Light dark-matter window in $K^{+}\to\pi^{+}$$+$$\not{\!\!E}$

Recently the NA62 collaboration has reported the first observation of kaon decay $K^{+}\to\pi^{+}$+$\not{E}$, with missing energy ($\not{E}$) carried away by emitted invisible particles, its branching ratio measured to be  ${\cal B}(K^{+}\to\pi^{+}$+$\not{E})_{\rm exp}=\big{(}13.0_{-3.0}^{+3.3}\big{)}\times 10^{-11}$ [1], making this mode the rarest one discovered to date. In the standard model (SM), neutrinos ($\nu\bar{\nu}$) act as the invisibles, and the $K^{+}\to\pi^{+}\nu\bar{\nu}$ prediction is remarkably precise [2] but still suffers from significant parametric uncertainty arising from the input values of the Cabibbo-Kobayashi-Maskawa (CKM) angles. This is illustrated by the three different branching-ratio numbers quoted in Ref. [1]. Conservatively, the one with the largest quoted uncertainty,  ${\cal B}(K^{+}\to\pi^{+}\nu\bar{\nu})_{\rm SM}=(8.4\pm 1.0)\times 10^{-11}$ [3], may be adopted for discussion purposes. It is less than the measurement by  $\Delta{\cal B}_{K}=(4.6^{+3.4}_{-3.2})\times 10^{-11}$.

The preceding $\Delta{\cal B}_{K}$ value can be regarded as a potential window into new physics beyond the SM. If  $\Delta{\cal B}_{K}\neq 0$  persists with increasing statistical significance as more data are collected in the future, it might be due to dark matter (DM) particles contributing to $\not{E}$. Analogous ideas have lately been proposed [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] to explain the recent Belle II measurement [27] on the $b$-meson decay $B^{+}\to K^{+}$+$\not{E}$  with a rate that is higher than the SM expectation. It is therefore interesting to consider the possibility that the same new physics can be responsible for both  $\Delta{\cal B}_{K}\neq 0$  and the  $B^{+}\to K^{+}$+$\not{E}$  anomaly, with the obvious condition that the DM mass not exceed  $(m_{K^{+}}-m_{\pi^{+}})/2=177$ MeV in order that the $K^{+}$ channel with the DM could occur. The DM candidate must additionally be able to reproduce the observed relic abundance and comply with bounds inferred from DM direct, indirect, and collider searches. Here we carry out a model-independent analysis of this kind of scenario to look for the parameter space at the level of low-energy effective field theory (LEFT) where all these requisites can be fulfilled. We arrive at interesting allowed parameter regions which can be tested further with future data, especially from Belle II, NA62, and the various DM searches, and which were not covered in the earlier $B^{+}\to K^{+}$+$\not{E}$  studies.

The organization of the paper is as follows. In the next section, we discuss the effective interactions of the DM with SM quarks. In Sec. III, we address in greater detail the implications for the $B^{+}$ and $K^{+}$ transitions of interest. In Sec. IV, we examine the restrictions on the parameter space from various DM searches. In Sec. V, we give our conclusions.

We suppose that the DM particle is a real scalar boson $\phi$ which is a SM-gauge singlet and interacts only in pair with SM quarks due to it being charged under some symmetry of a dark sector beyond the SM or odd under a $\mathbb{Z}_{2}$ symmetry which does not influence SM fields. This ensures the stability of $\phi$. In the LEFT-like framework that we adopt, the lowest-dimension operators describing the $\phi$-quark couplings of interest are given by [28, 29]

where summation over $(kl)=(dd,ss,sd,sb)$ is implicit and the $C$s are generally complex, independent constants. The Hermiticity of this Lagrangian implies that  $C_{d\phi}^{S(P),kl}=C_{d\phi}^{S(P),lk*}$  and $C_{u\phi}^{S(P),uu}$ is real. Only the $(\bar{d}_{k}d_{l})\phi^{2}$ term in Eq. (1) can supply the $bs\phi^{2}$ and $sd\phi^{2}$ couplings that contribute to  $B^{+}\to K^{+}$+$\not{E}$  and  $K^{+}\to\pi^{+}$+$\not{E}$,  respectively, as the matrix element of a pseudoscalar quark bilinear between two pseudoscalar mesons vanishes.⁵⁵5If  $C_{d\phi}^{P,bs}\neq 0$  in Eq. (1), the $(\bar{b}i\gamma_{5}s)\phi^{2}$ operator could affect the yet-unobserved  $B\to K^{*}$+$\not{E}$  at levels similar to what is required of $(\bar{b}s)\phi^{2}$ to explain the $B^{+}\to K^{+}$+$\not{E}$ excess without violating their experimental bounds [30]. If  $C_{d\phi}^{P,sd}\neq 0$,  the $(\bar{s}i\gamma_{5}d)\phi^{2}$ operator could contributes to  $K\to\pi\pi$+$\not{E}$  and  $K\to\not{E}$ [29], their SM branching ratios being of order $10^{-13}$ or less [31, 32, 33, 34, 35, 36], as well as to analogous hyperon decays [37], all of which still have relatively weak empirical limits [38, 39, 40, 41, 42], implying that these kaon and hyperon modes each offer a potentially wide window into new physics [37, 40, 43]. For $\phi$ to be the cosmological DM candidate, we demand that its annihilation generate the observed relic abundance via the freeze-out mechanism and include the $(\bar{u}u)\phi^{2}$ and $(\bar{d}d)\phi^{2}$ operators. Since the $\phi$ mass must be  $m_{\phi}<177$ MeV  and the latest results of DM direct-detection experiments have translated into strict bounds on sub-GeV DM, especially when the Migdal effect is taken into consideration, it turns out that the impact of the latter can be alleviated by turning on the $(\bar{s}s)\phi^{2}$ term as well, which does not affect the relic density. These are discussed in more detail below.

In 2023 the Belle II experiment [27] came up with  ${\cal B}(B^{+}\to K^{+}$+$\not{E})_{\rm exp}=(2.3\pm 0.7)\times 10^{-5}$, which is 2.7$\sigma$ bigger than the SM prediction  ${\cal B}(B^{+}\to K^{+}\nu\bar{\nu})_{\rm SM}=(4.43\pm 0.31)\times 10^{-6}$ [44, 7].⁶⁶6This does not include the tree-level contribution mediated by the tau lepton [27]. Combining this new measurement with the earlier findings by BaBar [45, 46], Belle [47, 48], and Belle II [49] yields the weighted average  ${\cal B}(B^{+}\to K^{+}$+$\not{E})_{\rm exp}=(1.3\pm 0.4)\times 10^{-5}$ [27], which is larger than the SM value by 2.1$\sigma$. This could be interpreted as attributable to the presence of the exotic channel  $B^{+}\to K^{+}\phi\phi$  induced by the $(\bar{b}s)\phi^{2}$ operator with  $C^{S,bs}_{d\phi}\neq 0$  in Eq. (1) and DM mass  $m_{\phi}<(m_{B^{+}}-m_{K^{+}})/2$.  It has been shown [7, 16] that this can indeed explain the Belle II anomaly if, for example,  $C^{S,bs}_{d\phi}\in[3,7]/(10^{5}$ TeV)  and $m_{\phi}\in[0.1,1]$ GeV.

If the same underlying new physics is to induce  $K^{+}\to\pi^{+}\phi\phi$  as well and fill the potential new-physics window $\Delta{\cal B}_{K}$ in the aforementioned  $K^{+}\to\pi^{+}$+$\not{E}$  data from NA62, the DM needs to be sufficiently light with mass  $m_{\phi}<(m_{K^{+}}-m_{\pi^{+}})/2=177$ MeV.  In this range, the  $B^{+}\to K^{+}$+$\not{E}$ anomaly can be accounted for with  $C_{d\phi}^{S,bs}\sim(2.5-6)/(10^{5}\,\rm TeV)$ [7, 16].

For such low-mass DM, we can deal with  $K\to\pi\phi\phi$  and the DM annihilation into light hadrons brought about by the LEFT operators in Eq. (1) by means of chiral perturbation theory. This leads to the $\phi$ interactions with light mesons given by

where  $B_{0}=m_{\pi^{+}}^{2}/(m_{u}+m_{d})=2.8$ GeV  with $m_{u,d}$ values from Ref. [30]. From the $C_{d\phi}^{S,sd}$ terms, we derive the differential rates of  $K\to\pi\phi\phi$ [50]

where $q^{2}$ is the squared invariant-mass of the $\phi\phi$ pair and  $\lambda(x,y,z)=(x-y-z)^{2}-4yz$.

In the left panel of Fig. 1 we have displayed the $\big{|}C_{d\phi}^{S,sd}\big{|}$ values versus $m_{\phi}$ (the unshaded area under the green solid curve) allowed by the NA62 [1] measurement on  $K^{+}\to\pi^{+}$+$\not{E}$  in two signal regions (SR), the first one (SR1) being specified by   $q^{2}=[0.000,0.010]\,\rm GeV^{2}$  and  $|\textbf{{p}}_{\pi}|=[15,35]\rm\,GeV$ and the second (SR2) by  $q^{2}=[0.026,0.068]\,\rm GeV^{2}$  and  $|\textbf{{p}}_{\pi}|=[15,45]\rm\,GeV$.  The right panel of Fig. 1 is the same as the left, except that only the total branching-ratio result from NA62 has been used. In the left (right) panel, the unshaded area between the solid and dashed green (purple) curves corresponds to the aforementioned possible new-physics window $\Delta{\cal B}_{K}$ in the NA62 data. The plots in Fig. 1 also indicate the only mass interval,  $m_{\phi}=[110,146]$ MeV,  not yet ruled out by the DM direct and indirect searches, to be discussed in Sec. IV. Additionally, we have drawn on these plots the hatched areas which, if $C_{d\phi}^{S,sd}$ is purely real, are excluded by the limit from the KOTO search for the isospin-related neutral-kaon mode:  ${\cal B}(K_{L}\to\pi^{0}$+$\not{E})<2.2\times 10^{-9}$ at 90% confidence level [51]. If  Re$\,C_{d\phi}^{S,sd}=0$  instead, the KOTO constraint would be absent.

Clearly, the fact that NA62 has to exclude some kinematic regions from their analysis affects significantly the region allowed for new physics that its latest data can accommodate. In particular, when combined with the DM constraints, it leaves a modest window for our scenario, namely that $m_{\phi}$ has to be between 110 MeV and 130 MeV, as can be viewed in the left panel of Fig. 1. Nevertheless, on the upside, this means that the $\phi$ interactions in the 130-146 MeV interval remain unrestrained.

To understand why the NA62 measurement produces the exclusion region in the left panel of Fig. 1 instead of the one in the right panel, we have depicted in Fig. 2 the kinematic distributions with respect to the squared missing-mass, $q^{2}$, and charged-pion momentum, $|\textbf{{p}}_{\pi}|$, in the lab for the SM and for three different values of DM mass $m_{\phi}$, along with vertical dashed lines indicating the experimental cuts imposed by NA62. The left plot in Fig. 2 reveals that the peaking of the new-physics distributions puts them outside the signal region for mass values  $m_{\phi}\gtrsim 130$ MeV,  in which case as remarked above they would not be constrained by this measurement and cannot contribute to any observed enhancement over the SM rate. Similarly, our scenario does not contribute to this mode in SR1 for the DM masses of interest. The shape of these two distributions could be used in the future to distinguish between the SM and dark scalars. However, it turns out that within the relevant SR2 the possible dark scalar models have a kinematic $|\textbf{{p}}_{\pi}|$-distribution remarkably similar to its SM counterpart.

The pion-momentum distribution seen in Fig. 2 changes significantly when the $q^{2}$ limits of the signal region are applied, as displayed in Fig. 3. This result is surprising at first but can be easily understood because the matrix element squared for the new physics is a constant. This implies that the Dalitz plot distribution for the final state with the pion and a pair of dark scalars simply follows phase space, and all of the features seen in Fig. 2 simply reflect the kinematic limits. The NA62 signal region SR2 (which is the relevant one for us) is away from the kinematic limits leading to the result seen in Fig. 3. The immediate consequence is that the experimental efficiencies will have minimal impact on our signal as it closely resembles the SM within SR2.

We have explored the possibility that the recently measured excess of  $B^{+}\to K^{+}$+$\not{E}$  rate relative to its SM expectation originates from DM-quark interactions that can also saturate the potential new-physics window in the newly observed  $K^{+}\to\pi^{+}$+$\not{E}$.  Starting with effective operators involving quarks and the real scalar boson $\phi$ acting as the DM, we investigate the ranges of their couplings that can satisfy the preceding requirements, taking into account the constraints from the DM sector. In particular, restrictions from the relic density data and from DM direct, indirect, and collider searches turn out to restrain the $\phi$ mass to be between 110 MeV and 146 MeV. We further find that the NA62 data on  $K^{+}\to\pi^{+}$+$\not{E}$,  including the information on its signal regions, reduce this range to  $110{\rm~{}MeV}\;\mbox{\small$\lesssim$}\;m_{\phi}\;\mbox{\small$\lesssim$}\;130$ MeV.  On the upside, this implies that the $\phi$ interactions for  $130{\rm~{}MeV}\;\mbox{\small$\lesssim$}\;m_{\phi}\;\mbox{\small$\lesssim$}\;146$ MeV  remain unconstrained. Improved data from future measurements of  $B\to K$+$\not{E}$  by Belle II,  $K^{+}\to\pi^{+}$+$\not{E}$  by NA62, and  $K_{L}\to\pi^{0}$+$\not{E}$  by KOTO and from various upcoming DM searches can be expected to test our new-physics scenario more stringently. It goes without saying that more precise predictions for these decay modes in the SM and better estimation of the Migdal effect are desirable for a sharper comparison of experiment and theory.

X.-G.H. was supported by the Fundamental Research Funds for the Central Universities, by the National Natural Science Foundation of the People’s Republic of China (Nos. 12090064, 11735010, 12205063, 11985149, 12375088, and W2441004), and by MOST 109–2112-M-002–017-MY3. X.-D.M. was supported by Grant No. NSFC-12305110. J.T. and G.V. thank the Tsung-Dao Lee Institute, Shanghai Jiao Tong University, for kind hospitality and support during the preparation of this paper.