The calculation of the dispersion ratio of the real parts of the forward scattering amplitudes .. rr p and K-p is carried out assuming constant total cross sections in the Serpukhov energy range. A connection between high-energy Reggae amplitude behavior and low-energy scattering is established by means of analyticity in the energy variable V (see Figure 2). On the other hand, nN, KN, and NN both have low-energy cross sections that fall roughly like this.
According to FESR, it does not have it even at high energies.. the total cross-section, which is proportional to the imaginary part of the amplitude, is purely diffraction, i.e. level. Cross symmetry implies amplitude invariance, up to sign, under a formal permutation of quantum numbers. The elimination of the right cut for the exotic s channel is a statement of the analytic structure of the amplitude.
If we are to preserve analyticity, the real part of the amplitude can no longer be described by the Regge approximation (1). In the same way, the new cut affects the rules for continuous sum of moments18), which we have not considered. parameterization of the imaginary part of the amplitude below 30 BeV), but not of the real part. It's just an intuition), which can serve as a basis for a search for the underlying theory.
Although the experimental data on the real parts of the K+ -p amplitudes are not very . correct) they favor the existence of an additional non-Regge term.
Functions of type a have a cutoff from threshold to infinity in the first variable and are integral in the second. We have not been able to find the weakest assumptions necessary to imply this, but we conjecture that they may be quite general, due to the different structure of the intercept of each function in the product of the complex planes s and t. It is worth looking for the more general basis of functions of type F for the three irreducible representations of s.
20 for a review of properties of finite groups and their representations, with particular emphasis on s3• We. In the next two sections, we use this technique to study I-spin invariant nn scattering and SU(3) invariant pseudoscalar-pseudoscalar scattering. Using (7), one can therefore write the most general F-function form for A .• After a little renormalization of the arbitrary functions,.
Mande ls tam's well-known result on cuts in the J. plane 8) thus eludes one in advance for this amplitude. We present his results in the appendix and relate them to Rosner's quark diagrams 27). The linear relation (29) between the couplings f·f implies arbitrariness in B .• This is removed by the constraint.
We can now write down the most general form of the F-function of invariant amplitudes} and explore the consequences of eliminating the exotic. Omitting the contributions of the third dual spectral function, we obtain the following couplings in the non-exotic channels. The singlet and the symmetric octet turn out to be degenerate because they share the same symmetric pattern, viz. they both are. symmetric under the exchange of the two outer octets into which they decay.
We are looking for a mode with a total I = 1 and with two of the three pions also in the I = 1 state. The Pauli principle (or cross . symmetry, if we are dealing with A. 2 exchange) allows only a certain combination of the two modes. The physics is contained in the observation that decay appears to take place via an intermediate state of two particles.
When one of the latter decays, it no longer remembers the resonance from which it originated. There may be a final state interaction, but it should be considered a small perturbation of the two-particle decay scheme.
REAL PARTS OF FORWARD M ESON-NUCLEON SCATTERING AMPLITUDES
We want to see what the predictions of these assumptions are for the real part. It avoids the emphasis on the low energy input and thus increases the errors in the calculation. We are, in fact, able to reproduce at low energies (e.g. below 4 BeV) the same results previously obtained using (2) with any reasonable decreasing adjustment to 0(-). To illustrate the changes brought about by the assumptions about the behavior of the overall cross-section, let's discuss an rnathe analysis.
It is now interesting to note that although Eq. 5) and (7) are very different from each other, it is still possible to find a value of c that will show a similar behavior for low V. It is therefore possible that although Im B. real parts of the different amplitudes may agree approximately for V < /1. Any violation of the Pomeranchuk theorem leads to a logarithmic rise of the real part of the amplitude, especially of A(-)(V)ll,l9). Above it was pointed out that the strength of the logarithmic term in the real part of the amplitude is proportional to 0(-).
We adopted the latter in order to test the sensitivity of the calculation to the possible violation of the Pomeranchuk theorem. If we attribute the discrepancy to rotation violation I of the electromagnetic amplitude}, we find that it is 20 percent of the total amplitude A(-). The errors involved here are much larger than in "P· Uncertainties in the subthreshold singularities do not allow a good determination of the real parts at low energies.
In case II, we can explain the discrepancy using the subtraction term. However, currently energies seem to come the bulk of the real part from the subtraction term; and not from the logarithmic one. The difference between the pion and kaon amplitudes lies in the energy range below the cutoff point.
In the latter case, the existence of an additional real term seems to be implied by the data. Qyo,a:13 is the projection operator for an intermediate octet with f coupling at the initial state and d at the end or vice versa. This means physically that the o particle quark: annihilates the 5 particle antiquark, the 5 quark, the y antiquark, and so on.
The connection between the Rosner scheme and ours is straightforward with the use of . the following identities. fa~·fy5 = - t \ [Aa,A~ ] [Ay,A5]). The discrepancy between the fit and the data is an indication of the amount of I spin that violates the electromagnetic effect.
