tHWZ - latest formulation

During a discussion at PF, I found the following interesting way to think of these quantities:

m_H ~ √2 m_Z
m_t ~ 2 m_Z
H_vev ~ 2 √2 m_Z
m_W ~ √7 / 3 m_Z

The last one may look a little odd, but it allows us to approximate sin² of the Weinberg angle as 2/9.

The impetus was a comment by @arivero in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies

(m_W² - m_H²) / (m_Z² - m_t²) = 3/8

Now in many GUTs, at the GUT scale, we have that

m_W² / m_Z² = 3/8

So it's as if (m_W² - m_H²) / (m_Z² - m_t²) is almost invariant under renormalization group flow, with m_H = m_t = 0 at the GUT scale.

We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)

Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.

Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that m_Z is very close to the standard model's μ parameter.