H and c from Newton y Fermi

In summary, the conversation discussed the redefinition of the top mass and Fermi constant in terms of the Planck mass and Newton constant. The equations showed a connection between the two sets of constants, but there is no theory to justify the exact equality between the two. This suggests that there may be something beyond the Standard Model at play. The conversation also touched on the didactical aspects of using natural units and how the value of the top mass can affect the equations.
  • #1
arivero
Gold Member
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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

[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 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

[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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  • #2
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 [tex]G_F[/tex] we had used [tex]m_W[/tex] 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[tex]\hbar=c=1[/tex] 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

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

and we have used the empirical fact [tex]y_t^2=.9816 \pm 0.026 [/tex] 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 [tex] y_t = 1 [/tex] 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 [tex] y_t = 1 [/tex] . As said before, Sugra can break about the order of magnitude, but not exactly there.
 
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  • #3
So it is also beyond the current understanding of gravity.

my response to this content would be to acknowledge the interesting and potentially significant relationship between the Fermi constant and the mass of the top quark in the context of gravity and the Standard Model. However, I would also point out that this relationship is currently not fully understood or explained by any existing theories or extensions of the Standard Model. It may require further research and investigation to fully understand the implications of these equations and their potential impact on our understanding of gravity and particle physics. Additionally, I would caution against drawing any definitive conclusions until more evidence and theoretical support can be provided.
 

1. What is "H and c from Newton y Fermi"?

"H and c from Newton y Fermi" refers to a mathematical equation that combines two fundamental constants in physics: Planck's constant (h) and the speed of light (c). It was created by physicists Isaac Newton and Enrico Fermi.

2. What is the significance of "H and c from Newton y Fermi"?

"H and c from Newton y Fermi" is significant because it helps to explain the relationship between energy and mass in the universe. It is a key equation in quantum mechanics and has been used to make important discoveries in the field of particle physics.

3. How is "H and c from Newton y Fermi" used in scientific research?

"H and c from Newton y Fermi" is used in a wide range of scientific research, from understanding the behavior of subatomic particles to studying the properties of light and energy. It is also used in the development of technologies such as lasers and semiconductors.

4. Can "H and c from Newton y Fermi" be applied to everyday life?

While "H and c from Newton y Fermi" may seem abstract and complex, its principles can be seen in everyday life. For example, the equation helps to explain the behavior of electrons in electronic devices and the way light interacts with matter. It also has practical applications in fields such as medicine and energy production.

5. Are there any controversies surrounding "H and c from Newton y Fermi"?

There are no major controversies surrounding "H and c from Newton y Fermi." However, there have been ongoing debates and discussions about the interpretation and implications of the equation, particularly in the field of quantum mechanics.

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