New Sum Rule for the Nuclear Magnetic Polarizability

Abstract

I extend the well-known photonuclear sum rule that relates the strength of the photoexcitation of the giant dipole resonance in a nucleus to the number of elementary scatterers-nucleons to the case of virtual photons. The new sum rule relates the size of the magnetic polarizability of a nucleus to the slope of the transverse virtual photoabsorption cross section integrated over the energy in the nuclear range. I check this sum rule for the deuteron where necessary data is available, discuss possible applications and connection with other sum rules postulated in the literature.PACS numbers: 11.55. Hx, 25.20.Dc, 25.30.Fj, 13.60.Fz Keywords: dispersion relations, Compton scattering, sum rules Scattering of light off a composite object has long been used to study its structure. At low frequencies, electromagnetic waves scatter without absorption and solely probe its mass and electric charge, the classical Thomson result. With the photon energy raising above the absorption threshold internal structure is revealed. Kramers and Kronig related the photoabsorption spectrum of a material to its index of refraction by means of a dispersion relation [1, 2] based on the probability conservation and causality. Dispersion relations and sum rules have been among the main tools for studying the electromagnetic interactions in atomic, nuclear and hadronic physics domains. These domains roughly correspond to keV, MeV and GeV photon energies, respectively, and this scale hierarchy indicates that dynamics in each domain can be clearly identified. Thomas-Reiche-Kuhn sum rule equated the sum of oscillator strengths in an atom to the number of electrons [3][4][5]. For nuclei, Levinger-Bethe [6] and Gell-Mann, Goldberger and Thirring [7] related the integrated photoabsorption cross section to the number of elementary scatterers, protons and neutrons in a nucleus. For GeV energy photons that resolve the nucleon structure, Gorchtein, Hobbs, Londergan and Szczepaniak [8] observed that the integrated strength of the nucleon resonances may be explained by counting the constituent quarks. These sum rules are an economic, albeit approximate way to express duality, the transcendence of higher energy degrees of freedom in the low-energy phenomena [9]. In this letter I extend the Thomas-Reiche-KuhnLevinger-Bethe sum rule to the case of virtual photons, obtain a sum rule for the nuclear magnetic polarizability, and discuss further applications.The spin-averaged, forward Compton tensor T µν is expressed in terms of two scalar amplitudes T 1,2 (ν, Q 2 ),with the invariants defined in terms of the nucleus and photon four-momenta p, q as ν = (p · q)/M T , Q 2 = −q µ q µ = −q 2 ≥ 0, and p 2 = M 2 T , with M T the target nucleus mass. In this letter I concentrate on the transverse amplitude T 1 . Its imaginary parts is related to the unpolarized structure function F 1 as ImT 1 = (πα em /M T )F 1 , with α em ≈ 1/137 the fine structure constant. T 1 satisfies a once subtracted dispersion relation (DR),where the integral is understood in terms of its princ...

Two-photon exchange correction to2S−2Psplitting in muonicHe3ions

“…Excepting Q 2 = 0, this is unphysical kinematics and not directly related to scattering data. Instead we obtain it from a sum rule [52] based on the dispersion relation for T 1 and several additional observations.…”

Dispersive approach to two-photon exchange in elastic electron-proton scattering

“…Using the parametrization of the TPE amplitude M γγ in terms of the generalized form factors F 1 , F 2 and G a , we note that these TPE amplitudes have the symmetry properties [18,19] …”

“…The real and imaginary parts can be related through dispersion relations [18,19], which forms the basis of the dispersive method discussed in this section. Our discussion in this section follows the formalism of Tomalak and Vanderhaeghen [21,39].…”

“…The dispersive method involves the exclusive use of on-shell transition form factors, thereby avoiding the problem of unphysical violation of unitarity in the high energy limit. The dispersive approach to TPE was developed at forward angles by Gorchtein [18], and at non-forward angles by Borisyuk and Kobushkin [19,20,29,30], and more recently by Tomalak and Vanderhaeghen [21]. A feature of the latter two analyses has been the use of monopole form factor parametrizations, which allowed the computations to be performed semi-analytically.…”

“…A number of approaches have been adopted to computing the TPE corrections to elastic scattering, including direct calculation of the loop contributions in terms of hadronic degrees of freedom [4,[8][9][10][11][12][13][14][15], modeling the high energy behavior of box diagrams at the quark level through generalized parton distributions [16,17], or more recently dispersion relations [18][19][20][21]. Each of these methods has its own advantages as well as limitations (for reviews, see Refs.…”