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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>{{cite journal |last1=Arkani–Hamed |first1=Nima |last2=Dimopoulos |first2=Savas |last3=Dvali |first3=Gia |date=1998-06-18 |title=The hierarchy problem and new dimensions at a millimeter |journal=Physics Letters B |volume=429 |issue=3 |pages=263–272 |doi=10.1016/S0370-2693(98)00466-3 |issn=0370-2693|doi-access=free |arxiv=hep-ph/9803315 |bibcode=1998PhLB..429..263A }}</ref> occurs when the fundamental value of some physical parameter, such as a [[coupling constant]] or a mass, in some [[Lagrangian mechanics|Lagrangian]] is vastly different from its effective value, which is the value that gets measured in an experiment. This happens because the effective value is related to the fundamental value by a prescription known as [[renormalization]], which applies corrections to it.
A hierarchy problem occurs when the fundamental value of some physical parameter, such as a [[coupling constant]] or a mass, in some [[Lagrangian mechanics|Lagrangian]] is vastly different from its effective value, which is the value that gets measured in an experiment. This happens because the effective value is related to the fundamental value by a prescription known as [[renormalization]], which applies corrections to it. 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|fine-tuning problem]]s and problems of [[naturalness (physics)|naturalness]]. Over the past decade many scientists<ref>{{Cite journal |last1=Fowlie |first1=Andrew |last2=Balazs |first2=Csaba |last3=White |first3=Graham |last4=Marzola |first4=Luca |last5=Raidal |first5=Martti |date=17 August 2016 |title=Naturalness of the relaxion mechanism |journal=Journal of High Energy Physics |volume=2016 |issue=8 |pages=100 |arxiv=1602.03889 |bibcode=2016JHEP...08..100F |doi=10.1007/JHEP08(2016)100 |s2cid=119102534}}</ref><ref>{{Cite journal |last=Fowlie |first=Andrew |date=10 July 2014 |title=CMSSM, naturalness and the ?fine-tuning price? of the Very Large Hadron Collider |journal=Physical Review D |volume=90 |issue=1 |pages=015010 |arxiv=1403.3407 |bibcode=2014PhRvD..90a5010F |doi=10.1103/PhysRevD.90.015010 |s2cid=118362634}}</ref><ref>{{Cite journal |last=Fowlie |first=Andrew |date=15 October 2014 |title=Is the CNMSSM more credible than the CMSSM? |journal=The European Physical Journal C |volume=74 |issue=10 |arxiv=1407.7534 |doi=10.1140/epjc/s10052-014-3105-y |s2cid=119304794}}</ref><ref>{{Cite journal |last1=Cabrera |first1=Maria Eugenia |last2=Casas |first2=Alberto |last3=Austri |first3=Roberto Ruiz de |last4=Marzola |first4=Luca |last5=Raidal |first5=Martti |year=2009 |title=Bayesian approach and naturalness in MSSM analyses for the LHC |journal=Journal of High Energy Physics |volume=2009 |issue=3 |page=075 |arxiv=0812.0536 |bibcode=2009JHEP...03..075C |doi=10.1088/1126-6708/2009/03/075 |s2cid=18276270}}</ref><ref>{{Cite journal |last=Fichet |first=S. |date=18 December 2012 |title=Quantified naturalness from Bayesian statistics |journal=Physical Review D |volume=86 |issue=12 |pages=125029 |arxiv=1204.4940 |bibcode=2012PhRvD..86l5029F |doi=10.1103/PhysRevD.86.125029 |s2cid=119282331}}</ref> argued that the hierarchy problem is a specific application of [[Bayesian statistics]].▼
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 naturalness.
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|shortest-distance theory of physics]], 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.▼
▲
==Overview==▼
▲Studying
Scientists might wonder how such figures arise. But in particular, might be especially curious about a theory where three values are close to one, and the fourth is so different; in other words, the huge disproportion we seem to find between the first three parameters and the fourth. We might also wonder if one force is so much weaker than the others that it needs a factor of 4×10<sup>29</sup> to allow it to be related to them in terms of effects, how did our universe come to be so exactly balanced when its forces emerged? In current particle physics, the differences between some parameters are much larger than this, so the question is even more noteworthy.▼
▲== Overview ==
{{refimprove|date=April 2024}}
▲
A second possible answer is that there is a deeper understanding of physics that we currently do not possess. There might be parameters that we can derive physical constants from that have less unbalanced values, or there might be a model with fewer parameters.▼
One explanation given by philosophers is the [[anthropic principle]]. If the universe came to exist by chance and vast numbers of other universes exist or have existed, then lifeforms capable of performing physics experiments only arose in universes that, by chance, had very balanced forces. All of the universes where the forces were not balanced did not develop life capable of asking this question. So if lifeforms like [[human being]]s are aware and capable of asking such a question, humans must have arisen in a universe having balanced forces, however rare that might be.<ref>{{cite web |date=2024-02-08 |title=Anthropic principle {{!}} Cosmology, Physics & Philosophy {{!}} Britannica |url=https://www.britannica.com/science/anthropic-principle |access-date=2024-04-01 |website=www.britannica.com |language=en}}</ref><ref>{{citation |last1=Dimopoulos |first1=Savas |title=The anthropic principle, dark energy and the LHC |date=2007 |work=Universe or Multiverse? |pages=211–218 |publication-date=5 July 2014 |editor-last=Carr |editor-first=Bernard |url=https://www.cambridge.org/core/books/universe-or-multiverse/anthropic-principle-dark-energy-and-the-lhc/1FEAD231F875FB51E8A01EF68541A9D8 |access-date=2024-04-01 |place=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-14069-0 |last2=Thomas |first2=Scott|bibcode=2007unmu.book..211D }}</ref>
▲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]]]]
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 (physics)|fine-tuning]] cancellation between the quadratic radiative corrections and the bare mass.
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
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>▼
<math display="block">\Delta m_{\rm H}^{2} = - \frac{\left|\lambda_{f} \right|^2}{8\pi^2} [\Lambda_{\mathrm{UV}}^2+ \dots].</math>
▲====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>{{Cite journal |last1=Barbieri |first1=R. |last2=Giudice |first2=G. F. |year=1988 |title=Upper Bounds on Supersymmetric Particle Masses |url=http://cds.cern.ch/record/180560 |journal=Nucl. Phys. B |volume=306 |issue=1 |page=63 |bibcode=1988NuPhB.306...63B |doi=10.1016/0550-3213(88)90171-X}}</ref> This still leaves open the [[mu problem]], however. Currently, the tenets of supersymmetry are being tested at the [[Large Hadron Collider|LHC]], although no evidence has been found so far for supersymmetry.
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:▼
▲Each particle that couples to the Higgs field have a [[Yukawa coupling]] λ<sub>f</sub>. The coupling with the Higgs field for fermions gives an interaction term <math>\mathcal{L}_{\mathrm{Yukawa}}=-\lambda_f\bar{\psi}H\psi</math>, with <math>\psi</math> being the [[Dirac field]] and <math>H</math> 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 diagram#Feynman rules|Feynman rules]], one gets the quantum corrections to the Higgs mass squared from a fermion to be:
▲The <math>\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:
▲:<math>\lambda_S= \left|\lambda_f\right|^2</math> (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 ====
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
====
{{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.
===
▲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>
=== Cosmological constant ===
{{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}}
{{DEFAULTSORT:Hierarchy Problem}}
[[Category:
[[Category:Physics beyond the Standard Model]]
[[Category:Unsolved problems in physics]]
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