Feasibility study of the $K^{+} d \to K^{0} p p$ reaction in killing/saving the"$\Theta ^{+}$"pentaquark

We investigate theoretically the $K^{0} p$ invariant mass spectrum of the $K^{+} d \to K^{0} p p$ reaction and scrutinize how the signal of the"$\Theta ^{+}$"pentaquark, if it exists, emerges in the $K^{0} p$ spectrum. The most prominent advantage of this reaction is that we can clearly judge whether the"$\Theta ^{+}$"exists or not as a direct-formation production without significant backgrounds, in contrast to other reactions such as photoproduction and $\pi$-induced productions. We show that while the impulse or single-step scattering process can cover the"$\Theta ^{+}$"energy region with an initial kaon momentum $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ in the laboratory frame, the contributions from double-step processes may have a potential possibility to reach the"$\Theta ^{+}$"energy region with a higher kaon momentum $k_{\mathrm{lab}} \sim 1 \, \mathrm{GeV} / c$. Assuming that the full decay width of the"$\Theta ^{+}$"is around $0.5\, \mathrm{MeV}$, we predict that the magnitude of the peak corresponding to the"$\Theta^+$"is around a few hundred $\mu \mathrm{b}$ to $1\, \mathrm{mb}$ with the momentum of the kaon beam $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ while it is around $\lesssim 1\, \mu \mathrm{b}$ with $k_{\mathrm{lab}} \approx 0.85\, \mathrm{GeV} / c$. Thus, the"$\Theta^+$"peak is more likely to be seen at $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ than at $k_{\mathrm{lab}} \approx 0.85 \,\mathrm{GeV} / c$.


Introduction
The physics of pentaquarks, which are the baryons consisting of four valence quarks and one anti-quark, has been renewed very recently, as the LHCb Collaboration announced the new findings of three heavy pentaquarks, P c 's [1][2][3][4]. The LHCb Collaboration also found the five excited Ω c 's in the channel of the Ξ + c K − invariant mass [5]. The four of them were confirmed by the Belle Collaboration [6]. Since these newly found excited Ω c 's have very small decay widths, several theoretical works have suggested that at least some of them may be identified as the singly heavy pentaquarks [7][8][9][10]. On the other hand, the discussion of the light pentaquarks became dormant, which was once triggered by the theoretical prediction [11] and the first measurement of the "Θ + " [12], since the null results of the "Θ + " baryon reported by the CLAS Collaboration [13][14][15]. Moreover, both the KEK-PS E533 Collaboration [16] and the J-PARC E19 Collaboration [17,18] searched for the "Θ + " using the pion beam but found no significant peak corresponding to the "Θ + " pentaquark. The Belle Collaboration looked for isospin partners of the "Θ + " in the first observed process γγ → ppK + K − but again has no significant evidence for them [19]. All these negative results make the existence of the "Θ + " rather skeptical, so that both experimental and theoretical investigations on the "Θ + " ebbed away.
In the meanwhile, the LEPS Collaboration and DIANA Collaboration continued to report the evidence for the existence of the "Θ + " [20][21][22][23]. Some years ago, Amaryan et al. analyzed the data from the CLAS Collaboration [24], using the interference method with φ-meson photoproduction. They found the peak around ∼ 1.54 GeV, which corresponds to the "Θ + ". The statistical significance of this peak was 5.3 σ [24]. In Ref. [25], the SELEX data on hadro-nucleus collisions at Fermilab were analyzed in searching for formation of the "Θ + ". A narrow enhancement near 1539 MeV was observed in the mass spectrum of the pK 0 S system emitted at small x F from hadron collisions with copper nuclei, where x F denotes the Feynman variable defined as the ratio of the momentum p * L /p * max (For details see Ref. [25]). However, the results from Ref. [25] show definite dependence on the kinematics.
After the LHCb Collaboration reported the existence of the heavy pentaquarks, interest in light pentaquarks seems to be renewed. For example, the existence of a narrow nucleon resonance N * (1685) has been announced by a series of experiments in η photoproduction off the quasi-neutron [26][27][28][29][30][31][32][33][34][35]. More recently, a similar narrow peak was observed in the γp → pπ 0 η reaction [36]. Though the identification of this narrow resonance is still under debate, one possible interpretation is that it can be regarded as a pentaquark nucleon, which is a member of the baryon antidecuplet [37][38][39].
Based on previous experimental studies on the "Θ + ", we can draw at least one conclusion: the "Θ + " is most unlikely to exist. Even though it might exist, it is elusive to observe it. However, we want to mention that almost all previous experiments have utilized indirect methods such as the photon and pion beams, which suffer from large backgrounds. Moreover, we know that the "Θ + ", if it exists, decays into K 0 p or K + n. In particular, the K + n channel may be the most probable one to search for the "Θ + ". In fact, the DIANA Collaboration used the low-energy K + Xe reaction in the xenon bubble chamber [22,23,[40][41][42], though there is also a theoretical criticism on the DIANA results in 2003 [43]. Nevertheless, the K + beam may provide an ultimate smoking gun whether the "Θ + " exists or not, since it will create the "Θ + " by direct formation and will be seen in the differential and total cross sections, if it exists. Thus, measuring the K + d → K 0 pp reaction is the final experiment to put a period to the existence of the "Θ + " pentaquark. This process, compared with other reactions such as photoproduction and π-induced productions, is not hampered by significant backgrounds. This means that the experiment of the K + d → K 0 pp will clearly judge the existence of the "Θ + ".
The K + d → K 0 pp reaction was already investigated theoretically [44] with the width of the "Θ + " being assumed to be 1-20 MeV. In doing so, Sibirtsev et al. considered the single-step process or the impulse scattering process in which the proton in the deuteron was regarded as a spectator and the neutron interacts with the K + to produce the proton and the neutral kaon. When the K + momentum lies in the range of 0.47-0.64 GeV/c, the peak corresponding to the "Θ + " was seen in the K 0 p invariant mass spectra. In the vicinity of 0.47 GeV/c, the "Θ + " peak was also shown in the total cross sections. In the present work, we include both the single-step and double-step processes and scrutinize the feasibility of the K + d → K 0 pp reaction in observing the "Θ + " pentaquark. In the double-step processes, a kaon is exchanged in the course of the interaction between the proton and the neutron. We will show that in the present work the single-step process can cover the energy region corresponding to the "Θ + " peak with an initial kaon momentum k lab ≈ 0.4 GeV/c in the laboratory (Lab) frame while the double-step processes provide a potential possibility to reach the "Θ + " energy region with a higher kaon momentum k lab ≈ 1 GeV/c. In the present work, thus, we will carefully investigate the K + d → K 0 pp reaction in the context of a possible existence of the "Θ + ", considering both the single-and double-step processes.
This paper is organized as follows. In Sec. 2, we formulate the cross section of the K + d → K 0 pp reaction. The KN → KN scattering amplitude is also shown in this section. In Sec. 3, we give numerical results on the cross section of the K + d → K 0 pp reaction and investigate strength of a peak signal in the K 0 p spectrum, which will provide a good guideline to conclude whether the "Θ + " exists or not. Section 4 is devoted to the summary of this study.

Formulation
First of all, we formulate the cross section of the K + d → K 0 pp reaction. Since we are interested in the K 0 p invariant mass spectrum, in which we search for the "Θ + " signal, it is convenient to calculate the differential cross section as a function of the K 0 p invariant mass together with the scattering angle for the other proton. In this respect, we can express the differential cross section of this reaction as [45][46][47][48], Before we explain Eq. (1), let us distinguish the two protons in the final state. We will call the proton that is involved in producing the "Θ + " together with K 0 as the "first" proton, whereas the other one is called as the "second" proton. M K 0 p in Eq. (1) denotes the invariant mass of the K 0 and "first" p, θ ′ 2 stands for the scattering angle for the "second" proton in the center-of-mass (CM) frame of the K + d system, and Ω * K represents the solid angle for the K 0 in the rest frame of the K 0 and first p. W is the CM energy of the K + d system, and m d and m p correspond to the masses of deuteron and proton, respectively. The prefactor of the cross section contains the following momenta: the initial kaon momentum k cm and the final second proton momentum p ′ 2 are evaluated within the CM frame, while the final kaon momentum p * K is obtained in the rest frame of the K 0 and first p. They are calculated as where λ(x, y, z) ≡ x 2 + y 2 + z 2 − 2xy − 2yz − 2zx. |T | 2 denotes the squared scattering amplitude. When we calculate the K 0 p invariant mass spectrum or total cross section, we need a factor 1/2 to avoid the double counting of the two protons in the final state: 3/17 Fig. 1 Diagrams for the K + d → K 0 pp reaction. Momenta of particles are shown in parentheses.
2.2. Scattering amplitude of the K + d → K 0 pp reaction Next we construct the scattering amplitude of the K + d → K 0 pp reaction. Since the deuteron has spin 1, the scattering amplitude can be denoted by T a with a = 1, 2 and 3 to specify the deuteron spin component. As depicted in Fig. 1 together with the momenta of particles, the reaction mechanism consists of the three main diagrams: T a 1 stands for the impulse scattering process [44], and T a 2 and T a 3 represent the double-step scattering processes, where the intermediate kaons K + and K 0 propagate respectively between two nucleons 1 . Thus, the K + d → K 0 pp scattering amplitude is expressed as the sum of these three contributions: where the antisymmetric terms are required owing to the identical fermions, i.e., protons in the final state. We now derive the K + d scattering amplitude in the Lab frame, in which the deuteron three-momentum satisfies p d = 0. In particular, we evaluate each KN → KN amplitude in the target-baryon rest frame, as we will show below. The impulse scattering amplitude T a 1 , depicted in Fig. 1(a), is calculated as [48] T a Here, T K + n→K 0 p stands for the K + n → K 0 p scattering amplitude in a 2 × 2 matrix form, which is represented in the spin space of the nucleon, and the superscript t designates the transpose of a 2 × 2 matrix. The K + n → K 0 p amplitude depends on the CM energy and three-momenta of the initial and final kaons in the Lab frame, k and k ′ , respectively. The deuteron spin component is denoted by (S † ) a = −iσ 2 σ a / √ 2 (a = 1, 2, 3) in a 2 × 2 matrix form with the Pauli matrices σ a .φ is the deuteron wave function in momentum space, for which we neglect the d-wave component because it is negligibly small. An analytic parameterization of the s-wave component [50] facilitates the deutron wave function to be handled in an easy manner with C j and m j determined in Ref. [51]. As we mentioned previously, each part of the K + d scattering amplitude, i.e., the deuteron wave function and the K + n → K 0 p amplitude, is evaluated in the Lab frame. The expression of the K + n → K 0 p amplitude in the targetbaryon rest frame will be given in Sec. 2.3. The double-step scattering amplitudes, T a 2 and T a 3 , which are depicted respectively in Fig. 1(b) and (c), are calculated as [48] where F (q) represents a form factor for which we take a Gaussian form of F (q) = exp(−q 2 /Λ 2 ) with a cutoff Λ. The kaon energies in the propagators are fixed in the truncated Faddeev approach [47] as where the energy-momenta of the particles are fixed in the Lab frame and m N is the averaged nucleon mass m N = (m p + m n )/2. The CM energy for the first collision is given by w 2 = (q µ + p ′ µ 2 ) 2 , and hence it depends on the Fermi motion of bound nucleons as well as on the initial-kaon momentum.
As for the antisymmetric terms, we have to calculate the scattering amplitudes where the momenta and spins of the two protons are simultaneously exchanged, i.e., (p µ 1 , s 1 ) ↔ (p µ 2 , s 2 ) with s 1,2 being the spins of protons. This antisymmetrization for the present scattering amplitudes can be performed in the following manner: Finally, the squared scattering amplitude in Eq. (1) is obtained by the spin average and summation for the initial deuteron and final protons, respectively, which results in the following expression [48] |T | 2 = 1 3 the partial waves in the CM frame of the KN system, and then transform it to that in the target-baryon rest frame, taking the method developed in Ref. [48]. The KN amplitude is generally expressed in the KN CM frame as: where w denotes the CM energy and p * in (p * out ) stands for the three-momentum for the initial (final) kaon in the CM frame. Then, we can define p * out,in ≡ |p * out,in | and x * ≡ p * out · p * in /(p * out p * in ). The Pauli matrices σ act on the nucleon spinors, and g cm KN →KN and h cm KN →KN are expressed in terms of the partial waves as with the Legendre polynomials P L (x), P ′ L (x) ≡ dP L /dx, and orbital angular momentum L. Next we transform the above-given amplitudes to that in the target-baryon rest frame, which is in general written by where parameters in the target-baryon rest frame are expressed without asterisks in contrast to those in the CM frame. The amplitudes g KN →KN and h KN →KN are expressed in terms of g cm KN →KN and h cm KN →KN as [48]: where ω in , ω out , and E out denote respectively the initial kaon, final kaon and final nucleon energies in the target-baryon rest frame, and a out , a * in,out and b * in,out arise from Dirac spinors, which are defined as 6/17 We now construct the partial-wave amplitudes T cm KN →KN, L± , which should be in general off-shell amplitudes and thus functions of three independent variables: w, p * in , and p * out . In the present study, we assume that the partial-wave amplitudes depend on the momenta minimally required by the kinematics, i.e., the off-shell amplitudes are proportional to (p * out p * in ) L . Under this assumption, we have an advantage that the on-shell amplitudes can simulate the off-shell amplitudes by introducing the formula where p on-shell is the on-shell momentum for the KN system: with the isospin-averaged kaon mass m K . Since we need the KN amplitudes in the energy range from its threshold to w ∼ 2 GeV, we utilize the on-shell KN amplitude developed in the SAID program [53], in which they provide the on-shell KN amplitude in various partial waves. We here take the SAID partial-wave amplitudes up to the D waves and calculate the off-shell amplitudes by using Eq. (19). Finally we introduce the "Θ + " contribution, which is just added as an s-channel "Θ + " exchange term to the KN scattering amplitude in the present study. Here we assume the "Θ + " as an isosinglet, and examine four different cases of its spin/parity J P = 1/2 ± and 3/2 ± . The KN "Θ" coupling is governed by an effective Lagrangian as follows: in the spin 1/2 case, where Γ = 1 (iγ 5 ) for the negative (positive) parity and g KN Θ denotes the coupling constant, and in the spin 3/2 case. This provides us with the formula for the "Θ + " decay width: where M Θ stands for the "Θ + " mass, p * K designates the CM momentum of the final-state KN system, and E N (p) ≡ m 2 N + p 2 . Thus, using the "Θ + " mass M Θ and full decay width Γ Θ = Γ Θ→K + n + Γ Θ→K 0 p , we can fix the coupling constant g KN Θ . In the present study we use a presumable value of the mass M Θ = 1524 MeV [20] and a predicted value of the decay width Γ Θ = 0.5 MeV [52], which results in g KN Θ = 0.783 for J P = 1/2 + , g KN Θ = 0.101 for 1/2 − , g KN Θ = 0.352 for 3/2 + , and g KN Θ = 2.734 for 3/2 − . Then, the s-channel "Θ + " 7/17 exchange term enters into the partial-wave K + n → K + n amplitude as The "Θ + " contributions to the K + n → K 0 p and K 0 p → K 0 p amplitudes are evaluated with the isospin relation T

Numerical results and discussion
We are now in a position to present the numerical results and discuss their physical implications, in particular how the signal of the "Θ + " pentaquark, if it exists, emerges in the K 0 p spectrum.

Two mechanisms to reach the "Θ + " energy
Before we discuss the details of the K 0 p spectrum, we first examine which conditions are better to reach the "Θ + " energy region and to search for its peak. To this end, we calculate the K 0 p invariant mass of the K + d → K 0 pp reaction, assuming that nucleon Fermi motion is zero and that the reaction takes place only in the impulse scattering process. In this case, one proton is produced from a zero-momentum neutron in the initial state of the K + n → K 0 p reaction, while the other proton comes out just as a spectator. Then, we can calculate the K 0 p invariant mass in two ways: combining K 0 with the produced proton in the impulse scattering process, and doing K 0 with the spectator proton. Namely, in terms of the formulation in Sec. 2.1, the former (latter) case means that the produced proton is the "first" ("second") proton while the spectator is the "second" ("first") proton. The K 0 p invariant mass in these two ways is plotted in Fig. 2 in terms of the squared boxes (former) and curves (latter), respectively. We note that, in general, the possible K 0 p invariant mass discussed here is slightly smeared compared with those in Fig. 2 due to the Fermi motion of the nucleons inside a deuteron. As one can see, on the one hand, the squared boxes in Fig. 2 reaches around the "Θ + " energy region ∼ 1.52 GeV with the initial-kaon momentum k lab ≈ 0.40 GeV/c. This means that one can investigate the "Θ + " energy region directly in the impulse scattering process with the initial kaon momentum k lab ≈ 0.40 GeV/c and with the backward "second" proton. On the other hand, the curves in Fig. 2 suggest that even with higher kaon momenta one can reach the "Θ + " energy region by observing the forward "second" proton. In this case, although the impulse scattering process cannot produce the "Θ + ", the double-step scattering one can do it, where the intermediate kaon may lead to the formation of the "Θ + ", being combined with the "first" proton.
Therefore, the present study consists of two folds. Firstly, we check whether a possible "Θ + " signal will appear in the impulse scattering process with lower kaon momenta k lab ≈ 8/17  Fig. 2 Possible K 0 p invariant mass of the K + d → K 0 pp reaction with the impulse scattering process as a function of the scattering angle for the "second" proton in the global CM frame θ ′ 2 . The initial kaon momentum in the Lab frame k lab is taken from 0.30 GeV/c to 1.00 GeV/c in intervals of 0.05 GeV/c. We assume zero Fermi motion. The squared boxes represent the invariant mass of K 0 and produced proton of the impulse process, while the lines represent that of K 0 and spectator proton. 0.40 GeV/c. In fact, Ref. [44] already carried it out. Thus, in the first part of the present work, we extend Ref. [44] and perform a more detailed analysis of the study. Secondly, we investigate whether the double-step scattering contribution with higher kaon momenta k lab ∼ 1 GeV/c can generate the "Θ + " in the K + d → K 0 pp reaction.

Lower kaon momentum
Let us first consider the case of lower kaon momenta. In this case, we need to examine the impulse scattering process. This allows one to check how the results of the cross section for the K + d → K 0 pp reaction are affected when the "Θ + " is taken into account.   Fig. 3 Results for the differential cross section of the K + d → K 0 pp reaction in the impulse scattering process. The initial kaon momentum is fixed as k lab = 0.45 GeV/c. The "Θ + " with J P = 1/2 + is taken into account. 9/17 In order to see how the "Θ + " influences the cross section, we draw in Fig. 3 the results for the differential cross section d 2 σ/dM K 0 p cos θ ′ 2 with the initial kaon momentum k lab = 0.45 GeV/c. The "Θ + " with J P = 1/2 + is taken into account. Here we draw only the region cos θ ′ 2 ≤ −0.9 because there is no significant structure in the region cos θ ′ 2 > 0.9. We find two structures in Fig. 3: a sharp peak at M K 0 p = 1.524 GeV as a "Θ + " signal and a broad bump at M K 0 p = 1.54 GeV and cos θ ′ 2 = −1 corresponding to the squared boxes in Fig. 2, arising from the kinematical effects. The Fermi motion of the bound neutron due to the deuteron wave function, on the one hand, makes the peak at (M K 0 p , cos θ ′ 2 ) = (1.54 GeV, −1) broad. On the other hand, thanks to the same Fermi motion of the bound neutron, we can reach the "Θ" energy in the impulse scattering process even when the initial kaon momentum does not exactly match the kaon momentum that generates the two-body CM energy 1.524 GeV with a free nucleon at rest, i.e. the kaon momentum k lab ≈ 0.40 GeV/c. Therefore, we can observe the "Θ + " signal with k lab = 0.45 GeV/c as in Fig. 3. Integral range of the scattering angle is −1 < cos θ ′ 2 < −0.8. The "Θ + " with J P = 1/2 + is taken into account.
In Fig. 4 we plot the K 0 p invariant mass spectra of the K + d → K 0 pp reaction with three initial-kaon momenta k lab = 0.40 GeV/c, 0.45 GeV/c, and 0.50 GeV/c. Here we take into account the "Θ + " contribution with J P = 1/2 + , and we integrate with respect to the scattering angle in the range −1 < cos θ ′ 2 < −0.8. As one can see from Fig 4, on the one hand, the broad-peak structure, which corresponds to impulse scattering of the initial kaon and almost on-shell bound neutron, moves upward as k lab increases, as expected from the squared boxes in Fig. 2. On the other hand, the "Θ + " signal stays at 1.524 GeV with different values of k lab . Among three values of the kaon momenta, k lab = 0.40 GeV/c yields the highest peak at M K 0 p = 1.524 GeV for the "Θ + " signal on top of the broad peak. This is a consequence of the momentum matching in Fig. 2.
We then examine other spin/parity combinations of the "Θ + " pentaquark: J P = 1/2 − , 3/2 + , and 3/2 − . In Fig. 5 we show the K 0 p invariant mass spectrum of the K + d → K 0 pp reaction with the initial kaon momentum k lab = 0.40 GeV/c and with the "Θ + " of J P = 10/17   Fig. 5 K 0 p invariant mass spectrum of the K + d → K 0 pp reaction in the impulse scattering scattering with the "Θ + " spin/parity J P = 1/2 ± and 3/2 ± . The initial kaon momentum is fixed as k lab = 0.40 GeV/c. Integral range of the scattering angle is −1 < cos θ ′ 2 < −0.8.
1/2 ± and 3/2 ± . The integral range of the scattering angle is −1 < cos θ ′ 2 < −0.8. Figure 5 indicates that the peak height for the "Θ + " signal in different quantum numbers is similar to each other. These peaks generate the "Θ + " production cross section ∼ several hundred µb to 1 mb with k lab ≈ 0.40 GeV/c.   6 Total cross section of the K + d → K 0 pp reaction as a function of the initial kaon momentum in the Lab frame k lab . We take into account only the impulse scattering. The experimental data are taken from Refs. [54][55][56][57].
Finally we calculate the total cross section of the K + d → K 0 pp reaction with the "Θ + " contribution of spin/parity J P = 1/2 ± and 3/2 ± . The result is shown in Fig. 6 together with the old experimental data on K + d → K 0 pp scattering [54][55][56][57]. 2 Note that similar results were already obtained in Ref. [44] in which however the width of the "Θ + " was taken to be 1-20 MeV. As shown in Fig. 6, even if the decay width of the "Θ + " is as small as Γ Θ = 0.5 MeV, which is approximately two to forty times smaller than those in Ref. [44], one can observe a bump structure around the initial kaon momentum in the Lab frame k lab = 0.4 GeV/c. The height of the bump gives indeed a few hundred µb to 1 mb. While the old experiments have a lack of the data in the vicinity of k lab ≈ 0.4 GeV/c, new experiments at the J-PARC, if it is performed exclusively near this value of the initial kaon momentum in the near future, can judge the existence of the "Θ + ", because the size of the bump structure (a few hundred µb to 1 mb) is still strong enough to be seen.

Higher kaon momentum
We now focus on the case of higher kaon momenta. To reach the "Θ + " energy region with higher kaon momenta, we need to consider the double-step scattering process where the initial K + produces a proton from the deuteron in the first collision, losing some part of its momentum. Then, it interacts with the other nucleon in the second collision. In this process, the first collision corresponds to the K + p → K + p or K + n → K 0 p reaction of the forward proton emission. In this sense, the initial kaon momentum should be chosen such that the forward proton emission efficiently takes place in the first collision. In other words, we require a specific initial kaon momentum in such a way that the K + p → K + p and K + n → K 0 p cross sections with the forward proton emission should be large. In fact, this strategy was employed to search for aKN N quasi-bound state in the K −3 He → Λpn reaction in the J-PARC E15 experiment [58,59]. In the J-PARC E15 experiment, to generate aKN N quasi-bound state, they planned to prepare a slow antikaon and two of the three bound nucleons in a 3 He by using the K − n → K − n or K − p →K 0 n reaction with the fast forward neutron emission as the first collision, which eventually leads to theKN N quasi-bound state (see also a theoretical calculation of the K −3 He → Λpn reaction in Ref. [60]). To prepare a slow antikaon and fast forward neutron emission as much as possible, the initial K − momentum k lab = 1.0 GeV/c was selected in Refs. [58,59].
As shown in the top panels of Fig. 7 for the differential cross sections of theKN → KN reaction as functions of the initial antikaon momentum k lab and antikaon scattering angle θ K , theKN →KN differential cross section indeed reveals a peak structure at k lab ≈ 1 GeV/c and cos K ≈ −1, which is essential to obtain a slow antikaon with the forward neutron emission.
When it comes to the KN → KN case, the bottom panels of Fig. 7 illustrate the K + p → K + p and K + n → K 0 p reaction cross sections, which indicates that k lab = 0.8-0.9 GeV/c are the best values for the present study of the K + d → K 0 pp reaction. With these kaon momenta, we obtain the largest cross sections of the K + n → K 0 p and K + p → K + p reactions at cos θ K ≈ −1, which corresponds to the forward proton emission. Thus, we fix the initial kaon momentum to be k lab = 0.85 GeV/c and compute the K 0 p invariant mass spectrum 12/17 of the K + d → K 0 pp reaction. Note that we take into account here both the double-step scattering process and the impulse scattering one. In Fig. 8, we draw the results of the differential cross section in the "Θ + " energy region including the "Θ + " with J P = 1/2 + . Figure 8 exhibits three structures: a band at cos θ ′ 2 0.5, thin line at M K 0 p = 1.524 GeV, and sharp peak at (M K 0 p , cos θ ′ 2 ) = (1.524 GeV, −1). The first band structure corresponds to the line in Fig. 2 and originates from the impulse scattering contribution. Note, however, that it was parametrized in terms of the invariant mass of K 0 and spectator proton. The second one, the line structure, represents a signal of the "Θ + " in the double-step scattering process. The third one, which corresponds to the sharp peak around 1.524 GeV, arises from the "Θ + " production in the impulse scattering with a highly off-shell bound nucleon. This contribution, however, depends on the tail of the deuteron wave function in momentum space and thus contains large theoretical uncertainty. Therefore, we do not regard this third sharp peak structure at (M K 0 p , cos θ ′ 2 ) = (1.524 GeV, −1) as an important one. Now the very important task is to answer how much the signal of the "Θ + " in the doublestep scattering process, that is, the thin line structure in Fig. 8, is significant compared to the impulse scattering contribution parametrized in terms of the invariant mass of K 0 and spectator proton, band in Fig. 8. To do that, we integrate the differential cross section of Fig. 8 with the integral range 0 < cos θ ′ 2 < 1, which results in the K 0 p invariant mass 13 Fig. 8 Differential cross section of the K + d → K 0 pp reaction in the impulse plus double scattering processes. The initial kaon momentum is fixed as k lab = 0.85 GeV/c. The "Θ + " with J P = 1/2 + is taken into account.  Fig. 9 K 0 p invariant mass spectrum of the K + d → K 0 pp reaction in the impulse and double-step scattering processes with the "Θ + " spin/parity J P = 1/2 ± and 3/2 ± . The initial kaon momentum is fixed to be k lab = 0.85 GeV/c. The integral range of the scattering angle is given as 0 < cos θ ′ 2 < 1. The inset represents an enlarged figure. spectrum in Fig. 9. Here we introduced the "Θ + " contribution of spin/parity J P = 1/2 ± and 3/2 ± . As shown in Fig. 9, with every spin/parity of the "Θ + ", the impulse scattering contribution is dominant over that of the double-step processes, while the "Θ + " signal is tiny. By calculating the excess area of the spectrum in Fig. 9 on top of the background, i.e. impulse scattering contribution, we find that the "Θ + " production cross section turns out to be about 0.8 µb, 0.2 µb, 2 µb, and 5 µb for the spin/parity of the "Θ + " J P = 1/2 + , J P = 1/2 − , J P = 3/2 + , and J P = 3/2 − , respectively. Therefore, we expect that in the K + d → K 0 pp reaction with higher kaon momenta k lab ≈ 0.85 GeV/c the measurement of the production cross section 1 µb is required to save the "Θ + " pentaquark. 14/17

Summary and outlook
In the present work, we have investigated the K + d → K 0 pp reaction as a feasibility study to suggest the kinematical conditions for the most probable range of the initial kaon momentum and to judge the existence of the "Θ + " pentaquark in this reaction. We consider two different dynamical processes for the K + d → K 0 pp reaction, that is, the single-step or impulse scattering process and the double-step scattering processes. While the first one was already considered in a previous study, the latter was ignored. In the present work, we took into account both the processes and scrutinized the relevant kinematical conditions to each process and their relevances in the production of the "Θ + " pentaquark. We showed explicitly that, to produce the "Θ + ", the impulse scattering process is dominant over the double-step scattering process in lower momentum regions (k lab ≈ 0.40 GeV/c), whereas the double-step one takes over the impulse one in higher momentum regions (k lab ≈ 0.85 GeV/c). We found that the strength of the bump structure corresponding to the "Θ + " is about a few hundred µb to 1 mb in the lower momentum region, while it is about 1 µb in the higher momentum region.
The K + beam has a unique feature in investigating the existence of the "Θ + ", compared with almost all experiments done previously. This tentative pentaquark state "Θ + " is strongly coupled to either K + n or K 0 p. It implies that the charged K + beam provides a chance to produce the "Θ + " by diract formation. Thus, it is not required to resort to any complicated methods of experimental analyses to observe the "Θ + ", if it exists. In this sense, the J-PARC is the best place to perform the ultimate experiments with the K + beam to put a final period to the matter of the "Θ + " existence. It is physically worthwhile to carry out such experiments in the future. If the experiments at the J-PARC find that "Θ + " does not exist, it will bring any debate on the existence of the "Θ + " to an end. However, if the experiments yield any evidence for its existence, it will reignite interest in the physics of the light pentaquarks.