semi-leptonic beta-decay form factors and magnetic moments of octet baryons: Recoil effects and center-of-mass corrections in the cloudy bag model including gluonic effects

Nuclear Physics A500 (1989) 429-484 North-Holland. Amsterdam

SEMI-LEI’TONIC

BETA-DECAY

FORM FACTORS

AND MAGNETIC

MOMENTS OF OCTET BARYONS: Recoil effects and center-of-mass corrections in the cloudy bag model including gluonic effects Tetsuya

YAMAGUCHI,

Kazuo

TSUSHIMA,

Y. KOHYAMA

and K. KUBODERA

Department of Physics, Sophia University, Kioicho, Chiyoda-ku, Tokyo 102, Japan Received 24 October (Revised 27 January

1988 1989)

Abstract: In a previous article we made a detailed study of the weak interaction form factors and the magnetic moments of the octet baryons. The volume-type cloudy bag model was used as a possible framework for incorporating mesonic contributions in a systematic manner, and, furthermore, gluonic effects were included through evaluating one-gluon exchange diagrams. Despite the rather detailed treatment of these dynamical effects, the previous work was based on the static bag picture and fell short of including such kinematical effects as: (i) recoil corrections and (ii) center-of-mass corrections. One of the purposes of the present work is to investigate these kinematical effects and examine their influences on the results of the previous calculations based on the static bag picture. A second purpose of the present paper is to provide an extensive comparison of the calculated results with the existing experimental data. Regarding the weak interaction form factors of the hyperon beta-decays, it is customary to compare the theoretical values with the “semi-empirical” values obtained through Cabibbo-type fitting to the data, rather than directly dealing with the experimental observables. We make here a direct comparison of the theoretical prediction for various observables with the measured values. In particular, an attempt is made to determine the best-fit values of the Kobayashi-Maskawa matrix elements V,, and Vu,.

1. Introduction Magnetic moments and weak interaction form factors provide us with valuable information on the structure of baryons. The recent accumulation of experimental data on these quantities for the octet baryons is quite remarkable lm4), which has S-9,16,18,20,22-24,26-29,33-42)_

The

~~(31

invited a great many theoretical investigations model constitutes a useful starting point for correlating these data, whereas studies of the pattern of possible deviations from the SU(3) symmetry provide detailed information on the dynamics of quarks, mesons and gluons inside baryons. Thus, the study of SU(3) symmetry breaking in the physical baryons is of interest in its own right. It should be added, however, that the examination of the nature of SU(3) symmetry breaking is important also for translating information pertaining to the hadronic level into that of the quark level; this is exemplified by the recent attempt by Leutwyller and Roos ‘) and that by Donoghue et al. “) to determine the Kobayashi-Maskawa matrix elements, Vu, and Vu, [ref. ‘)I. 0375.9474/89/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division) August 1989

B.V.

430

T. Yamaguchi

ef al. / Semi-leptonic p-decay form factors

As regards to the magnetic moments, the achieved precision of the data is such that their systematics definitely shows significant deviations from the exact SU(3f symmetry; furthermore, there are indications of the breakdown of the additive quark model picture in general, On the other hand, the situation with the weak interaction form factors

is not yet so clear; there even exists a controversy

experimental simultaneous opinion that were gathered the treatment

data should be “‘favored”. The WA2 group “) that carried out a experimental study of several hyperon semi-leptonic decays is of the one should avoid taking the world average of the existing data that from separate experiments with varying experimental set-ups since of apparatus-dependent corrections can be highly delicate. From this

as to which set of

viewpont the&WA2 group, in testing the Cabibbo SU(3) scheme, used only those data that were obtained in their own experiments, and concluded that there is no signal for the breakdown of the Cabibbo model. This view, however, does not seem to be shared universally. For instance, Donoghue er al. ‘) recently carried out an overall fitting to the whole body of the existing data and concluded that the SU(3) broken picture is far superior to the assumption of perfect SU(3). Thus, it seems that the detailed verification of the Cabibbo SU(3) scheme is far from settled yet, and the SU(3)-symmetry breaking in the octet baryons deserves further detailed studies. the weak interaction form factors and the magnetic In refs. 8*9) we calculated moments of the octet baryons within the framework of the volume-type cloudy bag model supplemented by one-gluon exchange effects. The motivations for this approach were fully described there. Although the work of refs. *z9)contained rather elaborate treatments of dynamical effects such as mesonic and gluonic contributions, it was based on the static bag picture and did not take into account kinematical effects due to deviations from the static bag picture. One of the kinematical effects that need to be considered is a recoil correction ‘*-‘2). In calculating the weak interaction form factors and the magnetic moments, one needs to evaluate the matrix elements of the relevant current operators in the Breit frame. This implies that neither of the initial or final baryons are at rest in our reference frame. Therefore, if static bag wave functions are used for the initial and final states, the results should be corrected for the fact that the baryons are actually moving. This correction is called the recoil correction. In this connection, we remark that it was emphasized in ref. “) that the weak interaction form factors calculated without taking account of recoil corrections can be alarmingly sensitive to the choice of the reference frame, whereas the inclusion of the recoil corrections leads to the substantial reduction of this sensitivity. (We will later make a comment on this.) Another kinematical effect which needs to be considered is the center-of-mass correction 13-19). It is well known that, when a system of particles are described in terms of independent-particle wave functions, as in the bag model, the center-of-mass (c.m.) motion of the system is not in a momentum eigenstate. In fact, even the separability of the internal motion and the c.m. motion is violated in the independent-particle description, the only

T. Yamaguchi

exception

being

lator potential.

ground-state Thus,

et al. / Semi-leptonic

configurations

for a given

P-decay form factors

431

in the non-relativistic

independent-particle

model

harmonic

oscil-

wave function,

one

must specify how to project out a particular c.m. motion and an intrinsic wave function to be associated with this c.m. motion. Ideally speaking, once this specification

is made and the projection

accomplished,

we can calculate

physical

observables using thus obtained intrinsic wave functions. Corrections representing this procedure are called the center-of-mass (c.m.) correction. In nuclear physics, where the (model) hamiltonian for the whole system is more or less given, this program can in principle be carried out and it is known as the Peierls-Thouless double-projection method 43). It is, however, not trivial at all to extend the method to relativistic cases. Donoghue et al. “) recently pointed out that the inclusion of the center-of-mass correction can have an important influence on the evaluation of SU(3)-breaking. That is, these authors demonstrated that, by applying the center-ofmass correction (in a certain approximate method) to the results ofthe SU(3)-broken MIT bag model, the totality of the existing hyperon beta-decay data can be reproduced satisfactorily. All these indicate that, once the treatment of the static model reaches the level of refinement to include the mesonic and gluonic effects, it is desirable to consider the recoil and center-of-mass corrections at the same time. Unfortunately, there is no rigorous formalism available for applying the recoil correction to the bag model. The same is true with the c.m. correction. This is in a sense not too surprising, for, once we introduce such a phenomenological notion as a static bag nailed down at a fixed space point, there is a conceptual (not simply technical) difficulty in recovering the translational invariance of the theory. Thus, we are forced to rely on some approximations to estimate the influence of these kinematical corrections. There have been many attempts to develop approximate methods for applying the recoil correction “-12) and/or the c.m. correction 13-19) to the static bag model. One of the purposes of the present work is to estimate, using representative methods available in the literature, the effects of the recoil and c.m. corrections

to the weak-interaction

form factors

and the magnetic

moments

of the

octet baryons that were previously obtained within the framework of the static volume-type cloudy-bag model including the gluonic effects 8,9). Although our treatment here is not free from the basic limitation mentioned above, it will give us an idea on the relative importance of the kinematical corrections compared with dynamical effects such as mesonic and gluonic effects. The spirit of the present work is similar to that of ref. 23) in aiming at a simultaneous, coherent treatment of the mesonic and gluonic effects together with the kinematical corrections, as best as the existing formalisms allow. In the treatment of the mesonic effects, our approach (linearized volume-type cloudy bag model) relies on perturbative expansion, whereas a full account of the non-linearity of the meson fields is taken in ref. 23). However, the fact that we are using here, motivated by the Cheshire cat picture 44), a rather large bag radius may be expected to reduce substantially the importance of the

432

non-linearity. moments,

T. Yamaguchi

On the other whereas

et al. / Semi-leptonic

hand,

we calculate

Brown here,

P-decay form factors

et al. *‘) concentrate

in addition

to the

on the magnetic magnetic

moments,

all of the weak interaction form factors for all octet baryon semi-leptonic transitions. A second purpose of the present article is to give a detailed comparison of the theoretical results with the experimental data. As regards to the weak interaction form factors of the hyperon beta-decays, it is customary to compare the theoretical values with the “semi-empirical” values obtained through Cabibbo-type fitting to data, rather than directly dealing with experimental observables. We try here, following Garcia and Kielanowski 39), to go out of this routine; we estimate various experimental observables in terms of the weak interaction form factors obtained through the present detailed model calculations, and compare them directly with the available experimental data. This comparison involves the determination of the best-fit values of the Kobayashi-Maskawa (KM) matrix elements Vu, and Vu,. It is hoped that the present study will shed some light on the extent to which the determination of the KM matrix elements can be influenced by the details of the SU(3)-symmetry breaking pattern in the octet baryons. The organization of the paper is as follows. In sect. 2 we define notations for the weak-interaction form factors and the magnetic moments. The volume-type cloudy bag model and the one-gluon exchange effects are briefly recapitulated in sect. 3. After giving in sect. 4 a general discussion of kinematical corrections that describe deviations from the static bag picture, we discuss in sects. 5 and 6 the recoil correction; sect. 5 contains the explanation of boosted baryon states, while sect. 6 deals with recoil corrected matrix elements that are necessary for evaluating the weak-interaction form factors and magnetic moments. Subsequently, we discuss in sect. 7 another kinematical effect, the c.m. correction. The connection between the microscopic transition matrix elements and the observables is established in sect. 8. Numerical results on the weak-interaction form factors and the magnetic moments are given in sects. 9 and 10. For the reason explained in the text, we first present in sect. 9 the results for the case that includes only the recoil correction, and then give in sect. 10 those that include, in addition, the c.m. correction. Using the outcome of the present microscopic calculation of the weak-interaction form factors, we calculate in sect. 11 various beta-decay observables and make direct comparisons with the available data; the best-fit values of the Kobayashi-Maskawa matrix elements Vu, and V,,, will be deduced through these comparisons. Finally, sect. 12 will be devoted to discussion and summary.

2. Weak interaction

form factors

and magnetic

moments

We give here a brief summary of the definition of weak interaction form factors and magnetic moments of the octet baryons; the details can be found in refs. ‘,“).

T. Yamaguchi et al. / Semi-leptonic P-decay form factors 2.1. WEAK

INTERACTION

FORM

433

FACTORS

Consider the p decay between octet baryons A and B: B+ A-l-e + ~7~.The charge-raising current h”(x) responsible for the transition B+ A is expressed as h”(x)

= h&(x)+

h"as=l(x) ,

hi,=,(x)

= Vu,( VA (xl -A”(x))

h;,=,(x)

= V,,( V’*(x)

,

(2.2)

-A’“(x)),

where the V,, and Vu, are the Kobayashi-Maskawa of the form factors is as follows:

(Nphl V*(O)~B(pi))=~h)

(2.1)

(2.3)

matrix elements.

‘ThC1qp +* h(q*)Y*+--$& A

B

The definition

B

A

q*]U(pi) ) (2.4a)

(A(Pf)IA”(O)IB(Pi))=ii(P,)

P~Cq*)v*+~ aA”q,

A

+

B

(2.4b) where q =pf-pi, and mA (mB) 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 J 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-$ particles. The purpose of any microscopic calculation is to evaluate these global form factors starting from microscopic transition amplitudes. In ref. *), we obtained the microscopic transition amplitudes using the volume-type cloudy bag model supplemented by one-gluon exchange contributions. In the present work, we will the recoil as well as improve the evaluation of these matrix elements by including center-of-mass corrections. 2.2. MAGNETIC

MOMENTS

We next consider elements of J;“(O)

the magnetic moments of the octet baryons. for a baryon state A can be written as

The diagonal

ic2(q2)

(A(Pf) IJt.m.(O) IA(Pi)) = U(P~) Cl(q*)Y”+r with q = pf-pi.

The magnetic /~~=(A,s,=f

moment A ’ 2 {I

/1A of baryon d’rrx

J r.m.(d};

A

aAwqpU(Pi) 1

A is defined t A, sz =;) .

7

matrix

(2.5)

by (2.6)

In terms ofthe SU(3) octet vector components J:“. is expressed as: JF”’ = F’, +JfFt where F; and d$F”, are the isovector and isoscalar electromagnetic currents, respectively.

,

T. Yamaguchi et ai. / Semi-leptonic

434

3. Cloudy-bag We want interaction calculated

to examine form factors

model and gluonic effects

the effects

of the kinematical

and the magnetic

moments

on the base of the static cloudy-bag

contributions.

It will therefore

these quantities

P-decay form factors

model supplemented

be useful to recapitulate

in the static bag model.

corrections

on the weak

of the octet baryons

that are

with the gluonic

briefly the method to calculate

This will also help to introduce

notations

that will be used in what follows. 3.1. VOLUME-TYPE

CLOUDY

BAG MODEL

The lagrangian density of the volume-type linearized form is given by 8*9~20-22~25~26) T(x) TrL4,1.= [q(x)(id,Y

=&U-r -

cloudy

bag model

+~&+%“t+%J~,

m)q(x)-Ble,-~Q(x)q(x)As,

Lf+=&$“(x)a~#“(x) -~m~~“(x)#“(x) ,

(CBM)

in its

(3.1)

(3.2) (3.3) (3.4)

(3.5) We denote by H,, Hint and Hwr the hamiltonians associated with 2’M1T+2’+,, We assume that the quark motion can be restricted to %;,, and zwr, respectively. the ground-state configuration in the static spherical MIT bag. Thus,

(3.6)

where R is the bag radius, f2 is the lowest eigenvalue, and e is a flavor-spin wave function; (Y, = 1 for the u- and d-quarks, whose masses are assumed to be zero, whereas (Y, = v’( E f m,)/E for the s-quark with m, being the strange quark mass. The normalization constant N is so chosen that 5 d3rqt(r)q(r) = 1. Once quark configurations are restricted to be in the ground state configuration, the hamiltonian can be cast into a form in which the quantized meson fields interact with the lowest configurations IA), IB), . . . of three quarks 27*28).Thus, Ho= C m,,A~A,+~

r d’kw,(k)aL(k)a,(k),

(3.7)

(3.8)

with

V;:(k)

=-

iftB

u(kR)

rnn [2u,(k)(2sr)‘]“’

s^,“=

(-)SA-C42m

Tp=

( -)r~-‘47=J7

*’ e^*,. k)T;“, sm (

(3.10)

(3.11a) TA _-t

T,

TB

A

&x

fB 1.

(3.1 lb)

Here, mOA is the MIT bag mass, and a,,(k) (a:(k)) IS . an annihilation (creation) operator for an octet meson LYwith momentum k and energy w, = Jmi+k2; SA and s,.$ are the spin and its z-component of IA); T, and t, are the isospin and its third component of meson cy, &,, is the spherical representation of the unit vectors, and u(x) s&(x) +j,(x). The coupling constant f$” is given by

where the last factor stands for a matrix element reduced in isospin as well as spin spaces, & being a spherical tensor in isospin space. The rank T, is 1, $ and 0 for (Y= rr, K, n, respectively. The doubly reduced matrix elements in eq. (3.12) should be calculated with the use of the SU(6) wave functions for the octet baryons. The strength K is given by K = where

we have

suppressed

~~R”~~[.h(fUjI(fh)l,

the

a-dependence

of K and

corresponding to quark flavor u for all transitions. including the meson clouds can be written as jA>=@{l+(

(3.13)

The physical

mA’- H,-nHi,,A)-‘Hi,,}IAo)

~~Z~(l~(m,-~~)-‘Hi,t}~A~},

have

used baryon

N and

a

stae IA)

3 (3.14)

mass of particle A, I?“= C,m,+AiAo+C, f d”kw,( k) x is a projection operator that eliminates all states but those with at least one meson; 2;’ is the wave function renormalization constant. When the vertex correction is applied to V,,(k), the coupling constant is modified from f:” to fAB, but the change is found to be of minor numerical importance. We therefore assumeftR ---I*” -- (f~“/f~“)f”“. Although the quantity within the framework of CBM, it is known that the slight f NN can be calculated readjustment off”” leads to a better agreement with the TN scattering data. We therefore use here the “empirical” value .f”” = 3.03 instead of the calculated value f”N” = 2.20. where aL(k)

mA is the physical

and A = 1 -C,~~,,,b.hXB,/

T. Yamaguchi et al. / Semi-leptoniep-decay form factors

436

The vector current associated with vector transformation applied and meson fields in the lagrangian density eq. (3.1) is written as

The axial-vector

current

associated

A’” = @‘A\~q(A4+

with the chiral transformation

iA~)q6v+f~ah+4+i5

to the quark

is given by

f4+i5,jkgyhhjq4keV. (3.16b)

-$-

71 For J&.,

1 +i2 in eq. (3.15a)

should

be replaced

by 3 +4$ 8. Thus,

J:.,. = 4Y”i(A~+~A~)9e,+(/;,,+~t,k)OjaAgx_3

(f3jk+~f8ik)ZjYhYshjq~kev.

77

(3.17) We now describe a method to calculate microscopic matrix elements corresponding to the octet baryon p-decays. The treatment of the magnetic moment is much the same, so we do not give a separate explanation of it. We start with considering the matrix

element M,,s(J”)

=(As’l

W(J”(O))IBs),

(3.18)

where W(J’(t))=

d3xJ”(x,

t) e-iq’x

(J” = VA, A”,

V’”

and A”‘) .

(3.19)

I In eq. (3.18), IAs’) and /B s ) are the dressed baryon states, sf and s representing j,. The factor exp (-iq * x) with q =pf-pi in eq. (3.19) comes from the space-time dependence of the lepton wave functions. By inserting eqs. (3.15a)-(3.16b) (3.18) and by retaining terms only up to l/f:, we obtain M,~,(V”or

V’“)=(s’~l~s)(a,+a,+a,+a,-t-a,),

M,YP,(V or V’) = -g(s’/

into eq.

(3.20)

1 /s)(b, + b,+ b,+ b,+ b,)

-iqx(s’la/s)(b;+b;+b;+b;+b~), M,~,(A”orA’o)=-q~(s’~a~s)(c,+c,+c4+c6),

(3.21) (3.22)

M,.,(AorA’)=(s’~a(s)(d,+d,+d,) -&‘l(g-

4I+(e,+e,+Q+e,),

(3.23)

T. Yamaguchi

et al. / Semi-leptonic

-

431

/3-decay form factors

‘___’

(b) ,, ,

f

(d) ,*:/

..\

,’

,’

(c) +i

F”’ ,*;;,

‘.

‘\, \

(e)

%J--..,

,,/---..j’ (f)

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

where suffices j = 1,2,3,4,5 and 6 correspond, respectively, to diagrams (a), (b), (c), (d), (e) and (f) in fig. 1, and the explicit expressions for the quantities on the right-hand sides, which are somewhat lengthy, are given in ref. “)

3.2. ONE-GLUON

EXCHANGE

EFFECTS

We consider quark-gluon interactions inside the static spherical MIT bag, up to second order in g,; this implies that we can neglect the gluon self-interaction. As in sect. 2, we consider the p- decay between octet baryons: B+ A+e + Fe. The S-matrix element for this decay is written as 8,29)

dt(&(f)+

Kv(t))

I

1E&s), (3.24)

Hw = AGF

d’x(h”(x)l,(x)+h.c.),

(3.25)

438

7: Yumaguch~ et al. ,I Semi-le~tonjc

p-decay form factors

with h”(x) as defined in eq. (2.1), and &(x) being the ordinary current. The quark-gluon interaction Ho is written as

&=gc

J +f& JJ

charged

lepton

d3x : j”(x, t) - A”(x, t) :

V

d3x d3x’ G&x, x’) : p”(x, f) : : pa(x’, t) : ,

(3.26)

V

where

G&x, x’) is the color-Coulomb

Green

function

x,29) and

$(x, t) ,

(3.27)

$4&Y t) .

(3.28)

j”(x, r) = J(x, t)r?

P”b, f) = 40(x, f)Yo

Here, +4(x) is the quark field operator, and A”(x) the gluon field with color a; A,‘s are the Gell-Mann matrices for color SU(3), and V the bag volume. We have chosen here the Coulomb gauge for the gluon field. The S-matrix element up to second order in Hc is written as (for AS = 0 transitions) S(B~A+e-+~~)=~~G~~~(0)[{~(VA)-~(A~)}V”~].

(3.29)

For AS = 1 transitions, the quantity in the square bracket should be replaced [ {k( VA) -k(A’“)} Vu,]. Z?(J”)(J’ = V”, AA, V’*, A’*) is given by

Iz((J”)= s&P>+ S,(P), $(I”)

= (-i)‘(A,s’]

S,(JA)=;(-i)3(A,s’I

by

(3.30a)

J dt,c-kTCfG(4) W~(~A(f*~)) /%A, (3.30b)

J dt,dt,dt,~(&(t,)Hc(b) WqV’(cd) 1Bos), (3.3Oc)

where WJJ”( 1)) = j d3xJk(x, t)eviqx, with J;(x) representing the quark part of the current. Upon substituting the explicit expression for Hc and retaining the terms up to g:, various types of diagrams will appear. We consider here diagrams represented in figs. 2 and 3; the contribution of fig. 2 will be denoted by i’, and those of fig. 3 by Z?“. Thus, k’- [J?(J”)],,, 21r and g”= [~(J”)]~fi,, 3J. The difficulty that diagrams (a) and (b) in fig. 2, involving the color-Coulomb interaction, violate the color-singlet condition can be avoided by including diagrams of the quark seif-energy type shown in fig. 3 [ref. ‘“)I. The explicit expression for I?’ and I?” are given in ref. “f. The total contribution of one-gluon exchange processes is written _ as K = I?‘+I?“. We can show that AI,., of eq. (3.18) acquires from the gluonic

T. Yamaguchi et al. / Semi-lepionic P-decay form factors

439

(4

b)

(4

W

(Cl

(d)

(cl

(d)

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 transverse gluon propagators. The wavy line with a cross stands for a current.

effects an additional

piece Mf.,(J”) J? = -2rriS(E,-

defined

Fig. 3. Particular self-interaction contributions to be added to satisfy the color singlet requirement. The square on a quark line dictates that the intermediate quark state be limited to the lowest bag eigenmode.

by:

EA- E,-

E,)M:,,(J')

.

(3.31)

The calculation of the gluonic effects based on this equation was named type I in ref. “). We carried out in ref. “) another calculation, called type II, but the numerical results of type I and type II were found to be practically identical. We therefore concentrate here on type I. The final expression for Mf,,(J”), after the long wavelength expansion, is given by (see appendix C in ref. “): M$,(Por M~.,(Vor

M$,(A’or

V’“)=(s’Ills)a’,

(3.32)

V’)=-q(s’IlIs)Liqx(s’lals)b”‘,

(3.33)

A”) = -q.(s’lals)E,

(3.34)

M~~,(AorA’“)=(s’~a~s)d-q(s’~(q~a)~s)~.

The microscopic

matrix

the contributions C of ref. ‘).

of figs. 2 and 3, and their explicit

elements,

4. Kinematical

(3.35)

(?, 6, b”‘, ?, L? and 5, in eqs. (3.32)-(3.35)

corrections

expressions

for the static

represent

are given in appendix

hag model

It was mentioned in the introduction that there are two kinds of kinematical corrections to be applied to calculations based on the static bag picture: the center-of-mass (c.m.) correction r3-19) and the recoil correction ‘O-‘*). The CBM

440

T. Yamaguc~i et al. / semi-ie~tonic

calculation of the weak in the preceding section the matrix element of the static bag fixed at the one to be obtained

B-decay form factors

interaction form factors and the magnetic moments described is based on the static bag picture; namely, we have calculated a current using the CBM wave functions corresponding to the origin, and identified the resulting matrix element with in the Breit frame. Therefore, the c.m. and recoil corrections

are required. In the absence at present of exact formalisms for computing these corrections, we shall use some approximate methods available in the literature. Before going into the description of particular methods used here, we make some general remarks. First, although the above kinematical corrections should in principle be applied to the entire CBM wave functions, we consider only the kinematical corrections to the three-quark configurations, which are dominant components; the configurations involving mesons and gluons are expected to constitute significant, but not excessively large admixtures, so that the kinematical corrections to these configurations are considered to be higher order effects, which may be expected to be negligible. In fact, by limiting the kinematical corrections to the quark part, we are introducing a more fundamental ansatz; viz., we are assuming that the contribution of the bag itself to the kinematical effects can be completely ignored. It is an extremely difficult task to construct a bag model in which the bag itself has the correct translational property, and all existing formalisms (except for that of Rebbi I”)) deal only with the quarks inside the bag, leaving the bag itself outside of consideration. Our second remark is concerned with separating the kinematical correction into the c.m. and recoil corrections. In order to apply the recoil correction, one must first construct bag states which are eigenstates of the total momentum and then boost them to appropriate Lorentz frames. In this sense these two corrections should be treated SimultaneousIy. This is, however, so difficult a program to carry out that, in the usual treatment of the recoil correction, one assumes that the static bag wave function can be approximately identified with a zero-momentum eigenstate. In the present work, we will first study recoil corrections on the base of this assumption. The effect of deviations from this assumption will be estimated separately as cm. corrections. Provided these two corrections are not excessively large, which seems to be the case, it will not be unreasonable to consider the sum of these two corrections calculated separately as the total kinematical correction. In sects. 5 and 6, we deal with the recoil correction. The discussion of the c.m. correction will be presented in sect. 7. For evaluating the recoil correction, we use the formalism due to Betz and Goldflam “), which is to our kowledge the most systematic method available for this purpose. The original formulae given in ref. I’) need to be slightly generalized so that they can be applied to cases involving quarks of different masses. In order to evaluate the cm. correction, we use the formalism by Tegen et af. 16), in which an eigenstate of the total momentum is constructed non-relativistically by demanding that the sum of the Fourier components of three quarks inside the bag be equal to

T. Yamaguchi

a given total

momentum.

non-relativistic

methods,

ticularly

superior

et al. / Semi-leptonic

Although

there

are some arguments

we do not find the so-called

to the method

used here. Since none

In this situation,

we use the method

against

“relativistic”

the use of method

of the existing

address the fundamental problem of projecting out a genuine a given bag state, it is not easy to demonstrate the superiority others.

441

P-decay form factors

par-

formalism

intrinsic state from of one method over

of ref. 16) as a convenient

framework.

5. Recoil correction Betz and Goldflam “) proposed a systematic method to construct boosted bag states from a given static bag state. In their method, as in all others, only valence quarks are subject to Lorentz boosting, the bag itself being outside of the formalism. We have already mentioned that it is extremely difficult to get rid of this limitation, and the Betz-Goldflam formalism is the best available method for taking account of the recoil correction. One technical detail that must be taken care of in dealing with the hyperons is to extend the formalism to cases where the masses of quarks are not necessarily the same. This extension is described below. To begin with, we introduce IB( pR = 0)) and lpu) which denote, respectively, the eigenstate of a composite baryon B with four-momentum (m,, 0) and that with four-momentum (E( pB), pB). Here, mg is the mass of baryon B, and pR its threemomentum. The relation between IB(p,, = 0)) and 1~“) is given by IPR)= [(2~)‘~“‘(0)p(P,)l”2~(u,)IB(P,=0)),

where P(P~= E(~dlm~ transformation ref. ‘I)). Because normalization:

matrix

(5.1)

(E(P,) =J&+P&

corresponding

and U(u,) (uB=pB/E(pe)) is the to the boost velocity us (see eq. (5.7) and

of (B( pR = 0) (B( pB = 0)) - 1, eq. (5.1) guarantees

the conventional

(PI3IPA)= w-)3~(3’(PB-PA P(PA). From eq. (5.1) we can define the moving lB+,)-

baryon

(5.2)

bag state 1B.J by:

WB)~~(P,=W.

(5.3)

In the case at hand the baryon bag states are those of the volume-type CBM. Thus, a physical baryon state IB) is a “dressed” bag which has clouds of the octet meson

fields in addition

to the MIT bag configuration -

IB,). That is,

IB)=Jz,“IB”)+JZ,B(mg-~“)HintIBo) 3

(5.4)

where, as in eq. (3.14), Z,” is the wave function renormalization constant, and Hint is the effective transition potential of eq. (3.8). It was already mentioned, however, that we assume here that only the recoil correction for the main component corresponding to the three-quark configuration IB,) need to be considered. Another assumption we make here is that the MIT bag state IS,) can be approximately

T. Yumaguc~i et al. / Semi-leptonic

442

identified

with the zero-momentum

p-decay form factors

eigenstate tB%) = u(u,)

/B( pe = 0)). Thus, IB,) .

(W

Deviations from this second assumption will be treated separately as c.m. corrections later on. In order to evaluate the recoil correction arising from boosting IB,) from its rest frame to appropriate Lorentz frames, we have to relate the composite baryon state in eq. (5.5) to its elementary quark wave functions @j.f’(x) (f refers to quark flavors u, d and s, and i refers to the ith quark). With the definition (xl B,,) = W..(x), eq. (5.5) can be rewritten at the quark level as: YIuB(x)= h Ui(u,)Qi~/‘(coshwexll+xl)exp{i&~f’IxllIsinhw,} i=l

(at t=O), (5.6)

(sinh $w&&, . ai,

Ui(LIs) =coshfo,+ XII=

X .

i&i?, )

xI=x-x~f) IPel 2E IPsl/2

wg = arctanh ,p

=

[(myy+

(5.7) (5.8) (5.9)

’

(5.10)

(,py/py,

where &if’ represents the eigenenergy of the ith quark; R is the bag radius, 0:‘“’ the lowest eigenvalue, miU) the quark mass for flavor x and xif’ a flavor-spin wave function; the normalization constant N$” is so chosen that j d3x @jf’t(x)@~r’(x) = 1 (no summation overfor i). We note that, in eqs. (5.6) and (5.7), og for the composite baryon B is used instead of wjs), which depends on the mass of each quark.

6. Recoil corrections to current matrix elements We now amplitudes

consider obtained

how

to apply

in the volume-type

recoil

corrections

to microscopic

transition

CBM. We start with the form

~P~lJ~~~)/PB~=~P~/J~~O)lPB~ei4-X,

(6.1)

where q =pA-pB and J, is a generic symbol for the weak and electromagnetic currents derived from the volume-type CBM lagrangian density eq. (3.1); IpA) and IPs) should be understood as representing moving states given by eq. (5.1). Setting t = 0 in eq. (6.1), one obtains (PAI~~(0)IPB)=[P(PA)P(Ps)l”2

I d3x

xeiq’“(A(pA=O)I ut(U,)J,(x)U(u,)IB(P,=O)) (6.2)

T. Yamaguchi et al. / Semi-lepfonic /3-decay,form facrors

We calculate Hogaasen

443

eq. (6.2) in the Breit frame, for which PA = -pa = 49. Lie-Svendsen

‘*) used the “modified”

Breit frame, in which the initial

and

and final baryons

have velocities of the same magnitude and direction but opposite senses. Thus, in the modified Breit frame, the hyperbolic angles are the same (WA = w,), whereas pA = qm,/( m,+ m,),

pB = -qmB/( mA+ ms).

One

problem

with

the

use

of the

modified Breit frame is that the correspondence between the microscopic transition matrix elements and the global form factors becomes complicated because terms involving P = pA tp, need to be retained and rewritten in terms of q = pA -pB . On the other hand, owing to simplifications resulting from wA = wg, Lie-Svendsen and Hprgaasen could calculate terms of higher order in q. Since we are here interested in the form factors at q* = 0, we do not need to calculate these higher order terms. For the sake of comparison, we have carried out calculations with the use of both the ordinary and modified Breit frames. The results indicate that, as far as quantities of our present interest are concerned, the difference between the Breit and the modified Breit frames is negligible. In what follows, therefore, we use the ordinary Breit frame to secure more continuity with our previous work. Then, eqs. (5.3) and (6.2) will give (~4~~,(O)~-~q)=[p(~q)p(-~q)I”* Inserting eqs. (5.6)-(5.11) of different masses: (~4lJ,(O)I

-&)=

into eq. (6.3), we obtain the following

d3x exp {i(q -(sji’sinh

c

(6.3)

d’xei9‘“(A,,IJ,(x)lB.,). formula

wA+ aIf’sinh

wg)) lx,, I}

cyclic(i,,,k)

d’x exp {-i(

X

x @,@(,J{cosh

sji’

sinh WA+ &j-‘)sinh wg) 1XIII}

;(w~-w~)+

4. (Ysinhf(w,-o,)}

@jf’(xb)

x[Id’x(i-k)], where f and stands in eq.

for particles

1 (6.4)

q = 191,x’= cash (w)x,, +x,, and (i, j, k) refer to the ith, jth and kth quarks; f represent the initial and final quark flavor, respectively, and the bracket for a spin-flavor matrix element with respect to SU(6) wave functions. Tr’s (6.4) represent three types of matrices: %h’

r;

=

+

JLJYoY~

%A, + iAJy07, ( $(A1 +

where A, (I = 1 - 8) stands

i~2hoypy5

(for electromagnetic

current)

(for vector current) (for axial-vector

for a Gell-Mann

matrix

current)

in flavor space.

(6Sa) (6.5b)

,

(6.5~)

T. Yamuguchi

444

e? al. / Semi-feptunic p-decoyfotm

facfors

For strangeness-changing processes, 1 +i2 in eqs. (65b) replaced by 4+ i5. We note that, to order l/M (M: baryon are working,

the spectator

and (6.5~) should be mass) with which we

part in eq. (6.4) is just the normalization

integral,

which

implies that the recoil effects for the spectator part can be neglected. This completes the explanation of the generalization of the formalism of Betz and Goldflam to a case with different quark masses. Since we apply here recoil correction only to the leading MIT bag part in the CBM calculation, the consequence of the recoil correction can be stated by giving M yjT+REC(Jp), which represents the recoil corrected matrix elements to the MIT bag part. The explicit definition of M?JT+REC(Jp) is: M ~;r+REc(.Fp) = (Z;Z,“)“z(A,s’I

W&f,)

/ B,s)

corresponding

(6.6)

,

where (Aor’ / W, 1B,s) are equivalent to (44 1J,(O) 1-fq) in eq. (6.4). Then, neglecting the recoil effects for the spectator part in eq. (6.4), we obtain

W,=

C w(i),

(6.7)

i=1,2,3

(6.8)

w(i) = b.(~)iMi), where ha*(i) changes the flavor of the ith quark from b to a. The operator to be sandwiched between the quark-spin wave function is for v”

f ii% &b(i) =

flab(i)

-6abq-i7jhb(4x~q(i))

for V

--5%4~

for A0

o,(i)),

(6.9)

forA, u,(i)> I Sabua(i)-cL7(4where a, definition appendix follows:

is the Pauli spin operator acting on two-component quark spinors. The of the radial integrals - $$,, &,,
(i) MyJT,‘TfREC( p

or

(6.10)

V~)=(s’Ills)a^,, Gl = -~(z,Az,B)~‘~T*~(AIIIAII(B)~~~~~,

(ii) MFjT+REC( V

or

(6.10a)

V’)=-q(s’/lls)~,-i*x(s’/rrIs)~,

(6.11) (6.11a)

$1~ -~(Z~Z,B)“2(AiilAIllB~rj,~, 6: = -~~~(Z,AZ,B)1’2T*B(AI(IA*~llB)~bb (iii) M %‘T+REC(Ao or

A&)=-q*(s’lals)c^,

,

63,= -aJ~(2~228)“2TAB(AI(IAaljlB)~~ (iu) MyiTIREC(A

or

)

A’)=(s’fcr/s>a,-q(,‘~(q*u)~s)~,,

(6.1 lb) (6.12)

,

(6.12a) (6.13)

2, = -~~~(Z~Z-~)“2TAB(AIJlA~~(~B)~~h,

(6.13a)

3, = -~JS(Z~Z2B)1’2TAB(AIIIA~IIIB>~~b)

(6.13b)

T. Yamaguchi

et al. / Semi-leptonic

P-decay form factors

445

where TAB,

and A is a spherical

tensor

T TB A t tB >

_tTA

(-l)Ta-‘a

in isospin

space.

We can include the recoil corrections in the volume-type CBM a,, b,, b:, cl, d, and e, [eqs. (3.20)-(3.23)], respectively, with a^,, 6,, P, [eqs. (6.10a), (6.11a), (6.11b), (6.12a), (6.13a) and (6.13b)l. As can A eqs. (A.l) and (AS), a, = 6, and d, = d, to lowest order in 141 with working.

7. Center-of-mass

by replacing 6;) i?,, 2, and be seen from which we are

(c.m.) correction

There are several methods proposed for the c.m. correction i3-19). We use here the formalism due to Tegen et al. 16). The method consists in constructing a good eigenstate of the total momentum (in the non-relativistic sense) by demanding that the sum of the Fourier components of three quark wave functions inside the bag be equal to a given total momentum. This formalism was already used in ref. “), where the c.m. corrections to the results of the surface-type CBM were estimated. It is to be recalled that only the c.m. correction for the main component, i.e. the three-quark MIT bag state, was considered in ref. 27). Since the MIT bag states in the present volume-type CBM calculation and those of the surface-type CBM calculation are exactly the same, the results of the c.m. correction given in ref. 27) can be used unchanged here. For the sake of completeness, we reproduce here the calculation of the c.m. correction for the weak interaction form factors; the calculation for the magnetic moments follows the same pattern. Denoting by $m(r) the quark wave function of the ath kind orbiting in the bag, we introduce its Fourier transform 4=(p) by +u(p) = Then,

we define a three-quark

d3r e-ip"$a(r).

(7.1)

state qP( ri, r,, r3) by

(7.2) typ(r , , r,, r3) is translationally as

invariant

in the sense that its cm. motion

!PP = eZp-R~int ,

is factorized

(7.3)

where R = $(rl + r,+ r3) and Xint is an intrinsic wave function depending only on the relative distances between r 1, r2 and r,. We therefore assume that !Pp( r,, r2, r3) is a physical hadron wave function to be associated with the bag wave function

446

T Yamaguchi et al. / Semi-lepionic P-decay form factors

4a ( rl)&

( rz)&(

r3).

It is reasonable

to impose

!PL(r,,

d3r, d3r,d3r3 I

r,, r3)VQ(rl,

the normalization

condition

r,, r3)=(2n)36(P-Q),

(7.4)

which implies -l/2 d3r

N(P) =

e-iP.r pa(r)pp(rb,(r)

U

(7.5)

,

1

Pa(r) = (2~)~~

d3k eik”4L(k)+,(k).

(7.6)

I

If one uses !Pp( r, , r2, r3) instead of the bag wave function, then the matrix element T, corresponding to the quark transition diagram will be modified: Tl + fl, where ?, is obtained by applying c.m. correction to Tl. That is, (2~)‘fG’-Q-q)+(1U,(A,s’)l

W,I ~&h,s)).

(7.7)

Putting the expression for !Pp(&s’) and !Pp(BOs) and W,, we realize that the c.m.-corrected results can be obtained by replacing a,, b,, b:, c, and d, in the relevant expressions with 6i, b;, 6;) E, and d,, respectively. The quantities with bars are defined

by 27). a, = rSO,Z( T, O)(&lllAIIIBo) ,

(7.8)

6 = ~!:-,“,z(T, O)(&lll~lllBo) ,

(7.9)

1)(Aolll~~lllBo),

(7.10)

G = f:J(

T, 1)(&lllA ~lllBo> ,

(7.11)

& = &J(

T, 1~~&ll~dlIb.J.

(7.12)

b: = %J(T,

Here, Z(T, J) = -(Z~Z,“)“2TAB/(2~, ji$, = (~T)~N,N~N(O)

-

iiab

(7.134

~{~+P,(P)~~(P)+~-~(p)/,o)w,,(p),

I $:-II~ = i(4~)2W%N(0)2

and

i

(2r)3 [cu-{gU(p)f~-‘(p)+gb+‘fb(P)} d3p

cu+{f~-‘(P)gb(P)+f,(P)gb+‘(P)}l W,,(P) 7

= i(4~)2NaNbN(0)2

c2Tj3 d3p

[a-kz(df-)(P)

(7.13b)

+d%b(d)

I

+

‘tab =

a+tf:-)(Pkb(d

(4v)2NaNbN(0)2

+f,(h!b+)(~)~l

I

(2v)3 *{

w,,(P)

a+ ga(Pkb(P)

(7.13c)

2

-fad(p)_&(p))

w,,(P)

3

(7.13d)

T. Yamaguchi

et ai. / Semi-lepfonic &decay

447

form fackxs

(7.14)

(7.15)

with g?‘(p)

and j$’

(p) defined

by (7.16a)

The quantities with bars were already calculated in ref. 27), and we only have to substitute them into the relevant formulae in the present volume-type CBM Cal&ation [see eqs. (8.5a)-(8.6c)‘J The numerical results will be discussed in sect. 10.

8. Relation between gtobal form factors and microscopic matrix efements The derivation of the relation between the global form factors and the microscopic matrix elements was described in great detail in sect. 5 of ref. “). We will therefore sketch only the outline of the method using the weak-interaction form factors as an example. The magnetic moments can be treated in a similar manner. To establish the convection between M,,,(JA) in eq. (3.18) and the global form factors in eq. (2.4), we write down the non-relativistic reduction of eq. (2.4) in the Breit frame 27): (Alpf)s’tf”(O)IB(pi)s)~xX:,F~~, where xS (xSS) is the two-component F is given by (for

spinor

IA = Vo

(for

63+&l+

R

g2

f3.l)

for the initial

or

VrB) ,

J”=V

(t”%

(final) state. The operator

or

(for

V),

Jh = A0

(8.2b)

or

A’*) , (8.2e)

T Yamaguchi et al. / Semi-leptonic p-decay form factors

448

(for

J” = A

or

A’) ,

(82d)

where terms of second or higher order in ]q/, I/ mA and l/m, have been dropped except for q(rr . q). The identi~cation of h/l,.,(J”) in eqs. (3.20)-(3.23) with xifFLyS for each type of J” gives 8S9,26,27)

f,+~~~Bf;=a,+a,+u,+a,+a,, A

-~(A+h)+m A

B

A

:,

f3= -(b,-tb,+b,+b,+b,),

-!y&Q+n3,+---&

g,= A

B

g1+

mA-MB m +m A

-(c,+c2+c4+cd,

(8.3c)

(8.4a)

B

(8.4b)

g2=4+d2+4, B

1 =e,+e,fe,+e,.

g2 hmAmB

(8.3b)

B

~~~mmBR(~,+~~)=b;+b;+bi+b;ib;,

A

(8.3a)

E

(8.4~)

)1

The above expressions comprise the contributions of the (volume-type) suffices j(j = 1,2,3,4,5,6) on the right-hand side of eqs. (8.3a)-(8.4c)

CBM, and correspond,

respectively, to diagrams (a), (b), (c), (d), (e) and (f) in fig. 1. Thus, the microscopic matrix elements a,, b, , b; , c, , d, and e, represent the leading MIT bag contributions, while the remainder, Uj, bj, b,!, Cj, dj and e, (j = 2--6), correspond to the mesonic corrections; their explicit expressions are given in appendix B of ref. “). In order to incorporate the contributions of gluon exchange and the effects of recoil and c.m. corrections, we only need to apply the following modifications to the leading-order matrix elements, a,, b,, b;, cl, d, and e,. (8.5a) (8Sb) (8.5c) (8.6a) (8.6b) (8.6~)

T. Yamaguchi

et al. / Semi-leptonic

P-decay

form factors

449

In the above expressions, the matrix elements with tilde represent the gluonic 1 ?+ contributions (cf. sect. 3). Since the matrix elements, a*r, 6,) 6:) C,, d, and Z1 in eqs. (6.10)-(6.13b), related to the recoil effect were defined in such a way that they contain

the static MIT bag contribution

to obtain the increment

as well, we need to subtract

due to the recoil correction.

a,, 6,) 6;) C, and (s,, representing MIT bag contributions is required. in the above expressions have d,, b*,, b*i, c*,,8, and 8,) which we

Similarly,

them in order

for the matrix elements,

the c.m. correction, the subtraction of the static All the microscopic matrix elements appearing been calculated previously 8,9*27) except for newly calculate in the present work.

9. Numerical results for weak interaction form factors and magnetic moments-with recoil correction only We now present numerical results of calculations for the weak interaction form factors and the magnetic moments which include the recoil correction; the results of calculations that include c.m. corrections as well will be discussed in sect. 10. We deal with four cases of bag models: (i) Quark-only MIT bag model - abbreviated as MIT, (ii) MIT bag model plus one-gluon exchange effects - abbreviated

as MIT+

OGE,

(iii) Volume-type cloudy bag model with the clouds of all octet mesons abbreviated as CBM, (iv) Volume-type cloudy bag model with the clouds of all octet mesons plus one-gluon exchange effects - abbreviated as CBM + OGE. For each of these bag models, we distinguish two cases: one without the recoil correction, i.e., the static approximation, and the other that includes the recoil correction, the latter being appended with a qualifier “+REC”. Thus, in total, we shall deal with eight cases: CBM, CBM + REC, CBM+ sometimes generically refer and MIT+ OGE + REC) as CBM + REC, CBM + OGE,

MIT, MIT+ REC, MIT+OGE, MIT+OGE+ REC, OGE, and CBM+ OGE+ REC. For convenience, we to the first four cases (MIT, MIT+REC, MIT+OGE, “MIT-type models”, and to the last four cases (CBM, and CBM + OGE + REC) as “CBM-type models”.

For the parameters characterizing the models we use here the same values as in ref. ‘). Thus, we use R = 1 fm for the bag radius, and take m, = md = 0 and m, = 279 MeV for the quark masses. Concerning the CBM, the calculation of diagram (e) in fig. 1, which applies to the vector current, contains divergent loop integrals. They are regularized 22,3’) by introducing a momentum-dependent “damping” factor exp (-R:k*/3), where R, represents the “effective radius” of meson a(a = n, K, or 7). We ignore here the a-dependence of R, and treat R, as a phenomenological parameter to be adjusted so as to reproduce an appropriate observable, which is taken to be f2(n + P)~._, = 3.70. It is to be noted that R, is adjusted individually for each of the four cases of the CBM-type models. For the color electric coupling

450

T. Yamaguchi et al. / Semi-leptonic

p-decay form factors

TABLE 1 Vector current

form factors

Models W(3) MIT MIT+ REC MIT+ OGE MIT+OGE+REC CBM CBM + REC CBM+OGE CBM+OGE+REC exp.

SU(3) MIT MIT+ REC MIT+OGE MIT+OGE+ REC CBM CBM + REC CBM+OGE CBM+OGE+REC exp. SU(3) MIT MIT+ REC MIT+OGE MIT+OGE+REC CBM CBM+REC CBM + OGE CBM + OGE + REC t?xp.

BU(3) MIT MIT+ REC MIT+OGE MIT+OGE+ REC CBM CBM+REC CBM+OGE CBM+OGE+REC exp.

for AS = 0 (without

c.m. correction)

A

fi

f;

1.oo 1.oo 1.00 1.00 1.oo 1.oo 1.00 1.00 1.oo *

3.70 2.21 (3.70) 1.92 (3.70) 2.07 (3.70) 1.78 (3.70) 3.70 3.70 3.70 3.70 3.705887 f 1.23 x 1O-6

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 *

0.00

0.00 0.00 0.00 0.00 0.00 0.00 *

2.34 1.93 (2.83) 1.79 (2.88) 1.74 (2.66) 1.60 (2.70) 2.77 2.79 2.66 2.67 2.77ztOo.15

fi 1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 *

1.18 0.89 0.73 1.07 0.90 1.82 1.85 2.09 2.12 1.75

-1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 *

2.03 1.90 1.84 1.60 1.54 2.18 2.19 1.91 1.92 1.78

0.00 0.00

(1.96) (2.04) (2.39) (2.50)

* 0.05

(2.32) (2.36) (1.91) (1.91)

k 0.06

0.00 -0.06 -0.06 -0.06 -0.05 -0.04 -0.04 -0.04 -0.04 * 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 * 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 *

The calculated values off, , fi and jS are given for eight theoretical cases: (1) MIT- MIT bag model; (2) MIT+ REC-MIT bag model with recoil correction; (3) MIT+OGE-MIT bag model plus onegluon exchange effects; (4) MIT+OGE+ REC - MIT bag model plus one-&ion exchange effects with recoil correction; (5) CBM - volume-type CBM with octet meson clouds; (6) CBM -I-REC -volume-type CBM with octet meson clouds with recoil correction; (7) CBM+OGE -volume-type CBM with octet

T. Yamaguchi

constant

CY,appearing

in the calculation

2.2, a value consistent

9.1. WEAK

et al. / Semi-leptonic

with the hadron

INTERACTION

FORM

P-decay form factors

of the gluonic spectrum

451

effects, we use CX,= gf/4r

=

in the MIT bag model with R = 1 fm.

FACTORS

We represent in tables l-4 the calculated weak-interaction form factors; tables 1 and 2 refer to the AS = 0 transitions, while tables 3 and 4 to the AS = 1 transitions. The vector-current form factors, f,, fi and f3, are given in tables 1 and 3, whereas the axial-vector form factors, g, , g, and g,, in tables 2 and 4. All the form factors except for g, represent the values at q2 = 0. In addition to the theoretical results corresponding to the eight cases mentioned above, we give in the tables the semi-empirical values of the form factors obtained through fitting to the data assuming SU(3) symmetry. This fitting shares with the ordinary Cabibbo-type analysis ‘) the following features: (i) The flavor dependence of the form factors is given by the Wigner-Eckart theorem for SU(3); (ii) The form factors f3 and g, corresponding to the second-class current may be a Priori set equal to zero; (iii) g, can be ignored because its contribution is proportional to the lepton mass squared. In the present fitting, however, we treat the Kobayashi-Maskawa (KM) matrix elements, V,, and Vu,, as independent parameters without imposing the Cabibbo constraint: V,,, = cos Bc and Vu, = sin 13~. We shall refer to the present fitting as the KM-SU(3) fitting, whose explicit procedure can be summarized as follows. As is well known, V” = Fi + iFi, V’” = F: + iF:, A” = F:” + iFp and A’” = Fy + iF:” , where F;( F:*) is thejth component of SU(3)-octet vector (axial-vector) current. The matrix element of the current between the ith and kth components of the octet baryons is given by

(i(pf)

+%$$ ghpqp 1

IF:(O)Ik(pi))= U(Pf) F,(q2)YA

U(Pi) 3

(i(Pf)IFj”(0)Ik(Pi))=u(Pf)Gt(q2)yhygU(Pi)

(9.la) (9.lb)

3

where, to the

since we are dealing with the exact SU(3) limit, we fix m to be equal nucleon mass. If CVC is assumed for the vector current, F,(O) = iAjk,

F2(0)

l&f + 4&

=

with f = pu,+&,- 1, d = -&L,,.

AS

for

G, ,

we

obtain

meson clouds plus one-gluon exchange effects; (8) CBM + OGE+ REC -volume-type CBM with octet meson clouds plus one-&on exchange effects including recoil correction. The rows labeled SU(3) correspond to the results of semi-empirical KM-SU(3) fitting to data described in the text. The rows labeled “Exp” give the f2’s deduced from the experimental values of the magnetic moment using CVC; data are taken from ref ‘) except the one for B- + A, which is taken from ref. 3, Peterson et al. 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 (CBM), R, = 0.434 fm (CBM + REC), R, = 0.446 fm (CBM + OGE), and R, = 0.412fm (CBM+OGE+REC). For MIT, MIT+REC, MIT+OGE,and MIT+OGE+REC,therescaled values of fi are given in the parentheses, the method of resealing being described in the text.

T. Yamaguchi et al. ,/ Semi-leponic

4.52

P-decay

formfactors

TABLE 2 Axial-vector

Transitions

current form factors

Models SU(3) MIT MIT+ REC MIT+OGE MIT+OGE+REC CBM CBM+ REC CBM+OGE CBM+OGE+REC exp. SU(3) MIT MIT+REC MIT+ OGE MIT+OGE+REC CBM CBM + REC CBM+OGE CBM+OGE+REC SU(3) MIT MIT+REC MIT+OGE MIT+OGE+REC CBM CBM + REC CBM+OGE CBM+OGE+REC SU(3) MIT MIT+ REC MIT+OGE MIT+OGE+REC CBM CBM+REC CBM+OGE CBM f OGE+ REC

for AS = 0 (without

cm. correction)

g1

g2

(a) 1.254 (b) 1.254 1.09 (1.254) 1.09 (1.254) 1.12 (1.254) 1.12 (1.254) 1.26 (1.2543 1.26 (1.254) 1.29 (1.254) 1.29f1.254) 1.254*0.006

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 *

(a) 0.64 (b) 0.61 0.53 (0.61) 0.53 (0.61) 0.56 (0.63) 0.56 (0.63) 0.63 (0.63) 0.63 (0.63) 0.66 (0.64) 0.66 (0.64)

0.00 0.00 -0.10 -0.08 -0.17 -0.15 -0.15 -0.14 -0.22 -0.21

(a) 0.66 (b) 0.71 0.62 (0.71) 0.62 (0.71) 0.57 (0.64) 0.57 (0.64) 0.69 (0.68) 0.69 (0.68) 0.64 (0.62) 0.64 (0.62)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

(a) 0.32 (b) 0.25 0.22 (0.25) 0.22 (0.25) 0.27 (0.30) 0.27 (0.30) 0.27 (0.27) 0.27 (0.27) 0.32 (0.31) 0.32 (0.31)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

g,

* * 2.97 2.97 3.13 3.14 0.99+ 1.36Y 1.00+1.36Y 1.15+ 1.36Y 1.16+1.36Y * * * 2.33 2.36 2.70 2.72 0.9 1 + 0.67 Y 0.93 + 0.67 Y 1.28+0.67 Y 1.30+0.67Y * * 2.90 2.93 2.75 2.78 1.38 + 0.77 Y 1.40+0.77y 1.23iO.77Y 1.25 + 0.77 Y * * 1.28 1.29 1.91 1.93 0.52 + 0.27 Y 0.53 + 0.27 Y 1.15+0.27Y 1.17+0.27Y

For the meaning of the abbreviations for the various models, see the caption of table 1. The resealed values of g, are given in the parentheses, the method of resealing being explained in the text. For the g,‘s in SU(3), two values are given corresponding to two possible choices of the data to be used for fitting (a) the data compiled in Particle Data ‘) and (b) the CERN-WA2 data *). The method of semi-empirical KM-SU(3) fitting is described in the text. For g, = (g3)core+(gS)po,er which has an important q2 dependence due to (g3)po,e, the results are given in the form: (g3),,,+ Y(q)G,, where Y=(m,+m,)%(qR)/(mZ,-q’).

453

T. Yamaguchi et al. / Semi-leptonic P-decay form factors TABLE 3 Vector

current form factors

Transitions

for AS = 1 (without

cm.

correction)

Models

-4

SU(3)

-1.21

MIT MIT+

REC

MIT+OGE MIT+OGE+

REC

-2.19 -1.04

(-2.09)

0.00 -0.01

-1.21

-0.85

(-2.12)

-0.04

-1.21

-1.02

(-2.21)

-0.02

-1.21

-0.83

(-2.26)

-0.04

CBM

-1.21

-1.52

0.24

CBM+REC

-1.21

-1.52

0.23

CBM +OGE

-1.21

-1.59

0.24

CBM+OGE+REC

-1.21

-1.59

0.23

SU(3) MIT

-1.00

2.03

-1.01

1.64 ( 1.94)

-0.25

0.00

MIT+

REC

-1.02

1.60(1.96)

-0.29

MIT+

OGE

-1.01

1.34 (1.51)

-0.23

MIT+OGE+REC

-1.02

1.29 (1.49)

-0.26

CBM

-1.01

1.85

CBM+REC

-1.02

1.86

CBM+OGE

-1.01

1.57

CBM+OGE+REC

-1.01

1.58

SU(3) MIT

-4

1:43

-0.72

1.16 (1.37)

MIT+

REC

MIT+OGE MIT+

OGE + REC

0.03 -0.01 0.00 -0.18

-0.72

1.13 (1.39)

-0.20

-0.72

0.94 (1.07)

-0.16

-0.72

0.91 (1.05)

-0.19

CBM

-0.72

1.31

CBM + REC

-0.72

1.31

CBM+OGE

-0.72

1.11

CBM+OGE+REC

-0.72

1.12

./g

SU(3) MIT

0.02 -0.02

-0.15

0.01 -0.02 0.02 -0.00 0.00

1.22

-0.33

(0.08)

0.17

1.22

-0.39

(0.11)

0.18

MIT+OGE

1.22

-0.18

(0.38)

0.17

MIT+OGE+REC

1.22

-0.24

(0.44)

0.18

CBM

1.21

-0.14

-0.11

CBM + REC

1.21

-0.10 -0.10

MIT+

REC

CBM +OGE

1.21

-0.13 0.06

CBM + OGE + REC

1.21

0.06

SU(3) MIT

1.oo

3.70

0.00

0.99

2.77 (4.52)

0.02

-0.11

REC

0.98

2.51 (4.65)

0.00

MIT+OGE MIT+ OGE + REC CBM

0.99 0.99

2.60 (4.51)

0.02

MIT+

0.98

2.34 (4.65) 3.51

0.00 -0.18

CBM + REC

0.98

3.55

-0.20

CBM+OGE

0.98

3.50

-0.19

CBM+OGE+REC

0.98

3.53

-0.21

4

2.62

SU(3) MIT MIT+

REC

0.00

0.70

1.96 (3.19)

0.01

0.70

1.78 (3.29)

0.00

454

T. Yamaguchi

et al. / Semi-leptonic

P-decay form factors

TABLE 3-continued

z-+,p

For the meaning

f,

Models

Transitions

MIT+OGE MIT+OGE+REC CBM CBM + REC CBM + OGE CBM+OGE+REC

0.70 0.70 0.70 0.69 0.70 0.69

of the abbreviations

of the various

models,

.ii! 1.84 (3.19) 1.66 (3.29) 2.48 2.51 2.47 2.50 see the caption

f; 0.02 0.00 -0.13 -0.14 -0.13 -0.15 of table 1.

TABLE 4 Axial-vector

current

form factors

for AS = 1 (without

Models

Transitions SU(3)

MIT MIT+ REC MIT+OGE MIT+OGE+REC CBM CBM + REC CBM+OGE CBM+OGE+REC SU(3) MIT MIT+REC MIT+REC MIT+OGE+REC CBM CBM + REC CBM+OGE CBM + OGE + REC SU(3) MIT MIT+ REC MIT+OGE MIT+OGE+REC CBM CBM + REC CBM+OGE CBM+OGE+REC SU(3) MIT MIT+ REC

(a) (b) -0.87 -0.87 -0.88 -0.88 -0.93 -0.93 -0.95 -0.95

c.m. correction)

g1

g2

g3

-0.89 -0.92 (-1.00) (-1.00) (-0.98) (-0.98) (-0.93) (-0.93) (-0.92) (-0.92)

0.00

* *

0.00 0.04 0.05 0.03 0.05 0.07 0.08 0.06 0.07

(a) (b) 0.23 0.23 0.30 0.30 0.30 0.30 0.36 0.36

0.32 0.25 (0.27) (0.27) (0.33) (0.33) (0.30) (0.30) (0.35) (0.35)

0.00 0.00 -0.04 -0.03 -0.05 -0.04 -0.00 -0.00 -0.02 -0.01

(a) (b) 0.16 0.16 0.21 0.21 0.21 0.21 0.26 0.26

0.22 0.17 (0.19) (0.19) (0.24) (0.24) (0.21) (0.21) (0.25) (0.25)

0.00 0.00 -0.03 -0.02 -0.03 -0.03 -0.00 -0.00 -0.01 -0.01

(a) (b) 0.29 0.29

0.25 0.31 (0.33) (0.33)

0.00 0.00 -0.02 -0.03

-1.95 -1.87 -1.90 -1.82 -0.61-0.99Y -0.55 - 0.99 Y -0.56 - 0.99 Y -0.50 - 0.99 Y * * 0.57 0.55 0.87 0.85 0.02 + 0.27 0.01+0.27 0.32 + 0.27 0.30+0.27

Y Y Y Y

* * 0.41 0.39 0.61 0.60 0.02+0.19Y o.ol+o.l9Y 0.22+0.19Y 0.21+0.19 Y * * 0.97 0.94

455

T. Yamaguchi et al. / Semi-leptonic p-decay form factors TABLE 4-continued Models

Transitions

EO+z+

g1

g2

MIT+OGE MIT+OGE+REC CBM CBM + RX CBM + OGE CBM+OGE+REC

0.26 0.26 0.30 0.30 0.27 0.27

(0.29) (0.29) (0.30) (0.30) (0.26) (0.26)

SU(3) MIT MIT+ REC MIT+OGE MIT+OGE+REC CBM CBM+REC CBM+OGE CBM + OGE + REC

(a) (b) 1.19 1.18 1.22 1.21 1.30 1.29 1.33 1.32

1.25 1.25 (1.37) (1.37) (1.36) (1.35) (1.29) (1.29) (1.28) (1.28)

0.00 0.00 0.07 0.01 0.16 0.09 0.07 0.02 0.16 0.10

MIT MIT+REC MIT+ OGE MIT+OGE+REC CBM CBM+REC CBM + OGE CBM+OGE+REC

(a) (b) 0.84 0.84 0.86 0.86 0.92 0.91 0.94 0.94

0.89 0.89 (0.97) (0.97) (0.96) (0.96) (0.91) (0.91) (0.91) (0.91)

0.00 0.00 0.05 0.00 0.11 0.06 0.05 0.02 0.11 0.07

-0.05 -0.06 -0.07 -0.08 -0.10 -0.10

g3 0.76 0.73 0.44+0.33 0.42+0.33 0.23 + 0.33 0.21+0.33

Y Y Y Y

* * 4.27 4.16 4.48 4.37 1.67+1.36Y 1.59+1.36Y 1.88+1.36Y 1.79+1.36Y

* * 3.02 2.94 3.17 3.09 1.18+0.96Y 1.12+0.96Y 1.33 + 0.96 Y 1.27 + 0.96 Y

For the meaning of the abbreviations of the various models, see the caption of table 1. For g, = which has an important 9’ dependence due to (g,),,,,, the results are given in the (g3)core+(g3)poler + Y(q)G,, where Y~(m,+m,)2u(qR)/(mZ,-q2). form: k),,,,

G,(O) = ifjkF+ d,,D, with F+ D = g,(n+ p; q2 = 0) = g, = 1.254 [ref. I)]. Then, the form factors at q2 = 0 are completely determined up to one parameter, which may be chosen to be cr,, = D/( F-C D). Thus, if the KMSU(3) scheme is valid, we should be able to explain all transitions B + A + e- + Fe by adjusting gyp along with Vu, and Vu,. We note here that the data on O++ O+ nuclear beta decays put a stringent constraint on V,,: 0.97426 Vu, ~0.9756 and that the accuracy at present of the experimental data on hyperon decays is such that the x2 value of the KMSU(3) fitting will be dominated by the O++ O+ beta decay data (if these data are included in the fitting). We may therefore fix Vu, from the outset at the central value of the above-mentioned range; i.e., Vu, = 0.9749. Then, the remaining two parameters, Vu, octet-baryon and CY~,are adjusted so as to minimize the total x2 for the semi-leptonic decay rates including both AS = 0 and AS = 1 transitions. There are at present two sets of data on the octet baryon beta-decays: (a) the one compiled in Particle Data ‘)

456

T. Yamaguchi et al. / Semi-leptonic p-decay form factors

and (b) the other given by the CERN

WA2 group ‘). (These sets are reproduced

in

the column labelled Exp. in table 16.) In tables 1-4, the rows labelled (a) [(b)] give the results of the KM-SU(3) fitting to the data set (a) [(b)]. The best fit value of (Y,, turns out to be cub = 0.626 for (a), and (~b = 0.598 for (b). The results of the static treatment

without

the recoil correction

have already

been

discussed extensively in ref. “). So, our first concern here is whether and to what extent the recoil corrections modify the results of the static treatment. Upon examining tables 1-4, one will notice that the results which include the recoil correction are very close to those without it. In particular, for the leading-order form factors, f, and g, , the results with and without the recoil correction are practically identical. We intuitively expect that, sincef,tiy,u and g,Uy,y,u in eqs. (2.4a) and (2.4b) have no explicit q-dependence (we are here concerned with the form factors at q2 = 0), the effect of the recoil correction on fr and g, will be rather small; changes in fi and g, due to the recoil correction is a secondary effect that arises as a consequence of solving the simultaneous equations, eqs. (8.3a)-(8.4c) [with the substitutions eqs. (8Sa)-(8.6c) understood]. The present explicit calculation gives a quantitative confirmation of this expectation. The recoil correction, as considered here, arises through modifying the momentum transfer q in eqs. (2.4a) and (2.4b). Therefore, it will have a direct influence only on the form factors of non-leading order, f2, f3, g, and g, . The effect of recoil correction on these form factors can be directly seen by looking at the entries for the MIT-type models in tables l-4*. The most visible effect of the recoil correction is to change f2’s by the amount up to 0.3. Apart from this change, the present explicit calculation indicates that the effect of recoil correction on the weak interaction form factors is practically negligible. This result should be contrasted with the conclusion in ref. 12) that the recoil correction can influence significantly the values of g,, giving rise to the rather large values of g, for some decays. We summarize the main features of tables l-4 without distinguishing (except for f2’s for the MIT-type models) the results with and without the recoil correction. Since a fuller account of direct comparisons with the experimental data will be given in later sections, we shall be brief here in comparing the calculated form factors with the data. 9.1.1. Vector current form factors (tables 1 and 3). According to CVC, f, 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 transition satisfy this requirement. For non-isomultiplet transitions, the meson clouds and the gluonic contributions can in principle affect f,, but table 3 indicates that the effects are small. This is in conformity with the Ademollo-Gatto theorem. 32). l As mentioned earlier, the CBM-type models involve one adjustable parameter R,, which is readjusted for each case so that f ;xp (II+ p) = 3.70 should be reproduced. Therefore, the difference between CBM and CBM + REC or that between CBM + OGE and CBM + OGE+ REC does not directly represent the recoil effect itself but the combined effect of the recoil correction and the readjustment of R,.

T. Yamaguchi

et al. / Semi-leptonic

P-decay form factors

457

Although comparison of the theoretical and experimental values of f2could in principle provide a non-trivial test of theoretical models, there are at present no direct experimental data for f2.For the AS = 0 transitions, however, CVC allows us to deduce magnetic

the “experimental”

moments.

also given in table

value of

The “experimental”

f2 from the observed values of the relevant values of f2obtained in this manner are

1. One will see that they show appreciable

deviations

from the

SU(3) symmetry and that the pattern of deviation is reproduced reasonably well by CBM. With inclusion of the gluonic effects (CBM + OGE), agreement with the data improves for c”-+ so, but worsens for X+ 1’. We will later have more occasions to remark that the inclusion of the gluonic effects does not necessarily lead to the improvement of overall agreement with the totality of data. As far as the comparison between the CBM-type models and the MIT-type models is concerned, since the former have the parameter R,, which can be adjusted so as to reproduce fi(n + p)exp, it might be “fairer” to rescale f,(MIT) in such a way that f2(n+p)‘“P should be reproduced. We choose here to rescale the right-hand side of eq. (8.3~) by multiplication with an overall factor which is adjusted so that (f, +fZ)n+p = 4.70 should be reproduced. These resealed values are given in the parentheses in tables 1 and 3. We notice that after resealing the difference between MIT and MIT+ REC and that between MIT+ OGE and MIT+ OGE+ REC get substantially reduced so that, with the present experimental accuracy and in view of the precision of the present theoretical framework, it is not warranted to distinguish between the results with and without the recoil correction. Henceforth, we will concentrate on the four cases, (resealed) MIT+ REC, (resealed) MIT+ OGE + REC, CBM + REC and CBM + OGE+REC. (To simplify the notation, +REC may be omitted when there is no danger of confusion.) We observe in tables 1 and 3 that f2’s for these four cases exhibit noticeable differences. For the AS=0 transitions (table l), the CBM-type models tend to give a better fitting to the experimental data; for AS = 1 transitions (table 3), the resealed fi’s for MIT appreciably differ from the other cases but the lack of experimental data prevents us from making more discriminatory comments thereupon. For the ratio f2/ f, , there is direct experimental information for some transitions. We compare in table 5 the calculated ratios with the existing data. Although the available data with large error bars are in general not selective enough, we may remark that [ fJ filz-+,, for MIT+ OGE+ REC is outside the error bar. Regarding _f?, 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 f3’s for the AS = 1 transitions (table 3) are seen to vary appreciably depending on which model one uses, and f3’s for some transitions can be rather appreciable. The rather large values of f3 are also reported in ref 33). Unfortunately, it is beyond the present experimental feasibility to test this result. It should also be mentioned “) that, for transitions in which the initial and final masses are different, the roles of fi and f3 are partially intermingled. This implies that f3 can contain some amount of “contamination” of the first-class current.

458

T. Yamaguchi

et al. / Semi-leptonic

P-decay form factors

TABLE 5 Comparison

Transitions

of calculated

MIT+ REC

3.70+ 1.77 -1.92 0.09

and experimental MIT +OGE+REC 3.70+ 1.88 -1.46 0.37

values

of fi/ f, (without

CBM + REC

3.70+ 1.26 -1.83 -0.11

CBM +OGE+REC 3.70+ 1.32 -1.55 0.05

c.m. correction)

Exp.

3.705887 *0.00000123 2.43 f 1.49 “) -1.78+0.61”) -0.44* 0.46 “)

’ Fitted to data “) Ref. I’). For the meaning of the abbreviations for the various models, see the caption of table 1. Since the differences between the cases with and without the recoil correction are not significant, we give only the results including the recoil correction.

9.1.2. Axial-vector current form factors (tables 2 and 4). A remarkable feature with the g,‘s calculated in CBM is that their magnitudes are substantially larger than those of the MIT-type models. This feature is due to the contribution of diagram (f) in fig. 1, and characteristic of the volume-type CBM; the surface-type CBM, by contrast, decreases lgrl values 26,27). For the best known quantity g, - g,(n + p), the CBM value is the closest to the experimental value. Since it may make more sense to discuss the SU(3)-symmetry property of g,‘s when gyp is reproduced reasonably well, we also give in tables 2 and 4 the values of g,‘s resealed in such a manner that gyp be reproduced; the method of resealing is the same as in ref. “). It is to be seen from the tables that, upon resealing, the g,‘s obtained in CBM are the closest to the results of the semi-empirical KM-SU(3) fitting. Thus, as far as g,‘s are

concerned, the meson clouds in the volume-type CBM apparently play the role of “restoring” KMSU(3) symmetry, which, because of the different quark masses, is lost to a certain degree in the MIT bag model. We should mention, however, that the same meson clouds play a non-trivial role in explaining SU(3) breaking in the magnetic moments of the octet baryons 2”). By comparing CBM and CBM+OGE, one can see that the gluonic effects can influence g,‘s for some transitions rather appreciably; the relative changes for those cases where g,‘s are small can amount to as much as 25%. We compare in table 6 the calculated values of g,/f, with the available experimental data. The table indicates that, as far as the overall fitting to the data on g,/f, is concerned, the results of CBM without gluonic effects is at least as good as the models with the gluonic effects. Thus, although the gluonic effects are helpful in removing the “anomaly” in T(X+n+e-+ v,)/r(A +p+e-+ v,) (see sect. ll), they do not necessarily improve overall fitting to the data on g,. Turning now to g,, which represents the strength of the second-class axial-vector current, we note that (g2)cBM+oGE can be rather appreciable for a number of transitions; in particular (gJCBM+oGE(Z+ A) = -0.22. In general, the inclusion of

T. Yamaguchi et al. / Semi-leptonic

459

p-decay form factors

TABLE 6 Comparison

of calculated

and experimental Present

Transitions MIT+ REC

1.09 (1.254+) 0.44 (0.50) -0.22 (-0.25) 0.72 (0.83) -0.23 (-0.26) -0.23 (-0.26) 0.24 (0.27)

1.20 (1.39) 1.20 (1.39)

MIT + OGE + REC 1.12 (1.254+) 0.40 (0.45) -0.27 (-0.30) 0.73 (0.82) -0.29 (-0.33) -0.29 (-0.33) 0.21 (0.24) 1.23 (1.38) 1.23 (1.38)

’ Fitted to data. ‘I) Ref. ‘). ‘) Four theoretical cases including the CBM + REC, CBM + OGE + TEC. (The from the results given here.) The entries (with) applying the resealing described

values of g,/f,

(without

c.m. correction)

work CBM+

REC

1.26 (1.254+) 0.48 (0.48) -0.27 (-0.27) 0.77 (0.77) -0.29 (~0.29) -0.29 (-0.29) 0.25 (0.25) 1.32 (1.31) 1.32 (1.31)

CBM +OGE+REC 1.29 (1.254’) 0.45 (0.44) -0.32 (-0.31) 0.78 (0.76) -0.36 (-0.35) -0.36 (-0.35) 0.22 (0.22) 1.35 (1.30) 1.35 (1.30)

Exp.

1.254ztO.006”)

0.694 f 0.025 “) 0.70*0.03 h) -0.372 zt 0.050 “) -0.3410.05 h)

0.25 1- 0.05 “) 0.25 * 0.05 h,

Ref. 2). recoil correction are shown: MIT+REC, MIT+OGE+REC, results with no recoil correction are practically indistinguishable outside (inside) the parentheses are the values obtained without in the text.

the gluonic effects tend to increase the magnitudes of g,. The present results on g, should be contrasted with those of Carson et al. 33), who found small values of g, in a calculation that uses another method for including the one-gluon exchange corrections. As can be seen from eqs. (41), (42) and (B.24) of ref. ‘), g, has an important q2 dependence due to the meson pole. Therefore, in tables 2 and 4, the calculated g,‘s are given in the form: (g3)care+ Y(q)G3, where Y(q)= (mA+m,)2u(qR)/(m~ -q2) with (Y= v and K for AS = 0 and AS = 1 transitions, respectively, and G, is given in eq. (B.24’) of ref. “). The non-pole contribution (g3)core is defined by eqs. (40a’) and (40~‘) in ref. I). The presence of g, in eqs. (8.4a) and (8.4~) does affect g, and g,, but directly observable effects of g, are too small to be of current experimental relevance even for muonic decays of the hyperons 27). To summarize, the results of the present calculations incorporating the recoil correction indicate that the main features of the weak-interaction form factors described in ref. “) need not be modified when the recoil correction is included.

460

T. Yamaguchi

et al. / Semi-leptonic

Further discussion of the consequences factors will be given in sect. 11, where experimental

9.2. MAGNETIC

P-decay form factors

of the calculated weak-interaction form direct comparisons with all the existing

data are presented.

MOMENTS

Tables 7.1 and 7.2 show the calculated magnetic moments in units of the nuclear magneton pN= e/2m,. Table 7.1 refers to the MIT-type models (MIT, MIT+ REC, MIT+OGE and MIT+OGE+ REC), while table 7.2 to the CBM-type models (CBM, CBM+ REC, CBM+OGE and CBM+ OGE+ REC). In order to demonstrate the bag radius dependence of the results, we also show the results of calculations for several choices of R around the “standard” value R = 1 fm. Although (Y, is expected to increase as R becomes larger, we have used the common value cr, = 2.2 throughout the range of R shown in the tables since the gluonic effects are not too dominant in the present treatment. As mentioned before, the CBM-type models involve an adjustable parameter R,, which in the present work has been adjusted to reproduce &(n + p)exp = 3.70. Therefore, the isovector part of the nucleon magnetic moment, CL” =i(pppn), should be automatically reproduced by all of the CBM-type models; consequently, only the isoscalar part I_L’= f(pp+p,) is a prediction of the CBM-type models. On the other hand, there is no such adjustable parameter in the MIT-type models, and the results in table 7.1 indicate that the absolute magnitudes of the magnetic moments are all too small compared with the experimental values for the whole range of R considered here; the predicted transition magnetic moment pC”_,, is also far off the experimental value. Although the situation might be improved by introducing some type of resealing so as to reproduce, say, the nucleon magnetic moments, the principle of resealing is not as clear-cut as in the case of f2’s*. This is because the electromagnetic currents contain both the isoscalar and isovector components. We therefore will not try here to improve agreement with the data by introducing resealing. On the other hand, since table 7.1 does not involve resealing, the differences between the four cases of the MIT-type models give direct information on changes due to the recoil correction as well as the gluonic contribution. Thus, setting aside the afore-mentioned problem concerning the absolute values, we consider it worthwhile to point out the following two aspects of table 7.1: (i) The recoil correction universally reduces 1~1 by 5-10%. (ii) The gluonic effects have no influence on pLp but increases 1~1’s for the other charged particles; [PI’S for the neutral particles are reduced by the gluonic effects. There is a general argument 8,34) that, in the SU(3) limit, we can parametrize the gluonic contribution to the magnetic moments of the octet baryons in terms of one reduced matrix element, Y, whose magnitude and sign are calculable in the framework of the one-gluon exchange model. Thus, if we l Of course, this does not imply that the resealing absolutely profound meaning.

of fi’s employed

in the previous

subsection

has an

461

T. Yamnguchi et al. / Semi-lepronic p-decay form factors TABLE 7.1 Magnetic

moments

for “MIT-type”

models

R (fm) Exp.

Models 0.8

0.9

1.1

1.2

1.54

1.73

1.92

2.12

2.3 1

2.7928444

1.37

1.56

1.75

1.94

2.14

*0.0000011

MIT+OGE

1.54

1.73

1.92

2.12

2.31

MIT+OGE+REC

1.37

1.56

1.75

1.94

2.14

MIT MIT+

REC

-1.03

-1.15

-1.28

-1.41

-1.54

-1.91304308

-0.91

-1.04

-1.17

-1.30

-1.42

*0.00000054

MIT+OGE

-0.91

-1.03

-1.14

-1.26

-1.37

MIT+OGE+REC

-0.80

-0.91

-1.03

-1.14

-1.25

MIT MIT+

REC

1.51

1.69

1.87

2.06

2.24

1.38

1.56

1.74

1.93

2.11

2.379+

MIT+OGE

1.53

1.72

1.91

2.10

2.30

2.479 f 0.025 “)

MIT+OGE+REC

1.40

1.59

1.78

1.97

2.17

MIT MIT+

REC

0.48

0.54

0.59

0.65

0.70

0.44

0.50

0.55

0.6 1

0.66

MIT+OGE

0.43

0.48

0.54

0.59

0.64

MIT+OGE+REC

0.39

0.44

0.50

0.55

0.60

MIT MIT+

REC

0.020

*

-0.84

-0.55

-0.62

-0.69

-0.76

MIT+

REC

-0.50

-0.57

-0.64

-0.71

-0.79

-1.141*0.051

MIT+

OGE

-0.67

-0.75

-0.84

-0.93

-1.02

-1.166*0.017

MIT+OGE+REC

-0.62

-0.70

-0.79

-0.88

-0.97

MIT

MIT

-0.90

-0.99

-1.08

-1.17

-1.26

MIT+

REC

-0.83

-0.93

-1.02

-1.11

-1.20

MIT+

OGE

-0.79

-0.88

-0.96

-1.04

-1.11

MIT+OGE+REC

-0.73

-0.81

-0.90

-0.98

-1.06

MIT

-0.38

-0.42

-0.44

-0.47

-0.49

MIT+

REC

-0.36

-0.39

-0.42

-0.45

-0.47

MIT+

OGE

-0.46

-0.50

-0.53

-0.56

-0.59

MIT+OGE+REC

-0.44

-0.48

-0.51

-0.54

-0.57

MIT

-0.42

-0.46

-0.49

-0.53

-0.56

-0.38

-0.42

-0.50

-0.53

MIT+OGE

-0.36

-0.39

-0.46 -0.42

-0.45

-0.48

MIT+OGE+REC

-0.33

-0.36

-0.39

-0.42

-0.45

MIT

-0.89

-1.00

-1.11

-1.22

-1.33

-0.81

-0.92

-1.03

-1.14

-1.25

MIT+OGE

-0.80

-0.90

-1.00

-1.10

-1.20

MIT+OGE+REC

-0.72

-0.82

-0.92

-1.02

-1.12

MIT+

MIT+

“) Wilkinson

REC

REC

et al. 3).

For the explanation

b,

Zapalac

et al. ‘).

of the four theoretical

‘)

Peterson

cases, MIT,

REC, see the caption of table 1. The transition magnetic radius R is changed from 0.8 to 1.2 fm. The experimental stated.

1.0

MIT+

h,

-1.250+0.014

-0.693

* 0.040

-0.613

kO.004

-1.59*0.09

‘)

et al. 3). REC,

MIT+OGE,

and MIT+OGE+

moment for 8O-t A is also given. The bag data are taken from ref. ‘) unless otherwise

T. Yumaguthi

462

Magnetic

et at. / Semi-ieptonic

moments

p-decay form facror.9

for “CBM-type”

models

R (fm) Models

Exp. 0.8

0.9

1.0

1.1

1.2

CBM

2.62

2.64

2.67

2.68

2.70

2.7928444

CBM + REC

2.62

2.64

2.66

2.70

*0.0000011

CBM+OGE

2.69

2.71

2.75

2.68 2.77

CBM+OCE+REC

2.68

2.71

2.74

2.77

2.79

2.80

CBM

-2.09

-2.06

-2.04

-2.01

-1.99

-1.91304308

CBM + REC

-2.09

-2.06

-2.04

-2.02

-2.00

*0.00000054

CBM+OGE

-2.02

-1.98

-1.95

-1.92

-1.89

CBM+OGE+REC

-2.03

-1.99

-1.96

-1.93

-1.90

CBM

2.32

2.39

2.44

2.49

CBM+REC

2.33

2.40

2.46

2.51

2.55

2.379 f 0.020

CBM+OGE

2.41

2.49

2.56

2.62

2.68

2.479 f 0.025 “)

CBM+OGE+REC

2.42

2.50

2.58

2.64

2.70

2.53

0.69

CBM

0.58

0.62

0.64

0.67

CBM + REC

0.59

0.62

0.65

0.67

0.69

CBM+OGE

0.55

0.58

0.61

0.63

0.65

CBM+OGE+REC

0.56

0.61

0.59 -1.15

-1.15

-1.16

-1.16

-1.30

-1.32

CBM+OGE+REC

-1.30

CBM CBM + REC CBM+OGE

CBM

-1.15

CBM+REC CBM+OGE

CBM + OGE+

REC

0.64

0.66 -1.14

-1.17

-1.16

-1.16

-1.141*0.051

-1.34

-1.36

-1.37

-1.166*0.017h)

-1.33

-1.35

-1.37

-1.39

-1.25

-1.29

-1.32

-1.35

-1.36

-1.21

-1.31

-1.34

-1.37

-1.38

-1.19

-1.22

-1.25

-1.26

-1.27

-1.20

-1.24

-1.27

-1.28

-1.29

CBM

-0.46

-0.47

-0.49

-0.49

-0.49

CBM - REC

-0.46

-0.48

-0.50

CBMtOGE

-0.55

-0.58

-0.50 -0.60

-0.60

-0.50 -0.61

CBM+OGE+REC

-0.56

-0.59

-0.61

-0.62

-0.62

CBM

-0.59

-0.60

-0.60

-0.60

-0.60

CBM + REC

-0.59

-0.60

-0.61

-0.61

-0.61

CBM+OGE

-0.55

-0.55

-0.56

-0.55

-0.55

CBM+OGE+REC

-0.56

-0.56

-0.57

-0.56

-0.56

-1.25OztO.014

-0.693 f 0.040

-0.613

CBM

-1.57

-1.58

-1.59

-1.60

-1.60

CBM + REC

-1.58

-1.60

-1.61

-1.61

CBM+OGE

-1.52

-1.59 -1.52

-1.53

-1.53

- 1.53

CBM+OGE+REC

-1.53

-1.53

-1.54

-1.54

-1.54

12

10

8

9

9

12

9

9

9

10

CBM+REC CBM+OGE

8

8

9

12

16

8

8

10

13

1s

For the explanation see the caption

radius

R is changed

of the four theoretical of table from

of the experimental

cases, CBM,

1. The transition

data, see table 7.1.

CBM + REC, CBM +OGE, and CBM +OCE+

magnetic

0.8 to 1.2 fm. The “effective”

*0.004

-1.59*o.09c)

CBM+OGE+REC

REC,

*

-1.15

Total X2 CBM

sources

-

moment xz value,

for .Z’+A i’,

is also given.

is detined

The bag

in the text. For the

T. Yamague~i et al. / Semi-lepfonie p-decay form facrors

463

write its contribution as WY, w turns out to be 0, 3, +, 0, -4, -3, $ and -3 for p, n, A, Z+, E”, I-, E* and K, respectively; furthermore, Y - +0.2 (see table 10 of ref. “)). The above-mentioned understood in terms of this general amount

of deviation

features concerning the gluonic effect can be argument based on the exact SU(3) and a certain

from it.

Regarding table 7.2 for the CBM-type models, we emphasize again that, due to the resealing, the difference between CBM and CBM+OGE does not directly represent the effect of gluonic contributions. Similarly, the difference between CBM and CBM+REC does not directly correspond to the recoil correction, but, like in the case of the weak-interaction form factors, the difference between these two cases are practically negligible anyway. In what follows, therefore, we largely concenrate on the two cases, CBM and CBM + OGE. This means that some of the conclusions given below are the restatement of those in ref. “). We first deal with the two welI-known problems concerning the magnetic moments. The first one, which for short may be called the K - A problem, is that the additive quark picture invariably predicts ,u,,, < pg- whereas, experimentally, p,I > ps[ref. ““)I. Table 7.2 indicates that the non-additivity brought about by CBM cannot resolve this problem. By contrast, the inclusion of the gluonic effect can give y,% > pZ-, a remarkable feature first discovered by Ushio 36). We note, however, that, even if the gluonic effects are included, pn > EL,-- is never realized below R = 0.7 fm. The second problem is that the rather sizeable isovector nucleon magnetic moment due to mesonic (largely pionic) contributions destroys the successful prediction pJ+,,= -$ of the MIT bag model. Hprgaasen and Myhrer34) pointed out that the inclusion of the gluonic effect is also helpful in restoring this ratio. The present calculation demonstrates that these features survive the recoil correction. Now, apart from these desirable features of the calculations including the gluonic effects, it seems important to discuss the quality of overall agreement with the data. For this purpose

we give in table 7.2 the ‘“effective”

,$ value. The reason

why we

use here the “effective” x2 instead of the ordinary x2 value is that the achieved precision of experimental data for some cases is excessively (!!) high for the quality of available theoretical frameworks. Consequently, the use of the ordinary x2 value will not be too realistic. Therefore, in order to obtain semi-quantitative measure of the quality of general agreement with experiment, we arbitrarily assign a universal error bar of 0.10 II++, while retaining the experimental central values. The x2 value corresponding to this artificial case is called the effective x2 value, and denoted by 2”. By looking at iZ in table 7.2, we observe that the inclusion of the gluonic effects sometimes improves but sometimes worsens the overall agreement with the data, depending on the bag radius R; the changes, however, are never too large. Thus, the above-described phenomenological successes notwithstanding, the gluonic contributions are not particularly useful in reproducing the overall behavior of the octet magnetic moments. It is also to be noted that despite “generous” error bars arbitrarily assigned to the experimental data, the resulting 2’ values for all the cases treated

T. Yamaguchi

464

here are rather

et al. / Semi-leptonic P-decay form factors

large. As far as the R-dependence

R = 1.0 fm is not an unreasonable values of R are marginally favored We now compare

the present

is concerned,

choice although, for CBM, with slightly smaller i’.

results

with those of Brown

we may say that somewhat

et al. 23) based

smaller on the

chiral bag model. These authors added the gluonic effects to the chiral bag model and could reproduce the magnetic moments of the hyperons amazingly well for the choice of the bag radius R = 1.1 fm. Although the chiral bag model 37738)with the non-linearity of meson fields fully taken into account is conceptually very different from the linearized CBM used in the present work, the large value of R is expected to substantially reduce the difference. On the other hand, the treatment of ref. 23) involves one adjustable parameter, the strange-quark magnetic moment, which is adjusted to reproduce ~7;” *. Thus, we consider the difference between our results for CBM+OGE (and also CBM+OGE+ REC) and those of ref. 23) mainly due to the phenomenological fitting of ~7” involved in the latter.

10. Numerical results for the weak interaction form factors with c.m. corrections Although the c.m. correction 13-r9) may influence the pattern of SU(3) symmetry breaking, we have chosen to defer its discussion up to now. This is partly because the treatment of the c.m. correction is, as mentioned in sect. 1, is a highly delicate problem so that, in order to highlight characteristic features of dynamical effects such as mesonic and gluonic contributions, it may be convenient to deal with the c.m. correction separately. Another reason is that the inclusion of the c.m. correction, as calculated here, turns out to lead to an extremely poor agreement with data. In this situation, one may be tempted to leave out the c.m. correction altogether. We present here, however, the results of our calculations including the c.m. correction. This is partly for the sake of completeness, but our additional intention is to present a concrete calculation that will serve to demonstrate the delicate nature of the problem, which should be kept in mind in any microscopic model calculations of this type. We emphasize again that our treatment of c.m. correction is nothing more than one possible method. Tables 8- 11 give the weak-interaction form factors calculated by taking account of the c.m. correction with the method described in sect. 8; tables 8 and 10 correspond to the AS = 0 transitions, while tables 9 and 11 to the AS = 1 transitions. The vector-current form factors, fr, f2 and f3, are given in tables 8 and 10, whereas the axial-vector form factors, g, , g, and g,, in tables 9 and 11. All the form factors except for g, represent the values at q2 = 0. The eight theoretical cases considered in tables 8-11 are the same as those considered in tables l-4 except that they now include the c.m. correction; the qualifier “+ CM” in the abbreviations for the models * The present CBM calculation also involves one adjustable parameter R,, but this was already “used up” to fit a quantity in the non-strange sector (fi(n + p)); thus, as far as the strange sector is concerned, our treatment is parameter-free.

465

T. Yamaguchi et al. / Semi-leptonic P-decay form factors TABLE 8 Vector current Transitions

form factors

for AS = 0 (with c.m. correction)

Models

fi

MIT+ c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC~c.m. CBM + em. CBM+REC+c.m. CBM + OGE + em. CBM+OGE+REC+c.m. exp.

1.oo 1 .oo 1.oo 1.oo 1.00 1.oo 1.00 1.oo *

MlT+c.m. MIT+REC+c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + cm. CBM+REC+c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m. exp.

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 *

MIT+c.m. MIT+ REC + cm. MIT+ OGE + cm. MIT+OGEI REC+c.m. CBM + c.m. CBM + REC + c.m. CBM + OGE + c.m. CBM+OGE+REC+c.m. exp.

1.41 1.41 1.41 1.41 1.41 1.41 1.41 1.41 *

MIT+c.m. MIT+ REC + cm. MlT+OGE+c.m. MIT+OGE+REC+c.m. CBM +c.m. CBM+REC+c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m. exp.

-1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 -1.00 *

x 1.73 1.44 1.59 1.30 3.70 3.70 3.70 3.70 3.705887 i

h

(3.70) (3.70) (3.70) (3.70)

1.23 x 1O-6

1.64 (2.83) 1.50 (2.89) 1.45 (2.63) 1.3 1 (2.67) 2.75 2.77 2.64 2.66 2.77rtO.15 “)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 * -0.06 -0.05 -0.05 -0.04 -0.04 -0.04 -0.04 -0.04 *

It 0.05

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 *

1.77 (2.32) 1.71 (2.36) 1.46 (1.84) 1.40 (1.83) 2.17 2.1s 1.90 1.91 1.78*0.06

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 *

0.55 0.38 0.72 0.56 1.81 1.83 2.08 2.11 1.75

(1.96) (2.05) (2.46) (2.62)

The calculated values off, ,fz and .& are given for eight theoretical eases: (1) MIT- MIT bag model; (2) MIT+REC-MIT bag mode1 with recoil correction; (3) MIT+OGEMIT bag model plus onegluon exchange effects; (4) MIT+OGE+RECMIT bag model plus one-gluon exchange effects with recoil correction; (5) CBM -volume-type CBM with octet meson clouds; (6) CBM + REC - volume-type CBM with octet meson clouds with recoil correction; (7) CBM+OGE-volume-type CBM with octet meson clouds plus one-gluon exchange effects; (8) CBM + OGE+ REC - volume-type CBM with octet meson clouds plus one-gluon exchange effects including recoil correction. The qualifier “+c.m.” at the end of each abbrev~at;on means that the models now include the cm. correction. For the MIT-type models, the resealed values of fi are given in the parentheses, the method of resealing described in the text. The value of R, used here is: R, = 0.413 fm (CBM +c.m.), R, = 0.386 fm (CBM + REC+ cm.), R, =0.394fm (CBM+OGE+c.m.), and R, =0.370 fm (CBM+OGE+REC+c.m.).

466

T. Yamaguchi

et al. / Semi-leptonic

P-decay form factors

TABLE 9 Axial-vector Transitions *+p

current

form factors

for AS = 0 (with c.m. correction)

Models

g1

g2

g3

MIT+ c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM+c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM + OGE + REC + c.m. exp.

1.32 (1.254) 1.32 (1.254) 1.35 (1.254) 1.35 (1.254) 1.42 (1.254) 1.42 (1.254) 1.46 (1.254) 1.46 (1.254) 1.254 * 0.006

0.00 0.00 0.00 0.00

2.85 2.86 3.02 3.02 0.91+ 1.36 Y 0.91+ 1.36 Y 1.07+1.36Y 1.08+ 1.36Y *

MIT+c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + c.m. CBM+REC+c.m. CBM + OGE + c.m. CBM+OGE+REC+c.m.

0.64 0.64 0.67 0.67 0.71 0.71 0.75 0.75

(0.61) (0.61) (0.62) (0.62) (0.63) (0.63) (0.64) (0.64)

MIT+c.m. MIT+REC+c.m. MIT+ OGE + c.m. MIT+OGE+REC+c.m. CBM + c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

0.74 0.74 0.70 0.70 0.78 0.78 0.74 0.74

(0.71) (0.71) (0.65) (0.65) (0.69) (0.69) (0.63) (0.63)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

2.84 2.87 2.68 2.72 1.33+0.77y 1.36+0.77 Y 1.18+0.77Y 1.20+0.77Y

MIT+ c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM + OGE+ REC + c.m.

0.26 0.26 0.31 0.31 0.31 0.31 0.35 0.35

(0.25) (0.25) (0.29) (0.29) (0.27) (0.27) (0.31) (0.31)

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.25 1.27 1.89 1.91 0.50+0.27Y 0.51+ 0.27 Y 1.15+0.27Y 1.14+0.27Y

For the meaning of the abbreviations for the various of Y in the column for g, is: Y - (m,+ me)‘u(qR)/(mi

0.00 0.00

0.00 0.00 * -0.10 -0.08 -0.17 -0.15 -0.15 -0.14 -0.22 -0.21

models, see the caption - q2).

2.28 2.30 2.64 2.67 0.87 + 0.67 Y 0.89 + 0.67 Y 1.24 + 0.67 Y 1.25+0.67Y

of table 8. The definition

stands for the c.m. correction. As for g,‘s and f2’s, we also give the resealed values, the method of resealing being the same as in the previous section. The first comment on the results is that f,‘s remain practically unaffected by the inclusion of the cm. correction; this should be so because the vector charge should not be influenced by any kinematical corrections. Turning now to the non-trivial cases of g,‘s, we notice that, when the cm. correction is included, the magntidue of g, = g,(n + p) for all the eight cases largely overshoots

T. Yamaguchi et al. / Semi-leptonic p-decay TABLE Vector

current form factors

form

factors

467

10

for AS = 1 (with c.m. correction)

Models

fi

fi

f3

MIT+ cm. MIT+ REC+c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

-1.22 -1.21 -1.23 -1.21 -1.22 -1.22 -1.22 -1.22

-0.71 (-2.11) -0.54 (-2.17) -0.68 (-2.26) -0.52 (-2.34) -1.50 -1.51 -1.58 -1.59

-0.07 -0.09 -0.08 -0.10 0.20 0.20 0.21 0.20

MIT+c.m. MIT+ REC + c.m. MIT+ OGE + c.m. MIT+OGE+REC+c.m. CBM+c.m. CBM+REC+c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

-1.03 -1.01 -1.02 -1.01 -1.02 -1.03 -1.02 -1.02

1.57 1.51 1.26 1.20

(1.96) (1.97) (1.45) (1.40) 1.86 1.87 1.59 1.58

-0.27 -0.30 -0.24 -0.28 -0.02 -0.05 -0.04 0.00

MIT+ cm. MIT+REC+c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM+c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

-0.73 -0.72 -0.72 -0.72 -0.72 -0.73 -0.72 -0.72

1.11 1.07 0.89 0.85

(1.39) (1.39) (1.03) (0.99) 1.32 1.32 1.12 1.12

-0.19 -0.21 -0.17 -0.20 -0.01 -0.04 0.00 -0.02

Transitions

MIT+c.m. MIT+REC+c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM+c.m. CBM+REC+c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

1.21 1.22 1.23 1.22 1.22 1.22 1.22 1.22

-0.47 (0.08) -0.52 (0.14) -0.32 (0.44) -0.36 (0.54) -0.15 -0.14 0.04 0.05

0.22 0.23 0.21 0.22 -0.07 -0.06 -0.07 -0.06

MIT+c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBMfc.m. CBM + REC + cm. CBM+OGE+c.m. CBM+OGE+REC+c.m.

1.oo 0.98 1.00 0.98 0.99 0.99 0.99 0.99

2.23 (4.56) 1.99 (4.75) 2.06 (4.56) 1.82(4.75) 3.48 3.52 3.47 3.52

0.08 0.06 0.08 0.06 -0.15 -0.17 -0.16 -0.18

MIT+c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM+c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM + OGE + REC + c.m.

0.71 0.69 0.71 0.69 0.70 0.70 0.70 0.70

1.58 (3.23) 1.41 (3.36) 1.46 (3.22) 1.29 (3.36) 2.46 2.49 2.45 2.49

0.05 0.04 0.06 0.04 -0.11 -0.12 -0.11 -0.13

For the meaning

of the abbreviations

for the various

models,

see the caption

of table 8.

468

T. Yamaguchi et al. / Semi-leptonic P-decay form factors TABLE

Axial-vector Transitions

current

form factors

11 for AS = 1 (with c.m. correction)

Models MIT+ c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM-OGE+REC+c.m.

-1.02 -1.01 -1.03 -1.03 -1.04 -1.04 -1.06 -1.06

g1

g2

(-0.97) (-0.97) (-0.96) (-0.96) (-0.92) (-0.92) (-0.91) (-0.91)

0.01 0.03 0.01 0.02 0.05 0.06 0.05 0.06

g3 -1.87 -1.79 -1.83 -1.75 -0.55 -0.99 -0.50 - 0.99 -0.5 1 - 0.99 -0.45 - 0.99

Y Y Y Y

Y Y Y Y

MIT+c.m. MIT+ REC + c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + cm. CBM+REC+c.m. CBM+OGE+c.m. CBM + OGE + REC + c.m.

0.27 0.27 0.34 0.34 0.33 0.33 0.39 0.39

(0.26) (0.26) (0.31) (0.31) (0.29) (0.29) (0.34) (0.34)

-0.03 -0.03 -0.04 -0.04 -0.00 -0.00 -0.01 -0.01

0.55 0.53 0.85 0.83 0.01 + 0.27 0.01 + 0.27 0.30+ 0.27 0.29 + 0.27

MIT+c.m. MIT+ REC + cm. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM+c.m. CBM + REC + c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

0.19 0.19 0.24 0.24 0.23 0.23 0.28 0.28

(0.18) (0.18) (0.22) (0.22) (0.20) (0.20) (0.24) (0.24)

-0.02 -0.02 -0.03 -0.03 -0.00 0.00 -0.01 -0.01

0.39 0.38 0.60 0.59 0.00+0.19Y -0.01+0.19 Y 0.21+0.19Y 0.20+0.19Y

MIT+ c.m. MIT+REC+c.m. MIT+OGE+c.m. MIT+ OGE + REC + c.m. CBM+c.m. CBM + REC + c.m. CBM + OGE + c.m. CBM + OGE + REC + c.m.

0.34 0.34 0.31 0.31 0.34 0.34 0.31 0.31

(0.32) (0.32) (0.29) (0.29) (0.30) (0.30) (0.27) (0.27)

-0.01 -0.02 -0.04 -0.05 -0.06 -0.07 -0.09 -0.10

0.94 0.92 0.73 0.71 0.42 + 0.33 Y 0.40+ 0.33 Y 0.21+0.33 Y 0.19+0.33y

MIT+ c.m. MIT+REC+c.m. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + cm. CBM+REC+c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

1.39 1.39 1.42 1.42 1.45 1.45 1.48 1.48

(1.32) (1.32) (1.32) (1.31) (1.28) (1.28) (1.28) (1.27)

0.12 0.05 0.20 0.13 0.11 0.06 0.19 0.14

4.17 4.06 4.38 4.27 1.60+ 1.36Y 1.51+1.36Y 1.80+ 1.36Y 1.72+ 1.36Y

MIT+ c.m. MIT+ REC + cm. MIT+OGE+c.m. MIT+OGE+REC+c.m. CBM + c.m. CBM+REC+c.m. CBM+OGE+c.m. CBM+OGE+REC+c.m.

0.98 0.98 1.00 1.00 1.03 1.02 1.05 1.05

(0.94) (0.93) (0.93) (0.93) (0.91) (0.90) (0.90) (0.90)

0.08 0.04 0.14 0.09 0.08 0.04 0.13 0.10

2.95 2.87 3.10 3.02 1.13+0.96Y 1.07 + 0.96 Y 1.28 + 0.96 Y 1.21+0.96Y

For the meaning of the abbreviations for the various models, see the caption of Y in the column for g, is: Y = (m,+ n~a)~u(qR)/(m~ - q*).

of table 8. The definition

T. Yamnguchi

et al. / Semi-leptonic

469

P-decay form factors

TABLE 12 Comparison

of calculated

and experimental Present

Transitions

n+p

P-+X0

s-

+ s=‘o

A-+P

MIT REC + c.m.

MIT+ OGE + REC + c.m.

values of g,/f,

(with c.m. correction)

work CBM + REX + cm.

CBM+OGE +REC+c.m.

1.32 (1.254’)

1.35 (1.254+)

1.42 (1.254+)

1.46 (1.254’)

0.53 (0.50)

0.49 (0.46)

0.55 (0.49)

0.52 (0.45)

-0.26 (-0.25) 0.84 (0.81)

-0.31 (-0.29) 0.85 (0.80)

-0.3 1 (-0.27) 0.85 (0.75)

Exp.

1.254*0.006 “) 1.239 + 0.009 h,

-0.35 (-0.31) 0.86 (0.74)

X-n

-0.27 (-0.26)

-0.34 (-0.31)

-0.32 (-0.28)

-0.38 (-0.33)

BQP

-0.27 (-0.25)

-0.34 (-0.31)

-0.32 (-0.28)

-0.38 (-0.33)

=--+A

0.28 (0.26)

0.25 (0.24)

0.28 (0.24)

0.25 (0.22)

Ba+P+

1.41 (1.36)

1.44 (1.35)

1.46 (1.29)

1.49 (1.29)

E-+20

1.41 (1.36)

1.44 (1.35)

I .46 (1.29)

1.49 (1.29)

0.694iO.025 “) 0.70 * 0.03 b) -0.372 f 0.050 “) -0.34* 0.05 b)

0.25 f 0.05 “) 0.25 + 0.05 b,

+ Fitted to data. “) Ref. 1). b, Ref.*). Four theoretical cases including the recoil correction are shown: MIT+ REC+c.m., MIT+OGE+ REC + c.m., CBM + REC + c.m., CBM + OGE + REC + cm. Since the difference between the cases with and without the recoil correction is not significant, we give only the results with the recoil correction. The entries outside (inside) the parentheses are the results excluding (including) the resealing. The method of resealing is described in the text.

gyp = 1.254; thus, the c.m.-corrected g,‘s without resealing are a disaster. (A direct analysis in sect. 11 of the experimental data on the beta-decay observables also indicates that the use of the form factors calculated with cm. correction included but with no resealing leads to a disasterous situation.) We will therefore discuss only the resealed results. It is remarkable that, after resealing, each entry for the g,‘s in tables 9 and 11 is practically identical to its counterpart in tables 2 and 4. This feature can be attributed to the fact that the effect of the c.m. correction on g,‘s can be simulated reasonably well by multiplying them with an oveall constant. In table 12, we show the ratio g,/f, with c.m. corrections. Since these results are quite similar to those in table 6, the remarks we made on table 6 apply here too.

470

T. Yamaguchi

et ai. / Semi-leptonic

p-decay form factors

TABLE 13 Comparison

Transitions

n-tp A’P Z--+n z-+,4

of calculated

MIT + REC + c.m.

and experimental

MIT+ OGE +REC+c.m.

values of ,Q’f,

(with cm. correction)

CBM + REC + c.m.

CBM + OGE + REC + cm.

3.70’

3.70+

3.70’

3.70’

1.82

1.96

1.24

1.30

-1.93 0.11

-1.38 0.45

-1.82 -0.12

-1.55 0.04

Experiment

3.705887 *0.00000123 2.43 + 1.49 “) -1.78i0.61

“)

-0.44 zt 0.46 “)

’ Input data. “) Ref. I’). For the meaning of the abbreviations for the various models, see the caption of table 8. Since the difference between the cases with and without the recoil correction is insignificant, we only give the results with the recoil correction.

For fi’s also, we notice that the values after resealing are rather close to the corresponding values in tables 1 and 3. Table 13 shows the ratio f2/f, for the cm.-corrected and resealed cases; this should be compared with table 5. Here again, the large experimental errors keep us from making definite conclusions on the relative merit of the models. We therefore leave table 13 simply by stating that [fz/f,]s-+n for MIT+OGE+REC+CM is definitely outside of the experimental error bar.

11. Direct comparison with experimental data In order to study the systematics

of the weak-interaction

form factors of the octet

baryons without intermediate steps such as Cabibbo-SU(3) fitting or KM-SU(3) fitting, it is desirable 39) to evaluate various beta-decay observables using the calculated form factors and make direct comparisons with the experimental data. In this section, we will describe such an attempt. The general importance of this type of direct comparison was emphasized by Garcia and Kielanowski 39V4o),and an example of such analyses in the context of the quark model was presented by Beyer and Singh 24). There are two categories of beta-decay observables. To the first category belong correlation quantities that are independent of the overall strength of the currents, involving only information on the relative strengths of the weak interaction form factors. To the second category belong decay rates that depend not only on the form factors but also on the Kobayashi-Maskawa (KM) parameters, Vu, and V,, [ref. 7)]. In the present case, since all the form factors are already given through microscopic model calculations, there are definite predictions for correlation quantities to be directly compared with experiment, Formulae necessary for expressing correlation quantities in terms of the form factors can be found in the monograph

T. Yamaguchi

by Garcia

and Kielanowski

only as a function a set of calculated

et al. / Semi-leptonic

&decay

39)_ On the other hand,

471

form factors

a decay rate I’ can be predicted

of the KM parameters, V,, and V,,, but this means that, given weak-interaction form factors, one can use the data on the decay

rates to determine the best-fit values of the KM parameters. We will here carry out least-squares fitting to the experimental T’s and will determine the optima1 values the weak interacting form factors with decay of V,,, and V,,. Formulae connecting rates including radiative corrections are given in ref. 39); supplementary information on radiative corrections can be found in ref. 4’). The quality of overall fitting to data of the present microscopic calculations can be measured in terms of the total x2 value corresponding to twenty-six available data: seven data for the decay rates (see the column labeled Exp. in table 16) and nineteen data for correlation quantities and for gr/f, (see the column labeled Exp. in table 18). We give in tables 14.1, 14.2 and 14.3 the total ,$ value and its decomposition into individual contributions. The integrated correlation quantities appearing in the tables are defined in appendix B. The x2 values coming from the decay rates represent the values resulting from least-squares fitting to rexprs; the corresponding best-fit values of the KM parameters, V,, and V,,, will be discussed later. Table 14.1 refers to the models without c.m. correction, while table 14.2 to the models with c.m. correction. In both tables we have used the resealed form factors*. Since, as mentioned earlier, the results with and without recoil correction are quite similar, we show the results only. for the cases, MIT+ REC, MIT+ OGE + REC, CBM+ REC and CBM + OGE+ REC (see the beginning of sect. 8 for the explanation of these abbreviations). For the sake of comparison, the results of KM-SU(3) fitting are also shown. The x2 values have been calculated for the experimental values compiled in Particle Data ‘). Looking at tables 14.1 and 14.2, one will notice that the CBM-type models give appreciably smaller total x2 values than the MIT-type models; this holds for both the c.m.-corrected and c.m.-non-corrected cases. The total x2 values for the CBMtype models are comparable to that of the semi-empirical KM-SU(3) fitting. Between the cases

of the CBM-type

models,

there

is no significant

difference

between

CBM + REC and CBM +OGE+ REC; thus, as far as the overall fitting to the existing data are concerned, the gluonic effects do not play a particularly significant role. Finally, although it is amusing to note that the total x2 values for all the cases (with cm. correction) in table 14.2 are smaller than their respective counterparts (without cm. correction) in table 14.1, we probably should not attach too much significance to it. In order to illustrate the importance of resealing, we give in table 14.3 the results for the cases (without c.m. correction) in which no resealing has been applied. The general increase in the total x2 value is enormous, except for the case of CBM + REC. Therefore, from now on, we shall only be concerned with the resealed cases. In

l

Since the effects

of fX and g, are of no practical

relevance,

we have dropped

these two terms.

472

T. Yamaguchi et al. / Semi-leptonic

p-decay

form

factors

TABLE 14.1 Decomposition

Transitions

Total x2 For the difference the results B and g,/ A, and B,

of total x2 into individual

contributions

(without

MIT + OG E + REC

c.m. correction,

with resealing)

CBM + REC

CBM +OGE+REC

SU(3)

MIT+ REC

0.32 0.10 0.02 0.14 0.00

0.21 0.10 0.02 0.14 0.00

0.10 0.02 0.14 0.00

0.98 0.10 0.02 0.14 0.00

2.11 0.10 0.02 0.14 0.00

0.07 0.13

0.01 0.15

0.01 0.16

0.01 0.16

0.07 0.18

9.75 0.03 0.14 0.24

1.32 0.08 0.07 0.25

4.81 0.12 0.11 0.27

5.15 0.11 0.08 0.26

9.91 0.16 0.11 0.28

0.99 4.26 2.93 7.33 1.72 2.14

5.14 17.7 6.10 8.64 0.97 30.0

2.23 15.7 5.89 8.55 0.95 24.2

1.85 8.66 3.08 7.73 2.17 8.29

0.16 7.48 2.92 7.60 2.17 6.15

4.11 0.10 0.00 1.25

28.3 0.51 0.37 4.84

9.50 0.42 0.06 0.81

8.18 0.04 0.07 2.56

0.59 0.98 0.16 0.23

1.37 0.61 1.39 0.73

2.04 0.02 0.01 0.19

2.89 0.07 0.3 1 0.06

1.44 0.04 0.24 0.01

2.19 0.31 0.91 0.46

0.06

0.01

0.02

0.05

0.09

40

107

78

51

45

0.92

meaning of the abbreviations for the various models, see the caption of table 1. Since the between the cases with and without the recoil correction is practically negligible, we give only with the recoil correction. The eight physical observables, r (decay rate), LY,,, Q,, a,, B’, A, f, , are considered. The definition of the integrated correlation observables, aeV, cy,, (Y,, B’, is given in appendix B.

table 15 are given those values of the KM parameter, Vu, and V,,, which minimize the x2 value for the decay rates. (We repeat that the correlation quantities are independent of the KM parameters.) The corresponding x2 values contributed only by the data on T’s are also given in the table. Row 2 represents the results of KM-SU(3) fitting described in sect. 9. Rows 3-6 correspond to the cases without cm. correction which appear in table 14.1; rows 7-10 correspond to the cases with c.m. correction that appear in table 14.2. Two sets of experimental data are used:

T. Yamaguc~i et

al. /

Semi-ie~tonjc

p-decay

formfactors

473

TABLE 14.2 Decomposition

of total ,$ into individual

contributions

(with cm. correction,

with resealing)

MIT + REC + c.m.

MIT+OGE + REC + cm.

CBM + REC + cm.

CBM+OGE + REC + cm.

0.22 0.10 0.02 0.14 0.00

0.75 0.10 0.02 0.14 0.00

1.12 0.10 0.02 0.14 0.00

2.17 0.10 0.02 0.14 0.00

r %X.

0.01 0.15

0.01 0.16

0.02 0.16

0.08 0.18

r ffe* A B

1.31 0.08 0.07 0.25

4.00 0.12 0.12 0.27

5.79 0.11 0.08 0.26

10.2 0.16 0.11 0.28

1.10 14.0 5.38 8.44 1.09 21.2

2.04 12.6 5.34 8.37 1.01 17.6

1.54 6.84 2.66 7.50 2.35 5.57

0.16 5.90 2.54 7.38 2.33 4.00

26.1 0.76 0.49 5.48

8.69 0.07 0.01 1.54

6.75 0.15 0.13 3.24

0.67 0.40 0.06 0.67

1.90 0.00 0.03

2.18 0.08 0.34

0.08

1.07 0.06 0.28 0.02

1.66 0.30 0.87 0.44

0.07

0.04

0.06

0.10

88

65

46

40

Transitions -._

r %” *, a” Slifi

r %” % *, B’ g,/f, r (Yeri a,

St/f, r %” A

g,lfl r Total x2

0.08

For the meaning of the abbreviations for difference between the cases with and without the results with the recoil correction. The eight and g,/f, , are considered. The definition of and B, is given in appendix B.

the various models, see the caption of table 1. the recoil correction is practically negligible, we physical observables, F (decay rate), or,,,, cre, a,, the integrated correlation observables, a,,, oer

Since the give only B’, A, B, (Y,, B’, A,

(a) the decay rates given in Particle Data ‘), and (b) the CERN-WA2 data ‘). The most important difference between these two sets is that the former includes the data on the neutron decay. We first discuss case (a). It is noteworthy that the values of the KM parameters are stable against the change of microscopic models, and the results can be simply summarized as: V,, = 0.972 f 0.004 and V,, = 0.2 15 f 0.004 for the eight cases given in table 15. These values are very close to those resulting from the semi-empirical KM-SU(3) fitting given in row 2. Furthermore, we find

T. Yamogucbi

474

ef nl. / Semi-Zeptonic &decay jam

factors

TABLE 14.3 Decomposition

Transitions

Total x2 Same as in table

of total x2 into individual

MIT+

REC

contributions

(without

MIT +OGE+REC

c.m. correction,

CBM+REC

no resealing) CBM +OGE+REC

90.2 113 656 0.00 765

49.7 63.1 388 0.00 469

1.01 0.30 0.19 0.15 0.44

2.43 6.27 28.7 0.25 44.4

0.80 0.15

0.25 0.17

0.02 0.16

0.00 0.17

13.0 0.08 0.23 0.27

1.86 0.13 0.32 0.29

5.24 0.11 0.08 0.26

11.2 0.16 0.12 0.28

1.31 2.85 1.07 6.5 I 3.78 0.71

0.30 3.76 1.24 6.73 3.70 1.74

1.87 8.97 3.15 7.76 2.15 9.00

0.46 10.5 3.59 7.96 1.95 12.1

6.42 1.97 0.79 8.29

1.32 0.00 0.00 2.43

8.60 0.03 0.06 2.50

1.61 1.49 0.25 0.07

0.22 0.17 0.48 0.09

0.91 0.45 1.12 0.58

1.50 0.04 0.23 0.01

2.84 0.21 0.69 0.29

0.22

0.20

0.05

0.04

1674

997

53

138

14.1 but obtained

without

applying

the resealing

described

in the text.

that 1Vu,/’ + 1VuS12= 1 within 2%. Thus, although we have started from the twoparameter KM picture, we find ourselves led back to the Cabibbo-like picture. We should be warned, however, that the resulting x2 values, x2 = 14- 34, are rather large for the small-number of data N = 7. The same warning applies also to the semi-empirical KM-SU(3) fitting. In case (b) we notice that the reduced number of data, notably the omission of the neutron data, leads to the smaller x2 values, but the resulting values of Vu, are in glaring contradiction with the value dictated by the O++ Ot transitions: Vu, = 0.9749 (see sect. 9). Finally, we present individual comparisons of the calculated values of observables with the experimental data. Tables 16 and 17 give the decay rates obtained with the

T. Yamaguchi

et al. / Semi-leptonic

Best-fit values

of Kobayashi-Maskawa

415

P-decay form factou

TABLE 15 matrix

elements

Case (a)

Case (b)

Models V ud

V “I

X2

V ud

V “5

X2

SU(3)

0.9749’

0.222

16

0.9749+

0.226

6

MIT+ REC MIT+OGE+REC CBM + REC CBM+OGE+REC

0.976 0.97 1 0.971 0.967

0.211 0.211 0.219 0.218

34 19 16 15

0.943 0.917 0.916 0.900

0.218 0.218 0.224 0.222

10 10 5 10

MIT+ REC + c.m. MIT+OGE+REC+c.m. CBM + REC + c.m.

0.976 0.972 0.971 0.967

0.212 0.216 0.219 0.219

30 17 15 14

0.944 0.922 0.912 0.900

0.219 0.221 0.224 0.222

8 8 4 9

CBM + OGE + REC + c.m.

’ Input data. For the meaning of the abbreviations for the various models, MIT+ REC, MIT+ OGE+ REC, CBM + REC and CBE+OGE+REC, see the caption of table 1. For the meaning of the abbreviations, MIT+ REC+c.m., MIT+OGE+REC+c.m., CBM+REC+c.m. and CBM+OGE+REC+c.m., see the caption of table 8. In all the cases the resealed form factors have been used. Since the difference between the cases with and without the recoil correction is practically indistinguishable, only the results with the recoil correction are shown. (a) Data compiled in Particle Data ‘), (b) the CERN-WA2 data *). The column labelled x2 gives the sum of the ,$ values contributed by the decay rates (without including the correlation data).

use of the least-squares fitted values of the KM parameters given in table 15; 16 corresponds to the cases without c.m. correction and table 17 to the cases c.m. correction included. Tables 18 and 19 show the correlation observables; 18 is for the cases with no cm. correction, and table 19 for the cases with correction.

In all the cases the resealed

form factors

table with table c.m.

have been used. We can see

that in both of tables 16 and 17 the gluonic effect is most visible in the enhancement of r(Z‘-+ n), bringing the theoretical value closer to experiment. This improvement is closely related to the better agreement of g,/f,(X+ n) in tables 6 and 12. These features are reflected, as they should, in the appreciable reduction of the corresponding x2 in tables 14.1 and 14.2. We recall here the long-standing “2 -A problem”: the SU(3) model (or more generally, any additive quark model) gives the ratio T(X+ n)/r(A + p) significantly smaller than the experimental value. The results given in tables 14.1 and 14.2 indicate that the “2 -A problem” is resolved by the inclusion of the gluonic effect for both c.m. corrected and c.m. non-corrected cases (provided the resealing is applied). The integrated correlation observables are given in table 18 (without c.m. correction) and in table 19 (with c.m. correction). We notice as a common feature for both tables that there is very little model dependence and that agreement with data is satisfactory except for the observables pertaining to A + p: cx,,(A + p), cx,(A + p)

(b) 1.03 0.23 0.38 3.55 6.10 3.04 0.56 2.91 1.00

(4 1.10 0.25 0.41 3.32 5.70 2.84 OS2 2.72 0.94

(b)

1.10 0.24 0.40

3.49

6.10

3.21

0.52

2.91 0.94

1.10 0.27 0.44

3.24

6.48

2.93

0.50

3.09 0.91

MIT+ REC

(4 -

SU(3)

2.91 0.93

0.52

2.74

6.23

3.27

3.14 0.98

0.55

2.90

6.60

3.46

0.91

2.99

0.51

2.92

6.28

3.26

1.09 0.26 0.43

0.98

1.10 0.26 0.43 0.23 0.38

(4

(b)

(b)

3.13 0.95

0.53

3.06

6.57

3.41

0.97 0.23 0.38

CBM+REC

(4

MIT+ OGE+ REC

3.23 0.90

0.50

2.82

6.78

3.21

1.09 0.27 0.44

(4

3.34 0.93

0.52

2.92

7.02

3.32

0.94 0.23 0.38

(t-1

CBM+OGE+REC

1.114*0.020a) 0.250 i 0.063 “) 0.387 i: 0.018 “) 0.38 f 0.021 ‘) 3.180;t0.05Sa) 3.26rtO.14 ‘) 6.896 * 0.235 “) 6.48 * 0.34 ‘) 3.352*0.367 “) 3.44*0.19b) 0.53*0.10”) 0.53 *0.10 b)

Exp.

Decay rates corresponding to the microscopicaily calculated weak interaction form factors and the least-squares fitted values of Vu, and Vu, (given in table 15). The results of the semi~empi~cai KM-SU(3) fitting are also shown. For the meaning of the abbreviations for the models, MIT+REC, MIT+OGE+ REC, CBM+REC and CBM+OGE+ REC, see the caption of table 1. Since the difference between the cases with and without the recoil correction is insignificant, we give only the results with the recoil correction. The cases (a) and (b) in each model represent the nse of the data in ref. ‘) and those of ref. a), respectively; they are given in the column labeled “Exp.“. The unit is low3 s-r for the n -+pel’, rate, and 106s-’ for all the others.

Transitions

TABLE 16 Decay rates (without c.m. correction)

Q 2

?i

3.54

6.14

3.05

0.54

2.93 0.97

3.31

5.15

2.86

0.50

2.74 0.90

2.98 0.91

0.51

2.82

6.26

3.27

1.10 0.25 0.42

(aI

3.44

0.99 0.23 0.38

&I

included,

3.14 0.96

0.54

2.91

6.60

MIT+OGE+REC+c.m.

with the c.m. correction

1.03 0.23 0.38

1.10 0.25 0.41

Same as in table 16 but obtained

@I

(4

MIT+ REC+ cm.

TABLE

17

0.53

0.51

3.15 0.94

3.10

2.98

3.02 0.91

6.60

3.39

3.26 6.34

0.97 0.23 0.38

-

(b)

1.09 0.26 0.43

(8)

CBM + REC + c.m.

Decay rates (with c.m. correction)

3.33 0.92

2.98

2.89

3.23 0.90

6.98

6.17

0.51

3.31

3.21

0.50

0.94 0.23 0.38

(b)

1.08 0.27 0.44

(4

CBM + OGE + REC + c.m.

~.l14*0.020a) 0.250 * 0.063 “) 0.387 ztO.018 “) 0.38 kO.021 ‘) 3.180*0.058 “) 3.26icO.14 b, 6.896 IO.235 “) 6.48 * 0.34 h, 3.352 i 0.367 “f 3.44rt0.19b) 0.53 kO.10 “) 0.50+0.10 h)

Exp.

T. Yamaguchi

478

et al. / Semi-leptonic

p-decay form factors

TABLE 18 Integrated

Transitions

n+p

SU(3)

correlation

MIT+REC

observables

+OGyTRl_C

(without

c.m. correction)

CBM+REC

CBM + OGE + REC

Exp.

a, a”

-0.08 -0.08 0.99

-0.08 -0.08 0.99

-0.08 -0.08 0.99

-0.08 -0.08 0.99

-0.08 -0.08 0.99

-0.074 f 0.004 -0.083 f 0.002 0.998 f 0.025

P++A

%”

-0.40

-0.41

-0.41

-0.41

-0.41

-0.35 *0.15

Y-A

aev A B

-0.41 0.04 0.88

-0.42 0.05 0.88

-0.42 0.05 0.89

-0.42 0.05 0.89

-0.42 0.05 0.89

-0.404 * 0.044 0.07 * 0.07 0.85 zt 0.07

A+P

%V -0.04 a, 0.01 ; -0.59 0.98

-0.12 -0.03 1.oo -0.57

-0.09 -0.04 1.00 -0.57

-0.07 0.01 0.99 -0.60

-0.07 0.01 0.99 -0.60

0.036 0.125 0.821 -0.508

Z-+n

0.36 a.?” a, -0.57

0.43 -0.48

0.37 -0.62

0.40 -0.54

0.3 1 -0.64

0.382 f 0.07 -0.58 f 0.16

E-+/l

%” A

0.61 0.50

0.52 0.61

0.56 0.56

0.55 0.57

0.59 0.52

0.53*0.10 0.62*0.10

%”

zt 0.037 +0.066 f 0.060 f 0.065

For the explanation of the five cases, SU(3), MIT+REC, MIT+OGE+REC, CBM+REC and CBM + OGE + REC, see the caption of table 1. Since the difference between the cases with and without the recoil correction is insignificant, we give only the results with the recoil correction. The definition of the integrated correlation observables, (Y,,, (Y,, (Y,, B’, A and B, is given in appendix B. The experimental data are cited in ref. 24).

and a,(A +p). Although the CBM-type models tend to give somewhat better agreement, the resulting x2 values are still large. (Since we are using here the form factors that have been resealed so that g,(n + p) and f2(n + p) are reproduced, the results for the n + p transition are largely trivial.)

12. Discussion

and summary

An additional aspect of the gluonic effect we wish to discuss here is concerned with the value of or,= D/(F+ D). If one applies the semi-empirical KM-SU(3) fitting to the data compiled in ref. ‘), one obtains (Y,,= 0.626; a similar fitting to the et al. “) pointed slightly different set of data gives cx,,= 0.629 [ref. “)I. Donoghue out that the inclusion of a particular SU(3) breaking effect due to the mismatching of the initial and final wave functions of the P-decaying quark appreciably improves the quality of fitting to the data, and that the best fit is obtained with (Ye = 0.646 f O.OOS* All these results seem to indicate that the empirically favored value of a, l We mentioned earlier in the text that the analysis of the center-of-mass correction.

of ref. 6, involves a specific model for the treatment

T Yun~a~uchi et al. / Semi-lepto~ic TABLE

Integrated MIT + REC + cm.

Transitions

correlation

19

observables

MIT+OGE + RECfc.m.

%lS % a,

-0.08 -0.08 0.99

-0.08 -0.08

H’+.Ji

@ev

-5.43

x-+A

%I A B

*i + p

479

p-decay form factors

(with c.m. correction)

CBM +REC+c.m.

CBM + OGE +REC+c.m.

Exp.

-0.08 -0.08 0.99

-0.08

-0.41

-0.41

-0.41

-0.42 0.05 0.88

-0.42 0.05 0.89

-0.42 0.05 0.89

-0.42 0.05 0.89

-0.404 f 0.044 0.07 * 0.07 0.85 f 0.07

%‘. a, a” B’

-0.10 -0.03 1.00 -0.58

-0.10 -0.03 0.99 -0.57

-0.06 0.02 0.99 -0.61

-0.05 0.02 0.98 -0.61

0.036 * 0.037 0.125*0.066 0.821 * 0.060 -0.508 f 0.065

I”*n

CL a,

0.44 -0.47

0.36 -0.60

0.41 -0.52

0.34 -0.62

0.382 f 0.070 -0.58*0.16

s--t/j

ffW A

0.53 0.60

0.56 0.56

0.55 0.57

0.58 0.53

fl-+P

Same as in table

I8 but obtained

0.99

using the form factors

that inctude

-0.08

0.99

the cm.

-0.074 * 0.004 -0.083 *0.002 0.998 * 0.025 -0.3510.1s

0.53*0.10 0.62*0.10 correction.

shows a significant deviation from the SU(6) prediction: aD =$. In order to examine whether this tendency can be seen also in the calculated gi’s, we have deduced the values of an that would, within the SU(3) scheme, optimally reproduce the calculated values of g,‘s given in tables, 2,4,9 and 11. Although, in the original SU(3) picture, crD should be common for all the transitions, we have determined here the best-fit values of cyD for each of AS = 0 and AS = 1 spaces in view of the possibility 42) that this separate treatment might simulate a certain type of SUf3f breaking. The input of F-t D (=gA) = 1.254 was used in order to express g,‘s in the SU(3) formula as a function of “0 only. The results are given in table 20, which also gives the “standard deviation” ci. We observe that in all the cases studied the &b’s for AS = 0 and AS = 1 spaces are practically the same. This feature and the smallness of the standard deviation CTfor all the cases given in table 20 indicate that the SU(3) parametrization can reproduce very well the calculated values of g, for all the models listed in table 20. On the other hand, table 20 indicates that the cases without the gluonic effects give lyD--0.60 (for MIT) and -0.61 (for CBM), whereas the cases with the giuonic effects give on -0.62-0.63 (for MIT) and -0.63-0.64 (for CBM). Thus, the gluonic effects preserve the SU(3) symmetric pattern of g,‘s but generate the breakdown of SU(6) symmetry by increasing cyD from its SU(6) value 0.6; the resulting theoreticat value cyb-- 0.63 lies within the range of the em~i~cally favored value

480

T. Yamaguchi et al. / Semi-leptonic B-decay form factors TABLE

Systematics

20 of cxo AS=0

AS=1

Models

MIT MIT+ c.m. MIT+ OGE MIT+OGE+c.m. CBM CBM + c.m. CBM+OGE CBM+OGE+c.m.

0.600 0.600 0.627 0.622 0.612 0.611 0.635 0.63 1

0.000 0.000 0.019 0.016 0.006 0.004 0.020 0.015

0.596 0.597 0.619 0.617 0.613 0.611 0.633 0.629

0.066 0.041 0.064 0.040 0.022 0.016 0.024 0.017

Values of the 01p= D/(D+ F) that optimally reproduce the g,‘s calculated in the various microscopic models appearing in tables 2,4,9 and 11. The resealed g,‘s have been used. For the explanation of the abbreviations for the various models, see the caption of these tables. The standard deviation o is defined by: (T- J( l/N) Xi [g,( i: a,) - g,( i: models)]‘, where N is the number of the transitions to be dealt with; in the present case, N = 6 for both AS = 0 and or AS = 1 spaces; g,(i: (Y,,) and g,(i: models) are, respectively, the g,‘s obtained in the W(3)-parametrization and those calculated in the present microscopic models.

an= 0.626 - 0.646. One can see from table 20 that this feature is independent of the c.m. correction; furthermore, as discussed in sect. 9, the recoil correction does not affect g, in any significant manner. Thus, within the limited model calculations presented here, it seems that the gluonic effect plays an intriguing role of enhancing LYEover the SU(6) value, 0.6. We apologize to the reader for the length of the present article and hope that the following summary will facilitate the quick grasp of the main features of our work. We have carried out a detailed calculation of the weak-interaction form factors and the magnetic moments of the octet baryons, using the volume-type CBM. The combined effects of the mesonic and gluonic contributions and kinematical corrections such as recoil and c.m. corrections have been investigated. The recoil correction, as calculated here, turns out to be small. The cm. correction evaluated here with the use of the method in ref. ‘“) changes the results drastically (except for f,), and in fact in a wrong direction as far as agreement with data is concerned. However, when one applies resealing SO that the form factors for the nucleons should be reproduced, one almost recovers the results that do not include the c.m. correction. With the use of the calculated weak interaction form factors we have computed the decay rates and the integrated correlation observables, and have compared them directly with the experimental data (tables 14-19). When the resealed form factors

T. Yamaguchi et al. / Semi-leptonic P-decay form factors

481

are used, the overall agreement with the data is reasonable for the CBM-type models (CBM and CBM+OGE) with total x2 = 40 - 50 for the number of data N = 26 (tables 14.1 and 14.2). It is found that the inclusion of the gluonic effect (CBM+ OGE) does not lead to the significant lowering of the total x2. The MIT-type models (MIT and MIT+OGE)

give appreciably

larger total x2 values

than the CBM-type

models. Although the approximate nature of the present treatment does not allow us to make too strong a case here, this difference in the resulting x2 values seems at least worth one’s attention. It turns out that, if one uses the experimental data on the decay rates alone, the CBM-type models and the MIT+OGE (but not the MIT itself) would give practically the same quality of fitting; thus, the inclusion of the data other than those on the decay rates is responsible for the above-mentioned difference. In this connection, it should be remarked that Donoghue et al. “) could phenomenologically fit the decay-rate data remarkably well within the framework of the MIT bag model (including the specific SU(3) breaking effect due to the quark wave function mismatching) by adjusting the parameter (Y~. In the context of the present microscopic calculation, a possible explanation for the phenomenological success of ref. “) [we repeat that only the decay-rate data are considered in ref. “)I is that the phenomenological adjustment of (Y,,probably corresponds to the inclusion of the gluonic effects in the microscopic calculation. The approximate nature of the treatment of the gluonic effects in the present work should be kept in mind here again, though. The Kobayashi-Maskawa (KM) matrix elements, V,, and Vu,, have been deduced through least-squares fitting to the experimental decay rates. The resulting KM matrix elements (table 15) turn out to be almost independent of microscopic models used to evaluate the form factors in the present work. The results can be summarized as: V,, = 0.972&0.004 and Vu, = 0.215 hO.004. These values are close to those obtained through the semi-empirical KM-SU(3) fitting to the data I). They are also compatible with the results of Leutwyler and Roos *‘). The three known phenomenological successes of the gluonic effect pertain to: (1) (3) the r(X + n + eP + the “Z- - A problem” for the magnetic moment; (2) &p,; c,)/r(A + p + eP + c,). The present calculations demonstrate that all these successes survive such kinematical effects as recoil corrections and center-of-mass (c.m.) corrections. (For the latter, the appropriate resealing is needed, though.) Finally, we should repeat that the validity of the approximations used in treating various ingredients of the present calculation may be limited, and that the conclusions of this paper should be taken only in the light of these possible limitations.

One of the authors (K.K.) wishes to express his sincere thanks to Fred Myhrer and H. H$gaasen for discussions on ref. 34) and to Gerry Brown and Mannque Rho for discussions on ref. 23).

482

T. Yamaguchi et al. / Semi-leptonic P-decay form factors

Appendix A for ?jsb, jjab, 7jbb, &,

We give here explicit expressions

tab and tab in eqs. (6.10a)-

(6.13b).

64.1) 7jab

4nN(a)N(b)

s

dXX%(dl -k&)&)(X) R

I

(-) -mAB

dxx2_io(dl

-

(A.2)

E,b)x)&)(x)

0

dxx2jh(l dxx2joW

-

E",bbb$~)b)

&.b)h&)(X)

(A.3)

dxx*j,(q(l

gb = -~TN’“‘N’~’

-I-

iR

rnk-,'

dxx*jO(q(l

-

- &,,)x)c#&‘(x)

E,b)X)‘hb(X)

iab

E

(A.4)

,

0

I

R

&.j+a)N(b)

dxx2jo(q(l

,

&b)X)'hb(X)

-

I 0

R dxx21’“‘(x)lCb)(x)

2, = 4?.rNCa)NCb)

1

q(l_~ab)x_h(dl-~ab)X)-jO(dl-&b)X)

da’(x)r’“‘(x)+ dxx*j,(dl-

P(x)u’b’(x)

4m

kb)X)

A

dxx2jo(dl

The quantities

appearing

mkil

(A.5)

-

lzlB- mA 4m,m,

are defined

(+)_mB+mA

’

mABE-

4m,m,

I

(‘4.6)

&b)X)Kab(X)

in the above expressions

~

4mB

’

by

(A.7)

T Yamaguchi et al. / Semi-leptonic P-decay form factors

9 = 141

da)(x)= a+(a)j&2(“‘x/R), Pa)(x)= a-(a) j,(fl'"'x/R), &)(x) a &j(x)

(x) zk z’“‘(x)z’b’(x) )

ze

f,p)(x)Jb)

SE

u(a) (X)Fb’(X)*

K&(x) =

da’(x)db’(x) -+~ca)(x)~cb)(x) . Appendix

We

P’(x)u’“‘(x) )

483

(A-8) (A.9) (A.lO) (A.ll) (A.12) (A.13)

B

give

here the definition of the integrated correlation observables, in tables 14, 18 and 19. The e-v angular aevr a,, ff,, B’, A, and B 24339),appearing correlation coefficient (Y,, is defined by Lr

e”

=

2

N(B,,<$r)-N(B,“>$) N( 8,” -=C $7) + N(8,” > $7) ’

(B.1)

where N( 0_, c&r) or N( 8,, > $r) is the number of all events with e-v pairs emitted in such directions that the angle f3_ between e and Y is smaller or greater than 90”. The electron asymmetry coefficient (Y,, the neutrino assymetry coefficient cx,, and the emitted baryon asymmetry B’( = ags) are defined by

03.2) where Oe,u.BSis the angle between the polarization of the decaying baryon and the directions of the electron, Y and the outgoing baryon B, respectively. In order to define the asymmetry coefficients A and B, we first introduce the orthonormal basis iA and j& defined by PIA= NA( ie+iy), j& = NB( ie -by), with NA and N, normalization constants 24739).The asymmetry coefficient A is defined by eq. (B.2) in which, however, 8 refers to the angle between the polarization of the decaying baryon and iA; B is obtained by replacing iA with j& in the definition of A.

References 1) Particle

Data Group, Phys. Lett. B170 (1986) 1 et al. (WA2Collaboration), Z. Phys. Cl2 (1982) 307; C21 (1983) 1; 17; 27 P.C. Peterson et al., Phys. Rev. Lett. 57 (1986) 949; G. Zapalac er al., Phys. Rev. Lett. 57 (1986) 1526; C. Wilkinson et al., Phys. Rev. Lett. 58 (1987) 855 S.Y. Hsueh et a/., Phys. Rev. Lett. 54 (1985) 2399 H. Leutwyler and M. Roos, Z. Phys. C25 (1984) 91 J.F. Donoghue, B.R. Holstein and S.W. Klimt, Phys. Rev. D35 (1987) 934

2) M. Bourquin 3)

4) 5) 6)

484 7) M. Kobayashi 8) K. Tsushima,

T. Yamaguchi

et al. / Semi-leptonic

p-decay form factors

and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652 T. Yamaguchi, Y. Kohyama and K. Kubodera, Nucl. Phys. A489 (1988) 557

9) K. Tsushima, T. Yamaguchi, M. Takizawa, Y. Kohyama and K. Kubodera, Phys. Lett. 8205 (1988) 128 10) P.A.M. Guichon, Phys. Lett. B129 (1983) 108 11) M. Betz and R. Goldflam, Phys. Rev. D28 (1983) 2848 12) 0. Lie-Svendsen and H. Hogaasen, Z. Phys. C35 (1987) 239 13) J. Donoghue and K. Johnson, Phys. Rev. D21 (1980) 1975 14) C.W. Wong, Phys. Rev. D24 (1981) 1416 15) C. Carlson and M. Chachkhunashvili, NORDITA preprint (1981) 16) R. Tegen, R. Brockmann and W. Weise, Z. Phys. A307 (1982) 339 17) L. Mankiewicz and S. Tatur, Phys. Rev. D29 (1984) 1035 18) J.O. Eeg, H. Hogaasen and 0. Lie-Svendsen, Z. Phys. C31 (1986) 443 19) C. Rebbi, Phys. Rev. D12 (1975) 2407; D14 (1976) 2362 20) A.W. Thomas, J. of Phys. G7 (1981) L283; A.W. Thomas, Adv. in Nucl. Phys. 13 (1983) 1 21) A. Szymacha and S. Tatur, Z. Phys. C7 (1981) 311 22) M. Morgan, G. Miller and A. Thomas, Phys. Rev. D33 (1986) 817 23) G.E. Brown, S. Klimt, M. Rho and W. Weise, Z. Phys. in press 24) M. Beyer and S.K. Singh. Z. Phys. C31 (1986) 421 25) 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. Bl4Ii (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, B. Jennings and A. Thomas, Phys. Rev. D33 (1986) 1859 26) Y. Kohyama, K. Oikawa, K. Tsushima and K. Kubodera, Phys. Lett. B186 (1987) 255 27) K. Kubodera, Y. Kohyama, K. Oikawa and C.W. Kim, Nucl. Phys. A439 (1985) 695 28) S. Theberge, A. Thomas and G. Miller, Phys. Rev. D22 (1980) 2838; S. Theberge and A. Thomas, Nucl. Phys. A393 (1983) 252 29) K. Ushio and H. Konashi, Phys. Lett. B135 (1984) 468; K. Ushio, Z. Phys. C30 (1986) 115; K. Ushio, Ph.D. thesis (Osaka University, 1984) 30) T. DeGrand, R. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 31) G.A. Crawford and G.A. Miller, Phys. Lett. B132 (1983) 173 32) M. Ademollo and R. Gatto, Phys. Rev. Lett. 13 (1964) 264 33) L.J. Carson, R.3. Oakes and C.R. Willcox, Phys. Lett. B164 (1985) 155; Phys. Rev. D33 (1986) 1356; D37 (1988) 3197 34) H. Hogaasen and F. Myhrer, Phys. Rev. D37 (1988) 1950 35) H.J. Lipkin, Phys. Rev. D24 (1981) 1437; H.J. Lipkin, Nucl. Phys. B214 (1983) 136 36) K. Ushio, Phys. Lett. 8158 (1985) 71 37) C.G. Callan Jr., R.F. Dashen and D.J. Cross, Phys. Lett. B78 (1978) 307; 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 38) P. Gonzalez, V. Vento and M. Rho, Nucl. Phys. A395 (1983) 446 39) A. Garcia and P. Kielanowski, Lect. Notes in Phys. 222 (1984) 1 40) A. Garcia, Phys. Rev. D3 (1971) 2638; ibid D12 (1975) 2692. 41) J.M. Gaillard and G. Sauvage, Ann. Rev. Nucl. Part. Sci. 34 (1984) 351, and references cited therein 42) J. Donoghue and B. Holstein, Phys. Rev. D25 (1982) 2015 43) R.E. Peierls and D.J. Thouless, Nucl. Phys. 38 (1962) 154 44) S. Nadkami, H.B. Nielsen and I. Zahed, Nucl. Phys. B253 (1985) 308