A chiral confining model of the nucleon

0146-6410/93 $24.00 © 1993PergamonPressLtd

Prog. Part. Nucl. Phys., Vol. 31, pp. 77-157, 1993.

Prinladin C~eatBritain.Allrightsreserved.

A Chiral Confining Model of The Nucleon* MANOJ K. BANERJEE Department of Physics and The Center for Theoretical Physics, University of Maryland, College Park, MD 20742-4111, U. S. A.

ABSTRACT In this review article I describe a model of the nucleon whose ingredients are quark, meson and hybrid or meikton fields. The strategy has been to choose a suitable lagrangian and treat it in the mean field approximation. The model is easily extended to low-lying states of other baryons strange and nonstrange. The model is chiral invariant with small but appropriate chiral invariance breaking terms. It is based on the color dielectric model of Nielsen and Phtkos . The vanishing of the color dielectric function in the vacuum, an indispensable element of the model, has been justified on the basis of the area law of QCD. The model has essential connection to the two major characteristics of the QCD vacuum, viz, gluon and quark condensates. Specifically, fluctuations of the two condensates, hybrids and mesons, are ingredients of the model. Physical quantities of the model have the same large Nc behavior as the corresponding quantities in QCD. The last three features encourage us to hope that the model may have basis in QCD. The question of coexistence of quarks and bosons in an effective lagrangian has been considered with care. The conclusion is that in a confining model the quark field must be retained. But only tree graphs should be evaluated. The contributions of quark loops appear through the bosonic lagrangian. I argue that no special action needs to be taken in incorporating the role of the trace anomaly of QCD in an effective lagrangian. The trace of stress-energy tensor of our effective lagrangian is nonzero at the classical level. It should be interpreted as the manifestation of the QCD trace anomaly. A difficulty which plagues most confining models is that the product of the nucleon mass and the rms matter radius turns out to be ~ 6 or greater. This problem has been essentially eliminated by the inclusion of the 't Hooft interaction.

*Dedicated to the memory of my teacher, the late Professor Meghnad Saha, on the occasion of his 100th birthday. 77

M. K. Banerjee

78

Contents 1 Introduction 2 The MIT Bag Model 3 Large Nc QCD 3.1 Introduction 3.2 Scaling at Large Nc 3.3 't Hooft Diagrams 3.4 Meson Masses 3.5 Glueball Masses 3.6 Meson-Quark Coupling Constant 3.7 Baryon Masses and Radii 4 A Chiral Confining Model 4.1 Phenomenological View 4.2 The Nielsen-Pb,tkos Lagrangian 4.3 Role of the Current Quark Mass 4.4 The X Potential 5 The Glueball and the Meikton Fields 6

A Chiral Quark Meson Model 6.1 Introduction of Mesons 6.2 Average Field and Mesons

7

Quark Meson Interaction 7.1 The Canonical Quark Field 7.2 Coexistence of Mesons and Quarks 7.3 Chiral Invariance Breaking of the Lagrangian 7.4 PCAC 7.5 The Sigma Commutator 7.6 The Vector Mesons

8 Calculation in Mean Field T h e o r y 8.1 Mean Field Approximation 8.2 Hedgehog Ansatz 8.3 Virial Theorems 9 Chiral Confining Model Results 9.1 Chiral Confining Model with a and ~r Fields 9.2 Nucleon Properties 9.3 N-A Mass Splitting 9.4 Goldberger Treiman Relation 9.5 Inclusion of Vector Mesons 10 The Toy Model 11 Trace A n o m a l y and Effective Lagrangians

Chiral Confining Model 12 The 12.1 12.2 12.3 12.4

' t H o o f t Interaction The Instanton The CCM Version of C-,tHooyt The Extended Toy Model The Mean Field Results

13 Concluding R e m a r k s 13.1 Summary of the Model 13.2 Future Tasks 13.3 Acknowledgements 14 A p p e n d i x 14.1 Some Basic Results with the Hedgehog Spinor 14.2 Results from the Hedgehog Model 14.3 N and A States with Quarks only 14.4 Toy Model Results 15 R E F E R E N C E S

79

80 1

M.K. Banerjee Introduction

In this review article I describe the development of a particular model of baryons. It is called the chiral confining model (CCM). It is not formally derived from QCD. None of the models available to us are. But it is closely linked to QCD. The main features of the model are as follows: The model is based on a lagrangian which is treated in the mean field approximation. The model includes a confining mechanism based on the Nielsen-Pb,tkos (1982) version of the notion of color dielectric function. The model has two scalar fields. One of the fields is the ~r field. Together with the pion fields, it represents fluctuations of the quark condensate of the QCD vacuum. The four fields form a [~, ½] representation of the chiral SU(2) x SU(2) group. The other scalar field, labeled as X, is needed to represent the change in the color dielectric function produced by a baryon source. It represents a fluctuation of the gluon condensate of the QCD vacuum. The X field is a chiral singlet and belongs in the glueball/hybrid family. The model generates the bag dynamically. The model respects the chiral features of QCD. The model has correct large Nc behavior. The article is not a comprehensive review of all models. Several excellent reviews of various models are now in existence. Some of these will be cited in appropriate places. The MIT bag model (Chodos et al., 1974a, 1974b), justifiably the most famous of all quark models of hadrons, appeared in 1974. In 1975 De Rujula, Georgi and Glashow (1975) showed that the splittings between octet and decuplet baryons, e.g. N - A, A - E may be understood within a nonrelativistic constituent quark model in terms of the color magnetic dipole-dipole interaction term, S id a=l

of an effective one-gluon exchange. Here ~a is one of the 8 Gell-Mann SU(3) matrices. This basic idea, in its relativistic form, was incorporated successfully into the MIT bag model (DeGrand et al., 1975). Isgur and Karl (1977, 1978, 1979, 1980) developed a nonrelativistic model in which the confinement mechanism was included through the use of a harmonic oscillator interaction between each pair of quarks. They also extended the one-gluon exchange interaction by including the tensor and the spin-orbit terms. With these simple set of assumptions they were able to reproduce remarkably well the masses of a wide range of even and odd parity excited states of the baryon. Bardeen, Drell, Weinstein and Yan (Bardeen et al., 1975) and Huang and Stump (1976) introduced model lagrangians describing quarks interacting through a scalar field. cqo =

I

+ 1

-

(2)

where ~b is the 3(color)x2(isospin)×4(Dirac)= 24 component quark field and tr is the scalar field. The lagrangian was treated at the mean field level using the ansatz that the nucleon is made up of three quarks in the same spinor state without any quantum a particle. Of course, the three quarks generate a classical (time-independent) ~r field. These models do not have absolute confinement.

Chiral Conf'ming Model

81

In 1077 Friedberg and Lee (1977a, 1977b, 1978) introduced a model incorporating the notion of color dielectric function of the vacuum. They proposed a lagrangian of the form

I:F--L = ¢(i 1 ~ --ga -- g~[)¢ + ~9,aO~'o " - U(a)

°=,

(3)

where

A/j = ~z.., I~ , ~ ' A ,,,° . °~.l

. . 6 ~ , = O . A .. - . O~A,, + gj

/~--b--c

A,,A~.

(4)

"

The quantity ~(a) is the color dielectric function of the vacuum. Friedberg and Lee required that =

(5)

o.

It was already recognized that vanishing dielectric constant of the vacuum ensures absolute confinement of color (Kognt and Susskind, 1974, Hasenfratz and Kuti, 1978). The essential point is easily grasped if we ignore the nonlinear terms in Eq (4) or, equivalently, use an abelian gauge theory. For then we obtain V . D ° = p°, where/)° is the color displacement field and p° is the color charge density. The color electric field is given by ]~" =/~°/~(a). The color electric field energy is f dSr-~-~. The displacement field D(r) falls off with increasing r by some power law, i.e., N r -2-¢, where [ is the lowest multipolarity of the charge distribution. Therefore, as long as ~(r) goes to zero as r ~ oo faster than any power of r the color electric field energy will diverge. This ensures that a state, which is not a singlet of the color SU(3) group, cannot exist. Further studies of this model was carried out by Wilets and his collaborators and reviewed by Wilets (1989). It should be pointed out that in most treatments of the Friedberg-Lee model the quark spinors do not reflect absolute confinement. The fourier transform of the density of an absolutely confined quark should be an entire function of the square of the momentum transfer Q2. Any singularity for finite Q~ indicates that the quarks can be liberated. To remove such singularities the spinors must fall off faster than any exponential as r ~ oo. There are two versions of the model (Bayer et al., 1986, Fai et al., 1988) where this defect has been removed. The models, where the baryon is described with three quarks and classical meson fields, came to be called nontopological solitons. The presence of quarks can make such a model stable. In contrast, a purely mesonic model like the Skyrmion must have topological properties to make it stable. Otherwise, the configuration can tunnel to the vacuum state which is lower in energy (Cohen, 1988). The soliton models have been reviewed recently by Birse (1990). The QCD lagrangian is

:

O=1

where the gluon field ,% and antisymmetric field tensor G~,~have the same definitions as in Eq. (4). The current quark mass mq is believed to be 4MeV for the u-quark and 11MeV for the d-quark. In the idealized situation where the current quark mass m is set to zero, called the chiral limit, the QCD lagrangian is invariant under the chirai SUn(2) x SUL(2) transformation. To reflect this situation effective lagrangians are constructed to be almost invariant under chiral SUn(2) x SUL(2) with s small chiral invariance breaking term, £xlB. The Isgur-Karl model is non-relativistic and the notion of chiral transformation does not apply. It is well-known that the initial version of the MIT bag model (Chodos et al., 1974a, 1974b) was not chiral SUn(2) x SUL(2) invariant. Chiral invariance was restored by coupling meson fields to the quarks st the bag surface (Chodos and Thorn, 1975).

82

M.K. Banerjee

T h e Friedberg-Lee model and its successors are not chiral invariant. Unfortunately, there is no simple remedy in this case. The natural inclination is to interpret the scalar field a in Eq. (3) as the chiral partner of the pion field. Then the remedy is to extend the lagrangian to include the pion field in a chiral invariant manner. The result is the following lagrangian:

=

{i½

+

+

1

~

+

1

o

-

+

+ cx,B,

(7)

where we have added a chiral invariance breaking term, £xIB, explicitly. The requirements that the vacuum is stable against the fluctuations of the meson fields and that the vacuum expectation

value, = -F.,

(8)

where F~ = 93MeV, forces the lagrangian to take the form of the well-known Gell-Mann-LSvy (1960) linear sigma model lagrangian. In this form one can no longer arrange to make the quark effective mass small at short distance and large at large distance, which is a feature one must have to simulate confinement. Birse and Banerjee (1984, 1985) speculated that the confinement mechanism is operative at a length scale larger than the size of nucleon and A and, hence, plays a minor role in their structure. They built a model using specifically the Gell-Mann-L6vy (1960) linear sigma model lagrangian, interpreting the fermion field to be the quark field. Simultaneously and independently Kahana, Ripka and Soni (1984) proposed the same model. Broniowski and Banerjee (1985, 1986) extended the Birse-Banerjee model by including the vector mesons w , p and A1. This was done with the help of a lagrangian introduced by Ben Lee and Nieh (1968). The new model yielded nucleon properties which agreed with the experimental data far better than the earlier model. But as in the earlier model the quarks were not confined. Consequently, realistic calculation of excited states, such as the Roper resonance, could not be carried out without releasing a quark in the continuum. Prior to the introduction of these models Adkins, Nappi and Witten (1983) had given new life to the Skyrmion (Skyrme, 1961) as a model of non-strange baryons. This approach and the chiral invariant quark meson models discussed in the previous paragraph are mutually compatible. We do not discuss the Skyrmion in this article. For details the reader may consult the review articles by Zahed and Brown (1986) and by Oka and Hosaka (1992). In the mid-eighties groups at Heidelberg (Chanfrey et al., 1984, Schuh and Pirner, 1986), Adelaide (Williams and Thomas 1985, 1986) and Maryland (Banerjee, 1985, 1987, Broniowski et al., 1985, 1986, Broniowski, 1986, 1987) and Ian Duck (1986) independently proposed what may be generically described as the color dielectric model of bag formation. These models were all based on the Nielsen-Phtkos model (Nielsen and P£tkos, 1982) of the color dielectric function. The first two groups did not include mesons and the chiral symmetry breaking of the vacuum played no role in these models. The models studied by the Maryland group and by Duck used the Gell-MannL~vy lagrangian suitably modified to include the role of the color dielectric function. Thus these two models used lagrangians which are chiral invariant except for the £xlS = F , m ~ a term of the Gell-Mann-L6vy lagrangian. The Heidelberg group and the Maryland group have continued to improve their models. The work of the Heidelberg group has been reviewed recently by Pirner (1992). There is also a totally different approach to constructing a model of the nucleon with confinement. Instead of introducing an effective lagrangian with bosonic fields as new collective variables, one uses QCD directly, but often omitting suitably chosen sets of diagrams. The three body problem is tackled with the help of Faddeev equations. The confinement results from the ansatz that the gluon propagator behaves as .,- ~ in the momentum space. Cahill and his collaborators (Cahill et al., 1989, Burden et al., 1989) have pioneered this approach.

Chiral Confining Model

83

The models under discussion are not formally derived from QCD. Hence it is important that one imposes upon the models as many requirements of consistency with QCD as possible. Chiral invariance is obviously the most important requirement that one imposes. This requirement together with the presence of a chiral singlet scalar field X, mentioned earlier, produces a novel feature. The simplest, nontrivial, chiral invariant coupling of the field to the quarks is of the form gxX(bi~ ~ ¢. This unusual modification of the free quark term in the lagrangian constitutes an essential ingredient in the mechanism of bag formation. I also impose the additional requirement that the large Nc ('t Hooft , 1974a,Witten, 1979a) dependences of the physical quantities of the model be the same as those given by QCD. For example, according to large Nc QCD the nucleon mass grows as N~ while the nucleon radius is independent of Nc (Witten, 1979a). An acceptable model ought to reproduce these features. It should be noted that the requirement of proper large N~ behavior is quite distinct from the better known 1/N~ expansion. The latter suggests that the results for Nc = oo are reasonably close to those for Nc = 3. In contrast, the requirement of proper large N~ behavior is a diagnostic tool. The results of large N~ QCD reflect the nonabelian gauge theory character of the theory. The two main features of QCD - confinement and asymptotic freedom - are possible only for a nonabelian gauge theory. Most models of hadrons inspired by QCD are based on features following from its nonabelian gauge theory character rather than on the specific value of 3 for N~. Therefore it should be possible to generalize these models to large N~ and it is legitimate to demand that the results for large Nc agree with those of large Nc QCD. To avoid any possible misunderstanding I should stress that I am not suggesting that the large Nc results of either the model and or of QCD be taken as reasonable approximations to the physical results which are for N~ = 3. The anomalies of QCD must appear in an effective lagrangian at the tree level. The axial U(1) anomaly plays an important role in determining the spin content of the nucleon. The effective lagrangian must include appropriate terms reflecting the axial U(1) anomaly (Cohen and Banerjee, 1989) if one hopes to explain the spin content problem. However, these terms have little effect 1 on the structure of the baryon. ( One can verify that g~,NN "~glrNN. Furthermore the Tf mass is as high as 958 MeV. These two reasons prevent rf from playing an important role on nuclear structure.) So one may do the structure calculation without such terms and then add them on to obtain an improved U(1) axial-vector current of the model. In contrast, the trace anomaly has profound implication on the structure of a hadron. In Section 11 we discuss the issue in some detail. We conclude that nothing novel, like adding a particular term, is needed to reflect properly the trace anomaly of QCD. The model discussed in this review has an aspect which some may regard as controversial. The model has both quarks and mesons. This is, by no means, a feature exclusive to CCM. The models of Bardeen et al., (1975), Huang and Stump (1976), Friedberg and Lee (1977a, 1977b, 1978), Wilets (1989, Bayer et al., 1986, Fai et al., 1988, Birse and Banerjee 1984, et al., (1985), Kahana et al., (1984), Broniowski and Banerjee (1985), Duck (1986) have this feature. It is sometimes believed that the quarks can be integrated away completely leaving behind a theory of bosons only. This is correct, but the resulting theory may be extremely unwieldy. Nothing is gained by replacing computationally difficult QCD with an effective lagrangian which is also computationally difficult. A practical criterion for a simple theory is that it must not contain more than four powers of boson field derivatives. The question then is, can one have a practical theory with bosons only and without any quarks? We discuss this issue in detail in Section 7.2. For models based on the mean field approximation we conclude that the answer depends on the sign of the valence spinor eigenvalue. If it is negative, a practical theory without quarks may emerge. In a nonconfining model including vector mesons (Broniowski and Banerjee, 1985, 1986, Alkofer et al., 1992) the valence spinor eigenvalue is, indeed, negative and a purely mesonic theory may emerge (Alkofer et al., 1992). In a confining model neither we nor, to the best of our knowledge, anyone else has found ""

84

M.K. Banerjee

a case where the valence spinor eigenvalue is negative. In a certain situation, discussed in Section 8.3, the positivity of the valence spinor eigenvalue can be proved as a theorem. For the general case, we depend on our experience of numerical calculations to conclude that in these situations one must retain the quark field, but include only tree graphs. The contributions of all quark loops are included in the bosonic sector of the lagrangian. This observation has nontrivial implications in two areas. First, the current quark mass term must be retained over and above, F~rn~a, 2 the traditional chiral symmetry breaking term in the mesonic sector, introduced by Gell-Mann and L~vy (1960). Second, the 't Hooft interaction, which describes the role of instantons, must be retained over and above its mesonic manifestations. In the next section we review the MIT bag model. Section 3 is devoted to building the tools for large Nc analysis. The next six sections describe the development of the chiral confining model. The role of trace anomaly in an affective lagrangian is the content of Section 11. The next section describes the most recent development of the theory, namely, inclusion of the 't Hooft interaction in the CCM. The last section contains concluding remarks.

2

The MIT Bag Model

This section contains a brief review of those parts of the MIT bag model. For a more complete discussion the reader is referred to the original articles (Chodos et al., 1974a, 1974b, 1975) and the excellent review article by Thomas (1984). The spherical MIT bag may be described with the following lagrangian: :

where B is the bag pressure. The Euler-Lagrange equations are i•¢ i¢¢(R)

= =

0, r < R . ¢(R),

(10) (11)

where n = C0, ~) is a space-like 4-vector with n. n = -1. The quark field may be expanded in terms of the anticipated positive and negative energy eigenstates,

¢ = ~"~[e-iWX'uxbx + ei~X'v~dt~],

(12)

where we have utilized the fact that for this charge-conjugation invariant lagrangian the spectrum of the Dirac hamiltonian is symmetric around zero. Using Eq. (12) in the Euler-Lagrange Eqs. (10) and (11) one obtains the eigenvalue equation - i~. Vux = wxu~,

(13)

i~ux(R) = ux(R)

(14)

subject to the boundary condition The eigenfunctions may be labeled with the eigenvalues of the commuting operators f2, Jz and K =/~(~. L + 1). It is straightforward to check that K 2 = J'~ + ¼. Hence the eigenvalues of K are - ~ = q=(j + ½), or 1 j =] ~ I - 5 . (15) The simultaneous eigenspinors of J-~, J~ and K are denoted as q~ and are defined by the equations

J'~q~ = j ( j + 1)q~u, Jzq~ = /zq~, Kq~ = -nq~.

(16) (17) (18)

Chiral Confining Model

85

Note that j is not needed as a label because according to Eq. (15) r determines j uniquely. The general eigenspinor of the Dirac equation (13) is

q: = At~ [ j -t .i(~w. ~ ) ~je_~(~) 1 ~ ,

(19)

where j t ( w ~ ) is the spherical bessel function of order l. The factor At~ is the normalizing factor. The relationship between g~ and ~ is given by <0

e~ = l ~ l - 1

1

=j-~,

1 e_~ = ~ = j + ~ .

~>0

(20) (21)

The two-component spinor ~ is

~Z = 2E(e., m, 1/2, ~ - m I jr, . ) v ~ . - ~ ,

(22)

m

where r/u_,, is a Pauli spinor and (t~,m, 1/2, p - m [ j~,p)s are the Clebsch Gordan coefficients needed to couple the orbital angular momentum l~ to spin { to get the total angular momentum j. The boundary condition given by Eq. (14) gives the eigenvalue equation: je~(w) = -aign(,c).

(23)

je_~(~)

One needs a spinor with ,c = - 1 , j = 1/2, ! = O, i.e., a s½ spinor, for the construction of N and A. The eigenvalue equation becomes jo(w) = 1. (24)

jl(,~)

The lowest eigenvalue from Eq. (24) is

(25)

w0 = 2.0427870-... The normalizing factor for the ,c = - 1 spinor is given by

At_, = , [ - - ~

1

V2(~- 1) j o ( ~ ) R - S / ~

"

(26)

The spinor itself is q~l = A t - ,

jo(~)

i~.~j,(w~)

]

lh''

(27)

The weights of the upper and lower components of the spinor q~-i are G F

=

2~- i

I)' 2w-3 _ 4 ( ~ - 1)" 4(~-

(28) (29)

With w ,~ 2 these weights are ,~ 3/4 and 1/4. The energy of the baryon, N and A are given by

E =

N ~--

4~r

°R + T ~ B

(30)

86

M.K. Banerjee

Minimizing with respect R one obtains for the stable radius, Ro,

Ro (4~) '/~,

(31)

=

and for the minimized energy,

4New

Eo = 3---~"

(32)

The baryon energy receives a contribution of N c ~ from the quarks and the remainder ½Ne~ from the bag pressure. For the actual number of colors, Nc = 3 this works out to an equipartition of energy between the quarks and the bag pressure and the relationship is often referred to as the equipartition relation. The rms radius must be calculated numerically and turns out to be (r2)½ = 0.73Ro.

(33)

!
(34)

The dimensionless product It should be noted that the experimental value for the product of nucleon rms charge radius and nucleon mass is 3.86, significantly smaller than the MIT bag value. The difference is too large to be explained by pion cloud effects. Taking the nucleon rms radius to be 0.81fro, one gets from Eq. (33) the value Ro = 1.1fro. Using this value in Eq. (31) one obtains B --- (150MeV) 4. Eliminating Ro from Eqs. (31) and (32) one obtains 81 E04 B = 16r (New) 3"

(35)

In the MIT bag model the N and A are split by the dipole-dipole color magnetic interaction following the suggestion of deRujula, Georgi and Glashow (1975). This mechanism contributes equal and opposite amounts of energy to the two baryons. Accordingly, we take the mean of the two masses, 1085MeV, for Eo and obtain from Eq. (35) B = (314MeV) 4. The wide discrepancy between the two estimates reflects the rather large value of E0 given by the MIT bag. A central conjecture in the MIT bag model is that the space inside the bag is the perturbative vacuum, while the space outside is the QCD vacuum. The latter is characterized by two condensates, whose values, relative to those in the perturbative vacuum, are determined through the use of the QCD sum rules (Reinders et al., 1985): <: 2 ¢ :)~c 8

<-~~ : Ol

=

-2(225MeV) 3,

G.~G :).oc = (339Meg) 4. a

a t.tv

(36)

(37)

Shuryak (1988) obtained a value of the bag pressure in the following manner. The trace anomaly of QCD, which will be discussed later in Section 11, gives an expression (see Eq. (281)) for 0f, the trace of the stress-energy tensor. We can use this to write down the difference between the energy densities in the QCD vacuum and in the perturbative vacuum under two approximations, viz, the chiral limit and one-loop approximation. The result is

£qco vvac

pert = - ~--~( ~

- £~ac

N~ -

2 a L -~Nt)(-~ a=l

G~ G.u~ \ i.w

Ivac

"~ -(251MeV) 4. --

(38)

Chiral Confining Model

87

By definition B = _(E~co _ ~ep~t). Hence one finds B ,,~ (251MeV) 4. This value is ~ 8 times larger than the value obtained from the nucleon rms radius. It is ~ 2.4 smaller than the value obtained from the baryon mass. The reason for this difficulty of the MIT bag is clear. The bag must be created by the valence quarks. Thus the inside of the bag differs from the QCD vacuum only to the extent the latter is polarized (statically) by the valence quark. The mechanism of polarizing the gluonic sector of the vacuum cannot be introduced by some minor modification of the MIT bag model. One needs a 0 ++, chiral singlet field like the X field of our confining model. Linear, but nontrivial, coupling of this field to the quark must take the form gxx¢i~ ~ ¢. This fact coupled with the inevitable requirement that ( X ) ~ = 0 will alter the model radically. In fact, it will substitute the MIT bag model with a model very close to the one discussed in this article.

3 3.1

Large

Arc Q C D

Introduction

The idea of large Arc QCD was introduced by 't Hooft (1974a). The theory is a generalized version of QCD involving SU(Nc) color group with coupling constant

g(go) -- g

.

(39)

For large N~ one can replace the group SU(Nc) with U(No). He recognized that in such a theory physical quantities can be expanded in a power series of 1/Nc for large No. 't Hooft derived simple rules for determining how various Feynman graphs behave with N¢ for large Nc. In a subsequent paper (% Hooft 1974b) he presented the 1 + 1-dimensional QCD, which is exactly solvable. One of the interesting results of this work was to establish that meson masses become independent of N~ for large No. The case of t/' mass requires separate discussion (Witten 1979b). The field was enriched significantly by the work of Witten (Witten 1979a). He developed methods of determining the large Arc behavior of n-point functions of mesons. His studies lead to the following important results. The n-point functions of mesons behave as N(,2-')D. An important consequence of this result is that for large N~ one need consider tree graphs only, which, in turn, leads to the mean field theory. Equivalently, one may state that a mesonic lagrangian may be treated at the classical level for large Arc. All quantum loops are N~-suppressed. The OZI rule is exact for large Arc. All mesons, i.e., infinite number of them, must be included to obtain correct large momentum behavior of physical quantities. It should be made clear that in applying large Arc QCD analysis one assumes confinement. Furthermore, the only statement one can make is about the Arc dependence of various physical quantities for large N~. Nothing can be said about other multiplicative factors determining the size of a physical quantity. It is useful to introduce the following abbreviation: ~ ,,~ N~ means that for large Arc the quantity behaves as N~.

M. K. Bancrjcc

88 Feynman

't

t

.......

Hooft

I

r-

Nc-dependence

Nc-1/2

g(Nc)

I

1

Factor

II

• 1

:

II

Figure h N:dependent factors in a Feynman graph. 3,2

S c a l i n g a t L a r g e Nc

The dependence of the QCD coupling constant on the renormalization scale # is expressed through the ~-funetion: ~(g( gc) ) = ~ dgU-~-v~" c)," " (40)

d#

In the large N¢ limit the quantity /~(g) may be evaluated in the one-loop approximation. The expressions in the one-loop approximation are given below.

g(N~) 3 l l N c - 2n! ~(g(gc)) =

16z2

3

llN'g(Nc)3 ,,, N~"112. ~

(41)

48z 2

Here n/ is the number of flavors. The last part of the equation above shows that for large Nc fl(g(N¢)) and g(Nc) behave similarly, i.e., ,~ N71/2 . An examination of Eq. (40) tells us that for large N¢ the dependence of g(N¢) on the renormalization scale # becomes independent of N¢, Hence for large N~ we can choose some suitable value of ~u and keep it fixed there. The entire Nc dependence of physical quantities will come from the N:dependence of g(N~) and SU(N~) algebra. 3.3

't Hooft

Diagrams

In a U(Nc) theory the N~ gluons can be labeled with two colors. 't Hooft prescribed that all gluon lines in a Feynman graph be replaced by two quark lines with arrows in opposite directions each carrying a color label. This prescription automatically takes care of color counting at the vertices. We summarize below the few basic rules needed to obtain the large Nc behavior of any diagram. For more details the reader must consult the original papers ('t Hooft 1974a, Witten 1979a). The N:dependent vertices which appear in Feynman graphs are shown in Fig. 1 together with the corresponding 't Hooft vertices and their large N~ behavior. The only other rule which we need is a rather obvious one, namely, that for every quark loop of an 't Hooft graph there is a factor of No. Figs. 2, 3 and 4 show a set of vacuum to vacuum Feynman graphs, the corresponding 't Hooft graphs and their No-dependence for large N,. The diagrams of Figs. 2, 3 and 4 illustrate the following rule deduced by 't Hooft ('t Hooft 1974a) for the No-dependence of Feynman graphs. A Feynman graph with L quark loops and H handles ,,, N~ -L-2u.

89

Chiral Confining Model

Feynman Graph

a

b

C

d

e

f

't Hooft Graph

O e @ @ @ @

@ @ Figure 2: Purely gluon grnphs.

Nc-dependence

N

2 c

NC

2

NC

2

2

No

N

N

2 C

2 C

M. K. Banerjee

90

Feynman Graph

a

b

't Hooft Graph

G %Y

C

d

e

f

Figure 3: One quark loop graphs.

Nc-dependence

91

Chiral Conf'ming Model Feynman Graph

't Hooft Graph Nc-dependence

©

%0_-1

Nc0 = 1

Nc-I

Figure 4: Some No-suppressed graphs.

Whenever a quark or a gluon line cross over another line without intersecting via a vertex there is a handle and H = 1. Graphs with no handles, i.e.,H -- 0, are called planargraphs and those with handles, i.e.,H # 0, are called no,planargraphs. In graphs of Fig. 2 there are no handles (H = 0) and no quark loops (L = 0). All these graphs N~. Graphs of Fig. 3 contain one quark loop (L = 1) but no handles (H = 0) and they ,,,No. The graph of Fig. 4a is an example of a graph with two quark loops but no handles and it ,,,N °, i.e.,it is independent of Arc. The last two graphs of Fig. 4 show examples of graphs with a handle. The N~-dependence in each case is given by 't Hooft's rule. W e end this subsection by noting that gluon loops do not affect the No-dependence of a graph. But each additional quark loop decreases the power of Arc by 1. 3.4

Meson

Masses

't Hooft ('t Hooft 1974b) obtained the exact result that in the 1 + 1 dimensional QCD the meson masses are independent of No. In the process, he also had confinement. However, the confinement mechanism in this case is an artifact of 1 + 1 dimension and has nothing to do with the confinement mechanism in 3 + 1 dimensions. Here we give an alternative, simple quantum mechanical (fixed number of particle) treatment of the mass of a meson. It must be understood that what follows is not a theory of meson structure. The question of confinement is not discussed. It is merely a device to do the Nc counting. Let us conjecture that the state of a meson may be described as a superposition of states of the form I (af, af')c~l, where a stands for color and f and f ' stand for flavor. These qq p~irs all have the same total spin, isospin and parity, which are collectively indicated with a. To complete the definition some relative motion wavefunction must be chosen. We choose a wavefunction independent of the color label so that a color singlet state can be formed with these basis states. Eventually a variational calculation has to be done to determine the wavefunction. The q~ potentials are shown in Fig. 5. The standard one-gluon exchange potential is shown in Fig. 5a. Fig. 5b represents two successive one-gluon exchanges with the provision that the intermediate relative state is orthogonal to that in the initial state I (af, af)c~). This is indicated by the slashes on the inner quark lines. The first two graphs ~ N~-1. There are also a whole sequence of ladders of the form Fig. 5b involving increasing number of gluon exchanges with the provision that the intermediate relative state continues to remain orthogonal to that in the initial state I (af, ~f')a). All these graphs are N j 1. The last one involves a crossed gluon box. Being a non-planar graph, it ~ N~"2. So we will

92

M. K. Banerjee

1 .;

i-

.... X"

b

c

<_;;1

a

Figure 5: Examples of qq interactions via gluon exchanges ignore it. Thus the matrix element, ((a/, a f ' ) a I v I (bf, bf')a), of the one-gluon exchange potential is of the form g(N~)2F(a) = 3g2F(a)/No, where F ( a ) depends on the common space, spin and isospin structures of the pairs, but not on color. The initial ansatz ensures that the kinetic energy of a qq pair, is independent of color. Hence the hamiltonian matrix is of the form

((a f, af')a I v I (bf, bf')a} = ~°'gbb'(K + 2m) + 3g2F(a)/Ne, where m is the constituent mass of the quarks. The Ne × Ne matrix has one eigenvalue (K + 2m) + 3gXF(a) and N~ - 1 eigenvalues (K + 2m). The first eigenvalue is associated with the eigenstate V ~ ~ I (a f, a f ' ) a ) This is the only color singlet eigenstate and is the state of the meson in this simple picture. Of course, the quantity (K + 2m) + 3g2F(a) must be minimized by varying the space function. The minimized eigenvalue, which is the meson mass, is independent of Nc for large Ne.

3.5

Glueball

Masses

The minimum constituents of a glueball, in its rest frame, are just the minimum number of gluons needed to make up the specified quantum numbers. This number may be called the seniority of the glueball. Thus a 0+(0 ++) glueball has seniority two, while a 1 - ( 1 - - ) glueball has seniority three. We discuss only seniority two glueballs. The glueball may also be treated in a manner similar to that followed for the mesons. A glueball state may be written as [ (ab)(ba)A I where (ab) and (ha) axe two gluons of a U(N~) QCD coupled symmetrically to a state whose space-spin quantum numbers are denoted with A. Once again one must choose a suitable wavefunction which will be fixed later through a variational calculation. Some of the typical potential diagrams are shown in Figs. 6a through 6d. The interaction matrix is of the form

((cd)(dc)A [ v [ (ab)(ba)) =

(~ac + 6~d + 5be + ~bd) + B(),)

(42)

The last term arises from diagrams of the type shown in Fig. 6g and Fig. 6h. One can check that the matrix elements in which only one color changes ,,~ Arc-1, while those in which both colors change ~ N~-2. The full hamiltonian is obtained by adding the kinetic energy term ,~acS~di¢. It is e~rify that the only color singlet state which one can make out of the basic states, namely, ~ / ~ [ (ab)(ba)~} is an eigenstate of the hamiltonian with the eigenvalue x + 4A(~) + B($). Thus the mass of the glueball turns out to be independent of .Arcfor large No. A hybrid, sometimes called a meikton (Chanowitz 1991), is a color singlet boson made up mainly of a qq pair and a gluon. The qq pair is such that it can annihilate into a gluon. (Hence a hybrid has a small two-gluon component.) The qq pair is in a color octet, charge neutral and chiral singlet state. The task of analyzing the hybrid mass problem and reaching the final conclusion that the hybrid mass is independent of Nc for large N~ is left as an exercise for the reader.

Chiral Confining Model

a

93

b

Z

d "O000000GtOOOOOOOOOOOOOmO0000000"

"O0000001000000900~O00100000000" ...... ~b ab

ac

a.

)

e

a

a

<>

<>

c

f

<"

c>

a

a(

%

"O00000omoooooogoo~ooomo0oooooo ~ a

ab

. . . .

cd

a >

O000000~O00000~O0,OO' •00mO0000000a

•

Cl~

C

Figure 6: Examples of gluon-gluon interaction potentials. The first two rows show the Feynman graphs. Constructing the corresponding 't Hooft graphs are left to the reader. Each of the third and the fourth rows contains a Feynman graph and the corresponding 't Hooft graph. Here the two gluons in the intermediate states are in spatial states orthogonal to the initial spatial state. The diagram in the third row involve change of only one color label as in the diagrams of the first two r o w s . The diagram in the last row involve change of both color labels.

94

M.K. Banerjee

3.6

Meson-Quark Coupling Constant

Witten (1979a) has described how to obtain the large Arc behavior of meson-meson coupling constants. Here we describe brieflyhow one can obtain the large Arc behavior of meson quark coupling constant. The firststep is to define the meson field,~ by requiring that the amplitude, I ¢) ,

(¢ I ¢(0) I 0) = Z-½,

(43)

which connects the vacuum to the one-meson state, is independent of No. The coupling constant is extracted from the amplitude:

(q2 _ m2)i f e'~e-', ~ f e'y~'(,+,)'~(o I T(¢(~), ¢(0), ~(y)) I 0),

(44)

where m is the mass of the meson, by amputating the external quark propagators. The last procedure does not affect the Arc dependence of the meson-quark coupling constant as the quark dynamical mass

~'~ N eO.

The next step is to introduce an interpolating field, say, j¢(x) = ¢(x)ftO(x),

(45)

where f~is a product of gamma matrices and, where appropriate, isospin Pauli matrix such that it has the same transformation property as ¢ under Lorentz and SUR(2) x SUL(2) groups. In addition, it behaves exactly in the same way as ¢ under discrete transformations such as parity, time reversal, G-parity, etc. The third step is to connect the large Arc behaviors of ¢ and j~. This is done in the manner described by Witten (1979a). One evaluates the correlator - i f d4eiq'~(O I T(j~(x), j~(0)) I 0) in two different ways: (1) perturbatively, which is appropriate for large Arc and (2) by making a spectral representation. In the latter one observes that the ratio of the amplitudes (¢" I J~ I 0)/(¢ I J¢ I 0), where (¢* I is an excited state, can decrease with Arc, but can never increase with Arc. This should be true for all mesons. The perturbative evaluation tells us that the correlator ,,, Arc. Noting that the meson mass m ,,~ N ° one deduces that (¢ lYe I 0) ~ N).

(46)

In other words, J~ (47) ¢ ~ --T_" N2 Similar techniques have been used again in Section 5 where we discuss glueball and meikton fields. It then follows that

g '~ (q' - m2) i f d4xei":<0 I T(¢(x), ~ ,

¢(x)) I 0).

(48)

It is now straightforward to deduce, using perturbation theory, that _l

g ~ N¢'

(49)

The above result is true for all meson quark coupling constant. Using common sense one can deduce the following results for meson-nucleon coupling constants:

gpNN '~ N¢ ,

(50)

1

g,,NN,g,~NN "~ N2,

(51)

1

g,~NN, fpNN ~ N2. See Karl and Paton (1984) for the derivation of the last equation.

(52)

Chiral Confining Model

3.7

95

Baryon Masses and Radii

We do not present an analysis of the baryon mass problem in the same style as that followed for the boson masses. Neither do we repeat Witten's (1979b) analysis. Instead, we will be content with a set of plausibility arguments. The arguments themselves will recur in several of our subsequent discussions. We exploit results based on the scale transformation and the trace anomaly of QCD which are discussed later in Section 11. In the chiral limit the QCD lagrangian is invariant under an appropriately defined scale transformation. One can always choose a stress-energy tensor, 0 ~' (which may differ from the conventional definition through perfect derivative terms) such that

j~'. = =oe "~, and

a ""

~d,, = OZ"

(53) (54)

Naturally, at the classical level Of = 0 for QCD in the chiral limit. Collins, Duncan and Joglekar (1977) showed quantum effects change the value of the trace of the stress energy tensor: Of = (1 + ~ ) ¢ m ¢

+ ~(g---A)G G ~ 2g "~ "

(55)

This result is called the trace anomaly. From Eqs. (53) and (54) one can derive Eq. (279) for any eigenstate of the hamiltonian H = f d~zO °. Applying it to the state of a baryon we obtain for the baryon mass:

(B ] / d3ra, j~i, I B ) - (O I / dara, j~,, [0) = (B [ n l a >- ( 0 [ H I 0 ) .

(56)

Next, using the large Arc limit of the trace anomaly, given by Eq. (286), we obtain

M

=

(BIHIB)-(O[H]O),

The difference of expectation values (B I fl I B) - (0 I a I 0 ) of any operator, n, consists of graphs which are connected to the valence quarks. For the purpose of determination of large Nc behavior it is then sufficient to consider vacuum to vacuum graphs containing at least one quark loop. The quark propagator in the presence of a baryon differs from that in the QCD vacuum in having the pole in the complex energy plane corresponding to the 'valence' state shifted from below the real axis to above the real axis. The leading graphs will be of the form shown in Fig. 3. Note that the valence state is filled by Arc quarks. Hence the counting of powers of Nc is unaffected and we infer that M ,,, N¢. Our next task is to establish that the nucleon size is independent of Arc for large Arc. We note that the leading term in the energy functional must be of the form E = N~F(R), where R is a suitable length scale for the baryon. The stationary point of E with respect to R is the size of the baryon. It is clearly independent of N~.

96

M.K. Banerjee

4

A Chiral

4.1

Confining

Phenomenological

Model

View

We need a rising potential to confine a quark absolutely. Absolute confinement means that, regardless of how much energy we pump into it, the quark cannot come out of the confinement region, which is the bag. A harmonic oscillator potential or a linearly rising potential confines absolutely. A signature of an absolutely confined wavefunction u(r') is that the integral f dreXrr~u(~ exists for all X, real or complex, and for all v _> 0. Equivalently, the fourier transform of an absolutely confined spinor, u(~ = f d3rei¢'eu(~, is an entire function of q2, having singularities only at infinity. The reader may verify that the wavefunction for a potential rising as r n, with n > 0, behave asymptotically as exp(-~rl+~) and satisfies the requirements of an absolutely confined wavefunction. Unfortunately, we also know that rising potentials cannot be generated by exchanging bosons, massive or massless, between the quarks. An alternative scenario, worth exploring, is that the free quark term in the lagrangian, ¢i~ ~ ¢, goes to zero. An example would be the lagrangian £ ....

~,~

= g(x)~(x)i½ ~ ¢(x) - ,~(x)¢(x) +...,

(58)

supplemented by the requirement that

(0 I KCx) I O) = O.

(59)

Let us associate a field, X, with the dimensionless object K(x). g ( ~ ) = g,~x,

(60)

where the X field has the usual dimensions of mass and, hence, the coupling constant gx has the dimensions of length. The ellipsis in Eq. (58) hide the lagrangian, £x of this field. The details of this lagrangian are not crucial for the present. The lagrangian will be discussed later in this section. The Euler-Lagrange equation for the quark field is

K(z)i#¢(z) = #¢(x).

(61)

If # ~ 0 and K(x) = 0, it is clear that ¢(x) must vanish. In other words, quarks cannot exist in the vacuum. However, if there are valence quarks present, they can polarize the vacuum, through the coupling of the X field to the quark, and alter the value of K(x) away from zero. Quarks can exist in this region, which is, of course, the dynamically generated bag. As a quark tries to drift away from the bag it sees the ever-rising mass, --~K(x),.which keeps it absolutely confined. There are some obvious problems with the scenario. Let me list them. 1. Can one invent a K(x) which will satisfy Eq. (59)? 2. Can the current quark mass play the role g ? 3. If not, what can be the source of an effective mass term without violating the requirements of chiral invariance? We will see that the answer to the first question is - yes. The answer to the second question is no. Finally, we will find a satisfactory source for an effective # without violating the requirements of chiral invariance. This will be discussed in Section 7.

Chiral Confining Model

97

Figure 7: Links in a hypercube 4.2

The

Nielsen-Phtkos

Lagrangian

A basis for a lagrangian of the form of Eq. (58) was introduced by H. B. Nielsen and A. P~tkos (1982). Their objective was to develop a color dielectric function and a set of coarse-grained color fields as collective variables appropriate for long range physics. For this purpose they adopted a strategy similar to the method block spinning. Consider the hypercube of sides L, shown in fig. 7, and two points, z - e and z lying in it. Consider all paths connecting the two points and lying entirely inside the cube. Introduce the path-averaged link operator,

U(=, • - e) = Adge[Pexp{-ig

1: A(y). d~}l,

(62)

--C

where A~(y) = A~)~Q/2. Making a Taylor's expansion of U(z, z - e) in a power series of e one gets U(z, x - e) = U(z, z) - ige. B(z)....

(63)

Once again, B,(y) = B~)~Q/2. Nielsen and P~tkos introduced

(64)

K(z) = -~ TrU(z,z), and B~(z) as the collective variables.

The effective action in terms of these collective variables are obtained, in principle, by introducing these variable through appropriate delta functions in the QCD action and then integrating out the A fields.

= J D[K]/D[B]/D[¢,~]/D[A]e-f x6(K(z)

- - ~ T r U ( z , z))

i

, aU(z,z-

x n,.,,6(m~, - gtrA (

a,,,

~ffi''wz'

(65)

')) I,---o.

The quark sector of the resulting effective lagrangian may be visualized by considering the following bilocal form of the QCD lagrangian £'

-

.

a

.

r'

QcD = ¢( x )(-'7" ~'~e,) PexP{-gg Jr_,, A(y). dy } d2(z - e).

(66)

98

M.K.

Banerjee

One averages over all links in a hypercube of sides L containing the points z - ~ and x and then lets e go to zero. It is easy to see that the quark sector of the resulting effective lagrangian is

£Ne = ~b(x)(K(x)i2 ~ - gI](x))~b(x)+ ' " . The above form shows that

-~, and

not

B~,are the

(67)

gauge fields. Thus the covariant field tensors

are

a

F ~" = 0~-~

a

-

Bb ~c

O~-~ + g r, ° ~ -~---~. ~-'~

(68)

Finally, the leading terms of the effective lagrangian in a derivative expansion are

£NP = (b(Ki~ ~ - g ~ t ) ¢ - e(K) 4F:~F°'U~ + " ' .

(69)

The process of integrating out the gluon fields in favor of K and B , cannot give rise to negative powers of K in the effective lagrangian. Hence, to compensate for the factor K -a in the term Fu"y ~,a,uv must behave as K 4. The quantity a , there must appear a factor ¢(K) which for small K ¢(K) is the color dielectric function. Retaining only the leading term, i.e., ~(K) = ¢'K4, one can redefine g --* ~-lDg and B u --* ~l/ZB~ and rewrite the effective lagrangian as

Nielsen and Pgtkos conjectured that the vacuum expectation value of K(x) is zero.

(K(x))v,c = O.

(71)

This conjecture has been justified (Lee et al., 1989) from the point of view of the lattice gauge theory. Here I summarize very briefly the central physical idea behind the justification. Consider a lattice of spacing a such that there are N = L/a plaquettes on any edge of a hypercube of sides L. The hypercube has 6N 2 planes, each containing N 2 plaquettes. To form a loop covering an area t,a 2 we need to pick v plaquettes and then we can always find at least one way to run a loop going around these selected plaquettes and covering no more areas. In fact, in most cases we can find many ways to do this. Given a set of plaquettes we will pick one and only one of these loops. Since all of them will have identical value for any field configuration it does not matter which one we pick. Thus distinct loops are obtained by picking v distinct plaquettes. We would like to use for the value of these loops the area law e -~"2~, where a (,,, 1 G c V / f m ) is the string tension. Correlations among the fields through various plaquettes may prevent the validity of the area law. To avoid this difficulty we adopt a redundantly conservative approach. We pick the first plaquettes from any one of the 6N 4 available plaquettes. In picking the next one we avoid the columns which either contain the plaquette already picked or are tangential to it. There are 6 such columns each containing N 2 plaquettes. So there are 6N2(N 2 - 1) ways of picking the second plaquette. In picking the third plaquette we avoid all columns which contain any one of the two plaquettes or are tangential to them. This leaves 6NZ(N 2 - 2) plaquettes to pick the third plaquette from. Finally, •

2

the number of ways of picking v plaquettes in this manner is (6N2)V(~). It is reasonable to expect that the area law will apply to these loops. Taking the average over these loops we have ( 0 l K ( x ) 10)

= _

E~(6N2)~(~)e-"~ N2 E~(6N2)v( v )

(i + 6N% -"2~" 1 + 6N 2 )N~

e a~°" Na = e-L~(1 + "d-~) =

e-L2ae~eXp(a2¢).

(72)

Chiral Confining Model

99

In the last three steps we have kept L fixed but have allowed a to become arbitrarily small and hence N to become arbitrarily large. Finally in the continuum limit, i.e., a --* 0 we get

(0 I K ] 0) = e -L~.

(73)

To study the property of the vacuum we can make L arbitrarily large and the K becomes correspondingly small. We may consider this sufficient to justify the conjecture that (0 ] K I 01 = 0. In Lee et al., (1989) we have been somewhat less conservative and have obtained the result (0 ] K [ O) = e -3L~¢. If we insist on making L no larger than, say, 0.3fro, so that we can study the structure of the nucleon, then (0 I K I 0) is only as small as 0.20. We will see later that this means that asymptotically the quark sees a wall of height ~ , where g~ is the pion-quark coupling constant. The typical value of g~ is 4, which implies a wall height of 5.5 GeV. This does not produce absolute confinement, but the wall height is quite high. Our phenomenology will be based on the assumption that (0 I K I 0) = 0. 4.3

Role

of the

Current

Quark

Mass

Several authors (Chanfrey et al., 1984, Schuh and Piruer 1986, Williams and Thomas 1985, 1986) have used the current quark mass term for the interaction term p~b. We believe that there are three distinct problems associated with this approach. First, let us examine how the current quark mass term is expected to appear in the NielsenP~tkos lagrangian. There are two ways to introduce the current quark mass term. One may do it in the bilocal form given by Eq. (66) by simply adding the mass term to it,

This procedure leads to K(x)~(x)mq~p(z) as the current quark mass term in the Nielsen-P~tkos lagrangian. It vanishes when K(z) vanishes and cannot help generate the bag. The other alternative is to introduce the mass term in the usual local form, ~(z)mq~b(x). This strategy was followed in the references cited at the beginning of the discussion. It is then unaffected by the averaging procedure and appears in the Nielsen-P~tkos lagrangian in the same form. It can play the role of the ever-present interaction and will lead to generation of the bag. But by the same token, this approach retains the short distance physics present in the QCD lagrangian. In the first approach such short distance physics has been averaged out. We believe that the first approach is consistent with the rest of the Nielsen-Pktkos approach. Second, even if one believes that the mass term should appear in the Nielsen-Pktkos lagrangian without the factor K(z), it is likely that the constituent quark mass, which arises from the interaction of the valence quarks with the quark condensate of the vacuum, plays a much more important role. Finally, making the current quark mass term wholly responsible for the bag formation is likely to yield a sigma commutator which is -~ 1/4 of the nucleon mass--an unacceptably large value. 4.4

The

X Potential

We need a field associated with K(z) to describe the change due to the valence quarks. introduced such a field, X, through Eq. (60). Eq. (71) implies that

(0 I X I 0) -- 0.

We

(74)

100

M.K. Banerjee

We need to make a specific choice of £× in order to be able to calculate the static polarization of the vacuum induced by the valence quarks. One may write 1 Z x = ~O, xO X - U ( X ) ,

(75)

Knowing very little about the X potential, U(X), we are forced to make guesses. We consider two alternatives. Up,,. . . . . . (X) Uq~,,,c(X)

=

1 2 2 ~mxX,

(76)

1 2 ~- _ 2 ( 2 + r h ~ ) x + l ( l + ~ ) ( X ) , ] . ~ m x x tl m~ x0 x

(77)

=

The pure mass term, Up,. . . . . . (X), obviously contains no interaction among the X fields. It has a minimum at X = 0, which is the vacuum value. The value of the potential is zero at this minimum. The mass is m x. The quartic term, U q u a r t i e ( X ) , contains interactions among the X fields. It has two minima, one at X = 0, needed to describe the vacuum, and the other at X0 > 0. The masses at the two minima, given by the values of ~~ U quartie at the minima, are m x at X = 0 and rh x at X = X0. The height of the quartic potential at X = X0 is Uquc~rtie(Xo) = i~X0~7,,X 1 . 2/--2 -- rh~). Thus in this model rh x < m× always, otherwise the lowest state, the new vacuum, would have a value X = X0 for and not X = 0. We restrict ourselves to the case rh x = rex, which makes U(Xo) = O, i.e., the minima degenerate. Before ending this section a few remarks on the large N~ dependence of the baryon energy, resulting from the choice of t'×, are in order. The discussion of the large Arc dependence of boson masses in earlier section has shown that the masses of glueballs and meiktons are independent of Arc for large Arc . With the £x chosen the baryon energy functional is bound to contain terms of 1__2. 2 the form f dSr[~x • VX + ~,,,xx + "" "]- If the classical X field grows faster than N ]i. the model will predict that the baryon mass grows faster than N~, a patently unacceptable feature. So it is crucial to demonstrate that, indeed, the classical X field grows only as N~ for large Arc. This will be the subject of the next section.

5

T h e G l u e b a l l and t h e M e i k t o n Fields

In this section we study the large Arc dependence of the glueball and the meikton fields. The object 1

is to demonstrate that the X field of our confining model grows only as NP. The discussion is based on work done with T. D. Cohen and M. Li (Banerjee et al., 1989). Readers who are willing to grant the point may skip this section. For every N~ the lowest mass 0++ glueball state, I F), is dominated by its two-gluon component, while the lowest mass 0 ++ meikton state, I X), is dominated by its q~-gluon component. We introduce canonical fields F and X associated with the states I F) and [ X), respectively. We assume that it is always possible to choose these fields such that for large N~ their matrix elements between the vacuum and higher mass glueball and meikton states are No-suppressed compared to the amplitudes, -_t

Zr ~

=

(rlr(o)

I0),

(78)

=

(XIX(O) l 0),

(79)

--~

Zx

101

Chiral Confining Model

0

0

(a)

(b)

>

O

(c)

(d)

Figure 8: The leading graph of the correlator -i(O I f d4xe'q'~T(jr(x), jr(0)) I 0) shown in (a), of (0 I G~, G~ I 0) shown in (b), of the three-point function f d4ze -iq'~ f d4ye iq'u(0 I T(¢(x), F(0), ¢(y)) I 0) shown in (c) and of (B ] G,~G "~ [ B) - (0 [ G~G ~ I0) shown in (d) are shown. The requirement that the fields be canonical demands that we excercise our freedom of choice of the scales of F and X to ensure that the following is true: ~ N?.

(80)

jr = G..G "~, ix = g(Nc)qdq.

(81) (82)

We will elaborate this point later (after Eq. (96)). Next we introduce the interpolating fields

Notice the presence of g(Nc) in the definition of j×. If we do not include this factor then we must remember that all graphs involving j× has one quark-gluon vertex without the coupling constant and hence without the attendant factor of N71/2. The presence of the coupling constant in the definition makes counting easier. Let us begin by determining the large Arc behavior of C0 I F I 0) and gv, the quark-glueball coupling constant. We can only determine the large Nc behavior of matrix elements of jr. So the first task is to establish the relationship between the field F and the interpolating field jrHere we follow the methods developed by Witten (1979a). Consider the correlation function -i(O I f ~zelq'ffiY(Jr(z), jr(0)) I 0). We can always make a spectral representation of this correlator;

[ - i(O I jd4xeiq'~T(jr(x),jr(O)) I O) = q2 _Zrm~

""

(83)

It is legitimate to assume that for large Nc no other term has N~ dependence more dominant than that of the lowest mass term. For large Arc we can also evaluate the correlator perturbatively and the relevant diagram is shown in Fig. 8a. It is ~ N~ and we have 1 Z~ N N~. (84) In other words r ~ Jr.

No

(85)

102

M.K. Banerjee

From Fig. 8b and Eq. (81) we deduce the result:

(o I r I o) ~ (0 1 ~jr 10) ~

(86)

No.

Next let us calculate 9r which can be extracted from the three-point function given below.

/~ z e - ' " /d4ye""(O I T(¢(z), r(0), ,~(y)) I 0),

(87)

From the graph in Fig. 8c and using Eqs. (81) and (85) we determine that the three-point function ,-, N~"1. The correlator may also be expressed as 1

91"

- -

1

+....

(88)

Noting that the masses of the quark and the glueball are independent of Arc we infer that gr " g~-'.

(89)

For us the quantity (B I F I B) - (0 I F I 0), where I B) is a baryon state, is of particular importance. By definition this quantity is the sum of graphs connected to the valence quark line. As an example of such graphs is shown in Fig. 8d. Once again using Eq. (85) we find that

(BIFIB)-

(o l r I o) ~ ~.

(90)

The derivation of similar set of results for the X field follows the same pattern. The following results are left as exercises for the reader. Zx ~

NI/2 , ..~

(91)

J× rW,~/2,

(92)

(o Ix Io) ~ ..,N1/2,

(93)

X "

(BIxlB)-(OIxlO)

",, N 112 -'c

"

(94)

The apparent contradiction between Eel. (74) and Eq. (93) is removed when we remind ourselves that the latter equation can hide a multiplying factor on the right hand side which vanishes. The coupling constant gx can be extracted from the three-point function given below.

fd4ze-iq" fd4yeiq'"( 0 I T(¢(z), j×(O), ¢(y)) I 0>.

(95)

With the help of the Figs. 9a and 9b we can deduce that g~ ~ N : '/~.

(96)

In Section 3.1 we stated Witten's (Witten 1979a) result that the n-point functions of mesons behave as N (2-"}/2. Following his method and using Eqs. (91) and (92) one can establish the large Nc dependence of multipoint vertices of glueballs, meiktons and mesons. A gluon loop with n insertions of j r gives the result that an n-point function of glueballs .~ N~ -~. If the n-point function contains either meiktons or mesons the leading Nc is behavior is generated by a quark loop with appropriate insertions. One can derive t/he result that a multipoint function of nr F fields, nx X fields and nm meson fields ,~ N{c1-"r-(nx+n'~)/2}, provided n× + nm > 0. For a more detailed discussion the reader may consult Coleman 1985.

103

Chiral Confining Model

(a)

(b)

Figure 9: The leading Feynman graph for f d4ze-~q'=f d4ye~e'~(O [ T(¢(z), j×(0), ¢(y)) [ 0) is shown in (a) and the corresponding 't Hooft graph is shown in (b). These resultsshow that for large Arc the glueballand the meikton fieldsbehave as freefieldsand Z r ½, 2 ; ½ -~ 1.

(97)

We have assumed that it is possible to define these fields as canonical fields for every No. We can check, at least, that for large N¢

(0 I IF(O, ~), V(O)] [ O) = (0 I [~(0, ~), x(O)] I O) ~ -i6(~). Finally, we argue why X and not r is the field of interest in our phenomenology. Let us expand the K(z) field in terms of the r and the X fields. K =grr +gxX+....

(98)

By construction (0 I g I 0) ,~ N ° we can check from Eqs. (86), (89), (92) and (93) that this requirement is fulfilled. Of course, (O I K ] 0) = O and we are interested in describing the change ( B I K [ B ) - ( o [ g [0). We write

(a IK IB) - (0 1g I0) --sr{(a Ir IB) - (0 1r I0)}+gx{(a Ix IB) - (0 1x I0)}+.... (99) From Eqs. (89) and (90) we see that the glueball (r) contributionto the change is line, while from Eqs. (94) and (96) we see that the contributionof the meikton, X, is ,..N °. Therefore,for simplicity we drop the glueballfieldfrom our phenomenology.

6 6.1

A

Chiral

Quark

Meson

Model

Introduction of Mesons

Any realistic model of baryon structure must recognize that the chiral symmetry of the vacuum is broken. In QCD itself this is manifested through the nonzero value of quark condensate. Practitioners of the QCD sum rules (Shifman et al., 1979, Reinders et al., 1985) find that (flu) = (rid) ~_ - 2 f m -s.

(100)

In an effective lagrangian with mesons the symmetry breaking is incorporated at the classical level using a suitable mesonic lagrangian. The most famous example of such a lagrangian is the GellMann - l.hvy (1960) cr model in its linear and nonlinear forms. Indeed, the nonconfining models of Birse and Banerjee (1984) and Kahana et al., (1984) were based on the Gell-Mann l.~vy ~, model with the modification that the colored quark field was substituted for the nucleon field. It is useful to review briefly the Gell-Mann L~vy cr model even though confining models, based on the Nielson-Pktkos approach, include an important modification of the quark meson interaction

M.K. Bancrj~

104

terms. We begin with the chiral invariant form of the lagrangian. Therefore, the pion is a massless, pure Goldstone bosom In Section ?.3 we discuss chiral invariance breaking of our effective lagrangian. The chiral invariant linear u model lagrangian is: 1

1

1

£~Mr. = ~[i~ ~ +g¢(~r + i%~. ~)]~b 4- ~6~'~r0#o"4- ~'~O,~ -- U~..,~.(~r, #),

(101)

where the chiral invariant Mexican hat potential is:

~,.,o..(o,,,~) = ~(~2 + ,: _ F;)' The constants are

given by the

(102)

relations: 2

2F,~' The condition:

(103/

6U o

6~ l.=-F.,,=o=0,

(104)

ensures that the vacuum value of tr is - F . and not zero. This constitutes the realization of symmetry breaking of the vacuum at the classical level. The conditions: /f2U0 6o.2 lo,=-F, tf2Uo

=

~¢--T I.:-F.

= o.

' m~,,

(10.5) (106)

ensures that the ~r meson and the Goldstone pions have the correct masses. The constants are chosen to make the energy density of the vacuum zero. One can verify easily that under an infinitesimal chiral transformation •

ra

= -,e.@T%, (107) the lagrangian is invariant under chiral transformation. TheN~ther'scurrentsare

VECTOR: ~ AXIAL-VECTOR:

=

~

=

-

1..

1 ,k'fvy. ~?,h + ~0.,:r - a0.~.

(108) (109)

In a theory without vector mesons these will be taken as the definitions of the vector and the axial-vector currents. The charges are

Qt, = / daxVoa(W),

(110)

Q~ = / dazAg(z),

(111)

J

Chiral Confining Model

105

X:X H Figure 10: The hatched box is the gluon tower. Some of its simple constituents are shown on the right. which obey the commutation rules:

[Q°, Q~] = [Q~, Q~] = ieo~Q :~.

(112)

The combinations i1( Q 4- Q~) are the generators of the commuting groups SU(2)R and SU(2)L, respectively.

6.2

A v e r a g e Field and M e s o n s

Our objective is to propose an effective lagrangian which will describe, in the mean field approximation, the average field seen by a quark. It is appropriate to make a few general statements about building phenomenological models. In the first pass at the problem we need to make many simplifying approximations. As long as the approximations appear reasonable we should go forward. The correction terms deserve to be examined. Effects due to such corrections which can be simulated reasonably well by refitting parameters of the simple theory are not interesting. Effects which are qualitatively missing in the simple theory are the interesting ones. In QCD the average field is described with two-quark irreducible graphs which we may call twoquark interaction potential. These are generated by exchange of gluon towers of the form shown in Fig. 10. Because the quark is a color triplet we need consider only color singlet and color octet towers. The color singlet tower is described by exchanges of glueballs and hybrids. As we saw in Section 5 only the 0+(0 ++) members of the glueball family are of special interest . Glueballs and hybrids which transform like chiral singlet mesons are uninteresting as they play the same role as higher mass states of the mesons already included. We believe that in a simple phenomenology one should include only the lowest mass state of each species of mesons unless we discover some compelling evidence to include higher mass states. We also saw in Eqs. (89) and (96) that the hybrid-quark coupling is stronger than glueball-quark coupling. Hence in our approach only a single hybrid is considered and its role in the two quark potential is taken care of by the X field of the Nielsen-Pktkos lagrangian discussed in Section 4.1. We now discuss the exchange of octet towers and argue that these may be approximately represented by meson exchanges for the purpose of the generation of the average field. The underlying ideas have been discussed earlier (Broniowski 1986, Banerjee et al., 1987, Ball 1987). Let us examine the average field due to the valence quarks as seen by a test quark. In Fig. 11 we show explicitly only the color labels, a for the valence quarks and i for the test quark. The diagrams of Fig. l l a , which represents the direct matrix elements, will contain factors like (A~)~a, where )% is a Gell-Mann matrix. Since all color states are occupied in a baryon the direct matrix elements will sum to zero because the Gell-Mann matrices are traceless. In contrast the exchange matrix elements, shown in Fig. l l b , will contain factors like (A~)i~(~)ai = ()%)~i, which is non-zero. Thus only the exchange matrix elements of octet gluon tower exchanges contribute to the average field.

M. K. Banerjee

106 i

i

¢x~

a~ (a)

i

a (b)

i

p'~,i

¢z

D'#,G

p ,i

pz,ct {e)

Figure l h Direct and exchange matrix elements of octet gluon tower quarks (a) and a test quark (i).

exchangebetween valence

In Fig. l l c we put a box around the tower and the quark lines and regard it as a new system being exchanged between the quarks. It is obvious that the t-channel view of Fig. l l c is identical to the u-channel view of Fig. l l a . The newly defined exchanged system can be obtained by Fierzrearrangement of the content of Fig. l l a . The new system has both color singlet and color octet pieces. Only the former contributes to the average field and we drop the color octet part. The color singlet part contains hybrids and mesons. The role of the hybrids have been discussed before. 2 The amplitude shown in Fig. 11 c has the structure I- /x[P l 2, P2, Plt2 , P2t2 , s = (pl + p2)2, t = (p'~ - pl)2), upon ignoring the spin-isospin factors. The t-channel singularities occur precisely at the meson masses. As mentioned previously, we retain only the lowest mass. We regard the lowest resonances as poles, which is justifiable since the nucleon structure problem deals with t < 0. But representing V as a simple pole in the t variable with constant residues may seem too drastic in view of the other dependences on the quark momenta. Such dependences are often summarily referred to as nonlocality effects. Fortunately, in the mean field treatment of the baryon we. deal with exactly one spinor state of the quarks. In other words the nonlocality effects will be sampled by exactly one momentum distribution. Then it should be possible to use a simple pole representation for V with the fitted quark-meson coupling constants reflecting the effects of nonlocality. The resulting lagrangian should be usable for the study of ground state properties of the baryons studied at low momentum transfers. It may be usable for the study of a few low-lying excited states. It may not be usable to study, say, dense quark gas with Fermi momentum comparable to the scale of nonlocality of V. We therefore look for an effective quark-meson-x lagrangian which will produce one meson exchange potentials which approximate the V's discussed here. Before we leave this section we should stress a particular point. Our effective lagrangian must include, from the very beginning, meson dynamical terms. It is not practical to derive them from QCD. An attempt has been made by Ball (1987). The general question of introduction of mesons, referred to as bosonization, have been studied by Eguchi and his collaborators (Eguchi and Sugawara 1974, Eguchi 1976). The Nambu Jona-Lasinio (1961) model has provided a good basis for studying this question. There is extensive literature on this subject. There two excellent reviews: one by Vogel and Weise (1991), the other by Klevansky (1992). These studies are extremely instructive and have practical uses in the construction of QCD based models. However, in the final analysis one must simply write down the lagrangian one is proposing. The structure of the lagrangian must be constrained by known symmetries of the QCD lagrangian and the requirement of correct large Ne behavior. The parameters of the lagrangian must be fixed by fitting data. In the next section we proceed in this spirit. In section 7.2 we discuss the important question whether quarks and mesons need coexist in an effective lagrangian. The studies on bosonization help us find an answer to this question.

Chiral Confining Model

7

Quark

Meson

107

Interaction

For the chiral confining model let us propose the following lagrangian, transparently adapted from the Gell-Msnn L~vy ~ model lagrangian, £GML,defined in Eq. (101):

£.CCM= ~b[KiI ~ +K"g,(cr +iTs{" ~)]~ +

£ . . . . . + £x,

(113)

where K is the object introduced by Nielsen and Pgtkos and defined earlier by Eq. (64) in Section 4.2. The mesonic lagrangian £ . . . . . can be read off Eq. (101). The hybrid lagrangian, £x has been defined by Eq. (75) in Section 4.4. Since the ~, field has a vacuum expectation value of - F , we have the possibility of obtaining the ever-present interaction #, referred to in Section 4.1, that the quark must see in order to form a bag. It is then imperative that the interaction does not disappear when K goes to zero. We cannot assume, a priori, that this is the case. So we insert the multiplying factor K" with the quark-meson interaction term and set ourselves the task of showing that, at least, n < 1. We determine n by using an idea due to Eguchi (Eguchi and Sugawara 1974,Eguchi 1976). We take the Nielsen-Pgtkos lagrangian given by Eq. (70) in Section 4.2 and integrate out the coarsegrained gluon fields B~. Next we take the chiral confining model lagrangian given by Eq. (113) and integrate out the quantum meson field fluctuations around the classical fields which make the action stationary. In each case we get four-fermion interactions of highly nonlocal nature. We equate the nominal powers of K in the two four-fermion interactions. In making this comparison we do not distinguish between K's at different points and between K and its derivatives. The discussion about nonlocality in Section 6.2 justifies the strategy. The Euler-Lagrange equation for the B field obtained from Eqs. (70) and (68) is O.F=~

Bc

+ gf =boF b~. ~ . + ( a . l o g

K S ) F =.~ =

g - ~ ~)%~ 1 -K-~7

(114)

T h e r e are two points to note in the Eq. (114). First, Eq. (114) may be regarded as an equation for the gauge fields, -~-, rather than for the coarse-grained gluon fields B~,. The second point is that only the quark source term contains a factor which is a power of K. The Oflo9 K 3 term depends upon the rate at which K changes, but not on its value. Eq. (114) is a complicated non-linear equation. We will draw our conclusion based on the part that behaves as

B'~(~)

g

-

~1

K(x) = f #YD(x'Y) K---~I'(Y)7 ~A=¢(y)....

(115)

The ellipses hide terms which involve couplings to two or more quark sources. These terms deserves to be examined. It has not been done so far. Substituting the form given in Eq. (115) in the Nielsen-Pktkos lagrangian given by Eq. (70) we get a purely fermionic lagrangian of the form: 1 21 £~)p(z) -- ~b(oc)Ki1 ~ ~b(z) + ~g ~b(x)7,~)~=~b(x)K(x) /

g 1 d4vD(z, V)K-----~b(V)7" ~A=~b(V)....

(116) The discussions of Section 6.2 makes clear how we obtain effective meson exchange like pieces from an expression of the form of Eq. (116). Here we simply count the power of K. It is - 2 . Next we repeat the procedure with the chiral confining model lagrangian given by Eq. (113), i.e., we integrate out the meson fields and obtain nonlocal four-fermion forms. Here the ~ field has a vacuum expectation value of - F ~ and one integrates out the fluctuation around the.vacuum

M. K. Banerjee

108 value. The result is

£(1) tx~ OCM (

I

+ 2g~Kn(x)¢(x)¢(x)fd4yDo(x - u ) Un(Y)¢(U)¢(Y)+"" -

(117)

The net power of K is 2n. Equating this to the net power in Eq. (116) we obtain n = --1.

(118)

Using this value we obtain for our proposed effective lagrangian the form

£CCM = ¢[Ki ~1 "* O + ~g'~ ( ~ + i75¢. ~)1¢ + £ o

. . . .

7.1

The

Canonical

Quark

+ £x,

(119)

Field

An examination of the chiral confining model lagrangian shows that the field canonically conjugate to ¢ is not i¢t but iK¢ t. Thus all Noether's current will carry the factor K. For example the conserved fermion current is j, = K¢%¢. (120) The quark density operator is /~ = j0 = g c t ¢ .

(121)

Of course, one expects such a modification. We have seen that quark field cannot exist where K = 0. Naturally, the support of the density operator is confined to the region where the field can exist. The Eq. (121) has another useful content. A physically acceptable density operator, ~ must be positive definite. This can be ensured if and only if (algla)<

0,

(122)

where a is any state. This is a constraint which must he satisfied by any model based on the notion of color dielectric function of Nielsen and Phtkos . It is convenient to introduce ¢'(x) = ~ / ~ ¢ ( x ) , (123) as the canonical quark field. In terms of these fields the lagrangian T-,CCM given by Eq. (119) becomes

+ i~5~- ~)1¢' + c . . . . . + L;,, £VeM = 43'[/2 ~ +~-z(~ g~

(124)

where £x is, in principle, different from £x defined by Eq. (75) of Section 4.4. However, the potential U(X) was not derived, but chosen on the basis of phenomenological considerations such as the values of (0 I X [ 0) and mx. Thus we are free to make the same choices for t:~ and we do so because the phenomenological considerations remain the same. From hereon we will always use the canonical quark fields. We will also drop the primes on the ¢ field and/~x"

Chiral Confining Model

7.2

109

Coexistence of Mesons and Quarks

There is a very important question which we must face. Need quarks and mesons coexist in an effective lagrangian? Or, can one integrate away the quarks completely and be left with a theory of bosom? To make the question specific let us consider the generating functional Z[~, #1 = ~f f Dr(, ¢, X, a, r--']e z p ' f d'=l~ccM+~"+o¢],

(125)

where T/and ~ are Grassman numbers and the lagrangian is g~ ,CCCM = ~3[i~ + ~ . ( , , + i7,~- ~)1¢ + £ . . . .

+ £x,

(126)

Upon integrating out the quark fields one gets Z[~, ¢/]

where

f D[x,a,~]exp,{f d'=[~..... +c.x-~z~-',fl-iT,.to,V}, g~ :D = i# + ~--~(a + i7s¢. ~).

(127)

(128)

The new effective action is t

& = J d4x[£ . . . . . + £x] - iTrlog:D,

(129)

and it is purely bosonic. It is useful to separate the action into two parts: Sb = Sb.... + S~,,~, = ] d 4 z [ £ . . . . . + £×], Sq,,rk = -iTrlogT).

(130) (131) (132)

While we have a purely bosonic lagrangian, the formal result is of limited interest unless one has a practical method of calculating Sq,,rk. It is a divergent quantity and must be regulated in some appropriate manner. In the bosonized version of the NJL model one has

sqNJL

-iTrlog[i~ + g~(a + iTs'~ ~)].

(133)

Since the NJL model has a momentum cut-off there are no divergences. Several authors (Meissner et al., 1988, 1989, 1990, Reinhardt and Wfinsch 1989a and 1989b, Alkofer 1990, Blotz et al., 1990, Goeke et al., 1991, Alkofer and Reinhardt 1991) have evaluated Sq~,rk for the Nambu and JonaLasinio (1961) model and found field distributions which make it stationary. Our effective lagrangian is badly unrenormalizable. We may remind ourselves that Eq. (126) gives only the low energy-momentum sector of the effective lagrangian. It is hard to believe that making the action given by Eq. (129) stationary is a feasible numerical project. Thus the original question of coexistence of quark and boson fields must be restated. Must we have both quark and bason fields in a usable lagrangian? Our definition of an usable lagrangian is that it should contain only few derivatives, possibly not more than four. A usable lagrangian can be obtained by making a derivative expansion of the one-loop term - i T r l o g D and retaining only the lowest few terms. These are low energy-momentum

110

M.K. Banerjee

~xx

(a)

~

(b)

xxxt'~xx (c)

Figure 12: The pole distribution in the complex w plane of (w - h) -z in the vacuum is shown in (a), the distribution when one of the positive energy type state is filled (triply) by the valence quarks and e v = t ~ > 0 is shown in (b) and the distribution when e.=t.... < 0 is shown in (c). effects and can be calculated, in principle, with some suitable regulation scheme. The results will depend upon the regulation scheme. But even these problems are manageable. The real issue is that the derivative expansion can be made only after a Wick rotation to the Euclidean space and the question is whether this can be done without acquiring new terms. We will examine this question. Taking advantage of the invariance of the trace under transformation from time representation to energy representation we write

8q~=,k = -iTrlogD =

-iJd%trlog(z I DIx)

where D has been defined by Eq. (128) and the Dirac hamiltonian, h, is given by the equation: /---~'7o(O'+ i % ~ . ~).

(135)

Finally, integrating by parts one obtains

S,,,=,k = constant + i / d3x / d W -~t r ( x l

~w

I x)7o.

(136)

The discussion becomes clearer if we do not restrict ourselves to a Dirac hamiltonian of the form of Eq. (135) only, but also include the nonconfining variety, h = - i ~ . V - g~%(a + i%7~. if). The poles of (w - h) -1 are above the real axis if the corresponding eigenstates of h are filled. Otherwise they are below the real axis. In the vacuum of the nonconfining model, the action is stationary when the a field is constant and equal to - F ~ and the r field is zero everywhere. In this special case all the negative energy poles are above the real axis and all the positive energy poles are below the real axis, as shown in Fig. 12a. When a baryon is present the quark propagator in the nonconfining model is changed in several significant ways. First, the meson fields are changed from their values in the vacuum. Second, a positive energy type spinor, defined by the condition that f d3rut(-i~ • V)u > 0, is occupied by the valence quarks and the corresponding pole, located at w = e~=te,ce, moves from below the real axis to above. Figs. 12b and c exhibit the pole structure when e~,s.... is positive and negative, respectively. We can write the quark propagator for all three situations in the following manner:

SF(W) = Soea(w) + S~at.... (w),

(137)

Chiral Confining Model S,,(w) S,,.,..=(w)

= =

111

1

(138)

w - h + i~e(~o)' -27riO(e~,.t~=)6(w

-

e,~,..=).

(139)

where e(w) is +1(-1) when w > (<)0. The S,,, part is present in all cases, though with different field distributions. The S, tc,~e(w)is present only when the valence spinor eigenvalue is positive. For the So~, part we can rotate ( Wick rotation)the originalcontour of integration in Eq. (136), running from -co to +co, to a new one running from -ico to +ico. W e are now in a position to address the question : should quarks and mesons coexist in a usable effectivelagrangian? The answer liesin the contribution of the S,t~,~ part. The contribution of S°e, to the action may be described entirely in terms of the meson fields. But the contribution of S~,u,,~, when present, needs the valence spinors explicitly. It is true that this information is contained in the meson fieldsand ifone chooses to take the quark part of the action in the form of Eq. (132) and make it stationary one can claim to have a purely mesonic theory. But as we point out, this is not a practical theory, surely not when we include confinement. W e do not adopt it. W e restate the question in terms of a practicallagrangian containing limited number of derivatives of boson fields.The answer to the newly posed question is as follows: Whenever S~,~,¢~ is present, i.e.,whenever e~,t~,¢~ > 0, we must retain the quark field but include only treegraphs and no quark loops. The effectof the quark loop terms is in £ ..... + £×. In a nonconfining model it is possible to have e,,t.... < 0 if one includes vector mesons (Broniowski and Banerjee 1985, 1986) Alkofer et al.,(1992) have recently demonstrated this result for the bosonized version of the N a m b u Jona Lasinio theory. They have further shown that a pure mesonic theory emerges. In the Section 8.3 we will show that with £ C C M using the pure mass for of U(X) one can prove that the valence spinor eigenvalue e~,a~,~ > 0 always. This is true when all possible mesons are included. W e do not have a proof when X - X interactions are included, i.e.,when U(X) contains powers of X greater than 2. W e do not have a proof when we include the 't Hooft interactionfor any form of U(X). However, we have never been able to generate a solution in any variant of our model with confinement where e~,te,ce> 0. W e are not aware of anyone else finding such a result. W e may note that e~.te,~ > 0 in theMIT hag model. Therefore, we believe that in chiral confining models, treated at the mean fieldlevel,we must retain the quark fields,taking care to use only tree graphs. W e stressthat any one claiming to obtain a practicaleffectivelagrangian using only boson fields and incorporating confinement must carefullyreexamine any step involving the Wick rotation, ifit is employed in the derivation.

7.3

ChiralInvariance Breaking of the Lagrangian

The chiralinvariance of the Q C D lagrangian is broken by the current quark mass term. In Section 4.3 we have argued that the current quark mass term should appear in the Nielsen-P~tkos lagrangian in the form K C m q ¢ . Hence in £CVM it should appear as ¢ m ¢ , where we are now using the canonical quark fields.However the act of bosonization willproduce mesonic terms. W e take care of these by using the chiral invariance broken form of the Mexican hat potential (Gell-Mann and L~vy 1960).

UMe=ic.,, ~2

A2 = =

v2)2

"-~(a 2 + ~ -2 I% a -- m~.

~p2 + F,~m~(a + F,+) -

4 "- "+

_ v2)2'

(140)

2

mE.+

'

(141)

M. K. Banerjee

112

(a)

(b)

(c)

(d)

Figure 13: Some simple QCD graphs corresponding to the four terms of £×tB -

~

j,,/,_ 1_2~=~

3aa}

v2 _

-mq~

-

,~2 . - 2 + a a ) .

2

m~-

,~

3m~

(142)

- m]

It is only the term linear in a which breaks the chiral invaxiance of the lagrangian. The rest of the modifications are needed to ensure that (a) (0 [ a I 0) = - F , , (b) the a mass is m , and (c) now the pion mass is mr. Once again the constants are chosen to arrange that the energy density of the vacuum is zero. So the fully modified lagrangian is ~:ooM = ~[i 1~* # -mq +

g~

~-~(~+ i'~s¥ • ~)]¢ + £ . . . .

+ £×,

(143)

where the mesonic lagrangian with broken chiral invariance is 1

.

1

£ . . . . . = ~O.aO a + ~O.~O ~ - UM=,....

(144)

We have discussed £x in Section 4.4. Note that there are two terms in our lagrangian which break its chiral invariance. Specifically:

£ x t , = -mq(b¢ - F,m~(a + F,).

(145)

To appreciate the physical content of the two terms let us first write the classical Euler-Lagrange equation for a following from £°aM, given by Eq. (143).

(_~2 + m~)a = j,, J~ =

gr

(146) -

--

ma

2

¢ ~2

~ - 5 ¢ ¢ -e ~ - f ~ r

2

+ 3a r2} _ 2F~ a m ; '(,x'2 + ar2),

(147)

where a ' = a + F~, which is the additional a field generated by the valence quarks. Of course, the r field is generated entirely by the valence quarks. Next let m¢ be very large and approximate a' = j¢/m~. Using this in Eq. (145) we get £ x , - -~ -mq(p¢

g,F,m~ (p¢_ 1 2--2 m~ , . ~ 2 ~ 5m, i r + 3a '~} + ~ - a ( + an).

(148)

The expectation value of the QCD current quark mass term in the QCD vacuum , which equals (Gell-Mann et al., 1968) -F~m~ 2 2 has been subtracted out in the definition of UM~i~, given by Eq. (140). All other terms are connected to valence quarks. The first two terms are one-quark terms in the sense that they require at least one valence quark. In the same sense the third term is a two-quark term and the last one is a three-quark term. It is relatively easy to see that Figs. 13(a), 13(b), 13(c) and 13(d) are the simplest of the QCD graphs which correspond to the four terms of Eq. (148). We note, in particular that the second term of Eq. (148) is a radiative correction to the first term.

Chiral Confining Model

7.4

113

PCAC

From Eq. (145) we can derive the result that

8# A, = -m,~iTsr"~ - FemUr.

(149)

This expression disagrees with the conventional expectation that a"A# = F ~ m ~ . This was noted by Birse and McGovern (1992) and viewed as an undesirable feature. They also suggested that the third term of Eq. (148) constitutes double counting of effects already included in the second term and prescribed that the it be subtracted out, i.e., one should use

,

:xlB = - m , ~

g~F~-

- F . m 2 : - ~-~m2f~

,

where the two- and three-quark terms have been dropped. The discussion of Section 7.3 shows why the second term does not involve double counting. Instead, it represents radiative correction to the first term. Let us examine the situation regarding PCAC. A good theory should ensure that (0 [/~(0) [ ~(k)) = - i F , k#,

(150)

so that it can explain the leptonic decay rate of charged pions. Then translational invariance guarantees that (0 I 6#A,(0) [ ~(k)) = -F,m~. (151) Next let us write down the matrix element of the isovector axial-vector current between nucleon states: • 2 (N(p') [ .4#(0) I N(p)) = ~(p')[gA(q2)Ts% + :q#ha(q )%]~u(p), (152) where q = p' - p. As a function of q2 the left hand side should have a pole at q2 = m¢2 a n d a cut starting from q2 = 9m~. Presumably the discontinuity peaks around q2 = m~, the square of the mass of the A1 meson. It is clear that the induced pseudoscalar term hA(q 2) contains the pion pole. Let us make this point manifest by exhibiting the pole of hA(q 2) and writing

. . 2F,rG~NN(q 2) ,~ ~, (N(p') I .4#(0) I N(p)) = f~(p')[gA(q2)%% -1-,q# 2 2 %j~u(p), rn~ -- q

Z

(153)

where, by definition, G,NN(m~) ~ O. It follows from Eq. (153) that the matrix element of the divergence of the axial-vector current has the form:

_2F~G,cNN(q2)~-, ,. . . . . (N(p') I cO~.4#(O) I N(p)) = --[MgA(q ~) + ,1 -~--_ ~ .]utp p%rutp),

(154)

where M is the nucleon mass. In the chiral limit 0~A# = 0 and we deduce that

MgA(q 2) = F,G,NN(q2),

(155)

for

all q2. If an operator f~ connects the vacuum to a one particle state, say the pion, and we know the matrix element (0 I f/I g), then it is possible to claim that all matrix elements (a I N I b) have poles

~t (po - p~)2 = ,n2. g*ab

(a I N I b) = (0 [ n I ~ ) ( p . _ p b ) 2 _ m~ + ' " '

(156)

114

M.K. Banerjee

where g,~b is the relevant coupling constant on the pion mass shell. Using this fact and Eq. (151) we can state that as q2 ~ m~2

. 2F~G~NN(q2)I ,~, 2 glrNN [MgA(q2)-l-q ~ _ ~ - f l----,--r~m~m~_q2,

(157)

where g~NN is the pion nucleon coupling constant on the pion mass shell. This is the quantity which appears in dispersion relations for the lrN scattering amplitudes. The last fact makes it possible to extract it, in principle, from experimental data (H6hler 1990, Arndt et al., 1990). The quantity g~NN also determines the one-pion exchange N N potential at a large distance. Thus one can extract g~NN, in principle, from careful analysis of large partial waves N N phaseshifts (Bergervoet et al., 1990) as well. From Eq. (157) we find that G,NN(m:) = 9*NN. (158) The equations written down so far contain no dynamics. The dynamical content underlying the famous Goldberger-Treiman (1958) relation is as follows:

One conjectures that the matriz elements of the divergence of the isovector azial-vector current has the form (m~ - q~)-I × a slowly varying function of q2. That is, for I qZ I " m~2

or les8

(m: - q2)MgA(q2) + q2F~G~NN(q2 ) ~_ constant.

(159)

Equating the value of left hand side for q2 = 0 to that for q2 = m~ we obtain the GoldbergerTreiman relation: MgA(O) "~ F~g, NIV. (160) The conjecture described by Eq. (159) is quite reasonable. The left hand side of Eq. (159) is the coefficient of f~(p')iTs~u(p ) in the quantity (m~ - q2)(N(f) I 0uA~(0) I N(p)). The matrix element has singularities whenever q2 equals a possible mass of a state having the quantum number of the pion. The pion pole is killed by the multiplying factor (m~ - q2). So the first singularity is at 9ra~. One hopes that the absorptive part becomes large when q2 reaches the mass of ~r(1300). Thus one expects a ,,, 1% variation of the left hand side of Eq. (159) as qZ changes from the value 0 to m~. Currently the accepted values (Review of Particle Properties, 1992) of the quantities are as follows M = 2(Mn + Mp) = 938.9MeV, 1

gA(0) = 1.257, F~ = 92.5 MeV. Thus the left hand side of Eq. (160) is llSOMeV. According to Koch and Pietarinen (1980) g~lv = 13.39, which makes g~NNF, = 1239 M e V yielding a mismatch of ,~ 5%. It is hard to understand this large variation and the situation is often referred to as the Goldberger-Treiman anomaly. Recently groups at VPI (Arndt et al., 1990) and at Nijmegen (Bergervoet et al., 1990) have reported a value of g~Njv as low as 13.05. With this value g~NNF~ = 1207MeV and the discrepancy is a tolerable ,,~ 2%. It should be understood clearly that, at present, we do not have a theoretical value for the discrepancy. To obtain it one must calculate in QCD the quantity gOCO(q2) defined by the relation:

(N(p') I mq¢(O)iTse¢(O)lN(p)) = F,m~gOvN~(q2) f~(p')iTsr~u(p). rn; -- qo

(161)

Chiral Confining Model

115

It follows from Eq. (156) that 2 = g.NN. (162) g,O c D (m,) We should also note that G,NN(q 2) # gO~(q2). This observation should alert us to the fact that the object pion.nucleon form factor is different for different formulation. By definition, they are all

equal to g,NN at q2 = m~. The use of /~(z)

=

~,(=),

(163)

as an interpolating pion field (Adler and Dashen 1968) is well known. The soft pion approaches and many low energy theorems are based on it. The Gell-Mann and I~vy (1960) a model lagrangians not only realized current algebra but also had the feature that the interpolating field was proportional to the canonical pion field:

F1 2 0 , ~ " mw

=

¢.

(164)

In the literature this relation is often referred to as the PCAC relation. We alert the reader once again that the pion nucleon form factor of this theory is not going to be the same as either G~NN(q2) QCD 2 o r g ~ N N ( q ).

As we have already seen in the beginning of the section the PCAC relation is modified in our lagrangian: 0~A. -- -raq¢i75¥¢ - F , m ~ . (165) Recall that the quark fields in our effective lagrangian must be treated at the tree level only. Hence it does not contribute in the vacuum to one pion matrix element and Eq. (151) holds for our lagrangian. As pointed out by Birse and McGovem (1992) the Goldberger-Treiman relation takes a different form. Let us introduce the unusual form factor ~wNN by the equation _ 2 C w N N ( q 2) - , , . . . . . (Nip') I mq¢(0)iTs~¢(0) + F . r n ~ ( 0 ) I Nip)) -- F.¢',,. ~ - ' 2 " ~ u~p )r7sru(p). m= -

q

(166)

Then slow variation conjecture suggests that ~ , N N ( O ) e~__~wNN(WI~ ) = ~wNN"

(167)

From the fact that Eq. (149) is a consequence of our model lagranglan we must satisfy the relation: MgA(O) = ~wNN(0). (168) This should be verified as a check on our numerical procedure and approximations. In the works of Birse and Banerjee (1984, 1985), Broniowski and Banerjee (1985, 1986) and Ren (1991) the current quark mass terms was not included in the lagrangian and, instead of the equation above one had MgA(O) = g~NN(O). This was used to check the validity of the approximate projection method. The relation was also used in Ren and Banerjee (1991) to exhibit the improvement in the projection method upon the inclusion of quantum effects. In Sections 9.2 and 10 we present numerical results related to Eqs. (167) and (168) 7.5

The

Sigma

Commutator

With the help of the interpolating pion field/~(z) we can define a x - N forward scattering amplitude 2

2

Fo,~(q) = s"'qtl~-If1"tt ~ m'c'2 j / d4xei"=(N(O),is ] T(Do(x),Do(O)) I N(~),i3), lr

(169)

116

M.K. Banerjee

2 W h e n q0 "* where the pion four momentum is not necessarily on the mass shell i.e., q2 ~ m~. +V/-~'~ + ~ the amplitude corresponds to the physical forward scattering amplitude (with positive imaginary part) for a pion of momentum ~ and a nucleon of isospin ia at rest. The soft pion limit of the amplitude is obtained by first setting ~ = 0 and then letting q0 ~ 0. The result is:

1

1

F~,o(q) = ---ff~a,N( O)5o,O + -f~iqoeo,o,si3,

,,,,N(o)6o,o = i f d'x~(xo)(N(6)i3 I [A°~(z), D0(0)] I N(O)iz).

(170/ (171)

The quantity a~N(0), termed the pion-nucleon sigma-commutator, was introduced by Weinberg (1966). If the chiral invariance was exact D o = 0 and the sigma commutator would vanish. When the invariance is broken by an interaction lagrangian which belongs to the [1/2,1/2] representation of SUR(2) x SUL(2), which is the case for QCD and CCM, it is possible to show that O'rN(0)

=

(N(O)is [ 7"/xts [ N(6)is).

(172)

Using Eq. (1451 we find that for the CCM

a,N(O) = (N(6) [ m ¢ ¢ + F,m~(a + F~) ] Y(6))

(1731

For zero momentum transfer the Eq. (1481 is exact and we can write

#~N(0) = ( N ( 6 ) I - ~ ¢

41 z -2 + 3,, a} - 2m~ + g'~F'~m~d, ~ ~ _ ~m~{,~ F # ,(-2 ,~ + a,2) [ Y(O))

(174)

We will show the value of a~N(0) and the various contributors obtained with the CCM in Section 9.2. 7.6

The

Vector

Mesons

In principle, the vector mesons, p, A1 and w, can play important roles in the structure of N and A. This is, indeed, what was found in the nonconfining model of Broniowski and Banerjee (1985, 19861. For this reason and also for completeness we give here the lagrangian including these three vector mesons. The treatment of the vector mesons in a mean field theory has been discussed in detail in Broniowski and Banerjee (1985, 1986) and Ren (1991). We shall not repeat the material here. The lagrangian is constructed by taking the Lee and Nieh (19681 lagrangian and making two modifications. We modify the quark meson interaction by including the factor of K -2 and we add £× which describes the dynamics of the X field. The w meson is a chiral singlet. It appears in the lagrangian just like a massive photon which is coupled not to the conserved fermion current but to ¢'Tu¢/K 2. The combinations (p~ 4- A ~ ) transform like (1, 0) and (0, 1) representations of chiral SUR(2) x SUL(2). This is also the way the combination of vector and axial-vector currents, V~ 5= A~ behave. Therefore a chirally invariant lagrangian may be constructed by coupling the p meson to the vector current and the A1 meson to the axial-vector current. Lee and Nieh adopted the strategy of viewing the chiral symmetry as a local gauge symmetry with the p and the A1 as the gauge bosons. Thus only covariant derivatives appear in the lagrangian. The local symmetry is eventually reduced to a global symmetry by adding the vector meson mass terms. Finally, the quark-meson coupling terms are modified by the K -2 factor. Note that this modification does not affect the global SUR(2) x SUL(2) symmetry.

117

Chiral Confining Model The resulting lagrangian is

+

fD"a +D . ~ . D"~) - UM.,~,,,(~r,~) -- F,rra~(cr + F,~) 1 .

~u

4~

+

ra2

~"" + lra2~%'2

(m)

£x.

The modified Mexican hat potential is i ~ra~

Ub.,=,,(,.,,r) = i

-

=~

ra~. 2

ra,4 F,2~

2 ~,2~ral 2 - 3rail2

l" + ~2__

prrl,~,

-- try,

076)

The covariant derivatives are as follows: D/jo"

-~-

D.~

=

(17~)

o : + g,(:~ × ¢ - X~:).

The covariant field tensors are

g,. = o,,,a.-&¢.+g.(5, x : . + ~ , . X ~ l . ) , .~1.. = o,,~,, - ,9,X,,,, + g,(5,, x ~1, + ~1,, x :v), w,., = O,,w~, - (9,,~,.

(178)

An examination of the second of the Eqs. (177) shows that becauseof the symmetrybreaking of the vacuum, i.e., (0 [ ~ I O) = - F , , the A, and ~r fields get mixed. As a result the propagator ifd4xeiq'=(0 [ T(A](x),A1(O)) ] 0) as a function of q2 has poles at r%2 and ra~, where raA is the physical mass of the At meson. The latter is not equal to rap. The combination

X.h,. = A1. +

BgpF. rT~:~ ( O : + g . : . × ~), mp+~gpF 2,

(179)

produces a propagator free of pion pole. The procedure also alters the position of the pion pole in the pion propagator. To ensure that the pole occurs at ra,2 one must have

=

2

rap

(180)

- g,F~

Then the mass of the physical Al meson turns out to be

raA = vf~rap,

(181)

which is precisely the Weinherg (1967) sum rule result. The vector and the axial-vector Noether's currents are

~,

=

,p%~,,b + p" x ,~,.. + A~' x A,,,~, +/:i'~ × D.~,

(182)

A.

=

!b'ls'l.-~b +~ x

(183)

r~

The PCAC relation continues to be given by Eq. (165).

O.o~rD.~).

118

8

M.K. Baner.jee

Calculation

in Mean

Field

Theory

In this Section we present briefly the method of calculation of baryon structure and properties in the mean field approximation. For more details readers should consult Birse (1990), Broniowski and Banerjee (1986, Banerjee et al., (1987), Ren (1991), Ren and Banerjee (1991).

8.1

Mean

Field

Approximation

In the mean field approximation the quarks are the only quantum particles. The bosons appear as classical, i.e., time-independent, fields. These classical fields are expectation values of quantum field operators in a modified vacuua. In the present context, we must recognize three kinds of vacuua, in principle. i. Perturbative vacuum - Expectation value of every field operator is zero in this vacuum. 2. Physical vacuum - When distinct from the perturbative vacuum, the expectation value of, at least, one boson field operator is nonzero. In QCD, composite operators, such as, ~ b and GuvGuv, which are normal ordered with respect to the perturbative vacuum, have non-zero expectation values in the physical vacuum. 3. Modified vacuum - A source modifies the physical vacuum such that the expectation value of at least one of the boson field operators is changed. Any one of these vacuua may be written as a coherent state built on another vacuum. This is discussed in Birse (1990). We will not follow this strategy. We will discuss the mean field approximation in terms of the lagrangian £CCM, given by Eqs. (143) and (144), and repeated below:

Zcou = +

glr

~[i#- ,-,,, + ~--~(,, + i,v,¢. ,v)],~ I,

1 ,

~a ,.a.,, + ~a ,va,,~ - u~,o=,~,.(,., ,v) 1 ,.

+ 50 xOt.x- U(x),

(184)

where UM,,.,I,,,,,, is defined by Eq. (140) and U(X) is defined by Eq. (77) and K by Eq. (60). The corresponding hamiltonian density is g=

7~CCM = ~t[--ia. # +/~m~ -- ~-~/~(~ + i75~" e)]~

+ ~[a~ + (#~)~ + #~ + (#~)~1 + U~o.,0..(., g)

+ ½[~ + (X~x)~1+ V(x).

(18~)

By definition, the expectation value of all fields in the perturbative vacuum is zero. In the physical vacuum, which we indicate with I 0), the expectation values of the boson fields are as follows:

(01,' l 0) = -F,,, (01~10) = O, (0 IX I0) = O.

(186)

Chiral Confining Model

119

In the mean field approximation one takes the view that in the presence of a baryon, which serves as a source of boson fields, the physical vacuum is statically polarized into a new vacuum. The baryon, in the same approximation, is made up of three quarks of different colors occupying the same spinor state u(~). This spinor must be of the positive energy type, i.e., f d~zut(~)(-i6 • V)u(:F) > 0.

(187)

Let us introduce the quark creation operator bt.= = / ~zct(z)u(.'~)e -'0'='"."=',

(188)

where a = 1,2,3 is the color label. Both the spinor and the associated eigenvalue, e~=t=~, axe to be determined eventually in a self-consistent manner. The baryon state is written as

I B> = t,Ld,,,b~,,,,,,,,b~,=,,, 10)>,

(189)

where [ 0)) is the modified vacuum. The expectation values of the boson fields in the state I B) are, in principle, different from the corresponding values in the physical vacuum.

(B I ~r(=) [ B) = ao(~.), (B I ~(=) I B) = ~(~), (BIx(=)IB) = Xo(.~).

(190)

where the subscript c indicates that the right hand sides are time-independent classical fields. Of course, here the time-independence follows from the assumption that I B) is an eigenstate of the hamiltonian, HccM = f d3x?'lCCM. Let us decompose each boson field into a sum of a classical field, introduced in Eq. (190), and a quantum field: .(=)

=

.o(~)+~(=),

~=(=)

=

~o,o(~)+~(=),

x(=) = xo(~)+~(=),

(191)

where the hatted symbols represent the quantum fields. Next we expand the hamiltonian density, 7"/CCM,in a power series of the quantum fields arranging the terms in a manner appropriate for the introduction of the mean field approximation. It is convenient to introduce the notations ~b0 =

O",

S%=/~

¢~ = ri fli = if175ri.

With these notations we write

"Hccu = c t [ _ i ~ . ~ + / ~ m q _

gxXc)

'

+ 21(~¢,=)~ +uM.=,.=,(~= o) + ~(~×o)~ + U(xo) + [-(B I

~¢*n~¢ I B) - V'ck,~,, + ouu=.~=.]~,=

+ [(B I C t ~ ° , f l ° ¢ gxXc '

I B) - V2x, + dUx'" "~Xelx

(192) (193)

M. K. Banerjee

120 + +

1 2

1 a2Uu~..~ 2 O,/,o,o0~a,o

1 = 1.d~U,, I,/,tg,,/,o,or~°¢ I B)}.~= ~[~,, + (~.t)'] + ~"F~-X' + 6(z (Z,,Xo)'X~

+ +

[~*~¢'°'°f~°'/'-(B

I~t~*"°f~°e B)lX.q~xo ' I

+

(194)

The quantities =

8£ccm . ,

(195)

'~" =

a£CCM 0.~ '

(196)

:~o

are the variables canonically conjugate to the fields, ~o and ~. The first line of Eq. (194) describes the motion of a quark in a background potential. The second line represents the energy stored in the classical fields. The next two lines are one quantum terms. We will see later that, upon proper choice of the spinor and the classical fields, the one-quantum terms vanish. When evaluated with classical fields present in the physical vacuum, as specified by Eq. (186), the term ½0~UM,~i~,/O~o,cO~ba.c gives the meson mass matrix. The change in its value due to the classical fields of Eq. (190) is the background potential seen by the mesons. The fifth describes the motion of the meson quanta in this background potential. It should be noted that the background field may mix flavors. The sixth term describes the motion of the hybrid quantum in an anaiogous background potential. All three background fields will turn out to be due to the valence quarks. The next two terms mix the bosons with valence quark excitations. The ellipses hide all bosonboson interaction terms. They also contain terms involving quark bilinears and two or more boson quantum fields. In the mean field approximation the terms appearing in the last four lines are ignored. Thus we have a system of noninteracting quanta and a set of classical fields. In this approximation the boson field commute with the quark fields at all times. Then Eqs. (190) and (191) lead to the conditions:

((01a(=)10))

=

0

= o ((o1.~(=)1o)) = o

((O1~o(=)1o))

(197)

Let us denote the annihilation operators for the quarks and antiquarks with the symbols b's and d's, those for the mesons with a's and those for the hybrids with c's. Then the noninteracting mean field hamiltonian may be written in the diagonal form:

H,,.,.f = /d3x[2(V~bo)2 + UM~i,~.(~o,c)+ ~(X~) + U(X~)]

+

-

+E

k

,

~+~,,~+,

1

zo.

(19s)

Chiral Confining Model

121

The three terms in the last linerepresentsone-quark loop, one-meson loop and one-hybrid loop contributions. Following the point of view developed in Section 7.2 and the discussions at the end of the section we drop all loop terms. The definitionof the modified vacuum I 0)) is sharpened by requiring it to be the vacuum for the quanta introduced above. bA

10)) = d, l0)) =

ak 10))

=

% l0)) =

(199)

0.

We have supplied a large number of definitions which are all tied to the definition of the classical fields and the valence spinor u(~). These are determined by requiring that the energy functional: E ( u , ¢ .... X¢) =

]dar[Ncut(r-*){ - i a ' V +rn,

+

(g×~),,¢.,¢(r-')ft,~}u(~].._.

+

+

I

-

2

(Vxo) +

(200)

be stationary with respect to variations of the classical fields and the spinor, subject to the normalization constraint. The resulting equations are as follows:

{-i~. V -4-/3mq - (gx:~r_.,))2 ¢~,.c(~f~,}u(r-') -- e,,,a.... u(~ -v2¢"'c + OUM.xic.. = N c ~ u t ( r - ) f ~ . u ( r . )

-V2X~ + dUx dx ~ = - N c u t ( ~ ¢ . , ¢ ~ t . u ( r -')

(201) (202) (203)

Examination of Eqs. (202) and (203) show that when these are satisfied the coefficients of the one-quantum terms in Eq. (194) vanish. This is to be expected. Because making (B [ H,,/ ] B) stationary means that small admixtures of states like a~ [ B) or c~ ] B) will not change the energy in first order. If creation of a boson is accompanied by excitation of the quark from the valence state to a higher state then one can have a first order change in the energy. The mean field approximation does not produce a ground state stable against this kind of excitation. The only way to produce stability against this class of excitation is to include RPA correlation in the ground state. From now on we will drop the subscript c with the scalar fields. The context will always make clear whether we are referring to the full field or only its classical part.

8.2

Hedgehog

Ansatz

It has been known for a long time that the Eqs. (201), (202) and (203) can be solved using the hedgehog ansatz which was used in quark models firstby Chodos and Thorn (1975). For a fairly complete discussion of the use of the hedgehog ansatz see Broniowski and Banerjee (1986) and Banerjee et al., (1987). Here we present only the minimum that we need for the present. The hedgehog ansatz consistsof writing the spinor in the form:

u(r-') =

i~. ~F(r) ¢'

(204)

where the Pauli spinor ~ is given by the equation

¢=

(u

- d t),

(205)

122

M.K. Banerjee

with u and d indicating up and down flavor states. A set of results with the hedgehog spinor, essential for derivation of various results, is provided in Section 14.1 of the Appendix. The classical fields are written in the forms (remember that the subscript c has been dropped):

~(~

=

Ir=(~ = x(~

=

~(r), ~ r ( r ) ~ = a = l , 2,3,

(208)

x(r).

With the help of the results listed in Section 14.1 of the Appendix the stationarity equations (201), (202) and (203) can be translated into the following equations for G and F , the spinor radial functions and the boson fields a, r and X:

dG d-"~" =

1 1 - ..(gxx)zg"rG + (~-~g~a._^... - mq - e~=t.... )F,

d---r = (

g . a - mq + e . . t . . ~ ) a + (

d2x 2 dx dr 2 + r dr

)F.

dU× 2Nc9, d X = g2xx3 {(G2 - F ' ) a + 2 G F r } .

aaadr2 + 2rdadr OUM.~i.=.Oa d2r 2 dTr dr 2 -t r dr

g.r -

F.m~

=

2 OUMc~ic.. -r2- ~r -Orr

=

-N.(g~g.(G _N lg~2GF.

(207) (208)

2 - F2),

(209)

It is not difficult to see that the following asymptotic boundary conditions are self-consistent:

G(r), f ( r )

~

~ exp[- jU . ,~r (g~x(r,)), g.f. ,j,

(210)

x(r)

~

1 ~ - e -'~xr,

(211)

r

a(r) + F.

~

1 ~ -e .... ,

(212)

r

7r(r) ~

1

~-e .... . r

(213)

From Eqs. (210) and (211) we see that

G(r), F(r) --*~

exp(-~r2e2m~'),

(214)

where ~ is a dimensionful constant• This behavior assures us of two results• First, the fourier transform, f dare-lg'¢u(~ is an entire function of ~ which is a necessary property of an absolutely .~fht(r~ confined wavefunction. Second, the integral -ae a -~ar - u-t~( ix(r)" - - - ~ ' where fl is a Dirac matrix, is bounded for all n. With the help of the COLSYS (Ascher et al., 1981) software and nontrivial amount of skill and patience these boundary value equations can be solved, though not for all values of the various parameters. In Section 3.6 we derived the result that all quark-meson coupling constants, which includes g~, ,,~ Ng -~/2. Using Eqs. (209) we then obtain the important result that the classical fields: ~, ~ ~ N2/~.

(215)

• We have already seen from Eq. (94) that X ~ N~ 12. Using these results the reader can check that every term of the energy functional given by Eq. (200) ~ No, which ensures that the prediced baryon mass ,,~ Nc, while the baryon size ,,~ N °.

Chiral Confining Model

8.3

123

Virial Theorems

It is very useful to be aware of certain virial theorems which are obtained by exploiting the fact the energy functional is stationary with respect to distance scale and scales of the boson fields. The theorems are stated most conveniently in terms of the following quantities: '

Vq= Nc/ darutCr')(

(217)

g"

The two potentials, UMe=ic=. and Ux are both, at the most, quadratic functions of the relevant field variables. Let us express each as a sum:

uMo,,o=

vM~.i~.. Ux = ~ , U(~),

=

n

(218)

n

where each summand is a homogeneous polynomial of the boson fields of order n with n running from 1 to 4. Note that for a pure mass form of U× n = 2 only. Finally, we introduce

V.(") = / d~rU~.,~=~, Vx(")= / ~rU (").

(219)

From invariance with respect to distance scale we obtain the virial theorem:

- Tq + Tm . . . . + T× + 3 E ( V ~ ! o . + V(")) = 0.

(220)

tt

From invariance with respect to the scale of the meson fields we obtain the virlal theorem: Vq + 2Tm~,o~+ ~-"nV.meson (~) = 0.

(221)

tt

Finally, from inwriance with respect to the scale of the X field we obtain the virial theorem: - 2Vq + 2T× + E nV(x~) -- 0.

(222)

n

These equations are useful in checking the numerical work. If the coupled nonlinear equations are solved iteratively these equations are invaluable as checks of convergence of the procedure. For the special case of Ux = ]mxX 1 2, the pure mass form, the last virial theorem is particularly useful in giving us a very important result. In this case n = 2 only and this means that Vq = T× + Vxt2) > 0.

(223)

By the prescription specified by Eq. (187) we have Tq > 0. The valence spinor eigenvaiue is given by 1

Evalence --'- -~c(Tq Jr Vq) Jr Tnq J

d3r(G 2 - F2).

(224)

The quantity 3LT 1% q is typically ,,, 200 M e V . The last term of Eq. (224) has the lower bound of -m~, with mq ,~ 8 M e V . The positivity of Vq guarantees that

e~=t~.~ >

O.

(225)

From the nature of the discussion it is obvious that the result will hold when more varieties of mesons, such as the vector mesons, are included. We do not have an analytic proof of Eq. (225) for the case of a quartic potential. However, actual calculations have never yielded e~=t.... < 0 or even less than a few hundred M e V s . As mentioned in Section 7.2 this result allows us to conclude that we must retain the quark fields in the effective lagrangian, while taking care to use it only at the tree level.

M. K. Banerjee

124 1 . 0

•

i

,

i

•

i

,

i

•

i

'

i

•

i

'

[

•

i

'

i

'

0.9

O.l

0.8

0

0.7 0.6 0.5 0.4 0.3 0.2 0.I O0

0.2

G

-0.1 -0.2 -0.3 -0.4 -0.5 {7

-0.6 0.4

0.8

1.2 r(fm)

1.6

2.0

-0.7

,

i .

i .

i ,

0.4

i .

0.8

i .

i ,

i ,

1.2 r(fm)

i

.

1.6

I

,

q

.

2.0

(b) Figure 14: Results Plots of G, F and X obtained in the CCM (without vector mesons). Plots of G, F and X are in (a) and plots of a and ~r are in (b). All fields are in pion mass unit. The fixed line in (b) at ,,, - 0 . 6 7 m , represents - F ~ .

9 9.1

Chiral

Confining

Model

Results

Chiral Confining M o d e l w i t h a and ~ F i e l d s

In this section we present the main results of the chiral confining model with a representative set of parameters. The purpose is to appreciate the qualitative nature of the results. The calculations were done with the quartic potential, defined by Eq. (77). The values of the parameters used axe mq=7.bMeV

g~ =

ma = 1200 M e V , m x = 1400MeV, Xo =

4, g × = ( 3 0 M e V ) -1. m,~ = 139.6 MeV.

30MeV, V(x0) = 0.

(226)

Notice that in the quartic U(X) the height of second minimum was set equal to zero. In terms of the parameters appearing in Eq. (77) this means The = me. Using these parameters one solves the Eqs. (207), (208) and (209), subject to the boundary conditions specified by Eqs. (210), (211), (212) and (213). The results are exhibited in Fig. 14. The region in Fig. 14(a), where x(r) is nonzero, is the dynamically generated bag. The figure also shows the rapid drop of G and F consistent with the boundary condition specified by Eq. (210). In this example

fG 2

] G 2 = 0.886, ] F 2 = 0.114, ~

where f G 2 - f d3rG ~, etc. In the MIT bag model the ratio ~

-- 0.129,

(227)

= 0.25. The ratio of weights, F~

J

of a free spin 1/2 particle is 0.129 for v / c = 0.64 and 0.25 for v / c = 0.8, respectively. So we see that the quark motion is quite relativistic in both the MIT bag and the chiral confining model. The meson fields are shown in Fig. 14(b). The natural scale for these fields is F~ = 93 M e V = 0.666m~. The a field asymptotically goes to its vacuum value of - F ~ . The fields in Fig. 14(b) differ from their vacuum values by rather small amounts. These are also radically different from the fields obtained in the nonconfining model of Birse and Banerjee (1984, 1985). Both sets of fields

125

Chiral Confining Model

1.0

. . . . 6

.I

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

-~0" %,

0.5

0.0

.""'~-i-"2"~_~ • • os+~S_F~

"~-

~l÷~l-VJ

,~ o + ~

-F w

I ....

I ....

-0.5 -F

w

-1.0

.... 0

I .... 0.25

0.5

I ....

0.75

r (in mesic

1

I .... 1.25

1.5

units)

Figure 15: .The a and 7r fields (solid lines) of the CCM and of the nonconfining model Birse-Banerjee model (dashed lines). All fields are in pion mass unit. See text for details. are shown in Fig. 15. The fields of the nonconfining model are obviously much larger. The most significant difference is that the a field of the cbiral confining model does not change sign while that of the nonconfining model does. Since neither model uses the nonlinear sigma model there is no strict topological stability. Nevertheless, one can introduce the usual chiral angle 0(r) by the equation O(r) = tan -1 lr(r)

~(r)'

(228)

and use the winding number

B = 0(0) - O(0o)

,

(229)

TC

as a description of the feld distribution. From Fig. 15 we see that while B is 1 for the nonconfining model, it is 0 for the CCM. The fields obtained in the nonconfining model are very close to the fields in a Skyrmion. The main difference is that in the latter a 2 + r 2 = F~ always because of the use of the nonlinear a model. One can easily pass on to a Skyrmion like theory from the nonconfining model of Birse and Banerjee (1984, 1985). First, one introduces the vector mesons (Broniowski and Banerjee 1985, 1986) and then use the nonlinear a model for the a -~r sector. The winding number continues to remain equal to 1. The nonlinear a model ensures topological stability. As stated in Section 7.2 the e,t~ce is now negative and the quarks fields can be successfully integrated away in favor of a practical meson theory. In Table 1 we present the values of the various parts of the energy functional defined by Eq. (200) and the sigma commutator. Evaluation of these does not involve angular momentum projection. The quark-a interaction energy, given in the eighth row, is dominated by the vacuum value of a field. This quantity is the dynamical mass in the chiral confining model. The rest of the meson related terms add up to - 6 8 M e V corroborating the fact the meson fields generated by the valence quark themselves play a minor part in the structure of the baryon in the chiral confining model. The reason for the suppression of the meson fields is the presence of the factor 1 / K 2 in the meson quark interaction term. In the results presented here the value K 2 at r = 0 is ,,, 2. Let us

126

M.K. Banerjee Quantity

Value

(MeV)

~valence

Nc f ~rut(r-')(-i~ • ~)u(r-') Ncmq f d3rut( ~ u ( r -') f d3r(VX) 2 f d3rUx f d3r(~a) 2 f d3rUMe=ican

- N, f d3rut (~ K~~au( ~ -No f d%ut(~ K~-~i13%~.~u(~ Sigma Commutator Total Energy

371 640 17 116 129 63 24 576 -119 37.6 1447

Table 1: Decomposition of the energy functional for the chiral confining model with a and r mesons. consider the a field. The quark part of the a source current jo, given in Eq. (147), is proportional to G-a - F 2, the scalar density of the quarks. It is largest at the r = 0, because here G is maximum and F = 0. But this where K is maximum also and it quenches the source sufficiently to prevent a from differing significantly from - F ~ . If the a and the r are given the chance to move away from their vacuum values the meson-meson interactions given by the Mexican hat potential given by Eq. (140) cooperate to move the value of the a field to ~ +F~. This is what happens in the nonconfining model. A curiosity which may be noted is that the quark eigenvalue is nearly a quarter of the total energy. The situation is similar to that in the MIT bag model and is often mistermed as equipartition of energy. The quark rms radius is

(r2) ~q= / darr 2 ut(r-')u(r-') = 0.93 fm,

(230)

giving for the dimensionless quantity

E(r2) ~q= 6.86.

(231)

The tenth row gives the value of the pion nucleon sigma commutator, cr,N(O). Osypowski (1970) and Cheng and Dashen (1971) pointed out that the a,N(O) can be extracted from on-mass-shell rN scattering amplitude at the unphysical values, s = u, t = 2m~, provided corrections involving higher powers of the pion mass are neglected. The unphysical amplitude can be evaluated with the help of the dispersion relations and the most reliable value is due to HShler and his collaborators (HShler 1990). Gasser et al (1991) discovered that the correction due to the difference a,N(m~) -- a,N(O), which is in higher powers of the pion mass, is quite large. Combining these corrections with the basic result of HShler (1990) they give o'TrN(0) = 45 -4- 10 MeV.

(232)

This is currently the generally accepted value of the a~N(0). A dynamical theory of the low energy r N interaction was developed by myself and J. B. Cammarata (1977, 1978, 1979) based on the Low (Chew and Low 1956) expansion of the scattering amplitude and using the soft pion limit as a subtraction point. Thus, in this approach the a~N(0) is

Chiral Confining Model

127

Value (MeV)

Matrix Element

17.4 8.4

(N(~) I ~,,.,,., s_2_~ I N(I~))

(N(~) I ~__2z,,.~_-~I N(6)) (N(~) I -~,,~'(~ + o~) I N(6)) Total a,N(0)

1.5 10.9 -0.4

37.6

Table 2: Various contributors to a,N(O). the input and low energy S-wave phaseshifts are the outputs. Fitting the phaseshifts (Carter et al., 1973), then available, we obtained the very low value of 28 M e V for ~/v(0). Subsequently, using the Karlsruhe-Helsinki phaseshifts I obtained the slightly higher value of 30 M e V (Banerjee 1981). The breakdown of the value of the er,N(0) into its parts, as listed in Eq. (174) is shown in Table 2. It may be noted that ,-~ 70% of the contribution comes from the two one-quark terms in Eq. (174) and ~ 30% comes from the pion cloud which is a two-quark term.

9.2

Nucleon Properties

The baryon state obtained in the hedgehog ansatz is not an eigenstate of J-~ or I"~. The hedgehog state has its own symmetry group. It is a SU(2) group whose generators, /~ -- J + I, are called the grand spin. The lowest energy stationary solution obtained with the hedgehog ansatz belongs to the K -- 0 representation of the grand spin. It contains states with I = J only. So it is sumcient to project out either I or J. If we could treat the hedgehog baryon as a fixed particle number problem with only three quarks and no possibility of creating quantum mesons or quark-antiquark pair the task of projecting I or J would involve triviai Clebsch-Gordan algebra. With the help of the explicit expressions for the states of N and A provided in Eq. (343) of the Appendix we could then expand the hedgehog baryon in the following manner:

I--~,~

I B)-- ~~,~ (-)J-'~

¢2(2~--~1) II = J, is = -./s/.

(233)

Because of the nonspherical and nonisosymmetric nature of the hedgehog fields the quantum states of quarks and mesons do not have good angular momentum and isospin. As a result the the state J [ 01) contains quantum particles:

Yl 0)) --I ~em~so~)+ I twom~so~>+ I q{).

(234)

These states play a role in the construction of states of good I = J. Unlike in Eq. (233) the state ] B) contains many I = J states. This becomes possible because of the role of the quantum pions. Thus the quantum pions make the task of projection complicated. With the approach that has been described in Section 8.1 the state ] B) is a quantum mechanical state.

I B) = ~ c(I, ia) lB, I = J, i3 = -is), l,la

(235)

128

M.K. Banerjee

where c(l, i3) are the expansion coefficients. One can legitimately project out I or d. There are two principal approaches to the problem. In one the modified vacuum is described as a coherent state of plane wave boson quanta. Then the projection is carried out exactly. An excellent review of the method is given in Birse (1990). A new and faster method of projecting good angular momentum state in the coherent state approach has been developed by Rosina and his collaborators (~ibej et al., 1992). There is another approach to angular momentum projection which is approximate, but very easy to carry out. A full description of this approach is given in Ren and Banerjee (1991). There is yet a third method which is based on regarding the result of the mean field calculation not as a quantum mechanical state but a quantum lump. States of good J = I are generated by cranking. A comprehensive discussion of this method may be found in Cohen and Broniowski (1986) and Broniowski and Cohen (1986). The cranking method, sometimes called semiclassical projection method, should be good when the [ B) behaves like a classical rigid body. In a classical rigid body the variables describing internal motion can be separated from the collective variables describing the motion of the body as a whole. These latter variables are position of the center of mass (or any other fixed point in the body), orientation angles of the body fixed principal coordinate system and the associated linear and angular momenta. In a purely mesonic theory, like the Skyrmion, one deals not with a quantum mechanical state like I B) but with a classical solution of the Euler-Lagrange equations. Examination of time dependences of the meson fields in this case reveal the existence of zero modes associated precisely with the collective motions of a rigid body. Quantization of these is properly called semiclassical quantization. In the present case, where there are quarks and boson fields, zero modes appear as linear combination of boson fields and bilinears of quark fields (RPA modes). One cannot duplicate the procedure used in the purely mesonic case. We have to look for other indications of separation of 'internal' motion and the motion of the body as a whole. We may regard [ B) as describing the internal motion for fixed values of the collective variables, such as position and orientation of the body. We must be able to regard the two sets of variables, the internal variables and the collective variables, as mutually commuting sets. The state I B) can be regarded as an eigenstate of the collective position variables. A test of the validity of this notion is that should be orthogonal to another I B) obtained by changing at least one of the collective variables. Ideally, for rotation we should have (B I R(Ft) I B) = 6(9/). In practice, it should be, at least, a rapidly decreasing function of the angles of rotation. This situation is easily realized for a macroscopic object with ,~ 1023 constituents. We must examine what the situation is for I B). Without loss of generality we can choose R(Ft) = e id~O. If it is indeed a rapidly decreasing function of 0 it should be possible to write (B [ R(ft) [ B) "" exp(-O2(B [ j2 I B)/6). The quarks contribute only 9/4 to (B I J~ I B) which gives an angular spread of -,, V ~ "" 93°" In the nonconfining model the mesons contribute (Ren and Sanerjee 1991) --, 2 to (B I j2 I B) which reduces the angular spread to ,-, 67 °. The Table 3 shows that in the confining model the mesons contribution to (B I j2 I B) is only 0.14. We conclude that while one may apply the cranking method to the nonconfining model with some reservations, it should not be applied at all to the confining model. We may note in passing that the cranking method is justified in the large Arc limit in all hedgehog models - confining or nonconfining. This becomes apparent when we see from Eq. (337) of the Appendix that (B [ j2 [ B) ,,~ No. These different methods will not be discussed in this review article. The results presented here are based entirely on the method of projection described in Ren and Banerjee (1991). Later in this article we will argue for a strategy which makes angular momentum projection a simple task. In Table 3 we list the values of several properties of the nucleon which require projection. The details of the calculation can be found in Ren and Banerjee (1991). For each quantity two numbers

Chiral Confining Model Quantity (BIf2[B)

(r2)~ < f m 2 > (r2)~ < f m a > pp (nbm) pn(nbm)

gA(O) g, NN(O)~-~ 2 m_.z. g~NN(m~)2M

129

Quark 9/4

Meson 0.14

Total '2.39

Expt.

0.85

0.12

0.96

0.66

0.02 1.76 -1.19 1.38 0.89 0.96

-0.12 0.38 -0.38 0.05 0.16 0.16

-0.09 2.14 -1.57 1.43 1.04 1.12

-0.12 2.79 -1.91 1.26 1.00 1.00

Table 3: Chiral confining model results for some nucleon properties. are listed which add up to the total value in the fourth column. The numbers under 'quark' and 'meson' refer to the division of the relevant operator into its quark and meson parts. If quantum mesons are switched off both numbers will change. The first row is a property of the hedgehog baryon as a whole. The meson contribution of 0.14 to (B I J'~ I B) is an order of magnitude smaller than the meson contribution of 2.08 found in the nonconfining case (Birse and Banerjee 1984, 1985). In general, the mesonic contribution, which is entirely pionic, is small. The magnetic moments and g=NN~2M are the exceptions. In these cases the pionic contribution is ,,, 20%. We recall that the pionic contribution to a~N(O) is ,~ 30%. 9.3

N-A

Mass

Splitting

The only reliable way to calculate the N-A mass splitting in a hedgehog model is to carry out a full projection calculation (Birse 1990, (~ibej et al., , 1992). Such a calculation has not been done, even though the popular methods are particularly appropriate for the CCM situation where the quantum pions may be regarded as free with greater justification than in any other situation. An alternative approach is to regard the hedgehog as a stable, deformed object in the grand-spin space conjecture that the spectrum of the I = J members is rotational, i.e.,

Mz=j = Mo + J(J + 1)/22",

(236)

where :T is the moment of inertia. Inglis's (1954, 1955) method of calculating the moment of inertia in nonrelativistic nuclear physics has been extended to the present models by Cohen and Banerjee (1986). The expression for the moment of inertia is as follows:

Z = Zq + Z,~, l,

(237)

= 1No ] d°rutT3 h _ 1

T3U,

(238)

~valence :r,

=

2

f d°rTr2(r).

(239)

Here u, the hedgehog spinor defined by Eq. (204), is a solution of the eigenvalue equation (201). The Dirac hamiltonian h can be read off Eq. (201). The values of the moment of inertia in the BirseBanerjee (Cohen and Broniowski 1986) model and the CCM with quartic U(X) (Kim and Banerjee 1993) are listed in Table 4. The reduction in the value of 2"~ in going from the nonconfming to the confining model is easily understood in terms of the reduction of the size of the classical pion field. The radical increase in the value of 2"q is also due to the same reason. If there were no pion field

M.K. Banerjcc

130 Model Birse-Banerjee CCM

Zq (frn) 2". (fro) Z (fro) Ma - MN 0.44 0.72 1.16 255 (MeV) 4.77 0.13 4.90 60 (MeV)

Table 4: Moments of inertia for the nonconfining model of Birse and Banerjee (1984, 1985) and the CCM. The last column gives ( Ma - MN)rotatio,~at = 3_ 2Z" at all then the spinor u with grandspin 0 and the spinor rau with grandspin 1 would be degenerate eigenspinors of h and Eq. (238) would yield Zq = or. The small but non-zero pion field removes the degeneracy between the grandspin 0 valence state and the lowest grandspin 1 state. Specifically, the latter eigenvalue q = 432 MeV, while e~t .... = 371 MeV. The spinor r3u is no longer the grandspin 1 eigenspinor. But the overlap between the two is still close to 1. It is then easy to see from Eq. (238) that Ma - MN "~ q -- evaJe,~c~~-- 60 MeV. (240) The rotational energy content, (B I j2 I B)/2~, of the hedgehog solution is 368 M e V for the Birse-Banerjee model but only 48 M e V for the CCM. This result and the small value of (B I j2 I B) appear to suggest that it may not be a good idea to regard the CCM solution as a 'deformed' object giving rise to a rotational band. In view of the sharply reduced role of the meson, in general, and of the pion in particular, a more credible approach is to calculate the effects of the mesons perturbatively. Only the pion contributes to the N-A mass splitting mass. Here we present an estimate based on the method of Williams and Dodd (1988). The reader may also find the recent paper of Dodd and Driscoll (1993) useful. The one-pion exchange potential suitable for use (at the tree level) in configurations made up of the hedgehog spinor is .

4

vlj = -~-2",,~i. ~j~/. "~.

(241)

where

~offg2 Z, = ~ ] darl l dar2ff,(rl)Gl(ira~;rl,r2)ff~(r2), J~(r)Gl(im,;rl,r=)

G(r)F(r) K.(,)

'

(242)

(243)

2 r ~. j , ( q n ) j , ( q , , )

= 7 j q aq -~--~m2~

= m~jl(im,r<)h1(im~r>),

(244)

where r< = rain(r1, r2) and r> = max(r,, r2). In a spin-flavor symmetric state of Arc quarks, all occupying the same spinor state, the matrix element of the one-pion exchange potential is given by the expression

(Nc, I = J I ~-~vii I Nc, I = J} = -Z,[2Ne(Ne + 1) - l_~j(j + 1)]. i>j

(245)

This yields the result

(Mz~ - MN)~,,t -----162",.

(246)

The quark-pion interaction energy, listed in Table 1, may also be expressed in terms of 2",:

No / ~ruf(r-')-~.2i/375"Y. ~u(~ -- -(2N~)22",.

(247)

Chiral Confining Model

131

Using the value of -119 M e V listed in the table we get 2"~ = 3.3 MeV,

(248)

( M a - MN)pert = 53 MeV.

(249)

which, in turn, yields Both the rotational band and the perturbative approar~es give consistently small value for the N-A mass splitting. The natural conclusion is that, as it stands, CCM cannot account for the N-A mass splitting. Needless to say, the difficulty persists in the Toy model version of the CCM which is described in Section 10. In Section 3.3 we propose that the difficulties with the N - A mass splitting and the large value of M(r2)~, discussed in Section 9 can be resolved at the same time with the introduction of the 't Hooft interaction.

9.4

Goldberger Treiman Relation

Let us recall the PCAC relation for the CCM from Section 7.4. 0~',4~, = -m~biTsF~b - F,~m~.

(250)

Let us introduce the quantity % by the following equation:

(N(Pt) I m~(0)iTs~b(0) I N(p)) = %ft(f)iTs~u(p).

(251)

• Of course, the matrix element of the ~r field is defined in the usual manner: g~N/v(q2)

(N(p') ] ~(0) I N(p)) = m~ - q2 fi(f)i7s~u(p).

(252)

With the help of these two equations and Eq. (154) from Section 7.4 we obtain for the limit 1/--* p

MgA(O) = %(0) "t- F, gsNN(O), which we rewrite as

gA(0)

7,

2F~

2MF~

g,NN(0)

(253)

+ - -

(254)

= 0.02m~q.

(255)

2M

The calculated value of __2t_ 2MF~ is found to be %

• 2MF~

From the other numbers listed in the same table we see that in pion mass units the left hand side of Eq. (254) is 1.07 while the right hand side is 1.06. This is a check of of the accuracy of our calculations including the approximate projection method. The slow variation hypothesis of Goldberger and Treiman (1958) requires both sides of Eq. (254) to be close to the value 1.12 listed for n'ss¢"=') in Table 3. If we accept the lower value of g~lwv(m~) due to Arndt et al., (1990) and 2M Bergervoet et al., (1990) the experimental mismatch is 2%. Our calculated mismatch of 5% is not particularly bad.

M. K. Banerjee

132 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

'

00

i ,

i ,

i ,

0.4

J ,

0.8

i .

i .

1.2 r(fm)

i .

= .

1.6

i

r

j

.

2.0

Figure 16: Plots of G, F and X obtained with the CCM with vector mesons. The X field is in pion mass unit. 9.5

Inclusion

of Vector

Mesons

Although in this article much of the discussion of the chiral confining model has focussed on a theory with only two mesons - ~r and ~r, calculations including the vector mesons p, AI and oJ have been carried out (Ren 1991). In the hedgehog ansatz these fields are represented as

Ag(~ = O, ~o(~ = ~(r),

A ; ( ~ = As(r)~ °i + AT(r)(~a~ i -- ~=i), ~ i ( ~ = o.

(256)

In the equations above the superscript a stands for the isospin component, a = 1,2,3 and the subscripts stand for the space-time components. The calculations were done with the following parameters: m a = 773 McV, g,

=

5,

m~ = 783 McV, g,,, =

4,

(257)

with all other parameters having the same value as before. The resulting spinor components G and F and the X field are shown in Fig. 16 and the ~ and ~r fields are shown in Fig. 17(a). It is obvious that the inclusion of the vector mesons make no significant difference. The vector meson fields are shown in Fig. 17(b). The solid line is the p field and the dash-dot line is the u; field. The scalar (dashed) and tensor (dotted) components of ,41 are also shown in the same figure. In Section 7.6 we have discussed that the canonical -41 field has pion pole, while ,4~h~, cloes not. This explains the long tails of t h e / [ I field components. The vector meson fields may he compared with those obtained in the nonconfining model calculations of Broniowski and Baaerjee (1986). These are shown in Fig. 18. In comparing these fields with those shown in Fig. 17(b) one must keep in mind the fact that the nucleon generated in the nonconfining model has (r~)~ = 0.69 fro, while the confining model has I

Ir2)~ ~-- I fro. This means that the source current in the confining model is weaker by the factor of ~-VZ'J(°'sg~3,,, ~I due to size alone. For our qualitative purpose we should multiply the vector meson fields of Fig. 18 by this factor before comparing the two sets. Even with this scale factor the vector meson fields, just llke the ~ and Ir fields, are much smaller in the confining case.

133

Chiral Confining Model 0.2

0.14

O.t

0.12

•,.,.,.,.,.,.,.,.

, . , •

0

0.10

-O.t -0.2

0.08

-0.3

0.06

-0.4

",.,~a

'''",.

0.04

-0.5 -0.6

""

_

-0.7 0

_

o.oz

O"

i

' b . ' 4 " b ~ 8 . . . . .1. . .z. . . . 1.6

z.o

'",.,

.A_,~ ~ " ~ : " . ~ i ~ i

O0

0.4

r(fm)

0.8

1.2 r(fm)

........ A 1.6

2.0

(b)

(~)

Figure 17: Results from the CCM with vector mesons. The o(r) and rr(r) fields are in (a) and the vector meson fields are in (b). All fields are in pion mass unit.

3

2F\ G00

GS0

1,00

1,50

r in f m

Figure 18: The vector meson fields in units of F~ of the nonconfining model of Broniowski and Banerjee. See text for details. The A1 fields are the physical and not the canonical fields.

134

M.K. Banerjee Quantity E~lence

N, f d a r u t ( r - ' ) ( - i ~

Value (MeV) 372

• e)u(0

615 118 154 575 -25

f d%(~X) 2

f d~rU~ Other Mesonic Terms

Table 5: Decomposition of the energy functional for the chiral confining model including the vector mesons.

From the discussion of Section 7.6 it is apparent that the energy functional is more complicated when vector mesons are included. In table 5 we present an abbreviated breakdown of the energy functional into its parts. The last line represents the sum of parts of the energy functional not already listed in the previous lines. These are all dependent on the meson fields generated by the valence quarks. Once again we see that the meson fields do not play an important role in the structure of nucleon. The smallness of the changes in the meson fieldsin the interiorof the baryon and the resultant smallness of mesonic contributions to nucleon properties suggests that a simpler model is worth exploring. In this model, called the Toy model, all meson fields are set equal to their vacuum values. This means that a is fixed at -F~, while all other meson fields are fixed at 0. Naturally, £m~o~ is dropped from the lagrangian. W e discuss the Toy model in the next section.

10

The Toy M o d e l

In the Toy model one sets all meson fields to their vacuum values, viz., (a)~,~ = F~, (~)~=c - 0. The resulting lagrangian is (gxx )

The Toy model hamiltonian is tgxX) +

1

2

1

-

~lr x + ~(Vx) + U(X)]

(259)

Since the X field couples to the quark in a spin- and flavor-independent manner the Toy model gives the same mass for N and A. Hence, we choose a valence spinor with definite spin and isospin projection: u(~ =

i6.~F(r)

¢'

(260)

where the Pauli spinor ( is a member of the set [u T, u ~, d T, d ~]. As in the case of CCM, discussed in Section 8.1, we introduce creation operators: b~,u = / d3xCt~Cx)u(~)e-~°''°' .... ,

(261)

where a = 1,2,3 is the color label. A baryon state is constructed as follows:

I B) = b~d,,,b?9, .....b~,.... 10)),

(262)

135

Chiral Confining Model 1.0 0.9 0.8 0.7 0.6 0.5 0.40.3 0.2 0.1

1.0 • , . , . , . , . , 0.9 0.8 0.7

G

0.6

.,.,.,.,.

,.

G

0.5 0.40.3 0.2 0.1

0 0 '0'4.' ' b'a ' i 'e' f'/~ ' 2:0' r(fm)

0

0.4

0.8

1.2 r(fm)

(a)

1.6

2.0

(b)

Figure 19: G, F and X for the pure mass in (a) and for the quartic case in (b). The X fields are in pion mass unit. where I 0)) is the modified vacuum of the Toy model characterized by the fact that the expectation value of the field operator X(x) is not zero but is given by

(B [ X(x) l B) = Xc(£).

(263)

The baryon state I B) is, in general, a linear combination of N and A states of the specified charge. This may be checked with the Eqs. (342) of the Appendix. Writing the field operator as X(x) = X~(£) + ~(x), (264) we expand HT~ around X~. The subsequent steps parallel what was done in CCM. Eventually we obtain the mean field approximation to the energy of the baryon, which upon dropping the subscript c of the classical X field, has the form:

(B[HToy I B) = /dax[N~ut{-igt. V + ~ ) u

+ ~ ( ~ x ) ~ + V(x)]

(265)

Once again the unknown spinor u and the classical hybrid field X are determined from the condition of stationarity of the energy functional Eq. (265) subject to the normalization requirement, f dSr [ u(r-') [~ = 1. The resulting equations are given below.

/3g'~F'~lu = ~u, [-i,~. ~ + (gxx)~ - V2x +

(266)

dU(x) = 2No f~g'~F'~u. dx

g~x 3

(267)

The boundary conditions specified by Eqs. (210) and (211) are still valid. Virial theorems take particularly simple forms for the Toy model. They are obtained easily from Eqs. (220) and (222) by dropping all terms referring to mesons. The new theorems are: - T , + Tx + 3 ~ V(')

=

0,

(268)

n

- 2 v , + 2Tx + ~ , , V ~ (n) = 0. n

(269)

136

M.K. Bancrjcc Quantities e . . t ....

(MeV) N, f dZrut(r-')(-ia • V)u(r-')(MeV) N,m f darut(~u(r-')(MeY) f d%(Vx)2(MeV) f d%U(x)

Pure Mass 387 848 13.8

24.9

Quartic 406 617 16.7 137 159 585 1515

Exptal

E(total energy) (MeV)

274 299 1460

(r')~(fm)

0.88

1.03

0.8~

6.5 2.11 -1.41 1.24 0.97 0.24

7.9 2.11 -1.41 1.38 1.11 0.15

3.85 2.79 -1.91 1.26 1.00

Nof ~ , "Ksf~-~*(~u(~(MeV)

939

1

#n(nbra) #.(nbra)

ga(O) 2

g~rNN(mlr) 2M

f F2/ f G2

Table 6: Some results from the Toy model with two forms of U(X). As in the case of full CCM calculations, these virial theorems are used to look for programming errors and, more importantly, to check the convergence of the iterative procedure for solving the coupled nonlinears equations appearing above. Since there are no mesons and the quark states have good angular momentum and isospin projections, the states of N and A are constructed trivially using familiar procedures of fixed number particle physics. These states are listed in Eq. (342) of the Appendix. The illustrative calculations presented here were done with two forms of Ux - pure mass and quartic. The quartic potential is identical to the one described by Eq. (226) and used in the CCM. For both cases we used g~ = 4. The hybrid related parameters were as follows:

P U R E M A S S g× = (12.5 MeV) -x, QUARTIC g× = (30MeV) -~, m× = 1400MeV,

m x = 1200 MeV.

(270)

co = 30MeV, U(Xo) = 0.

(271)

Plots of G, F and X versus r are shown for the pure mass case and the quartic case in Fig. 19(a) and Fig. 19(b), respectively. One can see that with the quartic Ux one can make the bag much sharper. The various contributors to the nucleon mass and several properties of the nucleon, obtained with the two choice of U(X), are shown in Table 6. It is transparent that the results are not good. The values of gA(0) and gTrNN(mTr)2M2 ~ are satisfactory. The values of the magnetic moments are too small in magnitude. An examination of the meson cloud contributions to the magnetic moments in the CCM calculations, as listed in Table 3, suggests that the situation may improve upon inclusion of the same effect in the present case. One hopes that this can be done perturbatively using a hadronic lagrangian with form factors generated by our model. There are two major problems with the CCM and its Toy version. First, there is the problem of the nucleon mass and size. We note that the states generated in a mean field calculation do not have translational symmetry, i.e., these are not eigenstates of the space momentum operator. This is why we list it as the energy of the baryon in Tables 6 and earlier in Table 1. Leech and Birse (1989) have studied the problem of projecting out the state at rest from the mean field solution

Chiral Conf'ming Model

137

state obtained with a version of the Toy model and calculating its properties. Some of the issues are discussed further by Birse (1990). From their studies it appears that the rest mass may be ,,, 250 M e V less than the ener Lgivof the mean field state. If we use the experimental value for the rms radius, the product M(r2)$ remains as high as ,,, 5, while the eXperimental value is 3.85. We conclude that a short range £ttractive force is missing, i The second problem is with N-A mass splitting. The estimate of 53 M e V for the CCM discussed in Section 9.3 is applicable to the Toy model also. It falls far short of the experimental value of 294 M e V . Obviously we need another spin-isospin dependent force. We believe that the instanton-based 't Hooft interaction is the answer to both problems. We discuss it in the Section 12. Despite these limitations the Toy model is a very useful tool to explore ideas. It has been used by me to study what happens to a nucleon when it is placed in nuclear matter (Banerjee 1992). I found that not only the standard results that the effective mass, M*, decreases and the rms radius increases but also that the coupling constants change with density. Specifically, g,,NN/2M, g,,,NN, gp and fp increased with density.

11

Trace Anomaly

and Effective Lagrangians

The QCD lagrangian, in the chiral limit, r~r~a=¢(i

~ -g,J)¢-

l~7. G ~ G" "~

(272)

is invariant under an infinitesimal scale transformation

6¢(~) =

~(x.d-~+ 3)¢(x)

5A:(x) = ~(XOd-~zg+ l)A•(x),

(273)

at the classical level. It is possible (Coleman 1985) to define a stress-energy tensor, O "v such that the conserved Noether's current corresponding to this symmetry is

j~,, = x~,B ~'a.

(274)

With this form the divergence equation takes the form

O~,j~, = 0~,

(275)

which, of course, vanishes at the classical level for QCD in the chiral limit. It is useful to derive the following connection between f d3xO~,j~, and the hamiltonian H = / d3rOo°

(276)

Consider a localized (wave packet) physical state I a). The spatial width is arranged to be sufficiently large so that one can regard it as an eigenstate of the hamiltonian. The quantity (a I O~Jd~l I oL) -- (0 [ i)uj~i I [ 0) is nonzero only in the neighborhood of the localized state. We can then integrate the quantity over space and use integration by parts to obtain the result

J[ d 3 x { ( a ]O~,jd~l I s ) -- (O]Ozj~a 10)} = / d 3 x { ( a [ 0oj°a I s ) - (0 ]Ooj~i, [0)}.

(277)

138

M.K. Banerjee

Next we use Eq. (274) and the fact that the expectation value of any operator in an eigenstate of the hamiltonian is time-independent to obtain the result: f d3x{( c~ I O.J~,l c~) - (0 I O.Jd~, 10)} = (a I H I ~) -- (0 I H 10).

(278)

The presence of the current quark mass term, -~m~b, in the actual QCD lagrangian,

c~.=~(i

~-gJ-m)~-~

1 ~-.~~ ~ . ~~ , , ~

,

(279)

a=l

breaks the conservation of J~il at the classical level. Specifically

Classical:

O~jd~t = ~m~b.

However, the symmetry is broken in a much more significant way at the quantum level. This is referred to as the trace anomaly. According to the trace anomaly (Collins et al., 1977): O~ = (1 + 7"`)6m¢ + ~(g-----~)(7 G u~ 2g ~"~ '

(280)

where m and g are the renormalized current quark mass and coupling constant, respectively. The quantity fl(g) is the beta function fo QCD: (281)

~(g) = ~d~'

where # is the renormalization scale mass. The quantity 7,, is the anomalous dimension of the mass term and is defined by the relation: 7., = p dm (282) m dp " At this stage we digress a little and discuss the large N~ limit of the trace anomaly, because the results have been used in Section 3.7. In the large Nc limit the quantities ~(g) and 7,n may be evaluated in the one-loop approximation. The expressions in the one-loop approximation are given below.

~(g)

g(N~) a lINe -

=

16r 2

2n1

3

'

(283)

where n I is the number of flavors. 3g(Nc) 2 N~ - 1 7,-

=

16r2

(284)

Nc

Then, in the large N¢ limit, the trace anomaly has the form O~ = (1 + 4~)~m~b - I I ~ G , , , G u',

(285)

a = g(N~ -- 3)2

(286)

where 41r

Note that the numerical coefficients multiplying the operators are independent of No. Since the trace anomaly is a quantum effect, any effective lagrangian must exhibit its role at the tree level. This requiios, at the minimum, the following steps.

Chiral Confining Model

139

(i) One must introduce reasonable definition of scale transformation in the effective lagrangian. (ii) The effective lagrangian must not be invariant under the scale transformation at the classical level. We define scale transformation adhering very closely to the definitions used in QCD. ,¢(x)

=

e(Z~d--~+~)¢(z )

6¢(x) = ,(x.~x.+ 1)~b(x),

(28~)

where ~ stands for any boson field - the X field or any meson field. As in QCD, it is possible (Coleman 1985) to define a stress-energy tensor, O "v such that the conserved Noether's current corresponding to this symmetry is

j~. = x~o "~.

(288)

Writing the resulting change in the effective lagrangian as eA/~ we have 0uj~i , = O f = A £ .

(289)

When a symmetry of a lagrangian is broken slightly either explicitly or via an anomaly it is a sound idea to construct the effective lagrangian as the sum of two parts: £ = £o + £,b.

(290)

Here £0 is the main lagrangian and it fully respects the symmetry in question, while £,, is a small symmetry breaking interaction term. A good example is the treatment isospin symmetry in nuclear physics. At the basic level the strong interaction is taken to be isospin symmetric with the electromagnetic interaction as the major symmetry breaking term. Accordingly the hamiltonian used in nuclear physics is principally isospin symmetric. Thus the strong interaction potentials, Vt~, V,p and V~,, in I = 1 state are all set equal to each other. The coulomb interaction is then added as the principal source of symmetry breaking. ( Other effects of one photon exchange, r - T/and p - w mixings, etc. play interesting but quantitatively minor roles.) If the electric charge e had the value 3 instead of its actual value 0.3, we would not have proceeded in the manner just described. Charged and neutral pions would have had different coupling constants. Their masses would have differed by larger amounts. From the very beginning we would have set Vpp # V,p # V,,. The SUR(2) × SUL(2) invariance of the strong interaction lagrangian is also broken slightly. It occurs through the current quark mass term. Accordingly we construct effective lagrangians which has a major part £0 which is chiral invariant and add small terms, £°b = £×tB, to break the chiral invariance. This is manifest in the expression for £ W M given by Eq. (143). On the other hand the U(1) axial-symmetry is broken through the strong interaction. The magnitude of symmetry breaking is evident from the fact that the I/' mass is 958 M e V while the r/mass is 547MeV. In the axial-U(1) symmetric case they should have been equal. This suggests that in constructing an effective lagrangian we should not start with a £0 which is invariant under axial-U(1) transformation and then break it with one or two terms. From the very beginning we should assign r/' and its SUa(2) × SUL(2) partner, a0(980), meson their physical masses. They should be coupled to quarks in the form -~¢[iTsr/' + ¢" ff0]¢ to ensure SUn(2) × SUL(2) symmetry. But we should not require that g,~, = g,~. In addition we must add terms displaying the coupling of tf to vector mesons (Cohen and Banerjee 1989). The situation is the same with scale invariance. Eq. (278) shows that breaking of scale invariance as measured by the quantity f dZxAZ: is of the same scale as the hamiltonian itself. Use of a structure like Eq. (290) for the construction of the effective lagrangian to incorporate trace anomaly seems

140

M.K. Banerjee

inappropriate. It is hard to imagine how there can be a major and identifiable part of I'effective which is scale invariant. However precisely this suggestion was made by Campbell, Ellis and Olive (1990) using j-

e og o],

= BE¼( 0'-

(291)

following an earlier suggestion of Ellis (1970). The field ~ is of gluonic origin. Both suggestions have been used widely in the literature. We do not believe that this is a desirable way of constructing an effective lagrangian. We must also point out some associated difficulties with the lagrangian proposed in Campbell et at., (1990). From Eqs. (2.11) and (2.13) of this paper one can establish that for large N~ B ,,, N [ ~ and ~0 "" No. The last result is interesting because it identifies the ~ field as a gluebali field. In the presence of baryons the classical value of ~ shifts from ~o to ~0 - ¢~. One can verify that for large N~ ~' ,~ N °. Then the resulting contribution to energy from £,b is ,,, N °. However, the contribution of £,b to the vacuum energy density is ,-, N~ and we know, from the discussions of Section 2, that the change in the energy density due to the presence of baryons should be ,,, N~. Thus the proposed effective lagrangian fails to include physics which is of higher order in Ne for large N¢. The cure of this particular deficiency requires the introduction of a hybrid field. Of course, it will still not explain why the rest of the lagrangian should be scale invariant and why the burden of trace anomaly be carried by a single term. We believe that there is no need to take special steps to incorporate the effects of QCD trace anomaly in an effective lagrangian. Any effective lagrangian built from common sense will not be scale invariant. (Special steps, such as the ones taken by Campbell et al., (1990) have to be taken to make it scale invariant.) We should merely equate



-

<0lfa'x

(B [ f

-

(0

d3zO:(QCD) [ B)

xc..co.o.

10>

I f d3xO (QCD) I O)•

(292)

Here we have used the same symbol, I B), for a baryon eigenstate in both an effective lagrangian theory and QCD. This suggestion is as logical as anything else one can think of. It does not make any arbitrary separation of the lagrangian into t~0 and L:,b parts. Our particular lagrangian, £CCM, has the virtue that it generates a contribution to the baryon mass due to the (static) polarization of the vacuum which is ,~ Arc. No obvious term of leading order in Nc is missing. For the £CCM given by Eq. (143) the change due to scale transformation is as follows: ¢, OUMexiean

A£CCM = m(b¢ + 4UM,~i,,,,,(a, rr)- ~-~ ' a)lr

+

-~i

4 U ( x ) - X OU(X)" "

OX

+ 2g~ ~(`~ +(g~x), i.~,~. ~)¢

(293)

The reader may wander if the above definition is consistent with the Eq. (278). Using the virial theorems (220), (221) and (222) it is easy to verify that, at the stationary point, the energy functional, defined by Eq. (200) satisfy the condition:

E

= (B I f d3zA£ccM [m).

(294)

Note that by construction (0 [ f d3xA£ccM [ 0) = 0. In fact, the use of equations like (294) to check one's calculations was proposed by Rafelski (1987). The suggestion was based on the virial theorems discussed in Section sec-Vir and was not related to the trace anomaly.

Chiral Conf'ming Model

12

141

T h e 't H o o f t I n t e r a c t i o n

In this section we describe what is the 't Hooft interaction, why it should play an important role in a mean field theory and how the effect should be incorporated. The material presented here is based on work being done with Myunggyu Kim (Kim and Banerjee 1992) 12.1

The

Instanton

Belavin, Polyakov, Schwartz and Tyupkin (1975) discovered that euclidean QCD in the pure gluon sector possesses classical solutions. These were subsequently named instantons by 't Hooft (1976). Apart from the two papers cited, the reader may consult an excellent review of the subject by Coleman (1985). In the singular gauge the solutions are of the form:

2

A~, =

p2#~(z - z)~

g (x - z)2[(x- z) 2 + p21,

(295)

where z is the location of the instanton and p its size. The coefficients f/=,v are listed in 't Hooft (1976). Note that the superscript a takes only three values, say, 1, 2 and 3, spanning a SU(2) subspace of the full color group SU(3)~. Being the classical solutions of QCD, the instantons can make major contributions to the partition function. If, for a fixed gluon configuration, one integrates out the quark fields first the partition function acquires the factor det(i~, where ~ = ~ - ig~ in the chiral limit (i.e., mq -- 0). Considered as a matrix, the operator i]9 has real eigenvalues. Since 7s anticommutes with i~), it follows that for every eigenvector ~ with eigenvalue A there must be another eigenvector 7s~ with eigenvalue -A. However, it is possible that in some cases % ~ = -;-~. In these cases the corresponding eigenvalue must be zero. These are the only situations where the eigenvalues do not appear in matched positive and negative value pairs. If the gluon field A is an instanton field then, indeed, there is one unmatched zero eigenvalue ('t Hooft 1976). This will produce a hi-fold zero in det(i~, where n! is the number of flavors of quark. Let us digress briefly and note a few interesting properties of the instanton before returning to the main topic. The number of unmatched zero eigenvalues is equal to a topological measure of the A field (Atiyah and Singer 1968). It is the winding number g2

=

v

-

-

[d4xG~G~a,

32~2 J

(296)

where G~. = ~e#.oaG~,~. The winding number of an instanton is +I. The instanton is also self~a a dual, i.e.,G~. = G~.. This means that an instanton contributes the amount - ~ to the action and an amount 8 to the gluon condensate (0 [ -~-tG~ 4~2 ~ /~v ]~2 ] 0). There are also classical solutions of euclidean QCD which are called anti-instantons. These ~a ~ a fields are anti-self-dual, i.e., G,~ -G,~. The winding number of an anti-instanton is -1. Its contributions to the action and the gluon condensate are the same as those of an instanton. Returning to the original point, we note that the gluon field configurations with nonzero winding number does not contribute to the partition function. The situation is not materially altered if we introduce the current quark mass which will make the contribution m~~1~I, a negligible quantity. It is interesting to consider field configurations comprising of equal numbers of instantons and anti-instantons. The winding number is zero for these configurations and the zero eigenstate of the quark field is no longer present. If the density of the instantons is small we may regard these field configurations as approximate classical solutions. Thus, they will make major contributions to the QCD action. The new eigenstates of i~)may be treated as the zero modes around individual

142

M.K. Banerjee

instantons with perturbative shifts due to all other instantons and anti-instantons. 't Hooft (1976) pointed out that such shifts could be calculated replacing the instantons and anti-instantons with a 2ny-fermion interaction. We restrict ourselves to n/ = 2 which makes the 't Hooft interaction a four-fermion interaction. The thrust of 't Hooft's suggestion is that we pass from £OCD to an effective lagrangian, £efj'ective, which contains a new term,/~tHooft, the four-fermion interaction terms. It carries the instruction that the gluon field configurations be restricted to quantum fluctuations around the classical solutions - exact and approximate ones. The classical solutions themselves have already been taken into account in obtaining l~,,Hooyt. There is an alternative derivation of l~{tttooyt by Shifman, Valnshtein and Zakharov (1980) which includes the role of the current quark mass. Their expression for Nc = 3 and ny = 2 is given below: 4 2 3z:',.oos,(z ) = fdn(p){ 1-1 ( m i p - -~x p ~)iR(Z)~iL(Z)) i=1,2 3

4

2

3

a = + ~ ( ~ p )2 [~IR(Z)A ~IL(Z)~2R(Z)~ ~2L(Z) 3-

/~

=

-

a

+ -~!b,R(z)~r A ~b,L(Z)~b2R(z)~,~A ~b2L(Z)]} + h.a.

(297)

Here ,tbL = ~(1 + 7s)~b and ~bR = ½(1 - %)~b. This is the effective lagrangian due to instantons and anti-instantons , all centered at z. The quantity dn(p) is the density of instantons of size in the range p to p + dp. This quantity is the same for anti-instantons. Thus the entire anti-instanton contribution is obtained by adding the hermitian adjoint terms to the terms exhibited explicitly. The integration over the position z gives the contribution to the action. Shuryak (1982) has suggested that the size distribution of the instantons is sharply peaked and has proposed the form:

dn(p)sh~,v~k 1

p, "" -~ f m

= dpn~6(p- p~). n, "~ 1 f m -4.

(298)

From Eq. (297) we see that £'~,Hooft has three parts - a constant, which we throw out, a term which modifies the quark mass term of QCD and a term which survives in the chiral limit:

~'"tHooyt =" --Cmmq~b~) T £'tHooyt,

(299)

where, as in our earlier discussions, we have ignored the mass difference of the u and d quarks. The coefficient C,, is given by

C,, = / dn(p)~Tr2p4.

(300)

We should note that the modification of the quark mass from rnq to (1 + Cm)rnq occurs at the level of the effective lagrangian, /~.t.t~cti~, and has no bearing on the Gell-Mann Oakes Rennet (1968) relation. If we use Shuryak's density function given in Eq. (298) we obtain:

(7,sh'v=k-,.

4 24 4 24 = Jf d n(p)sh,,y.k-~r p = -~Tr p~n~ ,~ 7 %.

(301)

Since we cannot claim to know mq to anywhere near an accuracy of 7%, we drop the instanton induced correction and retain only £'tHooyt.

Chiral Confuting Model

143

The definition of £,txoojt is obvious. With the help of Fierz rearrangement (Nowak et al.,, 1989) we rewrite:

+ 10(~5~s~

-

~.~.~)

where r " s ( a = 1, 2, 3) are SU(2)/Pauli matrices, and

c'. = ,~' ] dn(p)p s

(303)

The quantity C,t has the dimensions of [length] 2. Using Shuryak's density function we get

Co,Sh~ry=k 7r4nC~_sc =

(0.37 fro) 2.

(304)

Each group of terms in Eq. (302) is invariant under SUn(2) x SUL(2), but they are not invariant under axial U(1). 't Hooft (1986) suggested that the U(1) breaking terms should be regarded as manifestation of the U(1) axial anomaly. He showed how the large mass of ~/' can be explained with the help of £'tHoo/t. When we introduce mesons, most of the effects of £'tHooyt will appear in the mesonic sector and primarily through terms connected to the t/' field. The latter has very little role in the structure of the baryons. However, they do play important role in the spin content of the proton (Cohen and Banerjee 1989). But we have also learnt from the discussions of Section 7.2 that we must retain £'tHooyt for evaluation at the tree level. This is how the 't Hooft interaction affects the structure of the baryons in a mean field theory. It should be noted that several authors have studied the role of the 't Hooft interaction in the Nambu-Jona Lasinio model (Bernard et al., 1987, 1988, Reinhard and Alkofer 1988, Kunihiro and Hatsuda 1988, Alkofer and Reinhardt 1989, Takizawa 1990, 1991). Dorokhov and Kochelev (1990) included the 't Hooft interaction in a modified and extended version of the MIT bag. They obtain an expression for the square of the baryon mass. The quark condensate, the strong interaction constant a, and pc, the instanton size, are some of the parameters of the theory. Curiously, the instanton density does not appear as a direct parameter. Finally, they make the square of the mass stationary with respect to the bag radius. The product of mass and the rms matter radius turns to be 4.3. The instanton contribution to the nucleon mass is -223 MeV. Both results are very satisfactory. However, we are unable to establish a connection between their method and ours. The four-fermion interactions appearing in Eq. (302) imply ultraviolet divergence which is not present in the original evaluation of the action ('t Hooft 1976). Therefore, at some stage we must introduce a regulator. Our next step will be to subject £o//ecti~e to the averaging procedure of Nielsen and P~tkos. The subsequent transition to the CCM is simple. We discuss these topics in the next subsection. In this article we present calculations based on the addition of 't Hooft interactions to the Toy model only. 12.2

The

CCM

Version

o f ~'tHooft

In the previous section we have discussed how the effective lagrangian, £~txoo/t, arises from £¢CD. Our next step is to examine how the 'block spinning' procedure of Nielsen and P~tkos modifies the effective lagrangian. We propose a procedure which is a natural extension of the discussions of Section 4.2.

144

M.K. Banerjee

We confine ourselves to the two flavor situation. Thus we deal with four-fermion interactions only. In every four-fermlon interaction we split the adjacent ,~ and ¢, and connect them with a link operator to preserve gauge invariance, i.e., 6 ( x ) f ~ ¢ ( x ) ¢ ( x ) ~ ' ~ b ( x ) ~ ¢ ( x ) ~ P e -ig f:-~ dy.A(y)~)(X -- £)@(x)~'~tPe -ig f:_.t dyt'A(Y')~)(x _ ~,)

(305)

where ~ and ~ ' are matrices in the color-flavor-Dirac space. The two paths are independent of one another, but the end points and the paths are confined within the Nielsen-P~tkos hypercube of sides The first step is to average over all possible paths. Then the end points are brought together, i.e., e, e' --* O. Finally, one projects out the color singlet part. It is not possible to extend to the present case the method of Lee et al., (1989) which connected the result, (K)~c = 0, to the area law of QCD. So we have to guess what the result is. The naivest view is to suggest that - -1t r , a-v- ' g e t"r'--iaJ~'d~'a(U)Pe-iaf~-,'d¢'a(¢)) r~ = g2(x).

(306)

N~

But we also know that the average of the square is larger than the square of the average. W e express our ignorance by introducing a multiplicative factor Ce and write: ~b(x)n¢(x)~b(x)~'¢(x) 4.4 C ~ K ' ( x ) ¢ ( x ) n ¢ ( x ) ~ b ( x ) n ' ¢ ( x ) .

(307)

We recall that in the Nielsen-Phtkos lagrangian the quark fields are not canonical. They are made canonical by the transformation v / K ¢ ~ ¢. Thus, in terms of the canonical field,

£~tHooft(~r)QCD, ~)QCD) 4"4 Ce£~tHooft(@N-P, ~)N-P),

(308)

keeping the same functional form. The mesons are introduced by integrating out the coarse grained gluon fields. This procedure has no effect on the form of £'tHooyt. So in terms of the quark field of CCM we have the same result as in Eq. (308). To be specific, NP C, 1 0 ( ¢ ¢ ¢ ¢ £'tHoolt = rCCM ""'tHooft = -~[ +

lO(~s¢~-y~¢

¢7%'¢¢75r"¢) -

~,-o¢~o¢)

where C , = C,C'.

(310)

We do not know the value of the strength parameter C,. It must be fixed by fitting some data. We fix it by fitting the N - A mass splitting. 12.3

The

Extended

Toy

Model

We use the name 'extended Toy model' for the version of the Toy model which includes rCCM ~"tHooft" The lagrangian is: g,~r ,~ rCCM (311) £To ~ = ~ [ i O -- mq t^ ..~2]¢ + t~× + "~'tHoolt, (g×x) The need for a regulator has been mentioned in Section 3.3. A translationally invariant method of regulation is to use momentum cutoff. Since the gluons have been integrated away there is no problem with gauge invariance in doing so. However, the lagrangian is designed for use in a mean field calculation which inherently breaks translational invariance. Momentum cutoff is not a

Chiral Contrming Model

145

convenient regulator for such a calculation and we also see that it is not essential. We introduce a regulator which is not translationally invariant. But as long as the prescription is used only in a mean field calculation no harm is done. Some corrections will have to be made when taking account of the motion of the center of mass of the localized baryon. An examination vL ^¢ ~'~tHoaJt, rCCM given by Eq. (309), shows that for every term without r matrices there is a term with r's appearing in both quark hilinear in the form of an isoscalar object. Thus, if we consider the interaction between two quarks, 1 and 2, ~'tHoo.tt r a a M will contain the factor (1 - ~1" ~2). We recall that ¼(1 - ~1" ~2) is the projection operator for the isospin singiet state. The lagrangian is used in calculations where the three quarks are put in identical ~+, j = ~, even parity, spinor states with three different colors which are combined to form a color singiet state. Total antisymmetry demands that the state be fully symmetric in isospin and angular momentum. This means that when a pair of quarks is in isospin singlet state the spin state must also be singiet. In other words the :CCM A"'tHooft is operative only in the singiet-singiet state of quark pairs. In a A all quark pairs are in triplet-triplet state. So ~'tHooyt raaM does not contribute at all to the mass of A. In a nucleon the quark pairs are equally distributed among singiet-singiet and triplet-triplet pairs and the nucleon mass is reduced by the attraction generated k. :OCM u y "'tHoo.ft" Because of the spin- and isospin-dependence .^¢ . . .:OOM tHoo/t mean field calculations cannot be done with spinors of pure spin and isospin projection. Once again the hedgehog ansatz is the answer. There are no mesons in the theory. Hence, only the spinor need be in the hedgehog form. For convenience we repeat the defining equations which appeared in Section 8.2. u(~ =

i~.~F(r)

¢'

(312)

where the Pauli spinor ¢ is given by the equation --

(u ~, - d T),

(313)

with u and d indicating up and down flavor states. Other useful properties and results related to the hedgehog spinor may he found in Section 14.1 of the Appendix. The hedgehog baryon is constructed in exactly the same manner as in Section 8.1. First, we introduce the quark creation operator

b~Cu) =/dax¢~(x)u(£)e -'=°'''"~,

(314)

where a = 1,2,3 is the color label. The baryon state is written as

I B)

t t .... = b,~d(u)b'~

t (,,)b~o(,,)lO)).

(315)

Here the vacuum,I 0)), is defined as the vacuum for states which do not have good spin or isospin projections. Instead, they have hedgehog symmetry characterized by a SU(2) group whose generators are the grand spin/~ = J + f. The definition of [ 0)) is supplemented by the relation: (B

I X(x) I B) = ((0 I X(x) I 0)) = Xo(£),

(316)

where Xc(£) is the classical X field produced by by the baryon. Let us write the hamiltonian which follows from £~'ovas

H~o¢ = Hro~ + H,,Hooy,,

(317)

146 where

M.K. Banerjee

HToy has been defined by Eq. (259) of Section 10 and CCM H'~Hoolt = -- . / darZ(r)£,tHoolv

(318)

Notice the appearance of the factor Z(r). This is the regulating factor we had promised before. The function Z(r) must vanish as r ~ 0 and must become unity as r becomes large. In QCD also the 't Hooft interaction introduces an ultraviolet singularity which is not present in the original problem. One may hope that a study of this problem may help us choose a form for Z(r). Unfortunately the two divergence problems have different origins. In QCD the divergence occurs because the ¢ and the ¢ fields of the same flavor are at the same space-time point in the 't Hooft interaction. In our mean field calculation the divergence Occurs because fields of different flavors are at the same space-time point. Although the root cause is having all fields at the same space-time point, the cure for one disease will not cure the other. At present, we have no choice but to guess the form. The results presented in this article are based on the use of the following form: Z ( r ) = 1 - e- : # ~ .

(319)

The mean field approximation is developed in a manner quite similar to that followed for the Toy model. The difference is that now we have four fermion interaction terms. Of course, these must be evaluated at the tree level. The energy functional in the mean field approximation is given by the expression:

(B I H~o~ I B) -- f dar[Sut{-i~ • V +/~mq + / ~ } u (gxX ) + E.,Hoop(B),

-I- I(VX)2 + V(x)] (320)

where

E,tHoo/,(B) = (B IH,,noo/, IB),

(321)

E, mool,(B) = - / darZ(r)~[lO( ~ueu - ~75:u~75,%1 -

10(er=ufiv=u-eTSue7~u)

-

Oq~Uufia,~u + fiaU~r%fiq,:'au].

(322)

One can verify from the expressions for the energy functional given above that the hedgehog ansatz is a self-consistent ansatz. Let us add some remarks on the role of the regulator specific to the mean field calculation. Without the regulator the four-fermion interaction is an attractive 6-function interaction. Thus E',uoolt(B) will behave like -~/(length scale) a, where ~ > 0. The only quantity, which grows with decreasing length scale, is the quark kinetic energy. But it grows only as 1/(length scale). So the structure would collapse. This is how the ultraviolet divergence shows up in the mean field calculation. Naturally, a regulator which chops off the short distance contribution will prevent collapse. With the help of the formulas appearing in Section 14.1 of the Appendix one can evaluate E'tttoolt(B). The result is extremely simple:

E,,Hooit(B) = -2C. / darZ(r)(G2(r) + F2(r)) 2.

(323)

As usual, the equations for G, F and X are obtained by making the energy functional stationary with respect to these quantities subject to the normalization condition, f d3r(G 2 + F ~) = 1. The resulting equations are:

dG ( -~r = -mq

g.F, ) ( ' ~ 2 - e, F

4C, Z(r) 2 -~ (G + F2)F,

(324)

147

Chiral Confining Model ~..2

,

t

,

i

•

u

•

i

•

u

.

u

.

u

•

u

.

i

•

i

12 1.l 1.0

•

1.1 1.0 0.9 .......... 0.8

'""'..

0.4

0.8

i

•

i

•

r

•

i

.

n

,

u

.

i

,

u

.

i

•

i

•

0.9 0.8

G

0.7 0.6 0.5 0.40.3 0.2 0.1 00

'

1.2 r(fm)

1.6

0.7 0.6 0.5 0.4 0.3 02 0.1 0

2.0

'"'"'""'-..

G

',r...

0

0.4

0.8

12

l.O

2.0

r(fm)

Figure 20: G, F and X for the pure mass potential case in Fig. (a) and for the quartic potential case in Fig. (b) with (full lines) and without (dotted lines) rOOM ~*'tHoo]t"

d-~ +

r

=

-mq

(gxX)2

2 dx

d2x +

+ ev G ± ~ ( ~

OU(x) -

- 6 ~ ( G tg

± F2)G,

2 - f2).

(325) (326)

xJ x

The boundary conditions of Eqs. (210) and (211) of Section 8.2 are still valid. Because of the nonlinear dependence on G and F some special cares are needed to solve these equations. The virial theorem following from stationarity with respect to the length scale also undergoes change because of the presence of the regulator Z(r). These details are to be found in Kim and Banerjee (1992).

12.4

The

Mean

Field Results

Calculations were carried out for both pure mass and quartic forms of U(X). All parameters were the same as in the Toy model calculation and listed in Eqs. (270) and (271) in Section 10. The exceptions are the parameters appearing in ~'tuoo/t, rCCM i.e.,the strength C, and the cut-offparameter re. For each choice of rc one can find a C0 which will give the desired N - A mass splittingof 294 M e V . Only the resultsfor r~ = 0.25 f m are presented here. As long as r~ is significantlysmaller than the sizeof the nucleon, the nucleon properties are weakly dependent on its precise value. The values of Ca for the two choices of U x are given in Table 7. In Fig. 20 we show G, F and X obtained with and without e~-,tnoo/, ccM using both forms of U(X), viz, the pure mass form and the quartic form. r C C M 'o ;o obvious. It is interestingto note that for both choices of The reduction in size due to ~'tnooyt U(X) the X fieldsare nearly the same, with and without e •~'~noolv CCM They differnear the tailbecause of the differencein size. The numerical results appear in Table 7. The following abbreviations have been used:

T, = Nof

T.. = Nomf ,er,,*(Oa,,(O,

Despite the absence of mesons in the present model the task of projection is not exactly as simple as it is for the Toy model without rvcM *"'¢Hoo]t" The vacuum I 0)) is n o t invariant under rotation. The

148

M.K. Banerjee Quantities

1 2 3 4 5

C,(fm) 2 e~at.... (MeV) Tq(MeV) Tcq,~(MeV) Tx(MeV )

6

V×

7 8 9

Vq(MeY) (B]H~ou[B)(MeV) E, tHooft(B)(MeY)

10 11

(r2)$(fm) MN(MeV)

Pure Mass with without 1.85 324 387 1033 848 11.8 13.8 22.0 24.9 200 274 222 299 1 3 4 2 1460 -147

Quartic with without 2.19 348 406 881 617 13.9 16.7 101 137 124 159 442 585 1415 1515 -147

Exptal

0.77 0.88 1195 1460

0.83 1268

1.03 1515

0.81 939

4.66 294 1.98 -1.32 1.14 0.88 0.31

5.34 294 1.98 -1.32 1.24 0.91 0.24

7.9 0 2.11 -1.41 1.38 1.11 0.15

3.85 294 2.79 -1.91 1.26 1.00

1

1

12 MN (r2)~q 13 Ma - MN(MeV) 14 gp(nbm) 15 l~(nbm) 16 gA(O) 2 17 g,rNN(m,)2M 18 f F 2 / f G 2

6.5 0 2.11 -1.41 1.24 0.97 0.24

Table 7: Values of various quantities in the Toy model with the two forms of U(X) and with and without A.',tHooft. rCCM act of rotation creates qq pairs. The corresponding graphs are not tree graphs, but loops. Recalling the instruction that we should only count quark tree graphs, we ignore the loops, even though they are convergent. Then the task of projection reduces to that in a fixed number of particle problem, exactly as in the Toy model without rCCM "~"'tHooft" Once again we use the states listed in Eq. (342) in the appendix. The expansion of the hedgehog baryon state in the form: z,i3 I B) = ~(-)J-'312(-~+

1) IX = J' i3 = - j 3 ) .

(328)

is now valid. The reader may recall that the expansion, which appeared earlier as Eq. (233) in Section 9.2, was not valid in CCM because of the presence of the mesons. The values of C, appear in the 1st row. Let us examine these values in the following manner. We ignore the enhancement factor C, introduced in Eq. (307), use Shuryak's density distribution given by Eq. (298), but replace Pc with Pef/eetive and treat it as a free parameter. Then we have: 4

S

7r n~p~ss~a~¢= Co.

(329)

The two values of C, translate to Peffective = 0.517fro for the pure mass case and to Pelleetive = 0.531 f m for the quartic case. Alternatively, we could use Shuryak's Pc ~ 1/3 f m and convert the Co's into values of C,'s. We obtain C~(puremass) = 13.8 and C~(quartic) = 16.3. These appear to be on the large side. Two points should be noted in this connection. First, we have so far omitted the contribution of the pions to the N - A mass splitting. One pion exchange gives rise to a force ,~ - ~ , • g2q, • ~6(r-*)

Chiral Confuting Model

149

i.e., the attraction is 9 times stronger in the singlet-singlet state than in the triplet-triplet state. When this effect is included a smaller value of C - e will be obtained. Second, we should note that Shuryak's estimate of nc is based on the assumption that the entire gluon condensate of the vacuum arises from the instanton liquid. In principle, quantum fluctuations must also contribute. In calculating the latter one must subtract from the contribution of quantum fluctuations around the nomperturbative instanton solutions the contribution of quantum fluctuation in the perturbative vacuum. Thus the sign of the quantum fluctuation is not known a priori. It is necessary to do the calculation. A negative contribution from the quantum fluctuations will increase the value of nc and lead to a smaller value of C~. The 't Hooft interaction reduces the values of e~at~n~, but they are still comfortably positive. Thus our conclusion (see Section 7.2) that the quark fields must be retained and treated at the tree level still holds. Noting that I B) is an equal admixture of N and A and recalling the earlier observation that H, tHoo/t does not contribute to the A energy we find that:

MN =-.(N IH~o,~ lN) =

(B IH~. IB) "4-E'tuoolt(B),

MA = (A lH~,,,~lA) = (B I H~'o, IB) - E,tHoo:t(B), MA -- MN = -2E.tHoop(B)

(330)

The numbers appearing in rows 8, 9, 11 and 13 illustrate these results. The values of the quark rms radius appear in row 10. These values reflect what is obvious from the Figs. 20, namely, that the size of the baryon is reduced when rC'CM ~'tHooft is included. The reduction in size explains the enhanced quark kinetic energies, shown in row 3. Increased average momentum makes the quarks more relativistic. This is apparent from the ratio of f F 2 / f Ga, shown in row 18. With the help of Eq. (345) of the Appendix we can correlate the values of gA(0), listed on row 16, with the changes in f F 2. From Eq. (168) of Section 7.4 and the rest of the discussions therein it is clear that the quantity g,~NN(m,~)2 2 M must follow the trend of gA(O) and it does. The reduction in the value of the magnetic moments due to ~CCM "-'~Hoo]~can be traced principally to the reduction in size. The biggest accomplishment of the extended version of the Toy model is that it produces sig1

nificant reduction in the value of the product MN (r2)~q. As discussed in Section sec-Toy both the mass and the quark rms radius will be reduced by corrections due to the motion of the center of mass. The experimental value of 3.85 is the product of nucleon mass and the charge radius. The charge radius has contribution from the pion cloud around the nucleon, a piece of physics which is missing in the extended Toy model. The pion cloud will also contribute to the magnetic moment. 2 The effect of the pion cloud o n g~NN(m~r)2M is somewhat smaller. The qUantity gA(O) is affected only slightly by the pion cloud. At present, there is no calculation on either the corrections due to the motion of the center of mass or due to the pion cloud. However, there are calculations (Birse and McGovern 1992, Leech and Birse 1989) based on the Toy model without r6'CM ~'tHooft" These were discussed in Section 10.

13

Concluding Remarks

In this review article I have described the development of a model of the nucleon whose ingredients are quark, meson and hybrid (also called meikton) fields. The development is not complete, but considerable progress has been made. 13.1

Summary

of the

Model

The salient points of the model are as follows:

150

M.K. Banerjee

1. It is a chiral invariant model with small, but appropriate, chiral invariance breaking terms. 2. The confining bag is generated dynamically using the Nielsen-P~tkos color dielectricmodel. 3. The single most crucial ingredient in the color dielectricmodel is the vanishing of the color dielectric function in the vacuum. W e have shown (Lee ctal., 1989) that this follows from the area law of QCD. 4. The model has essential connection to the two major characteristicsof the Q C D vacuum, viz, gluon and quark condensates. Specificallyfluctuations of the two condensates, hybrids and mesons, are ingredients of the model. 5. The model has correct large Arc behavior. This by itselfdoes not guarantee that the model is good. But not having it makes a model unacceptable. 6. The last three items strengthens the idea that the model described in this article may have basis in QCD. 7. The question of coexistence of quarks and bosons in an effectivelagrangian, designed for mean fieldtreatment, has been considered with care. The conclusion is that the quark fieldmust be retained if e~:~c~, the valence quark eigenvalue, is positive. But only tree graphs should be evaluated. The contributions of quark loops appear through the bosonic lagrangian. For the C C M and the Toy model without the 't Hooft interaction and without any X - X interaction we can prove that e~t~, > 0. W e do not have a theorem for the more general case, but so far as we know no example of negative e~tence has been found for any confining model treated at the mean fieldlevel. The other side of the story for a confining model is that any one claiming to be able to eliminate the quark fieldscompletely and obtain a practical lagra~gian of boson fields only must examine carefully the consequences of a Wick rotation if it has been used anywhere in the theory. 8. The earliernonconfining models (Birse and Banerjee 1984, Broniowski and Banerjee 1985) and the present confining models differin major respects. In the nonconfining models the valence quark eigenvalue, e~l.... is either very small (without vector mesons) or negative (with vector mesons). Thus the quarks can be essentiallyintegrated out, leaving a purely mesonic theory. The meson field distributions differ only slightly from that of the Skyrmion. The difference is solely due to the use of the linear sigma model. Without implying topological stabilityone can assign a winding number to the fielddistribution (see Section 9.1) and obtain the value 1 for it. In contrast, the quarks cannot he integrated away in a confining model because ~ is positive and large. The ~ and lr fields are significantly different from those of the Skyrmion or the nonconfining models. The cr field never changes sign. The winding number of the field distribution is 0. 9. The instruction that the quark terms in the lagrangian must be retained for treatment at the tree level has had two operational consequences. First, the chiral invariance breaking term of the lagrangian includes, in addition to the traditional F~rrz~cr term, the current quark mass term. 10. The second consequence is that it requires us to include the 't Hooft interaction in the mean field calculation. Its role in the mesonic sector of an effective lagrangian or in loops in the Nambu-Jona Lasinio model have been implemented by several authors (Bernard et al., 1987, 1988, Reinhard and Alkofer 1988, Kunihiro and Hatsuda 1988, Alkofer and Reinhardt 1989, Takizawa 1990, 1991).

Chiral Confining Model

151

11. We discuss the role of the trace anomaly and reach two conclusions. Since the breaking of the dilatational symmetry is of the same scale as the hamiltonian density, one should not propose an effective lagrangian which consists of a major part which has dilatational symmetry and an additional term which breaks the symmetry. This style is appropriate for SUR(2) x SUL(2) which is broken slightly. We propose that one should construct the best possible effective lagrangian without any attention to to the question of trace anomaly. The resulting stressenergy tensor will almost certainly have a nonvanishing trace. The last quantity should be interpreted as the manifestation of the QCD trace anomaly. 13.2

Future

Tasks

Needless to say there are many aspects of the model which require further work. An obvious area is the correction due to the motion of the center of mass. Probably the biggest question is the treatment of the role of mesons, particularly the pions. We have suggested in Section 10 that a possible approach is to start with the extended Toy model, which includes the instanton effects, and then include the effects of pion and other mesons perturbatively through a suitable hadronic lagrangian. Being significantly lighter than any other meson, the pion is the most important contributor to most properties of the nucleon. The chiral perturbation theory (xPT) (Weinberg 1979, Gasser and Leutwyler 1984) was developed for this task. It deals only with the pion and the nucleon fields. In xPTthe pion is derivative-coupled to the nucleon. As a result most graphs of interest are ultra-violet divergent and require counter terms. If the pion was massless, these diagrams could have infra-red divergences also. The thrust in the applications of xPTis to isolate these singularities and terms of lowest powers of m~/M. Instead of introducing the requisite number of counter terms, one often calculates not the quantity of interest, but its derivative with respect to m~. The derivative is of the lowest order necessary to make the integral convergent. This procedure trades off certain number of unknown counter terms in favor of equal number of unknown integration constants. Recently Cohen and Broniowski (1992) have pointed out that the contribution of internal A in these graphs are by no means negligible. Despite such improvements the fact remains that xPT does not have full predictive power for many nucleon properties. Another approach, found in the literature (Thomas et al., 1981, Thomas 1984, Williams and Dodd 1988, Birse and McGovern 1992), introduces form factors in the effective lagrangian and thus controls the divergences of the Feynman graphs. The form factors are obtained from a quark model of the baryon. This approach gives finite results. At the same time, the results are completely consistent with the xPT results in the limit m ~ / M --, O. One may even argue that this approach is in the same spirit as that underlying the quarks models used. We feel that, at present, this is the desirable approach to calculation of nucleon properties with confining quark models. The question of the strength of the 't Hooft interaction in the confining model is completely open. In the extended Toy model we fix it by fitting the N-A mass difference. The required strength appears to be on the high side. Pionic effects, when included, will also contribute to the mass splitting and thus reduce the estimated strength. This question must be examined further. Finally, we must include two-quark correlations a la Isgur, Karl and their collaborators (Isgur et al., 1978, Isgur et al., 1981). In their work, the magnetic part of the one giuon exchange potential (deRujula et al., 1975) was responsible for the correlation. The interaction is attractive in spin singlet state and repulsive in spin triplet state. Thus, a quark pair in a baryon in the singlet-singlet state will be more compact, while the triplet-triplet state will be more spread apart. In a chiral effective lagrangian the main effect of one gluon exchange appears through exchange of mesons. One pion exchange gives rise to a force ,,~ -~1" ~2¥1 • ~2~(~ i.e., the attraction is 9 times stronger in the singiet-singiet state than in the triplet-triplet state. The differential effect on the two spin-isospin

152

M.K. Banerjcc

states is similar to that due to one-gluon exchange. The addition of the 't Hooft interaction further accentuates the differential effect. I have no doubt that the reader can add to this list of things to be done.

13.3

Acknowledgements

I must thank those who collaborated with me in developing the model described in this article. They are Mike Birse, Wojtek Broniowski, Tom Cohen, Ching-Yun Pen and Myunggyu Kim. Su Houng Lee and Ming Li made important contributions. Hilmar Forkel has been particularly helpful in the development of the 't Hooft interaction and on all issues related to the instantons. Comments by John Collins have clarified several issues relevant to this article. Joe Milana's comments and criticisms have lead to many improvements. I thank Loretta Robinette's help in all aspects of production of this article.

14 14.1

Appendix Some

Basic

Results

the Hedgehog

with

Spinor

The hedgehog spinor has been defined by by Eqs. (204) and (205). The Panli spinor, ~, appearing in it has the following properties: (~+~)~ = 0 ~ta~ = 0, ~t'7~" = 0.

(331)

From these properties the following results can be derived:

14.2

flu = G 2 - F 2, f i % r ° u

=

-2iGFP,

fia°ir~u = 0, fiaiJu

=

0, fiaiJr~u = eijk[--6ak(G 2 + F 2) + 2F2P~k].

Results

from

the

fia°~u = - 2 G F ~ i

(332)

Model

Hedgehog

Some of main results of the projection method, described in Ren and Banerjee (1991), are listed here without derivation. The quantum correction involves only meson quantum of even parity and grand spin K = 1. It is a pure pion state with / -- 1 coupled to the isospin to form K -- 1. The radial wavefunction, ~b+(p, r), satisfies the equation:

ida__ 2 d dr 2

r dr

2 r2

m~~ - m 2, , 2 -- 92 _ F2) + p2l~+(p, r) -2F~

( a "l-

=

0.

(333)

Here a and r are the classical pion fields defined in Eq. (206). At the level of approximation employed in (Ren and Banerjee 1991) the quantum effects in the matrix element of an operator which involves time derivative of the pion field will involve the wavefunction ¢+(p, r). Two such results are listed below: . . . .

(334) tr )o-~,,n

= i ( B I Y2 I B)

~p[

dr:¢'+(P'r)'~(r)][ d"rZ¢+'(P'r)~(r)]

(335)

Chiral Confining Model

153

Using Eq. (215) we find that

( B I ~ I B) ~ No, ({r2~].,-#p,. \ raejon ." N-0~.

(336) (337)

The quantum effects in the matrix elements of operators not involving time derivative of the pion field do not refer to to the quantum wavefunction directly, but does so through (B ] J'~ I B). Some of the more frequently used results are listed below with the quark and the meson contributions separated with square brackets. pp/e

=

[(I + 4(B

I fi3 I B) ) f d3r rGF]

+ [ (1 + 2(B I Y21n))/dzr:(~)]" p,,/e

=

1 [( 3

1

(338)

f d3r rGF]

4(BlfllB) ) 3

-

gA(O)

-

=

(339)

[I(i+ 2(B i f2 I B)) f a~r"~(~)]• [(i+ 2(BI fl IB))(1-

- [4 / d°r-~.(r)(1 + 2(BI 1

F2)] 3

fllB ))]

(340)

. f d3r rGF.

g,rNN/2M = [g,(2+ ( S i f l i B ) ) ]

--K-i-]

2 2 [~lm= - m " f ~ r r ( : + : -

3 F:)r(r)(l + 2(B i J21B)

(341)

The quantities G and F have been defined in Eq. (204). The magnetic moments need to converted into nuclear Bohr magneton, nbm = ~-~, where M = 939 MeV, the actual nucleon mass.

14.3

N a n d A S t a t e s w i t h Q u a r k s only

In a world of fixed number of quarks, it is trivial to construct the states of N and A. Some of the states are listed below:

IA+, m = 3 / 2 )

= V~-[iuTuTdT)+a.p.],

IA+,m=l/2)

=

l[{luTuTdJ,)+a.p.}+{luTuldT)+a.p.}]

,

?._,=_

Ip, m = l / 2 )

=

71[2{luTuTdJ,)+a.p.}-{luTu,~dT)+a.p.}],

(342)

where a. p. means all permutations. The states of I A-, ra) and I n, m) are obtained by interchanging u ~ d. The value of m may also be flipped in a similar manner.

154 14.4

M.K. Banerjec Toy

Model

Results

The results of the Toy model are derived easily by omitting all meson contributions. For convenience they are listed explicitly.

#,/e = g

d3rraF,

p./e = - 9 / d3rrGF'

10 f ,~ rGF g~NN/2M = g~--~J a - r K2 .

(343) (344)

(346)

Chiral Confining Model

15

155

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