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 $s\to d\nu\bar\nu$ 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 $K\to\pi\nu\bar\nu$ do not permit sizable NP impact on the hyperon decays via underlying operators with mainly parity-even quark parts. In contrast, NP operators with primarily parity-odd quark parts are much less restricted by the existing bounds on $K\to\rm invisible$ and $K\to\pi\pi\nu\bar\nu$ and consequently could give rise to substantially amplifying effects on the hyperon modes. Their NP-enhanced branching fractions could reach levels potentially observable in the ongoing BESIII experiment.


I. INTRODUCTION
The strangeness-changing quark transition s → d / E, with missing energy / E in the final state, is of great interest because it serves as an environment in which to test the standard model (SM) and therefore also to look for signals of possible new physics (NP) beyond it. Within the SM, this process is predominantly due to s → dνν arising from Z-penguin and box diagrams with up-type quarks and the W boson in the loops [1], the neutrino pair (νν) being undetected. In the presence of NP, this SM contribution could be altered [2][3][4][5][6][7][8][9][10][11], and there might be invisible nonstandard states which are light enough and have sd couplings to give rise to new s → d / E channels [12][13][14][15][16].
Currently there are ongoing efforts to observe s → dνν via the kaon decays K + → π + νν and K L → π 0 νν by the NA62 [17] and KOTO [18] Collaborations, respectively. These measurements might then probe for hints of s → d / E beyond the SM as well. Additional kaon modes worth pursuing are K L → / E and K → ππ ′ / E, as only moderate bounds on K → ππ ′ νν from direct searches are available [19]. Hence improved data on these extra modes would also be desirable. In the baryon sector, their counterparts are the strangeness-changing (|∆S| = 1) decays of light hyperons into another baryon plus missing energy. Interestingly, measuring such processes in the BESIII experiment has recently been proposed and may be realized in the near future [8].
Here we explore these rare hyperon decays to see how much they may be affected by different NP possibilities, taking into account restrictions from the kaon sector. Initial studies on the hyperon modes with νν being the invisible particles have been carried out in refs. [8,11]. In this paper, we investigate a more general scenario in which the invisible pair could be nonstandard fermions. It turns out that in this case the hyperon branching fractions may be significantly raised with respect to their SM values and even reach potentially discoverable levels at BESIII. Our results illustrate that these rare processes can offer useful information on s → d / E which is complementary to that gained from their kaon counterparts.
The organization of the paper is as follows. In section II 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 III 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 IV we provide the formulas for the rare kaon decays of concern. In section V 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 enhance the hyperon rates to values that may be observable by BESIII. In section VI, we give our conclusions. We collect extra formulas and information in a couple of appendixes.

II. INTERACTIONS
Beyond the SM, there could be new ingredients which induce modifications to the SM transition s → dνν 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 sd ff interactions described by the low-energy effective Lagrangian where in our model-independent approach C V,A,S,P f andc v,a,s,p f are free parameters which are in general 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 As will be seen later, in the hyperon decay rates C V,A,S,P f , which belong to terms with parity-even quark parts, do not interfere withc v,a,s,p f , which belong to terms with parity-odd quark parts.
In what follows we address how new interactions entering L f 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 will also need to ensure that the applied values of C V,A,S,P f andc v,a,s,p f are compatible with the available relevant data.
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. (1). We estimate the matrix elements with 1 For f being a particle from a dark sector beyond the SM, the terms in L f would constitute a subset of the independent operators detailed in [12], which include those containing dark particles of spin 0, 1, or 3/2. 2 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 [19]. 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 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 I, the us are Dirac spinors, Q = p B ′ − p B , with p X denoting the four-momentum of X, and the other quantities are defined in appendix A. For whereq = p Ω − − p Ξ − and u 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 IV, fulfill the relations Y |dγ η s|X p η X − p η Y = (m s −m) Y |ds|X and Y |dγ η γ 5 s|X p η Y − p η X = (m s +m) Y |dγ 5 s|X based on the free Dirac equation. With eq. (2), we obtain the amplitude for B → B ′ ff to be where the vs are Dirac spinors forf , This leads to the differential decay rate where The rate results from integrating the differential rate over 4m 2 In eq. (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 where In eq. (2) there are form-factor effects not yet taken into account. To incorporate them, in numerical work we modify , respectively, with M V = 0.97 GeV and M A = 1.25 GeV, following the commonly used parametrization in experimental analyses of hyperon semileptonic decays [20][21][22][23] and assuming isospin symmetry. Analogously, sinceq 2 in Ω − → Ξ − ff has a significantly wider range than Q 2 in B → B ′ ff , in the Ω − decay rate we implement the change C → C/ 1 −q 2 /M 2 A 2 . These modifications turn out to translate into increases of the rates by up to ∼ 16 percent.
It follows that the amplitudes for K L → ff and K S → ff induced by L f are leading to the decay rates Thus, K L,S → ff are not sensitive to C V,A,S,P f andc v f . The amplitude for K → πff has the form We put the resulting differential rates of K − → π − ff and K L,S → π 0 ff in appendix B, which also shows that they, 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 Their differential rates are also relegated to appendix B.
We notice from eqs. (5) and (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.
w and C S,P ν l =c s,p ν l = 0. Similarly, we can determine B(Ω − → Ξ − νν) sm using eq. (9). 3 Thus, with the central values of the input parameters, we arrive at the entries in the second row of table II. 4 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 predictions is about 50%.
Decay mode Expected BESIII sensitivity [8] 3 × 10 −7 4 × 10 −7 8 × 10 −7 9 × 10 −7 -2.6 × 10 −5 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 II we quote its estimated sensitivities [90% confidence level (CL)] for their branching fractions [8]. 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.
Turning to the kaon sector, we see that eqs. (15) and (16) imply Γ sm K L,S →νν = 0 due to the neutrinos' masslessness in the SM. If it is supplemented with nonzero neutrino masses, their highest one from the direct limit m exp ντ < 18.2 MeV [19] translates into the maximal values B(K L → νν) sm ≃ 1.0 × 10 −10 and B(K S → νν) sm ≃ 1.6 × 10 −14 . Therefore, observations of B(K L,S → / E) ≫ 10 −10 would constitute evidence of NP. Although to date there are still no measurements on K L,S → / E, from the available data [19] on the visible decay modes of K L,S one can extract indirect upper bounds on their invisible branching fractions [26]: both at 95% CL. Hence there is still plenty of room for NP to influence these decays, specifically via the couplingsc a,s,p f , as eq. (16) indicates.

B. 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 II.
Therefore, hereafter we concentrate on the possibility that NP can generate sizable effects only viac v,a,s,p leading to and for the four-body decays Evidently, all the interference terms with different couplings have vanished as m f → 0.
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. (20) are more stringent than the K → ππ ′ / E ones in eq.
and smaller numbers for B Ξ 0,− → Σ 0,− ff . In view of table II, these results far exceed the SM values, but are still roughly two orders of magnitude beyond the expected BESIII reach.
If onlyc v,a f are nonzero, then according to eq. (25) the K → / E constraints no longer apply asc v,a f 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 This now translates into B Λ → nff < 6.6 × 10 −6 , B Σ + → pff < 1.7 × 10 −6 , most of which have upper values exceeding the corresponding estimated BESIII sensitivity levels quoted in table II. This suggests that BESIII might discover NP hints in these processes or, if not, come up with improved restrictions onc v,a f . If we let Imc v,a f = 0, bigger branching fractions than those in eq. (29) 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 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.

VI. 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. d γ η γ 5 s ⇔ For K − → π − ff and K L → π 0 ff , from eqs. (1) and (12), we find the S and P terms in eq. (17) to be For K S → π 0 ff , the S and P formulas are equal to S K L π 0 f and P K L π 0 f but with ReC f and −iImC f interchanged. The differential rate of the K L decay is then where βς = 1 − 4m 2 π ς ,ς = (p 0 + p − ) 2 = (p 1 + p 2 ) 2 ,λ = m 4 K − 2 ŝ +ς m 2 K + ŝ −ς 2 , (B5)