Heavy QCD axion in $b\to s$ transition: Enhanced limits and projections

We study a “heavy” QCD axion whose coupling to the standard model is dominated by $aG\tilde{G}$ but with $m_a\gg m_{\pi }{f}_{\pi }/f_a$. This is well motivated as it can solve the strong $CP$ problem while evading the axion quality problem. It also poses interesting challenges for its experimental search due to its suppressed couplings to photons and leptons. Such axion with mass around a GeV is kinematically inaccessible or poorly constrained by most experimental probes except $B$-factories. We study $B\to Ka$ transitions as a powerful probe of the heavy QCD axion by performing necessary 2-loop calculations for the first time, together with some improvement on the existing analysis strategy. We find some of the existing limits are enhanced by at least an order of magnitude. We also demonstrate that the bounds are robust against unknown UV physics. For forthcoming data sets of the Belle II experiment, we provide a projection that $f_a$ of a few TeV is within its future reach, which is relevant to the quality problem.

Figure 1

Leading 1-particle-irreducible diagrams for $b\to sa$ from the Lagrangian (1).

Figure 2

The diagram that generates the $C_{qq}$ term of Eq. (2).

Figure 3

$C_1$, $C_2$, and $C_3$ refers to the first, second, and third contribution in $C_W$ respectively, for different UV scales [see Eq. (7)].

Figure 4

We portray the constraints from different $B$-decay measurements in the $m_a-f_a$ plane. Three curves are drawn for each constraint corresponding to different initial conditions [see Eq. (3)], i.e., the strongest $(A=+3,B=-3)$, weakest $(A=-3,B=+3)$, and central constraints $(A=B=0)$. We choose the UV scale ${\mathrm{\Lambda }}_{\mathrm{UV}}$ to be 1 and 10 TeV for the top and the bottom plot, respectively. See main texts about systematic uncertainties from the form factor of Eq. (10) and the data-driven calculation of the axion branching fractions. The grey shaded regions comprise bounds from [13, 15, 43, 47, 52, 57, 58, 59]. For $B\to Ka$, we use [72] for inclusive analysis and [73, 74, 75] for exclusive channels $a\to 3\pi ,\eta \pi \pi ,KK\pi ,\varphi \varphi$. For the projection at Belle II (dashed lines), $5\times {10}^{10}$ $\overline{B}B$ pair is assumed, and the band shows the dependence on the different initial conditions. The right vertical axis is labeled using the notation of Ref. [43] for comparison.

Figure 5

In this figure, the dark red colored regions are obtained with our two loop amplitude. This shows a factor of 5 or 6 enhancement from the axion production rate compared to the previous estimation obtained in [43] (shown by the dashed lines). Both of these results use single bin analysis as mentioned in the text. Additional enhancement of factor 2, shown by the light red region, comes from our multiple bin analysis.