In the broad class of models Dark Matter (DM) consists of particles similar to those in SM, with the following properties:

• The neutral DM particle with mass and spin or has a new conserved quantum number, which we call the -parity. All known particles are -even, while the is -odd.

• In addition to, other -odd particles exist: a charged and (sometimes) a neutral, with the same spin and with masses. The -parity conservation ensures stability of the lightest -odd particle.

• These -particles interact with the SM particles via the Higgs boson, , and via the covariant derivative in the kinetic term of the Lagrangian. These are the gauge interactions, , , , with the standard electroweak couplings, and (coupling to can be added by a mixing factor, deviation from 1 appears due to possible mixing of with other -odd neutrals).

The first example of such model provides well known MSSM (see e.g. [1] -[6] ) for specific set of para- meters. Here our term -parity means -parity. For the considered set of parameters, is the lightest neutralino, the heavier neutralino can play role and the next in mass -odd particle is the lightest chargino, spin of these -particles. The other -odd particles (in particular, sleptons and squarks) are supposed to be heavier than the ILC beam energy.

The second example of such models provides the Inert Doublet Model (IDM) (see e.g. [7] -[13] and Appendix A). That is the symmetric Two Higgs Doublet Model, containing two scalar doublets and. The “standard” scalar (Higgs) doublet is responsible for electroweak symmetry breaking and the masses of fermions and gauge bosons just as in the Standard Model (SM). The second scalar doublet doesn’t receive vacuum expectation value and doesn’t couple to fermions. In this model the -parity con- servation is ensured by a symmetry, four degrees of freedom of the Higgs doublet are the same as in the SM: three Goldstone modes and the standard Higgs boson. All the components of the scalar doublet are realized as the massive -particles: two charged and two neutral ones, with masses, , respectively with. IDM contains no other -odd particles. All -particles have spin.

A possible value of mass is limited by stability of during the Universe existence [14] -[20] . The non-observation of the processes and at LEP gives GeV and GeV (at GeV) [11] -[13] . Limitations for masses of neutralino and chargino can be found in [20] . For IDM, limitations for parameters of -peak, and results in ([11] -[13] [20] )

with expectation value of Higgs field. Further, we will have in mind and assume GeV.

The neutral and stable can be produced and detected via production or and subsequent decay, with either on shell (real) or off shell or. The off shell emerges as a pair (dijet¹) or, having the same quantum numbers as but with an effective mass. From now on, or refers to any of these two cases.

To discover the DM particle, one needs to specify such processes with a clear signature. The Collider ILC/CLIC provides an excellent opportunity for this task (see, e.g., [21] [22] ) in the process with a clear signature, see Equations (6) and (7) below. The cross section of this process is a large fraction of the total cross section of annihilation, Section 3.3.

The masses and could be found via the edges of the energy distribution of dijets, originating from from decay, section 2.2, 2.4 (see [5] [6] for MSSM and [11] -[13] for IDM). However, this method cannot provide a good accuracy in measuring the mass. Indeed, the individual jet energy measurement suffers from a sizable uncertainty. In particular, this uncertainty smoothes the lower edge in the dijet energy spectrum.

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 and with good precision (Sections 2.3 and 2.4). In contrast to [5] -[6] [11] -[13] , our description is suitable for different models.

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. First of all, signature of this process is quite different from that one in our problem (6), (7). Second, the energy of observable lepton decay product of slepton is measurable well in each individual event, in difference with our case, when similar product of decay, , is seen as dijet or lepton plus neutrino with badly measurable energy in each individual event. Therefore, the approach used in the analysis of slepton production (cf. [23] -[25] ) cannot be applied directly to our problem.

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 for similar problems, for completeness. In contrast with previous studies, we find that this potential is not too high.

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 annihilation at ILC for GeV is. The annual integrated luminosity for the ILC project [22] gives

The process represents a significant fraction of all annihilation events-see (19), (20), Figure 2 and Table 2. With the luminosity (3), the annual number of events of discussed type will be, depending on and, and about of them (in the mode with or plus dijet) are suitable for our analysis.

If or is absent, once produced, particles decay fast (with a unit probability) to,

The observable states are decay products of with a large missing transverse energy carried away by the invisible -particle, and the missing mass of particles escaping observation is large. In contrast to the LHC, where a large flux of low particles demands an additional cut off, at LC such particles are absent.

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

At GeV, the branching ratios for different channels of decay are roughly identical for on-shell [20] and off-shell. In particular, the fraction of events with signature (6a) is. The fraction of events with signature (6b) is (here 0.17 is a fraction of or from the decay of). At GeV, and increase, while the dijet becomes a set of a few hadrons.

If, when analysing the main process, one more decay channel is added,. Its branching ratio is typically less than 0.5 (see discussion in section 2.4). Particle decays fast to, creating new cascades,. As a result, the signature of the processes in the modes suitable for observation contains both (6) and processes with decay’s or’s in the mentioned cascades:

Note that the processes with invisible decay (we denote these states as, their) have signature (6).

2.2. Energy Distribution in

Here we consider the energy distribution of with an effective mass. At each value of, we have in the rest frame of a two-particle decay with²

Denoting by the escape angle in the rest frame with respect to the direction of motion in the laboratory frame and using, we find the energy of in the laboratory frame as. Therefore, at given, the energy of lies within the interval.

In particular, at we have, and the kinematical edges of the energy distribution are

At we have, and obtain similar edges, which are different for each value of. The absolute upper and lower bounds on the energy distribution of are attained at, they are equal to

At the highest value we have, and an interval (9) is reduced to a point, where the entire energy distribution has a maximum (peak) of

The fraction of such events for each separate lepton, , , or, is about 0.08, their sum is about 0.33 of the total cross section of the process. We will speak, for definiteness, and neglect the muon mass.

Note that in the laboratory frame, for a with some energy, its -factor and the velocity are and.

We study the distribution³ of muons over its energy,. We show that this distribution has singular points, whose positions are kinematically determined, i.e. model independent.

a) If we have, and the muon energy and momentum in the rest frame of are. Just as above, denoting by the escape angle of relative to the direction of the in the laboratory frame and using, we find that the muon energy in the laboratory frame is. Therefore, for these muons where

and.

It is easy to check that the interval corresponding to energy is located entirely within the interval, correspondent to energy. Therefore, all muon energies lie within the interval determined by the highest value of energy:

(Note that.)

With a shift of from these boundaries inwards, the density of states in the distribution grows monotonically due to contributions of smaller values up to values, corresponding to the lowest value of energy:

In these points the energy distributions of muons has kinks. Between these kinks, the -distribution is approximately flat.

Figure 1, the left plot, shows the energy distribution of muons for the case of the matrix element independent of. Since positions of kinks are kinematically determined, it is not surprising that calculations for distinct models (containing different angular dependence) demonstrate variations in shapes but do not perturb the position of kinks.

b) If, the decays to, where is an off-shell with an effective mass. The calculations for each similar to shown above demonstrate that the muon energies are within the interval appearing at:

Similarly to the preceding discussion, the increase of shifts the interval boundaries inwards. Therefore, the muon energy distribution increases monotonously from the outer bounds up to the maximum (peak) at (cf. (11)):

To get an idea about the shape of the peak, we use the distribution of’s (dijets or pairs) over the effective masses which is given by the spin-dependent factor:

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

Figure 1, right plot. One can see that the discussed peak is sharp enough for both values of spin and.

Characteristic values for singular points in the energy distributions of muons (kink and peak) together with similar points for the energy distributions of (dijets) are given in Table 1.

The cascade modifies the spectra just discussed. The energy distribution of produced in the decay is the same as that for or, discussed above (within the accuracy of). Once produced, decays to in 17% of cases (the same for decay to). These muons are added to those discussed above.

In the rest frame, the energy of muon is with. The energy spectrum of muons is (see textbooks). This spectrum and the distributions obtained above are converted into the energy distribution of muons in the Lab frame. It is clear that this contribution is strongly shifted towards the soft end of the entire muon energy spectrum.

The resulting distribution retains the upper boundary of the energy distribution of muons (12), (14). Numerical examples show that here the upper kink is smeared, while lower kink become even more sharp without shift from position (13) in wide region of masses and. The position of peak (15) is also shifted weakly.

At the decay become possible and the processes, etc. with signature (7) should be taken into account.

The total probability of decay to and equals 1. The decay is described by the same equation as, but with other kinematical factors since. In the IDM the probability of this new decay is lower than that without due to smaller final phase space, i.e.. In the MSSM value of depends additionally on the mixing angles. We assume that in general case.

Below we limit ourself by the study of processes with signature (6b), (7a). Unfortunately, some of new processes with intermediate look as those with signature (6) since large fraction (20%) of decays of is invisible (final states). We denote these states of as.

Let us consider in more detail production of an observed state with signature (6b), (7a) dijets nothing. This state can be obtained from two different cascades.

1) The cascade. The energy distribution of here reproduces, discussed in Section 2.3 with an additional factor.

2) Cascade. Since couplings and differ by a phase factor only (and perhaps mixing angle factors), the energy distribution of in this case is described by the same de- pendence but with the change, the corresponding contribution to the entire energy distri- bution is. For brevity we will write and

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

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

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 resulting energy distribution is

The shape of the distribution is similar to that for, but with different positions of kinks and (or) peak. As, these new kinks and (or) peak are situated below similar points for. Since this contribution is much smaller than the main contribution (with the overall ratio at), it only results in a weak reshaping of the full energy distribution as compared with distributions.

Note that in the case the distributions and are close to each other, and. In the opposite degenerate case, the quantity and the influence of the intermediate state on the result is negligible. (Such very cases are widely discussed in context of MSSM).

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 and (and) leads to a blurring singularities derived. This effect in- creases with the growth 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 and. In a well known approach, one measures edges in the energy distributions of dijets, representing in the decay [5] [6] . However, the individual jet energies and consequently, effective masses of dijets cannot be measured with a high precision. The observed lower edge of the energy distribution in the dijet mode and the position of a peak in this distribution (11) are smeared by this uncertainty. One can only hope for a sufficiently accurate measurement of the upper edge of the energy distribution, (9), (10).

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 particle is absent or, the results (12)-(15) describe the energy distributions completely. The shape of the energy distribution of leptons (with one peak or two kinks) allows to determine which case is realized, or.

At, the positions of upper edge in the dijet energy distribution (9) and the lower kink in the muon energy distribution (13) give us two equation necessary for determination of and. We reproduce these equations for clarity

At, two similar equations are provided by the position of the upper edge in the dijet energy distribution (10) and the peak in muon energy distribution (15).

In both cases the position of the upper edge in the dijet energy distribution or should be extracted from all events with signature (6), (7), the position of the lower kink in the muon energy distribution or peak can be extracted from events with signature (6b) only.

M2) The signal of realization of the inequality will be observation of the process, having signature (26). In this case the position of the upper edge in the dijet energy distribution is the same as in previous case. The position of lower edge in the dijet energy distribution is either shifted or smeared, in this case the method of [5] [6] becomes completely inapplicable. The entire energy distribution of muons in the observed state or dijet nothing was described in the Section 2.4. It was shown there that taking into account a new decay channel changes the position of the main singularities in the muon energy spectrum very weakly. Therefore the above mentioned procedure for finding and can be used in this case as well.

The opportunity to extract new singularities from the data, related to (and giving additionally), requires a separate study (see also analysis in Appendix B).

3.3. Spin of -Particles

The amplitude of the process is the sum of model-independent QED diagram (the photon annihilation), the annihilation diagram and in some models -channel exchange by other -odd particles. We start with the description of cross section in the minimal approximation, taking into account only photon and annihilation diagrams. Neglecting terms (described interference) we have:

where, factor is expressed via parameters of possible

mixing, etc. Figure 2 and Table 2 represent dependence of (19) on beam energy for.

The cross section of the process is reduced by contribution of the diagram with -channel exchange by other -odd particle. This decrease is not so strong if mass of is high enough. For example, if mass of selectron is more than 250 GeV (condition 2 in Section 1 and [20] ), the cross section for is reduced by a factor,. Combining with numbers from Figure 2 and Table 2 we

Table 2. Some values of.

Table 2Table 2. Some values of.

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 at a given beam energy):

The experimental value of the cross section is obtained by summing over all processes with signature (6), (7) (that is about 3/4 of the total cross section). By taking into account the known BR’s for decay the accuracy of this restoration of can be improved.

When masses become known, the cross section is calculated with reasonable precision with Equation (19). The strong inequality (20) allows to determine spin from the obtained values of cross sections even with a handful of well-reconstructed events.