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Operator product expansion coefficient functions in terms of composite operators only. Nonsinglet case

A new method for calculating the coefficient functions of the operator product expansion is proposed which does not depend explicitly on elementary fields. Coefficient functions are defined entirely in terms of composite operators. The method is illustrated in the case of QCD nonsinglet operators.


Introduction
The structure functions of deep inelastic lepton-proton scattering (DIS) are related with an absorptive part of the matrix elements Here J em µ (x) is an electromagnetic current and | p means a proton state. In its turn, an expansion of a chronological product of two electromagnetic currents near the light-cone in terms of composite operators O a, m looks like: where a = NS, S, g. The dots denote other Lorentz structure, constant term as well as gradient terms. The latter give no contribution to DIS structure functions. The index a runs all allowed types of the composite operators. The expansion (2) is a generalization of the operator product expansion (OPE) at small distances [1]. It is known as light-cone OPE [2] (see also [3,4]). The quantity C (i) a, m is called an OPE coefficient function (CF). The composite operators in Eq. (2) need a regularization. After a renormalization of the operators, there arises a dependence of CF's on a renormalization scale µ.
On the other hand, the DIS structure functions can be represented in a factorizable form [5]. For instance, we have Here f a (x, M 2 ) is a distribution of a parton of type a inside the nucleon, while quantities C a (x, Q 2 /M 2 ) are known as DIS coefficient functions. In Eq. (3), M is a factorization scale. It is usually identified with the renormalization scale µ. The m-th moment of the DIS coefficient function, C a (m, Q 2 /µ 2 ), is identified with the Fourier transform of the corresponding OPE CF's in (2).
The parton distributions f a (x, µ 2 ) are related with matrix elements of the composite operators between one-nucleon states (for details, see [3]).
The coefficient functions C a (x, Q 2 /M 2 ) were calculated in perturbative QCD. For example, contributions of the first and second orders in α s can be found in Refs. [6] (where light quarks were considered), and in Refs. [7] (where both light and heavy quarks were accounted for). Note that a very definition of DIS coefficient functions applies to the diagram (perturbative) technique, not to the operator formalism. Moreover, since nucleon wave function is yet unknown, one has to deal with diagrams which describe a lepton scattering off (nonphysical) quark or gluon off-shell state.
The light-cone OPE for the scalar theory was studied in Refs. [8] in which rules for calculating CF's were presented. Note that the T-product of two scalar currents near the light-cone was defined in term of so-called bi-local light-ray composite fields [8]. The local light-cone expansion can be obtained by performing a Taylor expansion of the non-local one [9]. The non-local expansion is more general, but we restrict ourselves to considering local OPE. In Refs. [10] a problem of finding n-loop contributions to the OPE CF's were reduced to evaluating of propagator type (n + 1)-loop Feynman diagrams.
The goal of the present paper is to present a derivation of a closed representation for the OPE CF's in term of composite operator Green functions which does not lean on the perturbation theory. This will be done in the next Section. In Section 3 we check a validity of our results in a free scalar field theory. In Section 4 we calculate nonsinglet CF's in perturbative QCD in order to demonstrate that our main formulae not only reproduces wellknown expressions for the quark CF's, but enables us to obtain CF's of the gradient operators in the OPE. The finite renormalization of CF's is considered in Section 5. In Appendix A a number of useful mathematical formulae is collected. In Appendix B we show that our scheme does result in a set of homogeneous renormalization group equations for the OPE CF's.

OPE coefficient functions and matrix elements of composite operators
Let us define a quark electromagnetic current where Ψ(x) is a quark field. The electric charge operator in (4), obeys the equations: Here λ a (a = 1, 2, . . . 8) are the Gell-Mann matrices, Sp(λ a λ b ) = 2δ ab , and λ 0 is the identity matrix. The operator product expansion (OPE) for the T-product of two electromagnetic currents looks like (see, for instance, [4]) where the dots denote contributions from other Lorentz structures and singlet quark and gluon operators. It is clear from (7) that this expansion should contain nonsinglet (triplet and octet) and singlet composite operators. Near the light-cone, a leading contribution comes from twist-2 operators. For instance, quark twist-2 (traceless) operator is of the form (operator S means a complete symmetrization in Lorentz indices): Here is a covariant derivative, and A a µ (x) is a gluon field. If the OPE (8) is applied to deep inelastic scattering, only operators of the type are important, since forward matrix elements of the operators with 1 l m−1 are zero. 1 For non-forward Compton scattering, all operators contribute where Z is a triangle matrix. In particular, we find that the composite operator O m,1 is multiplicatively renormalized, while the composite operator O m,m , which is relevant to DIS, is not. In perturbative QCD in the first order of α s = g 2 /(4π), elements of the matrix Z are given by the following expressions [4] (with non-significant finite terms omitted): where C F = (N c − 1)/2N c , with N c being a number of colors. In deriving (13), a dimensional regularization [11] was used, and ε = 2 − D/2, where D is the number of dimensions. Here and belowμ denotes the regularization mass, 2 while µ denotes the renormalization scale.
In what follows, we will be interested in one of nonsinglet quark operators, namely, in O 3, m,l N S, µ 1 ...µm (x). 3 Let us introduce brief notations: 2 The regularization scaleμ arises in dimensional regularization, when one changes an integration volume, d 4 k →μ 4−D d D k. 3 It is obviously that the operator O 8, m,l N S, µ1...µm (x) can be treated in the same manner. The singlet case is technically more complicated, since the singlet quark and gluon operators have to be considered simultaneously.
After multiplying both sides of Eq. (8) by the operator O n,k ν 1 ...νn (with arbitrary n, and k = 1, 2 . . . n), we get the following relation for T-products of the composite operators between the vacuum states: 4 Here dots mean other Lorentz structures. It is necessary to stress that propagator of only singlet quark operator has to appeared in (16), due to relation (7). As usual, we assume that C m,l (x 2 ) are tempered generalized functions (this is explicit in perturbative calculations) so the symbolic relation holds in connection with the Fourier transform in (16). Then we obtain that Let n µ to be a light-cone 4-vector which is not orthogonal to 4-momentum p µ : Throughout the paper, we will work in the limit Let us now convolute our matrix elements with the projector In particular, we can define the following invariant structure, which depends on variables p 2 , Q 2 = −q 2 , and The propagator of the composite operator has the following Lorentz structure: Equation (25) means: Note that F n,k (ω, Q 2 , p 2 ) and O m,l O n,k (p 2 ) are dimensionless. Let us note that at p 2 → 0 propagators of composite operators of higher twists are suppressed by powers of p 2 with respect to the propagators of twist-2 operators. Thus, our approach enables us to isolate a contribution from twist-2 operators.
At fixed Q 2 and p 2 , 3-point Green function TJ em µ J em ν O n,k has a discontinuity in the variable (q + p) 2 for (q + p) 2 0 (that is, for ω 1). By using the unsubtracted dispersion relation for F n,k (ω, Q 2 , p 2 ), As we will see in the next Sections, both propagator of the composite operator and matrix element F n,k need a renormalization already in zero order in strong coupling constant [12]. Thus, we have O m,l O n,k = O m,l O n,k (p 2 /µ 2 ), F n,k = F n,k (x, Q 2 /p 2 , p 2 /µ 2 ), and, consequently,C m,l =C m,l (Q 2 /µ 2 ), where µ is a renormalization scale. Remember that all these quantities are dimensionless. The Eq. (28) are valid for all integer k 1 (of course, k n, but we can choose O n,k with any n). Let us put n m and define the matrixes: and where Lorentz invariant part of 3-point Green function is implied. 6 Then from (28) we obtain m equations for CF's (for k = 1, 2, . . . m): where OO −1 kl is the inverse of the matrix OO lk . The formulae (28), (31) is our main theoretical result. The equation (31) gives an operator definition of the OPE CF's in term of vacuum matrix elements of the composite operators. 7 It is important to stress that our definition does not lean on a notion of quark distributions.

Coefficient functions in free scalar field theory
As a simple example, let us consider an expansion of T-product of two composite operators φ 2 (z) ≡ : φ(z)φ(z) : in free scalar field theory. The scalar fields φ(z) are assumed to be real and massless. Using the Wick theorem, one can find where is a causal propagator of the field φ(z). By expanding bilocal operator in powers of variable (x − y) and putting then y = 0, we get (here and below a constant term is omitted): As usual, the symbol (34) denotes a derivative which acts on the left (right) standing field φ in the composite operator φ 2 (0).
Using an explicit symmetry of x µ 1 x µ 2 . . . x µm in indices, one can write Then we obtain from Eqs. (34), (35): Here (−q 2 + i0) −1 is a Fourier transform of D c (x), and a brief notation, is introduced. The OPE CF's in Eq. (36) are the following dimensionless quantities: Let us demonstrate that our main formula (28) gives the same result (38). To do this, one has to calculate both the propagator of the composite operator, φ 2 m,l φ 2 n,k , and 3-point Green function φ 2 φ 2 φ 2 n,k . The corresponding diagrams are presented in Fig. 1. The vertex which corresponds to the composite operator φ 2 n,k is equal to (pn) n−k (kn) k , with k and (k−p) being ingoing and outgoing 4-momentum, respectively. The result of our calculations is the following: and The quantity B(x, y) is the beta-function, B(x, y) = B(y, x).
In order to get a finite result for free scalar propagator (39), we had to make a renormalization. As is well known, any Green function with an insertion of one composite operator is multiplicatively renormalized [13,14].
The renormalization of Green functions with the insertion of two (or more) composite operators needs additive counterterms [12]. The details can be found in Appendix B.
Form formulae (39), (40) and (28) (taking into account the factor in front of the sum in Eq. (36)), we obtain a set of equations for CF's: It is easy to check that C-numbers C m,l (38) do obey equations (41) for all m, k 0 (for this purpose, formula (A.1) from Appendix A should be used).

Calculations of coefficient functions in perturbative QCD
In this section we will use our formula (28) for calculations of the OPE CF's in perturbative QCD. For the composite operator, we will often use a brief notation O m,l instead of O m,l µ 1 ...µm (0). Contrary to the matrix element of O m,l between one-particle (quark) states, p | O m,l | p , which defines nonsinglet quark distributions in DIS, a propagator O m,l O n,k is divergent already in zero order in the strong coupling α s . The matrix element J J O n,k (Q 2 , x, p 2 ) is also divergent, whereas its discontinuity in variable (p + q) 2 is not. The 2-point Green function O m,l (x)O n,k (y) , is renormalized by a subtraction of a contact term of the form ( where Z is the renormalization matrix of the composite operators. The details can be found in Appendix B. We work in the dimensional regularization and use the MS-scheme to renormalize ultra-violet divergences. Although all calculations will be done in the Feynman gauge, our results are gauge invariant since we sum all diagrams in each order of perturbation theory. Remember that in order to find the OPE CF's, we have to retain only leading terms in the limit p 2 → 0. This simplifies our calculations significantly. We will restrict ourselves by considering leading terms in ln(Q 2 /µ 2 ), although our formula (28) enables one to calculate subleading terms as well. In other words, along with the limit p 2 → 0, we are interested in large values of Q 2 .
Let us start from the leading (zero) order in α s . 8 The corresponding Feynman diagrams are shown in Fig. 4. All diagrams have logarithmic singularity at p 2 → 0. They can be easily calculated with the use of Feynman rules for the composite operators presented in Fig. 2. Note that a vertex which corresponds to the composite operator is not symmetric with respect to ingoing and outgoing momenta. For a particular case with k = n, p = 0, we reproduce the well-known expressions (see the first paper in Refs. [15]).
The result of our calculations in the leading order in the coupling constant is the following: 9 and where e 2 q is an electric charge of a quark inside a loop. The terms in square brackets in Eq. (43) correspond to two bottom diagrams in Fig. 4 with opposite directions of the quark momentum inside the loop. As for two top diagrams in Fig. 4, they are identical. So, only one of these diagrams was taken into account in Eq. (42). The formula (42) gives a finite (singular at p 2 → 0) part of the composite operator propagator (see our comments after Eq. (40)).
Equating factors in front of ln(−p 2 ) in both sides of Eq. (42), (43), and putting k = 1, 2, . . . m, we obtain the following set of m equations: 8 It corresponds to a theory of free quarks (which interact only via electromagnetic forces). 9 Here and in what follows we omit terms which are sub-leading in the limit p 2 → 0. The superscript (n) means that a corresponding quantity is calculated in n-th order in strong coupling constant α s .
Using formulae (A.2)-(A.5) from Appendix A, we find the solution of these equations in the form 10 where n k denotes a binomial coefficient. Let us stress, the quantitiesC m,l (46) satisfy equation (44) for all integer k 1, although for our purpose it was enough to take m values of k.
One can check that  Fig. 3, we obtain the following contribution of the diagrams in Fig. 5a: 10 Let us stress, we didn't demand from the very beginning thatC m,m should be equal to zero for odd m. It is a consequence of the fact that electromagnetic interactions conserve P-parity. Remember that DIS structure function F 2 (x, Q 2 ) is an even function of variable x, and its non-zero moments are F 2 (n, Q 2 ) =C n,n (Q 2 /µ 2 ) p |Ô n,n | p (µ 2 ), with n = 2k.
where C F = (N 2 c − 1)/2N c , with N c being a number of colors. For the diagrams in Fig. 5b, we get the expression: Both diagrams in Fig. 6 are equal to zero since they are proportional to n 2 µ . Thus, the sum of diagrams which make a contribution to the propagator of the composite operator is given by Note, that O n,k O m,l (1) = 0 for k = 1 or l = 1. Equation (50) can be presented in the form: For our further purposes, it is useful to present (50) in a form which has no explicit symmetry in l and k: In deriving Eq. (52) from Eq. (50), the formulae from the Appendix A were used. Taking into account (46), we find As one can see from (53) The expression (53) can be simplified and presented as 11 With the use of Eq. (A.15) from Appendix A, one can rewrite (54) in the form which has an explicit symmetry in k, m: Now let us calculate the Green function which contains one composite 11 We used the relation B(m + 1, 2) − [(m + 1)(m + 2)] −1 = 0.
operator and two electromagnetic currents. The diagrams in Fig. 7a give The equation (56) can be rewritten as follows: with zero order expression given by (43). The contribution from the sum of the diagrams in Fig. 7b is equal to Note again that Eq. (58) can be rewritten in terms of an absorptive part of the 3-point Green function calculated in the leading order in α s : The diagram in Fig. 8 is sub-leading at p 2 → 0, since it has no singularities at p µ = 0. The diagrams in Figs. 9a and 9b describe a renormalization of the electromagnetic current J em which is conserved and, therefore, has no anomalous dimensions. As a result, these diagrams are suppressed by the inverse of ln(µ 2 /(−p 2 )) with respect to (56) and (58). Thus, we have to sum only expressions (56) and (58): .
Making use of equation (A.7), one can easily calculate the sum in the last line of equation (60) and rewrite it in the form: The set of equations for the OPE CF'sC (1) for m different values of k = 1, 2 . . . m. In particular, we obtain for k = 1: This equation is similar to formula (47) which was obtained in the leading order. However, in order to findC (1) m,l , it is much more convenient to use equivalent form of the expression in the RHS of Eq. (63) For this purpose, one should compare (54) with (53), and then exploit the above demonstrated symmetry in k, l(k, m) (55): Thus, we get m equations (corresponding to k = 1, 2 . . . m) for m coefficient functions. The solution of these equations (65) is rather easy to find, the result isC for l = 1, 2, . . . , m − 1. 12 We used formulae for summation in l ((A.5), (A.6)) and j ((A.11)) from Appendix A. In particular, we get from (67) (m 2): As one can see from (66), "major" coefficient functionC (1) m,m is defined by well-known anomalous dimension of O m,m [4],  (67)). Let us emphasize that our operator definition of the coefficient functions C m,l (Q 2 /µ 2 ) (31) results in a homogeneous renormalization group equation for C m,l (Q 2 /µ 2 ) with respect to µ, in spite of the fact that composite operator Green functions O n,k O m,l need an additional renormalization. The details are discussed in Appendix B.

Finite renormalization of composite operators and rescaling of coefficient functions
With accounting for equations (45), (46) and (66), (67), the expressions for the OPE CF's can be rewritten in the following form: for l = 1, 2, . . . , m − 1. Let us consider a sum of products of renormalized composite operators and corresponding CF's which enter the OPE of two electromagnetic currents (4). According to (12), the renormalized composite operator O m,l R (µ 2 1 ) is changed under rescaling µ 1 → µ 2 as follows where µ i denotes a renormalization scale of the composite operators in the MS-scheme, 13 andẐ is a matrix of a finite renormalization. Let us emphasize that due to Eq. (B.9) since additive coefficients in Eq. (B.9) do not depent on the renormalization scale µ.
In the first order of strong interaction, the matrix looks likê In a general case we find: Thus, we obtain from (76) for l = 1, 2, . . . , m−1. It was taken into account thatẐ kl = 0 for k < l. Thus, if one changes the renormalization scale µ, the "major" coefficient functioñ C m,m is simply multiplied by the factorẐ mm , while the other CF's mix with each other. It is a consequence of the fact that the matrixẐ kl has a triangle form with zero elements above its diagonal. The coefficient functionsC m,l (µ 2 2 /µ 2 1 ) are normalized so thatC m,l (1) = C m,l , where numbersC m,l are "bare" coefficient functions which were obtained for the case with no strong interactions (see Eqs. (45), (46)). We conclude from explicit expression ofẐ in the first order in α s (74) that our perturbative result, equations (70) and (71), is a particular case of general formulae (77) and (78). 14

Conclusions
As was shown above, the OPE coefficient functions can be expressed in terms of the Green functions of the corresponding composite operators without explicit use of the elementary (quark and gluon) field operators. We would like to specially stress that our general expression (31) for the OPE coefficient functions holds in any renormalization scheme in contrast with previous prescriptions (see, for instance, [10]). We believe that such a representation is useful in proving general properties of the coefficient functions and for non-perturbative calculations (e.g. in lattice QCD).
We would like to stress that if one assumes the light-cone expansion in the framework of axiomatic approach (see, for instance, Ref. [18]), then our formulae do not lean on perturbative expansions at all and deal with v.e.v. of T-products of local Heisenberg operators.
In this paper we limited ourselves by the QCD nonsinglet operator only. The singlet case needs a special treatment and will be considered in a forthcoming paper.
. (A.9) Now let us consider summation in index j (k, l 1 is assumed everywhere): Note that the last equation is a particular case of Eq. (A.10) for 1 ≤ l ≤ m. Let us present other useful formulae: The last relation is a consequence of Eqs. (A.13) and (A.14). It demonstrates that the sum in (A.15) is symmetric under replacement l ⇄ k.
The last useful formula contains sums in l and j:

Appendix B
The goal of this Appendix is to derive a renormalization group equation (see, for instance, [16]) for the CF's stating from our formula ( where n means a sum in a complete set of elementary (quark and gluon) states |n . Let us underline that each state |n should include at least one quark-antiquark pair (in non-singlet state). It is well-known that a matrix element with an insertion of one composite operator is multiplicatively renormalized (see, for instance, [14]). Therefore, one has the following relation: Since electromagnetic current is conserved, its renormalization constant, Z J , does not depend on the renormalization scale, (d/dµ)Z J = 0. , Now we will show that the matrix element n|JO n,k R in (B.2) is also multiplicatively renormalized. Consider arbitrary diagram G which contributes to a matrix element with an insertion of two composite operators, J and O k . We assume that divergences of all sub-diagrams of G are already removed. Let G to have (p + 2) external lines (p 0), and q internal vertexes. The index of this diagram, ω(G), is defined by [14] Here ω max i is a maximal index of internal vertex of a type i, and r l is a power of a polynomial corresponding to an external field of a type l (r q = 1 for a quark line). The quantities ω J is the dimension of the electromagnetic current, while ω O n,k is the dimensions of our composite operators. Note that ω max i = 0 for all QCD vertexes, and ω J = 3. As for ω O n,k , it is formally equal to (k + 2) (only derivatives with respect to quark line momenta have to be taken into account, not total derivatives). In particular, ω O n,1 = 3. Let us show, however, that effectively ω O n,k does not depend on k, and ω O n,k = ω O n,1 . Let {r i } (i = 1, . . . p + 4) to be a set of external momenta (p, q including), while {k i } (i = 1, . . . m) to be a set of loop momenta of G (with m being a number of loops). After using the Feynman parametrization and linear transformations from {k i } to {l i }, an analytical expression for the diagram can be written in the form: where x i are the Feynman parameters. The change of variables {k i } → {l i } is done so that the denominator in Eq. (B.5) depends on l i only via l 2 i . Notice, the momentum l m is a linear combination of r i , and k m , ingoing quark momentum for the vertex O n,k . The numerator in Eq. (B.5) is a sum of polynomials of (l i l j ), (l i r j ) and (r i r j ). Each polynomial includes at least one scalar product of the vector n, namely, (l i n) or (r i n).
After integrating in d D l 1 , . . . d D l m−1 , we get (we denoted l m = l). Note that dimD − dimP = 5. Let us analyze possible terms inP which is a polynomial of l 2 , (lr i ), (r i r j ), and depends linearly on (ln) or (r i n). The term (ln)(l 2 ) (dimP −1)/2 results in zero (for even r) or an expression proportional to n 2 µ = 0 (for odd r) in the dimensional regularization. 17 All other terms proportional to (ln) drop for the same reason (if r 1) or give finite integrals which converge in the ultra-violet region. One can verify that terms proportional to (r i n) give non-zero convergent integrals only for r = 0.
Thus, we have to put ω O n,k = ω O n,1 = 3 in (B.4). In its turn, it means that ω(G) −1, with ω(G) = −1 for p = 1. In other words, there is no need to introduce a new counterterm : J(x)O(y) : δ(x − y), and, consequently, It follows from (B.3), (B.7) that This result is in agreement with perturbative QCD calculations from Section 4. Now let us turn to the propagators of the composite operators. The renormalization properties of Green functions with insertions of more than one composite operator were studied in details in Ref. [17]. In particular, The divergent coefficients f kl B are, in general, non-zero even in the free theory, as it is the case for our quark operators Q n,k .
The renormalization matrix of the composite operator Z depends on a renormalization scale µ, while the coefficients f kl B do not. The renormalized Green function OO kl R is also µ-dependent, but it does not depend on the regularization scaleμ. All physical (measurable) quantities should be, of course,μ and µ-independent.
As one can see, Eq. (B.9) which relates renormalized and unrenormalized propagators has an additive term. Nevertheless, renormalization group equations for the propagators and for CF's have no additive terms. 18 Indeed, acting on both sides of Eq. (B.1) by the operator µd/dµ = µ∂/∂µ + β(g s )∂/∂g s , and taking into account Eqs. (B.8), (B.9), we obtain: is the matrix of anomalous dimensions of the composite operators. Equation (B.10) can be represented in the form: Since (B.12) is valid for arbitrary integer k 1, we derive a set of renormalization group equations for the coefficient functionsC m,l (l = 1, 2, . . . m): In particular, the coefficient functionC m,m , which is relevant for DIS, obeys the following closed equation: The analysis of renormalization properties of the composite operator propagator is simplified, if we choose a light-cone axial gauge n µ A µ = 0 in which only diagrams shown in Fig. 10 contribute to OO kl . Correspondingly, the renormalization of the composite operator in this gauge is given by diagram in Fig. 11. The blob in these figures is a sum of all possible QCD diagrams (with a disconnected part included). A specific character of the diagrams in Fig. 10, 11 is the following: they can be divided in two parts by cutting two quark lines. It enables us to derive the following relation: where the matrix Z B (ε,μ 2 /p 2 ) is a bare quark loop with an insertions of two composite operators: