Mixed signals in the IR: Positivity bounds with indefinite species

TL;DR

This work develops analytic positivity bounds for inelastic 2-to-2 scattering with multiple scalar species of unequal mass by exploiting indefinite superposition states and backwards (fixed-angle) dispersion relations. It derives mass-difference dependent bounds that reduce to the well-known equal-mass results in the appropriate limit, and introduces generalized superposition amplitudes with two independent center-of-mass energies to obtain complementary constraints. The authors quantify corrections due to unequal masses, showing they are typically modest (order 10–20%) and can be further mitigated via improved bounds that subtract low-energy IR cuts within an EFT framework. The framework provides concrete, analytic dispersion-based constraints applicable to EFTs with composite states, pions, and potentially SM-like sectors, and sets the stage for extensions to spins and more complex multi-field setups.

Abstract

In theories with multiple particle species standard fixed-t positivity bounds do not directly apply to 2-to-2 definite species scattering amplitudes when the initial and final state are not the same (inelastic processes). These inelastic amplitudes are nevertheless constrained by positivity bounds indirectly, by considering scattering states which are arbitrary superpositions of definite species two-particle states. While these `superposition bounds' have been studied and utilised extensively in the past, earlier analyses typically consider cases insensitive to relative particle masses and IR branch cuts. Here we derive new families of bounds that take account and depend explicitly on mass differences between species making no assumption of weak-coupling. We emphasise unusual non-analyticities induced by the IR mass difference within the superposition amplitude and use fixed (backwards) angle dispersion relations to prove our bounds. We then discuss extensions of our results to `improved bounds', with implications worth exploring for pions and other EFTs of the Standard Model and Beyond, particularly where IR branch cuts are non-negligible.

Figures (9)

• Figure 1: Real $s$ branch cut structure of the three elastic forward limit scattering amplitudes, assuming maximal analyticity. All cuts lie on the real $s$ axis and the vertical separation is purely for visual aid. The orange line denotes branch cuts of forward limit $\chi\chi\xrightarrow[]{}\chi\chi$, the purple of $\phi\chi\xrightarrow[]{}\phi\chi$ and the blue of $\phi\phi\xrightarrow[]{}\phi\phi$ scattering respectively. We see that a sum of these three functions is analytic in the gap between $4\Delta$ and $4m_\phi^2$.
• Figure 2: Plot of $h(s)=\Sigma-s-\frac{\Delta^2}{s}$. The asymptote shown in orange is given by $\Sigma-s$.
• Figure 3: Analytic structure of $B(\mu)\equiv \mathcal{A}_{\phi\chi}(\mu;-1)$. Thick blue lines are branch cuts and each lettered segment of the contour is identified by an arrow at its mid-point. The radius of the circular contour is $\Delta \equiv m_\chi^2-m_\phi^2$.
• Figure 4: As $s$ (horizontal axis) increases between $4\Delta$ and $4$ in units of $m_\phi^2$, the value of $x(s)$ (vertical axis) decreases monotonically. When $\Delta=0$ the curve lies flat along the $s$-axis. These plots are in fact discrete and are generated by fixing evenly spaced values of $s$ between $4\Delta$ and $4m_\phi^2$ and numerically maximising the functions $r_{b,t}$ over the range of the integral. The plot points are joined by a curve that we expect to be piecewise smooth since the underlying functions $r_{b,t}$ are smooth in $\mu$ -- however due to the discrete maximisation in Equation \ref{['eq:x_maxx']} there is a visible kink in the curves where $x_t$ grows larger than $x_b$.
• Figure 5: The allowed parameter spaces according to the rigorous unequal mass bound (\ref{['boundEFTresult_1']}) (in purple) compared to that predicted by the equal mass bound (\ref{['tightestsec3']}) (in red), for the choice $\lambda_{\chi} = \lambda_\phi$ (also using $s = 4m_\phi^2$ and $\Delta = m_\phi^2$ and $x(s) \simeq 0.16$) applied to the tree-level EFT amplitude. The unequal mass bound is slightly weaker than the equal mass bound however requires no weak-coupling assumptions.