Observation of a narrow pentaquark state, $P_c(4312)^+$, and of two-peak structure of the $P_c(4450)^+$

A narrow pentaquark state, $P_c(4312)^+$, decaying to $J/\psi p$ is discovered with a statistical significance of $7.3\sigma$ in a data sample of ${\Lambda_b^0\to J/\psi p K^-}$ decays which is an order of magnitude larger than that previously analyzed by the LHCb collaboration. The $P_c(4450)^+$ pentaquark structure formerly reported by LHCb is confirmed and observed to consist of two narrow overlapping peaks, $P_c(4440)^+$ and $P_c(4457)^+$, where the statistical significance of this two-peak interpretation is $5.4\sigma$. Proximity of the $\Sigma_c^+\bar{D}^{0}$ and $\Sigma_c^+\bar{D}^{*0}$ thresholds to the observed narrow peaks suggests that they play an important role in the dynamics of these states.

smooth parametrization of the background.Here, background refers to Λ * reflections, small non-Λ 0 b contributions (which comprise 6.4% of the sample), and possibly additional broad P + c structures.Many different background parametrizations are considered (discussed below), each of which is found to produce negligible bias in the P + c parameters in the validation fits.Each fit component is multiplied by a phase-space factor, p • q, where p (q) is the break-up momentum in the Λ 0 b → P + c K − (P + c → J/ψ p) decay.Since the signal peaks are narrow, all fit components are convolved with the detector resolution, which is 2-3 MeV in the fit region (see the Supplemental Material).Finally, the detection efficiency has negligible impact on the signal m J/ψ p distributions, and therefore, is not considered in these fits.
In the nominal fits, the BW contributions are added incoherently.The results of these fits are displayed in Fig. 5 for two parametrizations of the background: one using a high-order polynomial; and another using a low-order polynomial, along with an additional wide P + c BW term whose mass and width are free to vary in the fits.For both background parametrizations, a range of polynomial orders is considered.The lowest order used for each case is the smallest that adequately describes the data, which is found to correspond to the minimum order required to obtain unbiased P + c estimators in the fit-validation studies in the absence of interference.The highest orders are chosen such that the background model is capable of describing any structures that could be produced by either non-P + c or broad-P + c contributions. Figure 6 shows the fit from which the central values of the P + c properties are obtained, while the background-model-dependent variations observed in these properties are included in the systematic uncertainties.The fits with and without the broad P + c state both describe the data well.Therefore, these fits can neither confirm nor contradict the existence of the P c (4380) + state.
To determine the significance of the P c (4312) + state, the change of the fit χ 2 when adding this component is used as the test statistic, where the distribution under the null hypothesis is obtained from a large ensemble of pseudoexperiments.The p-value, expressed in Gaussian standard deviations, corresponds to 7.6σ (8.5σ) for the fits to the m Kp > 1.9 GeV (cos θ P c -weighted) distribution, ignoring the look-elsewhere effect.To account for this effect, the m J/ψ p distribution in each pseudoexperiment is scanned to find the most significant narrow and isolated peak (excluding the 4450 MeV peak region).This method lowers the P c (4312) + significance to 7.3σ (8.2σ).To evaluate the significance of the two-peak structure versus the one-peak interpretation of the 4450 MeV region, the null hypothesis uses just one BW to encompass both the P c (4440) + and P c (4457) + peaks (the fit also includes the P c (4312) + BW), which gives P c (4450) + mass and width values that are consistent with those obtained from the amplitude analysis of Ref. [1].Pseudoexperiments are again used to determine the ∆χ 2 distribution under the null hypothesis.The significance of the two-peak structure is 5.4σ (6.2σ) for the m Kp > 1.9 GeV (cos θ P c -weighted) samples.This significance is large enough to render the single peak interpretation of the 4450 MeV region obsolete.Therefore, the results presented here for this structure supersede those previously presented in Ref. [1].
To investigate the systematic uncertainties on P + c properties due to interference, which can only be important for P + c resonances with the same spin and parity, fits to the cos θ P c -weighted distribution are repeated using various coherent sums of two of the BW amplitudes.Each of these fits includes a phase between interfering resonances as an extra free parameter.None of the interference effects studied is found to produce a significant ∆χ 2 relative to the fits using an incoherent sum of BW amplitudes.However, substantial shifts in the P + c properties are observed, and are included in the systematic uncertainties.For example, in such fit the P c (4312) + mass increases, while its width is rather stable, leading to a large positive systematic uncertainty of 6.8 MeV on its mass.
As in Ref. [1], the Λ 0 b candidates are kinematically constrained to the known J/ψ and Λ 0 b masses [25], which substantially improves the m J/ψ p resolution and determines the absolute mass scale with an accuracy of 0.2 MeV.The mass resolution is known with a 10% relative uncertainty.Varying this within its uncertainty changes the widths of the narrow states in the nominal fit by up to 0.5 MeV, 0.2 MeV, and 0.8 MeV for the P c (4312) + , P c (4440) + , and P c (4457) + states, respectively.The widths of all three narrow P + c peaks are consistent with the mass resolution within the systematic uncertainties.Therefore, upper limits are placed on their natural widths at the 95% confidence level (CL), which account for the uncertainty on the detector resolution and in the fit model.
A number of additional fits are performed when evaluating the systematic uncertainties.The nominal fits assume S-wave (no angular momentum) production and decay.Including P-wave factors in the BW amplitudes has negligible effect on the results.In addition to the nominal fits with three narrow peaks in the 4.22 < m J/ψ p < 4.57 GeV region, fits including only the P c (4312) + are performed in the narrow 4.22-4.44GeV range.Fits are also performed using a data sample selected with an alternative approach, where no BDT is used resulting in about twice as much background.
The total systematic uncertainties assigned on the mass and width of each narrow P + c state are taken to be the largest deviations observed among all fits.These include the fits to all three versions of the m J/ψ p distribution, each configuration of the P + c interference, all variations of the background model, and each of the additional fits just described.The masses, widths, and the relative contributions (R values) of the three narrow P + c states, including all systematic uncertainties, are given in Table 1.
To obtain estimates of the relative contributions of the P + c states, the Λ 0 b candidates are weighted by the inverse of the reconstruction efficiency, which is parametrized in all six dimensions of the Λ 0 b decay phase-space (Eq.( 68) in the Supplemental Material to Ref. [26]).The efficiency-weighted m J/ψ p distribution, without the m Kp > 1.9 GeV requirement, is fit to determine the P + c contributions, which are then divided by the efficiency-corrected and background-subtracted Λ 0 b yields.This method makes the results independent of the unknown quantum numbers and helicity structure of the P + c production and decay.Unfortunately, this approach also suffers from large Λ * backgrounds and from sizable fluctuations in the low-efficiency regions.In these fits, the P + c terms are added incoherently, absorbing any interference effects, which can be large (see, e.g., Fig. S2 in the Supplemental Material), into the BW amplitudes.Therefore, the R ≡ B(Λ state differ from the fit fractions typically reported in amplitude analyses, since R includes both the BW amplitude squared and all of its interference terms.Similar fit variations are considered here as above, e.g., different background models and selection criteria are all evaluated.The resulting systematic uncertainties on R are large, as shown in Table 1. The narrow widths of the P + c peaks make a compelling case for the bound-state character of the observed states.However, it has been pointed out by many authors [16][17][18][19] that peaking structures in this J/ψ p mass range can also be generated by triangle diagrams.The P c (4312) + and P c (4440) + peaks are unlikely to arise from triangle diagrams, due to a lack of any appropriate hadron-rescattering thresholds as discussed in more detail in the Supplemental Material.The P c (4457) + peaks at the Λ + c (2595)D 0 threshold (J P = 1/2 + in S-wave) [18], and the D s1 (2860) − meson is a suitable candidate to be exchanged in the corresponding triangle diagram.However, this triangle-diagram term does not describe the data nearly as well as the BW does (Fig. S5 in the Supplemental Material).This possibility deserves more scrutiny within the amplitude-analysis approach.
Narrow approximately 5 MeV and 2 MeV below the Σ + c D 0 and Σ + c D * 0 thresholds, respectively, as illustrated in Fig. 6, making them excellent candidates for bound states of these systems.The P c (4440) + could be the second Σ c D * state, with about 20 MeV of binding energy, since two states with J P = 1/2 − and 3/2 − are possible.In fact, several papers on hidden-charm states created dynamically by charmed meson-baryon interactions [31][32][33] were published well before the first observation of the P + c structures [1] and some of these predictions for Σ + c D 0 and Σ + c D * 0 states [28][29][30] are consistent with the observed narrow P + c states.Such an interpretation of the P c (4312) + state (implies J P = 1/2 − ) would point to the importance of ρ-meson exchange, since a pion cannot be exchanged in this system [10].
In summary, the nine-fold increase in the number of Λ 0 b → J/ψ pK − decays recon-structed with the LHCb detector sheds more light onto the J/ψ p structures found in this final state.The previously reported P c (4450) + peak [1] is confirmed and resolved at 5.4σ significance into two narrow states: the P c (4440) + and P c (4457) + exotic baryons.A narrow companion state, P c (4312) + , is discovered with 7.3σ significance.
The minimal quark content of these states is duucc.Since all three states are narrow and below the Σ + c D 0 and Σ + c D * 0 ([duc][uc]) thresholds within plausible hadron-hadron binding energies, they provide the strongest experimental evidence to date for the existence of bound states of a baryon and a meson.The Σ + c D 0 (Σ + c D * 0 ) threshold is within the extent of the P c (4312) + (P c (4457) + ) peak, and therefore virtual [34] rather than bound states are among the plausible explanations.In simple tightly bound pentaquark models, the proximity of these states to baryon-meson thresholds would be coincidental, and furthermore, it is difficult to accommodate their narrow widths [35].A potential barrier between diquarks, which could separate the c and c quarks, has been proposed to solve similar difficulties for tetraquark candidates [36].An interplay between tightly bound pentaquarks and the Σ c D, Σ c D * thresholds may also be responsible for the P + c peaks [37][38][39][40].Therefore, such alternative explanations cannot be ruled out.Proper identification of the internal structure of the observed states will require more experimental and theoretical scrutiny.
Observation of a narrow pentaquark state, P c (4312) + , and of two-peak structure of the P c (4450 The Λ 0 b → J/ψ pK − sample analyzed in the Letter is selected by requiring that the invariant mass of J/ψ pK − candidates is in the 5605-5635 MeV range.To determine the Λ 0 b signal yield within this range, the background density is linearly interpolated from the 5480-5580 MeV and 5660-5760 MeV sidebands to the signal region, as illustrated in Fig. S1.There are 246k Λ 0 b decays with 6.4% background contamination in the analyzed sample. After selecting candidates in the Λ 0 b signal region indicated in Fig. S1, the Λ 0 b mass constraint is imposed on all Λ 0 b candidates, in addition to the J/ψ mass and vertex constraints, to improve the m J/ψ p resolution.To a good approximation, the mass resolution is Gaussian with a standard deviation (σ m ) given by σ m (m J/ψ p ) = 2.71 − 6.56 • 10 −6 (m J/ψ p / MeV − 4567) 2 MeV. (S1) 2 Example fit with interference  c state is added coherently to the P c (4312) + amplitude.In this fit model, the magnitude of the P c (4312) + peak in the data is dominated by its interference with the broad P + c state.Each P + c contribution is displayed as the BW amplitude squared (the interference contributions are not shown individually).

Study of triangle diagrams
The narrow widths of the P + c peaks make a compelling case for the bound-state character of the observed states.However, it has been pointed out by many authors [16][17][18][19] that peaking structures in this J/ψ p mass range can also be generated by triangle diagrams (see Fig. S3).In these processes, the Λ 0 b baryon (of mass m 1 ) decays into two nearly on-mass-shell hadrons, one of which (of mass m h ≡ √ t) is an excited strange meson or baryon (denoted here as h) that subsequently emits a kaon (of mass m 2 ) and a non-strange decay product (of mass m 4 ) that rescatters with the other Λ 0 b child (of mass m 3 ) into the J/ψ p system (of m J/ψ p ≡ √ s).Such triangle-diagram processes are known to peak when all three hadrons in the triangle are nearly on their mass shells.Since the overall probability across coupled channels must be conserved, a peak in the J/ψ p channel is only possible if there is a corresponding depletion in the final state composed of the particles that rescatter in Fig. S3 to form the J/ψ p [41].
The triangle-diagram contribution often peaks at a threshold, given by the sum of the masses of the rescattering hadrons (m 3 + m 4 ) creating a cusp.For a fine-tuned BW resonance mass of the intermediate hadron h (M 0 ), the rate can peak above (but never below) the corresponding threshold.The amplitude for a triangle-diagram process, which incorporates a finite width for the exchanged particle (Γ 0 ), is given by where all quantities are defined in Fig. S3.The BW term corresponds to the exchanged h hadron.The Feynman triangle-amplitude formula is expressed in terms of a onedimensional integral over a single Feynman parameter x as follows: where Here, λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc and is a small real number.The 4457 MeV structure peaks near the Λ + c (2595)D 0 threshold (J P = 1/2 + ) [18].The best h candidate for the corresponding triangle diagram is the D s1 (2860) − meson, which has a mass of 2859 ± 27 MeV and a width of 159 ± 80 MeV [25].The P c (4312) + is not far from the Λ + c D * 0 threshold (J P = 1/2 − or 3/2 − ).Exchanging an excited 1 − D * * s meson with M 0 = 3288 MeV produces the peak at 4312 MeV in the narrow width approximation (Γ 0 → 0).The P c (4440) + is well above the χ c0 p threshold (1/2 + ).Exchanging an excited 1/2 + Λ * with M 0 = 2153 MeV produces a peak at the right mass when Γ 0 → 0. In fact, a good quality fit to all three P + c peaks is obtained when Γ 0 is small, as illustrated in Fig. S4 (top).However, this interpretation is unrealistic for the P c (4312) + and P c (4440) + peaks.When more plausible widths for the excited hadrons are used, such as Γ 0 = 50 MeV, no narrow peaks can be created above the thresholds, as shown in Fig. S4 (bottom).The triangle-diagram hypothesis is more plausible for the P c (4457) + state.An example fit using two BW terms and one triangle-diagram amplitude is shown in Fig. S5.The fit quality is lower than that obtained using three BW amplitudes.However, further investigation of this interpretation of the P c (4457) + state is warranted within an amplitude analysis, which will provide greater discrimination between the triangle-diagram and BW amplitudes.

Figure 1 :
Figure 1: Distribution of (left) m J/ψ p and (right) m Kp for Λ 0 b → J/ψ pK − candidates.The prominent peak in m Kp is due to the Λ(1520) resonance.

Figure 3 :
Figure 3: Distribution of m J/ψ p from Λ 0 b → J/ψ pK − candidates after suppression of the dominant Λ * → pK − contributions with the m Kp > 1.9 GeV requirement.The inset shows a zoom into the region of the narrow P + c peaks.

Figure 5 :
Figure 5: Fits to the m J/ψ p distributions of the (top row) inclusive, (middle row) m Kp > 1.9 GeV, and (bottom row) cos θ P c -weighted samples with three incoherently summed BW amplitudes representing the narrow P + c signals on top of a (left column) high-order polynomial function or (right column) lower-order polynomial plus a broad P + c state represented by a fourth BW amplitude.

Figure 6 :
Figure 6: Fit to the cos θ P c -weighted m J/ψ p distribution with three BW amplitudes and a sixth-order polynomial background.This fit is used to determine the central values of the masses and widths of the P + c states.The mass thresholds for the Σ + c D 0 and Σ + c D * 0 final states are superimposed.

Figure S1 :
Figure S1: Invariant mass spectrum of J/ψ pK − candidates.The Λ 0 b signal region is between the vertical red lines.A linear interpolation of the background, determined from the sideband regions (bounded by the shorter vertical blue lines), to the signal region is shown by the dashed blue line.

Figure
Figure S2 shows an example fit with interfering resonances.

Figure S2 :
Figure S2: Fit to the cos θ P c -weighted m J/ψ p distribution with four BW amplitudes and a linear background.The broad P +c state is added coherently to the P c (4312) + amplitude.In this fit model, the magnitude of the P c (4312) + peak in the data is dominated by its interference with the broad P + c state.Each P + c contribution is displayed as the BW amplitude squared (the interference contributions are not shown individually).

Figure S3 :
Figure S3: Triangle diagram for the Λ 0 b → J/ψ pK − decay.The figure defines the symbols used in the formulae in the text.

Figure S4 :Figure S5 :
Figure S4: Fit of three triangle-diagram amplitudes and a quadratic background to the cos θ P cweighted distribution.The widths of the excited particles exchanged in the triangles is (top) an unrealistic value of 1 MeV or (bottom) a more plausible value of 50 MeV.Individual triangle diagram contributions are also shown.The dashed vertical lines are the Λ + c D * 0 , χ c0 p and Λ + c (2595)D 0 thresholds.

Table 1 :
Summary of P + c properties.The central values are based on the fit displayed in Fig.6.
Università di Roma Tor Vergata, Roma, Italy k Università di Roma La Sapienza, Roma, Italy l AGH -University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland m LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain n Hanoi University of Science, Hanoi, Vietnam o Università di Padova, Padova, Italy p Università di Pisa, Pisa, Italy q Università degli Studi di Milano, Milano, Italy r Università di Urbino, Urbino, Italy s Università della Basilicata, Potenza, Italy t Scuola Normale Superiore, Pisa, Italy Sezione INFN di Trieste, Trieste, Italy z School of Physics and Information Technology, Shaanxi Normal University (SNNU), Xi'an, China aa Physics and Micro Electronic College, Hunan University, Changsha City, China ab Lanzhou University, Lanzhou, China ac Thomas Jefferson National Accelerator Facility, Newport News, United States † Deceased b Laboratoire Leprince-Ringuet, Palaiseau, France c P.N.Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia d Università di Bari, Bari, Italy e Università di Bologna, Bologna, Italy f Università di Cagliari, Cagliari, Italy g Università di Ferrara, Ferrara, Italy h Università di Genova, Genova, Italy i Università di Milano Bicocca, Milano, Italy j u Università di Modena e Reggio Emilia, Modena, Italy v H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom w MSU -Iligan Institute of Technology (MSU-IIT), Iligan, Philippines x Novosibirsk State University, Novosibirsk, Russiay