Impact of CP phases on stop and sbottom searches

We study the decays of top squarks (stop_{1,2}) and bottom squarks (sbottom_{1,2}) in the Minimal Supersymmetric Standard Model (MSSM) with complex parameters A_t, A_b, mu and M_1. We show that including the corresponding phases substantially affects the branching ratios of stop_{1,2} and sbottom_{1,2} decays in a large domain of the MSSM parameter space. We find that the branching ratios can easily change by a factor of 2 and more when varying the phases. This could have an important impact on the search for stop_{1,2} and sbottom_{1,2} and the determination of the MSSM parameters at future colliders.


Introduction
The experimental studies of supersymmetric (SUSY) particles will play an important role at future colliders. Studying the properties of the 3rd generation sfermions will be particularly interesting because of the effects of the large Yukawa couplings. The lighter of their mass eigenstates may be the lightest charged SUSY particles and they could be investigated at an e + e − linear collider [1]. Moreover, they could also be copiously produced in the decays of heavier SUSY particles. Several phenomenological studies on SUSY particle searches have been performed in the Minimal Supersymmetric Standard Model (MSSM) [2] with real SUSY parameters. Analyses of the decays of the 3rd generation sfermionst 1,2 ,b 1,2 ,τ 1,2 andν τ in the MSSM with real parameters have been made in Refs. [3,4], and phenomenological studies of production and decays of the 3rd generation sfermions at future e + e − linear colliders in Ref. [5].
In general, however, some of the SUSY parameters may be complex, in particular the higgsino mass parameter µ, the gaugino mass parameters M 1,2,3 and the trilinear scalar coupling parameters A f of the sfermionsf . The SU(2) gaugino mass parameter M 2 can be chosen real after an appropriate redefinition of the fields. The experimental upper bounds on the electric dipole moments (EDMs) of electron, neutron and the 199 Hg and 205 Tl atoms may pose severe restrictions on the phases of the SUSY parameters, though they are model dependent. In the constrained MSSM the phase of µ turns out to be restricted to the range |ϕ µ | < ∼ 0.1 if all SUSY masses are in the TeV range [6]. On the other hand, in more general models, such as those with lepton-flavour violation, no restriction on the phase ϕ µ from the electron EDM at one-loop level is obtained [7]. Restrictions arising due to two-loop contributions to the EDMs are less severe [8].
Therefore, in a complete phenomenological analysis of production and decays of the SUSY particles one has to take into account that A f , µ and M 1 can be complex. The most direct and unambiguous way to determine the imaginary parts of the complex SUSY parameters could be done by measuring relevant CP-violating observables. For example, for the case of sfermion decays CP-violating [9,10] and T-violating [11] observables have been proposed. On the other hand, also the CP-conserving observables depend on the phases of the complex parameters, because in general the mass-eigenvalues and the couplings of the SUSY particles involved are functions of the underlying complex parameters. For example, the decay branching ratios of the Higgs bosons depend strongly on the complex phases of thet andb sectors [12,13,14], while those of the stausτ 1,2 and τ -sneutrinoν τ can be quite sensitive to the phases of the stau and gaugino-higgsino sectors [15]. Also the Yukawa couplings of the third generation sfermions are sensitive to the SUSY phases at one-loop level [16]. Furthermore, explicit CP violation in the Higgs sector can be induced byt andb loops if the parameters A t , A b and µ are complex [12,17,18]. It is found [12,14,17,19] that these loop effects could significantly influence the phenomenology of the Higgs boson sector.
In this article we study the effects of the phases of A t , A b , µ and M 1 on the decay branching ratios of thet 1,2 andb 1,2 withq 1 (q 2 ) being the lighter (heavier) squark. We take into account the explicit CP violation in the Higgs sector. We will show that the influence of the phases can be quite strong in a large domain of the MSSM parameter space. This could have an important impact on the search fort 1,2 andb 1,2 and on the determination of the MSSM parameters at future colliders.
In section 2 we discuss the SUSY CP phase dependences of masses, mixings and couplings. Section 3 contains our numerical investigation on the CP phase dependence of the branching ratios oft 1,2 andb 1,2 decays. In section 4 we present our conclusions.

SUSY CP Phase Dependences of Masses, Mixings and Couplings
In the MSSM the squark sector is specified by the mass matrix in the basis (q L ,q R ) with q =t orb [20] = |a q m q | e iϕq (−π < ϕq ≤ π).
Here I q 3 is the third component of the weak isospin and e q the electric charge of the quark q. MQ ,Ũ,D and A t,b are soft SUSY-breaking parameters, µ is the higgsino mass parameter, and 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 ). As the relative phase ξ between v 1 and v 2 is irrelevant in our analysis, we adopt the ξ = 0 scheme [17]. We take A q (q = t, b) and µ as complex parameters: A q = |A q | e iϕ Aq and µ = |µ| e iϕµ with −π < ϕ Aq,µ ≤ π. Diagonalizing the matrix (1) one gets the mass eigenstatesq 1 andq 2 with the masses mq 1 and mq 2 (mq 1 < mq 2 ), and the mixing angle θq Theq L −q R mixing is large if |m 2 q L − m 2 q R | < ∼ |a q m q |, which may be the case in thet sector due to the large m t and in theb sector for large tan β and |µ|. From Eqs. (4), (7) and (8) we see that m 2 q 1,2 and θq depend on the phases only through a term cos(ϕ Aq + ϕ µ ). This The properties of charginosχ ± i (i = 1, 2; mχ± 1 < mχ± 2 ) and neutralinosχ 0 j (j = 1, ..., 4; mχ0 1 < ... < mχ0 4 ) are determined by the parameters M 2 , M 1 , µ and tan β. We assume that the gluino mass mg is real. We write the U(1) gaugino mass M 1 as M 1 = |M 1 |e iϕ 1 (−π < ϕ 1 ≤ π). Inspired by the gaugino mass unification we take |M 1 | = (5/3) tan 2 θ W M 2 and mg = (α s (mg)/α 2 )M 2 . In the MSSM Higgs sector with explicit CP violation the mass-eigenvalues and couplings of the neutral and charged Higgs bosons, ) and H ± , including Yukawa and QCD radiative corrections, are fixed by m H + , tan β, µ, m t , m b , MQ, MŨ , MD, A t , A b , |M 1 |, M 2 , and mg [17]. The neutral Higgs mass eigenstates H 0 1 , H 0 2 and H 0 3 are mixtures of CP-even and CP-odd states (φ 1,2 and a) due to the explicit CP violation in the Higgs sector. For the radiatively corrected masses and mixings of the Higgs bosons we use the formulae of Ref. [17].
Here we list possible important decay modes oft 1,2 andb 1,2 : The decays into a gauge or Higgs boson in (9)- (12) are possible in case the mass splitting between the squarks is sufficiently large [3,4]. The explicit expressions of the widths of the decays (9)-(12) in case of real SUSY parameters are given in [21]. Those for complex parameters can be obtained by using the corresponding masses and couplings (mixings) from Refs. [17,20,21] and will be presented elsewhere [22].
The phase dependence of the widths stems from that of the involved mass-eigenvalues, mixings and couplings among the interaction-eigenfields. Here we summarize the most important features of the phase dependences.
This sector is independent of the phases. (b) For the decays into Higgs bosons in Eqs.(9)-(12), theq L -q R (q L -q L ,q R -q R ) couplings are dependent on (independent of) the phases ϕ At , ϕ A b and ϕ µ ; with Here φ i = O ij H 0 j (i = 1, 2) and a = O 3j H 0 j are the CP-even and CP-odd neutral Higgs bosons, respectively [17].
According to items (I)-(V) we expect that the widths (and hence the branching ratios) of the decays (9)-(12) are sensitive to the phases (ϕ At , ϕ A b , ϕ µ , ϕ 1 ) in a large region of the MSSM parameter space.

Numerical Results
Now we turn to the numerical analysis of thet 1,2 andb 1,2 decay branching ratios. We calculate the widths of all possibly important two-body decay modes of Eqs.(9)- (12). Three-body decays are negligible in the parameter space under study. In order to improve the convergence of the perturbative expansion [4,23] we calculate the tree-level widths by using the corresponding tree-level couplings defined in terms of "effective" MSSM running quark masses m run t,b (i.e. those defined in terms of the effective running Yukawa couplings h run t,b ∝ m run t,b ). For the kinematics, e.g., for the phase space factor we use the on-shell masses obtained by using the on-shell (pole) quark masses M t,b . We take In order not to vary too many parameters we fix |A t | = |A b | ≡ |A| and M 2 = 300 GeV, i.e. mg = 820 GeV. In the following we will assume that mg > mt 2 ,b 2 so that the decayst i → tg andb i → bg are kinematically forbidden. In our numerical study we take tan β, mt 1 , mt 2 , mb 1 , |A|, |µ|, ϕ At , ϕ A b , ϕ µ , ϕ 1 and m H + as input parameters, where mt 1,2 and mb 1 are the on-shell squark masses. Note that for a given set of the input parameters we obtain two solutions for (MQ, MŨ ) corresponding to the two cases mt L ≥ mt R and mt L < mt R from Eqs.
Condition (i) is imposed to satisfy the experimental mass bounds from LEP [27]. (ii) is the approximate necessary condition for the tree-level vacuum stability [28]. (iii) constrains µ and tan β (in the squark sector). For the calculation of the b → sγ width we use the formula of [29] including the O(α s ) corrections as given in [30].
As alreaday mentioned the experimental upper limits on the EDMs of electron, neutron, 199 Hg and 205 Tl strongly constrain the SUSY CP phases [6]. One interesting possibility for evading these constraints is to invoke large masses (much above the TeV scale) for the first two generations of the sfermions [31], keeping the third generation sfermions relatively light ( < ∼ 1 TeV). In such a scenario (ϕ 1 , ϕ µ ) and the CP phases of the third generation (ϕ At , ϕ A b , ϕ Aτ ) are practically unconstrained [31]. We adopt this scenario. Furthermore, we have checked that the electron and neutron EDM constraints at two-loop level [8] are fulfilled in the numerical examples studied in this article.

From item (I)(c) [(IV)(b) and (V)(b)] we expect that the widths of the decayst
Forb 1,2 decays we have obtained results similar to those for thet 1,2 decays. Here we show just a few typical results for them. In Fig.5 we show the ϕ A b dependence of theb 1 [b 2 ] decay branching ratios for tan β=30, (mt 1 , mt 2 , mb 1 )=(200,700,400) GeV [(175,500,350) GeV], (|A|, |µ|)=(800,700) GeV [(600,500) GeV], ϕ At =ϕ µ =π, ϕ 1 =0, and m H + = 180 GeV in the case mt L ≥ mt R . For mt L < mt R our results are similar. We find that theb 1,2 decay branching ratios are very sensitive to ϕ A b as expected from items (IV)(b) and (V)(b). The main reason is that the decay widths forb 1,2 →t 1 H − strongly depend on ϕ A b (and ϕ µ ) for large tan β (see Eqs. (13,14)). This explains also the tendency of the ϕ A b dependence of the branching ratios for the other decays shown in Fig.5. For small tan β ∼ 8 we expect that theb 1,2 decay branching ratios can be somewhat sensitive to ϕ A t,b and ϕ µ (see items (I) and (V)(b)). Similarly, we expect that the decay widths oft 1,2 H − can be fairly sensitive to ϕ A t,b and ϕ µ . We have confirmed this.

Conclusions
We have calculated the branching ratios of the two-body decays oft 1,2 andb 1,2 and studied their CP phase dependence within the MSSM with complex parameters A t , A b , µ and M 1 . We have shown that the effect of the SUSY CP phases on the branching ratios can be quite strong in a large domain of the MSSM parameter space. The ϕ A b dependence of thẽ b 1,2 decay branching ratios is mainly due to the ϕ A b dependence of their couplings to the Higgs bosons. In the case of thet 1,2 decays the branching ratios depend on ϕ At also via thet-mixing phase ϕt ≈ ϕ At for |A t | ≫ |µ|/ tan β, in addition to the ϕ At dependence of the mixing angle θt and their couplings to the Higgs bosons. Some of the branching ratios can change by a factor of 2 or more when varying the phases. This CP phase dependence of the branching ratios could have an important impact on the search fort 1,2 andb 1,2 and on the determination of the MSSM parameters at future colliders.  ) and B(t 1 → bχ + 1 ) (c,d) in the |A| − |µ| plane for ϕ At = 0 (a,c) and π/2 (b,d) with tan β = 8, (mt 1 ,mt 2 ,mb 1 ) = (400,700,200) GeV, ϕ A b =ϕ 1 =0, ϕ µ =π, and m H + = 600 GeV in the case mt L ≥ mt R . The blank areas are excluded by the conditions (i) to (iv) given in the text.