and momentum q can polarize the nuclear medium through nucleon-hole and A-hole excitations as depicted in Figure 30. These contributions to the pion self-energy, through iterations of the particle-hole excitations to all orders, are treated by Cohen and Eisenberg [CE83] by renormalizing the entire production amplitude in the manner The diamesic function, e(q, u>), can be expressed in terms of Lindhard functions [FW71], the one pion propagator and the Migdal parameter g'. A value of g' of 0.4 corresponds to the case where pion condensation exists in the nucleus, while g' = 0.7 corresponds to a minimal condensation effect. The Migdal parameter is equivalent to the Lorentz-Lorenz parameter, A, used to describe nuclear correlation effects in low energy pion elastic scattering. Numerically, the equivalence is given by g' = A/3. The currently accepted range for g' is 0.6-0.7. For high momentum transfer, but low energy transfer, the diamesic function can be written as [o \u00E2\u0080\u00A2 < f r j renormali. 'zed e(q,0) = l + W(q,0)U(q,0) where u2 \u00E2\u0080\u0094 q2 \u00E2\u0080\u0094 rn2 99 h - + - + 11 - + - + + + \"1 - + + Figure 30: A diagrammatic representation of nucleon-hole and A-hole ex-citations in the nuclear medium contributing to the pion self-energy. ;~ 100 is the pion polarization propagator describing the longitudinal part of the spin-isospin nuclear interaction. The Lindhard function is written as / V ) U(q,u>) = mi [UN(q,u) + 4UA(q,u)] where f2(q2) is the monopole form factor multiplied by the coupling /2/47r = 0.08 given by / V ) = ' A 2 - m 2 \ 2 f2. \ A2 + q2 , The cutoff A is taken to be 1000 MeV/c. The factor of 4 multiplying the A Lindhard function is the square of the ratio of nucleon to A coupling constants as given by Chew-Low theory. The nucleon Lindhard function can be written as UN(q,uj) --7T' i + y+ i - y + - | ( l - \u00E2\u0080\u009E ! ) l n l + y-i - y -and the A Lindhard function as The arguments in the functions are y\u00C2\u00B1 PF q u?& = m A \u00E2\u0080\u0094 m^v = 2.2mff. The effective nucleon mass, m*, is taken as 0.8m ,^ pp is the Fermi momentum, p is the matter density normalized to one and A is the atomic weight of the nucleus under consideration. The local Fermi momentum is determined by '3 - N 1 / 3 PF = ( T ^ M O ) where the nuclear density form taken is the two parameter Fermi form p(r) = 47TC3 1 + 1 + exp ( T-~j -1 101 with c the half density radius and At ln 3 is the 10% to 90% density distance. The (7r,27r) reaction will produce a wide range of excited states in the residual nucleus. To reduce the calculational difficulty, a closure approximation is used by [CE83] to truncate the possible Fermi gas states of the residual nucleus. To do this, a mean nuclear excitation must be chosen and this energy can then be be taken outside of the integrals required to evaluate the cross section. Assuming a quasi-free peak contribution dominates the cross section, the closure energy is taken to be _ Q2 e \u00E2\u0080\u0094 -2m N where Q ~ q is taken as the average momentum transfer. A correction term measuring the error of the closure approximation was estimated and found to be small above 500 MeV/c but at 400 MeV/c (284 MeV), the calculated sample values of [CE83] indicate that it is roughly 50% of the size of the closure term. The correction term becomes even larger for lower momenta. The cross section calculation is expanded into partial waves for the three pions. A form of the local density approximation (LDA) is used to evaluate the diamesic function through the nuclear volume. That is, it is evaluated for the local Fermi momentum calculated from the local density. This assumes that the nuclear density is varying slowly enough so that it is meaningful to assign a local Fermi momentum. The pion waves are distorted in the calculation of [CE83] using the computer code DWPI [EM76] to calculate the distorted partial waves. For the incoming wave, optical distortion (quasi-elastic scattering) as well as true absorption (proportional to p2) are used. For the outgoing channel, Coulomb distortion and true absorption are used. The optical distortion in the final state reduces the pion energy but does not eliminate it, so it is not included in the final state. The 102 optical potential parameters used to evaluate the distortions are taken from [SMC79]. For pion momenta less than 50 MeV/c, the pionic atom parameters are used while for momenta from 50-130 MeV/c, the Set 1 parameters are used. For higher momenta, the optical potential parameters are extrapolated from the Set 1 values following a prescription from [SMC79]. The cross section is obtained after numerically integrating over the partial waves for all three pions. In order to reduce the computation time, some approximations are made in the integration by estimated average values for some variables for which the matrix element depends only weakly upon. Results from the calculation for the 1 6 0 ( 7 r + , 7 r + 7 T ~ ) reaction are shown in Figure 31, as x's, for the values of G' of 0.40, 0.55 and 0.70, where the larger cross sections correspond to smaller G' values. The results for 1 6 0 ( 7 r + , 2TT+) reaction are found to be about 20% larger than the values for 1 6 0(7r + , 7r + 7r~) . However, a factor of | has been omitted in the calculation that accounts for the identity of the 7r+'s in the final state so that the ( 7r + , 27r + ) cross sections quoted are a factor of two too large. It should be noted that the cross sections calculated for low G' already exceed the inclusive DCX cross section measured at 270 MeV [Woo84] and can thus be ruled out. Several comments can be made concerning calculations of [CE83]. Cohen and Eisenberg point out that the closure approximation introduces into the cross section calculation terms corresponding to the single nucleon contribution and to a, two nucleon spin-isospin correlation contribution, the latter of which is left out of the calculation. This two nucleon term is distinct from direct two nucleon mechanisms which would have their own T-matrix contibutions in the calculation. The correlation term is examined by [CE83] and estimated to be roughly 10% the size of the single nucleon contribution. Also, the choice of the closure energy is not unique. The quasi-free peak has a large width, and different choices of the closure energy can shift the cross section up to 30%. This uncertainty is what is used in 103 [CES3] for a measure of the error in the calculation. The closure approximation is apparently inappropriate for low incoming pion energies as shown by the size of the correction term, however the threshold approximation chosen to model the underlying free process becomes worse as energy increases. If we empirically try to compensate the calculation for underestimating the free reaction, the calculated results only become larger, becoming uncomfortably high. Another estimate of the (7r,27r) process is given by [Roc83] for the 1 8 0(7r _ ,2TT~) which, with some assumptions, we can relate to the 1 6 0(T T + , n+ir') process. Rockmore uses a Fermi gas description of the nucleus which allows a convenient way to sum over the final nuclear states and avoids tedious details of the nuclear structure. The approach was motivated by earlier applications to the threshold pion electroproduction process in nuclei [CW64]. The forms of the threshold approximation taken by Rockmore allow the phase-space calculation to be viewed as a Fermi averaging over phase-space rather than an integration over a response function requiring a lengthy Monte Carlo process. Approaching the problem in this manner allows Rockmore to relate the emission of two pions as encountered in the (7r,27r) process to one pion emission as found in the electroproduction process. The one-body pion pole term is chosen as input into the Fermi gas model. Rockmore makes the threshold approximations 1 1 uxu>2 ml and ux + w2 ~ 2mv + 2mv The cross section can then be written as 104 180 220 260 300 Tn (MeV) Figure 31: Predictions for the 1 6 0 ( 7 r + , 7 r + 7 r ~ ) reaction by the three au-thors discussed above compared to the experimentally evalu-ated total cross section. Note that the Rockmore curve (lower solid curve) contains the corrections discussed in the text. The square point is the measured datum. 105 where dk dq[ dq~2 1' (k + q-ql-q^y p mi 1m* 2m* 2mw 6 (jk + q- q[ -q\"2\- PF) KPF - k). The nuclear size is described by which represents a hard sphere nucleus of radius Al^r0 with r0 = 1.2 fm. The effective nucleon mass, m\", is taken as 0.6mjv and uq - 2m\u00E2\u0080\u009E rj = mv With the additional threshold approximation of \u00E2\u0080\u0094\u00E2\u0080\u00A2 \u00E2\u0080\u0094\u00E2\u0080\u00A2 k + q- ql -