We extract the magnetic polarizability from the quadratic response of a
hadron's mass shift in progressively small static magnetic fields. The
calculation is done on a 24x12x12x24 lattice at a = 0.17 fm with an improved
gauge action and the clover quark action. The results are compared to those
from experiments and models where available.Comment: 3 pages, 3 figures, contribution to Lattice 2002 (spectrum

Bellucci

Bernard

Christensen

Ebert

F.X. Lee

Federspiel

Fiebig

Hallin

Hornidge

J. Christensen

Kossert

L. Zhou

Levchuk

Liebl

MacGibbon

W. Wilcox

Zieger

English

arXiv

Crossref

Magnetic polarizability of hadrons from lattice QCD

We extract the magnetic polarizability from the quadratic response of a hadron's mass shift in progressively small static magnetic fields. The calculation is done on a 24x12x12x24 lattice at a = 0.17 fm with an improved gauge action and the clover quark action. The results are compared to those from experiments and models where available

Zhou, L.

Lee, F.X.

Wilcox, W.

Christensen, J.

UNT (University of North Texas) Digital Library

arXiv:hep-lat/0209128 v1   16 Sep 20021Magnetic polarizability of hadrons from lattice QCD ∗ †L. Zhoua, F.X. Leeab, W. Wilcoxc, J. ChristensendaCenter for Nuclear Studies, George Washington University, Washington, DC 20052, USAbJefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USAcDepartment of Physics, Baylor University, Waco, TX 76798, USAdDepartment of Physics, McMurry University, Abilene, TX 79697, USAWe extract the magnetic polarizability from the quadratic response of a hadron’s mass shift in progressivelysmall static magnetic fields. The calculation is done on a 24 × 12 × 12 × 24 lattice at a = 0.17 fm with animproved gauge action and the clover quark action. The results are compared to those from experiments andmodels where available.Polarizabilities are important fundamentalproperties of particles. They determine dynami-cal response of a bound system to external per-turbations, and provide valuable insight into in-ternal strong interaction structure. We discuss alattice calculation for static magnetic polarizabil-ity β in the quenched approximation. This workis an extension of a lattice calculation for the elec-tric polarizability [1]. For an updated calculationof the electric polarizability, see [2].For small external magnetic fields, β can bedefined via the energy (mass) shift of a particlein the fields∆m = −12βB2 (1)We determine the magnetic polarizability by cal-culating the mass shift of hadrons∆m = m(B) −m(0) (2)where m(B) is the hadron mass in the presenceof an magnetic field B. We fit the data with thepolynomial∆m(B) = c1B +12c2B2 (3)∗presented by L. Zhou at Lattice 2002.†This work is supported in part by U.S. Department ofEnergy under grant DE-FG02-95ER40907, and by NSFgrant 0070836. The computing resources at NERSC andJLab have been used.We calculate mass shifts both in the field B andits inverse −B, then average them. The magneticpolarizabilities are the negative quadratic coeffi-cientsβ = −c2 (4)The gauge action is the zero-loop, tadpole-improved Lüscher-Weiz action with a=0.17 fm (or1/a=1159 MeV) set from the string tension. Thetadpole factor is determined by the average pla-quette as u0 = 0.877. The clover quark action wasused to put fermions on the lattice,Sf = SW −κu40∑x∑µ<ν[ψ(x) iσµνFµνψ(x)], (5)where SW is the tadpole-improved Wilsonfermion action.In order to place an magnetic field on the lat-tice, we construct an analogy to the continuumcase. There, the fermion action is modified by theminimal coupling prescription ∂µ → ∂µ + iqAµ,where q is the charge of the fermion field andAµ is the vector potential describing the externalfield. On the lattice, the prescription amountsto a phase factor eiaqAµ to the link variables [1].Choosing A2 = Bx1 and A0 = A1 = A3 = 0a constant magnetic field F12 = ∂1A2 − ∂2A1 =B3 = B can be introduced in the z-direction. We2Figure 1. Mass shifts of the neutron as a functionof the magnetic field B at the six quark masses.introduce two dimensionless parameters: one isη = qBa2, (6)where q = Qe with Q = ±1/3,±2/3. The otheris the integer lattice length,ρ = x1/a. (7)In terms of these two parameters the phase factorbecomes:eiaqA2 = eiηρ → (1 + iηρ), (8)where we have linearized the phase factor tomimic the continuum prescription. Therefore,the field strength we use will be chosen to satisfythe linearization requirement. To summarize, ourmethod to place the external magnetic field on thelattice is to multiply each gauge field link variableUµ(x) in the z-direction with a x-dependent fac-tor:U3(x) → (1 + iηρ)U3(x) (9)This factor is applied only to the SW part of theaction, the clover term is untouched.Periodic boundary conditions were imposed onthe fermion fields in y and z directions. In thex and t directions, we have employed fixed (orDirichlet) boundary conditions, i.e. the fermionfields are not allowed to propagate across theedges of x and t. To minimize the non-periodicboundary effects, we use a lattice size of 24 ×Figure 2. Magnetic polarizability of the neutronand proton as a function of m2π in physical units.12 × 12 × 24 with the fermion source location(x, y, z, t) = (12, 1, 1, 3). We analyzed 100 con-figurations.Fermion propagatorsM−1 were constructed atsix κ values, κ=0.1219, 0.1214, 0.1209, 0.1201,0.1194, 0.1182 corresponding to mπ=450 to 820MeV. The critical value κc = 0.1232(1). Weuse six different values of the parameter η =0.0, +0.00036, −0.00072, +0.00144, −0.00288,+0.00576. The η values in this sequence are re-lated by a factor of −2. Thus we are able tostudy the response of a hadron composed of bothup and down (strange) quarks to four differentnonzero magnetic fields.Typical mass shifts are displayed in Fig. 1 inthe case of neutron. There is good parabolicbehavior going through the origin, an indicationthat contamination from the linear term has beeneffectively eliminated by averaging results fromBand −B. The factore2a3 = 0.358× 10−43cm3 (10)was used to convert the lattice numbers into phys-ical polarizability in units of 10−4fm3. The curvesin Fig. 2 and Fig. 3 represent our preliminary re-sults for neutron, proton, π0, and K±. They aresmooth functions of the quark mass. We did notattempt an extrapolation in the quark mass.In Table 1, we show some experimental and3Figure 3. Magnetic polarizability for pion andkaon.theoretical values for the magnetic polarizabili-ties β of neutron, proton, pion and kaon. Theβ for neutron is not well-determined. A recentexperimental value [3] is 2.7 with a big error bar,while an earlier experimental data analysis [7] hasit at 11. The range from models and experimentsis from 1.22 to 14. Our lattice simulation resultis at the higher end of this range. For the proton,the range is 1.6 to 4.4, while our lattice result liesin the range. For π0, there is scant data. Thenumber from Chiral Perturbation Theory calcu-lation to O(p6) is 0.5 to 1.7. Our result is quitehigh, about 20. For K+, there is one theoreticalvalue −6.2 from the NJL model. Our result isaround -20.In summary, we have computed the magneticpolarizability of hadrons for the first time on thelattice using the external field method. In addi-ton to the particles described here, we have com-puted β for other mesons, the other members ofthe octet baryon, and the entire decuplet baryonfamily. We plan to present the final results else-where. Also under way is the extraction of mag-netic moments from the same data set.REFERENCES1. H.R. Fiebig, W. Wilcox, R.M. Woloshyn,Nucl. Phys. B324 (1989) 47.Table 1A compilation of magnetic polarizabilitiy for se-lected hadrons from experiments or models. Thenumbers are in units of 10−4 fm3.Ref. Particle β Approach[3] n 2.7 ∓1.8+0.6−1.1 ∓ 1.1 Theo.[4] n 6.7+1.3−0.7 Theo.[5] n 7.8 Theo.[6] n 10.3 ∓ 2.0 Exp.[7] n 11 ± 3 Theo.[8] p 3.3 ∓ 2.2 ∓ 1.3 Exp.[9] p 2.1 ∓ 0.8 ∓ 0.5 Exp.[5] p 3.5 Theo.[10] p 4.4 ± 0.4 ± 1.1 Exp.[11] p 3.58+1.19−1.25+1.03−1.07Exp.[12] π0 0.50 Theo.[12] π0 1.50 ± 0.20 Theo.[13] K+ -6.2 (-2.9) Theo.2. J. Christensen, F.X. Lee, W. Wilcox, L. Zhou,these proceedings. heplat/0209043.3. K. Kossert et al., nucl-ex/0201015, Phys.Rev. Lett. 88 (2002) 162301.4. H. Liebl, G.R. Goldstein, hep-ph/9411230,Phys.Lett. B343 (1995) 363.5. V. Bernard, N. Kaiser, A. Schmidt, and U.-G.Meiβner, hep-ph/9309211, Phys.Lett. B319(1993) 269.6. D. L. Hornidge et al., nucl-ex/9909015, Phys.Rev. Lett. 84 (2000) 2334.7. M.I. Levchuk, A.I. L′vov, nucl-th/0010059,Nucl. Phys. A684 (2001) 490.8. F.J. Federspiel et al., Phys. Rev. Lett. 67,(1991) 1511.9. B. E. MacGibbon et al., Phys. Rev. C52(1995) 2097.10. E.L. Hallin et al., Phys. Rev. C48 (1993)1497.11. A. Zieger et al., Phys. Lett B278 (1992) 34.12. S. Bellucci, hep-ph/9508282. Workshop onChiral Dynamics, M.I.T., July 25-29, 1994.13. D.Ebert, M.K.Volkov, Phys. Atom. Nucl. 60(1997) 796; Yad. Fiz. 60N5 (1997) 891-899.

Magnetic polarizability of hadrons from lattice QCD

Abstract

Similar works

Full text

Available Versions

Crossref

UNT (University of North Texas) Digital Library