Muon Capture on 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^3$$\end{document}H

The μ-+3H→νμ+n+n+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ^- + ^3\mathrm{H} \rightarrow \nu _\mu + n + n + n$$\end{document} capture reaction is studied under full inclusion of final-state interactions with the AV18 nucleon-nucleon potential and the Urbana IX three-nucleon force. We employ the single nucleon weak current operator comprising the dominant relativistic corrections to obtain first estimates of the total capture rates based on realistic forces. Our results are compared with older theoretical predictions.


Introduction
Muon capture reactions on light nuclei have been investigated for many years. Earlier experimental and theoretical work is described in Refs. [1][2][3]. More recent theoretical studies, presented for example in Refs. [4][5][6], has concentrated on the μ − + 2 H → ν μ + n + n and μ − + 3 He → ν μ + 3 H reactions, representing the so-called phenomenological approach, the "hybrid" chiral effective field theory (χ EFT) approach and the "non-hybrid" χ EFT approach. The results obtained within different approaches agreed very well and provided a good description of available experimental data.
In Ref. [7] we combined our experience from the momentum space treatment of electromagnetic processes [8,9] as well as from the potential model approach developed in Ref. [4] and initiated systematic studies of all the A = 2 and A = 3 muon capture reactions. First we compared results of calculations carried out in the momentum space for the μ − + 2 H → ν μ + n + n and μ − + 3 He → ν μ + 3 H reactions with those of Ref. [4] performed in the coordinate space. These two types of calculations employed the AV18 nucleonnucleon (NN) potential [10] and the Urbana IX three-nucleon (3N) force [11]. In the momentum space the weak current operator was taken in the impulse approximation but it was also supplemented by the mesonexchange currents from Ref. [12] [Eqs. (4.16)-(4.39), without Δ-isobar contributions]. A very good agreement was found for the two considered models of the weak current operator. The two break-up channels in muon capture on 3 He were also studied in Ref. [7]. The differential and total capture rates were calculated with the same forces and the single nucleon current operator, providing first reliable estimates of these observables based on a modern nuclear Hamiltonian.
Muon capture on 3 H has not been studied so intensively. This reaction, with all uncharged particles in the final-state, would be very difficult to measure because of the radioactivity of the target and due to the meso-molecular complications [1]. Theoretical studies of this reaction were initiated in the 1970s [13][14][15] and continued in the 1980s [16]. Those early calculations were performed predominantly in the configuration space, using two-nucleon potential models available at that time. Table I in Ref. [16] nicely summarized all the early theoretical predictions.
Very recently, we also investigated this reaction in Ref. [17], since its study was for us the natural next step after the μ − + 3 He → ν μ + n + n + p reaction had been considered. The μ − + 3 H → ν μ + n + n + n process possesses very interesting features: it allows one to study the neutron-neutron interaction and the three-neutron force acting exclusively in the total isospin T = 3/2 state. Our calculations in Ref. [17] were again performed using the AV18 [10] NN potential and the Urbana IX 3N force [11]. Within our momentum space Faddeev framework we incorporated all final-state interactions, retaining only single-nucleon contributions in the weak current operator.
In this contribution we thus present only selected results for the differential and total capture rates in muon capture on 3 H. For the details of our formalism and more complete discussion of the results we refer the reader to Ref. [17].

Results and Outlook
Our calculations are based on the nonrelativistic kinematics and dynamics. That is why we first checked in Refs. [7,17] that for all the three break-up processes nonrelativistic treatment of kinematics provides a very good approximation. Thus one may hope that also the nonrelativistic dynamical framework is fully justified.
Our results for the μ − + 3 H → ν μ + n + n + n process are presented in Fig. 1. First in its left panel we show final-state interaction effects, which turn to be very important. They not only change the plane wave predictions by a factor of 2 but affect also the shapes of the curves and their peak positions. We thus could confirm the findings of Refs. [15,16] obtained with completely different frameworks and much simpler forces. Next, like in Ref. [7], we also study the 3N force effects. They are most visible in the peak area, where the predictions including the 3N force drop by approximately 20%. We checked, however, in Ref. [17] that the 3N force effects come mainly from the initial bound state. The corresponding values of the integrated capture rate are Γ = 36.5 s −1 (calculated without the 3N force) and Γ = 32.6 s −1 (calculated with the 3N force).
Undeniably, further theoretical work and new precision measurements in the whole kinematical region are needed to improve our knowledge about break-up channels in muon capture on 3 He and 3 H. In the near future we intend to combine the presently used computational techniques with improved dynamical ingredients [18][19][20][21] to perform complete chiral EFT calculations at high orders.