Castillejo, Dalitz, and Dyson ambiguity in field theoretic models of scattering

We examine the numerical solutions of a field theoretic model to better understand the role played by the poles associated with the ambiguity identified by Castillejo, Dalitz, and Dyson (CDD). By analytically analyzing the numerical solutions to the crossing-symmetric Chew-Low model previously found, we show that the solutions are unique. This is done by requiring a second solution that would reproduce the original threshold and resonance. This uniqueness applies to the family of solutions, found by varying the parameters, that produces a resonance. The $T$ matrix for this model is not a generalized $R$ function. and no analogy can be made to the CDD analysis. Adding a CDD pole to the solution would constitute a new model, so there is no CDD ambiguity. None of the family of solutions from this new model will reproduce the original threshold and resonance. We also find that one cannot treat an individual channel as an $R$ function, nor can one add a CDD pole to an individual channel alone. Both a pole and a crossed pole must be added to all channels to maintain crossing symmetry. Crossing symmetry analytically connects all of the channels. The Chew-Low model being phenomenological is more pertinent to modern field theoretic models.

Figure 1

The phase of $H_T(z\to \omega +i\eta )$, $\theta (\omega )$, as a function of $k_{c.m.}$ for the solution of the Chew-Low model with $\beta =752\mathrm{MeV}$ as found in Ref. 16.