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.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]

\hat m_{top} \equiv \sqrt 2 \; m_{top}

[/tex]

[tex]

\hat G_F \equiv \sqrt 2 \; G_F

[/tex]

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

[tex] m_P^2 G_N = \hbar c [/tex]

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

the quotient between the RHS of both equations is [tex](\hbar/c)^2[/tex] 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 lenght 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

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

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

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.

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# H and c from Newton y Fermi

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