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The Baryon Isgur-Wise Function

arXiv:hep-ph/9208248v1 26 Aug 1992The Baryon Isgur-Wise Functionin the Large Nc LimitElizabeth Jenkins and Aneesh V. ManoharDepartment of Physics, University of California at San Diego, La Jolla, CA 92093Mark B. WiseCalifornia Institute of Technology, Pasadena, CA 91125AbstractIn the large Nc limit, the Λb and Λc can be treated as bound states of chiral solitons andmesons containing a heavy quark. We show that the soliton and heavy meson are boundin an attractive harmonic oscillator potential.

The Isgur-Wise function for Λb →Λc e−νedecay is computed in the large Nc limit. Corrections to the form factor which depend onmN/mQ can be summed exactly (mN and mQ are the nucleon and heavy quark masses).We find that this symmetry breaking correction at zero recoil is only 1%.UCSD/PTH 92-27CALT-68-1809hep-ph/9208248August 19921

1. IntroductionIn the heavy quark limit, the form factors for semileptonic Λb →Λc e−νe decay [1]are characterized by a single universal function η(v · v′)⟨Λc(v′, s′)| c γµ (1 −γ5) b |Λb(v, s)⟩= η(v · v′) u(v′, s′)γµ (1 −γ5) u(v, s),(1.1)where vµ and v′µ are the four-velocities of the Λb and Λc, respectively.

The Isgur-Wisefunction η(v · v′) [2] has logarithmic dependence on the heavy b and c quark masses whichis calculable using perturbative QCD methods. The quark mass dependence can be putinto a multiplicative factor [3]η(v · v′) = Ccb(v · v′) η0(v · v′),(1.2)whereCcb(v · v′) =αs(mb)αs(mc)−6/25 αs(mc)αs(µ)aL(v·v′),(1.3)andaL(v · v′) = 827 [v · v′ r(v · v′) −1] ,(1.4)r(v · v′) =1p(v · v′)2 −1lnv · v′ +p(v · v′)2 −1.

(1.5)For very large heavy quark masses (and µ of order the QCD scale), Ccb(v · v′) has arapid dependence on v · v′. The function η0(v · v′) is determined by low-momentum stronginteraction physics.It depends on the subtraction point µ in a way that cancels thesubtraction point dependence of Ccb(v · v′).

At zero recoil, i.e. v · v′ = 1, η0 is independentof µ and is normalized to unity [2][4][5] by heavy quark flavor symmetry,η0(1) = 1.

(1.6)Some of the low momentum properties of QCD are determined by its symmetries (e.g.chiral symmetry and heavy quark symmetry). Those nonperturbative aspects of the theorywhich are not determined by symmetries cannot be treated using perturbation theory inthe strong coupling constant.

QCD, however, does have an expansion parameter whichcan be used to study low momentum features of the strong interactions analytically. Inthe limit that the number of colors Nc is large, the theory simplifies and many predictions2

are possible [6]. The main purpose of this paper is to examine η0(v · v′) in the large Nclimit.In the large Nc limit, baryons containing light u and d quarks can be viewed as soli-tons [7][8] of the nonlinear chiral Lagrangian for pion self-interactions.

Baryons containinga single heavy c (or b) quark and light u and d quarks are then described as bound states ofthese solitons with D and D∗mesons (or B and B∗mesons) [9][10]. The large Nc behaviorof η0(v · v′) can be determined using the bound state wavefunctions of the Λb and Λc.For large Nc, the function η0(v · v′) is strongly peaked about zero recoil since any velocitychange must be transferred to ∼Nc light quarks.

Independent of the details of the boundstate approach, we find thatη0(v · v′) = exp[−λN 3/2c(v · v′ −1)],(1.7)where λ is a constant of order unity. This equation is valid for v · v′ −1 of order N −3/2c.In this kinematic region, η0 falls from unity to a very small quantity.

The derivation ofthis result is the main purpose of this paper. In addition, the effect on η0 of correctionsto the heavy quark limit that depend on mN/mQ is examined.Some features of thesoliton picture of heavy baryons not discussed in previous work on this subject will also bederived here.

In Refs. [10][11] it was shown that the leading term in the chiral Lagrangianfor heavy-meson–pion interactions gives rise to a heavy-meson–soliton potential that isattractive at the origin in the ΛQ, ΣQ and Σ∗Q channels.

We show in this paper that thecurvature of the soliton–heavy-meson potential is positive, indicating that the origin is astable minimum of the potential energy for these channels. The curvaturve is negative forthe exotic channels.2.

ΛQ as a Heavy Meson-Soliton Bound StateThe starting point for discussing soliton–heavy-meson bound states is the chiral La-grangian for the interactions of mesons containing a heavy quark Q with pions [12]. Inthe limit mQ →∞, the total angular momentum of the light degrees of freedom, ⃗Sℓ, is asymmetry generator.

The lowest mass mesons with Qqa (q1 = u, q2 = d) flavor quantumnumbers have sℓ= 1/2 and form a degenerate doublet consisting of pseudoscalar and vec-tor mesons. In the case Q = c, these are the D and D∗mesons, and in the case Q = b,these are the B and B∗mesons.3

It is convenient to combine the fields Pa and P ∗aµ for the ground state sℓ= 1/2 mesonsinto the bispinor matrixHa = (1 + v/)2P ∗aµγµ −Paγ5,(2.1)where vµ is the heavy quark four velocity, and v2 = 1. The vector meson field is constrainedto satisfy vµP ∗aµ = 0.In this section, we work in the rest frame of the heavy meson,vµ = (1,⃗0).

Under the heavy quark spin symmetry,Ha →SHa,(2.2)where S ∈SU(2)v is the heavy quark spin transformation. The transformation propertyof H under SU(2)L × SU(2)R chiral symmetry has an arbitrariness associated with fieldredefinitions.

We will use the basis chosen in Ref. [10],* with the transformation ruleHa →(HR†)a,(2.3)under SU(2)L × SU(2)R, where R ∈SU(2)R. It is also convenient to introduce the fieldHa = γ0H†aγ0 =P ∗†aµγµ + P †aγ5 (1 + v/)2.

(2.4)The Goldstone bosons occur in the fieldΣ = exp2iMf,(2.5)whereM =π0/√2π+π−−π0/√2,(2.6)and f ≈132 MeV is the pion decay constant. Under SU(2)L × SU(2)RΣ →LΣR†,(2.7)with L ∈SU(2)L and R ∈SU(2)R. Under parity,Σ(x0, ⃗x) →Σ†(x0, −⃗x),(2.8)since M(x0, ⃗x) →−M(x0, −⃗x).

If Ha transforms under chiral symmetry as in Eq. (2.3),then the parity transform of H must transform under chiral symmetry with a factor of L†.

* The computations are repeated for the ξ basis in the appendix.4

Consequently, in the basis we are using, the action of parity on the H field is somewhatunusual,Ha(x0, ⃗x) →γ0Hb(x0, −⃗x) γ0 Σ†ba(x0, −⃗x). (2.9)The chiral Lagrangian density for heavy meson-pion strong interactions isL = −i Tr Hv · ∂H + i2 Tr HHvµΣ†∂µΣ + ig2 Tr HHγµγ5Σ†∂µΣ + .

. .

,(2.10)where the ellipsis denotes the contribution of terms containing more derivatives or factorsof 1/mQ. The coefficient of the second term in the Lagrangian density Eq.

(2.10) is fixed(relative to the first) by parity invariance. The coupling g determines the D∗→Dπ decayrate.

Present experimental information on the D∗width and the D∗→Dπ branchingratio implies that g2 < 0.4 [13]. The constituent quark model predicts that g is positive.The soliton solution of the SU(2)L×SU(2)R chiral Lagrangian for baryons containingu and d quarks isΣ = A(t) Σ0(⃗x)A−1(t),(2.11)whereΣ0 = exp (iF(r) ˆx · ⃗τ) ,(2.12)and r = |⃗x|.

A(t) contains the dependence on the collective coordinates associated withrotations and isospin transformations of the soliton solution. For solitons with baryonnumber one, F(0) = −π and F(∞) = 0.

The detailed shape of F(r) depends on the chiralLagrangian for pion self interactions including terms with more than two derivatives. Weexpect that Σ0(⃗x) has a power series expansion in ⃗x.

Consequently, the even powers ofr must vanish when F(r) is expanded in a power series in r, e.g. F ′′(0) = 0.

The chiralLagrangian for pion self interactions is of order Nc. However, the chiral Lagrangian forheavy-meson–pion interactions is only of order one.

Thus, to leading order in Nc the shapeof the soliton F(r) is not altered by the presence of the heavy meson.In the large Nc limit, baryons containing light u and d quarks are very heavy and timederivatives on the Σ field can be neglected. Consequently, it is the interaction HamiltonianHI = −ig2Zd3⃗x Tr HHγjγ5Σ†∂jΣ + .

. .

,(2.13)with Σ given by Eqs. (2.11) and (2.12) that determines the potential energy of a configu-ration with a heavy meson at the origin and a baryon at position ⃗x.

Neglecting operators5

with more than one derivative (the ellipsis in Eq. (2.13)), and expanding the interactionpotential operator in ⃗x givesˆVI(⃗x) = g SjℓH IkH Tr Aτ iA−1τ k nδij hF ′(0) −23r2 [F ′(0)]3 + 16r2F ′′′(0)i+xixj h23 [F ′(0)]3 + 13F ′′′(0)i+ ǫijmxm [F ′(0)]2o+ O(x3),(2.14)where SjℓH denotes the angular momentum of the light degrees of freedom of the heavymeson, and IkH denotes the isospin of the heavy meson.

The interaction potential has termswhich superficially have the wrong parity, e.g. the term involving the ǫ symbol.

However,these terms are required because of the Σ† factor in the parity transformation of H inEq. (2.9).The ΛQ baryon has isospin zero and total angular momentum of the light degreesof freedom equal to zero.

In the large Nc limit, it arises from a bound state of nucleonswith P and P ∗mesons. Baryons with I > 1/2 such as the ∆cannot produce a heavybaryon bound state with I = 0.

On nucleon states, Tr Aτ iA−1τ k is equal to −8SiNIkN/3where SiN is the spin of the nucleon, and IkN is the isospin of the nucleon [14]. Using thissimplification, the potential operator becomesˆVI(⃗x) = ˆV (0)I+ ˆV (1)I+ ˆV (2)I+ O(x3),(2.15)where ˆV (n)Idenotes the term of order rn in the potentialˆV (0)I= −83 gF ′(0) ⃗IH · ⃗IN ⃗SℓH · ⃗SN,ˆV (1)I= 83 g [F ′(0)]2 ⃗IH · ⃗IN ⃗x · (⃗SℓH × ⃗SN),ˆV (2)I= −83 g ⃗IH · ⃗INn⃗SℓH · ⃗SNh−23r2 [F ′(0)]3 + 16r2F ′′′(0)i+ (⃗SℓH · ⃗x)(⃗SN · ⃗x)h23 [F ′(0)]3 + 13F ′′′(0)io.

(2.16)The potential ˆVI(⃗x) commutes with the total angular momentum of the light degrees offreedom, ⃗L + ⃗SℓH + ⃗SN, where ⃗L is the orbital angular momentum.To find the ΛQ wavefunction ΨΛ(⃗x) and its potential energy VΛ(⃗x), the potentialenergy operator must be diagonalized at each point ⃗x on the product space of nucleonheavy meson states (e.g. |p, ↑⟩|P ∗, −⟩).

It is convenient to consider linear combinationsof these product states that have definite isospin ⃗I = ⃗IH + ⃗IN, spin of the light degrees of6

freedom, ⃗Sℓ= ⃗SℓH + ⃗SN, and total spin ⃗S = ⃗SQ + ⃗Sℓ. These states are labeled |I, s, sℓ⟩.It is straightforward to diagonalize ˆVI(⃗x) in this basis, and we find thatΨΛ(⃗x) =h1 −F ′(0) ⃗x · (⃗SℓH × ⃗SN)i 0, 12, 0φ(⃗x),(2.17)is an eigenstate of ˆVI(⃗x) with eigenvalueVΛ(⃗x) = −32gF ′(0) + gr2 h16 [F ′(0)]3 −512F ′′′(0)i= −32gF ′(0) + 12κr2,(2.18)where κ is defined byκ = gh13 [F ′(0)]3 −56F ′′′(0)i.

(2.19)The form of the wavefunction in Eq. (2.17) can also be found using Eq.

(2.9) and demandingthat it has definite parity. The factor in square brackets in Eq.

(2.17) compensates for thefactor of Σ† in Eq. (2.9).

In Eqs. (2.17) and (2.18), rΛQCD is treated as a small quantity*and the wavefunction is given to linear order in rΛQCD, while the potential energy isgiven to quadratic order in rΛQCD.

The spatial part of the wavefunction φ(⃗x) has a morerapid dependence on r which will be computed later in this article. As r goes from zero toinfinity, F(r) goes from −π to 0.

Consequently, we expect that F ′(0) is positive and F ′′′(0)is negative. This is true for example for the solution given in Ref.

[14] where a particularfour derivative term in the chiral Lagrangian for pion self-interactions is used to stabilizethe soliton. Furthermore, the constituent quark model suggests that g is positive.

Thus,Eq. (2.18) implies that the ΛQ is bound by a harmonic oscillator potential with κ > 0.In the limit Nc →∞, the nucleon is infinitely heavy and terms in the Hamiltonian in-volving the nucleon momentum are neglected.

The lowest energy state then has the spatialwavefunction φ(⃗x) = δ3(⃗x) corresponding to the minimum energy classical configurationwhere the nucleon is located at ⃗x = 0. The 1/Nc terms in the Hamiltonian involving thenucleon momentum give the spatial wavefunction a finite spread and it is this finite extentwhich is responsible for the Isgur-Wise function η0(v · v′).

Terms of order 1/Nc involv-ing the momentum are found by writing Σ = Σ(⃗x −⃗r(t)) and quantizing the collectivecoordinate ⃗r (t). This procedure yieldsHkin =⃗p 22mN−43F ′(0)⃗IH · ⃗IN ⃗SN · ⃗pmN,(2.20)* ΛQCD denotes a nonperturbative strong interaction scale that is finite as Nc →∞.7

where mN is the mass of the nucleon. The first term is the usual kinetic energy of thesoliton and the second term in Eq.

(2.20) results from the second term in Eq. (2.10).

TheSchr¨odinger equation is[Hkin + VΛ(⃗x)] ΨΛ(⃗x) = EΨΛ(⃗x),(2.21)which implies that φ(⃗x) obeys the differential equation"−⃗∇22mN+ VΛ(⃗x)#φ(⃗x) = E φ(⃗x),(2.22)in the rest frame of the bound state.Note that the term linear in ⃗p in Eq. (2.20) isnecessary for the wavefunction in Eq.

(2.17) to obey the Schr¨odinger equation (2.21). Thisresult is not surprising because both the term linear in ⃗p and the part of the wavefunctionlinear in ⃗x arise from the peculiar definition of parity in Eq.

(2.9). In deriving Eqs.

(2.21)and (2.22) we have treated the term linear in ⃗p in Hkin as a perturbation and neglectedits action on the “small” part of the wavefunction ΨΛ(⃗x) (i.e. the piece proportional to⃗x).

As we shall see shortly, for large Nc the term proportional to ⃗x in ΨΛ(⃗x) and the termlinear in ⃗p in Hkin are subdominant and can be neglected in the calculation of η0(v · v′).In general, for large Nc we expect the potential VΛ(⃗x) to have the harmonic oscillatorformVΛ(⃗x) = V0 + 12κ⃗x2. (2.23)The absence of a term linear in r requires F ′′(0) = 0 which is a consequence of the derivativeexpansion of the chiral Lagrangian for pion self-interactions.

Quantum corrections caninduce nonanalytic behavior in ⃗x, but because of an explicit factor of 1/Nc such terms areless important than those we have kept. The particular expression for κ in Eq.

(2.19) is,however, model dependent and arises from keeping only terms with one derivative in thechiral Lagrangian for heavy meson-pion interactions.The model independence of the results of this paper follows from an analysis of largeNc power counting. In the limit mQ →∞, the typical size (or momentum) of the boundstate wavefunction occurs when the kinetic energy ⃗p 2/2mN and potential energy κ⃗x 2/2of the bound state contribute equally to the total energy E −V0,r ∼(κmN)−1/4 ,p ∼(κmN)1/4 .

(2.24)Since κ is of order Λ3QCD and mN is of order ΛQCDNc, the Nc dependence of the typicalbinding energy is given byE −V0 ∼ΛQCDN −1/2c. (2.25)8

It is now straightforward to see that higher order terms in the effective Lagrangian canbe neglected. Any term in the effective Lagrangian is a function of ⃗r = ⃗rN −⃗rH, ˙⃗rN,and ˙⃗rH.

Terms involving only the soliton field carry an overall factor of Nc. These termsare independent of the relative coordinate ⃗r and depend only on ˙⃗rN.In the effectiveHamiltonian, the dependence of these terms on ˙⃗rN enters through powers of the nucleonmomentum.

A term with n powers of the nucleon momentum has the following large Ncbehavior,ΛQCDNc pMNn∼ΛQCDNcN −3n/4c. (2.26)Thus, all terms with higher powers of the nucleon momentum than the leading order kineticenergy term p2/2mN are suppressed by more powers of 1/Nc and can be neglected in thelarge Nc limit.

Terms in the Lagrangian which involve the interaction between the solitonand the heavy meson can depend on ⃗r, but they are at most of order one in the large Nclimit. The typical scale of the r dependent interaction is ΛQCD, so higher order interactionterms have the formΛQCD (ΛQCDr)m pMNn∼ΛQCDN −m/4cN −3n/4c.

(2.27)Hence, all the higher order interaction terms in the effective Lagrangian other than theharmonic potential are higher order in 1/Nc and can be neglected (including the termlinear in ⃗p in Hkin Eq. (2.20)).There is also a class of 1/mQ corrections which can be summed exactly.

So far, wehave concentrated on the order of limits mQ →∞followed by Nc →∞. We now switchto the situation in which mQ, Nc →∞simultaneously, with the ratio ΛQCDNc/mQ heldfixed.

The only additional term in the effective Lagrangian which must be included in thisdouble scaling limit is the kinetic energy of the heavy meson,p2mH∼mNmQp2mN∼ΛQCDNcmQ p2mN,(2.28)which is of the same order as a term we have included, since ΛQCDNc/mQ is of order one.Higher order terms in 1/mQ such asp4m3Q∼mNmQ3 p4m3N,(2.29)are of order one times terms which can be neglected by the power counting arguments ofthe previous paragraph, so they too can be neglected. Additional 1/mQ effects arise from9

1/mQ operators in the heavy quark effective theory. The P ∗−P mass difference is oforder Λ2QCD/mQ ∼(NcΛQCD/mQ) ΛQCD/Nc which is subleading in Nc.

Higher derivativeoperators in the current are also suppressed. For example, the leading correction1mcTr H(c) ←/D γµ (1 −γ5) H(b) ∼ΛQCDN 1/4cmc∼ΛQCDNcmQN −3/4c,(2.30)since the typical momentum of the heavy quark in the baryon is of order ΛQCDN 1/4c.

Thus,no additional terms other than the kinetic energy of the heavy meson are relevant.3. Λb →Λc e−νe DecayFor non-relativistic Λc recoil, the matrix element of the weak current Eq.

(1.1) in theΛb rest frame is⟨Λc(v′, s′)| c γµ (1 −γ5) b |Λb(v, s)⟩=Zd3⃗p ′Zd3⃗p φ∗c(⃗p ′ ) φb(⃗p )14 ⟨N(−⃗p ′ + mN⃗v ′ , s′)| N(−⃗p , s)⟩⟨D(⃗p ′ + mD⃗v ′ )| c γµ (1 −γ5) bB(⃗p )+ . .

. (3.1)where the ellipsis represents terms involving at least one vector meson.

φc and φb are theFourier transform of the ground state wave functionφc,b(⃗p ) =1(π2µc,b κ)3/8 exp−⃗p 2/2√µc,b κ,(3.2)where κ is defined in Eq. (2.19), and µc,b is the reduced mass µ = mN mH/(mN + mH)of the bound state, with mH = mD, mB for the c and b subscripts, respectively.

The ΛQstate is a superposition of products of N with P and P ∗states. However, we do not needthe details of the Clebsch-Gordan structure* of the ΛQ state, as will become clear soon.The nucleon matrix element vanishes unless⃗p ′ = ⃗p + mN⃗v ′,(3.3)at which point, for the term explicitly displayed in Eq.

(3.1), the required heavy mesonmatrix element is⟨D (⃗p + (mN + mD)⃗v ′ )| c γµ (1 −γ5) b |B (⃗p )⟩. (3.4)* The factor of 1/4 is the square of the appropriate Clebsch-Gordan coefficient for the termdisplayed explicitly in Eq.

(3.1).10

This matrix element can be evaluated in terms of heavy meson form factors. In the ΛQheavy meson-nucleon bound state, the typical momentum is of order ⟨p⟩∼(mNκ)1/4and so form factors for semileptonic Λb →Λc e−νe decay are smooth functions ofmN⃗v ′/ (mNκ)1/4 which is of order N 3/4c⃗v ′.

Consequently, we are interested in the kine-matic region v′ of order N −3/4c, and so for large Nc we can replace the heavy meson formfactors by their rapidly varying part, Ccb(v · v′), and neglect the slow variation in themeson Isgur-Wise function. This is why we do not need the details of the Clebsch-Gordanstructure of the ΛQ bound state.

The Λb →Λc e−νe form factor in the large Nc limit canbe written in terms of an Isgur-Wise function, as in Eq. (1.1), withη0 =Zd3⃗p φ∗c(⃗p + mN⃗v ′) φb(⃗p )="2 (µc µb)1/4√µb + √µc#3/2exp−m2N⃗v ′ 2/2√κ√µb + √µc.

(3.5)Since for non-relativistic recoils ⃗v ′ 2 ≈2(v · v′ −1), the expression for η0 in a general framehas the formη0(v · v′) ="2 (µc µb)1/4√µb + √µc#3/2exp−m2N(v · v′ −1)/√κ√µb + √µc. (3.6)The result we have obtained is valid in the heavy quark and large Nc limits, where wehave included all terms of order (mN/mQ)n ∼(NcΛQCD/mQ)n. The baryon form factorsfor Λb →Λce−νe are still written in terms of a single function in this limit, even thoughwe have included a class of 1/mQ corrections.

In the large Nc limit, the function η0(v · v′)falls offrapidly away from zero recoil. Derivatives of η0 at zero recoil diverge as Nc →∞.Eq.

(3.6) indicates that the mth derivative is of order N 3m/2c, and includes all contributionsof this order neglecting less divergent pieces. For example, there could be corrections toη0 of the form N 1/2c(v · v′ −1) in the exponent.

This term is small compared with theleading term N 3/2c(v · v′ −1), but is significant when v · v′ −1 is of order N −1/2c. ThusEq.

(3.6) is valid in the large Nc limit in the region where v · v′ −1 <∼O(N −3/2c), i.e. inthe region where η0 falls from about unity to a very small quantity.At zero recoil in the limit that mb →∞, η0 has the formη0(1) ="2 (µc mN)1/4√mN + √µc#3/2= 1 −364mNmD2+ .

. .,(3.7)11

where we have expanded the exact expression in a power series in mN/mD. The termlinear in mN/mD vanishes, which is consistent with Luke’s theorem [15].

For physicalvalues of mN/mD, the correction to the symmetry limit prediction η0(1) = 1 is only 1%.The parameter κ can be determined to be (530 MeV)3 in the Skyrme model usingthe shape function used in Ref. [14], and the value of g obtained in Ref.

[11]. With thisvalue of κ, we find that the orbitally excited ΛQ state should be about 400 MeV above theground state, and that the form factor η0 of Eq.

(3.6) isη0(v · v′) ∼0.99 exp [−1.3 (v · v′ −1)] ,(3.8)using the known values of mN, mD and mB.The Skyrme model prediction for κ issensitive to the precise shape of the soliton solution. A better way to determine κ is touse the experimentally measured excitation energy of the orbitally excited ΛQ, which ispκ/µQ.The large Nc predictions of this paper rely on the number of light quarks in the heavybaryon being large.

For Nc = 3 there are only two light quarks in ΛQ, so we expect ourresults to only be qualitatively correct. Nevertheless, it is interesting that the baryon formfactors are calculable in the large Nc limit of QCD.

There are other results that can becomputed using the methods developed here. For example, the Isgur-Wise functions fortransitions to excited states are also calculable.

It should also be possible to derive ourresults using the methods of Witten [16]. It would be interesting to explore that approach.AcknowledgementsWe would like to thank J. Hughes for several discussions which stimulated our in-terest in the subject.

E.J. and A.M. would like to thank the Fermilab theory group forhospitality.

This work was supported in part by DOE grant #DOE-FG03-90ER40546 andcontract #DEAC-03-81ER40050, and by a NSF Presidential Young Investigator awardPHY-8958081.Appendix A. The ξ BasisIn this appendix, we briefly discuss the computation of the binding potential in the ξbasis discussed in Ref.

[10]. The notation is the same as found in Ref.

[10]. The ξ basis issingular at the origin, but has a simple transformation rule for the H field under parity,H(x0, ⃗x) →γ0H(x0, −⃗x)γ0.12

The interaction Hamiltonian in this basis isHI = −ig2Zd3x Tr HHγjγ5ξ†∂jξ −ξ∂jξ†. (A.1)Expanding the Goldstone boson fieldξ0 = exp (iF(r) ˆx · ⃗τ/ 2),(A.2)in a power series about ⃗x = 0, and using ξ = A ξ0 A−1, we get the interaction potentialˆVI(⃗x) = 2g Tr Aτ jA−1τ k nF ′(0)SℓH · ˆx IkH ˆxj −12SjℓHIkH+r2 112 [F ′(0)]3 −112F ′′′(0)SjℓH IkH + SℓH · ⃗x IkH xj −112 [F ′(0)]3 + 13F ′′′(0)o.

(A.3)Note that only even powers of x occur in Eq. (A.3) because the parity transformationis trivial in the ξ basis.

To leading order in Nc, the kinetic term of the soliton can beneglected, so that states with a definite value of A are eigenstates of the Hamiltonian. Itis convenient to consider the soliton state that is an eigenstate of A with A = 1.

Stateswith other values of A can be obtained by an isospin transformation, and so have the sameenergy. On states with A = 1, the interaction potential reduces toˆVI(⃗x) = 4gF ′(0)SℓH · ˆx IH · ˆx −12SℓH · IH+r2 112 [F ′(0)]3 −112F ′′′(0)SℓH · IH + SℓH · ⃗x IH · ⃗x−112 [F ′(0)]3 + 13F ′′′(0)o(A.4)This Hamiltonian is singular at the origin because of the coordinate singularity in the ξbasis, and the eigenstates of the Hamiltonian will also be singular at the origin.

We knowthat the singularity at the origin has the form ⃗τ · ˆx, because that is the transformationfunction from the singular ξ basis to the non-singular basis used earlier in this article. Wetherefore write the eigenstates of the interaction potential Eq.

(A.4), |ψ⟩in terms of neweigenstates |φ⟩which are related by the unitary transformation|ψ⟩=2 ⃗IH · ˆx|φ⟩. (A.5)The interaction potential in the |φ⟩basis isˆV ′I(⃗x) =2 ⃗IH · ˆxˆVI(⃗x)2 ⃗IH · ˆx= 2gnF ′(0) ⃗SℓH · ⃗IH + r2 −19 [F ′(0)]3 + 518F ′′′(0)⃗SℓH · ⃗IH+16 [F ′(0)]3 + 13F ′′′(0) ⃗SℓH · ⃗x ⃗IH · ⃗x −13r2⃗SℓH · ⃗IHo,(A.6)13

where we have used the relations⃗IH · ˆx2= 14,⃗IH · ˆx ⃗SℓH · ⃗IH ⃗IH · ˆx= 12⃗SℓH · ˆx ⃗IH · ˆx−14⃗SℓH · ⃗IH,(A.7)that follow from the anticommutation relationIjHIkH + IkHIjH = 12δjk,(A.8)for the isospin-1/2 operators IH. It is convenient to define the operator ⃗KH = ⃗IH + ⃗SℓH,in terms of which Eq.

(A.6) can be rewritten asˆV ′I(⃗x) = gnF ′(0)⃗K 2H −32+ r2 −19 [F ′(0)]3 + 518F ′′′(0) ⃗K 2H −32+16 [F ′(0)]3 + 13F ′′′(0) ⃗KH · ⃗x ⃗KH · ⃗x −13r2 ⃗K 2Ho. (A.9)The allowed values of KH for the H field are KH = 0 and KH = 1 obtained by combiningIH = 1/2 with SℓH = 1/2.

The states with KH = 0 are the bound physical baryon statescontaining a heavy quark. On the KH = 0 states, Eq.

(A.9) reduces to the potentialˆV KH=0I(⃗x) = −32gnF ′(0) + r2 −19 [F ′(0)]3 + 518F ′′′(0)o= −32gF ′(0) + 12κr2,(A.10)where κ is defined in Eq. (2.19).

Thus the states with KH = 0 are bound in an attractiveharmonic oscillator potential, and the potential at the origin has the value −3gF ′(0)/2.The potential is more complicated for exotic states which have KH = 1. The last term inEq.

(A.9) is an irreducible tensor operator with KH = 2 and contributes a spin-orbit termto the interaction potential (but note that it does not cause any mixing between KH = 0and KH = 1 states). The spherically averaged potential for KH = 1 states is simple tocompute,ˆV KH=1I(⃗x) = 12gnF ′(0) + r2 −19 [F ′(0)]3 + 518F ′′′(0)o= 12gF ′(0) −16κr2.

(A.11)Thus the KH = 1 states are unbound since the potential at the origin is positive, and theinteraction potential is a repulsive inverted harmonic oscillator potential. The ΛQ state isobtained from the KH = 0 state by applying a projection operator [11], so the interactionpotential for the ΛQ state is Eq.

(A.10). This is precisely the potential given in Eq.

(2.18)of the text.14

The kinetic term in the ξ basis on the A = 1 soliton states has the formL = 12MN ˙⃗x2 −2r2 ˙⃗x ·⃗x × ⃗IH. (A.12)The first term is the usual soliton kinetic energy, and the second term is from the expansionofL = i2 Tr HHvµ ξ†∂µξ + ξ∂µξ†.

(A.13)The kinetic Hamiltonian in the ξ basis obtained from the Lagrangian Eq. (A.12) isHkin =12MN⃗pN + 2r2 ⃗x × ⃗IH2.

(A.14)Transforming from the singular basis using Eq. (A.5) gives the kinetic energyH′kin =2 ⃗IH · ˆxHkin2 ⃗IH · ˆx=⃗p 22MN.

(A.15)The bound state problem in the ξ basis reduces to that of a three-dimensional harmonicoscillator with a conventional kinetic term.15

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