Evidence for exotic hadron contributions to $\Lambda_b^0 \to J/\psi p \pi^-$ decays

A full amplitude analysis of $\Lambda_b^0 \to J/\psi p \pi^-$ decays is performed with a data sample acquired with the LHCb detector from 7 and 8 TeV $pp$ collisions, corresponding to an integrated luminosity of 3 fb$^{-1}$. A significantly better description of the data is achieved when, in addition to the previously observed nucleon excitations $N\to p\pi^-$, either the $P_c(4380)^+$ and $P_c(4450)^+\to J/\psi p$ states, previously observed in $\Lambda_b^0 \to J/\psi p K^-$ decays, or the $Z_c(4200)^-\to J/\psi \pi^-$ state, previously reported in $B^0 \to J/\psi K^+ \pi^-$ decays, or all three, are included in the amplitude models. The data support a model containing all three exotic states, with a significance of more than three standard deviations. Within uncertainties, the data are consistent with the $P_c(4380)^+$ and $P_c(4450)^+$ production rates expected from their previous observation taking account of Cabibbo suppression.

From the birth of the quark model, it has been anticipated that baryons could be constructed not only from three quarks, but also four quarks and an antiquark [1, 2], hereafter referred to as pentaquarks [3,4]. The distribution of the J/ψ p mass (m J/ψ p ) in Λ 0 b → J/ψ pK − , J/ψ → µ + µ − decays (charge conjugation is implied throughout the text) observed with the LHCb detector at the LHC shows a narrow peak suggestive of uudcc pentaquark formation, amidst the dominant formation of various excitations of the Λ [uds] baryon (Λ * ) decaying to K − p [5,6]. It was demonstrated that these data cannot be described with K − p contributions alone without a specific model of them [7]. Amplitude model fits were also performed on all relevant masses and decay angles of the six-dimensional data [5], using the helicity formalism and Breit-Wigner amplitudes to describe all resonances. In addition to the previously well-established Λ * resonances, two pentaquark resonances, named the P c (4380) + (9 σ significance) and P c (4450) + (12 σ), are required in the model for a good description of the data [5]. The mass, width, and fractional yields (fit fractions) were determined to be 4380 ± 8 ± 29 MeV, 205 ± 18 ± 86 MeV, (8.4 ± 0.7 ± 4.3)%, and 4450 ± 2 ± 3 MeV, 39 ± 5 ± 19 MeV, (4.1 ± 0.5 ± 1.1)%, respectively. Observations of the same two P + c states in another decay would strengthen their interpretation as genuine exotic baryonic states, rather than kinematical effects related to the so-called triangle singularity [8], as pointed out in Ref. [9].
In this Letter, Λ 0 b → J/ψ pπ − decays are analyzed, which are related to Λ 0 b → J/ψ pK − decays via Cabibbo suppression. LHCb has measured the relative branching fraction B(Λ 0 b → J/ψ pπ − )/B(Λ 0 b → J/ψ pK − ) = 0.0824 ± 0.0024 ± 0.0042 [10] with the same data sample as used here, corresponding to 3 fb −1 of integrated luminosity acquired by the LHCb experiment in pp collisions at 7 and 8 TeV center-of-mass energy. The LHCb detector is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, described in detail in Refs. [11,12]. The data selection is similar to that described in Ref. [5], with the K − replaced by a π − candidate. In the preselection a larger significance for the Λ 0 b flight distance and a tighter alignment between the Λ 0 b momentum and the vector from the primary to the secondary vertex are required. To remove specific B 0 and B 0 s backgrounds, candidates are vetoed within a 3 σ invariant mass window around the corresponding nominal B mass [13] when interpreted as B 0 → J/ψ π + K − or as B 0 s → J/ψ K + K − . In addition, residual long-lived Λ → pπ − background is excluded if the pπ − invariant mass (m pπ ) lies within ±5 MeV of the known Λ mass [13]. The resulting invariant mass spectrum of Λ 0 b candidates is shown in Fig. 1. The signal yield is 1885 ± 50, determined by an unbinned extended maximum likelihood fit to the mass spectrum. The signal is described by a double-sided Crystal Ball function [14]. The combinatorial background is modeled by an exponential function. The background of Λ 0 b → J/ψ pK − events is described by a histogram obtained from simulation, with yield free to vary. This fit is used to assign weights to the candidates using the sPlot technique [15], which allows the signal component to be projected out by weighting each event depending on the J/ψ pπ − mass. Amplitude fits are performed by minimizing a six-dimensional unbinned negative log-likelihood, −2 ln L, with the background subtracted using these weights and the efficiency folded into the signal probability density function, as discussed in detail in Ref. [5].
Amplitude models for the Λ 0 b → J/ψ pπ − decays are constructed to examine the  possibility of exotic hadron contributions from the P c (4380) + and P c (4450) + → J/ψ p states and from the Z c (4200) − → J/ψ π − state, previously reported by the Belle collaboration in B 0 → J/ψ K + π − decays [16] (spin-parity J P = 1 + , mass and width of 4196 +31

−29 +17
−13 MeV and 370 ± 70 + 70 −132 MeV, respectively). By analogy with kaon decays [17], pπ − contributions from conventional nucleon excitations (denoted as N * ) produced with ∆I = 1/2 in Λ 0 b decays are expected to dominate over ∆ excitations with ∆I = 3/2, where I is isospin. The decay matrix elements for the two interfering decay chains, Λ 0 b → J/ψ N * , N * → pπ − and Λ 0 b → P + c π − , P + c → J/ψ p with J/ψ → µ + µ − in both cases, are identical to those used in the Λ 0 b → J/ψ pK − analysis [5], with K − and Λ * replaced by π − and N * . The additional decay chain, Λ 0 b → Z − c p, Z − c → J/ψ π − , is also included and is discussed in detail in the supplemental material. Helicity couplings, describing the dynamics of the decays, are expressed in terms of LS couplings [5], where L is the decay orbital angular momentum, and S is the sum of spins of the decay products. This is a convenient way to incorporate parity conservation in strong decays and to allow for reduction of the number of free parameters by excluding high L values for phase-space suppressed decays. Table 1 lists the N * resonances considered in the amplitude model of pπ − contributions. There are 15 well-established N * resonances [13]. The high-mass and high-spin states (9/2 and 11/2) are not included, since they require L ≥ 3 in the Λ 0 b decay and therefore are unlikely to be produced near the upper kinematic limit of m pπ . Theoretical models of baryon resonances predict many more high-mass states [18], which have not yet been observed. Their absence could arise from decreased couplings of the higher N * excitations to the simple production and decay channels [19] and possibly also from experimental difficulties in identifying broad resonances and insufficient statistics at high masses in scattering experiments. The possibility of high-mass, low-spin N * states is explored by including two very significant, but unconfirmed, resonances claimed by the BESIII collaboration in ψ(2S) → ppπ 0 decays [20]: 1/2 + N (2300) and 5/2 − N (2570). A nonresonant J P = 1/2 − pπ − S-wave component is also included. Two models, labeled "reduced" (RM) and "extended" (EM), are considered and differ in the number of resonances and of LS couplings included in the fit as listed in Table 1. The reduced model, used for the central values of fit fractions, includes only the resonances and L couplings that give individually significant contributions. The systematic uncertainties and the significances for the exotic states are evaluated with the extended model by including all well motivated resonances and the maximal number of LS couplings for which the fit is able to converge.
All N * resonances are described by Breit-Wigner functions [5] to model their lineshape and phase variation as a function of m pπ , except for the N (1535), which is described by a Flatté function [21] to account for the threshold of the nη channel. The mass and width are fixed to the values determined from previous experiments [13]. The couplings to the nη and pπ − channels for the N (1535) state are determined by the branching fractions of the two channels [22]. The nonresonant S-wave component is described with a function that depends inversely on m 2 pπ , as this is found to be preferred by the data. An alternative description of the 1/2 − pπ − contributions, including the N (1535) and nonresonant components, is provided by a K-matrix model obtained from multichannel partial wave analysis by the Bonn-Gatchina group [22,23] and is used to estimate systematic uncertainties.
The limited number of signal events and the large number of free parameters in the amplitude fits prevent an open-ended analysis of J/ψ p and J/ψ π − contributions. Therefore, the data are examined only for the presence of the previously observed P c (4380) + , P c (4450) + states [5] and the claimed Z c (4200) − resonance [16]. In the fits, the mass and width of each exotic state are fixed to the reported central values. The LS couplings describing P + c → J/ψ p decays are also fixed to the values obtained from the Cabibbo-favored channel. This leaves four free parameters per P + c state for the Λ 0 b → P + c π − couplings. The nominal fits are performed for the most likely (3/2 − , 5/2 + ) J P assignment to the P c (4380) + , P c (4450) + states [5]. All couplings for the 1 + Z c (4200) − contribution are allowed to vary (10 free parameters).
The fits show a significant improvement when exotic contributions are included. When all three exotic contributions are added to the EM N * -only model, the ∆(−2 ln L) value is 49.0, which corresponds to their combined statistical significance of 3.9 σ. Including the systematic uncertainties discussed later lowers their significance to 3.1 σ. The systematic uncertainties are included in subsequent significance figures. Because of the ambiguity between the P c (4380) + , P c (4450) + and Z c (4200) − contributions, no single one of them makes a significant difference to the model. Adding either state to a model already containing the other two, or the two P + c states to a model already containing the Z c (4200) − contribution, yields significances below 1.7 σ (0.4 σ for adding the Z c (4200) − after the two P + c states). If the Z c (4200) − contribution is assumed to be negligible, adding the two P + c states to a model without exotics yields a significance of 3.3 σ. On the other hand, under the assumption that no P + c states are produced, adding the Z c (4200) − to a model without exotics yields a significance of 3.2 σ. The significances are determined using Wilks' theorem [24], the applicability of which has been verified by simulation.
A satisfactory description of the data is already reached with the RM N * model if either the two P + c , or the Z − c , or all three states, are included in the fit. The projections of the full amplitude fit onto the invariant masses and the decay angles reasonably well reproduce the data, as shown in Figs. 2-5. The EM N * -only model does not give good descriptions of the peaking structure in m J/ψ p observed for m pπ > 1.8 GeV ( Fig. 3(b)). In fact, all contributions to ∆(−2 ln L) favoring the exotic components belong to this m pπ region. The models with the P + c states describe the m J/ψ p peaking structure better than with the Z c (4200) − alone (see the supplemental material).
The model with all three exotic resonances is used when determining the fit fractions. The sources of systematic uncertainty are listed in Table 2. They include varying the masses and widths of N * resonances, varying the masses and widths of the exotic states, considering N * model dependence and other possible spin-parities J P for the two P + c states, varying the Blatt-Weisskopf radius [5] between 1.5 and 4.5 GeV −1 , changing the angular momenta L in Λ 0 b decays that are used in the resonant mass description by one or two units, using the K-matrix model for the S-wave pπ resonances, varying the fixed couplings of the P + c decay by their uncertainties, and splitting Λ 0 b and J/ψ helicity angles into bins when determining the weights for the background subtraction to account for correlations between the invariant mass of J/ψ pπ − and these angles. A putative Z c (4430) − contribution [16,25,26] hardly improves the value of −2 ln L relative to the EM N * -only model, and thus is considered among systematic uncertainties. Exclusion of the Z c (4200) −   [GeV] state from the fit model is also considered to determine the systematic uncertainties for the two P + c states. The EM model is used to assess the uncertainty due to the N * modeling when computing significances. The RM model gives larger significances. All sources of systematic uncertainties, including the ambiguities in the quantum number assignments to the two P + c states, are accounted for in the calculation of the significance of various contributions, by using the smallest ∆(−2 ln L) among the fits representing different systematic variations.
The fit fractions for the P c (4380) + , P c (4450) + and Z c (4200) − states are measured to be (5.1 ± 1.5 +2.6 −1.6 )%, (1.6 +0.8 −0.6 +0.6 −0.5 )%, and (7.7 ± 2.8 +3.4 −4.0 )% respectively, and to be less than 8.9%, 2.9%, and 13.3% at 90% confidence level, respectively. When the two P + c states are not considered, the fraction for the Z c (4200) − state is surprisingly large, (17.2 ± 3.5)%, where the uncertainty is statistical only, given that its fit fraction was measured to be only (1.9 +0.7 [16]. Conversely, the fit fractions of the two P + c states remain stable regardless of the inclusion of the Z c (4200) − state. We measure the relative branching fraction −0.010 ± 0.009 for P c (4450) + , respectively, where the first error is statistical, the second is systematic, and the third is due to the systematic uncertainty on the fit fractions of the P + c states in J/ψ pK − decays. The results are consistent with a prediction of (0.07-0.08) [27], where the assumption is made that an additional diagram with internal W emission, which can only contribute to the Cabibbo-suppressed mode, is negligible. Our measurement rules out the proposal that the P + c state in the Λ 0 b → J/ψ pK − decay is produced mainly by the charmless Λ 0 b decay via the b → uus transition, since this predicts a very large value for R π/K = 0.58 ± 0.05 [28].
In conclusion, we have performed a full amplitude fit to Λ 0 b → J/ψ pπ − decays allowing for previously observed conventional (pπ − ) and exotic (J/ψ p and J/ψ π − ) resonances. A significantly better description of the data is achieved by either including the two P + c states observed in Λ 0 b → J/ψ pK − decays [5], or the Z c (4200) − state reported by the Belle collaboration in B 0 → J/ψ π − K + decays [16]. If both types of exotic resonances are included, the total significance for them is 3.1 σ. Individual exotic hadron components, or the two P + c states taken together, are not significant as long as the other(s) is (are) present. Within the statistical and systematic errors, the data are consistent with the P c (4380) + and P c (4450) + production rates expected from their previous observation and Cabibbo suppression. Assuming that the Z c (4200) − contribution is negligible, there is a 3.3 σ significance for the two P + c states taken together.
We thank the Bonn-Gatchina group who provided us with the K-matrix pπ − model. We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: [rad] φ Yields Figure 5: Background-subtracted data and fit projections of decay angles describing the N * decay chain, which are included in the amplitude fit. The helicity angle of particle P , θ P , is the polar angle in the rest frame of P between a decay product of P and the boost direction from the particle decaying to P . The azimuthal angle between decay planes of Λ 0 b and N * (of J/ψ ) is denoted as φ π (φ µ ). See Ref. [5] for more details.

Dalitz plot distributions
In Fig. 6, we show the Dalitz plots using the invariant mass squared, m 2 pπ vs m 2 J/ψ p , and m 2 pπ vs m 2 J/ψ π as independent variables. As expected, significant N * contributions are seen, especially in the region around m 2 pπ ≈ 2 GeV 2 . There is no visible narrow band around the P c (4450) region at about 20 GeV 2 in the J/ψ p mass-squared. However many events concentrate in a limited window around m 2 pπ = 6 GeV 2 and m 2 J/ψ p between 18 to 20 GeV 2 , which could be due to the P c contribution. Distributions of efficiency and background on the Dalitz plane are shown in Fig. 7, where the background consists of events from the Λ 0 b candidate mass sideband of 5665 − 5770 MeV. The efficiency is obtained from MC simulation with data-driven corrections applied for the particle identification of the pion and proton.     2 Additional fit results 2.1 Additional fit displays Figure 8 shows the m pπ distribution with all individual fit components overlaid. In Fig. 9 we show the same m pπ distribution but with a linear scale. The projections from the reduced model fit with the two P + c states are shown in Figs. 10-12. The projections from the reduced model fit with the Z c (4200) − state are shown in Figs. 13-15. The models with the P + c states describe the m J/ψ p peaking structure better than with the Z c (4200) − alone; the m J/ψ p distribution is better described in Fig. 11 (b) than that in Fig. 14 (b).

Fit fractions
The fit fraction of any component R is defined as f R = |M R | 2 dΦ/ |M| 2 dΦ, where M R is the matrix element, M, with all except the R amplitude terms are set to zero. The phase space volume dΦ is equal to p q dm pπ d cos θ Λ 0 b d cos θ N * d cos θ J/ψ dφ π dφ µ , where p is the momentum of the pπ system (i.e. N * ) in the Λ 0 b rest frame, and q is the momentum of π − in the N * rest frame. In Table 3, we show the fit fractions from the "reduced" and "extended" model fits.
[GeV]   [GeV]  [GeV]   3 Details of the matrix element for the decay amplitude 3.1 Helicity formalism and notation For each two-body decay A → B C, a coordinate system is set up in the rest frame of A, withẑ being 1 the direction of quantization for its spin. We denote this coordinate system as ( , where the superscript "{A}" means "in the rest frame of A", while the subscript "0" means the initial coordinates. For the first particle in the decay chain (Λ 0 b ), the choice of these coordinates is arbitrary. 2 However, once defined, these coordinates must be used consistently between all decay sequences described by the matrix element. For subsequent decays, e.g. B → D E, the choice of these coordinates is already fixed by the transformation from the A to the B rest frames, as discussed below. Helicity is defined as the projection of the spin of the particle onto the direction of its momentum. When the z axis coincides with the particle momentum, we denote its spin projection onto it (i.e. the m z quantum number) as λ. To use the helicity formalism, the initial coordinate system must be rotated to align the z axis with the direction of the momentum of one of the child particles, e.g. the B. A generalized rotation operator can be formulated in three-dimensional space, R(α, β, γ), that uses Euler angles. Applying this operator results in a sequence of rotations: first by the angle α about theẑ 0 axis, followed by the angle β about the rotatedŷ 1 axis and then finally by the angle γ about the rotatedẑ 2 axis. We use a subscript denoting the axes, to specify the rotations which have been already performed on the coordinates. The spin eigenstates of particle A, |J A , m A , in the (x where D J m, m (α, β, γ) * = J, m|R(α, β, γ)|J, m * = e i mα d J m,m (β) e i m γ , and where the small-d Wigner matrix contains known functions of β that depend on J, m, m . To achieve the rotation of the originalẑ . This is depicted in Fig. 16, for the case when the quantization axis for the spin of A is its momentum in some other reference frame. Since the third rotation is not necessary, we set γ = 0. 3 The angle θ {A} B is usually called "the A helicity angle", thus to simplify the 1 The "hat" symbol denotes a unit vector in a given direction. 2 When designing an analysis to be sensitive (or insensitive) to a particular case of polarization, the choice is not arbitrary, but this does not change the fact that one can quantize the Λ 0 b spin along any well-defined direction. The Λ 0 b polarization may be different for different choices. 3 An alternate convention is to set γ = −α. The two conventions lead to equivalent formulae. notation we will denote it as θ A . For compact notation, we will also denote φ {A} B as φ B . These angles can be determined from 4 Angular momentum conservation requires Each two-body decay contributes a multiplicative term to the matrix element The helicity couplings H A→B C λ B , λ C are complex constants. Their products from subsequent decays are to be determined by the fit to the data (they represent the decay dynamics). If the decay is strong or electromagnetic, it conserves parity which reduces the number of independent helicity couplings via the relation where P stands for the intrinsic parity of a particle. After multiplying terms given by Eq. direction (this is the z axis in the rest frame of A after the Euler rotations; we use the subscript "3" for the number of rotations performed on the coordinates, because of the three Euler angles, however, since we use the γ = 0 convention these coordinates are the same as after the first two rotations). This is visualized in Fig. 16, with B → D E particle labels replaced by A → B C labels. This transformation does not change vectors that are perpendicular to the boost direction. The transformed coordinates become the initial coordinate system quantizing the spin of B in its rest frame, The processes of rotation and subsequent boosting can be repeated until the final-state particles are reached. In practice, there are two equivalent ways to determine theẑ {B} 0 direction. Using Eq. (7) we can set it to the direction of the B momentum in the A rest frameẑ Alternatively, we can make use of the fact that B and C are back-to-back in the rest frame of A, p Since the momentum of C is antiparallel to the boost direction from the A to B rest frames, the C momentum in the B rest frame will be different, but it will still be antiparallel to this boost direction Figure 16: Coordinate axes for the spin quantization of particle A (bottom part), chosen to be the helicity frame of A (ẑ 0 || p A in the rest frame of its parent particle or in the laboratory frame), together with the polar (θ , is aligned with the B momentum; thus the rotated coordinates become the helicity frame of B. If B has a sequential decay, then the same boost-rotation process is repeated to define the helicity frame for its decay products.
Then we obtainŷ 0 . If C also decays, C → F G, then the coordinates for the quantization of C spin in the C rest frame are defined byẑ i.e. the z axis is reflected compared to the system used for the decay of particle B (it must point in the direction of C momentum in the A rest frame), but the x axis is kept the same, since we chose particle B for the rotation used in Eq. (5).

Matrix element for the Z − c decay chain
The decay matrix elements for the two interfering decay chains, Λ 0 b → J/ψ N * , N * → pπ − and Λ 0 b → P + c π − , P + c → J/ψ p with J/ψ → µ + µ − in both cases, are identical to those used in the Λ 0 b → J/ψ pK − analysis [5], with K − and Λ * replaced by π − and N * . We now turn to the discussion of the additional interfering decay chain, Λ 0 b → Z cf p, Z cf → ψπ − decays (denoting J/ψ as ψ), in which we allow more than one tetraquark state, f = 1, 2, . . . . Superscripts containing the Z c decay chain name without curly brackets, e.g. φ Zc , will denote quantities belonging to this decay chain and should not be confused with the superscript "{Z c }" denoting the Z c rest frame, e.g. φ {Zc} . With only a few exceptions, we omit the N * decay chain label. The angular calculations for the Z − c decay chain are analogous to that for P + c by interchange of p and π − , except for the angles to align the proton helicity.
The weak decay Λ 0 b → Z cf p is described by the term, where H Λ 0 b →Zc f p λ Zc ,λ Zc p are resonance (i.e. f ) dependent helicity couplings. As for |λ Zc − λ Zc p | ≤ 1 2 , there are two different helicity couplings for J Zc = 0, and four for higher spin. The above mentioned φ Zc , θ Zc Λ 0 b symbols refer to the azimuthal and polar angles of Z c in the Λ 0 b rest frame (see Fig. 17).
With the direction of Λ 0 b in the lab framep , and the direction of Z c in the Λ 0 b rest frame, the Λ 0 b helicity angle in the Z c decay chain can be calculated as, The φ Zc angle cannot be set to zero, since we have already defined thex The φ Zc angle can be determined in the Λ 0 b rest frame from where φ Zc ψ , θ Zc are the azimuthal and polar angles of the ψ in the Z c rest frame (see Fig. 17). Theẑ The azimuthal angle of the ψ can now be determined in the Z c rest frame (see Fig. 17) Thex {Zc} 0 direction is defined by the convention that we used in the Λ 0 b rest frame. Thus, we have α Zc p = atan2 (ẑ where all vectors are in the p rest frame,x L Zc f ≤ J Zc f + 1, which is further restricted by the parity conservation in the Z cf decays, P Zc f = (−1) L Zc f . We assume the minimal values of L and of L Zc f in R Zc f (m ψπ ). The electromagnetic decay ψ → µ + µ − in the Z c decay chain contributes a term D 1 λ Zc ψ , ∆λ Zc µ (φ Zc µ , θ Zc ψ , 0) * .
The azimuthal and polar angle of the muon in the ψ rest frame, φ Zc µ , θ Zc ψ , are different from φ µ , θ ψ introduced in the N * decay chain. The ψ helicity axis is along the boost direction from the Z c to the ψ rest frames, which is given bŷ and so cos θ Zc The x axis is inherited from the Z c rest frame (Eq. (11) as well asx Collecting terms from the three subsequent decays in the Z c chain together, and adding them coherently to the N * and the P c matrix elements, via appropriate relations