Tensor polarizability of the vector mesons from SU(3) lattice gauge theory
E.V. Luschevskaya, O.V. Teryaev, D.Yu. Golubkov, O.V. Solovjeva, R.A.Ishkuvatov
aa r X i v : . [ h e p - l a t ] N ov Prepared for submission to JHEP
Tensor polarizability of the vector mesons from SU (3) lattice gauge theory E.V. Luschevskaya a,b
O.V. Teryaev c,a
D.Yu. Golubkov a O.V. Solovjeva a R.A.Ishkuvatov a,b a Institute for Theoretical and Experimental Physics named by A.I.Alikhanov of NRC “KurchatovInstitute”, 117218, Bolshaya Cheremushkinskaya 25, Moscow, Russia b Moscow Institute of Physics and Technology, Dolgoprudnyj, Institutskij lane 9, Moscow Region141700, Russia c Joint Institute for Nuclear Research, Dubna, 141980, Russia
E-mail: [email protected] , [email protected] , [email protected] Abstract:
The magnetic dipole polarizabilities of the vector ρ and ρ ± mesons in SU (3)pure gauge theory are calculated in the article. Based on this the authors explore thecontribution of the dipole magnetic polarizabilities to the tensor polarization of the vectormesons in external abelian magnetic field. The tensor polarization leads to the dileptonasymmetry observed in non-central heavy ion collisions and can be also estimated in latticegauge theory. Keywords:
Strong magnetic field, quantum chromodynamics, lattice gauge theory, spin,magnetic dipole polarizability ontents m eff and its statistical errors 55 Magnetic polarizabilities of the ρ ± mesons 66 Magnetic polarizability of the ρ meson 107 Tensor magnetic polarizability 138 Conclusion 14 The influence of strong magnetic fields on quark-hadron matter represents rich and fullof surprising effects area of science. These fields could exist in the Early Universe, theycan influence the physics of neutron stars, and lead to non-trivial effects in non-centralheavy-ion collisions in terrestrial laboratories. Behaviour of the hadron energy in externalmagnetic field may provide information about the particle internal structure. In strongmagnetic field the hadronic wave function deforms. This deformation is defined both byQED and QCD interactions inside the hadron. The magnetic polarizability and hyperpo-larizabilities are quantities describing the response of the hadron to the external magneticfield, which we have explored in our previous work [1]. Apparently, this non-linear responsearises solely due to the strong QCD interaction binding quarks together.The magnetic field effect on the hadronic energy and structure was explored by theoret-ical models [2–8] as well as lattice calculations [9–15]. The polarizabilities of hadrons werealso investigated by analytical methods using dispersion relations in [16, 17], the magneticmoments were studied in [18–22].Below we discuss the calculations in lattice gauge theory with chiral invariant Diracoperator without dynamical quarks. Our method enables to calculate the hadronic energiesfor different spin projections on the magnetic field axis. Magnetic polarizabilities and mo-ments can be extracted as the fit parameters from the magnetic field value dependence ofthe energy. The values of the magnetic polarizabilities depend on the meson spin projectionon the field axis. The physical meaning of this phenomenon is related to different deforma-tions of a hadron in the various space direction. Therefore, the presence of a magnetic field– 1 –reates a kind of anisotropy in space, which can lead to tensor polarization (alignment) ofthe vector meson and, after its decay, to dileptonic asymmetry in collisions of heavy ions.Dilepton anisotropy [23] is an important physical characteristic which is sensitive to dif-ferent channels of particle’s decay and can be utilized to disentangle contributions of somechannels. Dilepton asymmetries provide information about the evolution of quark-gluonplasma in non-central heavy-ion collisions [24]. Here we study the influence of magneticfield on the tensor polarization of ρ mesons which was detected in angular distribution oftheir decay products. This requires a more accurate calculation of polarizability, so, werepeat we repeat, improve and extend our previous analysis [1, 11]. We used ensembles of statistically independent SU (3) gauge field configurations. For thegeneration of these configurations the improved L¨uscher-Weisz action [25] was used: S = β imp X pl S pl − β imp20 u X rt S rt , (2.1)where S pl , rt = (1 / − U pl , rt) is the plaquette and rectangular loop terms respec-tively, u = ( W × ) / is defined by the relation W × = h (1 / U pl i calculated at zerotemperature [26].The calculations were carried out in SU (3) lattice gauge theory without dynamicalquarks. We consider the lattice volumes N t × N s = 18 and 20 and a set of latticespacings a = { . , . , . } fm. In Table 2 we show the lattice volume N t × N s , thelattice spacing a , the corresponding β imp values and the number of configurations. N t × N s β imp a, fm N conf18 D − ( x, y ) = X k 1) = h O ( t ) ¯ O (0) i A + h O ( t ) ¯ O (0) i A ± i ( h O ( t ) ¯ O (0) i A − h O ( t ) ¯ O (0) i A ) (3.4)give the energies of vector mesons with the spin projections equal to +1 and − O = ψ † d,u ( x ) γ ψ u,d ( x ) , O = ψ † d,u ( x ) γ ψ u,d ( x ) are the interpolation operatorsof the ρ ± mesons. The interpolation operators for the ρ case are constructed similarly to(3.4) taking into account (3.3).The correlation function can be expanded in a series over the eigenstates of the Hamil-tonian b H h O i ( t ) ¯ O j (0) i T = 1 Z X m,n h m | e − ( T − t ) b H b O i | n ih n | e − t b H b O † j | m i = (3.5)= 1 Z X m,n e − ( T − t ) E m h m | b O i | n i e − tE n h n | b O † j | m i , where i, j = 1 , , E m and E n are the energies of the excited states with the numbers m and n , and Z = P n h n | e − T b H | n i = P n e − T E n is the partition function.In expression (3.5) we take out the factor e − T E , as a result, we obtain the followingrelation h O i ( t ) ¯ O j (0) i T = P m,n e − ( T − t )∆ E m h m | b O i | n i e − t ∆ E n h n | b O † j | m i e − T ∆ E + e − T ∆ E + ... , (3.6)– 4 –here ∆ E n = E n − E . In the thermodynamical limit of the theory T → ∞ from (3.6) weobtain h O i ( t ) ¯ O j (0) i T →∞ = X n h | b O i | n ih n | b O † j | i e − tE n . (3.7)In our case we got the following expression C ( n t ) = h ψ † ( , n t ) γ i ψ ( , n t ) ψ † ( , γ j ψ ( , i A = X k h | b O i | n ih n | b O † j | i e − n t aE n . (3.8)One can see from (3.8) that at-large n t the main contribution comes from the ground state.Taking into account periodic boundary conditions on the lattice we obtain the final formula C fit ( n t ) = A e − n t aE + A e − ( N T − n t ) aE =2 A e − N T aE / cosh(( N T − n t ) aE ) , (3.9)where A is the constant, E is the ground state energy. We use this formula for fittingthe correlation functions obtained from the lattice propagators. From this fits the groundstate energies can be obtained. m eff and its statistical errors The ρ -meson effective masses m eff were extracted from χ -fits to the standard asymptoticparametrization (3.9), with fit parameters being m eff = aE and A . The effective masscan be also found from the following equation: C ( n t ) C ( n t + 1) = cosh( m eff ( n t − N T / m eff ( n t + 1 − N T / , (4.1)in which the boundary conditions on the lattice are taken into account. We have checkedthat the results for the energies coincide with the results obtained from (4.1), taking intoaccount the correlation matrix between the adjacent points of the plateau.In order to control the stability of resulting m eff values, i.e. to determine the m eff ( n t )plateau, the fits were performed in four ranges of n t = h N T − k, N T + k i , k = 1 .. 4, wherethe fit quality could be satisfactory, χ /d.o.f. ∼ 1. Despite that the fit quality was oftenacceptable already at k ≃ 4, the values of m eff at k = 4 could still be systematicallyoverestimated.Usually, the m eff plateau was reached at k = 3, i.e. n t = 6 ÷ 12 (7 ÷ 13) for latticevolume 18 (20 ). In some cases a narrower n t range was conservatively chosen after avisual inspection of the fits to ensure that m eff was well within the plateau. A typicalgraph of m eff ( ρ ) depending on the fit range n t is shown in Fig. 1 for the lattice volume18 , the lattice spacing a = 0 . 115 fm, spin projections | s z | = 0 , ρ -meson effective masses m eff are statistical-only, as determined by the fit in the chosen n t range.– 5 – m e ff a ( ρ ) , a=0.115 fm: s z =0, eB=0.34 GeV s z =0, eB=0.51 GeV |s z |=1, eB=0.34 GeV |s z |=1, eB=0.51 GeV Figure 1 . The m eff stability plot of the ρ -meson for the spin projections s z = 0 and | s z | = 1on the magnetic field axis depending on the n t fit ranges. The results were obtained on the latticewith the volume 18 , the lattice spacing 0 . 115 fm and the pion mass m π = 541 MeV. ρ ± mesons In the magnetic field the energy levels of the point-like charged particle are described bythe following dependency [32] E = p z + (2 n + 1) | qB | − gsqB + m , (5.1)where p z is the momentum in the magnetic field direction, n is the principal quantumnumber, q is the particle electric charge, g is the g-factor, s and m are the particle spinand mass respectively.As we discussed earlier [1] the magnetic field can affect the internal structure of vectormesons if it is sufficiently strong. This influence is characterized by non-zero magneticpolarizabilities and hyperpolarizabilities which depend on the projection of the meson spin s z on the direction of the magnetic field. It means that the response of the fermioniccurrents inside the meson is defined by the mutual orientation of the quarks and theexternal magnetic field. These phenomena are interesting and can give a contribution tothe polarization of the emitted charged particles in the strong magnetic field which wediscuss below.According to the parity conservation for the spin projection s z = 0 the energy squaredgets corrections from the non-linear terms of even powers in a magnetic field. For the spinprojections s z = +1 and s z = − qs z = +1 is not larger than 20% and– 6 –ompatible with errors at eB ∈ [0 , . 2] GeV . It may be seen from Fig. 9 presented in ourprevious work [1]. Similarly, a correction of the fourth power does not give a significantcontribution to the square of the energy for the case s z = 0.Therefore, we obtain the dipole magnetic polarizability for the spin projection s z = 0from the fit of lattice data by the following relation E s z =0 = | eB | + m − πmβ m ( eB ) (5.2)at eB ∈ [0 , . 2] GeV , where eB is the magnetic field in GeV , β m is the dipole magneticpolarizability and fit parameter, m is the fit parameter also. The lattice results togetherwith the fitting curves are shown in Fig. 2 which shows the energy squared increasing withthe magnetic field value. At eB ∼ . ÷ . we observe a hump which leads to increaseof the errors in determination of β m ( s z = 0) value. It can be a result of lattice spacing andlattice volume effects or have some physical explanation. We cannot perform a rigorousanalysis due to high errors of the calculations, but we note that this hump is absent for thelattice with smaller lattice spacing a = 0 . 095 fm. Therefore it is more probably a latticespacing artefact.The values of the β m with the values of χ /n d.o.f. and lattice parameters are shown inTable 1. E ( ρ s z = ) , G e V eB, GeV : a=0.095 fm, m π =596 MeVa=0.105 fm, m π =574 MeVa=0.115 fm, m π =395 MeVa=0.115 fm, m π =541 MeV Figure 2 . The energy squared of the charged ρ meson for the projection s z = 0 depending on themagnetic field value for various lattice sets of data. The solid lines correspond to the fits of thelattice data obtained with the use of formula (5.2) The behaviour of the energy squared for the case qs z = +1 (which corresponds to ρ − at s z = − ρ + at s z = +1) can be described by the dependency E qs z =+1 = | eB | − g ( eB ) + m − πmβ m ( eB ) (5.3)at eB ∈ [0 , . 2] GeV . The magnetic dipole polarizability was obtained from the fit oflattice data by the relation (5.3), where m , g and β m are the fit parameters.– 7 – m π (MeV) a (fm) β m (GeV − ) χ /d.o.f. ± . 105 0 . ± . 01 6 . ± . 115 0 . ± . 006 0 . ± . 115 0 . ± . 004 1 . Table 1 . The magnetic dipole polarizability β m of the charged ρ meson with spin projection s z = 0obtained from the fits (5.2) of the lattice data sets. The pion masses are represented in the secondcolumn, the lattice spacings in the third column, the values of χ /d.o.f are shown in the last one. We represent this energy component in Fig.3 for the lattice volume 18 , lattice spacings0 . 105 fm, 0 . 115 fm and for the lattice volume 20 , lattice spacing 0 . 115 fm. The values ofpion mass are also shown. E ( ρ q s z = ) , G e V eB, GeV : a=0.095 fm, m π =596 MeVa=0.105 fm, m π =574 MeVa=0.115 fm, m π =395 MeVa=0.115 fm, m π =541 MeV20 : a=0.115 fm, m π =535 MeV Figure 3 . The energy squared of the charged ρ meson for the case qs z = +1 depending on themagnetic field value for various lattice data sets. The solid lines correspond to the fits of the latticedata obtained with the use of formula (5.3) The values of the magnetic dipole polarizability are represented in Table 2. Theseresults are obtained from the 3-parameter fit in comparison with the results of our previouspaper (see Table 3 and Fig.9), where we fixed the g -factor value and used 2-parametric fit.The β m values are in a good agreement with our previous ones within the error range.In Fig.4 we represent the energy squared for different meson spin projections on thefield axis for the lattice volume 18 , the lattice spacing a = 0 . 115 fm and the pion massequal to 541 MeV. The energy errors for the case qs z = − qs z = − E qs z = − = | eB | + g ( eB ) + m − πmβ m ( eB ) . (5.4)– 8 – m π (MeV) a (fm) g -factor β m (GeV − ) χ /d.o.f. ± . 105 2 . ± . − . ± . 010 2 . ± . 115 2 . ± . − . ± . 006 2 . ± . 115 2 . ± . − . ± . 006 1 . ± . 115 2 . ± . − . ± . 006 1 . Table 2 . The magnetic dipole moment and the magnetic dipole polarizability of the charged ρ meson with qs z = +1 for the lattice spacings 0 . 105 fm, 0 . 115 fm, the lattice volume 18 , variouspion masses and for the lattice spacing 0 . 115 fm, the lattice volume 20 and the pion mass m π =535(4) MeV with their errors and χ /d.o.f values. The results were obtained with the use of3-parametric fit (5.3) at eB ∈ [0 , . 2] shown in Fig. 3. E ( ρ ) , G e V eB, GeV a=0.115 fm, 18 , m π =541 MeV: s=0qs z =+1qs z =-1 Figure 4 . The energy squared of the charged ρ meson for the lattice volume 18 , lattice spacing0 . 115 fm and the pion mass m π = 541 MeV in dependence on the meson charge and spin projectionon the magnetic field axis. To estimate the contribution of the magnetic dipole polarizability to the lepton asym-metry it is needed to know the dipole magnetic polarizability for the s z = +1 and s z = − s z = +1 if we consider ρ − or from s z = − ρ + . Nevertheless, the parity conservation demands the equality of the magneticdipole polarizabilities for s z = +1 and s z = − 1. Then it follows from (5.3) and (5.4) E qs z = − − E qs z =+1 = 2 g ( eB ) . (5.5)In order to make sure that the relation (5.5) is satisfied for our data we represent thevalue of ( E qs z = − − E qs z =+1 ) / g -factor obtained from these– 9 – ( E s z = - - E s z =+ ) / , G e V eB, GeV : a=0.105 fm, m π =574 MeVa=0.115 fm, m π =541 MeV Figure 5 . The value of ( E qs z = − − E qs z =+1 ) / , lattice spacings 0 . 105 fm and 0 . 115 fm. fits are equal to 2 . ± . a = 0 . 105 fm and 2 . ± . 04 for the spacing a = 0 . 115 fm. The errors of the g -factor determination are underestimated because theeffective mass plateau is very noisy for the qs z = − 1. We have determined the g -factorfrom the lowest energy sub-level in our previous work [1]. ρ meson We have discussed the magnetic polarizabilities of the neutral ρ mesons previously in [11],where we considered the toy model with one type of quarks. In an external magnetic fieldthe u- and d-quarks couple differently to the magnetic field. It has to be taken into accountwhen we calculate the physical observables. We repeat the analysis represented in [11] withthe difference that we find the energies from the correlation function (3.3) and increase thestatistics.In the relativistic case the energy of the neutral ρ meson with spin projection s z = 0is described by the following dependency E = m − πmβ m ( eB ) − πmβ h m ( eB ) − πmβ h m ( eB ) − πmβ h m ( eB ) − ... , (6.1)where β h m , β h m and β h m are the various magnetic hyperpolarizabilities of higher orders, m is the mass of the meson at zero field.In Fig. 6 we represent the energy squared of the ρ meson with s z = 0 versus the fieldsquared for various lattices and pion masses at ( eB ) ∈ [0 : 0 . 5] GeV . The lattice data aredepicted by points. We observe the energy decrease rapidly for all the lattice data. It alsostrongly depends on the lattice spacing, the lattice volume, and the pion mass. The linesare the fits of these data obtained with the use of formula (6.1) when we also include terms– 10 – E ( ρ s z = ) , G e V (eB) , GeV : a=0.105 fm, m π =574 MeVa=0.115 fm, m π =541 MeVa=0.115 fm, m π =395 MeV20 : a=0.115 fm, m π =535 MeV Figure 6 . The energy squared of the ρ ( s z = 0) ground state versus the field value squared. Thedata are shown by points for various lattice spacings, pion masses and two lattice volumes 18 and20 with the fits obtained using formula (6.1). ∼ ( eB ) and ∼ ( eB ) . For the lattice with spacing a = 0 . 105 fm the fits were performedat ( eB ) ∈ [0 : 1 . 7] GeV , for the other lattices we use ( eB ) ∈ [0 : 1 . 5] GeV . The termsof higher powers of field begin to contribute significantly at low fields. So, it is difficult toextract the magnetic polarizability, because the quantization condition imposes a limitationon the minimal field value. We do not perform an extrapolation to the chiral limit sincewe are interested only in the qualitative predictions at this stage of investigations.In Table 3 one can find the magnetic polarizability β m and hyperpolarizability β hm obtained from the fits. The lattice volume V , the lattice spacing a , the pion mass m π ,the interval of fields selected for the fitting procedure and χ /n.d.o.f. are also shown. Theresults agree with each other within the errors. V a (fm) m π (MeV) β m (GeV − ) β hm (GeV − ) n ( eB ) (GeV ) χ /d.o.f. . 105 574 ± . ± . − . ± . 98 10 [0 : 1 . 7] 1 . . 115 541 ± . ± . − . ± . 59 12 [0 : 1 . 5] 2 . . 115 535 ± . ± . − . ± . 60 12 [0 : 1 . 5] 2 . . 115 395 ± . ± . − . ± . 74 12 [0 : 1 . 5] 3 . Table 3 . The value of the magnetic dipole polarizability β m and the magnetic hyperpolarizability β hm for the ρ with spin projection s z = 0 are shown for the various lattice volumes V , latticespacings a and pion masses m π . The degree of polynomial n , field range used for the fitting and χ /d.o.f values are represented in columns sixth to eighth correspondingly. At eB ∈ [0 : 1 . 2] GeV for the spin projection | s z | = 1 on the field axis the energy– 11 – E ( ρ | s z | = ) , G e V eB, GeV : a=0.095 fm, m π =596 MeVa=0.105 fm, m π =574 MeVa=0.115 fm, m π =541 MeV a=0.115 fm, m π =395 MeV20 : a=0.115 fm, m π =535 MeV Figure 7 . The energy squared of the ρ ( | s z | = 1) meson versus magnetic field for various latticespacings, pion masses, and two lattice volumes 18 and 20 . The lines correspond to the fits of thelattice data obtained using formula (6.2). squared of the neutral vector meson can be described by the following relation: E = m − πmβ m ( eB ) − πmβ h m ( eB ) . (6.2)In Fig. 7 the energy squared is shown for the ρ meson with the spin projection | s z | = 1,the energies of the neutral vector meson for the s z = +1 and s z = − β m is obtained from the fit of thelattice data by formula (6.2) for the lattices with spacings 0 . 105 fm and 0 . 115 fm, where m , β m and β h m are the fit parameters. The lattice data for a = 0 . 084 fm and a = 0 . 095 fmdo not allow to extract statistically significant β m values, but we show them to checkthe lattice volume and lattice spacing effects. The β m values with the errors and otherparameters are shown in Table 4. The results agree with each other within the errors. V a (fm) m π (MeV) β m (GeV − ) β hm (GeV − ) eB, GeV χ /d.o.f. . 105 574 ± − . ± . 02 0 . ± . 02 [0 : 1 . 1] 0 . . 115 541 ± − . ± . 02 0 . ± . 03 [0 : 1 . 1] 0 . . 115 535 ± − . ± . 02 0 . ± . 03 [0 : 1 . 1] 1 . . 115 395 ± − . ± . 03 0 . ± . 03 [0 : 1 . 1] 0 . Table 4 . The value of the magnetic dipole polarizability β m and the magnetic hyperpolarizability β hm are shown for ρ with | s z | = 1, for the lattice volumes 18 and 20 , the lattice spacings 0 . 105 fmand 0 . 115 fm and various pion masses. The fourth and the last columns contain the intervals of themagnetic field used for the fit and χ /d.o.f. correspondingly. – 12 – Tensor magnetic polarizability In the previous sections we have found that the energy of the ρ meson with s z = ± ρ meson with s z = 0diminishes quickly versus the magnetic field value. For the ρ ± the energy decreases for thecase qs z = +1 and increases for the qs z = 0 and qs z = − ρ meson correspondingto s z = 0 has to dominate in the collisions. Therefore, in the non-central heavy-ion collisionthe magnetic field favours longitudinal polarization of the ρ mesons.The dilepton asymmetries in non-central heavy ion collisions depend on the energybehaviour of the vector mesons. According to the vector dominance principle the vectormesons can directly convert to virtual photons. In turn, the electromagnetic decay of virtualphotons is one of the main sources of dilepton production in heavy-ion collisions. Whendileptons are produced in non-central heavy-ion collisions, their anisotropy depends on theresponse of the spin structure of intermediate resonances, such as ρ to the magnetic fieldof the collision. Therefore, it is of particular interest to identify and distinguish betweenvarious sources of dileptons emitted from non-central heavy-ion collisions.The shape of the dilepton distribution is characterized by the following differentialcross section: dσdM d cos θ = A ( M )(1 + B cos θ ) , (7.1)where M = ( p + p ) is the energy of the lepton pair in their rest frame, p and p arethe four-momenta of the leptons, θ is the angle between the momenta of the virtual photonand the lepton. The asymmetry coefficient B is defined by the polarization of the virtualphotons produced in the collisions: B = γ ⊥ − γ k γ ⊥ + γ k , (7.2)where the γ ⊥ , k are the contributions of the transverse and longitudinal polarizations of thevirtual intermediate photon.Our lattice calculations make possible to get the estimation of the asymmetry factorfor such processes. For the vector particle in Cartesian basis the polarization tensor hasthe following form P ij = 32 h s i s j + s j s i i − δ ij . (7.3)If w s z =+1 , w s z = − and w s z =0 are the probabilities that the ρ meson has a spin projection onthe field direction equal to +1, − P can be represented in terms of these probabilities in the following form: P = w s z =+1 + w s z = − − w s z =0 = N s z =+1 + N s z = − − N s z =0 N s z =+1 + N s z = − + N s z =0 , (7.4)where N s z =+1 , N s z = − and N s z =0 are the numbers of particles with different spin projec-tions. – 13 –s w s z =+1 + w s z = − + w s z =0 = 1, therefore, P = 1 − w s z =0 and − ≤ P ≤ 1. Thetensor polarizability describes the effect of the magnetic field for the spin states of the ρ meson, in turn, the spin states and tensor polarization is revealed in its decay to leptonpair, see [23]. In a strong magnetic field the spin of the particle tends to align along thefield direction, but the non-zero temperature leads to the spin-flipping.The differential cross section for the decay of ρ meson to the lepton pair may be writtenas dσdM d cos θ = N ( M )(1 + 14 P (3 cos θ − . (7.5)Comparing with (7.1) one can see that B = 3 P − P . (7.6)For the transversely polarized ρ meson B = 1 and for the longitudinally polarized B = − β t = β s z =+1 + β s z = − − β s z =0 β s z =+1 + β s z = − + β s z =0 , (7.7)which is the measure of the magnetic field effect on a vector meson, in particular for highmagnetic fields and temperature P ∼ β t .We calculate β t on the lattice taking into account the equality β s z =+1 = β s z = − . Theresults are presented in Table 5 for the neutral ρ -meson and in Table 6 for the charged ρ -meson. The large negative values of β t suggest the dominating longitudinal polarizationof the ρ -meson. The dileptons are mainly emitted in the directions perpendicular to themagnetic field axis. This is a convincing result, as we clearly see from Figure 6 and 7,thatthe energy of the state with s z = 0 decreases, and the energy of | s z | = 1 increases.It was found previously that for soft dileptons the longitudinal polarization also dom-inates [33]. This result was obtained by comparing the formation of soft dileptons with anonzero component of the conductivity of the strongly interacting matter, parallel to theexternal magnetic field [34]. V a (fm) m π (MeV) β t . 105 574 ± − . ± . . 115 541 ± − . ± . . 115 535 ± − . ± . . 115 395 ± − . ± . Table 5 . The tensor polarizability β t of the ρ meson is shown in the last column for the latticevolume V , the lattice spacing a and the pion mass m π . In this paper the calculation of the magnetic dipole and tensor polarizabilities have beenpresented for the charged and neutral ρ mesons on the lattice volume. We perform a– 14 – a (fm) m π (MeV) β t . 105 574 ± . ± . . 115 541 ± . ± . . 115 395 ± . ± . Table 6 . The tensor polarizability β t of the ρ ± is represented for various lattices and pion masses. thorough analysis of the behaviour of the effective mass plateau depending on the numberof points used for the fitting by hyperbolic cosine function.In Section 5 the magnetic dipole polarizability and g -factor for the charged mesons havebeen represented. We include the additional sets of data into consideration and increasestatistics with the lattice spacing a = 0 . 105 fm and volume 18 . This allows one to obtainthe more statistically significant values of the magnetic dipole polarizabilities for the spinprojection s z = 0, see Table 3. In contrast to the previous results [1], we have calculated thedipole magnetic polarizability for the case qs z = 1 from the 3-parametric fit (5.3), wherethe g -factor is a free parameter of the fit. Obviously, we could find the g -factor values atsmaller field interval and use these values for the β m determination, but the purpose ofthis work was to obtain the qualitative predictions.In Section 6 the dipole magnetic polarizability have been extracted for the case of theneutral ρ meson. The magnetic dipole polarizability was calculated, taking into accountthat the u and d -quarks couple differently to the magnetic field. In [11] the magneticdipole polarizability of ρ was calculated in the toy model with only one type of quark.Section 7 is devoted to the discussion of the polarization of the dileptons which resultfrom the decays of ρ mesons. We have found that the longitudinal polarization of the ρ mesons dominates in the collisions, because the low energy is more preferential than highenergy. Therefore the dileptons occurring due to decays of the ρ mesons will be emittedperpendicular to the direction of the magnetic field. This result reinforces the previousresults obtained in [34], i.e. the nonzero conductivity of the quark-hadronic matter in astrong magnetic field. We introduce new characteristics of the meson magnetic properties- the tensor palarizability. This quantity has been suggested to be related to the coefficientof asymmetry in the differential cross section for the dilepton production. Acknowledgments The authors are grateful to FAIR-ITEP supercomputer center where these numerical calcu-lations were performed. This work is completely supported by the grant from the RussianScience Foundation (project number 16-12-10059). References [1] E.V. Luschevskaya, O.E. Solovjeva, O.V. Teryaev, Determination of the properties of vectorsmesons in external magnetic field by quenched SU(3) lattice QCD, JHEP (2017) 142 – 15 – 2] M.A. Andreichikov, B.O. Kerbikov, V.D. Orlovsky and Yu.A. Simonov, Meson Spectrum inStrong Magnetic Fields, Phys.Rev. D (2013) 094029 [arXiv: 1304.2533][3] V.D. Orlovsky and Yu.A. Simonov, Nambu-Goldstone mesons in strong magnetic field,JHEP (2013) 136 [arXiv:1306.2232][4] S. Cho, K. Hattori, S.H. Lee, K. Morita and Sho Ozaki, Charmonium Spectroscopy in StrongMagnetic Fields by QCD Sum Rules: S-Wave Ground States, Phys.Rev. D (2015) 045025[arXiv:1411.7675][5] H. Taya, Hadron Masses in Strong Magnetic Fields, Phys.Rev. D (2015) 014038[arXiv:1412.6877][6] M. Kawaguchi and S. Matsuzaki, Vector Meson Masses from Hidden Local Symmetry inConstant Magnetic Field, Phys.Rev. D (2016) 125027 [arXiv:1511.06990][7] K. Hattori, T. Kojo and N. Su, Mesons in strong magnetic fields: (I) General analyses,Nucl.Phys. A (2016) 1 [arXiv:1512.07361][8] Ph. Gubler, K. Hattori, S.H. Lee, M. Oka, S. Ozaki and K. Suzuki, D mesons in a magneticfield, Phys.Rev. D (2016) 054026 [arXiv:1512.08864][9] G. Martinelli, G. Parisi, R. Petronzio and F. Rapuano, The proton and neutron magneticmoments in lattice QCD, Phys. Lett. (1982) 434[10] H. Liu, L. Yu and M. Huang, Charged and neutral vector meson under magnetic field, Phys.Rev. D (2015) 014017 [arXiv:1408.1318][11] E.V. Luschevskaya, O.E. Solovjeva, O.E. Kochetkov and O.V. Teryaev, Magneticpolarizabilities of light mesons in SU(3) lattice gauge theory, Nucl. Phys. B (2015) 627[arXiv: 1411.4284][12] S.R. Beane, E. Chang, W. Detmold, K. Orginos, A. Parreno, M.J. Savage and B.C. Tiburzi, Ab initio calculation of the np → dγ radiative capture process, Phys.Rev.Lett. (2015)132001 [arXiv:1505.02422][13] E.V. Luschevskaya, O.E. Kochetkov, O.V. Teryaev and O.E. Solovjeva, π ± and ρ , ± mesonsin a strong magnetic field on the lattice, JETP Letters no.10 (2015) 674[14] G. Bali, B.B. Brandt, G. Endrodi and B. Glaessle, QCD spectroscopy and quark massrenormalization in external magnetic fields with Wilson fermions, The 33rd InternationalSymposium on Lattice Field Theory, PoS LATTICE2015 (2016) [arXiv:1510.03899][15] E.V. Luschevskaya, O.E. Solovjeva and O.V. Teryaev, Magnetic polarizability of pion,Phys.Lett.B (2016) 393[16] A.M. Baldin, Polarizability of nucleons, Nucl. Phys. (1960) 310[17] L.V.Fil’kov and V.L.Kashevarov, Determination of π + -meson polarizabilities from γγ → π + π − process, Phys. Rev. C (2006) 035210 [nucl-th/0512047][18] A. Samsonov, Magnetic moment of the rho-meson in QCD sum rules: perturbativecorrections, JHEP (2003) 061 [hep-ph/0308065][19] T.M. Aliev, A. ¨Ozpineci and M. Savc, Magnetic and quadrupole moments of light spin-1mesons in light cone QCD sum rules, Phys. Lett. B (2009) 470[20] D. Djukanovic, E. Epelbaum, J. Gegelia and U.-G. Meissner, The magnetic moment of the ρ -meson, Phys. Lett. B (2014) 115 [arXiv:1309.3991] – 16 – 21] F.X. Lee, S. Moerschbacher and W. Wilcox, Magnetic moments of vector, axial, and tensormesons in lattice QCD, Phys. Rev. D (2008) 094502[22] B. Owen, W. Kamleh, D. Leinweber, B. Menadue, and S. Mahbub, Light Meson FormFactors at near Physical Masses, Phys. Rev. D (2015) 074503 [arXiv:1501.02561][23] E.L. Bratkovskaya, O.V. Teryaev, V.D. Toneev, Anisotropy of dilepton emission from nuclearcollisions, Physics Letters B (1995) 283[24] G. Baym, T. Hatsuda and M. Strickland, Structure of virtual photon polarization inultrarelativistic heavy-ion collisions, Nucl. Phys. A (2017) 712[25] M. L¨uscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. (1985) 59[26] V.G. Bornyakov, E.-M. Ilgenfritz and M. M¨uller-Preussker, Universality check of AbelianMonopoles, Phys. Rev. D (2005) 054511 [hep-lat/0507021][27] H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B (1998) 141[hep-lat/9707022][28] G.’t Hooft, A property of electric and magnetic flux in non-abelian gauge theories, Nucl.Phys. B (1979) 141[29] H. Zainuddin, Group-theoretic quantization of a particle on a torus in a constant magneticfield , Phys. Rev. D (1989) 636[30] G.-H. Chen, Degeneracy of Landau levels and quantum qroup sl q (2) , Phys. Rev. B (1996)9540[31] M.H. Al-Hashimi and U.J. Wiese, Discrete Accidental Symmetry for a Particle in a ConstantMagnetic Field on a Torus, Annals Phys. (2009) 343 [arXiv: 0807.0630][32] Dmitri E. Kharzeev, Karl Landsteiner, Andreas Schmitt, and Ho-Ung Yee, Stronglyinteracting matter in magnetic fields: an overview, Lecture Notes in Physics Volume (2013)[33] P.V. Buividovich, M.I. Polikarpov, O.V. Teryaev, Lattice studies of magnetic phenomena inheavy-ion collisions, Lecture Notes in Physics (2013) 377[34] P.V. Buividovich, M.N. Chernodub, D.E. Kharzeev, T.K. Kalaydzhyan, E.V. Luschevskaya,M.I. Polikarpov, Magnetic-Field-Induced insulator-conductor transition in SU(2) quenchedlattice gauge theory, Phys. Rev. Lett. (2010) 132001(2010) 132001