4 Polarizabilities

The polarizabilities --- electric , magnetic --- of a particle, like its charge radius, reflect its response to an electromagnetic stimulus [28,21,29]. They are related to how the Compton amplitudes approach the Born limit at threshold. Thus the combination is determined by the slope of the S-wave amplitude at the cross-channel threshold :

where for the charged pion case is the one-pion-exchange Born amplitude, , , and for neutral pions , with our definition of Eq. (4). In terms of the dispersive representation of Eqs. (6-9), these polarizabilities are simply related to the subtraction constants [6] and, of course, the exchange contributions at s=0 :

Moreover, these combinations are sensitively related to the position of any sub-threshold zeros. In the neighbourhood of s = 0, the charged and neutral S-wave amplitudes can be parametrized as

then

Note that this approximation is only valid for , since the amplitudes have a cusp at threshold, which is particularly marked in the amplitude, see Fig. 25 of Ref. 6 or Fig. 8 of Bellucci et al. [24] in 2 loop standard PT or Fig. 6 of Knecht et al. [30] in 1 loop generalized PT . Of course PT has a definite expectation for the values of and . However, we have seen from Fig. 7 that present (or even future) data cannot tell without a model parametrization whether the sub-threshold zero in the neutral channel is at , or . This variation leads to a factor of four difference in the neutral polarizability from to cm. These values are merely illustrative , in practice the range of uncertainty of a model-independent extraction is still larger. Moreover, present data even allow to be zero, so could be zero, also with a very large uncertainty.

Rather than use the sub-threshold behaviour around as we have done in Sect. 3 to fix the S-wave subtraction constants, , Kaloshin and Serebryakov [31] have attempted a closely related dispersive analysis in which the subtraction constants for the S-wave are parametrized directly in terms of the polarizabilities . They find in units of cm : ; . As just remarked a value of = 0 is perfectly consistent with the charged channel data, so why have Kaloshin and Serebryakov [31] excluded this by many standard deviations ? This is because though they only fit the Mark II data [7] below 400 MeV, they include in their fit data on [10] up to 850 MeV and yet assume they know the form of all the cross-channel exchanges , , very far from their t and u-channel poles. The existence of the and , the former markedly affects data at 800 MeV, because of its large width (Fig. 12), require many more exchanges than and . Moreover, even in going from the and poles at , where the couplings are, of course, determined by the measured rates, to the Compton threshold at , the couplings can change by a factor of 2 --- a simple Veneziano-like model with towers of resonances, not just the and gives a factor of . To then assume at 850 MeV, where , the pure and exchange amplitudes have the same couplings as at is a grossly over-simplified model leading to far too small an estimate of the uncertainties on the supposedly determined polarizabilities. That the details of the cross-channel exchanges become increasingly important at higher masses can be seen from Fig. 3 and from the band shown in Fig. 6.

We now turn to the determination of combinations of polarizabilities. While the are related to how the S-waves approach their Born term as , Eq. (16), the are determined by the way the helicity two D-waves approach their Born term at the same Compton threshold. These are even more difficult to determine from data. As we have stressed, it is only in the very low energy region that cross-channel exchange contributions are accurately calculable. Below 500 MeV, the D-wave amplitude is overwhelmingly () controlled by its Born component. It is the residual that has to be extrapolated to s=0 to determine the combinations of polarizabilities, clearly an impossible task with present data that only cover a very limited part of the angular range. More accurate separation of higher waves with helicity two will become possible using the azimuthal information that the DANE experiment should provide [32,33,34]. However, this has not deterred Kaloshin, Persikov and Serebryakov [35] from attempting a first estimate of the with amazing results. They have once again, even up to 1.4 GeV in mass, assumed only elementary , and exchange determine the left hand cut effects. No obvious t, u-dependence is included in the pole numerators, even though t, u-channel unitarity demand these. They then add an direct channel contribution assuming this to be wholly helicity two with no S-wave background under this. While present data are not incompatible with this, they are equally consistent (see Fig. 12) with 30% of the I=0 cross-section from 1--1.4 GeV being S-wave [11,8] and possibly 30% of the D-wave having helicity zero [11]. This provides at least a 40--50% uncertainty in the -contribution to the helicity two D-wave cross-section. Yet Kaloshin et al. [35] quote values of with 5% errors : , far away at s = 0.

In the dispersive treatment discussed above, the description of the region in terms of Eq. (12), embodies the expected Breit-Wigner shape of the resonance, while provides a smooth modulation of this over the peak in much the same way as the -line shape in differs smoothly from that in elastic scattering. The exact form of depends, of course, on the structure of the left hand cut discontinuity embodied in the model for defining in Eq. (12). Kaloshin et al. take their analogue of to be a constant, as far as its s-dependence is concerned. However, is really built from many t and u-channel exchanges. While these do generate a smooth form over the -region from 1 to 1.4 GeV, for almost any couplings, it is the exact values of these individually that determines the extrapolation to s = 0 (cf. the analogue of Eq. (17)). Consequently, the analysis of Kaloshin, Persikov and Serebryakov [35] is misguided. Fitting in the -region can in no way determine the polarizability charged or neutral. This is obvious from the structure of Fig. 3, where exchanges crowd in near the physical region and only their collective effect is measured, whereas in extrapolating to s=0, it is the individual exchange contributions that matter.

All this means that the only way to measure the pion polarizabilities is in the Compton scattering process near threshold and not in . Though low energy scattering is seemingly close to the Compton threshold ( to ) and so the extrapolation not very far, the dominance of the pion pole (for final state interaction effects, for example) means that the energy scale for this continuation is . Thus the polarizabilities cannot be determined accurately from experiments in a model-independent way and must be measured in the Compton scattering region [29].