In physics, naturalness is the property that
the dimensionless ratios between free parameters
or physical constants appearing in a physical
theory should take values "of order 1" and
that free parameters are not fine-tuned. That
is, a natural theory would have parameter
ratios with values like 2.34 rather than 234000
or 0.000234.
The requirement that satisfactory theories
should be "natural" in this sense is a current
of thought initiated around the 1960s in particle
physics. It is an aesthetic criterion, not
a physical one, that arises from the seeming
non-naturalness of the standard model and
the broader topics of the hierarchy problem,
fine-tuning, and the anthropic principle.
However it does tend to suggest a possible
area of weakness or future development for
current theories such as the Standard Model,
where some parameters vary by many orders
of magnitude, and which require extensive
"fine-tuning" of their current values of the
models concerned. The concern is that it is
not yet clear whether these seemingly exact
values we currently recognize, have arisen
by chance (based upon the anthropic principle
or similar) or whether they arise from a more
advanced theory not yet developed, in which
these turn out to be expected and well-explained,
because of other factors not yet part of particle
physics models.
The concept of naturalness is not always compatible
with Occam's razor, since many instances of
"natural" theories have more parameters than
"fine-tuned" theories such as the Standard
Model. Naturalness in physics is closely related
to the issue of fine-tuning, and over the
past decade many scientists argued that the
principle of naturalness is a specific application
of Bayesian statistics.
== 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
larger 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 some 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 asks such a question, it
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 what we do find.
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.
== Introduction ==
In particle physics, the assumption of naturalness
means that, unless a more detailed explanation
exists, all conceivable terms in the effective
action that preserve the required symmetries
should appear in this effective action with
natural coefficients.In an effective field
theory, Λ is the cutoff scale, an energy
or length scale at which the theory breaks
down. Due to dimensional analysis, natural
coefficients have the form
h
=
c
Λ
4
−
d
,
{\displaystyle h=c\Lambda ^{4-d},}
where d is the dimension of the field operator;
and c is a dimensionless number which should
be "random" and smaller than 1 at the scale
where the effective theory breaks down. Further
renormalization group running can reduce the
value of c at an energy scale E, but by a
small factor proportional to ln(E/Λ).
Some parameters in the effective action of
the Standard Model seem to have far smaller
coefficients than required by consistency
with the assumption of naturalness, leading
to some of the fundamental open questions
in physics. In particular:
The naturalness of the QCD "theta parameter"
leads to the strong CP problem, because it
is very small (experimentally consistent with
"zero") rather than of order of magnitude
unity.
The naturalness of the Higgs mass leads to
the hierarchy problem, because it is 17 orders
of magnitude smaller than the Planck Mass
that characterizes gravity. (Equivalently,
the Fermi Constant characterizing the strength
of the Weak Force is very very large compared
to the Gravitational Constant characterizing
the strength of gravity.)
The naturalness of the cosmological constant
leads to the cosmological constant problem
because it is at least 40 and perhaps as much
as 100 or more orders of magnitude smaller
than naively expected.In addition, the coupling
of the electron to the Higgs, the mass of
the electron, is abnormally small, and to
a lesser extent, the masses of the light quarks.
In models with large extra dimensions, the
assumption of naturalness is violated for
operators which multiply field operators that
create objects which are localized at different
positions in the extra dimensions.
== Naturalness and the gauge hierarchy problem
==
A more practical definition of naturalness
is that for any observable
O
{\displaystyle O}
which consists of n independent contributions
O
=
a
1
+
⋯
+
a
n
,
{\displaystyle O=a_{1}+\cdots +a_{n},}
then all *independent* contributions to
O
{\displaystyle O}
should be comparable to or less than
O
{\displaystyle O}
.
Otherwise, if one contribution, say
a
1
≫
O
{\displaystyle a_{1}\gg O}
, then some other independent contribution
would have to be fine-tuned to a large opposite-sign
value
such as to maintain
O
{\displaystyle O}
at its measured value. Such fine-tuning is
regarded as unnatural and indicative of some
missing ingredient in the theory.
For instance, in the Standard Model with Higgs
potential given by
V
=
−
μ
2
ϕ
†
ϕ
+
λ
(
ϕ
†
ϕ
)
2
{\displaystyle V=-\mu ^{2}\phi ^{\dagger }\phi
+\lambda (\phi ^{\dagger }\phi )^{2}}
the physical Higgs boson mass is calculated
to be
m
h
2
=
2
μ
2
+
δ
m
h
2
{\displaystyle m_{h}^{2}=2\mu ^{2}+\delta
m_{h}^{2}}
where the quadratically divergent radiative
correction is given by
δ
m
h
2
≃
3
4
π
2
(
−
λ
t
2
+
g
2
4
+
g
2
8
cos
2
⁡
θ
W
+
λ
)
Λ
2
{\displaystyle \delta m_{h}^{2}\simeq {\frac
{3}{4\pi ^{2}}}{\Bigl (}-\lambda _{t}^{2}+{\frac
{g^{2}}{4}}+{\frac {g^{2}}{8\cos ^{2}\theta
_{W}}}+\lambda {\Bigr )}\Lambda ^{2}}
where
λ
t
{\displaystyle \lambda _{t}}
is the top-quark Yukawa coupling,
g
{\displaystyle g}
is the SU(2) gauge coupling and
Λ
{\displaystyle \Lambda }
is the energy cut-off to the divergent loop
integrals. As
δ
m
h
2
{\displaystyle \delta m_{h}^{2}}
increases (depending on the chosen cut-off
Λ
{\displaystyle \Lambda }
), then
μ
2
{\displaystyle \mu ^{2}}
can be freely dialed so as
to maintain
m
h
{\displaystyle m_{h}}
at its measured value (now known to be
m
h
≃
125
{\displaystyle m_{h}\simeq 125}
GeV).
By insisting on naturalness, then
δ
m
h
2
<
m
h
2
{\displaystyle \delta m_{h}^{2}<m_{h}^{2}}
.
Solving for
Λ
{\displaystyle \Lambda }
, one finds
Λ
<
1
{\displaystyle \Lambda <1}
TeV.
This then implies that the Standard Model
as a natural effective field theory is only
valid up to the 1 TeV energy scale.
Sometimes it is complained that this argument
depends on the regularization scheme introducing
the cut-off
Λ
{\displaystyle \Lambda }
and perhaps the problem disappears under dimensional
regularization.
In this case, if new particles which couple
to the Higgs are introduced, one once again
regains the quadratic divergence now in terms
of the new particle squared masses.
For instance, if one includes see-saw neutrinos
into the Standard Model, then
δ
m
h
{\displaystyle \delta m_{h}}
would blow up to near the see-saw scale, typically
expected in the
10
13
{\displaystyle 10^{13}}
GeV range.
== Naturalness, supersymmetry and the little
hierarchy ==
By supersymmetrizing the Standard Model, one
arrives at a solution to the
gauge hierarchy, or big hierarchy, problem
in that supersymmetry guarantees
cancellation of quadratic divergences to all
orders in perturbation theory.
The simplest supersymmetrization of the SM
leads to the
Minimal Supersymmetric Standard Model or MSSM.
In the MSSM, each SM particle has a partner
particle known as a super-partner or
sparticle. For instance, the left- and right-electron
helicity components
have scalar partner selectrons
e
~
L
{\displaystyle {\tilde {e}}_{L}}
and
e
~
R
{\displaystyle {\tilde {e}}_{R}}
respectively whilst the eight colored gluons
have eight colored spin-1/2 gluino
superpartners. The MSSM Higgs sector must
necessarily be expanded to include two
rather than one doublets leading to five physical
Higgs particles
h
,
H
,
A
{\displaystyle h,\ H,A}
and
H
±
{\displaystyle H^{\pm }}
whilst three of the eight
Higgs component fields are absorbed by the
W
±
{\displaystyle W^{\pm }}
and
Z
{\displaystyle Z}
bosons to make them massive.
The MSSM is actually
supported by three different sets of measurements
which test for the presence of
virtual superpartners: 1. the celebrated weak
scale measurements of
the three gauge couplings strengths are just
what is needed for gauge coupling
unification at a scale
Q
≃
2
×
10
16
{\displaystyle Q\simeq 2\times 10^{16}}
GeV, 2. the value of
m
t
≃
173
{\displaystyle m_{t}\simeq 173}
GeV falls squarely in the range needed to
trigger a radiatively-driven
breakdown in electroweak symmetry and
3. the measured value of
m
h
≃
125
{\displaystyle m_{h}\simeq 125}
GeV falls within the narrow window of allowed
values for the MSSM.
Nonetheless, verification of weak scale SUSY
(WSS, SUSY with superpartner masses at or
around the weak scale as characterized by
m
(
W
,
Z
,
h
)
∼
100
{\displaystyle m(W,Z,h)\sim 100}
GeV) requires
the direct observation of at least some of
the superpartners at
sufficiently energetic colliding beam experiments.
As recent as 2017, the CERN Large Hadron Collider,
a
p
p
{\displaystyle pp}
collider operating at center-of-mass energy
13 TeV,
has not found any evidence for superpartners.
This has led to mass limits on the
gluino
m
g
~
>
2
{\displaystyle m_{\tilde {g}}>2}
TeV and on the lighter top squark
m
t
~
1
>
1
{\displaystyle m_{{\tilde {t}}_{1}}>1}
TeV (within the context of certain simplified
models
which are assumed to make the experimental
analysis more tractable).
Along with these limits, the rather large
measured value of
m
h
≃
125
{\displaystyle m_{h}\simeq 125}
GeV
seems to require TeV-scale highly mixed top
squarks.
These combined measurements have raised concern
now about an emerging Little Hierarchy
problem characterized by
m
W
,
Z
,
h
≪
m
s
p
a
r
t
i
c
l
e
{\displaystyle m_{W,Z,h}\ll m_{sparticle}}
.
Under the Little Hierarchy, one might expect
the now log-divergent light Higgs mass to
blow up to the sparticle mass scale unless
one fine-tunes. The Little Hierarchy
problem has led to concern that WSS is perhaps
not realized in nature, or at least not
in the manner typically expected by theorists
in years past.
== Status of naturalness and the little hierarchy
==
In the MSSM, the light Higgs mass is calculated
to be
m
h
2
=
μ
2
+
m
H
u
2
+
m
i
x
i
n
g
+
l
o
o
p
s
{\displaystyle m_{h}^{2}=\mu ^{2}+m_{H_{u}}^{2}+{\rm
{mixing}}+{\rm {loops}}}
where the mixing and loop contributions are
<
m
h
2
{\displaystyle <m_{h}^{2}}
but where in most
models, the soft SUSY breaking up-Higgs mass
m
H
u
2
{\displaystyle m_{H_{u}}^{2}}
is driven to large,
TeV-scale negative values (in order to break
electroweak symmetry). Then, to maintain
the measured value of
m
h
=
125
{\displaystyle m_{h}=125}
GeV, one must tune the superpotential
mass term
μ
2
{\displaystyle \mu ^{2}}
to some large positive value.
Alternatively, for natural SUSY, one may expect
that
m
H
u
2
{\displaystyle m_{H_{u}}^{2}}
runs to small negative values
in which case both
μ
{\displaystyle \mu }
and
|
m
H
u
|
{\displaystyle |m_{H_{u}}|}
are of order 100-200 GeV.
This already leads to a prediction: since
μ
{\displaystyle \mu }
is supersymmetric and feeds mass to both SM
particles (W,Z,h)
and superpartners (higgsinos), then it is
expected
from the natural MSSM that light higgsinos
exist nearby to the 100-200 GeV scale.
This simple realization has profound implications
for WSS collider
and dark matter searches.
Naturalness in the MSSM has historically been
expressed in terms of the
Z
{\displaystyle Z}
boson mass, and indeed this approach leads
to more stringent upper bounds
on sparticle masses. By minimizing the (Coleman-Weinberg)
scalar potential of the
MSSM, then one may relate the measured value
of
m
Z
=
91.2
{\displaystyle m_{Z}=91.2}
GeV to the
SUSY Lagrangian parameters:
m
Z
2
2
=
(
m
H
d
2
+
Σ
d
d
(
i
)
)
−
tan
2
⁡
β
(
m
H
u
2
+
Σ
u
u
(
j
)
)
tan
2
⁡
β
−
1
−
μ
2
≃
−
m
H
u
2
−
Σ
u
u
(
i
)
−
μ
2
{\displaystyle {\frac {m_{Z}^{2}}{2}}={\frac
{(m_{H_{d}}^{2}+\Sigma _{d}^{d}(i))-\tan ^{2}\beta
(m_{H_{u}}^{2}+\Sigma _{u}^{u}(j))}{\tan ^{2}\beta
-1}}-\mu ^{2}\simeq -m_{H_{u}}^{2}-\Sigma
_{u}^{u}(i)-\mu ^{2}}
Here,
tan
⁡
β
∼
5
−
50
{\displaystyle \tan \beta \sim 5-50}
is 
the ratio of Higgs field vacuum expectation
values
v
u
/
v
d
{\displaystyle v_{u}/v_{d}}
and
m
H
d
2
{\displaystyle m_{H_{d}}^{2}}
is the down-Higgs soft breaking
mass term. The
Σ
d
d
(
i
)
{\displaystyle \Sigma _{d}^{d}(i)}
and
Σ
u
u
(
j
)
{\displaystyle \Sigma _{u}^{u}(j)}
contain
a variety of loop corrections labelled by
indices i and j, the most important of
which typically comes from the top-squarks.
In the renowned review work of P. Nilles,
titled "Supersymmetry, Supergravity and Particle
Physics", published on Phys.Rept. 110 (1984)
1-162, one finds the sentence "Experiments
within the next five to ten years will enable
us to decide whether supersymmetry as a solution
of the naturalness problem of the weak interaction
scale is a myth or a reality".
== See also ==
Fine tuning
Hierarchy problem
Large extra dimensions
Split supersymmetry
