CP violation in chargino production and decay into sneutrino

We study CP odd asymmetries in chargino production E+ E- -->~Chi(+-)(1) \~Chi(-+)(2) and the subsequent two-body decay of one chargino into a sneutrino. We show that in the Minimal Supersymmetric Standard Model with complex parameter \mu the asymmetries can reach 30 %. We discuss the feasibility of measuring these asymmetries at a linear collider with \sqrt{s} = 800 GeV and longitudinally polarized beams.


Introduction
In the the chargino sector of the Minimal Supersymmetric Standard Model (MSSM) [1] the Higgsino mass parameter µ can be complex [2]. It has been shown that in the production of two different charginos, e + e − →χ ± 1χ ∓ 2 , a CP violating phase ϕ µ of µ causes a non-vanishing chargino polarization perpendicular to the production plane [3,4]. This polarization leads at tree level to triple product asymmetries [5,6,7], which might be large and will allow us to constrain ϕ µ at a future e + e − linear collider [8]. Usually it is claimed that this phase has to be small for a light supersymmetric (SUSY) particle spectrum due to the experimental upper bounds of the electric dipole moments (EDMs) [9]. However, these restrictions are model dependent [10]. If cancellations among different contributions occur and, for example, if lepton flavor violating phases are present, the EDM restrictions on ϕ µ may disappear [11].
We define the triple product and the T odd asymmetry of the cross section σ for chargino production (1) and decay (2). The asymmetry A T ℓ is not only sensitive to the phase ϕ µ , but also to absorptive contributions, which could enter via schannel resonances or final-state interactions. In order to eliminate the contributions from the absorptive parts, which do not signal CP violation, we will study the CP asymmetry whereĀ T ℓ is the asymmetry for the CP conjugated process In this context it is interesting to note that in chargino production it is not possible to construct a triple product and a corresponding asymmetry by using transversely polarized e + and e − beams [3,12], therefore, one has to rely on the transverse polarization of the produced chargino.
In Section 2 we give our definitions and formalism used, and the analytical formulae for the chargino production and decay cross sections. In Section 3 we discuss some general properties of the CP asymmetries. In Section 4 we present numerical results for A ℓ and the cross sections. Section 5 gives a summary and conclusions.

Lagrangians and couplings
The MSSM interaction Lagrangians relevant for our study are (in our notation and conventions we follow closely [1,13]): with the couplings: with i, j = 1, 2. Here P L,R = 1 2 (1 ∓ γ 5 ), g = e/ sin θ W is the weak coupling constant, and e ℓ and T 3ℓ denote the charge and the third component of the weak isospin of the lepton ℓ. The τ -Yukawa coupling is given by where v 1,2 are the vacuum expectation values of the two neutral Higgs fields. The chargino mass and h ± the two-component spinor fields of the W-ino and the charged Higgsinos, respectively. The complex unitary 2 × 2 matrices U mn and V mn diagonalize the chargino

Cross section
We choose a coordinate frame such that in the laboratory system the four momenta are: with the beam energy E b = √ s/2, the scattering angle θ ( p e − , pχ− j ) and the azimuth φ is chosen zero. For the description of the polarization of charginoχ + i we choose three spin vectors in the laboratory system: Together with p μ χ + i /mχ+ i they form an orthonormal set. For the calculation of the cross section for the combined process of chargino production (1) and the subsequent two-body decay ofχ + i (2), we use the spin-density matrix formalism as in [13,14]. The amplitude squared, is composed of the (unnormalized) spin-density production matrix ρ P (χ + i ) and the decay matrix ρ D (χ + i ), with the helicity indices λ i and λ ′ i of the chargino. The propagator is given by ∆ ]. The production matrix ρ P (χ + i ) can be expanded in terms of the Pauli matrices σ a , a = 1, 2, 3: With our choice of the spin vectors s ã χ + i , Eq. (15), Σ 3 P /P is the longitudinal polarization ofχ + i in the laboratory system, Σ 1 P /P is the transverse polarization in the production plane and Σ 2 P /P is the polarization perpendicular to the production plane. The analytical formulae for the expansion coefficients P and Σ a P are given in [13]. The coefficient Σ 2 P is non-zero only for production of an unequal pair of charginos, e + e − →χ ± 1χ ∓ 2 , and obtains contributions from Z-exchange and Z-ν interference only [13]: with The propagators are defined by and the longitudinal electron and positron beam polarizations, P e − and P e + , respectively, are included in the coefficients The contribution (19) from Z-exchange is non-zero only for ϕ µ = 0, π, whereas the Z-ν interference term, Eq. (20), obtains also absorptive contributions due to the finite Z-width which do not signal CP violation. These, however, will be eliminated in the asymmetry A ℓ , Eq. (5).
Analogously to the production matrix, the chargino decay matrix can be written as For the chargino decay (2) into an electron or muon sneutrino the coefficients are: where the sign in parenthesis holds for the conjugated processχ − i → ℓ −ν ℓ . For the decay into the tau sneutrino the coefficients are given by where the sign in parenthesis holds for the conjugated processχ − i → τ −ν τ .
Inserting the density matrices (17) and (24) in Eq. (16) leads to: The cross section and distributions in the laboratory system are then obtained by integrating |T | 2 over the Lorentz invariant phase space element, where we use the narrow width approximation for the chargino propagator.

CP asymmetries
Inserting the cross section (28) in the definition of the asymmetry (4) we obtain: In the numerator only the CP sensitive contribution Σ 2 P Σ 2 D from chargino polarization perpendicular to the production plane remains, since only this term contains the triple product T ℓ = ( p e − × pχ+ i ) · p ℓ , Eq. (3). In the denominator only the term P D remains, since all spin correlations a Σ a P Σ a D vanish due to the integration over the complete phase space. For chargino decay into a tau sneutrino,χ + i → τ +ν τ , the asymmetry is reduced, which follows from the expressions for D and Σ 2 D , given in Eq. (26). With S ℓ the standard deviations, the relative statistical error of the asymmetry A T ℓ is given by δA T [7], where N = L · σ is the number of events with L the integrated luminosity and the cross section σ = σ P (e + e − →χ + iχ − j )×BR(χ + i → ℓ +ν ℓ ). For the CP asymmetry A ℓ , defined in Eq. (5), we have ∆A ℓ = ∆A T ℓ / √ 2. Taking δA ℓ = 1 it follows S ℓ = |A ℓ | √ 2L · σ. Note that in order to measure A ℓ the momentum ofχ + i , i.e. the production plane, has to be determined. This could be accomplished by measuring the hadronic decay of the other charginoχ − j , if the masses of the charginos and the sneutrinos are known. Also it is clear that detailed Monte Carlo studies taking into account background and detector simulations are necessary to predict the expected accuracy. However, this is beyond the scope of the present work.

Numerical results
We present numerical results for the asymmetries A ℓ for ℓ = e, µ, Eq. (5), and the cross sections σ = σ P (e + e − →χ + 1χ − 2 ) × BR(χ + 1 → ℓ +ν ℓ ). We study the dependence of the asymmetries and cross sections on the MSSM parameters µ = |µ| e i ϕµ , M 2 and tan β. We choose a center of mass energy of √ s = 800 GeV and longitudinally polarized beams with beam polarizations (P e − , P e + ) = (−0.8, +0.6), which enhanceν e exchange in the production process. This results in larger cross sections and asymmetries.
The M 2 -ϕ µ dependence of the CP asymmetry A ℓ for ℓ = e or µ is shown in Fig. 1b. The asymmetry can be as large as 10% and it does, however, not attain maximal values for ϕ µ = 0.5π, which one would naively expect. The reason is that A ℓ is proportional to a product of a CP odd (Σ 2 P ) and a CP even factor (Σ 2 D ), see Eq. (29). The CP odd (CP even) factor has as sine-like (cosine-like) dependence on ϕ µ . Thus the maximum of A ℓ is shifted towards ϕ µ = ±π in Fig. 1b. Phases close to the CP conserving points, ϕ µ = 0, ±π, are favored by the experimental upper limits on the EDMs. For example in the constrained MSSM, we have |ϕ µ | < ∼ π/10 [9]. However, the restrictions are very model dependent, e.g., if also lepton flavor violating terms are included [11], the restrictions may disappear. In order to show the full phase dependence of the asymmetries, we have relaxed the EDM restrictions for this purpose.   The area A is kinematically forbidden by mν ℓ + mχ0 1 > mχ+ 1 .
For M 2 = 200 GeV, we show the tan β-ϕ µ dependence of σ and A ℓ in Figs. 2a,b. The asymmetry can reach values up to 30% and shows a strong tan β dependence and decreases with increasing tan β. The feasibility of measuring the asymmetry depends also on the cross section σ = σ P (e + e − →χ + 1χ − 2 ) × BR(χ + 1 → ℓ +ν ℓ ), Fig. 2a, which attains values up to 20 fb.

Summary and conclusions
We have studied CP violation in chargino production with longitudinally polarized beams, e + e − →χ + iχ − j , and subsequent two-body decay of one chargino into the sneutrinoχ + i → ℓ +ν ℓ . We have defined the T odd asymmetries A T ℓ of the triple product ( p e − × pχ+ i ) · p ℓ . The CP odd asymmetries A ℓ = 1 2 (A T ℓ −Ā T ℓ ), whereĀ T ℓ denote the CP conjugated of A T ℓ , are sensitive to the phase ϕ µ of the Higgsino mass parameter µ. At tree level, the asymmetries have large CP sensitive contributions from spin correlation effects in the production of an unequal pair of charginos. In a numerical discussion for e + e − →χ + 1χ − 2 production, we have found that A ℓ for ℓ = e or µ can attain values up to 30%. By analyzing the statistical errors, we have shown that, even for of e.g. ϕ µ ≈ 0.9π, the asymmetries could be accessible in future e + e − collider experiments in the 800 GeV range with high luminosity and longitudinally polarized beams.