aa r X i v : . [ h e p - ph ] F e b Exotic pentaquarks as Gamov–Teller resonances
Dmitri Diakonov , Petersburg Nuclear Physics Institute,Gatchina 188300, St. Petersburg, Russia Institut f¨ur Theoretische Physik II,Ruhr-Universit¨at, Bochum 44780, Germany (Dated: December 16, 2009)
Abstract
If the number of colors N c is taken large, baryons and their excitations can be considered in amean-field approach. We argue that the mean field in baryons breaks spontaneously the sphericaland SU (3) flavor symmetries, but retains the SU (2) symmetry of simultaneous rotations in spaceand isospace. The one-quark and quark-hole excitations in the mean field, together with the SU (3) rotational bands about them determine the spectrum of baryon resonances, which turns outto be in satisfactory accordance with reality when one puts N c = 3. A by-product of this schemeis a confirmation of the light pentaquark Θ + baryon uudd ¯ s as a typical Gamov–Teller resonancelong known in nuclear physics. An extension of the same large- N c logic to charmed (and bottom)baryons leads to a prediction of a anti-decapenta (15)-plet of charmed pentaquarks, two of which, B ++ c = cuud ¯ s and B + c = cudd ¯ s , may be light and stable with respect to strong decays, and shouldbe looked for [1].Keywords: mean field, baryon resonances, exotics, charmed baryons, bottom baryonsPACS: 12.39.Ki, 14.20.Dh, 14.20.Jn . RELATIVISTIC MEAN FIELD It has been argued 30 years ago by Witten [2] that if the number of colors N c is large, the N c quarks of a baryon can be viewed as moving in a mean field. It is helpful to understandhow baryons look like in the large- N c limit, before 1 /N c corrections are considered.At the microscopic level quarks experience only color interactions, however large N c donot suppress gluon fluctuations: the mean field can be only ‘colorless’. An example howoriginally color interactions are Fierz-transformed into interactions of quarks with mesonicfields are provided by the instanton liquid model [3].We shall thus assume that quarks in the large- N c baryon obey the Dirac equation in abackground mesonic field since there are no reasons to expect quarks to be non-relativistic,especially in excited baryons. In a most general case the background field couples to quarksthrough all five Fermi variants. If the background field is stationary in time, it leads to theeigenvalue equation for the u, d, s quarks in the background field: Hψ = Eψ,H = γ (cid:18) − i∂ i γ i + S ( x ) + P ( x ) iγ + V µ ( x ) γ µ + A µ ( x ) γ µ γ + T µν ( x ) i γ µ γ ν ] (cid:19) , (1)where S, P, V, A, T are the mean fields that are matrices in flavor. In fact, the one-particleDirac Hamiltonian (1) is generally nonlocal, however that does not destroy symmetries inwhich we are primarily interested. We include dynamically-generated quarks masses intothe scalar term S .The key issue is the symmetry of the mean field. From the large- N c point of view, thecurrent strange quark mass is very small, m s = O (1 /N c ) [4], therefore a good startingpoint is exact SU (3) flavor symmetry. A natural assumption, then, would be that the meanfield is flavor-symmetric, and spherically symmetric. This assumption, however, leads to toomany “missing resonances” in the spectrum. In addition, we know that baryons are stronglycoupled to pseudoscalar mesons ( g πNN ≈ N c it is a classical mean field. There is no way of writing downthe pseudoscalar field that would be compatible with the SU (3) flav × SO (3) space symmetry.The minimal extension of spherical symmetry is to write the “hedgehog” Ansatz “marrying”2he isotopic and space axes: π a ( x ) = n a F ( r ) , n a = x a r , a = 1 , , , , a = 4 , , , , . (2)This Ansatz breaks the SU (3) flav symmetry. Moreover, it breaks the symmetry under inde-pendent space SO (3) space and isospin SU (2) iso rotations, and only a simultaneous rotationin both spaces remains a symmetry, since a rotation in the isospin space labeled by a , canbe compensated by the rotation of the space axes. Therefore, the Ansatz (2) breaks sponta-neously the original SU (3) flav × SO (3) space symmetry down to the SU (2) iso+space symmetry.It is analogous to the spontaneous breaking of spherical symmetry by the ellipsoid form ofmany nuclei. II. QUARKS IN THE ‘HEDGEHOG’ MEAN FIELD
We shall call the SU (2) iso+space symmetry of the mean field the “hedgehog symmetry”.What mesonic fields S, P, V, A, T in Eq. (1) are compatible with this symmetry? Since SU (3)symmetry is broken, all fields can be divided into three categories:I. Isovector fields acting on u, d quarkspseudoscalar : P a ( x ) = n a P ( r ) , (3)vector : V ai ( x ) = ǫ aik n k P ( r ) , axial : A ai ( x ) = δ ai P ( r ) + n a n i P ( r ) , tensor : T aij ( x ) = ǫ aij P ( r ) + ǫ bij n a n b P ( r ) . II. Isoscalar fields acting on u, d quarksscalar : S ( x ) = Q ( r ) , (4)vector : V ( x ) = Q ( r ) , tensor : T i ( x ) = n i Q ( r ) . III. Isoscalar fields acting on s quarksscalar : S ( x ) = R ( r ) , (5)vector : V ( x ) = R ( r ) , tensor : T i ( x ) = n i R ( r ) . SU (2) symmetryand/or the needed discrete C, P, T symmetries. The 12 ‘profile’ functions P , , , , , , Q , , and R , , should be eventually found self-consistently from the minimization of the mass ofthe ground-state baryon. However, even if we do not know those profiles, there are importantconsequences of this Ansatz for the baryon spectrum.Given the
Ansatz , the Hamiltonian (1) actually splits into two: one for s quarks andthe other for u, d quarks. The former commutes with the angular momentum of s quarks, J = L + S , and with the inversion of spatial axes, hence all energy levels are characterizedby half-integer J P and are (2 J + 1)-fold degenerate. The latter commutes only with the‘grand spin’ K = T + J and with inversion, hence the u, d quark levels have definite integer K P and are (2 K + 1)-fold degenerate. The energy levels for u, d quarks on the one handand for s quarks on the other are completely different, even in the chiral limit m s → e.g. Q ( x ) ∼ R ( x ) ∼ σr .]According to the Dirac theory, all negative -energy levels, both for s and u, d quarks, haveto be fully occupied, corresponding to the vacuum. It means that there must be exactly N c quarks antisymmetric in color occupying all (degenerate) levels with J from − J to J , or K from − K to K ; they form closed shells that do not carry quantum numbers. Filling inthe lowest level with E > N c quarks makes a baryon [4, 5], see Fig. 1. E =0 u , d sK P = 0 + P =1/2 + J ... ... FIG. 1: Filling u, d, s shells for the ground-state baryons: ( , / + ) , ( , / + ). The mass of a baryon is the aggregate energy of all filled states, and being a functionalof the mesonic field it is proportional to N c since all quark levels are degenerate in color.Therefore quantum fluctuations of mesonic field in baryons are suppressed as 1 /N c so that4he mean field is indeed justified.Quantum numbers of the lightest baryons are determined from the quantization of therotations of the mean field, leading to specific SU (3) multiplets that reduce at N c = 3 to theoctet with spin and the decuplet with spin , see e.g. [6]. Witten’s quantization condition Y ′ = N c [7] follows trivially from the fact that there are N c u, d valence quarks each with thehypercharge [8]. Therefore, the ground state shown in Fig. 1 entails in fact 56 rotationalstates. The splitting between the centers of the multiplets ( ,
12 + ) and ( ,
32 + ) is O (1 /N c ),and the splittings inside multiplets can be determined as a perturbation in m s [8]. III. EXCITED STATES IN THE MEAN FIELD
The lowest baryon resonance beyond the rotational excitations of the ground state is thesinglet Λ(1405 , − ). Apparently, it can be obtained only as an excitation of the s quark,and its quantum numbers must be J P = − [4], see transition in Fig. 2.The existence of an − level for s quarks automatically implies that there is a particle-hole excitation of this level by an s quark from the
12 + level. We identify this transition with N (1535 , − ) [4]. It is predominantly a pentaquark state u ( d ) uds ¯ s (at N c = 3). Thisexplains its large branching ratio in the ηN decay [9], a long-time mystery. We also seethat, since the highest filled level for s quarks is lower than the highest filled level for u, d quarks, N (1535 , − ) must be heavier than Λ(1405 , − ): the opposite prediction of the non-relativistic quark model has been always of some concern. Subtracting 1535 −
12 + s -quark level is approximately 130 MeV lower in energy than the valence0 + level for u, d quarks.The low-lying Roper resonance N (1440 ,
12 + ) requires an excited one-particle u, d statewith K P = 0 + [4], see transition . Just as the ground state nucleon, it is part of the excited( ′ ,
12 + ) and ( ′ ,
32 + ) split as 1 /N c . Such identification of the Roper resonance solves anotherproblem of the non-relativistic model where N (1440 ,
12 + ) must be heavier than N (1535 , − ).In our approach they are unrelated.Given that there is an excited 0 + level for u, d quarks, one can put there an s quark aswell, taking it from the s -quark
12 + shell, see transition . It is a particle-hole excitation withthe valence u, d level left untouched, its quantum numbers being S = +1 , T = 0 , J P =
12 + .At N c = 3 it is a pentaquark state uudd ¯ s , precisely the exotic Θ + baryon predicted in5 IG. 2: All baryon resonances below 2 GeV follow from this scheme of one-quark levels. Thetransitions shown by arrows correspond to: : Λ(1405 , / − ), : N (1535 , / − ), : N (1440 , / + ), : Θ + (1530 , / + ), : Λ(1520 , / − ), : N (1650 , / − ?), : N (1710 , / + ), : N (1680 , / + ). Otherresonances belong to SU (3) multiplets obtained as rotational excitations of these one-particle andparticle-hole excitations. Ref. [10] from other considerations. The quantization of its rotations produces the antide-cuplet ( ,
12 + ). In our original prediction the O (1) gap between Θ + and the nucleon wasdue to the rotational energy only, whereas here the main O (1) part of that gap is due tothe one-particle levels, while the rotational energy is O (1 /N c ). Methodologically, it is moresatisfactory.In nuclear physics, excitations generated by the axial current j ± µ , when a neutron fromthe last occupied shell is sent to an unoccupied proton level or v.v. are known as Gamov–Teller transitions [12]. Thus our interpretation of the Θ + is that it is a Gamov–Teller-typeresonance long known in nuclear physics.An unambiguous feature of our picture is that the exotic pentaquark is a conse-quence of the three well-known resonances and must be light. Indeed, the Θ + masscan be estimated from the sum rule [4]: m Θ ≈ − ≈ O ( m s ) corrections to this equation.To account for higher baryon resonances one has to assume that there are higher one-particle excitations, both in the u, d - and s -quark sectors, shown in Fig. 2. It is easy toobtain that order of levels under mild assumptions about the profile functions (3)–(5).6 V. BARYON RESONANCES FROM ROTATIONAL BANDS
The original SU (3) flav × SO (3) space symmetry is restored when flavor and space rotationsare accounted for. Each transition in Fig. 2 generally entails “rotational bands” of SU (3)multiplets with definite spin and parity. The short recipe of getting them is: Find thehypercharge Y ′ from the number of u, d, s quarks involved; only those multiplets are allowedthat contain this Y ′ . Take an allowed multiplet and read off the isospin(s) T ′ of particles atthis value of Y ′ . The allowed spin of the multiplet obeys the angular momentum additionlaw: J = T ′ + J + J + K + K where J , and K , are the initial and final momentaof the s and u, d shells involved in the transition, respectively. The mass of the center of amultiplet does not depend on J but only on T ′ according to the relation [11] M = M + C ( p, q ) − T ′ ( T ′ + 1) − Y ′ I + T ′ ( T ′ + 1)2 I (6)where C ( p, q ) = ( p + q + pq ) + p + q is the quadratic Casimir eigenvalue of the multiplet, I , = O ( N c ) are moments of inertia. After the rotational band for a given transition isconstructed, one has to check if the rotational energy of a particular multiplet is O (1 /N c )and not O (1), and if it is compatible with Fermi statistics at N c = 3: some a priori possiblemultiplets drop out. One gets a satisfactory description of all baryon resonances up to about2 GeV, to be published separately. V. CHARMED AND BOTTOM BARYONS
If one of the u, d quarks in a light baryon is replaced by a heavy b or c quark, there are still N c − u, d quarks left. At large N c , they form the same mean field as in light baryons, withthe same sequence of Dirac levels (up to 1 /N c corrections). The heavy quark contributes tothe mean SU (3)-symmetric field but it is a 1 /N c correction, too.The filling of Dirac levels for the ground-state c (or b ) baryon is shown in Fig. 3: there isa hole in the 0 + shell for u, d quarks. Quantizing rotations of this state leads to the following SU (3) multiplets: ( ¯3 , / + ), ( , / + ) and ( , / + ). The last two are degenerate whereasthe first is split from the rest by O (1 /N c ). The splitting inside multiplets is O ( m s N c ).There are good candidates for those ground-state multiplets: Λ c (2287) and Ξ c (2468)for ( ¯3 , / + ); Σ c (2455), Ξ c (2576) and Ω c (2698) for ( , / + ); finally Σ c (2520), Ξ c (2645)7 IG. 3: Filling u, d, s shells for the ground-state charmed baryons, ( ¯3 , / + ), ( , / + ) and( , / + ). The arrow shows the Gamov–Teller excitation leading to charmed pentaquarks forming( , / + ). and Ω c (2770) presumably form ( , / + ). There are ¯3 ’s and ’s with parity minus arisingfrom exciting the 1 / − s -quark level. The lightest are the degenerate singlets, presumablyΛ c (2595 , / − ) and Λ c (2625 , / − ?).Our new observation is that there is a Gamov–Teller-type transition when axial currentannihilates a strange quark in the
12 + shell, and creates an u or d quark in the 0 + shell, likein the case of the Θ + . In heavy baryons it is even more simple as there is a hole in the u, d + valence shell from the start. Filling in this hole means making charmed pentaquarkswhich we name “beta baryons”, B + c = cudd ¯ s and B ++ c = cuud ¯ s . Quantizing rotations tellsus that these pentaquarks are members of the anti-decapenta-plet ( , / + ), Fig. 4. In fact,there must be two additional (nearly degenerate) multiplets, one with spin 1 / + and theother with spin 3 / + .Charmed pentaquarks have been considered by Wu and Ma in another approach [13];however, they get far larger masses and in addition pentaquarks with ¯ c quarks appearalmost degenerate with those made of c quarks. In our picture the lightest ¯ c pentaquarksΘ c probably arise from putting the fourth ( s ) quark at the − level; they form a quadruplet,have parity minus, and are much heavier.Since we know the separation between the 1 / + level for s quarks and the 0 + level for u, d quarks from fitting the light baryon resonances, and assuming that it does not change forheavy baryons (as it would be at N c → ∞ ), we estimate the mass of the B ++ , + c pentaquarksat about 2420 MeV! The corresponding bottom pentaquarks are about m (Λ b ) + 130 MeV =5750 MeV. Such light charmed and bottom pentaquarks have no strong decays. Theirweak decays, for example B + c → pφ → pK + K − , have clear signatures especially in a vertex8 IG. 4: Decapenta-plet of charmed pentaquarks. detector, and should be looked for at LHC, Fermilab and B-factories. A cautionary remark,though, is that the production rate is expected to be quite low.A detailed elaboration of the ideas presented here will be published elsewhere.I am grateful to Victor Petrov, Maxim Polyakov and Alexei Vladimirov for their help.I thank Ben Mottelson and Semen Eidelman for useful discussions and Harry Lipkin fora correspondence. This work has been supported in part by Russian Government grantsRFBR-06-02-16786 and RSGSS-3628.2008.2, and by Mercator Fellowship (DFG, Germany). [1] An extended version of the invited talk at
Quark Nuclear Physics - 2009 , Beijing, Sep. 21-26,2009, to be published in Chinese Physics C.[2] E. Witten, Nucl. Phys.
B160 (1979) 57.[3] D. Diakonov, Prog. Part. Nucl. Phys. (2003) 173, arXiv:hep-ph/0212026.[4] D. Diakonov, JETP Letters (2009) 407 [Pis’ma v ZHETF, (2009) 451], arXiv:0812.3418[hep-ph]; Nucl. Phys.
A827 (2009) 264C, arXiv:0901.1373 [hep-ph].[5] D. Diakonov, V. Petrov and P. Pobylitsa, Nucl. Phys.
B306 (1988) 809.[6] D. Diakonov and V. Petrov, arXiv:0812.1212 [hep-ph], to be published in
The MultifacetedSkyrmion , G. Brown and M. Rho, eds., World Scientific.[7] E. Witten, Nucl. Phys.
B223 (1983) 433.[8] A. Blotz, D. Diakonov, K. Goeke, N.W. Park, V. Petrov and P. Pobylitsa, Nucl. Phys.
A355 (1993) 765.
9] B.-S. Zou, Eur. Phys. J.
A35 (2008) 325, arXiv:0711.4860 [nucl-th].[10] D. Diakonov, V. Petrov and M. Polyakov, Zeit. Phys.
A359 (1997) 305, arXiv:hep-ph/9703373.[11] D. Diakonov and V. Petrov, Phys. Rev
D69 (2004), 056002, arXiv:hep-ph/0309203.[12] A. Bohr and B. Mottelson. Nuclear structure. New York: W. A. Benjamin (1998) vol. 1.[13] B. Wu and B.-Q. Ma, Phys. Rev.
D70 (2004) 034025, arXiv: hep-ph/0402244.(2004) 034025, arXiv: hep-ph/0402244.