pieces stemming from the one-loop corrections. These conditions are 1 2v where V vac is the one-loop vacuum piece of the effective potential. The counterterms δa 2 and δλ are given by The magnetic field-correction to the vacuum expectation value is obtained by finding the magnetic field displaced new minimum, v B 0 . It turns out that in the weak field limit, v B 0 ≃ v ′ 0 , that is, the corrections are negligible. This is shown in Fig. 4.

VI. DISCUSSION AND CONCLUSIONS
Since the corrections to the magnetic field-displaced position of the minimum are negligible, the significant correction to M 2 π (B) in Eq. (28) comes from the effective boson self-coupling and thus, the magnetic field-modified  pion mass can be written as Substituting Eq. (35) in Eq. (43), simplifying and keeping terms up to O(eB) 2 , the magnetic field-modified pion mass is given by (44) Figure 5 shows the neutral pion mass dependence on the the magnetic field, obtained from Eq. (44) when varying the sigma and pion vacuum masses. Figure 6 shows the ratio of the neutral pion mass dependence on the magnetic field, when working with the numerical solution of Eq. (24) (M 2 π ) to the case where we use the approximate solution (M 2 π ), given by Eq. (25). Figure 6a shows the case for a fixed vacuum value m σ = 450 MeV, for two values of the vacuum m π = 100, 160 MeV. Figure 6b shows the case for a fixed vacuum value m π = 140 MeV, for two values of the vacuum m σ = 400, 550 MeV. Notice that using the exact and the approximate solutions of Eq. (24) does not make a difference. We stress that the calculation is limited to the weak field case and thus it has to be regarded as the trend for the neutral pion mass obtained as the magnetic field is turned on.
In conclusion, we have used the LSMq with spontaneous and explicit symmetry breaking to compute the magnetic field dependence of the neutral pion mass, including magnetic corrections to the one-loop pion selfenergy, to the boson self-coupling and to the sigma field vacuum expectation value, in the weak field limit. Although the latter is a negligible correction, we found that the correction to the boson self-coupling produces that the pion mass decreases as a function of |eB|, which is opposite to the naive calculation obtained when ignoring self-coupling corrections.
The results in this work should be regarded as the setting of the trend for magnetic field dependence of the neutral pion mass as the magnetic field is turned on. In order to compare this calculation to recent LQCD data, we require to extend the validity of the calculation to include the region of intermediate and large magnetic field compared to the vacuum pion mass. Work along these lines is currently being performed and will be reported elsewhere.

ACKNOWLEDGEMENTS
where we introduced the spin-projection operators that satisfy the identities In order to compute the traces, we name the two terms in Eq. (A1) according to the power of |q f B| in each of the propagators of Eq. (17). The first term comes from the product of the linear terms in |q f B| Using the properties of O ± , we have and simplifying The second term in Eq. (A1) comes from the product of the vacuum term times the quadratic term in |q f B| Substitution of the traces in Eqs. (A7) and (A8) into Eq. (A1), leads to − iΠ We now introduce an integration over Feynman parameters. Equation (A9) contains two terms. We name them K 1 and K 2 . These are given by [35]