Connection BetweenFπ(t)and the Amplitudes forππ→ππand

Abstract

We have studied this background process since it provides a test of charge independence. We have found that both the energy dependence of the total cross sec-tion and the angular distributions are in excellent agreement with existing data for n +p -*d + 7r° andp +p -*-d + TT + c I. Hammerman, thesis, Princeton University, Palmer Physical Laboratory Report No. PURC-2137-9 (unpublished). 6 Monte Carlo program based on technique of J. Butcher and H. Messell, NucL Phys. 20, 15 (1960). We have checked this calculation by using y rays from the reaction n+p-^d + ir 0 . 7 To Devlin, P. Shepard, R" Mischke, and J. Solomon, private communication 0 8 A possible source of systematic error could arise if the neutron beam were polarized. Reciprocity requires that the angular distributions for Reactions (1) and (2) be the same if initial-state spins are averaged and final-state spins summed. We have measured the polarization of the incident neutrons in our beam and found it to be 0.04 ±0.02 (T n & 600 MeV), a value too small to yield significant asymmetries. Our detector is insensitive to the spin of either the deuteron or the photon. A parallel situation holds for the measurement of Reaction (2).The current-algebra connection between the pion electromagnetic form factor F v (t) and the amplitudes for 7T7r-*7T7T and Tnr-*TrA i is examined by explicitly extrapolating off the pion mass shell the matrix element (^{A^l TTTT) which is taken to be dominated by ir and A t poles and possible subtractions. It is found that the connection is broken by the presence of an almost arbitrary subtraction function. In particular the Apir interaction remains arbitrary as well as the form for F^(t). The results are applied briefly to the Veneziano model. Several authors 1 have recently explored the connection between the amplitudes 7T7T -^-mr and TTTT •*irA 1 that follows from the assumptions that the matrix element <7T7T lA M | 7r> is dominated by ir and A x poles and that the scattering amplitudes are given by the Veneziano model. 2 Oyanagi 3 has extended the analysis by extrapolating to zero the momentum of one of the pions in the matrix element and thereby obtaining a Veneziano-like expression for F^(t). One result of this work, and that of a subsequent analysis, 4 has been to propose a vanishing D-wave Apu interaction, either to yield a more convergent F^it) for large t or to eliminate satellites from the int -7nr and -nil -^,nA 1 scattering amplitudes. In all these treatments, however, the extrapolation of one pion off the mass shell of the amplitude <7T7T IA M |7T> has not been done in a consistent way. 5 We intend to summarize the results of such a consistent treatment in this Letter and show that the use of the Veneziano model for this example imposes no restriction on the Z>-wave Apir interaction and that one is free to obtain almost any asymptotic behavior for F^(t).Consider the amplitude for Tr + {p) + 7r~ (q)-A^(k) + TT~ (q') with the TT + extrapolated off the mass shell,

Review of Recent Work on Narrow Resonance Models

“…al. (1969 and Geffen (1969) In the next section we will see that if we construct directly a form for (5. 27) where one current is linked to a hadron amplitude, then it is necessary to include nontrivial q2 dependence.…”