Open questions from the double phase representation: constant total cross sections, elastic unitarity, and singularities in the complex angular-momentum plane

The double phase representation (DPR) of Sugawara and Nambu (Phys. Rev. 131, 2335 (1963)) writes the elastic scattering amplitude as $A(s,t,u) = [\Pi(s,t,u)/\Pi_s(s,t,u)]\,Q(s,t,u)$, with a finite polynomial $\Pi$ absorbing the amplitude's zeros and $Q$ reconstructed from phase functions along the cuts. Section 6 of that paper left four issues open, the central one being whether the "no-shrinkage" asymptotic form $A(s,t) \to i s\,\beta(t)$ — an asymptotically constant total cross section with a diffraction peak of fixed slope — is compatible with unitarity, given that in the authors' judgment both Gribov's and Oehme's proofs against it contain unjustified steps. The headline physics is settled at the level of consensus: Gribov's theorem (elastic unitarity in the $t$-channel elastic strip forbids an absorptive part $A_s(s,t)$ — the imaginary part of the amplitude — growing linearly in $s$, $A_s(s,t) \sim s f(t)$) is the accepted answer (Martin and Richard 2001; Kupsch 2008), and experimentally total cross sections rise while the diffraction peak shrinks, so the no-shrinkage scenario is empirically dead. The mathematical core, however, is open: no amplitude is known that satisfies crossing symmetry, elastic unitarity, and all inelastic unitarity constraints together with Mandelstam analyticity and prescribed high-energy asymptotics — a problem Kupsch (2008) states is "still unsolved"; the exact complex-angular-momentum singularity structure required (fixed poles forbidden; rising Regge poles allowed; cut-protected fixed poles proposed by Oehme (1967); multi-Reggeon cuts accumulating at $j = 1$) is not classified; and the DPR-internal questions — incorporating unitarity into the representation, and proving or relaxing the finite-polynomial zeros condition (iv) — were never addressed (the DPR program itself ended, with Grunberg (1977) its only later use). The surviving open core is therefore the rigorous existence/classification question under full unitarity, together with the dormant representation-internal questions.

Background

Setting and terminology (for a reader from a neighboring subfield). In two-body scattering the invariant amplitude $A(s,t,u)$ is a function of the Mandelstam variables: $s$ is the square of the total center-of-mass energy, $t$ the square of the momentum transfer, and $u$ the crossed energy variable, with $s + t + u$ fixed by the external masses. Mandelstam analyticity postulates that $A$ is the boundary value of a function analytic in a prescribed domain of these variables and satisfies a double dispersion relation (a double integral representation over two spectral functions supported on the double spectral regions). Crossing symmetry is the statement that one and the same analytic function $A(s,t,u)$ describes all scattering processes related by exchanging the roles of the variables; for identical particles $A$ is symmetric under permutations of $s$, $t$, and $u$. The $t$-channel partial waves $a_j(t)$ are defined by expanding the amplitude in Legendre polynomials of the crossed-channel scattering angle and then analytically continuing the angular momentum $j$ to complex values (the complex angular-momentum, or Regge, plane). The singularities of $a_j(t)$ in this plane control the high-energy behavior: a Regge pole at $j = \alpha(t)$ contributes $A(s,t) \sim s^{\alpha(t)}$; a fixed pole sits at a $t$-independent value of $j$; Regge cuts are moving branch points generated by the simultaneous exchange of several Regge poles (multi-pomeron exchange; the pomeron is the leading Regge trajectory carrying vacuum quantum numbers, i.e. the exchange responsible for diffractive scattering). Elastic unitarity in the $t$-channel elastic strip is the statement that for $t$ in the interval where only the lowest two-particle intermediate state is open, unitarity of the S-matrix implies $\mathrm{Im}\, a_j(t) = \rho(t)\,|a_j(t)|^2$ ($\rho$ the elastic phase-space factor); above inelastic thresholds the right-hand side acquires additional non-negative inelastic contributions. The diffraction peak is the forward ($t \approx 0$) peak of the differential cross section $d\sigma/dt$; by the optical theorem, $\sigma_{\mathrm{tot}}(s) \propto \mathrm{Im}\, A(s, 0)/s$. "No shrinkage" denotes the asymptotic form $A(s,t) \to i s\,\beta(t)$: an asymptotically constant $\sigma_{\mathrm{tot}}$ with a peak of fixed slope; "shrinkage" denotes a peak whose width in $t$ decreases as $s$ grows. The Froissart–Martin bound $\sigma_{\mathrm{tot}} \le C \ln^2 s$ is the model-independent upper bound implied by analyticity and unitarity; "saturation" means achieving this maximal growth. The double phase representation (Sugawara and Tubis 1962, 1963; Sugawara and Nambu 1963) writes $A(s,t,u) = [\Pi(s,t,u)/\Pi_s(s,t,u)]\,Q(s,t,u)$, where $\Pi$ is a finite polynomial carrying the zeros (condition (iv)), $\Pi_s$ a denominator polynomial, and $Q$ is reconstructed from phase functions along the cuts; the representation holds under conditions (i)–(v): (i) $A$ is analytic everywhere except for three cuts $\infty>s\ge s_0$, $\infty>t\ge t_0$, $\infty>u\ge u_0$ (with $s_0$, $t_0$, $u_0$ real positive thresholds) and a finite number of real poles; (ii) $A^*(s,t,u)=A(s^*,t^*,u^*)$ (reality); (iii) $A$ is bounded by finite polynomials at infinity; (iv) a finite real polynomial $\Pi$ accommodates all the zeros of $A$ except for the one at infinity and no others; (v) $A$ has even or odd crossing symmetry under interchange of a pair of variables, e.g. $A(s,t,u)=\pm A(u,t,s)$. Prior results the question rests on. Gribov (Nucl. Phys. 22, 249 (1961)) argued that elastic unitarity in the $t$-channel forbids the no-shrinkage asymptotics $A_s(s,t) \sim s f(t)$ (here and below $A_s$ denotes the absorptive part, i.e. the imaginary part of the amplitude taken on the $s$-channel physical region); Oehme (Phys. Rev. Lett. 9, 358 (1962)) argued compatibly that $t$-independent fixed poles are excluded. Sugawara and Nambu criticized both proofs: Gribov's continuation of the unitarity condition to amplitudes growing linearly in $s$ is not justified ("his proof breaks down in the case of actual interest"), and Oehme's implication from the asymptotic form to a fixed pole in the partial-wave amplitude is unproven; they noted that an essential singularity in the angular-momentum plane "may be the case, though this question needs further study", that they did "not know how to use rigorously the unitarity condition" within the representation, and that the finite-polynomial zeros condition was adopted for practical reasons with "no indication that the double phase representation may be too restrictive".

Problem Statement

Let $A(s,t)$ be an elastic two-body amplitude assumed to satisfy Mandelstam analyticity (double dispersion relation), crossing symmetry, elastic unitarity in the $t$-channel elastic strip, and full ($s$-channel elastic plus inelastic) unitarity. The surviving open core of the Sugawara–Nambu question is two-fold. The sub-deliverables listed under Q1 and Q2 are independently resolvable: for Q1 these are the existence of an admissible asymptotically constant-$\sigma_{\mathrm{tot}}$ amplitude, the classification of the required complex-angular-momentum singularity structure (conditional on existence), closure of the two Gribov-proof gaps, and the proton–proton case; for Q2 they are imposing unitarity inside the representation and deciding whether condition (iv) is a genuine restriction. A partial result settles only the sub-deliverables it addresses, and the task is resolved when both Q1 and Q2 are settled. (Q1 — existence and classification) Determine rigorously which asymptotic behaviors of $A$ are admissible: prove or disprove the existence of an amplitude with asymptotically constant $\sigma_{\mathrm{tot}}(s) \to \mathrm{const} > 0$ (with or without shrinkage) that satisfies all of the above constraints simultaneously, and — if such an amplitude exists — classify the complex-angular-momentum singularity structure it must possess (rising Regge pole; cut-protected fixed pole in the sense of Oehme 1967; accumulating multi-Reggeon cuts; or essential singularity); a rigorous disproof of existence settles this existence-and-classification part with no classification to supply. This includes completing Gribov-type theorems: closing the gaps identified already by Sugawara and Nambu — the analytic continuation of the elastic-unitarity condition to amplitudes growing linearly in $s$ is unjustified, and the implication from the asymptotic form to a fixed pole in the $j$-plane is unproven — as well as treating the proton–proton case, where Martin and Richard (2001) note additional technical problems from the $t$-channel unphysical region. (Q2 — representation-internal) Decide whether the unitarity condition can be imposed at all, for a nonzero amplitude, within the double phase representation $A = [\Pi/\Pi_s]\,Q$; and determine whether condition (iv) — absorption of all the zeros of $A$ except for the one at infinity into a finite polynomial $\Pi$ — is a genuine restriction (exhibit a nonzero function satisfying conditions (i)–(iii) and (v) whose zeros are not captured by any finite polynomial) or provable in generality. Scope note: the headline physical question — compatibility of the no-shrinkage form $A \to i s\beta(t)$ with elastic unitarity — is considered settled negatively at the level of consensus (Gribov's theorem; Martin–Richard 2001; Kupsch 2008) and is empirically excluded; the genuinely open part is the rigorous classification under full unitarity (Q1) and the dormant representation-internal questions (Q2).

References

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