- Email: [email protected]
Elastic photon–proton scattering: the polarizabilities of the proton
Radiation Physics and Chemistry 56 (1999) 113±123
www.elsevier.com/locate/radphyschem
Elastic photon±proton scattering: the polarizabilities of the proton Alan M. Nathan Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
Abstract A review is presented of the elastic photon scattering data from the proton at energies below 150 MeV, where the important physics to be learned is the electric and magnetic polarizabilities of the proton. The relationship between the polarizabilities and the Compton scattering cross section is established. The experimental data are reviewed. Techniques for extracting the polarizabilities from the data are discussed and then used to obtain new values. Particular attention is paid to the theoretical uncertainty in this procedure. A brief theoretical review is given with the principal emphasis on dispersion relations. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The electric and magnetic polarizabilities, labeled a and b, respectively, measure the ease with which a static external electric or magnetic ®eld can induce an electric or magnetic dipole moment in a composite system.1 The proton itself is a composite system, and its
E-mail address: [email protected] (Alan M. Nathan) It has been conventional to designate Compton polarizabilities with ``bars'' in order to distinguish them from ``static'' polarizabilities. That convention is largely obsolete and is not followed here. Often the notations aE and bM is also used as a reminder of the electric and magnetic nature of these polarizabilities. Perhaps a better notation would be aE1 and bM1 to emphasize that the multipolarity is electric or magnetic dipole and to distinguish them from polarizabilities of dierent multipolarity, which are discussed later in this review. Neither of those notations will be used here. 2 Nuclear and particle physicists generally refer to elastic photon scattering as Compton scattering, and that terminology will be used here. This is dierent from atomic physics, where coherent elastic scattering from the atomic electrons is called Rayleigh scattering and incoherent quasi-elastic scattering from the electrons is called Compton scattering. 1
polarizabilities constitute fundamental structure constants which are as important as the charge and magnetic radii. In this paper, the current experimental status of the proton polarizabilities is reviewed. The link between the polarizabilities and Compton scattering is given in Section 2. The experimental data are reviewed in Section 3 and the results of global ®ts to extract the polarizabilities from the data are given in Section 4. The results obtained from this analysis are put into theoretical perspective in Section 5. A summary and conclusions are given in Section 6.
2. Polarizabilities and Compton scattering 2.1. Classical picture The relationship between low-energy Compton scattering2 on a composite system and the static electric and magnetic polarizabilities of that system is essentially a classical concept. It is instructive to examine a simple classical model in which two particles of charge and mass q1, q2 (total charge Q=q1+q2) and M1, M2 (total mass M=M1+M2) are on opposite ends of a
0969-806X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 9 9 ) 0 0 2 9 0 - X
114
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
spring of force constant k. If a uniform electric ®eld is applied, the whole system will accelerate and the spring will stretch, giving rise to an induced electric dipole moment. The constant of proportionality between the induced moment and the applied ®eld is de®ned to be the electric polarizability, which for this system is easily shown to be3 a
q2eff , 4pmo 20
where qe=(q1M2ÿq2M1)/M, m is the reduced mass, and o 20=k/m. Using classical antenna theory (Jackson, 1961), the photon scattering cross section for this system can be derived: 2 ds Q2 1 cos 2 y , ÿ o 2 a o dO 2 4pM
ing from a composite system with Q 2/M$ 0 that the cross section is non-zero at o=0 and initially decreases. For a composite system with Q 2/M = 0 (such as a neutron or a neutral atom or molecule), there is no Thomson term and the leading term in the cross section is proportional to a 2o 4, leading to familiar eects such as the blue sky. The classical model presented here only includes the eects of the electric polarizability, but one could similarly construct a model exhibiting the eects of the magnetic polarizability. The total photoabsorption cross section sg for this simple system is exhausted by the single resonant excitation of the spring:
sg o 2p
2
q2eff 4pm
!2 d o ÿ o 0 ,
where y is the scattering angle and a o
q2eff 4pm o 20 ÿ o 2
is the dynamic (frequency-dependent) electric polarizability, which reduces to the static polarizability a in the limit o 4 0. For low frequencies (i.e., o<
leading to the following sum rule for a:
a
1 2p2
1 0
sg o do : o2
The physical origin of this sum rule is familiar from optics and relies on very fundamental principles of physics. First, causality leads to a dispersion relation relating the real part of the scattering amplitude to an integral over the imaginary part. Second, unitarity leads to a unique relationship between the imaginary part of the forward amplitude and sg (the Optical Theorem). These two ingredients combined with the low-energy form of the scattering amplitude lead uniquely to the sum rule. We close this section by remarking that this classical model can be used to obtain an approximate value of a for the proton. Suppose one end of the spring has two u quarks (q1=4e/3; M1 1 600 MeV) and the other end has one d quark (q2=ÿe/3; M2 1 300 MeV). With o0 adjusted to reproduce the approximate energy of dipole excitations in the proton (o0 1 500 MeV), we obtain a 1 5 10ÿ4 fm3, which is within about a factor of 2 of the experimental value. Interestingly, if the identity of u and d quarks are interchanged, thereby turning a proton into a neutron, the same value of a is obtained. 2.2. Compton scattering on the proton For Compton scattering from the proton, one can similarly expand the cross section in powers of o, leading to the following Low Energy Expansion (LEX) (Petrun'kin, 1981):
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
ds dO
LEX o ,y
ds dO
115
Born o ,y
0 2 e2 o ab 0 ÿ 1 z2 o o 2 4pM o aÿb 2 1 ÿ z , 2
1
where 0
o
o o 1 1 ÿ cos y M
is the energy of the photon scattered by an angle y and z = cos(y ). The term (ds/dO)Born is the exact cross section for a proton with an anomalous magnetic moment k but no other structure (Powell, 1949)
ds dO
Born
2 0 2 0 1 e2 o o o 1 z2 2 4pM M2 o 2 2 2 9z 1 2k 1 ÿ z k ÿ 5z 2 2 3 2 4 3ÿz k 3 ÿ 2z ÿ z k , 4
2
with k=1.7928 being the anomalous magnetic moment of the proton. These expressions have the same general structure as the classical expression. For the proton, the leading term is the Born cross section, which is the relativistic counterpart of the Thomson cross section for a spin 1/2 particle with an anomalous magnetic moment. For the special case k=0, it is precisely the Klein±Nishina cross section. At zero energy, it reduces to the Thomson cross section. Moreover, as with the classical expression, the leading correction to the Born cross section is the interference between the Thomson amplitude and terms linear in the polarizabilities. The interference term is quadratic in the energy and is negative. It can be rigorously shown that no additional structure-dependent terms appear in the Compton cross section to this order in the expansion. Since the Born cross section is known precisely, the only unknowns to this order are a and b. The particular linear combination of a and b that enters into the cross section depends on the scattering angle. From Eq. (1) it is easy to see that the forward and backward cross sections are sensitive mainly to a+b and aÿb, respectively, whereas the 908 cross section is sensitive only to a. As with the classical system, a forward dispersion re4 The natural units for the nucleon polarizabilities is 10ÿ4 fm3 and will be understood hereafter.
Fig. 1. Calculations of the Compton scattering cross sections at 1358. The Born, LEX and ELEX curves are the expressions of Eqs. (2), (1), and (4), respectively, where the latter uses parameters A, B, C, D estimated using dispersion relations. The DR curve was calculated using ®xed-t dispersion relations and is valid up to about 300 MeV.
lation can be written leading to the following modelindependent sum rule, due originally to Baldin (Baldin, 1960): ab
1 2p2
1 0
sg o do 14:220:5 10ÿ4 fm3 , 3 o2
where sg(o ) is the total photoabsorption cross section on the proton. The numerical value4 is obtained using both the available experimental data and a reasonable theoretical ansatz for continuing the integral to in®nite energy (Damashek and Gilman, 1970). The uncertainty on this value (L'vov et al., 1979) includes a substantial systematic error. More recent analyses (Babusci et al., 1998a) lead to a slightly smaller number but still within the error quoted above. This sum rule has enormous practical implications, since it implies that the sum a+b is completely constrained by the total photoabsorption cross section. It can be tested by measuring the Compton scattering cross section in the forward direction. On the other hand, the dierence aÿb is not similarly constrained in a model independent way by sg (L'vov et al., 1979); indeed, the only direct way to determine aÿb is from Compton scattering in the backward direction. We will return to this point in Section 4.3.
116
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
Table 1 Data sets used in the present analysisa Data set
Energies (MeV)
Angles (deg)
Technique
Moscow-1960 (Gol'danski et al., 1960) Moscow-1975 (Baranov et al., 1975) Illinois-1991 (Federspiel et al., 1991 Mainz-1992 (Zieger et al., 1992) SAL-1993 (Hallin et al., 1993) Illinois/SAL-1995 (MacGibbon et al., 1995)
56 225 70±110 32±72 98, 132 136±150 70±150
75±150 [90] 60, 135 [180] 24±135 90, 135
Untagged Untagged Tagged Untagged/recoil Untagged Tagged & untagged
Syst. Err. (%) 6.0 ± 3.0 4.3 4.0 3.0
a
SAL refers to the Saskatchewan Accelerator Laboratory. The experiments utilize beams of tagged and/or untagged photons. The Moscow-1975 experiment combines statistical and systematic errors point by point in their tabulation of the cross sections and does not quote a separate systematic error. Moreover, the 1508 data from this experiment has been excluded, as discussed in the text.
The polarizabilities are determined by measuring the deviation of the cross sections from the Born values. A plot of the Born and LEX cross sections is shown in Fig. 1. The curves are very revealing in that they show that the cross sections are small, typically a few 10s of nb/sr. Moreover they show that the eect of the polarizabilities, which is proportional to the dierence between the Born and LEX curves, is even smaller, thereby placing great demand on the statistical precision and systematic accuracy of the measurements in order to obtain precise values for the polarizabilities. For example, at about 100 MeV and 1358, a change in aÿb by 1.0 changes the cross section by only 2%. Those demands can be relaxed by going to a higher photon energy, since the sensitivity of the cross section to the polarizabilities increases with energy. However if the energy becomes too large, the LEX breaks down and theoretical uncertainty is introduced into the extraction of the polarizabilities from the measured cross sections. Guided by the classical model, the natural energy scale for discussing the validity of the LEX is the threshold (o 1 mp ) for photoproduction of a pion, gp 4 pp, since no photoabsorption occurs below that energy. Therefore, one expects the LEX to be valid at energies suciently low compared to 150 MeV. This is demonstrated in Fig. 1, where the curve labeled DR is a model calculation of the cross section using dispersion relations and is valid, in principle, up 5 There has been a recent eort (Tonnison et al., 1998) to reconcile the Compton scattering data below the pion threshold with the data in the 150±500 MeV range. However, as of this writing there are outstanding issues (L'vov and Nathan, 1999a) that may soon be resolved by new experiments that will investigate the scattering of circularly polarized photons on polarized protons. Our preference for this review is to avoid these problems and use only the data below 150 MeV, since the polarizabilities can be extracted from those data with relatively little model dependence, as will be shown in Section 4.
to many hundreds of MeV. This curve shows that the LEX breaks down above about 90 MeV. Also shown is a curve labeled ELEX (or Extended Low Energy Expansion), which extends the expansion to order (oo ' )2 and which is valid below about 150 MeV. The use of the ELEX to extract the polarizabilities from the cross sections, as well as the model dependence associated with this technique, are discussed at length in Section 4.
3. The experimental data The discussion in the preceding section suggests that the experimental challenge for determining the proton polarizabilities is to measure an absolute cross section over an energy and angular range that is appropriately balanced between sensitivity to the polarizabilities and insensitivity to any theoretical model. With this in mind, the experiments considered in this review (Gol'danski et al., 1960; Baranov et al., 1975; Federspiel et al., 1991; Zieger et al., 1992; Hallin et al., 1993; MacGibbon et al., 1995) are summarized in Table 1. The experiments reported prior to 1990 were reviewed earlier by Petrun'kin (Petrun'kin, 1981) and the newer ones by MacGibbon (MacGibbon et al., 1995). Not shown are a number of recent Compton scattering measurements (Hallin et al., 1993; Molinari et al., 1996; Peise et al., 1996; Blanpied et al., 1996) at energies in the 150±500 MeV range, since these experiments are outside the range where the polarizabilities can be extracted from the cross sections without signi®cant model dependence.5 Also not shown are earlier works by B. Govorkov (Govorkov et al., 1956), Hyman (Hyman et al., 1959), and Oxley (Oxley, 1958) since the combined statistical and systematic error in these data preclude a meaningful constraint on a and b. The ®rst experiment to attempt speci®cally to determine the polarizabilities is the pioneering Moscow-
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
117
Fig. 2. The world Compton scattering cross sections on the proton at 908, including data from Moscow-1960 (closed square), Moscow-1975 (closed triangles), SAL-1993 (open squares), and Illinois/SAL-1995 (closed circles). The curve is the overall best ®t to all the scattering data up to 150 MeV using the ELEX.
1960 work of Gol'danski (Gol'danski et al., 1960) who used a bremsstrahlung photon beam with a 75 MeV endpoint and a detector with poor energy resolution. Nevertheless, a clever technique was developed that allowed the cross section to be normalized to the wellknown Klein±Nishina cross section for Compton scattering on the electron. The original result quoted in (Gol'danski et al., 1960) was a=92 2, where the error does not include the systematic error. Unfortunately, despite the normalization technique, the systematic error was quoted to be only 26%, suggesting that the overall uncertainty on a is 25. Therefore, like the earlier experiments, these data do not provide meaningful constraints on the polarizabilities. Nevertheless, the remarkable agreement between the published result for a and the results of the more recent measurements suggest that perhaps the authors overestimated their systematic errors. The Moscow-1975 experiment (Baranov et al., 1975) uses a bremsstrahlung beam and the normalization technique developed by Gol'danski. Data are reported at 908 and 1508 in the energy range 70±110 MeV, with a reported combined statistical and systematic uncertainty of 23%, leading to small uncertainties on the reported polarizabilities (a=10.7 2 1.1 and b=ÿ0.7 2 1.6). These numbers were extracted based on a ®t to the cross sections using the LEX, despite the fact that the maximum energy is outside the range of validity of the LEX. If one re®ts the cross sections
using techniques described in Section 4, one ®nds very dierent results (a=11.8 2 2.0 and b=ÿ5.8 2 1.8). The value of a, which is largely based on the 908 data, is in excellent agreement with the subsequent world data. However the value of aÿb, which is based on the 1508 data, is not comparably consistent. The net eect is that the value of a+b is badly inconsistent with the dispersion sum rule (Eq. 3). A closer look (MacGibbon et al., 1995) indicates that the 908 data are consistent with the subsequent world data (see Fig. 2) but the 1508 data are not (see Fig. 3). Therefore, in the global ®ts discussed below, only the 908 data are used. The Illinois-1991 experiment (Federspiel et al., 1991) had two very desirable features. First, it was done at both a forward and backward scattering angle and at low energies, so that model-independent determinations of both a+b (testing the dispersion sum rule and/or the systematics of the experiment) and aÿb were possible. Second, a tagged photon beam was used, thereby considerably improving the ability to measure absolute cross sections accurately. Unfortunately, the combined eects of low energy (implying low sensitivity) and the counting rate limitations inherent in a tagged photon experiment resulted in reduced statistical precision in the extracted polarizabilities. The Mainz-1992 experiment (Zieger et al., 1992) measured the Compton cross section at 1808, where
118
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123 Table 2 Results of global ®ts to the experimental cross sections, using the theoretical method given in Column 1 up to the maximum energy given in Column 2. Columns 3 and 4 are the ®tted values obtained without imposing the sum-rule constraint, Eq. (3). The ®nal column is the ®tted value of aÿb obtained using the sum-rule constraint for a+b. The ®rst quoted error is the combined statistical and systematic error and the second is an estimate of the model uncertainty inherent in the dispersion relation approach Theory
Emax (MeV)
LEX 88 ELEX 150 CELEX-a 150 CELEX-b 150
Fig. 3. The world Compton scattering cross sections on the proton at 1358, including data from Moscow-1960 (closed square), Illinois-1991 (open circles), SAL-1993 (open squares), and Illinois/SAL-1995 (closed circles). The closed triangles are the Moscow-1975 data at 1508 that have been extrapolated to 1358 cross sections using correction factors calculated with the ELEX. These data are not included in the ®t. The curve is the overall best ®t to the scattering data up to 150 MeV using the
the cross section is uniquely sensitive to aÿb, by detecting the recoil proton at 08 in a magnetic spectrometer, normalizing to the Compton cross section on the electron. The energy, 132 MeV, was a good compromise between sensitivity to the polarizabilities and model independence. Both the statistical and systematic errors were excellent, although aÿb was determined by just a single cross-section measurement at one energy and angle.6 The SAL-1993 experiment (Hallin et al., 1993) used a high duty-factor photon beam and a high-resolution NaI(Tl) detector to measure an extensive set of angular distributions via the bremsstrahlung endpoint technique. The statistical and systematic quality of the data were excellent. Although data were reported in the energy range 130±289 MeV, only the data below 150 MeV will be used in the analysis here in order to 6 Actually cross sections were measured at both 132 and 98 MeV, but the latter datum has poor statistical quality and does not provide a serious constraint on the polarizabilities. 7 Since the Moscow-1975 data were reported with their statistical and systematic errors combined point by point, the normalization for these data was ®xed.
a+b
aÿb
13.122.7 8.122.6 13.523.2 9.022.9
aÿb 7.62 2.3 8.82 2.7 9.62 2.5 9.121.1 21.0
avoid large uncertainties in the polarizabilities due to model-dependent eects. The Illinois/SAL-1995 experiment used an experimental technique in which measurements were done simultaneously using tagged photons (70±100 MeV) and untagged photons (100±148 MeV). Data were taken simultaneously at 908 and 1358 with a pair of large-volume NaI(Tl) detectors. The reported statistical and systematic quality of the data were excellent. A summary of the world data is given in Figs. 2 and 3 for the 908 and 1358 cross sections, respectively. It is gratifying to see that overall the data are very consistent with each other, with the exception noted above.
4. Extraction of polarizabilities from the data The polarizabilities are determined by ®tting the experimental cross sections to theoretical expressions, taking full account of the statistical and systematic errors. One example of a theoretical expression is the LEX, Eq. (1), whose validity is limited to low energies. In this section, other expressions will also be described that will allow the analysis to be extended to about 150 MeV, albeit with some model dependence. The inclusion of the systematic errors in the ®t is important since a 1% change in the overall normalization of the cross section results in a change in the extracted value of aÿb by approximately 0.5. A standard technique is used (D'Agostini, 1994) in which the normalization of each data set is allowed to vary in the ®t within the constraints allowed by the quoted systematic error for that data set.7 The net result is that each data set is properly weighted based on its systematic error, taking full account of the correlations in those systematic errors, and the uncertainties in the polarizabilities include contributions from both the statistical and the systematic errors. Fits are performed both with and
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
without imposing the sum rule constraint, Eq. (3). The latter is achieved by including an additional data set consisting of a single datum, whose experimental value and uncertainty are 14.2 2 0.5 and whose corresponding calculated value is equal to a+b. The results of our global ®ts to the world data is given in Table 2. 4.1. The low energy expansion As remarked above, a and b are determined by ®tting the experimental cross sections to Eq. (1), treating a and b as adjustable parameters, with or without imposition of the sum rule constraint Eq. (3). Only cross sections up to a maximum energy of 88 MeV are used in the ®tting. The results are given in line labeled LEX in Table 2. The sum and dierence of the polarizabilities are determined to about 22.3 (combined statistical and systematic uncertainties).
119
higher order polarizabilities and outlines a procedure for measuring them, including Compton scattering with unpolarized as well as polarized beams and/or targets. In this review, only the unpolarized cross sections are addressed. Comparison with the DR curve in Fig. 1 shows that the expansion to this order is accurate for energies below about 150 MeV, although that energy depends somewhat on the scattering angle (Babusci et al., 1998b). By ®tting this expression to the experimental cross section with six adjustable parameters (a, b, and A, B, C, D ), the polarizabilities can be extracted model-independently from the data. The result is given in the line labeled ELEX in Table 2. Despite the improved sensitivity to a and b implied by the higher energy, the uncertainty in the parameters is actually worse than with the LEX since additional parameters have to be determined from the data. In order to improve the situation, it is necessary to digress momentarily for a brief introduction to dispersion relations.
4.2. The extended low energy expansion
4.3. Dispersion relations
As shown originally by Guias° u and Radescu (Guias° u and Radescu, 1978) and more recently by Babusci (Babusci et al., 1998b), the expansion of the cross section can be extended to order (oo ' )2, resulting in a formula referred to as the ELEX:
Dispersion relations can be used to help constrain some of the higher order polarizabilities that appear in the ELEX formula, thereby improving the use of that technique for extracting a and b from the data. However, this necessarily introduces theoretical uncertainties into a and b which we will attempt to quantify in the next section. The theoretical approach and the computer code used for the numerical work are due to L'vov (L'vov, 1981; L'vov et al., 1997). The Compton scattering cross section is a function of six independent invariant amplitudes, Ai (n, t ), where n=(sÿu )/4M=(o+o' )/2, and s, t, and u are the usual Mandelstam kinematic variables. The low-energy behavior of the Compton cross section is completely determined by the lowenergy behavior of the Ai. Indeed, by doing such an expansion, one can derive the ELEX, Eq. (4), and relate the 10 polarizabilities to various combinations of the Ai and their derivatives at n=t = 0 (Babusci et al., 1998b). For example, the dierence of polarizabilities is given by
ELEX LEX p0 0 2 ds ds ds o 0 o o 2 dO dO dO o 04 A Bz Cz2 Dz3 O o 4 o ,
4
where the term labeled LEX is de®ned in Eq. (1). The 0 term (ds/dO)p is the contribution due to the t-channel exchange of a p 0 which is calculated exactly (Guias° u and Radescu, 1978; Babusci et al., 1998b) and explicitly separated out in order to extend the validity of the expansion to somewhat higher energy. The parameters A, B, C, D are new structure constants which themselves may be interesting to measure. As shown by Babusci (Babusci et al., 1998b), these are linear combinations of eight higher order polarizabilities which are classi®ed as follows: . Electric and magnetic quadrupole polarizabilities aE2 and bM2 . Dispersion corrections to the dipole polarizabilities aEn and bEn . Dipole and quadrupole spin polarizabilities, gE1, gM1, gE2 and gM2. These quantities along with a and b completely determine the Compton scattering cross section on the proton to order (oo ' )2. Babusci (Babusci et al., 1998b) gives classical and quantum interpretations of the
aÿbÿ
ANB 1 0, 0 , 2p
5
whereas the dierence of the quadrupole polarizabilities is given by aE2 ÿ bM2
6 @ ANB 1 ÿ , p @t nt0
6
where NB means the non-Born part (see discussion below).
120
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
A forward (i.e., at t = 0) dispersion relation can be written for Ai, which takes the following form at n=0: Re Ai 0, 0 ABorn 0, 0 i
Aasympt 0, i
2 p
o M o th
Im Ai o , 0
do o
7
0:
A similar equation can be written for the derivatives of the Ai. As we shall see, these equations lead to sum rules for the polarizabilities. The Born part of the amplitude can be calculated exactly and leads to the Born cross section, Eq. (2). The remaining two terms, the integral and asymptotic parts, constitute the nonBorn part of the amplitude, the low-energy limit of which is intimately connected with the polarizabilities. The integral part can be evaluated using unitarity to relate the imaginary part of the Ai to the multipole amplitudes for total photoabsorption, where the integral is taken from the pion photoproduction threshold oth 1 150 MeV to some large energy (typically oM=1.5 GeV). The integral part can be determined quite reliably. It is dominated by single-pion photoproduction, for which the important multipole amplitudes are well measured (Arndt et al., 1996). For multi-pion photoproduction, which dominates the photoabsorption cross section for energies greater than o0 600 MeV, the multipole amplitudes are poorly known experimentally and therefore need to be treated in the context of a model (L'vov, 1981). Fortunately, this contribution to the dispersion integral is suppressed by a large energy denominator so the integral part should depend only weakly on the model assumptions (MacGibbon et al., 1995). The asymptotic part, A asympt (0,0), takes into i account contributions of higher energies into the dispersion relations and is not constrained by our knowledge of the photoabsorption cross section. According to Regge phenomenology, only the amplitudes A1 and A2 can have a nonvanishing part at high energies and thus have a large asymptotic contribution. The physical origins of this contribution are the Feynman diagrams involving the t-channel exchange of a scalar or pseudoscalar meson for A1 and A2, respectively. For the other amplitudes (A3,4,5,6), the dispersion integrals provide a very reliable estimate of the corresponding non-Born parts. The combination of expressions like Eq. (5) and dispersion relations like Eq. (7) lead to sum rules for the polarizabilities. For 7 of the 10 polarizabilities contributing to the ELEX, there is no asymptotic part and therefore the corresponding sum rules are well constrained by the photoabsorption cross section. An example of a well-constrained quantity is the sum rule for a+b, Eq. (3). Similar but more complicated sum rules exist for other combinations of the 10 polarizabil-
ities (Babusci et al., 1998b; Drechsel et al., 1998). However, there are three combinations of polarizabilities, aÿb, aE2ÿbM2, and gp 0 ÿgE1+gM1+gE2ÿgM2, that have signi®cant asymptotic contributions so that their sum rules are not well constrained by the photoabsorption cross section. These are therefore new quantities to be determined from low-energy Compton scattering measurements. For example, the sum rule for aÿb reads aÿbÿ
o M o th
Im Ai o , 0
Aasympt 0 do ÿ 1 , 2 p o 2p
8
where the integral part is well constrained but the asymptotic part is not. As we shall see in Section 5, the asymptotic part dominates. In summary, forward dispersion relations can be used to predict some of the polarizabilities that appear in the ELEX formula, thereby oering the possibility to ®x those parameters in order to obtain an improved determination of aÿb, the primary quantity of interest. We next turn to an application of this procedure, which we call the Constrained Low Energy Expansion, or CELEX. 4.4. Constrained ELEX We ®t the ELEX formula, Eq. (4), to the data imposing constraints in the polarizabilities from the dispersion relations, after which there are only three unknown quantities: aÿb, aE2ÿbM2, and gp. Interestingly, all three quantities contribute to the Compton cross section primarily in the backward direction. In fact, aE2ÿbM2 and gp are essentially inseparable in the ELEX, since they both contribute to the angular distribution coecients in the combination AÿB+CÿD. Therefore, a constrained ELEX ®t can be done with essentially two free parameters, aÿb and AÿB+CÿD, with the remaining parameters (including a+b ) ®xed from the dispersion relations. The result of this analysis is given in the line labeled CELEX-a of Table 2. In fact, the precision of aÿb does not improve signi®cantly over the unconstrained ELEX ®t, since both it and AÿB+CÿD are sensitive primarily to cross sections in the backward direction and are therefore highly correlated. Indeed an inspection of the error matrix for this ®t shows a high degree of correlation. Although neither aE2ÿbM2 nor gp are well constrained by the forward dispersion relations, they can be constrained from other considerations. Recently a new sum rule was derived and evaluated for gp based on a 1808 dispersion relation (L'vov and Nathan, 1999a), with the result gp=(ÿ39.52 2.4) 10ÿ4 fm4. A similar sum rule can be written for aE2ÿbM2 (L'vov and Nathan, 1999b), with the result
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
121
Fig. 4. Integrands to the sum rule for a int (solid curve) and b int (dotted curve).
aE2ÿbM2=(50 26) 10ÿ4 fm5. In eect, these two quantities are better determined by the sum rules than they are by the Compton scattering data. Therefore, we ®x them (and thus all of A, B, C, D ) to their sum rule values and ®t the ELEX top the data with aÿb as the only free parameter. The result, which is listed as CELEX-b in Table 2, is a signi®cantly improved determination of aÿb with a precision of 21.1. However, it is now necessary to quantify how the uncertainties in the higher order polarizabilities (and therefore in A, B, C, D ) propagate to an uncertainty in aÿb. Therefore a series of ®ts were done with dierent sets of parameters derived by making adjustments to the higher order polarizabilities consistent with their uncertainties. By inspecting the spread of values of aÿb derived from this process, one can establish a model uncertainty, which we estimate to be 21.0. Most of this (20.7) comes from the uncertainty in aE2ÿbM2, with smaller amounts coming from gp and experimental uncertainties in the multipole amplitudes that enter into the integral contributions. We obtain as our ®nal result
Proton World Average:
a 11:720:620:5 b 2:530:630:5
9
The ®rst error is the combined statistical and systematic error and the second is the estimate of the model uncertainty. The errors in a and b are anticorrelated because of the precise sum rule determination of the
sum a+b. The curves in Figs. 2 and 3 were calculated with these values of the polarizabilities. 5. Discussion of results In recent years, various theoretical approaches have been used to calculate the polarizabilities of the nucleon, including non-relativistic quark models, bag models, chiral quark models, chiral perturbation theory, soliton models, and dispersion relations. An excellent review has been given by L'vov (L'vov, 1993). Here we elaborate on the forward dispersion relations, which we have seen allow sum rules to be established for the polarizabilities. These sum rules are completely rigorous yet semi-phenomenological, since they relate the polarizabilities to features of the photoabsorption cross section. This is physically appealing, since it helps identify the physics that gives rise to a and b, such as the contribution of a particular nucleon resonance. Two sum rules have already been presented: Eq. (3) for a+b and Eq. (8) for aÿb. It is illuminating to combine these to obtain sum rules for a and b separately, each of which can be written as a sum of an integral part and an asymptotic part: 1 a aint a ÿ basymp , 2
1 b bint ÿ a ÿ basymp 2
10
The integral part can be directly evaluated using the single-pion multipole amplitudes and the model for the
122
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
multi-pion multipole amplitudes, as discussed in the preceding section. The asymptotic part is just the dierence between the measured polarizability and the integral part. The values thus obtained are
ternal structure of the pion may contribute signi®cantly. However, these issues are very uncertain at present.
aint 16 bint 18 a ÿ basymp 111 ÿ 12
6. Summary and conclusions
11
We ®rst comment on the integral parts, the integrands for which are shown in Fig. 4. For a int the integral is dominated by multipoles involving nonresonant pion photoproduction,8 except for a negative contribution (0ÿ3) which comes from the excitation of the D resonance near 300 MeV. Otherwise, there is apparently very little contribution from degrees of freedom associated with excitations of the valence quarks. Indeed, a qualitative calculation in the context of the chiral bag model (Weiner and Weise, 1985) shows that both the electric polarizability and the negative or diamagnetic part of the magnetic polarizability are dominated by the polarization of the pion cloud relative to the quark core and have little to do with the polarization of the core itself. This notion is con®rmed by calculations using chiral perturbation theory at the oneloop order (Bernard et al., 1994). For b int the integral is dominated by the D resonance (Mukhopadhyay et al., 1993). We next comment on the asymptotic parts of the polarizabilities, which are neither small nor well constrained by the photoabsorption cross section. Indeed, the combination aÿb is almost entirely due to asymptotic contributions. As discussed in Section 4.3, these asymptotic contributions arise primarily from the tchannel exchange of a scalar meson or a correlated pair of pions in a relative 0+ state (for example, the s meson, which is responsible for the medium-range part of the NN interaction). Various attempts have been made to provide estimates of (aÿb )asympt using either experimental data on the linear polarization asymmetry in Compton scattering at 3.5 GeV or on estimates of the sNN and sgg couplings (L'vov et al., 1979; Petrun'kin, 1981). These estimates usually are low compared to the experimental value. Alternate estimates come from a sum rule based on a backward dispersion relation (Bernabeu and Tarrach, 1977; Holstein and Nathan, 1994) and the physical amplitudes for the process gg 4 pp 4 NN. Although not quantitatively successful, it is consistent with the picture that the pion cloud dominates and that the in8 The peak in the integrand near 200 MeV is not a resonance but an eect due to the opening of the single-pion photoproduction threshold near 150 MeV. Similarly, the broad bump in the vicinity of 600 MeV is associated with the opening of the two-pion threshold and not with a nucleon resonance.
A review of Compton scattering on the proton below the pion photoproduction threshold has been presented, with the primary emphasis on the electric and magnetic polarizabilities of the proton. The relationship between the polarizabilities and the Compton scattering cross section was established. The world experimental data were reviewed. Techniques were discussed for extracting the polarizabilities from the data based upon low-energy expansions of the cross section and sum rules based on dispersion relations. These techniques were then utilized to determine new values for a and b. A brief theoretical review was given with the principal emphasis on dispersion relations.
Acknowledgements It is a pleasure to recognize my many fruitful discussions and collaborations with Dr Anatoly L'vov and to thank him for a careful reading of this manuscript as well as helpful suggestions. The work was supported in part by the U.S. National Science Foundation under Grant No. 94-20787.
References Arndt, R.A., Strakovsky, I.I., Workman, R.L., 1996. Updated resonance photo-decay amplitudes to 2 GeV. Phys. Rev. C 53, 430. Babusci, D., Giordano, G., Matone, G., 1998a. A new evaluation of th Baldin sum rule. Phys. Rev. C 57, 291. Babusci, D., Giordano, G., L'vov, A.I., Matone, G., Nathan, A.M., 1998b. Low-energy Compton scattering of polarized photons on polarized nucleons. Phys. Rev. C 58, 1013. Baldin, A.M., 1960. Polarizability of nucleons. Nucl. Phys 18, 310. Baranov, P.S., et al., 1975. Elastic scattering of low-energy photons by protons. Sov. J. Nucl. Phys 21, 355. Bernabeu, J., Tarrach, B., 1977. Electromagnetic polarizabilities of the proton and the scalar-isoscalar gg 4 pp amplitude. Phys. Lett 69B, 484. Bernard, V., et al., 1994. Aspect of nucleon Compton scattering. Z. Phys. A 348, 317. Blanpied, G., et al., 1996. Polarized Compton scattering from the Proton. Phys. Rev. Lett 76, 2097. D'Agostini, G., 1994. On the use of the covariance matrix to ®t correlated data. Nucl. Instrum. Methods Phys. Res. Sect. A 346, 306.
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123 Damashek, M., Gilman, F.J., 1970. Forward Compton scattering. Phys. Rev. D 1, 1319. Drechsel, D., Krein, G., Hanstein, O., 1998. Dispersion relations and the spin polarizabilities of the nucleon. Phys. Lett. B 420, 248. Federspiel, F.J., et al., 1991. Proton Compton eect: a measurement of the electric and magnetic polarizabilities of the proton. Phys. Rev. Lett 67, 1511. Gol'danski, V.I., et al., 1960. Elastic g-p scattering at 40 to 70 MeV and polarizability of the proton. Sov. Phys. JETP 11, 1223. Govorkov, B.B., et al., 1956. Communications from the Academy of Sciences, USSR 111, 988. Guias° u, I., Radescu, E.E., 1978. Note on low-energy proton Compton scattering. Phys. Rev. D 18, 651. Hallin, E., et al., 1993. Compton scattering from the proton. Phys. Rev. C 48, 1497. Holstein, B.R., Nathan, A.M., 1994. Dispersion relations and the nucleon polarizability. Phys. Rev. D 49, 6101. Hyman, L.G., et al., 1959. Scattering of 50±140 MeV photons by protons and deuterons. Phys. Rev. Lett 3, 93. Jackson, J.D., 1961. Classical Electrodynamics. Wiley, New York. L'vov, A.I., 1981. Compton scattering on the proton at energies up to 400 MeV and ®nite-energy sum rules. Sov. J. Nucl. Phys 34, 597 and private communication. L'vov, A.I., 1993. Theoretical aspects of the polarizability of the nucleon. Int. J. Mod. Phys. A 8, 5267. L'vov, A.I., Nathan, A.M., 1999a. Sum rule for the backward spin polarizability of the nucleon from a backward dispersion relation. Phys. Rev. C 59, 1064.
123
L'vov, A.I., Nathan, A.M. 1999b, in preparation. L'vov, A.I., Petrun'kin, V.A., Startsev, S.A., 1979. Finiteenergy sum rule for the nucleon polarizability dierence. Sov. J. Nucl. Phys 29, 651. L'vov, A.I., Petrun'kin, V.A., Schumacher, M., 1997. Dispersion theory of proton Compton scattering in the ®rst and second resonance regions. Phys. Rev. C 55, 359. MacGibbon, B.E., et al., 1995. Measurement of the electric and magnetic polarizabilities of the proton. Phys. Rev. C 52, 2097. Molinari, C., et al., 1996. Scattering of photons by the proton through ycms=908 in the D-resonance region. Phys. Let. B 371, 181. Mukhopadhyay, N., Nathan, A.M., Zhang, L., 1993. D contributions to the paramegnetic polarizability of the proton. Phys. Rev. D 47, R7. Oxley, C.L., 1958. Scattering of 25±87 MeV photons by protons. Phys. Rev 110, 733. Peise, J., et al., 1996. Compton scattering of the proton through ycms=758 in the D-resonance region. Phys. Lett. B 384, 37. Petrun'kin, V.A., 1981. Electric and magnetic polarizabilities of hadrons. Sov. J. Part. Nucl 12, 278. Powell, J.L., 1949. Note on the bremsstrahlung produced by protons. Phys. Rev 75, 32. Tonnison, J., et al., 1998. First extraction of a spin polarizability of the proton. Phys. Rev. Lett 80, 4382. Weiner, R., Weise, W., 1985. Electromagnetic polarizability of the nucleon and chiral quark models. Phys. Lett 159B, 85. Zieger, A., et al., 1992. 1808 Compton scattering by the proton below the pion threshold. Phys. Lett 278B, 34.
www.elsevier.com/locate/radphyschem
Elastic photon±proton scattering: the polarizabilities of the proton Alan M. Nathan Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA
Abstract A review is presented of the elastic photon scattering data from the proton at energies below 150 MeV, where the important physics to be learned is the electric and magnetic polarizabilities of the proton. The relationship between the polarizabilities and the Compton scattering cross section is established. The experimental data are reviewed. Techniques for extracting the polarizabilities from the data are discussed and then used to obtain new values. Particular attention is paid to the theoretical uncertainty in this procedure. A brief theoretical review is given with the principal emphasis on dispersion relations. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The electric and magnetic polarizabilities, labeled a and b, respectively, measure the ease with which a static external electric or magnetic ®eld can induce an electric or magnetic dipole moment in a composite system.1 The proton itself is a composite system, and its
E-mail address: [email protected] (Alan M. Nathan) It has been conventional to designate Compton polarizabilities with ``bars'' in order to distinguish them from ``static'' polarizabilities. That convention is largely obsolete and is not followed here. Often the notations aE and bM is also used as a reminder of the electric and magnetic nature of these polarizabilities. Perhaps a better notation would be aE1 and bM1 to emphasize that the multipolarity is electric or magnetic dipole and to distinguish them from polarizabilities of dierent multipolarity, which are discussed later in this review. Neither of those notations will be used here. 2 Nuclear and particle physicists generally refer to elastic photon scattering as Compton scattering, and that terminology will be used here. This is dierent from atomic physics, where coherent elastic scattering from the atomic electrons is called Rayleigh scattering and incoherent quasi-elastic scattering from the electrons is called Compton scattering. 1
polarizabilities constitute fundamental structure constants which are as important as the charge and magnetic radii. In this paper, the current experimental status of the proton polarizabilities is reviewed. The link between the polarizabilities and Compton scattering is given in Section 2. The experimental data are reviewed in Section 3 and the results of global ®ts to extract the polarizabilities from the data are given in Section 4. The results obtained from this analysis are put into theoretical perspective in Section 5. A summary and conclusions are given in Section 6.
2. Polarizabilities and Compton scattering 2.1. Classical picture The relationship between low-energy Compton scattering2 on a composite system and the static electric and magnetic polarizabilities of that system is essentially a classical concept. It is instructive to examine a simple classical model in which two particles of charge and mass q1, q2 (total charge Q=q1+q2) and M1, M2 (total mass M=M1+M2) are on opposite ends of a
0969-806X/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 9 9 ) 0 0 2 9 0 - X
114
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
spring of force constant k. If a uniform electric ®eld is applied, the whole system will accelerate and the spring will stretch, giving rise to an induced electric dipole moment. The constant of proportionality between the induced moment and the applied ®eld is de®ned to be the electric polarizability, which for this system is easily shown to be3 a
q2eff , 4pmo 20
where qe=(q1M2ÿq2M1)/M, m is the reduced mass, and o 20=k/m. Using classical antenna theory (Jackson, 1961), the photon scattering cross section for this system can be derived: 2 ds Q2 1 cos 2 y , ÿ o 2 a o dO 2 4pM
ing from a composite system with Q 2/M$ 0 that the cross section is non-zero at o=0 and initially decreases. For a composite system with Q 2/M = 0 (such as a neutron or a neutral atom or molecule), there is no Thomson term and the leading term in the cross section is proportional to a 2o 4, leading to familiar eects such as the blue sky. The classical model presented here only includes the eects of the electric polarizability, but one could similarly construct a model exhibiting the eects of the magnetic polarizability. The total photoabsorption cross section sg for this simple system is exhausted by the single resonant excitation of the spring:
sg o 2p
2
q2eff 4pm
!2 d o ÿ o 0 ,
where y is the scattering angle and a o
q2eff 4pm o 20 ÿ o 2
is the dynamic (frequency-dependent) electric polarizability, which reduces to the static polarizability a in the limit o 4 0. For low frequencies (i.e., o<
leading to the following sum rule for a:
a
1 2p2
1 0
sg o do : o2
The physical origin of this sum rule is familiar from optics and relies on very fundamental principles of physics. First, causality leads to a dispersion relation relating the real part of the scattering amplitude to an integral over the imaginary part. Second, unitarity leads to a unique relationship between the imaginary part of the forward amplitude and sg (the Optical Theorem). These two ingredients combined with the low-energy form of the scattering amplitude lead uniquely to the sum rule. We close this section by remarking that this classical model can be used to obtain an approximate value of a for the proton. Suppose one end of the spring has two u quarks (q1=4e/3; M1 1 600 MeV) and the other end has one d quark (q2=ÿe/3; M2 1 300 MeV). With o0 adjusted to reproduce the approximate energy of dipole excitations in the proton (o0 1 500 MeV), we obtain a 1 5 10ÿ4 fm3, which is within about a factor of 2 of the experimental value. Interestingly, if the identity of u and d quarks are interchanged, thereby turning a proton into a neutron, the same value of a is obtained. 2.2. Compton scattering on the proton For Compton scattering from the proton, one can similarly expand the cross section in powers of o, leading to the following Low Energy Expansion (LEX) (Petrun'kin, 1981):
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
ds dO
LEX o ,y
ds dO
115
Born o ,y
0 2 e2 o ab 0 ÿ 1 z2 o o 2 4pM o aÿb 2 1 ÿ z , 2
1
where 0
o
o o 1 1 ÿ cos y M
is the energy of the photon scattered by an angle y and z = cos(y ). The term (ds/dO)Born is the exact cross section for a proton with an anomalous magnetic moment k but no other structure (Powell, 1949)
ds dO
Born
2 0 2 0 1 e2 o o o 1 z2 2 4pM M2 o 2 2 2 9z 1 2k 1 ÿ z k ÿ 5z 2 2 3 2 4 3ÿz k 3 ÿ 2z ÿ z k , 4
2
with k=1.7928 being the anomalous magnetic moment of the proton. These expressions have the same general structure as the classical expression. For the proton, the leading term is the Born cross section, which is the relativistic counterpart of the Thomson cross section for a spin 1/2 particle with an anomalous magnetic moment. For the special case k=0, it is precisely the Klein±Nishina cross section. At zero energy, it reduces to the Thomson cross section. Moreover, as with the classical expression, the leading correction to the Born cross section is the interference between the Thomson amplitude and terms linear in the polarizabilities. The interference term is quadratic in the energy and is negative. It can be rigorously shown that no additional structure-dependent terms appear in the Compton cross section to this order in the expansion. Since the Born cross section is known precisely, the only unknowns to this order are a and b. The particular linear combination of a and b that enters into the cross section depends on the scattering angle. From Eq. (1) it is easy to see that the forward and backward cross sections are sensitive mainly to a+b and aÿb, respectively, whereas the 908 cross section is sensitive only to a. As with the classical system, a forward dispersion re4 The natural units for the nucleon polarizabilities is 10ÿ4 fm3 and will be understood hereafter.
Fig. 1. Calculations of the Compton scattering cross sections at 1358. The Born, LEX and ELEX curves are the expressions of Eqs. (2), (1), and (4), respectively, where the latter uses parameters A, B, C, D estimated using dispersion relations. The DR curve was calculated using ®xed-t dispersion relations and is valid up to about 300 MeV.
lation can be written leading to the following modelindependent sum rule, due originally to Baldin (Baldin, 1960): ab
1 2p2
1 0
sg o do 14:220:5 10ÿ4 fm3 , 3 o2
where sg(o ) is the total photoabsorption cross section on the proton. The numerical value4 is obtained using both the available experimental data and a reasonable theoretical ansatz for continuing the integral to in®nite energy (Damashek and Gilman, 1970). The uncertainty on this value (L'vov et al., 1979) includes a substantial systematic error. More recent analyses (Babusci et al., 1998a) lead to a slightly smaller number but still within the error quoted above. This sum rule has enormous practical implications, since it implies that the sum a+b is completely constrained by the total photoabsorption cross section. It can be tested by measuring the Compton scattering cross section in the forward direction. On the other hand, the dierence aÿb is not similarly constrained in a model independent way by sg (L'vov et al., 1979); indeed, the only direct way to determine aÿb is from Compton scattering in the backward direction. We will return to this point in Section 4.3.
116
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
Table 1 Data sets used in the present analysisa Data set
Energies (MeV)
Angles (deg)
Technique
Moscow-1960 (Gol'danski et al., 1960) Moscow-1975 (Baranov et al., 1975) Illinois-1991 (Federspiel et al., 1991 Mainz-1992 (Zieger et al., 1992) SAL-1993 (Hallin et al., 1993) Illinois/SAL-1995 (MacGibbon et al., 1995)
56 225 70±110 32±72 98, 132 136±150 70±150
75±150 [90] 60, 135 [180] 24±135 90, 135
Untagged Untagged Tagged Untagged/recoil Untagged Tagged & untagged
Syst. Err. (%) 6.0 ± 3.0 4.3 4.0 3.0
a
SAL refers to the Saskatchewan Accelerator Laboratory. The experiments utilize beams of tagged and/or untagged photons. The Moscow-1975 experiment combines statistical and systematic errors point by point in their tabulation of the cross sections and does not quote a separate systematic error. Moreover, the 1508 data from this experiment has been excluded, as discussed in the text.
The polarizabilities are determined by measuring the deviation of the cross sections from the Born values. A plot of the Born and LEX cross sections is shown in Fig. 1. The curves are very revealing in that they show that the cross sections are small, typically a few 10s of nb/sr. Moreover they show that the eect of the polarizabilities, which is proportional to the dierence between the Born and LEX curves, is even smaller, thereby placing great demand on the statistical precision and systematic accuracy of the measurements in order to obtain precise values for the polarizabilities. For example, at about 100 MeV and 1358, a change in aÿb by 1.0 changes the cross section by only 2%. Those demands can be relaxed by going to a higher photon energy, since the sensitivity of the cross section to the polarizabilities increases with energy. However if the energy becomes too large, the LEX breaks down and theoretical uncertainty is introduced into the extraction of the polarizabilities from the measured cross sections. Guided by the classical model, the natural energy scale for discussing the validity of the LEX is the threshold (o 1 mp ) for photoproduction of a pion, gp 4 pp, since no photoabsorption occurs below that energy. Therefore, one expects the LEX to be valid at energies suciently low compared to 150 MeV. This is demonstrated in Fig. 1, where the curve labeled DR is a model calculation of the cross section using dispersion relations and is valid, in principle, up 5 There has been a recent eort (Tonnison et al., 1998) to reconcile the Compton scattering data below the pion threshold with the data in the 150±500 MeV range. However, as of this writing there are outstanding issues (L'vov and Nathan, 1999a) that may soon be resolved by new experiments that will investigate the scattering of circularly polarized photons on polarized protons. Our preference for this review is to avoid these problems and use only the data below 150 MeV, since the polarizabilities can be extracted from those data with relatively little model dependence, as will be shown in Section 4.
to many hundreds of MeV. This curve shows that the LEX breaks down above about 90 MeV. Also shown is a curve labeled ELEX (or Extended Low Energy Expansion), which extends the expansion to order (oo ' )2 and which is valid below about 150 MeV. The use of the ELEX to extract the polarizabilities from the cross sections, as well as the model dependence associated with this technique, are discussed at length in Section 4.
3. The experimental data The discussion in the preceding section suggests that the experimental challenge for determining the proton polarizabilities is to measure an absolute cross section over an energy and angular range that is appropriately balanced between sensitivity to the polarizabilities and insensitivity to any theoretical model. With this in mind, the experiments considered in this review (Gol'danski et al., 1960; Baranov et al., 1975; Federspiel et al., 1991; Zieger et al., 1992; Hallin et al., 1993; MacGibbon et al., 1995) are summarized in Table 1. The experiments reported prior to 1990 were reviewed earlier by Petrun'kin (Petrun'kin, 1981) and the newer ones by MacGibbon (MacGibbon et al., 1995). Not shown are a number of recent Compton scattering measurements (Hallin et al., 1993; Molinari et al., 1996; Peise et al., 1996; Blanpied et al., 1996) at energies in the 150±500 MeV range, since these experiments are outside the range where the polarizabilities can be extracted from the cross sections without signi®cant model dependence.5 Also not shown are earlier works by B. Govorkov (Govorkov et al., 1956), Hyman (Hyman et al., 1959), and Oxley (Oxley, 1958) since the combined statistical and systematic error in these data preclude a meaningful constraint on a and b. The ®rst experiment to attempt speci®cally to determine the polarizabilities is the pioneering Moscow-
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
117
Fig. 2. The world Compton scattering cross sections on the proton at 908, including data from Moscow-1960 (closed square), Moscow-1975 (closed triangles), SAL-1993 (open squares), and Illinois/SAL-1995 (closed circles). The curve is the overall best ®t to all the scattering data up to 150 MeV using the ELEX.
1960 work of Gol'danski (Gol'danski et al., 1960) who used a bremsstrahlung photon beam with a 75 MeV endpoint and a detector with poor energy resolution. Nevertheless, a clever technique was developed that allowed the cross section to be normalized to the wellknown Klein±Nishina cross section for Compton scattering on the electron. The original result quoted in (Gol'danski et al., 1960) was a=92 2, where the error does not include the systematic error. Unfortunately, despite the normalization technique, the systematic error was quoted to be only 26%, suggesting that the overall uncertainty on a is 25. Therefore, like the earlier experiments, these data do not provide meaningful constraints on the polarizabilities. Nevertheless, the remarkable agreement between the published result for a and the results of the more recent measurements suggest that perhaps the authors overestimated their systematic errors. The Moscow-1975 experiment (Baranov et al., 1975) uses a bremsstrahlung beam and the normalization technique developed by Gol'danski. Data are reported at 908 and 1508 in the energy range 70±110 MeV, with a reported combined statistical and systematic uncertainty of 23%, leading to small uncertainties on the reported polarizabilities (a=10.7 2 1.1 and b=ÿ0.7 2 1.6). These numbers were extracted based on a ®t to the cross sections using the LEX, despite the fact that the maximum energy is outside the range of validity of the LEX. If one re®ts the cross sections
using techniques described in Section 4, one ®nds very dierent results (a=11.8 2 2.0 and b=ÿ5.8 2 1.8). The value of a, which is largely based on the 908 data, is in excellent agreement with the subsequent world data. However the value of aÿb, which is based on the 1508 data, is not comparably consistent. The net eect is that the value of a+b is badly inconsistent with the dispersion sum rule (Eq. 3). A closer look (MacGibbon et al., 1995) indicates that the 908 data are consistent with the subsequent world data (see Fig. 2) but the 1508 data are not (see Fig. 3). Therefore, in the global ®ts discussed below, only the 908 data are used. The Illinois-1991 experiment (Federspiel et al., 1991) had two very desirable features. First, it was done at both a forward and backward scattering angle and at low energies, so that model-independent determinations of both a+b (testing the dispersion sum rule and/or the systematics of the experiment) and aÿb were possible. Second, a tagged photon beam was used, thereby considerably improving the ability to measure absolute cross sections accurately. Unfortunately, the combined eects of low energy (implying low sensitivity) and the counting rate limitations inherent in a tagged photon experiment resulted in reduced statistical precision in the extracted polarizabilities. The Mainz-1992 experiment (Zieger et al., 1992) measured the Compton cross section at 1808, where
118
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123 Table 2 Results of global ®ts to the experimental cross sections, using the theoretical method given in Column 1 up to the maximum energy given in Column 2. Columns 3 and 4 are the ®tted values obtained without imposing the sum-rule constraint, Eq. (3). The ®nal column is the ®tted value of aÿb obtained using the sum-rule constraint for a+b. The ®rst quoted error is the combined statistical and systematic error and the second is an estimate of the model uncertainty inherent in the dispersion relation approach Theory
Emax (MeV)
LEX 88 ELEX 150 CELEX-a 150 CELEX-b 150
Fig. 3. The world Compton scattering cross sections on the proton at 1358, including data from Moscow-1960 (closed square), Illinois-1991 (open circles), SAL-1993 (open squares), and Illinois/SAL-1995 (closed circles). The closed triangles are the Moscow-1975 data at 1508 that have been extrapolated to 1358 cross sections using correction factors calculated with the ELEX. These data are not included in the ®t. The curve is the overall best ®t to the scattering data up to 150 MeV using the
the cross section is uniquely sensitive to aÿb, by detecting the recoil proton at 08 in a magnetic spectrometer, normalizing to the Compton cross section on the electron. The energy, 132 MeV, was a good compromise between sensitivity to the polarizabilities and model independence. Both the statistical and systematic errors were excellent, although aÿb was determined by just a single cross-section measurement at one energy and angle.6 The SAL-1993 experiment (Hallin et al., 1993) used a high duty-factor photon beam and a high-resolution NaI(Tl) detector to measure an extensive set of angular distributions via the bremsstrahlung endpoint technique. The statistical and systematic quality of the data were excellent. Although data were reported in the energy range 130±289 MeV, only the data below 150 MeV will be used in the analysis here in order to 6 Actually cross sections were measured at both 132 and 98 MeV, but the latter datum has poor statistical quality and does not provide a serious constraint on the polarizabilities. 7 Since the Moscow-1975 data were reported with their statistical and systematic errors combined point by point, the normalization for these data was ®xed.
a+b
aÿb
13.122.7 8.122.6 13.523.2 9.022.9
aÿb 7.62 2.3 8.82 2.7 9.62 2.5 9.121.1 21.0
avoid large uncertainties in the polarizabilities due to model-dependent eects. The Illinois/SAL-1995 experiment used an experimental technique in which measurements were done simultaneously using tagged photons (70±100 MeV) and untagged photons (100±148 MeV). Data were taken simultaneously at 908 and 1358 with a pair of large-volume NaI(Tl) detectors. The reported statistical and systematic quality of the data were excellent. A summary of the world data is given in Figs. 2 and 3 for the 908 and 1358 cross sections, respectively. It is gratifying to see that overall the data are very consistent with each other, with the exception noted above.
4. Extraction of polarizabilities from the data The polarizabilities are determined by ®tting the experimental cross sections to theoretical expressions, taking full account of the statistical and systematic errors. One example of a theoretical expression is the LEX, Eq. (1), whose validity is limited to low energies. In this section, other expressions will also be described that will allow the analysis to be extended to about 150 MeV, albeit with some model dependence. The inclusion of the systematic errors in the ®t is important since a 1% change in the overall normalization of the cross section results in a change in the extracted value of aÿb by approximately 0.5. A standard technique is used (D'Agostini, 1994) in which the normalization of each data set is allowed to vary in the ®t within the constraints allowed by the quoted systematic error for that data set.7 The net result is that each data set is properly weighted based on its systematic error, taking full account of the correlations in those systematic errors, and the uncertainties in the polarizabilities include contributions from both the statistical and the systematic errors. Fits are performed both with and
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
without imposing the sum rule constraint, Eq. (3). The latter is achieved by including an additional data set consisting of a single datum, whose experimental value and uncertainty are 14.2 2 0.5 and whose corresponding calculated value is equal to a+b. The results of our global ®ts to the world data is given in Table 2. 4.1. The low energy expansion As remarked above, a and b are determined by ®tting the experimental cross sections to Eq. (1), treating a and b as adjustable parameters, with or without imposition of the sum rule constraint Eq. (3). Only cross sections up to a maximum energy of 88 MeV are used in the ®tting. The results are given in line labeled LEX in Table 2. The sum and dierence of the polarizabilities are determined to about 22.3 (combined statistical and systematic uncertainties).
119
higher order polarizabilities and outlines a procedure for measuring them, including Compton scattering with unpolarized as well as polarized beams and/or targets. In this review, only the unpolarized cross sections are addressed. Comparison with the DR curve in Fig. 1 shows that the expansion to this order is accurate for energies below about 150 MeV, although that energy depends somewhat on the scattering angle (Babusci et al., 1998b). By ®tting this expression to the experimental cross section with six adjustable parameters (a, b, and A, B, C, D ), the polarizabilities can be extracted model-independently from the data. The result is given in the line labeled ELEX in Table 2. Despite the improved sensitivity to a and b implied by the higher energy, the uncertainty in the parameters is actually worse than with the LEX since additional parameters have to be determined from the data. In order to improve the situation, it is necessary to digress momentarily for a brief introduction to dispersion relations.
4.2. The extended low energy expansion
4.3. Dispersion relations
As shown originally by Guias° u and Radescu (Guias° u and Radescu, 1978) and more recently by Babusci (Babusci et al., 1998b), the expansion of the cross section can be extended to order (oo ' )2, resulting in a formula referred to as the ELEX:
Dispersion relations can be used to help constrain some of the higher order polarizabilities that appear in the ELEX formula, thereby improving the use of that technique for extracting a and b from the data. However, this necessarily introduces theoretical uncertainties into a and b which we will attempt to quantify in the next section. The theoretical approach and the computer code used for the numerical work are due to L'vov (L'vov, 1981; L'vov et al., 1997). The Compton scattering cross section is a function of six independent invariant amplitudes, Ai (n, t ), where n=(sÿu )/4M=(o+o' )/2, and s, t, and u are the usual Mandelstam kinematic variables. The low-energy behavior of the Compton cross section is completely determined by the lowenergy behavior of the Ai. Indeed, by doing such an expansion, one can derive the ELEX, Eq. (4), and relate the 10 polarizabilities to various combinations of the Ai and their derivatives at n=t = 0 (Babusci et al., 1998b). For example, the dierence of polarizabilities is given by
ELEX LEX p0 0 2 ds ds ds o 0 o o 2 dO dO dO o 04 A Bz Cz2 Dz3 O o 4 o ,
4
where the term labeled LEX is de®ned in Eq. (1). The 0 term (ds/dO)p is the contribution due to the t-channel exchange of a p 0 which is calculated exactly (Guias° u and Radescu, 1978; Babusci et al., 1998b) and explicitly separated out in order to extend the validity of the expansion to somewhat higher energy. The parameters A, B, C, D are new structure constants which themselves may be interesting to measure. As shown by Babusci (Babusci et al., 1998b), these are linear combinations of eight higher order polarizabilities which are classi®ed as follows: . Electric and magnetic quadrupole polarizabilities aE2 and bM2 . Dispersion corrections to the dipole polarizabilities aEn and bEn . Dipole and quadrupole spin polarizabilities, gE1, gM1, gE2 and gM2. These quantities along with a and b completely determine the Compton scattering cross section on the proton to order (oo ' )2. Babusci (Babusci et al., 1998b) gives classical and quantum interpretations of the
aÿbÿ
ANB 1 0, 0 , 2p
5
whereas the dierence of the quadrupole polarizabilities is given by aE2 ÿ bM2
6 @ ANB 1 ÿ , p @t nt0
6
where NB means the non-Born part (see discussion below).
120
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
A forward (i.e., at t = 0) dispersion relation can be written for Ai, which takes the following form at n=0: Re Ai 0, 0 ABorn 0, 0 i
Aasympt 0, i
2 p
o M o th
Im Ai o , 0
do o
7
0:
A similar equation can be written for the derivatives of the Ai. As we shall see, these equations lead to sum rules for the polarizabilities. The Born part of the amplitude can be calculated exactly and leads to the Born cross section, Eq. (2). The remaining two terms, the integral and asymptotic parts, constitute the nonBorn part of the amplitude, the low-energy limit of which is intimately connected with the polarizabilities. The integral part can be evaluated using unitarity to relate the imaginary part of the Ai to the multipole amplitudes for total photoabsorption, where the integral is taken from the pion photoproduction threshold oth 1 150 MeV to some large energy (typically oM=1.5 GeV). The integral part can be determined quite reliably. It is dominated by single-pion photoproduction, for which the important multipole amplitudes are well measured (Arndt et al., 1996). For multi-pion photoproduction, which dominates the photoabsorption cross section for energies greater than o0 600 MeV, the multipole amplitudes are poorly known experimentally and therefore need to be treated in the context of a model (L'vov, 1981). Fortunately, this contribution to the dispersion integral is suppressed by a large energy denominator so the integral part should depend only weakly on the model assumptions (MacGibbon et al., 1995). The asymptotic part, A asympt (0,0), takes into i account contributions of higher energies into the dispersion relations and is not constrained by our knowledge of the photoabsorption cross section. According to Regge phenomenology, only the amplitudes A1 and A2 can have a nonvanishing part at high energies and thus have a large asymptotic contribution. The physical origins of this contribution are the Feynman diagrams involving the t-channel exchange of a scalar or pseudoscalar meson for A1 and A2, respectively. For the other amplitudes (A3,4,5,6), the dispersion integrals provide a very reliable estimate of the corresponding non-Born parts. The combination of expressions like Eq. (5) and dispersion relations like Eq. (7) lead to sum rules for the polarizabilities. For 7 of the 10 polarizabilities contributing to the ELEX, there is no asymptotic part and therefore the corresponding sum rules are well constrained by the photoabsorption cross section. An example of a well-constrained quantity is the sum rule for a+b, Eq. (3). Similar but more complicated sum rules exist for other combinations of the 10 polarizabil-
ities (Babusci et al., 1998b; Drechsel et al., 1998). However, there are three combinations of polarizabilities, aÿb, aE2ÿbM2, and gp 0 ÿgE1+gM1+gE2ÿgM2, that have signi®cant asymptotic contributions so that their sum rules are not well constrained by the photoabsorption cross section. These are therefore new quantities to be determined from low-energy Compton scattering measurements. For example, the sum rule for aÿb reads aÿbÿ
o M o th
Im Ai o , 0
Aasympt 0 do ÿ 1 , 2 p o 2p
8
where the integral part is well constrained but the asymptotic part is not. As we shall see in Section 5, the asymptotic part dominates. In summary, forward dispersion relations can be used to predict some of the polarizabilities that appear in the ELEX formula, thereby oering the possibility to ®x those parameters in order to obtain an improved determination of aÿb, the primary quantity of interest. We next turn to an application of this procedure, which we call the Constrained Low Energy Expansion, or CELEX. 4.4. Constrained ELEX We ®t the ELEX formula, Eq. (4), to the data imposing constraints in the polarizabilities from the dispersion relations, after which there are only three unknown quantities: aÿb, aE2ÿbM2, and gp. Interestingly, all three quantities contribute to the Compton cross section primarily in the backward direction. In fact, aE2ÿbM2 and gp are essentially inseparable in the ELEX, since they both contribute to the angular distribution coecients in the combination AÿB+CÿD. Therefore, a constrained ELEX ®t can be done with essentially two free parameters, aÿb and AÿB+CÿD, with the remaining parameters (including a+b ) ®xed from the dispersion relations. The result of this analysis is given in the line labeled CELEX-a of Table 2. In fact, the precision of aÿb does not improve signi®cantly over the unconstrained ELEX ®t, since both it and AÿB+CÿD are sensitive primarily to cross sections in the backward direction and are therefore highly correlated. Indeed an inspection of the error matrix for this ®t shows a high degree of correlation. Although neither aE2ÿbM2 nor gp are well constrained by the forward dispersion relations, they can be constrained from other considerations. Recently a new sum rule was derived and evaluated for gp based on a 1808 dispersion relation (L'vov and Nathan, 1999a), with the result gp=(ÿ39.52 2.4) 10ÿ4 fm4. A similar sum rule can be written for aE2ÿbM2 (L'vov and Nathan, 1999b), with the result
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
121
Fig. 4. Integrands to the sum rule for a int (solid curve) and b int (dotted curve).
aE2ÿbM2=(50 26) 10ÿ4 fm5. In eect, these two quantities are better determined by the sum rules than they are by the Compton scattering data. Therefore, we ®x them (and thus all of A, B, C, D ) to their sum rule values and ®t the ELEX top the data with aÿb as the only free parameter. The result, which is listed as CELEX-b in Table 2, is a signi®cantly improved determination of aÿb with a precision of 21.1. However, it is now necessary to quantify how the uncertainties in the higher order polarizabilities (and therefore in A, B, C, D ) propagate to an uncertainty in aÿb. Therefore a series of ®ts were done with dierent sets of parameters derived by making adjustments to the higher order polarizabilities consistent with their uncertainties. By inspecting the spread of values of aÿb derived from this process, one can establish a model uncertainty, which we estimate to be 21.0. Most of this (20.7) comes from the uncertainty in aE2ÿbM2, with smaller amounts coming from gp and experimental uncertainties in the multipole amplitudes that enter into the integral contributions. We obtain as our ®nal result
Proton World Average:
a 11:720:620:5 b 2:530:630:5
9
The ®rst error is the combined statistical and systematic error and the second is the estimate of the model uncertainty. The errors in a and b are anticorrelated because of the precise sum rule determination of the
sum a+b. The curves in Figs. 2 and 3 were calculated with these values of the polarizabilities. 5. Discussion of results In recent years, various theoretical approaches have been used to calculate the polarizabilities of the nucleon, including non-relativistic quark models, bag models, chiral quark models, chiral perturbation theory, soliton models, and dispersion relations. An excellent review has been given by L'vov (L'vov, 1993). Here we elaborate on the forward dispersion relations, which we have seen allow sum rules to be established for the polarizabilities. These sum rules are completely rigorous yet semi-phenomenological, since they relate the polarizabilities to features of the photoabsorption cross section. This is physically appealing, since it helps identify the physics that gives rise to a and b, such as the contribution of a particular nucleon resonance. Two sum rules have already been presented: Eq. (3) for a+b and Eq. (8) for aÿb. It is illuminating to combine these to obtain sum rules for a and b separately, each of which can be written as a sum of an integral part and an asymptotic part: 1 a aint a ÿ basymp , 2
1 b bint ÿ a ÿ basymp 2
10
The integral part can be directly evaluated using the single-pion multipole amplitudes and the model for the
122
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123
multi-pion multipole amplitudes, as discussed in the preceding section. The asymptotic part is just the dierence between the measured polarizability and the integral part. The values thus obtained are
ternal structure of the pion may contribute signi®cantly. However, these issues are very uncertain at present.
aint 16 bint 18 a ÿ basymp 111 ÿ 12
6. Summary and conclusions
11
We ®rst comment on the integral parts, the integrands for which are shown in Fig. 4. For a int the integral is dominated by multipoles involving nonresonant pion photoproduction,8 except for a negative contribution (0ÿ3) which comes from the excitation of the D resonance near 300 MeV. Otherwise, there is apparently very little contribution from degrees of freedom associated with excitations of the valence quarks. Indeed, a qualitative calculation in the context of the chiral bag model (Weiner and Weise, 1985) shows that both the electric polarizability and the negative or diamagnetic part of the magnetic polarizability are dominated by the polarization of the pion cloud relative to the quark core and have little to do with the polarization of the core itself. This notion is con®rmed by calculations using chiral perturbation theory at the oneloop order (Bernard et al., 1994). For b int the integral is dominated by the D resonance (Mukhopadhyay et al., 1993). We next comment on the asymptotic parts of the polarizabilities, which are neither small nor well constrained by the photoabsorption cross section. Indeed, the combination aÿb is almost entirely due to asymptotic contributions. As discussed in Section 4.3, these asymptotic contributions arise primarily from the tchannel exchange of a scalar meson or a correlated pair of pions in a relative 0+ state (for example, the s meson, which is responsible for the medium-range part of the NN interaction). Various attempts have been made to provide estimates of (aÿb )asympt using either experimental data on the linear polarization asymmetry in Compton scattering at 3.5 GeV or on estimates of the sNN and sgg couplings (L'vov et al., 1979; Petrun'kin, 1981). These estimates usually are low compared to the experimental value. Alternate estimates come from a sum rule based on a backward dispersion relation (Bernabeu and Tarrach, 1977; Holstein and Nathan, 1994) and the physical amplitudes for the process gg 4 pp 4 NN. Although not quantitatively successful, it is consistent with the picture that the pion cloud dominates and that the in8 The peak in the integrand near 200 MeV is not a resonance but an eect due to the opening of the single-pion photoproduction threshold near 150 MeV. Similarly, the broad bump in the vicinity of 600 MeV is associated with the opening of the two-pion threshold and not with a nucleon resonance.
A review of Compton scattering on the proton below the pion photoproduction threshold has been presented, with the primary emphasis on the electric and magnetic polarizabilities of the proton. The relationship between the polarizabilities and the Compton scattering cross section was established. The world experimental data were reviewed. Techniques were discussed for extracting the polarizabilities from the data based upon low-energy expansions of the cross section and sum rules based on dispersion relations. These techniques were then utilized to determine new values for a and b. A brief theoretical review was given with the principal emphasis on dispersion relations.
Acknowledgements It is a pleasure to recognize my many fruitful discussions and collaborations with Dr Anatoly L'vov and to thank him for a careful reading of this manuscript as well as helpful suggestions. The work was supported in part by the U.S. National Science Foundation under Grant No. 94-20787.
References Arndt, R.A., Strakovsky, I.I., Workman, R.L., 1996. Updated resonance photo-decay amplitudes to 2 GeV. Phys. Rev. C 53, 430. Babusci, D., Giordano, G., Matone, G., 1998a. A new evaluation of th Baldin sum rule. Phys. Rev. C 57, 291. Babusci, D., Giordano, G., L'vov, A.I., Matone, G., Nathan, A.M., 1998b. Low-energy Compton scattering of polarized photons on polarized nucleons. Phys. Rev. C 58, 1013. Baldin, A.M., 1960. Polarizability of nucleons. Nucl. Phys 18, 310. Baranov, P.S., et al., 1975. Elastic scattering of low-energy photons by protons. Sov. J. Nucl. Phys 21, 355. Bernabeu, J., Tarrach, B., 1977. Electromagnetic polarizabilities of the proton and the scalar-isoscalar gg 4 pp amplitude. Phys. Lett 69B, 484. Bernard, V., et al., 1994. Aspect of nucleon Compton scattering. Z. Phys. A 348, 317. Blanpied, G., et al., 1996. Polarized Compton scattering from the Proton. Phys. Rev. Lett 76, 2097. D'Agostini, G., 1994. On the use of the covariance matrix to ®t correlated data. Nucl. Instrum. Methods Phys. Res. Sect. A 346, 306.
A.M. Nathan / Radiation Physics and Chemistry 56 (1999) 113±123 Damashek, M., Gilman, F.J., 1970. Forward Compton scattering. Phys. Rev. D 1, 1319. Drechsel, D., Krein, G., Hanstein, O., 1998. Dispersion relations and the spin polarizabilities of the nucleon. Phys. Lett. B 420, 248. Federspiel, F.J., et al., 1991. Proton Compton eect: a measurement of the electric and magnetic polarizabilities of the proton. Phys. Rev. Lett 67, 1511. Gol'danski, V.I., et al., 1960. Elastic g-p scattering at 40 to 70 MeV and polarizability of the proton. Sov. Phys. JETP 11, 1223. Govorkov, B.B., et al., 1956. Communications from the Academy of Sciences, USSR 111, 988. Guias° u, I., Radescu, E.E., 1978. Note on low-energy proton Compton scattering. Phys. Rev. D 18, 651. Hallin, E., et al., 1993. Compton scattering from the proton. Phys. Rev. C 48, 1497. Holstein, B.R., Nathan, A.M., 1994. Dispersion relations and the nucleon polarizability. Phys. Rev. D 49, 6101. Hyman, L.G., et al., 1959. Scattering of 50±140 MeV photons by protons and deuterons. Phys. Rev. Lett 3, 93. Jackson, J.D., 1961. Classical Electrodynamics. Wiley, New York. L'vov, A.I., 1981. Compton scattering on the proton at energies up to 400 MeV and ®nite-energy sum rules. Sov. J. Nucl. Phys 34, 597 and private communication. L'vov, A.I., 1993. Theoretical aspects of the polarizability of the nucleon. Int. J. Mod. Phys. A 8, 5267. L'vov, A.I., Nathan, A.M., 1999a. Sum rule for the backward spin polarizability of the nucleon from a backward dispersion relation. Phys. Rev. C 59, 1064.
123
L'vov, A.I., Nathan, A.M. 1999b, in preparation. L'vov, A.I., Petrun'kin, V.A., Startsev, S.A., 1979. Finiteenergy sum rule for the nucleon polarizability dierence. Sov. J. Nucl. Phys 29, 651. L'vov, A.I., Petrun'kin, V.A., Schumacher, M., 1997. Dispersion theory of proton Compton scattering in the ®rst and second resonance regions. Phys. Rev. C 55, 359. MacGibbon, B.E., et al., 1995. Measurement of the electric and magnetic polarizabilities of the proton. Phys. Rev. C 52, 2097. Molinari, C., et al., 1996. Scattering of photons by the proton through ycms=908 in the D-resonance region. Phys. Let. B 371, 181. Mukhopadhyay, N., Nathan, A.M., Zhang, L., 1993. D contributions to the paramegnetic polarizability of the proton. Phys. Rev. D 47, R7. Oxley, C.L., 1958. Scattering of 25±87 MeV photons by protons. Phys. Rev 110, 733. Peise, J., et al., 1996. Compton scattering of the proton through ycms=758 in the D-resonance region. Phys. Lett. B 384, 37. Petrun'kin, V.A., 1981. Electric and magnetic polarizabilities of hadrons. Sov. J. Part. Nucl 12, 278. Powell, J.L., 1949. Note on the bremsstrahlung produced by protons. Phys. Rev 75, 32. Tonnison, J., et al., 1998. First extraction of a spin polarizability of the proton. Phys. Rev. Lett 80, 4382. Weiner, R., Weise, W., 1985. Electromagnetic polarizability of the nucleon and chiral quark models. Phys. Lett 159B, 85. Zieger, A., et al., 1992. 1808 Compton scattering by the proton below the pion threshold. Phys. Lett 278B, 34.