Mixing and Decay in the NMSSM with the Flavour Expansion Theorem
In this paper, motivated by the observation that the Standard Model predictions are now above the experimental data for the mass difference , we perform a detailed study of mixing and decay in the -invariant NMSSM with non-minimal flavour violation, using the recently developed procedure based on the Flavour Expansion Theorem, with which one can perform a purely algebraic mass-insertion expansion of an amplitude written in the mass eigenstate basis without performing diagrammatic calculations in the interaction/flavour basis. Specifically, we consider finite orders of mass insertions for neutralinos but general orders for squarks and charginos, under two sets of assumptions for the squark flavour structures (i.e., while the flavour-conserving off-diagonal element is kept in both cases, only the flavour-violating off-diagonal elements and () are kept in cases I and II, respectively). Our analytic results are then expressed directly in terms of the initial Lagrangian parameters in the interaction/flavour basis, making it easy to impose experimental bounds on them. It is found numerically that the NMSSM effects in both cases can accommodate the observed deviation for , while complying with the experimental constraints from the branching ratios of and decays.
It is known that Supersymmetric (SUSY) extensions of the Standard Model (SM) are well motivated by being able to provide a unification of the SM gauge couplings at high scales, a solution of the hierarchy problem, and a viable dark matter candidate [1, 2]. As one of the low-energy realizations of SUSY, the Minimal Supersymmetric Standard Model (MSSM) , carrying the minimal field content consistent with observations, has received over years of continuous attentions; see e.g. Refs. [4, 5] for reviews. Despite having many advantages [4, 5], the MSSM needs to be extended because of the following two main motivations. The first one is due to the existence of the “-problem” in the MSSM , where is a dimensionful parameter set by hand at the electroweak (EW) scale before spontaneous symmetry breaking. The second one is driven by the recent discovered GeV SM-like Higgs boson [7, 8], which has imposed strong constrains on the parameter space of MSSM [9, 10, 11]. Among the various non-minimal SUSY models, the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [12, 13, 14, 15], the simplest extension of the MSSM with a gauge singlet superfield, not only has the capability to fix these shortcomings of the MSSM, but also alleviates the tension implied by the lack of any evidence for superpartners below the EW scale . Specifically, this model can solve the “-problem” of the MSSM elegantly by generating an effective -term at the SUSY breaking scale [17, 18]. Restrictions on the Higgs sector can also be relaxed in the NMSSM, because the Higgs field can acquire a larger tree-level mass with a low SUSY scale, making accordingly quantum corrections to the observed GeV Higgs boson small [14, 15].
Along with the dedicated direct searches at the Large Hadron Collider (LHC) for SUSY particles , it is also interesting and even complementary to investigate virtual effects of these hypothesized particles on low-energy processes, such as the neutral-meson mixings as well as the CP violation and rare decays of various hadrons [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In this paper, we shall focus on the -invariant NMSSM [14, 15], a simpler scenario of NMSSM with a scale-invariant superpotential, and study its effects on mixing, and decays.
With the latest lattice inputs [34, 35] for the bag parameters that are used to quantify the non-perturbative hadronic matrix elements between and states, the SM predictions for the mass difference between the mass eigenstates of neutral meson read [34, 36]
This observation has profound implications for New Physics (NP) models [36, 39]. As detailed in Refs. [32, 39, 27], this implies particularly that the constrained minimal flavour violating (CMFV) models, in which all flavour violations arise only from the Cabibbo-Kobayashi-Maskawa (CKM) matrix [40, 41], have difficulties in describing the current data on . Thus, in order to relax such a tension, one has to resort to scenarios with non-minimal flavour violation (NMFV), which involve extra sources of flavour- and/or CP-violation, and can provide therefore potential negative contributions to [42, 43, 44]. This motivates us to investigate whether the -invariant NMSSM with NMFV, in which the extra flavour violations arise from the non-diagonal parts of the squark mass matrices related to the soft SUSY breaking terms, can accommodate the observed deviation for .
In SUSY models, a transition amplitude is more conveniently calculated in the interaction/flavour basis in which gauge interactions are flavour diagonal and flavour-changing interactions originate from the off-diagonal entries of the mass matrices in the initial Lagrangian before diagonalization and identification of the physics states, than in the mass eigenstate (ME) basis in which the transition amplitude is expressed in terms of the physical masses and mixing matrices. This can be achieved using two different methods. The first one is based on the well-known diagrammatic technique called the Mass Insertion Approximation (MIA) [45, 46, 47, 48]. Here diagonal elements of the mass matrices are absorbed into the definition of (un-physical) massive propagators and the amplitude is, at each loop order, expanded into an infinite series of the off-diagonal elements of the mass matrices, commonly referred to as mass insertion (MI). The second one is based on the Flavour Expansion Theorem (FET) , according to which an analytic function about zero of a Hermitian matrix can be expanded polynomially in terms of its off-diagonal elements with coefficients being the divided differences of the analytic function and arguments the diagonal elements of the Hermitian matrix. As a purely algebraic method, it offers an alternative derivation of the MIA result directly from the amplitude calculated in the ME basis, without performing tedious and error-prone diagrammatic calculations with MIs in the interaction/flavour basis . Even in the case where there is no clear diagrammatic picture, the FET expansion can still give a consistent MIA result. This method has also been automatized in the package MassToMI , facilitating the expansion of an ME amplitude to any user-defined MI order. See e.g. Refs. [49, 51, 52, 43, 53] for recent applications of this method.
In the -invariant NMSSM with NMFV, we further assume that the third-generation squarks can mix with the other two generations simultaneously, but leaving the latter two immune to each other. Such a choice is motivated by the flavour mixing effects observed in mixing, and decays [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. With such a specific squark flavour structures, we shall then adopt the FET procedure  to calculate the mass difference and the branching ratio of decay, by considering general MI orders for squarks and charginos but finite MI orders for neutralinos. While the general MI orders have also been considered in Refs. [54, 55], only one kind of MI parameter is kept in the whole “fat propagators”. In our case, however, there exist two kinds of MI parameters in each line and the mixed arrangement of them is required. For concreteness, We call our procedure the FET expansion with different MI order and, by checking if the FET results agree with the ones calculated numerically in the ME basis, test our estimation for the optimal cutting-off MI orders. For the branching ratio of decay, the public code SUSY_FLAVOR [56, 57, 58] is used.
Our paper is organized as follows. In Sec. 2, after specifying the flavour structures assumed in our scenario, we introduce the FET procedure with different MI order, which is then used to calculate the mixing and decay in Sec. 3. Detailed numerical results and discussions are then presented in Sec. 4. Our conclusions are finally made in Sec. 5. For convenience, the block terms of squarks and charginos are listed in the appendix.
2 FET with different MI order in -invariant NMSSM
2.1 Lagrangian of -invariant NMSSM
where is the MSSM superpotential but without the term [61, 62], and denotes the Higgs singlet superfield, while and are the two Higgs doublet superfields, with the convention . The dimensionless parameters and can be complex in general, but are real in the CP-conserving case. After the scalar component of gets a non-zero vacuum expectation value (VEV), , the second term in Eq. (2.1) generates an effective term, with , which then solves the “-problem” of the MSSM [14, 15].
where corresponds to the MSSM part but with the -related term removed [61, 62]. The soft SUSY breaking mass parameter is real, while the soft SUSY breaking trilinear couplings and are complex in general, but are also taken to be real in the CP-conserving case, as is assumed throughout this paper.
2.2 Flavour structures of -invariant NMSSM
Firstly, we focus on the up- and down-squark mass squared matrices and , which can be written in their most general -block form as 
in the so-called super-CKM basis . In the NMFV paradigm [29, 30], these two mass matrices are not yet diagonal and can introduce general squark flavour mixings that are usually described by a set of dimensionless parameters , with A, B=L, R referring to the left- and right-handed superpartners of the corresponding quarks and the generation indices .
Throughout this paper, we assume that the third-generation squarks can mix with the other two generations simultaneously, but leaving the latter two immune to each other, so as to comply with the severe constraints from flavour and precision data [30, 63, 64]. This promotes us to consider the following two sets of squark flavour structures: while the flavour-conserving off-diagonal element is kept in both cases, only the flavour-violating off-diagonal elements and () are kept in cases I and II, respectively. In addition, to avoid the occurrence of dangerous charge and colour breaking minima and unbounded from below directions in the effective potential [65, 66], we set all the flavour-violating off-diagonal elements in the LR and RL sectors to be zero. Then, the two mass squared matrices and in case I are given, respectively, by
where and . In the LL sectors, which satisfy the relation (with being the CKM matrix) due to the gauge invariance , we have neglected safely the terms in Eq. (2.5) (and also in Eq. (2.7)), where is one of the CKM parameters. Here we have also assumed that the first two generations of squarks are nearly degenerate in mass .
In case II, on the other hand, the two mass squared matrices are given, respectively, by
where and . In both of these two cases, a non-zero is kept to reproduce the GeV SM-like Higgs boson [30, 63, 64]. Here, for simplicity, we assume that all the parameters are real and hence , due to hermiticity of the squark mass matrices.
The mass matrix for charginos in the interaction basis reads 
where is the wino mass, and is the mixing angle of the two Higgs doublets, defined in terms of their VEVs and . The squared masses , and the MI parameters , are defined, respectively, by
where and the summation is not applied for the same index here.
where is the bino mass, and is the weak mixing angle. Such a mass matrix indicates that the singlino couples only to the Higgsinos and , but not to the gauginos and . The squared masses and the MI parameters are defined, respectively, by
2.3 FET expansion with different MI order
Before performing the FET expansion, one has to write down the transition amplitude in the ME basis . All the relevant Feynman rules are taken from Refs. [62, 15, 14, 67, 68]. Then, the procedure of FET expansion with general/finite MI order includes the following three steps.
Step 1. One transforms the amplitude written in the ME basis into the intermediate result expressed in terms of the blocks , which are defined, respectively, as 
for the up and down squarks behaving as scalar fields,
for the charginos that behave as Dirac fermions, and
for the neutralinos behaving as Majorana fermions. Here represents symbolically part of the transition amplitude that depends on the mass of an internal physical particle in a Feynman diagram, at tree or loop level; for example, can be a propagator, with being the momentum of particle . The unitary transformation matrices , , , , and are introduced to diagonalize the Hermitian mass squared matrices , , , , and , respectively. We use and to represent the flavour indices of field multiplets in the ME and the interaction/flavour basis, respectively.
After applying the transformation rules specified by Eqs. (2.13)–(2.21), one can see that the blocks depend only on the matrix elements of some functions with arguments being the Hermitian mass squared matrices, and can be given by the expansion 
where represents the -th term in the MI order of the blocks .
Step 2. During our calculation, we also encounter the case in which a Feynman diagram contains two lines involving the same particle. In such a case, the product of two blocks with MI orders specified respectively by and , , should be firstly combined into a single term with fixed MI order, such as , where denotes the -th term in the MI order of the block , with
All the non-zero block terms and for squarks and charginos, with the flavour structures specified in Sec. 2.2, can be easily derived and are listed in the appendix.
As the neutralino mass matrix , given by Eq. (2.11), has many non-zero elements, one can use the following recursive formulas to represent the corresponding blocks
where , and can be re-expressed in terms of using the “divided difference” method .
Step 3. One now needs to perform the loop-momentum integration over the products of blocks , , and introduced in the last step. As only one-loop amplitudes are involved throughout this paper, this can be done using iteratively the operation , where denotes symbolically the squared mass, equaling to for scalars and to or for fermions, and is the -point one-loop integrals in the Passarino-Veltman (PV) basis , such as , and introduced in Ref. .
2.4 MI-order estimation
Although the FET procedure can provide with us the result expanded to any MI order, an optimal cutting-off should be applied due to the time costing of the programme running. Here we illustrate an efficient MI-order estimation method to get the appropriate order which was usually set by hand (order 2 or 4) in most of recent works [51, 43, 52, 53].
Taking the block term in case I,
where , as an example, and using the inequality 
satisfied by the PV-integrals with vanishing external momenta that are defined by
with the assumption that to avoid divergent integrals, we can obtain
where represent the blocks related to the other types of particles, such as . Then, for a given small constant , only when
can the terms starting from the -th MI order in the series expansion of the block be safely neglected. Thus, the cutting-off MI order for should be at least. The same method can be applied for other blocks, and the final cutting-off MI orders for squarks and charginos can be determined accordingly.
For the neutralino blocks, Eq. (2.26) still works for estimating the required MI order. In this case, we obtain
So, when and for fixed indices and , the summation over the MI index can be terminated to the -th order.
3 mixing and decay
In this section, we shall apply the procedure of FET expansion with general/finite MI order to the mixing and decay, within the -invariant NMSSM with NMFV.
The strength of mixing is described by the mass difference , defined by 
where denotes the off-diagonal element in the neutral -meson mass matrix, and the effective weak Hamiltonian can be written in a general form as 
Within the -invariant NMSSM with NMFV, the following eight operators, as defined in Ref. , are all found to be relevant:
where and are the colour indices, , and .
To the lowest order in the EW theory, the corresponding Wilson coefficients , at the matching scale, of the operators are obtained by evaluating the various one-loop box diagrams mediated by heavy particles appearing in the SM and beyond555Here we do not consider the double-penguin diagrams, which involve the exchange of CP-even and CP-odd scalars, and can give significant contributions only for large values of [22, 71, 72]. This is justified by our choice of the two sets of SUSY parameters collected in Table 2, with being fixed at and , respectively.. Within the SM, only gets a non-negligible contribution from the one-loop box diagrams with up-type quarks and bosons circulating in the loops , and the perturbative two-loop QCD corrections to are also known . In the context of -invariant NMSSM with NMFV, on the other hand, all the eight Wilson coefficients can get non-zero contributions from the additional one-loop box diagrams mediated by: 1) charged Higgs, up-quarks; 2) chargino, up-squarks; 3) gluinos, down-squarks; 4) neutralinos, down-squarks; 5) mixed gluino, neutralino, down-squarks [27, 74, 22]. With the aid of FeynArts  and FeynCalc  packages, all these Feynman diagrams can be calculated and the resulting Wilson coefficients are expressed in terms of the rotation matrices , , , , and , as well as the blocks and . Our results for the Wilson coefficients agree with the ones given in Refs. [43, 29, 27]. Then, following the procedure detailed in Sec. 2.3, we can transform these Wilson coefficients given in the ME basis into the FET results. Here we have made full use of the hierarchies among the CKM parameters to simplify the final results. For example, when calculating in case I, we encounter a term
As , with , does not vanish only when or , and because of , we can safely neglect the term with to get