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Spin Correlations in Production and Subsequent Decay of Neutralinos

We study the process e^{-} e^{+} ->chi^0_1 chi^0_2 with the subsequent decay chi^0_2 ->chi^0_1 l^{+} l^{-} taking into account the complete spin correlations. We give the analytical formulae for the differential cross section and present numerical results for the lepton angular distribution and the distribution of the opening angle between the outgoing leptons for the LEP2 energy of sqrt{s}=193 GeV and for sqrt{s}=500 GeV. We examine three representative mixing scenarios in the MSSM and also study the influence of the common scalar mass parameter m_0 on the shape of the distributions. For the lepton angular distribution the effect of the spin correlations amounts to up to 20% for lower energies. We find that the opening angle distribution is suitable for distinguishing between higgsino-like and gaugino-like neutralinos. The shape of the lepton angular distribution is very sensitive to the mixing in the gaugino sector and to the value of m_0. For higher energies it is also suitable for distinguishing between higgsino-like and gaugino-like neutralino.


I. INTRODUCTION
Although the Standard Model (SM) is extraordinarily successful for describing the electroweak phenomena, it has several theoretical shortcomings. The most severe of these deficiencies, the hierarchy problem, can satisfactorily be solved by the concept of supersymmetry (SUSY) broken at the TeV scale. This concept received a fresh impetus by the result that in a SUSY-GUT model the measured coupling constants evolved to high energies meet at a single point.
The most economical candidate for a realistic SUSY model with minimal gauge group SU (3) × SU (2) × U (1) and with minimal content of particles is the Minimal Supersymmetric Extension of the Standard Model (MSSM). In this paper we consider its simplest version with the conserved quantum number R parity. Its implications include that SUSY particles can only be produced in pairs and the lightest supersymmetric particle (LSP) is stable and escapes detection. As usual we assume that this particle is the lightest neutralinoχ 0 1 . Among the new particles the charginos (the supersymmetric partners of the charged gauge and Higgs bosons) and the neutralinos (the partners of the neutral gauge and Higgs bosons) are of particular interest. As they are expected to be lighter than the gluino and in most scenarios lighter than the squarks and sleptons [1], [2], the next-to-lightest neutralinoχ 0 2 and the lightest chargino could be first observed in future experiments at e + e − -colliders. In particular the production ofχ 0 1χ 0 2 pairs allows to study a wide region of the SUSY parameter space. Although in general chargino production is favoured by larger cross sections, in certain regions of the parameter space sizeable cross sections for the neutralino process can be expected [3]. Moreover it might be possible to discover SUSY by neutralino production if charginos are not accessible.
If new particles are discovered which are possible neutralino candidates, for a clear identification the complete investigation of their decay characteristics is indispensable. Neutralino decay widths and branching ratios have been thoroughly studied [4]. Angular distributions and angular correlations of the decay products can give valuable additional information on their composition from photino, zino and higgsino components.
Another interesting question is to see if angular distributions of decay products allow separation of neutralino production from chargino production.
Moreover, from decay angular distributions with complete spin correlations of the decaying particle one can infer the spin of the new particles.
Finally the identification of neutralinos would be completed by ascertaining their Majorana character. In Ref. [5,6] it is demonstrated that this is possible by means of the energy distributions of the decay leptons if the neutralinos are produced in collisions of polarized e + e − beams. The angular distributions of the decay products might, however, offer the possibility to prove the Majorana character if polarized beams are not available. Furthermore the angular distributions of the final leptons are suitable observables for studying CP-violation in supersymmetric models [6], [7].
The above mentioned reasons motivate a study of angular distributions in associated production of neutralinos and the subsequent decay of the next-to-lightest neutralino. Since angular distributions depend on the polarization of the parent particles one has to take into account spin correlations between production and decay.
In general quantum mechanical interference effects between various polarization states of the parent particles preclude simple factorization of the differential cross section into a production factor and a decay factor [8] unless the production amplitude is dominated by a single spin component [2]. However, in Ref. [9] it is shown that the factorization property holds for particles with spin provided that suitable Lorentz invariant variables are used. But this is not the case for angular distributions of the neutralino (or chargino) decay products in the laboratory frame. For energy distributions of the final particles in the laboratory frame the spin correlations are nevertheless usually ignored.
Heavy fermion production with subsequent decay was considered in Ref. [10]. Recently a Monte-Carlo generator for chargino production and decay including spin correlations was developed in Ref. [11].
In this paper we calculate the cross section of the process e + e − →χ 0 1χ 0 2 and the subsequent direct leptonic decay, χ 0 2 →χ 0 1 ℓ + ℓ − . We give the analytical formulae of the differential cross section with complete spin correlations of the decaying neutralino. We study numerically the influence of these spin correlations on energy and angular distributions. The energy distributions are independent of spin correlations. However, for the lepton angular distributions for lower energies the effect of the spin correlations amounts to up to 20%. The shape of the lepton angular distribution is very sensitive to the mixing character in the gaugino sector and to the value of m 0 . For higher energies it is suitable for distinguishing between higgsino-and gaugino-like neutralino. We also analyze the opening angle distributions and show that at lower energies they are suitable for distinguishing between higgsino-and gaugino-like neutralinos. This will allow one to constrain the parameter space of the MSSM. We also consider the influence of the scalar mass parameter m 0 on the shape of distributions.
In Sec. 2 the general formalism is shown. In Sec. 2.1 the Lagrangian, couplings and Feynman diagrams are given, in Sec. 2.2 the spin-density formalism is presented, and in Sec. 2.3 the kinematics is given. In Sec. 3 we present the formulae for the differential cross section with complete spin correlations. Numerical results for the LEP2 energy √ s = 193 GeV and for √ s = 500 GeV and a discussion are presented in Sec. 4.

A. The Feynman diagrams
In this section we show the Feynman diagrams and give the Lagrangian for the production process, e − (k 1 )e + (k 2 ) → χ 0 1 (q 1 )χ 0 2 (q 2 ), and for the direct leptonic decay,χ 0 2 (q 2 ) →χ 0 1 (p 1 )ℓ + (p 2 )ℓ − (p 3 ). The arguments k 1 , k 2 , q 1 , q 2 and p 1 , p 2 , p 3 denote the momenta of the incoming electron, positron, the produced neutralinosχ 0 1 ,χ 0 2 and the outgoing neutralinoχ 0 1 and leptons ℓ + , ℓ − from theχ 0 2 decay. Both the production and the decay process contain contributions from Z 0 exchange in the direct channel (s-channel) and fromẽ L andẽ R exchange in the crossed channels (t-, uchannel). We introduce the kinematic variables: Channels referring to the decay are marked by a dash. The Feynman diagrams are shown in FIG.1.
From the interaction Lagrangian of the MSSM ( in our notation and conventions we follow closely [12]), we obtain the couplings: In our case i = 1, j = 2 and we shall write O L,R 12 = O L,R . Here P L,R = 1 2 (1 ∓ γ 5 ), g is the weak coupling constant (g = e/ sin θ W , e > 0), and e ℓ and T 3ℓ denote the charge and the third component of the weak isospin of the lepton ℓ. Furthermore tan β = v2 v1 , v 1,2 are the vacuum expectation values of the two neutral Higgs fields and N ij is the unitary 4 × 4 matrix which diagonalizes the neutral gaugino-higgsino mass matrix in the basisγ,Z,H 0 a ,H 0 b . Since we disregard in this paper CP violating phenomena the elements N ij of the diagonalized matrix and the couplings can be chosen real. Then some of the neutralino mass eigenvalues may be negative and we shall write them in the form η i m i with m i > 0, η i = ±1 [13]. The respective amplitudes for the Feynman diagrams are taken from Ref. [14].

B. Spin density matrix for production and decay
The differential cross section for the production e − (k 1 )e + (k 2 ) →χ 0 1 (q 1 )χ 0 2 (q 2 ) and the subsequent decayχ 0 2 (q 2 ) → χ 0 1 (p 1 )ℓ + (p 2 )ℓ − (p 3 ) is given by where s is the cms-energy squared of the incoming e − and e + and λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2xz − 2yz is the kinematical triangle function. All lepton masses are neglected. In Eq. (7) λ e − , λ e + , λ1 and λ 1 , λ + , λ − label the helicities of the electron, positron andχ 0 1 of the production process and the helicities of theχ 0 1 , ℓ + , ℓ − of the decay process, respectively. dlips(q 1 , p 1 , p 2 , p 3 ) = d 3 q1 is the Lorentz invariant phase space element of the four final particles. The helicities λ e − , λ e + of the initial particles are averaged because in this paper we consider the case of unpolarized beams. Since the polarization of the outgoing particles will not be measured, the helicities of the final particles λ1 and λ 1 , λ + , λ − have to be summed over. Therefore we suppress these helicity indices in the following and only display the helicity λ 2 ofχ 0 2 . The total width Γ 2 of the decaying neutralinoχ 0 2 is small compared to its mass m 2 . Therefore the amplitude T of the combined process is a coherent sum over all polarization states of the helicity amplitude P λ2 for the production process times the helicity amplitude D λ2 for the decay process and a pseudopropagator ∆ 2 = 1 q 2 2 −m 2 2 +im2Γ2 ofχ 0 2 : The amplitude squared for the combined process is thus composed from the unnormalized spin density matrix of the neutralinoχ 0 2 and the decay matrix for the respective decay channel. As in FIG.2 all helicity indices but that of the decaying neutralino are suppressed. Repeated indices are summed over. Squaring the total amplitude one obtains interference terms between various helicity amplitudes. These terms preclude factorization in a production factor λ2 |P λ2 | 2 times a decay factor λ2 |D λ2 | 2 as for the case of spinless particles.
We use the general formalism to calculate the helicity amplitudes for production and decay of a particle with fourmomentum p and mass m. Therefore we introduce three spacelike four-vectors s a µ , (a = 1, 2, 3) which together with p/m form an orthonormal set: A convenient choice for the explicit form of s a is in a coordinate system where the direction of the three-momentum isp = (sin Θ cos Φ, sin Θ sin Φ, cos Θ) [15]: Then in this frame of reference s (1),(2) and s (3) describe transverse and longitudinal polarization of the particle. When computing the density matrices for production and decay, Eqs. (10), (11), we make use of the Bouchiat-Michel formulae [15]: In the amplitude squared (Eq. (9)) the spin vectors s a linearly enter the matrices ρ P (Eq. (10)) and ρ D (Eq. (11)). They induce by Eq. (14) the above mentioned quantum mechanical correlations between production and decay.
If the total width of the decaying particleχ 0 2 is much smaller than its mass, Γ 2 ≪ m 2 , one can make the narrow width approximation and one obtains for the differential cross section in the e + e − -cms, i.e. in the laboratory frame: Θ is the production angle ofχ 0 2 , Φ describes the rotation around the electron beam axis, and dΩ + (dΩ − ) is the solid angle of ℓ + (ℓ − ).
In consequence of momentum conservation the energy E + is determined by E − , the angles θ 2− (θ 2+ ) between ℓ − (ℓ + ) andχ 0 2 , and the opening angle Θ +− between the leptons: The possible region of the lepton energy E − depends on the lepton decay angle θ 2− : so that E + = 0, see Eq. (22), at the upper bound.

III. ANALYTICAL FORMULAE
In this section we give the analytical expressions for the product W = ρ (9)) of the density matrices for the production process, e − (k 1 )e + (k 2 ) →χ 0 1 (q 1 )χ 0 2 (q 2 ), and the direct leptonic decay, . Both the production and the decay process contain contributions from Z 0 exchange in the direct channel and fromẽ L andẽ R exchange in the crossed channels ( compare FIG.1).
The product W in Eq. (9) is a sum of contributions from the different production and decay amplitudes and their interference terms, where the first pair a, b of indices refers to the production process, and the second pair c, d refers to the decay process. The pairs (ab) and (cd) run through all combinations of the values Z, L t , L u , R t , R u , where Z and L, R denote the exchanged particles Z 0 andẽ L ,ẽ R andl L ,l R , respectively. In the case of slepton exchange the indices t, u denote the channel. Thus W ZLt,RtRu is the contribution of the interference term to the production process from Z 0 -exchange andẽ L -exchange in the t-channel and the interference term to the decay process froml R -exchange in thet-channel andl R -exchange in theū-channel (FIG.1). There are altogether 121 contributions W ab,cd which can be classified into 16 groups. We give one representative of each group and list the indices of the possible combinations. The other terms W ab,cd of this group are then obtained by substituting the propagator combination according to the pair of indices (see Eqs. (65)-(70)) and by substituting momenta and couplings and sign factors as given in TABLE I below. The substitutions of the momenta k 1 , k 2 , p 2 , p 3 and of the couplings O L , O R , L, R also have to be performed in Eqs. (41)-(64). In the representative term the sign factors µ, ν, ω, τ, υ have the value +1.
For illustration we consider as an example W LtLu,LtLt , which is the representative of Group 12. In order to get the contribution of W RtRu,LuLu one has to change the propagator combination ∆ and one has to substituteẽ L →ẽ R in the production process and one has to substitutē t →ū in the decay process. When substitutingẽ L →ẽ R , one has to change the sign factors ν, ω, υ and the couplings according to the first line of TABLE I. Moreover, when substitutingt →ū one has to change the sign factors τ, υ, the couplings and the momenta p 2 → p 3 according to the fourth line of TABLE I.
These terms, i.e. Eqs. (52)-(64), would be missing if we had assumed factorization of the differential cross section in production and decay.
For a better transparency all couplings originating from the decay process are marked by a dash. The indices are used in order to emphasize the symmetry between e − and e + in the initial state and ℓ − and ℓ + in the final state, respectively.
We have introduced the following products of propagators and coupling constants:

A. Scenarios
Neutralinos are linear superpositions of the photinoγ, the ZinoZ and the two higgsinosH 0 a andH 0 b . Theγ and Z components only couple to the selectrons whereas the higgsino components couple to the Z 0 . The composition and the masses of the neutralino states depend on the three SUSY mass parameters M, M ′ (sometimes also called M 2 and M 1 ) and µ, whose values follow from the specific SUSY breaking mechanism, and on the ratio tan β = v 2 /v 1 of the vacuum expectation values of the Higgs fields. In order to reduce the number of parameters we shall assume M ′ = 5 3 M tan 2 θ W as suggested by grand unification [12]. The gaugino mass parameter M is related to the gluino mass by M ≈ 0.3mg [17] and the gluino mass is roughly given by mg ≈ 2.4m 1/2 [18], where m 1/2 is the common gaugino mass at M GUT (M ≈ 0.72m 1/2 ).
The masses of the sleptons are related to the SUSY parameters M and tan β and to the common scalar mass parameter m 0 at the unification point [17]: In order to illustrate the influence of the neutralino mixing and of the scalar mass m 0 we shall consider three representative scenarios which differ significantly in the nature of the two lowest mass eigenstatesχ 0 1 andχ 0 2 . The selectron masses are calculated for two values of the scalar mass m 0 , m 0 = 90 GeV and m 0 = 200 GeV. For the parameters of the Standard Model (SM) we take m Z = 91.19 GeV, Γ Z = 2.49 GeV, sin 2 θ W = 0.23 [19] and α = 1/128. We choose tan β = 2. The parameters of our scenarios and the mass eigenvalues of the two lightest neutralinos, the light chargino and the selectrons are given in TABLE II. The width of theχ 0 2 has been computed with the program of [20] (see TABLE II).
Notice that also in scenarios with µ < 0 the branching ratio for the radiative decay is less than 0.5% in the examined region of parameter space [21].
Theγ andZ are mixtures of theB and theW 3 gauginos,γ = cos θ WB + sin θ WW3 ,Z = − sin θ WB + cos θ WW3 . Therefore in TABLE III the components of the neutralino states are given in the basis (γ,Z,H 0 a ,H 0 b ) and in the basis (B,W 3 ,H 0 a ,H 0 b ). In scenario Aχ 0 1 has a dominating photino component andχ 0 2 has a dominating zino component, whereas in scenario B both neutralinos are nearly equal photino-zino mixtures. In the two scenarios A and B theχ 0 1 is almost a pure B-ino and the neutralinoχ 0 2 is nearly a pureW 3 -ino. In scenario C both neutralino states are dominated by strong higgsino components (TABLE III).
For the phase space integration (chosen relative accuracy 10 −3 ) we used the Monte-Carlo routine Vegas. The evaluation was made for cms-energies of √ s = 193 GeV and √ s = 500 GeV. It can be derived from [9] that the total cross section factorizes, i.e. it is independent of the spin correlations. This fact has been used in our numerical calculations as a check for the phase space integration and the total cross sections are given in TABLE IV.

B. Lepton angular distributions
In this section we give numerical results for the angular distributions of ℓ − with respect to the electron beam axis computed with complete spin correlations according to Eqs. (52)-(64). The angular distributions of ℓ + are obtained by substituting cos Θ M− → − cos Θ M+ .
In order to demonstrate the significance of the spin correlations we compare our results with those obtained from the assumption of factorization of the differential cross section into production and decay. As can be seen from  .4 (FIG.5). In both cases the contribution of the spin correlation is negative in the backward direction and positive in the forward direction.
A comparison with the results for the gaugino-like scenario B (FIG.6 and FIG.7) shows how sensitively the spin correlation effect depends on the gaugino-higgsino mixing and on the value of m 0 . It is noteworthy that although in both scenarios A and B theχ 0 1 is B-ino-like andχ 0 2 isW 3 -ino-like with, however, different phases of theW 3 -ino and B-ino admixture. For √ s = 193 GeV and m 0 = 90 GeV (FIG.6) the magnitude of the spin effect is only 2% in the forward and in the backward direction whereas for m 0 = 200 GeV (FIG.7) it amounts to 10%. For higher energy √ s = 500 GeV it is negligible (FIG.8).
It is remarkable that in the case of gaugino-like neutralinos the scalar mass m 0 crucially determines the shape of the angular distributions. This is most obvious for scenario B and √ s = 193 GeV. For the smaller value m 0 = 90 GeV the angular distribution has a maximum nearly perpendicular to the beam direction and is almost FB symmetric, A F B = −0.8% (compare TABLE V). The contribution of the spin correlations is positive in the backward direction and negative in the forward direction (FIG.6). Increasing the value of m 0 completely changes the shape of the angular distribution and for m 0 = 200 GeV it has a minimum in the backward hemisphere and the forward direction is favoured, A F B = +5.9% (FIG.7). Now the contribution of the spin correlations is negative in the backward direction and positive in the forward direction.
In the higgsino-like scenario C both production and decay are dominated by Z 0 -exchange. Therefore the dependence on m 0 is considerably smaller and we give only numerical results for m 0 = 90 GeV. Here the contribution of spin correlations is negligible and the angular distribution is practically flat for √ s = 193 GeV (FIG.9) and FB-symmetric, A F B = 0.07%, and for √ s = 500 GeV with a minimum perpendicular to the beam direction and A F B = 0.11% (FIG.10).
Notice however that a negligible FB-asymmetry is not an unequivocal signature for higgsino-like neutralinos. As can be seen from FIG.6 for scenario B with m 0 = 90 GeV the FB-asymmetry may also be small for gaugino-like neutralinos (see TABLE V). From FIGs.5, 8 and 10 we conclude that for higher energies far enough from threshold the lepton angular distribution is suitable for distinguishing between gaugino-like and higgsino-like neutralinos. For √ s = 500 GeV the angular distribution is practically FB-symmetric with however a maximum for gauginos but a minimum for higgsinos perpendicular to the beam direction.

C. The lepton opening angle distribution
In contrast to the lepton angular distribution, the distribution of the opening angle between both leptons factorizes. Due to the Majorana character of the decaying neutralino the spin correlation terms are just cancelled by this partial phase space integration.
It is also noteworthy that in contrast to the lepton angular distributions, the distributions of the lepton opening angle are similar for both gaugino-like scenarios A and B (FIGs.11 and 12). Especially for √ s = 193 GeV they differ, however, distinctively from that for higgsino-like neutralinos in scenario C (FIG.13).
In the case of gaugino-like neutralinos the distributions for √ s = 193 GeV are rather flat (FIGs.11 and 12). Changing the scalar mass from m 0 = 90 GeV to m 0 = 200 GeV results in a reduction of the cross sections by a factor of approx. 3 in scenario A and by a factor of approx. 4 in scenario B. The shape of the distribution, however, remains essentially unchanged.
One should note that for the shape of the opening angle distributions the influence of varying the value of m 0 is much smaller than for the lepton angular distributions.
For higgsino-like neutralinos (FIG.13), the shape of the opening angle distribution for √ s = 193 GeV is completely different from those of gaugino-like neutralinos. Here the lepton pairs are preferably emitted with small angles between them, approximately 66% of them with an opening angle between 0 and π/2. With increasing energy the opening angle distribution shrinks more and more and for √ s = 500 GeV it displays for both higgsino-and gaugino-like neutralinos a rather narrow peak at or near cos Θ +− = 1. Obviously the distribution of the opening angle between the leptons is suitable to distinguish between gaugino-and higgsino-like neutralinos at lower energies.

D. Energy Distributions
Just as the opening angle distribution the energy distribution of the outgoing lepton factorizes. As a consequence of CP invariance (Sec. 2.1) and the Majorana character of the neutralinos the energy spectra of both leptons, ℓ − and ℓ + , are identical [5]. In

E. Summary
In this paper we have calculated the analytical expression for the differential cross section of the associated production of neutralinos, e − + e + →χ 0 1 +χ 0 2 , and the subsequent direct leptonic decay,χ 0 2 →χ 0 1 + ℓ + + ℓ − , with complete spin correlations between production and decay. The angular and the energy distribution of the outgoing lepton as well as the distribution of the opening angle between both leptons have been computed for cms-energies of √ s = 193 GeV and √ s = 500 GeV. These distributions have been examined with regard to their dependence on spin correlations, on the neutralino mixing character and on the scalar mass parameter m 0 .
The quantum mechanical interference terms between the various polarization states of the decaying neutralino give rise to a strong effect in the lepton angular distribution with respect to the beam axis, whereas the opening angle distribution and the energy distribution are independent from these spin correlations.
For energies not too far above the threshold ( √ s = 193 GeV) the opening angle distribution turns out to be suitable for distinguishing between higgsino-like and gaugino-like neutralinos. However, it is rather insensible to variable mixing in the gaugino sector. Here the shape of the opening angle distribution only slightly depends on the scalar mass m 0 .
The lepton angular distribution, on the other hand, is for lower energies not only very sensitive to the mixing in the gaugino sector but also to the actual value of m 0 .
For energies far above threshold the shape of the lepton angular distribution is rather insensible to the mixing in the gaugino sector and to the value of m 0 . It is, however, extremely different for gaugino-and higgsino-like neutralinos and suitable for distinguishing between them.
If the neutralinos are gaugino-like, the effect of spin correlations in the angular distributions can be large amounting to about 20%. For higgsino-like neutralinos, on the contrary, the contribution of spin correlations is practically negligible.
The energy distributions and the distributions of the opening angle, finally, are independent from the spin correlations. Apart from the magnitude of the cross sections they are rather insensitive to the actual value of m 0 .
The clear structure of the analytical formulae presented here allows to include hadronic decays and to extend the investigations to cascade decays and to production and decay of, for instance,χ 0 2χ 0 2 pairs. Concerning the hadronic decays,χ 0 2 →χ 0 1 qq, where the outgoing quarks develop two jets, we expect a similar shape of the opening angle distribution between quark and antiquark. Thus, the two jets would be better separated for gaugino-like neutralinos than for higgsino-like neutralinos which prefer small opening angle. These investigations as well as the inclusion of beam polarization and the results for the chargino process angular distributions will be discussed in a forthcoming paper with regard to the determination of SUSY parameters.

V. ACKNOWLEDGEMENT
We thank A. Bartl and W. Majerotto for many useful discussions. We are grateful to V. Latussek for his support in the development of the numerical program. This work was also supported by the Deutsche Forschungsgemeinschaft under contract Fr 1064/2-2 and the 'Fond zur Förderung der wissenschaftlichen Forschung' of Austria, Project No. P10843-PHY.  Production