In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the
weak force is 10
24 times stronger than
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
.
Technical definition
A hierarchy problem occurs when the fundamental value of some physical parameter, such as a
coupling constant
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...
or a mass, in some
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
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
, 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 problems and problems of naturalness.
Throughout the 2010s, many scientists argued that the hierarchy problem is a specific application of
Bayesian statistics
Bayesian statistics ( or ) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about ...
.
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
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
, 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
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 . 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 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
Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, the differences between some actual parameters are much larger than this, so the question is noteworthy.
One explanation given by philosophers is the
anthropic principle
In cosmology, the anthropic principle, also known as the observation selection effect, is the proposition that the range of possible observations that could be made about the universe is limited by the fact that observations are only possible in ...
. 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 beings are aware and capable of asking such a question, humans must have arisen in a universe having balanced forces, however rare that might be.
A second possible answer is that there is a deeper understanding of physics that we currently do not possess. There may be parameters from which we can derive physical constants that have fewer unbalanced values, or there may be a model with fewer parameters.
Examples in particle physics
Higgs mass
In
particle physics
Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, the most important hierarchy problem is the question that asks why the
weak force is 10
24 times as strong as
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. Both of these forces involve constants of nature, the
Fermi constant for the weak force and the
Newtonian constant of gravitation
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's theory of general relativity. It is also known as t ...
for gravity. Furthermore, if the
Standard Model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
is used to calculate the quantum corrections to Fermi's constant, it appears that Fermi's constant is surprisingly large and is expected to 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.

More technically, the question is why the
Higgs boson
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field,
one of the field (physics), fields in particl ...
is so much lighter than the
Planck mass (or the
grand unification energy The grand unification energy \Lambda_, or the GUT scale, is the energy level above which, it is believed, the electromagnetism, electromagnetic force, weak interaction, weak force, and strong interaction, strong force become equal in strength and un ...
, 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.
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 is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
. 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
Barbieri–
Giudice criterion. This still leaves open the
mu problem, however. The tenets of supersymmetry are being tested at the
LHC, although no evidence has been found so far for supersymmetry.
Each particle that couples to the Higgs field has an associated
Yukawa coupling . The coupling with the Higgs field for fermions gives an interaction term
, with
being the
Dirac field
In quantum field theory, a fermionic field is a quantum field whose Quantum, quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relation ...
and
the
Higgs field
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field,
one of the field (physics), fields in particl ...
. 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
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
, one gets the quantum corrections to the Higgs mass squared from a fermion to be:
The
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
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
is an extension of this that creates 'superpartners' for all Standard Model particles.
Conformal
Without supersymmetry, a solution to the hierarchy problem has been proposed using just the
Standard Model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
. 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
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 and is currently under scrutiny. But if no further excitation is observed beyond the one seen so far at
LHC, this model would have to be abandoned.
Extra dimensions
No experimental or observational evidence of
extra dimensions
In physics, extra dimensions or extra-dimensional spaces are proposed as additional space or time dimensions beyond the (3 + 1) typical of observed spacetime — meaning 5-dimensional or higher. such as the first attempts based on the K ...
has been officially reported. Analyses of results from the
Large Hadron Collider
The Large Hadron Collider (LHC) is the world's largest and highest-energy particle accelerator. It was built by the CERN, European Organization for Nuclear Research (CERN) between 1998 and 2008, in collaboration with over 10,000 scientists, ...
severely constrain theories with
large extra dimensions.
However, extra dimensions could explain why the gravity force is so weak, and why the expansion of the universe is faster than expected.
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
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the s ...
. Note that Newton's constant can be rewritten in terms of the
Planck mass.
If we extend this idea to extra dimensions, then we get:
where
is the -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 than normal dimensions. If we let , then we get (2). However, if we let , then we get our usual Newton's law. However, when , 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 because this is the flux in the extra dimensions. The formula is:
which gives:
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.
Braneworld models
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
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
relative to the other forces.
This theory requires that the fields of the
Standard Model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
are confined to a four-dimensional
membrane
A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. Bi ...
, while gravity propagates in several additional spatial dimensions that are large compared to the
Planck scale
In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: '' c'', '' G'', '' ħ'', and ''k''B (described further below). Expressing one of ...
.
In 1998–99
Merab Gogberashvili published on
arXiv
arXiv (pronounced as "archive"—the X represents the Chi (letter), Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly pee ...
(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
A synonym is a word, morpheme, or phrase that means precisely or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are a ...
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
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
and Universe thickness, and thus to solve the hierarchy problem. It was also shown that four-dimensionality of the Universe is the result of a
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Exponential stability
** Linear stability
**Lyapunov stability
** Marginal s ...
requirement since the extra component of the
Einstein field equations
In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
giving the localized solution for
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
fields coincides with one of the conditions of stability.
Subsequently, there were proposed the closely related
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 quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
could resolve the hierarchy problem. In 2021, another group of researchers showed that UV/IR mixing could resolve the hierarchy problem in string theory.
Cosmological constant
In
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
, current observations in favor of an
accelerating universe imply the existence of a tiny, but nonzero
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
. 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 its calculation is complicated by the necessary involvement of
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
in the problem. Proposed solutions to the cosmological constant problem include modifying and/or extending gravity,
[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.] adding matter with unvanishing pressure,
and UV/IR mixing in the Standard Model and gravity.
Some physicists have resorted to
anthropic reasoning to solve the cosmological constant problem, but it is disputed whether such anthropic reasoning is scientific.
[ See also thi]
news story.
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See also
* CP violation
* Quantum triviality
* Weak gravity conjecture
References
{{DEFAULTSORT:Hierarchy Problem
Standard Model
Physics beyond the Standard Model
Unsolved problems in physics