Boundedness of the full two-particle amplitude system under ordinary unitarity

Determine whether imposing the ordinary unitarity constraints on the full system of two-to-two scattering amplitudes involving particles A and B, namely AA→AA, BB→BB, AB→AB, and AA→BB, suffices to obtain two-sided bounds on all observables.

Background

The paper studies only the reduced crossing-closed system consisting of AB→AB and AA→BB. In the unequal-mass case, the authors find that several observables are unbounded because the pseudo-physical region is not directly constrained by ordinary physical unitarity.

The full system additionally includes AA→AA and BB→BB, whose unitarity constraints couple the amplitudes in the neutral sector. The authors note that these constraints might control the pseudo-physical discontinuities sufficiently to restore two-sided bounds, but they do not establish whether this occurs. They also discuss analytically continued, or extended, unitarity as a possible mechanism for obtaining boundedness.

References

It is not clear to us whether ordinary unitarity constraints in~eq:unitarity-AB-mixed and~eq:unitarity-ABtoAB would suffice to obtain two-sided bounds on all observables.

eq:unitarity-AB-mixed:

(101−iTAA→AAℓ(s)∗−iTAA→BBℓ(s)∗01−iTAA→BBℓ(s)∗1−iTBB→BBℓ(s)∗1+iTAA→AAℓ(s)iTAA→BBℓ(s)10iTAA→BBℓ(s)1+iTBB→BBℓ(s)01)⪰0 .\begin{pmatrix} 1 & 0 & 1 - i T_{AA \to AA}^{\ell}(s)^{*} & - i T_{AA \to BB}^{\ell}(s)^{*}\\ 0 & 1 & -i T_{AA \to BB}^{\ell}(s)^{*} & 1 - i T_{BB \to BB}^{\ell}(s)^{*} \\ 1 + i T_{AA \to AA}^{\ell}(s) & i T_{AA \to BB}^{\ell}(s) & 1 & 0 \\ i T_{AA \to BB}^{\ell}(s) & 1 + i T_{BB \to BB}^{\ell}(s) & 0 & 1 \end{pmatrix} \succeq 0 \,.

eq:unitarity-ABtoAB:

(11−iTAB→ABℓ(s)∗1+iTAB→ABℓ(s)1)⪰0.\begin{pmatrix} 1 & 1 - i T_{AB \to AB}^{\ell}(s)^{*} \\ 1 + i T_{AB \to AB}^{\ell}(s) & 1 \end{pmatrix} \succeq 0.

— Bounds on scattering amplitudes of non-identical scalar particles in 4d  (2609.09282 - Ferretti et al., 8 Sep 2026) in Section "Discussion," paragraph "Full system"