In theoretical physics, a hierarchy problem occurs when the fundamental parameters (couplings or masses) of some Lagrangian are vastly different (usually larger) than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization. Typically the renormalization parameters are closely related to the fundamental parameters, but in some cases, it appears that there has been a delicate cancellation between the fundamental quantity and the quantum corrections to it. Hierarchy problems are related to fine-tuning problems and problems of naturalness.

Studying the 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 shortest-distance theory of physics (quantum gravity), we cannot even address how this delicate cancellation between two large terms occurs. Therefore, researchers postulate new physical phenomena that resolve hierarchy problems without fine tuning.

The Higgs Mass

In particle physics, the most important hierarchy problem is the question that asks why the weak force is 1032 times stronger than gravity. Both of these forces involve constants of nature, Fermi's constant for the weak force and Newton's constant for gravity. Furthermore if the Standard Model is used to calculate the quantum corrections to Fermi's constant, it appears that Fermi's constant is unnaturally large and should be closer to Newton's constant, unless there is a delicate cancellation between the bare value of Fermi's constant and the quantum corrections to it.
Cancellation of the Higgs boson quadratic mass renormalization between fermionic top quark loop and scalar stop squark tadpole Feynman diagrams in a supersymmetric extension of the Standard Model

More technically, the question is why the Higgs boson is so much lighter than the Planck mass (or the grand unification energy, or a heavy neutrino mass scale): one would expect that the large quantum contributions to the square of the Higgs boson mass would inevitably make the mass huge, comparable to the scale at which new physics appears, unless there is an incredible fine-tuning cancellation between the quadratic radiative corrections and the bare mass.

It should be remarked that the problem cannot even be formulated in the strict context of the Standard Model, for the Higgs mass cannot be calculated. 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. Implicit in the reasoning that leads to the fine-tuning concern is the unsubstantiated assumption that little physics other than renormalization group scaling exists between the Higgs scale and the grand unification energy. As these two scales are separated by at least 11 orders of magnitude, this "big desert" assumption is seen as unlikely to be true by most physicists outside the string discipline.

If one accepts the big-desert assumption and thus the existence of a Hierarchy Problem, some new mechanism at Higgs scale becomes necessary to avoid the fine-tuning.

The most popular theory—but not the only proposed theory—to solve the hierarchy problem is supersymmetry. This explains 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; however, there is no understanding of why the Higgs mass is so small in the first place which is known as the mu problem. Furthermore, there is no natural way to break supersymmetry so far below the grand unification energy, so what one gets is basically substituting the original Hierarchy Problem of Higgs with a new Hierarchy Problem of supersymmetry breaking.
Supersymmetric Solution

Each particle that couples to the Higgs field has a Yukawa coupling λf. The coupling with the Higgs field for fermions gives an interaction term \( \mathcal{L}_{Yukawa}=-\lambda_f\bar{\psi}H\psi, \psi \) being the Dirac Field and H the Higgs Field. Also, the mass of a fermion is proportional to its Yukawa coupling, meaning that the Higgs boson will couple most to the most massive particle. 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:

\( \Delta m_{H}^{2} = - \frac{\left|\lambda_{f} \right|^2}{8\pi^2} [\Lambda_{UV}^2+ ...]. \)

The \( \Lambda_{UV} \) is called the ultraviolet cutoff and is the scale up to which the Standard Model is valid. If we take this scale to be the Planck scale, then we have the quadratically diverging Lagrangian. However, suppose there existed two complex scalars (taken to be spin 0) such that:

λS= |λf|2 (the couplings to the Higgs are exactly the same).

Then by the Feynman rules, the correction (from both scalars) is:

\( \Delta m_{H}^{2} = 2* \frac{\lambda_{S}}{16\pi^2} [\Lambda_{UV}^2+ ...]. \)

(Note that the contribution here is positive. This is because of the spin-statistics theorem, which means that fermions will have a negative contribution and bosons a positive contribution. This fact is exploited) This gives a total contribution to the Higgs mass to be zero if we include both the fermionic and bosonic particles. Supersymmetry is an extension of this that creates 'superpartners' for all Standard Model particles.

This section adapted from Stephen P. Martin's "A Supersymmetry Primer" on arXiv.[1]
Solution via Extra Dimensions

If we live in a 3+1 dimensional world, then we calculate the Gravitational Force via Gauss' law for gravity:

\( \mathbf{g}(\mathbf{r}) = -Gm\frac{\mathbf{e_r}}{r^2} (1) \)

which is simply the Newton's law of gravitation. Note that Newton's constant G can be rewritten in terms of the Planck Mass.

\( \frac{1}{M_{Pl}^{2}} \)

If we extend this idea to \delta extra dimensions, then we get:

\( \mathbf{g}(\mathbf{r}) = -m\frac{\mathbf{e_r}}{M_{Pl_{3+1+\delta}}^{2+\delta}r^{2+\delta}} \) (2)

where \( M_{Pl_{3+1+\delta}} \) is the \( 3+1+ \delta \) dimensional Planck mass. However, we are assuming that these extra dimensions are the same size as the normal 3+1 dimensions. Let us say that the extra dimensions are of size n <<< than normal dimensions. If we let r << n, then we get (2). However, if we let r >> n, then we get our usual Newton's law. However, when r >> n, the flux in the extra dimensions becomes a constant, because there is no extra room for gravitational flux to flow through. Thus the flux will be proportional to \( n^{\delta} \) because this is the flux in the extra dimensions. The formula is:

\( \mathbf{g}(\mathbf{r}) = -m\frac{\mathbf{e_r}}{M_{Pl_{3+1+\delta}}^{2+\delta}r^2 n^{\delta}} \)

\( -m\frac{\mathbf{e_r}}{M_{Pl}^2 r^2} = -m\frac{\mathbf{e_r}}{M_{Pl_{3+1+\delta}}^{2+\delta}r^2 n^{\delta}} \)

which gives:

\( \frac{1}{M_{Pl}^2 r^2} = \frac{1}{M_{Pl_{3+1+\delta}}^{2+\delta}r^2 n^{\delta}} \Rightarrow
M_{Pl}^2 = M_{Pl_{3+1+\delta}}^{2+\delta} n^{\delta}. \)

Thus the fundamental Planck Mass (the extra dimensional one) could actually be small, meaning that gravity is actually strong, but this must be compensated by the number of the extra dimensions and their size. Physically, this means that gravity is weak because there is a loss of flux to the extra dimensions.

This section adapted from "Quantum Field Theory in a Nutshell" by A. Zee.[2]
Braneworld models
Main article: Brane cosmology

In 1998 Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali proposed the ADD model, also known as the model with large extra dimensions, is an alternative scenario to explain the weakness of gravity relative to the other forces.[3][4] 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.[5]

In 1998/99 Merab Gogberashvili published on Arxiv (and subsequently in peer-reviewed journals) a number of articles where showed that if the Universe is considered as a thin shell (a mathematical synonym for "brane") expanding in 5-dimensional space then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional cosmological constant and Universe thickness, and thus to solve the hierarchy problem.[6][7][8] It was also shown that four-dimensionality of the Universe is the result of stability requirement since the extra component of the Einstein equations giving the localized solution for matter fields coincides with the one of the conditions of stability.

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

Until now, no experimental or observational evidence of extra dimensions has been officially reported. An analysis of results from the Large Hadron Collider in December 2010 severely constrains theories with large extra dimensions.[9]

The Cosmological Constant

In physical cosmology, current observations in favor of an accelerating universe imply the existence of a tiny, but nonzero cosmological constant. This is a hierarchy problem very similar to that of the Higgs boson mass problem, since the cosmological constant is also very sensitive to quantum corrections. It is complicated, however, by the necessary involvement of General Relativity in the problem and may be a clue that we do not understand gravity on long distance scales (such as the size of the universe today). While quintessence has been proposed as an explanation of the acceleration of the Universe, it does not actually address the cosmological constant hierarchy problem in the technical sense of addressing the large quantum corrections. Supersymmetry does not permit to address the cosmological constant problem, since supersymmetry cancels the M4Planck contribution but not the M2Planck one (quadratically diverging).
See also

Little hierarchy problem


^ Stephen P. Martin, A Supersymmetry Primer
^ Zee, A. (2003). Quantum field theory in a nutshell. Princeton University Press. Bibcode
^ N. Arkani-Hamed, S. Dimopoulos, G. Dvali (1998). "The Hierarchy problem and new dimensions at a millimeter". Physics Letters B429: 263–272. arXiv:hep-ph/9803315. Bibcode 1998PhLB..429..263A. doi:10.1016/S0370-2693(98)00466-3.
^ N. Arkani-Hamed, S. Dimopoulos, G. Dvali (1999). "Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity". Physical Review D59: 086004. arXiv:hep-ph/9807344. Bibcode 1999PhRvD..59h6004A. doi:10.1103/PhysRevD.59.086004.
^ For a pedagogical introduction, see M. Shifman (2009). "Large Extra Dimensions: Becoming acquainted with an alternative paradigm". Crossing the boundaries: Gauge dynamics at strong coupling. Singapore: World Scientific. arXiv:0907.3074.
^ M. Gogberashvili, Hierarchy problem in the shell universe model, Arxiv:hep-ph/9812296.
^ M. Gogberashvili, Our world as an expanding shell, Arxiv:hep-ph/9812365.
^ M. Gogberashvili, Four dimensionality in noncompact Kaluza-Klein model, Arxiv:hep-ph/9904383.
^ CMS Collaoration, "Search for Microscopic Black Hole Signatures at the Large Hadron Collider,"

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