PHYSICAL REVIEW VOLUME 105, NUMBER 3 FEBRUARY 1, 1957 Pion Production in Pion-Nucleon Scattering* JERROLD FRANKLINJ University of Illinois, Urbana, Illinois (Received October 25, 1956) The cross section for the production of one additional pion in a pion-nucleon scattering is calculated using the Chew-Low theory of P-wave pion-nucleon scattering. The transition matrix element for the scattering of one meson into two mesons is defined in terms of exact eigenstates of the total Hamiltonian and an approxi- mate expression is derived which expresses the one- to two-meson matrix element as a product of an elastic scattering matrix element and a meson-emission matrix element. Experimental elastic scattering phase shifts are used in calculating the two-meson cross section. The results of this calculation are used to estimate the effect of two-meson states on elastic scattering. The contribution to the effective range for elastic pion- nucleon scattering is small. Cross sections are also obtained for all possible two-meson charge states from either a x + or a x~ meson incident on a proton. Comparison is made with experiment and the recent theo- retical work of Barshay. I. INTRODUCTION T HE problem of elastic scattering of pions by a nucleon has been approached recently by a new method 1-3 which involves the use of exact eigenstates of the total meson-nucleon Hamiltonian taken in the static limit (fixed nucleon). In this paper this method, in the form introduced by Wick for one-meson states, is used to investigate the two-meson eigenstates of the Hamil- tonian and to calculate the transition matrix for scat- tering from a one-meson state to a two-meson state. Barshay 4 has made a recent calculation similar to this one using Low's method, but the results are quite different from those obtained here because of different approximations introduced by Barshay in evaluating the T matrix. Barshay introduces sums over two com- plete sets of eigenstates and then makes the approxima- tion of limiting one sum to include states with up to one real meson and the other to include only states with no real mesons (physical nucleon). The approximation is also made by Barshay of neglecting the energy of one of the outgoing mesons in certain energy denominators. In the present paper the procedure used corresponds to summing exactly one of Barshay's complete sets and including only the physical nucleon states in the other while all energy denominators are treated exactly. II. TWO MESON EIGENSTATES In the static limit the pion-nucleon Hamiltonian has the form 5 H=Ho-E 0 +H I , (1) where Ho = Y,ka^a k o}k, (2) Eo is a constant energy subtracted to make the self- * Based on a dissertation submitted in partial fulfillment of the requirements for the Ph.D. degree at the University of Illinois. t Now at Columbia University, New York, New York. 1 G. C. Wick, Revs. Modern Phys. 27, 339 (1955). 2 F. E. Low, Phys. Rev. 97, 1392 (1955). 3 G. F. Chew and F. E. Low, Phys. Rev. 101, 1570 (1956). The notation used in this paper follows that of Chew and Low. 4 Saul Barshay, Phys. Rev. 103, 1102 (1956). 5 -h and c have been set equal to unity. energy of a single nucleon zero, H r =j: k V k ^a k +V k Wa k \ (3) and V k ^ = i(4,w)Kf (0) /^'^r k v(k)/(2c)K (4) Here a^ and a k are, respectively, creation and annihila- tion operators for single, bare mesons, 00= (/x 2 +£ 2 )*, / (0) is the unrenormalized coupling constant, o- is the nucleon spin vector, T k is the £th component of the nucleon isotopic spin operator and v(k) is a cutoff function which approaches zero for large momenta. In the notation used here the meson quantum numbers are all described by a single symbol (k) which includes the three components of momentum and the isotopic spin. A two-meson eigenstate corresponding to one meson of type pi and one meson of type p 2 can be found from the assumed form *Pi,P2 (±) = ta P1 ia P 2^o+x (±) l/^, (5) which represents a state with two plane-wave mesons produced by the two creation operators acting on the eigenstate, ^0, corresponding to a physical nucleon and an (outgoing/incoming)-wave scattered part, x (±) - If this form of the two-meson eigenstate were used in matrix elements, however, the two creation operators appearing in the first term would lead to two energy denominators requiring expansions in two complete sets of states. For this reason, it is better for most calcula- tions to use a two-meson eigenstate of the form ¥ P1P2 (±> = [ api t^ P2 (±)+ x (±)]/v2, (6) where > J r p 2 (±) is a one-meson eigenstate. Equations (5) and (6) will be shown to represent identical states, but the effect of using the latter is to sum exactly one of the expansions required by the use of (5) and condense the resulting equations. Because of this, (6) will lead to a simpler and more accurate result than (5) for the same level of approximation. The normalization of the eigenstates has been chosen 1101