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Baryons Containing a Heavy Quark as Solitons

arXiv:hep-ph/9205243v3 20 Aug 1992CALT-68-1783UCSD/PTH 92-17DOE RESEARCH ANDDEVELOPMENT REPORTBaryons Containing a Heavy Quark as Solitons⋆(REVISED VERSION)Elizabeth Jenkins and Aneesh V. ManoharDepartment of Physics, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093-0319Mark B. WiseCalifornia Institute of Technology, Pasadena, CA 91125AbstractThe possibility of interpreting baryons containing a single heavy quark as boundstates of solitons (that arise in the nonlinear sigma model) and heavy mesons isexplored. Particular attention is paid to the parity of the bound states and to therole of heavy quark symmetry.⋆Work supported in part by the U.S. Dept.

of Energy under Contract no. DEAC-03-81ER40050and under grant no.

DE-FG03-90ER40546, and by the National Science Foundation under aPresidential Young Investigator award no. PHY-8958081.

Many properties of baryons B containing light u and d quarks suggest that theycan be represented as solitons.For example, they have a mass of order 1/α andthe cross section for e+e−→B ¯B is of order e−1/α, where α = 1/Nc is taken as asmall quantity.1 Skyrme originally suggested that baryons are solitons in the chiralLagrangian (i.e.nonlinear sigma model) used to describe pion self interactions.2The solitons of the chiral Lagrangian have the right quantum numbers to be QCDbaryons,3 provided one includes the Wess-Zumino term.4 The model of QCD baryonsas solitons has been used to compute many of their properties.5,6 The same large Ncpower counting that suggests that baryons containing only light quarks are solitons,also suggests that baryons containing a single heavy quark Q (mQ ≫ΛQCD) can berepresented as solitons.Recently there has been considerable progress in understanding the properties ofhadrons containing a single heavy quark. In the limit mQ →∞, QCD has a heavyquark spin-flavor symmetry7,8 that determines many properties of hadrons containinga single heavy quark.

In the mQ →∞limit, the total angular momentum of the lightdegrees of freedom (i.e. light quarks and gluons),⃗Sℓ= ⃗S −⃗SQ ,(1)is conserved.

Consequently mesons and baryons containing a single heavy quark canbe labeled by sℓ, and provided sℓ̸= 0, they come in degenerate doublets9 with totalspinss± = sℓ± 1/2(2)formed by combining the spin of the heavy quark with the angular momentum of thelight degrees of freedom.Callan and Klebanov originally suggested an interpretation of baryons containinga heavy quark as bound states of solitons of the pion chiral Lagrangian with mesonscontaining a heavy quark.10,11 In this paper we examine this model for heavy baryons.An important aspect of this work is that the consequences of heavy quark symmetry1

are taken into account. Heavy quark symmetry implies that a doublet of mesons con-taining the heavy quark must be considered and that both members of the doubletcan occur in the bound state.

In this regard the work presented here differs from thatof Callan and Klebanov. An important feature of the work of Callan and Klebanovwas that the ground state of the heavy baryon is in an L = 1 partial wave, and thusthe lowest mass baryon containing a heavy quark has positive parity in accord withexperiment.

In this paper both the soliton and meson are taken as infinitely heavyand it is assumed that the soliton-meson potential is minimized at the origin. Conse-quently, the spatial wavefunction for the ground state is a Dirac delta function δ3(⃗x)and corresponds to the lowest energy classical configuration which has the soliton andheavy meson located at the same spatial point.

We give a simple explanation of theCallan-Klebanov parity flip appropriate to this description of the bound state.The lowest lying heavy mesons with Q¯qa (q1 = u, q2 = d) flavor quantum numbershave sℓ= 1/2 and come in a doublet containing spin-zero and spin-one mesons. ForQ = c these are the D and D∗mesons.

The interactions of these heavy mesons withpions are described by a chiral Lagrangian which is invariant under both heavy quarkspin symmetry and chiral SU(2)L × SU(2)R symmetry.⋆Chiral SU(2)L × SU(2)Rsymmetry is spontaneously broken to the vector SU(2)V subgroup, so the Gold-stone boson manifold is SU(2)L × SU(2)R/SU(2)V . A general Goldstone boson fieldconfiguration is obtained by making a space-time dependent chiral transformation(gL(x), gR(x)) on a standard vacuum state.

The Goldstone boson fields Ξ(x) in aG/H sigma model are defined by12g(x) = Ξ(x)h(x)(3)where g(x) ∈G describes the locally rotated vacuum, h(x) ∈H, and Ξ(x) is generatedby ng independent broken generators, where ng is the number of Goldstone bosons.⋆We will restrict the analysis to the case of two light flavors; the generalization to three flavorsis straightforward.2

Different choices for the ng broken generators lead to different (but equivalent) non-linear realizations of the spontaneously broken chiral symmetry.The standard Σbasis for the QCD chiral Lagrangian is defined by(gL(x), gR(x)) ≡(Σ(x), 1) · (VΣ(x), VΣ(x)) = (Σ(x)VΣ(x), VΣ(x)) ,(4)where the broken generators are chosen to be the SU(2)L generators T AL . The fieldΣ(x) is equal to gL(x)g†R(x), and transforms under chiral SU(2)L × SU(2)R asΣ(x) →L Σ(x) R† ,(5)since gL(x) →LgL(x), gR(x) →RgR(x).

The pion fields occur in the 2 × 2 matrixM,M ="π0/√2π+π−−π0/√2#,(6)whereΣ = exp2iMf,(7)and f is the pion decay constant. Experimentally f ≃132 MeV.

In the large Nc limitf ∼1/√α (recall that α = 1/Nc). For considering the couplings of heavy mesons tothe pseudo-Goldstone bosons, it is useful to introduce the ξ basis defined by(gL(x), gR(x)) ≡ξ(x), ξ†(x)·Vξ(x), Vξ(x)=ξ(x)Vξ(x), ξ†(x)Vξ(x)(8)where the broken generators are chosen to be the axial generators T AL −T AR .

Underchiral SU(2)L × SU(2)R, ξ transforms asξ(x) →L ξ(x) U†(x) = U(x) ξ(x) R† ,(9)where U is a complicated function of L, R and the pion fields and in general dependson the space-time coordinates. However, for an unbroken SU(2)V transformation,3

V = L = R, U(x) is equal to V , and is constant. Comparing eq.

(8) with eq. (4), wesee that ξ and Σ are related byΣ(x) = ξ2(x).

(10)The Goldstone boson manifold SU(2)L ×SU(2)R/SU(2)V is diffeomorphic to thegroup manifold SU(2), which is diffeomorphic to the three-sphere S3. The standardidentification of Σ with S3 is obtained by writingΣ = a + i⃗b · ⃗τ,a2 + |⃗b|2 = 1,(11)and identifying Σ with the point (a,⃗b) ∈S3.

The Σ basis provides a single-valuedcoordinate system that covers the entire surface of SU(2), because(gL, gR) = (ΣVΣ, VΣ) =Σ′V ′Σ, V ′Σ(12)implies that Σ = Σ′.Unlike the Σ basis, the ξ basis is not a well-defined coordinate system on SU(2).Two fields ξ and ξ′ denote the same Goldstone boson configuration if(gL, gR) =ξ(x)Vξ(x), ξ†(x)Vξ(x)=ξ′(x)V ′ξ(x), ξ′†(x)V ′ξ(x). (13)This equality is satisfied if and only ifξ′ = ξh,hξh = ξ,h ∈SU(2).

(14)It is straightforward to solve this equation for h ∈SU(2). Denote the general SU(2)matrix ξ byξ = α + i⃗β · ⃗τ,α2 + |⃗β|2 = 1.

(15)If α ̸= 0, then ξ and −ξ are equivalent. However, if α = 0, all configurations ofthe form i⃗β · ⃗τ are equivalent, irrespective of the value of ⃗β.

If the vector (α, ⃗β)4

is considered to be a point on the three sphere S3, then we see that points on thenorthern and southern hemisphere of S3 obtained by inversion through the origin areequivalent, and the entire equator is identified to a point. (The resulting manifold isstill equivalent to S3.) Thus the ξ basis gives a coordinate system for the Goldstonebosons which is well-defined except near the equator of S3, ξ = i⃗β ·⃗τ.

In the Σ basis,the points on the entire three sphere S3 correspond to inequivalent field configurations,and the Σ basis gives a well-defined coordinate system on all of S3.The chiral Lagrangian for heavy mesons is easy to construct in the ξ basis since theheavy meson fields transform in a simple way under parity. Heavy quark symmetryis incorporated into the chiral Lagrangian by defining a heavy meson field whichcontains both pseudoscalar and vector mesons.

The heavy meson field for the groundstate Q¯qa mesons is written as a 4 × 4 matrix13Ha = (1 + /v)2[P ∗aµγµ −Paγ5] ,(16)where vµ is the heavy quark four-velocity, v2 = 1. We shall work in the rest frame ofthe heavy mesons where vµ = (1,⃗0).

The fields Pa, P ∗aµ destroy the heavy pseudoscalarand vector particles that comprise the ground state sℓ= 1/2 doublets. The vectormeson field is constrained by vµP ∗aµ = 0.

Under SU(2)Q heavy quark spin symmetryHa →SHa ,(17)where S ∈SU(2)Q, and under chiral SU(2)L × SU(2)RHa →HbU†ba . (18)It is also convenient to introduce¯Ha = γ0H†aγ0 = [P ∗†aµγµ + P †aγ5](1 + /v)2.

(19)5

Under parityH(x0, ⃗x) →γ0H(x0, −⃗x)γ0(20)andξ(x0, ⃗x) →ξ†(x0, −⃗x) . (21)The chiral Lagrangian density for heavy meson-pion interactions is14,15,16L = −iTr ¯Havµ∂µHa + i2Tr ¯HaHbvµ[ξ†∂µξ + ξ∂µξ†]ba+ig2 Tr ¯HaHbγνγ5[ξ†∂νξ −ξ∂νξ†]ba + ... ,(22)where the ellipsis denotes terms with more derivatives and repeated a, b indices aresummed over 1, 2.

Eq. (22) is the most general Lagrangian density invariant underchiral SU(2)L×SU(2)R, heavy quark spin symmetry and parity.

It is easy to general-ize this Lagrangian density to include explicit SU(2)L × SU(2)R symmetry breakingfrom u and d quark masses and explicit SU(2)Q symmetry breaking from ΛQCD/mQeffects.The coupling g determines the D∗→Dπ decay width,Γ(D∗+ →D0π+) = 16πg2f2|⃗pπ|3 . (23)The present experimental limit,17 Γ(D∗+ →D0π+) <∼72 keV, implies that g2 <∼0.4.In this work the sign of g plays an important role.

Applying the Noether procedureto the Lagrangian density in eq. (22) and to the QCD Lagrangian density gives¯qaγµγ5Tabqb = −gTr ¯HaHbγµγ5Tba + ... ,(24)where T is a traceless 2 × 2 matrix and the ellipsis denotes terms containing thepion fields.

Treating the quark fields in eq. (24) as constituent quarks and using the6

nonrelativistic quark model to evaluate the D∗matrix element of the l.h.s. of eq.

(24)gives16 g = 1. (A similar estimate for the pion nucleon coupling gives gA = 5/3.) Inthe chiral quark model18 there is a constituent-quark pion coupling.

Using the chiralquark model gives g ≃0.75.The soliton solution of the SU(2)L × SU(2)R chiral Lagrangian for pion selfinteractions isΣ = A(t)Σ0(⃗x)A−1(t) ,(25)whereΣ0 = exp (iF(|⃗x|) ˆx · ⃗τ) ,(26)and A(t) contains the dependence on collective coordinates associated with rotationsand isospin transformations of the soliton solution,A = a0 + i⃗a · ⃗τ ,(27)where a20+|⃗a|2 = 1. The soliton has the quantum numbers of a baryon containing lightu and d quarks.

For solitons with baryon number one, the function F(|⃗x|) satisfiesF(0) = −π and F(∞) = 0.⋆Its detailed shape depends on higher derivative terms inthe chiral Lagrangian for pion self interactions. Since F increases as |⃗x| goes from zeroto infinity, it is expected that F ′(0) is positive.

The baryons containing u and d quarkshave wavefunctions that are functions of the aµ. For the neutron and proton states;|p ↑⟩= (1/π)(a1 + ia2), |p ↓⟩= −(i/π)(a0 −ia3), |n ↑⟩= (i/π)(a0 + ia3) and |n ↓⟩=−(1/π)(a1 −ia2) .

The soliton Σ0(x) is a winding number one map from spacetimeinto the Goldstone boson manifold. In the ξ basis, the soliton solution has the form(up to an overall sign)ξ0 = exp (iF(|⃗x|) ˆx · ⃗τ/2) .

(28)As F varies from F = 0 at spatial infinity to F = −π at the origin, ξ0 variesfrom ξ0 = 1, the north pole of S3, to ξ0 = −i⃗τ · ˆx, the equator, with intermediate⋆There appears to be a sign error in the formula for the baryon number current used in Ref. 5.7

points covering the northern hemisphere of S3. However, we have already seen thatthe Goldstone boson manifold in the ξ basis is the northern hemisphere of S3, withthe equator identified to a point.

Thus the soliton solution eq. (28) is a windingnumber one map.

It looks singular at |⃗x| = 0 because the ξ coordinate system for theGoldstone boson manifold is singular at the equator of S3. However, this singularityis a reflection of the coordinate singularity on the Goldstone boson manifold in the ξbasis and is not a physically singular field configuration.The form of the Lagrangian density for heavy meson-pion interactions in eq.

(22)is not very convenient for discussing heavy meson-soliton interactions because of thecoordinate singularity in ξ. It is convenient when dealing with matter fields interactingwith solitons to redefine the fields so that the Lagrangian density is expressible interms of Σ.

For example, introducing new heavy meson fields P ′a and P′∗aµ, defined byH′a = Hbξba ,(29)the SU(2)L × SU(2)R transformation law in eq. (18) becomesH′a →H′bR†ba ,(30)and the Lagrangian density in eq.

(22) becomesL = −iTr ¯H′avµ∂µH′a + i2Tr ¯H′aH′bvµ(Σ†∂µΣ)ba+ig2 Tr ¯H′aH′bγνγ5(Σ†∂νΣ)ba + ...(31)The parity transformation rule for the primed heavy meson fields is a little morecomplicated,H′a(x0, ⃗x) →γ0H′b(x0, −⃗x)γ0 Σ†ba(x0, −⃗x) . (32)Note that in a background Goldstone boson field configuration of a soliton locatedat the same spatial point as the heavy meson the factor of Σ† becomes −1, whereas8

Σ† = 1 for a heavy meson infinitely far from the soliton. This relative minus signis the source of the parity flip that gives positive parity heavy meson-soliton boundstates.In the large Nc limit, the baryon solitons B containing light u and d quarks arevery heavy and time derivatives on the pion fields can be neglected.

Consequentlythe interaction HamiltonianHI = −ig2Zd3⃗x Tr ¯H′aH′bγjγ5[Σ†∂jΣ]ba + ... ,(33)with Σ given by eq. (25) determines the potential energy of a configuration with abaryon B at the origin and a heavy P ′ or P ′∗meson at position ⃗x.

Using eq. (16)and taking the trace in eq.

(33) givesHI = gZd3⃗x [P ′∗j†a(⃗x)P ′b(⃗x) + P ′†a (⃗x)P ′∗jb (⃗x) + iǫikjP ′∗i†a(⃗x)P ′∗kb(⃗x)]×"A xj|⃗x| ˆx · τF ′ −sin(2F)2|⃗x|+ τj2|⃗x| sin(2F) + ǫjmkxkτmsin2(F)|⃗x|2A−1#ba+ ...(34)Assuming that in attractive channels the potential energy is minimized at ⃗x = ⃗0 wherethe heavy meson and baryon soliton are located at the same point, it is eigenvaluesof the potential operator at the origin that are needed to determine the spectrum ofbound states. Using eq.

(34)VI(⃗0) = gF ′(0)[P ′∗j†aP ′b + P ′†a P ′∗jb+ iǫikjP ′∗i†aP ′∗kb]×[(a0 + i⃗a · ⃗τ)τj(a0 −i⃗a · ⃗τ)]ba + ... ,(35)where the ellipsis denotes the contribution of terms in the chiral Lagrangian with morethan one derivative. It is easy to diagonalize the potential (at the origin) matrix that9

arises from taking the truncated basis of nucleon-heavy meson product states (i.e.,|p ↑⟩|P ′1⟩, etc.). On this space it is straightforward to show that⋆VI(⃗0) = −23gF ′(0)(⃗S 2ℓ−3/2)(⃗I 2 −3/2) + ... ,(36)where ⃗Sℓdenotes the total angular momentum vector of the light degrees of freedom(soliton and heavy meson combined) and ⃗I denotes the total isospin vector.

Theeigenstates of VI(⃗0) have definite isospin, spin and angular momentum for the lightdegrees of freedom. They are denoted by |I, s, sℓ⟩, where I is the total isospin, s thetotal spin and sℓis the angular momentum of the light degrees of freedom.The spatial wavefunctions of the eigenstates of eq.

(36) are δ3 (⃗x), and are parityeven. The parity of the meson-soliton bound state is also even, because the primedheavy meson fields are odd under parity at infinity, and so are even under parity atthe origin, as noted below eq.

(32). The unprimed heavy meson fields have a simpletransformation law under parity, and do not have a relative minus sign between theparity at infinity and parity at the origin.

However, in the ξ basis, the wavefunctionof the bound state contains a factor of the form ⃗τ · ˆx near the origin. Thus the parityof the soliton is also even in the ξ basis because the negative parity of the spatialwavefunction is combined with the negative intrinsic parity of the heavy meson field.The factor ⃗τ · ˆx appears singular at the origin, but that is because of the coordinatesingularity in the ξ basis.

Any physical quantity is well-defined at the origin.It is interesting to see if the interaction in eq. (33) gives a reasonable spectrum ofbaryons containing a heavy quark when the effects of operators with more than onederivative are neglected.

The first column of energies in Table 1 gives the eigenstatesand eigenvalues of VI(⃗0) in the truncated basis. Only the |0, 1/2, 0⟩, |1, 1/2, 1⟩and|1, 3/2, 1⟩states are bound if g is taken positive.

For the case Q = c these states havethe right I, s, sℓand parity quantum numbers to be the Λc, Σc and Σ∗c respectively.⋆More correctly this gives the matrix elements of the potential operator in eq. (35) divided bythe normalization of states.

The conventions for the field operators and state normalizationsare the same as those in Ref. [14].10

In the large Nc limit, the N and ∆are degenerate and the space of states shouldbe enlarged to include products of ∆-baryons with the P ′ and P ∗′ mesons. Suchstates can have the same quantum numbers as the Σc and the Σ∗c.

Because the spin-spin interaction is suppressed by 1/Nc, including the ∆causes the Σc and Σ∗c to bedegenerate with the Λc and gives the energies presented in the last column of Table1.19 The real world is intermediate between the Nc →∞limit with a degenerate Nand ∆, and the limit where the ∆is neglected because it is considered to be muchheavier than the nucleon. Including the ∆with a finite ∆−N mass difference will leadto a Σc state which is heavier than the Λc with a binding energy that is intermediatebetween the values in the two columns of Table 1.

Thus the interaction in eq. (33)gives a reasonable qualitative description of heavy baryons.

However it is importantto remember that anti-baryon-meson bound states are described by taking F →−Fso the interaction in eq. (33) gives rise to exotic states.

These states can be removedby including the effects of higher derivative terms in the chiral lagrangian. This is notneccessarily in conflict with the derivative expansion, since the normal state bindingenergy is three times the exotic state binding energy.In the Nc →∞and mQ →∞limit, the most general potential at the originincluding all possible higher dimension operators has the formV = V0(F) −23V1(F)(⃗S 2ℓ−3/2)(⃗I 2 −3/2) ,(39)on the truncated meson-nucleon subspace.

In eq. (39) V0 and V1 are functionals ofthe soliton shape function F(|⃗x|).

The coefficients V0 and V1 have no definite sym-metry under F →−F, so that the meson-nucleon and meson-antinucleon scatteringpotentials are not related to each other.The work presented in this paper is similar in spirit to that of Callan andKlebanov.10 The main difference is that constraints imposed by heavy quark sym-metry are taken into account. The bound state approach of Callan and Klebanovcannot be taken over to baryons containing a heavy charm or bottom quark withoutimposing heavy quark symmetry (e.g., there is no reason to expect eq.

(6.2) of Ref.11

[10] to hold). The mechanism for the parity flip presented in this paper is the sameas that of Callan and Klebanov.

For example, if the kinetic energy of the nucleon isincluded then the ground state is an L = 0 partial wave,⋆instead of a Dirac delta func-tion. (Some of the excited negative parity baryons containing a heavy quark appearas bound states with odd orbital angular momentum.) However, since the unprimedand primed heavy meson fields are related (near the origin) by a factor of ˆx · ⃗τ theS-wave primed heavy meson-nucleon bound state wave functions get multiplied bya linear combination of Y1,m spherical harmonics in the unprimed basis and henceappear as P-waves.

Similar changes in quantum numbers also occur in bound statesinvolving monopoles.20Mark B. Wise thanks N. Seiberg for suggesting that it would be interesting toexamine how baryons containing a heavy quark arise as solitons.Table 1StatesEnergies neglecting the ∆Energies including the ∆|0, 1/2, 0⟩−3gF ′(0)/2−3gF ′(0)/2|1, 3/2, 1⟩, |1, 1/2, 1⟩−gF ′(0)/6−3gF ′(0)/2|1, 1/2, 0⟩gF ′(0)/2gF ′(0)/2|0, 3/2, 1⟩, |0, 1/2, 1⟩gF ′(0)/2gF ′(0)/2⋆With a radial wave function that is strongly peaked about |⃗x| = 0.12

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