Holographic charm and bottom pentaquarks I: Mass spectra with spin effects

We revisit the three non-strange pentaquarks $[\frac 12\frac 12^-]_{S=0,1}$ and $[\frac 12\frac 32^-]_{S=1}$ predicted using the holographic dual description, where chiral and heavy quark symmetry are manifest in the triple limit of a large number of colors, large quark mass and strong $^\prime$t Hooft gauge coupling. In the heavy quark limit, the pentaquarks with internal heavy quark spin $S$ are all degenerate. The holographic pentaquarks are dual to an instanton bound to heavy mesons in bulk, without the shortcomings related to the nature of the interaction and the choice of the hard core inherent to the molecular constructions. We explicitly derive the spin-spin and spin-orbit couplings arising from next to leading order in the heavy quark mass, and lift totally the internal spin degeneray, in fair agreement with the newly reported charmed pentaquarks from LHCb. New charm and bottom pentaquark states are predicted.


I. INTRODUCTION
Recently the LHCb collaboration has revisited its analysis of the pentaquark states using a nine-fold increase in reconstructed Λ 0 b → J/ΨpK − decays from the LHCb Run-2 data batch at 13 TeV [1]. The LHCb new high statistics analysis shows that the previously reported P + c (4450) [2] splits into two narrow peaks P + c (4440) and P + c (4457) below the Σ + cD * 0 threshold, with the appearance of a new and narrow P + c (4312) state below the Σ + cD 0 . The evidence for the previously reported state P + c (4380) [2] has weakened. We regard the new LHCb data as evidence that supports the three lowest non-strange pentaquarks with spin-isospin assignments [ 1 − ] S=1 predicted by holography [3], in the triple limit of a large number of colors, strong t Hooft gauge coupling and a large quark mass. More importantly, we will show below that the degeneracy in the internal heavy quark spin S = 1, is lifted by spin-orbit effects at next to leading order in the heavy quark mass as heavy quark symmetry is broken, in fair agreement with the new data. Furthermore, we regard the closeness of the pentaquarks P + c (4457) and P + c (4312) to the Σ + section VI we derive the holographic mass formula for the heavy-light baryons and their exotic pentaquarks including the spin contributions. By adjusting the chief Kaluza-Klein scale used in [3,29], a more refined heavy baryon spectrum emerges, including the newly reported charmed pentaquarks by LHCb. Our conclusions are in section VII. A number of Appendices are added to support the various results.

II. HOLOGRAPHIC HEAVY-LIGHT EFFECTIVE ACTION
The D4-D8-D8 set-up for light flavor branes is standard [26]. The minimal modification that accommodates heavy mesons makes use of an extra heavy brane as discussed in [3,28,29]. It consists of N f light D8-D8 branes (L) and one heavy (H) probe brane in the cigar-shaped geometry that spontaneously breaks chiral symmetry. We assume that the L-brane world volume consists of R 4 × S 1 × S 4 with [0 − 9]-dimensions. The light 8-branes are embedded in the [0 − 3 + 5 − 9]-dimensions and set at the antipodes of S 1 which lies in the 4-dimension. The warped [5 − 9]-space is characterized by a finite size R and a horizon at U KK .

A. Dirac-Born-Infeld (DBI) action
The effective action on the probe L-branes consists of the non-Abelian DBI and Chern-Simons action. After integrating over the S 4 , the leading contribution in 1/λ to the DBI action is S DBI ≈ −κ d 4 xdz Tr (f (z)F µν F µν + g(z)F µz F νz ) . (1) The warping factors are with U 3 z = U 3 KK + U KK z 2 , and κ ≡ aλN c and a = 1/(216π 3 ) [26]. All dimensions are in units of M KK (Kaluza-Klein scale) unless given explicitly. Our conventions are (−1, 1, 1, 1, 1) with A † M = −A M and the labels M, N running over µ, z only in this section. The effective fields in the field strengths are [3,28] The matrix valued 1-form gauge field is For N f = 2, the naive Chern-Simons 5-form is We note that for only N f > 2 it fails to reproduce the correct transformation law under the combined gauge and chiral transformations [35]. In particular, when addressing the N f = 3 baryon spectra, (5) does not reproduce the important hypercharge constraint [35], but can be minimally modified to do that. For N f coincidental branes, the Φ multiplet is massless, but for separated branes they are massive with the additional contribution The value of m H is related to the separation between the light and heavy branes, which is about the length of the HL string. It is related to the heavy meson masses M D = 1870 MeV (charmed) and M B = 5279 MeV (bottomed) through [28] Given M KK and M D,B , the mass parameter m H is therefore totally fixed.

B. Light fields
In the coincidental brane limit, light baryons are interchangeably described as a flavor instanton or a D4 brane wrapping the S 4 . The instanton mass is M 0 = 8π 2 κ in units of M KK . The instanton size is small with ρ ∼ 1/ √ λ after balancing the order λ bulk gravitational attraction with the subleading and of order λ 0 U(1) induced topological repulsion [26]. The bulk instanton is described by the O(4) gauge field From hereon M, N run only over 1, 2, 3, z unless specified otherwise. If ρ ∼ 1/ √ λ is the typical size of these tunneling configurations, then it is natural to recast the DBI action using the rescaling The rescaled fields satisfy the equations of motion with the use of the Hodge dual notation.

C. Heavy-light fields
Let (Φ 0 , Φ M ) be the pair of heavy quantum fields that bind to the tunneling configuration above. If again ρ ∼ 1/ √ λ is their typical size, then it is natural to recast the heavy-light part of the DBI action using the additional rescaling The interactions between the light gauge fields (A 0 , A M ) and the heavy fields (Φ 0 , Φ M ) to quadratic order split to several contributions [3,28] L = aN c λL 0 + aN c L 1 + L CS .
which are quoted in (A1). We start by recalling the leading contributions in 1/m H stemming from (12) as thoroughly discussed in [3,29]. For that, we split Φ M = φ M e −im H x 0 for particles (m H → −m H for anti-particles). The leading order contribution takes the form subject to the constraint equation while the subleading contributions in (12) to order λ 0 m H simplify to For self-dual light gauge fields with F M N = F M N , the last contribution in (13) vanishes, and the minimum is reached for f M N = f M N . This observation when combined with the transversality condition for D M φ M = 0, amounts to a first order equation for the as noted in [3,28]. In a self-dual gauge configuration, the heavy spin-1 meson transmutes to a massless spin-1 2 spinor that is BPS bound in leading order.
To account for the spin effects and the breaking of heavy quark symmetry we need to account for the 1/m H contributions to (12)(13)(14)(15). This will be sought by restricting the quantum and heavy fields to the quantum moduli. More specifically, we choose to parametrize the fields using which is equivalent to after gauge transformation. The Φ is parameterized as where Φ a diagonalizes D cl M D cl M Φ a and where are expressed in terms of the collective variables a ∈ SU(2) for a rigid SU(2) rotation. The temporal component Φ 0 satisfies the constraint and in leading order in 1/m H can be ignored. Inserting the expansion (18) in (A1) yields the quadratic 1/m H contributions, HereL CS contains only Φ M . Each of the contribution in (22) is discussed in Appendix A.
Combining the results (A21,A50,A56) we have for the quadratic contributions to order In the mean time, one has also to take into account the Chern-Simons term contribution which is where m y = 16π 2 a. The above analysis ignores the Coulomb back reaction (repulsion from the bound charged fields) as we discussed in [3,28] and can lead to instabilities. In Appendix B we detail the back-reaction from the Coulomb field with the final result for (23) to order with This is the first major result of this paper. We now study the quantization of the (27) and the ensuing heavy-light baryonic spectra.

IV. QUANTUM SPIN EFFECTS
A. Spin-orbit effect The first major spin contribution occurs through the spin-angular momentum coupling χ a χ † τ a χ. Recall that χ a in modular variables is with a 2 4 + 3 a=1 a 2 a = 1 parametrizing the SU (2) ∼ S 3 moduli. Thus, the canonical momenta for y I read Therefore the spin-orbit contribution to the Hamiltonian is with the orbital angular momentum The integer-valued spin of the heavy-light doublet translates to a half-integer spin on the moduli a remarkable transmutation induced by the binding of the zero mode to the instanton in bulk [28]. To leading order in 1/m H only the first two contributions in (32) will be retained. The last contribution in (32) is the induced spin-spin interaction of the heavy mesons and is suppressed by 1/m 2 H .

B. Spin effects
The leading spin effects to order 1/m H stem from the quadratic and quartic χcontributions detailed above. The terms with a first order time-derivative of χ are and imply the equation of motion Therefore to second order in 1/m H one has from which the Hamiltonian can be easily extracted.

C. Hamiltonian
With the above in mind and to obtain the Hamiltonian in leading order in 1/m H , it is sufficient to perform the following substitution and add to the the spin-independent Hamiltonian [28]. More specifically, for a single heavy-quark with n = 1, the total Hamiltonian to order 1/m H now reads (N c = 3) The change in the Laplacian is due to theρ 2 ρ 2 + a 2 I term following from the new line element on the moduli with a change in the small ρ behavior. For the penta-quark states where NQ = N Q = 1, the corresponding Hamiltonian is (N c = 3) Below we solve the corresponding Schroedinger equation numerically.

V. INDUCED QUANTUM POTENTIALS
A. The effective potential for single-heavy quark: l = 0 state For l = 0, the spin-orbit coupling vanishes, i.e. L · S = 0, and the induced effective potential simplifies to Although the sign of the 1 ρ 2 is negative, the m H = ∞ system is still stable due to the uncertainty principle. Indeed, for small ρ, the kinetic contribution is of order 1 ρ 2 and compensates the negative sign to maintain stability. In this case, the 1/m H term implies additional repulsion that further stabilizes the system. As m H → ∞, the spectrum approaches the infinite mass limit smoothly.
B. The effective potential for single-heavy quark: l > 0 state For l = 2, 4, .., one has J = (l ± 1)/2. We first consider the J = (l − 1)/2 case. Again for N Q = 1 and N c = 3, the effective potential reads with α = 13 10 . The 1/m 2 H term due to the spin-orbit coupling is kept to maintain stability at small ρ. The change of the potential as one increases m H tends to decrease for larger l. For l = 2, the shapes of the potential at m H = 2 and m H = ∞ differ moderately, but for l = 2 the difference is already quite small.
Similarly, in the J = l+1 2 case the effective potential is Again, the 1/m H contribution further stabilizes the system and pushes the spectrum a little bit higher.

C. The effective potential for penta-quark state
Here we focus on the pentaquark states with N Q =N Q = 1 state or hidden Q = c, b, with S = 0, 1. For S = 0, the potential reads For S = 1 we can have J = l − 1, l, l + 1, and the potential in this case reads with β = 13 10 and More specifically, for l = 1 we have ∆(J = 1/2) = −2 and ∆(J = 3/2) = 1.

VI. SPECTRA
Given the Hamiltonian and the explicit induced quantum potentials, we can now obtain the spectra of the holographic heavy-light hadrons. Our strategy is the following: we treat the warping contribution as a small perturbation, while solving the radial part numerically. For the warping part, using the average we obtain in unit of M KK . To obtain the radial part, we need to solve the Schroedinger equation with the warped normalization condition For this purpose we perform the transformation Ψ → u and useρ 2 = m y ρ 2 Ψ = u to simplify (51) with the normalization condition Notice that the normalization condition actually requires the u n,l to vanish nearρ. In this case one can show that although the additional term ( large bracket in (51) ) becomes negative at smallρ, the spectrum E n,l is still bounded from below. The above equation for u n,l can be diagonalized numerically and below we present the results for different states.  spectra [32], but about half the value of M KK ∼ 1 GeV used originally in [26] and adopted in [3,28,29]. In this case we have m H = (1.87 − 0.168) GeV= 3.66M KK . In Fig. 1 we show the radial wavefunctions for the first and second excited states following from (51) for a single heavy-baryon (top) and doubly heavy-baryon or pentaquark state (bottom). Note the rapid decay of the wavefunctions near the instanton core as ρ → 0.
The corresponding charm and bottom states for single-and double-heavy hadrons are listed in Table I and Table II respectively. Note that while m Λc = 2.286 GeV is fitted to fix the Kaluza-Klein scale M KK = 0.475 GeV, m Λ b = 5.608 GeV is a holographic prediction which is remarkably close to the experimental value of 5.620 GeV. The details of the mass budgets for each of the states in terms of the three holographic parameters, are given in Appendix C. The results for the single-heavy baryon spectrum are remarkable given the small number of parameters used in this holographic approach. The spin contributions improve considerably the predictions for the masses and their hierarchy. In particular, the empirical mass ordering Σ c − Λ c < Λ * c − Λ c is obtained contrary to the claim in [31]. The mass splitting between Σ c and Σ b is higher than observed due to the sizable repulsion from the l = 2 intrinsic angular momentum assignment. assignment since the instanton core carries equal spin-isospin [3]. The splitting between the different pentaquark states are somehow smaller than expected, due to the strength of the spin-orbit coupling to order 1/m 2 H . Additional contributions are expected to order 1/m 3 H . This construction supports additional Roper-like and odd-parity-like pentaquark states which we have denoted by P * c,b , although heavier and mor susceptible to decay.

VII. CONCLUSIONS
In the holographic construction presented in [3,28,29], heavy hadrons are described in bulk using a set of degenerate N f light D8-D8 branes plus one heavy probe brane in the cigar-shaped geometry that spontaneously breaks chiral symmetry. This construction enforces both chiral and heavy-quark symmetry and describes well the low-lying heavylight mesons and baryons. Heavy baryons are composed of heavy-light mesons bound to a core instanton in bulk. Remarkably, the bound heavy-light mesons with spin-1 transmute to heavy quarks with spin-1 2 , an amazing spin-statistics transmutation by geometry. In [3,28,29] the analysis of the bound states and spectra was carried to order m 0 H where the spin effects are absent. In this work and for N f = 2, we have now carried the analysis at next to leading order in 1/m H where the spin-orbit and spin corrections are manifest. By refining the Kaluza-Klein scale M KK from 1 GeV used in [3,29] to 0.475 GeV used here, a rich spectrum with single-and double-heavy baryons emerges with fair agreement with the empirically observed states, including the newly reported charm pentaquark states by LHCb. This is remarkable, given that only three parameters were used in the holographic construction: Finally, the present holographic description can be regarded as the holographic dual of the chiral soliton construction of heavy-light baryons [39,40] (and references therein). However, in the latter the uncertainties in combining chiral and heavy quark symmetry strongly limit their predictive range, especially when addressing the spin corrections. This is not the case for the holographic description as we have shown, as both symmetries are geometrically embedded in the bulk brane construction with just three parameters. The dual approach is vastly superior.  with each contribution given by

Acknowledgements
We now use the expansion (17) to explicitly derive the various contributions in (A1) in leading order in 1/m H . The net result has manifest heavy quark symmetry to order m 0 H , with the spin-orbit and spin-spin contributions breaking this symmetry to order 1/m H .

Kinetic contribution: L kin
The explicit form of the kinetic contribution is which contains derivative of χ. With the help of the identity for Weyl matrices σ M τ aσ M = 0, (A3) reads which can be further simplified by using the explicit relations after integration over space.

Chern-Simons contribution: L CS
The Chern-Simons term is where in the second line we have performed a partial integration with the help of the Bianchi identity DF = 0. More explicitly, we have which is seen to contain χ † χ as well as linear terms in derivatives. Recall that the electric field F 0M after solving Gauss constraint reads The linear terms inρ,Ẋ N vanish due to parity and translational invariance, but there are terms of the form which couple to isospin. Again, using the identity for Weyl matrices all terms that require anti-symmetrization vanish but the more involved one does not. Using the identity σ M τ iσ M = 0, the second term vanishes, while the first and the third term read Since we finally have One should also consider the contribution fromF 0N = − 1 The final Chern-Simons contribution to order 1/m H after rescaling is In fact, the first term can be obtained from the leading order result by noticing that ∂ t → −im H + ∂ t and requiring gauge in variance. Usinĝ and performing the spatial integration we finally have This is the most difficult term to unravel to order 1/m H . The equation of motion for Φ 0 reads after using the self-dual condition for F . Using the standard relations forσ M N , we have for the last two contributions in (A22) For the first contribution in (A22) we have or more explicitly Inserting (A27-A28) into (A22) we have with the source for Φ 0 . In this equation the Abelian part of F N 0 has been included. Since one finally has with g n (X 2 , ρ 2 ) = 1 4(n + 1)X 2 Here ∆(n, X 2 ) reads ∆(n, with the limit subsumed. As X → 0, f n and g n are all regular. With the above in mind, the explicit solution for Φ 0 follows where we have used the zero-mode profile with c = √ 2ρ/π. In terms of (A41), the Φ 0 contribution to the Lagrangian is Using the fact that χ a is anti-hermitian, all the mixing terms vanish, with the exception of 6i which couples the spin of the nucleon core and the heavy-quarks. After the spatial integration, it reads i 32m H π 2 aρ 2 χ † τ a χχ a . (A45) The diagonal terms give 6144m H π 4 a 2 ρ 4 + i 32m H π 2 aρ 2 χ † τ a χχ a .
The warping contribution stems fromS 1 and does not have any derivative coupling. More specifically, we have After spatial integration, (A49) gives rise to a Z 2 ρ 2 χ † χ term as well as a χ † χ term, namely Notice that the Z 2 contribution is negative, which is consistent with an instability at large Z.
6. The contribution: L 0 To leading order in λ, this contribution vanishes since Φ M satisfies the equation of motion. However, there are contributions toÂ M at order 1/λ, To linear order in χ a , we need the explicit solution toÂ M With this in mind and using the identities we have which after spatial-integration reduces to where ρ cl is the source without the heavy-quark field and we have Notice that originates purely from the Chern-Simions contribution. Given the action for A 0 , at the minimum we have which is a complicated function in χ † χ and always leads to positive energy. In fact, the f 2 m H term in the denominator plays the role of a screening mass which can be seen after certain coordinate transformation.
To estimate how good the first order expansion is, one can consider the simplest case where the inversion is acting only on the ρ 0 ∝ f 2 . To keep track of the dependence on ρ and m H , it is useful to perform the re-scaling As a result we have which can be exactly solved as with b = 32χ † χ m Hρ . Therefore, one has Notice that although the 1 b appears to be at variance with power-counting, the Taylor expansion formally converges for any b. However, for the case whereρ = 1 and χ † χ = 1, one has b = 32 m H ≈ 8 for charm and ≈ 3.2 for bottom, the convergence is poor for the first few terms. To perform an estimate, one can consider the ratio which is shown in Fig. 2. One can actually show that R(b) is always positive and goes to zero as b → ∞ orρ → 0, which implies a weaker repulsion compared to the Leading order Coulomb one. However, expanding to leading order in b, the potential becomes unbounded from below at large b or small ρ. Apparently, this instability is caused by the breakdown of the small b expansion near the core. To fix the instability, we can include the second order term in the expansion. In fact, in Fig. 2 we note that after including the second-order term, the difference between the full result is around 10% for 1/m H ≈ 4 at ρ ≈ 1 for the charm quark. It is even better for the bottom quark.
Using the explicit form of the inversion Here we detail the various contributions to the mass spectra recorded in Table I and  Table II. For completeness, we recall that we fix M D = 1.87 GeV to reproduce the Dmeson mass in (7) and fix M KK = 0.475 GeV to reproduce the M Λc = 2.286 GeV. As result, we have for the charmed heavy-light hadrons recorded in Table I