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Scheme dependence of the QCD R ratio in O(α3s) perturbation theory
Volume 213, number 2
PHYSICS LETTERS B
20 October 1988
SCHEME DEPENDENCE OF THE QCD R RATIO IN O(o~3) PERTURBATION THEORY C.J. M A X W E L L and J.A. N I C H O L L S Department of Physics University of Durham, Durham DH1 3LE, UK Received 3 June 1988
We use a recent calculation of the QCD R ratio to O (a 3) in the MS scheme, to investigate the differences between the PMS, PMS, and effective charge approaches for fixing the renormalization scheme. Estimates for A ~ are given by fitting to combined PETRA and PEP data.
Recently a calculation of the Q C D R ratio, R = a~o,(e+e - --,hadrons) a(e+e-__,tx+~t - ) , to O ( a 3 ) in the MS scheme has been completed by Gorishny et al. [ 1 ]. This constitutes the first next-tonext-to-leading order perturbative Q C D calculation o f a physical quantity. Removing the zeroth-order parton-model part of the result and defining /~_R-
3 YQF 3YQ{ '
(1)
we have a perturbation series / ~ = a ( 1 +r~a+r2a2+...) ,
(2)
where a-= as/zr. For Nr massless quark flavours, and in the MS renormalization scheme, Gorishny et al. [ 1 ] obtain
/~(N)=a(N)
X(l+rla~')+r2a(X)Z+...+rN_la~N)N-l).
(4)
The couplant a ~N) in Nth order is taken to satisfy the truncated//-function equation
rMS = (1.986--0.115 N r ) , r Ms = (70.985-- 1.2 N f - 0 . 0 0 5 N~) (yQf)2 3~Q~ 1.679.
to worry about the renormalization scheme dependence of the result. There is nothing sacrosanct about the MS scheme; it is just calculationally convenient. In this letter we wish to investigate the results obtained using other methods for fixing the renormalization scheme. It is not our purpose here to motivate or promote any of these methods. We shall consider the Principle o f Minimal Sensitivity ( P M S ) criterion proposed by Stevenson [2] the modified PMS ( P M S ) p r o p o s a l o f Maxwell [ 3 ], and the effective charge (EC) method of Grunberg [4]. We briefly outline these methods before giving numerical results. For a quantity having a perturbation series o f the form of eq. (2) we define the Nth order perturbative approximant
b oa Oz = - b a 2 ( l + c a + c z a 2 + . . . + c ~ (3)
The terms proportional to (YQr) 2 arise from upsilon decay-like diagrams involving a quark loop and three exchanged gluons, such that the final quark flayour is uncorrelated with that o f the loop. The large r Ms coefficient will give a sizeable correction to the MS prediction, and so it is important 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
.- l a N - I ) '
(5)
where z = b l n ( l z / A ) , b = ( 3 3 - 2 N f ) / 6 and [5] c = ( 1 5 3 - 19 N f ) / 2 ( 3 3 - 2 N O . Eq. (5) can be integrated to yield a transcendental equation for a t~') For N = 2 l'2 = ~
1
+cln
(ca'Z) ~ 1 +ca~2)J '
(6)
217
Volume 213, number 2
PHYSICS LETTERS B
and for N = 3 with 4c2 > c 2
20 October 1988
Table 1 /~(2,, /q,3, in the M-~, PMS~,PMS EC schemes, x/]=34GeV, Nf= 5,/IM--S= 100 MeV.
r3 = l / a(3) +c ln ca(3)-½c lnl l + cat3) +c2a (3)2]
m
2C,-C2[ -
(2c2aA3)+c)
+ --:7--- [arctan A= (4c2 - c 2) 1/2
-arctan (A)], (7)
The boundary condition applied to eq. (5) corresponds to a definition of the Q C D A parameter (A) which differs by a factor ( 2 c / b ) - ' / b from the conventional one [ 2 ], A~g = A ~ g ( 2 c / b y /b .
(8)
The N = 3 result for all cases is given in ref. [ 3 ]. The N = 2 result/~(2~ depends only on the choice of renormalization point r, and the N = 3 result on z and c2. For given values of z and c2, a (2 ~and a ~3) are obtained by solving the transcendental equations (6) and (7). It is useful to define a set of scheme invariant quantities p, [ 2 ]. po = z - - r l , Pl ~C
,
P2 "~C2 -I-r2 - r , c - r {
.
N
MS
PMS
PMS
EC
2 3
0.0402 0.0439
0.0404 0.0412
0.0404 0.0453
0.0404 0.0454
0R (3) 0R (3) 0~--- 0c~ =0.
(13)
These are equivalent to solving the simultaneous transcendental equations (c2 +2rl c + 3r2) + a ( 2 r l c2 + 3r2c) +a2(3r2c2) = 0 , (14)
io ( l + c x + c 2 x 2 ) 2 -
a
l+(c+2r~)a'
(15)
for c2 and a. The r, above are implicit functions of a and c2 via equations ( 7 ) and (9). The PMS procedure [3] sets the scheme-dependent B-function coefficients to zero (c2=c3 . . . . . 0) and optimizes the value of r alone, as in eq. (10). For N = 2 PMS and PM'-~-~are identical. For N = 3 the condition requires solving
(9)
( 1 +ca)rz(r) + ~crl (z) = 0 . Using the scheme invariance of these quantities, rt and re can be calculated for given choices of z and c> The PMS criterion of Stevenson [ 2 ] requires that one picks the scheme in which the result is least sensitive to small changes in the renormalization scheme. At second order, N = 2, the condition is dR~2)(r) = 0 . dr
(10)
This corresponds to solving the transcendental equation
The effective charge approach of Grunberg [4] picks the scheme so that the couplant a is the physical quantity/~. That is/~ ~:) --"EC-" (~v) in all orders. This can be achieved by setting z=po for N = 2 , and z=po, c2 =P2 for N = 3. We tabulate in tables 1 and 2 the MS, PMS, PMS and EC results for /~(2) and ,g(3~. We shall take x / s = 3 4 G e V , N r = 5 and A ~ - g = 1 0 0 M e V (table 1) and A~-s = 5 0 0 MeV (table 2). For the MS predictions one takes r~-g = b In (x/s/A~g) and [ 6 ] cMS
a1 + c l n
ca
+ 21+ca=P°"
(1 + c a )
2857 5 0 3 3 R]- ..I.. 325 R]- 2 2 - - 18 l V l ' ~ 54 1* f
16(ll-~Nf) Table 2 As table 1 but A~s = 500 MeV.
(12)
At third order, N = 3, the two conditions to be satisfied are 218
~
(11)
The solution a = a then gives Rp~is-
(16)
N
MS
PMS
PMS
EC
2 3
0.0543 0.0631
0.0550 0.0470
0.0550 0.0694
0.0549 0.0704
Volume 213, number 2
PHYSICS LETTERS B
F o r completeness we give in table 3 the couplant a, and perturbative coefficients rj, r2 for each scheme. One expects the PMS and EC results to be very close to each other, and indeed this is seen to be the case. This expectation m a y be m o t i v a t e d by a p p r o x i m a t ing the rather c o m p l i c a t e d transcendental equations ( 12 ) a n d ( 1 4 ) , ( 15 ) by e x p a n d i n g t h e m in a to one order b e y o n d the o r d e r o f p e r t u r b a t i o n theory being studied. This was first suggested by Pennington and elaborated by Wrigley [ 7 ]. The result o f this approximation is that the perturbation series coefficients are proportional to the fl-function coefficients in the P M S scheme,
1 Gl 2. )
r,,= - -
n+l
r,,
(17)
F o r N = 2 one has r = P o - ½cand r~ = - ½c which for Po >> c will be close to the EC result which corresponds to ZEc=P0 and rEC = 0 . F o r N = 3 one has ~= Po and (2=3/2p2 to be c o m p a r e with ~Ec=P0 a n d c~ c =P2. Thus, if the a p p r o x i m a t i o n o f Pennington a n d Wrigley is believable, the N = 3 P M S a n d EC results should be close to each other. However, for application ( , ~ = 3 4 G e V , A ~ - ~ = 1 0 0 M e V ) , c2= 93.77 >> 1 a n d so the expansion o f e q . ( 15 ) inherent in the P e n n i n g t o n - W r i g l e y a p p r o x i m a t i o n should be unreliable. Nevertheless, the results in the tables, which we stress are based on exactly solving eqs. (14) and ( 15 ) on a computer, do show that the P M S and EC a p p r o x i m a n t s r e m a i n close for N = 3 , and give
20 October 1988
values ( ~ - g = 1 0 0 M e V ) o f f = 2 2 . 2 9 , g2=95.57 which are near to the a p p r o x i m a t e estimates ofpo, 3 / 2p2. The MS, PMS, EC m e t h o d s give positive 10-20% corrections to the N = 2 result for/~. The N = 3 P M S condition, eq. (16), which corresponds to d t ~ - s / d r = 0 with c2 = 0 does not have a solution. Due to the shape o f the N = 3 P M S approxi m a n t as a function o f r one would require a p o i n t o f inflexion, which is rather unlikely to occur. One seeks instead to m i n i m i z e the slope, and looks for a zero o f the second derivative. A zero corresponding to a m i n i m u m o f the slope does exist for Po > 20 and for such values ofpo the correction in going from R~Z~s to/~ is positive ( 2 - 4 % ) and smaller than the correction in the other schemes. A ~ = 1 0 0 M e V (Po = 20.933) falls in this range. F o r Po < 20 the second derivative ceases to have a zero and one is forced to resort instead to seeking zeros o f third and higher derivatives. One might doubt in this case whether opt i m i z a t i o n o f z has any meaning. The ,4M~= 500 MeV P M S result quoted in table 2 corresponds to a zero o f fourth derivative and so should p r o b a b l y be disregarded. We note, however, that at LEP energies P M S will have a m i n i m u m o f the slope for the whole range o f A ~ up to 500 MeV, and the corrections going from N = 2 to N = 3 will be small 0 ( 2 3%) and positive. As far as PM'--'~is concerned ~ / s = 34 G e V appears to be too small an energy for perturbation theory for the R ratio to be reliably optimized. We shall finally perform some fits for A m using
Table 3 The couplant a, and perturbative coefficients r~, r2 for each scheme. A~ (MeV)
N
MS
PMS
PMS
EC
100
2
a r~
0.0381 1.411
0.0415 -0.5990
0.0415 -0.5990
0.0404 0
540
2
a r~
0.0507 1.411
0.0569 -0.5882
0.0569 -0.5882
0.0549 0
100
3
a rl r2
0.0382 1.411 64.81
0.0301 8.120 138.7
0.0452 1.363 -29.48
0.0454 0 0
500
3
a rl r,
0.0509 1.411 64.81
0.0258 19.607 471.7
0.0690 1.988 -27.59
0.0704 0 0
219
Volume 213, number 2
PHYSICS LETTERS B
combined P E P / P E T R A data. Using the quoted [8 ] % ( 3 4 G e V ) = 0 . 1 4 5 +0.019 in the MS scheme with /~=x/s fitted to the O(oq2) result, one can infer /~cxp= 0.049 + 0.007. We then find the corresponding Afi-g for each of the renormalization scheme choices, working to second and third order. For N = 3 we cannot perform a PM~~---Sfit at x / s = 34 GeV as discussed above. We have converted our A~--g to the usual A~-~ using eq. (8). For N = 2 and MS, P M S / P M S , EC respectively we find A ~ - 2- - ~t:+233 MeV, 9~a+219 MeV, uu-152 ~' J-- 144 +~'~O 255_Ta5 MeV. For N = 3 and MS, PMS, EC we find A~-g= 1~2 +123 MeV, 1. ".} .£ +. 9 1 67 MeV, l1~-,_65 '~IA+88 MeV. --' - - 8 4 We note that the A~g estimate is halved by including the O (tx 3) corrections. The A~g estimates for M S / P M S / E C are in reasonable accord in each order of perturbation theory. We have found that to O(c~3), MS, PMS, EC choices for the renormalization scheme give 10-20% corrections to the O (ce ~) result for the quantity/~ defined in eq. ( 1 ). This corresponds to a correction o f less than 1% to the value of R, which has a large ze_roth order parton model contribution. The PMS scheme for sufficiently large po ( , ~ ) , where the slope
220
20 October 1988
of the N = 3 approximant has a minimum, gives smaller O ( 2 - 4 % ) corrections to/~ (2) than the other schemes. The EC and PMS results are extremely close to each other in O ( a 3). We would like to thank Patrick Aurenche for bringing the calculation o f Gorishny et al. to our attention and James Stirling for useful discussions. We thank Paul Stevenson for useful comments and for explaining the origin of the (Y~Qf) 2 terms in eq. (3).
References [ 1] S.G. Gorishny, A.L. Kataev and S.A. Larin, Proc. Hadron Structure '87 Conf. (Bratislava, 1987). [2] P.M. Stevenson, Phys. Rev. D 23 ( 1981 ) 2916. [3] C.J. Maxwell, Phys. Rev. D 28 (1983) 2037. [4] G. Grunberg, Phys. Lett. B 95 (1980) 70. [5] D.R.T. Jones, Nucl. Phys. B 75 (1974) 531; W. Caswell, Phys. Len. B 33 (1974) 244. [6] O.V. Tarasov, A.A. Vladimirov and A.Yu. Zharkov, Phys. Len. B 93 (1980) 429. [ 7 ] M.R. Pennington, Phys. Rev. D 26 (1982) 2048; J.C. Wrigley, Phys. Rev. D 27 (1983) 1965. [8] W, De Boer, SLAC preprint SLAC PUB-4428 (1987).
PHYSICS LETTERS B
20 October 1988
SCHEME DEPENDENCE OF THE QCD R RATIO IN O(o~3) PERTURBATION THEORY C.J. M A X W E L L and J.A. N I C H O L L S Department of Physics University of Durham, Durham DH1 3LE, UK Received 3 June 1988
We use a recent calculation of the QCD R ratio to O (a 3) in the MS scheme, to investigate the differences between the PMS, PMS, and effective charge approaches for fixing the renormalization scheme. Estimates for A ~ are given by fitting to combined PETRA and PEP data.
Recently a calculation of the Q C D R ratio, R = a~o,(e+e - --,hadrons) a(e+e-__,tx+~t - ) , to O ( a 3 ) in the MS scheme has been completed by Gorishny et al. [ 1 ]. This constitutes the first next-tonext-to-leading order perturbative Q C D calculation o f a physical quantity. Removing the zeroth-order parton-model part of the result and defining /~_R-
3 YQF 3YQ{ '
(1)
we have a perturbation series / ~ = a ( 1 +r~a+r2a2+...) ,
(2)
where a-= as/zr. For Nr massless quark flavours, and in the MS renormalization scheme, Gorishny et al. [ 1 ] obtain
/~(N)=a(N)
X(l+rla~')+r2a(X)Z+...+rN_la~N)N-l).
(4)
The couplant a ~N) in Nth order is taken to satisfy the truncated//-function equation
rMS = (1.986--0.115 N r ) , r Ms = (70.985-- 1.2 N f - 0 . 0 0 5 N~) (yQf)2 3~Q~ 1.679.
to worry about the renormalization scheme dependence of the result. There is nothing sacrosanct about the MS scheme; it is just calculationally convenient. In this letter we wish to investigate the results obtained using other methods for fixing the renormalization scheme. It is not our purpose here to motivate or promote any of these methods. We shall consider the Principle o f Minimal Sensitivity ( P M S ) criterion proposed by Stevenson [2] the modified PMS ( P M S ) p r o p o s a l o f Maxwell [ 3 ], and the effective charge (EC) method of Grunberg [4]. We briefly outline these methods before giving numerical results. For a quantity having a perturbation series o f the form of eq. (2) we define the Nth order perturbative approximant
b oa Oz = - b a 2 ( l + c a + c z a 2 + . . . + c ~ (3)
The terms proportional to (YQr) 2 arise from upsilon decay-like diagrams involving a quark loop and three exchanged gluons, such that the final quark flayour is uncorrelated with that o f the loop. The large r Ms coefficient will give a sizeable correction to the MS prediction, and so it is important 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
.- l a N - I ) '
(5)
where z = b l n ( l z / A ) , b = ( 3 3 - 2 N f ) / 6 and [5] c = ( 1 5 3 - 19 N f ) / 2 ( 3 3 - 2 N O . Eq. (5) can be integrated to yield a transcendental equation for a t~') For N = 2 l'2 = ~
1
+cln
(ca'Z) ~ 1 +ca~2)J '
(6)
217
Volume 213, number 2
PHYSICS LETTERS B
and for N = 3 with 4c2 > c 2
20 October 1988
Table 1 /~(2,, /q,3, in the M-~, PMS~,PMS EC schemes, x/]=34GeV, Nf= 5,/IM--S= 100 MeV.
r3 = l / a(3) +c ln ca(3)-½c lnl l + cat3) +c2a (3)2]
m
2C,-C2[ -
(2c2aA3)+c)
+ --:7--- [arctan A= (4c2 - c 2) 1/2
-arctan (A)], (7)
The boundary condition applied to eq. (5) corresponds to a definition of the Q C D A parameter (A) which differs by a factor ( 2 c / b ) - ' / b from the conventional one [ 2 ], A~g = A ~ g ( 2 c / b y /b .
(8)
The N = 3 result for all cases is given in ref. [ 3 ]. The N = 2 result/~(2~ depends only on the choice of renormalization point r, and the N = 3 result on z and c2. For given values of z and c2, a (2 ~and a ~3) are obtained by solving the transcendental equations (6) and (7). It is useful to define a set of scheme invariant quantities p, [ 2 ]. po = z - - r l , Pl ~C
,
P2 "~C2 -I-r2 - r , c - r {
.
N
MS
PMS
PMS
EC
2 3
0.0402 0.0439
0.0404 0.0412
0.0404 0.0453
0.0404 0.0454
0R (3) 0R (3) 0~--- 0c~ =0.
(13)
These are equivalent to solving the simultaneous transcendental equations (c2 +2rl c + 3r2) + a ( 2 r l c2 + 3r2c) +a2(3r2c2) = 0 , (14)
io ( l + c x + c 2 x 2 ) 2 -
a
l+(c+2r~)a'
(15)
for c2 and a. The r, above are implicit functions of a and c2 via equations ( 7 ) and (9). The PMS procedure [3] sets the scheme-dependent B-function coefficients to zero (c2=c3 . . . . . 0) and optimizes the value of r alone, as in eq. (10). For N = 2 PMS and PM'-~-~are identical. For N = 3 the condition requires solving
(9)
( 1 +ca)rz(r) + ~crl (z) = 0 . Using the scheme invariance of these quantities, rt and re can be calculated for given choices of z and c> The PMS criterion of Stevenson [ 2 ] requires that one picks the scheme in which the result is least sensitive to small changes in the renormalization scheme. At second order, N = 2, the condition is dR~2)(r) = 0 . dr
(10)
This corresponds to solving the transcendental equation
The effective charge approach of Grunberg [4] picks the scheme so that the couplant a is the physical quantity/~. That is/~ ~:) --"EC-" (~v) in all orders. This can be achieved by setting z=po for N = 2 , and z=po, c2 =P2 for N = 3. We tabulate in tables 1 and 2 the MS, PMS, PMS and EC results for /~(2) and ,g(3~. We shall take x / s = 3 4 G e V , N r = 5 and A ~ - g = 1 0 0 M e V (table 1) and A~-s = 5 0 0 MeV (table 2). For the MS predictions one takes r~-g = b In (x/s/A~g) and [ 6 ] cMS
a1 + c l n
ca
+ 21+ca=P°"
(1 + c a )
2857 5 0 3 3 R]- ..I.. 325 R]- 2 2 - - 18 l V l ' ~ 54 1* f
16(ll-~Nf) Table 2 As table 1 but A~s = 500 MeV.
(12)
At third order, N = 3, the two conditions to be satisfied are 218
~
(11)
The solution a = a then gives Rp~is-
(16)
N
MS
PMS
PMS
EC
2 3
0.0543 0.0631
0.0550 0.0470
0.0550 0.0694
0.0549 0.0704
Volume 213, number 2
PHYSICS LETTERS B
F o r completeness we give in table 3 the couplant a, and perturbative coefficients rj, r2 for each scheme. One expects the PMS and EC results to be very close to each other, and indeed this is seen to be the case. This expectation m a y be m o t i v a t e d by a p p r o x i m a t ing the rather c o m p l i c a t e d transcendental equations ( 12 ) a n d ( 1 4 ) , ( 15 ) by e x p a n d i n g t h e m in a to one order b e y o n d the o r d e r o f p e r t u r b a t i o n theory being studied. This was first suggested by Pennington and elaborated by Wrigley [ 7 ]. The result o f this approximation is that the perturbation series coefficients are proportional to the fl-function coefficients in the P M S scheme,
1 Gl 2. )
r,,= - -
n+l
r,,
(17)
F o r N = 2 one has r = P o - ½cand r~ = - ½c which for Po >> c will be close to the EC result which corresponds to ZEc=P0 and rEC = 0 . F o r N = 3 one has ~= Po and (2=3/2p2 to be c o m p a r e with ~Ec=P0 a n d c~ c =P2. Thus, if the a p p r o x i m a t i o n o f Pennington a n d Wrigley is believable, the N = 3 P M S a n d EC results should be close to each other. However, for application ( , ~ = 3 4 G e V , A ~ - ~ = 1 0 0 M e V ) , c2= 93.77 >> 1 a n d so the expansion o f e q . ( 15 ) inherent in the P e n n i n g t o n - W r i g l e y a p p r o x i m a t i o n should be unreliable. Nevertheless, the results in the tables, which we stress are based on exactly solving eqs. (14) and ( 15 ) on a computer, do show that the P M S and EC a p p r o x i m a n t s r e m a i n close for N = 3 , and give
20 October 1988
values ( ~ - g = 1 0 0 M e V ) o f f = 2 2 . 2 9 , g2=95.57 which are near to the a p p r o x i m a t e estimates ofpo, 3 / 2p2. The MS, PMS, EC m e t h o d s give positive 10-20% corrections to the N = 2 result for/~. The N = 3 P M S condition, eq. (16), which corresponds to d t ~ - s / d r = 0 with c2 = 0 does not have a solution. Due to the shape o f the N = 3 P M S approxi m a n t as a function o f r one would require a p o i n t o f inflexion, which is rather unlikely to occur. One seeks instead to m i n i m i z e the slope, and looks for a zero o f the second derivative. A zero corresponding to a m i n i m u m o f the slope does exist for Po > 20 and for such values ofpo the correction in going from R~Z~s to/~ is positive ( 2 - 4 % ) and smaller than the correction in the other schemes. A ~ = 1 0 0 M e V (Po = 20.933) falls in this range. F o r Po < 20 the second derivative ceases to have a zero and one is forced to resort instead to seeking zeros o f third and higher derivatives. One might doubt in this case whether opt i m i z a t i o n o f z has any meaning. The ,4M~= 500 MeV P M S result quoted in table 2 corresponds to a zero o f fourth derivative and so should p r o b a b l y be disregarded. We note, however, that at LEP energies P M S will have a m i n i m u m o f the slope for the whole range o f A ~ up to 500 MeV, and the corrections going from N = 2 to N = 3 will be small 0 ( 2 3%) and positive. As far as PM'--'~is concerned ~ / s = 34 G e V appears to be too small an energy for perturbation theory for the R ratio to be reliably optimized. We shall finally perform some fits for A m using
Table 3 The couplant a, and perturbative coefficients r~, r2 for each scheme. A~ (MeV)
N
MS
PMS
PMS
EC
100
2
a r~
0.0381 1.411
0.0415 -0.5990
0.0415 -0.5990
0.0404 0
540
2
a r~
0.0507 1.411
0.0569 -0.5882
0.0569 -0.5882
0.0549 0
100
3
a rl r2
0.0382 1.411 64.81
0.0301 8.120 138.7
0.0452 1.363 -29.48
0.0454 0 0
500
3
a rl r,
0.0509 1.411 64.81
0.0258 19.607 471.7
0.0690 1.988 -27.59
0.0704 0 0
219
Volume 213, number 2
PHYSICS LETTERS B
combined P E P / P E T R A data. Using the quoted [8 ] % ( 3 4 G e V ) = 0 . 1 4 5 +0.019 in the MS scheme with /~=x/s fitted to the O(oq2) result, one can infer /~cxp= 0.049 + 0.007. We then find the corresponding Afi-g for each of the renormalization scheme choices, working to second and third order. For N = 3 we cannot perform a PM~~---Sfit at x / s = 34 GeV as discussed above. We have converted our A~--g to the usual A~-~ using eq. (8). For N = 2 and MS, P M S / P M S , EC respectively we find A ~ - 2- - ~t:+233 MeV, 9~a+219 MeV, uu-152 ~' J-- 144 +~'~O 255_Ta5 MeV. For N = 3 and MS, PMS, EC we find A~-g= 1~2 +123 MeV, 1. ".} .£ +. 9 1 67 MeV, l1~-,_65 '~IA+88 MeV. --' - - 8 4 We note that the A~g estimate is halved by including the O (tx 3) corrections. The A~g estimates for M S / P M S / E C are in reasonable accord in each order of perturbation theory. We have found that to O(c~3), MS, PMS, EC choices for the renormalization scheme give 10-20% corrections to the O (ce ~) result for the quantity/~ defined in eq. ( 1 ). This corresponds to a correction o f less than 1% to the value of R, which has a large ze_roth order parton model contribution. The PMS scheme for sufficiently large po ( , ~ ) , where the slope
220
20 October 1988
of the N = 3 approximant has a minimum, gives smaller O ( 2 - 4 % ) corrections to/~ (2) than the other schemes. The EC and PMS results are extremely close to each other in O ( a 3). We would like to thank Patrick Aurenche for bringing the calculation o f Gorishny et al. to our attention and James Stirling for useful discussions. We thank Paul Stevenson for useful comments and for explaining the origin of the (Y~Qf) 2 terms in eq. (3).
References [ 1] S.G. Gorishny, A.L. Kataev and S.A. Larin, Proc. Hadron Structure '87 Conf. (Bratislava, 1987). [2] P.M. Stevenson, Phys. Rev. D 23 ( 1981 ) 2916. [3] C.J. Maxwell, Phys. Rev. D 28 (1983) 2037. [4] G. Grunberg, Phys. Lett. B 95 (1980) 70. [5] D.R.T. Jones, Nucl. Phys. B 75 (1974) 531; W. Caswell, Phys. Len. B 33 (1974) 244. [6] O.V. Tarasov, A.A. Vladimirov and A.Yu. Zharkov, Phys. Len. B 93 (1980) 429. [ 7 ] M.R. Pennington, Phys. Rev. D 26 (1982) 2048; J.C. Wrigley, Phys. Rev. D 27 (1983) 1965. [8] W, De Boer, SLAC preprint SLAC PUB-4428 (1987).