Probing new physics with the kaon decays $K\to\pi\pi\!\not\!\!E$

The most recent search for the rare kaon decay $K^+\to\pi^+\nu\bar\nu$ by the NA62 experiment has produced evidence for it with a branching fraction consistent with the prediction of the standard model. The new result implies that in this decay, with the $\nu\bar\nu$ pair appearing as missing energy ($\not\!\!E$), the room for possible new physics is no longer sizable and that therefore its contributions to $s\to d\!\not\!\!E$ operators with parity-even $ds$ quark bilinears have become significantly constrained. Nevertheless, we point out that appreciable manifestations of new physics via operators with mainly parity-odd $ds$ quark bilinears may still occur in $K\to\pi\pi\!\not\!\!E$ modes, on which there are only minimal empirical details at present. We find in particular that new physics may enhance the branching fractions of $K^+\to\pi^+\pi^0\!\not\!\!E$ and $K_L\to\pi^0\pi^0\!\not\!\!E$ to their current experimental upper limits and of $K_L\to\pi^+\pi^-\!\not\!\!E$ to the level of $10^{-5}$. Thus, quests for these decays in existing kaon facilities, such as NA62 and KOTO, could provide valuable information complementary to that gained from $K\to\pi\!\not\!\!E$.


I. INTRODUCTION
One of the potentially promising avenues to discover new physics (NP) beyond the standard model (SM) is to look for processes that are expected to be very rare in the SM. An observation of such a process having a rate much greater than what the SM predicts would then be a compelling indication of NP effects. Among places where this may be realized are the flavor-changing neutral current (FCNC) decays of strange hadrons with missing energy ( / E), which are known to be dominated by short-distance physics [1][2][3][4][5][6] and arise mainly from the quark transition s → d / E. In the SM, it proceeds from loop-suppressed diagrams [2] and the final state contains undetected neutrinos (νν). Beyond the SM, there could be additional ingredients which alter the SM component and/or give rise to extra channels with one or more invisible nonstandard particles carrying away the missing energy.
As it turns out, of the possible underlying s → d / E operators [13], these decays are sensitive to only a subset. Specifically, they can probe the operators having parity-even ds quark bilinears but are not affected by the ones with exclusively parity-odd ds bilinears [6,13,14]. However, the latter can contribute to kaon reactions with no or two pions, namely K → / E and K → ππ / E, as well as to analogous decays in the hyperon sector [6,14]. This implies that, since at the moment there are precious few data on these processes [15], searches for them might still come up with substantial manifestations of NP or at least yield information about it complementary to that supplied by K → π / E measurements.
In this paper, we adopt a model-independent approach to explore how large the branching fractions of the various K → ππ / E modes might be, taking into account the available pertinent constraints. We assume especially that the invisibles comprise a pair of spin-1/2 fermions or spinless bosons, all of which are singlets under the SM gauge groups. The results of our study will hopefully motivate renewed attempts to pursue these decays as NP tests.
The organization of the rest of the article is the following. In Sec. II, we describe the quarklevel operators responsible for the interactions of interest. In Sec. III, we derive the amplitudes for the aforementioned kaon decay modes and calculate their rates. We also write down the corresponding numerical branching fractions. In Sec. IV, we compare the SM predictions for these transitions with their current data. In Sec. V, we address the allowed maximal branching fractions of K → ππ / E due to NP and present our conclusions.

II. INTERACTIONS
Depending on the types of particles carrying away the missing energy, the effective s → d / E operators are generally subject to different sets of restrictions. If the invisible particles are SM neutrinos, which have charged-lepton partners because of the SM SU(2) L -gauge invariance, the operators would have to face stringent restraints from lepton-flavor violation data. Since these do not apply if the invisibles are SM-gauge singlets, hereafter we consider a couple of cases involving them.
The missing energy is carried away by a spin-1/2 Dirac fermion f and its antiparticlef in the first scenario, whereas it is due to a pair φφ of complex spin-0 bosons in the second one. 1 At low energies, the relevant quark-level operators need to respect the strong and electromagnetic gauge symmetries and are obtainable from the literature [13,16]. We can express the interaction Lagrangians as and for the two scenarios, respectively, where the Cs,cs, and cs are generally complex coefficients, which have the dimension of inverse squared mass, except for c S,P φ , which are of inverse-mass dimension. These are free parameters in our model-independent approach and will be treated phenomenologically in our numerical work later on. In L f , there are merely two tensor operators due to the identity 2iσ αω γ 5 = ǫ αωβψ σ βψ . We note that L f and L φ could originate from effective Lagrangians that are invariant under all the SM gauge groups [13].

III. DECAY AMPLITUDES AND RATES
To examine the amplitudes for the kaon decays of concern, we need the mesonic matrix elements of the quark portions of the operators in Eqs. (1) and (2). They can be estimated with the aid of 1 In the recent literature covering the impact of NP on K → ππ / E, there are other possibilities for what carries away the missing energy. In particular, it could alternatively be due to a single particle such as a massless dark photon [17][18][19] or an invisible axion [20].
flavor-SU(3) chiral perturbation theory at leading order [6,13,21]. For K L,S → / E, the relevant hadronic matrix elements are with f K ≃ 156 MeV [15] denoting the kaon decay constant, p X being the momentum of X, and B 0 = m 2 K /(m + m s ) ≃ 2.0 GeV involving the average kaon mass and the light-quark mass combinationm + m s ≃ 124 MeV at a renormalization scale of 1 GeV, while for K → π / E, where a T is a constant. Assuming isospin symmetry and making use of charge conjugation, we further have π 0 |d(γ η , 1, Although the matrix elements in Eqs. (4)-(6) generally involve momentum-dependent form factors, to investigate the NP influence on these processes in this study we do not need a high degree of precision and therefore can ignore form-factor effects. We also neglect π 0,− π + |sγ η d|K +,0 , and their charge conjugates, which arise from small contributions derived from the anomaly Lagrangian, which occurs at next-to-leading order in the chiral expansion [3,13].
We now apply these matrix elements to kaon decays induced by L f in Eq. (1), taking the f mass to vanish, i.e. m f = 0. Thus, with the approximate relations √ 2 K L,S = K 0 ± K 0 , we obtain the amplitudes for K L,S → ff to be from which follow the decay rates whereS We see that For K − → π − ff and K L → π 0 ff, the S and P terms are These lead to the differential rates with respect to the invariant mass squared,ŝ, of the ff pair where Evidently, K → πff, unlike K → ff, are sensitive to C V,A,S,P,T,T′ f , but not toc v,a,s,p f . For K − → π 0 π − ff and K L → (π + π − , π 0 π 0 )ff, we find whereq with m K ink being the average kaon mass. We then arrive at the double differential rates whereς is the invariant mass squared of the pion pair, For obtaining the rates, theŝ andς integration ranges are 0 ≤ŝ ≤ (m K − ,K 0 − 2m π ) 2 and 4m 2 π ≤ς ≤ m K − ,K 0 −ŝ 1/2 2 for the K − and K L channels, respectively. In the formulas above for the mode with the π 0 π − (π + π − or π 0 π 0 ) pair, m π refers to the isospin-average (charged or neutral) pion mass. The corresponding expressions for the K S → (π + π − , π 0 π 0 )ff rates equal their K L counterparts in Eq. (16) except that the imaginary (real) parts of the coefficients are replaced by their real (imaginary) parts. Clearly, K → ππff, as opposed to K → πff, can probec v,a,s,p f besides C T,T′ f but not C V,A,S,P f in our approximation of the hadronic matrix elements.
For the processes induced by L φ in Eq. (2), we also take the φ mass to vanish, m φ = 0. Accordingly, the amplitudes for K L,S → φφ are [14] and hence, the decay rates For the three-body modes, we find [14] M with p (p) standing for the momentum of φ (φ), and consequently whereŝ = (p +p) 2 . For K → ππφφ, we derive from which we arrive at the double-differential rates As the last paragraph shows, K L,S → φφ are sensitive exclusively to c P φ , whereas K → ππφφ can probe solely the two parity-odd couplings, c A φ and c P φ , in our approximation of the mesonic matrix elements. By contrast, the K → πφφ amplitudes depend on the parity-even coefficients, c V φ and c S φ , but are independent of c A,P φ . In Table I, we list the contributions of the different constants in Eqs. (1) and (2) to the kaon decays of interest according to the preceding discussion. We remark that for f having a Majorana nature, instead of Dirac one, f γ η f = 0, causing the C V f andc v f parts to disappear. Moreover, for φ being a real field, rather than complex one, the c V,A φ terms would be absent. For later convenience, from the K → / E and K → ππ / E rate formulas above, here we write down the corresponding numerical branching fractions in terms of the coefficients, employing the central values of the measured kaon lifetimes and meson masses from Ref. [15] as well as a T = 0.658(23)/GeV from lattice QCD work [22]. Thus, Eq.

IV. SM PREDICTIONS AND EMPIRICAL INFORMATION
As mentioned earlier, the latest NA62 measurement on K + → π + νν has turned up evidence for it that is fully consistent with the SM expectation [12]. In view of Table I, this implies that the couplings C V,A,S,P,T,T′ f and c V,S φ originating from possible NP cannot by sizable anymore. 2 To explore how much the other coefficients shown in Table I may be affected by NP to amplify the K → / E and K → ππ / E rates with respect to their SM values, we need to know the latter. 2 A preliminary report by KOTO [23] has revealed that their most recent data contain a couple of K L → π 0 νν events suggesting an anomalously high rate, which still needs confirmation from further measurements. If this anomaly persists in the future, it may be due to NP, as discussed in e.g. [24,25] and the references therein, but its effects would not be large enough to modify our conclusions for K → ππ / E.
As for the four-body channels, employing Eq. (25) we obtain B(K − → π 0 π − νν) sm ∼ 4 × 10 −15 and B(K L → (π + π − , π 0 π 0 )νν) sm ∼ (8, 5)×10 −14 . These are in rough agreement with more refined evaluations in the literature [3,4,26,27]: with the most recent CKM matrix elements [15]. The estimates for K S → ππνν are about three orders of magnitude less than their K L counterparts. The two sets of K − and K L numbers above indicate the level of uncertainties in our K → ππ / E predictions in the next section.
On the experimental side, only two of these modes have been looked for [15], with negative outcomes which led to the limits [28,29] B(K − → π 0 π − νν) exp < 4.3 × 10 −5 , both at 90% CL. These exceed the corresponding SM numbers in Eq. (29) by several orders of magnitude. As regards K L,S → / E, there have been no direct searches for them yet. Nevertheless, from the existing data [15] on the visible decay channels of K L,S one can extract indirect upper bounds on their invisible branching fractions [30]. Thus, one can infer [19] at the 2σ level, which are far away from the aforesaid B(K L,S → νν) sm values. Comparing Eqs. (30)- (31) with Eqs. (24)- (27), as well as Table I, we conclude that currently there remains potentially plenty of room for NP to enhance the rates of these decays viac v,a,s,p f and c A,P φ .

V. NP EXPECTATIONS AND CONCLUSIONS
Based on the considerations made in the previous section, we hereafter entertain the possibility that among the couplings listed in the table NP manifests itself exclusively viac v,a,s,p f or c A,P φ and demand that they fulfill the conditions Moreover, as remarked earlier, the same underlying interactions induce analogous decays of hyperons (Λ, Σ + , Ξ 0 , Ξ − , Ω − ) and their data turn out to supply additional constraints. Although these transitions have never been searched for, they are among the hyperons' yet-unobserved modes whose branching fractions have maxima which can be inferred from the available data on the observed channels [15]. These upper bounds have been estimated in Ref. [19], which we adopt here to impose The expressions for the corresponding hyperon rates in terms ofc v,a,s,p f c A,P φ have been derived in Ref. [6] ( [14]). To illustrate the ramifications that may arise for the various K → ππ / E modes if NP occurs in these couplings, we can look at several simple examples.
If it impactsc s,p f alone, we learn from Eqs. (24)-(25) that these parameters need to satisfy the kaon restraints specified in the last paragraph and that Eq. (32) is stricter than Eq. (33). Then, assumingc s,p f to be complex, from their allowed ranges we find the maximal values B K L → π 0 π 0 ff < 9.9 × 10 −11 , B K S → π + π − ff < 5.8 × 10 −11 , If now onlyc v,a f are influenced by NP, it is clear from Eq. (24) that Eq. (32) is no longer relevant but Eq. (33) still matters. In this case, if these couplings are real, the K L → π 0 π 0 / E requirement is the stronger and yields Rec v 2.8 × 10 −10 1.3 × 10 −9 9.9 × 10 −11 5.8 × 10 −11 3.9 × 10 −11 Rec v,a For the preceding two instances the hyperon limitations in Eq. (34) are unimportant. In contrast, ifc v,a f are permitted to be complex, the above requisite on their real parts still applies, but Eq. (34) needs to be taken into account as well, leading to Imc v f 2 + Imc a f 2 < 1.4 × 10 −11 GeV −4 , and so these mostly much bigger results may be achieved: only the second bound in Eq. (33) being saturated as in Eq. (36).
In Table II we collect our findings in Eqs. (35)-(39) and the associated coefficients. We note that in the cases seen in this table with the largest branching fractions the corresponding predictions for their hyperon counterparts are also large and might therefore be within the sensitivity reach of the BESIII experiment [32,33], as discussed in Refs. [6,14,31].
To conclude, motivated by the latest NA62 measurement on K + → π + νν, which is in good agreement with the SM and consequently implies stringent constraints on NP which might be hiding in K → π / E, we have explored how other types of FCNC kaon decays with missing energy might shed additional light on potential NP in the underlying s → d / E transition. We have argued that there are other operators contributing to s → d / E which are not restricted by K → π / E and accordingly could still significantly affect K → / E and K → ππ / E, on which the empirical details are currently meager. We have demonstrated especially that the branching fractions of K → ππ / E could yet be amplified far higher than their SM expectations, to levels which might be within the reaches of ongoing or near-future experiments, such as KOTO and NA62. Our results, which are illustrated with the instances summarized in Table II, will hopefully help stimulate new quests for these decays as NP probes. Lastly, we have pointed out that similar kinds of enhancement would occur in the hyperon sector, which may be detectable by BESIII and thus could provide complementary information.