Minimal Supersymmetric Standard Model

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The Minimal Supersymmetric Standard Model (MSSM) is the minimal extension to the Standard Model that realizes supersymmetry (non-minimal extensions) exist. Supersymmetry pairs bosons with fermions, therefore every Standard Model particle has a partner that has yet to be discovered. If these supersymmetric partners exist, it is likely that they will be observed at the Large Hadron Collider, which is planned to begin running in 2007. If the superparticles are found, it is analogous to discovering antimatter and depending on the details of what is found, it could provide evidence for grand unification and might even in principle provide hints as to how string theory describes nature.

The MSSM was originally proposed in 1981 to stabilize the weak scale, solving the hierarchy problem. The Higgs 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 has a fermionic superpartner, called the Higgsino, that would have the same mass as itself if supersymmetry was an exact symmetry. Because fermion masses are radiatively stable, the Higgs mass inherits this stability.

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. Currently the Tevatron is the highest energy particle collider and is actively looking for evidence of the production of supersymmetric particles. 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, gluinos, charginos, neutralinos, and sleptons. These superparticles have their interactions and subsequent decays described by the MSSM and each has characteristic signatures.

The MSSM imposes R-parity to explain the stability of the proton. It adds supersymmetry breaking by introducing explicit soft supersymmetry breaking operators into the Lagrangian that is communciated 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 unnacceptable phenomenology such as large flavor changing neutral currents or large electric dipole moments for the neutron and electron. To avoid these problems, the MSSM takes all of the soft susy breaking to be diagonal in flavor space and for all of the new CP violating phases to vanish.

[edit] Theoretical Motivations

There are three principle motivations for the MSSM over other theoretical extensions of the Standard Model, namely:

Naturalness
Gauge coupling unification
Dark Matter

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 or the LHC.

[edit] Naturalness

Cancellation of the Higgs boson quadratic mass renormalization between fermionic top quark loop and scalar top squark Feynman diagrams in a supersymmetric extension of the Standard Model

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 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 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. This mass scale is being probed currently at the Tevatron and will be more extensively explored at the LHC.

[edit] 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 functions for the MSSM gauge couplings are given by

Gauge Group	$\alpha^{-1}(M_{Z^0})$	$b_0^{MSSM}$
SU(3)	8.5	$- 3$
SU(2)	29.6	$+ 1$
U(1)	59.2	$+6\frac{3}{5}$

where $\alpha^{-1}_{1}$ is measured in SU(5) normalization -- a factor of $\frac{3}{5}$ different than the Standard Model's nomalization and predicted by Georgi-Glashow SU(5) .

The condition for gauge coupling unification at one loop is whether the following expression is satisfied $\frac{\alpha^{-1}_3 - \alpha^{-1}_2}{\alpha^{-1}_2-\alpha^{-1}_1} = \frac{b_{0\,3} - b_{0\,2}}{b_{0\,2} -b_{0\,1}}$ .

Remarkably, this is precisely satisfied to experimental errors. 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. It should be noted that 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.

[edit] Dark Matter

If R-parity is preserved, then the lightest superparticle (LSP) of the MSSM is stable and is a weakly interacting massive particle (WIMP) — i.e. it does not have electromagnetic or strong interactions. This makes the LSP a good dark matter candidate and falls into the category of cold dark matter (CDM) particle.

[edit] Discovery of the MSSM at Hadron Colliders

The Tevatron and LHC have active experimental programs searching for supersymmetric particles. Since both of these machines are hadron 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 gluinos. Since the MSSM has R-parity, 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' signal from these particles leaving the detector.

[edit] Neutralinos

There are four Neutralinos that are fermions and are electrically neutral, the lightest of which is typically stable. They are typically labeled $\tilde{N}_1^0, \ldots, \tilde{N}_4^0$ . These four states are mixtures of the Bino, neutral Wino, and neutral Higgsinos. 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 conserving models, the lightest neutralino is stable and all supersymmetric cascades 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 $Z 0$ to a lighter neutralino or through a $W^\pm$ to chargino. Thus a typical decay is

$\tilde{N}^0_2 \rightarrow \tilde{C}_1^\pm W^\mp \rightarrow \tilde{N}_1^0 W^\pm W^\mp \rightarrow$ Missing energy + $\ell^+\ell^-$
$\tilde{N}^0_2 \rightarrow \tilde{N}^0_1 Z^0\rightarrow$ Missing energy + $\ell^+ \ell^-$

The mass splittings betweent the different Neutralinos will dictate which patterns of decays are allowed.

[edit] Charginos

There are two Charginos that are fermions and are electrically charged. The heavier chargino can decay through $Z 0$ to the lighter chargino. Both can decay through a $W^\pm$ to neutralino.

[edit] 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 stops and sbottoms.

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. Squarks are typically pair produced and therefore a typical signal is

$\tilde{q}\tilde{\bar{q}} \rightarrow q \tilde{N}^0_1 \bar{q} \tilde{N}^0_1 \rightarrow$ 2 jets + Missing energy
$\tilde{q}\tilde{\bar{q}} \rightarrow q \tilde{N}^0_2 \bar{q} \tilde{N}^0_1 \rightarrow q \tilde{N}^0_1 \ell \bar{\ell} \bar{q} \tilde{N}^0_1 \rightarrow$ 2 jets + 2 leptons + Missing energy

[edit] Gluinos

Gluinos are Majorana fermionic partners of the gluon 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

$\tilde{g}\tilde{g}\rightarrow (q \tilde{\bar{q}}) (\bar{q} \tilde{q}) \rightarrow (q \bar{q} \tilde{N}^0_1) (\bar{q} q \tilde{N}^0_1) \rightarrow$ 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

$\tilde{g}\tilde{g}\rightarrow (\bar{q} \tilde{q}) (\bar{q} \tilde{q}) \rightarrow (q \bar{q} \tilde{C}^+_1) (q \bar{q} \tilde{C}^+_1) \rightarrow (q \bar{q} W^+) (q \bar{q} W^+) \rightarrow$ 4 jets+ $\ell^+ \ell^+$ + Missing energy

This is a distinctive signature because it has same-sign di-leptons and has very little background in the Standard Model.

[edit] Sleptons

Sleptons are the scalar partners of the leptons of the Standard Model. They are not strongly interacting and therefore are not produced very often at hadron colliders unless they are very light. They will typically be found in decays of a charginos and neutralinos if they are light enough to be a decay product

$\tilde{C}^+\rightarrow \tilde{\ell}^+ \nu$
$\tilde{N}^0 \rightarrow \tilde{\ell}^+ \ell^-$

[edit] MSSM Fields

Fermions have bosonic superpartners, and bosons have fermionic superpartners. For most of the Standard Model particles, doubling is very straight forward. However, for the Higgs boson, it is more complicated.

A single Higgsinos (the fermionic superpartner of the Higgs boson) would lead to a gauge anomaly and would cause the theory to be inconsistent. However if a pairs of Higgsinos are added, there is no gauge anomaly. The simplest theory is one with a single pair of Higgsinos and therefore a pair of scalar Higgs doublets. In addition to this previous argument, a pair of Higgs doublets (called the up-type Higgs and the down-type Higgs) is desired in order to have renormalizable Yukawa couplings between the Higgs and all the Standard Model fermions because the couplings have to be holomorphic.

SM Particle type	Particle	Symbol	Spin	R-Parity	Superpartner	Symbol	Spin	R-parity
Fermions	Quark	$q$	$\begin{matrix} \frac{1}{2} \end{matrix}$	+1	Squark	$\tilde{q}$	0	-1
	Lepton	$\ell$	$\begin{matrix} \frac{1}{2} \end{matrix}$	+1	Slepton	$\tilde{\ell}$	0	-1
Bosons	W	$W$	1	+1	Wino	$\tilde{W}$	$\begin{matrix} \frac{1}{2} \end{matrix}$	-1
	B	$B$	1	+1	Bino	$\tilde{B}$	$\begin{matrix} \frac{1}{2} \end{matrix}$	-1
	Gluon	$g$	1	+1	Gluino	$\tilde{g}$	$\begin{matrix} \frac{1}{2} \end{matrix}$	-1
Higgs bosons	Higgs	$h u, h d$	0	+1	Higgsinos	$\tilde{h}_u, \tilde{h}_d$	$\begin{matrix} \frac{1}{2} \end{matrix}$	-1

[edit] MSSM Superfields

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 superfields associated with the Standard Model gauge groups which contain the vector bosons and associated gauginos. It also contains chiral superfields for the Standard Model fermions and Higgs bosons (and their respective superpartners).

field	multiplicity	representation	Z₂-parity
Q	3	$(3,2)_{\frac{1}{6}}$	–
U^c	3	$(\bar{3},1)_{-\frac{2}{3}}$	–
D^c	3	$(\bar{3},1)_{\frac{1}{3}}$	–
L	3	$(1,2)_{-\frac{1}{2}}$	–
E^c	3	$(1,1) 1$	–
H_u	1	$(1,2)_{\frac{1}{2}}$	+
H_d	1	$(1,2)_{-\frac{1}{2}}$	+

[edit] The MSSM Lagrangian

The Lagrangian for the MSSM contains several pieces.

The first is the Kahler 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 for the matter and Higgs fields. These produce the Yukawa couplings for the Standard Model fermions and also the mass term for the Higgsinos. After imposing R-parity, the renormalizable, gauge invariant operators in the superpotential are

$W = μ H u H d + y u H u Q U c + y d H d Q D c + y l H d L E c$

The constant term is unphysical in global supersymmetry.

[edit] 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

$\mathcal{L} \supset m_{\frac{1}{2}} \tilde{\lambda}\tilde{\lambda} + h.c.$

Where $\tilde{\lambda}$ are the gauginos and $m_{\frac{1}{2}}$ is different for the wino, bino and gluino.

The next are the soft masses for the scalar fields

$\mathcal{L} \supset m_0 \phi^\dagger \phi$

where $φ$ are any of the scalars in the MSSM and $m 0$ are $3\times 3$ hermitean matrices for the squarks and sleptons of a given gauge quantum numbers.

Finally there are the $A$ and $B$ terms which are given by

$\mathcal{L} \supset B_{\mu} h_u h_d + A h_u \tilde{q} \tilde{u^c}+ A h_d \tilde{q} \tilde{d^c} +A h_d \tilde{l} \tilde{e^c} + h.c.$

The $A$ terms are $3\times 3$ complex matrices much as the scalar masses are.

[edit] The CMSSM

There is a particular ansatz for the soft supersymmetry breaking that is very popular in the literature known as the 'Constrained MSSM' (fomerly called mSugra). In this ansatz, all of the squark and slepton soft masses are assumed to be the same at the GUT scale and to not violate flavor. Similarly all of the A-terms are also taken to be flavor independent and universal at the GUT scale as well. Finally all of the gaugino masses are taken to the same at the GUT scale. With this ansatz, the parameters are RGE evolved to the TeV scale and masses and interactions of the particles are studied. The useful aspect of this parameterization of supersymmetry breaking is that it results in phenomologically acceptable parameters and only has 4 continuous parameters to vary and one sign. The down side is that no known theory of supersymmetry breaking is known to give this exact pattern of supersymmetry breaking.

[edit] Electroweak Symmetry Breaking

Electroweak symmetry is broken using the Higgs mechanism where a Higgs doublet acquires a vacuum expectation value (vev). The MSSM contains two Higgs doublet, $h u$ and $h d$ where the subscripts indicate whether the Higgs couples to up-type fermions or down-type fermions (down quarks and charged leptons):

$\mathcal{L} \supset y_u\, h_u q u^c + y_d\, h_d q d^c+ y_e\, h_d l e^c + h.c.$

In order to have all the Standard Model fermions acquire mass, both Higgs doublets must acquire a vev

$\langle h_u\rangle = v_u/\sqrt{2} \;\; \langle h_d\rangle= v_d/\sqrt{2}$

(we can use the freedom to rescale the Higgs superfields by a complex phase to ensure that the VEVs are positive real) Usually these are rewritten in terms of the effective electroweak vev and the ratio of the two vev

$v^2 = v_u^2 + v_d^2$

$\tan \beta = \frac{v_u}{v_d}$ .

[edit] The Higgs Mass

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 divergences to 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 Higgs boson was not found at LEP II and the four experiments placed a lower limit on the Higgs mass of 114.4 GeV. This lower limit is significantly above where the MSSM would typically predict it to be, and while it does not rule out the MSSM, the non-discovery of the Higgs makes proponents of the MSSM nervous. If the Higgs is found above 125 GeV (along with the other superparticles) at the LHC, then this will strongly hint at new dynamics beyond the MSSM such as the 'Next to Minimal Supersymmetric Standard Model' (NMSSM).

[edit] Neutralinos and Charginos

After electroweak symmetry breaking, the Higgsinos, Binos and Winos will mix with each other. The mass eigenstates are called Neutralinos and Charginos depending on whether the particles are electrically neutral or charged. Typically the lightest neutralino is the lightest supersymmetric particle and makes up the dark matter of the universe.

[edit] Problems with the MSSM

There are several problems with the MSSM — most of them falling into the understanding the parameters.

The mu Problem
Flavor universality of soft masses and A-terms
Smallness of CP violating phases

More recently physicists have become concerned about the non-discovery of the Higgs boson, or any superpartner at LEP II or the Tevatron.

[edit] Theories of Supersymmetry Breaking

A large amount of theoretical effort has been spent trying to understand the mechanism for soft supersymmetry breaking that produces the desired properties in the superpartner masses and interactions. The three most extensively studied mechanisms are

Gravity Mediated Supersymmetry Breaking
Gauge Mediated Supersymmetry Breaking (GMSB)
Anomaly Mediated Supersymmetry Breaking (AMSB)

[edit] External links

A Supersymmetry Primer by Stephen P. Martin
Particle Data Group review of MSSM and search for MSSM predicted particles

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Categories: Supersymmetry | Particle physics