Two-Sided Positivity Bounds

• Two-sided positivity bounds are rigorous constraints derived from analyticity, unitarity, Lorentz invariance, and crossing symmetry, ensuring both upper and lower limits on effective field theory coefficients.
• They are formulated via dispersion relations and recursive derivative structures in EFT and S-matrix bootstrap, creating convex allowed regions for Wilson coefficients.
• These bounds have practical applications in chiral perturbation theory, cosmological correlators, and stochastic processes, while guiding refined numerical and analytic methods in modern QFT.

Two-sided positivity bounds are rigorous constraints derived from analyticity, unitarity, Lorentz invariance, and crossing symmetry in quantum field theory (QFT) and related mathematical contexts. These bounds typically manifest as linear or nonlinear inequalities on operator coefficients, Wilson coefficients, or physical densities, where both upper and lower limits are imposed instead of merely restricting signs. The concept is central in effective field theory (EFT), S-matrix bootstrap, PDE theory, and stochastic processes, and has seen significant recent development via dispersion relations, convex geometry, and dual spectrahedral constructions.

1. Foundational Principles: Analyticity, Unitarity, and Crossing

Two-sided positivity bounds generally arise when one considers twice-subtracted dispersion relations for $2\to2$ amplitudes $T(s, t)$ or analogous correlators. Key axiomatic requirements include:

• Local, Lorentz-invariant QFT, with crossing symmetry $s \leftrightarrow t \leftrightarrow u$,
• Analyticity of $T(s, t)$ in the complex $s$-plane at fixed $t$, with only physical cuts for $s\geq 4M_\pi^2$, $u\geq 4M_\pi^2$,
• Froissart–Martin bound: $\lim_{|s|\to\infty}|T(s,t)|<C|s|^{1+\epsilon(t)}$ with $\epsilon(t)<1$,
• Optical theorem and partial-wave unitarity: $\Im\,t_\ell(s)\ge|t_\ell(s)|^2$ for $s\ge4M_\pi^2$.

Dispersion relations, often twice-subtracted at a symmetry point $\mu_p = -t/2$, transform physical amplitudes into subtraction-removed forms $B(s,t)$, which are then expanded in derivatives. Unitarity ensures that certain combinations of derivatives must always be strictly positive for all physical $s$. The signature feature of two-sided bounds is that, depending on the derivative structure and external states, alternating sign pieces and crossing considerations generate upper as well as lower inequalities on the coefficients of the amplitude's low-energy expansion (Wang et al., 2020).

2. Recursive Formulation: The $Y$-Bounds and Generalized Derivative Structures

The generalized two-sided positivity bounds are encapsulated in recursively defined $Y^{(2N,M)}(t)$, which form linear combinations of even-order $v$ and $t$ derivatives,

$$Y^{(2N,M)}(t) = \sum_{r=0}^{\lfloor M/2\rfloor}c_r B^{(2(N+r),M-2r)}(t) + \frac{1}{\mathcal M^2} \sum_{k=0}^{\lfloor (M-1)/2\rfloor}(2(N+k)+1)\,\beta_k Y^{(2(N+k),M-2k-1)}(t)$$

where $B^{(2N,M)}(t)$ are derivatives at the s↔u symmetric $v=0$, and the recursion coefficients $c_r, \beta_k$ are generated by expansions of hyperbolic secant and tangent, respectively. Inductive proofs establish that for all $N\geq1, M\geq0, 0\leq t < 4M_\pi^2$,

$Y^{(2N,M)}(t) > 0.$

In EFT settings, expansion in local operators implies that $Y^{(2N,M)}(t)$ are explicit linear functions of Wilson coefficients, and for each $(N,M,t)$, one obtains

$a_0 + \sum_i a_i c_i > 0,$

where $a_0$ may flip sign for odd $M$, producing upper and lower bounds simultaneously. Collective imposition of these inequalities carves out convex polyhedral allowed regions in Wilson coefficient space, with detailed implications at NLO (e.g., constraints on $\bar\ell_1, \bar\ell_2$) and NNLO (constraints on $b_i$) in SU(2) chiral perturbation theory (Wang et al., 2020).

3. Convex Geometry and Spectrahedral Duality in Multi-Field Systems

For multi-field EFTs, especially with $n\geq3$ modes, two-sided bounds demand a full geometric analysis. The amplitude's quad-indexed second derivatives $\mathcal{M}^{ijkl}$ span a convex cone generated by all rank-2 positive semi-definite tensors respecting crossing symmetry. The space of admissible $\mathcal{M}$ is the convex cone $\mathcal{C}^{n^4}$, and positivity is encoded as non-negativity of $Q\cdot \mathcal{M}$ for all $Q$ in the dual spectrahedron $Q^{n^4}$ (Li et al., 2021): $$Q\cdot \mathcal{M} = Q^{ijkl} \mathcal{M}^{ijkl} \geq 0 \quad \forall Q\in\text{Ext}(Q^{n^4})$$ where $\text{Ext}(Q^{n^4})$ are the extremal rays of the spectrahedron, parameterized as crossing-symmetric positive semidefinite matrices.

Semidefinite programming (SDP) enables systematic computation of all optimal linear bounds:

• Elastic bounds on factorized states generate only lower bounds,
• Inelastic functionals corresponding to entangled combinations generate strictly two-sided constraints, appearing only in the spectrahedral approach,
• General multi-field systems (scalars, vectors, fermions, gravitons) are accommodated by assembling the forward amplitude and imposing crossing; explicit construction of extremal rays yields both new and tighter bounds (Li et al., 2021).

4. Improved Bounds: Subtractions, Loop Effects, and Robustness

The analytic machinery is sensitive to loop corrections and IR singularities. One key improvement is the subtraction of the low-energy part of the dispersion integral up to EFT cutoff $\Lambda_{\rm EFT}$, yielding

$$B_\Lambda(s,t) = B(s,t) - \int_{4M_\pi^2}^{\Lambda^2} d\mu \frac{\cdots}{(\mu-\mu_p)^2(\mu-s)},$$

and analogously improved recursion relations for $Y^{(2N,M)}(\Lambda)$. The induced bounds are systematically tighter since the hard scale $\mathcal M^2$ shifts to large values. Empirically, lower bounds on combinations such as $\bar\ell_1 + 2\bar\ell_2$ increase from $5.57$ toward $6.3$ as $\Lambda$ increases, and allowed regions in $b_i$ space contract (Wang et al., 2020).

Loop corrections can introduce IR singularities (e.g., $t^2\ln(-t)$ near $t=0$), which compromise lower bounds in the $m \to 0$ limit, particularly in Galileon scenarios. The bounds are restored by working at fixed finite negative $t$ (“large-angle” dispersion) or smearing in the impact-parameter variable $b$ via Bessel kernel convolution, ensuring all integrals and derived bounds remain finite and robust at loop level (Bellazzini et al., 2021).

5. Concrete Examples in EFT, SDEs, and Cosmology

Chiral Perturbation Theory

Two-sided bounds at NLO and NNLO tightly constrain the $\bar\ell_i$ and $b_i$ constants, with the allowed regions forming convex polygons or polyhedra closely tracking Roy-equation fits. The improved bounds further restrict the domain to scales below broad resonances, e.g., $f_0(500)$ (Wang et al., 2020).

Massive Galileons

Explicit two-sided bounds such as

$0 < \frac{g_4}{g_3^2} \leq \frac{3}{4}$

arise, and higher derivative corrections (e.g., $d_1+d_4$ combinations) require additional positivity. Weakly coupled UV completions are explicitly constructed to satisfy all constraints, and nonzero mass is essential for strictly positive leading coefficients (Rham et al., 2017).

Cosmological Correlators

In de Sitter EFTs, an infinite tower of two-sided bounds on propagator coefficients $c_n, \gamma'_n$ rules out unphysical correlators and ties the allowed size of residual non-Gaussianity in primordial cosmological collider signals to damping rates. For example, $c_2$ can be negative only if accompanied by sufficiently positive $\gamma'$, thus tightly correlating oscillatory and non-oscillatory bispectrum shapes (Lee et al., 23 Dec 2025).

Degenerate Stochastic Processes

For SDEs satisfying weak Hörmander conditions, sharp two-sided density bounds are rigorously proved via Harnack chains (lower) and Malliavin calculus (upper). Asymptotic regimes (Gaussian, heavy-tail, boundary) are precisely characterized, and the bounds match the exact kernels in classical cases (Kolmogorov, quadratic integrals) (Cinti et al., 2012).

6. Limitations, Special Cases, and Extensions

• Bounds are most cleanly derived for identical scalars; spinning and charged states require careful extension.
• Systematic higher derivative expansions may require inclusion of all possible heavy-induced operators, cannot set their coefficients identically to zero.
• Padé unitarization, while restoring partial-wave unitarity, can spoil analyticity and generate spurious singularities that weaken the dispersion-based bounds.
• The convexity of the allowed parameter space is generic—intersections of linear inequalities from each $Y$-bound or SDP constraint define compact polytopes; the empirical theory must remain inside this “positivity island.”
• At higher energies or scales $\Lambda_{\rm EFT}$ exceeding the first resonance, the EFT ceases to saturate positivity requirements and new degrees of freedom must be included (Wang et al., 2020).

7. Broader Significance and Outlook

Two-sided positivity bounds are fully nonperturbative, model-independent tests for effective theories' consistency with QFT axioms. They underpin EFT parameter estimation, S-matrix bootstrap approaches, and mathematical probability models for degenerate diffusions. Their application controls the admissibility of models, rules out inconsistent constructions, and tightly constrains allowed phenomenology—especially in low-energy hadron physics, fundamental scalar EFTs, collider phenomenology, and cosmological correlators. Numerical methods (SDP, geometric optimization) and analytic improvements (large-angle, impact-parameter techniques) have rendered computation of these bounds feasible and efficient, enabling their deployment across fields with multiple interacting modes, loop-level corrections, and in strongly correlated systems (Wang et al., 2020, Li et al., 2021, Lee et al., 23 Dec 2025, Rham et al., 2017, Cinti et al., 2012, Bellazzini et al., 2021).