Further Theoretical Analysis on the $K^{-} {}^{3} \text{He} \to \Lambda p n$ Reaction for the $\bar{K} N N$ Bound-State Search in the J-PARC E15 Experiment

Based on the scenario that a $\bar{K} N N$ bound state is generated and it eventually decays into $\Lambda p$, we calculate the cross section of the $K^{-} {}^{3} \text{He} \to \Lambda p n$ reaction, which was recently measured in the J-PARC E15 experiment. We find that the behavior of the calculated differential cross section $d ^{2} \sigma / d M_{\Lambda p} d q_{\Lambda p}$, where $M_{\Lambda p}$ and $q_{\Lambda p}$ are the $\Lambda p$ invariant mass and momentum transfer in the $(K^{-} , \, n)$ reaction in the laboratory frame, respectively, is consistent with the experiment. Furthermore, we can reproduce almost quantitatively the experimental data of the $\Lambda p$ invariant mass spectrum in the momentum transfer window $350 \text{ MeV} /c<q_{\Lambda p}<650 \text{ MeV} /c$. These facts strongly suggest that the $\bar{K} N N$ bound state was indeed generated in the J-PARC E15 experiment.


Introduction
Because the chiralKN interaction is strongly attractive and dynamically generates the Λ(1405) [1][2][3][4][5][6], it is natural to extend the idea from theKN bound state [Λ(1405)] to theKNN bound state. Thē KNN bound state is a good ground to apply techniques of few-body calculations and to investigate further properties of theKN interaction. Until now a lot of effort has been put into theoretical predictions and experimental searches for theKNN bound state (see Ref. [7] for the status up to 2015).
Recently, in the J-PARC E15 experiment the cross section of the K −3 He → Λpn reaction was measured and a peak structure was found in the Λp invariant mass spectrum around the K − pp threshold [8,9]. One of the biggest advantages of this reaction is to putK directly into the nucleus to generate theKNN bound state. Thanks to that, we can theoretically trace the behavior of theK in the K −3 He → Λpn reaction [10]. As a result, the first run data of the J-PARC E15 experiment [8] were well reproduced in the scenario that theKNN bound state is generated.
In this manuscript we perform further theoretical analysis on the K −3 He → Λpn reaction measured in the J-PARC E15 experiment, in particular focusing on its second run data [9].

Formulation
We calculate the cross section of the K −3 He → Λpn reaction in the same manner as in our previous paper [10] except for one thing: the treatment of theKN →KN scattering amplitude at the first collision denoted by T 1 . In Ref. [10] we fixed T 1 as a real number by using the experimental values of theKN →KN differential cross sections only at an initial kaon momentum 1.0 GeV/c in the laboratory frame. Namely, we neglected the Fermi motion of the nucleons and fixed the invariant mass of the first-collisionKN, w 1 , to a unique value. In contrast, now we take into account the Fermi motion of the nucleons to evaluate w 1 and treat T 1 in a 2 × 2 matrix form in terms of the partial waves: where p out and p in are outgoing and incoming momenta ofK in theKN rest frame, p out,in ≡ |p out,in |, x ≡ p out · p in /(p out p in ), σ are the Pauli matrices acting on the baryon spinors, and g and h are expressed in terms of the partial-wave amplitudes T L± as with the Legendre polynomials P L (x), P ′ L (x) ≡ dP L /dx, and orbital angular momentum L. Because theKN →KN scattering at the first collision takes place with bound nucleons, it is better to treat the partial-wave amplitudes T L± as functions of three independent variables w 1 , p out , and p in , i.e., as off-shell amplitudes. We here assume that the partial-wave amplitudes depend on the momenta minimally required by the kinematics, i.e., they are proportional to (p out p in ) L . Under this assumption, the partial-wave amplitudes can be easily evaluated from the on-shell ones in the formula where p on-shell (w) is the on-shell momentum for theKN system with the center-of-mass energy w. In the present study, we utilize the on-shellKN amplitudes T on-shell L± (w) of Ref. [11], where the authors calculatedKN amplitudes up to the F wave (L = 4) based on a dynamical coupled-channels model with phenomenological SU(3) Lagrangians, to evaluate the amplitudes T L± (w, p out , p in ).

Numerical results
We now calculate the differential cross section d 2 σ/dM Λp d cos θ cm n of the K −3 He → Λpn reaction, where M Λp is the Λp invariant mass and θ cm n is the neutron scattering angle in the global center-of-mass frame, in the scenario that aKNN bound state is formed, decaying eventually into a Λp pair. The formulation is the same as in the previous calculation [10] except for theKN →KN amplitude at the first collision, which now takes the Fermi motion into account. We then multiply d 2 σ/dM Λp d cos θ cm n by |∂ cos θ cm n /∂q Λp |, where q Λp is the momentum transfer in the (K − , n) reaction in the laboratory frame, to obtain the differential cross section d 2 σ/dM Λp dq Λp . Note that, for a fixed M Λp , the momentum transfer reaches its minimum at cos θ cm n = 1 and increases as cos θ cm n decreases. In Fig. 1 we show the numerical result of the differential cross section d 2 σ/dM Λp dq Λp in thē KNN bound-state scenario. Here we take two options A and B for the evaluation of the energy carried by theK (for the details see Ref. [10]). From Fig. 1, in both options A and B, we can see that the structure near the K − pp threshold (2.37 GeV) is generated dominantly in the lower q Λp region, which corresponds to the condition of forward neutron scattering. In addition, we observe two trends in d 2 σ/dM Λp dq Λp ; one goes from M Λp = 2.35 GeV at q Λp = 0.25 GeV to the upward direction, and the other goes from the K − pp threshold at q Λp = 0.2 GeV to the upper-right direction. As discussed  in Ref. [10], the former is the signal of theKNN bound state, and the latter is the contribution from the quasi-elastic scattering of theK at the first collision. Interestingly, the behavior of the two trends in d 2 σ/dM Λp dq Λp was indeed observed in the second run data of the J-PARC E15 experiment [9], and hence our result is consistent with the experimental result. Next we integrate the differential cross section d 2 σ/dM Λp dq Λp to obtain the invariant mass spectrum dσ/dM Λp , which is shown in Fig. 2. When we take into account the whole region of the momentum transfer, we obtain lines "Full" in Fig. 2. In this case, we observe a two-peak structure around the K − pp threshold as we found in Ref. [10]; the peak below (above) the K − pp threshold originates from theKNN bound state (quasi-elastic scattering of theK at the first collision). Then, we restrict the momentum transfer to the region 350 MeV < q Λp < 650 MeV, as done in the experimental analysis [9], where they aimed at reducing the kinematic peak above the K − pp threshold corresponding to the quasi-elasticK scattering. Our mass spectrum with the momentum-transfer cut is plotted as lines "Cut" in Fig. 2. With this cut, only the peak for the signal of theKNN bound state survives. In this sense, our result supports the validity of the experimental cut of the momentum transfer.
Finally, we compare the calculated invariant mass spectrum dσ/dM Λp in the momentum transfer window 350 MeV < q Λp < 650 MeV with the experimental one in the same window. The result is shown in Fig. 3. As for the experimental data, we subtract the background contribution in the analysis of the experiment [9], because we do not take into account background but rather the generation of thē KNN bound state. From Fig. 3 Fig. 3. Comparison between theoretical and experimental results of the Λp invariant mass spectrum dσ/dM Λp for the K −3 He → Λpn reaction in the momentum transfer window 350 MeV/c < q Λp < 650 MeV/c. For the experimental data we subtract the background contribution in the experimental analysis [9]. spectrum strongly suggests that theKNN bound state was indeed generated in the J-PARC E15 experiment.