Nucleon Polarizabilities: from Compton Scattering to Hydrogen Atom
NNucleon Polarizabilities: from Compton Scattering to Hydrogen Atom
Franziska Hagelstein a , Rory Miskimen b , Vladimir Pascalutsa a a Institut für Kernphysik and PRISMA Excellence Cluster, Johannes Gutenberg-Universität Mainz, D-55128 Mainz, Germany b Department of Physics, University of Massachusetts, Amherst, 01003 MA, USA
Abstract
We review the current state of knowledge of the nucleon polarizabilities and of their role in nucleon Comptonscattering and in hydrogen spectrum. We discuss the basic concepts, the recent lattice QCD calculations andadvances in chiral effective-field theory. On the experimental side, we review the ongoing programs aimedto measure the nucleon (scalar and spin) polarizabilities via the Compton scattering processes, with real andvirtual photons. A great part of the review is devoted to the general constraints based on unitarity, causality,discrete and continuous symmetries, which result in model-independent relations involving nucleon polariz-abilities. We (re-)derive a variety of such relations and discuss their empirical value. The proton polarizabilityeffects are presently the major sources of uncertainty in the assessment of the muonic hydrogen Lamb shift andhyperfine structure. Recent calculations of these effects are reviewed here in the context of the “proton-radiuspuzzle”. We conclude with summary plots of the recent results and prospects for the near-future work.
Keywords:
Proton, Neutron, Dispersion, Compton scattering, Structure functions, Muonic hydrogen, ChiralEFT, Lattice QCD
Contents1 Introduction 3 a r X i v : . [ nu c l - t h ] F e b .3.1 Linearly Polarized Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3.2 Circularly Polarized Photons and Transversely Polarized Target . . . . . . . . . . . . . . 284.3.3 Circularly Polarized Photons and Longitudinally Polarized Target . . . . . . . . . . . . . 314.4 Virtual Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4.1 Response Functions and Generalized Polarizabilities . . . . . . . . . . . . . . . . . . . . 324.4.2 Radial Distribution of the Electric Dipole Polarizability . . . . . . . . . . . . . . . . . . . 344.5 Timelike Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5.1 Spin-Independent Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5.2 Spin-Dependent Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.6 Empirical Evaluations of Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Acknowledgements 68Appendix A Born Contribution in RCS and VVCS amplitudes 69Appendix B Derivation of a Dispersion Relation 69Appendix C Collection/Index of Formulae 71 . Introduction The concept of polarizabilities , common in optics and classical electrodynamics, was extended to the nucleonin the 1950s [1, 2], together with the first observations of Compton scattering (CS) on the proton [3–8]. Sincethen, the CS process, with real (RCS) or virtual (VCS) photons, became the main experimental tool in studyingthe nucleon polarizabilities, with dedicated experiments completed at: Lebedev Institute (Moscow) [8, 9],MUSL (Illinois) [10], SAL (Saskatoon) [11, 12], LEGS (Brookhaven) [13], Bates (MIT) [14], MaxLab (Lund)[15], MAMI (Mainz) [16–22], and Jefferson Laboratory (Virginia) [23].In recent years, the nucleon polarizabilities have advanced to the avantgarde of hadron physics. Theyare a major source of uncertainty in the muonic-hydrogen determination of the proton charge radius [24] andZemach radius [25], and hence are a prominent part of the “proton-radius puzzle” [26]. They play an importantrole in the controversy of the electromagnetic (e.m.) contribution to the proton-neutron mass difference [27–29]. Several issues involving the nucleon polarizabilities have emerged from the ongoing ‘spin physics program’at the Jefferson Laboratory (JLab), which is mapping out the spin structure functions of the nucleon [30–32].The various moments of these structure functions are related to the forward spin polarizabilities , with one ofthem, δ LT , being notoriously difficult to understand within the chiral effective-field theory ( χ EFT) [33, 34].The currently operating photon beam facility MAMI has established a dedicated experimental program todisentangle the nucleon polarizabilities through the low-energy RCS with polarized beams [35, 36] and targets[37, 38]; a complementary program, at even lower energy, is planned at HIGS (Duke) [39, 40]. A newexperimental program is being developed for the upcoming high-intensity electron beam facility MESA (Mainz).The recent theory advances include: (partially) unquenched lattice QCD calculations [41–47]; novel χ EFTcalculations of CS [48–54] and of the polarizability effects in hydrogenic atoms [55, 56]; development andevaluation of model-independent relations involving the nucleon polarizabilities [57–60]. These are the topicsof this review. The paper is organized as follows. Sect. 2 outlines the basic concepts as well as discusses the current ef-forts to calculate the nucleon polarizabilities from first principles: lattice QCD (Sect. 2.3) and χ EFT (Sect. 2.4).Sect. 3 describes the way polarizabilities appear in the CS processes, while Sect. 4 discusses the way they are ex-tracted from the CS experiments. Sect. 5 is devoted to dispersive sum rules, i.e., a variety of model-independentrelations derived from general properties of the forward doubly-virtual CS amplitude. They involve the wealthof inelastic electron-scattering data into the polarizability studies, and their data-driven evaluations are dis-cussed in Sect. 5.6. In Sect. 6 we present an overview of nucleon structure contributions to the hydrogenLamb shift and hyperfine structure. The reviewed results for nucleon polarizabilities and their effect in muonichydrogen are collected in the summary plots in Sect. 7. The reader interested in only a brief survey of thefield may skip to that section. Finally, the Appendices contain the expressions for the Born contribution to CSamplitudes (Appendix A), a derivation of generic dispersion relations (Appendix B), and a collection of themost important formulae (Appendix C).The remainder of this section contains the notations and conventions used throughout the paper. • We use the natural units, (cid:126) = c = 1 , and the following notation for the well-established parameters, alongwith their Particle Data Group (PDG) values [71]: α the fine-structure constant, α = 1 / . . (cid:126) c conversion constant, (cid:126) c = 197 . . m lepton mass, ( m e , m µ ) (cid:39) (0 . , . . m π pion mass, ( m π , m π ± ) (cid:39) (134 . , . . For other recent reviews (more focused on a subset) of these topics see: Drechsel et al. [61] (sum rules and fixed- t dispersionrelations for CS), Schumacher [62] (RCS experiments), Kuhn et al. [63] (spin structure functions and sum rules), Phillips [64] (few-nucleon χ EFT, neutron polarizabilities), Grießhammer et al. [65] ( χ EFT and RCS experiments), Guichon and Vanderhaeghen [66](VCS and generalized polarizabilities), Holstein and Scherer [67] (pion, kaon, nucleon polarizabilities), Pohl et al. [68], Carlson [69],Karshenboim et al. [70] (proton-radius puzzle). nucleon mass, ( M p , M n ) (cid:39) (938 . , . . κ nucleon anomalous magnetic moment, ( κ p , κ n ) (cid:39) (1 . , − . . f π pion decay constant, f π = 92 . . g A nucleon axial charge, g A = 1 . . • Other frequently used notation: s , t , u Mandelstam variables. ν , ω B , ω photon energy in the lab, Breit, and center-of-mass reference frames. ϑ , θ B , θ scattering angle in the lab, Breit, and center-of-mass reference frames. d Ω L , d Ω cm element of the solid angle in the lab and center-of-mass reference frames. Q = − q momentum transfer, photon virtuality. τ = Q / M dimensionless momentum-transfer variable. x = Q / M ν
Bjorken variable. F ( Q ) , F ( Q ) Dirac and Pauli form factors. G E ( Q ) , G M ( Q ) electric and magnetic Sachs form factors. G E = F − τ F , G M = F + F . f , ( x, Q ) unpolarized structure functions. g , ( x, Q ) polarized (or spin) structure functions. λ γ , λ (cid:48) γ helicities of the incident and scattered photon. λ N , λ (cid:48) N helicities of the incident and scattered nucleon. σ ( ν ) , d σ/ d Ω unpolarized total and differential cross sections. σ T , σ L unpolarized absorption cross section of the transverse ( T ) or longitudinal ( L ) photon. σ T T , σ LT doubly-polarized photoabsorption cross sections [see below Eq. (5.6)]. Z e , κ charge and anomalous magnetic moment of the nucleon ( Z = 1 for proton, Z = 0 for neutron). Ze , κ charge and anomalous magnetic moment of the nucleus (for hydrogen: Z = 1 , κ = κ p ). µ magnetic moment of the nucleon or nucleus, µ = Z + κ = Z (1 + κ ) , in units of nuclear magneton. a , m r Bohr radius and reduced mass, a − = Zαm r , m r = mM/ ( m + M ) . • Salient conventions: (cid:3)
Metric: g µν = diag(+1 , − , − , − . Levi-Civita symbol: (cid:15) = +1 = − (cid:15) . (cid:3) Scalar products: p · q = p i q i , p · q = p q − p · q , p · T · q = p µ T µν q ν , /p = p · γ . (cid:3) Pauli and Dirac matrices: = (cid:32) (cid:33) , σ = (cid:32) (cid:33) , σ = (cid:32) − ii (cid:33) , σ = (cid:32) − (cid:33) ,γ = (cid:32) − (cid:33) , γ i = (cid:32) σ i − σ i (cid:33) , γ = iγ γ γ γ = (cid:32) (cid:33) , (1.1) γ µν = [ γ µ , γ ν ] = − i / (cid:15) µναβ γ α γ β γ , γ µνα = ( γ µ γ ν γ α − γ α γ ν γ µ ) = − i(cid:15) µναβ γ β γ ,γ µναβ = [ γ µνα , γ β ] = i(cid:15) µναβ γ , (1.2)satisfying: [ σ i , σ j ] = iε ijk σ k , { γ µ , γ ν } = 2 g µν , { γ µ , γ } = 0 .4 Helicity spinors: u λ ( p ) = (cid:32) (cid:112) E p + M λ (cid:112) E p − M (cid:33) ⊗ χ λ ( θ, ϕ ) , (1.3)with E p = (cid:112) M + p ; λ = ± / the helicity (i.e., the spin projection onto p ); θ, ϕ the sphericalcoordinates of p ; and the two-component Pauli spinors: χ / ( θ, ϕ ) = (cid:32) cos( θ/ e iϕ sin( θ/ (cid:33) , χ − / ( θ, ϕ ) = (cid:32) − e − iϕ sin( θ/ θ/ (cid:33) . (1.4)The helicity spinors satisfy the following relations, for p = ( E p , p ) : ¯ u λ (cid:48) ( p ) u λ ( p ) = 2 M δ λ (cid:48) λ , (cid:88) λ u λ ( p ) ¯ u λ ( p ) = /p + M, ( /p − M ) u λ ( p ) = 0 . (1.5) (cid:3) Photon polarization vector, ε λ γ ( q ) , for a photon with four-momentum q and helicity λ γ = − , , .a) for real photon moving along the z -axis, q = ( ν, , , ν ) , there are only transverse polarizations,i) circularly polarized photons: ε µ ± = √ (cid:0) , ∓ , − i, (cid:1) , (1.6a)ii) linearly polarized photons: ε µ ( φ ) = (cid:0) , cos φ, sin φ, (cid:1) , (1.6b)b) for virtual photon moving along the z -axis, q = ( ν, , , | q | ) , there is, in addition, the longitudinalpolarization: ε µ = 1 (cid:112) q (cid:0) | q | , , , ν (cid:1) , with | q | = (cid:112) ν − q . (1.7)The transversality, orthonormality and completeness conditions are: q · ε λ γ ( q ) = 0 , ε ∗ λ (cid:48) γ ( q ) · ε λ γ ( q ) = − δ λ (cid:48) γ λ γ , (cid:88) λ γ = ± , ε ∗ µλ γ ε νλ γ = − g µν + q µ q ν q . (1.8)Note that for a spacelike photon ( q = − Q < ), the longitudinal polarization vector is antiher-mitian, ε ∗ = − ε , and as the result the above orthonormality and completeness conditions are notsatisfied. This is why one often defines the longitudinal polarization vector for a spacelike photon as ε (cid:48) ≡ iε = / Q ( | q | , , , ν ) . The two definitions are connected by a gauge transformation.5 . Basic Concepts and Ab Initio Calculations π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + π + + + + + + + + − − − − − − − + + + + − − − − E Figure 2.1: Naive view of the proton, consisting of apion cloud and a quark core, placed between the platesof a parallel plate capacitor. The left (right) figureshows the capacitor discharged (charged). Plot cour-tesy of Phil Martel. π + π + u du π + π + π + π + u du π + π + N N N N N N N
S S S S S S S S
N N N N
S S S S H ParamagneticDiamagnetic
Figure 2.2: Naive view of the proton, consisting of apion cloud and a quark core, placed between the polesof a magnet. The left (right) figure shows the externalmagnetic field turned off (on). Plot courtesy of PhilMartel.A polarizability, by definition, quantifies the re-sponse of a system to an external electromagnetic(e.m.) field, or more precisely, the e.m. moments in-duced in response to a moderate e.m. field. An evi-dent picture is provided by an atom immersed in a ho-mogeneous electric field. The atomic nucleus and theelectron cloud displace in opposite directions, thuscreating an electric dipole moment proportional tothe field strength, with the proportionality coefficient α E , the electric dipole polarizability. The polariz-ability mechanism in the nucleon is less obvious, but,very roughly, one can replace the electron cloud bythe “pion cloud” and the nucleus by a “quark core”to have a similar picture, see Fig. 2.1. An analogousrepresentation of the magnetic polarizability, β M , isdisplayed in Fig. 2.2.This naïve interpretation is realized, in a way,in χ EFT where the (renormalized) pion loops canbe thought of as the effect of the pion cloud, whilethe ∆(1232) -resonance excitation and the low-energyconstants (LECs) are the effect of the quark core. Inthe case of the magnetic dipole polarizability β M , the diamagnetic contribution of the pion cloud is com-peting against the paramagnetic contribution of thequark-core excitation, see Fig. 2.2. The two contri-butions are largely canceling each other, leaving thenucleon with a relatively small magnetic polarizabil-ity, cf. Sect. 2.4 for details.Other intuitive pictures of the nucleon polarizabil-ities emerge in quark models [72–76], the Skyrmemodel [77–82], and the Nambu-Jona-Lasinio model[83]. All of them point out the large paramagneticcontribution due to the nucleon-to- ∆ (1232) M transition.While for the atoms the polarizabilities are of order of the atomic volume, the nucleon being much tighterbound (nearly of its mass coming from the binding force) has polarizabilities which are about three ordersof magnitude smaller than its volume. It is customary to use the units of − fm for the dipole polarizabilitiesof the nucleon.The critical electric field strength needed to induce any appreciable polarizability of the nucleon can beestimated as the ratio of the average energy level spacing in the nucleon to the size of the nucleon, i.e.,E crit . ≈
100 MeV / ( e fm) = 10 Volt / m . Static electric field strengths of this intensity are not available in alaboratory, and will never be available. However, a classical estimate of the electric field strength of a
100 MeV photon Compton scattering from the nucleon is approximately Volt / m . Given the absence of static e.m.fields of the required immensity, the CS process is currently the best available tool for accessing the nucleonpolarizabilities experimentally, cf. Sect. 4.In the rest of this section we introduce the nucleon polarizabilities and discuss their calculation from firstprinciples. We shall focus on describing the efforts to compute the nucleon polarizabilities in lattice QCD andchiral EFT. In the latter case, calculations of the CS observables will be discussed too.It is worthwhile noting that is a number of sophisticated theoretical approaches, other than lattice QCDand chiral EFT, applied to the nucleon polarizabilities and low-energy CS. They include: the fixed- t dispersion6elations [84–87], effective-Lagrangian models with [88–91] and without [92–94] causality constraints, theDyson–Schwinger equation approach to QCD [95]. The first one in this list is very popular in the extractions ofpolarizabilities from CS data, and will be mentioned frequently in other chapters of this review. The response in the energy of the system due to polarizability effects is described by an effective Hamilto-nian, which usually is ordered according to the number of spacetime derivatives of the e.m. field A µ ( x ) [96, 97], H (2)eff = − π (cid:0) α E E + β M H (cid:1) , (2.1a) H (3)eff = − π (cid:16) γ E E σ · ( E × ˙ E ) + γ M M σ · ( H × ˙ H ) − γ M E E ij σ i H j + γ E M H ij σ i E j (cid:17) , (2.1b) H (4)eff = − π (cid:16) α E ν ˙ E + β M ν ˙ H (cid:17) − π (cid:0) α E E ij + β M H ij (cid:1) , (2.1c)where the electric ( E ) and magnetic ( H ) fields are expressed in terms of the e.m. field tensor, F µν = ∂ µ A ν − ∂ ν A µ , as: E i = F i , H i = (cid:15) ijk F jk . Furthermore, the following shorthand notation is used: E ij = ( ∇ i E j + ∇ j E i ) , H ij = ( ∇ i H j + ∇ j H i ) . (2.2)The 3 rd -order term depends on the nucleon spin via the Pauli matrices σ , and the corresponding polariz-abilities are called the spin polarizabilities [98]. They have no analog in classical electrodynamics, but evidentlythey describe the coupling of the induced e.m. moments with the nucleon spin. Unlike the scalar polarizabili-ties, they are not invoked by static e.m. fields.The above Hamiltonian is quadratic in the e.m. field. This means that the polarizabilities can directly beprobed in the CS process. The expansion in derivatives of the e.m. field translates then into the low-energyexpansion. The polarizabilities thus appear as coefficients in the low-energy expansion of the CS amplitudes,cf. Sect. 3.As noted above, the scalar dipole polarizabilities are measured in units of − fm . In general, the nucleonpolarizabilities are measured in units − fm n +1 , where n is the order at which they appear. Presently all of the lattice QCD calculations of nucleon polarizabilities use the background-field method [99,100], which amounts to measuring the shift in the mass spectrum upon applying a classical background field.On a given configuration, one multiplies the SU (3) gauge fields by a U (1) gauge field. The U (1) links are givenby U µ ( x ) = exp [ ie q a A µ ( x )] , (2.3)where e q is the quark charge and a is the lattice spacing.The case of a constant magnetic field is the simplest to illustrate. For the field with a magnitude H pointingin the + z -direction, the usual choice is A µ ( x, y, z, t ) = a Hx δ µy . The problem with this choice is that due tothe condition that the gauge links U µ must be periodic, the field is continuous only if e q a H = 2 πn/L , withinteger n . The minimal value of H is thus severely limited by the size of the lattice, although an improvementto H ∼ /L behavior is easily achieved (see, e.g., Ref. [101]).One can calculate a baryon two-point function which behaves for large time in the usual manner C ( t ) ∼ e − M ( H ) t + . . . , (2.4)but with the exponential damping governed by a field-dependent mass [102] M ( H ) = M − µ z H − β M H + O ( H ) , (2.5)where M is the mass with no field and µ z is the projection of the magnetic moment. One may cancel theodd terms by considering M ( H ) + M ( − H ) and fit the remaining H -dependence by adjusting the value of themagnetic polarizability. 7mplementation of the electric field is somewhat more tricky and has led to an overall sign mistake in thevalue of the electric polarizability (which affects, e.g., Ref. [41] as well as many of the earlier calculations).In the proton case, one in addition needs to take care of the Landau levels, which thus far has only beendone by the NPLQCD collaboration [47]. Implementation of the varying fields needed to compute the spinpolarizabilities is considered in Ref. [103].In the background field method one obviously assumes that the Taylor expansion in the field strength isquickly convergent. The non-analyticity due to the pion production induced by the background field mayhowever become a problem. This problem is similar to the one encountered in experiment, where to see thesignal in the CS observables one needs energies approaching the pion-production threshold.Another difficulty of this method is the inclusion of the background-field effect on the “sea”. Most of thecalculations to date assume the sea quarks to be neutral. Studies of the charged sea-quark contributions havebeen done in, e.g., Refs. [41, 104].
200 300 400 500 600 700 80002468101214 m Π (cid:64) MeV (cid:68) Α n (cid:64) (cid:45) f m (cid:68) HB Χ PT (cid:45) NNLOB Χ PT (cid:45) NLODetmoldEngelhardtEN2EN1Expt.
Figure 2.3: Lattice QCD results for the electric polarizability of the neutron, at unphysical values of pion mass:Engelhardt [41], Detmold et al. [42, 43], and Lujan et al. [46] on two different ensembles (’EN1’ and ’EN2’).The curves with error band are the predictions of the baryon [49] and heavy-baryon [105] χ PT. The HB χ PTresult is a fit at the physical pion mass. Plot courtesy of Andrei Alexandru.Figure 2.4: Results of Hall et al. [45] for the magneticpolarizability of the neutron. Plot courtesy of JonathanHall.Despite these concerns, the recent lattice resultsare very encouraging. Most of the results are ob-tained for the neutron electric polarizability, seeFig. 2.3. The lightest-pion results, indicated as EN1and EN2 therein, are from the GWU group [46].They have recently received substantial finite-volumecorrections, moving them upwards, right onto theHB χ PT curve [106].For the magnetic polarizability we refer to therecent work of the Adelaide CSSM group [44, 45]which used the PACS-CS flavor gauge fieldconfigurations [107] and performed an extrapolationto the physical pion mass and infinite volume, seeFig. 2.4. Their extrapolated result is [45]: β ( n ) M =1 . stat (11) sys × − fm , and can be directlycompared with experiment and other theoretical re-sults in Fig. 7.2. Very recently, the NPLQCD Collabo-ration obtained results for the electric and magneticpolarizabilities of both proton and neutron (and a 8umber of light nuclei), albeit at a relatively large pion mass of 800 MeV [47].As an alternative to the background-field method, one may consider a four-point function calculation of theCompton tensor for spacelike photons. At present this has only been done for the light-light system (light-by-light scattering) [108]. This method would avoid the problem with the non-analytic behavior of the mass spec-trum in the background field, but would require an extrapolation to the real-photon point. On the other hand,it would be a direct calculation of the polarizabilities at a finite photon virtuality Q , which experimentally isaccessed through the dispersive sum rules (cf. Sect. 5), and is required in the atomic and two-photon-exchangecalculations (cf. Sect. 6). In practice, such a direct calculation of doubly-virtual CS on the nucleon would bevery challenging due to the usual problems of the noise-to-signal ratio and excited-state contamination. TheCS on the pion would certainly be a better place to start. The chiral effective-field theory ( χ EFT), also referred to as the chiral perturbation theory ( χ PT) is a low-energy EFT of QCD, see Refs. [109–111] for the seminal papers. It is a quantum field theory with the Lagrangianwritten in terms of hadronic fields, in contrast to QCD which is written in terms of quark and gluon fields. The χ EFT, however, holds the promise to match the QCD description of the low-energy phenomena, where bylow-energy one assumes the relative energy in the hadronic system to be well below 1 GeV. In our case ofCS processes, this means that the energy ν and the momentum transfer Q with which the photon probes thenucleon are much smaller than 1 GeV. At these energies the pion interactions are suppressed. More precisely,the interaction goes with pion 4-momenta, which are relatively small for small pion energy. A perturbativeexpansion of scattering amplitudes in pion momenta is called the chiral expansion.In contrast to QCD, χ EFT is non-renormalizable in the usual sense. However, to any finite order in theEFT expansion, all the divergencies are absorbed by renormalizations of a finite number of parameters, calledlow-energy constants (LECs). The renormalized LECs are to be matched to QCD: in practice, either extractedfrom the lattice QCD results or fit to experimental data.The number of unknown LECs grows quickly with the order (or precision) to which one wants to compute.For this reason, one might be tempted to dismiss such theory as having practically no predictive power. Thisindeed would be the case if the LECs are treated as entirely free parameters, i.e., allowed to take arbitraryvalues. They are not — their effect must be of natural size [112], which simply speaking means that the LECsmay only have an effect consistent with the estimate based on power counting (i.e., in most cases the naivedimensional analysis ). When a certain LEC effect is unnaturally large, hence exceeds the expectation and/orrequires the “promotion to a lower order”, the EFT should be revised to include the missing low-energy physics.On the other hand, when naturalness is implemented, the EFT is predictive and the uncertainty due to neglectof the higher-order corrections can be estimated.The calculations with no divergencies, and/or no new constants to be fit, are genuine predictions of χ EFT.Such examples are quite rare, however the calculation of nucleon polarizabilities presents one of them. Theleading-order [ O ( p ) ] contribution to nucleon polarizabilities is predictive, as there are no LECs renormalizingthe polarizabilities [until O ( p ) ]. This case therefore presents a great testing ground of the χ EFT framework.To begin with, one can clearly see here that rather different predictions are obtained depending on whetherthe so-called heavy-baryon (HB) expansion [113] is employed [114, 115] or not [116, 117]. For example, forthe proton dipole polarizabilities { α E , β M } one obtains { . , . } in HB χ PT, versus { . , − . } in χ PT. Theuncertainty on such a leading order prediction can be quite large and hence this discrepancy might not lookas bad at first. The discrepancy deepens at the next order, i.e., with the inclusion of the ∆(1232) as an explicitdegree of freedom. The ∆ contributions to the nucleon polarizabilities come out to be large in HB χ PT [118]and ought to be canceled eventually by the LECs which are “promoted” from higher orders, cf. Grießhammeret al. [65], Hildebrandt et al. [119]. This problem is discussed at length in Refs. [49, 120]. For a more generaloverview of the χ PT in the single-baryon sector (B χ PT) and the current status of the theory, see Geng [121].The ∆(1232) -resonance plays a prominent role in the modern formulation of χ PT in the baryon sector. Itsexcitation energy, ∆ = M ∆ − M N (cid:39) MeV , (2.6)is relatively low, and the ∆ must be included explicitly in the χ PT Lagrangian. The construction of HB χ PT(semi-relativistic) Lagrangians with ∆ ’s, and decuplet fields in general, was considered in Refs. [113, 114].9he manifestly Lorentz-invariant B χ EFT Lagrangians with the spin-3/2 ∆ fields were reviewed in [122, Sect.4]. Concerning the power counting of the ∆ contributions, Hemmert et al. [123] coined the term “ (cid:15) -expansion”,whereas a different counting (“ δ -expansion”) was subsequently proposed by Pascalutsa and Phillips [124]. Thetwo power-counting schemes differ in how much the excitation energy ∆ weighs in, compared with the pionmass m π . In the (cid:15) -expansion they are the same ( ∆ ∼ m π ), while in the δ -expansion m π (cid:28) ∆ . The mainadvantage of the latter is that it provides a systematic counting of the ∆ -pole contributions, which go as / ( p − ∆ ) where p is the typical energy or momentum. Indeed, as p is of the same order as m π and ∆ , the (cid:15) counting implies that these contributions are always overwhelmingly important. In practice, however, the (cid:15) -expansion counts these propagators as /p . This works for the energies well below the resonance, but inthe ∆ -resonance region these contributions are dominating and the power counting should reflect that. The δ counting does just that transition. When p ∼ m π , the propagator / ( p − ∆ ) counts as /∆ . When p ∼ ∆ (theresonance region), the ∆ -pole contributions are summed yielding the dressed propagator / ( p − ∆ − Σ ) , with Σ the self-energy of the ∆ , and the resulting dressed propagator counts as / Σ ∼ /p , since usually Σ ∼ p .Thus, the δ -expansion is an EFT with a hierarchy of two low-energy scales and as such has two regimes, p ∼ m π (low-energy) and p ∼ ∆ (resonance), where the ∆ contributions count differently. For definiteness,the following powers are assigned to the low-energy scales in the two regimes:1. low-energy: m π ∼ p , ∆ ∼ p / .2. resonance: m π ∼ p , ∆ ∼ p .Hence, e.g., the propagator / ( p − ∆ − Σ ) is of O ( p − / ) in the first region and of O ( p − ) in the second.The present state-of-the-art calculations of CS observables, within HB χ PT [51, 65] and B χ PT [48, 49, 54],employ the δ -expansion. Other applications of the δ -expansion include: the forward VVCS [34, 125], pion-nucleon scattering [126], pion photo- [127] and electro-production [128, 129], radiative pion photoproduction[130, 131]. For the VVCS case, there is a significant discrepancy between two B χ PT calculations, based on the (cid:15) [33] and the δ [34, 125] counting schemes, for some of the forward spin polarizabilities, see Fig. 7.4 andFig. 7.6. This discrepancy is not yet completely understood.Coming back to RCS, the B χ PT calculation of Lensky and Pascalutsa [49] are done to next-to-next-to-leadingorder (NNLO), i.e. to O ( p / ) , in the low-energy region. To this order, these are genuine predictions of B χ PT inthe sense that all the parameters are determined from elsewhere; the LECs intrinsic to polarizabilities do notenter until O ( p ) . A very good description of the CS experimental data is nevertheless observed — a typicaldescription of the unpolarized angular distribution is shown in Fig. 2.5. d ! / d " c . m . ( nb / s r) c.m. (deg) E $ = 149 MeV NNLO B % PTMAMI 01SAL 93
Figure 2.5: Results of the B χ PT calculation [49]for the differential cross section of proton Comp-ton scattering (red curve with the band), com-pared with the experimental data from SAL [11]and MAMI [17]. The numerical composition of the various contributionsto the dipole polarizabilities of the proton is given in B χ PTby (in units of − fm ): α ( p ) E = 6 . (cid:124)(cid:123)(cid:122)(cid:125) O ( p ) + ( − . . (cid:124) (cid:123)(cid:122) (cid:125) O ( p / ) ) = 11 . , (2.7a) β ( p ) M = − . (cid:124) (cid:123)(cid:122) (cid:125) O ( p ) + (7 . − . (cid:124) (cid:123)(cid:122) (cid:125) O ( p / ) ) = 3 . , (2.7b)where the first number in O ( p / ) comes from the ∆ -resonance excitation while the second comes from the π ∆ loops. One sees that the ∆ -excitation is mainly affectingthe magnetic polarizability and is of paramagnetic nature(i.e., positive contribution to β M ). This is expected fromthe first nucleon excitation which predominantly is of M type. The pion loops, playing here the role of the “pioncloud”, induce the diamagnetic effects (i.e., negative β M ).These B χ PT results can be contrasted with the corre-10able 2.1: Predictions for the proton static dipole, quadrupole, and dispersive polarizabilities, in units of − fm (dipole) and − fm (quadrupole and dispersive), compared with the χ PT-based fits dipole po-larizabilities to RCS database. The latest PDG values for dipole polarizabilities are shown too.
Source α E β M α E β M α E ν β M ν HB χ PT fit [65] . ± .
50 3 . ± . · · · · · · · · · · · · B χ PT fit [52] . ± . . ± . · · · · · · · · · · · · B χ PT NNLO [54] . ± . . ± . . ± . − . ± . − . ± . . ± . PDG [71] . ± . . ± . · · · · · · · · · · · · sponding HB χ PT calculation [118]: α ( p ) E ( HB ) = 12 . ( π N loop) + 0 ( ∆ pole) + 8 . ( π ∆ loop) = 20 . , (2.8a) β ( p ) M ( HB ) = 1 . ( π N loop) + 12 ( ∆ pole) + 1 . ( π ∆ loop) = 14 . . (2.8b)Here the chiral loops give a much larger (than in B χ PT) effect in the electric polarizability, while in the magneticthey even have an opposite sign. As the result, both polarizabilities come out to be way above their empiricalvalues. As noted above, this discrepancy is usually corrected by promoting the higher-order [ O ( p ) ] LECs, atthe expense of violating the naturalness requirement.Recently, the B χ PT framework of Ref. [49] has been extended to the ∆ -resonance region [54], with anupdate on the predictions for α E and β M (included in the above numbers for the proton). Predictions forthe spin and higher-order polarizabilities have also been obtained. The results for the scalar polarizabilities ofthe proton are presented in Table 2.1, where they are compared with the χ PT results obtained by fitting theexperimental RCS cross sections. The fitting in Refs. [52, 65] is done using LECs from the orders beyond NNLO.The fact that the B χ PT fit [52] and prediction [54] agree, within the uncertainties, indicates that the LEC effect(which is the only difference between the two calculations) is of natural size.A number of predictions for polarized CS observables, emphasizing the role of the chiral loops, are givenin Ref. [54] as well, see e.g., Fig. 4.8 below. The corresponding results for the proton spin polarizabilities areshown in Table 4.2.A brief summary of the χ PT results for the nucleon polarizabilities is given in Sect. 7. The HB χ PT calcula-tions therein have recently been reviewed by Grießhammer et al. [65].11 . Polarizabilities in Compton Scattering
The CS processes, represented by Fig. 3.1, can be classified according to the photon virtualities, q and q (cid:48) ,while the target particle (hereby the nucleon) is on the mass shell: p = p (cid:48) = M . The Mandelstam variablesfor this two-body scattering process are: s = ( p + q ) = M + 2 p · q + q = ( p (cid:48) + q (cid:48) ) , (3.1a) u = ( p (cid:48) − q ) = M − p (cid:48) · q + q = ( p − q (cid:48) ) , (3.1b) t = ( p − p (cid:48) ) = 2 M − p · p (cid:48) = ( q − q (cid:48) ) . (3.1c)Figure 3.1: The Compton scattering off thenucleon: γ ( q ) + N ( p ) → γ ( q (cid:48) ) + N ( p (cid:48) ) . Their sum is as usual given by the sum of invariant massessquared: s + t + u = 2 M + q + q (cid:48) . Throughout the paperwe use the following kinematical invariants, ν = p · q/M, ν (cid:48) = p · q (cid:48) /M, (3.2)which in the lab frame become the energy of, respectively, theincoming and outgoing photon.In the most general case, the initial and final photons are vir-tual, with different virtualities, q (cid:54) = q (cid:48) . In reality, this situationmay occur in the dilepton electro-production, e − N → e − N e + e − ,the N ¯ N production in e + e − collisions, or in the two-photon-exchange contribution to lepton-nucleon scatter-ing, discussed in Sect. 6 in the context of atomic calculations.Denoting the photon helicity by λ γ = ± , and the nucleon helicity by λ N = ± / , there are obviously × × × helicity amplitudes, T λ (cid:48) γ λ (cid:48) N λ γ λ N ( s, t ) , describing this process. Discrete symmetries, such asparity and time reversal, reduce the number of independent helicity amplitudes by more than a half, as will bediscussed in more detail below.The Feynman amplitude T µν describing this process is a rank-2 tensor-spinor which depends on the four-momenta q , q (cid:48) , p , p (cid:48) . Due to momentum conservation, three of them are independent, e.g.: q , q (cid:48) , and P = ( p + p (cid:48) ) . The helicity amplitudes are expressed in terms of the Feynman amplitude as: T λ (cid:48) γ λ (cid:48) N λ γ λ N = ¯ u λ (cid:48) N ( p (cid:48) ) ε ∗ λ (cid:48) γ ( q (cid:48) ) · T ( q (cid:48) , q, P ) · ε λ γ ( q ) u λ N ( p ) , (3.3)with the nucleon spinors and photon polarization vectors defined in Sect. 1.1. A consequence of the e.m. gaugeinvariance is q (cid:48) µ T µν ( q (cid:48) , q, P ) = 0 = q ν T µν ( q (cid:48) , q, P ) , (3.4)valid for on-shell nucleons and arbitrary photon virtualities. The Lorentz decomposition of the Feynman am-plitude in terms of the invariant amplitudes A i , T µν ( q (cid:48) , q, P ) = e (cid:88) i O µνi A i ( ν (cid:48) , q (cid:48) , ν, q ) , (3.5)contains terms, after the constraints due to parity, time reversal and gauge invariance are taken into account[132]. For off-forward VVCS with q (cid:48) = q , this number reduces to ; for forward VVCS, to 4. For the restof this section we restrict ourselves to the RCS, i.e., the case where both photons are real ( q (cid:48) = q = 0 ). Thecase where one of the photons is virtual (VCS and timelike CS) is briefly discussed in Sect. 4. The forwarddoubly-virtual CS appears prominently in Sect. 5 and 6. Consider the classic CS: the elastic scattering of a real photon, or real CS (RCS). The RCS on a spin- / target is described by the helicity amplitudes, T λ (cid:48) γ λ (cid:48) N λ γ λ N ( s, t ) , subject to the following parity ( P ) and time-reversal ( T ) constraints: P : T − λ (cid:48) γ − λ (cid:48) N − λ γ − λ N ( s, t ) = ( − λ (cid:48) γ − λ (cid:48) N − λ γ + λ N T λ (cid:48) γ λ (cid:48) N λ γ λ N ( s, t ) , (3.6a) T : T λ γ λ N λ (cid:48) γ λ (cid:48) N ( s, t ) = ( − λ (cid:48) γ − λ (cid:48) N − λ γ + λ N T λ (cid:48) γ λ (cid:48) N λ γ λ N ( s, t ) . (3.6b)12hese constraints reduce the number of independent amplitudes from × × × to . The amplitudesin this case depend only on the combined (or total) helicities, H = λ γ − λ N , H (cid:48) = λ (cid:48) γ − λ (cid:48) N , which run through ± / , ± / , and the above constrains read: T H (cid:48) H ( s, t ) P = ( − H (cid:48) − H T − H (cid:48) − H ( s, t ) T = ( − H (cid:48) − H T HH (cid:48) ( s, t ) . (3.7)The six independent amplitudes are usually chosen as follows [84]: (8 πs / ) Φ ≡ T − / − / = T + / + / , (8 πs / ) Φ ≡ T − / + / = − T + / − / , (8 πs / ) Φ ≡ T − / + / = T + / − / = T + / − / = T − / + / , (8 πs / ) Φ ≡ T − / − / = − T + / + / = T + / + / = − T − / − / , (8 πs / ) Φ ≡ T + / + / = T − / − / , (8 πs / ) Φ ≡ T − / + / = − T + / − / . (3.8)Their normalization is chosen such that, given the unpolarized cross section element, d σ unpol. = d t π ( s − M ) (cid:88) HH (cid:48) (cid:12)(cid:12) T H (cid:48) H (cid:12)(cid:12) , (3.9)the unpolarized angular distribution in the center-of-mass frame is: d σ unpol. d Ω cm = 164 π s (cid:88) HH (cid:48) (cid:12)(cid:12) T H (cid:48) H (cid:12)(cid:12) = | Φ | + | Φ | + | Φ | + | Φ | + | Φ | + | Φ | . (3.10)Incidentally, the conversion to the lab frame goes as: d σ d Ω L = d Ω cm d Ω L d σ d Ω cm = (cid:16) ν (cid:48) ν (cid:17) sM d σ d Ω cm . (3.11)As a simple illustration consider the RCS in tree-level QED, described by the following Feynman amplitude:Figure 3.2: Tree-level CS graphs. T (1) µν = − e (cid:18) γ µ /p + /q + Ms − M γ ν + γ ν /p (cid:48) − /q + Mu − M γ µ (cid:19) . (3.12)Using Eq. (3.3) and definitions from Sect. 1.1, one easily obtains thefollowing expressions for the helicity amplitudes: πs / Φ (1)1 ≡ T (1)+1 + / +1 + / = − πα ( M η − tν ) η / M ν ν (cid:48) , πs / Φ (1)2 ≡ T (1) − − / +1 + / = πα ( − t ) / ν ν (cid:48) , πs / Φ (1)3 ≡ T (1) − / +1 + / = πα tη / ν ν (cid:48) , πs / Φ (1)4 ≡ T (1)+1 − / +1 + / = πα ( − t ) / ην ν (cid:48) , πs / Φ (1)5 ≡ T (1) − / − / = − πα η / ν ν (cid:48) , πs / Φ (1)6 ≡ T (1) − / +1 − / = − πα ( − t ) / sM ν ν (cid:48) , (3.13) This, up to a phase convention which flips the sign of Φ , , , agrees with Tsai et al. [133, Eq. (5)]. α , and the kinematical invariantsare: η = M − suM , ν = s − M M , ν (cid:48) = M − u M . (3.14)Substituting these expressions in Eq. (3.10), one obtains the Klein-Nishina cross section [134]: d σ KN d Ω L = α M (cid:16) ν (cid:48) ν (cid:17) (cid:18) ν (cid:48) ν + νν (cid:48) − sin ϑ (cid:19) , (3.15)where sin ϑ = 1 − (1 + t/ νν (cid:48) ) = − tη/ (2 νν (cid:48) ) .The same steps can be done for polarized observables. For instance, for the linearly polarized photon beamwe have: d σ d Ω cm Σ ≡ (cid:18) d σ || d Ω cm − d σ ⊥ d Ω cm (cid:19) = − Re (cid:2) (Φ + Φ )Φ ∗ + (Φ − Φ )Φ ∗ (cid:3) , (3.16)where Σ is the beam asymmetry. The tree-level QED result is given by: d σ || d Ω L − d σ ⊥ d Ω L = − α M (cid:18) ν (cid:48) ν (cid:19) sin ϑ. (3.17)This result holds (in tree-level QED) for RCS on a particle with spin , / , , and it might hold for higher spinsas well. Other polarized observables are considered in Sect. 3.7. A better use of the rotational and discrete symmetries is made by the partial-wave expansion in the center-of-mass system: T H (cid:48) H ( ω, θ ) = ∞ (cid:88) J = / (2 J + 1) T JH (cid:48) H ( ω ) d JHH (cid:48) ( θ ) , (3.18a) T JH (cid:48) H ( ω ) = 12 ˆ − d(cos θ ) T H (cid:48) H ( ω, θ ) d JHH (cid:48) ( θ ) , (3.18b)where J is total angular momentum and d J ( θ ) are the Wigner d -functions. The partial wave-amplitudes T J and the d -functions satisfy the symmetry relations (3.7) separately.Assuming the parity to be a good quantum number, it is convenient to form the partial-wave amplitudeswith definite J P : A J p = Φ J + p Φ J = 18 πs / (cid:16) T J + / + / − p T J + / − / (cid:17) , (3.19a) B J p = − p Φ J − Φ J = 18 πs / (cid:16) T J + / + / − p T J + / − / (cid:17) , (3.19b) C J p = Φ J + p Φ J = 18 πs / (cid:16) T J + / + / − p T J + / − / (cid:17) , (3.19c)where p = ± is the parity eigenvalue. Note that in the above partial-wave expansion neither H nor H (cid:48) exceed J , hence the amplitudes B and C are only defined for J ≥ / .The conventional multipole amplitudes ( multipoles for short) are denoted as [135, 136] f (cid:96) ∓ ρ (cid:48) ρ ( ω ) , with (cid:96) = J ± / , and ρ (cid:48) , ρ = E, M . (3.20)The combination of ρ and (cid:96) reflects the photon multipolarity (e.g., E the electric dipole). The multipoles areexpressed in terms of the partial-wave amplitudes as follows. For the d -functions we use the conventions of Edmonds (also used by Davydov, or Varshalovich). In the other popular convention(Rose, Wigner or Landau and Lifshitz) the sign of θ is the opposite and hence, due to the property d JHH (cid:48) ( − θ ) = d JH (cid:48) H ( θ ) , the helicitiesappearing on the d -functions would be interchanged. J = / : f − EEMM = 12 A J = / ± , f ρ (cid:48) ρ = 0 . (3.21a)For J ≥ / : f ( J + / ) − EEMM = 2(2 J + 1) (cid:16) A J ± − (cid:113) J − / J + / B J ± + J − / J + / C J ± (cid:17) , (3.21b) f ( J − / )+ EEMM = 2(2 J + 1) (cid:16) A J ∓ + 2 (cid:113) J + / J − / B J ∓ + J + / J − / C J ∓ (cid:17) , (3.21c) f ( J − / )+ EMME = 2(2 J + 1) (cid:18) −A J ∓ − √ ( J + / )( J − / ) B J ∓ + C J ∓ (cid:19) . (3.21d)The inverse relation can be written as (in shorthand notation, f EE ± MM = f EE ± f MM ): Φ J = (cid:8) ( J + / ) f ( J + / ) − EE ± MM ± ( J − / ) f ( J − / )+ EE ± MM ∓ J + / )( J − / ) f ( J − / )+ EM ± ME (cid:9) , (3.22a) Φ J = (cid:112) ( J + / )( J − / ) (cid:8) ( J + / ) f ( J + / ) − EE ∓ MM ± ( J − / ) f ( J − / )+ EE ∓ MM ∓ f ( J − / )+ EM ∓ ME (cid:9) , (3.22b) Φ J = ( J + / )( J − / ) (cid:8) f ( J + / ) − EE ± MM ± f ( J − / )+ EE ± MM ± f ( J − / )+ EM ± ME (cid:9) . (3.22c)As an illustration we once again consider the tree-level QED, and for greater simplicity take the zero-energylimit, ω = 0 . According to the low-energy theorem (LET) [137–139], the tree-level result in this limit is exactand we may omit the label indicating the order of α . The zero-energy helicity amplitudes are thus given by: Φ (0 , θ ) = ∓ αM (cid:0) ± cos θ (cid:1) / , Φ (0 , θ ) = αM (cid:0) ∓ cos θ (cid:1) (cid:0) ± cos θ (cid:1) / , Φ (0 , θ ) = ± Φ (0 , θ ) , (3.23)while the non-vanishing partial-wave amplitudes are: Φ J = / (0) = ∓ α M ˆ − d(cos θ ) (cid:18) ± cos θ (cid:19) / d / + / ± / ( θ ) = − α M , Φ J = / (0) = ∓ α M , (3.24a) Φ J = / (0) = ∓ α M ˆ − d(cos θ ) 1 ∓ cos θ (cid:18) ± cos θ (cid:19) / d / + / ∓ / ( θ ) = ∓ α √ M , (3.24b) Φ J = / (0) = − α M ˆ − d(cos θ ) (cid:18) ± cos θ (cid:19) / d / + / ± / ( θ ) = ∓ α M . (3.24c)The parity-conserving amplitudes assume the following values: A J = / ± = αM (cid:18) − (cid:19) , A J = / ± = αM (cid:18) − (cid:19) , B J = / ± = αM (cid:32) − √ (cid:33) , C J = / ± = αM (cid:18) − (cid:19) , (3.25)and as the result, there are only two non-vanishing multipoles which happen to be equal (at ω = 0 ): f − EE (0) = f EE (0) = − α M . (3.26)
For various microscopic calculations, as well as for the general low-energy expansion (LEX), it is convenientto isolate the Lorentz structure of the amplitude by decomposing it into a set of tensors. There are several neatdecompositions described in the literature, we only consider two of them here. The first — perhaps the earliestone — is the following decomposition into a non-covariant set of (minimal number) tensors: ¯ u (cid:48) ( ε (cid:48) · T · ε ) u = 2 M e ˆ A T ( s, t ) χ (cid:48) ε (cid:48) i ˆ O ij ε j χ, (3.27a)15ith ˆ A and ˆ O being respectively the arrays of the scalar complex amplitudes and tensors: ˆ A ( s, t ) = (cid:8) A , · · · , A (cid:9) ( s, t ) , (3.27b) ˆ O ij = (cid:8) δ ij , n i n (cid:48) j , i(cid:15) ijk σ k , δ ij i(cid:15) klm σ k n (cid:48) l n m , i(cid:15) klm σ k ( δ il n m n (cid:48) j − δ jl n i n (cid:48) m ) , i(cid:15) klm σ k ( δ il n (cid:48) m n (cid:48) j − δ jl n i n m ) (cid:9) , (3.27c)where n and n (cid:48) are the directions of the incoming and outgoing photons.The second decomposition considered here is a covariant, overcomplete set of 8 tensors [124]: ¯ u (cid:48) ( ε (cid:48) · T · ε ) u = e ˆ A T ( s, t ) ¯ u (cid:48) ˆ O µν u E (cid:48) µ E ν , (3.28a)with ˆ A ( s, t ) = (cid:8) A , · · · , A (cid:9) ( s, t ) , (3.28b) ˆ O µν = (cid:8) − g µν , q µ q (cid:48) ν , − γ µν , g µν ( q (cid:48) · γ · q ) , q µ q (cid:48) α γ αν − γ αµ q α q (cid:48) ν , q µ q α γ αν − γ αµ q (cid:48) α q (cid:48) ν ,q µ q (cid:48) ν ( q (cid:48) · γ · q ) , − iγ (cid:15) µναβ q (cid:48) α q β (cid:9) , (3.28c) E µ = ε µ − P · εP · q q µ , E (cid:48) µ = ε (cid:48) µ − P · ε (cid:48) P · q q (cid:48) µ , P µ = ( p + p (cid:48) ) µ , P · q = P · q (cid:48) = M ξ. (3.28d)This decomposition is manifestly gauge-invariant, because the vectors E are. It can be reduced [54] to anyof the covariant sets with the minimum number of tensors, such as that of Hearn and Leader [84] or L’vov[96]. Nevertheless, the overcomplete set is better suited for practical calculations of Feynman diagrams usingcomputer algebra, since simple Gordon-like identities are sufficient for the decomposition. Another advantageis that it readily applies to the forward VVCS case, see Sect. 5.1.The correct relation between the amplitudes A ,..., and A ,..., was given by McGovern et al. [51]: A = (cid:15) B M A + ω B t M A ,A = (cid:15) B ω B M A + ω B M (cid:0) A + A − t A (cid:1) ,A = (cid:15) B M A − M η t M − t (cid:18) A + A M ( (cid:15) B + M ) − A (cid:19) − ω B t M A ,A = ω B A ,A = ω B A + ω B M ( (cid:15) B + M ) (cid:104) A + M η M − t ( A + A ) (cid:105) − ω B ( ω B + t ) A + ω B M A ,A = ω B A − ω B M ( (cid:15) B + M ) (cid:104) A + M η M − t ( A + A ) (cid:105) + ω B A − ω B M A , (3.29)where (cid:15) B and ω B are the nucleon and photon energies in the Breit frame (defined by p (cid:48) = − p ). Thesekinematical variables, along with η , can be expressed in terms of Mandelstam invariants, cf. Appendix C. Thus,although obtained in the Breit frame, this relation is Lorentz invariant.Both sets of amplitudes have a definite parity under the photon crossing (i.e., ε ↔ ε (cid:48) , q ↔ q (cid:48) , hence s ↔ u ,etc.). Writing the amplitudes as functions of ξ and t , the crossing symmetry implies: A , ( − ξ, t ) = A , ( ξ, t ) , A ,..., ( − ξ, t ) = − A ,..., ( ξ, t ) , (3.30a) A , , ( − ξ, t ) = A , , ( ξ, t ) , A ,..., ( − ξ, t ) = − A ,..., ( ξ, t ) . (3.30b) Derived from unitarity, the optical theorem establishes the relation between the imaginary part of theforward CS amplitude and the total photoabsorption cross section, and in our case of the nucleon target reads: Im T H (cid:48) H ( ν,
0) = ν σ H ( ν ) δ HH (cid:48) , (3.31) Here we correct the typos of Refs. [51, 124] made in the expressions for O and O , respectively. δ HH (cid:48) is the Kronecker symbol. The cross section σ H corresponds with the absorption of a circularlypolarized photon on a longitudinally polarized target, with their combined helicity given by H . In terms of theinvariant amplitudes we have: Im A ( ν,
0) = Im A ( ν,
0) = ν πα σ T ( ν ) , (3.32a) Im A ( ν,
0) = Im A ( ν,
0) = ν πα σ T T ( ν ) , (3.32b)where σ T = ( σ / + σ / ) / is the unpolarized cross section and σ T T = ( σ / − σ / ) / . The remainingamplitudes do not contribute in the forward scattering, and as such are not constrained by the optical theorem. At low energies there are further unitarity constraints for the nucleon CS. They are less strict, since theyhold in a limited energy range and to leading order in α . At the same time, they are more stringent, since theyapply to all the multipole and partial-wave amplitudes, and hence are not limited to the forward kinematics.Below the two-pion threshold, one is limited to the channel space spanned by the πN and γN states, andhence the following four processes: πN → πN, πN → γN,γN → πN, γN → γN. (3.33)To have exact unitarity, in this channel space, we set up a linear coupled-channel integral equation: (cid:32) T ππ T πγ T γπ T γγ (cid:33) = (cid:32) V ππ V πγ V γπ V γγ (cid:33) + (cid:32) V ππ V πγ V γπ V γγ (cid:33) (cid:32) G π G γ (cid:33) (cid:32) T ππ T πγ T γπ T γγ (cid:33) , (3.34)where T and V are suitably normalized amplitudes and potentials of pion-nucleon scattering ( ππ ), pion pho-toproduction ( πγ ), absorption ( γπ ), and nucleon CS ( γγ ). The propagators G π and G γ are, respectively, thepion-nucleon and photon-nucleon two-particle propagators. With the assumption of hermiticity of the potentialand time-reversal symmetry, which relates the γπ and πγ amplitudes, the above equation leads to the unitary S -matrix, S fi = δ fi + 2 iT fi .Neglecting the iterations of the potential involving photons (which amount to small radiative corrections),the coupled-channel equation reduces to: T ππ = V ππ + V ππ G π T ππ , (3.35a) T πγ = V πγ + T ππ G π V πγ , (3.35b) T γπ = V γπ + V γπ G π T ππ , (3.35c) T γγ = V γγ + V γπ G π T πγ . (3.35d)Only the first of these is an integral equation, the rest are obtained by a one-loop calculation.After the partial-wave expansion, the solution for the πN amplitude can be written as: T IJ p ππ = K IJ p − i k K IJ p = 1 k e iδ I(cid:96) p sin δ I(cid:96) p , (3.36)where K IJ p is the ‘ K -matrix’ with definite isospin I , total angular momentum J , and parity p ; the correspond-ing πN phase-shift is δ I(cid:96) p = arctan( k K IJ p ) , which is a function of the πN relative momentum k . The latter isgiven by k = 12 s / (cid:2)(cid:0) s − ( M + m π ) (cid:1) (cid:0) ( s − ( M − m π ) (cid:1)(cid:3) / . (3.37)Note that we have neither specified V ππ , nor solved Eq. (3.35a); we have merely written it in the manifestlyunitary form.Continuing to the pion photoproduction channel, we obtain the statement of the celebrated Watson’s theo-rem [140]: T IJ p γπ = (cid:12)(cid:12) T IJ p γπ (cid:12)(cid:12) e iδ I(cid:96) p , (3.38)17ith (cid:96) = J − p / . The phase of the photoproduction amplitudes is thus identical to the πN phase-shift, for eachset of good quantum numbers.Extending these arguments to the Compton channel, we obtain T J p γγ = V J p γγ + / (cid:88) I =1 / (cid:12)(cid:12) T IJ p γπ (cid:12)(cid:12) (cid:0) − tan δ I(cid:96) p + i (cid:1) . (3.39)There are two interesting results here. The first is that the imaginary part of the partial-wave RCS amplitudeis given by the isospin sum of the photoproduction amplitudes squared. In this case the sum over the isospin isequivalent to the sum over the charged states, hence, e.g., for the proton Im T J p γp → γp = / (cid:88) I =1 / (cid:12)(cid:12) T IJ p γπ (cid:12)(cid:12) = (cid:12)(cid:12) T J p γp → π p (cid:12)(cid:12) + (cid:12)(cid:12) T J p γp → π + n (cid:12)(cid:12) . (3.40)The second result concerns the ∆(1232) resonance, which is the only resonance occurring between the one-and two-pion production thresholds. Recall that, in the πN scattering, this resonance occurs in the P partialwave (i.e., I = 3 / J , (cid:96) = 1 , p = + ). The position, M ∆ , of such an elastic resonance is identified with thephase-shift crossing ◦ . This means the tangent terms in Eq. (3.39) blow up and can only be canceled by asingularity in V γγ . Near the resonance position the K -matrix takes the form K P33 ≈ M ∆ Γ ∆ k ( s − M ) , (3.41)where Γ ∆ is the resonance width, and hence the cancellation is achieved when lim s → M (cid:2) ( s − M ∆ ) V / γγ (cid:3) = M ∆ Γ ∆ lim s → M (cid:12)(cid:12) T P33 γπ (cid:12)(cid:12) . (3.42)Thus, unitarity provides a stringent relation among the ∆(1232) -resonance parameters occurring in the differ-ent processes. As a consequence, the ∆(1232) -resonance contribution to polarizabilities is constrained too.These results apply as well to the multipole amplitudes. In particular, the imaginary parts of the Comptonmultipoles, between the one- and two-pion production thresholds, are given by the pion photoproductionmultipoles: Im f (cid:96) ± EE = k (cid:88) c (cid:12)(cid:12) E ( c )( (cid:96) ± ∓ (cid:12)(cid:12) , Im f (cid:96) ± MM = k (cid:88) c (cid:12)(cid:12) M ( c ) (cid:96) ± (cid:12)(cid:12) , (3.43a) Im f ( (cid:96) ± ∓ EM = Im f (cid:96) ± ME = ∓ k (cid:88) c Re (cid:0) E ( c ) (cid:96) ± M ( c ) ∗ (cid:96) ± (cid:1) , (3.43b)where the sum is over the charged πN states, i.e: c = π p, π + n and c = π n, π − p for the proton and neutronRCS, respectively. As mentioned above, an equivalent result is obtained by summing over the isospin states. The celebrated LET for RCS [137–139] can be extended to include higher-order terms, parametrized interms of polarizabilities [1]. It is customary to separate out the Born contribution by writing T = T Born + T , (3.44)such that T Born is the Born contribution specified in Appendix A. In the low-energy limit, it yields the classicLET. The rest (non-Born), T , is expanded in powers of energy with coefficients given by static polarizabilities.For example, the LEX of the non-Born part of the 6 invariant amplitudes of the decomposition (3.27) goes as18 (cid:64) MeV (cid:68) Α E1 (cid:45) Β M1 (cid:45) Α E2 (cid:45) (cid:45) (cid:45) (cid:45) Β M2 Figure 3.3: The scalar dipole and quadrupole dynamic polarizabilities of the proton , in units of − fm and − fm , respectively. The curves are the results of the B χ PT calculation of Lensky et al. [54] (red bands),compared with the results of the DR calculation of Hildebrandt et al. [119] (black dot-dashed) and with theresults of Aleksejevs and Barkanova [142, 143] (green dotted, not shown for the quadrupole polarizabilities).(in the Breit frame): α ¯ A ( ω B , t ) = ω B (cid:2) α E + β M + ω B ( α E ν + β M ν ) (cid:3) + t (cid:0) β M + ω B β M ν (cid:1) + ω B ( α E + β M ) + t (4 ω B + t ) β M + O ( ω B ) ,α ¯ A ( ω B , t ) = − ω B (cid:0) β M + ω B β M ν (cid:1) + ω B ( α E − β M ) − tω B β M + O ( ω B ) ,α ¯ A ( ω B , t ) = − ω B (cid:2) γ E E + γ E M + z ( γ M E + γ M M ) (cid:3) + O ( ω B ) ,α ¯ A ( ω B , t ) = ω B ( γ M E − γ M M ) + O ( ω B ) ,α ¯ A ( ω B , t ) = ω B γ M M + O ( ω B ) ,α ¯ A ( ω B , t ) = ω B γ E M + O ( ω B ) . (3.45)Certainly, the convergence radius of such a Taylor expansion is limited by the first singularity, which in thenucleon case is set by the pion-production branch cut (neglecting the small effects from radiative corrections).An expansion which extends beyond the pion-production threshold is the multipole expansion. The relationbetween the two expansions (i.e., polarizability vs. multipole) is as follows.One can divide out the Born contribution in the multipole amplitudes, f = f Born + ¯ f . The non-Born part ofthe multipoles is then used to define the dynamic polarizabilities as [141]: (cid:32) α E(cid:96) ( ω ) β M(cid:96) ( ω ) (cid:33) = [ (cid:96) (2 (cid:96) − ω (cid:96) (cid:20) ( (cid:96) + 1) ¯ f (cid:96) + EEMM ( ω ) + (cid:96) ¯ f (cid:96) − EEMM ( ω ) (cid:21) , (3.46a) γ E(cid:96)E(cid:96)M(cid:96)M(cid:96) ( ω ) = 2 (cid:96) − ω (cid:96) +1 (cid:20) ¯ f (cid:96) + EEMM ( ω ) − ¯ f (cid:96) − EEMM ( ω ) (cid:21) , (3.46b) γ E(cid:96)M ( (cid:96) +1) M(cid:96)E ( (cid:96) +1) ( ω ) = 2 − (cid:96) (2 (cid:96) + 1)!! ω (cid:96) +1 ¯ f (cid:96) + EMME ( ω ) . (3.46c)Given that the low-energy behavior of the non-Born part of multipoles goes as ¯ f (cid:96) ± EEMM ∼ ω (cid:96) , ¯ f (cid:96) + EMME ∼ ω (cid:96) +1 , (3.47)the low-energy limit of (3.46a) and (3.46c) is straightforward and corresponds with the static polarizabilities.The limit of (3.46b) needs more care, but the matching to the static polarizabilities is possible as well [54]. Asan illustration, Fig. 3.3 shows the plots of the scalar dynamic polarizabilities of the proton from Ref. [54].19 .7. Polarized Observables Besides the unpolarized differential cross section, given by Eq. (3.9), and the linearly-polarized photonbeam asymmetry Σ , Eq. (3.16), there is a number of observables that depend on polarization of the (nucleon)target. Here we only consider the case when the polarizations of the final particles (scattered photon andrecoiled nucleon) are not observed.We consider the photon, traveling along the z -axis, with a linear polarization and P γT at the angle φ withrespect to the scattering plane xz , and the right-handed circular polarization P γR . The degree of the targetpolarization along the x -, y -, z -direction is denoted as P x , P y , P z respectively. In this case the polarized cross-section element is given by d σ = d σ unpol . (cid:2) P γT Σ cos 2 φ + P x (cid:0) P γR Σ x + P γT Σ x sin 2 φ (cid:1) + P y (cid:0) Σ y + P γT Σ y cos 2 φ (cid:1) + P z (cid:0) P γR Σ z + P γT Σ z sin 2 φ (cid:1)(cid:3) , (3.48)where d σ unpol . stands for the unpolarized cross section and Σ ’s denote the various asymmetries. This notationfor asymmetries is motivated by Babusci et al. [96]. The conversion to the standard notation adopted in mesonphotoproduction (see, e.g., Appendix A of Worden [144]) is as follows: Σ = − Σ , Σ x = F, Σ x = − H, Σ y = T, Σ y = − P, Σ z = − E, Σ z = G. (3.49)Similar to Eq. (3.16) for Σ , we may express these asymmetries in terms of the specific polarized crosssections and in terms of the helicity amplitudes: d σ d Ω cm Σ x ≡ (cid:18) d σ Rx d Ω cm − d σ Lx d Ω cm (cid:19) = Re[Φ (Φ − Φ ) ∗ − (Φ + Φ )Φ ∗ ] , (3.50a) d σ d Ω cm Σ z ≡ (cid:18) d σ Rz d Ω cm − d σ Lz d Ω cm (cid:19) = − (cid:0) | Φ | + | Φ | − | Φ | − | Φ | (cid:1) , (3.50b) d σ d Ω cm Σ x ≡ (cid:18) d σ π / x d Ω cm − d σ − π / x d Ω cm (cid:19) = Im[Φ ∗ Φ + Φ ∗ Φ ] , (3.50c) d σ d Ω cm Σ z ≡ (cid:18) d σ π / z d Ω cm − d σ − π / z d Ω cm (cid:19) = − Im[Φ (Φ − Φ ) ∗ − (Φ + Φ )Φ ∗ ] , (3.50d) d σ d Ω cm Σ y ≡ (cid:18) d σ y d Ω cm − d σ − y d Ω cm (cid:19) = − Im[Φ (Φ + Φ ) ∗ + (Φ − Φ )Φ ∗ ] , (3.50e) d σ d Ω cm Σ y ≡ (cid:18) d σ y d Ω cm − d σ π / y d Ω cm (cid:19) = d σ d Ω cm Σ + Im[Φ ∗ Φ + 2Φ ∗ Φ − Φ ∗ Φ ] . (3.50f)The superscript on σ indicates here the photon polarization: right ( R ), left ( L ) for the circular polarization, orthe value of φ for the linear polarization. The subscript indicates the nucleon spin polarization. The expressionsin terms of the helicity amplitudes assume parity conservation. Terms proportional to the imaginary part arenegligible below the pion-production threshold (because the imaginary part of the amplitudes is suppressed byan α ).Some of these polarized observables have been measured for the proton RCS, which brings us to the fol-lowing section. Here it is important that, in the center-of-mass frame, the nucleon travels in the − z direction. Hence, its polarization along z -axiscorresponds with helicity − / , and so on. . Compton Scattering Experiments The CS processes remain to be the only method of accessing the nucleon polarizabilities experimentally.The nucleon probed by long-wave photons reveals its structure in the manner of a multipole expansion. Atfirst, one distinguishes the electric charge and magnetic dipole moment, contributions which comprise the low-energy theorem for CS [138, 139]. Further terms (in multipole or energy expansion) can be described in termsof polarizabilities, of which α E and β M play the leading role. For instance, the low-energy expansion (LEX)of the unpolarized CS cross section, truncated at O ( ν ) , is given by: d σ d Ω L = d σ Born d Ω L − νν (cid:48) (cid:18) ν (cid:48) ν (cid:19) παM (cid:104) ( α E + β M ) (1 + z ) + ( α E − β M ) (1 − z ) (cid:105) , (4.1)where z = cos ϑ , and ν ( ν (cid:48) ) is the energy of the incoming (scattered) photon.The simplicity of the formalism is appealing. However, the region of its applicability is unclear a priori. Itis only clear that the convergence radius of such a LEX is limited by the nearest singularity, which in this caseis at pion production threshold (neglecting the small e.m. corrections). This is illustrated by the dynamicalpolarizabilities in Fig. 3.3. Their LEX begins with the static value, which is the value at zero energy ( ω = 0 ).Clearly, approximating the dynamic polarizabilities by a constant (the static value) only works well at energiesfar below the pion-production threshold ( ω (cid:28) m π ).Figure 4.1: CS cross sections calculated in the Born approx-imation (solid), the leading-order LEX (dotted) and a dis-persion model calculation (dashed). Plot reproduced fromMacGibbon et al. [12].Further insight can be obtained by compar-ing the LEX of the cross section with experi-mental data and the results of calculations thatextend beyond the pion threshold. Figure 4.1shows a calculation of the Born and LEX crosssections for CS on the proton at fixed angleand function of beam energy [12]. There isincreased sensitivity to the polarizabilities athigher incident energies. However, at higherenergies there is also increased sensitivity interms of order O ( ν ) and higher. The impor-tance of the higher order terms is indicated bythe dashed curve, which is a dispersion modelcalculation valid to all orders in ν , albeit withunclear model-dependencies. One sees thatabove
100 MeV , the lowest-order LEX is not ad-equate. The higher-order terms may extend itsapplicability, but will depend on the spin polar-izabilities and other, higher-order polarizabili-ties.Nonetheless, the LEX is instructive for understanding how the polarizabilities affect the observables. Onecan see, for example, that at forward angles the polarizability effect enters as α E + β M , while at the backwardangles as α E − β M . Hence, one can in principle use the angular distribution to disentangle α E and β M .The leading LEX expressions for CS with linearly polarized photons are obtained by Maximon [145] whofinds [rearranging his Eqs. (19) and (20)]: (cid:18) d σ ⊥ d Ω L − d σ Born ⊥ d Ω L (cid:19) − (cid:32) d σ || d Ω L − d σ Born || d Ω L (cid:33) = − παM (cid:18) ν (cid:48) ν (cid:19) νν (cid:48) α E sin ϑ (4.2a) cos ϑ (cid:18) d σ ⊥ d Ω L − d σ Born ⊥ d Ω L (cid:19) − (cid:32) d σ || d Ω L − d σ Born || d Ω L (cid:33) = 8 παM (cid:18) ν (cid:48) ν (cid:19) νν (cid:48) β M cos ϑ sin ϑ (4.2b)where d σ ⊥ / d Ω L and d σ || / d Ω L are the differential cross sections for CS perpendicular and parallel to the planeof incident photon polarization. By taking weighted differences of σ ⊥ and σ || , it is possible to measure β M Target Energy range Theoretical model ReferenceProton 2 π threshold Fixed- t dispersion calculation Drechsel et al. [86]Proton ∆(1232) region Chiral Lagrangian Gasparyan et al. [91]Proton ∆(1232) region HB χ PT with ∆(1232)
McGovern et al. [51]Proton ∆(1232) region B χ PT with ∆(1232)
Lensky and Pascalutsa [49]Deuteron ≈
130 MeV χ EFT Grießhammer et al. [65] He ≈
130 MeV χ EFT Shukla et al. [151] separately from α E . Hence, in separating the electric and magnetic polarizabilities the beam polarization canserve as an alternative to the angular distribution.The issue here is that the cross section differences in parentheses on the left sides of the above equations arerelatively small, at the level compared to the cross sections. To measure polarizabilities with a precisionof ≈ requires measurement of the absolute cross sections σ ⊥ and σ || with uncertainties of ≈ . This is asevere challenge for CS experiments, both in statistics and systematic errors.It is much easier in this respect to measure the linear polarization asymmetry, Σ = d σ || − d σ ⊥ d σ || + d σ ⊥ . (4.3)The asymmetry measurements do not critically depend on knowing the incident photon flux, target thickness,and photon and recoil proton detection efficiencies. To O ( ν ) this asymmetry depends only on β M and isindependent of α E [36]: Σ = Σ Born3 − M ω B cos θ B sin θ B (1 + cos θ B ) α − β M (4.4)where ω B and θ B are the photon energy and scattering angle in the Breit (brick-wall) frame, see Eqs. (C.1) and(C.3) for their relations to invariants. Calculations at NNLO in B χ PT indicate that the range of applicabilityof Eq. (4.4) is as high as
100 MeV . However, the sensitivity to β M in Eq. (4.4) is weak, at the order of ∆ Σ / ∆ β M ≈ . . To measure β M at the level of ± . (in the usual units) would require uncertainty in Σ toreach ± . . To attain this statistical precision, photon intensities at least an order of magnitude greater thanthose available with standard photon tagging techniques would be required.Nevertheless, the LEX formula (4.4) paves the way for a more sophisticated analysis, awaiting the first datafor Σ below the pion production threshold. The LEX formula for the doubly-polarized asymmetries, Σ x and Σ z can be found in Krupina [146].To summarize, a model-independent extraction of polarizabilities from CS observables based on LEX isnearly impossible at the existing facilities. On one hand, the sensitivity must be substantial enough for a goodsignal at the given level of experimental accuracy, which drives the experiment to higher energy. On the otherhand, at higher energies the LEX applicability is compromised. Therefore, although the LEX gives a valuableinsight on the sensitivity of observables, it has been impractical in quantitative extractions.A more practical and common approach is to extract polarizabilities by fitting the CS data using a systematictheoretical framework, such as χ PT [48–52, 65, 147, 148] or fixed- t dispersion relations (DRs) [85–87, 149,150]. Table 4.1 presents a listing of several calculations that have been or could be used for fitting cross sectionand asymmetry data from pion threshold up to the ∆(1232) region. By fitting data with several theoreticalmodels, it is possible to obtain an estimate for the model dependence of the result. This was, for example, theapproach taken by Martel et al. [38] for their analysis of double-polarized CS in the ∆(1232) region. A limitation in fitting large numbers of data points with a fitting program such as MINUIT is that many recursive calls are made tothe subroutine calculating the observable. If the calculation of the observable is based upon numerical integrations, such as the fixed- t .2. Determination of Scalar Polarizabilities4.2.1. Proton The first CS measurement of the proton polarizabilities using a tagged photon beam was by Federspiel etal. [10] at the University of Illinois MUSL-2 microtron. Experiments prior to this used bremsstrahlung beams,and were limited by systematic errors due to uncertainties in the incident photon energy. The most recentpublished results for CS on the proton are by the LEGS group [13], and the TAPS at MAMI setup [17], both in2001. Their results for α E and β M are in agreement.After publication of the LEGS and TAPS results, activity in this area slowed. Then in 2010-2012 new χ PTcalculations of CS [49, 65] showed that β M is larger than the PDG average of that time, β ( p ) M = (1 . ± . × − fm [152], by +1 to +3 of the standard deviations. In 2014 the CS global analysis of McGovern et al. [51]was included in the PDG average, and the current (2014) PDG values are [71]: α ( p ) E = (11 . ± . × − fm , β ( p ) M = (2 . ∓ . × − fm . (4.7)Nevertheless, the uncertainty quoted in the PDG average should still be taken with a grain of salt because thedata sets are not treated consistently: e.g., results from the analysis of specific experiments are averaged withresults from global analyses. The summary plots in Sect. 7 show the real state of affairs for α E and β M of theproton, cf. Figs. 7.1 and 7.3 (left panel). It not as certain as the PDG average portrays it. There is certainly aroom for improvement.The thrust of new proton polarizability measurements is to use linearly polarization as an analyzer tomeasure α E and β M separately, and independently of the Baldin sum rule value for α E + β M . Programsto measure the proton polarizabilities with linearly polarized photons are currently underway at the MainzMicrotron (MAMI) and at the High Intensity Gamma-Ray Source (HIGS).Preliminary data [35] with linearly polarized incident photons have been taken by the A2 collaboration atthe tagged photon facility [153] at MAMI [154]. In this experiment a diamond radiator is used to producelinearly polarized coherent bremsstrahlung [155] with a peak polarization of approximately . The targetis a
10 cm long liquid hydrogen target, and Compton scattered photons are detected in the Crystal Ball [156]and TAPS detectors [157], both of which are outfitted with charged particle identification systems [158]. TheCrystal Ball, TAPS, and the charged particle system internal to the Crystal Ball (the PID scintillator array andMWPC), are shown in Fig. 4.2. The solid angle coverage for Compton-scattered photons is approximately of π . The incident photon energies for the measurement range from to
140 MeV . Recoil proton detection
DR code of Drechsel et al. [86], then execution times can stretch into days. Martel et al. [38] handled this problem assuming a lineardependence of the observable on the polarizabilities: O i ( { P } ) = O i ( { P } ) + (cid:88) j =1 ∂ O i ( { P } ) ∂P j ( P j − P j ) , (4.5)where O i is the observable, { P } is the set of six polarizabilities (two scalar and four spin polarizabilities), and { P } is the set of starting"guesses" for the polarizabilities. The standard expression for χ is given by: χ = N data (cid:88) i =1 (cid:18) O data i − O i ( { P } ) σ i (cid:19) + (cid:88) j =1 (cid:32) P data j − P j σ P j (cid:33) , (4.6)where the second term in χ allows for the possibility of introducing constraints on the polarizabilities (e.g., the sum-rule constraintson α E + β M and γ ). Substituting Eqs. (4.5) into (4.6), and setting ∂χ /∂P j = 0 leads to a linear equation, C i = D ij P j , with C i = P data i σ P i + N data (cid:88) j =1 σ j (cid:104) O data j − O j ( { P } ) + (cid:88) k =1 ∂ O j ( { P } ) ∂P k P k (cid:105) ∂ O j ( { P } ) ∂P i , i = 1 , . . . , ,D ij = δ ij σ P i + N data (cid:88) k =1 σ k ∂ O k ( { P } ) ∂P i ∂ O k ( { P } ) ∂P j , i, j = 1 , . . . , . One can solve it for { P } , as P = D − C . After the first iteration the substitution { P } → { P } is made, O i ( { P } ) , ∂ O i ( { P } ) /∂P j and { P } are reevaluated, and the fit repeated until the set { P } differs from { P } by less than one standard deviation. It was found thatthis methodology is very efficient in fitting large data sets with computationally intensive codes. d σ ⊥ and d σ || , the asymmetry Σ , and the unpolarized cross sectionby running on an amorphous photon radiator.At HIGS [39] there are plans [40] to take CS data on the proton with linearly polarized photons in 2016.The incident photon beam at HIGS is exceptionally well suited for a LEX based CS analysis for α E and β M ;energies up to
100 MeV are available, the beam energy is monochromatic with a spread of ≈ . , photonintensities on the target are ≈ γ ’s / s , and the linear polarization is approximately [159]. Figure 4.4shows the HINDA NaI array that has been developed for CS experiments. Each module has a . indiameter and . long NaI core, surrounded by active NaI shields that are . thick and . long. As a “free” neutron target does not exist, there are no truly model-independent means to determine theneutron’s polarizabilities. Obtaining neutron polarizabilities from the analysis of deuteron CS data requiresaccurate effective-field theory calculations. The favored approach is to use elastic scattering data, not quasi-freed ( γ, γ (cid:48) n ) p data, because (i) the elastic process is theoretically less complicated than the quasi-free scattering,and treatable through effective-field theory calculations, and (ii) elastic scattering has a larger cross sectionand greater sensitivity to the polarizabilities than the quasi-free process. The latter point can be understood bynoting that for CS on a free neutron there is no Thomson term in the scattering amplitude, because the neutronis neutral, and the polarizability effect goes as O ( ν ) . For elastic scattering there is a Thomson term, becausethe deuteron is charged, and the polarizability effect goes as O ( ν ) . The disadvantage of elastic scattering isthat it places a premium on the utilization of large NaI detectors with sufficient energy resolution to resolveelastic and inelastic scattering. Furthermore, the elastic scattering is sensitive to the isoscalar polarizabilities,and the proton contribution must be subtracted.Until recently there has been a paucity of CS data on the deuteron. For example, the analysis of Grießham-mer et al. [65] was based on three data sets with a total of 37 data points. The focus of new studies has beento stage experiments that can obtain relatively high statistics with wide kinematic coverage, utilize large NaIdetectors for optimal energy resolution, and to use targets with A > . In this section recent progress in thisarea is outlined.New data from Lund has recently been published for elastic CS on the deuteron [15], nearly doubling theeffective number of world data points and extending the energy range by
20 MeV to higher energies. Thelarge-volume, segmented NaI(Tl) detectors, BUNI, CATS, and DIANA were used to detect Compton-scatteredphotons in the experiment. These detectors are composed of a NaI(Tl) core surrounded by optically isolated,annular NaI(Tl) segments. The cores of the BUNI and CATS detectors each measures . in diameter, whilethe core of the DIANA detector measures . . The depth of all three detectors is greater than radiationlengths. The annular segments are
11 cm thick on the BUNI and CATS detectors and thick on the DIANAdetector. Figures 7.2 and 7.3 (right panel) show the present state of affairs for α E and β M of the neutron.24
00 950 10000100020003000
Missing Mass [MeV] C oun t s Figure 4.3: Missing mass distribution for the Mainzproton scalar polarizability measurement for incidentphoton energies from to
140 MeV , and angularrange ◦ < ϑ < ◦ [35]. Data with parallel photonpolarization is blue, and perpendicular polarization isred. The black curve represents the simulated distri-bution. Plot courtesy of Vahe Sokhoyan. Figure 4.4: Diagram of the HINDA NaI array at HIGS.There are compelling reasons to consider CS on Z > nuclei as an attractive route to the neutron polariz-abilities; the Thomson cross section goes at Z , and there is a better ratio of elastic to incoherent scattering for A > . As a test of the effective interaction theories used to analyze the data, it is also important to demon-strate that polarizabilities obtained from deuterium are in agreement with results from other nuclei. Figure 4.5shows the results of a NLO χ EFT calculation without ∆(1232) degrees of freedom at
60 MeV (left panel) and
120 MeV (right panel) for CS on He. The curves show appreciable sensitivity to the neutron polarizabilities,especially at the higher energy.There are plans [160] to measure elastic CS on He and He with an active, gaseous helium target usingthe Crystal Ball and TAPS detectors at the Mainz microtron MAMI. The operating principle of the target is thationizing particles produce UV scintillation light in the helium, and by the addition of small amounts of N as awavelength shifter, and a photo-detector coupled to the target cell, the target can operate as a detector. Elasticscattering is separated from incoherent processes by detecting the recoil helium nucleus in coincidence withthe Compton-scattered photon. Development of this target is in progress, and test runs are anticipated in 2016. Compared to the situation for the proton scalar polarizabilities, relatively little is known experimentallyabout the spin polarizabilities. Prior to the advent of single-polarized and double-polarized CS asymmetrymeasurements, only two linear combinations of the polarizabilities were known. One combination is the for-ward spin polarizability: γ = − γ E E − γ E M − γ M M − γ M E , (4.8)fixed by the GTT sum rule (5.24). The results of the GTT sum rule evaluation are summarized in Table 5.2.The other combination is the backward spin polarizability γ π : γ π = − γ E E − γ E M + γ M M + γ M E . (4.9)25igure 4.5: Sensitivity of the differential cross sections for CS on He in the c.m. frame at NLO in χ EFT withoutexplicit ∆(1232) [151] at
60 MeV (left panel) and
120 MeV (right panel). Solid (black) curve: central value α nE = 12 . ; long-dashed (blue): α nE − ; dot-dashed (red): α nE − ; dotted (magenta): α nE + 2 ; dashed(green): α nE + 4 ; in units of − fm . Plots reproduced from Shukla et al. [151].The forward (backward) spin polarizability, according to its name, appears in the spin-dependent CS am-plitude at forward (backward) kinematics. More specifically, in both kinematics the CS amplitude splits into aspin-independent and spin-dependent part, i.e.: πM T ( ν, ϑ = 0) = f ( ν ) ε (cid:48)∗ · ε + g ( ν ) i σ · ε (cid:48)∗ × ε , (4.10a) πM T ( ν, ϑ = π ) = 1 (cid:112) ν/M (cid:104) ˜ f ( ν ) ε (cid:48)∗ · ε + ˜ g ( ν ) i σ · ε (cid:48)∗ × ε (cid:105) , (4.10b)and the low-energy expansion for the scalar amplitudes goes as follows [96]: f ( ν ) = − Z αM + ( α E + β M ) ν + O ( ν ) , (4.11a) g ( ν ) = − α κ M ν + γ ν + O ( ν ) , (4.11b) ˜ f ( ν ) = (cid:16) νM (cid:17)(cid:104) − Z αM + ( α E − β M ) ν ν/M + . . . (cid:105) , (4.11c) ˜ g ( ν ) = (cid:2) − κ + ( Z + κ ) (cid:3) ανM + γ π ν ν/M + . . . (4.11d)Hence, γ and γ π appear at O ( ν ) in the LEX of the spin-flip amplitude at, respectively, the forward andbackward scattering angle.Figure 4.6 shows the sensitivity of backward angle CS cross sections in the ∆(1232) region to γ π . The mostwidely accepted value for γ π is actually an average of three measurements at MAMI performed with differentdetector configurations: TAPS [17], LARA [18, 19], and SENECA [20]. All three of the measurements agreewithin their statistical and systematic errors, and the average value is [20]: γ π = (8 . ± . × − fm , (4.12)where the error includes statistical and estimated model uncertainties. Here we use the standard conventionof excluding the t-channel π -pole contribution . The latter is evaluated as (e.g., Ref. [161]): γ π pole π = − αg A (2 πf π ) m π = ( − . ± . × − fm . (4.13) In the literature the result with the π -pole excluded is sometimes referred to as ¯ γ π (see, e.g., Ref. [13]). π -pole contribution is also excluded from each of the spin polarizabilities: γ E E , γ M M , γ E M and γ M E [96]. However, this contribution cannot affect γ and other forward polarizabilities, because the π -polediagram vanishes in the forward direction.The LEGS result [13], γ π = (cid:0) . ± . stat+syst + (cid:2) +2 . − . (cid:3) model (cid:1) × − fm , (4.14)is in disagreement with the Mainz result, despite the good agreement between LEGS and Olmos de León et al.[17] for α E and β M . As seen in Fig. 4.6, the main cause for this disagreement is the discrepancy betweenmeasured cross sections at backward angles at energies above pion threshold. Figure 7.5 summarizes theresults for the backward spin polarizability of the proton.Figure 4.6: Differential cross section of the proton RCS forthe center-of-mass angle of ◦ in the ∆(1232) region [20].The curves correspond to different values of γ π , which hereinclude the t-channel π -pole contribution. Plot reproducedfrom Camen et al. [20]. Figure 4.7: CS configuration for the LEGS detec-tors.Single and double-polarized CS asymmetries have sensitivity to the spin polarizabilities [87, 146]. Measure-ments of this type provide essentially the only means by which the four lowest order spin polarizabilities γ E E , γ M M , γ E M and γ M E can be individually separated. The linear polarization asymmetry Σ in the ∆(1232) region was measured at LEGS [13], and a new program of single and double polarized CS measurements iscurrently underway at MAMI [37]. The linear polarization asymmetry, Σ , is defined in Eq. (4.3), see also Eq. (3.16). The first measurementsof this asymmetry were by the LEGS collaboration [13] in the ∆(1232) region, see Fig. 4.8. Their experimentalsetup, shown in Fig. 4.7, is fairly typical of CS experiments in the ∆(1232) region, where the Compton photonis detected in a NaI detector, and the recoil proton is detected in a spectrometer arm specifically designed forrecoil detection. Detecting the recoil proton is necessary to suppress background from π → γγ ; the ratio of π photoproduction to CS in the ∆(1232) region is approximately : . In the LEGS measurement precisionwire chambers are used to define the trajectory of the proton, and time-of-flight over and energy loss inscintillator paddles are used to establish particle type and momentum.In the LEGS analysis two spin polarizability combinations are fit to their data: γ ≡ − γ E E + γ E M = (cid:0) . ± . stat+syst + (cid:2) +0 . − . (cid:3) model (cid:1) × − fm , (4.15a) γ ≡ − γ E E − γ E M − γ M M = − (cid:0) . ± . stat+syst + (cid:2) +0 . − . (cid:3) model (cid:1) × − fm . (4.15b)New data for the Σ asymmetry in the ∆(1232) region have recently been taken by the Mainz A2 collabora-tion at the microtron MAMI using the Crystal Ball and TAPS detectors [162]. In this experiment π events are27 Ω lab (cid:61)
324 MeV0 30 60 90 120 150 Θ (cid:64) deg (cid:68) Ω lab (cid:61)
298 MeV0 30 60 90 120 1500.00.20.40.6 Ω lab (cid:61)
276 MeV Ω lab (cid:61)
265 MeV Ω lab (cid:61)
245 MeV (cid:45) Γ (cid:61)
224 MeV
Figure 4.8: The CS beam asymmetry Σ as function of the c.m. angle at different values of the beam energy E γ . The experimental data are from LEGS [13] (open diamonds) and MAMI [162] (cyan squares). The bandsrepresent the NNLO B χ PT result of Lensky et al. [54]. The blue dashed lines represent their calculation withonly the nucleon-, pion-, and Delta-pole graphs included (chiral loops are switched off).suppressed by making use of the hermeticity of the detector, requiring that only one neutral and one chargedtrack are present in the event. Additional background suppression is provided by imposing a co-planarity andopening angle cut of ◦ on the direction of the recoil proton relative to the momentum transfer direction q defined by the incident and final photons. A missing mass distribution with cuts applied is shown in Fig. 4.9for an incident photon energy of
277 MeV . In contrast to the missing mass distribution shown in Fig. 4.3, CSexperiments in the ∆(1232) region typically have prominent background due to γp → π p . For the Mainzanalysis a conservative cut is placed on the missing mass distribution to limit π background to the few percentlevel. The relevant double-polarized CS asymmetry is defined as Σ x = d σ Rx − d σ Lx d σ Rx + d σ Lx , (4.16)where d σ R ( L ) x is the differential cross section for transverse target polarization in the x -direction, and for right(left) circularly polarized photons. Data for the Σ x asymmetry in the ∆(1232) region have recently beenpublished by the Mainz A2 collaboration [38]. Figure 4.10 shows the expected sensitivity of Σ x to the spinpolarizabilities. The left figure shows significant sensitivity to γ E E , and the right figure shows little sensitivityto γ M M . Based on measurements of Σ x at angles ϑ ≈ ◦ , it is possible to uniquely identify γ E E [163].The target for this experiment was a frozen spin butanol target [164], approximately long, where theprotons in the butanol are polarized by dynamic nuclear polarization [165]. Proton polarizations were typically with relaxation times on the order of hours. To remove systematic effects, the direction of polarizationwas reversed several times, typically once per week of experiment running time. To remove backgrounds frominteractions of the photon beam with the material of the cryostat and non-hydrogen nucleons in the butanoltarget and He bath, separate data were taken using a carbon foam target, POCOFoam [166], with density .
55 g / cm inserted into the cryostat. The density of the carbon foam was such that a cylinder of identicalgeometric size to the butanol target provided a close approximation to the number of non-hydrogen nucleonsin the butanol target, allowing for a simple : subtraction accounting only for differences in luminosity.Event selection is similar to that described for the Mainz Σ analysis. Even with the exclusivity selection,accidental subtraction, and opening angle requirement, backgrounds persist into the missing-mass spectrum,28 issing mass [MeV]
850 900 950 1000 1050 1100 C oun t s Data (Dec. 2012) p CS + Compton scattering photoproduction p Figure 4.9: Missing mass distribution for the Mainz Σ experiment for incident photon energy ±
10 MeV ,and ◦ − ◦ . The blue histogram is data, the green curve shows the simulated response for CS, the blackcurve is a simulation of γp → π p events that satisfy the exclusivity requirements for CS, and the red curve isthe sum of simulated CS and π background. Plot courtesy of Cristina Collicott.similar if not worse than those shown in Fig. 4.9. Typically these backgrounds are from π events, where a low-energy decay photon escaped detection by passing up or down the beam-line, or through the gap between theCB and TAPS. An estimate of the π background was made by measuring the rate of good π events where bothdecay photons are detected, but one of the photons is detected in a detector region adjacent to a region withreduced or zero acceptance. The subtraction of backgrounds is done separately for each helicity state, as the π backgrounds themselves result in non-zero asymmetries. After removing the background contributions, thefinal missing-mass distribution is shown in Fig. 4.11. A simulation of the CS lineshape shows good agreementbetween data and calculation for the Compton peak. A relatively conservative integration limit of
940 MeV wasused in the analysis.Table 4.2: Predictions for proton spin polarizabilities compared with experimental extractions (in units of − fm ). O ( p ) b O ( (cid:15) ) O ( p ) a K-matrix HDPV DR L χ HB χ PT B χ PT Experiment[167] [168] [169] [89] [97] [96] [91] [51] [54] γ E E − . − . − . − . − . − . − . − . ± . th − . ± . − . ± . [38] γ M M . . . . . . . . ± . ± . th . ± . . ± . [38] γ E M . . . − . − . − . . − . ± . th . ± . − . ± . [38] γ M E . . . . . . . . ± . th . ± . . ± . [38] γ − . . − . . − . − . − . − . − . ± . − . ± . ± . [170] γ π . . . . . . . . . ± . . ± . [20] The measured asymmetries are plotted in Fig. 4.12. The curves are from the fixed- t DR model of Pasquiniet al. [87] for values of γ E E ranging from − . to − . , but with γ M M fixed at . [97]. The width of eachband represents the propagated errors using α E = 12 . ± . and β M = 1 . ± . , as well as γ and γ π from Table 4.2, combined in quadrature. The curves graphically demonstrate the sensitivity of the asymmetriesto γ E E , showing a preferred solution of γ E E ≈ − . ± . .Martel et al. [38] performed a global analysis of single and double-polarized CS data in the ∆(1232) region Whenever the units are omitted, it is understood that the scalar (spin) polarizabilities are measured in units of − fm ( fm ). (deg) lab q Compton 0 20 40 60 80 100 120 140 160 180 S -0.6-0.4-0.200.20.40.6 = -3.3 E1E1 g = -4.3 E1E1 g = -5.3 E1E1 g (deg) lab q Compton 0 20 40 60 80 100 120 140 160 180 S -0.6-0.4-0.200.20.40.6 = 3.9 M1M1 g = 2.9 M1M1 g = 1.9 M1M1 g Figure 4.10: The asymmetry Σ x for E γ = 290 MeV . The curves are from a dispersion theory calculation [87]with α E , β M , γ , and γ π held fixed at their experimental values. The left plot has γ M M fixed at . , and thered, black, and blue bands are for γ E E equal to − . , − . , and − . , respectively. The width of each bandrepresents the propagated errors from α E , β M , γ , and γ π combined in quadrature. The right plot has γ E E fixed at − . , and the red, black, and blue bands are for γ M M equal to . , . , and . , respectively (in unitsof − fm ). Plot courtesy of Phil Martel.[13, 38] using the DR model of Pasquini et al. [87], and the NNLO B χ PT calculation of Lensky and Pascalutsa[49] amended with some higher-order LECs which are then fitted to the data. Only Σ asymmetry pointsbelow double-pion photoproduction threshold were used in the analysis. In the fitting procedure, α E − β M , α E + β M , γ E E , γ M M , γ , and γ π were fitted to the asymmetry data sets, and to known constraints on α E + β M , α E − β M , γ , and γ π . The constraint α E − β M = (7 . ± . × − fm is taken from the analysisof Ref. [65].Table 4.3: Results from fitting Σ x and Σ asymmetries using either a dispersion model calculation (Disp) [87]or a B χ PT calculation [49]. Spin polarizabilities are given in units of − fm . Σ x [38] Σ [13] Model γ E E γ M M (cid:88) Disp − . ± . − ± (cid:88) Disp − . ± . . ± . (cid:88) (cid:88) Disp − . ± . . ± . (cid:88) (cid:88) B χ PT − . ± . . ± . Table 4.3 shows results from data fitting. The first two columns give the data sets used for fitting, the thirdcolumn shows the model used, and the fourth and fifth columns show the results for γ E E and γ M M . Thefirst data row confirms the graphical analysis of Figures 4.10 and 4.12, that the Σ x data prefer a solution γ E E ≈ − . × − fm , and the data by themselves have little predictive power for γ M M . The second datarow confirms the graphical analysis of Fig. 4.8, that the Σ data have reasonable sensitivity to γ M M , andmarkedly less sensitivity to γ E E . The third row shows the results from the combined fit of Σ x and Σ datausing the dispersion model [87], and the fourth row shows the combined fit using the B χ PT calculation [49].Within the uncertainties, the results for γ E E and γ M M from the two model fits are in agreement, indicatingthat the model dependence of the polarizability fitting is comparable to, or smaller than, the statistical errors.The last column of Table 4.2 displays the results of Martel et al. [38] for all four spin polarizabilities,obtained from the combined analysis of Σ x , Σ using the DR model of Pasquini et al. [87]. The empiricalresults for γ and γ π shown therein have also been used in the analysis of Martel et al. [38]. There is a generallygood agreement between the extracted spin polarizabilities and the χ PT calculations [51, 52] shown in thetable. The other calculations lack an uncertainty estimate, which makes them harder to judge. Nevertheless,it is rather clear from the table that the fixed- t dispersive framework [87, 97] and causal K-matrix modeling[89, 91] agree rather well with the experiment as well.30 issing Mass (MeV)860 880 900 920 940 960 980 1000 C oun t s Figure 4.11: Missing-mass spectrum after removalof backgrounds for E γ = 273 −
303 MeV , and ϑ = 100 − ◦ . The solid line is the CS lineshapedetermined from simulation. The dashed line indi-cates the upper integration limit used in the analy-sis. Plot courtesy of Phil Martel. (deg) lab θ Compton 0 20 40 60 80 100 120 140 160 180 Σ Figure 4.12: Σ x for E γ = 273 −
303 MeV . The curvesare from a dispersion theory calculation [87] with α E , β M , γ , and γ π held fixed at their experimental values,and γ M M fixed at . [97]. From bottom to top, thegreen, blue, brown, red and magenta bands are for γ E E equal to − . , − . , − . , − . , and − . , respectively.The width of each band represents the propagated errorsfrom α E , β M , γ , and γ π combined in quadrature. Plotcourtesy of Phil Martel. We finally consider the following double-polarized CS asymmetry: Σ z = d σ Rz − d σ Lz d σ Rz + d σ Lz , (4.17)where d σ R ( L ) z is the differential cross section for right (left) circularly polarized photons to scatter from anucleon target polarized in the incident beam direction. Note that the value of Σ z at the zero scattering angleis well-known from the sum rules for the forward CS amplitudes, see Sect. 5.6.Figure 4.13 shows the sensitivity of Σ z to the spin polarizabilities. The definition of the curves is identicalto that of Fig. 4.10. The left panel shows little sensitivity to γ E E , while the right panel shows significantsensitivity to γ M M . A measurement of Σ z will thus compliment the information obtained from the Σ x asymmetry. Data taking on the Σ z asymmetry started at MAMI in 2014, and continued in 2015. (deg) lab q Compton 0 20 40 60 80 100 120 140 160 180 z S E1E1 g = -4.3 E1E1 g = -5.3 E1E1 g (deg) lab q Compton 0 20 40 60 80 100 120 140 160 180 z S M1M1 g = 2.9 M1M1 g = 1.9 M1M1 g Figure 4.13: The asymmetry Σ z for E γ = 290 MeV . The curves are explained in Fig. 4.10. Plot courtesy ofPhil Martel. Although the electric, magnetic and even the spin polarizabilities of the proton are now known with rea-sonable accuracy from CS experiments, relatively little is known about the distribution of polarizability density31 += VCS Born VCS non-BornBethe-Heitler e γ e ′ p epe ep p p peep ′ Im ... Ν (p) (p’) Νγ∗ (q) γ (q’) Ν γπ N Νγ∗
Figure 4.14: The photon electroproduction process giving access to the VCS amplitude.inside the nucleon. To measure a polarizability density it is necessary to use the VCS reaction [171], where theincident photon is virtual. The VCS reaction is sensitive to the generalized electric and magnetic polarizabilities α E ( Q ) and β M ( Q ) . The relationship between VCS cross sections and the polarizabilities is most easily seen in the LEX of theunpolarized VCS cross section [171]: d σ V CS = d σ BH+Born + | q (cid:48) | Φ Ψ ( | q | , (cid:15), θ, φ ) + O ( | q (cid:48) | ) , (4.18)where | q | ( | q (cid:48) | ) is the absolute value of the incident (final) photon three-momentum in the photon-nucleon c.m.frame, (cid:15) is the photon polarization-transfer parameter, θ ( φ ) is the c.m. polar (azimuthal) angle for the outgoingphoton, and Φ is a phase space factor. Note that d σ BH+Born is the cross section for the Bethe-Heitler + Bornamplitudes only, i.e., no polarizability information is contained, and it is exactly calculable from QED and thenucleon form factors (FF). The Bethe-Heitler and Born diagrams for the VCS reaction are shown in Fig. 4.14.The polarizabilities enter the cross section expansion at order O ( | q (cid:48) | ) through the term Ψ , given by [66]: Ψ ( | q | , (cid:15), θ, φ ) = V (cid:20) P LL ( | q | ) − P T T ( | q | ) (cid:15) (cid:21) + V (cid:112) (cid:15) (1 + (cid:15) ) P LT ( | q | ) , (4.19)where P ’s are the response functions of unpolarized VCS, and V ’s are functions of kinematical variables, the (cid:15) -dependence is written out explicitly. In the limit of | q | → , P LL ∝ α E , P T T ∝ γ E M , and P LT ∝ β M .Therefore, response function P LL ( | q | ) is proportional to α E ( Q ) , P LT ( | q | ) is proportional to β M ( Q ) plus aspin polarizability term, and P T T ( | q | ) is proportional to spin polarizabilities.There have been two analysis techniques utilized to obtain the response functions P LL ( | q | ) − P T T ( | q | ) /(cid:15) and P LT ( | q | ) from VCS cross sections. The first technique is the LEX, cf. Eqs. (4.18) and (4.19). In the LEXanalysis, the response functions are fitted to the θ , φ and | q (cid:48) | dependence of VCS cross sections at fixed | q | and (cid:15) . To find the generalized polarizabilities α E ( Q ) and β M ( Q ) , a theoretical calculation of P T T and thespin polarizability contributions to P LT must be utilized, and the predictions subtracted from the experimentalresults for P LL − P T T /(cid:15) and P LT . VCS experiments have generally operated in kinematic regions where thespin polarizability contributions are small, but not negligible. For example, in the kinematics of the MIT-BatesVCS experiment [14], it is estimated that the spin polarizability contribution to P LL − P T T /(cid:15) is , and thecontribution to P LT is [174, 175].The second technique uses the VCS dispersion model [61]. In this analysis, the VCS amplitudes obtainedfrom the MAID γ ∗ p → πN multipoles [176] are held fixed, and the two unconstrained asymptotic contributionsto the VCS amplitudes are fit to experimental data at fixed Q . For data fitting, a dipole ansatz has traditionally Note that the connection between the scalar GPs (i.e., the VCS polarizabilities discussed in this section), and the VVCS polariz-abilities (e.g., the generalized Baldin sum rule of (5.34)) is not known at finite Q . A low- Q relation for some of the spin GPs exists,cf. Eq. (5.45). Alternative expansions of VVCS and VCS in generalized polarizabilities were proposed in Refs. [172] and [173]. (if one dipole)example L a = 0.7 GeV ChPT o(p ) ( e = 0.645) P LL - P TT / e ( G e V - ) RCSBates MAMI JLab JLab -20-15-10-50 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2DR (if one dipole)example L b = 0.63 GeV Q (GeV ) P LT ( G e V - ) -20-15-10-50 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 RCSBates MAMI JLab JLab
LEX analysisdirect DR fit
Figure 4.15: VCS response functions from: [14],Mainz 2000 [21], Mainz 2008 [22], and JLab [23].RCS points correspond with the older values for po-larizabilities [62]. The solid curves are O ( p ) HB χ PTcalculation [175] with (cid:15) = 0 . . The dashed curve isa dispersion-model fit [61] to the RCS and MIT-Batesdata points. Plot courtesy of Helene Fonvieille. α E (10 -4 fm )RCSBates MAMI JLab JLab024 0 0.5 1 1.5 2Q (GeV )024 0 0.5 1 1.5 2 β M (10 -4 fm ) LEX minus spin(DR)direct DR fit
DISPERSION RELATION MODEL, B.Pasquini et al. (IF ONE DIPOLE) example Λ α = 0.70 GeV example Λ β = 0.63 GeV Figure 4.16: The generalized polarizabilities α E ( Q ) (upper panel) and β M ( Q ) (lower panel). The curvesand data references are the same as in Fig. 4.15. Thearrows indicate Q of forthcoming MAMI data. Plotcourtesy of Helene Fonvieille.been used [61] to parametrize the asymptotic contributions: α E ( Q ) − α πNE ( Q ) = α E − α πNE (1 + Q / Λ α ) , (4.20a) β M ( Q ) − β πNM ( Q ) = β M − β πNM (1 + Q / Λ β ) . (4.20b)In the first equation, α E is the experimental electric polarizability from RCS, α πNE is the calculated πN contri-bution to the electric polarizability at Q = 0 , and α πNE ( Q ) is the calculated πN contribution to the electricpolarizability at finite Q . The definitions are the same for the magnetic polarizability. The only free param-eters in Eq. (4.20) are Λ α and Λ β . The advantage of parameterizing the generalized polarizabilities this wayis that once the parameters Λ α and Λ β are fixed from fitting VCS cross sections, the formalism has predictivepower for the response functions and generalized polarizabilities at other Q values, provided of course thedipole assumption in Eq. (4.20) is valid. Once the parameters Λ α and Λ β are determined, the generalizedpolarizabilities are calculated from Eq. (4.20), and the response functions P LL − P T T /(cid:15) and P LT are found bysumming the asymptotic terms with calculated spin polarizability contributions.The VCS response functions P LL ( | q | ) − P T T ( | q | ) /(cid:15) and P LT ( | q | ) are plotted in Fig. 4.15 for the three lowest Q VCS experiments: MIT-Bates [14], Mainz [21] [22], and JLab [177]. The dashed curves in Fig. 4.15 are thedispersion model calculations assuming the dipole choice of Eq. (4.20), and the fitted values for Λ α and Λ β thatby construction make the dispersion calculations go directly through the RCS and JLab data points. As shownin Fig. 4.15, the dipole ansatz of Eq. (4.20) allows for a unified description of all low- Q VCS response functionmeasurements, with the exception of the Mainz P LL − P T T /(cid:15) , and to a lesser extent P LT , measurements at Q = 0 .
33 GeV .The generalized polarizabilities α E ( Q ) and β M ( Q ) are shown in Fig. 4.16. The dashed curves in thefigures are from the dispersion model calculation using the same Λ α and Λ β as shown in Fig. 4.15. Similar to33he situation shown in Fig. 4.15, the dipole assumption of Eq. (4.20) also allows for a unified description of alllow- Q polarizability measurements, with the exception of the Mainz measurements at Q = 0 .
33 GeV .The πN contribution to the electric polarizability is positive at low Q , but quickly decreases and crosses0 at Q ≈ . . Having a negative contribution to electric polarizability is not unphysical. A negativeelectric polarizability occurs in a class of materials known as ferroelectrics, where the internal electric field ofthe material is stronger than the applied external field. The πN contribution to the magnetic polarizabilityis paramagnetic (positive) in HB χ PT, as discussed in Sect. 2.4. The asymptotic contribution is diamagnetic(negative) and mimics the effect of the LECs.
Figure 4.17: The induced polarization of the pro-ton for a transverse electric field. White denotespositive induced polarization, and black denotesnegative induced polarization. The upper panelis for the dispersion fit for α E ( Q ) shown inFig. 4.16. The lower panel is for α E ( Q ) adjustedto fit the Mainz data. Plot courtesy of Marc Van-derhaeghen. The mean-square electric polarizability radius, definedas (cid:104) r (cid:105) α E = − α E d α E ( Q )d Q (cid:12)(cid:12)(cid:12)(cid:12) Q =0 , (4.21)was extracted for the proton by Bourgeois et al. [14] usingthe DR fit to the MIT-Bates data. Their result, (cid:104) r (cid:105) α E = (cid:0) .
02 + (cid:2) +0 . − . (cid:3)(cid:1) fm , (4.22)where the error is statistical only, is in good agreement withthe HB χ PT prediction [178] of . .The mean-square polarizability radius is thus signifi-cantly larger than the proton mean-square charge radius(which is about .
77 fm ) demonstrating the dominance ofmesonic effects in the electric polarizability. The additionale.m. vertex in the polarizability diagram relative to the FFdiagram increases the range of the interaction by approxi-mately a factor of two as compared to the charge FF. Also ofinterest is the uncertainty principle estimate for the meansquare radius of the pion cloud, (cid:104) r (cid:105) ≈ (1 /m π ) = 2 fm ,which is in better agreement with (cid:104) r (cid:105) α E .As seen in Fig. 4.16, there is poor agreement at Q =0 .
33 GeV between the measured values of α E ( Q ) andthe dispersion model fit to the RCS and MIT-Bates datapoints. This discrepancy has been analyzed by Gorchteinet al. [179] using a light-front interpretation of the gen-eralized polarizabilities. This formalism provides a wayto calculate deformations of the quark charge densitieswhen an external e.m. field is applied. They found thatadding a Gaussian term to Eq. (4.20) to improve agree-ment with data at Q = 0 .
33 GeV , also gives the protona pronounced structure in its induced polarization at largetransverse distances, . to . This is vividly shown inFig. 4.17, where the bottom panel shows the effect of ad-justing α E ( Q ) to fit the Mainz data points. Clearly, addi-tional VCS data in the low to intermediate Q region . to . are needed to confirm this prediction. NewVCS data have recently been taken by the Mainz A1 collaboration at Q ≈ . , . and . , and this dataare currently under analysis [180]. 34 epton universality test in the photoproduction of e e + versus µ µ + pairs on a proton target Vladyslav Pauk and Marc Vanderhaeghen
Institut f¨ur Kernphysik, Cluster of Excellence PRISMA,Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany (Dated: March 5, 2015)In view of the significantly different proton charge radius extracted from muonic hydrogen Lamb shift mea-surements as compared to electronic hydrogen spectroscopy or electron scattering experiments, we study in thiswork the photoproduction of a lepton pair on a proton target in the limit of very small momentum transfer as away to provide a test of the lepton universality when extracting the proton charge form factor. By detecting therecoiling proton in the p ! l l + p reaction, we show that a measurement of a ratio of e e + + µ µ + over e e + cross sections with a relative precision of around 2 %, would allow for a test to distinguish between thetwo different proton charge radii currently extracted from muonic and electronic observables. Recent extractions of the proton charge radius from muonichydrogen Lamb shift measurements [1, 2] are in strong con-tradiction, by around 7 standard deviations, with the valuesobtained from energy level shifts in electronic hydrogen [3]or from electron-proton elastic scattering experiments [4, 5].This so-called ”proton radius puzzle” has triggered a large ac-tivity and is the subject of intense debate, see e.g. [6–8] forrecent reviews, and references therein. One important ele-ment in this puzzle is that the proton charge radius extrac-tions from both electron scattering and electronic hydrogenspectroscopy seem to be in agreement with each other. Fur-thermore, the theory to extract the proton radius from hydro-gen spectroscopy has been scrutinized and improved in recentyears and seems to be well under control, see [9] for a recentreview. Lepton universality requires the same radius to en-ter in the electronic and muonic observables. If the differentcharge radius extractions cannot be explained by overlookedcorrections, it would point to a violation of electron-muonuniversality. Several scenarios of new, beyond the StandardModel, physics have been proposed by invoking new particleswhich couple to muons and protons, but much weaker to elec-trons, see e.g. [10–14]. Such models would also lead to largeloop corrections to the muon’s anomalous magnetic moment, ( g µ , which presently displays a deviation between ex-periment and its Standard Model prediction, see e.g. [15] fora recent update. Explaining both the ( g µ discrepancy andthe proton radius puzzle by new particles coupling mainly tomuons seems an attractive perspective. It does however re-quire a significant fine-tuning, especially for larger values ofthe conjectured new particle masses, see e.g. [8] for a re-cent review. To further test the electron-muon universality, ithas been proposed by the MUon proton Scattering Experiment(MUSE) [16] to make a simultaneous measurement of both µp and ep elastic scattering. The MUSE experiment aims to testthe proton charge form factor extractions from µp versus ep scattering, by a comparison of absolute cross section measure-ments, with a required precision on each absolute cross sec-tion at the level of 1 %. Besides the plans to measure µp elas-tic scattering, several new experiments are underway to extendthe ep scattering to lower momentum transfer values, down to GeV , and to cross-check its systematics [17, 18].In this work, we present a complementary cross-check inthis field through a comparison of the photo-production of e e + versus µ µ + pairs on a proton target. The photo-production of a lepton pair is a well studied process, see e.g. Ref. [19] for an older review. In the present work, we showthat by detecting the recoiling proton in the p ! l l + p re-action, a measurement of the ratio of the e e + cross sectionbelow µ µ + threshold versus the e e + + µ µ + cross sec-tion sum above µ µ + threshold with a relative precision ofaround 2 % would allow to distinguish between the differentproton charge radius extractions from muonic and electronicobservables. Furthermore, we show that the linear photon po-larization asymmetry has a discriminatory power between the e e + and µ µ + channels in the µ µ + threshold region. p’p k l − l + p’p k l − l + FIG. 1: Bethe-Heitler mechanism to the p ! l l + p process, wherethe four-momenta of the external particles are: k for the photon, p ( p ) for initial (final) protons, and l , l + for the lepton pair. We will consider the lepton pair production on a protontarget, p ! l l + p , in the limit of very small momentumtransfer, defined as t = ( p p ) , with four-momenta as in-dicated on Fig. 1. Furthermore, we will use in the followingthe Mandelstam invariant s = ( k + p ) = M + 2 M E , with M the proton mass and E the photon lab energy, as wellas the squared invariant mass of the lepton pair, defined as M ll ⌘ ( l + l + ) . In the limit of small t , the Bethe-Heitler(BH) mechanism, shown in Fig. 1 totally dominates the crosssection of the p ! l l + p reaction. As the momentum trans-fer t is the argument appearing in the form factor (FF) in theBH process, a measurement of the cross section in this kine-matic regime will allow to access the proton electric FF G Ep at small spacelike momentum transfer, with a very small con-tribution of the proton magnetic FF G Mp . As the differentialcross section for the BH process is strongly peaked for lep-tons emitted in the incoming photon direction, and as we aimto maximize the BH contribution in this work in order to ac-cess G Ep , we will study the p ! l l + p process when (only)detecting the recoiling proton’s momentum and angle, thus ef-fectively integrating over the large lepton peak regions. The lab momentum of the proton is in one-to-one relation withthe momentum transfer t : | ~p | lab = 2 M p ⌧ (1 + ⌧ ) , with a r X i v : . [ h e p - ph ] M a r Figure 4.18: Bethe-Heitler process in dilepton photo-production, γp → l − l + p .
50 100 150 200 250 300 350 400050100150200250300 Ν ! MeV " d Σ d r ! pb sr " VCSBHTotal
Figure 4.19: The differential cross-section of e + e − photoproduction off the proton as a function of thephoton beam energy for forward-recoil kinematics. The process of dilepton photoproduction from protons by cosmic microwave background radiation, γp → p(cid:96) + (cid:96) − , is one of the main mechanisms for depletion of cosmic-ray energy in the universe. It is dominatedby the Bethe-Heitler (BH) process, Fig. 4.18, which can be accurately calculated from QED alone. In certainkinematics, however, the nuclear component—represented by timelike VCS—dominates. Until now there hasbeen only one lab-based experiment dedicated to dilepton photoproduction off the proton [181]. Conducted inthe 70s, it aimed to measure the forward CS amplitude f ( ν ) at ν ≈ . , and to verify the Kramers-Kronigrelation for the proton, cf. Sect. 5.6. Recently, Pauk and Vanderhaeghen [182] made a proposal to study theratio between photoproduction of e − e + and µ − µ + pairs on a proton target in the limit of very small momentumtransfer. This measurement would serve as a test of lepton universality and is of interest to the proton-radiuspuzzle.Without distinguishing electrons from positrons, the cross-section element (for the e + e − photoproduction)can be written as an incoherent sum of BH and VCS cross-sections, the interference terms drop out because ofthe charge-conjugation symmetry: d σ = d σ BH + d σ VCS . (4.23)In this case VCS can be accessed only in the regions where the BH mechanism is suppressed. In thedistribution over the solid angle of the recoil proton, Ω r , the leading contribution to BH is given by the twodiagrams shown in Fig. 4.18, which are known to cancel in the forward kinematics [183]. That is to say that d σ BH / d Ω r is smallest when ϑ r = 0 , where for pointlike proton it takes a particularly simple form: d σ BH d Ω r ( ν, ϑ r = 0) = α π ν (cid:26)(cid:2) m e + ν (cid:3) K (cid:18)(cid:113) − m e ν (cid:19) − ν E (cid:18)(cid:113) − m e ν (cid:19)(cid:27) , (4.24)with E and K being the elliptic integrals, and m e the electron mass. On the other hand, if we neglect for amoment the momentum-transfer dependence of the VCS process, the VCS cross section factorizes into the RCScross section and a factor responsible for the pair production: d σ d Ω r d M ee = d σ RCS d Ω r α πM ee (cid:115) − m e M ee (cid:18) m e M ee (cid:19) (cid:34)(cid:18) M ee M ν (cid:19) − M ee ( M + 2 M ν ) M ν (cid:35) / , (4.25)where M ee is the invariant mass of the lepton pair.The resulting angular distributions for BH and VCS, as well as their sum, are shown as functions of photonenergy in Fig. 4.19. The figure clearly shows that for beam energies above
200 MeV the CS off proton is thedominant mechanism. Any substantial deviation from these predictions can be interpreted as the timelikemomentum-transfer dependence of the Compton process, and hence attributed to the aforementioned effectsof the timelike e.m. structure of the nucleon. 35 . Sum Rules
The fundamental relation between light absorption and scattering, encompassed for example in the cele-brated Kramers–Kronig relation, is manifested in a variety of model-independent relations. They allow us toexpress certain linear combinations of polarizabilities in terms of weighted energy integrals of total photoab-sorption cross sections, or equivalently, in terms of the moments of structure functions [2, 61, 63, 184]. Theyall are derived from the analyticity, unitarity and symmetry properties of the forward CS amplitude, depictedin Fig. 5.1. In general, the photons are virtual, with spacelike virtuality q < . The corresponding amplitudeis then referred to as the forward doubly-virtual Compton scattering (VVCS) amplitude. In what follows weconsider its properties, sketch the derivation of the sum rules, and discuss their empirical consequences.Figure 5.1: Forward Compton scattering: N ( p ) + γ ( q ) → N ( p ) + γ ( q ) , with either real or virtual photons. In the forward kinematics ( t = 0 ), the Lorentz structure of the VVCS amplitude is decomposed in fourindependent tensor forms: T µν ( q, p ) = (cid:18) − g µν + q µ q ν q (cid:19) T ( ν, Q ) + 1 M (cid:18) p µ − p · qq q µ (cid:19) (cid:18) p ν − p · qq q ν (cid:19) T ( ν, Q ) − M γ µνα q α S ( ν, Q ) − M (cid:0) γ µν q + q µ γ να q α − q ν γ µα q α (cid:1) S ( ν, Q ) , (5.1)with T , the spin-independent and S , the spin-dependent invariant amplitudes, functions of ν = ( s − u ) / M and Q = − q . This decomposition is explicitly gauge invariant and splits into symmetric and antisymmetricparts, T µν = T µνS + T µνA , (5.2)which, respectively, do not and do depend on the nucleon spin. Given that the vector indices are to be contractedwith either the polarization vector, satisfying q · ε = 0 , or with another gauge-invariant tensor, the termscontaining q µ or q ν can be omitted, hence, T µνS ( q, p ) = − g µν T ( ν, Q ) + p µ p ν M T ( ν, Q ) , (5.3a) T µνA ( q, p ) = − M γ µνα q α S ( ν, Q ) + Q M γ µν S ( ν, Q ) . (5.3b)One immediate observation is that the symmetry under photon crossing translates into the following conditions,for real ν : T ( − ν, Q ) = T ( ν, Q ) , T ( − ν, Q ) = T ( ν, Q ) , (5.4a) S ( − ν, Q ) = S ( ν, Q ) , S ( − ν, Q ) = − S ( ν, Q ) . (5.4b) It is customary to write the spin-dependent amplitude with the help of the nucleon spin four-vector s α , satisfying s = − and s · p = 0 : T µνA ( q, p ) = iM (cid:15) µναβ q α s β S ( ν, Q ) + iM (cid:15) µναβ q α ( p · q s β − s · q p β ) S ( ν, Q ) . S is odd with respect to the sign reflection of ν , the other amplitudes are even. We will often considerthe combination νS , such that it has the same crossing properties as the other amplitudes.The Born contribution to these amplitudes is well known (cf. Appendix A) and given by [61]: T Born1 ( ν, Q ) = 4 παM (cid:26) Q (cid:2) F ( Q ) + F ( Q ) (cid:3) Q − M ν − F ( Q ) (cid:27) , (5.5a) T Born2 ( ν, Q ) = 16 παM Q Q − M ν (cid:26) F ( Q ) + Q M F ( Q ) (cid:27) , (5.5b) S Born1 ( ν, Q ) = 2 παM (cid:26) M Q (cid:2) F ( Q ) + F ( Q ) (cid:3) F ( Q ) Q − M ν − F ( Q ) (cid:27) , (5.5c) S Born2 ( ν, Q ) = − παM νQ − M ν (cid:2) F ( Q ) + F ( Q ) (cid:3) F ( Q ) , (5.5d)where F and F are the elastic Dirac and Pauli FFs of the nucleon, which are normalized to F (0) = Z and F (0) = κ . Introducing the Bjorken variable, x = Q / M ν , we recall that the physical region in electronscattering corresponds with x ∈ [0 , . Obviously the Born graphs exhibit the pole at ν = Q / M ≡ ν el , orequivalently x = 1 . This nucleon-pole part of the Born contribution is isolated below, see Eq. (5.13). Thenon-Born part of the full amplitudes will be denoted as T i , S i . The optical theorem relates the absorptive parts of the forward VVCS amplitudes to the nucleon structurefunctions , or equivalently, to the cross sections of virtual-photon absorption γ ∗ N → X : Im T ( ν, Q ) = 4 π αM f ( x, Q ) = ν σ T ( ν, Q ) , (5.6a) Im T ( ν, Q ) = 4 π αν f ( x, Q ) = Q νν + Q [ σ T + σ L ] ( ν, Q ) , (5.6b) Im S ( ν, Q ) = 4 π αν g ( x, Q ) = M ν ν + Q (cid:20) Qν σ LT + σ T T (cid:21) ( ν, Q ) , (5.6c) Im S ( ν, Q ) = 4 π αMν g ( x, Q ) = M νν + Q (cid:20) νQ σ LT − σ T T (cid:21) ( ν, Q ) , (5.6d)where the cross sections are defined as: σ T = / ( σ / + σ / ) and σ T T = / ( σ / − σ / ) for transverselypolarized photons, and σ L = / ( σ / + σ − / ) for longitudinal photons, where the subscript on the right-handside ( rhs ) indicates the total helicity of the γ ∗ N state. The cross section σ LT corresponds with a simultaneoushelicity change of the photon (from longitudinal to transverse) and the nucleon (spin-flip) such that the totalhelicity is conserved. These unitarity relations hold in the physical region, where the Bjorken variable isconfined to the unit interval.Figure 5.2 illustrates the photoabsorption process. The structure functions describing the purely elasticscattering, shown in the left panel, are given in terms of the elastic FFs: f el1 ( x, Q ) = 12 G M ( Q ) δ (1 − x ) , (5.7a) f el2 ( x, Q ) = 11 + τ (cid:2) G E ( Q ) + τ G M ( Q ) (cid:3) δ (1 − x ) , (5.7b) g el1 ( x, Q ) = 12 F ( Q ) G M ( Q ) δ (1 − x ) , (5.7c) g el2 ( x, Q ) = − τ F ( Q ) G M ( Q ) δ (1 − x ) , (5.7d) The unpolarized structure functions f and f are the standard F and F . However, the latter notation is reserved here for theDirac and Pauli FFs respectively. The flux factor for virtual photons which goes into these cross sections is rather arbitrary, cf. [61] for common choices. Ourexpressions correspond to the flux factor choice K = ν . Expressions in terms of the structure functions are not affected by the choiceof the flux factor. γ ∗ N (a) γ ∗ N X (b)Figure 5.2: (a) ‘Elastic’ and (b) ‘inelastic’ part of the photoabsorption cross section. (a) is related to the ‘pole’contribution, whereas (b) is related to the ‘non-pole’ contribution.where τ = Q / M , and the electric and magnetic Sachs FFs are linearly related to the Dirac and Pauli FFs as: G E ( Q ) = F ( Q ) − τ F ( Q ) , G M ( Q ) = F ( Q ) + F ( Q ) . (5.8)Furthermore, δ ( x ) is the Dirac delta-function, such that δ (1 − x ) = ν el δ ( ν − ν el ) , with ν el ≡ M τ. (5.9)In the limit, Q → ∞ , fixed x , the structure functions are related to the parton distribution functions. Weare, however, interested in the limit where Q and ν are small. In this case the VVCS amplitudes can on onehand be expanded in terms of polarizabilities and e.m. radii, and on the other in terms of moments of structurefunctions. This expansion and the resulting relations between the static e.m. properties of the nucleon and themoments of structure functions will be discussed further below. Before that, we need to establish the dispersionrelations (DRs) for the forward VVCS amplitudes. We next consider the analytic structure of the VVCS amplitudes T i and S i in the complex plane of ν . We havealready seen that the Born contribution contains the nucleon pole at the kinematics of elastic scattering, ν = ν el .The inelastic processes are manifested in the branch cuts, starting at the first particle-production threshold ν and extending to infinity. Due to crossing symmetry, the analytic structure for negative real ν is similar. In anycase, the physical singularities are confined to the real axis. Elsewhere in the complex plane the amplitudesare analytic (or, holomorphic) functions. The latter requirement is inferred by micro-causality, a fundamentalpostulate of special relativity which states that all the signals propagate strictly within the light-cone.For the subsequent discussion it is important that the VVCS amplitudes do not have a pole in the limit Q → , then ν → . Such a pole can only come from the nucleon propagator entering the Born contribution.From Eq. (5.5) we see that the pole is absent from all the amplitudes except S . We therefore will write downthe DR for νS , which is pole free for real photons.These analytic properties, together with the crossing symmetry conditions from Eq. (5.4), are well-knownto result in the following DRs (cf. Appendix B for the derivation): T i ( ν, Q ) = 2 π ˆ ∞ ν el d ν (cid:48) ν (cid:48) Im T i ( ν (cid:48) , Q ) ν (cid:48) − ν − i + , (5.10a) S ( ν, Q ) = 2 π ˆ ∞ ν el d ν (cid:48) ν (cid:48) Im S ( ν (cid:48) , Q ) ν (cid:48) − ν − i + , (5.10b) νS ( ν, Q ) = 2 π ˆ ∞ ν el d ν (cid:48) ν (cid:48) Im S ( ν (cid:48) , Q ) ν (cid:48) − ν − i + , (5.10c)where + is an infinitesimally small positive number. As emphasized in the derivation of these relations, theyare only valid provided the “good” behavior of these amplitudes for large ν . It turns out that for T the above38 nsubtracted DR is not warranted and at least one subtraction is required. We postpone a detailed discussionof this issue till Sect. 5.4 and Sect. 5.5 while continuing to deal here with the unsubtracted DR.Substituting the unitarity relations, Eq. (5.6), into Eq. (5.10) we have: T ( ν, Q ) = 8 παM ˆ d xx f ( x, Q )1 − x ( ν/ν el ) − i + = 2 π ˆ ∞ ν el d ν (cid:48) ν (cid:48) σ T ( ν (cid:48) , Q ) ν (cid:48) − ν − i + , (5.12a) T ( ν, Q ) = 16 παMQ ˆ d x f ( x, Q )1 − x ( ν/ν el ) − i + = 2 Q π ˆ ∞ ν el d ν (cid:48) ν (cid:48) [ σ T + σ L ]( ν (cid:48) , Q )( ν (cid:48) + Q )( ν (cid:48) − ν − i + ) , (5.12b) S ( ν, Q ) = 16 παMQ ˆ d x g ( x, Q )1 − x ( ν/ν el ) − i + = 2 Mπ ˆ ∞ ν el d ν (cid:48) ν (cid:48) (cid:2) Qν (cid:48) σ LT + σ T T (cid:3) ( ν (cid:48) , Q )( ν (cid:48) + Q )( ν (cid:48) − ν − i + ) , (5.12c) νS ( ν, Q ) = 16 παM Q ˆ d x g ( x, Q )1 − x ( ν/ν el ) − i + = 2 M π ˆ ∞ ν el d ν (cid:48) ν (cid:48) (cid:2) ν (cid:48) Q σ LT − σ T T (cid:3) ( ν (cid:48) , Q )( ν (cid:48) + Q )( ν (cid:48) − ν − i + ) . (5.12d)Substituting here the elastic structure functions, Eq. (5.7), we obtain the nucleon-pole contribution: T pole1 ( ν, Q ) = 4 παM ν G M ( Q ) ν − ν − i + , (5.13a) T pole2 ( ν, Q ) = 8 πα ν el ν − ν − i + G E ( Q ) + τ G M ( Q )1 + τ , (5.13b) S pole1 ( ν, Q ) = 4 πα ν el ν − ν − i + F ( Q ) G M ( Q ) , (5.13c) [ νS ] pole ( ν, Q ) = − πα ν ν − ν − i + F ( Q ) G M ( Q ) . (5.13d)These pole terms vanish in the limit Q → , then ν → , as required.We are now in a position to derive the various sum rules arising from low-energy and/or low-momentumexpansion of the CS amplitudes. The above DRs clearly show that the expansion in energy ν is an expansion inthe moments of structure functions. For example, the Burkhardt-Cottingham (BC) sum rule [185] arises fromtaking the low-energy limit, ν → , of the relation (5.12d) for νS : ˆ d x g ( x, Q ) , (5.14)valid for any Q > . Note that, although the unitarity relations are valid in the physical region only, the DRscan be valid outside of the physical region. The photon virtuality must nevertheless be spacelike, Q > , inorder to exclude the particle production off the external photons.Subtracting the DR (5.12d) at ν = 0 , and using the BC sum rule, we obtain: S ( ν, Q ) = 64 παM νQ ˆ d x x g ( x, Q )1 − x ( ν/ν el ) − i + = 2 M νπ ˆ ∞ ν el d ν (cid:48) ν (cid:48) (cid:2) ν (cid:48) Q σ LT − σ T T (cid:3) ( ν (cid:48) , Q )( ν (cid:48) + Q )( ν (cid:48) − ν − i + ) . (5.15)This expression could be obtained immediately by writing the DR for S , rather than νS , but then we wouldnot have established the BC sum rule. Substituting in here the elastic g , we find that the pole and Born partof S coincide, see Eq. (C.5). Using that, with x = ν el / ν , ν el = Q / M , the change of the integration variable from ν to x goes as: ˆ ∞ ν el d νν n f ( ν, Q ) = (cid:18) MQ (cid:19) n − ˆ d x x n − f ( x, Q ) . (5.11) .4. Sum Rules for Real Photons We start with considering the model-independent of (5.12) for the case of Q = 0 . The amplitudes T and S drop out, and so do the cross sections containing longitudinal photons. We thus have: T ( ν,
0) = 2 π ˆ ∞ d ν (cid:48) ν (cid:48) σ T ( ν (cid:48) ) ν (cid:48) − ν − i + , (5.16a) S ( ν,
0) = 2 Mπ ˆ ∞ d ν (cid:48) ν (cid:48) σ T T ( ν (cid:48) ) ν (cid:48) − ν − i + . (5.16b)The cross sections σ T and σ T T are, respectively, the unpolarized and helicity-difference photoabsorption crosssections: / ( σ / ± σ / ) . The amplitudes T ( ν, and S ( ν, are (up to overall factors) identical to the RCSamplitudes A ( ν, and A ( ν, introduced in Sect. 3, and hence the above DRs apply to the latter amplitudestoo.The low-energy expansion of the amplitudes goes as: π T ( ν,
0) = − Z αM + ( α E + β M ) ν + (cid:2) α E ν + β M ν + / ( α E + β M ) (cid:3) ν + O ( ν ) , (5.17a) π S ( ν,
0) = − α κ M + M γ ν + M ¯ γ ν + O ( ν ) , (5.17b)where the O ( ν ) terms represent the low-energy theorem (LET) [138, 139]; the scalar polarizabilities α E and β M are introduced in Sect. 2; the forward spin polarizabilities γ , ¯ γ are linear combinations of spinpolarizabilities, e.g.: γ = − ( γ E E + γ M M + γ E M + γ M E ) . (5.18)The rhs of Eq. (5.16) can also be Taylor expanded in ν and each term matched to the low-energy expansionof the amplitude on the left-hand side ( lhs ). We however run immediately into the following difficulty. At ν = 0 (the 0 th order in ν ), the relation for T yields an apparently wrong result: − Z α/M = (2 /π ) ˆ ∞ d ν σ T ( ν ) . (5.19)The lhs is negative definite whereas the rhs is positive definite. The empirical knowledge of the photoabsorptioncross section for the nucleon shows in addition that the integral on the rhs diverges. This invalidates theunsubtracted DR for T . A common choice is to make a subtraction at ν = 0 , and use the LET to obtain: T ( ν,
0) = − π Z αM + 2 ν π ˆ ∞ d ν (cid:48) σ T ( ν (cid:48) ) ν (cid:48) − ν − i + . (5.20)The integral now converges and its evaluation for the proton will be discussed in Sect. 5.6.Matching the low-energy expansion of T at O ( ν ) , one obtains the Baldin sum rule [2]: α E + β M = 12 π ˆ ∞ d ν σ T ( ν ) ν . (5.21)At O ( ν ) , we obtain a sum rule for a linear combination of the energy slope of the dipole polarizabilities ( α E ν , β M ν ) and the quadrupole polarizabilities ( α E , β M ) [60]: α E ν + β M ν + / ( α E + β M ) = 12 π ˆ ∞ d ν σ T ( ν ) ν , (5.22)referred to as the 4 th -order Baldin sum rule.Considering the low-energy expansion of S , at the 0 th order one obtains the celebrated Gerasimov-Drell-Hearn (GDH) sum rule [186–188]: αM κ = − π ˆ ∞ d ν σ T T ( ν ) ν , (5.23)40hich expresses the anomalous magnetic moment κ in terms of an energy-weighted integral of the helicity-difference photoabsorption cross section. This is probably the best studied sum rule. It directly demonstratesthe idea of expressing a purely quantum effect, which is the anomalous magnetic moment, in terms of aclassical quantity, which is the cross section. The perturbative verifications of the GDH sum rule in QED andother quantum field theories provide further insight into quantum dynamics (see, e.g., Refs. [189–193]).At O ( ν ) one arrives at the forward spin polarizability (FSP) sum rule, also referred to as the Gell-Mann,Goldberger and Thirring (GGT) sum rule [184]: γ = 12 π ˆ ∞ d ν σ T T ( ν ) ν , (5.24)while at O ( ν ) one obtains the higher-order FSP sum rule [170]: ¯ γ = 12 π ˆ ∞ d ν σ T T ( ν ) ν . (5.25)The numerical evaluation of these sum rules based on empirical photoabsorption cross sections is discussedin Sect. 5.6. Q The main idea in the derivation of sum rules is to use the unitarity relations in combination with theDRs, Eq. (5.12), and then expand the left- and right-hand sides in the photon energy ν and the virtuality Q .In so doing, one expresses the static e.m. properties of the nucleon (e.g., magnetic moment, charge radius,polarizabilities), which appear as coefficients in the low-momentum expansion of the VVCS amplitudes, interms of the moments of its structure functions. In what follows, we derive a number of such sum rules andrelations for the spin-independent and spin-dependent properties of the nucleon. As we have established, for Q = 0 the convergence properties of the T amplitude are such that its DRrequires one subtraction. It is customary to choose ν = 0 as the subtraction point, leading to: T ( ν, Q ) = T (0 ,
0) + 2 π (cid:26) ˆ ∞ ν el d ν (cid:48) ν (cid:48) Im T ( ν (cid:48) , Q ) ν (cid:48) − ν − i + − ˆ ∞ d ν (cid:48) Im T ( ν (cid:48) , ν (cid:48) − i + (cid:27) . (5.26)The subtraction term, in accordance with the classic LET of Low [138], Gell-Mann and Goldberger [139], isgiven by the Thomson term: T (0 ,
0) = − π Z α/M, (5.27)while the rest of the amplitude T ( ν, Q ) could completely be determined by an integral of Im T = (4 π α/M ) f .That would be quite remarkable, because we could calculate T (0 , Q ) and then for instance take its non-Bornpiece which goes as: T (0 , Q ) = 4 πβ M Q + O ( Q ) , (5.28)and extract the magnetic polarizability. In other words, we could have a sum rule for β M and for α E sepa-rately, rather than together as in the Baldin sum rule.Unfortunately, this appears to be not possible, as the DR requires a subtraction at each Q , cf. [194–196].In this case T (0 , Q ) is an unknown subtraction function, and the corresponding DR reads: T ( ν, Q ) = T (0 , Q ) + 2 ν π ˆ ∞ ν el d ν (cid:48) Im T ( ν (cid:48) , Q ) ν (cid:48) ( ν (cid:48) − ν − i + ) , = T (0 , Q ) + 32 παM ν Q ˆ d x x f ( x, Q )1 − x ( ν/ν el ) − i + , (5.29) = T (0 , Q ) + 2 ν π ˆ ∞ ν el d ν (cid:48) σ T ( ν (cid:48) , Q ) ν (cid:48) − ν − i + . ∂∂Q T ( ν, Q ) (cid:12)(cid:12)(cid:12) Q =0 = T ( ν, − T (0 , ν = 2 π ˆ ∞ d ν (cid:48) σ T ( ν (cid:48) ) ν (cid:48) − ν − i + . (5.30)For finite Q , the pole part of the amplitudes, T pole i in Eq. (5.13), satisfies the DR with the elastic part of thestructure functions. We therefore consider just the non-pole parts and their determination from the inelasticstructure functions.The low-energy, low-momentum expansion of the spin-independent amplitudes is given by: π (cid:104) T − T pole1 (cid:105) ( ν, Q ) = − Z αM + (cid:18) Z α M (cid:104) r (cid:105) + β M (cid:19) Q + ( α E + β M ) ν + (cid:2) α E ν + β M ν + / ( α E + β M ) (cid:3) ν + . . . , (5.31a) π (cid:104) T − T pole2 (cid:105) ( ν, Q ) = ( α E + β M ) Q + (cid:2) α E ν + β M ν + / ( α E + β M ) (cid:3) Q ν + . . . . (5.31b)where, in case of T , the Thomson term and the Dirac radius (cid:104) r (cid:105) = − d / d Q F ( Q ) | Q =0 , come from thenon-pole part of the Born contribution.The fact that the same combination of polarizabilities enters in both amplitudes follows from Eq. (5.30).We thus see for instance that the Baldin sum rule can equivalently be written as: α E + β M = lim Q → αMQ ˆ x d x x f ( x, Q ) = lim Q → αMQ ˆ x d x f ( x, Q ) , (5.32)where x is the inelastic threshold. At the next order in ν , we have the 4 th -order sum rule: α E ν + β M ν + / ( α E + β M ) = lim Q → αM Q ˆ x d x x f ( x, Q ) = lim Q → αM Q ˆ x d x x f ( x, Q ) . (5.33)These sum rules can be generalized to finite Q , and the usual choice is to do that using f . For instance, thegeneralization of the Baldin sum rule reads [61]: α E ( Q ) + β M ( Q ) = 8 αMQ ˆ x d x x f ( x, Q ) . (5.34)It was evaluated in Ref. [197], and more recently in Refs. [57, 58] using an improved empirical parametrizationof the structure function f .In general, the relation in Eq. (5.30) implies that the longitudinal structure function: f L ( x, Q ) = − xf ( x, Q ) + f ( x, Q ) , (5.35)which is known to vanish for asymptotically large Q (Callan–Gross relation), also vanishes for low Q , and itsmoments go as: lim Q → Q − − n ´ d x x n f L ( x, Q ) = 0 .It is natural to consider the combination, ˜ f L ≡ f L + (2 M x/Q ) f = Q σ L ( ν, Q ) / π α , and define alongitudinal polarizability as: α L ( Q ) = 4 αMQ ˆ x d x ˜ f L ( x, Q ) = 12 π ˆ ∞ ν d ν σ L ( ν, Q ) Q ν . (5.36)At low Q this quantity is easily described in B χ PT, but not in HB χ PT, cf. [34, Fig. 3]. As all the quantitiesinvolving the longitudinal polarization, it is fairly insensitive to the ∆(1232) -resonance excitation. This definition differs from the original one [61] by a factor /Q , and as the result, α L (0) is not vanishing here. .5.2. Spin-Dependent Relations The sum rule derivation for the spin-dependent amplitudes proceeds in the same steps. As noted earlier,the DR for νS in the limit ν → leads to the BC sum rule [185], see Eq. (5.14). This sum rule implies thefollowing relation between the elastic and inelastic part of S : I ( Q ) ≡ M Q ˆ x d x g ( x, Q ) = 14 F ( Q ) G M ( Q ) . (5.37)We next consider the simultaneous expansion of the non-pole parts in ν and Q [59, 198]: π (cid:104) S − S pole1 (cid:105) ( ν, Q ) = α M κ (cid:20) − Q (cid:104) r (cid:105) (cid:21) + M γ ν + M Q (cid:110) γ E M − M α (cid:2) P (cid:48) ( M ,M (0) + P (cid:48) ( L ,L (0) (cid:3)(cid:111) + O ( ν , ν Q , Q ) , (5.38a) ν π (cid:104) S − S pole2 (cid:105) ( ν,
0) = − M ν (cid:110) γ + γ E E − M α (cid:2) P (cid:48) ( M ,M (0) − P (cid:48) ( L ,L (0) (cid:3)(cid:111) + O ( ν ) , (5.38b)where κ is the nucleon anomalous magnetic moment; (cid:104) r (cid:105) = − / κ d / d Q F ( Q ) | Q =0 is the mean-squarePauli radius; γ E M and γ E E are the spin polarizabilities; γ is the forward spin polarizability; and P s arethe generalized polarizabilities (GPs) coming from the VCS, see Eq. (5.46) below. In case of S , the firstterm originates from the difference between the Born and pole amplitudes, whereas polarizabilities affect thenon-Born part of the amplitudes only.On the rhs of the DRs for S and S we have an expansion in terms of moments of the spin structurefunctions g and g . The 0 th moment of g is related to the generalized GDH integrals: I ( Q ) = 2 M Q ˆ x d x g ( x, Q ) , (5.39a) I A ( Q ) = 2 M Q ˆ x d x g T T ( x, Q ) = M π α ˆ ∞ ν d νν σ T T ( ν, Q ) , (5.39b)with g T T = g − (4 M x /Q ) g . In the limit, Q → , ν → , they yield the GDH sum rule of (5.23): − κ = I (0) = I A (0) . (5.40)The 2 nd moments appear in the following generalization of the forward spin polarizabilities [61]: γ ( Q ) = 16 αM Q ˆ x d x x g T T ( x, Q ) = 12 π ˆ ∞ d νν σ T T ( ν, Q ) , (5.41) δ LT ( Q ) = 16 αM Q ˆ x d x x [ g + g ]( x, Q ) = 12 π ˆ ∞ d νν Q σ LT ( ν, Q ) , (5.42)which evidently satisfy the following relations at Q = 0 : γ = lim Q → αM Q ˆ x d x x g ( x, Q ) , (5.43) δ LT = γ + lim Q → αM Q ˆ x d x x g ( x, Q ) . (5.44)The first of these is simply the GTT sum rule given in Eq. (5.24). At large Q , where the Wandzura–Wilczekrelation [199] [quoted in Eq. (6.45) below] is applicable and the elastic contributions can be neglected, onecan show that [61]: δ LT ( Q ) = γ ( Q ) .From the Q term in the expansion of S , and the ν term in the expansion of S , one obtains the followingrelations involving the GPs [59]: αI (cid:48) (0) = α κ (cid:104) r (cid:105) + M γ E M − αM (cid:2) P (cid:48) ( M ,M (0) + P (cid:48) ( L ,L (0) (cid:3) , (5.45a) δ LT = − γ E E + 3 αM (cid:2) P (cid:48) ( M ,M (0) − P (cid:48) ( L ,L (0) (cid:3) . (5.45b)43he momentum derivatives of the GPs are given by: P (cid:48) ( M ,M (0) ± P (cid:48) ( L ,L (0) ≡ dd q (cid:104) P ( M ,M ( q ) ± P ( L ,L ( q ) (cid:105) q =0 , (5.46)with q being the initial photon c.m. three-momentum squared. The superscript indicates the multipolarities, L M denoting electric (magnetic) dipole transitions of the initial and final photons, and ‘ ’ implies thatthese transitions involve the spin-flip of the nucleon, cf. [66, 171]. An empirical implication of these relations,in the context of the so-called “ δ LT -puzzle", is briefly considered in Sect. 7.Another combination of the 2 nd moments of spin structure functions, i.e.: ¯ d ( Q ) = ˆ x d x x (cid:2) g ( x, Q ) + 2 g ( x, Q ) (cid:3) , (5.47)is of interest in connection to the concept of color polarizability [200]. In terms of the above-introducedquantities it reads: ¯ d ( Q ) = Q M (cid:26) M Q α δ LT ( Q ) + (cid:2) I ( Q ) − I A ( Q ) (cid:3)(cid:27) , (5.48)and goes as Q for low Q . Recall that the forward RCS is described by two scalar amplitudes, denoted here [and in Eq. (4.10a)] as: f ( ν ) ≡ T ( ν, π = √ s M (cid:0) Φ + Φ (cid:1)(cid:12)(cid:12) θ =0 , g ( ν ) ≡ νS ( ν, πM = √ s M (cid:0) Φ − Φ (cid:1)(cid:12)(cid:12) θ =0 , (5.49)Figure 5.3: Amplitude f ( ν ) for the proton obtained from Eq. (5.50a) using different fits of the total photoab-sorption cross section [17, 60, 201, 202] (fit I & II refer to the results of Ref. [60]). The experimental point isfrom DESY [181]. 44igure 5.4: Spin-dependent amplitude g ( ν ) obtained from Eq. (5.50b). The lower panel shows also the B χ PTpredictions for this amplitude [49, 54].where the helicity amplitudes Φ i are introduced in Sect. 3.2. The corresponding DRs, Eqs. (5.20) and (5.16b),read then as follows: f ( ν ) = − Z αM + ν π ˆ ∞ d ν (cid:48) σ T ( ν (cid:48) ) ν (cid:48) − ν − i + , (5.50a) g ( ν ) = ν π ˆ ∞ d ν (cid:48) ν (cid:48) σ T T ( ν (cid:48) ) ν (cid:48) − ν − i + . (5.50b)Therefore, given the total unpolarized cross section σ T and the helicity-difference cross-section σ T T , the for-ward CS can be completely determined. The cross sections for the proton are fairly well known. Their mostrecent fits and the evaluation of the integrals are performed by Gryniuk et al. [60]. The corresponding resultsfor the amplitudes are displayed in Fig. 5.3 and Fig. 5.4. The first figure shows also the results of previous eval-uations and an experimental point from the DESY 1973 experiment [181]. In the second figure the upper panelshows the fit to Im g together with the corresponding result for the real part. The lower panel shows a compar-ison of these results with a B χ PT calculation at lower energy. Given these amplitudes, one can determine thetwo non-vanishing (in the forward limit) observables: d σ d Ω L θ =0 = | f | + | g | , Σ z θ =0 = − f g ∗ + f ∗ g | f | + | g | . (5.51)The obtained Σ z [203], compared with the B χ PT predictions, demonstrates the importance of chiral dynamicsin this observable, cf. [54, Fig. 16].One can also evaluate the various sum rules presented in Sect. 5.4. Evaluations of the sum rules derivingfrom f ( ν ) (i.e., Baldin sum rule, etc.) are gathered in Table 5.1 for the proton and neutron, respectively. Theseresults are summarized and compared to the state-of-art χ PT results in Figures 7.1 and 7.2.Damashek and Gilman [201] initiated a study of the high-energy behavior of the amplitude f ( ν ) for theproton. In addition to the Regge prediction, they found a constant contribution comparable in sign and mag-nitude to the Thomson term: − α/M (cid:39) − . µ b GeV . This extra constant is assumed to correspond to a45xed J = 0 Regge pole ( α i ( t ) = 0 ) [204, 205], originating from local photon interactions with the constituentquarks. Based on newer photoabsorption data, Gorchtein et al. [206] have obtained a significantly smallervalue f ( ∞ ) = ( − . ± . µ b GeV .Recently, Gasser et al. [29] have made a sum rule determination of the proton-neutron difference (isovectorcombination) of the electric dipole polarizability: α ( p − n ) E = − . × − fm . (5.52)Their calculation is based on a Reggeon dominance assumption, which means there is no fixed pole in theisovector CS amplitude. It could be that the fixed pole, which is likely to be present for the proton, is canceledexactly by the one of the neutron. However, this is yet to be verified. The fact that the above value is inagreement with the empirical information on the isovector polarizability is certainly encouraging.Table 5.1: Empirical evaluation of spin-independent sum rules for the proton and neutron. Baldin SR th -order SR th -order SR Re f ( ν = 2 . )[ − fm ] [ − fm ] [ − fm ] [ µ b GeV ]Proton Neutron Proton Neutron Proton ProtonGryniuk et al. [60] . ± .
20 6 . ± .
03 4 . ± . − . Armstrong et al. [202] − . Damashek and Gilman [201] . ± . Schröder [207] . ± . . ± . . . Babusci et al. [208] . ± .
14 14 . ± . Levchuk and L’vov [209] . . . .
5) 15 . ± . Olmos de León et al. [17] . ± . − . MAID ( π channel) [210] .
63 13 . SAID ( π channel) [211] . . Alvensleben et al. [181] − . ± . Table 5.2: Empirical evaluation of the GDH and GTT sum rules for the proton and neutron.
GDH SR [ µ b] γ [10 − fm ] Proton Neutron Proton NeutronSum rule value ( lhs )
205 233
GDH-Coll. [212–215] ±
17 225 − . ± . Gryniuk et al. [203] . ± . − . ± . Pasquini et al. [170] ± − . ± . Babusci et al. [96] − . − . Schumacher and Levchuk [216] − . ± .
20 0 . ± . MAID ( π channel) [210] − . − . − . − . SAID ( π channel) [211] − − − . − . The evaluations of the GDH and GTT sum rules, deriving from g ( ν ) , are presented in Table 5.2 and Fig. 7.4.These results became largely possible due to the GDH-Collaboration data for the helicity-difference photoab-sorption cross section, in the region from . to . [212, 214, 217, 218].For the proton, the running GDH integral, I GDHrun ( ν max ) = ˆ ν max ν d νν (cid:2) σ / ( ν ) − σ / ( ν ) (cid:3) , (5.53)with ν being the lowest particle-production threshold, effectively set by the pion-production threshold ν π = m π + ( m π + Q ) / (2 M ) , evaluates to [212]: I GDHrun (2 . ± stat ± syst µ b , (5.54)46here the unmeasured low-energy region is covered by MAID [210] and SAID [211] analyses. The extrapolatedresult, I GDHrun ( ∞ ) = 212 ± stat ± syst µ b , (5.55)is in agreement with the GDH sum rule value (obtained by substituting the proton anomalous magneticmoment): µ b . The negative contribution to the integrand at higher energies is supported by a Reggeparametrization of the polarized data, as well as by fits of deep-inelastic scattering (DIS) data [219, 220].The neutron cross section is extracted from the difference of deuteron and proton cross sections, see Aren-hovel [221] for critical discussion. Presently, the GDH integral for the neutron is estimated to be µ b , whichcompares well to the sum rule value of µ b . Table 5.2 summarizes the GDH sum rule results for the protonand neutron.In future, one would like to measure the neutron cross sections based on He targets. Since the proton spinsare paired in the ground state, a polarized He target is a good alternative to the non-existent free neutrontarget. The neutron spin structure is quite similar to the one of He. Therefore, in contrast to deuterium,the magnetic moment of He is comparable to that of the neutron, with the GDH integral equal to: − µ b .Below the pion-production threshold, the dominant channels are the two- and three-body breakup reactions: (cid:126) He ( (cid:126)γ, n ) d and (cid:126) He ( (cid:126)γ, n ) pp. The latter has been experimentally accessed at HIGS [222]. At MAMI, thehelicity-dependent total inclusive He cross section is measured with circularly polarized photons in the energyrange: < ν <
500 MeV [223]. An estimate of the GDH sum rule for the neutron based on He experimentshas not yet been done. 47 . Proton Structure in (Muonic) Hydrogen
An exciting development in the field of nucleon structure has come recently from atomic physics. TheCREMA collaboration discovery of the P − S transitions in muonic hydrogen ( µ H) has led to a precisionmeasurement of the proton charge radius [24, 25]. The resulting value is an order of magnitude more precisethan that from hydrogen spectroscopy (H) or electron-proton ( ep ) scattering. It also turned out to be sub-stantially ( σ ) different from the CODATA value [224], which had been the standard value based on H and ep scattering. The latter discrepancy is known as the proton-radius puzzle (see Sect. 7.3 for more details andreferences).In this section we examine the proton structure effects in hydrogen-like atoms. While all the followingformulae are applicable to both H and µ H, the numerics will only be worked out for µ H. As far as protonstructure is concerned, all the effects are much more pronounced in µ H. The proton charge radius, forexample, is the second largest contribution to the µ H Lamb shift (after the vacuum polarization due to theelectron loop in QED), cf. Fig. 6.1.The proton structure effects are naturally divided into two categories:(i) Finite-size (or ‘elastic’) effects, i.e., the effect of the elastic FFs, G E and G M .(ii) Polarizability (or ‘inelastic’) effects, which basically is everything else. Assuming the proton e.m. structure is confined within a femtometer radius, the finite-size effects can beexpanded in the moments of charge and magnetization distributions, ρ E ( r ) and ρ M ( r ) , which are the Fouriertransforms of the elastic FFs G E and G M , respectively. To O ( α ) , the finite-size effects in the hydrogen Lambshift and hyperfine splitting (HFS) are found as (omitting recoil) [225]: E LS ≡ E (2 P / ) − E (2 S / ) = − Zα a (cid:2) R E − (2 a ) − R (cid:3) + O ( α ) , (6.1a) E HFS ( nS ) ≡ E ( nS F =11 / ) − E ( nS F =01 / ) = E F ( nS ) (cid:2) − a − R Z (cid:3) + O ( α ) , (6.1b)where a = 1 / ( Zαm r ) is the Bohr radius, Z is the nuclear charge ( Z = 1 for the proton), E F is the Fermi energyof the nS -level: E F ( nS ) = 8 Zα a κmM n , (6.2)and the radii are defined as follows (for other notations, see Sect. 1.1): (cid:3) Charge radius (shorthand for the root-mean-square (rms) radius of the charge distribution): R E = (cid:112) (cid:104) r (cid:105) E , (cid:104) r (cid:105) E ≡ ˆ d r r ρ E ( r ) = − Q G E ( Q ) (cid:12)(cid:12)(cid:12) Q =0 ; (6.3a) (cid:3) Friar radius (or, the 3 rd Zemach moment): R F = (cid:113) (cid:104) r (cid:105) E (2) , (cid:104) r (cid:105) E (2) ≡ π ˆ ∞ d QQ (cid:104) G E ( Q ) − R E Q (cid:105) ; (6.3b) (cid:3) Zemach radius: R Z ≡ − π ˆ ∞ d QQ (cid:20) G E ( Q ) G M ( Q )1 + κ − (cid:21) . (6.3c)A derivation of these formulae will be given in Sect. 6.2.The proton polarizability effects begin to contribute at order ( Zα ) m r . The usual way of calculating theseeffects is through the two-photon exchange (TPE) diagram, see Sect. 6.3. The elastic effects beyond the chargeradius (i.e., the contributions of Friar and Zemach radii), together with some recoil corrections, are sometimesreferred to as the ’elastic TPE’. Therefore the TPE effect is split into the ‘elastic’ and ‘polarizability’ contribution(see, e.g., Fig. 6.1). In layman’s terms, because the Bohr radius of µ H is about 200 times smaller than that of H, the muon comes much closer tothe proton, thus having a “better view” (or, more precisely, spending considerably more time “inside the proton”, thus “feeling” lessCoulomb attraction). In the literature polarizability is sometimes called polarization . We prefer to reserve the latter for the proton spin polarization. In exceptional cases, ‘inelastic’ may refer to only a part of the polarizability effect, as explained in Sect. 6.4.1. ddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd dddddddddddddddddddddddddddddddddddddddddddddd µ ddddddd µ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd dddddd ddddd dddd dd ddd dddd dddddddddd Figure 6.1: The budget of the µ H Lamb shift [226].The TPE is displayed in blue; we give estimates forthe elastic and polarizability contributions (unfilledbars), as well as for the total TPE contribution (solidbar). The proton radius discrepancy (shown in red)amounts to .
31 meV . The theoretical uncertainty isestimated as . , cf. Eq. (6.6a). A. Antognini et al. / Annals of Physics 331 (2013) 127–145
Fig. 1.
2S and 2P energy levels. The measured transitions ⌫ t [9] and ⌫ s [71] are indicated together with the Lamb shift, fineand hyperfine splittings, and finite-size effects. The main figure is drawn to scale. The inset zooms in on the 2P states. Here, themixing of the 2P ( F =
1) levels shifts them by ± (see Eq. (7)). • All corrections are mixed as ↵ x ( Z ↵ ) y ( m / M ) z r t E . • There are large finite-size and recoil ( m / M ⇡ /
9) corrections. • One cannot develop the calculation in a systematic way, such as in g • Widely different scales are involved: the masses, the three-momenta, and the kinetic energies ofthe constituents. • Different authors use different terminologies for identical terms. • Different methods are being used: Schrödinger equation + Breit corrections versus Dirac equation,Grotch- versus Breit-type recoil corrections, all-order versus perturbative in ( Z ↵ ) and finite-size,non-relativistic QED (NRQED), etc.In this study, we summarize all known terms included in the Lamb shift and the 2S-HFS predictionswhich are used in [71] to determine the proton charge radius and the Zemach radius. The majorityof these terms can be found in the works of Pachucki [10,11], Borie [12], and Martynenko [72,73].These earlier works have been reviewed in Eides et al. [74,75]. After the publication of [9], a number ofauthors have revisited the theory in muonic hydrogen, e.g. Jentschura [76,77], Karshenboim et al. [78],and Borie [79]. Note that the arXiv version 1103.1772v6 of Borie’s article [79] contains corrections tothe published version, which is why we refer to ‘‘Borie-v6’’ here. In addition, Indelicato [80] checkedand improved many of the relevant terms by performing numerical integration of the Dirac equationwith finite-size Coulomb and Uehling potentials. Carroll et al. started a similar effort [81].
2. The experimental Lamb shift and 2S-HFS
The 2S and 2P energy levels in muonic hydrogen are presented in Fig. 1. Because the measurementsof the Lamb shift involve only n = u r ⇡ . Thus, thevarious contributions to the Lamb shift should be calculated to better than ⇠ .
001 meV to be able toexploit the full experimental accuracy.
Figure 6.2: S and P energy levels in µ H. The twomeasured transition frequencies, ν t [24] and ν s [25],are indicated together with the Lamb shift, fine andhyperfine structure, and finite-size effects. The mainfigure is drawn to scale. The inset zooms in on the P states. Here, the mixing of the P F =1 -levels shiftsthem by ± δ . Plot courtesy of Aldo Antognini. Figure 6.2 shows the n = 2 energy-level scheme of µ H and the measured transitions, i.e.: hν t = E (2 P F =23 / ) − E (2 S F =11 / ) , (6.4a) hν s = E (2 P F =13 / ) − E (2 S F =01 / ) . (6.4b)The obtained experimental values for the Lamb shift and the HFS [24, 25, 226], E expLS = / hν s + / hν t − E FS (2 P ) − / E HFS (2 P / ) − / δ = 202 . , (6.5a) E expHFS (2 S ) = hν s − hν t + E HFS (2 P / ) − δ = 22 . , (6.5b)thus rely on the theoretical calculation of the fine and hyperfine splittings of the P -levels [227]: (cid:3) P fine structure splitting: E FS (2 P ) = 8 . , (cid:3) P / hyperfine structure splitting: E HFS (2 P / ) = 3 . , (cid:3) P F =1 level mixing: δ = 0 . .Furthermore, the extraction of charge and Zemach radii from µ H relies on the following theoretical descrip-tion of the ( P − S ) Lamb shift and the S HFS [226] (in units of meV ): E thLS = 206 . − . R E / fm) + E TPELS , with E TPELS = 0 . , (6.6a) E thHFS (2 S ) = 22 . − . R Z / fm) + E polHFS (2 S ) , with E polHFS (2 S ) = 0 . , (6.6b)where E TPELS contains the Friar radius, recoil finite-size effects, and the polarizability effects; E polHFS (2 S ) is theHFS polarizability effect only. The precise numerical values of these TPE effects will be considered in Sect. 6.4.For review of the QED effects we refer to Refs. [225, 228, 229].49itting the theory to experiment thus allows one to extract both the proton charge radius: R E = 0 . ,and the Zemach radius: R Z = 1 . . One caveat here, as pointed out by Karshenboim [232, 233], isthat this extraction relies on the Friar radius obtained from empirical FFs, which in turn have a different R E than extracted from the µ H Lamb shift. This issue will be discussed in Sect. 6.2.2.
Figure 6.3: One-photon exchangegraph with nucleonFFs, giving rise toFSE. The usual derivation of the finite-size effects (FSE) is done in terms of the chargeand magnetization densities (see, e.g., Friar [234]), which makes it difficult to derivethe relativistic corrections. We choose a different path [235] and derive the Breitpotential from the manifestly Lorentz-invariant expression for the Feynman diagram ofFig. 6.3, with the e.m. vertex of the nucleus given by: Γ µ = Zeγ µ F ( Q ) − Ze M γ µν q ν F ( Q ) . (6.7)The Dirac and Pauli FFs are then assumed to fulfill the once-subtracted DRs: (cid:32) F ( Q ) F ( Q ) (cid:33) = (cid:32) κ (cid:33) − Q π ˆ ∞ t d tt ( t + Q ) Im (cid:32) F ( t ) F ( t ) (cid:33) , (6.8)with t being the lowest particle-production threshold. The use of the DRs makes therest of the derivation to be very much analogous to Schwinger’s method of calculating the Uehling (vacuumpolarization) effect [236]. The Breit potential for the Uehling effect was considered in, e.g., Refs. [228, 237].At leading order (in /c ) we obtain the following terms for the Breit potential in momentum space: V e FF ( Q ) = 4 Zα ˆ ∞ t d tt Im G E ( t ) t + Q , (6.9a) V l =0 m FF ( Q ) = 4 πZα mM (cid:2) F ( F + 1) − (cid:3) (cid:26) κ − Q π ˆ ∞ t d tt Im G M ( t ) t + Q (cid:27) , (6.9b)where the magnetic (spin-dependent) part is only given for the S -states ( l = 0 ). The imaginary part (discon-tinuity along the branch cuts) of the electric and magnetic Sachs FFs is straightforwardly related to the one ofDirac and Pauli FFs: Im G E ( t ) = Im F ( t ) + t (2 M ) Im F ( t ) , (6.10a) Im G M ( t ) = Im F ( t ) + Im F ( t ) . (6.10b)The potential in coordinate space is obtained via Fourier transform, V ( r ) = 4 π (2 π ) r ˆ ∞ d Q Q V ( Q ) sin Qr, (6.11)with the following result: V e FF ( r ) = Zαπr ˆ ∞ t d tt e − r √ t Im G E ( t ) , (6.12a) V l =0 m FF ( r ) = 4 πZα mM (cid:2) F ( F + 1) − (cid:3) (1 + κ ) ρ M ( r ) , (6.12b) The first µ H measurement [24] had only determined ν t , and hence needed theory input for the S HFS too: E HFS (2 S ) =22 . [230] (using R Z = 1 .
022 fm [231]). Here we introduce the e.m. FFs of a spin-1/2 nucleus. For hydrogen they are of course identical to the proton FFs. The Comptonscattering formalism of the previous sections is applicable to spin-1/2 nuclei with
Z > , provided we replace the FFs as: F i → ZF i ,and set F (0) = 1 , F (0) = κ , rather than Z , κ of the nucleon case. Likewise, we replace the structure functions: f i → Z f i , g i → Z g i . ρ M (the Fourier transform of G M ) is a Laplace-type of transform of Im G M : ρ M ( r ) = 1(2 π ) r ˆ ∞ t d t Im G M ( t )1 + κ e − r √ t . (6.13)The latter definition shows explicitly that a spherically symmetric density is a Lorentz invariant quantity. Ananalogous definition, but in terms of Im G E , applies to the charge density ρ E ( r ) .The FF effect can now be worked out using time-independent perturbation theory. For example, the energyshift of the nl -level due to a spherically symmetric correction V δ ( r ) to the Coulomb potential V C ( r ) = − Zα/r is to 1 st order given by: E (cid:104) δ (cid:105) nl ≡ (cid:104) nlm | V δ | nlm (cid:105) = 12 π ˆ ∞ d Q Q w nl ( Q ) V δ ( Q ) = ˆ ∞ d r r R nl ( r ) V δ ( r ) , (6.14)where the momentum-space expression contains the convolution of the momentum-space wave functions, w nl ( Q ) = ˆ d p ϕ ∗ nlm ( p + Q ) ϕ nlm ( p ) , (6.15)while the coordinate-space one contains the radial wave functions R nl ( r ) . The explicit forms of the wavefunctions can be found in, e.g., Bethe and Salpeter [238]. For completeness we give here the expressions for S , S , and P states: R ( r ) = 2 a / e − r/a ,R ( r ) = 1 √ a / (cid:16) − r a (cid:17) e − r/ a ,R ( r ) = 1 √ a ) / ra e − r/ a , w S ( Q ) = 16 (cid:0) aQ ) (cid:1) ,w S ( Q ) = (cid:0) − ( aQ ) (cid:1)(cid:0) − aQ ) (cid:1)(cid:0) aQ ) (cid:1) ,w P ( Q ) = 1 − ( aQ ) (cid:0) aQ ) (cid:1) . (6.16)For the following discussions it is useful to note the asymptotic behavior of w for large Q : w nS ( Q ) Q →∞ = 16 πaQ φ n , (6.17)where φ n = 1 / √ πa n is the coordinate-space wave function at the origin, r = 0 . Consider first the correction due to the electric FF ( G E ), as given by Eq. (6.12a). At 1 st order, it yields thefollowing correction to the P − S Lamb shift: E (cid:104) e FF (cid:105) LS = − Zα πa ˆ ∞ t d t Im G E ( t )( √ t + Zαm r ) , (6.18a) = − πZα a ˆ ∞ d r r e − r/a ρ E ( r ) , (6.18b) = − Zα a ∞ (cid:88) k =0 ( − Zαm r ) k k ! (cid:104) r k +2 (cid:105) E = − ( Zα ) m r (cid:16) (cid:104) r (cid:105) E − Zαm r (cid:104) r (cid:105) E (cid:17) + . . . , (6.18c)where in the last steps we have expanded in the moments of the charge distribution: (cid:104) r N (cid:105) E ≡ π ˆ ∞ d r r N +2 ρ E ( r ) = ( N + 1)! π ˆ ∞ t d t Im G E ( t ) t N/ . (6.19)One should keep in mind though that the expansion in moments is not necessarily convergent. For instance,Eq. (6.18a) tells us that the expansion is applicable when the nearest particle-production threshold is wellabove the inverse Bohr radius, i.e.: Zαm r (cid:28) √ t . 51ncidentally, one of the early proposals for solving the proton-radius puzzle [239] does not work out pre-cisely because the expansion in moments is not applicable for the choice of ρ E proposed therein. The fine-tuningof ρ E affected mainly the region r > a , enhancing the Friar radius by almost a factor of 3, and thus achievinga huge impact on the Lamb shift, according to Eq. (6.1a). On the other hand, according to the exact formula(6.18b), the region above the Bohr radius makes a negligible impact on the Lamb shift, which was verifiedexplicitly by us [235] for the model of Ref. [239].It is also useful to have an expression in terms of G E itself: E (cid:104) e FF (cid:105) LS = − Zαπ ˆ ∞ d Q w P − S ( Q ) G E ( Q ) , with w P − S ( Q ) = 2( Zαm r ) Q ( Zαm r ) − Q [( Zαm r ) + Q ] . (6.20)Using the expression for the weighting function, w , in terms of the wave functions, cf. Eq. (6.15), it is easy tosee that for G E ( Q ) = const the effect vanishes. That is, the charge normalization drops out. Note also thatthis FF effect is still of O ( α ) , despite the O ( α ) overall prefactor. The naive expansion of w in α does not work,as the resulting integral is infrared divergent. As seen below more explicitly, this infrared enhancement yieldsin the end the correct charge radius contribution of O ( α ) .To complete the derivation of the standard formulae for the FSE in the Lamb shift to O ( α ) , we takethe potential to the 2 nd order in perturbation theory. The 2 nd -order contribution at O ( α ) comes from thecontinuum states only and amounts to: E (cid:104) e FF (cid:105)(cid:104) e FF (cid:105) LS = Zαa π ˆ ∞ d QQ (cid:104) G E ( Q ) − (cid:105) + O ( α ) = − Zα a (cid:104) (cid:104) r (cid:105) E − (cid:104) r (cid:105) E (2) (cid:105) + O ( α ) . (6.21)Adding this result to the O ( α ) term from the 1 st order, Eq. (6.18c), we can see that the 2 nd -order effect replacesthe 3 rd charge radius (cid:104) r (cid:105) E by the 3 rd Zemach moment (cid:104) r (cid:105) E (2) , resulting in Eq. (6.1a). The consistency problem, recently addressed by Karshenboim [232], is basically that in order to computethe Friar radius (6.3b), and hence its contribution to the Lamb shift (6.1a), one must know the charge radius R E ,which in turn needs to be extracted from the Lamb shift. Presently the µ H extraction of R E uses as input thevalue of R F obtained from the empirical FF, which has a different R E . This obviously is not consistent andleads to a systematic uncertainty.To see the origin of this problem, let us examine the exact (unexpanded in moments) FSE to 2 nd order inperturbation theory: E FSELS ≡ E (cid:104) e FF (cid:105) LS + E (cid:104) e FF (cid:105)(cid:104) e FF (cid:105) LS = − Zαπ ˆ ∞ d Q w P − S ( Q ) G E ( Q ) , (6.22)where the weighting function is given in Eq. (6.20). This form clearly shows that the Lamb shift is a functionalof the FF. Ideally, one needs to find G E which fits the ep and atomic data simultaneously. This, however, hasnot yet been realized.Figure 6.4: Poles of w l , and the con-tour in the complex Q plane. Let us evaluate the integral by the residue method. For this weconsider the complex Q plane, Fig. 6.4. The poles of w ( Q ) are at Q = ± iZαm r . (6.23)The singularities of G E ( Q ) also lie on the imaginary axis, since itobeys the DR of the type (6.8), see Eq. (C.6). The integrand is even in Q and hence we may extend the integration to negative Q . We thenclose the contour in the upper half-plane, use the residue theorem,and neglect the integral over the big semicircle to arrive at ˆ ∞ d Q w ( Q ) G E ( Q ) = πi Res (cid:2) w ( Q ) G E ( Q ) (cid:3) Q = iZαm r + . . . , (6.24)52here the dots stand for the residue of G E poles. Those can be computed using the DR or an explicit anzatz(e.g., an empirical parametrization) for G E . When using the DR of (C.6) one simply obtains the Friar radiuscontribution written in terms of Im G E , cf. Eq. (C.9).The residue over the pole of w evaluates to:Res [ w ( Q ) g ( Q )] Q = iZαm r = i ( Zαm r ) (cid:2) / g (cid:48) ( Q ) + Q g (cid:48)(cid:48) ( Q ) + / Q g (cid:48)(cid:48)(cid:48) ( Q ) (cid:3) Q = − ( Zαm r ) , (6.25)where we have introduced for a moment g ≡ G E , and the primes denote the derivatives over Q . Obviously,the first term dominates (lowest in α ) and yields the usual R E contribution of O (( Zα ) m r ) , cf. Eq. (6.1a).It is interesting to observe that Eq. (6.25) is only dependent on the derivatives of G E , and thus does notinvolve any of the odd moments, which are integrals of G E , see Eq. (C.8). The contribution of odd moments,and in particular the one of the Friar radius, comes from the singularities of G E . Thus, the consistency problemin question is absent if the rms charge radius and the poles of G E are uncorrelated . We, however, are not awareat the moment of an empirical parametrization in which the charge radius and the poles are not correlated.Quite the opposite, the correlation is usually strong. The simplest example is provided by the dipole form, G E ( Q ) = (1 + Q / Λ ) − . Both the radius and the pole positions are given by the mass parameter Λ : theradius is / Λ , while the pole is at Q = ± i Λ . An empirical parametrization with weak correlation betweenthe value of G (cid:48) E (0) and the position of its poles would be preferred from this point of view.General constraints on the FF parametrizations have recently been discussed at length by Sick et al. [240–243]. One finds in particular that certain parametrizations have unphysical poles which result in weird chargeor magnetization distributions. To take the advantage of studying the r -space simultaneously, it has beensuggested to parametrize the FF in a basis with analytic Fourier transform, e.g., with a sum of Gaussians. Itwould be interesting to see if these kind of parametrizations lead to weaker correlation between the rms radiusand the FF poles. The HFS, introduced in Eq. (6.1b), receives at first only the magnetic contribution: E (cid:104) m FF (cid:105) HFS ( nS ) = 4 Zα πmM ˆ ∞ d Q Q w nS ( Q ) G M ( Q ) , (6.26a) = 8 πZα mM (1 + κ ) ˆ ∞ d r r R n ( r ) ρ M ( r ) , (6.26b)where w nS is given by Eq. (6.15). Setting in this expression G M = 1 + κ , or equivalently ρ M ( r ) = δ ( r ) / πr ,yields the Fermi energy, E F ( nS ) , given in Eq. (6.2). Here, the expansion in α works straightforwardly (we canexpand under the integrals), and we obtain: E (cid:104) m FF (cid:105) HFS ( nS ) E F ( nS ) = 1 − a (cid:104) r (cid:105) M + O ( α ) , (6.27)where the first moment of ρ M ( r ) can equivalently be written as: (cid:104) r (cid:105) M = − π ˆ ∞ d QQ (cid:20) G M ( Q )1 + κ − (cid:21) . (6.28)At the 2 nd order in perturbation theory, we obtain the interference between the potentials of the electricand magnetic term: E (cid:104) e FF (cid:105)(cid:104) m FF (cid:105) HFS ( nS ) E F ( nS ) = 8 aπ ˆ ∞ d QQ (cid:2) G E ( Q ) − (cid:3) G M ( Q )1 + κ + O ( α ) . (6.29)Adding up the 1 st - and 2 nd -order contributions, we obtain the well-known FSE in the HFS given by Eq. (6.1b).Taking V l =0 m FF to 2 nd order gives rise to E (cid:104) m FF (cid:105)(cid:104) m FF (cid:105) , which is a higher-order recoil effect. This, and someother, recoil effects are treated more properly within the approach we consider next.53a) (b)Figure 6.5: (a) TPE diagram in forward kinematics: the horizontal lines correspond to the lepton and theproton (bold), where the ‘blob’ represents all possible excitations. (b) Elastic contribution to the TPE. Thecrossed diagrams are not drawn. Having obtained the standard FSE of (6.1) using the Breit potential, we consider here a different approach.We consider the correction, to the Coulomb potential, due to the TPE, see Fig. 6.5 (a). A great advantageof this approach is that one can access the inelastic effects of the proton structure [244, 245]. On the otherhand, it is less systematic and cannot be used without matching to a systematic expansion. We shall only useit to 1 st -order perturbation theory and match the elastic part, Fig. 6.5 (b), with the FSE derived from the Breitpotential.Let us note right away that the TPE contains an iteration of the Coulomb potential present in the wavefunctions. However, we need not to worry about the double-counting. The Coulomb interaction by itself isharmless, as it gives no contribution to the Lamb shift or HFS.To O ( α ) it will be sufficient to evaluate the TPE amplitude at zero energy ( p · (cid:96) = mM ) and momentumtransfer ( t = 0 ). The resulting amplitude yields a constant potential in momentum space, which of coursetranslates to a δ -function potential in coordinate space. The energy shift is thus proportional to the wave-function at the origin, and hence only the S -levels are affected.The forward TPE amplitude is a one-loop integral involving the doubly-virtual Compton scattering (VVCS)amplitude. The latter has been discussed in Sect. 5. According to Eq. (5.1), its tensor structure decomposes intoa symmetric, spin-independent part parametrized by the scalar amplitudes T , ( ν, Q ) , and an antisymmetric,spin-dependent part parametrized by S , ( ν, Q ) . The HFS obviously depends on the latter, while the Lambshift depends on the former.More specifically, the shift of the nS -level is given by: ∆ E ( nS ) = 8 παm φ n i ˆ ∞−∞ d ν π ˆ d q (2 π ) (cid:0) Q − ν (cid:1) T ( ν, Q ) − ( Q + ν ) T ( ν, Q ) Q ( Q − m ν ) , (6.30)with φ n = 1 / ( πn a ) the wave function squared at the origin, and ν = q , Q = q − q . The correction to theHFS is given by: E HFS ( nS ) E F ( nS ) = 4 mµ i ˆ ∞−∞ d ν π ˆ d q (2 π ) Q − m ν (cid:40) (cid:0) Q − ν (cid:1) Q S ( ν, Q ) + 3 νM S ( ν, Q ) (cid:41) . (6.31)To proceed further, one performs a Wick rotation, i.e. changes the integration over q to Q = − iq . Notethat this is only possible at vanishing external energy (threshold) of lepton-proton scattering. At finite energyone needs to take care of the poles moving across the imaginary q axis (see, e.g., Ref. [246]).After the Wick rotation, the integration four-momentum is Euclidean and we can evaluate it in hyperspher-ical coordinates, Q µ = Q (cos χ, sin χ sin θ cos ϕ, sin χ sin θ sin ϕ, sin χ cos θ ) . The integrals over θ and ϕ can be done right away, yielding a factor of π . The integral over ν = iQ cos χ can be done after substituting the DRs for the VVCS amplitudes, Eq. (5.12). Introducing the “lepton velocity”54 l = (cid:112) m /Q , we obtain the following expression for the S -level shift: ∆ E ( nS ) = 16( Zα ) mM φ n ˆ ∞ d QQ ˆ d x v l + √ x τ − × (cid:40) f ( x, Q ) x − f ( x, Q )2 τ + 1(1 + v l )(1 + √ x τ − ) (cid:18) f ( x, Q ) x + f ( x, Q )2 τ (cid:19)(cid:41) , (6.32)and the HFS: E HFS ( nS ) E F ( nS ) = 16 ZαmMπ (1 + κ ) ˆ ∞ d QQ ˆ d x v l + √ x τ − × (cid:26)(cid:20) v l + 1)(1 + √ x τ − ) (cid:21) g ( x, Q ) + 3 g ( x, Q ) (cid:27) . (6.33)These are the master formulae containing all the structure effects to O ( α ) . One ought to be careful thoughin matching the contribution of the elastic structure functions (5.7) to the standard FSE of (6.1). In the non-relativistic (heavy-mass) limit we obtain: ∆ E el ( nS ) = − Zα ) m r φ n ˆ ∞ d QQ G E ( Q ) , (6.34a) E elHFS ( nS ) = 64( Zα ) m r mM φ n ˆ ∞ d QQ G M ( Q ) G E ( Q ) . (6.34b)The correct matching is achieved by regularizing the infrared divergencies with the convoluted wave functions,i.e., w ( Q ) in Eq. (6.15). For example, to obtain the charge radius contribution to the P − S Lamb shift oneshould replace φ n in Eq. (6.34a) with: − aw P − S ( Q ) Q / π . For the HFS, the replacement in Eq. (6.34b) is: φ n → aw nS ( Q ) Q / π , yielding the correct Fermi energy and Zemach radius contributions. The infrared-safecontributions, such as the recoil and polarizability corrections, need no regularization. In what follows we onlyconsider those kind of effects. The O ( α ) effects of proton structure in the Lamb shift are usually divided into the effect of (i) the Friarradius, (ii) finite-size recoil, and (iii) polarizabilities. The first two are sometimes combined into (i’) the ‘elastic’TPE contribution. The ‘polarizability’ effect is often split between (ii’) the ‘inelastic’ TPE and (iii’) a ‘subtraction’term, i.e., the contribution of T (0 , Q ) .The elastic and inelastic TPE contributions are well-constrained by the available empirical information on,respectively, the proton FFs and unpolarized structure functions, whereas the subtraction contribution must bemodeled. It certainly helps to know that lim Q → T (0 , Q ) /Q = 4 πβ M , (6.35)but otherwise, the Q behavior of this amplitude leaves room to imagination. For example, Pachucki [247] andlater Martynenko [248] use: T (0 , Q ) = 4 πβ M Q / (cid:0) Q / Λ (cid:1) , (6.36)with Λ = 0 .
71 GeV , whereas Carlson and Vanderhaeghen [249] and Birse and McGovern [250] use moresophisticated forms, inspired by chiral loops. The leading-order [ O ( p ) ] χ PT calculation contains a genuineprediction for the subtraction function, as well as for the whole polarizability effect, see Sect. 6.5 for moredetails. For the expansion we use, / ( v l + v ) (cid:39) Q m + M ) (cid:0) − Q / mM (cid:1) , [2( v l + 1)( v + 1)] − (cid:39) Q / mM , where v = √ τ − . S -level in µ H. Energyshifts are given in µ eV , β M is given in − fm . Pachucki Martynenko Carlson & Birse & Gorchtein et al. [254] a [247] [248] Vanderhaeghen [249] b McGovern [250] β M . [257] . [258] . . [147, 148] . [65] ∆ E subt (2 S ) 1 . . . .
9) 4 . . − . . E inel (2 S ) − . [259, 260] − . − . [261, 262] − . c − . [261–263] ∆ E pol (2 S ) − − . . − . . − . . − . . E el (2 S ) − . . [264] − . [265] − . ( . ) [266] − . [267, 268] − . . d − . . [265–267] ∆ E (2 S ) − . . − . . − − . . a Adjusted values; the original values of Ref. [254], ∆ E subt (2 S ) = 3 . and ∆ E el (2 S ) = − . , are based on a different decomposi-tion into the elastic and polarizability contributions. b In this work a separation of the amplitude into ‘pole’ and ‘non-pole’, rather than ‘Born’ and ‘non-Born’, was chosen. It is pointed outin Ref. [250] that the ‘pole’ decomposition applied in [249] is inconsistent with the standard definition of the magnetic polarizabilityused ibidem. c Value taken from Ref. [249]. d Result taken from Ref. [249] (FF [266]) with reinstated ‘non-pole’ Born piece.
An early study of the electric polarizability effect on the S -level shift in electronic and muonic atoms canbe found in Ref. [251]. That work exploited an unsubtracted DR for the longitudinal amplitude T L ( ν, Q ) =(1 + ν /Q ) T ( ν, Q ) − T ( ν, Q ) , as introduced in Ref. [252]. As we have discussed in the previous section,such a DR is not valid for the proton.A first standard dispersive calculation of the TPE effect was done by Pachucki [247], see also [228, 253].The most recent updates can be found in Refs. [249, 250]. Presently, the recommended value is that of Birseand McGovern [250]. A somewhat different dispersive evaluation has recently been done by Gorchtein et al.[254]. There, the high-energy behavior of the subtraction function is related to the fixed J = 0 Regge pole[204] through a finite-energy sum rule (see Eq. (29) in Ref. [254]).Table 6.1 summarizes the dispersive evaluations of the TPE effects in the µ H Lamb shift, while the χ PTpredictions can be found in Table 6.3. The corresponding ‘polarizability’ and ‘elastic’ TPE results are representedin the summary plots, see Fig. 7.9 and Fig. 7.10. Table 6.1 also shows the value of the magnetic polarizabilityused in the evaluations, since this is the main source of discrepancy among them.Other frameworks, different from DR and χ PT, for calculating the TPE effects in the Lamb shift can be foundin [255] and [256]. The values obtained in these works are generally in agreement with the dispersive results.For example, Mohr et al. [256] quote: ∆ E inel (2 S ) = − µ eV , ∆ E el (2 S ) = − µ eV . (6.37) The leading-order HFS is given by the Fermi energy of the nS -level, cf. Eq. (6.2). The full HFS is dividedinto the following contributions: E HFS ( nS ) = [1 + ∆ QED + ∆ weak + ∆ structure ] E F ( nS ) . (6.38)We are interested in the proton-structure correction, which is split into three terms: Zemach radius, recoil, andpolarizability contribution, ∆ structure = ∆ Z + ∆ recoil + ∆ pol . (6.39) A review of polarizability corrections to the hydrogen HFS can be found in Ref. [269, Sect. 3]. A detailed formalism of thestructure-dependent corrections to the S HFS in both H and µ H is given in Ref. [270], with comments on various conventions. S and S , which satisfy the DRs of Eqs. (5.12c) and (5.15). This means the entire TPE contribution is givenby the spin structure functions g and g . To separate out the polarizability contribution, one can write the DRsfor the non-Born (polarizability) part of the amplitudes only: S ( ν, Q ) = 2 πZ αM F ( Q ) + 16 πZ αMQ ˆ x d x g ( x, Q )1 − x ( ν/ν el ) , (6.40a) νS ( ν, Q ) = 64 πZ αM ν Q ˆ x d x x g ( x, Q )1 − x ( ν/ν el ) . (6.40b)One should not be perplexed by the Pauli FF term, F , appearing in S . Its purpose is to cancel the elasticcontribution of the GDH integral I ( Q ) , such that S is indeed proportional to polarizabilities alone.Let us now specify the decomposition of the structure-dependent correction into the three terms of Eq. (6.39).The first one is the Zemach contribution [274]: ∆ Z = 8 Zαm r π ˆ ∞ d QQ (cid:20) G E ( Q ) G M ( Q )1 + κ − (cid:21) ≡ − Zαm r R Z . (6.41)The second one is the remaining elastic TPE contribution, which is a recoil-type of correction to the Zemachterm: ∆ recoil = Zαπ (1 + κ ) ˆ ∞ d QQ (cid:26) G M ( Q ) Q mMv l + v (cid:18) F ( Q ) + F ( Q ) + 3 F ( Q )( v l + 1)( v + 1) (cid:19) − m r G M ( Q ) G E ( Q ) Q − m F ( Q ) M v l (1 + v l ) (cid:27) (6.42) ≈ Zαm r π (1 + κ ) mM ˆ ∞ d Q (cid:20)(cid:16) mM (cid:17) F ( Q ) + F ( Q ) (cid:21) F ( Q ) , with v = √ τ − . Finally, the polarizability contribution is written as: ∆ pol = Zαm π (1 + κ ) M [ δ + δ ] = ∆ + ∆ , (6.43a)with the separate contributions due to g and g given by: δ = 2 ˆ ∞ d QQ (cid:18) v l ( v l + 1) (cid:2) I ( Q ) + F ( Q ) (cid:3) + 8 M Q ˆ x d x g ( x, Q ) × (cid:26) v l + √ x τ − (cid:20) v l + 1)(1 + √ x τ − ) (cid:21) − v l ( v l + 1) (cid:27)(cid:19) , (6.43b) δ = 96 M ˆ ∞ d QQ ˆ x d x g ( x, Q ) (cid:26) v l + √ x τ − − v l + 1 (cid:27) . (6.43c)As emphasized before, our decomposition into ∆ recoil and ∆ pol corresponds with the decomposition intothe Born and non-Born part. In this way, the decomposition is consistent with Pachucki [228], Carlson et al.[270], Bodwin and Yennie [275] and different from Martynenko [230], Faustov et al. [231]. In the latterworks, the F term was shared differently between the elastic and polarizability contributions. The conversionbetween the two decompositions can be found in Ref. [270].Let us now consider the numerical results. Early works mainly studied the proton structure corrections tothe ground-state HFS in H [244, 245, 279, 280]. More recent evaluations of the polarizability contribution to The validity of the unsubtracted DRs is based on Regge theory [271], see also Refs. [272, 273] for a discussion of the no-subtractionassumption. S HFS of µ H. Reference R Z [ fm ] ∆ Z [ ppm ] ∆ recoil [ ppm ] ∆ pol [ ppm ] ∆ [ ppm ] ∆ [ ppm ] ∆ structure [ ppm ] E S HFS [ meV ]Carlson et al. [276] a . − − − . [270]Faustov et al. [277] − Martynenko et al. [231] . − −
58 22 . Experiment [25] . b . a QED and structure-independent corrections are taken from Martynenko [230]. The Zemach term includes radiative corrections: ∆ Z = − αm r R Z (1 + δ rad Z ) , with δ rad Z of Refs. [275, 278]. b Extraction based on the recoil and polarizability corrections from Ref. [276] (1st row of the Table). the H HFS can be found in Refs. [272, 273, 281–283], radiative corrections are calculated in Ref. [278]. Themost recent calculations of the polarizability contribution to the HFS in H are:Carlson et al. [276] : ∆
Hpol = 1 . ± .
64 ppm , (6.44a)Faustov et al. [277] : ∆ Hpol = 2 . ± . . (6.44b)The available dispersive calculations for the S TPE correction to the HFS in µ H are listed in Table 6.2 and,in a more illustrative form, Fig. 7.11. Some of the results are given in terms of the S Fermi energy in µ H: E F (2 S ) = 22 . meV.Most of the calculations show a relatively small effect from g , see ∆ in Table 6.2. It seems to be wellwithin the uncertainty of the g contribution (∆ ) . However, it is important to note here that the spin structurefunction g of the proton has not been measured experimentally in the low- Q region, relevant to the atomiccalculations. The above evaluations are either modeling g , or make use of the Wandzura-Wilczek relation[199] to express it in terms of g : g WW2 ( x, Q ) = − g ( x, Q ) + ˆ x d x (cid:48) x (cid:48) g ( x (cid:48) , Q ) . (6.45)The latter relation is for asymptotically large Q . It is certainly violated for low Q — it’s only a question of howbadly. The ongoing JLab measurement of proton g [284] is extremely important for answering that question.Information on the structure function g is available for momentum-transfers larger than Q . ∼ .
05 GeV [285]. Below this threshold, the Q -integrand of Eq. (6.43b) is interpolated by exploiting the sum rules. In thecase of H, where the electron mass can safely be neglected, the slope of the integrand is fixed by the GDH sumrule (5.23). In µ H the dependence on the muon mass is not negligible, and the GTT sum rule (5.24) proves tobe useful, cf. Carlson et al. [276].
Below O ( p ) , χ PT provides a genuine prediction for the TPE effects. At O ( p ) there is a number oflow-energy constants (LECs), entering through the effective lepton-lepton-nucleon-nucleon ( (cid:96)(cid:96)N N ) coupling,whose values are presently unknown. Therefore, the predictive power is lost at this order. Here we only con-sider the "predictive orders", i.e., O ( p ) and O ( p / ) . These will be called the leading (LO) and next-to-leading(NLO) order, respectively. The leading-order [ O ( p ) ] calculations of the µ H Lamb shift have been done in both heavy-baryon (HB χ PT)and baryon (B χ PT) frameworks [55, 286]. The diagrams arising in these calculations are shown in Fig. 6.6. This relation automatically satisfies the BC sum rule, i.e., ´ d x g WW2 ( x, Q ) = 0 , as easily seen via the Fubini rule. Technically, the leading order is O ( p ) , but it is included in the Coulomb interaction. ur. Phys. J. C (2014) 74:2852 Page 3 of 10 The two-photonexchange diagrams of elasticlepton–nucleon scatteringcalculated in this work in thezero-energy (threshold)kinematics. Diagrams obtainedfrom these by crossing andtime-reversal symmetry areincluded but not drawn (b) (c)(a)(d) (e) (f)(g) (h) (j) of two scalar amplitudes: T µ ν ( P , q ) = − g µ ν T ( ν , Q ) + P µ P ν M p T ( ν , Q ), (5)with P the proton 4-momentum, ν = P · q / M p , Q = − q , P = M p . Note that the scalar amplitudes T , are evenfunctions of both the photon energy ν and the virtuality Q .Terms proportional to q µ or q ν are omitted because theyvanish upon contraction with the lepton tensor.Going back to the energy shift one obtains [12]: " E nS = α em φ n π m ℓ i ! d q ∞ ! d ν × ( Q − ν ) T ( ν , Q ) − ( Q + ν ) T ( ν , Q ) Q [ ( Q / m ℓ ) − ν ] . (6)In this work we calculate the functions T and T byextending the B χ PT calculation of real Compton scatter-ing [26] to the case of virtual photons. We then split theamplitudes into the Born (B) and non-Born (NB) pieces: T i = T ( B ) i + T ( NB ) i . (7)The Born part is defined in terms of the elastic nucleon formfactors as in, e.g. [13,27]: T ( B ) = πα em M p " Q ( F D ( Q ) + F P ( Q )) Q − M p ν − F D ( Q ) , (8a) T ( B ) = πα em M p Q Q − M p ν " F D ( Q ) + Q M p F P ( Q ) . (8b)In our calculation the Born part was separated by subtract-ing the on-shell γ N N pion loop vertex in the one-particle-reducible VVCS graphs; see diagrams (b) and (c) in Fig. 1. Focusing on the O ( p ) corrections (i.e., the VVCS amplitudecorresponding to the graphs in Fig. 1) we have explicitly ver-ified that the resulting NB amplitudes satisfy the dispersivesum rules [28]: T ( NB ) ( ν , Q ) = T ( NB ) ( , Q ) + ν π ∞ ! ν d ν ′ σ T ( ν ′ , Q ) ν ′ − ν , (9a) T ( NB ) ( ν , Q ) = π ∞ ! ν d ν ′ ν ′ Q ν ′ + Q σ T ( ν ′ , Q ) + σ L ( ν ′ , Q ) ν ′ − ν , (9b)with ν = m π + ( m π + Q )/( M p ) the pion-productionthreshold, m π the pion mass, and σ T ( L ) the tree-level crosssection of pion production off the proton induced by trans-verse (longitudinal) virtual photons, cf. Appendix B. Wehence establish that one is to calculate the ‘elastic’ con-tribution from the Born part of the VVCS amplitudes andthe ‘polarizability’ contribution from the non-Born part,in accordance with the procedure advocated by Birse andMcGovern [13].Substituting the O ( p ) NB amplitudes into Eq. (6) weobtain the following value for the polarizability correction: " E ( pol ) S = − . µ eV . (10)This is quite different from the corresponding HB χ PT resultfor this effect obtained by Nevado and Pineda [11]: " E ( pol ) S ( LO-HB χ PT ) = − . µ eV . (11)We postpone a detailed discussion of this difference tillSect. 4. = Figure 6.6: The TPE diagrams of elastic lepton-nucleon scattering to O ( p ) in χ PT. Diagrams obtained fromthese by crossing and time-reversal symmetry are not drawn. ∆ Figure 6.7: TPE diagram with ∆(1232) , represented by the double line.The LO HB χ PT result for the polarizability contribution to the S -level shift is well described by the followingsimple formula [55]: ∆ E polHB χ PT (2 S ) = α m r g A πf π ) m µ m π (cid:16) − G + 6 ln 2 (cid:17) (cid:39) − . µ eV , (6.46)where G (cid:39) . is the Catalan constant; other parameters are defined in Sect. 1.1. The LO B χ PT resultis somewhat smaller in magnitude, see Table 6.3. This is mainly because of the smaller value of the protonelectric polarizability α E arising in B χ PT at leading order, cf. Sect. 2.4. The π ∆ loops at O ( p / ) are expectedto correct this situation.At next-to-leading order [ O ( p / ) ], the diagrams with the ∆(1232) -isobar arise, of which the one in Fig. 6.7dominates the magnetic polarizability of the proton. In the Lamb shift, however, the magnetic polarizability issuppressed, and this particular diagram is suppressed too [55].The calculations including the ∆ have thus far been done in HB χ PT only [56]. The resulting NLO polar-izability contribution is larger in magnitude than the LO, see Table 6.3 and Fig. 7.9. This is simply because inHB χ PT the ∆ gives too large of a correction to the polarizabilities, cf. Eq. (2.8). On the other hand, the HB χ PTunderpredicts the elastic TPE contribution, see Table 6.3 and Fig. 7.10, because the Friar radius comes out tobe smaller than the empirical value. The total value for the TPE effect in HB χ PT happens to be in agreementwith the empirical expectations.Table 6.3: Summary of available χ PT calculations for the TPE effect in the S -level shift of µ H (in µ eV ). Nevado & Pineda Alarcón et al. Peset & PinedaLO HB χ PT [286] LO B χ PT [55] NLO HB χ PT [287] ∆ E pol (2 S ) − . . − . +1 . − . ) − . . E el (2 S ) − . . − . . The LO B χ PT calculation of the HFS should in addition to the diagrams in Fig. 6.6 include the neutral-pion exchange, Fig. 6.8. The latter effect, however, turned out to be consistent with 0, at least for the µ H 2S59 N π Figure 6.8: Pion-exchange in hydrogen.HFS [288]: E (cid:104) π (cid:105) HFS (2 S ) = 0 . ± . µ eV , (6.47)where the uncertainty comes from the experimental error of the π → e + e − decay width. In retrospect this isnot so surprising, since the pion-exchange vanishes in the forward kinematics, and as such becomes suppressedby an additional α .A substantially larger pion-exchange effect has recently been found in Refs. [289, 290]. The calculation ofZhou and Pang [289] suffers from a technical mistake, as pointed out in [290]. On the other hand, Huonget al. [290] neglect the Q -dependence of the pion coupling to leptons which is not a good approximation forthe reason explained below. The non-relativistic limit of the pion-exchange potential reads (for the S -waves): V l =0 π ( Q ) = − F ( F + 1) − / mM Q Q + m π g πNN F π(cid:96)(cid:96) ( Q ) , (6.48)where F = 0 or 1 is the eigenvalue of the total angular momentum, m π is the neutral-pion mass, g πNN is thepion-nucleon coupling constant, and F π(cid:96)(cid:96) is the FF describing the pion coupling to leptons. The latter satisfiesthe well-known (once-subtracted) DR [291]: F π(cid:96)(cid:96) ( Q ) = F π(cid:96)(cid:96) (0) − Q π ˆ ∞ d ss Im F π(cid:96)(cid:96) ( s ) s + Q , with Im F π(cid:96)(cid:96) ( s ) = − α m arccosh( √ s/ m )2 πf π (cid:112) − m /s , (6.49)where f π is the pion-decay constant, m is the lepton mass. This decomposition into the subtraction constantand the effect of the γ loop is illustrated in Fig. 6.8. The subtraction constant can be extracted from theexperimental value of the π → e + e − decay width, which in terms of the FF is given by: Γ( π → e + e − ) = m π π (cid:115) − m e m π (cid:12)(cid:12) F πee ( − m π ) (cid:12)(cid:12) . (6.50)Now, the point is that the FF in Eq. (6.49) does not admit a good Taylor expansion around Q = 0 , becauseof the branch cut starting at 0. Hence, in contrast to the πN N FF, we cannot neglect its Q -dependence. Astraightforward calculation yields the following result for the HFS effect: E (cid:104) π (cid:105) HFS ( nS ) = − E F ( nS ) g πNN m r π (1 + κ ) m π (cid:20) F (0) + α m π f π I (cid:0) m π m (cid:1)(cid:21) , (6.51)where we introduce the following integral, I ( γ ) ≡ ˆ ∞ d ξ ξ/γ ) arccos ξ (cid:112) − ξ . (6.52)For H, γ (cid:29) , and one can make use of the expansion: I ( γ ) = 7 π /
12 + ln (2 γ ) + O (1 /γ ) . For the more generalsituation, γ = sin θ ≥ , we have: I (sin θ ) = tan θ [ Cl (2 θ ) − π ln tan( θ/ , (6.53) Note added in proof: The Q -dependence is taken into account in the revised version of Ref. [290]. Their revised value is inagreement with Eq. (6.47). ( θ ) = − ´ θ d t ln (cid:0) t / (cid:1) = i (cid:2) Li (cid:0) e − iθ (cid:1) − Li (cid:0) e iθ (cid:1)(cid:3) is the the Clausen integral; Li ( x ) is the Eu-ler dilogarithm. The numerical values for the electron and muon, respectively, are: I ( m π / m e ) (cid:39) . , I ( m π / m µ ) (cid:39) . .We find that in H and µ H alike, there is a large cancellation between the two terms in Eq. (6.51), or equiv-alently between the two diagrams in Fig. 6.8. The resulting µ H value is the one quoted above, in Eq. (6.47).A preliminary calculation [288] shows that in total the LO B χ PT effects amount to the following polariz-ability contribution to the S HFS of µ H: O ( p ) : E polHFS (2 S ) = 0 . ± . µ eV . (6.54)This is about an order of magnitude smaller than the effect obtained in the empirical dispersive calculations,cf. Table 6.2. However, it can be expected that the ∆ -excitation mechanism of Fig. 6.7 can play an importantrole here. It remains to be seen whether this NLO effect restores the agreement between the χ PT and dispersiveresults. 61 . Summary Plots and Conclusions
To summarize and conclude we have compiled the following summary plots surveying the recent results fornucleon polarizabilities and for their contribution to the S -levels of muonic hydrogen. Figures 7.1 and 7.2 present the situation for α E + β M and β M of the proton and neutron, respectively. Inthe top of the left panels we have the results of the Baldin sum-rule evaluations considered in Table 5.1. Theorange band indicates the weighted-average of these evaluations. ddd ddd ddd ddd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d dddd ddd ddd ddd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d dddd ddd ddd ddd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d dddd ddd ddd ddd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d dddd ddd ddd ddd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d d dd dd dd dd ddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd ddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd ddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd ddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd ddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d d Figure 7.1: Left panel: sum of the electric and magnetic dipole polarizabilities of the proton. Right panel: themagnetic dipole polarizability of the proton. The orange band is the weighted average over the Baldin sumrule evaluations listed in Table 5.1. The DR prediction for β ( p ) M can be found in the review of Schumacher[62]. “Lensky-Pascalutsa ’15” refers to Ref. [34, 125], whereas “Lensky et al. ’15” refers to Ref. [54]. All otherreferences and declarations are given in the text. ddd ddd ddd ddd ddd ddd ddd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d d ddd ddd ddd ddd ddd ddd ddd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d d ddd ddd ddd ddd ddd ddd ddd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d d ddd ddd ddd ddd ddd ddd ddd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddd α ddddd ddd β ddddd dddd dd ddd d d dd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd β ddddd dddd dd ddd d d Figure 7.2: Left panel: sum of the electric and magnetic dipole polarizabilities of the neutron. Right panel:the magnetic dipole polarizability of the neutron. The orange band is the weighted average over the Baldinsum rule evaluations listed in Table 5.1. The experimental results for β ( n ) M are from Refs. [292, 293] and [62].Other references are given in the text.It appears that there is a substantial tension in the value of the proton magnetic polarizability, cf. the rightpanel of Fig. 7.1. An emerging objective in this area is to reduce the uncertainty on β ( p ) M by approximately ����������� �� �� ��� ��� ��� ��� � � ��� � ����� � � � � � ���� ���� � ��� � � � � � � ��� � � � � � � ��� �� � � �� � � � � ����� ����� �� ��� � ����� � � � � � � � � ������������ �� �� ��� ��� ��� ��� � � � � � � �� �� � � �� � � � � ����� ���� �� ��� � � � � � � � � � � � � �� � � � ����� ������������������� �� �� ��� ��� ��� ��� � � � � � � �� �� � � �� � � � � ����� ���� �� ��� � � � � � � � � � � � � �� � � � ����� ������������������� �� �� ��� ��� ��� ��� � � � � � � �� �� � � �� � � � � ����� ���� �� ��� � � � � � � � � � � � � �� � � � ����� ������������������� �� �� ��� ��� ��� ��� � � � � � � �� �� � � �� � � � � ����� ���� �� ��� � � � � � � � � � � � � �� � � � ����� ������������������� �� �� ��� ��� ��� ��� � � � � � � �� �� � � �� � � � � ����� ���� �� ��� � � � � � � � � � � � � �� � � � ����� ������� Figure 7.3: Plot of α E versus β M for the proton (left panel) and neutron (right panel), respectively. Theorange band is the average over the Baldin sum rule evaluations listed in Table 5.1. References and declarationsare given in the text, cf. also Fig. 7.1 and 7.2.through a measurement technique that is ideally independent of the Baldin sum rule. The utilization of photonbeams with high intensity and high linear polarization will be a key part of these investigations. Exploratorymeasurements are currently underway at HIGS and Mainz.In the area of the neutron scalar polarizabilities, the recent Lund publication [15] of elastic CS on thedeuteron is an important milestone. For the first time, relatively high statistics and wide kinematic coverageelastic data are available, and the data are analyzable with state-of-the-art effective-field theory calculations.With the unfortunate discontinuation of the CS program at Lund, the focus will now likely shift to other labsand different nuclear targets. At Mainz an experiment to measure elastic CS on He is in preparation.Another graphical representation of the experimental and theoretical results for the dipole polarizabilities, α E and β M , is shown in Fig. 7.3. The orange band again represents the constraint by the Baldin sum rule. Thelight green bands show experimental constraints on the difference of dipole polarizabilities, i.e., α E − β M , cf.Kossert et al. [292, 293] and Zieger et al. [16]. For the proton, other experimental constraints are shown byblack lines: Federspiel et al. [10], MacGibbon et al. [12] and TAPS [17]. The B χ PT constraint is from Ref. [54].The HB χ PT constraint is from Ref. [51], in case of the proton, and [65], in case of the neutron. Obviously theknowledge of the neutron polarizabilities is less precise than for the proton. This is mainly due to the lack offree neutron targets.Note that in these plots we have used the most recent PDG values [71]: α ( p ) E = (11 . ± . × − fm , β ( p ) M = (2 . ∓ . × − fm , (7.1a) α ( n ) E = (11 . ± . × − fm , β ( n ) M = (3 . ∓ . × − fm . (7.1b)They differ for the proton from the 2012 and earlier editions by inclusion of the global data fit analysis [51]. Concerning VCS, what has emerged from the low- Q studies, see Fig. 4.16, is interesting and provocative;there may well be a non-dipole-like structure in α E ( Q ) at Q ≈ .
33 GeV . If correct, this would indicate thatthe proton has a pronounced structure in its induced polarization at large transverse distances, . to , cf.Fig. 4.17. New data are required to confirm this. The Mainz A1 collaboration have taken VCS data at Q ≈ . , . and . , and this data is currently under analysis. Formulating a connection between VCS and VVCSpolarizabilities at finite Q is a future task and could be of interest in this context. The 2015 PDG online edition has also changed the values for the neutron. d dddd dd dddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddd ε d dddddddddddddddddddddddddddddddddddddddd γ dddd dddd dd ddd d ddd dddd dd dddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddd ε d dddddddddddddddddddddddddddddddddddddddd γ dddd dddd dd ddd d ddd dddd dd dddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddd ε d dddddddddddddddddddddddddddddddddddddddd γ dddd dddd dd ddd d ddd dddd dd dddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddd ε d dddddddddddddddddddddddddddddddddddddddd γ dddd dddd dd ddd d d dd dddd ddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd γ dddd dddd dd ddd d ddd dddd ddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd γ dddd dddd dd ddd d ddd dddd ddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd γ dddd dddd dd ddd d d Figure 7.4: Forward spin polarizability, γ , of the proton (top panel) and neutron (bottom panel). Shown arethe experimental value from the GDH-Collaboration [214], the DR result of Pasquini et al. [170], the HB χ PTfit of McGovern et al. [51], the B χ PT predictions of Lensky et al. [34, 125] and Bernard et al. [33]. All otherreferences and declarations are given in the text or the Tables 4.2 and 5.2.
The theoretical and experimental results for the proton spin polarizabilities have been presented in Ta-ble 4.2. Figure 7.4 summarizes the situation for the forward spin polarizability of the proton and neutron. Thesum rule evaluations therein are from Table 5.2. Results for the backward spin polarizability of the proton areshown in Fig. 7.5. dd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd γ π ddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd γ π ddd dddd dd ddd d ddd dd dd dd dd dd dd ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddd χ ddddddddddddddddddddddddd γ π ddd dddd dd ddd d d Figure 7.5: Backward spin polarizability, γ π , of the pro-ton. We show the experimental value from Camen [20],cf. Eq. (4.12), the fixed- t DR result of Babusci et al. [96],the HB χ PT fit of McGovern et al. [51] and the B χ PT pre-diction of Lensky et al. [54].A milestone in this area has been the recentMainz publication of double polarized CS datafor the Σ x asymmetry, and the global analysis ofCS asymmetry data, leading to the first measure-ment of all four spin polarizabilities, γ E E , γ M M , γ E M , and γ M E , cf. Table 4.2 [38]. At Mainznew data have been taken on the linear polar-ization asymmetry Σ , and the double polariza-tion asymmetry with longitudinally polarized tar-get Σ z .An attainable goal in this area is to reduce theuncertainties in spin polarizabilities, currently at ≈ ± × − fm , by approximately . Given thatCS count rates for a long frozen spin butanoltarget are very low compared to a
10 cm liquid hy-drogen target, long running times on polarized tar-gets may not be the best approach to drive down er-rors. Another strategy is to combine a global anal- 64 ddd dddd ddddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd δ ddddd dddd dd ddd d ddddd dddd ddddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd δ ddddd dddd dd ddd d ddddd dddd ddddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd δ ddddd dddd dd ddd d d dd dddd dddd dddd dddd ddddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd δ ddddd dddd dd ddd d ddd dddd dddd dddd dddd ddddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd δ ddddd dddd dd ddd d ddd dddd dddd dddd dddd ddddddddddddd χ ddd ε d ddddddddddddddddddddd χ ddddddddddddddddddddddddd δ ddddd dddd dd ddd d d Figure 7.6: Longitudinal-transverse polarizability, δ LT , for the proton (left panel) and neutron (right panel),respectively. We show the B χ PT predictions of Lensky et al. [34, 125] and Bernard et al. [33], and a resultfrom MAID [210].ysis of the asymmetry data, Σ , Σ x and Σ z , with measurements of backward angle CS cross sections in the ∆(1232) region. Because the Σ z asymmetry for backward angle CS approaches (see Fig. 4.13), unpolarizedCS preferentially selects one initial target polarization, and there is reasonable sensitivity to the spin polariz-abilities in the unpolarized cross sections.Figure 7.7: Spin polarizabilities, γ E E versus δ LT , for the proton. Results for γ E E (horizontal bands) arefrom: the experiment of Martel et al. [38] (beige), the B χ PT calculation of Lensky et al. [54] (red), and thefixed- t DR calculation of [96, 97] (purple). Results for δ LT (vertical bands) are from: MAID [210] (dashedline), Lensky et al. [34] (red), and Bernard et al. [33] (gray). The line across is based on the relation of (5.45b)using the values of GPs from the DR calculation of Pasquini et al. [150].The longitudinal-transverse polarizability, δ LT , and the forward spin polarizability, γ , deserve a specialattention. Their values at the real-photon point are shown in Fig. 7.6 and 7.4 for the proton and neutron,respectively. On the theory side, the baryon χ PT yields genuine predictions for the spin polarizabilities, cf.Bernard et al. [33] and Lensky et al. [34, 125]. On the empirical side, we have for instance the results fromthe latest version of the MAID partial-wave analysis (MAID’07), which is based on the empirical knowledgeof the single-pion photoproduction cross section σ LT . Especially for δ ( p ) LT , one B χ PT result is in significantcontradiction with MAID, while the other one is in agreement. The two B χ PT calculations are done in differentcounting schemes for the ∆ -isobar contributions, cf. Sect. 2.4. As result, the (cid:15) calculation includes in additionthe graphs with several ∆ -propagators, and in particular the one where the photons couple minimally to the ∆ inside the chiral loop. The latter graph is allegedly making up all the difference [294]. This would mean the π ∆ channel is extremely important for this quantity and that is why the MAID estimate would be inadequate.New JLab data [284] for δ ( p ) LT down to virtualities of .
02 GeV are currently at a final stage of analysis and willshed a further light on this “ δ LT puzzle”. Complementary, as a check one could simply study the effect of this65raph for the sum of the scalar polarizabilities, α E + β M . There the empirical number is known very wellfrom the Baldin sum rule. At the moment, however, this discrepancy is an open problem of B χ PT and is yetanother reincarnation of the “ δ LT puzzle”.Another view of this problem is presented in Fig. 7.7. The vertical lines clearly show the discrepancy in δ LT . The red bands are from the NNLO calculations of Lensky et al. [34, 54] using the δ -expansion. Theyare consistent with the DR approach. It would be interesting to see the (cid:15) -expansion result of Bernard et al.[33] for γ E E too, because it seems it would contradict with the empirical results in either the GP’s slope[cf. Eq. (5.45b)] or γ E E itself. Currently, the value of the proton rms charge radius extracted from H spectroscopy disagrees with the µ Hvalue by nearly five standard deviations:
Sick
BernauereHLorenzH Μ R E ! fm " R M ! f m " Figure 7.8: Determination of the proton’s elec-tric and magnetic radii. The shown values aregiven in the text. The green lines display theBernauer fit with TPE corrections: TPE,a (solid),TPE,b (dashed). The different uncertainties givenin Ref. [295] are added in quadrature. (cid:3)
H [224]: R E = 0 . ; (cid:3) µ H [25, 226]: R E = 0 . .On the other hand, the elastic electron-proton ( ep )scattering, which is the classic way of accessing the chargeradius, yields conflicting results on the charge and mag-netic rms radii: (cid:3) Sick [241]: R E = 0 . , R M = 0 . ; (cid:3) Lorenz et al. [296]: R E = 0 .
840 [0 . . . . . , R M = 0 .
848 [0 . . . . . ; (cid:3) Bernauer et al. [295] (world data): R E = 0 . stat (4) syst (2) model (4) group fm ,R TPE , a E = 0 . stat (4) syst (2) model (5) group fm ,R TPE , b E = 0 . stat (4) syst (2) model (5) group fm ,R M = 0 . stat (9) syst (5) model (2) group fm ,R TPE , a M = 0 . stat (9) syst (5) model (3) group fm ,R TPE , b M = 0 . stat (9) syst (5) model (3) group fm , where the superscript refers to the set of applied TPEcorrections: TPE,a [297], TPE,b [298, 299].The current situation is illustrated in Fig. 7.8. The CODATA 2010 recommended value, which combines theH and some of the ep scattering results, is [224]: R E (H + ep ) = 0 . , (7.2)which is in σ disagreement with the µ H result. This value does not include the interpretation of the ep scattering data based on dispersive approaches [296, 300, 301].Further details can be found in dedicated reviews [69, 70]. A nice overview of the current and futureexperimental activities called to resolve the puzzle has recently been given by Antognini et al. [302]. Figures 7.9 and 7.10 display the various results of the dispersive and χ PT calculations for the ‘polarizability’and ‘elastic’ contributions of the TPE correction to the Lamb shift in µ H. The corresponding values for the Based on H and D spectroscopy. From H alone (neglecting the isotope-shift measurements) R E = 0 . . dd ddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddddddddddddddddddddddddd χ ddddddddddddddddddddddddd ∆ d ddddd dd µ dddddd ddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddddddddddddddddddddddddd χ ddddddddddddddddddddddddd ∆ d ddddd dd µ dddddd ddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddddddddddddddddddddddddd χ ddddddddddddddddddddddddd ∆ d ddddd dd µ dddddd ddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ ddddddddddddddddddddddddd χ ddddddddddddddddddddddddd ∆ d ddddd dd µ ddd Figure 7.9: Proton polarizability effect in the S -level shift of µ H.dispersive calculations are listed in Table 6.1, whereas the B χ PT and HB χ PT predictions are summarized inTable 6.3. In Fig. 7.10, we also show the contribution of the Friar radius (3 rd Zemach moment) from Jentschura[303] and Borie [229]. We also quote the result from the bound-state QED approach of Mohr et al. [256], cf.Eq. (6.37).Figure 7.9 shows an overall agreement among the dispersive and B χ PT calculations of the proton polariz-ability correction. The dispersive results involve the modeling of the ‘subtraction’ contribution which rely onthe empirical value of proton β M , cf. Table 6.1, and Eqs. (6.35), (6.36). Given this model dependence, theagreement with the leading-order B χ PT prediction is quite remarkable. ddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ dddddddddddddddddddddddd ∆ d dddd ddd µ dddddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ dddddddddddddddddddddddd ∆ d dddd ddd µ dddddd ddd ddd ddd ddd dd dddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd χ dddddddddddddddddddddddddd χ dddddddddddddddddddddddd ∆ d dddd ddd µ ddd Figure 7.10: ‘Elastic’ TPE effect in the S -level shift of µ H. dd dd dd dd dd ddd ddd dddd χ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd d dddddd dddddd µ ddddd dd dd dd dd ddd ddd dddd χ ddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddddd d dddddd dddddd µ ddd Figure 7.11: Proton polarizability effect in the S HFSof µ H.Figure 7.10 shows the situation for the ‘elastic’ contribution. Since this contribution is completely de-termined by empirical FFs, the dispersive calculations agree very well. The bound-state QED approach ofRef. [256] gives a compatible result. The HB χ PT results are not in good agreement with the empirical evalua-tions because the FFs are not well reproduced in these calculations.Concerning the HFS, the polarizability corrections to the S HFS are summarized in Fig. 7.11. The leading-order χ PT prediction is quoted from Eq. (6.54). It is rather small compared with the dispersive calculations,which are taken from the ∆ pol column of Table 6.2 and converted to µ eV (multiplying the number in the67olumn with E F (2 S ) × − (cid:39) . µ eV ). Account of the ∆(1232) -resonance excitation in χ PT is expectedto improve the agreement. The different dispersive calculations are in agreement with each other and serve asinput for the recent extraction of the proton Zemach radius [25].Precise calculations of proton structure effects will be required to enable a direct measurement of the µ Hground-state HFS (see, e.g., Sect. 6 of Ref. [304]). The corresponding transition is much narrower than theobserved S − P transitions, and hence is much harder to find. Quantitative theory guidance will be veryimportant here. Once found, the HFS transition will greatly amplify the precision of our understanding of theproton structure in general, and of proton polarizabilities in particular.Until then, in the words of the title of this paper, the nucleon polarizabilities are taken from Comptonscattering and serve as input to hydrogen atom . We look forward to the times when the reverse is possible. Acknowledgements
We thank A. Alexandru, A. Antognini, C. Collicott, J. M. M. Hall, H. Fonvieille, P. P. Martel, and V. Sokhoyanfor providing figures and useful insights into their work. We thank Jeremy Green, Misha Gorchtein, VadimLensky, Anatoly L’vov, Barbara Pasquini, Randolf Pohl and Marc Vanderhaeghen for reading the manuscriptand their valuable remarks. R. M. thanks and acknowledges the Mainz A2 Compton collaboration for theircritical support. V. P. gratefully acknowledges the inspiring discussions with Aldo Antognini, Carl Carlson,Misha Gorchtein, Savely Karshenboim, Marc Vanderhaeghen, and Thomas Walcher, which found their vaguereflection in the pages of this manuscript.This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Re-search Center SFB 1044 [The Low-Energy Frontier of the Standard Model], and the Graduate School DFG/GRK1581 [Symmetry Breaking in Fundamental Interactions], and by the U.S. Department of Energy under grantDE-FG02-88ER40415. 68 ppendix A. Born Contribution in RCS and VVCS amplitudes
For RCS, the Born term is given by the tree-level graphs with the photon coupling to the nucleon charge Z e and theanomalous magnetic moment κ . The invariant amplitudes of the overcomplete tensor decomposition of Eq. (3.28) aregiven by [124]: A Born i ( s, t ) = (cid:40) A si ( ν, t ) + A si ( − ν (cid:48) , t ) , for i = 1 , , A si ( ν, t ) − A si ( − ν (cid:48) , t ) , for i = 3 , . . . , , (A.1)with A si being the contribution of the s -channel graph: A s ( ν, t ) = − M (cid:20) Z + t M ν ( Z + κ ) + 12 κ (cid:18) νM + t M (cid:19)(cid:21) , A s ( ν, t ) = κ M ν (cid:20) Z + 12 κ (cid:18) − ν M − t M (cid:19)(cid:21) , A s ( ν, t ) = A s ( ν, t ) , A s ( ν, t ) = − M ν (cid:104) ( Z + κ ) + ν M κ (cid:105) , A s ( ν, t ) = ( Z + κ ) M ν , A s ( ν, t ) = − Z ( Z + κ )4 M ν , A s ( ν, t ) = κ M ν , A s ( ν, t ) = − A s ( ν, t ) , (A.2)Adding it up, we obtain: A Born1 = − Z M − ( Z + κ ) ξ M ( ξ − ξ ) , A Born2 = − κ M + 2 M (2 Z κ + κ ) ξ − κ ξ M ( ξ − ξ ) , A Born3 = − M ξ (cid:20) κ M + ( Z + κ ) ξ M ( ξ − ξ ) (cid:21) = − M ξ A Born8 , A Born4 = − ( Z + κ ) ξ M ( ξ − ξ ) = − A Born5 , A Born6 = − Z ( Z + κ ) ξ M ( ξ − ξ ) , A Born7 = κ ξ M ( ξ − ξ ) , (A.3)where ξ = − q · q (cid:48) / M = t/ M and ξ − ξ = νν (cid:48) for real photons.To obtain the Born contribution to the forward VVCS amplitudes, Eq. (5.5), one may use T = e A , T = e Q ν (cid:0) A + Q A (cid:1) , (A.4a) S = e Mν (cid:2) A + Q (cid:0) A + A (cid:1)(cid:3) , S = e M (cid:0) A + A (cid:1) , (A.4b)with ξ = ν and ξ = − q / M = Q / M , and replace Z → F ( Q ) , κ → F ( Q ) . (A.5)Often, the π -exchange contribution is considered to be a part of the Born contribution. The only non-vanishingamplitude for the π -exchange graph is: A ( π )8 = − (2 Z − g A (2 πf π ) Mm π − t . (A.6) Appendix B. Derivation of a Dispersion Relation
Consider f ( ν ) , an analytic function in the entire complex ν plane except for the branch cut on the real axis, starting at ν and extending to infinity, as shown in Fig. B.1 (a). In the case when there are left- and right-hand branch cuts, locatedsymmetrically around ν = 0 , corresponding to Fig. B.1 (b), we can assume that f ( ν ) = f s ( ν ) ± f s ( − ν ) , (B.1) a) (b) Figure B.1: Contours in the complex ν plane for (a) a single cut along the positive real axis, and (b) branchcuts along the positive and negative real axes. where f s has only the right-hand cut. Hence, for our purpose it is sufficient to only consider the case of Fig. B.1 (a).The starting point is Cauchy’s formula for analytic functions: f ( ν + ) = 12 πi ˛ C d ξ f ( ξ ) ξ − ν + , (B.2)where the contour C avoids the branch cut on the real axis, as seen in Fig. B.1, and ν + is in the region of analyticity (i.e.,inside the contour). We choose ν + = ν + iδ , where ν and δ are real. We will take the limit δ → in the end.Next, we assume that f drops to zero for all large | ν + | , and it does so sufficiently fast, such that the integral over thebig semicircle can be neglected. We are then left with only the integrals around the cut: f ( ν + ) = lim (cid:15) → + πi ˆ ∞ ν d ξ (cid:20) f ( ξ + i(cid:15) ) ξ − ν + + i(cid:15) − f ( ξ − i(cid:15) ) ξ − ν + − i(cid:15) (cid:21) , (B.3)where (cid:15) is the gap between the contour and the real axis. The integral over the small semicircle around ν vanishes in thelimit (cid:15) → , since f has no poles at ν .Employing the Schwarz reflection principle for analytic functions: f ∗ ( s ) = f ( s ∗ ) , (B.4)and substituting ν + = ν + iδ , we have f ( ν + ) = lim (cid:15) → + πi ˆ ∞ ν d ξ (cid:20) f ( ξ + i(cid:15) ) ξ − ν − i ( δ − (cid:15) ) − f ∗ ( ξ + i(cid:15) ) ξ − ν − i ( δ + (cid:15) ) (cid:21) . (B.5)With (cid:15) < δ , the limit (cid:15) → can now be taken explicitly, since δ takes over the signs of the imaginary parts, hence f ( ν + ) = 12 πi ˆ ∞ ν d ξ f ( ξ ) − f ∗ ( ξ ) ξ − ν − iδ = 1 π ˆ ∞ ν d ξ Im f ( ξ ) ξ − ν − iδ , = 1 π ˆ ∞ ν d ξ (cid:20) ξ − ν ( ξ − ν ) + δ + iδ ( ξ − ν ) + δ (cid:21) Im f ( ξ ) . (B.6)Taking the limit to the real axis, δ → , we obtain the sought DR: f ( ν ) = lim δ → + f ( ν + ) = 1 π ˆ ∞ ν d ξ Im f ( ξ ) ξ − ν − i + = 1 π ∞ ν d ξ Im f ( ξ ) ξ − ν + i Im f ( ν ) . (B.7) When this is not the case, we could replace f ( ν + ) with f ( ν + ) /ν n + , provided we do not introduce another pole by doing that. So, forsmall ν + , f should go as ν n + . If f does not have any obvious zeros, one can make a subtraction. The subtracted function, by definition,has a zero at the subtraction point. oming back to the case when f is given as in Eq. (B.1), with f s satisfying the above DR, we then obviously have f ( ν ) = 1 π ˆ ∞ ν d ξ (cid:20) ξ − ν − i + ± ξ + ν − i + (cid:21) Im f s ( ξ ) = 2 π ˆ ∞ ν d ξ (cid:26) ξν (cid:27) Im f ( ξ ) ξ − ν − i + , (B.8)where in the last step we have used Im f ( ξ ) = Im f s ( ξ ) , which is true if ν ≥ . The situation with negative ν is inprinciple treatable but is beyond the present scope. Appendix C. Collection/Index of Formulae (cid:3)
Kinematical invariants and relations for real Compton scattering (RCS) η = M − suM , ν = s − M M , ν (cid:48) = M − u M . (3.14) ω = s − M √ s = M ν √ s , ω B = s − u √ M − t , (cid:15) B = 12 (cid:112) M − t. (C.1) ξ = s − u M = ν + ν (cid:48) ω B (cid:15) B M , ξ − t M = η − t = 4 νν (cid:48) . (C.2) t = 2 M ( ν (cid:48) − ν ) = − νν (cid:48) (1 − cos ϑ ) = − ω (1 − cos θ ) = − ω B (1 − cos θ B ) , d t = 2 M d ν (cid:48) = (1 /π ) ν (cid:48) d Ω L = (1 /π ) ω d Ω cm , cos ( ϑ/
2) = η/ (4 νν (cid:48) ) , sin ( ϑ/
2) = − t/ (4 νν (cid:48) ) , sin ϑ = √− tη/ (2 νν (cid:48) ) , cos ( θ/
2) = η/ (2 ν ) , sin ( θ/
2) = − t/ (2 ω ) , sin θ = √− tηs/ (2 M ν ) , cos ( θ B /
2) = η/ (2 ξ ) , sin ( θ B /
2) = − t/ (2 ω B ) , sin θ B = √− tη/ (2 ξω B ) , tan( ϑ/
2) = ( − t / η ) / = M / √ s tan( θ/
2) = M / (cid:15) B tan( θ B / . (C.3) (cid:3) Relations for the forward doubly-virtual Compton scattering (VVCS) ν = ω B = ξ = η / = s − u M = s − M + Q M = Q M x = 2
M τx , ν el = Q M = 2 M τ (C.4) ˆ ∞ Q M d νν n f ( ν, Q ) = (cid:18) MQ (cid:19) n − ˆ d x x n − f ( x, Q ) (5.11) (cid:3) Elastic structure functions f el1 ( x, Q ) = 12 G M ( Q ) δ (1 − x ) , (5.7a) f el2 ( x, Q ) = 11 + τ (cid:2) G E ( Q ) + τ G M ( Q ) (cid:3) δ (1 − x ) , (5.7b) g el1 ( x, Q ) = 12 F ( Q ) G M ( Q ) δ (1 − x ) , (5.7c) g el2 ( x, Q ) = − τ F ( Q ) G M ( Q ) δ (1 − x ) . (5.7d) (cid:3) Nucleon-pole and Born contributions to the forward VVCS amplitudes T pole1 ( ν, Q ) = 4 παM ν G M ( Q ) ν − ν − i + = T Born1 ( ν, Q ) + 4 παM F ( Q ) , (5.13a) T pole2 ( ν, Q ) = 8 πα ν el ν − ν − i + G E ( Q ) + τ G M ( Q )1 + τ = T Born2 ( ν, Q ) , (5.13b) S pole1 ( ν, Q ) = 4 πα ν el ν − ν − i + F ( Q ) G M ( Q ) = S Born1 ( ν, Q ) + 2 παM F ( Q ) , (5.13c) S pole2 ( ν, Q ) = − πανν − ν − i + F ( Q ) G M ( Q ) = S Born2 ( ν, Q ) . (C.5) Sum rulesBaldin: α E + β M = 12 π ˆ ∞ ν d νν σ ( ν ) (5.21) th -order: α E ν + β M ν + / ( α E + β M ) = 12 π ˆ ∞ ν d νν σ ( ν ) (5.22)Gerasimov–Drell–Hearn (GDH): − αM κ = 12 π ˆ ∞ ν d νν (cid:2) σ / ( ν ) − σ / ( ν ) (cid:3) (5.23)Gell-Mann–Goldberger–Thirring (GTT): γ = 14 π ˆ ∞ ν d νν (cid:2) σ / ( ν ) − σ / ( ν ) (cid:3) (5.24) th -order: ¯ γ = 14 π ˆ ∞ ν d νν (cid:2) σ / ( ν ) − σ / ( ν ) (cid:3) (5.25)Burkhardt–Cottingham (BC): ˆ d x g ( x, Q ) = τ (cid:2) I ( Q ) − F ( Q ) G M ( Q ) (cid:3) (5.14)spin-GP sum: dd q (cid:2) P ( M ,M ( q ) + P ( L ,L ( q ) (cid:3) q =0 = γ E M αM − M dd Q (cid:2) F ( Q ) + I ( Q ) (cid:3) Q =0 (5.45a)spin-GP difference: dd q (cid:2) P ( M ,M ( q ) − P ( L ,L ( q ) (cid:3) q =0 = γ E E + δ LT αM (5.45b) (cid:3) Dispersion relations for the Sachs FFs [for Dirac and Pauli FFs, see Eq. (6.8)] (cid:32) G E ( Q ) G M ( Q ) (cid:33) = (cid:32)
11 + κ (cid:33) − Q π ˆ ∞ t d tt ( t + Q ) Im (cid:32) G E ( t ) G M ( t ) (cid:33) . (C.6) (cid:3) Moments of the (spherically-symmetric) charge distribution, for any N : (cid:104) r N (cid:105) E ≡ π ˆ ∞ d r r N +2 ρ E ( r ) = Γ( N + 2) π ˆ ∞ t d t Im G E ( t ) t N/ , (6.19)with the normalization (cid:104) r (cid:105) E = 1 . Equivalently, for integer N : (cid:104) r N (cid:105) E = ( − N (2 N + 1)! N ! G ( N ) E (0) , (C.7) (cid:104) r N − (cid:105) E = ( − N (2 N )! 2 π ˆ ∞ d QQ N (cid:104) G E ( Q ) − N − (cid:88) k =0 Q k k ! G ( k ) E (0) (cid:105) , = ( − N (2 N )! 2 π ˆ ∞ d QQ N (cid:104) G E ( Q ) − N − (cid:88) k =0 ( − Q ) k (2 k + 1)! (cid:104) r k (cid:105) E (cid:105) . (C.8)The moments of the magnetization distribution, (cid:104) r N (cid:105) M , are defined similarly, replacing G E with G M / (1 + κ ) . (cid:3) Friar radius (or, the 3 rd Zemach moment of the charge distribution) R F = (cid:113) (cid:104) r (cid:105) E (2) , (cid:104) r (cid:105) E (2) ≡ π ˆ ∞ d QQ (cid:104) G E ( Q ) − R E Q (cid:105) , (6.3b) = 2 (cid:104) r (cid:105) E + 24 π ˆ ∞ t d t ˆ ∞ t d t (cid:48) Im G E ( t ) Im G E ( t (cid:48) )( t (cid:48) t ) / ( √ t (cid:48) + √ t ) . (C.9) (cid:3) Zemach radius R Z ≡ − π ˆ ∞ d QQ (cid:20) G E ( Q ) G M ( Q )1 + κ − (cid:21) , (6.3c) = (cid:104) r (cid:105) E + (cid:104) r (cid:105) M − π ˆ ∞ t d tt Im G M ( t (cid:48) )1 + κ ˆ ∞ t d t (cid:48) t (cid:48) Im G E ( t (cid:48) ) √ t (cid:48) + √ t . (C.10) (cid:3) Finite-size effects ) P − S Lamb shift: E LS = − Zαπ ˆ ∞ d Q w P − S ( Q ) G E ( Q ) , (6.22) = − Zα a (cid:2) R E − (2 a ) − R (cid:3) + O ( α ) . (6.1a)b) nS hyperfine splitting (HFS): E HFS ( nS ) = 4 Zα πmM ˆ ∞ d Q Q w nS ( Q ) G E ( Q ) G M ( Q ) , (6.34b) = E F ( nS ) (cid:2) − a − R Z (cid:3) + O ( α ) , (6.1b)with the convolution of momentum-space wave functions: w S ( Q ) = 16 (cid:0) aQ ) (cid:1) , w S ( Q ) = (cid:0) − ( aQ ) (cid:1)(cid:0) − aQ ) (cid:1)(cid:0) aQ ) (cid:1) , w P − S ( Q ) = 2( aQ ) (cid:0) − ( aQ ) (cid:1)(cid:0) aQ ) (cid:1) , (6.16)and the Fermi energy: E F ( nS ) = 8 Zα a κmM n . (6.2) eferences [1] A. Klein, Phys. Rev. 99 (1955) 998–1008.[2] A. M. Baldin, Nucl. Phys. 18 (1960) 310–317.[3] C. L. Oxley, V. L. Telegdi, Phys. Rev. 100 (1955) 435–436.[4] C. L. Oxley, Phys. Rev. 110 (1958) 733–737.[5] G. E. Pugh, R. Gomez, D. H. Frisch, G. S. Janes, Phys. Rev. 105 (1957) 982–995.[6] L. G. Hyman, R. Ely, D. H. Frisch, M. A. Wahlig, Phys. Rev. Lett. 3 (1959) 93–96.[7] G. Bernardini, A. O. Hanson, A. C. Odian, T. Yamagata, L. B. Auerbach, I. Filosofo, Nuovo Cim. 18 (1960) 1203–1236.[8] V. I. Goldansky, et al., Nucl. Phys. 18 (1960) 473 – 491.[9] P. Baranov, G. Buinov, V. Godin, V. Kuznetzova, V. Petrunkin, L. Tatarinskaya, V. Shirthenko, L. Shtarkov,V. Yurtchenko, Yu. Yanulis, Phys. Lett. B 52 (1974) 122–124.[10] F. J. Federspiel, et al., Phys. Rev. Lett. 67 (1991) 1511–1514.[11] E. L. Hallin, et al., Phys. Rev. C 48 (1993) 1497–1507.[12] B. E. MacGibbon, et al., Phys. Rev. C 52 (1995) 2097–2109.[13] G. Blanpied, et al., Phys. Rev. C 64 (2001) 025203.[14] P. Bourgeois, et al., Phys. Rev. C 84 (2011) 035.[15] L. S. Myers, et al., Phys. Rev. Lett. 113 (2014) 262506.[16] A. Zieger, et al., Phys. Lett. B 278 (1992) 34–38 [Erratum: Phys. Lett. B 281 (1992) 417].[17] V. Olmos de León, et al., Eur. Phys. J. A 10 (2001) 207–215.[18] G. Galler, et al., Phys. Lett. B 503 (2001) 245.[19] S. Wolf, et al., Eur. Phys. J. A 12 (2001) 231.[20] M. Camen, et al., Phys. Rev. C 65 (2002) 032202.[21] J. Roche, et al., Phys. Rev. Lett. 85 (2000) 708–711.[22] P. Janssens, et al., Eur. Phys. J. A 37 (2008) 1–8.[23] G. Laveissière, et al., Phys. Rev. Lett. 93 (2004) 122001.[24] R. Pohl, et al., Nature 466 (2010) 213–216.[25] A. Antognini, F. Nez, K. Schuhmann, F. D. Amaro, F. Biraben, et al., Science 339 (2013) 417–420.[26] J. C. Bernauer, R. Pohl, Sci. Am. 310 (2014) 18–25.[27] A. Walker-Loud, C. E. Carlson, G. A. Miller, Phys. Rev. Lett. 108 (2012) 232301.[28] F. B. Erben, P. E. Shanahan, A. W. Thomas, R. D. Young, Phys. Rev. C 90 (2014) 065205.[29] J. Gasser, M. Hoferichter, H. Leutwyler, A. Rusetsky, Eur. Phys. J. C 75 (2015) 375.[30] K. Slifer, in: Proceedings, 21st Conference on New Trends in High-Energy Physics (Experiment, Phenomenology,Theory), Yalta, Crimea, Ukraine, September 15–22, 2007 [nucl-ex/0711.4411].[31] J. P. Chen, Eur. Phys. J. ST 162 (2008) 103–116.[32] J. P. Chen, A. Deur, S. Kuhn, Z. E. Meziani, J. Phys. Conf. Ser. 299 (2011) 012005.
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