Chiral perturbation theory makes definitive predictions for the extrinsic
behavior of hadrons in external electric and magnetic fields. Near the chiral
limit, the electric and magnetic polarizabilities of pions, kaons, and nucleons
are determined in terms of a few well-known parameters. In this limit, hadrons
become quantum mechanically diffuse as polarizabilities scale with the inverse
square-root of the quark mass. In some cases, however, such predictions from
chiral perturbation theory have not compared well with experimental data.
Ultimately we must turn to first principles numerical simulations of QCD to
determine properties of hadrons, and confront the predictions of chiral
perturbation theory. To address the electromagnetic polarizabilities, we
utilize the background field technique. Restricting our attention to
calculations in background electric fields, we demonstrate new techniques to
determine electric polarizabilities and baryon magnetic moments for both
charged and neutral states. As we can study the quark mass dependence of
observables with lattice QCD, the lattice will provide a crucial test of our
understanding of low-energy QCD, which will be timely in light of ongoing
experiments, such as at COMPASS and HI\gamma S.Comment: 3 pages, talk given by B. C. Tiburzi at PANIC 201

Detmold, W.

Tiburzi, B. C.

Walker-Loud, A.

English

arXiv

W. Detmold

B. C. Tiburzi

A. Walker-Loud

Crossref

Electromagnetic polarizabilities: Lattice QCD in background fields

Chiral perturbation theory makes definitive predictions for the extrinsic behavior of hadrons in external electric and magnetic fields. Near the chiral limit, the electric and magnetic polarizabilities of pions, kaons, and nucleons are determined in terms of a few well-known parameters. In this limit, hadrons become quantum mechanically diffuse as polarizabilities scale with the inverse square-root of the quark mass. In some cases, however, such predictions from chiral perturbation theory have not compared well with experimental data. Ultimately we must turn to first principles numerical simulations of QCD to determine properties of hadrons, and confront the predictions of chiral perturbation theory. To address the electromagnetic polarizabilities, we utilize the background field technique. Restricting our attention to calculations in background electric fields, we demonstrate new techniques to determine electric polarizabilities and baryon magnetic moments for both charged and neutral states. As we can study the quark mass dependence of observables with lattice QCD, the lattice will provide a crucial test of our understanding of low-energy QCD, which will be timely in light of ongoing experiments, such as at COMPASS and HI gamma S

UNT (University of North Texas) Digital Library

arXiv:1108.1698v1  [hep-lat]  8 Aug 2011Electromagnetic Polarizabilities:Lattice QCD in Background FieldsW. Detmold∗,†, B. C. Tiburzi∗∗ and A. Walker-Loud‡∗College of William and Mary, Williamsburg, VA†Thomas Jefferson National Accelerator Facility, Newport News, VA∗∗Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA‡Lawrence-Berkeley National Laboratory, Berkeley, CAAbstract. Chiral perturbation theory makes definitive predictions for the extrinsic behavior of hadrons in external electric andmagnetic fields. Near the chiral limit, the electric and magnetic polarizabilities of pions, kaons, and nucleons are detrminedin terms of a few well-known parameters. In this limit, hadrons become quantum mechanically diffuse as polarizabilitiesscale with the inverse square-root of the quark mass. In somecases, however, such predictions from chiral perturbationtheoryhave not compared well with experimental data. Ultimately we must turn to first principles numerical simulations of QCDto determine properties of hadrons, and confront the predictions of chiral perturbation theory. To address the electromagneticpolarizabilities, we utilize the background field technique. Restricting our attention to calculations in backgroundelectricfields, we demonstrate new techniques to determine electricpolarizabilities and baryon magnetic moments for both chargedand neutral states. As we can study the quark mass dependenceof observables with lattice QCD, the lattice will providea crucial test of our understanding of low-energy QCD, whichwill be timely in light of ongoing experiments, such as atCOMPASS and HIγS.Keywords: background field, chiral symmetry breaking, electric polarizability, Monte Carlo calculations, Schwinger’s proper time trickPACS: 12.38.Gc, 12.39.FeMOTIVATIONBeyond intrinsic properties of hadrons, which characterize their internal structure, there are extrinsic propertieswhichcharacterize the response of a hadron to external conditions. Electric and magnetic polarizabilities are examples of suchproperties, and are the only measured extrinsic propertiesof hadrons listed in the PDG. The electric polarizabilityαE ,for example, characterizes the strength of the induced electric dipole moment,~pE , when the hadron is subjected to anexternal field,~pE =−αE~E. For a hadron,h, dimensional analysis gives usαE(h) = N(h)α f s(43π [fm3]), (1)whereN(h) is a pure number that is hadron dependent, andα f s is the fine-structure constant.The MIT bag model [1], for example, provides a way to compute nucleon polarizabilities. The electric and magneticpolarizabilities one computes are the right order of magnitude [2]. Presumably this is due to the mechanism ofconfinement in the model which has as input a natural-sized hadronic length scale. For this model calculation, thequarks are taken to be massless. Thisc ral limit, however, is one in which the behavior of QCD can be understoodusing an effective theory [3]. Chiral perturbation theory results from considering the pattern of spontaneous andexplicit symmetry breaking in low-energy QCD. This theory makes simple predictions for pion, kaon, and nucleonpolarizabilities, which are generically of the form [4, 5]αχE (h) = Nχ(h)α f sfπ [mq〈ψψ〉]1/2, (2)where fπ is the pion decay constant,〈ψψ〉 is the chiral condensate, andmq is the quark mass. The pure numbersNχ(h) are determined within chiral perturbation theory. The theory is only effective when there is a power counting toorder the infinite tower of operators contained in the chiralLagrangian. In general, the expansion is controlled by thesmallness of the light quark masses compared to the chiral symmetry breaking scale. Despite numerous successes, thepion polarizability prediction is a factor of two differentthan the most recent experimental determination [6].BACKGROUND FIELD METHODTo determine polarizabilities and confront the discrepancies with experiment, we turn to lattice QCD. Unfortunatelydirect evaluation of the Compton scattering tensor,Tµν(ω ,ω ′), with current state-of-the-art lattice computations isout of reach. One particular challenge that would need to be addressed is the quantization of momentum in units of2π/L, due to the periodicity of the lattice. As such, one requiresvery large lattices,ω ,ω ′ = 2π/L ≪ mπ , to access thezero-momentum limit of the Compton tensor. A promising alternative is to use background fields [7, 8, 9].The basics of the background field method are basic: (i) studyQCD in the presence of external fields by measuringcorrelation functions, and (ii) study the behavior of correlations functions to determine response parameters. To achievethe former, quarks are coupled to external electromagneticfi lds through their charges. In practice, this amounts tomultiplication of the color gauge links by a classicalU(1) link.1 Only in the continuum limit are the quarks minimallycoupled to the electromagnetic field. With gauge links, the uniform field strengths allowed on a lattice are subjectto certain quantization conditions [10], else there is a considerably sized field gradient. Such huge gradients lead toundesirable effects, even when located far from the latticemeasurements [11].To perform measurements in lattice QCD, one chooses an interpolating fieldχh(x) having the quantum numbers ofa desired hadron,h (here we presume the ground state hadron is of interest). First, let us start with a neutral spin-lesshadron operator. One computes the two-point correlation function in the applied field,E ,GE (τ) = ∑~x〈χh(~x,τ)χ†h (0)〉E = ∑nZn(E )e−En(E )τ . (3)The long Euclidean time limit of the correlation function can be used to determine the ground-state hadron’s energyE0(E ) = Mh +12αhEE2, where we have kept only terms at second-order in the external field, which is presumedperturbatively small, and the sign of the second-order terma ises from treatment in Euclidean space. Measurementof the two-point correlation function for several values ofthe external electric field, will allow one to determine theelectric polarizability. We carried out such studies for the neutral pion (connected part) and neutral kaon [12].When one considers charged particles, this simple spectroscopic method will no longer work. We suggested thatone could still extract useful information from the two-point correlation functions of charged particles by matchingonto the behavior predicted from single-particle effective actions [13]. For a charged spin-less particle subjected to thefield Aµ = (0,0,−E x4,0), we have for~p = 0,G−1 =−∂ 2∂x24+E 2x24+E2(E ) ⇒ G =12H +E2(E ), (4)whereH is the harmonic oscillator Hamiltonian of the auxiliary quantum mechanics. The two-point function canthen be computed as a function ofτ using a method well known to those who know it well: Schwinger’s p oper-timetrick [14, 15]. The result is, of course, not a simple exponential in time; but, the predicted form of the correlationfunction matches well against lattice data [12], and can be used to extract the polarizability.When one considers spin-12 hadrons, an additional ingredient appears, namely the magnetic moment interaction.This is relevant even for an external electric field because the interaction has the formσµνFµν and does not disap-pear from the effective action when the velocity is projected to zero. An unpolarized neutron correlation function,Tr[GE (τ)] is described at long times by an exponential fall-off, but governed byE(E ) = Mn + 12E2[αnE −µ2n4M3n]. Onemust separate out the magnetic moment contribution to determin the electric polarizability. This can be achievedwith so-called boost projection [16], which utilizes the information contained in the various spin-components of thecorrelation function. For a magnetic field, one hasσµν Fµν = ~σ ·~B and one uses spin projection to separate out thetwo energy levels. On the other hand, for an electric field,σµν Fµν = ~K ·~E, where~K are the boost generator matrices.Tracing with boost projectors gives a way to separate out themagnetic momentTr[(1±K3)GE (τ)] = ZE(1±µnE2Mn)e−E(E )τ , (5)but it is done from the differing amplitudes. The boost-projection method has also been extended to proton correlationfunctions [16].1 This must be done for both valence quarks (affecting propagators) and sea quarks (affecting gauge field configurations).Due to currentcomputational restrictions, the sea quarks in our simulations are electrically neutral. We are investigating techniques to cure this malady.OUTLOOKThe background field method is a practical current-day technique to compute the polarizabilities of hadrons. Ourstudy was not limited to spin-less neutral hadrons. On the contrary, we developed new techniques to handle chargedhadrons, and spin-12 hadrons. Although we pursued computations in background electric fields, such methods areeasily generalized to magnetic fields. Lattice sizes are nearly large enough to support perturbatively small values ofquantized magnetic fields for the study of magnetic moments and magnetic polarizabilities. There are a number of areasfor improvement in our calculations. We intend to study the pion mass dependence of the extracted polarizabilities tomake contact with chiral perturbation theory. This can be done even with electrically neutral sea quarks, as partiallyquenched chiral perturbation theory has been employed to determine the sea quark charge dependence of nucleonand pion polarizabilities [13, 17]. Another notable area ofimprovement and concern is the effect of finite volume onthe extraction of polarizabilities. On a periodic lattice,virtual pions that propagate around the worldn-times lead tofinite volume artifacts that scale ase−nmπ L. For our background field simulations, contributions from virtual chargedpions wrapping around the worldn-times (plusn-anti-times) are accompanied by Wilson lines leading to modifiedfinite volume effects of the forme−nmπ L cos(nΦ) [18, 19]. HereΦ is the holonomy of the gauge field, which, forus, readsΦ =∫ L0 A3dz = −E Lx4. The presence of such time-dependent interactions complicates the extraction ofthe desired infinite volume physics from two-point functions. We are investigating means to handle such volumecorrections using single-particle effective actions in time-dependent perturbation theory.2 Nonetheless, the lattice canmake a fundamental contribution to hadron physics through the computation of polarizabilities. Such computationswill be timely because improved experimental measurementsare anticipated. The results, moreover, will allow us totest our understanding of low-energy QCD from first principles.ACKNOWLEDGMENTSWork supported in part by Jefferson Science Associates, LLCunder U.S. Dept. of Energy contract No. DE-AC05-06OR-23177 (W.D.), and the U.S. Dept. of Energy, under GrantNos. DE-SC000-1784 (W.D.), DE-FG02-94ER40818(B.C.T.), and DE-AC02-05CH11231 (A.W.-L.).REFERENCES1. A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, V. F. Weisskopf, Phys. Rev.D9, 3471-3495 (1974).2. A. Schäfer, B. Müller, D. Vasak, W. Greiner, Phys. Lett.B143, 323-325 (1984).3. J. Gasser, H. Leutwyler, Annals Phys.158, 142 (1984).4. B. R. Holstein, Comments Nucl. Part. Phys.A19, 221-238 (1990).5. V. Bernard, N. Kaiser, U. G. Meissner, Phys. Rev. Lett.67, 1515-1518 (1991).6. J. Ahrens, V. M. Alexeev, J. R. M. Annand, H. J. Arends, R. Beck, G. Caselotti, S. N. Cherepnya, D. Drechselet al., Eur.Phys. J.A23, 113-127 (2005). [nucl-ex/0407011].7. C. W. Bernard, T. Draper, K. Olynyk, M. Rushton, Phys. Rev.Lett. 49, 1076 (1982).8. G. Martinelli, G. Parisi, R. Petronzio, F. Rapuano, Phys.Lett. B116, 434 (1982).9. H. R. Fiebig, W. Wilcox, R. M. Woloshyn, Nucl. Phys.B324, 47 (1989).10. G. ’t Hooft, Nucl. Phys.B153, 141 (1979).11. W. Detmold, B. C. Tiburzi, A. Walker-Loud, PoSLATTICE2008, 147 (2008). [arXiv:0809.0721 [hep-lat]].12. W. Detmold, B. C. Tiburzi, A. Walker-Loud, Phys. Rev.D79, 094505 (2009). [arXiv:0904.1586 [hep-lat]].13. W. Detmold, B. C. Tiburzi, A. Walker-Loud, Phys. Rev.D73, 114505 (2006). [hep-lat/0603026].14. J. S. Schwinger, Phys. Rev.82, 664-679 (1951).15. B. C. Tiburzi, Nucl. Phys.A814, 74-108 (2008). [arXiv:0808.3965 [hep-ph]].16. W. Detmold, B. C. Tiburzi, A. Walker-Loud, Phys. Rev.D81, 054502 (2010). [arXiv:1001.1131 [hep-lat]].17. J. Hu, F. -J. Jiang, B. C. Tiburzi, Phys. Rev.D77, 014502 (2008). [arXiv:0709.1955 [hep-lat]].18. B. C. Tiburzi, Phys. Lett.B674, 336-343 (2009). [arXiv:0809.1886 [hep-lat]].19. W. Detmold, B. C. Tiburzi, A. Walker-Loud, [arXiv:0908.3626 [hep-lat]].20. M. Engelhardt [ LHPC Collaboration ], Phys. Rev.D76, 114502 (2007). [arXiv:0706.3919 [hep-lat]].2 An alternate way to handle such effects is to perturbativelyexpand the background field method from the start. This giveson direct control overWilson loop contributions which appear only as∝ Φ2, and can be separated out from the polarizability signal by var ing the reference time [20].

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Electromagnetic Polarizabilities: Lattice QCD in Background Fields

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