---

DOE/ER/40322-155, U. of MD PP #92-193

arXiv:hep-ph/9208256v1 27 Aug 1992DOE/ER/40322-155, U. of MD PP #92-193Response of nucleons to external probes in hedgehog models:I. Electromagnetic polarizabilitiesWojciech Broniowski∗and Thomas D. CohenDepartment of Physics and Astronomy, University of MarylandCollege Park, Maryland 20742-4111Electromagnetic polarizabilities of the nucleon are analyzed in a hedgehog modelwith quark and meson degrees of freedom. Semiclassical methods are used (linearresponse theory, quantization via cranking).

It is found that in hedgehog models(Skyrmion, chiral quark models, Nambu–Jona-Lasinio model), the average electricpolarizability of the nucleon, αN, is of the order Nc, and the splitting of the neutronand proton electric(proper) polarizabilities, δα = αn −αp, is of the order 1/Nc. Wepresent a general argument why one expects δα > 0 in models with a pionic cloud.Our model prediction for the sign and magnitude of δα is in agreement with recentmeasurements.

The obtained value for αN, however, is roughly a factor of three toolarge. This is because of two problems with our particular model: a too strong piontail, and the degeneracy of N and ∆states in the large-Nc limit.

This degeneracyalso results in a very strong Nc-dependence of the paramagnetic part of the magneticpolarizability, β, which is of the order N 3c . We compare the large-Nc results to theone-loop chiral perturbation theory predictions, and show the importance of ∆-effectsin pionic loops.

We also investigate the role of non-minimal substitution terms in theeffective lagrangian on the polarizabilities of the nucleon.PACS numbers: 12.38.Lg, 12.40.Aa, 14.20.Dh, 14.60.FzTypeset Using REVTEX1

I. INTRODUCTIONPolarizabilities are important fundamental properties of particles – they determine dy-namical response of a bound system to external perturbations, and provide valuable insightinto internal strong-interaction structure. Recent measurements of electromagnetic polariz-abilities of the proton [1,2] and neutron [3] renewed theoretical interest in these quantities[4].

Many calculations of the electric, α, and magnetic, β, polarizabilities of the nucleoncan be found in the literature in models ranging from various quark models and bags [5–9]cloudy bags [10] to Skyrmions [11–13]. A quenched lattice calculation for α has also beendone [14].

Extensive reviews of the subject are available [15,16].In this paper we analyze the nucleon polarizabilities in hedgehog models, using theframework of the linear response theory. Our analysis differs significantly from previousworks by allowing the soliton to deform ( dispersive effects).

Our semiclassical methodsare described in detail in the following paper [17], referred to as (II). The reader who is notfamiliar with the basics of hedgehog models or semiclassical methods used in their treatment,is urged to read (II) before this paper.In recent years numerous hedgehog models, such as the Skyrme model [18–21], chiralquark models [22–27], hybrid bag models [28], chiral models with confinement [26,29–31], orthe Nambu–Jona-Lasinio model [32] in the solitonic treatment [33–39], were quite success-ful in describing the phenomenology of nucleon structure.

The basis for these hedgehogsmodels is the large-Nc limit [40,41] of QCD. Thus, 1/Nc should consistently be treated asan expansion parameter in this approach.

Previous analyses of polarizabilities in hedgehogmodels [11–13] have not emphasized the need for consistent Nc-counting. In particular, afrozen approximation in the treatment of the response to electromagnetic perturbations wasmade, and the effects of distortions of the soliton, which occur at a relevant Nc-level, werenot included.

In this paper we develop consistent Nc-counting rules for α and β, and eval-uate these observables in a specific model. We point out serious difficulties arising in thecase of the magnetic polarizabilities.

Special attention is paid to the issue of neutron-protonsplitting of the polarizabilities, which has already been reported elsewhere [42].Throughout this paper we use the so-called proper (or static) polarizabilities, α and β,defined via the energy shift of an object in constant electric, E, or magnetic, H, fields [43]:δE = −12αE2 −12βH2. (1.1)These are directly related (see Sec.

III F) to the coefficients α and β encountered in theCompton amplitude [44–47,15,16,12]α = α + ∆α, ∆α = Q2⟨r2⟩E3M, β = β,(1.2)where the recoil term ∆α involves the charge, Q, mass M, and electric mean square radiusof the particle. Although the experimental errors are still substantial [48], the new measure-ments indicate that the proper electric polarizability is larger for the neutron than for theproton,αn = (12.0 ± 1.5 ± 2.0) × 10−4fm3,αp = (7.2 ± 1.0 ± 1.0) × 10−4fm3,(1.3)2

and that the magnetic polarizabilities are positive, and equal within experimental errors:βn = (3.1 ∓1.5 ∓2.0) × 10−4fm3,βp = (3.4 ∓1.0 ∓1.0) × 10−4fm3. (1.4)A number of results presented in this paper are generic to any hedgehog model, e.g.

theNc-counting rules and the role of the intermediate π −∆states. Our numerical results areobtained using the simple model of Ref.

[22] (Sec. II of (II)).

Since this model possessesboth quark and meson degrees of freedom, it is in this respect representative of a generalclass of hedgehog models. Our calculations can be repeated in any hedgehog model withminor modifications.The organization of this article is as follows: In Sec.

II we write down the gauged σ-model lagrangian, introduce collective coordinates in the usual way (Sec. III F of (II)),and classify various electromagnetic perturbations according to the hedgehog symmetries.We discuss the sea-gull and dispersive contributions to the Compton amplitude (Sec.

II B),and present the Nc-counting rules (Sec. II D).

For the electric polarizabilities we find thatthe neutron-proton splitting effect is two powers of Nc suppressed compared to the averagenucleon value:αN = (αn + αp)/2 ∼Nc,δα = αn −αp ∼1/Nc. (1.5)For the magnetic case the Nc-counting is more complicated (Sec.

IV).In Sec. III we obtain numerical results for the electric polarizabilities.

For αN, the sea-gull contribution (Sec. III A) is dominant, and the valence quark effects enter at the levelof 10 %.

This is due to the long-range nature of the pion. Our model prediction is about afactor of three too large than the experimental number.

We discuss this discrepancy, whichis due to two resons. Firstly, our specific model has a pionic tail which is too strong (i.e.the value of gπNN is too large compared to nature).

This enhances the model predictionby about a factor of 2.A more fundamental reason is discussed in Sec.V, where wepoint out that hedgehog predictions for some observables largely overestimate results due tothe implicit treatment of the ∆resonanse as degenerate with the nucleon. Our predictionfor δα = αn −αp are less sensitive on the strength of the pionic tail, and they are notaffected by problems discussed in Sec.

V. Numerically, we obtain δα = 5.4 × 10−4fm3 [42],in agreement with the sign and magnitude given by the experiment. This splitting arisesnaturally in hedgehog models when dispersive effects are taken into account.

The effect isdominated by the distortion of the pionic cloud. We also present a classical argument whythe sign of δα is expected to be positive in models with pionic clouds (Sec.

III E).In Sec III F, we show that the standard retardation correction (Eq. 1.2) strictly holdsin our linear response method.

In Sec. IV we discuss the issue of magnetic polarizability inhedgehog models.

We show that the degeneracy of the ∆and nucleon masses in the large-Nclimit precludes the use of linear response theory to determine β. The N −∆paramagneticcontribution, βN∆(given by the Born term with the intermediate ∆state) dominates theNc behavior of β, and leads toβN = (βn + βp)/2 ∼N3c ,(1.6)3

In our view, this invalidates the claims of Refs. [11–13] that the diamagnetic sea-gull inter-action cancels the paramagnetic ∆term, since these terms occur at different Nc-levels InSec.

IV C we also show the relevance of dispersive contributions in the calculation of themagnetic polarizability.In Sec. V we compare our results to predictions of the chiral perturbation theory (χPT).We find, that in the chiral limit the hedgehog predictions and the χPT predictions for αand β agree up to a factor which can be attributed to the role of the ∆in pionic loops.We discuss interesting physical implications of this issue, and point out that while the naivehedgehog predictions overestimate the role of the ∆resonance in hadronic loops, the naiveapproach to χPT drops these important contributions altogether.

We discuss the effects offinite N −∆mass splitting in these approaches, and propose how the hedgehog and χPTresults should be corrected.In Sec. VI we study the effects of the pionic substructure on nucleon polarizabilities.These effects are introduced via non-minimal substitution terms L9 and L10 [49,50] in theeffective lagrangian.

At the mean-field level, we obtain simple expressions involving pionpolarizability and pion mean squared radius. Numerically, these pion structure contributionsenter at the level of ∼1 −2 × 10−4fm3, which is a small but non-negligible effect.In Sec.

VII we make a few comments relevant to calculations in other hedgehog models.Throughout the paper we use the units ¯h = 1, c = 1, e2/(¯hc) = 1/137.II. ELECTROMAGNETIC PERTURBATIONSOur basic methods of treatment of external perturbations in hedgehog models are de-scribed in (II).

The approach developed in (II) is a rather straightforward adaptation of theRPA method of traditional many-body physics. The diference with the traditional nuclearphysics case is the presence of mesonic degrees of freedom, relativistic dynamics, and specialsymmetries (Sec.

II B in (II)).Polarizabilities are defined as second-order energy shifts due to (static) external elec-tromagnetic fields, Eq. (1.1), or, equivalently, via Compton amplitude (Sec.

II B). Theyarise from two types of terms: sea-gulls, which are due to quadratic terms in E or B in thelagrangian, and from dispersive terms.

Sea-gull effects result in local expressions involvingthe pion mean field [11–13], while the dispersive terms are given by the usual second-orderperturbation theory expressions, which result from linear perturbations in the lagrangian.These dispersive effects distort the soliton. They are particularly important in the neutron-proton splitting effects, as well as in the magnetic response, and have not been consideredin earlier works.

More importantly, these effects enter at the same Nc-level as the sea-gullterms, and thus should be included in order to comply with the basic organizational principleof the hedgehog approach, namely, the 1/Nc-perturbation expansion. The dispersive termsin the Compton amplitude correspond, in our approach, to diagrams in which the propa-gator between the two photon insertions is the RPA (or linear response) propagator (Fig.1).

Relevant expressions are obtained using equations-of-motion method, which amountsto solving linear differential equations with potentials and driving terms determined by thesolitonic profile. In addition, electromagnetic perturbations are carried in the presence ofcranking, which ensures projection on states with good quantum numbers (Sec.

II G in(II)).4

A. Lagrangian in presence of electromagnetic interactionsThe first step is to identify perturbations resulting from coupling to electromagneticinteractions, which are introduced by gauging the σ-model lagrangian (II.2.1):L = ¯ψ [˙ı/∂+ g (σ + ˙ıγ5τ · π)] ψ+ 12(∂µσ)2 + 12(∂µπ)2 −U (σ, π) . (2.1)Through this procedure one generates a dispersive interaction, as well as a covariantizingsea-gull term:Ldisp.

= −eAµψ 12Nc+ 12τ3γµψ + (π × ∂µπ)3,Lsea−gull = 12e2AµAµπ21 + π22,(2.2)where ψ is the quark field, π is the pion field, and Aµ is the photon field. It is hoped thatthis minimal substitution procedure gives the bulk of electromagnetic interactions in oureffective hadronic lagrangian, however, non-minimal substitution terms may a priori play animportant role.

In Section VI we discuss the role of simple non-minimal substitution termsin effective lagrangians, and find that their effects enter (in the electric polarizability) at thelevel of 10 %, compared to the terms resulting from (2.2).To identify the appropriate perturbations, we make a transformation of the full la-grangian (II.2.1, 2.2) to the isorotating frame, according to Eqs. (II.3.22).

This is donebecause we are interested in the linear response of the nucleon, which is “projected out” ofthe hedgehog (see Sec. III in (II) for details of projection via cranking, and linear responsein presence of cranking).For a more compact notation, we also replace the quark fieldbilinears by Nc times the bilinears of the valence quark, q.

The Lorentz gauge is used, andA0 = −r · E, A = −12r × B. We obtainL →L0 + L1λ + L2λ + L1E0 + L1E1 + L2E1 + L2λE1+ L1B0 + L1B1 + L2B1,(2.3)L0 = Nc q (˙ı/∂+ g (σ + ˙ıγ5τ · π)) q+ 12(∂µσ)2 + 12(∂µπ)2 −U (σ, π) ,(2.4)L1λ = −Nc q† 12λ · τq −λ · (π × ˙π) ,(2.5)L2λ = 12(λ × π)2,(2.6)L1E0 = e r · E 12q†q,(2.7)L1E1 = e r · ENc q† 12c · τq + c · (π × ˙π),(2.8)L2E1 = 12e2(r · E)2(c × π)2,(2.9)L2λE1 = −e r · E (c × π) · (λ × π),(2.10)5

L1B0 = −14e (r × B)iq†αiq,(2.11)L1B1 = −12e (r × B)i(Nc q† 12c · ταiq−c · (π × ∇iπ)),(2.12)L2B1 = −18e2 (r × B)2(c × π)2,(2.13)where λ is the cranking velocity, and the collective vector c = 12Tr[τ3BτB†] is defined inApp. B in (II).

In the above equations, subscript λ refers to term arising upon cranking, E0and E1 denote isoscalar and isovector electric perturbations, and B0 and B1 denote isoscalarand isovector magnetic perturbations. Indices i in Eqs.

(2.11 - 2.12) are spatial Cartesianindices. Superscript 1 denotes dispersive terms, which lead to linear shifts in fields accordingto perturbation theory.

Superscript 2 denotes terms which are quadratic in perturbations.They include a term quadratic in λ (Eq. 2.6), sea-gull terms (Eqs.

2.9,2.13), and a mixedterm (Eq. 2.10), which will play an essential role in the splitting of the neutron and protonpolarizabilities.

The classification of various dispersive terms, as well as the explicit formsof the corresponding sources arising in linear response equations, are given in Table I. Thegrand spin (sum of spin and isospin) is denoted by K, parity is denoted by P, and thegrand-reversal symmetry by R (see Sec. II B in (II) for details).B.

Compton amplitude and polarizabilitiesThe Compton amplitude corresponding to lagrangian (2.3) reflects the presence of bothdispersive and sea-gull terms. It can be written as [12]˜Mµν(p, q) = Mµν(p, q) + Sµν(q),Mµν(p, q) =˙ıZd4x e˙ıq·x⟨N(p′)|TJe.m.µ(x)Je.m.ν(0)|N(p)⟩,(2.14)where Mµν is the dispersive T-product part, and Sµν is the sea-gull part.

Correspondingly,the polarizabilities have dispersive as well as sea-gull parts, and can be written as [12]α = αdisp. + αsea−gull,β = βdisp.

+ βsea−gull,αdisp. = 2Xb′|⟨N|R d3x cE · rJe.m.0(r)|b′⟩|2Eb′ −EN,αsea−gull = ⟨N|Zd3x (cE · r)2S00|N⟩,βdisp.

= 2Xb′|⟨N|R d3x 12cB · (r × J e.m.(r))|b′⟩|2Eb′ −EN,βsea−gull = 14⟨N|Zd3x ǫmni bBmrnǫklj bBkrlSij|N⟩. (2.15)Hats denote unit vectors in the direction of E or B fields, and |b′⟩is an intermediate statewith energy Eb′.6

The sea-gull parts of expressions (2.15) can be readily identified from Eq. (2.9,2.13).One getsS00 = e2(c × π)2,Sij = −e2δij(c × π)2.

(2.16)Evaluating these in the collective nucleon state one obtains (App. B in (II), [12,13])αsea−gull = −2βsea−gull = 8πe29Zdr r4π2h,(2.17)where πh is the hedgehog pion field profile.In Eq.

(2.17) we recognize the hedgehogmodel relation between the seagull contributions to the electric and magnetic polarizabilities[12,13]. The radial integral in Eq.

(2.17) is quadratic in the pion field. In the Skyrme modeladditional terms arise due to higher-order terms in the lagrangian [11–13].C.

Linear responseIn our treatment of dispersive contributions, the intermediate states |b′⟩in Eq. (2.15)correspond to RPA excited states, and the energy denominators involve energies of theseexcitations (Fig.

1). These contributions can be written in the general form given in Eqs.(II.3.27).

The electric and magnetic dispersive polarizabilities result from either isoscalardipole or isovector dipole transitions, and will be labeled by E0 and E1, or B0 and B1,respectively.These dispersive contributions, as well as the sea-gull contributions (2.17)contribute equal amounts to the neutron and the proton.This is because the resultingcollective operators in Eqs. (II.3.27) are isoscalar (bilinear in the collective vector c), andmatrix elements are equal for the proton and the neutron.

The neutron-proton splittingeffects arise from the mechanism described in Sec. III G in (II).

This can be briefly describedin the following way: We perform electromagnetic perturbations not on hedgehogs, but onnucleons. These are obtained from hedgehogs via projection (cranking), which in turn mayalso be viewed as linear response.

It is sufficient to work to linear order in the crankingvelocity, λ, and we can write down symbolically|N⟩= (1 + GVcr)|H⟩,(2.18)where G is the RPA propagator, Vcr is the cranking interaction, and |H⟩is the hedgehogstate. Using these “cranked” states in our perturbation theory leads to expressions of theform of Eqs.

(II.3.29), with one cranking, and two electromagnetic interactions. As discussedin Sec.

III G in (II), the KP numbers of interactions in Eqs. (II.3.29) have to be additive toKP = 0+, since the hedgehog has KP = 0+.

From Table I we can see that we can composecranking, one isoscalar electric (magnetic), and one isovector electric (magnetic) interactionto KP = 0+. Since Vcr carries a collective operator λ, the appropriate collective operator isisovector, and neutron-proton splitting of polarizabilities is generated.From a slightly different but equivalent point of view, we may understand the isospineffect in electric polarizabilities by inspecting the term (2.10).

It is linear in cranking, andlinear in isovector electric (E1) perturbation. It leads to the following contribution to α:7

αmes.λ= −e ⟨N|Zd3r (cE · r)[(c × π) · (λ × π)]|N⟩. (2.19)If the pion field in the above equation were taken to be just the hedgehog profile, then αmes.λwould vanish by parity.

However, the isoscalar electric perturbation (Section III C) distortsthe meson fields, and π = πh + δπE0, where δπE0 has S- and D-wave components. As aresult, the integral is non-zero.

The corresponding collective operator is c · λ = I3/Θ, whereΘ is the moment of inertia (see App. B in (II)), and we obtainαmes.λ= −e I3Θ 8/3Zdr (cE · r)(πh · δπE0), .

(2.20)From the quark parts of Eqs. (2.3) we getαquarkλ= 2⟨N|Zd3r [−δq†E0 12λ · τδqE1 + δq†E1 ecE · r2Ncδqλ+ δq†E0 12ecE · rc · τδqλ]|N⟩+ h.c.,(2.21)where various terms in the interaction are sandwiched with shifts corresponding to otherinteractions.

Eqs. (2.20,2.21) can be straightforwardly obtained from Eq.

(II.3.29). Terms(2.20,2.21) are proportional to the nucleon isospin, I3, thus are responsible for splitting ofthe neutron and proton electric polarizabilities, δα.For the splitting of magnetic polarizabilities, mesons do not contribute at the linear-response level, since there is no mesonic analog of the term (2.10) in lagrangian (2.3).

Onlythe magnetic analog of the quark part (Eq. 2.21) is present.

However, the linear responsecalculation of the magnetic polarizability encounters fundamental problems, which will bediscussed in Sec. IV.D.

Nc - countingIn Eqs. 2.3 we list explicitly occurrences of Nc.

We also recall [41] that in the large-Nclimit the meson fields scale as √Nc, the quark fields scale as 1, and the moment of inertiascales as Nc. Note that the sources for isoscalar electromagnetic perturbations are one powerdown compared to the corresponding sources for the isovector interactions.

We easily arriveat the following Nc-rules for electric polarizabilities:αsea−gull ∼Nc, αE0 ∼1/Nc, αE1 ∼Nc, αλ ∼1/Nc. (2.22)This leads directly to the following rules for the nucleon polarizabilities:αN = αn + αp2= αsea−gull + αE1 ∼Nc,δα = αn −αp = αλ ∼1/Nc.

(2.23)Note, that since the αE0 part of the dispersive term contributes to αN at a subleadinglevel, it should be dropped according to Nc-rules. Other physical effects affect our resultsat this level, e.g.centrifugal stretching, center-of-mass corrections to the soliton mass,Nc-suppressed terms in the effective lagrangian, etc.

Numerically, the αE0 contribution isnegligible, confiring the validity of Nc-counting for polarizabilities.Naively, one would write down expressions analogous to Eqs. (2.23) for the magneticpolarizability, β.

We will show in Sec. IV that this is not correct.8

III. ELECTRIC POLARIZABILITYIn this section we present our numerical results for α.

Numerical methods, explicit formsof the differential equations, and other details are given in (II). The model parameters forour numerical calculations are taken from Refs.

[22,51]: Fπ = 93MeV , gFπ = 500MeV ,mπ = 139.6MeV , and mσ = 1200MeV .A. Sea-gull contributionBecause the pion field is long ranged, the nucleon electric polarizability, αN, is dominatedby the pionic sea-gull contribution (2.17).

Numerically, in our model we getαsea−gull = 28 × 10−4fm3,(3.1)which is roughly three times larger than the experimental value in (1.3). There are twobasic reasons for this discrepancy.

The first one is discussed below, and the other in Sec.V. Figure (2) shows the radial density of the integrand of (2.17) (solid line).

It is clearthat the sea-gull term acquires most of its value from the asymptotic region, r > 1fm.Recall, that the asymptotic behavior of the pion tail in hedgehog models is determined bythe pion-nucleon coupling constant, gπNN [20,51]. Using the Goldberger-Treiman relation,we can write downπasympt.h= (3gA)/(8πFπ)ˆx(mπ + 1/r)exp(−mπr)/r.

(3.2)Our model predicts gA = 1.86 [51], hence the tail contribution is overestimated by a factorof (gmodelA/gexp.A)2 ∼2. The dotted line in Fig.

(2) shows the integrand of Eq. (2.17) with thepion field having the form (3.2) in the whole region of r. This corresponds to the chiral limitcase, and will be discussed in detail in Sec.

V. We can see from Fig. (2) that the size ofαsea−gull is controlled by how fast the solid line departs from the dashed line at lower values ofr.

In other words, the results are sensitive to the profile of the pion field in the intermediateregion of about 1fm. Note, that πh enters the expression (2.17) quadratically.

Therefore,we expect substantial model sensitivity in hedgehog predictions of αsea−gull, e.g. modelswhich have a suppressed pion field, such as hedgehog models with confinement [26,29–31],or models with vector mesons, e.g.

[52,53], are expected to predict lower values for αsea−gull.We stress that to compare fairly the model predictions with experiment, the model shouldpredict correctly the quantity gA/Fπ, which enters the asymptotic form (3.2).In Sec. V we show how the sea-gull contribution is additionally suppressed when N −∆mass splitting effect is taken into account.

This results in another factor of ∼2 reduction.With these corrections we note that it is possible to put the sea-gull contribution in theright experimental range. The model uncertainty, however, is big.B.

Isovector electric perturbationIn Sec. II D we have shown that the dispersive electric isovector (E1) contribution topolarizability, αE1, is of the order Nc, the same order as for αsea−gull.

The E1 perturbation9

in lagrangian (2.8) has KPR = 0−−, 1−−and 2−−(Table I). Since it is odd under grand-reversal R, only the valence quark fields acquire shifts (Sec.

III A in (II)), and we solveequations of the form of Eqs. (II.3.7):(h −ε)qE1 = 12Nce r · cE c · τqh.

(3.3)These equations are decomposed into K = 0, 1 and 2 components, as described in App. Ain (II), and solved numerically.

The corresponding polarizability is calculated according tothe expressionαE1 = −2Nce ⟨N|Zd3xq†E1r · cEc · τqh|N⟩,(3.4)where |N⟩is the nucleon collective state (App. B in (II)).

Note that with our definition, thespinor qE1 carries the collective variable c, such that Eq. (3.4) contains a matrix element ofa collective operator quadratic in c, similarly to the case of the sea-gull part in Sec.

II B.Numerically, we get in our modelαE1 = 3.49 × 10−4fm3,(3.5)with K = 0, 1, and 2 components of qE1 contributing −12%, 26%, and 86%, respectively.The dashed line in Fig. 2 depicts the radial density of αE1.

Because the quark mass inour model is 500MeV , the purely quark αE1 contribution is strongly suppressed comparedto the pion sea-gull contribution. However, the value of αE1 is non-negligible compared toexperimental numbers (Table II), and quark effects are substantial.In models with vector mesons [52,53], the E1 perturbation involves shifts in the ρ and ωmesons, and these effects must be included in calculations of the electric polarizability [54].They enter at the leading-Nc level, and should contribute comparably to the valence quarkeffects.C.

Isoscalar electric perturbationNow we turn to a more difficult case of the isoscalar electric perturbation, E0, whichhas KPR = 1−+.These are quantum numbers of the translational zero mode, and, asexplained in Sec. III A in (II), we have to deal with full RPA equations of the form (II.3.3).The appearance of the zero mode is easy to understand.

The hedgehog soliton possessesisoscalar electric charge, QI=0 = 12e. Thus, in a constant electric field, the soliton acceleratesin the direction of E — the translational mode is excited.

In addition to this zero-modemotion, the soliton is getting distorted (as viewed from the center of mass).Below weshow that the excitation of the translational motion corresponds to the Thompson limit ofthe Compton scattering, and the deformation corresponds to the polarizability. In orderto separate the zero mode from the physical modes, instead of a constant electric field weconsider a slowly time-dependent field of the formE(t) = 12E0e−˙ıωt + E∗0e˙ıωt(3.6)The resulting sources in Eq.

(II.3.3) are10

jX = jY = 12 e N−1c (r · cE)qh,jZ = jP = 0. (3.7)Using the technique described in Sec.

III B in (II), we find that the total isoscalar electricpolarizability consist of the translational zero-mode part, and a “physical” part:αtot.E0 = αzeroE0 + αphys.E0,(3.8)where the zero-mode part diverges in the limit ω →0,αzeroE0= −(QI=0)2Mω2 ,(3.9)and the physical part, αphys.E0, has a finite ω →0 limit.The quantity M in Eq. (3.9)is the soliton mass.

In the expression for the forward Compton scattering amplitude, αis multiplied by ω2 [47,44,16]. We note immediately that ω2αzeroE0= −(QI=0)2Mis just theThompson term in scattering of a particle with charge QI=0 and mass M. Thus, the zero-mode part is responsible for the Thompson limit.

Since RPA leads to small fluctuationequations of motion, analogously to the case of classical physics [44], it is clear it leads tothe correct Thompson limit.In the case of the E1 perturbation, the quantum numbers precluded excitation of a zero-mode on top of the hedgehog solution. This is a manifestation of the fact that the hedgehogdoes not have any isovector electric charge, QI=1 = 0.

This charge arises only upon cranking,but the organization of our perturbation theory is such, that we treat cranking, E0 and E1perturbations separately (Sec. III G and V in (II)).

If linear response were performed on acranked soliton, then the full charge Q = QI=0+QI=1 would appear in Thompson scattering.Earlier works [12,13] on electric polarizabilities in hedgehog models did not take intoaccount the effects of zero modes. In fact, this is justified by the 1/Nc expansion.

The E0contribution is suppressed by two powers of Nc compared to the E1 contribution (Eq. (2.22).Therefore, if one is interested in the leading Nc behavior of αN, it is sufficient to considerthe isovector electric perturbation only, where the issue of the translational zero mode doesnot arise.Since we are interested in the splitting of the neutron and proton polarizabilities, wehave to calculate the quantities αmes.λ(Eq.

(2.20)) and αquarkλ(Eq. (2.21)).

Thus, we haveto find the shifts in the fields due to the isoscalar perturbation, E0. We need to extract the“physical” parts of the solution.

The numerical procedure has been described in Sec. IIIC in (II).

Here we only remark, that very good numerical accuracy is necessary in order toseparate the zero mode from the physical mode. This is because at small values of ω thesolution can be written asξ =˙˙ıathQ0Mω2 ξ0 −Q0Mωξ1 + ξphys + O(ω),(3.10)where ξ0 is the translational mode, ξ1 is the conjugated mode (“boost mode”), and ξphys.is the physical mode which we want to extract (Sec.

III B in (II)). Solutions of equations(II.3.3) in the limit ω →0 give full ξ, from which we subtract the pieces divergent in ω.This can easily be done, since we know the exact forms of ξ0 and ξ1 (App.

C in (II)). Thepresence of the zero mode provides a useful algebraic check of equations from App.

A in(II), since the divergent parts of Eq. (3.10) have known coefficients.11

D. Neutron-proton splitting of electric polarizabilitiesHaving solved the linear response equations for the E0 perturbation, we use qphys.E0=Xphys. +Y phys and φphys.

= 2Zphys. (App.

A in (II)) in equations (2.20, 2.21). The dominantpart to αλ comes from the pionic contribution.

The pion shift has S- and D-wave components(Table I), which we denote by πS and πD, respectively.The explicit expression for themesonic contribution of δαmes has the formδαmes = −16 e3Θs4π3Zdr r3πh(q1/3πS −q2/3πD. (3.11)The quark contribution is obtained analogously from Eq.

(2.21). The numerical results giveδα = −5.6 × 10−4fm3,(3.12)with the quarks carrying −5% of the total.Hence, as in the E1 case, we observe thedominance of the pionic contribution.

Figure 3 shows the radial density of expression (3.11).The long-range nature of the pion is evident. As discussed in Sec.

III B, our predictions areenhanced due to the fact that gA has a too large value in our model. In the present case,however, we estimate this effect at the level of gmodelA/gexp.A∼1.4, rather than a factor of 2 inthe pionic sea-gull term.

This is because Eq. (3.11) is linear in πh (recall that asymptoticallyπh is proportional to gA, Eq.

(3.2)), and πS and πD are not dependent on the strength ofthe pionic tail in πh. The reason is that the sources in the linear response equations comefrom the quarks, which are short-ranged.

In the asymptotic region, where Eq. (3.11) getsmost of its contribution, the shifts in the pion field, πS and πD, depend on the total strengthof the source (QI=0 = 12), and the pion mass, which enters the Green’s function, but not onthe strength of the tail of πh.

Consequently, our overestimate of δα should be at the levelof 40% only.Our results are summarized in Table II. We note, that the sign of the neutron-protonsplitting effect, and its magnitude, agree with the recent experimental numbers.E.

Pionic cloud and sign of δαIn the previous section the sign of δα = αn −αp was found to be negative. The followingplausibility argument can be given as to why we expect this behavior in models with pionicclouds.

The hedgehog models imply that the nucleon consists of a quark core, carryingisoscalar charge, and a pion cloud, carrying isovector charge. In the proton, these chargeshave the same sign, and the electric field distorts both the core and the cloud, but it does notdisplace them relative to each other.

In the case of the neutron, the cloud and the core, inaddition to being deformed also get displaced, since their charges are opposite. This resultsin additional polarizability.

Calculations based on other methods also give αn > αp [55,56].F. Retardation termOur calculation was performed with a spatially constant electric field.

Now considerplane-wave photons, as in Compton scattering, i.e.12

E = E0exp(˙ık⊥· x) ≃E0(1 + ˙ık⊥· x −12(k⊥· x)2),k⊥2 = ω2. (3.13)The linear-response source, j, is modified accordingly, and the charge Q = ξ0†j (Sec.

III in(II)) is replaced by an effective charge Qeff. = Q(1−16ω2⟨r2⟩E).

Substituting this expressionto the expression for the zero-mode part of the polarizability, Eq. (3.9), we obtain the usualretardation term, exactly as in the classical derivation of ref.

[44], as well as in more generalderivations [16,47].IV. MAGNETIC POLARIZABILITYAlthough naively one would think that the analysis of the magnetic polarizabilities wouldparallel the electric case, it turns out that there are fundamental difficulties.

The reasonis the non-commutativity of the ω →0 limit of the Compton scattering, and the large-Nclimit, in the magnetic case. As stressed earlier, since the large-Nc limit is the basic principlebehind the hedgehog approach, one has to work consistently in the Nc counting.

We pointout that with the linear response approach to the magnetic polarizabilities it is not possible.We also show importance of pionic dispersive terms in the magnetic polarizability, whichenter at the same level as the pionic sea-gull contribution. Such terms were neglected inprevious works [11–13].A.

The N −∆Born termThe importance of the Born term with intermediate ∆state in the estimates for β is awell recognized fact. The hedgehog possesses isovector magnetic moment, and the magneticinteraction term in the effective hamiltonian has the form [11,13]Hmagn.

= 3µI=1c · B,(4.1)where µI=1 is the isovector magnetic moment, and the collective vector c is defined in App.B in (II). Hamiltonian (4.1) leads to N −∆transitions, and, according to perturbationtheory, the contribution to the magnetic polarizability isβN∆= 18(µI=1)2|⟨N|c0|∆⟩|2M∆−MN=4(µI=1)2M∆−MN,(4.2)where in the last equality we have used Eq.

(II.B.9). Now, recalling the Nc-counting rules[19,51]: µI=1 ∼Nc, M∆−MN ∼1/Nc, we immediately obtain the resultβN∆∼N3c .

(4.3)This surprising “superleading” behavior is the result of collective effects in the hedgehogwave function. In the orthodox approach to Nc-counting, one should stop the analysis atthis point, and conclude that as far as the magnetic polarizability is concerned, one is notclose to the Nc →∞limit: according to Eq.

(4.3) β should be much larger than α, which13

contradicts the experiment. Using the physical values for various magnitudes in Eq.

(4.2)we obtain the numerical valueβN∆∼12 × 10−4fm3. (4.4)Note, that to compare fairly to experiment, a given model has to predict properly thevalue of µN∆.A related fundamental point is that the polarizabilities are defined in ω →0 limits in theCompton amplitude, and in the physical world the resonances are at finite values of ω, wellseparated from the ω = 0 region.

In the large-Nc limit, the ∆-resonance occurs at ω = 0.Thus, the meaning of the polarizability is in fact lost, unless the ω →0 limit is taken beforethe Nc →∞limit. In the linear response method, the limits are implicitly taken in thereversed order.In principle, a consistent analysis of the Nc-subleading physics is possible, but this wouldrequire a fully quantum-mechanical (not semiclassical) treatment, e.g.

one could use theKerman-Klein method [57]. Technically, this would be a tremendous effort, and of question-able merit in a simple effective theory such as hedgehog models.In order to be able to make some estimates, we relax the strict Nc-counting requirementin the rest of this section, and try to examine the subleading terms in Nc in a more “flexible”approach.B.

Sea-gull termIn hedgehog models, the magnetic seagull term is related to the electric sea-gull term bythe relation (2.17). In enters at the level N1c .

Numerically,βsea−gull = −14 × 10−4fm3,(4.5)which would almost exactly cancel the βN∆term, in apparent rough agreement with exper-iment. This is, however, not the full story at the level Nc.

In the next section we show, thatthe dispersive contributions to β arise of the same order as βsea−gull, but have positive sign,and the desired cancellation is largely suppressed.C. Dispersive termsThe calculation of the magnetic dispersive terms goes along the same lines as the calcu-lation of the electric polarizability.

There are, however, a few important differences. Firstly,for the ever-R B1 perturbation the pionic term in Eq.

(2.12) leads to pionic sources in thelinear response equations. Since these are long-ranged (proportional to πh), we get strongdispersive effects.

The sources are listed explicitly in Table I. For the K = 0++ and 2++cases we obtain equations of the form (II.3.8).

The only difference is that in the K = 0++the valence quark eigenvalue, ε, changes, and the resulting quark equation in (II.3.8) hasthe form(h −ε)δq+ −δεqh −2gMqhZ = j+q ,(4.6)14

where δε is the shift in the quark eigenvalue.This shift may be viewed as a Lagrangemultiplier ensuring the orthogonality of the shifted quark spinor, δq+, and the hedgehogquark spinor, qh.The numerical methods of treating the K = 0++ and 2++ cases arestraightforward.In the K = 1++ case, however, the perturbation excites the rotational zero-mode. Thisis a similar mechanism as in the case of the excitation of the translational zero mode inthe E0 case — KPR = 1++ are the quantum numbers of the rotational mode in spin, orisospin (these are equivalent, since the hedgehog soliton is a KPR = 0++ object).

Thus, thehedgehog baryon placed in a constant magnetic field starts to rotate, and continues to spinup indefinitely. Quantum-mechanically, if time-dependent perturbation theory were used, itwould correspond to transitions to higher spin and isospin states.There are a few subtle issues here, concerning the order of limits.

Suppose we perform(exact) projection first (e.g. suppose we have done a calculation using the Kerman-Kleinmethod [57]), and have states with good angular momentum and isospin on which we applythe magnetic perturbation.

Then, the rotational zero modes no longer appear, since theyarise only if the solution breaks the symmetry of the lagrangian (Sec. III B in (II)).

Thus,the appearance of the rotational zero mode in the KPR = 1++ is the effect of starting fromthe unprojected hedgehog. In fact, this problem is another manifestation of the noncom-mutativity of the ω →0 and the Nc →∞limits, described in Sec.

(IV A). One may hope,that in an improved treatment, the zero-mode physics is described by the N −∆term, Eq.(4.2).

With this in mind, in order to estimate the size of the dispersive effects, we projectout the divergent zero-mode part from the K = 1 contribution, and retain the physical part.The total result for β consists of βN∆, βsea−gull, and βdisp. with the zero-mode contributionprojected out.

As mentioned before, this is not a consistent procedure, but it allows us toestimate the importance of various effects in the magnetic polarizability.The numerical values obtained for the K = 0, 1 and 2 parts of the B1 magnetic pertur-bation are 3.6, −0.6 and 3.3 × 10−4fm3, respectively. The total dispersive contribution toβN isβdisp = 6.5 × 10−4fm3,(4.7)The quark contribution is negative, and carries −15% of the total.The isoscalar magnetic perturbation, B0, which is a 1/Nc effect, and comes entirely fromthe quarks, contributes 0.2 × 10−4fm3, which is negligible, as expected from Nc counting.We also expect the neutron-proton splitting of magnetic polarizabilities, δβ, to be small.

Itcomes entirely from the quarks, since there is no magnetic analog of the electric term (2.10),and is expected to be much smaller than δα, which had a pionic contribution.At this point it might seem that adding up contributions (4.5,4.7,4.4) we still get apartial cancellation. However, the sea-gull and dispersive pieces are reduced if the pion fieldhas the correct strength by a factor of ∼2, as in the electric sea-gull case.

The effectsdiscussed in Sec. V further reduce these values, by another factor of 2.

These effects causethe cancellation to disappear, and we are left with a large value of βN. As stressed before,rigorous estimates could only be done in a calculation beyond linear response.15

V. ROLE OF THE ∆IN HEDGEHOG MODELS AND CHIRAL PERTURBATIONTHEORYIn this section we compare the hedgehog model predictions to the results obtained inchiral perturbation theory χPT. At first, it may seem awkward, since the two methods arebased on two different limits: Nc →∞, and mπ →0, which are known not to commute,and give different results for various observables [20].

However, as we show in Ref. [58], forquantities which are divergent in the chiral limit as m−1π , there is a simple connection betweenthe hedgehog predictions and the χPT predictions for scalar-isoscalar and vector-isovectorquantities (electromagnetic polarizabilities αN and βN are scalar-isoscalar).Let us evaluate the polarizabilities using our methods in the chiral limit.

For αN, thedominant part comes from the sea-gull term. In the chiral limit the quantity diverges as m−1π ,and all of the contribution comes from the pionic tail, which has the form (3.2).

Evaluatingintegral (2.17) with the profile (3.2 we obtainαN =5e2g2A32πM2mπ. (5.1)Similarly, including both the sea-gull and dispersive pieces for the magnetic polarizability,we obtain in the chiral limitβN =e2g2A64πM2mπ.

(5.2)These expressions are exactly a factor of 3 larger than the analogous expressions followingfrom χPT [55]. This difference comes from the different treatment of the ∆isobar.

In χPTthe ∆contributions in pionic loops (Fig. 4) are not included, since it is implicitly assumedthat the N −∆mass splitting is much larger than the pion mass, and consequently loopswith the ∆do not contribute to the leading singularity.

They are counted as effects of orderlog(mπ), 1, .... In hedgehog models, on the contrary, the N −∆mass splitting is a N−1ceffect, much smaller than the pion mass, which is of the order N0c .

Therefore, hedgehogmodels include the ∆on equal footing with the nucleon. As shown below, spin and isospinClebsch factors account for the difference by a factor of 3 between Eqs.

(5.1,5.2), and χPTpredictions.One may ask, how the physics of pionic loops, such as in Fig. 4, is present in hedgehogmodels.After all, the treatment of the pion field in hedgehog models is classical.Todemonstrate how hedgehog expressions in the chiral limit may be viewed as hadronic loopdiagrams such as in Fig.

4, let us evaluate this diagram in the chiral limit. First, we performa calculation of the self-energy due to an insertion of the electric sea-gull interaction in thepion loop.

In momentum space, we obtainΣ∗= ( gA2Fπ)2Zd4k(2π)4Zd4q(2π)4 τaγ5(/k + 12/q)SF(p −k)τbγ5(/k −12/q)D(k −12q) ˜V (q)D(k + 12q)Tab,(5.3)where SF is the Feynman propagator of the nucleon, D(k) = ˙ı/(k2−m2π+˙ıε) is the propagatorof the pion field, ˜V (q) = 12e2(2π)4δ4(q) 12(E · ∇q)2 is the Fourier transform of the electric16

sea-gull interaction in the coordinate space, V (r) = 12e2(E · r)2, and Tab = δab −δa3δb3. Nowwe proceed as follows: we first carry out the integral over k0 in (5.3).

For the leading chiralsingularity piece, m−1π , the contribution comes from the poles in the pionic propagators. Thepoles in the nucleon propagators contribute to less singular terms.

We can then perform thenonrelativistic reduction of the nucleon propagator and the pion-nucleon vertices, and takethe expectation value of Σ∗in positive energy spinors, in order to extract the energy shift.Transforming back to coordinate space, we obtain the expressionα = e2Zd3x φasym.a(r)φasym.b(r)Tab,(5.4)which has the form of the expression obtained in hedgehog models.We now go back to the question of the factor of 3. When collective coordinates areintroduced, expression (5.4) becomesα = e2Zd3x (φasym.

× c)2 . (5.5)In Sec.

III A, to obtain the sea-gull contribution to the αN we evaluated the collective matrixof expression (5.5) in nucleon collective wave functions. Now we repeat the procedure, butwe write the resulting matrix element of the integrand in Eq.

(5.5) asXiXa⟨N|(φasym. × c)a|i⟩⟨i|(φasym.

× c)a|N⟩,(5.6)where |i⟩is a collective nucleon or ∆intermediate state. If the sum over i is unrestricted,then we just recover our previous expressions, (5.1,5.2).

If, however, we restrict i to runonly over nucleon collective states, then using Eq. (B10) in (II), we obtain a result smallerby a factor of 3, and this result agrees with the χPT prediction.

Conversely, had the χPTcalculation included the ∆in diagrams of Fig. 4, with M∆= MN, it would predict a result3 times larger than quoted in Ref.

[55].Let us introduced = (M∆−MN)/mπ. (5.7)These two cases, hedgehog models and χPT, correspond to two limits: d →0, and d →∞.In nature, d ≃2, which is between the two limits, and we do not have the separation of thepion mass and M∆−MN scales.

In this case it seems most appropriate to treat both scalesas small, and keep d as an unconstrained parameter. This is the spirit of the approach ofRefs.

[59,60]. To estimate the contribution of the ∆at the physical value of d, we evaluatethe diagram of Fig.

4, starting from expression (5.3), with SF describing the ∆propagatorwith the physical mass, and with the vertices modified appropriately. The m−1πcontributionis easily obtained, since, as in the nucleon calculation presented above, we can performthe non-relativistic reduction.

The result for the ratio of the ∆to nucleon contribution indiagram 4 isα∆αN = 2S(d),(5.8)where the 2 comes from Clebsch factors, and the “mass suppression” function has the simpleform17

S(d) = 4πArctanq1−d1+d/√1 −d2for d ≤1Arctanhqd−11+d/√d2 −1for d > 1(5.9)For degenerate N and ∆, S(d = 0) = 1. In the limit of d →∞, S(d) ∼log d/d.

Forthe physical value of d we find S(2.1) = 0.47, and from Eq. (5.8) we find, that the ∆contribution in pionic loops of Fig.

4 to the electric polarizability is roughly equal to thenucleon contribution. The results for the magnetic polarizability are analogous.This important feature of large ∆contributions in pionic loops (for scalar-isoscalar op-erators) is also present in other quantities [58].

In the above discussion we have tacitlyassumed that the ratio gπN∆/gπNN = 3/2 [20,51], as predicted by hedgehog models. Exper-imental numbers agree with this prediction to within a few percent.

The correction may beintroduced to expression (5.8) [58].We note that the above analysis and the comparison of the hedgehog model predictionsand the χPT predictions can be done only for observables which do not depend on cranking(independent of the cranking frequency, λ). Observables, which do depend on cranking donot have the same chiral singularities, e.g.

the electric mean squared radius diverges as m−1πin hedgehog models, and as log mπ in χPT. The same is true of the splitting of electricpolarizabilities, δα.

In this case the issue of nonocommutativity of the large-Nc and chirallimits is much more complicated.The results of this section indicate how one should try to improve the hedgehog pre-dictions by subtracting the amounts by which these models overestimate the leading chiralsingularity terms. There is uncertainty in such a non-rigorous procedure, but in our view itis required by physics.

As already mentioned at the end of Sec. IV A, the correct treatmentof the ∆could be done in a quantum calculation with the inclusion of the Nc-subleadingterms.On the other hand, the predictions of χPT will be largely effected by the diagrams ofFig.

4 with ∆intermediate states. For polarizabilities, at the leading singularity level, thesediagrams are as important as the diagrams with the nucleon intermediate states.VI.

EFFECTS OF THE PIONIC SUBSTRUCTUREThe minimal substitution prescription in an effective lagrangian cannot produce all in-teractions of a hadronic system with the electromagnetic field. It is possible to write downterms which are by themselves gauge-invariant, and thus not obtainable by gauging la-grangian (2.1).

In this section we focus on terms L9 and L10 of reference [50]:L9 = −˙ıL9 Tr[F LµνDµUDνU† + F RµνDµU†DνU],(6.1)L10 = L10 Tr[F LµνUF µν,RU†],(6.2)where F L,Rµνare the left and right chiral field strength tensors. For the case of electromagneticfield we have F L,Rµν= e12τ3Fµν = e12τ3(∂µAν −∂νAµ).

In the linear σ-model, U correspondsdescribes the chiral field, U = F −1π (σ + ˙ıτ · π). The constants L9 and L10 can be expressedthrough measurable quantities, namely the pion electric mean square radius, ⟨r2⟩πE, and thepion polarizability απ = −βπ [50,49]:18

L9 = F 2π⟨r2⟩πE12,(6.3)L10 = mπF 2παπ4−L9 = mπF 2παπ4. (6.4)In the chiral perturbation theory treatment, one considers pionic loops, and through renor-malization L9 and L10 acquire chiral logarithms.

In our approach, we simply treat the terms(6.2) in the mean-field approximation, replacing the meson field operators by classical fields.As before, the E and B fields are constant, and we find from the L9 termZd3x L9 = 4e2L9F −2πZd3x F iνAν(c × πh) · (c × ∂iπh)= 2L9(E2 −B2)Zd3x(c × πh)2,(6.5)where i runs over spatial values. The L10 term leads toZd3x L10 = 4e2L10(E2 −B2)F −2πZd3x(c × πh)2 −12(σ2h + π2h).

(6.6)The last term under the integrand is canceled by the vacuum subtraction, which is implicit.It becomes −(σ2h + π2h) + σ2vac = −(σ2h + π2h) + F 2π, which is zero in the nonlinear sigmamodel, but also vanishingly small in our case due to proximity of the hedgehog solution tothe chiral circle. We can thus drop this term in our estimate.

Using (6.4, II.3.21) we readofffrom (6.5, 6.6) the following contribution to the nucleon polarizabilities:απN = −βπN = mπΘmesαπ. (6.7)The physical interpretation of this contribution is clear.

The pion, having electromagneticstructure, is polarizable. Since the nucleon is surrounded by a pion cloud, this pion polar-izability results in additional polarizability of the nucleon.

Note the opposite signs of theelectric and magnetic polarizabilities in (6.7), reflecting the fact that απ = −βπ. Also notice,that since απ is of order 1 in Nc-counting, the contributions 6.7 are of order Nc.

The terms(6.2) do not lead to additional neutron-proton splitting of polarizabilitiesWe can call Nπ = mπΘmes in 6.7 the “number of pions” in the nucleon seen in theCompton scattering process. Numerically, Nπ ≃0.5 in our model.

Using the relation of themoment of inertia to the N −∆mass splitting, we can writeNπ = 32mπM∆−MNΘmesΘ ,(6.8)where the last factor is the fraction of the total moment of inertia carried by the pion. Thisquantity is ∼1 in hedgehog models: 60% in our model with quarks, 100% in the Skyrmion.We thus have a quasi-model-independent result Nπ ∼0.5 −0.7.The value of απ can be determined experimentally, however existing experimental data[61] do not seem reliable, and are in contradiction with a low-energy theorem due to Holstein[49], which gives απ = 2.8 × 10−4fm3.

With this value we get19

απN = −βπN ∼1.3 × 10−4fm3,(6.9)which is a few times smaller compared to the minimal substitution terms, but non-negligible,especially for the magnetic case, where we expect cancellations to occur. If experimentalnumbers for απ were used [61], then a three times larger result would follow.Other non-minimal substitution terms can also be considered, but there is no knowledgeof the phenomenological low-energy constants which have to be introduced, and predictivepower is lost.VII.

OTHER MODELSThe analysis of this paper can be straightforwardly applied to other chiral models. Inpurely mesonic Skyrmions [21], the RPA approximation corresponds to linearizing the smallfluctuation equations, hence the non-linearity constraint is not imposed at the quantum level.Appropriate linear response equations can be derived along the lines of App.

B in (II). Therole of the isoscalar source is carried by the topological (Goldstone-Wilczek) current, andeffects of the higher order terms in the equations are present.In purely quark models (NJL) [32], upon minimal substitution, the lagrangian in presenceof the electromagnetic field Aµ, and after introducing collective degrees of freedom, becomesSNJL(Aµ) = −˙ı Tr log˙ı/∂−g(σ + ˙ıγ5τ · π) −12γ0τ · λ+ 12/A(N−1c+ τ · c)−(vac),(7.1)where (vac) denotes the vacuum subtraction, and an NJL cut-offis understood.

The traceaccounts for the occupied (valence) levels. This expression may be expanded to second-orderin Aµ, and expressions for polarizabilities can be easily derived.

The dispersive pieces, asin the model treated in this paper, lead in case of even-R interactions to distortions in theprofile functions σ and π. This is likely to result in numerical complications, in particular incases with zero-mode excitations.

One can use gradient expansion techniques [62] instead ofsolving the model exactly. Then, for example, the isovector electric responce of the quarksfrom the Dirac sea generates the pionic sea-gull contribution.All our generic hedgehog model conclusions described in this paper hold for these, andother hedgehog models.VIII.

CONCLUSIONThe main results obtained in this paper can be summarized as follows:1) We show the Nc-counting rules for electromagnetic polarizabilities in hedgehog models.For electric polarizabilities the basic experimental pattern is reproduced (αN ∼Nc > δα ∼1/Nc). For the magnetic case the rules show the inapplicability of linear response (βN ∼N3c ).2) Dispersive terms lead to deformation of hedgehog solitons, and the resulting con-tributions to polarizabilities enter at the same Nc-level as the sea-gull contributions.

Inthe magnetic case, we expect that the presence of dispersive terms leads to large positivecontributions to βN. There is no cancellation mechanism which can bring β down to the20

experimental value. In fact, all model calculations with the ∆degree of freedom have thisproblem, since the paramagnetic N-∆term is large, and there is no simple mechanism tocancel this effect.3) Hedgehog models provide a mechanism of splitting of the neutron and proton polar-izabilities.

An explicit calculation gives reasonable numbers, and the sign of δα = αn −αpis expected to be positive in models with pionic clouds.4) Concerning numerical results in hedgehog models, because of the sensitivity of theresults to the pion tail, the value of gA in a model should be well reproduced. Also, for themagnetic case, good prediction for µN∆, as well as M∆−MN, is necessary to reproduce theN −∆paramagnetic term.5) The ∆resonance plays an important role.Hedgehog models largely overestimatethese contributions, since they neglect the effects of N −∆mass splitting in projection.We show how to estimate the ∆effects in pionic loops in a modified chiral perturbationtheory, and find these effects are at the level of 100 % in calculation of the electromagneticpolarizabilities.

One should also try to improve the hedgehog predictions by subtracting theamounts by which these models overestimate the leading chiral singularity terms.6) The effects of non-minimal substitution terms L9 and L10 [50] in the effective la-grangian enter at the level of 1 −2 × 10−4fm3.ACKNOWLEDGMENTSSupport of the the National Science Foundation (Presidential Young Investigator grant),and of the U.S. Department of Energy is gratefully acknowledged. We thank Manoj Banerjeefor many useful suggestions and countless valuable comments.

One of us (WB) acknowledgesa partial support of the Polish State Committee for Scientific Research (grants 2.0204.91.01and 2.0091.91.01).21

REFERENCES∗OnleaveofabsencefromH.Niewodnicza´nskiInstituteofNuclearPhysics,ul. Radzikowskiego 152, 31-342 Cracow, POLAND.

[1] F. Federspiel et al., Phys. Rev.

Lett. 67, 1511 (1991).

[2] A. Zieger et al., Phys. Lett.

B 278, 34 (1992). [3] J. Schmiedmayer, P. Riehs, J. Harvey, and N. Hill, Phys.

Rev. Lett.

66, 1015 (1991). [4] For a review of experiments on electric polarizability see for example J. Schmiedermayer,H.

Rauch, and P. Riehs, Nucl. Instrum.

Methods. Phys.

Res. A 284, 137 (1989).

[5] For a review of earlier calculations in quark models see G. Dattoli, G Matone and D.Prosperi, Lett. Nuovo Cimento 19, 601 (1977).

[6] P. Hecking and G. F. Bertsch, Phys. Lett.

B 99, 237 (1981). [7] H. Krivine and J. Navarro, Phys.

Lett. B 171, 331 (1986).

[8] A. Sch¨afer, B. M¨uller, D. Vasak, and W. Greiner, Phys. Lett.

B 143, 323 (1984). [9] S. Capstick and B. D. Keister, Technical Report CEBAF-TH-91-22, CEBAF, 1991.

[10] R. Weiner and W. Weise, Phys. Lett.

159B, 85 (1985). [11] E. M. Nyman, Phys.

Lett. 142B, 388 (1984).

[12] M. Chemtob, Nucl. Phys.

A473, 613 (1987). [13] N. N. Scoccola and W. Weise, Phys.

Lett. B232, 495 (1989); Nucl.

Phys. A517, 495(1990).

[14] H. R. Fiebig, W. Wilcox, and R. M. Woloshyn, Nucl. Phys.

B 324, 47 (1989). [15] V. A. Petrun’kin, Sov.

J. Part.

Nuc. 12, 278 (1981).

[16] J. L. Friar, in Proc. Workshop on electron-nucleus scattering, EIPC, Marciana, Italy,edited by A. Fabrocini et al., World Scientific, Singapore, 1989.

[17] W. Broniowski and T. D. Cohen, Technical Report DOE/ER/40322-155, U. of MD PP# 92-193, Univ. Of Maryland, 1992, The following article.

[18] T. H. R. Skyrme, Proc. R. Phys.

Soc. London A260, 127 (1961); Nucl.

Phys. 31, (1962)556.

[19] G. S. Adkins, C. R. Nappi, and E. Witten, Nucl. Phys.

B228, 552 (1983). [20] G. S. Adkins and C. R. Nappi, Nucl.

Phys. B233, 109 (1984).

[21] I. Zahed and G. E. Brown, Phys. Rep. 142, 1 (1986), and references therein.

[22] M. C. Birse and M. K. Banerjee, Phys. Lett.

136B, 284 (1984); Phys. Rev.

D 31, 118(1985). [23] G. R. S. Kahana and V. Soni, Nucl.

Phys. A415, 351 (1984).

[24] S. Kahana and G. Ripka, Nucl. Phys.

A419, 462 (1984). [25] G. Kalberman and J. M. Eisenberg, Phys.

Lett. 139B, 337 (1984).

[26] M. K. Banerjee, W. Broniowski, and T. D. Cohen, in Chiral Solitons, edited by K.-F.Liu, World Scientific, Singapore, 1987. [27] M. C. Birse, Progress in Part.

and Nucl. Physics 25, 1 (1990).

[28] L. Vepstas and A. D. Jackson, Phys. Rep. 187, 109 (1990), and references therein.

[29] M. K. Banerjee, in Proc. of Workshop on Nuclear Chromodynamics, Santa Barbara,1985, edited by S. Brodsky and E. Moniz, World Scientific, Singapore, 1985.

[30] W. Broniowski, Fizika 19 Supplement 2, 38 (1987), Proceedings of Workshop onMesonic Degrees of Freedom in Nuclei, Bled/Ljubljana, Yugoslavia, 1987. [31] I.

Duck, Phys. Rev.

D 34, 1493 (1986). [32] Y. Nambu and G. Jona-Lasinio, Phys.

Rev. 122, 345 (1961).22

[33] D. I. Dyakonov and V. Y. Petrov, JETP Lett. 43, 57 (1986).

[34] D. I. Dyakonov, V. Yu. Petrov, and P. V. Pobylitsa, Nucl.

Phys. B306, 809 (1988).

[35] Th. Meissner, E. Ruiz Arriola, F. Gr¨umer, K. Goeke and H. Mavromatis, Phys.

Lett.B214, 312 (1988). [36] H. Reinhardt and R. W¨unsch, Phys.

Lett. B215, 577 (1988).

[37] Th. Meissner, F. Gr¨umer and K. Goeke, Phys.

Lett. B227, 296 (1989).

[38] D. I. Dyakonov, V. Yu. Petrov, and M. Prasza lowicz, Nucl.

Phys. B323, 53 (1989).

[39] R. Alkofer, Phys. Lett.

B236, 310 (1990). [40] G. ’t Hooft, Nucl.

Phys. B75, 461 (1974).

[41] E. Witten, Nucl. Phys.

B160, 57 (1979). [42] W. Broniowski, M. K. Banerjee, and T. D. Cohen, Technical Report DOE/ER/40322-144, U. of MD PP # 92-130, Univ.

Of Maryland, 1991, To appear in Phys. Lett.

B. [43] Strictly speaking, this definition is for charge-neutral particles only, but can be easilymodified for the case of charged objects (see discussion in Sec.

III C.[44] T. E. O. Ericson and J. H¨ufner, Nucl. Phys.

B57, 808 (1961). [45] A. M. Baldin, Nucl.

Phys. 18, 310 (1960).

[46] A. Klein, Phys. Rev.

99, 998 (1955). [47] V. A. Petrun’kin, Sov.

Phys. JETP 13, 278 (1981).

[48] The error estimates are from a talk by A. Nathan presented at the Baryons ’92 confer-ence, New Haven, June 1-4, 1992. [49] B. R. Holstein, Comments Nucl.

Part. Phys.

A 19, 221 (1990). [50] J. Gasser and H. Leutwyler, Ann.

Phys. (N.Y.) 158, 142 (1989).

[51] T. D. Cohen and W. Broniowski, Phys. Rev.

D34, 3472 (1986). [52] M. Bando, T. Kugo, and K. Yamawaki, Phys.

Reports 164, 217 (1988). [53] W. Broniowski and M. K. Banerjee, Phys.

Lett. 158B, 335 (1985); Phys.

Rev. D 34,849 (1986).

[54] Note that vector mesons acquire shifts also upon cranking, which is an odd grand-reversal perturbation: W. Broniowski and T. D. Cohen, Phys. Lett.

177B, 141 (1986). [55] V. Bernard, N. Kaiser, and U.-G. Meissner, Phys.

Rev. Lett.

67, 1515 (1991); Nucl.Phys. B 373, 346 (1992).

[56] U. E. Schr¨oder, Nucl. Phys.

B166, 103 (1980). [57] A. Kerman and A. Klein, Phys.

Rev. 132, 1326 (1963).

For soliton models this methodwas introduced by J. Goldstone and R. Jackiw, Phys. Rev.

D 11, 1486 (1975). [58] T. D. Cohen and W. Broniowski, Technical Report DOE/ER/40322-154, U. of MD PP# 92-191, Univ.

of Maryland, 1992. [59] E. Jenkins and A. Manohar, Phys.

Lett. B255, 558 (1991); E. Jenkins, Nucl.

Phys.B368, 190 (1992). [60] E. Jenkins and A. Manohar, Technical Report UCSD/PTH 91-30, 1991.

[61] For references see Ref. [49].

[62] See Refs. (49-51) in (II).23

FIGURESFIG. 1.

Dispersive contributions to the Compton scattering amplitude. In our treatment, G isthe RPA (linear response) propagator.FIG.

2. Radial densities of the leading-Nc contributions to αN (in units of 10−4fm2): Sea-gull(solid line), and quark E1 contribution (dashed line).

The dotted line shows the sea-gull termevaluated with the asymptotic profile, Eq. (3.2).FIG.

3. Radial density of the pion contribution to the splitting of the neutron and protonelectric polarizabilities, δα (in units of 10−4fm2).FIG.

4. Hadronic one-pion-loop diagrams giving the leading chiral contribution to nucleonpolarizabilities.

The vertex corresponds to the sea-gull interaction e2 R d3x (E · x)(c × π)24

TABLESTABLE I. KP R classification of various dispersive perturbations, and sources in the corre-sponding linear response equations (App. A in (II)).

Quantum numbers (L, Λ) label quark shifts,Lπ and Lσ are the orbital numbers of meson shifts. Null entry in columns for Lπ and Lσ meansthat the fluctuation does not arise.

For cases with zero modes (electric isoscalar and magneticisovector, K=1), the sources listed are twice the jX, jY or jZ sources from equations in App. A in(II).

For other cases they are the sources entering Eqs. (II.3.7,II.3.8).

Radial functions Gh and Fhare the upper and lower components of the hedgehog valece spinor, and πh in the hedgehog pionprofile. See (II) for details.QuarksδπδσinteractionKPR(L,Λ)jG/√4πjF /√4πLπjπ/√4πLσ jσ/√4πcranking1+−(0,1)−12Gh−12Fh(2,1)00electric isovector 0−−(1,1)12rGh/√312rFh/√31−−(1,0)00(1,1)12rGh/√312rFh/√32−−(1,1)12rGh/√312rFh/√3(3,1)00electric isoscalar 1−+(1,0)000010(1,1)12N −1crGh/√312N −1crFh/√320magn.

isovector0++(0,0)12rFh/√312rGh/√31−2πh/√3 001++(0,1)rFh/√18rGh/√181−πh/√3(2,1)−(1/12)rFh−(1/12)rGh2++(2,0)12rFh/√3012rGh/√301πh/√320(2,1)12rFh/√2012rGh/√2030magn. isoscalar1+−(0,1)−(1/6)N −1crFh−(1/6)N −1crGh(2,1)−(√2/12)N −1crFh −(√2/12)N −1crGh25

TABLE II. Electric polarizability.The model predictions for αN are expected to be larglyreduced by effects disussed in the text.αN (10−4fm3)sea-gull28.5dispersive E1 (quarks)3.5total32experiment9.6 ± 1.8 ± 2.2δα (10−4fm3)pion5.6quarks-0.2total5.4experiment4.8 ± 1.8 ± 2.226

---

출처: arXiv:9208.256 • 원문 보기