Hierarchy problem: Difference between revisions

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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>{{cite web |date=1614 August 2011 |title=The Hierarchy Problem {{pipe}} |website=Of Particular Significance |url=http://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-hierarchy-problem/ |access-date=13 December 2015 |websitefirst=Profmattstrassler.comMatt |last=Strassler}}</ref> There is no scientific consensus on why, for example, the [[weak force]] is 10²⁴ times stronger than [[gravity]].

== Technical definition ==

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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.

OverThroughout the past decade2010s, 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 |pagesarticle-number=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 |pagearticle-number=3105 |arxiv=1407.7534 |doi=10.1140/epjc/s10052-014-3105-y |bibcode=2014EPJC...74.3105F |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 |pagesarticle-number=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]].

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.

== 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 a value near {{val|4|e=29}}. One might wonder how such figures arise. In particular, one might be especially curious about a theory where three values are close to one, and the fourth is so different; i.e., the huge disproportion we seem to find between the first three parameters and the fourth. If one force is so much weaker than the others that it needs a factor of {{val|4|e=29}} to allow it to be related to the others in terms of effects, we might also wonder how our universe come to be so exactly balanced when its forces emerged. In current [[particle physics]], the differences between some actual parameters are much larger than this, so the question is noteworthy.

One 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 {{val|4|e=29}} 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.

One answerexplanation given by philosophers is the [[anthropic principle]]. If the universe came to exist by chance, and perhaps vast numbers of other universes exist or have existed, then lifelifeforms 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 mightmay be parameters thatfrom which we can derive physical constants from that have lessfewer unbalanced values, or there mightmay be a model with fewer parameters.{{Citation needed|reason=Reliable source needed for the whole sentence|date=April 2024}}

== Examples in particle physics ==

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==== 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=https://cds.cern.ch/record/180560 |journal=Nucl.Nuclear Phys.Physics 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. The tenets of supersymmetry are being tested at the [[Large Hadron Collider|LHC]], although no evidence has been found so far for supersymmetry.

Each particle that couples to the Higgs field has an associated [[Yukawa coupling]] λ<submath display="inline">f\lambda_f</submath>. The coupling with the Higgs field for fermions gives an interaction term <math display="inline">\mathcal{L}_{\mathrm{Yukawa}}=-\lambda_f\bar{\psi}H\psi</math>, with <math display="inline">\psi</math> being the [[Dirac field]] and <math display="inline">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:

: <math>\Delta m_{\rm H}^{2} = - \frac{\left|\lambda_{f} \right|^2}{8\pi^2} [\Lambda_{\mathrm{UV}}^2+ ...].</math>

: <math display="block">\Delta m_{\rm H}^{2} = - \frac{\left|\lambda_{f} \right|^2}{8\pi^2} [\Lambda_{\mathrm{UV}}^2+ ...\dots].</math>

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).

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:

<math display="block">G \lambda_S= \frac{left|\hbar c}{M_{lambda_f\mathrm{Pl}}right|^{2}}</math>

: <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:

: <math display="block">\Delta m_{\rm H}^{2} = 2 \times \frac{\lambda_{S}}{16\pi^2} [\Lambda_{\mathrm{UV}}^2+ ...\dots].</math>

(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.)

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==== 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">{{cite journal |last1=Aad |first1=G. |last2=Abajyan |first2=T. |last3=Abbott |first3=B. |last4=Abdallah |first4=J. |last5=Abdel Khalek |first5=S. |last6=Abdinov |first6=O. |last7=Aben |first7=R. |last8=Abi |first8=B. |last9=Abolins |first9=M. |last10=Abouzeid |first10=O. S. |last11=Abramowicz |first11=H. |display-authors=296 |year=2014 |title=Search for Quantum Black-Hole Production in High-Invariant-Mass Lepton+Jet Final States Using Proton-Proton Collisions at {{sqrt|s}} = 8 TeV and the ATLAS Detector |journal=Physical Review Letters |volume=112 |issue=9 |pages=091804 |arxiv=1311.2006 |bibcode=2014PhRvL.112i1804A |doi=10.1103/PhysRevLett.112.091804 |pmid=24655244 |last12=Abreu |first12=H. |last13=Abulaiti |first13=Y. |last14=Acharya |first14=B. S. |last15=Adamczyk |first15=L. |last16=Adams |first16=D. L. |last17=Addy |first17=T. N. |last18=Adelman |first18=J. |last19=Adomeit |first19=S. |last20=Adye |first20=T. |last21=Aefsky |first21=S. |last22=Agatonovic-Jovin |first22=T. |last23=Aguilar-Saavedra |first23=J. A. |last24=Agustoni |first24=M. |last25=Ahlen |first25=S. P. |last26=Ahmad |first26=A. |last27=Ahmadov |first27=F. |last28=Aielli |first28=G. |last29=Åkesson |first29=T. P. A. |last30=Akimoto |first30=G.|s2cid=204934578 }}</ref> However, extra dimensions could explain why the gravity force is so weak, and why the expansion of the universe is faster than expected.<ref>{{cite web |date=20 January 2012 |title=Extra dimensions, gravitons, and tiny black holes |url=http://home.web.cern.ch/about/physics/extra-dimensions-gravitons-and-tiny-black-holes |access-date=13 December 2015 |websitepublisher=Home.web.cern.chCERN}}</ref>

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

<math display="block">\mathbf{g}(\mathbf{r}) = -Gm\frac{\mathbf{e_r}}{r^2} \qquad (1)</math>

which is simply [[Newton's law of gravitation]]. Note that Newton's constant {{mvar|G}} can be rewritten in terms of the [[Planck mass]].

<math display=block>G = \frac{\hbar c}{M_{\mathrm{Pl}}^{2}}</math>

<math display="block">G = \frac{\hbar c}{M_{\mathrm{Pl}}^{2}}</math>

If we extend this idea to {{mvar|&delta;}} extra dimensions, then we get:

<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|3+1+<math display="inline">\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 {{math|''n'' ≪}} than normal dimensions. If we let {{math|''r'' ≪ ''n''}}, then we get (2). However, if we let {{math|''r'' ≫ ''n''}}, then we get our usual Newton's law. However, when {{math|''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 {{mvar|n{{sup|&delta;}}}} because this is the flux in the extra dimensions. The formula is:

<math display=block>\begin{align}

<math display="block">\begin{align}

\mathbf{g}(\mathbf{r}) &= -m \frac{\mathbf{e_r}}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} r^2 n^\delta} \\[2pt]

-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}

\end{align}</math>

which gives:

<math display="block">\begin{align}

\frac{1}{M_\mathrm{Pl}^2 r^2} &= \frac{1}{M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} r^2 n^\delta} \\[2pt]

\implies \quad M_\mathrm{Pl}^2 &= M_{\mathrm{Pl}_{3+1+\delta}}^{2+\delta} n^\delta

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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 "''Quantum Field Theory in a Nutshell"'' by A. Zee.<ref>{{cite book |last=Zee |first=A. |title=Quantum field theory in a nutshell |publisher=Princeton University Press |year=2003 |isbn=978-0-691-01019-9 |bibcode=2003qftn.book.....Z}}</ref>

==== Braneworld models ====

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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">{{cite journal |last1=Arkani-Hamed |first1=N. |last2=Dimopoulos |first2=S. |last3=Dvali |first3=G. |year=1998 |title=The Hierarchy problem and new dimensions at a millimeter |journal=[[Physics Letters]] |volume=B429 |issue=3–4 |pages=263–272 |arxiv=hep-ph/9803315 |bibcode=1998PhLB..429..263A |doi=10.1016/S0370-2693(98)00466-3 |s2cid=15903444}}</ref><ref name="ADD2">{{cite journal |last1=Arkani-Hamed |first1=N. |last2=Dimopoulos |first2=S. |last3=Dvali |first3=G. |year=1999 |title=Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity |journal=[[Physical Review]] |volume=D59 |issue=8 |page=086004 |arxiv=hep-ph/9807344 |bibcode=1999PhRvD..59h6004A |doi=10.1103/PhysRevD.59.086004 |s2cid=18385871}}</ref> This theory requires that the fields of the [[Standard Model]] are confined to a four-dimensional [[membrane (M-Theory)|membrane]], while gravity propagates in several additional spatial dimensions that are large compared to the [[Planck scale]].<ref>For a pedagogical introduction, see {{cite conference |last=Shifman |first=M. |author-link=Mikhail Shifman |year=2009 |title=Large Extra Dimensions: Becoming acquainted with an alternative paradigm |journal=International Journal of Modern Physics A |volume=25 |issue=2n03 |pages=199–225 |conference=Crossing the boundaries: Gauge dynamics at strong coupling |___location=Singapore |publisher=World Scientific |arxiv=0907.3074 |bibcode=2010IJMPA..25..199S |doi=10.1142/S0217751X10048548}}</ref>

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>{{cite journal |last1=Gogberashvili |first1=Merab |last2=Ahluwalia |first2=D. V. |year=2002 |title=Hierarchy Problem in the Shell-Universe Model |journal=International Journal of Modern Physics D |volume=11 |issue=10 |pages=1635–1638 |arxiv=hep-ph/9812296 |bibcode=2002IJMPD..11.1635G |doi=10.1142/S0218271802002992 |s2cid=119339225}}</ref><ref>{{cite journal |last=Gogberashvili |first=M. |year=2000 |title=Our world as an expanding shell |journal=Europhysics Letters |volume=49 |issue=3 |pages=396–399 |arxiv=hep-ph/9812365 |bibcode=2000EL.....49..396G |doi=10.1209/epl/i2000-00162-1 |s2cid=38476733}}</ref><ref>{{cite journal |last=Gogberashvili |first=Merab |year=1999 |title=Four Dimensionality in Non-Compact Kaluza–Klein Model |journal=Modern Physics Letters A |volume=14 |issue=29 |pages=2025–2031 |arxiv=hep-ph/9904383 |bibcode=1999MPLA...14.2025G |doi=10.1142/S021773239900208X |s2cid=16923959}}</ref> It was also shown that four-dimensionality of the Universe is the result of a [[Stability theory|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 model|Randall–Sundrum]] scenarios which offered their solution to the hierarchy problem.

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=== 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 itits calculation is complicated by the necessary involvement of [[general relativity]] in the problem. Proposed solutions to the cosmological constant problem include modifying and/or extending gravity,<ref name="dark universe">Bull, Philip, Yashar Akrami, Julian Adamek, Tessa Baker, Emilio Bellini, Jose Beltrán Jiménez, Eloisa Bentivegna et al. "Beyond ΛCDM: Problems, solutions, and the road ahead." Physics of the Dark Universe 12 (2016): 56-99.</ref><ref>{{cite journal|last=Ellis |first=George F. R. |authorlink=George F. R. Ellis |title=The trace-free Einstein equations and inflation |journal=[[General Relativity and Gravitation]] |year=2014 |volume=46 |issue=1 |pagesarticle-number=1619 |doi=10.1007/s10714-013-1619-5 |arxiv=1306.3021|bibcode=2014GReGr..46.1619E |s2cid=119000135 }}</ref><ref>{{cite journal|last=Percacci |first=R. |title=Unimodular quantum gravity and the cosmological constant |doi=10.1007/s10701-018-0189-5 |arxiv=1712.09903 |year=2018 |journal=[[Foundations of Physics]] |volume=48 |number=10 |pages=1364–1379|bibcode=2018FoPh...48.1364P |s2cid=118934871 }}</ref> adding matter with unvanishing pressure,<ref name="Luongo Muccino pp. 2-3">{{cite journal |last1=Luongo |first1=Orlando |last2=Muccino |first2=Marco |title=Speeding up the Universe using dust with pressure |journal=Physical Review D |volume=98 |issue=10 |date=2018-11-21 |issn=2470-0010 |doi=10.1103/physrevd.98.103520 |pages=2–3|arxiv=1807.00180 |bibcode=2018PhRvD..98j3520L |s2cid=119346601 }}</ref> and UV/IR mixing in the Standard Model and gravity.<ref>{{cite journal|title=Effective Field Theory, Black Holes, and the Cosmological Constant|last1=Cohen|first1=Andrew|last2=Kaplan|first2=David B.|last3=Nelson|first3=Ann|journal=Physical Review Letters|volume=82|issue=25|date=21 June 1999|pages=4971–4974|doi=10.1103/PhysRevLett.82.4971|arxiv=hep-th/9803132|bibcode=1999PhRvL..82.4971C|s2cid=17203575}}</ref><ref>{{cite arXiv|title=Densities of States and the CKN Bound|author1=Nikita Blinov|author2=Patrick Draper|eprint=2107.03530|date=7 July 2021|class=hep-ph}}</ref>

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 ==