Peaks within peaks and the possible two-peak structure of the Pc(4457): the effective field theory perspective

The LHCb pentaquarks -- the $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ -- have been theorized to be $\Sigma_c \bar{D}$ and $\Sigma_c \bar{D}^*$ S-wave molecules. Here we explore the possibility that two of these pentaquarks -- the $P_c(4440)$ and $P_c(4457)$ -- contain in addition a $\Lambda(2595) \bar{D}$ component in P-wave. We will analyze the effects of this extra channel within two effective field theories: the first one will be a standard contact-range effective field theory and the second one will include the non-diagonal pion dynamics connecting the $\Sigma_c \bar{D}^*$ and $\Lambda(2595) \bar{D}$ channels, which happens to be unusually long-ranged. The impact of the coupled-channel dynamics between the $\Sigma_c \bar{D}^*$ and $\Lambda(2595) \bar{D}$ components is modest at best for the $P_c(4440)$ and $P_c(4457)$, which will remain to be predominantly $\Sigma_c \bar{D}^*$ molecules. However, if the quantum numbers of the $P_c(4457)$ are $J^P = \frac{1}{2}^-$, the coupled-channel dynamics is likely to induce the binding of a $\Lambda(2595) \bar{D}$ S-wave molecule (coupled to $\Sigma_c \bar{D}^*$ in P-wave) with $J^P = \frac{1}{2}^+$ and a mass similar to the $P_c(4457)$. If this is the case, the $P_c(4457)$ could actually be a double peak containing two different pentaquark states.

In this manuscript we will explore a modified molecular interpretation of the P c (4440) and P c (4457) pentaquarks and the consequences it entails. Of course the fundamental idea will still be that these two pentaquarks are hadronic bound states, but besides the standard Σ cD * interpretation we will also consider the existence of a Λ c (2595)D (Λ c1D from now on) component for the P c (4440) and P c (4457). In the isospin-symmetric limit the Σ cD * and Λ c1D threshold are located at 4462.2 and 4459.5 MeV, respectively, very close to the masses of the P c (4440) and P c (4457). Thus it is natural to wonder whether the Λ c1D channel plays a role in the description of the pentaquarks.
This idea was originally proposed by Burns [8], who conjectured that the Λ c1D component might be important for the binding of molecular pentaquarks. Later it * mpavon@buaa.edu.cn was realized that the pion-exchange dynamics mediating theD * Σ c →DΛ c1 transition is unusually long-ranged and in practice takes the form of a 1/r 2 potential [22]. This is indeed a really interesting potential in the sense that it can display discrete scale invariance when attractive enough [23][24][25], which in turn opens the possibility of the existence of hadronic molecules for which there is a geometric spectrum reminiscent of the Efimov effect in the three-boson system [26]. For the hidden charm pentaquarks the strength of the 1/r 2 potential is probably not enough to trigger a geometric molecular spectrum [22], yet this might very well happen in other twohadron molecular systems. Recently, Burns and Swanson have considered theD * Σ c →DΛ c1 pion-exchange dynamics beyond its long-distance behavior, leading to the conclusion that the P c (4457) might not be a 1 2 −D * Σ c S-wave molecular state but a 1 2 +D Λ c1 one instead [8].
The present manuscript delves further into the consequences that aDΛ c1 component will have for the pentaquark spectrum. For this we formulate two effective field theories (EFTs): a pionless EFT and a halfpionful EFT. By half-pionful we denote an EFT which includes the unusually long-ranged pion dynamics of thē D * Σ c →DΛ c1 transition, for which the characteristic length scale is between 10 and 20 fm, but does not include the pion dynamics of theD * Σ c system, which has a range in between 1 and 2 fm. We find that the addition of theDΛ c1 channel is inconsequential if the quantum numbers of the P c (4440) and P c (4457) molecular pentaquarks are 1 2 − and 3 2 − , respectively. However, if the quantum numbers of the P c (4457) pentaquark are 1 2 − instead, then the existence of a partner state with a similar mass and quantum numbers 1 2 + is very likely. That is, the P c (4457) might be a double peak, as happened with the original P c (4450) pentaquark discovered in 2015.
The manuscript is structured as follows. In Sect. II we explain how to describe theDΣ c ,DΣ * c andDΛ c1 interactions within a pionless contact-range EFT. In Sect. III we introduce the half-pionful theory, in which we include the pion exchange transition potential in theDΣ * c -DΛ c1 channel. In Sect. IV we revisit the description of the LHCb pentaquark trio within the previous two EFTs. Finally in Sect. V we present our conclusions.

II. PIONLESS THEORY
In this section we will derive the lowest-order contactrange interaction for theD * Σ c -DΛ c1 system. For this we will find convenient to use the light-quark notation explained in detail in Ref. [27], but which has been previously used in the literature, e.g. in Refs. [16,28]. In contrast with the standard superfield notation (see for instance Ref. [29] for a clear exposition) in which we combine heavy hadrons with the same light-quark spin into a unique superfield, in the light-quark notation we simply write the interactions in terms of the light-quark spin degrees of freedom within the heavy hadrons. Of course both notations are equivalent, but for non-relativistic problems the light-quark notation is easier to use.
A. TheDΣc andD * Σc channels TheD andD * charmed antimesons areQq states where the light-quark q and heavy antiquarkQ are in Swave. From heavy-quark spin symmetry (HQSS) we expect the heavy-antiquark to effectively behave as a static color source, which in practical terms means that the wave function of the light quark is independent of the total spin of the S-wave heavy meson. That is, the lightquark wave function (the "brown muck") of theD and D * charmed antimesons is the same (modulo corrections coming from the heavy-antiquark mass m Q , which scale as Λ QCD /m Q , with Λ QCD ∼ (200 − 300) MeV the QCD scale). Two possible formalisms to express this symmetry are the standard heavy-superfield notation and the light-subfield notation. In the former, we combine theD andD * field into a single superfield [29] where the superfield is well-behaved with respect to heavy-antiquark rotations with S H representing the heavy-antiquark spin operator and θ the rotation axis and angle. Thus the combination of H † Q and HQ superfields in the Lagrangian effectively results in invariance with respect to heavy-antiquark rotations, i.e. to heavy-antiquark spin. Conversely, in the light-subfield (or light-quark) notation, we prescind of writing down the heavy antiquark explicitly and instead express everything in terms of the effective light-quark degrees of freedom within the charmed antimeson and the light quark spin operator: where q L represents an effective light-quark subfield, i.e. a field with the quantum numbers of the light quark within the charmed antimeson 1 . Then we write down explicit rules for transforming the light-quark spin operator into charmed antimeson spin operators where ǫ 1 is the polarization vector of theD * meson and S 1 the spin-1 matrices.
Regarding the Σ c and Σ * c charmed baryons, their quark content is Qqq where the qq diquark has light spin S L = 1 and the system is in S-wave. The structure of the S-wave charmed baryons is independent of whether the baryon spin is S = 1 2 (Σ c ) or 3 2 (Σ * c ). In the standard heavysuperfield notation this is taken into account by defining the superfield [30] With these ingredients the interaction between aD charmed antimeson and a Σ c charmed baryon can be easily written as which leads to the non-relativistic contact-range potential This potential can be particularized for the two cases of interest for us in the present work, theDΣ c andD * Σ c systems which we will use for the P c (4312) and theD * Σ c component of the P c (4440) and P c (4457), respectively.
B. TheD * Σc-DΛc1 transition Now we will consider theD ( * ) Σ c1 transitions, which are necessary for the description of theDΛ c1 component in the P c (4440) and P c (4457) pentaquarks. First we will consider the structure of the Λ c1 and Λ * c1 Pwave charmed baryons, which are Qqq states in which the spin of the light-quark pair is S L = 0 and their orbital angular momentum is L L = 1, yielding a total angular momentum of J L = 1. In practice this means that there is no substantial difference (except for parity) between the description of the Σ c , Σ * c and Λ c1 , Λ * c1 charmed baryons either in terms of heavy-superfield or light-subfield notations. In the superfield notation, we will write [32] while in the light-quark notation we use with v L representing the light-diquark pair (with quantum numbers J P = 1 − , i.e. a vector field) and L L the spin-1 matrices, where we use a different notation than in the S-wave charmed-baryon case to indicate that the angular momentum comes from the orbital angular momentum of the light-quark pair. This does not entail any operational difference, with the translation rules being which are analogous to these of the Σ c , Σ * c baryons, see Eqs. (9)(10)(11).
With these ingredients we are ready to write thē c1 transition Lagrangian. We find that at lowest order there are two independent operators mediating the transition, which for convenience we write as with and where J L refers to the spin-1 matrices as applied between the light-diquark axial and vector fields within the S-and P-wave charmed baryons. The translation rules for the J L operator happen to be which are analogous to Eqs. (9)(10)(11) and (18)(19)(20), except that now the initial and final baryon states are different (either the S-to P-wave baryon transition or vice versa).
Other operators choices are possible in the Lagrangian of Eq. (21), but the present one is particularly useful because the D a term is pion-like, while the D b term is ρ-like: they are similar to what we could get from the exchange of a pion and a ρ respectively, as we explain in Appendix A. The potential we obtain is while in the other direction it is where D a and D b are real in the convention we have used to write the potentials. It is important to notice that V C2 is a non-diagonal potential and can be redefined by a phase in which case the potential is still self-adjoint. In the convention above, the p-space partial wave projection is purely real while the r-space partial wave projection is purely imaginary. To avoid the inconveniences originating from this fact, when working in coordinate space we will automatically add the phase φ = ±π for the nondiagonal potential to be real. Phenomenologically we expect the D a and D b couplings to represent the exchange of a pseudoscalar and vector mesons, respectively. However there is no shortrange contribution directly attributable to a pseudoscalar meson: pion exchange is excessively long-ranged as to be included in the contact-range potential. For taking this into account, we will devise a power counting in which the D a coupling is a subleading order contribution, while D b remains leading. Thus the effective potential we will use from now on will be C. TheDΛc1 channel Finally we consider theD ( * ) Λ ( * ) c1 system, which enters the description of the P c (4440) and P c (4457) as an additional (P-wave) component of the wave function. Yet this meson-baryon system is particularly relevant for a theoretical pentaquark with quantum numbers J P = 1 2 + , for which the most important meson-baryon component of the wave function will beDΛ c1 in S-wave.
The lowest order interaction in theD * Λ * c1 system happens to be formally identical to the one for theD ( * ) Σ ( * ) c system, that is which leads to the potential If we particularize to theDΛ c1 molecule, we will end up with which is a really simple potential, where the coupling E a is unknown.

D. Partial-Wave Projection
For the partial-wave projection of the contact-range potentials (and the OPE potential later on), we will use the spectroscopic notation 2S+1 L J to denote a state with spin S, orbital angular momentum L and total angular momentum J. For the pentaquarks states we are considering -P c , P c ′ , P * c -the relevant partial waves are where we indicate the relevant meson-baryon channels within parentheses.

E. Momentum-Space Representation
For the momentum-space representation, we simply project the relevant contact-range potential into the partial waves of interest. For the P c (DΣ c ) pentaquark we simply have Next, for the two P * c configurations (J = 1 2 , 3 2 ) we have Finally for the P ′ c (DΛ c1 ) pentaquark we have which can be simplified to a two-channel form if we take into account that the two P-waveD * Σ c components can adopt the configuration which maximizes the strength of the transition potential and we end up with Notice that this simplification is only possible for the pionless theory at LO: if we include pion-exchanges or other effects we will have to revert to the original threechannel representation.

F. Coordinate Space Representation
We obtain the r-space contact-range potential from Fourier-transforming the p-space one which in the case of the V C1 and V C3 potentials leads to For the V C2 potential, which contains one unit of orbital angular momentum, the transformation is a bit more involved, resulting in which can be further simplified by rewriting leading to This last expression is particularly useful because the partial wave projection of the σ L1 ·r andr · σ L1 × J L2 is identical to their p-space versions. Finally we redefine V C2 (1 → 2) by a phase to end up with a purely real potential: With the previous conventions and the power counting we use (for which D a is a subleading order effect), we end up with the r-space potentials V ( r; P * c , where for the P ′ c pentaquark we have written the simplified two-channel version of the potential.

G. Regularization and Renormalization
The contact-range potentials we are using are not welldefined unless we include a regulator to suppress the unphysical high-momentum components of the potential. For the p-space version of the potential this is done with the substitution with f (x) a regulator function, for which we will choose a Gaussian, f (x) = e −x 2 . For the r-space version of the potential we will use a delta-shell regulator with R c the coordinate space cutoff, where the 3/R c factor in the derivative of the delta is chosen for its Fouriertransform to be either p or p ′ in the R c → 0 limit after the partial wave projection.

H. Dynamical Equation
For finding the location of the bound states we have to iterate the r-or p-space potentials that we have obtained within a dynamical equation. For the r-space potential, we will solve the reduced Schrödinger equation where a, b are indices we use to represent the different channels in the molecules we are considering as detailed in Eqs. (33)(34)(35)(36), while V ab is the potential between two channels, see Eqs. (50-53), which is regularized as in Eqs. (55) and (56). The reduced mass, angular momentum and wave number of a given channel a are represented by µ a , L a and γ a . In turn the wave number is given by γ a = 2µ a (M th(a) − M ), with M th(a) the mass of the two-hadron threshold for channel a and M the mass of the molecular pentaquark we are predicting. For the p-space potential we will solve the Lippmann-Schwinger equation as applied to the pole of the Tmatrix, that is: where a, b represent the channel, φ a is the vertex function for channel a (where the vertex function is related to the residue of the T-matrix), V ab is the potential between two channels, see Eqs. (37)(38)(39)(40), which is regularized according to Eq. (54), and M is the mass of the molecular pentaquark, while M th(a) and µ a are the two-hadron threshold and the reduced mass for a given channel a.

III. HALF-PIONFUL THEORY
The exchange of one pion between theD * Σ c andDΛ c1 channels has the particularity that its range is extremely enhanced. The reason is that the pion in the Σ c Λ c1 π and D * Dπ vertices can be emitted or absorbed almost on the mass-shell, resulting in an improved range. Besides, owing to the opposite parity of the Σ c and Λ c1 baryons, the pion exchange in this vertex is in S-wave. In combination with the standard P-wave pion in the vertex involving the charmed mesons, the outcome is that instead of having a central and tensor forces with orbital angular momentum L = 0 and 2 respectively, we end up with a vector force with L = 1. The long-range behavior of the vector force is 1/r 2 , i.e. an inverse square-law potential, which can trigger a series of interesting theoretical consequences when the strength of the potential is above a certain critical value [22]. Yet, as explained in Ref. [22], this is probably not the case for the LHCb pentaquarks as hadronic molecules. Now, we begin by writing the pion-exchange Lagrangian for the Σ c to Λ c1 transition in the heavy superfield notation: which are obtained from the non-relativistic limits of the Lagrangians of Refs. [29,32]. The light-quark notation version happens to be trivial From the previous Lagrangians we can derive the OPE potential in momentum space, which reads as follows where we are indicating that this is the transition potential in theD ( * ) Σ c1 direction. The operator τ 1 · t 2 = √ 3 for total isospin I = 1 2 and 0 otherwise. The equivalent expression in coordinate space can be obtained by Fourier-transforming the previous expression, where in addition we include a phase to follow the convention of having a purely real transition potential with W E defined as For the couplings we have taken g 1 = 0.59 (as deduced from the D * → Dπ and D * → Dγ decays [33,34]), h 2 = 0.63 (from the analysis of Ref. [35], where h 2 is extracted from Γ(Λ c1 → Σ c π) as measured by CDF [36]), f π = 130 MeV and ω π ≃ (m(Λ c1 ) − m(Σ 1 )) ≃ (m(D * ) − m(D)) ≃ m π , with m π = 138 MeV. Finally µ π = m 2 π − ω 2 π ≃ 0, a value we will further discuss in the following lines.

A. Infrared regularization
In the µ π → 0 limit, which is close to the physical situation we are dealing with and will probably represent a good approximation of it, the previous OPE potential becomes a 1/r 2 infinite-range potential. In particular the p-space potential reads while for the r-space potential we can take this approximation into account within the function W E Of course this is merely an approximation. What is actually happening is that the modulus of the effective pion mass |µ π | will be in general considerably smaller than the pion mass m π (or any other hadronic scale for that matter). We have |µ π | ∼ (10 − 20) MeV, its concrete value depending on the specific particle channel under consideration. In a few particle channels µ π is purely imaginary, indicating the possibility of decay into theDΣ c π channel, and in others it is real. A detailed treatment of these difference is however outside the scope of the present manuscript.
Here we will opt for the much easier treatment we were describing above, that is, to assume that µ π = 0. For taking into account that the range of the OPE potential is actually not infinite we will include an infrared cutoff. For the partial-wave projection of OPE in momentum space, we will introduce an infrared cutoff Λ IR in the following way with q − = p − p ′ and q + = p + p ′ , with the infrared cutoff chosen within the cutoff window Λ IR = (10 − 20) MeV, which corresponds with the size of the modulus of the effective pion mass. In coordinate space the inclusion of the infrared cutoff R IR will be considerably simpler where we will take R IR = (10 − 20) fm. Actually the effect of this infrared cutoff is only important if the strength of the 1/r 2 potential is equal or larger to the critical triggering a geometric spectrum. This does not happen for any of the pentaquarks we are considering, at least with the currently known values of the couplings g 1 and h 2 . However in the P ′ c pentaquark the strength is not far way to that critical value [22], indicating that in this case the results will have a larger dependence on the infrared cutoff.

B. Partial-wave projection
The partial-wave projection of the OPE potential is trivial for its coordinate space representation: owing to its clear separation into a radial and angular piece -Eq. (64) -it merely requires to consider the partial wave projection of the vector operator σ L1 · r, which we already showed in Table I. For the momentum-space representation of the potential the partial-wave projection is a bit more complex, yet it can be written as where the matrix elements of the vector operator are again to be found in Table I, to which we have to add the partial wave projection of the 1/| q| potential: Of course, we still supplement the previous expressions with the infrared cutoff of Eq. (68).

IV. THE PENTAQUARK TRIO REVISITED
In this Section we consider the description of the P c (4312), P c (4440) and P c (4457) pentaquarks within the EFTs proposed in this work. We will begin by reviewing their standard molecular interpretations asDΣ c and DΣ * c bound states and then we will move to the novel molecular interpretation in which theDΛ c1 channel is included as an explicit degree of freedom for the P c (4440) and P c (4457) pentaquarks. The prediction of aDΛ c1 bound state is contingent on an unknown coupling constant, E a . For dealing with this issue we will consider two different estimations of the value of this coupling and the predictions they will entail.

A. The standard molecular interpretation
We begin by reviewing the standard molecular interpretation of Ref. [5], in which the pentaquarks were considered to beDΣ c andD * Σ c molecules (without anȳ DΛ c1 component) described by a pionless EFT. This pionless EFT is equivalent to using the V C1 contact-range potential of Eq. (13), which contains two independent couplings C a and C b . The original procedure [5] for determining these two couplings was as follows: (i) use the P c (4440) and P c (4457) asD * Σ c molecules to determine the C a and C b couplings; (ii) postdict the P c (4312) as aDΣ c molecule and compare with its experimental location.
For convenience we will modify the previous procedure in this manuscript: (i) use the P c (4312) and P c (4457) asDΣ c andD * Σ c molecules to determine the C a and C b couplings; (ii) postdict the P c (4440) as aD * Σ c molecule and compare with its experimental location.
This choice guarantees that the prediction of the pentaquark trio remain all below their respective mesonbaryon thresholds: the later inclusion of theDΛ c1 channel can in a few instances move the P c (4457) a bit above the threshold for hard cutoffs if we fit the couplings as in Ref. [5]. Now for theD * Σ c molecules there are two spin configurations, J = 1 2 and 3 2 , but we do not know which one corresponds with each of the pentaquarks. As a consequence we consider two scenarios, A 0 and B 0 : (a) in scenario A 0 the P c (4440) has J = 1 2 , while the spin of the P c (4457) is J = 3 2 , (b) in scenario B 0 the P c (4440) has J = 3 2 , while the spin of the P c (4457) is J = 1 2 , where we use the subscript "zero" to indicate that this is the base case in which theDΛ c1 channel is not included.
for the r-space delta-shell regulator with R = 1.0 (0.5) fm. These numbers are to be compared with the experimental value M = (4440.3 ± 1.3 +4.1 −4.6 ) MeV, which indicates that scenario A 0 is marginally preferred over scenario B 0 (particularly for softer cutoffs). This coincides with the conclusions of the previous pionless EFT of Ref. [5].

B. The novel molecular interpretation
Now we explore the novel molecular interpretation we propose, in which the P c (4312) is aDΣ c molecule while the P c (4440) and P c (4457) areD * Σ c -DΛ c1 molecules. The contact-range piece of the potential for the pionless and half-pionful EFTs is given by Eqs. (37-39), which contain three independent coupling constants (C a , C b and D b ). Finally we conjecture the existence of aDΛ c1 S-wave molecule, which we call the P ′ c and for which the contact-range piece of the potential is given by Eq. (40), which includes a new coupling (E a ).
Of these four couplings, we can determine three of them -C a , C b and D b -from the masses of the three pentaquarks. The procedure we will follow is: (i) use the P c (4312) as aDΣ c molecule to determine the C a coupling, (ii) use the P c (4440) and P c (4457) asD * Σ c -DΛ c1 molecules to determine the C b and D b couplings, (ii') if there is no solution for the previous procedure, we will set D b = 0 and, as in the uncoupled-channel case, we will determine C b from the condition of reproducing the P c (4457) pole, (iii) finally we determine for which values of E a the P ′ c (the conjectured S-waveDΛ c1 molecule) binds and compare these values with expectations from naive dimensional analysis (NDA).
As in the standard molecular interpretation, we have two possible scenarios which we now call A 1 and B 1 , where A 1 (B 1 ) corresponds to the P c (4457) being a J = 3 2 ( 1 2 ) molecule. We will further subdivide the scenario A 1 (B 1 ) into a pionless and a half-pionful version, which we will denote A / π 1 (B / π 1 ) and A π 1 (B π 1 ), respectively. It happens that the couplings can be compared with NDA, in particular D b and E a : the D b comparison can provide an indirect estimation of the likelihood of scenarios A 1 and B 1 , while E a will provide the binding likelihood of the P ′ c pentaquark.
To illustrate this idea, we can consider the pionless pspace calculation, which for Λ = 0.5 GeV in scenario A This translates into the following condition for the P ′ c to bind which in scenario A / π 1 requires the coupling E a to be attractive, while scenario B  [37].
A complete list of the couplings can be consulted in Table III for the different EFTs and regulators considered in this work. Independently of the choice of regulator and cutoff, the binding of the P ′ c pentaquark is much more probable in scenario B 1 (pionless or half-pionful). In scenario A 1 there is no pair of values for the C b and D b couplings that simultaneously reproduces the P c (4440) and P c (4457) pentaquarks, and thus we have set D b = 0 and followed the same procedure as in scenario A 0 to determine C b . We will further comment on why this happens later on this section. Now we can compare the previous numbers with the NDA estimation of the expected size of a contact-range coupling where l, l ′ are the angular momenta of the initial and final states that the contact-range potential couples, while M is the hard-scale of the theory. For hadrons we expect M ∼ 1 GeV, which gives us the following estimations for an S-wave and S-to-P wave counterterms |C NDA S | ∼ 0.49 fm 2 and |C NDA SD | ∼ 0.29 fm 3 . (84) From this we see that C a is unnatural (see Table II), which is to be expected for the coupling of a two-body TABLE IV. The mass of the P ′ c pentaquark as deduced from the NDA estimate of the Ea coupling (assuming it is attractive) in scenario B1, both in the pionless and halfpionful theory. For reference, theDΛc1 threshold is located at 4459.5 MeV in the isospin-symmetric limit.
system that binds [38,39], while C b and D b are closer to natural, though this depends on the cutoff (particularly for D b , see Table III). In addition we can appreciate that in scenario A 1 the binding of the P ′ c pentaquark is possible but not particularly probable, as the size of the coupling E a that is required to bind is larger than the NDA expectation. In contrast, in scenario B 1 the coupling E a required to bind falls well within what is expected from NDA. Thus in this second case binding seems to be much more likely.
Regarding the P ′ c pentaquark, we can deduce its probable mass from the NDA estimation of the E a coupling, provided this coupling is attractive: Within scenario B 1 , this estimation of the coupling consistently generates a shallow P ′ c close to theDΛ c1 threshold, where the concrete predictions can be consulted in Table IV. Of course the question is whether it is sensible to assume that the E a coupling is attractive. We will examine the validity of this assumption in the next few lines.
C. Can we further pinpoint the location of the P ′ c pentaquark?
Regarding E a , it will be useful not only to determine its sign but also its size beyond the NDA estimation we have already used to argue the existence of the P ′ c pentaquark. From arguments regarding the saturation of contact-range couplings by light-mesons [31,40], the light-meson contributions to E a can be divided into two components which correspond to the scalar (σ) and vector (ω) meson contributions. The scalar and vector contributions are attractive and repulsive (E S a < 0 and E V a > 0), respectively. At first sight this ambiguous result seems to indicate that we cannot determine the sign of E a , yet this would be premature. As a matter of fact the same situation would arise had we applied this argument to the two-nucleon system, but it happens that the deuteron binds. The reason is that the scalar meson contributions have a longer range than the vector meson ones, leading to net attraction.
This seems to be the case not only in the two-nucleon system, but also in theDΛ c case: according to a recent calculation in the one-boson-exchange model [41], theDΛ c system is not far away from binding. In fact, had we adapted the recent one-boson exchange model of Ref. [42] (originally intended for theD ( * ) Σ ( * ) c molecules) to theDΛ c1 system, the system will not bind, yet its twobody scattering length a 2 would probably be unnaturally large where the errors are computed as in Ref. [42] and which are compatible with binding 2 (the lower error indicates that the scattering length changes sign, hence the −∞, and that in that case its value would be +9.1 fm). This reinforces the conclusions derived from Ref. [41] for thē DΛ c case. That is, we expect E a < 0 and close to the value required to have a shallow bound state in the absence of coupling with theD * Σ c channel. All this makes the P ′ c pentaquark very likely in scenario B 1 , as we will now show with explicit calculations.
If we now describe theDΛ c1 two-body system in a pionless EFT, the coupling E a can be determined from the value of the scattering length that we have already computed within the OBE model, leading to for R c = 1.0 (0.5) fm in r-space. As already explained, this extracted value of the coupling is enough as to guarantee binding in scenario B 1 , both in the pionless and pionful versions. This would lead to a P ′ c that is bound by (4 − 9) MeV depending on the case. The predicted locations can be found in Table V, where we have only consider scenario B 1 (for which binding is more probable). We can appreciate that the predictions are very  similar, independently of the cutoff or whether the calculation has been done in r-or p-space. For a more graphical comparison we have included Fig. 1, which shows the dependence of the binding energy on the coupling E a for the half-pionful theory in momentum space (we have chosen this particular calculation as the representative case, as the other three calculations in scenario B 1 would yield pretty similar results). Notice that in Fig. 1 we indicate the most probable values of E a and the binding energy of the P ′ c within a square.

D. Can scenario A be discarded?
A preliminary examination of the different determinations of the couplings presented in Table III reveals that D b = 0 in scenario A. The reason for this is that in general it is not possible to reproduce the twoD * Σ c pentaquarks in this scenario. This seems counter intuitive at first, but actually there are good reasons for this to be the case, which have to do with coupled-channel dynamics and which we will explain below.
First, we will consider a molecular pentaquark P Q in the heavy-quark limit, in which the masses of the charmed hadrons diverge and we can ignore the kinetic energy of the hadrons. In this limit the binding energy of a molecular pentaquark is given by where V S is the expected value of the S-wave potential and where we have taken the convention that the binding energy B PQ is a positive number, thus the minus sign in front of V S . Now we consider the case where the molecular pentaquark contains an additional P-wave component, for which the coupled-channel potential reads with V SP the S-to-P wave transition potential and λ a number describing the strength of the transition potential. If λ is small, the effect of the coupled-channel dynamics on the binding of the pentaquarks can be estimated in perturbation theory, leading to where G 0 is the two-hadron propagator, which in the static limit (infinitely heavy hadrons) reduces to with M PQ the mass of the heavy pentaquark, M P th the location of the P-wave threshold and ∆ P the energy gap. This simplifies the S-to-P wave contribution to which will increase the binding energy provided that ∆ P < 0, which happens to be the case 3 . The parameter λ is useful because it is proportional to the non-diagonal elements of the potentials Eqs. (38)(39)(40). Thus we have for the P * c (1/2), P * c (3/2) and P ′ c pentaquarks, respectively. The actual effect of the P-wave channel also depends on the inverse of the mass gap, i.e.
which implies that the impact of theDΛ c1 channel will be larger in the P c (4457) pentaquark than in the P c (4440) one (∆ P = −2.2 and −19.2 MeV respectively). However, once we take into account the finite mass of the hadrons, the effect of the mass gap on the P c (4457) will diminish in relative terms as it will be softened owing to the kinetic energy contributions.
In scenario A the P c (4457) receives a large boost to its binding energy: the coupling to theDΛ c1 channel is 4 times stronger than for the P c (4440) pentaquark. Besides, the P c (4457) is also considerably closer to thē DΛ c1 threshold. Thus the P c (4457) receives much more attraction from the coupled-channel dynamics than the P c (4440), to the point that it is not possible to reproduce the two of them simultaneously. That is, the P c (4457) as a J = 3 2D * Σ c molecule is only possible if the coupling D b is much smaller than its NDA estimation. In scenario B this does not happen because the P c (4457) couples much more weakly to theDΛ c1 channel than in scenario A. But this does not imply that scenario A can be discarded though: we simply do not know the size of the coupling D b and it is within the realm of possibilities that its size is considerably smaller than the NDA estimation.
Here it is important to mention that the two theoretical scenarios we have presented (A and B) are but a subset of all the possible scenarios. We have three molecular explanations (P * c (1/2), P * c (3/2) and P ′ c ) for two pentaquarks, which gives a total of six possible scenarios instead of the two we are considering. But with the exception of scenarios A and B, it is not possible to determine the value of the couplings in other cases. For instance, had we assumed that the P c (4440) is the J = 3 2D * Σ c molecule and P c (4457) the J = 1 2D Λ c1 one, i.e. the scenario originally proposed in Ref. [8], we would have ended with three unknown couplings (C b , D b and E a ) for two pentaquarks. Though this limitation can indeed be overcome by invoking NDA, the resulting analysis is considerably more involved than in scenarios A and B and thus we have decided not to consider them in this work.
Another factor that we have not taken into account in the present analysis is the effect of theDΛ * c1 channel, which lies about 20 MeV above theDΛ c1 threshold. Thē DΛ * c1 channel can mix with the J = 3 2D * Σ c one, inducing a bit of extra attraction in this later case. However, from Eq. (96) and the larger mass gap for theDΛ * c1 channel (∆ P = −55.0 MeV versus −19.2 MeV for theDΛ c1 one for scenario B), we expect this effect to be fairly modest.

V. CONCLUSIONS
In this manuscript we have considered the impact of theDΛ c1 channel for the description of the P c (4440) and P c (4457) pentaquarks. Within the molecular picture, the standard interpretation of the P c (4440) and P c (4457) states is that they areD * Σ c bound states. This is motivated by the closeness of theD * Σ c threshold to the location of the two pentaquark states. But the same is true for theDΛ c1 threshold, which naturally prompts the question of what is the contribution of this channel to the description of the pentaquarks [8,22,43].
For answering this question we have analyzed the inclusion ofDΛ c1 channel from the EFT perspective. We find that the importance of theDΛ c1 channel depends on which are the quantum numbers of the P c (4440) and P c (4457) pentaquarks: in the standard molecular interpretation (D * Σ c ) their quantum numbers can be either J P = 1 2 − or 3 2 − , but we do not know which quantum numbers correspond to which pentaquark. There are two possibilities: that the P c (4440) and P c (4457) are respectively the J P = 1 2 − and 3 2 −D * Σ c molecules, or vice versa. The first possibility, which we call scenario A, corresponds to the standard expectation that hadron masses increase with spin. The second possibility, scenario B, represents the opposite pattern, which has recently been conjectured to be a property of hadronic molecules [40].
In scenario A the inclusion of theDΛ c1 channel is inconsequential for the description of the molecular pentaquarks: theDΛ c1 can effectively be ignored, as the transition potential between theDΛ c1 →D * Σ c channels is weak. However this is not the case in scenario B, where the inclusion of theDΛ c1 channel can potentially have important consequences on the pentaquark spectrum. In this case the coupling between theDΛ c1 and D * Σ c channels is strong enough as to facilitate the binding of theDΛ c1 system in S-wave, as happened in Ref. [8]. Right now there is no experimental determination of the quantum numbers of the pentaquarks, with different theoretical explorations favoring different scenarios. We see a preference towards A in Refs. [5,6] and towards B in Refs. [7,9,44,45], though other scenarios are possible: for instance in Ref. [8] the 1 2 −D * Σ c pentaquark does not bind. Within the molecular picture there seems to be a tendency for pionless theories to favor A, while theories that include pion exchange effects tend to fall into scenario B.
If scenario B happens to be the one preferred by nature, the prospects for the J P = 1 2 +D Λ c1 molecule to bind are good: though the fate of this bound state is ultimately contingent on the unknown short-distance details of the interaction, phenomenological arguments indicate a moderate attraction between theD meson and Λ c1 baryon at short distances. If this is the case and this molecule binds, it might very well be that the P c (4457) is a double peak, containing both aD * Σ c and aDΛ c1 molecule with opposite parities. If scenario A is the one that actually describes the pentaquarks, the J P = 1 2 + pentaquark cannot be discarded either -there is the possibility that it binds even without coupling to thē D * Σ c channel -but is less likely to exist nonetheless.
Yet we stress the exploratory nature of the present manuscript: the EFT framework requires experimental input and a series of assumptions for it to be able to generate predictions. In this regard it would be very welcome to have phenomenological explorations of thē DΛ c1 interaction and theDΛ c1 →D * Σ c transition.
Hence the decision to consider that D a = 0 at lowest order.
For the D b coupling the situation is different, because theD * Σ c →DΛ c1 transition can happen via rhoexchange. The relevant Lagrangians read where q L , a L and v L are the light subfields of the D ( * ) , Σ ( * ) c and Λ ( * ) c1 charmed hadrons, ρ aµ is the rho meson field, with µ a Lorentz index (i is used to indicate µ = 1, 2, 3) and a and isospin index, t a and τ a are isospin matrices, g ρ1 , f ρ1 and f ρ2 are coupling constant, and M is a mass scale for the magnetic and electric dipole terms (i.e. the piece proportional to f ρ1 and f ρ2 , respectively). The charge-like term (i.e. the one proportional to g ρ1 ) can contribute toDΣ c →DΛ c1 andD * Σ c →D * Λ c1 transitions, but not to theD * Σ c →DΛ c1 one which is of interest for this work. The magnetic and electric and dipole terms of these Lagrangian lead to the potential where ω ρ ≃ (m(Λ c1 ) − m(Σ c )) ≃ (m(D * ) − m(D)), µ 2 ρ = m 2 ρ − ω 2 ρ and the rest of the terms have the same meaning as in Eqs. (25) and (63). Finally the saturation of the D b coupling by the rho will lead to a value proportional to This is why we keep D b as a leading-order effect, but consider D a to be subleading.