Magnetic field dependence of the neutral pion mass in the linear sigma model with quarks: The strong field case

Electromagnetic fields play a relevant role for the dynamics of strongly interacting systems. For instance, it is well known that an external magnetic field helps to catalyze the breaking of chiral symmetry, producing a stronger quark-antiquark condensate catalysis . On the other hand, when temperature is taken into account, magnetic fields inhibit the condensate formation producing the opposite effect whereby the pseudocritical temperature for the chiral phase transition is reduced, it is the so-called inverse magnetic catalysis LQCD ; Bruckmann ; Farias ; Ferreira ; Ayala0 ; Ayala1 ; Ayala2 ; Ayala3 ; Avancini ; Ayala4 ; vertex1 ; vertex2 ; qcdcoupling ; Mueller ; imcreview . In this context, the properties of hadron degrees of freedom in the presence of magnetic fields have become a subject of intense study bali01 ; iranianos ; simonov03 ; aguirre02 ; tetsuya ; dudal04 ; kevin ; gubler ; noronha01 ; morita ; morita02 ; sarkar03 ; band ; Ayalachi ; nosso1 ; nosso03 ; zhuang ; iran ; scoccola01 ; huang01 ; scoccola02 ; scoccola03 ; luch ; farias01 ; mao01 ; sarkar ; sarkar02 ; zhang01 ; huan02 ; simonov01 ; fraga01 ; aguirre ; taya ; shinya ; andersen01 ; kojo ; simonov02 ; ghoshrho ; AMMmesons ; TBspectralprop ; ghoshEPJA ; Avila ; Avila:2020ved ; dudal ; dudal02 ; andrei02 ; he ; nucleon ; barionslattice ; Sheng .

Given that, from the hadron sector, the dynamics of chiral symmetry breaking is dominated by pions, it becomes important to study the influence of magnetic fields on pion properties such as masses and form factors. On general grounds, charged and neutral pions behave differently under the influence of an external magnetic field. Charged pions with mass $m_{0}$ and at rest in the direction of the magnetic field, have an energy spectrum given by $E^{2}=m_{0}^{2}+(2n+1)|eB|$, where $|eB|$ is the field strength and $n$ labels the $n$-th Landau level. The lowest energy state can be interpreted as the magnetic field-dependent mass, which is then given by $m_{B}^{2}=m_{0}^{2}+|eB|$. In contrast, neutral pions do not experience directly the effects of a magnetic background and thus their mass remains at first sight unaffected. Interactions with other particles that populate the strongly interacting vacuum can change this picture. In fact, the magnetic field driven modifications of the neutral pion mass were first computed by lattice QCD (LQCD) calculations in Refs. Hidaka ; Luschevskaya . These works found contradictory results: whereas Ref. Luschevskaya obtains a neutral pion mass that monotonically decreases with the field strength, Ref. Hidaka finds a dip at an intermediate value and then an increase for larger field strengths. This discrepancy was analyzed in Refs. Endrodi ; Ding where the monotonically decrease of the pion mass as a function of the field strength was confirmed.

The problem has been also addressed from the point of view of effective models. Working within the linear sigma model with quarks (LSMq) in the weak field limit, Ref. Ayala1 has shown that the neutral pion mass starts off decreasing as a function of the field strength. An important element for this finding is to account for the magnetic field driven modification of the boson self-coupling. Within the same model, and also with magnetic field modified couplings, Ref. Das finds that the neutral pion mass starts off decreasing to then increase at an intermediate value of the field strength, becoming even larger than the mass at zero field strength. A similar behavior is found in Refs. Li ; Scoccola using the NJL model.

The results of Ref. Das , which are in contrast with the most recent LQCD results Endrodi ; Ding , may be due to the use of a procedure, advocated in Ref. Scoccola , to remove the vacuum, which in Ref. Ayala1 has been shown to not describe the large magnetic field strength limit, generating a discrepancy with the well established LLL result. Moreover, since the effective boson-fermion coupling in Ref. Das is computed from the magnetic field corrections to the quark mass, it is not clear what, if any, is the role of Schwinger’s phase factor when this effective coupling is computed from the one-loop triangle perturbative correction.

In order to assess whether the use of one-loop magnetic field-modified couplings can account for the behavior of the recent LQCD results for the magnetic field dependence of the neutral pion mass, in Ref. AHHFZ we made a detailed study of the magnetic field modifications to the boson self-coupling and boson-fermion coupling in the LSMq. We found that the couplings experience a monotonic decrease as a function of the field strength. The pending question is thus to clarify what are the overall ingredients that can explain the neutral pion mass monotonic decrease as a function of the field strength. In this work we address this question within the LSMq, showing that the elements driving the neutral pion mass behavior are the properly combined effects of the magnetic field corrections to the couplings together with the contribution from charged particles to the one-loop pion self-energy and to the vacuum expectation value of the sigma field.

The work is organized as follows: In Sec. II, we introduce the linear sigma model with quarks. In Sec. III we make a quick survey of the way magnetic field effects are introduced into the propagators of charged bosons and fermions. In Sec. IV, we compute the necessary elements to obtain the magnetic modification of the pion mass to one-loop order, namely, the neutral pion self-energy, the vacuum expectation value of the sigma field from the effective potential, and the correction to the couplings. In Sec. V we compute the magnetic corrections to the neutral pion mass and compare to recent LQCD calculations, showing that the monotonic decrease with the field strength can be reproduced. We finally summarize and conclude in Sec VI. We reserve for the appendices the explicit calculation details for the one-loop corrections to both, the neutral pion self-energy and the effective potential.

The LSMq is an effective model that describes the low-energy regime of QCD, incorporating the spontaneous breaking of chiral symmetry. The Lagrangian for the LSMq can be written as

	$\displaystyle\mathcal{L}$	$\displaystyle=$	$\displaystyle\frac{1}{2}(\partial_{\mu}\sigma)^{2}+\frac{1}{2}(\partial_{\mu}\vec{\pi})^{2}+\frac{a^{2}}{2}(\sigma^{2}+\vec{\pi}^{2})-\frac{\lambda}{4}(\sigma^{2}+\vec{\pi}^{2})^{2}$		(1)
		$\displaystyle+$	$\displaystyle i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-ig\gamma^{5}\bar{\psi}\vec{\tau}\cdot\vec{\pi}\psi-g\bar{\psi}\psi\sigma.$

Pions are described by an isospin triplet, $\vec{\pi}=(\pi_{1},\pi_{2},\pi_{3})$. Two species of quarks are represented by an $SU(2)$ isospin doublet, $\psi$. The $\sigma$ scalar is included by means of an isospin singlet. Also, $\lambda$ is the boson self-coupling and $g$ is the fermion-boson coupling. $a^{2}>0$ is the mass parameter.

To allow for spontaneous symmetry breaking, we let the $\sigma$ field develop a vacuum expectation value $v$

	$\displaystyle\mathcal{L}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma+\frac{1}{2}\partial_{\mu}\pi_{0}\partial^{\mu}\pi_{0}+\partial_{\mu}\pi_{-}\partial^{\mu}\pi_{+}$		(2)
		$\displaystyle-$	$\displaystyle\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{2}m_{0}^{2}\pi_{0}^{2}-m_{0}^{2}\pi_{-}\pi_{+}+i\bar{\psi}\not{\partial}\psi$
		$\displaystyle-$	$\displaystyle m_{f}\bar{\psi}\psi+\frac{a^{2}}{2}v^{2}-\frac{\lambda}{4}v^{4}+\mathcal{L}_{int},$

where the charged pion fields can be expressed as

$$\pi_{\pm}=\frac{1}{\sqrt{2}}(\pi_{1}\pm i\pi_{2}),$$

(3)

and the interaction Lagrangian is defined as

$$\begin{split}\mathcal{L}_{int}&=-\frac{\lambda}{4}\sigma^{4}-\lambda v\sigma^{3}-\lambda v^{3}\sigma-\lambda\sigma^{2}\pi_{-}\pi_{+}-2\lambda v\sigma\pi_{-}\pi_{+}\\ &-\frac{\lambda}{2}\sigma^{2}\pi_{0}^{2}-\lambda v\sigma\pi_{0}^{2}-\lambda\pi_{-}^{2}\pi_{+}^{2}-\lambda\pi_{-}\pi_{+}\pi_{0}^{2}-\frac{\lambda}{4}\pi_{0}^{4}\\ &+a^{2}v\sigma-g\bar{\psi}\psi\sigma-ig\gamma^{5}\bar{\psi}\left(\tau_{+}\pi_{+}+\tau_{-}\pi_{-}+\tau_{3}\pi_{0}\right)\psi.\end{split}$$

(4)

In order to include a finite vacuum pion mass, $m_{0}$, one adds an explicit symmetry breaking term in the Lagrangian of Eq. (2) such that

$$\mathcal{L}\rightarrow\mathcal{L^{\prime}}=\mathcal{L}+\frac{m_{0}^{2}}{2}v(\sigma+v).$$

(5)

As can be seen from Eqs. (2) and (4) there are new terms which depend on $v$ and all fields develop dynamical masses

	$\displaystyle m_{\sigma}^{2}$	$\displaystyle=3\lambda v^{2}-a^{2},$
	$\displaystyle m_{0}^{2}$	$\displaystyle=\lambda v^{2}-a^{2},$
	$\displaystyle m_{f}$	$\displaystyle=gv.$		(6)

Using Eqs. (2) and (5), the tree-level potential is given by

$$V^{\text{tree}}(v)=-\frac{a^{2}+m_{0}^{2}}{2}v^{2}+\frac{\lambda}{4}v^{4}.$$

(7)

This potential develops a minimum, called the vacuum expectation value of the $\sigma$ field, namely

$$v_{0}=\sqrt{\frac{a^{2}+m_{0}^{2}}{\lambda}}.$$

(8)

Therefore, the masses evaluated at $v_{0}$ are

	$\displaystyle m_{f}(v_{0})$	$\displaystyle=g\sqrt{\frac{a^{2}+m_{0}^{2}}{\lambda}},$
	$\displaystyle m_{\sigma}^{2}(v_{0})$	$\displaystyle=2a^{2}+3m_{0}^{2},$
	$\displaystyle m_{0}^{2}(v_{0})$	$\displaystyle=m_{0}^{2}.$		(9)

Finally, an external magnetic field, uniform in space and constant in time, can be included in the model introducing a covariant derivative in the Lagrangian density, Eq. (2), namely

$$\partial_{\mu}\to D_{\mu}=\partial_{\mu}+iqA_{\mu},$$

(10)

where $A^{\mu}$ is the vector potential corresponding to an external magnetic field directed along the $\hat{z}$ axis. In the symmetric gauge, this is given by

$$A^{\mu}(x)=\frac{1}{2}x_{\nu}F^{\nu\mu},$$

(11)

and couples only to the charged pions and to the quarks.

Notice that, in order to consider the propagation of charged particles, one can resort to introduce Schwinger propagators which can be expressed either in terms of their proper time representation or as a sum over Landau Levels. For completeness of the presentation, we now proceed to briefly discuss the properties of these propagators.

In order to consider the propagation of charged particles within a magnetized background, we use Schwinger’s proper time representation. The fermion propagator can be written as schwinger

$$S_{f}(x,x^{\prime})=e^{i\Phi(x,x^{\prime})}S_{f}(x-x^{\prime}),$$

(12)

where $\Phi(x,x^{\prime})$ is the Schwinger’s phase given by

$\displaystyle\Phi(x,x^{\prime})=q\int_{x}^{x^{\prime}}d\xi_{\mu}\left[A^{\mu}(\xi)+\frac{1}{2}F^{\mu\nu}(\xi-x^{\prime})_{\nu}\right],$

(13)

where $q$ is the particle electric charge. $\Phi(x,x^{\prime})$ corresponds to the translationally non-invariant and gauge dependent part of the propagator. On the other hand, $S_{f}(x-x^{\prime})$ is translationally and gauge-invariant and can be expressed in terms of its Fourier transform as

$$S_{f}(x-x^{\prime})=\int\frac{d^{4}p}{(2\not{p})^{4}}S_{f}(p)e^{-ip\cdot(x-x^{\prime})},$$

(14)

where

	$\displaystyle iS_{f}(p)$	$\displaystyle=$	$\displaystyle\int_{0}^{\infty}\frac{ds}{\cos(|q_{f}B|s)}e^{is\left(p_{\parallel}^{2}-p_{\perp}^{2}\frac{\tan(|q_{f}B|s)}{|q_{f}B|s}-m_{f}^{2}+i\epsilon\right)}$		(15)
		$\displaystyle\times$	$\displaystyle\left[\Big{(}\cos(|q_{f}B|s)+\gamma_{1}\gamma_{2}\sin(|q_{f}B|s)\text{sign}(q_{f}B)\Big{)}\right.$
		$\displaystyle\times$	$\displaystyle\left.\left(m_{f}+\not{p}_{\parallel}\right)-\frac{\not{p}_{\perp}}{\cos(|q_{f}B|s)}\right].$

In a similar fashion, for a charged scalar field we have

	$\displaystyle D(x,x^{\prime})$	$\displaystyle=$	$\displaystyle e^{i\Phi(x,x^{\prime})}D(x-x^{\prime}),$
	$\displaystyle D(x-x^{\prime})$	$\displaystyle=$	$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}D(p)e^{-ip\cdot(x-x^{\prime})},$		(16)

with

$\displaystyle iD(p)$

$\displaystyle=$

$\displaystyle\int_{0}^{\infty}\frac{ds}{\cos(|q_{b}B|s)}e^{is\left(p_{\parallel}^{2}-p_{\perp}^{2}\frac{\tan(|q_{b}B|s)}{|q_{b}B|s}-m_{b}^{2}+i\epsilon\right)},$

where the boson and fermion masses and electric charges are $m_{b}$, $q_{b}$ and $m_{f}$, $q_{f}$, respectively.

The propagators in Eqs. (15) and (LABEL:bosonpropagatormomentumspace) can also be expanded as a sum over Landau levels. In this case, the expressions for the charged fermion and scalar propagators are given by Ayala-Sahu ; Shovkovy

$$iS_{f}(p)=ie^{-\frac{p_{\perp}^{2}}{|q_{f}B|}}\sum_{n=0}^{\infty}\frac{(-1)^{n}D_{n}(p)}{p_{\parallel}^{2}-m_{f}^{2}-2n|q_{f}B|+i\epsilon},$$

(18)

$$iD_{b}(p)=2ie^{-\frac{p_{\perp}^{2}}{|q_{b}B|}}\sum_{n=0}^{\infty}\frac{(-1)^{n}L_{n}^{0}\left(\frac{2p_{\perp}^{2}}{|q_{b}B|}\right)}{p_{\parallel}^{2}-m_{b}^{2}-(2n+1)|q_{b}B|+i\epsilon},$$

(19)

respectively, where

	$\displaystyle D_{n}(p)$	$\displaystyle=$	$\displaystyle 2(\not{p}_{\parallel}+m_{f})\mathcal{O}^{+}L_{n}^{0}\left(\frac{2p_{\perp}^{2}}{|q_{f}B|}\right)$		(20)
		$\displaystyle-$	$\displaystyle 2(\not{p}_{\parallel}+m_{f})\mathcal{O}^{-}L_{n-1}^{0}\left(\frac{2p_{\perp}^{2}}{|q_{f}B|}\right)$
		$\displaystyle+$	$\displaystyle 4\not{p}_{\perp}L_{n-1}^{1}\left(\frac{2p_{\perp}^{2}}{|q_{f}B|}\right),$

and $L_{n}^{m}(x)$ are the generalized Laguerre polynomials. Also, in Eq. (20) the operators ${\mathcal{O}^{\pm}}$ are defined as

$\displaystyle{\mathcal{O}^{\pm}}=\frac{1}{2}\left(1\pm i\gamma_{1}\gamma_{2}\,{\mbox{sign}}(qB)\right).$

(21)

We now use these ingredients to compute the elements necessary to obtain the magnetic modification of the neutral pion mass.

In order to compute the magnetic field-induced modification to the neutral pion mass, the starting point is the equation defining its dispersion relation in the presence of the magnetic field, namely

$\displaystyle q_{0}^{2}-|\vec{q}|^{2}-m_{0}^{2}(B)-{\mbox{Re}}[\Pi(B,q;\lambda_{B},g_{B},v_{B})]=0,$

(22)

where $\Pi$ is the neutral pion self-energy and $\lambda_{B},g_{B},v_{B}$ represent the magnetic field dependent boson-self coupling, boson-fermion coupling and vacuum expectation value, respectively. The computation requires knowledge of each of these elements as functions of the field strength. $v_{B}$ can be computed finding the minimum of the magnetic field-dependent one-loop effective potential. This can be analytically computed using the full magnetic field dependence of the charged particle propagators. For the neutral pion self-energy and the magnetic field corrections to the couplings, we work in the large field limit and thus resort to use propagators in the lowest Landau level (LLL) approximation.

With all the elements at hand, we can now find the magnetic field-dependent neutral pion mass from the dispersion relation, Eq. (22), in the limit where $\vec{q}\to 0$ and $q_{0}\to m_{B}$, namely

$$m_{B}^{2}=m_{0}^{2}(B)+\Pi(B,q_{0}=m_{B},\vec{q}=0;\lambda_{B},g_{B},v_{B}),$$

(51)

where, in order to incorporate the magnetic field-dependent boson self-coupling and vacuum expectation value in the three-level pion mass, we write

$\displaystyle m_{0}^{2}(B)=\lambda_{B}v_{B}^{2}-a^{2}.$

(52)

To reduce the parameter space, we consider that in the absence of baryons, the constituent quark mass is such that $m_{0}=2m_{f}$. With this choice, the only free parameters are $\lambda$ and $g$. We have explored a large range for these parameters and hereby we show the results for the set that best describes simultaneously the LQCD data of Refs. Endrodi ; Ding . Moreover, the values we use as initial inputs for $g$ and $\lambda$ produce, for the lowest vacuum pion mass considered, values of $v_{0}$ larger than $f_{\pi}$ only by a factor $\sim 1.5$, which we take as an indication of consistency within the limitations of an effective theory such as the LSMq. Since Refs. Endrodi ; Ding report their findings for different values of the vacuum pion mass, we also vary this mass and consequently the rest of the dependent parameters have to be changed to suit these choices. In particular, a larger vacuum pion mass implies a larger $\sigma$ mass. Thus, in the strong field limit, our results are restricted to the domain where $|eB|>m_{\sigma}^{2}$.

Figure 6 shows the magnetic field-dependent neutral pion mass as a function of the field strength computed for two cases: with (black dots) and without (blue diamonds) magnetic field-dependent couplings, using as inputs $m_{0}=140$ MeV, $\lambda=3.67$, $g=0.46$, and correspondingly $m_{\sigma}=435$ MeV, $a=256$ MeV and $v_{0}=152$ MeV. Notice that, whereas the former shows a monotonic decrease, the latter starts off decreasing to later on increase as a function of the field strength. This result signals the importance of including magnetic field corrections to the couplings in the calculation of the magnetic field-dependent neutral pion mass.

In order to compare with LQCD simulations, which are implemented for different values of the vacuum pion mass, Fig.  7 shows the magnetic field-dependent neutral pion mass as a function of the field strength when varying the input vacuum pion mass. Shown are three cases: $m_{0}=140$ MeV (black dots), $m_{0}=220$ MeV (blue triangles) and $m_{0}=415$ MeV (red diamonds). Notice that, as the vacuum pion mass decreases, the corresponding magnetic field-dependent pion mass also decreases and that all cases show a monotonic decrease as a function of the field strength, in agreement with the LQCD findings.

To make direct contact with LQCD data, Fig. 8 shows the results for the magnetic field-dependent neutral pion mass as a function of the field strength using as input $m_{0}=415$ MeV and with $\lambda=3.67$ and $g=0.46$, compared to the results from Ref. Endrodi . The data points correspond to the $\pi_{d}$ (blue diamonds) and $\pi_{u}$ (red diamonds) masses computed also using as input $m_{0}=415$ MeV. Notice that our calculation does a nice description of the data average, particularly for the largest field strengths. Figure 9 shows also a comparison of our calculation with the LQCD calculation of Ref. Ding , this time computed with $m_{0}=220$ MeV as input together with $\lambda=3.67$ and $g=0.46$, The data points correspond to the $\pi_{d}$ (blue diamonds) and $\pi_{u}$ (red diamonds) masses computed also using as input $m_{0}=220$ MeV. Once again, we notice that our calculation does a nice job describing the average of the LQCD masses, particularly for large values of the field strength.