We report measurements of the flux-integrated $\overline{\nu}_\mu$ and $\overline{\nu}_\mu+\nu_\mu$ charged-current cross-sections on water and hydrocarbon targets using the T2K anti-neutrino beam with a mean beam energy of 0.86 GeV. The signal is defined as the (anti-)neutrino charged-current interaction with one induced $\mu^\pm$ and no detected charged pion or proton. These measurements are performed using a new WAGASCI module recently added to the T2K setup in combination with the INGRID Proton Module. The phase space of muons is restricted to the high-detection efficiency region, $p_{\mu}>400~{\rm MeV}/c$ and $\theta_{\mu}<30^{\circ}$, in the laboratory frame. An absence of pions and protons in the detectable phase spaces of $p_{\pi}>200~{\rm MeV}/c$, $\theta_{\pi}<70^{\circ}$ and $p_{\rm p}>600~{\rm MeV}/c$, $\theta_{\rm p}<70^{\circ}$ is required. In this paper, both the $\overline{\nu}_\mu$ cross-sections and $\overline{\nu}_\mu+\nu_\mu$ cross-sections on water and hydrocarbon targets and their ratios are provided by using the D’Agostini unfolding method. The results of the integrated $\overline{\nu}_\mu$ cross-section measurements over this phase space are $\sigma_{\rm H_{2}O}=(1.082\pm0.068(\rm stat.)^{+0.145}_{-0.128}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$, $\sigma_{\rm CH}=(1.096\pm0.054(\rm stat.)^{+0.132}_{-0.117}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$, and $\sigma_{\rm H_{2}O}/\sigma_{\rm CH} = 0.987\pm0.078(\rm stat.)^{+0.093}_{-0.090}(\rm syst.)$. The $\overline{\nu}_\mu+\nu_\mu$ cross-section is $\sigma_{\rm H_{2}O} = (1.155\pm0.064(\rm stat.)^{+0.148}_{-0.129}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$, $\sigma_{\rm CH}=(1.159\pm0.049(\rm stat.)^{+0.129}_{-0.115}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$, and $\sigma_{\rm H_{2}O}/\sigma_{\rm CH}=0.996\pm0.069(\rm stat.)^{+0.083}_{-0.078}(\rm syst.)$.
We have studied the contribution of high-scale SUSY to the |$K_L \to \pi ^0 \nu {\bar {\nu }}$| and |$K^+ \to \pi ^+ \nu {\bar {\nu }}$| processes by correlating with the CP-violating parameter |$\epsilon _K$|. Taking account of the recent LHC results for Higgs discovery and SUSY searches, we consider high-scale SUSY at the |$10$|–|$50$|TeV scale in the framework of non-minimal squark (slepton) flavor mixing. The Z penguin mediated chargino dominates the SUSY contribution for these decays. At the |$10$|TeV SUSY scale, the chargino contribution can enhance the branching ratio of |$K_L \to \pi ^0 \nu {\bar {\nu }}$| by eight times compared with SM predictions, whereas the predicted branching ratio |${\textit {BR}}(K^+ \to \pi ^+ \nu {\bar {\nu }})$| increases by up to three times that of the SM. The gluino box diagram dominates the SUSY contribution of |$\epsilon _K$| up to |$30{\%}$|. If down-squark mixing is neglected compared with up-squark mixing, the Z penguin mediated chargino dominates both SUSY contributions of |${\textit {BR}}(K_L \to \pi ^0 \nu {\bar {\nu }})$| and |$\epsilon _K$|. Then, a correlation between them is found, but the chargino contribution to |$\epsilon _K$| is at most |$3{\%}$|. Even if the SUSY scale is |$50$|TeV, the chargino process still enhances the branching ratio of |$K_L \to \pi ^0 \nu {\bar {\nu }}$| from the SM prediction by a factor of two, and |$\epsilon _K$| is deviated from the SM prediction by |$\mathcal {O}(10{\%})$|. We also discuss the chargino contribution to the |$K_L \to \pi ^0 e^+e^-$| process.
Many particle quantum hydrodynamics based on the Darwin Hamiltonian (the Hamiltonian corresponding to the Darwin Lagrangian) is considered. A force field appearing in corresponding Euler equation is considered in details. Contributions from different terms of the Darwin Hamiltonian in the Euler equation are traced. For example, the relativistic correction to the kinetic energy of particles leads to several terms in the Euler equation, these terms have different form. One of them has a form similar to a term appearing from the Darwin term. Hence, the two different mechanisms give analogous contributions in wave dispersion. Microscopic analog of the Biot-Savart law, called the current-current interaction and describing an interaction of moving charges via the magnetic field, is also included in our description. The semi-relativistic generalization of the quantum Bohm potential is obtained. Contribution of the relativistic effects in the spectrum of plasma collective excitations is considered.
