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The hierarchy problem is often stated in two ways: First, the divergent corrections to the Higgs bare mass, second, why is gravity so much weaker than the other three forces.

The solution to the divergent corrections is explained many sources, but I don't know how does SUSY explain the hierarchy of all the three forces over gravitational force

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second, why is gravity so much weaker than the other three forces.

There is NO "second"! The "natural" Higgs mass should have been Planck mass, which means that the gravity between fundamental particles would have been as strong as the other three forces.

If the "unnatural" Higgs mass is explained, the weakness of the gravity is automatically explained: small Higgs mass is translated to small masses of fundamental particles (via small Higgs VEV in the Yukawa terms), hence "weak" gravity between them.

There is a fundamental difference between gravity and other forces: gravitational constant is dimensional whereas the other coupling constants are dimensionless. Thus yapping about the "weakness" of gravity vs other forces is comparing apples to oranges. The correct way is to always specify the MASS of the objects in your comparison.

Wiki quote (en.wikipedia.org/wiki/Dimensionless_quantity) "$\alpha_G ≈1.75×10^{−45}$, the gravitational coupling constant which is the square of the ratio of the mass of the electron to the Planck mass, which characterizes the magnitude of the gravitational interaction between electrons. It is because, fundamentally, this number is so small that it is meaningful to say Gravity is an extremely weak fundamental force in comparison to either the electromagnetic force or the strong nuclear force".

Notice the definition of "gravitational coupling constant" $\alpha_G$ above has to involve electron mass to make it dimensionless! Basically, "weak" gravity is all about the ratio between say, electron mass and Planck mass. Therefore, we are looping back to your first hierarchy problem, since the masses of fundamental particles are all tied to the electroweak symmetry breaking scale.

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  • $\begingroup$ OK, I have the same question again. In MSSM Higgs has the mass at the electroweak scale. What is the status of gravity and other forces in MSSM ? $\endgroup$ – 23rduser Aug 28 '19 at 16:48
  • $\begingroup$ SUSY by the introduction of partner particles explains why the natural Higgs can be at EWK scale. So the explanation of 'why Higgs mass is not at plank scale' is not needed, in fact, this is no more a problem. But the weak gravity as compared to other forces is still not explained. $\endgroup$ – 23rduser Nov 20 '19 at 16:40
  • $\begingroup$ @23rduser, There is a fundamental difference between gravity and other forces: gravitational constant is dimensional whereas the other coupling constants are dimensionless. Thus yapping about the "weakness" of gravity vs other forces is comparing apples to oranges. The correct way is to specify the MASS of the objects in your comparison. Our conversation is going nowhere if you fail to appreciate this crucial point, $\endgroup$ – MadMax Nov 20 '19 at 16:59
  • $\begingroup$ Oh yes, I have been missing this crucial point, can you please point to some proper text, wherein I can understand things more clearly. Thanks! $\endgroup$ – 23rduser Nov 20 '19 at 18:15
  • $\begingroup$ @23rduser, wiki quote (en.wikipedia.org/wiki/Dimensionless_quantity) "$\alpha_G ≈ 1.75 × 10^{-45}$, the gravitational coupling constant which is the square of the ratio of the mass of the electron to the Planck mass, which characterizes the magnitude of the gravitational interaction between electrons. It is because, fundamentally, this number is so small that it is meaningful to say Gravity is an extremely weak fundamental force". Notice the definition of "gravitational coupling constant" above has to involve electron mass to make it dimensionless! $\endgroup$ – MadMax Nov 20 '19 at 18:26

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