ISSN 10637796, Physics of Particles and Nuclei, 2014, Vol. 45, No. 1, pp. 207–210. © Pleiades Publishing, Ltd., 2014.

Precise Measuring Mass and Spin of Dark Matter Particles at ILC via Singularities in the Single Lepton Energy Spectrum1 I. F. Ginzburg Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia Abstract—We consider models, in which stability of Dark Matter particles D is ensured by the conservation of the new quantum number, called Dparity here. Our models contain also charged Dodd particle D±. We propose method for precision measuring masses and spin of Dparticles via the study of energy distribution of single lepton (e or μ) in the process e+e– → D+D– → DDW+W– with the observable state dijet + μ (or e) + nothing. It is shown that this distribution has kinematically determined singular points (upper edge and kinks or peak). Measuring of their positions allow to determine precisely masses of D and D±. After this, even a rough measuring of corresponding cross section allows to determine the spin of D particles. DOI: 10.1134/S106377961401033X 1

We consider a wide class of models, in which Dark Matter (DM) consists of particles D similar to those in SM, with the following properties (the examples are: MSSM where D is the lightest neutralino with spin 1/2 [1], and inert doublet model IDM [2] where D is the Higgslike neutral). 1. Neutral DM particle D with mass MD and spin sD = 0 or 1/2 has new conserved discrete quantum number. I call it Dparity. All known particles are Deven, while the DM particle is Dodd (for MSSM Dparity means Rparity). The Dparity conservation ensures stability of the lightest Dodd particle D. 2. In addition to D, a charged Dodd particle D± exist, with the same spin sD and with masses M+. (In MSSM D± is the lightest chargino, in the IDM D± is similar to the charged Higgs.) Other Dodd par ticles, if they exist, are very heavy. (The case with another neutral DA, lighter than D± is considered in [3]). 3. These Dparticles interact with the SM particles only via the Higgs boson and the covariant derivative in the kinetic term of the Lagrangian—gauge interac tions with the standard electroweak gauge couplings g, g' and e (for coupling to Z—with possible reducing mixing factor), D+D–γ, D+D–Z, D+DW–. A possible value of mass MD is limited by stability of Dparticles during the age of the Universe [4, 5]. We assume 4 GeV ⱗ MD ⱗ 80 GeV. The nonobser vation of processes e+e– → D+D– at LEP gives M+ > 90 GeV [6]. The neutral D can be produced and detected via production D± and its decay D± → DW±. To discover DM particle, one needs to specify such processes with clear signature. The e+e– Collider ILC/CLIC at s =

2E > 200 GeV provides excellent opportunity for these tasks in the process e+e– → D+D– [7, 8]. Main process e+e– → D+D–. Energies, γfactors and velocities of D± are s/2, γ + = E/M + , β + =

E± = E =

2

2

1 – M + /E .(1)

Neglecting terms ∝ (1/4 – sin2θW), the cross sec tion of process is a sum of model independent QED term (photon exchange) and axial Z exchange term— Fig. 1: ⎧ ⎪ β+ ⎪ σ = σ0 ⎨ ⎪ 3 ⎪ β+ ⎩

2

2M 2 1 + + + r Z β + ⎛ s D = 1⎞ , ⎝ 2⎠ s 2 1 + r Z cos ( 2θ W ) ( s D = 0 ); 4

μM 0.124μ M ;  =  r Z =  2 4 2 2 ( 2 sin ( 2θ W ) ) ( 1 – M Z /s ) ( 1 – M Z /s ) + –

+

–

2

σ 0 ≡ σ ( e e → γ → μ μ ) = 4πα /3s. Here μM ≤ 1 is model dependent mixing factor.

1 The article is published in the original.

207

1.2 1.0 0.8 0.6 0.4 0.2 0

100 200 300 400 500

Fig. 1. The dependence σ(e+e– → D+D–)/σ0 on E at M+ = 150 GeV and μM = 1, upper curve for sD = 1/2, lower curve for sD = 0.

(2)

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The cross section of the e+e– annihilation in all final states at ILC for s > 200 GeV is ~ 10 σ0. The cross Section (2) is a large fraction of the total cross section of e+e– annihilation, it makes this observation very realistic task. The annual luminosity integral  for the ILC project [8] gives σ0 ~ 105. Signature. Particles D± decay completely to DW±, so we have process e+e– → D+D– → DDW+W– with either on shell (real) or off shell W±, the latter is q q pair (dijet) or ν, having the same quantum numbers as W but effective mass M* < MW. The observable states are decay products of W with large missing transverse energy E T carried away by the neutral and stable Dparticle + nothing, the missing mass of parti cles escaping observation M( E T ) is large. Therefore, the signatures of the process in the modes, suitable for observation, is (A) One dijet plus e or μ or ( B ) two dijets with large E T and large M ( E T ) + nothing, total (3) energy of each dijet or lepton less than E.

W energy distribution. Here we denote by W the dijet (q q ) or ν pair, obtained from W decay, with the effective mass M*. At M+ – MD > MW we have M* = MW (on shell W), at M+ – MD > MW possible values of M* are within interval (0, M+ – MD) (off shell W). At each M* in the rest frame of D± we have 2particle r

r

decay with known energy E W* and momentum p W* of W [5]. Denoting by θ the W– escape angle in D– rest frame with respect to the direction of D– motion in the Lab system and using c ≡ cosθ, we find the energy of W– in r γ+( E W*

r p W* ).

the Lab system as = + cβ + There fore, at given M* the energy is limited from both above and below. In particular, at M+ – MD > MW the kine L, ±

matical limits of W energy we denote by E W, on . At M+ – MD < MW similar edges are different for each value of M*. In particular, at the highest value M* = r

M+ – MD we have p W = 0, and interval, similar to (4) L

L, ±

r

r

E W, on = γ + ( E W ( M W ) ± β + p W ( M W ) ); L, ±

L

E W, p ≡ E W | ( M* = M+ – MD ) = E ( 1 – M D /M + ); L, ±

2

(4)

2

E W, off = E ( 1 ± β + ) ( 1 – M D /M + )/2. e+e– → D+D– → DDW+W– → DDq q ν (process with signature (3A), single lepton energy distribution. We consider, for definiteness,  = μ–. (a) If M+ – MD > MW, the muon energy and momentum in the rest frame of W are MW/2. In the L

Lab system for W with some energy E W the γfactor L

and the velocity of W are γWL = E W /MW and βWL ≡ –2

1 – γ WL . Just as above, denoting by θ1 the escape angle of μ relative to the direction of the W in the Lab system and c1 = cosθ1, we find that in the Lab system the muon energy ε = γWL(1 + c1βWL)(MW/2). There L

L

fore, for these muons ε+( E W ) ≥ ε ≥ ε–( E W ) where L

L

L

ε±( E W ) = E W (1 + βWL)/2 = ( E W ±

L 2

2

( E W ) – M W /2. L

The fraction of events with signature (A) and (B) is calculated with known BR’s for W decay [5]. The observation of events with this signature will be a clear signal of candidates for DM particles.

L EW

L, ±

lower bounds E W, off on the W energy distribution are achieved at M* = 0:

reduces to a point E W, p , where entire W energy distri bution has maximum (peak). Absolute upper and

The interval, corresponding to energy E 1W , is located entirely within the interval, correspondent to L energy E W . Therefore, all muon energies lie within the interval determined by the highest value of W energy ε+ ≥ ε ≥ ε–, where: +

+

L, +

L, +

L, +

2

2

ε ≡ ε ( E W, on ) = ( E W, on + ( E W, on ) – M W )/2, –

ε =

(5)

2 + M W / ( 4ε ).

Contributions of W with intermediate energies are added into the entire distribution of muons in the energy, and it increases monotonically from the outer limits to kinks at energies k, corresponding to the low est value of W energy: ±

±

L, –

L, –

L, –

2

2

ε k ≡ ε ( E W, on ) = ( E W, on + ( E W, on ) – M W )/2. (6) The energy distribution of muons for the case of matrix element, independent on θ1, is shown in Fig. 2left. (b) If M+ – MD < MW, the D± decays to D plus off shell W with effective mass M* ≤ M+ – MD. The calcu lations, similar to above, for each M* shows that the muon energies are within the interval, appearing at M* = 0: –

+

L, +

{ ε = 0; ε = E W, off }.

(7)

Similarly to the preceding discussion, the increase of M* shifts the interval boundaries inside. Therefore, the muon energy distribution increases monotonically

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PRECISE MEASURING MASS AND SPIN OF DARK MATTER PARTICLES –

+

⑀k

dN/d⑀

⑀k

0.008 0.006 0.004 0.002

∈+ 100

50

150

dN/d⑀ 0.012 0.010 0.008 0.006 0.004 0.002

209

∈p

⑀+

200

50

⑀

⑀

150

100

Fig. 2. Distributions (1/σ)dσ/dε at E = 250 GeV, MD = 50 GeV for M+ = 150 GeV—left, for M+ = 120 GeV—right (upper peak: sD = 0, lower: sD = 1/2).

from outer bounds up to the maximum (peak) at M* = M+ – MD (cf. (4)): ε p = E ( 1 + β + ) ( 1 – M D /M + )/2.

(8)

Characteristic values for singular points in energy distributions of muons (kink and peak) together with similar points for energy distributions of W (dijets) at MD = 50 GeV are given in the table. To get an idea about the shape of the peak, we use the distribution of W*'s (dijets or ν pairs) over the effective masses M* which is given by the spin depen dent factor R sD p*dM*

2

2 / ( MW

2 2

– M* ) : 2

2

2

2

R 0 = p* , R 1/2 = [ ( M + + M D – M* ) 2

2

2

2

2

2

× (2M W + M + + M D ) – 4M + M D ]/M W .

(9)

Neglecting angular dependence of the matrix elemen, we obtain result in form of Fig. 2right. We see that the discussed peak is sharp enough for both values of spin sD = 0 and 1/2. Summary I. Masses M+ and MD can be determined from energy distributions of observable products of reaction. Well known approach is to measure edges in the energy distributions of dijets [8]. However, the individ ual jet energies cannot be measured with high preci sion. The lower edge of energy distribution of W in dijet mode and position of peak in this distribution are smeared by this inaccuracy in the measuring of energy of an individual jet. One can hope only to measure with satisfactory precision the upper bound of energy L, + distribution of W in dijet mode E W (4). The lepton energy is measurable with higher accuracy. We find above that the singular points of the energy distribution of the leptons in the final state one or two dijets +  + nothing are kinematically deter mined, and—therefore—can be used for measuring of masses. With mentioned luminosity the 1year number of events of this type (for e and μ) will be ~(1–3) × 104. PHYSICS OF PARTICLES AND NUCLEI

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The shape of energy distribution of leptons (with one peak or two kinks) allows to determine what case is realized, M+ – MD > MW or M+ – MD < MW. At M+ – MD > MW the position of upper edge of the muon +

energy ε+ (5) and upper kink ε k (6) give us two equa tions necessary for determination of MD and M+. At M+ – MD < MW two similar equations are given by the position of upper end point of the muon energy ε+ (7) and peak εp (8). L, +

The upper edges of dijet energy distribution E W and muon energy distribution ε+ contains identical L, + 2 + information. We have E W = ε+ + M W /4ε at M+ – L, +

MD > MW (cf. (4), (5)) and E W = ε+ at M+ – MD < MW (cf. (7)). Summary II. Spin of Dparticles sD. The experi mental value of cross section of the process e+e– → D+D– is obtained by summation over all processes with signature (3) taking into account the known BR’s for W decay. When masses M+ become known, this cross section is given by Eq. (2). Its main part is given by model independent QED contribution of photon exchange, whereas the model dependent contribution of Z exchange at s > 200 GeV contributes less than 30%. For identical masses σ(sD = 1/2) > 4σ(sD = 0) (cf. Fig. 1). This strong difference in the cross sections for differ

No. 1

L, +

E

M+

ε+

εk

+

εp

E W, p

L

EW

250

150

186.3

77.8

–

–

195.4

250

200

184.9

46.3

–

–

193.6

250

80

148.3

–

91.3

93.75 148.3

100

80

78

–

30

37.5

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ent sD allows to determine spin of D particle even at low accuracy in the measuring of cross section. ACKNOWLEDGMENTS This work was supported by Grants RFBR 1102 00242, NSh3802.2012.2, Program of Dept. of Phys. Sc. RAS and SB RAS “Studies of Higgs boson and exotic particles at LHC” and Polish Ministry of Sci ence and Higher Education Grant N202 230337. I am thankful A.E. Bondar, A.G. Grozin, I.P. Ivanov, D.Yu. Ivanov, D.I. Kazakov, J. Kalinowski, K.A. Kani shev, P.A. Krachkov and V.G. Serbo for discussions. REFERENCES 1. For example, D. Hooper, hepph/0901.4090; M. Maniatis, hepph/0906.0777; D. I. Kazakov, hep ph/1010.5419; J. Ellis. hepph/1011.0077.

2. N. G. Deshpande and E. Ma, Phys. Rev. D 18, 2574 (1978); R. Barbieri, L. J. Hall, and V. S. Rychkov, Phys. Rev. D 74, 015007 (2006), hepph/0603188; I. F. Gin zburg, K. A. Kanishev, M. Krawczyk, and D. Sokolowska, Phys. Rev. D 82, 123533 (2010); hep ph/1009.4593. 3. I. F. Ginzburg, arXiv:1211.2429 hepph. 4. See e.g. E. M. Dolle and S. Su, Phys. Rev. D 80, 055012 (2009) arXiv:0906.1609 hepph. 5. Particle Data Group, Journ. of Phys. G 37 (7A), 075021 (2010). 6. E. Lundstrom, M. Gustafsson, and J. Edsjo, Phys. Rev. D 79, 035013 (2009) arXiv:0810.3924 hepph. 7. R. D. Heuer et al., TESLA Technical Design Report, DESY 2001011, TESLA Report 200123, TESLA FEL 200105 (2001). 8. M. Asano, K. Fujii, R. S. Hundi, H. Itoh, S. Matsu moto, N. Okada, T. Saito, T. Suehara, Y. Takubo, and H. Yamamoto, Phys. Rev. D 84, 115003 (2011), arXiv:1007.2636; 1106.1932 hepph.

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