Pis'ma v ZhETF, vol. 101, iss. 10, pp. 750-754 © 2015 May 25
7Z±- and p^-mesons in a strong magnetic field on the lattice
E.V.Luschevskaya+*1\ O. A. Kochetkov+x<1\ O.V.Teryaev°V1^ , O. E. Solovjev,a+vi> + JVRC "Kurchatov Institute" SSC RF Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia * School of Biornedicine, Far Eastern Federal University, 690950 Vladivostok, Russia x Institut fur Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany ° Joint Institute for Nuclear Research, 141980 Dubna, Russia vNational Research Nuclear University "MEPhI" 115409 Moscow, Russia
Submitted 24 November 2014 Resubmitted 16 March 2015
We calculated the correlators of pseudoscalar and vector currents in external strong abelian magnetic field in SU(3) gluodynamics. From the correlation functions we obtain the ground state energies (masses) of neutral Jo°-meson and charged 7r±- and Jo±-mesons. The energy of the Jo°-meson with zero spin projection on the axis of the field decreases, while the energies with non-zero spins increase with the field value. The mass of charged 7r±-mesons increases with the field. We observe the agreement between Landau level picture and behaviour of charged Jo±-mesons for moderate magnetic fields. There are no evidences in favour of charged vector meson condensation or tachyonic mode existence at large magnetic fields. The (/-factor of Jo± is estimated in the chiral limit.
DOI: 10.7868/S0370274X15100045
1. Introduction. Researching QCD in the external electromagnetic field plays the important role in understanding the structure of hadrons. Today the strong magnetic fields of hadronic scale can be created in terrestrial laboratories like ALICA, RHIC, NIC A, and FAIR. In non-central heavy ion colisions the magnetic field value in the moment of collision can reach up to 15m^ ~ 0.27GeV2 [1]. Such a strong magnetic field can modify the properties of strongly interacting matter. Many interesting effects have been observed in experiment and discovered theoretically, for example inverse magnetic catalysis [2], chiral magnetic effect [3, 4], enhancement of the chiral symmetry breaking [5-9].
The investigations related to QCD phase diagram in strong magnetic field are presented in [10-17]. Numerical simulations in QCD with Nf = 2 and 2 + 1 show that the strongly interacting matter in strong magnetic field posses paramagnetic properties in the confinement and deconfinement phases [18-20].
In this work we explore the splitting of ground state energy of neutral p° and charged vector mesons p^ depending on its spin projection on the axis of the external abelian magnetic field. This exploration is important because such splitting can lead to the asymme-
e-mail: luschevskaya@itep.ru; oleg.kochetkov@physik.uni-r.de; teryaev@theor.jinr.ru; oesolovjeva@gmail.com
try of emitted neutral and charged particles above and under reaction plane and contribute to the chiral magnetic effect. We also give a preliminary estimation of ^-factor of charged ^-mesons Articles [21-25] are also devoted to the behaviour of hadron masses in the external abelian magnetic field. The magnetic moments of p-mesons have been explored in [26-30]. Our value of ^-factor is in agreement with the previous lattice calculations [31].
2. Details of calculations. The technical details of our calculations are presented in [32]. We generate 200—300 SU(3) statistically independent lattice gauge configurations for lattice volumes 164, 184 and lattice spacings a = 0.105, 0.115, and 0.125 fm. The U( 1) external magnetic field is included only into the Dirac operator which is used for the calculation of eigenfunctions ipk and eigenvectors A & of a test quark in a background gauge field A^. This field is a sum of non-abelian SU(3) gluonic field and U( f) abelian constant magnetic field,
Apij —> A^ij + A^Sij, (f)
= -^-(^l'W - (2)
To take into account periodic boundary conditions for fermions the twisted boundary conditions are superim-
7r±- and p0'^-mesons in a strong magnetic Geld on the lattice
751
posed [33]. Magnetic field is directed along ¿-axis and its value is quantized
qB =
2tt k (alp
k G
(3)
where q = —1/3 e. Eq. (3) leads to a minimal value of magnetic field (eB)1/2 = 380 MeV for lattice volume 184 and lattice spacing a = 0.125 fm. For each meson we construct interpolation operators with given quantum numbers. Then we calculate correlation functions of these operators in Euclidean space
{^(x^o^-Wiy^Ky^A,
(4)
where we use Oi, Oo = 7M, for the vector particle and 75 for pion, yit, v = 1,..., 4 are Lorenz indices. The Dirac propagator for the massive quark can be approximated by its eigenvectors and eigenvalues
D~1(x,y)= J2
k<M
V'fcWV'fcCy)
iXh + m
(5)
Cfit(nt) =A0e-ntaEo + ,4oe-(№r-nt)aiSo
'NT
= 2Aoe~NTaE°/2 cosh
- nt aE\
(7)
and exclude the contribution of excited states we take various values of nt from the interval 5 < nt < Nt — 5. We also use a smeared gaussian source and point sink for our calculations.
The correlation functions for various spatial directions are given by the following relations
Gj J = (^(0,nt)7iV(0,nt)^(0,0)7iV(0,0)>, (8)
C™ = (0(0, nt)72^(0, n,m0, 0)72-0(0, 0)), (9) = (^(0,^)730(0,^)^(0,0)730(0,0)). (10)
The form of the density matrix for vector particle with spin s = 1 gives the formulas for energies of meson with various spin projections on the axis of the external magnetic field.
For the sz = 0 one can obtain the energy of the ground state from the C}}' correlator. The combinations of correlators
, .1 \ 1 \ , .1 V 1 , .1 V 1 -f/~iV V , .1
C (sc=±l) = Cra +C ±«(C -C
iVV
(11)
where M = 50 is the number of the lowest eigenmodes. The correlator (4) is a sum of connected and disconnected contributions. The disconnected parts equal to zero because we consider the isovector states. We make 3-dimensional Fourier transformation of correlators and consider zero momentum p = 0 because we are interested in the ground energy state.
For particles with zero momentum their energy is equal to its mass Eq = mi in zero magnetic field. The expansion of correlation function to exponential series has the form
C(nt) = (0t(O, nt)C>iV>(0, nt)^(0,0)O20(O, 0))A =
= (6)
k
where a is the lattice spacing, nt is the number of nodes in the time direction, Eu is the energy of the state with quantum number k. At large nt the main contribution in (6) comes from the ground state. On the lattice due to the periodic boundary conditions the main contribution to the ground state has the following form
gives the energies of mesons with = +1 and —1.
3. Results. In Fig. 1 we depict the mass of the p°-meson with spin projections sz = ±1 increasing with the
9
<D
o
1.8 1.6 1.4 1.2 1.0
18 , a = 0.125 fill, m = 34.26 MeV
164, a = 0.125 fin, m =31.52 MeV
16 ,a = 0.125 fin, m = 47.28 MeV
16\,a = 0.115 fill, m = 34.26 MeV
18 , a = 0.115 fill, m = 34.26 MeV
t
i
k tf
0
0.5
1.0
1.5
2.0
2.5
eB (GeV )
where An is a constant, Eq is the energy of the ground state. Therefore we fit our data for the correlators to the (7) function and find the ground state energy as a fit parameter. In order to minimize the errors
Fig. 1. The ground state energy of the neutral p°-meson with spin s, = il as a function of magnetic field for lattice volumes 164 and 184, lattice spacings a = 0.115 fm and 0.125 fm and various bare quark masses
magnetic field value. The masses for = —1 and +1 are equal which is a manifestation of definite C-parity of p°-meson. Fig. 1 demonstrates small lattice spacing and lattice volume artefacts.
We do not present the neutral pion and p°-meson with zero spin projection on the field axis, because there is a contribution of pion to correlators of vector currents
nucbMa B >K3TO TOM 101 BHH.9-10 2015
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E. V. Luschevskaya, O. A. Kochetkov, O. V. Teryaev, O. E. Solovjev.a
in the external magnetic field due to abelian anomaly. This problem requires more detailed investigation and will be studied in the future.
The energy levels of free charged pointlike particle in a background magnetic field is described by the formula
E2 = \qB\-gszqB + m2, (12)
where ^-factor characterizes magnetic properties of the particle, q is the charge of the particle, is the spin projection on the field direction, m is the particle mass at B = 0. The Eq. (12) is true only for pointlike particle and doesn't take into account polarizabilities of mesons. If the particle is not pointlike then the magnetic polar-izabilty is not zero. In the relativistic case the meson energy levels has the following form
E2 = \qB\ - gszqB + m2 - Airm(i{qB)2, (13)
where /3 is the magnetic polarizability, the charge of the particle q = —e for 7r~, or q = +e for 7r+, p+ and m is its mass at B = 0. We consider p~-mesons while p+ corresponds to the reversal of the direction of magnetic field.
In Fig. 2 the energy of charged 7r±-meson is depicted.
Fig. 2. The squared energy of the ground state of the charged 7r±-meson with spin s = 0 as a function of the magnetic field for lattice volume 184, various lattice spac-ings a = 0.084, 0.095, 0.115, 0.125 fm and the bare quark mass equal to rnq = 34.26 MeV. The fit by solid curve corresponds to the data at lattice spacing a = 0.084 fm, the dashed curve is for a = 0.095 fm, the dashed-dotted curve describes data at a = 0.115 fm and dashed-dotted-dotted curve corresponds to a = 0.125 fm
The fitting curves correspond to the fit function E2 = \qB\ + m2 — 47Tmfi(qB)2, where m and ¡3 are the fit parameters. The masses of the 7r± increase with the field value. The nonzero values of /3 indicate a not pointlike
184, m„ = 34.26 MeV, p" ,a = 0.084 fm >- -B—1
a = 0.095 fin i- -e—i
a - 0.115 fill ' -e—i
ci= 0.125 fin >- A
©Ju^"?"
----------- ----------
r" i
0 0.5 1.0 1.5 2.0 2.5 eB (GeV2)
Fig. 3. The squared energy of the ground state of the charged p~-meson with spin s, = 0 as a function of the magnetic field for lattice volume 184, various lattic
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