Toward an understanding of discrete ambiguities in truncated partial-wave analyses

It is well known that the observables in a single-channel scattering problem remain invariant once the amplitude is multiplied by an overall energy- and angle-dependent phase. This invariance is called the continuum ambiguity and acts on the infinite partial-wave set. It has also long been known that, in the case of a truncated partial wave set, another invariance exists, originating from the replacement of the roots of partial-wave amplitudes with their complex conjugate values. This discrete ambiguity is also known as the Omelaenko-Gersten-type ambiguity. In this paper, we show that for scalar particles, discrete ambiguities are just a subset of continuum ambiguities with a specific phase and thus mix partial waves, as the continuum ambiguity does. We present the main features of both continuum and discrete ambiguities and describe a numerical method which establishes the relevant phase connection.

Figure 1

The geometrical picture of the general continuum ambiguity (3) is depicted here. The differential cross section (2) constrains the true solution for the amplitude $A(W,\theta )$ (blue solid arrow) to be located on a circle of radius $+\sqrt{{\sigma }_0}$ in the complex plane (thin orange-colored circle). A rotation of the amplitude $A(W,\theta )$ to $\tilde{A}(W,\theta )$ (red dashed arrow and thick dashed curved line) does not alter the cross section.

Figure 2

Three schematics are shown in order to illustrate the meaning of the terms discrete and continuum ambiguities. The gray-colored box represents in each case the higher dimensional space furnished by the partial-wave amplitudes, be it for infinite partial-wave models or for truncated ones. Top: One-dimensional (for instance circular) arcs can be traced out by continuum ambiguity transformations, both for infinite and truncated models. Center: More general connected continua in amplitude space, containing an infinite number of points belonging to the same cross section, can be generated by use of the continuum ambiguity (3). However, this phenomenon is only present once the partial-wave series goes to infinity. The connected patches are also referred to as islands of ambiguity [1, 3]. Bottom: Discrete ambiguities refer to cases where the cross section is the same for disconnected, discretely located points in amplitude space. These ambiguities are most prominent in TPWAs [1, 14] (see Sec. 3 below). However, twofold discrete ambiguities can also appear for infinite partial-wave models, where elastic unitarity is employed [1].

Figure 3

These plots show the real (blue solid line) and imaginary parts (red dashed line) of the phase rotations (39) extracted from the toy-model amplitude (33) defined in the main text. The individual figures are labeled via the respective ambiguity ${\mathbf{\pi }}_p$ belonging to each phase $e^{i{\phi }_p\left(x\right)}$.

Figure 4

The convergence process of the functional minimization procedure as described in the main text is demonstrated here. For the phase rotations $e^{i{\phi }_0\left(x\right)}$ and $e^{i{\phi }_1\left(x\right)}$, generating the discrete ambiguities ${\mathbf{\pi }}_0$ and ${\mathbf{\pi }}_1$ of the toy model (33), two randomly drawn initial functions have been picked from the applied ensemble. These initial conditions have, in the process of minimization, converged to these two respective phases. Minimizations have been performed by starting always at the same initial function, but applying different numbers for the maximal number of iterations $N_{\max }$ of the minimizer, as indicated in the headers of the plots. Values range from $N_{\max }=5$ (minimizer has barely changed the initial function) up to $N_{\max }=500$ (convergence condition fulfilled for any of the minimizations). In all plots, the real and imaginary parts of the precise Gersten ambiguity are drawn as thin blue solid lines and thin red finely dashed lines, respectively. The results of the functional minimizations up to $N_{\max }$ are drawn as thick blue coarsely dashed lines for the real parts and thick red finely dashed lines for the imaginary parts.

Figure 5

This is the continuation of Fig. 4. Convergence of the minimization of the functional (23) is illustrated for the phases $e^{i{\phi }_2\left(x\right)}$ and $e^{i{\phi }_3\left(x\right)}$, which generate discrete symmetries for the toy model (33).