10
Goldstone bosons decoupling from high-lying hadrons L. Ya. Glozman Institute for Physics, University of Graz, Universita ¨tsplatz 5, A-8010 Graz, Austria A. V. Nefediev Institute of Theoretical and Experimental Physics, 117218, B. Cheremushkinskaya 25, Moscow, Russia (Received 3 March 2006; published 20 April 2006) In this paper, we discuss a decoupling of the Goldstone bosons from highly excited hadrons in relation to the restoration of chiral symmetry in such hadrons. We use a generalized Nambu-Jona-Lasinio model with the interaction between quarks in the form of an instantaneous Lorentz-vector confining potential. This model is known to provide spontaneous breaking of chiral symmetry in the vacuum via the standard self-energy loops for valence quarks. For highly excited hadrons, where the typical momentum of valence quarks is large, the loop contributions represent only a small correction to the chiral-invariant classical contributions and asymptotically vanish. Consequently, the chiral symmetry violating Lorentz-scalar dynamical mass of quarks vanishes. Then the conservation of the axial-vector current in the chiral limit requires, via the Goldberger-Treiman relation, that the valence quarks decouple from the Goldstone boson. As a consequence, the whole hadron decouples from the Goldstone boson as well, which implies that its axial constant also vanishes. DOI: 10.1103/PhysRevD.73.074018 PACS numbers: 12.38.Aw, 12.39.Ki, 12.39.Pn I. INTRODUCTION An approximate restoration of SU2 L SU2 R and U1 A symmetries in excited hadrons has recently become a subject of a significant theoretical effort [1–14]. This effective restoration of chiral symmetry requires, in par- ticular, that highly excited hadrons should gradually de- couple from the Goldstone bosons [4,7,14]. There is an indirect phenomenological hint for such a decoupling. Indeed, the coupling constant for the process h ! h decreases for high-lying resonances, because the phase- space factor for such a decay increases with the mass of a resonance much faster than the decay width. A coupling of the Goldstone bosons to the valence quarks is regulated by the conservation of the axial current (we consider for simplicity the chiral limit). This conser- vation results in a Goldberger-Treiman relation [15], taken at the ‘‘constituent quark’’ level, giving g / m eff q ; (1) where m eff q is the quark Lorentz-scalar dynamical mass which appears self-consistently due to spontaneous break- ing of chiral symmetry (SBCS) in the vacuum. Appearance of such a dynamical mass is a general feature of chiral symmetry breaking and has been studied in great detail in the context of different models like the Nambu and Jona- Lasinio model [16,17] and the instanton liquid model [18], within the Schwinger-Dyson formalism with the quark kernel formed by the QCD string [19], by the perturbative gluon exchange [20] or by the instantaneous Lorentz- vector confinement [21–23]. A general feature of this dynamical mass is that it results from quantum fluctuations of the quark field and vanishes at large momenta where the classical contributions dominate [5,13]. Then, since the average momentum of valence quarks higher in the spec- trum increases, the valence quarks decouple from the quark condensate and their dynamical Lorentz-scalar mass de- creases (and asymptotically vanishes), so that chiral sym- metry is approximately restored in the highly excited hadrons [1,12,13]. This implies, via the Goldberger- Treiman relation (1), that valence quarks, as well as the whole hadron, decouple from the Goldstone bosons [4]. This, in turn, requires the axial coupling constant of the highly excited hadrons to decrease and to vanish asymptotically. While this perspective was shortly outlined in the past, this has never been considered in detail microscopically. However, it is important to clarify this physics, especially because the origins of this phenomenon cannot be seen at the level of the effective Lagrangian approach, where the coupling constant of the Goldstone boson to the excited hadron is an input parameter and the decoupling is not intuitive [14]. Even though the role of different gluonic interactions in QCD, which could be responsible for chiral and U1 A symmetries breaking, is not yet clear, the most fundamen- tal reason for the restoration of these symmetries in excited hadrons is universal [5]. Namely, both SU2 L SU2 R and U1 A symmetries breaking results from quantum fluctuations of the quark fields (that is, loops). However, for highly excited hadrons, where the action of the intrinsic motion is large, a semiclassical regime necessarily takes place. Semiclassically, the contribution of quantum fluctu- ations is suppressed, relative to the classical contributions, by a factor @=S, where S is the classical action of the intrinsic motion in the hadron in terms of the quark and gluon degrees of freedom. Since, for highly excited had- rons, S @, contributions of the quantum fluctuations of PHYSICAL REVIEW D 73, 074018 (2006) 1550-7998= 2006=73(7)=074018(10)$23.00 074018-1 © 2006 The American Physical Society

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PHYSICAL REVIEW D 73, 074018 (2006)

Goldstone bosons decoupling from high-lying hadrons

L. Ya. GlozmanInstitute for Physics, University of Graz, Universitatsplatz 5, A-8010 Graz, Austria

A. V. NefedievInstitute of Theoretical and Experimental Physics, 117218, B. Cheremushkinskaya 25, Moscow, Russia

(Received 3 March 2006; published 20 April 2006)

1550-7998=20

In this paper, we discuss a decoupling of the Goldstone bosons from highly excited hadrons in relationto the restoration of chiral symmetry in such hadrons. We use a generalized Nambu-Jona-Lasinio modelwith the interaction between quarks in the form of an instantaneous Lorentz-vector confining potential.This model is known to provide spontaneous breaking of chiral symmetry in the vacuum via the standardself-energy loops for valence quarks. For highly excited hadrons, where the typical momentum of valencequarks is large, the loop contributions represent only a small correction to the chiral-invariant classicalcontributions and asymptotically vanish. Consequently, the chiral symmetry violating Lorentz-scalardynamical mass of quarks vanishes. Then the conservation of the axial-vector current in the chiral limitrequires, via the Goldberger-Treiman relation, that the valence quarks decouple from the Goldstone boson.As a consequence, the whole hadron decouples from the Goldstone boson as well, which implies that itsaxial constant also vanishes.

DOI: 10.1103/PhysRevD.73.074018 PACS numbers: 12.38.Aw, 12.39.Ki, 12.39.Pn

I. INTRODUCTION

An approximate restoration of SU�2�L � SU�2�R andU�1�A symmetries in excited hadrons has recently becomea subject of a significant theoretical effort [1–14]. Thiseffective restoration of chiral symmetry requires, in par-ticular, that highly excited hadrons should gradually de-couple from the Goldstone bosons [4,7,14]. There is anindirect phenomenological hint for such a decoupling.Indeed, the coupling constant for the process h� ! h�� decreases for high-lying resonances, because the phase-space factor for such a decay increases with the mass of aresonance much faster than the decay width.

A coupling of the Goldstone bosons to the valencequarks is regulated by the conservation of the axial current(we consider for simplicity the chiral limit). This conser-vation results in a Goldberger-Treiman relation [15], takenat the ‘‘constituent quark’’ level, giving

g� / meffq ; (1)

where meffq is the quark Lorentz-scalar dynamical mass

which appears self-consistently due to spontaneous break-ing of chiral symmetry (SBCS) in the vacuum. Appearanceof such a dynamical mass is a general feature of chiralsymmetry breaking and has been studied in great detail inthe context of different models like the Nambu and Jona-Lasinio model [16,17] and the instanton liquid model [18],within the Schwinger-Dyson formalism with the quarkkernel formed by the QCD string [19], by the perturbativegluon exchange [20] or by the instantaneous Lorentz-vector confinement [21–23]. A general feature of thisdynamical mass is that it results from quantum fluctuationsof the quark field and vanishes at large momenta where theclassical contributions dominate [5,13]. Then, since the

06=73(7)=074018(10)$23.00 074018

average momentum of valence quarks higher in the spec-trum increases, the valence quarks decouple from the quarkcondensate and their dynamical Lorentz-scalar mass de-creases (and asymptotically vanishes), so that chiral sym-metry is approximately restored in the highly excitedhadrons [1,12,13]. This implies, via the Goldberger-Treiman relation (1), that valence quarks, as well as thewhole hadron, decouple from the Goldstone bosons [4].This, in turn, requires the axial coupling constant of thehighly excited hadrons to decrease and to vanishasymptotically.

While this perspective was shortly outlined in the past,this has never been considered in detail microscopically.However, it is important to clarify this physics, especiallybecause the origins of this phenomenon cannot be seen atthe level of the effective Lagrangian approach, where thecoupling constant of the Goldstone boson to the excitedhadron is an input parameter and the decoupling is notintuitive [14].

Even though the role of different gluonic interactions inQCD, which could be responsible for chiral and U�1�Asymmetries breaking, is not yet clear, the most fundamen-tal reason for the restoration of these symmetries in excitedhadrons is universal [5]. Namely, both SU�2�L � SU�2�Rand U�1�A symmetries breaking results from quantumfluctuations of the quark fields (that is, loops). However,for highly excited hadrons, where the action of the intrinsicmotion is large, a semiclassical regime necessarily takesplace. Semiclassically, the contribution of quantum fluctu-ations is suppressed, relative to the classical contributions,by a factor @=S, where S is the classical action of theintrinsic motion in the hadron in terms of the quark andgluon degrees of freedom. Since, for highly excited had-rons, S � @, contributions of the quantum fluctuations of

-1 © 2006 The American Physical Society

FIG. 1. Graphical representation of the equations for thedressed-quark propagator, Eq. (5), and for the quark massoperator, Eq. (7).

L. YA. GLOZMAN AND A. V. NEFEDIEV PHYSICAL REVIEW D 73, 074018 (2006)

the quark fields are suppressed relative to the classicalcontributions. Consequently, both chiral and U�1�A sym-metries are approximately restored in this part of thespectrum.

Although this argument is quite general and solid, it doesnot provide one with any detailed microscopic picture ofthe symmetry restoration. Then, in the absence of control-lable analytic solutions of QCD, such an insight can beobtained only through models.

It is instructive to outline the minimal set of require-ments for such a model. It must be (i) relativistic,(ii) chirally symmetric, and (iii) able to provide sponta-neous breaking of chiral symmetry; in addition, (iv) itshould contain confinement and (v) it must explain therestoration of chiral symmetry in excited states. There isa model which does incorporate all required elements. Thisis the generalized Nambu and Jona-Lasinio (GNJL) modelwith the instantaneous Lorentz-vector confining kernel[21–23]. In this model, confinement of quarks is guaran-teed due to instantaneous, infinitely rising (for example,linear) potential. Then chiral symmetry breaking can bedescribed by the standard summation of the valence-quarkself-interaction loops giving rise to the Schwinger-Dysonequation for the quark self-energy [21,22]. Alternatively,the model can be considered in the Hamiltonian approachusing the Bardeen, Cooper, and Schrieffer (BCS) formal-ism [24]. In this case, chiral symmetry breaking in thevacuum happens via condensation of the 3P0 quark-antiquark pairs and dressed quarks appear from theBogoliubov-Valatin transformation applied to bare quarks.The mass-gap equation ensures absence of anomalousBogoliubov terms in the Hamiltonian [23]. Finally, mesonsare built using the Bethe-Salpeter equation for the quark-antiquark bound states [22,23] or by a generalized bosonicBogoliubov-like transformation applied to the operatorscreating/annihilating quark-antiquark pairs [25].

It was demonstrated in Ref. [12] that, for the low-lyingstates, where the typical momentum of valence quarks isnot large and chiral symmetry breaking is important, thismodel leads to an effective Lorentz-scalar binding poten-tial, while for high-lying states, such an effective potentialbecomes a pure Lorentz spatial vector. Then the above-discussed quantum nature of chiral symmetry breaking inQCD as well as the transition to the semiclassical regimefor excited states, with loop effects being suppressed rela-tive to the classical contributions, has been illustratedwithin the same model in Ref. [13]. As a result, the modeldoes provide chiral symmetry restoration for excitedhadrons.

The purpose of this paper is to give an insight intophysics of decoupling of the Goldstone bosons from theexcited hadrons. Such a decoupling happens in line and forthe same reason as the approximate restoration of chiralsymmetry in these excited hadrons.. We resort to the GNJLmodel in view of its obvious advantage as a tractable model

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for QCD which can be used as a laboratory to get a micro-scopical insight into the restoration of chiral symmetry forexcited hadrons.

II. GENERALIZED NAMBU–JONA-LASINIOMODEL

A. Some generalities

In this section, we overview the GNJL chiral quarkmodel [21–23] which is described by the Hamiltonian

H �Zd3x � � ~x; t���i ~� ~5�m� � ~x; t�

�1

2

Zd3xd3yJa�� ~x; t�K

ab��� ~x� ~y�Jb�� ~y; t�; (2)

with the quark current-current [Ja�� ~x; t� �� � ~x; t���

�a2 � ~x; t�] interaction parametrized by an instan-

taneous confining kernel Kab��� ~x� ~y� of a generic form. In

this paper, we use the simplest form of the kernel compat-ible with the requirement of confinement,

Kab��� ~x� ~y� � g�0g�0�

abV0�j ~x� ~yj�: (3)

We do not dwell at any particular form of the confiningpotential V0�j ~x� ~yj�, though, if needed for illustrationpurposes, we employ a powerlike confining potential[21,26],

V0�j ~xj� � K��10 j ~xj�; 0 � 2; (4)

for qualitative analysis, concentrating mostly on the case ofthe linear confinement (� � 1) or, for numerical studies,resorting to the harmonic oscillator potential (� � 2) [21–23].

B. Chiral symmetry breaking

Spontaneous breaking of chiral symmetry in the class ofHamiltonians (2) is described via the standard Dyson seriesfor the quark propagator which takes the form, schemati-cally (see Fig. 1),

S � S0 � S0�S0 � S0�S0�S0 � . . . � S � S0 � S0�S;

(5)

with S0 and S being the bare- and dressed-quark propaga-tors, respectively; � is the quark mass operator. TheDyson-Schwinger equation (5) has the solution

S�1�p0; ~p� � S�10 �p0; ~p� � �� ~p�; (6)

-2

0 1 2 30,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6 ϕp

p

FIG. 2. Nontrivial solution to the mass-gap equation (12) withm � 0 and for the linear confinement (see, for example,Refs. [26,37] for the details). The momentum p is given in theunits of

�����p

; � is the fundamental string tension.

GOLDSTONE BOSONS DECOUPLING FROM HIGH-LYING . . . PHYSICAL REVIEW D 73, 074018 (2006)

where the mass operator independence of the energy p0

follows from the instantaneous nature of the interaction.The expression for the mass operator through the dressed-quark propagator (see Fig. 1) reads

i�� ~p� � CFZ d4k

�2��4V0� ~p� ~k��0S�k0; ~k��0;

CF �N2C � 1

2NC;

(7)

with both quark-quark-potential vertices being baremomentum-independent vertices �0. This corresponds tothe so-called rainbow approximation which is well justifiedin the limit of the large number of colors NC. We assumethis limit in what follows. Then all nonplanar diagramsappear suppressed by NC and can be consecutively re-moved from the theory. Equations (6) and (7) togetherproduce a closed set of equations, equivalent to a singlenonlinear equation for the mass operator,

i�� ~p� �Z d4k

�2��4V� ~p� ~k��0

1

S�10 �k0; ~k� ��� ~k�

�0; (8)

where the fundamental Casimir operator CF is absorbed bythe potential, V� ~p� � CFV0� ~p�.

To proceed, we use the standard parametrization of themass operator �� ~p� in the form

�� ~p� � �Ap �m� � � ~� ~p��Bp � p�; (9)

so that the dressed-quark Green’s function (6) becomes

S�1�p0; ~p� � �0p0 � � ~� ~p�Bp � Ap; (10)

where, due to the instantaneous nature of the interquarkinteraction, the time component of the four-vector p� isnot dressed.

It is easily seen from Eq. (10) that the functions Ap andBp represent the scalar and the space-vectorial part of theeffective Dirac operator, respectively. Notice that it is thescalar part Ap that breaks chiral symmetry and hence it canbe identified with the dynamical mass meff

q of the valencequark appearing in the Goldberger-Treiman relation (1). Inthe chiral limit, Ap vanishes, unless chiral symmetry isbroken spontaneously. It is convenient, therefore, to intro-duce an angle, known as the chiral angle ’p, according tothe definition

tan’p �ApBp; (11)

and varying in the range� �2 <’p

�2 , with the boundary

conditions ’�0� � �2 , ’�p! 1� ! 0.

The self-consistency condition for the parametrization(9) of the nonlinear equation (8) requires that the chiralangle is subject to a nonlinear equation—the mass-gapequation,

074018

Ap cos’p � Bp sin’p � 0; (12)

with

Ap � m�1

2

Z d3k

�2��3V� ~p� ~k� sin’k;

Bp � p�1

2

Z d3k

�2��3� ~p ~k�V� ~p� ~k� cos’k:

(13)

For the given chiral angle ’p the dispersive law of thedressed quark can be built as

Ep � Ap sin’p � Bp cos’p; (14)

and it differs drastically from the free-quark energy E�0�p �������������������p2 �m2

pin the low-momentum domain; Ep approaches

this free-particle limit as p! 1.It was demonstrated in the pioneering papers on the

model (2) [21] that, for confining potentials, the mass-gap equation (12) always possesses nontrivial solutionswhich break chiral symmetry, by generating a nontrivialmasslike function Ap, even for a vanishing quark currentmass. In Fig. 2, as an illustration, we show the numericalsolution to the mass-gap equation (12) for the linearlyrising potential V�r� � �r in the chiral limit. The chiralangle depicted in Fig. 2 shares all features of chiral sym-metry breaking solutions to the mass-gap equation (12) forvarious confining quark kernels (for a comprehensiveanalysis of powerlike potentials see Refs. [21,26]), namely,it is given by a smooth function which starts at �

2 at theorigin, with the slope inversely proportional to the scale ofchiral symmetry breaking, generated by this solution. At

-3

FIG. 3. Graphical representation of the Bethe-Salpeter equa-tion (15) for the quark-antiquark bound state, in the ladderapproximation.

L. YA. GLOZMAN AND A. V. NEFEDIEV PHYSICAL REVIEW D 73, 074018 (2006)

large momenta it approaches zero fast. The latter propertyallows the reader to anticipate the principal conclusion ofthis paper. Indeed, since the Fourier transform of thepotential is peaked at ~p ’ ~k, whereas sin’k decreaseswith the increase of k, then, as p! 1, Ap is a decreasingfunction of the momentum p. In the chiral limit it vanishesasymptotically. Therefore, for highly excited hadrons, withthe typical momentum of valence quarks being large, thedynamical Lorentz-scalar mass of such valence quarksdecreases, and so does the coupling constant g�, due tothe Goldberger-Treiman relation (1). Hence highly excitedhadrons decouple from the Goldstone bosons. Below weprove this general conclusion by a detailed analysis of theamplitude of the pion emission process h! h0 � � with h(and perhaps h0) being highly excited hadrons.

III. PROPERTIES OF THE GOLDSTONE MODE

A. Mesonic Salpeter vertex; Bethe-Salpeter equationfor the chiral pion

In this section we remind the reader about the main stepsto take in order to derive the Bethe-Salpeter equation forthe generic quark-antiquark bound state, paying specialattention to the case of the chiral pion. We follow the linesof Refs. [21,23,25,27,28].

We start from the homogeneous Bethe-Salpeter equa-tion, depicted in Fig. 3 in the graphical form,

� ~p; ~P� � �iZ d4k

�2��4V� ~p� ~k��0S�k0; ~k�

� � ~k; ~P�S�k0 �M; ~k� ~P��0; (15)

for the mesonic Salpeter amplitude � ~p; ~P�; M is the massof the bound state and ~P is its total momentum.Equation (15) is written in the ladder approximation for

074018

the vertex which is consistent with the rainbow approxi-mation for the quark mass operator and which is welljustified in the large-NC limit. For future convenience weintroduce the matrix wave function of the meson as

�� ~p; ~P� �Z dp0

2�S�p0; ~p�� ~p; ~P�S�p0 �M; ~p� ~P�;

(16)

and use the standard representation for the dressed-quarkpropagator via Dirac projectors,

S�p0; ~p� ���� ~p��0

p0 � Ep � i�

��� ~p��0

p0 � Ep � i; (17)

� � ~p� � TpP Typ ; P �

1 �0

2;

Tp � exp��

1

2� ~� ~p�

��2� ’p

��:

(18)

We also consider the bound state in its rest frame, setting~P � 0 and skipping this argument of the bound-state wavefunction for simplicity. It is easy to perform the integrationin energy in Eq. (15) explicitly. Then, for the Foldy-rotatedwave function ~�� ~p� � Typ�� ~p�Typ , the Bethe-SalpeterEq. (15) reads

~�� ~p� � �Z d3k

�2��3V� ~p� ~k�

�P�

TypTk ~�� ~k�TkTyp

2Ep �MP�

� P�TypTk ~�� ~k�TkT

yp

2Ep �MP�

�: (19)

It is clear now that a solution to Eq. (19) is to have the form

~�� ~p� � P�AP� � P�BP�; (20)

and, due to the obvious orthogonality property of theprojectors P , P�P� � P�P� � 0, only matrices anti-commuting with the matrix �0 contribute to A and B. Theset of such matrices is f�5; �0�5; ~�; �0 ~�g which can bereduced even more, down to f�5; ~�g, since the matrix �0

can always be absorbed by the projectors P . The matrixwave function �n� ~p� for the nth generic mesonic state canbe parametrized through the positive- and negative-energycomponents of the mesonic wave function ’ n �p�, and abound-state equation in the form

� �2Ep �Mn�’�n �p� �

R k2dk�2��3�T��n �p; k�’

�n �k� � T

��n �p; k�’

�n �k��

�2Ep �Mn�’�n �p� �

R k2dk�2��3�T��n �p; k�’

�n �k� � T

��n �p; k�’

�n �k��

(21)

can be derived for such wave functions. The interestedreader can find the details in Refs. [21,23] or inRef. [25], where a Hamiltonian approach to the quark-

antiquark bound-state problem is also developed (as ageneralization of the method applied to the ’t Hooft modelfor QCD in two dimensions in Ref. [28]) which, after a

-4

GOLDSTONE BOSONS DECOUPLING FROM HIGH-LYING . . . PHYSICAL REVIEW D 73, 074018 (2006)

generalized Bogoliubov-like transformation, leads to thesame Eq. (21); ’ n �p� play the role of the bosonicBogoliubov amplitudes, so that their normalization condi-tion,

Z p2dp

�2��3�’�n �p�’�m�p� � ’�n �p�’�m�p�� � �nm;

Z p2dp

�2��3�’�n �p�’�m�p� � ’�n �p�’�m�p�� � 0;

(22)

should not come as a surprise.Let us consider the case of the chiral pion in more detail.

In this case only �5 contributes to A and B in Eq. (20) andone has

A � � �5’�� �p�; B� � ��5’

�� �p�: (23)

It is an easy task now to extract the amplitudes T � [seeEq. (21)] from Eq. (19) using the explicit form of thematrix wave function ~��� ~p� and of the operator Tp. Theresult reads

T��� �p; k� � T��� �p; k�

� �Zd�kV� ~p� ~k�

�cos2

’p � ’k2

�1� � ~p ~k�

2cos’p cos’k

�;

T��� �p; k� � T��� �p; k�

�Zd�kV� ~p� ~k�

�sin2

’p � ’k2

�1� � ~p ~k�

2cos’p cos’k

�: (24)

In the chiral limit, ’�� �p� � �’�� �p� � ’��p�, so thatthe bound-state equation (21) for the pion reduces to asingle equation,

2Ep’��p� �Z k2dk

�2��3�T��� �p; k� � T��� �p; k��’��k�

� �Z d3k

�2��3V� ~p� ~k�’��k�; (25)

or, in the coordinate space, one arrives at the Schrodinger-like equation,

�2Ep � V�r��’� � 0: (26)

It is instructive to notice that Eq. (26) reproduces the mass-gap equation (12) for ’��p� � sin’p. This is the wavefunction of the chiral pion whose dual nature is clearly seenfrom this consideration. Indeed, as a Goldstone boson, thepion appears, through the mass-gap equation, already at thelevel of the quark dressing, whereas the same entity reap-pears as the lowest pseudoscalar solution to the quark-

074018

antiquark bound-state equation (21). Below we study theproperties of the pionic matrix wave function (16).

B. Bound-state equation in the matrix form:Gell-Mann-Oakes-Renner relation

The mesonic bound-state equation (21) admits a matrixform for the wave function �� ~p� and it can be deriveddirectly from Eq. (19). Skipping the details of this deriva-tion (the interested reader can find them in Ref. [28], for thetwo-dimensional ’t Hooft model, whereas the generaliza-tion to the model (2) is trivial), we give it here in the finalform [25]:

M�� ~p� � �� ~� ~p� � �m��� ~p� ��� ~p��� ~� ~p� � �m�

�Z d3q

�2��3V� ~p� ~k�f��� ~k��� ~p����� ~k�

���� ~p��� ~k����� ~p� ���� ~k��� ~p����� ~k�

���� ~p��� ~k����� ~p�g: (27)

For the chiral pion, the explicit form of ��� ~p� followsfrom Eqs. (20) and (23) and reads

��� ~p� � Tp�P��5’�� � P��5’

�� �Tp

� �5G� � �0�5T2pF�; (28)

where G� �12 �’

�� � ’�� �, F� �

12 �’

�� � ’�� �.

In order to normalize the pion wave function in its restframe, one is to go slightly beyond the chiral limit and toconsider pionic solutions to the bound-state equation (21)in the form [23,25,28]

’ � �p� �N �

1�������m�p sin’p �

�������m�p

�p

�;

N �2� � 4

Z 10

p2dp

�2��3�p sin’p;

(29)

where all corrections of higher order in the pion mass areneglected and the function �p obeys a reducedm�-independent equation (see, for example, Ref. [23] orthe papers [28] where such an equation for �p is discussedin two-dimensional QCD).

Furthermore, the pion norm N � can be easily related tothe pion decay constant f�. To this end, we multiply thematrix bound-state equation (27) by �0�5, integrate bothits parts over d3p=�2��3, and, finally, take the trace. Theresulting equation reads

m�

Z d3p

�2��3F� sin’p � 2m

Z d3p

�2��3G�; (30)

and it is easy to recognize the celebrated Gell-Mann-Oakes-Renner relation [29] in Eq. (30). Indeed, using theexplicit form of the pionic wave function beyond the chirallimit, Eq. (29), one can see that G� � �N �=

�������m�p

� sin’pand F� � �N �

�������m�p

��p which, after substitution into

-5

L. YA. GLOZMAN AND A. V. NEFEDIEV PHYSICAL REVIEW D 73, 074018 (2006)

Eq. (30), give the sought relation

m2�

�NC�2

Z 10dpp2�p sin’p

�� �2mh �qqi; (31)

where the definition of the chiral condensate [21,23],

h �qqi � �NC�2

Z 10dpp2 sin’p; (32)

was used. Therefore,

N � �

�������������2�NCp

f�: (33)

Finally,

��� ~p� � Tp�P��5’�� �p� � P��5’

�� �p��Tp

�1

f�

�������������2�NCm�

s��5 sin’p �O�m���; (34)

where the properties Tp�5 � �5Typ and TypTp � TpT

yp � 1

of the Foldy operator Tp [see Eq. (18) for its definition]were taken into account.

C. Goldberger-Treiman relation and the pionemission vertex

The standard Goldberger-Treiman relation is the relationbetween the pion-nucleon coupling constant and the axialconstant of the nucleon axial-vector current, and its deri-vation is present in any textbook on hadronic physics (see,for example, Ref. [30]). This relation can be derived ana-lytically for the GNJL model as well [31]. A similarrelation holds in the GNJL models for the pion couplingto dressed quarks. Indeed, the model admits two represen-tations: the dressed-quark and the mesonic representationswhich are interchangeable up to corrections suppressed bythe large NC number [25], so that, in the leading approxi-mation, one has two equivalent representations for theaxial-vector current complying with the PCAC theorem[for the sake of simplicity, we stick to the one-flavor theoryand ignore the axial anomaly; generalization to the flavornonsinglet axial-vector current is trivial]1:

�J5��x��� � f�@����x�; (35)

through the Goldstone boson, with ���x� being the chiralpion wave function, and, generically,

1It was noticed long ago [21] that, in the given model, the piondecay constant in the temporal and in the spatial parts of thecurrent in Eq. (35) may differ. This is a consequence of anexplicit breaking of Lorentz covariance by the instantaneousinteraction in the Hamiltonian (2). Although some improvementsof the model can be made in order to get rid of this discrepancy(see, for example, Ref. [32]), its underlying origin cannot beremoved by simple amends. Thus we consider the model (2) as itis, making emphasis on its qualitative predictions. Everywherethroughout this paper, as f� we denote the temporal constantextracted from the Gell-Mann-Oakes-Renner relation (31).

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�J5��x��q � �q�x��gA���5 � hA@��5�q�x�; (36)

through the dressed-quark states, where gA � 1 and hA �0. We evaluate now the matrix element of this currentdivergence between dressed-quark states,hq�p�j@�J

5��x�jq�p

0�i, in the two given representations,arriving at

hq�p�j�@�J5��x���jq�p

0�i � f�m2�hq�p�j���x�jq�p

0�i

/ f�g��q2�� �up�5up0 �;

q � p� p0; (37)

where the pion-quark-quark effective form factor g��q2� isintroduced, and

hq�p�j�@�J5��x��qjq�p0�i / meff

q � �up�5up0 �; (38)

where it is assumed that dressed quarks obey an effectiveDirac equation with the dynamically generated effectiveLorentz-scalar mass meff

q . Obviously this dynamical massmust be identified with the function Ap—see Eqs. (10) and(13)—and we discuss this issue in detail below.

Finally, Eqs. (37) and (38) together yield the neededrelation

f�g� � meffq ; g� � g��m

2��; (39)

where, for the sake of simplicity, we absorb all coefficientsinto the definition of g�.

This derivation of the Goldberger-Treiman relation (39)is based on quite general considerations of chiral symmetrybreaking and the PCAC theorem. The only input is therequirement of the (partial) conservation of the total axial-vector current in QCD and the appearance of the Lorentz-scalar dynamical mass of quarks as a consequence ofSBCS. It leaves a number of questions, such as the micro-scopic picture for the pion-quark-quark vertex and thedependence of the effective quark mass meff

q on the quarkmomentum. Below we give a microscopic derivation of theGoldberger-Treiman-like relation (39) for the pion-quark-quark vertex in the framework of the quark model (2).

Consider a hadronic process with an emission of a softpion, for example, a decay h! h0 � �, with h and h0

being hadronic states (below we consider them to bemesons) with the total momenta ~p and ~p0, respectively.The amplitude of this process is given by the sum of twotriangle diagrams depicted in Fig. 4 and can be written as

FIG. 4. The amplitude of the decay h! h0 � �.

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GOLDSTONE BOSONS DECOUPLING FROM HIGH-LYING . . . PHYSICAL REVIEW D 73, 074018 (2006)

M�h!h0 ����Z d4k

�2��4Sp�h� ~k; ~p�S�k�p�

� �h0 � ~k� ~p; ~p0�S�k�q� ��� ~k; ~q�S�k��

�Z d4k

�2��4Sp�h� ~k; ~p�S�k�p� ��� ~k; ~q�

�S�k�q� �h0 � ~k� ~p; ~p0�S�k��; (40)

where the pion momentum is q � p� p0. Every vertex inthese diagrams contains a Salpeter amplitude ( � foroutgoing vertices) which obeys a Bethe-Salpeter equationof the form of Eq. (15) which can be rewritten as

� ~p; ~P� �Z d3k

�2��3V� ~p� ~k��0�� ~k; ~P��0: (41)

The incoming and outgoing matrix vertices for the givenhadron are related to one another as

�� ~p; ~P� � �0y� ~p; ~P��0: (42)

Thence, considering Eqs. (34), (41), and (42) together,one arrives at the relation between the soft-pion ( ~P� �~q! 0) emission vertex and the dressed-quark effectivedynamical mass:

f� ��� ~p� �

�������������2�NCm�

s�5

Z d3k

�2��3V� ~p� ~k� sin’k

� const� �5Ap; (43)

where the definition of the function Ap, Eq. (13), was used.Thus, for the sake of transparency, introducing the formfactor g��p� such that

�u p ��� ~p�up � const� g��p�� �up�5up�; (44)

with the same constant as in Eq. (43), one finally arrives atthe Goldberger-Treiman relation,

f�g��p� � Ap; (45)

which explicitly relates the pion coupling to the dressedquarks with the effective chiral symmetry breakingLorentz-scalar quark mass Ap. Notice an important differ-ence between the naive Goldberger-Treiman relation (39)and the microscopic relation (45). In the former case, thepion coupling constant depends only on the momentumtransfer in the pionic vertex (that is, the pion total momen-tum ~q) squared, so that, for the on-shell pion, g� is aconstant. On the contrary, the pion emission vertex��� ~p; ~q� in the microscopic GNJL model depends on twoarguments: one, as before, being the pion total momentum~q—as mentioned before, we treat it in the same manner asin the standard derivation of the Goldberger-Treiman rela-tion and continue the soft-pion vertex to the point ~q � 0—whereas the other argument is the momentum flow in theloop, which also plays the role of the momentum of thedressed quark that the pion couples to. Without SBCS and

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beyond the chiral limit, Ap � m and thus the couplingg��p� turns into a constant. On the contrary, in the chirallimit and after SBCS, Ap is a rapidly decreasing function ofthe momentum p. Indeed, using the definition of Ap andthat of the dressed-quark dispersive law, one can find therelation

Ap � Ep sin’p: (46)

At large momenta, Ep � p, whereas the chiral angle ap-proaches zero fast—see Fig. 2. For a powerlike confiningpotential (4) ’p / 1=p4�� as p! 1 (see, for example,Ref. [26]).

The decay amplitude of the process h! h0 � � is givenby an overlap of the three vertices—see Fig. 4 andEq. (40). Each of them depends on the momentum circu-lating in the loop, and the maximal overlap is achieved forall three meson wave functions localized at comparablevalues of this momentum. Clearly the pion vertex (and thepion wave function sin’p) is dominated by the low mo-menta and it decreases fast with the increase of the latter(see Fig. 2). In the meantime, for highly excited hadrons,the momentum distribution in these hadrons as well as inthe corresponding vertices in Fig. 4 is shifted to largemomenta. Therefore, the amplitude (40) vanishes withthe increase of the hadron h and/or h0 excitation numberand so does the effective coupling constant of the pion tosuch highly excited hadrons. We emphasize that it is thepion wave function sin’p that suppresses the Goldstoneboson coupling to highly excited hadrons.

It was demonstrated recently that chiral symmetry res-toration for excited hadrons happens for the same reason—the effective interaction responsible for the splitting withina chiral doublet is also proportional to sin’p [12].Therefore, if an effective constant is introduced, in par-ticular, for the pion transition within the approximateparity doublet, this constant must be proportional to thesplitting �M�� � M� �M� within the doublet and it hasto vanish with �M�� for highly excited hadrons.

D. Numerical estimates

In this section we present some quantitative estimatesrelated to the chiral pion and its coupling to excited had-rons. For the sake of simplicity, we stick to the harmonicoscillator potential corresponding to the marginal choice of� � 2 in Eq. (4). The mass-gap equation reduces in thiscase to a second-order differential equation studied in de-tail in Refs. [21–23],

p3 sin’p �12K

30�p

2’00p � 2p’0p � sin2’p� �mp2 cos’p;

(47)

where the dressed-quark dispersive law is

Ep � m sin’p � p cos’p � K30

�’02p2�

cos2’pp2

�: (48)

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L. YA. GLOZMAN AND A. V. NEFEDIEV PHYSICAL REVIEW D 73, 074018 (2006)

The bound-state equation for the chiral pion, Eq. (21),takes the form���K3

0

d2

dp2 � 2Ep

� 1 0

0 1

" #� K3

0

�’02p2�

cos2’pp2

�1 1

1 1

" #�m�

1 0

0 �1

" #� ��� �p�

��� �p�

" #� 0; (49)

where the radial wave functions � � �p� � p’ � �p� wereintroduced for convenience, which are rescaled so as toobey the one-dimensional normalization [23],Z

dp���2� �p� � ��2

� �p�� � 1: (50)

As noticed before, Eq. (49) reduces to the mass-gap equa-tion (47) in the chiral limit, so that ’ � �p� � sin’p and thequalitative form of the solution for the chiral angle appearsto be the same as for the linear confinement depicted inFig. 2.

In Fig. 5, we plot the ratio g��p�=g��0� defined with thehelp of Eqs. (45) and (46). This ratio also describes theactual decrease of the pion coupling to the hadron with theincrease of the average dressed-quark momentum in it.Furthermore, for the sake of transparency, we measurethis momentum in physical units, in MeV, using the chiralcondensate to fix the value of the mass parameter K0. Tothis end, for the given solution to the mass-gap equation(47), we calculate the chiral condensate according toEq. (32) which gives (we use the numerical results ofRefs. [12,25])

h �qqi � ��0:51K0�3 (51)

and, therefore, we fix K0 � 490 MeV to arrive at thestandard value of the chiral condensate. From Fig. 5 onecan see a fast decrease of the pion coupling to hadrons for

FIG. 5. The ratio g��p�=g��0� as a function of the dressed-quark momentum.

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large p’s, as was discussed before using general qualitativearguments.

Now we estimate the average momentum of the valencequarks in excited mesons. We consider a radially excitedS-wave light-light meson consisting of two light (massless)quarks. For highly excited bound states of such a system(n� 1), the negative-energy component of the wave func-tion ’�n �p� vanishes and the bound-state equation reducesto a single equation for the positive-energy component’�n �p�. This equation can be roughly approximated bythe Schrodinger equation with the Hamiltonian

H � 2jpj � K30r

2; (52)

and we identify the corresponding self-energies Mn asMn �

12 �Mn� �Mn��, with Mn being the masses for

the chiral doublet partners. Although the given naiveSalpeter Hamiltonian is unable to correctly discriminatebetween Mn� and Mn� and a more sophisticated approachis required for this purpose (see Ref. [12] for a detaileddiscussion), it is sufficient to estimate the typical quarkmomentum in excited states. To make things simpler, weuse the einbein field method (see the original paper [33] forthe details and, for example, Ref. [34] for einbeins treatedusing the Dirac constraints formalism [35]) rewriting theHamiltonian (52) as

H �p2

���� K3

0r2; (53)

with the einbein � treated as a variational parameter [36].The spectrum of the Hamiltonian (53) minimized withrespect to the parameter � reads Mn � 3K0�2n� 3=2�2=3

and the mean quark momentum can be easily estimatedthen, using the virial theorem for the oscillatorHamiltonian (53), to be

hpi � K0�2n�32�

2=3; (54)

which gives an approximate analytic dependence of hpi onthe radial excitation number n. It is obvious, therefore, thatfor n� 1 the quark momentum already appears largeenough and, as seen from Fig. 5, the corresponding cou-pling of the Goldstone boson to this hadron is suppressedas compared to its naive value of g��0�. An accuratesystematic evaluation of the pion coupling to excited had-rons goes beyond the scope of the present qualitativepaper—this work is in progress now and will be reportedelsewhere.

IV. CONCLUSIONS

In this paper, we considered in detail physics of theGoldstone bosons decoupling from highly excited hadrons,where chiral symmetry is approximately restored. Thisinteresting physics can be summarized in a few words.The coupling of the Goldstone bosons to the valencequarks is regulated, via the Goldberger-Treiman relation,by the dynamical Lorentz-scalar mass of quarks. This

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GOLDSTONE BOSONS DECOUPLING FROM HIGH-LYING . . . PHYSICAL REVIEW D 73, 074018 (2006)

dynamical mass which arises via the loop dressing ofquarks represents effects of chiral symmetry breaking inthe vacuum. A key feature of this dynamical mass is that itis strongly momentum dependent and vanishes at largequark momenta. Hence at large momenta the valencequarks decouple from the quark condensates and fromthe Goldstone bosons. In this regime chiral symmetrybreaking in the vacuum barely affects observables andphysics is essentially as if there had been no such chiralsymmetry breaking in the vacuum. It is clear, therefore,that, for a given hadron, the influence of SBCS on itsproperties is determined by the typical momentum ofvalence quarks in it. For low-lying hadrons this momentumlies below the chiral symmetry breaking scale; the valencequarks acquire a significant Lorentz-scalar dynamicalmass, which results in a strong coupling of the low-lyinghadrons to the Goldstone bosons. On the contrary, for high-lying hadrons, a typical momentum of valence quarks islarge; their dynamical chiral symmetry breaking massbecomes small (and asymptotically vanishes), which im-plies that these high-lying hadrons decouple from theGoldstone bosons. Consequently the axial-vector constant,gA, of the highly excited hadrons appears suppressed andasymptotically vanishes.

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We illustrate this physics in the framework of the GNJLmodel for QCD with the only interaction between quarksbeing the instantaneous Lorentz-vector confining potential.Such a model is tractable and contains all the requiredelements such as the spontaneous breaking of chiral sym-metry via quantum fluctuations of the quark fields andconfinement. While it is only a model, it neverthelessprovides a significant insight into physics of chiral sym-metry restoration in excited hadrons.

ACKNOWLEDGMENTS

L. Ya. G. thanks A. Gal for his hospitality at the HebrewUniversity of Jerusalem where this paper was prepared,and T. Cohen, M. Shifman, and A. Vainstein for correspon-dence. He acknowledges the support of the P16823-N08project of the Austrian Science Fund. A. V. N. would like tothank Yu. S. Kalashnikova and E. Ribeiro for reading themanuscript and for critical comments and to acknowledgesupport of Grant No. DFG 436 RUS 113/820/0-1,No. RFFI 05-02-04012-NNIOa, and No. NSh-843.2006.2,as well as of the Federal Programme of the RussianMinistry of Industry, Science, and TechnologyNo. 40.052.1.1.1112.

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