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The order αs2 energy-energy correlation function at small angles
Nuclear Physics B214 (1983) 513-518 <©North-Holland Publishing Company
T H E O R D E R et2 E N E R G Y - E N E R G Y C O R R E L A T I O N F U N C T I O N AT
SMALL ANGLES Shen-Chang CHAO and Davison E. SOPER
Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA John C. COLLINS
Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616. USA Received 9 November 1982
The coefficients that give the small angle (back-to-back) behavior of the energy-energy correlation function in e + e - annihilation are calculated at order a~ from previous all order, small angle formulas. The purpose is to facilitate comparison with all angle, order a~ calculations.
In recent years there has been progress in understanding the two-hadron inclusive cross section in e+e - annihilation, e + + e ~ A + B + X, in the back-to-back region [1-7]. This cross section is important because it is sensitive to the transverse momentum imparted by gluon bremsstrahlung from the highly virtual initial quarks. Since copious bremsstrahlung is characteristic of gauge theories, the measurement of this cross section constitutes a test of the gauge theoretic nature of the strong interaction. Particular attention has been paid to the energy-energy correlation function [8], d~'
1 r!
rl
do
dcos0=~J0 dxAxAJ° d x . x . hadron F_, dxAdx.dcosO'
(1)
flavors
where 0 is the angle between PA and --PB and XA, XB are the hadrons' momentum fractions. The back-to-back region of interest in this paper is the small 0 region. Recently two groups have reported calculations of d ~ / d c o s 0 in perturbation theory at order a s2 [9, 10]. The purpose of this paper is to use the all orders, small 0 results of [4-6] to examine the small 0 behavior that is to be expected in the result of a complete order a s2 calculation. Let us be more specific. It is convenient to consider the energy-energy correlation function integrated from zero angle up to an angle 0: ~i(cosO)=f l dcosO d~ "cos0 dcosO " 513
(2)
514
S.-C. Chao et al. / Energy-energy correlation function
If we want the energy correlation function itself, we have only to differentiate Xi(cos 0). The perturbation expansion of ~ has the following form at small 0 [1-7]: 2i(cos0)-
½a~T°) 1 +
~
Cumlnm(1/sin2(lO
N=I m=0
+
RN(0)
,
(3)
N=O
0 The rewhere o~°)= 4¢raZ(~,e2)/Q 2 is the total e+e cross section at order a s. mainders RN(O ) tend to zero like a power of 0 (or a power times logarithms) as 0 ---, 0. The coefficients CNma r e independent of 0, but may depend on the ratio of the c.m. energy Q to the renormalization mass #. It is these coefficients, for N = 1 and particularly for N = 2, that are the subject of this paper. There are two reasons for our interest in these coefficients. First, the evaluation of these coefficients in perturbation theory can provide some checks on the correctness of the results in refs. [1-7]. Second, there is an important but as yet uncalculated* coefficient called y~2) that appears in [4]. The value of 1,~2) can be extracted if the value of C22 is known from a perturbative calculation. The order cts2 results reported in [9, 10] are obtained with the aid of a numerical integration. We have learned that the authors of [9] are working to extract the small 0 coefficients Czm by fitting the numerically generated curve at very small values of 0. Their intention is to extract y~r2) in this way. We hope that this method works. Alternatively, we present the results of this paper in the hope that they can be compared to a future analytic calculation of the C2m. We now begin our analysis by recalling the small 0 formula from refs. [4, 5]. The energy-energy correlation function is given by
dZ
dcos0
Q2 [dbe-ik't'°T'iZ j ,,z~tb, Q~+oTYzIO, J ~ Q ~, I
167r
(4)
where k 2 = Q2sin2(½0). The function Y is a finite 0 correction that will not concern us here. The small 0 coefficients CN,,,are all contained in 1~. It has the structure
OTl7Vz(b, Q) = o~°)IH(Q; g( C2Q), C2O)12(j(b; g( Cl/b ), C,/b ) × T(b; g(C,/b), CJb)Zexp[-S(Q,
b)],
(5)
where
S(Q,b)= fC~Q2d~2 :c?/b2 /if2 { ½1n ( 2 ~ ) yr(g(fi))-K(b;g(Ct/b),CJb) -~(~/C2;
g(/~), ~) +
½1'v(g(~)) + 2 ) , , ( g ( / ~ ) ) } . t
*See, however, ref. [7].
(6)
S.-C. Chao et al. / Energy-energy correlation function
515
Here C~ and C 2 are arbitrary " s c h e m e - d e p e n d e n c e " parameters introduced in ref. [4]. The functions H(Q; g(/0,/~), etc. are defined by calculational rules discussed in ref. [4]; the functions 7 ( g ( ~ ) ) are anomalous dimensions associated with these functions. The function T(b) is defined in terms of the Wilson coefficients Ta/j(~/z, b) of ref. [4] by
T(b) = E l ' a
d z z T ~ / j ( 1 / z , b).
(7)
0
We shall m a k e use of the following perturbative results for the functions appearing in eqs. (5) and (6):
2.(2
I H ( Q ; g, P,)[ 2 = 1 + ~ - ~ In
)
Q~ e v+~'2-4 + . . . 4rr~2
4 -ags ln(/x2b2~.eV ) + . . . O ( b ; g, t,) = 1 - ~-
1% T(b; g, it; C,/C2) = 1 + -~ --~ {31n(/x2b2,n.eV)
+ l . ~r2 . .
.ln2( C2 e2V-I )} + . . . , 4c?
4
K ( b ; g,/z)
3 er
~(Q;g,~)
" " "ln//.t2b2weV/, ,+
4 a~l n
3 ';'r
8 as ~K(g) = ~ ;
[ °2 k 4'/r, tt2
...
1
eV_ ~ + . . - ,
}
.,,(2) 0/2 + .K ~2 + '
80~ s v,.,-(s) = 3 ~ - + " ' =---
~7
+....
(8)
Notice that a numerical value for the order a s2 contribution to the a n o m a l o u s dimension 7K(g) is l a c k i n g - even though it is i m p o r t a n t because it multiplies an extra logarithm in eq. (6). n value for y~2), namely y~2) = ~ _ ~20N t -- 2~2, can be
S.-C. Chaoet al. / Energy-energycorrelationfunction
516
extracted from the results of ref. [7]*. However, these results rest on several unproved assumptions. Thus we do not consider this value to be completely reliable. The result for T at first order is taken from ref. [6]. T h e rest of the results are from ref. [4], except that we have corrected an error in the order a S value of H. Our task is to extract the coefficients CNm (for N = 1 and for N = 2, M = 2, 3, 4) from these results. We first rewrite ln(oxlYgz/o~°) ) in the form that it would take in ordinary perturbation theory: In
°T
17Vz(b,Q) - Y'~ ~, DNmlnm(lQZb2)
(9)
N=I m=0
Here a s is the coupling as(/~ ) evaluated with an arbitrary (minimal subtraction) renormalization scale /t. The b dependence is displayed explicitly, but the coefficients DNm contain logarithms of Qz///L2. It is a remarkable feature of the result of ref. [4] that there are n o ONto coefficients for m > N + 1. The required coefficients ONto c a n be calculated as follows: (a) replace all running couplings as(/~ ) and as(Cl/b ) by a s ( ~ ) using
a ~ ~r (~)
a ~ ~r ( ~ ) ~- ~
ln[ ~-~
(10)
where Nf is the n u m b e r of fermion flavors; (b) set the arbitrary parameters C l and C 2 to C l = 2I~/Q, C2 = tx/Q; (c) p e r f o r m the integral over g in S(Q, b); (d) take the logarithm of (ox/o~x°))lTv perturbatively. F r o m the perturbative information given in (8) one can c o m p u t e O12 , D | I , D I 0 and all of the Dum with m > / N [although one needs to use the further terms in (10), including those generated by the two-loop fl function, to go b e y o n d N = 2]. T h e results for N ~< 2 are D12=
_2,
DII=
2 -
Dt °
_ 2,77.2
=
8
33 D23
D2 2
=
-
-
8 + 4"y - 8 2 ~ y
,
2N r 27 '
__ !. ~ . ~, (42t K )
33 - 2Nf ~ [21n(Q2/4~r/t2) + 3 - 6 7 ]
(11)
3 ,(2)• We have verified that the coefficient of Nr in V~2) as given in * The Constant called K in ref. [7] is ~rK
ref. [7] is correct.
S.-C. Chao et al. / Energy-energy correlation function
517
We can now transform b-space to angle space using the formula [11] 2"T F ( 1 -- d / d L )
exp
( ~DumLm
,
(12)
where L = In(I/sin2(½0)). We use F(1 + d / d L )
F ( 1 - d / d L ) = I - 2 Y ~ L + 2 Y 2 ( ~ _d~ ) 2 - 2 ( 2 3 ' 3 + g(3)) ( ~ L ) 3 +
+""
.
(13)
This gives the desired coefficients Cure in the expansion (3) of Ei(cos 8): C12 =
2 - 7 ,
CI1 =
2,
C~0
- 2¢~ 2
3
C24 = 2
C23 = _ 4
33 - 2Nf 27 ' 3 3 - 2Nf (
C22 =
-- !"v(2' + 4IK
C21 = Dzt +
1-2-
Q2 ) 1 + ~ln( ,--ZSY--2eV ] + ~ + 2477r2' \ 4~M s
]
D~ID1o- 27[2022 + D~l + 2012D,o ]
+ 12]/2[D,2D11 + D23] - 8(23, 3 + ~'(3)) D~2, C20 = 0 2 0 +
, 2 _ 23'[D2, + 7Dlo
DI,D,o] + 2y212D22 + D~1 + 2DIzDIo ]
- 4 ( 2 3 '3 + ~'(3))[D,2D,1 + D23 ] + 8]/(]/3 -1- 2[(3)) D122.
(14)
All of the Dum coefficients that appear in the last two formulas, except for D2~ and 020 , are given in eq. (11). Let us comment on the comparison of these results based on refs. [4-6] to the results of other small 0 methods. In the leading double logarithmic approximation one obtains all of the coefficients Cure for rn = 2N. This leading result was first obtained by Parisi and Petronzio [1]. Ellis and Sterling [3] were the first to obtain
518
s.-c. Chaoet al. / Energy-energycorrelationfunction
also the next-to-leading coefficients, Cure with m = 2 N - 1. These authors' results agree [5] with the m = 2 N and 2 N - 1 coefficients obtained from the formulas used here, including the coefficients C]2, C24 and C11, C23 listed above. The formula of Kodiera and Trentadue [7] reproduces all of the b-space coefficients Dm~ with m >~ N that are obtained from the formula used here. In addition they give a value for 022 and thus for 7~K2). The values given above for C12, C1], and C]0 can also be c o m p a r e d to those obtained from the original order a]s calculation of d ~ / d c o s 0 by Basham, Brown, Ellis and Love [8]. They agree. In the case of C~0, this agreement is a new result since it requires the inclusion of the first order term in T(b), obtained from [6]. This comparison provides the only independent check of the third to leading coefficient D10 , which appears in C]0 and also appears, along with ~/~2) in C22. We conclude by reviewing some of the things that can be learned from an order a 2 calculation. The coefficient .yk2) can be extracted if C22 is computed. If C2~ could be calculated, it would allow one to extract the less important but still interesting and as yet u n k n o w n coefficient D21. Similarly, a calculation of C20 (which requires the completely virtual two-gluon graphs) would give D20. It is also important to extract the derivative of the remainder function R2(0 ) defined in eq. (3), since - d R z ( O ) / d c o s O is the order as2 contribution to the function of oTYz(O, Q) that appears in eq. (4). Finally, the values of C24 and C23, if verified in an order a S2 calculation, would help to check the small 0 formulas from which they were extracted. It would be shocking if these values were not confirmed, since they are leading double log and next-to-leading double log coefficients, which have been obtained by several independent methods. It is a pleasure to thank J. Stirling and L. Trentadue for helpful conversations. This work was supported in part by the US D e p a r t m e n t of Energy under contracts DE-AT06-76ER-70004 and DE-AC02-80ER-10712.
References [1] G. Parisi and R. Petronzio, Nucl. Phys. B154 (1979) 427; G. Curchi, M. Greco and Y. Shrivastava, Phys. Rev. Lett. 43 (1979) 834; Nucl. Phys. B159 (1979) 451 [2] Y.L. Dokshitzer, D.I. D'Yakanov and S.I. Troyan, Proc. 13th Winter School of the LPNI, Leningrad (1978) (Institute of Nuclear Physics, Leningrad, 1978); Phys. Reports. 58 (1980) 271 [3] S.D. Ellis and W.J. Stirling, Phys. Rev. D23 (1981) 214 [4] J.C. Collins and D.E. Soper, Nucl. Phys. B193 (1981) 381 [5] J.C. Collins and D.E. Soper, Nucl. Phys. B197 (1982) 446 [6] S-C. Chao and D.E. Soper, Nucl. Phys. B214 (1983) 405 [7] J. Kodaira and L. Trentadue, Phys. Lett. I I2B (1982) 66 [8] C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love, Phys. Rev. D19 (1979) 2018; L.S. Brown and S.D. Ellis, Phys. Rev. D24 (1981) 2383 [9] D.G. Richards, W.J. Stirling and S.D. Ellis, Phys. Lett. 119B (1982) 193 [10] A. Ali and F. Barreiro, Phys. Lett. l18B (1982) 155 [11] J.P. Ralston and D.E. Soper, Nucl. Phys. B172 (1980) 445
T H E O R D E R et2 E N E R G Y - E N E R G Y C O R R E L A T I O N F U N C T I O N AT
SMALL ANGLES Shen-Chang CHAO and Davison E. SOPER
Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403, USA John C. COLLINS
Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616. USA Received 9 November 1982
The coefficients that give the small angle (back-to-back) behavior of the energy-energy correlation function in e + e - annihilation are calculated at order a~ from previous all order, small angle formulas. The purpose is to facilitate comparison with all angle, order a~ calculations.
In recent years there has been progress in understanding the two-hadron inclusive cross section in e+e - annihilation, e + + e ~ A + B + X, in the back-to-back region [1-7]. This cross section is important because it is sensitive to the transverse momentum imparted by gluon bremsstrahlung from the highly virtual initial quarks. Since copious bremsstrahlung is characteristic of gauge theories, the measurement of this cross section constitutes a test of the gauge theoretic nature of the strong interaction. Particular attention has been paid to the energy-energy correlation function [8], d~'
1 r!
rl
do
dcos0=~J0 dxAxAJ° d x . x . hadron F_, dxAdx.dcosO'
(1)
flavors
where 0 is the angle between PA and --PB and XA, XB are the hadrons' momentum fractions. The back-to-back region of interest in this paper is the small 0 region. Recently two groups have reported calculations of d ~ / d c o s 0 in perturbation theory at order a s2 [9, 10]. The purpose of this paper is to use the all orders, small 0 results of [4-6] to examine the small 0 behavior that is to be expected in the result of a complete order a s2 calculation. Let us be more specific. It is convenient to consider the energy-energy correlation function integrated from zero angle up to an angle 0: ~i(cosO)=f l dcosO d~ "cos0 dcosO " 513
(2)
514
S.-C. Chao et al. / Energy-energy correlation function
If we want the energy correlation function itself, we have only to differentiate Xi(cos 0). The perturbation expansion of ~ has the following form at small 0 [1-7]: 2i(cos0)-
½a~T°) 1 +
~
Cumlnm(1/sin2(lO
N=I m=0
+
RN(0)
,
(3)
N=O
0 The rewhere o~°)= 4¢raZ(~,e2)/Q 2 is the total e+e cross section at order a s. mainders RN(O ) tend to zero like a power of 0 (or a power times logarithms) as 0 ---, 0. The coefficients CNma r e independent of 0, but may depend on the ratio of the c.m. energy Q to the renormalization mass #. It is these coefficients, for N = 1 and particularly for N = 2, that are the subject of this paper. There are two reasons for our interest in these coefficients. First, the evaluation of these coefficients in perturbation theory can provide some checks on the correctness of the results in refs. [1-7]. Second, there is an important but as yet uncalculated* coefficient called y~2) that appears in [4]. The value of 1,~2) can be extracted if the value of C22 is known from a perturbative calculation. The order cts2 results reported in [9, 10] are obtained with the aid of a numerical integration. We have learned that the authors of [9] are working to extract the small 0 coefficients Czm by fitting the numerically generated curve at very small values of 0. Their intention is to extract y~r2) in this way. We hope that this method works. Alternatively, we present the results of this paper in the hope that they can be compared to a future analytic calculation of the C2m. We now begin our analysis by recalling the small 0 formula from refs. [4, 5]. The energy-energy correlation function is given by
dZ
dcos0
Q2 [dbe-ik't'°T'iZ j ,,z~tb, Q~+oTYzIO, J ~ Q ~, I
167r
(4)
where k 2 = Q2sin2(½0). The function Y is a finite 0 correction that will not concern us here. The small 0 coefficients CN,,,are all contained in 1~. It has the structure
OTl7Vz(b, Q) = o~°)IH(Q; g( C2Q), C2O)12(j(b; g( Cl/b ), C,/b ) × T(b; g(C,/b), CJb)Zexp[-S(Q,
b)],
(5)
where
S(Q,b)= fC~Q2d~2 :c?/b2 /if2 { ½1n ( 2 ~ ) yr(g(fi))-K(b;g(Ct/b),CJb) -~(~/C2;
g(/~), ~) +
½1'v(g(~)) + 2 ) , , ( g ( / ~ ) ) } . t
*See, however, ref. [7].
(6)
S.-C. Chao et al. / Energy-energy correlation function
515
Here C~ and C 2 are arbitrary " s c h e m e - d e p e n d e n c e " parameters introduced in ref. [4]. The functions H(Q; g(/0,/~), etc. are defined by calculational rules discussed in ref. [4]; the functions 7 ( g ( ~ ) ) are anomalous dimensions associated with these functions. The function T(b) is defined in terms of the Wilson coefficients Ta/j(~/z, b) of ref. [4] by
T(b) = E l ' a
d z z T ~ / j ( 1 / z , b).
(7)
0
We shall m a k e use of the following perturbative results for the functions appearing in eqs. (5) and (6):
2.(2
I H ( Q ; g, P,)[ 2 = 1 + ~ - ~ In
)
Q~ e v+~'2-4 + . . . 4rr~2
4 -ags ln(/x2b2~.eV ) + . . . O ( b ; g, t,) = 1 - ~-
1% T(b; g, it; C,/C2) = 1 + -~ --~ {31n(/x2b2,n.eV)
+ l . ~r2 . .
.ln2( C2 e2V-I )} + . . . , 4c?
4
K ( b ; g,/z)
3 er
~(Q;g,~)
" " "ln//.t2b2weV/, ,+
4 a~l n
3 ';'r
8 as ~K(g) = ~ ;
[ °2 k 4'/r, tt2
...
1
eV_ ~ + . . - ,
}
.,,(2) 0/2 + .K ~2 + '
80~ s v,.,-(s) = 3 ~ - + " ' =---
~7
+....
(8)
Notice that a numerical value for the order a s2 contribution to the a n o m a l o u s dimension 7K(g) is l a c k i n g - even though it is i m p o r t a n t because it multiplies an extra logarithm in eq. (6). n value for y~2), namely y~2) = ~ _ ~20N t -- 2~2, can be
S.-C. Chaoet al. / Energy-energycorrelationfunction
516
extracted from the results of ref. [7]*. However, these results rest on several unproved assumptions. Thus we do not consider this value to be completely reliable. The result for T at first order is taken from ref. [6]. T h e rest of the results are from ref. [4], except that we have corrected an error in the order a S value of H. Our task is to extract the coefficients CNm (for N = 1 and for N = 2, M = 2, 3, 4) from these results. We first rewrite ln(oxlYgz/o~°) ) in the form that it would take in ordinary perturbation theory: In
°T
17Vz(b,Q) - Y'~ ~, DNmlnm(lQZb2)
(9)
N=I m=0
Here a s is the coupling as(/~ ) evaluated with an arbitrary (minimal subtraction) renormalization scale /t. The b dependence is displayed explicitly, but the coefficients DNm contain logarithms of Qz///L2. It is a remarkable feature of the result of ref. [4] that there are n o ONto coefficients for m > N + 1. The required coefficients ONto c a n be calculated as follows: (a) replace all running couplings as(/~ ) and as(Cl/b ) by a s ( ~ ) using
a ~ ~r (~)
a ~ ~r ( ~ ) ~- ~
ln[ ~-~
(10)
where Nf is the n u m b e r of fermion flavors; (b) set the arbitrary parameters C l and C 2 to C l = 2I~/Q, C2 = tx/Q; (c) p e r f o r m the integral over g in S(Q, b); (d) take the logarithm of (ox/o~x°))lTv perturbatively. F r o m the perturbative information given in (8) one can c o m p u t e O12 , D | I , D I 0 and all of the Dum with m > / N [although one needs to use the further terms in (10), including those generated by the two-loop fl function, to go b e y o n d N = 2]. T h e results for N ~< 2 are D12=
_2,
DII=
2 -
Dt °
_ 2,77.2
=
8
33 D23
D2 2
=
-
-
8 + 4"y - 8 2 ~ y
,
2N r 27 '
__ !. ~ . ~, (42t K )
33 - 2Nf ~ [21n(Q2/4~r/t2) + 3 - 6 7 ]
(11)
3 ,(2)• We have verified that the coefficient of Nr in V~2) as given in * The Constant called K in ref. [7] is ~rK
ref. [7] is correct.
S.-C. Chao et al. / Energy-energy correlation function
517
We can now transform b-space to angle space using the formula [11] 2"T F ( 1 -- d / d L )
exp
( ~DumLm
,
(12)
where L = In(I/sin2(½0)). We use F(1 + d / d L )
F ( 1 - d / d L ) = I - 2 Y ~ L + 2 Y 2 ( ~ _d~ ) 2 - 2 ( 2 3 ' 3 + g(3)) ( ~ L ) 3 +
+""
.
(13)
This gives the desired coefficients Cure in the expansion (3) of Ei(cos 8): C12 =
2 - 7 ,
CI1 =
2,
C~0
- 2¢~ 2
3
C24 = 2
C23 = _ 4
33 - 2Nf 27 ' 3 3 - 2Nf (
C22 =
-- !"v(2' + 4IK
C21 = Dzt +
1-2-
Q2 ) 1 + ~ln( ,--ZSY--2eV ] + ~ + 2477r2' \ 4~M s
]
D~ID1o- 27[2022 + D~l + 2012D,o ]
+ 12]/2[D,2D11 + D23] - 8(23, 3 + ~'(3)) D~2, C20 = 0 2 0 +
, 2 _ 23'[D2, + 7Dlo
DI,D,o] + 2y212D22 + D~1 + 2DIzDIo ]
- 4 ( 2 3 '3 + ~'(3))[D,2D,1 + D23 ] + 8]/(]/3 -1- 2[(3)) D122.
(14)
All of the Dum coefficients that appear in the last two formulas, except for D2~ and 020 , are given in eq. (11). Let us comment on the comparison of these results based on refs. [4-6] to the results of other small 0 methods. In the leading double logarithmic approximation one obtains all of the coefficients Cure for rn = 2N. This leading result was first obtained by Parisi and Petronzio [1]. Ellis and Sterling [3] were the first to obtain
518
s.-c. Chaoet al. / Energy-energycorrelationfunction
also the next-to-leading coefficients, Cure with m = 2 N - 1. These authors' results agree [5] with the m = 2 N and 2 N - 1 coefficients obtained from the formulas used here, including the coefficients C]2, C24 and C11, C23 listed above. The formula of Kodiera and Trentadue [7] reproduces all of the b-space coefficients Dm~ with m >~ N that are obtained from the formula used here. In addition they give a value for 022 and thus for 7~K2). The values given above for C12, C1], and C]0 can also be c o m p a r e d to those obtained from the original order a]s calculation of d ~ / d c o s 0 by Basham, Brown, Ellis and Love [8]. They agree. In the case of C~0, this agreement is a new result since it requires the inclusion of the first order term in T(b), obtained from [6]. This comparison provides the only independent check of the third to leading coefficient D10 , which appears in C]0 and also appears, along with ~/~2) in C22. We conclude by reviewing some of the things that can be learned from an order a 2 calculation. The coefficient .yk2) can be extracted if C22 is computed. If C2~ could be calculated, it would allow one to extract the less important but still interesting and as yet u n k n o w n coefficient D21. Similarly, a calculation of C20 (which requires the completely virtual two-gluon graphs) would give D20. It is also important to extract the derivative of the remainder function R2(0 ) defined in eq. (3), since - d R z ( O ) / d c o s O is the order as2 contribution to the function of oTYz(O, Q) that appears in eq. (4). Finally, the values of C24 and C23, if verified in an order a S2 calculation, would help to check the small 0 formulas from which they were extracted. It would be shocking if these values were not confirmed, since they are leading double log and next-to-leading double log coefficients, which have been obtained by several independent methods. It is a pleasure to thank J. Stirling and L. Trentadue for helpful conversations. This work was supported in part by the US D e p a r t m e n t of Energy under contracts DE-AT06-76ER-70004 and DE-AC02-80ER-10712.
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