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The Gauge Hierarchy Problem
---------------------------
The gauge hierarchy problem is perhaps the main motivation
for supersymmetry. To understand what the problem is, it
is necessary to look more closely at the mass of the Higgs
boson.
Consider the following 1-loop Feynman diagrams:
(a) Represents the bare Higgs mass.
(b) Represents the Higgs SELF INTERACTION.
(c) Represents the Yukawa interaction.
The corresponding Lagrangian is:
_{ } _
L = ∂_{μ}φ∂^{μ}φ - μ^{2}φ^{2}/2 + λφ^{4}/4 - g_{Y}ψψφ
Case (b). Applying Feynman rules we get:
I = λ∫d^{4}p/(2π)^{4} 1/(p^{2} - m_{H}^{2})
Where i/(p^{2} - m_{H}^{2}) is the massive bosonic propagator.
From the discussion on Regularization and Renormalization
we had (using dimensional regularization):
I = (iλΔ/16π^{2})[2/ε - ln(μ^{2}/Δ) + ln(4π) - γ_{E} + 1]
__
= (iλm_{H}^{2}/16π^{2})(1 + ln(μ^{2}/m_{H}^{2}) after MS
Which is quadratically divergent.
Likewise, case (c) is also calculated in the notes on
Regularization and Renormalization. The 1 loop Yukawa
correction to the Higgs mass is given by:
Π(k) = -4g^{2}(-i)[1/8π^{2}(k^{2}/6 + m^{2}) - 3/16π^{2}(ln(μ^{2}/Δ))]
Which, again, is quadratically divergent.
Aside: We can schematically see this as follows:
Π = -g_{Y}^{2}∫ d^{4}k/(2π)^{4} (γ^{μ}k + m)^{2}/(k^{2} - m^{2})^{2}
(γ^{μ}k + m)^{2} = k^{2} + 2km + m^{2}
Therefore,
I = -g_{Y}^{2}∫ d^{4}k/(2π)^{4} (k^{2} + 2km + m^{2})/(k^{2} - m^{2})^{2}
= -g_{Y}^{2}∫ d^{4}k/(2π)^{4} (k^{2} + m^{2})/(k^{2} - m^{2})^{2}
Write the numerator as k^{2} - m^{2} + 2m^{2} and the integral
becomes:
I = -g_{Y}^{2}∫ d^{4}k/(2π)^{4} 1/(k^{2} - m^{2}) + 2m/(k^{2} - m^{2})^{2}
Note: There is a -ve in front of the g^{2} term because it
represents a fermionic loop. The sign difference between
fermionic and bosonic loops can be seen in the following
diagram.
Bosons Fermions
---------- ----------
A B AB A B AB
- - -- - - --
+ + + + - -
- - + - + -
Therefore, bosonic loops make a positive contribution
to the vacuum energy while fermionic loops make a
negative contribution. The diagram also illustrates
the SPIN-STATISTICS THEOREM. If A and B swap positions
there is no change in sign for the bosons but there is a
change for the fermions. Thus,
φ(x_{1})φ(x_{2}) = φ(x_{2})φ(x_{1})
While,
ψ(x_{1})ψ(x_{2}) = -ψ(x_{2})ψ(x_{1})
The Problem
-----------
We can write the physical Higgs mass with loop corrections
(neglecting all factors of π etc.) as:
m_{H}^{2}/2 = m_{0}^{2}/2
+ λm_{H}^{2}(1 + log terms)
- g_{Y}^{2}(p^{2} + 2m_{H}^{2} + log terms)
The log terms are of lesser importance because they are
very nuch smaller than the squared terms. Again, we can
write the above schematically as:
m_{H}^{2}/2 ~ m_{0}^{2}/2 + λm_{H}^{2} - g_{Y}^{2}m_{H}^{2}
Unfortunately, these quadratically divergent terms
cannot be removed by renormalization. They are
considered to be unphysical or 'unnatural'. As a
result, one would expect that these large quantum
corrections to m_{H}^{2} would make the Higgs mass huge as
m_{H} approaches the scale at which new physics appears
(the Planck mass). In fact, unless there is an incredible
amount of fine tuning cancellation between the quadratic
terms and the bare mass, the 1-loop correction to m_{H}^{2}
would be more than 30 orders of magnitude larger than
the measured m_{H} of a few hundred GeV (~(10^{19})^{2}/(100)^{2}).
To compound issues, there are many, many combinations
of possible Feynman diagrams leading to additional
corrections to the mass, some positive and some negative
with different powers of the coupling constant. For
example, every fermion in the SM has a Yukawa coupling
to the Higgs field and will quadratically diverge. In
addition there are contributions from the gauge bosons.
All of these terms would need to collectively cancel
each other to achieve the physical mass!
The Mass of the Higgs
---------------------
The Higgs field is described by:
V(φ) = -μ^{2}φ^{2}/2 + λφ^{4}/4
This has a minimum (VEV) at dV(φ)/dφ = 0. Therefore,
-μ^{2}φ = λφ^{3}
or φ_{MIN} = f = μ/√λ
We know that f ~ 246 GeV and μ_{H} ~ 125 GeV. Therefore,
λ < 1.
Supersymmetry to the Rescue
---------------------------
What if the Higgs had a fermionic superpartner, the Higsino,
~
H, that couples to the field with a coupling constant, λ_{S}.
This would be represented by (b) in the following diagram:
The cancellations would be exact if the mass and charge
of the partner/superpartner pair were the same and
λ_{S} = λ_{Y}^{2}.
This is exactly the idea behind supersymmetric theories.
However, because they have yet to be observed in the
laboratory, superpartners must be heavier than their
partners. The fact that there is not a perfect symmetry
implies that supersymmetry is not an exact, unbroken
symmetry. The implication is that the true symmetry is
somehow spontaneously broken.
The difference in masses jeopardizes the solution to
the hierarchy problem because exact cancellation is
spoiled - but not totally. The prediction is that
the mass difference is small enough to still provide
a high degree of cancellation.
R-Parity
--------
R-parity is a set of rule that define possible partner/
superpartner interactions. If we define 'normal' SM
particles, N as + and supersymmetric particles, S as -1,
the rules are:
N -> NN
N -> SS
S -> NS
For 2-body particle transitions the rules are:
NN -> NN
NN -> SS
SS -> SS
NS -> NS
Therefore, R parity is a multiplicative quantum number.
For example
~ ~~
H -> H -> H is not allowed while H -> FF -> H or
~~
H -> HH -> is allowed.
(a), (b), (c) and (d) are valid. However, a diagram like
the one below would not be.
Dark Matter
-----------
As we said before, superpartners can only be pair produced
from SM particles and a superpartner can only produce a
another superpartner and a normal partner. Thus,
S
/
/
N ------
\
\
S
and,
S
/
/
S ------
\
\
N
Because of their large mass, superpartners decay and are
not freely floating around in the Universe. Superpartners
will decay in a chain until the lightest superpartner (LSP)
is produced (cascade decay). Thus,
LSP
/
/
------ N
/
.
.
S /
/
------ N
S /
/
S ------
\
\
N
The resulting LSP) is thought to be stable and electrically
neutral. If it were charged the theory would be cosmologically
disfavored. In addition, LSPs must interact weakly with
ordinary matter and are not seen in detectors since they
interact only by exchange of heavy virtual particles. These
are exactly the characteristics required for Dark Matter,
thought to make up most of the matter in the universe and to
hold galaxies together.
Fermionic and Gauge Boson Mass
------------------------------
The Hierachy problem does not impact fermions or the gauge
bosons. In both cases the self energy varies logarithmically
and are not quadratically divergent. (see the discussion on
regularization and renormalizationfor the fermionic self
energy calculation). More fundamentally, for fermions it is
the chiral symmetry that protects the mass. The Lagarangian
has a symmetry as the fermionic masses goes to 0. This
guarantees that all corrections to the mass must also vanish
as the mass goes to 0. We can see this from the discussion
on regularization and renormalization. There we showed that
the fermionic self energy correction to the mass is given by
(schematically):
δm ~ αmln(μ^{2}/Δ) for 1 loop.
δm is very small. For example, for μ^{2}/m^{2} = 10^{30} the log = 69
Now, α = 1/137 ∴ δm ~ (1/137)* 69 ~ 0.5*m and as m -> 0 δm -> 0
For the gauge bosons, it is the gauge invariance that protects
the mass. A mass term for a gauge field is m^{2}A_{μ}A^{μ}. However,
such terms are not invariant and break the local symmetry.
Therefore, m^{2} must be zero.
The Vacuum Catastrphe
---------------------
While Supersymmetry offers a possible solution to the Gauge
Hierarchy problem. It does not explain the vacuum catastrophe
problem that is discussed in another note.
The Fine Tuning Problem
-----------------------
The problem with the Higgs mass being so finely tuned is
part of a wider question as why the Universe in generai
is so finely tuned. The current standard model of particle
physics has 25 freely adjustable parameters. In addition
gravity has the cosmological constant. A small change in
several of the dimensionless physical constants would make
the Universe radically different from what we observe today.
The famous scientist Stephen Hawking has stated, "The laws of
science, as we know them at present, contain many fundamental
numbers, like the size of the electric charge of the electron
and the ratio of the masses of the proton and the electron.
The remarkable fact is that the values of these numbers seem
to have been very finely adjusted to make possible the
development of life." If any of several fundamental constants
were only slightly different, the Universe would be unlikely
to be conducive to the establishment and development of matter,
astronomical structures, elemental diversity, or life as we
know it. For example, if the coupling constant for the strong
force was 2% larger, while the other constants were unchanged,
the physics of stars would be drastically different.
Possible explanations for fine-tuning are the subject
of vigorous discussions among philosophers, scientists,
theologians, and proponents of creationism.