Heavy Baryons and their Exotics from Instantons in Holographic QCD
HHeavy Baryons and their Exotics from Instantons in Holographic QCD
Yizhuang Liu ∗ and Ismail Zahed † Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA (Dated: April 28, 2017)We use a variant of the D D always bind in the form of a flavor instantonzero mode in the fundamental representation. The ensuing instanton moduli for the heavy baryonsexhibits both chiral and heavy quark symmetry. We detail how to quantize it, and derive modelindependent mass relations for heavy bayons with a single-heavy quark in leading order, in overallagreement with the reported baryonic spectra with one charm or bottom. We also discuss thelow-lying masses and quantum assignments for the even and odd parity states, some of whichare yet to be observed. We extend our analysis to double-heavy pentaquarks with hidden charmand bottom. In leading order, we find a pair of doube-heavy iso-doublets with IJ π =
12 12 − ,
12 32 − assignments for all heavy flavor combinations. We also predict five new Delta-like pentaquark stateswith IJ π =
32 12 − ,
32 32 − ,
32 52 − assignments for both charm and bottom. PACS numbers: 11.25.Tq, 11.15.Tk, 12.38.Lg, 12.39.Fe, 12.39.Hg, 13.25.Ft, 13.25.Hw
I. INTRODUCTION
In QCD the light quark sector (u, d, s) is dominated bythe spontaneous breaking of chiral symmetry. The heavyquark sector (c, b, t) is characterized by heavy-quarksymmetry [1]. The combination of both symmetries is atthe origin of the chiral doubling in heavy-light mesons [2,3] as measured by both the BaBar collaboration [4] andthe CLEOII collaboration [5].Recently the Belle collaboration [6] and the BESIIIcollaboration [7] have reported many multiquark ex-otics uncommensurate with quarkonia, e.g. the neutral X (3872) and the charged Z c (3900) ± and Z b (10610) ± .These exotics have been also confirmed by the DO col-laboration at Fermilab [8], and the LHCb collabora-tion at CERN [9]. LHCb has reported new pentaquarkstates P + c (4380) and P + c (4450) through the decays Λ b → J Ψ pK − , J Ψ pπ − [10]. More recently, five narrow andneutral excited Ω c baryon states that decay primarily toΘ + c K − were also reported by the same collaboration [11].These flurry of experimental results support new phe-nomena involving heavy-light multiquark states, a priorioutside the canonical classification of the quark model.Some of the tetra-states exotics maybe understood asmolecular bound states mediated by one-pion exchangemuch like deuterons or deusons [12–19]. Non-molecularheavy exotics were also discussed using constituent quarkmodels [21], heavy solitonic baryons [22, 23], instan-tons [24] and QCD sum rules [25]. The penta-statesexotics reported in [10] have been foreseen in [26] andsince addressed by many using both molecular and di-quark constructions [27], as well as a bound anti-charmto a Skyrmion [28]. String based pictures using string ∗ Electronic address: [email protected] † Electronic address: [email protected] junctions [29] have also been suggested for the descrip-tion of exotics, including a recent proposal in the contextof the holographic inspired string hadron model [30].The holographic construction offers a framework foraddressing both chiral symmetry and confinement inthe double limit of large N c and large t (cid:48) Hooft coupling λ = g N c . A concrete model was proposed by Sakai andSugimoto [31] using a D D N f D N c D D S , and are dual to Skyrmions on theboundary [36, 37]. Remarkably, this identification pro-vides a geometrical description of the baryonic core thatis so elusive in most Skyrme models [38]. A first principledescription of the baryonic core is paramount to the un-derstanding of heavy hadrons and their exotics since theheavy quarks bind over their small Compton wavelength.The purpose of this paper is to propose a holographicdescription of heavy baryons and their exotics that in-volve light and heavy degrees of freedom through a vari-ant of the D D ± , ± ) heavy-light multiplets, their pertinent vector and axial corre- a r X i v : . [ h e p - ph ] A p r lations, and leads reasonable estimates for the one-pionaxial couplings and radiative decays in the heavy-lightsector.In this construction, the heavy baryons will be soughtin the form of a bulk instanton in the worldvolume of D − , − ) multiplet. This approach will extend thebound state approach developed in the context of theSkyrme model [28, 40] to holography. We note that al-ternative holographic models for the description of heavyhadrons have been developed in [33, 34] without the dualstrictures of chiral and heavy quark symmetrty.The organization of the paper is as follows: In sec-tion 2 we briefly outline the geometrical set up for thederivation of the heavy-light effective action through thepertinent bulk DBI and CS actions. In section 3, wedetail the heavy-meson interactions to the flavor instan-ton in bulk. In section 4, we show how a vector mesonwith spin 1 binding to the bulk instanton transmutes to aspin . In section 5, we identify the moduli of the boundzero-mode and quantize it by collectivizing some of thesoft modes. The mass spectra for baryons with single-and double-heavy quarks are explicitly derived. Some ofour exotics are comparable to those recently reported byseveral collaborations, while others are new. Our conclu-sions are in section 6. In the Appendix we briefly reviewthe quantization of the light meson moduli without theheavy mesons. II. HOLOGRAPHIC EFFECTIVE ACTIONA. D-brane set up
The D D N f − D
8- ¯ D N f = 3 is shown in Fig. 1.We assume that the L-brane world volume consists of R × S × S with [0 − − − S which lies in the 4-dimension. Thewarped [5 − R and a horizon at U KK . B. DBI and CS actions
The lowest open string modes streched between the H-and L-branes are attached to a wrapped S in D M FIG. 1: N f − L light branes, and one 8 H heavy brane shown in the τ U plane, with a bulk SU (2) in-stanton embedded in 8 L and a massive HL -string connectingthem. and longitudinal modes Ψ, both fundamental with re-spect to the flavor group SU ( N f − /λ ex-pansion, the effective action on the probe L-branes con-sists of the non-Abelian DBI (D-brane Born-Infeld) andCS (Chern-Simons) action. After integrating over the S ,the leading contribution to the DBI action is S DBI ≈ − κ (cid:90) d xdz Tr ( f ( z ) F µν F µν + g ( z ) F µz F νz ) (1)Our conventions are ( − , , ,
1) with A † M = − A M . Thewarping factors are f ( z ) = R U z , g ( z ) = 98 U z U KK (2)with U z = U KK + U KK z , κ = ˜ T (2 πα (cid:48) ) = aλN c and a =1 / (216 π ) [31]. All dimensions are understood in unitswhere the Kaluza-Klein mass M KK ≡ M, N run over ( µ, z )) F MN = (cid:32) F MN − Φ [ M Φ † N ] ∂ [ M Φ N ] + A [ M Φ N ] − ∂ [ M Φ † N ] − Φ † [ M A N ] − Φ † [ M Φ N ] (cid:33) (3)The CS contribution to the effective action is (form no-tation used) S CS = N c π (cid:90) R Tr (cid:18) AF − A F + 110 A (cid:19) (4)where the normalization to N c is fixed by integrating the F RR flux over the S . The matrix valued 1-form gaugefield is A = (cid:18) A Φ − Φ † (cid:19) (5)For N f coincidental branes, the Φ multiplet is massless.However, their brane world-volume supports an adjointand traceless scalar Ψ in addition to the adjoint gaugefield A M both of which are hermitean and N f × N f val-ued, which we have omitted from the DBI action in so farfor simplicity. They are characterized by a quartic po-tential with finite extrema and a vev v for the diagonal ofΨ [35]. As a result the Φ multiplet acquires a Higgs-likemass of the type12 m H Tr (cid:16) Φ † M Φ M (cid:17) ∼ v Tr (cid:16) Φ † M Φ M (cid:17) (6)The vev is related to the separation between the lightand heavy branes [35], which we take it to be the massfollowing from the length of the streched HL string, andwhich we identify as the mass of the heavy-light (0 − , − )multiplet for either charm ( D, D ∗ ) or bottom ( B, B ∗ ).In the heavy quark limit, the radial spectra, axial andvector correlations, and the one-pion radiative decaysof the (0 − , − ) multiplet are fairly reproduced by thismodel [39]. III. HEAVY-LIGHT-INSTANTONINTERACTIONS
In the original two-flavor D D ρ ∼ / √ λ , as a result of bal-ancing the large and leading attraction due to gravity inbulk (large warpings) and the subleading U(1) Coulomb-like repulsion induced by the Chern-Simons term. In the geometrical set up described in Fig. 1, the smallsize instanton translates to a flat space 4-dimensional in-stanton [36] A clM = − ¯ σ MN x N x + ρ ,A cl = − i π ax (cid:18) − ρ ( x + ρ ) (cid:19) (7)after using the rescalings x → x , x M → x M / √ λ, √ λρ → ρ ( A , Φ ) → ( A , Φ ) , ( A M , Φ M ) → √ λ ( A M , Φ M ) (8)in (1). From here and throughout the rest of the pa-per, M, N run only over 1 , , , z . To order λ therescaled contributions describing the interactions be-tween the light gauge fields A M and the heavy fields Φ M to quadratic order split in the form S = aN c λS + aN c S + S CS (9)with each contribution given by S = − ( D M Φ † N − D N Φ † M )( D M Φ N − D N Φ M )+2Φ † M F MN Φ N S = +2( D Φ † M − D M Φ † )( D Φ M − D M Φ ) − † F M Φ M − † M F M Φ − m H Φ † M Φ M + ˜ S S CS = − iN c π ( d Φ † Ad Φ + d Φ † dA Φ + Φ † dAd Φ) − iN c π ( d Φ † A Φ + Φ † A d Φ + Φ † ( AdA + dAA )Φ) − iN c π Φ † A Φ + S C (Φ , A ) (10)and ˜ S = + 13 z ( D i Φ j − D j Φ i ) † ( D i Φ j − D j Φ i ) − z ( D i Φ z − D z Φ i ) † ( D i Φ z − D z Φ i ) − z Φ † i F ij Φ j + 2 z (Φ † z F zi Φ i + c . c . ) (11) IV. BOUND STATE AS A ZERO-MODE
We now show that in the double limit of large λ fol-lowed by large m Q , a heavy meson in bulk always binds tothe flavor instanton in the form of a 4-dimensional (123 z )flavor zero-mode that effectively is a spinor. This holo-graphic zero-mode translates equally to either a boundheavy flavor or anti-heavy flavor in our space-time (0123).This is remarkable to holography, as the heavy boundstates in the Skyrme-type involve particles but with dif-ficulties anti-particles [40, 41]. Indeed, in the Skyrmemodel, the Wess-Zumino-Witten term which is time-odd, carries opposite signs for heavy particles and anti-particles that are magnified by N c in comparison to theheavy-mesonic action. As a result the particle state isattractive, while the anti-particle state is repulsive. A. Field equations
We now consider the bound state solution of the heavymeson field Φ M in the (rescaled) instanton background7). We note that the field equation for Φ M is independentof Φ and reads D M D M Φ N + 2 F NM Φ M − D N D M Φ M = 0 (12)while the contraint field equation (Gauss law) for Φ de-pends on Φ M through the Chern-Simons term D M ( D Φ M − D M Φ ) − F M Φ M − (cid:15) MNP Q π a K MNP Q = 0 (13)with K MNP Q defined as K MNP Q = + ∂ M A N ∂ P Φ Q + A M A N ∂ P Φ Q + ∂ M A N A P Φ Q + 56 A M A N A P Φ Q (14)In the heavy quark limit it is best to redefine Φ M = φ M e − im H x for particles. The anti-particle case followsthrough m Q → − m H with pertinent sign changes. Asa result, the preceding field equations remain unchangedfor φ M with the substitution D φ M → ( D ∓ im H ) φ M understood for particles ( − ) or anti-particles (+) respec-tively. B. Double limit
In the double limit of λ → ∞ followed by m H → ∞ ,the leading contributions are of order λm H from the lighteffective action in (1), and of order λ m H from the heavy-light interaction term S in (10). This double limit isjustified if we note that in leading order, the mass ofthe heavy meson follows from the straight pending stringshown in Fig 1, with a value [39] m H λM KK = 29 π ( M KK u H ) (15)where u H is the holographic height of the heavy brane.The double limit requires the ratio in (15) to be para-metrically small. With the above in mind, we have S ,m aN c = 4 im H φ † m D φ m − im H ( φ † D M φ M − c . c . ) (16)and from the Chern-Simons term in (10) we have m H N c π (cid:15) MNP Q φ † M F NP φ Q = m H N c π φ † M F MN φ N (17)The constraint equation (13) simplifies considerably toorder m Q , D M φ M = 0 (18)implying that φ M is covariantly transverse in leading or-der in the double limit. C. Vector to spinor zero-mode
The instanton solution A M in (7) carries a fieldstrength F MN = 2 ¯ σ MN ρ ( x + ρ ) (19)We now observe that the heavy field equation (12) incombination with the constraint equation (18) are equiv-alent to the vector zero-mode equation in the fundamen-tal representation. To show that, we recall that the fieldstrength (19) is self-dual, and S in (10) can be writtenin the compact form S = − f † MN f MN + 2 φ † M F MN φ N = − f † MN f MN + 2 (cid:15) MNP Q φ † M D M D Q φ N = − f † MN f MN + f † MN (cid:63) f MN = −
12 ( f MN − (cid:63)f MN ) † ( f MN − (cid:63)f MN ) (20)after using the Hodge dual (cid:63) notation, and defining f MN = ∂ [ M φ N ] + A [ M φ N ] (21)Therefore, the second order field equation (12) can bereplaced by the anti-self-dual condition (first order) andthe transversality condition (18) (first order), f MN − (cid:63)f MN = 0 D M φ M = 0 (22)which are equivalent to σ M D M ψ = Dψ = 0 with ψ = ¯ σ M φ M (23)The spinor zero-mode ψ is unique, and its explicit matrixform reads ψ aαβ = (cid:15) αa χ β ρ ( x + ρ ) (24)which gives the vector zero-mode in the form φ aM = χ β ( σ M ) βα (cid:15) αa ρ ( x + ρ ) (25)or in equivalent column form φ M = ¯ σ M χ ρ ( x + ρ ) ≡ ¯ σ M f ( x ) χ (26)Here χ α is a constant two-component spinor. It can bechecked explicitly that (26) is a solution to the first orderequations (22). The interplay between (24) and (25) isremarkable as it shows that in holography a heavy vectormeson binds to an instanton in bulk in the form of avector zero mode that is equally described as a spinor.This duality illustrates the transmutation from a spin 1to a spin in the instanton field. V. QUANTIZATION
Part of the classical moduli of the bound instanton-zero-mode breaks rotational and translational symmetry,which will be quantized by slowly rotating or translat-ing the bound state. In addition, it was noted in [36]that while the deformation of the instanton size and holo-graphic location are not collective per say as they incurpotentials, they are still soft in comparison to the moremassive quantum excitations in bulk and should be quan-tized as well. The ensuing quantum states are vibrationaland identified with the breathingh modes (size vibration)and odd parity states (holographic vibration).
A. Collectivization
The leading λN c contribution is purely instantonic andits quantization is standard and can be found in [37]. Forcompleteness we have summarized it in the Appendix.The quantization of the subleading λ m H contributioninvolves the zero-mode and is new, so we will describeiin more details. For that, we let the zero-mode slowlytranslates, rotates and deforms throughΦ → V ( a I ( t ))Φ( X ( t ) , Z ( t ) , ρ ( t ) , χ ( t ))Φ → δφ (27) Here X is the center in the 123 directions and Z is thecenter in the z directon. a I is the SU(2) gauge rotationmoduli. We denote the moduli by X α ≡ ( X, Z, ρ ) with − iV † ∂ V = Φ = − ∂ t X N A N + χ a Φ a χ a = − i Tr (cid:0) τ a a − I ∂ t a I (cid:1) (28) a I is the SU(2) rotation which carries the isospin andangular momentum quantum numbers. The constraintequation (13) for φ has to be satisfied, which fixes δφ at sub-leading order − D M δφ + D M ¯ σ M ( ∂ t X i ∂ X i f χ + ∂ t χ )+ i ( ∂ t X α ∂ α Φ M − D M Φ)¯ σ M χ + δS cs = 0 (29)The solution to (29) can be inserted back into the actionfor a general quantization of the ensuing moduli. B. Leading heavy mass terms
There are three contributions to order λ m H , namely16 im H χ † ∂ t χf + 16 im H χ † χA f − m H f χ † σ µ Φ¯ σ µ χ (30)with the rescaled U(1) field A , and the Chern-Simonsterm im H N c π φ † M F MN φ N = i m H N c π f ρ ( x + 1) χ † χ (31)with the field strength given in (19). Explicit calculationsshow that the third contribution in (30) vanishes owingto the identity σ µ τ a ¯ σ µ = 0.The coupling χ † χA term in (30) induces a Coulomb-like back-reaction. To see this, we set ψ = iA and collectall the U(1) Coulomb-like couplings in the rescaled effec-tive action to order λ m H S C ( A ) aN c = (cid:90) (cid:18)
12 ( ∇ ψ ) + ψ ( ρ [ A ] − m H f χ † χ ) (cid:19) ρ [ A ] = 164 π a (cid:15) MNP Q F MN F P Q (32)The static action contribution stemming from the cou-pling to the U(1) charges ρ and χ † χ is S C aN c → S C [ ρ ] aN c + 16 m H χ † χ (cid:90) f ( − iA cl ) − (16 m H χ † χ ) π (33)The last contribution is the Coulomb-like self-interactioninduced by the instanton on the heavy meson throughthe U(1) Coulomb-like field in bulk. It is repulsive andtantamount of fermion number repulsion in holography. C. Moduli effective action
Putting all the above contributions together, we obtainthe effective action density on the moduli in leading orderin the heavy meson mass L = L [ a I , X α ] + 16 aN c m H (cid:18) iχ † ∂ χ † (cid:90) d x f − χ † χ (cid:90) d x f (cid:18) iA cl − aπ ρ ( x + ρ ) (cid:19)(cid:19) − aN c (16 m H χ † χ ) π ρ (34)with L referring to the effective action density on themoduli stemming from the contribution of the light de-grees of freedom in the instanton background. It is iden-tical to the one derived in [36] and to which we referthe reader for further details. In (34) We have made ex-plicit the new contribution due to the bound heavy mesonthrough χ . To this order there is no explicit coupling ofthe light collective degrees of freedom a I , to the heavyspinor degree of freedom χ , a general reflection on heavyquark symmetry in leading order. However, there is acoupling to the instanton size ρ through the holographicdirection which does not upset this symmetry. After us-ing the normalization (cid:82) d x f = 1, inserting the explicitform of A cl from (7), and rescaling χ → χ/ √ aN c m H ,we finally have L = L [ a I , X α ] + χ † i∂ t χ + 332 π aρ χ † χ − ( χ † χ ) π aρ N c (35)Remarkably, the bound vector zero-mode to the instan-ton transmutes to a massive spinor with a repulsiveCoulomb-like self- interaction. The mass is negative which implies that the heavy meson lowers its energy inthe presence of the instanton to order λ . We note thatthe preceding arguments carry verbatum to an anti-heavymeson in the presence of an instanton, leading (35) witha positive mass term. This meson raises its energy in thepresence of the instanton to order λ . These effects orig-inate from the Chern-Simons action in holography. Theyare the analogue of the effects due to the Wess-Zumino-Witten term in the standard Skyrme model [40, 41].While they are leading in 1 /N c in the latter causing theanti-heavy meson to unbind in general, they are sublead-ing in 1 /λ in the former where to leading order the boundstate is always a BPS zero mode irrespective of heavy-meson or anti-heavy-meson. D. Heavy-light spectra
The quantization of (35) follows the same argumentsas those presented in [36] for L [ a I , X α ] and to which we refer for further details in general, and the Appendix forthe notations in particular. Let H be the Hamiltonianassociated to L [ a I , X α ], then the Hamiltonian for (35)follows readily in the form H = H [ π I , π X , a I , X α ] − π aρ χ † χ + ( χ † χ ) π aρ N c (36)with the new quantization rule for the spinor χ i χ † j ± χ † j χ i = δ ij (37)The statistics of χ needs to be carefully determined. Forthat, we note the symmetry transformation χ → U χ and φ M → U Λ MN φ N (38)since U − ¯ σ M U = Λ MN ¯ σ N . So a rotation of the spinor χ is equivalent to a spatial rotation of the heavy vectormeson field φ M which carries spin 1. Since χ is in thespin representation it should be quantized as a fermion.So only the plus sign is to be retained in (37). Also, χ carries opposite parity to φ M , i.e. positive. With this inmind, the spin J and isospin I are then related by (cid:126) J = − (cid:126) I + (cid:126) S χ ≡ − (cid:126) I + χ † (cid:126)τ χ (39)We note that in the absence of the heavy-light meson J + I = 0 as expected from the spin-flavor hedgehogcharacter of the bulk instanton (see also the Appendix).The spectrum of (36) follows from the one discussed indetails in [36] with the only modification of Q enteringin H as given in the Appendix Q ≡ N c aπ → N c aπ (cid:18) − N c χ † χ + 5( χ † χ ) N c (cid:19) (40)The quantum states with a single bound state N Q = χ † χ = 1 and IJ π assignments are labeled by | N Q , JM, lm, n z , n ρ (cid:105) with IJ π = l (cid:18) l ± (cid:19) π (41)with n z = 0 , , , .. counting the number of quanta associ-ated to the collective motion in the holographic direction,and n ρ = 0 , , , .. counting the number of quanta associ-ated to the radial breathing of the instanton core, a sortof Roper-like excitations. Following [36], we identify theparity of the heavy baryon bound state as ( − n z . Using(40) and the results in [36] as briefly summarized in theAppendix, the mass spectrum for the bound heavy-lightstates is M NQ = + M + N Q m H + (cid:32) ( l + 1) N c (cid:32) − N Q N c + 5 N Q N c (cid:33)(cid:33) M KK + 2( n ρ + n z ) + 2 √ M KK (42)with M KK the Kaluza-Klein mass and M /M KK = 8 π κ the bulk instanton mass. The Kaluza-Klein scale is usu-ally set by the light meson spectrum and is fit to repro-duce the rho mass with M KK ∼ m ρ / √ . ∼ M KK through model independentrelations for fixed N Q .We note that the net effect of the heavy-meson isamong other thinghs, an increase in the iso-rotationalinertia by expanding (42) in 1 /N c . The negative N Q /N c contribution in (42) reflects on the fact that a heavy me-son with a heavy quark mass is attracted to the instantonto order λ . As we noted earlier, a heavy meson with aheavy anti-quark will be repelled to order λ hence a sim-ilar but positive contribution. The positive N Q /N c con-tribution is the repulsive Coulomb-like self-interaction.Note that it is of the same order as the rotational contri-bution which justifies keeping it in our analysis.(42) is to be contrasted with the mass spectrum forbaryons with no heavy quarks or N Q = 0, where the nu-cleon state is idendified as N Q = 0 , l = 1 , n z = n ρ = 0and the Delta state as N Q = 0 , l = 3 , n z = n ρ = 0 [36].The radial excitation with n ρ = 1 can be identified withthe radial Roper excitation of the nucleon and Delta,while the holographic excitation with n z = 1 can be in-terpreted as the odd parity excitation of the nucleon andDelta. E. Single-heavy baryons
Since the bound zero-mode transmuted to spin , thelowest heavy baryons with one heavy quark are charac-terized by N Q = 1 , l = even, N c = 3 and n z , n ρ = 0 , M X Q = + M + m H (43)+ (cid:18) ( l + 1) − (cid:19) M KK + 2( n ρ + n z ) + 2 √ M KK (44)
1. Heavy baryons
Consider the states with n z = n ρ = 0. We identifythe state with l = 0 with the heavy-light iso-singlet Λ Q with the assignments IJ π = 0
12 + . We identify the statewith l = 2 with the heavy-light iso-triplet Σ Q with theassignment 1
12 + , and Σ (cid:63)Q with the assignment 1
32 + . Bysubtracting the nucleon mass from (43) we have M Λ Q − M N − m H = − . M KK M Σ Q − M N − m H = − . M KK M Σ ∗ Q − M N − m H = − . M KK (45)Hence the holographic and model independent relations M Λ Q (cid:48) = M Λ Q + ( m H (cid:48) − m H ) M Σ Q (cid:48) = 0 . m N + m H (cid:48) + 0 .
16 ( M Λ Q − m H ) (46)with Q, Q (cid:48) = c, b . Using the heavy meson masses m D ≈ m B = 5279 MeV and m Λ c = 2286 Mev wefind that M Λ b = 5655 MeV in good agreement with themeasured value of 5620 MeV. Also we find M Σ c = 2725Mev and M Σ b = 6134 Mev, which are to be compared tothe empirical values of M Σ c = 2453 Mev and M Σ b = 5810Mev respectively.
2. Excited heavy baryons
Now, consider the low-lying breathing modes R with n ρ = 1 for the even assignments 0
12 + ,
12 + ,
32 + , andthe odd parity excited states O with n z = 1 for theodd assigments 0 − , − , − . (43) shows that the R-excitations are degenerate with the O-excitations. Weobtain ( E = O, R ) M Λ EQ (cid:48) = +0 . M Λ Q + 0 . m N − . m H + m H (cid:48) M Σ EQ (cid:48) = − . M Λ Q + 1 . m N + 0 . m H + m H (cid:48) (47)We found M Λ Oc = 2686 MeV which is to be comparedto the mass 2595 MeV for the reported charm 0 − state,and M Λ Ob = 6095 MeV which is close to the mass 5912MeV for the reported bottom 0 − state. (47) predictsa mass of M Σ Oc = 3126 MeV for a possible charm 1 − state, and a mass of M Σ Ob = 6535 MeV for a possiblebottom 1 − state. F. Double-heavy baryons
For heavy baryons containing also anti-heavy quarkswe note that a rerun of the preceding arguments usinginstead the reduction Φ M = φ M e + im H x , amounts tobinding an anti-heavy-light meson to the bulk instantonin the form of a zero-mode also in the fundamental rep-resentation of spin. Most of the results are unchangedexcept for pertinent minus signs. For instance, whenbinding one heavy-light and one anti-heavy-light meson,(35) now reads L = + L [ a I , X α ]+ χ † Q i∂ t χ Q + 332 π aρ χ † Q χ Q − χ † ¯ Q i∂ t χ ¯ Q − π aρ χ † ¯ Q χ ¯ Q + ( χ † Q χ Q − χ † ¯ Q χ ¯ Q ) π aρ N c (48)As we indicated earlier the mass contributions are oppo-site for a heavy-light and anti-heavy-light meson. Thegeneral mass spectrum for baryons with N Q heavy-quarks and N ¯ Q anti-heavy quarks is M ¯ QQ = + M + ( N Q + N ¯ Q ) m H + (cid:18) ( l + 1)
6+ 215 N c (cid:18) − N Q − N ¯ Q )4 N c + 5( N Q − N ¯ Q ) N c (cid:19)(cid:19) M KK + 2( n ρ + n z ) + 2 √ M KK (49)
1. Pentaquarks
For N Q = N ¯ Q = 1 we identify the lowest state with l = 1 , n z = n ρ = 0 with pentaquark baryonic states withthe IJ π assignments
12 12 − and
12 32 − , and masses given by M ¯ QQ − M N − m H = 0 (50)Amusingly the spectrum is BPS as both the attractionand repulsion balances, and the two Coulomb-like selfrepulsions balance against the Coulomb-like pair attrac-tion. Thus we predict a mass of M ¯ cc = 4678 MeV forthe
12 32 − which is close to the reported P + c (4380) and P + c (4450). We also predict a mass of M ¯ bc = 8087 MeVand M ¯ bb = 11496 MeV for the yet to be oberved pen-taquarks. Perhaps a better estimate for the latters is totrade M N in (54) for the observed light charmed pen-taquark mass M ¯ cc = 4678 MeV using instead M ¯ QQ = M ¯ Q (cid:48) Q (cid:48) + 2 ( m H − m H (cid:48) ) (51)Using (51) we predict M ¯ bc = 7789 MeV and M ¯ bb = 11198MeV, which are slightly lighter than the previous esti-mates. The present holographic construction based onthe bulk instanton as a hedgehog in flavor-spin spacedoes not support the
12 52 + assignment suggested for theobserved P + c (4450) through the bound zero-mode for thecase N f = 2.
2. Excited pentaquarks
For N Q = N ¯ Q = 1 we now identify the lowest statewith l = 1 , n z = 1 , n ρ = 0 with the odd parity pen-taquarks O with assignments
12 12 + and
12 32 + , and the l = 1 , n z = 0 , n ρ = 1 with the breathing or Roper R pentaquarks with the same assignments as the groundstate. The mass relations for these states are ( E = O, R ) M E ¯ QQ − M N − m H = 0 . M KK (52)which can be traded for model independent relations M E ¯ QQ = 1 . m N + 2 m H + 0 .
51 ( m H (cid:48) − M λ Q (cid:48) ) (53)by eliminating M KK using the first relation in (45). Us-ing (53) we predict M E ¯ cc = 4944 MeV, M E ¯ bc = 8353MeV, M E ¯ bb = 11762 MeV as the new low lying exci-tations of heavy pentaquarks with the preceding assign-ments.
3. Delta-like pentaquarks
For N Q = N ¯ Q = 1, the present construction allowsalso for Delta-type pentaquarks which we identify with l = 3 , n z = n ρ = 0. Altogether, we have one
32 12 − , two
32 32 − , and one
32 52 − states, all degenerate to leading order,with heavy flavor dependent masses M ∆ ¯ QQ − M N − m Q = 0 . M KK (54)Again we can trade M KK using the first relation in (45)to obtain the model independent relation M ∆ ¯ QQ = 1 . m N + 2 m H + 0 .
57 ( m H (cid:48) − M Λ Q (cid:48) ) (55)In particular, we predict M ∆¯ cc = 4976 MeV, M ∆¯ cb =8385 MeV, and M ∆¯ bb = 11794 MeV, which are yet to beobserved. VI. CONCLUSIONS
We have presented a top-down holographic approachto the single- and double-heavy baryons in the variant of D D λm Q , the heavy baryons emerge from the zero modeof a reduced (massless) vector meson that transmutesboth its spin and negative parity, to a spin with positiveparity in the bulk flavor instanton. Heavy mesons andanti-mesons bind on equal footing to the core instantonin holography in leading order in λ even in the presenceof the Chern-Simons contribution. This is not the casein non-holographic models where the anti-heavy mesonbinding is usually depressed by the sign flip in the Wess-Zumino-Witten contribution [40]. Unlike in the Skyrmemodel, the bulk flavor instanton offers a model indepen-dent description of the light baryon core. The binding ofthe heavy meson over its Compton wavelength is essen-tially geometrical in the double limit of large λ followedby large m Q .We have shown that the bound state moduli yieldsa rich spectrum after quantization, that involves cou-pled rotational, translational and vibrational modes.The model-independent mass relations for the low-lyingsingle-heavy baryon spectrum yield masses that are inoverall agreement with the reported masses for the cor-responding charm and bottom baryons. The spectrumalso contains some newly excited states yet to be ob-served. When extended to double-heavy baryon spectra,the holographic contruction yields a pair of degenerateheavy iso-doublets with IJ π =
12 12 − ,
12 32 − assignments.The model gives naturally a charmed pentaquark . It alsopredicts a number of new pentaquarks with both hiddencharm and bottom, and five new Delta-like pentaquarkswith hidden charm. The hedgehog flavor instanton whencollectively quantized, excludes the IJ π =
12 52 + assign-ment for N f = 2.The shortcomings of the heavy-light holographic ap-proach stem from the triple limits of large N c and strong (cid:48) t Hooft coupling λ = g N c , and now large m H as well.The corrections are clear in principle but laborious inpractice. Our simple construct can be improved througha more realistic extension such as improved holographicQCD [45]. Also a simpler, bottom-up formulation follow-ing the present general reasoning is also worth formulat-ing for the transparency of the arguments.Finally, it would be interesting to extend the currentanalysis for the heavy baryons to the more realistic caseof N f = 3 with a realistic mass for the light strangequark as well. Also, the strong decay widths of the heavybaryons and their exotics should be estimated. They fol-low from 1 /N c type corrections using the self-generatedYukawa-type potentials in bulk, much like those stud-ied in the context of the Skyrme model [46]. We ex-pect large widths to develop through S-wave decays, andsmaller widths to follow from P-wave decays because ofa smaller phase space. Also the hyperfine splitting in theheavy spectra is expected to arise through subleadingcouplings between the emerging spin degrees of freedomand the collective rotations and vibrations. The perti-nent electromagnetic and weak form factors of the holo-graphically bound heavy baryons can also be obtainedfollowing standard arguments [36, 37]. Some of these is-sues will be addressed next. VII. ACKNOWLEDGEMENTS
We thank Rene Meyer for a discussion. This workwas supported by the U.S. Department of Energy under Contract No. DE-FG-88ER40388.
VIII. APPENDIX
In this Appendix we summarize some of the essentialsteps for the quantization of the instanton moduli devel-oped in [36], and fill up for some of the notations usedin the main text. In the absence of the heavy mesons,we also take the large λ limit using the same rescaling tore-write the contributions of the light gauge fields as S = aN c λS Y M ( A M , ˆ A M ) + aN c S ( A , ˆ A , A M , ˆ A M )(56)Here A refers to the SU(2) part of the light gauge field,and ˆ A to its U(1) part. The equation of motion for A M , ˆ A M are at leading order of λD N F NM = 0 and ∂ N ˆ F NM = 0 (57)They are solved using the flat instanton A M and 0 forˆ A M . The equation of motion for the time componentsare subleading D M F M + 164 π a (cid:15) MNP Q ˆ F MN F P Q = 0 ∂ M ˆ F M + 164 π a (cid:15) MNP Q trF MN F P Q = 0 (58)They are solved using A = 0 and a non-zero ˆ A as de-fined in the main text.To obtain the spectrum we promote the moduli of thesolution to be time dependent, i.e.( a I , X α ) → ( a I ( t ) , X α ( t )) (59)Here a I refers to the moduli of the global SU(2) gaugetransformation. In order to satisfy the constraint equa-tion (52) (Gauss’s law) we need to impose a further gaugetransformation on the field configuation A VM = V † ( A M + ∂ M ) V and A V = V † ∂ t V (60)Inserting the transformed field configuration in the con-straint equation, we find that V is solved by − iV † ∂ t V = Φ = − ∂ t X N A N + χ a Φ a (61)with χ a [ a I ] as defined in the main text. 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