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NONABELIAN BERRY PHASES IN BARYONS

arXiv:hep-ph/9301242v1 15 Jan 1993May 1, 2018NTG-92-20NONABELIAN BERRY PHASES IN BARYONSH.K. Leea∗, M.A.

Nowakb‡†, Mannque Rhoc and I. Zahedd‡a Department of Physics, Hanyang UniversitySeoul 133-791, Koreab Institute of Physics, Jagellonian UniversityPL-30059, Krakow, Polandc Service de Physique Th´eorique, C.E. SaclayF-91191 Gif-sur-Yvette, Franced Department of Physics, State University of New YorkStony Brook, New York 11794, USAABSTRACTWe show how generic nonabelian gauge fields can be induced in baryons when a hierarchy offast degrees of freedom is integrated out.

We identify them with nonabelian Berry potentialsand discuss their role in transmuting quantum numbers in bag and soliton models of baryons.The resulting baryonic spectra both for light and heavy quark systems are generic andresemble closely the excitation spectrum of diatomic molecules. The symmetry restorationin the system, i.e.

the electronic rotational invariance in diatomic molecules, the heavy-quark symmetry in heavy baryons etc. is interpreted in terms of the vanishing of nonabelianBerry potentials that otherwise govern the hyperfine splitting.∗Supported in part by the KOSEF under Grant No.91-08-00-04 and by Ministry of Education(BSRI-92-231) at Hanyang University and through the Center for Theoretical Physics, Seoul National University.†Supported in part by the KBN under Grant No PB 2675/2.‡Supported in part by the Department of Energy under Grant No.

DE-FG02-88ER40388 with the StateUniversity of New York at Stony Brook.1

1IntroductionWhenever a quantum system with a hierarchy of length scales is truncated, inducedgauge potentials are naturally generated reflecting on the degrees of freedom that are inte-grated out. A natural setting for discussing these issues has been discovered by Berry [1] insimple quantum systems responding to slowly varying external parameters.

He showed thatabelian magnetic monopoles naturally arise in the space of the slow variables due to degen-eracy points. This concept has been generalized by Wilczek and Zee [2] to the nonabeliancase.

They have shown that if a set of degenerate energy levels depends on adiabaticallyvarying external parameters, nonabelian gauge potentials are induced, affecting dynamicsin a nontrivial way. Such gauge potentials have attracted a lot of attention in recent yearsbecause of their fundamental character and growing importance in quantum systems.The concept of induced gauge fields has led to important understanding of subtleeffects ranging from condensed matter to elementary particle physics [3].

Whenever theunderlying dynamics can be separated into “slow” and “fast” degrees of freedom, inducedgauge fields are generally expected. They are generic and embody the essence of the ge-ometrical symmetries in a given problem.

One may therefore ask whether similar gaugestructures are encountered in models of strong interaction physics, and to what extent theybear on our understanding of hadronic physics.In a series of recent papers [4], we have shown that nonabelian Berry structures canand do appear naturally in topological chiral bags [5] that model spontaneously brokenchiral symmetry and confinement of QCD 1. The distinction between “fast” and “slow”degrees of freedom is somehow blurred in the topological bag model.

However, if we were toassume that the external pion field can be decomposed into a classical and quantum part,then a semiclassical delineation is possible in which the “slow” degrees of freedom refer tothe large component of the fields and “fast” degrees of freedom refer to the small componentof the field. In the semiclassical limit the bag is composed of a classical pion field that wrapsvalence quarks (bound fermions) and polarizes the Dirac sea.

In this limit neither isospinnor angular momentum are good quantum numbers. Corrections to this limit are down by¯h and correspond to quantum external pions and (multi) quark-antiquark excitations eachof which have good isospin and angular momentum assignment.In the semiclassical quantization, the classical bag is adiabatically rotated generatingstates of good spin and isospin.

Even when ignoring the quantum corrections the adiabatic1While the topological bag model has confinement and the bag boundary condition plays an essentialrole in [4], we suspect that confinement is not really necessary for generating gauge structure and that it isonly the symmetry that is relevant.2

quantization does not reduce simply to the quantization of a spinning particle in isospinand spin space as for ordinary classical fields. Indeed, the degeneracy of the Dirac spec-trum, following the symmetries of the classical field, implies that under any rotation (evenif infinitesimally small) the quarks in the valence orbitals and the Dirac sea mix inside de-generate bands and between crossing levels.

Adiabatic quark mixing is at the origin of theBerry phases in models of strong interactions. Below we will analyze their occurrence andphysical relevance in the context of the topological bag model.This paper is organized as follows.

In section 2, we discuss the general setting forBerry phases in Born-Oppenheimer approximation and we demonstrate explicitly the Berryphase for the case of diatomic molecule. We show how the integration of the electronic (fast)degrees of freedom leads to the extra gauge potential-like term in the effective Hamiltonianfor the nuclear (slow) degrees of freedom.

This term causes splittings of the energy levelsand changes quantum numbers of the system.In section 3, we show how these concepts extend to the topological bag model. Sub-section 3.1 includes the toy model displaying all the features of the topological bag model.Subsection 3.2 is devoted to the bag model itself.

The role of the fast variables is played bythe sea quarks inside the bag, whereas the adiabatic rotation of the solitonic cloud surround-ing the bag constitutes the slow motion. We detail the explicit Hamiltonian construction forexcited baryons in the light-quark sector and discuss in subsection 3.3 model-independentmass relations.

In section 4, we show how the analogous construction of the Berry phasescan be made in the context of soliton-heavy meson system. Integrating out heavy mesondegrees of freedom we end up with the usual Skyrme-like rotor term, however, submitted tothe influence of a non-trivial magnetic field (non-abelian Berry phase).

This monopole-likefield is responsible for the spin-isospin transmutation of the quantum numbers and for thestructure of the hyperfine splittings. The independent mass relations are identical to theones obtained in the framework of the Callan-Klebanov model of hyperon skyrmions [6].In section 5, we discuss what we believe happens to the skyrmion structure associatedwith induced gauge fields when the heavy meson becomes infinitely heavy at which therecently discovered heavy-quark symmetry [7] is operative.Our major conclusions andprospects are relegated to section 6.

In Appendix A, we give a heuristic reason based onan argument by Aharanov et al [8] why nonabelian Berry potentials cannot vanish in light-quark systems in contrast to diatomic molecules and heavy-quark baryons. In AppendixB, an argument is provided as to how heavy mesons decouple from the Wess-Zumino termresponsible for the binding of heavy mesons to the soliton.3

2Berry Phases and the Born-Oppenheimer ApproximationTo define the general setting for Berry phases and help understand their emergence inthe context of the models of elementary particles, we will first present, following [9, 10] thepedagogical example of the induced gauge fields (Berry phases) in the Born-Oppenheimerapproximation [11]. This approximation is usually described as a separation of slow (nu-clear) and fast (electronic) degrees of freedom.

This separation is motivated by the fact thatthe rotation of the nuclei does not cause the transitions between the electronic levels. Inother words, the splittings between the fast variables are much bigger than the splittings be-tween the slow ones.

We will demonstrate how the integration of the fast degrees of freedomleads to the induced vector potential of the Dirac monopole affecting the dynamics of theslow motion. To make our analysis more quantitative, we define the generic Hamiltonian.Generically, the Hamiltonian is given by :H =⃗P 22M + ⃗p22m + V (⃗R,⃗r)(1)where we have reserved the capitals for the slow variables and lower-case letters for thefast variables.

We expect the electronic levels to be stationary under the adiabatic (slow)rotation of the nuclei. We split therefore the Hamiltonian into the fast and slow part,H=⃗P 22M + hh(⃗R)=⃗p22m + V (⃗r, ⃗R)(2)where the fast Hamiltonian h depends parametrically on the slow variable ⃗R.

The snapshotHamiltonian (for fixed ⃗R) leads to the Schr¨odinger equation:hφn(⃗r, ⃗R) = ǫn(⃗R)φn(⃗r, ⃗R) . (3)The wave function for the whole system isΨ(⃗r, ⃗R) =XnΦn(⃗R)φn(⃗r, ⃗R) .

(4)Substituting the wave function into the full Hamiltonian and using the equation for the fastvariables we getXn" ⃗P 22M + ǫn(⃗R)#Φn(⃗R)φn(⃗r, ⃗R) = EXnΦn(⃗R)φn(⃗r, ⃗R)(5)4

where E is the energy of the whole system. Note that the operator of the kinetic energyof the slow variables acts on both slow and fast part of the wavefunction.We can nowintegrate over the fast degrees of freedom.

A simple algebra leads to the following effectiveSchr¨odinger equationXmHeffnm Φm = EΦn(6)where the explicit form of the matrix-valued Hamiltonian (with respect to the fast eigen-vectors) isHeffmn =12MXk⃗Πnk⃗Πkm + ǫnδnm(7)where⃗Πnm = δnm ⃗P −i < φn(⃗r, ⃗R)|⃗∇r|φm(⃗r, ⃗R) >≡δmn ⃗P −⃗Anm . (8)The above equation is exact.

We see that the fast variables act like a gauge field. Thevector part couples minimally to the momenta, and the fast eigenvalue acts like the scalarpotential.In the adiabatic approximation one may neglect the off-diagonal transition terms inthe induced gauge potentials, which leads to the simpler HamiltonianHeffn=12M (⃗P −⃗An)2 + ǫN(9)where we denote the diagonal component of the Berry phase (or more precisely Berrypotential) Ann by An.

If the electronic eigenvalues are degenerate, i.e., to the particulareigenvalue ǫn correspond Gn eigenvectors, instead of one Berry phase we obtain the wholeset of the Gn × Gn Berry phases, forming the matrixAk,k′n= i < n, k|∇|n, k′ >k, k′ = 1, 2, ...Gn . (10)The gauge field so generated is in this case non-abelian and corresponds to the gauge groupU(Gn).

In practical calculations, one truncates the infinite sum in (4) to a few finite terms.Usually the sum is taken over the degenerate subspace corresponding to the particulareigenvalue ǫn. This is so-called Born-Huang approximation, which we will use throughoutthis paper.5

Let us finally note that the above formalism may be rewritten in the Lagrangianlanguage. The corresponding effective Lagrangian is then equal to2Leffnm = 12M ˙⃗R(t)2δmn + i ⃗Amn[⃗R(t)] · ˙⃗R(t) −ǫmδmn.

(11)Let us see how this scenario works for the case of the simple diatomic molecule. Thefast variable describes the motion of the electron around the internuclear axis.

The slowvariables are the vibrations and rotations of the internuclear axis. This case correspondsto the situation when the energy of the spin-axis interaction is large compared with theenergy splittings between the rotational levels.

This case is usually called “Hund case a.”We follow the standard textbook notation of Ref.[14]. Let ⃗N be the unit vector along theinternuclear axis.

We can define then the following quantum numbersΛ=eigenvalue of ⃗N · ⃗LΣ=eigenvalue of ⃗N · ⃗SΩ=eigenvalue of ⃗N · ⃗J = |Λ + Σ|(12)so Λ, Σ, Ωare the projections of the orbital momentum, spin and total angular momentumof the electron on the molecular axis, respectively.Let us analyze the simple case of Σ = 0, Λ = Ω= 1. The fast eigenstates are| ± Ω, θ, φ >S= e−iφJ3e−iθJ2e+iφJ3| ± Ω, 0, 0 >(13)where the index S denotes the parametrization singular on the south pole (θ = π).

Alter-natively, we may use the parametrization| ± Ω, θ, φ >N= e−iφJ3e−iθJ2e−iφJ3| ± Ω, 0, 0 >(14)which is singular on the north pole (θ = 0). The Berry connection is, in our case, a 2 × 2matrix with the following structureAΩΩ′S=iS⟨±Ω′, θ, φ|d| ± Ω, θ, φ⟩S=iS⟨±Ω′, θ, φ| ∂∂θ| ± Ω, θ, φ⟩Sdθ+iS⟨±Ω′, θ, φ| ∂∂φ| ± Ω, θ, φ⟩Sdφ .

(15)2One can avoid the matrix-valued Lagrangian by using Grassmannian variables [12]. This formalism wasused in [13] for describing the same molecular system.6

We can use the orthonormal basis ⃗AS = ar⃗er + aθ⃗eθ + aφ⃗eφ. A simple calculation showsthat only the φ component is different from zero, and has the quasi-abelian formaΩ,Ω′φ= −Ω1 −cos θsin θ(σ3)Ω,Ω′(16)where σ3 denotes the third Pauli matrix, and their components are numbered by Ω, Ω′ = ±1.An identical calculation based on the parametrization (14) leads to the expressionaΩ,Ω′φ= +Ω1 + cos θsin θ(σ3)Ω,Ω′.

(17)We can now calculate the curvature of the Berry connection, i.e. F = dA + A ∧A.

In oursimple case the field tensor is quasi-abelian. We can use any of the gauge fields to calculatethe field tensor.

The answer is given by⃗B = rot ⃗A = −Ω⃗NR3σ3 . (18)This is nothing else but the magnetic field of the Dirac monopole with the charge eg =−Ω.

We know that the monopole leads to the observable effects. The kinematical angularmomentum operator gets modified due to the angular momentum stored in the field ofthe monopole.

A short calculation allows us to extract from the canonical form (9) therotational part of the spectrum. The effective Hamiltonian readsHeff =12MR2 ( ⃗J2 −Ω2) + · · ·(19)where angular momentum operator is given by⃗J = ⃗R × ⃗Π −12⃗RǫabcRaFbc = ⃗R × ⃗Π −Ω⃗N(20)and the ellipsis denotes the vibrational terms.Of course, the traditional calculation (as one sees in e.g.

[14] after correcting amisprint of the factor of 2 in eq. (83.7)) leads to the identical result, modulo some phe-nomenological assumptions about the possible spin structure.We presented the abovecalculation for two reasons.

Firstly, we believe that this example explains the basic featuresof the Berry phase, and while providing a new insight into the structure of the spectrum ofthe diatomic molecules, allows us to understand the modification of the rotator spectrumin terms of simple physical properties of the Dirac monopole. Secondly, in the followingchapters, we will basically exploit the same strategy to construct a tower of excited statesin the bag model and to make a model-independent analysis of the soliton-heavy meson7

bound systems. Since the generic structure of the obtained spectra for elementary parti-cles is basically similar to the diatomic mass formulae – modulo some generalizations dueto truly non-abelian character of the phase which we will shortly sketch– we will use theabove example as a guide in developing the framework for describing the technically morecomplicated systems one encounters in strong interaction physics.Before leaving this section, we briefly discuss how the above discussion can be gener-alized to a nonabelian situation.

The abelian monopole spectrum corresponds to a specialcase of diatomic molecule when one restricts the consideration to the degenerate Π doubletwith ±Ω, Ω= 1. For small internuclear distance R, the potential energy curve for thesinglet Σ with Ω= 0 lies higher than that for the Π for which the quasi-abelian approxima-tion is reliable.

However if R is sufficiently large, then the two potential energy curves cansubstantially overlap in which case one must treat the triplets (Π, Σ) together, as pointedout by Zygelman [10]. The resulting Berry potential is then truly nonabelian.

The resultingspectrum can then be written in a generic form as [13]Heff =12MR2 ⃗JR + (1 −κ) ⃗Jg2 −12MR2 (1 −κ)2(21)where ⃗JR is the rotor (“dumb-bell”) angular momentum ⃗R × ⃗Π and ⃗Jg the angular momen-tum stored in the nonabelian gauge field, none of which is conserved separately and theconstant κ defined byκ =< Π| 1√2(Lx −iLy)|Σ >(22)where ⃗L is the electronic orbital angular momentum, measures how much the rotationalsymmetry is restored, e.g., κ = 1 corresponding to the full restoration of the symmetry.The conserved angular momentum is ⃗J = ⃗JR + ⃗Jg which as shown first by Jackiw [11] isindependent of the charge [1 −κ]. The limit κ →0 (small R) corresponds to the quasi-abelian magnetic monopole spectrum (19) with Ω= 1.

In the limit R →∞, the singlet Σbecomes degenerate with the doublet Π and hence κ →1. Zygelman shows that in thatlimit1 −κ ∼C/R4(23)where C a constant.

In this limit, one can show that the field strength tensor vanishes (puregauge). Nonetheless as noted above, there is an angular momentum associated with theelectronic degrees of freedom which however decouples from the spectrum.

What happensis that the electronic rotational symmetry, broken for small R, is restored for large R sothat the electronic angular momentum becomes a good quantum number. This point willbe relevant when discussing the analogy with the heavy quark limit below.8

3Berry Phase in the Topological Chiral Bag3.1Toy ModelTo fully appreciate the generic structure of the Berry potentials that we will exhibit,it is useful to reformulate a well-studied case in a way suitable to our strong-interactionmodel. Consider a system of slowly rotating solenoid coupled to a fast spinning object (callit “electron”) described by the (Euclidean) action [15]SE =ZdtI2˙⃗n2 + ψ†(∂t −µˆn · ⃗σ)ψ(24)where na(t), a=1,2,3, is the rotator with ⃗n2 = 1, I its moment of inertia, ψ the spinningobject (“electron”) and µ a constant.

We will assume that µ is large so that we can make anadiabatic approximation in treating the slow-fast degrees of freedom. We wish to calculatethe partition functionZ =Z[d⃗n][dψ][dψ†]δ(⃗n2 −1)e−SE(25)by integrating out the fast degree of freedom ψ and ψ†.

This system in the space of therotating solenoid gives precisely the same abelian monopole spectrum (19) with Ω= 1/2.We will solve this problem first in the standard way used by Stone and then by the methodwe shall use. The procedure used by Stone goes as follows.

Imagine that ⃗n(t) rotates slowly.At each instant t = τ, we have an instantaneous Hamiltonian H(τ) which in our case is just−µ⃗σ · ˆn(τ) and the “snap-shot” electron state |ψ0(τ)⟩satisfyingH(τ)|ψ0(τ)⟩= ǫ(τ)|ψ0(τ)⟩. (26)In terms of these “snap-shot” wave functions, the solution of the time-dependent Schr¨odingerequationi∂t|ψ(t)⟩= H(t)|ψ(t)⟩(27)is|ψ(t)⟩= eiγ(t)−i R t0 ǫ(t′)dt′|ψ0(t)⟩.

(28)Note that this has, in addition to the usual dynamical phase involving the energy ǫ(t), anontrivial phase γ(t) – known as Berry phase – which substituted into (27) is seen to satisfyidγdt + ⟨ψ0| ddtψ0⟩= 0. (29)9

This allows us to do the fermion path integrals to the leading order in adiabaticity and toobtain (dropping the trivial dynamical phase involving ǫ)Z = constZ[d⃗n]δ(⃗n2 −1)e−Seff ,(30)Seff(⃗n) =Z Leff =Z[I2˙⃗n2 −i ⃗A(⃗n) · ˙⃗n]dt(31)wherei ⃗A(⃗n) = −⟨ψ0(⃗n)| ∂∂⃗nψ0(⃗n)⟩(32)in terms of which γ isγ =Z⃗A · d⃗n. (33)A so defined is the Berry potential or connection and γ is the Berry phase.

A is a gaugefield with coordinates defined by ⃗n.We can obtain the same result by defining S(τ) in (24) asˆn(τ) · ⃗σ = S(τ)σzS†(τ). (34)We now rotate the electron field asψ →Sψ.

(35)Then Eq. (24) can be writtenSE = SS=1 +Zdtψ†S†i∂tSψ.

(36)When the electron field is integrated out, the second term of this action gives rise to theBerry potential term of (31) given in terms of the matrix element taken with the basis thatdiagonalizes the fermion term in SS=1. If we call this basis |σz > (i.e., eigenstate of σz),theni ⃗A(⃗n) = −⟨σz|S† ∂∂⃗nS|σz⟩.

(37)This is the procedure that we will use for the more complicated case of the topological chiralbag.10

3.2Bag ModelThe simple description outlined above carries through in spirit to a system of quarksconfined inside a cavity wrapped by a strong pion field. In the bag, the monopole fieldis substituted by an induced instanton-like field in isospin space (the slow variable space)and the heavy fermion is played by valence quarks.

In what follows, we will use the actionformulation.Inside the bag, the quarks are described by free QCD and confined by fiat at the bagsurface. To prevent explicit chiral symmetry breaking a pion field surrounds the bag.

In thetopological bag model the pion field has the structure of the Skyrme hedgehog ansatz. Thelatter is invariant under a grand-spin rotation (angular momentum plus isospin), ⃗K = ⃗J +⃗I.As a result, the quarks inside the cavity are polarized in a level structure that dependsexplicitly on the strength of the pion field at the bag surface (denoted F and referred to as“chiral angle”).

The level degeneracy is 2K + 1. Thorough discussions on the topologicalbag model can be found in Ref.

[5, 16].Suppose that we adiabatically rotate the bag in space. Because of the degeneracyof the Dirac spectrum, mixing between quark levels is expected no matter how small therotation is.

This mixing takes place in each quark K-band and leads to a nonabelian Berryor gauge field.Indeed, an adiabatically rotating bag can be described by the followingactionSS =ZVψiγµ∂µψ −12Z∆s ψ S eiγ5⃗τ·ˆrF (r) S† ψ + SM(SU0S†)(38)where V is the bag volume (which we shall suppress below unless ambiguity arises), F is thechiral angle appearing in U0 = ei⃗τ·ˆrF (r), ∆s is a surface delta function and the space rotationhas been traded in for an isospin rotation (S(t)) due to the hedgehog symmetry consideredhere. The purely mesonic terms outside the bag are described by SM.

Presently we shalldiscuss the effect of rotations on quarks only, relegating the discussion of the mesonic cloudto the second part of this chapter. The massless quarks inside the bag are assumed to befree.

The rotation at the boundary can be unwound by the redefinition ψ →Sψ, leading toSS = SS=1 +Zψ† S†i∂tS ψ . (39)The effect of the rotation on the fermions inside the bag is the same as a time dependentgauge potential.

This is the origin of the induced Berry potential analogous to the solenoid-electron system , Eq. (36).11

To understand the physics behind this term, we expand the fermionic fields in thecomplete set of states ψKM with energies ǫK in the unrotating bag corresponding to theaction SS=1 in (39), and M labels 2K + 1 projections of the grand spin K. Generically,ψ(t, x) =XK,McKM(t)ψKM(x)(40)where the c’s are Grassmannians, so thatSS =XKMZdt c†KM(i∂t −ǫK)cKM +XKMK′NZdt c†KMAKK′MN cK′N(41)whereAKK′MN =ZVd3x ψ†KMS†i∂tSψK′M . (42)No approximation has been made up to this point.

If the A of (42) were defined in thewhole K space, then A takes the form of a pure gauge and the field strength tensor wouldbe identically zero 3. However we are forced to truncate the space.

As in the precedingchapter, we can use now the adiabatic approximation and neglect the off-diagonal termsin K, i.e., ignore the effect of adiabatic rotations, which can cause the jumps betweenthe energy levels of the fast quarks. Still, for every K ̸= 0 the adiabatic rotation mixes2K + 1 degenerate levels corresponding to the particular fast eigenenergy ǫK.

In this formwe clearly see that the rotation induces a hierarchy of Berry potentials in each K-band, onthe generic form identical to Eq.(10). This field is truly a gauge field.

Indeed, any localrotation of the ψKM →DKMNψKN where DK is a 2K + 1 dimensional matrix spanning therepresentation of rotation in the K-space, can be compensated by a gauge transformationof the Berry potentialAK →DK(∂t + AK)DK†(43)leaving SS invariant [17].The structure of the Berry potential depends on the choice of the parametrization ofthe isorotation S (gauge freedom). For the parametrization S = a4 + i⃗a ·⃗τ with the unitary3As we saw in the case of the diatomic molecule, the vanishing of the field tensor does not imply thatthere is no effect.It describes the restoration of certain symmetry.See later a similar phenomenon inheavy-quark baryons.12

constraint a · a = 1 (unitary gauge), we haveAK = T aK AaK = T aKgKηaµν aµdaν1 + a2(44)where η is the t’Hooft symbol and gK the induced coupling to be specified below. TheT’s refer to the K-representation of SU(2), the group of isorotations.

In the unitary gaugethe Berry potential has the algebraic structure of a unit size instanton in isospace, i.e., thespace of the slow variables. It is not the Yang-Mills instanton, however, since the aboveconfiguration is not self-dual due to the unitarity gauge constraint.

This configuration is anon-abelian generalization of the monopole-like solution present in the diatomic molecularcase.To make our analogy more quantitative, let us refer to the Grassmannians c in thevalence states by α’s and those in the Dirac sea by β’s. Clearly (41) can be trivially rewrittenin the formSS=XKMNZdt α†KMh(i∂t −ǫK)1MN −(AK)MNiαKN+XKMNZdt β†KMh(i∂t −ǫK)1MN −(AK)MNiβKN .

(45)Integrating over the Dirac sea in the presence of valence quarks yields the effective actionSS =XKMNZdt α†KMh(i∂t −ǫK)1MN −(AK)MNiαKN+iTr ln(i∂t −ǫK)1MN −(AK)MN(46)where the Trace is over the Dirac sea states. The latter can be Taylor expanded in theisospin velocities ˙aµ in the adiabatic limit,iTr ln ((i∂t −ǫk)1MN −(AKµ )MN ˙aµ) =Zdt Iq2 ˙aµ ˙aµ + · · ·(47)We have exposed the velocity dependence by rewriting the form AKMN = (AKµ )MN ˙aµ.

Linearterms in the velocity are absent since the Berry phases in the sea cancel pairwise in theSU(2) isospin case under consideration. For SU(3) they do not and are at the origin of theWess-Zumino term.

The ellipsis in (47) refers to higher derivative terms. Iq is the momentof inertia of the bag.

We do not need the explicit form of this term for our considerations.We would like to point out that this term includes implicitly the valence quark effect,because the levels of the Dirac sea are modified due to the presence of the valence quarks.13

To see the general motivation for studying the excited states via Berry phases let usconsider the case of the bag containing one valence quark in the K = 1 state. The actionfor the adiabatic motion of this quark is obtained from the above formulae and yieldsSS =Zdt [iα†1M ˙α1M −ǫ1α†1Mα1M + 12Iq ˙aµ ˙aµ + ˙aµ(A1µ)MNα†1Mα1N] .

(48)As we will see below, when canonically quantized, the generic structure of the resultingHamiltonian is identical to (19) and shows that the excited quark system in the slow variablespace behaves as a spinning charged particle coupled to an instanton-like gauge field centeredin an S3 sphere in the four dimensional isospin space.This once again illustrates theuniversal character of the Berry phases.Let us now quantize the system. Since S3 is isomorphic to the group manifold ofSU(2), it is convenient to use the left or right Maurer-Cartan forms as a basis for thevielbeins (one-form notation understood)S†idS = −ωaτa = −vca(θ)dθcτ a(49)where we expressed the “velocity” forms ω in the basis of the vielbeins vca, and θ denotessome arbitrary parametrization of the SU(2), e.g.

Euler angles. In terms of the vielbeins,the induced gauge potential simplifies toAc = −gK vca(θ)T a(50)where T are the generators of the Berry potential in the K representation and gK is thecorresponding charge [4]gK = 1K11 + y−1K + 1y1 + y(51)wherey = j2K+1 + j2K −2(K + 1)jK+1jK/xj2K−1 + j2K −2KjK−1jK/x·jK(1 + sin F2K+1) −jK−1 cos FjK(1 −sin F2K+1) + jK+1 cos Fand jK are the spherical Bessel functions calculated at x = ωR – the lowest energy solutionfor fixed K and parity P = (−1)K+1 in a spherical bag.

A qualitative behavior of the Diracspectrum and the induced charge versus F are shown in Figs. 1 and 2.

(Note that we couldhave equally well used the right-invariant Maurer-Cartan form instead of the left-invariantMaurer-Cartan form (49)).The field strength can be written in terms of A defined ineq. (50)FK = dAK −iAK ∧AK = −gK(1 −gK/2)ǫmijT mK vi ∧vj.

(52)14

FK vanishes for gK = 0 (trivial case) and for gK = 2, i.e. the Berry potential becomes apure gauge.The vielbeins – and hence A and F – are frame-dependent, but to quantize thesystem, no specific choice of framing is needed.

The canonical momenta are pa = ∂L/∂˙θa.Our system lives on S3 and is invariant under SO(4) ∼SU(2) × SU(2). Right and leftgenerators are defined asRa=uca pcLa=Dab(S) Rb(53)where uai vic = δac and D(S) spans the adjoint representation of the SU(2).

Following theprocedure described in [4], we get our Hamiltonian in terms of the generators4H∗= ǫK 1 + 18I (Rj −gK TKj) (Rj −gK TKj) . (54)This resembles closely the nonabelian molecular Hamiltonian (21).

In fact, it is identical toit with a suitable reinterpretation of the charge gK to be explained below. As a result, theHamiltonian for a singly excited quark5 takes the simple formH∗= ǫK1 + 18I⃗R2 −2gK ⃗R · ⃗TK + g2K ⃗T 2K.

(55)The spectrum can be readily constructed if we notice that (55) can be rewritten solely interms of the independent CasimirsH∗= ǫK1 + 12I+gK2⃗JK2 + (1 −gK2 )⃗I2 −gK2 (1 −gK2 ) ⃗TK2(56)where ⃗JK = −⃗R/2 + ⃗TK and ⃗I = ⃗L/2 are the angular momentum and isospin respectively.The identification of the quantum numbers follows from the original symmetries ofthe action. Indeed, under an isospin transformationS →e−iT·αSψ →ψ(57)4Canonical quantization for this system goes much like that of eq.

(11) except that here we carry alongGrassmanians which play an inert role of specifying the quark states involved, ı.e., equivalent to projectionoperators. On the other hand, one can also get eq.

(54) following the quantization procedure described in[12] for a system with Grassmanian variables.5If we were to add a second quark to this band (doubly excited state) then we could no longer have anirreducible representation of TK but a reducible representation instead.15

following the redefinition ψ →S†ψ (isospin co-moving frame). The isospin operator is givenby the standard Noether constructionIa = Dab(S)Iωb +Zd3xψ†T bψ= Dab(S)Iωb + gKT aK2.

(58)The second term in (58) is the induced Berry phase. The term in bracket is the momentumcanonically conjugate to the velocity ωa, referred to as pa above in the canonical frame.Under a rotation,S →S e−iT·βψ →eiT·βe−iJ·βψ.

(59)Again, the angular momentum is given by the conventional Noether constructionJa = −Iωa +Zd3xψ†La + σa2ψ = −Iωa + gKT aK2+Zd3xψ†Kaψ. (60)Since the states are eigenstates of K, the last term in (60) is just the representation of theSU(2) algebra spanned by K,Ja = −Iωa + gKT aK2+ T aK.

(61)The angular momentum gets an extra contribution due to the induced non-abelian Berryphase. This is the reason why we are able to avoid the Skyrme constraint I = J - isospinhidden in the K structure of the rotated degenerate levels gets transmuted into an extracomponent of the angular momentum.For gK = 0, we have the rotor spectrum H∗= ⃗I2/2I.

This happens for any valueof the chiral angle only for the K = 0 level and corresponds to the known case of thenucleon and delta. For K > 0, gK vanishes for some specific values of the chiral angle (seeFig.2), most probably connected with the additional level crossings in the spectrum (seeFig.1).

But these may be artifacts and may not be physically meaningful. For gK= 2,the Berry field strength vanishes but the gauge field nontrivially affects the spectrum, i.e.,H∗∼⃗J2K/2I.

The spectrum may look analogous to the quasi-abelian case of the diatomicmolecule but because of the vanishing field strength, the analogy is not significant. In oursystem, however, this situation is never reached as the charge gK in Eq.

(51) cannot reach2. The reason is that there is no limit in which the adiabatic rotation would be induced by16

the K-spin, and not by the isospin only. Indeed, if that was the case, we see immediatelyfrom (61) that the angular momentum of the system would reduce to the inertial part I⃗ωcarried solely by the hedgehog core.

More discussions on this difference will be given inAppendix A.There is an amusing analogy with the diatomic molecule above and the heavy quarksystem below, if we were to consider the fictitious situation of two quarks in the (1−, 2−),(2+, 3+), etc. states.

These multiplets, correspond respectively to a core with angular mo-mentum 32−, 52+, etc. coupled to isospin 1/2.

They are the equivalent of the heavy quarkmultiplets to be discussed below. As the bag radius is increased (MIT limit) angular mo-mentum becomes a good quantum number.Thus the isospin triplet and singlet statesbecome degenerate.

In this limit, the Berry phase stemming from the singlet exactly bal-ances the Berry phase from the triplet at the MIT point (F = 0) since g−2 = −g−1 = 1/2,g+3 = −g+2 = 1/3, etc.. This cancellation does not occur in the lowest multiplet (0+, 1+)with an angular momentum core 12+.

The reason is that the Berry phase vanishes identi-cally in the K = 0 state for all values of the pion field F. Since they interpolate betweenpositive and negative energy levels, these states are not allowed to carry a Berry phase.To summarize: We see that the role of the induced gauge potential is to lift thedegeneracy between angular momentum and isospin, and leads naturally to the descriptionof excited states. The Hamiltonian (56) allows a simple description of the even/odd parityexcitations of the nucleon and ∆, in terms of the original splittings in the topological bagmodel and the induced Berry charge gK.

For that we have to add two quarks in the inactiveband K = 0 each with energy ǫ0 and recall that the parity assignment follows from the parityof the excited quark in the active band K = 1, which is assumed to describe the low-lyingexcited states.3.3Light-quark spectrumA number of relations among the low-lying excited states of baryons follow from (56).Here we will only quote some model-independent results 6, obtained by elimination of boththe Berry charge g1 and the moment of inertia I. For instance, in the Roper channel, itfollows from (56) thatM(P11) −M(N) = M(P33) −M(∆).

(62)Empirically, the left-hand side is 502 MeV and the right-hand side is 688 MeV. In the6In deriving these formulae we have assumed that the pion cloud outside the bag is not substantiallydistorted by the excitation of a single quark inside the bag.17

odd-parity channelM(D13) −M(D35) + M(∆) −M(N) = −14(M(D35) −M(S31)). (63)From the data, we get 116 MeV for the left-hand side and 76 MeV for the right-hand side.AlsoM(S31) −M(S11) = 52(M(∆) −M(N)) −32(M(D35) −M(D13)).

(64)Empirically, the left-hand side gives 85 MeV and the right-hand side gives 125 MeV.We recall that the above formulae were obtained for the quark sector only (the inte-rior of the bag), i.e., till now we were ignoring the pionic cloud outside the bag, describedby SM in (38). We expect that the detailed analysis of the pionic sector should give thesame structure of the mass formula.

The argument is as follows. Description of the reso-nances in the Skyrme model is obtained by studying phase shifts of the pionic fluctuationsin the background of the static soliton.

The adiabatic rotation (cranking) of the solitoncorresponds to slow variables. The pionic fluctuations are fast and are equivalent to the“particle-hole” vibrations in the quark bag.

Again, the generic Born-Oppenheimer scenariotells us that the evolution of the Skyrme cloud outside the bag will be influenced by thepresence of the magnetic force coming from the integrated-out pionic fluctuations. Thecounterpart of the charge gK and moment of inertia will of course depend on the version ofthe Skyrme Lagrangian used, but the generic formula should be identical.

If we neglect theanharmonicities coming from the vibration and higher order terms (O(1/N 2C)) coming fromthe collective rotations we are at the same level of accuracy in the 1/Nc expansion on bothsides of the bag wall, i.e., in the quark sector inside as well as in the pionic sector outsidethe bag. The analysis done recently in [18] for the pure skyrmion case supports this pointof view.Pure skyrmion may be viewed as the limiting case of the shrinking bag.

The formulaepresented in [18] for the S-wave pion-nucleon scattering have the same generic form as ourmass formula. We would like to stress that the proper inclusion of the rotational effects iscrucial for the solution of the long standing problem in the Skyrme like models of the S11and S31 degeneracy.

Explicit calculations in [18] (although not relating explicitly to Berryphases) and our formula (64) confirm the role of the Berry phase for splitting the degeneracybetween these two levels. It was noted recently by Masak et al [19] that incorporation ofthe vector mesons ρ and ω in a way consistent with hidden gauge symmetry of chiralLagrangians [20], in particular in the intrinsic-parity odd sector, improves markedly thephase shifts for S11 and S31.

The corresponding processes inside the bag would require18

additional structure than what we have been considering and will bring modification to thespectrum, particularly to (64).Finally, let us speculate how bag-radius independent the above formulae are. In otherwords, does the Cheshire Cat Principle [21] (“physics is independent of the bag radius”)holds for the excited states?

The structure of the energy levels in the bag as a function of theskyrmion profile is very complicated. When changing the bag radius, several level-crossingsare expected to generate additional contributions to the induced potential.

It can be shown(see Appendix A) using the reasoning of [8] that modulo a phase the same field strengthtensor can be obtained either from the Berry potentials constructed within one K-subspace– as in our case – or from the off-diagonal potentials, connecting different K-subspaces.Therefore, in principle, for a large bag the spacing between the energy levels becomesincreasingly small, so that some off-diagonal contributions from e.g. the K = 0+, 2+ levelscrossing could play an important role.

The point we wish to make is that the universalcharacter of the Berry phases leaves some hope that if all the contributions to the gaugepotentials are taken into account on both sides of the bag to the same order of Nc expansion,one might expect to obtain an approximate Cheshire Cat picture for the excited states atthe level of the accuracy of the 1/Nc expansion.4Berry Phase in Strange SolitonsAnother interesting application of the above concept is to a system composed ofa soliton and a strange meson. Strange quarks play a very distinctive role in the stronginteraction, being neither heavy nor light compared with the typical scale of QCD.

A simplebut subtle example of the interplay of strange-light degrees of freedom is provided by theCallan-Klebanov description [6] of strange baryons. In this version of the Skyrme modelone assumes ab initio that SU(3) flavor symmetry is so badly broken by the massive kaons,that the usual perturbation theory applied to the mass term in the Hamiltonian is nolonger justified.

Kaons are therefore described as the chiral excitations in the backgroundof the non-strange, SU(2) topological soliton. The hyperons are then described as molecule-like states composed of the kaon bound to the soliton.

The identification of the quantumnumbers is provided by the usual collective rotations of the soliton. Adiabatic rotation ofthe soliton corresponds to the slow variables, and the kaonic excitations correspond to thefast ones.

We therefore could expect a Berry phase, which may influence the dynamics ina non-trivial way.Here we will describe a simplified model for a system composed of a heavy meson19

coupled to a soliton, with an overall isospin invariance. In the adiabatic limit, the systemmay be schematically described bySA =Zdt −MH −I4 Tr(S† ˙S)2 +Zd3xK†(t, ⃗x)"i∂t +∇22MK−SV (⃗x)S†#K(t, ⃗x)!

(65)where I is the moment of inertia of the meson-soliton bound state, MK is the meson massand V is the soliton induced potential, all of which are model-dependent. Their detailedstructure will not be necessary for our discussion.

We will only mention that the potentialdistinguishes between the kaons and anti-kaons in the solitonic background. This is due tothe Wess-Zumino term, which acts as a magnetic like force attracting kaons to the solitonand repulsing anti-kaons, providing in this way a mechanism for eliminating spurious stateswith B = 1, S = 1 from the spectrum.

The Wess-Zumino term itself can be traced backas an abelian Berry phase coming from the Dirac sea of the fermionic description of theoriginal system, but here we would like to concentrate on the Berry phase coming from the“heavy” collective quark-antiquark state (i.e. meson) as described above.Again, the rotating meson background in (65) can be unwound through K →S(t)Kinducing a Berry type termZdtK†(S†i∂tS)K .Using the decompositionK(t, ⃗x) =Xnan(t)Kn(⃗x)(66)in the unrotated basis, we can rewrite (65) in the formSA =Zdt −MH −I4 Tr(S† ˙S)2 +Xmnam†(i∂t −ǫm)1mn +Zdx Km†(S†i∂tS)Knan!.

(67)The latter form is totally identical to (46) with (47) except that the a’s now are c-numbersrather than Grassmannians. The role of the Berry potential is to induce hyperfine splittingin the rotor spectrum.

If we denote the eigenenergy of the kaon (or more generally, theheavy pseudoscalar meson P = K, D as we will discuss later) as ǫ, then the skyrmionwith a bound heavy P has the fine-structure and hyperfine-structure splitting given by theHamiltonianH = ǫ + 12I ⃗JR + c⃗T2 = ǫ + 12I ⃗J + (c −1)⃗T2(68)where ⃗JR is the angular momentum of the rotor (related to ⃗R/2 of eq. (55)), ⃗T the isospincarried by the meson (or vibration) and ⃗J = ⃗JR + ⃗T is the total angular momentum of the20

bound state. I is the moment of inertia of the rotor and c is a constant analogous to thecharge (1 −κ) in diatomic molecules or to the charge gK/2 of the light-quark in the chiralbag.

We may immediately write the model-independent formula for this Lagrangian. It isequivalent to our formulae (62-64) and reads13 (2M(Σ∗) + M(Σ)) −M(Λ) = 23 (M(∆) −M(N)) .

(69)Experimentally, the left hand side is 304 MeV and the right hand side is 293 Mev. Originally,this formula was obtained by [6] without reference to Berry phases.5Berry Phase in Heavy SolitonsSuppose that the strange quark mass becomes so large that it can no longer beconsidered as a chiral quark.

The question is: As the s-quark mass increases, say, beyondthe chiral symmetry breaking scale, does the concept of skyrmion with its induced gaugestructure still hold? This is a relevant question since it appears now that the skyrmionpicture holds even when the heavy quark becomes infinitely massive [22, 23, 24].Thecorrect description, however, requires starting ab initio with a Lagrangian that satisfiesboth the chiral symmetry of the light quarks and the Isgur-Wise (IW) symmetry [7, 25, 26]of the heavy quarks.

The heavy-quark symmetry implies that the pseudoscalar meson Pwhich plays a key role in the Callan-Klebanov model and the corresponding vector mesonP ∗of the quark configuration Q¯q (where Q denotes heavy quark and q = u, d light quark)become degenerate.Our starting point is the effective action for heavy-light mesons in the infinite quarkmass limit. If we denote byH = 1 + γ02(−γiP ∗i + iγ5P)andH = γ0H+γ0(70)the (0−, 1−) degenerate doublet in the rest frame of the heavy quark, then to leading orderin the derivative expansion the effective action follows from [26, 24]LH = −iTr(∂tH ¯H) + TrHV 0 ¯H −gHTrHAiσi ¯H + mHTrH ¯H(71)Here the vector and axial currents are entirely pionic and readVµ =+ i2ξ∂µξ† + ξ†∂µξ,Aµ =+ i2ξ∂µξ† −ξ†∂µξ.

(72)21

The pion field ξ = exp(i⃗τ · ⃗nF(r)/2) is described by the usual Skyrme type action. Alter-native formulations involving light vector mesons are also possible in which case a term ofthe form∼Tr ¯HHvµBµ(73)where Bµ is the topological baryon current can be generated [24] and provide a bindingmechanism as discussed in [23].

In (71) the parameter mH is a mass of order m0Q. For otherconventions we refer to [24].The effective action following from (71) is invariant under local SU(2)V symmetry(h), in which V transforms as a gauge field, A transforms covariantly and H →Hh† andH →hH.

It is also invariant under heavy-quark symmetry SU(2)Q (S), H →SH andH →HS†. This symmetry mixes the vectors (1−) with the pseudoscalars (0−).

Under theinfinitesimal transformation,δ ⃗P ∗= ⃗αP + (⃗α × ⃗P ∗)δP = −⃗α · ⃗P ∗(74)In the soliton sector the pion field is in the usual hedgehog configuration. In this caseit is useful to organize the H field in K-partial waves.

GenericallyH(x, t) =XKMaKM(t)HKM(x)(75)where the a’s annihilate H particles with good K-spin where K = I + J ≡KL + SQ with Iand J the total isospin and angular momentum of the H-soliton system, KL the K spin ofthe light antiquark in H and SQ the spin of the heavy quark. In the original approach ofCallan and Klebanov [6], the Kπ = 12+ state in the kaon channel was found to bind to thesoliton.

Can this binding persist in the infinite mass limit?To answer this question, first let us recall the essential feature of the Callan-Klebanovscheme which we have argued above is closely connected to the gauge field hierarchiesinduced dynamically. In the Callan-Klebanov scenario the Wess-Zumino term plays a centralrole in lifting the degeneracy between the strangeness S = ±1 states and assigning thecorrect quantum numbers to the physical states.

The presence of the Wess-Zumino termcauses P-wave kaons to bind to the soliton to order N 0c . The bound state carries the goodgrand spin K = I + J (12+) and heavy-flavor quantum number.

However states with goodisospin (I) and angular momentum (J) emerge after “cranking” (or rotating) the kaon-soliton bound state as a whole.The origin of the Wess-Zumino term goes back to the underlying fermionic characterof all hadronic excitations. In Appendix B, we argue that as the mass of Q increases, the22

topological Wess-Zumino term decouples from the heavy sector and hence one of the prin-cipal (if not the principal) agents for the binding needed for a skyrmion with a heavy mesonis lost. The reason can be easily understood.

As the mass of the strange quark is increased(say, to that of the charm or bottom quark), the Wess-Zumino term truncates to the two-flavor sector which is identically zero. This is because the heavy mesons can no longer beviewed as angle excitations of the chiral order parameter in the QCD vacuum.

The rateat which the Wess-Zumino term disappears depends on detailed dynamics. Our qualitativearguments suggest that the rate is controlled by the ratio of the induced constituent massesσ.

For strange quarks this ratio is : σ/σS ∼0.47 while for bottom quarks this ratio is :σ/σB ∼0.32.What is the fate of the heavy-meson-skyrmion bound state when the Wess-Zuminoterm vanishes? Two mechanisms providing classical binding were proposed [24, 23].

Inthe first approach the binding depends on the form of background pionic potential, in thesecond it is strengthened by the vector-meson induced term (73). If the binding persistseven in the heavy quark limit then our previous discussion carries through nicely.

Indeedas the mass of the heavy quark is raised, the Berry phase receives contribution from boththe P and P ∗. GenericallyH = ǫ + 12I ⃗JR + c⃗T + c∗⃗T∗2(76)where ⃗T∗is the isospin contribution of P ∗to the induced Berry phase.

More explicitly, thisformula can be rewritten as [27]H = ǫ + 12I(⃗J −⃗SH) −(1 −CP )Tr(P⃗IP +) −(1 −C∗P )Tr(P ∗j ⃗IP ∗+j)2(77)where I, J are isospin and angular momentum operators, respectively, and SH is the totalspin of the H-particle. In the K = 12+ shell it reduces to ⃗σ/2, i.e.

spin 1/2 representation.This shows that the spin of the heavy quark and the light quark fractionate in the K-representation.The H particle bound to the skyrmion resembles a heavy fermion withspin 1/2. A similar transmutation occurs in the Callan-Klebanov construction [6].

Thisfermionization of the original bosonic degrees of freedom through the hedgehog structure iswhat makes skyrmions so remarkable. This result carries to higher K-shells.In general CP and C∗P are complicated functions of the heavy quark mass.

However,in the heavy quark limit CP = −C∗P = 1 and one recovers the rotor spectrum.Thiscancellation is guaranteed by two facts : heavy quark symmetry that implies the samestrength for CP and C∗P and the underlying hedgehog character of the skyrmion that forces23

the isospin in P ∗to be antiparallel to the spin, flipping the sign of CP compared to C∗P . Inthe infinitely heavy quark limit Hamiltonian for the K = 12+ shell takes the form [23, 27](to order m0QN −1c)H12 = J2R2I = (⃗J −⃗SH)22I= I22I .

(78)Thus to order m0QN −1cthe Σ and Σ∗are degenerate. Note that the situation is totallyanalogous to the non-abelian molecular case for R →∞discussed in Section 2.The above Hamiltonian implies the following mass relation in the heavy hyperonspectrum(M(Σ∗) −M(Λ)) = 23 (M(∆) −M(N))(79)If we were to ignore P ∗for any finite mQ then (77) reduces toH1 = 12Ω⃗J −(1 −CP)Tr(P⃗IP +)2 .

(80)The latter reduces to (the incorrect) H = J2/2Ωas opposed to (the correct) H = I2/2Ωinthe heavy quark limit.7It is interesting to ask which of the mass formulae, (69) or (79), works better for thecharm sector. The direct comparison is impossible at the moment, since the mass of the Σ∗is not yet measured.

The mass spectrum predicted according to the Callan-Klebanov scheme– and generalized for more than one heavy mesons – for the charm baryons is given in [29].There the Ξ’s and Ω’s are described by binding the K’s and D’s without interactions, thatis to say, in quasiparticle approximation. The prediction of [29] which does not manifestlyrespect the Isgur-Wise symmetry is nonetheless surprisingly close to that of quark models,suggesting that perhaps the mass of the charm quark is not large enough to see clearly theeffect of the Isgur-Wise symmetry at the level of mass formulae.

The effective hyperfinecoefficient c as defined in (68) comes out to be 0.62 for the strange hyperons and 0.14 forthe charmed hyperons. The latter is small, but certainly not near zero as would be the caseif the charm quark were massive enough to satisfy the Isgur-Wise symmetry.Let us finally note that an approach to the heavy solitons similar in spirit to what wasdiscussed above was suggested recently by Manohar and collaborators [22].

The differenceis that we have insisted on the mechanism of binding at the classical level (in [22] binding7 The heavy-meson limit of the Callan-Klebanov model with the Skyrme quartic term or with vectormesons as studied in [28, 29] do not go to H = J2/2Ωsince part of the P ∗contribution is included in thetreatment. It does not go to the correct heavy limit either.24

has quantum mechanical nature) and we rely on the concept of the Berry phases.Inour approach the P and P ∗are defined in the (isospin) co-moving frame making theirquantization simpler for the bound state problem since they do not carry good isospin(they carry good K-spin). The dressed P and P ∗used by Manohar and collaborators aredefined in the laboratory frame and their quantization is simpler for the scattering problemsince they carry good isospin (as asymptotic P and P ∗do).

The two descriptions are relatedby a global isospin rotation. Since both descriptions have built in heavy quark symmetry,they yield similar physical predictions.

Indeed it is not difficult to also formulate Manohar’sapproach in such a way that the Isgur-Wise symmetry is realized as the vanishing of theBerry potential defined in the laboratory frame [30].6ConclusionsThe topological bag model offers a suitable setting for discussing Berry phases. Theanalogy with the fermion-monopole system is striking.

In the bag, the strong pion fielddistorts the Dirac spectrum causing the emergence of Berry phases under any adiabaticrotation. While the Dirac sea produces no net Berry (Wess-Zumino) contribution due topairwise cancellations in the sea, the valence states do.

The net effect is similar to a spinningcharged particle coupled to an instanton-like gauge field in isospin space.The role of the Berry phase is to induce hyperfine splitting in the rotor spectrum.This effect can be used to describe excited baryons in the light-quark nonstrange sector.The model-independent relations discussed here are in fairly good agreement with the data.Given the simplicity of the description this is striking.We have argued that the features displayed in the context of the topological bagmodel are in fact generic. They can be easily extended to strange baryons as discussed byCallan and Klebanov and even to heavier systems as the ones discussed by Manohar andcollaborators [22] and also by Min and collaborators [23].

This is hardly a surprise giventhe generic character of Berry phases.Our work is certainly far from complete. We have not investigated systematicallythe relevance of the Cheshire Cat description for the excited states, nor have we exploredtotally the heavy light systems.

Moreover, we should be also able to address exotic issuesrelated to photoproduction mechanisms and dibaryon systems where excited quarks arenaturally triggered. We hope, however, that our initiative will spur more excitement inthese directions.25

AcknowledgmentsOne of us (MR) is grateful for useful discussions with Y. Aharonov, in particular for ex-plaining his approach of Ref.[8]. This work has been supported in part by a DOE grantDE-FG02-88ER40388 and by KBN grant PB 2675/2.26

Appendix A: The Nonvanishing of Nonabelian Berry Potentials in theChiral BagIn this Appendix, we wish to explain the difference in structure between the in-duced field in light-quark systems and the one in heavy-quark systems and also in diatomicmolecules. We noted in the main text that while the structure of Berry potentials and theirphysical effects are generic, the field tensor behaved differently.

To be specific, in heavybaryons and diatomic molecules, both the Berry potential and its field strength vanished incertain limit while they did not in the chiral bag modeling the light-quark baryons 8.We first note that when the condition of adiabaticity is satisfied, there are two waysof describing Berry potentials. One is the standard way used by Berry [1] which is to definethe potential within a diagonal subspace (denoted A) and the other proposed by Aharanovet al.

[8] is to define it in off-diagonal subspaces (denoted ˜A). To define these quantitiesprecisely, we use the Hamiltonian formalism of Ref.[31].

Let the fast variable Hamiltonianparametrized by aµ at a given time t be written in the formH(aµ) =XKǫK(aµ)ΠK(aµ)(A.1)where ΠK(aµ) is the projection operator onto the subspace (labeled by the index K) spannedby the ‘snap-shot’ energy eigenstate of ǫK(aµ),H(aµ)|K, aµ⟩= ǫK(aµ)|K, aµ⟩. (A.2)The quark action in eq.

(38) discussed in ref. [4] can clearly be quantized to take this genericform.

In adiabatic approximation the standard form of the Berry potential that is inheritedfrom the fast space can be written asA =XKΠKS†dSΠK(A.3)where the dependence of the projection operator on the coordinates aµ is suppressed and|K, aµ(t)⟩= S(t)|K, aµ(0)⟩. (A.4)Let us now define formally the off-diagonal field, ˜A, that connects different subspaces˜A =XK̸=K′ΠKS†dSΠK′.

(A.5)8It is possible to construct a chiral bag that includes heavy mesons for which case one should also havea vanishing Berry potential in heavy-meson limit. See [33] for a recent discussion on this.27

This is the gauge potential of Aharanov et al. [8] up to a unitary transformation.

Calculatingthe field strength with ˜A we getF ˜A =XK”ΠK” ˜A ∧˜AΠK”. (A.6)Thus as discussed in Ref.

[8, 32], although we have used the field ˜A that only mixes differentspaces, the field tensor is diagonal. Now let us calculate the field tensor with the diagonalfield A (A.3).

Using the properties of the projection operator, one can readily verify thatone obtains exactly the same expression as (A.6) except for a minus signFA = −F ˜A. (A.7)This is of course a direct consequence of the fact that F originates from the diagonal ofS†dS, and the latter is a pure gauge in the full Hilbert space.The way this relation might impact on our discussion is as follows.

In subsections(3.2) and (3.3) we focused on the K = 1+ band which is the first excited K band abovethe ground band K = 0+. However, the K = 1 band can be connected by the adiabaticrotation operator S to not only the K = 0 band but also to the K = 2 band.

As the bagradius increases or equivalently the chiral angle F tends to zero, the K = 0, 1 bands crosseach other (following the restoration of the angular momentum into the Dirac spectrum).The K = 2 band never crosses any of them at any point of the chiral angle since it carriesdifferent angular momentum. This should be contrasted with the molecular case or withthe heavy-baryon case.

In the diatomic molecule, the doubly degenerate Π states cross thesinglet Σ at R = ∞at which point the rotational symmetry is restored in L = 1. What(A.7) says is that sufficiently far away from the triple degeneracy point, one can describethe spectrum either with the diagonal field or with the off-diagonal field.

An analogoussituation holds for the heavy-baryon case where the singlet P “crosses” the triplet P ∗inthe IW limit. Thus what is different in the chiral bag case is that there is no point at whichall the relevant K states, namely K = 0, 1, 2, become degenerate.To make the above statements more quantitative, Let us make an ansatz for a Berrypotential that captures the essence of the above structure.

We take in K spaceAKK′ = A δKK′ + ρKK′ ˜AKK′(A.8)where we have introduced the “suppression factor” ρKK′ for K ̸= K′ for which we makethe simplest possible assumption,ρKK′=1,for |ǫK −ǫK′| ≪∆,=0,otherwise. (A.9)28

Here the ∆represents the scale of the adiabaticity of the slow-variable system. The stan-dard Berry potential is recovered when the adiabatic change of state (i.e., the completesuppression of off-diagonal transitions) is applicable, that is to say, ρ = 0.

Now we calculatethe field strength of A using eq. (A.8),FKA=ΠKA ∧AΠK=XK′̸=K1 −|ρKK′|2 ˜AKK′ ∧˜AK′K(A.10)where the superscript K on the field strength means that we are focusing on a particular Kspace.

Although it is obtained with a specific ansatz, we believe (A.10) to be generic. To seethat it is quite general, consider the diatomic molecular case [10, 13].

As the internucleardistance R becomes large, the energies of the Π and Σ levels become degenerate and henceρΣΠ = 1. ThereforeFΣ = 0 = FΠ,(A.11)implying the vanishing of the induced interaction9.Let us now use eq.

(A.10) to show that in contrast to the diatomic molecule, there isno such limit in the chiral bag for light-quark baryons for either the gauge field or the fieldtensor to vanish. Suppose such a limit existed in the topological bag model.

Then from theabove discussion, we should expect all the relevant energy levels connected to the referenceK level by the adiabatic rotation operator S to become degenerate. But in the chiral bagmodel with the charges given by (51), this cannot happen.

What happens is that when thechiral angle F(R) goes to 0, the K = 0, 1 levels become degenerate. So from eq.

(A.10), forK = 1,FK=1A=1 −|ρ10|2 ˜A10 ∧˜A01 +1 −|ρ12|2 ˜A12 ∧˜A21,(A.12)=1 −|ρ12|2 ˜A12 ∧˜A21(A.13)since ρ10 = 1 from our ansatz (A.9). However ρ12 = 0 since the K = 2 state is still splitfrom K = 0, 1 states.

Therefore (A.13) need not vanish. A similar observation can be madeas the bag radius goes to zero, although the nature of level crossings is somewhat different.In the derivation of the Hamiltonian for the excited states, eq.

(56), the adiabaticapproximation has been assumed to be valid. This has led to the bag-radius-independent(“Cheshire Cat”) mass relations among the excited baryons as discussed in the text.

The9In fact, if the Π and Σ levels are degenerate, then the induced gauge potential is really a pure gaugewhich can be gauged away so that A = 0 and FA = 0.29

repeated level crossings, however, may invalidate some of these approximations. Also, asthe bag radius is increased the level spacing decreases suggesting also the breakdown of theadiabatic approximation.

This seems to suggest that the so-called “bag-radius-independent”mass relations cannot hold and hence the Cheshire Cat Principle must be breaking downfor the excited states. Nonetheless the mass formulas in subsection 3.3 worked fairly well.How do we understand this?The answer may lie in the fact that there is no limit at which the field tensor (or thegauge potential) vanishes.

At the bag radius at which the adiabaticity condition presumablyfails to hold, the off-diagonal contributions could significantly modify the charge g1 fromthe value implied by (51) in a way suggested by eq.(A.12). However since (gK)R cannot 2,the structure of eq.

(56) from which the same mass relations follow will remain unmodified.30

Appendix B: The Vanishing of the Wess-Zumino Term for Heavy QuarkConsider QCD with two massless quarks and a heavy quark of mass mQ. Generically,the effective action in the single gluon-exchange approximation to QCD can be rewrittenas follows (using Euclidean conventions)S[S, P] = −NCTr Ln (/∂+ m + S + iP)(A.1)where S and P are scalar and pseudoscalar 3 × 3 hermitian matrices in flavor space and mis short for m = mQ(1 −√3λ8)/3.

A comprehensive discussion of (A.1) can be found inRef.[34]. Without loss of generality, we can use the decompositionS + iP = Σ eiγ5φaT a ≡Σ U †5(A.2)Standard arguments show that the φ’s could be interpreted as pseudoscalar mesons andthat Σ can be related to the dynamically generated (or “constituent”) quark mass in thevacuum [34].

Since the argument in (A.1) is nonhermitian operator, the effective actiondevelops both a real (SR) and imaginary part (SI). The latter follows fromδSIδφa = 12 (/∂+ m + Σ U †5)−1ΣδU †5δφa −h.c.!

(A.3)and gives rise “usually” to the Wess-Zumino term. If the mass of the heavy quark becomeslarge the Wess-Zumino term vanishes in the three flavor case.Indeed, let us assume that the constituent masses are triggered by the quark con-densation in the vacuum.

The details by which this occurs is certainly model-dependent,however, the generic trend is not. Generically the quark condensate is given by< ΨΨ >= −iZdλ1λ + imρ(λ)(A.4)where ρ(λ) is the distribution of the eigenvalues of the Dirac operator in Euclidean space.For massless quarks it reduces to< qq >= −πsgn m ρ(0),(A.5)whereas for heavy quarks mQ >> κ – the half width of ρ(λ), typically of the order of Λ inQCD –, we have< QQ >= −1mQZdλρ(λ) .

(A.6)31

Typically, the eigenvalues have a Gaussian distribution (following the randomness prevailingin the QCD vacuum) so thatρ(λ) ∼ρ(0) exp −λ24κ2(A.7)leading to< QQ >< qq > =r 2πκmQ(A.8)This shows that the heavy quark condensate in the vacuum vanishes as 1/mQ. Up to thispoint, the arguments are general.To be able to relate (A.8) to the “constituent” masses, Σ = diag(σ, σ, σQ), we needto resort to a model description of the vacuum.

Sum rule arguments combined with theconstituent quark model [35] suggest thatσQσ = < QQ >< qq >!1/3= r 2πκmQ!1/3(A.9)which shows that the “constituent” mass vanishes as the inverse cubic root of the heavycurrent quark mass in the limit where mQ >> κ ∼Λ. It is possible that other models leadto a somewhat different scaling, but we believe the estimate (A.9) is good enough to gainsome idea how things might go.

In the heavy quark limit σQ ∼0.With the above in mind, we can evaluate the Wess-Zumino term through the standardderivative expansion, using for the propagator(/∂+ m + Σ)−1 =/∂−σ∂2 −σ2 12 + /∂−mQ∂2 −mQ13(A.10)where 12 = diag(1, 1, 0) and 13 = diag(0, 0, 1). Since 12 13 = 0 it is straightforward to showthat the heavy quark contribution drops from the Wess-Zumino term, and one is left onlywith the two-flavor (chiral quark) Wess-Zumino term that is known to vanish.To summarize, we have shown that as the quark mass mQ becomes considerablylarger than κ ∼Λ – the width of the eigenvalues distributions of the Dirac operator inthe vacuum –, the heavy quark decouples and the Wess-Zumino term truncates to the two-flavor Wess-Zumino term which is identically zero.

Our reasoning sketched above couldpresumably allow one to make an explicit calculation of the rate at which the Wess-Zuminoterm vanishes with the heavy-quark mass.32

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FIGURE CAPTIONSFigure 1Schematic quark spectrum ǫKR (where the subscript K stands for the grandspin of the quark level) in the chiral bag wrapped by hedgehog pions as functionof the chiral angle F(R). Note that F(0) = −π.

For a realistic spectrum, seeMulders in Ref. [16].Figure 2Schematic plot of the “Berry charge” gK where K is the grand spin of the quarklevel as function of the chiral angle F(R).

For a more realistic plot, see Ref. [4].36

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