Relativistic center-of-mass variables and the harmonic oscillator quark model calculation of the nucleon magnetic moment and the axial-vector coupling constant

ANNALS

OF PHYSICS

168. 181-206 (1986)

Relativistic Center-of- Mass Variables and the Harmonic Oscillator Quark Model Calculation of the Nucleon Magnetic Moment and the Axial-Vector Coupling Constant A. ILAKOVAC AND D. D. TADI~‘ Zavod

za Teorijsku Fiziku, Prirodoslovno-Matematifki Fakultet. University of Zagreb. Zagreb, Croatia, Yugoslavia

F. A. B. COUTINHO Instituro

de Fisica,

Universidade

de Sao Paula,

Sao Pa&,

Brazil

AND

F. KRMPOTI~: Far&ad

de Ciencias

E.xactas, Universidad La Plata, Argentina

National

de la Plala,

Received April 13, 1985

We study the introduction of the internal dynamical variables for constituent quarks. These variables are related to the center-of-mass of a nucleon. The problem is connected with the description of spinorial properties of the quarks. The spinors must be artificially introduced in a harmonic oscillator (HO) model. Experimental values of the magnetic moment and the axial-vector coupling constant of a nucleon can be easily reproduced. The theoretical results are not sensitive to the theoretical details; they follow from the general properties of the quark structure of baryons. The connections with the relativistic HO models are also discussed. The .I’ 1986 Academic Press. Inc case of a very small confinement radius is explored in the Appendix.

I. INTRODUCTION

Quark models have shown notable successes in calculating dynamic properties of baryons such as magnetic moment p and axial-vector coupling constant g, [ 1J. This has been achieved either by using models in which quarks move inside a confinement without experiencing any mutual correlation [2-71 or by using models with two-body forces acting among quarks [8-143. This paper will deal with the second group of the models, selecting those in which the interaction among quarks is of the harmonic oscillator (HO) form [8-143. Such 181 oaO3-4916/86 $7.50 Copyright .f 1 1986 by Acadermc Press. Inc. All rights of reproductmn in any form reserved

182

ILAKOVAC

ET AL.

models are especially intriguing as they allow explicit separation of the center-ofmass coordinates of a hadron so that the recoil effects can be studied. Such models actually depict scalar quarks. Only Schroedinger or Klein-Gordan equations with HO potential, which describe scalar particles, are explicitly solvable in a multipartitle case. (A two-body Dirac equation already leads to considerable difficulties [ 151.) Description of the spinorial property of the quarks has to be introduced “by hand” in a form of additional (i.e., outside of the original dynamical equation) assumptions and conditions [8-141. This problem is closely connected with the realization [ 16-191 that the internal motion of quarks, with respect to the centerof-mass of a composite system (hadron), has to be relativistic. There are various possibilities for implementing the desired relativistic motion of spin + quarks [8-141 which will be either used or discussed in the following text. It would be gratifying if one finds that the desired good physical results (i.e., correct predictions for ,u and gA) are stable against model variations. That would indicate that “good” predictions are due to the quark structure of baryons, having SU(6) spin-flavor symmetry, and not to some accidental dynamical quirks in a particular variation of a model. At the present stage of our knowledge such a conclusion is to be expected, and by itself would not present a very strong reason for the following investigations. However, detailed studies of the quark models of hadrons might lead to some insights which could be helpful in the construction of the composite models of leptons and quarks [20-221. Some preliminary investigations in that matter have already been done using the bag model. The calculations presented here will be based on the simplest Schroedinger HO Hamiltonian [ 12, 13, 231. Comparisons will be made with various other models [S-11]. The quark models under discussion are quantized composite systems. All physical quantities, i.e., operators, of the composite system can be expressed in terms of the constituent particle variables {ri, pi, $} where ri, pi, and !$ are the coordinate, momentum, and spin of the ith particle of the composite system. Any quark model with harmonic forces between quark pairs leads naturally to the introduction of the internal dynamical variables which give coordinate, momentum, and spin of each constituent (i.e., quark) relative to the center-of-mass [16-181. This is easily accomplished in the nonrelativistic limit. One defines center-of-mass radius R and the total momentum of the composite system P as R=+$nir, I

(1.1)

P=CP* where M and mj are the total mass and the constituent relative coordinates are defined by

mass, respectively. The

RELATIVISTIC

VARIABLES

r; = pi + R,

AND

MAGNETIC

MOMENT

p,=k,+$P

(1.2)

si = si Cm,p,=O, I

183

Ck,=-0. I

This second set of variables {R, P, pi, ki, si) will be referred to as CM (center-ofmass) variables. A simple, but nevertheless quite successful, nonrelativistic quark model [ 121 was successfully extended by including some momentum-dependent effects [ 131. Quarks were described by free particle, on-mass-shell, Dirac spinors U( p ;), whose momentum dependence was weighted by a Gaussian G(k, , k?, k,) (to be specified later) indicated by the HO Hamiltonian. In such a model the wave function in the impulse space has a general form p=fV’)

W,,

k,, k,) VP,)

WP,)

UP,).

(1.3)

Here -f(P) is a plane wave which describes motion of the composite object as a whole. The momenta k, are correlated by (1.2). The momentum pi in the Dirac spinor is connected to ki also by (1.2). This formula implies a completely nonrelativistic dynamics. Yet one has to retain all orders in k/m in spinors to obtain gA which agrees with the measured value [13]. Each of the existing ingredients in such a “hybrid” model seems justified on an intuitive physical basis. However, as has been realized sometime ago 1124, 16--l S], the inconsistencies could arise if nonrelativistic variables (1.2) were used to describe the relativistic interactions of a composite system. Thus, it is obviously reasonable and justified to reexamine the whole scheme by using the relativistic CM variables. Analogous problems have already been studied in nuclear and atomic physics [ 16-18, 241. The main difference with the atomic case is that the internal dynamics of the constituents, i.e., quarks, is relativistic. Only the motion of the whole composite particle (hadron) can be nonrelativistic. Thus one cannot just copy the existing expressions [16-l 8) for CM variables with relativistic corrections. One has to deal with a novel situation in which the internal motion of the constituents is relativistic. This means that one is not allowed to make an expansion in Ikl/m [l&18] but that all matrix elements have to be calculated to all orders in Ikl/m. The task is manageable if one assumes to be dealing with a weakly bound system, for which the required CM variabIes have been found in a closed form [ 17, 181. The employment of such relativistic CM variables does not seem unreasonable as one is using free quark spinors U(p) anyhow. The usage of the free spinors inside the continement, here provided by the gaussian G, makes this model somehow related to the bag model. The model is also closely connected to the relativistic versions of the HO model. As long as one uses on-mass-shell spinors U(p) the relativistic and the nonrelativistic HO models lead

184

ILAKOVACETAL.

to the same values for 1 and g,. Detailed illustrations of those statements and other comparisons are relegated to Section IV. The next sections will deal with various selections of CM variables with a general description of the model, and with two possible couplings to photons and intermediate weak bosons. The last section will present and discuss numerical results. As already hinted above the overall conclusion does not depend on the variations in the particular species of the model. The Appendix deals with the case of the very small confinement radius, which might be of interest when attempting to build composite models of leptons and quarks [2&22]. II. RELATIVISTIC CENTER-OF-MASS INTERNAL VARIABLES The HO models are naturally formulated using internal coordinates {pi, k,, si} (i= l,..., N; N number of constituents) which are defined relative to the center-ofmass (R, P, J} of the system. This set {R, P, J, pi, kj, si} will be referred to as CM variables. The HO model wave functions, which are Gaussians for the ground state, are functions of either pi or ki variables. In this case the latter will be used. The impulse space is a natural selection if one has to introduce spinors U(p). The problem of relativistic CM variables has been intensively studied. There exist approximate expressions for CM variables [ 16-1 S] in the form of the expansion in powers of l/c, where c is the speed of light. There is useful for composite systems in which both center-of-mass motion and internal motion are almost nonrelativistic. As already discussed in the Introduction quarks have relativistic internal motion. Yet, from some point of view, they behave as almost free particles inside the confinement region. For weakly bound systems an exact expression has been found [17, 181, valid in any Lorentz frame, relating the internal variable ki with the momentum pi 1 k,=p,+ (Pp,+ (E+M)E;)P. M(E + M) Here M, E, and P refer to the composite system. Furthermore $=kf+mf Ef=pf+mf E2=P2+M2

(2.2)

M=Cq. The formula (2.1) can be easily inverted for [PI/Me One finds

1, which is valid in our case. (2.3a)

RELATIVISTIC

VARIABLES

AND

MAGNETIC

MOMENT

185

which is an expansion in (PI/M but not in the internal variables k, and E,. The first two terms in (2.3a) are quite sufficient for the present application. The last term in (2.3a) was kept in order to show that this expression agrees with formula (2.27b) of Krajcik and Foldy [16] or formula (14) of Ref. [18]. One has to keep next to the leading terms in the expansion in Ikil/mi. Thus: ,tl==Cm,+O(k)

M-+Af+O(k);

+

&

(P. ki).P

m?kf- m’k2 J

1

2mim,,i2J

y+‘WP,.

The nonrelativistic case (1.2) follows if all terms proportional to either kf or P’ are neglected. One has also to connect a spin operator Oi of a general system with the CM system spin operator s,. To do that one can expend formula ( 5) of Ref. [ 1S] to order JP(/M. One finds s,+;(pixP)x$+ I

.‘.

a, = (E, + mi)(E, + m,)(E+

M) M (2.4)

b,=(E,+m,)(E+M)-P.p, 1 (E,+M)M’

b ‘= a,

The expression (2.4) is trivially

inverted to

$=s,+H,(k,xP)xs,+ H;=

1

... (2.3b)

M(E, + mi)’

The substitutions E, + mj and A4 -+ A lead to a full agreement with formulas (14) from Ref. [lS] and (2.27~) from Ref. [16]. Formulas (2.3a) and (2.3b) (referred to later as (2.3)) define CM variables which will be used in numerical calculations. Those formulas are not the unique choice for the relativistic variables [24, 251. They are not quite compatible (see Section IV) with the relativistic HO models [8-11, 141 which employ Klein-Gordan (KG) equations. However, they do correctly introduce important relativistic effects in the quark model based on the nonrelativistic HO Hamiltonian.

186

ILAKOVAC

ET AL.

Some useful physical insights might be gained if formulas (2.3) are used and interpreted in a flexible way. One possibility is to use a fixed composite mass M, which in our case should be equal to the real experimental mass of the proton M,,. Such a choice is suggested by a possibility that a baryon mass is not simply a sum of the single-particle energies of the constituents. It can contain contributions due to the effective two-particle interactions, which were not absorbed in HO Hamiltonian [ 121. The selection M= M, requires slight modification of the formula (2.3) in order to satisfy the relation c P, = P. One has to use (2.5)

It is probably more in the spirit of formula (2.1) to assume that a composite mass M should be determined by a sum C, E, of the internal energies. As the internal energies (2.2) are relativistic, this sum depends on the internal momenta ki. In the first approximation, neglecting residual two-body interactions, A4, can be determined as an expectation value of the sum of the internal energies

M, = j d3p,d3p,G2(p,pn) (c Q) I = 167~~I 4,

dpi. P; P:G~(P, PJCQ +

f2 = $[(a + b)3’2- (a - b)3’2] E=Jm

~1 (2.6)

a=rn2+4p~+~p: b=lpppi. fi

Here G(p,, p,.) is HO model Gaussian which is defined by (3.2) below. The formulas connecting CM variables with variables defined in any slowly moving ([PI/M, 6 1) Lorentz frame have a general form pi=k,+LiP iji = ci + H,(k, x P) x 6,. The term proportional

to Hi is known as Wigner rotation.

(2.7)

RELATIVISTIC

Numerical

investigations

187

VARIABLES AND MAGNETIC MOMENT

were carried out for the following cases:

Case A. t:, = c1 = CJ = c;

M=3&,

L, = f,

H,= [3~(c+m)]

’

(2.8)

h4, = formula (2.4). Case B. H,= [M(e+m)]--‘,

(2.9)

M= M,.

Casec. M = c El, I

H,=[@)

(E;+W]

’ (2.10)

M, = formula (2.6). Casr D. Li=

(

Hi= [M(ei+m)lmM=M,.

Cases C and D correspond to physical assumptions which were discussed above. The somewhat artificial Cases A and B serve as additional tests of the stability of general conclusions irrespectively of the variations in theoretical inputs.

III.

Our intention baryon states

CURRENT MATRIX

ELEMENTS

is to calculate matrix elements of single-particle (0

= (W’)I

40) lB,W).

The calculation will be carried out in the momentum wave function is a Gaussian of the form [ 131 qp,,

operators between

(3.1)

space where HO model

p;,) = Ne (li2&lP;, +$1 1 NC------(,1/2,)3'

(3.2)

188

ILAKOVAC

The momenta

ET AL.

pp and pi are connected with the internal variables ki as follows:

h=

-$,,+$,i

k,=

-a

(3.3 1

pn.

Wave functions (3.2) are appropriate for three scalar quarks so the spinors have to be introduced in addition. A “mock hadron” prescription of Ref. [ 131 assigns to each quark a free particle spinor U(p,), which satisfies

(3.4) WPJ

=

N

Here 2 are Pauli spinors. In Ref. [13] (1.2). Relativistically corrected formulas ferences with Ref. [ 131 will be clarified with some other HO models [8-l l] is discussed in Section IV. It is important particles on the mass shell, i.e.,

pi were expressed in CM coordinates by (2.7) will be used here. Some additional difin the following text. The main difference due to the choice of quark spinors, which is to note that spinors (3.4) describe spin $

p’+mf=E:.

(3.5)

One can only hope that (3.4) and (3.5) approximately reflect the real dynamics of the interacting quarks. In the model building one is forced to approximations and assumptions of this kind since there are no explicit solutions of the multiparticle Dirac equation. In the mock hadron language of Ref. [15] one describes protons by lP~Pp?PLhoton=~

c Permutations (i= 1.2.3)

.-

G(P,, PA) p,

C@$(PJ

b:;(P,)-Q;(P,)

%gPJ)

K;(P3)1

IO> (3.6)

as a collection of free quarks weighted over the wave function G. Here b’s are quark

RELATIVISTIC

VARIABLES

ANO

MAGNETIC

189

MOMENT

creation operators of a given flavor (u, d), color (a, h, c), and spin orientation. In the same spirit the single-particle operator J is expressed through free quark fields &(@=~~d3kd3kz Ll

F$k,l-,;!?‘y(kZ) (3.7) li3~(u;[‘(k)h~..(k)+a;(k)d;“(k)). I

Here I- symbolizes an operator containing y matrices and impulses. Alternatively (instead of using n-representation (3.6), (3.7)) one can work with a wave function (3.2) in the configuration space multiplied by the product of spinors U( p, ) U( pz ) U(p,) (1.3) as was done in Ref. [S]. Such formulation immediately turns attention on the definition of the single-particle operator in the space of three quarks. One has to understand [S] I-+

(3.8)

C T,x ljx 1,. PWIIl. i. 1.h

In the n-representation (3.8) would be multiplied by bilinear products P!K There is a real distinction between (3.7) and (3.8). In the former approach factors 2, (defined in (3.9) below) are absent, which leads to noticeable numerical differences in the intermediate steps, although the final results, as discussed in Section V, are not significantly different. In a sense, these two choices can be interpreted as alternative model variants. It is useful to write down a general form of the matrix element (3.1) before this is discussed any further. Since baryon wave functions are symmetrized the result has only one term:

.Z,(P;,

pI)-G(p;,

~2)

h;VWp,W(&

P;)G(P,,

PA) (3.9)

q=P’-P z;tp:7 P,) = ad)

WP,).

Here we have kept d-functions explicitly as in the integration over, say, ~7; and p>,, one has to be careful with spinors U(p,). The results are simplified if one selects a particular coordinate system: P= -9. (3.10) P’ = 0, For that case one finds

-Z,(k,

- $3 k, - L, q) Z,(k,

. O(k, + fq) f U(k, - Lq).

- $4. k, - Lzq)

190

ILAKOVAC

ET AL.

In order to take properly into account the Wigner rotation (2.7) one has to keep in mind that spinors I!? correspond to P’= 0 while the spinors U are in the system where P = -9. The overlap factors Zj, introduced in Ref. [S] and used by others [9, 10, 143, are best discussed for the time component of the electric current Jo when r=yO. The charge conservation requires (Jo(q))

z

(3.12)

1.

This does not follow from the formula (3.11). One finds (J(O~~~))=~~~~p~~~~G*(~,~~pi)Zl(~l~k,)Z2(k,~k,) (3.13) = Q. In order to obtain physically correct results one has to renormalize, suitable adjustment factors [S] or by redefinition [9],

(J(q, YO))p,,ys= e-‘
Yo)).

either by (3.14)

If one used (3.11) with Z, = 1 [ 131, the condition (3.12) would be immediately valid. However, (3.11) with Zi = 1 and f = yfl is no longer Lorentz vector in the space of three Dirac spinors [9, 10, 143 but some tensor. Even a choice Zi # 1 is not completely consistent as Q itself transforms as a component of a four-vector. Thus the redefinition (3.14) is not introduced in some relativistic quark models [ 111. All these differences and distinctions vanish in the exact nonrelativistic limit. One way to explain the form (3.14) is to understand it as a result of redefinition of the quark model wave function (1.3). The factors Q ~ ‘I* can be absorbed in a redefined function G in (3.6). That makes the formalism consistent with relativistic invariance. Such a point of view is supported by the fact that baryon magnetic moment has contributions coming from Z,‘s (see Eqs. (3.22)-(3.25) below). Although in (3.11) the electromagnetic field is explicitly coupled to a third quark line (or any quark line, as the problem is symmetrical in three quarks), the composite structure of a baryon leads in a relativistic case to additional contributions. The quark models in which spinors do not depend on internal variables [S-lo, 143 do not give such contributions. In a sense free particle spinors, being functions of internal variables (2.7) which depend on the parameters of a bound state system, are no longer really free. In order to be consistent any physical current (r= yp, yfly5, b’y” + y”$,...) must be defined as (3.15) (4% 0 > phys=
assuming that the quark couples to the elecY’A,,

(3.16)

RELATIVISTIC

VARIABLES

AND

MAGNETIC

where A,, is a four-vector electromagnetic potential. constituents suggests an alternative coupling [8]

191

MOMENT

The KG equation of motion for

(3.17) This coupling goes into (3.16) for free-on-mass shell spinorial quarks. The couplings (3.16) and (3.17) are not quite equivalent when using relativistic CM variables. However, it turns out that the time-like component of (3.17) leads to the same Q as has been obtained using (3.16). (As explained in Section IV Ref. [S] gave (3.17) a somewhat different interpretation.) The differences between (3.16) and (3.17 ) are apparent in the magnetic moment calculation to which we turn next, starting from (3.16). The required combination of spinors and y-matrices is up to the leading order in q: x=0

=-

i

k+iq

1 2x

>

yU(k-Lq)

i(oxk)J(k.q)(-A+B)l+i(axq)

(3.18a) with A=----.-

L

H=

E(c+m) B=-

1 M(c + 772)

-213 1(E + m)

(3.18b)

E, = Jkf

+ m

E3= c,

k,=k

and Z,=

U k;-;q i

>

U(k,-

L,q)

m+(l/3-L,-H;kf)(iaxq).k, =x + 2E,(E;+m) Et

i= 1, 2.

Here x’s are Pauli spinors. Final expression for the magnetic moment

(3.19)

p is

(3.20)

192

ILAKOVAC

ET AL.

The magnetic moment ,U (expressed here in nucleon magnetons pN = e/2&I,) is determined by the integral I which can easily be extracted from (3.11) as the relevant terms are labelled by the combination (r x q. The first and third terms in (3.18) also contribute. The integration over d3pi (i.e., d3k) extracts the desired G x q vectorial product. The last term in (3.18) combines with the last term in (3.19). (Note that 1 +fx =f if f does not contain any o matrices.) The first term in (3.19) gives the factor m/si which contributes both in I and QP’. The final expression for Q does not depend on the particular choice of CM variables:

(3.21)

a=m2+$p~+~p: h-f-

ppp*. Js

The expressions for Z(p, . pi = pp pAx) are listed below for all selections of variables defined by (2.8~(2.11). Case A. I= J dpp dp, P; p34n12 IG(P, PA)I~

-1

2E+m

2E-m

--3 6E2

m -

-&P:(

m2

6E2

‘EIEZ

.s2-m

El-m

~~~~~~~~~~~~

3E(EI

E1-m

m)

3E(E1 +

-

i-m)S3E(E2$m)

E2-m )]I. + m)

(3.22)

3&(E2

Case B. I= J 4,

4, P: ~:(479~ UP,, 2Etm

-&P:(

2E-m

E1-m

M(E,

PAIN

+m)

-

E2-m

M(E*+m)

)]}.

(3.23)

RELATIVISTIC

VARIABLES

AND

MAGNETIC

193

MOMENT

Case C.

I= i‘ dp,, &A pi p2,(4nJ2 IG(P,, PA)\*

i El+E2+E-3m

El+E2+&-3m

--

3(~, +E~+E)(E,

+wI)-~(E,+E~+E)(E~+~~)

’

(3.24) Case D. I= J”dpp dP, P: P:(~Jc)’ IG(P,, Pi)I* 2&+m __---

SE-m

I

3 2M

.- m2 E,EZ

E,+Ez+E-3m +

-___

E,+E?+E-3m

3M(c,+m)

3M(c2

E,+E?+E-3m

-

3M(c,+m)

.

+ m)

(3.25 )

The first term in the first square parentheses in (3.22) is the one obtained by Ref. [13]. The second term is due to the Wigner rotation. The factor m2/E,c, is produced by the overlap Zi factors; it was not considered by Ref. [ 131. As discussed and explained above Zi factors also give an additive contribution to the magnetic moment. This is contained in the last term in the formula (3.22). All formulas (3.22)-(3.25) are written in the same format so that the corresponding terms can be easily identified. With the coupling (3.17) one has to expand up to the leading orders in q the expression

-H &-$(ia.kxq)k.

(3.26)

194

ILAKOVAC

ET AL.

Going through the same calculations as for X (3.18a) one gets the following expressions for the integrals J which replace integral I in the expression (3.20). We again have Cases A, B, C, and D defined by formulas (2.8~(2.11). Case A.

J= [ dp, dp, p;p:(471)*

IG(pp, P,#

2E+m+(E-mm)* 6E2 - 18c2m

1

m* -E,E2

.5,-m

1

E2--m

(3.27)

Case B. J= j 4,

dp, P;P:(~~I’

lG(pp, PJI*

--

(3.28)

Case C.

J= j dppdp, P;

~:(47c)* MP,,

m ( Ii-t,+~~+iz

i- [ - ~JSE,E2~XPyPA m

m

~~E,QE

-- 1

1

m

--

PJ*

3

E,+E~+E

>(

1

c2+m+8,+m

--- 1

1

)( .z,+m

c2+m

>

111 ’

(3.29)

RELATIVISTIC

VARIABLES

AND

MAGNETIC

195

MOMENT

Case D. J= [ dPp &I. I:’

IG(Pp, P>.)I*

m

- (i&$ppp~-&p:

-+-

3M

&,+.z2+c-3m 3M

(&h+-

1 E, Sm >

g2 +m )I1 ’

(3.30)

The ordering of terms corresponds to the one used in formulas (3.22)-(3.25). For example, the Wigner rotation term is the second term in the first square parentheses; the overlap term is in the second square parentheses. Two sets of formulas, i.e., (3.22k( 3.25) and (3.27)-( 3.30), lead to numerically different predictions for the nucleon magnetic moment, as will be discussed in Section V. In the Appendix we will discuss the behaviour of these two alternative sets for the case of the very small confinement radius (01%). The Feynman set (3.27)-( 3.30) has in the limit LY$ physically acceptable behaviour. The axial-vector coupling construct g, is defined through fqg,

=x+q;(Q-‘K).

(3.31 )

Spin-flavor symmetry of the baryon states leads to the factor 3 [ 11. In order to find K one has to consider the combination

y= wL+SqhUkL4) =X+CX3r.

H-2m

(3.32)

In analogy with (3.17) one can also use instead of (3.32) the expression

+YY5W3-L4)I Uk,-bq).

(3.33)

However, in the q -+ 0 limit both (3.32) and (3.33) lead to the same integral K. Its value does not depend on a particular choice (2.8)-(2.11) of the CM variables either:

.5+2mm2 h - arc sin -. a 36 h

595,168.l.13

(3.34)

196

ILAKOVACETAL.

In Section V we will present a complete numerical functions of model parameters c1and m.

study of p, g,, and M,

as

IV. CONNECTIONS AND COMPARISONS WITH OTHER MODELS In this section we first discuss transformation introducing internal dynamical variables related to the center-of-mass as applied to the relativistic (i.e., four dimensional) HO models. Second, we consider relativistic HO model predictions for p and g, and their relationship with our formulas. In the relativistic HO models described by the Klein-Gordan equation with a two-body harmonic potential u, = W2(Xi - Xi)2 [S-11, 14) one can separate the external four-momentum of the whole system P= of internal four-vectors k;, i.e.,

(P,, P) by a simple introduction

pi=;P+ki.

The relativistic form [16]

(4.1)

CM variables used here are based on the Hamiltonian (p?+m~)1’2+j3U.

H=x

H of the (4.2)

Here U is an internal interaction potential and /I is a parameter. The form (4.2) is not compatible with the usual Klein-Gordan operator OKG on which the relativistic versions of HO model are based. One can write 0 KG

--O

KG0

-

OKGl

OKGO=~“k~O=C

(POi-c

0 KG1

=I

OkGl i

uij)2

i

i =I

(4.3)

i (Pf

+

m?>.

I

The identity (4.4)

does not lead to (4.2). One has to use instead an ad hoc relation

(4.5)

RELATIVISTIC

VARIABLES

AND

MAGNETIC

197

MOMENT

This inconsistency does not appear for the free particle case (C, U,i= ,QU= 0). Relativistic spinors, used for quarks, satisfy a free particle Dirac equation and thus also a free particle KG equation. With free particle on-mass-shell spinors used for quark description, the difference between relativistic and nonrelativistic HO quark models is not of any practical consequence in the small q (momentum transfer) limit, in which one calculates p and g,. The model explored here is actually a relativised “nonrelativistic” HO model as it uses nonrelativistic Gaussian G (3.2) as a weighting function in the integral involving relativistic four-component Dirac spinors U(p). The relativistic HO model leads to the change of the Gaussians G, appearing in the expression (3. lo), by introducing the time-like components G(p,, PiI + G(P,, P”,, Pj.7 PY).

However, there is also additional

c

integration

(4.6)

over those components.

4,o +io IG(P,, pi, ppo> P;.o)I~ = F(P,,

One has,

pJ12.

By using (3.11), in the limit q + 0, one can show that there is no additional contribution to either p or g,. A particular choice of Gaussians G would be of importance only if one wanted to calculate q2 dependence of the electromagnetic and weak formfactors [S-lo]. The low q (i.e., static) quantities p and g, depend very much on the particular choice of spinors. As the HO model corresponds to either the KG equation or its nonrelativistic approximation, the spinorial character of the quarks has to be added into the theory. In the nonrelativistic HO to each quark one simply assigns a twocomponent Pauli spinor x. The Pauli spinors can be boosted into Dirac spinors so that they acquire “small” components. Reference [S] chose to boost to the speed of the center-of-mass, so the Dirac spinors depend on P alone. In Ref. [ 131 and in this paper spinors are boosted to a general reference frame. They depend on pi’s defined by (2.3), so that the small components do influence the value of the integral (3.11) and one finds modified values (see Tables I and II) for p and g,. Our factors Zi are analogous to the factors g in Ref. [S]. One finds g -+ 1 for equal quark masses and for q + 0. The same limit Zj --t 1 for q -+ 0 is obtained if one neglects dependence of spinors (3.4) on the internal variables k,, i.e., by substituting

In that limit all products of the spinors can be shifted in front of the integral (3.11) and one obtains nonrelativistic static values pp = e/2m, p,, = -e/3m, and g, = 2 for the formfactors. (One needs m = MJ3 for a rough agreement with experimental values.) This results from the SU(6) spin-flavor symmetry of the baryon (hadron) states which is the same as in the nonrelativistic HO model [ 11.

198

ILAKOVAC

ET AL.

The alternative couplings to photon field (3.17) and to intermediate-vector-boson field (3.33) which were used by Ref. [8] do not change the above conclusions. In the discussed limit one coupling, (3.17) or (3.33), goes into an alternative coupling (3.16) or (3.32) as can be easily shown by using Dirac equation (3.4). The model of Ref. [8] is actually somewhat more complicated. We will discuss here only its low q limit. (The model’s dependence on any q is quite involved.) There are two types of the four-momenta used to describe constituents and/or quarks. The model has four-momenta p, (a = 1,2,3) connected with the three constituents for which the KG equation is solved. The four-moentum p, is connected with the four-momentum of the system P through p,=$P-g. Electrodynamic in which

interactions

(4.9a)

are described by using a particular

coordinate

system

p = (MP, 0) (4.9b)

Pa= -4s p: = $4, - f(,. The quark spinors in that model [8] are boosted to the momentum means

P, which (4.10)

P’+M;=E*

From (4.9b) one can conclude that the constituents interacting by HO forces behave as having an effective mass A4,/3, while spinors (4.10) behave as having a mass M,. The impulse which explicitly enters the interaction with electromagnetic field (3.17) is p,. With conventions (4.9) and (4.10) the formulas (3.16) and (3.17) are no longer equivalent. (In the former discussion, we have assumed that effective masses in (4.9) and (4.10) are equal.) The end result depends on those specific details of the interaction. It is easy to see, by comparing formulas (35) and (47) of Ref. [S], that a correct value of the proton charge (see matrix element (3.21)) is obtained if the factor fMp is taken out. That leads to 3e p(P=2Mp3

/AL,=

(4.11)

+. P

This result has nothing to do with the internal spinors are not connected to the internal motion

dynamics, as in that model of constituents.

[S]

RELATIVISTIC

VARIABLES

AND

MAGNETIC

MOMENT

199

For the axial-vector coupling constant g, too large a SU(6) value 3 is retained in the model of Ref. [8]. This is not surprising. The magnetic moment value which is an effect proportional to q (i.e., n x q) is influenced by the details in the selection of CM variables. The value of g,, which is associated with the zeroth power of q, is always 5, if spinors do not depend an internal (ki) variables. It seems that somehow disjointed selections (4.9) and (4.10) are forced by the inherent contradiction of the model, which was discussed in the beginning of this section. The low q characteristics of the model [8] are retained in a very similar, but explicitly relativistically covariant, model of Ref. [ 111. We do not advocate here that the model of Ref. [ 131, which was generalized in Sections II and III, is free of contradictions. In that case Gaussian (3.2) is a wave function of nonrelativistic scalar constituents and the relativistic internal dynamics is forced in through the selection of spinors. As there is no explicit solution of the three-body Dirac equation, one is always compelled to add spinors in an essentially arbitrary way, which constitutes an additional intrinsic characteristic of a given model. In Ref. [ 111, for example, spinors are made to be functions of the four-momenta P:

Y/,=N

(4.12)

Here m,, the quark mass, is an additional parameter of the model and spinors are no longer on-mass-shell. Reference [ 1l] was dealing with high q effects, so it had not required (Jo) = 1. It has chosen completely relativistic normalisation. There are a variety of other relativistic models [lo, 14, 26). Their conventions and approaches are more or less similar to those of Refs. [8] and [9] so it would be tedious to go into all the particular details and choices. Model [ 141, for example, introduces a magnetic moment operator (4.13 ) where mj are quark masses. The factor g, comes from a generalisation of the coupling (3.17) by an addition of a Pauli-like term proportional to #” = (yr, r,1/2i. The coupling (3.17) corresponds to gM = 1. The last sum in (4.13) has to be evaluated between SU(6) baryon states. It is 1 for the proton and f for the neutron. When all quark masses are equal (m,= m) one finds the results (4.11) up to a factor g,: 3e lP=2M,

it?,,

pN=

-+

g,. P

(4.14)

200

ILAKOVAC

ET AL.

Unequal quark masses, together with g, # 1, improve the fit of the octet baryon magnetic moments [ 141. For the sake of completeness let us mention that concerning the spin part two types of extension have been proposed [14]: the Bargmann-Wigner (BW) scheme uses a Dirac spinor and was also known as the 8(12) scheme [27]. The Pauli spin scheme uses a suitably boosted Pauli spinor [28]. The first case was used by Ref. [9] while the conventions of Ref. [S] correspond to the second choice. In any of these approaches, spinors do not depend on the internal momenta, which always give g, = 2. V.

NUMERICAL

RESULTS

AND

DISCUSSION

There are three interesting numerical aspects of the outlined model. First, one has to find how numerically important are all complicated relativistic terms. It is also useful to find out whether one can simultaneously lit p, g,, and Mp (Cases A and C) with the same set of parameters. Finally, one can ask about model behaviour for a very large a, which is equivalent to a very small confinement radius. The first question is best answered by the study of Table I, which is concerned with the (3.16) form of the coupling. Five columns in Table I correspond to the case where 2 factors are given by formula (3.19), i.e., Q # 1, as displayed in the eighth column. The fourth to seventh columns display the contribution (i) to magnetic moment without Wigner rotation term and overlap (first term in the first square bracket in formulas (3.22)-(3.25)), &o; (ii) the Wigner rotation contribution, &io; (iii) additive overlap contribution, &; and (iv) total magnetic moment, p’. As seen from the table the Wigner rotation term decreases &o by about 8%. The TABLE Axial-Vector Case

a

Constants

and Magnetic

I Moments

for Coupling

(3.16)

l&G

m

P’

g‘4

M

A

0.262 0.271 0.267

0.170 0.1765 0.1745

3.056 2.950

-0.259 - 0.249

5.45. 10m4 2.797 2.89. 1O-4 2.701 2.739

4.143. 4.728.

lo-’ 1O-3

2.942 2.841

2.678 2.587

1.249 0.9046 1.250 0.9368 1.251 0.9241

C

0.262 0.271 0.267

0.170 0.1765 0.1745

3.048 2.943

- 0.272 -0.261

2.17. 1O-5 1.68. lo-“

2.777 2.682

4.143. 4.728.

lo-’ 1O-3

2.902 2.803

2.633 2.544

1.249 0.9046 1.250 0.9368 1.251 0.9241

B

0.262 0.271 0.267

0.170 0.1765 0.1745

3.056 2.950

-0.259 -0.249

10m5 2.805 lo-“ 2.700

4.143. 4.728.

1O-3 lo-’

2.842 2.841

2.678 2.578

1.249 1.250 1.251

D

0.262 0.271 0.267

0.170 0.1765 0.1745

3.058 2.952

-0.251 -0.250

4.143. 4.728.

1O-3 1O-3

2.9025 2.8019

2.639 2.539

1.249 1.250 1.251

“p is in nuclear

magnetons,

-7.12. -2.26.

1.94. 1O-4 3.94. 1O-4

a and M are in GeV.

2.799 2.694 2.735

RELATIVISTIC

VARIABLES

AND

MAGNETIC

201

MOMENT

additive overlap term is negligible in comparison with other terms as it is about lo4 times smaller. The next two columns correspond to the Z= 1 case. The ninth column displays values for magnetic moment without Wigner rotation, &o (the formula from which this quantity is (in Case A) calculated is the same as that in Isgur’s paper [ 131). The tenth column corresponds to the magnetic moment where Wigner rotation is included, p’. As can be seen from Table I, &o is larger than the corresponding p”z by about 5 %. On the other hand, p’ is smaller than $, and also smaller than ,&. It has to be emphasized that p (Z# 1) terms are equal to the product I”Q ‘? where Q is quite a small quantity so that the integral is also quite small. This means that one has to compare ZzQP’ (and not Zz) with I’ values. The last two columns in Table I show values of the axial-vector coupling constant (gA ) and of the mass (M). Table I shows only those values of m and CI for which quantities pz, g,, and M can be simultaneously approximately fitted. The fitting procedure has been carried out by seeking for (a, m) pairs of values for which either pz or g, or M are close to the experimental values. Those sets of (a, m) values gave curves for II”, g,, and M which are shown in Fig. 1. All three curves do not intersect in the same point but m(GeV)

10'

20

/

1

.22

.2L

.26

26

30

a (GeV I FIG. 1. Lines show pairs of values a, m for which p = 2.783, pN, g, = 1.25, and and B for coupling (3.16) and Case A for coupling (3.17) are shown.

M = M,.

Cases A

202

ILAKOVAC

ET AL.

TABLE

II

Axial-Vector Coupling Constants and Magnetic Moments for Coupling (3.17)” 0.298 0.199 2.800 2.790 2.800 2.793 1.248

0.289 0.1872 2.864 2.883 2.868 2.877 1.248 0.9968

1.028 ‘p is in nuclear magnetons,

a

and m are in GeV.

two-curve intersections are not too far. The values for magnetic moment that are displayed in Table II are calculated at various two-curve intersection points. As seen from Table I, quantities ,uz, gA, and M can be fitted within a few percent. For instance, for (a, m) = (0.269,0.166) we get in Case A M = 0.917, pz = 2.755, and g, = 1.235. Table I also shows that Cases A and C behave almost identically (this is illustrated in Figs. 1 and 2), and all values for all quantities are nearly equal. 35

PAN!%) r

51

1 .I

I .2

mlGeV)

I .3

I .A

2. Magnetic moment ,u is shown as a function of m for a given a. The curves correspond to Case A, coupling (3.16). FIG.

RELATIVISTIC

VARIABLES

AND

MAGNETIC

MOMENT

203

Cases B and D (M = const) also give almost identical numerical results. There is not much difference between groups (A, C) and (B, D) as can be seen from Table I and Fig. 1. In Fig. 2 we show, for Cases A and C, dependence of p on m and a. For a given x the maximum of p-curve depends on the quark mass m. The maximum value of p decreases with the increase of a. Such a behaviour means that one cannot build HO models of the composite leptons, as is already known [20, 211. The curves in Fig. 3 show the variation of g, with quark mass m for a fixed value of a. All curves coincide for m = 0. This behaviour of g, is very similar to that found using the MIT bag model [l-3]. It should also be stressed that the theoretical expression for g, in the HO model does not depend on the choice of internal variabl& i.e., it is the same for Cases A, B, C, or D. Thus it can be said that the theoretical value of g, depends on the most general characteristics of the quark model, which will be discussed below. An alternative coupling to the electromagnetic field (3.17) does not lead to any dramatic changes, as can be seen from Table II and Fig. 1. Thus, for example, in Case C one finds g, = 1.25 and p = 2.790 n.mn. for a =0.298 GeV and m =0.193 GeV with (3.17) while with (3.16) one finds g, = 1.25 and p = 2.682 n.mn. for a = 0.274 GeV and m = 0.177 GeV. While model parameters rn and z differ by

L

/ 1

I

2

/ 3

I L

m (GeV) FIG. 3. Axial-vector coupling constant g, is shown as a function of M for a given a.

204

ILAKOVACETAL.

about 11% the values of the magnetic moment p differ by about 4%. Only in the limit of a very large a, which is discussed in the Appendix, do douplings (3.16) or (3.17) lead to a markedly different theoretical prediction for CL. Expression (2.6) for the average internal energy can be interpreted as a mass M of the composite object only in the framework of a nonrelativistic HO Hamiltonian. It does not transform as a Lorentz scalar so its theoretical justification is weaker than is the case with the formulas for ~1and g,. The relativistic HO models have an entirely different mass operator [g-11, 143. For these reasons not too much weight should be attached to the columns labelled M in Tables I and II. Nevertheless, it is amusing that an almost correct M can be found using parameters m and a which were needed to reproduce p and g,. Investigations presented in this paper show that the theoretical values for ,u and g, depend only weakly on the particular details of an HO model such as choice of internal variables or form of the coupling with the electromagnetic field. As long as quarks have internal dynamics connected with Dirac spinors HO model predictions are qualitatively and even quantitatively (within 10-15 %) in good agreement with bag model predictions [l-3, 131. This is certainly correct as long as the confinement radius is not much smaller than 1 fermi, which for the HO model means a
presented here unambiguously

sup-

VERY SMALL CONFINEMENT RADIUS

If leptons and quarks are not elementary but have some structure, the corresponding constituents must have a very small confinement radius. In the case of the electron this radius R should be of the order of TeV-’ [2&22]. In the language of our model, where R - c(-I, this means u > 0.1 TeV. If the HO model can aspire to describe such a system, it should be able to reproduce a very large magnetic moment PW 1 Bohr magnetons (pB) and a very small mass (or internal energy). Thus when a increases for about lo3 times, ,u should also proportionally increase, while internal energy must decrease. Even a cursory study of formulas (2.6) and (3.20)-(3.25) shows that the above tasks are impossible. For a large a, expectation values of internal energies si and impulses pP, pi increase with ct. Thus

M(2.6)~cc;

a%

(A.11

RELATIVISTIC

VARIABLES

AND

MAGNETIC

205

MOMENT

which is exactly opposite to the required behaviour. The theoretical expressions for magnetic moment either decrease with CI (Cases A (3.22) and C (3.24)) or change sign (Cases B (3.23) and D (3.24)). Numerical calculation confirms these quantitative conclusions. This totally unacceptable behaviour of quark models is well known [20,21]. In the case of the bag model it can be repaired by a suitable reformulation of the model [22]. For the present model the large c( behaviour of ,U depends on the coupling with the electromagnetic field. With the choice (3.17) p increases with a(. For Cases B (3.28) and D (3.30) this increase is enormous, p- 10’ pLg. However, for Cases A (3.27) and C (3.29) one finds p of order pg for CYof the order of TeV. This difference in behaviour is connected with the Wigner rotation, as explained in Section III of this paper. It does not depend on particular definitions and normalisations. Even neglecting 2,‘s one finds using formula (3.27) P=

I

d3Ppd3Pi G'(P,,

PiI

2&+NI+(&-mm)’

6~2

18E',n

(A.2)

The magnetic moment does not vanish with the increase of x. Some interesting values for large but final a are listed in Table III. The results in Table III would be encouraging for the case of the composite theories if one could satisfactorily deal with the problem of internal energy or predicted mass. The escape via fixed mass, i.e., Cases B and D, is closed. Even if it worked, such a solution would not be aesthetically appealing. Cases A and C, which could lead to acceptable p, seem to be associated with formula (2.6). According to that formula the composite object would have a mass about three orders of magnitude larger than the proton mass. However, formula (2.6) is appropriate for scalar constituents described by a nonrelativistic HO Hamiltonian. From a relativistic point of view it does not even transform as a Lorentz scalar. Obviously when o! becomes very large, one has to use the relativistic KG equation with HO potential [8-11, 141. In that case the composite mass of the ground state is determined by an arbitrary constant. TABLE Magnetic d m A.\ P&Y P&Y M “g is in Bohr

Moments

for Very

0.17 I 10-7 4.56. lo5 4.56. to5 0.565 5.11.10-’ magnetons,

a and M are in TeV.

III Large

a for Coupling 0.34 1.10-' 9.24. IO5 9.24. 10' 0.572 5.11 lo--'

(3.17)” 0.17 2.10.' 2.283.105 2.283.105 0.284 5.11' 10 --'

206

ILAKOVAC

ET AL.

One would have to reconsider the whole structure of the model in view of its schizophrenic character; i.e., a composite particle made out of HO potential bound scalars which are given spinorial features by an ad hoc introduction of Dirac spinors.

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