Production of hyperon resonances induced by kaon on a deuteron target

The $K^- d \to \pi YN$reaction is theoretically studied so as to investigate the $K^{-}$-induced production of the hyperon resonances $\Sigma (1385)$ and $\Lambda (1405)$. For this purpose we take into account the p-wave amplitudes for the meson-baryon two-body scatterings as well as the s-wave amplitudes. Due to the fact that the hyperon resonances are selectively produced from the $\bar{K} N$ channel in this reaction, the $\Lambda (1405)$ peak appears at 1420 MeV, which implies that $\Lambda (1405)$ and $\Sigma (1385)$ could be separately seen in the missing mass spectrum of the emitted nucleon in the $K^{-} d \to n X$ reaction. The $\pi Y$ invariant mass spectrum in this study is consistent with experimental data both for $\Sigma (1385)$ and $\Lambda (1405)$. The pion exchange contributions are also included and are found to give smooth background without distorting the peak structure of the hyperon resonances.


Introduction
The properties of hyperon resonances are important for understanding the role of strangeness in hadron physics. In particular, the understanding of hyperon resonances located below theK N threshold, such as (1405) and (1385), is essential for the physics of strangeness in nuclear matter and nuclei. In addition, because baryonic resonances decay into mesons and a baryon in strong interactions, one can learn about basic interactions between hadrons with strangeness from the properties of hyperon resonances.
Among the various hyperon resonances, recent attention is particularly focused on the (1405) hyperon resonance with J P = 1/2 − and I = 0. The (1405) resonance has long been considered a quasibound state ofK N [1,2], being an extremely important resonance for understandingK N interaction. Theoretically, (1405) is successfully reproduced as dynamically generated states in the coupled-channel approach based on chiral dynamics [3][4][5][6][7][8][9][10][11][12][13][14][15], and this approach has confirmed that (1405) is predominantly described by meson-baryon components [16]. Recently, it has also been pointed out in Ref. [10] that (1405) is composed of two resonance states with different coupling natures and that the one that dominantly couples to theK N channel is located at 1420 MeV instead of nominal 1405 MeV. Since the resonance position of (1405) in theK N channel is strongly related to the strength of theK N interaction, it is very important to observe (1405) spectra in theK N → π channel and pin down the resonance position of (1405) seen below theK N threshold.

Formulation
In this study, we consider the K − d → π 0 n, (π ) 0 n, π − p, and (π ) − p reactions so as to discuss the production of the hyperon resonances (1385) and (1405) initiated by theK N channel. The kinematical variables are given in Fig. 1. We discuss the kinematical aspects of the reactions in Sect. 2.1, followed by the dynamics of the reaction including the scattering amplitudes for the K − d reaction in Sect. 2.2 and the meson-baryon scattering amplitudes in the Appendix.

Reaction kinematics
The reaction K − d → π Y N requires five variables to completely fix the phase-space of the threebody final state [31]. In this study we are interested in the mass spectra of the π Y systems, thus we choose as the five variables the π Y invariant mass M π Y , the solid angle of the final N in the total center-of-mass frame N , and the solid angle of the final π in the π Y center-of-mass * π . Then, the cross section of the reaction is calculated by where M d , M Y , and M N are the initial-state deuteron and final-state hyperon and nucleon masses in the reaction, respectively, E c.m. is the total center-of-mass energy, k c.m. is the K − momentum in the total center-of-mass frame, θ N is the scattering angle of N with respect to K − in the total centerof-mass frame, and T is the T -matrix of the reaction. In this form the azimuthal angle of N is integrated. Now let us pin down the reaction mechanism. Since we want to produce hyperon resonances in the final state, diagrams in which π and Y come from the same vertices are essential. Here, following Ref. [21], we evaluate the impulse and double scattering amplitudes for the production of the hyperon resonances.
First, we consider the K − d reaction with neutron emission in the final state. The relevant diagrams for this reaction are given in Fig. 2. Diagram 1 in Fig. 2 corresponds to the impulse process for hyperon resonance production, whereas diagrams 2 and 3 are double scattering processes. We emphasize that the conservation of strangeness in the strong interaction guarantees the production of hyperon resonances from theK N channel both in the impulse and double scattering processes. The π Y systems are fixed as charge zero ones, π 0 , π + − , π − + , and π 0 0 . The π 0 (π 0 0 ) mass spectrum will be dominated by (1385) ( (1405)), since it is a purely I = 1 (0) channel. The π + − and π − + mass spectra can contain (1385) in addition to (1405), since (1385) has a branching ratio of 12% to the π channel.
Second, we consider the K − d reaction with proton emission in the final state. The relevant diagrams are given in Fig. 3. In this case we have only two diagrams so as to emit the final-state π and Y from the same vertex; one is the impulse diagram and the other the double scattering diagram.  Fig. 2. Diagrams for the K − d → π Y n reaction. In the diagrams, T 1 and T 2 denote the scattering amplitudes for K − N →K n andK N → π Y , respectively.
In the diagrams, T 1 and T 2 denote the scattering amplitudes π Y systems are π − , π − 0 , and π 0 − , and all of them contain only the − (1385) contribution as hyperon resonances of interest. Now that we have determined the reaction diagrams, all we have to do is the evaluation of the scattering amplitudes, which contain the dynamics of the hyperon resonance production.

K − d scattering amplitudes
In this section, we formulate the scattering amplitudes of the K − d reaction given in Figs 2 and 3. These amplitudes are composed ofK N →K N and π Y amplitudes, a kaon propagator, and a deuteron wave function. We study both (1385) and (1405) in the reaction by including s-and p-wave contributions in theK N amplitudes. It is important that the p-wave contribution requires a nucleon spin component in the deuteron wave function, which is not needed in the case of (1405) production with the s-waveK N amplitudes [21].
First, we formulate the amplitude of the impulse approximation given in Figs 2(1) and 3(1). According to Ref. [21] and taking into account the spin component inside the deuteron, we can write the scattering amplitude for the impulse approximation as Here, TK N →M B is theK N → M B amplitude in a 2 × 2 matrix form, which compensates the Pauli spinor for baryons, and is a function of the center-of-mass energy M π Y and the anglek ·p π in the meson-baryon center-of-mass frame, (S † ) a = −iσ 2 σ a / √ 2 (a = 1, 2, 3) is the spinor component for each nucleon inside the deuteron in a 2 × 2 matrix form, andφ( p) is the momentum representation of the deuteron wave function with momentum p, for which we neglect the d-wave component 4 and use a parameterization of the s-wave component given by an analytic function [32] as with C j and m j determined in Ref. [33]. Second, let us consider the double scattering amplitudes. According to Ref. [21] and taking into account the spin component inside the deuteron, the double scattering amplitudes are formulated as where q 0 and W are fixed as q 0 = k 0 + M 1 − p 0 N and W = (M 1 + k 0 ) 2 − k 2 , with M 1 the first-scattered nucleon mass and p 0 N the energy of the final-state nucleon. This prescription of q 0 takes account of the N N potential in the deuteron nonperturbatively, as suggested in the Watson formulation [23]. The superscript t denotes the transposition of the matrix in the spin space.
In our approach, theK N →K N and π Y amplitudes are essential to the production of the hyperon resonances. For the meson-baryon amplitudes we apply the so-called chiral unitary approach [3][4][5][6][7][8][9][10]. In the chiral unitary approach, (1405) is dynamically generated by the unitarized coupled-channel method based on chiral dynamics without explicit poles [16]. In contrast, (1385) can be included as an explicit pole in the p-wave kernel interactions in the coupled-channel approach [34]. The details of the description of theK N →K N and π Y amplitudes for (1405) and (1385) are given in Appendix A.
The chiral unitary amplitude has been calculated in the center-of-mass frame of the two-body meson-baryon system. Since the deuteron wave function has been calculated in the rest frame of the deuteron, we calculate the K − d → π Y N amplitude in the laboratory frame, in which the target deuteron is at rest. Thus, we make a transformation of the amplitude obtained in the two-body centerof-mass frame to the baryon rest frame using the method shown in Appendix B. For the p-wave amplitude, we define the off-shell behavior by using a form factor Fig. 4. Diagrams of the π meson exchange contribution. T 2 and T 3 denote the scattering amplitudes for K N → π Y and π N → π N , respectively. Table 1. Possible exchange pions for the diagram shown in Fig. 4.
The total amplitude for K − d → π Y n and π Y p is given by the coherent sum of the impulse and double scattering contributions as respectively. Then, the squared amplitude spin-summed for the final state and averaged in the initial state (deuteron) is given as denoting the trace of the 2 × 2 spin space matrix by "tr".

Estimation of the pion exchange contributions
It is instructive to estimate the pion exchange contribution to the production of the hyperon resonances, which mainly comes from the amplitudes without π Y correlation in the final state. In the estimation of the pion exchange amplitude, we evaluate the diagrams given in Fig. 4. The particles, which are correlated with the pion exchange process, are listed in Table 1. They contain the π N → π N amplitude, which does not appear in the usual diagrams for the production of hyperon resonances given in Figs 2 and 3. For the π N → π N amplitude, we take an empirical amplitude [35] up to the p-wave including the (1232) resonance, which is obtained based on the observed scattering data. The off-shell extrapolation is done in the same way for the kaon exchange. We show a Dalitz plot in order to see the energy range of the π N amplitude. In Fig. 5, we have found that the π N energy range, which is considered from (1405) and (1385) production, includes (1232) resonance. Therefore, the contribution of (1232) production seems to be mainly smooth background from π N scattering in (1405) and (1385) production. 6

Numerical results
In this section we show our numerical results for the production of hyperon resonances (1385) and (1405) in the K − d reactions. Using Eq. (1), we can evaluate the differential cross section as with the K − d → π Y N scattering amplitude T , which includes the pion exchange process. Then, integrating cos θ N , one can obtain the π Y mass spectrum: Further, integrating M π Y with an appropriate range (M min , M max ) for the hyperon resonances, the production cross section for the hyperon resonances is obtained as It is also interesting to calculate the angular dependence of the production by In this study, we evaluate the K − d amplitudes in Figs 2 and 3 by using the chiral unitary approach for the description of the meson-baryon scattering amplitudes. In this approach, (1405) is dynamically generated without introducing explicit poles in the s-wave amplitude, whereas (1385) is included as an explicit pole in the p-wave amplitude.
First, we compare our results with experiments in Sect. 3.1. Next, in Sect. 3.2, theoretical studies of the production of the hyperon resonances are given.

Production of hyperon resonances: Comparison with experimental data
(1385) production. First of all, we show the results of the (1385) production and make a comparison with the experimental data [25]. For the (1385) production, it is better to see the K − d → π − p reaction, in which (1405) does not contribute to the π − mass spectrum. Here, we include the pion exchange contribution in addition to the kaon exchange contribution for the hyperon resonance production.  Using Eq. (13), we show the π − mass spectrum in the K − d → π − p reaction in Fig. 6, together with the experimental data [25], which includes the background contribution. Here, the kaon momentum is fixed at 800 MeV/c in our calculation, whereas the range of the initiated kaon momentum is from 686 to 844 MeV/c in the experiment [25]. As seen in the figure, we can well reproduce the experimental mass spectrum with the (1385) peak around 1385 MeV of the π − invariant mass.
Next, let us evaluate the cross section of the (1385) production. In principle, such cross sections for the resonances can be evaluated theoretically by taking the residue of the scattering amplitude at the resonance pole. Here we use a more phenomenological technique for the comparison with experimental data, as follows. Since the pion exchange contribution gives only the non-resonant background, we take only the kaon exchange contribution for the production cross section, and integrate it in the range of M min = 1370 MeV and M max = 1400 MeV: where factor 1/0.88 comes from the branching ratio of (1385) → π , 88%, and σ F.G. is the cross section by the foreground process (kaon exchange contribution). We obtain the (1385) production cross section as 179 μb with 800 MeV/c incident K − , whereas the experimental value observed in the K − d → π − p reaction is reported to be 252 ± 30 μb at 778 MeV/c of the incident K − momentum [25]. As one can see, our cross section is consistent with the experimental value. There is, however, a small difference between our theoretical production cross section and the experimental one. This difference may come from the ways of subtracting the background contribution of the resonance. In Ref. [25], the background contributions have been estimated by fitting the mass spectrum in a sum of Legendre polynomials together with a relativistic Breit-Wigner form for the resonance. In the theoretical side, we have estimated the production cross section by using the diagrams in which π − and are emitted from the same vertex and we have not subtracted the non-resonant background appearing in the K − n → π − amplitude from (1385), which is clearly seen above the 1400 MeV of the invariant mass π − . We also note that the value of the production cross section depends on the choice of the range of the invariant-mass integration.  Finally, we should note that above the 1420 MeV of the π − invariant mass, the mass spectrum is not suppressed, although this is far above the (1385) energy region. This comes from the I = 1 non-resonant background in theK N scattering, which can interfere with (1405).

3.1.2.
(1405) production. Next we show the results of (1405) production in the K − d reaction. For (1405) production we choose the K − d → π + − n reaction and calculate the π + − mass spectrum. Here we note that the π + − spectrum comes not only from (1405) but also from 0 (1385), whose branching ratio to π is about 12%.
First, we show the π + − mass spectrum of the K − d → π + − n reaction in Fig. 7, together with the experimental data [25]. In the previous paper [21], the K − d reaction with only the s-wave mesonbaryon amplitude was calculated, whereas in this study we include both s-and p-wave meson-baryon amplitudes. In Fig. 7, we plot the full calculation with a coherent sum of s-and p-wave contributions with the solid line, whereas the s-(p-) wave contribution to theK N → π + − amplitude (T 2 in Figs 2 and 3) is shown by the dashed (dotted) line. The K − N →K N amplitudes T 1 in Figs 2 and 3 are fixed to be a coherent sum of the s-and p-wave contributions in every case. As can be seen from the figure, we reproduce well the experimentally observed mass spectrum of the reaction. We can see that the (1405) peak appears at 1420 MeV in π + − invariant mass instead of nominal 1405 MeV as in Ref. [21], without contamination from (1385) or the p-wave background contribution. Furthermore, it is important that the (1385) contribution appears as a shoulder around 1390 MeV π + − invariant mass, which may explain the bump structure in the empirical data.
Next, let us consider the (1405) production cross section from the kaon exchange contribution, which is calculated by the formula with the range M min = 1400 MeV and M max = 1440 MeV. The factor 3 comes from the branching ratio of (1405) → π + − , 33%. The (1405) production cross section is consistent with the empirical value [25], as obtained previously in Ref. [21].

Production of hyperon resonances: Theoretical studies
For the understanding of the production of the hyperon resonances in the K − d reaction, it is important to investigate theoretically the production mechanism of the reaction. Therefore, we undertake theoretical studies of the K − d reaction in this subsection. (1405) properties, it is important to understand the behavior of each π channel spectrum. Hence, we plot the π + − , π − + , and π 0 0 mass spectra in the K − d → (π ) 0 n reaction in Fig. 8. Here the initial kaon momentum is fixed at 800 MeV/c. From Fig. 8, one can see that the behavior of the π mass spectra differs slightly between the spectra. This is due to the interference between the (1405) contribution (I = 0) and the I = 1 non-resonant contribution. The significant non-resonant contribution with I = 1 around the (1405) energy can be seen in the π − spectrum shown in Fig. 6. As a consequence of the interference, the π + − spectrum shows the largest contribution of (1405) production and has a peak position at a higher energy than the π − + spectrum, which is consistent with the results obtained in Ref. [21]. The interference between the (1405) and I = 1 contributions has been, indeed, important in the photoproduction of (1405) [36][37][38]. Therefore, experimental data on the π ± ∓ spectrum will bring us further information on the (1405) structure.

π channel components. For clarification of the
We also note that the π 0 0 spectrum does not show the (1385) contribution, because the π 0 0 channel does not contain an I = 1 component. Hence, it will be important to observe all three (π ) 0 spectra in the K − d reaction and compare them in the experiment to obtain understanding of the (1405) structure.

Diagram contributions.
For the understanding of the K − d reaction, it is helpful to investigate each diagram contribution to the production of hyperon resonances. Here we show the π 0 0 and π 0 invariant-mass spectra separately plotted in each diagram contribution in Fig. 9. The incident kaon momentum is 800 MeV/c. As can be seen, both (1405) in the π 0 0 spectrum and (1385) in the π 0 spectrum show that diagram 1 (impulse contribution) in the reaction (Fig. 2) makes quite a small contribution, whereas diagram 2 in the reaction (Fig. 2) has the largest contribution. The reason that diagram 1, corresponding to the impulse production of the hyperon resonances, makes a small contribution is that the nucleons inside the deuteron hardly have higher momentum component as discussed in Ref. [21]. Namely, in order to produce the hyperon resonances below theK N threshold, one needs to create 10 an energetic nucleon in the final state of the K − d → π Y N reaction. Such an energetic nucleon is, however, scarcely produced in the impulse process of the K − d reaction, because the large Fermi momentum of the nucleon is highly suppressed by the deuteron wave function due to the small binding energy. As a consequence, the impulse production of the hyperon resonances makes quite a small contribution. Compared with the impulse process, the double scattering process (diagrams 2 and 3) is kinematically favored. Namely, in the double scattering process, the transferred energy can be taken from the incident kaon so that the exchanged kaon has less energy than that on-shell, which is favorable for the production of hyperon resonances below theK N threshold. Among the double scattering, since the K − n → K − n amplitude takes a larger value than that of the K − p →K 0 n amplitude in the energy region considered, diagram 2 makes the largest contribution to both (1405) and (1385) production (see also the discussion in Ref. [21]).

3.2.
3. Angular dependence of production. Now let us see the angular dependence of the production of the hyperon resonances. First of all, we show the double-differential cross section d 2 σ/d M π d cos θ n in the K − d → π + − n reaction in Fig. 10, in which the incident kaon momentum is fixed to be 800 MeV/c. In this reaction we can see the two peaks coming from two resonances, (1385) around 1385 MeV and (1405) around 1420 MeV, for 60 • ≤ θ n ≤ 150 • . As one can see, the (1405) peak has strong angular θ n dependence. The (1405) peak takes its largest value at θ n = 0 • , which corresponds to the forward neutron emission in the total center-ofmass frame, and it becomes about 10 times smaller in the region θ n ≥ 60 • . The reason for this is as follows. The momentum transfer by the exchanged kaon becomes smaller in the double scattering process in the region θ n ≤ 60 • , because in this case the incident kaon kicks out a neutron in the direction of the incident kaon and simply gives most of its momentum to the neutron in the first double scattering step. This small momentum transfer makes the exchanged kaon close to being on the mass shell, in which the double scattering amplitude takes a larger value. Hence, due to the kinematical reasons of dominant double scattering (discussed in the previous section) and of suitable momentum transfer, the (1405) peak has a large angular dependence and consequently backward (1405) production is dominant. The (1385) peak, on the other hand, shows only small angular dependence in the K − d → π + − n reaction. This is caused by the p-wave nature of (1385) in meson-baryon scattering. Namely, since the p-wave amplitude depends linearly on the transferred momentum, (1385) cannot be greatly produced with the neutron forward angle, in which the momentum transfer becomes smaller (as discussed above). This momentum transfer becomes larger as the angle θ n becomes larger, since the incident kaon has to kick out a neutron in the opposite direction of the kaon momentum in the first double scattering step. Therefore, as a combination of the p-wave nature of (1385) and a suitable region for momentum transfer in the process, (1385) production shows small angular dependence although, in the middle θ n region, the production is not favored by the kinematics.

Missing-mass spectrum.
As we have seen in the previous sections, the (1405) peak appears at 1420 MeV instead of nominal 1405 MeV in the π invariant mass of the K − d → π n reaction, thanks to the selective (1405) production by theK N channel. This fact implies that there is a possibility that the (1385) and (1405) peaks are seen separately in the π mass spectrum of the K − d reaction; they are usually mixed together due to their similar peak energies. Here, in order to investigate the behavior of the peak structures, we plot the missing-mass spectrum of the reaction K − d → n X. In the energy region around 1400 MeV, the missing-mass spectrum is expected to be dominated by the (π Y ) 0 system. Hence, evaluating the relevant diagrams shown in Figs 2 and 3 and neglecting the amplitudes in which π and Y do not correlate with each other, which will give no structure in the missing-mass spectrum, we can plot the missing-mass spectrum in Fig. 12. As one can see, a double-peak structure appears, coming from (1385) (around 1385 MeV) and (1405) (around 1420 MeV). In the missing-mass spectrum, the (1385) peak comes from the π 0 and the (1405) peak from the (π ) 0 channel. The reason why we can separate the (1385) peak and the (1405) one is, as discussed above, because we can produce (1405) from theK N initial channel, pinning down the production process via conservation of strangeness. 13

Summary
We have undertaken a study of the (1385) and (1405) production induced by K − on a deuteron target by calculating the K − d → π Y N reaction. We have taken into account both the s-and p-wave contributions of meson-baryon scattering in the K − d reaction; in particular, the contribution of the (1385) resonance is newly considered in this study. In the K − d → π Y N reaction, the hyperon resonances are created selectively by theK N channel. Thanks to this fact, the higher pole state of (1405) is produced and the π spectrum has a peak at around 1420 MeV. This finding implies that (1405) and (1385) could be seen separately in the missing-mass spectrum of the emitted nucleon in the K − d → n X reaction. We have found that the invariant-mass spectrum of π Y is consistent with experimental data [25], both for the (1385) and (1405) cases. We have investigated the angular dependence of the hyperon resonances and found that (1405) production mainly takes place in the backward direction, while (1385) production does not strongly depend on the angle of the emitted nucleon. We studied the production mechanisms of (1385) and (1405) from the theoretical side. We have also estimated the pion exchange contributions coming from diagrams in which π and Y are not correlated with each other and found that these contributions give a smooth background and do not spoil the peak structure of the hyperon resonances.
Substituting Eqs. (B5) and (B6) into Eqs. (B8) and (B9), we obtain where x = cos θ and x = cos θ with the angles between the mesons in the c.m. and baryon rest frames, respectively. These are related by the boost transformation: where β = | p 1 |/E 1 and γ = (1 − β 2 ) −1/2 . The amplitudes F (W, x ) and G (W, x ) can be decomposed by a partial wave in the laboratory frame: where (B20) Here we have used