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 sin

^{2}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

_{W}

^{2}- m

_{H}

^{2}) / (m

_{Z}

^{2}- m

_{t}

^{2}) = 3/8

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

m

_{W}

^{2}/ m

_{Z}

^{2}= 3/8

So it's as if (m

_{W}

^{2}- m

_{H}

^{2}) / (m

_{Z}

^{2}- m

_{t}

^{2}) 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.