Nucleon polarizabilities from low-energy Compton scattering

An effective field theory is used to give a model-independent description of Compton scattering at energies com to the pion mass. The amplitudes for scattering on the proton and the deuteron, calculated to fourth order in small in chiral perturbation theory, contain four undetermined parameters that are in one-to-one correspondence with the polarizabilities. These polarizabilities are extracted from fits to data on elastic photon scattering on hydrogen and de For the proton we find: αp = (12.1±1.1)+0.5 −0.5 × 10−4 fm3, βp = (3.4±1.1)+0.1 −0.1 × 10−4 fm3. For the isoscalar polarizabilitie we obtain:αN = (9.0± 1.5)+3.6 −0.8 × 10−4 fm3, βN = (1.7± 1.5)+1.4 −0.6 × 10−4 fm3.  2003 Published by Elsevier B.V. Open access under CC BY license. tal mic ge acthe obons tron rD. se the rks ecn emrst in k 2,3]. in leon ) set pt that Electromagnetic polarizabilities are a fundamen property of any composite object. For example, ato polarizabilities contain information about the char and current distributions that result from the inter tions of the protons, neutrons, and electrons inside atom. Protons and neutrons are, in turn, complex jects composed of quarks and gluons, with interacti governed by QCD. It has long been hoped that neu and proton polarizabilities will give important info mation about the strong-interaction dynamics of QC For example, in a simple quark-model picture the polarizabilities contain averaged information about E-mail address: phillips@phy.ohiou.edu (D.R. Phillips). 0370-2693  2003 Published by Elsevier B.V. doi:10.1016/j.physletb.2003.06.040 Open access under CC BY lice charge and current distribution produced by the qua inside the nucleons [1]. In this Letter we use an eff tive field theory (EFT) of QCD to extract both proto and neutron polarizabilities in a consistent and syst atic manner from Compton scattering data—the fi EFT extraction of all these important quantities with the same framework. 1 The background to this wor 1 The predictions of an EFT of QCD for γp scattering have been compared with cross-section data, for instance, in Ref. [ However, no extraction of proton polarizabilities was made either paper. There is also recent work in which isoscalar nuc polarizabilities are obtained via an EFT analysis of a (restricted of γ d data [4]. However, the authors of Ref. [4] did not attem an analysis of proton scattering in the low-energy EFT used in work.

Electromagnetic polarizabilities are a fundamental property of any composite object. For example, atomic polarizabilities contain information about the charge and current distributions that result from the interactions of the protons, neutrons, and electrons inside the atom. Protons and neutrons are, in turn, complex objects composed of quarks and gluons, with interactions governed by QCD. It has long been hoped that neutron and proton polarizabilities will give important information about the strong-interaction dynamics of QCD. For example, in a simple quark-model picture these polarizabilities contain averaged information about the E-mail address: phillips@phy.ohiou.edu (D.R. Phillips). charge and current distribution produced by the quarks inside the nucleons [1]. In this Letter we use an effective field theory (EFT) of QCD to extract both proton and neutron polarizabilities in a consistent and systematic manner from Compton scattering data-the first EFT extraction of all these important quantities within the same framework. 1 The background to this work 1 The predictions of an EFT of QCD for γp scattering have been compared with cross-section data, for instance, in Ref. [2,3]. However, no extraction of proton polarizabilities was made in either paper. There is also recent work in which isoscalar nucleon polarizabilities are obtained via an EFT analysis of a (restricted) set of γ d data [4]. However, the authors of Ref. [4] did not attempt an analysis of proton scattering in the low-energy EFT used in that work.
will be found primarily in two papers, Ref. [3] for the proton, and Ref. [5] for the deuteron. Further details of interest to the specialist will be published elsewhere [6].
In an atomic or molecular system the polarizabilities are measured with static fields. Nuclear polarizabilities can analogously be determined by the scattering of long-wavelength photons. Experimental facilities which accurately measure the energy of a photon beam using photon tagging have made possible a new generation of experiments which probe the lowenergy structure of nucleons and nuclei. In particular, photon tagging can be used to measure Compton scattering on weakly-bound systems, since it facilitates the separation of elastic and inelastic cross sections. At sufficiently low incoming (outgoing) photon energy ω (ω ) and momentum k ( k ), the spin-averaged Compton scattering amplitude for any nucleus is, in the nuclear rest frame: where and are the polarization vectors of the initial and final-state photons. The first term in this series is a consequence of gauge invariance, and is the Thomson limit for low-energy scattering on a target of mass m A and charge Ze. The coefficients of the second and third terms are the target electric and magnetic polarizabilities, α and β, respectively. The polarizabilities can be separated by the angular dependence: for example, at forward (backward) angles the amplitude depends only on α + β (α − β). Other terms, represented by ". . .", include higher powers of energy and momentum and relativistic corrections. 2 Hydrogen targets are used to determine proton polarizabilities α p and β p [8,9]. By contrast, the absence of dense, stable, free neutron targets requires that the neutron polarizabilities α n and β n be extracted from scattering on deuterium (or other nuclear) targets. Data exist for coherent γ d → γ d from 49 to 95 MeV [10][11][12], and for quasi-free γ d → γpn from 200 to 400 MeV [13]. The coherent process is sensitive to the 2 Our definition of the polarizabilities is identical to that used in recent dispersion-relation extractions-see Ref. [7]. They are the leading terms in the expansion of the non-Born amplitudes. isoscalar nucleon polarizabilities-α N ≡ (α p + α n )/2, β N ≡ (β p + β n )/2-via interference with the larger Thomson term. The extraction of these polarizabilities from data requires a consistent theoretical framework that clearly separates nucleon properties from nuclear effects. In the long-wavelength limit pertinent to polarizabilities EFT provides a model-independent way to do exactly this [14][15][16].
The EFT of QCD relevant to the low-energy interactions of a single nucleon with any number of pions and photons is known as chiral perturbation theory (χ PT). The Lagrangian is constrained only by approximate chiral symmetry and the relevant spacetime symmetries. S-matrix elements can be expressed as a simultaneous expansion in powers of momenta and the pion mass (collectively denoted by Q) over the characteristic scale of physics not included explicitly in the EFT. Many processes have been computed in this EFT to non-trivial orders and it has proven remarkably successful [15]. In this, as in any EFT, detailed information about short-distance physics is absent. The shortdistance physics relevant to low-energy processes appears in the theory as constants whose determination lies outside the scope of the EFT itself. In the purest form of the theory they are determined by fitting experimental observables. In many cases the sparsity of low-energy single-nucleon data makes such a determination problematic-and for free neutrons data is nonexistent.
When the amplitude for unpolarized Compton scattering is expanded in powers of Q, we obtain Eq. (1) with α and β given as functions of the EFT parameters. To O(Q 3 ), no parameters appear apart from those fit in other processes, so predictions can be made. At this order the proton and neutron polarizabilities are given by pion-loop effects [17]: Here g A 1.26 is the axial coupling of the nucleon and f π 93 MeV is the pion decay constant. At O(Q 4 ) there are new long-range contributions to these polarizabilities. Four new parameters also appear which encode contributions of short-distance physics to the spin-independent polarizabilities. Thus, minimally, one needs four pieces of experimental data to fix these four short-distance contributions, but once they are fixed χ PT makes model-independent predictions for Compton scattering on protons and neutrons.
The amplitude for Compton scattering on the nucleon has been computed to O(Q 4 ) in Ref. [3]. The shifts of α p and β p from their O(Q 3 ) values were not fitted directly to the data in that work. Instead the Particle Data Group values for the polarizabilities were used [18]. These were originally extracted using a dispersion-theoretic approach that incorporates model-motivated assumptions regarding the asymptotic behavior of the amplitude. The differential cross sections which result from this procedure are in good agreement with the low-energy data [8].
Thus the calculation of Ref. [3] is not, strictly speaking, model independent: there is model input in the values used for α p and β p . Using the same amplitude but fitting the polarizabilities, very good fits of the proton data in the low-energy regime (ω, √ |t| < 200 MeV) can be obtained. The central values for α p and β p are similar to those employed in Ref. [3], but the uncertainty is larger: where statistical (1-σ ) errors are inside the brackets, and an estimate of the contribution from higher-order terms is given outside. These numbers are based on varying the upper bound on which data are fit from 160 MeV to 200 MeV, and on estimates of the O(Q 5 ) effect on γp scattering. A sample of the best-fit results is shown, together with data, in Fig. 1. The fit has χ 2 /d.o.f. = 170/131. These results are fully compatible with other extractions, though the central value of β p is somewhat higher [18,19]. They are also consistent with the values predicted in Refs. [17,20], where resonance saturation was used to estimate the The values (2) can be compared with the recent reevaluation of the Baldin sum rule by Olmos de León et al. [8], which gives: This result overlaps the 1-σ error ellipse of the fit (2), as shown in Fig. 2. Including the constraint (3) in our fit leads to values for α p and β p consistent with (2), but with smaller statistical errors, namely: EFT best fit (solid line) to the differential cross sections for Compton scattering on the proton at four different angles, compared to the experimental data [8,9]. (Symbols as in Ref. [3].) The grey area is the region excluded from the fit (ω, In (4) we have left the systematic error unchanged, but have now included a second error inside the brackets, whose source is the error on the sum rule evaluation (3). The smaller overall statistical error in (4) is achieved at the expense of additional (reasonable) assumptions about the high-energy behaviour of hadronic amplitudes, and also by using higher-energy data on the total proton photo-absorption cross section, since these are the two ingredients entering the Baldin Sum Rule result for α p + β p .
The amplitude for Compton scattering on a nuclear target can also be calculated in the EFT, although the unraveling of scales is more subtle when more than one nucleon is present [16]. In the single-nucleon sector a typical intermediate state has an energy denominator of O(Q), but in multi-nucleon processes "reducible" intermediate states can have small energy denominators of O(Q 2 /m N ). The resulting infrared enhancement complicates the perturbative expansion [21]. Furthermore, the very existence of nuclei implies a breakdown of perturbation theory; hence the leading two-nucleon interactions should be summed to all orders, thereby building up a nuclear wave function |ψ . In the case of Compton scattering at energies of O(m π ), this is the only resummation necessary [5], and the full amplitude, T , can be written in terms of a kernel K γ γ which contains all irreducible (in the above sense) γ NN → γ NN graphs: The kernel can be calculated in χ PT. Several other reactions involving deuterium have been successfully analyzed using analogous approaches [16]. However, for Compton scattering at energies of O(m 2 π /m N ) further resummations are necessary. In particular, such resummations restore the Thomson limit for Compton scattering from deuterium [5,22,23]. In this energy regime it seems more appropriate to use a lowerenergy EFT where pions have been integrated out [4,24].
At momentum transfers of O(m π ), a formallyconsistent power counting is emerging [25] which organizes the nuclear interactions that give rise to the wave function |ψ . This power counting is an improvement over Weinberg's original proposal [21], which has led to fairly accurate wave functions [16]. To the order we are working in the EFT, we require a deuteron wave function calculated to O(Q 2 ) in the chiral expansion. We employ a wave function generated using the O(Q 2 ) χPT potential of Ref. [26]. (The Λ = 600 MeV wave function was chosen, but choosing Λ = 540 MeV instead produces very similar results.) In order to test the consistency of our error estimates, we have also computed results with other "realistic" wave functions, such as that obtained from the Nijm93 OBE potential [27]. 3 Below we will quote results only for the O(Q 2 ) wave function of Ref. [26] and for the Nijm93 wave function. The results for differential cross sections generated using |ψ 's obtained from other NN potentials almost invariably lie between these two extremes.
Meanwhile, the kernel K γ γ is the sum of the single-nucleon Compton amplitude, in which one nucleon is a "spectator" to the Compton scattering, and "two-nucleon" contributions in which both nucleons are involved in the scattering of the photon. The amplitude for coherent Compton scattering on the deuteron was computed to O(Q 3 ) in Ref. [5]. There are no free parameters to this order. The corresponding cross section is in good agreement with the Illinois data [10] at 49 and 69 MeV, but under-predicts the SAL data [11] at 95 MeV. This calculation yields cross sections which agree well with the recent Lund data [12]. It also agrees qualitatively with potentialmodel calculations of γ d scattering [28,29].
We have now extended our calculation to O(Q 4 ). The calculation contains both single-nucleon and twonucleon contributions. For the first class of diagrams, we employ the single-nucleon amplitude of Ref. [3]. The amplitude given there must be boosted from the Breit frame to the γ d center-of-mass frame, which is straightforward once the γp amplitude has been decomposed into six invariant functions multiplying structures that transform in well-defined ways under boosts [5,6].
Less straightforward are technical issues associated with the fashion in which nucleon recoil is included in the γ d scattering calculation. These occur because the heavy-baryon formulation of χ PT employed in this work expands about the limit of static nucleons, with nucleon recoil treated as a perturbative correction. The first issue has to do with the treatment of the verylow-energy region, ω ∼ m 2 π /M. At these energies the nucleons inside the deuteron are not static on the time-scales relevant to Compton scattering, and so a resummation of nucleon-recoil corrections must be performed. Below we will invoke this resummation when we calculate γ d scattering for these low photon energies. The second issue has to do with ensuring that particle-production cuts appear at the correct position. In the absence of nucleon recoil the channel γ d → πd will open at a position which is in error by an amount m 2 π /(2M d ). Since the opening of this channel can result in rapid variation of the Compton cross section with energy, this problem must be dealt with when we describe γp scattering for ω close to m π [3,20]. However, even at the highest energy for which Compton scattering on deuterium is calculated below, this small error in the position of the πd cut is a relatively minor effect. We emphasize that these subtleties are artifacts which arise due to the fact that nucleon recoil is treated as a perturbation in the γ N amplitude we have employed here [3]. Such a treatment is in accord with the usual practice in the chiral EFT known as heavy-baryon χ PT, but a perturbative treatment of nucleon recoil is not mandatory in the EFT. Thus these two problems in no way represent true limitations of a chiral EFT of Compton scattering.
In addition, the two-nucleon diagrams at O(Q 4 ) are added to the two-nucleon O(Q 3 ) diagrams shown in Ref. [5]. These O(Q 3 ) graphs were generated by the leading chiral Lagrangian. In going to O(Q 4 ) we include diagrams with one insertion from the sub-leading chiral Lagrangian. The coefficients of these vertices are essentially determined by relativistic invariance, and so these O(Q 4 ) two-body effects are suppressed by Q/m N . No unknown parameters occur in these graphs. However, two free parameters associated with the neutron polarizabilities do appear in the single-nucleon contribution: the shifts in the isoscalar polarizabilities from their O(Q 3 ) values. Indeed, of all the additional graphs which enter our new, O(Q 4 ), calculation only this effect from singlenucleon Compton scattering changes the cross section significantly. (Details can be found in Ref. [6].) We have fitted these two free parameters to the existing γ d scattering data. As in the proton case, we impose a cutoff on the data that we fit when we extract α and β. With a cutoff of ω, √ |t| < 200 MeV all of the 29 world data points [10][11][12] are included. We also use a cutoff of 160 MeV, in which case all but the two backward-angle SAL points must be fitted. In both cases we float the experimental normalization for each experimental run within the quoted systematic error [9], resulting in 22 (20) degrees of freedom for the fit [6].
Using the wave function of Ref. [26], we fit to data with ω, √ |t| < 160 MeV. This produces the isoscalar nucleon polarizabilities and the χ 2 per degree of freedom given in the first line of Table 1. The χ 2 is unacceptably large. This is driven mainly by the 49 MeV data from Illinois, and occurs because, as discussed above, for ω ∼ m 2 π /m N further resummations are necessary in the EFT. Here we adopt the strategy of including the dominant contributions that need to be added to our standard O(Q 4 ) result in this very-lowenergy region in order to restore the Thomson limit for the γ d amplitude [5]. This yields the results on the second line of Table 1. An alternative strategy, namely dropping the 49 MeV data altogether, produces very similar central values for α N and β N .
When the upper limit on which data is fitted is increased the central value of α N -β N changes markedly, with a slightly higher χ 2 /d.o.f. (third line of Table 1). Another key test involves examining the impact of the choice of deuteron wave function. The variability in the differential cross section due to the choice of wave function is ∼10%. When the Nijm93 wave function is used the results are as shown in the fourth line of Table 1. The high χ 2 is again due to a failure to reproduce the 49 MeV data-this time even with the very-low-energy contributions included. While the Nijm93 wave function is not consistent with Fig. 3. Results of the O(Q 4 ) EFT best fit to the (lab. and c.m. as appropriate) differential cross sections for Compton scattering on deuterium at four different photon energies: 49 MeV lab. and 55, 66, and 95 MeV c.m. The data are from Illinois [10] (circles), Lund [12] (diamonds) and SAL (squares) [11]. The solid line is the O(Q 4 ) calculation with α N = 9.0 × 10 −4 fm 3 , β N = 1.7 × 10 −4 fm 3 . The grey area is the region excluded from the fit (ω, √ |t| > 160 MeV). The dashed line is the (parameter-free) O(Q 3 ) calculation.
χ PT, it does have the correct long-distance behavior. Such sensitivity to the choice of |ψ is worrisome and merits further study.
Adding statistical errors in quadrature and including the theory errors linearly we infer values of α n ranging from 0.6 to 16.8 (10 −4 fm 3 ) and β n between −4.5 and 6.1 (same units). The values of α n and β n extracted from data on the quasi-free process γ d → γ np using a theoretical model fall within this range [13]. We find that a wide range of neutron polarizabilities is consistent with a model-independent analysis of the current low-energy γp and coherent γ d data. Narrower ranges for the neutron polarizabilities can be obtained from this data at the expense of introducing model dependence.
In conclusion, we have determined nucleon polarizabilities from a model-independent fit to low-energy Compton scattering on the proton and the deuteron. Our results are consistent, within error bars, with the recent extraction of α N ± β N from the Lund data using the detailed model of Levchuk and L'vov [12,29]. (But see also the values found using the data of Refs. [10,11] and this model [11,29].) They are also consistent with the Baldin sum rule results for α p + β p and α n + β n [30]. The EFT can be improved by the introduction of an explicit -isobar field, a complete treatment of the very-low-energy region, and a better understanding of the dependence of the results on the choice of deuteron wave function. An EFT study of the quasi-free deuteron process is also an important future step.