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

Rui Wen<EMAIL_ADDRESS>School of Nuclear Science and Technology, University
of Chinese Academy of Sciences, Beijing, 100049, P.R. China  Shi Yin  School
of Physics, Dalian University of Technology, Dalian, 116024, P.R. China  Wei-
jie Fu  School of Physics, Dalian University of Technology, Dalian, 116024,
P.R. China  Mei Huang<EMAIL_ADDRESS>School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing, 100049, P.R.
China

###### Abstract

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.

##  I Introduction

Studying Quantum Chromodynamics (QCD) matter under strong external magnetic
field and vortical field have attracted many attentions in recent years.
Relativistic heavy-ion collisions provide us a platform to study QCD matter
under extreme conditions in the laboratory. In non-central heavy-ion
collisions, the collision of two high-speed nuclei moving in opposite
directions could create strong magnetic fields of order $\sim 10^{18}$ Gauss
[1, 2]. Strong magnetic fields also exist in the early universe and magnetars
[3, 4, 5]. Understanding the strongly interacting matter in background
magnetic fields requires a combination of the QCD and QED theories, which has
brought about plenty of novel phenomena of magnetized quark matter, such as
the chiral magnetic effect (CME) [6, 7], magnetic catalysis (MC) [8, 9, 10],
inverse magnetic catalysis (IMC) [11, 12, 13], diamagnetism at low temperature
and paramagnetism at high temperature [14]. These rich phenomena have
attracted theoretical investigations in lattice Monte-Carlo simulations [15,
16, 17, 18, 19, 20, 21], as well as model calculations, such as Nambu-Jona-
Lasinio (NJL) [22, 23, 24, 25, 26, 27, 28, 28], quark-meson (QM) model [29,
30, 31] and AdS/QCD [32, 33], within mean-field approximation or functional
methods [34, 35, 36, 37, 38], see e.g., [39, 40, 41, 42] for reviews.

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, 45, 17, 46,
47, 48, 18, 49, 50, 51]. Recent Lattice calculation in [20] showed that at
zero temperature, the mass of the neutral $\pi$ 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 first-principle 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
low-energy 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]

$\displaystyle\Gamma_{k}=$
$\displaystyle\int_{x}\bar{q}\gamma_{\mu}(\partial_{\mu}-iQA_{\mu})q+\operatorname{Tr}(D_{\mu}\phi\cdot
D_{\mu}\phi^{\dagger})$
$\displaystyle+h\bar{q}(T^{0}\sigma+i\gamma_{5}\vec{T}\cdot\vec{\pi})q+V_{k}(\rho)-c\sigma,$
(1)

with $\int_{x}=\int d^{4}x$, $Q=diag(2/3,-1/3)e$ and $q=(u,d)^{T}$. Here,
$\phi$ denotes the meson fields:

$\displaystyle\phi$
$\displaystyle=T^{0}\sigma+\vec{T}\cdot\vec{\pi}=\frac{1}{2}\begin{pmatrix}\sigma+\pi^{0}&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&\sigma-\pi^{0}\end{pmatrix}\,.$ (2)

In Equation 1, the potential $V(\rho)$ is chiral symmetric with
$\rho\equiv\text{Tr}[\phi^{\dagger}\phi]=\frac{1}{2}(\sigma^{2}+\vec{\pi}^{2})$,
and $c\sigma$ is the linear sigma term, which explicitly breaks the chiral
symmetry and accounts for the pion masses. The covariant derivative of meson
fields reads

$\displaystyle D_{\mu}\phi=\partial_{\mu}-iA_{\mu}[Q,\phi].$ (3)

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.
$A_{\mu}=(0,0,xB,0)$. For convenience, we define $p_{\perp}=(p_{1},p_{2})$ and
$p_{\parallel}=(p_{0},p_{3})$.

The curvature masses are defined as the two-point correlation function at
vanishing external momentum

$\displaystyle
m_{\phi,\text{cur}}^{2}=\Gamma_{\phi\phi}^{(2)}(p_{0}=0,\vec{p}=0),$ (4)

and for the $\pi$ and $\sigma$ meson, they are given as

$\displaystyle m_{\pi}^{2}=V^{\prime}(\rho)\quad
m_{\sigma}^{2}=V^{\prime}(\rho)+2\rho V^{\prime\prime}(\rho).$ (5)

The light quark mass is

$\displaystyle m_{q}=\frac{1}{2}h\sigma_{0}.$ (6)

Here $\sigma_{0}$ 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:

$\displaystyle f_{\pi}=\sigma_{0}.$ (7)

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.

##  III Flow equations and regulators

Figure 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.

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 $\Lambda_{\mathrm{UV}}$, 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:

$\partial_{t}\Gamma_{k}=\frac{1}{2}\text{Tr}[G_{k}^{\phi}(p)\partial_{t}R_{k}^{B}]-\text{Tr}[G_{k}^{q}(p)\partial_{t}R_{k}^{F}].$
(8)

Here $R_{k}$ denotes the regulators and $G^{\phi/q}_{k}(p)$ are scale-
dependent 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

$\displaystyle R_{k}^{B}$
$\displaystyle=p_{\parallel}^{2}r_{B}(p_{\parallel}^{2}/k^{2}),$
$\displaystyle R_{k}^{F}$
$\displaystyle=ip_{\parallel}\cdot\gamma_{\parallel}r_{F}(p_{\parallel}^{2}/k^{2}),$
(9)

with $p_{\parallel}^{2}=p_{0}^{2}+p_{3}^{2}$ and the shape functions

$\displaystyle r_{B}(x)$
$\displaystyle=\bigg{(}\frac{1}{x}-1\bigg{)}\Theta(1-x)$ $\displaystyle
r_{F}(x)$ $\displaystyle=\bigg{(}\frac{1}{\sqrt{x}}-1\bigg{)}\Theta(1-x).$
(10)

Here $\Theta(x)$ 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 $\zeta$-function [40]

$\displaystyle\zeta(s,q)=\sum_{n}\frac{1}{(q+n)^{s}}.$ (11)

In this work, we use a transverse momentum cutoff
$\Lambda_{\perp}=5\text{GeV}$ to calculate the $u$-$d$ quark mixed threshold
functions. We have checked that our results show no obvious dependence on the
choices.

###  III.1 propagators and flow equations

The quark propagator in magnetic fields in the Schwinger scheme reads

$\displaystyle
G(x,y)=e^{i\Phi(x_{\perp},y_{\perp})}\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\tilde{G}(p).$
(12)

Where prefactor
$\Phi(x_{\perp},y_{\perp})=s_{\perp}(x^{1}+y^{1})(x^{2}-y^{2})|q_{f}B|/2$ with
$s_{\perp}\equiv\text{sign}(q_{f}B)$ 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]:

$\displaystyle\tilde{G}_{k}^{q}(p)=\exp(-\frac{p_{\perp}^{2}}{|q_{f}B|})\sum_{n=0}^{\infty}\frac{(-1)^{n}D_{n}(p_{\parallel,R_{F}},p_{\perp})}{p_{\parallel,R_{F}}^{2}+2n|q_{f}B|+m_{f}^{2}}\,,$
(13)

with $p_{\parallel,R_{F}}\equiv p_{\parallel}(1+r_{F})$ and

$\displaystyle D_{n}(p_{\parallel},p_{\perp})$ $\displaystyle=$
$\displaystyle(-i\gamma_{\parallel}p_{\parallel}+m_{f})\Bigg{[}(1+i\gamma_{1}\gamma_{2}s_{\perp})\mathcal{L}_{n}\bigg{(}\frac{2p_{\perp}^{2}}{|q_{f}B|}\bigg{)}$
$\displaystyle-(1-i\gamma_{1}\gamma_{2}s_{\perp})\mathcal{L}_{n-1}\bigg{(}\frac{2p_{\perp}^{2}}{|q_{f}B|}\bigg{)}\Bigg{]}$
$\displaystyle+4i\gamma_{\perp}p_{\perp}\mathcal{L}_{n-1}^{1}\bigg{(}\frac{2p_{\perp}^{2}}{|q_{f}B|}\bigg{)}.$
(14)

Here $\mathcal{L}_{n}^{a}(x)$ are the generalized Laguerre polynomials with
$\mathcal{L}_{-1}^{a}(x)=0$. Similarly, the translationally invariant part of
the scale-dependent meson propagator reads

$\displaystyle\tilde{G}_{k}^{\phi}(p)=$ $\displaystyle
2\exp(-\frac{p_{\perp}^{2}}{|q_{\phi}B|})$
$\displaystyle\times\sum_{n=0}^{\infty}\frac{(-1)^{n}\mathcal{L}_{n}\big{(}\frac{2p_{\perp}^{2}}{|q_{\phi}B|}\big{)}}{p_{\parallel,R_{B}}^{2}+(2n+1)|q_{\phi}B|+m_{\phi}^{2}}.$
(15)

with $p_{\parallel,R_{B}}\equiv p_{\parallel}(1+r_{B})^{\frac{1}{2}}$.

With the aforementioned setup, one is led to the flow equations of the
effective potential:

$\displaystyle\partial_{t}V_{k}=$
$\displaystyle\frac{1}{2}\big{[}l_{B}(m_{\sigma})+l_{B}(m_{\pi^{0}})+2\,l_{B}(m_{\pi^{\pm}})\big{]}$
$\displaystyle-4N_{c}\big{[}l_{F}(m_{f},q_{u})+l_{F}(m_{f},q_{d})\big{]}.$
(16)

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

$\displaystyle\partial_{t}\Gamma^{(2)}_{\pi^{0}\pi^{0},k}=$
$\displaystyle\frac{1}{2}\big{[}V_{2\pi^{0}2\sigma}\mathcal{J}_{B}(\sigma)+V_{4\pi^{0}}\mathcal{J}_{B}(\pi^{0})$
$\displaystyle+2V_{2\pi^{0}2\pi^{\pm}}\mathcal{J}_{B}(\pi^{\pm})\big{]}-V_{2\pi^{0}\sigma}^{2}\mathcal{J}_{2B}(\pi^{0},\sigma)$
$\displaystyle+V^{2}_{\bar{u}u\pi^{0}}\mathcal{J}_{F}(u)+V^{2}_{\bar{d}d\pi^{0}}\mathcal{J}_{F}(d),$
(17)

and the flow equation of two-point correlation function of charged pions,

$\displaystyle\partial_{t}\Gamma^{(2)}_{\pi^{\pm}\pi^{\pm},k}=$
$\displaystyle\frac{1}{2}\big{[}V_{2\pi^{\pm}2\sigma}\mathcal{J}_{B}(\sigma)+V_{2\pi^{0}\pi^{\pm}}\mathcal{J}_{B}(\pi^{0})$
$\displaystyle+2V_{4\pi^{\pm}}\mathcal{J}_{B}(\pi^{\pm})\big{]}-V_{2\pi^{\pm}\sigma}^{2}\mathcal{J}_{2B}(\pi^{\pm},\sigma)$
$\displaystyle+V^{2}_{\bar{u}d\pi^{\pm}}\mathcal{J}_{2F}(u,d).$ (18)

Here $V_{[\cdots]}$ denote different vertices listed in Appendix A, and
$\mathcal{J}_{B},\mathcal{J}_{2B},\mathcal{J}_{F},\mathcal{J}_{2F}$ are
threshold functions, which are defined in Appendix B. The corresponding
Feynman diagrams are shown in the second line of Figure 1. It can be readily
verified that the neutral pion flow equation Equation 17 coincides with the
flow equation of first order derivative of the potential, i.e.,

$\displaystyle\partial_{t}\Gamma^{(2)}_{\pi^{0}\pi^{0},k}=\partial_{t}V_{k}^{\prime}(\rho).$
(19)

###  III.2 weak-field expansion

$\lambda_{1}[\text{MeV}]^{2}$ | $\lambda_{2}$ | $h$ | $c\,[\text{MeV}]^{3}$ | $m_{\pi}\,[\text{MeV}]$ | $m_{\sigma}\,[\text{MeV}]$ | $m_{q}\,[\text{MeV}]$ | $f_{\pi}\,[\text{MeV}]$
---|---|---|---|---|---|---|---
$(740)^{2}$ | -5.0 | 6.4 | $4.5\times 10^{6}$ | 220 | 475 | 295 | 92
$(775)^{2}$ | 6.0 | 6.4 | $1.6\times 10^{7}$ | 416 | 675 | 295 | 92
Table 1: Parameters for the initial conditions in Equations 25 and 1 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
$m_{\pi}=220$ MeV.

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]

$\displaystyle\tilde{G}_{k}^{q}(p)$ $\displaystyle=$
$\displaystyle\frac{-ip_{\mu,R_{F}}\gamma_{\mu}+m_{f}}{p_{R_{F}}^{2}+m_{f}^{2}}+i\frac{\gamma_{1}\gamma_{2}(m_{f}-i\gamma_{\parallel}p_{\parallel,R_{F}})}{(p_{R_{F}}^{2}+m_{f}^{2})^{2}}q_{f}B$
$\displaystyle+2\frac{p_{\perp}^{2}(m_{f}-i\gamma_{\parallel}p_{\parallel,R_{F}})+i\gamma_{\perp}p_{\perp}(m_{f}^{2}+p^{2}_{\parallel,R_{F}})}{(p_{R_{F}}^{2}+m_{f}^{2})^{4}}(q_{f}B)^{2}$
$\displaystyle+\mathcal{O}(q_{f}B)^{3}.$ (20)

Thus, one arrives at the quark loop function for the two-point correlation
function of charged pions, as follows

$\displaystyle\mathcal{J}_{2F}(u,d)=-\frac{k^{4}N_{c}}{2\pi^{2}}\bigg{[}\frac{\Lambda_{\perp}^{2}}{(k^{2}+m_{f}^{2})(k^{2}+m_{f}^{2}+\Lambda_{\perp}^{2})}$
$\displaystyle+\Big{(}\frac{1}{4(k^{2}+m_{f}^{2})^{3}}+\frac{5k^{2}+5m_{f}^{2}+8\Lambda_{\perp}^{2}}{12(k^{2}+m_{f}^{2}+\Lambda_{\perp}^{2})^{4}}\Big{)}(q_{u}B)(q_{d}B)\bigg{]}$
$\displaystyle+\mathcal{O}(B)^{4}.$ (21)

In the same way, the quark loops for the two-point correlation function of
neutral pions read

$\displaystyle\mathcal{J}_{2F}(q_{f})=-\frac{k^{4}N_{c}}{2\pi^{2}}\bigg{[}\frac{\Lambda_{\perp}^{2}}{(k^{2}+m_{f}^{2})(k^{2}+m_{f}^{2}+\Lambda_{\perp}^{2})}$
$\displaystyle+\Big{(}\frac{1}{4(k^{2}+m_{f}^{2})^{3}}+\frac{5k^{2}+5m_{f}^{2}+8\Lambda_{\perp}^{2}}{12(k^{2}+m_{f}^{2}+\Lambda_{\perp}^{2})^{4}}\Big{)}(q_{f}B)^{2}\bigg{]}$
$\displaystyle+\mathcal{O}(B)^{4}.$ (22)

The weak-field expansion for the meson propagator reads [83, 29]

$\displaystyle\tilde{G}_{k}^{\phi}(p)=$
$\displaystyle\frac{1}{p_{R_{B}}^{2}+m_{\phi}^{2}}+\frac{p_{\perp}^{2}-p_{\parallel,R_{B}}^{2}-m_{\phi}^{2}}{(p_{R_{B}}^{2}+m_{\phi}^{2})^{4}}(q_{\phi}B)^{2}$
$\displaystyle+\mathcal{O}(q_{\phi}B)^{4}.$ (23)

Then the weak-field expansions of the charged pion loop function
$\mathcal{J}_{B}(\pi^{\pm})$ and the pion-sigma loop function
$\mathcal{J}_{2B}(\pi^{\pm},\sigma)$ 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 $\Lambda_{\perp}\rightarrow{\infty}$, the leading term in $B$ reads
$1/(4(k^{2}+m_{f}^{2})^{3})q_{f_{1}}q_{f_{2}}B^{2}$. 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 $q_{u}^{2}$ or $q_{d}^{2}$ are positive.
Consequently, the flow is increased and the mass of neutral pion is decreased
in comparison to that in vacuum.

##  IV numerical results

In this work, we solved the flow equation of effective potential by employing
the Taylor expansion method around a fixed point, i.e.

$\displaystyle
V_{k}(\rho)=\sum_{n}^{N_{v}}\frac{\lambda_{n,k}}{n!}(\rho-\kappa)^{n}.$ (24)

Here $\kappa$ 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 [84, 77]. We have also checked the physical-point
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 $\Lambda_{\mathrm{UV}}=700$ MeV, where the initial condition
of the effective potential reads

$\displaystyle
V_{\mathrm{UV}}(\rho)=\lambda_{1}\rho+\frac{\lambda_{2}}{2}\rho^{2}.$ (25)

Here, the parameters of the initial conditions and the corresponding physical
observables at $B=0$ are listed in Table 1. In order to compare with the
lattice QCD results, $m_{\pi}=220$ MeV and $m_{\pi}=416$ MeV are chosen. Note
that if not mentioned explicitly, most of the results are calcluated with
$m_{\pi}=220$ MeV.

Figure 2: Neutral pion mass $m_{\pi^{0}}$ as a function of the strength of
magnetic field. The lattice QCD results are taken from Ref [20].

In Figure 2, we show the neutral pion mass $m_{\pi^{0}}$ 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 $eB<0.05[\text{GeV}^{2}]$, 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. $\bar{u}u$ or
$\bar{d}d$, the neutral pion mass decreases with the magnetic field, as
discussed in Section III.2. 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.

Figure 3: Left panel: Charged pion mass $m_{\pi^{\pm}}$ as a function of the
strength of magnetic field with $m_{\pi}(0)=220$ 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 $m_{\pi^{\pm}}(B)/m_{\pi}(0)$ as a function of magnetic fields with
$m_{\pi}(0)=416$ MeV in comparison to the relevant Lattice QCD [17].

The charged pion mass $m_{\pi^{\pm}}$ is defined as the lowest energy of
quantum states for the charged pion [25], i.e.
$m_{\pi^{\pm}}(B)=E_{\pi^{\pm}}|_{p_{z}=0,n=0}$. For the point particle of the
charged pion, the mass is given by
$m_{\pi^{\pm}}(B)=\sqrt{m_{\pi^{\pm}}^{2}(B=0)+eB}$. According to the
definition, we need to calculate the two-point correlation function
$\Gamma_{\pi^{\pm}\pi^{\pm}}^{(2)}(p_{\parallel}=0,p_{\perp}=|eB|)$. Note,
however, that it is challenging to integrate the loop functions
$\mathcal{J}_{2F}(u,d)$ and $\mathcal{J}_{2B}(\pi^{\pm},\sigma)$ with finite
external momenta. In our calculation, we use the approximation as follows

$\displaystyle
m_{\pi^{\pm}}(B)=\sqrt{\Gamma_{\pi^{\pm}\pi^{\pm}}^{(2)}(p_{\parallel}=0,p_{\perp}=0)+eB}.$
(26)

We also calculate
$\Gamma_{\pi^{\pm}\pi^{\pm}}^{(2)}(p_{\parallel}=0,p_{\perp}=|eB|)$ 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 $m_{\pi}(B=0)=220$ MeV. In order to
compare with the Lattice QCD results [20], where the computation is done with
$m_{\pi}(B=0)\sim 220\text{MeV}$. We use lattice results of
$m_{\pi^{\pm}}^{2}(B)-m_{\pi}^{2}(B=0)$ and construct

$\displaystyle
m_{\pi^{\pm}}(B)=\sqrt{m_{\pi^{\pm}}^{2}(B)-m_{\pi}^{2}(B=0)+(220\text{MeV})^{2}},$
(27)

to be compared with FRG calculations. In the right panel of Figure 3, we use
the initial conditions in the second line in Table 1, corresponding to
$m_{\pi}(B=0)=416$ MeV, and compare the normalized charged pion mass
$m_{\pi_{\pm}}(B)/m_{\pi}(0)$ 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.2, 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] $a\simeq 0.117$ 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.

Figure 4: Pion decay constant as a function of the strength of magnetic field.
The lattice QCD results are taken from Ref [20].

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
$f_{\pi^{0}_{d}}$ in lattice QCD.

Figure 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 $\sigma$ meson mass is also plotted.

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 $f_{\pi}$, 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.

##  V Conclusion

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 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 $eB<1.2{\rm GeV^{2}}$. However, the
charged pion mass is in agreement with the lattice results in [17] but no non-
monotonic mass behavior for charged pion 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.

###### Acknowledgements.

We thank Chuang Huang, Jie Mei, Yang-yang Tan and Kun Xu for their valuable
discussions. This work is supported in part by the National Natural Science
Foundation of China (NSFC) Grant Nos: 12235016, 12221005, 12175030 and the
Strategic Priority Research Program of Chinese Academy of Sciences under Grant
No XDB34030000, the start-up funding from University of Chinese Academy of
Sciences(UCAS), and the Fundamental Research Funds for the Central
Universities.

##  Appendix A Vertexes

As mentioned above, we need the n-point vertexes to calculate the neutral and
charged pion two-point correlation function Equations 17 and 18. The n-point
vertexes are defined as

$\displaystyle
V_{\phi_{1},\phi_{2}\cdots\phi_{n}}=\frac{\partial^{n}\Gamma_{k}}{\partial\phi_{1}\partial\phi_{2}\cdots\partial\phi_{n}}.$
(28)

The quark-meson interaction in the 2-flavor QM model reads

$\displaystyle V_{\bar{u}d\pi^{+}}$
$\displaystyle=V_{\bar{d}u\pi^{-}}=\frac{\sqrt{2}}{2}hi\gamma_{5}$ (29)
$\displaystyle V_{\bar{u}u\pi^{0}}$
$\displaystyle=V_{\bar{d}d\pi^{0}}=\frac{1}{2}hi\gamma_{5}$ (30)
$\displaystyle V_{\bar{u}u\sigma}$
$\displaystyle=V_{\bar{d}d\sigma}=\frac{1}{2}h.$ (31)

The nonvanishing mesonic three-point and four-point vertexes are

$\displaystyle V_{2\pi^{\pm}\sigma}$ $\displaystyle=V_{2\pi^{0}\sigma}=\sigma
V^{\prime\prime}(\rho)$ (32) $\displaystyle V_{3\sigma}$
$\displaystyle=3\sigma
V^{\prime\prime}(\rho)+\sigma^{3}V^{\prime\prime\prime}(\rho)$ (33)
$\displaystyle V_{2\pi^{\pm}2\sigma}$
$\displaystyle=V_{2\pi^{0}2\sigma}=V^{\prime\prime}(\rho)+\sigma^{2}V^{\prime\prime\prime}(\rho)$
(34) $\displaystyle V_{2\pi^{\pm}2\pi^{0}}$
$\displaystyle=V^{\prime\prime}(\rho)$ (35) $\displaystyle V_{4\pi^{0}}$
$\displaystyle=3V^{\prime\prime}(\rho)$ (36) $\displaystyle V_{4\pi^{\pm}}$
$\displaystyle=2V^{\prime\prime}(\rho).$ (37)

##  Appendix B loop functions

The threshold functions of the effective potential for neutral meson
$\pi^{0},\sigma$ in Equation 16 read

$\displaystyle
l_{B}(m_{\phi})=\frac{k^{4}}{16\pi^{2}}(\log(k^{2}+m_{\phi}^{2}+\Lambda_{\perp}^{2})-\log(k^{2}+m_{\phi}^{2})).$
(38)

For the charged pion under magnetic fields, the threshold function is

$\displaystyle
l_{B}(m_{\phi})=\frac{k^{4}}{8\pi^{2}}|q_{\phi}B|\sum_{n=0}^{\Lambda_{\perp,n}}\frac{1}{k^{2}+m_{\phi}^{2}+(2n+1)|q_{\phi}B|}.$
(39)

The quark loop function for the effective potential in the vacuum:

$\displaystyle
l_{F}=\frac{k^{4}}{16\pi^{2}}\Big{(}\log(k^{2}+m_{f}^{2}+\Lambda_{\perp}^{2})-\log(k^{2}+m_{f}^{2})\Big{)},$
(40)

and the quark loop threshold function under magnetic fields reads

$\displaystyle
l_{F}=\frac{k^{4}}{16\pi^{2}}|q_{f}B|\sum_{n=0}^{\Lambda_{\perp,n}}\sum_{s=\pm
1}\frac{1}{k^{2}+m_{f}^{2}+|q_{f}B|(2n+1+s)}.$ (41)

The loop function of the tadpole diagram in weak-field expansion reads

$\displaystyle\mathcal{J}_{B}(\phi)$
$\displaystyle=\frac{k^{4}}{8\pi^{2}}\Bigg{[}-\frac{\Lambda_{\perp}^{2}}{(k^{2}+m_{\phi}^{2})(k^{2}+m_{\phi}^{2}+\Lambda_{\perp}^{2})}$
$\displaystyle+\bigg{(}\frac{1}{3(k^{2}+m_{\phi}^{2})^{3}}-\frac{k^{2}+m_{\phi}^{2}-5\Lambda_{\perp}^{2})}{3(k^{2}+m_{\phi}^{2}+\Lambda_{\perp}^{2})^{4}}\bigg{)}(q_{\phi}B)^{2}$
$\displaystyle+\mathcal{O}(q_{\phi}B)^{4}.$ (42)

For the loop functions of neutral meson, we just need to set $q_{\phi}=0$. And
loop functions of the charge pion in Landau leval representation reads

$\displaystyle\mathcal{J}_{B}(\phi)=-\frac{k^{4}|q_{\phi}B|}{4\pi^{2}}\sum_{n=0}^{\Lambda_{\perp,n}}\frac{1}{k^{2}+(2n+1)|q_{\phi}B|+m_{\phi}^{2}}.$
(43)

The $\sigma-\pi$ loop functions in weak-field expansion read

$\displaystyle\mathcal{J}_{2B}(\pi,\sigma)$ $\displaystyle=$
$\displaystyle\frac{k^{4}}{8\pi^{2}}\Bigg{[}\bigg{(}\frac{1}{(k^{2}+m_{\sigma}^{2}+\Lambda_{\perp}^{2})(k^{2}+m_{\pi}^{2}+\Lambda_{\perp}^{2})}$
$\displaystyle-\frac{1}{(k^{2}+m_{\sigma}^{2})(k^{2}+m_{\pi}^{2})}\bigg{)}$
$\displaystyle-\int_{0}^{\Lambda_{\perp}^{2}}\bigg{(}\frac{5p_{\perp}^{2}-3m_{\pi}^{2}-3k^{2}}{(k^{2}+m_{\pi}^{2}+p_{\perp}^{2})^{5}(k^{2}+m_{\sigma}^{2}+p_{\perp}^{2})}$
$\displaystyle+\frac{p_{\perp}^{2}-k^{2}-m_{\pi}^{2}}{(k^{2}+m_{\pi}^{2}+p_{\perp}^{2})^{4}(k^{2}+m_{\sigma}^{2}+p_{\perp}^{2})^{2}}\bigg{)}dp_{\perp}^{2}(q_{\pi}B)^{2}$
$\displaystyle+\mathcal{O}(q_{\pi}B)^{4}$ (44)

with neutral pion $q_{\pi^{0}}=0$ and charged pion $q_{\pi^{\pm}}=\pm e$.

The $\sigma-\pi^{\pm}$ loop function in Landau leval representation reads

$\displaystyle\mathcal{J}_{2B}(\pi^{\pm},\sigma)=$
$\displaystyle-\frac{k^{4}}{4\pi^{2}}\sum_{n=0}^{\Lambda_{\perp,n}}\bigg{(}\frac{1}{(k^{2}+(2n+1)|eB|+m_{\pi}^{2})^{2}}$
$\displaystyle\int_{0}^{\infty}\frac{e^{-y}\mathcal{L}_{n}(2y)dy}{y^{2}+(k^{2}+m_{\sigma}^{2})/|eB|}$
$\displaystyle+\frac{1}{(k^{2}+(2n+1)|eB|+m_{\pi}^{2})|eB|}$
$\displaystyle\int_{0}^{\infty}\frac{e^{-y}\mathcal{L}_{n}(2y)dy}{(y^{2}+(k^{2}+m_{\sigma}^{2})/|eB|)^{2}}\bigg{)}$
(45)

The weak-field expansion of quark loop of the two-point 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

$\displaystyle\mathcal{J}_{2F}(q_{f})=\frac{k^{4}N_{c}}{2\pi^{2}}\sum_{n=0}^{\Lambda_{\perp,n}}\sum_{s=\pm
1}\frac{1}{(k^{2}+m_{f}^{2}+|q_{f}B|(2n+1+s))^{2}}$ (46)

For the charged pion two-point correction, it contains a $u-d$ quark loop. The
threshold function of this diagram reads

$\displaystyle\mathcal{J}_{2F}(u,d)=$
$\displaystyle\frac{N_{c}k^{4}B}{\pi^{2}}\sum_{n_{1},n_{2}}^{\Lambda_{\perp,n}}(-1)^{(n_{1}+n_{2})}[((\bar{G}^{u}_{n_{1}})^{2}\bar{G}^{d}_{n_{2}}$
$\displaystyle+\bar{G}^{u}_{n_{1}}(\bar{G}^{d}_{n_{2}})^{2})((k^{2}+m_{f}^{2})(LL(n_{1},n_{2}-1)$
$\displaystyle+LL(n_{1}-1,n_{2}))-8BL^{1}L^{1}(n_{1}-1,n_{2}-1))$
$\displaystyle-\bar{G}^{u}_{n_{1}}\bar{G}^{d}_{n_{2}}(LL(n_{1},n_{2}-1)+LL(n_{1}-1,n_{2}))$
(47)

here

$\displaystyle\bar{G}^{q_{f}}_{n}\equiv\frac{1}{k^{2}+m_{f}^{2}+2n|q_{f}B|)}$
(48)

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

$\displaystyle LL(n_{1},n_{2})\equiv\int_{0}^{\infty}$ $\displaystyle
dx\exp(-x(\frac{1}{|q_{u}|}+\frac{1}{|q_{d}|}))$ (49)
$\displaystyle\mathcal{L}_{n_{1}}(\frac{2x}{|q_{u}|})\mathcal{L}_{n_{2}}(\frac{2x}{|q_{d}|})$
$\displaystyle L^{1}L^{1}(n_{1},n_{2})\equiv\int_{0}^{\infty}$ $\displaystyle
dxx\exp(-x(\frac{1}{|q_{u}|}+\frac{1}{|q_{d}|}))$ (50)
$\displaystyle\mathcal{L}^{1}_{n_{1}}(\frac{2x}{|q_{u}|})\mathcal{L}^{1}_{n_{2}}(\frac{2x}{|q_{d}|}),$

with $\mathcal{L}_{n}^{a}(x)$ are the generalized Laguerre polynomials.

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