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Operator product expansion in the minimal subtraction scheme
Volume 119B, number 4,5,6
PHYSICS LETTERS
23/30 December 1982
OPERATOR PRODUCT EXPANSION IN THE MINIMAL SUBTRACTION SCHEME K.G. CHETYRKIN a, S.G. GORISHNY b and F.V. TKACHOV a a Institute for Nuclear Research, USSR Academy o f Sciences, 60th October Prospect 7a, Moscow 117312, USSR b Joint Institute for Nuclear Research, Dubna, USSR
Received 9 August 1982
We formulate a new approach to evaluating the coefficient functions of the operator product expansion, which ensures that the coefficient functions for the T-product of two arbitrary composite operators can be analytically calculated at the three-loop level in the MS-scheme without limitations on the model, spin, twist, etc.
1. Introduction. Invented by Wilson in 1969 [1] the operator product expansion (OPE) has become by now an efficient method of studying various problems where an explicit separation of short and long distances is necessary [2]. Schematically, the OPE for two operators looks as T(J(x)J(O))
= ~ cn,r(x2,g) x ~ 0 n,r
X x (ul ... Un) O(Ul ... Un) .
(1)
Here ( 0 (ul "'" Un)} is a full set of local spin n operators, x(ul ... Un) = (xUl ... xUn - traces) is the irreducible rank n tensor formed with the vector x U , and cn'r(x2, g) represents the coefficient function (CF) corresponding to the operator 0 (n) (for brevity we sometimes condense ~ 1 '"/~n) to (n)). As usual, g stands for the coupling constant (s). In most practical applications of the OPE the/3function governing the behaviour of the effective coupling g, the anomalous dimensions of the operators entering into eq. (1), and the CF's are evaluated perturbatively, while the matrix elements of O n contain long distance contributions and should be extracted from the experimental data. As for the renormalization group (RG) functions (i.e./3- functions and anomalous dimensions), at present there exist powerful methods to compute them at least within the dimensional regularization [3] and the minimal subtraction (MS-) scheme [4]. We mean various methods of rearranging 0 031-9163/82/0000-0000/$02.75
© 1982 North:Holland
the IR structure of the Feynman integrals [ 5 - 7 ] , the Gegenbauer polynomial x-space technique [6] and the integration by parts method [8,9] which allowed for the first time to perform three-loop analytical calculations of the RG functions in some nonabelian models [ 10,11 ] and opened the possibility of doing this at the four-loop level [8,9]. At the same time no regular method to compute the CF's o f e q . (1) was developed. This is why almost all the earlier performed calculations of CF's were done by a "brute force" method and, as a result, never went beyond the one-loop approximation. (As far as we know there are only two papers dealing with the two-loop CF. The first one relies on numerical computation of complicated diagrams [ 12], while the second one deals with the OPE for three-quark (baryonic) currents and makes an essential use of particular properties of the corresponding diagrams [13].) Below we formulate a regular method which reduces the calculation of CF's at the L-loop level to the problem of finding the ultraviolet counterterms for some properly constructed (L + 1)-loop Feynman integrals. 2. F o r m u l a t i o n o f the method. For definiteness and brevity we will consider the OPE for the forward spin average scattering amplitude of a photon off a fermion in a toy massless model with the interaction L 1 = g~ff~p and rewrite eq. (1) as ~1
* ! We always work with euclidean Green (i.e. Schwinger) functions so that (3'/z,3'v} = 28Uv,~r~ = 3'/z and Q2 =q2=q,qu > O. 407
Volume 119B, number 4,5,6
PHYSICS LETTERS
f q(Ul ...Un) q(al ...a~) dDgl
T(p, q) =f exp(-iqx)(pi T(Ju(x ) Ju(0))l p) dOx Q22~ ' Z~
q(Ul...Un)
n>>-O(Qz)n+r/2-1 r~O t =/22/Q2 .
23130 December 1982
cn,r(t)(p[O(rUl...Un)(O)lp ) (2)
Here Ju = ~ ( - i T u ) qJ,D = 4 - 2e is the space-time dimension, # is the renormalization parameter in the MS-scheme [4]. For simplicity we assume that there is only one operator with a given spin n and twist r. What is the structure of the functions cn,r(t)? It is well known that in the limit e -* 0 they are nothing but polynomials of degree L in In(t):
=
6nhn ! Q2n
S E (al ""an) I'(n + ~D) 2 n D - 1 (u, ...Un) '
where S D _ 1 = 27rD/Z/P(D/2) is the surface of the unit hypersphere in RD (recall that dDq = ~ ( Q 2 ) l - e d(Q2) X dDh~ (al"''an) on the sP ace of •-t/-~ndE (/'tl --"Un) is the P ro'ector J all symmetric traceless tensors of rank n. Eq. (6) is easily deduced from the well-known connection between the convolution (q • p)(n) -_ q(at ... an) pal ... pan, and the Gegenbauer polynomial Cln-e(p. q/tPl" Iql) (see, e.g. refs. [6,9]). Substituting asymptotic expansion (2) for T(p, q) in eq. (5) and using eq. (6) we have
L lim cne,r(t) = cS'r(t) = i=0 f/(In t) i e~0
(3)
where L is the maximal number of loops in diagrams contributing to T(p, q) in the given order in g. It will be convenient to keep D 4= 4 during all calculations, so that eq. (3) can be cast into the following form:
L cne,r(t) : ~ Kn'r(e)(te)i '
eq.(5)=N(h,e)
~
K~,'/r e ] .
The angular integration in eq. (5) is performed by means of the orthogonality relation 408
2 hr dO Ce' (t)
(7)
where N(n, e) is a known number. Performing then the Q2 integration in each term of the sum on the rhs of eq. (7) we get eq. (5) = N(h, e)(pl o(h)[ p)
(4)
These formulae must be accompanied with two remarks. (i) Ansatz (4) is only a generalization of calculational experience. Hence, we take its validity for granted but there is no doubt that it is true in general case. (ii) Eqs. (4) remain valid even if some of the fields involved are massive, provided the coefficients Kn,r i,j acquire an extra dependence on masses. To extract the term with fixed n = fi and r = from the rhs of eq. (2), consider the integral
q(al...ah) (/22)(l+ 1)e dDq f T(q,p) (Q2)3 -e/2+le Q2 ;~ A2 (2rr)D
0o ~a ["
X Oa2/Q2) (l+l)e (plO (al "'"a'91p) + O ( 1 ) ,
L
Kn'r(e)=
(6)
X~
(/22 ](i+/+X)e
i>~o(i+l+ 1) e \ A 2 /
+ 0(1).
(8)
We conclude that it is sufficient to know the pole part (in e) of integral (5) at fixed h, ~ and arbitrary l to find the product ceh,e(t)(plO(h)lp) with an error O(e). This, in turn, allows one to obtain Ctd'r(t) (provided that the matrix element (plO~(h)lp) has been computed independently). It is worth noting that the pole part of eq. (5) does not depend on the value of A 2 so that one can put A 2 = 0 on the lhs of eq. (8), A 2 =/22 on the rhs of the same equation and rewrite it as eq. (5)1A2 =0 =N(h, e)(plO(h)lp)
L (5)
Kh'r(e)
X ~
K~,~(e)
i>>.O( i + l ÷ 1)e
+ O(1),
provided no IR divergences appear in eq. (5) when A 2 ~ 0. For the OPE under consideration this only happens if (h, F) = (0, 2). In what follows we will
(9)
Volume 119B, number 4,5,6
work with eq. (9) and consider only the case when (h, ~) ~ (0, 2). A general method of suppressing IR divergences induced by putting A 2 = 0 will be discussed elsewhere. So, we have reduced the problem of finding the Lloop CF's to evaluating the pole part of some propagator type (L + 1)-loop Feynman diagrams (i.e. diagrams depending on one external momentum). A more thorough analysis of the pole structure of the involved integrals has allowed us to represent eq. (5) as (poles in e) times (plO n Ip). Leaving a formal proof for a more substantial publication we will illustrate the point by computing the one-loop correction to the coefficient function C~' 2(0 of the twist 2 operators 0(2h) = (-i)nt~7(a, 8a2...aah) ~ from expansion (2). To save space occupied by figures we shall work within the so-called ladder approximation. The corresponding diagrams are shown in fig. 1. In diagrammatic language integral (5) can be considered as obtained by means of "gluing" two external fermion-photon vertices with a new line corresponding to the "propagator" [q(a, ... ah)/(Q2)3-g/2] (gZ/Q2)le
Writing down the identity/~R(c + d) = 0 and collecting all the terms induced by the R-operation in the rhs we get (see fig. 2) /~(c + d) =/£((e + f)(g + h)},
(b)
(10)
or
K(c+d)=(e+f)Zh(plO(2h)lp)+O(1).
(11)
Here Z h = 1 + (g2/l&r2)[e. h(h + 1)] -1 is the renormalization constant of the operator oh2 chosen so that Z~-1 • (g + h) remains finite at e ~ 0 and equals to the MS-renormalized matrix element (p[O(2n)lp) in the one-loop ladder approximation. On combining eqs. (9) and (1 I) we come to the following equation to fred C~'2(t) L
No(h, e) Z.~
- 2(e + f ) Z h + O(1). (i + l + 1) e
(12)
(The factor 2 on the rhs of eq. (12) takes into account the contribution of the crossed diagrams.) Now C~'2(t) can be obtained as follows:
c '2(0
(see figs. 1c, 1d). Our next aim consists in transforming the expression 2/~(c + d) to the form: (poles in e) times (renormalized matrix element
(a)
23/30 December 1982
PHYSICS LETTERS
L
= 2h 16rr2(h + 1) lira e ~ e--r0
lim
{(k + 1 +l)
k=0/~-(k+l)
× tek[2 • (e + f) Z a ] ) .
(13)
A straightforward calculation of the counterterms corresponding to the diagrams (e) and (f) gives
+2 Formal definitions of the R-operation and counterterms in the MS-scheme can be found e.g. in ref. [6].
(c)
(~)
Fig. 1. (a) The Born contribution to the amplitude T(p, q). (b) The ordy ladder, type diagram contributing to T(p, q) in the oneloop approximation. Inclusion of the crossed diagrams is understood. (c) and (d) are the diagrams obtained from (a) and (b) after performing integration over q with the weight function q(al...ah) (~2)(1 + 1) e / (Q2)3 -~'/2 +le_
409
Volume 119B, number 4,5,6
PHYSICS LETTERS
(e)
(f)
23•30 December 1982
(g)
(h)
Fig. 2. An equation obtained by means of disentangling identity/~R(c + d) = 0. Encircled graphs denote polynomials in e-1 which, multiplied by appropriate momentum structures, give counterterms for these graphs. (g) and (h) are the diagrams for the (unrenormalized) matrix element (pl ~'r (al aa2 ... ac~h)q~lp) in the oneqoop ladder approximation.
(e) = 4(16rr2)-1/e(l + 1)(h + 1), 4g 2
[
(-1)
( f ) - (16~r2)2[ e2(l + 2 ) h ( h + 1) 2. (h + 1)[~'h(1 ) +~i] - - h 2 - 5h - 3
+
e(t+ -2h(h
+
1)h(h + I) 3
+ 1)[~h(1 ) + ,51 - 2h 2 - 2h + 1 \
(14)
e(l+ 2 ) h 2 ( h + 1) 3
)'
where ;t > 0 is even, 6 = (2 - 7E + In 4rr) and ~h(1) = ~n= 1 1/i. The attentive reader should not worry about the appearance o f the Euler constant "YE and In 4rr in the result. The well-known theorem which states that these constants never enter into the MScounterterms is no longer operative for diagram (f) because the latter includes a nonstandard propagator with the e-dependent exponent. Note that ~E and In 4rr go away if one puts I = 0. Finally, combining eqs. (13) and (14) we get for n even:
cn,2(t) = 2n+3 _ [2n+3gZJ(47r)2 l x(n[lnt+~n(1)+6] +n 2 +6nn2(n + 1) and
1)
c2n+ l,2(t) = O.
3. Discussion. Analogous considerations dealing with objects like (5) plus a bit o f combinatorics a la Anikin and Zavialov [15] lead to a derivation of OPE in any model (including massive ones) together with explicit formulae for CF's generalizing eq. (13). Moreover, the main feature o f eq. (11), namely, the neces410
sity o f computing only MS-counterterms with the number o f loops not exceeding (L ÷ 1) to find the Lloop correction to the CF's goes through the generalization untouched. And since in the MS-scheme counterterms depend on masses polynomially [16] we come to the conclusion that in this scheme CF's depend on masses also polynomiaUy. This unique feature o f the MS-scheme allows one to compute CF's for arbitrary massive models. Finally, recalling the theorem on the analytical calculability of the MS-counterterms up to the four-loop level we conclude that the formulated technique ensures analytical calculability o f one-, two- and threeloop corrections to CF's in any model within the MSscheme. We are grateful to Professor V.A. Matveev and Professor A.N. Tavkhelidze for the continuing support and to S.A. Larin for valuable comments.
References [1] K. Wilson, Phys. Rev. D2 (1969) 1473. [2] A. Peterman, Phys. Rep. 53 (1979) 159. [3] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; C.G. BoUini and J.J. Giambiagi, Nuovo Cimento 12B (1972) 20; G. Cicuta and E. Montaldi, Nuovo Cimento Lett. 4 (1972) 329. [4] G. 't Hooft, Nucl. Phys. B61 (1973)475. [5] A.A. Vladimirov, Teor. Mat. Fiz. 43 (1980) 210. [6] K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Nucl. Phys. B174 (1980) 345. [7] K.G. Chetyrkin and F.V. Tkachov, Phys. Lett., to be published. [8] F.V. Tkachov, Phys. Lett. 100B (1981) 65.
Volume 119B, number 4,5,6
PHYSICS LETTERS
[9] K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159. [10] O.V. Taxasov, A.A. Vladimirov and A.Yu. Zharkov, Phys. Lett. 93B (1980) 429. [ 11 ] L.V. Avdeev, O.V. Tarasov and A.A. Vladimirov, Phys. Lett. 96B (1980) 94. [12] D.W. Duke, J.D. Kimel and G.A. Sowell, Phys. Rev. D25 (1982) 71.
23•30 December 1982
[13] A.V. Smilga, Yad. Fiz. 35 (1982) 473. [ 14] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Nauka, Moscow, 1976). [ 15] S.A. Anikin and O.I. Zav'yalov, Theor. Math. Phys. 26 (1976) 105. [16] J.C. Collins, Nucl. Phys. B92 (1975) 477.
411
PHYSICS LETTERS
23/30 December 1982
OPERATOR PRODUCT EXPANSION IN THE MINIMAL SUBTRACTION SCHEME K.G. CHETYRKIN a, S.G. GORISHNY b and F.V. TKACHOV a a Institute for Nuclear Research, USSR Academy o f Sciences, 60th October Prospect 7a, Moscow 117312, USSR b Joint Institute for Nuclear Research, Dubna, USSR
Received 9 August 1982
We formulate a new approach to evaluating the coefficient functions of the operator product expansion, which ensures that the coefficient functions for the T-product of two arbitrary composite operators can be analytically calculated at the three-loop level in the MS-scheme without limitations on the model, spin, twist, etc.
1. Introduction. Invented by Wilson in 1969 [1] the operator product expansion (OPE) has become by now an efficient method of studying various problems where an explicit separation of short and long distances is necessary [2]. Schematically, the OPE for two operators looks as T(J(x)J(O))
= ~ cn,r(x2,g) x ~ 0 n,r
X x (ul ... Un) O(Ul ... Un) .
(1)
Here ( 0 (ul "'" Un)} is a full set of local spin n operators, x(ul ... Un) = (xUl ... xUn - traces) is the irreducible rank n tensor formed with the vector x U , and cn'r(x2, g) represents the coefficient function (CF) corresponding to the operator 0 (n) (for brevity we sometimes condense ~ 1 '"/~n) to (n)). As usual, g stands for the coupling constant (s). In most practical applications of the OPE the/3function governing the behaviour of the effective coupling g, the anomalous dimensions of the operators entering into eq. (1), and the CF's are evaluated perturbatively, while the matrix elements of O n contain long distance contributions and should be extracted from the experimental data. As for the renormalization group (RG) functions (i.e./3- functions and anomalous dimensions), at present there exist powerful methods to compute them at least within the dimensional regularization [3] and the minimal subtraction (MS-) scheme [4]. We mean various methods of rearranging 0 031-9163/82/0000-0000/$02.75
© 1982 North:Holland
the IR structure of the Feynman integrals [ 5 - 7 ] , the Gegenbauer polynomial x-space technique [6] and the integration by parts method [8,9] which allowed for the first time to perform three-loop analytical calculations of the RG functions in some nonabelian models [ 10,11 ] and opened the possibility of doing this at the four-loop level [8,9]. At the same time no regular method to compute the CF's o f e q . (1) was developed. This is why almost all the earlier performed calculations of CF's were done by a "brute force" method and, as a result, never went beyond the one-loop approximation. (As far as we know there are only two papers dealing with the two-loop CF. The first one relies on numerical computation of complicated diagrams [ 12], while the second one deals with the OPE for three-quark (baryonic) currents and makes an essential use of particular properties of the corresponding diagrams [13].) Below we formulate a regular method which reduces the calculation of CF's at the L-loop level to the problem of finding the ultraviolet counterterms for some properly constructed (L + 1)-loop Feynman integrals. 2. F o r m u l a t i o n o f the method. For definiteness and brevity we will consider the OPE for the forward spin average scattering amplitude of a photon off a fermion in a toy massless model with the interaction L 1 = g~ff~p and rewrite eq. (1) as ~1
* ! We always work with euclidean Green (i.e. Schwinger) functions so that (3'/z,3'v} = 28Uv,~r~ = 3'/z and Q2 =q2=q,qu > O. 407
Volume 119B, number 4,5,6
PHYSICS LETTERS
f q(Ul ...Un) q(al ...a~) dDgl
T(p, q) =f exp(-iqx)(pi T(Ju(x ) Ju(0))l p) dOx Q22~ ' Z~
q(Ul...Un)
n>>-O(Qz)n+r/2-1 r~O t =/22/Q2 .
23130 December 1982
cn,r(t)(p[O(rUl...Un)(O)lp ) (2)
Here Ju = ~ ( - i T u ) qJ,D = 4 - 2e is the space-time dimension, # is the renormalization parameter in the MS-scheme [4]. For simplicity we assume that there is only one operator with a given spin n and twist r. What is the structure of the functions cn,r(t)? It is well known that in the limit e -* 0 they are nothing but polynomials of degree L in In(t):
=
6nhn ! Q2n
S E (al ""an) I'(n + ~D) 2 n D - 1 (u, ...Un) '
where S D _ 1 = 27rD/Z/P(D/2) is the surface of the unit hypersphere in RD (recall that dDq = ~ ( Q 2 ) l - e d(Q2) X dDh~ (al"''an) on the sP ace of •-t/-~ndE (/'tl --"Un) is the P ro'ector J all symmetric traceless tensors of rank n. Eq. (6) is easily deduced from the well-known connection between the convolution (q • p)(n) -_ q(at ... an) pal ... pan, and the Gegenbauer polynomial Cln-e(p. q/tPl" Iql) (see, e.g. refs. [6,9]). Substituting asymptotic expansion (2) for T(p, q) in eq. (5) and using eq. (6) we have
L lim cne,r(t) = cS'r(t) = i=0 f/(In t) i e~0
(3)
where L is the maximal number of loops in diagrams contributing to T(p, q) in the given order in g. It will be convenient to keep D 4= 4 during all calculations, so that eq. (3) can be cast into the following form:
L cne,r(t) : ~ Kn'r(e)(te)i '
eq.(5)=N(h,e)
~
K~,'/r e ] .
The angular integration in eq. (5) is performed by means of the orthogonality relation 408
2 hr dO Ce' (t)
(7)
where N(n, e) is a known number. Performing then the Q2 integration in each term of the sum on the rhs of eq. (7) we get eq. (5) = N(h, e)(pl o(h)[ p)
(4)
These formulae must be accompanied with two remarks. (i) Ansatz (4) is only a generalization of calculational experience. Hence, we take its validity for granted but there is no doubt that it is true in general case. (ii) Eqs. (4) remain valid even if some of the fields involved are massive, provided the coefficients Kn,r i,j acquire an extra dependence on masses. To extract the term with fixed n = fi and r = from the rhs of eq. (2), consider the integral
q(al...ah) (/22)(l+ 1)e dDq f T(q,p) (Q2)3 -e/2+le Q2 ;~ A2 (2rr)D
0o ~a ["
X Oa2/Q2) (l+l)e (plO (al "'"a'91p) + O ( 1 ) ,
L
Kn'r(e)=
(6)
X~
(/22 ](i+/+X)e
i>~o(i+l+ 1) e \ A 2 /
+ 0(1).
(8)
We conclude that it is sufficient to know the pole part (in e) of integral (5) at fixed h, ~ and arbitrary l to find the product ceh,e(t)(plO(h)lp) with an error O(e). This, in turn, allows one to obtain Ctd'r(t) (provided that the matrix element (plO~(h)lp) has been computed independently). It is worth noting that the pole part of eq. (5) does not depend on the value of A 2 so that one can put A 2 = 0 on the lhs of eq. (8), A 2 =/22 on the rhs of the same equation and rewrite it as eq. (5)1A2 =0 =N(h, e)(plO(h)lp)
L (5)
Kh'r(e)
X ~
K~,~(e)
i>>.O( i + l ÷ 1)e
+ O(1),
provided no IR divergences appear in eq. (5) when A 2 ~ 0. For the OPE under consideration this only happens if (h, F) = (0, 2). In what follows we will
(9)
Volume 119B, number 4,5,6
work with eq. (9) and consider only the case when (h, ~) ~ (0, 2). A general method of suppressing IR divergences induced by putting A 2 = 0 will be discussed elsewhere. So, we have reduced the problem of finding the Lloop CF's to evaluating the pole part of some propagator type (L + 1)-loop Feynman diagrams (i.e. diagrams depending on one external momentum). A more thorough analysis of the pole structure of the involved integrals has allowed us to represent eq. (5) as (poles in e) times (plO n Ip). Leaving a formal proof for a more substantial publication we will illustrate the point by computing the one-loop correction to the coefficient function C~' 2(0 of the twist 2 operators 0(2h) = (-i)nt~7(a, 8a2...aah) ~ from expansion (2). To save space occupied by figures we shall work within the so-called ladder approximation. The corresponding diagrams are shown in fig. 1. In diagrammatic language integral (5) can be considered as obtained by means of "gluing" two external fermion-photon vertices with a new line corresponding to the "propagator" [q(a, ... ah)/(Q2)3-g/2] (gZ/Q2)le
Writing down the identity/~R(c + d) = 0 and collecting all the terms induced by the R-operation in the rhs we get (see fig. 2) /~(c + d) =/£((e + f)(g + h)},
(b)
(10)
or
K(c+d)=(e+f)Zh(plO(2h)lp)+O(1).
(11)
Here Z h = 1 + (g2/l&r2)[e. h(h + 1)] -1 is the renormalization constant of the operator oh2 chosen so that Z~-1 • (g + h) remains finite at e ~ 0 and equals to the MS-renormalized matrix element (p[O(2n)lp) in the one-loop ladder approximation. On combining eqs. (9) and (1 I) we come to the following equation to fred C~'2(t) L
No(h, e) Z.~
- 2(e + f ) Z h + O(1). (i + l + 1) e
(12)
(The factor 2 on the rhs of eq. (12) takes into account the contribution of the crossed diagrams.) Now C~'2(t) can be obtained as follows:
c '2(0
(see figs. 1c, 1d). Our next aim consists in transforming the expression 2/~(c + d) to the form: (poles in e) times (renormalized matrix element
(a)
23/30 December 1982
PHYSICS LETTERS
L
= 2h 16rr2(h + 1) lira e ~ e--r0
lim
{(k + 1 +l)
k=0/~-(k+l)
× tek[2 • (e + f) Z a ] ) .
(13)
A straightforward calculation of the counterterms corresponding to the diagrams (e) and (f) gives
+2 Formal definitions of the R-operation and counterterms in the MS-scheme can be found e.g. in ref. [6].
(c)
(~)
Fig. 1. (a) The Born contribution to the amplitude T(p, q). (b) The ordy ladder, type diagram contributing to T(p, q) in the oneloop approximation. Inclusion of the crossed diagrams is understood. (c) and (d) are the diagrams obtained from (a) and (b) after performing integration over q with the weight function q(al...ah) (~2)(1 + 1) e / (Q2)3 -~'/2 +le_
409
Volume 119B, number 4,5,6
PHYSICS LETTERS
(e)
(f)
23•30 December 1982
(g)
(h)
Fig. 2. An equation obtained by means of disentangling identity/~R(c + d) = 0. Encircled graphs denote polynomials in e-1 which, multiplied by appropriate momentum structures, give counterterms for these graphs. (g) and (h) are the diagrams for the (unrenormalized) matrix element (pl ~'r (al aa2 ... ac~h)q~lp) in the oneqoop ladder approximation.
(e) = 4(16rr2)-1/e(l + 1)(h + 1), 4g 2
[
(-1)
( f ) - (16~r2)2[ e2(l + 2 ) h ( h + 1) 2. (h + 1)[~'h(1 ) +~i] - - h 2 - 5h - 3
+
e(t+ -2h(h
+
1)h(h + I) 3
+ 1)[~h(1 ) + ,51 - 2h 2 - 2h + 1 \
(14)
e(l+ 2 ) h 2 ( h + 1) 3
)'
where ;t > 0 is even, 6 = (2 - 7E + In 4rr) and ~h(1) = ~n= 1 1/i. The attentive reader should not worry about the appearance o f the Euler constant "YE and In 4rr in the result. The well-known theorem which states that these constants never enter into the MScounterterms is no longer operative for diagram (f) because the latter includes a nonstandard propagator with the e-dependent exponent. Note that ~E and In 4rr go away if one puts I = 0. Finally, combining eqs. (13) and (14) we get for n even:
cn,2(t) = 2n+3 _ [2n+3gZJ(47r)2 l x(n[lnt+~n(1)+6] +n 2 +6nn2(n + 1) and
1)
c2n+ l,2(t) = O.
3. Discussion. Analogous considerations dealing with objects like (5) plus a bit o f combinatorics a la Anikin and Zavialov [15] lead to a derivation of OPE in any model (including massive ones) together with explicit formulae for CF's generalizing eq. (13). Moreover, the main feature o f eq. (11), namely, the neces410
sity o f computing only MS-counterterms with the number o f loops not exceeding (L ÷ 1) to find the Lloop correction to the CF's goes through the generalization untouched. And since in the MS-scheme counterterms depend on masses polynomially [16] we come to the conclusion that in this scheme CF's depend on masses also polynomiaUy. This unique feature o f the MS-scheme allows one to compute CF's for arbitrary massive models. Finally, recalling the theorem on the analytical calculability of the MS-counterterms up to the four-loop level we conclude that the formulated technique ensures analytical calculability o f one-, two- and threeloop corrections to CF's in any model within the MSscheme. We are grateful to Professor V.A. Matveev and Professor A.N. Tavkhelidze for the continuing support and to S.A. Larin for valuable comments.
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PHYSICS LETTERS
[9] K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159. [10] O.V. Taxasov, A.A. Vladimirov and A.Yu. Zharkov, Phys. Lett. 93B (1980) 429. [ 11 ] L.V. Avdeev, O.V. Tarasov and A.A. Vladimirov, Phys. Lett. 96B (1980) 94. [12] D.W. Duke, J.D. Kimel and G.A. Sowell, Phys. Rev. D25 (1982) 71.
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[13] A.V. Smilga, Yad. Fiz. 35 (1982) 473. [ 14] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Nauka, Moscow, 1976). [ 15] S.A. Anikin and O.I. Zav'yalov, Theor. Math. Phys. 26 (1976) 105. [16] J.C. Collins, Nucl. Phys. B92 (1975) 477.
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