Search for sub-eV scalar and pseudoscalar resonances via four-wave mixing with a laser collider

The quasi-parallel photon-photon scattering by combining two-color laser fields is an approach to produce resonant states of low-mass fields in laboratory. In this system resonances can be probed via the four-wave mixing process in the vacuum. A search for scalar and pseudoscalar fields was performed by combining 9.3 $\mu$J/0.9 ps Ti-Sapphire laser and 100 $\mu$J/9 ns Nd:YAG laser. No significant signal of four-wave mixing was observed. We provide the upper limits on the coupling-mass relation for scalar and pseudoscalar fields, respectively, at a 95\% confidence level in the mass region below 0.15~eV.

Quasi-parallel colliding system by combining two-color laser fields [2], where beam diameter d, focal length f , and the incident angle ϑ takes 0 < ϑ ≤ ∆θ which is unavoidable due to ambiguity of the wave vectors of incident photons by the nature of focused lasers.

I. INTRODUCTION
Uncovering the nature of dark energy and dark matter is one of the most crucial problems in modern physics. Low-mass and weakly coupling fields predicted by theoretical models in cosmology and particle physics can be candidates for such dark components. For instance, based on the scalar-tensor theory with the cosmological constant Λ (STTΛ) [1], dark energy is interpreted as decaying Λ while the universe becomes older due to the gravitational coupling between extremely light dilatons, a kind of scalar fields (φ), and matter fields. Observing the γγ → φ → γγ process with extremely high intensity laser fields can be a method of searching for φ in laboratory [2]. The same approach can also be applied to searches for low-mass pseudoscalar fields (σ), if the photon spin states are properly chosen [3]. Axion [4,5], a pseudoscalar field associated with breaking of Peccei-Quinn symmetry [6], is a suitable candidate to which this method is directly applicable. Axion is supposed to be one of the most reasonable candidates for cold dark matter [7,8]. Therefore, these theoretical models strongly motivate us to search for such fields in laboratory in general.
With the schematic view of QPS in Fig. 1, we briefly explain the essence of our method as follows. By using variables defined at QPS, the center of mass system (CMS) energy between a randomly selected photon pair is expressed as where ω is the energy of incident photons and ϑ is half of the incident angle of the photon pair. Extremely low collision energies are realizable at QPS by focusing a laser field because small values of ϑ can be automatically introduced.
In order to overcome low scattering amplitudes of γγ → φ/σ → γγ processes due to weak coupling, we first utilize the character of the integrated resonance effect by capturing E CM S within ∆E CM S via ∆θ prepared by a creation laser field. Secondly we let another laser field propagate into the optical axis common to the creation laser. This laser induces decay of resonance states into a specific energy-momentum space by the coherent nature of the inducing field. The scattering probability is thus proportionally increased by the number of photons in the inducing laser field [2,3,23,24].
Energies of decayed photons are defined by the following energy conservation where u is an arbitrary number which satisfies 0 < u < 1. We re-define the energies of final state photons as following where ω 3 and ω 4 are energies of signal photon and inducing photons, respectively.
In the case of the scalar field exchange, the relation of linear polarization states between initial and final state photons when the wave vectors are on the same reaction plane are expressed as follows: where {1} and {2} are linear polarization states orthogonal to each other. In the pseudoscalar filed exchange, the polarization relation are expressed as We emphasize that above relations are limited only to the theoretically ideal case where all four photons are on the same reaction plane within the treatment based on plane waves. In the focused QPS, however, we must accept independent rotations of the incident p 1 −p 2 plane and the outgoing p 3 − p 4 plane as illustrated in Fig. 13 with respect to an experimentally given linear polarization plane. This implies that even if we supply ω as the pure {1}-state by a polarizer at the moment of plane wave propagation in advance of focusing, mixing of {1} and {2} states for randomly selected incident photon pairs is unavoidable while lasers are focused. Therefore, the focused QPS with a fixed initial linear polarization plane has sensitivity to both scalar and pseudoscalar fields simultaneously. We discuss about this nature in detail in Appendix A.
The relation in Eq.(2) is similar to "four-wave mixing" in matter corresponding to the third order non-linear quantum optical process in atoms [25,26]. Therefore, the observation of the four-wave mixing process in the vacuum may be interpreted as a replacement of the atomic nonlinear process by the exchange of unknown scalar or pseudoscalar fields. The observation of four-wave mixing in the vacuum is also used as a method for testing higher-order QED effect [27][28][29][30].
Photons produced via the atomic four-wave mixing process can be the main background source for this search. The first search for scalar fields at QPS [22] was performed with weak intensity lasers, thus, the effect of the four-wave mixing process in atoms was negligible. In this experiment, however, the four-wave mixing photons originating from the residual gas are anticipated due to much higher beam intensities. In this paper the method to obtain the exclusion limits in the search at QPS sensitive to both scalar and pseudoscalar fields is provided under the circumstance where a finite amount of background photons must be evaluated.

II. THE COUPLING-MASS RELATION
The effective interaction Lagrangians coupling between two photons and φ / σ are expressed where M has the dimension of energy and g is the dimensionless constant. The yield of signal photons, Y, is expressed with experimental parameters relevant to lasers and optical elements as follows: where the subscripts c and i indicate creation and inducing laser, respectively, λ is wavelength, τ is pulse duration, f is focal length, d is beam diameter, u and u are upper and lower values on u determined by the spectrum width of ω 4 , respectively, m is mass of the exchanging field, W is the numerical factor relevant to the integral of the weighted resonance function which is refined in Eq.(50) in Appendix B compared to W ∼ π/2 in Ref. [22], G is the incident-plane-rotation factor described in Appendix A, F S is the polarization dependent axially asymmetric factor for outgoing photons [3], C mb is the combinatorial factor originating from selecting a pair of photons among multimode frequency states and N is the average numbers of photons in the coherent state. The detail of the formulation of the signal yield is summarized in Appendix of Ref. [22]. The coupling constant g/M is expressed

III. EXPERIMENTAL SETUP
We explain the experimental setup to detect signals of four-wave mixing in the vacuum.
The schematic view of the setup is shown in Fig. 2.   "only creation laser is incident (C)", "only inducing laser is incident (I)", and "neither of lasers are incident (P)". The digital oscilloscope recorded waveform data from the PMT and two photo-diodes synchronized with the 20 Hz data acquisition trigger. The recorded waveform data from the PMT are sorted into four types of trigger patterns S, C, I and P. The four trigger patterns are classified by checking the charge correlations between the waveform data from the two photo-diodes for intensity monitoring.

IV. METHOD OF THE WAVEFORM ANALYSIS
The observed photon counts are estimated by analyzing the waveform data from the PMT.
The individual waveform consists of 500 sampling data points within a 200 ns time window.
We search for negative peaks of which amplitude exceed a given threshold. We then calculate charge sums of the peak structures. Figure  There are some accidental noisy events among recorded waveform data. In our analysis method, these noise structures could be misidentified as large photon-like peak structures.
Therefore, it is necessary to remove such noisy events from analyzed waveform data before counting photon-like peaks. We can identify noisy events by analyzing the frequencies of waveforms. Noisy waveforms tend to have lower frequencies than those of normal waveforms.  (18) and (19) in Ref. [22]) with the same shot statistics as the vacuum data are evaluated as follows:  N gas1 = 1.7 ± 1.1 × 10 −5 , We confirmed that the expected value of four-wave mixing photons from the residual gas are negligibly small in the vacuum data for a given total statistics.

VI. SEARCH FOR FOUR-WAVE MIXING SIGNALS IN THE VACUUM
We acquired data at 2.3 × 10 −2 Pa for the search for the resonant states of φ and σ fields.

DOSCALAR FIELDS
There is no significant four-wave mixing signal in this search from the result in (12). We thus evaluate the exclusion regions on the coupling-mass relation as follows. We estimate the upper limit on the sensitive mass range as based on values summarized in Table II, where ϑ in Fig. 1 varies from 0 to ∆θ defined by a focal length f and a beam diameter d.
The number of efficiency-corrected {1}-polarized signal photons N S1 and that of {2}polarized signal photons N S2 are evaluated from the following relations with the experimental parameters where ǫ opt1 and ǫ opt2 are the attenuation ratios of the signal photons propagating from the interaction point through Path{1} and Path{2}, respectively.   detection efficiency of the PMT mainly caused by the quantum efficiency of the device. ǫ D is evaluated using a 532 nm pulse laser in advance of the search. We evaluate the absolute detection efficiency by splitting the 532 nm beam equally and taking the ratio between these energies. The one is measured by a calibrated beam energy meter and the other is measured by that PMT with neutral density filters with measured attenuation factors. We then corrected the difference of the quantum efficiencies between 532 nm and 641 nm lights by taking the relative quantum efficiencies provided by HAMAMATSU into account.
We then evaluate upper limits on the coupling-mass relation at a 95% confidence level on the basis that the fluctuation of the number of signal yields forms a Gaussian distribution.
We define δN S as the one standard deviation of N S . It is evaluated from the quadratic sum of statistical and systematic errors in Eq.(12) and 2.24δN S is the upper limit of N S when we obtain a 95% confidence level [31]. The upper limit of signal yields per shot Y sc (for the scalar field exchange) and Y ps (for the pseudoscalar field exchange) are evaluated as follows: As we briefly mention in Introduction and in detail in Appendix A, even though we fix linear polarization planes for creation and inducing laser fields by the polarizers at the moment of plane wave propagation, mixing of {1} and {2}-polarization states is unavoidable in the focused QPS. By this effect, the focused system has sensitivity to both scalar and pseudoscalar fields simultaneously.
We obtain the coupling-mass relation from Eq. (8). The exclusion limits for scalar and pseudoscalar fields at a 95% confidence level are shown in Fig. 11 and Fig. 12 , respectively.

VIII. CONCLUSIONS
A search for scalar and pseudoscalar fields via the four-wave mixing precess at QPS has been performed by focusing 10 µJ/0.9 ps pulse laser and 100 µJ/9 ns pulse lasers. The number of shaded area shows the excluded region by our previous search, which is renewed from the black dotted line obtained from Ref. [22] by taking the incident-plane-rotation factor G and the massdependent W factor in Appendix B into account. The blue shaded area represents the excluded region for scalar fields by light shining through a wall experiment "ALPS" [19] (For the mass region above 10 −3 eV, the sine function part of the sensitivity curve is simplified to unity for drawing purposes). The green shaded areas indicate the limits given by non-Newtonian force searches by torsion balance experiments "Irvine" [32], "Eto-wash" [33,34], "Stanford1" [35], "Stanford2" [36] and Casimir force measurement "Lamoreaux" [37].
The cyan band indicates the expected coupling-mass relation of QCD axion predicted by KSVZ model [38,39] (1) and (2). It also depicts relations between p 1 − p 2 and p 3 − p 4 planes with respect to the x − z plane where the theoretically allowed coupling of an exchanged field to the linear polarization states can be evaluated in the clearest way. In Ref. [3], we have assumed the incident photons p 1 and p 2 are both plane waves with different wave vectors on the same reaction plane which always ensures the clearest condition. In the general 3-dimensional incident case such as a focused Gaussian beam, however, a p 1 − p 2 plane can rotate with respect to the x − z plane, which results in a deviation from the theoretically clearest condition. We, therefore, introduce a weighted averaging factor G over the clockwise rotation angle Φ of the incident reaction plane with respect to the x-axis as follows.
As we have discussed in ref. [3], the Lorentz invariant s-channel scattering amplitude for Lagrangian defined in Eq.(6) have the following basic form where S ≡ abcd with a, b, c, d = 1 or 2, respectively, denotes a sequence of four-photon polarization states and m is the mass of scalar or pseudoscalar field. With vectors defined below, the vertex factors for the scalar case (SC) are expressed as and these for the pseudoscalar case (PS) are expressed as We must first take into account the clockwise rotation angle ϕ of p 3 − p 4 plane with respect to the given x − z plane independent of the p 1 − p 2 plane, because these two planes are not coplanar in QPS contrary to the situation where the coplanar condition of p 1 through p 4 is always satisfied in CMS. This implies that the simple summation factor 2π on the azimuthal degree of freedom of solid angle cannot be applied to QPS, instead, the ϕ-dependent squared transition amplitude must be summed over the possible rotation ϕ from 0 to 2π. We have already introduced this axially asymmetric factor F S with respect only to the incident reaction plane at Φ = 0 in [3]. This factor essentially depends only on the second vertex factors above, while the incident-plane-rotation factor G is relevant only to the first vertex factors. We thus define the incident-plane-rotation factor as a weighted average with respect to F S at Φ = 0 as follows because experiments cannot fix the incident reaction plane and intensity of the creation laser field must be shared over possible incident reaction planes.
We note here that we cannot rotate polarization vectors because the experiment must introduce fixed polarization vectors. This implies that the clear distinction between scalar and pseudoscalar couplings cannot be stated due to non-zero rotation angles because nonidentical linear polarization planes between photon 1 and 2 or photon 3 and 4 are implicitly introduced.
This yields the following averaging factor on the incident reaction plane We also provide the case of ab = 12 for the pseudoscalar exchange as follows. Based on the first of Eq. (18), the first vertex factor with vector definitions above is expressed as = p 2ρ p 10 ǫ 0yρx + p 1z ǫ zyρx cos ϑ + p 10 ǫ 0yρz + p 1x ǫ xyρz sin ϑ = p 2ρ −ωǫ 0yρx + ω cos ϑǫ zyρx cos ϑ + −ωǫ 0yρz + ω sin ϑ cos Φǫ xyρz sin ϑ = −ωǫ 0yzx p 2z + ω cos ϑǫ zy0x p 20 cos ϑ + −ωǫ 0yxz p 2x + ω sin ϑ cos Φǫ xy0z p 20 sin ϑ = ω 2 (− cos ϑ + cos ϑ) cos ϑ + (− sin ϑ − sin ϑ) cos Φ sin ϑ = −2ω 2 sin 2 ϑ cos Φ. (28) This yields the following averaging factor on the incident reaction plane Appendix B: Refinement of the weight factor W In Ref. [3,22], we approximated W as a constant π/2 for the mass region much smaller than that covered by ∆θ as a conservative estimate. This is because we rather respected simplicity of the parametrization than accuracy. However, once we need to compare the sensitivity for the higher mass region with the other search methods, the validity of the approximation applicable only to the smaller mass region must be reconsidered. In the following, we first exactly repeat the relevant part of Ref. [22] and then refine W as a function of sensitive mass regions by quoting necessary equations.
We first express the squared scattering amplitude for the case when a low-mass field is exchanged in the s-channel via a resonance state with the symbol to describe polarization combinations of initial and final states S.
where χ = ω 2 − ω 2 r with the resonance condition m = 2ω r sin ϑ r for a given mass m and a is expressed as with the resonance decay rate of the low-mass field The resonance condition is satisfied when the center-of-mass system (CMS) energy between incident two photons E CM S = 2ω sin ϑ coincides with the given mass m. At a focused geometry of an incident laser beam, however, E CM S cannot be uniquely specified due to the momentum uncertainty of incident waves. Although the incident laser energy has the intrinsic uncertainty, the momentum uncertainty or the angular uncertainty between a pair of incident photons dominates that of the incident energy. Therefore, we consider the case where only angles of incidence ϑ between randomly chosen pairs of photons are uncertain within 0 < ϑ ≤ ∆ϑ for a given focusing parameter by fixing the incident energy. The treatment for the intrinsic energy uncertainty is explained in Appendix B later. We fix the laser energy ω at the optical wavelength while the resonance condition depends on the incident angle uncertainty. This gives the expression for χ as a function of ϑ where dϑ = ϑ r 2ω 2 We thus introduce the averaging process for the squared amplitude |M S | 2 over the possible uncertainty on incident angles With the variance ∆ϑ 2 = 2( 1 4 ∆θ 2 ), the pair angular distribution ρ(ϑ) is then approximated as for 0 < ϑ < π/2 (45) where the coefficient 2 of the amplitude is caused by limiting ϑ to the range 0 < ϑ < π/2, and ϑ ∆θ 2 ≪ 1 is taken into account because ∆θ in Eq.(43) also corresponds to the upper limit by the focusing lens based on geometric optics. This distribution is consistent with the flat top distribution applied to Ref. [3,24] except the coefficient.
We now re-express the average of the squared scattering amplitude as a function of χ ≡ aξ where we introduce the following constant W ≡ In Eq.(47) the weight function W (ξ) is the positive and monotonic function within the integral range and the second term is the Breit-Wigner(BW) function with the width of unity. Note that |M S | 2 is now explicitly proportional to a but not a 2 . This gives the enhancement factor a compared to the case |M S | 2 ∝ a 2 where no resonance state is contained in the integral range controlled by ∆θ experimentally. The integrated value of the pure BW function from ξ = −1 to ξ = +1 gives π/2, while that from ξ = −∞ to ξ = +∞ gives π.
The difference is only a factor of two. The weight function W (ξ) of the kernel is almost unity for small aξ, that is, when a is small enough with a small mass and a weak coupling.
Therefore, we will consider only the region of ξ ± 1 as a conservative estimate. By taking only this integral range, we can be released from trivial numerical modifications originating from ξ = −∞ and the behavior of W (ξ) at ξ = ω 2 opt a {1 − (ϑ r /(π/2)) 2 } which are not essential due to the strong suppression by the Breit-Wigner weight.
We now refine W in order to apply it more accurately even to the case for ϑ r /∆θ ∼ 1 where, exactly speaking, the second approximation in Eq.(45) is not valid. In this case, by using the first of Eq.(45) with substitution of the relation between χ ≡ aξ and ϑ expressed in Eq.(34), Eq.(48) is modified as follows where the last approximation is based on a/ω 2 opt ≪ 1 with respect to the integral range ξ ± 1 in Eq.(47) for the conservative estimate. This is justified in the mass-coupling range we are interested in via the first relation in Eq. (31), for instance, a/ω 2 opt ∼ 10 −29 for m ∼ 0.1 eV and g/M ∼ 10 −4 GeV −1 . By substituting Eq.(49) into Eq.(47), the conservative evaluation on W over ξ ± 1 is expressed as This factor is dependent of ϑ r , equivalently dependent of mass, especially for larger ϑ r close to ∆θ while it is almost π/2 for smaller ϑ r .