Methods of calculation of higher power corrections in QCD

arXiv:hep-ph/9412238v2 6 Dec 1994

OUT–4102–54 hep-ph/9412238 December 1994

Methods of calculation of higher power corrections in QCD

A. G. Grozin1 Physics Department, Open University, Milton Keynes MK7 6AA, UK

Abstract Although the methods of calculation of power corrections in QCD sum rules are well known, algebraic complexity rapidly grows with the increase of vacuum condensates’ dimensions. Currently, state–of–the–art calculations include dimension 7 and 8 condensates. I summarize and extend algorithms of such calculations. First, I present all the formulae necessary for application of the systematic classification of bilinear quark condensates proposed earlier, and extend this method to the case of gluon condensates. Then I apply these systematic procedures to expansions of bilinear and noncollinear quark and gluon condensates in local ones, and of noncollinear condensates in bilocal ones. The formulae obtained can be used for calculation of correlators involving nonlocal condensates, and for inventing consistent anzatzs for these condensates. Finally, I briefly summarize the methods of calculation of heavy– and light–quark currents’ correlators. This paper is aimed both to present new results on gluon and nonlocal condensates and to be a self–contained handbook of formulae necessary for calculation of power corrections in QCD sum rules.

1

[email protected]; on leave of absence from Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia 1

1

Introduction

The Operator Product Expansion (OPE) is widely used for investigation of correlators in quantum field theory. In particular, it is the basis of the QCD sum rules method. Let Π(p) be a correlator of two local currents in the momentum space. Then OPE reads Π(p) =

X i

ci (p2 ) hOi i ,

(1.1)

where hOi i are vacuum averages of local operators of various dimensions d (called also vacuum condensates), and ci (p2 ) are coefficient functions. They are perturbative series in αs . In order to make sum rules more precise and to establish their applicability regions it is necessary to calculate both higher perturbative corrections (higher terms in αs expansions of coefficient functions) and higher non–perturbative (power) corrections. I shall not discuss physical applications of QCD sum rules here. Methods of calculation of higher loop diagrams (perturbative corrections) form an established field of research, and many articles and reviews are devoted to them. I shall not discuss this subject; instead I shall concentrate on the methods of calculation of higher–dimensional terms in the OPE (1.1) in the leading order in αs . There already exists an excellent review on technical problems of QCD sum rules calculations [1]. The present article can be considered as an addendum to it. In all cases when a question is discussed in [1], I shall refer to this review and the literature cited in it. I should like to apologize to the authors of these articles for not citing them directly. I summarize and extend the methods of calculation of higher power corrections to correlators of heavy– [2, 3] and the light–quark currents [4–9]. Full technical details (partly omitted from the original papers) will be included for the reader’s convenience. In many cases, even if an idea of a calculation technique have appeared in the literature, it is very tedious and error–prone to rederive all the necessary formulae for each specific application. A set of reliable formulae can provide an invaluable help for many calculations of power corrections to QCD sum rules. I shall formulate the algorithms of calculation of higher power corrections in a form suitable for a Computer Algebra implementation. Unfortunately, there is no complete package for such calculations in conventional Computer Algebra systems (REDUCE, Mathematica,. . . ). Such a package would be very useful. I have written several REDUCE procedures and Modula2 programs (producing a REDUCE readable output) which implement some of the discussed algorithms. They are useful as a set of tools though they can’t be a substitute for a complete package. The plan of the article is following. In Section 2 I briefly introduce the fixed–point (Fock– Schwinger) gauge on which the modern methods of calculation of power corrections are based; for more details and proofs see [1]. Section 3 is devoted to the systematic classification of vacuum condensates. There exist many relations between condensates, therefore one has to choose a basis of independent condensates and formulate an algorithm of reducing an arbitrary condensate to this basis. A systematic classification of bilinear quark condensates has been proposed in [9]. A carefully checked set of formulae implementing this method is presented in Section 3.1; this makes its application to any particular case very easy. In Section 3.2, I extend this method to the case of gluon condensates. Section 3.3 contains some useful techniques of averaging two– and three–point correlators. Any calculation of power corrections to a correlator naturally involves nonlocal condensates. In the standard approach, they are expanded in series in local ones. Section 4 is devoted to a detailed study of some simple types of nonlocal condensates using the systematic methods introduced in Section 3. Tree–level calculations of two–point correlators usually involve gauge– invariant bilocal condensates. Their expansions in local ones are considered in Section 4.1. 2

Three–point correlators, as well as correlators involving bilocal currents (which are used in sum rules for hadron wave functions) require more complicated trilocal objects—noncollinear condensates (Section 4.2). In some applications, such as sum rules for wave function moments, it is convenient to expand noncollinear condensates in series in one of the spacings. Coefficients in these series are bilocal condensates. Such expansions are constructed in Section 4.3. Expansions obtained in this Section can be used for calculation of correlators involving nonlocal condensates, and for inventing consistent anzatzs for these condensates. Finally, I briefly summarize methods of calculation of heavy quark currents’ correlators (including heavy quark condensates) in Section 5, and of light quark currents’ correlators— in Section 6.

2

Fixed–point gauge

The QCD vacuum has a non–trivial structure. In order to calculate a correlator in QCD, we should first calculate it in a background (vacuum) gluon and quark field. Then we should average the expression for a correlator via these fields over the vacuum. After that we obtain the expression for a correlator via vacuum condensates (1.1). Correlators of colourless currents are gauge invariant. Therefore we can use any gauge for the background gluon field. It is convenient to use the fixed–point (Fock–Schwinger) gauge xµ Aaµ (x) = 0.

(2.1)

In this gauge the Taylor expansions for Aaµ (x) and q(x) can be written in a gauge–covariant form [1] Aaµ (x) =

1 1 xν Gaνµ (0) + xα xν Dα Gaνµ (0) 2 · 0! 3 · 1!

1 xβ xα xν Dβ Dα Gaνµ (0) + O(x4 ), (2.2) 4 · 2! 1 q(x) = q(0) + xα Dα q(0) + xβ xα Dβ Dα q(0) + O(x3 ) 2! ← − (and hence q(x) = q(0) + q(0) D α xα + · · ·). Here Dµ q = (∂µ − iAµ )q is a covariant derivative b in the fundamental representation, Aµ = gAaµ ta ; Dµ Gaρσ = (∂µ δab + Aab µ )Gρσ is a covariant acb Ac . Hence only gauge covariant quantities derivative in the adjoint representation, Aab µ µ = gf are used at all stages of calculations. In this gauge the theory is not translational invariant. A translation should be accompanied by a gauge transformation to another fixed point gauge. Correlators become gauge invariant and hence translationally invariant after vacuum averaging. We can choose the gauge fixed point (origin in (2.1)) at any vertex of the correlator to simplify calculations. In the momentum space, d4 ki a background gluon lines have incoming momenta ki and contribute Aaµ (ki ) (2π) 4 where Aµ (ki ) is the Fourier transform of (2.2) and is the series in derivatives of δ(ki ). The vertex chosen as the gauge fixed point provides a common sink for all vacuum momenta ki . The quark propagator (Fig. 1) can be written as a sum over the number of background gluon lines: +

S(p) =

∞ X

Sk (p),

k=0 ∞ X

Aµ = −i

l=0

S0 (p) =

pˆ + m , p 2 − m2

ˆ k−1 (p), Sk (p) = −S0 (p)AS

(−i)l Dα . . . Dαl Gaνµ ta ∂α1 . . . ∂αl ∂ν . (l + 2)l! 1

3

(2.3)

iS(p) r

x

r





p

 

p

r =

iS0 (p)

r

r +

 

0

r

p

) ( ) k1 ( ? ) (  

p − k1

 

p

) ( ) k2 ( ? ) (  

) ( ) k1 ( ? ) (  

r +

p − k1

r +···

p − k1 − k2

Figure 1: Quark propagator in background gluon field It is convenient to use recurrent relations for Sk (p): Sk = S0

∞ X Akl l=0

l+2

,

Ak0 = iGµν ∂µ γν Sk−1 ,

i Akl = − ∂α Dα Ak,l−1 . l

(2.4)

Here ∂µ = ∂/∂pµ acts on all propagators to the right, and Dµ —only on the nearest Gµν = gGaµν (0)ta . For example, retaining gluon operators with d ≤ 4 (Fig. 1) we have 1 i S = S0 + Gµν S0 ∂µ γν S0 + Dα Gµν S0 ∂α ∂µ γν S0 2 3 i 1 − Dβ Dα Gµν S0 ∂β ∂α ∂µ γν S0 − Gρσ Gµν S0 ∂ρ γσ S0 ∂µ γν S0 . 8 4

(2.5)

I have written a Modula-2 program for simplification of expressions constructed from S0 , ∂µ , and γµ . It applies ∂µ to all S0 to the right using ∂µ S0 = −S0 γµ S0 , collects similar terms, and produces a REDUCE readable output, which can be used, for example, to expand these expressions in basis γ matrices. These formulae can be used for both massive and massless quarks. For a light quark we can expand S0 in m. For a massless quark the equation (2.5) gives S(p) =

1 1 e µν γν γ5 pˆ − 4 pµ G p2 p

(2.6)

"

 2 2ˆ 1 e µν γν γ5 p J − pµ Jµ pˆ + pλ Dλ pµ Gµν γν − 2ipλ Dλ pµ G + 6 − p 3 "

#

    1 + 8 2 p2 γµ Gµλ Gλν pν − pµ Gµλ Gλν pν pˆ − 2ipλ Dλ p2 Jˆ − pµ Jµ pˆ p 

2

2

+ 4(pλ Dλ ) − p D P

2



# i h e e pµ Gµν γν γ5 − i pµ Gµλ , Gλν γν γ5 .

Here Jµ = gJµa ta , Jµa = Dν Gaµν = g q′ q′ γµ ta q ′ . Note that the last term in (2.6) is missing in [1]. In deriving this formula, we have used the relations D 2 Gµν = −Dµ Jν + Dν Jµ + 2i[Gµλ , Gλν ], 4

e ρσ Gρσ , Gµν G e µν = G e µν Gµν . In e λν + G e νλ Gλµ = − 1 δµν G Dν Dλ Gµν = Dλ Jµ − i[Gνλ , Gµν ], Gµλ G 2 practical calculations it is often more convenient to substitute symbolic expressions for propagators like (2.5) to the diagrams and to simplify the complete answers rather than to use (2.6). In order to consider the gluon propagation in the background gluon field we should substitute Aaµ + aaµ into the QCD lagrangian, where Aaµ is the background field and aaµ is the quantum gluon field. The lagrangian becomes

1 1 1 1 b L = − Gaµν Gaµν − (Dµ aaν )(Dµ aaν ) + (Dµ aaν )(Dν aaµ ) + aaµ Gab µν aν + · · · 4 2 2 2

(2.7)

acb Gc . We choose Here Gaµν and Dµ include only the background field Aaµ , and Gab µν µν = gf a the fixed–point gauge for the background field Aµ . This lagrangian is still gauge invariant with respect to aaµ . We should add a gauge fixing term and a corresponding ghost term. It is convenient to use a generalization of the Feynman gauge fixing term − 21 (Dµ aaµ )2 . Then the 0 (p) = δ /p2 , and the vertices of the gluon’s interaction with the free gluon propagator is Dµν µν ab ab ab background field (Fig. 2) are iAab (p 1 + p2 )λ δµν − iAµ (p1 − p2 )ν − iAν (p2 − p1 )µ − iGµν and λ ac cb cb ac cb −Aac λ Aλ δµν − Aµ Aν + Aµ Aν .

) ( ) µ ν r ⌣⌢⌣⌢⌣⌢⌣ ⌢⌣⌢⌣⌢⌣⌢⌣(⌢ a b

p1

p2

 

  µ ν r ⌣⌢⌣⌢⌣⌢⌣

⌢⌣⌢⌣⌢⌣⌢⌣ ⌢

a

b

Figure 2: Vertices of gluon’s interaction with background gluon field

ab (p) µa −iDµν

νb

) ( k1 ) ? r ⌣⌢⌣⌢⌣(⌢ r ⌣⌢⌣⌢⌣r + ⌢  

ab (p) −iD0µν

r ⌣⌢⌣⌢⌣⌢⌣r + r ⌣⌢⌣⌢⌣⌢⌣r = ⌢ ⌢  

x

p

0

) ) ( ( k1 k2 ) ) ? ? r ⌣⌢⌣⌢⌣(⌢ r ⌣⌢⌣⌢⌣r + r ⌣⌢⌣⌢⌣(⌢ ⌢   

p

p − k1

p

p

p − k1

 

  r ⌣⌢⌣⌢⌣ + · · · r ⌣⌢⌣⌢⌣

⌢ ⌢

p − k1 − k2

Figure 3: Gluon propagator in background gluon field Retaining terms with d ≤ 4, we have the gluon propagator (Fig. 3) 

1 1 2 1 ipµ Jµ δαβ + 4ipλ Dλ Gαβ Dαβ = 2 δαβ + 4 2Gαβ + 6 p p p 3 "   1 + 8 − 2pλ Dλ pµ Jµ δαβ − 2 4(pλ Dλ )2 − p2 D 2 Gαβ p 5



(2.8)

#

 1 2 + p Gµν Gµν + 4pµ Gµλ Gλν pν δαβ + 4p2 Gαλ Gλβ . 2

Here the matrix notations are used for colour indices.

3

Vacuum condensates’ classification

Vacuum condensates can be divided into classes according to the number of quark fields in them. Those without quark fields are gluon condensates. Those with two quark fields are bilinear quark condensates; they have d ≥ 3. Four–quark condensates have d ≥ 6, and so on. The unit operator is the gluon operator with d = 0, according to this classification. As is clear from the previous Section, bilinear quark condensates of the form hq(D . . . DG)(D . . . DG) . . . γ . . . γqi appear in calculations of bilinear quark currents’ correlators at the tree level. Analogously, gluon condensates of the form hTr(D . . . DG)(D . . . DG) . . .i appear in the one–loop approximation (there are at least two (D . . . DG) groups because otherwise the colour trace vanishes). In Section 3.1 I discuss the systematic classification of bilinear quark condensates following [9]. All formulae necessary for the practical use of this method are presented. In Section 3.2 I apply similar methods to the gluon condensates. Techniques of vacuum averaging of expressions for correlators via quark and gluon fields are discussed in Section 3.3.

3.1

Bilinear quark condensates

Using the formulae Gµν = gGaµν ta = i[Dµ , Dν ], (Dµ A) = (Dµab Ab )ta = [Dµ , A] (where Aa is in the adjoint representation of the colour group, and A = Aa ta ), we can easily reduce any bilinear quark condensate of dimension d = m + 3 to a combination of terms of the form hqDµ1 . . . Dµm Γµ1 ...µm qi, where Γµ1 ...µm is constructed from γµ and δµν . Choosing the terms with the largest number of γ matrices, we permute them in such a way that their indices are in the same order as in Dµ1 . . . Dµm . Arising additional terms have fewer γ matrices. We repeat this procedure going to terms with fewer γ matrices until we reach terms with one γ matrix or without them at all. As a result, a bilinear quark condensate reduces to a linear combination ˆ and Dµ . Due to the equations of of terms of the form hqOi qi, where Oi are constructed from D ˆ is adjacent to q or q reduce to lower–dimensional condensates motion, those terms in which D multiplied by m. ˆ Having written down all the condensates Bid = hqOim qi where Oim are constructed from D ˆ is adjacent to q and q, we obtain a certain set of d–dimensional bilinear quark and Dµ and no D condensates. We have just demonstrated that any d–dimensional condensate can be systematically expressed via Bid , mBid−1 ,. . . This means that they form a basis of quark condensates with dimensions ≤ d. But this basis is extremely inconvenient for use because its condensates contain a maximum number of derivatives acting on q. So we should better choose a basis of the most convenient d–dimensional condensates Qdj . We can express Qdj via Bid , mBid−1 ,. . . Solving this linear system, we obtain the expressions for Bid via Qdj , mQd−1 ,. . . Having these expressions, we j ,. . . can easily reduce any d–dimensional bilinear quark condensate to the basis ones Qdj , mQd−1 j d A general guideline for choosing good Qj is to have a minimum number of derivatives acting on the quark field, and hence a maximum dimension of gluon operators in Qdj . It allows to expand propagators to a minimum dimension in the background gluon field. Among such operators one should first of all choose those containing Jµ because they are really four–quark ones and are more easily calculable. They may be suppressed in some vacuum models (of the instanton type) in which vacuum gluon fields may be strong but Dν Gµν ≈ 0. 6

For dimensions d ≤ 6 we have

D

E

B 5 = − qD 2 q ,

B 3 = hqqi , We choose the basis condensates

Q3 = hqqi ,

D

E

ˆ µq . B 6 = −i qDµ DD

Q5 = i hqGµν σµν qi ,

D

E

ˆ , Q6 = q Jq

(3.1)

(3.2)

where σµν = γ[µ γν] , and square brackets mean antisymmetrization (Q5 is usually denoted m20 Q3 ). These condensates are expressed via Bid as Q3 = B 3 , Q5 = −2B 5 + 2m2 B 3 ,

(3.3)

B 3 = Q3 , 1 B 5 = − Q5 + m 2 Q3 , 2 1 m B 6 = − Q6 − Q5 + m 3 Q3 . 2 2

(3.4)

6

6

5

Q = −2B + 2mB .

Solving this linear system we obtain

For d = 7 we have E

D

B17 = qD 2 D 2 q ,

D

E

B27 = qDµ D 2 Dµ q ,

B37 = hqDµ Dν Dµ Dν qi ,

D

E

ˆ 2 Dµ q . B47 = qDµ D

We choose basis condensates D

E

e µν γ5 q , Q72 = i qGµν G

Q71 = hqGµν Gµν qi ,

Q73 = hqGµλ Gλν σµν qi ,

Q74 = i hqDµ Jν σµν qi .

(3.5)

(3.6)

e µν = 1 εµνρσ Gρσ and γ5 = i εαβγδ γα γβ γγ γδ is understood as a Here the condensate containing G 2 4! short notation for the expression from which both ε tensors are eliminated using εµνρσ εαβγδ = µ ν ρ σ −4!δ[α δβ δγ δδ] , and this expression is valid at any space dimension D. These condensates are expressed via Bid as

Q71 = 2B27 − 2B37 ,

Q72 = −2B17 − 2B27 + 2B37 + 2B47 − 4mB 6 + 6m2 B 5 − 2m4 B 3 ,

Q73 = 2B27 − 2B37 − B47 + 2mB 6 − m2 B 5 ,

Q74

=

−2B17

−

2B27

+

4B37

6

2

(3.7)

5

− 4mB + 4m B .

Solving this linear system we obtain 1 1 B17 = Q71 − Q72 − Q73 − m2 Q5 + m4 Q3 , 2 2 3 7 1 7 1 7 B2 = Q1 − Q2 − Q73 + Q74 − mQ6 − m2 Q5 + m4 Q3 , 2 2 2 1 1 B37 = Q71 − Q72 − Q73 + Q74 − mQ6 − m2 Q5 + m4 Q3 , 2 2 m2 5 B47 = Q71 − Q73 − mQ6 − Q + m 4 Q3 . 2 7

(3.8)

Similarly, for d = 8 we have E

D

ˆ µ D 2 + D 2 Dµ DD ˆ µ )q , B18 = i q(Dµ DD D

E

ˆ 2 Dµ + Dµ D 2 DD ˆ µ )q , B28 = i q(Dµ DD D

E

ˆ ν Dµ Dν + Dν Dµ Dν DD ˆ µ )q , B38 = i q(Dµ DD D

E

ˆ 2q , B48 = i qD 2 DD D

D

(3.9) E

ˆ ν Dµ q , B58 = i qDµ Dν DD E

ˆ µ Dν q , B68 = i qDµ Dν DD

D

E

ˆ 3 Dµ q . B78 = i qDµ D

8 but with a minus sign between two terms are C–odd, and Operators similar to those in B1...3 their vacuum averages vanish. An interesting new phenomenon similar to the axial anomaly arises for bilinear quark condensates of dimensions d ≥ 8. At a space dimension D 6= 4 we can construct the condensate

E

D

A = i qDα Dβ Dγ Dδ Dε γ[α γβ γγ γδ γε] q .

(3.10)

All coefficients in the expansion of this condensate in gluon condensates (see Section 5.1) contain traces vanishing in the 4–dimensional space. Momentum integrals in coefficients at d > 8 gluon condensates converge, and at d = 8 they contain divergencies 1/ε. As a result, the quark condensate A is equal to a combination of d = 8 gluon condensates. Choosing the anomalous condensate A as one of the basis ones, we can choose 6 proper quark d = 8 basis condensates as Q81 = i hq[[Gµλ , Gλν ]+ , Dµ ]+ γν qi , D

E

e ], Dµ ]+ γν γ5 q , Q82 = − q[[Gµλ , G λν D

E

ˆ µν , Gµν ]q , Q83 = i q[DG Q85 = i hq[Gµν , Jµ ]γν qi ,

D

E

ˆ , Q84 = qD 2 Jq D

(3.11) E

e µν , Jµ ]+ γν γ5 q . Q86 = q[G

e µν and γµ γ5 = i ε Here again the expressions containing G 3! µαβγ γα γβ γγ are understood as a short notation for the expressions in which both ε tensors are eliminated. All operators in (3.11) (as well as in (3.6), (3.2)) are purely C–even. C–conjugation permutes ← − all Dµ and γµ in the opposite order, and changes their signs (because Dµ is transformed to D µ ). Therefore every commutator gives a factor −1 to the C–parity, and every anticommutator gives +1; we should not use ordinary products if we want to obtain operators with a definite C–parity. According to these rules, Gµν and all its derivatives Dλ Gµν ,. . . (including Jµ ) are C–odd; γµ and σµν are C–odd, while 1, γ5 , and γµ γ5 are C–even. The condensates (3.10), (3.11) are expressed via (3.9) as

A = −B18 − B28 + B38 + B48 + B58 − B68 + B78

+ 2mB17 + 2mB27 − 2mB37 − 2mB47 + 3m2 B 6 − 4m3 B 5 + m5 B 3 ,

Q81 = B18 − B28 + 2B58 − 2B68 − 2mB17 + 2mB37 ,

Q82 = 2B18 − 2B28 − 2B48 + 2B58 − 2B68 + 2B78 Q83 Q84 Q85 Q86

− 2mB17 + 2mB27 + 2mB37 − 2mB47 − 2m2 B 6 + 2m3 B 5 ,

= −4B58 + 4B68 + 4mB27 − 4mB37 , = 2B18 + 2B28 − 2B48 − 4B58 − 2mB17 , = B18 + B28 − 2B38 − 2mB17 − 2mB27 + 4mB37 , = 3B18 + B28 − 2B38 − 2B48 − 4mB17 − 2mB27 + 4mB37 − 4m2 B 6 + 4m3 B 5 . 8

(3.12)

Solving this linear system we obtain 1 1 1 1 B18 = A + Q81 − Q82 + Q83 − Q85 + Q86 2 2 4 2 m + mQ71 − Q72 − 2mQ73 − m2 Q6 − 2m3 Q5 + 2m5 Q3 , 2 1 1 1 1 B28 = A − Q81 − Q82 − Q83 − Q85 + Q86 2 2 4 2 m 7 7 7 + 3mQ1 − Q2 − 2mQ3 + mQ74 − 3m2 Q6 − 2m3 Q5 + 2m5 Q3 , 2 1 8 8 B3 = A − Q2 − Q85 + Q86 2 m 7 + 2mQ1 − Q72 − 2mQ73 + mQ74 − 3m2 Q6 − 2m3 Q5 + 2m5 Q3 , 2 1 8 1 8 1 8 1 8 m 7 8 B4 = A + Q1 − Q2 + Q3 + Q6 + Q1 − mQ73 − m3 Q5 + m5 Q3 , 2 2 4 2 2 1 1 1 1 1 1 3 B58 = A − Q81 − Q82 − Q83 − Q84 − Q85 + Q86 2 4 4 8 4 2 4 3 m 7 m 7 7 7 2 6 + mQ1 − Q2 − mQ3 + Q4 − 2m Q − m3 Q5 + m5 Q3 , 2 4 2 1 1 1 1 1 1 3 B68 = A − Q81 − Q82 + Q83 − Q84 − Q85 + Q86 2 4 4 8 4 2 4 m 7 m 7 7 7 2 6 3 5 + mQ1 − Q2 − mQ3 + Q4 − 2m Q − m Q + m5 Q3 , 4 2 1 8 1 8 8 B7 = A − Q 1 + Q 6 2 2 3 m3 5 m Q + m 5 Q3 . + mQ71 + Q72 − mQ73 − m2 Q6 − 2 2 2

(3.13)

I have implemented the discussed algorithm as a Modula-2 program. It accepts an arbitrary bilinear quark condensate and expresses it via Bid producing a REDUCE readable output, which can be used to express the condensate via Qdj using the formulae (3.4), (3.8), (3.13). The problem of four–quark condensates’D classificationE is Even at D much moreEdifficult. D E a d = 6 there exist infinitely many condensates qγ[µ1 . . . γµn ] q qγµ1 . . . γµn ] q , qt γ[µ1 . . . γµn ] q

D

E

qta γµ1 . . . γµn ] q . Those with n > 4 are anomalous and can be eliminated.

3.2

Gluon condensates

In this Section we apply similar methods to the classification of gluon condensates. For each dimension d = 2n we introduce a linear space. We shall call it the extended space. It is generated by the basis of formal sequences Eid constructed from Dµ in which each index is contained twice. Sequences obtained from each other by renaming indices, cyclic permutations (trace cyclicity), and reversing (C–parity) are considered equivalent. Physical gluon condensates can be systematically expressed via this basis, and form a subspace of the extended space. We can choose a basis in this subspace composed from the most convenient condensates Gdj , and extend it to a basis in the whole space using a subset of Eid . After that we can expand any condensate in Eid and then reexpress it via Gdj and the selected Eid . In fact Eid should not appear if we have indeed selected a complete basis in the physical subspace. General guidelines for choosing good basis condensates are similar to the quark case. First of all, we should include all independent condensates containing Jµ . The number of derivatives should be kept minimum. There is only one gluon condensate with d = 4, namely G4 = hTr Gµν Gµν i. The extended 9

space is 2–dimensional: E14 = D2 D2 , E24 = Dµ Dν Dµ Dν ; G4 = 2E14 − 2E24 . But all this is of no use here. At d = 6 the extended space is 5–dimensional: E16 = D 2 D2 D 2 ,

E26 = D 2 Dµ Dν Dµ Dν ,

E46 = Dλ Dµ Dλ Dν Dµ Dν ,

E36 = D2 Dµ D2 Dµ ,

E56 = Dλ Dµ Dν Dλ Dµ Dν .

(3.14)

There are 2 linearly independent physical condensates: G61 = i hTr Gλµ Gµν Gνλ i ,

They are expressed via Ei6 as

G61 = E16 − 3E26 + 3E46 − E56 ,

6 : We can exclude E4...5

G62 = hTr Jµ Jµ i .

(3.15)

G62 = −2E16 + 8E26 − 2E36 − 4E46 .

(3.16)

1 1 1 E46 = − G62 − E16 + 2E26 − E36 , 4 2 2 3 1 3 E56 = −G61 − G62 − E16 + 3E26 − E36 . (3.17) 4 2 2 6 ; After that we can expand any d = 6 condensate in (3.14) and express it via (3.15) and E1...3 the last terms should not appear. I have verified this for all d = 6 condensates. At d = 8 the extended space is 17–dimensional: E18 = D 2 D2 D 2 D2 ,

E28 = D2 D 2 Dµ Dν Dµ Dν ,

E38 = D 2 D2 Dµ D 2 Dµ ,

E48 = D 2 Dλ Dµ Dλ Dν Dµ Dν ,

E58 = D 2 Dλ Dµ Dλ D 2 Dµ ,

E68 = D 2 Dλ Dµ Dν Dλ Dµ Dν ,

E78 = D 2 Dλ Dµ Dν Dλ Dν Dµ , E98 = D 2 Dµ Dν D2 Dµ Dν , 8 E11 8 E13 8 E15 8 E17

E88 = D2 Dλ Dµ Dν Dµ Dν Dλ ,

8 E10 = D 2 Dµ Dν D2 Dν Dµ ,

= Dµ Dν Dµ Dν Dρ Dσ Dρ Dσ , = Dµ Dν Dµ Dρ Dσ Dν Dρ Dσ , = Dµ Dν Dρ Dµ Dσ Dν Dρ Dσ ,

8 E12 8 E14 8 E16

(3.18)

= Dµ Dν Dµ Dρ Dν Dσ Dρ Dσ , = Dµ Dν Dµ Dρ Dσ Dν Dσ Dρ , = Dµ Dν Dρ Dµ Dσ Dρ Dν Dσ ,

= Dµ Dν Dρ Dσ Dµ Dν Dρ Dσ .

A set of independent d = 8 gluon condensates was found in [3]. We choose one condensate in a different way [9]: G81 = hTr Gµν Gµν Gαβ Gαβ i ,

G83 = hTr Gµα Gαν Gνβ Gβµ i ,

G85 = i hTr Jµ Gµν Jν i ,

G87

D

2

E

G82 = hTr Gµν Gαβ Gµν Gαβ i ,

G84 = hTr Gµα Gαν Gµβ Gβν i ,

G86 = i hTr Jλ [Dλ Gµν , Gµν ]i ,

(3.19)

= Tr Jµ D Jµ .

All condensates are written in an explicitly C–even form (even number of commutators). They are expressed via Ei8 as 8 8 G81 = −8E88 + 4E10 + 4E11 ,

8 8 8 G82 = −8E15 + 4E16 + 4E17 ,

8 8 8 G83 = E18 − 4E28 + 4E48 + 2E11 − 4E12 + E16 ,

8 8 8 8 G84 = 2E68 − 4E78 + E98 + 2E12 − 4E13 + 2E14 + E15 ,

G85 G86 G87

=

= =

E18

8 8 − 5E28 + 2E38 + 4E48 − 2E58 + 4E78 − 4E88 − E98 + E10 + 4E11 − 8 8 8 8 8 8 8 8 8 −4E2 + 4E3 + 8E6 − 8E7 − 12E8 + 4E10 + 8E11 − 8E13 + 8E14 , 8 −2E18 + 8E28 − 6E38 − 8E48 + 12E58 − 16E78 + 4E98 + 8E14 .

10

(3.20) 8 4E12 ,

8 : We can exclude E11...17

1 8 8 , E11 = G81 + 2E88 − E10 4 1 1 8 E12 = G81 − G85 4 4 1 1 3 8 1 8 5 8 1 8 , + E1 − E2 + E3 + E48 − E58 + E78 + E88 − E98 − E10 4 4 2 2 4 4 1 1 1 8 E13 = G81 − G86 + G87 4 8 8 1 8 3 8 5 8 3 1 1 1 8 + E1 − E2 + E3 + E48 − E58 + E68 + E78 + E88 − E98 − E10 , 4 2 4 2 2 2 2 1 1 3 3 1 8 E14 = G87 + E18 − E28 + E38 + E48 − E58 + 2E78 − E98 , 8 4 4 2 2 1 8 1 8 1 8 1 8 8 8 E15 = G1 + G4 + G5 − G6 + G7 2 2 2 4 3 1 8 3 8 5 8 , − E2 + E3 − 2E58 + 2E68 + 2E78 − E98 − E10 2 2 2 2 1 8 8 E16 = G81 + G83 − G85 − E28 + 2E38 − 2E58 + 4E78 − E98 − E10 , 2 1 1 1 8 E17 = G81 + G82 − G83 + 2G84 + 2G85 − G86 + G87 2 4 2 − 2E28 + 3E38 − 2E58 + 4E68 − 2E98 .

(3.21)

8 ; After that we can expand any d = 8 condensate in (3.18) and express it via (3.19) and E1...10 the last terms should not appear. I have written a Modula-2 program that expands any gluon condensate in Eid and produces a REDUCE readable output, which can be used for expressing this condensate via Gdj and selected Eid . Using this program, I have verified that all d = 8 gluon condensates indeed can be expanded in the basis (3.19).

3.3

Vacuum averaging

After obtaining an expression for a correlator via background fields we should average it over the vacuum. The first possible way is to write down the most general expressions for vacuum averages with free indices and to find unknown coefficients by solving linear systems. For the quark condensates with d ≤ 6 we have Q3 1ρσ , 22 imQ3 hq σ Dα qρ i = − 4 (γα )ρσ , 2" #   i 1 2 3 5 , hq σ Dα Dβ qρ i = 5 Q δαβ + σαβ − 2m Q δαβ 2 3 ρσ

hq σ qρ i =

(3.22)

"

1 hq σ Dα Dβ Dγ qρ i = 6 3 Q6 (δαβ γγ + δβγ γα − 5δαγ γβ − 3iεαβγδ γδ γ5 ) 2 3 − 3mQ5 (δαβ γγ + δβγ γα + δαγ γβ − iεαβγδ γδ γ5 ) 3

#

3

+ 6m Q (δαβ γγ + δβγ γα + δαγ γβ )

. ρσ

One can easily obtain similar formulae with Gαβ and its derivatives by antisymmetrization over corresponding indices. 11

For the gluon condensates with d ≤ 6 we have D

E

G4 δab (δµρ δνσ − δµσ δνρ ), N CF D(D − 1) D E δab g2 Gaµν Dα Dβ Gbρσ = N CF D(D − 1)(D2 − 4) g2 Gaµν Gbρσ =

"

× − (D + 2)G61 (((δσµ δνα δβρ − (µ ↔ ν)) − (ρ ↔ σ)) − (α ↔ β)) h

i

+ (D − 4)G61 + (D − 2)G62 (((δσµ δνα δβρ − (µ ↔ ν)) − (ρ ↔ σ)) + (α ↔ β)) h

− 4(D − D

1)G61

+ 2(D −

E

2)G62

i

#

δαβ (δµρ δνσ − (ρ ↔ σ)) ,

(3.23)

2G61 f abc − 1)(D − 2) × (((δσµ δνα δβρ − (µ ↔ ν)) − (ρ ↔ σ)) − (α ↔ β)).

g3 Gaµν Gbαβ Gcρσ = −

N 2 CF D(D

We can substitute these relations to expressions for correlators and obtain scalar expressions. But these formulae become very complicated in higher dimensions. An alternative and simpler way is to average an expression for a correlator over p directions. This is done using the recurrent relation (n is even) pµ1 pµ2 . . . pµn =

p2 (δµ µ pµ . . . pµn + δµ1 µ3 pµ2 . . . pµn + · · · + δµ1 µn pµ2 . . . pµn−1 ) (3.24) D+n−2 1 2 3

or the explicit formula

pµ1 pµ2 . . . pµn =

(p2 )n/2 (δµ µ δµ µ . . . δµn−1 µn + · · ·), D(D + 2) . . . (D + n − 2) 1 2 3 4

(3.25)

where the sum is taken over all (n − 1)!! methods of paring n indices. Here D is the space dimension; the formulae can be used at any dimension including non–integer one (the dimensional regularization). After such an averaging we obtain scalar vacuum condensates for which the methods of the previous Sections can be directly used. In calculations of three–point correlators we encounter the averages pµ1 . . . pµm qν1 . . . qνn over orientations of the pair p, q with a fixed relative orientation. These averages can be calculated 2 = (p2 q 2 − (pq)2 )/p2 . First we average over using the decomposition q = (qp)p/p2 + q⊥ , q⊥ orientations of q⊥ orthogonal to p in (D − 1)–dimensional space using (3.25) with δµν → δ⊥µν = δµν − pµ pν /p2 . Then we average over orientations of p. It is clear that if there is only one vector q then the averages are given by the formula (3.25) in which one p2 is replaced by pq. Some less trivial averages are pα pβ q µ q ν =

h 1 ((D + 1)p2 q 2 − 2(pq)2 )δαβ δµν (D − 1)D(D + 2) i

+ (D(pq)2 − p2 q 2 )(δαµ δβν + δαν δβµ ) , pq pα pβ pγ q λ q µ q ν = (D − 1)D(D + 2)(D + 4) h

× ((D + 1)p2 q 2 − 2(pq)2 )(δαβ δλµ δγν + · · ·) i

+ ((D + 2)(pq)2 − 3p2 q 2 )(δαλ δβµ δγν + · · ·) , pα pβ pγ pδ q µ q ν =

p2 (D − 1)D(D + 2)(D + 4) 12

(3.26)

h

× ((D + 3)p2 q 2 − 4(pq)2 )δµν (δαβ δγδ + · · ·) i

+ (D(pq)2 − p2 q 2 )(δαβ δγµ δδν + · · ·) . Of course, these formulae can be also obtained by writing down the most general forms with unknown coefficients and solving linear systems.

4

Nonlocal condensates

As a sample application we consider expansion of nonlocal vacuum condensates in local ones. Nonlocal condensates naturally appear in calculations of correlators [10–14]; their expansion in spacings lead to the usual OPE (1.1). Expansions derived in this Section can be used either for easy obtaining of OPE of correlators involving nonlocal condensates, or for inventing consistent anzatzs for nonlocal condensates.

4.1

Bilocal condensates

The gauge–invariant bilocal quark condensate [11, 12] has the structure hq σ (0)E(0, x)qρ (x)i = 

E(0, x) = P exp −i

In the fixed–point gauge E(0, x) = 1, and

Zx 0





hqqi iˆ x fS (x2 ) − fV (x2 ) 4 D 

,

(4.1)

ρσ

Aµ (y)dyµ  .

D 

 E

q 1 + 2!1 xα xβ Dα Dβ + 4!1 xα xβ xγ xδ Dα Dβ Dγ Dδ + · · · q hq(0)q(x)i 2 = fS (x ) = hqqi hqqi 5 7 7 7 B B + B2 + B3 4 =1− x2 + 1 x + O(x6 ) 3 2!DB 4!D(D + 2)B 3 1 5 Q − m 2 Q3 2 =1+ 2 x 2!DQ3 3Q71 − 23 Q72 − 3Q73 + Q74 − 2mQ6 − 3m2 Q5 + 3m4 Q3 4 x + O(x6 ), (4.2) + 4!D(D + 2)Q3 iD hq(0)ˆ xq(x)i fV (x2 ) = hqqi x2 =

D



x xα Dα + iD qˆ

1 3! xα xβ xγ Dα Dβ Dγ

1 5! xα xβ xγ xδ xε Dα Dβ Dγ Dδ Dε hqqi x2

+

B 6 + 2mB 5 2 x 3!(D + 2)B 3 B 8 + B28 + B38 + B48 + B58 + B68 + 2m(B17 + B27 + B37 ) 4 + 1 x + O(x6 ) 5!(D + 2)(D + 4)B 3 1 6 Q + 32 mQ5 − 3m3 Q3 2 x =m+ 2 3!(D + 2)Q3

=m−

h

+ 5A − 52 Q82 + 41 Q83 − 12 Q84 − 3Q85 + 5Q86 + 5m(3Q71 − Q72 − 3Q73 + Q74 ) i

− 15m2 (Q6 + mQ5 − m3 Q3 )

x4 + O(x6 ) 5!(D + 2)(D + 4)Q3 13

 E

+ ··· q

On the first step we use the averaging over x directions (3.25) and the definitions of Bid (3.1), (3.5), (3.9), and on the second step—the expressions for Bid via Qdj (3.4), (3.8), (3.13). The gauge–invariant bilocal gluon condensate [13, 14] has the structure D

E

Gaµν (0)E ab (0, x)Gbρσ (x) =

D

Gaµν Gaµν

E

Kµν,ρσ (x), D(D − 1) Kµν,ρσ (x) = D(D − 1) hTr Gµν (0)Gρσ (x)i /G4 = (D(x2 ) + D1 (x2 ))(δµρ δνσ − (ρ ↔ σ)) dD1 (x2 ) ((xµ xρ δνσ − (µ ↔ ν)) − (ρ ↔ σ)), + dx2

(4.3)

where E ab (0, x) is the P –exponent in the adjoint representation equal to δab in the fixed–point gauge. We obtain the linear system A(x2 ) 2x2 ′ 2 D1 (x ) = , D G4 B(x2 ) D(x2 ) + D1 (x2 ) + x2 D1′ (x2 ) = , G4

D(x2 ) + D1 (x2 ) +

(4.4)

where A2 x4 A1 x2 + + O(x6 ), (4.5) 2D 4!D(D + 2) Dxµ xν B1 x2 B2 x 4 4 B(x2 ) = hTr G (0)G (x)i = G + + + O(x6 ). µλ νλ x2 2(D + 2) 4!(D + 2)(D + 4) A(x2 ) = hTr Gµν (0)Gµν (x)i = G4 +

Here E

D

A1 = Tr Gµν D 2 Gµν , B1 = A1 + hTr Gµλ (Dµ Dν + Dν Dµ )Gνλ i ,

E

D

A2 = Tr Gµν (D 2 D2 + Dα D 2 Dα + Dα Dβ Dα Dβ )Gµν , B2 = A2 + h Tr Gµλ (Dµ Dν D 2 + Dν Dµ D 2 + Dµ D 2 Dν + Dν D 2 Dµ + D 2 Dµ Dν + D 2 Dν Dµ + Dµ Dα Dν Dα + Dν Dα Dµ Dα

+ Dα Dµ Dν Dα + Dα Dν Dµ Dα + Dα Dµ Dα Dν + Dα Dν Dα Dµ )Gνλ i. Using the method described in Section 3.2, we obtain A1 = −4G61 − 2G62 ,

B1 = −6G61 − 4G62 , A2 =

B2 =

3G81−2 5G81−2

+

+

(4.6)

24G83−4 50G83−4

− 36G85 − 84G85

+ 10G86 + 24G86

−

−

6G87 , 18G87 ,

where G81−2 = G81 − G82 , G83−4 = G83 − G84 . The solution of (4.4) is #

"

1 (2A1 − B1 )x2 (3A2 − B2 )x4 D(x ) = 4 + + O(x6 ) G 2(D − 2) 2 · 4!(D 2 − 4) 2

=−

2G81−2 + 11G83−4 − 12G85 + 3G86 x4 G61 2 x + + O(x6 ), (D − 2)G4 (D 2 − 4)G4 4!

1 D D1 (x2 ) = 1 + 4 G D−2

"

B1 A1 − D+2 D



x2 + 2

14



B2 A2 − D+4 D



(4.7) #

x4 + O(x6 ) 2 · 4!(D + 2)

=1− h

(D − 4)G61 + (D − 2)G62 2 x (D2 − 4)G4

+ (D − 6)G81−2 + (13D − 48)G83−4 − 24(D − 3)G85 + (7D − 20)G86 − 6(D − 2)G87

i

x4 + O(x6 ). 4!(D + 4)(D2 − 4)G4

In the abelian case, only D1 (x2 ) survives [13]; this is the reason why D(x2 ) (4.7) contains only purely nonabelian condensates G61 , G81−2 , G83−4 , G85,6 .

4.2

Noncollinear condensates

Three–point correlators, sum rules for hadron wave functions, and some other kinds of calculations naturally involve trilocal objects which we shall call noncollinear condensates. The noncollinear quark condensate has the structure hq σ (y)E(y, 0)E(0, x)qρ (x)i =

#

"

i hqqi Q5 fT (x, y)[ˆ x, yˆ] − (fV (x, y)ˆ x − fV (y, x)ˆ y) fS (x, y) + 3 4 4D(D − 1)Q D

(4.8) .

ρσ

In the fixed–point gauge E(y, 0) = E(0, x) = 1. The difference hq(y)q(x)i − hq(0)q(x − y)i hqqi

 2 2 2 ∆ q(2D D − Dµ D Dµ − Dµ Dν Dµ Dν )q = + O(x3 , y 3 ) 12D(D − 1) hqqi
∆ 3Q71 + 2Q74 − 4mQ6 =− + O(x3 , y 3 ) 24D(D − 1)Q3

fS (x, y) − fS ((x − y)2 ) =

(4.9)

(where ∆ = x2 y 2 − (xy)2 ) vanishes when x and y are collinear, as expected. The coefficient fT (x, y) multiplies the structure [ˆ x, yˆ] vanishing in the collinear limit, therefore it need not vanish itself: D(D − 1) hq(y)[ˆ y, x ˆ]q(x)i =1 (4.10) 5 Q ∆

6Q71 − 3Q72 − 6Q73 + 2Q74 − 3mQ6 − 3m2 Q5 (x − y)2 − 3Q73 + 2Q74 xy + + ··· 6(D + 2)Q5

fT (x, y) =

For the vector matrix elements we find xq(x)i = −i hq(y)ˆ y q(x)ix↔y i hq(y)ˆ =

Q3 (x2 − xy) ∆ fV ((x − y)2 ) − g(x, y) D 12D(D − 1)

(4.11)

where g(x, y) = Q6 +

h 1 − 20A(x2 − y 2 ) − 30Q81 (x2 − xy) + 10Q82 (x2 − y 2 ) 40(D + 2)

− Q83 (13x2 − y 2 − 9xy) − 2Q84 (2x2 + y 2 − 6xy) − 2Q85 (7x2 + 6y 2 − 11xy)

+ 20Q86 (y 2 − xy) + 60mQ71 (x2 − xy) − 10mQ72 (x2 − y 2 ) + 40mQ74 (x2 − xy) i

− 20m2 Q6 (5x2 + y 2 − 6xy) + O(x3 , y 3 ). 15

(4.12)

Solving the system of two linear equations, we finally obtain fV (x, y) = fV ((x − y)2 ) −

y 2 g(x, y) + xy g(y, x) . 12(D − 1)Q3

(4.13)

The noncollinear gluon condensate has the structure D

E

Gaµν (y)E ab (y, 0)E bc (0, x)Gcρσ (x) =

D

Gaµν Gaµν

E

D(D − 1)

Kµν,ρσ (x, y),

Tµνρσ = G4 (Kµν,ρσ (x, y) − Kµν,ρσ (x − y))

(4.14)

= c1 (δµρ δνσ − (ρ ↔ σ))

+ c2 ((xµ xρ δνσ − (µ ↔ ν)) − (ρ ↔ σ))

+ c3 ((yµ yρ δνσ − (µ ↔ ν)) − (ρ ↔ σ))

+ c4 ((xµ yρ δνσ − (µ ↔ ν)) − (ρ ↔ σ))

+ c5 ((yµ xρ δνσ − (µ ↔ ν)) − (ρ ↔ σ))

+ c6 (xµ yν − (µ ↔ ν))(xρ yσ − (ρ ↔ σ)) with c3 (x, y) = c2 (y, x). We find  ∆ −3G81−2 + 2G86 , 12 Tλµ,λν xµ xν = (Tλµ,λν yµ yν )x↔y

Tµν,µν =

(4.15)

h   i ∆ (4x2 + xy) −G81−2 + G86 − (x2 − 2xy)G83−4 − 4xy G85 , 12(D + 2) h ∆ 12(D + 2)G61 − (2x2 + 2y 2 + xy)G81−2 Tλµ,λν xµ yν = 12(D + 2)

=

i

− (11x2 + 11y 2 − 20xy)G83−4 + 12(x − y)2 G85 − (3x2 + 3y 2 − 10xy)G86 ,

Tλµ,λν yµ xν =

h ∆ − 12(D + 2)G61 + (x2 + y 2 − 7xy)G81−2 12(D + 2)

i

+ (13x2 + 13y 2 − 22xy)G83−4 − 8(2x2 + 2y 2 − 3xy)G85 + 2(2x2 + 2y 2 − xy)G86 , Tµν,ρσ xµ xρ yν yσ =

  ∆2 −5G81−2 + 4G83−4 + 12G86 . 12(D + 1)(D + 2)

Solving the system of 6 linear equations, we finally obtain 



∆ −D(3D − 7)G81−2 + 4(D + 3)G83−4 + 2(D − 2)(D − 3)G86 , c1 = 12(D + 1)(D + 2)(D − 2)(D − 3) n 1 c2 = 4(D + 1)(D − 3)xyG85 12(D + 1)(D + 2)(D − 2)(D − 3) h

i

+ (D − 3) D(2y 2 − xy) − 4y 2 − xy G86 h

i

− D 2 (y 2 − xy) − 2D(3y 2 − xy) + 3(y 2 + xy) G81−2 h

i

o

− (D 2 − 3)(y 2 + 2xy) + 2D(3y 2 − 2xy) G83−4 ,

c3 = c4 (x ↔ y), h 1 24(D + 2)G61 − 3(x − y)2 G81−2 c4 − c5 = 12(D 2 − 4)

(4.16) i

− 6(4x2 + 4y 2 − 7xy)G83−4 + (7x2 + 7y 2 − 12xy)(4G85 − G86 ) , 16

c4 + c5 =

n 1 − 4(D + 1)(D − 3)(x2 + y 2 )G85 12(D + 1)(D + 2)(D − 2)(D − 3) h

i

+ (D − 3) D(x2 + y 2 − 4xy) + x2 + y 2 + 8xy G86 h

i

− D 2 (x − y)2 − 2D(x2 + y 2 − 6xy) − 3(x + y)2 G81−2 h

i

o

+ 2 (D 2 − 3)(x2 + y 2 + xy) − 2D(x2 + y 2 − 3xy) G83−4 , c6 =

(D − 4)G81−2 + (D − 1)2 G83−4 + (D − 2)(D − 3)G86 . 2(D + 1)(D + 2)(D − 2)(D − 3)

In the abelian case Kµν,ρσ (x, y) = Kµν,ρσ (x − y) has only the D1 ((x − y)2 ) term (4.7), therefore Tµν,ρσ contains only purely nonabelian condensates. All calculations in this Section were performed using REDUCE (with some REDUCE code generated by the Modula-2 programs described in Section 3). Some of them require a lot of memory: I could not complete the gluonic calculations on a PC with 8 megabytes.

4.3

Expansion in bilocal condensates

In some applications, such as sum rules for wave function moments, it is convenient to expand noncollinear condensates in series in x keeping y finite. Coefficients in these series are bilocal condensates (Section 4.1). Expanding a noncollinear quark condensate hq(y)E(y, 0)ΓE(0, x)q(x)i in x, we obtain terms like hq(y)E(y, 0)Oq(0)i where O involves derivatives and gluon fields at x = 0. Therefore we encounter the problem of classification of such bilocal condensates. Similarly to Section 3.1, we can easily reduce any condensate of this kind to a linear combination of terms with O constructed ˆ (yD), and Dµ , maybe multiplied by yˆ from the left. Terms with D ˆ at the right reduce from D, to lower–dimensional condensates multiplied by m. Terms with (yD) at the left reduce to derivatives of lower–dimensional condensates: expanding the equality hq(y)E(y, 0)(Oq)x=αy i = hq((1 − α)y)E((1 − α)y, 0)(Oq)x=0 i in α we obtain hq(y)E(y, 0)(yD)Oq(0)i = −2y 2

d hq(y)E(y, 0)Oq(0)i . dy 2

(4.17)

There are 2 basis dimension 3 bilocal condensates: B13 (y 2 ) = hq(y)E(y, 0)q(0)i ,

B23 (y 2 ) = −i hq(y)E(y, 0)ˆ y q(0)i .

(4.18)

They can be simply expressed via 2 bilocal functions considered in Section 4.1: B13 (y 2 ) = Q3 fS (y 2 ),

B23 (y 2 ) = Q3 fV (y 2 )

y2 . D

(4.19)

Expansions of these functions at small y 2 are given by equation (4.2). There are 4 basis dimension 5 bilocal condensates: D

E

B15 (y 2 ) = − q(y)E(y, 0)D2 q(0) , D

E

ˆ y D(yD)q(0) , B25 (y 2 ) = q(y)E(y, 0)ˆ D

E

y D 2 q(0) , B35 (y 2 ) = i q(y)E(y, 0)ˆ D

E

ˆ B45 (y 2 ) = i q(y)E(y, 0)D(yD)q(0) .

17

(4.20)

Expanding in y and averaging over y directions (Section 3.3) we obtain the series B17 y 2 + O(y 4 ), 2D −2B 5 + m2 B 3 2 2B17 + B47 + mB 6 − m2 B 5 4 y + y + O(y 6 ), B25 (y 2 ) = D 2D(D + 2) mB 5 2 B18 + 2B48 + 2mB17 4 B35 (y 2 ) = y − y + O(y 6 ), D 12D(D + 2) B 6 2 B18 + B28 + B38 4 B45 (y 2 ) = y − y + O(y 6 ). D 12D(D + 2) B15 (y 2 ) = B 5 −

(4.21)

We introduce 4 new bilocal functions Q5 f1 (y 2 ) = i hq(y)E(y, 0)Gµν (0)σµν q(0)i , iDyµ yν Q5 f2 (y 2 ) = hq(y)E(y, 0)Gλµ (0)σλν q(0)i , y2 2D Q6 f3 (y 2 ) = 2 hq(y)E(y, 0)yµ Gµν (0)γν q(0)i , y E D 2iD e µν (0)γν q(0) . Q6 f4 (y 2 ) = 2 q(y)E(y, 0)yµ G y

(4.22)

Q5 f1 (y 2 ) = −2B15 (y 2 ) + 2m2 B 3 fS (y 2 ),   D Q5 f2 (y 2 ) = 2 B25 (y 2 ) + mB 3 fV (y 2 ) + 2y 2 fV′ (y 2 ) , y 2D Q6 f3 (y 2 ) = − 2 B45 (y 2 ) − 4DmB 3 fS′ (y 2 ), y  2D  Q6 f4 (y 2 ) = 2 B35 (y 2 ) − B45 (y 2 ) − 4DmB 3 fS′ (y 2 ) − 2m2 B 3 fV (y 2 ). y

(4.23)

They are expressed via the basis bilocal condensates (4.20) as

Solving this linear system we obtain 1 B15 (y 2 ) = − Q5 f1 (y 2 ) + m2 Q3 fS (y 2 ), 2  i 2 y h 5 Q f2 (y 2 ) − mQ3 fV (y 2 ) + 2y 2 fV′ (y 2 ) , B25 (y 2 ) = D  i y2 h 6  B35 (y 2 ) = Q f4 (y 2 ) − f3 (y 2 ) + 2m2 Q3 fV (y 2 ) , 2D i y2 h 6 Q f3 (y 2 ) + 4DmQ3 fS′ (y 2 ) . B45 (y 2 ) = − 2D Substituting (4.21) and (4.2) into (4.23) we arrive at the small y 2 expansions

Q71 − Q72 − 2Q73 − m2 Q5 2 y + O(y 4 ), 2DQ5 2Q71 − Q72 − 3Q73 − mQ6 − m2 Q5 2 y + O(y 4 ), f2 (y 2 ) = 1 + 2(D + 2)Q5 6A − 3Q82 − 4Q85 + 6Q86 + 3mQ72 − 6m2 Q6 2 f3 (y 2 ) = 1 + y + O(y 4 ), 12(D + 2)Q6 mQ5 h f4 (y 2 ) = 1 − + − 6Q81 − 3Q83 − 6Q85 + 4Q86 Q6  i y2 + O(y 4 ). + 12m −Q71 + Q72 + 2Q73 − mQ6 + m2 Q3 24(D + 2)Q6

(4.24)

f1 (y 2 ) = 1 +

18

(4.25)

Now it is easy to expand the noncollinear quark condensate (4.8) in small x keeping y fixed. We expand q(x), substitute x = (xy/y 2 )y + x⊥ (with x2⊥ = ∆/y 2 ), and perform the (D − 1)– dimensional averaging (Section 3.3) over x⊥ directions orthogonal to y. Then we express the result in terms of the basis bilocal condensates (4.19), (4.22), and substitute (4.19), (4.24). Finally, we arrive at the result Q5 ∆ f1 (y 2 ) 4(D − 1)Q3 y 2  x2 y 2 − D(xy)2  ′ 2 m2 ∆ 2 ′′ 2 − fS (y 2 ) + O(x3 ) f (y ) + 2y f (y ) − S S (D − 1)y 2 2(D − 1)y 2

fS (x, y) = fS (y 2 ) − 2(xy)fS′ (y 2 ) +

#

"

∆ 1 Q5 m2 2 ′ 2 2 ′′ 2 = fS ((y − x) ) + f (y ) − Df (y ) − 2y f (y ) − fS (y 2 ) + O(x3 ), 1 S S (D − 1)y 2 4 Q3 2 2

(

m Q3 fV (y 2 ) fT (x, y) = 4D 5 fS′ (y 2 ) + Q 2D "

(4.26)

 xy m  1 Q5 2 ′ 2 2 ′′ 2 2 2 ′ 2 − 2 − f (y ) + f (y ) + 2y f (y ) + f (y ) + 2y f (y ) 2 V S S V y 4D Q3 2D 



h

#)

+ O(x2 ), i

g(y, x) = 3Q6 f4 (y 2 ) − f3 (y 2 ) + 6Q3 2(D + 2)fV′ (y 2 ) + 4y 2 fV′′ (y 2 ) + m2 fV (y 2 ) + O(x), g(x, y) = −

 12Q3  2 2 ′ 2 2 Df (y ) + 2y f (y ) − Dmf (y ) V S V y2

 3xy n 6  2 2 Q f (y ) − 2f (y ) 4 3 y2 h

io

+ 2Q3 6(D + 2)fV′ (y 2 ) + 12y 2 fV′′ (y 2 ) − 4DmfS′ (y 2 ) + m2 fV (y 2 )

+ O(x2 ).

If y is also small, we substitute the y 2 expansions of the bilocal condensates (4.2), (4.25) and obtain expansions consistent with (4.9), (4.10), (4.12). This provides a strong check of our results. Expanding a noncollinear gluon condensate in x, we obtain terms like hTr Gµν (y, 0)Oµν (0)i where Gµν (y, 0) = gGaµν E ab tb and Oµν (0) is constructed from Gµµ (0) and Dµ . A straightforward generalization of the systematic classification procedure of Section 3.2 to the case of such bilocal condensates seems impossible. Fortunately, it is not difficult to find all basis bilocal condensates up to dimension 6. Two dimension 4 condensates were introduced in Section 4.1: A(y 2 ) = hTr Gµν (y, 0)Gµν (0)i ,

B(y 2 ) = hTr Gλµ (y, 0)Gλν (0)i

Dyµ yν . y2

(4.27)

There is one dimension 5 bilocal condensate C(y 2 ) = hTr Gµν (y, 0)Jµ (0)i

Dyν , y2

(4.28)

and 5 dimension 6 ones F1 (y 2 ) = i hTr Gµν (y, 0)Gνλ (0)Gλµ (0)i , Dyρ yσ , F2 (y 2 ) = i hTr Gµν (y, 0)Gνρ (0)Gσµ (0)i y2 Dyρ yσ F3 (y 2 ) = i hTr Gµρ (y, 0)Gσλ (0)Gλµ (0)i y2 Dyρ yσ , = i hTr Gρν (y, 0)Gνλ (0)Gλσ (0)i y2 19

(4.29)

F4 (y 2 ) = hTr Gµν (y, 0)Dµ Jν (0)i , Dyρ yσ F5 (y 2 ) = hTr Gµρ (y, 0)Dµ Jσ (0)i y2 (the equivalence of the two definitions of F3 (y 2 ) follows from the C–conjugation symmetry). Small y 2 expansions (4.5), (4.6) of A(y 2 ), B(y 2 ) were obtained in Section 4.1; the other expansions are 8G85 − 3G86 + 6G87 2 y + O(y 4 ), 12(D + 2) −4G83−4 + 4G85 − G86 2 y + O(y 4 ), F1 (y 2 ) = G61 + 4D −G81−2 − 4G83−4 + 4G85 − G86 2 F2 (y 2 ) = G61 + y + O(y 4 ), 4(D + 2) −3G83−4 + 4G85 − G86 2 y + O(y 4 ), F3 (y 2 ) = G61 + 2(D + 2) 2G85 − G86 + G87 2 F4 (y 2 ) = G62 + y + O(y 4 ), 2D −3G86 + 2G87 2 y + O(y 4 ). F5 (y 2 ) = G62 + 4(D + 2) C(y 2 ) = G62 +

(4.30)

Now we can derive the expansions in the bilocal gluon condensates Tµν,µν = −

 D∆  2 2 ′ 2 2 ′′ 2 2F (y ) + F (y ) + DA (y ) + 2y A (y ) + O(x3 ), 1 4 y2 "

1 1 Tλµ,λν yµ yν = −∆ 2F3 (y 2 ) + F5 (y 2 ) + C(y 2 ) + y 2 C ′ (y 2 ) 2 2 #

A(y 2 ) − B(y 2 ) + DB (y ) + 2y B (y ) + D + O(x3 ), y2 2

′

2

2

′′

A(y 2 ) − B(y 2 ) Tλµ,λν yµ xν = ∆ C(y ) + 2B (y ) − D y2 2

2

′

"

!

3 1 ∆xy + 2 − 3F3 (y 2 ) − F5 (y 2 ) − C(y 2 ) − 3y 2 C ′ (y 2 ) y 2 2

#

A(y 2 ) − B(y 2 ) − 2D + 2DA′ (y 2 ) − (3D + 2)B ′ (y 2 ) − 6y 2 B ′′ (y 2 ) + O(x4 ), y2 A(y 2 ) − B(y 2 ) Tλµ,λν xµ yν = ∆ 2B (y ) − DA (y ) − D y2 ′

2

2

′

"

!

(4.31)

∆xy 1 A(y 2 ) − B(y 2 ) 1 + 2 F2 (y 2 ) − 2F3 (y 2 ) − F5 (y 2 ) − C(y 2 ) − y 2 C ′ (y 2 ) − 2D y 2 2 y2 2

∆xy Tλµ,λν xµ xν = 2 y "

′

2

2

′′

2

2

′′

2

#

+ 3DA (y ) − (3D + 2)B (y ) + 2Dy A (y ) − 6y B (y ) + O(x4 ), ′

A(y 2 ) − B(y 2 ) C(y 2 ) − DA′ (y 2 ) + 4B ′ (y 2 ) − 2D y2

!

  ∆ 2 2 + − D∆ 3F (y ) + 2F (y ) 1 4 (D + 1)y 4   1 2 2 + x y − (D + 2)(xy)2 −2F2 (y 2 ) + 6F3 (y 2 ) + F5 (y 2 ) + 3C(y 2 ) + 6y 2 C ′ (y 2 ) 2

20



+ D(D + 1) x2 y 2 − 4(xy)2 

 A(y 2 ) − B(y 2 )

y2



− D (D + 3)x2 y 2 − 2(3D + 4)(xy)2 A′ (y 2 ) 



+ (D + 4)x2 y 2 − (5D 2 + 10D + 8)(xy)2 B ′ (y 2 ) 



− 2D 2x2 y 2 − (D + 3)(xy)2 y 2 A′′ (y 2 ) 

2 2

2

+ 2 3x y − (5D + 8)(xy) Tµν,ρσ yµ xν yρ xσ = −



2

2

#

y B (y ) + O(x5 ), ′′

h ∆2 6F3 (y 2 ) + 3F5 (y 2 ) + C(y 2 ) + 2y 2 C ′ (y 2 ) 2(D + 1)y 2 i

+ 2(D + 2)B ′ (y 2 ) + 4y 2 B ′′ (y 2 ) + O(x5 ).

Solving the system of 6 linear equations, we finally obtain "

2(D − 2)(−DF1 (y 2 ) + 2F3 (y 2 )) + 2F2 (y 2 ) ∆ c1 = (D + 1)(D − 2)y 2 D−3 − DF4 (y 2 ) + F5 (y 2 ) + C(y 2 ) + 2y 2 C ′ (y 2 ) 2

′

′

2

2

′′

2

′′

#

2

− (D + 2)(DA (y ) − 2B (y )) − 2y (DA (y ) − 2B (y )) + O(x) , "

(D − 5)(−DF1 (y 2 ) + 2F3 (y 2 )) − (D − 1)F2 (y 2 ) 1 c2 = (D + 1)(D − 2) D−3 − DF4 (y 2 ) + F5 (y 2 ) + C(y 2 ) + 2y 2 C ′ (y 2 ) + D(D + 1)

A(y 2 ) − B(y 2 ) y2 #

− DA′ (y 2 ) + 2B ′ (y 2 ) − 2y 2 (DA′′ (y 2 ) − 2B ′′ (y 2 )) + O(x) , (

xy 1 c3 = D − 2 y2

!

A(y 2 ) − B(y 2 ) + DA′ (y 2 ) − 4B ′ (y 2 ) −C(y ) + 2D y2 2

"

1 x2 4D(D − 1)F1 (y 2 ) − 4F2 (y 2 ) − 2(D + 1)(2D − 5)F3 (y 2 ) + 2(D + 1) y 2 D−3

+ 2DF4 (y 2 ) − DF5 (y 2 ) − (D + 2)(C(y 2 ) + 2y 2 C ′ (y 2 )) A(y 2 ) − B(y 2 ) + 2D(D + 2)A′ (y 2 ) − 2(D 2 + 2D + 2)B ′ (y 2 ) − 2D(D + 1) y2 #

+ 4Dy 2 A′′ (y 2 ) − 4(D + 2)y 2 B ′′ (y 2 ) "

(4.32)

1 (xy)2 + − 2D(3F1 (y 2 ) + 2F4 (y 2 )) + (D + 2)(−2F2 (y 2 ) 2(D + 1) y 4 + 6F3 (y 2 ) + F5 (y 2 ) + 3C(y 2 ) + 6y 2 C ′ (y 2 )) + 8D(D + 1) − 4D(3D + 4)A′ (y 2 ) + 2(5D 2 + 10D + 8)B ′ (y 2 ) 2

′′

2

2

′′

2

#

3

)

A(y 2 ) − B(y 2 ) y2

− 4D(D + 3)y A (y ) + 4(5D + 8)y B (y ) + O(x ) , "

1 xy  c4 − c5 = − C(y 2 ) − DA′ (y 2 ) + 2 F2 (y 2 ) + F3 (y 2 ) D−2 y 21

2

2

2

′

2

′

2

′′



2

2

#

+ C(y ) + 2y C (y ) + D(A (y ) + 2y A (y )) + O(x ) , (

1 A(y 2 ) − B(y 2 ) c4 + c5 = − DA′ (y 2 ) + 4B ′ (y 2 ) C(y 2 ) − 2D D−2 y2 "

1 xy 2D(D − 5)F1 (y 2 ) + (D 2 − 5)F2 (y 2 ) − (D2 + 2D − 23)F3 (y 2 ) + D + 1 y2 D−3

+ 2DF4 (y 2 ) − 2F5 (y 2 ) − (D + 3)(C(y 2 ) + 2y 2 C ′ (y 2 )) A(y 2 ) − B(y 2 ) + D(5D + 7)A′ (y 2 ) − 4(D 2 + 2D + 2)B ′ (y 2 ) − 4D(D + 1) y2 2

′′

2

2

′′

2

#

2

)

+ 2D(D + 3)y A (y ) − 8(D + 2)y B (y ) + O(x ) , "

1 (D − 5)(DF1 (y 2 ) − (D − 1)F3 (y 2 )) + (D − 1)F2 (y 2 ) c6 = (D + 1)(D − 2) D−3 2

2

2

′

2

′′

2

#

+ DF4 (y ) − (D − 1)F5 (y ) + D(A (y ) + 2y A (y )) + O(x) . Substituting the y 2 expansions of the bilocal gluon condensates (4.5), (4.6), (4.30) into (4.31), (4.32), we obtain expansions consistent with (4.15), (4.16); this provides a strong check of our results.

5

Heavy quarks

5.1

Heavy quark condensates

If the quark mass is large, all correlators can be expressed via gluon condensates only. This is also true for quark condensates (one–current correlators): Qk =

X

ckn (m)Gn .

(5.1)

n

This is an expansion in 1/m. p'$ s &%

Figure 4: One-loop diagram for a heavy-quark condensate In the one-loop approximation (Fig. 4) quark condensates are given by the formula Qk = hqOk [Dµ ]qi = −i

Z 

dp 2π



D

hTr Ok [−ipµ − iAµ ]S(p)i ,

(5.2)

where Aµ is given by (2.3). We use the MS regularization: the space dimension is D = 4 − 2ε, 

dp 2π



D

≡

µ2 eγ 4π 22

!ε

dD p , (2π)D

(5.3)

µ is the normalization point, γ is the Euler’s constant. We substitute the quark propagator (2.3), (2.4) into (5.2) and average the integrand over p directions in the D–dimensional space. The result is expressed via the integrals In = I1 = In =

Z 

dp 2π

im2 (4π)2



D

1 i(−1)n (m2 )2−n = (p2 − m2 )n (4π)2 (n − 1)! !

1 m2 − log 2 + 1 + O(ε), ε µ

m2 µ2

!−ε

=m

dk −dn

"

γkn

+ O(ε),

(n > 2).

(5.4)

Coefficients ckn with dn ≤ dk contain ultraviolet divergences: cbare kn

!

1 m2 − log 2 ε µ

i I2 = (4π)2

i(−1)n + O(ε) (4π)2 (n − 1)(n − 2)(m2 )n−2

eγε Γ(n − 2 + ε),

!

1 m2 − log 2 ε µ

+

c′kn

#

.

(5.5)

We define the quark condensate Qk (µ) renormalized at the point µ as the sum of the series (5.1) from which 1/ε poles are removed. We obtain the following results [9, 6] "

1 1 1 (G6 − 6G62 ) − 6m3 (L + 1)N + G4 − Q =− 24π 2 m 15m3 1 3



1 1 + (19G81 + 16G82 − 20G83 − 78G84 + 84G85 − 4G86 − 18G87 ) + O 5 210m m7 "

1 1 3mLG4 − (G61 − G62 ) Q =− 12π 2 m 5



1 1 + (3G81 + 2G82 − G83 − 9G84 + 9G85 − 54 G86 − G87 ) + O 3 10m m5 "



1 1 1 Q =− (18G85 − 12 G86 − 4G87 ) + O LG62 + 2 2 12π 20m m4 6

"



#

#

#

,

#

,

,

1 1 8 1 G1 + O − 3m3 (L + 1)G4 + , Q71 = − 2 12π 2m m3   1 1 8 8 8 Q72 = (G + G2 − 4G4 ) + O , 16π 2 m 1 m3 "  # 1 1 1 8 8 8 6 7 1 8 (G − G4 + 2G5 − 2 G6 ) + O , Q3 = 2 − mLG1 + 4π 6m 3 m3 "



1 1 1 Q74 = − 2 mLG62 + (4G85 − 21 G86 − G87 ) + O 4π 6m m3 Q81

"

#

,



1 1 3m4 (L + 32 )G4 + L(G81 − 2G83 − 2G84 ) + O =− 2 12π m2 "

(5.6)

#

, 

1 1 Q82 = 12m2 (L + 1)G61 − L(G81 − G82 − 2G83 + 2G84 − 4G85 + G86 ) + O 2 24π m2   1 L Q83 = − G8 + O , 12π 2 6 m2   1 L 8 G + O Q84 = − , 12π 2 7 m2 23

#

,





L 8 1 G5 + O , 2 6π m2   1 , Q86 = 0 + O m2 Q85 =

2

µ where L = log m 2 . As we have already mentioned, the series (5.1) for the anomalous condensate A includes only d = 8 gluon condensates:

1 (G8 + G82 − 4G84 ). (5.7) 32π 2 1 Many of these results can be obtained using various physical considerations [8, 9] instead of the described “brute force” method. A=−

5.2

Heavy quark currents’ correlators -'$ p-

k

p-

 &% -

p−k

Figure 5: One-loop diagram for a correlator Correlators of heavy quark currents can be expressed via gluon condensates: Π(p) =

X

an (p2 , m)Gn .

(5.8)

n

The one–loop diagram (Fig. 5) contains two propagators of the type (2.3), (2.4) (don’t forget that one of them has the vacuum momenta sink on the other side, and is given by the formulae mirror symmetric to (2.3), (2.4)!). After differentiations we obtain a formula with the integrals Z 

dk 2π



D

(k2

−

P (k) . − p)2 − m2 )m

(5.9)

xn−1 (1 − x)m−1 dx . [xa + (1 − x)b]n+m

(5.10)

m2 )n ((k

The denominators can be combined using the Feynman’s formula Γ(n + m) 1 = n m a b Γ(n)Γ(m)

Z1 0

After the shift of the integration momentum k → k + xp the denominator becomes k2 + x(1 − x)p2 − m2 ; we may average the numerator P (k + xp) over k directions. The integrals reduce to the form (5.4) with m2 → m2 − x(1 − x)p2 . We are left with one–dimensional integrals over the Feynman parameter x of the form [3] Jn (ξ) =

Z1 0

dx [1 + ξx(1 − x)]n

(2n − 3)!! = (n − 1)!

"

a−1 2a

n

(5.11) √

# √ X (k − 1)!  a − 1 n−k a + 1 n−1 + a log √ , a − 1 k=1 (2k − 1)!! 2a

where ξ = −p2 /m2 , a = 1 + 4/ξ. Using these formulae, one can calculate any gluon contribution to a heavy quark currents’ correlator. The simplest example of d = 4 is considered in [1]; contributions with d = 6 and d = 8 were calculated in [3]. 24

6

Light quarks

6.1

Limit m → 0 in heavy quark correlators

Let us consider the heavy–quark correlator (5.8) (Fig. 5) at p2 ≫ m2 . We can express it via both gluon and quark condensates: Π(p) = ΠG (p) + ΠQ (p), ΠG (p) =

X

a′n (p2 , m)Gn ,

ΠQ (p) =

n

X

(6.1) bk (p2 , m)Qk .

k

Here ΠG (p) corresponds to the contribution of the region where the virtualities of both quark and antiquark in Fig. 5 are large (k2 ∼ p2 ), and ΠQ (p) corresponds to the contribution of the regions where either the quark or the antiquark has a small virtuality (k2 ∼ m2 ). These contributions are usually depicted as the diagrams of Fig. 6.

6

x

?

 

0

p

?

+



x

p

6

0

Figure 6: Quark condensates’ contribution to a correlator The quark condensates’ contribution (Fig. 6) in the coordinate space is ΠQ (x) = hq(x)ΓS(x, 0)Γq(0)i + c.c.,

(6.2)

where Γ is a γ matrix, and c.c. means charge conjugate. In the momentum space 



1← ← − − ← − ΠQ (p) = q(1 − i D α ∂α − D α D β ∂α ∂β + · · ·)ΓS(p)Γq + c.c. 2!

(6.3)

We substitute S(p) (2.3), (2.4) and average over p directions, and finally reduce (6.3) to the basis quark condensates following the Section 3.1. The contribution of A is omitted because this condensate is a gluon one. The quark contributions with d = 7, d = 8 were obtained in [8, 6]. Having obtained ΠQ (p), we can find ΠG (p) from the heavy-quark correlator Π(p) using the expansions (5.6): ΠG (p) = =

X n

X n

a′n (p2 , m)Gn = Π(p)|G − ΠQ (p)|G

an (p2 , m)Gn −

X

bk (p2 , m)ckn (m)Gn .

(6.4)

k,n

The coefficients an (p2 , m) have singularities at m → 0 arising from the regions k2 ∼ m2 . These singularities have to cancel in (6.4) giving a′n (p2 , m) finite at m → 0. Using the results of [3], the gluon contribution with d = 8 was obtained in [9, 6].

6.2

Minimal subtraction of mass singularities

The method of the previous Section is good when the heavy quark correlator is already known. But when we want to calculate a light quark correlator from scratch, there should be a simper way than to solve a more difficult heavy quark problem first. Such a method was proposed in [5], and used in [6] for d = 8 calculations.

25

Now we want to go to the limit m → 0 before D → 4. The difference of a renormalized quark condensate and a bare one is (see (5.5)) Qk − Qbare =− k

1 X 1 X mdk −dn γkn Gn → − γkn Gn . ε d ≤d ε d =d n

n

k

(6.5)

k

At m = 0 all loop integrals for Qbare vanish because they contain no scale (ultraviolet and k infrared divergences cancel each other). Therefore we obtain for the gluon contribution to a correlator (see (6.4)) 1 X bk (p)γkn Gn . (6.6) ΠG (p) = Π(p)|G + ε d =d n

k

Here Π(p)|G is calculated in the MS scheme with m = 0, and γkn are mixing coefficients of quark condensates Qk with gluon condensates Gn of the same dimension (coefficients at L in (5.6)). Omitting 1/ε poles we finally obtain ΠG (p) = Π(p)|G +

X dbk

dn =dk

dε

γkn Gn .

(6.7)

The quark condensates’ coefficient functions bk should be calculated at m = 0 up to linear terms in ε (using D–dimensional tensor and γ matrix algebra, and in particular D–dimensional averaging, see Section 3.3). Acknowledgements. I am grateful to Yu. F. Pinelis for the collaboration in writing the papers [8, 9], to D. J. Broadhurst for useful discussions of the works [4–7], and to S. V. Mikhailov for fruitful discussions of nonlocal condensates. Various stages of this work were supported by grants from International Science Foundation, Russian Foundation of Fundamental Research, Royal Society, and PPARC.

26

References [1] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, V. I. Zakharov. Fortschr. Phys. 32 (1984) 585. [2] S. N. Nikolaev, A. V. Radyushkin. Phys. Lett. B110 (1982) 476; (E) B116 (1982) 469; Nucl. Phys. B213 (1983) 285. [3] S. N. Nikolaev, A. V. Radyushkin. Phys. Lett. B124 (1983) 243; Sov. J. Nucl. Phys. 39 (1984) 91. [4] S. C. Generalis, D. J. Broadhurst. Phys. Lett. B139 (1984) 85. [5] D. J. Broadhurst, S. C. Generalis. Phys. Lett. B142 (1984) 75. [6] D. J. Broadhurst, S. C. Generalis. Phys. Lett. B165 (1985) 175. [7] S. C. Generalis. Open University thesis OUT-4102-13 (1984); J. Phys. G15 (1990) L225; G16 (1990) 367, 785, L117. [8] A. G. Grozin, Yu. F. Pinelis. Phys. Lett. B166 (1986) 429. [9] A. G. Grozin, Yu. F. Pinelis. Zeit. Phys. C33 (1987) 419. [10] V. N. Baier, Yu. F. Pinelis. Preprint INP 81-141, Novosibirsk (1981); D. Gromes. Phys. Lett. B115 (1982) 482; E. V. Shuryak. Nucl. Phys. B203 (1982) 116. [11] S. V. Mikhailov, A. V. Radyushkin. JETP Lett. 43 (1986) 712; Sov. J. Nucl. Phys. 49 (1989) 494; Phys. Rev. D45 (1992) 1754. [12] S. V. Mikhailov, A. V. Radyushkin. Sov. J. Nucl. Phys. 52 (1990) 697; A. V. Radyushkin. Phys. Lett. B271 (1991) 218; A. P. Bakulev, A. V. Radyushkin. Phys. Lett. B271 (1991) 223; A. G. Grozin, O. I. Yakovlev. Phys. Lett. B285 (1992) 254; B291 (1992) 441; A. P. Bakulev, S. V. Mikhailov. Preprint JINR E2–94–208, Dubna (1994); hep-ph/9406216. [13] H. G. Dosh, Yu. A. Simonov. Phys. Lett. B205 (1988) 338; Yu. A. Simonov. Nucl. Phys. B307 (1988) 512; B324 (1989) 67; Sov. J. Nucl. Phys. 50 (1989) 134; 54 (1991) 115. [14] S. V. Mikhailov. Phys. Atom. Nucl. 56 (1993) 650.

27