EFT-hedron: Geometry of Low-Energy EFT Coefficients

• EFT-hedron is a positive geometry that organizes Wilson coefficients in low-energy effective field theories based on analyticity, causality, unitarity, locality, and crossing symmetry.
• It defines a convex region using dispersive integrals, moment problems, and positivity constraints, thereby mapping the ratios of low-energy expansion coefficients.
• Recent generalizations include supersymmetric, string-oriented, and Goldstino versions that impact S-matrix bootstrap and UV completions in high-energy physics.

The EFT-hedron is the positive geometry formed by the Wilson coefficients of a low-energy effective field theory that are compatible with analyticity, causality, unitarity, locality, and crossing. In its original formulation for four-particle scattering, the low-energy expansion coefficients of the amplitude are constrained to lie in a convex region whose boundary is determined by positivity of dispersive integrals, total positivity of moment data, and crossing-induced relations among coefficients (Arkani-Hamed et al., 2020). Later work generalized the construction to fully crossing-symmetric variables, supersymmetric amplitudes, non-projective absolute bounds, loop-level deformations, and Goldstino EFTs, where the geometry can be a convex body, a polytope, or a loop-deformed region depending on the assumptions and kinematics (Raman et al., 2021, Chiang et al., 2022, Berman et al., 2023, Peng et al., 16 Jan 2025, Bellazzini et al., 16 Jul 2025).

1. Definition and scope

In the original four-point formulation, one expands the low-energy amplitude as

$$M(s,t)\Big|_{s,t\ll M^2}=\sum_{k,q\ge0} a_{k,q}\,s^{k-q}t^q,$$

and for each derivative order defines the coefficient vector

$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$

The corresponding “$s$-channel” EFT-hedron at fixed $k$ is the convex hull

$U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$

where $\mathbf V_\ell$ is built from Gegenbauer-Taylor coefficients extracted from partial waves (Arkani-Hamed et al., 2020). In this projective presentation, the geometry organizes the ratios of Wilson coefficients at a given derivative order.

A crossing-symmetric formulation for identical-scalar $2\to2$ scattering instead uses

$$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$

Here the “space of EFTs” is the set of coefficients $\{W_{p,q}\}$ compatible with causality, unitarity, and crossing. Geometrically, this is a convex body in infinite-dimensional coefficient space, whose flat facets arise from linear positivity constraints, curved faces from nonlinear determinant bounds, and vertices from extremal “pure” amplitudes such as tree-level exchange of a single resonance (Raman et al., 2021).

The same terminology is used in more specialized settings. In the Goldstino EFT of “(Super)$\,$Gravity from Positivity,” the $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$0 amplitude is expanded as

$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$1

and the parameter vector

$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$2

is constrained to lie inside a convex polytope in parameter space. In that context, the EFT-hedron is explicitly identified with the intersection of positivity half-spaces and null-constraint hyperplanes (Bellazzini et al., 16 Jul 2025).

2. Dispersive representation and positivity

The geometric construction is rooted in dispersion relations and positivity of partial-wave spectral data. For fixed $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$3, the amplitude admits a dispersive representation with non-negative spectral densities,

$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$4

with $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$5 (Arkani-Hamed et al., 2020). In partial-wave language, unitarity implies $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$6, and the EFT coefficients become moments of a positive measure. This is the origin of the convexity of the allowed coefficient region.

In planar four-dimensional $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$7 SYM with higher-derivative corrections, the same logic appears in a supersymmetric setting. After stripping off the simple tree pole, the most general weak-coupling, local, crossing-symmetric low-energy expansion is

$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$8

with the supersymmetric crossing condition $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$9. Analyticity plus a mass gap then gives

$s$0

which immediately implies $s$1 and, more generally, $s$2 for $s$3 (Berman et al., 2023).

A related description views the EFT-hedron as a moment problem. In the projective setting, the Hankel or moment matrices built from the EFT coefficients are totally positive; in the crossing-symmetric setting, Toeplitz determinant positivity supplies nonlinear constraints; and in the non-projective setting the unitarity upper bound $s$4 upgrades the problem to a generalized $s$5-moment problem (Arkani-Hamed et al., 2020, Raman et al., 2021, Chiang et al., 2022). This common structure is what makes the geometric language possible across otherwise different EFTs.

3. Boundary structure, crossing constraints, and extremality

The boundary of the EFT-hedron is determined by linear and nonlinear constraints. In the original projective formulation, the row-space geometry is governed by “Gegenbauer” facets: because

$s$6

the boundary of $s$7 is cut out by linear inequalities built from consecutive-spin minors. In parallel, for each fixed $s$8, the sequence $s$9 must satisfy “Hankel” constraints, such as positivity of the infinite Hankel matrix $k$0, whose $k$1 principal minor gives the forward-limit inequality

$k$2

(Arkani-Hamed et al., 2020).

Crossing symmetry adds further relations that reduce the allowed region. In planar $k$3 SYM, plugging the dispersive representation into the supersymmetric crossing condition $k$4 yields linear “null constraints,” and an additional set of sum rules follows from a fixed-$k$5 dispersion relation and the $k$6 parity of $k$7 (Berman et al., 2023). In the crossing-symmetric typically-real approach, the Bieberbach–Rogosinski inequalities give linear two-sided bounds on combinations of Wilson coefficients, while positivity of Toeplitz determinants such as

$k$8

produces curved facets; examples include $k$9 and $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$0 (Raman et al., 2021).

The extremal structure is correspondingly interpretable. In the original construction, the vertices are the $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$1 or their mixed analogues, and the geometry is completely determined by sign patterns of minors (Arkani-Hamed et al., 2020). In the crossing-symmetric setting, the vertices correspond to extremal “pure” amplitudes, such as single-resonance exchange (Raman et al., 2021). This supports the broader interpretation, stated explicitly in the early work, that proximity to particular facets encodes the dominant low-spin exchanges in the UV (Arkani-Hamed et al., 2020).

4. Supersymmetric and string-oriented EFT-hedra

The supersymmetric version developed in planar four-dimensional $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$2 SYM studies the convex region of higher-derivative coefficients $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$3 consistent with unitarity, analyticity, locality, and crossing. Equivalently, one may work with projective ratios $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$4, in which case the EFT-hedron is a convex subset of the hypercube $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$5 (Berman et al., 2023). The numerical problem is formulated as a linear optimization problem and implemented with SDPB and CPLEX, using positivity of the spectral densities together with all null constraints.

A central numerical observation is “flattening.” As the highest derivative order $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$6 is increased and null constraints are imposed up to that order, the allowed region shrinks in many directions, and as $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$7 roughly one-third of the directions “collapse” to measure zero (Berman et al., 2023). Cross-sections such as $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$8 or $U_k=\Conv\{\mathbf V_\ell\mid \ell=0,1,2,\dots\},$9 display islands that shrink towards points as $\mathbf V_\ell$0 grows.

Motivated by this behavior, the amplitude is reorganized into a partially resummed expansion,

$\mathbf V_\ell$1

where the “monovariables” $\mathbf V_\ell$2 multiply simple degree-$\mathbf V_\ell$3 symmetric polynomials and the complementary coefficients $\mathbf V_\ell$4 multiply infinite towers beginning at degree $\mathbf V_\ell$5. Numerically, up to $\mathbf V_\ell$6 order, these towers can be resummed into

$\mathbf V_\ell$7

with $\mathbf V_\ell$8 fully symmetric in $\mathbf V_\ell$9 (Berman et al., 2023).

This framework is tied directly to the search for string amplitudes. Imposing linear monodromy constraints fixes about two-thirds of the coefficients to the even-zeta combinations of the Veneziano amplitude, while the remaining one-third, involving odd zeta values, are left free. Combining those monodromy null constraints with positivity and analyticity yields islands that shrink to the string values $2\to2$0 as $2\to2$1, with sub-per-mille or better agreement by $2\to2$2–$2\to2$3 (Berman et al., 2023). In geometric terms, the intersection of the EFT-hedron with the “monodromy plane” collapses to the unique point associated with the tree-level Veneziano open-string amplitude.

5. De-projecting and running the geometry

One limitation of the original projective analyses is that they bound only ratios of couplings. To obtain absolute, non-projective bounds on individual Wilson coefficients, one must supplement the lower positivity condition with the unitarity upper bound

$2\to2$4

This recasts the problem as a generalized $2\to2$5-moment problem, with each spin sector contributing a convex region and the full non-projective EFT-hedron given by their Minkowski sum (Chiang et al., 2022). In this picture the geometry acquires an infinite number of non-linear facets.

The resulting bounds are absolute rather than projective. For the leading-derivative operator, imposing positivity, unitarity, and the first null constraint yields

$2\to2$6

while for general operators of mass dimension $2\to2$7 the optimal bound found in the $2\to2$8 slice shows a large-$2\to2$9 behavior with a $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$0 fall-off (Chiang et al., 2022). The paper emphasizes that, upon normalizing by $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$1 and the cutoff scale, the leading derivative operators are bounded by unity up to $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$2.

A further extension studies loop effects from massless intermediate states. In the “running EFT-hedron,” one-loop massless cuts generate logarithmic branch cuts in the EFT amplitude,

$$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$3

with

$$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$4

These terms produce IR singularities in the arc integrals, so one must construct finite combinations of null constraints before taking the forward limit (Peng et al., 16 Jan 2025). The essential conclusion is that loops do not merely run tree-level couplings; they deform the null constraints into non-zero “$$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$5-constraints.”

This deformation changes the shape of the allowed region. In the scalar example discussed in detail, the pair $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$6 defines an $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$7 Hausdorff moment problem. At tree level, $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$8 gives $$M(s,t)=\sum_{p,q\ge0}W_{p,q}\,x^p y^q,\qquad x=-(st+tu+us),\qquad y=-stu.$$9, while the loop-corrected $\{W_{p,q}\}$0-constraint gives the stronger bound $\{W_{p,q}\}$1 (Peng et al., 16 Jan 2025). Likewise, the projective region in the $\{W_{p,q}\}$2 plane rotates and shrinks as $\{W_{p,q}\}$3 or the IR cutoff is varied, and loop effects can allow $\{W_{p,q}\}$4, which is forbidden at tree level. This directly addresses a common simplification in earlier positivity analyses: the EFT-hedron need not remain fixed once massless loops are included.

6. Goldstino EFT-hedron and supergravity bootstrapping

In “(Super)$\{W_{p,q}\}$5Gravity from Positivity,” the EFT-hedron is developed for the longitudinal polarizations of massive spin-$\{W_{p,q}\}$6 particles, namely Goldstinos. To make positivity manifest under $\{W_{p,q}\}$7, the construction introduces

$\{W_{p,q}\}$8

and defines at fixed $\{W_{p,q}\}$9 the contour integral

$\,$0

Expanding $\,$1 in powers of $\,$2 generates both sum rules for the EFT couplings and an infinite tower of null constraints forcing certain linear combinations of high-spin partial waves to vanish (Bellazzini et al., 16 Jul 2025).

For the Goldstino coefficients, positivity and the null constraints take the form of linear inequalities and hyperplane conditions,

$\,$3

so the allowed region is the intersection of half-spaces and hyperplanes. Under mild conditions this is a closed, convex, polyhedral region, and a convex polytope when only finitely many matter multiplets are assumed (Bellazzini et al., 16 Jul 2025). In practice, the boundary is explored numerically, for example through semidefinite programming.

The vertices of this Goldstino EFT-hedron correspond to simple UV amplitudes that saturate multiple bounds simultaneously. The scalar-exchange model of O’Raifeartaigh type has only $\,$4 exchange in the $\,$5 channel and amplitude

$\,$6

The vector-exchange model of Fayet–Iliopoulos type has only $\,$7 exchange in the $\,$8 channel and amplitude

$\,$9

Higher-spin “$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$00” models, with all $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$01 states at a single mass and tuned residues, form the left edge of the polytope, while a string-like model in the $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$02 sector is represented by a Lovelace–Shapiro amplitude dressed with the little-group factor

$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$03

(Bellazzini et al., 16 Jul 2025).

The broader significance of this construction lies in the spin-$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$04 consistency problem. The paper finds no solution to positivity constraints for isolated, massive, and weakly interacting spin-$$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$05 particles, except when gravitons are also present and couple in a nearly supersymmetric way. For two such particles forming a $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$06-charged state, a graviphoton gauging the symmetry is also required, with couplings characteristic of supergravity and consistent with both the no global symmetry and weak gravity conjectures (Bellazzini et al., 16 Jul 2025). In the summary given there, the strongest bounds force the appearance of a massless graviton and matter fields with SUSY-like couplings, including $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$07, $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$08, and WGC saturation. In this sense, the Goldstino EFT-hedron is not merely a catalog of allowed low-energy coefficients; it functions as a bootstrap mechanism that reconstructs the minimal additional states and couplings required by $$\mathbf a_k=(a_{k,0},a_{k,1},\dots,a_{k,k})\in\mathbb P^k.$$09-matrix consistency.