## 1 Introduction  
Photon-photon scattering is a nonlinear interaction between photons, intermediated by a virtual electron loop at the lowest order of quantum electrodynamics (QED) (Figure 1, left). The contribution of this diagram ubiquitously appears in radiative corrections to charged particles at higher orders (Figure 2a), and its agreement with measurements constitutes a solid basis for the Standard Model of particle physics at low energies [1; 2; 3]. In addition, some of these photons have been changed into real-state photons (Figure 2b-d) and observed in experiments [4; 5; 6]. These processes occur under a Coulomb field of pre-existing, real-state charges, and therefore can be classified as \"real-charge processes\". By contrast, there are many attempts to observe this diagram in a vacuum by using a PW-class high-power laser or using a high-field magnet. This kind of \"vacuum process\" occurs without such charge sources and cannot be removed by any effort, thus representing the _core_ of all physical processes. The vacuum process is currently studied in two streams. The first is based in high field science, where the behavior of dense plasma is studied using high-power lasers. They predict many kinds of vacuum phenomena occurring at the focal spot of an intense laser [7; 8]. An efficient probe on these phenomena is provided by the recent progress in X-ray free-electron lasers (XFELs) due to their short wavelength, peak brilliance, and more particle nature compared to laser photons [9]. The second is an experimentally ongoing stream that studies the optical birefringence of a magnetized vacuum (Figure 1, right). The measurement of vacuum magnetic birefringence (VMB) provides the current best sensitivity to the observation of the vacuum process [10]. The theoretical basis describing these vacuum processes can be formulated in terms of the Euler-Heisenberg Lagrangian, which effectively contains only electromagnetic fields. With assumptions that photon energies are well below the electron mass \(m_{\mathrm{e}}\) and fields much smaller than the QED critical values (\(E\ll 1.3\times 10^{18}\) V m\({}^{-1}\), \(B\ll 4.4\times 10^{9}\) T), the Lagrangian density is written in natural units (\(\hbar=c=1\)) as [11]  
\[\mathcal{L}=\frac{1}{2}\left(E^{2}-B^{2}\right)+\frac{2\alpha^{2}}{45m_{ \mathrm{e}}^{4}}\left[\left(E^{2}-B^{2}\right)^{2}+7\left(\mathbf{E}\cdot \mathbf{B}\right)^{2}\right]\,, \tag{1}\]  
with the fine-structure constant \(\alpha\) and electric and magnetic fields \(\mathbf{E}\) and \(\mathbf{B}\), respectively. The second term of Equation (1) describes four-field interactions, and theoretical studies on these effects have been summarized in References [7; 8]. In addition to these nonlinear QED contexts, searches for vacuum processes concern a wider range of research fields, including astrophysics and particle physics. For instance, a practical effect of VMB has been suggested by the recent observation of a strongly magnetized neutron star [12; 13; 14]. Furthermore, the vacuum process may be easily affected by the contribution of new physics beyond the Standard Model (BSM) [15]. The conventional benchmark in a low-energy scale is the axion, and more generally, axion-like particles (ALPs) that have an effective coupling to two photons [16]. If such light particles exist, their mass spectrum would increase the diversity of vacuum interactions ascribed to electrons at low energies. To our best knowledge, we are currently the only group that experimentally studies the vacuum physics in both streams. In this article, we review the status and results of our first-phase experiments along with their technical aspects. Section 2 describes experiments probing the vacuum process; two experiments using the XFEL, SACLA (SPring-8 Angstrom Compact Free-Electron Laser) [17; 18],  
Figure 1: Vacuum processes of photon–photon interaction at the lowest-order quantum electrodynamics (QED). The time goes from left to right. (**Left**) Photon–photon scattering.