# Determining the space-time structure of bottom-quark couplings to spin-zero particles

## TL;DR

This paper discusses the challenges in distinguishing scalar versus pseudoscalar bottom-quark couplings to spin-zero particles at high-energy colliders, highlighting the role of chiral rotations and energy dependence near production thresholds.

## Contribution

It demonstrates that energy dependence near threshold can differentiate scalar and pseudoscalar couplings, despite the invariance under chiral rotations at high energies.

## Key findings

- Energy dependence near threshold can distinguish scalar vs pseudoscalar couplings.
- Chiral rotations can transform between scalar and pseudoscalar interactions without observable effects at high energies.
- Sensitivity to the space-time structure diminishes for heavy spin-zero particles with mass much larger than the top quark mass.

## Abstract

We present a general argument that highlights the difficulty of determining the space-time structure of the renormalizable bottom quark Yukawa interactions of the Standard Model Higgs boson, or for that matter of any hypothetical spin-zero particle, at high energy colliders. The essence of the argument is that, it is always possible, by chiral rotations, to transform between scalar and pseudoscalar Yukawa interactions without affecting the interactions of bottom quarks with SM gauge bosons. Since these rotations affect only the $b$-quark mass terms in the Standard Model Lagrangian, any differences in observables for scalar versus pseudoscalar couplings vanish when $m_b \rightarrow 0$, and are strongly suppressed in high energy processes involving the heavy spin-zero particle where the $b$-quarks are typically relativistic. We show, however, that the energy dependence of, for instance, $e^+e^- \rightarrow b\bar{b} X$ (here $X$ denotes the spin-zero particle) close to the reaction threshold may serve to provide a distinction between the scalar versus pseudoscalar coupling at electron-positron colliders that are being proposed, provided that the $Xb\bar{b}$ coupling is sizeable. We also note that while various kinematic distributions for $t \bar{t} h$ are indeed sensitive to the space-time structure of the top Yukawa coupling, for a spin-0 particle $X$ of an arbitrary mass, the said sensitivity is lost if $m_{X} >> m_t$.