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Response of nucleons to external probes in hedgehog models:

arXiv:hep-ph/9208257v1 27 Aug 1992Response of nucleons to external probes in hedgehog models:II. General formalismWojciech Broniowski∗and Thomas D. CohenDepartment of Physics and Astronomy, University of MarylandCollege Park, Maryland 20742-4111Linear response theory for SU(2) hedgehog soliton models is developed in analogy to a standardmethod in many-body physics.

In this framework, we discuss response of baryons to external probes,and develop expressions for polarizabilities. We discuss isospin effects (neutron-proton splitting) inpolarizabilities.

Methods for cases with zero modes are presented, including numerical techniques.Our approach is based on the 1/Nc-expansion scheme. We work in a model with quark and mesondegrees of freedom, but the basic method is valid in any hedgehog model, such as the Skyrmionor the Nambu–Jona-Lasinio model in the solitonic treatment.

The equations of motion for coupledRPA quark-meson fluctuations are classified according the hedgehog symmetries, and are writtendown explicitly in the grand-spin basis.PACS numbers: 12.38.Lg, 12.40.Aa, 14.20.Dh, 14.60.FzI. INTRODUCTIONIn recent years various hedgehog models (ChiralQuark Meson (CQM) models [1,2,3,4,5,6], Skyrme mod-els [7,8,9,10],hybrid bag models [11],chiral mod-els with confinement [5,12,13,14] or the Nambu–Jona-Lasinio (NJL) model [15] in the solitonic treatment[16,17,18,19,20,21,22]) were extensively applied to de-scribe the physics of low-energy baryons.

Semiclassicalmethods for treatment of these models, such as variousprojection methods [23,24,25,26], or RPA method [27]were developed. Masses, various charges, π −N phaseshifts [28], were calculated, with quite reasonable agree-ment with experiment, depending on the specific model,number of included fields, etc.

In this article we developthe linear response formalism for hedgehog models. Wework in the framework of a CQM model, since it hasboth quark and meson degrees of freedom, and in this re-spect has the essential features of both the purely mesonicSkyrme model, and the NJL model, which involves quarkdegrees of freedom only.

Our methods and final expres-sions can be modified straightforwardly to be applicablein these models.Hedgehog models can be used to describe response ofnucleons to external probes, and to calculate correspond-ing polarizabilities. A natural approach is the linear re-sponse method of many-body physics [29].

The under-lying picture is as follows: A current interacts with thenucleon, creates an intermediate state which is an RPAphonon excitation on top of the soliton. This state in-teracts with another current and de-excites back into anucleon state.

The RPA phonon states are constructedfrom one-particle–one-hole excitations of the quarks, as1

well as from quantum meson excitations. Quark and me-son fluctuations are coupled, and the resulting equationsof motion for the fluctuations are solved.

An example ofphysically important two-current observables which canbe calculated in this way are the electromagnetic polar-izabilities of the nucleon [30]. This topic is extensivelystudied in the preceding paper [31], henceforth referred as(I).

The present article is devoted to development of thenecessary formalism, and contains many technical butnecessary details of linear response in hedgehog models.while (I) concentrates on physical aspects.This article is organized as follows: In Sec. II we verybriefly review a CQM model [32], its soliton solutions(Sec.

II A), as well as hedgehog symmetries (Sec. II B).One of the discrete symmetries, the grand-reversal sym-metry [33,23], will be particularly useful in classifyingvarious perturbations.

Section III is the core of the pa-per, and describes the equations-of-motion approach [34]to linear response in hedgehog models. We start from de-riving small-fluctuation equations of motion (Sec.

III A)for coupled quark-meson systems driven by an externalperturbation. These equations are classified according tohedgehog symmetries.

In the static limit the grand re-versal symmetry, R, decouples the equations into odd-Requations, involving quarks only, and even-R equations,involving both quarks and mesons.We discuss in de-tail the problem of zero modes (Sec. III B).

These zeromodes arise from braking of the symmetries of the la-grangian by the soliton solution. In our applications, wewill have to deal with rotational zero modes (cranking)and translational zero modes (isoscalar electric pertur-bation in (I)).

We describe a numerical method to dealwith the excitation of zero modes whose amplitude di-verges as the frequency of the perturbation goes to zero(Sec. III C).

We discuss stability of solitons (Sec. III D).In Sec.

III E we present cranking in the linear responseformalism. Quantization via cranking is reviewed in Sec.III F. In Sec.

III G we describe the calculation of polar-izabilities in states of good spin and isospin, and obtainour basic formulas.Section IV illustrates the methodby presenting the standard calculation of the N-∆masssplitting, as well as the evaluation of the neutron-protonhadronic mass difference. The issues of Nc-counting arediscussed in Sec.

V . We show how to apply the linear re-sponse in a way consistent with 1/Nc-expansion scheme.Finally, Sec.

VI contains remarks relevant to other mod-els (Skyrme model, NJL).Appendices contain some details of the grand-spin al-gebra, derivation of the explicit forms of the equations ofmotion for fluctuations in of the grand-spin basis (App.A), and a glossary of useful formulas with collective ma-trix elements (App. B).

We also give a simple proof ofequality of the soliton mass and the inertial mass param-eter (App. C), and discuss the issue of Pauli blocking ofthe Dirac sea in chiral quark models (Sec.

D).2

II. HEDGEHOG MODELSIn this paper most of the derivations will be done inthe framework of the chiral quark-meson model (CQM)of Ref.

[1]. For the details, description of solutions, andthe resulting phenomenology obtained with the crankingprojection method, the reader is referred to Ref.

[23].The reason of choosing this particular model over othermodels, e.g. the Skyrmion or the NJL model, is that itcontains both quark and meson degrees of freedom, andformally has all essential features of a generic hedgehogmodel with two flavors.

At the same time, it is free ofthe non-linear complications of the Skyrme model, or theDirac-sea complications of the NJL model.A. Soliton solutionsThe lagrangian of the model is the Gell-Mann–L´evylagrangian [35], with ψ denoting the quark operator, andσ and π denoting the meson fields:L = ¯ψ [˙ı/∂+ g (σ + ˙ıγ5τ · π)] ψ+ 12(∂µσ)2 + 12(∂µπ)2 −U (σ, π) .

(2.1)The Mexican Hat potential,U (σ, π) = λ24σ2 + π2 −ν22 + mπ2Fπσ,λ2 = mσ2 −mπ22Fπ2,ν2 = mσ2 −3mπ2mσ2 −mπ2 ,(2.2)leads to the spontaneous breaking of the chiral symmetryin the usual way [35,1].Our convention for the piondecay constant is Fπ = 93MeV . At the (time-dependent)mean-field level, only valence quarks, denoted by q, areretained in the expansion of the quark fields, and themeson fields are treated as classical, c-number fields [23](see also App.D).The time-dependent equations ofmotion have the form(h[φ] −˙ı∂t)q = 0,(2.3)−✷φ = δU[φ]δφ−gNc¯qMq,(2.4)where φ = (σ, π) denotes the meson fields, M = (β, ˙ıγ5τ)describes the quark-meson coupling, and the Dirac hamil-tonian is h[φ] = −˙ıα · ∇−gMφ.

Equations (2.4) have astationary solution of the formσ(r, t) = σh(r),π(r, t) = brπh(r),q(t) = qh(r)e−˙ıεt,qh =Gh(r)˙ıσ · brFh(r)(|u ↓⟩−|d ↑⟩)/√2,(2.5)where ε is the quark eigenvalue. For discussion of thissolution, plots of the radial functions σh, πh, Gh, and Fh,and other details, the reader is referred to Refs.

[1,23].3

B. Hedgehog symmetriesThe solution (2.5) has the hedgehog form, which breaksthe spin, J, and isospin, I, symmetries of the lagrangian(2.1), leaving as a good symmetry the grand spin, K =I + J.There are also two discrete symmetries whichare very useful in classifying solutions and perturbations.One is parity, P, the other is the “grand-reversal” sym-metry, R, discussed in Refs. [33,27].

Formally, R is de-fined as the time-reversal, followed by an isorotation byangle π about the 2-axis in isospin. Explicitly, it trans-forms the quark spinors and mean meson fields as follows[23]:q(r, t) →σzτ2q∗(r, −t),σ(r, t) →σ∗(r, −t),π(r, t) →π∗(r, −t).

(2.6)We denote the action of R on an object by the super-script R. The soliton solution has KPR = 0++.III. LINEAR RESPONSE IN HEDGEHOGMODELSIn this section the basic formalism of linear response inhedgehog models is developed.

We use the equation-of-motion approach [34], which is based on solving equationsof motion for small oscillation on top of the ground statesolutions. This method is equivalent to the particle-holeformalism [29], in which one introduces a quantum RPAstate, quasi-boson RPA phonon operators, etc.Meth-ods such as cranking, projection, or quantization of zeromodes, can be described in this framework, and have def-inite quantum-mechanical interpretation.

For simplicityof notation, we present our formalism in the equations-of-motion method.A. Equations of motion for small fluctuationsLet us consider a small oscillation problem in our sys-tem.

We introduce shifts in the valence quark spinor andin the meson fields,δq(r, t) =X(r)e−˙ıωt + Y R(r)e˙ıωte−˙ıεt,δφa(r, t) = Za(r)e−˙ıωt + ZRa (r)e˙ıωt,(3.1)where X and Y describe the shift in the valence quarkspinor, and δφ0 and δφ are the shifts in the σ and πfields, respectively. Note, that in Eqs.

(3.1) the R trans-formation has taken the place of the usual [29] complexconjugation.This is because in hedgehog systems thegrand-reversal replaces the usual time-reversal symme-try. According to definition (2.6), the meson shifts δφare even under grand-reversal, but the quark shifts havein general both even and odd components.

We linearizeequations (2.4) about the solitonic solution (2.5), and4

obtain the quark-meson RPA equations. When externalperturbations are present, these equations are in generaldriven by a quark source, jq, and a meson source, jφ,jq = jXe−˙ıωt + jRY e˙ıωt,jφ = jZe−˙ıωt + jRZ e˙ıωt,(3.2)Again, the meson source is even under R, whereas thequark source has in general even and odd-R components.Using the fact that h[φh], M and qh are even under R (infact they are KPR = 0++ objects), we obtain a generalform of the linear response equations for our hedgehogsystem:(h[φh] −ε) X −gXaMaqhZa −ωX = jX,(h[φh] −ε) Y −gXaMaqhZa + ωY = jY ,−∇2Za +Xbδ2Uδφaδφbφ=φhZb−gNcq†hMaX + Y †Maqh−ω2Za = jZ.

(3.3)Introducing auxiliary meson momentum variables Pa =−˙ıωZa, we observe that Eqs. (3.3) can be written in thesymplectic form [29]Hξ −ωΛξ = j,(3.4)where H is the RPA hamiltonian, and Λ is the symplecticRPA metric, satisfying Λ2 = 1.

In the grand-spin basis(App.A), H is real.Our problem (3.3) can then bewritten asH =Nc(h −ε)0−gNcMqh 00Nc(h −ε) −gNcMqh 0−gNcq†hM −gNcq†hM −∇2 + U ′′ 00001,(3.5)Λ =10000 −100000˙ı00−˙ı 0,ξ =XYZP,j =NcjXNcjYjZjP. (3.6)Note the appearance of an odd-R momentum componentin the source, jP , which arises in some cases (cranking).We are interested in the limit of vanishing frequencyof the external perturbation, ω →0.

If zero modes areexcited by an even-R perturbation (Sec. III B), then thefull equations Eq.

(3.3) have to be solved. Otherwise,one can set ω = 0 and deal with simplified cases.

Atthis point the grand-reversal classification becomes veryuseful. Acting with R on Eq.

(3.4) effectively replacesX ↔Y , jX ↔jY , Z ↔Z, jZ ↔jZ, P−↔P, andjP −↔jP . Let us introduce odd and even grand-reversal5

combinations: δq± = X ± Y , j±q = jX ± jY , and rewriteEq. (3.4) by adding and subtracting the first two equa-tions.

We get for the case of an odd-R perturbation(h −ε)δq−= j−q ,P = jP ,(3.7)and for the case of an even-R perturbation(h −ε)δq+ −2gMqhZ = j+q ,(−∇2 + U ′′)Z −Ncgq†hMδq+ = jZ(3.8)The odd-R equations (3.7) involve a quark field equation,and a trivial equation for P. The even-R equations (3.8)involve coupled quark and meson fluctuations. Equations(3.4), or (3.7,3.8) are further decomposed by grand-spin,K, and parity, P (App.

A).In models with vector mesons, such as [32], the odd-Requations may also involve mesonic shifts. For example,the space components of the ω meson and the time com-ponenent of the ρ meson enter into the cranking equa-tions of motion [36].B.

Zero modesFirst consider the undriven problem (3.4), with j = 0,which determines the RPA spectrum and eigenmodes. Acomplication arises whenever a continuous symmetry ofthe lagrangian is broken by the solitonic solution, e.g.translation, or rotational symmetry.For each brokensymmetry the small fluctuation equations have a pair ofzero-modes [29]: ξ0, the symmetry mode, obtained byacting with a symmetry generator on the solitonic solu-tion, and a conjugate zero mode, ξ1.

They satisfy theequationsHξ0 = 0,(3.9)Hξ1 = −˙ıΛξ0. (3.10)The remaining “physical” modes, ξi, satisfy the equationsHξi = ωiΛξi.

(3.11)One can easily show that the symplectic norms satisfyconditionsξ†0Λξ0 = ξ†1Λξ1 = 0,ξ†0Λξ1 = −˙ı4M,ξ†i Λξj = 14δijNi,ξ†0Λξi = ξ†1Λξi = 0,i = 2, 3, ...(3.12)where M is the appropriate inertia parameter (mass, mo-ment of inertia) parameter, and Ni are the symplecticnorms of the physical modes. The factors of 14 are con-ventional, and factors of ˙ı are inserted for convenience.Expanding the solution of Eq.

(3.4) in RPA eigenmodes,6

ξ =Xµ=0,1,2,...cµξµ,(3.13)introducing “charges”: Q0 = 4˙ıξ†0j, Qµ = 4j, µ =1, 2, ..., and using Eqs. (3.12), we find thatωc1M + Q0 = 0,˙ıωc0M + Q1 −c1M = 0,ci(ωi −ω)Ni = Qi.

(3.14)We consider two cases which arise in practical applica-tions: 1) Q1 = 0, and 2) Q1 ̸= 0, Q0 = 0.1. Case Q1 = 0Using Eq.

(3.14) we findc0 =˙ıQ0Mω2 ,c1 = −Q0Mω ,ci =QiNi(ωi −ω). (3.15)The second-order energy shift, κ, corresponding to agiven perturbation (a “polarizability” is equal to 2κ) isgiven by the usual perturbation theory resultκ = 2ξ†j =Xµc∗µQµ = κzero + κphys.,κzero = −12Q20Mω2 ,κphys = 12XiQ2iNi(ωi −ω).

(3.16)In the limit ω →0, the coefficients c0, c1 and the zero-mode part of κ diverge, as long as Q0 ̸= 0. This has aphysical interpretation: for instance in the case of trans-lation the center of mass of the system moves, and theamplitude of this motion, c0, as well as “velocity”, c1,diverge.

In (I) we show how this feature of the linearresponse formalism leads to the Thompson limit of theCompton scattering amplitude.2. Case Q1 ̸= 0, Q0 = 0In this case we can take the limit ω →0 on the outset,and from Eq.

(3.14) we getc1 = Q1M,ci =QiNiωi. (3.17)The amplitude of the symmetry mode, c0, remains un-dermined.

The second-order energy shift is:κ = 12Q21M + 12XiQ2iNiωi. (3.18)7

C. Numerical methods in presence of diverging zeromodesNumerically, the excitation of amplitude-growing zeromodes (Sec.III B 1) creates special difficulties in ex-tracting the “physical” parts of observables, e.g.elec-tromagnetic polarizabilities. The problem can be reme-died as follows: We solve Eqs.

(3.4) for a small valueof ω. Next, we project out the zero-mode part from ξ,obtaining ξphys.

= ξ −c0ξ0, and calculate physical partsof observables. The procedure is repeated with decreas-ing ω, until the results no longer change.

In practice,a very high accuracy of the soliton solution, as well asthe fluctuation solutions, is required for this procedureto be feasable. A better method is to project the part ofthe source, j, which couples to the zero mode, and solveequationsHξphys.

−ωΛξphys. = jphys.,(3.19)where jphys.

= j −(Q0/M)Λξ1, and ξ1 is obtained bysolving Eq. (3.10) first.

Equations (3.19) do not excitethe zero mode, and directly lead to the physical partof the solution. The advantage of the method with theprojected source over the direct method described previ-ously follows from the fact that in numerical solutions toEqs.

(3.19) the admixtures of the zero mode arise onlyfrom numerical noise. Their amplitude is small, such thatwe can easily control numerical precision in the physi-cal mode.

Because of these admixtures, a small nonzerovalue of ω should be kept as a regulator in Eqs. (3.19),and the zero-mode contamination has to be projected outafter the numerical solution is found.D.

Stability of solitonsSince in our problem H and Λ are hermitian, one findsthat H2ξi = ω2i ξi is a hermitian eigenvalue problem.Therefore in our case ω2i are real, and ωi can either bepurely real, or purely imaginary. The modes appear inconjugated pairs (ξi, ξj), with ωi = −ωj.

If the spectrumcontains an imaginary eigenvalue, we have to instability(in the Lyapunov sense [37]) of the ground-state (soliton)solution [38,39], and of course linear response on top ofan unstable system makes no sense. In Ref.

[27] we haveshown that the soliton of Ref. [1] is stable with respect tobreathing modes, i.e.

the KP = 0+ excitations. With theexplicit forms of the equations in App.

A, stability couldbe checked numerically for any KP vibrational mode. Itis generally believed that the hedgehog solitons are in-deed stable, although it has not been proved analyticallyor numerically.8

E. Cranking as linear responseCranking [23] may be viewed as linear response. Ina frame iso-rotating with a small angular velocity λ, wediscover equations of the form (3.10), with ω = 0 andj = −˙ıλΛξ0(bλ).

In this case ξ0(bλ) is the symmetry modeobtained by acting on the soliton fields with the generatorof isorotation about the axis bλ:ξ0(bλ) = 12˙ı/2 τ · bλqh˙ı/2 τ · bλqh−bλ × πh0. (3.20)Next, we have to find the conjugated mode, by solvingthe second of Eqs.

(3.10). We notice, that this is an odd-R case (3.7).

We immediately get P = 12λ × πh. For thequark shift, δqcr, a differential equation of the form (3.7)is solved [23].

The problem is of the type discussed inSec. III B 2, where M is the moment of inertia, Θ, andthe “charges” are: Q1 = λΘ, Qµ = 0 for µ ̸= 1.

Thesecond-order energy shift is: κ = 12λ2Θ. Explicitly, onefindsΘ = Θm + Θq,Θm =Zd3x(bλ × πh)2 = (8π/3)Zdrr2π2h,Θq = 2Zd3xδq†crbλ · τqh(3.21)F. QuantizationThe simplest approach to quantization via cranking, isto recognize that in the frame isorotating with velocity λ,in which we solve the cranking equations of motion(Sec.III E), we still have the freedom of (iso)rotating the soli-ton by an arbitrary (time-independent) angle.This isan example of the freedom of choice in the c0 coefficientin Sec.

III B 2, which in this case corresponds to threeEuler angles, or, in the commonly used Cayley-Klein no-tation [8], to the matrix B = b0 + b · τ [23].In ourmean-field approach, the corresponding fields carry these(time-independent) B matrices, and in the rotating framethey assume the form:σ →σ,π →BπB†,q →Bq. (3.22)Matrix B plays the role of coordinate variables conju-gated to λ, which upon quantization becomes a differ-ential operator [8,23].The quantization is straightfor-wardly implemented in two steps: 1) one identifies thecollective spin and isospin operators, as done in Ref.

[23].ThenλΘ = J,Ia = cabJb,(3.23)9

where J and I are the spin and isospin operators, sat-isfying appropriate commutation relations, and cab, de-fined in App. B, has the meaning of the transformationmatrix from the body-fixed to the lab frame [29].2)Corresponding collective wave functions are introduced.Expectation values of operators are calculated by firstidentifying in the semiclassical expression for an opera-tor its collective part (dependent on λ, cab, etc.

), and anintrinsic part (dependent on the meson and quark fieldsσ, π, q). Then, the matrix element factorizes into a col-lective matrix element in the wave functions of App.

B(this is an integral over the collective coordinates, viz.Euler angles, or (b0, b)), and an intrinsic matrix element,which is a space integral over the quark and meson fields.For details, see Ref. [23].G.

External perturbationsThe quark and meson field profiles in Eq. (3.22) arein general not equal to the hedgehog profiles.

We havedemonstrated in Sec. III E that the quarks develop shiftsupon cranking.If some other (external) interaction ispresent, then the profiles are additionally shifted.

Theseshifts are obtained by solving the linear response equa-tions, as described in Sec. III.

We introduce a resolventfor the H −ωΛ operator in Eq. (3.4) (RPA propagator)and solve formally Eq.

(3.4), obtainingξ = Gj,G = (H −ωΛ)−1. (3.24)In the presence of cranking and some other external per-turbation, we haveξ = ξcr + ξext = G(jcr + jext),(3.25)where subscripts cr and ext refer to cranking, and an ex-ternal perturbation, respectively.

The second-order en-ergy shift corresponding to a perturbation can be writtenasκ = 2ξ†j = 2j†Gj. (3.26)The difference between this expression, and the genericexpression (3.16) is that in the present case the sourcecarries collective degrees of freedom, j = jcolljintr.

Thus,the matrix element of κ in a baryon state |b⟩is (see ex-ample in Sec. IV A):κb = 2⟨coll|jcolljcoll|coll⟩Zd3x d3x′ jintr†(x)G(x, x′)jintr(x′),= 2⟨coll|jcolljcoll|coll⟩Zd3x ξintr†(x)jintr(x), (3.27)where |coll⟩represents the collective wave function (App.B) associated with the baryon state |b⟩.It is possible to have isospin-dependent effects in linearresponse of the nucleon.For example, if the external10

interaction has KP = 1+ (the same quantum numbers asin cranking), we pick up cross terms between cranking,and the external perturbation (see example in Sec. IV B):κb = 2⟨coll|jcollcr jcollext |coll⟩Zd3x d3x′ jintr†cr(x)G(x, x′)jintrext (x′) + h.c.= 2⟨coll|jcollcr jcollext |coll⟩Zd3x ξintr†cr(x)jintrext (x) + h.c.(3.28)Expressions (3.27, 3.28) are just second-order pertur-bation results.

We may formally continue to higher orderin perturbation theory, which leads to chains of the formκi1,...,in = 2⟨coll|jcolli1 Vcolli2 ...jcollin |coll⟩Zd3x1 ... d3xn jintr†i1(x1)G(x1, x2)Vi2(x2)G(x2, x3) ... Vin−1(xn−1)G(xn−1, xn)jintrin(xn),(3.29)where Vik is interaction of kth type. The total energyshift is the sum over all possible orderings of (i1, ..., in)in (3.29).

Because the ground state has KP = 0+, thematrix element in Eq. (3.29) is non-zero only if one cancompose the KP quantum numbers of ji1, Vi2, ..., jin toKP = 0+.

In (I) we show an application of Eq. (3.29)with two RPA propagators in the analysis of the neutron-proton splitting of electromagnetic polarizabilities.InSec.

V we discuss in what cases going to a higher orderin perturbation theory is consistent with Nc-counting,which is our basic principle in organizing the perturba-tion expansion in hedgehog models.IV. SIMPLE EXAMPLESIn this section we give some simple application of thedescribed formalism.

A more advanced and physicallyimportant case of electromagnetic polarizabilities is givenin (I).A. N-∆mass splittingAs an illustration of application of Eq.

(3.27), considerthe N-∆mass splitting. In this case κb is the energy shiftof the baryon |b⟩due to the cranking perturbation.

FromEqs. (3.23, 3.21, 3.27) we obtain immediately the usualexpression for the N-∆mass splitting:M∆−MN = 12(⟨∆|λ2|∆⟩−⟨N|λ2|N⟩)Θ = 32Θ(4.1)11

B. Hadronic p - n mass splittingAs an example of an isospin-dependent effect, considerthe neutron-proton mass difference due to the differenceof the up and down quark masses. The perturbation inthe lagrangian has the form Lm = 12(md −mu)ψτ3ψ.

Ithas KPR = 1+−, exactly as cranking, hence a mixedperturbation of the form (3.28) appears. Passing to anisorotating frame, we find the source corresponding to thequark mass splitting, which arises in Eqs.

(3.7): jm =12(md −mu)Ncγ0c · τqh, where c is defined in App. B.Since we have already solved the cranking equation, wedo not have to solve the new equation with source jm.We simply calculate the overlap of jm with the shift inthe fields due to cranking, δqcr, according to Eq.

(3.28).Using the fact that ⟨N|λ · c|N⟩= −⟨N|I3|N⟩, we obtainthe following expression for the hadronic splitting of theneutron and proton masses:(Mn −Mp)hadr. = (md −mu)Zd3x d3x′[⟨n|j†mδqcr|n⟩−(n ↔p)]= md −muΘZd3x jintr†mδqintrcr.

(4.2)The numerical value, obtained for the solution of Ref. [23] gives (Mn −Mm)hadr.

= 0.4 × (md −mu), whichfor typical values of (md −mu) gives a number around2MeV . The electromagnetic mass difference can also bestudied in hedgehog models models [40].V.

NC-COUNTINGThe basic organizational principle behind hedgehogmodels is the 1/Nc expansion of QCD [41,42,43]. In theNc →∞limit, masses of baryons diverge as Nc, andcan be calculated using mean-field theory [42].

It shouldbe noted that the assumption of the spin-isospin corre-lated wave function, which is essential in hedgehog mod-els, does not follow from the large-Nc limit alone — itis an additional assumption of the hedgehog approach.By analogy to nuclear physics, in systems with many nu-cleons we may have nuclei with intrinsic deformations,but we may also have spherically symmetric nuclei, andit is the dynamics which determines whether the wave-function is deformed or not.In hedgehog models thehedgehog wave function is assumed to be deformed inthe spin-isospin space, and the nucleon the ∆masses,which are of the order Nc, are degenerate in the leading-Nc order.When cranking is used, these masses split as ∼N −1c.In fact, cranking becomes an exact projection methodin the large-Nc limit, since it may be viewed as aPeierls-Yoccoz projection with δ-function overlaps be-tween rotated wave functions [29].Thus we obtainthe hedgehog result for the mass splitting, Eq. (4.1).12

It would not be consistent, however, to conclude thatthe nucleon or ∆masses individually are given by thehedgehog soliton mass plus the cranking piece.Thereare other effects (center-of-mass correction, centrifugalstretching, etc. )which enter at the same level as thecranking term.Also, the effective lagrangian may besupplemented by subleading terms in Nc, which wedid not have to include to obtain the leading piece inthe hedgehog mass.Therefore, it is useless to writedown MJ = Mh + J(J + 1)/(2Θ) + O(N −1c), since thelast term, which we do not calculate, enters at the samelevel as the cranking term.

We can only trust the lead-ing piece, MJ = Mh + O(N −1c), and, in order to main-tain consistency with the Nc-counting, the mass for-mula should not be “improved” by adding the crank-ing term. The mentioned effects of center-of-mass cor-rections, centrifugal stretching, etc., are at the leadinglevel the same for the nucleon and for the ∆, there-fore for the N −∆mass splitting we get the formulaM∆−MN = 3/(2Θ) + O(N −2c).The prescription, which we tried to illustrate above,is that with semiclassical methods we can only get theleading-Nc term for a given observable.

The power of Ncvaries, depending on the quantity we are investigating.The same is true for the calculation of polarizabilities,described in this paper.We can easily obtain the Ncbehavior of various terms in Eqs. (3.27,3.28,3.29), butonly the leading-Nc piece corresponding to a particularpolarizability should be retained.

As an illustration, con-sider the electric polarizability of the nucleon, discussedextensively in [30] and in (I). The electric field polarizesthe hedgehog.

The electric charge of the quark has anisoscalar component, of order N −1c, and isovector com-ponent, of order 1. We immediately see from Eq.

(3.27)that the leading part of the electric polarizability of thenucleon is obtained from interactions with two isovec-tor sources, and the term with two isoscalar sources istwo powers of Nc suppressed. The non-dispersive seagulleffects also enter at the level of Nc (I), hence the nu-cleon polarizability goes as Nc.

Quite analogously to theproblem of the N −∆mass splitting, the neutron-protonsplitting of the electric polarizability is a N −1ceffect (I),and we can calculate it consistently only to this order.In principle, one might try to perform a calculationwhich consistently takes into account the subleadingpieces. The appropriate scheme would be the Kerman-Klein method [44], but its application would involve acomplicated fully quantum-mechanical calculation.VI.

OTHER MODELSTechniques described in this paper are applicable toother model after straightforward modifications. In theSkyrme model, the described RPA method involves fluc-tuations of the meson fields which do not satisfy the non-linear constraint for the σ and π field operators.

This13

linearization may be viewed as an approximation to thefully nonlinear dynamics. The RPA dynamics, obviously,involves mesons only, and the higher-derivative terms aremanifest in the equations of motion for the fluctuations.In the case of the (partly bosonized) NJL model [45],the mesonic potential has the simple form 12µ2(σ2 + π2).The sea quarks are present explicitly, and the number ofquark equations is infinite.

Standard methods of solv-ing these equations numerically may encounter problemsfor the case when the translational zero mode is excited,since extremely good accuracy is necessary in this case.VII. CONCLUSIONWe have presented the linear response method inhedgehog soliton models.We have shown that themethod is consistent with the basic philosophy of thesemodels, namely, the 1/Nc-expansion, if its applicationis restricted to obtaining the leading-Nc order of a givenquantity.

We have discussed many technical points whichare encountered in practical calculations, especially thetreatment of zero-modes, which create special problems.Appropriate equations of motion have been classified ac-cording to hedgehog symmetries, and derived explicitlyfor the model of Ref. [1].

Our method, after straightfor-ward modifications, is directly applicable to other hedge-hog models. A physical application of the approach isdescribed in the preceding paper, (I), where we studythe electromagnetic polarizabilities of the nucleon.ACKNOWLEDGMENTSSupport of the the National Science Foundation (Pres-idential Young Investigator grant), and of the U.S. De-partment of Energy is gratefully acknowledged.Wethank Manoj Banerjee for many useful suggestions andcountless valuable comments.One of us (WB) ac-knowledges a partial support of the Polish State Com-mittee for Scientific Research (grants 2.0204.91.01 and2.0091.91.01).14

APPENDIX A: EQUATIONS OF MOTION FORSMALL FLUCTUATIONS IN THE GRAND-SPINBASISWe compose the basis of Dirac spinors with good Kquantum numbers using the coupling scheme in whichthe isospin, I =12, and spin S =12, are first coupledto a quantum number Λ, and then orbital angular mo-mentum, L, and Λ are coupled to K. Since there is noconfusion concerning the value of K or K3, we use thenotation|L, Λ >= |K, (L, Λ(I = 12, S = 12)), K3 > . (A1)States with parity P = (−)K (or P = −(−)K) are callednormal (abnormal) parity states.The basis of Diracspinors isqL,Λ =GL,Λ(r)˙ıσ · brF L,Λ(r)|L, Λ > .

(A2)Spinors X and Y are expressed in states (A2). The quarksources are decomposed into (L, Λ) components:jL,Λ =jL,ΛG(r)˙ıσ · brjL,ΛF(r)|L, Λ >,(A3)Tables I - III list the matrix elements which arise in deriv-ing the quark parts of perturbation equations.

It is clearfrom Table III that unless K = 0, the kinetic term mixesthe Λ = 0 and Λ = 1 components of the L = K states(normal parity case). Diagonalization is made throughthe substitutionGa =rK + 12K + 1GK,0 −rK2K + 1GK,1,Gb =rK2K + 1GK,0 +rK + 12K + 1GK,1,(A4)and similarly for the F-components, and the sources.The basis for the meson fluctuations is composed bycoupling isospin to L. For a given value of K, the σ andπ fluctuations can be expressed through functionsσL(r)|K, (L, 0), K3 >, πL(r)|K, (L, 1), K3 >,(A5)Obviously, L = K for σ, and L = K −1, K, K + 1 for π,such that for a given K and K3δσ = σK(r)|K, (L, 0), K3 >,δπ =XL=K−1,K,K+1πL(r)|K, (L, 1), K3 > .

(A6)Using standard Racah algebra, it is straightforward toderive the general equations (3.4) for a given K pertur-bation. In the notation of this appendix, Ga{X,Y }, etc.,correspond to the X and Y spinors from Eq.

(3.4) ,15

and Ga{X+Y } = GaX + GaY , etc.. The functions describ-ing meson fluctuation, σL, πL, have the meaning of theZ-functions of Eq.

(3.4).For normal parity equation we get∂rGa{X,Y } = Kr Ga{X,Y } + (gσh −ε ∓ω)F a{X,Y }+ gπh(−12K + 1Ga{X,Y } −2pK(K + 1)2K + 1Gb{X,Y })+ g(rK + 12K + 1FhσK + GhπK+1) −jaF,{X,Y },∂rGb{X,Y } = −K + 1rGb{X,Y } + (gσh −ε ∓ω)F b{X,Y }+ gπh(−2pK(K + 1)2K + 1Ga{X,Y } +12K + 1Gb{X,Y })+ g(rK2K + 1FhσK −GhπK−1) −jbF,{X,Y },∂rF a{X,Y } = −K + 2rF a{X,Y } + (gσh + ε ± ω)Ga{X,Y }+ gπh(12K + 1F a{X,Y } + 2pK(K + 1)2K + 1F b{X,Y })+ g(rK + 12K + 1GhσK −FhπK+1) + jaG,{X,Y },∂rF b{X,Y } = K −1rF b{X,Y } + (gσh + ε ± ω)Gb{X,Y }+ gπh(2pK(K + 1)2K + 1F a{X,Y } −12K + 1F b{X,Y })+ g(rK2K + 1GhσK + FhπK−1) + jbG,{X,Y },(A7)∂2r + 2r ∂r −(K −1)Kr2πK−1 = λ2(σ2h + π2h −ν2 −ω2)πK−1+ 2λ2 K22K + 1π2hπK−1 +rK2K + 1σhπhσK −pK(K + 1)2K + 1π2hπK+1!−gNc(FhGb{X+Y } + GhF b{X+Y }) + jK−1π,∂2r + 2r ∂r −K(K + 1)r2σK = λ2(σ2h + π2h −ν2 −ω2)σK+ 2λ2 rK2K + 1σhπhπK−1 + σ2hσK −rK + 12K + 1σhπhπK+1!−gNc Gh rK + 12K + 1Ga{X+Y } +rK2K + 1Gb{X+Y }!−Fh rK + 12K + 1F a{X+Y } +rK2K + 1F b{X+Y }! !+ jKσ ,∂2r + 2r ∂r −(K + 1)(K + 2r2πK+1 = λ2(σ2h + π2h −ν2 −ω2)πK+116

+ 2λ2 −pK(K + 1)2K + 1π2hπK−1 −rK + 12K + 1σhπhσK −K + 12K + 1π2hπK+1!−gNc(−FhGa{X+Y } −GhF a{X+Y }) + jK+1π,(A8)The abnormal parity equations have the form∂rGK−1,1{X,Y } = K −1rGK−1,1{X,Y } + (gσh −ε ∓ω)F K−1,1{X,Y }+ gπh(12K + 1GK−1,1{X,Y } + 2pK(K + 1)2K + 1GK+1,1{X,Y })−grK + 12K + 1GhπK −jK−1,1F,{X,Y },∂rGK+1,1{X,Y } = −K + 2rGK+1,1{X,Y } + (gσh −ε ∓ω)F K+1,1{X,Y }+ gπh(2pK(K + 1)2K + 1GK−1,1{X,Y } −12K + 1GK+1,1{X,Y })−grK2K + 1GhπK −jK+1,1F,{X,Y },∂rF K−1,1{X,Y } = −K + 1rF K−1,1{X,Y } + (gσh + ε ± ω)GK−1,1{X,Y }+ gπh(−12K + 1F K−1,1{X,Y } −2pK(K + 1)2K + 1F K+1,1{X,Y } )+ grK + 12K + 1FhπK + jK−1,1G,{X,Y },∂rF K+1,1{X,Y } = Kr F K+1,1{X,Y } + (gσh + ε ± ω)GK+1,1{X,Y }+ gπh(−2pK(K + 1)2K + 1F K−1,1{X,Y } +12K + 1F K+1,1{X,Y } )+ grK2K + 1FhπK + jK+1,1G,{X,Y },(A9)∂2r + 2r ∂r −K(K + 1)r2πK = λ2(σ2h + π2h −ν2 −ω2)πK−gNc Gh rK + 12K + 1F K−1,1{X+Y } +rK2K + 1F K+1,1{X+Y }!+ Fh rK + 12K + 1GK−1,1{X+Y } +rK2K + 1GK+1,1{X+Y }! !+ jKπ .

(A10)For the ω = 0, odd-R case, meson fluctuations van-ish, and appropriate equations have the form of Eqs. (A7,A9), with the meson fluctuations set to zero.

TheX and Y equations can be combined to a single equationof the form (3.7).In the case of an even-R source which does not excitea zero mode (case Q0 = 0 in Sec. III B 1), we can setω = 0, in the above equations.

We can combine the Xand Y equations, and obtain the form (3.8). If the zeromode is excited (Q0 ̸= 0), we have to solve full equations(A7-A8), or (A9-A10), depending on parity.17

For the special case of K = 0, Ga = G0,0, Gb = 0, etc.,and only equations for the a components in Eqs. (A7)remain.

Fields with negative (i.e. K −1) superscripts,and equations for these fields are eliminated.The boundary conditions in Eqs.

(A7-A10) are suchthat the solutions are everywhere finite. At the origin,radial derivatives of S-wave fields vanish, and the valuesof higher-L fields vanish.At r →∞, the appropriateboundary conditions follow from solutions of the equa-tions in the asymptotic region.APPENDIX B: COLLECTIVE MATRIXELEMENTSSuppose a space rotation, R, is described by Euler an-gles α, β and γ:R = e−˙ıαJze−˙ıβJye−˙ıγJz.

(B1)Then, matrix B from Sec. III F is given byB = e˙ıγτ3/2e˙ıβτ2/2e˙ıατ3/2.

(B2)The matrix transforming from the body-fixed to labframe, cab, is defined ascab = 12T r[τaBτbB†] = D1ba(α, β, γ),(B3)where the first (second) subscript in the Wigner D-matrixis connected to the spin (isospin) space. It follows thatPb cabJb = −Ia.

The spin operator and the matrix ccommute, [cab, Jk] = 0.We also introduce a vector cdefined asc = 12T r[τ3BτB†];cµ = D1µ0. (B4)The collective baryon states with spin J, isospin I = J,and projections m and I3 are|J = I; m, I3⟩=r2J + 18π2DJm,−I3.

(B5)In formulas below we do not display m or I3 in labels ofthe states, and use notation |N⟩= | 12; m, I3⟩, and |∆⟩=| 32; m, I3⟩.The following useful formulas can be easilyderived (no implicit summation over repeated indices):(cµ)∗cµ = 13 + ( 23 −µ2)D200,Xµ(cµ)∗cµ = 1,(B6)from which follows that⟨N|(cµ)∗cµ|N⟩= 13;any µ,⟨∆|(cµ)∗cµ|∆⟩= 13 +23 −µ25+1 ; |m| = |I3|−1 ; |m| ̸= |I3| ,⟨N|(cµ)∗cµ|∆⟩=√2( 23 −µ2)5+1 ; m = I3−1 ; m = −I3 . (B7)18

One also finds⟨N| ((Jµ)∗cµ + (cµ)∗Jµ) |N⟩= −23I3;any µ. (B8)One also derives⟨N|c0|∆⟩=√23 .

(B9)For our analysis of the ∆states in hadronic loops in (I)the following formulas are important:Xµ,m′,I′3⟨N|(cµ)∗|N; m′, I′3⟩⟨N; m′, I′3|cµ|N⟩= 13,Xµ,m′,I′3⟨N|(cµ)∗|∆; m′, I′3⟩⟨∆; m′, I′3|cµ|N⟩= 23. (B10)The sum of the above formulas gives unity, in accordanceto the sum rule (B6).APPENDIX C: EQUALITY OF INERTIAL ANDSOLITON MASSESFor the case of translations, the inertia parameter, M,is equal to the soliton mass, Msol.

This result, requiredby Lorentz invariance, can be verified explicitly as fol-lows: Consider a boost in the z-direction, with small ve-locity v. The fields transform asφ →φh(r −vt),e−˙ıεt →e−˙ıε(t−vz)+ 12 vαzqh(r −vt),(C1)which lead to the following shifts linear in the velocity:δφ = −vt∂zφh,δqh = ve−˙ıεt(˙ıεz + 12αz −t∂z)qh. (C2)Using identities [h, z] = −˙ıαz and {h, αz} = −2˙ı∂z, weeasily derive the equation[h −ε](−εz + 12 ˙ıαz)qh = ∂zqh.

(C3)After integrating by parts we get the expression for theenergy shift of a moving soliton:δE = 12v2( 13Tq + 23Tφ + Ncε),(C4)where Tq and Tφ are kinetic energies in the soliton, car-ried by the quarks and mesons, respectively.Next, we use a virial relation. Consider scale change ofthe radial coordinate, r →sr.

The soliton energy scalesas E(s) = Tq/s+ Vqφ + sTφ + s3Vφ, where Vqφ and Vφ arethe quark-meson, and meson-meson interaction energies.Stationarity of the solution imposes ∂sE|s=1 = 0, which,together with the relation Ncε = Tq + Vqφ, leads to thevirial relationMsol = 13Tq + 23Tφ + Ncε. (C5)Comparing Eq.

(C4) and Eq. (C5) completes the proofthat δE =12v2Msol.Using similar methods, one canshow the equality of inertial and soliton masses in othermodels [46], also in non-local theories, such as the NJLmodel [47].19

APPENDIX D: BOSONIZATION AND PAULIBLOCKING OF THE DIRAC SEAIn this section we return to the question whether theDirac sea should be “Pauli-blocked” in our model. Effec-tive chiral models are believed to result from bosonizingQCD, which, of course, can only be done approximately.For definiteness, we discuss the issue of Pauli blocking inthe framework of the partly-bosonized [45] NJL model,but the result is more general.

In presence of an externalsource, J, the action of the model is:SNJL = −˙ıT r log[˙ı/∂−gU −J] −vac,gU = g(σ + ˙ıγ5τ · π),(D1)A cut-offis understood, T r denotes functional trace,and vac means the vacuum subtraction.For simplic-ity, we assume the nonlinear constraint σ2 + π2 = Fπ2.The source J may represent interactions with externalprobes (e.g.electromagnetic) or result from cranking(Sec.III E).For definiteness, let us evaluate the mo-ment of inertia. In this case J = 12λ · τ, and expandingthe action to second order in λ we obtain∆SNJL = 12λ2ΘZdt,(D2)where the moment of inertia, Θ, is given byΘ = ˙ı4NcZ dω2π Sp1ω −hτ31ω −hτ3,(D3)where Sp denote the trace over space, spin and isospin,and h is the Dirac hamiltonian.

The pole structure andthe contour of the ω integration in Eq. (D3) is given inFig.

(1). Note that the contour goes above the occupiedvalence state, as well as above all the negative-energy seastates.

Performing the integration over ω in Eq. (D3),we obtain the usual spectral expression for:Θ = 12NcXi∈occ.j∈unocc.|⟨i|τ3|j⟩|2εi −εj,(D4)where occ.

denotes all occupied states, i.e. the valence aswell as the sea states, and unocc.

denotes the unoccupiedpositive energy states (see Fig.1 for the meaning oflabels). The expression under the sum is antisymmetricwith respect to exchanging i and j, therefore the sumas in Eq.

(D4) over i and j belonging to the same setof indices vanishes. Using this trick we can replace theranges of summation indices as follows:Xi∈occ.j∈unocc.=Xi∈occ.j∈all′ =Xi∈val.j∈all′ +Xi∈seaj∈all′ =Xi∈val.j∈all′ +Xi∈seaj∈pos.en.,(D5)20

where the prime means the exclusion of i = j term, anydenotes all states, and pos.en. denotes the positive energystates.

According to Eq. (D5), the moment of inertia canbe decomposed into the valence and sea parts:Θ = Θval.

+ Θsea,Θval. = 12NcXi∈val.j∈all′ |⟨i|τ3|j⟩|2εi −εj, Θsea = 12NcXi∈seaj∈pos.en.|⟨i|τ3|j⟩|2εi −εj.

(D6)Note that the “full” expression (D4) obeys the Pauli ex-clusion principle, hence using Eq. (D5) we have brokenthe the original expression into two parts, each of whichviolates the Pauli principle.In fact, an analogous de-composition is used in the treatment of the relativisticfermion propagator in fermion matter [48].

Below we ex-plain why this is useful. Firstly, the expression for Θvalcorresponds to our quark part of the moment of inertiacalculated in Sec.

III E. Secondly, the sea part of themoment of inertia can be simply approximated only if itis written as in Eq. (D6).

Indeed, we can write downΘsea = ˙ı4NcZ dω2π Sp1ω −hτ31ω −hτ3,(D7)where the contour of integration is given in Fig. 2.

Thiscontour can be Wick-rotated without picking up any polecontributions, and we obtain an expression in Euclideanspace. We can then perform standard gradient expansionmethods [49,50,51,17] to rewrite Θsea as an integral overthe classical pion field.

The first term, with no deriva-tives, is just our expression for the pion part of the mo-ment of inertia, Eq. (3.21).

Furthermore, this term doesnot depend on the NJL cut-off, since the normalizationfactor is the same as in the pion wave function normal-ization [15]. Further terms in the gradient expansion dodepend on the cut-off.

If we tried to perform the Wickrotation on the original expression (D3), we would pickup a pole contribution from the occupied valence level,and our final expression (D6) would also follow.∗On leave of absence from H. Niewodnicza´nski Institute ofNuclear Physics, ul. Radzikowskiego 152, 31-342 Cracow,POLAND.

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FIG. 1.

Contour of integration, C, for the total (sea- and valence-quark) contribution: C cannot be Wick-rotated withoutpicking up the valence quark contribution. Notation for various labels used in the text is visualized.FIG.

2. Contour of integration for the sea-quark contribution: C can be Wick-rotated to the contour C’.

Upon bosonization,the sea-quark effects can be described by mesonic degrees of freedom.24

TABLE I. Matrix elements of τ · br|K, 0 >|K, 1 >|K −1, 1 >|K + 1, 1 >< K, 0|00qK2K+1−qK+12K+1< K, 1|00−qK+12K+1−qK2K+1< K −1, 1|qK2K+1−qK+12K+100< K + 1, 1|−qK+12K+1−qK2K+100TABLE II. Matrix elements of σ · br|K, 0 >|K, 1 >|K −1, 1 >|K + 1, 1 >< K, 0|00−qK2K+1qK+12K+1< K, 1|00−qK+12K+1−qK2K+1< K −1, 1|−qK2K+1−qK+12K+100< K + 1, 1|qK+12K+1−qK2K+100TABLE III.

Matrix elements of σ · L|K, 0 >|K, 1 >|K −1, 1 >|K + 1, 1 >< K, 0|0−pK(K + 1)00< K, 1|−pK(K + 1)-100< K −1, 1|00K −10< K + 1, 1|000−K −225

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출처: arXiv:9208.257 • 원문 보기