Write down the equations you have in mind.

Are you referring to how mass is shown in the Standard Model table? Mass can be written in terms of gram (natural unit) or electron-Volt (dimensional unit) or electron-mass (relative unit).

Related: What is the full QED Lagrangian with physics units written out?

As a practical matter, why would you ever need this? Normally a physicist would calculate something — a differential cross section or a decay time or whatever — in natural units and only at the very end convert just that measurable quantity into SI units.

I am interested in the equations of the standard model not in natural units, but in the dimensional notation of the equations of the standard model. To begin with, I mean the dimensional relations of the weak interaction, but preferably also the strong interaction. I say in advance that the notation in natural units does not suit me. You can refer to the literature, but not in natural units.

It isn’t clear what you mean by “the equations of the standard model”. Are you asking for the 17 coupled partial differential equations for the 17 fields? Or the action, from which the field equations can be derived? Or something else?

Landau-Lifshitz books use natural units, but in complex cases they provide dimensional formulas. I have such a case, natural units lie on the proportionality coefficient, so I want to get formulas in dimensional units. This will be useful for you too, you will get rid of errors of natural units. The proportionality coefficient in physical formulas is not predictable, so the count in natural units can lie, it does not take into account, for example, the coefficient of the frequency $2*\pi$ and many other proportionality coefficients.

In the more than 50 years that I’ve been doing physics, I have never once had any problem calculating in natural units. They do not “lie”. I think you just have some confusion about them, but I don’t know what it is. No physicist I’ve ever known uses SI units to do QFT calculatations.

When everyone calculates using the same formulas, everyone gets the same result, but this does not mean that this result is correct, I repeat, it is necessary to take into account the proportionality coefficient, which is different for different formulas. In the dimensional form, this is manifested.

You’re suggesting that everyone doing QFT has been calculating wrongly for decades. This is an extremely fringe view shared by approximately no one in the physics community. “I am right and everyone else is wrong” is however very common among people outside the physics community. Are you by any chance an engineer?

It’s a little hard for everyone doing QFT to be wrong when they get results that agree with experiment. If you agree with Nature, you’re not wrong.

I suggest that you post another question with a clear and explicit example where you think natural units “lie”. Then perhaps we can figure out what you are really talking about.

And what about the proportionality coefficients, or do you think that they are all equal to 1, including $2\pi$? You just don't want to think, but I can't do anything about it. Continue counting in natural units, and I just can't imagine what that will lead to.

OK, this is unproductive. I think you’re babbling gibberish because you simply don’t understand natural units, and you think I don’t want to think. We’re not getting anywhere, so I’m done.

Related: Misunderstanding the dimension of QCD

You have tables common to all sciences, according to which you calculate all disciplines, or am I wrong, and you calculate different disciplines differently. Too many coefficients need to be entered to calculate different disciplines, and this will create confusion. And what about related sciences?

Pick no more than four of these, and ask to have them redimensionalized.

To use the natural system of units correctly, you need to know the proportionality coefficients for each formula, which requires a formula in the dimensional system of units. I was convinced of this using the standard model as an example.

You made a numerical mistake! Recall, in natural HEP units, $\alpha=e^2/4\pi$, fixing your mismatch.

Oh, you need to take the square root of $4\pi$, you get 74GeV, which means the standard model error is 10%

Still wrong. Follow Wikipedia, as linked, for crying out loud…

Not a single one of your formulas is even dimensionally consistent! You’re using the usual system where the units of $m_W$ are GeV and the units of $G$ are $\text{GeV}^{-2}$. This means that the left sides of your mass formulas are $\text{GeV}^2$ but the right sides are $\text{GeV}^4$. This is nonsense. The $G$’s in the three denominators are not supposed to be squared, as is obvious from Wikipedia.

I calculate in the CGS system. You’re doing no such thing. If you were you would be computing $m_W$ in grams, not GeV, and your formulas would need some $\hbar$’s and $c$’s.

Sorry, I wrote the formula down incorrectly. I calculated using the formula $m_w^2=\frac{4\pi \alpha}{2^{2.5}Gsin^2(\theta_w)}$. I just made a mistake in writing the formula.

But I repeat, to get the correct proportionality coefficient, it is necessary to use dimensional formulas, so multiplication by the value $4\pi\alpha$ is obtained from dimensional formulas. It turns out that a coefficient must be added to formulas in natural units, which creates problems when writing new formulas. This coefficient can only be determined from dimensional formulas. And still, the formula is valid with an accuracy of 1%, which leaves its mark on all weak interaction formulas.

In dimensional formulas, the dimensionless coefficient $4 \pi \alpha$ is determined.

But I didn't get an answer to the main question, what is the formula in dimensional form, and then what is the value of G

Dear scientists, how did it happen that with an accuracy of 1%, when calculating the mass of a vector boson, a result was obtained for the anomalous magnetic moment of a muon with 9 significant digits, and what should we think about this?

You very much did get your answer, here, at tree level. More refined determinations, of course, are given in the PDG...

You suggested me an approximate formula with an accuracy of 1%. But I did not receive this formula in dimensional form to make sure what the dimensional formula gives. It can give a better or worse result. Moreover, the formula in the SI system gives a different result - an error of 10% with the same Fermi constant. This raises suspicions about the correctness of the calculation in natural units. In general, I want to receive the formula in dimensional form and participate in the creation of new physics.

I am the creator of the solution of the Navier-Stokes equations in the complex plane, describing turbulent processes. I also developed quantum mechanics in the complex plane. The imaginary part of the solution is the mean-quadratic deviation, and the real part of the solution is the mean value. The imaginary part satisfies the uncertainty relation. The mean real part can be determined.

In the literature I have looked through, there are correction formulas to the basic idea of ​​calculating the mass of particles of the standard model, weak interaction. But they use corrections. They do not have a basic formula for the mass of particles. This is a drawback of the whole theory. If there is a general formula for mass, it will solve many problems. True, there is a problem with the angle, but it can be solved based on experiment.