Experimental Search for Non-Newtonian Forces in the Nanometer Scale with Slow Neutrons

The standard model of particle physics has been developed for almost half a century and provides a solid basis for understanding nature from the view point of its fundamental constituents, matter and interactions. It has succeeded in describing various phenomena of particle physics observed in a wide energy range up to the electroweak scale. However, there are still some unnatural points left, such as the so-called fine-tuning problem in radiative corrections to the Higgs mass, or the metastability of the electroweak vacuum Espinosa (1995); Isidori, Ridolfi, and Strumia (2001); Giudice (2013). Those imply extensions to the standard model or new theoretical frameworks which may appear at an energy below the Grand Unified Theory’s (GUT’s) scale Georgi and Glashow (1974).

We focus particularly on searching for new bosons of gravity-like coupling fields, which are naturally involved in some beyond-the-standard-model theories based on supersymmetry or spatial extra dimensions Fayet (1996); Arkani-Hamed, Dimopoulos, and Dvali (1998); Randall and Sundrum (1999). They are also predicted by scalar-tensor theories of gravitation that treat gravity as a field theory Brans and Dicke (1961); Khoury and Weltman (2004). In many cases, new fields would become manifest in additional gravity-like forces between test objects.

A sensitive method to search for these effects evaluates the neutron scattering processes with xenon atomic gas. The experiment has been performed using a monochromatic cold neutron beam on the D22 small-angle neutron scattering instrument at the Institut Laue-Langevin (ILL) Kamiya, Cubitt, and Zimmer (2016), and a momentum transfer dependence ($q$-dependence) in the scattering processes has been analyzed. We have selected a slightly larger neutron wavelength than in a previous measurement Kamiya et al. (2015), allowing us to investigate a region of lower $q$. In addition, we have collected around $1.4\times 10^{8}$ scattering events. Along with this improvement of statistical error, systematic control on the xenon gas pressure has been carefully considered. We follow the evaluation scheme which was widely utilized in previously performed experiments Pokotilovski (2006); Haddock et al. (2018); Kamiya et al. (2015); Nesvizhevsky, Pignol, and Protasov (2008); Borkowski et al. (2019); Bordag, Mohideen, and Mostepanenko (2001). Therein, the Yukawa-type scattering potential in natural units,

is used as a model of new physics, where $g^{2}$ is a coupling strength, $Q_{i}$ are coupling charges, and $\mu$ is the mass of the boson mediating the new interaction. This additional potential causes a new $q$-dependent term in neutron scattering distributions. The differential scattering cross-section can in total be written as

where the constant $b_{c}$ ($\sim 5$ fm) is the coherent scattering length. The dominating first term describes ordinary nuclear scattering. The second term represents the well-known $q$-dependent electromagnetic interactions between the neutron and intra-atomic fields. Here, $\chi_{em}$ is on the order of $10^{-2}$. $f(q)$ is the atomic form factor, which is modeled as $f(q)=[1+3(q/q_{0})^{2}]^{-0.5}$ with $q_{0}=6.86$ Å^-1 for xenon gas within $10^{-4}$ accuracy Sears (1986). The third term is due to the additional $q$-dependent Yukawa-type scattering potential. Here $\chi_{new}\equiv m_{n}g^{2}Q_{1}Q_{2}/2\pi b_{c}\mu^{2}$ where $m_{n}$ is the neutron mass. Since $\chi_{em}$ and $\chi_{new}$ are small enough, we neglect terms where products of the small amplitudes occur. We assume that the coupling charges are masses $M_{i}$ to keep compatibility with discussions on the Cavendish type experiments Bordag, Mohideen, and Mostepanenko (2001); Chen et al. (2016); Geraci et al. (2008); Kapner et al. (2007); Tan et al. (2016); Yang et al. (2012), which verify the inverse-square law of Newtonian gravity using macroscopic test masses. Reviews of such experiments, and connections to results from the large hadron collider can be found in Ref. Murata and Tanaka (2015).

The beam line consists of a neutron velocity selector, a neutron flux monitor, a collimation system, a scattering chamber which contains xenon atomic gas of 167 kPa pressure, and a 2 dimensional neutron detector with 1 m $\times$ 1 m sensitive area. We use an unpolarized neutron beam supplied from the ILL research reactor of 58 MW power. A wavelength of 6 Å was selected with 10% FWHM fractional resolution. Circular neutron apertures of diameters 20 mm was located 2.80 m and 0.22 m upstream from the center of the xenon-filled chamber, defining the beam divergence to be around 8 mrad. The scattering chamber had a cylindrical shape of length 250 mm and diameter 130 mm. It was mounted on a leveling stage with a thermal insulating plate. The cylinder axis was aligned along the beam traveling line using a laser light. Outgas rate of the chamber had been measured before the beam time to be $0.5$ Pa/h, which corresponds to contaminations of less than 75 ppm in our successive measurements. The chamber was connected to a getter-based gas purifier, a vacuum system, and a high-pressure xenon gas bottle. The vacuum system could evacuate the chamber volume down to at least $10^{-4}$ Pa level. The main detector on the D22 beam line consists of 128 ³He filled tube counters arranged vertically and horizontally spaced by 8 mm. The neutron position along the tube axis could be determined with 4 mm resolution. Therefore the detector enabled us to measure the 2 dimensional scattering distribution image of 8 mm (horizontal) times 4 mm (vertical) pixels. The detector was located $3005\pm 3$ mm downstream from the center of the scattering chamber.

Data sets are categorized to five: background data (cat.0), data of transmission through the evacuated chamber (cat.1), data of scattering from the evacuated chamber (cat.2), data of transmission through the xenon gas filled chamber (cat.3), and data of scattering from the xenon gas filled chamber (cat.4). The transmission data are taken with an attenuated direct beam, where a beam stopper is moved out of the beam axis. Total accumulation times are 20 minutes, 10 minutes, 16 hours, 50 minutes, and 24 hours for cat.0, 1, 2, 3, and 4, respectively. The cat.3 and cat.4 data are taken alternately to monitor the stability of xenon gas pressure. Time sequence of the taken data is illustrated in the Fig 1(a). Figure 1(b) shows a variation of the neutron transmission through the xenon gas. The gas pressure varied during the first period of cat.4 (#1), so we use the data from cat.4 (#2) to (#8) for the new-force evaluations. The transmission is measured to be $0.921\pm 0.001$ during the stable period.

Figure 2(a) shows the measured scattering contributions with and without xenon gas. They have been normalized by the transmission and integrated counts on the neutron flux monitor for each data category. The difference between the two distributions, which is reflected by the $q$-dependence of the neutron-xenon scattering processes, is shown in Fig. 2(b) with the best fit curve of the fitting function ${\rm d}\tilde{\sigma}/{\rm d}\Omega(\theta;\beta)=(1-\alpha^{*})(1-\beta)h_{c}(\theta)+\alpha^{*}(1-\beta)h_{em}(\theta)+\beta h_{new}(\theta;\mu)$ for $1/\mu=$ 1 nm. Here, ${\rm d}\tilde{\sigma}/{\rm d}\Omega(\theta;\beta)$ represents a smoothened differential cross-section evaluated by convoluting the Eq. (2) with the finite beam size, the length of the scattering target, neutron attenuation in xenon gas, and the thermal motion of xenon gas atoms at 293 K using the Monte Carlo method. The quantities $h_{c}(\theta)$, $h_{em}(\theta)$, and $h_{new}(\theta;\mu)$ are convoluted distributions corresponding to the first, second, and the third term in the equation (2), respectively. The Monte Carlo estimation assumes isotropic thermal motion of Maxwellian distributed atoms. This thermal motion leads to a slight forward-backward asymmetry and hence inclination of the expected distribution. This effect mimics an additional $q$-dependence in the scattering processes, which becomes notable for lighter targets, such as for argon gas. The measured distribution is normalized to $\int_{S}d\tilde{\sigma}/d\Omega(\theta)d\theta=1$ over the signal acceptance $S$ of 28 mrad $<\theta<$ 150 mrad, providing a high sensitivity to ranges of a new interaction around the nanometer scale. In the fitting function, $\alpha^{*}$ is a constant calculated from measurements of the neutron mean-square charge radius Tanabashi (2018), and $\beta$ is a fitting parameter, which represents the strength of new forces $g^{2}$. Residuals from the best fit curve are shown in the Fig. 2(c). The best fit values of $\beta$ are, for example, $(-1.7\pm 1.6)\times 10^{-3}$ and $(-9.6\pm 5.6)\times 10^{-4}$ for new forces of 0.4 nm and of 1 nm range, respectively. Using the Feldman and Cousins interpretation for the estimation around the physics boundary Feldman and Cousins (1998), we set improved limits for new gravity-like interactions as shown in the Fig. 2(d). Estimations of systematic effects on beta are dominated by the uncertainties of the neutron transmission $\gamma$ and the sample-to-detector distance. A shift by the one-sigma uncertainties of these quantities would shift the best fit value of beta for new forces of 1 nm range by around 4% and 6%, respectively. Other effects due to a non-Gaussian component of neutron wavelength distribution, temperature of the xenon gas, calibration factors for each pixel, and uncertainty of the atomic form factor are estimated to be much smaller.

The obtained limits are re-interpreted within a model in which a charge of new interaction is expressed as a linear combination of the baryon number $B$ and the lepton number $L$. This assumption was discussed in references Fischbach et al. (1988); Klapdor-Kleingrothaus and Staudt (1995) and experimental constraints in a hundreds micron range on this picture are reported in Adelberger et al. (2007). Here we follow the notation of the charge of new interactions as in Klapdor-Kleingrothaus and Staudt (1995), $Q=B\sin\theta_{5}+(B-2L)\cos\theta_{5}~{},$ where the mixing angle $\theta_{5}$ expresses a preference of the new force to couple to baryons or leptons. The leptophilic forces would imply $\theta_{5}=-\pi/4$.

We measured a neutron-xenon scattering distribution and obtained improved limits for gravity-like non-Newtonian forces in the nanometer scale. The results were re-interpreted with a model with new charges which are expressed as a linear combination of the baryon and the lepton number.