Chiral symmetry breaking in hidden local symmetry

IL NUOVO CIMENTO VOL. 108A, N. 9 Settembre 1995

Chiral Symmetry Breaking in Hidden Local Symmetry(*).

S. FURUI(1), R. KOBAYASHI(2) and M. NAKAGAWA(3)

(1) Department of Physics, Ibaraki University - Mito 310, Japan (~) Department of Mathematics, Science University of Tokyo - Noda 278, Japan (3) Department of Physics, Meijo University - Tempaku-ku, Nagoya 468, Japan

(ricevuto il 19 Gennaio 1995; approvato il 16 Febbraio 1995)

Summary. - - Chiral breaking for baryon and meson interaction is formulated in the framework of the hidden local symmetric theory of the non-linear sigma model on the manifold U(3)LXU(3)R/U(3)v . The chiral symmetry-breaking term transforms as (3, 3*) under U(3)L x U(3)R at the f'u'st order. Several low-energy theorems are derived for the electromagnetic interaction of baryon. The symmetry-breaking term preserves SU(2)x U(1) symmetry; we have confirmed that the Sakurai formula of the electromagnetic interaction of vector meson holds and directly shown that the photon mass vanishes. The fundamental gauge coupling constant g and the mixing angle 0 of vector meson are fLxed by independent phenomena of vector meson masses and ee decay of vector meson. Using these values of g and 0, we extend our analysis to the nucleon form factor and indicate a breaking of the OZI rule and a sizable contents of s-quark pair in the nucleon.

PACS 12.90 - Miscellaneous theoretical ideas and models.

1. - I n t r o d u c t i o n .

The hidden local symmet ry [1] may now be a promising candidate for the effective low-energy of the QCD. This theory has a fundamental coupling constant g of vector meson as the dynamical gauge boson and an adjustable pa ramete r a in the meson interaction. The only crucial assumption is that the kinetic par t of the dynamical gauge vector meson is of dynamical origin. Low-energy theorems in the meson sector such as KSFR relation, etc., were shown to be satisfied independently of the parameter a at the symmetrical limit.

In a previous paper [2], we constructed an effective baryon interaction of the lowest-dimension operators in the hidden local theory based on the U(3)L X X U(3)R/U(3)v non-linear sigma model, and successfully reproduced the S-wave pion nucleon scattering lengths, the conserved vector current (CVC) of fl-decays of

(*) The authors of this paper have agreed to not receive the proofs for correction.

1051

1052 S. FURUI, R. KOBAYASHI and M. NAKAGAWA

baryons and the Sakurai scenario of the electromagnetic interactions (hereafter, we abbreviate as EMI) of hadrons. The effective interaction contains an adjustable parameter a' with one more parameter a~ due to a non-vanishing U(1) component of /7(3) trace in addition to the fundamental coupling constant g. The low-energy theorems in baryon sector have been shown to be satisfied independently of a' and as.

In this paper, we introduce the chiral-symmetry-breaking interaction in a most simple way according to ref. [3] and construct an effective theory of hadrons including octet baryons and mesons interacting with external gauge bosons. Owing to the nonet scheme of the vector meson, we have two more meson interactions with adjustable parameters c and d to the original one of ref. [1]. The effective interaction with chiral breakings enables us to evaluate real values of a' and as as well as to fLX g and mixing ancle 0 of ~o and ~ mesons.

We analyze the mass splitting of vector mesons, the electron pair decays of vector mesons and the electromagnetic form factor of nucleon. In this analysis, we take into account the leading order of the symmetry-breaking term with the neglect of loop corrections. In our treatment of the symmetry breaking, it is to be noted, the SU(2) x x U(1) symmetry is preserved; we check the universal EMI, the vanishing of fictitious ,,photon mass- and the maintainance of the Sakurai formula of vector meson-photon interaction.

This paper is organized as follows. The effective Lagrangian of baryon and meson with chiral symmetry breaking is formulated up to the first order of the current quark mass in sect. 2. In sect. 3, we deal with the vector meson mass formula to estimate g, 0 and d / a . In sect. 4, the EMI of vector meson is studied; values of g and 0 are obtained independently of the other estimation. In sect. 5, the EMI of nucleon is constructed, and is shown to satisfy the minimal EMI. Also we establish low-energy theorems for both the isoscalar and isovector channels. In sect. 6, the electromagnetic form factor of nucleon is analyzed on the basis of the low-energy theorems, which are used to extract values of a' and a~ in sect. 7. Finally results and discussions are given in sect. 7, where we summarize the result on g and 0, and derive a new sum rule for the vector meson pole contribution to the nucleon form factor, and discuss the validity of the OZI rule and the rate of the content of the s-quark pair in the nucleon.

2. - L a g r a n g i a n .

In this section we will build the chiral Lagrangian with the chiral-symmetry- breaking parts in the hidden local symmetric theory of the manifold U(3)L X X U ( 3 ) R / U ( 3 ) v . The same notations as those in ref. [1] and [2] are used without any notice. The other necessary notations are presented in the appendix. The symmetrical Lagrangians consist of two kinds of building blocks as

(2.1) 1 D ~ L . ~ L __ D,~R" ~R), 2i

where

(2.2) { ~L = exp [ i u / f ~ ] exp [ - i z / f , ] ,

~R = exp [ i ,~/f ,] exp [ i~/ f~] .

CHIRAL SYMMETRY BREAKING IN HIDDEN LOCAL SYMMETRY 1053

The ~'s transform as

(2.3) { ~L(x) = h(X)~L(X)g*L,

~ (X) = h(x) ~R(X)gR*,

where generators are ga e UL(3), gne UR(3) and h e Uv(3). The covariant derivatives under Uv(3) are defined as

(2.4) { D ~ L = 9 ~ L -- iV~,~L + i~LBLI, ,

D l , ~ = 9~,~n -- iVz~R + i~RBRF, .

For local Uv(3), we fix the gauge as a = 0 and then have ~L(R) a s ~L(R) = exp[ T- i z f i ] . We introduce the symmetry-breaking terms in accordance with the paper by Bando, Kugo and Yamawaki [3]. We replace two blocks of (2.1) by new building blocks as

(2 .5) v 1

= " + O ~ R ' ~ v ~ L ) , a~ ~_ ( D ~ L ~t L + Dz~R.~tR + D~,~L.SV~tR t t

(2.6) a A - 1 (D~,~L'~ - D ~ R ' ~ + D ~ L ' S A ~ - D.~R'StA~tL), "/4

where ~V(A) transforms as (3, 3*) under U(3)L • U(3)R and the explicit expression will be given later. The meson part of the Lagrangian L M is given in terms of _V(A) of eqs. (2.5) and (2.6) as

(2.7) LM ---- LA + aLv + eL A, + dLv, ,

where

(2.8)

LA = f2((aA)2>,

a L v 2 v 2 = a f ((a~,) },

cLA, = cf 2 (aA'> 2 ,

alLy, = d f z ( a Y') 2 .

In eqs. (2.8), the mass term of the pseudoscalar meson and the kinetic term of the external gauge boson are omitted. -V'(A') are defined by eqs. (2.5), (2.6) with replacing tt/~

V(A) by s ~(A). The assumption of the existence of the kinetic term of dynamical gauge vector mesons is essential in the hidden local symmetric theory. In eqs. (2.8), we

1054 s. FURUI, R. KOBAYASHI and M. NAKAGAWA

reparametrize the breaking ~ terms as

2af2ev = Corn,

(2.9) 2df2e~ Dora.

where m is expressed in terms of the current quark masses as follows:

(2.10) m =

(; 0} m d

ms

The eA terms are irrelevant in the following analysis and we do not redefine them. Lagrangian LB of the baryon is given as follows:

(2.11)

where

(2.12)

and

LB = L<s ~ + a'L(B ') + L{ 2) + L(B 3) + a:L~ 4) + L(B m) ,

L~ ~ = <Biy~'D~,B) = <Biy" 3~ B> + g<By~'[V~, B]),

a' L~ 1' = a'<Biy~'[a~, B]),

L~ 2) = - br<BT~'rs[a~, S]) - bD@y~'ys {a~, B}),

L~ 3) - bs (/~y ~ Y5 B)(a~'),

asL(B4) = as<BT~'B><aV~ >,

<[1 ]> (2.13) L(B 'n)= -Mo</~B> + Kp B (~im~ta + ~Rm~[), B +

1 nu/('D (B { ~ (~Lm~R ~- ~Rm~)a), B}) �9

In eqs. (2.12), we replace e term as we did in eq. (2.9)

(2.14) I a ' e v = CBm , [ ase{~ DBm .

LB has essentially only one parameter a' which corresponds to the parameter a in LM. a~L~ 4) should in general be added because a m has non-vanishing U(1) component of U(3) trace. This interaction has an important role for the electromagnetic form factor of nucleon as vS_ll be shown later.


3. - M a s s e s o f v e c t o r m e s o n .

The mass p a r t of the L a g r a n g i a n equat ions (2.8) is given as follows up to the f irst o rder m (*):

(3.1) L(M ~n) = - ag2f2(V#W '} - g2Co(V#mW'} - g2Do(Vf, m)(W' }.

The masses of the non-mixing par t ic les a re obtained by eq. (3.1) as

(3.2)

1 2 = ag2f2 + 2g2Co(mu + md) -- ag2f 2 + mQ*- ms

m~.o = ag2f 2 + l g ~ c o ( m d + m~) =- ag2f 2 + l ( m 3 + 3m8) - ~ 2 4 2

1 1 V3 m~.• = ag2f 2 + - g 2 C o ( m u + m~) =- ag2f 2 + =-(ms + 3 m s) + - -

2 4 2

me 8 ,

mDs �9

The mass mat r ix of the mixed part icle is given as follows:

1 G m V ~ (3.3) L~~ - 2 '

where

(3.4) Voro~ = (p, ~os, wl)z �9

The mass matr ix m is

(3.5)

ag2f + m3 me8 me1 )

m = ~ mQ~ ag 2f2 + ms ms ~ ,

\ rng: ms1 ag2f 2 + m I

(*) The authors of ref. [3] keep m 2 term, but take only aLvwithout Lv.. Even when the m 2 term is added to our mass term equation (3.1), it can be later shown that the same low-energy theorems are also satisfied and that under mu = md = 0 a mass fitting to all the vector mesons gives constraints on parameters consistent with ours in, e.g., eqs. (3.13) and (3.16). It should be noted, in any case, that dL v, needs to exist though it is much smaller than aLy. So we retain only the first order of m term throughout this paper for simplicity.

1056 s. FURUI, R. KOBAYASHI and M. NAKAGAWA

where

(3.6)

m~ - g2Com , ms - l g 2 c o ( m + 2ms),

1 g2 2 3_ m l - ~ ( C o + 3 D o ) ( 2 m + 2 m s ) - dg2f 2,

rest 3 V~ Co + ~ Do (rnu - m~),

1 = g2 Co (mu - md) ,

mes 2 V~

-- - - Co + Do (mu -- rod)

1 - ~ (m, + md).

Under the approximation (SU(2) invariant)

(3.7) mos = mQ~ = 0,

we obtain the masses of the Q and K* mesons as

(3.8)

2 _ mo2 = m 2 = a g 2 f 2 + m s , mo

1 1 m~r =- : ( m ~ . o + m~.- ) = agZf 2 + -7(ms + 3ms).

g 4

The to and (p mesons are mixing states of ws and W l. We define the mixing angle 0 as

(3.9) ws = r + oJ sin 0,

1 = - r + wcos0 .

The mass matrix of the mesons w s and (/)1 is diagonalized with the mixing angle 0,

(3.10) tg20 = 2ms1/(ml - ms),

and eigenmass values of to and (p mesons are given as

(3.11) 2 m,o = agZf 2 + ms + msx / tg 0,

m~ agZff -]- m 8 -]- m s x tg 0.

The input experimental data are four mass values of m~, mKz., m z and m~. The mass


matr ix has six paramete rs ; ag2f 2, ms, ms, m81, ml and 0, of which one p a r a m e t e r is f'Lxed by the diagonalization condition. Thus we have one free paramete r . The number of pa rame te r s in the mass matr ix is jus t equal to the number of pa r ame te r s involved in the Lagrangian L~ ~) of eq. (3.1); ag2f 2, dg2f 2, g2Com, g2Dom, ras/m and 0. I t should be noticed tha t 0 and Do~Co are numerically fixed independently of the free p a r a m e t e r and they are

(3.12)

2 sin2 0 = 3m~ - 4mi2. + mQ

3(m~ - m~) '

2 2 2 3m 2) Do _ 2 ~V/(3m~ - 4m~, + mo)(4mK, - m e -

Co 3 2 V~(m~, - m~) 1]

Using the experimental mass values of the vector mesons, we have

(3.13) 0 = 39.16 ~ , Do/Co = 0.042.

Equations (3.13) shows the value of 0 nearly equal to the ideal mixing angle (0 = 35.3~ Equat ions (3.12) is nothing but the Gell-Mann-Okubo mass formula for nonet scheme(*). We choose the free p a r a m e t e r as x - ms/~t, and then ag2f 2 is expressed as

1 (3.14) ag2f 2 - - -

x - 1 [m~(x + 1) - 2m~, ] .

As the value of x, we use x = 25.7 which is given by QCD analysis [4]. Using a = 2 [1] and f = 92.5 MeV, we obtain the gauge coupling constant g of the vector meson as

(3.15) g = 5.76.

The values of the other pa r ame te r s are fixed as follows:

(3.16)

ms - 0.030, ms - 0.52, ag2f 2 ag2f 2

ml msl - 0.37, - 0.37,

ag~ f 2 ag2 f 2

d - 0 . 0 3 8 .

a

Equations (3.16) show tha t the order of the symmet ry -b reak ing coupling to the

(*) As was noticed before, if the m 2 term is added to the mass term, this does not hold [3]. Under mu =md = 0, we can obtain 0 = 32.97 ~ g = 5.87 and d/a = -0.110 from the mass fitting.


symmetric one is up to above 50%. The magnitudes of the neglected mQ8 and mQ~ are

(3.17) me8 - 0.0081, me, = - 0.012. ag2f 2 ag2f 2

Equations (3.16) and (3.17) show that the magnitude me1 is close to that of m3. However, it should be noted that the contribution of m, arises from the mass difference of u and d quarks and the EMI should come mto an effect. Accordingly, as an approximation, we neglect mQ~ and mob in the diagonalization of the mass matrix.

4. - Electromagnet ic interaction of the vector meson and V-~ e~ decay.

In this section, we estimate the gauge coupling constant g of the vector meson and the mixing angle 0 from the analysis of the decay V--* e~. At first, we confunn that our Lagrangian preserves the U(1) symmetry of EMI. The explicit interaction Lagrangian of the vector meson and the electromagnetic field is given from eq. (2.8) as

(4.1) L(y-V) = eg (af 2 + Com)~ ~ + af 2 + Co ~ + 2ms ] [

3 V~ Co + ~ Do (ms - m) co it t A",

[ 1 1 L('I-V)= g[(ag2f2+ms)~~ -~(ag2f2-t-ms)egs~+ --~mslwl~]A ~ ,

e [ 2 o - ~ 3 m ~ s i n O w ~ , + 1 ] :

Here we have used eqs. (3.8) and (3.11). Equation (4.1) shows that our Lagrangian gives the Sakurai formula

(4.2) L(~-V) = e E m~ V~,A ~

with

V~g V~g (4.3) fQ = g, fo, - sin 0 ' fr = cos 0

Equation (4.3) leads to f~,/f,, = tg0. This relation is also followed from only SU(3) argument. Our results show that such relations hold even in the presence of the chiral symmetry breaking�9 It should be noted that the all fy are given in terms of the fundamental coupling constant g and only the mixing 0. Now we show the vanishing of the fictitious ~photon mass, which means that our Lagrangians preserve the electromagnetic U(1) symmetry even in the presence of the breaking effects. The


Lagrangian equation (2.8) contains the fictious -photon mass, term such as

(4.4)

On the other hand, the exchange process of the vector meson by the interaction equation (4.2) gives at the low-energy limit

(4.5)

AL(A~) = 1 ee[ m~ m~ m~ ]A2

2 e 2 AL(A~)=[-3ae2f2+~g-~g2(ms+lms)]A~ �9

This AL(A~) cancels L(A~) exactly and thus the U(1) symmetry is preserved. Next, we proceed to estimate the coupling constant g and the mixing angle 0 from

the decay process V ~ ee. By the well-known formula f~ = 4za2mv/(3F(V ~ e6)), we obtain the following values of the fv as:

(4.6) fo = 5.03, f~ = 17.1, fr = 12.9.

Equations (4.3) and (4.6) give us the following values of g and 0:

(4.7) [ Q decay: g = 5.03,

~, ~ decay: g = 5.93, 0 = 37.08 ~ .

The value of 0 is consistent with the result obtained in sect. 3 from the masses of the vector mesons. The values of g may be consistent under the three diagram approximation and the neglection of mQ8 and mQ1. The equality of the values of 0 in

eq. (4.7) with the one of eq. (3.13) was known as an ~accidental agreement, [5] but this result shows the theoretical consistency of our framework.

5. - E lec tromagnet i c interact ion o f the octet baryons.

We investigate the electromagnetic interaction of the octet baryon in our framework at first and then show that the minimal photon interaction of the baryon holds even in the presence of the symmetry breaking at the low-energy limit. Next we derive the low-energy electromagnetic effective coupling constants for the I = 0 and I = 1 channels.


The direct interaction of the baryon and photon is obtained by eq. (2.12) as

(5.1)

L(u = ea'(PT~,P + Y,+ y~,X + - X- y#X- - F.- ~,#~- )A ~' +

+eCB[3(ms+2m)PY~,P+

1 - - + + - 1 - - o o +-~(ms-m)NT~,N +m(X 7~,X - X - 7 ~ , X - ) - ~ ( m s - m ) - ~ 7~,F-. -

t 1 - - ~, 1 - + ~(ms - 2 rh)~- 7~,.~- ]A - eDa -~ (ms - ~n)(BT~,B)A~',

ea' [(/~7~S)x + (BT~,B)~] A ~' + eCB [m(By~B)l + L(u

+ 1 (2ms + ~)(/~y~B)e ]A ~ - eDs 1 (m~- ~)(BT, B)A ~' , J

where

(5.2)

(ByaB) - / 57~P + NyaN + .4y~A + 2 + yaZ + + 2 ~ ~ + 2 - ya2- +

+ ~o r~ z ~ + ~- r~ ~ - ,

- 1 [(t57, P - .~7,X) + 2(2: § 7 ,X + - ~ - 7~X-) + (BY. B)I

+ (~o 7~ ~o _ ~. - 7~ ~ - )],

(Br~B)2 = 1 [ ( P r , P + 2Vr~N) - (~o~, zo + ~-- ~ , ~ - ) ] .

The interaction of the neutral vector boson and baryon is given as

(5.3) L(V~ = G~O~ + G~w~ + GJw~ ,

where

(5.4)

G~ -g [ (1 - a') - CBrh](/~y~B)x,

GS~-g V ~ ( 1 - a ' ) + - ~ C B ( 2 m s + ~ t ) (/~7.B)2

2 - ( m s - -

2 1 gDB(ms + 2m)(BT,B) - ~ asg(Byl, B).

1 + - ~ gDB (ms + vh)(By~ B),

The interaction of the vector meson and photon equation (4.2) gives the following

CHIRAL SYMMETRY BREAKING IN HIDDEN LOCAL SYMMETRY

a) b)

1061

Fig. 1.

~ _ V

effective baryon-photon interaction as shown in fig. 1

(5.5) AL(tree) =

: g ( G ~ e

1 1 ] + (G~ s sin 0 + GJ cos 0) - ~ sin 0 + (G~ s cos 0 - G~ sin 0) - ~ cos 0 A" =

1 s) + - ~ G A z =

r - - = e[ {ByzB} l + {/~7,B}2] A ~ - e a ({By~,B)I + (By~,B}2) +

-3DB(ms - m)(BT~,B) A ~' �9

It should be noted that GJ and 0 are dropped out at the second line in eq. (5.5). Finally, we have the following effective EMI of the baryon and photon as

(5.6) L(yBB) + AL(tree) = e [(BynB}l + (ByF, B}2] A ~' =

= e [(/SyzP + ~:+ 7zX- - Y.- 7~Z- - ~ - y ~ - ] A ~ .

This is the minimal EMI of the baryon. The derivation of eq. (5.6) does not depend on the parameters a' and a~ and the breaking effects in eqs. (5.1) and (5.5) cancel each other to get eq. (5.6).

Now we proceed to extract the effective EMI of the nucleon for each isospin channels I = 0 and I = 1. These interactions may be used to make the analysis of the electromagnetic form factor of the nucleon in the next section. Equation (2.12) gives the direct EMI of the nucleon as

1 (5.7) L(y3F3~) = e [ 2 (a' + CBm)(fiy~P - Ny~N) +

+ ~ a + - -CB(2m~+n~) -3 ~ D B ( m ~ - m ) ( P ~ P + N y ~ N ) A ~


The interaction of the neutral vector meson and the nucleon is obtained in the same

way as

L(V~162162 = g~xo ~ (['y~,P - NT~,N) + (g~.~.~,.~o~ + gl~,.cw~)(PT~,P + fi[7~,N), (5.8)

where

(5.9)

1 = 2 [ ( 1 - a ' ) - CB r~]g, g~o~

1 2 2 gl~cx= -~[-Vrda~ + --~ Ca(m~-~t)- --~ DB(m, + 2m)lg.

Rewriting eq. (5.8) in terms of mixing eigenstates of co and (p fields, we have

(5.10) L(V~ = gQ~0 ~ (i6y~P - A~?~N) + (gsxxa~ ~ + g ~ ' ) ( P i , F,P + ,~?~N),

where

(5.11) { g ~ = gs~' sin 0 + glgg COS O,

gcN~ = gsx~' cos 0 - gl~x sin O.

The effective EMI of the nucleon at the zero momentum transfer consists of two kinds of process shown in fig. 1, one is the direct interaction (fig. la)) and the other (fig. lb)) is the process mediated by the vector mesons. We have the following low-energy effective coupling constants in units of e as follows:

(5.12) g~__~x + l ( a , + CB~h ) 1 fQ 2 2

for the isospin 1 = 1 channel, and

1[ (5.13) g"~------L~ + g ~ + a

f~ f~

1 2 ] 1 + --CB(2ms3 + rn) - ~ D s ( m 8- m) =-~

for the isospin I = 0 channel.

6. - Electromagnetic form factor of the nucleon.

On the electromagnetic form factors, it has been known, it is not adequate to use only the simple Breit-Wigner shape for the peak of the Q-meson and simple tree diagram but it is necessary to take into account the pion-nucleon phase shifts for the I = 1 channel [6, 7]. On the other hand, the form factor for the I = 0 channel may be understood by the contribution of co- and d~-mesons[5]. Here we use only the


information of the form factor for the I = 0 channel and write it as [7]

al (aD al (~) al (s) ( 6 . 1 ) F ~ ( t ) - + - - + - - + C~ F ~ ~ ,

t + m~ t + m~ t + m[

where a~ (V) is the pole residue and C1 is a constant. The a~ (s) term corresponds to a pole of S-particle with higher mass 1.68 GeV. In our theory this term and the asymptotic term F ~ may be considered as being frozen into the direct interaction equation (5.7). Furuichi and Watanabe [7] gave the best-fit values as

(6.2) al (co) = 0.7627 GeV 2, al (~) = - 0.7665 GeV 2,

with the positions

(6.3) too) = 0.783 GeV, m, = 1.02 GeV.

The normalization conditions of the form factors are

(6.4) F[ (0) = FlY(0) = 1//2,

which corresponds to our low-energy effective coupling constants by eqs. (5.12) and (5.13). In our framework, we have the following expressions for al (w) and a~ (~) by eqs. (4.3) and (5.9):

} m~ V~(1 - a ') 2ms - n~ 2(m~ - n~) a l ( w ) = g ~ n ~-~ = V~ CB + V~ DB s i n e -

{ 6a 2(ms-~t) 2(ms+2m) } ] sinO 2 (6.5) - V~ s ~ CB + V6 DB cos0 g - ~ g m~,

f , = V~ CB + V~ DB' COS 0 +

+ [ V 6 a s - @ CB+ @ DB sin0 g--~ggm~.

By eq. (6.5) with the values of al (V) of eq. (6.2), we obtain independently of g and 0,

(6.6) a' + - CB (2ms + ~ ) - DB (ms - ~ ) = - 0.0146. 3

By using eq. (6.6) and 0 = 37.08 ~ from the analysis of the V ---> ee decay, we have the following result from eq. (6.5) independently of g:

(6.7) a ~ - ~ C B ( m s - N ) + D s ( m s + 2 ~ ) = -3 .116 .

The pole residue al (V) = m~gvxx/fv with the values of eqs. (4.6), (6.2) and (6.3) gives us g~s" and g,xx as

(6.8) g ~ = 21.21, g , ~ = - 9.492.

1064 s. FURUI, R. KOBAYASHI a n d M. NAKAGAWA

If we go back to eqs. (5.9) and (5.11) and insert the results eqs. (6.6), (6.7) and (6.8) into eq. (5.11), we have g = 5.934 which reproduces again the value of eq. (4.7) obtained from the decay (~o, ~) --, e6. But this is not the independent evaluation of the value of g, because we have used the expression eq. (4.3) in eq. (6.5) and the values of 0 and fy in the evaluation of eqs. (6.7) and (6.8), respectively. Main results of this section are eqs. (6.6) and (6.7) and they will be used in the evaluation of the s-quark pair content of the nucleon in the following section.

7. - R e s u l t s a n d d i s c u s s i o n .

We have constructed the effective Lagrangian for both the meson and the baryon in the symmetry-breaking hidden local theory. In our treatment, the symmetry breaking comes from the mass of the current quarks. We have shown that the low-energy theorem of the EMI holds. Our symmetry-breaking term transforms as (3, 3*) under U(3)L • UR(3) and preserves SU(2) • U(1) symmetry. We have directly shown that the Sakurai formula of y-V interaction holds and that our effective theory preserves the vanishing photon mass. We evaluated the fundamental coupling constant g and the mixing angle 0 by the analysis of the independent phenomena, the masses of vector mesons, the decay of V--~ e6 and used them in the analysis of the electromagnetic form factor. These evaluations of g and 0 show that our effective theory gives consistently the values of g and 0. The symmetry-breaking parameter given by eq. (3.5) are all consistently small except ms comparing with the symmetry limit parameter ag2f 2 as shown in eqs. (3.16) and (3.17). Here we list up the results of the evaluation of g and 0.

The value of the g estimated by Q --~ e~ decay is about 20% smaller than the others; this may be due to the other electromagnetic interaction and the mass splitting among the Q-mesons and the higher-order effects which are all neglected in our calculation. This may be called ,,Q-meson problem, together with the one of Q-pole contribution to the nucleon form factor as mentioned in the previous section. We leave this problem to a future study. It may be said that our theory shows that 0 = (37-39) ~ and g = 5.8-6.0 and this value of 0 indicates that ideal mixing is rather good and stable under the symmetry-breaking effects.

Equations (6.6) and (6.7) suggest the following values of the parameters a' and as:

(7.1) a' ~ 0.0, as ~ - 3.1,

if the breaking effects are small and negligible. It should be noted that this result supports the VDM and a Q-universality scenario (a' = 0) as was shown in ref. [2]. This

TABLE I.

Input Estimated values Comment

mass of V 0 = 39.16 ~ ms/m = 25.7 is assumed g = 5.76

V ~ ee 0 = 37.08 ~ r r decays g = 5.93

g = 5.03 Q decay


also gives an interesting sum rule for the vector meson pole contribution to the nucleon form factor as

(7.2) _ [ a I (0)) al (q~) 1 a' 1-2 ~ + - - = = 0 [ m~ m~ J '

under the neglect of symmetry-breaking effect. Equation (7.1) is a breaking of the OZI rule. In our interaction, the condition of the OZI rule is as follows. In eqs. (5.8) and (5.9), the interaction of co and ~) may be written as

(7.3) L(co, ~ ,2 r NT~'N],

where w s and ~ol states are

(7.4) 1 u 7)

I(A) I) -- ~ ( ]

+ ]dd) - 21s }),

+ I dd) +

Thus the vanishing of the s~ component in the interaction requires a constraint in terms of our parameters as

(7.5) as - 1. 1 - a'

Hence our result eq. (7.1) is a breaking of the 0ZI rule. We express the coupling constant of q~ pair to nucleon as

(7.6) t [uu+dd (NJ~)~=0 1 g[ V~ J= ~ (gs~wv + V~glx~),

1

Using eqs. (5.9), (6.6) and (6.7), we obtain

(7.7) g[s~(Jr = o] = V~ --2"101 = 0.41,

g [ ((u~ + dd)/V~)(X~)~ = o] 7.246

which is consistent with the result by Genz and HShler [5]. This result would also be interesting in view of the recent measurement of the spin structure of proton [8], which indicates a sizable s~ component in the nucleon.

One of the authors (SF) would like to express his sincere thanks to the financial support of Ibaraki Coop and to T. Fujiwara for helpful discussions.

1066

APPENDIX

S. FURUI, R. KOBAYASHI a n d M. NAKAGAWA

Nota t ions . - The U(3) matrices for baryon octet, pseudoscalar and vector meson nonet are as follows:

(A.1) B =

X ~ A - - X + p

I: o A X- - - - + - - N

v~ v~ ~- _ ~o V ~ - - - A

(A.2) 7 =

z o r/ + +

~ + v ~ ~ K y~o

n - - - - + ~/ K ~ v~ v~

K - ko V2

v~ '7

8

1 a~__l~a~a, = - ~ _

(A.3) 1 y=-~

0 ~ u s u.......L1 +

_ _ 0__~ ~ u s

K * - [C *~

u 1 + - -

v~

K*+

g *0

V~ ul

1 8 = _ ~ o V a + l z o u l .

2 a = 1 2

External gauge bosons are defined as follows:

(A.4) { BL~ = eQ(A~, - tg 0wZz) + 2gz T ~ Z~ + 2gwW~, ,

BR~ eQ(A~, tg 0wZ~),

where

(A.5)

e g~ - V 2 G F gz - , , 2 COS 0W sin 0w m~

e g~v _ GF

gw - 2 V~ sin 0w m~ V~

CHIRAL SYMMETRY BREAKING IN HIDDEN LOCAL SYMMETRY

(A.6) W, - -

0

cos 0 c W7

sin 0 c Wz

o ) cos cW~ s i n 0 c W J

0 0

0 0

and

(A.7) 1 TL = 1 Q = -~ - 1 , z -~ - 1 .

0 - 0 -

1067

R E F E R E N C E S

[1] BANDO M., KUGO T. and YAMAWAKI K., Phys. Rep., 164 (1988) 217; Prog. Theor. Phys., 73 (1985) 1541.

[2] FURUI S., KOBAYASHI R. and NAKAGAWA M., The gauge interaction of the baryons in hidden local symmetry, to be published in Nuovo Cimento A.

[3] BANDO M., KUGO T. and YAMAWAKI K., Nucl. Phys. B, 259 (1985) 493. [4] GASSER J. and LEUTWYLER H., Nucl. Phys. B, 250 (1985) 46. [5] GENZ H. and H(~HLER G., Phys. Lett. B, 61 (1976) 389; JOFFE R. J., Phys. Lett. B, 229 (1989)

275. [6] HOHLER G. and PIETARINEN E., Yucl. Phys. B, 95 (1975) 210. [7] FURUICHI S. and WATANABE K., Prog. Theor. Phys., 82 (1989) 581; 83 (1990) 565; see,

however, Prog. Theor. Phys., 92 (1994) 339. [8] THE EUROPEAN MUON COLLABORATION (J. ASHMAN et aL), Nucl. Phys. B, 328 (1989) 1.

Documents

Chiral symmetry breaking in hidden local symmetry