Flavor-changing hyperon decays with light invisible bosons

We consider the strangeness-changing decays of hyperons into another baryon and missing energy which is carried away by a pair of invisible spinless bosons. Pursuing a model-independent approach and taking into account constraints from the kaon sector, we find that these hyperon modes can have large rates, especially if the bosons have parity-odd effective couplings to light quarks. In that case the rates could attain values which are well within the expected reach of the currently running BESIII experiment. Since the branching fractions of hyperon decays with missing energy are tiny in the standard model, observing them at ongoing or upcoming facilities would likely be indicative of new physics. In addition, we show that the kaon restrictions may be weakened or absent if the bosons have nonnegligible mass. Specifically, if the bosons interact with the quarks via an axial-vector current, in the mass region from about 115 MeV to 175 MeV the constraints from rare kaon decays no longer apply and a couple of the hyperon modes become the only direct probes of such interactions. We also make comparisons with the previously explored scenario in which the invisible particles are fermions.


I. INTRODUCTION
In the standard model (SM) the flavor-changing neutral current (FCNC) decay of a strange meson or baryon with missing energy ( / E) in the final state proceeds mainly from the quark transition s → dνν generated by loop diagrams [1], with / E being carried away by the unobserved neutrino pair (νν). Beyond the SM there could be extra ingredients which cause changes to the SM process and/or give rise to additional channels with one or more invisible nonstandard particles contributing to / E. As such decay modes of strange hadrons have very suppressed rates in the SM, the modifications due to new physics (NP) may translate into effects big enough to be discoverable. These rare modes can then be expected to serve as sensitive probes of NP.
Experimentally, there has been much activity to study such processes in the meson sector. Particularly, a while ago the E949 Collaboration [2] reported seeing the kaon mode K + → π + νν, and currently there are running programs to measure it more precisely and observe the neutral channel K L → π 0 νν by the NA62 [3] and KOTO [4] Collaborations, respectively. As for other FCNC kaon decays with missing energy, the only data available are upper limits on the branching fractions of two of the K → ππ ′ νν modes [5][6][7]. Indirectly, the corresponding limits on K → / E can be inferred from the existing data on the neutral kaon visible decays [8].
In the baryon sector, the analogous transitions are the strangeness-changing (|∆S| = 1) hyperon decays into another baryon plus missing energy. However, there has never been any search for them as far as we know. Interestingly, there is a recent proposal to measure them in the BESIII experiment [9]. Hence new data on these rare hyperon modes may be forthcoming.
It is therefore timely to investigate them theoretically to see whether NP might impact them significantly in light of the constraints from the kaon sector. Initial works on the hyperon modes being affected by NP via s → dνν interactions with the same chiral structure as in the SM have been done in Refs. [9,10]. Subsequently, Ref. [11] explores a more general scenario in which the underlying NP operators involve other Lorentz structures and the pair of invisible particles could be nonstandard spin-1/2 fermions. In this more general possibility, it turns out that the hyperon decays could have rates which are greatly increased relative to the SM expectations, even up to levels potentially detectible by BESIII in the near future [11].
In the present paper, we extend the analysis of Ref. [11] and entertain the possibility that the invisible pair consists of a spin-0 boson, φ, and its antiparticle,φ. We assume that φ does not carry any SM gauge charge but is either charged under some symmetry of a dark sector beyond the SM or odd under a Z 2 symmetry which does not influence SM particles. As a consequence, φ andφ always appear together in their interactions with SM particles. Scenarios in which such nonstandard bosons have ds couplings that contribute to kaon processes have been addressed before in the literature under specific NP contexts [12][13][14][15]. Here we pursue a model-independent approach, starting from leading-order operators containing dsφφ couplings in the most general effective Lagrangian that respects SM gauge invariance.
We will show that the rates of |∆S| = 1 hyperon decays with missing energy can be sizable in the presence of the new interactions, similarly to what happens in the invisible fermion case studied in Ref. [11]. Such a possibility is partly attributable to the fact that these hyperon transitions and their kaon counterparts do not probe the same portions of the underlying dsφφ operators. In particular, K → π / E is sensitive exclusively to parity-even operators and K → / E to parity-odd ones, while K → ππ ′ / E and the spin-1/2 hyperon modes can probe both. Another factor is the kaon data situation at the moment, which has a bearing on limiting mainly parityeven dsφφ operators, due to the existing empirical information on K → πνν decays [3][4][5]. By contrast, the restraints from K → / E and K → ππ ′ / E are relatively far weaker and still permit parity-odd dsφφ operators to bring about considerable amplifying effects on the hyperon rates.
Besides the similarities, there are differences between the bosonic and fermionic cases. Due to the invisible particle having spin 1/2, the latter scenario involves more operators with their associated coupling parameters, leading to more phenomenological possibilities, especially if the invisible particle's mass is not negligible. Our examples will illustrate some of the differences and similarities.
The structure of the remainder of the paper is as follows. In the next section we write down the NP operators of interest, treating them in a model-independent manner. In Sect. III we first deal with the FCNC hyperon decays with missing energy, deriving their differential rates, and subsequently provide the formulas for the corresponding kaon decays of concern. In Sect. IV we present our numerical results. We evaluate the hyperon modes, taking into account restrictions from the kaon sector, and demonstrate that the hyperon rates can be substantial and within the proposed BESIII reach. We also make comparisons with the fermionic case examined in Ref. [11]. In Sect. V we give our conclusions.

II. INTERACTIONS
At lowest order, the effective operators responsible for the SM-gauge-invariant interactions of φ as specified above are of mass dimension six [14]. The Lagrangian L np containing them is given by where Λ np is a heavy mass scale representing the underlying NP interactions, C (′,′′) jk in our modelindependent approach are dimensionless free parameters which are generally complex, q j and d j denote a left-handed quark doublet and right-handed down-type quark singlet, respectively, H stands for the Higgs doublet, and summation over family indices j, k = 1, 2, 3 is implicit. The Hermiticity of L np implies C To deal with the contributions of L np to hyperon and kaon decays, it is convenient to rewrite the dsφφ operators in it explicitly separating their parity-even and -odd parts. Thus, choosing to work in the mass basis of the down-type quarks, we can express where with v ≃ 246 GeV being the Higgs vacuum expectation value. 1 As we will see below, in the decay rates the contributions of c V,A,S,P φ do not interfere. In numerical calculations, we will adopt the phenomenological viewpoint that these free parameters can have any values compatible with the empirical constraints and perturbativity, implying that some of them may be taken to be vanishing or much less than the others.
To derive the amplitudes for these decays, we need the baryonic matrix elements of the quark portions of the operators in Eq. (2). They can be estimated with the aid of flavor-SU(3) chiral perturbation theory at leading order [11,17]. For B → B ′ φφ, from Ref. [11] we have where For Ω − → Ξ − φφ, the hadronic matrix elements that do not vanish at lowest order in chiral perturbation theory are [11] whereq = p Ω − − p Ξ − and u η Ω is a Rarita-Schwinger spinor. The constants D, F , and C above come from the lowest-order chiral Lagrangian and can be fixed from baryon decay data. In numerical analysis, we will employ the same (central) values of these and other input parameters as those given in Ref. [11].
With Eqs. (4) and (5), we derive the amplitudes for B → B ′ φφ and Ω − → Ξ − φφ to be where p andp denote the momenta of φ andφ, respectively. Subsequently, we arrive at the differential decay rates where To get the decay rates, we integrate the differential rates in Eq.
respectively. Following Ref. [11], in our numerical treatment of these hyperon decay rates we will incorporate form-factor effects not yet taken into account in Eqs. (4) and (5). Specifically, we will make the changes (4) GeV and M A = 1.25 (15) GeV, which are in line with the commonly used parametrization in experimental studies of hyperon semileptonic decays [18][19][20][21][22]. Moreover, for the Ω − decay we apply the modification
From Eq. (9), we obtain the amplitudes for K L,S → φφ induced by L φ to be with the approximate relations √ 2 K L,S = K 0 ± K 0 . The corresponding decay rates are For the three-body modes, we derive These lead to the differential decay rates and the rates result from integrating them over 4m 2 φ ≤ŝ ≤ (m K − ,K 0 − m π − ,π 0 ) 2 , respectively. For the four-body kaon decays, the amplitudes are from which we arrive at the double-differential decay rates 2 where and m π in the K − (K L ) formula is the isospin-average (neutral) pion mass. Their integration ranges are 4m 2 φ ≤ŝ ≤ (m K − ,K 0 − 2m π ) 2 and 4m 2 π ≤ς ≤ m K − ,K 0 −ŝ 1/2 2 , respectively. Evidently, K L,S → φφ are sensitive to c P φ alone, while K → ππ ′ φφ and Ω − → Ξ − φφ can probe only the two parity-odd coefficients, c A φ and c P φ , in our approximation of the hadronic matrix elements. On the other hand, K → πφφ are sensitive to both of the parity-even coefficients, c V φ and c S φ , but not to c A,P φ . In contrast, the spin-1/2 hyperon modes B → B ′ φφ can probe all the coefficients, according to Eq. (7). It follows that quests for B → B ′ / E as well as Ω − → Ξ − / E are beneficial because the acquired data could reveal information on dsφφ interactions which complements that gained from kaon measurements. Furthermore, a couple of the hyperon modes, namely Σ + → pφφ and Ω − → Ξ − φφ, can cover broader m φ ranges than K → ππ ′ φφ and therefore provide the only window into c A φ for certain m φ values, as we discuss later on.

IV. NUMERICAL RESULTS
Although there is still no empirical information on the rare hyperon decays with missing energy, first data on them may become available from BESIII not long from now. Their branching fractions in the SM have been estimated to be less than 10 −11 in Ref. [11] and hence are several orders of magnitude below the estimated sensitivity levels of BESIII [9], quoted in Table I.
To explore how much NP can enhance the hyperon rates, we consider first the possibility that the φ mass can be ignored relative to the pion mass or vanishes, m φ = 0. In this case, integrating the differential rates of the baryon decays, we arrive at the branching fractions All these results have incorporated the form factors mentioned in section III A.
As mentioned above, the current K → πνν data do not leave ample room for NP to affect c V,S φ significantly. More specifically, if these coefficients are the only ones being nonzero, comparing their contributions, for m φ = 0, with the corresponding bounds in Eq. (21), we find that the resulting hyperon branching fractions in Eqs. (24) and (25) are below 10 −11 . This scenario would therefore be out of BESIII reach according to Table I.
Therefore, hereafter we concentrate on the possibility that NP can generate considerable effects solely through c A,P φ . With this assumption and m φ = 0, for the two-body kaon decays we get implying and for the four-body decays To illustrate the potential impact of sizable c A,P φ on the hyperon modes, we can look at a couple of examples with different choices of the nonvanishing coupling, which we take to be real to avoid any new CP -violation source.
Thus, if only c P φ = 0, the K → / E restraints in Eq. (22) turn out to be more stringent than the K → ππ ′ / E ones in Eq. (23) and, with the aid of Eq. (29), lead to c P The upper ends of these ranges greatly exceed the corresponding SM values quoted in Table I but are still below the expected BESIII sensitivity levels, mostly by at least an order of magnitude.
If instead c A φ is nonvanishing and real, whereas c V,P,S φ = 0, then the K → / E bounds do not matter any more but the K → ππ ′ / E ones in Eq. (23) still do, the K L → π 0 π 0 / E one being the stronger and yielding Re c A φ 2 < 3.8 × 10 −13 GeV −4 . This now translates into B Λ → nφφ < 6.6 × 10 −6 , B Σ + → pφφ < 1.7 × 10 −6 , Their upper values are larger than the corresponding BESIII sensitivity levels quoted in Table I. Allowing Im c A φ = 0, one could get higher branching fractions with c A φ being complex, as its imaginary part would escape the K L → π 0 π 0 / E restriction and be subject only to the milder K − → π 0 π − / E one. The numbers in Eq. (32) are the same as their counterparts in the case treated in Ref. [11] where the invisible particles are spin-1/2 fermions, highlighting the similarities of the two scenarios when the masses of the invisible particles can be neglected.
We consider next the possibility that m φ is nonnegligible and let it vary over the allowed kinematical range of each of the hyperon decays. As in the second example, we suppose that c A φ is nonzero and real, while c V,S,P φ = 0, and so the K L → π 0 π 0 / E restriction again needs to be taken into account. Now, as m φ grows this restriction increasingly weakens until it no longer applies at m φ ≃ 114 MeV when K L → π 0 π 0 / E becomes kinematically closed. Accordingly, for m φ > 0 the hyperon rates can be substantially bigger than their m φ = 0 values, and for m φ > 114 MeV the kaon decay constraint is no more. However, as the restraint on Re c A φ continues to loosen, its rising upper bound may no longer be compatible with the perturbativity of the φ interactions. To ensure that they remain perturbative, inspired by the form of the standard effective weak interaction, we impose Re c A φ < G F / √ 2 ≃ 8.2/TeV 2 , where G F is the Fermi constant. Since this corresponds to the requisite B K L → π 0 π 0 φφ < 8 × 10 −7 at m φ ≃ 76 MeV, we then set Re c A φ = G F / √ 2 for m φ > 76 MeV. In Fig. 1 we depict the resulting maximal branching fractions of the hyperon modes versus m φ varying from 0 to 175 MeV, which is approximately its highest value in Ω − → Ξ − φφ. Evidently, over the majority of their respective m φ ranges, four of the modes (the ones with BB ′ = Λn, Σ + p, Ξ 0 Λ and the Ω − one) have maximal branching fractions that are significantly greater than their m φ = 0 values, implying that their m φ > 0 predictions are easier to test experimentally.
In the scenario examined in Ref. [11], where the invisible particles are spin-1/2 fermions, ff , the situation may be different if their mass, m f , is not negligible. In that case, as Table II   doscalar) quark bilinears. 3 As a consequence, forc a,s,p f , the K → / E restrictions become more important than the K → ππ ′ / E ones over most of the m f > 0 region, with the implication that the resulting hyperon rates tend to be much smaller than their m f = 0 values. This is illustrated in Fig. 2, for which Rec a f is taken to be the only nonzero coupling. In this plot, we see that the K L → / E constraint is stricter than the K L → π 0 π 0 / E one if m f > 5 MeV. It also reveals striking differences from Fig. 1   from the data on the observed ones [5]. Therefore, for this m f region we additionally require Rec v f to yield B Ω − → Ξ − ff < 1%. We display the resulting upper limits on the hyperon branching fractions in Fig. 3 which shows stark differences from Fig. 2 for m f > 5 MeV. On the other hand, Fig. 3 resembles Fig. 1 in important ways, especially their depictions of the amplified branching fractions over much of the m φ,f > 0 ranges compared to their values at m φ,f = 0. These two figures also have dissimilarities, one of which is that the curves in Fig. 3 tend to be higher at larger masses.

V. CONCLUSIONS
We have explored the strangeness-changing decays of hyperons into another baryon plus missing energy carried away by a pair of invisible spin-0 bosons, φφ, which are SM gauge singlet. Adopting a model-independent approach, we start from an effective Lagrangian respecting SM gauge invariance and containing dimension-six dsφφ operators. Thus, they contribute also to FCNC kaon decays with missing energy and are subject to restrictions from their data. Although the existing K → π / E constraints do not permit NP to generate sizable effects via parity-even dsφφ operators, we demonstrate that the restrictions from K → / E and K → ππ ′ / E are much weaker on the parity-odd operators. Specifically, the one with the axial-vector quark bilinear needs to fulfill only the mild K → ππ ′ / E restraints and could give rise to considerable hyperon rates which may be within the reach of BESIII or future measurements. Moreover, if NP enters predominantly through the coefficient c A φ of this operator and m φ > 0, the kaon decay constraints become weakened and as a result the hyperon rates may be greatly enhanced with respect to their m φ = 0 predictions. Interestingly, these kaon constraints on c A φ are absent if m φ > 114 MeV but two of the hyperon modes, especially Ω − → Ξ − φφ, can still serve as direct probes of c A φ until m φ reaches 175 MeV. We also perform comparisons with the scenario in which the invisible particles are spin-1/2 fermions. We discuss special instances in which the bosonic and fermionic cases may be discriminated experimentally, particularly if their masses are nonvanishing. The results of our analysis illustrate well that these rare hyperon decays and their kaon counterparts are complementary to each other in providing access to possible NP in s → d / E transitions.