1. Introduction
Axion-like particles (ALPs) [1] are generically classified as the pseudo Nambu–Goldstone bosons (PNGBs) associated with the spontaneous breaking of global chiral symmetry that arise in many well-motivated models beyond the standard model (SM). ALPs might address several open problems in particle physics, such as the evidence of dark matter [2–4] as well as the flavor [5] and the weak-scale hierarchy [6] problems. Unlike quantum chromodynamics (QCD) axions, introduced to solve the strong charge parity (CP) problem via the Peccei–Quinn mechanism [7, 8], ALPs are unrelated to the solution of the strong CP problem and their mass and coupling are independent to the QCD anomaly. The mass of an ALP may come from the non-perturbative instanton effect in a strongly interacting hidden sector like the QCD axion or from explicit symmetry breaking. Consequently, a generic ALP may interact with various particles and have masses in a wide range [1], whose couplings are not tightly constrained by either cosmological or astrophysical bounds [9–11]. In general, ALPs do not have any specific relation between their masses and decay constants. Therefore, ALPs can generate rich phenomenology in low-energy and high-energy experiments.
There are a wide range of new physics models that can predict ALPs that can be produced from meson decays. Strict constraints on ALP couplings with SM particles can be derived using the corresponding experimental data. For example, the couplings of ALP to the SM particles can be constrained by the decay K → πa via the recent data from MicroBooNE [12] and NA62 experiments [13]. In principle, ALPs are produced from mesonic rare decays in two different scenarios. One scenario is induced by flavor-violation (FV) ALP couplings with quarks at tree-level, and the other is induced by the effective ALP couplings that appear due to the renormalization group effects. In this note, we will systematically consider the decays $M\to M^{\prime} a$, with ${M}^{(^{\prime} )}$ being a pseudoscalar meson (B, D, K or π) in the second scenario. In our calculation, the ALP is assumed to be an invisible particle, the latest experimental results for the decays $M\to M^{\prime} \nu \bar{\nu }$ are used, and the updated bounds on the ALP–quark couplings for the ALP mass Ma in the sub-GeV range are obtained.
In the next section, we give our calculation framework from the most general effective ALP Lagrangian in the dimension-5 order. By comparing our numerical results for the decays $M\to M^{\prime} a$ (with ALP escaping the detector or decaying into an ‘invisible’ sector) to the corresponding latest data of the decays $M\to M^{\prime} \nu \bar{\nu }$, the constraints on the ALP–quark couplings are given in section 3 . A discussion and conclusions are presented in section 4 .
2. Theoretical framework
It is well known that the ALP a regarded as a CP-odd boson is a singlet under the SM gauge group. The effective interactions of an ALP with the SM particles can be described by the effective Lagrangian. The general effective Lagrangian, including operators of dimension up to five, is given by [14, 15]
$\begin{eqnarray}\begin{array}{l}{{ \mathcal L }}_{{\rm{eff}}}^{a}=\displaystyle \frac{1}{2}\left[{\left({\partial }_{\mu }a\right)}^{2}-{M}_{a}^{2}{a}^{2}\right]\\ \,-\,\displaystyle \frac{{\partial }_{\mu }a}{2{F}_{a}}\displaystyle \sum _{{ij}}{\bar{f}}_{i}{\gamma }^{\mu }\left({V}_{{ij}}^{f}-{a}_{{ij}}^{f}{\gamma }_{5}\right){f}_{j}\\ \,-\,\displaystyle \frac{{\partial }_{\mu }a}{{F}_{a}}\displaystyle \sum _{\chi }{C}_{\chi }{\chi }_{\mu \nu }^{a}{\tilde{\chi }}^{\mu \nu \alpha }.\end{array}\end{eqnarray}$
Here, Ma denotes the ALP mass, f stands for three generations of the SM fermions, i and j are flavor indices and Fa is the ALP decay constant. Also, ${\chi }_{\mu \nu }^{a}$ indicates any SM gauge boson field strength with ${\tilde{\chi }}^{\mu \nu a}\equiv \tfrac{1}{2}{\chi }_{\alpha \beta }^{a}{\varepsilon }^{\alpha \beta \mu \nu }$ and Cχ being a coupling coefficient, while Vijf and aijf stand for the vector and axial-vector ALP couplings to SM fermions, respectively.As carried out in [16–19], the low-energy CP and flavor-conserving (FC) effective Lagrangian for ALP–fermion interactions can be written as
$\begin{eqnarray}{{ \mathcal L }}_{{\rm{fermion}}}^{a}=-\displaystyle \frac{{\partial }_{\mu }a}{2{F}_{a}}{g}_{i}{\bar{f}}_{i}{\gamma }^{\mu }{\gamma }^{5}{f}_{i}={\rm{i}}\displaystyle \frac{{M}_{i}}{{F}_{a}}{g}_{i}a{\bar{f}}_{i}{\gamma }^{5}{f}_{i}.\end{eqnarray}$
The index i includes all of the SM fermions, with gi being real ALP–fermion couplings, and massless neutrinos are implied. According to the idea of the minimal FV hypothesis [20], all FV ALP couplings with SM fermions are loop induced and CKM is suppressed [16].In general, ALPs with mass in few GeV can be produced from the decays $M\to M^{\prime} a$, which are induced at tree-level and one-loop level. It has been shown that the contributions of the one-loop penguin diagram are much larger than those of the tree-level diagram when M and $M^{\prime} $ are mesons B, D, K or π [16, 21, 22]. Therefore, in this note, we only consider the decays $M\to M^{\prime} a$ mediated at the one-loop level. Then, the branching ratio ${Br}(M\to M^{\prime} a)$ can be written as [16–19, 21–25]4 ), which are suppressed by at least an extra factor ${M}_{f}^{2}/{M}_{W}^{2}$ with respect to the penguin contributions.
$\begin{eqnarray}\begin{array}{l}{Br}(M\to M^{\prime} a)=\displaystyle \frac{{M}_{M}^{3}{\left|{\left({V}_{{ij}}\right)}_{{MM}^{\prime} }\right|}^{2}}{64\pi {F}_{a}^{2}{{\rm{\Gamma }}}_{M}}\\ \,\times \,{\left(1-\displaystyle \frac{{M}_{M^{\prime} }^{2}}{{M}_{M}^{2}}\right)}^{2}{\left[{f}_{0}^{{MM}^{\prime} }({M}_{a}^{2})\right]}^{2}{\lambda }_{{MM}^{\prime} }^{1/2}\end{array}\end{eqnarray}$
with $\begin{eqnarray}{\left({V}_{{ij}}\right)}_{{MM}^{\prime} }=\displaystyle \frac{{\alpha }_{e}}{4\pi {S}_{W}^{2}}\displaystyle \sum _{f}{V}_{{fi}}{V}_{{fj}}^{* }\,{g}_{f}{\chi }_{f}\mathrm{ln}\displaystyle \frac{{F}_{a}^{2}}{{M}_{f}^{2}}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{\lambda }_{{MM}^{\prime} }=\left[1-\displaystyle \frac{{\left({M}_{M^{\prime} }+{M}_{a}\right)}^{2}}{{M}_{M}^{2}}\right]\\ \,\times \,\left[1-\displaystyle \frac{{\left({M}_{M^{\prime} }-{M}_{a}\right)}^{2}}{{M}_{M}^{2}}\right].\end{array}\end{eqnarray}$
Here, ${M}_{{M}^{(^{\prime} )}}$ and ΓM respectively denote the mass and total decay width of the pseudoscalar meson ${M}^{(^{\prime} )}$, the sum of quark f for all quarks that run in the loop and f indicates all the quarks in the circle diagram, ${\chi }_{f}={M}_{f}^{2}/{M}_{W}^{2}$, ${f}_{0}^{{MM}^{\prime} }\left({m}_{a}^{2}\right)$ stands for the $M\to M^{\prime} $ scalar form factor from the hadronic matrix element. The values of the CKM matrices Vijf, the weak mixing angle θW and the fine structure constant αe are taken from the particle data group (PDG) [26]. In this paper, we focus our attention on the constraints on the ALP–fermion coupling gf from the relevant experimental data; thus we have neglected the contributions for the ALP emitted from the initial/final quarks in equation (3. The constraints on the FC ALP–quark couplings from the decays $M \rightarrow M^{\prime} \nu \bar{\nu }$
The semi-leptonic decays $M\to M^{\prime} \nu \bar{\nu }$, with M and $M^{\prime} $ being B, D, K or π mesons, are the cleanest flavor-changing neutral current processes because of the absence of photonic penguin contributions and strong suppression of light quark contributions, which play an important role when seeking the underlying mechanism of flavor mixing and CP violation. They are frequently considered as a useful tool for probing new physics beyond the SM. Thus, many experimental programs are working for these decays, such as B-factories, including Belle and BaBar, CHARM, BESIII, MicroBooNE and NA62 experiments. The existing experimental measured values for the branching ratios ${Br}(M\to M^{\prime} \nu \bar{\nu })$ are summarized in table 1, and the corresponding SM predictions are also shown for comparison.
Table 1. The experimental measurement and SM prediction values for the branching ratios ${Br}(M\to M^{\prime} \nu \bar{\nu })$. |
| Br | SM | Exp |
|---|---|---|
| ${B}^{0}\to {K}^{0}\nu \bar{\nu }$ | (3.91 ± 0.52) × 10−6 [27] | ≤2.6 × 10−5 [26] |
| ${B}^{+}\to {K}^{+}\nu \bar{\nu }$ | (4.23 ± 0.56) × 10−6 [27] | (2.3 ± 0.7) × 10−5 [28] |
| ${B}^{0}\to {\pi }^{0}\nu \bar{\nu }$ | 1.2 × 10−7 [29] | ≤9 × 10−6 [26] |
| ${B}^{+}\to {\pi }^{+}\nu \bar{\nu }$ | (2.49 ± 0.25) × 10−9 [29] | ≤1.3 × 10−5 [30] |
| ${K}_{L}^{0}\to {\pi }^{0}\nu \bar{\nu }$ | (2.94 ± 0.15) × 10−11 [31] | ≤2.0 × 10−9 [32] |
| ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ | (8.6 ± 0.42) × 10−11 [33] | (1.14 ± 0.4) × 10−10 [26] |
From equation (3 ) one can see that the branching ratios ${Br}(M\to M^{\prime} a)$ depend not only on the ALP free parameters gf and Ma, but also on the relevant meson parameters ${M}_{{M}^{(^{\prime} )}}$, ${{\rm{\Gamma }}}_{{M}^{(^{\prime} )}}$ and ${f}_{0}^{{MM}^{\prime} }\left({M}_{a}^{2}\right)$ . The masses and total decay widths of the pseudoscalar mesons (B, D, K or π) are given in table 2.
Table 2. The masses and total decay widths of the pseudoscalar mesons given by PDG [26]. |
| Meson | Mass (GeV) | Γtotal (GeV) |
|---|---|---|
| B+(B0) | 5.27934 (5.27966) | 4.01 × 10−13(4.33 × 10−13) |
| D+(D0) | 1.86966 (1.86484) | 6.39 × 10−13(1.6 × 10−12) |
| ${K}^{+}({K}_{L}^{0})$ | 0.49368 (0.49761) | 5.31 × 10−17(1.29 × 10−17) |
| π+(π0) | 0.13957 (0.13498) | 2.53 × 10−17(7.8 × 10−9) |
The form factors ${f}_{0}^{{MM}^{\prime} }\left({M}_{a}^{2}\right)$ parameterize the relevant hadronic matrix elements, which can be obtained from lattice-QCD (LQCD) calculations and are transition specific. For the B → π transition, we use the combined results made by FLAG [34], and for B → K we use the results given in [35, 36]. For D → π, the form factor ${f}_{0}^{D\pi }\left({M}_{a}^{2}\right)$ is taken from the LQCD results of [37], while ${f}_{0}^{K\pi }\left({q}^{2}\right)$ is close to unity for $0\lt {q}^{2}\leqslant {\left({M}_{K}-{M}_{\pi }\right)}^{2}$ [38].
If we assume that the ALP lifetime is sufficiently long to escape the detector or the ALP is mainly decaying into ‘invisible’ particles, then its signature is missing energy/momentum, just as for neutrinos. Based on this hypothesis, we can use equations (3 )–(5 ) to calculate the contributions of ALPs to the decays $M\to M^{\prime} \nu \bar{\nu }$ and compare our numerical results with the relevant experimental data given in table 1. Then, the constraints on the FC ALP–quark couplings can be obtained, as shown in figure 1 for Fa = 1 TeV. The penguin contributions do not dominate anymore over tree-level contributions for the decay D → Ma [16]: equation (3) cannot be applied to us. Therefore, we do not show the results for $D\to \pi \left(K\right)a$. For the decays ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$, ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$ and ${K}^{+}/{K}_{L}^{0}\to {\pi }^{+,0}\nu \bar{\nu }$, the Cabibbo–Kobayashi–Waskawa (CKM) matrices ${\left({V}_{{ij}}\right)}_{{MM}^{\prime} }\,={\left({V}_{{bs}}\right)}_{{B}^{+,0}{K}^{+,0}}$, ${\left({V}_{{bd}}\right)}_{{B}^{+,0}{\pi }^{+,0}}$ and ${\left({V}_{{sd}}\right)}_{{K}^{+,0}{\pi }^{+,0}}$, respectively. While for all of these decay processes, the upper-type quarks (u, c, and t) of quark f of circle diagram is the dominant. Certainly, the contributions mainly come from the top quark. For convenience, we define a coupling factor gu for up-type quarks. The differences in each upper-type quark come from its mass, as shown in equation (2 ). From figure 1 we can see the most severe constraint on the FC ALP coupling factor gu given by the decay ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ for Ma ≤ 0.35 GeV, which demands 5.57 × 10−5 ≤ gu ≤ 3.58 × 10−4 for Fa = 1 TeV and 0.05 GeV ≤ Ma ≤ 0.35 GeV. While the constraints on gu come from the decays ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$, which are 5.06 × 10−3 ≤ gu ≤ 5.33 × 10−3 and 7.30 × 10−3 ≤ gu ≤ 7.53 × 10−3 for 0.45 GeV ≤ Ma ≤ 4.5 GeV, respectively. For the decays ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$, they are 3.16 × 10−2 ≤ gu ≤ 2.0 × 10−2 and 2.71 × 10−2 ≤ gu ≤ 1.69 × 10−2 for 0.45 GeV ≤ Ma ≤ 4.5 GeV, respectively.
Figure 1. Upper limits on the coupling parameter gu (u being up-type quarks) from the pseudoscalar meson decay of ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$ (a), ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$ (b) and ${K}^{+}/{K}_{L}^{0}\to {\pi }^{+,0}\nu \bar{\nu }$ (c) as a function of the mass parameter Ma for Fa = 1 TeV. |
4. Conclusions
The absence of evidence for new physics at the TeV scale so far has led to renewed attention to lighter new particles. In this work, we have considered the production of light ALPs from the decays $M\to M^{\prime} a$ for the ALP mass in the sub-GeV range. After assuming the ALP to be an invisible particle, we compare our numerical results with the corresponding experimental measurements for the decays ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$, ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$ and ${K}^{+}/{K}_{L}^{0}\to {\pi }^{+,0}\nu \bar{\nu }$, and the upper limits on the FC ALP–quark coupling parameter gu are obtained. We find that the most severe constraint on the coupling gu comes from the decay ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ for 0.05 GeV ≤ Ma ≤ 0.35 GeV, while the decays ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$ and ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$ can also generate significant constraints on gu. We hope that our results will be helpful for the discovery of ALPs in future experiments.