Studying Exclusive Semi-leptonic $b\to (s,d) \nu \bar{\nu}$ Decays in the MSSM without R-parity

We present a complete study of R-parity violating supersymmetric effects in thirteen exclusive and inclusive semi-leptonic $b \to (s,d) \nu \bar{\nu}$ decays, including $B^+_u\to K^{(*)+}\nu\bar{\nu}$, $B^0_d\to K^{(*)0}\nu\bar{\nu}$, $B^0_s\to \phi\nu\bar{\nu}$, $B^0_d\to \pi^0(\rho^0)\nu\bar{\nu}$, $B^+_u\to \pi^+(\rho^+)\nu\bar{\nu}$, $B^0_s\to K^{(*)0}\nu\bar{\nu}$ and $B \to X_{s,d} \nu \bar{\nu}$ decay modes, and we find those thirteen modes are very sensitive to the constrained R-parity violating couplings. We derive stringent bounds on relevant R-parity violating couplings, which are based on all existent experimental upper limits of involved semi-leptonic decays. In addition, we also investigate the sensitivities of the branching ratios and di-neutrino invariant mass spectra to the survived R-parity violating coupling spaces. Since the experimental bounds would become much better soon through Super-B, we expect that future experiments will greatly strengthen our bounds.

In the minimal supersymmetric standard model (MSSM) [19,20] with R-parity conservation, the new contributions to the b → sνν transition have been discussed (for instance, see Refs. [21,22,23]). In this work, we will concentrate on RPV effects in the exclusive and inclusive semi-leptonic b → (s, d)νν decays. From the latest experimental data given in Eqs. (1)(2) and the theoretical parameters with uncertainties, we will derive the new conservative upper limits on the relevant RPV coupling products. Moreover, we will also investigate how survived RPV coupling spaces can affect on the branching ratios and di-neutrino invariant mass (i.e. missing mass) spectra in these semi-leptonic b → (s, d)νν decays. We find these observables are still very sensitive to survived RPV coupling spaces.
Our letter is organized as follows: In Sec. 2, we review the effective Hamiltonian for b → (s, d)νν transitions and define the observables that can in principle be measured in these decays. In Sec. 3, we deal with the numerical results. We display the constrained parameter spaces which satisfy all the available experimental upper limits of the b → (s, d)νν, and then, we investigate the sensitivities of the branching ratios and di-neutrino invariant mass spectra to the survived RPV coupling spaces in those decays. We conclude in Sec. 4.

Theoretical Framework
The b → d j ν i ′ν i (j = 1, 2 and i, i ′ = e, µ, τ ) transitions can be described by the effective Hamiltonian, In the SM, b → d j ν i ′ν i proceeds via W box and Z penguin diagrams, therefore only purely [24], where G F is the Fermi constant, α e is the fine structure constant, θ W is the Weinberg angle, and V ij are the CKM matrix elements. Function X(x t ) is dominated by the short-distance dynamics associated with top quark exchange [9], and has the theoretical uncertainty due to the error of top quark mass, whose explicit form can be found in Refs. [25,26].
In supersymmetric models without R-parity [15,16], extra trilinear RPV terms 1 λ ′ ijkL iQjD c k are allowed in the superpotential [27]. Both left-handed and right-handed currents are present in b → d j ν i ′ν i transition at the tree level in these models. Then the corresponding coefficients in Eq. (3) are written as 1L andQ are the SU(2) doublet lepton and quark superfields, respectively,D c are the singlet superfields, while i, j and k are generation indices and the superscript c denotes a charge conjugate field.
From the theoretical point of view, the inclusive semi-leptonic b → qνν (q = s, d) decays are very clean proceses, since both the perturbative α s and the non-perturbative 1/m 2 b corrections are known to be small. Their dineutrino invariant mass distributions are given as following where , and κ(0) = 0.83 represents the QCD correction to the b → qνν matrix element [6,28,29]. We have summed over the neutrino flavors in Eq. (5).
In order to compute branching ratios of the exclusive semi-leptonic b → (s, d)νν decays, we need the matrix elements of the effective hamiltonian between the states of the initial B particle and the final particles M, ν,ν. The hadronic matrix elements for B → P transition (P is a pseudoscalar meson, π or K) can be parameterized in terms of the form factors f P + (s B ) and f P 0 (s B ) as where the factor c P accounts for the flavor content of particles (c P = √ 2 for π 0 , and c P = 1 for can be written in terms of five form factors where In terms of the effective Hamiltonian shown in Eq. (3) and the relevant form factors given in Eqs. (6)(7), the di-neutrino invariant mass distributions for B → P νν and B → V νν decays can be written as [7,30] where , and we have summed over the neutrino flavors.
For our numerical results, we use the relevant B → P (V ) form factors given in [31]. However, B s → K form factors are not given in LCSR results [31]. After discussions with authors of Ref. [31], we obtain them as . The uncertainties of form factors at s B = 0 induced by F (0) are considered, to be conservative, we adopt these uncertainties for full s B range. The CKM matrix elements are taken from [32], and masses and lifetimes are from Ref. [33].

Numerical Results and Discussions
In this section, we summarize our numerical results and analysis in the semi-leptonic b → (s, d)νν decays. To be conservative, we use all input parameters which are varied randomly within 1σ ranges in our numerical results. We use the average τ B = (τ B + + τ B 0 )/2 for the inclusive decays. When we study the RPV effects, we consider only one RPV coupling product contributions at one time, neglecting the interferences between different RPV coupling products, but keeping their interferences with the SM amplitude. We assume the masses of sfermions are 500 GeV. For other values of the sfermion masses, the bounds on the couplings in this paper can be easily obtained by scaling them by factorf 2 ≡ ( mf 500 GeV ) 2 . The transitions b → (s or d)ν i ′ν i involve the same set of the RPV coupling products for every generation of neutrinos: , which come from left-handed and right-handed squark exchanges, respectively. For arise from left-handed and righthanded squark exchanges, respectively. We use the latest experimental upper limits from Refs. [1,2,3,4,5], which are listed in Eqs. (1)(2), to constrain the relevant RPV coupling products.
The decay b → sγ in the MSSM without R-parity has been shown in [34] to give weak constraints on relevant RPV coupling combinations, |λ ′ * i3k λ ′ i2k | ≤ 2.25 and |λ ′ * ik2 λ ′ ik3 | ≤ 0.87 with 500 GeV sfermion masses. The RPV effects in b → (s, d)ℓ + ℓ − processes have been studies in Refs. [35,36,37,38,39,40,41,42], and some upper limits of their RPV coupling combinations are about one order of magnitude stronger than ours from b → (s, d)νν. Here, we list the stronger upper limits from b → (s, d)ℓ + ℓ − processes with 500 GeV sfermions: [39]. In addition, single bounds of λ ′ ijk are obtained by many authors (for instance, see Refs. [40,41,42,43,44,45,46,47]). We also note that some of the single λ ′ couplings can generate sizable neutrino masses [41,43]. Allanach et al. have obtained quite strong upper bound |λ ′ ijj | < 10 −2 with 500 GeV sfermions in the RPV mSUGRA model, and Barbier et al. have gotten |λ ′ i33 | < 4.4 × 10 −3 . Furthermore, the λ ′ 111 coupling has been constrained as low as |λ ′ 111 | < 1.8 × 10 −2 by neutrino-less double beta decay [47]. If we now compare our combined bounds with the products of the single bounds, we find that our combined bounds are weaker one or two order(s) of magnitude than the products of the single bounds. However, it also should be noted that the parameter spaces of λ ′ from neutrino masses can be evaded since several other parameters are usually involved in the extraction of the constraints [48]. Furthermore, the constraints on λ ′ from neutrino masses would depend on the explicit neutrino masses models with trilinear couplings only, bilinear couplings only, or both [41].
Next, we will first explore the RPV MSSM effects by using our constrained RPV parameter spaces, and then discuss the RPV effects after also considering previous stronger bounds in the semi-leptonic b → (s, d)νν decays. Now using the survived RPV parameter spaces shown in Fig.   1, we explore the RPV MSSM effects in the semi-leptonic b → (s, d)νν decays, which satisfy all experimental upper limits given in Eqs. (1)(2). Our RPV MSSM predictions within the theoretical uncertainties of input parameters are given in Table 2, together with experimental upper limits and the SM predictions for a convenient comparison. In Table 2, the second and third columns give the experimental upper limits and the SM predictions, respectively, the forth column lists the effects of left-handed squark exchange coupling λ ′ * i3k λ ′ i ′ jk , and the last column summaries the effects of coupling λ ′ * i ′ kj λ ′ ik3 due to right-handed squark exchange. Main theoretical uncertainties of the SM predictions arise from the CKM matrix elements, Wilson coefficient and hadronic transition form factors(only for the exclusive decays). Comparing with experimental upper limits and the SM predictions, we find some salient features of numerical results of the RPV effects listed in Table 2. 1 RPV coupling λ ′ * i3k λ ′ i ′ 2k is only constrained by the experimental upper limit of B(B + u → K + νν), and bounds on this coupling constant obtained from other exclusive b → sνν decays and inclusive B → X s νν are weaker than one obtained from B + u → K + νν decay. Comparing with the SM predictions, we find contributions of λ ′ * i3k λ ′ i ′ 2k coupling could enlarge the allowed ranges of all relevant branching ratios, their upper limits are increased two or three times, and their lower limits are reduced more than one order. 2 The restrictions of λ ′ * i ′ k2 λ ′ ik3 come from the experimental upper limits of B(B + u → K + νν) and B(B + u → K * + νν). The λ ′ * i ′ k2 λ ′ ik3 coupling effects are same as the effects of coupling could obviously increase the allowed upper limits of these branching ratios.
Next we want to illustrate briefly the sensitivities of relevant observables to RPV couplings.
To this end, for each RPV coupling product, we can present the correlations of di-neutrino invariant mass spectra and branching ratios within the constrained parameter space displayed in Fig. 1 by two-dimensional scatter plots. The RPV coupling λ ′ * i3k λ ′ i ′ jk or λ ′ * i ′ kj λ ′ ik3 contributions to these semi-leptonic B d , B u and B s decays are very similar to each other. So we will take an example for B → X s,d νν, K + (K * + )νν, π + (ρ + )νν decays to illustrate the sensitivities of quantities to RPV couplings.

Summary
In this letter we have performed a brief study of the RPV coupling effects in supersymmetry from the exclusive and inclusive semi-leptonic B decays with a νν pair, which include B + u → K ( * )+ νν, B 0 d → K ( * )0 νν, B 0 s → φνν, B → X s νν, B 0 d → π 0 (ρ 0 )νν, B + u → π + (ρ + )νν, B 0 s → K ( * )0 νν and B → X d νν thirteen decay modes. Considering the theoretical uncertainties, we have obtained conservatively constrained parameter spaces of RPV coupling constants from the latest experimental upper limits. We found, at present, the strongest bounds on the relevant RPV couplings come from the exclusive decays. Furthermore, we also investigated the sensitivities of the di-neutrino invariant mass spectra and branching ratios to the survived R-parity violating coupling spaces.
We have found that, after satisfying all the current experimental upper limits, both lefthanded and right-handed squark exchange RPV couplings still have significant effects on these di-neutrino invariant mass spectra and branching ratios. The RPV contributions are not easily distinguishable from the SM predictions in the di-neutrino invariant mass spectra of the semi-leptonic b → sνν decays, nevertheless, the di-neutrino invariant mass spectra of the semileptonic b → dνν decays are very useful to distinguish the RPV coupling effects at all kinematic regions. The branching ratios of the semi-leptonic b → sνν decays are sensitive to both moduli and phases of relevant RPV coupling products, and the branching ratios of the semi-leptonic b → dνν decays are only very sensitive to the moduli of relevant RPV coupling products.
However, observing rare B decays with a νν pair is experimentally very challenging because of the two missing neutrinos and (many) hadrons, and these decays can be searched for through the large missing energy events in B decays. With an advent of Super-B facilities [49], the prospects of measuring the branching ratios of the semi-leptonic b → sνν decays in next decade could be highly realistic, and it's also possible to observe B + → π + νν decay. We expect that future experiments will significantly strengthen the allowed parameter spaces for RPV couplings. Our predictions of RPV effects on related observables could be very useful for probing RPV supersymmetric effects in future experiments.