On the structure observed in the in-flight ${}^{3}\text{He} ( K^{-} , \, \Lambda p ) n$ reaction at J-PARC

A theoretical investigation is done to clarify the origin of the peak structure observed near the $K^{-} p p$ threshold in the in-flight ${}^{3}\text{He} ( K^{-}, \, \Lambda p ) n$ reaction of the J-PARC E15 experiment, which could be a signal of the lightest kaonic nuclei, that is, the $\bar{K} N N (I=1/2)$ state. For the investigation, we evaluate the $\Lambda p$ invariant mass spectrum assuming two possible scenarios to interpret the experimental peak. One assumes that the $\Lambda (1405)$ resonance is generated after the emission of an energetic neutron from the absorption of the initial $K^-$, not forming a bound state with the remaining proton. This uncorrelated $\Lambda (1405) p$ system subsequently decays into the final $\Lambda p$. The other scenario implies that, after the emission of the energetic neutron, a $\bar{K} N N$ bound state is formed, decaying eventually into a $\Lambda p$ pair. Our results show that the experimental signal observed in the in-flight ${}^{3}\text{He} ( K^{-} , \, \Lambda p ) n$ reaction at J-PARC is qualitatively well reproduced by the assumption that a $\bar{K} N N$ bound state is generated in the reaction, definitely discarding the interpretation in terms of an uncorrelated $\Lambda (1405) p$ state.


Introduction
The study of theKN interaction with coupled channels has been a traditional test field for chiral dynamics in its unitarized version. Since the pioneer works of [1,2], many works have been done in this field including also the contribution of the higher order chiral Lagrangians (see recent review in [3]). One of the unexpected results was the finding of two poles for the Λ(1405) resonance [4,5], which recently became official in the Particle Data Group [6] [see note on the Λ(1405) [7]]. The other issue that became topical was the possibility of having kaonic nuclei, in particular a boundKN N system. The study of this system has been thoroughly addressed theoretically [8][9][10][11][12][13][14][15][16][17][18][19] (see review paper in [20]), obtaining a binding energy that varies from a few MeV to 100 MeV. There are also discrepancies in the width of the state, which varies from 10 to 90 MeV. One step forward in the evaluation of the width was given in [21], where two-nucleonK absorption was explicitly considered. In that study a binding of 15-30 MeV was found, together with a width of the order of 80 MeV. The fact that the width is larger than the binding energy is shared by most of the theoretical approaches. One interesting point of view was to consider this system as a bound state of Λ(1405)N [22]. This allows one to make some qualitative pictures that help understanding some of the results obtained when studying the possible formation of this system in different reactions.
There have been previous claims of the formation of this state in some experiments, but soon the experimental signals were interpreted in terms of unavoidable conventional mechanisms (see overview in [20,23]). Yet, the experimental search has continued [24][25][26][27][28] with often contradictory views (see overview in [29]).
In this line, very recently, a peak structure was observed near the K − pp threshold in the Λp invariant mass spectrum of the in-flight 3 He(K − , Λp)n reaction of the J-PARC E15 experiment [29]. According to their analysis, this peak can be described by the Breit-Wigner formula with mass M X = 2355 +6 −4 (stat.) ± 12(sys.) MeV and width Γ X = 110 +19 −17 (stat.) ± 27(sys.) MeV. This structure could be a signal of theKN N (I = 1/2) bound state with a binding of ∼ 15 MeV from the K − pp threshold.
In this paper we theoretically investigate the origin of the peak structure observed in the J-PARC E15 experiment. For this purpose, we take into account two possible mechanisms for producing a peak in the mass spectrum of the 3 He(K − , Λp)n reaction. One corresponds to assuming the formation of a Λ(1405) resonance that does not form a bound state with the remaining proton, while the other considers the formation of a bound state of theKN N system. We evaluate the cross section of the 3 He(K − , Λp)n reaction assuming theKN → KN scattering around threshold and the Λ(1405) resonance to be described by the chiral unitary approach [1,2,4,5], while the description of theKN N bound state is done in terms of the so-called fixed center approximation to the Faddeev equation [17,21].
As a result, we can unambiguously interpret the experimental spectrum in the scenario of ā KN N broad bound state, obtained from the interaction of theK with a pair of nucleons [21]. In addition, in this scenario, we obtain a two peak structure of the mass spectrum near thē KN N threshold. The peak below the threshold is the signal of theKN N bound state, while the peak above the threshold originates from the quasi-elastic scattering of the kaon in the first collision emitting a fast nucleon, in processes of the type K − n → K − n escape or K − p →K 0 n escape . This paper is organized as follows. In Sec. 2 we develop our formulation to obtain the cross section of the 3 He(K − , Λp)n reaction, describing the details that allow us to obtain the scattering amplitude for the uncorrelated Λ(1405)p mechanism and for theKN N bound state one. Next, in Sec. 3 we show our results and discuss the origin of the peak structure observed in the in-flight 3 He(K − , Λp)n reaction in the J-PARC experiment. Section 4 is devoted to the conclusions of this study.

Formulation
In this section we formulate the cross section and scattering amplitude of the in-flight 3 He(K − , Λp)n reaction. After showing the expression of the cross section in Sec. 2.1, we construct the scattering amplitude of the reaction in Sec. 2.2 and Sec. 2.3. In Sec. 2.2 we consider the case of an uncorrelated Λ(1405)p system, i.e., the Λ(1405)p system is generated without binding after emission of a fast neutron, and in Sec. 2.3 we take into account the multiple scattering ofK between two nucleons to generate aKN N quasi-bound state.
According to the experimental condition, we concentrate on the three-nucleon absorption of K − , i.e., we do not allow a spectator nucleon. Throughout this study, we take the global center-of-mass frame when we calculate the phase space for the cross section, while we evaluate the scattering amplitude in the 3 He rest frame so as to omit the center-of-mass momentum of 3 He in the wave function. Throughout this work the physical masses for 2/27 the hadrons are used, except in the evaluation of the 3 He wave function and in the kaon propagators of the multiple scatteringKN N amplitude.

Cross section of the 3 He(K − , Λp)n reaction
First we formulate the cross section for the K − (k) 3 He(P ) → Λ(p ′ Λ )p(p ′ p )n(p ′ n ) reaction, where the momenta of these particles are shown in parentheses. Since we are interested in the Λp spectrum as a function of its invariant mass M Λp , we fix the final-state phase space with the invariant mass M Λp , the solid angle for the neutron momentum in the global center-of-mass frame Ω n , and the solid angle for Λ in the Λ-p rest frame Ω * Λ . With these quantities, the differential cross section can be expressed as [6]: where we have performed the integral with respect to the azimuthal angle for the neutron momentum, which is irrelevant to the present formulation. In the expression, θ cm n is the neutron scattering angle in the global center-of-mass frame, M3 He , m Λ , m p , and m n are the masses of 3 He, Λ, proton and neutron, respectively, and p cm and E cm are the center-of-mass momentum and energy for the initial state: the initial kaon momentum in the laboratory frame k µ , and ω K − (k) ≡ k 2 + m 2 K − . The momenta p ′ n and p * Λ correspond to that of the neutron in the global center-of-mass frame and that of the Λ in the Λ-p rest frame, respectively, evaluated as By means of the summation symbols in Eq. (1), we perform the average and sum of the squared scattering amplitude, |T | 2 , for the polarizations of the initial-and final-state particles, respectively. From the double differential cross section d 2 σ/dM Λp d cos θ cm n , we can evaluate the mass spectrum dσ/dM Λp and the differential cross section with respect to the neutron angle dσ/d cos θ cm n by performing the integral with respect to cos θ cm n and M Λp , respectively: where M min and M max are the lower and upper bounds of the invariant mass M Λp , respectively, which become M min ∼ 2.1 GeV and M max ∼ 2.9 GeV for an initial kaon momentum in the laboratory frame of k lab = 1 GeV/c, as that employed in the J-PARC E15 experiment [29]. In this study, however, we restrict those values to M min = 2.2 GeV and M max = 2.6 GeV, since we are interested only in the physics leading to a peak around thē KN N threshold and we are ignoring other contributions which only play a background role in this region. We can also evaluate the total cross section as 3/27 Two Feynman diagrams most relevant to the three-nucleon absorption of K − via an uncorrelated Λ(1405)p system. We also show momenta of particles in parentheses.

Scattering amplitude: generating an uncorrelated Λ(1405)p system
Next we construct the scattering amplitude of the 3 He(K − , Λp)n reaction with three-nucleon absorption of K − . In this subsection we consider a case that an uncorrelated Λ(1405)p system, without binding, is generated after the emission of a fast neutron. This process has a possibility of making a peak structure around the K − pp threshold in the Λp invariant mass spectrum, since the Λ(1405) resonance appears below the K − p threshold. In Fig. 1 we show the two diagrams which implicitly contain the Λ(1405) resonance in the intermediate state.
In these diagrams, a fast neutron is emitted after the first collision of the K − with a nucleon (T 1 ). Then, the second collision of the K − p or theK 0 n (T 2 ) is enhanced at the energy of the Λ(1405) resonance, producing an enhancement around the K − pp threshold. Finally, the rescattered K − is absorbed into a proton to become a Λ particle. An important point is that one expects a kaon, rather than another meson like the η or the pion, to be absorbed by the last nucleon because 1) the propagating kaon after the first collision is almost on its mass shell, hence the amplitude T 2 covers the region of the Λ(1405) resonance, 2) the Λ(1405) is expected to be aKN (I = 0) bound state [30][31][32], hence the process is dominated by the isospin I = 0 component of theKN interaction, and 3) the coupling strength of the K − pΛ vertex is strong while that of ηpp is very weak in flavor SU(3) symmetry [33,34]. 1 Therefore, the most relevant diagrams for the reaction are the two shown in Fig. 1. The contribution from the uncorrelated Λ(1405)p system will be observed as a peak in the Λp invariant mass spectrum. Due to the antisymmetrization for the two protons in 3 He, we have two contributions from each diagram in Fig. 1. Therefore, the scattering amplitude of the reaction T can be expressed as The antisymmetrized 3 He wave function is given in Appendix A, and T 1,2,3,4 come from the first, second, third, and fourth terms of the 3 He wave function in Eq. (A18), respectively. 2 In the following we give the explicit form of each amplitude T i . Let us first fix the amplitude T 1 , which comes from the left diagram in Fig. 1. The momenta of the three nucleons in 3 He are denoted as p µ 1 , p µ 2 , p µ 3 from left to right. Thus, we have From these momenta, we can construct the momenta in Jacobi coordinates as in Appendix A, and here we show only the relevant expressions to the present formulation: Then, by using a scheme similar to that developed in Refs. [35][36][37][38] and the 3 He wave function summarized in Appendix A, we can evaluate the scattering amplitude T 1 as where Γ K is the kaon absorption width by two nucleons in theKN N system, whose value is fixed to be Γ K = 15 MeV so as to reproduce the kaon absorption width of theKN N bound state in the fixed center approximation [21] (see Fig. 4 and related discussions below). We note that the exchanged kaon after the secondary scattering (q µ ) goes highly off its mass shell in the present kinematics, so the term im K − Γ K is unnecessary in the denominator of the corresponding propagator. The energies of the intermediate kaons are fixed in two ways: one employs the Watson approach [39] and the other one relies on the truncated Faddeev approach [40], to which we refer as options A and B, respectively. Namely, in option A we have [38] with B3 He = 7.7 MeV being the 3 He binding energy, while in option B we have [40] where E p (q) ≡ m p + q 2 /(2m p ). The Watson approach contains more contributions from diagrams of the rescattering of nucleons via the N N interaction, while the truncated Faddeev approach can give a correct threshold behavior (see Ref. [38] for details).
The spinors χ, χ Λ , χ p , and χ n stand for initial-state 3 He, and final-state Λ, proton, and neutron, respectively, all of which being either χ ↑ = (1, 0) t or χ ↓ = (0, 1) t . Since we assume that the spin direction of 3 He equals that of the bound neutron, we take the same spinor for both of them. The 3 He wave functionΨ is evaluated with the harmonic oscillator potential, and its explicit form is given in Appendix A. 3 The Yukawa K − pΛ vertex gives rise to Pauli matrices σ and a coupling constantṼ where f is the meson decay constant, taken to be f = 93 MeV, while D = 0.795 and F = 0.465 are adjusted to the weak decay of baryons. We also introduce a form factor for this vertex. We take a typical cutoff value Λ = 0.8 GeV, but the cutoff dependence of the cross section will be discussed later on. The K − n → K − n scattering amplitude, T , is a function of the center-of-mass energy for the initial kaon-bound neutron system, w 1 , and the scattering angle in their center-of-mass frame θ 1 , both of which are evaluated by neglecting the Fermi motion of the bound neutron. As a result, we have where We note that the value of right-hand side in Eq. (17) may become larger than 1 or smaller than −1 because the bound nucleons actually have an energy and momentum distribution different to the free one. In such a case we take cos θ 1 = 1 or −1, respectively. Now that w 1 and θ 1 are fixed by the momenta of the initial-and final-state particles, we can put T outside of the integral. In addition, for this amplitude T 1 we neglect the spin flip contribution, and hence we can factorize the spinor part χ † n χ. The amplitude T , is a function of the center-of-mass energy for the exchanged kaon (q µ ) and final-state proton, w 2 : Since the relevant energies to our study are those near the K − p threshold, we only consider the s-wave part of the amplitude T . We calculate this K − p → K − p amplitude in 3 We would have a factor 1/ √ 6 from the 3 He wave function as in Eq. (A18), but this factor will be compensated in the cross section by the identical contributions of six diagrams of different topology. These correspond to having the first scattering, T 1 , in either of the three nucleons and the second scattering, T 2 , in either of the remaining two nucleons. The final states have the triplet npΛ produced in different order and these contributions add incoherently in the cross section. the so-called chiral unitary approach [1,2,4,5], where the Λ(1405) resonance is dynamically generated from the meson-baryon degrees of freedom. For the chiral unitary amplitude we take into account ten channels: K − p,K 0 n, π 0 Λ, π 0 Σ 0 , π + Σ − , π − Σ + , ηΛ, ηΣ 0 , K 0 Ξ 0 , and K + Ξ − . The formulation of the amplitude T 2 in the chiral unitary approach is summarized in Appendix C. An important point is that in this amplitude we take into account the kaon absorption by two nucleons in theKN N system effectively via the inclusion of a width Γ K in the kaon propagator [see Fig. C1(a) in Appendix C].
As a consequence, the explicit form of T 1 finally becomes In a similar manner, we can write the formulas of the other scattering amplitudes. Here we note that, although one needs to antisymmetrize the momentum and spin of the nucleons in 3 He, the wave functionΨ(p λ , p ρ ) in Eq. (A15) is unchanged for the exchange of momenta p i ↔ p j (i, j = 1, 2, and 3), since the global argument of the Gaussian functions . Therefore, we have to consider the antisymmetrization of the spin variables only, and we have Here ωK0(p) ≡ p 2 + m 2K 0 with theK 0 mass mK0 and w ′ 1 and cos θ ′ 1 are fixed in the same manner as in Eqs. (16) and (17), respectively, but for the K − p →K 0 n reaction instead of the K − n → K − n one. The energy p ′ 0 is fixed as B3 He in option A, with E n (q) ≡ m n + q 2 /(2m n ), where p ′ 0 is different from p 0 as the former contains the neutron mass or energy. The amplitudes T are evaluated phenomenologically in Appendix B, while the amplitudes T are taken from a chiral unitary approach in s wave, as described in Appendix C.

Scattering amplitude: generating aKNN quasi-bound state
In this subsection we consider the case of the formation of aKN N quasi-bound state in the 3 He(K − , Λp)n reaction, which would be the origin of the peak structure seen in the J-PARC E15 experiment. TheKN N quasi-bound state is generated as the multiple scattering of the kaon between two nucleons after emission of a fast neutron in the reaction. The most relevant diagrams are shown in Fig. 2.
Taking into account the antisymmetrization for three nucleons, we have six contributions to the scattering amplitude of the reaction: The amplitudes T 1,2,3,4,5,6 come from the first, second, third, fourth, fifth, and sixth terms of the 3 He wave function in Eq. (A18) in Appendix A, respectively. Let us consider the first term T 1 . We can use the same form as in Eq. (20), keeping the scattering amplitude of the first step, T , the K − propagator of p µ , and the 3 He wave function. Then, the most important part of the scattering amplitude, i.e. the part where thē KN N quasi-bound state is generated and the kaon is absorbed, remains to be implemented. This is represented by the diagrams shown in Fig. 3, which are calculated as follows. First, we do not consider spin flips during the multiple scattering since the process takes place near theKN N threshold. Therefore, the spinor factor χ † p χ ↑ χ † Λ σχ ↓ is the same as in Eq. (20). Second, the multiple scattering amplitude of Fig. 3 is calculated employing the so-called fixed center approximation to the Faddeev equation [17,21], and we denote this part as T FCA . Third, the kaon absorption takes place after the multiple scattering, which is Fig. 2 The three most relevant Feynman diagrams depicting the three-nucleon absorption of a K − implementing the multiple kaon scattering between two nucleons, which is represented by the shaded rectangles (see Fig. 3). be evaluated in the same manner as the K − pΛ vertex in Eq. (20). Here, we have two types of vertices, K − pΛ andK 0 nΛ, and both have the same structure and coupling constant as in Eq. (14). As a consequence, the scattering amplitude T 1 can be obtained by replacing T where everything, except for the amplitude of the multiple scattering T FCA , is evaluated as in Eq. (20). In particular, the energies of the kaons, p 0 and q 0 , are fixed by the option A or B.
The multiple scattering amplitude T FCA ij is labeled by channel indices i and j. Since we have theKN N system decaying into Λp, we consider six channels in the order K − pp,K 0 np, K 0 pn, ppK − , npK 0 , and pnK 0 . In this study, we neglect diagrams with pion exchange between two nucleons in the T FCA ij amplitude, since these contributions are found to be small (see Appendix D). We note that we distinguish the ordering of the kaon and the two nucleons; in channels where the kaon appears first (last), the kaon interacts with the first (second) nucleon of the ordering at the first or last of the scatterings. In this sense, the T represent K − pp and ppK − →K 0 pn scatterings, respectively, and theK 0 is absorbed via theK 0 nΛ vertex. Since we fix the ordering of the final state in the reaction as npΛ from left to right, the final-state channel of the multiple scattering should be 1 or 3, where the kaon is absorbed by the second nucleon of the pair after the interaction with the first one which emits the proton. We also note that the multiple scattering amplitude in the fixed center approximation is a function of the invariant mass 9 Fig. 4 Absolute values of the kaon multiple amplitudes evaluated in the fixed center approximation.
M Λp . The formulation of the fixed center approximation is given in Appendix D, and the details are given in Refs. [17,21].
In a similar manner, we can calculate the amplitudes T 2,3,4,5,6 as: 10/27 In Fig. 4  + T FCA 63 , which are evaluated in the fixed center approximation as functions of M Λp . As one can see from the figure, the amplitudes T FCA show a peak around M Λp = 2.35 GeV, which corresponds to a signal of theKN N quasi-bound state in our approach. Therefore, in the Λp invariant mass spectrum of the 3 He(K − , Λp)n reaction, we will observe such a peak structure if the strength of the peak for the signal of theKN N quasi-bound state is strong enough. The amplitude |T FCA 11 + T FCA 41 | has the strongest signal peak, since theKN N quasi-bound state is generated dominantly by the K − p → K − p scattering in the K − pp configuration. Here we note that the kaon absorption width Γ K is fixed as 15 MeV, so that the amplitude T F CA reproduces the width of theKN N bound-state signal in [21]. We also note that we cannot see a clearKN N threshold around 2.37 GeV in the amplitude T FCA in Fig. 4 because we introduce a finite width for the kaon propagators of the amplitude. In contrast, the cusp at 2.25 GeV is caused by the πΣN threshold, where the πΣ degrees of freedom are intrinsically contained in the two-body chiralKN →KN amplitudes employed in the construction of the T FCA amplitudes.

Numerical results
In this section we show the numerical results of our calculations for the cross section of the 3 He(K − , Λp)n reaction. We fix the initial kaon momentum in the laboratory frame to be k lab = 1 GeV/c, as in the experiment [29]. We note that the momentum of the intermediate kaon after the first K − n → K − n escape or K − p →K 0 n escape collision becomes 50 MeV/c for an initial kaon momentum of k lab = 1 GeV/c, if the escaping neutron goes forward. This means that the initial K − energy chosen favors the production of low energy kaons and is suitable to see a possibleKN N state around its threshold.
As mentioned before, we consider two scenarios to reproduce a peak structure in the Λp invariant mass spectrum of the 3 He(K − , Λp)n reaction. One consists of generating the Λ(1405) but not allowing it to form a bound state with the remaining proton. The uncorrelated Λ(1405)p system eventually decays into a Λp pair. The other consists of generating aKN N quasi-bound state which subsequently decays into Λp. The numerical results of the spectrum in these two scenarios are shown in Sec. 3.1 and Sec. 3.2, respectively.
Our numerical results are compared with the experimental data [29] and in particular to their analysis in terms of a Breit-Wigner amplitude with parameters M X = 2355 +6 −4 (stat.) ± 12(sys.) MeV and Γ X = 110 +19 −17 (stat.) ± 27(sys.) MeV [29]. 4 The comparison between our results and this mass spectrum helps us to discuss the origin of the peak structure in the invariant mass spectrum. 11 Fig. 1) with the option A. The experimental (E15) data for the mass spectrum dσ/dM Λp and its fit [Eq. (32)] are taken from Ref. [29] and are shown in arbitrary units.

Generating an uncorrelated Λ(1405)p system
First, we consider the case that the uncorrelated Λ(1405)p system is generated in the in-flight 3 He(K − , Λp)n reaction, as formulated in Sec. 2.2. The numerical results of the differential cross section are shown in Figs. 5 and 6 for the options A and B, respectively. As seen from Figs. 5 and 6, both in the options A and B, we find the peak of the invariant mass spectrum dσ/dM Λp at around the K − pp threshold, M Λp ≈ m K − + m p + m p = 2.370 GeV. This corresponds to the signal of the uncorrelated Λ(1405)p. Here we emphasize that the uncorrelated Λ(1405)p gives a peak not at the Λ(1405)p threshold, which is about 4 We note that the parameters for this mass spectrum as well as the experimental data shown in the figures were obtained without acceptance corrections.  2.355 GeV in our model, but just at the K − pp threshold in the option A or at 2.4 GeV in the option B. This is because the Λ(1405) is slowly moving due to the momentum p carried by the kaon after the first scattering, and hence the moving Λ(1405) and the proton (the third nucleon of 3 He) have an invariant mass larger than the Λ(1405)p threshold. However, regardless of the option A or B, the peak position is not consistent with the experimental data and their fit [Eq. (32)], which are shown in Figs. 5 and 6 in arbitrary units. The peak position in the experiment is more than 20 MeV lower than that of our result in the uncorrelated Λ(1405)p case. In addition, we cannot reproduce the behavior of the tail of the peak below the K − pp threshold, say at M Λp = 2.3 GeV. This fact indicates that the experimental signal observed in the in-flight 3 He(K − , Λp)n reaction at J-PARC [29] is definitely not the uncorrelated Λ(1405)p state.

12/27
In Figs. 5 and 6, we also show the behavior of the angular distribution of the cross section dσ/d cos θ cm n as well as the double differential cross section d 2 σ/dM Λp d cos θ cm n . Here we show the results only in the region cos θ cm n ≥ 0, since there is no significant contribution in 13/27 the region cos θ cm n < 0. From the figure, one can see that the uncorrelated Λ(1405)p signal dominantly comes from the condition of forward neutron emission, i.e. θ cm n ≈ 0 degrees. This is caused by both kinematic and dynamical reasons. As far as kinematics is concerned, we note that for finite scattering angle, θ cm n > 0 degree, the kaon momenta in the intermediate states |p| and |q| become large and are comparable to the initial kaon momentum 1 GeV/c in the laboratory frame. Therefore, the kaon propagator 1/[(q 0 ) 2 − q 2 − m 2 K ] and the form factor F(q) suppress the cross section for θ cm n > 0 degree 5 . With respect to the dynamics, we observe that the cross sections of K − p →K 0 n and K − n → K − n at k lab = 1 GeV/c have a local maximum for forward neutron emission, as seen in Fig. B1 in Appendix B, and hence the forward neutron emission is also favored in the 3 He(K − , Λp)n reaction. Because of these reasons, the uncorrelated Λ(1405)p signal favors the condition of forward neutron emission, θ cm n ≈ 0 degrees 6 . We also note that the peak shifts slightly upwards as the scattering angle θ cm n increases, which means that the Λ(1405) gets more momentum from the kaon after the first scattering. This contribution can be seen as the band from theKN N threshold at cos θ cm n = 1 to the lower-right direction in the d 2 σ/dM Λp d cos θ cm n plot of the figures, although its strength is very weak due to the kinematic and dynamical reasons explained above.
As for the dependence of the cross section on the cutoff of the form-factor employed in the K − pΛ vertex (15), we have found that Λ values in the range [0.7 GeV, 1.0 GeV] produce only quantitative differences in the corresponding mass spectra and angular distributions, while their shape is preserved. In particular, with the cutoff value Λ = 1.0 GeV the height of the mass spectrum becomes about 1.3 times larger than that with Λ = 0.8 GeV in both options A and B, and likewise the size of the spectra gets reduced by about 20% for a cutoff value of Λ = 0.7 GeV. Let us discuss the difference between the results in the options A and B. In the option A (B), the peak height of the mass spectrum dσ/dM Λp is about 110 µb/GeV (80 µb/GeV), and the peak position is about M Λp = 2.37 GeV (2.4 GeV). This difference of the peak structure is caused by the treatment of the kaon energies in the intermediate state and could be interpreted as a theoretical ambiguity in calculating the mass spectrum of this reaction in the present formulation.
In order to see the structure created by the underlying kinematic features of the amplitudes rather than by the uncorrelated Λ(1405)p system, we take the amplitude T 2 as a constant: The result of the mass spectrum with this constant amplitude is plotted in Fig. 7. As one can see, even if we do not take into account the Λ(1405) contribution, a peak appears in the M Λp mass spectrum just above 2.4 GeV. This is due to the quasi-elastic scattering of the kaon in the first collision of the process. Namely, the intermediate kaon after the neutron emission at T 1 goes almost to its on mass shell, where the denominator of the propagator 1/[(p (′)0 ) 2 − p 2 − m 2 K + im K Γ K ] gets close to zero. Then, the peak position and its height is slightly different in the options A and B. Since the energy p (′)0 contains the kinetic energies 5 Since (q 0 ) 2 − m 2 K is always negative in the present kinematics, the square of this propagator, 1/[(q 0 ) 2 − q 2 − m 2 K ] 2 , monotonically decreases as |q| increases. 6 This means that the neutron goes in opposite direction to the original one in the K − n → K − n center-of-mass frame, and equivalently the kaon goes backward in that frame.  Fig. 7 Mass spectrum for the Λp invariant mass of the in-flight 3 He(K − , Λp)n reaction with a constant T 2 .
of the nucleons with negative signs in the option B, the denominator, (p (′)0 ) 2 − p 2 − m 2 K + im K Γ K , gets close to zero for a larger value of p ′ 0 Λ + p ′ 0 p in this option. This kinematic fact makes the peak in option B to appear at a higher energy compared to that in A. We emphasize that this shift of the peak position makes a significant difference in the signal region. Actually, at theKN N threshold the mass spectrum in Fig. 7 can be twice as large in option A than in option B. As a consequence, the uncorrelated Λ(1405)p contribution, which can be calculated essentially by the product of the squared amplitude |T 2 | 2 and the mass spectrum in Fig. 7 and whose peak position eventually appears around theKN N threshold, shows a non-negligible difference in both options.
The other lesson that we learn from this exercise is that the peaks in Fig. 7 are shifted upwards by about 30 MeV with respect to those in Figs. 5 and 6. This reflects the fact that the excitation of the Λ(1405) in Figs. 5 and 6 puts strength to the left of the quasielastic kaon peak, which makes it appear at lower energies merged with the signal of the uncorrelated Λ(1405)p pair.

Generating aKNN quasi-bound state
In the previous subsection we have shown that the experimental signal observed in the inflight 3 He(K − , Λp)n reaction at J-PARC [29] is definitely not the uncorrelated Λ(1405)p state. Here we consider the case that aKN N quasi-bound state is generated after the first kaon scattering T 1 , as formulated in Sec. 2.3. The numerical results are shown in Figs. 8 and 9.
An important thing to be noted is that the peak structure in our mass spectrum, regardless of the option A or B, is consistent with the experimental results. In particular, we can reproduce qualitatively well the behavior of the tail of the peak below the K − pp threshold, say at M Λp = 2.3 GeV. We also note that the width of our peak is similar to that in the experiment. Therefore, our mass spectrum supports the explanation that the peak structure observed in the experiment of Ref. [29] is indeed a signal of theKN N quasi-bound state.
Besides, interestingly, we observe a two-peak structure of the mass spectrum around thē KN N threshold from the figures in both options A and B. The peak just below theKN N   Fig. 2) in the option A. The experimental (E15) data and its fit in the mass spectrum dσ/dM Λp are taken from Ref. [29] and are shown in arbitrary units.
threshold corresponds to theKN N quasi-bound state, while the second peak just above theKN N threshold comes from the quasi-elastic scattering of the kaon, as seen in Fig. 7. Note that, since the mass spectrum is essentially a product of the mass spectrum in Fig. 7 and the square of the multiple scattering amplitude, |T FCA | 2 , the obtained mass spectrum has the peak associated to that of |T FCA | 2 and another peak corresponding to that of the quasi-elastic kaon scattering, which is shifted to slightly lower energies by the effect of the energy dependence of |T FCA | 2 above its peak. We emphasize that we cannot generate such a two-peak structure when we consider the case of the uncorrelated Λ(1405)p system in the previous subsection. In principle, we would always have a possibility of having such a two-peak structure even in the uncorrelated Λ(1405)p system: one associated with the quasi-elastic scattering of the kaon, i.e., the first 16  intermediate antikaon practically on-shell, and another structure linked with the "Λ(1405) signal" of T 2 . However, the location of these structures and their particular shapes in the uncorrelated Λ(1405)p case make them merge into just one broader peak. In contrast, for the case of theKN N quasi-bound state the locations of the peaks from the two origins are separated enough such that a two-peak structure remains. The fact that the spectrum in Fig. 7 Fig. 4 falls fast to the right, also help in creating a dip in between the two peaks. This discussion indicates that, if such a two-peak structure would be observed in experiments, this could be a strong evidence that there should be a certain state originating from a dynamical factor, such as aKN N bound state, in addition to the quasi-elastic scattering of the kaon.
Note that the peak height of the mass spectrum in the case of theKN N bound state becomes about half of that of the uncorrelated Λ(1405)p case. This is due to the combination of the structure tied to quasi-elastic kaon scattering in the first collision, shown in Fig. 7, with that of the scattering amplitude, which is either T 2 , in the uncorrelated Λ(1405)p case, 17  or T FCA , in theKN N case. Hence, the peak produced by the T 2 or T FCA amplitude will be multiplied by the quasi-elastic kaon scattering structure of Fig. 7. Actually, although the peak heights of T 2 and T FCA are similar, their peak positions are displaced energetically, and this fact produces a drastic effect in the final value of the corresponding spectrum. Indeed, the peak of T 2 for the uncorrelated Λ(1405)p case appears, due to some intrinsic momentum of the generated Λ(1405), at invariant M Λp masses above theKN N threshold and close to 2.4 GeV, which is the position of the kinematical peak associated to quasi-elastic scattering, hence producing an enhanced effect in the resulting mass spectrum. This is opposite in the correlated case producing theKN N bound state, since the T FCA amplitude peaks around 2.35 GeV, a region where the strength of the kaon quasi-elastic structure has fallen down appreciably with respect to its peak. This explains why the mass spectrum obtained in the case of theKN N bound state becomes half of that for the uncorrelated Λ(1405)p case. In Figs. 8 and 9, we also plot the angular distribution of the cross section dσ/d cos θ cm n as well as the double differential cross section d 2 σ/dM Λp d cos θ cm n . Again there is no significant contribution in the region cos θ cm n < 0. From these results, we can see that the structure at theKN N threshold is generated dominantly in the condition of forward neutron scattering. The reason is the same as that discussed in the previous subsection. In addition, we may observe two bands in the d 2 σ/dM Λp d cos θ cm n plot of the figures, although their strength is weak; one goes from M Λp ≈ 2.35 GeV at cos θ cm n = 1 to the lower direction, and the other goes from theKN N threshold at cos θ cm n = 1 to the lower-right direction. Actually, the former is the signal of theKN N quasi-bound state, and the latter is the contribution from the quasi-elastic scattering of the kaon.
Similarly to what we find in the uncorrelated Λ(1405)p case, the use of different cutoff values only affects the size of the mass spectrum and angular distribution of theKN N bound state signal. More specifically, using a cutoff value of Λ = 1.0 GeV (0.7 GeV) gives rise to a mass spectrum which is about 1.3 times higher (1.2 times lower) than that with Λ = 0.8 GeV in both options A and B.
Finally, we compare the results of the mass spectrum employing the options A and B in Fig. 10. As one can see, the strength of the peak is different in the two options, although 18/27 their distributions become similar at the tails. In fact, the total cross section, calculated by integrating the mass spectrum dσ/dM Λp with respect to the invariant mass M Λp , gives 7.6 µb for option A and 5.6 µb for option B. This difference of the strength can be explained from the results displayed in Fig. 7: the different location of the quasi-elastic kaon scattering peak produces a larger strength for option A than for option B.
In Fig. 10 we also plot the J-PARC E15 data and its fit, which is scaled so as to reproduce the empirical value of 7 µb [29]. We find that the absolute value of our mass spectrum is qualitatively consistent with the experimental one. 7

Conclusion
We have studied the reaction 3 He(K − , Λp)n measured recently in the J-PARC E15 experiment for a kaon beam of 1 GeV/c. This momentum is suited to produce secondary kaons nearly stopped when the neutrons go fast and forward in the laboratory system, which is guaranteed by a form factor that suppresses high momentum kaons in the intermediate state and K − n → K − n or K − p →K 0 n cross sections that peak at backward kaon angles in the center of mass frame.
We have concentrated our study on mechanisms which involve the participation of the three nucleons in 3 He, allowing us to compare our results with the part of the experimental spectrum where there are no spectator nucleons. Our approach relies on a first collision of the K − with a nucleon in 3 He, leading to the production of a slowK that is absorbed by the remaining N N pair. However, before the absorption takes place, the kaon is allowed to interact repeatedly with this pair of nucleons, hence providing a source of binding for thē KN N system.
Technically, we employ a fully antisymmetrized 3 He wave function, which leads to many different combinations of first and second scattering processes, and we use Jacobi coordinates to describe the 3 He system. TheKN interaction around threshold is obtained within a chiral unitary approach, and the interaction of the kaon with the two nucleons is treated in terms of the fixed center approximation to the Faddeev equation, which has proved to be fair enough to deal with this problem.
The results that we obtain are interesting. We observe that the Λp invariant mass distribution is clearly shifted to higher energies compared to experiment when the interaction of the slow kaon with the remaining two nucleons is not taken into account. The situation changes drastically when this interaction is considered, allowing for the formation of aKN N quasibound state, and producing a spectrum which reproduces the shape, position, and width of the experimental distribution. The total cross section obtained is also in good agreement with experiment within uncertainties.
A novel aspect of our approach is the presence of a two-peak structure in the Λp invariant mass spectrum, for which we provide a justification in terms of a quasi-elastic peak forK production in the first collision of the reaction and a peak associated to the production of theKN N bound state that decays into Λp. The present data, which have large error bars, does not permit to infer this behavior. It will be interesting to see what the coming update of the experiment, with larger statistics, provides.
In any case, from the analysis of this experiment and comparison with the data, we can claim that the peak seen is indeed tied to the production of a quasi-boundKN N state, with properties very similar to those obtained in the work of [21] and to the average of other theoretical works.

A. The wave function of 3 He
In this study, we evaluate the wave function of the 3 He nucleus with the harmonic oscillator potential governed by the following Hamiltonian: wherep i and r i are the momentum operator and coordinate for the ith nucleon, respectively, and k is the spring constant taken as a parameter of the system. The mass of the nucleon is fixed as m 1 = m 2 = m 3 = m N . In order to separate the center-of-mass motion, we introduce the Jacobi coordinates and their conjugate momentâ P ≡p 1 +p 2 +p 3 , respectively. With these Jacobi coordinates, we can rewrite the kinetic part of the Hamiltonian, regardless of values of the masses, as On the other hand, in the potential term the λ and ρ modes decouple when m 2 = m 3 , which is the case for 3 He, and its expression is where the oscillator frequency is defined as As a consequence, the Hamiltonian can be rewritten aŝ Now let us omit the center-of-mass motion and evaluate the wave function in terms of the λ and ρ modes. For the nucleons in 3 He, we assume that all of them are in the s-wave state and neglect contributions from higher partial waves such as d wave, which are known to be small. In this condition, the wave function is expressed as the product of harmonic oscillator wave functions for the λ and ρ modes, both in the ground state: with λ ≡ |λ| and ρ ≡ |ρ| 2 . The normalization of the wave function is From the wave function, we can calculate the mean squared radius of the 3 He nucleus as the expectation value of r 2 i for the ith nucleon measured from the center of mass. Namely, A straightforward calculation gives As expected, one obtains the same value of the mean squared radius for each nucleon with m 1 = m 2 = m 3 = m N . Then the parameter k can be fixed with the empirical value of the 21/27 mean squared radius of 3 He, r 2 i = 3.2 fm 2 . The result is In this study we use the 3 He wave function in momentum space to calculate the scattering amplitude. For the harmonic oscillator potential, we can easily evaluate the wave function in momentum space as where p λ ≡ |p λ | and p ρ ≡ |p ρ |. This is related to the wave function in coordinate space as and its normalization is Finally, taking into account the antisymmetrization for the nucleons in 3 He, we express the full wave function of the 3 He nucleus as for the spinor of the 3 He, χ = χ ↑ = (1, 0) t or χ ↓ = (0, 1) t . In this study the spin direction of 3 He is taken to be the same as that of the neutron.
B. K − p →K 0 n and K − n → K − n scattering amplitudes at k lab = 1 GeV/c In this Appendix we formulate the K − p →K 0 n and K − n → K − n scattering amplitudes at k lab = 1 GeV/c, which is needed to emit the fast neutron in the final state of the 3 He(K − , Λp)n reaction. In this study, we neglect the spin flip contribution and estimate the scattering amplitude TK N →KN of these reactions at k lab = 1 GeV/c from the differential cross section dσK N →KN /dΩ with the following formula where θ is the scattering angle for the kaon, w 1 ≡ (pK + p N ) 2 , and m N is the nucleon mass. When we theoretically calculate the differential cross section, we always fix the initial kaon momentum as k lab = 1 GeV/c. We parametrize the differential cross section dσK N →KN /dΩ by the Legendre polynomials P l (x) as dσK N →KN dΩ = l c l P l (cos θ), with constants c l , which are fixed so as to reproduce the experimental data. For the proton target reaction, we have many experimental data points for the differential cross section at 22/27 Table B1 Parameter sets for the K − p →K 0 n and K − n → K − n scattering amplitudes at k lab = 1 GeV/c. All the parameters are given in units of mb/sr. 0.76 k lab = 1 GeV/c [41,42], so we take the polynomials up to l = 4. From the best fit we obtain the parameters c l listed in Table B1. For the neutron target reaction, on the other hand, only the data at k lab = 1.138 GeV/c [43] and at k lab = 0.862 GeV/c [44] are available, so we take the polynomials up to l = 2 and make a rough fit to the cross sections at these momenta. As a result, we obtain the parameters in Table B1. For both reactions, the fitted differential cross sections are shown in Fig. B1 together with the experimental data.

C.KN →KN scattering amplitude around threshold
In this Appendix we briefly introduce theKN →KN scattering amplitude around threshold, which appears in the secondary scattering of the 3 He(K − , Λp)n reaction and in the multiple scattering of theKN N system. For this amplitude we employ the so-called chiral unitary approach [1,2,4,5], modified to take into account the kaon absorption by two nucleons in theKN N system in a simple way. In this study we introduce ten meson-baryon channels: K − p,K 0 n, π 0 Λ, π 0 Σ 0 , π + Σ − , π − Σ + , ηΛ, ηΣ 0 , K 0 Ξ 0 , and K + Ξ − . Since we are interested in the amplitude around theKN threshold, we only consider its s-wave part.
In the chiral unitary approach, theKN →KN scattering amplitude T ChUA ij , where i and j are channel indices, is the solution of the coupled-channels Lippmann-Schwinger equation  [41,42] at k lab = 1 GeV/c for a proton target, and from Ref. [43] at k lab = 1.138 GeV/c and from [44] at k lab = 0.862 GeV/c for a neutron target. 23 in an algebraic form: Here, w is the center-of-mass energy, V ij is the interaction kernel taken from chiral perturbation theory, and G k is the meson-baryon loop function. In this study the interaction kernel V ij is fixed to be the leading order term of chiral perturbation theory for the s-wave meson-baryon scattering, i.e., the Weinberg-Tomozawa term, whose expression is given in Ref. [5]. We note that in this scheme we can effectively take into account the kaon absorption by two nucleons, whose diagram is shown in Fig. C1(a), in a simple way by just adding the imaginary part of that diagram by means of an empirical width Γ K in the kaon propagator. The loop function is then evaluated with the cutoff scheme as