Neutral and charged (pseduo)scalar mesons and diquarks under magnetic fields

We investigate both (pseudo)scalar mesons and diquarks in the presence of external magnetic field in the framework of the two-flavored Nambu--Jona-Lasinio (NJL) model, where mesons and diquarks are constructed by infinite sum of quark-loop chains by using random phase approximation. The polarization function of the quark-loop is calculated to the leading order of $1/N_c$ expansion by taking the quark propagator in the Landau level representation. We systematically investigate the masses behaviors of scalar $\sigma$ meson, neutral and charged pions as well as the scalar diquarks, with respect to the magnetic field strength at finite temperature and chemical potential. It is shown that the numerical results of both neutral and charged pions are consistent with the lattice QCD simulations. The mass of the charge neutral pion keeps almost a constant under the magnetic field, which is preserved by the remnant symmetry of QCD$\times$QED in the vacuum. The mass of the charge neutral scalar $\sigma$ is around two times quark mass and increases with the magnetic field due to the magnetic catalysis effect, which is an typical example showing that the polarized internal quark structure cannot be neglected when we consider the meson properties under magnetic field. For the charged particles, the one quark-antiquark loop contribution to the charged $\pi^{\pm}$ increases essentially with the increase of magnetic fields due to the magnetic catalysis of the polarized quarks. However, the one quark-quark loop contribution to the scalar diquark mass is negative comparing with the point-particle result and the loop effect is small.

issue of QCD, which is deeply related to hadrons' properties, such as the pion mass and the pion decay constant via the GellCMann-Oakes-Renner (GOR) relation [40]. It means that, exploring the modification of hadrons' properties in the magnetized hot and/or dense medium, will help to understand the effects of magnetic fields on the chiral phase transition at finite temperature and chemical potential. Thus, the behaviors of the mass spectrum and weak decay constant of pions have then been extensively studied recently [19,21,[35][36][37][38][39]. However, most of them only focused on the neutral pion due to the difficulty of treating different charged quark propagators under magnetic fields. Therefore, in this paper, we will study not only the pseudoscalar neutral pion and scalar σ meson but also the charged pions in the two-flavored NJL model under magnetic fields. In addition, since the mass generation of nucleon is also an important feature of dynamical chiral symmetry breaking, we further analyze the scalar color3 diquark channel in the magnetic field, which will help to probe the properties of nucleons in the magnetized medium in the future. This paper is organized as following: in the next section, we give a general expression of the two-flavor NJL model including the scalar and pseudo-scalar channels, and then derive the polarization functions of both (pseudo)scalar meson channels and the scalar color3 diquark channel in a magnetic field at finite temperature and chemical potential. In Sec. III we show our numerical results and analysis. Finally, the discussion and conclusion part is given in Sec. IV.

II. FOMALISM
A. The two flavor NJL model in an external magnetic field We investigate the (pseudo)scalar mesons and the diquark in the scalar channel by using a low energy approximation of QCD effective model SU(2) NJL model under an external magnetic field, the Lagrangian is given by [41,42] L =ψ(i D −m + µγ 0 )ψ + G S (ψψ) 2 + (ψiγ 5 Where ψ represents the quark field of two light flavors u and d, the current mass matrixm = diag(m u , m d ), we can assume that m u = m d = m 0 . τ a = (I, τ ) with τ = (τ 1 , τ 2 , τ 3 ) corresponding to the isospin Pauli matrix. G S is the coupling constants corresponding to the (pseudo)scalar channel. The covariant derivative, D µ = ∂ µ − iQeA ext µ , couples quarks to an external magnetic field B = (0, 0, B) along the positive z direction via a background field, for example, A ext µ = (0, 0, Bx, 0) and the field strength tensor is defined by F µν = ∂ [µ A ext ν] , Q = (−1/3, 2/3) is a diagonal matrix in the flavor space which respects to the electric charge of the quark field (u, d).
Semi-bosonizing the above Lagrangian and the Eq.(1) can be rewritten as where the Euler-Lagrange equation of motion for the auxiliary fields leads to the constraints as follows: For each flavor, by introducing the chirality projector P L = 1+γ 5 2 and P R = 1−γ 5 2 , we have and for two-flavor spinorψ = (ūd), the auxiliary neutral scalar and pseudo scalar σ and π 0 can be written as follows: It is obviously to find that the neutral scalar σ is symmetric in the flavor space, while the neutral π 0 is anti-symmetric in the flavor space and keeps as a pseudo Goldstone mode. The charged π ± can be represented as: with τ ± = 1 √ 2 (τ 1 ± iτ 2 ). The quark-antiquark condensation which gives quark the dynamical mass and the constituent quark mass is obtained as We should minimize the effective potential in order to obtain the dynamical quark mass, i.e., the σ condensation. The one-loop effective potential in this model is given as follows: where β = 1 T , α p = 2 − δ p,0 represents the spin degeneracy.

B. The scalar meson σ
In the framework of the NJL model, the meson isqq bound state or resonance, it can be obtained from the quarkantiquark scattering amplitude [43,44]. The meson is constructed by summing up infinite quark-loop chains in the random phase approximation(RPA), the quark loop of the σ meson polarization function is calculated to the leading order of 1/N c expansion, the one-loop polarization function of σ meson Π σ (q ⊥ , q || ) in the magnetic field takes the form of [45] where q ⊥ = (0, q 1 , q 2 , 0), q || = (q 0 , 0, 0, q 3 ). p = k + q σ corresponds to the momentum conservation and the Landau level representation of the quark propagator S(k) is given by [13,46] with Here, L α n are the generalized Laguerre polynomials and L n = L 0 n . In the rest frame of σ meson, i.e., q µ σ = (M σ , 0), by using the quark propagator in the Eq. (14), the polarization function of σ is given as follows, Here, we have used the relations Moreover, where the functions I 1 and I 2 are given in appendix A. We can get the mass of σ meson by solving the gap equation [45] 1 C. The pseudoscalar meson π For neutral π 0 meson, the one-loop polarization function is Similarly, p = k + q π 0 and in the rest frame of the π 0 meson, i.e., q µ π 0 = (M π 0 , 0), the Π π 0 is written as For charged π ± meson, the one-loop polarization function is (the notation is only for π + meson, and there is a similar notation for π − meson) where p = k + q π + . In the rest frame of π + meson, the polarization function is given by Here, the functions I ′ 1 , I ′′ 1 and I ′ 2 are represented in appendix A. Because of electric charge difference, we could not get a simple form as in Eq.(17) for σ meson, so we perform the numerical calculation for the integral of k ⊥ . Also we will do the same operation in the diquark case.
Similarly, we use the following gap equation to obtain the mass of π meson D. The diquark in the scalar channel For the diquark channel, the interaction term in the Lagrangian L I consists the term of the form (ψAψ T )(ψ T Bψ) and the A,B are matrices antisymmetric in Dirac, isospin and color indices. Here we only consider the scalar channel in the color3 channels, the interaction term L I,D is given by [48][49][50] with ψ C = Cψ T ,ψ C = ψ T C are charge-conjugate spinors, C = iγ 2 γ 0 is the charge conjugation matrix (the superscript T denotes the transposition operation), the quark field ψ ≡ ψ iα with i = 1, 2 and α = 1, 2, 3 is a flavor doublet and color triplet, as well as a four-component Dirac spinor, (ε) ik ≡ ε ik , (ǫ b ) αβ ≡ ǫ αβb are totally antisymmetric tensors in the flavor and color spaces. In this work, we choose G D = 3 4 G S . Then we introduce the auxiliary diquark fields ∆ b and ∆ * b which are color antitriplet and (isoscalar) singlet under the chiral SU (2) L × SU (2) R group. The gap equation for the diquark in the color antitriplet scalar channel is given by [48] 1 where By introducing the functions I ′ 1 , I ′ 2 defined in Appendix, we can express the diquark loop polarization function as: Here, p = k + q ∆ b and we assume q µ ∆ b = (M ∆ b , 0) (i.e., in the rest frame of diquark).

III. NUMERICAL RESULTS
In the numerical calculation, we use the soft cut-off functions [47] f Λ = Λ 10 Λ 10 + k 2 * 5 , for zero magnetic field and nonzero magnetic field respectively. In order to reproduce the pion mass M π = 140 MeV, pion decay constant f π = 92.3 MeV, constituent quark mass M = 336 MeV in the vacuum, we choose the following model parameters: Λ = 616 MeV, G S Λ 2 = 2.02 and the current quark mass m 0 is 5 MeV. In Fig.1, we show the magnetic field dependence of quark mass with different fixed temperatures and chemical potentials. We can see that the quark mass increases when the magnetic field strength increases, which is the magnetic catalysis effect obviously. It is noticed that in the regular NJL model, there is no mechanism for the inverse magnetic catalysis around the critical temperature region, therefore, all results in this work are taken below the critical temperature.
A. Numerical results for neutral σ and π 0 We firstly check the mass behavior of charge neutral σ and π 0 under magnetic fields. The numerical result for the mass square of the σ meson is shown in Fig.2, and the constant mass of the pointparticle model is also listed for comparison. It is observed that when the quark-loop polarization under magnetic fields is considered, similar to the behavior of the quark mass, the mass of the σ meson increases with magnetic field for different temperatures and chemical potentials. Comparing with the constant mass of σ meson as a point-particle, we find that the quark-loop contribution or quark polarization effect is very essential. On the other hand, at zero temperature and below the critical chemical potential, we find that the chemical potential does not affect the mass of the σ meson, and for µ = 0 MeV case, the mass of σ meson decreases when temperature increases at fixed magnetic field. In Fig.3, we show the magnetic field dependence of the mass square for neutral pion π 0 , and compare with the result from the point-particle model. The neutral pion mass keeps as a constant as a function of the magnetic field in the point-particle model. It is observed that in the vacuum when T = 0, µ = 0, the quark-loop contribution has tiny effect on neutral pion mass, which almost keeps a constant with the increasing of the magnetic field. This is preserved by the remnant symmetry and π 0 is the only pseudo Goldstone boson in the system. But even it is a tiny effect, we still can see that the π 0 mass decreases a little firstly and then increases a little with the magnetic field, which is in agreement with the lattice result in Ref. [33]. At zero temperature when T = 0 MeV, the mass of π 0 meson does not change with different chemical potentials in the chiral symmetry breaking phase, and at zero baryon density, the increasing temperature lowers the mass of neutral π 0 from vacuum mass 140MeV to almost 0 around the critical temperature.
In the point-particle model, the mass of a neutral meson either for scalar σ or peudo-scalar π 0 should keep a constant as the function of the magnetic field. However, when the polarized quark-loop effect is considered, the results for scalar σ or peudo-scalar π 0 are quite different. Similar to the quark mass, the mass of neutral scalar σ linearly rises as the magnetic field and one has M σ (eB) ≃ 2M (eB). However, as the only pseudo Goldstone meson, the neutral π 0 mass keeps as an almost constant value, which is preserved by the remnant symmetry of the system. Then we analyze the mass behavior for charged π ± with electric charge ±1 and ∆ b with electric charge 1/3 under the magnetic field.
By solving the gap equation Eq.(24) numerically, the mass square of charged π + meson is given in Fig.4 where we also list the result of the point-particle model. The mass of charged π + increases with increasing magnetic field in different temperature T and chemical potential µ. The chemical potential µ has no effect on the mass of the charged π + with zero temperature in the chiral symmetry breaking phase, the increasing temperature shows tiny effect on the mass of the π + meson. Obviously, the quark-loop contribution plays an important role by comparing with the results of the one-loop polarization function and the point-particle model. We also investigate the mass square of scalar diquark in color anti-triplet channel in magnetic field with different temperature and chemical potential in Fig.5. The mass of scalar diquark increases with the magnetic field. At zero temperature, the chemical potential affects the mass of diquark in the scalar channel slightly. Moreover, at zero chemical potential, the mass of diquark in the scalar channel decreases with the increasing temperature. Generally, the one-loop contribution has important effect on the mass of scalar diquark. However, it is important to notice that different from charged π ± , including the quark-loop contribution, the mass of the scalar diquark increases more slowly with magnetic fields comparing with the point-particle model result when eB < 0.9 GeV 2 , and only at high magnetic field when eB > 0.9 GeV 2 , the mass of scalar diquark increases faster with magnetic fields comparing with the result of the point-particle model.

C. Quark-loop contribution
It is observed that except for the charge neutral π 0 , which is the only pseudo Goldstone boson of the system preserved by the remnant symmetry of QCD×QED, for all other scalar mesons including the charge neutral σ, ±1 charged pseudo scalar π ± and 1/3 charged scalar diquark ∆ b , their masses including the one quark-antiquark loop contribution under magnetic fields are quite different from the point-particle results. We explicitly show the one quark-loop contribution to σ, π ± and ∆ b as the function of the magnetic field at T = 0, µ = 0 in Fig.6. The one quark-loop contribution to the charge neutral σ and charged π ± increases essentially with the the increase of magnetic fields due to the magnetic catalysis of the polarized quarks. At strong magnetic fields, when eB > 0.5GeV 2 , the one quark-loop contribution to the mass of charge neutral σ meson becomes more essential than the charged π ± , and at eB = 1GeV 2 , the one quark-loop contribution to the mass of charge neutral σ meson can reach 8 times of the point-particle results. However, the one quark-quark loop contribution to the scalar diquark mass is negative and less than 50% of the point-particle result below eB < 0.9GeV 2 .
FIG. 6: The one quark-loop contribution to the mass of the σ, π + and the diquark ∆ b in the scalar channel.

IV. DISCUSSION AND CONCLUSION
In this paper, we have studied the masses of charge neutral scalar σ and pseudo-scalar π 0 , charged pseudo-scalar meson π ± and scalar diquark ∆ b in an external magnetic field in the NJL model, and the mesons and diquark are constructed by summing up infinite quark-loop chains in the random phase approximation (RPA), the quark-loop of mesons polarization function is calculated to the leading order of 1/N c .
It is found that the mass of the charge neutral π 0 keeps almost constant under the magnetic field and the one quarkloop contribution to π 0 is tiny. This is because that π 0 is the only pseudo Goldstone boson of the system preserved by the remnant symmetry of QCD×QED. From the point-particle model, the mass of the charge neutral scalar σ should keep as a constant value under the magnetic field, however, considering the one quark-loop contribution, the σ mass is around two times quark mass and increases with the magnetic field due to the magnetic catalysis effect. This is an typical example showing that the polarized quark structure cannot be neglected when we consider the meson properties under magnetic field, and the quark-loop contribution to σ mass plays dominant role at strong magnetic field, e.g., the quark-loop contribution to σ mass can reach 8 times of the point-particle results at eB = 1GeV 2 .
For the charged particles, the contribution from the one quark-loop are quite different to the masses of pseudoscalar π ± and scalar diquark ∆ b under magnetic fields. The one quark-antiquark loop contribution to the charged π ± increases essentially with the the increase of magnetic fields due to the magnetic catalysis of the polarized quarks. However, the one quark-quark loop contribution to the scalar diquark mass is negative and less than 50% of the point-particle result below eB < 0.9GeV 2 .
As mentioned in the Introduction that one of the motivations of this work is to check whether the method we used to calculate the charged vector meson [23][24][25] is correct. From the results of pseudo-scalar π by using the same method, we are confident that the results of charged vector meson in [23][24][25] is correct at least in the framework of the NJL model. Firstly, we have reproduced the result for the charge neutral π 0 , its mass is almost a constant under the magnetic fields, and this is preserved by the remnant symmetry and π 0 is the only pseudo Goldstone boson in the system. Secondly, the charged pion mass increases with the magnetic field, which is in agreement with lattice results in [33,34].
At last, we have to mention that when this paper is almost finished, there is a similar paper published [51], where the authors have pointed out that the method we are using in [23][24][25] only take the translation invariance part of the Schwinger phase in quark propagator. In fact, we only keep the first order of local expansion for the charged field and the phase in our paper, which leads to a cancellation of the translation variance part. Compare the result in both papers, we can see, the first-order expansion already gives a very good result and physical quantities.