Volume 63B, number 4 PHYSICS LETTERS 16 August 1976 THE OCCURENCE OF A CDD SINGULARITY IN THE NEUTRON-DEUTERON DOUBLET SCATTERING AMPLITUDE ~ A.T. STELBOVICS Universitllt Mtlnster, Mflnster, West Germany Received 26 May 1976 It is shown that for the Amado model of elastic neutron-deuteron scattering in the s-wavedoublet channel the single channel N/D solution is not equivalent to the exact amplitude, Equivalenceof the N/D solution is demonstrated under the insertion of a CDD pole which simultaneouslygenerates the triton pole and the effective range anomaly, thus establishing a link between these apparently unrelated features of the doublet amplitude. In a previous communication by Stelbovics and Dodd [1 ] a comparison was presented between the on-shell N/D method and the direct solution of the off-sheU Amado model [2] equations for s-wave neutron-deu- teron scattering. Single channel N/D equations of the "R-method" type [3] were used in both the singlet and doublet channels. Their work was concerned mainly with the scattering phase shifts and did not study the triton bound state explicitly. The purpose of this letter is to examine the equivalence problem for the triton pole and to point out a subtle inconsis- tency in the interpretation given in ref. [ 1] of the re- suits for the doublet channel. We will show that the correct interpretation leads to the requirement of a CDD pole in the doublet channel N[D equations. One of the features of their calculation was that they checked the dispersion relations by computing exact left-hand inputs and inelasticities from the solu- tion of the off-shell equations and compared them with the exact model values. This entailed solving the follow- ing equations: N(s) = B(s) + n¢l f B(s') - B(sl o(s'lN(s'l R 1 Fp (s')JV(s') ds' O(s) = 1 --~-d s'- s ' R with 1 ('Im T(s' + i0) B(s) =~3 L 7-- s ds'. (1) Work supported by Deutsche Forschungsgemeinsehaft. In the s-wave doublet channel they calculated the "exact" input from the formula 1 p/" Im T(s' + i0) "~(s)= Re T(s+iO) --~ JR -Sr-S- s ds' , (2a) and the exact inelasticity or phase-space factor accord- ing to p(s) = Im [T--1 (s + i0)], (2b) with T taken from the exact off.shell solution. This requires only a knowledge of the model amplitude for real energies on the right-hand cut but differs from the scheme given by eqs. (1) in that the left-hand input also includes a contribution from the triton bound state pole (in this model at -11.0 MeV), that is R T ~¢(s) =B(s) + ~ , ST= -11.0MeV. (3) s - s T The N/D solution with this input gives complete agree- ment with the exact model phase shifts and absorption coefficients; on the other hand the D function has no zero corresponding to the triton bound state (see fig. 1). In ref. (1) it was reasoned that the N function carries a pole at this position so accounting for the lack of a zero in D, which is intuitively justified by the observa- tion that B'(s) itself contains the triton pole. However this assumption is not correct as can be seen from the following argument. Suppose the N/D equations are solved with the exact input B(s) and that the resultant D function has a zero at the triton bound state energy, so D(ST) = 0. Then using eq. (3) one can rewrite the N integral equation in 374