$\Delta S=2$ nonleptonic hyperon decays as probes of new physics

Hyperon nonleptonic decays that change strangeness by two units, such as $\Xi\to N\pi$ and $\Omega^-\to nK^-,\Lambda\pi^-,\Sigma^{(*)}\pi$, are highly suppressed in the standard model. Only a few of them have been searched for to date, leading to experimental upper bounds which are many orders of magnitude above the expectations of the standard model. This leaves ample opportunity to look for indications of new physics in these processes. At the same time, most, but not all, $\Delta S=2$ interactions beyond the standard model are severely constrained by kaon-mixing data. We present two scenarios where new physics satisfying the kaon-mixing constraints can enhance the hyperon decay rates to levels that can be probed in future quests by BESIII and LHCb and at the proposed Super Tau-Charm Factory. Both scenarios require significant fine-tuning.


I. INTRODUCTION
The nonleptonic decays of light hyperons that modify the strangeness quantum number by two units have very small rates in the standard model (SM).Hence such ∆S = 2 processes could serve as an environment in which to search for hints of new physics beyond the SM.This was first investigated in Ref. [1], focusing on the Ξ , which has spin 1/2, turning into a nucleon N and a pion π.In the spin-3/2 sector, the Ω − hyperon can also be used to test for new physics in ∆S = 2 interactions, via the decays Ω − → nK − , Λπ − , Σ π, Σ * π.
The latest quests for Ξ → N π were conducted decades ago [2,3] and came up empty, implying the branching-fraction bounds B(Ξ 0 → pπ − ) exp < 8 × 10 −6 and B(Ξ − → nπ − ) exp < 1.9 × 10 −5 [4] both at 90% confidence level (CL).In the Ω − case, only Ω − → Λπ − has been searched for [3], also with a null outcome, which translated into B(Ω − → Λπ − ) exp < 2.9 × 10 −6 at 90% CL [4].As these results are far above the SM expectations, by up to ten orders of magnitude, the window to discover new physics in such ∆S = 2 decays is wide open.Efforts to pursue this may be made in ongoing experiments, such as LHCb and BESIII.The former, short of discovery, could better the preceding limits by 3 to 4 orders of magnitude after upcoming upgrades [5].At e + e − facilities, BESIII [6] might be able to improve on the Ξ bounds, and farther in the future the Super Tau-Charm Factory [7] would expectedly have much enhanced sensitivity to both the Ξ and Ω − channels [8].All this has prompted us to revisit these rare processes in hopes to learn new information about them.
There are relations among several of them, and we identify the independent ones here.For the hyperons in the octet of ground-state spin-1/2 baryons, the ∆S = 2 nonleptonic decays into two-body final states that are kinematically allowed are Ξ 0 → pπ − , nπ 0 , and Ξ − → nπ − .Within or beyond the SM, the leading operators contributing to these flavor-changing neutral-current processes are of dimension six and consist of four light-quark fields, which can only be the downtype ones.Thus, the operators entail the conversion of two s-quarks into two d-quarks, altering isospin by ∆I = 1.It follows that, in light of isospin symmetry of the strong interactions, the invariant amplitudes for Ξ → N π satisfy As a consequence, it suffices to examine the amplitudes for just two of them, which we choose to be Ξ 0 → pπ − and Ξ − → nπ − .
The structure of the paper is as follows.In Sec.II we address the ∆S = 2 nonleptonic hyperon decays (NLHD) within the SM.Specifically, we start by updating the short-distance predictions for Ξ → N π and subsequently treat their Ω − counterparts.Moreover, we explicitly look at longdistance effects brought about by ∆S = 1 operators acting twice, which turn out to be numerically important.Since these processes have relatively low rates already, we do not consider modes with three or more particles in the final states, which have less phase-space.Beyond the SM, in Secs.III A and III B we explore how a Z ′ boson and leptoquarks, respectively, may give rise to substantially amplified contributions to the ∆S = 2 NLHD.We present our conclusions in Sec.IV.In three appendices we summarize the numerical values we use for input parameters, collect the rate formulas for the Ω − modes, and provide further details of the Z ′ model.

A. Short-distance contributions
In the SM the effective Hamiltonian for ∆S = 2 transitions among light quarks is approximately given by [9] which involves a QCD-correction factor η cc , the Fermi constant G F , the charm-quark mass m c , the elements V mn of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and with P L = (1 − γ 5 )/2, the subscripts k, l, n, o = 1, 2, 3 being implicitly summed over, t kl,no = 0 except for t 22,33 = 1, and the light-quark fields ψ 1,2,3 = u, d, s.In Eq. ( 3) we have retained only the charm-quark portion, as it dominates the SM short-distance (SD) predictions for the hyperon decays of interest and the neutral-kaon mass difference ∆M K = Re⟨K 0 |H ∆S=2 | K0 ⟩/m K 0 , the correction from the top and charm-top contributions being merely at the percent level [9,10].
To deal with the hyperon amplitudes generated by H sm ∆S=2 requires the hadronized form of Q LL .It transforms like (27 L , 1 R ) under the chiral-symmetry group SU(3) L × SU(3) R and has a leadingorder hadronic realization [1,11] expressible as where Λ χ is the scale of chiral-symmetry breaking, f π denotes the pion decay constant, β27 and δ27 are parameters to be fixed below, B and ξ stand for 3×3 matrices incorporating the fields of the lowest-mass octet-baryons and -mesons, respectively, r, v, w, x, z = 1, 2, 3 are also summed over, and (T rvw ) α is a Rarita-Schwinger field [12] for the spin-3/2 decuplet baryons and has completely symmetric SU(3) indices (r, v, w), the components being explicitly listed in Ref. [11].
The amplitude for a spin-1/2 baryon, B, converting into another one, B ′ , and a pion can be put in the general form iM B→B ′ π = ūB ′ (A BB ′ − γ 5 B BB ′ )u B comprising, in succession, parity-odd S-wave and parity-even P-wave portions [4].For the former in the ∆S = 2 case, H sm ∆S=2 in Eq. ( 3) with Q LL changed to O LL brings about the diagram depicted in Fig. 1 (a), leading to [1] A (SM,SD) where The corresponding B pieces are calculated from pole diagrams, displayed in Fig. 1 (b), which depend on c sm and also have a vertex furnished by the leading-order strong-interaction chiral Lagrangian [16,17] where D, F , H, and C are constants and B (SM,SD) where m N and m Ξ are isospin-averaged nucleon and Ξ −,0 masses, respectively.
The mode Ω − → Bϕ, with ϕ being a pseudoscalar meson, is made up of P-wave and D-wave transitions.In the SM, the SD contribution to the former proceeds from the pole diagrams exhibited in Fig. 1 (c) which include not only a weak coupling produced by H sm ∆S=2 but also a strong vertex from the C term of Eq. ( 7).The D-wave piece arises from a higher order in the chiral expansion and hence will be neglected.Writing the amplitude accordingly as iM Ω − →Bϕ = C Bϕ ūB u α Ω pα , with p  3).Here and in Fig. 2, a dashed line represents a pseudoscalar meson, a single (double) solid-line a spin-1/2 (spin-3/2) baryon, and a thick dot a strong vertex from L s in Eq. (7).
being the four-momentum of ϕ, we then have where As for Ω − → Σ * 0 π − , it is described by S-, P-, D-, and F-wave amplitudes.The first two of them can be expressed as iM , and in the SM the SD ones are determined from the leading-order diagrams in Fig. 1 (d,e), respectively.Thus, we find Ã(SM,SD) The D-and F-wave terms occur at higher chiral orders and will therefore be ignored.
Nevertheless, there is one exception, namely that the S-wave amplitude for Σ + → nπ + receives no (8 L , 1 R ) contribution in chiral perturbation theory up to second order in external momentum or meson mass [17,21] . Equating these As, assuming that higher chiral orders can be neglected, and using 0.64 ≤ ĉ1 + ĉ2 ≤ 0.72 computed in Ref. [9] at leading order (for the renormalization scale of 1 GeV and QCD scales of 215-435 MeV) and the f π , G F , and V * ud V us values collected in Appendix A, we then extract β27 = 0.076 (15) .
As for δ27 , at the moment it cannot be estimated unambiguously from experiment because its role in the observed Ω − transitions is minor compared to those of the (8 L , 1 R ) parameters.Since, like β27 , it belongs to (27 L , 1 R ) interactions, to illustrate how δ27 may influence the Ω − channels of interest, we will set δ27 = β27 or − β27 .
, where p ′ (E ′ ) is the three-momentum (energy) of B ′ in the rest frame of B. We can employ this to evaluate the contributions of Eqs. ( 6) and ( 8) to Ξ → N π, with the central values of β27 above and of η cc , m c , |V cd V cs |, D, and F quoted in Appendix A, leading to the branching fractions For the Ω − transitions, from the aforementioned amplitudes it is straightforward to obtain the rates written in Eqs.(B3)-(B4).With Eqs. ( 9)- (10) and the central values of the input parameters, including C and H from Appendix A, we then find where the two entries for Ω − → nK − correspond to δ27 = (1, −1) β27 , respectively.

B. Long-distance contributions
These Ξ and Ω − modes are also affected by the pole diagrams depicted in Fig. 2, with two couplings from the lowest-order ∆S = 1 chiral Lagrangian [17,22] which transforms as (8 L , 1 R ) under SU(3) L × SU(3) R and contains parameters h D,F,C and a 3×3 matrix κ with elements κkl = δ 2k δ 3l .The diagrams for the Bs and Cs, in Fig. 2 (b,c,e), again include a strong vertex from Eq. ( 7) as well.Accordingly, for Ξ → N π we derive the long-distance (LD) contributions B (SM,LD) where m Σ is the average of the Σ +,0,− masses, and for the Ω − channels Ã(SM,LD) where m Ξ * is the isospin-averaged mass of the Ξ (1530) resonances, which are of spin-3/2 and also members of the baryon decuplet.
The unknowns here are h D,F,C , but they can be evaluated from the available data on the ∆S = 1 processes Λ → N π, Σ → N π, Ξ → Λπ, and Ω − → ΛK − , Ξ π [17,22].Thus, performing a leastsquares fit of the octet-hyperon S-wave and Ω − P-wave decay amplitudes at leading order to their empirical values [4] yields the numbers in Eq. (A2).Subsequently, combining the central values of h D,F with Eqs. ( 17)- (18), we arrive at B Ξ 0 → pπ − ld sm = 2.7×10 −15 , B Ξ 0 → nπ 0 ld sm = 4.8×10 −16 , and B Ξ − → nπ − ld sm = 1.5 × 10 −15 , which exceed their SD counterparts in Eq. ( 14) by up to ∼ 20 times, implying that we need to put together the LD and SD amplitudes.Since the relative phase between the two is undetermined, we simply subtract one from the other or add them up to find In the case of Ω − → Bϕ, the LD contributions turn out to be significantly bigger than the SD ones, but the two are not highly disparate in Ω − → Σ * π, similarly to Ξ → N π.Explicitly, neglecting the SD ones in Ω − → Bϕ, with the central values of h D,F,C in Eq. (A2) we have where the first three results surpass the ones in Eq. ( 15) by over 3 orders of magnitude.
Although the preceding h D,F numbers give rise to a good fit to the S-wave ∆S = 1 NLHD, they translate into a poor representation of the P waves.On the other hand, it is possible to come up with a satisfactory account of the P waves, but end up with a disappointing description of the S waves.This is a well-known longstanding problem [17,[20][21][22], which lies beyond the scope of our analysis.Here we would merely like to see how different possible picks of h D,F,C might alter the ∆S = 2 predictions.Particularly, fitting to the ∆S = 1 octet-hyperon and Ω − P-waves produces the entries in Eq. (A3).These cause the LD components in Ξ → N π to be much greater than the SD ones, which now impact the branching fractions by no more than 15%, whereas the Ω − outcomes, are roughly comparable to those in Eq. (22).
To understand the parametric uncertainty of these SM predictions and their correlations, we quote the 90%-CL intervals for each observable at a time in Table I  selected to have the same sign as β27 .For the third column (labeled s), we have incorporated the LD components, taking them to have the same phase as the SD ones and including the correlations between the values of h D,F,C as obtained from fitting the S waves of octet-hyperon nonleptonic decays and P waves of Ω − → Bϕ in the ∆S = 1 sector.For the fourth column (labeled p), we have repeated this exercise but with δ27 and β27 having different signs, the SD and LD parts being opposite in phase, and h D,F,C from fitting the P waves of both the ∆S = 1 octet-hyperon and Ω − decays.As anticipated, for the last column the SD terms are, on the whole, numerically insignificant relative to the LD ones.
We complementarily show a number of pairwise 90%-CL regions of quantities induced by the SM SD contributions alone in Fig. 3, with δ27 and β27 having the same sign, and of the total SM branching-fractions in Fig. 4, after applying the procedure delineated at the end of Appendix A. For the top (bottom) plots in Fig. 4 the parameter choices are the same as those for the s (p) column in Table I specified in the previous paragraph.
In view of the smallness of the SM predictions in Table I, it is unlikely that they will be testable any time soon.On the upside, the striking dissimilarity between Eqs. ( 21) and (23), and between the corresponding entries in the third and fourth columns of Table I, implies that future observations of Ξ → N π with branching fractions at the level of 10 −12 or below could offer extra insight for dealing with the S-wave/P-wave problem in the ∆S = 1 nonleptonic decays of the octet hyperons.Furthermore, given that the measured bounds on these ∆S = 2 decays are scanty and fairly weak at the moment, the room for potential new-physics hiding in them is still substantial.
It is unfortunate that hadronic uncertainty plagues a good number of hyperon decay modes, making it difficult to tease out new-physics effects even in supposedly simpler semileptonic modes such as Σ + → pµ + µ − [23][24][25][26] or weak radiative modes [27,28].This implies that it is essential to keep pursuing processes which in the SM are either forbidden, such as those not conserving lepton flavor/number [29][30][31] and decays into a final state containing a dark boson/fermion [31][32][33][34], or very rare, such as the ∆S = 2 ones investigated here and flavor-changing neutral-current decays with missing energy carried away by a pair of invisibles [26,29,31,[35][36][37][38].It is therefore exciting that there are ongoing and proposed quests for some of them at running facilities [5,29,31].It is also encouraging that a couple of channels that have been searched for experimentally [39][40][41] are now under consideration by the lattice community [42].In addition, the aforementioned problem of ∆S = 1 NLHD and other aspects of them continue to receive theoretical attention [43][44][45][46].

III. ∆S = 2 NONLEPTONIC HYPERON DECAYS FROM NEW PHYSICS
The study of ∆S = 2 processes within the SM presented in the last section serves to guide us about what can be expected with new physics (NP).An effective theory at the weak scale required to satisfy the gauge symmetries of the SM will in general contain four-quark operators of definite chiral structure.The ∆S = 2 ones will then contribute to both K 0 -K0 mixing and hyperon decays, and if the Wilson coefficients are constrained by the former, the latter can generally be anticipated to occur at most near SM levels.
Nevertheless, the currently huge window between the SM predictions for the hyperon modes and their empirical upper-limits invites an exploration of NP scenarios that could populate it.It should be clear that, in order to achieve this, fine-tuning will be necessary.
We have found two ways in which NP can avoid the restriction from K 0 -K0 mixing.The first one relies on fine-tuning of model parameters that results in a cancellation among different contributions to the mixing.This is feasible because a four-quark operator comprising purely leftor right-handed fields leads to a K 0 -K0 matrix-element which is different than that of an operator consisting of chirally mixed fields.In Sec.III A we sketch a model exemplifying how this could happen.
The second scenario was already pointed out in Ref. [1] and involves NP which gives rise to |∆S| = 2 four-quark operators that exclusively violate parity and therefore do not contribute to K 0 ↔ K0 transitions.This also entails fine-tuning because SM gauge symmetries force any new particles to have chiral couplings to quarks at the weak scale.Cancellations between different operators are then needed to eliminate the parity-conserving ones.In Sec.III B we illustrate how this can be accomplished with two leptoquarks.

A. Z ′ contributions
We entertain the possibility that there exists a spin-1 massive gauge field Z ′ which is associated with a new Abelian gauge group U(1) ′ and couples to SM quarks in a family-nonuniversal manner, but has negligible mixing with SM gauge bosons.After the quark fields are rotated to the mass eigenstates, the Z ′ gains flavor-changing interactions at tree level with generally unequal left-and right-handed couplings [47].Here we focus on the dsZ ′ sector specified by the Lagrangian with g L and g R being constants and P R = (1 + γ 5 )/2.We suppose that additional fermionic interactions that the Z ′ may possess already fulfill the empirical restraints to which they are subject, but on which we do not dwell in this paper.
With the Z ′ mass, m Z ′ , assumed to be big, from Eq. ( 25) one can come up with tree-level Z ′ -mediated diagrams contributing to the s d → sd reaction and described by at an energy scale µ ≲ m Z ′ , with To examine the effects of H Z ′ ∆S=2 on hadronic transitions, we need to take into account the QCD renormalization-group running from the m Z ′ scale down to hadronic scales.This modifies Eq. ( 26) into [48,49] where η LL = η RR and η (′) LR are QCD-correction factors and Q ′ LR = dP L s dP R s.The chiral realization of Q LL for hyperons is already given in Eq. ( 5).Hence, since Q RR transforms like (1 L , 27 R ) under SU(3) L × SU(3) R rotations and the strong interaction is invariant under a parity operation, the lowest-order chiral realization of Q RR is For LR , which belongs to (8 L , 8 R ) and is even under parity, the leading-order baryonic chiral realization relevant to the decays of interest is where β(′) 88 and δ(′) 88 will be estimated shortly.Being parity even, O LR at tree level impacts only the P waves of Ξ → N π and Ω − → Bϕ, Σ * π.
Before calculating the hyperon rates, we also need to pay attention to potential restrictions implied by kaon-mixing data.This is because the interactions in Eq. ( 28) affect the neutral-kaon mass difference ∆M K = 2 Re M K K and the CP -violation parameter |ϵ| ≃ |Im where ⟨Q⟩ ≡ ⟨K 0 |Q| K0 ⟩.Numerically ⟨Q LL ⟩ = 0.002156(34) GeV 4 , ⟨Q LR ⟩ = −0.0482(28)GeV 4 , and ⟨Q ′ LR ⟩ = 0.0930(30) GeV 4 computed at µ = 3 GeV in Ref. [52].In Eq. ( 37) we additionally use η LL = 0.74, η LR = 0.89, and η ′ LR = −2.07,all at µ = 3 GeV as well.With these numbers, it turns out that M Z ′ K K goes to zero for certain values of g L /g R where one of the two couplings is small relative to the other.In Appendix C we look at an illustrative Z ′ model that shows in some detail how this can be realized.
More generally, we may let g L and g R vary freely under the experimental requisites.In the instance that these couplings are real, since the latest SM estimate ∆M sm K = 5.8(2.4)× 10 −12 MeV from lattice-QCD studies [53] is still much less precise than its measurement ∆M exp K = 3.484(6) × 10 −12 MeV [4], we may impose −1 < ∆M Z ′ K /∆M exp K < 0.5, which is consistent with the two-sigma range of ∆M exp K − ∆M sm K , but there is no constraint from ϵ.For an example of this case, we pick the first option in Eq. ( 36) and δ27 = − β27 , as well as m Z ′ /g L ≥ 5 TeV, which reflects our assuming |g L | ≤ 1 to guarantee perturbativity, with m Z ′ = 5 TeV.This results in the allowed (blue and red) regions of m Z ′ /g L versus g R /g L displayed in Fig. 5. 3 The vertical span of the red area in this figure corresponds to These are far greater than their SM counterparts in Eqs. ( 23)-( 24) and might be sufficiently sizable to be within reach of LHCb [5] and BESIII [8] in their future quests and of the proposed Super Tau-Charm Factory [8].It should be pointed out, however, that in specific Z ′ models the hyperon rates may be comparatively less enhanced due to various restraints on the Z ′ couplings, such as the model discussed in Appendix C, which yields B(Ξ 0 → pπ − ) Z ′ ∼ 4 × 10 −10 .

B. Leptoquark contributions
By introducing more than one leptoquark (LQ) it is possible to generate an effective fourquark ∆S = 2 interaction that is parity violating and hence eludes the kaon-mixing requirement.The LQs of interest here, with their SM gauge-group assignments SU(3) C , SU(2) L , U(1) Y , are S1 ∼ (3, 1, 4/3) and R 2 ∼ (3, 2, 7/6) in the nomenclature of Ref. [54].They can have renormalizable interactions with SM fermions according to where y and Y are Yukawa coupling matrices, q j and d j represent a left-handed quark doublet and right-handed down-type-quark singlet, respectively, and e x is a right-handed charged-lepton singlet.Working in the mass basis of the down-type fermions, we rewrite Eq. ( 40) as where j, k, x = 1, 2, 3 here denote family indices and are summed over, the superscripts of S1 and R 2 indicate the electric charges of their components, and U 1,2,3 = u, c, t, D 1,2,3 = d, s, b, and ℓ 1,2,3 = e, µ, τ refer to the mass eigenstates.Although these LQs could have other couplings with SM fermions or engage in scalar interactions [54], for our purposes we do not entertain such possibilities, considering only the minimal ingredients already specified in L lq above.
From Eq. ( 41), with the LQs taken to be heavy, we can derive box diagrams which lead to the effective Hamiltonians where Q LL,RR have been written down in Eqs. ( 4) and (27).Evidently L lq affects not only ∆M K via ∆M (LQ) K also suppressed or vanishing.In such instances the ∆M K limitation can be evaded. 4In the remainder of this section, we explore this scenario and for simplicity set m S1 = m R 2 ≡ m LQ and with y dτ and y sτ being real constants, ensuring that x y rr * 1x y rr It follows that now Since Q LL − Q RR = −dγ α s dγ α γ 5 s is parity odd, H LQ ∆S=2 no longer influences K 0 -K0 mixing.On the other hand, the contribution to ∆M D is still present, but this will be avoided if one of the brackets in H LQ ∆C=2 is zero.Thus, we opt for V ud y dτ + V us y sτ = 0, which causes H LQ ∆C=2 = 0 and at a scale µ ≲ m LQ .Moreover, given that V ud and V us are real in the standard parametrization, y dτ and y sτ stay real as well, and with V ud /V us = 4.33 from Ref. [4] the perturbativity condition |y dτ,sτ | < √ 4π implies the requisite |y dτ | < 0.819.
It is worth remarking that in general, below the high scale (µ NP ) at which new physics is integrated out, the effects of QCD renormalization-group running on the Wilson coefficients C LL and C RR of Q LL and Q RR in the effective Hamiltonian H eff containing them are known to be the same [48,56,57], which reflects the fact that the strong interaction conserves parity.This means that the QCD-evolution factors, η LL and η RR , which accompany these operators in H eff are also the same, η LL = η RR .Then, in the case where C LL = −C RR at µ NP , at lower energies Accordingly, in our particular LQ scenario, Eq. ( 47) translates into ⟨K 0 |H LQ ∆S=2 | K0 ⟩ = 0 at any scale µ < m LQ .From the last two paragraphs and the chiral realizations of Q LL,RR in Eqs. ( 5) and (29), we get the S-wave amplitude terms for Ξ → N π and In contrast, being parity odd, H LQ ∆S=2 in Eq. (47) does not modify the P-wave parts, and consequently Σ * π = 0.With β27 = 0.076 as before, η LL = 0.68 for m LQ = 1 TeV, and |y dτ | < 0.8, from Eq. ( 48) we arrive at the upper values exceeding the corresponding SM predictions in Eqs. ( 23)- (24) and Table I by five orders of magnitude or more.Some of these enhanced results might soon be probed by LHCb [5] and BESIII [8].
Finally, we comment that although the LQs considered here influence various other low-energy processes, such as s → dγ, dg and the anomalous magnetic moment of the τ lepton, we have checked that the effects are not significant with the parameter choices we made.These include the special textures of the Yukawa matrices in Eq. ( 43) which also help the LQs evade the constraints from collider quests [4].

IV. CONCLUSIONS
We have explored the ∆S = 2 nonleptonic decays of the lowest-mass hyperons within and beyond the SM.Concentrating on two-body channels, we first updated the SM predictions for Ξ → N π and subsequently addressed those for Ω − → nK − , Λπ − , Σ π, Σ * π.Furthermore, we investigated the impact on these processes of long-distance diagrams involving two couplings from the ∆S = 1 Lagrangian in the SM.The LD contributions turned out to be much bigger than the SD ones on the whole, but can raise the branching fractions of the majority of these decay modes merely to the 10 −12 level, making the SM predictions unlikely to be tested in the near future.Beyond the SM, new physics may bring about substantial amplifications, although restrictions from kaon mixing play a consequential role.We showed that a Z ′ boson possessing family-nonuniversal interactions with quarks can give rise to rates of the ∆S = 2 hyperon transitions which greatly surpass the SM expectations and a few of which could be within reach of BESIII and LHCb.We also demonstrated that a model with two leptoquarks can achieve similar outcomes.Although these two cases are very distinct in their details, both require some degree of fine-tuning to make the hyperon modes potentially observable not too long from now.
where Σ12 is the numerically estimated variance-covariance matrix for the pair.Then we select the 90% of points closest to their mean.
where C Bϕ , ÃΣ * π , and BΣ * π are constants, p stands for the momentum of ϕ, and the D-wave term in M Ω − →Bϕ and the D-wave and F-wave ones in M Ω − →Σ * 0 π − have been neglected.To calculate the corresponding rates, we need the sum over polarizations, ς, of a spin-3/2 particle of momentum k and mass m given by5 3/2 After averaging (summing) the absolute squares of the amplitudes over the initial (final) spins, we arrive at where μ2 ± = (m Ω ± m Σ * 0 ) 2 − m 2 π − and E (P) is the energy (three-momentum) of the daughter baryon in the Ω − rest-frame.
Appendix C: Simple Z ′ possibility For a particular example of the Z ′ scenario considered in Sec.III A, we suppose that under the U(1) ′ gauge group the left-and right-handed quarks in the first (second) family carry charge q = 1 (−1) whereas the other SM fermions are singlets.It is straightforward to see that with these charge assignments the model is free of gauge anomalies.Accordingly, with the covariant derivative of fermion f having the form D α f ⊃ ∂ α + iǵ qf Z ′ α f, the Z ′ interactions with the quarks are described by L are linked to the CKM matrix by V d L = V u L V ckm , the expression for V d L is fixed once V u L has been specified and vice versa, but this does not apply to V u,d R and there is freedom to pick their elements.This is because M u,d are arbitrary as long as they satisfy the abovementioned diagonalization equations and can be arranged to have the textures by introducing the appropriate Higgs sector.To suppress other effects of the new Higgs particles, including flavorchanging neutral currents which might be associated with them, they are assumed to be sufficiently heavy.
Thus, for our purposes, we can choose with which Eq. (C1) becomes where the θs and ω are real quantities, C f X = cos 2θ f X , and S f X = sin 2θ f X .Taking θ u L and θ u R to be tiny or vanishing then leads to where the ucZ ′ part has dropped out, avoiding the limitation from D 0 -D0 mixing.Comparing the dsZ ′ portion of Eq. (C4) with Eq. ( 25), we identify g L = ǵ V * ud V us − V * cd V cs and g R = e iω ǵ S d R .Selecting ω = Arg V * ud V us − V * cd V cs and a suitable θ d R , we can then acquire the special g L /g R ratio which renders M Z ′ K K in Eq. ( 37) vanishing.
It is interesting to point out that, after the CKM parameters from Ref. [4] are incorporated, the terms L qZ ′ ⊃ −ǵ (0.011 − 0.003i) d L / Z ′ b L + ǵ (0.040 + 0.0008i) s L / Z ′ b L + H.c. can be shown to elude B d,s -Bd,s mixing constraints if ǵ/m Z ′ ≲ 0.1/TeV, as new-physics effects of order ∼ 10% in the mass differences ∆M d,s are still permitted [59].Moreover, although a flavor-changing coupling and a flavor-diagonal one from Eq. (C4) can translate into operators contributing to four-quark penguin interactions [9], the impact can be demonstrated to be weaker than that of the SM by at least an order of magnitude if ǵ/m Z ′ ≲ 0.1/TeV.In addition, the flavor-conserving couplings in Eq. (C4) can escape the restraints from Z ′ searches in hadronic final-states at colliders provided that the Z ′ mass is around 5 TeV or more [4].
Lastly, from Eq. (C4) one can derive long-distance contributions to ∆S = 2 transitions involving two ∆S = 1 Z ′ -mediated couplings or one of them and one ∆S = 1 coupling from the SM.One can deduce from the preceding two paragraphs, however, that such LD effects are unimportant relative to the SD interactions in Eq. ( 28).
and m Σ * is the isospin-averaged mass of the Σ (1385) resonances.

FIG. 3 .
FIG. 3. Distributions (top) of B Ξ 0 → pπ − and B Ω − → Λπ − versus ∆M K and (bottom) of the branching fractions of three pairs of ∆S = 2 nonleptonic hyperon decays, all arising solely from the shortdistance interactions in the SM.The blue thick vertical lines in the top graphs indicate the experimental value, ∆M exp K .The large black dots mark the central values of our estimates.

FIG. 4 .
FIG. 4. Distributions of the SM branching fractions of different pairs of ∆S = 2 nonleptonic hyperon decays from the summed SD and LD amplitudes, as explained in the text.The large black dots mark the central values.
Appendix B: Rates of Ω − decaysThe amplitudes for Ω − → Bϕ and Ω C1) where ǵ denotes the U(1) ′ gauge coupling constant, the primed quark fields are in the flavor basis, U and D represent column matrices with elements (U 1 , U 2 , U 3 ) = (u, c, t) and (D 1 , D 2 , D 3 ) = (d, s, b) in the mass basis, and V u L,R and V d L,R are 3×3 unitary matrices which connect the fields in the two bases and also diagonalize the quark mass matrices M u and M d via diag(m u , m c , m t ) = V u † L M u V u R and diag(m d , m s , m b and therefore offers possibly the cleanest way to assess β27 .From Σ + → nπ + , after implementing the steps outlined in Appendix A. The second column of the table lists only the SD contributions, with δ27 × 10 −15 (0.01, 2.6) × 10 −14 (0.7, 8.2) × 10 −13

TABLE I .
The 90%-CL intervals of branching fractions of ∆S = 2 nonleptonic hyperon decays from the short-distance and complete contributions of the SM, as explained in the text.
also its charmed-meson analog, ∆M D .It is interesting to notice that, since y rr ix and Y LR ix besides the LQ masses are free parameters, the model parameter space contains regions in which