Minimal super symmetrical standard model

The minimal supersymmetric standard model ( MSSM ) is the smallest possible choice with regard to the particle content to extend the existing standard model of elementary particle physics (SM) to a supersymmetric physics model.

In the extension of the SM to MSSM the field contents for another Higgs - doublet expanded and then each field / particle exactly one Super partner associated. Since the calibration symmetries remain unchanged compared to the SM, the calibration interactions in the MSSM are already determined by the SM for the newly emerging particles.

In addition, however, there can be a large number of interaction terms that do not originate from a calibration symmetry. The existence and strength of these terms is initially unknown, making the MSSM in the most general case a model with many new and unknown parameters .

Interactions

The MSSM has the same calibration interactions as the SM, so . Since, apart from the new Higgs doublet, whose charges are required, the charges of all fields are already known from the SM, the calibration interactions of all particles (including the new superpartners) are already determined by the SM. To be a realistic physics model, the MSSM must also break the electroweak symmetry group to electromagnetic symmetry . As in the SM, this happens spontaneously through non-vanishing vacuum expected values ​​of the Higgs fields. ${\displaystyle U(1)_{Y}\times SU(2)_{L}\times SU(3)_{C}}$ ${\displaystyle U(1)_{Y}\times SU(2)_{L}}$ ${\displaystyle U(1)\!\,_{Q}}$

In addition to the calibration interactions, other possible non-gauge interactions can occur, in particular terms that give the particles their mass. These terms initially have unknown factors, but all are permitted in the MSSM - with the exception of the R-parity violating terms (see below) - as long as they are physically meaningful in the sense of a renormalizable gauge theory. The new terms increase the number of independent parameters of the theory by more than one hundred (!) Compared to the SM. The MSSM is thus minimal in the sense that the number of particles occurring in the theory increases by the minimal possible, but maximal in the sense that many theoretically permitted new interactions are taken into account.

Masses of super partners

The super partners of the Standard Model particles initially have the same mass as the original particles. But since z. B. no boson electron was discovered, one assumes that the super partners have a much higher mass: ~ (1 TeV / c²); for comparison: the mass of the proton is 0.94 GeV / c². This leads to initial restrictions of the corresponding parameters in the interaction terms. ${\displaystyle \mathrm {O} }$

There are approaches to relate the masses of the particles occurring in the MSSM to one another. These are extended physics models ( large unified theory , superstring theories ) that behave like the MSSM in the limit case of low energies ("low" is to be seen here in relative terms, it should include at least the TeV / c² energy range).

R parity conservation

Some interaction processes with an uneven number of super partners allow the spontaneous decay of free protons . Since this process was not observed in various experimental searches, the associated parameters must be very small. Often processes with an odd number of super partners are simply forbidden by defining a new quantum number to be preserved . In this case one speaks of R-parity conservation .

The R parity is a discrete, multiplicative symmetry and is defined as

${\displaystyle \!\,P_{R}=(-1)^{3B+L+2s}}$

with B = baryon number , L = lepton number , s = particle spin .

The R parity is +1 for standard model particles (3B + L + 2s even) and −1 for supersymmetric particles (3B + L + 2s odd).

In models with R-parity conservation, the lightest supersymmetric particle (LSP) is stable. Since such a particle has not yet been observed, it can only be a weakly interacting , electrically neutral particle, which is why it is also seen as a possible candidate for dark matter .

Eigenstates

Interaction eigenstates

Interaction eigenstates
(yellow: extension of the SM to the MSSM)

Calibration group	SM-like fields	Super partner
${\displaystyle U(1)_{Y}}$	B ( ) ${\displaystyle B\!\,}$	Bino ( ) ${\displaystyle {\tilde {B}}}$
${\displaystyle SU(2)_{L}}$	W ( ) ${\displaystyle W^{0},W^{1},W^{2}\!\,}$	Winos ( ) ${\displaystyle {\tilde {W}}^{0},{\tilde {W}}^{1},{\tilde {W}}^{2}}$
${\displaystyle SU(3)_{c}}$	Gluons ( ) ${\displaystyle g\!\,}$	Gluinos ( ) ${\displaystyle {\tilde {g}}}$
-	Higgs fields ( ) ${\displaystyle H\!\,}$	Higgsinos ( ) ${\displaystyle {\tilde {H}}}$
-	Leptons ( ) ${\displaystyle e,\mu ,\tau ,\nu _{e},\nu _{\mu },\nu _{\tau }\!\,}$	Sleptonen ( ) ${\displaystyle {\tilde {e}},{\tilde {\mu }},{\tilde {\tau }},{\tilde {\nu }}_{e},{\tilde {\nu }}_{\mu },{\tilde {\nu }}_{\tau }}$
-	Quarks ( ) ${\displaystyle d,u,s,c,b,t\!\,}$	Squarks ( ) ${\displaystyle {\tilde {d}},{\tilde {u}},{\tilde {s}},{\tilde {c}},{\tilde {b}},{\tilde {t}}}$

The field content of the MSSM (the amount of the existing types of particles in interaction representation) results from the field content of the SM through the following steps:

• The set of fields is expanded by a Higgs doublet, which results in the existence of four further Higgs bosons ( see below mass eigenstates ). These additional fields are regarded as standard model- like because they have not yet been created by supersymmetry transformations.
• The calibration group (group of calibration interactions / local symmetries) remains unchanged.
• The global transformation of the spacetime components is extended by a set of supersymmetry transformations. This increases the number of fields in the model, since each SM-like field is assigned a super partner that initially only differs from the original field in terms of the spin .

Energy dependence of the coupling constants for as well as for and for ${\displaystyle \alpha _{1}}$ ${\displaystyle U(1)_{Y}}$ ${\displaystyle \alpha _{2}}$ ${\displaystyle SU(2)_{L}}$ ${\displaystyle \alpha _{3}}$ ${\displaystyle SU(3)_{C}}$

The following convention applies to the names of the numerous new fields : the fermions corresponding to the conventional bosons end with -ino (Wino, Bino, Gluino and Higgsino), the bosons corresponding to the fermions receive an S- (selectron to electron, corresponding to Squarks to Quarks).

The coupling constants of the three fundamental forces for high energies can be extrapolated on the basis of the MSSM. A standardization is   calculated for an energy of around 10 ¹⁶ GeV.

Mass eigenstates

Mass eigen-states

group	SM-like fields	Super partner
Neutrals	Photon ( ), Z ( ), neutral Higgs ( ) ${\displaystyle \gamma \!\,}$ ${\displaystyle Z^{0}\!\,}$ ${\displaystyle h^{0},\!\,H^{0},A^{0}}$	Neutralinos ( ) ${\displaystyle {\tilde {\chi }}_{1}^{0}\dots {\tilde {\chi }}_{4}^{0}}$
Invited	W ( ), loaded Higgs ( ) ${\displaystyle W^{\pm }}$ ${\displaystyle H^{\pm }}$	Charginos ( ) ${\displaystyle {\tilde {\chi }}_{1}^{\pm },\,{\tilde {\chi }}_{2}^{\pm }}$
${\displaystyle SU(3)}$	Gluons ( ) ${\displaystyle g\!\,}$	Gluinos ( ) ${\displaystyle {\tilde {g}}}$
Leptons	Electron ( ) ${\displaystyle e\!\,}$	Selectrons ( ) ${\displaystyle {\tilde {e}}_{L},{\tilde {e}}_{R}}$
	Muon ( ) ${\displaystyle \mu \!\,}$	Smyons ( ) ${\displaystyle {\tilde {\mu }}_{L},{\tilde {\mu }}_{R}}$
	Dew ( ) ${\displaystyle \tau \!\,}$	Traffic jam ( ) ${\displaystyle {\tilde {\tau }}_{L},{\tilde {\tau }}_{R}}$
	Neutrinos ( ) ${\displaystyle \nu _{e},\!\,\nu _{\mu },\nu _{\tau }}$	Sneutrinos ( ) ${\displaystyle {\tilde {\nu }}_{1L,R},{\tilde {\nu }}_{2L,R},{\tilde {\nu }}_{3L,R}}$
Quarks	Up, Down ( ) ${\displaystyle u,d\!\,}$	Sup, Sdown ( ) ${\displaystyle {\tilde {u}}_{L},{\tilde {u}}_{R},{\tilde {d}}_{L},{\tilde {d}}_{R}}$
	Charm, Strange ( ) ${\displaystyle c,s\!\,}$	Scharm, Sstrange ( ) ${\displaystyle {\tilde {c}}_{L},{\tilde {c}}_{R},{\tilde {s}}_{L},{\tilde {s}}_{R}}$
	Top, Bottom ( ) ${\displaystyle t,b\!\,}$	Stop, sbottom ( ) ${\displaystyle {\tilde {t}}_{L},{\tilde {t}}_{R},{\tilde {b}}_{L},{\tilde {b}}_{R}}$

The mass eigenstates ( observable momentum eigenstates) can - as already in the SM - be mixed from different interaction eigenstates. After breaking the electroweak symmetry, the mixing particles must have identical spin and identical electrical and color charges.

The concrete mixing ratios depend on the choice of the free parameters ; in particular, the mixing ratio for standard model-like and supersymmetrical particles can be different. Therefore, it is no longer possible to maintain the simple nomenclature described above . The resulting mass eigenstates are partially numbered in ascending order of mass.

• When the electroweak symmetry is broken, only three degrees of freedom of the Higgs fields are absorbed by the gauge bosons , as in the SM . This has the consequence that in the MSSM not only a single ( scalar ) Higgs particle remains as a mass eigenstate compared to the SM , but a total of five different ones:
  • a relatively light scalar Higgs particle that resembles the Higgs boson of the SM, ${\displaystyle h^{0}\!\,}$
  • a heavy scalar Higgs particle , ${\displaystyle H^{0}\!\,}$
  • a heavy pseudoscalar Higgs particle and ${\displaystyle A^{0}\!\,}$
  • a pair of charged Higgs particles and . ${\displaystyle H^{+}\!\,}$ ${\displaystyle H^{-}\!\,}$

• The electrically neutral Wino (partner des ), the bino (partner des ) and the two electrically neutral Higgsinos mix to form the neutralinos . ${\displaystyle W^{0}}$ ${\displaystyle B^{0}}$

• The electrically charged Winos (partner of the charged W-fields W ^± ) and the charged Higgsinos mix to form Charginos .

• The quarks there is also the mixture of their super affiliates, the Squarks , each of which has two Quark: a partner for the right-handed spinor - component and one for the left-handed.
  • Because of the low mass of the quarks of the first two generations (up / down, charm / strange), their super partner fields mix to form eigenstates of mass without a name.
  • The super partners of the heavy quarks top and bottom can be distinguished from them due to their large mass. But here there is a notable 'right-left mix' of right- and left-handed stops and sbottoms . So for the stop you have: ${\displaystyle {\tilde {t}}}$

${\displaystyle {\tilde {t}}_{1}=e^{+i\phi }\cos(\theta ){\tilde {t_{L}}}+\sin(\theta ){\tilde {t_{R}}}}$

${\displaystyle {\tilde {t}}_{2}=e^{-i\phi }\cos(\theta ){\tilde {t_{R}}}-\sin(\theta ){\tilde {t_{L}}}}$.

The same applies to the Sbottom with individual parameters and . ${\displaystyle {\tilde {b}}}$ ${\displaystyle \phi }$ ${\displaystyle \theta }$

• The super partners of the leptons are called sleptons . As with quarks, there are two scalar super partners for each lepton.
  The right-left mix also occurs with the super partner of the heaviest lepton dew , the traffic jam . The above relationship applies accordingly here. ${\displaystyle {\tilde {\tau }}}$

Desired experimental evidence

An important class of experiments for the search for supersymmetry are experiments at future particle accelerators , in particular at the Large Hadron Collider (LHC) of the European Nuclear Research Center ( CERN ). The most frequently examined supersymmetric model is the MSSM.

To obtain before the experiments information about what you hope to see the experiments are Monte Carlo event generators (z. B. Pythia ) ahead simulated . However, since it is practically impossible to examine the entire 105-dimensional space of the additional parameters of the MSSM, extended models with fewer free parameters are usually used, cf. Benchmark scenario . In order to be able to compare the simulations, it was agreed on certain parameter points (Snowmass Points and Slopes, SPS), which are each characteristic of certain parameter regions of the extended models and should thus represent the entire possible parameter space well.

Studies show that one should be able to detect supersymmetrical particles well if they exist in the mass range up to about 1 TeV / c² (as of 2006).

It is assumed (in most models) that the lightest supersymmetric particle (LSP) is stable and leaves the detector undetected. This would lead to the typical signal of a lack of energy perpendicular to the incoming particle beam (the amount of energy parallel to the particle beam often cannot be determined for technical reasons). A typical process is given in the Feynman diagram above.

Extension: NMSSM

Non -minimal supersymmetric standard model , English: Next-to-Minimal Supersymmetric Standard Model or Non-minimal supersymmetric SM

In order to eliminate certain difficulties of the MSSM (µ problem, see WP: Mu problem ), an additional chiral superfield singlet N with super partner Ñ is introduced. This is particularly necessary in GUT models. In addition to the physical Higgs bosons, a scalar (s ⁰ ) and a pseudoscalar (a ⁰ ) singlet are added.

In the MSSM, the lightest Higgs h ^{0 is} always SM-like, so its possibilities of generation and decay are essentially clear. In the NMSSM, however, the additional singlets s ⁰ and a ^{0 are also} candidates for the lightest Higgs particle. This means that the other Higgse could predominantly disintegrate into these states, which would dramatically change the whole appearance of the Higgse.

Web links

• Janusz Rosiek: Complete set of Feynman rules for the minimal supersymmetric extension of the standard model (Erratum) . arxiv : hep-ph / 9511250 . Feynman rules for calculating particle physical processes in the MSSM; Corrected version by J. Rosiek: Complete set of Feynman rules for the minimal supersymmetric extension of the standard model . In: Phys. Rev. D . tape 41 , p. 3464 , doi : 10.1103 / PhysRevD.41.3464 .

References and footnotes

1. ↑ This is identical to , since the difference of the exponents with 4L is even. In many models, BL is a preserved quantity, even if B and L themselves are not preserved. ${\displaystyle R_{P}=(-1)^{3(B-L)+2s}}$
2. ↑ In the SM, the down-like fermions get their mass from the Higgs field, the up-like fermions from the complex conjugated Higgs field. In the context of extensions such as SUSY, this second field can no longer be associated with the first in this way. Hence H ₁ ≡ H _d , H ₂ ≡ H _u . See Marc Hohlfeld, JGU Mainz: Search for final states with two leptons and missing transverse energy in p p collisions with a center of mass energy of 1.96 TeV , May 2004.
3. ↑ For a list of the mixtures in the MSSM see, for example, J. Rosiek: Complete Set of Feynman Rules for the MSSM - erratum. hep-ph / 9511250
4. ↑ A. Bartl. including: Impact of SUSY CP Phases on Stop and Sbottom Decays in the MSSM. Postscript , PDF , Lectures at DESY, 2003.
5. ↑ Allanach include: The Snowmass points and slopes: Benchmarks for SUSY searches . In: Eur. Phys. J. C . tape 25 , 2002, pp. 113–123 , doi : 10.1007 / s10052-002-0949-3 , arxiv : hep-ph / 0202233 .
6. ↑ Jörg Ziethe, RWTH Aachen: Theoretical studies on the production of (heavy) neutral Higgs bosons in hadron collisions (PDF; 1.4 MB), 2005 and theoretical studies on the production of heavy, neutral Higgs bosons in hadron collisions (PDF; 283 kB), 2004.
7. ↑ Fabian Franke, JMU Würzburg: Production and Decay of Neutralinos in the Non-Minimal Supersymmetric Standard Model (PDF; 4.7 MB), 1995.