Study of $s\to d\nu\bar{\nu}$ rare hyperon decays within the standard model and new physics

FCNC processes offer important tools to test the standard model, and to search for possible new physics. In this work, we investigate the $s\to d\nu\bar{\nu}$ rare hyperon decays within the standard model and beyond. The hadronic matrix elements, parametrized in terms of form factors, are calculated in the light-front approach. We find branching ratios for these rare hyperon decays range from $10^{-14}$ to $10^{-11}$ within the standard model. And after taking into account the contribution from the new physics, the generalized SUSY extension of the SM and the minimal 331 model, the decay widths for these channels can be enhanced by a factor of 2 and 7, respectively. Uncertainties are discussed.

the SM. The new physic contribution, the generalized SUSY extension of the SM and the minimal 331 model, are considered. We also discuss the possible uncertainties from the form factors. The last section contains a short summary.
Here λ (′) denotes the helicity of the parent (daughter) baryon in the initial (final) state, and λ V is the helicity of the virtual intermediate vector particle. It can be shown that the helicity amplitudes H V,A λ ′ ,λV have the following neat forms [16]: In the above, Q ± = (M ± M ′ ) 2 − q 2 , and M (M ′ ) is the parent (daughter) baryon mass in the initial (final) state. The amplitudes for the negative helicities are obtained in terms of the relations The complete helicity amplitudes are obtained by The differential decay width of B → B ′ is Here dΓ L /dq 2 and dΓ T /dq 2 are the longitudinal and transverse parts of the decay width, and their explicit expressions are given by In Eqs. (9) and (10), p ′ = Q + Q − /2M is the magnitude of the momentum of B ′ in the rest frame of B, and Note that we have neglected the electron and muon masses. One can then obtain the decay width

A. Calculation within the Standard Model
The following inputs are used [17,18]: The masses and lifetimes of baryons in the initial and final states are listed as follows [17]: The following CKM parameters are used [17]: With the above inputs and the formulae given in the last section, we have for µ c = 1GeV, µ t = 100GeV and µ c = 3GeV, µ t = 300GeV the LO and NLO results listed in Table I. One can see from the corresponding results in Table I that: • The branching fractions of s → dνν rare hyperon decays range from 10 −14 to 10 −11 .
• For µ c = 1GeV, µ t = 100GeV the NLO results are smaller than the corresponding LO ones by about 30%, while for µ c = 3GeV, µ t = 300GeV the NLO results are larger than the corresponding LO ones by about 10%.
• The branching fraction of Ω − → Ξ − νν is the largest among those of the 6 channels. And it is in the same order of K + → π + νν and K L → π 0 νν.
The effective Hamiltonian for s → dνν in a generalized SUSY extension of the SM is given in Eq.
(1) with X(x t ) replaced by [19] Here x tH = m 2 t /m 2 H ± and X H (x tH ) corresponds to the charged Higgs contribution. The C χ and C N denote the chargino and neutralino contributions where X i χ and X N depend on SUSY masses and respectively on chargino and neutralino mixing angles. The explicit expressions of X H (x), C χ and C N can be found in Ref. [19]. The R parameters are defined in terms of the mass insertions, and their upper limits are listed in Table II [19]. It should be mentioned that the phase φ of R U sLtR and R U tRdL is a free parameter which ranges from 0 to 2π. We set φ = 0 as a central result.
The following parameters are adopted [20]: where all mass parameters are in GeV. The assumption M 1 ≈ 0.5M 2 has been made [21]. Our calculation results are listed in Table I. Comparing the results of NLO+SUSY with the corresponding ones of NLO, one can see that after taking account of the contribution from new physics, all the branching fractions are roughly enhanced by a factor of 2.

C. Contribution from minimal 331 model
The so-called minimal 331 model is an extension of the SM at the TeV scale, by extending the weak gauge group of the SM SU (2) L to a SU (3) L . In this model, a new neutral Z ′ gauge boson could give a by far important additional contributions, for it can transmit FCNCs at tree level. In Table I, we denote this model as M331. Many more details of this model could be found in Ref. [22]. For the effective Hamilton, the minimal 331 model leads to a new term in the following form [23]: with M Z ′ = 1 TeV, Re[(Ṽ * 32Ṽ31 ) 2 ] = 9.2 × 10 −6 and Im[(Ṽ * 32Ṽ31 ) 2 ] = 4.8 × 10 −8 . And the other parameters are the same as the SM inputs [17,18]. Then the function X(x t ) in Eq. (1) could be redefined as X(x t ) = X SM (x t ) + ∆X with Using the modified function X(x t ) and considering the NLO contribution, we calculate the branching ratios of rare hyperon decays in the minimal 331 model as shown in Table I. We find that the numerical results of the minimal 331 model are 7 times larger than the NLO results in the SM and three times larger than the NLO+SUSY results.

D. Form factors uncertainties
In the last section, we have set f 1 = g 1 = 1 and f 2,3 = g 2,3 = 0 in Eq. (2) as an approximation for q 2 small. In fact, it can be shown that f 3 and g 3 do not contribute to the decay width since the neutrino's mass is negibible. To explore the errors caused by this approximation, we turn to the Light-Front Quark Model (LFQM) [24]. This approach has been adopted in Refs. [16,25,26] to study baryon decays using a quark-diquark picture, and we refer the reader to these references for the explicit expressions of transition form factors. For simplicity, we only consider the process Λ → nνν. The following input parameters in this approach are used: Several comments are in order: • Constituent quark masses m s = 0.37 GeV and m d = 0.25 GeV are widely used in Refs. [27][28][29][30][31][32][33][34][35][36].
• In this calculation, we consider the two spectator quarks u and d to be a diquark with J P = 0 + , which is denoted by [ud]. According to Refs. [16,37], the mass of the diquark [ud] should be close to the mass of s quark, so we take m [ud] = 0.37 GeV.
The explicit expressions of the form factors f 1,2 and g 1,2 are given in Ref. [16]. Using those formulae and adopting the following fit formula the parameters (F (0), m fit , δ) are given in Table III. One can see from Table III that: • f 1 , which ranges from f 1 (0) = 0.98 to f 1 (q 2 max ) = 1.02, agrees well with our approximation f 1 = 1. Here q 2 max = (m Λ − m n ) 2 ≈ 0.03GeV 2 .
• g 1 , which ranges from g 1 (0) = 0.75 to g 1 (q 2 max ) = 0.77, deviates from the approximation g 1 = 1 by about 1/4. • f 2 , which ranges from f 2 (0) = −0.71 to f 2 (q 2 max ) = −0.75, was considered to be small. • g 2 , which ranges from g 2 (0) = −0.0037 to g 2 (q 2 max ) = −0.0049, agrees well with our approximation g 2 = 0.  From the above analysis, it is clear that the dominant deviations come from g 1 and f 2 . Thus in the following analysis, we will only consider the uncertainties from them.
Instead of the simplified Eq. (6), the full expressions for the helicity amplitudes are given as [16]: We neglect the terms proportional to q 2 /M since the momentum transfer q 2 is small compared to the baryon mass.
As a good approximation we set f 1 = 1, g 1 = 0.76, f 2 = −0.73, g 2 = 0. Using the above approximations, we find that the branching fraction deviates by about 30% from the original one.

IV. CONCLUSIONS
FCNC processes offer important tools to test the SM, and to search for possible new physics. The two decays K + → π + νν and K L → π 0 νν have been widely studied while the corresponding baryon sector has not been explored.
In this work we have studied the s → dνν rare hyperon decays. We adopt the leading order approximations for the form factors for the q 2 is small, and finally we arrive at our expression of decay width. After that, we apply the decay width formula to both the SM and the new physics contribution. Different energy scales are considered. The branching fractions within the SM range from 10 −14 to 10 −11 , and the largest one is at the same order as those of K + → π + νν and K L → π 0 νν. After taking into account the contribution from the new physics, the generalized SUSY extension of SM and the minimal 331 model, the branching fractions can be enhanced by a factor of 2 and 7 respectively. The uncertainties coming from the form factors are also discussed using the light-front approach under a diquark picture. We find that the uncertainties are large, which signifies some more efficient computing methods are needed to improve the evaluation of the form factors.