Magnetic polarizability of pion

# Magnetic polarizability of pion

## Abstract

We explore the energy dependence of mesons off the background Abelian magnetic field on the base of quenched SU(3) lattice gauge theory and calculate the magnetic dipole polarizability of charged and neutral pions for various lattice volumes and lattice spacings. The contribution of the magnetic hyperpolarizability to the neutral pion energy has been also found.

###### keywords:
lattice QCD, SU(3) gluodynamics, magnetic field, magnetic polarizability, pseudoscalar meson

## 1 Introduction

Quantum Chromodynamics in strong magnetic fields is a promising topic for research. Magnetic fields of hadronic scale could exist in the Early Universe Hector:00 () and could be formed in cosmic objects like magnetars and neutron stars. They can also be achieved in terrestrial laboratories (RHIC, LHC, FAIR, NICA) Skokov:2009 (). The external electromagnetic fields can be utilized as a probe of QCD properties, the recent progress obtained on this way in lattice gauge theories is discussed in Massimo:2015 (). The energy levels of hadrons in external magnetic field can be usefull for the calculation of the cross sections Savage:2015 (). The energies of mesons at the nonzero magnetic field were calculated in various phenomenological approaches Simonov:2013 (); Liu:2015 (); Taya:2015 (); Kawaguchi:2016 (); Hattori:2016 (), within the QCD sum rules Cho:2015 (); Gubler:2015 () and in the lattice gauge theories Bali:2015 (); Luschevskaya:2015a (); Luschevskaya:2015b ().

Background magnetic Abelian fields also enable to calculate the magnetic polarizabilities of hadrons. In order to get the dipole magnetic polarizability and the hyperpolarizability we measure the energy of a meson as a function of the magnetic field. The magnetic polarizabilities are important physical characteristics describing the distibution of quark currents inside a hadron in an external field. For the first time the concept of polarizability for the nuclear matter was used by A.B. Migdal in the analysis of the scattering of low energy gamma quanta by atomic nuclei Migdal (). For hadrons the notion of the polarizability was discussed in papers klein (); baldin ().

There were some discrepancies between the experimental obtained value of the magnetic and the electric polarizabilities of the charged mesons and some theoretical predictions based on the chiral perturbation theory Gasser:2005 (); Aleksejevs:2013 ().

Measurement of the electrical and the magnetic polarizabilities of mesons was performed on the spectrometer SIGMA-AJAX in Serpukhov, on the electron synchrotron Pakhra (LPI) in Moscow, on the MarkII detector at SLAC, at COMPASS (CERN) and other experiments.

According to the obtained data from these experiments, the value of the polarizability of the charged mesons is positive. The most precise value of the charged pion electric polarisability has been obtained experimentally by the COMPASS experiment Adolph:2015 () under the assumption . Comparison with the successful predictions of the experiments and the chiral perturbation theory Gasser:2005 (); Aleksejevs:2013 () is interesting for fundamental science.

In this work we consider a behaviour of the ground state energy of pions in the strong magnetic fields on the base of the SU(3) lattice gauge theory. The details of the calculations are briefly sketched in section 2. We discuss the dipole magnetic polarizabilities of and mesons in sections 3 and 4 accordingly. The magnetic hyperpolarizability of the neutral pion is calculated in section 5.

## 2 Details of calculations

### 2.1 The improved gauge action

For the generation of quenched lattice configurations we used the tadpole improved Lüscher-Weisz action Luscher:1985 (), which reduces the ultraviolet lattice artifacts. The action has the form

 S=βimp∑plSpl−βimp20u20∑rtSrt, (1)

where is the plaquette (denoted by pl) or 12 rectangular loop term (rt), is the input tadpole factor computed at zero temperature Bornyakov:2005 (). Our simulations have been carried out on the symmetrical lattices. The parameters of the lattice ensembles and number of the configurations are listed in the Table 1.

### 2.2 Fermionic spectrum

We solve the Dirac equation numerically

 Dψk=iλkψk,  D=γμ(∂μ−iAμ) (2)

and found the eigenfunctions and the eigenvalues for a test quark in an external gauge field .

For this goal we use the massive overlap operator  Neuberger:1997 (). It has the following form

 Mov=(1−amq2ρ)Dov+mq, (3)

where is the quark mass, is the parameter in our calculations, is the massless overlap Dirac operator, which preserves chiral invariance even at the finite lattice spacing . It may be written as

 Dov=ρa⎛⎜ ⎜⎝1+DW√D†WDW⎞⎟ ⎟⎠=ρa(1+γ5sign(H)), (4)

where is the Wilson-Dirac operator with the negative mass term , is the Wilson hopping term, is the Hermitian Wilson-Dirac operator.

We construct the polynomial approximation for the function

 sign(H)=H/√H†H. (5)

This approximation should be valid on the entire spectrum of matrix . Since

 sign(H)≡sign(H/∥H∥)=sign(W) (6)

and , then . The MinMax polynomial approximation is used for the function at , where .

The polynomial of a degree can be the best approximation, if it minimizes the maximal relative error

 δ=max|h(H2)| (7)

where . The polynom is represented by the series

 Pn(H2)=n∑k=0ckTk(z),  z=2H2−1−ϵ1−ϵ, (8)

where are the Chebyshev polynomials defined in the range . The detailed description of the algorithm used for the calculation of can be found in Giusti:2003 (). The resulting polynomial of the matrix has the same eigenfunctions as the original matrix. The eigenvalues can be found from the eigenfunctions using the formula for a some operator Neff:2001 (). From the polynomial approximation for the sign function we get the approximation for the overlap Dirac operator. We find its eigenfunctions and eigenvalues, which are used for the calculation of the propagators and the correlators.

As we consider pure lattice gauge theory, the Abelian magnetic field is introduced only into the overlap Dirac operator. In the symmetric gauge the magnetic field parallel to ’z’ axis has the form

 ABμ=B2(xδμ,2−yδμ,1). (9)

The total gauge field is the sum of the non-Abelian gluonic field and the Abelian field of magnetic photons

 Aμij=Aglμij+ABμδij, (10)

where are the colour indices, are the Lorentz indices. Quark fields obey periodic boundary conditions in space and antiperiodic boundary conditions in time. In order to match (10) with the periodic boundary conditions we apply the additional x-dependent boundary twist for fermions Al-Hashimi:2009 ().

In the finite lattice volume the magnetic flux trough any two-dimensional face of the hypecube is quantized. So the magnetic field value is

 eB=6πnB(aNs)2,  nB∈Z, (11)

where is the elementary charge and is the numbers of lattice sites in spatial directions.

### 2.3 Calculation of correlation functions

To observe the ground state energy for a meson we construct the interpolating operator creating the state with the corresponding quantum numbers. In case the pseudoscalar charged meson the interpolating operator is described by the equations

 O(π+)=¯ψd(n)γ5ψu(n),   O(π−)=¯ψu(n)γ5ψd(n) (12)

The interpolating operator for the neutral pion is

 O(π0)=(¯ψu(n)γ5ψu(n)−¯ψd(n)γ5ψd(n))/√2. (13)

It should be mentioned that in Eucledean space . We are interested in the 2-point lattice correlation function of the interpolating operators

 ⟨O1O2⟩A=⟨ψ†(n)Γ1ψ(n)ψ†(n′)Γ2ψ(n′)⟩A, (14)

where are Dirac gamma matrices, and are lattice coordinates. The spatial lattice coordinate and are the numbers of lattice sites in time direction.

The correlation function can be represented as the sum of connected and disconnected parts

 ⟨O1O2⟩A=−Tr[Γ1D−1(n,n′)Γ2D−1(n′,n)]
 +Tr[Γ1D−1(n,n)]Tr[Γ2D−1(n′,n′)], (15)

where is the Dirac propagator. For the isovector currents the disconnected part of the correlation function has to be zero due to cancelation of and quarks contributions, and we verify this on the lattice (see Sec. 4). The massive Dirac propagator is calculated using the lowest Wilson-Dirac eigenmodes

 D−1(n,n′)=∑k

We perform a discrete Fourier transformation of (15) numerically and set because we are interested in the ground state energy. To obtain the masses we expand the correlation function to the exponential series

 ⟨O1O2⟩A=∑k⟨0|^O1|k⟩⟨k|^O†2|0⟩e−ntaEk, (17)

where and are the corresponding Hilbert space operators.

For the large the main contribution to the correlator (17) comes from the term corresponding to the ground state. Due to the periodic boundary conditions the correlator in the leading order has the following form

 ~C(nt)=2A0e−NTaE0/2cosh((NT2−nt)aE0), (18)

where is a constant, is the ground state energy, is the lattice spacing. We found the ground state energies fitting the lattice correlators by the equation (18) at .

## 3 Dipole polarizability of π± meson

The energy levels of a free charged pointlike particle in a constant Abelian magnetic field parallel to z axis are described by the following equation:

 E2=p2z+(2n+1)|qH|−gszqH+E2(H=0), (19)

where is the particle momentum along ’z’ axis, is the principal quantum number, is the particle charge, is its spin projection, is the dimensionless quantity, that characterizes the magnetic moment of the particle and is the energy at zero magnetic field (the mass).

For the charged pion in the ground state in the rest frame it is necessary to utilize , , and . In the strong external magnetic field the charged pion is not a pointlike particle anymore and its internal structure can be described by the magnetic polarizabilities. If we take into account the dipole magnetic polarizability and the first order magnetic hyperpolarizability , then in the relativistic case the pion energy squared has the following form

 E2=|qH|+m2−4πmβmH2−4πmβ1hmH4,  H=eB. (20)

The energy of a charged pion was calculated from the correlation function

 CPSPS=⟨¯ψd(→0,nt)γ5ψu(→0,nt)¯ψu(→0,0)γ5ψd(→0,0)⟩. (21)

The energies squared are shown in Fig.1 for the lattice volumes , , , lattice spacings , , , quark masses and . We fit our lattice data by formula (20) at , where , and are the fit parameters. The fitting curves are also depicted in Fig.1. At the magnetic fields we observe a linear dependence of the energy squared versus the magnetic field and it increases with the field for the all sets of lattice data. At higher fields the nonlinear terms in the magnetic field contribute to the pion energy.

Fitting the lattice data we do not consider the term with the second order magnetic hyperpolarizability () in (20), because from the data fits it follows that its relative contribution is small in comparison with the terms proportional to and .

In Table 2 we show the values of the magnetic dipole polarizability and hyperpolarizability for several lattice data sets. The best values obtained are for the lattice volume , lattice spacing and for the lattice volume and lattice spacing .

In two loops of the chiral perturbation theory it was predicted that the magnetic polarizability of equals to Aleksejevs:2013 (); Ivanov:2015 (), that is close to our result. The polarizabilities for the and lattice at quark mass and are not represented in Table 2 because of the poor relative accuracy. It is a very subtle effect but the work in this direction is carried out, including the smaller bare quark masses.

In 2015 the COMPASS collaboration at CERN has investigated the pion Compton scattering Adolph:2015 (). They have found the pion electric polarizability equal to under the assumption of the ChPT, which is true in the exact chiral limit Donoghue:1989 (). We have observed agreement of this value with our lattice results. This is also in accordance with the earlier analysis of MARK II group data of the cross section of the process Boyer:1990 () which was made in Babusci:1992 () and gave the value .

The Serpukhov group have found the value exploring the radiative pion nucleon scattering in 1983 Antipov:1983 (). Without using the relation they obtained .

The analysis of the experimental data on helicity amplitudes of the reaction gave the value Filkov:2006 ().

## 4 Dipole polarizability of π0 meson

We calculate the energy of neutral pion from the correlation function

 CPSPS=(⟨¯ψd(→0,nt)γ5ψd(→0,nt)¯ψd(→0,0)γ5ψd(→0,0)⟩
 +⟨¯ψu(→0,nt)γ5ψu(→0,nt)¯ψu(→0,0)γ5ψu(→0,0)⟩)/2. (22)

It is taken into account that the and quarks interact with the external magnetic field differently because of their non-equal charges.

We have checked on the lattice that the disconnected part of the correlator (15) is zero within the error range. In Fig.3 it is shown for three values of the magnetic field. In Fig.4 the connected correlators are represented with the fits (18) for comparison. In the case of exact isospin symmetry the disconnected part has to be zero so in what follows we neglect it at all.

Fig.2 shows the ground state energy of the meson depending on the field value squared for ensembles at the quark mass and at the quark masses , (see Table 1). If the neutral pion would consist of one type of quark, (), then its energy decreases slower (faster) than for the real pion, see Fig.2.

For the pion each lattice data set is fitted at using the function

 E=E(B=0)−2πβm(eB)2, (23)

where we determine and as the fit parameters. is the energy at zero magnetic field and is the magnetic polarizability of neutral pion which is presented in Table 3. We observe the linear energy dependence on the magnetic field squared at small values of the field.

In Fig.5 we represent the values of the meson as a function of the quark mass for ensemble , the values for ensembles , at quark mass are also depicted.

The value of diminishes with the quark mass and extrapolation to the chiral limit gives the number for the lattice volume and lattice spacing . It is a bit higher than the value of magnetic polarizability obtained in our previous work Luschevskaya:2015a (), because we considered meson consisting of one type of quarks . In the framework of the chiral perturbation theory the value of the is equal to in one (two) loops Gasser:2005 ().

Our result coincides in sign with the prediction of the ChPT and differs in value, and there can be various reasons for that, including the contributions of higher chiral loops. In turn, from the lattice side, the dynamical quark loops may be relevant.

## 5 Magnetic hyperpolarizability of neutral pion

The terms of the higher degrees of the magnetic field give contribution to the energy of the pseudoscalar meson at very high magnetic field. Fig.6 shows the ground state energy depending on the magnetic field squared at . The lattice data fits are described by the formula

 E=E(B=0)−2πβm(eB)2−2πβhm(eB)4−k(eB)6, (24)

where is the magnetic hyperpolarizability and is the hyperpolarizability of higher order, , and are the fit parameters. The term proportional to is parity forbidden: the physical interpretation is that can’t decay to three (magnetic) photons.

The decay is allowed, so the term presents in formula (24). The values are negative for the all sets of the lattice data and showed in Table 4.

We found the value at lattice spacing and at for the lattice volume , and quark mass .

At the lowest quark mass we obtained using for fitting formula (24).

Note also that the energy shows the qualitative tendency of flattening at large magnetic field so that it should not turn to zero and cause the phase transition, in complete similarity to meson case studied earlier Luschevskaya:2015b (). The decrease at very large fields is likely to be compensated by higher hyperpolarizabilities. Still, the quantitative analysis of this behaviour requires more investigations.

## 6 Conclusions

We have calculate the magnetic dipole polarizability of the charged pion for several lattice spacings, lattice volumes and bare lattice quark mass . For the lattice volume and the lattice spacing we obtain the value . At finest lattice with the lattice spacing and the volume we get . We found the agreement of this result with the observation of COMPASS collaboration.

We have explored the energy dependence of neutral pion off the external Abelian magnetic field and have found its dipole magnetic polarizability and hyperpolarizability. The dipole magnetic polarizability depends on the lattice quark mass and the chiral extrapolation was performed. In the chiral limit we have obtained the magnetic dipole polarizability of the is equal to for the lattice volume and the lattice spacing , that is close to the prediction of the ChPT Aleksejevs:2013 ().

The contribution of the hyperpolarizability have been also revealed to the energy at considered strong magnetic field. This is very tiny effect and for the all sets . For the finest lattice with spacing we found . We have observed qualitative behaviour of energy which is not favouring the appearance of tachyonic mode and phase transition.

## 7 Acknowledgements

O.T. is indebted to A. Guskov and M. Ivanov for useful discussions.

The authors are grateful to FAIR-ITEP supercomputer center where these numerical calculations were performed. This work is completely supported by a grant from the Russian Science Foundation (project number 16-12-10059).

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