The Minimal Supersymmetric Standard Model (MSSM) is an extension to the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
that realizes
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. MSSM is the minimal supersymmetrical model as it considers only "the
inimumnumber of new particle states and new interactions consistent with "Reality".
Supersymmetry pairs
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s with
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s, so every Standard Model particle has a superpartner yet undiscovered. If discovered, such superparticles could be candidates for
dark matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...
,
[
] and could provide evidence for
grand unification or the viability of
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
. The failure to find evidence for MSSM using the
Large Hadron Collider
The Large Hadron Collider (LHC) is the world's largest and highest-energy particle collider. It was built by the European Organization for Nuclear Research (CERN) between 1998 and 2008 in collaboration with over 10,000 scientists and hundred ...
has strengthened an inclination to abandon it.
Background
The MSSM was originally proposed in 1981 to stabilize the weak scale, solving the
hierarchy problem
In theoretical physics, 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 1024 times stronger than gravit ...
.
[
] 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 quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the Stand ...
mass of the Standard Model is unstable to quantum corrections and the theory predicts that weak scale should be much weaker than what is observed to be. In the MSSM, 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 quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the Stand ...
has a fermionic superpartner, the
Higgsino
In particle physics, for models with N=1 supersymmetry a higgsino, symbol , is the superpartner of the Higgs field. A higgsino is a Dirac fermionic field with spin and it refers to a weak isodoublet with hypercharge half under the Standard Mo ...
, that has the same mass as it would if supersymmetry were an exact symmetry. Because fermion masses are radiatively stable, the Higgs mass inherits this stability. However, in MSSM there is a need for more than one Higgs field, as described
below.
The only unambiguous way to claim discovery of supersymmetry is to produce superparticles in the laboratory. Because superparticles are expected to be 100 to 1000 times heavier than the proton, it requires a huge amount of energy to make these particles that can only be achieved at particle accelerators. The
Tevatron
The Tevatron was a circular particle accelerator (active until 2011) in the United States, at the Fermilab, Fermi National Accelerator Laboratory (also known as ''Fermilab''), east of Batavia, Illinois, and is the second highest energy particle ...
was actively looking for evidence of the production of supersymmetric particles before it was shut down on 30 September 2011. Most physicists believe that supersymmetry must be discovered at the
LHC if it is responsible for stabilizing the weak scale. There are five classes of particle that superpartners of the Standard Model fall into:
squarks,
gluino In supersymmetry, a gluino (symbol ) is the hypothetical supersymmetric partner of a gluon.
In supersymmetric theories, gluinos are Majorana fermions and interact via the strong force as a color octet. Gluinos have a lepton number 0, baryon number ...
s,
chargino
In particle physics, the chargino is a hypothetical particle which refers to the mass eigenstates of a charged superpartner, i.e. any new electrically charged fermion (with spin 1/2) predicted by supersymmetry. They are linear combinations of t ...
s,
neutralino
In supersymmetry, the neutralino is a hypothetical particle. In the Minimal Supersymmetric Standard Model (MSSM), a popular model of realization of supersymmetry at a low energy, there are four neutralinos that are fermions and are electrically ...
s, and
sleptons. These superparticles have their interactions and subsequent decays described by the MSSM and each has characteristic signatures.
The MSSM imposes
R-parity
R-parity is a concept in particle physics. In the Minimal Supersymmetric Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings in the theory. Since baryon number and lepton number conservatio ...
to explain the
stability of the proton. It adds supersymmetry breaking by introducing explicit
soft supersymmetry breaking In theoretical physics, soft SUSY breaking is type of supersymmetry breaking that does not cause ultraviolet divergences to appear in scalar masses.
Overview
These terms are relevant operators—i.e. operators whose coefficients have a positive ...
operators into the Lagrangian that is communicated to it by some unknown (and unspecified) dynamics. This means that there are 120 new parameters in the MSSM. Most of these parameters lead to unacceptable phenomenology such as large
flavor changing neutral current
In particle physics, flavor-changing neutral currents or flavour-changing neutral currents (FCNCs) are hypothetical interactions that change the flavor of a fermion without altering its electric charge.
Details
If they occur in nature (as refl ...
s or large
electric dipole moment
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The ...
s for the neutron and electron. To avoid these problems, the MSSM takes all of the soft supersymmetry breaking to be diagonal in flavor space and for all of the new
CP violating phases to vanish.
Theoretical motivations
There are three principal motivations for the MSSM over other theoretical extensions of the Standard Model, namely:
*
Naturalness
*
Gauge
Gauge ( or ) may refer to:
Measurement
* Gauge (instrument), any of a variety of measuring instruments
* Gauge (firearms)
* Wire gauge, a measure of the size of a wire
** American wire gauge, a common measure of nonferrous wire diameter, es ...
coupling unification
*
Dark Matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...
These motivations come out without much effort and they are the primary reasons why the MSSM is the leading candidate for a new theory to be discovered at collider experiments such as the
Tevatron
The Tevatron was a circular particle accelerator (active until 2011) in the United States, at the Fermilab, Fermi National Accelerator Laboratory (also known as ''Fermilab''), east of Batavia, Illinois, and is the second highest energy particle ...
or the
LHC.
Naturalness
The original motivation for proposing the MSSM was to stabilize the Higgs mass to radiative corrections that are quadratically divergent in the Standard Model (
hierarchy problem
In theoretical physics, 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 1024 times stronger than gravit ...
). In supersymmetric models, scalars are related to fermions and have the same mass. Since fermion masses are logarithmically divergent, scalar masses inherit the same radiative stability. The Higgs
vacuum expectation value (VEV) is related to the negative scalar mass in the Lagrangian. In order for the radiative corrections to the Higgs mass to not be dramatically larger than the actual value, the mass of the superpartners of the Standard Model should not be significantly heavier than the Higgs VEV – roughly 100 GeV. In 2012, the Higgs particle was discovered at the
LHC, and its mass was found to be 125–126 GeV.
Gauge-coupling unification
If the superpartners of the Standard Model are near the TeV scale, then measured gauge couplings of the three gauge groups unify at high energies.
[
][
][
] The
beta-function
In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory.
It is ...
s for the MSSM gauge couplings are given by
where
is measured in SU(5) normalization—a factor of
different
than the Standard Model's normalization and predicted by
Georgi–Glashow SU(5) .
The condition for gauge coupling unification at one loop is whether the following expression is satisfied
.
Remarkably, this is precisely satisfied to experimental errors in the values of
. There are two loop corrections and both TeV-scale and GUT-scale
threshold corrections that alter this condition on gauge coupling unification, and the results of more extensive calculations reveal that gauge coupling unification occurs to an accuracy of 1%, though this is about 3 standard deviations from the theoretical expectations.
This prediction is generally considered as indirect evidence for both the MSSM and SUSY
GUTs.
[Gordon Kane, "The Dawn of Physics Beyond the Standard Model", '']Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
'', June 2003, page 60 and ''The frontiers of physics'', special edition, Vol 15, #3, page 8 "Indirect evidence for supersymmetry comes from the extrapolation of interactions to high energies." Gauge coupling unification does not necessarily imply grand unification and there exist other mechanisms to reproduce gauge coupling unification. However, if superpartners are found in the near future, the apparent success of gauge coupling unification would suggest that a supersymmetric grand unified theory is a promising candidate for high scale physics.
Dark matter
If
R-parity
R-parity is a concept in particle physics. In the Minimal Supersymmetric Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings in the theory. Since baryon number and lepton number conservatio ...
is preserved, then the lightest superparticle (
LSP) of the MSSM is stable and is a
Weakly interacting massive particle
Weakly interacting massive particles (WIMPs) are hypothetical particles that are one of the proposed candidates for dark matter.
There exists no formal definition of a WIMP, but broadly, a WIMP is a new elementary particle which interacts via gra ...
(WIMP) – i.e. it does not have electromagnetic or strong interactions. This makes the LSP a good
dark matter
Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...
candidate, and falls into the category of
cold dark matter
In cosmology and physics, cold dark matter (CDM) is a hypothetical type of dark matter. According to the current standard model of cosmology, Lambda-CDM model, approximately 27% of the universe is dark matter and 68% is dark energy, with only a sm ...
(CDM).
Predictions of the MSSM regarding hadron colliders
The
Tevatron
The Tevatron was a circular particle accelerator (active until 2011) in the United States, at the Fermilab, Fermi National Accelerator Laboratory (also known as ''Fermilab''), east of Batavia, Illinois, and is the second highest energy particle ...
and
LHC have active experimental programs searching for supersymmetric particles. Since both of these machines are
hadron
In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the ele ...
colliders – proton antiproton for the Tevatron and proton proton for the LHC – they search best for strongly interacting particles. Therefore, most experimental signature involve production of
squarks or
gluino In supersymmetry, a gluino (symbol ) is the hypothetical supersymmetric partner of a gluon.
In supersymmetric theories, gluinos are Majorana fermions and interact via the strong force as a color octet. Gluinos have a lepton number 0, baryon number ...
s. Since the MSSM has
R-parity
R-parity is a concept in particle physics. In the Minimal Supersymmetric Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings in the theory. Since baryon number and lepton number conservatio ...
, the lightest supersymmetric particle is stable and after the squarks and gluinos decay each decay chain will contain one LSP that will leave the detector unseen. This leads to the generic prediction that the MSSM will produce a '
missing energy In experimental particle physics, missing energy refers to energy that is not detected in a particle detector, but is expected due to the laws of conservation of energy and conservation of momentum. Missing energy is carried by particles that do no ...
' signal from these particles leaving the detector.
Neutralinos
There are four
neutralino
In supersymmetry, the neutralino is a hypothetical particle. In the Minimal Supersymmetric Standard Model (MSSM), a popular model of realization of supersymmetry at a low energy, there are four neutralinos that are fermions and are electrically ...
s that are fermions and are electrically neutral, the lightest of which is typically stable. They are typically labeled , , , (although sometimes
is used instead). These four states are mixtures of the
Bino and the neutral
Wino
Wino may refer to:
* WINO (band), a Japanese rock band
* WINO (FM), a radio station (91.9 FM) licensed to serve Wakins Glen, New York, United States
* WRFI (FM), a radio station (89.7 FM) licensed to serve Odessa, New York, which held the call ...
(which are the neutral electroweak
Gaugino
In supersymmetry theories of particle physics, a gaugino is the hypothetical fermionic supersymmetric field quantum (superpartner) of a gauge field, as predicted by gauge theory combined with supersymmetry. All gauginos have spin 1/2, except for ...
s), and the neutral
Higgsinos
In particle physics, for models with N=1 supersymmetry a higgsino, symbol , is the superpartner of the Higgs field. A higgsino is a Dirac fermionic field with spin and it refers to a weak isodoublet with hypercharge half under the Standard Mo ...
. As the neutralinos are
Majorana fermions, each of them is identical with its
antiparticle
In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
. Because these particles only interact with the weak vector bosons, they are not directly produced at hadron colliders in copious numbers. They primarily appear as particles in cascade decays of heavier particles usually originating from colored supersymmetric particles such as squarks or gluinos.
In
R-parity
R-parity is a concept in particle physics. In the Minimal Supersymmetric Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings in the theory. Since baryon number and lepton number conservatio ...
conserving models, the lightest neutralino is stable and all supersymmetric cascade decays end up decaying into this particle which leaves the detector unseen and its existence can only be inferred by looking for unbalanced momentum in a detector.
The heavier neutralinos typically decay through a to a lighter neutralino or through a to chargino. Thus a typical decay is
:
Note that the “Missing energy” byproduct represents the mass-energy of the neutralino ( ) and in the second line, the mass-energy of a
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
-
antineutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is ...
pair ( + ) produced with the lepton and antilepton in the final decay, all of which are undetectable in individual reactions with current technology.
The
mass splittings between the different Neutralinos will dictate which patterns of decays are allowed.
Charginos
There are two
Chargino
In particle physics, the chargino is a hypothetical particle which refers to the mass eigenstates of a charged superpartner, i.e. any new electrically charged fermion (with spin 1/2) predicted by supersymmetry. They are linear combinations of t ...
s that are fermions and are electrically charged. They are typically labeled and (although sometimes
and
is used instead). The heavier chargino can decay through to the lighter chargino. Both can decay through a to neutralino.
Squarks
The
squarks are the scalar superpartners of the quarks and there is one version for each Standard Model quark. Due to phenomenological constraints from flavor changing neutral currents, typically the lighter two generations of squarks have to be nearly the same in mass and therefore are not given distinct names. The superpartners of the top and bottom quark can be split from the lighter squarks and are called ''stop'' and ''sbottom''.
In the other direction, there may be a remarkable left-right mixing of the stops
and of the sbottoms
because of the high masses of the partner quarks top and bottom:
:
:
A similar story holds for bottom
with its own parameters
and
.
Squarks can be produced through strong interactions and therefore are easily produced at hadron colliders. They decay to quarks and neutralinos or charginos which further decay. In R-parity conserving scenarios, squarks are pair produced and therefore a typical signal is
:
2 jets + missing energy
:
2 jets + 2 leptons + missing energy
Gluinos
Gluinos are
Majorana fermionic partners of the
gluon
A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
which means that they are their own antiparticles. They interact strongly and therefore can be produced significantly at the LHC. They can only decay to a quark and a squark and thus a typical gluino signal is
:
4 jets + Missing energy
Because gluinos are Majorana, gluinos can decay to either a quark+anti-squark or an anti-quark+squark with equal probability. Therefore, pairs of gluinos can decay to
:
4 jets+
+ Missing energy
This is a distinctive signature because it has same-sign di-leptons and has very little background in the Standard Model.
Sleptons
Sleptons are the scalar partners of the
lepton
In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neutr ...
s of the Standard Model. They are not strongly interacting and therefore are not produced very often at hadron colliders unless they are very light.
Because of the high mass of the tau lepton there will be left-right mixing of the stau similar to that of stop and sbottom (see above).
Sleptons will typically be found in decays of a charginos and neutralinos if they are light enough to be a decay product.
:
:
MSSM fields
Fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s have
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
ic superpartners (called sfermions), and bosons have fermionic superpartners (called
bosinos). For most of the Standard Model particles, doubling is very straightforward. However, for the Higgs boson, it is more complicated.
A single Higgsino (the fermionic superpartner of the Higgs boson) would lead to a
gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e. of a gauge theory.
All gauge anomal ...
and would cause the theory to be inconsistent. However, if two Higgsinos are added, there is no gauge anomaly. The simplest theory is one with two Higgsinos and therefore
two scalar Higgs doublets.
Another reason for having two scalar Higgs doublets rather than one is in order to have
Yukawa couplings between the Higgs and both
down-type quarks and
up-type quarks; these are the terms responsible for the quarks' masses. In the Standard Model the
down-type quarks couple to the Higgs field (which has Y=−) and the
up-type quarks to its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
(which has Y=+). However, in a supersymmetric theory this is not allowed, so two types of Higgs fields are needed.
MSSM superfields
In supersymmetric theories, every field and its superpartner can be written together as a
superfield
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply ...
. The superfield formulation of supersymmetry is very convenient to write down manifestly supersymmetric theories (i.e. one does not have to tediously check that the theory is supersymmetric term by term in the Lagrangian). The MSSM contains
vector superfield
In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion.
Gauge theory
A gauge theory is a mathematical framework for analysing gau ...
s associated with the Standard Model gauge groups which contain the vector bosons and associated gauginos. It also contains
chiral superfield
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply ...
s for the Standard Model fermions and Higgs bosons (and their respective superpartners).
MSSM Higgs mass
The MSSM Higgs mass is a prediction of the Minimal Supersymmetric Standard Model. The mass of the lightest Higgs boson is set by the Higgs ''
quartic coupling''. Quartic couplings are not soft supersymmetry-breaking parameters since they lead to a quadratic divergence of the Higgs mass. Furthermore, there are no supersymmetric parameters to make the Higgs mass a free parameter in the MSSM (though not in non-minimal extensions). This means that Higgs mass is a prediction of the MSSM. The
LEP II and the IV experiments placed a lower limit on the Higgs mass of 114.4
GeV GEV may refer to:
* ''G.E.V.'' (board game), a tabletop game by Steve Jackson Games
* Ashe County Airport, in North Carolina, United States
* Gällivare Lapland Airport, in Sweden
* Generalized extreme value distribution
* Gev Sella, Israeli-Sou ...
. This lower limit is significantly above where the MSSM would typically predict it to be but does not rule out the MSSM; the discovery of the Higgs with a mass of 125 GeV is within the maximal upper bound of approximately 130 GeV that loop corrections within the MSSM would raise the Higgs mass to. Proponents of the MSSM point out that a Higgs mass within the upper bound of the MSSM calculation of the Higgs mass is a successful prediction, albeit pointing to more fine tuning than expected.
Formulas
The only
susy-preserving operator that creates a quartic coupling for the Higgs in the MSSM arise for the
D-term
In theoretical physics, one often analyzes theories with supersymmetry in which D-terms play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coor ...
s of the
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
and
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
gauge sector and the magnitude of the quartic coupling is set by the size of the gauge couplings.
This leads to the prediction that the Standard Model-like Higgs mass (the scalar that couples approximately to the VEV) is limited to be less than the Z mass:
.
Since supersymmetry is broken, there are radiative corrections to the quartic coupling that can increase the Higgs mass. These dominantly arise from the 'top sector':
where
is the
top
A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect.
Once set in motion, a top will usually wobble for a few ...
mass and
is the mass of the top
squark. This result can be interpreted as the RG
running
Running is a method of terrestrial locomotion allowing humans and other animals to move rapidly on foot. Running is a type of gait characterized by an aerial phase in which all feet are above the ground (though there are exceptions). This is ...
of the Higgs quartic
coupling
A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mov ...
from the scale of supersymmetry to the top mass—however since the top squark mass should be relatively close to the top mass, this is usually a fairly modest contribution and increases the Higgs mass to roughly the LEP II bound of 114 GeV before the top squark becomes too heavy.
Finally there is a contribution from the top squark A-terms:
where
is a dimensionless number. This contributes an additional term to the Higgs mass at loop level, but is not logarithmically enhanced
by pushing
(known as 'maximal mixing') it is possible to push the Higgs mass to 125 GeV without decoupling the top squark or adding new dynamics to the MSSM.
As the Higgs was found at around 125 GeV (along with no other
superparticles) at the LHC, this strongly hints at new dynamics beyond the MSSM, such as the 'Next to Minimal Supersymmetric Standard Model' (
NMSSM); and suggests some correlation to the
little hierarchy problem.
The MSSM Lagrangian
The Lagrangian for the MSSM contains several pieces.
*The first is the Kähler potential for the matter and Higgs fields which produces the kinetic terms for the fields.
*The second piece is the gauge field superpotential that produces the kinetic terms for the gauge bosons and gauginos.
*The next term is the
superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
for the matter and Higgs fields. These produce the Yukawa couplings for the Standard Model fermions and also the mass term for the
Higgsinos
In particle physics, for models with N=1 supersymmetry a higgsino, symbol , is the superpartner of the Higgs field. A higgsino is a Dirac fermionic field with spin and it refers to a weak isodoublet with hypercharge half under the Standard Mo ...
. After imposing
R-parity
R-parity is a concept in particle physics. In the Minimal Supersymmetric Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings in the theory. Since baryon number and lepton number conservatio ...
, the
renormalizable
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similarity, self-similar geometric structures, that are used to treat infinity, infinities arising in calculated ...
,
gauge invariant
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
operators in the superpotential are
The constant term is unphysical in global supersymmetry (as opposed to
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
).
Soft Susy breaking
The last piece of the MSSM Lagrangian is the soft supersymmetry breaking Lagrangian. The vast majority of the parameters of the MSSM are in the susy breaking Lagrangian. The soft susy breaking are divided into roughly three pieces.
*The first are the gaugino masses
Where
are the gauginos and
is different for the wino, bino and gluino.
*The next are the soft masses for the scalar fields
where
are any of the scalars in the MSSM and
are
Hermitian matrices for the squarks and sleptons of a given set of gauge quantum numbers. The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of these matrices are actually the masses squared, rather than the masses.
*There are the
and
terms which are given by
The
terms are
complex matrices much as the scalar masses are.
*Although not often mentioned with regard to soft terms, to be consistent with observation, one must also include Gravitino and Goldstino soft masses given by
The reason these soft terms are not often mentioned are that they arise through local supersymmetry and not global supersymmetry, although they are required otherwise if the Goldstino were massless it would contradict observation. The Goldstino mode is eaten by the Gravitino to become massive, through a gauge shift, which also absorbs the would-be "mass" term of the Goldstino.
Problems with the MSSM
There are several problems with the MSSM — most of them falling into understanding the parameters.
* The
mu problem: The
Higgsino
In particle physics, for models with N=1 supersymmetry a higgsino, symbol , is the superpartner of the Higgs field. A higgsino is a Dirac fermionic field with spin and it refers to a weak isodoublet with hypercharge half under the Standard Mo ...
mass parameter μ appears as the following term in the
superpotential
In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...
: μH
uH
d. It should have the same order of magnitude as the
electroweak scale
In particle physics, the electroweak scale, also known as the Fermi scale, is the energy scale around 246 GeV, a typical energy of processes described by the electroweak theory. The particular number 246 GeV is taken to be the vacuum expectation ...
, many orders of magnitude smaller than that of the
Planck scale, which is the natural
cutoff scale. The soft supersymmetry breaking terms should also be of the same order of magnitude as the electroweak scale. This brings about a problem of
naturalness: why are these scales so much smaller than the cutoff scale yet happen to fall so close to each other?
* Flavor universality of soft masses and A-terms: since no
flavor mixing additional to that predicted by the
standard model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
has been discovered so far, the coefficients of the additional terms in the MSSM Lagrangian must be, at least approximately,
flavor
Flavor or flavour is either the sensory perception of taste or smell, or a flavoring in food that produces such perception.
Flavor or flavour may also refer to:
Science
*Flavors (programming language), an early object-oriented extension to Lis ...
invariant (i.e. the same for all flavors).
* Smallness of CP violating phases: since no
CP violation
In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be the ...
additional to that predicted by the
standard model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
has been discovered so far, the additional terms in the MSSM Lagrangian must be, at least approximately, CP invariant, so that their CP violating phases are small.
Theories of supersymmetry breaking
A large amount of theoretical effort has been spent trying to understand the mechanism for soft
supersymmetry breaking
In particle physics, supersymmetry breaking is the process to obtain a seemingly non-supersymmetric physics from a supersymmetric theory which is a necessary step to reconcile supersymmetry with actual experiments. It is an example of spontaneous s ...
that produces the desired properties in the superpartner masses and interactions. The three most extensively studied mechanisms are:
Gravity-mediated supersymmetry breaking
Gravity-mediated supersymmetry breaking is a method of communicating supersymmetry breaking to the supersymmetric Standard Model through gravitational interactions. It was the first method proposed to communicate supersymmetry breaking. In gravity-mediated supersymmetry-breaking models, there is a part of the theory that only interacts with the MSSM through gravitational interaction. This hidden sector of the theory breaks supersymmetry. Through the supersymmetric version of the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
, the
gravitino
In supergravity theories combining general relativity and supersymmetry, the gravitino () is the gauge fermion supersymmetric partner of the hypothesized graviton. It has been suggested as a candidate for dark matter.
If it exists, it is a f ...
, the supersymmetric version of the graviton, acquires a mass. After the gravitino has a mass, gravitational radiative corrections to soft masses are incompletely cancelled beneath the gravitino's mass.
It is currently believed that it is not generic to have a sector completely decoupled from the MSSM and there should be higher dimension operators that couple different sectors together with the higher dimension operators suppressed by the Planck scale. These operators give as large of a contribution to the soft supersymmetry breaking masses as the gravitational loops; therefore, today people usually consider gravity mediation to be gravitational sized direct interactions between the hidden sector and the MSSM.
mSUGRA stands for minimal supergravity. The construction of a realistic model of interactions within ''N'' = 1
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
framework where supersymmetry breaking is communicated through the supergravity interactions was carried out by
Ali Chamseddine
Ali H. Chamseddine ( ar, علي شمس الدين, link=no, born 20 February 1953) is a Lebanese physicist known for his contributions to particle physics, general relativity and mathematical physics. , Chamseddine is a physics Professor at t ...
,
Richard Arnowitt
Richard Lewis Arnowitt (May 3, 1928 – June 12, 2014) was an American physicist known for his contributions to theoretical particle physics and to general relativity.
Arnowitt was a Distinguished Professor (Emeritus) at Texas A&M University, whe ...
, and
Pran Nath in 1982.
[
] mSUGRA is one of the most widely investigated models of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
due to its predictive power requiring only 4 input parameters and a sign, to determine the low energy phenomenology from the scale of Grand Unification. The most widely used set of parameters is:
Gravity-Mediated Supersymmetry Breaking was assumed to be flavor universal because of the universality of gravity; however, in 1986 Hall, Kostelecky, and Raby showed that Planck-scale physics that are necessary to generate the Standard-Model Yukawa couplings spoil the universality of the supersymmetry breaking.
[
]
Gauge-mediated supersymmetry breaking (GMSB)
Gauge-mediated supersymmetry breaking is method of communicating supersymmetry breaking to the supersymmetric Standard Model through the Standard Model's gauge interactions. Typically a hidden sector breaks supersymmetry and communicates it to massive messenger fields that are charged under the Standard Model. These messenger fields induce a gaugino mass at one loop and then this is transmitted on to the scalar superpartners at two loops. Requiring stop squarks below 2 TeV, the maximum Higgs boson mass predicted is just 121.5GeV.
With the Higgs being discovered at 125GeV - this model requires stops above 2 TeV.
Anomaly-mediated supersymmetry breaking (AMSB)
Anomaly-mediated supersymmetry breaking is a special type of gravity mediated supersymmetry breaking that results in supersymmetry breaking being communicated to the supersymmetric Standard Model through the conformal anomaly.
[
][
] Requiring stop squarks below 2 TeV, the maximum Higgs boson mass predicted is just 121.0GeV.
With the Higgs being discovered at 125GeV - this scenario requires stops heavier than 2 TeV.
Phenomenological MSSM (pMSSM)
The unconstrained MSSM has more than 100 parameters in addition to the Standard Model parameters.
This makes any phenomenological analysis (e.g. finding regions in parameter space consistent
with observed data) impractical. Under the following three assumptions:
* no new source of CP-violation
* no Flavour Changing Neutral Currents
* first and second generation universality
one can reduce the number of additional parameters to the following 19 quantities of the phenomenological MSSM (pMSSM):
The large parameter space of pMSSM makes searches in pMSSM extremely challenging and makes pMSSM difficult to exclude.
Experimental tests
Terrestrial detectors
XENON1T
Xenon is a chemical element with the symbol Xe and atomic number 54. It is a dense, colorless, odorless noble gas found in Earth's atmosphere in trace amounts. Although generally unreactive, it can undergo a few chemical reactions such as the ...
(a dark matter WIMP detector - being commissioned in 2016) is expected to explore/test
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
candidates such as CMSSM.
See also
*
Desert (particle physics)
In the Grand Unified Theory of particle physics (GUT), the desert refers to a theorized gap in energy scales, between approximately the electroweak energy scale–conventionally defined as roughly the vacuum expectation value or VeV of the Higgs fi ...
References
External links
MSSM on arxiv.org *
Particle Data Group review of MSSM and search for MSSM predicted particles*
{{Standard model of physics
Supersymmetric quantum field theory
Physics beyond the Standard Model