In theoretical physics, the hierarchy problem
is the large discrepancy between aspects of
the weak force and gravity. There is no scientific
consensus on why, for example, the weak force
is 1024 times stronger than gravity.
== Technical definition ==
A hierarchy problem occurs when the fundamental
value of some physical parameter, such as
a coupling constant 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,
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 and over the past decade many
scientists 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 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 ==
A simple example:
Suppose a physics model requires four parameters
which allow it to produce a very high quality
working model, calculations, and predictions
of some aspect of our physical universe. Suppose
we find through experiments that the parameters
have values:
1.2
1.31
0.9 and
404,331,557,902,116,024,553,602,703,216.58
(roughly 4 x 1029).We might wonder how such
figures arise. But in particular we 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 x 1029 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 answer given by physicists is the anthropic
principle. If the universe came to exist by
chance, and perhaps vast numbers of other
universes exist or have existed, then life
capable of physics experiments only arose
in universes that by chance had very balanced
forces. All the universes where the forces
were not balanced, didn't develop life capable
of the question. So if a lifeform 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. So when we look, that is what we
would expect to find, and if done properly,
it is acquired.
A second answer is that perhaps there is a
deeper understanding of physics, which, if
we discovered and understood it, would make
clear these aren't really fundamental parameters
and there is a good reason why they have the
exact values we have found, because they all
derive from other more fundamental parameters
that are not so unbalanced.
== Examples in particle physics ==
=== The Higgs mass ===
In particle physics, the most important hierarchy
problem is the question that asks why the
weak force is 1024 times as strong as gravity.
Both of these forces involve constants of
nature, Fermi's constant for the weak force
and Newton's constant for gravity. Furthermore,
if the Standard Model 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 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 cancellation
between the quadratic radiative corrections
and the bare mass.
It should be remarked that 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.
One proposed solution, popular amongst many
physicists, is 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
Barbieri–Giudice criterion. This still leaves
open the mu problem, however. Currently the
tenets of supersymmetry are being tested at
the LHC, although no evidence has been found
so far for supersymmetry.
=== Theoretical solutions ===
==== Supersymmetric solution ====
Each particle that couples to the Higgs field
has a Yukawa coupling λf. The coupling with
the Higgs field for fermions gives an interaction
term
L
Y
u
k
a
w
a
=
−
λ
f
ψ
¯
H
ψ
{\displaystyle {\mathcal {L}}_{\mathrm {Yukawa}
}=-\lambda _{f}{\bar {\psi }}H\psi }
, with
ψ
{\displaystyle \psi }
being the Dirac field and
H
{\displaystyle H}
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 rules, one gets the quantum corrections
to the Higgs mass squared from a fermion to
be:
Δ
m
H
2
=
−
|
λ
f
|
2
8
π
2
[
Λ
U
V
2
+
.
.
.
]
.
{\displaystyle \Delta m_{H}^{2}=-{\frac {\left|\lambda
_{f}\right|^{2}}{8\pi ^{2}}}[\Lambda _{\mathrm
{UV} }^{2}+...].}
The
Λ
U
V
{\displaystyle \Lambda _{\mathrm {UV} }}
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:
λS= |λf|2 (the couplings to the Higgs are
exactly the same).
Then by the Feynman rules, the correction
(from both scalars) is:
Δ
m
H
2
=
2
×
λ
S
16
π
2
[
Λ
U
V
2
+
.
.
.
]
.
{\displaystyle \Delta m_{H}^{2}=2\times {\frac
{\lambda _{S}}{16\pi ^{2}}}[\Lambda _{\mathrm
{UV} }^{2}+...].}
(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.
==== Conformal solution ====
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 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.
==== Solution via extra dimensions ====
If we live in a 3+1 dimensional world, then
we calculate the Gravitational Force via Gauss'
law for gravity:
g
(
r
)
=
−
G
m
e
r
r
2
{\displaystyle \mathbf {g} (\mathbf {r} )=-Gm{\frac
{\mathbf {e_{r}} }{r^{2}}}}
(1)which is simply Newton's law of gravitation.
Note that Newton's constant G can be rewritten
in terms of the Planck mass.
G
=
ℏ
c
M
P
l
2
{\displaystyle G={\frac {\hbar c}{M_{\mathrm
{Pl} }^{2}}}}
If we extend this idea to
δ
{\displaystyle \delta }
extra dimensions, then we get:
g
(
r
)
=
−
m
e
r
M
P
l
3
+
1
+
δ
2
+
δ
r
2
+
δ
{\displaystyle \mathbf {g} (\mathbf {r} )=-m{\frac
{\mathbf {e_{r}} }{M_{\mathrm {Pl} _{3+1+\delta
}}^{2+\delta }r^{2+\delta }}}}
(2)where
M
P
l
3
+
1
+
δ
{\displaystyle M_{\mathrm {Pl} _{3+1+\delta
}}}
is the 3+1+
δ
{\displaystyle \delta }
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
n
δ
{\displaystyle n^{\delta }}
because this is the flux in the extra dimensions.
The formula is:
g
(
r
)
=
−
m
e
r
M
P
l
3
+
1
+
δ
2
+
δ
r
2
n
δ
{\displaystyle \mathbf {g} (\mathbf {r} )=-m{\frac
{\mathbf {e_{r}} }{M_{\mathrm {Pl} _{3+1+\delta
}}^{2+\delta }r^{2}n^{\delta }}}}
−
m
e
r
M
P
l
2
r
2
=
−
m
e
r
M
P
l
3
+
1
+
δ
2
+
δ
r
2
n
δ
{\displaystyle -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 }}}}
which gives:
1
M
P
l
2
r
2
=
1
M
P
l
3
+
1
+
δ
2
+
δ
r
2
n
δ
⇒
{\displaystyle {\frac {1}{M_{\mathrm {Pl}
}^{2}r^{2}}}={\frac {1}{M_{\mathrm {Pl} _{3+1+\delta
}}^{2+\delta }r^{2}n^{\delta }}}\Rightarrow
}
M
P
l
2
=
M
P
l
3
+
1
+
δ
2
+
δ
n
δ
.
{\displaystyle M_{\mathrm {Pl} }^{2}=M_{\mathrm
{Pl} _{3+1+\delta }}^{2+\delta }n^{\delta
}.}
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 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 relative to the other forces. This
theory requires that the fields of the Standard
Model are confined to a four-dimensional membrane,
while gravity propagates in several additional
spatial dimensions that are large compared
to the Planck scale.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. It
was also shown that four-dimensionality of
the Universe is the result of 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 scenarios which
offered their solution to the hierarchy problem.
===== Finite Groups =====
It has also been noted that the group order
of the Baby Monster group is of the right
order of magnitude, 4×1033. It is known that
the Monster Group is related to the symmetries
of a particular bosonic string theory on the
Leech lattice. However, there's no physical
reason for why the size of the Monster Group
or its subgroups should appear in the Lagrangian.
Most physicists think this is merely a coincidence.
Another coincidence is that in reduced Planck
units, the Higgs mass is approximately
48.
|
M
|
−
1
/
3
=
125.5
G
e
V
{\displaystyle 48.|M|^{-1/3}=125.5\;\mathrm
{GeV} }
where |M| is the order of the Monster group.
This suggests that the smallness of the Higgs
mass may be due to a redundancy caused by
a symmetry of the extra dimensions, which
must be divided out. There are other groups
that are also of the right order of magnitude
for example
Weyl
(
E
8
×
E
8
)
{\displaystyle {\text{Weyl}}(E_{8}\times E_{8})}
.
===== Extra dimensions =====
Until now, 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. However, extra
dimensions could explain why the gravity force
is so weak, and why the expansion of the universe
is faster than expected.
=== The cosmological constant ===
In physical cosmology, current observations
in favor of an accelerating universe imply
the existence of a tiny, but nonzero cosmological
constant. This 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. It is complicated,
however, by the necessary involvement of general
relativity in the problem and may be a clue
that we do not understand gravity on long
distance scales (such as the size of the universe
today). While quintessence has been proposed
as an explanation of the acceleration of the
Universe, it does not actually address the
cosmological constant hierarchy problem in
the technical sense of addressing the large
quantum corrections. Supersymmetry does not
address the cosmological constant problem,
since supersymmetry cancels the M4Planck contribution,
but not the M2Planck one (quadratically diverging).
== See also ==
CP violation
Quantum triviality
