^{+}e

^{-}Linear Collider

_{1}

^{*}

In many models stability of Dark Matter particles D is ensured by conservation of a new quantum number referred to as D -parity. Our models also contain
charged D -odd particles *D*^{±} with the same spin as D. (For more information,please refer to the PDF.)

In the broad class of models Dark Matter (DM) consists of particles

• The neutral DM particle

• In addition to

• These

A possible value of mass

with

The neutral and stable ^{1}) or

To discover the DM particle, one needs to specify such processes with a clear signature. The

The masses

On the contrary, the lepton energy can be measured much more precisely. In this paper we show, first, that the energy distribution of leptons has singular points whose positions are kinematically determined. Measuring positions of these singularities will allow, in principle, to determine masses

Moreover, we present a simple method for measuring spin of DM particles in these very experiments.

The discussed problem differs strongly from that for the case when the lightest charged D-odd particle is slepton (another set of parameters of MSSM). In the latter case DM particles are produced via slepton pair

The overall picture is summarized in Section 3. Short conclusion is given in Section 4.

In the Appendix B we discuss the potential of the process

In the Appendix C we consider possible background processes and show that the most of them can be neglected at the analysis.

We express discussed cross sections via

The total cross section of

The process

Note before all that the energies,

The observable states are decay products of

Therefore, the signatures of the process in the modes suitable for observation are

At

Note that the processes with invisible decay

2.2.

Here we consider the energy distribution of ^{2}

Denoting by

In particular, at

At

At the highest value

The fraction of such events for each separate lepton,

Note that in the laboratory frame, for a

We study the distribution^{3} of muons over its energy

a) If

It is easy to check that the interval corresponding to energy

(Note that

With a shift of

In these points the energy distributions of muons has kinks. Between these kinks, the

b) If

Similarly to the preceding discussion, the increase of

To get an idea about the shape of the peak, we use the distribution of

The density of muon states in energy is calculated by convolution of kinematically determined distribution with distribution (16). Neglecting the angular dependence of the matrix element, we obtain the result in form of

Distributions at GeV, GeV for GeV—the case with (the right plot) and for GeV—the case with (the left plot). In the latter case, the higher and lower peaks are for and, respectively

Characteristic values for singular points in the energy distributions of muons (kink and peak) together with similar points for the energy distributions of

The cascade

In the

The resulting distribution retains the upper boundary of the energy distribution of muons

At

The total probability of

Below we limit ourself by the study of processes with signature (6b), (7a). Unfortunately, some of new processes with intermediate

Let us consider in more detail production of an observed state with signature (6b), (7a)

1) The cascade

2) Cascade

Table 1. The singular point energies of lepton and dijet in (in GeV) at GeV.

250 | 150 | 186.3 | 20.8 | 77.8 | - | - | 195.4 |

250 | 200 | 184.9 | 34.9 | 46.3 | - | - | 193.6 |

250 | 80 | 148.3 | - | - | 91.3 | 93.8 | 148.3 |

100 | 80 | 78 | - | - | 30 | 37.5 | 78 |

The shape of the distribution

Note that in the case

Observation of events with signature (6), (7) will be a clear signal for DM particle candidates. The non- observation of such events will allow to find lower limits for masses

At

A more detailed analysis reveals two sources of distortion of the obtained results (we neglect them in our preliminary analysis).

1. The final width of

2. The energy spectra under discussion will be smoothed due to QED initial state radiation (ISR), final state radiation (FSR) and beamsstrahlung (BS). The ISR and FSR spectra are machine independent, while BS spectrum is specific for each machine (but well known during operations). This smoothing decreases accuracy in measuring of masses. However, the precise knowledge of mentioned spectra allows to solve the problem about restoration original accuracy by means methods of deconvolution in so called “incorrect inverse problem”. This work and the estimates of the range where masses and spins can be determined with reasonable accuracy will be the subject of the forthcoming paper.

Masses

We suggest to extract the second quantity for description of masses from the lepton energy spectra. The lepton energy is measurable with a high accuracy. We found above that the singular points of the energy distribution of the leptons in the final state with signature (6a) are kinematically determined, and therefore can be used for a mass measurement.

M1) If a

At

At

In both cases the position of the upper edge in the dijet energy distribution

M2) The signal of realization of the inequality

The opportunity to extract new singularities from the data, related to

3.3. Spin of

where

mixing, etc.

The cross section of the process is reduced by contribution of the diagram with

The upper curve for, the lower for; GeV

Table 2. Some values of.

, GeV | 100 | 250 | 250 | 250 |
---|---|---|---|---|

80 | 80 | 150 | 200 | |

0.066 | 0.245 | 0.162 | 0.062 | |

0.84 | 1.107 | 1.02 | 0.82 |

obtain (for identical masses

When masses

We consider models in which stability of dark matter particles

This method is in several aspects superior to the standard approaches discussed elsewhere.

1) It uses leptons which are copious and can be accurately measured in contrast with jets that individual energy can be measured only with lower precision.

2) These singularities are robust and survive even when superimposed on top of any smooth background.

In addition, even a rough measurement of cross sections with a very clean signature allows us to determine spin of DM particles based on the results of mentioned kinematical measurements.

This work was supported in part by grants RFBR and NSh-3802.2012.2, Program of Dept. of Phys. Sc. RAS and SB RAS “Studies of Higgs boson and exotic particles at LHC” and Polish Ministry of Science and Higher Education Grant N202 230337. I am thankful to A. Bondar, E. Boos, A. Gladyshev, A. Grozin, S. Eidelman, I. Ivanov, D. Ivanov, D. Kazakov, J. Kalinowski, K. Kanishev, P. Krachkov and V. Serbo for discussions.