How does nonlinear unitarity extend the EFThedron geometry of the S-matrix?

From positivity to the unitarity disk

The S-matrix bootstrap bounds low-energy couplings without specifying the high-energy dynamics. In the standard EFThedron picture, dispersion relations write each coupling as an integral of the imaginary part of the amplitude over high energies, and unitarity makes that absorptive part positive. Each coupling is then a moment of one positive but otherwise unknown spectrum, and the allowed couplings form a classical moment problem whose geometry is fixed by Hankel-matrix positivity and crossing relations.

That picture uses only part of unitarity. Writing a partial wave as S = 1 + i f, conservation of probability requires (Re f)^2 + (Im f − 1)^2 ≤ 1, with equality for purely elastic scattering. Keeping only 0 ≤ Im f ≤ 2 gives the deprojected EFThedron, a moment problem with bounded density. The full disk is stronger because it ties the real part to the imaginary part, and the usual coefficient sum rules involve only the imaginary part.

The paper's hierarchy makes the difference concrete. Positivity restricts f to a half-plane, linearized unitarity restricts it to a strip, and full unitarity restricts it to a disk. The extremal spectra change accordingly: delta functions for positivity, steps between b = 0 and b = 2 for linearized unitarity, and pure phases on the unitarity circle for full unitarity.

Fractional kernels make the real part a moment

The technical move is to multiply the amplitude by K_α(x) = (1 − x^2)^α with real α before deforming the contour. The discontinuity of this product mixes the two components of the amplitude: it equals (x^2 − 1)^α [cos(πα) Im A − sin(πα) Re A]. At α = 0 this reduces to the ordinary relation involving only Im A; at α = 1/2 it isolates the real part.

After a partial-wave expansion, every coefficient obtained this way becomes a linear functional of the complex partial waves. The couplings take the form of complex moments M_k = Σ_ℓ ∫ Im(q_kℓ f_ℓ) dν_ℓ with known complex weights q_kℓ and positive reference measures ν_ℓ, subject to |f_ℓ − i| ≤ 1. This is the analogue of the classical moment problem in which the positive density is replaced by a complex one confined to the unitarity disk.

Because different kernels reconstruct the same coefficients, their results must agree. The paper calls the resulting linear conditions analyticity null constraints, by analogy with the crossing null constraints. A useful family compares ordinary and square-root kernels: the first harmonic reconstructs the subtraction constant a_0, while higher harmonics give nulls that vanish. The authors stress that these are not a new principle but a different way to implement analyticity, closely related to the real-part dispersion relation and to the Roy equations.

Nulls cut linear slices through the moment region

Unitarity alone defines a convex body of complex moments. The crossing and harmonic nulls cut this body with linear planes, and the resulting geometry is the Real EFThedron. With finitely many nulls the procedure gives an outer bound: for a given set of nulls, the Real EFThedron is the set of couplings for which there exist partial waves in the unitarity disk that reproduce those couplings and satisfy all the nulls. The set is convex, shrinks as nulls are added, and contains the physical region.

The paper computes bounds in two complementary ways. In the extremal-measure approach, nulls are added with adjustable multipliers to the combination of couplings being bounded, and the integrand U Re f + V Im f is maximized at each energy and spin. Its maximum, V + √(U^2 + V^2), lies on the unitarity circle at a definite phase, so pure phases take over the role played by delta functions in the positive moment problem and by steps in linearized unitarity. Integrating that local maximum, summing over spins, and minimizing over multipliers yields the bound.

In the moment-matrix approach, full unitarity is written as positivity of a 2×2 matrix W_ℓ with entries Im f_ℓ, Re f_ℓ, Re f_ℓ, and 2 − Im f_ℓ, with trace 2. Taking moments of W_ℓ in energy produces positive block Hankel matrices whose absorptive principal submatrices are the familiar EFThedron Hankel matrices of Im f and 2 − Im f, now linked by the real moments. Dispersion relations and nulls become linear equations among the entries, and enlarging the matrices gives a systematic hierarchy of constraints.

Two dimensions solve exactly with CDD factors

In two dimensions there is no spin sum, and the coefficient region can be determined exactly. For identical scalars of mass m, the paper considers the pole-free, crossing-symmetric class in which S is analytic on the crossing-identified disk, and defines couplings at s = 2m^2 through an expansion of T(s). Using kernels with α = 0, −1/2, and −1, one coupling combination is reconstructed from both the imaginary and the real part, so analyticity requires the first null N_1 = ⟨sin θ Re f − cos θ Im f⟩_θ = 0.

Bounding Λ_2 + (1 − μ)Λ_0/4 for each μ with the null added through a multiplier, the optimal choice λ = (1 − μ^2)/2 makes U^2 + V^2 a perfect square. The extremal partial wave is then a rational function of cos θ and sin θ, and the corresponding S-matrix on the cut has |S| = 1 and reads S(θ) = (−μ + e^{−iθ})/(1 + μ e^{−iθ}), the boundary value of a one-factor Castillejo–Dalitz–Dyson amplitude. Eliminating μ reproduces the known exact lens: −8 ≤ Λ_0 ≤ 0 and Λ_0^2/32 ≤ Λ_2 ≤ −Λ_0/2 − Λ_0^2/32.

At higher orders the nulls N_j = ⟨sin(jθ) Re f − cos(jθ) Im f⟩_θ = 0 for j = 1, …, n, combined with unitarity, determine exactly the region of the first n + 1 couplings. Its boundary consists of products of n CDD factors. In this two-dimensional setting the Real EFThedron is not merely an outer bound but the exact region.

Four dimensions bring the subtraction constant into play

In four dimensions the paper uses twice-subtracted dispersion relations, assuming M(s,t)/s^2 → 0 as |s| → ∞ at fixed t. For the massless EFThedron model, the tree-level EFT amplitude holds up to a cutoff, heavy states produce cuts starting at s = 1, and loops of light particles are neglected; unitarity is imposed only above the cutoff, along the heavy cut.

The real part gives access to the constant coupling g_0, which positivity cannot bound. Ordinary absorptive sum rules determine g_2 and g_3 in terms of Im f, but g_0 is a subtraction constant not fixed by those relations. The square-root kernel at order z^4 does contain it: at t = 0, the harmonic moments evaluate to g_2/16 for j = 0, g_0/128 for j = 1, and zero for j ≥ 2. The higher modes give the harmonic nulls, and standard crossing arguments add the first crossing null X_4 = ⟨L(L − 8) Im f / s^2⟩ = 0 with L = ℓ(ℓ + 1).

For this massless model, bounds from full unitarity had been obtained before with the primal bootstrap. The paper instead provides dual bounds that follow from twice-subtracted dispersion relations, which it describes as new for this model. The bounds are presented as closed-form inequalities and as stronger numerical bounds, with the constraint sets used far from exhaustive.

Massive scalar bounds narrow the dual-primal gap

For massive scalars with the full physical threshold, the paper combines fixed-transfer averages with smeared Roy equations and harmonic nulls. The fixed-transfer averages also test external energies beyond the compact window used in earlier dual calculations, which matters because the nulls constrain the real part at all energies rather than only inside that window.

The numerical output is a set of 25 inequalities. Together with positivity they give −7.751502 ≤ g_0^m ≤ 2.705955 and 0 ≤ g_2^m ≤ 1.052351. The full enclosure is the intersection of these half-planes and is an outer bound on the amplitude region. The published dual and primal curves use the same normalization, (c_{0,0}, c_{1,0}) = (g_0^m, g_2^m), so the comparison is like for like.

The comparison is directional rather than uniform. For 24 inequalities the paper requires the outward-rounded constant to be smaller than the published support in the same direction, using the observed digitized dual branches; the remaining inequality g_0^m ≤ 2.705955 is compared separately with the published extrapolated scalar bound g_0^m ≤ 2.7272. Each retained inequality improves at least one direction, but their intersection need not improve the entire published curve uniformly. The authors state that these certified inequalities improve on portions of the published dual boundaries, narrowing the gap to the primal results while relying only on the proven analyticity requirements.

A forward Gram family with a known limit

The forward Gram family gives a concrete closed-form example. Using the variable x = 2/(s − 2) running from 0 to 1 along the cut, the spectral average becomes an integral with the measure dμ_m = (4/π) x(1 + x)√(1 − x^2) dx, whose moments are expressed through Euler beta functions. With weights adapted to the harmonics, the Gram matrix takes a block form in which the mixing entries equal the forward moment because harmonics with j ≥ 2 vanish.

Eliminating that moment yields the two-sided inequality −(128/27) g_2^m − √2 K_N g_2^m ≤ g_0^m ≤ −(64/27) g_2^m + √2 K_N g_2^m, where K_N = (1^T T_N^{−1} 1)^{−1}. Increasing the matrix size strengthens both bounds. The first constants are K_2 = 496.5422085…, K_4 = 286.0222386…, and K_32 = 235.8219144…, and the analytic limit is K_∞ = 233.7516883972….

The authors are explicit that this limiting bound is an analytic outer bound, not the exact region, because the bounds W_ℓ ⪯ 2I and a Legendre-polynomial inequality were used. In particular, it does not bound g_2^m from above. In this family the number of nulls and the size of the matrix grow together with N, so the hierarchy improves in a controlled way rather than converging to the physical region outright.

Where the Real EFThedron changes the priority

The paper's distinctive claim is not that unitarity, analyticity, and crossing can be combined, but that they can be combined inside one moment geometry by putting the real and imaginary parts of the amplitude on an equal footing. That is what brings the subtraction constant g_0 into the bounded region, and it is what turns analyticity into linear nulls that cut slices through the complex moment body rather than into a separate reconstruction step.

The comparison that matters most is between the two-dimensional and four-dimensional outcomes. In two dimensions, finitely many nulls give the exact coupling region, and the extremal measures are CDD amplitudes. In four dimensions, every bound reported is an outer bound, and the improvement over published dual bounds is certified direction by direction rather than as a uniform shrinking of the whole curve.

For a reader deciding what to take from this work, the operative distinction is therefore between exactness and certified improvement. The two-dimensional result establishes that the method can be exact when the null set is complete. The four-dimensional results establish that selected null families already tighten portions of the dual boundary while using only proven analyticity requirements, and the authors note that the constraint sets used are far from exhaustive, so a systematic optimization including crossing-symmetric dispersion relations should strengthen these bounds further.

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