Can Gravity and the Standard Model Explain the Fermi-Top Mass Relationship?

[arivero]

I am detaching this from the "All the lepton masses..." because there are not leptons anymore, and it is sort of Gravity plus Beyond Standard Model.

First let me to put some square roots under the carpet by redefining

<br /> \hat m_{top} \equiv \sqrt 2 \; m_{top}<br />

<br /> \hat G_F \equiv \sqrt 2 \; G_F <br />

from the Fermi constant and the mass of the top.Now take Planck mass and Newton Constant as usual. We have the following pair of equations

m_P^2 G_N = \hbar c

\hat m_{top}^2 \hat G_F = \hbar^3 / c

the quotient between the RHS of both equations is (\hbar/c)^2 the square of the product of an (arbitrary) mass times its Compton lenght. This can be partly understood because Fermi force and Newton force have different shapes: One does not depend of masses, the other does: so a mass square term is needed to adjust. One depends of r^-4, the other goes as r^-2: so a length square term is needed.

Now the funny thing is that we can use the pair of equations to solve for h and c. We have

c^4= {(m_P^2 G_N)^3 \over (\hat m_{top}^2 \hat G_F) }

\hbar^4= (m_P^2 G_N) (\hat m_{top}^2 \hat G_F)

Actually, while the equations work empirically, there is not any theory justifying the fermi times top mass equality to be exact (sugra justifies the order of magnitude). So it is beyond the SM and beyond known extensions. On the gravity side, the Newton times Planck mass combination has the issue of being really a definition of Planck mass; it is not a measured mass of a known particle.

 

[arivero]

Spoiler :-)

Of course, (almost) all the trick is that the definitions of Fermi constant and Planck mass are a good place to hide h and c. If instead of G_F we had used m_W and electroweak couplings, nothing happens.

Still, I think that the example is didactical at a pregraduate level; on one hand it shows you that natural units\hbar=c=1 can hide some equations from view. From other side, it shows how Fermi and Newton constant, albeit they have the same high energy [mis]behaviour, are not the same thing because they differ in h and c constants factors, so for instance perturbative expansions on h, c, or some combination of them, can differ.

Last, and this is pehaps didactical at the graduate level, the second equation in the first pair should be

\hat m_{top}^2 \hat G_F = y_t^2 \hbar^3 / c

and we have used the empirical fact y_t^2=.9816 \pm 0.026 to set this factor to be exactly 1. This is true for a top mass of 174.11 GeV, so the current precision on top mass measurements implies this result is here to stay, and any deviation -it it happens- should be focused in similar ways to the giromagnetic ratio of electron going away from 2. (Of course, assuming ten years of Fermilab research are not just faking a result extracted from fermi constant; G_F was already at the current level of precision in 1995).
The didactical part here is that a theory for y_t = 1 is lacking; if coming from GUT, it should either keep the running at almost null levels (thus M_W, m_t and g running in a conspiratorial way) or to have a way to break Electroweak Symmetry exactly when the running down reaches y_t = 1 . As said before, Sugra can break about the order of magnitude, but not exactly there.