NUCLEAR OPTICAL MODEL FOR VIRTUAL PIONS* J. S. Bell f. Stanford Linear Accelerator Center Stanford University, Stanford, California

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1 NUCLEAR OPTICAL MODEL FOR VIRTUAL PIONS* J. S. Bell f Stanford Linear Accelerator Center Stanford University, Stanford, California (To be submitted to Physical Review Letters) * Suppor;ed by the U. S. Atomic Energy Commission. f On leave of absence from CERN, Geneva.

2 Cross sections for incident virtual pions are sometimes needed, for example in the one pion exchange approximation..' A particularly nice example has been given recently by Adler.2 He studies neutrino reactions v+g!-4+a*. w'nere t is a charged lepton and G? an aggregate of strongly interacting particles. Assuming conserved vector and almost conserved axial vector currents, he finds that for forward-going leptons & (q2 = 0) the differential cross section has the form a2 0 aq2aw2 =K 2P(W)W Ma dw, 3") (1) Here. q is the difference of b-momenta between v and &, W is the mass of the aggregate C?, K is a given kinematical factor, and M, is the target mass; finally o(w,mflt would be the total cross section for the reaction at total center-of-mass energy W, with p(w) the corresponding center-ofmass pion momentum. If the difference between q2 = 0 and rnz can be ignored in Eq. (1) we have a relation between observable quantities. Sucn a result would be particularly valuable for neutrino reactions, where nuc1e.l are much more convenient targets than nucleons, because the complications of nuclear physics are apparently eliminated by the use of observed cross sections for pions on nuclei. However, the following difficulty presents itself. Because of absorption, pion cross sections depend on the size oz large nuclei roughly as A213. But neutrinos penetrate to all parts of nuclei; -l-

3 for them cross sections should contain at least a part proportional to A. This indicates for large nuclei a critical dependence of a(w,-q2) on- q2. It could be discussed in terms of anomalous thresholds. However, we use here a more traditional method of the nuclear many-body problem, the optical model potential. As in a previous discussion3 of the nuclear optical model, we ignore the motion of the heavy target cx and regard it as fixed in space. Then in terms of the renormalized pion field operator at zero time, ~r(x,o) we have ho,-s, = 2P(W)W M a dw, -q'> = ($. + q2)' II< nl~d~ei~%(~,o)la > 1' 2rtG(En - E, - qo) (2) n where the summation is over all final states I n >. Equivalently I? = 2(mz + q2)2re s d$p ii&* x<(z,;, qo)eip Y (3) where4 G(XY q > = M.2n.A' 0 s dtemiqot < Q. 1 T[lc(c,o),Ir(_y,t)]la > P-4 As usual the propagator G satisfies an integral equation ho) = Gob,) - igo(qo)v(qo)g(qo) (5) where space arguments and integrations are suppressed in a conventional way. GO is the propagator in vacua, and the integral operator V represents -2-

4 the effect of nuclear matter. We will approximate5 G o by the pole term Goo(yJko) = -i s dj.5 -ee ikh-y)o k2 _ ie)-l II (2d3 (6) where k2 = k2 - kz. "b The optical model wave function defined by Y(L) = i(mz + q") s dy, G(;T,z,qo)ei%Y (7) then satisfies Y(x) = ei%5+ (ie + d" + q2 - rnz)-'vy 0 (8) From Eqs. (3>, (51, (6), and (8) r = -21m s dxdye-i%'%(x y,q )Y(y).A+... -',w. 0 nt... (9) This relates the total transition rate to an off-the-mass shell forward- scattering amplitude. It can further be shown6 that i? is the sum of "elastic" and "absorptive" parts: r = re + ra with ra = -2Im s d~~y*(x)v(~,~,qo)y(y) (11). The integrand of ra gives the relative probability for final states con- sisting of a pion of momentum,$ and the original unexcited nucleus; Pa gives the probability for all other states. When the b-vector q is -3-

5 I. on the mass shell, Eqs. (8), (p), (lo), and (11) are equivalent to those customary in the optical model, with a potential V that is in general non-local and energy dependent. They are seen to remain relevant off the mass shell. The critical effect of going off the mass shell is revealed by writing Eq. (8) as a differential equation: (+ - rnz + QY = VY - (mz + q2)ei%l (12) On the mass shell, rnz + q2 = 0, there is no inhomogeneous term and the wave is absorbed in the surface of the nucleus toward the incident beam. But off the mass shell there is a driving term carrying the wave to all parts of the nucleus. Beep enough inside a sufficiently large nucleus the solution is Y = rnz + q2 m ;+q2+? iq-x e -- (13) where y= s d(x - de i4(:-&$(y, x) cu P.-h l-h* (14) Thus the vacuum propagator is replaced by that for nuclear matter. For + q2 # 0 substitution of Eq. (13) in Eqs. (p), (lo), and (11) gives mz volume proportional terms which are equal for r and Pa and zero for re' To obtain a first orientation let us make the approximation, which appears to work fairly well for real pions of high energy, = -pr, (15) -4-

6 where p is nucleon density and Pl. the value of r appropriate to a single nucleon target. From Fqs. (11) and (13) the volume contribution is then 2 1 m2?( + s2 repa= 1 I 2 - i ml im II + q2 + v i (16).Ll ff "v could be ignored in the first factor we would have simply the summed cross sections of individual nucleons. This would be correct for sufficiently dilute matter, but not for nuclear matter. Giving rl. its mass shell value 2 i/q: - rnz ul = 2ka,, rnz + q2 2 <- m2 ' 7l m ; + q2 + v - [ ka,o (u) for q2,< rnz. With U,> 3 X 10-26cm2, p = 1.7 X 1038cm'3 one finds 1 mz + 92 I2 2 m _/ 52-f rnz + q2 + V I 0 (18) This decreases with increasing virtual pion energy, and is already small compared with unity at a few hundred MeV. Estimation of surface effects depends ona more accurate solution of Eq. (8) or Eq. (12). However, the above conclusion about the volume effect Is already of interest. As applied to neutrinos, by Eq. (l), it implies t-at highly inelastic events wit'n small q2 should be much less probable than migkit be expected by counting nucleons. It is interesting to conaider if other contributions to the neutrino reaction (e.g., p-exchange) should - 5 -

7 show similar effects at appropriately higher energies. For useful discussion I am indebted to Dr. M.,Bander, Professor S. M. Berman, and Professor S. D. Drell. -6-

8 LIST OF REFERENCES 1. S. D. Ikell, Rev. Mod. Phys. 33, 458 (1961). E. ierrari and F. Sellerl, Nuovo Cimento, Suppl. 2, 453 (1962). 'i"r.ese papers give earlier references s. -'. --..A-.--,.ATz,,- -A >.-, ^&.bc ""A- -"::;;->-~ys~~y pxpr;st. 3. J. S. Bell and E. J. Squires, Pays. Rev. Letters 2, 96 (1959). J. S. Bell, in Ikctl:r~~s on the Many Body Problem (Academic Press, 3Ie:c York and London, 1962); p Instead of the time-ordered product, the retarded commutator could be used. %is is advantageous for the enera averaging problem This approximation can be avoided by accepting the existence of short range integral operators M and M-l such that MGoM = Goo. Then G' = MGM satisfies G =G 00 - igoovig1 with V' = M'lVM-1. The subsequent development proceeds with G' and V' replacing G and V. Equations (9), (lo), and (11) finally contain V' instead of V and their right-hand sides are multiplied by i'f2(q2), a factor which for q2 = -mz is unity and for q2 > -mz is real, positive, non-zero, non-singular, and independent of nuclear size. 6. To prove this "off-the-mass shell optical theorem" write X = Y - exp(iq.2) so that Eq. (12) becomes (V2 - rnz + qz)x=vy. whence 2 s- ds * Im~"Vy( =2Im s d&(y*vy - 2%%Y)

9 I where the first integral is over a distant surface and the second over the enclosed volume. The last term is I' of Eq. (9); the other tern? on the right is -Fa of Eq. (11); on using the asymptotic form of % the right-hand side is seen to equal Fe of Eq. (10). 7. J. W. Cronin, R. Cool, and A. Abashian, Whys Rev. 107, 1121 (1957). M. J. Long0 and B. J. Moyer, Phys. Rev. 3, 701 (1962).