Chiral pions in a magnetic background

We investigate the modification of the pion self-energy at finite temperature due to its interaction with a low-density, isospin-symmetric nuclear medium embedded in a constant magnetic background. To one loop, for fixed temperature and density, we find that the pion effective mass increases with the magnetic field. For the $\pi^{-}$, interestingly, this happens solely due to the trivial Landau quantization shift $\sim |eB|$, since the real part of the self-energy is negative in this case. In a scenario in which other charged particle species are present and undergo an analogous trivial shift, the relevant behavior of the effective mass might be determined essentially by the real part of the self-energy. In this case, we find that the pion mass decreases by $\sim 10%$ for a magnetic field $|eB|\sim m_\pi^2$, which favors pion condensation at high density and low temperatures.

We investigate the modification of the pion self-energy at finite temperature due to its interaction with a low-density, isospin-symmetric nuclear medium embedded in a constant magnetic background. To one loop, for fixed temperature and density, we find that the pion effective mass increases with the magnetic field. For the π − , interestingly, this happens solely due to the trivial Landau quantization shift ∼ |eB|, since the real part of the self-energy is negative in this case. In a scenario in which other charged particle species are present and undergo an analogous trivial shift, the relevant behavior of the effective mass might be determined essentially by the real part of the self-energy. In this case, we find that the pion mass decreases by ∼ 10% for a magnetic field |eB| ∼ m 2 π , which favors pion condensation at high density and low temperatures.

I. INTRODUCTION
The behavior of hadronic matter in a medium under the influence of a strong external magnetic field can be very rich and subtle, and has been the subject of intense investigation in the last few years. In fact, in-medium strong interactions under extreme magnetic fields are of experimental relevance in heavy ion collisions and in astrophysics, exhibit a rich new phenomenology and are amenable to lattice simulations. (For comprehensive reviews, see Ref. [1].) Even if every model calculation has predicted that large enough magnetic fields, typically of the order of a few times m 2 π , could bring remarkable new features to the thermodynamics of strong interactions, from shifting the chiral and the deconfinement crossover lines in the phase diagram [2][3][4][5][6][7][8][9][10][11][12][13][14][15] to transforming the vacuum into a superconducting medium via ρ-meson condensation [16,17], essentially all models fail to describe coherently the available lattice data [18][19][20][21]. The reasons for that are still unclear, although there are some indications that confinement plays a relevant role [15,22], which is not captured in the usual low-energy effective chiral models of QCD [23]. In any case, the situation calls for theoretical investigations in more controlled setups, with less freedom and parameters to adjust. This approach has proved to be fruitful in the large-N c [22] and perturbative [24] limits of QCD: in the former, the behavior of the critical temperature for deconfinement was found to be in qualitative agreement with lattice data; in the latter, a trivial chiral limit for the two-loop contribution to the QCD pressure in a strong magnetic background was revealed.
Following this line of action, a natural extension is the study of hadronic matter in the complementary, lowenergy sector, in the presence of a strong magnetic field, in a controlled setup. Thus, since we are interested in the low-density, low-temperature sector of the phase diagram of nuclear matter, we adopt the framework of chiral perturbation theory, which represents a powerful tool to study the low-energy regime of the pion-nucleon physics [25].
It is the purpose of this work to investigate some properties of isospin-symmetric nuclear matter in the limit of low density and temperature, embedded in a strong magnetic background. In particular, we study the modifications of the spectrum of the lowest energy degree of freedom, the pion, due to the interaction with nucleons and the constant magnetic field. More specifically, we compute the pion effective mass in the presence of a constant magnetic field to one loop. (Even if we do not address the phase diagram here, it should be mentioned that the inclusion of nucleons, and pion-nucleon interactions, proved to be necessary for a satisfactory description of the behavior of the deconfinement critical temperature as a function of the pion mass and isospin [26].) For this purpose, we consider fully relativistic chiral perturbation theory as a framework for our computation. This is needed to define consistently the fermion propagators in a magnetic background. At the same time, this work extends a previous treatment on the calculation of the fermion self-energy in relativistic chiral perturbation theory [27].
In-medium pion properties have been extensively investigated, both in finite systems, i.e. pionic atoms [28,29], and in infinite nuclear matter. In the latter, an interesting aspect of pion phenomenology is represented by pion condensation at high densities, introduced by Migdal [30], which is a consequence of the fact that at high density the electron chemical potential grows until it is favorable for a neutron on the top of the Fermi sea to turn into a proton and a (negatively charged) pion. On other hand, the interaction of the pion with the background matter can enhance its self-energy and consequently the pion condensation threshold density. This issue is still open and requires more investigation because of its implications in the context of compact stars phenomenology [31][32][33]. We shall see in the sequel that the in-medium modification of the (negatively charged) pion self-energy due to the presence of a strong magnetic background might lead to relevant phenomenological consequences.
The paper is organized as follows. In Section II, we consider the relativistic formulation of the theory, since in this framework it is possible to define the Green's function of the theory in the presence of a constant magnetic background in a consistent fashion. In Section III, we compute the lowest order pion self-energy for the three charge eigenstates in isospin symmetric nuclear matter.
In Section IV, we compute the in-medium effective mass of the pion and its dependence on the value of the applied magnetic field. Finally, in Section V, we summarize our conclusions. We use natural units = c = k B = 1. Four-vectors are denoted by capital letters, for instance P µ = (p 0 , p).

II. REMINDER OF THE PION EFFECTIVE MASS
The low-energy phenomenology of pions in nuclear matter is well described in terms of a chirally invariant pion-nucleon interaction Lagrangian, expanded in powers of the low-energy scale of the theory, i.e. the ratio of the pion momentum or mass over (4π times) the pion decay constant: where the leading order, L (2) Here τ is the vector of Pauli matrices in isospin space, π is the isotriplet of pions, f π the pion decay constant and g A is the axial-vector coupling. The diagrams contributing to the pion self-energy from the Lagrangian (2) are shown in Fig. 1. The former is obtained from the Weinberg-Tomozawa term, while the latter comes from the one-pion exchange Lagrangian. Due to the coupling of the charge to the vector potential, in the case of a constant magnetic background we need to compute separately those diagrams for different pion and nucleon charge eigenstates.
Formally, the self-energy can be defined from the pion Schwinger-Dyson equation [34], displayed in Fig. 2.

FIG. 2: Pion Schwinger-Dyson equation.
Here D0 is the free pion propagator and D is the full one. The diagram in the previous equation denotes the sum of all one-particle irreducible (1PI) diagrams. Q µ = (ω, q) is the pion four momentum.
Pionic modes of excitation in nuclear matter are obtained as solutions ω(q) of the following equation and in the limit of vanishing momenta this solution corresponds to the effective pion mass In absence of a magnetic background, it can be shown that the lowest-order (LO) contribution to the effective mass in (4) vanishes in isospin symmetric nuclear matter [35].
In asymmetric nuclear matter, the LO self-energy of the (negatively charged) pion receives a contribution from the Weinberg-Tomozawa diagram, given by [36] In the presence of a magnetic background, the pion charge eigenstates Eq. (4) have to be modified (due to the Landau level quantization) to where B is the magnetic field and n is the index of the Landau level.
In what follows we focus on the case of symmetric nuclear matter in the presence of a constant magnetic background. Thus, any deviation from zero of the LO pion self-energy will give a contribution to the effective pion mass in a magnetic background. Since we are dealing with dilute nuclear matter at low temperatures, we neglect the contribution of anti-nucleons. Moreover, we choose the x 3 −axis to be parallel to the magnetic field and |eB| = eB, e being the proton electric charge. In order to simplify the calculation, we assume the regime of strong magnetic fields, in which one can apply the lowest-Landau-level (LLL) approximation to simplify the propagators. We neglect the effect of the anomalous magnetic moment of protons and neutrons. The calculation is carried out in the Landau gauge.

III. PION SELF-ENERGY IN A CONSTANT MAGNETIC FIELD
For the negatively charged pion, the first diagram in Fig. 1 leads to the following contribution: where the first term is the proton loop contribution, which by using the Furry representation at finite temperature for the proton propagator [37] reads whereas the neutron loop contribution is not affected by the presence of the magnetic field In Eq. (8) n F (x) = (e x/T + 1) −1 is the Fermi distribution, E 3 = p 2 3 + m 2 , m being the proton mass, and the subscript L indicates that the vectors live in the twodimensional subspace defined by the time component and the space component that is aligned with the magnetic field, i.e. p L = (p 0 , p 3 ).
The WT self-energy for the positively charged pion will be just the opposite of Eq. (7). Finally, the π 0 does not receive any one-loop contribution from the WT interaction term. The contribution to the pion self-energy from the onepion exchange term in the Lagrangian (2) is, on the other hand, quite involved. For the charged pion one has that the two nucleons in the diagram correspond to two different isospin states, one being a proton and the other a neutron. For the proton we choose now a more convenient form for the propagator [38], which in the LLL approximation reads S (p) LLL (X, Y ) = dp 0 dp 2 dp 3 (2π) 3 The self-energy in the coordinate space is given by where Q µ and Q ′ ν are defined in the momentum space as the pion momenta whereΠ and Notice that, in Eq. (14), we used the fact that the δfunction and the on-shell condition for the pion lead to q 1 + q ′ 1 = 0. Since the pole structure of the propagators was not modified, the self-energy at finite temperature is obtained by replacing the time component of the four momenta by the appropriate (fermionic or bosonic) Matsubara frequency, i.e. P µ = (p 0 , p), with p 0 = ip n +µ, ip n = 2nπiT for the pion and ip n = (2n+1)πiT for the nucleon, n ∈ Z and T being the temperature.
Performing the sum over the fermionic Matsubara frequency, the retarded self-energy reads where E L pq = (p L − q L ) 2 + m 2 . Notice that in this case, due to the presence of a neutral field in the loop, one does not have the dimensional reduction which takes place in the case of the gluon self-energy in QCD [39][40][41]. This is instead the case for the π 0 , as we shall see in the following.
We separate the imaginary and real parts of the selfenergy in Eq. (16) by means of the Sokhotski-Plemelj formula. Since the imaginary part of the self-energy satisfies the hypothesis of the Kramers-Kronig dispersion relation, the real part can be computed as As before, the result for the positively charged pion will be the opposite as compared to the π − . The one-loop contribution to the neutral pion selfenergy is the sum of the proton and the neutron loops. The neutron loop is not affected by the presence of the magnetic field and vanishes in the limit of q → 0. The proton loop can be computed in a way that is very similar to the charged pion self-energy computation. The result reads where As already stated previously, in this case we recover the dimensional reduction, from (3+1) to (1+1), where the only spatial dimension that is relevant for the dynamics is determined by the direction of the magnetic field.

IV. RESULTS
We compute numerically the integrals appearing in Eqs. (7), (16) and (18). In the limit of q → 0 the only nonvanishing contribution to the self-energy is given by the WT contribution in Eq. (7). This is due to the fact that in symmetric nuclear matter the poles in Eq. (16) and (18) vanish for q → 0, leading thus to a vanishing self-energy. So, the neutral pion, whose self-energy comes only from the one-pion exchange term in the Lagrangian, has its mass unaltered as is also the case when no magnetic background is present. Fig. 3 shows the solution of Eq. (6) in the LLL approximation for the negatively charged pion, in the case of isospin symmetric nuclear matter, as a function of the density. The upper panel corresponds to the case of zero temperature. Clearly, the main contribution to the effective mass is given by the term proportional to |eB| in Eq. (6) that represents the trivial Landau quantization shift. The variation with the density is almost negligible. Nevertheless, one can notice that in the case of low magnetic fields the effective mass slightly increases with the density, while in the case of extremely strong magnetic fields the mass decreases. Therefore, if we would plot only the self-energy contribution to the effective mass, it would exhibit an appreciable drop for high magnetic fields. We should, of course, stress that the extremal values of the magnetic field magnitude shown in Fig. 3 might bring some inconsistency due to the fact that for too low fields one can not apply the LLL approximation, whereas for extremely strong fields one has to treat more carefully the chiral power counting. Yet, we believe that Fig. 3 illustrates clearly the qualitative trend as one comes from low to high magnetic fields. In the lower panel the effect of the temperature is included. The main features of the plot remain the same but, at equivalent values of the magnetic field, one can see that these curves lie above the corresponding curves at zero temperature. This is also shown in Fig. 4, in which we fix the density at nuclear saturation, ρ = 0.16 fm −3 , and vary the magnetic field. The dashed curves correspond to the zero temperature case, while the full lines to T = 50 MeV, and one can see that, for fixed density and temperature, the pion mass increases steeply with the external magnetic field.
As remarked previously, the result for very strong magnetic fields (|eB| ≥ m 2 π ) has to be considered as an extrapolation. Indeed, for such high values of the magnetic field, the chiral power counting does not hold anymore. Since the low energy scale of the theory (the pion mass) and the hard scale become comparable, one should take into account higher order diagrams that become, in principle, relevant in this case. Nonetheless, the trend seems clear from within the region of validity of our approach and connects smoothly to the regions of higher and lower fields, which is encouraging.
To unveil the role played by the real part of the selfenergy contribution to the mass of the π − , we compute its effective mass having subtracted the trivial shift due to the presence of the magnetic background (Landau quantization), namely we solve Fig. 5 displays our results for m π as a function of the magnetic field, and shows a significant decrease in the effective mass of the π − as a function of the magnetic field. This effect is, of course, diminished as the temperature is increased. The phenomenological motivation comes from physical systems with different charged particle species in the presence of moderately strong magnetic fields, as can be found e.g. in compact stars. In this case, one has to take into account the contributions coming from higher Landau levels, at least the first nontrivial ones. Thus, the trivial shift in the spectrum of different (charged) fermions and bosons will be of the same order (∼ |eB|), and the contribution that will be relevant for the behavior of the effective mass will possibly be determined essentially by the real part of the self-energy. In this context, we find that for the negatively charged pion the effec-tive mass is lowered by the presence of a strong magnetic background. Due to this feature, properties of a system involving dense nuclear matter and leptons, as can be found in compact stars and supernovae [33,42], might change significantly, depending on the behavior of the spectrum of charged fermions.

V. SUMMARY
We have investigated hadronic matter in the lowenergy sector in the presence of a strong magnetic field in the controlled setup of chiral perturbation theory. More specifically we have studied the modification of the pion self-energy at finite temperature due to its interaction with a low-density, isospin-symmetric nuclear medium in the presence of an external magnetic field.
To one loop, for fixed temperature and density, we found that the pion effective mass increases steeply with the magnetic field, a result that is enhanced when temperature is included, as expected. As a function of the density, on the other hand, the behavior of the effective mass is quite flat for different values of the field. However, even keeping in mind the caveat of our method when considering too low or too high fields, it seems clear that there is a qualitative change in the overall behavior: augmentation of the effective mass for low fields and depletion for high fields. The latter effect is, of course, hampered as we increase the temperature.
A subtle point that can play a relevant role in some actual physical systems, with different charged particle species in the presence of moderately strong magnetic fields, is the fact that the increase in the effective mass of the π − with the magnetic field is due solely to the trivial Landau quantization shift ∼ |eB|, since the real part of the self-energy is negative in this case. As was shown in the previous section, if we subtract the former trivial effect, the effective mass of the negatively charged pion drops considerably with the magnetic field. If all charged particles undergo an approximately equivalent trivial shift of this sort, the modifications that may be relevant, phenomenologically, are those brought about by the real part of the respective self-energies. In this case, we find that the pion mass decreases by ∼ 10% for a magnetic field |eB| ∼ m 2 π , which favors pion condensation at high density and low temperatures. Such scenario may take place in neutron star matter and supernovae and requires further investigation.

Acknowledgments
The work of GC was supported by the HGS-HIRe graduate program at Frankfurt University. The work of ESF was financially supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse. GC thanks S. Benić for useful discussions.