The role of the triangle singularity in $\Lambda(1405)$ production in the $\pi^-p\rightarrow K^0\pi\Sigma$ and $pp\rightarrow pK^+\pi\Sigma$ processes

We have investigated the cross section for the $\pi^-p\rightarrow K^0\pi\Sigma$ and $pp\rightarrow pK^+\pi\Sigma$ reactions paying attention to a mechanism that develops a triangle singularity. The triangle diagram is realized by the decay of a $N^*$ to $K^*\Sigma$ and the $K^*$ decay into $\pi K$, and the $\pi\Sigma$ finally merges into $\Lambda(1405)$. The mechanism is expected to produce a peak around $2140$ MeV in the $K\Lambda(1405)$ invariant mass. We found that a clear peak appears around $2100$ MeV in the $K\Lambda(1405)$ invariant mass which is about $40$ MeV lower than the expectation, and that is due to the resonance peak of a $N^*$ resonance which plays a crucial role in the $K^*\Sigma$ production. The mechanism studied produces the peak of the $\Lambda(1405)$ around or below 1400 MeV, as is seen in the $pp\rightarrow pK^+\pi\Sigma$ HADES experiment.

In Ref. [29], the role of the triangle singularity (TS) on the angle and the energy dependence of the Λ(1405) photoproduction was studied. The triangle singularity was first pointed out in Ref. [42]. The correspond- * Electronic address: melahat.bayar@kocaeli.edu.tr ing Feynman diagram is formed by a sequential decay of a hadron and a fusion of two of them, and the amplitude associated with the diagram has a singularity if the process has a classical counterpart, which is known as Coleman-Norton theorem [43]. The studies of many processes including the triangle singularity elucidate the possible effect of the triangle singularity on the hadron properties, e.g., the η(1405) decay into π 0 a 0 or π 0 f 0 [44][45][46], the possible origin of Z c (3900) [47][48][49], the speculation on the pentaquark candidate P c [50][51][52] (see Ref. [53] for a critical discussion to the light of the preferred experimental quantum numbers [54]), the B s decay into Bππ [55] and B − decays [56,57]. Here, we note that the strength of the triangle peak is tightly connected with the coupling strength of the two hadrons merging into a third one. For example, in the study of the B − decay into K − π − D + s0 (D + s1 ) [56], the DK (D * K) in the triangle loop goes into D s0 (D s1 ), which is dynamically generated from the DK (D * K) and has a large coupling to this channel [58,59]. Then, the observation of the peak from the triangle mechanism would give an additional support for the hadronic molecular picture of these states.
For further understanding of the nature of the Λ(1405) and triangle mechanisms, in this paper we investigate the π − p → K 0 πΣ and pp → pK + πΣ processes including a triangle diagram. In both processes, the triangle diagram is formed by a N * decay into K * Σ followed by the decay of K * into πK and the fusion of the πΣ to form the Λ(1405), which finally decays into πΣ. In this process, the K * πΣ loop generates a triangle singularity around 2140 MeV in the invariant mass of KΛ(1405) from the formula given by Eq. (18) of Ref. [53]. The corresponding diagram is shown in Fig. 1. The N * resonance which strongly couples to K * Σ is obtained in Ref. [60] based on the hidden lo- FIG. 1: Triangle diagram for the Λ(1405) production from a N * resonance.
cal symmetry and the chiral unitary approach, and the analysis of the KΣ photoproduction off nucleon around the K * Λ threshold energy suggests that the resonance is responsible for the observed cross section [61].
As the result of our calculation, we found a peak in the KΛ(1405) mass distribution around 2100 MeV in both reactions, which is lowered with respect to the 2140 MeV given by the TS master formula [53] by the initial N * resonance which peaks around 2030 MeV. The experimental study on the Λ(1405) production from the π − p is reported in Refs. [62,63], but the energy is too small for the triangle singularity from the K * πΣ loop to be observed. The production of the Λ(1405) from the proton-proton collision is studied in Refs. [64][65][66]. The future observation of the inevitable peak from the triangle mechanism induced by the Λ(1405) would give further support for the molecular nature of the Λ(1405).
In this subsection we will study the effects of the triangle loop in the following decays: π − p → K 0 π + Σ − , π − p → K 0 π 0 Σ 0 and π − p → K 0 π − Σ + . The diagrams where the triangle singularity can appear for those reactions are shown in Fig. 2. To evaluate the differential cross section associated with this diagram we will use with m inv the invariant mass of the final πΣ, the momentum of the initial π − in the π − p center-ofmass frame (CM), the momentum of the final K 0 in the π − p CM, and the momentum of the final π in the πΣ CM. The resonance N * (2030) could have J P = 1 2 − or J P = 3 2 − as stated in Refs. [60,61] with a width Γ N * ≃ 125 MeV, but in order to have J = 3/2 in π − p we need p-wave and then we have positive parity. Hence, we take π − p in L = 0 and hence J P N * = 1 2 − . In the isospin basis the π − p → N * vertex has then the form (II.5) To estimate the g πN,N * we assume that Γ N * ,πN is of the order of 70 MeV and then use the formula, with M N * the mass of N * (2030). Here, | p π | is the momentum of π that results from the decay of N * and is evaluated using Eq. (II.2), s = M 2 N * . Finally, we obtain g Since we will have different amplitudes if we change the charge of the intermediate πΣ particles, it is convenient to go from the isospin basis (|I, I 3 ) to the charge basis. Using the Clebsch-Gordan coefficients, we have This means that the coupling of π − p to N * will be The N * (2030) → K * Σ process occurs in s-wave, then the amplitude is written as N * ,K * Σ = 3.9 + i0.2, and since we have both Σ − K * + and Σ 0 K * 0 (Figs. 2a and 2b respectively), then, using we get FIG. 2: Diagrams for the reaction π − p → K 0 πΣ that contain the triangle mechanism, where πΣ can be π − Σ + , π 0 Σ 0 and π + Σ − .
Then, for the amplitude of the π − p → ΣK * reaction through N * (2030), we have Now, the K * + → K 0 π + vertex can be calculated using the chiral invariant Lagrangian with local hidden symmetry given in Refs. [67][68][69][70], (II.13) The symbol ... here represents the trace over the SU (3) flavor matrices, and the coupling is g = m V /2f π , with m V = 800 MeV and f π = 93 MeV. The SU(3) matrices for the pseudoscalar and vector octet mesons Φ and V µ are given by where in the last step we made a nonrelativistic approximation, neglecting the ǫ 0 K * component. This is very accurate when the momentum of the K * is small compared to its mass. We shall evaluate the triangle diagram in the ΣK * CM, where the on-shell momentum of the K * is about 250 MeV/c at M inv (ΣK * ) ≃ 2140 MeV where the triangle singularity appears. In Ref. [56] it is shown that the effect of neglecting the ǫ 0 component goes as ( The final vertex that we need to calculate in the diagrams of Fig.1 is t Σπ,Σπ which is given by the Σπ → Λ(1405) → Σπ amplitude studied in Ref. [6] based on chiral unitary approach. There, the authors use the lowest order meson-baryon chiral lagrangian (II.21) The Bethe-Salpeter equation is then used to calculate the meson-baryon amplitude, where t, V , and G are the meson-baryon amplitude, interaction kernel, and meson-baryon loop function, respectively. For the evaluation of t, we use the momentum cutoff q max = 630 MeV for the loop function G, and f = 1.15f π with the pion decay constant f π = 93 MeV as done in Ref. [6].
Thus, the amplitude associated with the diagram in Fig. 2a, that we call t 1 , is given by (II.23) Using the following property, with f ( q, k) the three propagators in the integrand of Eq. (II.23), and the formula in the nonrelativistic approximation, pol.
Eq. (II.23) becomes Integrating t T over q 0 , we get [53,71], We regularize the integral in Eq. (II.26) by using the same cutoff of the meson loop in Eq. (II.22), θ(q max − | q * |), where q * is the Σ momentum in the final πΣ CM [53] and q max = 630 MeV. The width of K * is taken into account by replacing ω * with ω * − i Γ K * 2 .
Calculating the square of the amplitude and summing and averaging over the spins we get Now we will study the effects of the triangle loop in the following decays: pp → pK + π + Σ − , pp → pK + π 0 Σ 0 and pp → pK + π − Σ + . For this, we will first start analysing the diagram in Fig. 3. For this diagram, the differential cross section is calculated using the formula in Ref. [72], with t = (p a − p 1 ) 2 , M inv the invariant mass of particles 2 and 3, p 2 the momentum of particle 2 in the 23 CM, such that p a the momentum of the particle a in the initial state, and Π F (2M F ) means that we multiply 2M F for each fermion in Fig. 3, where M F is the mass of the respective fermion. This factor appears because we use the normalization of Ref. [73].
The complete diagrams for our reaction are shown in Fig. 4. The triangle part of the diagrams is very similar to the last case, except that because of charge conservation the particles in the loop will be different. Thus, instead of Eqs. (II.10a) and (II.10b), we will have where, to match the sign convention of the Φ and B matrices, we used |Σ + = − |1 1 (see Ref. [74] for further discussion). Then, we get g N * ,Σ + K * 0 = − 2/3g I= 1 2 N * ,ΣK * and g N * ,Σ 0 K * + = − 1/3g I= 1 2 N * ,ΣK * . The vertices K * 0 → K + π − and K * + → K + π 0 are calculated using Eq. (II.13), which gives To calculate the cross section for the diagrams in Fig. 4, we proceed as done in Ref. [72]. In Fig. 3, the t matrix found in Eq. (II.31) is given by with C ′ a parameter that carries the dependence of the amplitude on the variable t as well as information about the pp → pR transition. Substituting where particle 3 is assumed to be a baryon, into Eq. (II.31), we get Now we can take into account the complete reaction by substituting Γ R,23 for Γ N * →K + πΣ , where (II.39) with | p K | the momentum of K in the rest frame of N * , and | p π | the π momentum in the πΣ CM given by Eq. (II.4).
Then, from Eq. (II.38) we obtain (II.41) The transition amplitude t ′ in Eq. (II.41) is (II.42) which is constructed in a similar way to what was done in the previous subsection to obtain Eq. (II.28) but now changing the following variables in Eq. (II.26), Putting Eq. (II.42) into Eq. (II.41) we get where | p π | is the π momentum in the πΣ CM, and FIG. 4: Diagrams for the reaction pp → pK + πΣ that contain the triangle mechanism, where πΣ can be π − Σ + , π 0 Σ 0 and π + Σ − .
which is a function of s = (p a + p b ) 2 and t = (p a − p 1 ) 2 .
Using now the relation which follows from t = (p a − p 1 ) 2 , then we obtain This last step is important to account for the phase space of this process that depends on | p 1 |, which is tied Finally, we should integrate out the cos θ in Eq. (II.48) but C ′ in C ′′ depend on it. The resultant factor of the cos θ integration is denoted by C ′′′ and since we do not know the expression for C ′ , we take C ′′′ = 1. This means that from now on we will use arbitrary units (a.u.) for the cross section.
Thus, we end up with (II.50)

III. RESULTS
In Fig. 5, we show the real, imaginary part and absolute value of the amplitude t T of Eq. (II.26) as a function of the invariant mass of the KΛ(1405), M inv , by fixing the invariant mass of πΣ, m inv , at 1400 MeV. The absolute value of t T has a peak around 2140 MeV as expected from the condition for a triangular singularity by Eq. (18) of Ref. [53], and the peak is dominated by the imaginary part of the amplitude. As mentioned in The d 2 σ pK + πΣ dM inv dm inv as a function of m inv (π 0 Σ 0 ), m inv (π + Σ − ) and m inv (π − Σ + ) for the pp → pK + πΣ scattering with several fixed values of M inv and √ s = 3179 MeV.
of [75], where the peak showed at 1420 MeV. This is due to the fact that with the TS the Λ(1405) is formed by πΣ, rather thanKN , and this channel couples mostly to the lower mass state of the two Λ(1405) states [10]. For the case of the π − p → K 0 πΣ reaction, we in- the isospin-averaged mass of π and Σ, and we obtain the cross section of π − p → K 0 πΣ, σ K 0 πΣ , as a function of M inv . The results are represented in Fig. 8. There are peaks around 2100 MeV for all cases. though the expected value of triangular singularity is 2140 MeV. This is because the N * resonance in the K * Σ production has a peak around 2030 MeV (the term 1/|M inv − M N * + i Γ N * 2 | 2 in Eq. (II.48)). For the case of the pp → pK + πΣ reaction, integrating now the d 2 σ pK + πΣ dM inv dm inv over the m inv we obtain dσ pK + πΣ dM inv which are shown in Fig. 9 as a function of M inv for π 0 Σ 0 , π + Σ − and π − Σ + . Similarly we get peaks around 2100 MeV for the three cases.
In the π − p and pp reactions, the strength is largest for the π + Σ − and π − Σ + final state, respectively, reflecting the strength before the integration shown in Figs. 6 and 7.

IV. SUMMARY
We have carried out a study of contributions of a triangle diagram to the the π − p → K 0 πΣ and pp → pK + πΣ processes. In both reactions, the triangle diagram is formed by the N * decaying first to K * and Σ, the K * decays into πK, and then the Σ and the π merge to give Λ(1405), which finally decays into πΣ. In this process, the K * πΣ loop generates a triangle singularity around 2140 MeV in the invariant mass of KΛ(1405) from the formula of Eq. (18) of Ref. [53]. We evaluate the real, imaginary part and absolute value of the amplitude t T and find a peak around 2140 MeV.
We calculate the dσ K 0 πΣ dm inv with some values of M inv in the π − p → K 0 πΣ reaction and d 2 σ pK + πΣ dM inv dm inv with some values of M inv and fixed √ s = 3179 MeV as a function of m inv (π 0 Σ 0 ), m inv (π + Σ − ) and m inv (π − Σ + ). In these distributions, we see peaks around 1400 MeV, representing clearly the Λ(1405). Integrating over the m inv we obtain σ K 0 πΣ and dσ pK + πΣ dM inv and these distributions show a clear peak for M inv (N * (2030)) around 2100 MeV. The peak of the singularity shows up around 2140 MeV. This peak position of the triangular singularity is lowered by the initial N * resonance peak around 2030 MeV in the K * Σ production.
Thus, our results constitute an interesting prediction of the triangle singularity effect in the cross sections of these decays. The work done here could explain why in the experiments of Refs. [65,66] the invariant mass distribution of πΣ for the Λ(1405) are found at lower invariant masses than in other reactions. It would also be interesting to see if the predictions done here concerning the triangle singularity are fulfilled by the experimental data, an issue that has not been investigated so far. This work also can serve as a warning to future experiments that measure these interactions, that they should be careful when associating peaks in this energy region to resonances.