Magnetic polarizabilities of light mesons in $SU(3)$ lattice gauge theory

We investigate the masses (ground state energies) of neutral pseudoscalar and vector meson in $SU(3)$ lattice gauge theory in strong abelian magnetic field. The energy of $\rho^0$ meson with zero spin projection $s_z=0$ on the axis of the external magnetic field decreases, while the energies with non-zero spins $s_z=-1$ and $+1$ increase with the field. The energy of $\pi^0$ meson decrease as a function of the magnetic field. We calculated the magnetic polarizabilities of pseudoscalar and vector mesons for lattice volume $18^4$. For $\rho^0$ with spin $|s_z|=1$ and $\pi^0$ meson the extrapolations to zero lattice spacing have been done. We do not observe any evidence in favour of tachyonic mode existence.


INTRODUCTION
Quantum Chromodynamics in abelian magnetic filed of hadronic scale is a reach area for exploration. The investigation of strongly interacting quark-hadronic matter in such field has a deep fundamental meaning. In a present days to create a strong magnetic field of 15m 2 π ∼ 0.27 GeV 2 [1] turns to be possible in a terrestrial laboratories (ALICA, RHIC, NICA, FAIR) during noncentral heavy ion collisions. Our studies have a goal to shed light on the effects that can appear in such experiments. The properties of fundamental particles related to their internal structure are also very important for understanding of the effects observed at the experiments.
Let us mention the most famous results concerning QCD physics in strong magnetic fields. The charge asymmetry of emitted charged particles [2][3][4] in non-central collisions of gold ions at RHIC was explained by chiral magnetic effect [4][5][6]. Many phenomenological studies have been devoted to the understanding of QCD phase structure in strong magnetic fields. Chiral perturbation theory predicts decrease of transition temperature with increasing Abelian magnetic field [7], and this result is consistent with the expectation of the lattice [8]. External magnetic fields also lead to strengthening of the chiral symmetry breaking [9][10][11][12][13]. In the framework of the Nambu-Jona-Lasinio model it was shown that QCD vacuum becomes a superconductor in sufficiently strong magnetic field (B c = m 2 ρ /e ≃ 10 16 Tl) [14][15][16][17][18] along the direction of the magnetic field. This transition to superconducting phase is accompanied by a condensation of charged ρ mesons. The strong magnetic fields physics can also change the order of the phase transition from confinement phase to deconfinement phase [19][20][21][22][23].
Lattice studies reveals such interesting effect as an inverse magnetic catalysis [24]. According to the calculations on the lattice with two types of valence quarks in QCD, the critical temperature of the transition from confinement phase to deconfinement phase increases in a strong magnetic field [8]. Calculations in the theory with N f = 2 + 1 [25] with dynamical quarks showed that T c decreased with increasing magnetic field. Lattice simulations with dynamical overlap fermions in two-flavor lattice QCD also showed the decrease of the critical temperature of confinement -deconfinement transition when the field strength grows [26].
Numerical simulations in QCD with N f = 2 and N f = 2 + 1 show that strongly interacting matter in strong magnetic field posses paramagnetic properties in the confinement and deconfinement phases [27][28][29]. Equation of state of quark-gluon plasma was investigated in [30].
Here we continue our previous work where we studied light mesons in SU (2) lattice gauge theory [31]. We extend this analysis to the SU (3) lattice gauge theory which is more realistic, and calculate the ground state energies of the light mesons as functions of magnetic field depending on their spin. Our previous results are in the qualitative agreement with the results of this work. We also calculate several hadronic characteristics such as magnetic polarizabilities of light neutral pseudoscalar and vector mesons. The magnetic polarizability is an important physical quantity which reveals the internal structure of a particle in external magnetic field. We also made the extrapolation to zero lattice spacing where it was possible. Our approach is numerically expensive so we do not take into account dynamical quarks. The quark mass extrapolations for the vector meson masses with various spins also have been done.
Several articles have been devoted to the study of meson masses in strong magnetic field. The masses of ρ mesons have been calculated according to the relativistic quark-antiquark model in [32]. Lattice study is given in [33] in the approach with dynamical quarks and agree with our data for eB < 1 GeV 2 . Phenomenological study have been done in [34]. For the case of non-zero spin our results in a qualitative agreement with the results [32][33][34]. For zero spin our data agree with these results only for small magnetic fields.

DETAILS OF CALCULATIONS
For generation of SU (3) gauge configurations the tadpole improved Symanzik action was used where S pl,rt = (1/2)Tr (1 − U pl,rt ) is lattice plaquette (denoted as pl) or 1×2 rectangular loop (rt), u 0 = (W 1×1 ) 1/4 = (1/2)Tr U pl 1/4 -is a tadpole factor, calculated at zero temperature [35]. This action suppresses ultraviolet dislocations that lead to creation of non-physical near zero modes of Wilson-Dirac operator. Next, we solve Dirac equation numerically and find eigenfunctions ψ k and eigenvectors λ k for test quark in the external gauge field A µ . Eigenmodes of the Dirac operator are used for calculation of correlators, which are used for calculation of the energy of the ground state of mesons. For the calculation of the fermion spectrum Neuberger overlap operator was used [36]. This operator allows to investigate the theory in the limit of massless quarks without breaking of chiral symmetry and can be written in this form where D W = M − ρ/a is a Wilson-Dirac operator with negative ρ/a, a is the lattice spacing in physical units, M is a Wilson term. Fermion fields obey periodical boundary conditions in space and antiperiodical boundary conditions in time. Sign function is calculated using minmax polynomial approximation, where H W = γ 5 D W is hermitian Wilson-Dirac operator. We investigate behaviour of the meson ground energy state in background gauge field, which is a sum of non-abelian SU (3) gluon field and U (1) abelian uniform magnetic field. Abelian gauge fields interact only with quarks. In our calculations we have neglected the contribution of dynamical quarks. Therefore, we added the magnetic field only in overlap Dirac operator. For this reason we use the following ansatz: where In order to make this substitution consistent with fermion boundary conditions, one should use the twisted boundary conditions [37]. Magnetic field is directed along z axis and its value is quantized where q = −1/3 e. The quantization condition implies that the magnetic field has a minimal value √ eB min = 380 MeV for 18 4 lattice volume and a = 0.125 fm. We are far from saturation regime, where k/(L 2 ) is not small because we use k between 0 and 32. For the inversion of overlap Dirac operator we use Gaussian source (with radius r = 1.0 in lattice units in space and time direction) and point receiver ( the quark position smoothed with gauss profile). Our simulations have been carried out on symmetrical lattices with lattice volume 16 4 with lattice spacing 0.105 fm and 18 4 with lattice spacings a = 0.105 fm, 0.115 fm and 0.125 fm . We use statistically independent configurations of gluon field for the every value of the quark mass.

OBSERVABLES
We calculate the following observables in coordinate space and background gauge field A where O 1 , O 2 = γ 5 , γ 5 γ µ , γ µ are Dirac gamma matrices, µ, ν = 1, .., 4 are Lorenz indices, x = (na, n t a) and y = (n ′ a, n ′ t a) are coordinates on the lattice, spatial lattice coordinate n, n ′ ∈ Λ 3 = {(n 1 , n 2 , n 3 )|n i = 0, 1, ..., N − 1} , n t , n ′ t are numbers of the lattice sites in the time direction. In the Euclidean space ψ † =ψ. In order to calculate the observables (8) we calculate quark propagators in coordinate space. For M lowest eigenmodes massive Dirac propagator is represented by the following sum: In our calculations we use M = 50. For the observables (8) the following equation is valid First term in (10) is connected part, second term is disconnected part. We have checked that in SU (3) theory without dynamical quarks the disconnected part contribution to correlators is zero. To explore meson states at definite spatial momentum p we perform Fourier transformation numericallỹ The momenta p has components For particles with zero momentum their energy is equal to mass E 0 = m 0 . As we are interested in the meson ground state, we choose p = 0. For the calculation of the masses we perform the expansion of correlation function to exponential series A 0 , A 1 are constants, E 0 is the energy of the ground state. E 1 is the energy of first exited state, a is the lattice spacing, n t is the number of nodes in the time direction. From expansion (13) one can see that for large n t the main contribution origins from the ground state. Because of the periodic boundary conditions the main contribution to the ground state has the following form Mass of the ground state can be evaluated fitting the correlator (12) with (14) function. In order to minimize the errors and exclude the contribution of the exited states we take different values of n t from the interval 5 ≤ n t ≤ N T − 5. Masses of the ρ mesons have been obtained from correlator (8) In our calculations u and d quarks are degenerate.
The value of the ground state mass can be obtained by fitting the function (14) to the lattice correlator (12). To minimize errors we take various n t values from the interval 4 < n t < N T − 4.  Fig. 1 shows the ground state energy of the neutral pion obtained from the correlator C P SP S = ψ (0, n t )γ 5 ψ(0, n t )ψ(0, 0)γ 5 ψ(0, 0) . On this plot we present the data for the smallest bare quark mass m q = 34.26 MeV, lattice volume 18 4 with lattice spacings 0.105 fm, 0.115 fm, 0.125 fm and lattice volume 16 4 with lattice spacing 0.115 fm. The mass of π 0 decreases for the all sets of lattice data.

THE GROUND STATE ENERGIES OF MESONS IN STRONG MAGNETIC FIELD
At moderate values of magnetic field the energy of the pion slightly depends on the lattice volume and lattice as a function of the magnetic field. The lattice data are the same that in Fig.2 spacing. With increase of the field value the lattice effects become more strong, because the wave function of light pion becomes of the order or exceeds the lattice size. This decrease of pion mass is compensated by higher powers of eB for its values larger than 1 GeV or so. These effects are preventing, in particular, from mass turning to zero and possible emergence of tachyonic mode. The same will be true also for the energies of all spin states of neutral and charged [38] ρ mesons. Still, the observed non-occurrence of tachyonic modes requires further investigation.
To obtain the energies of neutral vector mesons with various spin projections on the axis of the external magnetic field we use the combinations of the correlators in various spatial dimensions.
C V V yy = ψ (0, n t )γ 2 ψ(0, n t )ψ(0, 0)γ 2 ψ(0, 0) , The mass of ρ 0 meson with s z = 0 spin projection is obtained from the C V V zz correlator. The combinations of correlators give the ground state energies with spins s z = +1 and s z = −1. Fig. 3 represents the mass of ρ 0 with non-zero spin projection on the axis of external magnetic field. The energy increases with the field for the all sets of lattice data, at |eB| 2.5 GeV 2 the energy increase quadratically and at large magnetic fields it looks like a plateau. The imaginary terms iC V V xy and iC V V yx in (18) are zero for the case of neutral particles so the ρ 0 masses with s = −1 and s z = +1 coincide that is the consequence of C-parity. In Fig. 3 we also see that at some value of magnetic field ∼ 2.5 GeV 2 the data becomes tightly dependent on the lattice spacing and lattice volume. Simple estimates give the following values of magnetic fields corresponding to the lattice spacing cut-off: eB ∼ 3.5 GeV 2 for a = 0.105 fm, 2.9 GeV 2 for lattice spacing a = 0.115 fm and 2.5 GeV 2 for a = 0.125 fm.
In Fig. 2 we see the ground state energy of ρ 0 meson with s z = 0 for various lattice spacings and volumes. In spite of mixing between ρ 0 (s z = 0) and π 0 (s = 0) having the same quantum number we expect that at magnetic fields eB < 2 GeV 2 such correlator gives ρ 0 meson. Therefore the mass of the state diminishes as a function of the magnetic field at eB < 2 GeV 2 . In Fig.4 we show the sum of three energy components of vector neutral meson.
For the magnetic fields eB < 2 GeV 2 the lattice spacing effects exist. However with the decrease of the lattice spacing the mass of unpolarized meson becomes closer to a constant value, so the mixing between ρ 0 (s z = 0) and π 0 (s = 0) states has to be weak at such magnetic fields. If we suggest that the energy of unpolarized meson is constant then the raising energy shift at larger fields in Fig.4 demonstrates the increase of branching for the decay ρ 0 → π 0 γ at strong magnetic fields.
From the assumption of constant energy of unpolarized meson and Fig.3 we can conclude that there is no tachyonic mode for the explored range of magnetic fields, i.e. the mass of ρ 0 (s = 0) doesn't turn to zero. As we see the lattice volume effects are small and do not change this conclusion at large magnetic fields. Therefore our calculations show that there is a splitting of ground state energy of neutral vector meson in a strong abelian magnetic field that is an interesting physical effect.

MAGNETIC POLARIZABILITIES
The polarizability of a particle is an important physical quantity for understanding of its internal hadronic structure. The magnetic polarizability of meson shows how it responds to the external magnetic field. In this section we talk about magnetic polarizabilities of pseudoscalar π 0 and vector ρ 0 mesons. In Fig.5 we show the ground state energy of pion as a function of squared magnetic field (eB) 2 for fields (eB) 2 < 0.4 GeV 4 . We fit the data at (eB) 2 ∈ [0, 0.2 GeV 4 ] by the following function where E(B = 0) and β m are the parameters which we find from the fit. We choose this interval for the fit because we consider the terms ∼ (eB) 4 give small contribution to the pion energy at such magnetic field values.
The obtained values of polarizabilities for various lattice spacings is summarized in Table 1. We do not see any dependence on the lattice size so make the lattice extrapolation to the limit of zero lattice spacing by a constant function, which is shown in Fig.6. As a result we obtain the magnetic polarizability of π 0 meson β m (π 0 ) = (0.044 ± 0.005) 1/GeV 3 or (3.4 ± 0.4) · 10 −4 fm −3 . This number agrees with sign with the result of work [39] but in approximately in 2.7 times larger.   Table 2 and shown in Fig. 8.
We see the strong dependence of the results on the lattice spacing so we make an extrapolation to continuum limit. The extrapolation gives the value of magnetic polarizability β  In Fig.9 the mass of ρ 0 meson with zero spin is depicted for the small magnetic fields. We observe the linear in (eB) 2 behaviour only for (eB) 2 ∈ [0, 0.   (20) to the pion mass mπ = 135 MeV.
In Fig.11 we show the quark mass extrapolation of the neutral vector ρ meson spin |s z | = 1 for lattice volume 18 4 and lattice spacing a = 0.115 fm. The mass of ρ 0 meson was calculated for several m q values in the interval m q a ∈ [0.02, 0.06]. Then we perform a fit by a linear function m ρ = a 0 + a 1 m q (20) and find the coefficient a 0 and a 1 and its errors from the fit by χ 2 method. Then we extrapolate m ρ (m q ) to physical value m ρ (m q0 ) at m q = m q0 corresponding to the pion mass m π = 135 MeV. We perform this procedure for every spin projection, lattice spacing and magnetic field value. The result of such extrapolation is presented in Fig.11. The masses of ρ 0 with zero spin smoothly decrease with magnetic field while the energies with masses spin increase with the field value. In Table 1 we also represent the values of magnetic polarizability obtained from the fits to the data after quark mass extrapolation and compare its values with the magnetic polarizability for m q = 34.26 MeV. The value of β |s|=1 m after quark mass extrapolation (−0.0136±0.0005) 1/GeV 3 agrees with the value (−0.0142 ± 0.0008) 1/GeV 3 obtained for the bare quark mass m q = 34 MeV and lattice spacing for lattice a = 0.115 fm. The results for lattice spacing a = 0.125 fm doesn't cohere so excellent, we have to increase statistics in this point.

CONCLUSIONS
In this work we explore the ground state energies of π 0 and ρ 0 mesons. The mass of pseudoscalar meson diminishes with the field value. We observe that the energies of the ground state of neutral vector ρ meson with zero spin projection on the axis of the external magnetic field decrease while the energies with non-zero spin increase as a function of magnetic field. The magnetic polarizability of ρ 0 meson with s z = 0 differs from the magnetic polarizability of ρ 0 meson with |s z | = 1. We consider this phenomena to be the result of the anisotropy created by the strong magnetic field. The energies of ρ 0 with spin s z = +1 and s z = −1 coincides which is the consequence of C-parity. For vector meson the magnetic polarizability β |s|=1 m (ρ 0 ) = (−0.029 ± 0.08) 1/GeV 3 after extrapolation to zero lattice spacing a = 0. For zero spin β s=0 m (ρ 0 ) is opposite in sign than for nonzero spin case and much larger in absolute value. The magnetic polarizability of π 0 meson β m (π 0 ) = (0.044 ± 0.005) 1/GeV 3 . The mixing between π 0 and ρ 0 (s = 0) states doesn't manifest itself at eB < 2 GeV 2 . We also do not observe any evidence in favour of tachionic mode existence.

ACKNOWLEDGMENTS
This work was carried out with the financial support of Grant of President MK-6264.2014.2 and FRRC grant of Rosatom SAEC and Helmholtz Assotiation. The authors are grateful to FAIR-ITEP supercomputer center where these numerical calculations were performed. O.T. is supported by RFBR grants 12-02-00613, 14-01-00647 and in part by Heisenberg-Landau program.