An Analytic Prescription for $t$-channel Singularities

TL;DR

The paper tackles the problem of $t$-channel singularities that arise when an intermediate stable particle goes on-shell during scattering. It introduces an analytic prescription that rewrites the $t$-channel integral with $X=\varepsilon\bar{X}$ to extract and remove the divergent $1/\varepsilon$ term, leaving a finite scattering contribution $I_0$. Applied to a Beyond-Standard-Model $U(1)_{L_\mu-L_\tau}\times U(1)_L$ framework with a $Z'$ and a Majoron, the method enables a consistent Boltzmann treatment of Majoron production, revealing that scattering processes can dominate over inverse-decay at high temperatures. This analytic approach avoids ad hoc cutoffs or finite-beam assumptions, reducing computational cost and allowing broad exploration of parameter space and application to various cosmological and collider contexts, including potential interpretations of finite-beam-size effects in collider settings. $1/\varepsilon$ terms are systematically removed to separate scattering from decays, yielding robust, scalable predictions for early-Universe particle production and related phenomenology.

Abstract

The $t$-channel singularity is a divergence in the scattering amplitude which occurs when a stable particle propagating in $t$-channel scattering process becomes an on-shell state. Such situations appear either in the system of collider experiments or in the context of the cosmological particle production. No scheme which is generally applicable is known. In this work, we propose a new formulation to identify and remove the source of the divergence. The scheme is fully analytical and various applications can be expected. This work provides a valuable tool in this research field.

Figures (4)

• Figure 1: A $t$-channel diagram for scattering process $\phi_3\phi_4\to \phi_1\phi_2$ for scalar fields $\phi_i$ with momenta $p_i ~ (i=1\sim4)$. There is a stable scalar particle $\Phi$ with momentum $p$ in the intermediate state. The coupling among $\phi_2, \phi_3,$ and $\Phi$ ($\phi_1, \phi_4$, and $\Phi$) is represented by $y_1 (y_2)$.
• Figure 2: Decomposition of the $t$-channel scattering process. The left-hand side is regarded as a sum of the two diagrams in the right-hand side when the intermediate state becomes on shell. The first(second) diagram in the right-handed side corresponds to the decay(inverse-decay). We denote the momenta of incoming and outgoing particles by $(p_3,p_4)$ and $(p_1,p_2)$, respectively.
• Figure 3: $s$-channel (left) and $t$-channel (right) diagrams of the Compton-like process $Z'\nu_\alpha \leftrightarrow \phi\bar{\nu}_\beta$. Similar diagrams appear in the process $Z' \bar{\nu}_\alpha \leftrightarrow \phi \nu_\beta$.
• Figure 4: Comparison between the production efficiencies of the inverse decay and the scattering. Horizontal axis is the temperature of the Universe normalized by the Majoron mass, and the vertical axis shows the interaction rate. The scattering rate can dominate over the inverse decay rate at high temperatures. We fix the Majoron mass $m_\phi=0.1$ MeV. We limit our discussion to the regime after the QCD phase transition.