Quark solitons from effective action of QCD

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B424 (1994) 3—17

Quark solitons from effective action of QCD a,* Amihay Hanany a,** Marek Karliner b,*** Department of Particle Physics, Weizmann Institute of Science,. Rehovot, 76100, Israel b School of Physics and Astronomy, Beverly and Raymond Sackler, Faculty of Exact Sciences,

Yitzhak Frishman a

Ramat Aviv Tel-Aviv, 69987, Israel Received 11 November 1993; revised 7 February 1994; accepted 22Apr11 1994

Abstract We derive an effective low-energy action for QCD in 4 dimensions. The low energy dynamics is described by chiral fields transforming nontrivially under both color and flavor.

We use the method of anomaly integration from the QCD action. The solitons of the theory have the quantum numbers of quarks. They are expected to be the constituent quarks of hadrons. In two dimensions our result is exact, namely the bosonic gauged action of WZW.

1. Introduction—Effective action from anomaly integration The connection between the phenomenologically successful non-relativistic constituent-quark model (NRQM) and QCD’s fundamental degrees of freedom continues to be one of the open and puzzling questions in particle physics. The constituent quarks which are the basic building blocks of the NRQM have the same quantum numbers as the much lighter and highly relativistic QCD currentquark fields. In addition, recent data from polarized deep inelastic lepton—nucleon scattering [1—3]and the ensuing conclusions about spin structure of the nucleon [4] indicate that the constituent quarks are composite objects with internal structure. The emergence of constituent quarks as low-energy effective excitations of QCD is clearly due to a nonperturbative mechanism. Finding a theoretical description of this mechanism is an important and interesting challenge. Some steps in this direction have already been made. A model combining some

* **

~

E-mail: [email protected]. E-mail: [email protected]. E-mail: [email protected].

Elsevier Science B.V. SSDI 0550-3213(94)00177-G

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Y Frishman et al. /Nuclear Physics B424 (1994) 3—17

of the features of the chiral-quark [51and skyrmion [6—8]approaches has been constructed by Kaplan [9]. In this model it is postulated that at distances smaller than the confinement scale but large enough to allow for nonperturbative phenomena the effective dynamics of QCD is described by chiral dynamics of a bosonic field which takes values in U(NC X Ne). This effective theory admits classical soliton solutions. Assuming that they are stable and may be quantized semiclassically, one then finds that these solitons are extended objects with spin and that they belong to the fundamental representation of color and flavor. Their mass is of order A0~~ and radius of order l/AQCD [101. It is very tempting to identify them ~-,

as the constituent quarks. Thus the constituent quarks in this model are “skyrmions” in color space. This is a very attractive idea, but some crucial elements of the puzzle are still missing, most importantly a derivation of the effective dynamics from QCD. However, recent work on QCD in 1 + 1 dimensions (QCD2) provides evidence in support of this picture [11]. Exact non-abelian bosonization allows the lagrangian of QCD2 to be re-expressed in terms of a chiral field U(x) E U(N~X Nf) and of the usual gauge field. This bosonized lagrangian has topologically nontrivial static solutions that have the quantum numbers of baryons and mesons, constructed out of constituent quark and anti-quark solitons. When N~solitons of this type are combined, a static, finite-energy, color singlet solution is formed, corresponding to a baryon. Similarly, static meson solutions are formed out of a soliton and an anti-soliton of different flavors. The stability of the mesons against annihilation is ensured by flavor conservation. These results can be viewed as a derivation of the constituent-quark model in QCD2. Thus the idea of constituent quarks as solitons of a lagrangian with colored chiral fields becomes exact in D = 1 + 1. In this paper we will provide a constructive scheme for deriving from QCD the relevant effective lagrangian. In two dimensions the procedure will yield the exact bosonic action of QCD2, namely the gauged WZW lagrangian. In four dimensions the resulting action turns out to be a chiral lagrangian whose leading terms are very similar to the action conjectured by Kaplan. The construction of the effective chiral action for bosonic variables is based on integration of the anomaly equation [12—14].The idea is to construct an action which will reproduce the anomaly under chiral transformations. One begins with the anomaly equation as the starting point and integrates it with respect to the chiral rotation parameter. The resulting action contains terms in the form of Chern—Simons action which are non-local. It also contains additional terms which are anomaly free. These terms are analogous to an integration constant in an ordinary indefinite integral. Therefore their coefficients are arbitrary. In order to resolve this freedom, we adopt a strategy analogous to computing a definite integral. The first step involves rotating the external fields by a finite chiral rotation U. The effective action is vector invariant by choice of scheme. When the change in external fields is compensated by a change in the chiral phase, the rotated action is also axial invariant. The next step is taking the difference of the rotated and unrotated action. The result is the effective chiral action for bosonic

Y Frishman et al. /Nuclear Physics B424 (1994) 3—17

5

fields U. The latter action includes a term which is nonlocal, namely the Wess— Zumino term, which locally is a total derivative. The chiral phase U which is now promoted to a dynamical field turns out to describe the low-energy nonlinear dynamics of the color-carrying chiral condensate, in analogy with the nonlinear sigma model which describes the low-energy dynamics of the usual pseudoscalar mesons. The topologically nontrivial configurations of the chiral field turn out to describe constituent quarks and baryons, with baryon number given by the topological winding number. The remaining part of the action, which is there in 4 dimensions and is vector as well as axial invariant, does not contain the chiral field by construction and therefore does not affect its dynamics at the classical level. It does include, however, important physical information which is relevant beyond the low-energy approximation. The procedure we are using cannot determine the U-independent part of the action, and therefore its analysis is beyond the scope of this paper. The hope that the low-energy dynamics is well approximated by the U-dependent part of the action is based on experience with the usual chiral lagrangians. The latter give a very good description of low-energy dynamics of QCD in terms of a chiral field which is an element of the flavor symmetry group. We anticipate that this remains true in the present scheme, where the chiral field represents the colored and flavored chiral condensate. This belief is strongly reinforced by the fact that in 2 dimensions this procedure generates the exact gauged WZW action. In order to carry out our program, we need to deal with several preliminary issues. First, we have to define the scheme in which the anomaly equation will be discussed. Having chosen the scheme, we also need to choose a suitable regularization prescription. In a given scheme the choice of the regularization cannot influence the results of an exact calculation. As we shall see, however, in 4 dimensions the effective action contains an infinite number of terms, corresponding to expansion in terms of powers of external momenta divided by the cutoff. If we truncate the expansion at a finite number of terms and take a finite cutoff, the result will be regularization dependent. We cannot take the infinite coupling limit directly, so as to get rid of those higher order terms, as the kinetic term is multiplied by cut-off squared. On the other hand, in 2 dimensions the limit of infinite cutoff reproduces the exact bosonized action of WZW. Given an anomaly equation one can redefine the axial current and add a counterterm which is an axial variation of a local action [15]. This creates an ambiguity in the definition of the quantum axial current. So we need some principles which remove this ambiguity. The first one is our choice of the vector scheme; we want to remain with a vector-conserving theory. The second one is to keep the right relation between the axial and the vector currents. In two dimensions the axial current is the dual of the vector current .J~’ ~ In 4 dimensions no such simple relation exists. We choose the heat kernel regularization [16]. In 2 dimensions this procedure is well defined and the end result is a complete bosonization for gauged fermions. Unfortunately in 4 dimensions this cannot be done and the kinetic term is proportional to the cutoff squared. We therefore keep the UV cutoff finite and look for its physical meaning. The price we =

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Y Frishman et a!. /Nuclear Physics B424 (1994) 3—17

pay is that we need to deal with regularization dependent terms and a low-energy expansion. The UV cutoff of the regularization prescription shows up as an undetermined coefficient of the kinetic term of the U field. Its value determines the range of energies below which this approximation is valid. In the case of the usual chiral lagrangian whose target space is in the flavor group, the cutoff is just the scale of the spontaneous chiral symmetry breaking. It determines the mass of the nucleon which appears as a soliton of the usual chiral field. We think the same happens in the present case, where U carries both flavor and color degrees of freedom. One expects that the cutoff of this theory is reduced [9] by a factor of as compared with the usual chiral lagrangian and that it determines the mass of the lowest topologically nontrivial soliton excitation. In carrying out this program we are encouraged by the remarkable success of the anomaly integration approach in the derivation of the usual chiral lagrangians in the case of flavor symmetry. The coefficients of the various terms in the effective lagrangian calculated in the heat kernel regularization [13,17] turn out to be in good agreement with the experimentally measured values [18]. The outline of the paper is as follows. In Section 2 we introduce the notation and set up the general formalism. In Section 3 we apply the formalism to derive the gauged WZW action in 2 dimensions. In Section 4 we derive the low energy-effective action for the chiral field in 4 dimensions. In Section 5 we discuss various schemes of choosing the group in which the chiral field is embedded, new classical solutions in the “U-scheme”, conserved currents including the baryon number current, and finally the Polyakov—Wiegmann formula extended to 4 dimensions. Its low-energy approximation, up to dimension-4 operators, is given explicitly.

2. General formulation In this section we set the notation and describe the general procedure. Those features which are special to 2 or 4 dimensions will be discussed in the relevant sections. The fermionic part of the QCD lagrangian with external sources coupled to vector and axial currents, scalar and pseudoscalar densities labeled V~,A~,S, P respectively, has the form SF =

—

V’—4y5

—

S

—

iy5P)cli,

(2.1)

where J/~Ta,AF A~Ta, S SQTa, p papa 7~E G are the generators of some group G, normalized through Tr(T~Tb) ~ab In the generic case we think of G as the group U(N~x Ne), where N~and N~are the number of colors and flavors, respectively. Other groups can also be chosen, as will be discussed below. Since we are integrating over fermionic degrees of freedom the gauge field can be considered as an external source. At the end of the calculation, V~ will simply =

Y Frishman eta!. /Nuclear Physics B424 (1994) 3—17

7

include the dynamical gauge field GF. For later convenience we introduce the chiral fields LF

=

V~—AF,

RF

=

V~+A,~.

(2.2)

At this stage we do not include a mass term, as this breaks the chiral symmetry explicitly, on top of the breaking due to anomaly. The variation of S and P, on the other hand, is compensated by a rotation of i/i (see below). Since (2.1) is quadratic in the fermion fields, the fermions can be integrated out explicitly, yielding the determinant Z[V, A, 5, P]

=

f[D~r][D~Iexp[iSF].

(2.3)

At this point we have some freedom of choice. The axial current cannot be conserved. We need to choose a scheme of conservation. Here, since at the end V will contain the dynamical field for gluons, we choose Z to be invariant under vector gauge transformations, ~V~=D,,~w=äFw+i[V~,

to],

~

~AF =i[A~, ~S=i[S to] ~P=i[P, w],

(2.4)

=

+

i[LF, to],

=

+

i[RF, to].

On the other hand, the variation of the determinant under the axial transformation i[A~, Al, =

5AF =DFA

8/LA +i[V/L, A],

35= —{P,A}, 8P={S,A}, ~L/L= —ö/LA—i[L/L,Aj,

(2.5)

3R~=3/LA +i[R/L, A] is anomalous. This noninvariance of Z under axial transformation can be written in a compact form as 5 log Z —i

=

_da[V, A, 5, pl,

(2.6)

where d is the anomaly. We integrate this equation using the chain rule

—=—D-—-———i A

8Z

( 6P\ ( 3Z~ +~S —~—~P ~ 6P) ~ 5SJ —~

(2.7)

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Y Frishman et al. /Nuclear Physics B424 (1994) 3—17

to get an action Z1 which obeys the same anomaly equation as the full action Z, ~ log Z1 —i ~A’~ _da[V, A, 5, P]. (2.8) Clearly, at this stage log Z1 is defined up to an integration constant, so that Z1 is determined up to a chirally invariant multiplicative factor. Another way of putting this is by observing that Z/Z1 is chirally invariant and cannot be calculated via this procedure. We need to choose a well-defined prescription, in order to fix the normalization of Z1. The prescription we adopt is analogous to fixing the integration constant by making a specific choice of the lower limit of an integration. To that effect, we now introduce a chiral field U and perform a right rotation on the external fields, =

L~=L/L, R~L~=U~RU-U’iôU, F

(2.9)

F

(S + jp)U= (S + iP)U, (S jp)U U’(S iP). —

—

In the following we will denote the set of the external fields L/L, R/L, 5, P as cP: {L R/L, S, P}, and the set of “rotated” fields in (2.9) by cT~’:I” {L~,R~, 5U PL~i.Similarly, ZU Z[’1~’]. ,

We will show shortly that it is sufficient to consider the right rotation only, thanks to the vector invariance. It is also worth pointing out that in (2.9) one can consider special cases where U is chosen to lie in a subgroup of U(N~X Nf), rather than the full group. Using the rotated fields we can now fix the normalization of Z1 by using log Z1[cP~’] as the integration constant. It is important to note that the “rotated” action log Z~’is invariant under a combined chiral rotation U, over both the external fields and the U variables,

I ~ ~!‘ 1U.

(2.10)

~U-’ U of Z~’is simply due to the invariance of I~’itself under the The invariance transformation (2.10), (~U)UU

(2.11)

=

Therefore, by construction Z and Z 1 differ only by an axially invariant factor. Note, however, that Z1[i] does not depend directly on U, and the anomaly can be reproduced by applying the variation of ‘1 in log Z1. Our goal now is to reproduce the anomaly through an action depending on the field U, which is presently promoted to the status of a dynamical variable. To this effect we rewrite the action Z as an integral over U, implicitly defining Z2 through Z

=

z1[~] z2f [DU] z [~pu] z2f [DU] =

exp(iSeff),

(2.12)

Y Frishman et aL /Nuclear Physics B424 (1994) 3—17

9

where (2.13) —i log Z1[TI~] +i log Z1[cP11] is the U-dependent action which reproduces the anomaly, as follows from Eqs. (2.8) and (2.11). Thus Z 5eff 0 at 2 is the chirally In our normalization U 1. Seff is invariant under vectorinvariant. transformation ~eff

=

=

R/L

-~

U’R/LU-J/L,

L/L

-*

U1LU-J/L,

(2.14)

S±iP-*U1(S±iP)U U-* U1UU, where iU 3/LU• At this point several observations are in order. First, even though the anomaly is given by Z 1, Z2 does contain important physical information. In principle it includes all powers of the external sources, therefore it contains nontrivial information on their correlation functions. Second, depend U. There5eff~Z2 Wedoes will not argue lateronthat soliton fore all the U physics is determined by quarks and to baryons. Finally, we note solutions of 5eff correspond to constituent that in 2 dimensions our prescription yields the exact bosonized action and therefore in that case Z 2 becomes trivial. In the forthcoming sections we will 5eff only. discuss We now turn to explain why it is sufficient to consider right rotation only in (2.9). The generalization of (2.9) to arbitrary left and right phases g~and g~is =

L~=gj~’L/Lg~—gj~’i0/Lg~, R~=g~’R/Lg~—gj~’i3/Lg~, (S+zP) g =gL (S+zP)g~,

215

_~

(5_jp)~=gj~1(5_jp)g~,

where g denotes transformation under both g~and both left and right rotations is 5’eIf[~”

A, 5,

~,

g~,g~]

=

g~.

The effective action with

—i log Z 1[V, A, 5, P] 5, Li’, ~e pg]• +i log Z1[R

Vector invariance (2.14) implies Seff[V, A, 5, P, g~,g~]

~eff[

V, A, 5, P, 1,

and we can define U=g 1g~to obtain (2.13). 1

gj~1g~],

(2.16)

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Y Frishman et a!. /Nuclear Physics B424 (1994) 3—17

3. The action in 2 dimensions In this section we calculate the effective action in 2 dimensions. In this case the limit of the UV cutoff going to infinity is well defined. We use Tr(ysy/LYV) = 2c/L~, = 1, g 00 = —g11 = 1. The axial variation of the action in 2 dimensions is given by

~

~

=

~[EF~(F/L~

—

2i[A/L, A~])~ + 2(D/LAFy

—

2{S,

p}aj•

(3.1) F/Li. is given by F/Lu = 0/LI/c,

—

3~V/L+ i[I/c~, I/c,]

+

i[A/L, A~]= ~(L/LV+ R/L~),

(3.2)

and we have introduced the left and right field strengths L/L~=0/LLV—0~L/L+i[L/L,Lu], R/L~=0/LR~—0~R/L+i[R/L, Ru].

(3.3)

Also, D/LAV = 0/LAp + i[V/L, Ar]. Note the term D/LA~= g~”D/LA~ in (3.1), it is proportional to the metric tensor rather than the antisymmetric tensor E/L~. This term can be calculated by recognizing that for A = 0 the anomaly is (1/2~)E~”F/LV and, for A 0, F is replaced by the field strength of the new vector field l/c~+ E/LPA”. Using the chain rule Eq. (2.7) we have an action which has the variation (3.1), —i log Z1=S~~[R]—S~~ —

+

~_~fF~Tr(V/LAV)

—

~_fTr(A/LAF)

+

~_fTr(s2), (3.4)

where scs is the Chern—Simons action (an extension to 2 S~5[AI =

+

1 dimensions)

Tr(A/L0VAU+ 3~/LA~A~),

(3.5)

which has the variation ~S~~[R]

=

~—fE~

and when using t~R/L= ~S~~[RI

=

~fEF”

Tr[(öR/L)R~] + 0/Lw

+

i[R/L,

to]

-~--—J~ Tr[R/L~RU)],

(3.6)

the variation is

Tr[(3/LR~)w].

(3.7)

We use the Chern—Simons anomaly contribution

Scs[RUI —S~~[R]=_~fEF” Tr(J/LR~)+ ~feF~

Tr(J/LJ~J~), (3.8)

Y Frishman et al. /Nuclear Physics B424 (1994) 3—17

where

=

U’id/LU and

=

Seff= ~fTr(J/LJ/L) +

~—J(g~

—

~Tr{[U1(S

—

UJ/LU’, to calculate the effective action (2.13),

fc/Lvu

—

+

~ta~)

—

11

Tr(J/LJ~J~)

Tr[J/LR~ iP)j2

+

—

[(S

L/LU’R~U+L/LJ~+L/LR~] +

iP)U]2

+

2(P2

(3.9)

_52))

Note that it is R that couples to the right fermion current, and L to the left one. As can be seen, the action (3.9) is the gauged WZW action of level 1 [19].We learn that by anomaly integration, the effective action is an exact bosonization of Dirac fermions. Hints for this were given in Ref. [20]. Note that when P 0 and S M, we get a U2 + U2 term, but not the expected U + U’ term which corresponds to a fermionic mass term. This is because we included only effects that come from the anomaly, through the jacobian of the change of variables in the fermion sector, but not the explicit breaking coming from the mass term. +

—‘

—*

4. The action in 4 dimensions We now turn to 4 dimensions. We follow exactly the same procedure as in the 2-dimensional case but here we encounter some difficulties. While the result of Eq. (3.9) is an exact bosonization of fermions coupled to vector and axial sources, its 4-dimensional analogue is an approximation, valid only at low energies. We use an expansion in powers of the cutoff, assuming the higher terms are suppressed by powers of the cutoff squared. The scale at which the approximation breaks down is determined by A, the ultraviolet cutoff in the regularization procedure. Let us now go over to the derivation. The variation of the determinant under the axial transformation (2.5), including terms up to zeroth power of the cutoff, is given by [21,12] ~logZ —i ~Aa

1 =

~

-~-H/L~H

5,~,— ~(A/LA~FAJ+AFFVAA~ 2D/LA~+ ~[D/LF/L~,A~] + ~A/LA~AAA,,] + 16A + ~[F/L~,DFA~] + ~{D~A/L,APA~} + 2A/LDPAVA/L +i[A/L,A~l,H~”l A”A~

+~(D/LA~ A~A”+A/LA~D”A~)

+A/LA~DFA”}+ O(A2),

(4.1)

where H/Lu = ~(R/L~ L/L~).In (4.1) we have terms which are contracted with the metric rather than with the antisymmetric tensor. Their coefficients are determined from a heat kernel regularization procedure [12,16]. We do not have here a relation like in 2 dimensions between vector and axial currents, and therefore we —

Y Frishman et a!. /Nuclear Physics B424 (1994) 3—17

12

do not see simple relations among the various terms in (4.1). Using the chain rule (2.7), an action which has the above variation is given by 2 A2 —i log Z 1 = —~--fTr(A/LA/L)+ S~~[R] S~~[L] —

—

+

Tr[i(R/L~ +L/L~){LA,R~}

~JE/L”A~

+(L/LL~—~L/LR,,+R/LRz,)LARffI l2~2fTr{4 +F[A/L, A~] 2— (A/LA/L)2}, (4.2) +~(D/LA~) where S~s[RI is the Chern—Simons action in 5 dimensions (see for example the review in Ref. [22]), —

S~~[R]=

24~2

—

J

~/LPA~~

Tr(R/L0~R 50~R~ + ~iR/LR~RA0URP

~R/LR~RARURP).

(4.3)

After rotation by U we get the final result Eq. (2.13),

1R~U+ 2L/LJ~+ 2L/LR/LJ

-

Seff = ~—~fTr[J/LJ/L + 2J/LR/’ —

240~2

—

2L/LU

fe~~v Tr(J/LJ~JAJ~JP) + 48~2 fE/L

—R/LR~RAJU+ ~R/LJ~RAJ~+ J/LJ~JARff)+ X

Tr[i(R~

+

48~2

4~2

L/L~){L 5,Rff}

—

Tr(iR/L~{RA,i~}

fE

Tr[i(U

+

(L/LL~ ~L/LR~ + R/LR~)LAR~J —

‘R/LVU + L/L~){LA,R~} 2

f

~L/LR~ +R~R~)LAR~] + 192~2 Tr[_2(D/LU_1 D/LU) x(D/Lu’ D~U)2+2(D/LD/LU’)D~D~U —

+4(R/L~D/LU’D~U+ L/L~D/LUD~U’) + U ‘L~UR~

—

+

16[FM~[A/L, A~]+ ~(D/LA~)2

—

(A/LA/L)2],

LF~R/L~I (4.4)

where D/LU = 0/LU + iR/LU iUL/L, D/LU1 = 0/LU1 + iL/LU’R/L. This action is the low-energy approximation to bosonized QCD, including coupling to external sources. We can now replace the external source J/c~ by l/c~+ G/L, to include the —

Y Frishman et al. /Nuclear Physics B424 (1994) 3—17

13

coupling to the dynamical gauge field G/L. This construction is general, in the sense that one can choose to gauge any subgroup of U(NC X Nf). The choice which is relevant for the real world is obviously to gauge the SU(N~)subgroup. As the final step, in order to make contact with the usual formulation of QCD, we write down the effective action setting the external sources to zero, i.e. by taking L/L = R/L = G/L in Eq. (4.4), to obtain 2 1 Seff ~A Tr(D/LUDt’~U’) 240~2 Tr(J/LJ~J

f

=

—

fc~’~

5J,7J~)

+

4~2f

—

48~2

Tr{iG/L~{GA,i~} (G/LG~ —

fe~’~ Tr[i(U’G/LPU+

~

—

~G/LJ.~

+J/LJ~)GAJ~}

G/L~){GA,G~} + 19~2fTr[_2(D/LU~1DvU)2

2+ 2(D/LD/LU1)D~D~U+

4(G/L~D/LU’

D~U

+(D/LU’ D~U) +G/LVD~UD”U’)+ U’G/L~UG~~G/L~G~1. —

(4.5)

The action (4.5) is similar to, although not identical with the action proposed by Kaplan [91on the basis of a rather different qualitative reasoning. The static properties of constituent quarks obtained from Kaplan’s action have been worked out in Ref. [101. It would be very interesting to carry out a similar analysis in order to examine the quantitative consequences of the action (4.5).

5. Bosonization schemes and solutions 5.1. Different schemes In this section we introduce several schemes of the effective action. In the previous section we derived the effective low-energy action for a general coset G/H with H a subgroup of G. The chiral phase takes values in G, while the gauge field takes values in H. In QCD, H is SU(NC). Depending on the physics one is interested in, there are various choices for G. We list here those which we consider the most important. (1) The “U-scheme”. In this case we choose U E U(NC X Ne). (2) The “product scheme”. The group G is SU(NC) X SU(Nf) X U(1). One can parameterize U by U = h g, where h E SU(NC) and g E U(N~). (3) The effective U(1) action. It is an effective action for the s~’.It is related to det U [23]. .

14

Y Frishman et a!. /Nuclear Physics B424 (1994) 3—17

5.2. Classical solutions in the U-scheme The classical soliton solutions are embedded in an SU(2) subgroup of U(NC X Ni). They have the form U~= exp[zf(r)iwhere r are the generators of this SU(2) subgroup and f(r) is a radial shape function with boundary conditions f(0) = ir, f(cc) = 0. There are three choices for 1•. (1) 7 = x Ef, where Ef is a matrix in flavor space which satisfies the relation E~= Ef. This is a colored soliton. This kind of soliton has been discussed in detail by Kaplan [9]. Its baryon number is Trf(Ef). (2) i- = E~ x o~.If E~is the identity, this is a color singlet flavored soliton. This is the familiar Skyrme-type soliton [6,7], widely discussed in the literature. Its baryon number is Tr~(E~). (3) A mixed colored flavored soliton. From the general discussion in Ref. [9] and from the calculations in 2 dimensions [11] we expect that this new type of soliton corresponds to a constituent quark. This case actually includes (1) and (2) above, in cases where E~is diagonal but not unity. Of course, the quantum fluctuations will mix these three. The solitons of the theory can be taken in color space using the following ansatz: Uc’=exp[if(r)(i•c.r)E~], (5.1) where i-~ is in an SU(2) subgroup of SU(N~)and E~is a matrix in flavor space with 1 in the (i, i) entry and zeroes elsewhere. In the normalization convention in which a nucleon has baryon number N~,the baryon number of this soliton is 1 and its flavor quantum number is i, i.e. it has the quantum numbers of a quark. It is important to keep in mind, however, that the classical solution with the quantum numbers of a constituent quark has a finite energy. This is in clear contrast with the expectation that there are no finite-energy colored states. This is a clear deficiency of the present treatment, and it can be traced back to the fact that in the anomaly integration approach, confinement has to be put in by hand, as it comes from the terms which we do not treat. One may also consider solutions which carry baryon number different from 1. In view of the results in 2 dimensions [11] we expect to find color singlet solutions constructed out of the soliton—anti-soliton pair and solutions which are constructed from N~solitons. It is natural to interpret such solutions as mesons and baryons and their constituent solitons as constituent quarks. ~,

5.3. Conserved currents The right and left currents are given by

= ~j/L_ = —P’ ~

~2

+

2~ ~e’J 48~2

~

J5J ~

~ —

2 —({J 48ir

“

J~ JM} —J J/LJ~),

(5.2) (5.3)

Y Frishman et al. /Nuclear Physics B424 (1994) 3—17

15

and the baryon number current by the trace of their sum,

~=

24~2eTrj~~

(5.4)

These are approximate low-energy bosonization formulas for the corresponding currents. Eqs. (5.2) and (5.3) can be conveniently written in a compact form,

967~2’~)I’

~5effT[1Y5/LJ(1t

(5.5)

96~T2~’~’)I’

effT[Y5/LJ(

(5.6)

5.4. The Polyakov—Wiegmann formula [24] We want to find the effective action for 2 chiral phases. The basic identity 5eff (2.13), follows from the definition of the effective action ‘~eit[1~,R, U,, U 2] = ~~eff[L, RU1, U2] + ~~eff[L, R, U1]. (5.7) For zero sources we have Seff[0, 0, U,, U2] = Seff[0, 0, U,] + S’eti[O, 0, U2] which defines si [Ui, (12] as S,[U1, U~]= Seff[0, —Jf’~,U2] 5’eff[O, 0, U2].

+

S,[U1, U2], (5.8)

—

For 2 dimensions we have [241 Seff[0, 0, U]

=

~—fTr(J/LJ/L)

—

-j~--fE~ Tr(J/LJVJ~)

(5.9)

and Sl=[U,,U2]=_~f(g/LV+c/L~)Tr(J~ir). For 4 dimensions we have 2 A Seff[0, 0, U] = Tr(J/LJ~)

~—~f

—

+ 192~2 fTr[

(5.10)

1

~

240~r2

Tr(J/LJc,J~J,,J~)

—2(J/LJ~)2+ (J/LJj2 + 2(0/L0/LU1)3~dVU}, (5.11)

A2 5

-

1 2

1[U1, U2]

=

~f Tr(J~J~) + 48ir X Tr(J~J~J~J~ + ~J~J~JjV~ J~’J~’i~J~’). —

—

(5.12)

16

Y Frishman et a!. /Nuclear Physics B424 (1994) 3—17

6. Summary and outlook We have derived an effective low-energy action for QCD in 4 dimensions. We have shown that the solitons in the theory have quantum numbers of quarks. We interpret these solitons as constituent quarks. An interesting plan might be to calculate the equations of motions numerically and compare the results to experimental values. Other quantities such as the g,,~of the constituent quarks might be calculated. In this case g~ has a classical contribution too, in analogy with the Skyrme model. Multi-soliton solutions can also be considered as baryons and mesons constructed out of quark solitons.

Acknowledgements The authors would like to thank John Ellis for useful discussions. Y.F. would like to acknowledge interesting discussions with W. Bardeen. A.H. would like to thank Adam Schwimmer, Jacob Sonnenschein and Yigal Shamir for useful discussions. The research of Y.F. and A.H. was supported in part by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities. The research of M.K. was supported in part by the Einstein Center at the Weizmann Institute, by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities and by grant No. 90-00342 from the United States— Israel Binational Science Foundation (BSF), Jerusalem, Israel. Note added Upon the completion of this work we have received a draft of the paper by Damgaard and Sollacher [25] in which the effective action in four dimensions is derived by closely related techniques. We thank the authors for communicating their results to us prior to publication.

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