# All Possible Lightest Supersymmetric Particles in R-Parity Violating
Minimal Supergravity Models ^{1}^{1}1Preprint: BONN-TH-2008-14

###### Abstract

We investigate, which lightest supersymmetric particles can be obtained via a non-vanishing lepton- or baryon-number violating operator at the grand unification scale within the R-parity violating minimal supergravity model. We employ the full one-loop renormalization group equations. We take into account restrictions from the anomalous magnetic moment of the muon and , as well as collider constraints from LEP and the Tevatron. We also consider simple deformations of minimal supergravity models.

###### keywords:

Renormalization group, mSUGRA, R-parity violation, Lightest supersymmetric particle###### Pacs:

11.10.Hi, 04.65.+e, 12.60.Jv, 14.80.Ly## 1 Introduction

In the minimal supersymmetric Standard Model (MSSM) Martin:1997ns , the lightest supersymmetric particle (LSP) is stable, guaranteed by the discrete symmetry proton hexality, Dreiner:2005rd or R-parity Barbier:2004ez . This also ensures the stability of the proton. For cosmological reasons the LSP must then be the lightest neutralino Ellis:1983ew ; Hebbeker:1999pi and it has been widely studied as a very promising cold dark matter candidate Blumenthal:1984bp . However, if we drop P then there are further renormalizable interaction operators in the superpotential Allanach:2003eb

(1) | |||||

In order to ensure the stability of the proton, we must prohibit
either the first three set of terms which violate
lepton-number, or the last set of terms which violate
baryon-number.^{2}^{2}2See also Refs. Lee:2007fw ; Lee:2007qx
for a solution to the proton decay problem with R-parity
violation. These terms violate P () and thus
the LSP is no longer stable. It is then also not restricted to be the
lightest neutralino and can in principle be any supersymmetric (SUSY)
particle
Barbier:2004ez

(2) |

Here we have the lightest neutralino and chargino (), a left-/right-handed charged slepton (), a sneutrino (), a left-/right-handed squark (), and a gluino (). We have separately listed the lightest stau , sbottom , and stop , as they have possibly large Yukawa couplings and left-right mixing and are thus promising LSP candidates. Potential other dark matter candidates are the axino Chun:1999cq , the gravitino Buchmuller:2007ui and the lightest U-parity particle Lee:2007fw ; Lee:2007qx ; Lee:2008pc .

In the search for supersymmetry at colliders, it is essential to know the nature of the LSP, because SUSY particles, if produced, normally cascade decay down to the LSP within the detector. The LSP then decays promptly or with a detached vertex if is violated. It is thus a central ingredient of almost all SUSY signatures.

In Eq. (2), we have a bewildering array of potential LSPs. We thus need a guiding principle in order to perform a systematic phenomenological analysis. A well motivated restricted framework for detailed studies of the MSSM is minimal supergravity (mSUGRA) Martin:1997ns . The 124 free MSSM parameters are reduced to only five,

(3) |

where () is the universal supersymmetry breaking scalar (gaugino) mass and is the universal supersymmetry breaking trilinear scalar interaction; all given at the grand unification (GUT) scale: . is the ratio of the two vacuum expectation values and is the Higgs mixing parameter. We obtain the masses of the SUSY particles (sparticles) by running the renormalization group equations (RGEs) for the SUSY parameters, from to the electroweak scale using SOFTSUSY rpv_softsusy . In most of the mSUGRA parameter space the or the , is the LSP Allanach:2003eb ; Allanach:2006st ; Ibanez:1984vq .

In Ref. Allanach:2003eb the effects were taken into account in the RGEs: giving the mSUGRA model. Here, one additional coupling beyond Eq. (3) is assumed:

(4) |

We thus have a simple well-motivated framework, in which we can systematically investigate the nature of the LSP. It is the purpose of this letter to determine all possible LSPs in the mSUGRA model. We also briefly discuss simple deformations of mSUGRA. This is very important for SUSY searches at the LHC.

## 2 Non- LSP parameter space of mSUGRA models

If a sparticle directly couples to the operator corresponding
to , the dominant contributions to the RGE of the
running sparticle mass are
Allanach:2003eb ; Snu_LSP :^{3}^{3}3For third
generation sparticles we also need to take into account the
contributions from the Higgs-Yukawa interactions. Their effect is
similar to and in
Eq. (5), see Ref. Allanach:2003eb .

(5) | |||||

(6) |

Here (), , are the gauge couplings (soft breaking gaugino masses). with the renormalization scale and , , are constants of . and are linear functions of products of two softbreaking scalar masses and are given explicitly in Refs. Allanach:2003eb ; Snu_LSP .

coupling | LSP | coupling | LSP |
---|---|---|---|

,, | |||

The sum of the first two P-conserving terms in Eq. (5) is negative and thus increases when running from to . In contrast, the last two terms proportional to and , are always positive and therefore decrease . We thus expect new LSP candidates beyond , and if these latter terms contribute substantially. As we shall see below, this is the case if , i.e. Snu_LSP . We can strengthen the (negative) contribution of , by choosing a negative with a large magnitude; for moderate positive there is a cancellation in the RGE evolution of Snu_LSP . The other terms are not significantly affected by . Note that we also need () large (small) enough to avoid a () LSP.

We now investigate the non- LSP parameter space of mSUGRA, i.e. with one non-vanishing coupling . We discuss the obtainable LSPs for the various -operators. We shall focus here on the and operators. In previous work in Refs. Allanach:2003eb ; Allanach:2006st ; Snu_LSP it was shown that can lead to a LSP. In general operators can only lead to a LSP beyond a or a , in mSUGRA. We thus refer to Ref. Snu_LSP for a discussion of the LSP via . We also note that Ref. Allanach:2003eb ; Jack:2005id gives the example of an LSP via .

In searching for LSP candidates, we take into account the constraints from the decay Barberio:2008fa , as well as the anomalous magnetic moment of the muon Stockinger:2007pe . In the figures, we show (dashed yellow) contour lines corresponding to the window for the BR Barberio:2008fa ,

(7) |

and (solid green) contour lines corresponding to the window for the SUSY contributions to the anomalous magnetic moment of the muon Stockinger:2007pe

(8) |

See also Ref. Snu_LSP and references therein. We employ the LEP exclusion bound on the light Higgs mass Barate:2003sz , but we reduce it by 3 GeV to GeV, to account for numerical uncertainties of SOFTSUSY Allanach:2006st ; Allanach:2003jw . We use microOMEGAs1.3.7 Belanger:2001fz to calculate BR and .

Our results are summarized in Table 1 and are explained in the following. We only consider operators for which is consistent with existing experimental bounds Barbier:2004ez ; Allanach:1999ic . We argue that Table 1 gives a complete list of all possible non- and non- LSP candidates in mSUGRA. In Sect. 3 below, we shall also consider simple deformations of mSUGRA.

### 2.1 Non- LSPs via LLE

The least constrained couplings of the operators, Eq. (1), are Barbier:2004ez ; Allanach:2003eb ; Allanach:1999ic

(9) |

where the bounds apply at . Note, that is reduced by roughly a factor of 1.5 when running from to Allanach:1999ic . There is no scaling factor for the first bound as it is derived from the neutrino mass bound via the RGEs Allanach:2003eb .

We give in Fig. 1 the LSP region in the – plane for a -coupling. We show the mass difference, , between the NLSP and LSP. We have employed a lower bound of 190 GeV on the mass to fulfill the strong bound on . The remaining SUSY particles are then so heavy within mSUGRA, that other collider constraints from LEP and the Tevatron are automatically fulfilled.

We see that the LSP exists in an extended region of mSUGRA. We find a LSP for all GeV, because increases the mass of the (bino-like) faster than the mass of the Snu_LSP . The complete LSP region in Fig. 1 agrees with BR() at . But only a tiny region is consistent with at , i.e. lies above the solid green line. The mass spectra are rather heavy and thus is suppressed.

If we use , or instead of in our parameter scans, we obtain a as the LSP. We can not obtain a as the LSP in mSUGRA with . On the one hand, the -conserving contributions to the RGEs of have a larger magnitude compared to those for . On the other hand, the (negative) contributions to are smaller in magnitude compared to those for Allanach:2003eb .

### 2.2 Non- LSPs via UDD

The following baryon-number violating couplings, , are only constrained by perturbativity Barbier:2004ez ; Allanach:1999ic ; bounds_UDD

(10) |

The corresponding operators only affect SU(2) singlet squarks directly, we can thus only obtain LSPs via these couplings.

We assume that the weak- and mass-eigenstates of right-handed quarks are the same Agashe:1995qm . With this assumption we avoid the RGE generation of additional couplings at out of , which might be in contradiction with experiment Barbier:2004ez ; Allanach:1999ic . We have also checked that there are then no new contributions (at one-loop) in the RGEs which generate off-diagonal squark mass matrix elements. The right-handed squark weak-eigenstates are therefore approximately equal to their mass eigenstates at . We thus avoid large flavour changing neutral currents. Note that we only have experimental information about mixing in the left-handed quark sector (CKM matrix).

We show in Fig. 2 the LSP region via in the
– plane. The and are
degenerate in mass, because both sparticles interact the same via the
gauge interactions and via Allanach:2003eb . We
conservatively impose a lower bound of 380 GeV on the
mass, consistent with the non-observation of
the in resonance searches in the dijet
channel at the Tevatron CDFnote .^{4}^{4}4It is not clear if
Ref. CDFnote can exclude
GeV. They did not search for single squark resonances. A more detailed
analysis is required, including NLO corrections to single
production.

We can not get a LSP via . The contributions to the RGEs of the
, and mass are the same. But
the couples stronger to the U(1) gaugino than the
and and is therefore always heavier than
and .^{5}^{5}5The can in principle be lighter than
the and if GeV due to
different D-term contributions. However, the LSP
parameter space is in that case excluded by constraints from LEP
Heister:2002jc ; Barate:2003sz .
For example, the in Fig. 2 is roughly 60 GeV
heavier than the .

Due to GeV, we need , as can be seen in Fig. 2, to obtain also a heavy . This results in such a heavy mass spectrum that lies beyond the experimental window. However the complete LSP region in Fig. 2 is consistent with BR() at .

Only small intervals are allowed in Fig. 2, because at increases very rapidly with increasing . The dependence on is weaker, i.e. intervals up to 100 GeV (for constant ) are allowed in Fig. 2. These are general features of most of the squark LSP regions. We thus concentrate on and in what follows. is important, because increasing increases [decreases] [BR()], cf. Ref. Allanach:2006st .

We give in Fig. 3 the LSP region via in the – plane. The LSP mass lies between 77 GeV and 180 GeV. The lower value corresponds to the strongest LEP bound Heister:2002jc . Note, that there is no bound on the LSP mass from Tevatron searches. The single production cross section via lies below the exclusion limits for a dijet resonance, cf. Ref. CDFnote , due to the small incoming parton luminosity.

Most of the LSP region in Fig. 3 is also consistent with BR (below the upper dashed yellow line) and (above the lower solid green line) at the level. We observe that is vital to obtain a LSP. Increasing , i.e. decreasing the magnitude, reduces the (negative) effect of on the running of the mass and we re-obtain the or LSP, cf. Ref Snu_LSP .

We can also obtain a LSP, if we use . But now there might be additional constraints from the Tevatron on di-jet resonances CDFnote . The couplings and unlike allow for single production via a valence quark or antiquark, which enhances the hadronic cross section. Note, that these three couplings can only lead to a LSP, because the mass (compared to the masses of the first two generations) is further reduced by the large bottom Yukawa coupling and by larger left-right mixing.

For , we obtain a LSP
as shown in Fig. 4 for the –
plane. The LSP mass ranges from 94 GeV to 200 GeV. The
lower bound corresponds to the LEP bound on
Heister:2002jc .^{6}^{6}6Unlike the LSP, the LSP has a
large left-handed component due to left-right mixing. As a
conservative approach, we take the (stronger) mass bounds from
Ref. Heister:2002jc for purely left-handed up-type squarks.
The LSP region below the upper dashed yellow line
[above the solid green line] is also consistent with BR() [] at .

We need in general a smaller coupling to obtain a LSP than LSP, because the mass is further reduced by the large top Yukawa coupling. This effect is enhanced by a negative with a large magnitude. also leads to large left-right mixing, which further reduces the mass. For the same reasons we can not obtain another squark LSP than the via .

The complete LSP region in Fig. 4 should be testable at the Tevatron Choudhury:2005dg . (See also Ref. Dreiner:1991dt .) The authors found that masses up to 190 GeV (210 GeV) can probably be explored at the Tevatron for an integrated luminosity of 2 (8 ). However, this analysis has not yet been performed by the Tevatron collaborations.

## 3 LSP candidates in simple deformations of mSUGRA

Up to now, we have considered the restricted framework of mSUGRA. In this section, we want to briefly comment on how the nature of the LSP can change when we relax some of the mSUGRA boundary conditions, Eqs. (3) and (4). Throughout this section, we assume that , Eq. (4), is . terms then have no significant impact on the RGE running of the sparticle masses, cf. Eq. (5). We thus effectively explore the various corners of deformed, -conserved mSUGRA parameter space. This has hitherto not been done, since it leads to cosmologically not viable LSPs in the case. The additional effects on the low-energy mass spectrum of the -operators for larger have been discussed in the previous sections, and apply here correspondingly. In Table 2, we give examples for the scenarios which we now discuss.

scenario | universal masses | non-universal masses | LSP |
---|---|---|---|

1 | GeV, GeV, | GeV | |

GeV, , sgn()=+1 | |||

2 | GeV, GeV, | GeV | |

GeV, , sgn()=+1 | |||

3 | GeV, GeV, | GeV | |

GeV, , sgn()=+1 | |||

4 | GeV, GeV, | GeV, | |

GeV, , sgn()=+1 | GeV | ||

5 | GeV, GeV, | GeV, | |

GeV, , sgn()=+1 | GeV | ||

6 | GeV, GeV, | GeV, | |

GeV, , sgn()=+1 | GeV | ||

7 | GeV, GeV, | GeV | |

GeV, , sgn()=+1 |

First, we consider non-universal gaugino masses, i.e. at . Here, , , and are the masses of the bino, the winos, and the gluinos, respectively. For at , we can obtain a LSP if the scalar sparticles and the are heavy enough, i.e. if , , and and are large enough. As an example, we present scenario 1 in Table 2. If we reduce but maintain , we get a as the LSP, see scenario 2. The squarks are relatively light for small and . The mass is further reduced by the effect of the large top Yukawa coupling on the running and due to left-right mixing. For at , we can get a wino-like LSP (instead of a bino-like as in mSUGRA) which is nearly degenerate in mass with the . A LSP is in principle possible but difficult to obtain, see Ref. Kribs:2008hq for details.

Next, we consider non-universal sfermion masses at . For small right-handed slepton soft breaking masses, , we can obtain a or a LSP (beyond the LSP in mSUGRA) if is large enough, scenario 3. For small left-handed slepton soft breaking masses, , and , we can get a LSP. We show the example of a LSP scenario in Table 2 (scenario 4). A LSP is not possible, because it is always heavier than the due to the different D-terms Snu_LSP . is vital for the to be the LSP, because increases faster with then the mass with . Finally, squark LSPs are possible for small and non-universal squark soft breaking parameters and . In these scenarios a and LSP is preferred, because their masses are additionally reduced by large Yukawa couplings affecting the RGE running and by left-right mixing; see scenario 5 in Table 2. We obtain non- and non- squark-LSPs, if we assume non-universal masses for different squark flavours, cf. scenario 6 with a LSP.

Choosing soft breaking Higgs mass parameters different from the universal scalar mass has the following impact on the sparticle mass spectrum. On the one hand, the RGE running of third generation masses is affected due to terms proportional to the Higgs-Yukawa couplings. We then obtain, for example, a LSP, cf. scenario 7 in Tab. 2. On the other hand, the Higgs mixing parameter and the physical Higgs masses can be changed. If is small we can get a Higgsino-like LSP. Note that depends on the Higgs soft breaking masses via radiative electroweak symmetry breaking Ibanez:1982fr .

The mass spectra of the scenarios described above can significantly change if a large coupling is present at , i.e. . The masses are then modified according to the discussion in the previous sections. For example, if we assume in scenario 2 in Tab. 2 an additional coupling , we obtain a scenario with a LSP, cf. Sect. 2.1, and a as the NLSP.

## 4 Conclusion

We have investigated for the first time all possible non- and non- LSPs in R-parity violating mSUGRA models; see Table 1. We found that a non-vanishing operator at the GUT scale can lead to a () or () LSP; cf. Fig. 1. A non-vanishing operator can lead to a LSP; cf. Ref. Snu_LSP . We can also obtain squark LSPs, namely the , , and via a non-vanishing operator; see Fig. 2, Fig. 3 and Fig. 4, respectively. We found , , and LSP scenarios consistent with the observed anomalous magnetic moment of the muon. All LSP candidates found here can be consistent with as well as with collider constraints from LEP and the Tevatron. According to Ref. Choudhury:2005dg , LSPs up to a mass of 190 GeV can be tested at the Tevatron with 2 of data. We therefore want to encourage the Tevatron collaborations to investigate the LSP parameter space of R-parity violating mSUGRA, as well as to look for squark LSP resonances in dijet events.

We have also discussed simple deformations of mSUGRA; see Tab. 2 for explicit examples. We have first assumed that the R-pariy violating coupling at the GUT scale is small, i.e. . We have found scenarios with a and LSP if . We can obtain a LSP for small right-handed slepton (soft breaking) masses. A LSP is possible for small left-handed slepton masses as long as . These scenarios will be significantly affected by R-parity violating terms in the RGEs when as described in Sect. 2.

Due to the simplicity of the framework, we have in this first study restricted ourselves mainly to the mSUGRA case. It would be interesting to extend this work to other supersymmetry breaking models such as gauge mediation Giudice:1998bp or anomaly mediation anomaly_break .

## Acknowledgments

We thank Benjamin Allanach for help with the version of SOFTSUSY and Volker Büscher for helpful discussions on the Tevatron searches. SG thanks the ‘Deutsche Telekom Stiftung’ and the ‘BCGS of Physics and Astronomy’ for financial support. The work of HD was supported by the SFB TR-33 ‘The Dark Universe’.

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