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{{Short description|Unsolved problem in physics}}
{{for|the supersymmetric anomaly|Little hierarchy problem}}
{{Use dmy dates|date=December 2024}}
{{MOS|article|date=July 2025| [[MOS:FORMULA]] - avoid mixing {{tag|math}} and {{tl|math}} in the same expression}}
{{Beyond the Standard Model|expanded=Evidence}}
In [[theoretical physics]], the '''hierarchy problem''' is the problem concerning the large discrepancy between aspects of the weak force and gravity.<ref>{{
== Technical definition ==
A hierarchy problem<ref>{{
Typically the renormalized value of parameters are close to their fundamental values, but in some cases, it appears that there has been a delicate cancellation between the fundamental quantity and the quantum corrections. Hierarchy problems are related to [[Fine-tuning (physics)|fine-tuning problem]]s and problems of
Studying
== Overview ==
{{refimprove|date=April 2024}}
Suppose a physics model requires four parameters to produce a very high-quality working model capable of generating predictions regarding some aspect of our physical universe. Suppose we find through experiments that the parameters have values: 1.2, 1.31, 0.9 and
One
A second possible answer is that there is a deeper understanding of physics that we currently do not possess. There
== Examples in particle physics ==
===
In [[particle physics]], the most important hierarchy problem is the question that asks why the [[weak force]] is 10<sup>24</sup> times as strong as [[gravity]].<ref>{{
[[File:Hqmc-vector.svg|thumb|300px|right|Cancellation of the [[Higgs boson]] quadratic [[mass renormalization]] between [[fermion]]ic [[top quark]] loop and [[scalar field|scalar]] stop [[squark]] tadpole [[Feynman diagram]]s in a [[supersymmetry|supersymmetric]] extension of the [[Standard Model]]]]
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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.
=== Theoretical solutions ===
There have been many proposed solutions by many experienced physicists.
==== Supersymmetry ====
Some physicists believe that one may solve the hierarchy problem via [[supersymmetry]]. 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 [[Riccardo Barbieri|Barbieri]]–[[Gian Francesco Giudice|Giudice]] criterion.<ref>{{
Each particle that couples to the Higgs field has an associated [[Yukawa coupling]]
The <math display="inline">\Lambda_{\mathrm{UV}}</math> 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:
(the couplings to the Higgs are exactly the same). Then by the Feynman rules, the correction (from both scalars) is:
(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.<ref>{{
==== Conformal ====
Without supersymmetry, a solution to the hierarchy problem has been proposed using just the [[Standard Model]]. The idea can be traced back to the fact that the term in the Higgs field that produces the uncontrolled quadratic correction upon renormalization is the quadratic one. 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 [[Coleman–Weinberg potential|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]]<ref>{{
==== Extra dimensions ====
No experimental or observational evidence of [[extra dimensions]] has been officially reported. Analyses of results from the [[Large Hadron Collider]] severely constrain theories with [[large extra dimensions]].<ref name="ATLAS_blackholes">{{
If we live in a 3+1 dimensional world, then we calculate the gravitational force via [[Gauss's law for gravity]]:
which is simply [[Newton's law of gravitation]]. Note that Newton's constant ''G'' can be rewritten in terms of the [[Planck mass]].▼
▲which is simply [[Newton's law of gravitation]]. Note that Newton's constant
If we extend this idea to <math>\delta</math> extra dimensions, then we get:▼
where <math>M_{\mathrm{Pl}_{3+1+\delta}}</math> is the {{nowrap|3+1+<math>\delta</math>}} 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 <math> n^{\delta} </math> because this is the flux in the extra dimensions. The formula is:▼
:<math>\mathbf{g}(\mathbf{r}) = -m\frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^2 n^{\delta}}</math>▼
<math display=block>\mathbf{g}(\mathbf{r}) = -m\frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^{2+\delta}} \qquad (2)</math>
▲where <math display="inline">M_{\mathrm{Pl}_{3+1+\delta}}</math> is the {{
<math display="block">\begin{align}
▲
\end{align}</math>
▲:<math>-m\frac{\mathbf{e_r}}{M_{\mathrm{Pl}}^2 r^2} = -m\frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^2 n^{\delta}}</math>
which gives:
<math display="block">\begin{align}
▲:<math> \frac{1}{M_{\mathrm{Pl}}^2 r^2} = \frac{1}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta}r^2 n^{\delta}} \Rightarrow </math>
\implies \quad M_\mathrm{Pl}^2 &= M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} n^\delta
\end{align}</math>
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 is adapted from
==== 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]], an alternative scenario to explain the weakness of [[gravity]] relative to the other forces.<ref name="ADD1">{{
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.<ref>{{
Subsequently, there were proposed the closely related [[Randall–Sundrum model|Randall–Sundrum]] scenarios which offered their solution to the hierarchy problem.
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In 2019, a pair of researchers proposed that [[IR/UV mixing]] resulting in the breakdown of the [[effective field theory|effective]] [[quantum field theory]] could resolve the hierarchy problem.<ref>{{cite journal|title=IR dynamics from UV divergences: UV/IR mixing, NCFT, and the hierarchy problem|first1=Nathaniel|last1=Craig|first2=Seth|last2=Koren|journal=Journal of High Energy Physics|doi=10.1007/JHEP03(2020)037|date=6 March 2020|volume=2020|issue=37|page=37|arxiv=1909.01365|bibcode=2020JHEP...03..037C|s2cid=202540077}}</ref> In 2021, another group of researchers showed that UV/IR mixing could resolve the hierarchy problem in string theory.<ref>{{cite journal|title=Calculating the Higgs mass in string theory|first1=Steven|last1=Abel|first2=Keith R.|last2=Dienes|journal=Physical Review D|volume=104|issue=12|date=29 December 2021|page=126032|doi=10.1103/PhysRevD.104.126032|arxiv=2106.04622|bibcode=2021PhRvD.104l6032A|s2cid=235377340}}</ref>
===
{{main article|Cosmological constant problem}}
In [[physical cosmology]], current observations in favor of an [[accelerating universe]] imply the existence of a tiny, but nonzero [[cosmological constant]]. This problem, called the '''cosmological constant problem''', 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, but
Some physicists have resorted to [[anthropic reasoning]] to solve the cosmological constant problem,<ref>{{cite journal|last1=Martel|first1=Hugo|author2-link=Paul R. Shapiro|last2=Shapiro|first2=Paul R.|last3=Weinberg|first3=Steven|title=Likely Values of the Cosmological Constant|journal=The Astrophysical Journal|date=January 1998|volume=492|issue=1|pages=29–40|doi=10.1086/305016|arxiv=astro-ph/9701099|bibcode=1998ApJ...492...29M|s2cid=119064782}}</ref> but it is disputed whether such anthropic reasoning is scientific.<ref>{{cite book | author = Penrose, R. |author-link = Roger Penrose | title = The Emperor's New Mind | url = https://archive.org/details/emperorsnewmindc00penr | url-access = registration | publisher = Oxford University Press | isbn = 978-0-19-851973-7 | date =1989}} Chapter 10.</ref><ref>{{cite journal | author = Starkman, G. D. | author2 = Trotta, R. | title = Why Anthropic Reasoning Cannot Predict Λ | journal = Physical Review Letters | volume = 97 |page = 201301 | date = 2006 | doi = 10.1103/PhysRevLett.97.201301 | pmid = 17155671 | issue = 20 | bibcode=2006PhRvL..97t1301S|arxiv = astro-ph/0607227 | s2cid = 27409290 }} See also this [http://www.physorg.com/news83924839.html news story.]</ref> == See also ==
{{wikiquote}}
* [[CP violation]]
* [[
* [[
== References ==
{{
{{Standard model of physics}}
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