$t\to cV$ via SUSY FCNC couplings in the unconstrained MSSM

We recalculate the branching ratios for $t\to cV$ ($V=g,\gamma,Z$) induced by SUSY FCNC couplings within the general unconstrained MSSM framework using mass eigenstate approach. Our results show that the branching ratios for these decays are larger than ones reported in previous literatures in the MSSM with R-parity conservation, and they can reach $\sim 10^{-4}$, $\sim 10^{-6}$, and $\sim 10^{-6}$, respectively, for favorable parameter values allowed by current precise experiments. Thus, the branching ratios for $t\to cg$ and $t\to c\gamma$ may be measurable at the LHC.


Introduction
The top quark flavor changing neutral current (FCNC) processes t → cV (V = g, γ, Z) have tiny branching ratios in the standard model (SM) [1], and are too small to be measurable in the future colliders, and thus any detected signal of these rare decay events definitely indicates some new physics beyond the SM. Actually, t → cV (V = g, γ, Z) have been studied in various new physics models beyond the SM in detail, such as the two-Higgs-doublet model (2HDM) [1,2], the technicolor model (TC) [3], the top-color-assisted technicolor model (TC2) [4], the models with extra vector-like quark singlets [5], the minimal supersymmetry (SUSY) extension of the SM (MSSM) with R-parity conservation [6]- [11] and without R-parity conservation [12]. The decay branching ratios for t → cV (V = g, γ, Z) are enhanced in general several orders of magnitude in these new physics models. The MSSM, which is believed as one of the most attractive candidates of new physics model, has gotten many attentions, and the investigation of t → cV (V = g, γ, Z) in the MSSM is a long story. C.S. Li et al. studied one-loop SUSY-QCD and SUSY-EW contributions in Ref. [6], subsequently G. Couture et al. recalculated and generalized the SUSY-QCD corrections to include the left-hand (LH) squark mixing in Ref. [7] and the right-hand (RH) squark mixing in Ref. [8]. All works above are within the framework of the MSSM with flavor-universal soft SUSY breaking terms. Later J.L. Lopez et al. further generalized the SUSY-EW corrections to the case of including neutralino-quarksquark loops in Ref. [9]. and G.M. de Divitiis et al. reinvestigated t → cV (V = g, γ, Z) in the universal case as well as non-universal case in Ref. [10], and obtained different results from Ref. [6,7] due to the calculation of the relevant SUSY mixing angles and diagrams involving a helicity flip in the gluino line, which was confirmed by J. Guasch et al. in a RG-based framework for t → cg decay [11]. All the above results of the MSSM are summarized in Table 1 (for comparing, we also list the results of the SM and new physics models mentioned above), one can find that they are all below 5 × 10 −5 , which is the roughly estimated sensitivities for the measurements of top rare decay at the LHC with 100f b −1 of integrated luminosity [11].  Since the contributions to the top FCNC decays mediated by the charged current interactions (from W ± , H ± , G ± andχ ± ) are invisibly small as shown in the previous works [6]- [11] and can not be enhanced in this framework, in this paper we will reinvestigate the t → cV (V = g, γ, Z) only via SUSY FCNC couplings in the unconstrained MSSM, and try to show what are the maximal branching ratios for t → cV (V = g, γ, Z) in the MSSM using SUSY parameters allowed by current data, and whether they can be detected at the LHC.

FCNC
In the super-CKM basis [13], in which the mass matrices of the quark fields are diagonal by rotating the superfields, the up squark mass matrix M 2 U is a 6 × 6 matrix, which has the form: where θ W is the Weinberg angle, 1l 3 stands for the 3 × 3 unit matrix, the angle β is defined by tan β ≡ v 2 /v 1 , the ratio of vacuum expectation values of the two Higgs doublets, µ is the Higgs mixing parameter in the superpotential, and (M 2 U ) LL , (M 2 U ) RR , and (M 2 U ) LR contain the flavor-changing entries, which are given by respectively. Here M 2 Q , M 2 U,D and A U,D are the soft broken SU(2) doublet squark mass squared matrix, the SU(2) singlet squark mass squared matrix and the trilinear coupling matrix, respectively. They are directly related to the mechanism of SUSY breaking, and are in general not diagonal in the super-CKM basis. Furthermore, (M 2 U ) LR , arising from the trilinear terms in the soft potential, namely A U,ij H U U i U c j , is not hermitian. The matrix M 2 U can further be diagonalized by an additional 6 × 6 unitary matrix Z U to give the up squark mass eigenvalues Thus, we get new sources of flavor-changing neutral current: neutralino-quark-squark coupling and gluino-quark-squark coupling, which arise from the off-diagonal elements of (M 2 U ) LL , (M 2 U ) LR and (M 2 U ) RR , and can be written as (I = 1, 2, 3,i = 1, ..., 6, j = 1, 2, 3, 4) Here For the aim of this paper, the following strategy in the numerical calculations of the decay branching ratios of t → cV will be used: first we deal with the LL, LR, RL and RR blocks of the matrix M 2 U separately and in each block we only consider the effects of individual element on the top quark rare decays, and then we investigate the interference effects between some different entries within one block and the interference effects between different blocks. In order to simplify the calculation we further assume that all diagonal RR are set to be equal to the common value M 2 SUSY , and then normalize the off-diagonal elements to M 2 SUSY [14,15], Thus (M 2 U ) LL can be written as follows: and analogously for all the other blocks.
The related Feynman diagrams for t → cV (V = g, γ, Z) induced by the SUSY FCNC are shown in Fig.1. Neglecting the charm quark mass, the amplitude of the decay process is given by where p t , p c , and k are the momenta of the top-quark, charm-quark, and gauge boson respectively, and ǫ µ (k, λ) is the polarization vector for the gauge boson. The vertex V µ can be written as where F L(R) V 1 (2) are the form factors, and their explicit expressions through the SUSY-QCD FCNC (g a −q i − q I ) are: and the explicit expressions through the SUSY-EW FCNC (χ 0 k −q i − q I ) are: Here , m 2 q l , m 2 q l ′ ) are 2 and 3-point one-loop integrals [16]. And the relevant couplings are: After squaring the decay amplitude and multiplying by the phase space factor, one obtains the decay width of t → cV (V = g, γ, Z): and we define the branching ratio as Ref. [1]: which will be the main object of our numerical study.  [15,14,18,19], in which (δ 12 U ) LL also is constrained by the chargino contributions to K-K mixing [20], and D 0 -D 0 mixing makes constraints on (δ 12 U ) LL , (δ 12 U ) LR and (δ 12 U ) RL [21]. And (δ 31 U ) LL , (δ 32 U ) LL , (δ 31 U ) RL and (δ 32 U ) RL are constrained by the chargino contributions to B d -B d mixing [18]. Finally, there also are constraints on the up squark mass matrix from the chargino contributions to b → sγ [22,14]. Taking into account above constraints, in our numerical calculations, we use the following limits: first is that the branching ratio increases rapidly with the mixing parameters increasing, and the second is that the branching ratio depends strongly on the gluino mass mg, and the last is that the dependence of branching ratio on the M SUSY is medium comparing with above two parameters.

Numerical calculation and discussion
For each decay modes t → cV (V = g, γ, Z), in Figs. 2 and 3 we show the dependence of the decay branching ratios on RR and LR off-diagonal elements, respectively (We do not show the results for LL off-diagonal elements as their contributions are similar to the ones for RR off-diagonal elements). We find that for t → cV the largest results come from the LR block, which arises from the soft trilinear couplings A U . We also give the results of the interference effects on the branching ratios for t → cg in Figs. 2(c)and 2(d).
In general, these interference effects increase the decay branching ratios, and since the interference effects between different blocks are similar, we do not show them for the space of this paper. The results of decay t → cγ, cZ are about two orders of magnitude smaller than ones of decay t → cg, as shown in Figs. 2 and 3. From these figures, we can find that the decay branching ratios for t → cV (V = g, γ, Z) induced by the SUSY FCNC couplings can reach ∼ 10 −4 , ∼ 10 −6 and ∼ 10 −6 , respectively, for the favorable parameter values allowed by current precise experiments, and they are larger than all the previous ones in the MSSM with R-parity conservation (it should be pointed out that the results of Ref. [6], [7] and [8] in Table 1 are obtained at mg = 100 GeV, which is disfavored by current data).
According to the analysis of T. Han et al. [23], the sensitivities for t → cγ and t → cZ at the LHC with 100f b −1 integrated luminosity are 5 × 10 −6 and 2 × 10 −4 , respectively, and our results show that the rare decay t → cγ may be detectable. Later T. Han et al. [24] and M. Hosch et al. [25] studied the sensitivities to the top quark anomalous FCNC couplings at the LHC for single top quark and direct top quark productions, respectively, and the corresponding decay t → cg branching ratios transferred from their results are 4.9 × 10 −5 and 2.7 × 10 −5 , respectively. Thus, our results of the branching ratios for t → cg indicate that the top quark FCNC production processes (both for single top and direct top) may be measurable at the LHC. But if we use the 5 × 10 −5 as the sensitivity for the FCNC decay t → cV at the LHC with 100f b −1 integrated luminosity as shown in Ref. [11], our results show that the rare decay t → cg are also potentially measurable at the LHC.
In conclusion, we have calculated the top quark rare decay t → cV (V = g, γ, Z) induced by SUSY-FCNC couplings in the general unconstrained MSSM using mass eigenstate approach. Our results show that the branching ratios for these decays are larger than ones reported in previous literatures in the MSSM with R-parity conservation, and especially, the branching ratios for the rare decay modes t → cg, cγ we calculated are very hopefully to be measurable at the LHC for the favorable parameter values allowed by current precise experiments. Moreover, we find that the decay branching ratios for t → cV (V = g, γ, Z) strongly depend on the soft trilinear couplings A U , and it is possible to get some valuable information of soft SUSY breaking parameters by measuring the branching ratio for the top quark rare decay at the LHC.