Two-body decays of gluino at full one-loop level in the quark-flavour violating MSSM.

We study the two-body decays of the gluino at full one-loop level in the Minimal Supersymmetric Standard Model with quark-flavour violation (QFV) in the squark sector. The renormalisation is done in the [Formula: see text] scheme. The gluon and photon radiations are included by adding the corresponding three-body decay widths. We discuss the dependence of the gluino decay widths on the QFV parameters. The main dependence stems from the [Formula: see text]-[Formula: see text] mixing in the decays to up-type squarks, and from the [Formula: see text]-[Formula: see text] mixing in the decays to down-type squarks due to the strong constraints from B-physics on the other quark-flavour-mixing parameters. The full one-loop corrections to the gluino decay widths are mostly negative and of the order of about -10%. The QFV part stays small in the total width but can vary up to -8% for the decay width into the lightest [Formula: see text] squark. For the corresponding branching ratio the effect is somehow washed out by at least a factor of two. The electroweak corrections can be as large as 35% of the SUSY QCD corrections.

Introduction

After the discovery of the Higgs particle in 2012 [1, 2], a task with high priority of the LHC is the search for new physics, beyond the framework of the Standard Model (SM). One of the most favoured candidates to be discovered are the supersymmetric (SUSY) particles. Their decay chains have been, therefore, extensively studied during the last two decades. Especially relevant are the decays of strongly interacting SUSY particles, squarks and gluinos. At tree-level, the leading gluino decays are those into a quark and a squark. Only when these processes are kinematically forbidden, more-body and loop-induced gluino decays become important.

The decays of the gluino in the Minimal Supersymmetric Standard Model (MSSM) were previously studied with general quark-flavour violation (QFV) in the squark sector at tree level [3-5] or including one-loop corrections with no QFV in the squark sector [6, 7]. In this paper we study the two-body decays of the gluino into a scalar quark and a quark at full one-loop level with general quark-flavour mixing in the squark sector of the MSSM. Such a study has been performed in detail in [8]. The analytical results obtained therein, as well as the developed numerical package FVSFOLD, will be used in the current paper. Since the experiments on K-physics disfavour mixing between the first two squark generations [9], we only consider mixing between the second and the third generations of squarks. More concrete, we consider scenarios where the gluino only decays into the lightest up- and down-type squarks, and , which can be mixtures of and and and , respectively, and all the other decays into are kinematically forbidden. There exist constraints from B-physics on such mixing as well, which we take into account. The mass limits on SUSY particles as well as the theoretical constraints on the soft-SUSY-breaking trilinear coupling matrices from the vacuum stability conditions are also taken into account.

In Sect. 2 we give the formulae for the QFV mixing squark system. In Sect. 3 the tree-level partial two-body decay widths are derived and then the used renormalisation scheme is explained. In order to cure the infrared (IR) divergences, we include the widths of the real gluon/photon radiation process, introducing a small regulator gluon/photon mass. In Sect. 4 we perform a detailed numerical analysis on the dependences of the two-body decay widths and branching ratios (BRs) on the quark-flavour-mixing parameters and and on the gluino mass. Appendix A contains the Lagrangian for the gluino–squark–quark interaction. In Appendix B all constraints we obey are summarised and Appendix C gives the detailed formulae for the hard radiation of a gluon or a photon.

QFV parameters in the squark sector of the MSSM

We define the QFV parameters in the up-type squark sector of the MSSM as follows: where denote the quark flavours u, c, t, and . Analogously, for the down-type squark sector we have where the subscripts denote the quark flavours d, s, b, and . are the hermitian soft-SUSY-breaking squark mass matrices and are the soft SUSY-breaking trilinear coupling matrices of the up- and down-type squarks. These parameters enter the left–left, right–right and left–right blocks of the squark mass matrix in the super-CKM basis [10],with . The different blocks in Eq. (6) are given bywhere is the higgsino mass parameter, is the ratio of the vacuum expectation values of the neutral Higgs fields , and are the diagonal mass matrices of the up- and down-type quarks. Furthermore, and , where and are the isospin and electric charge of the quarks (squarks), respectively, and is the Weinberg mixing angle. is the Cabibbo–Kobayashi–Maskawa matrix, which we approximate with the unit matrix. The squark mass matrix is diagonalised by the unitary matrices , such thatwith , and . The physical mass eigenstates are given by .

In this paper we study –, –, –, and – mixing, which is described by the QFV parameters , , , and , respectively, as well as –, –, –, and – mixing, which is described by the QFV parameters , , , and , respectively. Note that describes the left–left mixing in both and sectors. The – mixing is described by the quark-flavour conserving (QFC) parameter . All parameters mentioned are assumed to be real.

Two-body decays of gluino at full one-loop level in the general MSSM

We study two-body decays of gluino into a squark and a quark, . The tree-level partial decay widths , with , , and the subscript g being the quark-generation index, are given bywhere is the average factor for the incoming . The tree-level amplitude squared readswith , no summation over i, and the tree-level couplings are given by (see also Appendix A)where T are the generators of the SU(3) colour group, and , with are the up- and down-squark mixing matrices defined by Eq. (8). By inserting Eq. (11) into Eq. (10) and using tr we can write Eq. (9) in the explicit formIn order to obtain an ultraviolet (UV) convergent result at one-loop level we employ the dimensional reduction () regularisation scheme, which implies that all tree-level input parameters of the Lagrangian are defined at the scale . Since in this scheme the tree-level couplings are defined at the scale Q, they do not receive further finite shifts due to radiative corrections. The physical scale independent masses and fields are obtained from the ones using on-shell renormalisation conditions.

We can write the renormalised one-loop partial decay widths aswhere is the one-loop amplitude. The complete list of diagrams can be found in the appendix of [8]. The one-loop shifts to the coupling constants, and , receive contributions from all vertex diagrams, the amplitudes arising from the wave-function renormalisation constants and the amplitudes arising from the coupling counter-terms,1 where is due to all vertex radiative corrections, and is due to the coupling counter-terms. The wave-function induced corrections are given bywith i and j fixed, , . Note that denote the couplings. The explicit expressions for the renormalisation constants in (15) can be found in [8].

To cure the infrared (IR) divergences, in addition to (13), we include the widths of the real gluon/photon radiation processes, , assuming a small regulator gluon/photon mass . The explicit formulae for the hard radiation widths are given in Appendix C.

The full one-loop contribution to the total two-body decay width, see (13), is due to SUSY-QCD and electroweak corrections, includes loops with gluon and gluino, and includes loops with EW gauge bosons, photon, Higgs bosons and EWinos. In the numerical analyses performed in [8], as well as in [6], it was shown that in the considered scenarios the electroweak corrections are not negligible, but necessary for a correct one-loop evaluation. As you will see, in our numerical analysis we will come to a similar conclusion.

Hereafter we will use the notation . In our case where CP is conserved this is equivalent with .

Numerical results

In order to demonstrate quantitatively our results on the gluino decay widths and branching ratios we first fix a reference scenario and then vary the QFV parameters within the allowed region. Our reference scenario fulfils all relevant theoretical and experimental constraints, which we discuss in more detail in Appendix B. The input parameters and the physical output parameters are shown in Tables 1 and 2, respectively. The flavour decomposition of the and squarks is shown in Table 3. For calculating the mass and the low-energy observables, especially those ones in the B-sector (see Table 4), we use the public code SPheno v3.3.3 [11, 12]. The gluino two-body widths and branching ratios at full one-loop level in the MSSM with QFV are calculated with the numerical code FVSFOLD, developed in [8]. For building FVSFOLD the packages FeynArts [13, 14] and FormCalc [15] were used. Furthermore, we use LoopTools [15] based on the FF package [16], and SSP [17]. In order to have simultanuosly a UV and IR finite result we first calculate the total result by using only parameters for the one-loop partial width including the real hard radiation. Then we use on-shell masses, which are calculated within FVSFOLD, in the kinematic two-body prefactor , see Eq. (12).

Table 1

QFV reference scenario: all parameters are calculated at , except for which is the pole mass of , and  GeV (corresponding to ). All other squark parameters are zero

M_1	M_2	M_3	μ
500 GeV	1000 GeV	3000 GeV	500 GeV

Table 2

Physical masses of the particles in GeV for the scenario of Table 1

m_χ~10	m_χ~20	m_χ~30	m_χ~40	m_χ~1+	m_χ~2+
460	500	526	1049	493	1049

Table 3

Flavour decomposition of and for the scenario of Table 1. Shown are the squared coefficients

	u~_L	c~_L	t~_L	u~_R	c~_R	t~_R
u~_1	0	0.004	0	0	0.38	0.61
u~_2	0	0.001	0.99	0	0.006	0

Table 4

Constraints on the MSSM parameters from the B-physics experiments relevant mainly for the mixing between the second and the third generations of squarks and from the data on the mass. The fourth column shows constraints at 95% CL obtained by combining the experimental error quadratically with the theoretical uncertainty, except for

Observable	Exp. data	Theor. uncertainty	Constr. (95% CL)
ΔM_B_s [ps^-1]	17.757±0.021 (68% CL) [19]	±3.3 (95% CL) [20, 21]	17.757±3.30
10^4×B(b→sγ)	3.41±0.155 (68% CL) [22]	±0.23 (68% CL) [23]	3.41±0.54
10^6×B(b→sl^+l^-)	1.60-0.45+0.48 (68% CL) [24]	±0.11 (68% CL) [25]	1.60-0.91+0.97
(l=eorμ)
10^9×B(B_s→μ^+μ^-)	2.8-0.6+0.7 (68% CL) [26]	±0.23 (68% CL) [27]	2.80-1.26+1.44
10^4×B(B^+→τ^+ν)	1.14±0.27 (68% CL)  [22, 28]	±0.29 (68% CL) [29]	1.14±0.78
m_h^0 [GeV]	125.09±0.24(68%CL) [30]	±3 [31]	125.09±3.48

The scenario shown in Table 1 violates quark flavour explicitly in both up- and down-squark sectors. The values of the parameters are chosen to satisfy approximately the GUT relations (). The Higgs boson is SM-like with and all other Higgses are much heavier in mass and degenerate. The ratio of the vacuum expectation values of the neutral Higgs fields is taken relatively small, . The value of the parameter is also chosen small for naturalness reasons. The flavour decompositions of the and squarks are shown in Table 3. In this scenario the squark is a strong mixture of and , with a tiny contribution from , and the squark is mainly , with a tiny contribution from . The is a mixture of and , and is a pure .

At our reference parameter point the gluino decays into , , and are kinematically allowed, with branching ratios B, B, B, B and the others being very small. The total two-body width including the full one-loop contribution, , where the tree-level width ,2 and the SUSY-QCD and the electroweak corrections are negative, and , giving about −6.4 and −0.7% of the total two-body gluino width , respectively. Note that the SQCD contribution includes gluon and gluino, and the EW contribution includes also the photon. In the same scenario with no quark-flavour violation, i.e. when all QFV () parameters listed in Table 1 are set to zero, we have .

The QFV left–right mixing, described by the parameters , is constrained from the vacuum stability conditions (see Sect. B) and is required to be rather small. On another hand, a sizeable value of is not possible because it violates B-physics constraints such as the B() constraint. However, large right–right mixing in both and sectors is allowed (see also [18]) and therefore, in the following, we show only plots with dependences on the and parameters. In Fig. 1 we show dependences on the QFV parameter . In Fig. 1a the tree-level, the SQCD and total full one-loop widths and in Fig. 1b the relative contributions of the one-loop SQCD and the full one-loop part in terms of the tree-level result are shown. The partial decay widths as well as the branching ratios of the kinematically allowed two-body channels at full one-loop level are shown in Fig. 1c, d, respectively. In Fig. 1a it is seen that is quite sensitive to the parameter . The dependence of the tree-level width and the full one-loop corrected width is similar and their difference becomes a little more important for large absolute values of . This means that the QFV parameter dependence is mainly due to the kinematic factor, see Sect. 3. The SQCD correction shown in Fig. 1b is only weakly dependent on and is about −8%. The EW correction can become −3% for large and negative values of . In Fig. 1c the partial widths of the modes coincide because . The same holds for the branching ratios in Fig. 1d. For the width of becomes tiny because then is mainly as all the other QFV ’s are relatively small.

Fig. 1

a Total two-body decay width at tree level, SQCD one-loop and full one-loop corrected as functions of the QFV parameter ; b being the SQCD one-loop and the full one-loop corrections to relative to the tree-level width; c partial decay widths and d branching ratios of the kinematically allowed individual two-body channels at full one-loop level as functions of . All the other parameters are fixed as in Table 1, except

Figure 2 shows the relative contribution of the one-loop SQCD and the full one-loop part in terms of the tree-level result for the partial decay width (Fig. 2a) and the branching ratio (Fig. 2b) of the decay as a function of . We see in Fig. 2a that the SQCD corrections vary in the range of −8 to −10%. The EW correction is much stronger dependent on varying between 1% down to −8%. The effects are similar in the branching ratio (b), but weaker. Out of the squark masses only is strongly dependent on . In the whole range of no additional channel opens but those visible in Fig. 1c, d. Therefore, the wiggles stem from the complex structures of the QFV one-loop contributions.

Fig. 2

and BR denote the SQCD one-loop and the full one-loop corrections relative to the tree-level result for the decay as a function of ; a, b is for the partial width and the branching ratio, respectively. The other parameters are fixed as in Fig. 1

In Fig. 3 we show dependences on the QFV parameter . In Fig. 3a the tree-level, the SQCD and total full one-loop widths and in Fig. 3b the relative contribution of the one-loop SQCD and the full one-loop part in terms of the tree-level result are shown. The partial decay widths as well as the branching ratios of the kinematically allowed two-body channels are shown in Fig. 3c, d, respectively. A comparison of Fig. 3 with Fig. 1 demonstrates the equal importance of QFV mixing in both and sector. But in the sector all plots are more symmetric around compared to these in sector around . This stems from the fact that in the mass matrix  GeV but in the mass matrix  GeV is taken and is relatively small, see Eq. (7). The SQCD corrected width in Fig. 3a seem to coincide with the full one-loop corrected width, which we see in detail in Fig. 3b. There the SQCD correction is about −7.5% and varies only within 1% around this value. The EW part varies between −0.5 to −1.5%. In the Fig. 3c, d we see that for large absolute values of the  right–right mixing parameter the decay modes become much more important than the ones since the mass becomes smaller due to the mixing effect. As are mainly bottom squarks, the EW corrections to the modes are small, mainly controlled by the rather small bottom-quark Yukawa coupling for . On the other hand, as are mainly top squarks, the EW corrections to the modes are significant, mainly controlled by the large top-quark Yukawa coupling . This explains the smallness of the EW corrections in Fig. 3a, b, especially for large .

Fig. 3

a Total two-body decay width at tree level and full one-loop corrected (which coincides with the SQCD one-loop corrected one) as functions of the QFV parameter ; b being the SQCD one-loop and the full one-loop corrections to relative to the tree-level width; c partial decay widths and d branching ratios of the kinematically allowed individual two-body channels at full one-loop level as functions of . All the other parameters are fixed as in Table 1, except

Figure 4 shows the relative contribution of the one-loop SQCD and the full one-loop part in terms of the tree-level result for the partial decay width Fig. 4a and the branching ratio Fig. 4b of the decay as a function of in the phenomenologically allowed region. The interesting point is that the dependence of this channel on comes mainly from the gluino wave-function correction term with in the loop. The SQCD correction varies between −8 and −9.5% and the EW correction is about constant and is % for the width (Fig. 4a). For the branching ratio (Fig. 4b), the effect is much smaller for the SQCD correction, between −0.5 and −1.5%. The EW part is maximal −3%.

Fig. 4

and BR denote the SQCD and the full one-loop contribution in terms of the tree-level result for the decay as a function of , a to the partial width, b to the branching ratio, respectively. The parameters are fixed as in Fig. 3

Figure 5 shows the relative contribution of the one-loop SQCD and the full one-loop part in terms of the tree-level result for the partial decay width (Fig. 5a) and the branching ratio (Fig. 5b) of the decay as a function of in the phenomenologically allowed region. The SQCD correction varies between −6.5 and −8% and the EW correction can become 1% for the width (Fig. 5a). For the branching ratio (Fig. 5b), the effects are again smaller, the SQCD correction is less than 1% and the EW part maximal 3%. As in Fig. 2 the wiggles stem from the complex structures of the QFV one-loop contributions.

Fig. 5

and BR denote the SQCD and the full one-loop contribution in terms of the tree-level result for the decay as a function of , a to the partial width, b to the branching ratio, respectively. The parameters are fixed as in Fig. 3

In Fig. 6 a simultaneous dependence on the right–right mixing parameters of both and sectors is shown. It is clearly seen that the total two-body decay width can vary up to 70 GeV in the allowed parameter region due to QFV.

Fig. 6

Total two-body decay width at full one-loop level as a function of the QFV parameters and . All the other parameters are given in Table 1, except

In Fig. 7a the full one-loop part in terms of the tree-level result and in Fig. 7b the EW contribution relative to the SQCD contribution are shown for the total two-body gluino decay width as a function of and . We see in Fig. 7a a constant QFC one-loop contribution of and % variation due to QFV. The EW part can become up to 35% of the SQCD one (Fig. 7b) for large where the mode becomes important, since the mass becomes smaller due to the -sector right–right mixing effect. Furthermore, as is mainly a top squark, the EW corrections to the mode are significant, mainly controlled by the large top-quark Yukawa coupling .

Fig. 7

denotes in a the full one-loop contribution in terms of the total tree-level width, in b the EW contribution relative to the SQCD contribution. Both plots are given as a function of the QFV parameters and . All the other parameters are given in Table 1, except

Figure 8 shows the dependence of the total two-body decay width on the gluino mass in our reference scenario (Fig. 8a) and in a quark-flavour conserving scenario, setting all QFV () parameters of Table 1 to zero (Fig. 8b). It is seen that in the QFV scenario (Fig. 8a) is somewhat enhanced. Because of the large () the mass difference between and ( and ) is bigger. Consequently, and are lighter and decays into these particles are already possible for smaller gluino masses.

Fig. 8

Dependence of the total two-body decay width at tree level (dashed) and full one-loop level (solid) on the gluino mass. a QFV scenario with the parameters as given in Table 1; b QFC scenario with the parameters as given in Table 1, but with all QFV () parameters set to zero

We have compared our numerical results in the flavour conserving limit with the results obtained in [6]. For their reference scenario with  GeV assuming their input parameters to be ones, we get a total width of 379 GeV. We agree with them within 2%. For the relative size of the full one-loop correction we get −2% compared to their result of −2.5%.

Conclusions

We have studied all two-body decays of the gluino at full one-loop level in the Minimal Supersymmetric Standard Model with quark-flavour violation in the squark sector. We have discussed a scenario where only the decays to and are kinematically open and is a mixture of and controlled by , and is a mixture of and controlled by . All other QFV parameters are small in order to fulfil the constraints from B-physics. The LHC constraints for the masses of the SUSY particles are also satisfied, especially that one for and the vacuum stability conditions are fulfilled.

The full one-loop corrections to the gluino decay widths are mostly negative. For the total decay width they are in the range of −10% with a weak dependence on QFV parameters for both SQCD (including gluon loops) and electroweak (including also photon loops) corrections. For the decay width into we can have a total correction up to −18%, with the EW part up to −8%, strongly depending on the QFV parameters. For the corresponding branching ratio the effect is somehow washed out. For the decay into we have maximal corrections of −8%. In general, it turns out that the EW corrections can be in the range of up to 35% of the SQCD corrections due to the large top-quark Yukawa coupling. The full one-loop corrections to the total width are of the order of about −10% in the gluino mass range of 2.3–4.0 TeV.