Weak interaction form factors and magnetic moments of octet baryons: Chiral bag model with gluonic effects

Nuclear Physics A489 (1988) 557-611 North-Holland, Amsterdam

WEAK AND

Kazuo

INTERACTION

FORM

FACTORS

MAGNETIC MOMENTS OF OCTET BARYONS: Chiral bag model with gluonic effects

TSUSHIMA,

Tetsuya

YAMAGUCHI,

Y. KOHYAMA

and K. KUBODERA

Department of Physics, Sophia University, Kioicho, Chiyoda-ku, Tokyo 102, Japan Received 27 April 1988 (Revised 7 July 1988) Abstract: The weak interaction

form factors and the magnetic moments of the octet baryons are calculated in the volume-type cloudy bag model including clouds of all the octet mesons. In addition, one-gluon exchange effects are also evaluated. The theoretical predictions including both the mesonic and gluonic effects are compared, wherever possible, with experimental data.

1. Introduction Magnetic moments and weak interaction form factors are useful and important observables for understanding the detailed structure of baryons. We have recently witnessed a remarkable improvement in the measurement of these quantities for the octet baryons lm4), an d serious interest is shown in a systematic framework in which to correlate these experimental data 5-‘9X44*45).One of the central issues regarding the magnetic moments and the weak interaction form factors of the octet baryons is to what extent they obey the SU(3)-symmetric pattern or, more generally, the prediction of the additive quark model. As for the magnetic moments, the achieved precision of data ‘) is such that their systematics definitely shows significant deviations from the simple SU(3) symmetry, or even from the generalized additive quark

picture

‘). Since

the Cabibbo

theory 20) of the octet baryon

semi-leptonic

decays is based on the assumption of perfect SU(3) symmetry, one expects that, at a certain level, data will show significant deviations from it. Experimental information available at present, however, is still limited, and only the leading-order form factor g, (see sect. 2 for its precise definition) can be deduced directly from data. There are split views on how to treat the existing data for the weak-interaction form factors. The WA2 group ‘) that carried out experimental studies of several hyperon semi-leptonic decays simultaneously, maintains the view that there is little meaning in taking the world average of the existing data obtained in separate measurements with different experimental set-ups because the treatment of apparatus-dependent corrections is highly delicate. From this viewpoint, the WA2 group, in testing the Cabibbo theory, used only those hyperon semi-leptonic decay data that were 03759474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

558

K. ~s~~jrna

et ai / Weak jnteraetja~ formfactors

obtained in their own experiments “) The least-square fit of the WA2 data within the framework of the Cabibbo theory gives typically x2 = 8.8 for DOF (degree of freedom) = 6. From this result it was concluded in ref. *) that there is no need to modify the original Cabibbo SU(3) scheme. We may, however, raise here the following questions. First, can the obtained value of x2 be considered really small enough to conclude that the Cabibbo lit is satisfactory? In this connection, we recall that the ratio f(.Z- + n i-e- + ;i;,)/l”(A -+ p + e-+ %,) predicted by the Cabibbo scheme (or more generally, by the additive quark model) tends to be signi~~antiy smaller than the observed ratio ‘1. This discrepancy, which we may call for short the “2 -A problem”, still persists in the analysis of the WA2 data. Secondly, how compelling is the argument that it is better not to average the worId data but use the WA2 data alone? Donoghue et al. “) recently analyzed the world average of the available data I) and suggested that the SUf3)-broken picture is much superior to the assumption of perfect SU(3). (Later in the text, however, we will discuss the issue of center-of-mass correction involved in this analysis.) Thus, it seems that the detailed verification of the Cabibbo scheme is far from settled yet. We believe that the problem of the SU(3)-symmetry breaking in the octet baryons deserves further detailed studies, both experimental and theoretical. Although quantum chromodynamics (QCD) is expected to provide a basic theoretical framework for this purpose, it is at present practically impossible to carry out necessary ~l~~la~io~s starting from the first principle. Thus we are obliged to resort to some ph~~om~nological approaches. One of the most widely used approaches is the MIT bag model 2’). This model, however, contains a number of problems. First, it violates chiral symmetry, which is one of the most fundamental symmetries of strong interactions **). This causes the non-conservation of the axial current on the bag surface, which is a disturbing feature in the calculation of the weak-interaction form factors. Secondly, it turns out that, within the quark-only MIT bag model, one cannot reproduce the experimental value of the nucleon magnetic moments if the parameters of the model are chosen consistentty with the mass spectrum “). Thirdly, as was emphasized by the Stony Brook group 24>,the rather large bag radius (R - 1 fm) predicted by the MIT bag model does not, as it stands, fit well into the standard picture of nuclei consisting of largely free nonoverlapping nucleons. In order to avoid these di~culties, the chiraf bag model was deveIoped, in which the discontinuity of the axial current on the bag surface is considered to be compensated by the emission of pseudo-scalar mesons. There are two versions of the chiral bag model. One version is the Princeton-Stony Brook mode1 23*24)in which mesons can exist onfy outside the bag boundary, the interior of the bag being assumed to be in the chiral-symmetric Wigner mode with no Goldstone bosons. Mesons outside the bag can exert pressure on the bag surface, giving a substantially reduced bag radius. This “little” bag model, which resolves the size problem of the original MIT model, has been extensively investigated by the Stony Brook group 24). In the other version, called the cloudy bag model

K. ~~~shj~a

(CBM) 25), mesons

are allowed

of this penetration

is that mesons

excitations. but

to exist inside the bag boundary. within

Which of these two versions

its answer

confinement

will

probably

the bag surface is closer to nature

require

and chiral-symmetry

We have investigated

formfactors

et al. / Weak interactive

a more

breaking

based

in a series of papers

559

One interpretation

simulate

the effects of qq

is an interesting

fundamental

question,

understanding

of

on QCD.

r”-r2) the consequences

of CBM on

the SU(3) properties of the weak interaction form factors and the magnetic moments of the octet baryons. The first paper lo) in the series was based on the surface-type CBM 26), whereas subsequent calculations 1’,‘2) were done with the use of the over the surfacevolume-type CBM 27*28),which is considered to be an improvement type CBM for a number of reasons (see below). Some results of these volume-type CBM calculations were briefly reported in two short communications 1’*12).One of the objectives of the present article is to give a fuller and more systematic account of the results obtained in the volume-type CBM. We remark that the CBM calculations reported here include (as in refs. ‘O-r*)) the clouds of all the octet mesons. A second

objective

of this article

is to present

a systematic

calculation

on the

effects of gluon exchange. (Some results on the vector-current form factors and the magnetic moments were briefly reported in ref. “).) As mentioned above, the “2 - A problem” in the hyperon decay has been with us for a while. On the other hand, regarding the magnetic moments of hyperons, there is the “A - H problem”; viz., as was pointed out by Lipkin 5), the additive quark picture invariably predicts which contradicts the experimental fact: p,, > I”~-. Ushio and P-n
to the MIT bag model, both of these difficulties can be removed. It conclusion in refs. ‘3-15) that, before the final quantitative

was emphasized, however, can be drawn, one needs mensurately

calculations

with the gluon-exchange

that include

meson-exchange

effects. Ushio “) presented

effects com-

an example

of this

type of calculation for the magnetic moments using the surface-type CBM including the pion cloud. We wish to report here a systematic study of the combined effects of the mesonic and gluonic effects on the weak interaction form factors as well as on the magnetic moments, the mesonic effects being calculated in the volume-type CBM including all the octet mesons. The organization of this article is as follows. In sect. 2 are defined the weakinteraction form factors and the magnetic moments. We give in sect. 3 a brief summary of the volume-type CBM, insofar as it is relevant for the present purpose. In sect. 4 we describe methods for calculating the CBM matrix elements needed for evaluating the weak-interaction form factors and the magnetic moments. The connection between the CBM matrix elements and the observables is established in sect. 5. Sect. 6 deals with the evaluation of the gluon exchange effects. Numerical results are presented in sect. 7; the problem of center-of-mass correction will be briefly dealt with at the end of the section. Finally, sect. 8 will be devoted to discussion and summary.

560

K. Tsushima

er al. / Weak interaction form factors

2. Global form factors and magnetic moments

2.1. WEAK

Let

us

1NTERACTI~N

consider

FORM

the

/3-

FACTORS

decay

A( pr) + e-+

Fe. The charge-raising

is expressed

as

between

current

octet

h”(x)

baryons

responsible

h&l(x)=

B;

B(pi) +

for the transition

B -+ A

)

(1)

= &I( VA(xl - A”(x)) ,

(2)

h”(x) = h&+0(X) + h&,(X)

hL,(x)

A and

(31

K&J”“(X)-A’“(x)),

where the Kij are the Kobayashi-Maskawa matrix elements. With the present experimental precision, the use of the six-quark expressions, I(,, = cos 0, and kI,, = sin 8, cos tY2,is not warranted, and therefore we might as well stay with the traditional Cabibbo parametrization; K, i = cos Bc and K12 = sin Bc. The definition of the form factors is as follows:

.h(q’)~*+~~~~qw+~ 4* A B A B 1

(A(pf)lV”(O)IB(pi)>=u(Pf)

U(Pi)

ig2(q2) (A(P,)IA”(O)IB(Pi))=

C(P~>

g,(q2)Y”

+m

(44

hlr

(+ A

9

4,

+

B (4b)

where

q =pr--pi7

and mA(mn) is the mass of the final (initial)

baryon.

For AS = 1

transitions, V” and A” should be replaced by V’” and A”‘, respectively. The form factors fi and gi (i = 1,2,3) may be called the “global” form factors because they represent the general structure of matrix elements of the currents between spin-4 particles. In the present work, we calculate these global form factors in terms of microscopic

transition

amplitudes

which are obtained

with the volume-type

CBM,

and investigate the dependence of these global form factors on the flavors of A and B. Furthermore, we will make a detailed study of the influence of one-gluon exchange contributions added to the CBM transition matrix elements.

2.2. MAGNETIC

MOMENTS

We next consider the magnetic element of J&(O) f or a baryon

moments

of the octet baryons.

state A can be written

The diagonal

matrix

as

(5)

K. Tsushima et al. / Weak interaction form factors

with

q = pf-pi.

proportional

Note

that

because

of the current

to qh in eq. (5). The magnetic

pa/, =

(A, S, = 41;

conservation

moment

d3rrxJ..,.(r)

/1* of baryon

561

there

is no term

A is defined

by

IA, S,=$. z

The Dirac form factor c1 and the Pauli form factor c2 are related to puA as 2m,pu, = c, + c2. In terms of the SU(3) octet vector components J&,,. is expressed as: Jk.,. = 9; 4-G-S:, where 9: and x+9; are the iso-vector and iso-scalar electromagnetic currents, respectively.

3. Cloudy bag model (CBM) There are two different versions of the CBM 25). In the surface-type CBM 2”), the mesons can interact with quarks only on the bag surface, while in the volume-type CBM 27*28)they are allowed to interact throughout the whole bag volume. The unitary transformation relating the surface-type CBM to the volume-type CBM was first discussed by Thomas 27), and also by Szymacha and Tatur 28). Although the two versions in their exact treatments are equivalent to each other, the equivalence is lost when one uses, as one is compelled to, perturbative expansion. Thus, lower-order terms in one version might contain more physics than those in the other version. In particular, it was emphasized that the volume-type CBM generates the Weinberg-Tomozawa term 29) in a natural way, a feature that gives an improved description of the TN S-wave scattering. Furthermore, the volume-type CBM has a better convergence property with respect to the excitation of quark configurations 30). Many interesting outcomes of the volume-type CBM have been investigated so far 3’). Motivated by these developments, we carry out here a systematic study of the consequences of the volume-type CBM for the weak interaction form factors for all possible p- decays between the octet baryons, as well as for the magnetic moments of the octet baryons. The clouds of all the octet mesons will be taken into account. It was pointed out in ref. lo) that the clouds of the kaon and the r] play non-negligible The lagrangian

density

roles for the AS = 1 beta decays. of the linearized

volume-type

CBM is written

ax) = =%ur +~r$+~ie,,t+y*T~ .GIT=

[q(x)(ia-

m)q(x)-

as 32) (7)

Bl~v-%(xMx)4,

(8)

562

K. Tsushima

et al. / Weak interaction form factors

We denote by Ho, Hi,, and HWT the hamiltonians associated with Z~rrf2’+, 2i”t and zwl, respectively. Hwf represents the Weinberg-Tomozawa term mentioned before. We have used here pe~urbative expansion with respect to the meson fields and kept terms up to l/f’,. In Iine with this approximation we assume that the quark motion can be restricted to the ground-state configuration in the static spherical MIT bag 8). Then

(11) where R is the bag radius, 0 is the lowest energy eigenvalue and e is a flavor-spin wave function; cy, = 1 for u- and d-quarks, which are assumed to be massless, whereas IY, = \/( E ztrm,)/E for the s-quark. The normalization constant N is so chosen that 5 d3r q+(r)q(r) = 1. With the restriction on quark configurations, the hamiltonian can be cast into a form in which the quantized meson fields interact with the lowest configurations IA), IB), . . . of three quarks ““). (There are 56 such configurations comprising an SU(3) octet and a decuplet.) Furthermore, in the same approximation, High in the volume-type CBM can be identified with the Hint in the surface-type CBM (see sect. 4). Thus,

(12)

with

V,,(k) = C A:,V,A,B(k)f-%, A,,&

ifoAB

V&f(k) =L

m,

u(kR) [2w,(k)(2r)3]1/2

sA,=(-)SA-S+/‘~

TtB = (-)

=A-fAm

ABe^*,.k)T:“, sm (

(14)

(15)

(16a)

(16b)

Here, ??roAis the MIT bag mass, and a,(k) (al(k)) is an annihilation (creation) operator for an octet meson LYwith momentum k and energy w, = (m”, + k2)‘/‘. S, and sA are the spin and its z-projection of /A). T, and f, are the isospin and its third component of meson cr, G,,,is the spherical representation of the unit vectors,

K. Tsushimn et al. / Weak interaction form factors

and u(x) =j,(x) +j,(x).

563

The coupling constant f,“n” is given by

where the last factor stands for a matrix element reduced in both spin and isospin spaces, A, being a spherical tensor in isospin space. The rank T, of A, is 1,; and 0 for a! = rr, K and 7, respectively. The explicit form of the A, is lo)

A, =

=&A, * ihz), A3 r&A,+ ih5), -&h,*

ih,)

1 A8

for a! = 7~ for LY= K for ‘~=r).

(18)

The doubly-reduced matrix element in eq. (17) is to be calculated using the SU(6) wavefunctions for the octet baryons. The strength Ka is given by

where a(b) is the flavor of the final (initial) quark pa~~cipating in the transition. We consider here the difference between Km’s for various octet mesons as a higherorder effect and use I(a=+,,, for all the mesons. This allows us to drop the CYdependence off o”,”and write it simply as f $” [ref. “)I. The physical baryon state /A) “dressed” with meson clouds can be written as “) (A) = (Z,“)“‘{ 1 + ( mA

-

Ho-

AHintA

~(Z~)1’2{1+(m*-~~)-‘Hi,,}~~),

)-lHintII&)

3

(20)

(20’)

where MA is the physical mass, I?o=~‘,m,A~Ao+X~ 5 d”kaL(k)a,(k), and A = 1-X e,Es6]Bo)(Bolis a projection operator that eliminates all states but those with at least one meson. The wavefunction renormalization constant 2,” can be calculated from the self-energy z*(E) as (Z,^)-‘= l-[XXA(E)/?IE],=,,, where T*(E) = (AoIHi”,(E - r?,)-‘H~~~]Ao). The explicit expression in lowest order is

When the vertex correction is applied to V,(k), the coupling constant is modified from f$” to f AB,but the change turns out to be of minor numerical importance “). We therefore assume St” =fAB = {~~B/~~~)~NN. The quantity f”” is calculable in CBM, but it is known 3’) that the slight readjustment of f NN leads to a better agreement with the TN scattering data. We therefore use here the “empirical” value f”” = 3.03 [ref. ““)I rather than the calculated value f NN=

X. Tsushima et af. / Weak interaction form fuctors

564

The vector current and axial-vector eq. (7) as Noether mations

currents

current

corresponding,

for the quark and the meson

are derived from the lagrangian

respectively,

density

to vector and chiral transfor-

fields 29):

(25) T’he last term propo~ionai to l/fn in each of the above expressions represents extra term which characterizes the volume-type CBM. Although the expressions eqs. (22)-(25) have apparent SU(3) symmetry, the use of the physical masses

an of for

the octet mesons partially destroys this symmetry. For J,^,,., we change 1 + i2 into 3 +Ji+ 8 in eq. (22); thus, J:.rn. = @”

f!h3+~~As!yB”+(f?,k+yTif8jk)~‘ahdtk

(26) 4. Microscopic

matrix elements

We now describe a method to calculate microscopic matrix elements corresponding to the octet-baryon p- decays. The treatment of the magnetic moments being quite analogous, we suppress its description. Let us consider the matrix element M,r,(f”)

=(As’( W(JA(0))IBs)

,

(27)

where W(J”(t))

d3XJh(X, t) e-iq’X

= I

(J” = V”, A”, V’” and A”‘).

(28)

In eq. (27), IAs’) and IBs ) are the dressed baryon states (eq. (20’)) with their J, quantum numbers written explicitly. The factor exp f-iq. x) with q =pf--pi in eq. (28) comes from the space-time dependence of the lepton wave functions. It is Wf J” (t)) as W(J” (t)) = Wq(JA( t)) + W+(J’( t)) + convenient to decompose W,,(J”(t)), where W,(J”(t))=Sd3xJh,(5t)e-iq’x (a=q,+,q+); W,(J”(t)) and W+(J” (t)) operate on quarks and mesons, respectively, whereas W,,(J” (t)) operates on both quarks and mesons. A method to relate the global form factors to &&,s(J”) will be described in sect.5. By putting eq. (20’) and eqs. (22)-(25) into eq. (27),

I(. Tsu~h~m~ ei al. / Weak interaction form factors

M,~,(J”) can be written up to l/f:

565

as

M,.,(J”)=T,+T,-l_T,+T,+7;+T,,

(29)

where Tj (j = 1 - 6) are given by +-2~i~(E,-EA-Ee-E,e)?; i; =

(-i)(&s’l

-r;=(-i)2(&s’l

(j=l-6),

(30)

(314

dt,

j

dt,

df2

T(f&vr(~J

~&“02))#3,~),

(31e)

6~ (-i)2(&~‘I dt, dt2Tfffint(l,) W~~(~A(f2)))IBoS) 1

(3lf)

where T implies a time-ordered product. Tj (j = 1,2,3,4,5,6) correspond, respectively, to diagrams (a), (b), (c), (d), (e) and (f) in fig. 1. Explicit expressions for T,- T6 are: TI = (Z,^Z,“)“*(A,s’~ W,IB,s)

(32a)

(32bf

(32e)

566

K Tsus~i~a

et al. / Weak interaction form facfors

T

(a)

7.y‘,

,,-.yp( (f)

Fig. 1. Processes contributing to M,.,(J*). Quarks are represented by solid lines and mesons by broken lines; the wavy lines with crosses attached indicate the current. The contribution of diagram (e) exists includes all the octet only for the vector current; i.e., J” = V” or V’“. Note that the present calculation mesons.

with wa,@= Jrn& + k2. Since the expression for T3 is somewhat complicated, it is given in appendix A. We note here that diagrams (a), (b), (c) and (d) in fig. 1 have their counterparts in the surface-type CBM *-lo), whereas diagrams (e) and (f) represent new contributions characteristic of the volume-type CBM. One can prove ““) that Hint of the surface-type CBM is equivalent to that of the volume-type CBM*. * Strictly speaking, this equivalence holds only in the limit of zero quark masses. However, since the mesonic effects are corrections due to the finite strange quark in the present context.

to the leading bag-model term, the change in the mesonic corrections mass may be considered as higher order effects which can be neglected

K. Tsushima et al. / Weak interaction formfacfors

This means matrix

that,

elements

CBM, even though objects related elements

as long as quarks

are confined

of the same type of diagram

to the lowest

are identical

the quark Iines in the two versions

to each other through

for the processes

(a)-(d)

a unitary calculated

567

bag orbital,

the

in the two versions

stand for somewhat

transformation.

Therefore,

with the surface-type

of

different the matrix

CBM lo) can be

taken over to the present case with no change. This saves us a great deal of labor, the calculation of these terms being extremely tedious. Thus, only diagrams (e) and (f) need to be newly calculated. In eqs. (32a-f), the intermediate sum over CO and D,, is limited to the fifty-six lowest q3 configurations. As mentioned earlier, this truncation is expected to be safer in the volume-type CBM than in the surface-type CBM 30). The matrix elements MSL,7(JA)of eq. (29) are given as follows: for JA = P or V’O,

M,~*(JO)=(s’~l~s)(a,+a,+a,+a,+a,), M,,,(J)

= -q(s’llls)(b,

+ b,+ b,+ b,+ b6) forJ”=Vor

-iqx(s’lols)(b:+b;+b;+b;+b;),

= W&W,+ -4(s’l(9.

V’,

for J” = A0 or A”,

~~~,(J”)=-q~(s’~a(s)(c,+c2+cq+cs), M,,,(J)

(33)

(34) (351

dz+ de) a)ls>(er

+ e,+ e,+ e6),

for J”=A

or A’.

Here, a,., bi, bi, ci, di and ei come from the processes represented (32a-f), and their explicit expressions are given in appendix B.

(36)

by 7; in eqs.

5. Relation between global form factors and microscopic matrix elements To establish

the connection

between

Msr,(J" ) of eq. (27) and the global

factors in eq. (4), we write down the non-relativistic frame lo);

of eq. (4) in the Breit

(A(P,)s’lJ”(O)IB(pi)s)~.XS’FX,,

(37)

where xS (xSf) is the two-component F is given by

F=~+~~~B~, A

form

reduction

spinor

for the initial

(final) state. The operator

WW

for J” = V” or V”,

B

-E(f,+f2)+ A

B

forJ”=Vor

v’,

for J’ = A0 or A”,

Wb) (38c)

K. Tsushima et at. / Weak interachm

568

form factors

*A-*B

g*+m+m2

except for q(a. gives ‘“,‘l)

q).

The

(38d)

B

or higher order in 141,

where terms of second

for J” = A or A’,

(4*-h,

A

identification

l/m,

of M,,,(J”)

and l/m,

have been dropped,

with xl,Fxs

for each type of J*

fi +

f,=-(b,+b,-tb,+b,+b,),

(39b)

~~~~~(~,+~)=b~+b~+b~+b~+b~,

(@)a)

-~(g*+g3)+&g2=-(CI+C2+Cq+C6), A

B

A

0

g1+

(4Ob)

Eg2=dl+d2+d6, A

B

(4Oc)

=e,+e2+e,+e,.

For the vector form factors fi , fi and f3,one can immediately invert eqs. (39a), (39b) and (39~) to write down the global form factors in terms of the microscopic transition matrix elements. The situation is not so straightforward for the axial-vector form factors, g,, g2 and g,. The problem lies with the c, and e,, appearing in eqs. (40a) and (40~). These terms corresponding to diagram (d) in fig. 1 involve a meson pole, and therefore their treatment in the long-wavelength expansion requires a special consideration. It is useful to recall here that the PCAC argument dictates that g, should contain a piece (actually a dominant piece) which comes from a meson-pole diagram. This suggests the decomposition of g, into two pieces of different origins lo): g3

=

(&Lre

+

k3)pole

(41)

.

Upon inserting this expression into eqs. (40a) and (~OC), we identify (g3)poie with the contribution of c, or e4, and subtract these quantities from each side of eq. (40a) and (40~). Then, we will be left with equations involving only (gJcore. Thus, instead of eqs. (40a) and (~OC), we have: -

y;-,“”

(8, A

B

+

(g3Lord

+

*

& A

B

=

-(cl

+

c2+

cc4

,

(40a’)

K. Tsushima et al. / Weak interaction form factors

569

(4Oc’)

(42) By solving

the simultaneous

equations,

eqs. (40a’), (40b) and (~OC’), one can obtain

gl, g2 and (g3LFe. We recall that the axial-vector current quark sector satisfies the PCAC relation &A”(x)

massless

-j$n%(x) .

=

This leads to the Goldberger-Treiman

of eq. (24) for the SU(2)

(u-d)

(43)

relation

-firgAB

= (m* + +Jg*(O)

,

(44)

3

(45)

where g,a(q’)

s (A(pr)Ij(O)IB(R))

with (3,a” + vnt)& = j(x), On the other hand, the dominant the pion-pole diagram (diagram (d) in fig. 1) is given by

part of g, coming

from

Eqs. (44) and (46) give the PCAC result

cm*+ m3)*fdq2)

k3(q2))pole =

m2,

_

q2

~

fLd0)

g

(o)

l

-

(47)

With the reasonable assumption that g~~(q*)/g~~~O) = u(qR), and g,(O) = G, (eq. (B.24’)), the PCAC result is equivalent to the “microscopic” expression eq. (42), or eq. (B.24)*. The reason why we use eq. (42) in the present work instead of the global PCAC relation eq. (47) is twofold lo). First, eq. (42) can be used for both AS = 0 and AS = 1 transitions whereas it is not obvious how to extend the PCAC argument to the AS = 1 sector, which involves m, # 0. Secondly, as discussed before, we are using in the present

work the meson-baryon

coupling

constant

that has been

re-adjusted from the value given in eq. (17). This implies that there will be some deviation from the Goldberger-Treiman relation eq. (44), which should entail a deviation from eq. (47) as well. In ref. 6), the f3 and g, terms were dropped from the outset on the base of the argument that the contributions of these form factors to beta-decay observables are small. As was pointed out in ref. ‘*), however, it is in general not justifiable to ignore f3 and g, in eqs. (39a)-(40c), because their presence can affect the other form factors, l Our treatment here should be contrasted and g, also receive the pion-pole contribution. ref. ‘?.

with that of Lie-Svendsen and Hdgaasen 16) in which The role of PCAC is not transparent in the treatment

g, of

570

K. Tsushima et al. f Weak interaction form .&actors

which are obtained by solving these simultaneous equations. It was demonstrated in ref. lo) that the retention of the g, can indeed modify g2 rather appreciably. 6. One-gluon exchange effects 6.1. GENERAL The

REMARKS

mesonic effects contained in CBM can modify the simple additivity inherent in any independent quark models. Introducing one-giuon exchange effects can give rise to another deviation from the additivity. One may naively expect that the two-body effects due to the exchange of gluons, which are flavor blind, are less efficient in producing SU(3) breaking than those due to flavor-dependent meson exchanges. That this is not necessarily so was pointed out by Ushio and Konashi r3) who demonstrated that the two-body effects due to meson exchange and those due to gluon exchange have different selection rules, and that the latter can play a unique role for certain observables. The giuonic effects on gi/f, ratios in the MIT bag model were studied in ref. 13),whereas Ushio i4) calculated the octet baryon magnetic moments by adding the gluonic ~ont~bution to the results of the surface-type CBM. The results indicate that the one-gluon exchange contributions can resolve the “A -F problem” for the magnetic moments along with the “.E - A problem” for the beta decays. Furthermore, Hdgaasen and Myhrer 17)pointed out that the problem of the ratio &p/p” caused by the isovector pionic correction can also be resolved by the inclusion of the one-gluon exchange effect. We present here the results of a systematic calculation in which the volume-type CBM supplemented by one-gluon exchange effects are used to evaluate all six weak-interaction form factors for each of the possible beta decays among the octet baryons, and also the magnetic moments of the octet baryons. The evaluation of one-gluon exchange effects involves a delicate aspect, as was first discussed by DeGrand et af. **) in their study of the gluonic effects on the hadron spectrum. In considering one-gluon exchange, the QCD interaction hamiltonian is expanded up to second order in E&. Then, there will be diagrams that involve the color Coulomb interaction, like diagrams (a) and (b) in fig. 2. These diagrams, however, will lead to color non-singlet hadron states. This difficulty arises because one cannot construct the color Coulomb Green function which satisfies both the bag boundary condition and the color-singlet requirement. DeGrand et al. proposed to include certain types of quark self-energy diagrams in order to satisfy the color singlet requirement. Ushio and Konashi 13) developed a practical method to carry out this prescription. On the other hand, QCD for quarks and gluons confined within a bag was studied in great detail by Lee 34) and by Hansson and .JafIe “). Lee 34) emphasized that those terms responsible for color non-singlet states should be considered as unphysical and are to be discarded. From this viewpoint, one should simply leave out the color Coulomb interaction altogether.

K. Tsushima et al. / Weak interaction form factors

571

(b)

(a>

-_L

L

T

4

(cl

W

Fig. 2. Processes treated as one-gluon exchange effects. The broken-line helices in (a) and (b) represent the color-Coulomb interaction, while the solid-line helices in (c) and (d) represent the tranverse gluon propagators. The wavy line with a cross stands for a current.

In the present study, we carry out calculations in the two approaches, one based on ref. i3) and the other based on ref. 34), without attempting to determine which one is fundamentally better. The calculation based on the method of Ushio and Konashi r3) will be referred to as type I, whereas that based on ref. 34) will be called type II. Fortunately it will turn out that the difference between type I and type II is not numerically important for the quantities of our present interest. Maxwell and Vento 36) made a careful study of one-gluon exchange effects on g,, gA and the nucleon magnetic moments. In particular, they examined in great detail the dependence of these quantities on the QCD cut-off parameter, the quark wavefunction renormalization, and the vertex correction. One of the main conclusions of ref. 36) is that applying one-body vertex corrections to the bag model based on the static cavity approximation may lead to physically ambiguous results. In the present

work, however,

we will not consider

these basic questions,

but limit

ourselves to a very pragmatic question. That is, in view of the impressive phenomenological success of the one-gluon exchange effects in refs. 13-15,17,18),we wish to investigate how well the one-gluon exchange effects combined with the volume-type CBM can account for the totality of the data on the octet baryon beta-decays and the magnetic moments.

6.2. S-MATRIX

ELEMENTS

FOR

ONE-GLUON

EXCHANGE

We consider quark-gluon interactions inside the static spherical MIT bag, up to second order in gc; this implies that we can neglect the gluon self-interaction and apply the usual method of QED.

K. Tsushima ef al. / Weak inieracrion form factors

512

As in sect. 2, we consider the p- decay between octet baryons: B(pi)+ A( pr) + e-( p,) + c,( p,). The S-matrix element for this decay is written as 13) m

S(B+ A+e-+

F,) =(A&;

e-?,/Texp

-i {

J -m

d@&(r)+

H&r))

/BOX), (48)

where d3x(~A(x)~~(x)~h.~.~,

Hw = &G, J

(4%

with h A(x) as defined in eq. (I), and h(x) being the ordinary charged lepton current. The quark-gluon interaction H, is written as

J

ir,=gc

d3x: ja(x, t) . A”(x, t): +$g;:

V

JJ

d3x d3x’ G,(x, x’): p”(x, t): :p’(x’, t):

,

V

(50)

where

where A, are chosen second

$(x) is the quark field operator, and A”(x) the gluon field with color a; the the Gell-Mann matrices for color SU(3), and V the bag volume. We have here the Coulomb gauge for the gluon field. The S-matrix element up to order in H, is written as (for AS = 0 transitjons) (53)

For AS = 1 transitions, the quantity in the square bracket should be replaced by (R(V’“)-g(A’“))sin &. K(i”) (J” = V”, A”, V’“, A”) is given by Z(P)

= s,(J”)+s,(J”)

S,(J”)

5

(-i)‘(A,s’I

)

(544

J dt,dt,T(H,(t,)Ws(JAtt2)))tB,S>, Wb)

s2(JA) =f(-i)3(bs’l J df,dt,&T(Hc(hWc(h) ~q(J”(~dNbW,(54c) where W,(J”(t))=jd3xJA(~ t) ePiq’x. Upon substituting the explicit expression for H, and retaining the terms up to gf will give various types of diagrams. Diagrams

K. Tsushima et al. / Weak interaction form factors

of the two-body shown

exchange-current

type are depicted

573

in fig. 2, and all others

are

in fig. 3. We follow the conventional

have already

been included

ansatz **,r3) that the processes in fig. 3 in the bag model parameters, such as the quark masses.

This leaves us with the processes

in fig. 2, whose contributions

we denote

by K’, i.e. (55)

We must recall here, however, that diagrams (a) and (b) in fig. 2, involving the color Coulomb interaction, violate the color-singlet condition. One method to avoid diagrams of the quark this difficulty is to include, following refs. 13,22), additional self-energy type shown in fig. 4, and regard the sum of the contributions of figs. 2 and 4 as one-gluon diagrams in fig. 4:

exchange

effects.

We denote

I?= [K(.P)](fi,

by kU the contribution

4).

Fig. 3. Processes considered to be already included in the bag-model and solid-line helices represent the color Coulomb interaction respectively.

of the

(56)

parameters. The broken-line helices and transverse gluon propagators,

K. Tsushima

574

et al. / Weak interaction form factors

(al

(b)

(cl

(d)

Fig. 4. Particular self-interaction contributions to be added to satisfy the color-singlet requirement. The square on a quark line is a reminder that the intermediate quark state is limited to the lowest bag eigenmode.

The explicit expression for k’ of eq. (55) is given by Z’=C

fj

(j=a,b,candd),

(57)

i where ?:, = (&s’lgT.

I

d3x d3x’ d3x” dt dt” G,(x, x’) e-iq*X”

[ ?b =(&s’/g2, I x :~(x)~‘~

i&(x, x”)J”!x”)~(x”)li;(x’)rOS

+(x’):

1

(58a)

1

Wb)

/B,s),

d3x d3x’ d3x” dt dt” G,(x, x’) eWigw”

x :~(x”)J”(x”)iSF(xn,

x)

~“2~(x)~(xf)~‘~

@(xl): ]B,s),

daX’ ddX” jG;;(X, $) ‘1”: =(&s’lg:. IddX [

e--i9’X”

1 dJX’ ddXjf jGz\(x, xt) e-i¶‘x” IF& =&)s’lg: IdaX :~(x”).P(x”)iS#‘, x)y”’ ; JI(*)li;(W~ +(x7: s 1lBo.9 x

:&(x)+

I.

i&(x,

x”)J”(x”)(k(x”)3(x’)y”~

$(x’):

IB”s),

(58~)

(58d)

I(. ~susk~~a ef al. / Weak inreraction

The color-Coulomb

formfactors

57s

Green function GE(x) x’) is 37)

G&r, x’) =

(59)

whereas the static gluon propagator iG”,b,(x, x’) =

in the Coulomb gauge is given by

m d(r - t’)
= i13”b[G~n(~, n’)+ G::(x,

x’)] ,

(60)

where the transverse electric and magnetic Green functions for gluons are given, respectively, by 37): GT,E,(q x’) =:

GTyX

x,)

FnR

-9

=2_(V

(61)

x kz(V’x 1’)” l2

I

I

C52)

d3x” h(x, x”)h(x”, x’) ,

with h (x, x’) =

1 47&x’/

P[(f. ;‘) ,

(63)

In the above, 2 = (--ix x V), P,(& 2’) the Legendre polynomial, and R the bag radius. The Feynman propagator i&(x, JJ) for quarks confined within the bag is given by u,(x)ii,(Y)+v,fx)~~~Y) dw e -i*(x”--y”) w+w,-i& w-w,iiE

I ’

(64)

where a: denotes the assembly of the principal quantum number, Dirac’s K the third component of the total angular momentum and the quark flavor. The spinor U,(X) is the charge conjugation of the u,(x). (See appendix C.) The initial and final baryon states, IB,s) and /A&) in eq. (58) are assumed to be in the lowest MIT bag configuration. This assumption imposes stringent constraints on the gluon propagator, restricting gluon eigenmodes that can be exchanged between the quarks; only those modes carrying no energy can be exchanged between quarks. Eqs. (%a-d) can be rewritten as ‘-~;:~-~~T~S(E~-E~-E,-E~~)T:, Tj z (:;g,)”

&

&X6 &$t ei4’x”fj

(651

(i=a,b,cand

d),

(66)

K. Tsushimo et at. / Weak

576

inreraction

form factors

where ua (x)%X (x? + % (xk

t:, = G,(x, x’>Q(x)r*Aa

Ef+

&f--%

tb = G,(x,

J”(x”)q(x”)p”(x’)

0,

u,(+“)a,(X)+V,tX”)ii,(X)

x’)q(x”)J’(x”)

&i+W,

&i-W,

xJA(x”)q(xn)~(x’)y”h,q(x’) t& =

1

(x”)

1

r*&&)P”(x’)

)

(674

(67b)

9

)

G’,,(x, x’)q(x”)Jh(Xv)

(67~)

u~tx”)ii*(x)+~~tx”)~~(I) &i+W,

Ei+CO,

[

x r”A,s(x)4(x’)r”~,q(x’)

1

(j=TEorTM),

(67d)

In the above, Es, E, are the energies of baryons B and A, and E, and EC, those of the electron and the anti-neutrino; sr(ei) equals the lowest bag eigenenergy for the final (initial) quark, and w, represents bag eigenenergies for the inte~ediate states. (Summation over (Yis understood.) The quark wave function q(x) is as defined in eq. (ll), and p”(x)= q’(x)(h,/2)q(x). (See appendix C for details.) Similarly, l? of eq. (56) is given as follows.

where ~~=-21Tis(EB-EA-E,-E~~T:!,

(6%

with T; s $g;

d3x d3xI d3,/ ei4’X”t:

t:: = G,(x, x’)p”(x’)q(x)y

,&I -y

(i=a,b,cand ua(x~~~~x”)+

ti; =

t,”= G,(x, x’)q(x)y

R(,(x)~~(xa)+v,(x)~~(x”) &f-w2

t; = G,(x, ~‘)q(x”)J~(x”)

&f+

we

u,(x”)~~(x)+~*(x”)ii,(xf &i--W, &j-i-W, i

G&x, x’)~~(x~)~(x”)~~(x”~

Ef+

1 tx> I

J”(x”)q(x”),

(714

rO$q(x),

t71b)

I

I

rA(X”)q(x”)pa(x’)

WC2

%x(-e17,(xl + u, W’) ij, &i-W,

(70)

%(x>%s,(x”)

Q-S

[

d),

&i+0,

)

(714

(714

K. Tsushima

The total contribution

et al. / Weak interaction form factors

of the one-gluon

exchange

processes

577

is given by

K=K’+&“,

(72)

with I?’ and 8” given by eqs. (57) and (68). We can show that M,,,(J”) acquires

from the gluonic

effects an additional

i? = -2d(E,-E,The calculation

of the gluonic

piece M$,(J”)

E,-

effects based

E,)Mt.,(J")

defined .

of eq. (27) by: (73)

on eq. (73) will be called type I; this

is the method used by Ushio and Konashi r3). On the other hand, if we are to adopt the prescription of ref. 34), we should simply drop the color Coulomb interaction. Thus, we only have to evaluate the contribution of diagrams (c) and (d) in fig. 2. The calculation in this scheme will be referred to as type II. Let 2”’ denote the one-gluon exchange contribution in the calculation of type II; 2”‘s [k(J”)] (diagrams (c) and (d) in fig. 2). Then, instead of eq. (57), we obtain ,,,,= In this case, the additional defined by

f:+

contribution

f&.

to M,,,(J”)

(74) of eq. (27) is given by M’f,,(J”)

~“‘=-~T~~(E,-E,-E,-E,)M~~,~(J*).

6.3. MICROSCOPIC

We summarize

MATRIX

ELEMENTS

(75)

M:.,(J”)

here the final expressions

for M$,(J*),

details to appendix C. M$,(J*) defined by eq. (73) or eq. (75) can be rewritten, expansion,

into the following

relegating

after the long-wavelength

form.

M$,( V” or V”) = (s’(l(s)a’, Mf.,( V or V’) = -q(s’~l~s)b”Mf,,(A’ M$,(A

the calculational

(76) iq x (s’la[s)b”‘,

or A”) = -q*(s’la[s)E, or A’) = (s’laJs)d” -q(s’l(q.a)ls)t?.

(77) (78) (79)

In the case of type II, the microscopic matrix elements 6-e” in the above equations receive contributions only from diagrams (c) and (d) in fig. 2; E, b: dl, E, d” and e” in this case will be denoted, respectively, by cl, b;, 6, &, d, and k, . Their explicit expressions are given in appendix C. In the case of type I, we need to include in addition the contributions of diagrams (a) and (b) in fig. 2 and those of the diagrams in fig. 4. The additional contribution

K. Tsushima et al. j Weak interaction form factors

578

of these processes

for each type of the current

;;2. Their

explicit

expressions

elements

for type I are given

a=&+&

is denoted

by Z,, &, &

E2, & and

are given

in appendix C. The microscopic matrix Q=@,+& E=c’,+&, by: a’ = &1+ a”*, b”=&+b”,

and e”=e”,+&.

The connection

between

the microscopic

transition

amplitudes

eqs. (76)-(79)

and

the global form factors, eqs. (39a)-(40c), can be established simply by adding the gluonic contributions to the CBM expressions in the following manner. Add G to the r.h.s (right-hand side) of eq. (39a); add 6 to the r.h.s. of eq. (39b); add 5’ to the r.h.s. of eq. (39~); add E to the r.h.s. of eq. (40a’); add 2 to r.h.s. of eq. (40b); add e”to the r.h.s. of eq. (40~‘).

7. Numerical

results

In order to characterize the volume-type CBM completely, we must specify the parameters involved in the model. First, we discuss the choice of the bag radius R. Although the little bag model ‘) suggested R - 0.5 fm for the nucleons, the subsequent development of the Cheshire Cat picture 38) seems to indicate that larger values of R are not excluded. For the hyperons, recent studies ‘8S39)indicate that their radii should be around 1 fm or somewhat large?. On the other hand, R - 1 fm is commonly used in CBM for non-strange as well as strange baryons “). In the present study, therefore, we use R = 1 fm as a representative value. For the quark masses, we choose: m, = md = 0 and m, = 279 MeV. This choice gives reasonable results for the mass spectrum *l). The calculation of diagram (e) in fig. 1, which is relevant with the vector current, involves divergent loop integrals. We regularize the integrals, following refs. 2g,40), by introducing represents the a-dependence adjusted so as

a momentum-dependent “damping” factor exp (-$Rzk2), where R, “effective” radius of meson a (a = rr, K and n). We ignore here the of R, and treat R, as a purely phenomenalogical parameter to be to reproduce an appropriate observable. In the present work we use

fi(n+ P)~~,, = 3.70 for this purpose. The values of R, determined this way will be described later. We have already mentioned that the flavor dependence of the meson-baryon interaction is assumed to be given by the SU(6) quark model with no additional symmetry breaking. In the same spirit the flavor dependence of the radial integrals appearing in aj, bi, bf, ci, di, and e, (i = 5 and 6) in eqs. (39a)-(40c) are regarded as higher-order effects and will be ignored. As for the color electric coupling constant (Y, appearing in the calculation of the gluonic effects, we use LX,= ga/4?r = 2.2, a value consistent with the hadron spectrum in the MIT bag model with R = 1 fm. * For example, in a chiral-bag model treatment of ref. 39), R,- is required to be very large to reproduce the experimental value of CL?-; if the s-quark mass is fixed to be, say m,= 150 MeV, then R,-a 1.3 fm.

K. Tsushima et al. / Weak interaction form factors

In sect. 6, we described effects: type I (based be discussed

at the end of this section,

out to be practically in tables

l-7,

in detail

the two methods

on ref. “)) and type II (based

indistinguishable.

we give only the results

the numerical Therefore,

579

for calculating

the gluonic

on ref. ‘“)). However,

as will

results for types I and II turn

in presenting

the numerical

results

for type I.

Some comments are in order here on kinematical corrections, which have not been touched upon so far. Center-of-mass (c.m.) corrections (and, probably to a lesser degree, recoil corrections as well) might modify the magnitude and pattern of symmetry breaking 16,i9). As a matter of fact, the c.m. correction plays an important role in the argument of Donoghue et al. 19), mentioned in sect. 1, that the SU(3) broken picture is superior to the assumption of perfect SU(3) in fitting the data on the semi-leptonic hyperon decays. These authors used a specific model for estimating the c.m. correction in the MIT bag model. Although the validity of this particular model might be subject to questions, their work indicates that the issue of SU(3) symmetry breaking in the octet baryons may be interrelated to the problem of c.m. correction. On the other hand, we estimated in ref. lo) c.m. correction to the weak interaction form factors that were evaluated in the surface-type CBM. A specific formalism for c.m. correction 42) was used there. One of the motivations to include the c.m. corrections in ref. lo) was a phenomenological one; viz., the g,‘s in the surface-type CBM tend to be too small compared with the experimental values while the c.m. correction helps to increase the g,‘s. By contrast, as will be discussed below, the volume-type CBM in general gives values of g,‘s rather close to experiment, whereas the inclusion of c.m. correction drastically deteriorate agreement with data. This feature weakens the phenomenological motivation to introduce the c.m. correction in the volume-type CBM. From a calculational point of view, it should be remarked that there is unfortunately at present no universally accepted prescription for implementing c.m. corrections. The method in ref. lo) is just one of the possible approximations. In the present work, therefore, we have chosen not to include cm. corrections in the main body of discussions. This limitation should be kept in mind throughout the whole presentation of the numerical results in this section. At the end of the section we will briefly come back to the problem of the c.m. correction and discuss some results obtained by applying the c.m. correction using the formalism of ref. 42).

7.1. WEAK INTERACTION

FORM

FACTORS

We present in tables 1, 2, 3 and 4 the calculated weak-interaction form factors; tables 1 and 2 refer to the AS = 0 transitions, while tables 3 and 4 refer to AS = 1 transitions. The results are given for the following cases: (1) MIT bag node1 with no meson clouds - (abbreviation) MIT; (2) Volume-type CBM with octet meson clouds - (abbreviation) CBM; (3) Volume-type CBM with octet meson clouds plus one-gluon exchange contributions of type I - (abbreviation) CBM+OGE.

K. Tsushima et al. / Weak interaction form factors

580

TABLE

Vector current Transitions

n+p

from factors

1

for AS = 0 transitions

Models

fl

MIT CBM CBM+OGE

1.00 1.00 1 .oo 1.oo

2.21 (3.70) 3.70 3.70 3.70 3.70588748 * 1.23 x 1O-6

0.00 0.00 0.00 0.00

1.93 (2.83) 2.77 2.66 2.34 2.77ztO.15 “)

1.41 1.41 1.41 fi

0.89 (1.96) 1.82 2.09 1.18

0.00 0.00 0.00 0.00

1.90 (2.32) 2.18 1.91 2.03 1.78ztO.06

0.00 0.00 0.00 0.00

SU(3) exp. MIT CBM CBM+OGE SU(3) exp. MIT CBM CBM+OGE SU(3) MIT CBM CBM + OGE SU(3) exp.

-1.00 -1.00 -1.00 -1.00

h

h 0.00 0.00 0.00 0.00

-0.06 -0.04 -0.04 0.00

Theoretical results for three cases are given: (1) MIT bag model - (abbreviation) MIT; (2) volume-type CBM with octet meson clouds - (abbreviation) CBM; (3) volume-type CBM with octet meson clouds plus one-gluon exchange effects (type I) - (abbreviation) CBM + OGE. The rows labelled SU(3) correspond to the results of semi-empirical Cabibbo-SU(3) fitting to data 2). The rows Iabelled “exp” give the fi's deduced from the magnetic moment data using CVC. As described in the text, the volume-type CBM involves one adjustable parameter R,. The value of R, used here is: R, = 0.475 fm for CBM, and R, = 0.446 fm for CBM + OGE. (For the method to fix R,, see the text.) For MIT, the resealed values of f2are given in the parentheses, the method of resealing being described in the text. “) Peterson et al. 4).

Regarding comparison with the experimental data, we mention that the quality of the existing data is such that their theoretical implications can be analyzed only in conjunction with the following assumptions ‘): (i) There is no SU(3) symmetry breaking in the leading-order vector form factor f,; (ii) the flavor dependence of f2 obeys the SU(3) prediction; (iii) the form factors f3and g, corresponding to the second-class current may be a priori set equal to zero; (iv) g, can be ignored because its contribution to electronic P-decays involves rnz. Fitting to data with the abovementioned assumptions is referred to as CabibboSU(3) fitting *l), the results of which are also given in tables 1-4. We first look at the vector current form factors shown in tables 1 and 3. The above-mentioned assumptions that go into the analysis of the P-decay data prevent us from making a direct comparison of the calculated vector-current form factors

K. Tsushima

et al. / Weak interaction form factors TABLE

Axial-vector Transitions

current

n+p

z--n

for AS = 0 transitions

g,

g,

MIT CBM CBM + OGE SU(3)

1.09 1.26 1.29 1.23

0.00 0.00 0.00 0.00

2.97 0.99 + 1.36 Y 1.15+1.36Y

MIT CBM CBM+OGE SU(3)

0.53 0.63 0.66 0.62

-0.10 -0.15 -0.22 0.00

2.33 0.91+ 0.67 Y 1.28+0.67Y

MIT CBM CBM+OGE SU(3)

0.62 0.68 0.64 0.67

0.00 0.00 0.00 0.00

2.90 1.38+0.77Y 1.23+0.77Y

MIT CBM CBM + OGE

0.22 0.27 0.32 0.28

0.00 0.00 0.00 0.00

1.28 0.52 + 0.27 Y 1.15+0.27Y

MIT, CBM, of Y in

CBM+OGE, the column

and SU(3), see the Y= for g, is:

SU(3) For the meaning caption of table

2

form factors

Models

581

of the abbreviations, 1. The definition

g3

(m,+ma)*u(qR)l(m2,-q2).

with the experimental data. We therefore discuss whether and to what extent the calculated vector form factors deviate from the SU(3) symmetry pattern. Starting with the leading term fr, we recall that CVC demands that the fi for a transition within an isomultiplet should not be influenced by the meson clouds or by the gluonic effects. Our results for the AS = 0 transitions in table 1 satisfy this requirement, providing a check for the numerical calculations. For non-isomultiplet transitions the meson clouds and the gluonic contributions can in principle affect fi , but table 3 indicates that the effects are small. On the other hand, the AdemolloGatto theorem demands that there should be no first-order SU(3) breaking in f,‘s irrespective of microscopic models used to calculate them. The f,‘s given in tables 1 and 3 indeed show very small deviations with this theorem*.

from the SU(3) pattern

in conformity

l In the bag model, f, is given in terms of a radial integral that measures the overlap between the initial and final quark wave functions. Therefore, for a A.5 = 1 transition, which involves a radial integral with different quark masses, the magnitude offi is expected to be smaller than the SU(3) limit. A similar argument will hold true also for CBM calculations. The magnitudes of some of the f,‘s in table 3, however, are slightly larger than the corresponding SU(3) values. We must recall here, however, that, in transitions for which the initial and final baryon masses are different, the roles off, and f3 are “intermingled”; thus, it is not f, itself, but a linear combination off, and f, that is related to the radial integral of the normalization type (see eq. (39a)). This is the reason why our results do not agree with the above-mentioned simple expectation.

582

K. Tsushima et al. / Weak interaction form factors TABLE

3

Vector current

The comparison

of the theoretical

and experimental

values of f2 could provide

a

non-trivial test of theoretical modes. As mentioned above, however, there are at this moment no direct experimental data for f2. For the AS = 0 transitions, however, invoking CVC allows us to deduce the “experimental” value off2 from the observed values of the relevant magnetic moments I); the explicit formulae are: (fi +fJ_,, = PP - P*,J%fi)z~+* = ((mn + m~)/2m,)(~9-~Ln), Jz(fi+.L)~-+x0= (mJm,> x ps-). (The magnetic moments are (PZ+ - PZ-) and (fi +.L)z- _,zo= (ma/m,)(p,-oin units of the nuclear magneton.) The “experimental” values of f2 obtained in this manner are given in table 1. One will see that they show appreciable deviations from the SU(3) pattern and that the pattern of deviation is reproduced reasonably well in case (2)CBM. With the inclusion of the gluonic effects, case (3)CBM+OGE agreement with the data improves for S-+ 8’, but worsens for X --f 2’. In comparing case (1)MIT with the other models, we need to recall that we have adjusted R, in cases (2) and (3) so as to reproduce fJn+ p)_. Therefore, the comparison will

K. Tsushima et al. / Weak interaction form factors

583

TABLE 4 Axial-vector Transitions

current

Models

.??+p

s-+n

go+

_.z+

z--tzyO

0.57 0.02 + 0.27 Y 0.32 + 0.27 Y

SU(3)

-0.04 0.00 -0.02 0.00

MIT CBM CBM+OGE SU(3)

0.16 0.21 0.26 0.20

-0.03 0.00 -0.01 0.00

0.41 0.02 + 0.19 Y 0.2210.19Y

MIT CBM CBM+OGE

-0.02 -0.07 -0.10 0.00

0.97 0.44 -t-0.33 Y 0.23 40.33 Y

SU(3)

0.29 0.30 0.27 0.28

MIT CBM CBM+OGE W(3)

1.19 1.30 1.33 1.23

0.07 0.07 0.16 0.00

4.27 1.674 1.36Y 1.88+-1.36Y

MIT CBM CBM+OGE

0.84 0.92 0.94 0.87

0.05 0.05 0.11 0.00

3.02 1.18+0.96Y 1.33 + 0.96 Y

SW31 For the meaning caption of tabie

g3

0.23 0.30 0.36 0.28

MIT CBM CBM+OGE

of the abbreviations, 1. The definition

-0.87 -0.93 -0.95 -0.89

R2

-1.95 -0.61 - 0.99 Y -0.56 - 0.99 Y

SU(3) H_+n

g1

0.04 0.07 0.06 0.00

MIT CBM CBM+OGE

A’P

for AS = 1 transitions

form factors

MIT, CBM, of Y in

CBM+OGE the column

and SU(3), see the Y= for gtr is:

(m~+ma)~u(4R)/(m:-q’).

be more meaningful if we rescale f,(RIIT) in such a way that fi(n + p)__ should be reproduced. We choose here to rescale the right-hand side of eq. (39~) by multiplication with an overall factor; the factor is adjusted so that (f, +f&,_, = 4.70 should be reproduced. These rescaIed values are given in the parentheses in tables 3 and 3. One will see that even after this resealing the agreement with experiment is poorer in case (I)MIT than in other cases. Also, in the fi’s for the AS = 1 transitions (table 3), there are appreciable differences between the resealed f,(MIT) and the other results. Unfortunately, however, this feature cannot be tested experimentally at present. Regarding f3, which corresponds to the second-class vector current, the first obvious comment is that the f3’s for the AS=0 transitions (table 1) vanish in conformity to the CVC requirement. The.f;‘s for the AS = I transitions (table 3) are seen to vary appreciably, depending on which model one uses, and for some

584

transitions,

K. Ts~sh~~u et al. / Weak interacfio~ form factors

f3’s can be rather

in ref. 4’). Unfortunately

large. The rather

it is beyond

We now turn to the discussion

the present

experimental

of the axial-vector

term g, , a remarkable

and 4. Concerning

the leading

in the volume-type

CBM [given as case (2)CBM

are substantially

large values

of f3 are also reported feasibility

form factors

to test this.

given in tables

2

feature with the g,‘s calculated

in the tables]

is that the magnitudes

Iarger than those of the MIT bag model [case (l)MIT].

Concerning

g, = g,(n+ p), it was already known that the MIT bag model gives too small a value and that the surface-type CBM makes gA even smaller, whereas the volume-type CBM gives reasonable results “). This is due to the contribution from diagram (f) in fig. 1. The results shown in tables 2 and 4 indicate that there is a similar trend for all the transitions among the octet baryons ‘I). (For the results with the surfacetype CBM, see ref. lo)). Furthermore, the g,‘s obtained in the volume-type CBM are very close to the values determined through the semi-empirical Cabibbo SU(3) fitting to the data*). Thus, as far as g,‘s are concerned, the apparent role of the meson clouds in the volume-type CBM is to “recover” SU(3) symmetry, which, because of the different quark masses, is lost to certain degree in the MIT bag model. It must be emphasized, however, that the same meson clouds play a nontrivial role in explaining SU(3) breaking in the magnetic moments of the octet baryons I’), When the gluonic effects are added to the CBM results, the g,‘s vary rather appreciably for some transitions; the relative change for those cases where g,‘s are small can amount to as much as 25%. In order to demonstrate how these changes affect agreement with experiment, we compare in table 5 the theoretical values of g,(O)/f,(O) with the available experimental data. Five theoretical cases are shown. The first three cases are the results of the present calculation. As explained earlier, MIT, CBM and CBM+OGE represent the MIT bag model, the (volume-type) CBM, and the (volume-type) CBM plus one-gluon exchange effects (type I), respectively. The column labelled “Ushio, MITSOGE” gives the results of ref. 15>, in which the one-gluon exchange effects are added to the MIT bag model; in this work, the bag radius and the quark masses were adjusted so as to reproduce the experimental value (gi/f,),.+,, = 1.23. The column labelled “Carson et al., MITIOGE’” gives the results of ref. 4’), in which the one-gluon

exchange

effects including

vertex

corrections are added to the MIT bag model; the experimental value gJf, = 1.254 was used as one of the input quantities to fix the parameters in the model. As described in sect. 1, there are at present two possible sets of experimental data. In table 5, the set given in ref. ‘) is referred to as (a), whereas that due to Bourquin et ai. ‘) as (b). For each set of data, and for each theoretical case, we have evaluated the x2 defined by x2= C,R:, with Rf = [(xtheo’- x~xp)/8xexp]2~ It should be emphasized that none of the calculations in table 5 includes c.m. corrections, and therefore discussions in what follows are to be taken with some reservation. At the end of this section we will briefly discuss the effects of c.m. corrections on table 5. For the data set (a), the x2 for CBM is by far the smallest. In this sense, although the inclusion of the gluonic effects has the remarkable phenomenological consequences

K. Tsushima

et al. / Weak interaction form factors

585

TABLE 5 Comparison

cl)+ MIT

g,lf,(n + P)

1.09 747.1 274.1

R:(a) R:(b) g,/f,W+ R?(a) R:(b)

n)

g,/fi(E-+ R:(a) R:(b)

‘4)

g/K,‘” R:(b)

x*(a) x’(b) ,?(a) 2(b)

-0.23 9.4 4.8

+ P)

of theoretical

(a+ CBM

1.26 1.0 5.4 -0.30 2.1 0.6

values of g,(O)/f,(O)

(3)+ CBM +OGE

with experimental (4)

MI;;h$E

1.29 36.0 32.1

1.23* 16.0 0.0

-0.36 0.0 0.2

-0.33 0.6 0.0

(5) Experiment

;;;I;;;

1.25* 0.0 1.5 -0.29 2.8 1.0

0.25 0.0 0.0

0.22 0.4 0.4

0.23 0.2 0.2

0.29 0.6 0.6

0.72 1.1 0.4

0.77 9.2 5.4

14.8 0.79 9.0

14.8 0.79 9.0

55.4 0.88 36.0

51.1 41.6 15.8 10.8

31.5 9.2 15.7 9.2

58.8 39.1 58.8 37.7

12.3 11.5 11.3 6.3

1.254 * 0.006 “) 1.239rtO.009 “)

-0.362 * 0.043 “) -0.34* 0.05 h)

0.24 0.0 0.0

757.6 279.4 23.8 16.3

data

0.25 + 0.05 “) 0.25 * 0.05 b,

0.694 * 0.025 “) 0.70 f 0.03 b)

Five theoretical cases are shown. The first three cases, (1) MIT, (2) CBM, and (3) CBM+OGE are the results of the present calculations. For the explanation of these three cases, see the caption of table 1. Case (4) Ushio, MIT+OGE is the results of ref. 15), in which the one-gluon exchange effects are added to the MIT bag model. Case (5) Carson et al, MIT+ OGE corresponds to the results of ref. 4’), in which the one-gluon exchange effects including vertex corrections are added to the MIT bag model. For the two different sets, (a) and (b), of experimental data are given Rf [(x~“‘“‘-x~p)/BxpXp]2 for each transition, and x2 E 1, R:. The symbol i’ stands for the “effective” x2 defined in the text. t Present calculation. “) Ref. ‘). ‘) Ref. ‘). * Fitted to data.

mentioned

earlier,

it does not necessarily

lead to improvement

in overall

fitting to

the data on g,(O)/f,(O). If we use the data set (b), the x2’s for CBM and MIT+OGE are significantly smaller than the other cases. Although x2 for MIT + OGE is slightly smaller than x2 for CBM, we should recall that (gl/fi)_r was fitted in the former case. If we normalize x2 by DOF (the degree of freedom), the normalized x2 becomes smaller for CBM than for MIT+ OGE; DOF= 4 for CBM, while DOF= 3 for MIT+OGE. Thus, again, the inclusion of the gluonic effects does not improve the overall fitting to the data. We may remark here that, due to the fact that the experimental error for the n + p transition is “disproportionately” small compared with those of the hyperon decays, the x2 tends to be dominated by the contribution from the n + p transition. On the other hand, no theoretical treatment described here is expected to be elaborate

586

I(. Tsushim~

et ni. / Weak jnferac~io~jbrm facturs

enough to reproduce (gl)n+p up to this extremely high precision. The quality of a modei, therefore, should probably be judged not in terms of the x2 in the usual sense, but in terms of a certain “effective” x2. The “effective” x2 is a somewhat ambiguous notion, but we consider it reasonable to use the effective x2 defined as follows. In view of the fact that the relative experimental errors for g,(O)/~,(O) for the hyperons range 4 - 20%) we arbitrarily “degrade”” the quaiity of the data for the n -+p transition and assign the relative error of 4% to (g,(O)/f,(O)),,,. The x2 is defined to be the effective xz and obtained after applying this “degradation” denoted by i”. Table 5 also gives i’. If we judge the quality of fitting to the data in terms of i2, the CBM gives the best result for both (a) and (b). The above discussion should probably be taken only semi-quantitatively, but it seems that, as far as the overall fitting to the data on gr{O)/f,(O) is concerned, the CBM without gluonic effects is at least as good as the models with the gluonic effects. Another noteworthy feature in table 5 is that the simplest MIT bag model gives the best result for (g,(O)/~~(O)),~_~; the inciusion of the mesonic effects and/or the gluonic effects worsens agreement with the data. (It must be mentioned, however, that the effects of the 2’ -A mixing are left out here.) Next, we discuss g;, which represents the strength of the second-class axial-vector current. It is to be noted that (gZ)eBMCoGEcan be rather appreciable for a number of transitions; in particular, (g&aM+oGE (.Y+ A) = -0.22. A genera1 tendency is that the inclusion of the gluonic effects tend to increase the magnitudes of g,. The difference between the (g2)M1r in the present catculation and (g,),iT of Donoghue and Holstein6) arises mainly because the g, terms were dropped from the outset in ref. 6). As mentioned earlier, it is not justifiable to drop g, (or, more precisely, (g3)core) in eqs. (40a’) and (40~‘). The present results on g, should be contrasted with those of Carson et al. 4’), who found small values of g, in a calculation that uses another method for including the one-gluon exchange corrections. For g,, which has an important q2-dependence due to (g3)poler eq. (42) or eq. (B.24), the results are given in the form: (g3)core+

K. Tsushima ef al. / Weak interaction form factors TABLE

Calculated

P n A I’

and experimental

587

6

values of octet baryon

MIT

CBM

1.92 -1.28 -0.49 1.87

2.67 -2.04 -0.60 2.44

2.75 -1.95 -0.56 2.56

0.59 -0.69

0.64 -1.15

0.61 -1.34

-1.08 -0.44 -1.11

-1.32 -0.49 -1.59

-1.25 -0.60 -1.53

magnetic

CBM+OGE

moments

Experiment 2.7928444* 1.1 x 1O-6 -1.91304308*5.4x lo-’ -0.613 *0.0044 2.379 f 0.020 2.479 i 0.025”) -1.141*0.051 -1.166*0.017 b, -1.250*0.014 -0.693 zt 0.040 -1.59*0.09”)

For the explanation of the three theoretical cases, (1) MIT, (2) CBM, and (3) CBM + OGE, see the caption of table 1. The transition magnetic moment for X0+ A is also given. The experimental data are taken from ref. ‘) unless otherwise stated. Note that the difference between case (ii) CBM and case (iii) CBM+OGE should not be simply identified with the gluonic contribution, for the parameter R, involved in the volume-type CBM was readjusted in going from case (2) to case (3). See the text and the caption for table 1 for details. “) Wilkinson er al. 4). b, Zapalac et al. 4). “) Peterson et al. “).

feature that helps to resolve the long-standing “A -H problem”. As mentioned earlier, it has been known since the work of Ushio 14) that introducing the one-gluon exchange effects to the MIT bag model or to the surface-type CBM can remove the “A -E problem”. The present results demonstrate that the same feature persists when the mesonic

effects are included the volume-type

through

the volume-type

CBM gives a substantially

CBM. For the proton

magnetic

moment,

MIT bag exchange the change related to

model, and furthermore, the volume-type CBM including one-gluon effects is closer to the experimental value. We must emphasize here that from *P = 2.67 (CBM) to pP = 2.75 (CBM + OGE) should not be directly the gluonic effect, which is in fact zero for the proton (see table 8). This

larger value than the

change comes about as an indirect consequence of readjusting the “effective” meson radius R, for CBM+ OGE; viz., in order to reproduce the (f&_, = 3.70, R, must be varied from 0.475 fm (case (2)CBM) to 0.446 fm (case (3)CBM+OGE), which modifies the contribution of diagram (e) in fig. 1 to the magnetic moments. (See eqs. (B.43)-(B.45) in appendix B.)

7.3. FURTHER

CONSIDERATION

OF GLUONIC

EFFECTS

It should be remarked that the contributions from the color Coulomb interaction denoted by GZ, b;, 6;, &, & and .& [see eqs. (C.36a-e) in appendix C] are identically

K. Tsushima

588

ef ai. / Weak interaction

form factors

zero in the SU( 3) limit. Because of this feature the cont~butions of the color Coulomb interactions are expected to be rather small even for realistic cases that involve a certain amount of SU(3) breaking. This implies that the distinction between type I and type II, defined and discussed in detail in sect. 6, disappears in the SU(3) limit, and that it will not be too large even in realistic cases. The one-gluon exchange contributions to the weak-current matrix elements and those to the magnetic moments are given in table 7 and table 8, respectively. By comparing the results for type I and type II given in these tables, one can confirm that the numerical differences between type I and type II are indeed practically negligible. Finally, we discuss the SU(3) limit of the gluonic effects. Table 9 shows one-gluon exchange contributions, in the SU(3) limit, to the weak-current matrix elements, or more specifically, to the microscopic matrix elements, aT,g, h”‘,E, d’ and c, defined in eqs. (76)-(79). One will see that each column in the table can be characterized in terms of one dynamical factor (I,, a = V or S [see eq. (C.37)], or IF’, a = M, TABLE One-giuon

exchange a^

Transitions

contributions

6 (GeV-r)

7

to the weak-current

matrix

elements

& (GeV’)

F (GeV-‘)

d

e’(GeV2)

-0.075 -0.075

0.0

0.0

0.036 0.036

-0.051 -0.051

I II

0.0 0:o

0.0 0.0

I II

0.0 0.0

0.0 0.0

0.085 0.092

0.024 0.024

-0.034 -0.044

0.072 0.063

I II

0.0 0.0

0.0

0.0

0.073 0.064

0.0 0.0

-0.044 -0.03 1

0.031 0.041

I II

0.0 0.0

0.0 0.0

0.12 0.12

0.0 0.0

-0.048 -0.056

0.095 0.088

I II

0.0 0

0.0

0.010 0

0.0 0

0.015 0

I II

0.0 0.0

0.0 0.0

-0.15 -0.14

0.015 0.015

0.068 0.064

-0.073 -0.076

I II

0.0 0.0

0.0 0.0

-0.10 -0.10

0.011 0.01 I

0.048 0.045

-0.052 -0.054

I II

0.0 0.0

0.0 0.0

-0.064 -0.064

0.019 0.019

0.026 0.026

-0.038 -0.038

I II

0.0 0.0

0.0 0.0

-0.068 -0.068

0.037 0.037

0.027 0.027

-0.035 -0.035

I II

0.0 0.0

0.0 0.0

-0.048 -0.048

0.026 0.026

0.019 0.019

-0.025 -0.025

0

0.009 0

The rows labelled I (II) correspond to the calculation of type I (II). For the definition of types I and II, see sect. 6. The microscopic matrix elements C, 5, g, E, d and e’ appearing in eqs. (76)-(79) are given. As described in the text, a^= cI, + LI,, g= 6, + &, etc., for type I, whereas a’ = ci,, 6= &, etc., for type II. Entries smaller than 0.001 are represented by 0.0.

K. Tsushima

et al. / Weak interaction form factors TABLE

589

8

One-gluon exchange contributions to the octet baron magnetic moments and I’+ A Ml transition magnetic moment for the two types of calculation, type I and type II, which are defined in sect. 6. Also given is the breakdown into the isoscalar and the isovector parts of the magnetic moments. The unit is the nuclear magneton Isoscalar

Isovector I II

-0.07 -0.07

0.07 0.07

0.00 0.00

I II

0.07 0.07

0.07 0.07

0.14 0.14

I II

0.00 0.00

0.07 0.06

0.07 0.06

I II

0.10 0.08

-0.06 -0.06

I II

0.00 0.00

-0.06 -0.06

-0.06 -0.06

-0.06 -0.06

-0.16 -0.14

-0.10 -0.08

I II

0.02 0.01

0.11 0.11

I II

0.02 0.01

-0.11 -0.11

I II

ID+‘4

Total

I II

0.11 0.12

0.00 0.00

0.04 0.02

0.13 0.12 -0.09 -0.10 0.11 0.12

T, A or P [see eq. (C.38)]) and geometrical coefficients. This feature was noticed by Ushio and Konashi 13) for 2 (in our notation) for a limited number of transitions. The present

results

generalize

their

observation

to all transitions

and to all the

microscopic transition matrix elements. In table 10 are given, again in the SU(3) limit, the contributions of the one-gluon exchange diagrams to the octet baryon magnetic moments and their decomposition into the iso-scalar and iso-vector components.

Here again,

the one-gluon

exchange

effects can be parametrized

in terms

of one dynamical factor I(+) M , given in eq. (C.38), and geometrical coefficients. checks with the results of Ushio 14) and Hogaasen and Myhrer I’).

7.4. CENTER-OF-MASS

(cm.)

This

CORRECTION

As has been repeatedly emphasized, the c.m. correction may influence the pattern of SU(3) symmetry breaking. Nevertheless, we have chosen in the present work to separate the discussion of such dynamical effects as the mesonic and gluonic contribution from the kinematical effects such as the c.m. correction. As mentioned

K. Tsushima et al. / Weak interaction form factors

590

TABLE

One-gluon Transitions

exchange a’

21, 0.0

-&zv

6 (GeV-‘) -21, 0.0

to the weak-current @ (GeV-‘1 -$J(fJ) -0.075

matrix

elements

in the SU(3) limit d’

e’ (GeV-‘)

0.036

-0.051

F (GeV-‘)

-;p

-jp

&I, 0.0

fvJQ$ 0.092

$%I$+’ -0.044

y6rji*’

fir, 0.0

j⁢’ 0.053

+%y’ -0.026

s&lw(,+) 0.036

41s 0.0

4 (“1 51, 0.15

3r6”

0 0

0 0

41” 0.0

-41, 0.0

-0.15

2JsI” 0.0

-2vB1, 0.0

-0.11

-&BIg” 0.05 1

-$l?!l$+’ -0.072

&I, 0.0

-&l, 0.0

-$/%I$’ -0.092

+$I~) 0.044

+&I$+’ -0.063

21, 0.0

-2I, 0.0

-@(M+)

-$a’)

-0.075

0.036

J%, 0.0

-dI, 0.0

-gGp -0.053

-+&I~) 0.026

0.0

-4% 0.0 -41, 0

0

The expressions

contributions

9

-.$?[,c)

-&jjl’,”

-0.073 0 0 -$T) 0.073

for 1, (i = V, S, M, T, A, P) and Ii”’ (i = M, T, A, P) are given in appendix

0.063

4 (+I SIP 0.10 0 0 -$$(p+) -0.10

-Q(p+) -0.051 -$VQ$+’ -0.036 C.

in sect. 1, this is partiy because the treatment of the cm. correction is a delicate problem so that, in order to delineate characteristic features of the dynamical effects, it may be convenient to deal with the c.m. correction separately. It was also mentioned that, as far as agreement with the data is concerned, the inclusion of the c.m. correction in the volume-type CBM turns out to be rather unwelcome. To exemplify this last statement, we present here some results we would obtain if the c,m. correction is applied. We discuss here g),/&, which was dealt with in connection with table 5. In order to estimate c.m. correction, we use here the method of ref. 42). As a matter of fact, the same method was already used when we estimated the effect of c.m. correction in the surface-type CBM ‘O). It is expected that the results of ref. lo) can be used unchanged in the volume-type CBM as well. This is because the c.m. correction is probably required only for the main term, the quark transition term, represented by diagram (a) in fig. 1. The contribution of this diagram, however, is identical in both the surface-type and volume-type CBM 29). Thus, the estimation in ref. ‘“) of c.m. correction can be used also in the volume-type CBM. Table 11

K. Tsushima et al. / Weak interaction form factors

591

TABLE 10 One-gluon exchange contributions to the octet baryon magnetic and X0+ A Ml transition magnetic moment in the W(3) Isovector

Isoscalar

-;I(,‘)

f&l

0

-0.071

0.071

0

fI(M+)

f&,+

$I(+$

0.07 1

0.071

0.14

0 0

f&’

f&’

0.071

0.071

-f[(M+)

0.071

-0.071

0 0

0 0

-fIG:’

-$IG’

-0.071

-0.071

-+I(;)

-$I(;+)

-$I(;)

-0.071

-0.071

-0.14

0.14 -f@ -0.14 &G’ 0.12

gives the results

of applying

Total

;rC$

f&1

The expression magneton.

moments limit

0 0

;I(,‘)

0 0

-21(+)

0 0

&,+) 0.12

0.14

-:.;”

for I, (+) is given in appendix

the c.m. correction

C. The unit is the nuclear

to the MIT bag model, the volume-

type CBM without the gluonic contributions, and the volume-type CBM with the gluonic contributions. It is to be seen from table 11 that the inclusion of the c.m. correction (as calculated here) in the volume-type CBM drastically increases the x2 values; the disagreement with the data on g,/fi(n+p) is particularly glaring. The simple MIT bag model gives a better result in this regard although the resulting x2 value is rather large*. This feature together with the delicate nature of the c.m. correction formalism itself makes us feel that we may be advised to leave out cm. correction altogether. Needless to say, the conclusions we state in this paper should be taken

in the light of this restriction. 8. Discussion

and summary

We have calculated the weak-interaction form factors and octet-baryon magnetic moments (plus the ,X0-A transition magnetic moment). The volume-type CBM l The reason why the present result of the MIT bag model differs from that in ref. 19) is two-fold: (i) The methods of applying c.m. correction are not identical and (ii) There is one fudge factor in the analysis of ref. I’).

K. Tsushima et al. / Weak interaction form factors

592

TABLE

11

Effect of c.m. correction

(1)

on g,/f,

(2)

MfT+c.m.

CBM + cm.

(3)

g,/f,tn+P) R:(a) R:(b)

1.32 121.0 81.0

1.42 756.3 400.0

I.46 1133.4 581.3

gIlfiW-+n) R:(a) R:(b)

-0.27 4.6 2.0

-0.32 1.0 0.2

-0.39 0.3 0.9

g,lf,(E_ R?(a) R:(b)

“d{U,‘“‘P) R;(b)

x’(a) x’(b) 2%) x7*(b)

+ A)

Experiment

CBM+OGE+c.m.

1.254*0.006~) 1.239*0.009 b,

-0.362 i 0.043 “f -0.34 f 0.05 b)

0.28 0.4 0.4

0.28 0.3 0.3

0.25 0.0 0.0

0.25 i 0.05 “) 0.25 * 0.05 b)

34.1 0.84 21.8

39.9 0.85 25.1

45.7 0.86 29.5

0.694i 0.025 “) 0.70 * 0.03 b)

160.1 105.1 41.2 21.4

191.4 426.1 54.4 42.4

1179.4 611.7 66.0 54.1

The c.m. correction is applied to the results of the three calculations: (1) MIT, (2) CBM and (3) CBM+OCE. For the meaning of these abbreviations, see the caption for table 1. For the two different sets, (a) and (b), of experimental data are given Rf - f (x:h’“’ - xyp )/SxrxpJ2 for each transition, as well as x2 = xi Rf. The symbol f* represents the “effective” x2 defined in the text. “) Ref. ‘), h, Ref. 2).

including

the octet meson

clouds

has been

used,

and furthermore

the one-gluon

exchange effects have been taken into account. The salient features of the present results may be summarized as follows: (I) The volume-type CBM gives substantially larger values of g, than the MIT bag modei and the surface-type

CBM. In particular,

the vohtme-type CBM gives gA = (g,),+, = 1.26, in good agreement with the experimental value, without introducing any adjustable parameters. (2) The one-gluon exchange effects on the weak-interaction form factorsfi, g1, g, and g, are appreciable for some transitions, while their effects on f, and f3 are negligible. The gluonic effects on the magnetic moments are at most - 10%. (3) The three major phenomenological successes of the gluonic effects have so far been reported. That is, the addition of the one-gluon exchange contribution to the MIT bag model can remove (i) the “Z: -A problem” in the weak decay and (ii) the “A - 6 problem” in the magnetic moment 13-15) and furthermore it can provide a right amount of isoscalar magnetic moment for the nucleons “). These successes still remain when we include the mesonic effects through the volume-type CBM. (4) The inclusion of the gluonic effects, however, does not necessarily improve overall fitting to the available data on g,/f, . (5) The magnitudes of f3 and g, representing the second-class currents can be rather appreciable for some transitions.

K. Tsushima

We briefly

touched

values of gl/fl

obtained

of ref. 42). It was found results

worsens

correction

upon

c.m. correction,

by including

form factors

showing

with the data.

to the MIT bag model

results,

the

with the use of the method

applied

to the volume-type

It was argued plays

593

as representative

the c.m. correction

that the c.m. correction

agreement

applied

et al. / Weak interaction

CBM

in ref. 19) that the c.m.

an essential

role in bringing

the

brokenSU(3) picture into agreement the data on hyperon semi-leptonic decays. Our present results, however, seem to indicate that the treatment of c.m. correction is still an open problem. Depending on which model c.m. correction is applied, agreement with data sometimes improves and sometimes deteriorates. Another kinematical effect which was left out in the present work is recoil correction 46). Lie-Svendsen and Hogaasen 16) studied the recoil corrections extensively for various quark models. Their results indicate that g, is reduced roughly by a factor of 3 when recoil effects are included*. It will be of interest to examine whether this feature persists in the volume-type CBM as well. A detailed account of recoil corrections applied to the volume-type CBM including the gluonic effects will be given elsewhere 47). Maxwell and Vento 36) pointed out the danger of calculations in which only a subset of possible gluonic diagrams is dealt with. A related difficulty is that, if one starts with a phenomenological model in which parameters are adjustea to reproduce appropriate observables, one can encounter the problem of separating perturbative gluonic effects from those already contained in the phenomenological model. Also pointed out in ref. 36) are highly delicate aspects of the renormalization of the weak current vertex due to gluons. Therefore, the gluonic effects evaluated in the present work should be taken with caution. Although we have not addressed here any of the basic questions raised in ref. 36), it is our hope that the present calculation of the gluonic effects still has merits in that it gives information on whether the remarkable phenomenological success of the one-gluon exchange effects reported in the literature is specific to the MIT bag model or something more universal. To study the SU(3) symmetry breaking effects without intermediate steps such as the Cabibbo-SU(3) fitting, it is desirable to evaluate the decay rate and other observables for each semileptonic decay using the calculated form factors, and compare them directly with experimental data 44). To this end, one must take into account radiative corrections with a precision that matches the experimental accuracy. A direct analysis of the data using the results of the present detailed calculations with the radiative elsewhere 47).

corrections

taken

into

account

will be reported

We are deeply obliged to Dr. K. Ushio for generously providing us with a copy of his thesis, which was extremely useful for the present investigation. One of the authors (K.K.) wishes to express his sincere thanks to G. A. Miller, Fred Myhrer, * As commented some ambiguity.

upon earlier,

however,

the treatment

of the pion-pole

term in ref. 16) seems to have

594

K. Tsus~imff

et al. / Weak interuc?iQ~ form faciors

H. Mgaasen and Stuart Freedman for illuminating discussions at various stages of this work. The numerical calculations reported here were done at the Computer Center of University of Tokyo with a partial financial support by the Institute for Nuclear Study, University of Tokyo.

Appendix A

The method we use here to calculate Ij of eq. (29) is exactly the same as in ref. *‘). We therefore simply give the final expressions. We define matrix elements $*I, +!?i2’,s”$,“’and s(3) by

where Qa is the charge of the meson Q, and g(l)

=

CWAB

+w,(k’)-t-w,(k))-’

xc

(Ai

tk-

WiWCiv;Wb)+

(Ai

V;(k)lWCj

V%-f4lB) ,

C

qi(k)

wCa+

w,(k’)

(AlV,(k’)tC)(ClV~(k)lB>

gcz,_c 1-

wB+

c

64.2) (A.3)

(WcA+W,(k’))(Wca+OP(k))’

(A.41 $3) =

w,+~,W)+qG>)-’

(@J

(AlV-,(-k)lC)(CI~(k’)tB),(AIV,(k’)IC)(CtV...,(-k)tB) wzA++(k)

@CA+%(k')

(A.9

Appendix B

Explicit expressions for ai, b<, b: ( i = 1,2,3,5,6), 1,2,4,6) in eqs. (39a)-(40c). Vector currents

(i) it&,(f”),

Jo= Vo or V0

ci, di (i = 1,2,6)

and ei (i =

K. Tsushima et al. / Weak interaction form factors U, =

595

(B.1)

-~(z~A~~B)*'~T*B(AJJ~AJIJB)~]O~~ ,

u~=-~(Z~AZ~B)“~~A~~BT*~~A~B cg Pe(D, C)Y’O’(D, , ,a

C)(Cj(lAJIID)ICD(~), (B.2)

a3 =J7;?*FBTAB

P) ,

,;, &,(C)H%, , 1

(B-3) (for As = 0),

a5=0, I a5 =aTAB(Al((Aal(lB)170,b[L~,(A)+L~K(A)+

(A-B)]

(for As=

03.4)

l),

(B-5)

a6=0, (ii) Mss,(J),

V or V’

J=

1’2(AlllAIllW% ,

b, = -;(Z:Z;)

b, = -~(Z,“Z,“)“2~A~BTAB~~SIB

03.6) C Sa(D, C)Y”‘(D, GDP

C)(C~~lA~~~D)I”“(~), (B.7)

b,=

-JjTA?B~AB c %Km(d% C&P

b,= -~~*“(~lll~lll~)~,,[~~~=(~)

(B.8) + M&(A)

- (A-B)]

b, = -~T*“(A~~~A~(~B)~~~[M~~(A)+M~~(A) - (A-B)] b, = -v?(Z,“Z,“)“~

TAB C

4s

(for AS = o), (for AS=

bl = -&

(Z;\Z~)1’2TAB(All(A~l((B)~,~,

b;= -&

(Z;Zf)1’2fA?BTABiAf?B

(B.lO) (B.11)

1 .Fm(D, C)Y”‘(D, CJ’,*

C) (B.12)

x(Clll~~Il~WCDW, A

(B.13)

~ATBT*~~~,~~~~(C)Y(C)H~(~,P),

b; = -&

(B.9)

~~~~~~~~~ill~~~lll~>~~lll~p~lll~>~~~~~

-~~,(~>(All~~,~~~~~>~Blll~,~~~~C>~~(C>I,

b;=+

I),

T*“(A~~~Au~~~B)&,[~N~~(A)+ N&(A)

+ (A-B)]

(for AS = 0),

I

I

b; = -&

T”“(A~~~Au~~~B)S~~[N~,(A) + N:,(A)

+ (A-B)]

(for AS = l), (B.14)

b:, = -fi(Z,“Z;)“’ +

TAB C [~~a(C)(AI~l~a~lllC)(Bill~p~lllC>~~(C) 0.C

~~p~~~~~Ill~p~lll~>~~lll~~~lll~>~~~~~I~~~~ ,

(B.15)

596

K. Ts~~imu et at. j Weak inferacf~o~ form facrors

Axial-vector current:

(iii) MS~,(Jo), Jo= A0 or A” (B.16) c2= -~(242~)‘/‘~*~*TAHSA~*

C

CPP

Sa(D,C)Y”‘(D,

C)

jB.17)

(B.18) (iv) M,.,(J),

J = A or A’ (B.19)

d,=-1 2J5

(.Z;“Z,“)“‘f~‘?BT”“t$A~B

C Sa(D, C,D,a

CjY”‘(D,

C)

(B.20)

(B.21) (B.22)

e2=-&(~~z,“)“2~AfBTAB~A&2

SO(D, C)L@‘(D, C)

C,D,a

(B.23) (B.24f (B.24’) e6 =

-&(Z$Z~) “* TAB C [~~ru(C)(Alll~n~ll~~>(BIII~p~lll~)R~~~)

~,P,C

+ ~~a(C)~Alll~a~IllC>(BII~~~~~l~C~~~(C)IY(C) ,

(B.25)

The various quantities appearing in the above expressions are defined below.

K. Tsushi~a et al. ,I’Weal jnt~raction form factors

597

(B.26)

f=V2TT-i, TUB=

(_)

r~-c~

7;\ _t A

7.

TB

t

tB

(B.27)

3

(B.28)

~~(D,C)=(_)78”T~‘T+T~

(B.29) (B.30) (3.31) (B.32) (B.33) (B.34)

(B.35a) (B.35b)

,

with (B.36) (B.37) (B.38)

k”u*( kR) ((W,(k)+Wp(k))2-W~A}(W,(k)+WCA)(Wp(k)fWC*) W23 ---

WCA

%(k)

@$3(k)

+ >(

q?(k) --m,(k)

w,(k) #p(k)

*

(B.39)

1

Xw,tk)wp(k){(w,(k)+~~(k))2-w~,}(w,Ik)+wc~)(oat (B.40)

(o,(k)+up(k))(w,(k)+wS(k)~ocla)+oeAw,(k)

Xw,(k)optk){(w,(k)+up(k))2-WZBA)(W,(k)+uCA)

co J

L%(A) =&

M:,(A)

=-

(B.41)

dk k2 exp (-$R2k2)

?T a

(up(k) - tie(k))’ w,tk)oa(k)tw,(k)-t-w,(k)+w,,)’ (B.42)

o2

1 96r2f2,

J

dk k” exp ( -$R2,k2)

,,

1 w,(k)qAk)(u,(k)

1 +@W-

(w,(k) - dW2 ~~(k)~~~k)(~~(k)

-t w@(k) + #.a)

dk k4 exp (-fR;k2)

L;(C) = -

M;(c)

= -

op(k))2

w,(k)qs(k)

>( -i-

ifNN 14472m,f, if”” l~~zm=f~

mdk Jo

J

;odk

0

+ tip(k)+ @AB)

I

(B.43)

’ 1

w,(k)op(k)(w,(k)+wp(k)+WAB)

k3$‘(k)u(kR)

qstk)(wzrs+qAk))

(B.45) ’

k3~~-)(k)~(kR) qs(k)(w,,+

am)

* (B.44)

(B.46) ’

K. Tsushima et al. / Weak interaction form factors

599

with

with

&+‘(k) = 4rr

drr2fr2+~‘(r)j,(kr),

(B.52)

Appendix C

We give here calculational details for the one-gluon exchange contributions the weak-current matrix elements, eqs. (76)-(79). C.l.

DEFINITIONS

AND

SOME

to

FORMULAE

We denote by t&(x) the spin and spatial part of a quark wavefunction q=(x) confined in the spherical static MIT bag, where (Y= (n, K, ,u,flavor). In the notation of ref. 48),

The spin and spatial part of an anti-quark conjugated state of r&(x)) is given by

The radial functions g,,(r)

wavefunction

(defined as a charge-

and fnK are expressed as

(C.4)

600

K. Tsushima et al. / Weak interaction form factors

The normalization

whereas

constant

N, is given by

the bag eigenenergy

for the quarks

are determined

by solving

(‘35) For later convenience, we cast eqs. (61) and (62) for the traverse gluon Green functions into slightly different forms. For any constant vectors A and B in 3dimensional

space, A”‘B”G$,(x,

A”B”G~(x,

x’) (j = TE, TM) can be written

x’) = 2

c

I=1

In=-I

(Y,,,(x^)*A)(-)“(

as

Y,~_,(i?)d?)gTE(r,

r’) ,

(C.7)

(C.8) with Dj+‘=~3/~3r--l/r

and D$“=a/ar+(l+l)/r.

Here,

g;fE( r, r’) = g;f”( r, r’) =

whereas

l/r { l/r’

(r 3 r’) (r’>

(C.9)

r),

for 12 1 (rzr’) (r’>

(c 10a) r),

(r 3 r’) (r’)

r) ,

(C.lOb)

R

hT”( r,

r’) =

I

dr”( r”)2g:M( r, r”)gT”( r”, r’) .

(C.11)

0

C.2. CALCULATION

OF MATRIX

ELEMENTS

IN EQS.

(66) AND

(70)

C.2.1. Modes contributing to gluon Green functions and quarkpropagator. In describing the calculation of Tj’s in eq. (66) and Ti’s in eq. (70), we first consider the

K. Tsushima et al. / Weak interaction form factors

601

transverse gluon contributions eqs. (67~) and (67d); the color Coulomb contributions, eqs. (67a) and (67b), will be discussed later together with the contribution of the color-Coulomb self-energy type, eqs. (71a-d). We start with examining what multipoles will contribute upon muItipole-expanding the transverse gluon Green functions appearing in eqs. (67~) and (67d). Let us take eq. (67~) as an example; the computation of eq. (67d) is completely analogous. Omitting for the moment the color-flavor part of the quark wavefun~tion q(x), which is not essential for our argument, we consider G-‘,,(x, x’)&x’)fq(x’)cc G’,,(x, 1~~)J,._,~~(x’)y”~,_~~(x’) contained in eq. (67~). Identifying B” in eqs. (C.7) and (C.8) with _ u,_~~,(x’)Y~u~_~~(x’), we find that eq. (67~) contains the factor Y,,_,(x^‘).(u”,__,,,(x’)yf,_,,(x’)) forj=TE or Y,,+,_,(;‘).(u”l_,,,(x’)ru”l_l,(x’)) and YN+,_m(x^‘) * (al-,,,(x’)ru”l-,,(x’)) forj = TM. We therefore look into selection rules for the matrix element Urr_r~(x’) k;r,(.?) .yii-Ifi(x’). In the two-component spinor description we have

(C.12) Noting that x-rr is an S-wave whereas xly is a P-wave, one can conclude that I in eq. (C.12) must be equal to 1. For these restricted multipoles, one can derive the formula

I

d~‘~,-,,,(x’)uj,,(;‘)‘y~,_,,(x’)

=

-&

N:--lgl--1(~‘lfi--l(~‘)~IU’lET,ILL~~J;I.

(C.13)

Thus, in eq. (C.12), only YIlm will survive the angular integration (to be carried out in eq. (66)). This means that the only TE mode with I = 1 will give a transverse one-gluon exchange contribution to eq. (67~). The same selection rule is obtained for eq. (67d). We next consider the color-Coulomb contributions in eqs. (67a) and (67b) as well as those in eq. (71). The color-Coulomb Green function, eq. (59), can be written as (C.14) with (C.15a)

602

K. Tsushima er aL ,I Weak inferaction form factors

(r k r’) (C.lSb) (r’> r). We focus on the factor G,(x, x’)p”(x’) which is common to eqs. (67a), (67b) and (71a-d). This factor involves ~~-~~Y~~a~,-~~, which can be written as ;;l_l,,(r’)yOY~(R’)z”il-l,(x’)

and therefore only the f = 0 multipoie will contribute. Thus, all the gluon Green functions needed for calculating the one-gluon exchange have been obtained. We next determine what mode will contribute to the quark propagator in the one-gguon exchange processes. Again, Iet us consider eq. (UC) as an exampie. Here, we will take advantage of the above-obtained restriction that only the TE mode with I = 1 will contribute to the transverse one-gluon exchange processes. We examine G’,,(x, x’)ii,_,,$‘%~ and G’,,(x, x’)~,_,,$“~~ appearing in eq. (67~). Angular integration will give

al-,,~(x)Y,,,(x^),yu”,,,(x) Jdf2 (C.lSa)

(CSSb) Therefore, we need to retain only K = ml and *I2 states for the ~ropagat~u~ quarks. The treatment of eq. (67d) is similar, and the same selection rule, K = il or *2, is obtained.

K. Tsushima

The color Coulomb handle;

since,

terms

et al. / Weak interaction form factors

in eqs. (67a),

as mentioned

above,

(67b),

603

and (71a-d)

G,(x, x’) brings

is much

simpler

in only a monopole

to

term,

K

must be -l(+l) for l?,(x)(;,(x>>. C.2.2. One-gluon exchange matrix elements. We are in a position to present explicit procedures to calculate the one-gluon exchange matrix elements TI (i = a, b, c, d) appearing in eq. (66), and T:’ (i = a, b, c, d) appearing in eq. (70). We illustrate the calculational details taking Ti as an example. With the use of eqs. (C.lSa) and (C.l8b), we obtain dr dr’ dr” r’(r’)“(r”)‘gT”(r,

T;=-

x

X(r;

1, -1;

1, -1;

1, -1)

n, K) (2&,-r -JZ&,J@(r”)(U

[ +

r’)X(r’;

&f--W,

Y(r; 1, -1, n, K) q+wl

I

(2%,,-d%-2)@dr’X0 b) a#(n=l,

, (C.19)

K=--1,/J),

where

d;(r)(l)

_+,

= (a,)?(-)‘/‘-@’

where the suffix /3 differentiates have also used in eq. (C.19)

various

dfii&(x)J”(x)

eiq’x~l-,P(x),

(C.20a)

d0r?,(x)J”(x)

eiq’xc,_,,(x),

(C.20b)

choices

of J”; /3 = VA, A”, VA, or A’“. We

X(6 n, K; n’, K’) = N,,N,,,,(g,,(rlf,,,,(r)+f,,(r)g,,,,(r)) Y(r; n,

K;

n’,

K’)

=

N,,N,,,,(gn,(r)g,,,,(r)+f,,(r)f,,,,(r)).

,

(C.21a) (C.21b)

In eq. (C.19), IA,&) and (B0~) are color-flavor wave functions. We introduce long-wavelength expansion into J” (x) eiq’x appearing in Oi( r)( 1) and d>(r)(Z) and retain only lowest-order terms necessary for evaluating the matrix elements, ci; 6”, 61, 5, d” and t?, in eqs. (76)-(79). For those operators that needed to be retained, we summarize their matrix elements between the two-component spinors.

604

(I)

K. Tsushima et al. / Weak interaction form factors

Vector current (p = V” or VA)

(i) time component (C.22a)

b’f’,p’(K,K') =j+&. ,

(C.22b)

(ii) space component

(xKw~i~(*~~)~x,~,t)= C (-)j*-” j. m,

(jx mjj jdP”’)

M:@‘(K,K';jmj),

(C.23a)

-CL

M:“(K, K'; jmj) = -3Jzi(-)‘Ki,i,,j^,j^2j^,.

(II) Axial-vector current (/3 = A” or A’*) (i) time component (C24a)

(C.24b) (ii) space component

(-)j~-“( Ap f, i:)MLp)(~, K’; m) ,

(xKplu(xKsJI.)=

(C.25a)

(C.26a)

(C.26b)

K. Tsushima et al. / Weak interaction form factors

with C, = -& utilizing

t-1

for 1 = 0 (I = 2). Using

($)

the quantities

defined

605

above,

and also

a formula

1/2-m;+j,-w

1 2

1

jK

j,

-4

m

P

-p

(

>(

1

t (dJ1 m,

>

ml

(C.27) 3

we can write down explicit (I)

expressions

for O:(r)(l)

and d;(r)(l)

as follows:

Vector current (p = VA or V'" )

O~(r)(l)=~~(ui.uj)S,,Ig,,(r)g,-~(r)+f,,(rlfi-l(r)}b,-,

,

(C.28)

[(c+J)i@ujl!n,x r{g,,(rlfi-l(r)N~P’(K, 1; Jh)

Op(r)(O=C J

-fn~(r)gl-l(r)N~~‘(-K,

@‘(K,

(II)

-1;

ih’fj”(K,

K’;

(C.29a)

Jh)),

K’;

h,)

.

(C.29b)

Axial-vector current (p = A” or A’*)

[((T’)i@~jlh8~,I

@(r)(l)=1 J

x r{gnK(r>h-l(r)NY’(K, 1; Jm>-fnK(rhl(r)~:B’(-K, -1; Jm>l, (C.30a) @‘(K,

K’;

h)

0 Pr( )(I)=

=

(C.30b)

(-)‘-

O(“(r)(l)+O~)(r)(Z) P

(C.31)

9

[(aJ>i@ujl!n~~,I

@‘(r)(0=C J

x{g,,(r)gl-,(r)NjP’(K,

-l;Jm)+fn~(r)fi-l(r)N~~‘(-K,

l;Jm)l,

(C.32a) (C.32b) [(cTJ)iOUj]fn,

@?(r)(O=C J

ib$“(K,

K’;

&ZZ)=

it@‘(K,

K’;

hZZ) .

(C.33b)

The corresponding expressions for @(r)(l)‘s can be obtained by exchanging, eqs. (C.28)-(C.33b), the roles of g,, andf,,, and furthermore by replacing K by except for the K’S that appear as a quantum number in g,, or f,,.

in -K

X. Tsushima

606

et al. / Weak interaction form factors

The computation of TI’s other than T: and T’:‘s can be made in much the same manner and will not be described here. Having obtained Ti’s and T;‘s, we now relate these quantities with M$,(JX) in eq. (73) [for type I], or eq. (75) [for type II]. Referring to eqs. (57) and (68) for type I, or referring to eq. (74) for type II, we can identify M$,(J”) =

C

(T:+ T!:)

(C.34)

i=a,b,c,d

for type I, whereas M:,(J’)

= T;+ T&

(C.3.5)

for type II. For each type of J’, we compare the coefficients of (s’(lls), q(s’/l(s), (s’lol.Q, 4’ (+I& iclx (s’Ir+) or cl(&* a Is > on each side of (C.34) or (C.35). This procedure allows us to determine I;, b”,b”‘,I?, 2 and &?in eqs. (76)-(79), the results of which are given below. C.3. EXPLICIT EXPRESSIONS OF CURRENT MATRIX ELEMENTS DUE TO ONE-GLUON EXCHANGE

We give here explicit expressions for the current matrix elements ri, 6, b”‘,E, d” and e”of eqs. (76)-(79). As described in text, Q’= Zr + (i2, 6=. . . , c?= k, + C2for type I, whereas a”=a”,,g= . . . . e”=e’,, for type II. The ii, b”i,b”:, &, Ji and & (i = 1,2) are now given by: (C.36a) (C.36b)

(C.36b’)

(C.36~)

(C.36d)

(C36e)

(C.37a)

(C.37b)

K. Tsushima et al. / Weak interaction form factors

607

& =&-ABWIIIJ~(f; f’, f”b(Ql (AIIIB) - (Alll(Ji& f’, f’> + Jdf, f’, f ))u( i>lllWl,

(C.37b’)

f ', f"b(i)1(.dlllB>

c” = ~~~AB~(AIIIJ~(f;

- (4ll(J~(Af’, f’>+ Mt f’, f)b(i)lllB)I , & =

-d7’*“I(AIIlJA(f;f’, f”)di)lWlllB> -(AIII(JA(~, f',f')+J~I,(_t

4 =

(C.37c)

f',f)>4W919

(C37d)

-~~~*“~LNMJ;f’, f%(i) 1WlllJ9 - (4ll(JdJ;f’, f’) +Jdf; f’, f)hdi>llb9),

(C.37e)

with TAB=

(_)%-‘A

TA

_t

A

The definition For a=V

of I,, which appears

T

TB

t

tB >.

in the calculation

of the TE mode, is as follows

and S, L(f;

, h-l; g-1, LA; f”l@‘b,

f’, f”) = Uf’(g,-,

-1)

x ~,Lf’(&-,, “t-1); Sk-1 2“fH)l +W-‘k-,,

h-1; _&I,gn,); f”l@%,

1)

x ~~rf’cL1, &I);.&-1, h-J1 +F,LI-‘k-1, h-J; fkn-1, fn-J@(n, x Km%-1, “6-l; a-J-1); +FAf’k-,,

h-h; fk,

f”1 gnJ@!‘b, 1)

x aKL* 3IL,; a-If,-,); 7’1 , For (Y=M, I!,%

((2.38)

T, A and P, f’, f”) = K[f’(g,-,,

h-1;

g,-, , fn-1);

f”l@“h

-1)

x ~Lf’kn-1, L-1); fk-1, &Al +KCf’k-, , A-1; .L,gn,); f”l@“h x~ceu-‘CL, &A fk-I,&,)I

-1)

1)

K. Tsushima et al. / Weak interaction form factors

608

+

(“:>

h-1; L-2, gn-2);f”l@-%, -2)

KLf’k-,,

x co-‘(fn-2, g,-2);“G-l, fi-111 +

(‘:>

FaLf’(g,-,, h-J; fkn-1, L-AlG’%,

x ml-(&I,

+

0; 0;

g1-I,

h-1); f”1

.f-,I; fknz, .LW%,

CLf’k-1,

x afk,,, +

L-1;

-1)

2)

xX2;g1-I h-1); .I-“1 7

, fi-d; SK2, gn-dlG%,

F&Lf’k-,

x w-(fn-2,

gn-2; fi!-1,

fi-1);f”l

-2) (C.39)

7

where the upper (lower) members in the column vectors refer to I’+‘(l’-‘). J,, (Y= V, S, M, T, A or P, which appears in the calculation of the color Coulomb interaction

is given by: .L(Jf’,

f”) = HU-‘k-,,

h-1;

g,--1, L-I);

f”l@%,

-1)

x ~&-‘kn-I, fn-1); fk-1, h-d1 +NI-‘(a-,,

h-1; A,,, gnd; f”l@“h

x Fau-‘(fnl, gnd; fk-I

1)

3L-111

+FJf’k-,, h-d; fkn-1, _L>lG%, x fmgn-1, h-1; f”1 g1-1,

+~&-‘k-,,h-,I; x ffU&, The quantities

newly appearing

@(n,

K) =

-1)

h-1);

f&,

gndlGk%, 1)

%li g1-I7h-1); 7’1 9

in the above equation

are defined

(C.40) below:

E,_,&(4 K) # (1, -1)) cfj

KMg,, h; g,, fi>; f’l =z 2

(C.41)

dr dr’ r2(r’)2gTE(r, r’>

x [NI-“NY’(s~‘(rlfV’(r)+f~‘(r)gy’(r))l x[2(N~‘,)2gfA?l(r’)f$!Y’,(r’)]

,

(C.42)

609

K. Tsushima et al. / Weak interaction form factors

(rB

r’)

(r’>

r)

(C.43)

dr dr’ r2(r’)*d?(~,r’) x [~~‘~~‘(g~‘(r)gY’(r)+f~‘(rlf~‘(r))l

x[N-l(g~-l(r’)

(rB

xxv?={;;;,

h(f(g,,

h);

&Mg,,

_fJ

f’k2,

9

f’(g2,

(C.44)

+.Cl(r')>l, r’)

(C.45)

(rl>r),

.a>

=

jww

.a)

=

4NWP

J -f?(r)&)(r)) , Jdrr3(g%rlfY’)(r) r”(&‘(r>fY”(r) +f?(r)&“(r)) 3 Jdr dr r’(gy’(

r)gy’(

r) +flf’(

r)fr’(

(C.46)

r)) ,

(C.47)

hl(f(g1,

.A);

f’k2,

.a>

=fWNf)

(C.48)

&(f(g,

3A>;

f’k2,

FA(f(gl> .fl>; f’k2,

Fo-(g,,

%4(f(g*,

.a)

=

Mfkl,

.A>>

=

WW’

_flk f’(gz, “f-2))-iwY’N’

f1);

f’(g2,

.f-2))

-iwYwT’

“flh

f’k2,

x2))

(C.49)

f

J

dr r*(gy)( r)g$“( r) -i/y)‘<

r)fr’(

r)) ,

(C.50)

J 3 Jdrr3(g~‘(rlfY”(r)+f dr r4ftfy)( r)fy)(

(C.51)

r) 2

(C.52)

%(f(g,,

A>;

f’k2,

_fi>)

=

-bWNP

%(f(gl,

h); f’(g2, _A>)= --i@‘Ni!”

S(f(g,,

fl);

f’k2,

.a)

=

-aww

J

dr r’(gif’(

r)fy’(

r) -fZf’(

r)g$“(

r)) 3

(C.53)

J J

dr r2f?(rlfY”(r> dr r”(gy’( r)gy’(

(C.54)

, r) + $y’(

r)fy’(

r)) 3

(C.55)

610

I(. Tsushima et al. ,J Weak jnteractio~ form factors

where f, f’ and f” represent the quark flavors (u, d, s); .cJf? in G!$(n, K) is the eigenenergy of a quark with flavor f in the MIT bag model. References 1) Particle Data Group, Phys. Lett. B170 (1986) 1 2) M. Bourquin et a/. (WAZ-Collaboration), 2. Phys. C21 (1983) 27; J.M. Gaillard and G. Sauvage, Ann. Rev. Nucl. Part. Sci. 34 (1984) 3.51, and references therein 3) S.Y. Hsueh et al., Phys. Rev. Lett. 54 (1985) 2399 4) P.C. Peterson ef al., Phys. Rev. Lett. 57 (1986) 949; G. Zapalac et a!., Phys. Rev. Lett. 57 (1986) 1526; C. Wilkinson et at, Phys. Rev. Lett. 58 (1987) 855 5) H.J. Lipkin, Phys. Rev. D2.4 (1981) 1437; H.J. Lipkin, Nucl. Phys. B214 (1983) 136 6) J. Donoghue and B. Holstein, Phys. Rev. D25 (1982) 206; D25 (1982) 2015 7) G.E. Brown, Nucl. Phys. A374 (1982) 63c, and references therein; G.E. Brown and F. Myhrer, Phys. Lett. B128 (1983) 229 8) S. Theberge and A. Thomas, Nucl. Phys. A393 (1983) 252 9) P. ienczykowski, Phys. Rev. D29 (1984) 577 10) K. Kubodera, Y. Kohyama, K. Oikawa and C.W. Kim, Nucl. Phys. A439 (1985) 695 11) Y. Kohyama, K. Oikawa, K. Tsushima and K. Kubodera, Phys. Lett. B186 (1987) 255 12) K. Tsushima, T. Yamaguchi, M. Takizawa, Y. Kohyama and K. Kubodera, Phys. Lett. B205 (1988) 128 13) K. Ushio and H. Konashi, Phys. Lett. B135 (1984) 468 14) K. Ushio, Phys. Lett. BIJS (1985) 71 15) K. Ushio, 2. Phys. CM (1986) 115; K. Ushio, Ph.D. thesis (Osaka University, 1984) 16) 0. Lie-Svendsen and H. Hdgaasen, 2. Phys. C35 (1987) 239 17) H. Hdgaasen and F. Myhrer, Phys. Rev. D37 (1988) 1950 18) GE. Brown, S. Klimt, M. Rho and W. Weise, Stony Brook preprint, 1987 19) J.F. Donoghue, B.R. Holstein and S.W. Klimt, Phys. Rev. D35 (1987) 934 20) N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531 21) A. Chodos, R. Jaffe, K. Johnson, C. Thorn and V. Weisskopf, Phys. Rev. D9 (1974) 3471; A. Chodos, R. Jaffe, K. Johnson and C. Thorn, Phys. Rev. DlO (1974) 2599 22) T. DeGrand, R. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 23) C.G. Callan Jr., R.F. Dashen and D.J. Gross, Phys. Lett. B78 (1978) 307 24) G.E. Brown and M. Rho, Phys. Lett. B82 (1979) 177; G.E. Brown, M. Rho and V. Vento, Phys. Lett. B&l (1979) 383 25) A.W. Thomas, Adv. Nuct. Phys. 13 (1983) 1 26) S. Theberge, A. Thomas and G. Miller, Phys. Rev. D22 (1980) 2838 27) A.W. Thomas, J. of Phys. G7 (1981) L283 28) A. Szymacha and S. Tatur, 2. Phys. C7 (1981) 311 29) M. Morgan, G. Miller and A. Thomas, Phys. Rev. D33 (1986) 817 30) B. Jennings and 0. Maxwell, Nucl. Phys. A422 (1984) 589 31) G.A. Miller, Quarks and nuclei, Int. Review of nuclear physics, ed. W. Weise (World Scientific, Singapore, 1984) p. 190; F. Myhrer, ibid. p. 326 32) E.A. Veit, B.K. Jennings, R.C. Barrett and A.W. Thomas, Phys. Lett. B137 (1984) 415; B.K. Jennings, E.A. Veit and A.W. Thomas, Phys. Lett. B148 (1984) 28; E. Veit, B. Jennings, A. Thomas and R. Barrett, Phys. Rev. D31 (1985) 1033; E. Veit, A. Thomas and B. Jennings, Phys. Rev. D31 (1985) 2242; E. Veit, 8. Jennings and A. Thomas, Phys. Rev. D33 (1986) 1859 33) A.W. Thomas, S. Thcberge and G.A. Miller, Phys. Rev. D24 (1981) 216 34) T.D. Lee, Phys. Rev. D19 (1979) 1802

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611

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