Rare hyperon decays with missing energy

We explore the strangeness-changing decays of the lightest hyperons into another baryon plus missing energy within and beyond the standard model (SM). In the SM these processes arise from the loop-induced quark transition s→dνν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ s\to d\nu \overline{\nu} $$\end{document} and their branching fractions are estimated to be less than 10−11. In the presence of new physics (NP) the rates of these hyperon decays with missing energy could increase significantly with respect to the SM expectations because of modifications to the SM process or contributions from additional modes with new invisible particles. Adopting a model-independent approach and taking into account constraints from the kaon sector, we find that the current data on K→πνν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K\to \pi \nu \overline{\nu} $$\end{document} do not permit sizable NP impact on the hyperon decays via underlying operators having mainly parity-even quark parts. In contrast, NP operators with primarily parity-odd quark parts are much less restricted by the existing bounds on K → invisible and K→ππνν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K\to \pi \pi \nu \overline{\nu} $$\end{document} and consequently could produce substantially amplifying effects on the hyperon modes. Their NP-enhanced branching fractions could reach levels potentially observable in the ongoing BESIII experiment.


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structure as in the SM have been carried out in refs. [7,9]. In this paper, we explore a more general scenario in which the underlying NP operators might involve other Lorentz structures and the invisible pair could be nonstandard fermions. It turns out that in this more general case the hyperon rates may be significantly enlarged with respect to their SM values and even reach potentially discoverable levels at BESIII.
As detailed later on in this analysis, such a possibility has to do with the fact that these kaon and hyperon decays do not probe the same portions of the underlying s → d / E operators and with the kaon data situation at the moment. Particularly, K → π / E is sensitive exclusively to the terms in the operators which have parity-even quark parts and K → / E to the terms having parity-odd quark parts, whereas K → ππ / E and the hyperon modes can probe both. Given that the latest measurements of K → πνν decays [20][21][22] have left only little room for NP to influence them, it follows that NP cannot raise the hyperon rates considerably above their SM expectations if it enters via s → d / E operators with mainly parity-even quark parts. On the other hand, the constraints from the existing data on K → / E and K → ππ / E are relatively much weaker, implying that NP operators having primarily parity-odd quark parts are still allowed to yield sizable enhancing effects on the hyperon modes.
The organization of the paper is as follows. In section 2 we write down a number of effective low-energy operators contributing to s → d / E which may be generated by NP. Without getting into model specifics, we treat the operators in a model-independent manner. In section 3 we first deal with the baryonic matrix elements pertinent to the corresponding hyperon decays with missing energy and subsequently derive their differential rates. Similarly, in section 4 we provide the formulas for the rare kaon decays of concern. In section 5 we present our numerical results. We begin by evaluating the SM predictions for the hyperon modes and comparing them to the proposed sensitivity reach of BESIII. Next we look at the kaon sector and examine its restraints on NP impacting the operators. We then show that some of the allowed NP couplings can amplify the hyperon rates to values that may be observable by BESIII. In section 6, we give our conclusions. We collect extra formulas and further details in a couple of appendices.

Interactions
Beyond the SM, there could be new ingredients which induce modifications to the s → dνν transition in the SM and/or bring about additional s → d / E channels with one or more invisible light nonstandard bosons or fermions emerging in the final states. These new particles could be stable or sufficiently long-lived to escape detection. Among such possibilities, in this study we focus on the s → d / E scenario in which the missing energy is due to ff being emitted where f is an electrically neutral, uncolored, and invisible Dirac fermion having spin 1/2. Thus, f could be the SM neutrino ν, which is not detected, or a nonstandard fermion.
We consider sdff interactions described by the low-energy effective Lagrangian

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where in our model-independent approach C V,A,S,P f andc v,a,s,p f are free parameters which are generally complex and have the dimension of inverse squared mass. The terms in L f are Lorentz invariant and respect the unbroken SU(3) color ×U(1) em gauge symmetry. 1 The grouping of the different operators in L f according to the parity of their quark bilinears is convenient because in the decay rates to be examined the contributions of C V,A,S,P f , belonging to L f terms with parity-even quark bilinears, do not interfere with the contributions of c v,a,s,p f , belonging to the terms with parity-odd quark bilinears. As we will concentrate on exploring the potential implications of NP encoded in L f for the transitions of light baryons and mesons at low hadronic energies, here we will not concern ourselves with how the parameters in L f evolve from their high-energy values. It is nevertheless worth mentioning that for effective flavor-changing operators involving light quarks and SM-gauge-singlet dark particles the running of their coefficients from high to low energies has been estimated to be negligible [23].
In what follows we address how the NP may enlarge the rates of |∆S| = 1 hyperon decays with missing energy compared to the SM expectations. Since L f influences kaon decays as well, we need to ensure that the applied ranges of C V,A,S,P f andc v,a,s,p f are compatible with the available relevant data. In numerical work, we will assume the phenomenological viewpoint that these free parameters can have any values consistent with the empirical restrictions and perturbativity, and so some of them may be taken to be vanishing or much smaller than the others. This will allow us to look for parameter ranges that would translate into the maximal hyperon rates permitted by the kaon constraints. Although this may entail substantial differences among the NP parameters that appear rather unnatural, we optimistically suppose that models could be devised to accommodate them. 2
To calculate the amplitudes for the hyperon decays, we need to know the baryonic matrix elements of the quark parts of the operators in eq. (2.1). We estimate the matrix elements with the aid of flavor-SU(3) chiral perturbation theory at leading order. Their derivation from the leading-order chiral Lagrangian is outlined in appendix A. We express 1 For f being a particle from a dark sector beyond the SM, the terms in Lf would constitute a subset of the independent operators detailed in [13], which include those containing dark particles of spin 0, 1, or 3/2.
2 For instance, a model possessing a heavy Z boson which has family-nonuniversal purely-vector interactions with SM quarks [24] might be responsible for the dγ η s terms in Lf . If the Z couplings to SM quarks are purely axial-vector instead [24][25][26], it might give rise to the dγ η γ 5 s terms. 3 We do not include Σ 0 → nff because its branching fraction is expected to be comparatively very suppressed due to the Σ 0 width being overwhelmingly dominated by the electromagnetic channel Σ 0 → Λγ [22].

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The constants D and F come from the lowest-order chiral Lagrangian.
the results pertaining to B → B ff as where V B B and A B B are constants whose values for the aforementioned B B pairs are collected in table 1, the us are Dirac spinors, Q = p B − p B , with p X denoting the momentum of X, and the other quantities are defined in appendix A. For η Ω is a Rarita-Schwinger spinor. Accordingly, in our approximation the amplitude for Ω − → Ξ − ff , to be given below, does not contain the couplings C V,A,S,P f . It is worth noting that the preceding baryonic matrix elements, and their mesonic counterparts to be discussed in section 4, fulfill the relations Y |dγ η s|X p η With eq. (3.1), we obtain the amplitude for B → B ff to be where the vs are Dirac spinors forf , (3.5)

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This leads to the differential decay rate The rate results from integrating the differential rate over 4m 2 In eq. (3.6) we observe that the C V,A,S,P f terms do not interfere with thec v,a,s,p f ones, which is also the case in the kaon decays to be examined later on.
For the Ω − decay we find .
with f K = 155.6(4) MeV [22] being the kaon decay constant, while for K → πff where f + and f 0 represent form factors which are functions of q 2 Kπ . In addition, assuming isospin symmetry, we have π 0 d (γ η , 1) We can then adopt f +,0 = f + (0) 1 + λ +,0 q 2 Kπ /m 2 π + with λ + = 0.0271(10) and λ 0 = 0.0142(23) from K L → π + µ − ν measurements [22] as well as f + (0) = 0.9681(23) from lattice computations [31]. For K − → π 0 π − ff and K L → π 0 π 0 ff , from the results in appendix A we obtain In the K − case, there is additionally a small contribution involving one of the parity-even quark transitions, π 0 π − |dγ η s|K − = 0, which arises from the anomaly Lagrangian [13], at next-to-leading order in the chiral expansion, and which we have therefore neglected. Since the existing empirical limits on K → ππ ff are not very stringent, we also ignore form-factor effects in calculating their rates. It follows that the amplitudes for K L → ff and K S → ff induced by L f are leading to the decay rates

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We put the resulting differential rates of K − → π − ff and K L,S → π 0 ff in appendix B, which also shows that these modes, in contrast to K L,S → ff , can probe C V,A,S,P f , but notc v,a,s,p f . For K − → π 0 π − ff and K L → π 0 π 0 ff , we get (4.8) Their differential rates are also relegated to appendix B. We notice from eqs. (3.5) and (3.6) that, unlike these kaon modes, B → B ff are sensitive to both C V,A,S,P f andc v,a,s,p f . It is therefore advantageous to measure B → B / E, as the acquired data could supply information on s → d / E which is complementary to that from the kaon sector.

SM predictions and empirical information
Within the SM, our hyperon decays of interest are induced by effective short-distance sdν lνl interactions, with l = e, µ, τ , described by [1] where α e 1/128 and G F are the usual fine-structure and Fermi constants, s 2 w ≡ sin 2 θ w = 0.231 with θ w being the Weinberg angle, V qq are Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, X t = 1.481(9) comes from t-quark loops [32], and X e c = X µ c 1.0 × 10 −3 and X τ c 7 × 10 −4 are c-quark contributions [1]. To evaluate B(B → B νν) sm = l B(B → B ν lνl ) sm we can apply eq. (3.6) with 8πs 2 w and C S,P ν l =c s,p ν l = 0. Similarly, we can determine B(Ω − → Ξ − νν) sm using eq. (3.9). 4 Thus, with the central values of the input parameters, we arrive at the entries in the second row of table 2. 5 The CKM factors and X t,c contribute an uncertainty of almost 10% to the SM predictions, the estimation of the baryonic matrix elements has an uncertainty of ∼ 20%, and so the total uncertainty of the branching-fraction predictions is about 50%. 4 These decays also receive long-distance contributions mediated by the Z boson, such as B → B Z * → B νν, but their size is estimated to be small compared to the short-distance SM contribution. 5 Our numbers are roughly comparable to the ones given in [9], but therein the flavor-SU(3) properties of the baryon interactions were not taken into account and different momentum-dependences of the baryonic matrix elements were used.
At present there are no data available on these hyperon transitions, but this situation may change in the near future if BESIII performs a quest for them. In the last row of table 2 we quote its estimated sensitivities [90% confidence level (CL)] for their branching fractions [7]. Clearly, it is unlikely that the SM predictions will be tested anytime soon. Nevertheless, as we demonstrate in the next section, it is possible for NP to amplify the branching fractions to levels potentially reachable by BESIII.

Beyond SM
As mentioned in the preceding subsection, the current K → πνν data do not leave ample room for NP to affect C V,A,S,P f greatly. More specifically, our numerical scans reveal that the allowed values of these couplings alone cannot produce B B → B ff above 10 −11 , and so this scenario would be out of BESIII reach according to table 2.

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Therefore, hereafter we concentrate on the possibility that NP can generate sizable effects only viac v,a,s,p f . For simplicity, we assume that f is a nonstandard fermion which is sufficiently light compared to the mass difference between the initial and final baryons, so that we can approximately set m f to zero in numerical work. It follows that only the constraints on K → ff and K → ππ ff need to be addressed when dealing with the hyperon decays. Moreover, since the NP contributions do not interfere with s → dνν, the tiny SM contributions to these processes with missing energy can be ignored.
Integrating the differential rates of the baryon decays, for m f = 0 we arrive at the branching fractions

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We can now look at a couple of representative instances with different choices of nonvanishing couplings, which we take to be all real to ignore any new source of CP violation. If onlyc s,p f are nonzero, we find that the K → / E restrictions in eq. (5.2) are more stringent than the K → ππ / E ones in eq. (5.3) and, with the aid of eq. (5.9), lead to c s  do not affect these decays in the m f = 0 limit, but the K → ππ / E bounds still matter, the K L → π 0 π 0 / E one being the stronger and yielding Rec v  = 0, bigger branching fractions than those in eq. (5.12) could be achieved with purely imaginaryc v,a f , as they would escape the K L → π 0 π 0 / E restraint and be subject only to the weaker K − → π 0 π − / E one, implying the mild limit Imc v f 2 + Imc a f 2 < 6.4 × 10 −11 GeV −4 . This serves to indicate further the benefit of measuring these hyperon decays, which may test some of the NP couplings more stringently than the kaon decays.

Conclusions
We have explored the possibility that new physics contributes to the strangeness-changing transition s → d / E, with missing energy in the final state. Depending on the sizes of the NP couplings involved and the masses of the emitted invisible particles, various changes could occur to the SM predictions for rare kaon and hyperon |∆S| = 1 decays with missing energy. We have learned that the current data on K → πνν do not allow NP to influence the hyperon decays considerably if the underlying operators have mostly parity-even quark portions. On the other hand, if the NP operators have predominantly parity-odd, especially axial-vector, quark parts, the restraints implied by the K → invisible and K → ππ νν data are comparatively weaker. We have demonstrated that NP with the latter kind of interactions could cause the hyperon rates to be substantially amplified with respect to their SM expectations and have large values potentially testable in the ongoing BESIII experiment. This well illustrates that these rare hyperon decays and their kaon counterparts are complementary to each other as probes of possible NP in s → d / E.
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