---
title: 'EFT-hedron: Geometry of Low-Energy EFT Coefficients'
url: https://www.emergentmind.com/topics/eft-hedron
type: topic
---

# EFT-hedron: Geometry of Low-Energy EFT Coefficients

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 [2012.15849]. 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 [2107.06559] [2204.07140] [2310.10729] [2501.09717] [2507.12535].

## 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 [2012.15849]. 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 [2107.06559].

The same terminology is used in more specialized settings. In the Goldstino EFT of “(Super)$\,$Gravity from Positivity,” the $2\to2$ amplitude is expanded as
$$
f(s,t)=\sum_{n\ge2k} g_{n,k}\,s^{\,n-2k}(tu)^k,
$$
and the parameter vector
$$
G=(g_{1,0},g_{2,0},g_{2,1},\dots)
$$
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 [2507.12535].

## 2. Dispersive representation and positivity

The geometric construction is rooted in dispersion relations and positivity of partial-wave spectral data. For fixed $t\ll M_{\rm gap}^2$, the amplitude admits a dispersive representation with non-negative spectral densities,
$$
M(s,t)=M_{\rm sub}(t)+\int_{M_s^2}^\infty d\mu^2\,\frac{\rho_s(\mu^2,t)}{s-\mu^2}
+\int_{M_u^2}^\infty d\mu^2\,\frac{\rho_u(\mu^2,t)}{u-\mu^2},
$$
with $\rho_{s,u}(\mu^2,t)\ge0$ [2012.15849]. In partial-wave language, unitarity implies $\Im f_\ell(s)\ge0$, 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 $\mathcal N=4$ 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
$$
M(s,t)\equiv A[zz\bar z\bar z](s,u)
= -\frac{s}{u}+s^2\sum_{k=0}^\infty\sum_{q=0}^k a_{k,q}\,s^{k-q}u^q,
$$
with the supersymmetric crossing condition $a_{k,k-q}=a_{k,q}$. Analyticity plus a mass gap then gives
$$
a_{k,q}
=\sum_{\ell=0}^\infty\int_0^1 dx\,p_\ell(x)\,x^k\,v_{\ell,q},
\qquad p_\ell(x)\ge0,
$$
which immediately implies $a_{k,q}\ge0$ and, more generally, $a_{k',q}\le a_{k,q}$ for $k'\ge k$ [2310.10729].

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 $0\le \rho_\ell\le2$ upgrades the problem to a generalized $L$-moment problem [2012.15849] [2107.06559] [2204.07140]. 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
$$
\det[\mathbf V_{\ell_1},\dots,\mathbf V_{\ell_{k+1}}]>0
\qquad (\ell_1<\cdots<\ell_{k+1}),
$$
the boundary of $U_k$ is cut out by linear inequalities built from consecutive-spin minors. In parallel, for each fixed $q$, the sequence $\{a_{k,q}\}$ must satisfy “Hankel” constraints, such as positivity of the infinite Hankel matrix $H_q=[a_{i+j,q}]$, whose $2\times2$ principal minor gives the forward-limit inequality
$$
a_{q,0}a_{q+2,0}-a_{q+1,0}^2\ge0
$$
[2012.15849].

Crossing symmetry adds further relations that reduce the allowed region. In planar $\mathcal N=4$ SYM, plugging the dispersive representation into the supersymmetric crossing condition $a_{k,k-q}=a_{k,q}$ yields linear “null constraints,” and an additional set of sum rules follows from a fixed-$t$ dispersion relation and the $\pm\ell$ parity of $P_\ell(1+2t/s)$ [2310.10729]. 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
$$
D_k\equiv \det_{0\le i,j\le k-1}[a_{i-j}-\delta_{i,j}\,2]\ge0
$$
produces curved facets; examples include $D_3\ge0\Rightarrow a_3\ge a_2^2$ and $D_4\ge0\Rightarrow a_4a_2\ge a_3^2$ [2107.06559].

The extremal structure is correspondingly interpretable. In the original construction, the vertices are the $\mathbf V_\ell$ or their mixed analogues, and the geometry is completely determined by sign patterns of minors [2012.15849]. In the crossing-symmetric setting, the vertices correspond to extremal “pure” amplitudes, such as single-resonance exchange [2107.06559]. 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 [2012.15849].

## 4. Supersymmetric and string-oriented EFT-hedra

The supersymmetric version developed in planar four-dimensional $\mathcal N=4$ SYM studies the convex region of higher-derivative coefficients $\{a_{k,q}\}$ consistent with unitarity, analyticity, locality, and crossing. Equivalently, one may work with projective ratios $\bar a_{k,q}\equiv a_{k,q}/a_{0,0}$, in which case the EFT-hedron is a convex subset of the hypercube $0\le \bar a_{k,q}\le1$ [2310.10729]. 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 $k_{\max}$ is increased and null constraints are imposed up to that order, the allowed region shrinks in many directions, and as $k_{\max}\to\infty$ roughly one-third of the directions “collapse” to measure zero [2310.10729]. Cross-sections such as $(a_{1,0},a_{3,0})$ or $(a_{2,0},a_{2,1})$ display islands that shrink towards points as $k_{\max}$ grows.

Motivated by this behavior, the amplitude is reorganized into a partially resummed expansion,
$$
A[zz\bar z\bar z]
= -\frac{s}{u}+s^2\Bigl(\sum_{k,i}r_i^{(k)}\,P_i^{(k)}(s,u)
+\sum_{k,i}A_i^{(k)}\,Q_i^{(k)}(s,u)\Bigr),
$$
where the “monovariables” $r_i^{(k)}$ multiply simple degree-$k$ symmetric polynomials and the complementary coefficients $A_i^{(k)}$ multiply infinite towers beginning at degree $k$. Numerically, up to $20^{\rm th}$ order, these towers can be resummed into
$$
\sum_{k,i}A_i^{(k)}Q_i^{(k)}(s,u)
=\frac{\sin(\pi t)}{\pi}\sum_{k,i}\tilde A_i^{(k)}\,\mathcal S_i^{(k)}(s,t,u),
$$
with $\mathcal S_i^{(k)}$ fully symmetric in $(s,t,u)$ [2310.10729].

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 $a_{k,q}=a_{k,q}^{\rm str}$ as $k_{\max}\to\infty$, with sub-per-mille or better agreement by $k_{\max}\approx8$–$20$ [2310.10729]. 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
$$
0\le \rho_\ell(z)\le2.
$$
This recasts the problem as a generalized $L$-moment problem, with each spin sector contributing a convex region and the full non-projective EFT-hedron given by their Minkowski sum [2204.07140]. 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
$$
\frac{g_{2,0}}{(4\pi)^2}\le \frac{1.58}{M^4},
$$
while for general operators of mass dimension $2k$ the optimal bound found in the $(n_4,g_{k,0})$ slice shows a large-$k$ behavior with a $1/k$ fall-off [2204.07140]. The paper emphasizes that, upon normalizing by $(4\pi)^2$ and the cutoff scale, the leading derivative operators are bounded by unity up to $\mathcal O(1)$.

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,
$$
A_{\rm EFT}\supset b_1\bigl[s^4\ln(-s)+t^4\ln(-t)+u^4\ln(-u)\bigr]
+b_2\bigl[s^2tu\ln(-s)+\cdots\bigr],
$$
with
$$
b_1=-\frac{21\,g_2^2}{240\pi^2},\qquad b_2=\frac{g_2^2}{240\pi^2}.
$$
These terms produce IR singularities in the arc integrals, so one must construct finite combinations of null constraints before taking the forward limit [2501.09717]. The essential conclusion is that loops do not merely run tree-level couplings; they deform the null constraints into non-zero “$\beta$-constraints.”

This deformation changes the shape of the allowed region. In the scalar example discussed in detail, the pair $(g_2,n_4)$ defines an $L=2$ Hausdorff moment problem. At tree level, $n_4=0$ gives $g_2\lesssim124.7$, while the loop-corrected $\beta$-constraint gives the stronger bound $g_2\lesssim37.8$ [2501.09717]. Likewise, the projective region in the $(\tilde g_3,\tilde g_4)$ plane rotates and shrinks as $g_2$ or the IR cutoff is varied, and loop effects can allow $\tilde g_4<0$, 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)$\,$Gravity from Positivity,” the EFT-hedron is developed for the longitudinal polarizations of massive spin-$3/2$ particles, namely Goldstinos. To make positivity manifest under $t\leftrightarrow u$, the construction introduces
$$
p^2\equiv \frac{tu}{s},\qquad 0\le p^2\le \frac{M_s^2}{4},
$$
and defines at fixed $p^2>0$ the contour integral
$$
A_n(p^2)\equiv \oint_{C_{\rm low}}\frac{ds}{2\pi i}\,
\frac{M^{++\to--}(s,t(s,p^2))}{s^n}.
$$
Expanding $A_n(p^2)$ in powers of $p^2$ 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 [2507.12535].

For the Goldstino coefficients, positivity and the null constraints take the form of linear inequalities and hyperplane conditions,
$$
a\cdot G\ge0,\qquad a\cdot G=0,
$$
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 [2507.12535]. 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 $J=0$ exchange in the $++$ channel and amplitude
$$
M(s,t,u)=-\frac{g_{++}^2}{m_S^2}\,\frac{s^2}{s-m_S^2}.
$$
The vector-exchange model of Fayet–Iliopoulos type has only $J=1$ exchange in the $+-$ channel and amplitude
$$
M(s,t,u)= -\frac{g_{+-}^2}{m_V^2}
\left[\frac{s\,u}{u-m_V^2}+\frac{s\,t}{t-m_V^2}\right].
$$
Higher-spin “$s/tu$” models, with all $J\ge2$ states at a single mass and tuned residues, form the left edge of the polytope, while a string-like model in the $+-$ sector is represented by a Lovelace–Shapiro amplitude dressed with the little-group factor
$$
M(s,t,u)= -\frac{s}{M^2}\,
\frac{\Gamma(\tfrac12-\tfrac{t}{2M^2})\Gamma(\tfrac12-\tfrac{u}{2M^2})}
{\Gamma(-\tfrac{t+u}{2M^2})}
$$
[2507.12535].

The broader significance of this construction lies in the spin-$3/2$ consistency problem. The paper finds no solution to positivity constraints for isolated, massive, and weakly interacting spin-$3/2$ particles, except when gravitons are also present and couple in a nearly supersymmetric way. For two such particles forming a $U(1)$-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 [2507.12535]. In the summary given there, the strongest bounds force the appearance of a massless graviton and matter fields with SUSY-like couplings, including $F^2=3m^2M_P^2$, $g=2$, 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 $S$-matrix consistency.

Source: https://www.emergentmind.com/topics/eft-hedron