Large-N baryons: Collective coordinates of the topological soliton in the SU(3) chiral model

Nuclear Physics B258 (1985) 713-725 (~ North-Holland Publishing Company

L A R G E - N BARYONS: C O L L E C T I V E C O O R D I N A T E S OF T H E T O P O L O G I C A L S O L I T O N IN T H E SU(3) C H I R A L M O D E L Sanjay JAIN and Spenta R. WADIA

Tata Instituteof FundamentalResearch,HomiBhabhaRoad, Bombay400005,India Received 17 February 1984 (Revised 6 November 1984) The collective coordinate method of Gervais and Sakita is used to quantise the large-N topological soliton in the SU(3) chiral model. The leading-order hamiltonian is that of a symmetrical top. The Wess-Zumino term plays a role in determining the flavour and spin quantum numbers of the baryon multiplets. 1. Introduction

The chiral model of mesons with three flavours is described by an SU(3) valued field U(x) whose action is given by

s( u) = N[So( U)+ r ( u)],

(1)

where

So( U)= f {~6f2 Tr(L~L~')+3~e2 Tr[L~,, L~]2} d4x ,

(2)

L, = ia~ UU -~, F~ =-xf'-Nf~ is the pion decay constant, e is a dimensionless number and N is the n u m b e r of colours. S0(U) was considered by Skyrme [1] and NF(U) is the W e s s - Z u m i n o term as constructed by Witten [2, 3], to be described later. This model is known to reproduce the phenomenology of low-energy m e s o n - m e s o n scattering including anomalous conservation laws when electro-weak interactions are introduced. It also possesses a soliton solution (3) to its classical equations of motion. This p a p e r deals with the old idea of Skyrme that the soliton be associated with baryons. Aspects of this association have been studied by m a n y authors [4]. The connection between Q C D and a weakly coupled mesonic model has been made plausible in the context of the 1/N expansion by the work of Witten [5]. In the l a r g e - N limit Q C D can be considered a local field theory of mesons in which meson interactions are of order 1/N. Further, baryons occur as solitons of this local field theory with masses and interaction energies of order N. In the model (1), provided the pion decay constant F~ is of order ~ (as can be seen from large-N QCD), N scales out and a semiclassical expansion about the soliton solution in powers o f 1/N can be made. Analogous to h, 1/N is a WKB parameter in the l a r g e - N limit. It turns out that Witten's qualitative conclusions about the large-N dependence of meson couplings, soliton masses, etc. are reproduced in this expansion of the model (1). 713

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S. Jain, S.R. Wadia / Large-N baryons

It must be pointed out, however, that the connection between QCD and the chiral model can only be approximately true in the broken ~ymmetry phase of QCD. This is because QCD is known to undergo a finite-temperature continuous transition to a phase where chiral symmetry is restored. A phenomenological model which incorporates the description of both phases and which in a sense is more fundamental than (1), is a recently proposed Nambu-Jona-Lasinio type model [6]. From this starting point a hydrodynamic lagrangian can be written in the long wavelength limit in terms of the velocity fields L, = iO,UU-~ and the density field H. U(x) and H(x) are the phase and density of the gauge-invariant chiral order parameter ~(I + 75)~b. The topological soliton that describes the hadronic bag in the broken symmetry phase involves both L,,(x) and H(x). The action (1) in which only the velocity field enters provides a description of the bag only far away from its core, where H(x) is frozen to be a constant. The variation of H(x) is important for the bag structure. However, the approximate action (1) is accurate for describing global characteristics of the baryon like its quantum numbers and multiplet structure. This is a consequence of the fact that the Wess-Zumino term which carries "global" and "length scale independent" information is independent of the density field. This paper is devoted to the quantum numbers and the multiplet structure of the Skyrme soliton solution (3) using the collective coordinate method of Gervais and Sakita [7]. The Wess-Zumino term, being linear in the time derivative of U, imposes a crucial boundary condition on the baryon wave functions. For N = 3 the particle states that emerge fall into triality trivial representations of flavour SU(3) (i.e. octet, decuplet, etc.) and half-integral spin representations of the rotation group. These are the representations in which baryons are known to exist. We also present a mass formula for baryons in the chiral limit. 2. The collective coordinates

To quantise the theory about a classical solution the field expansion is broken up into modes generated by the symmetries of the hamiltonian and modes orthogonal to the symmetry modes. Fluctuations in the symmetry modes are large - it costs little energy to excite them. To leading order only symmetry modes (collective coordinates) are considered and treated quantum mechanically, exactly. The hamiltonian of the collective coordinates is the unperturbed part of the full hamiltonian; perturbations consist of the orthogonal modes. The particle states to be associated with the classical solution are just the eigenstates of this hamiltonian. In the present case the classical solution is

Uo(x) =

,

(3)

0 where f ( 0 ) = m

f(r)~O as r ~ c c , and rl,

T2, "/'3

are Pauli matrices. Uo is also a

S. .lain, S.R. Wadia / Large-N baryons

715

topological soliton with topological charge 1. Topological charges of configurations arise in the following way: an arbitrary classical configuration U(t, x) to have finite energy must, at every t, map the boundary S2 of real space to a single, constant point of SU(3). Since R 3 with its boundary S2 pinched to a single point is topologically just S3, this implies that for a fixed t, U defines a map from S 3 to SU(3). As t changes, while this map from S 3 to SU(3) defined by U may change, its homotopy class remains the same because time evolution is continuous. The topological charge of U is defined as the integer which characterises this homotopy class and is given by

n(U) = ~

'I

e •k Tr (L,LjLk) d3x,

Lj = iOjU(t, x) U - ' ( t , x ) ,

j=1,2,3.

(4) It can be shown that n(Uo) = 1. The symmetry o f ( l ) i s SU(3)L xSU(3)R,i.e. U ~ V U W -~, V, W e SU(3). A general expansion of U about Uo in the symmetry modes would be U ( x ) = V ( t ) U o ( x ) W - ~ ( t ) . But we are interested in the broken phase, i.e. only in those fluctuations for which U reduces to the classical vacuum (the identity matrix) at r = co. Since the remaining symmetry is only diagonal SU(3), we have

U(t, x) = 12(0 Uo(x)12-~(t),

O(t) c SU(3).

(5)

/2(t) are the collective coordinates. Note that as r ~ c o , U(t, x ) ~ 1. I f / 2 ' ( t ) differs from 12(0 by a right hypercharge multiplication, i.e.

IT( t) = l'l( t)h( t) , h ( t ) = e i~
e(t) real,

Y = I diag (1, 1, - 2 ) ,

(6)

(7)

then 12'(t) Uo(x)12'-l(t) = 1-2(t) Uo(x)12-l(t) = U ( x ) , because h commutes with Uo. Thus the space of collective coordinates is SU(3), but with the constraint that matrices which differ only by a right multiplication by a hypercharge rotation be identified. This is the 7-dimensional manifold SU(3)/U(1). We will, however, treat the space of collective coordinates as the 8-dimensional SU(3) manifold. The constraint will naturally reappear as a local symmetry of the action of the collective coordinates. We express SU(3) matrices in terms of the exponential coordinates 0 a, a = 1, 2 . . . . . 8 as O = e i°°to, ta = x/~ Aa, ha being the usual Gell-Mann matrices. With this choice the normalization is Tr (tatb) = 6ab, and Y = x/~ ts. The structure constants are defined by [to, tb] = if~bctc.

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3. Action for the collective coordinates

The action for 12 is obtained by substituting (5) into the action (1). Substitution in So yields (the constant term Met = energy of classical solution, is omitted) So = -½ y Icd Tr (tctoR) Tr (tdtoR) dt,

(8)

where t o R - l - / - ~ ( t ) d l 2 ( t ) / d t is the "right angular velocity" of the collective coordinates and the constants

NfTr{ ~f~[Uo, '~ t~][ta, Uo ~]

Icd =--

1

+-~e 2 (tJ, tal, + tcr,tar, + tcUo~l~tdUo + tcUo--1tdli2 Uo- tctd(12i+ r 2)-2t~liUotdUo~li)

}

d3x

(9)

constitute the "moment of inertia tensor" of the collective coordinates. It(x) and ri (x) are defined in (14). When contracted with Tr (tctoR) Tr (tdtoR) only the symmetric part of Icd contributes; henceforth I~a denotes only its symmetric part. Also I~s = Is~ = 0, because [ Uo, ts] = 0. We now proceed to calculate the Wess-Zumino term for the collective coordinates. It is given by the expression 1

F(U)=

240Ir2 J- Tr (LiL~LkLtL,,) dE ;jktm

(10)

where Lj = idjUU -~, j = 0, 1, 2, 3, 5 and U is a map from a 5-dimensional disc D 5 that reduces to the given map U on the boundary S4 of D 5. F was constructed by Witten [3] from the Wess-Zumino equation of motion*. The prescription to compute F for a non-solitonic configuration is as follows: a configuration U that maps all of spacetime infinity to a single point in SU(3) defines a map from S4 into SU(3). Look at the image of S4 in SU(3) under the given U. Construct a 5-dimensional disc D in SU(3) with this image of S4 as boundary. The integral of the 5-form to (defined below) over D is F ( U ) . F depends (modulo 2rr) only upon the boundary of D and not on the particular choice of D. The 5-form to expressed in components in terms of the exponential coordinates is

to = to~bcd~dO~ ^ dob ^ dO~ ^ don ^ dO~, to.bed,=

i

(OVv_,

2407r2Tr ~

O V v _ I OV

~-g

10Vv_,

~-~V- ~

OVv_x)

00--7

antisymmetrised in abcde, where V = C °°'o ~ SU(3). * Ramadas [8] has recently presented a hamiltonian formulation of Witten's work.

,

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717

Now for solitons (configurations with non-zero topological charge) the construction of the 5-dimensional surface has to be different. We must ask the question: what kind of image of spacetime does a time-dependent solitonic configuration generate in SU(3). Can this image be topologically the same as $4? The answer is no. The image of spacetime under a m a p U is S 4 only if the whole boun0ary of spacetime is m a p p e d by U to the same point in SU(3). The boundary of spacetime includes not only all points (t, Ix[---oo) for all t but also the points (t = +oo, x) for all space. If U maps all the points (t = -oo, x) to a single point in SU(3), at t --- -oo, U defines a m a p from S 3 to SU(3) in the trivial h o m o t o p y class. Under time evolution U(t, x) changes, but at any given t, the m a p from S 3 to SU(3) defined by U must continue to fall in the same trivial h o m o t o p y class. In other words if U maps all the points (t = -oo, x) to the same point in SU(3), it cannot have non-zero topological charge. Thus if a configuration U is known to be solitonic, the image of spacetime that it generates in SU(3) is not S4. The image of spacetime that (5) generates in SU(3) is topologically the same as S ~x S 3, Uo being the m a p from S 3 (space) and 12 being the map from S 1 (time). We assume that in this case F ( U ) is again given by the integral of w over a 5-dimensional surface M 5 constructed over this image of S ~× S 3, but now the construction of M 5 is as follows: we think of S ~ as the boundary of a 2-dimensional disc D 2 and extend 12 to the whole of D 2. Points on D 2 are characterised by two coordinates t and s; t labels the boundary and s is a radial coordinate. With g2 denoting the extension of 12,

U(x, s)=-~2(t, s)Uo(x)~-'(t, s)

(11)

becomes a m a p from D 2 x S 3 into SU(3) that reduces to (5) at its boundary S 1 x S 3. The image of D 2 × S 3 under /.~ is M 5. In this case again the integral of to over M 5 is independent (modulo 2 ~ ) of the particular choice of the extension ~ , and depends only on 12. We now substitute (11) in (10) to get the W e s s - Z u m i n o term for 12. With some algebra which involves identifying a 3-divergence term in the integrand it can be shown that (10) reduces to r ( u ) = 4 - 1~

f ~ Tr([o~R,~K) dtds,

(12)

where

O;R------~-l(t, S) -~- (t, S),

foR=-- ~-'(t, s) ~s" (t, s),

(13)

K -- i f eqk(lJ~lk+ rirjrk) d3x, l, =- O,Uo(x) Uo~(X),

r, =- Uo~(X)O,Uo(x),

i = 1, 2, 3.

(14)

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S. Jain, S.R. Wadia / Large-N baryons

To write down a lagrangian for g2 the integral fD ~Tr ([03R, 03~]K) dt ds - X

(15)

~'hould be written as a line integral over the boundary of D 2, using Stokes theorem. To do this we write (1 3) as the components of a 2-dimensional vector 03~ = (03~, 032)= (03R, 03~). The labels I and 2 stand for t and s respectively. Then since 03~ is "pure gauge", it is curl free, i.e. 0,o32- 0203,+[03. 032]= 0.

(16)

Using the abelian projection of (16) in the direction of K we have X = -f

(0, Tr (032K) - 0 2 T r (03,K)) dx'

dx 2 ,

dD 2

which, by Stokes theorem, reduces to X = - - f o D ~Tr ( 0 3 ~ K ) d x ° = - f

Tr (03RK)dt.

(17)

03R=tOR on the boundary of D 2. Using this in (12) and using (8) and (1) the total action of the collective coordinates can be written as S = f L(/2, ~ ) dt,

(18)

L = -½I~d Tr (tcOgR)Tr (td~0R) -- ~--g~N 2 Tr (OJRK). /-I.~ "W

(19)

4. Hamiltonian It is convenient to work with exponential parameters 0 a as canonical coordinates instead o f / 2 . To write the lagrangian (19) in terms o f 0 ~ and 0a we use the identity tOR=

ivC~(O)O~tc,

where

vco(o) -~ ~----~ a (-o, ~)1.=o, 0~ a

and q~ is the function that defines the group composition law ei't'a(ot,o2)q ~ eiO~t~ eiO~tb. Then (19) can be written as Iv' c d AaAb iN L=~lcdv av bY V --4--~ 2 vcaO a T r ( t c K ) .

(20)

S. Jain, S.R. Wadia / Large-N baryons

719

The canonical momenta Pa =--OL/O~a turn out to be

iN Pa = Icdv CavdbOb-- 4--~2 v C~Tr ( tcK ) .

(21)

We now define some new quantities. The matrix (Icd) is singular, its eighth row and column being zero. We denote

( I ~ a ) = ( ( ~ °)

00) ,

c~,/3 = 1, 2 . . . . , 7 .

The matrix ( ~ ) is invertible and we denote its inverse as (.¢~). The inverse of (vC, (0)) is denoted (u ~, (0)). The generator of right rotations is defined by E~ = u ~aPc and we denote i N Tr (t~K)/48¢r 2 by Ac. The matrix K can be exactly calculated because the Skyrme solution is invariant under combined spin + isospin rotations. This implies that [K, t;] = 0. ti are the first 3 generators of SU(3), which form an SU(2) subgroup. Also the third row and column of K are zero. Hence K = i24zr 2 diag (1, 1, 0). Because Icd is singular the velocities 0~ cannot be expressed in terms of O's and p's. We have a constrained hamiltonian system. Multiplying (21) by u~8 we get the hamiltonian constraint u"spa ~ E8 = ( - i N / 4 8 ¢r2) Tr (tsK), which reduces to

Es(O, p) - ~ N = O.

(22)

(21) gives v ~ 0 ~ = 3 ~ E a . Substituting this in the hamiltonian H = p~/}~-L, we get

H=½N~rJE~Eo,

a, 3 = 1, 2 , . . . , 7 .

In the quantum theory the right generators E~ are replaced by o p e r a t o r s / ~ . The representation E~=-iuC~(O)O/OO ~ reproduces the required SU(3) commutation relations [/~,/~b] =/fab~/~ and the right-generator property [/~, O] =/2t~. We define the quantum hamiltonian by ^

- 2. . . .

8,

(23)

with /~,~ given by the above-mentioned differential operator. It will be seen later that ( ~ ) is actually diagonal, and the choice (23) in fact reduces to a simple sum of quadratic casimirs.

5. S y m m e t r i e s

5.1. FLAVOUR

The action (1) is invariant under diagonal SU(3) transformations U ( x ) ~ U'(x) = VU(x) V -1, V e S U(3 ). With U = / 2 Uo/2 -1 we have U' = ( V/2 ) Uo( V/2 ) -1 =/2, Uo/2 ,-1 w h e r e / 2 ' = V/2. Thus the action of V on U is produced in the space of collective coordinates by left multiplication of/2 by V. It follows that/2 --*/2' = V/2 is a symmetry

s. Jain, S.R, Wadia / Large-N baryons

720

of (18) and this can be explicitly checked to be the case. To obtain the expression for the generators of this symmetry we define as usual V ( t ) = l+iea(t)t~ and O ' ( t ) = V(t) O (t). Then the generators Q, are given by 0L(O'(t), g)'( t))/0~ ~1~=o.~:=o. The result is /v

Qa

=

- i Tr ( 0 -1 t~Otc)( I~d Tr ( tdtoR) + ,-'~2 Tr ( t~K ) ) 48~r

= Tr (O-1 taOt~)E~.

(24)

5.2. ROTATION (1) is invariant under x ~ R x , R being a space rotation. With U = O U o 0-~, U(t, Rx)= g2(t)Uo(Rx)O-~(t). But because of the special spin-isospin mixed form of Uo(x) we can write Uo(Rx)= VRUo(x)VR 1 with VR =

,

~ R = ( ~ , , ~2, ~ 3 ) ,

0

Z (~,)2 =

~,

i=l

Rij = 6~j cos 2a +2(aic~j/c~ 2) sin 2 a - eijk(C~k/a) sin 2a. Thus the ettect of a space rotation R is reflected in terms of the collective coordinates in the right multiplication of O by VR. So the generators of right SU(2) transformations on O (VR is an SU(2) matrix embedded in SU(3)) are in fact the generators of the rotation. O-~ OVR can be explicitly checked to be a symmetry of (18) and the p r o o f depends upon the fact that K and Icd are constructed out of Uo which has spin-isospin symmetry. The generators of rotation turn out to be and

a~

satisfying

U(t, Rx)= (O(t)VR)Uo(x)(O(t)Va)-l;

J, = -x/~ i(I,d Tr (tdtoR)) = x/~ E,,

i = 1, 2, 3.

(25)

We mention here that we do not consider the collective coordinate arising from the translational symmetry of (1) as we are interested only in the properties of the baryon at rest. Consequently (25) is the rotation generator in the rest frame of the soliton, i.e. in the space of states with zero linear momentum. For non-zero linear m o m e n t u m P (25) would be modified by the addition of the orbital angular m o m e n t u m L = X × P, where X(t) is the collective coordinate for translation. (25) would then become Ji = L~+4~ El. 5.3. LOCAL U(1) SYMMETRY As mentioned earlier it turns out that O ( t ) ~ O(t)h(t) with h(t) defined by (6) and (7) is a local symmetry of (18). This corresponds to right hypercharge rotations on the space of collective coordinates. The generator of these is YR = --i(Icd Tr (toY)Tr (tdtOR)+ (N/48~r 2) Tr ( Y K ) ) , but because y o z t8 and lsd = 0 this

S. Jain, S.R. Wadia / Large-N baryons

721

reduces to YR = ½N. In the hamiltonian formalism YR can be identified with x/] E8 and the constraint (22) itself reads

YR=½N.

(26)

In the quantum theory (26) has the content that it imposes a constraint on the state space. It restricts the physical states to eigenstates of the right hypercharge generator I~R with eigenvalue ~N. Thus (26) is like the Gauss constraint on physical states in gauge theories. In both cases the constraint is a consequence of local symmetry. A local symmetry has arisen in the present case because we have described the system with variables whose number exceeds the number of degrees of freedom (eight instead of seven). One can now see why N must be an integer. For a wavefunction ~b(12), (26) implies I~'R0(12) = ~Nq'(12).

(27)

I)R, being the generator of right hypercharge rotations also satisfies

eiaf'R~(12) =

~(12 e i~Y)

for all real a ,

(28)

where Y is defined by (7). Since e i 6 ~ Y : 1, (27) and (28) imply that ~b(12)= C2"N~/,(12). The quantisation of N then follows from the single-valuedness of $. 6. Quantum numbers The space of variables o f the system is SU(3) and the transformation 1-2~ V12W is a symmetry, where V belongs to SU(3) and W must lie in the SU(2) or U(1) subgroup of S U(3) mentioned above. In this respect this system is essentially identical to the symmetrical top. The latter is a rigid body whose moment of inertia tensor is such that two of the three principal moments of inertia are equal to each other and different from the third one. The coordinates 12 in this case are SU(2) matrices constructed out of Euler angles that describe the position of the top. In this again the transformation 12 -~ V12W is a symmetry of the hamiltonian but here V is an SU(2) matrix that depends upon a general space rotation; left multiplication by V is a symmetry because of isotropy of space, and W is an element of a U(1) subgroup of SU(2) and is of the type e i°'3, right multiplication by W is a symmetry because of the symmetric nature of the top. Landau and Lifshitz [9] exhibit what this means for the eigenspaces of the hamiltonian. Energy eigenfunctions as functions of the Euler angles (a,/3, y) are given, (up to normalisation), by the matrix elements Dm,,,(a, J [3, y) of the matrices DJ(a,/3, y), which form the ( 2 J + 1)-dimensional irreducible representation of the rotation group. The index m in the wave function Dm,,,( J a, /3, y) is the eigenvalue of the z-component of ordinary (space-fixed) angular momentum and the index m' is the eigenvalue of the generator of right U(1) symmetry (this generator is in fact the component of the body-fixed angular momentum along the symmetry axis of

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S. Jain, S.R. Wadia / Large-N baryons

the top). D~,m(a, [3, 7) is the simultaneous eigenfunction of the latter two operators, j2 and the hamiltonian. The present system is the same with SU(2) replaced by SU(3) and the right symmetry U(1) replaced by S U ( 2 ) x U ( 1 ) . In this case the eigenfunctions will be matrix elements of SU(3) representation matrices. Representations of SU(3), denoted D(p, q) are labelled [10] by two integers p and q. The eigenfunctions here will be matrix elements DPq~,~(~-2) of the representation matrices DPq(~'~). The indices m and m' in the SU(2) case were the eigenvalues of symmetry generators that could be simultaneously diagonalised with the hamiltonian; the indices at and /3 mean the same here. In the SU(3) case i 2, I3 ( I is the isospin) and the hypercharge ~" can be simultaneously diagonalised so that tx or /3 stands for the three indices (I,/3, Y) collectively. The index m in Dm'm was the eigenvalue of the diagonal generator of a left symmetry; similarly for the index/3 in DPq,~o. Since in the present problem the left symmetry is flavour S U(3),/3 - (/, 13; Y) gives the flavour quantum numbers of the s t a t e DPqo,~, Analogous to m' in Dim,m, the eigenvalue of the diagonal generator of a right symmetry, a =- (I', I~, Y'), stands for eigenvalues of the diagonal generators of the right SU(2) and right U(1) symmetry in the s t a t e DPqc~#.Since the right SU(2) has been identified with spatial rotations I' and I~ are just the spin quantum numbers of the state DPq~o. We now notice the crucial difference between the present system and the symmetrical top. The right U(1) symmetry is a local symmetry and imposes a constraint on the physical state space. Because of the Wess-Zumino term this constraint takes the form (27). Thus in DPq,~ the only allowed values of a are those for which Y ' =~N. 1 One may now ask: which SU(3) representations D(p, q) allow the value ½N for Y'? The answer is: p and q must satisfy

p - q = N + 31,

l an integer.

(29)

Also, if in a -~ (I', I~, Y') Y' is fixed to ~N, what possible values can I' or I~ have? Again, from the representation theory of SU(3) it can be seen that

I'3=½N+k,

k an integer.

(30)

Thus the particle states are fermionic if N if odd and bosonic if N is even. (29) means that the triality of the SU(3) flavour representations these states can carry equals N modulo 3. In particular if there are 3 colours, only triality trivial representations of SU(3) are allowed. The first of these is the octet, in which Y' = 1 implies that the spin is ½. The next is the decuplet in which I ' =~.3 Thus with N = 3, the quantisation of Skyrme soliton reproduces the known SU(3) representations and spins of baryons. 7. Diagonal form of hamiltonian

The hamiltonian (23) can be simplified using the properties of the moment of inertia matrix I = (lab). The spin-isospin property of the Skyrme solution implies

S. Jain, S.IL Wadia / Large-N baryons

723

that I = RIR T or [/, R] = 0, where R = (R.b) is given by Rab = Tr

(taVRtbVtR),

VR =

0 It is in fact this very property of I that allows 12 ~ 12VR to be a symmetry of the collective coordinate lagrangian. The matrices R form an orthogonal representation of SU(2):

°

where R3×s and R4×4 form respectively 3- and 4-dimensional irreducible representations of SU(2). From Schur's lemma it follows that I has the simple form I = diag (C,, C1, C1, (?2, C2, C2, (?2, 0).

(32)

CI and C2 are determined from (9) in terms of the function f occurring in Uo: 4Nrr fo ~ sin2f[r2+4(r2f'2+sinEf)] dr, C1 - 3e3f~

(33)

N~r (?2 - 2e3f,~

(34)

Io~sin2f[r2+ r~f~ +2

sin2f] dr,

f,=_df dr" The hamiltonian (23) now takes the simple form /~

1

=~

A

2

A

2

^

2

1

(Et + E2+ E3) + ~

2C 1

^2

^

2

A

2

A

2

(E4+ E s + E6+ E7)-

21_. 2

Using the hamiltonian constraint /~.=4~ N for physical wave-functions and the equality ~ . E~ ^ 2 _- ~ a Q, ^2 which follows From the operator version of (24), this reduces to I~I-2~C1 C 2 ] i=l

2C2

a=l

(35)

NOW ½~=~ 3 E^ 2 and ~Y.a=l 1 8 Q. A2 are the quadratic casimirs of the rotation group and flavour group respectively. Hence in view of our previous discussion of the baryon quantum numbers, the eigenvalues of the hamiltonian in the state Din.t3 are given by E(p,q;I')

(1_,1 1'~'2/

1) + ~--~2(Cp,q -~2N2 ) .

(36)

I' is the baryon spin and Cp.q is the casimir of the D(p, q) representation of SU(3).

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We recall that the c-number contribution of the classical soliton energy Mcj to So was omitted in (8). This must be added to E ( p , q; I') to get the total energy of the static baryon. This gives for the "nucleon" and "delta" baryons the masses

=

\ C1

C2] '

\ C~

C:/

(37)

Note that Mc~ is of order N and 1/C1 and 1/(72 of order 1/N.

Notes added (i) While this work was in progress we became aware of the published version of Witten's paper [4] which carried a "note added in prooF' referring to the baryon spectrum in the SU(3) chiral model. (ii) After the completion of this work we learnt that the spectrum of the SU(3) chiral soliton has been independently discussed by other authors. These are Guadagnini [11], Balachandran, Barducci, Lizzi, Rodgers and Stern (to appear) and Mazur, Nowak and Praszalowicz [12]. (iii) The connection between large-N baryon dynamics and the static strong coupling theory has been recently discussed by Gervais and Sakita and by Bardakci [13]. This observation has also been made by Virendra Singh. We would like to thank Virendra Singh for a critical comment and useful discussions. We also acknowledge helpful discussions with Avinash Dhar, P.P. Divakaran, L.K. Pandit, T.R. Ramadas and R. Shankar. One of us (S.J.) would like to thank Kapil Paranjape, Ajit Sanzgiri and T.N. Venkataramana of the School of Mathematics for help in some mathematical questions, and Samir Mathur for discussions.

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