The Rise of Linear Trajectories

Abstract

In this letter, we consider constraints on the low-energy spectrum of amplitudes with higher-spin exchange. Assuming unitarity, crossing symmetry, and super-convergent high energy behavior, reminiscent of the scattering of spin-1 and spin-2 massless helicity states, we demonstrate that the spectrum that maximizes the leading higher spin couplings of the second and third resonances is consistently given by a linear trajectory. Furthermore, for gravitational theories, the optimal spectrum is the linear trajectory defined by the mass and spin of the graviton and the lightest spin-4 resonance.

Figures (9)

• Figure 1: Maximization of $\lambda_{2,2}/g_0m_2^2$ in a system with two resonances below the threshold $M$, for different values of $m_1$ and $M$. Vertical lines at critical values $m_1=m_{1,c}$ saturating Eq. (\ref{['lineq']}). All our results are in $D=10$ dimensions. See App. \ref{['app:sdpb1']} for details on numerics (the small rise at $m_1\to 0$ is a numerics artifact). Orange star for DBI. Green star for accumulation-point amplitude: $\left[(s{-}M^2)(t{-}M^2)(u{-}M^2)\right]^{{-}1}$.
• Figure 2: Spectra that maximize the coupling of Fig. \ref{['fig:2state']} (shown only states with $\lambda_{i,\ell}\geq 10^{-5}g_0m_1^2$). The yellow shaded area represents the UV, for $M^2=1.2m_2^2$. For this value of $M$, the solid yellow line is extremal $m_1^2= m_{1,c}^2=0.8m_2^2$, and has linear trajectories that emerge and persist in the UV, just like the sub-extremal light and dark blue lines at $m_1^2<m_{1,c}^2$. Lines are plotted with slopes fixed by the two lighter states. In red, $m_1>m_{1,c}$ for which the linear trajectory is not allowed by the large $M$, and does not appear in the UV. With higher $M^2=1.5m_2^2$ the blue solid line becomes extremal $m_{1,c}^2=0.5 m_2^2$ (UV region in shaded blue). The light blue spectrum is unaffected by these choices of $M$. Light blue dotted: the exponential spectrum saturating the analytic bounds Berman:2024kdhBerman:2024owc.
• Figure 3: Maximal coupling of the $\ell=4$ state, as a function of $m_2$ for $m_3^2=5m_1^2$, and for different values of $M^2/m_1^2=6,7,8$, with $7$ the reference DBI value, and the DBI result denoted by a star. Analytic lower bound shown by the dashed line Berman:2024kdhBerman:2024owc.
• Figure 4: TOP: A cartoon of linear spectra, for different choices of $M^2$. Blue/ black/red lines represent lower-than-critical/ critical/larger-than-critical slopes. BOTTOM: maximization of $\lambda_{2,2} m_1^6/8\pi G m_2^2$ in a system with two resonances below threshold, for different values of $m_1^2$ and $M^2$. The vertical black arrow correspond to the amplitude in Eq. (\ref{['eq: DeformAmp']}), with the star indicating the $\epsilon=0$ case.
• Figure 5: Maximum of $\lambda_{3,4} m_1^6/8\pi G m_3^2$, as a function of $(m_2/m_1)^2$, for $m_3^2=3 m_1^2$ and different values of $M^2/m_1^2$.