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The pion evolution kernel beyond the leading order
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
THE P I O N E V O L U T I O N KERNEL B E Y O N D T H E L E A D I N G O R D E R
M.H. S A R M A D I Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA and International Centre for Theoretical Physics, Trieste, Italy l Received 4 October 1982 Revised manuscript received 5 March 1984
We have calculated the order a 2 QCD correction to the evolution kernel for pion "distribution amplitude". It can be used to analyze a number of wide angle exclusive processes beyond the leading order. The eigenvalues of the kernel agree with the anomalous dimensions of the non-singlet operators in deep inelastic scattering. The numerical values for some of the off-diagonal terms are given.
In the last few years there has been progress in our understanding of large angle exclusive processes involving hadrons. Perturbative QCD has been used [1-4] to make predictions about the asymptotic behavior of a number of such processes to the leading order. But the study of the higher order correlations to inclusive processes has shown that the significance of the leading order results depends on the magnitude of the higher order terms. In this letter we compute the next to the leading order QCD corrections for the pion form factor. The process independent helicity dependent anomalous dimension matrix that we obtain can be used in analyzing the asymptotic behavior of other amplitudes which involve pions. One needs only to do a straightforward calculation of the hard scattering part for the process in question. The approach we use is based on the factorization of mass singularities [5], These mass singularities determine the evolution of the distribution amplitudes. Curci et al. [5] used this method to find the evolution of parton densities to order (a2). We follow their suggestion about subtracting spurious ultraviolet singularities, which in the axial gauge (n • A = 0) arise from the simplifying choice n 2 = 0. There are also gauge dependent singularities coming from a 1/(l. n) term which are regularized by the principal value prescription [5-7]: 1 / ( t . n) -~ (l. n ) / [ ( l . n ) 2 + 8 2 ( p . n)2].
(1)
This will introduce In 8 and ln28 terms which cancel in the final gauge independent result. We have used this cancellation as one of the intermediate checks on our result. We use dimensional regularization [8] to regularize the mass singularities and UV divergences. (The space-time dimension is continued to D = 4 2c.) The pion form factor at large Q2 is given by the diagram in fig. 1 in terms of the bare distribution amplitude ~0 and the scattering amplitude for y*qcl --+ qcl- The scattering amplitude T is expanded in terms of the two-particle irreducible (2PI) kernel K 0 and the 2PI connected five-point function TOas shown in fig.
1 Current address.
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
471
Volume 143B, number 4, 5, 6
,,.7,°
PHYSICS LETTERS
\.-2°
\ /-'~
(-l+'q)p
16 August 1984
;¢s~
I
T
=
+
(_r+C)q +
F=(Q2)
Fig. 1. The diagram giving the pion form factor at large Q2 = 8p • q in terms of q'o, the distribution amplitude of quarks inside the pion, and T, the scattering ampfitude for Y*q¢l --* qq.
+ ,.,
Fig. 2. The expansion of T in terms of the 2PI kernel K 0 and the 2PI connected five-point function T0.
2. In axial gauges both K 0 and TO are free of mass singularities [9]. The mass singularities in T arise when the momentum of the internal propagators between K0's or K 0 and TObecome collinear with p. These can be factorized [3] in the form:
T(~, ~; Q2/~2, g, l / e ) = f dudvV(n, v; g, 1 / c ) T ( o , u; Q2/#2)v(~, u; g, l / e ) .
(2)
The moments ~j(Q2/#2, g) of T with respect to the eigenfunctions fi(m g) of V satisfy a renormalization group equation which has the solution:
~j(Q2/tx2, g)= Q-2~j(l, g(Q2)) exp(- fgg(Q2)dg'Yi(g') )' B ( g+ ' , cYJ(g') )
(3)
where the yi(g) are anomalous dimensions given by F~(g, l/e), the eigenvalues of V: y , ( g ) = - f l ( g , c)a In I',(g, 1/c)/ag.
(4)
The asymptotic form factor is then given by the expression: F=(Q 2) =
Eai(Q2)--~Tij(I,g(Q2))aj(Q2), i,j ~-
(5)
where
a ' ( Q 2 ) = exp( - Jo/'g(Q2)"' Yi(g')ag/3~g',7) ) a~°)(g'l/c)'
(6)
and a}°) is the coefficient of the expansion of the bare distribution amplitude q~00/, g, l / e ) in terms of the eigenfunctions f~01, g)At the one-loop level, V0/, v; g, l / e ) is obtained by evaluating the one-gluon exchange diagram and the residue of the pole of the quark propagator computed to order a. The eigenfunctions of V to this order are known to be the Gegenbauer polynomials C y 2 0 / ) , and the eigenvalues are the same as the O(a) anomalous dimensions of lowest twist flavor non-singlet operators which enter the calculation of deep inelastic scattering [1,3,4]. To this order T,7(1, g(Q2)) is given by the Born diagram. The order a 2 correction to V is written as ~0)(~/, v) = V(1)(~/, v) + 8(7 - v)z °), where V (t) is given by 472
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
the diagrams ( a ) - ( g ' ) of fig. 3 and z (1) is the O(ct 2) correction to the residue of the pole in the quark propagator, Z(I)
~ ! 26
F CG (17-[-~'z/'2k6 12ff(3) -
(268 - - - ~4 r2
)
fldBB)
3--9
_
+490
-
(7)
Here N v is the n u m b e r of flavors, ~'(n) is the R i e m a n n zeta function, and for SU(3)c, C v = 4 / 3 and Cc = 3. In fig. 3 diagram (g') subtracts the mass singularity coming from the inner loop of (g). To shorten the expression for V °), we define x = (1 + 7 ) / 2 , 2 = (1 - "0)/2, y = (1 + v)/2, ~ = (1 - v)/2, F = F(x, y) = yx-l[1 + 1 / ( x - y ) ] and i f = F(2, .~) and write the result as:
V(1) ( ~1, v) = 1CvCcVG ( n, v) = C2 Vv ( n, y) + ½NFCvV N ( ~1, v ),
(8)
where
Vo(x,y)=(~-r+~y/x+1311n(y/x)F+Zlnxln~F)O(x-y)+G(x,y)+(x~Z,y~), V v ( x , y) = [ - ½rr2F+ y / x - l n ( y / x ) ( 3 r
- ½y/Y) - l n ( y / x ) ln(1 - y / x ) ( F
+ l n 2 ( y / x ) ( r + ½ y / Y ) - 2 l n y In x ff] O ( x - y ) -
½1ny(1 + l n y -
(9)
- if)
G(x, y)
+ (x ~ Y , y ~,~),
2 ln~)y/Y
(10)
and
VN(X, y ) = [-- ~ - F - - ~ y / x - 2 l n ( y / x ) f ] O ( x - -
y ) + ( x *-',Y, y ~-~y).
(11)
Here G(x, y) is defined by
G ( x , y ) = [2 Li2(1 - y / x ) ( F +2 Liz(~)r[O(x-~)-
if) + l n Z x ( F - F ) O(x-y)]
2 In x l n y F ] O ( x - y )
- 2 Li2(y)F[O(x-y)-
O(y- x)].
(12)
The function Liz(x ) is the Spence function. The x = y singularities in Vo and VN are cancelled with the corresponding fl = 1 singularities in Z m. Thus f~o) is well defined at ~ = v. Using these results we have c o m p u t e d the first ten eigenvalues numerically; again they agree with the anomalous dimensions of the corresponding operators for deep inelastic scattering [10]. Such agreement is
(o)
(9)
(b}
(c)
(d}
re)
if}
19")
Fig. 3. The diagrams contributing to V(71, o; g, 1 / ( ) at the order ct2. The crossed diagrams corresponding to (a), (d) and (e) should also be included.
473
Volume
143B,
number
Table 1 The numerical values of j
16 August 1984
PHYSICS LETTERS
4, 5, 6
Aij for i, j
~<10.
i 0
0 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
-- 1.667 - 2.20 -2.533
- 0.867 - 1.362
- 2.762
- 0.543
-
1.728
- 2.928
_ 0.941
-3.005
- 2.008 - 1.270
- 0.377 --
-3.155
-- 2.227
0.696 - 0.979
- 0.279
- 3.236
- 2.403
- 1.540
- 2.547
- 1.763
e x p e c t e d b e c a u s e t h e a n o m a l o u s d i m e n s i o n m a t r i x is t r i a n g u l a r : A f l a v o r n o n - s i n g l e t o p e r a t o r ~a~,5O(n • o n l y i n t o t h e o p e r a t o r s ~ a y s ~ i ( n • ~ ) h k b o f l o w e r d i m e n s i o n ( j < i ) [3,11]. T o c a l c u l a t e t h e c o r r e c t i o n s t o t h e l e a d i n g o r d e r e i g e n f u n c t i o n s (i.e. t h e G e g e n b a u e r p o l y n o m i a l s ) , t h e
"~)i~b m i x e s
off-diagonal moments d i j = f_l 1d'0 f _ 1 1 d v ( 1
o f I~(a)('0, v):
(13)
_ '02 ) l~(1) ( "0, u )Ci3/2( 71)g3/2 (o)
are needed. We have found that the term 21nxlnp FO(x-y)+G(x,y) w h i c h is i n Vc a n d VF, contributes only to the diagonal elements of the anomalous dimension matrix. Thus the off-diagonal moments
Aij =
of
½CFCcVG + ½CvNFV N
are
f11d'0fl_1dv(1--'02)
CFfloAij
ln[(1 + v)/(1
+
where/3 o = ~Co
"0)]FO('0-v)Ci3/2('0)Cj3/2(u).
In table 1 we give the numerical values of Aij for Bij=
f_l d ' 0 f _ l l d o ( 1 -
Table 2 The numerical values of j
474
i,j <~10.
(14)
In table 2 the off-diagonal moments
o f VF:
(15)
v)Ci3/2('0)Cj3/2(u)
Bij for i,j <<.10
i 0
0 1 2 3 4 5 6 7 8 9 10
'02)Vv('0 ,
- ~N v and
1
2
3
4
5
6
7
8.333 6.111 4.011
6.067 6.446
2.053 6.108
4.511 5.625
6.029
3.496 4.802 2.805
0.254 - 1.394
5.427 4.592
5.529
-
2.908 - 4.303
3.694
6.010 5.742
2.778
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
f o r i, j ~< 10 are given. W e d i s a g r e e w i t h the c l a i m in ref. [12] t h a t t h e c o e f f i c i e n t s Bij vanish. I n refs. [13,14] t h e n e x t to l e a d i n g o r d e r c o r r e c t i o n s to t h e p i o n f o r m f a c t o r w e r e s t u d i e d b y c o n s i d e r i n g o n l y the c o r r e c t i o n s to t h e h a r d s c a t t e r i n g a m p l i t u d e . I n a f u t u r e p u b l i c a t i o n w e p l a n to use the results p r e s e n t e d h e r e to a n a l y z e t h e c o r r e c t i o n s to the p i o n f o r m factor. ( W h i l e this p a p e r was b e i n g r e v i s e d for p u b l i c a t i o n , we r e c e i v e d a p a p e r f r o m D i t t e s a n d R a d y u s h k i n [15] w h i c h c o n f i r m s the results o f this analysis.) I t h a n k P r o f e s s o r R o b e r t C a r l i t z for s u g g e s t i n g this p r o b l e m a n d his e n c o u r a g e m e n t a n d s u g g e s t i o n s t h r o u g h o u t the w o r k . I also t h a n k P r o f e s s o r R a l p h R o s k i e s a n d P e n n y S a c k e t t for their a s s i s t a n c e in the use o f A S H M E D I .
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15]
G.P. Lepage and S.J. Brodsky, Phys. Lett. 87B (1979) 359. G.P. Lepage and S.J. Brodsky, Phys. Rev. D22 (1981) 2157. A. Duncan and A. Mueller, Phys. Rev. D21 (1980) 1636. A.V. Efremov and A.V. Radyushkin, Phys. Lett. 94B (1980) 245. G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B175 (1980) 27. W. Kainz, W. Kummer and M. Schweda, Nucl. Phys. B79 (1974) 484; W. Konetschny and W. Kummer, Nucl. Phys. B100 (1975) 106; W. Kummer, Acta. Phys. Austriaca 41 (1975) 315. D.J. Pritchard and W.J. Stirling, Nu¢l. Phys. B165 (1980) 237. G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Nucl. Phys. B152 (1979) 285. E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B129 (1977) 66; B139 (1978) 545 (E). S.J. Brodsky, Y. Frishman, G.P. Lepage and C. Sachrajda, Phys. Lett. 91B (1980) 239. P. Damgaard, Ph.D. Dissertation, University of Cornell (1982). R.D. Field, R. Gupta, S. Otto and L. Chang, Nucl. Phys. B186 (1981) 429. F.M. Dittes and A.V. Radyushkin, Sov. J. Nucl. Phys. 34 (1981) 293. F.M. Dittes and A.V. Radyushkin, Phys. Lett. 134B (1984) 359.
475
PHYSICS LETTERS
16 August 1984
THE P I O N E V O L U T I O N KERNEL B E Y O N D T H E L E A D I N G O R D E R
M.H. S A R M A D I Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA and International Centre for Theoretical Physics, Trieste, Italy l Received 4 October 1982 Revised manuscript received 5 March 1984
We have calculated the order a 2 QCD correction to the evolution kernel for pion "distribution amplitude". It can be used to analyze a number of wide angle exclusive processes beyond the leading order. The eigenvalues of the kernel agree with the anomalous dimensions of the non-singlet operators in deep inelastic scattering. The numerical values for some of the off-diagonal terms are given.
In the last few years there has been progress in our understanding of large angle exclusive processes involving hadrons. Perturbative QCD has been used [1-4] to make predictions about the asymptotic behavior of a number of such processes to the leading order. But the study of the higher order correlations to inclusive processes has shown that the significance of the leading order results depends on the magnitude of the higher order terms. In this letter we compute the next to the leading order QCD corrections for the pion form factor. The process independent helicity dependent anomalous dimension matrix that we obtain can be used in analyzing the asymptotic behavior of other amplitudes which involve pions. One needs only to do a straightforward calculation of the hard scattering part for the process in question. The approach we use is based on the factorization of mass singularities [5], These mass singularities determine the evolution of the distribution amplitudes. Curci et al. [5] used this method to find the evolution of parton densities to order (a2). We follow their suggestion about subtracting spurious ultraviolet singularities, which in the axial gauge (n • A = 0) arise from the simplifying choice n 2 = 0. There are also gauge dependent singularities coming from a 1/(l. n) term which are regularized by the principal value prescription [5-7]: 1 / ( t . n) -~ (l. n ) / [ ( l . n ) 2 + 8 2 ( p . n)2].
(1)
This will introduce In 8 and ln28 terms which cancel in the final gauge independent result. We have used this cancellation as one of the intermediate checks on our result. We use dimensional regularization [8] to regularize the mass singularities and UV divergences. (The space-time dimension is continued to D = 4 2c.) The pion form factor at large Q2 is given by the diagram in fig. 1 in terms of the bare distribution amplitude ~0 and the scattering amplitude for y*qcl --+ qcl- The scattering amplitude T is expanded in terms of the two-particle irreducible (2PI) kernel K 0 and the 2PI connected five-point function TOas shown in fig.
1 Current address.
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
471
Volume 143B, number 4, 5, 6
,,.7,°
PHYSICS LETTERS
\.-2°
\ /-'~
(-l+'q)p
16 August 1984
;¢s~
I
T
=
+
(_r+C)q +
F=(Q2)
Fig. 1. The diagram giving the pion form factor at large Q2 = 8p • q in terms of q'o, the distribution amplitude of quarks inside the pion, and T, the scattering ampfitude for Y*q¢l --* qq.
+ ,.,
Fig. 2. The expansion of T in terms of the 2PI kernel K 0 and the 2PI connected five-point function T0.
2. In axial gauges both K 0 and TO are free of mass singularities [9]. The mass singularities in T arise when the momentum of the internal propagators between K0's or K 0 and TObecome collinear with p. These can be factorized [3] in the form:
T(~, ~; Q2/~2, g, l / e ) = f dudvV(n, v; g, 1 / c ) T ( o , u; Q2/#2)v(~, u; g, l / e ) .
(2)
The moments ~j(Q2/#2, g) of T with respect to the eigenfunctions fi(m g) of V satisfy a renormalization group equation which has the solution:
~j(Q2/tx2, g)= Q-2~j(l, g(Q2)) exp(- fgg(Q2)dg'Yi(g') )' B ( g+ ' , cYJ(g') )
(3)
where the yi(g) are anomalous dimensions given by F~(g, l/e), the eigenvalues of V: y , ( g ) = - f l ( g , c)a In I',(g, 1/c)/ag.
(4)
The asymptotic form factor is then given by the expression: F=(Q 2) =
Eai(Q2)--~Tij(I,g(Q2))aj(Q2), i,j ~-
(5)
where
a ' ( Q 2 ) = exp( - Jo/'g(Q2)"' Yi(g')ag/3~g',7) ) a~°)(g'l/c)'
(6)
and a}°) is the coefficient of the expansion of the bare distribution amplitude q~00/, g, l / e ) in terms of the eigenfunctions f~01, g)At the one-loop level, V0/, v; g, l / e ) is obtained by evaluating the one-gluon exchange diagram and the residue of the pole of the quark propagator computed to order a. The eigenfunctions of V to this order are known to be the Gegenbauer polynomials C y 2 0 / ) , and the eigenvalues are the same as the O(a) anomalous dimensions of lowest twist flavor non-singlet operators which enter the calculation of deep inelastic scattering [1,3,4]. To this order T,7(1, g(Q2)) is given by the Born diagram. The order a 2 correction to V is written as ~0)(~/, v) = V(1)(~/, v) + 8(7 - v)z °), where V (t) is given by 472
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
the diagrams ( a ) - ( g ' ) of fig. 3 and z (1) is the O(ct 2) correction to the residue of the pole in the quark propagator, Z(I)
~ ! 26
F CG (17-[-~'z/'2k6 12ff(3) -
(268 - - - ~4 r2
)
fldBB)
3--9
_
+490
-
(7)
Here N v is the n u m b e r of flavors, ~'(n) is the R i e m a n n zeta function, and for SU(3)c, C v = 4 / 3 and Cc = 3. In fig. 3 diagram (g') subtracts the mass singularity coming from the inner loop of (g). To shorten the expression for V °), we define x = (1 + 7 ) / 2 , 2 = (1 - "0)/2, y = (1 + v)/2, ~ = (1 - v)/2, F = F(x, y) = yx-l[1 + 1 / ( x - y ) ] and i f = F(2, .~) and write the result as:
V(1) ( ~1, v) = 1CvCcVG ( n, v) = C2 Vv ( n, y) + ½NFCvV N ( ~1, v ),
(8)
where
Vo(x,y)=(~-r+~y/x+1311n(y/x)F+Zlnxln~F)O(x-y)+G(x,y)+(x~Z,y~), V v ( x , y) = [ - ½rr2F+ y / x - l n ( y / x ) ( 3 r
- ½y/Y) - l n ( y / x ) ln(1 - y / x ) ( F
+ l n 2 ( y / x ) ( r + ½ y / Y ) - 2 l n y In x ff] O ( x - y ) -
½1ny(1 + l n y -
(9)
- if)
G(x, y)
+ (x ~ Y , y ~,~),
2 ln~)y/Y
(10)
and
VN(X, y ) = [-- ~ - F - - ~ y / x - 2 l n ( y / x ) f ] O ( x - -
y ) + ( x *-',Y, y ~-~y).
(11)
Here G(x, y) is defined by
G ( x , y ) = [2 Li2(1 - y / x ) ( F +2 Liz(~)r[O(x-~)-
if) + l n Z x ( F - F ) O(x-y)]
2 In x l n y F ] O ( x - y )
- 2 Li2(y)F[O(x-y)-
O(y- x)].
(12)
The function Liz(x ) is the Spence function. The x = y singularities in Vo and VN are cancelled with the corresponding fl = 1 singularities in Z m. Thus f~o) is well defined at ~ = v. Using these results we have c o m p u t e d the first ten eigenvalues numerically; again they agree with the anomalous dimensions of the corresponding operators for deep inelastic scattering [10]. Such agreement is
(o)
(9)
(b}
(c)
(d}
re)
if}
19")
Fig. 3. The diagrams contributing to V(71, o; g, 1 / ( ) at the order ct2. The crossed diagrams corresponding to (a), (d) and (e) should also be included.
473
Volume
143B,
number
Table 1 The numerical values of j
16 August 1984
PHYSICS LETTERS
4, 5, 6
Aij for i, j
~<10.
i 0
0 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
-- 1.667 - 2.20 -2.533
- 0.867 - 1.362
- 2.762
- 0.543
-
1.728
- 2.928
_ 0.941
-3.005
- 2.008 - 1.270
- 0.377 --
-3.155
-- 2.227
0.696 - 0.979
- 0.279
- 3.236
- 2.403
- 1.540
- 2.547
- 1.763
e x p e c t e d b e c a u s e t h e a n o m a l o u s d i m e n s i o n m a t r i x is t r i a n g u l a r : A f l a v o r n o n - s i n g l e t o p e r a t o r ~a~,5O(n • o n l y i n t o t h e o p e r a t o r s ~ a y s ~ i ( n • ~ ) h k b o f l o w e r d i m e n s i o n ( j < i ) [3,11]. T o c a l c u l a t e t h e c o r r e c t i o n s t o t h e l e a d i n g o r d e r e i g e n f u n c t i o n s (i.e. t h e G e g e n b a u e r p o l y n o m i a l s ) , t h e
"~)i~b m i x e s
off-diagonal moments d i j = f_l 1d'0 f _ 1 1 d v ( 1
o f I~(a)('0, v):
(13)
_ '02 ) l~(1) ( "0, u )Ci3/2( 71)g3/2 (o)
are needed. We have found that the term 21nxlnp FO(x-y)+G(x,y) w h i c h is i n Vc a n d VF, contributes only to the diagonal elements of the anomalous dimension matrix. Thus the off-diagonal moments
Aij =
of
½CFCcVG + ½CvNFV N
are
f11d'0fl_1dv(1--'02)
CFfloAij
ln[(1 + v)/(1
+
where/3 o = ~Co
"0)]FO('0-v)Ci3/2('0)Cj3/2(u).
In table 1 we give the numerical values of Aij for Bij=
f_l d ' 0 f _ l l d o ( 1 -
Table 2 The numerical values of j
474
i,j <~10.
(14)
In table 2 the off-diagonal moments
o f VF:
(15)
v)Ci3/2('0)Cj3/2(u)
Bij for i,j <<.10
i 0
0 1 2 3 4 5 6 7 8 9 10
'02)Vv('0 ,
- ~N v and
1
2
3
4
5
6
7
8.333 6.111 4.011
6.067 6.446
2.053 6.108
4.511 5.625
6.029
3.496 4.802 2.805
0.254 - 1.394
5.427 4.592
5.529
-
2.908 - 4.303
3.694
6.010 5.742
2.778
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
f o r i, j ~< 10 are given. W e d i s a g r e e w i t h the c l a i m in ref. [12] t h a t t h e c o e f f i c i e n t s Bij vanish. I n refs. [13,14] t h e n e x t to l e a d i n g o r d e r c o r r e c t i o n s to t h e p i o n f o r m f a c t o r w e r e s t u d i e d b y c o n s i d e r i n g o n l y the c o r r e c t i o n s to t h e h a r d s c a t t e r i n g a m p l i t u d e . I n a f u t u r e p u b l i c a t i o n w e p l a n to use the results p r e s e n t e d h e r e to a n a l y z e t h e c o r r e c t i o n s to the p i o n f o r m factor. ( W h i l e this p a p e r was b e i n g r e v i s e d for p u b l i c a t i o n , we r e c e i v e d a p a p e r f r o m D i t t e s a n d R a d y u s h k i n [15] w h i c h c o n f i r m s the results o f this analysis.) I t h a n k P r o f e s s o r R o b e r t C a r l i t z for s u g g e s t i n g this p r o b l e m a n d his e n c o u r a g e m e n t a n d s u g g e s t i o n s t h r o u g h o u t the w o r k . I also t h a n k P r o f e s s o r R a l p h R o s k i e s a n d P e n n y S a c k e t t for their a s s i s t a n c e in the use o f A S H M E D I .
References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15]
G.P. Lepage and S.J. Brodsky, Phys. Lett. 87B (1979) 359. G.P. Lepage and S.J. Brodsky, Phys. Rev. D22 (1981) 2157. A. Duncan and A. Mueller, Phys. Rev. D21 (1980) 1636. A.V. Efremov and A.V. Radyushkin, Phys. Lett. 94B (1980) 245. G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B175 (1980) 27. W. Kainz, W. Kummer and M. Schweda, Nucl. Phys. B79 (1974) 484; W. Konetschny and W. Kummer, Nucl. Phys. B100 (1975) 106; W. Kummer, Acta. Phys. Austriaca 41 (1975) 315. D.J. Pritchard and W.J. Stirling, Nu¢l. Phys. B165 (1980) 237. G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Nucl. Phys. B152 (1979) 285. E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B129 (1977) 66; B139 (1978) 545 (E). S.J. Brodsky, Y. Frishman, G.P. Lepage and C. Sachrajda, Phys. Lett. 91B (1980) 239. P. Damgaard, Ph.D. Dissertation, University of Cornell (1982). R.D. Field, R. Gupta, S. Otto and L. Chang, Nucl. Phys. B186 (1981) 429. F.M. Dittes and A.V. Radyushkin, Sov. J. Nucl. Phys. 34 (1981) 293. F.M. Dittes and A.V. Radyushkin, Phys. Lett. 134B (1984) 359.
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