Rigorous partial-wave positivity classification of the hypergeometric amplitude: the positive-mass-squared islands, the root-spacing conjecture, and the analytic–numerical gap

The hypergeometric amplitude of Cheung and Remmen is a crossing-symmetric, ultraviolet-soft, two-parameter deformation of the Veneziano amplitude that has become a central test case for the stringy S-matrix bootstrap. Mansfield and Spradlin proved that its residue partial-wave coefficients are nonnegative throughout a large region of the $(r, m^2, D)$ parameter space and negative throughout another large region, but their analytic map still disagrees with the numerical phase diagram in four precise ways that define this problem: an infinite family of potentially unitary islands at positive $m^2$ remains unsettled; the root-spacing conjecture on the zeros of derivatives of rising factorials, whose proof would extend the analytic non-unitarity exclusion to all of $m^2 > r+1$, is unproven; the proven positive region falls far short of the numerical boundary, with a sharper convolution bound (the source's Lemma 2) or a low-spin dominance principle as the proposed mechanisms; and the critical amplitude at $(r, m^2) = (-1/2, 0)$ is proven positive in $D = 4$ but inaccessible to the contour method in its critical dimension $D = 5$. The problem asks for the complete rigorous sign classification of the partial-wave coefficients $B^a_{n,j}(r, m^2, D)$, each gap to be settled by a proof or an explicit negative coefficient. No resolution of any of these gaps is known in the literature to date: the two later positivity theorems (Mansfield 2025; Chen and Yin 2026) settle only the undeformed Veneziano points at $r = 0$.

Background

Tree-level, or meromorphic, scattering amplitudes — functions of the Mandelstam invariants $s$ and $t$ (the squared total energy and the squared momentum transfer in a $2 \to 2$ collision) with simple poles but no branch cuts — are the natural laboratory of the modern S-matrix bootstrap: the program of determining which amplitudes are consistent with Poincaré invariance, crossing symmetry (invariance under permutations of the external momenta, for example $s \leftrightarrow t$), soft ultraviolet behavior, and unitarity. The Veneziano amplitude $A_{\rm ven}(s,t) = \Gamma(-\alpha(s))\Gamma(-\alpha(t))/\Gamma(-\alpha(s)-\alpha(t))$ describes tree-level open-string scattering and satisfies all of these; a guiding question of the field is how unique it is among consistent amplitudes.

Cheung and Remmen introduced a two-parameter answer: the hypergeometric amplitude, a crossing-symmetric, ultraviolet-soft amplitude for the scattering of four identical scalar particles of mass $m^2$, with an evenly spaced spectrum of poles at $s = m^2 + n$ for $n = 0, 1, 2, \ldots$. In the conventions of Mansfield and Spradlin the bosonic variant is

$\mathcal{A}^1(s,t) = -\frac{1}{s-m^2}\;{}_3F_2\!\left[1,\; -s+m^2,\; 1+t+r-m^2;\; 1-s+m^2,\; 1+r;\; 1\right],$

where ${}_3F_2$ is a generalized hypergeometric series which terminates on these parameters, so each residue is a polynomial in $t$. The residue at the pole $s = m^2 + n$ is $R_n^1(t) = (1+t+r-m^2)_{(n)}/(r+1)_{(n)}$, where $(x)_{(n)} = x(x+1)\cdots(x+n-1)$ is the rising factorial. A type-I variant $\mathcal{A}^0$ is related to $\mathcal{A}^1$ by an exact shift identity that transposes positivity results between the two, so the two families share one analysis. At $(r, m^2) = (0, -1)$ the amplitude reduces to the bosonic Veneziano amplitude, so the pair $(r, m^2)$ deforms away from the open-string point.

Tree-level unitarity is tested through the partial-wave expansion: each residue is expanded in $D$-dimensional Gegenbauer polynomials $G_j^D$ — the angular eigenfunctions that generalize Legendre polynomials to $D$ spacetime dimensions —

$R_n(t) = \sum_j B_{n,j}\, G_j^D(\cos\theta),$

and the existence of a unitary spectrum of exchanged particles requires $B_{n,j} \ge 0$ for every spin $j$ and every mass level $n \ge 3m^2$ (the exchanged particle of mass $m^2 + n$ must at least be kinematically accessible to the external particles of mass $m^2$). This positivity condition is necessary but not sufficient for full unitarity, which would additionally require a consistently factorizing amplitude with more than four external particles; no such generalization is known for this family, so unitarity throughout means four-point partial-wave positivity. For the undeformed string amplitudes positivity follows indirectly from the no-ghost theorem — the statement that the string spectrum contains no negative-norm states — but for deformations it must be established directly.

Before Mansfield and Spradlin, rigorous results were sparse. Cheung and Remmen mapped the phase diagram numerically. Rigatos derived an explicit formula for the partial-wave coefficients at $m^2 = -1$ and used it to exclude regions analytically. Rigatos and Wang, and Wang, expressed the coefficients of the $q$-deformed (Coon-type) amplitude in a basis of $q$-deformed harmonic numbers and proved positivity for a set of low-spin coefficients, conjecturing that the positive region is determined by these low-spin coefficients alone — the low-spin dominance principle, echoing the mechanism identified by Bern, Kosmopoulos and Zhiboedov for gravitational effective field theory. Mansfield and Spradlin then generalized the double-contour representation of the Veneziano amplitude due to Arkani-Hamed, Eberhardt, Huang and Mizera to nonzero $r$, and used it to prove two kinds of theorem: exclusions — positivity fails in all $D \ge 4$ if $2r + m^2 + a < 0$, or $r < -1$ with $\lfloor r \rfloor$ even, or $m^2 > r + 1$ with $\lceil 6m^2 \rceil$ odd, together with a further excluded band — and positivity of all $B^a_{n,j}$ throughout a large region of $(r, m^2, D)$. The remaining disagreement between these analytic results and the numerical phase diagram is the subject of the problem below.

Problem Statement

Determine, rigorously and completely, the sign structure of the partial-wave coefficients $B^a_{n,j}(r, m^2, D)$ of the hypergeometric amplitude — with $a \in \{0, 1\}$ labeling the bosonic ($a = 1$) and type-I ($a = 0$) variants — over its parameter space $r, m^2 \in \mathbb{R}$, $D \ge 4$; in other words, complete the unitarity phase diagram begun by Mansfield and Spradlin. The open core left by their paper consists of four linked gaps, each to be settled by a proof or by an explicit counterexample:

1. Root-spacing conjecture. Prove or refute that the roots $\rho_k^j$, ordered $\rho_0^j > \rho_1^j > \rho_2^j > \cdots$ (all at most $0$), of $p^{(j)}(y) = d^j y_{(n)}/dy^j$ satisfy $\rho_1^j - \rho_2^j \le \rho_0^j - \rho_1^j$ for all $j$ and $n$ with $n - j \ge 3$. A proof removes the auxiliary parity constraint $\lceil 6m^2 \rceil$ odd and thereby extends the analytic non-unitarity exclusion to the entire region $m^2 > r + 1$.

2. The islands at positive $m^2$. Numerics exhibit an infinite family of small regions at positive $m^2$ not excluded by any theorem. Settle each island: either exhibit an explicit negative coefficient $B^a_{n,j} < 0$ with $n \ge 3m^2$ at parameters inside the island, or prove positivity across it.

3. The analytic–numerical gap. The proven positive region falls far short of the numerical positivity boundary already at low dimensions. Close this gap at $r \ne 0$, for example by strengthening the source's combinatorial bounds (Lemma 2) on the convolution coefficients $b_{\alpha\beta}$ of the double-contour representation, or by proving the low-spin dominance principle for this family — that nonnegativity of the low-spin coefficients implies nonnegativity of all coefficients.

4. The critical amplitude in $D = 5$. The amplitude at $(r, m^2) = (-1/2, 0)$, whose numerical critical dimension is exactly $D = 5$, is proven positive in $D = 4$, but its status in the critical dimension $D = 5$ is beyond the contour method, which degenerates there. Settle the $D = 5$ case.

The problem is complete when every point of the parameter space with $r, m^2 \in \mathbb{R}$ and $D \ge 4$ is either covered by a positivity theorem for all $B^a_{n,j}$ (with $n \ge 3m^2$) or excluded by a rigorous argument such as an explicit negative coefficient or a general exclusion theorem; an answer may resolve any subset of the four gaps by proof and the rest by explicit counterexample, noting that refuting the root-spacing conjecture classifies no parameter point by itself, so a gap-1 counterexample must be accompanied by a classification of the band $m^2 > r + 1$ with $\lceil 6m^2 \rceil$ even by other means for the per-point criterion to be met there.

References

1. Gareth Mansfield and Marcus Spradlin, "On Unitarity of the Hypergeometric Amplitude", JHEP 02 (2025) 145, DOI 10.1007/JHEP02(2025)145, arXiv:2409.09561, https://arxiv.org/abs/2409.09561
2. Clifford Cheung and Grant N. Remmen, "Stringy dynamics from an amplitudes bootstrap", Phys. Rev. D 108 (2023) 026011, DOI 10.1103/PhysRevD.108.026011, arXiv:2302.12263, https://arxiv.org/abs/2302.12263
3. Nima Arkani-Hamed, Lorenz Eberhardt, Yu-tin Huang and Sebastian Mizera, "On unitarity of tree-level string amplitudes", JHEP 02 (2022) 197, DOI 10.1007/JHEP02(2022)197, arXiv:2201.11575, https://arxiv.org/abs/2201.11575
4. Konstantinos C. Rigatos, "Positivity of the hypergeometric Veneziano amplitude", Phys. Rev. D 109 (2024) 086008, DOI 10.1103/PhysRevD.109.086008, arXiv:2310.12207, https://arxiv.org/abs/2310.12207
5. Konstantinos C. Rigatos and Bo Wang, "Coon unitarity via partial waves or: how I learned to stop worrying and love the harmonic numbers", Phys. Rev. D 110 (2024) 126024, DOI 10.1103/PhysRevD.110.126024, arXiv:2401.13031, https://arxiv.org/abs/2401.13031
6. Bo Wang, "Positivity of the hypergeometric Coon amplitude", JHEP 04 (2024) 143, DOI 10.1007/JHEP04(2024)143, arXiv:2403.00906, https://arxiv.org/abs/2403.00906
7. Zvi Bern, Dimitrios Kosmopoulos and Alexander Zhiboedov, "Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude", J. Phys. A 54 (2021) 344002, DOI 10.1088/1751-8121/ac0e51, arXiv:2103.12728, https://arxiv.org/abs/2103.12728
8. Gareth Mansfield, "Positivity of the Veneziano amplitude in ten dimensions", JHEP 07 (2025) 179, DOI 10.1007/JHEP07(2025)179, arXiv:2502.20372, https://arxiv.org/abs/2502.20372
9. Qi Chen and Yuan Yin, "Exact All-Level Positivity of the Bosonic Veneziano Amplitude", arXiv:2608.25183, https://arxiv.org/abs/2608.25183 (preprint, 2026)
10. Shota Komatsu and Pronobesh Maity, "Worldsheet for Generalized Veneziano Amplitudes", Phys. Rev. Lett. 137 (2026) 021601, DOI 10.1103/bplf-h1vx, arXiv:2511.16280, https://arxiv.org/abs/2511.16280
11. Christian Baadsgaard Jepsen, "Veneziano amplitude in any dimension and a Virasoro-Shapiro partial amplitude", Phys. Rev. D 112 (2025) 066011, DOI 10.1103/3xb3-5yyc, arXiv:2506.05253, https://arxiv.org/abs/2506.05253
12. Federico Ambrosino and Nathan Haouzi, "Meromorphic amplitudes from 3-dimensional supersymmetry", arXiv:2606.18331, https://arxiv.org/abs/2606.18331 (preprint, 2026)