Searching for axion-like particles at future electron-positron colliders

We investigate the prospects for discovering axion-like particles (ALPs) via a light-by-light (LBL) scattering at two colliders, the future circular collider (FCC-ee) and circular electron-positron collider (CEPC). The promising sensitivities to the effective ALP-photon coupling $g_{a\gamma\gamma}$ are obtained. Our numerical results show that the FCC-ee and CEPC might be more sensitive to the ALPs with mass 2 GeV $\sim$ 10 GeV than the LHC and CLIC.


I. INTRODUCTION
At present, the "strong-CP" is still one of the theoretical problems that the standard model (SM) doesn't explain. To solve it, many well-motivated extensions of the SM have been proposed. An attractive solution was proposed by Peccei-Quinn which is called Peccei-Quinn mechanism and predicts the existence of QCD axions [1][2][3]. The axion-like particle (ALP) is a generalization of the QCD axion, which is predicted by new physics models with the breaking of global U(1) symmetry [4][5][6][7]. The ALP interactions with the SM fermions and gauge bosons arise through five-dimensional operators, and their masses can be treated independently of their couplings [8]. While their couplings to the Higgs bosons are given by dimension-6 operators and higher [9,10]. This property makes ALPs have much wider parameter space and hence generate rich phenomenology at low-and high-energy experiments.
The constraints on the couplings of ALPs to the SM fermions or bosons have been widely studied using various experimental data from particle physics, astroparticle physics and cosmology for example see Refs. [11][12][13] and their references. Generally, the bounds depend on the ALP mass range considered. For example, the electroweak gauge boson Z decaying into two or three photons at e + e − colliders and hadron colliders can generate severe constrains on the ALP parameter space [14]. The mass of ALP and its coupling to two photons have already been constrained using LEP data [15,16] via the process e + e − → γ * → aγ → 3γ [17]. The couplings of ALPs to the massive gauge bosons are constrained mainly due to loop induced processes [18][19][20]. The current limits provide valuable information for direct searching ALPs in running or future high-and low-energy experiments.
The properties of ALPs in high-energy collider experiments have been extensively studied [10,21,22]. For example, ALPs can be effectively detected at high energy colliders in a light-by-light (LBL) scattering [23][24][25][26]. Using the proton tagging technique, the LHC generally is more sensitive to the heavy ALP searched by LBL scattering than other processes.
Especially, in the mass region 0.6 TeV ∼ 2 TeV, the lowest values of the coupling of ALP with a pair of photons in the range of 0.4 TeV −1 ∼ 0.06 TeV −1 [24]. It has been shown that the CLIC with the initial unpolarized Compton backscattered (CB) photons and polarized CB photons can also be used to search ALPs [25,26]. The CLIC searches with the unpolarized photon-photon scattering obtain the bounds on ALP-photon coupling about between 1×10 −2 TeV −1 and 3×10 −4 TeV −1 for the ALP masses between 1 TeV ∼ 2.4 TeV [25], and the polarized bounds are about 1.5 times stronger than the unpolarized ones in the same mass interval [26].
It is well known that, compared to hadron colliders such as the LHC, lepton colliders have higher luminosity and more clean experimental environment. The future circular collider (FCC-ee) [27][28][29] and the circular electron-positron collider (CEPC) [30,31] are the unprecedented luminosity and energy frontier e + e − colliders, which can not only study the SM observables at unprecedented accuracy, but will be very useful to discover the evidence In this paper, we will investigate the possibility of detecting ALP via considering its single production induced by two-photon fusion at the FCC-ee and CEPC and focus on comparing the respective sensitivities to the ALP free parameters. This paper is structured as follows: In Sec. II, we briefly review the couplings of ALPs with the SM particles and show the cross sections of the LBL scattering induced by ALPs at e + e − colliders. In Secs. III and IV, we analyze the possibility of detecting ALPs at FCC-ee and CEPC by Monte Carlo simulation.
Finaly, Sec. V includes our conclusions and a simple discussion.

II. EFFECTIVE COUPLINGS OF ALP AND ITS INDUCED LBL SCATTERING
As ALPs come from the breaking of a global symmetry at high energy scale, their interactions with ordinary particles can be suitably described via an effective Lagrangian and studied in effective field theory framework [8,10,19]. At low-energies, the general dimension-5 effective Lagrangian describing the interactions of ALP with ordinary fermions and the gauge bosons is given by where X µν denotes the field strength tensor,X µν = 1 2 ε µναβ X αβ with ε 0123 = 1 and X ∈ {G, B, W }. The coupling constants g s , g and g correspond the groups SU (3) C , SU (2) L and U (1) Y , respectively. Λ is the characteristic scale of global symmetry breaking and M a is the ALP mass.
After electroweak symmetry breaking, Eq. 1 generates the aγγ, aγZ and aZZ couplings, which are related our calculation with the Wilson coefficients reading where s w = sin θ w , c w = cos θ w with θ w being the weak mixing angle, and F µν and Z µν are the field strength tensors of photon and Z boson, respectively. In the following, the coupling parameter g aγγ is defined as g aγγ = 4e 2 Cγγ Λ . The relationship between g aγγ and the ALP-photon coupling f used in Refs. [24][25][26] is gaγγ 4 = f −1 . In this paper, we consider the LBL scattering process induced by ALPs via e + e − collision to search for ALPs. The LBL scattering is characterized by the presence of a pair of photons and two forward electrons, which is shown in Fig. 1. Although this process at the CLIC has been studied in Refs. [25,26], which shows the possibility of detecting ALPs at TeV. While ALPs have the potential to be found at the GeV scale. Thus, we focus our attention on the prospects for discovering ALPs at the FCC-ee and CEPC. We use FeynRules [32] to generate the Feynman rules corresponding to the effective Lagrangian. The cross sections of the process e + e − → γγe + e − induced by ALPs at the FCC-ee and CEPC are calculated by Madgraph5-aMC@NLO [33] with the following set of basic cuts in Table I, which are shown in Fig. 2. We choose the c.m. energies of FCCee as 365 GeV and 91 GeV [27][28][29], and that of CEPC as 240 GeV and 91 GeV [30, 31], respectively. From Fig. 2, we can see that the values of the production cross sections decrease with the increase of the ALP mass M a and increase rapidly with the increase of the coupling parameter g aγγ .

III. THE POSSIBILITY OF DETECTING ALPS AT THE FCC-EE
We first study the feasibility of probing ALPs at the FCC-ee which will operate at √ s = 365 GeV with the integrated luminosity L = 1.0 ab −1 and √ s = 91 GeV with L = 150 ab −1 [28]. Compared with the LBL process, the contribution of the process e + e − → aγ * → γγe + e − induced by ALP is small. However, it can also contribute to this signal. So this process is considered as a supplement of ALP signal in the following analysis. The dominant Feynman diagrams of the SM background are shown in Fig. 3, which are mainly induced by electroweak interaction. The background also might be contaminated by the γγ initial state and ZZ-induced processes [25]. However, these possible backgrounds can safely be ignored since their contributions are estimated less than 1% [34,35].  In Table II,  GeV −1 , the value of the signal cross section is smaller than 10 fb for M a > 60 GeV, therefore we take the signal benchmark points as M a = 6, 8, 10, 50 GeV. From Table II, we can see that the background is suppressed very effectively, while the signal still has well efficiency after all cuts applied. To estimate the SS, we use the following Poisson formula [38]: where S and B respectively denote the effective cross sections of the signal and background after all cuts applied.      [15][16][17], the same final states was studied at CDF (magenta) [39] and LHC (peach) [40]. The green and light-shaded green regions respectively depict the results from searches for ALP via the LBL scattering at the LHC, ultraperipheral heavy-ion (Pb-Pb) collisions [41,42] and the CLIC [26]. The exclusion regions also include the results from the central exclusive diphoton production at the LHC (light-shaded grey) [24], and the exotic decays h → Za, h → aa and Z → γa with a decaying into a pair of charged lepton or two photons (lavender) [10]. From Fig. 6 we can see that the detectable mass ranges of the LHC [24] and CLIC [25,26] are much larger than those of the FCC-ee. This is because the higher collision energy can improve the detectability of new physics. For 8 GeV ≤ M a ≤ 300 GeV, our FCC-ee bounds on the ALP coupling with photons are stronger than those given by the LBL scattering at the LHC, while are weaker than those of the polarized LBL scattering at the CLIC. Certainly, the ALP parameter space for this sector has been excluded by other experiments. However, for 2 GeV ≤ M a ≤ 8 GeV, the sensitivity bounds of 365  Table I) on the signal and background, and show the normalized distributions of η(e ± ), θ(γγ), ∆θ(e + e − ) and p T (γγ) of the signal and background in Fig. 7.
We can see that the normalized distribution of θ(γγ) and ∆θ(e + e − ) in Fig. 7 are similar with those in Fig. 4. Therefore, for θ(γγ) and ∆θ(e + e − ), we choose the same cuts as in Sec. III. The p T (γγ) distribution shows larger shift to the left compared to Fig. 4 (e), so we do not ignore the impact on cut efficiency and impose the new cut on p T (γγ). According to In Table III, we summarize the numerical results of signal and background after imposing the above cuts at the CEPC. From Table III, we can see that the cross section of signal is reduced more than that of the background after imposing Cut-1 at √ s = 91 GeV. The signal reduces almost by a factor of 4-8 for low mass points (in the range of 2 GeV < M a < 8 GeV), while the background gets reduced by a factor of 1/2 (approximately). Therefore, we appropriately extend the limits of η(e + ) and η(e − ). Through calculation, Cut-1 is modified to 0.5 < η(e + ) < 2 and −2 < η(e − ) < −0.5. After imposing the modified Cut-1, the cross section of signal reduces by about 35%, and that of the background is reduced by 80%. The SS can be improved 3-11 times with the optimized choice of cuts (0.5 < η(e + ) < 2 and −2 < η(e − ) < −0.5) compared to the existing choice of cuts for low mass ALPs at √ s = 91 GeV. Even though the limits of η(e + ) and η(e − ) are extended, after imposing all cuts the SS at √ s = 91 GeV is still less than that at the CEPC with √ s = 240 GeV. In Fig. 8, we plot the 3σ and 5σ curves in the plane of M a − g aγγ at the CEPC with √ s = 240 GeV, L = 5.0 ab −1 and √ s = 91 GeV, L = 16 ab −1 , respectively. From Fig. 8, we can obtain that the sensitivity bounds as 1.8×10 −4 GeV −1 < g aγγ < 2.1×10 −2 GeV −1 (2.1×10 −4 GeV −1 < g aγγ < 2.4×10 −2 GeV −1 ) in the ALP mass interval 1 GeV ∼ 220 GeV with √ s = 240 GeV and 7.0×10 −4    the M a − f −1 plane. Compared with the results of the LHC [24] and CLIC [25,26], the CEPC has better potential for detecting the ALP coupling with photons at GeV scale. For 2 GeV ≤ M a ≤ 8 GeV, the sensitivity bounds of the 240 GeV CEPC with L = 5.0 ab −1 are in the range of 0.05 TeV −1 ∼ 0.4 TeV −1 , which are even stronger than those of the FCC-ee slightly. Thus, the CEPC might be the most sensitive to the ALPs with mass 2 GeV ∼ 8 GeV than other colliders.

V. CONCLUSIONS AND DISCUSSIONS
As pseudo-Goldstone bosons, ALPs are one of the most potential particles to actually exist. There will be great expectations for discovering them in future electron-positron colliders. In this paper, we have investigated the observability of the ALP diphoton signal through the process e + e − → γγe + e − at the FCC-ee and CEPC. Our results show that the detectable mass ranges of the FCC-ee and CEPC are much smaller than those of the LHC and CLIC. For 8 GeV ≤ M a ≤ 300 GeV, our FCC-ee and CEPC bounds on the ALP coupling with photons are stronger than those given by the LBL scattering at the LHC,