Chiral solitons in the spectral quark model

Chiral solitons in the spectral quark model

Enrique Ruiz Arriola*Departamento de Fısica Atomica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain

Wojciech Broniowski†

The H. Niewodniczanski Institute of Nuclear Physics, PL-31342 Krakow, PolandInstitute of Physics, Swietokrzyska Academy, ul. Swietokrzyska 15, PL-25406 Kielce, Poland

Bojan Golli‡

Faculty of Education, University of Ljubljana, 1000 Ljubljana, SloveniaJ. Stefan Institute, 1000 Ljubljana, Slovenia

(Received 23 October 2006; published 24 July 2007)

Chiral solitons with baryon number one are investigated in the spectral quark model. In this model thequark propagator is a superposition of complex-mass propagators weighted with a suitable spectralfunction. This technique is a method of regularizing the effective quark theory in a way preserving manydesired features crucial in analysis of solitons. We review the model in the vacuum sector, stressing thefeature of the absence of poles in the quark propagator. We also investigate in detail the analytic structureof meson two-point functions. We provide an appropriate prescription for constructing valence states inthe spectral approach. The valence state in the baryonic soliton is identified with a saddle point of theDirac eigenvalue treated as a function of the spectral mass. Because of this feature the valence quarksnever become unbound nor dive into the negative spectrum, hence providing stable solitons as absoluteminima of the action. This is a manifestation of the absence of poles in the quark propagator. Self-consistent mean-field hedgehog solutions are found numerically and some of their properties aredetermined and compared to previous chiral soliton models. Our analysis constitutes an involved exampleof a treatment of a relativistic complex-mass system.

DOI: 10.1103/PhysRevD.76.014008 PACS numbers: 12.38.Lg, 12.38.�t

I. INTRODUCTION

The original proposal by Skyrme of the early sixtiesforesaw that baryons could be described as classical topo-logical solitons of a specific nonlinear chirally symmetricLagrangean in terms of meson fields with nontrivial bound-ary conditions [1]. Within the accepted QCD frameworkthe large-Nc analysis under the assumption of confinement[2,3] supports many aspects of this view. Moreover, theidentification of the topological winding number as theconserved baryon number of quarks from the occupiedDirac sea in the background of large and spatially extendedpion fields [4] suggested the underlying necessary fermi-onic nature of the soliton at least for odd Nc [5]. The formof the Lagrangean remains unspecified besides the require-ment of chiral symmetry, leaving much freedom on theassumed meson-field dynamics needed for practical com-putations of low-energy baryonic properties [6,7], usuallyorganized in terms of a finite number of bosons of increas-ing mass. One appealing feature of the Skyrme model (forreviews see e.g. Ref. [8–11]) is that confinement appears tobe explicitly incorporated, since the baryon number topol-ogy of the meson fields cannot be changed with a finiteamount of energy and the soliton is absolutely stable

against decay into free quarks. This has also made possiblethe study of excited baryons (see e.g. Ref. [12] and refer-ences therein). However, when the partonic content of thebaryon is analyzed in deep inelastic scattering, in theSkyrme model the Callan-Gross relation is violated, hencethe partons turn out not to be spin one-half objects [13].Therefore, making credible nonperturbative estimates ofhigh-energy properties is out of reach of the standardSkyrme model. This also suggests that the relevant degreesof freedom should explicitly include constituent quarkschirally coupled to mesons [14–22] motivating the use ofchiral-quark models with quarks and to search for solitonicsolutions to describe baryons. The approach is very muchin the spirit of the Skyrme model but with the importantfeature that the partonic interpretation corresponds to spinone-half constituents, despite the subtleties of the regulari-zation [23]. The considerable effort exerted to describelow-lying baryons as solitons of effective chiral-quarkmodels has been described in detail in the reviews [24–28].

The chiral-quark models that arise naturally in severalapproaches to low-energy quark dynamics, such as theinstanton-liquid model [29] or the Schwinger-Dyson re-summation of rainbow diagrams [30], are nonlocal, i.e. thequark mass function depends on the quark virtuality. Forthe derivations and applications of these models see, e.g.,[31– 47]. A major success was the finding of baryon hedge-hog solitons in nonlocal models, which turn out to be stablealso in the linear version of the model. Moreover, the

*[email protected]†[email protected]‡[email protected]

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nonlocal models have nice features as compared to thelocal variants, in particular, they use a uniform regulariza-tion of both the normal and abnormal parity processes, thusreproducing properly the anomalies in the presence ofregularization. In addition, the theory is made finite to allorders of perturbation theory. The price to pay for thenonlocality is a complicated nature of interaction verticeswith currents, where the gauge invariance imposes non-local contributions. Their presence leads to technical com-plications in the treatment of nonlocal models.

A relevant issue for chiral-quark solitons is related to theconfinement and stability of the mean-field solution. Thechiral-quark soliton model builds baryons as bound statesof valence constituent quarks in nontrivial meson fields.Indeed, such a model interpolates between the nonrelativ-istic quark model and the Skyrme model in the limit ofsmall solitons and large solitons, respectively [50]. As amatter of fact, from the view point of the chiral-quarkmodel, the apparent confinement in the Skyrme model isreinterpreted as a strong binding effect, and indeed deeplybound states are not sensitive to the confinement propertiesof the interaction. Nevertheless, it is notorious that depend-ing on the details, the mean-field soliton may in fact decayinto three free constituent quarks. Moreover, in local mod-els for small enough solitons the valence state becomes anunbound free quark at rest. Actually, to our knowledgethere is no satisfactory calculation of excited baryon statesin the chiral-quark soliton model precisely because of thelack of confinement [51]. These features happen for real-istic parameter values and despite the alleged compatibilityof the chiral-quark model with the large-Nc limit. As al-ready mentioned the soliton description of baryons in thelarge-Nc limit is based on the assumption of confinement.

Let us define the scope of the present work. After manyyears, the issue of color confinement remains a crucial andintriguing subject for which no obvious solution exists yet.In particular, its realization implies two relevant conse-quences. In the first place, there exists a spectrum of colorsinglet excited states. Second, quarks cannot become onshell and hence quark propagators merely cannot havepoles on the real axis. This restrained meaning of confine-ment is often called analytic confinement in the literatureand in that sense is adopted in the present work. We hastento emphasize that although our work obviously does notsuggest how the problem of color confinement might beresolved, we manage to set up a framework where thecalculation of color singlet states such as excited baryonsbecomes possible as a matter of principle within a chiral-quark soliton approach.

In this paper we show how a recently proposed versionof the chiral-quark model, the spectral quark model (SQM)[52–55] (see also Ref. [56] for the original insights on thespectral problem), not only allows for solitonic solutions,but due to its unconventional and indeed remarkable ana-lytic properties yields a valence eigenvalue which never

becomes unbound. Thus the soliton is absolutely stable.The model is based on a generalized Lehmann representa-tion where the spectral function is generically complex,involving a continuous superposition of complex masses.The subject of defining a well-founded quantum fieldtheory of fixed complex masses is an old story [57–61].Our approach differs from these works, since our complexmass is an integration variable so that many standardobjections are sidestepped by choosing an appropriateintegration contour. In practical terms, the spectral functionacts as a finite regulator fulfilling suitable spectral condi-tions but with many desirable properties, in particular, thesimple implementation of chiral symmetry and gauge in-variance. One of its outstanding features is the uniformtreatment of normal and abnormal parity processes ensur-ing both finiteness of the action and a simultaneous im-plementation of the correctly normalized Wess-Zumino-Witten term.1 This uniform treatment of regularization hassome impact on soliton calculations even in the SU(2) case(where the Wess-Zumino-Witten term vanishes) since thevalence and sea contributions to the soliton energy areindeed related to the abnormal and normal parity separa-tion of the effective action, respectively. The basics ofSQM are described in detail in Refs. [53,55] and the readeris referred there for the description of the method andnumerous applications to the pion phenomenology. TheSQM approach bares similarities to nonlocal models, how-ever the construction of interaction vertices with currents isvery simple in SQM, as opposed to almost prohibitivecomplications of the nonlocal approach.

The purpose of this paper is twofold: First, we show thatabsolutely stable baryon solitons in SQM exist and discusstheir properties. Second, and more generally, we study aninstance of an involved complex-mass system, and showhow to treat valence states. Despite the complexity, theresulting prescription turns out to be very simple and easyto implement in practical calculations. It amounts to locat-ing the saddles of the valence eigenvalue, �0, as a functionof the complex mass !, based on the condition

d�0�!�d!

� 0: (1)

The outline of our paper is as follows. In Sec. II weprovide an operational justification for the need of someuniform regularization for the full action without an ex-plicit separation between normal and abnormal parity con-tributions. Certain a priori field-theoretic consistencyconditions are discussed. SQM is shown to fulfill theone-body consistency condition in contrast to previouslocal versions of chiral-quark models. We also discuss insome detail the analytic structure of the meson correlators,

1In the standard treatment of local models the somewhatartificial and certainly asymmetric prescription of regularizingthe real part of the Euclidean action and not regularizing theimaginary part has been used.

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showing that despite analytically-confined quarks theypossess cuts at large values of q2, as requested by (asymp-totic) quark unitarity. However, the meromorphic structureexpected from general large Nc considerations is violated.This point is analyzed in the light of a large-Nc Reggemodel. Next, we pass to the analysis of solitons. First, wediscuss the connection between the baryon and topologicalcurrents in the limit of large solitons, which holds in thepresence of the spectral regularization. Following the stan-dard field-theoretic approach the construction of thebaryon state is pursued in Sec. III. The unconventionalappearance of complex-masses requires a demandingmathematical treatment of both the valence and sea con-tributions. Nevertheless, ready-to-use formulas for the totalsoliton mass are derived and analyzed for several solitonprofiles showing the existence of chiral solitons withbaryon number one. An alternative derivation is providedin Appendix E based on computing the spectral integralexactly. Dealing with complex-mass Dirac operators bothfor bound states and continuum states is involved and someof the features may be studied in a somewhat comprehen-sive toy model in Appendix D. The techniques are close tothe more familiar complex potentials in nonrelativisticquantum mechanics which are reviewed for completenessin Appendix C. In Appendix G we show that a linearextension of SQM leads to instability of the vacuum, henceSQM can only be constructed in the originally proposednonlinear realization. In Sec. IV we look for self-consistenthedgehog solutions and determine their properties both inthe chiral limit as well as for finite pion masses. Finally, inSec. V we come to the conclusions.

II. THE SPECTRAL QUARK MODEL ANDCONSISTENCY RELATIONS

We begin with some basic expressions of the generalfield-theoretic treatment of chiral-quark models, which arethe groundwork for our treatment of solitons and themethod of including valence states in SQM discussed inSec. III. In this section we highlight an important consis-tency condition which was not fulfilled in the hithertoextensively used chiral-quark soliton models based onlocal interactions. Remarkably, this condition happens tobe satisfied in SQM. The analytic properties of the quarkpropagator and in particular the lack of poles are reviewed.We also analyze some important aspects of the mesonsector, and, in particular, the analytic structure of two-pointcorrelators in the complex q2 plane. Finally, we show howthe correct topological current arises in the presence ofregularization.

A. A consistency condition

The vacuum-to-vacuum transition amplitude in the pres-ence of external bosonic �s; p; v; a� and fermionic ��; ��fields of a chiral-quark model Lagrangian can be written inthe path-integral form as

Z�j; �; �� h0jT exp�iZd4x� �qjq� ��q� �q��

�j0i; (2)

where the compact notation

j � v6 � a6 �5 � �s� i�5p� (3)

has been introduced. The symbols s, p, v�, and a� denotethe external scalar, pseudoscalar, vector, and axial flavorsources, respectively, given in terms of the generators ofthe flavor SU�3� group

s �XN2F�1

a�0

sa�a2; . . . (4)

with �a representing the Gell-Mann matrices. Any physi-cal matrix element can be computed by functional differ-entiation with respect to the external sources.

Let us consider the calculation of a bilinear quark op-erator, such as, e.g., the quark condensate (for a singleflavor) h �qqi. We can think of two possible ways of makingsuch a computation, namely, via coupling of an externalscalar source s�x� (a mass term) and differentiating withrespect to s�x�, or by calculating the second functionalderivative with respect to the Grassmann external sources��x� and ��x� taken at the same spatial point. The consis-tency of the calculation requires the following trivial iden-tity for the generating functional:

h �q�x�q�x�i � i1

Z�Z�s�x�

��0� lim

x0!xh �q�x0�q�x�i

� limx0!x��i�2

1

Z�2Z

��x� ��x0�

��0; (5)

where j0 means all external sources set to zero. Thisrequirement can be generalized to any quark bilinearwith any bosonic quantum numbers and thus we call itthe one-body consistency condition. In the traditional treat-ment of local chiral-quark models the left-hand side (l.h.s.)of the above formula corresponds to a closed quark line andis divergent, calling for regularization, whereas for x0 � xthe right-hand side (r.h.s.) is finite and corresponds to anopen quark line, thus no regularization is demanded. Thisposes a consistency problem which actually becomes cru-cial in the analysis of high-energy processes and introducesan ambiguity in the partonic interpretation as well asconflicts with gauge invariance and energy-momentumconservation (see, e.g., Ref. [62] for a further discussionon these subtle but relevant issues). Obviously, the pre-vious argument could be equally applied to sources withany quantum numbers suggesting that any consistent regu-larization should be applied to the full action. As alreadymentioned in Sec. I, in the traditional approach to localmodels the treatment of singularities first requires us toseparate the normal and abnormal parity contributions and

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to regularize only the normal parity piece. In the Euclideanspace this separation corresponds to the real and imaginaryparts of the action.

In the next section we show that SQM fulfills the one-body consistency condition. More generally, one mightwant to extend the condition (5) to any number of quarkbilinears contracted to bosonic quantum numbers, namely,the N-body consistency condition

h �q�x1��1q�x1� � � � �q�xN��1q�xN�i

� limx0i!xih �q�x01��1q�x1� � � � �q�x0N��1q�xN�i; (6)

where the l.h.s. is evaluated after functional derivativeswith respect to bosonic sources and the r.h.s. with respectto fermionic sources, respectively. Here �i are generalspin-flavor matrices. We have found that this is not possiblein the SQM scheme (see the discussion in Sec. II B below).

B. The spectral quark model and the quark propagator

In SQM the regularization is imposed already at thelevel of one open line through the use of a generalizedLehmann representation for the full quark propagator inthe absence of external sources,

S�p6 � �ZCd!

��!�p6 �!

� p6 A�p2� � B�p2�; (7)

where C is a suitable contour in the complex-mass plane[54] (see also Fig. 1 below). Chiral and electromagneticgauge invariance can be taken care of through the use of thegauge technique of Delbourgo and West [63,64], whichprovide particular solutions to chiral Ward-Takahashi iden-

tities, or through the use the standard effective actionapproach, applied in this paper. As a result, one can‘‘open the quark line’’ from one closed loop and computehigh-energy processes with a partonic interpretation, suchas, e.g., the structure function and the light-cone wavefunction of the pion, etc. [54], or the photon and�-meson light-cone wave functions [65]. The generaliza-tion of the (one body) consistency condition for scalarsources, Eq. (5), to all possible bosonic quantum numberscan be achieved by taking the generating functional to be

Z��; ��; s; p; . . .� �ZDUe�ih ��;S�U;s;p;v;a��iei��U;s;p;v;a�;

(8)

where the quark propagator and the effective action aregiven by

hx0jS�U; s; p; v; a�aa0 jxi �ZCd!��!�hxj�D��1

aa0 jx0i (9)

and

��U; s; p; v; a� � �iNcZCd!��!�Tr log�iD�; (10)

respectively. The Dirac operator has the form

iD � i@6 �!U5 � m0 � �v6 � a6 �5 � s� i�5p�:

For a bilocal (Dirac- and flavor-matrix valued) operatorA�x; x0� we use the notation

TrA �Zd4x trhA�x; x�i; (11)

with tr denoting the Dirac trace and h i the flavor trace. Thematrix

U5 � ei�5

��2p

�=f � 12�1� �5�U�

12�1� �5�Uy; (12)

while U � u2 � ei��2p

�=f is the flavor matrix representingthe pseudoscalar octet of mesons in the nonlinear repre-sentation,

� �

1��2p �0 � 1��

6p � �� K�

�� 1��2p �0 � 1��

6p � K0

K� �K0 � 2��6p �

0BB@1CCA: (13)

The matrix m0 � diag�mu;md;ms� is the current quark

w−Complex Plane

x

−Mv / 2 Mv / 2

FIG. 1. The contour C in the complex-! plane for the calcu-lations in the vacuum sector in the meson-dominance variant ofSQM. MV denotes the �-meson mass. The two segments shownin the figure are connected at infinity with semicircles, notdisplayed.


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mass matrix and f � 86 MeV denotes the pion weak-decay constant in the chiral limit, ensuring the propernormalization condition of the pseudoscalar fields. For atwo-flavor model it is enough to consider � � ~ � ~�=

��2p

.In Eq. (8) the Dirac operator appears both in the fermion

determinant as well as in the quark propagator and could,in principle, be treated independently. The one-body con-sistency condition is fulfilled precisely because we use thesame spectral function ��!� for both. We have refrained onpurpose from writing a Lagrangean in terms of quarksexplicitly since anyhow chiral-quark models are definedin conjunction with the regularization and the approxima-tion used. In our case we work in the leading order of thelarge-Nc expansion, which amounts to a saddle-point ap-proximation in the bosonic U-fields and use the spectralregularization which is most explicitly displayed in termsof the generating functional presented above. The newingredient of SQM compared to earlier chiral-quark mod-els is the presence of the quark spectral function ��!� inEqs. (9) and (10) and the integration over ! along asuitably chosen contour C in the complex-! plane. Ourapproach extends the early model of Efimov and Ivanov

[56] by including the gauge invariance, the chiral symme-try, and the vector meson dominance, as well as applyingthe model to both low- and high-energy processes.

The above SQM construction implements the one-bodyconsistency condition, as follows

i�Z

�j�x�

�� 0� lim

x0!x��i�2

��x�

��

��x0�Zj�� 0;

(14)

which is more general than Eq. (5) since it is valid in thepresence of nonvanishing external bosonic sources withany quantum numbers and nontrivial background pionfield, U. This represents a clear improvement on the pre-vious local chiral-quark models where this requirementwas violated. Equation (14) has direct applicability onthe soliton sector as we will discuss in Sec. III. It shouldbe mentioned, however, that similarly to many other chiral-quark models, the two-body and higher consistency con-ditions, Eq. (6) are not satisfied in SQM. For instance, forthe two-body consistency condition one gets

h �q�x1��1q�x1� �q�x2��2q�x2�i �1

Z

ZDU

Zd!��!� tr��1hx1jD�1�!�jx1i� tr��2hx2jD�1�!�jx2i�; (15)

while

limx0i!xih �q�x1��1q�x

01� �q�x2��2q�x

02�i � lim

x0i!xi

1

Z

ZDU

Zd!1��!1� tr��1hx

01jD

�1�!1�jx1i�

Zd!2��!2� tr��2hx01jD

�1�!2�jx1i�

� h �q�x1��1q�x1� �q�x2��2q�x2�i: (16)

Likewise, we do not know if this one-body consistencycondition is fulfilled beyond the leading large-Nc approxi-mation. At present the only known way to fulfill all con-sistency conditions is by returning to nonlocal versions ofthe chiral-quark model. Therefore, we must provide aprescription of how higher functional derivatives shouldbe handled in case were more than a single quark line couldbe opened. Equation (14) suggests to use always themethod based on the bosonic sources for operators involv-ing any number of quark bilinear and local operators. Foroperators involving one single bilocal and bilinear operatorq�x� �q�x0� we may use the method based on fermionicsources, since Eq. (14) guarantees the consistency in thecoincidence limit x0 ! x. Of course, this prescription doesnot yield a unique result when more then one bilocal andbilinear quark operator is involved or equivalently whenmore than one quark line is opened.

Using the standard variational differentiation for thegenerating functional the Feynman rules for SQM followfrom the action (10). They have the form of the usual

Feynman rules for a local theory, amended with the spec-tral integration associated to each quark line, according toEq. (10). This resembles very much the well known Pauli-Villars regularization method (however with a continuoussuperposition of complex masses) and allows for veryefficient computations, see Refs. [52–55].

The basic paper [53] explains the general constructionand the conditions for the moments of the spectral function��!� coming from physics constraints. In particular, nor-malization requires

ZCd!��!� � 1; (17)

while observables are related to the log-moments andinverse moments of ��!� [53]. The full spectral functionconsists of two parts of different parity under the change!! �!, i.e. ��!� � �V�!� � �S�!�, with the scalarpart, �S, even and the vector part, �V , odd. A particularimplementation of SQM is the meson-dominance model,where one requests that the large-Nc pion electromagnetic


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form factor has the monopole form of the vector-mesondominance (VMD),

F��q2� �M2V

M2V � q

2 ; (18)

where MV denotes the �-meson mass.2 The matching ofthe model predictions to this form yields the rather unusualspectral function

�V�!� �1

2�i1

!1

�1� 4!2=M2V�

5=2; (19)

with the pole at the origin and cuts starting at MV=2,where MV is the mass of the vector meson. Similar con-siderations for the photon light-cone wave function [65]and matching to VMD yield for the scalar part an analo-gous form

�S�!� � �1

2�i48�2h �qqi

NcM4S�1� 4!2=M2

S�5=2: (20)

where h �qqi is the single flavor quark condensate, andMS �MV [55]. The contour C for the ! integration to be usedwith formulas (19) and (20) is shown in Fig. 1 for �V (for�S one has the same contour withMV ! MS). This contouris applicable in the vacuum (i.e., no baryon number) sectorof the model. The extension to baryons is described in thenext section. From Eqs. (19) and (20) one gets the quarkpropagator functions from Eq. (7),

A�p2� �ZCd!

�V�!�

p2 �!2 �1

p2

�1�

1

�1� 4p2=M2V�

5=2

�;

B�p2� �ZCd!

!�S�!�

p2 �!2 �48�2h �qqi

M4SNc�1� 4p2=M2

S�5=2:

(21)

These functions do not possess poles (the alleged pole atp2 � 0 in A�p2� is canceled), but only cuts starting at p2 �M2V=4, reflecting the structure of ��!�. Actually, the !

integral can be evaluated using the integral transformationsdisplayed in the first lines of Eqs. (21). For instance, takingthe limit p2 ! 1 one gets the moments

�k;0 �ZCd!�V�!�!2k; k � 0; 1; 2; . . . ; (22)

0 �ZCd!�S�!�!

2k�1; k � 0; 1; 2; . . . ; (23)

which imply, in particular, the vanishing of all positivemoments for the spectral function ��!�.

The pion weak-decay constant in the chiral limit comesout to be [53]

f2 �NcM

2V

24�2 ; (24)

a relation which works well phenomenologically. Thisrelation will be used as an identity in the rest of the paper.Compared to the standard field-theoretic case, each quarkline is supplemented with a spectral integrationRC d!��!�. This makes calculations very straightforward

and practical for numerous hadronic processes involvingmesons and photons. We stress that despite a rather ‘‘ex-otic’’ appearance of the quark spectral function, SQMleads to proper phenomenology for the soft pion, includingthe Gasser-Leutwyler coefficients [55], the soft matrixelements for hard processes, such as the distribution am-plitude, transition form factor, or structure functions [53],the generalized forward parton distribution of the pion[66], the photon distribution amplitude and light-conewave functions [65], or the pion-photon transition distri-bution amplitude [67]. In addition, the calculations arestraightforward, leading to simple analytic results. In-terestingly, the quark propagator corresponding to (19)and (20) has no poles, only cuts, in the momentum space.The evaluated mass function of the quark, M�Q2�, displaysa typical dependence on the virtuality Q2 in the Euclideanregion and at the qualitative and quantitative level com-pares favorably to the nonlocal quark models and to thelattice calculations [68]. As a sample calculation, we ex-tend in the next subsections the previous considerations tothe calculation of vacuum properties and two-pointcorrelators.

C. The vacuum sector

As mentioned above the functional (one-body) consis-tency conditions guarantee the unambiguous calculation ofobservables obtained from quark fields, either as doubleGrassmann functional derivatives or as single bosonicones. These identities are generically formally satisfiedbut become tricky under regularization. Here we use thefirst possibility for the quark condensate and the vacuumenergy density. The second method based on the effectiveaction and yielding identical results is outlined inAppendix A.

The (single flavor) quark condensate h �qqi can directly becomputed from the quark propagator

Nfh �qqi � �iNcZ d4p

�2��4TrS�p6 �

� 4NcNf1

i

Z d4p

�2��4B�p2�; (25)

which through the use of Eq. (21) becomes an identity. The

2This example shows in a transparent way a peculiar feature ofthe model. In the standard constant mass case the pion formfactor, a three-point function, due to Cutkosky’s rules, displays acut in the form factor for sufficiently large energy due to asuperposition of poles in the quark propagator. In SQM themechanism is just the opposite; the cuts conspire to build apole in the form factor.


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vacuum energy density can be computed by using the traceof the energy-momentum tensor, yielding

h��i � h �q�x�i4��@

$�� @

$��q�x�i

� �iNcZ d4p

�2��4Tr�

1

2�p�� p��S�p6 �

�

� �4iNcNfZ d4p

�2��4p�p�A�p2�

�1

4g�� h��i0; (26)

where the subscript 0 refers to the trivial vacuum (A � 1and B � 0). The bag constant is defined by the difference� � 1

4 h��i � 1

4 h��i0 and after computing the integral be-

comes

� � �M4VNfNc

192�2 : (27)

The formula implies that it costs energy to dig a hole in thevacuum with a nonvanishing pion weak-decay constant[see Eq. (24)], as one might expect for the chirally sym-metric broken phase.

D. Two-point correlators in the meson sector

In SQM the quark propagator has no poles but cuts atp2 � M2

V=4 preventing the occurrence of quarks on themass shell. This complies to the notion of analytic con-finement, a necessary but certainly not sufficient conditionfor color confinement. This nonstandard behavior alreadysuggests a possible departure from the standard treatment.However, to justify the claim of confinement one shouldcheck, in addition, the absence of cuts in physical correla-tors. Note that this question is in principle unrelated to theanalytic confinement of bound quarks in a soliton withbaryon number one which will be discussed in furthersections below, but it is still interesting to see the analyticproperties of the model. We analyze this issue below insome detail for the two-point mesonic correlators definedas

�AB�q� � iZd4xe�iq�xh0jTfJA�x�JB�0�gj0i; (28)

where JA�x� and JB�x� are interpolating currents with therelevant meson quantum numbers. In the standard path-integral approach [see Eq. (8)] the time-ordered productsof currents in the vacuum can be evaluated by suitablefunctional derivatives of the generating functional (10)with respect to external bosonic currents, and in thelarge-Nc limit the path integral is driven to the saddle pointof the path integral in the presence of those currents. On theother hand, at large Nc any two-point mesonic correlationfunction should have the general structure [2,3]

�J�q2� �

Xn

f2J;n

M2J;n � q

2 (29)

due to confinement. This is a very stringent test, since itimplies, in particular, a meromorphic analytic structure inthe q2 complex plane. Amazingly, this rather simple re-quirement has never been accomplished in chiral-quarkmodels, either local or nonlocal.3

A straightforward consequence of the large-Nc repre-sentation of the two-point correlator in Eq. (29) is positiv-ity, �J�q2�> 0, in the Euclidean region q2 � �Q2 < 0and quark unitarity at large values of q2. This ensures that,for instance, the inclusive e�e� ! hadrons total crosssection is proportional to the imaginary part of the polar-ization operator in the vector channel.4

In our previous work [53], we evaluated the VV and AAcorrelators corresponding to the conserved vector and axialcurrents

J�;aV �x� � �q�x��a2q�x�; (30)

J�;aA �x� � �q�x��5�a2q�x� (31)

for a general spectral function ��!� using solutions to theWard-Takahashi identities based on the gauge technique.In this paper we use the effective action (10) to evaluate thecorrelation functions. Details of the calculation are pro-vided in Appendix A. We find

��a;�bVV �q� � 1

2�ab�g��q2 � q�q��T

V�q2�; (32)

with5

�TV�q

2� �2Nc3q2

Z��!�d!

�2!2�I�q2; !� � I�0; !��

� q2

�1

48�2 � I�q2; !�

��; (33)

and

��a;�bAA �q� � 1

2�ab�g��q2 � q�q��T

A�q2�; (34)

with

3Instead, calculations at large values of Q2 have only beenconfronted to QCD sum rules (see, e.g., Ref. [47] and referencestherein.)

4This asymptotic quark unitarity holds for external currentsand should not be confused with the S-matrix hadron unitaritywhich is violated at any finite order of the large Nc expansiondue to the 1=

��Ncp

suppressed behavior of the meson-mesoncouplings.

5We correct here a typo in our previous work Ref. [53],Eq. (4.3). Our conventions here are �T

V�q2� � �2��q2�=q2.


014008-7

�TA�q

2� � �TV�q

2� � 4NcZd!!2��!�I�q2; !�: (35)

In the particular case of the meson-dominance model forthe spectral function, Eq. (19), up to an (infinite) constantone has a remarkably simple result

�TV�q

2� �f2V

M2V � q

2 �Nc

24�2 log�1�

q2

M2V

�;

�TA�q

2� � �f2

q2 �Nc

24�2 log�1�

q2

M2V

�;

(36)

where f2V � f2. The above formulas clearly display the �

meson pole in the vector channel and the pion pole in theaxial channel. Also, we see that f2

A � 0, i.e., there is noaxial meson dominance. Our expressions fulfill the firstWeinberg sum rule

limq2!0

q2��TV�q

2� ��TA�q

2�� f2: (37)

Similarly as in other local quark models, the secondWeinberg sum rule is not satisfied,

limq2!�1

q4��TV�q

2� ��TA�q

2�� M2Vf

2 � 0; (38)

as noted in our previous work [53]. Since the secondWeinberg sum rule is a high-energy feature of the theory,one hopes that it is not crucial for the low-energy phe-nomena studied in this paper and, in particular, for thesoliton properties which probe Euclidean momenta corre-sponding to a soliton size �

��6p=MV [see, e.g., Eq. (98)

below]. This assumption has implicitly been made in state-of-the-art chiral-quark soliton models [24–28].

The presence of the log pieces in the correlators guar-antees the fulfillment of quark unitarity since6

Im �TV�q

2� � Im�TA�q

2� �Nc

24�; q2 >M2

V: (39)

The appearance of these quark unitarity cuts can be in-ferred from the general structure of the quark propagator[see Eq. (21)] for any quark momentum p2 >M2

V=4, inspite of the fact that there are no poles in the quarkpropagator. A more detailed analysis is presented inAppendix B. Note that in QCD one obtains these partonlikerelations for q2 ! 1. We stress that the coefficients of thelog terms are precisely such as in the one-loop QCDcalculation, complying to the parton-hadron duality.Moreover, despite the unconventional features of SQMinvolving complex masses, positivity is preserved bothfor the pole and for the cut contributions to the imaginary

parts of the considered correlators, as can be seen fromEq. (36). This is a virtue of the spectral model, not easilyfulfilled in other chiral-quark models.7 It is important torealize that this quark unitarity relation is hidden in thelarge-Nc meromorphic representation of Eq. (29) at largeq2 through the asymptotic density of �qq states [70].

To see this we may compare to the large-Nc Reggemodels (see e.g. Ref. [71] and references therein), wherethe meson spectrum is chosen to be a tower of infinitelymany radially excited states with masses M2

n;V � M2V �

2� n, where is the string tension and the residue istaken to be constant f2

n;V � Nc =�12�� precisely to imple-ment quark-hadron duality at large q2. As we see, SQMcorresponds to keeping one pole in the vector channel andapproximating the higher states by a logarithm. Moreexplicitly, one has

�V�q2� ��V�0� �Nc 12�

Xn�0

�1

M2V � 2� n� q2

�1

M2V � 2� n

�

�Nc

24�2

��M2V � q

2

2�

��

�M2V

2�

��;

where the digamma function, �0�z� � �0�z�=��z� has beenintroduced. SQM corresponds to representing the infiniteRegge sum

��1� z� ��1� � �X1n�1

�1

n� z�

1

n

�; (40)

where ��1� � � is the Euler-Mascheroni constant, bythe approximation of taking explicitly the first pole. Thehigher poles lead to the asymptotic behavior

� 1=�z� 1� � 1� log�1� z�; (41)

approximated by the cut in SQM. The accuracy is betterthan 20% for 0< z <1. We also recall that the large-Ncanalyses restricted to a finite number of resonances [72,73]provide a meromorphic structure but fail to give thelarge-q2 parton-hadron duality conditions.

For completeness we also discuss the scalar and pseu-doscalar channels, where the currents are given by

JaS�x� � �q�x��a2q�x�; (42)

JaP�x� � �q�x�i�5�a2q�x�: (43)

No Ward-Takahashi identities may be written for thesecurrents and thus they are not directly amenable to the

7For instance, the Pauli-Villars regularization spoils positivitydue to subtractions, while dispersion relations are fulfilled. Theproper-time regularization preserves positivity but does not ful-fill dispersion relations due to a plethora of complex poles [69].

6We define the discontinuity as

Disc ��q2� � ��q2 � i0�� q2 � i0�� 2i Im��q2�:


014008-8

gauge technique. The effective action approach used hereis superior to the gauge technique since it also allows tocompute the two-point correlators not directly related toconserved currents. The result is given up to two subtrac-tion terms as follows (see Appendix A). Defining

�abSS�q� �

12�

ab�S�q�; �abPP�q� �

12�

ab�P�q� (44)

one gets

�S�q2� �

Nc8�2 q

2 log�1�

q2

M2V

�;

�P�q2� �Nc

8�2 q2 log

�1�

q2

M2V

��

2f2M2V

M2V � q

2

�2h �qqi2

f2

1

q2

M4S�M

2V � q

2�

M2V�M

2S � q

2�2;

(45)

where the residue of the pion pole is 2f2B20, with B0 �

�h �qqi=f2. We see that fS � 0, i.e., there is no excitedpseudoscalar meson dominance. Again, the emergence ofthe quark unitarity cuts is manifest. Similarly to the vectorand axial correlators, the first SS� PPWeinberg-like sumrule is verified

limq2!0

q2��S�q2� ��P�q2�� 2B20f

2; (46)

while the second SS� PP sum rule is not:

limq2!�1

q2��S�q2� ��P�q2�� 2M2Vf

2 � 0: (47)

Our analysis of both the second Weinberg sum rules forVV � AA and SS� PP correlators agrees with the ob-served mismatch of the low-energy coefficients L8 andL10 between SQM [55] and the large-Nc evaluation inthe single-resonance approximation (SRA) [72,73] (whereboth sum rules are enforced). In Ref. [55] it was also shownthat matching L3 of both SQM and the large-Nc SRAyieldsthe identity between the scalar and meson masses, MS �MV (as we assume for the rest of the paper). It is interestingthat the same condition also follows from the requirementof having a single pole in the PP correlator, Eq. (45).Moreover, at small q2 one has

�V�q� ��A�q� �f2

q2 � 4L10 � . . . ; (48)

�S�q� ��P�q� � �2B20

�f2

q2 � 16L8 � . . .�: (49)

Using Eqs. (36) and (45) and taking MS � MV yields

L8 �Nc

384�2 �f6

16h �qqi2; L10 � �

Nc92�2 ; (50)

in agreement with results from the derivative expansioncarried out in Ref. [55]. The large-Nc SRA yields L8 �

3f2=�32M2S� and L10 � �3f2=�8M2

V� [72,73], while thelarge-Nc Regge models produce L10 � �Nc=�96

��3p��.8

To summarize this section, SQM provides meson two-point correlators which carry poles as well as cuts, whilestrictly speaking largeNc requires only having poles. Thus,despite the quark propagator not having poles, the mesoncorrelations do not exhibit true confinement. Although wecannot prove it in general for any correlation function, alltwo-point correlators we have considered obey positivityand analyticity, i.e. dispersion relations. This is required bythe (asymptotic) quark unitarity and causality, i.e. thecurrent commutators must vanish outside the causal cone,�j�x�; j�0�� 0 for x2 < 0. The coefficients of the logterms are in agreement with the parton-hadron duality.On the other hand, the second Weinberg sum rule is notsatisfied, as in other local chiral-quark models. This callsfor caution, in particular, when analyzing processes sensi-tive to high momenta.9

E. The topological current

In a previous work [55] it was shown how the Wess-Zumino-Witten term arises for SQM in the presence of thespectral regularization. As a sample calculation illustratingthe consistency condition of Sec. II A we compute thebaryon current in the limit of spatially large backgrounds.This also shows how the regularization effects cancel in thefinal result, yielding the well known Goldstone-Wilczekcurrent [4]. Taking the appropriate functional derivative inEq. (2) we find

h �q�x��q�x�i � i1

Z�Z

�v��x�

��0

� limx0!x��i�2

1

Z�

��x��

�Z� ��x0�

��0

�ZCd!��!� tr

��hxj

�i

i@6 �!U5jxi�:

(51)

The first two lines display the consistency condition. Thederivative expansion of the Dirac operator can be neatlydone with the help of the identity

8At the mean-field level in a gradient expansion the L8 termcorresponds to O�m4� corrections to the soliton energy, whileL10 couples to external axial and vector currents. So a slightmismatch in these values should not influence strongly the mean-field soliton properties. We estimate the corresponding correc-tion to the energy as �E8 � �L84m4

�Rd3x�cos�2��r�� 1� �

�3 MeV, with �L8 � 10�3 —a negligible number.9We remind that nonlocal models do indeed fulfill this high-

energy constraint [39] and the violation of the second Weinbergsum rule in local models is probably related to the violation ofthe two-body consistency conditions discussed above (seeSecs. II A and II B). On the other hand it is uncertain if nonlocalmodels do indeed satisfy analyticity.


014008-9

hxj1

i@6 �!U5jxi � tr

Z d4k

�2��41

k6 � i@6 �!U5; (52)

where the differential operator acts to the right on thefunction f�x� � 1. This formula can be justified throughthe use of an asymmetric version of the Wigner trans-formation presented in Ref. [74]. Formal expansion inpowers of @ yields

hxj1

i@6 �!U5jxi �

X1n�0

trZ d4k

�2��4

�1

k2 �!2

�n�1

�k6 �!U5y��i@6 ��k6 �!U5y��n:

(53)

The leading nonvanishing term is

h �q�x��q�x�i �ZCd!��!�

Z d4k

�2��4!4

�k2 �!2�4

tr��U5y�i@6 U5�3� � �0B�; (54)

where B��x� is the topological Goldstone-Wilczek current

B� �1

24�2 ��h�Uy@U��U

y@�U��Uy@�U�i; (55)

and

�0 � ��96�2i�ZCd!��!�

Z d4k

�2��4!4

�k2 �!2�4

�ZCd!��!� � 1; (56)

where in the first line we have computed the momentumintegral and in the second line used the normalizationcondition (17). Thus the proper normalization of the spec-tral function is responsible for the correct topologicalproperties in SQM, preservation of anomalies, etc. Infact, we view the uniform treatment of the anomalousprocesses as one of the main advantages of SQM overother local chiral-quark models.

III. BUILDING THE BARYON

In this section we show how valence orbits are con-structed in SQM. The issue is far from trivial, as unlikethe case of local models, there is no obvious Fock-spacerepresentation. Already the experience of nonlocal modelsshowed that the construction of valenceness is an involvedissue [48,49]. There a bound-state pole of the propagatorwas found in the background of hedgehog chiral fields, andthis state was occupied with Nc � 3 valence quarks. Thefull contribution to the baryon current (local and nonlocal)yielded a correct (and quantized) baryon number of thesoliton. In the present case we face a different situation,with the spectral density and inherent complex massespresent. Thus we start our derivation from very basicfield-theoretic foundations, arriving in the end at a very

simple prescription holding under plausible assumptionsconcerning the analyticity properties of the valence eigen-value as a function of the complex mass.

A. Interpolating baryon fields

In the standard field-theoretic approach a baryon can bedescribed in terms of the corresponding correlation func-tion

�B�x; x0� � h0jTfB�x� �B�x0�gj0i (57)

with B�x� denoting an interpolating baryonic operator interms of anticommuting quark fields. We take the simplestcombination

B�x� �1

Nc!�1;...;Nc�a1;...;aNc q1a1

�x� � � � qNcaNc �x�;

(58)

where (1; . . . ; Nc) are the color indices, (a1; . . . ; aNc) thespinor-flavor indices, and �a1;...;aNc is the appropriate com-pletely symmetric spinor-flavor amplitude. Inserting acomplete set of baryon eigenstates gives

�B�x;x0� � ��t� t0�Xn

h0jB�0�jBn; ~kihBn; ~kj �B�0�j0i

e�i�x�x0�k��1�Nc��t0 � t�

Xn

h0j �B�0�j �Bn; ~ki

h �Bn; ~kjB�0�j0ie�i�x�x0�k; (59)

where Bn ( �Bn) are the baryon (antibaryon) states withmomentum ~k. Next we take the limit t� t0 � T ! �i1.That way the lightest baryon at rest is selected in the sum

�B�x; x0� � h0jB�0�jBihBj �B�0�j0ie�iTMB: (60)

In the large-Nc limit, we first rewrite the time-orderedproduct as a path integral over the fermionic degrees offreedom with the weight exp�iSSQM�. The resulting expres-sion, in turn, can be obtained by the appropriate functionaldifferentiation of the generating functional Z�s; p; v;a; �; �� with respect to the external quark fields ��x� and��x�, yielding

�B�x; x0� � �a1;...;aNc ��a01;...;a

0Nc

ZDUei��U�

YNci�1

iSaia0i�x; x0�:

(61)

Here the one-particle Green function in SQM is given by

Saa0 �x; x0� �

ZC0d!��!�hxj�i@6 �!U5��1

aa0 jx0i (62)

and the one-loop effective action has the form

��U� � �iNcZC0d!��!�Tr log�i@6 �!U5�: (63)

Note the presence of the spectral integration, however, thecontour C0 to be used in the baryon sector is yet to be


014008-10

determined. There is no a priori reason why it should beequal to the vacuum contour C.10 The limit Nc ! 1 drivesthe functional integral over the bosonic U fields into asaddle point. In the vacuum sector one then obtains themean-field vacuum. In the baryon sector, due to the pres-ence of Nc factors Sa;a0 in the numerator, the dominatingsaddle-point configuration is of course different from thatof the vacuum and depends nontrivially on x and x0. Thelimit of a large evolution time T selects the minimum-energy stationary configuration. For stationary configura-tions the Dirac operator can be written as

i@6 �!U5 � �0�i@t �H�!��; (64)

with the Dirac Hamiltonian equal to

H�!� � �i � r �!U5: (65)

Note that the spectral mass ! appearing here is complexand the situation is unconventional, since H�!� is notHermitian. In such a case one must distinguish betweenthe right and left eigenvectors, H Rn � �n Rn and Hy Ln ��n Ln , not related by the Hermitian conjugation, i.e.� Rn �y � Ln . The orthogonality relation is h Ln ; Rmi ��nm and completeness is given in terms of the left-rightidentity

Pj Rn ih

Ln j � 1. We will not use this fancy nota-

tion, but will implicitly understand that is the righteigenvector, while y is in fact the complex-conjugatedleft eigenvector. The corresponding eigenvalue problem isthen

H�!� n� ~x;!� � �n�!� n� ~x;!�: (66)

For our particular Hamiltonian one has the followinguseful properties

H�!�y � H�!��;

�5H�!��15 � H��!�;

�0H�!��10 � �H��!��y;

��0�5�H�!��0�5��1 � �H�!�;

trH�!� � 0:

(67)

where the trace is in the Dirac sense. Some properties ofthe eigenvalues deduced from the properties (67) are

�n�!�� n�!

��; �n�!� � �n��!�: (68)

One can now use the spectral representation of the propa-gator

iSaa0 � ~x; t; ~x0; t0� �

ZC0d!��!�

Z�

d�2�

Xn

ei��t�t0�

�� n�!�

na� ~x;!� � na0 � ~x0;!� (69)

with na� ~x;!� ( � na� ~x;!�) and �n�!� denoting the right(left conjugated) eigenfunctions and eigenvalues of H�!�evaluated at the stationary bosonic configuration. The con-tour for the energy integration, discussed in the following,is denoted by �. For the calculation of the baryon state notethat the propagator given by Eq. (69) and the fermiondeterminant given by Eq. (63) have to be evaluated at largeEuclidean times.

B. Hedgehog ansatz

The stationary solutions of chiral-quark models have thefamiliar hedgehog form:

U5� ~x� � exp�i�5 ~ � ~�� ~x��; (70)

with ~� denoting the chiral phase field. In the hedgehogansatz

~� � r��r�; (71)

with ��r� being a radial function. In the following twosubsections we analyze the Dirac problem (66) for theexponential profile as an example,

��r� � �e�r=R: (72)

In particular our figures are obtained for that case. Otherpopular cases include the linear profile

��r� � ��1� r=R��R� r� (73)

and the arctan profile [20]

��r� � 2 arctan�R=r�2: (74)

The parameter R is a generic size scale of the solitonproportional to the baryon rms radius. The profile (72) isin fact a fairly good approximation to the fully self-consistent profile found numerically in Sec. IV.

In the treatment of hedgehog systems it is relevant toconsider symmetries such as the grand spinG � I � J, thesum of isospin and spin. The reader is referred toRefs. [24–27] for necessary details.

C. The valence contribution

Now we resort to the numerical calculation of the spec-trum in the chiral soliton, described in detail in the pro-ceeding sections. What we need for the present discussionis the behavior of the spectrum when we vary the spectralmass! as an independent variable. We focus on the grand-spin 0 parity� state, GP � 0�, where G � I � J is thesum of isospin and spin, appropriate to classify states in thehedgehog background (71). In Fig. 2 we show the depen-dence of the lowest 0� eigenvalue on the mass parameter

10Note by analogy that in the standard many-body quantumfield theory the contour in the complex energy variable selectsthe orbits to be occupied and is clearly different for states withdifferent baryon number, or baryon density, where it crosses thereal axis at the value of the chemical potential.


014008-11

M for real masses [in our notationM is the same as real!].Subsequent curves are obtained for the profile (72) with afixed radius R. In fact, all the displayed curves are relatedto each other via simple scaling. Indeed, introducing ~� �~r=R and � � !R we can rewrite the Dirac equation in theform

��i � r ~� ��U5� ~��n� ~�; �� En��n� ~�; ��;

(75)

which depends on a single scale �. By identification

�n�!� �En�!R�R

: (76)

As can be promptly seen from Fig. 2, for any value of Rwe have for the valence 0� level the limiting behavior

�0�M� �M at M ! 0; (77)

�0�M� � �M� a at M ! 1; (78)

where a is a constant. Correspondingly, at fixed M in thelimit of low R (small solitons) we have �0�M� ’ M, i.e. thelevel goes to the top of the gap, whereas in the limit of highR (large solitons) the behavior �0�M� ’ �M� a showsthat the level goes to the bottom of the gap. The small Mbehavior, together with the reflexion property Eq. (68)suggests a branch point behavior at the origin

�0�!� !��!2

p; !! 0; (79)

which is no mystery, since it corresponds to a free particlewith the complex mass and with energy

��k2 �!2p

at zeromomentum. Unfortunately, in our study we only haveaccess to the chiral soliton spectrum for real M, thus wedo not possess the information on analyticity properties ofthe eigenvalues as functions of!, which is fully accessibleonly in exactly soluble problems.

Indeed, due to the inherent numerical complications,finite-size discrete basis used, etc., such information wouldbe difficult to obtain numerically in a reliable way. Thus,we proceed motivated by the real-mass results and theanalogy to a similar exactly-solvable model presented inAppendix D. In other words, our assumptions made in thegeneral hedgehog case appear to be justified.

Let us now review the calculation of the Dirac propa-gator for the standard case of a real mass. If we firstcompute the � integral in Eq. (69), we have to considerthe � contour in the complex-� space which for real massesand hence real eigenvalues has the standard form of goingbelow the real axis for states with energy below the gap andgoing above the real axis for states above the gap. Thisyields Z�

d�2�i

ei��t�t0�

�� n�M�� t� t0��n�M��

� ��t0 � t��n�M��e�i�n�M��t�t0�;

(80)

hence the positive (negative) energy states propagate for-ward (backward) in time. This is equivalent to a damped(exploding) imaginary-time behavior for t� t0 ! i1.This real-mass result coincides with the standard one. Ifwe changeM the valence level may change sign [because itis equivalent to changing the soliton size R and the leveldives into the sea according to (78)]. Thus, according toEq. (80) one keeps the contour. If we deformed the contourdeciding that we occupy the negative eigenvalue then theimaginary-time behavior implying the standard particle(antiparticle) interpretation would be violated.

Let us now analyze the case of a complex mass. For thatpurpose we go slightly off the real axis, ! � M� i�, andexpand

�n�M� i�� n�M� � i��0n�M� �

12�

2�00n�M� � � � � :

(81)

The edges of the bound-state gap and consequently theDirac eigenvalues wander into the complex-� plane mov-ing upwards or downwards, depending on the sign of�0n�M�. The eigenvalues fulfilling �0n�M� � 0 stay station-ary on the real axis. According to Fig. 2, we have �000 �M�<0 for any M> 0. The question is whether or not we shouldalso deform the contour � in the evaluation of the propa-gator (69). On the one hand, the eigenvalues lying in thecomplex plane should not cross the contour, as this wouldlead to discontinuities, thus some deformation must be

-200

-100

0

100

200

300

400

0 200 400 600 800 1000 1200

ε 0 [M

eV]

M [MeV]

R = 0.3 fmR = 0.4 fmR = 0.6 fmR = 1.0 fmR = 2.0 fm

FIG. 2. The lowest JP � 0� eigenvalue �0 as a function ofmass M for several soliton size parameters R in the profile (72).The curves are arranged with the lowest value of R at the top.The two straight solid lines indicate the boundaries of the gap, Mand �M. Each curve leaves the positive continuum at somecritical value of M, assumes a maximum, and then decreases,asymptotically becoming parallel to the �M line.


014008-12

done. On the other hand, if �00�M� � 0 we move along thereal axis in the positive direction by the amount� 1

2 �2�000 �M�, Eq. (81), hence crossing the contour onlyin the case where also �0�M� � 0. According to Fig. 2 thissituation never happens unless M � 0 or M is much largerthan the position of the maximum (which, as we will see, isof no relevance to our construction). Hence, we deform thecontour in such a way that it continues to pass through the� � 0 point, but never allows a complex eigenvalue tointersect. Such a contour yields Z�

d�2�i

ei��t�t0�

�� n�!�� t� t0��Re�n�!��

��t0 � t��Re�n�!��e�i�n�!��t�t0�:

(82)

We note that the large imaginary-time evolution is damped,as we might have expected. In the positive t� t0 > 0branch, where a baryon (and not an antibaryon) propagates,we are thus left with the integral

ZC0d!��!��Re�n�!��e��n�!�: (83)

for the imaginary time ! 1. Note that we only have thepositive section of the contour, as implied by the conditionRe�n�!�> 0. Recall that we wish to occupy the lowest 0�

orbit with the eigenvalue �0�!�. We proceed by using thesaddle-point method. First we have to locate stationarypoints of �0�!�, i.e. the points !m where

�00�!m� � 0: (84)

From Fig. 2 we find, that for any value of R there exists asaddle located at the maximum of the curve �0�M�. Wedenote the position of the saddle as M0.11 We call M0 thesaddle mass. As we move left from the saddle, decreasingM, we enter at some critical value Mc into the uppercontinuum. On the basis of the toy model results ofAppendix D, we expect that �0�!� has a branch cut at ! �Mc running downwards. We observe from Fig. 2 that thesaddles are located right of Mc for any value of the solitonsize R.

In order to compute the integral in Eq. (83) we deformthe original contour C used in the vacuum sector into thecontour C0 shown in Fig. 3. It contains the segment C0 � C

(dashed line), which runs across ! � M0 parallel to theimaginary axis. Note that it is not necessary to change thepath globally but only in the vicinity of the stationary point.Making the change of variables ! � M0 � i�, where M0

is the saddle mass, �0�M0� � 0 we get Z

C0�Cd!��!��Re�n�!��e��n�!�

!Z 1�1

id��M0�e��M0�e�1=2��00n�M0��

2

� i��M0�

��2�

��000 �M0�

se��M0�

� Z0e��M0�; (85)

where the wave function renormalization factor Z0 hasbeen included. Finally, we compare Eq. (85) to Eq. (60)and read off the valence contribution to the energy as

Eval � Nc�0�M0�: (86)

The necessary conditions are �00�M0� � 0, �0�M0�> 0, and�000 �M0�< 0, all satisfied in our case, as shown in Fig. 2. Inaddition, the solution is only admissible with the condi-tions Mc <M0 <MV=2, which guarantee a real residue.As we have already discussed, Mc <M0 holds for allvalues of R. For M0 >MV=2 we would have a complexresidue, contradicting the spectral decomposition of thepropagator, Eq. (60). According to Fig. 2, for the ansatz(72) this occurs for R< 0:6 fm, where no bound-statenucleon can be constructed. Thus, in the large imaginary-time limit t! �i1 and t0 ! �i1 we get [see, e.g.,Eq. (85)]

S�x0; x� ! Z0�0�x� ��0�x0�e�i�t�t

0��0 ; (87)

where �0 is the valence Dirac spinor.

w−Complex Plane

x

−Mv / 2 Mv / 2

C

C’

FIG. 3. The deformed contour C0 in the complex ! plane forthe saddle-point evaluation of the valence contribution.

11A priori we should admit multiple saddles !m. We choose thebranch, denoted !0, with the lowest possible real part of theeigenvalue. In addition, the complex reflexion property impliesthat if !0 is a saddle, then !�0 is also a saddle. As a consequence,for complex saddles the large Euclidean time behavior would notbe purely exponential, but also some oscillations would set in.This obviously contradicts the spectral decomposition (60),which allows only for real stationary points. Therefore, althoughcomplex solutions may a priori exist, they should be considereda spurious result of the model. Similar problems occur also in thelocal models [69].


014008-13

To summarize, our prescription for the valence orbitconsists of the following very simple steps:

(1) Obtain (numerically) the spectrum for the valenceorbital as a function of the real positive mass M.

(2) Look for the saddle mass, i.e., a maximum withrespect to M, with the help of conditions �00�M0� �0 and �000 �M0�< 0. From analyticity, this is thesaddle point which corresponds to the saddle massM0.

(3) If Mc M0 <MV=2, the valence contribution tothe soliton mass is Nc�0�M0�, otherwise there is nobaryon state in the model.

There is an essential difference between the standard va-lence prescription based on the Fock-space decomposition,used in local quark models, and our prescription for SQMpresented above. In the first case the size of the profile, R,and the quark mass, M, can be fixed independently in theDirac operator, whereas in the present case we actually finda correlation between them. In the quark propagator wehave a superposition of masses ! which are integratedover,12 thus no additional independent scale is present inthe problem. An immediate consequence can be seen inFig. 4, where we compare the valence eigenvalue for theprofile function (60) along the manifold where �00�M� � 0(for a given R) to the fixed-mass result of the local chiral-quark model, obtained with MQ � 300 MeV. From thescaled Dirac Eq. (76) and our valence prescription it fol-lows that

�0 � k=R; (88)

where k is a constant depending on the particular profile. Inparticular, for R! 0 in SQM one has �0 ! 1=R (a signa-ture of a ‘‘repulsive force’’), as opposed to the behavior�0 ! MQ of a free particle at rest in the case of the fixed-mass model. This means in practice that the bound statenever becomes unbound in the presence of a chiral fieldbackground (see the discussion of Sec. III F) and reflectsthe absence of on-shell quarks in the vacuum (analyticconfinement) in SQM. Another feature of SQM with ourvalence prescription is that the valence eigenvalue neverdives into the negative part of the spectrum. The prescrip-tion for fixed-mass models allows the valence contributionto become negative, and this happens with a finite slopewhich originates from nonanalyticity. This entering of thevalence level into the negative part of the spectrum wasinterpreted within the chiral-quark soliton model as enter-ing the ‘‘Skyrme’’ model regime. In our approach such aregime never arises. However, the 1=R dependence of thevalence contribution behaves very much like the Skyrmestabilizing term.

Similarly, the saddle mass scales as

M0 � k0=R: (89)

Numerically, for the exponential profile (72) one gets k �123 MeV fm and k0 � 151 MeV fm, for the linear profile(73) k � 234 MeV fm and k0 � 257 MeV fm, and forthe arctan profile (74) k � 100 MeV fm and k0 �119 MeV fm.

D. The soliton baryon density and the baryon number

A crucial point is to show that our soliton corresponds toa system with baryon number equal to one. This requiresthe separation between the valence and sea contributions tothe baryon number. To analyze this point we have tocompute the following time-ordered product,

��B �x; x

0; y� � h0jTfB�x� �q�y��q�y� �B�x0�gj0i

��

�v��y��vB�x; x

0�j0 (90)

with B�x� denoting the interpolating baryonic operator,Eq. (58) and �v

B�x; x0� the baryon correlator (57) in the

presence of a external vector field v��y� for which theresult (61) follows if the external source is included in thepropagator

-100

0

100

200

300

400

500

600

700

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

ε 0 [M

eV]

R [fm]

ε0 (M = M0)

εv(M = 300 MeV)

FIG. 4. Solid line: the valence eigenvalue �0 and the saddlemass, M0, for the 0� state (maxima of Fig. 2), plotted as afunction of the soliton size R. Dotted line: the 0� eigenvalue � atthe fixed value of M � 300 MeV, as used in traditional fixed-mass models. We note that the SQM prescription has �0 �const=R, whereas �! const at R! 0 and becomes negativeat R! 1.

12This reminds of the familiar distinction between the phaseand group velocities of wave packets made out of a continuoussuperposition of plane waves.


014008-14

Svaa0 �x; x0� �

ZC0d!��!�hxj�i@6 � v6 �!U5��1

aa0 jx0i: (91)

For the open line propagator the functional derivative canbe readily evaluated yielding

�Sv�x0; x��v��y�

��0�ZCd!��!�

hxj1

i@6 �!U5jyi��hyj

1

i@6 �!U5jx0i

! ��0�y��0�y�S�x; x0�; (92)

where in the last line the limit t! �i1 and t0 ! �i1 hasbeen taken along the line of reasoning developed inSec. III C [see Eq. (87)]. The determinant contribution tothe baryon current can be deduced from Eq. (51).Collecting all results we obtain for the three-point corre-lator in the asymptotic limit the factorized form

��B �x; x

0; y� ! B��y��B�x; x0� (93)

with the total baryon current given by

B��x� � ��0�x��0�x�

�ZCd!��!� tr

��hxj

�i

i@6 �!U5jxi�; (94)

which is our result for the baryon current and in general forany observable based on a one-body operator. In Fig. 5 thesea and the valence contributions to the baryon density forseveral soliton radii in the SQM are displayed. As we see,

the Dirac-sea contribution gives a vanishing contribution tothe baryon number; only the valence quarks contribute tothe total normalization. This is consistent with our findingthat the valence level never dives into the negative-energyregion.

It is worth mentioning that the previous calculation canbe extended mutatis mutandis to any one-body observableand, in particular, to the energy-momentum tensor. In thefollowing section we analyze the sea contribution as itarises from Eq. (60) onwards. A nontrivial check whichresults from the one-body consistency condition is thatsuch a calculation coincides with the one based on theenergy-momentum tensor in the soliton background.

E. The sea contribution

To identify the sea contribution we compare, as done forthe valence case, the spectral decomposition of the two-point correlator at large Euclidean times, Eq. (60), with thepath-integral representation, Eq. (61). Using Eq. (63) forstationary configurations, the Dirac-sea contribution to theenergy is obtained by the spectral integration along theoriginal contour C of Fig. 1,

Esea �iTNc

ZCd!��!�Tr log�i@6 �!U5�

� �NcZCd!��!�

1

2

Xn

��n�!�2

q

� NcZCd!��!�

Xi2sea

�i�!�; (95)

where T is the time and n indicates all (positive- andnegative-energy) states of the Dirac HamiltonianH definedin Eq. (65) and the frequency integral has been carried out.In the last line Eq. (67) has been used and summation i runsover negative-energy (sea) states only. A full-fledgedevaluation of this quantity is presented in Sec. IV. Tounderstand the general trend we will present an estimatebased on the so-called two-point function approximation(TPA) proposed originally in Ref. [20] and further ex-ploited in Refs. [75,76]. This approximation has the virtueof reproducing both the limit of large and small solitonsizes and is based in the identity for the normal paritycontribution to the action

ZCd!��!�Tr log�i@6 �!U5�jn:p:

�1

2

ZCd!��!�Tr log�@2 � i!@6 U5 �!2� (96)

(deduced by commuting the �5 matrix across the Diracoperator) and further expanding the logarithm to secondorder in the field U5, yielding

-0.5

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

r [fm]

R=0.2 fm

R=0.5 fm

R=2.0 fm

FIG. 5. Sea and valence contribution to the radial baryondensity for several soliton radii in the SQM (the valence partshave positive sign at all r).


014008-15

ETPAsea �

�iNc4T

ZCd!��!�Tr

��1

@2 �!2 i!@6 U5

�2�

��iNcT

ZCd!��!�

Z d4q

�2��4q2I�q2; !�!2hjU�q�j2i;

(97)

where the functional trace and Dirac traces have beenevaluated, h:i indicates the isospin trace, and U�q� is theFourier transform of the chiral field U�x� (see Appendix Bfor notation). The one-loop integral I�q2; !� is introducedand calculated in Appendix B. A straightforward calcula-tion in the meson-dominance version of the SQM for staticfields yields

ETPAsea �

1

4f2�

Z d3q

�2��3~q2hjU� ~q�j2i

M2V

~q2 �M2V

; (98)

where

U� ~q� �Zd3xU� ~x�ei ~q� ~x: (99)

The factor appearing in Eq. (98) is the pion form factor,Eq. (18) in the spacelike region, q2 � � ~q2 < 0. Actually,this is a general feature, static soliton profiles probe theEuclidean region of mesonic correlation functions. At largevalues of the soliton size R, small values of q dominate inEq. (98) and one gets

E� af2�R; (100)

where a is a numerical constant. For the exponential profile(72) one gets numerically a � 30:99. In the opposite limitof small soliton sizes, large q values dominate and Eq. (98)yields

E� bf2�R�MVR�

2; (101)

with b � 28:11. This behavior is different from theR3 logR short distance behavior documented in Ref. [20]for the proper-time regularized fermion determinant; it isalso free of the R logR behavior reported in Refs. [77,78]for the renormalized sea energy, which generated a vacuumLandau instability.

F. Existence of absolute minima

The total soliton energy is the valence plus the seacontribution as

EB � Eval � Esea: (102)

At small radii the valence contribution (88) dominates andE� 1=R, while at large radii the Dirac-sea contributiondominates and E� R. Since the function E�R� is continu-ous, on these simple grounds we prove to have a minimum.The behavior is illustrated in Fig. (6), where we show theenergy dependence as a function of the profile size R forthe profile (72). For this restricted configuration one clearlysees the occurrence of an absolute minimum as a conse-

quence of the fact that the valence quarks never becomeunbound. In contrast, in the standard approach to chiral-quark solitons the valence quarks become unbound at smallR and one has instead a local minimum, which becomesunstable when the total soliton energy exceeds that of Ncfree quarks at rest, E> NcM. This instability has been anobvious cause of concern since this situation correspondsto bound but not confined solitons. In practice, it would notbe a problem if valence states were deeply bound.However, most calculations have produced solutions whichlie on the unstable branch. The fact that our solutionscorrespond to an absolutely stable state is a remarkableproperty of SQM together with our construction of thevalence state. Results for other profiles of the chiral fieldare qualitatively similar to the case of Fig. 6.

Another interesting feature that can be seen from a directcomparison of Figs. 4 and 6 is that the minimum takesplace at a soliton size R where the valence state for ourmodel and the fixed-mass models produce a similar depen-dence. This suggests that the constituent fixed mass insoliton models may indeed correspond to a given saddlemass in the spectral construction and that the shallow (andunstable) minima found there may indeed be identified asthe absolute and stable minima obtained here.

IV. RESULTS FOR THE SELF-CONSISTENTSOLITON

In this section we describe the numerical properties ofself-consistent chiral solitons in SQM, as well as thecorresponding observables.

A. Evaluation of observables

As is well known, observables in hedgehog solitons fallinto two categories: independent of and dependent oncranking [79]. The second case is much more complicated,

0

500

1000

1500

2000

2500

3000

3500

4000

0.2 0.4 0.6 0.8 1 1.2 1.4

E [M

eV]

R [fm]

EtotEseaEval

FIG. 6. Total energy of the soliton (solid line) and its valenceand sea components for the profile (72), plotted as functions ofR. An absolute minimum exists.


014008-16

hence in this paper we deal only with the quantities that donot involve cranking. Observables independent of crankingcan be written in terms of single spectral sums of the form

A �Xi

Ai; (103)

where A is the value of the observable in the soliton, and Aiis the single-particle contribution which includes a valenceorbit and the Dirac sea. In the spectral approach this isgeneralized, in the sense that an extra integration over thespectral variable ! is present. With our method of treatingthe valenceness we also have a separated contribution fromNc occupied valence states

A � Aval � Asea; Aval � NcA�!val�;

Asea �ZCd!��!�Asea�!�;

(104)

where val and sea indicate valence and sea, !val � M0 isthe saddle mass, ��!� is the spectral function consisting ofthe vector and scalar parts given in Eq. (20), and Asea�!� �Pi2seaAi�!� with Ai�!� denoting the !-dependent single-

particle value of the observable in the orbit i. Because ofthe hedgehog symmetries for the observables in questionone may replace the sum over the negative-energy states,Pi2sea, with 1=2 of the sum over all states, 1

2

Pn2all.

The contour C is invariant under the transformation!! �!, see Fig. 1. Therefore

Asea �1

2

�ZCd!��!�Asea�!� �

ZCd!��!�Asea��!�

�;

�ZCd!�V�!�

Asea�!� � Asea��!�2

�ZCd!�S�!�

Asea�!� � Asea��!�2

; (105)

where �V and �S are the odd and even parts of the spectralfunction �, respectively, see Eq. (20). As we have stressedbefore, the contour C in Fig. 1 is complex. This is acomplication, since in a numerical calculation we obvi-ously do not have access to A�!� in the complex ! plane.For that reason we use a method which allows us to carryon the calculation along the real axis in the ! plane, asexplained in Appendix F. According to the formalism ofSec. III, the energy of the soliton is

E � NcEval �d2

du2 ��EV�u� � �ES�u��ju�1=M2

V;

�EV;S�u� � 2Z 1

1=�2��up�d!disc� ��V;S�!��

Xi

�i�!;m� �i��!;m�2

: (106)

With the help of Eq. (F14) we may also write

�EV;S�u� � 2Z 1

1=�2��up�d!disc� ��V;S�!��

Xi

�i�!;m� �i�!;�m�2

: (107)

In the chiral limit

�EV�u� � 2Z 1

1=�2��up�disc� ��V�!��

Xi

�i�!; 0�;

�ES�u� � 0:(108)

All observables not involving cranking can be evaluatedwith the technique described above and with the help of‘‘standard’’ formulas at each !. The derivatives with re-spect to u are carried out numerically.

B. Self-consistent equations

Now we present some details of our self-consistentprocedure. We use the linear representation of the hedge-hog field, U � s� ir � ~p, and impose the nonlinear con-straint s2 � p2 � 1 by renormalizing the fields after eachnumerical iteration. The Euler-Lagrange equations for theradial s and p fields are obtained from Eq. (G1) and havethe form

s�r�

g2� Nc � 0� ~x;M0� 0� ~x;M0�

�ZCd!��!�

Xi

� i� ~x;!� i� ~x;!�;

p�r�

g2 � Nc � 0� ~x;M0�i�5 ~ � r 0� ~x;M0�

�ZCd!��!�

Xi

� i� ~x;!�i�5 ~ � r i� ~x;!�;

(109)

where g2 is treated as a Lagrange multiplier. The first termson the r.h.s. are the valence contributions, evaluated ac-cording to the SQM prescription. The second terms are theDirac-sea contributions, where the sum over the negative-energy states i is carried out, or, equivalently, it can bereplaced by 1=2 of the sum over all states. With the help offormulas (F10) and (F11) the spectral integration inEq. (109) can be performed as a real-valued integral. Thespinors i� ~x; !� are obtained by solving the Dirac equationin the background of the fields s�r� and p�r� at a givenvalue of !.

The code used to find numerically the self-consistentsolutions is a modification of the method used in solvingdifferent versions of chiral-quark models. The quark orbitsare calculated by diagonalizing H for each value of ! inthe discrete Kahana-Ripka basis [16]. The Euler-Lagrangeequations are solved by iteration. The numerical effortinvolved in the calculation is similar to that in the case ofsolitons in nonlocal models [48,49].


014008-17

C. The self-consistent solution

For the sake of obtaining physical properties of thesoliton, we use two versions of the model. The first one(model I) is just the standard meson-dominance SQM, withthe single vector-meson mass, MV � m� � 769 MeV.This model (in the chiral limit) has only one scale, thusall observables are proportional to the value of MV inappropriate power. In particular, f2

� � M2V=�8�

2�. Thesecond model (model II) includes also the excited � state,�0�1465�. The spectral function is taken to be the weightedsum of Eq. (20), containing 90% of the ground-state �, and10% of excited �. All quantities are distributive over thespectral density, e.g. f2

� � 0:9m2�=�8�2� � 0:1m02� =�8�2�.

Model II contains two scales, and produces somewhatheavier and more compact solitons, as expected on simplescaling grounds. We work with the physical pion, m� �139:6 MeV, and the current quark mass is adjusted suchthat the Gell-Mann–Oakes–Renner relation is fulfilled,mh �qqi � m2

�f2�.

The results of the self-consistent calculation for the casewith physical pion mass are displayed in Table I and in

Fig. 7. The chiral corrections are small and do not alter ourbasic conclusions. We note that for both models the valueof the saddle mass M0 is well below the critical value 1

2MV

discussed in Sec. III.Comparing the soliton energy to the experimental nu-

cleon mass we should be aware that our soliton is a mean-field solution with grand spin 0 and should not be identifiedwith the nucleon, but rather with the average of the massof the nucleon and the ��1232� isobar 1

2 �MN �M��

1174 MeV. Moreover, quantization of the collective coor-dinates, or projection of the soliton wave function on thesubspace with good quantum numbers [80,81], reduces itsenergy by eliminating the spurious rotational and transla-tional energy. We note that the contribution to the energyarising from the scalar spectral density, Esea;S, is muchsmaller than the vector part, Esea;V . Furthermore, we noticethat the model gives approximate equipartition of energybetween each valence quark and the Dirac sea, in agree-ment to the 1=R behavior of the valence contribution andthe approximate �R dependence of the sea part.

D. Results for observables

As discussed in Sec. IVA, in the present approach wedeal only with those observables that do not involve crank-ing. Inclusion of cranking, which requires a linear-response calculation, is numerically involved and is out-side of the scope of this study. In Table II we display somecharacteristic observables of the self-consistent solutionfor the choice of parameters of Table I. First, we notethat the isoscalar rms charge radius of the soliton is toolarge as compared to the experiment. This quantity isdominated by the valence contribution, as can also beseen from Fig. 8. The quark spinors of the valence wavefunction exhibit a long tail which is a feature related to ourprescription for constructing the valence orbit. The large r

behavior is of the form exp��M2

0 � �20

qr�, and in our case

TABLE II. Properties of the self-consistent solution of Table I:the rms isoscalar charge radius, the axial-vector charge gA, theisovector magnetic moment �I�1, and the sigma commutator �N . The valence contribution (val) is given separately. Thecontribution from the sea (not given explicitly) is dominated bythe vector part, similarly as in the case of the energy. Theexperimental value of �N is taken from Ref. [82].

Model I Model II Experiment��hr2iI�0;val

q[fm] 1.23 1.07��

hr2iI�0;total

q[fm] 1.24 1.08 0.79

gA (val) 0.79 0.79gA (total) 0.93 0.95 1.26�I�1(val) 2.40 2.07�I�1 (total) 2.96 2.67 4.71 �N [MeV] 36 33 45 8

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

r [fm]

FIG. 7. The s � cos� and p � sin� fields as a function of theradius r for the two self-consistent solutions of Table I: the solidlines for the model I and the dashed lines for model II.

TABLE I. Model parameters and the soliton energy in the self-consistent calculation with the physical pion mass.

Model I Model II

f� [MeV] 86.5 97.3m� [MeV] 139.6 139.6��h �qqi�1=3 [MeV] 243 243m [MeV] 5.04 6.37M0 [MeV] 267 304"0 [MeV] 233 263Esea;V [MeV] 285 351Esea;S [MeV] 36 35Etotal [MeV] 1019 1174


014008-18

the value of � ��M2

0 � �20

qis small, with � � 1=�1:5 fm�

in model I and � � 1=�1:3 fm� in model II. The largesoliton radius can perhaps be reduced by subtracting thespurious center-of-mass motion of the soliton. Includinghigher vector mesons in the model tends to reduce thesoliton size, as expected by scaling arguments, however,on the expense of increasing f� and Etotal.

In the evaluation13 of gA and the isovector magneticmoment �I�1 we use the semiclassical projection coeffi-cients as obtained in the large-Nc limit [79]. Both valuesare lower than the corresponding experimental values. Butsince our solution is dominated by the valence state wecould as well, instead of the semiclassical projection, usethe quark model wave functions for evaluation of thevalence parts of observables. Then the expectation valueof the operator for the proton and the neutron yields thecoefficient 5

9 instead of 13 . That would result in higher

values, in better agreement with the experiment. The valueof the sigma commutator, which in our model is equal to

�N � �m2�f

2�

Zd3r�s�r� � 1� (110)

is reasonably well reproduced, assuming values somewhatsmaller than the experimental number, showing that thespatial extent of our chiral profile is not too large.

The results for the observables turn out to be quitesimilar to those obtained in other chiral models. In particu-

lar, they are strikingly close to the predictions of the modelwith the nonlocal regulators [48,49].

V. CONCLUSION

For many years it has been accepted that baryons arise assolitons of a chiral Lagrangean in the large-Nc limit ofQCD. However, the precise realization of this attractiveand fruitful idea is still unclear. Many model Lagrangeanshave been proposed, emphasizing different aspects of theproblem and incorporating as many known features of theunderlying quark-gluon dynamics as possible. While onthe one hand the Skyrme soliton models incorporate con-finement but not the spin-1=2 partons, on the other hand,chiral-quark soliton models account for spin-1=2 partonsbut lack the confinement property. This unsatisfactorysituation has been calling for improvement. As a first andhopefully useful step, we have considered a model wherequark poles on the real axis are absent, be it in the vacuumor in the soliton. Unfortunately, confinement does not holdsince basic large-Nc requirements on the analytic structureof meson correlators are violated, similarly to other effec-tive quark models. On the other hand the spectral quarkmodel provides a framework with spin-1=2 partons in thevacuum which simultaneously allows for baryonic solitonsolutions corresponding to absolute minima of the actionand hence cannot decay into three free quarks, unlikeprevious local chiral-quark soliton models. This is due tothe rather peculiar features of the model characterized by acontinuous superposition of masses in the complex planewith a suitable spectral function. As a necessary comple-mentary study we have examined an instance of an in-volved complex-mass relativistic system.

The solitons we find fit nicely into the phenomenologi-cal expectations of more standard chiral-quark solitonswith full inclusion of the polarized Dirac sea. In contrastto constituent chiral-quark models where one is allowed totune the constituent quark mass as a free parameter, thespectral quark model does not have this freedom. In ourcase the vector-meson mass determines one unique solu-tion in the chiral limit and thus all properties scale with thismass. Actually, due to the superposition of complex spec-tral masses !, in the soliton calculation there appears asaddle mass, M0, defined as a stationary point of thevalence quark eigenvalue as a function of ! in a fixedsoliton profile. In the chiral limit M0 also scales with thevector-meson mass and thus is a fixed number. To someextent the saddle mass behaves as a constituent quark massdetermined uniquely by the soliton, and its numerical valuecomes out within the expected range M0 � 300 MeV forthe typical solitons minimizing the total energy. Moreover,the close resemblance of the valence contribution to theenergy around the minimum suggests that the phenome-nology and results of previous calculations within constitu-ent chiral-quark models with a constant mass arereproduced, at least for the low-lying baryon states. The

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

r [fm]

FIG. 8. The valence and the sea contributions to the baryondensity (multiplied by 4�r2) for the self-consistent solutions: thesolid lines for model I and the dashed lines for model II.

13Explicit expressions for these quantities may be found, e.g.,in Refs. [26,27].


014008-19

new aspect unveiled by our calculation is the possibility ofstudying the monopole vibrations and their quantizationwhich are the traditional candidate for describing excitedbaryonic states such as, e.g., the Roper resonance. Whilestudies of this sort are routinely carried out in the Skyrmemodel, the lack of confinement in the traditional chiral-quark soliton model has prevented, as a matter of principle,calculations of excited states.

One distinguishing feature of the spectral quark modelconstruction is the verification of an important consistencyrelation involving quark two-point functions which werenot fulfilled for fixed constituent mass chiral-quark models.This feature, in addition to the uniform treatment of regu-larization removes theoretical doubts on the proper com-putation of the high-energy processes as regards theirpartonic interpretation and normalization. As we havementioned, one motivation for using chiral-quark solitonmodels is the spin 1=2 nature of the constituents, whichmakes the calculation of partonic distributions in the nu-cleon possible as a matter of principle. The results found inthe present paper pave the way for such a calculation in thesoliton picture, where the interplay between the analyticityenforced by the lack of on-shell quarks and the chiralsymmetry may simultaneously be tested for the low-lyingbaryon states.

ACKNOWLEDGMENTS

This research is supported by the Polish Ministry ofEducation and Science, Grants No. 2P03B 02828, by theSpanish Ministerio de Asuntos Exteriores and the PolishMinistry of Education and Science, Project No. 4990/R04/05, by the Spanish DGI and FEDER funds with GrantNo. FIS2005-00810, Junta de Andalucıa GrantsNo. FQM225-05, EU Integrated Infrastructure InitiativeHadron Physics Project Contract No. RII3-CT-2004-506078 and by the Bilateral Program for Scientific andTechnological Cooperation of the Ministries of Science,Technology, and Higher Education of Poland and Slovenia.

APPENDIX A: THE EFFECTIVE ACTION TOSECOND ORDER IN THE FIELDS

When using the path integral (8) with the effective action(10), it is enough to expand the action to second order inthe external sources and the dynamical pion field

U � 1�if~ � ~��5 �

1

2f2 ~� � ~�� . . . : (A1)

This can be done by noting that the Dirac operator inEq. (11) can be decomposed as a free part plus a perturba-tion

iD � i@6 �!� V; (A2)

where

V � !�if~ � ~��5 �

1

2f2 ~� � ~�� . . .�

� m0 � �v6 � a6 �5 � s� i�5p�: (A3)

Note that both the free propagator and the potential V maydepend on the spectral mass !. Then the perturbativeexpansion of the fermion determinant can be readilydone, yielding

i� � NcZCd!��!�Tr log�i@6 �!�

� NcZCd!��!�

X1n�1

1

nTr�

1

i@6 �!V�n

� i��0� � i��1� � i��2� � . . . ; (A4)

which can be classified according to the number of externalsources as well as dynamical fields.

The zeroth order contribution yields the vacuum energy

i��0� � NcZCd!��!�Tr log�i@6 �!�

� NcNfZd4x

ZCd!��!�

Z d4p

�2��4tr log�p6 �!�

�Zd4x

�� NcNf

Z d4p

�2��4tr log�p6 �

�; (A5)

where the vacuum energy density, �, defined relative to thecase of massless quarks, has been explicitly separated. Thisquantity was computed in Ref. [53] and in Sec. II C of thepresent paper through the use of the energy-momentumtensor. It is instructive to check the calculation in theeffective action formalism. We use the reflection symmetryp! �p and the standard identity for matrices, tr logA�tr logB� log detA� log detB� log det�AB� � tr log�AB�for the commuting Dirac matrices A � p6 �! and B �p6 �! (in four dimensions detA � det��A�). The vacuumenergy density can be written as one half of any of thesecontributions. The Dirac trace becomes trivial and afterinterchanging the order in the momentum and spectralintegrals one gets

� � �2iNcNfZ d4p

�2��4ZCd!��!�

�log�p2 �!2� � log�p2��: (A6)

To evaluate first the !-integral we write

log�p2 �!2� �Z p2 dk2

k2 �!2 : (A7)

Thus, we readily get from direct application of Eq. (21),


014008-20

� � �2iNcNfZ d4p

�2��4Z p2

dk2

�A�k2� �

1

k2

�

� iNcNfZ d4p

�2��4�p2A�p2� � 1�; (A8)

after going to the Euclidean space and integrating by parts.The surface term has been discarded, as the conditionlims!1s

3A��s� � 0, holds in the meson-dominance real-ization Eq. (21). Thus the final result is finite

� � �NcNfM4

V

192�2 � ��0:2 GeV�4 �Nf � 3�

� �3Nf�

2f4

Nc; (A9)

in agreement with Eq. (27) based on the energy-momentum tensor.

Returning to Eq. (A4), the first nonvanishing correctioninvolves only the scalar source, yielding

i��1� � NcZCd!��!�Tr

�1

i@6 �!s�

� 4NcZ d4p

�2��4ZCd!

��!�

p2 �!2

Zd4xhs�x�i (A10)

(h:i means the flavor trace), whence the quark condensatemay be obtained as

Nfh �qqi � �4iNcNfZ d4p

�2��4B�p2�; (A11)

which becomes an identity according to Eq. (21).

Finally, the second order contribution can be written as

��2� � �SS � �PP � �VV � �AA � �� P

� ��A � �PA: (A12)

The calculation is straightforward. For the bilinears in thefields, generically written as ’�x� � ’A�x��

A with A de-noting Lorentz-flavor index, we have

i��2� � �1

2Nc

Zd!��!�Tr

�’A�A

1

i@6 �!’B�B

1

i@6 �!

�

�i2

Z d4q

�2��4�’A�q� �’B��q�KAB�q2�; (A13)

where

��i�KAB�q� � �NcZCd!��!�

Z d4p

�2��4Tr�

�Ai

p6 � q6 �!�B

ip6 �!

�:

(A14)

The Fourier-transformed fields are defined through therelation

’�x� �Z d4q

�2��4eiq�x �’�q�: (A15)

The flavor trace is trivial, h�a�bi � �ab=2. After using thevanishing condition of the positive moments, Eq. (23), thecalculation of the Dirac trace and momentum integrationyields

��i�KabSS�q� � �Nc

ZCd!��!�

Z d4p

�2��4Tr��a

2

ip6 � q6 �!

�b

2

ip6 �!

�

� �abNcZCd!��!��2I�0; !�!2 � I�q2; !��4!2 � q2��; (A16)

��i�KabPP�q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�

ip6 � q6 �!

i�5�a

2

ip6 �!

i�5�b

2

�

� �abNcZCd!��!��2I�0; !�!2 � I�q2; !�q2�; (A17)

��i�Ka�;b�VV �q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�

ip6 � q6 �!

��a

2

ip6 �!

��b

2

�

�1

3�abNc

��g��

q�q�

q2

�ZCd!��!�

�4I�0; !�!2 �

q2

24�2 � 2I�q2; !��2!2 � q2�

�; (A18)

��i�Ka�;b�AA �q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�

ip6 � q6 �!

��5�a

2

ip6 �!

��5�b

2

�

�1

3�abNc

��g��

q�q�

q2

�ZCd!��!�

�4I�0; !�!2 �

q2

24�2 � 2I�q2; !��4!2 � q2�

�

� �ab4Ncq�q�

q2

ZCd!��!�I�q2; !�!2; (A19)


014008-21

��i�Ka�;bAP �q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�

ip6 � q6 �!

��5�a

2

ip6 �!

i�5�b

2

�� ab2iNcq�

ZCd!��!�!I�q2; !�;

(A20)

��i�Kab��q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�i!

f2 �ab ip6 �!

�i

p6 � q6 �!!f�5�

a ip6 �!

!f�5�

b�

� ��ab4Ncq2

f2

ZCd!��!�!2I�q2; !�; (A21)

��i�Ka�;bA� �q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�

ip6 � q6 �!

��5�a

2

ip6 �!

i�5�b !f

�

� ��ab4iNcq�

f

ZCd!��!�I�q2; !�!2; (A22)

��i�KabP��q� � �Nc

Zd!��!�

Z d4p

�2��4Tr�

ip6 � q6 �!

i�5�a

2

ip6 �!

i�5�b!f

�

� �ab2Nc1

f

ZCd!��!��2I�0; !�!3 � I�q2; !�q2!�: (A23)

where the basic one-loop two-point integral I�q2; !� isdefined in Eq. (B1).

To compute the correlation functions as functional de-rivatives we must take into account the contributions fromthe pion pole. This is accomplished in a standard way byeliminating the pion field at the mean-field level throughthe equations of motion, ��=��a�x� � 0, yielding at low-est order in the field

�a�q�Kab��q� � p

a�q�Kab�P�q� � a

a;��q�Ka;�;bA� �q� � 0;

(A24)

and reinserting the pion field into the effective action ��2�.This contribution exactly cancels the nontransverse pieceof the AA correlator, reproducing the gauge techniqueresult of Eq. (36) and an additional pion pole contributionto the PP correlator yielding Eq. (45).

APPENDIX B: USEFUL INTEGRALS

The basic two-point integral is given by14

I�q2; !� �1

i

Z d4k

�2��41

k2 �!2

1

�q� k�2 �!2

�1

i

Z d4k

�2��4Z 1

0

dx

�k2 �!2 � q2x�1� x��2

� �1

�4��2Z 1

0dx log�!2 � x�1� x�q2�: (B1)

For the meson-dominance case one can use the momentsmethod, based on expansion in powers of q2 and resum-

mation of the series. This method provides a useful checkto our computations. However, for the purpose of illustrat-ing the issues of quark unitarity with our unconventionalpropagator it is enlightening to provide general formulas interms of the vector and scalar components of the quarkpropagator. The ! integrals can be evaluated first by usingEqs. (21). We get the useful identities

A0�p2� � �ZCd!

��!�

�p2 �!2�2;

B0�p2� � �ZCd!

!��!�

�p2 �!2�2;

A�p2� � p2A0�p2� � �ZCd!

!2��!�

�p2 �!2�2;

B�p2� � p2B0�p2� � �ZCd!

!3��!�

�p2 �!2�2:

(B2)

The normalization of the pion field yields

f2 � 4NciZ d4k

�2��4d

dk2 �k2A�k2��

M2VNc

24�2 : (B3)

We also have Z

Cd!��!�I�q2; !� � i

Z 1

0dxZ d4k

�2��4

A0�k2 � x�1� x�q2�;ZCd!��!�I�q2; !�! � i

Z 1

0dxZ d4k

�2��4

B0�k2 � x�1� x�q2�;

(B4)

and so on. The integrals can be evaluated by passing to the14There is a typo in Eq. (A4) of Ref. [53].


014008-22

Euclidean space k2 ! �k2E and ��i�d4k! �2k2

Edk2E �

�2sds. We may then shift the integration variable in orderto get, e.g.,

J�q2� � �iZ 1

0dxZ d4k

�2��4F�k2 � x�1� x�q2�

�1

16�2

Z 10sds

Z 1

0dxF��s� x�1� x�q2�

�1

16�2

Z 10dSF��S�

Z 1

0dx�S� x�1� x�q2��;

(B5)

where we have introduced the distribution �x�� x��x�.Let us assume that for definiteness q2 < 0. Then, the argu-ment of the step function vanishes for x < x� and x > x�,where

x �1

2

��1�

4S

q2

s: (B6)

We have

Z 1

0dx�S� x�1� x�q2�� S�

q2

6� ��q2 � 4S�

��1�

4S

q2

s�q2 � 4S�: (B7)

Thus

J�q2� � �1

16�2

�Z 10dSF��S�

�S�

q2

6

�

�Z �q2=4

0dSF��S�

��1�

4S

q2

s�q2 � 4S�

�: (B8)

This formula can be analytically continued to any complexq, and the second integral becomes a line integral in thecomplex S plane. Clearly, if the function F��S� has a cut atsay S � �M2

V=4, the result of the line integral becomespath dependent and will develop an imaginary part dis-continuity for q2 >M2

V . This is the way how a quarkpropagator with no poles generates the unitarity cuts inthe two-point correlators.

We list the final results: Z

Cd!��!�I�q2; !� �

ZCd!��!�I�0; !�

�1

�4��2

�� log

�1�

q2

M2V

�

�2

3

q2

M2V � q

2

�;

ZCd!��!�I�q2; !�! � �

h �qqi2Nc

1

M2S � q

2 ;

ZCd!��!�I�q2; !�!2 �

M2V

96�2

M2V

M2V � q

2 ;

ZCd!��!�I�q2; !�!3 � �

h �qqi8Nc

�M2S

M2S � q

2

� 3MS

qtanh�1 q

MS

�:

(B9)

APPENDIX C: QUANTUM-MECHANICALEXAMPLES OF COMPLEX-MASS SYSTEMS

Establishing analytic properties of the Dirac operatoreigenvalues for a complex mass is an involved mathemati-cal problem. To gain some insight and develop someintuition, in this Appendix we consider a few cases ofnonrelativistic quantum-mechanical system with complexcoupling potentials. Our aim is to define what we mean bythe energy eigenvalues for a complex coupling, assumingthat we know the definition for the real coupling, and todetermine its analytic properties in the complex plane. Themost natural and obvious way to do so is in terms ofanalytic continuation in the coupling from the real case.

As a first example let us consider the harmonic oscillatorwhich has the reduced potential U � 2mV � m2!2x2 andthe ground-state energy is given by E0�!� � !=2 for real!> 0. The problem is obviously invariant under thechange !! �!, and we should write E0�!� � j!j=2for real !. Clearly, we have a branch cut at ! � 0 as afunction of the coupling !2, hence the right way to writethe energy for a complex coupling is E0�!� �

��!2p

=2.Written this way the bound-state wave function is given

by 0 � Ce��!2p

x2=2. The analytic continuation of the de-creasing exponential becomes an increasing exponentialon the second Riemann sheet, !2 ! e2�i!2. For negative!2 we have a negative energy. So, the energies on the firstand second Riemann sheets differ only in the sign.

As a second example let us consider the hydrogen atomwhere we have V � �Z=r with Z > 0 and the energy isE0�Z� � �Z

2=2. This suggests an analytic behavior in Z,and, in particular, having a continued bound-state solutionfor repulsive potentials (Z < 0). Again, the bound-statefunction u�r� � re�Zr transforms into a positive exponen-tial for negative Z. So, we have a cut at Z � 0. For negative


014008-23

Z we have a real energy but it is on the second Riemannsheet. In this case the energy on the first and the secondRiemann sheets coincide.

As a final example, more directly related to the morecomplicated case of the toy model for the Dirac equation inAppendix D, we analyze the complex square well potentialfor which we have U�r� � �U0��R� r�, with U0 com-plex. For real U0 the s-wave bound-state solution is givenby (we take 2m � 1)

u�r� � ��R� r�Ae��r ��r� R�B sinKr; (C1)

where

� ��Ep

; K ��U0 � E

p: (C2)

The continuity condition for the logarithmic derivativeyields the bound-state relation

� � � K cotKR; (C3)

which defines an implicit function E0�U0� which we wantto extend to complex U0. For real U0 there is a criticalvalue U0;c � ��=2R�2 above which the equation has realsolutions. Actually, close to the threshold we have forU0 >U0;c the ground state

E0�U0� � �1

4R2�U0 �U0;c�

2

�1

8R4

�1�

4

�2

��U0 �U0;c�

3 � � � � ; (C4)

which suggests an analytic behavior of the energy. Again,it is from the wave function at large distances where we seethat �! �� if we loop once about the critical pointU0;c � ��=2R�2 in the complex U0 plane. In this particularcase, this corresponds to the well known fact that a squarewell potential with a subcritical coupling has a virtual stateand no bound state. As U0 ! 0� one gets �! �1 orE0 ! �1 on the second Riemann sheet. Similar featuresshould appear also for excited states, En, with the corre-sponding critical values. Thus, we may define the continu-ous function E0�U0� for all real values of U0. ForU0 >U0;c it corresponds to a bound state, whereas forU0 <U0;c it describes a virtual state.

If we go now to the complex plane in U0 we get theimplicit function defined through

A�k;U� � 0: (C5)

If we have a solution A�k0; U0� � 0 and go close to it, thenby Taylor expanding we get

0 � A�k0 � �k;U0 � �U�

� A�k0; U0� �@A�kc; Uc�

@k�k�

@A�kc; Uc�

@U�U: (C6)

This way we can define a one to one relation unless eitherderivative vanishes at a critical point

@A�kc; Uc�

@k� 0 or

@A�kc;Uc�

@U� 0: (C7)

If this is the case we have a square root branch pointassuming

@2A�kc; Uc�

@k2� 0: (C8)

In our case we have the critical points located at

� � � K cot�KR�; (C9)

� 1 ��R

sin�KR�2��K

cot�KR�: (C10)

Combining both equations we get

�c � �1

R; Ec � �

1

R2 ; (C11)

1 � KR cot�KR�; Uc �1� x2

n

R2 ; (C12)

with xn � 0;4:49;7:72;10:9; . . . . Moreover, wehave

@2A�kc; Uc�

@k2�

R3Uc

R2Uc � 1; (C13)

which diverges forUc � 1=R2. Note that the critical pointsare located in the second Riemann sheet. They generatebranch points of second order, i.e. looping twice around thepoint the function returns to its original value. TheRiemann surface is obtained by joining all critical pointswith a line. Since they are infinitely many, the cut dividesthe complex U plane into two disjoint pieces. The com-plete and extensive study of Riemann sheets of this par-ticular problem can be looked up at Ref. [83].

The main outcome is that the analytic structure of a theanalytically continued eigenvalues of a complex couplingpotential can be determined by the study of the criticalpoints which generate branch cuts. Otherwise, the functionis analytic in the complex coupling plane. This situationappears also in Appendix D for the Dirac operator with acomplex mass.

APPENDIX D: ANALYTIC PROPERTIES OF THEVALENCE EIGENSTATE IN A TOY MODEL

The discussion in Sec. III made explicit use of the factthat an integration path in the complex-mass plane of theDirac Hamiltonian can be deformed without pinching anysingularities. Although we cannot prove this in general, thecomplex-mass coupling Dirac systems may have unusualproperties. In this Appendix we investigate a model baringsimilarity to the full chiral model where our assumptionsare verified. The main issue is both to define the meaning ofan eigenvalue as a function of the complex mass ! as wellas to determine its analytic properties in the complex-mass


014008-24

plane. To our knowledge this topic is not discussed in theliterature at all. However, strong similarities are found withthe analytic properties of the energy eigenvalues of com-plex potentials, which have been motivated in the contextof optical models [84,85] and for review purposesAppendix C includes some warm-up quantum-mechanicalproblems which may be helpful in the understanding ofanalyticity properties that arise in studies of our type. Theprescription for valenceness derived in Sec. III holds underspecific conditions. In the chiral soliton model we have noprecise knowledge of the analytic properties, hence it is notmathematically proven that the prescription can actually beused. The model of this section is much simpler andsolvable semianalytically. It shows that the desired analyticproperties are fulfilled, providing support for the method inapplication to the chiral soliton model described in thispaper.

Let us consider the Dirac equation for the state with 0grand-spin and positive parity, GP � 0�, with the uppercomponent u and the lower component v. It has the form

u0 � �uM sin�� v��M cos��

v0 � u��M cos�� v��

2

r�M sin�

� (D1)

with the usual boundary conditions for a normalizable state

u0�0� � 0; v�0� � 0; u�1� � 0; v�1� � 0:

(D2)

Following Ref. [86] we look for a solution with the profilefunction

��r� � ��R� r�; (D3)

where ��x� the standard Heaviside step function. Theprofile (D3) has the winding number ��0� � ��1��=� �1. The analytic solution is

u�r� � N��R� r�

�3

��Msinh�rr

��r� R��3

M� �e��r

r

�;

v�r� � N��R� r�

�

r2 ��r cosh��r� � sinh��r��

��r� R��r� 1�e��r

r2

�;

(D4)

where � ��M2 � �2p

and N is the normalization factorchosen such that N2

Rd3x�u2 � v2� � 1. Using the match-

ing condition expressing the continuity of u�r�=v�r� atr � R we get the eigenvalue equation

�R coth��R� ��R�M� �� 2M

��M; (D5)

which can be written it terms of dimensionless variables

x � �R and x0 � MR as

x coth�x� �x�x0

��x2

0 � x2

q� � 2x0

��x2

0 � x2

q� x0

; (D6)

where corresponds to the positive (negative) energystates, � > 0 (� < 0). The value x � 0 is formally alwaysa solution of Eq. (D6), but it does not correspond to a zeromomentum state �0 � M since the solution is

u�r� �1

R2 ��R� r� �2Mr

��r� R�; (D7)

v�r� � ��R� r�2Mr

3R2 ��r� R�1

r2 ; (D8)

which are regular both at r � 0 and r � 1 but fail to fulfillthe matching condition since v=u � 2Mr=3 for r > Rwhereas v=u � 2Mr for r > R.

The numerical dependence of the eigenvalue as a func-tion of the soliton size for the 0� state in the toy model isdepicted in Fig. 9. For large values of the soliton size atfixed mass we get

�0�R� !1

R; (D9)

which means according to the scaling property that thisbehavior also holds true for large mass and fixed radius.The solution is always in the positive energy region andnever dives into the sea (despite having topological num-ber 1—this was actually the point of Ref. [86]). For R!1 one gets �0�R� ! 0�. The reason is related to the factthat always sin� � 0 for the profile, and we have a scalarcoupling for which H�0�5 � ��0�5H and the spectrumis symmetric.

The threshold value for having a bound state is �MR�2 >3=4. Note that since �R � x changes sign across thethreshold value, the bound state becomes virtual (exponen-

FIG. 9. Valence eigenvalue �0 for the toy model (D1) and (D3)as a function of the soliton size R in units of 1=M.


014008-25

tially growing). Thus, the analytic continuation of the un-bound valence state for R2M2 < 3=4 is an exponentiallygrowing state.

Equation (D6) defines implicitly a relation between xand x0. Actually, we can compute the inverse function

x20 �

x4�coth�x� � 1�2

4�x� 1��x coth�x� � 1�: (D10)

For x > 0 the function on the r.h.s. is a monotonouslyincreasing function and thus the inverse is uniquely de-fined. The minimum takes place at x � 0 and assumes thevalue of 3=4. Hence there are no positive energy bound-state solutions for x2

0 3=4. For x20 � 3=4 we have a

negative-energy solution �R � x ’ �0:848 and � ’�0:175=R, of no concern here. For 0:512 x2

0 3=4we have a positive energy state with � < 0. For 0:512>x2

0 the value of � becomes complex. For x0 ! 0 we getx! �1.

In summary, for the real-mass case we have(i) For 0:866< x0 we have one bound-state solution.(ii) For 0:797< x0 < 0:866 we have one virtual state

solution.(iii) For 0:712< x0 < 0:797 we have two virtual states.

For x0 � 0:7968 one at x � �0:797 and the otherone at x � �0:196, the higher one being a continu-ation of the previous case.

(iv) For x0 < 0:716 the two solutions collide at x ��0:574 and bifurcate into the complex plane.

So clearly for x0 < 0:797 we have a cut in the � plane. Thesituation for the real-mass case has been displayed inFig. 10. This plot should be compared with the exponentialsoliton profile of Fig. 2.

We can now proceed to the complex plane in !. Theobvious way to define the eigenvalue �0�!� as a function ofthe complex variable ! is by analytic continuation, sincewe want to follow the evolution of the state in the complex-mass plane. For illustration purposes we plot in Fig. 11 thecorresponding analytically continued eigenvalues when! � ei for several soliton sizes R. Equivalently, we con-sider the function of the variable x2

0. Thus, we take thesolutions which were defined for real M fulfilling theboundary conditions and arrive at the transcendentalEq. (D6), which now defines implicitly �0�!�.

15 Thus, ifwe differentiate with respect to x the implicit function weget the equation for the critical points

2t4 � t3 � 4t2 � 2t� 1� �2t� 3��1� t2p

t��1� t2p

�1��1� t2p

�2� 0; (D11)

where for simplicity the variable t � x=x0 has been intro-duced. This is an algebraic equation with solutions t � tn.We find numerically

FIG. 11. Analytically continued complex valence eigenvalue�0�R;w� in the toy model plotted as a function of the complexmass ! � ei for in the first quadrant, 0 �=2. Theaxes represent Re��0� and Im��0�. Subsequent curves, from rightto left, correspond to increasing values of R � 2, 2.5, 3, 3.5, 3.9.The points on the real axis are for � 0, and the value of parametrizes the curves, which evolve away from the real axis.The other quadrants are obtained by reflection about the realaxis.

0.5 1 1.5 2 2.5 3MR

-2

-1

1

0 R

FIG. 10. Solid straight lines are � � M, the solid curve is thebound state, the dashed curve is the virtual state. Blobs indicatethe position of the saddle, the emergence of a bound state, andthe bifurcation point of two real virtual states into two complexstates. For the bound state �0R approaches unity at MR! 1,while for the virtual branch it approaches the negative contin-uum.

15According to the implicit function theorem the equation y �f�x� can be inverted around the solution y0 � f�x0� if f0�x0� �0. Critical points for which f0�x0� � 0 define bifurcationbranches if f00�x0� � 0.


014008-26

tn � x=x0

� �1:2741 i0:1174; 0:9092; 1:3898 i0:4832; . . . ;

(D12)

which forms a bundle of solutions

tnx0 coth�tnx0� �x0tn�1�

��1� t2n

p� � 2tn��

1� t2np

� 1: (D13)

We see that for any value of tn there are infinitely manysolutions for x0. Roughly, they are located along a straightline. For instance, for x=x0 � 0:9092 the critical points inx0 are located at about x0 � 0:5� in� for large N. So, thewhole critical points structure looks like five straight linesrunning in all directions. An alternative technique to ana-lyze this kind of problems is by using the modular land-scape of the exponential as explained in detail inRefs. [84,85].

The upshot of this study is that ��!� has only criticalpoints on the second Riemann sheet. This means thatwhenever the bound state becomes unbound we are cross-ing the branch cut, which happens for sufficiently smallvalues of !. For the spectral integral that would set theconditionMc <MV=2 not to have cut intersection betweenthe spectral function and the energy function.16

APPENDIX E: COMPUTING THE SPECTRALINTEGRAL FOR THE VALENCE CONTRIBUTION

FIRST

In this Appendix we show how our basic result for thevalence contribution, Eq. (86), can be reproduced by eval-uating the spectral integral first. In fact, experience withperturbative calculations of quantities such as f� or thepion form factor suggest that there is no danger in comput-ing the! integral first. At finite temperature this is actuallythe only way to get analytic results [88]. To compute thetroublesome spectral integral first in the soliton case, wetransform the scalar-pseudoscalar coupling into a vector-axial derivative coupling and a constant mass. We write

�i@6 �!U5� � u5�i@6 � u�5i@6 u�5 �!�u5; (E1)

where u2 � U. This is a field dependent axial rotation andat the level of the determinant of the Dirac operator gen-erates the anomalous Wess-Zumino term in the SU(3) case.Then, we get

Saa0 �x;x0��

Zd!��!�

hxj�u�5�i@6 �u�5i@6 u�5�!��1u�5�aa0 jx0i

�

�hxju�5

�Zd!

��!�

i@6 �u�5i@6 u�5�!

�u�5jx0i

�a;a0

� �u�x��5hxjS�i@6 �u�5i@6 u�5�jx0iu�x0��5�a;a0 :

(E2)

Now, in the meson-dominance model the propagator func-tion is given by

S�p6 � �ZCd!

��!�p6 �!

� 2�i��p6 � �1

p6: (E3)

This relation is obtained for the closed contour under theassumption that ��!� has a pole only at ! � 0, which isclearly the case of the meson-dominance model (20).Equation (E3) is just a functional relation, and does notdepend on the fact that the variable p6 is a matrix. We haveindeed computed the spectral integral, but for this formulato be of any use, we have to look for the eigenvalueproblem of the chirally rotated Dirac operator

�i@6 � u�5i@6 u�5��n � Mn�n: (E4)

This eigenvalue problem may be tricky in practice sincethe operator is not normal, i.e. the adjoint operator does notcommute with the original operator even if we go to theEuclidean space (where the derivative and the vector partof the coupling are anti self-adjoint and the axial part isself-adjoint). This means that Mn may be complex ingeneral.

Undoing field dependent chiral rotation we get

�i@6 �MnU5��n � 0; (E5)

where

�n � u�5�n; �yn � �ynu�5: (E6)

Thus, the eigenvalue problem corresponds to the search forthe (eigen)masses of the original Dirac operator whichyield a zero mode of the four dimensional Dirac operator.This equation looks as an on-shell condition for the boundquarks. In the stationary field case we have, �n� ~x; t� �e�i�t n� ~x�. At this point we must make a choice on thevalues of �. A natural condition would be to request anexponential damping in the Euclidean space. Thus, we take� real and positive for forward propagating states, t > 0,and negative for backward propagating states. (Anotherpossibility would be to take a contour with conditions onRe�),

��i � r �Mn�U5� n � � n: (E7)

Thus, here we encounter just the opposite problem to thecase of the spectral mass problem; for a given energy wehave to look for the eigenmasses,Mn�� (instead of �n�!�).

16Obviously, the smaller Mc the better. Now, it would beinteresting to see if there are chiral angles ��r� for which onealways has a bound state. In such a case cuts would not intersect.This is not such a strange situation, since e.g. in one dimensionalquantum mechanics, any tiny attractive potential generates im-mediately a bound state. Also, in 1� 1 dimensions the chiralangle ��x� � sign�x� has always a bound state � � M cos��[87].


014008-27

Actually, this looks like an isospectral problem; given theenergy how many chiral couplings possess the same en-ergy. Thus, the eigenvectors of this problem are not exactlythe eigenvectors of the standard Dirac Hamiltonian prob-lem. Note also that because we are searching for zeromodes of the Dirac operator its eigenvectors are alsoeigenvectors of the Dirac Hamiltonian, which is not thecase for nonzero eigenvalues, i.e. �0�i@t �H��n � �n�nwith �n � 0 cannot be solved by a eigenstate ofH becauseof the �0. Clearly, multivaluedness issues may becomequite relevant when discussing the possible equivalenceof both problems. On the other hand,Hy � H if and only ifMn�� is real. For such a case if both Mn and � are real,then bound states happen for �Mn < � <Mn. In generalMn is complex, since even in the free case,U � 1, we have

Mn�� 2 � k2

n

p, with kn denoting the momentum quan-

tized due to box boundary conditions. Then, we are lead to

Saa0 �x; x0� � u�5�x�hxjS�i@6 � u�5i@6 u�5�jx0iu�5�x0�

� u�5�x�Xn

�n�x�S�Mn��yn �x0�u�5�x0�

�Xn

�n�x�S�Mn��yn �x0�

�Z 1�1

d�2�

Xn

n�x��ei��t�t0��t0 � t�

� e�i��t�t0��t� t0��S�Mn��

yn �x0�:

(E8)

In this representation, the states propagating forward intime have been chosen to be those of positive energy andwe are led to the basic integral

Z 10

d�2�

e��S�Mn�� Z 1

0

d�2�

e��logS�Mn��; (E9)

with ! 1. The function S�M� of Eq. (E3) has no polesbut branch cuts starting at M � MV=2. The stationarypoints are determined from the equation

�@ logS�Mn��

@�� M0n��

S0�Mn��S�Mn��

: (E10)

Note that a real S�M� requires Mn <MV=2, thus forMn�� MV=2 we must have a divergent M0n�� ! 1.Obviously, the smallest positive � (the valence orbit) ful-filling 1=M0n�� 0 dominates, and we get the relation

�0 � min�jM0n��1;�>0; (E11)

in agreement with our result of Sec. III, which is the resultequivalent to (84) written in terms of the inverse function.

APPENDIX F: EVALUATION OF THE DIRAC-SEACONTRIBUTION TO SOLITON OBSERVABLES

Dirac-sea contributions to observables involve thecomplex-mass integral. However, it is possible to rewrite

this contribution as a real-mass distribution. The vectorspectral density �V can be written as

�V�!� �d2

du2 ��V�!�ju�1=M2V; (F1)

where

�� V�!� �1

2�i1

12!5

1

�1� 4u!2�1=2: (F2)

The function ��V�!� has the property that its integrals withsmooth functions vanish along the parts of the contour Cencircling the branch points of Fig. 1, as well as are finitealong the cut. Similarly, for the scalar part, �S, we have

�S�!� �d2

du2 ��S�!�ju�1=M2V; (F3)

with

�� S�!� �1

2�i1

12!4

12�03M4S�1� 4u!2�1=2

: (F4)

We now define

�A sea;V�u� � 2Z 1

1=�2��up�d!disc� ��V�!��

Asea�!� � Asea��!�

2;

disc� ��V�!�� 1

12�!5��4u!2 � 1p ;

(F5)

(the factor of 2 comes from the two sections of C) and,similarly,

�Asea;S�u��2Z 1

1=�2��up�d!disc� ��S�!��

Asea�!��As��!�2

;

disc� ��S�!��03

�M4S!

4��4u!2�1p : (F6)

The Dirac-sea contributions to observables are now ob-tained from

Asea �d2

du2 ��Asea;V�u� � �Asea;S�u��ju�1=M2

V: (F7)

Thus, we have managed to ‘‘put the model on the realaxis,’’ at the only expense of the need of differentiationwith respect to u at the end of the calculation. As a bonuswe also get a very high degree of convergence at large !,since disc� ��V�!�� 1=!6 and disc� ��S�!�� 1=!5. It isuseful to get rid of the integrable singularities in Eqs. (F5)and (F6) by means of introducing the variable

z ��4u!2 � 1

p; ! �

��z2 � 1

p2��up : (F8)

Then


014008-28

�A sea;V�u� �8u2

3�

Z 10

dz

�z2 � 1�31

2

�Asea

� ��z2 � 1

p2��up

�

� Asea

��

��z2 � 1

p2��up

��; (F9)

and

�A sea;S�u� � �8�h �qqi

NcM4V

�4u�3=2Z 1

0

dz

�z2 � 1�5=2

1

2

�Asea

� ��z2 � 1

p2��up

�� Asea

��

��z2 � 1

p2��up

��:

(F10)

The Dirac Hamiltonian has the form

H�!;m� � �i � r � �m� �!U5; (F11)

with the corresponding Dirac equation

H�!;m�ji;!;mi � �i�!;m�ji;!;mi: (F12)

It has the property

�5H�!;m��5 � H��!;�m�;

�5ji;!;mi � ji;�!;�mi:(F13)

Thus �5 flips the sign of the current quark mass m and !.Obviously, the spectrum of the operator is unchangedunder this similarity transformation. Nevertheless, it isconvenient in the numerical work to deal with positivevalues of ! only. Thus, any one-body observable at nega-tive ! can be transformed according to

Asea�!� �Xi

hi;!;mja�!;m�ji;!;mi

�Xi

hi;!;mj�5�5a�!;m��5�5ji;!;mi

�Xi

hi;�!;�mj�5a�!;m��5ji;�!;�mi:

(F14)

Some examples of �5a�!;m��5 encountered in the evalu-ation of observables are �5H�!;m��5 � H��!;�m�,�5��5 � ��, etc. The formula for the energy followingfrom the above formulation is given in Eqs. (106) and(107).

APPENDIX G: INSTABILITY OF THE LINEARMODEL

Although SQM is only constructed in the nonlinear case,it looks tempting to extend it in the spirit of the originalbosonized Nambu–Jona-Lasinio (NJL) model to a linearversion where the fields may depart from the chiral circle

I � iZCd!��!�Tr log�i@6 �m� w�s� i�5� � p��

�1

2g2

Zd4x�s2 � p2�: (G1)

Here s and p denote the scalar-isoscalar and pseudoscalar-isovector fields, and g is a coupling constant. The mesonfields s and p are dimensionless. In the chiral limit and inthe vacuum (s � 1, p � 0) we find from the Euler-Lagrange equation for the s field the condition

s

g2 � �iZCd!��!�Tr

!i@6 �!s

� �4iNcNfZCd!��!�

Z d4k

�2��4!2s

k2 � �!s�2: (G2)

With the explicit form of the meson-dominance spectralfunction (20) and the techniques of Ref. [53] we find

1

g2 �NcNfM4

V

48�2 : (G3)

The effective potential assumes the form

V �IRd4x�NcNfM4

V

192�2 �2s2 � s4�; (G4)

where the s4 term originates form the first term in Eq. (G1).This corresponds to an inverted Mexican hat, and clearlydisplays instability. Therefore the linear version of themodel (G1) does not make sense.

It is worthwhile to mention that this feature should notbe regarded as specific to the spectral regularization. Theinverted potential also arises when renormalizing the fer-mion determinant with the help of the �-function regulari-zation in the local NJL model. Therefore one needs to usefrom the outset the nonlinear realization of chiral symme-try on its own, and it cannot be treated as an approximationto the linear model. In that regard we also note that thesoliton instability in linear NJL models has been found inRef. [89].

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Chiral solitons in the spectral quark model