Neutral pion mass in the linear sigma model coupled to quarks at arbitrary magnetic field

We calculate the neutral pion mass in the presence of an external magnetic field of arbitrary strength in the framework of the linear sigma model coupled to quarks at zero temperature. We find nonmonotonic behavior of the pion mass as a function of magnetic field. We are also able to reproduce existing results for the weak-field approximation.


I. INTRODUCTION
In heavy-ion collision (HIC) experiments, a very strong anisotropic magnetic field (∼ 10 19 G) is generated in peripheral collisions due to the relative motion of the colliding ions [1][2][3][4][5]. The direction of the generated magnetic field is perpendicular to the reaction plane. Apart from HIC experiments, finite magnetic fields also involved in the interior of dense astrophysical objects like compact stars, magnetars [6], and in the early Universe. The effects of such magnetic fields on fundamental particles cannot be neglected and a detailed understanding of their effects on elementary particles is essential.
One such effect is the behavior of meson masses as a function of the strength of the magnetic field. The study of magnetic -field-dependent meson masses is the subject of active research. In Ref. [7], the authors studied the masses of light mesons -namely, the charged and neutral pions (π ± , π 0 ) and rho mesons (ρ ± , ρ 0 ) in the presence of an external electromagnetic field in the framework of lattice quantum chromodynamics (LQCD) and showed that the magnetic-field-dependent neutral pion mass decreases with the magnetic field strength.
The linear sigma model (LSM) is one of the oldest and simplest model in pre-QCD era, and was originally proposed by Gell-Mann and Lévy [28] to describe pionnucleon interactions. Many global symmetries of QCD are exhibited in the LSM. Later, this simple model was also used to study chiral phase transitions [29], magnetic and thermomagnetic corrections to the π-π scattering length [30,31], pion condensates in the presence of a magnetic field [32], and many more.
The addition of light quarks to the LSM Lagrangian density has given a new dimension to the existing model. It is called the linear sigma model coupled to quarks (LSMq). The study of QCD phase diagrams has been carried out within the framework of the LSMq in Refs. [33][34][35]. In Ref. [36], the authors not only studied the thermodynamics of the QCD phase diagram and chiral transition, but also compared it with that obtained from the NJL model. Also, the LSMq was recently used to study, the magnetized QCD phase diagram [37] and inverse magnetic catalysis [38].
In Ref. [39] the authors studied the magnetic-fielddependent neutral pion mass within the weak magnetic field approximation. In this article, we generalize the calculation which is valid at any value of the magnetic field strength.
The paper is organized as follows. In Sec. II, we review the LSMq with an explicit symmetry-breaking term to account for the nonzero pion mass. In Sec. III we compute the magnetic field correction of the neutral pion self-energy at one-loop order that comprises the quarkantiquark contribution (Sec. III A), the charged meson contribution (Sec. III B). In Sec. IV we compute the neutral pion mass by solving the pion dispersion relation, which is the sum of the quark-antiquark and meson contributions. In Sec. V the pion mass in the weak-field limits is investigated and finally we conclude in Sec. VI.

II. LINEAR SIGMA MODEL COUPLED TO QUARKS
The Lagrangian density for the LSMq reads arXiv:1908.10323v2 [hep-ph] 28 Apr 2020 The first four terms on the rhs of Eq. (1) constitute the LSM part and the last two terms are the quark part of the Lagrangian density. Here π = (π 1 , π 2 , π 3 ). The charged and neutral pion fields are defined as respectively, σ is the sigma meson of the LSM, and ψ is the light quark doublet with is the i th Pauli spin matrix. Also, a 2 is the mass parameter of the theory but, unlike in the usual convention, we take a 2 < 0 in the symmetry-unbroken state. Finally, λ is the coupling within σ-σ, π-π, and σ-π, g is the coupling between the degrees of freedom (d.o.f) of the LSM and those of the quarks. When a 2 > 0, the O(4) symmetry of the Lagrangian is spontaneously broken and the σ field gets a nonzero vacuum expectation value (VEV). After the symmetry breaking, σ becomes where v is the VEV developed by the σ field. The Lagrangian density is written after the shift as follows: with The masses of the quarks, three pions, and sigma are given by Note that the minimum of the tree-level potential, obtained by solving dV tree (v) dv v=v0 = 0, is given by Therefore, the masses after symmetry breaking, evaluated at v 0 , are given by To incorporate a nonvanishing pion mass into the model, we add a small explicit symmetry-breaking term to the Lagrangian, with m π = 0.14 GeV. As a result, the tree potential V tree becomes V tree = − 1 2 (a 2 +m 2 π )v 2 + 1 4 λv 4 and the minimum is shifted to The masses, evaluated at this new minimum v 0 , are given by The value of a is given by solving Eq. (14) as We consider a time-independent homogeneous background magnetic field in the z direction B = Bẑ, which can be obtained from the symmetric four-potential A µ = B 2 (0, −y, x, 0). As a result, for the charged d.o.f (quarks and charged pions), the four-derivative ∂ µ is replaced by covariant four-derivative D µ = ∂ µ + iQA µ .
Here, Q = q f for quarks of flavor f and Q = e for π ± , respectively.

III. ONE-LOOP PION SELF-ENERGY
The following four terms contribute to the neutral pion self-energy: The one-loop diagram for the quark-antiquark contribution Π f f (B, P ) is depicted in Fig. 1, whereas that for the charged pion contribution Π π ± (B) is depicted in Fig. 2. Note that for the last two terms [Π π 0 ,σ ], there are no magnetic corrections as the particles in the loop are charge neutral. In the next section, we compute the first two contributions to the self-energy in Eq. (16).
In the presence of a magnetic field, the expression for the pion self-energy reads where S B f (K) is the charged quark propagator in momentum space in Schwinger's proper-time representation. It is given by where sgn is the sign function. We follow the following notations and conventions: • Four-vectors are denoted by capital letters.
• Also, for any four-vector A µ , we use the standard Feynman slash notation / A to indicate γ µ A µ .
After calculating the trace over Dirac matrices, Eq. (17) becomes where Q = K −P . Now, to carry out the four-momentum integration in Eq. (19), we switch to Euclidean spacetime by the usual replacement k 0 → ik 0 E and also with the substitution (s → −is, t → −it) as in Ref. [40]. The subscript E stands for momentum components in Euclidean spacetime. Now, we get Note that three-momentum in Minkowski and Euclidean space are the same. We also used the identities tan(−ix) = −i tanh(x) and sec(−ix) = sech(x). The momentum integration in Eq. (20) can be performed analytically: where Thus, we are left with only two proper-time integrations in the expression of Π ff : Since we are interested in calculating the pion mass modification given by Eq. (44), we take the limit p → 0 of Eq. (28) to get In the vanishing magnetic field limit, Π ff (B, p 0 E ) becomes After rotating back to Minkowski spacetime, Eq. (30) can also be expressed in terms of a four-momentum integration as Equation (31) is UV divergent and can be computed using the Feynman parametrization. The integrand can be regularized using the MS scheme and the result will contribute to the renormalized vacuum pion mass. As we are interested in the magnetic correction to the pion mass, we define the Vacuum part subtracted self-energy as Feynman diagram for the one-loop charged pion contribution to the π 0 self-energy The tadpole diagram, shown in Fig. 2, reads where D (B) (k) is the charged pion propagator in the presence of a magnetic field, given by We go to Euclidean spacetime and get where J (0) and J (0) ⊥ are given by The momentum integrations in Eqs. (36) and (37) can be performed analytically to obtain Plugging Eqs. (38) (39) into Eq. (35), we get For small eB, we can series expand the integrand in Eq. (40) around eB = 0 and carry out the integration over the proper time s. The terms with odd powers of eB do not appear since Π π± (B) is even in |eB|. In the vanishing magnetic field limit, Π π± (B) becomes Equation (41) can also be expressed in terms of an integration over four-momentum from Eq. (33) taking eB → 0 as Equation (42) is divergent and can be renormalized using the MS scheme and contribute to the renormalized vacuum pion mass. As in Sec. III A, we define the vacuum part subtracted self-energy as

IV. PION MASS
In order to compute the modified pion mass M π (B) in the presence of a magnetic field, we need to solve the equation in the limit p → 0 and p 0 = M π (B). The self-energy of π 0 has four contributions as mentioned in Eq. (16). Now, Π f f (B, P ) and Π π± (B) were calculated in Secs. III A and III B, respectively. But Π π 0 (P ) and Π σ (P ) do not receive any magnetic field corrections since the particles in the loop are neutral. So, the total self-energy Π(B, p 0 , p = 0) is obtained from Eqs. (29) and (43) as We can make a change of variable in Eq. (45) from (s, t) to (u, v) (as done in Refs. [40,41]) which leads to Using Eq. (47), the solution of p 0 from Eq. (44) gives the transcendental equation for the magnetic-field-dependent neutral pion mass as Equation (48) can be solved numerically to find the magnetic-field dependent pion mass, shown by the blue line Fig. 3. As is clear from the Fig. 3, the magnetic-field-dependent neutral pion mass increases with the field strength. But the calculation is not complete yet: we also need to incorporate the one-loop magnetic-field correction to the fermion coupling g, the boson self-coupling λ and v 0 .
So, the effective fermion mass becomes where v B 0 represents the magnetic-field-dependent minimum of the potential after symmetry breaking and g eff is the one-loop effective fermion vertex in the presence of the magnetic field.
Using Eq. (49) and replacing g eff with the other effective quantities, Eq. (48) becomes The effective magnetic-field-dependent quantities, namely, v B 0 , λ eff and M f,eff are calculated in Appendices A, B and C, respectively. The solution for M π (B) from Eq. (50) for fixed values of m σ and m π are plotted in the left and right panels of Fig. 4, respectively. Figure 4 shows the nonmonotonic behavior of neutral pion mass with magnetic field. It decreases with increasing magnetic field for a weak magnetic field in agreement with the Ref. [39]. But for large values of magnetic field, the neutral pion mass starts to increase. This could be due to the fact that the linear sigma model coupled to quarks is best at mimicking the low-energy region of QCD, but in the high-energy region (such as for large magnetic fields) this model is not reliable. The coupling constants λ = 0.86 and g = 1.11 are taken from Ref. [42].

V. ASYMPTOTIC SOLUTION
It is possible to find analytic expressions in asymptotic limits such as the weak-field limit as done in Ref. [39]. Expanding the integrand in Eq. (29) in powers of q f B and performing the u and v integrations analytically order by order in q f B, we get the vacuum -subtracted contribution from quark-loop in weak field as where M w f,eff represents weak-field expression of the quark mass up to O(eB) 2 . The coefficient of O(eB) 2 in Eq. (51) is different than that obtained in Ref. [39] as the authors of Ref. [39] used an approximation in which they considered p E 0 = 0. If we make the same approximation, we get the vacuum-subtracted contribution from the charged pion loop in weak field as This result matches that in Ref. [39] at O[(q f B) 2 ]. In a similar manner, in the weak magnetic field limit the charged pion tadpole diagram is obtained by expanding Eq. (43) in the small-eB limit and doing the proper-time integration, which gives Note that the O(eB) 2 term matches that in Ref. [39]. In weak-field limit, the transcendental Equation (47) can be solved for M 2 π (B) and we get where we have replaced M w f,eff with M f as the magnetic field correction in M w f,eff in the weak-field limit is O (eB) 2 . We have also approximated the magnetic-field-dependent minimum of the potential as v B 0 ≈ v 0 in Eq. (54). Using λ w eff from Eq. (B15), Eq. (54) becomes

VI. CONCLUSION AND OUTLOOK
In conclusion, we have studied modifications of neutral pion mass in the presence of an external background magnetic field under the framework of the LSMq. Our calculation is valid for weak to moderate external background magnetic fields. We also obtained an asymptotic solution of the dispersion equation of the neutral pion mass in the weak magnetic field limit. The calculation was carried out by taking into account the self-coupling of pions as well as the effective one-loop quark in the presence of a magnetic field. It matches that of Ayala et al. [39] in the low−|eB| regime. But, when the strength of the magnetic field is increased, we get a nonmonotonic behavior, unlike in Ref. [39]. Up to a moderate value of the magnetic field, our result also qualitatively agrees with LQCD studies as [7] in that the pion mass decreases with increasing magnetic field for low |eB|. Additionally, it is clear from Fig. 5 that the pion mass in a weak magnetic field as given in Eq. (55) is a good approximation for eB m 2 π . Moreover, the corrected weak-field result from Ref. [39] [as given in Eq. (B18)] is more or less the same as our weak-field pion mass [Eq. 55], which was obtained without approximating p 0 = 0 in the dispersion relation in Eq. (44) up to |eB| 0.85 m 2 π . For |eB| 0.85 m 2 π , the corrected weak-field expression [as given in Eq. (B18)] starts to deviates and our weak-field expression is more closer to the exact value of the magnetic-field-dependent pion mass.
Looking to the future, the present calculation can be extended to astrophysical objects, where the baryon density and magnetic field are very high. We can also extend the present calculation at finite temperature, which could be interesting for heavy-ion physics. Finally, one should keep in mind that the LSM is an effective model and it has a scalar degree of freedom (called sigma meson) which has not been confirmed yet but is still under consideration after the identification of f 0 (500) with J P C = 0 ++ . The nonlinear sigma model(NLSM) was proposed as an extension of the LSM long ago [43,44], where the σ field is removed by sending its mass to infinity and redefining the pion field Φ(x) by U (x) = exp(iτ · Φ(x)/f π ); the study of light mesons in the presence of a magnetic field in the NLSM might be interesting. Nevertheless, using the LSMq model, we can qualitatively capture the essential features obtained by much more involved and rigorous studies. The one-loop effective potential is where the tree-level potential after symmetry breaking is The one-loop effective potential for the boson fields is The one-loop effective potential for the fermion fields is After calculating the integrals, the renormalized potentials become and where ψ m (z) represents the derivative of the logarithm of the gamma function as ψ m (z) = d m+1 dz m+1 ln Γ(z). The total effective potential can be obtained from Eq. (A1) by adding the individual contributions. In the left panel of Fig. 6 we plot the effective potential as a function of the VEV v. It is clear from the figure that the position of the minimum varies with v. In the right panel of Fig. 6 we plot the scaled VEV as a function of the magnetic field.

Appendix B: Effective boson self-coupling vertex
In this section, we calculate the boson self-coupling up to one-loop order, as shown in the diagram in Fig. 7.
The effective vertex up to one-loop order is written as where ∆λ is given by [38] ∆λ = 24λ with i = π 0 , σ and j = π + , π − . Equations (B3) and (B4) are one-loop contributions from neutral and charged boson fields, respectively. Here D i (K) is the neutral boson propagator, and the charged boson propagator is given by Eq. (34). Now, I(B, P, m i ) does not contribute to the magnetic field corrections to the boson self-coupling, but J(B, P, m j ) does. Converting Eqs. (34) and (B4) to Euclidean space, we have where I (0) and I . (B7) In the limit that the three-momentum p → 0 and after changing variables from (s, t) to (u, v), Eq. (B7) is further simplified to In the vanishing magnetic field limit, Eq. (B8) becomes Equation (B9) can also be expressed in terms of the momentum integration from Eq. (B6) as which is UV divergent and can be regularized using the MS scheme. The regularized vanishing magnetic field contribution to the vertex will only contribute to the zero-magnetic-field pion mass. As before, we define the vacuum-subtracted self-interaction vertex as Thus, from Eq. (B2) we see that the term ∆λ receives corrections from the magnetic field through J only, and it is given by which leads to the one-loop neutral pion self-coupling vertex, In the weak magnetic field limit, the integrations over u and v in Eq. (B11) can be done analytically as and we can write the weak magnetic field effective vertex as In the limit p 0 It is worth mentioning here that the first term in Eq. (B16) differs from the corresponding expression in Ref. [39]. We have checked the calculation in Ref. [39] and we are able to get our expression as given in Eq. (B16). In Ref. [39], there is a mistake during the evaluation of the integral in Eq. (C6) over the Feynman parameter x. So, in light of assumptions that were used in Ref. [39], the correct expression for the self-coupling would be which leads to the following expression for the effective mass: Expanding the matrix structure in Eq. (C2) and simply writing it in terms of the LSM bosons (π + , π − , π 0 , and σ) and constituent light quark (u and d) fields, we get The contributions to the self-energy from u and d quarks are given in Figs Fig. 8(g)]. This leads to Each of the contributions is written separately as follows Now, using the anticommutator γ 5 γ µ + γ µ γ 5 = 0, the term γ 5 S (B) f (K)γ 5 is simplified as Substituting Eqs. (C6)-(C11) into Eqs. (C4) and (C5), we get We make the change of variable K µ → −K µ in the second and third terms on the right-hand sides of both the Eqs. (C13) and (C14) and get The propagators of the π 0 and σ fields are written in terms of Schwinger's proper-time parametrization as Before diving into explicit calculations using Feynman diagrams (Fig. 8), we note that the Dirac structure of the quark propagators along with its inverse (both bare and effective) and self-energies (which involve linear combinations of / P , 1, / P γ 1 γ 2 , γ 1 γ 2 and / P ⊥ ) are the same, but the structure coefficients will be different. Following the aforementioned argument, we write down the structure of S * f −1 as Now we proceed to compute the momentum integrals in the expression of self-energy for a particular flavor f which has three different channels, namely, the σ, π 0 , and π ± channels: The inverse of the propagator S −1 f (P ) has the same structure but different coefficients, and they are related to those of S f (p): where the "in" subscript in the above set of equations refers to the structure coefficients corresponding to Since we are only interested in the magnetic field corrections, we subtract from Eqs. (C22)-(C26) their vacuum counterparts. As a result of this, we have the following structure coefficients of S * f −1 (P ): The hat symbol indicates a vacuum-subtracted structure coefficient of the constituent quark self-energy. The inversion process is shown explicitly in Appendix C 2. To achieve the inversion of Eq (C18), we first have to note the number of Dirac matrix structures involved in the expression. It has five tensor structures: / P , / P ⊥ , / P γ 1 γ 2 , γ 1 γ 2 , and 1. The trick is to reduce the number of matrix structures as follows.
Suppose we have any matrix M whose inverse is desired. We multiply M by some matrix R of our choice and as a result we get some other matrix called U: Note that we have to choose R in such a way that matrix structure of U is simple and U −1 can be found easily. So, from Eq. (C47), we get Thus, the essence of this technique lies in properly choosing R.
In our case, we can identify S * f −1 as M and choose R as According to Eq. (C47), we have where we have defined P µ ≡ 1 B F µν P ν = (0, p 2 , −p 1 , 0) (C51) with F µν denoting Maxwell's electromagnetic fieldstrength tensor (F 21 = −F 12 = B and zero for others) in our background field configuration and The inversion of U is easy to perform and it is given as Finally, we get S * f as The one-loop effective quark mass M f,eff in the presence of a magnetic field is obtained by solving for the denominator of Eq. (C54), i.e., for p 0 in the static limit i.e., p → 0.