Radius-free causality bounds on the energy derivative of scattering phase shifts: removing the arbitrary channel radius from Wigner's 1955 inequality

Wigner (1955) proved that causality — the scattered wave cannot leave the scatterer before the incident wave has reached it — places a lower limit on the energy derivative of the elastic scattering phase shift, but only under the assumption that the interaction vanishes beyond a channel radius $a$ , and the bound becomes weaker as the assumed $a$ grows. Wigner himself identified the proper choice of $a$ as the principal difficulty in applying the inequality, e.g. to discriminate between the discrete ambiguous sets of pion–nucleon phase shifts of that era. Seventy years of descendants — the R-matrix tradition, the Hammer–Lee generalization of Wigner's causality bound to arbitrary dimension and angular momentum, Coulomb-modified bounds for charged particles, and effective-field-theory analyses of the Wigner bound on the effective range — all retain a range, cutoff, or channel-radius parameter playing exactly the role of $a$ ; the extensions to realistic interactions without a sharp cutoff (Yukawa/exponential tails, coupled channels, energy-dependent effective interactions) are qualitative at best, and relativistic zero-range scattering is known to surpass the bound while remaining causal. The open problem is to formulate — or refute the existence of — a causality constraint on the energy derivative of scattering phase shifts (equivalently, on the Wigner time delay) that is free of any arbitrarily chosen radius, with controlled behavior for interactions without a sharp finite range.

Background

In nonrelativistic quantum scattering, a particle scattered by a center of force is decomposed into partial waves, and each partial wave of angular momentum $l$ emerges with a phase shift $\delta_l(E)$ ; the set of phase shifts, not the cross section alone, is the primary object extracted from scattering experiments. The energy derivative of the phase shift has a direct physical meaning: a wave packet scattered in a given partial wave is delayed (or advanced) by the time $2\hbar\, d\delta_l/dE$ relative to free propagation, a quantity called the Wigner (or Wigner–Smith) time delay.

Wigner's 1955 paper asked what the principle of causality — in his formulation, that the scattered wave cannot leave the scatterer before the incident wave has reached it — implies about $d\delta/dE$ . A classical picture suggests that the advance of the scattered packet cannot exceed the time it takes light (or the particle) to cross the interaction region, so the time delay should be bounded below by roughly minus the traversal time of a sphere of radius $a$ enclosing the interaction. To make this rigorous, Wigner assumed that the interaction vanishes beyond a channel radius $a$ — the setting of the R-matrix method he had developed with Eisenbud. In that formalism one splits space into an internal region ( $r<a$ ) and an external region where the wave is free; the derivative matrix (R-matrix) is the reciprocal logarithmic derivative of the interior radial wave function at $r=a$ , and its analytic (pole) structure, $R(E)=\sum_n \gamma_n^2/(E_n-E)$ with positive residues, encodes the causality of the response. From this structure Wigner derived, for example for S-waves,

\frac{d\eta}{dk} \ge -a + \frac{1}{2k}\,\sin 2(\eta + ka),

where $\eta$ is the phase shift and $k$ the wave number: the time delay cannot be more negative than the free traversal time across the radius- $a$ sphere, up to an explicit oscillatory correction. The practical motivation was the phase-shift ambiguity problem in pion–nucleon scattering: in the early 1950s several distinct sets of phase shifts reproduced the same measured cross sections, and a model-independent inequality on $d\eta/dE$ promised to discriminate between them.

The difficulty, stated in the closing paragraphs of Wigner's own paper, is that the bound depends on the assumed channel radius. Realistic interactions do not vanish sharply: the pion–nucleon interaction is mediated by pion exchange and falls off exponentially. If $a$ is chosen larger than the true interaction range the bound is correct but weaker; if chosen too small, it is violated. Wigner noted that whether, for pion–nucleon scattering, $a=2\hbar/\mu c$ ( $\mu$ the pion mass) is a permissible choice, or whether one must assume $a=3\hbar/\mu c$ or larger, cannot be decided within the formalism — the estimate of $a$ requires 'an at least qualitative theory of the nature of the interaction'. He offered only a diagnostic: plot the R-function (the reciprocal logarithmic derivative at $a$ , computed from the data) versus energy and judge whether any irregularity can be blamed on a too-small $a$ . This radius ambiguity is the obstacle the present problem addresses.

The question re-emerged in modern language. At low energy the phase shift is expanded in the effective range expansion, $k\cot\delta = -1/a_s + r_e k^2/2 + \dots$ (with $a_s$ the scattering length and $r_e$ the effective range), and Wigner's inequality translates into an upper bound on the effective range of order the interaction range, $r_e \lesssim 2R$ , sometimes called the Wigner bound. In effective field theories (EFTs) of nuclear and atomic systems, where short-range physics is replaced by contact interactions cut off at momentum $\Lambda$ , this bound constrains the regulator/cutoff and the separation of low- and high-momentum scales, and it is routinely invoked to restrict permissible cutoffs and to test whether a zero-range or unitary limit (the limit of infinite scattering length) can be reached. In R-matrix analyses of nuclear resonance data, the channel radius remains an analyst-chosen parameter with known (but uncontrolled) influence on extracted reduced widths. In all of these settings the question Wigner left open persists in sharpened form: is there a causality constraint on the phase-shift energy dependence that does not require an arbitrarily chosen radius at all?

Problem Statement

Formulate — and establish, or refute the existence of — a causality constraint on the energy derivative of elastic scattering phase shifts (equivalently, on the Wigner time delay $2\hbar\, d\delta_l/dE$ ) that does not depend on any arbitrarily chosen channel radius, interaction range, or cutoff.

Such a constraint must (i) be established for a precisely stated class of interactions addressed — in particular whether interactions without a sharp cutoff are covered, including exponential/Yukawa tails (such as one-pion exchange), Coulomb tails, coupled channels with partial-wave mixing, and energy-dependent effective interactions — and a precisely stated notion of causality — macroscopic signal causality (Wigner's formulation: no scattered wave before the incident wave has reached the scatterer) versus S-matrix analyticity (the analytic properties of the scattering amplitude in the complex energy plane) — since these are known to diverge; (ii) be proved, with explicit control of the corrections introduced by the interaction tail, and with every parameter entering the bound either an observable or fixed by the interaction's declared long-distance behavior, so that no adjustable radius remains — that is, no cutoff or channel radius is left free, while a length scale fixed by the interaction's declared long-distance behavior does not reintroduce an adjustable radius; (iii) be consistent with the known limiting cases: recovery of Wigner's 1955 finite-range inequality when the interaction is made strictly finite-range, agreement with or controlled extension of the Coulomb-modified bounds of König, Lee, and Hammer, and explicit compatibility with the relativistic zero-range counterexample of Pascalutsa; and (iv) be non-vacuous, i.e. constrain physically relevant quantities (such as effective-range parameters or time delays) strongly enough to discriminate between phase-shift sets in at least one concrete system, as Wigner envisaged for pion–nucleon scattering.

Alternatively, the existence of such a constraint can be refuted by proving that no non-vacuous causality constraint of Wigner type exists without a range assumption.

References

E. P. Wigner, 'Lower Limit for the Energy Derivative of the Scattering Phase Shift', Physical Review 98, 145 (1955). DOI: 10.1103/PhysRev.98.145, https://doi.org/10.1103/PhysRev.98.145
E. P. Wigner and L. Eisenbud, 'Higher Angular Momenta and Long Range Interaction in Resonance Reactions', Physical Review 72, 29 (1947). DOI: 10.1103/PhysRev.72.29, https://doi.org/10.1103/PhysRev.72.29
Takesi Ogimoto, 'Macroscopic Causality and Lower Limit for the Momentum Derivative of the Scattering Phase Shift', Progress of Theoretical Physics 24, 949 (1960). DOI: 10.1143/ptp.24.949, https://doi.org/10.1143/ptp.24.949
Takesi Ogimoto, 'Macroscopic Causality and Lower Limit for the Energy Derivative of the Scattering Phase Shift' (subtitle: 'Relativistic Case'), Progress of Theoretical Physics 26, 483 (1961). DOI: 10.1143/ptp.26.483, https://doi.org/10.1143/ptp.26.483
S. J. Lindenbaum and R. M. Sternheimer, 'Application of the Spin-Flip Dispersion Relations to the Minami Ambiguity for the Pion-Nucleon Scattering', Physical Review 110, 1174 (1958). DOI: 10.1103/physrev.110.1174, https://doi.org/10.1103/physrev.110.1174
H.-W. Hammer and Dean Lee, 'Causality and universality in low-energy quantum scattering', Physics Letters B 681, 500 (2009). DOI: 10.1016/j.physletb.2009.10.033, arXiv:0907.1763, https://arxiv.org/abs/0907.1763
H.-W. Hammer and Dean Lee, 'Causality and the effective range expansion', Annals of Physics 325, 2212 (2010). DOI: 10.1016/j.aop.2010.06.006, arXiv:1002.4603, https://arxiv.org/abs/1002.4603
Serdar Elhatisari and Dean Lee, 'Causality bounds for neutron-proton scattering', European Physical Journal A 48, 110 (2012). DOI: 10.1140/epja/i2012-12110-x, arXiv:1206.1207, https://arxiv.org/abs/1206.1207
Sebastian König, Dean Lee, and H.-W. Hammer, 'Causality constraints for charged particles', Journal of Physics G: Nuclear and Particle Physics 40, 045106 (2013). DOI: 10.1088/0954-3899/40/4/045106, arXiv:1210.8304, https://arxiv.org/abs/1210.8304
Vladimir Pascalutsa, 'Surpassing Wigner's causality bound in relativistic scattering with zero-range interaction', Proceedings of Science, PoS(CD15), 114 (2016). DOI: 10.22323/1.253.0114, https://doi.org/10.22323/1.253.0114, arXiv:1402.4973, https://arxiv.org/abs/1402.4973
Saar Beck, Betzalel Bazak, and Nir Barnea, 'Removing the Wigner bound in non-perturbative effective field theory', Physics Letters B 806, 135485 (2020). DOI: 10.1016/j.physletb.2020.135485, arXiv:1907.11886, https://arxiv.org/abs/1907.11886
Silas R. Beane and Roland C. Farrell, 'Causality and dimensionality in geometric scattering', Annals of Physics 440, 168841 (2022). DOI: 10.1016/j.aop.2022.168841, arXiv:2112.02733, https://arxiv.org/abs/2112.02733
E. Di Salvo and G. A. Viano, 'Resonances and surface waves', Il Nuovo Cimento A 80, 317 (1984). DOI: 10.1007/BF02785804, https://doi.org/10.1007/BF02785804