Calculating fπ in the consistent ladder approximation

Physics Letters B 286 (1992) 355-364 North-Holland

P HY$1C $ I_ETTER$ B

Calculatingf in the consistent ladder approximation Taichiro Kugo and Mark G. Mitchard Department of Physics, Kyoto University, Kyoto 606, Japan

Received 12 May 1992

We recalculate the pion decay constant f~ and the vacuum expectation value (~q/) in a new ladder approximation scheme to the Schwinger-Dyson and Bethe-Salpeter equations which is consistent both with the axial Ward-Takahashi identity and Z2= 1 condition (or the vector Ward identity in the abelian case). We find that our previous numerical results remain qualitatively unchanged: in particular, the Pagels-Stokar formula is a good approximation to f~ which agrees with the ladder-exact value to within 5%-30%.

1. Introduction In two previous papers [ 1,2 ] we used the improved ladder approximation [3,4 ] to the Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations to study dynamical chiral symmetry breaking in QCD. We found that this approximation gives surprisingly good predictions for the pion decay constant f~, the vacuum expectation value (q)~,) and the masses of the lowest lying vector and axial-vector mesons. Further, we claimed that the Pagels-Stokar [ 5 ] formula forf~ is a good approximation to the ladder-exact value. Recently Jain and Munczek [ 6 ] performed a somewhat similar calculation, but with very different results. In their formalism the Pagels-Stokar formula overestimates the value off~ by a factor of two or three. Unfortunately neither of these calculations is completely self-consistent. As Jain and Munczek found numerically, the axial vector Ward-Takahashi ( W T ) identity is violated in the Higashijima-Miransky improved ladder approximation that we used in our calculation. This leaves two options for normalising the pion BS wavefunction: it could be fixed either by considering the contribution from the one pion intermediate state to the four point fermion Green's function, or by using the pion to saturate the axial Ward-Takahashi ( W T ) identity at q = 0. These two normalisations differ by some thirty percent. To remedy this, Jain and Munczek proposed a calculation using the gluon m o m e n t u m squared as the argument for the running coupling. In this case the axial W T identity is satisfied, but another problem appears. The fermion wavefunction renormalisation, A (p2), which is equal to one in the Higashijima-Miransky approximation, now deviates significantly from one in the infra-red. This is inconsistent with the use of the running coupling in the ladder approximation as we now explain. Values of A (p2) = 1.5-2 are typical in the region which gives the dominant contribution to f~: this implies that the wavefunction renormalisation factor Z2 = 1.5-2 if we renormalise at these scales. However, the running coupling constant is, by definition, the coupling strength between fields normalised with weight one at the scale being considered: in the improved ladder approximation we use the running coupling at all scales and so must use quark fields satisfying Z2 = 1 independent of the renormalisation scale. This requires A (p2) = 1. In our last paper [ 7 ] we carefully analysed the status of the axial W T identity and the Z2 = 1 condition in the ladder approximation. We showed that in order to satisfy the axial W T identity it is necessary to use a running coupling which is a function of the gluon momentum, and that to keep the Z2 = 1 condition we must then work in a given non-local gauge. In this paper we summarise the ladder-exact formalism for the pion BS equation, 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

355

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30 July 1992

recalculatef~ in the consistent ladder approximation and present our numerical results. We find that the PagelsStokar value differs from the ladder-exact f~ by a factor in the range 5%-30%, in qualitative agreement with our earlier work. We therefore stand by our conclusion that the Pagels-Stokar formula is a reliable and useful approximation. For completeness, we also give a brief discussion of the fixed coupling constant case: in this case we showed [ 7 ] that a simple cutoffon the loop momentum in the SD and BS equations violates the axial WT identity. Here we present a calculation using a regularised gluon propagator, and we find a remarkable agreement between the Pagels-Stokar formula and our ladder-exact result.

2. The Schwinger-Dyson equation The consistent ladder approximation involves using a gluon propagator

iDu,,(P) =(gu,, - rl(p2 ) ~

) ~2 ,

(2.1)

with the gauge parameter function t/(x) being given in terms of the rescaled running coupling,

3C2(F)g2(x) 2(X) =

4rr2

(2.2)

,

by 2 x q(x)2(x)=~

dy

[y2(y)-y22'(y)l.

(2.3)

0

In this gauge the Schwinger-Dyson (SD) equation for the fermion propagator,

S(p2)]} -',

iSFl(p) = ~ +

g2( (PE --kE)2)G(F)

iD~,(P-k)~'~iSv(k)~'" ,

SF(p)=i{A(p2) [~(2.4)

becomes S(p2) = where

f d4kE 4 - q ( l 2) X(k~) 1-i-~-~x22 (l~) l~k~+X2(k~),

l=p- k. We set p~ =x,

A(p~)=l,

(2.5)

k~ = y and PE 'kE = X / ~ COS0 SOthat d4kE = 2try dy sin20 dO and rewrite eq. (2.5)

as

r(y) Z(X) = fd y~ dy Ks(x' y) y+~-T~y)

,

(2.6)

where the kernel Kz(x, y) is given by

K~(x, y) =

i sin20 d02(z) 4t/(z..____~), o

(2.7)

Z

with z = x + y - 2 x / ~ cos 0. We calculate numerical values for S by solving the differential form of eq. (2.3), x(t/2)'= 2 2 - 2~/2-2x2', and numerically integrating eq. (2.7) to calculate values of the kernel Kz(x, y) at the discretised values o f x and y we require. Then we iteratively update ~old-~,~newaccording to 356

Volume 286, number 3,4

~Vnew(X)=

i

y dy

PHYSICS LETTERSB

S~ew(Y)

K~(x, y) y ~ y )

0

~ola(Y)

+

X~(x, y) y+S~o~(y ) ,

30 July 1992

(2.8)

x

where we calculate 27n~w(X) for successively increasing values of x starting from x = 0. This iteration converges rather rapidly to the required solution.

3. The pion Bethe-Salpeter equation We want to calculate the BS amplitude Z for the pion in QCD-Iike theories. Separating the flavour ( f f ') and colour (i,j) indices from the spinor indices (a, fl), we define

f d4r e ipr (0 [Tq/c~z(x+ ½r)~ffJ(x- ½r) I rta(q) ) = X ' (2")if' ~ e -i°xZ~(p; q) ,

(3.1)

where Y is a normalisation constant which we will fix later, [7~a(q)) is the pion state, normalised so ( n a (q) Irib(k) ) = (2•) 3. 2qo~ab~3(q--k), and 2a is the Gell-Mann flavour matrix, normalised to tr (,~a2b) = ½6ab. p is the relative momentum, and q is the total bound state momentum: here q2 = 0 because we are working in the chiral limit where the pions are the massless Nambu-Goldstone bosons. Introducing the amputated BS amplitude )~,p(P; q) = - [S~-~ (P + ½q)z(P; q)S~(P - ½q) l-B,

(3.2)

the ladder BS equation for the bispinor amplitude Z is )~(P; q ) = f ~

d4k

g2 ( (Pr -kE)2)C2( g ) iD~,~(P-k )Y~')~(k; q)Y~ .

(3.3)

To solve this equation, we use the following expansion of the bispinors X and ~ into invariant amplitudes:

Z(P; q)Y5 =S(p; q) + [P(p; q)(P'q)l~+ Q(p; q)~] + ½T(p; q) [~, ~] , ;~(P; q)Y5 =~(P; q) + [P(P; q) (P'q)~+O.(P; q)~] + ½7~(P; q) [~, ~] •

(3.4)

Using charge conjugation and parity it is easy to see that the invariant amplitudes X-= S, P, ..., 0, 7~are all even functions in p.q. Since q 2= 0 the amplitudes X(p; q) may be expanded in powers ofp. q as

X(p; q) - - X ( p

(3.5)

2 ) + (p'q)2X l (p2) + ....

We will need only the first terms S(p2), p(p2) ..... ~p(p2) to calculate f,. The decay constant f~ is defined as usual by (0[@2 aye,Y5~l 7~b(q) ) = if=q~,fi,b. Using the definition of the pion BS amplitude, eqs. (3.1) and (3.4), we find N ~I x dx f~qu = - Y - ~N¢ - f ~ d4p tr[yuYsZ(P; " q) ]=Sq"=-~ [4Q(x) - x P ( x ) ] .

(3.6)

0

We now fix the normalisation X by considering the contribution to the axial WT identity

iqUF~u(p; q) =S7 ~(p - ½q)Ray5"[- 2 a e 5 a y 1 (p + ½q)

(3.7)

from the single pion intermediate state in the limit q ~ 0. Using the definitions of the BS amplitude and f=, we find f~ X ~ ( x ) = 2 Z ( x ) .

( 3.8 ) 357

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We choose S = ( ~ ) - J so that we simply have S ( x ) = S ( x ) . Then from eq. ( 3.6 ) we obtain the following exact formula for f~:

f~d~

f ~ =Nc j 167r2 [4Q(x)

-xP(x) 1 •

(3.9)

0

On the other hand there is a well-known normalisation condition for the BS amplitude which follows from the inhomogeneous BS equation for the four point Green's function. In terms of our BS amplitude this condition reads (in the ladder approximation)

Z(P;-q)

~-j~tr

Sv'(p+½q) Z(p;q)Sy'(p-½q)

+S~-'(p+½q)z(p;q)(o~uSy'(p-½q))]}=2qU ,

(3.10)

which after substituting eq. (3.4) and ~f"= (-~f~) -~ gives the following equation:

f~ = -2Nc

j-x~

S(x) [S(x) ( 1 + x S ' 2 - x X S " - 2 £ £ ') +xP(x) ( S - 2xS') + 2Q(x) ( x £ ' - 2 S ) -3xT(x) ].

0

(3.11) We will explicitly calculate the RHS ofeq. (3.11 ) and confirm that it agrees with the formula forf~ given in eq. (3.9). This gives a check that normalisation determined by the axial WT identity, eq. (3.8), is satisfied in our ladder approximation. We now discuss the evaluation of the amplitudes S(x), P(x), Q(x) and T(x). To zeroth order in q the relation between Z and 2, eq. (3.2), becomes 1

S ( y ) - y+S2(y) S(Y),

(3.12)

whilst to zeroth order the BS equation, eq. (3.3), gives ,~(x) = f y dy ~.ztx, .~ .

y)S(y) ,

(3.13)

where the kernel Kz(x, y) is the same kernel that appears in the SD equation eq. (2.6). Comparing eqs. (3.13 ), ( 3.12 ) and ( 2.6 ) we verify that S (x) = X(x) gives a solution to the B S equation ( 3.13 ) as the axial WT identity requires. Now we expand the BS equation, eq. (Y3), as far as the first order in p.q: we obtain the following set of coupled integral equations: (.

-xP(x) +4(~(x) = |

d

ydy

[K l l (x, y)P(y) +K,2(x,

y)Q(y) ]

-xP(x) + O_(x) = f y dy [I£21(x, y)P(y) +1£22(x, y)Q(y) ] J 67c T(X)=

~ -Y- dY ~ K 33(X,y)T(y)

with the kernels K~, ...,/£33 given by 358

(3.14)

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Kll(x,y)=

PHYSICS LETTERSB

(

-

kt) 2)

Z

Z2

i

sin2OdO2(z) [ 2 - r / ( z ) ] y + 2 q ( z )

i

sin2OdOA(z) 2r/(Z)z- 8 ,

30 July 1992

J ~

0

K 1 2 ( X ' Y) =

o

K21(x,y)= i sin2OdO2(z) k_~( [ 2 - r / ( z ) ] __ k.p +2r/(z) (k'l)(p'l)'~ 0

(p.l)2"~ Kz2(X, y)= i sin20dO 2 ( z ) ( q ( ~ - 2 -2q(z) ~ j , 0

K33(x,y)= i sin2OdO'~(z)q(z) [4(p.l)(k.l)-z(k'p)] 3XZ2

(3.15)

o

where z = x + y - 2 , ~ c o s O, k ' p = . ~ cos 0, k.l=k.p-yandp'l=x-k.p. To the same order the relation between Z and 2, eq. (3.2), gives

- xP(x) + 4() (x) = { - x [x+S 2(x) ] P ( x ) + 2 [2S 2(x) - x] Q(x) + 6x27(x) T(x) } + -xP(x) + (~(x) = [ x + Z 2(x)] [ -xP(x) + Q ( x ) ] +

2xS(x)S'(x) - 4 Z Z ( x ) x+S,2(x)

2xZ(x)S'(x) -ZZ(x) x + 272(x)

T(x) = { - 2S(x)Q(x) + [ S 2 ( x ) - x ] T ( x ) } + x +Z(x) S2(x) ,

(3.16)

where we have already substituted for S(x) in terms of Z(x) using S = 27 and eq. (3.12 ). Eliminating the right hand sides of eqs. (3.14) and (3.16), we obtain an inhomogeneous linear equation of the form

\T(y)/

\T(x)/

4. Numerical calculations

In our numerical calculations, we discretise eq. (3.17 ) so that it becomes a multidimensional linear equation which we then solve by inverting the coefficient matrix. We perform our calculations under the following conditions. For the QCD-like case, we use the same form of running coupling constant as in our previous calculations [ 1,2 ]: 2(x) follows the one loop running coupling for large x but is smoothly continued to a constant value in the infra-red. In terms of the parameter ,=-

1

kA QCDJ'

(41)

d2/dt decreases linearly to zero as t decreases from an infra-red cutoff tw to a constant point to, resulting in the definition 359

Volume 286, number 3,4

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2(x)=B-~XC,

ift~
×C

30 July 1992

1

X 1 +~t'

if to ~
ift>~tw,

(4.2)

with

C = 1 (tiF--to) -t- 1 2(l+tw) 2

l+tw

where B is given in terms of the second Casimir of the gauge group, C2(F), and the lowest order coefficient of the beta function, flo, as B=

3o 12C2(F) "

(4.3)

For SU (3) colour QCD with three quark flavours, B = 9 . We vary the infra-red cutoff tiv, but fix to= - 3 (actually we have done some calculations with other values of to, and we find that a change in to can always be compensated by a small adjustment in tIV). The results of our numerical calculation are fairly insensitive to this smoothing. At the ultraviolet end we simply cutoff the loop momentum at A, taking the y-integration in eq. (3.17) as fA2 dy. We fix the ultraviolet cutoff as in (A 2 / A 6 c D ) = 21. For the fixed coupling constant case, however, we showed [7 ] that the axial WT identity is violated if we use a simple cutoffA on the loop momentum in the SD equation (essentially because this equation is linearly divergent and a shift of integration variable is not allowed). In order to preserve the WT identity we must regularise the gluon propagator. Here we use iDR(p)=

(

g~ _ q ( p 2 )

p2 /kp2

p2-A2

,

(4.4)

which corresponds to taking 2 ( x ) = 3g----~2C2(F) 1 47~ 2 1+x/A

2 -

20 1+x/A2 "

(4.5)

In this case there is no natural scale parameter; we therefore define log (A 2) = 20 and adjust 2o so that X(0) ~ 1. The value we use is 2o= 1.08. We make sure that the cutoff on the integration imposed by our numerical discretisation is much greater than A 2. Now we show the results of our numerical calculations. The colour factor Arc is always set equal to 3 (even for the fixed coupling case). In fig. 1 we show results for the three triplet QCD case (with the cutoff tlF= --0.5). Fig. la shows the coupling 2(x), the gauge parameter q(x) and the solution to the SD equation, X ( x ) . For comparison we also show the solutions A (x) and B ( x ) of the Landau gauge SD equation. The amputated solutions to the BS equation, P ( x ) , O ( x ) and T(x), are shown in fig. lb whilst the asymptotic scaling behaviour of the solutions P ( x ) , Q ( x ) and T ( x ) are given in fig. lc. In fig. ld we plot the three inlegrands forf~: the ladderexact integrand of eq. (3.9), the BS amplitude normalisation integrand of eq. (3.11 ) and the Pagels-Stokar integrand. Fig. 2 contains the same information as fig. 1 but for the case of the fixed coupling constant. Fig. 3 is an expanded version of figs. 1d and 2d, showing in greater detail the differences between the various integrands for f~. For the three triplet QCD case, changing the infra-red cutoff tlv we get the results shown in fig. 4 (the smaller 360

Volume 286, number 3,4

PHYSICS LETTERS B

I(E, B)-r~,

30 July 1992

I(E'B)T

~--a

4(~)-~ .......

I - > .......... .A.. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

~ E"-,

o.~

-0

o.oT(P,O)-]LS(Tp

b)

~

•f/•

o 0.014 f . ( e x ~

c)

2(T?

d)

0.022

:r,a P /.~

~---f.(PS)

~/

0

0.12 ~

o 0.4(P, Q) 7)

x~T/E

)

"'",B

.

---

-Q

b)

i

xaP/~

~

f~(ex~tct)

d)

~k--- f.(norm.) 0

o

- 0.04

0

In(z/A~CD/,,~,-- 1

Fig. 1. Solutions of the Schwinger-Dyson and Bethe-Salpeter equations; three triplet QCD case. We plot the solutions for the coupling ofeq. (4.2) with B= 9 and tlv=-0.5. The mass scale is fixed by the condition ln(A~cD) = - 1. (a) The running coupling constant 2(x), the gauge parameter q(x) and the solution to the SD equation Z(x) [for comparison we also show the solutions A(x) and B(x) to the Landau gauge SD equation]; (b) the amputated BS amplitudes P(x), ()(x) and T(x); (c) the quantities x3p(x)/A(x), x2Q(x)/A(x) and x2T(x)/Z(x), which display the asymptotic behaviour of the solution; and (d) the integrands [with respect to ln(x)] of (f~)J~da.......~ eq. (3.9), (fn) . . . . . lisation eq. (3.11 ), and (fn) Pagels-Stokar-

- 0.003 -5

(~(norm.) 0

20

lna-

30

Fig. 2. Solutions of the Schwinger-Dyson and Bethe-Salpeter equations; fixed coupling case. We plot the solutions for the coupling of eq. (4.5) with 2o= 1.08 and log(A2) = 20. We show the same quantities as in fig. 1, but the solution A(x) to the Landau gauge SD equation is very close to one and is not shown.

tw, the larger the infra-red coupling c o n s t a n t ) . The range of cutoff values we use is that for which the values o f (f~) . . . . . usatio, and (f~)~add....... , agree (see note 2 below). F o r tw below - 0 . 7 the coupling becomes very steep a n d our numerical procedure is not sufficiently accurate, whilst for tw much above 0.4 the infra-red value o f the coupling becomes close to the critical value for the SD equation. In fig. 4 we plot: the three values off~; 27(0); AQCD, defined as the m o m e n t u m scale corresponding to t--- - 1, the value at which the coupling would diverge in the absence o f the infra-red cutoff; the bare and renormalised fermion condensate expectation values, A2

u = -

x+272(x),

4n---5

<~q/)R------((@q/)lOeV>=(@~>V\),(1GeV2)]

(4.6)

0

and the constituent quark mass defined by m . . . . t = X ( 4 m .2. . . , ) . We set the scale by fixing f~ (ladder-exact) = 93 MeV, which is shown as a thick line in the plot. The corresponding numerical values are shown in table I. F o r the fixed coupling constant case we take in (A 2) = 20 and 20 = 1.08, which results in a mass unit set by 27(0) =0.721. We find (f~)~dd ....... t = 0 . 2 6 3 , (f~) . . . . . ~isatio,= 0.263, and (f~)Pag¢~s_Stokar=0.262. 361

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0.014 I

~

f~(exact)

f~(PS) f~(norm.) 1000 - c(o)

0 - 0.004 -4 0.022

0 b)

/

AQCD

4 f~(exact)

500

f~(PS) /

~:~,_~

0 0 0 @

o~QOO

f~(norm )

o° x

x

x

×

//

x X~x o

o

Q •

oo A( s2 93

0

- -

×

Q Q •

Q ~ m m

=

~-f~(norm.)

--5

0

0

-

Fig. 3. Integrands for f,. We plot an expanded view of the lower sections of figs, 1 and 2 showing the integrands of (f.)~aader-ex,c, eq. (3.9), (f~). . . . . . l i s a t i o n eq. (3.11 ), and (fn)Pagels-Stokar. The upper graph is the QCD case, the lower one fixed coupling.

0.8

0

0.5

Fig, 4, Value of physical quantities. The infra-red cutoff (by) dependence of various quantities are plotted for the three triplet QCD case. The ladder-exact f, is always used as an input parameter to fix the mass scale.

Table 1 Numerical results. B= 9 (three triplet QCD case), the mass unit is MeV, and the scale is fixed by setting (fn)ladaer_exact= 93.0 MeV. IlF

(fg)PS

(fn) .....

(fn) ....

~(0)

AQC D

(~7///) ~/3

(~ly) ~./3

m .....

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

122.6 116.0 112.1 109.3 107.2 105.4 103.9 102.5 101.3 100.2 99.2 98.4

93.0 93.0 93.0 93.0 93.0 93.0 93.0 93.0 93.0 93.0 93.0 93.0

92.4 92.4 92.4 92.4 92.4 92.4 92.4 92.4 92.4 92.4 92.4 92.4

897 851 799 742 685 633 586 544 508 475 446 420

481 506 531 555 579 606 635 669 708 755 813 883

346 352 356 360 364 369 374 380 388 396 407 419

233 234 235 235 235 235 235 235 234 232 228 217

387 360 343 331 321 312 304 297 290 283 277 271

We m a k e the following o b s e r v a t i o n s : ( 1 ) T h e P a g e l s - S t o k a r a p p r o x i m a t i o n r e m a i n s rather good, although for the Q C D - l i k e case the d i s c r e p a n c y b e t w e e n the P a g e l s - S t o k a r a n d ladder-exact values is rather greater t h a n in o u r p r e v i o u s calculation. N u m e r i cally we have

362

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(fn) Page|s-Stokar

PHYSICS LETTERS B

= ~ 1.0

fixed coupling,

=1.32

forB=9, hv=-0.7 ,

=1.06

forB= 9, hF=0.4.

30 July 1992

f n ) ladder-exact

(4.7)

(2) In our previous paper we noted that the axial Ward-Takahashi ( W T ) identity is satisfied up to a factor C2(F)

1-gZ(A2) ~

[ 2 + q ( A 2 ) ] = 1 - ~ 4 2 ( A 2) [ 2 + q ( A 2 ) ] ,

(4.8)

when the loop m o m e n t u m integration is cut off at a finite value k 2 = A 2. In our simulation we used IA= 20, SO that 2 (A 2 ) = ~ ( 1 + 2 0 ) - ~; taking r/(A 2) = 1, the factor in eq. (4.8) is 0.989. Since in our normalisation one side o f the W T identity is proportional to 0 c ~ )1add....... t and the other to ( f 2 ) . . . . . lisation we expect 2 ( f n) . . . . . lisation -- 0.989 . 0C~)ladd ....... t

(4.9)

This agrees well with the calculated value ( 9 2 . 4 / 9 3 ) 2 = 0 . 9 8 7 . (3) The renormalisation of ( q ~ ) remains good. We obtain a stable value for the renormalised VEV, ( ( ~ ¢ ) IGev ) = ( ~ ¢ ) R, almost independent o f the infra-red cutoff parameter tIF, while the unrenormalised VEV ( ( q g / ) n ) --- ( ~ ) u depends heavily on tiv. Numerically, the value for the three triplet Q C D case is -

(~u) ~/3 -

2.5,

(4.10)

A

stable over a wide range of hr. Puttingf~ = 93 MeV, - (q~q/) R = (235 MeV) 3 .

(4.11 )

This is slightly larger than we obtained previously, but is still in excellent agreement with the value given by Gasser and Leutwyler [ 8 ] - ( ~ u ) a = (225_+ 25 MeV) 3 .

(4.12)

(4) 27(0) varies by a factor of more than two as we vary fiF. m .... ~ is a little more stable: we obtain values within the range 330 + 60 MeV, a not unreasonable range for the constituent quark mass. (5) AQCDalso depends on tIV. In our previous calculation, the variation was negligible for the lower hE region, whilst now AQCD varies monotonically taking values between 480 and 880 MeV. However, note that what is actually relevant to our calculation is not the AQCD but rather the coupling constant in the low energy region around 700 MeV which gives the dominant contribution to f~. The Q C D coupling constant as (/t) at/~ = 700 MeV is a little more stable than AQCD, taking values around as(700 MeV) = 1.8-2.8 for hv in the range from - 0.7 to 0.4. These values are larger than obtained by extrapolating c~s(Mz) ~ 0.11 using the one loop or two loop RGE. But extrapolation to the region where a is large is unreliable, so we must wait for some other measurement to be performed in the future.

Acknowledgement

We wish to thank K.-I. Aoki and especially T. Maskawa for valuable discussions and comments. T.K. is sup363

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30 July 1992

p o r t e d in part by the G r a n t - i n - A i d for C o o p e r a t i v e R e s e a r c h ( # 0 2 3 0 2 0 2 0 ) a n d the G r a n t - i n - A i d for Scientific R e s e a r c h ( # 0 2 6 4 0 2 2 5 ) f r o m the M i n i s t r y o f E d u c a t i o n , Science a n d Culture. M . G . M . thanks the E u r o p e a n C o m m u n i t y for f i n a n c i a l support.

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