# The $\Upsilon(1S)$ leptonic decay using the principle of maximum conformality

## TL;DR

This paper applies the principle of maximum conformality (PMC) to accurately compute the leptonic decay width of the $
S$ bottomonium state, eliminating renormalization scale ambiguities and aligning well with experimental data.

## Contribution

It introduces the use of PMC scale-setting at NNNLO for $
S$ leptonic decay, providing a more reliable and scale-independent prediction.

## Key findings

- Decay width $
S o e^+ e^-$ calculated as 1.262 keV with uncertainties.
- Results agree with experimental measurements within errors.
- Demonstrates the effectiveness of PMC in quarkonium decay calculations.

## Abstract

In the paper, we study the $\Upsilon(1S)$ leptonic decay width $\Gamma(\Upsilon(1S)\to \ell^+\ell^-)$ by using the principle of maximum conformality (PMC) scale-setting approach. The PMC adopts the renormalization group equation to set the correct momentum flow of the process, whose value is independent to the choice of the renormalization scale and its prediction thus avoids the conventional renormalization scale ambiguities. Using the known next-to-next-to-next-to-leading order perturbative series together with the PMC single scale-setting approach, we do obtain a renormalization scale independent decay width, $\Gamma_{\Upsilon(1S) \to e^+ e^-} = 1.262^{+0.195}_{-0.175}$ keV, where the error is squared average of those from $\alpha_s(M_{Z})=0.1181\pm0.0011$, $m_b=4.93\pm0.03$ GeV and the choices of factorization scales within $\pm 10\%$ of their central values. To compare with the result under conventional scale-setting approach, this decay width agrees with the experimental value within errors, indicating the importance of a proper scale-setting approach.