# as a test case for quark flavor violation in the MSSM

Institut für Hochenergiephysik der Österreichischen Akademie der Wissenschaften, A-1050 Vienna, Austria

Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan

RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198, Japan

###### Abstract

We compute the decay width of in the MSSM with quark flavor violation (QFV) at full one-loop level adopting the renormalization scheme. We study the effects of mixing, taking into account the constraints from the B meson data. We show that the full one-loop corrected decay width is very sensitive to the MSSM QFV parameters. In a scenario with large mixing can differ up to from its SM value. After estimating the uncertainties of the width, we conclude that an observation of these SUSY QFV effects is possible at an collider (ILC).

HEPHY-PUB 942/14

UWThPh-2014-26

RIKEN-MP-101

###### Contents

- 1 Introduction
- 2 Definition of the QFV parameters
- 3 at full one-loop level with flavor violation
- 4 Numerical results
- 5 Observability of the deviation of from its SM value at the ILC
- 6 Conclusions
- A Interaction Lagrangian
- B Hard gluon/photon bremsstrahlung
- C Simplified formulas for the two- and three-point functions
- D Theoretical and experimental constraints

## 1 Introduction

The properties of the Higgs boson, discovered at the LHC, CERN, with a mass of (averaged over the values given by ATLAS [1, 2] and CMS [3, 4]) [5], are consistent with the prediction of the Standard Model (SM) [6]. Future experiments at LHC at higher energy ( TeV) and higher luminosity will provide more precise data on Higgs boson observables, as Higgs production cross sections, decay branching ratios etc.. Even more precise data can be expected at a future linear collider (ILC). This will allow one to test the SM more accurately and will give information on physics beyond the SM. The discovered Higgs boson could also be the lightest neutral Higgs boson of the Minimal Supersymmetric Standard Model (MSSM) [6, 7].

The decays of are usually assumed to be quark flavor conserving (QFC). However, quark flavor violation (QFV) in the squark sector may significantly influence the decay widths of at one-loop level. In particular, the rate of the decay into a charm-quark pair, , may be significantly different from the SM prediction due to squark generation mixing, especially that between the second and the third squark generations ( mixing). This possibility will be studied in detail in the present paper.

It is well known that the mixing between the first and the second squark generations is strongly suppressed by the data on K physics [8]. Therefore, we assume mixing between the second and the third squark generations, respecting the constraints from B physics. In the MSSM this mixing was theoretically studied for squark and gluino production and decays at the LHC [9, 10, 11, 12, 13, 14, 15, 16, 17].

The outline of the paper is as follows: In Section 2 we shortly give the definitions of the QFV squark mixing parameters. In Section 3 we present the calculation of the width of at full one-loop level in the renormalization scheme with quark flavor violation within the MSSM. In particular, we give formulas for the important one-loop gluino contribution. In Section 4 we present a detailed numerical analysis. In Section 5 we study the feasibility of observing the SUSY QFV effects in the decay at ILC by estimating the theoretical uncertainties. Section 6 contains our conclusions.

## 2 Definition of the QFV parameters

In the MSSM’s super-CKM basis of , with , one can write the squark mass matrices in their most general -block form [18]

(1) |

with . The left-left and right-right blocks in eq. (1) are given by

(2) |

where are the hermitian soft SUSY-breaking mass matrices of the squarks and are the diagonal mass matrices of the up-type and down-type quarks. Furthermore, and , where and are the isospin and electric charge of the quarks (squarks), respectively, and is the weak mixing angle. Due to the symmetry the left-left blocks of the up-type and down-type squarks in eq. (2) are related by the CKM matrix . The left-right and right-left blocks of eq. (1) are given by

(3) |

where are the soft SUSY-breaking trilinear coupling matrices of the up-type and down-type squarks entering the Lagrangian , is the higgsino mass parameter, and is the ratio of the vacuum expectation values of the neutral Higgs fields , with . The squark mass matrices are diagonalized by the unitary matrices , , such that

(4) |

with . The physical mass eigenstates are given by .

We define the QFV parameters in the up-type squark sector , and as follows [19]:

(5) | |||||

(6) | |||||

(7) |

where denote the quark flavors . In this study we consider , , , and mixing which is described by the QFV parameters , , , and , respectively. We also consider mixing described by the QFC parameter which is defined by eq. (7) with . All QFV parameters and are assumed to be real.

## 3 at full one-loop level with flavor violation

We study the decay of the lightest neutral Higgs boson, , into a pair of charm quarks (Figure 1) at full one-loop level in the general MSSM with quark flavor violation in the squark sector. The full one-loop decay width of was first calculated within the QFC MSSM by [20]. In [21, 22, 23] higher order SUSY corrections for the Higgs-fermion-fermion vertices were calculated in the generic MSSM in an effective-field-theory approach.

The decay width of the reaction including one-loop contributions can be written as

(8) |

The tree-level decay width reads

(9) |

where is the on-shell (OS) mass of and the tree-level coupling is

(10) |

Here is the mixing angle of the two CP-even Higgs bosons, and [24].

In the general MSSM at one-loop level, in addition to the diagrams that contribute within the SM, also receives contributions from diagrams with additional Higgs bosons and supersymmetric particles. The contributions from SUSY particles are shown in Figure 12, neglecting the contributions from scalar leptons. The flavor violation is induced by one-loop diagrams with squarks that have a mixed quark flavor nature. In addition, the coupling of with two squarks (see eq. (65) of Appendix A) contains the trilinear coupling matrices which for break quark flavor explicitly.

The one-loop contributions to contain three parts, QCD () corrections, SUSY-QCD () corrections and electroweak (EW) corrections. In the latter we also include the Higgs contributions. In the following we will mainly give details for the QCD and SUSY-QCD corrections.

### 3.1 Renormalization procedure

Loop calculations can lead to ultraviolet (UV) and infrared (IR) divergent result and therefore require renormalization. In order to get UV finite result we adopt in our study the renormalization scheme, where all input parameters in the tree-level Lagrangian (masses, fields and coupling parameters) are UV finite, defined at the scale , and the UV divergence parameter , where in a D-dimemsional space-time and is the Euler-Mascheroni constant, is set to zero. The tree-level coupling is defined at the given scale and thus does not receive further finite shifts due to loop corrections. In order to obtain the shifts from the masses and fields to the physical scale-independent masses and fields, we use on-shell renormalization conditions. To ensure IR convergence, we include in our calculations the contribution of the real hard gluon/photon radiation from the final charm quarks assuming a small gluon/photon mass .

The one-loop corrected width of the process including hard gluon/photon radiation is given by

(11) |

where read

(12) |

(13) |

Note that all parameters in the tree-level coupling , eq. (10), are running at the scale . The renormalized finite one-loop amplitude of the process is a sum of all vertex diagrams, the amplitudes arising from the wave-function renormalization constants and the amplitudes arising from the coupling counterterms. Note that in the renormalization scheme the counterterms contain only UV-divergent parts and have to cancel in order to yield a convergent result. The one-loop renormalized coupling correction can be written as

(14) |

where is the vertex coupling correction, is the wave-function coupling correction and is the coupling counter term. The tree-level interaction Lagrangian of the lightest Higgs boson and two charm quarks is given by eq. (63) in Appendix A. The renormalized Lagrangian is obtained after making the replacement , where describes all vertex-type interactions. The coupling correction due to wave-function renormalization is given by

(15) |

where is the coupling of the heavier neutral Higgs and the charm quark, . The charm quark wave-function renormalization constants read

(16) | |||||

and the Higgs wave-function renormalization constants for the case of mixing are given by

(17) |

(18) |

with the tadpole contribution

(19) |

where and . and are the loop corrections from the tadpole diagrams with and , respectively. In eqs. (16), (17) and (18) applied to the self-energies denoted by takes the real part of the loop integrals, but leaves the possible complex couplings unaffected. Finally, the coupling counter term is given by

(20) |

where the subindex means that only the part proportional to the UV divergence parameter is taken. The explicit expressions for the shifts of the parameters in (20) can be found in [25]. Note that is used.

### 3.2 One-loop gluon contribution

The one-loop virtual gluon contribution to is given by

(21) |

with . contains terms originating from the vertex correction, the wave-function correction and the coupling correction due to gluon interaction,

(22) |

The individual contributions in are given by

(23) |

(24) |

(25) |

where in the scheme and in the scheme. and are the two- and three-point functions

(26) |

(27) |

(28) |

with . Summing up eqs. (23)-(25) one can write in the form

(29) |

Furthermore, we will use the result for the hard gluon radiation, given in Appendix B. We can write eq. (69) in the form

(30) |

Combining (21), (29) and (30) for the gluon one-loop corrected convergent width we obtain

(31) |

where is the result of [26] and its explicit expression can be found therein or e.g. in [20, 27, 28]. Eq. (31) can be written in a compact form as

(32) |

where denotes the on-shell (OS) charm quark mass. Note that the result for the photon one-loop corrected convergent width is obtained from (32) by making the replacement :

(33) |

with .

For ()

(34) |

and from eq. (25) using eqs. (82) and (83) we get

(35) |

For in the limit we obtain

(36) |

where in (36) we have absorbed the logarithm of into

(37) |

Combining eq. (35) with eqs. (36) and (37) one can see that the one-loop level does not depend on the parameter . In the numerical evaluation of we follow the recipe given in [29], starting with eq. (4) and we use given therein. In all other cases we take from SPheno [30, 31], where it is calculated at two-loop level within the MSSM. In order to stay consistent, in our numerical calculations we have included in addition only the gluonic contributions, taken from [28]. With these, will be denoted as ,

(38) |

### 3.3 One-loop gluino contribution and decoupling limit

The one-loop gluino contribution to , Fig. 3 and Fig. 4, renormalised in the scheme reads

(39) |

acquires contributions from the vertex correction (Fig. 3), the wave-function correction (Fig. 4) and the coupling correction due to gluino interaction,

(40) |

In the following we will use the abbreviations and . Note that applying Einstein sum convention we get and . Neglecting the charm quark mass and the Higgs boson mass compared to the squark and gluino masses, one can write the individual contributions as

(41) |

(42) |

where the coupling is given in eq. (65) of Appendix A. For the following discussion of the gluino contribution in the large limit we give the charm mass counter term in the OS scheme, which has a UV divergent and a finite contribution,

(43) |

For the gluino contribution we have . Therefore, with eq. (42) we get

(44) |

In the scheme we need only the UV divergent part of (44) which is

(45) |

is the UV divergence factor. In eqs. (41)-(44) and are the two- and three-point functions

(46) |

(47) |

(48) |

The total correction (eq. (40)) is given by

(49) | |||||

(50) |

As and thus , (49) is UV convergent. As , also (50) is UV convergent.

In the limit , from (94) it follows and from (87) it follows . However, in this limit (78) and (79) become independent of the index and grow with . Therefore, guarantees decoupling of the gluino loop contribution in the OS scheme.

In the scheme for , we get

(51) |

At first sight it seems that the gluino contribution does not decouple for . However, the tree-level coupling (eq. (10)) contains a factor . We have

(52) |

where we take as input [32]. is due to the self-energy contributions with gluino (see Figs. 4 and 4). We get

(53) |

Thus the sum is indeed decoupling for . Analogously, this also holds for the chargino and neutralino contributions.

### 3.4 Total result for the width at full one-loop level

Finally, we want to sum up all contributions to get the total result for at full one loop level.

The one-loop result including gluino and EW contributions reads

(54) |

where , and are given by eqs. (9), (39) and (13), respectively. Note that eq. (54) is a series expansion around .

However, the improved result with gluon contribution (eq. (38)) given by

(55) |

is a series expansion around . In order to combine eqs. (54) and (55) in a consistent way we write: