Role of the triangle mechanism in the Λ b → Λ c π − f 0 ( 980 ) reaction

We investigate the Λb → Λcπf0(980) production with a f0(980) decay into π+π− via the K∗0K−K+ and K∗−K0K̄0 triangle loops. These loops produce a peak around 1.42 GeV in the πf0(980) invariant mass distribution, which is the same mechanism as the one considered to explain the a1(1420) peak. In the π +π− distribution obtained by fixing the πf0(980) invariant mass to some values, a clear peak of f0(980) is seen, and the π f0(980) distribution has a peak around Mπ−f0 = 1.42 GeV which is caused by the triangle mechanism of the K ∗K̄K loop. The branching ratio of Λb → Λcπf0(980) with f0(980)→ π+π− by the triangle mechanism, obtained by integrating the πf0(980) distribution from 1 GeV to 1.6 GeV, is estimated to be the order 10 −4. Future measurements of the Λb → Λcπf0(980) branching ratio and the πf0(980) invariant mass distribution predicted in this work would give further clues to clarify the nature of the a1(1420) peak. ∗ shsakai@mail.itp.ac.cn 1 ar X iv :2 00 2. 08 59 4v 1 [ he pph ] 2 0 Fe b 20 20


I. INTRODUCTION
Role of triangle singularities (TSs) in hadronic reactions has been investigated for a long time. A general discussion on the emergence of singularities from loop amplitudes was given by Landau [1], and a physical picture of the singularity from triangle loop was provided in Ref. [2] which is known as the Coleman-Norton theorem; the TSs can show up when all the internal particles are on shell and the reaction proceeds collinearly. One can find a refined formulation and the intuitive picture of the TS in Ref. [3] (see also Ref. [4] for a recent review of the TS).
An anomalously large production of the π 0 f 0 (980) decay mode of η(1405/1475), which is forbidden by isospin symmetry, was reported by the BESIII Collaboration [5], and that large amount production and the narrow f 0 (980) lineshape in the ππ distribution, which is the order of the mass difference of the charged and neutral kaons due to the isospin symmetry breaking, are explained well by the triangle mechanism [6][7][8][9] (see also Ref. [10] for a review article). The triangle diagram considered in the work is composed of K * K K and its charge conjugation shown in Fig. 1. The triangle loop diagram of Fig. 1 has a singularity around 1.42 GeV which is in the η(1405/1475) mass region. In the process, the K * K pair is produced by η(1405/1475) first, the K * decays into πK, and the KK couple to f 0 (980). The position of the singularity can be obtained with a formula given in Ref. [3]. The triangle singularity plays an essential role in this process because the position of the TS is sensitive to the masses of the particles and the mass difference of K and K * involved in the triangle loop introduces the isospin violation in this process. In practice, the singularity is turned into a peak by the width of the internal particles, and the detailed study on the width effect was done in Ref. [11]. In Refs. [12,13], some other processes were studied for further investigation of the anomalous enhancement of the isospin-forbidden π 0 f 0 (980) production by the triangle mechanism. The TS of the K * K K loop was mentioned in Ref. [14], and the possible role of the K * K K triangle loop has been investigated in many processes [6-9, 11-13, 15-22].
One important aim to study the TS is to clarify the origin of peaks in the invariant mass distribution. The peak of the TS has purely kinematical origin and cannot be associated with a resonant state. A peak of a 1 (1420), which is in the p-wave π − f 0 (980) mode in the π − p → π + π − π − p reaction, was found by the COMPASS Collaboration [23,24] followed by the studies on the properties of a 1 (1420) [25][26][27][28][29] and the work on the explanation of the peak focusing on its production mechanism [30] (see Ref. [31] for a recent review article, and see also a mini review for mesons in the 1400 MeV region in the Particle Data Group (PDG) [32]).
A possible understanding of the peak with the triangle mechanism was suggested in Refs. [17,18]. The position of the singularity around 1.42 GeV stemming from the K * K K loop coincides with the peak position of a 1 (1420). Despite the attempts to clarify the nature of the peak, significant difference of the resonance and TS scenarios of the a 1 (1420) peak could not be found in the partial wave analysis so far [31]. Some predictions based on the TS scenario of the a 1 (1420) peak were made in Refs. [20,21] in the B and τ decays.
In this work, we study the Λ b → Λ c π − f 0 (980) with f 0 (980) → π + π − via the triangle mechanism of the K * K K loop producing a peak around 1.42 GeV in the π − f 0 (980) distribution. We show in Fig. 2 the diagram of the K * K K triangle loop contributing to the Λ b → Λ c π − f 0 (980) process. Apart from the K * K production part, the mechanism producing a peak around 1.42 GeV is identical with the one considered in Refs. [17,18] for the a 1 (1420) peak. With known theoretical and experimental information, we make the predictions on the branching ratio and the invariant mass distribution in the Λ b → Λ c π − f 0 (980) reaction. Such predictions with including the triangle mechanism will be important for future experiments to clarify the nature of a 1 (1420) as an unavoidable peak of the kinematical effect.

II. SETUP
The diagrams we consider in this study are shown in Fig. 2. Here, we focus on the diagram Fig. 2(a) where a loop is formed by K * 0 K − K + ; Λ b first decays into Λ c K * 0 K − and subsequently K * 0 turns into the π − K + with a merging of K + and K − to give f 0 (980).
The f 0 (980) finally decays into a π + π − pair. Strictly speaking, this K * K K loop for the π − f 0 (980) production does not have the TS because the mass of f 0 (980) is slightly below the KK threshold, i.e., the KK in the loop cannot be on shell. However, in the distribution the remnant of the TS would be still expected due to the width of the particles. For example, by putting the f 0 (980) mass slightly above the KK threshold, with the formula in Ref. [3], the loop amplitude produces a TS around M π − f 0 = 1.42 GeV in the π − f 0 (980) invariant mass distribution.
In this section, the amplitude needed to evaluate the diagram in Fig. 2, the Λ b → Λ c K * K , K * → πK, and KK → π + π − transition amplitudes, will be explained, and the amplitude of the Λ b → Λ c π − f 0 (980) decay with f 0 (980) → π + π − will be given at the end of this section.
First, we consider the Λ b → Λ c K * 0 K − amplitude. Some possible diagrams for the Λ b → Λ c K * K in quark level are depicted in Fig. 3. In this calculation, we take account of the For the baryonic part of the Λ b → Λ c W − transition, B µ , we use a result of the QCD sum rule [34]. The Λ b → Λ c transition amplitude is parameterized as follows; where with F(0) and ξ i given in Ref. [34]. For later calculation, we give the spin sum and average of the baryonic partB µν = B µ B * ν (quantities with overline denote the spin summed and averaged ones in the following); Let us move to the mesonic part producing Here, we assume the a − 1 (1260) dominance for the K * 0 K − and K * − K 0 production. The [35] and a peak around 1.3 GeV which may be associated with a 1 (1260) is seen in the and we expect a large portion of Λ b → Λ c a − 1 (1260) in the K * K production by taking into account a fairly strong coupling of a 1 (1260) to K * K obtained theoretically in Ref. [37].
Then, J µ represents the amplitude of the W − → a − 1 (1260) → K * 0 K − or K * − K 0 transition here. The effect of a 1 (1260) on the π − f 0 (980) distribution will be checked later.
We write the conversion amplitude from W − to a − 1 (1260) as with V ud and g W being the element of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix and the coupling constant of the weak interaction, respectively, and the a − 1 (1260) → K * 0 K − amplitude is written as where we take into account the amplitude with the lowest angular momentum which gives the dominant contribution in low energies. In the case of the K * − K 0 pair in the final state, we just need an additional minus sign.
given by the external W − emission process is written as follows; by taking the leading order term of 1/m 2 is the Fermi coupling constant. The Λ b → Λ c K * − K 0 amplitude has the opposite sign relative to Eq. (8) which comes from the different sign of the a − 1 (1260) → K * 0 K − and K * − K 0 vertices. We fix the parameters in Eqs. (6) and (7) with the observed τ − → ν τ K * 0 K − partial width in the PDG [32] assuming the production is dominated by the a 1 (1260) meson. In the spectral function of τ → 3πν τ [38,39], one can see a significant peak at 1.2 GeV 2 . The in which a large amount of K + π − comes from K * 0 , is available [40], but the data is not enough for the analysis. The matrix element of the τ − → ν τ K * 0 K − decay is written as For the a 1 (1260) mass and width, the central value of the PDG [32] is used; m a 1 = 1.23 GeV and Γ a 1 = 0.425 GeV. 1 The spin sum and average of leptonic part L µ is given by with p τ and p ν being the momenta of the τ lepton and neutrino, respectively. Then, the matrix element squared with the spin sum and average is and the differential width is In practice, we do not need to fix f a 1 and g 1 independently because a product f a 1 m a 1 g 1 appears in the amplitude. The product f a 1 m a 1 g 1 is fixed to f a 1 m a 1 g 1 = 1.0 GeV 3 .
C. The KK → π + π − scattering amplitude and the applications to many reactions. In this work, we follow the setup of Ref. [55]; the f 0 (980) resonance is dynamically generated as a result of the nonperturbative meson-meson interaction, and it is found in Refs. [55][56][57] that the lineshape of the π + π − invariant mass distribution around 1 GeV is described fairly well. The amplitude is given by the scattering equation with i, j = π + π − , π 0 π 0 , K + K − , K 0K 0 , ηη. The interaction kernel v comes from the s-wave part of the leading-order chiral Lagrangian, and g is the meson-meson loop function with cutoff regularization given in Ref. [55], where the cutoff parameter Λ for g is chosen to be 0.6 GeV. 2 The scalar mesons around 1 GeV has been studied for a long time, and many studies were devoted for it from various view points, e.g., as done in Refs. [50][51][52][53] (see also Ref. [54] for a review article).
Combining the amplitudes given above, we can obtain the loop amplitude given by the diagram in Fig. 4, which is denoted by T µ . We can write the amplitude given by the diagram in Fig. 4 with the K * 0 K − K + triangle loop (a meson pair M M in the final state comes from the K + K − rescattering) as follows; with A library for the one-loop integrals, LoopTools, is used [58]. In Eq. (22), the momenta k 1 and k 2 are defined as in Fig. 4. Now, the renormalization scale, µ, associated with the divergence of the loop integral is fixed to be 1 GeV, and the change of this parameter to µ = 0.5 GeV or 1.5 GeV gives just a tiny difference. The width effect of the K * 0 meson in the loop is included by replacing the squared mass of the K * 0 , m 2 K * 0 , with m 2 K * 0 − im K * 0 Γ K * 0 in this study.
Then, with the part of the Λ b → Λ c K * 0 K − transition given in Eq. (8), the Λ b → Λ c π − f 0 (980); f 0 (980) → M M amplitude via the K * 0 K − K + triangle loop is written as The amplitude of the K * − K 0K 0 loop is obtained by just changing the label of the internal particles with the same sign relative to the K * 0 K − K + loop taking into account the minus sign of a − 1 K * K and K * πK vertices. Then, adding the contribution of the K * − K 0K 0 and In the following, we consider the case of M M = π + π − in the final state to see f 0 (980). In the isospin symmetric case, where the isospin averaged mass and width of the mesons are used, the amplitude is reduced as follows 3 ; This isospin averaged amplitude will be used in the following calculation.
Using the formula of the phase space volume in Ref. [22], the differential distribution is given by with The angles Ω Λc , Ω π − , and Ω π + are those of Λ c , π − , and π + in the Λ b rest frame, the π − f 0 (980) c.m. frame, and the π + π − c.m. frame, respectively.

III. RESULTS
We show in Fig. 5 the π + π − invariant mass distribution given by Eq. (26) normalized with the Λ b full width, Γ Λ b , with M π − f 0 = 1.3, 1.42, and 1.5 GeV. A peak of f 0 (980) is clearly seen at M π + π − = 0.98 GeV, and the largest strength is given with M π − f 0 = 1.42 GeV by the virtue of the K * K K triangle mechanism.
Integrating Eq. (26) over M π + π − in the range of M π + π − ∈ [0.9, 1.1] GeV, we obtain the Fig. 6. The distribution is normalized with Γ Λ b again. In the π − f 0 (980) distribution, a peak around 1.42 GeV with a width of the order of 0.1 GeV originating from the TS is clearly seen.
The distribution is similar to the ones in the a 1 (1260) decay and the τ decay calculated in Refs. [18,21]. Note that the π + π − pair in the final-state π − π + π − is produced by f 0 (980) in this calculation. The π − π + π − in the final state would be mainly produced by the s-wave πρ, which is a decay product of a − the branching ratio Br ∆ ; which is the same order of magnitude obtained in the τ − decay into ν τ π − f 0 (980) via the triangle mechanism [21].
To see the uncertainties from the Λ b → Λ c transition form factors, we show the plot in Fig. 7 with different parameter sets of the Λ b → Λ c form factors given in Refs. [34,59] which are denoted by the lines  stable. We note that, for the Λ b → Λ c transition amplitude, only the external W − emission diagram is taken into account, and the approximation gives some further uncertainties which are not addressed in this study.
In the amplitude Eq. (8), the a 1 (1260) dominance in the K * − K 0 and K * 0 K − production is assumed. For comparison, we show the plot without the intermediate a 1 (1260) resonance in the production. In this case, the decay amplitude of Λ b → Λ c π − f 0 (980) is given by replacing the a 1 (1260) propagator G µν a 1 with g µν in Eq. (25). The W − → K * K amplitude is given by Eq. (7) by replacing the a − 1 (1260) polarization vector with the W − one. The coupling constant of W − to K * K is fixed to be 1.1 GeV with the τ − → ν τ K * 0 K − partial width without the intermediate a 1 (1260) meson. The τ − → ν τ K * 0 K − decay amplitude without a 1 (1260) is given by replacing G µν a1 with g µν in Eq. (11). The π − f 0 (980) invariant mass distributions with and without a 1 (1260) are compared in Fig. 8. The peak around 1.42 GeV can be seen in both cases, and the distribution without a 1 (1260) has a longer To clarify the feature of the triangle mechanism in the Λ b → Λ c π − f 0 (980) process, we compare the invariant mass distribution of π − f 0 (980) produced with and without the triangle mechanism. The amplitudes of a 1 (1260) → π − f 0 (980) in p-wave and f 0 (980) → π + π − in s wave, which are needed for the amplitude of the direct production of the p-wave π − f 0 (980) pair from a − 1 (1260), are written as −it f 0 (980),π + π − =ig f 0 ,ππ .
The decay amplitude of Λ b → Λ c π − f 0 (980) followed by f 0 (980) → π + π − with π − f 0 (980) directly produced by a − 1 (1260) is given by For simplicity, we just use a Breit-Wigner amplitude of the f 0 (980) resonance with the mass In Fig. 9, the π − f 0 (980) invariant mass distribution with Eq. (34) is compared with the one with the triangle mechanism given by Eq. (25). The distribution with π − f 0 (980) directly produced by a 1 (1260) has no structure around 1.4 GeV and just increases as p wave. On the other hand, in the case with the triangle loop contribution, the peak of the TS is located around 1.42 GeV with the width about 0.2 GeV in the distribution. Thus, the triangle mechanism has the clear distinction from the other production mechanism we considered here.

IV. SUMMARY
We have studied the Λ b → Λ c π − f 0 (980) decay with f 0 (980) → π + π − . For the Λ b → Λ c K * K (KK * ) production part, the amplitude is factorized into the Λ b → Λ c transition and K * − K 0 (K − K * 0 ) production from the a − 1 (1260) resonance which are connected with a W − boson taking the leading contribution in terms of the color counting [33]. The Λ b → Λ c transition form factors are taken from the theoretical studies [34,59], and the chiral unitary approach is employed for the KK → π + π − transition amplitude [55]. A coupling constant related to the production of K * − K 0 is fixed with the τ − → ν τ K * − K 0 branching ratio assuming the a 1 (1260) dominance.
A peak of the f 0 (980) resonance is seen in the π + π − invariant mass distribution, and the peak has the largest strength when the π − f 0 (980) invariant mass is fixed to be 1.42 GeV.
Integrating the π + π − distribution, we obtain the π − f 0 (980) distribution which has a peak around 1.42 GeV due to the triangle singularity of the K * K K loop. With further integration over the π − f 0 (980) invariant mass in the range of M π − f 0 ∈ [1.0, 1.6] GeV, the branching ratio of Λ b → Λ c π − f 0 (980) with f 0 (980) → π + π − by the K * K K triangle mechanism is obtained as 2.2 × 10 −4 . Considering the uncertainties from the parameters appearing in this calculation, the renormalization scale for the loop regularization and the parameters in the Λ b → Λ c transition form factor, it is found that the branching ratio of Λ b → Λ c π − f 0 (980); f 0 (980) → π + π − is the order 10 −4 and the peak position originating from the triangle mechanism is not changed although more sophisticated treatment of the Λ b → Λ c transition part, in which the internal W − exchange processes are truncated in this calculation, may be needed for more definite predictions. The comparison of the distributions with and without the intermediate a 1 (1260) is also done, and it is found that the peak around 1.42 GeV is not changed even if the a 1 (1260) is omitted while some difference in the shape of the distribution can be seen. The branching ratio without the a 1 (1260) is still the order 10 −4 . The distribution of π − f 0 (980) directly produced by the a 1 (1260) meson without the triangle loop is also considered to compare it with the distribution including the triangle loop contribution, and it is found that the distribution without the triangle loop just increases without peak structures, which is quite different from the distribution with the K * K K triangle mechanism.
The part of the K * K K triangle loop is identical with the mechanism considered in Refs. [17,18] to explain the a 1 (1420) peak in π − p → π − π − π + p observed by the COM-PASS Collaboration [23]. Then, future measurements of the branching ratio of the Λ b → Λ c π − f 0 (980) and the π − f 0 (980) invariant mass distribution, particularly the peak structure around 1.4 GeV, which are the predictions made in this work, can provide a support of the a 1 (1420) peak as a manifestation of the triangle singularity, and they also provide further knowledge about the role of the triangle singularities in the hadronic reactions.