CP-violating asymmetry in chargino decay into neutralino and W boson

In the MSSM with complex parameters, loop corrections to $\cha_i^\pm \to \neu_j^0 W^\pm$ lead to a CP violating asymmetry $\ACP = (\Gamma(\cha_i^+ \to \neu_j^0 W^+) - \Gamma(\cha_i^- \to \neu_j^0 W^-))/ (\Gamma(\cha_i^+ \to \neu_j^0 W^+) + \Gamma(\cha_i^- \to \neu_j^0 W^-))$. We calculate this asymmetry at full one-loop level. We perform a detailed numerical analysis for $\cha_1^\pm \to \neu_1^0 W^\pm$ and $\cha_2^\pm \to \neu_1^0 W^\pm$ analyzing the dependence on the parameters and phases involved. Asymmetries of several percent are obtained. We also discuss the feasability of measuring these asymmtries at LHC.

It is well known that supersymmetric models contain new sources of CP violation if the parameters are complex. In the Minimal Supersymmetric Standard Model (MSSM), the U(1) and SU (2) gaugino mass parameters M 1 and M 2 , respectively, the higgsino mass parameter µ, as well as the trilinear couplings A f (corresponding to a fermion f ) may be complex. Usually, M 2 is made real by redefining the fields. Non-vanishing phases of M 1 and µ cause CP-violating effects already at tree-level in the chargino and neutralino production and decay [1,2,3]. In case the trilinear couplings of the third generation (A t , A b , A τ ) are complex not only the stop, sbottom, and stau sectors [4] are strongly affected but also the Higgs sector [5,6]. The three neutral Higgs bosons are no more CP eigenstates.
Although new phases in addition to the CKM in the Standard Model (SM) are desirable to explain baryogenesis, there are severe constraints on the phase of µ from the experimental limits on the electric dipole moments (EDMs) of the electron, neutron and Hg. For example, in the constraint MSSM |φ µ | has to be small [7,8] for a SUSY particle spectrum of the order of a few TeV.
In this note, we study CP violation in the decaysχ + i (k 2 ) →χ 0 j (k 1 ) + W + (−p) and χ − i (−k 2 ) →χ 0 j (−k 1 ) + W − (p) in the MSSM with complex parameters by calculating the CP-violating asymmetry at full one-loop order. The asymmetry is zero if CP is conserved and also vanishes at tree-level in case of CP violation. In Fig. 1 we show the graphs which contribute to this asymmetry at one-loop level. Of course, they give a contribution to A CP only if they have an absorptive part, i.e. some decay channels ofχ ± i must be open in addition to that intõ χ 0 j W ± . This asymmetry was already calculated in [9] considering only the third generation quarks and squarks in the vertex graphs. We have improved this calculation in several points. First, we performed a full one-loop calculation. In particular, we also calculated the contributions from self-energies of the charginos. It turns out, that these are important. (The self-energies of the neutralinos do not contribute due to their Majorana nature and the W ± -H ± transition vanishes for on-shell W -bosons.) In addition, we take the Yukawa couplings running, which also gives a sizeable effect. Moreover, we take into account that the neutral Higgs bosons (h 0 , H 0 , A 0 ) mix if the SUSY parameters mentioned are complex. In our case, this influence is, however, very small. As a loop-level quantity the asymmetry A CP depends on the phases of all complex parameters involved. One, however, expects that the dependence on the phases of M 1 and A t,b is strongest (taking µ real). There is even a strong correlation between them. Therefore, a measurement of this asymmetry represents not only a test of CP violation in chargino decay, but can also be used for the determination of the phases of M 1 and A t,b .  Figure 1: All one-loop graphs of the decayχ + i →χ 0 j + W + , which contribute to the CP asymmetry A CP defined in eq. (1), f ′ (f ′ ) denotes the isospin doublet partner of the fermion f (sfermionf ), e.g. f = t, f ′ = b, φ 0 = (H 0 1 , H 0 2 , H 0 3 , G 0 ), and φ + = (H + , G + ).
The widths Γ (±) can be written as Γ (±) ∝ |M Since |M tree | 2 , and assuming, that the one-loop contribution is small compared to the tree-level one, the CP-violating asymmetry A CP takes the form with the squared tree-level amplitude and the one-loop contributions with the kinematic factor λ = λ(m 2 . The chargino-neutralino-W coupling parameters O L,R , defined by the Lagrangian are where U, V , and Z are the matrices diagonalizing the chargino and neutralino system (see eqs. (A.16) and (A.17)). Λ and Π are form factors which are given in the Appendix A. We only give the form factors forχ + and not forχ − , so that Λ, Π always stands for Λ (+) , Π (+) . The form factors Λ (−) and Π (−) , belonging to theχ − decay, can be easily obtained by conjugating all couplings.
In Appendix A we present all formulas for the vertex contributions with thettb and bbt loops and the chargino self-energy contribution with thetb loop, see graphs SF 1 F 2 , F S 1 S 2 , and SF of Fig. 1. The complete analytical formulas will be given in [10]. All individual one-loop graphs were numerically checked using the packages FeynArts, FormCalc, and LoopTools [11], and FF [12]. We included the CP-violating mixing of the neutral Higgs bosons by writing our own FeynArts model file. For the numerical program we used FeynHiggs [13].

Numerical results
We present numerical results for the decay rate asymmetries A CP according to eq. (1) χ ± i →χ 0 j W ± , for i = 1, 2 and j = 1. A discussion of the other channels will be given in [10]. For the SM input parameters we take m Z = 91.1875 GeV, m W = 80.45 GeV, cos θ W = m W /m Z , α(m Z ) = 1/127.9, the on-shell parameters m t = 178 GeV, and m τ = 1.777 GeV. For the bottom mass, our input is the MS value m b (m b ) = 4.2 GeV. For the values of the Yukawa couplings of the third generation quarks (h t , h b ), we take the running ones at the scale of the decaying particle mass. In principle, the parameters A f , the U(1) gaugino mass parameter M 1 of the neutralino sector, and µ can be complex. We assume that |M 1 | = M 2 /2. In general, there are 15 independent sfermion breaking mass parameters. We take MQ as input and assume the MSUGRA inspired ratios mq : In order to reduce the number of input parameters further, we use In all figures we take M A 0 = 300 GeV, tan β = 10, and φ µ = π/10.
For the decayχ ± 1 →χ 0 1 W ± , the total one-loop asymmetry A CP is shown in Fig. 2a and the tree-level branching ratio (BR) in Fig. 2b, for M 2 = 500 GeV, |A| = 400 GeV, φ A = −π/4, φ M 1 = 3π/4, and three values of MQ as a function of |µ|. |A CP | increases for increasing values of |µ| because the tree-level decay width ofχ ± 1 →χ 0 1 W ± goes to zero, asχ 0 1 becomes almost a pure bino which does not couple to W ± . Therefore, for |µ| > ∼ 550 GeV the branching ratio drops below 1%. The higher the value of MQ the heavier becomes the stop mass. Hence A CP goes down but the branching ratio in (b) increases.   Now we discuss the asymmetry A CP forχ ± 2 →χ 0 1 W ± . Fig. 4a shows the dependence of the asymmetry A CP on the gaugino mass parameter M 2 for various values of |A|, φ M 1 = π, MQ = 300 GeV, φ A = −π/4, and |µ| = 200 GeV. For M 2 > 200 GeV, the lighter chargino and the two lighter neutralinos have dominating higgsino components and the heavier chargino is mostly gaugino-like (> 90%). The bigger |A|, the bigger is the mixing in the squark sector and hence A CP . Around M 2 ∼ 450 GeV theχ + 2 becomes massive enough so that the channels into bt 2 and tb 1,2 open. For M 2 > ∼ 250 GeV, the third generation (s)quark contributions clearly dominate the asymmetry, the self-energy contribution being bigger than the vertex contribution. For M 2 < 680 GeV, the vertex and the self-energy contributions for the third generation (s)quarks have opposite signs and cancel each other to a high degree. Nevertheless, they remain the dominant contributions in a large part of Fig. 4b.
Various pseudothresholds are visible in Fig. 5a, where the squark mass parameter MQ is varied. The parameter set M 2 = 450 GeV, φ M 1 = π, and |µ| = 200 GeV gives the masses mχ+ 2 = 468.55 GeV and mχ0 1 = 185.66 GeV. The strong dependence of A CP on the phase is clearly visible. Fig. 5b illustrates the dependence on the phase φ A for MQ = {230, 300, 400} GeV. That A CP does not factorize into a φ A dependent and a φ A independent part can be seen from the fact that the three curves do not meet in a single point. The other phases φ µ and φ M 1 distort the factorization.
The |µ| dependence of the decayχ ± 2 →χ 0 1 W ± is shown in Fig. 6 for the same parameter set as used in Fig. 2. The total one-loop asymmetry A CP is shown in Fig. 6a and the tree-level branching ratio (BR) in Fig. 6b. In the region |µ| ∼ 400 GeV to 600 GeV the character of theχ + 2 andχ 0 1 changes, forχ + 2 from gaugino to higgsino and forχ 0 1 from higgsino to mainly bino. Therefore, one has a strong dependence in A CP and BR there. The dependence on MQ is analogous to that in Fig. 2. For |µ| > ∼ 600 GeV, the mass ofχ + 2 ∼ |µ| andχ 0 1 ∼ M 2 /2 = 250 GeV. Therefore, the decay width ofχ ± 2 →χ 0 1 W ± (1)  (1); the chargino selfenergy contribution with the third generation (s)quarks in the loop in red (2); vertex and self-energy corrections with all other (s)fermions in the loop in blue (3); all remaining corrections in green (4). increases with |µ| and A CP goes to zero. The hump in Fig. 6b at |µ| ∼ 600 GeV for MQ = 500 GeV is due to the opening of thet 2 b channel.
It is known that the electric dipole moments (EDM) of the electron, the neutron and mercury strongly depend on the phase of µ for a light SUSY spectrum [14]. The experimental constraints for the EDMs of the electron [15], the neutron [16], and mercury [17] can  be fulfilled by heavy sfermions of the first generations [18] or if cancellations of different contributions occur [8]. We checked for all plots all three EDMs and found always (small) values of φ µ that fulfill all EDM constraints.
Finally, we want to comment on the measurability of this asymmetry. At LHC charginos are mainly produced in the cascade decays of gluinos and squarks so that the production rate strongly depends on their masses. If the gluino and squark masses are about the same, the gluino production cross section is far the dominant one. With mg ∼ mq = 750 GeV, we expect roughly 2.4 × 10 5 events containingχ ± 1 (one has the same amount ofχ + 1 andχ − 1 in the case where they originate from gluinos or from a gluon-gluon process), assuming a luminosity of 10 5 pb −1 and a branching ratio of a gluino decaying into aχ ± 1 of 40%. Taking into account the branching ratio forχ ± 1 →χ 0 1 W ± , one can measure A CP for this decay with a statistical significance of ∼ 2 (confidence level of 95%). For measuring A CP forχ ± 2 →χ 0 1 W ± , assuming 5 × 10 4 events containing aχ + 2 orχ − 2 , one gets a similar statistical significance.

Conclusions
We have calculated the CP-violating asymmetry between the partial decay rates Γ(χ + i → χ 0 j W + ) and Γ(χ − i →χ 0 j W − ) due to phases in the MSSM. It is a pure loop effect. We have calculated this asymmetry at full one-loop order. We have given numerical results forχ ± 1 →χ 0 1 W ± andχ ± 2 →χ 0 1 W ± . The respective asymmetries are of the order of several percent, depending on the values of parameters and phases involved. In order to have reasonable branching ratios for the decays theχ 0 1 must not be very bino like. We also discussed the feasability of measuring such an asymmetry at LHC. It might be possible to measured it with a confidence level of 95%.

A The (s)top/(s)bottom contributions to A CP
The relevant parts of the Lagrangian are with the coupling parameters to the W boson, the chargino, and the neutralino, with the gaugino components of the neutralino and the Yukawa couplings The charge and the isospin of the quark q are given by e q and I q 3L , g is the SU(2) coupling parameter. The mixing matrices are defined as with the mass matrices where the abbreviations s W = sin θ W , c W = cos θ W , s β = sin β, c β = cos β, and are introduced for a more convenient notation.
is the soft breaking mass parameter for the right stops (sbottoms).