Experimental constraints on the properties of $P_c$ states

Using new experimental data on photoproduction and $\Lambda_b^0$ decays, we derive constraints on the properties of the LHCb $P_c$ states. We conclude that $P_c(4312)$, $P_c(4380)$ and $P_c(4440)$ can be described as $\Sigma_c\bar D$, $\Sigma_c^*\bar D$ and $\Sigma_c\bar D^*$ molecules, but that $P_c(4457)$ does not fit into the same picture. Based on the apparent absence of additional partner states, and the striking disparity between $\Lambda_c^+\bar D^0$ and $\Lambda_c^+\bar D^{*0}$ decays, we conclude that $P_c(4440)$ has $3/2^-$ quantum numbers. Using heavy-quark symmetry we predict large experimental signals for $P_c$ states in $\eta_c p K^-$, $\Lambda_c^+\bar D^{*0}K^-$, and $\Sigma_c^{(*)}\bar D K^-$. We also argue that current experimental data on photoproduction is almost at the level of sensitivity required to observe $P_c$ states.


I. INTRODUCTION
The LHCb P c states have so far been observed only in the J/ψ p spectrum of Λ b → J/ψ p K − decays [1][2][3], but experimental data from other processes can also usefully constrain their properties. In this paper we explore the consequences of some recent experimental observations whose implications for the phenomenology of the P c states have not been recognised in the literature. In particular, we consider: tighter constraints on photoproduction and J/ψ p decays from the J/ψ-007 experiment, measurements of Λ 0 b decays to Λ + cD 0 K − , Λ + cD * 0 K − and η c pK − (including limits on P c fit fractions), and the apparent absence of P c partners with higher mass.
In the molecular scenario, we find that these new experimental results imply constraints on the quantum numbers of P c states, and give striking predictions for prominent production channels that can be tested in experiment. Our observations follow directly from experiment, with minimal theoretical assumptions. At most, we rely only on symmetry principles which are well-justified theoretically and empirically: heavy-quark symmetry, and the dominance of colour-favoured processes in weak transitions.
We begin with an overview of molecular scenarios for the P c states (Sec. II), and then review the recent experimental data which forms the basis of our analysis (Sec. III). We then consider the resulting phenomenology associated with various decay modes, namely J/ψ p (Sec. IV), η c p (Sec. V), Λ + cD 0 (Sec. VI), Λ cD ( * ) π (Sec. VII), Λ + cD * 0 (Sec. VIII) and Σ ( * ) cD (Sec. IX). We finally consider the implications of the apparent absence of heavier partners to the P c states (Sec. X), and summarise our main results (Sec. XI).

II. SCENARIOS
In the discovery paper [1], an amplitude analysis of Λ b → J/ψ p K − identified two states decaying to J/ψ p: the broad P c (4380) and narrower P c (4450). A model-independent analysis [2] confirmed the importance of including these contributions in adequately describing the data. A subsequent analysis, with an order of magnitude more data [3], discovered an additional new state P c (4312), and resolved the initial P c (4450) into two distinct, narrow peaks, P c (4440) and P c (4457).
The original P c (4380) did not feature in the newer analysis, which was sensitive only a genuine resonance, but with 1/2 + quantum numbers and Λ c (2595)D degrees of freedom.
We find that all of these scenarios give a very good fit to the Λ b → J/ψ p K − data [31].
It turns out to be very liberating to abandon the hypothesis that P c (4457) is a bound Σ cD * state, and not only because the alternative scenarios give a good fit to data. As we argue below, there are some phenomenological problems with the usual modelling, all of which can be traced to the assumption that P c (4457) is a bound Σ cD * state. By abandoning this assumption we avoid these problems.
Our alternative Scenario "C", also shown in the Table I, differs from the other two in no longer assuming that P c (4457) is a Σ cD * bound state. The assignments of the other states are consistent with Scenario B; the rationale for preferring 3/2 − quantum numbers for P c (4440) is explained below. Note that our Scenario C actually incorporates several possibilities, corresponding to different explanations for P c (4457).
Of course other scenarios are also discussed in the literature. In particular, we highlight refs. [33,34] in which non-resonant interpretations of P c (4312) have been discussed.

III. EXPERIMENTAL DATA
In Table II we show the experimental data which forms the basis of much of our discussion.
For each of four possible Λ b decays (indicated in the top row), we show the three-body branching fraction (B) and, where data are available, a measure (R) of the contribution of P c states in the fit, defined below. The data in the last three columns are quite new, and their significance has not yet been appreciated in literature.
The three-body branching fractions B for Λ 0 b → J/ψ pK − and Λ 0 b → η c pK − are from LHCb [35][36][37] [38], combining the measured ratios from LHCb data, with the Particle Data Group (PDG) value for the denominators [37] The quantity R is a measure of the contribution of the P c states in the fit, although the experimental procedures in how this number is computed differ. For Λ 0 b → J/ψ pK − and Λ 0 b → η c pK − , the numbers are from refs. [3,36], where they are described as "relative contributions", and are identified in terms of branching fractions as and The numbers for Λ 0 b → Λ + cD 0 K − , from the amplitude analysis of Piucci [39], are "fit fractions", defined in the usual way: the ratio, for a single resonance versus the full amplitude, of the phase space integrals of the square of the matrix element. (In the original P c discovery paper [1], the contributions of P c (4380) and P c (4450) were also quantified in terms of fit fractions.) We adopt the usual interpretation of the fit fraction, which is widespread in the experimental literature and the PDG [37], in terms of branching fractions, namely Note that the limits on the fit fractions depend on the assumed quantum numbers (1/2 − or 3/2 − ) of the P c states, as shown in the table.
The difference between the "relative contributions" of refs. [3,36] and the "fit fractions" of refs. [1,39] relates to the treatment of interference terms, and is discussed in ref. [3].
There is however no difference in the way the numbers are interpreted in terms of branching fractions, as shown in equations (4), (5) and (6). Hence we will use the generic term "fit fraction", and the same symbol R, for both quantities.
We remark, however, that for both the "relative contributions" and "fit fractions", the standard interpretation of these experimental numbers in terms of branching fractions has an implicit assumption, which is not necessarily justified. The factorisation of the numerator into a product of branching fractions is legitimate if the P c signal is due to a resonance, but not, for example, if it arises from a kinematical singularity or threshold cusp, in which case it is not possible (conceptually or mathematically) to separate the production (Λ 0 b → P + c K − ) and decay (such as P + c → J/ψ p). We discussed this point previously in relation to the X(2900) states [40], for which several competing interpretations are possible [41].
Much of our analysis below involves manipulations of measured R values, and in particular it relies on the standard factorisation of the branching fractions in the numerators.
In our preferred Scenario C, we can only assume this factorisation for P c (4312/4380/4440), which we are treating as Σ ( * ) cD ( * ) resonances. We cannot assume factorisation for P c (4457) because it is not justified for some of the scenarios applicable to this state (such as threshold cusp and triangle singularity); hence in much of the discussion below we will not consider P c (4457).
In computing the fit fractions R(Λ + cD 0 ), the masses and widths of P c states, as determined in previous experiments, are taken as inputs [39]. For this reason we only quote the results for P c (4312/4440/4457), rather than those of the precursor states P c (4380/4450), whose mass and width measurements are considered to be obsolete [3].
There is a caveat on the limits on Λ 0 b → Λ + The discussion so far has concentrated on P c states in 3-body Λ b decays. Another possibility, much discussed in the literature, is to search for the states in photoproduction [42][43][44][45][46][47].
Currently there is no evidence for P c states in γp → P c → J/ψ p, but the measured upper limits, which have recently become tighter, have interesting implications.
In particular, recall that the absence of the states in γp → P c → J/ψ p implies upper limits, via vector meson dominance, on the P c → J/ψ p branching fractions. After the initial discovery of the original LHCb states P c (4450) and P c (4380), analysis of photoproduction data available at the time implied an upper limit on the J/ψ p branching fractions at the percent level [42]. More recently, the GlueX experiment [45] obtained similar upper limits from their new data, using a variant of the JPAC model for the amplitude [46]. The GlueX results assume 3/2 − quantum numbers for all the states, however choosing a different assignment is not expected to change the results drastically. The results are depending on which of P c (4312/4440/4457) is being considered.
These stringent upper limits have recently become even tighter: preliminary results from another Jefferson Lab experiment, J/ψ-007, indicate that the J/ψ p upper limits are smaller than the GlueX limits by a factor of approximately 10 [47].
We may summarise the situation, rather roughly, as This is adequate for present purposes, as our arguments below only require estimates of the scales of the branching fractions, rather than their precise values. (We are also limited in not having an estimate of the systematic uncertainty associated with the assumed 3/2 − quantum numbers in the JPAC/GlueX analysis.)

IV. J/ψ p DECAYS
We now come to the main thrust of the paper, exploring the implications of the experimental data discussed in the previous section. We begin with J/ψ p decays, and note that the stringent upper limit (8) is in conflict with the vast majority of models, in some cases by orders of magnitude [4,5,9,[48][49][50][51][52][53][54][55][56][57][58]. Our own estimate [24], for P c (4440), is one of very few that is consistent with (8). Similarly, Dong et al. [59] obtain a P c (4312) branching fraction which is near to the upper limit (8). Eides et al. [60] computed J/ψ p branching fractions in both the molecular and hadrocharmonium pictures, noting a very different pattern in decays; their results for the molecular scenario, but not the hadrocharmonium scenario, are consistent with (8).
Superficially, the tight upper limit on J/ψ p decays may appear surprising, considering that the P c states were of course discovered as prominent peaks in the J/ψ p spectrum. This illustrates the point that in quantifying the prominence of a structure in the 2-body spectrum of a 3-body decay, the most relevant quantity is not the 2-body decay branching fraction itself, but the fit fraction R. Recalling equation (4), the P c fit fractions are a measure of the fraction of all J/ψ pK − events that are produced via Λ 0 b → P + c K − , P + c → J/ψ p. Obviously, the fit fractions therefore depend not only on the properties of the P c states themselves (the production and decay branching fractions in the numerator), but also on the sum total of all other production mechanisms resulting in the J/ψ pK − final state (the denominator). It  Fig. 2(b) and 2(c) are colour-suppressed [24]. On this basis we expect, very roughly, which is consistent with the numbers in Table II.
We conclude that the P c structures in Λ 0 b → J/ψ p K − are prominent (namely, have large fit fractions) in part because of the suppression of tree-level Λ 0 b → J/ψ p K − decays. Intriguingly, the situation is similar for several other exotic states observed in 3-body weak decays.
(Note that χ c1 (3872) has much larger branching fraction to D 0D * 0 , for example, but the tree-level diagram D 0D * 0 K is colour-enhanced, and much larger.) In ref. [40] we noted a similar mechanism enhancing the X(2900) signal in We return now to the tighter upper limit (8) on J/ψ p decays, and note that it has striking implications for P c production in Λ 0 b decays. Upper limits on B(P + c → J/ψ p) imply lower limits on B(Λ 0 b → P + c K − ) [63]. Following our previous paper [24], in Fig. 3 we plot B(Λ 0 b → P + c K − ) as a function of B(P + c → J/ψ p), using equation (4), and the experimental values (Table II) for B(Λ 0 b → J/ψ p K − ) and R(J/ψ p). We have also included P c (4380) in the plot, using the measured value [1] although as noted below, this part of the plot should be interpreted with some caution. than our previous estimates [24], because the limit on the J/ψ p branching fraction is now smaller by a factor of 10.) Strikingly, these numbers are comparable to the largest measured two-body branching fraction for Λ 0 b decays [37] The comparison is quite awkward because naively one would expect the production of multiquark states (whether molecular or compact) to be suppressed compared to that of conventional hadrons, and indeed that is what is observed in other sectors. The closest analogy is with χ c1 (3872), for which the measured branching fraction [37], is around a factor of fifty smaller than the corresponding branching fractions for conven- . Indeed, if anything, we would expect an even stronger relative suppression for the P c states because, unlike in the case of χ c1 (3872), their dominant wavefunction components cannot be produced in colour-favoured processes [24].
With this comparison in mind, the production branching fractions for P c (4380) seem implausibly large, but as noted previously, the numbers in this case should be treated with caution: we have used a measured fit fraction which, along with other properties of this state, are now considered to be obsolete [3].
But even the less dramatic numbers for P c (4312/4440/4457) present a challenge for models, and of course, the challenge becomes more acute as the limits on branching fractions B(P + c → J/ψ p) become tighter. This suggests that B(P + c → J/ψ p) cannot be much less than the current upper bound (8), otherwise B(Λ 0 b → P + c K − ) will become implausibly large. Indirectly, it suggests that the sensitivity required to observe P c states in photoproduction is not much more than that of the J/ψ-007 experiment.
Note that the production branching fractions implied by the above analysis are orders of magnitude larger than those predicted by the effective Lagrangian approach of ref. [64].
V. η c p DECAYS For 1/2 − states the S-wave decays to J/ψ p and η c p are related by heavy-quark symmetry [19,65]. In our Scenario C there is a single 1/2 − state, the P c (4312) with Σ cD constituents. Ignoring differences due to phase space, the relation among branching fractions We conclude, by comparison to equation (8), that Nevertheless, the experimental prospects in Λ b → η c pK − decays are encouraging: as noted previously, the relevant quantity is not the branching fraction B(P c → J/ψ p), but the fit fraction R(η c p). From equations (4) and (5) we find and thus, using the numbers from Table II, we predict a substantial fit fraction for P c (4312): This is large in comparison to the fit fractions for the discovery mode Λ 0 b → J/ψ p K − , suggesting the experimental prospects are encouraging. Note that the prediction is consistent with the experimental upper limit in Table II. As for the other states, for P c (4457) we cannot make any prediction for η c p decays in our Scenario C, without further modelling. (As noted previously, there are several viable scenarios for this state.) However for P c (4380) or P c (4440) we anticipate, on general grounds, negligible η c p signals, as they both have 3/2 − quantum numbers and thus do not couple to η c p in S-wave. This is quite different from Scenarios A and B, in which one of P c (4440) or P c (4457) is a Σ cD * bound state with 1/2 − quantum numbers, and thus couples to η c p in S-wave. Drawing on the results of ref. [65], we find This implies, in comparison to equation (8), that if P c (4440) or P c (4457) is a 1/2 − state, In terms of fit fractions, in Scenario A we predict for P c (4440) and negligible η c p for P c (4457), whereas for Scenario B we predict for P c (4457), and negligible η c p for P c (4440). The R(η c p) fit fractions in these scenarios are comparable to the measured R(J/ψ p) fit fractions in Table II, suggesting such measurements may be within reach in future analyses. Confronting these predictions with data can discriminate among Scenarios A, B and C.
We now turn to Λ + cD 0 decays, which in many models are expected to be prominent channels. As we will see, this is not borne out by experimental data.
For a given P c state, combining equations (4) and (6) yields a relation between the Λ + cD 0 and J/ψ p branching fractions Taking the experimental data from Table II we obtain depending on which of the states P c (4312/4440/4457) is considered, and its assumed quantum numbers. In combination with the photoproduction upper limit (8), we arrive at the surprising result We will show later that this limit argues in favour of Scenario C.
In previous sections we observed that the P c branching fractions to J/ψ p, η c p and Λ + cD 0 are tiny. This implies that their measured decay widths, must be dominated by other modes. One possibility is three-body decays which, in the molecular scenario, would arise via the decay of a constituent hadron. Given thatD is stable andD * has negligible decay width, molecular three-body decays are presumably dominated by Σ c → Λ c π or Σ * c → Λ c π [66]. Following refs. [24,65,67], we assume that the three-body width is determined by the width of the Σ ( * ) c constituent, resulting in the following predicted partial widths Evidently these three-body decays cannot account entirely for the measured P c decay widths.
Given the tiny J/ψ p branching fractions, we also disregard more esoteric possibilities like χ c0 p and J/ψN π.
We have established that the partial widths of P c states to J/ψ p, η c p and Λ + cD 0 are tiny, and that three-body modes, while not negligible, cannot account for the measured decay widths. The rest of the decay width must be accounted for by other modes. We assume that decays to exclusively light-flavoured hadrons are small; this is because of the OZI rule, and the apparent absence of prominent light-flavoured decays in other hidden-charm states.
In the case of P c (4312) we conclude, by a process of elimination, that the dominant decay must be Λ + cD * 0 , as this is the only two-body mode with hidden charm which is kinematically accessible. Ignoring the small contributions from J/ψ p, η c p and Λ + cD 0 , we estimate the Λ + cD * 0 partial width by taking the difference of equations (24) and (27), where we have combined the statistical and systematic uncertainties in (24) in quadrature.
We have not accounted for the uncertainties in equation (27), because of the difficulty in quantifying the systematic uncertainty on the underlying assumption, namely that the threebody width is equal to the width of the free constituents. Most of our arguments below relate to the scale of the experimental numbers, not their precise values, so we do not expect the conclusions to be compromised by the unquantified uncertainties.
One may question the validity of predicting prominent Λ + cD * 0 modes on the basis of a process of elimination. However we also note that our conclusion is supported by model calculations which confirm the prominence of Λ + cD * 0 decays in molecular models [9,13,25,51,52,59].
From our estimate (30), the corresponding branching fraction is enormous in comparison to the closely related Λ + cD 0 mode, equation (23). The challenge for models is to explain the striking disparity between these two modes which would naively be expected to be comparable. We will see that this issue argues in favour of our Scenario C.
Heavy quark symmetry implies relations among all couplings of the type Σ ( * ) cD ( * ) → Λ + cD 0( * ) . In Table III we show the relative matrix elements for S-wave transitions obtained assuming heavy-quark symmetry, taken from the potentials in ref. [30]. Note that the relative matrix elements apply not only to effective field theory models, where the S-wave potentials correspond to contact terms that are fit to data, but also to models where the transition is due to the central potential from one-pion exchange (since such models satisfy heavy-quark symmetry).
The first thing to notice is that for P c (4312), modelled as a 1/2 − Σ cD state, the striking disparity in the magnitudes of Λ cD and Λ cD * has a natural explanation, and confirms a selection rule predicted by Voloshin [65]. Although conservation of angular momentum allows for decays to both Λ cD (in S-wave) and Λ cD * (S-wave and D-wave), the Λ cD decay is forbidden by heavy-quark symmetry. In this sense the experimental data are nicely consistent with the molecular model for P c (4312). (The suppression of the Λ cD mode is also a feature of the chiral constituent quark model [59], and the chromomagnetic pentaquark model [68], but not the model of ref. [25].) The tight upper limit (23) on Λ cD decays also has a natural explanation for 3/2 − states, both P c (4380) (Σ * cD ) and P c (4440/4457) (Σ cD * ). In these cases the Λ cD decay would be D-wave, hence suppressed compared to the S-wave decay P c (4312) → Λ cD * .
On the other hand, for a 1/2 − Σ cD * state, the tight upper limit on Λ + cD 0 is a problem.
The previous argument relies on the assumption (which is very natural) that D-wave decays are suppressed compared to S-wave decays. In that case, the Λ cD and Λ cD * partial widths for the various P c states are dominated by the S-wave matrix elements, so we can estimate the relative strengths using the numbers in Table III. We notice in particular that for 1/2 − , the Σ cD * → Λ + cD 0 matrix element is identical to Σ cD → Λ + cD * 0 , whose scale is set by equation (30). This is a problem for Scenarios A and B, because it implies that if either of P c (4440) or P c (4457) were a 1/2 − Σ cD * state, then its Λ + cD 0 partial width would be 2.7 ÷ 12.5 MeV, indeed even larger because of the enhanced phase space compared to the P c (4312) decay. This is wildly inconsistent with the upper limit of eq. (23).
To avoid this problem, we have to abandon the assumption that there is a 1/2 − Σ cD * bound state, which leads us to Scenario C. Since for P c (4457) there are several viable alternatives, we assume that only P c (4440) is Σ cD * bound state, and that its quantum numbers are 3/2 − . We will argue later that there are other reasons to favour this over the alternative Scenarios A and B.
In summary, there is an apparent tension between the dominance of the P c (4312) → From now on we concentrate on Scenario C. We may estimate the Λ cD * partial widths of P c (4380) and P c (4440) using equation (30) and the numbers in Table III: Here we have ignored the effect of the mass differences between P c (4312), P c (4380) and P c (4440). (Phase space would enhance the decays of the heavier states, but this is to some extent mitigated by a corresponding form factor suppression in the matrix element.) For P c (4380) we assume the decay width is saturated by Λ + cD * 0 and, from equation (28), Λ cD π. This is because of the tight upper limit on J/ψ p, and the absence of other two-body S-wave decays. ( considerably smaller than the measured width [1] which, however, is considered to be obsolete [3]. Other approaches also find a narrower P c (4380) [15,30]. Assuming our estimated total width, the Λ cD * branching fraction is If neglected decays (notably the Σ cD D-wave) are significant, this will be an overestimate.
For P c (4440), the Λ cD * branching fraction is where we have used the measured total width, eq. (31), with statistical and systematic uncertainties combined in quadrature.
Given the predicted large Λ + cD * 0 branching fractions for P c (4312), P c (4380) and P c (4440), searching for these states in Λ 0 b → Λ + cD * 0 K − is warranted. We note that this decay has already been observed at LHCb [38], although an amplitude analysis has not been carried out.
To understand the experimental prospects in these channels, we consider now the fit We may eliminate the unknown production branching fraction B(Λ 0 b → P + c K − ) by taking a ratio with equation (4), to give Using the experimental numbers in Table II, (Table II). It suggests of course that there are strong prospects for the experimental observation of these states in Λ A corollary of this plot is that the branching fraction B[P c (4312) → J/ψ p] cannot be much less than the current bound in equation (8), otherwise the fit fractions R(Λ + cD * 0 ) become implausibly (or impossibly) large. We arrived at the same conclusion when interpreting Fig. 3. As discussed there, this also implies that observation of P c states in photoproduction requires not much more sensitivity than that of the J/ψ-007 experiment. cD final states, where phase space is available. The P c (4312) is too light, but P c (4380) could decay into Σ cD (in D-wave), and P c (4440) could decay into both Σ cD (in D-wave) and Σ * cD (in S-wave and D-wave), where we assume the 3/2 − quantum numbers of Scenario C. We further assume that S-wave decays dominate over D-waves, in which case the most prominent decay will be P c (4440) → Σ * cD . Note that having accounted for Λ + cD * 0 and Λ cD * π decays, in equations (29) and (33), respectively, the measured width (25) still allows for a considerable Σ * cD mode. We also note that various models find substantial Σ * cD decays, for example refs [25,51]. The experimental prospects for observing P c (4380) or P c (4440 Taking a ratio with equation (4), we get a relation between R(Σ ( * ) cD ) and R(J/ψ p), As far as we know, B(Λ 0 b → Σ ( * ) cD K − ) has not been measured, but since the tree-level contributions to this mode are colour suppressed (Fig. 2), we expect it to be comparable to B(Λ 0 b → J/ψ p K − ), as noted in equation (9). In this case, we expect the first ratio on the right-hand side of equation (40) to be a number of order 1. Conversely, because of the stringent upper limit (8) on P + c → J/ψ p, we expect the second ratio on the right-hand side to be large, at least for P c (4440) → Σ * cD , where the S-wave is expected to be prominent. We conclude that for P c (4440) + , the fit fraction R(Σ * cD ) ought to be large in comparison to the measured fit fraction R(J/ψ p), implying strong experimental prospects. Depending on the magnitude of the D-wave decays, by a similar argument we may also expect considerable (albeit smaller) fit fractions R(Σ cD ) for P c (4380) and P c (4440). This is another example of the general phenomenon discussed in Section IV. Where treelevel three-body decays are suppressed, two-body fit fractions can be large.
Finally, we remark that, assuming the P c states are isospin 1/2, the different charge Given the existence of states near the thresholds for Σ cD , Σ * cD and Σ cD * , the apparent absence of states near Σ * cD * threshold is conspicuous. In this section we show that this is a problem for Scenarios A and B, but not for Scenario C.
Interactions among Σ ( * ) cD ( * ) channels are constrained by heavy-quark symmetry. This applies both to models based on meson exchange, and also effective field theory approaches where the long-range contribution to the potential is due to pion-exchange, and the shortrange part is modelled via contact terms that are fit to data. The pattern of binding in such models can be understood qualitatively with reference to the simplest approach, where the potentials are due to S-wave contact terms only (no meson exchange). The elastic potentials, from ref. [69], are given in Table IV.
Clearly, binding in Σ cD and Σ * cD requires C a < 0. The potentials in these channels are identical, so apart from small differences due to their masses, and coupled-channel effects, binding in one channel implies binding in the other. Thus a model that accounts for P c (4312) inevitably implies a partner state P c (4380).
To get binding in both of the Σ cD * channels, and thus account for both P c (4440) and P c (4457), requires C a to be large enough (in magnitude) compared to C b . Scenarios A and B are then distinguished by having C b > 0 and C b < 0, respectively.
With C a large (in magnitude) compared to C b , on the basis of the potentials in Table IV it seems likely that all three of the Σ * cD * states (1/2 − , 3/2 − and 5/2 − ) will also bind, and that expectation is confirmed by calculation [22]. This immediately raises the question of why there is apparently no evidence for such states in the experimental data. The absence of the 5/2 − state may be understood from its lack of S-wave coupling to J/ψ p; it decays in D-wave, which may be expected to be suppressed. However the missing 1/2 − and 3/2 − states are more problematic.
It is possible that Scenarios A and B may be rescued with the idea that the extra Σ * cD * states do exist, but are just not seen in Λ b → J/ψ p K − because of suppressed production or decays. But in the model we propose [31] the situation is precisely the opposite. From heavy quark symmetry, the production and decay of the 3/2 − Σ * cD * state, in particular, would be enhanced compared to other observed states, so if there is binding in this channel, it ought to be visible as a prominent peak in Λ b → J/ψ p K − . Similarly, the 1/2 − Σ * cD * state would be expected to decay prominently Λ cD ; there is no obvious indication of structure in [39], but more detailed analysis with more statistics could be revealing.
The inevitability of binding in all three Σ * cD * channels (1/2 − , 3/2 − and 5/2 − ) is a problem that is intrinsic to Scenarios A and B. As noted, to get binding in both Σ cD * channels (1/2 − and 3/2 − ) implies that C a must be large in magnitude compared to C b , which in turn renders all of the Σ * cD * channels sufficiently attractive to cause binding.
It all plays out very differently if only one of the two Σ cD * were required to bind. In this case it is no longer necessary to have C a large (in magnitude) in comparison to C b , with the consequence that binding is not automatic in all channels. Indeed, this scenario was already considered after the initial LHCb paper, when the experimental data suggested one Σ cD * state, not two [69].
If only one of the Σ cD * channels binds, it is natural to associate this with P c (4440), since this is unambiguously below Σ cD * threshold. As discussed previously, there are several viable alternative scenarios for P c (4457), because its mass is consistent with Σ cD * and Λ c (2595)D thresholds. The next question is whether P c (4440) has 1/2 − or 3/2 − quantum numbers.
In Section VI we argued in favour of 3/2 − quantum numbers (what we called Scenario C), noting that the tight upper limits on Λ + cD 0 decays are not consistent with a 1/2 − Σ cD * state.
We will now show that the pattern of binding also argues in favour of the 3/2 − assignment.
With reference to Table IV, to achieve Σ cD * binding in 3/2 − , but not 1/2 − , implies C b < 0. This in turn suggests that out of the Σ * cD * states, there is binding in 5/2 − , but not 1/2 − or 3/2 − , and indeed this expectation is borne out in explicit calculation [69]. This spectrum of states is nicely consistent with the absence of Σ * cD * states in the data. The 1/2 − and 3/2 − states simply don't exist, and the 5/2 − state is not visible because its decay is suppressed. (In our model [31] we find that not only the decay, but also the production, of the 5/2 − state is suppressed.) The alternative scenario does not work so well. If instead we assume Σ cD * binding in 1/2 − (but not 3/2 − ), we need C b > 0, which implies Σ * cD * binding in 1/2 − and 3/2 − (but not 5/2 − ). In this scenario one needs an explanation for why the 1/2 − and 3/2 − Σ * cD * states are not visible in Λ b → J/ψ p K − . (Note that some authors do argue in favour of the C b > 0 pattern, for other reasons [70,71].) Another reason to prefer binding Σ cD * in 3/2 − is that it is also consistent with expectations from one-pion exchange. In Table V we show the central potentials due to pionexchange obtained from the quark model [24,69], where C(r) behaves as a Yukawa function (with positive sign) for large r (i.e. it is repulsive at large r). It follows that the attractive channels are those in which C(r) comes with a negative coefficient, namely Σ cD * in 3/2 − and Σ * cD * in 5/2 − ; this is consistent with the pattern of binding in our Scenario C.
Of course the arguments about attraction and repulsion in one-pion exchange are admittedly simplistic, particularly because they ignore the tensor potential, which is known to be important for binding. Nevertheless, explicit calculation including the tensor terms confirms that the channels that bind most easily (requiring the smallest form factor cutoff) are those identified above [18,24,69]. The pattern also applies to models where the potential is due to exchange of not only pions, but also other light mesons [17,27].
(Notice that the pion exchange potentials in Table V follow the same pattern as the "C b " contact terms in Table IV,

XI. CONCLUSIONS
We have shown that, with minimal theoretical assumptions, current experimental data on Λ b decays and photoproduction imply constraints on the quantum numbers of P c states, and lead to predictions for their decay branching fractions, and fit fractions in Λ 0 b decays to η c pK − , Λ + cD * 0 K − , and Σ ( * ) Unlike the usual Scenarios A and B, where P c (4440) and P c (4457) are both Σ cD * bound states, in our Scenario C only P c (4440) is a bound state, and its quantum numbers are 3/2 − .
Because the P c (4457) mass coincides with both Σ cD * and Λ c (2595)D thresholds, there are several viable alternatives for this state, which we explore in future work [31]. The cusp interpretation of P c (4457) is favoured by ref. [32].
One of the advantages of Scenario C is that, unlike Scenarios A and B, it is consistent with the striking observation that the P c states hardly decay to Λ + Another issue with Scenarios A and B is that they imply the existence of Σ * cD * partner states, with quantum numbers 1/2 − , 3/2 − and 5/2 − , which are not apparent in Λ b → J/ψ p K − data. This is an inevitable consequence of fixing the potential parameters in order to generate two Σ cD * bound states, with both 1/2 − and 3/2 − quantum numbers.
Our Scenario C naturally avoids this issue, as we only require a single Σ cD * state with 3/2 − quantum numbers. In this case the potentials are such that the 1/2 − and 3/2 − Σ * cD * partners do not bind, and while the 5/2 − Σ * cD * partner does bind, its absence in experiment can be understood as a consequence of D-wave suppression both in decay and, as discussed in our next paper [31], production.
A further advantage of Scenario C is that it reproduces the pattern of binding expected on the basis of one-pion exchange.
We have given many predictions for the branching fractions and Λ 0 b fit fractions of P c (4312), P c (4380) and P c (4440) in various modes. The most striking prediction is for strong experimental signals for all the states in Λ 0 b → Λ + cD * 0 K − . This is a particularly sharp prediction when juxtaposed against the absence of signals in the closely related decay Λ 0 b → Λ + cD 0 K − . The decay Λ 0 b → Λ + cD * 0 K − has been observed at LHCb [38], and we suggest an amplitude analysis to test for the existence of P c states in the Λ + cD * 0 spectrum.
In Λ 0 b → η c p K − , we predict a substantial P c (4312) fit fraction, which can be tested in future experimental analyses. (Our prediction is consistent with the current upper bound.) In our Scenario C, we do not expect prominent signals for the other P c states in Λ b → η c p K − . We showed that, by contrast, Scenarios A and B could be distinguished by the relative fit fractions of P c (4440) and P c (4457).
We also predict strong experiment signals for P c (4440) in Λ 0 b → Σ * cD K − , and smaller signals for both P c (4380) and P c (4440) in Λ 0 b → Σ cD K − .
We also pointed out that the branching fractions B(P + c → J/ψ p) cannot be much less than the current upper bound (8) interesting consequence is that existing experimental measurements are almost at the level of sensitivity required to observe P c states in photoproduction.
We want to emphasise that many of the constraints and predictions we have derived are very general in nature, and so if it turns out they are not satisfied by future experimental data, it could be an indication not of the failure of our particular realisation of the molecular model, but of the molecular approach all together. As an example, consider again the observation (Fig. 3) that if the upper limit on B(P + c → J/ψ p) is much less than equation (8), it would imply implausibly large production branching fractions B(Λ 0 b → P + c K − ). In this case some new mechanism would need to be invoked to explain how, contrary to intuition, molecular states are produced with comparable branching fraction to conventional hadrons.
Alternatively, since the observation is a direct consequence of the measured fit fractions, one could question whether or not it is appropriate, in the fit fraction, to factorise the production and decay branching fractions B(Λ 0 b → P + c K − ) and B(P + c → J/ψ p) in the numerator. The factorisation is legitimate in the case that P c is a resonance, but not if the P c signal is an effect which is uniquely associated with a particular production mechanism, such as a cusp or triangle singularity. Indeed we gave examples, in the case of the X(2900) states, where the factorisation is clearly not appropriate [40]. Many of our manipulations in this paper rely on this factorisation, and so our predictions should be interpreted with this in mind.