# Renormalization group: new relations between the parameters of the Standard Model

## TL;DR

This paper derives new exact relations between quark and lepton masses and gauge couplings in the Standard Model using renormalization group equations at one and two loops, revealing simplified parameter evolution.

## Contribution

It introduces novel relations between Standard Model parameters that remain approximately constant across energy scales, simplifying the understanding of their evolution.

## Key findings

- Identifies an exact constant involving quark masses and gauge couplings at one loop.
- Finds two relations between quark masses and gauge couplings at two loops with small variations.
- Derives a relation for charged lepton mass and gauge couplings with minimal variation.

## Abstract

We analyze the renormalization group equations for the Standard Model at the one and two loops levels. At one loop level we find an exact constant of evolution built from the product of the quark masses and the gauge couplings $g_{1}$ and $ g_{3}$ of the $U(1)$ and $SU(3)$ groups. For leptons at one loop level we find that the ratio of the charged lepton mass and the power of $g_{1}$ varies $\simeq 4\times 10^{-5}$ in the whole energy range. At the two loop level we have found two relations between the quark masses and the gauge couplings that vary $\simeq 4\%$ and $\simeq 1\%$, respectively. For leptons at the two loop level we have derived a relation between the charged lepton mass and the gauge couplings $g_ {1}$ and $g_{2}$ that varies $\simeq 0.1\%$. This analysis significantly simplifies the picture of the renormalization group evolution of the Standard Model and establishes new important relations between its parameters.