Single-Point Search for eV-Scale Axion-like Particles with Variable-Angle Three-Beam-Stimulated Resonant Photon Collider

Abstract

We report a laboratory search for axion-like particles (ALPs) in the eV mass range using a variable-angle three-beam-stimulated resonant photon collider. The scheme independently focuses and collides three laser beams, providing a cosmology- and astrophysics-independent test. By varying the angles of incidence, the center-of-mass energy can be scanned continuously across the eV range. In this work, we operated the collider in a vacuum chamber at a large-angle configuration, verified the spacetime overlap of the three short pulses, and performed a first search centered at $m_a\simeq 2.27\mathrm{eV}$. No excess was observed. Thus, we set a $95\%$ C.L. upper limit on the pseudoscalar two-photon coupling, with a minimum sensitivity of $g/M\simeq 4.2\times {10}^{-10}{\mathrm{GeV}}^{-1}$ at $m_a=2.27\mathrm{eV}$. This provides the first model-independent upper limit on the coupling that reaches the KSVZ benchmark in the eV regime and demonstrates the feasibility of eV-scale mass scans in the near future.

1. Introduction

The existence of dark matter is indispensable for understanding the formation and evolution of cosmic structures. Measurements of the cosmic microwave background indicate that dark matter accounts for approximately ${\mathrm{\Omega }}_{\mathrm{DM}}$ ∼ 0.27 of the total energy density of the Universe [1], yet its fundamental nature remains unknown.

In particle physics, the axion [2,3] and axion-like particles (ALPs) are well-motivated dark matter candidates, arising as pseudo Nambu–Goldstone bosons of spontaneously broken global symmetries [4,5,6]. In the ALP miracle scenario, the portion of the inflaton condensate that survives the post-inflationary reheating phase can account for the present dark matter abundance, predicting an ALP (inflaton) mass in the range of 0.01∼7.7 eV [7,8]. On the other hand, for cold–hot dark matter scenarios, it is discussed that a cold population of light bosons can be produced directly from the primordial thermal bath. Bose enhancement rapidly ‘bursts’ low-momentum modes, generating a light boson around the eV mass scale [9,10].

Also, phenomenologically there are indications favoring eV-scale ALPs. Recent measurements of the cosmic optical and infrared background (COB/CIB) have suggested possible excesses of diffuse emission in some bands. One intriguing interpretation attributes this excesses to two-photon decays of ALPs [11,12,13]. Independently, measurements of the optical depth of $\gamma$-rays from blazars show slightly stronger attenuation than that predicted by galaxy-count-based EBL models. Modeling the residual optical depth as photons from ALP two-photon decays favors eV-scale ALPs [14]. In the eV mass region, several indirect-detection searches have been carried out [15,16,17,18]. Among them, the WINERED experiment used high-resolution spectroscopy to probe the $1.8$–$2.7\mathrm{eV}$ window, setting stringent limits, and while the observing conditions were not ideal, several local $>5\sigma$ features were noted [18].

In this paper, as a laboratory-based search in the eV mass scale, we present the result of the search based on stimulated resonant photon collisions (SRPCs) [19,20,21,22,23] using three laser beams. Figure 1 shows a schematic view of the three-beam-stimulated resonant photon collider (^tSRPC). This method enables model-independent testing, free from astrophysical, astronomical and cosmological assumptions, by colliding two focused creation laser beams with a third focused inducing laser beam. The pair of creation lasers generates ALP-resonance states at the collision point. With the average energy of a single photon in the creation lasers, denoted by ${\omega }_c$, and subtended angles $2{\theta }_c$ between colliding two photons, the center-of-mass system energy is given by

$$E_{cms}=2{\omega }_{c}sin{\theta }_c.$$

(1)

The inducing laser stimulates the decay of the generated ALP. Denoting the average single-photon energy in the inducing field as ${\omega }_i$, one of the two decay photons can be extracted as the signal photon with energy ${\omega }_s=2{\omega }_c-{\omega }_i$, which is kinematically equivalent to the atomic four-wave-mixing (FWM) process. In practice, photons from each beam collide with a spread of angles and energies: focusing induces transverse-momentum (angular) spread, and the ultrashort pulse duration gives a finite energy bandwidth. As a result, the center-of-mass system energy is effectively modified to

$$E_{cms}=2\sqrt{{\omega }_1{\omega }_2}sin\left({\displaystyle \frac{{\vartheta }_1+{\vartheta }_2}{2}}\right),$$

(2)

and a fraction of the inducing field ${\omega }_4$ stimulates scattering into signal photons of energy ${\omega }_3$ that satisfy energy–momentum conservation. Consequently, the collision energy is not fixed but distributed, and by changing the incidence angle ${\theta }_c$ with a step matched to the mass resolution, we can perform a fast and continuous mass scan. We have, therefore, proposed [24], peformed the pilot search [25], and developed a variable-incidence three-beam collider that enables a continuous eV-scale scan [26] (see also sub-eV searches in quasi-collinear, coaxially focused two-beam configurations performed by the SAPPHIRES Collaboration [27,28,29]).

In our previous studies, we established a method to ensure precise spatiotemporal overlap at the focal point of the three-laser collision and performed searches using low-energy laser fields under atmospheric conditions [25]. Furthermore, we constructed a variable-angle three-beam-stimulated resonant photon collider and verified its operational feasibility [26]. In this paper, we present the result of the first point search with the variable-angle collider in the vacuum. The search was performed at the large angle of incidence between two creation beams, ${\theta }_c\pm 47.9°$, corresponding to ALP mass $m_a\sim 2.27\mathrm{eV}$, which is the most difficult angle to adjust to achieve the three-beam spatiotemporal overlap. The energies of the creation lasers and the inducing laser were each increased by more than three orders of magnitude compared to the previous search at the atmosphere.

This paper is organized as follows: We first describe the experimental setup designed for the search under vacuum conditions. Secondly, we detail the measurement. Thirdly, we present the analysis results and the null result. Finally, we provide the new upper limit based on the set of the experimental parameters and conclude the search.

2. Experimental Setup

Figure 2 shows a schematic view of the search setup in the large-angle configuration. From the left side of the figure, the creation laser (indicated in green) and the inducing laser (indicated in red) are introduced. We used a Ti:Sapphire laser (T6-system) with a pulse duration of ∼34 fs for the creation field and a Nd:YAG laser with a pulse duration of ∼7.1 ns for the inducing field. Both lasers are available at the Institute for Chemical Research, Kyoto University. The central wavelengths of these lasers were 811 nm and 1064 nm, respectively. On the right side of the figure, the variable-angle collider is configured at the large incidence angle. The collider comprises four layers combining rotation stages with aluminum plates, allowing independent control of the incidence angles for three beams (details in Ref. [26]). The angles of incidence were set to ${\theta }_c=\pm 47.9°$ for the creation beams, and ${\theta }_i=-66.4°$ for the inducing beam with respect to the bisecting angle between the two creation laser beams. Consequently, the resonance mass determined by the central wavelengths and the central angle of incidence corresponds to $m_a\sim 2.27$ eV. From energy and momentum conservation, the signal photon is expected to have a wavelength ${\omega }_s=655\mathrm{nm}$ and an emission angle of ${\theta }_s\sim 34.0°$.

The creation laser beam, with an initial diameter of approximately ∼40 mm, is split into two paths, c1 and c2, using an aperture plate (AP) with two ∼10 mm diameter apertures. The beam diameters of the three laser lines are defined by apertures ($A_{c1}, A_{c2}, A_i$) to be 5 mm for c1 and c2 and 7 mm for the inducing beam path, i. Each beam is independently focused by a parabolic mirror ($P{M}_{c1}, P{M}_{c2}, P{M}_i$) in the collider. The focal lengths are $f_c=101.6$ mm for the c1 and c2 beams and $f_i=203.2$ mm for the inducing beam. The optical paths of the creation lasers (c1 and c2) each include a 1.81 mm-thick half-wave plate ($\lambda$/2), and the polarizations are made mutually orthogonal at the focus to ensure sensitivity to a pseudoscalar field. The principle of the beam polarization measurement are given in Ref. [25]. In i-path for the inducing laser, a quarter-wave plate (λ/4) is inserted, and the polarization is set to right-handed circular as viewed from an observer at the collision point (CP). A mechanical shutter is placed at i-path and is used to generate the special trigger pattern described in the next section.

The spatiotemporal overlap of the three colliding laser beams is ensured by a specialized target system placed at the focus, with collision point monitors (CPM1, CPM2) and an array of photodetectors ($P{D}_c, P{D}_i$). CPM1, mounted on a motorized rotation stage with magenta layer, is used to check the focal position of each laser beam and to set the incidence angle with an accuracy of ∼$0.1°$. It is removed for the search runs in the vacuum. CPM2, mounted on an x-axis translation stage, is optically conjugate to the CP and enables three-beam focal-profile checks with a single camera. A target system is equipped with a cross-shaped wire of 10 μm diameter, a nonlinear crystal (NLC), and a “no target” option (vacuum), all of which can be precisely switched in the vertical positions using a motorized stage.

The left and middle panels in Figure 3 show a set of focal images of the three laser beams at CP, monitored by the CPM2. The spatial overlap of the three laser beams is ensured by imaging them onto the cross-shaped wire placed at the focal point and adjusting them to coincide relative to one another as shown in the most left three pannels. Along the beam focusing direction (the z-axis), the focal point is adjusted so that the vertical wire is in focus. Once the alignment with the wire is completed, the central position of the cross-shaped wire is mapped upon one camera pixel. By aligning the center of each laser beam to this defined pixel, spatial overlap can be consistently maintained as shown in middle three pannels obtained without wire-target. The right panel of Figure 3 shows an oscilloscope image captured when all the three laser pulses overlap. The temporal overlap at CP is verified using nonlinear optical phenomena via the NLC. A delay line (DL) composed of a retroreflector is installed at c2-path to allow fine adjustment of the optical path length. Second-harmonic generation (SHG:$2{\omega }_c\sim 405$ nm) indicated in yellow is generated only when the two green creation laser pulses overlap. By scanning the DL to maximize the SHG signal on a photomultiplier tube (PMT2), we ensure the temporal overlap of the two creation lasers. The overlap of three beams is ensured by observing a four-wave mixing (FWM) by a photomultiplier for the signal photon detection (PMT1). The red inducing laser is triggered by the creation laser system to ensure synchronization via electrical signals. FWM (cyan) is observed only when all three lasers are synchronized. In order to suppress noise components, optical filters are placed in front of each photomultiplier tube (PMT1, PMT2): one centered at 400 ± 5 nm to select SHG and another at 660 ± 10 nm to select FWM.

The NLC is mounted on a rotation stage with its surface perpendicular to the angle bisector of the two green creation beams.Taking the NLC surface as the CP for all three beams minimizes refractive-index-driven spatiotemporal offsets relative to the vacuum-target configuration to negligible levels.

A He:Ne laser (blue), with a wavelength of 633 nm chosen to be close to that of the expected signal photon is introduced from the left side of the setup for the signal acceptance measurement. During this measurement, inserting a calibration mirror (CM), c1-path is temporarily used to mimic the signal path from CP, ensuring the correct signal path. The laser fluxes are measured at two points at CP and at the position of PMT1 using the same camera, and the flux ratio gives the geometrical acceptance factor for signal photons. During acceptance measurements, the two bandpass filters: a long-pass filter transmitting wavelengths above 650 nm and a short-pass filter transmitting below 670 nm are removed. The acceptance measurement using the 633 nm He:Ne laser is conducted under a filtering condition that transmits the 570–690 nm wavelength range.

3. Measurement

The creation lasers (the T6-laser system) was operated at a repetition rate of $5\mathrm{Hz}$, whereas the inducing laser (Nd:YAG) was operated at $10\mathrm{Hz}$. For the ALP search, a mechanical shutter was inserted into the inducing laser beamline to form the trigger patterns in Figure 4.

The laser composition in each trigger pattern is as follows:

• S pattern: creation lasers plus inducing laser.
• C pattern: only creation lasers.
• I pattern: only inducing laser.
• P pattern: no laser (pedestal).

For the ALP search, we evaluated the number of photons observed by PMT1. Let us define $N_S,N_C,N_I,N_P$ as the observed numbers of photons per shot for the $S,C,I,P$ patterns, respectively, by assuming the following linear superpositions:

$$N_S=n_{\mathrm{sig}}+n_c+n_i+n_p,N_C=n_c+n_p,N_I=n_i+n_p,N_P=n_p.$$

(3)

The pedestal $n_p$ is present in all the patterns, whereas the noise photons originating from the creation and inducing lasers, $n_c$ and $n_i$, appear in their respective single-beam C, I patterns and also in the S pattern. The observed number of four-wave mixing (FWM) photons $n_{\mathrm{sig}}$ arises only in the S pattern. The number of signal photons can be extracted as

$$n_{\mathrm{sig}}=N_S-N_C-N_I+N_P$$

(4)

by taking the linear combinations of the number of photons observed in the four trigger patterns.

Figure 5 shows the arrival time distribution of FWM-like photons $N_S$ for the S pattern (500 shots) measured using PMT1 with NLC. The number of photons was reconstructed from the voltage–time relation of analog signals from PMT1 with a waveform digitizer. We also applied a peak-finding algorithm to simultaneously determine the number of photons and their arrival times from falling edges of amplitudes in the waveforms. The details of the peak-finding algorithm are given in the Section 4. The horizontal axis represents the photon arrival time with a bin width of $\Delta t=0.5$ ns, and the vertical axis shows the number of reconstructed photons per 0.5 ns. The arrival time of the FWM photons originating from the atomic process in NLC was used to define the time window corresponding to that for signal photon emission via ALP exchanges. With the creation and inducing lasers operating at low energy, the yields $N_C$ and $N_I$ in the C and I patterns were negligible. In the search experiment, we evaluated the number of photons within this red time window by counting the number of ALP-induced signal photons.

During the search, three beam focal spots can drift due to thermal variations in the laser system. To suppress the drift in focal position over time and to ensure good overlaps, we conducted 20,000 shots (5000 shots per trigger pattern: S, I, C, P) per run and performed three runs in total. Before and after individual runs, the focal positions of the laser beams were monitored with the CPM2 to assess the spatiotemporal misalignment. The measured drifts were used to evaluate the averaged three-beam overlaping factor as will be explained later.

4. Data Analysis

We evaluated the signal-photon yield with a PMT (R9880U-20) read out by a digital oscilloscope (Acqiris U1065A). A representative zoomed waveform is shown in Figure 6. The vertical axis is digitized with 10-bit resolution relative to the full-scale input voltage, and the horizontal axis covers $500\mathrm{ns}$ (full scale), sampled at period $\Delta t=0.5\mathrm{ns}$. For the analysis, we employ a peak-finding algorithm for two reasons: (1) to separate baseline electrical noise in the digital oscilloscope from photon-induced pulses; and (2) to use pulse arrival time to discriminate background/ambient light from the time-correlated signal.

We implement the peak-finding algorithm as follows:

(i)

Baseline and threshold. Let $V\left[t\right]$ denote the digitized PMT waveform (in volts) per shot as a function of time t. For each shot, define the initial baseline window as $t\in [0,100\mathrm{ns}]$. From the ensemble of baseline voltages over all shots, fit a Gaussian to obtain the baseline RMS ${\sigma }_v$. For each shot, compute the baseline mean $V_0$ and define the per-shot threshold level as

$$L_{5\sigma }\equiv V_0-5{\sigma }_v.$$

(5)

(ii)

Peak Identification. For each shot, count pulses that exceed $L_{5\sigma }$. For each pulse, identify the peak time $t_{\mathrm{pk}}$ as the most negative sample (the minimum voltage) within that pulse.

(iii)

Arrival time and integration range. Define two voltage threshold levels per shot:

$$\begin{array}{cc}\hfill L_{3\sigma }& \equiv V_0-3{\sigma }_v,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill L_{2\sigma }& \equiv V_0-2{\sigma }_v.\hfill \end{array}$$

(7)

Scanning backward in time from $t_{\mathrm{pk}}$, the arrival time $t_{\mathrm{arr}}$ and integration-start time $t_s$ are defined as the first upward crossings of $L_{3\sigma }$, and $L_{2\sigma }$, respectively. The index $t_{\mathrm{arr}}$ denotes the arrival time of pulses with threshold level at $L_{3\sigma }$ to mitigate false triggers from baseline fluctuations. The index $t_s$ marks the start time of charge integration for pulses using the milder threshold level $L_{2\sigma }$.

Set the integration-end time $t_e$ to be $3.5\mathrm{ns}$ after peak time,

$$t_e=t_{\mathrm{pk}}+3.5\mathrm{ns},$$

(8)

to include sufficient charge and to avoid contamination from PMT afterpulses, which typically occurs at $\sim 5\mathrm{ns}$ after the primary peak [30].

(iv)

Charge assignment. Compute the baseline-subtracted time-integral voltage between $t_s$ and $t_e$ with $\Delta t=0.5\mathrm{ns}$ as

$$Q=\sum _{t_s}^{t_e}\left(V\left[t\right]-V_0\right)\Delta t/R,$$

(9)

where $R=50\mathrm{\Omega }$ is the nominal input termination of the digitizer; since R cancels in the calibration ratio, its precise value does not affect the final results. The resulting charge Q is then assigned to the arrival timestamp $t_{\mathrm{arr}}$.

(v)

Photon conversion. Using the single-photo electron calibration of the PMT (Appendix A), performed with the same PMT, cabling, and front-end electronics as in the search data, we determine the single-photon-equivalent charge to be

$$Q_{1\mathrm{PE}}=-0.2337\pm 0.0004[\mathrm{pC}/\mathrm{photon}].$$

(10)

The corresponding number of detected photons is

$$N={\displaystyle \frac{Q}{Q_{1\mathrm{PE}}}}.$$

(11)

Applying steps (i)–(v) to every shot and the peak-finding algorithm enables time-resolved photon counting within the time window where signal photons are expected to arrive.

Figure 7 shows histograms in the upper left, upper right, lower left, and lower right corresponding to S, C, I, and P patterns of beam combinations, respectively. A total of $60,000$ shots were conducted across three runs, with $15,000$ shots in each trigger pattern. For each histogram, we use a time bin width of $\Delta t=0.5$ ns and sum the photon counts of all shots whose arrival times fall within each bin. In both the S and C patterns, a peak distribution of photon counts was observed mainly after the designated signal time (red window). Since no pressure dependence of the photon yield was observed and the similar distribution was also seen in the C pattern, this background is likely due to scattered lights from optical components encountered by the creation laser (c1, c2) during the propagation. Although identifying and suppressing this noise source remains a future task, the main noise distribution lies outside the arrival time window of the signal photons. Thus, we evaluate the number of signal photons based on Equation (4).

Figure 8 shows the result after $S-C-I+P$ subtraction with the statistical errors. The number of photons observed within the red time window was obtained as

$$n_{\mathrm{obs}}=-30.5\pm 52.4(\mathrm{stat}.)\pm 13.9(\mathrm{syst}.\mathrm{I})\pm 25.5(\mathrm{syst}.\mathrm{II})\pm 30.6(\mathrm{syst}.\mathrm{III}),$$

(12)

where the first uncertainty is statistical, evaluated from the Poisson fluctuations in the reconstructed photon numbers in the four trigger patterns, including the Fano-factor enhancement due to the Gamma-distributed single-photon charge response, as given by Equation (A20) and discussed in Appendix B, while the three systematic errors stem from the definition of the peak-finding algorithm as follows. The first systematic error (syst. I) was evaluated by varying the voltage threshold level in Equation (5) used to discriminate photon-peaks from baseline fluctuations,

$$L_{5\sigma }=V_0-5{\sigma }_v\pm 1{\sigma }_v.$$

(13)

The second systematic uncertainty (syst.II) was estimated by changing the voltage level of the arrival time threshold in Equation (6),

$$L_{3\sigma }=V_0-3{\sigma }_v\pm 1{\sigma }_v.$$

(14)

The third systematic uncertainty (syst.III) was evaluated by varying the voltage threshold level in Equation (7) that defines the initial bin of the charge-integration window,

$$L_{2\sigma }=V_0-2{\sigma }_v\pm 1{\sigma }_v.$$

(15)

The observed number of signal photons $n_{obs}$ was consistent with zero within the experimental uncertainty.

Other expected systematic errors arising from the degree of spatiotemporal overlap among the three beams during a single run will be discussed in the next section, where the effective overlap factors for individual runs will be evaluated and then incorporated to derive the upper limit from the null result.

5. Upper Limit in the Coupling–Mass Relation for ALP-Exchange

As no significant signal was observed in Section 4, we derive an exclusion region in the ALP coupling–mass plane. This is done using the formulation of the signal-photon yield presented in [24], together with the experimentally determined total uncertainty, as follows. The expected signal-photon yield per pulse collision in stimulated resonant scattering, ${\mathcal{Y}}_{c+i}$, is given by

$${\mathcal{Y}}_{c+i}\equiv N_1{N}_2{N}_4{\mathcal{D}}_{three}[s/L^3]{\overline{\Sigma }}_I[L^3/s],$$

(16)

where $N_1(=N_{c1}),N_2(=N_{c2})$ and $N_4(=N_i)$ are the average photon numbers in each laser, ${\mathcal{D}}_{three}$ represents the spatiotemporal overlap factor among the three focused beams at the collision [24], and ${\overline{\Sigma }}_I$ denotes the volume-wise interaction rate [22,23]. The corresponding units are expressed in [ ], with L representing length and s representing time.

The center-of-mass system energy $E_{\mathrm{cms}}$ of two colliding laser photons is intrinsically smeared by quantum uncertainty: each photon must be treated as a wave packet with finite temporal duration, which implies a finite spectral width via the Fourier transform relation between time and energy (set by the pulse width) and with a finite angular spread, which is Fourier-related to the beam waist (set by the focusing optics). In contrast, the natural ALP decay rate, $\Gamma ={(g/M)}^2{m}_a^3/\left(16\pi \right)$, that enters the Breit–Wigner propagator is extremely small in the coupling range of interest. For example, for $g/M={10}^{-10}{\mathrm{GeV}}^{-1}$ and $m_a=1\mathrm{eV}$, one finds $\Gamma \simeq 2\times {10}^{-40}\mathrm{eV}$. Compared to any realistic experimental window $\Delta E_{\mathrm{cms}}$ on $E_{\mathrm{cms}}$, the Breit–Wigner width is, therefore, almost delta-function-like. However, the integrated effect of the s-channel pole, including the off-shell part of the Breit–Wigner function within $\Delta E_{\mathrm{cms}}$, can drastically increase the sensitivity [23]. In our scheme, the observable lineshape is in practice governed by the experimental distribution of $E_{\mathrm{cms}}$ rather than by the tiny intrinsic ALP width. Given an accurately controlled $\Delta E_{\mathrm{cms}}$ in the experiment, if an ALP resonant mass lies within that range, we can claim the existence of a resonance through the averaged pole contribution.

Experimentally, the overlap factor ${\mathcal{D}}_{three}$ is expected to deteriorate due to variations in the incidence angle of each beam, $\Delta {\theta }_{x,j}, \Delta {\theta }_{y,j}, (j=c1, c2, i)$, which in turn causes shifts in peak positions at the focal point, $\Delta x_j, \Delta y_j, c\Delta t_j, (j=c1, c2, i)$. Therefore, in this study, the peak positions of the focal spot profiles of the three laser beams were measured before and after each run r = 1, 2, 3 using IPM2. The focal position measured before each of the three runs was treated as the ideal overlap factor with zero misalignment, denoted as ${\mathcal{D}}_0$. The overlap factor after each run, including the three beam offsets, is expressed as ${\mathcal{D}}_r(\Delta x_j, \Delta y_j, c\Delta t_j, \Delta {\theta }_{x,j}, \Delta {\theta }_{y,j})$ using the measured parameters. Here, the reduction in the overlap factor from timing and angular mismatches $c\Delta t_j, \Delta {\theta }_{x,j}, \Delta {\theta }_{y,j}$ is much smaller than that from the transverse offsets $\Delta x_j, \Delta y_j$. We, therefore, only retain the $\Delta x_j, \Delta y_j$ contributions. Finally, the merged and averaged overlap factor ${\overline{\mathcal{D}}}_{\mathrm{three}}$ is defined from the pre- and post-run values as

$${\overline{\mathcal{D}}}_{\mathrm{three}}={\displaystyle \frac{1}{N_{\mathrm{run}}}}\sum _{r=1}^{N_{\mathrm{run}}}{\displaystyle \frac{{\mathcal{D}}_0+{\mathcal{D}}_r(\Delta x_j, \Delta y_j)}{2}},$$

(17)

with $N_{\mathrm{run}}=3$ in this experiment. Typical transverse shifts were $\Delta x_j, \Delta y_j\simeq \pm 3\mathsf{\mu }\mathrm{m}$ for $j\in \{c1, c2, i\}$. For reference, the measured waist diameters at the CP were $2w_c\simeq 30\mathsf{\mu }\mathrm{m}$ (creation beams) and $2w_i\simeq 52\mathsf{\mu }\mathrm{m}$ (inducing beam). Further details of the three-beam overlap factor, ${\overline{\mathcal{D}}}_{\mathrm{three}}$, are given in Appendix C. Therefore, the experimental yield ${\mathcal{Y}}_{c+i}$ in Equation (16) is re-expressed as

$${\mathcal{Y}}_{c+i}\equiv N_1{N}_2{N}_4{\overline{\mathcal{D}}}_{\mathrm{three}}\left[s/L^3\right]{\overline{\Sigma }}_I\left[L^3/s\right].$$

(18)

Based on the set of experimental parameters P summarized in

Table 1. Experimental parameters used to obtain the upper limit.

Parameter	Value
Centeral wavelength of creation laser, ${\lambda }_c$	810.6 nm
Relative linewidth of creation laser, $\delta {\omega }_c$/<${\omega }_c$>	$1.5\times {10}^{-2}$
Duration time of creation laser (FWHM), $\sqrt{2ln2}{\tau }_c$	34.2 fs
Measured creation laser energy, $E_{c1}$	1.66 mJ
Creation energy fraction within 3 ${\sigma }_{xy}$ focal spot, ${\eta }_{c1}$	0.89
Effective creation energy within 3 ${\sigma }_{xy}$ focal spot	$E_{c1}{\eta }_{c1}$ = 1.48 mJ
Effective number of creation photons, $N_{c1}$	$6.0\times {10}^{15}$ photons
Beam diameter of creation laser beam, $d_{c1}$	5.0 mm
Polarization, ${\theta }_{c1}$, ${ϵ}_{c1}$	${\theta }_{c1}$ = 1.56 rad, ${ϵ}_{c1}$ = 0.61 rad
Measured creation laser energy, $E_{c2}$	1.26 mJ
Creation energy fraction within 3 ${\sigma }_{xy}$ focal spot, ${\eta }_{c2}$	0.88
Effective creation energy within 3 ${\sigma }_{xy}$ focal spot	$E_{c2}{\eta }_{c2}$ = 1.11 mJ
Effective number of creation photons, $N_{c2}$	$4.5\times {10}^{15}$ photons
Beam diameter of creation laser beam, $d_{c2}$	5.0 mm
Creation focal length, $f_c$	$f_c$ = 101.6 mm
Polarization, ${\theta }_{c2}$, ${ϵ}_{c2}$	${\theta }_{c2}$ = 0.04 rad, ${ϵ}_{c2}$ = 0.50 rad
Central wavelength of inducing laser, ${\lambda }_i$	1064 nm
Relative linewidth of inducing laser, $\delta {\omega }_i$/<${\omega }_i$>	$4.52\times {10}^{-5}$
Duration time of inducing laser beam (FWHM), $\sqrt{2ln2}{\tau }_{ibeam}$	7.11 ns
Measured inducing laser energy, $E_i$	60.55 mJ
Linewidth-based duration time of inducing laser, ${\tau }_i/2$	$\hslash /\left(2\delta {\omega }_i\right)$ = 6.24 ps
Inducing energy fraction within 3 ${\sigma }_{xy}$ focal spot, ${\eta }_i$	0.86
Effective inducing energy per ${\tau }_i$ within 3 ${\sigma }_{xy}$ focal spot	$E_i({\tau }_i/{\tau }_{ibeam}){\eta }_i$ = 107.8 μJ
Effective number of inducing photons, $N_i$	$5.8\times {10}^{14}$ photons
Beam diameter of inducing laser beam, $d_i$	7.0 mm
Inducing focal length, $f_i$	$f_i=203.2$ mm
Polarization	circular (right-handed state from observer)
PMT quantum efficiency, ${ϵ}_{QE}$	8.61%
Efficiency of optical path from IP to PMT, ${ϵ}_{opt}$	23.37%
Total detection efficiency, $ϵ$	2.0%
Total number of shots in trigger pattern S, $W_S$	14,934 shots
$\delta n_{obs}$	67

, the expected number of signal photons mediated by an ALP of mass $m_a$ and the coupling $g/M$ to two photons is expressed as

$$n_{obs}={\mathcal{Y}}_{c+i}\left(m_a,g/M;P\right)N_{shot}ϵ,$$

(19)

where $N_{shot}$ is the number of shots in S pattern and $ϵ$ is the overall detection efficiency. A coupling constant $g/M$ can be evaluated by numerically solving Equation (19) for an ALP mass $m_a$ and a given observed number of photons $n_{obs}$.

From Equation (12), since no significant signal was observed in the present search, an exclusion region is established. As a natural null hypothesis, we consider Gaussian fluctuations around zero. The confidence level for this null hypothesis is defined as

$$\mathrm{C}.\mathrm{L}.={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{\int }_{-\infty }^{\mu +\delta }exp\left[-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}\right]dx={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{\int }_0^{\mu +\delta }exp\left[-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}\right]dx,$$

(20)

where $\mu$ is the expected value of x under the hypothesis, and $\sigma$ is the standard deviation. In the search, the expected value x corresponds to the number of signal photons $n_{obs}$, and $\sigma$ is the one-standard deviation $\delta n_{obs}$. From Equation (12), the null result implies $\mu =n_{obs}=0$, and the acceptance-uncorrected uncertainty $\delta n_{obs}$ around $n_{obs}=0$ is estimated as the root-mean-square (RMS) of all relevant error components,

$$\delta n_{obs}=\sqrt{52.4^2+13.9^2+25.5^2+30.6^2}\simeq 67.$$

(21)

For a one-sided $95\%$ C.L., Equation (20) implies $\delta /\sigma \simeq 1.64$; accordingly, we exclude $x>\mu +\delta$ [29].

The upper limit in the parameter plane $\left(m_a,g/M\right)$ was estimated by numerically solving Equation (22) with the uncertainty $\delta n_{obs}$ defined in (21), using the set of experimental parameters P summarized in Table 1

$$1.64\delta n_{obs}={\mathcal{Y}}_{c+i}\left(m_a,g/M;P\right)N_{shot}ϵ,$$

(22)

where $N_{shot}=14,934$ and the overall efficiency $ϵ\equiv {ϵ}_{opt}{ϵ}_{QE}$ with the optical path acceptance from IP to PMT1, ${ϵ}_{opt}$, and the photocathode quantum efficiency of PMT1, ${ϵ}_{QE}$, were substituted. ${ϵ}_{opt}$ was estimated by aligning the c1 laser line with the signal line (${\theta }_s\sim 34°$) in the setup diagram in Figure 2 and measuring the energy at both the collision point and PMT1 positions using a common CMOS camera detector with a continuous calibration laser (He:Ne, 633 nm). The efficiency was determined as the ratio of the measured energy at these two locations. The He:Ne beam diameter was chosen to reproduce the expected signal photon angular distribution $\Delta {\theta }_s$, which was inferred from the error propagation via energy–momentum conservation. ${ϵ}_{QE}$ was estimated in a PMT calibration using a pulsed laser (wavelength 660 nm) as the ratio of the number of photoelectrons recorded by PMT1 to the number of incident photons, with the average photons per pulse held constant.

Figure 9 discusses theoretically and phenomenologically favored domains and existing upper limits in the ALP coupling–mass plane. For clarity, we first present the ALP-favored regions and theoretical benchmarks, and then overlay previously explored parameter spaces.

Figure 9 (top) shows the ALP-favored regions and theoretical models: the gray region shows the parameter space favored by $\gamma$-ray attenuation of blazar spectra when the optical-depth excess is modeled as ALP two-photon decays [14]. The cyan and yellow regions show the parameter spaces favored by COB excess (New Horizons/LORRI ’23,’24) [13]. The orange regions show the parameter spaces favored by HST + Spitzer near-IR anisotropy data when fitted with an ALP + intra-halo-light [12]. The beige shaded band corresponds to the benchmark QCD-axion model (KSVZ [31,32]) with $0.07<\left|E/N-1.95\right|<7$. The upper brown solid line shows the KSVZ model [31,32] with $E/N=0$, while the lower brown solid curve indicates the DFSZ model [33,34] with $E/N=8/3$. The cyan lines illustrate the predictions of the ALP miracle scenario [8] with the intrinsic parameters $c_{\gamma }=1, 0.1, 0.01$.

Figure 9 (bottom) shows the upper limit in the coupling–mass plane from this search, ^tSAPPHIRES01, enclosed by the red solid curve. The limit was set at a 95% confidence level for pseudoscalar-type ALP exchanges. The peak sensitivity was at $m_a=2.27\mathrm{eV}$ with $g/M\sim 4.2\times {10}^{-10}{\mathrm{GeV}}^{-1}$, using a central wavelength ${\lambda }_c=811\mathrm{nm}$ and an incidence angle ${\theta }_c=47.9°$. This represents the first model-independent exclusion/sensitivity reaching KSVZ line ($E/N=0$) [31,32]. The mass broadening at a relaxed coupling of $g/M\sim 4.2\times {10}^{-9}{\mathrm{GeV}}^{-1}$ was approximately $m_a=2.27\pm 0.07\mathrm{eV}$. The sensitivity spread arises from fluctuations in the center-of-mass system energy, driven by the finite momentum spread and bandwidth of the focused, short-pulse lasers [24]. The previous search ^tSAPPHIRES00 presents the excluded region based on the runs conducted in air at an incidence angle ${\theta }_c=30°$ (corresponding to $m_a=1.54,\mathrm{eV}$) [25]. We note that the label ^tSRPC00 used in Ref. [25] has been relabeled as ^tSAPPHIRES00 following reanalysis, to ensure consistency with ^tSAPPHIRES01 in the derivation of experimental parameters and the beam overlap factor.

The magenta shading shows the parameter space excluded by SRPC searches performed in a quasi-parallel collision setup (SAPPHIRES02) [29]. The purple-shaded regions correspond to constraints from Light-Shining-through-a-Wall (LSW) experiments, including results from ALPS [35] and OSQAR [36]. The gray region indicates the exclusion from the vacuum magnetic birefringence (VMB) measurement (PVLAS [37]). The light-cyan horizontal solid line shows the upper bound obtained by NOMAD using the SPS neutrino beam [38]. The green-shaded region denotes the CAST helioscope exclusion [39], obtained from searches for solar axions/ALPs via axion-photon conversion. The horizontal dotted line is disfavoured by globular-cluster constraints based on the impact of ALP emission on stellar evolution [40,41]. The light-green regions indicate exclusions from JWST/NIRSpec-IFU blank-sky spectra [15]. These limits overlap our search region but depend on background modeling (smooth continuum model) and astrophysical assumptions about the Milky Way DM halo (NFW), and are, hence, complementary. The purple regions indicate exclusions from high-dispersion WINERED spectroscopy of dSphs (Leo V, Tucana II adopting NFW profile) [18]. However, the authors also noted localized $>5\sigma$ excesses in that window, and its origin remains uncertain. The blue regions indicate exclusions from the VLT/MUSE-Faint observations of a dSph (Leo T adopting NFW profile) [16].

6. Conclusions

We have presented the results of a single-point ALP search using a variable-angle three-beam-stimulated resonant photon collider (^tSRPC). The energies of the creation lasers $c_1$ and $c_2$ ($1.4\mathrm{mJ}/34\mathrm{fs}$, Ti:Sapphire) and the inducing laser i ($60.6\mathrm{mJ}/7.1\mathrm{ns}$, Nd:YAG) were each increased by more than three orders of magnitude compared to the previous pilot search [25]. The experiment was performed in vacuum to suppress background photons. No statistically significant signal was observed. We, therefore, set exclusion limits, reaching a minimum coupling of $g/M=4.2\times {10}^{-10}{\mathrm{GeV}}^{-1}$ at $m_a=2.27\mathrm{eV}$, based on the formulation dedicated to ^tSRPC [24] and the experimentally determined overlap factor presented in this work. This is the first model-independent upper limit that reaches the KSVZ benchmark in the eV range. This single-point search demonstrates the feasibility of a smooth mass scan by varying the collision angle in near future searches.