Multipositivity Bounds in Scattering Theory

• Multipositivity bounds are higher-point positivity constraints derived from unitarity, analyticity, and factorization that correlate scattering amplitudes across varying external multiplicities.
• They replace isolated four-point positivity conditions with a hierarchy of principal-minor constraints linking 4-, 5-, and 6-point amplitudes or EFT Wilson coefficients.
• Applications include establishing lower bounds in chiral Lagrangians and probing deformation rigidity in open-string and higher-spin theories.

Multipositivity bounds are higher-point positivity constraints in scattering theory: nonlinear inequalities, derived from unitarity, analyticity, factorization, and suitable boundedness assumptions, that correlate amplitudes or EFT Wilson coefficients across different external multiplicities rather than within a single four-point process. In the recent amplitudes literature, they were formulated for planar tree-level theories as an infinite web of constraints on residues and low-energy coefficients, and then applied to large-$N_c$ QCD to show that certain nonanomalous chiral couplings are bounded from below by the chiral anomaly (Cheung et al., 8 May 2025).

1. Definition and scope

Standard positivity bounds are built from forward-limit dispersion relations for $2\to2$ scattering and constrain derivatives of a single four-point amplitude. Multipositivity bounds extend that logic to higher multiplicity. In one formulation, they are “positivity constraints on families of residues of tree-level scattering amplitudes with different numbers of external legs,” and they are “multi-” because they tie together the positivity properties of 4-, 5-, 6-, $\ldots$-point amplitudes at fixed mass level (Basile et al., 4 Mar 2026). In another formulation, they appear as positivity of residue matrices and Wilson-coefficient matrices,

$$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$

for planar scalar theories at tree level (Cheung et al., 8 May 2025).

The basic target is therefore not merely positivity of a single coefficient, but correlated positivity of an entire sequence of higher-point data. This replaces isolated inequalities by a hierarchy of principal-minor constraints. A simple example is the mixed 4–5–6 inequality

$$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$

and, in EFT language,

$$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$

which are already nonlinear in the low-energy data (Cheung et al., 8 May 2025).

The construction is intrinsically higher-point. This suggests that multipositivity probes consistency conditions that standard four-point positivity leaves invisible, especially odd-point or mixed-multiplicity sectors.

2. Factorization, complex forward limits, and the master inequalities

A direct derivation uses planar tree amplitudes with a single-trace ordering. For an $(n+m)$-point color-ordered amplitude with a factorization pole in a planar channel,

$s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$

the residue factorizes as

$$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$

In an even-point complex forward limit, such as

$A^{(2n)}(p_1,\dots,p_n,-\bar p_n,\dots,-\bar p_1),$

the residues become absolute squares and are therefore nonnegative. Defining vectors $2\to2$0, $2\to2$1 in the exchanged-state space, the relevant Wilson coefficients can be written as inner products,

$2\to2$2

Cauchy–Schwarz then yields the basic multipositivity inequality

$2\to2$3

together with the derivative-enhanced version

$2\to2$4

for kinematic parameters $2\to2$5 associated with opposite sides of the cut (Cheung et al., 20 May 2026).

The simplest nontrivial case is $2\to2$6, $2\to2$7, which relates 4-, 5-, and 6-point coefficients: $2\to2$8 Planarity is essential: with a single trace ordering, one can choose a channel $2\to2$9 so that only one factorization channel contributes, and the residue sum in that channel equals the Wilson coefficient defined by the small-$\ldots$0 expansion (Cheung et al., 20 May 2026).

3. Moment matrices, half-ladder residues, and EFT Wilson coefficients

A complementary formulation starts from planar half-ladder factorization. For a planar $\ldots$1-point amplitude, let $\ldots$2 denote the maximal-cut residue in the half-ladder topology. In soft kinematics, where the middle external legs are set to zero, all exchanged levels coincide, $\ldots$3, and the diagonal residue takes the form

$\ldots$4

with $\ldots$5 a Hermitian “soft operator” acting on the level-$\ldots$6 subspace. The associated Hankel matrix of moments is therefore positive semidefinite: $\ldots$7 and likewise

$\ldots$8

All principal minors are thus nonnegative (Cheung et al., 8 May 2025).

The same paper generalizes this by replacing the pure level-$\ldots$9 state with a density matrix $$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$0 and the soft operator with a weighted Hermitian operator $$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$1, producing weighted moments

$$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$2

Soft kinematics yields diagonal-in-level sums, while shock-wave kinematics with Mellin-transformed collinear external states realizes arbitrary transitions between levels and gives physical weighted sums of residues. In that setting,

$$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$3

for any positive semidefinite $$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$4 and Hermitian $$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$5 (Cheung et al., 8 May 2025).

Planar dispersion relations then convert these residue statements into EFT statements. For 4 points,

$$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$6

while for planar $$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$7-point amplitudes the coefficient of $$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$8 in the half-ladder channel is

$$\boldsymbol{R}_n^{(r)}(0)\succeq 0, \qquad \boldsymbol{g}_k^{(r)}(0)\succeq 0,$$9

With a specific choice of $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$0 and $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$1, one obtains

$$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$2

so multipositivity becomes a tower of mixed-multiplicity bounds on EFT Wilson coefficients (Cheung et al., 8 May 2025).

4. The chiral Lagrangian and the anomaly bound

In planar large-$$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$3 QCD with massless quarks, connected pion amplitudes decompose into single-trace partial amplitudes, and the low-energy EFT is the chiral Lagrangian on

$$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$4

At the derivative order relevant here, the parity-even sector contains $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$5 and $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$6, while the parity-odd sector is governed by the Wess–Zumino–Witten term with coefficient $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$7. The 5-pion anomalous vertex contributes only to odd-point amplitudes, so standard $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$8 positivity does not probe it (Cheung et al., 20 May 2026).

Multipositivity does. In the $$R_{6,n}^{(r)}(0)\,R_{4,n}^{(r)}(0)-R_{5,n}^{(r)}(0)^2\ge 0,$$9, $$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$0 case, one relates 4-, 5-, and 6-pion amplitudes in a kinematic chart where a single channel $$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$1 is analytic and all other invariants are independent of $$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$2. Expanding the EFT amplitudes at small $$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$3, one finds

$$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$4

$$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$5

$$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$6

Applying the derivative form of multipositivity and evaluating at $$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$7 gives

$$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$8

hence

$$g_{6,k}^{(r)}(0)\,g_{4,k}^{(r)}(0)-g_{5,k}^{(r)}(0)^2\ge 0,$$9

In the dimensionless variables of the large-$(n+m)$0 pion bootstrap literature,

$(n+m)$1

where the numerical value uses large-$(n+m)$2 lattice determinations of $(n+m)$3 and $(n+m)$4 (Cheung et al., 20 May 2026).

The conceptual content is the linkage between the anomalous and nonanomalous sectors. The anomaly coefficient $(n+m)$5 is fixed by UV matching, but multipositivity shows that it cannot float independently of the parity-even couplings: the 5-point anomalous amplitude forces the 4- and 6-point parity-even sector to be sufficiently strong. A plausible implication is that higher-point unitarity organizes the chiral EFT parameter space more tightly than ordinary four-point dispersion theory.

5. Open strings, higher-spin towers, and rigidity

Open-string amplitudes furnish a distinguished example. In the soft kinematic regime of the planar scalar construction, the diagonal residue at level $(n+m)$6 is

$(n+m)$7

and is independent of the multiplicity $(n+m)$8. The residue Hankel matrix is therefore rank one, so an infinite class of multipositivity bounds is exactly saturated. In EFT language, the corresponding Wilson coefficients at $(n+m)$9 are

$s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$0

again giving saturation of the mixed-multiplicity bounds (Cheung et al., 8 May 2025).

A related higher-spin formulation studies tree-level open-string–type amplitudes in a multi-soft regime with an infinite tower of massive resonances. At fixed mass level $s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$1, one defines soft residues $s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$2 and the Hankel matrix

$s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$3

whose principal minors must be nonnegative for all $s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$4. For undeformed Koba–Nielsen amplitudes,

$s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$5

independent of $s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$6, so the leading large-$s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$7 multipositivity matrix is again effectively rank one. Subleading $s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$8 corrections generate genuine inequalities across consecutive multiplicities, such as

$s = -(p_1+\cdots+p_n)^2 = -(q_1+\cdots+q_m)^2,$9

for $$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$0 fixed at large $$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$1 (Basile et al., 4 Mar 2026).

Applied to Gross’s simplest satellite deformation, this subleading bound reduces to

$$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$2

which, together with the low-level multipositivity constraints

$$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$3

forces $$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$4. Consequently the four-point deformation vanishes: $$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$5 The same paper summarizes the result as follows: any deformation of four-point string amplitudes of this type is forbidden by unitarity (Basile et al., 4 Mar 2026).

These results align with the broader higher-point program. Bespoke dual-resonant amplitudes with deformed spectra are constrained by multipositivity, including conditions such as $$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$6 from the level-1 4–5–6 determinant, and deformations of the worldsheet integrand are forced to agree with the string at low levels. This suggests that the infinite higher-spin tower is “dramatically more rigid than any finite number of species,” in the paper’s phrase (Basile et al., 4 Mar 2026).

6. Assumptions, limitations, and open directions

The current formulations are tree-level constructions. In the planar scalar and chiral-Lagrangian settings, the residue sums are discrete and rely on simple poles, while loop corrections would introduce branch cuts and a more complicated analyticity structure. The chiral application also uses the planar large-$$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$7 limit, where amplitudes are tree-level and single-trace, and assumes analyticity in one channel with Regge boundedness strong enough to neglect boundary terms in the dispersion relation (Cheung et al., 20 May 2026).

Planarity is not a technical ornament. In the EFT derivation it isolates the half-ladder channel, allowing a single set of residues to coincide with the Wilson coefficients extracted by multivariable contour integrals. Nonplanar amplitudes introduce additional channels and multi-trace structures, so the simple half-ladder representation of $$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$8 no longer follows in the same way (Cheung et al., 8 May 2025).

Several open directions are explicit in the literature. One is extension beyond tree level, possibly through a higher-point analogue of a Lehmann–Källén representation or some spectral integral over tree data. Another is extension beyond the planar limit, especially for finite-$$R^{(n+m)}_k(p_1,\dots,p_n\,|\,q_1,\dots,q_m) = A^{(n+1)}_k(p_1,\dots,p_n)\, A^{(m+1)}_k(q_1,\dots,q_m).$$9 QCD. A third is systematic inclusion of higher orders in the derivative expansion, which in the chiral case would generate an extended network of inequalities among $A^{(2n)}(p_1,\dots,p_n,-\bar p_n,\dots,-\bar p_1),$0, $A^{(2n)}(p_1,\dots,p_n,-\bar p_n,\dots,-\bar p_1),$1, and higher couplings. The same formalism is also proposed as input for numerical S-matrix bootstrap programs, where multipositivity could further shrink the allowed islands of amplitudes (Cheung et al., 20 May 2026).

In that sense, multipositivity bounds occupy a specific place in the modern amplitudes program. They do not replace ordinary positivity bounds; they refine them by importing higher-point factorization data into the positivity problem. The resulting constraints are mixed-multiplicity, nonlinear, and, in known examples, strong enough to relate anomalies to parity-even EFT data, to enforce coupling universality in the open string, and to obstruct deformations that would be invisible to four-point analyses alone.