Functional renormalization group study of neutral and charged pion under magnetic fields in the quark-meson model

We calculated the masses of neutral and charged pion and pion decay constants under an extra magnetic field at zero temperature. The quantum fluctuations are integrated through the functional renormalization group. We consider the quark and meson propagators in the Landau level representation and weak-field expansion, respectively. The neutral pion mass monotonically decreases with the magnetic field, while the charged pion mass monotonically increases with the magnetic field. The pion decay constant and the quark mass show the magnetic catalysis behavior at vanishing temperature. The neutral pion mass and pion decay constant are quantitatively in agreement with the lattice QCD results in the region of $eB<1.2 {\rm GeV}^2$, and no non-monotonic mass behavior for charged pion has been observed in this framework.

It is also valuable to study the meson spectrum of QCD under magnetic fields, which plays an important role in the understanding of the rich phenomena mentioned above.It is believed that the neutral pion is helpful to explain the inverse magnetic catalysis [43,44], and the charged pions can explain the diamagnetic around the pseudo-critical temperature [38].The meson spectra have been widely studied in lattice QCD and effective models [16][17][18][45][46][47][48][49][50][51].Recent Lattice calculation in [20] showed that at zero temperature, the mass of the neutral π meson decreases monotonously with the magnetic field, while that of the charged pions shows a non-monotonic behavior.Some efforts have been made to understand the pion mass behavior under magnetic field in low energy effective models [52][53][54][55][56].However, the mass behaviors of the neutral and charged pions under magnetic field have not been explained simultaneously.Besides, the lattice and effective model calculations are also extended to finite temperatures, see e.g., [21,[57][58][59][60].
In this work, we employ the quark-meson model, which is also called the linear sigma model coupled to quarks (LSMq) [29,61] to calculate the meson masses and decay constants under a magnetic field.This model is well used to study the QCD phase diagrams [62,63], Equation of State (EoS) [64,65] as well as the fluctuations of conserved charges [66,67].Note that it can be transformed from the NJL model through a Hubbard-Stratonovich transformation [68,69].The results of the mean-field approximation of the QM model coincide with the point-like particles.In this work, we include the quantum fluctuations through the functional renormalization group approach (FRG) [70,71], which is a functional continuum field approach.
This paper is organized as follows.In Section II, we introduce the low energy effective theory, i.e. the 2-flavor quark meson model.In Section III, the choice of the regulator, propagators under a magnetic field are discussed and the flow equations are presented.In Section IV, we show the numerical results in our calculation, including the meson masses, quark masses and decay constants as functions of the strength of magnetic field.In Appendix A, we show the vertexes of the 2-flavor quark meson model.In Appendix B, the threshold functions of the flow equations are given.

II. LOW ENERGY EFFECTIVE THEORY
At high renormalization group (RG) scale, the firstprinciple QCD system only includes the degrees of freedom of quarks and gluons.As the RG scale decreases, due to the finite mass gap, the gluons are decoupled from the system, and their dynamics are integrated out, left with gluonic background field and its potential.Consequently, composite degrees of freedom, e.g., mesons and baryons, emerge naturally from the dynamics of elementary degrees of freedom, see, e.g., [72][73][74].The degrees of freedom of the QCD system are transformed into those of quarks and hadrons, which can be described by lowenergy effective models, such as the QM model and NJL model.
The effective action of the two-flavor quark-meson model in Euclidean space reads [75] Here, ϕ denotes the meson fields: In Equation ( 1), the potential ), and cσ is the linear sigma term, which explicitly breaks the chiral symmetry and accounts for the pion masses.The covariant derivative of meson fields reads Without loss of generality, a homogeneous magnetic field of strength B is assumed along the z-direction and the Landau gauge is adopted, i.e.A µ = (0, 0, xB, 0).For convenience, we define p ⊥ = (p 1 , p 2 ) and p ∥ = (p 0 , p 3 ).The curvature masses are defined as the two-point correlation function at vanishing external momentum m 2 ϕ,cur = Γ (2) and for the π and σ meson, they are given as The light quark mass is Here σ 0 is the vacuum expectation value of the sigma meson field, which is located at the minimum of the effective potential.The mesonic decay constant is also related to the vacuum expectation value via: In this work, we employ the local potential approximation (LPA), which is the leading order of the derivative expansion.In other words, we ignore the mesonic and quark wave function renormalizations and the running of the Yukawa coupling.See [30] for a relevant discussion, where magnetic dependent wave function renormalizations beyond LPA are investigated in one-flavor case within the FRG approach.
FIG. 1: Feynman diagrams of the flow equations for the effective potential (upper) and the mesonic two-point correlation functions (lower).The solid lines and dashed lines denote the quark and meson propagators, respectively.The crossed circles donates the infrared regulators, as shown in Equation (9).

III. FLOW EQUATIONS AND REGULATORS
The evolution of the effective action with the RG scale is described by the Wetterich equation [76], where an infrared (IR) cutoff scale k, i.e., the RG scale, is used to suppress quantum fluctuations of momenta below the scale.Starting from a high ultraviolet (UV) scale, say Λ UV , with the classical action as the initial condition, one is able to integrate-in quantum fluctuations of different modes successively by evolving the RG scale k from UV to IR.The Wetterich equation for the effective action Equation (1) reads: Here R k denotes the regulators and G ϕ/q k (p) are scaledependent propagators of mesons and quarks.
In the vacuum, the effective action satisfies the O(4) space-time symmetry.When we consider an external magnetic field, the perpendicular (transverse) and parallel (longitudinal) directions to the magnetic field will split.Obviously, it will stay invariant in the temporal and z directions at zero temperature.A commonly used 3d regulator for the spatial momenta breaks the O(4) symmetry in the vacuum [77], while a regularization on the transverse momenta would give rise to non-physical artifacts [78].Therefore, in this work we adopt 2d regulators which regularize the temporal and longitudinal momenta, as follows with p 2 ∥ = p 2 0 + p 2 3 and the shape functions Here Θ(x) is the Heaviside step function.Notably, absence of regularization on the transverse momenta leads to a divergence for the flow equation of the potential V k .Fortunately, the two-point correlation functions stay finite [30].The summation of the Landau level can be calculated through the Hurwitz ζ-function [40] ζ(s, q) In this work, we use a transverse momentum cutoff Λ ⊥ = 5GeV to calculate the u-d quark mixed threshold functions.We have checked that our results show no obvious dependence on the choices.

A. propagators and flow equations
The quark propagator in magnetic fields in the Schwinger scheme reads Where prefactor Φ(x ⊥ , y ⊥ ) = s ⊥ (x 1 +y 1 )(x 2 −y 2 )|q f B|/2 with s ⊥ ≡ sign(q f B) is the Schwinger phase [79], which breaks the translational invariance.In this work, we ignore the Schwinger phase of the propagators under magnetic fields, and see, e.g., [25,80] for more discussions on the Schwinger phase with the Ritus scheme.Recently, it has been found that the Schwinger phase can be neglected when the meson masses are calculated [50].The translationally invariant part of the quark propagator in the representation of Landau levels in the Euclidean space with the regulator reads [30,40]: with p ∥,R F ≡ p ∥ (1 + r F ) and Here L a n (x) are the generalized Laguerre polynomials with L a −1 (x) = 0. Similarly, the translationally invariant part of the scale-dependent meson propagator reads . With the aforementioned setup, one is led to the flow equations of the effective potential: The relevant Feynman diagrams are presented in the first line of Figure 1.Here l B , l F are threshold functions given in Appendix B. By taking the second derivative of Equation (8) with the pion fields, one arrives at the flow equation of two-point correlation function of the neutral pion as follows and the flow equation of two-point correlation function of charged pions, Here V

B. weak-field expansion
The number of Landau levels increases significantly in the region of small magnetic field.We do the computation in this region by utilizing the weak-field expansion method.The weak-field expansion for the quark propagator in the Euclidean space reads [81,82] 1) and ( 25) and corresponding physical observables at B = 0.If not mentioned explicitly, most of the results are calculated with the parameters in the first line with m π = 220 MeV.
Thus, one arrives at the quark loop function for the twopoint correlation function of charged pions, as follows In the same way, the quark loops for the two-point correlation function of neutral pions read The weak-field expansion for the meson propagator reads [29,83] Gϕ Then the weak-field expansions of the charged pion loop function J B (π ± ) and the pion-sigma loop function J 2B (π ± , σ) can be readily obtained, and their explicit expressions are listed in Appendix B. We find that the quark loops, as the last diagram in the second line of Figure 1 shows, play the dominant role for the pion two-point correlation functions.When Λ ⊥ → ∞, the leading term in B reads 1/(4(k 2 +m 2 f ) 3 )q f1 q f2 B 2 .For the charged pion, the signs of q u , q d are opposite, so this term would make a negative contribution to the flow equation, implying that the contribution of quantum fluctuations to the charged pion mass is positive, which results in larger mass for charged pions in FRG than the point-like mass.On the contrary, for the neutral pion, the sign of q 2 u or q 2 d are positive.Consequently, the flow is increased and the mass of neutral pion is decreased in comparison to that in vacuum.

In
Here κ denotes the expanding point, located at the minimal value of the effective potential with k = 0. We choose the maximal order of the Taylor expansion to be N v = 5, and for more discussions on the convergence of the Taylor expansion see [77,84].We have also checked the physicalpoint expanding method, in which the expanding point is the minimal value of the effective potential at every value of the RG scale k.We find that these two methods coincide with each other and produce consistent results.The UV cutoff is chosen to be Λ UV = 700 MeV, where the initial condition of the effective potential reads Here, the parameters of the initial conditions and the corresponding physical observables at B = 0 are listed in Table I.In order to compare with the lattice QCD results, m π = 220 MeV and m π = 416 MeV are chosen.Note that if not mentioned explicitly, most of the results are calcluated with m π = 220 MeV.
In Figure 2, we show the neutral pion mass m π 0 as a function of the strength of magnetic field in comparison to the Lattice QCD results [20].In the region of small magnetic fields with eB < 0.05[GeV 2 ], we utilize the weak-field expansion method, while in other regions calculations are done through summation of the Landau levels.Our results are qualitatively or even quantitatively in agreement with the lattice results.If the neutral pion is regarded as a point particle, its masses will not change under magnetic fields.Due to the inner structure of the neutral pion, i.e. ūu or dd, the neutral pion mass decreases with the magnetic field, as discussed in Section III B. The neutral pion mass decreases monotonically with the increase of magnetic fields, and the rate of decrease is gradually reduced.Finally, it tends to saturate in large magnetic fields.
The charged pion mass m π ± is defined as the lowest energy of quantum states for the charged pion [25], i.e. m π ± (B) = E π ± | pz=0,n=0 .For the point particle of the charged pion, the mass is given by m π ± (B) = m 2 π ± (B = 0) + eB.According to the definition, we need to calculate the two-point correlation function Γ (2) π ± π ± (p ∥ = 0, p ⊥ = |eB|).Note, however, that it is challenging to integrate the loop functions J 2F (u, d) and J 2B (π ± , σ) with finite external momenta.In our calculation, we use the approximation as follows We also calculate Γ (2) π ± π ± (p ∥ = 0, p ⊥ = |eB|) at very large magnetic fields, and find that both results are consistent with each other.
In the left panel of Figure 3, we plot the charged pion masses as functions of the strength of magnetic field with m π (B = 0) = 220 MeV.In order to compare with the Lattice QCD results [20], where the computation is done with m π (B = 0) ∼ 220MeV.We use lattice results of m 2 π ± (B) − m 2 π (B = 0) and construct to be compared with FRG calculations.In the right panel of Figure 3, we use the initial conditions in the second line in Table I, corresponding to m π (B = 0) = 416 MeV, and compare the normalized charged pion mass m π± (B)/m π (0) with the lattice results with the same pion mass in the vacuum [17].The charged pion masses in our calculation increase monotonically with the magnetic field.Our results are larger than the point-like charged pion masses and in agreement with the lattice QCD results in [17].Similar results are also reported in the NJL calculation [25,53].However, for the lattice calculations in [20], the charged pion masses are smaller than the point-like results and exhibit nonmonotonic behaviors.This means our results receive an opposite contribution from the quantum fluctuation compared to the lattice QCD result in [20].The main contribution of quantum fluctuations comes from the u-d quark loop, as discussed in the last paragraph of Section III B, the leading order magnetic dependent quantum fluctuation of charged pion is opposite to that of the neutral pion, which could lead to the neutral pion masses smaller than point-like results and charged pion masses larger than point-like results in the region of weak magnetic field, as shown in the inlay in the left panel of Figure 3.The calculation results with the Landau level representation coincide with those of weak-field expansion.On the one hand, this discrepancy between the charged pion mass obtained in our calculations and that in lattice simulations in [20] also probably arises from the approximations used in our calculations, such as neglect of the magnetic dependence of the Yukawa couplings and the wave function renormalizations.Our calculation is based on an effective model, which only contains the scalar and pseudoscalar channels, and other tensor structure channels and gluon dynamics are not taken into account [54].On the other hand, the opposite quantum contribution could come from the lattice QCD calculation.Notably, the lattice cutoff in [20] a ≃ 0.117 fm and no continuum limit is done, while in [17] the continuum limit results are obtained, while the pion masses are much heavier than the physical value.Therefore, more detailed calculations and studies are required for both the lattice QCD and effective theories.
In Figure 4, we plot the pion decay constant as a function of the strength of magnetic field and compared it with the lattice QCD results [20].For the 2-flavor QM model, the pion decay constant is determined by the minimum of the effective potential in Equation (7).In the QM model, one cannot distinguish the u or d pion decay constants, and our results are close to that of f π 0 d in lattice QCD.
In Figure 5, we show the magnetic dependence of the sigma meson mass and light quark mass.The lattice QCD results are constructed from the quark chiral condensates in Ref [20].Similar to the pion decay constant f π , the light quark mass is close to the d quark results of the lattice QCD.Furthermore, due to the internal structure of mesons, the mass of sigma meson varies with the magnetic field.The sigma meson and light quark masses increase monotonically with the magnetic field.The decay constant, sigma meson mass, and light quark mass reflect chiral symmetry breaking increasing with magnetic fields, which is related to the magnetic catalysis.with the magnetic field, while the sigma meson mass increases monotonically due to their internal structure.
The decay constant and the light quark mass also increase with the magnetic field, reflecting the magnetic catalysis behavior at vanishing temperature.The neutral pion mass and pion decay constant are quantitatively in agreement with with the lattice QCD results especially in the range of eB < 1.2GeV 2 .However, the charged pion mass is in agreement with the lattice results in [17] but no non-monotonic mass behavior for charged pion Latt.QCD, Σ u (B) × m q (B = 0) Latt.QCD, Σ d (B) × m q (B = 0) FIG.5: Quark mass as a function of the strength of magnetic fields.The lattice QCD results are constructed from the quark chiral condensates in Ref [20].The σ meson mass is also plotted.
has been observed in this framework as shown in [20].This needs further investigation from both lattice QCD and functional methods.It is noteworthy that this is our first preliminary attempt to calculate meson masses and the pion decay constant in the QM model under strong magnetic fields within the FRG approach, and there are many things we need to do in the future.In the upcoming work, we will go beyond the LPA truncation, which includes the magnetic dependent Yukawa couplings and wave function renormalizations, and calculate the spectral functions of the mesons.After that, we will extend them into finite temperatures and chemical potential.The strange quark and vector meson will also be included in future work.
For the charged pion under magnetic fields, the threshold function is The quark loop function for the effective potential in the vacuum: and the quark loop threshold function under magnetic fields reads The loop function of the tadpole diagram in weak-field expansion reads 4 (q ϕ B) 2 + O(q ϕ B) 4 . (B5) For the loop functions of neutral meson, we just need to set q ϕ = 0.And loop functions of the charge pion in Landau leval representation reads The σ − π loop functions in weak-field expansion read + O(q π B) 4 (B7) with neutral pion q π 0 = 0 and charged pion q π ± = ±e.The weak-field expansion of quark loop of the twopoint correction of pion have shown in Equation (21) and Equation (22).If we set B = 0, they will come back to the representations of vacuum case.
In Landau leval representation, the quark loop threshold functions of the neutral pion become here where we also define the LL(n 1 , n 2 ) and L 1 L 1 (n 1 , n 2 ) as the integrations of perpendicular direction with L a n (x) are the generalized Laguerre polynomials.

4 eB
this work, we solved the flow equation of effective potential by employing the Taylor expansion method 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.FRG FRG, weak-field exp.

FIG. 2 :
FIG.2: Neutral pion mass m π 0 as a function of the strength of magnetic field.The lattice QCD results are taken from Ref[20].
This work calculates the meson masses and the pion decay constant at vanishing temperature under strong magnetic fields.The quantum fluctuations are successfully included using the FRG approach.The two-point correlation functions of neutral and charged pion are calculated.The neutral pion mass monotonically decreases 0

FIG. 3 :FRG, π 0 FIG. 4 :
FIG. 3: Left panel: Charged pion mass m π ± as a function of the strength of magnetic field with m π (0) = 220 MeV.The lattice QCD results are constructed based on data from Ref [20] and more details are shown in the text.In the inlay, we show the charged pion mass in the weak-field expansion with FRG subtracted by the point-like result.Right panel: Normalized charged pion mass m π ± (B)/m π (0) as a function of magnetic fields with m π (0) = 416 MeV in comparison to the relevant Lattice QCD [17].

TABLE I :
Parameters for the initial conditions in Equations (