Hierarchy problem

many scientists[3][4][5][6][7] argued that the hierarchy problem is a specific application of Bayesian statistics.

Studying renormalization in hierarchy problems is difficult, because such quantum corrections are usually power-law divergent, which means that the shortest-distance physics are most important.

Because we do not know the precise details of the quantum gravity, we cannot even address how this delicate cancellation between two large terms occurs.

Therefore, researchers are led to postulate new physical phenomena that resolve hierarchy problems without fine-tuning.

In current particle physics, the differences between some parameters are much larger than this, so the question is even more noteworthy.

All of the universes where the forces were not balanced did not develop life capable of asking this question.

[citation needed] In particle physics, the most important hierarchy problem is the question that asks why the weak force is 1024 times as strong as gravity.

In a sense, the problem amounts to the worry that a future theory of fundamental particles, in which the Higgs boson mass will be calculable, should not have excessive fine-tunings.

Supersymmetry can explain how a tiny Higgs mass can be protected from quantum corrections.

Supersymmetry removes the power-law divergences of the radiative corrections to the Higgs mass and solves the hierarchy problem as long as the supersymmetric particles are light enough to satisfy the Barbieri–Giudice criterion.

This means that the most significant corrections to the Higgs mass will originate from the heaviest particles, most prominently the top quark.

By applying the Feynman rules, one gets the quantum corrections to the Higgs mass squared from a fermion to be:

This gives a total contribution to the Higgs mass to be zero if we include both the fermionic and bosonic particles.

If the Higgs field had no mass term, then no hierarchy problem arises.

But by missing a quadratic term in the Higgs field, one must find a way to recover the breaking of electroweak symmetry through a non-null vacuum expectation value.

This can be obtained using the Weinberg–Coleman mechanism with terms in the Higgs potential arising from quantum corrections.

Mass obtained in this way is far too small with respect to what is seen in accelerator facilities and so a conformal Standard Model needs more than one Higgs particle.

This proposal has been put forward in 2006 by Krzysztof Antoni Meissner and Hermann Nicolai[13] and is currently under scrutiny.

No experimental or observational evidence of extra dimensions has been officially reported.

[14] However, extra dimensions could explain why the gravity force is so weak, and why the expansion of the universe is faster than expected.

[15] If we live in a 3+1 dimensional world, then we calculate the gravitational force via Gauss's law for gravity:

Note that Newton's constant G can be rewritten in terms of the Planck mass.

Physically, this means that gravity is weak because there is a loss of flux to the extra dimensions.

[16] In 1998 Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali proposed the ADD model, also known as the model with large extra dimensions, an alternative scenario to explain the weakness of gravity relative to the other forces.

[17][18] This theory requires that the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale.

[19] In 1998–99 Merab Gogberashvili published on arXiv (and subsequently in peer-reviewed journals) a number of articles where he showed that if the Universe is considered as a thin shell (a mathematical synonym for "brane") expanding in 5-dimensional space then it is possible to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant and Universe thickness, and thus to solve the hierarchy problem.

[20][21][22] It was also shown that four-dimensionality of the Universe is the result of a stability requirement since the extra component of the Einstein field equations giving the localized solution for matter fields coincides with one of the conditions of stability.

Subsequently, there were proposed the closely related Randall–Sundrum scenarios which offered their solution to the hierarchy problem.

In 2019, a pair of researchers proposed that IR/UV mixing resulting in the breakdown of the effective quantum field theory could resolve the hierarchy problem.

[23] In 2021, another group of researchers showed that UV/IR mixing could resolve the hierarchy problem in string theory.