A CP asymmetry in e^+e^- \to \chi^0_i\chi^0_j \to \chi^0_j \tau stau_k with tau polarization

We propose a CP-odd asymmetry in the supersymmetric process e^+e^- \to \chi^0_i \chi^0_j \to \chi^0_j \tau stau_k by means of the transverse \tau polarization. We calculate the asymmetry and cross sections at a future linear collider in the 500 GeV c.m.s. energy range with longitudinal polarized beams and high luminosity. We work in the Minimal Supersymmetric Standard Model with complex parameters \mu, M_1 and A_\tau. The asymmetry can reach values up to 60 %. We also estimate the sensitivity for measuring the \tau polarization necessary to probe the CP asymmetry.


Introduction
In supersymmetric (SUSY) extensions of the Standard Model (SM), some parameters can be complex. In the neutralino sector of the Minimal Supersymmetric Standard Model (MSSM), these are the higgsino mass parameter µ and the gaugino mass parameter M 1 , while M 2 can be chosen real by redefining the fields. In addition, in the scalar tau sector of the MSSM, also the trilinear scalar coupling parameter A τ can be complex. The non-zero phases ϕ µ , ϕ M 1 and ϕ Aτ of these parameters give rise to CP-odd observables, which are not present if CP is maintained. Measurements of such CP-odd observables will allow us to determine these phases, in particular also their signs.
In this Letter we consider neutralino production e + e − →χ 0 iχ 0 j ; i, j = 1, . . . , 4 (1) and the subsequent two-body decay of one neutralinõ for a fixed τ -polarization. We would like to stress that without measuring the transverse τ ∓ polarization no sensitivity to the phase of A τ , ϕ Aτ , can be obtained, because (2) is a two-body decay. When summing over the τ − polarization, we are sensitive only to CP violation in the production process [1,2]. The τ − polarization is given by [3] P = Tr(̺ σ σ σ) with ̺ being the hermitean spin density matrix of the τ − and σ i the Pauli matrices. We use a convention for P = (P 1 , P 2 , P 3 ) where the component P 3 is the longitudinal polarization and P 1 is the transverse polarization in the plane formed by p e − and p τ . The component P 2 is the polarization perpendicular to p τ and p e − and is proportional to the triple-product where s τ is the τ − spin 3-vector. Since under time reversal the triple-product changes sign, the transverse τ − polarization P 2 is a T-odd observable. Due to CPT invariance, P 2 is actually a CP-odd observable if absorbtive phases from final-state interactions are neglected.
In this Letter we study the asymmetry which is CP-odd, even if absorbtive phases can not be neglected. In Eq. (5), P denotes the τ − polarization vector in the decayχ 0 i →τ + m τ − andP denotes the τ + polarization vector in the decayχ 0 i →τ − m τ + . In Born approximation it follows from Eq. (5) that A CP = P 2 .
In Section 2 we briefly review stau mixing in the MSSM and define the part of the interaction Lagrangian which is relevant for our analysis. In Section 3 we define the τ spin density matrix ̺ and give the anlytical formulae. In Section 4 we discuss the qualitative properties of the asymmetry A CP . We present numerical results for e + e − →χ 0 1τ 1 τ in Section 5. We summarize and conclude in Section 6.

Stau mixing and Lagrangian
We give a short account ofτ L −τ R mixing for complex µ = |µ|e iϕµ , A τ = |A τ |e iϕ Aτ and M 1 = |M 1 |e iϕ M 1 . The masses and couplings of the τ -sleptons follow from the hermitian 2 × 2 mass matrix which in the basis (τ L ,τ R ) reads [4,5] with where m τ is the mass of the τ -lepton, Θ W is the weak mixing angle, tan β = v 2 /v 1 with v 1 (v 2 ) being the vacuum expectation value of the Higgs field H 0 1 (H 0 2 ), and ML, MẼ, A τ are the soft SUSY-breaking parameters of theτ i system. Theτ mass eigenstates are (τ 1 , and The mass eigenvalues are The part of the interaction Lagrangian of the MSSM relevant to study the decay (2) reads (in our notation and conventions we follow closely [7,8]): with and where Y τ = m τ / √ 2m W cos β, P L,R = 1/2(1 ∓ γ 5 ) and g is the weak coupling constant. N is the 4×4 unitary neutralino mixing matrix, which diagonalizes the neutral gaugino-higgsino mass matrix . The part of the Lagrangian for the neutralino production (1) can be found e.g. in [10,11].

Tau spin density matrix
The unnormalized, hermitean, 2 × 2 spin density matrix of the τ − is defined by: where M and dLips are the amplitude squared and the Lorentz invariant phase space element, respectively, for the whole process of neutralino production (1) and decay (2). The τ − helicities are denoted by λ k and λ ′ k . In the spin density matrix formalism (as used e.g. in [9,10]) the amplitude squared can be written as It is composed of the unnormalized spin density matrices ρ P for the production (1) and ρ D for the decay (2), the propagator ∆(χ 0 being the four-momenta, masses and widths of the decaying neutralino, respectively. ρ P and ρ D carry the helicity indices λ i , λ ′ i of the neutralinos and/or the helicity indices λ k , λ ′ k of the τ − . The factor 2 in Eq. (19) is due to the summation of theχ 0 j helicities, whose decay is not considered. We introduce a set of spin basis vectors s a χ i (a = 1, 2, 3) for the neutralinoχ 0 i , which fulfill the orthonormality relations s a χ i · s b χ i = −δ ab and s a χ i · p χ i = 0. Then the spin density matrices can be expanded in terms of the Pauli matrices: The analytical formulae of P and Σ a P can be found in [10]. Introducing also a set of spin basis vectors s b τ for the τ − , D λ k λ ′ k and (Σ a D ) λ k λ ′ k can be expanded: The expansion coefficient are given by with ǫ 0123 = 1. The last term in Eq. (26) contains the triple product (4). This term is proportional to Im(bτ mi * aτ mi ) and is therefore sensitive to the phases ϕ Aτ , ϕ µ and ϕ M 1 . Inserting the density matrices of Eq. (20) into Eq. (19) yields 4 Transverse tau polarization and CP asymmetry From Eqs. (3) and (27) we obtain for the transverse τ − polarization which follows because in the numerator we have used |∆(χ 0 i )| 2 P D 2 dLips = 0 and in the denominator we have used |∆(χ 0 i )| 2 Σ a P Σ a D dLips = 0. As can be seen from Eq. (28), P 2 is proportional to the spin correlation term Σ a2 D , Eq. (26), which contains the CP-sensitive part Im(bτ mi * aτ mi )ǫ µνρσ p τ µ pχ i ν s ã χ i ρ s 2 τ σ . In order to study the dependence of P 2 on the parameters, we expand using Eqs. (15)-(17) for m = 1. If CP violation is solely due to ϕ Aτ = 0 (mod π), we find from (29) that P 2 ∝ sin 2θτ sin ϕτ . We note that the dependence of ϕτ on ϕ Aτ is weak if |A τ | ≪ |µ| tan β, see Eq. (10). Thus, we expect that P 2 increases with increasing |A τ |.
Details concerning phase space and kinematics necessary for the calculation of P 2 from Eq. (28) can be found in [2]. The τ − spin vectors s b τ are chosen by: In order to measure P 2 and the CP asymmetry A CP , Eq. (5), the τ − from the decay (2) and the τ + from the subsequentτ + m decay,τ + m →χ 0 1 τ + , have to be distinguished. This can be accomplished by measuring the energies of the τ 's and making use of their different energy distributions [2].
A potentially large background may be due to stau production e + e − →τ + lτ − m → τ + τ −χ0 1χ 0 1 . However, these reactions would generally lead to "two-sided events", whereas the events from e + e − →χ 0 1χ 0 i → τ + τ −χ0 1χ 0 1 are "one-sided events". Moreover, the background reaction is CP-even and will not give rise to a CP asymmetry, because the staus are scalars with a two-body decay.
In Fig. 1 we show the contour lines for A CP in the ϕ Aτ -|A τ | plane. As can be seen A CP is proportional to sin 2θτ sin ϕτ , which is expected from Eq. (29). A CP increases with increasing |A τ | ≫ |µ| tan β, which also follows from Eq. (29). Furthermore, in the parameter region shown the cross section σ varies between 20 fb and 30 fb.
In Fig. 2 we show the dependence of A CP on tan β and ϕ M 1 . Large values up to ±20% are obtained for tan β ≈ 5. Note that these values are obtained for ϕ M 1 ≈ ±0.8π rather than for maximal ϕ M 1 ≈ ±0.5π. This is due to the complex interplay of the spin correlation terms in Eq. (28). In the region shown in Fig. 2, the cross section σ varies between 10 fb and 30 fb.
Figs. 3a and 3b show, for ϕ Aτ = 0.5π and ϕ M 1 = ϕ µ = 0, the |µ|-M 2 dependence of the cross section σ and the asymmetry A CP , respectively. The asymmetry reaches values up to −15% due to the large value of |A τ | = 1 TeV and the choice of the beam polarization (P e − , P e + ) = (−0.8, 0.6). This choice also enhances the cross section, which reaches values of more than 100 fb. The gray shaded area excludes chargino masses mχ± 1 < 104 GeV. In the blank area either the sum of the masses of the produced neutralinos exceeds √ s = 500 GeV or the two-body decayχ 0 2 →τ + 1 τ − is not open.
For ϕ M 1 = 0.5π and ϕ µ = ϕ Aτ = 0 we show in Figs. 4a,b the contour lines of σ and A CP in the |µ|-M 2 plane, respectively. It is remarkable that in a large region the asymmetry is larger than -10% and reaches values up to -40% while also the cross section is large. Unpolarized beams would reduce the largest values of σ by a factor 3, whereas A CP would only be marginally reduced.
For |µ| = 300 GeV and M 2 = 400 GeV, we show in Figs. 5a,b contour lines of σ and A CP , respectively, in the ϕ µ -ϕ M 1 plane. As can be seen the asymmetry A CP is very sensitive to variations of the phases ϕ M 1 and ϕ µ . Even for small phases, A CP can be sizable. Small values of the phases, especially of ϕ µ , are suggested by constraints on electron and neutron electric dipole moments (EDMs) [15] for a typical SUSY scale of the order of a few 100 GeV (for a review see, e.g., [16]).
The polarization of the τ is analyzed through its decay distributions. The sensitivities for measuring the polarization of the τ lepton for the various decay modes are         quoted in [17]. The numbers quoted are for an ideal detector and for longitudinal τ polarization and it is expected that the sensitivities for transversely polarized τ leptons are somewhat smaller. Combing informations of all τ decay modes a sensitivity of S = 0.35 [18] has been obtained. Following [17], the relative statistical error of P 2 (and ofP 2 analogously) can be calculated as δP 2 = ∆P 2 /|P 2 | = σ s /(S|P 2 | √ N), for σ s standard deviations, where N = σL is the number of events with integrated luminosity L and cross section σ = σ p (e + e − →χ 0 1χ 0 2 ) × BR(χ 0 2 →τ + 1 τ − ). Then for A CP , Eq. (5), it follows that ∆A CP = ∆P 2 / √ 2. We show in Fig. 6 the contour lines of the sensitivity S = √ 2/(|A CP | √ N) which is needed to measure A CP at 95% CL (σ s = 2) for L =500 fb −1 , for the parameters ϕ Aτ = 0.2π, ϕ M 1 = ϕ µ = 0. In Fig. 7 we show the contour lines of the sensitivity s for the parameters ϕ M 1 = 0.2π and ϕ µ = ϕ Aτ = 0. It is interesting to note that in a large region in the |µ|-M 2 plane in Figs. 6 and 7 we obtain a sensitivity S < 0.35, which means that the asymmetries can be measured at 95% CL.

Summary and conclusion
We have proposed and analyzed the CP odd asymmetry A CP in Eq. (5) in neutralino production e + e − →χ 0 iχ 0 j and the subsequent two-body decay of one neutralinoχ 0 i →τ ± k τ ∓ . The asymmetry is due to the transverse τ ∓ polarization, which is non-vanishing if CP-violating phases of the the trilinear scalar coupling parameter A τ and/or the gaugino and higgsino mass parameters M 1 , µ are present. The asymmetry occurs already at tree level and is due to spin effects in the neutralino production and decay process. In a numerical study for e + e − →χ 0 1χ 0 2 and neu- The blank area outside the area of the contour lines is kinematically forbidden since here either √ s < mχ 1 + mχ 2 or mτ 1 +m τ > mχ 2 . The gray area is excluded since mχ± 1 < 104 GeV. The blank area outside the area of the contour lines is kinematically forbidden since here either √ s < mχ 1 + mχ 2 or mτ 1 +m τ > mχ 2 . The gray area is excluded since mχ± 1 < 104 GeV.
tralino decayχ 0 2 →τ ± 1 τ ∓ we have shown that the asymmetry can be as large as 60%. It can be sizeable even for small phases of µ and M 1 , which is suggested by the experimental limits on EDMs. Depending on the MSSM scenario, the asymmetry should be accessible in future electron-positron linear collider experiments in the 500 GeV range. Longitudinally polarized electron and positron beams can considerably enhance both the asymmetry and the production cross section.