Search for sub-eV axion-like resonance states via stimulated quasi-parallel laser collisions with the parameterization including fully asymmetric collisional geometry

We have searched for axion-like resonance states by colliding optical photons in a focused laser field (creation beam) by adding another laser field (inducing beam) for stimulation of the resonance decays, where frequency-converted signal photons can be created as a result of stimulated photon-photon scattering via exchanges of axion-like resonances. A quasi-parallel collision system (QPS) in such a focused field allows access to the sub-eV mass range of resonance particles. In past searches in QPS, for simplicity, we interpreted the scattering rate based on an analytically calculable symmetric collision geometry in both incident angles and incident energies by partially implementing the asymmetric nature to meet the actual experimental conditions. In this paper, we present new search results based on a complete parameterization including fully asymmetric collisional geometries. In particular, we combined a linearly polarized creation laser and a circularly polarized inducing laser to match the new parameterization. A 0.10 mJ / 31 fs Ti:sapphire laser pulse and a 0.20 mJ / 9 ns Nd:YAG laser pulse were spatiotemporally synchronized by sharing a common optical axis and focused into the vacuum system. Under a condition in which atomic background processes were completely negligible, no significant scattering signal was observed at the vacuum pressure of $2.6 \times 10^{-5}$ Pa, thereby providing upper bounds on the coupling-mass relation by assuming exchanges of scalar and pseudoscalar fields at a 95 % confidence level in the sub-eV mass range.


I. INTRODUCTION
Spontaneous symmetry breaking is the key concept for understanding the fundamental laws of physics. In particular, when a symmetry is global, the appearance of a massless Nambu-Goldstone boson (NGB) [1] may be expected as a result of the broken symmetry.
This viewpoint can be a robust guiding principle for predicting new particle states based on various types of global symmetries in different theoretical contexts, including axion [2], dilaton [3], inflaton [4], and string-inspired models [5]. However, NGBs are observed as pseudo-NGB states (pNGB) with finite masses due to complicated quantum corrections, such as pions in the context of quantum chromodynamics (QCD). If pNGBs are coupled only very weakly to matter, they could be natural candidates to explain the dark components of the universe [6][7][8]. When pNGB masses are relatively small and the couplings to matter are feeble, they are referred to herein as axion-like particles (ALPs). Although ALP masses are supposed to be small, how small is not known theoretically. Therefore, experimental efforts to search for ALPs over wide low-mass and weak-coupling domains are valuable for discovering some of the dark components of the universe.
The XENON1T experiment recently reported an excess of electron recoil events compared with the defined background level [9]. Among three possible scenarios to explain this excess, the solar axion interpretation is the one that fits most nicely with the recoiled electron energy spectrum with 3.5 σ significance. The consistent axion mass range is O(0.1-10) eV in the photon-axion coupling range of O(10 −12 -10 −9 ) GeV −1 depending on the electron-axion coupling and the QCD axion models [10,11]. However, there is strong tension between this interpretation and the existing constraints from stellar cooling [12][13][14][15][16]. If the aforementioned significance level further increases through improved observations in the near future and the solar axion scenario remains the one that is most valid, then this tension becomes a real issue. To resolve this issue, model-independent observations of the direct production of axions and their decay in laboratory experiments would be indispensable.
In this paper, we focus on the coupling between sub-eV ALPs and laser photons. To describe the coupling of scalar (φ) or pseudoscalar (σ) ALPs to two photons, the following two effective Lagrangians are considered: where g is a dimensionless constant for a given energy scale M at which a relevant global symmetry is broken, and F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field strength tensor and its dualF µν ≡ ε µναβ F αβ with the Levi-Civita symbol ε ijkl .
FIG. 1: Concept of stimulated resonant photon-photon scattering in a quasi-parallel collision system (QPS) by focusing two-color laser fields in a vacuum. This figure is quoted from [19] with a slight modification. A coherent field with energy ω (solid green line) is combined with a different-color coherent field with energy uω (0 < u < 1) (dashed red line). The combined fields are focused by a lens element in a vacuum. Emission of signal photons with energy (2−u)ω (blue wavy line) is stimulated as a result of energy-momentum conservation in the scattering process ω + ω → φ/σ → (2 − u)ω + uω via a resonance state φ/σ. Given the focusing parameters of beam diameter d and focal length f , the incident angle ϑ is expected to vary over 0 < ϑ ≤ ∆θ. This unavoidable ambiguity of the wave vectors of the incident light waves provides a wide window for accessing different center-of-mass system energies at a given point in time.
Focusing on sub-eV ALPs, we have proposed to utilize quasi-parallel collision system (QPS) between two photon pairs with equal energy ω by combining and focusing two-color lasers along a common optical axis [17] as illustrated in Fig. 1. The corresponding centerof-mass system (CMS) energy in the QPS is expressed as where 2ϑ is the relative angle between a pair of incident photons. By controlling the beam diameter (d) and focal length (f ) experimentally, the QPS can be sensitive to ALP resonance states with the mass range of 0 < m < 2ω sin ∆ϑ, where m is the ALP mass and ∆ϑ can be adjusted by the focusing geometry determined with ∆ϑ ∼ (d/2)/f . The first key feature of the proposed method [17] is the resonant ALP production via the s-channel exchange within the E CM S uncertainty, which drastically enhances the production rate [17]. The second key feature is stimulated decays of produced ALPs to fixed final states via energy-momentum conservation between four photons in the initial and final states.
This stimulated resonant scattering rate eventually becomes proportional to the square of the number of photons in the creation laser beam and to the number of photons in the inducing laser beam. This cubic dependence on the number of photons in the beams offers opportunities to search for ALPs with extremely weak coupling when the beam intensity is high enough [18].
In past searches [19][20][21], we provided constraints on the coupling-mass relation based on a symmetric QPS interpretation in which the incident angles and energies of the two initial-state photons are symmetric (Fig. 1). Based on this parameterization, we provided conservative constraints, and we respected the simple analytic treatment with symmetric collisions because the initial search was made with narrow-bandwidth lasers. However, in a short-pulse laser that is close to the Fourier-transform limit, where the relation between laser frequency and time duration is governed by the wavelike nature of the system (i.e., the uncertainty principle), we must accept an energy spread in principle, and so the approximation of symmetric incident energies is not realistic. In addition, at the diffraction limit where the beam diameters reach their minimum values, the incident angles must also fluctuate greatly because of momentum-position uncertainties. Therefore, we must accept a situation in which the incident photon energies and angles are both asymmetric. Recently, we formulated the interaction rate based on the fully asymmetric collision system [18], where the non-coaxial geometry of the two-photon collisions and stimulated decays are explicitly included with respect to a given coaxial geometry of focused beams (Fig. 2b).
In general, linear polarization states in laser beams are supposed to be fixed precisely. Therefore, these fluctuations must be included in both the creation and inducing lasers around the focal points. In order for the stimulation due to the associated inducing field to be effective, one of the final-state light waves must coincide with the momentum and also with the linear polarization state of the inducing laser waves. Therefore, fluctuations of linear polarization directions make the evaluation of the stimulation effect in non-coaxial scattering events very complicated because coaxial symmetry of the inducing laser beam is no longer applicable. However, if a collection of light waves is circularly polarized, we can avoid this complication because even if the directions of the waves in three dimensions are changed, the individual light waves retain the circular polarization. Therefore, in [18] we evaluated the stimulation effect based on circularly polarized laser beams. However, there is an experimental constraint in that high-intensity laser pulses are provided as linearly polarized states, and changing their polarization from linear to circular adds a technical difficulty. In this paper, we report new results for sub-eV ALP searches by combining linearly and circularly polarized states for the creation and inducing lasers, respectively, based on the parameterization including the fully asymmetric collision geometry in QPS. This is in contrast to previous works in which we reported searches based on the symmetric QPS approximation with linear polarization states for both the creation and inducing lasers [19][20][21].
Stimulated resonant photon-photon scattering in the most general collisional geometry including asymmetric incidence and non-coaxial scattering was formulated in [18], and the full details can be found in the appendix of [18]. In the following subsections, we briefly explain how to relate the physical parameters of mass m and coupling g/M with the observed number of stimulated photon signals by reviewing only the relevant formulae to discuss the interaction rate dedicated for this search. We specifically replace the vertex factors in the scattering amplitude because we must change from the circular polarization formulated in [18] to the linear one for both scalar and pseudoscalar field exchanges in this search.
We address a search for signal photons p 3 for the following degenerate case in the generic QPS: where < > indicates that p 1 and p 2 are chosen stochastically from a single focused coherent beam whose central four-momentum is p c for the creation of ALPs via s-channel photonphoton scattering, while the focused coherent beam with the central four-momentum p i is co-moving to induce emission of signal photons p 3 when a fraction of the p i beam coincides with p 4 .
In symmetric incidence and coaxial scattering as illustrated in Fig. 2a, transverse momenta of stochastically selected wave pairs, p T , are constraint to be zero with respect to the common optical axis z. Given that the azimuthal angles of the final-state wave vectors are axially symmetric around the z-axis, the evaluation of the inducible momentum or angular range can be greatly simplified owing to the axial symmetric nature of the focused laser beams. In contrast, asymmetric incidence and non-coaxial scattering in Fig. 2b introduces a finite transverse momentum. In this case, a new zero-p T axis, referred to as the z ′ -axis, can be found for the pair of incident wave vectors. Based on the z ′ -axis, the axial symmetric nature of the azimuthal angles of the final-state wave vectors can be restored. However, the inducing coherent field is still physically fixed to the common optical axis z. This situation complicates the evaluation of the inducible momentum range depending on an arbitrarily formed z ′ -axis. To solve this complication, a numerical integration is required to express the number of signal photons in the scattering process (3) per pulse collision, Y c+i , as soon reviewed in the following subsections. With Y c+i and a set of laser beam parameters P , the number of stimulated signal photons, N obs , as a function of mass m and coupling g/M is eventually expressed as where t a is the data acquisition time, r is the repetition rate of the pulsed beams, and ǫ is the efficiency of detecting is a spatiotemporal overlapping factor in laboratory coordinates (see x, y, z in where w k are the beam radii as a function of time t whose origin is set at the moment when all the pulses reach the focal point, and τ k are the time durations of the pulsed laser beams with the speed of light c and the volume of the inducing beam V i defined as where w i0 is the beam waist (minimum radius) of the inducing beam. The actually used overlapping factor configured for the case of different beam diameters between creation and inducing lasers is summarized in the appendix of this paper.
Σ I in Eq. (5) is an integrated inducible volume-wise interaction rate that integrates the square of the scattering amplitude |M s (Q ′ )| 2 over an inducible variable set comprising energies ω i , polar angles Θ i , and azimuthal angles Φ i in laboratory coordinates for i = 1, 2, 4: with Gaussian distributions on energy G E and momentum G p , and also over an inducible Lorentz-invariant phase space in zero-p T coordinates: with two incident energies ω 1 and ω 2 , where the primed variables are converted from Q in laboratory coordinates to Q ′ in zero-p T coordinates via coordinate rotation. The inducing weight W (Q I ) takes care of the energy and momentum fractions of p 4 satisfying energymomentum conservation with respect to the energy and momentum distributions of the given inducing laser beam in laboratory coordinates. The essential element of Σ I is the Lorentz-invariant scattering amplitude defined in zero-p T coordinates, M S (Q ′ ), for the given polarization states S = abcd in a two-body interaction: p {d}. Unless confusion is expected, for simplicity, we omit the prime symbol associated with the momentum vectors in the following explanations. These polarization vectors are mapped on the rotating reaction planes in the case of asymmetric incident angles and energies in QPS, where the transverse momentum p T of the p 1 + p 2 pair is constraint to be zero as a specially simple case of general asymmetric collisions.

B. Vertex factors in scattering amplitude
The polarization information is normally useful for distinguishing whether ALPs are scalar or pseudoscalar fields. When a two-body photon-photon scattering process p 1 + p 2 → p 3 + p 4 in four-momentum space occurs on an identical reaction plane, namely, when the coplanar condition (Φ = ϕ = 0 in Fig. 3) is satisfied, the difference between scalar and pseudoscalar cases becomes distinct. Given an orthogonal set of linear polarization states {1} and {2}, the non-zero scattering amplitudes are limited to the following cases: for scalar field exchange and for pseudoscalar field exchange, where swapping {1} and {2} gives the same scattering amplitudes.
However, as illustrated in Fig. 3, the coplanar condition is, in most cases, not satisfied in QPS in contrast to CMS because the p 1 − p 2 plane and the p 3 − p 4 plane may differ from the x-z plane defined with the laboratory coordinates. Therefore, we must introduce Based on expansion of the electromagnetic field strength tensor F µν and its dualF µν , momentum-polarization tensors corresponding to the expanded coefficients are defined (see Eqs. (A.5) and (A.6) in [18] for details). With a polarization four-vector e i (λ p ) with an arbitrary polarization state λ p associated with a four-momentum p, and the symbol * indicating the complex conjugate, the momentum-polarization tensors are defined as for the tensor F µν andP Given the vector and tensor definitions above, the Lorentz-invariant scattering amplitude M S dedicated for scalar field exchange is expressed as (see Eq. (A.33) in [18] for the detailed derivation) where the factors (P i P j ) in the numerator correspond to the vertex factors reflecting polarization states in the initial and final states, respectively. (ST ) is the abbreviation for a momentum-polarization tensor product such as (ST ) ≡ S µν T µν for four-momenta s and t, that is, (P 1 P 2 ) corresponds to a momentum-tensor product for four-momenta p 1 and In the case of pseudoscalar exchange, we have only to replace the vertex factors with (P 1P2 )(P 3P4 ) using Eq. (13).
Hence, necessary momentum-polarization tensor products between four-momenta s and t with their polarization states λ s and λ t are summarized as for scalar field exchange and for pseudoscalar exchange.
The actually used vertex factors for scalar and pseudoscalar field exchanges dedicated for this search with the fixed left-handed circular polarization state, L, of the inducing laser are expressed as follows: where l i with i

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Step 1: A z ′ -axis of zero-p T coordinates is defined by finding a paring p 2 that satisfies the resonance condition with respect to the selected p 1 and to a finite energy segment from G E (ω 2 ) for a given mass parameter m. The kinematically possible ellipsoidal orbit is drawn with the purple belt on the angular distribution.
Step 2: Convert the polarization vectors e i (λ i )(i = 1, 2) from laboratory coordinates to zero-p T coordinates through coordinate rotation R(Q → Q ′ ).
Step 3: Calculate The voltages from the PMT and the two PDs as functions of time were recorded using a waveform digitizer with a time resolution of 0.5 ns. The digitizer was triggered by a basic 10-Hz laser oscillator clock to which the incident timing between the creation and inducing lasers was synchronized. The incident rate of the creation laser was reduced to equi-interval 5 Hz by a mechanical shutter (MS1), whereas that of the inducing laser was adjusted to nonequi-interval 5 Hz by a mechanical shutter (MS2) to produce four staggered trigger patterns for the offline waveform analysis. The four types of trigger were as follows: (i) two-beam incidence, "S"; (ii) only inducing-laser incidence, "I"; (iii) only creation-laser incidence, "C"; (iv) no beam incidence, "P". These were issued in order over a data acquisition run, which ensured equal shot statistics per trigger pattern and also minimized the systematic uncertainties associated with subtractions between trigger patterns as explained in the next section.

IV. DATA ANALYSIS
A. Counting number of photons by means of a peak finder The number of photons was evaluated based on the digitized waveform data from the PMT. In a waveform (sometimes referred to as a shot), voltage values V i were recorded with respect to individual sampling point i within a 500-ns time interval as shown in Fig. 6.
Because the time resolution (i.e., the width of a time bin) is 0.5 ns, i runs from 1 to 1000. shows the results measured at 10 Pa. The peak structures appeared in trigger patterns S and C. The peak seen in pattern C is expected because of plasma creation at the IP because the creation laser intensity is high enough to induce ionization of residual atoms. In contrast, the intensity of the inducing laser field is much lower because of the long time duration, as seen in pattern I where no peak is found. Meanwhile, the higher peak seen in pattern S is expected to be the sum between the atomic FWM and the plasma-origin photon yields.
The basic assumption that addition of the number of photons in individual trigger patterns corresponds to the number of photons in trigger pattern S is indeed supported by the following subtraction analysis. The acceptance-uncorrected number of atomic FWM photons, N S , can be obtained via where n i is the number of photons for trigger pattern i measured in the time interval subtended by the two red vertical lines corresponding to the signal generation timing window. Figure 8 shows the pressure dependence of the number of signal photons per shot, which is expected to be dependent upon the square of the pressure because the photon yield of the atomic FWM should be proportional to (χ (3) ) 2 ∝ (density) 2 ∝ (pressure) 2 . The dependence was thus fitted with where a and b are fitting parameters and P is pressure. The error bars are the quadratic sum of the statistical error propagation associated with the subtraction process between trigger patterns and systematic uncertainties of focal-point stability during a run period.
We explain these uncertainties in the following subsection. As expected, b = 1.85 ± 0.35 is close to the expected behavior N S ∝ P 2 in atomic physics [20,22]. Note that this pressure dependence itself is valuable as data because the special combination between linear and circular polarization state beams is a very rare case in atomic physics.

C. Focal-point stability
The systematic uncertainties due to focal-point fluctuations were estimated from overlaps between the two laser focal-spot profiles measured by the common CCD camera sensitive to both wavelengths. Figure 9 shows typical focal-spot images of the creation and inducing lasers.
With the local intensity per CCD pixel of the monitor camera, N(x, y), the overlap factor where the subscripts c and i specify the creation and inducing lasers, respectively. The summations were taken over the area framed by the full width at half maximum of the creation laser intensity profile. Fluctuations of the overlap factors with respect to the mean where O I,F are the overlap factors at the beginning and end, respectively, of a 2000-s unit run period.  Figure 10 shows the results of fitting the focal-plane intensity profiles of the creation and inducing beams with two-dimensional Gaussian distributions constrained by x-y symmetry.

D. Effective energy fraction in Gaussian beams
From the fitting results σ xy = 7 and 17 µm for the creation and inducing lasers, respectively, we evaluated the effective energy fraction contained in the region within 3 σ xy among the entire intensity profiles, including the peripheral diffraction parts that are assumed not to contribute to stimulated photon-photon scattering.

VI. UPPER LIMITS ON COUPLING-MASS RELATION FOR ALP EXCHANGES
From the result in (22), we conclude that no signal photons in the quasi-vacuum state were observed based on the total uncertainty. Indeed, this result is also consistent with the expected number of background photons per shot (efficiency-uncorrected) due to residual gases, estimated as by extrapolating to 2.6×10 −5 Pa with Eq. (19). In addition, for the given total statistics, the expectation value based on the QED photon-photon scattering process, which is the only possible process in the standard model, is negligibly low at E cms < 1 eV [23] even though the stimulation effect is taken into account [24]. Therefore, with respect to a null hypothesis following a Gaussian distribution, we provide the upper limits on the coupling-mass relation by assuming scalar and pseudoscalar field exchanges with the experimental parameters in Table I.
We note that the pulse duration of the Nd:YAG laser, τ ibeam , in Table I is not correspond-ing to that of the Fourier transform limit due to the different scheme to generate pluses from that of Ti:sapphire laser in which time duration close to reaching the Fourier transform limit is obtained. Thus, the effective time duration reaching the Fourier transform limit, τ i , which can overlap with the creation pulse duration, τ c , is evaluated from the spectrum linewidth of the Nd:YAG laser. This treatment is consistent with the basic assumption in [18] where the inducing effect is evaluated based on overlapping pulses individually reaching Fourier transform limits. In addition to the effective time durations, by considering the spatially overlapping regions within 3 σ xy focal spots which are consistent with the Gaussian shapes, the effective numbers of photons per pulses, N c and N i , were used for the following limit calculations. In this sense, the following results correspond to conservative upper limits, because the effective beam energies stored in pulses are very much reduced.
As for the upper mass range, because of the inclusion of general asymmetric collisions, this search is sensitive to a heavier mass range compared with the symmetric collision range, expressed as based on values in Table I with ∆θ ≡ d/(2f ) defined by the focal length f and beam diameter d of the creation laser in Fig. 1. Note, however, that this value is merely a reference mass at which the maximum sensitivity is expected.
A confidence level 1 − α to exclude a null hypothesis is expressed as where µ is the expected value of an estimator x following a hypothesis, and σ is one standard deviation. In this search, the estimator x corresponds to N S , and we assign the acceptanceuncorrected uncertainty δN S from the quadratic sum of all error components in the result (22) as the one standard deviation σ around the mean value µ = 0. In this search, the null hypothesis is produced from fluctuations of the number of photon-like signals following a Gaussian distribution whose expectation value, µ, is zero for the given total number of shots, W S = 2.9993 × 10 4 . This is because N S is calculated from subtractions between different trigger patterns whose baseline fluctuations, in principle, should follow Gaussian distributions individually. To obtain a confidence level of 95%, 2α = 0.05 with δ = 2.24σ is used, where a one-sided upper limit by excluding above x + δ [25] is applied. To evaluate  Wall experiments (LSWs)" (OSQAR [26] and ALPS [27]) with simplification of the sine-function part to unity above 10 −3 eV for drawing convenience. The gray area is the result from the "Vacuum Magnetic Birefringence (VMB)" experiment (PVLAS [28]). The green shaded areas are regions excluded based on non-Newtonian force searches ("Irvine" [29], "Eto-wash" [30], "Stan-ford1" [31], "Stanford2" [32]) and on Casimir force measurements ("Lamoreaux" [33]).
the upper limits on the coupling-mass relation, we then solved numerically based on Eq. (4) with respect to m and g/M for a set of experimental parameters P in Table I, where t a r = W S = 2.9993 × 10 4 and the overall efficiency ǫ ≡ ǫ opt ǫ d with the optical path acceptance ǫ opt to the p 3 detector position and the single p 3 -photon detection efficiency ǫ d were used. Figures 13 and 14 show the obtained upper limits on the couplingmass relations for scalar and pseudoscalar fields, respectively, at a 95% confidence level.
ǫ L is the acceptance factor with respect to left-handed circularly polarized photons measured from the IP to the p 3 -detection position. This is because both scalar and pseudoscalar fields can couple only to the same helicity state as that of the inducing field, which is provided as the left-handed state in the searching setup.