Unitarity Bounds

• Unitarity bounds are foundational constraints in quantum theories, ensuring probability conservation and limiting scattering amplitudes via methods like partial-wave analysis.
• They are derived using tools such as partial-wave, matrix, and spectral methods to impose strict limits on operator coefficients and model parameters.
• These bounds are crucial in effective field theory, collider phenomenology, and astrophysical contexts, tightly constraining new physics scenarios and parameter spaces.

A unitarity bound is a constraint on observable quantities—such as S-matrix elements, operator coefficients, or measurable spectra—arising from the requirement that time evolution in quantum theory (or more generally, probability conservation) is unitary. Across quantum field theory and related areas, unitarity bounds enforce fundamental limits on scattering amplitudes, effective theory parameters, form factors, spectrum structure, and even entropic data. These bounds play central roles in model-independent theory analysis, phenomenological consistency, and precision data interpretation.

1. General Formulation of Unitarity Bounds

Unitarity of the S-matrix ($S^\dagger S=1$) leads to constraints on amplitudes via the optical theorem and partial-wave decompositions. For a general process, the T-matrix satisfies

$$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$

Upon decomposing $2 \to 2$ scattering amplitudes into partial waves,

$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$

the unitarity condition translates for each partial wave to

$$|a_\ell(s)| \leq 1 \ , \qquad |\Re\, a_\ell(s)| \leq \frac{1}{2}$$

for elastic amplitudes, with analogous (matrix) bounds in coupled-channel situations (Bresciani et al., 17 Apr 2025, Bresciani et al., 3 Mar 2026). For amplitudes involving multiple initial and/or final state channels, the full partial-wave matrix must have all eigenvalues bounded by one.

In effective field theories and SMEFT, the scaling of new physics operators (e.g., with energy for $d>4$ operators) means tree-level amplitudes can grow with $s = (p_1 + p_2)^2$, making unitarity bounds a strong constraint on allowed Wilson coefficients at accessible energies (Bresciani et al., 3 Mar 2026, Cao et al., 2023).

2. Methodologies for Deriving Unitarity Bounds

The technical approach depends on the observable of interest:

• Partial-wave analysis: Amplitudes are projected onto angular momentum eigenstates; eigenvalues of the resulting matrix are constrained by $|a^J_{\max}| \leq 1$ (or $|\Re\, a^J_{\max}| \leq 1/2$ in standard normalization) (Bresciani et al., 17 Apr 2025, Bresciani et al., 3 Mar 2026, Bresciani et al., 15 Oct 2025).
• Matrix methods: In models with multiple scalar or vector fields (Higgs sector extensions, composite/EFT models), the full scattering matrix is block-diagonalized by conserved charges and symmetries; each block’s eigenvalues are analytically or numerically bounded (Bento et al., 2022, Lu et al., 2022, Lopes et al., 2 Oct 2025, Kanemura et al., 2015).
• Spectral and moment problem methods: For forward-limit S-matrix amplitudes and form factors, unitarity leads to positivity (and Hankel or Toeplitz-type) determinant inequalities constraining low-energy expansion coefficients (Chen et al., 2019, Ananthanarayan et al., 2012).
• Entropic or geometric approaches: In conformal field theories, unitarity and modular flow properties translate to lower bounds on operator dimensions via mutual information superadditivity and the Markov property (Casini et al., 2021).
• Dispersion/positivity constraints: Analyticity and crossing, together with unitarity, yield positivity bounds on EFT parameters—sometimes stronger than unitarity bounds alone (Bresciani et al., 15 Oct 2025, Bresciani et al., 3 Mar 2026).

3. Applications in Model Building and Phenomenology

3.1. Effective Field Theory and SMEFT

In the SMEFT, Wilson coefficients of higher-dimensional operators (e.g., $O_6$, $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$0) are bounded by requiring that $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$1, with $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$2 determined by the coupled-channel angular momentum analysis in the relevant sector. Explicit bounds for numerous operators, including four-fermion and bosonic channels, are provided in (Bresciani et al., 3 Mar 2026, Cao et al., 2023). For example, for four-fermion Class (LL)(LL) operators, the unitarity bound may read:

$$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$3

with detailed results for all operator classes given in tabular form in (Bresciani et al., 3 Mar 2026).

3.2. Scalar Sector Extensions

Multi-Higgs models, singlet/doublet/triplet-extended scalar sectors, and 3HDMs require diagonalization of large coupled-channel scattering matrices. Block diagonalization by $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$4 and irreducible representations results in a reduced set of analytic inequalities on quartic couplings, for instance in a 2HDM:

$$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$5

as derived in (Kanemura et al., 2015, Lopes et al., 2 Oct 2025, Lu et al., 2022).

3.3. Astrophysical Neutrinos

The flavor composition of TeV–PeV astrophysical neutrinos is constrained by unitarity of the $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$6 flavor propagation matrix $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$7. The transition probability matrix $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$8 constructed as $$S = 1 + iT \ ,\qquad S^\dagger S = 1 \implies T - T^\dagger = i T^\dagger T \ .$$9, with $2 \to 2$0, must satisfy explicit convex polytope bounds in Earth flavor space for any fixed source composition. The convex hull boundary—given by extremal values of linear functions of $2 \to 2$1—restricts the possible flavor shifts observed at detectors like IceCube, with all physically consistent flavor ratios lying within this region (Ahlers et al., 2018).

3.4. Composite and Excited Fermion Searches

Unitarity imposes strong model-independent bounds in EFT descriptions of composite (excited) states in high-energy collider settings. For dimension-6 contact interactions,

$2 \to 2$2

provides a lower limit on $2 \to 2$3 as a function of the mass $2 \to 2$4 and center-of-mass energy, strongly curtailing the parameter space for large $2 \to 2$5 even when $2 \to 2$6 (Biondini et al., 2019, Biondini, 2020).

3.5. Scalar Fields with Arbitrary Sound Speed

For scalar fields with non-standard dispersion relations ($2 \to 2$7), the partial-wave unitarity bound is modified:

$2 \to 2$8

for the local quartic coupling in $2 \to 2$9 scattering, as derived through a generalized treatment of the phase space and partial-wave expansion (Ageeva et al., 2022).

3.6. Quantum Gravity and Low-Energy Spectrum

Unitarity and analyticity impose an infinite hierarchy of positive-definiteness conditions (Hankel determinant inequalities) on low-energy expansion coefficients of four-point amplitudes. These S-matrix positivity constraints can, for instance, demand the existence of light neutral states (e.g., neutrinos) if a charged state exhibits a super-extremal charge-to-mass ratio—directly linking low-energy unitarity to the spectrum structure required for UV completeness in quantum gravity (Chen et al., 2019).

4. Unitarity Bounds and Positivity/Dispersion Relations

In many contexts, unitarity bounds closely interact with positivity and dispersion relations. Analyticity and crossing force positivity of certain forward scattering amplitude derivatives, leading to quadratic and linear constraints on EFT operator coefficients. In combined analyses, such as for ALP EFTs and SMEFT, the intersection of positivity (e.g., $$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$0, $$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$1) and unitarity regions yields a sharply restricted allowed parameter space, often significantly strengthening the theoretical constraints (with the intersection region as small as $$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$2–$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$3 of the unitarity-allowed disk) (Bresciani et al., 15 Oct 2025, Bresciani et al., 17 Apr 2025, Bresciani et al., 3 Mar 2026).

Table: Schematic interplay between unitarity and other analytic constraints:

Constraint Type	Example Inequality / Bound	Context
Partial-wave unitarity	$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$4	2$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$52, EFT, coupled channels
Positivity (dispersion)	$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$6	Forward elastic amplitudes
Hankel determinants	$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$7	Quantum gravity, S-matrix analyticity
Flavor transition	$$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$8	Astrophysical neutrino flavor bounds

5. Impact on Theory and Experiment

Unitarity bounds serve as critical consistency checks for new model proposals, effective operator analyses, and collider phenomenology. They:

• Delineate the validity domain of EFT descriptions, often stronger than naive $$\mathcal{M}(s, \theta) = 16\pi \sum_{\ell=0}^\infty (2\ell + 1) a_\ell(s) P_\ell(\cos\theta) \ ,$$9 limits.
• Supply robust ceilings for the physical masses, mixing angles, or coupling strengths of new scalars, HNLs, and sectors beyond the Standard Model (Kanemura et al., 2015, Lu et al., 2022, Urquía-Calderón et al., 2024).
• Rigorously restrict the allowed flavor compositions at neutrino observatories, serving as a test for new physics beyond standard oscillation paradigms (Ahlers et al., 2018).
• Intersect with positivity and analytic constraints to identify unique islands of parameter space in bootstrap and S-matrix studies, sometimes even isolating quantum-gravity-complete theories (e.g., string theory at U-duality symmetric points) (Bossard et al., 2023).

6. Extensions Beyond Standard Scenarios

Unitarity bounds are generalized to:

• Scenarios with non-trivial sound speeds (scalar-tensor gravities, multi-field inflation) (Ageeva et al., 2022).
• Arbitrary $$|a_\ell(s)| \leq 1 \ , \qquad |\Re\, a_\ell(s)| \leq \frac{1}{2}$$0 high-multiplicity processes and higher-spin amplitudes in gravity or beyond-the-Standard-Model effective theories, by advanced on-shell or spinor helicity amplitude methods (Bresciani et al., 17 Apr 2025).
• Nonperturbative regimes through entropic or information-theoretic inequalities in conformal field theory (Casini et al., 2021).
• The forward limit and moment problem settings, yielding infinite determinant sequences relevant for spectrum constraints and Weak Gravity Conjecture implications (Chen et al., 2019).

7. Summary and Outlook

Unitarity bounds, arising fundamentally from S-matrix unitarity and analyticity, constitute an essential nonperturbative control on quantum field theory, effective field theory, and quantum gravity conjectures. They underpin theoretical consistency, serve as model-independent bounds on observables and parameters, and, when combined with other analytic constraints, sharply restrict possible deviations from Standard Model expectations both in low- and high-energy settings. Advances in amplitude-based, entropic, and spectral techniques continue to expand the reach of unitarity bounds, making them increasingly central in both theoretical and experimental high-energy physics.

Key references: (Ahlers et al., 2018, Kanemura et al., 2015, Bresciani et al., 17 Apr 2025, Bresciani et al., 3 Mar 2026, Lopes et al., 2 Oct 2025, Ananthanarayan et al., 2012, Bresciani et al., 15 Oct 2025, Bento et al., 2022, Lu et al., 2022, Casini et al., 2021, Cao et al., 2023, Basso et al., 2010, Atkins et al., 2010, Chen et al., 2019, Biondini et al., 2019, Biondini, 2020, Ageeva et al., 2022, Bossard et al., 2023, Yamamoto, 2014, Urquía-Calderón et al., 2024).