A Discussion on Triangle Singularities in the $\Lambda_b \to J/\psi K^{-} p$ Reaction

TL;DR

This paper analyzes triangle singularities in the $ ext{Lambda}_b o J/ ext{psi} K^- p$ decay, deriving a formula for their evaluation and exploring their potential to explain narrow peaks observed in experiments.

Contribution

It introduces a simplified formula for evaluating triangle singularities and applies it to the $ ext{Lambda}_b$ decay process, identifying specific charmonium states and conditions affecting peak formation.

Findings

The $ ext{Lambda}(1890) ext{--} ext{chi}_{c1} ext{--}p$ loop produces a prominent peak at 4.45 GeV.

Loops with the same charmonium but different hyperons produce less dramatic peaks.

Certain quantum number configurations reduce the impact of triangle singularities on observed peaks.

Abstract

We have analyzed the singularities of a triangle loop integral in detail and derived a formula for an easy evaluation of the triangle singularity on the physical boundary. It is applied to the $\Lambda_b \rightarrow J/\psi K^{-} p$ process via $\Lambda^*$-charmonium-proton intermediate states. Although the evaluation of absolute rates is not possible, we identify the $\chi_{c1}$ and the $\psi(2S)$ as the relatively most relevant states among all possible charmonia up to the $\psi(2S)$. The $\Lambda(1890)\, \chi_{c1}\, p$ loop is very special as its normal threshold and triangle singularities merge at about 4.45 GeV, generating a narrow and prominent peak in the amplitude in the case that the $\chi_{c1}\, p$ is in an $S$-wave. We also see that loops with the same charmonium and other $\Lambda^*$ hyperons produce less dramatic peaks from the threshold singularity alone. For the case of $\chi_{c1}\, p \rightarrow J/\psi\, p$ and quantum numbers $3/2^-$ or $5/2^+$ one needs $P$- and $D$-waves, respectively, in the $\chi_{c1}\, p$, which drastically reduce the strength of the contribution and smooth the threshold peak. In this case we conclude that the singularities cannot account for the observed narrow peak. In the case of $1/2^+$, $3/2^+$ quantum numbers, where $\chi_{c1}\, p \rightarrow J/\psi\, p$ can proceed in an $S$-wave, the $\Lambda(1890)\,\chi_{c1}\,p$ triangle diagram could play an important role, though can neither assert their strength without further input from experiments and lattice QCD calculations.