mesons in strong abelian magnetic field in SU(3) lattice gauge theory
We explore the masses (ground states energies) of neutral and charged mesons in strong abelian magnetic field in gluodynamics. The energies of these particle in the external magnetic field depends on its spin projection on the axis of external magnetic field. The masses of meson with increase with the field. The masses of mesons with zero spin also grow with the magnetic field. The ground state energies of meson with and meson with decrease as a function of the field, while the energies of meson with and meson with rise with the field value.
mesons in strong abelian magnetic field in SU(3) lattice gauge theory
Joint Institute for Nuclear Research, Dubna, Russia
The exploration of hadronic matter in strong abelian magnetic field has a fundamental meaning. The strong magnetic fields of may be associated with the formation of the Early Universe . Now it is possible to observe such magnetic fields in non-central heavy-ion collisions in terrestrial laboratories (ALICA, RHIC, NICA, FAIR). The field value can reach up to , i.e. the order of energies at which the QCD effects appear.
Quantum chromodynamics in a strong magnetic field shows a lot of bright interesting effects, e.g. inverse magnetic catalysis . Calculations on the lattices are a good method for study of the phase diagram of QCD in external magnetic field [4, 5, 6]. Numerical simulations in QCD with and show that strongly interacting matter in strong magnetic field posses paramagnetic properties in the confinement and deconfinement phases [7, 8, 9]. Equation of state for quark-gluon plasma in strong magnetic field was also investigated in .
In the framework of the Nambu-Jona-Lasinio model it was shown that QCD vacuum becomes a superconductor in sufficiently strong magnetic field ( Tl) [11, 12] along the direction of the magnetic field. This transition to superconducting phase is accompanied by a condensation of charged mesons, their masses turns to zero.
We have calculated the ground state energies of neutral and charged vector mesons as a function of external magnetic field depending on the spin projections or on the axis of the magnetic field in lattice gauge theory without dynamical quarks. For not very large magnetic fields our data for charged particles confirm the picture of Landau levels for pointlike particles. Articles   and  are also devoted to the behaviour of meson masses in the external abelian magnetic field.
2 Observables and details of calculations
The technical details of our calculations are presented in . We generate statistically independent lattice gauge configurations with tadpole improved Symanzik action . Then we solve Dirac equation numerically
and find eigenfunctions and eigenvectors for a test quark in the external gauge field , which is a sum of non-abelian gluonic field and abelian constant magnetic field.
We add the abelian magnetic field only into Dirac operator, because our theory doesn’t contain dynamical quarks. Our simulations has been carried out on symmetrical lattices with lattice volumes , and lattice spacings , , . The number of configurations for the lattices with and spacings were , for the finest lattice with configurations was used.
We calculate the correlation functions in coordinate space
where are Dirac gamma matrices, are Lorenz indices.
In order to calculate the observables (2) the quark propagators have to be computed. The Dirac propagator for the massive quark is
where is the number of the lowest eigenmodes. For the correlators (2) the following equality is fulfilled
We calculate the correlators and the make its Fourier transformation. For the meson ground state we have to choose . For particles with zero momentum their energy is equal to its mass . The expansion of correlation function to exponential series has the form
where is the lattice spacing, is the number of nodes in the time direction, is the energy of the state with quantum number . From expansion (2) one can see that for large the main contribution comes from the ground state. Because of periodic boundary conditions on the lattice the main contribution to the ground state has the following form
where is a constant, is the energy of the ground state.
The correlators of vector currents in various spatial dimensions have the following form
The energy of the ground state or the mass of vector meson with spin projection could be obtained from the correlator. The combinations of correlators
correspond to the vector particles with quantum numbers and .
In Fig.1 we see the mass of the state with zero spin which was obtained from the correlator . We expect that at small magnetic field this state corresponds to the neutral meson with zero spin projection on the axis . The mass of the state diminishes with the magnetic field at small . In nature at strong magnetic field the branching for the decay have to be large. It is not easy to distinguish between and on the lattice because they have the same quantum numbers. We do not make it here.
Fig.2 shows the mass of the neutral meson with spin projections . The masses for and grow with the field value and coincide with each other because the imaginary part of (3) is zero. This is a manifestation of C-parity of meson. From Fig.1 and 2 we see that the data cohere for and lattice volumes and the same lattice spacings fm. Lattice spacing effects are also not large.
The results are in qualitative agreement with the calculations made in the lattice gauge theory in our previous work .
The energy levels of free charged pointlike particle in a background magnetic field parallel to axis
where is the momentum, is the number of energy level, -factor characterizes magnetic properties of the particle, is the charge of the particle, is the particle mass at . In our case , and and are the parameters of the fitting function. This equation doesn’t take into account polarizabilities of the particle and has not be valid at very large magnetic fields. The large fields at first sight do not manifest tachyonic mode although this requires further investigation.
Fig.3 depicts the mass of charged vector meson with . We cannot distinguish negative and positive vector mesons on the lattice because the exchange of the particle charge to the opposite is equivalent to the exchange of to which gives the same masses for . This is in accordance with (3). We make a fit of the data for lattice volume and fm lattice spacing, is a fit parameter. In spite of lack of statistics the data agree with the fit.
We shows the energies of charged with spin projections and in Figs. 4 and 5 correspondingly. The energy of the ground state with decreases with the field value. The data agree with fit for , at large magnetic field field the decrease became slower. We consider this effect as the result of non-zero polarizability of charged meson. The energy of ground state with increases with the field value. The function gives the excellent fits for the all presented data.
During calculations we are limited to a small lattice spacing and therefore we can not explore too large values of the magnetic fields. As a result of simple estimates we obtain the value of the magnetic field for lattice spacing and for , when the lattice spacing effects have to become appreciable.
We explore the behaviour of the masses of vector in lattice gauge theory. We found that masses of neutral vector mesons with zero and non-zero spin projection on the direction of the magnetic field differ from each other. Masses with decrease with increasing magnetic field, while the masses with increase under the same conditions. We consider this phenomenon as a result of the anisotropy produced by the strong magnetic field. We did not find condensation of neutral mesons, i.e. any indications of the existence of a phase of superfluidity in the confinement phase.
The masses of charged mesons cohere with the behaviour of Landau levels at not very large magnetic fields. We didn’t observe some evidences in favour of condensation of charged vector mesons in the considered range of fields. Condensation of charged mesons might indicate the existence of superconductivity in QCD at high magnetic fields. The existence of a superconducting phase in QCD at high values of the magnetic field B  is still a hot topic for discussions.
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.
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