Direct detections of the Axion-like particle Revisited

The strong CP problem finds a solution via the Peccei-Quinn mechanism Peccei and Quinn (1977a, b), which gives rise to a fascinating prediction: the existence of a new particle called the axion, as outlined in Refs Weinberg (1978); Wilczek (1978). These ultra-light, weakly interacting particles, such as axions or axion-like particles (ALPs), have also gained prominence as potential dark matter (DM) candidates within the diverse theories of DM Kim and Carosi (2010); Feng (2010); Tam and Yang (2011); Marsh (2016). Since they can couple to the Standard Model (SM) particles such as photon and electron, extensive experimental works Spector (2019); Anastassopoulos et al. (2017); Akerib et al. (2017); Aprile et al. (2022) have been devoted to searching for such particles through the axion-photon coupling or the axion-electron coupling, and stringent limits on these couplings have already been imposed.

The Sun is an excellent source of axions, offering an easily obtainable flux of axions generated in the center of the Sun via various processes Redondo (2013); Derbin et al. (2012). However, the endeavor to detect axions or ALPs is an arduous undertaking, primarily due to their extremely weak interactions with SM particles. Fortunately, Sikivie Sikivie (1983) made a breakthrough by showing that these elusive ALPs may be detected when they convert into photons in the magnetic fields in 1983. This discovery presented a viable method for the detection of ALPs. Currently, many experiments are carried out searching for ALPs, of which the constraint given by the CERN Axion Solar Telescope (CAST) is $g_{a\gamma}<6.6\times 10^{-11}(\rm GeV^{-1})$ Andriamonje et al. (2007); Anastassopoulos et al. (2017), with $g_{a\gamma}$ the coupling of the ALP to diphoton. Still, more stringent bounds arise from astrophysical observations, of which the stellar evolution, determined using the R2 parameter, the ratio of stellar populations on the asymptotic giant branch to Horizontal Branch stars in globular clusters, gives $g_{a\gamma}<4.7\times 10^{-11}(\rm GeV^{-1})$ Dolan et al. (2022).

In addition to searching for ALPs with a Helioscope, there is currently a significant focus on utilizing DM direct detection experiments based on underground laboratoriesFu et al. (2017); Knapen et al. (2017); Aad et al. (2021); Agnes et al. (2023) to either directly detect ALPs or set novel constraints on ALP couplings in projected direct detection experiments. Either solar axions or heavy axion DM with large kinetic energy Arisaka et al. (2013); Irastorza and Redondo (2018) can be detected in these experiments. Assuming that ALPs couple to neutrinos, the supernovae neutrinos can also boost the galactic DM ALPs via elastic scattering, resulting in relativistic ALPs Carenza and De la Torre Luque (2023). In this paper, we will consider the direct detection signals of the ALP arising from the following three sources: (1) supernova neutrino-boosted ALPs ($v\sim c$); (2) solar axion; (3) galactic DM ALP ($v\sim 10^{-3}c$) with large kinetic energy.

In this paper, we focus on the scattering process $a+e\to e+\gamma$, where $a$ is an ALP, $e$ and $\gamma$ are electron and photon, respectively. This process, if mediated by the dimension-five operator $g_{a\gamma}aF_{\mu\nu}\tilde{F}_{\mu\nu}$, is called inverse Primakoff (IP) scattering Buchmuller and Hoogeveen (1990); Creswick et al. (1998). Alternatively, the scattering mediated by the dimension-four operator $g_{ae}a\bar{e}i\gamma^{5}e$ is called the inverse Compton (IC) scattering Avignone et al. (1988); Chanda et al. (1988); Brodsky et al. (1986). As shown in Fig. 1 (a), (b) and (c), the coherent enhancement factor for the inverse Primakoff scattering is $Z^{2}$, with $Z$ the atomic number of the target nucleus, which is similar to the Coulomb scattering process, while the coherent enhancement factor for the inverse Compton scattering is $Z$. Usually these two processes are separately considered in the literature by setting either $g_{a\gamma}$ or $g_{ae}$ to be zero. However, the axion-photon and the axion-electron couplings may exist simultaneously in a real case, so one needs to account for the contribution to the scattering induced by the interference of two interactions, which is done in this paper. Our calculation shows that this contribution can be dominant for a large axion energy $E_{a}$, as can be seen in Fig. 2. The complete calculation helps to improve constraints on ALP couplings in projected DM direct detection experiments such as XENON, PandaX, etc.

Apart from the DM direct detection experiments mentioned above, an increasing number of researchers are interested in probing axions or ALPs using neutrino detectors, such as Super-KLi and Zhang (2022), JUNOLucente et al. (2022), and othersDent et al. (2020a); Bhusal et al. (2021). The strength of neutrino experiments lies in their substantial fiducial mass ($\sim$20kton), which helps to improve the exclusion limits. However, their drawback is that they can only search for ALPs with high energies due to the threshold of the detector. In this paper, we focus on the detectability of the supernova neutrino-boosted ALP, whose energy can reach MeV, using the JUNO detector. Projected constraints on the ALP coupling at the $90\%$ confidence level (C.L.) is presented in the Fig. 7.

The remaining of the paper is organized as follows: in Section II, we calculate the scattering cross section $a+e\to e+\gamma$ induced the inverse Primakoff process, the inverse Compton process and the interference between these two terms. We discuss three fluxes of incoming axions: supernovae neutrinos boosted axions, solar axions, and DM axions in Section III. Then, we study the limitations of the axion couplings $g_{a\gamma}$ and $g_{ae}$ in DM direct detection experiments. The constraints of JUNO are given in Section IV. Finally, summary remarks are given in the Section V.

ALPs have gained attention in particle physics as they may play an important role in addressing various problems in terrestrial experiments and astrophysical observations, including DM and the strong CP problem. ALPs are assumed to be super-light and weakly interacting particles, and are predicted in certain extensions to the SM with spontaneous breaking global symmetries. Despite the possibility for ALPs to interact with a wide range of particles in various forms, most ALP models actually propose interactions specifically with photons and electrons with Lagrangian of the following form:

where $a$ stands for the ALP. $F^{\mu\nu}$ is the electromagnetic field strength and its dual $\tilde{F}_{\mu\nu}=\dfrac{1}{2}\epsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}$. $g_{a\gamma}$ and $g_{ae}$ denote the axion-photon and axion-electron couplings, respectively. For the detection of ALPs, they can be probed via oscillations of ALP into the photon in a Helioscope with constraint put on the coupling $g_{a\gamma}$. They can also be detected by the Axio-electric effect by measuring the rate of atomic ionization induced by the absorption of ALP of the energies up to 100 keV Derevianko et al. (2010). Alternatively, they can be detected in direct detection experiments via the inverse Compton scattering Donnelly et al. (1978); Bellini et al. (2008); Dent et al. (2020a) or the inverse Primakoff process Avignone et al. (1998); Abe et al. (2021); Buchmuller and Hoogeveen (1990); Dent et al. (2020b); Gao et al. (2020) with the photon as the signal. In this paper, we focus on the photon signal of ALPs in underground detectors, so we pay special attention to the inverse Compton scattering and the inverse Primakoff process, as well as their interference effect assuming $g_{ae}$ and $g_{a\gamma}$ exist simultaneously.

Many experiments employ the inverse Primakoff scattering process of the ALP with atoms as the detection channel, where the incoming ALPs are converted into photons in the electromagnetic field of the atom. It utilizes a screened Coulomb potential to portray the electrostatic field in the target atom. This potential takes the form of the Yukawa potential, with the screening length denoted as $r_{0}$. For the inverse Compton scattering process as shown in the Fig. 1 (b) and (c), the collision of the ALP with the electron in the detector results in a photon in the final state. It should be mentioned that there is IP-like scattering process that also contribute to the $a+e\to e+\gamma$ scattering, as shown in the Fig. 1 (a), the interference of which with the traditional inverse Compton process hasn’t been explicitly calculated.

The differential scattering cross section of the IP process is directly governed by the parameters $g_{a\gamma}$ and $m_{a}$, which determine the strength and characteristics of the interaction. The cross section of this process is obtained as

where $\alpha=e^{2}/4\pi$ is the fine structure constant and $m_{e,N}$ is the electron or nucleus mass. For the incoming ALP of the mass $m_{a}$, its energy and momentum are $E_{a}$ and $\bm{p_{a}}$, respectively, $p_{a}=|\bm{p_{a}}|$. The energy and momentum of the outgoing photon are $E_{\gamma}$ and $\bm{p_{\gamma}}$. $\theta$ is the scattering angle and $\bm{q}=\bm{p_{\gamma}}-\bm{p_{a}}$ is the momentum transfer. An alternative method utilized in this paper to approximate the form factor $F_{a}(\bm{q}^{2})$ involves assuming a screened Coulomb potential arising from the electrostatic field, resulting in $F_{a}(\bm{q}^{2})=\frac{Z{p}_{a}^{2}}{\bm{q}^{2}+r^{-2}_{0}}$ Avignone et al. (1998); Abe et al. (2021). According to the energy conservation, one has

Taking Eq.(3) into Eq.(2) and then integrate over the solid angle, we can obtain $\sigma^{\rm{IP}}(g_{a\gamma},E_{a},m_{a})$.

In the traditional calculation of the IP process, the nucleus recoil is neglected resulting in $E_{\gamma}=E_{a}$. Furthermore, if we ignore the mass of the axion, the transfer momentum can be simplified as $|\bm{q}|=2p_{a}\sin{(\theta/2)}$ and the factor $(E_{a}+m_{e,N}-p_{a}\cos{\theta})$ can be approximated as $m_{e,N}$. In this way, our result in Eq. (2) can be simplified to the differential cross section which appear in most of the literature Avignone et al. (1998); Abe et al. (2021); Buchmuller and Hoogeveen (1990); Dent et al. (2020b); Gao et al. (2020),

The differential cross section for the inverse Compton scattering is given as Donnelly et al. (1978); Bellini et al. (2008); Dent et al. (2020a)

which is enhanced by the atomic number $Z$.

Most importantly, if there are axion-photon and axion-electron couplings simultaneously (i.e., both $g_{a\gamma}$ and $g_{ae}$ are non-zero), the total squared matrix elements of the combined process exhibit interference between the traditional inverse Compton scattering terms given in Fig. 1 (b) and (c) and the IP-like term given in Fig. 1 (a). To assess the contributions of the interference terms, we define the cross section arising from the interference terms (IT) as $\sigma^{\rm IT}$, which can be written as

It should be mentioned that $\sigma^{\rm IT}$ is only enhanced by the factor $Z$, which is because we only account the interference of the inverse Compton scattering and the IP-like process induced by the Coulomb potential of the single electron.

Finally, the total cross section is $\sigma^{\rm Tot}=\sigma^{\rm{IP}}+\sigma^{\rm{IC}}+\sigma^{\rm{IT}}$. We explore the contribution of each term, as depicted in Fig. 2, varies with the ALP energy ($E_{a}$). The depicted figure represents a scenario with parameters set as $g_{a\gamma}=10^{-10}~{}{(\rm{GeV}^{-1})}$, $g_{ae}=10^{-11}$, $Z=54~{}({\rm Xe})$, $r_{0}=2.45\mathring{{\rm{A}}}~{}({\rm LXe})$, $m_{a}=10^{-9}~{}{\rm{keV}}$. If we alter the coupling constants as well as the target material, the contributions of each process will undergo changes.

In this section, we discuss the photon signal of ALP in DM direct detection experiments by accounting the contribution of the interference given in the last section. We will first discuss the ALP flux used in the calculation, then we show constraints of current and projected direct detection experiments on ALP couplings.

The next-generation multi-purpose Jiangmen Underground Neutrino Observatory (JUNO) An et al. (2016); Djurcic et al. (2015); Abusleme et al. (2021a, b, 2022) is a state-of-the-art neutrino experiment primarily designed to investigate the properties of neutrinos, but it also has the potential to contribute to the search for new physics beyond the Standard Model, such as proton decay An et al. (2016), hidden sector particles D’Eramo et al. (2023) and axions Lucente et al. (2022), etc. In this paper, we employ the JUNO detection experiment to set boundaries on the coupling constants of the axion. Thanks to its extensive exposure, we can significantly restrict the allowable values of the coupling constants.

One of the main components of the JUNO is the central detector, which is a massive, spherical Acrylic sphere filled with a high-purity liquid scintillator(LS). The detection medium is a linear alkylbenzene(LAB) composed of 19 carbon atoms $\rm C_{19}H_{32}$, which has excellent transparency, high flash point, low chemical reactivity and good light yield. The LS also doped with 3 g/L of 2,5-diphenyloxazole(PPO) and 15 mg/L of pbis-(o-methylstyryl)-benzene(bis-MSB). The density of the LS is 0.859g/ml, with a total of 20 ktons in a spherical container with a radius of 17.7 meters. The central detector is submerged in a cylindrical pool to protect it from the radioactivity of the surrounding rock. On the top of the water pool, muon tracker will be installed. We refer the reader to Refs. An et al. (2016) for more details about the JUNO.

Due to the threshold of the detector, both the solar ALP flux and the DM ALP flux are almost undetectable in the JUNO, leaving the supernova neutrino boosted ALP as the only possible way out. The axion fluxes have been given in Section III Part A. Similar to the discussion in in the Section III Part B, we still use $\sigma^{\rm Tot}=\left(\sigma^{\rm{IP}}+\sigma^{\rm{IC}}+\sigma^{\rm{IT}}\right)$ to detect axions, and the expected number of events per unit time is

where $N_{e}\simeq 5.5\times 10^{33}$ is the number of electrons in the 16.2 kton fiducial volume. $\mathcal{R}$ and $\epsilon$ represent the detector energy resolution and the detector efficiency, respectively. The following discussion assumes that $\epsilon=1$ and over the energy threshold. Here, we use the likelihood function in Ref. Abusleme et al. (2022) to conduct numerical analysis. Ref. D’Eramo et al. (2023) shows the total number of events($S$) expected to be detected over a ten-year period, $S=97$, corresponding to a $90\%$ C.L. sensitivity. Therefore, we use $S_{\rm limit}=97$ counts for 10 years data. The number of events used to constrain the upper limit on the axion coupling constants satisfy $S=N_{\rm event}\cdot T\leq S_{\rm limit}$, and $T=10$ years. In this way, the numerical results we obtained are shown in the Fig.7.

We choose to forecast the sensitivity at the $90\%$ C.L. to make a direct comparison with constraints of DM direct detection experiments. Our result shows that the exclusion limits of the JUNO experiment are enhanced by approximately two orders of magnitude compared to direct detection experiments, owing to the substantial fiducial volume of JUNO. It is worth noting that even in the case of the next-generation PandaX-30T experiment, JUNO is likely to provide constraints on ALP couplings that are approximately one order of magnitude tighter. The most stringent constraints on coupling constants have been reached $g_{a\gamma}<7\times 10^{-10}\rm{GeV^{-1}}$ and $g_{ae}<7.5\times 10^{-13}$, which are better than constraints from astrophysical observations. It should be mentioned that these are very conservative limits, as we have considered small values for both $g_{ae}$(left-panel) and $g_{a\gamma}$(right-panel) in the Fig.7.

The direct detection of ALP is an important issue as ALP itself serves a promising DM candidate and may solve the strong CP problem. In addition to cavity experiments, which measure $g_{a\gamma}$ via oscillations of ALP into photon in electromagnetic fields, direct detection experiments may also detect ALP via the inverse compton process or inverse Primakoff process. In this paper, we pay special attention to the calculation of the process $a+e\to e+\gamma$, and show the interference between the inverse primakoff and inverse Compton amplitudes. We have shown constraints of projected direct detection experiments on couplings $g_{ae}$ and $g_{a\gamma}$. We further studied the constraints of the JUNO Cherenkov detector on these couplings. This study provides theoretical support for the detections of ALP.