Extraction of $ND$ scattering lengths from the $\Lambda_b\rightarrow\pi^-pD^0$ decay and properties of the $\Sigma_c(2800)^+$

The isovector and isoscalar $ND$ $s$-wave scattering lengths are extracted by fitting to the LHCb data of the $pD^0$ invariant-mass distribution in the decay $\Lambda_b\rightarrow\pi^-pD^0$, making use of the cusp effect at the $nD^+$ threshold. The analysis is based on a coupled-channel nonrelativistic effective field theory. We find that the real part of the isovector $ND$ scattering length is unnaturally large due to the existence of a near-threshold state with a mass around 2.8 GeV. The state is consistent with the $\Sigma_c(2800)^+$ resonance observed at Belle. Our results suggest that it couples strongly to the $ND$ channel in an $s$-wave, and that its quantum numbers are $J^P=1/2^-$. The strong cusp behavior at the $nD^+$ threshold can be verified using updated LHCb data.


Introduction
Properties of the ND system have been paid much attention to as an analogue to theKN system, where there exists the Λ(1405) resonance as aKN quasi-bound state near threshold (for reviews, see Refs. [1,2] and the review article dedicated to the Λ(1405) in the Reviews of Particle Physics [3]). Several studies are devoted to the ND system from the viewpoint of the description of the Λ * c or Σ * c charmed baryons [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Some reactions concerning the ND system and Λ * c resonances have been investigated [19][20][21] and extensions to systems with D and nuclei performed (see, for example, Ref. [22] and references therein). Most of these studies discuss possible hadronic molecules (see Ref. [23] for a recent review) in the charmed-meson-nucleon systems, analogous to the XYZ states in the heavy-quarkonia mass region and the hidden-charm pentaquarks. However, relevant experimental information is scarce and the abovementioned calculations are based on phenomenological models.
At low energies, the short-range interaction between two particles can be described by the effective-range ex-Email addresses: shsakai@itp.ac.cn (Shuntaro Sakai), fkguo@itp.ac.cn (Feng-Kun Guo), kubis@hiskp.uni-bonn.de (Bastian Kubis) pansion, 1 for s-waves, where f 0 (k) is the s-wave amplitude and k is the magnitude of the center-of-mass (c.m.) momentum. Threshold parameters, including the scattering length a 0 and the effective range r 0 , are important quantities as they govern the low-energy behavior of the scattering amplitude. Their values can be used to infer the structure of s-wave shallow bound states if there are any [23,24]. It is well-known that in invariant-mass distributions there must be cusps exactly at two-body s-wave thresholds of opening channels due to unitarity. Because the masses of the involved particles are fixed, the cusp strength is then determined by the interaction at threshold, and thus threshold cusps can be used to extract the corresponding scattering lengths (we refer to Ref. [25] for a recent review on this topic). In the case of ππ scattering, the role of a cusp at the π + π − threshold in the π 0 π 0 spectrum was discussed in Refs. [26,27], and a Figure 1: Diagrams for the decay Λ b → π − pD 0 taken into account in this study. Here ND represents both pD 0 and nD + .
possible way to extract the ππ scattering length from the threshold cusp was suggested in Refs. [28,29], followed by reformulations based on a nonrelativistic effective field theory (NREFT) [30,31] (cf. also Ref. [32]). The difference between the ππ(I = 0) and ππ(I = 2) scattering lengths is tied to the magnitude of the π + π − threshold cusp in the π 0 π 0 distribution in the decays K ± → π ± π 0 π 0 , and the ππ scattering lengths were determined from the cusp in the subsequent experimental studies [33,34]. This framework was applied to other processes in Refs. [35][36][37][38][39]. In the present study, we try to extract the s-wave scattering lengths of the ND system from the pD 0 invariant-mass distribution of the decay Λ b → π − pD 0 measured by the LHCb Collaboration [40], where a peculiar structure near the nD + threshold (about 6 MeV higher than the pD 0 threshold) is seen. The Letter is organized as follows. In Sec. 2, the NREFT formalism is introduced that can be used in analyzing the pD 0 invariant-mass distribution in the nearthreshold region. In Sec. 3, the results of fitting to the s-wave contribution extracted from the LHCb analysis are presented. A brief summary is given in Sec. 4.

Setup
In Fig. 1, we show the diagrams considered in this study. Diagram (a) of Fig. 1 represents the π − pD 0 production from Λ b without rescattering of particles, and diagram (b) takes into account the subsequent rescattering of ND → pD 0 where ND represents both pD 0 and nD + .
First, we explain the ND scattering part. The ππ lowenergy scattering is known to be rather weak due to chiral suppression; on the contrary, there is no suppression for the ND near-threshold interaction and it might be strong enough to generate a nearby pole. We employ the coupled-channel NREFT as developed in Refs. [41,42], which is based on the Lippmann-Schwinger equation and is adequate to treat the ND scattering in the nearthreshold region. There are two channels: pD 0 and nD + . The nonrelativistic T -matrix is given by the Lippmann-Schwinger equation as [41,42] where i = 1 and 2 denote the pD 0 and nD + channels, respectively, µ i denotes the reduced mass, and p i is the nonrelativistic momentum of D (or N) in the ND c.m. frame, with M ND being the ND invariant mass, and M i (m i ) the mass of D 0 or D + (p or n). With the above expression, the momentum p i is defined above threshold. Below threshold, it is analytically continued to i 2µ i (M i + m i − M ND ). The NREFT is based on an expansion in powers of the velocity of the particles in their c.m. frame. At the leading order of the NREFT, which is sufficient in the immediate vicinity of the thresholds given the current data quality, the interaction kernel v is a matrix for constant contact terms (the next-to-leading order terms are of O(p 2 i /µ 2 i )), and G is a 2 × 2 diagonal matrix with the diagonal matrix element G i given by the nonrelativistic two-body loop function, The ultraviolet (UV) divergence in the loop integral is regularized using a three-momentum cutoff Λ. In Eq. (2), the cutoff dependence of the loop function G i and the interaction kernel v are absorbed into the cutoffindependent parameters a i j (see Refs. [41,42] for details). The ND scattering lengths can be expressed in terms of these parameters. The scattering length a i in channel i is defined by the scattering amplitude at threshold as If the interaction is strong enough, a near-threshold pole of the T -matrix can be generated as a zero of the determinant defined in Eq. (3).
With the isospin phase convention and all other states taking a positive sign, the ND scattering amplitudes in the isospin basis and that in the particle basis are related to each other as Hence, the difference of the ND scattering lengths with I = 0 and I = 1 is proportional to t nD + ,pD 0 , At the lower (pD 0 ) threshold, the T -matrix elements are where κ = 2µ 2 (m n + M D + − m p − M D 0 ), and we have approximated a 22 a 11 using isospin symmetry. The more natural parameters are a c and a x instead of a 11 and a 12 . They are connected to the isoscalar and isovector scattering lengths as The parameters a 11 and a 12 can be expressed in terms of a c and a x : Second, let us consider the production from the Λ b decays. The leading contribution to the Figure 2: The W − emission mechanism for the Λ b → π − pD 0 decay. matrix and color counting is the W − emission mechanism, where the W − boson emitted from the b → cW − process becomes a π − , see Fig. 2. Because the ud pair in the Λ b is an isoscalar, the produced ND pair would be an isospin singlet as well. Thus, the production of the pD 0 should be approximately the same as that of the nD + .
Since parity is not conserved in the Λ b → π − ND weak decay, the ND pair can be produced not only in the s-wave, but also in higher partial waves. With the statistics of the current data, it is not possible to fix parameters with different partial waves included. Thus, we take the LHCb data and subtract the contributions from partial waves other than the ND s-wave. Several fits are presented in the LHCb analysis [40]. We take the analysis presented in Fig. 12 of Ref. [40], and subtract the contributions from the J P = 1/2 + and 3/2 ± partial waves from the measured pD 0 invariant-mass distribution. In this way, only the 1/2 − part, corresponding to the ND s-wave, is left.
For the energy region close to the thresholds, the s-wave production can be approximated by a constant contact term followed by the final-state interaction (FSI). The ND FSI can be described by the nonrelativistic T -matrix discussed above. Any possible singular behavior comes from the rescattering given in Eq. (2). We parametrize the s-wave contact term for the Λ b → π − ND production amplitude by a constant V P . A momentum factor p π − associated with the W − transition to π − is absorbed into V P as a constant because this factor is irrelevant to the nD + threshold cusp of interest, and we are focusing on a very small range of the ND invariant mass around the pD 0 and nD + thresholds.
Taking into account the ND rescattering described by Eq. (2) and the fact that ND produced from the Λ b weak decay has I = 0, the decay amplitude of Λ b → π − pD 0 in Fig. 1 for an ND s-wave can be written as follows: where V P has been rewritten as V Λ P to emphasize that it depends on the cutoff Λ. The second and third terms in the parentheses come from diagram (b) in Fig. 1 with s-wave ND rescattering. The rescattering of the other pairs (πN and πD) does not matter either because the pion moves much faster than the D-meson and the nucleon in the near-ND-threshold region. One notices that the T -matrix elements t pD 0 ,pD 0 and t nD + ,pD 0 are physical quantities and do not depend on the cutoff introduced in regularizing the UV divergence. The cutoff dependence of the loop functions in Eq. (13) needs to be absorbed by the production vertex V Λ P . Such a requirement is fulfilled by a multiplicative renormalization with V P ∝ 1/Λ and by keeping only the leadingorder term (in the expansion in powers of p i /Λ) in the loop function in Eq. (5). Therefore, we find the following cutoff-independent amplitude: where V P has been redefined to absorb the cutoff as well as other multiplicative constants with approximating µ pD 0 µ nD + in the cutoff term of G Λ i . Consequently, the 1/2 − part of the pD 0 invariantmass distribution for the decay Λ b → π − pD 0 can be fitted using where N is an overall normalization constant, p π − (p D 0 ) is the momentum of the π − (D 0 ) in the Λ b (pD 0 ) rest frame.
The ND scattering lengths can be extracted by fitting to the 1/2 − partial wave of the pD 0 invariant-mass distribution reported in Ref. [40] using Eq. (15) as a fitting function. There is one more complexity due to the inelasticity from coupling pD 0 and nD + to lower channels such as πΛ c and πΣ c . Since these channels are far away from the energy region of interest, the inelastic effects may be included by introducing imaginary parts to the parameters a x and a c . Unitarity is useful to constrain the parameter space: Im t ii = j ρ j |t i j | 2 ≥ 0 where ρ j ≥ 0 is the phase space factor of the intermediate states j. In order to obtain a stable fit, we further reduce the number of free parameters by approximating Im a x −Im a c .
Considering that the phase space for the lowest twobody isovector channel πΛ c is much larger than that for the isoscalar channel πΣ c , it is reasonable to assume Im a ND(I=0) Im a ND(I=1) . We therefore treat Im a ND(I=0) as a higher-order effect and neglect it in the leading approximation. This approximation is supported by the existing model calculations, listed in Ta best fit best fit (cont.) mn + MD+ LHCb data Figure 3: The best fit to the pD 0 distribution. The histogram shows the best fit with event numbers integrated in each bin, the dashed curve is the corresponding continuous distribution, which exhibit a clear cusp at the nD + threshold, and the band is the corresponding 1σ error region. The data points with error bars are taken from Fig. 12(b) in Ref. [40] with the contributions from the J P = 1/2 + and 3/2 ± partial waves subtracted. The vertical dashed line denotes the nD + threshold.  Hence, there are four free parameters in the fit, including Re a c , Im a c , Re a x , and N.

Results
As mentioned above, we subtract the 1/2 + and 3/2 ± contributions from the pD 0 invariant-mass distribution of the Λ b → π − pD 0 decay as reported in Fig.12(b) in Ref. [40]. Using the MINUIT algorithm [43,44], we fit to the 14 data points below 2.83 GeV by averaging the function in Eq. (15) for each bin to take into account the binning of the measured pD 0 invariant-mass distribution. 2 In this range, the nonrelativistic treatment of the ND systems is well justified. The best fit has a reduced chi-square of χ 2 /dof = 4.9/10. In Fig. 3, the best fit to the 1/2 − partial wave of the pD 0 distribution is compared to the data. The measured distribution indeed shows an evident change around the nD + threshold despite the low statistics, and the best fit curve has a clear 2 At 2.83 GeV, p 2 i /µ 2 i is smaller than 0.09 and thus the leadingorder NREFT treatment is sufficient, given the large uncertainties of the current data set. nD + threshold cusp. The resulting parameters from the fit are given in Table 1. We have checked that the values of a c and a x remain almost the same if we keep the loop functions in Eq. (13) and use Λ ranging from 0 to 10 GeV (the expression of G Λ with Λ = 0 GeV is formally the same as the one using the MS scheme of dimensional regularization as one does in the studies of the ππ cusp in NREFT, see, e.g., Refs. [30,31]). The change of Λ is absorbed by a corresponding change in the value of the normalization (V 2 P effectively). The isoscalar and isovector ND scattering lengths obtained from the fit follow from Eq. (11), and are listed in Table 2, where the errors are the 1σ uncertainties propagated from the statistical errors of the data. For comparison, we also show the scattering lengths obtained in a few phenomenological models [6][7][8]11] in Table 2. One sees that the results of the SU(4) DN model [7] and the meson-exchange model [11] are compatible with our findings. 3 It is worthwhile to notice that the real parts of both the isoscalar and isovector scattering lengths are negative, meaning that the interaction in each channel is either repulsive or strongly attractive such that there is a bound state below threshold. The large absolute value of Re a ND(I=1) within errors suggests the existence of a near-threshold isovector bound state; in contrast, no strong conclusion can be made in the isoscalar sector: within the uncertainties, one can find a pole, which can be near threshold or too far away to be valid in the NREFT framework. Indeed, keeping isospin symmetry breaking, we find two poles in the first Riemann sheet (RS-I) of the complex M ND plane 4 for the Tmatrix defined in Eq. (3), and one pole on the second 3 The nD 0 scattering length is evaluated to be −0.764 + i 0.615 fm in Ref. [45] (here the convention of the scattering length has been changed to be the same as ours), in which the parameters are chosen to reproduce the mass and width of the neutral Σ c (2800) measured by Belle [46]. 4 The first Riemann sheet is defined as the Riemann sheet with Im p 1 > 0 and Im p 2 > 0, and second Riemann sheet is defined as the one with Im p 1 < 0 and Im p 2 > 0. It is worthwhile to notice that because of the complexity of the a 11,12 parameters from the inelastic channels, there can be poles off the real axis on the first Riemann Riemann sheet (RS-II). Their locations computed using about 1000 parameter sets within 1σ are shown in Fig. 4. For the first pole on RS-I, the absolute value of its residue to the isovector channel is at least more than 6 times (up to two orders of magnitude) larger than that to the isoscalar channel, except when the pole is located on the real axis, in which case the residue sizes are comparable. The pole on RS-II couples more strongly to the isovector than to the isoscalar channel, with the ratio of the residue size ranging from 1.6 to 3.8. These two poles are at: MeV.

RS
From Fig. 5, one sees clearly that the 1st pole on RS-I produces a peak of t I=1 below the pD 0 threshold, while the RS-II pole would be partly responsible for the "bump" between the pD 0 and nD + thresholds. The two poles should be assigned to the Σ c (2800) + resonance discovered by the Belle Collaboration [46], which has a mass of 2792 +14 −5 MeV and a width of 62 +60 −40 MeV. 5 sheet, and the Schwarz reflection principle is not respected. 5 Apart from the literature on the molecular interpretation of the  Because of the isovector nature, the Σ c (2800) signal is much weaker than those of excited Λ c states in the Λ b → π − pD 0 decay. Yet, it leaves a footprint at the nD 0 threshold, producing a strong cusp. The threshold cusp effect can serve as a partial-wave filter since it shows up only for the s-wave. The quantum numbers of the Σ c (2800) are not known yet, and our analysis suggests them to be J P = 1/2 − . This feature is shared by the results in a meson-exchange model in Ref. [11]. Yet, we notice that a recent study of D ( * ) N interaction using chiral effective field theory and model inputs for the involved low-energy constants did not find a bound state in the isovector ND system [18]. The analysis here should also be useful for improving the input of such studies.
In contrast, the second pole on RS-I couples dominantly to the isoscalar channel except when it is located on the real axis; furthermore, this pole in most of the parameter space is much further away from the thresholds, and one cannot make a unique conclusion on whether there is a near-threshold isoscalar pole (or even whether the isoscalar interaction is attractive or repulsive). Correspondingly, the curve for |t I=0 | in most of the parameter space is quite smooth with two mild threshold cusps (as can be seen from the blue dashed best-fit curve in Fig. 5).

Summary
In this study, we have extracted the isoscalar and isovector ND scattering lengths from the pD 0 invariant-Σ c (2800) cited in Sec. 1, studies related to the Σ c (2800) at the quark level can be found in, e.g., Refs. [47][48][49][50][51][52][53][54][55][56][57]. Concerning its spin and parity, no consensus has been reached so far. mass distribution for the Λ b → π − pD 0 decay reported by the LHCb Collaboration [40]. In particular, there is a structure in the data around the nD + threshold that is likely due to the nD + threshold cusp. The strength of the cusp is sensitive to the ND interaction, and thus can shed light on resonances near the ND threshold. Our analysis is based on a low-energy nonrelativistic effective field theory with pD 0 and nD + coupled channels. The effects of lower channels are built in by introducing imaginary parts to the scattering lengths. From fitting to the 1/2 − partial wave of the pD 0 distribution, we find that the real parts of both the isoscalar and isovector ND scattering lengths are negative, implying that the interactions are either repulsive or have a bound state below threshold. Within uncertainties, the real part of the isoscalar ND scattering length ranges from −1.4 to −0.1 fm, and thus one is not able to conclude whether there must be a nearthreshold isoscalar Λ c excited state or the interaction is repulsive. In contrast, the absolute value of the real part of the isovector ND scattering length is always large, indicating the existence of a bound-state pole below the pD 0 threshold. We indeed find a bound state at 2801.8 +1.0 MeV. Both of them couple dominantly to the isovector channel, and in the isospin limit only one bound-state pole is left, suggesting they correspond to a two-pole structure of the same state due to coupled channels. This state could be assigned to the Σ c (2800) discovered by the Belle Collaboration [46], and its quantum numbers are suggested to be J P = 1/2 − . Data with better statistics would be helpful to further clarify this situation.