Georgi–Jarlskog mass relation

In grand unified theories of the SU(5) or SO(10) type, there is a mass relation predicted between the electron and the down quark, the muon and the strange quark and the tau lepton and the bottom quark called the Georgi–Jarlskog mass relations. The relations were formulated by Howard Georgi and Cecilia Jarlskog.^[1]

At GUT scale, these are sometimes quoted as:

$m_e \approx \frac{1}{3} m_d$

$m_{\mu} \approx 3 m_s$

$m_{\tau} \approx m_b$

and sometimes as:

$\frac{m_d}{m_s} \approx 9 \frac{m_e}{m_\mu}$

$\frac{m_s}{m_b} \approx \frac{1}{3} \frac{m_\mu}{m_\tau}$

Current values for Lepton and Quark masses

Symbol	Description	Renormalization scheme (point)	Value
me	Electron mass		511 keV
md	Down quark mass	μMS = 2 GeV	4.4 MeV
mu	Up quark mass	μMS = 2 GeV	1.9 MeV
mμ	Muon mass		105.7 MeV
ms	Strange quark mass	μMS = 2 GeV	87 MeV
mc	Charm quark mass	μMS = mc	1.32 GeV
mτ	Tau mass		1.78 GeV
mb	Bottom quark mass	μMS = mb	4.24 GeV
mt	Top quark mass	On-shell scheme	172.7 GeV

Inserting values from the table above for comparison:

$m_e \approx \frac{1}{3} m_d$ → $0.511 MeV \approx 1.5 MeV$

$m_{\mu} \approx 3 m_s$ → $105.7 MeV \approx 261 MeV$

$m_{\tau} \approx m_b$ → $1.78 GeV \approx 4.24 GeV$

$\frac{m_d}{m_s} \approx 9 \frac{m_e}{m_\mu}$ → $0.051 \approx 0.0435$

$\frac{m_s}{m_b} \approx \frac{1}{3} \frac{m_\mu}{m_\tau}$ → $0.021 \approx 0.0198$

References

1. Georgi, H.; Jarlskog, C. (1979). "A new lepton-quark mass relation in a unified theory". Physics Letters B 86 (3–4): 297–300. doi:10.1016/0370-2693(79)90842-6.