Editing Hierarchy problem (section)

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

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">\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.)

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>{{cite book |last=Martin |first=Stephen P. |title=Perspectives on Supersymmetry |year=1998 |isbn=978-981-02-3553-6 |series=Advanced Series on Directions in High Energy Physics |volume=18 |pages=1–98 |chapter=A Supersymmetry Primer |publisher=World Scientific |doi=10.1142/9789812839657_0001 |arxiv=hep-ph/9709356 |s2cid=118973381}}</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>{{cite journal |last1=Meissner |first1=K. |last2=Nicolai |first2=H. |year=2007 |title=Conformal Symmetry and the Standard Model |journal=[[Physics Letters]] |volume=B648 |issue=4 |pages=312–317 |arxiv=hep-th/0612165 |bibcode=2007PhLB..648..312M |doi=10.1016/j.physletb.2007.03.023 |s2cid=17973378}}</ref> and is currently under scrutiny. But if no further excitation is observed beyond the one seen so far at [[Large Hadron Collider|LHC]], this model would have to be abandoned.

==== 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=6 |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 |publisher=CERN}}</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>

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}
  \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
\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 ''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 ====
{{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">{{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.

==== UV/IR mixing ====
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>