epsf.sty

**Predictions from conformal algebra for the**

**deeply virtual Compton scattering.**

**A.V. Belitsky**

*Bogoliubov Laboratory of Theoretical Physics*

*Joint Institute for Nuclear Research*

*141980, Dubna, Russia*

*and*

*Institut für Theoretische Physik*

*Universität Regensburg*

*D-93040 Regensburg, Germany*

**D. Müller**

*Institute for Theoretical Physics*

*Center of Theoretical Science*

*Leipzig University*

*04109 Leipzig, Germany*

**Abstract**

Basing on the constraint equalities which arise from the algebra of the collinear conformal group and the conformal operator product expansion, we predict the solutions of the leading order evolution equations for the non-forward distributions, including transversity, in terms of the conformal moments; next-to-leading order flavour singlet coefficient functions for the polarized and unpolarized deeply virtual Compton scattering; as well as the contribution from renormalon chains to the eigenfunctions of the exclusive non-singlet evolution kernel.

Introduction.
=============

It is well known that the conformal twist-2 operators possess at leading order (LO) a diagonal anomalous dimension matrix $\hat{\gamma}=\{\gamma_{jj'}\}$, which is, in general, triangular due to Lorentz invariance. One may expect that in the conformal limit of the theory ($\beta = 0$) the former property will be fulfilled beyond one-loop order; however, this is not the case and the conformal operators mix already in the next-to-leading order (NLO) of perturbation theory. A closer look at the conformal Ward identities (CWI) in LO for the flavour non-singlet channel [@Mue91; @Mue94], which include both dilatation and special conformal transformations, shows that the conformal covariance is destroyed in the minimal subtraction scheme (${\overline{\rm MS}}$) by an off-diagonal special conformal anomaly matrix $\hat{\gamma}^c(l)=\{\gamma^c_{jj'}(l)\}$, which depends on the spin $l$ carried by the considered twist-2 operator. To restore it to all orders of perturbation theory one is forced to perform a finite renormalization of the conformal operators leading to the conformal subtraction (CS) scheme. Assuming a non-trivial fixed point $g^*$, i.e. that $\beta (g^*)$ is zero, this restoration is assured in any order of the perturbation theory by conformal constraints which arise from the algebra of the collinear conformal group: $$\begin{aligned}
\label{conf-constr-KD-1}
\left[ D , K_- \right]_- &=& iK_-
\hspace{2.7cm} \Longrightarrow \hspace{1cm} \left[ \hat a (l)
+ \hat \gamma ^c(l), \hat \gamma \right] = 0, \\
\label{conf-constr-KD-2}
\left[ P_+ , K_- \right]_- &=&
2i( D + M_{-+} )\hspace{1cm} \Longrightarrow \hspace{1cm}
\hat \gamma ^c (l+1) - \hat \gamma ^c (l) = - 2 \hat \gamma ,\end{aligned}$$ where $a_{jj'}(l)= a(j,l) \cdot \delta_{jj'}$ and $a(j,l)
\equiv 2(j-l)(j+l+3)$. For the case of $\phi^3_{(6)}$ theory, this constraints can be extended without any assumptions to nonvanishing $\beta$-function [@Mue91]: $$\begin{aligned}
\label{conf-constr-KD-full}
&&\left[\hat{a}(l)+\hat{\gamma}^c(l)
+2{\beta\over g}\hat{b}(l)
,\hat{\gamma}\right]=0,\end{aligned}$$ where the $\hat{b}$ matrix is defined as[^1] $$\begin{aligned}
\label{def-b}
b_{jj'}(l)= \theta_{jj'}
\left\{ 2(l+j'+3)\delta_{jj'} - [1 + (-1)^{ j - j'}] (2j' + 3)
\right\} .\end{aligned}$$ Hence, the contribution to the off-diagonal matrix elements of the anomalous dimension matrix due to $\beta$-function is predicted too. Under the conjecture that only gauge invariant composite operators contribute to the constraints for the gauge independent anomalous dimension matrix of the conformal operators, which turns out to be quite natural, formally the same equation can also be derived in Abelian gauge field theory for the flavour non-singlet channel [@Mue94] as well as for the quark-gluon singlet sector studied in the present paper. The knowledge of the conformal anomaly matrix allows to reconstruct the off-diagonal part of the anomalous dimension matrix for the corresponding channels.

Few remarks are in order at this point. Although the above mentioned commutator constraints are derived, for the reasons of technical simplification, in the theory with $U(1)$ symmetry group, nevertheless, they are valid in NLO approximation for $QQ$- and $QG$-channels we have considered, also in the non-Abelian theory. This is true since in the conformal limit, the symmetry breaking parts of the kernels do not contain the Casimir operator $C_A$ of the adjoint representation of $SU(3)$. Moreover, even in the theory with non-vanishing $\beta$ the correct results can be reconstructed by substituting the QED $\beta$-function by the corresponding QCD value [@Mue94] and, therefore, this treatment is sufficient to derive the reliable NLO predictions in QCD. Thus, some of the results of this paper can be regarded as a first step towards the complete derivation of the $\alpha_s$-corrections to the exclusive flavour-singlet evolution kernels which are presently completely unknown. Using the same technique, the purely gluonic case will be clarified in the nearest future [@BelMul98]. The knowledge of all other channels will allow one to check the result with a construction of the ($N=1$) SUSY Ward identities.

In this paper we mainly concern the predictions arising from the conformal algebra for the deeply virtual Compton scattering (DVCS) process, namely, the solution of the evolution equations, which govern the $Q^2$-dependence of the corresponding structure functions, and the NLO corrections to all singlet coefficient functions in the ${\overline{\rm MS}}$ renormalization scheme. The final section is devoted to the issue of the renormalon chains contribution to the Efremov-Radyushkin-Brodsky-Lepage (ER-BL) kernel which can be incorporated in a straightforward way in our approach.

Evolution of twist-2 conformal moments at LO.
=============================================

In the past [@muel88] and more recently [@ji97; @rad96; @RadPRD56] different authors introduced non-forward distributions, which describe the non-perturbative inputs in certain processes such as DVCS and exclusive electroproduction of mesons. In the consequent discussion we accept the following definition of the non-forward distribution functions [@rad96; @RadPRD56] $$\label{definition}
{^i{\cal O}^{\mit\Gamma}} (\lambda, \mu)
= \langle h' | \phi_i^* (\mu n) {\mit\Gamma}
\Phi \left[ \mu n, \lambda n \right]
\phi_i (\lambda n) |h \rangle
= \int d x e^{ i \mu (x - \zeta) - i \lambda x }\
{^i{\cal O}^{\mit\Gamma}} (x, \zeta),$$ in terms of the light-cone Fourier transformation. Here $i = Q,G$ runs over the parton species, ${\mit\Gamma}$ corresponds to different Dirac or Lorentz structures, depending on the spin of the constituents involved and $\Phi$ is a path ordered exponential. The momentum fractions of incoming $x$ and outgoing $x - \zeta$ partons are the Fourier conjugated variables of the light-cone positions $\lambda$ and $\mu$. Parameter $\zeta$ is a skewedness of the distribution defined as a $+$-component of the $t$-channel momentum $\Delta_+ =
\zeta$ (see next section for discussion of the kinematics of the DVCS process).

The scale dependence of the light-ray operators, standing on the left hand side of Eq. (\[definition\]), is governed by the renormalization group equation which is of the generic form ($\bar y \equiv 1 - y$) $$\begin{aligned}
\mu^2\frac{d}{d\mu^2}{\cal O}^{\mit\Gamma} (\lambda , \mu) &=&
\frac{\alpha_s}{2 \pi} \int_{0}^{1} dy \int_{0}^{\bar y} dz
{\cal K}^{\mit\Gamma} (y, z)
{\cal O}^{\mit\Gamma} (\lambda \bar z + \mu z , \mu \bar y + \lambda y).\end{aligned}$$ The evolution kernels for the parity-even ($V$) and parity-odd ($A$) chiral-even light-ray operators are known in the literature [@ji97; @rad96; @RadPRD56; @kern; @BB88]. Here we supplement this list by the kernel for the non-forward transversity ($T$) distribution [@CFS96]: $${\cal K}^T (y, z) = C_F
\left\{ \delta (y) \left[ \frac{1}{z} \right]_+
+ \delta (z) \left[ \frac{1}{y} \right]_+
- \delta (y) - \delta (z) + \frac{3}{2} \delta (y)\delta (z)
\right\}.$$ By exploiting the conversion formula derived in Ref. [@BM97] we can easily write the corresponding evolution equation in the momentum space $$\begin{aligned}
\label{MOMEE}
\mu^2\frac{d}{d\mu^2}{\cal O}^{\mit\Gamma} (x, \zeta) &=&
- \frac{\alpha_s}{2 \pi} \int d x ' K^{\mit\Gamma} (x , x ', \zeta)
{\cal O}^{\mit\Gamma} (x ', \zeta),\end{aligned}$$ with[^2] $$\begin{aligned}
\label{gen-ker-KT}
K^T (x , x ', \zeta)
= C_F \left\{\left[ \frac{x}{x - x'}
\Theta_{11}^0 (x , x - x ') + \frac{x - \zeta}{x - x'}
\Theta_{11}^0 (x - \zeta, x - x ') \right]_+
+ \frac{1}{2} \delta (x ' - x) \right\},\end{aligned}$$ where $\Theta^0_{11} (x, x') = (\theta(x) - \theta(x'))/(x - x')$. The corresponding ER-BL kernel ($\zeta=1$) $$\begin{aligned}
V^T ( x, x' ) \equiv - K^T (x , x', 1)
= C_F
\left\{
\left[
\theta( x' - x ) \frac{x}{x'} \frac{1}{x' - x}
+ \theta(x - x') \frac{\bar x}{\bar x'}\frac{1}{x - x'} \right]_+
+ \frac{1}{2} \delta (x - x')\right\},\end{aligned}$$ and the DGLAP kernel ($\zeta=0$ and $z=x/x'$) $$\begin{aligned}
P^T (z) \equiv - K^T (z x' , x', 0)
= C_F
\left\{
2 \left[ \frac{z}{\bar z} \right]_+
+
\frac{1}{2} \delta (\bar z)
\right\}\end{aligned}$$ follow immediately from the generalized one (\[gen-ker-KT\]).

In order to find the solution of the evolution equations of the type given in Eq. (\[MOMEE\]) we are looking for the conformal spin expansion in terms of reduced non-forward expectation values of conformal twist-2 operators which are related to the Gegenbauer moments of the corresponding distributions, namely, for quark function we have $$\begin{aligned}
\zeta^j
\int dx C_j^\frac{3}{2}\left( 2\frac{x}{\zeta} - 1 \right)
{^Q\!{\cal O}^{\mit\Gamma}} (x, \zeta) =
\langle h'| {^Q\!{\cal O}}^{\mit\Gamma}_{jj} |h \rangle .\end{aligned}$$ $C^\nu_j$ are the Gegenbauer polynomials [@BE53_2] with the normalization fixed below in Eq. (\[normalization\]). The gluon case differs only in the change of the index of the orthogonal polynomial by one unity and shifting $j$ to $j - 1$. Then, the complete sets of conformal twist-2 operators for the flavour singlet chiral-even un- and polarized as well as chiral-odd channels read: $$\label{treeCO}
\left\{\!\!\!
\begin{array}{c}
{^Q\!{\cal O}^V} \\
{^Q\!{\cal O}^A} \\
{^Q\!{\cal O}^T}
\end{array}
\!\!\!\right\}_{jl}
\!=
\bar{\psi} (i \partial_+)^l\!
\left\{\!\!\!
\begin{array}{c}
\gamma_+ \\
\gamma_+ \gamma_5 \\
\sigma_{+\perp} \gamma_5
\end{array}
\!\!\!\right\}
\!C^{\frac{3}{2}}_j\!
\left( \frac{\stackrel{\leftrightarrow}{D}_+}{\partial_+} \right)
\!\psi, \
\left\{\!\!\!
\begin{array}{c}
{^G\!{\cal O}^V} \\
{^G\!{\cal O}^A} \\
{^G\!{\cal O}^T}
\end{array}
\!\!\!\right\}_{jl}
\!=
G_{+ \mu} (i \partial_+)^{l-1}\!
\left\{\!\!\!
\begin{array}{c}
g_{\mu\nu} \\
\epsilon_{\mu\nu+-} \\
0
\end{array}
\!\!\!\right\}
\!C^{\frac{5}{2}}_{j - 1}\!
\left( \frac{\stackrel{\leftrightarrow}{D}_+}{\partial_+} \right)
\!G_{+\nu}.$$ Here $\partial \!= \stackrel{\rightarrow}{\partial}\!\!
+\!\! \stackrel{\leftarrow}{\partial}$ and $\stackrel{\leftrightarrow}{D}
= \stackrel{\rightarrow}{\partial}\!\! -\!\! \stackrel{\leftarrow}{\partial}
\!\!- 2i{\rm g}T^aB^a$ and we suppress the flavour and colour indices for simplicity. The operators we have introduced above mix under the renormalization and their anomalous dimension matrix $$\begin{aligned}
\hat{\bf \gamma}=
\left({
{^{QQ}\!\hat{\gamma}}\ {^{QG}\!\hat{\gamma}}\atop
{^{GQ}\!\hat{\gamma}}\ {^{GG}\!\hat{\gamma}}
}\right),\end{aligned}$$ possesses the square blocks ${^{ik}\!\hat{\gamma}}$ which are triangular matrices. Since the special conformal anomaly does not enter in the conformal constraint (\[conf-constr-KD-1\]) at LO, the latter takes the simplified form: $$\begin{aligned}
\label{conf-constr-KD-Sing}
\left[\left({\hat{a}(l)\quad 0\atop\;\;\; 0\quad \hat{a}(l)}\right),
\left({
{^{QQ}\!\hat{\gamma}}\ {^{QG}\!\hat{\gamma}}\atop
{^{GQ}\!\hat{\gamma}}\ {^{GG}\!\hat{\gamma}}
}\right)
\right]=0.\end{aligned}$$ The matrix $\hat{a}(l)$ is diagonal, then it follows from the above equality that all entries in $\hat{\bf \gamma}$ are diagonal too and coincide with the moments of the DGLAP evolution kernels[^3]. Therefore, we can easily write the solution of the renormalization group equation (\[MOMEE\]) in terms of the conformal moments, for instance, for the quark non-forward distribution in the flavour non-singlet channel: $$\label{solution}
{^{\rm NS}\!{\cal O}^{\mit\Gamma}} (x, \zeta)
= \sum_{j = 0}^{\infty} \frac{1}{N_j (\frac{3}{2}) \zeta^{j + 1}}
\frac{x}{\zeta} \left( 1 - \frac{x}{\zeta} \right)
C_j^{\frac{3}{2}} \left( 2 \frac{x}{\zeta} - 1 \right)
\left( \frac{\alpha_s(Q)}{\alpha_s(Q_0)} \right)^{-
{^{\rm NS}\gamma^{\mit\Gamma}_j}/\beta_0}
\left. \langle h' |
{^{\rm NS}\!{\cal O}^{\mit\Gamma}}_{jj}
| h \rangle \right|_{Q_0^2},$$ with $\beta_0 = \frac{2}{3}N_f - \frac{11}{3}C_A$ and the normalization factor $$\label{normalization}
N_j (\nu) = 2^{ - 4 \nu + 1 }
\frac{ \Gamma^2 (\frac{1}{2}) \Gamma ( 2 \nu + j )}
{( \nu + j ) \Gamma^2 (\nu) \Gamma (j + 1)}.$$ The quark–gluon mixing problem in the singlet channel can be easily solved by straightforward diagonalization in the same way as in the case of the deep inelastic scattering (DIS). Note that Eq.(\[solution\]) is valid only for the distributions with a support $0\le x\le \zeta$ [@RadPRD56].

NLO coefficient functions for DVCS.
===================================

In the previous section we have dealt so far with the LO analysis and have seen that the tree conformal operators (\[treeCO\]) diagonalize the anomalous dimension matrix. As we have mentioned in the Introduction, this property breaks down in the NLO in the ${\overline{\rm MS}}$ scheme even in the theory with $\beta = 0$. However, the conformal constraints (\[conf-constr-KD-1\]) and (\[conf-constr-KD-2\]) ensure the existence of the CS scheme, in which the conformal covariance is restored in the conformal limit of the theory to any order of perturbation theory. In this scheme the OPE in terms of the conformally covariant operators [@FGG73] holds true in the interacting theory and provides powerful restrictions on the corresponding Wilson coefficients and can be served, for instance, as a tool for the calculation of the scattering amplitude of two-photon processes in the generalized Bjorken region. To compare the predictions with the calculations performed in the ${\overline{\rm MS}}$ scheme, the following transformation for the operators is necessary $$\begin{aligned}
\label{transformation}
{\cal O}_{jl}=\sum_{j'=0}^{j} B_{jj'} \widetilde {\cal O}_{j'l}.\end{aligned}$$ In the CS scheme the operators $\widetilde O_{jl}$ in different conformal towers do not mix with each other up to terms proportional to $\beta$. The transformation matrix $B$, determined completely by the special conformal anomaly: $$\begin{aligned}
\label{blefdt-1}
\hat{B} = {\hat{1} \over \hat{1}+{\cal J}\hat{\gamma}^c}
= \hat{1}-{\cal J}\hat{\gamma}^c + {\cal J}(\hat{\gamma}^c
{\cal J}\hat{\gamma}^c) - \dots,\end{aligned}$$ with ${\cal J}\hat{A} = \theta_{jj' + 1} A_{jj'} / a( j, j' )$, specifies the partial conformal eigenfunctions of the evolution kernels. This matrix was calculated in Ref. [@Mue94] for the non-singlet parity-odd channel and used for a consistency check between the available NLO result for the coefficient function of the $\gamma^*\gamma \to \pi$ transition form factor and the forward NLO Wilson coefficients for polarized DIS [@Mue97].

Recently, the $\alpha_s$-correction to the non-forward coefficient function in the singlet channel has been calculated for unpolarized DVCS [@JiOs97]. In this section we evaluate the one-loop coefficient functions for the polarized $(T^A_{(1)})$ and non-polarized $(T^V_{(1)})$ cases as well: $$\begin{aligned}
{^i T}^{\mit\Gamma}( \omega, x, \zeta, Q^2 | \alpha_s )
= {^i T_{(0)}} ( \omega, x ) - \frac{\alpha_s}{2\pi}
{^i T_{(1)}^{\mit\Gamma}}( \omega, x, \zeta, Q^2 )
+ {\cal O} (\alpha_s^2).\end{aligned}$$ The predictions for the coefficient function can be obtained by employing the conformal OPE from the one side and the conformal constraints from the other. In NLO the symmetry breaking, which is proportional to the $\beta$-function, arises only in the off-diagonal part (in the basis of Gegenbauer polynomials) of the anomalous dimension matrix $\hat \gamma$, not in the coefficient function. So, we are able to make the predictions for the NLO coefficient functions without restrictions.

To overcome difficulties with gauge invariance and Lorentz decomposition of the hadronic tensor for the DVCS, we deal, in what follows, only with appropriate projections of the latter: $${\cal F}^{\mit\Gamma} ( \omega, \zeta )
=
i {\cal P}^{\mit\Gamma}_{\mu\nu}
\int d^4 z e^{iqz}
\langle h' |
T \left\{ J_\mu (0) J_\nu (z) \right\}
| h \rangle ,$$ where ${\cal P}^V_{\mu\nu} = g_{\mu\nu}$ corresponds to unpolarized scattering and ${\cal P}^{A}_{\mu\nu} = i \epsilon_{\mu\nu Qp}/(Qp)$ to polarized case. Here $1/\omega = -Q^2/2(Qp)$ and $Q = q - \Delta$.

On the one hand the amplitude for DVCS can be represented in the factorized form as convolution of the non-forward distribution (\[definition\]) and the perturbative coefficient function $$\label{DVCS}
{\cal F}^{\mit\Gamma} ( \omega, \zeta ) = \int d x
\sum_{i=Q,G} {^i T^{\mit\Gamma}}( \omega, x, \zeta, Q^2 | \alpha_s )
{^i {\cal O}^{\mit\Gamma}} ( x, \zeta ).$$ For the hand-bag diagram[^4] we have to LO in the coupling ${^Q T_{(0)}} ( \omega, x ) = \omega /
( x \omega - 1 )$.

On the other hand the conformal OPE [@FGG73; @Mue97] for the product of two electromagnetic currents sandwiched between the hadronic states, in the kinematics appropriate to the process in question, gives the following prediction: $$\label{COPE}
{\cal F}^{\mit\Gamma} ( \omega, \zeta ) =
\sum_{i} \sum_{j = 0}^{\infty}
{^iC_j^{\mit\Gamma}} (\alpha_s,Q^2/\mu^2) \
\omega^{j + 1}
{_2F_1} \left( \left.
{
1 + j + {^i\gamma^{\mit\Gamma}_j},\
2 + j + {^i\gamma^{\mit\Gamma}_j}
\atop
4 + 2j + 2\, {^i\gamma^{\mit\Gamma}_j}
}
\right| \omega \zeta \right)
\langle h' | {^i \widetilde {\cal O}}^{\mit\Gamma}_{jj} | h \rangle .$$ Here ${_2F_1}$ is confluent hypergeometric function [@BE53_1], the index $i$ sums up the eigenvectors ${^i\widetilde {\cal O}^{\mit\Gamma}}_{jj}$ of the singlet channel, which possess the diagonal anomalous dimensions ${^i\gamma^{\mit\Gamma}_j}$, and ${^iC_j^{\mit\Gamma}}$ is the Wilson coefficient of the forward scattering. We have the convention that in the interacting theory the scale dimension is shifted by the amount $2{^i\gamma^{\mit\Gamma}_j}$ compared to canonical one: $d_j = d_j^{\rm can} + 2{^i\gamma^{\mit\Gamma}_j}$. In Eq. (\[COPE\]) $\mu^2$ is a renormalization scale, while we have put the factorization scale $\mu^2_F$ equal to $Q^2$. Notice that for practical purposes ${\overline{\rm MS}}$ schemes will be used where the manifestly conformaly covariant form of the OPE (\[COPE\]) is hidden. Fortunately, the product of two electromagnetic currents is a renormalization group invariant, so that Eq. (\[DVCS\]) computed in any scheme is equivalent to Eq. (\[COPE\]) derived in the CS scheme.

Since the $\alpha_s$-corrections in Eq. (\[DVCS\]) come from the NLO result for the coefficient function and eigenfunctions of the evolution kernels we are able to predict the former from the combined use of the conformal OPE (\[COPE\]) and the special conformal anomaly which fixes precisely the corrections to the latter. The general conformal decomposition of the non-forward distributions looks like $${^i{\cal O}} (x, \zeta)
= \sum_{k = Q,G} \sum_{j=0}^{\infty}
{^{ik}\phi}_j (x, \zeta | \alpha_s)
\langle h' | {^k{\cal O}}_{jj} | h \rangle ,$$ where the partial conformal waves are generalized, beyond one-loop level, to non-polynomial functions which are the subject of the constraints. For the quark case, which is required for the present application, they have the form $${^{Qk}\phi}_j (x, \zeta | \alpha_s)
= \left\{
\delta_{Qk} \delta (x - y)
+ \frac{\alpha_s}{2\pi} {^{Qk}{\mit\Phi}} (x, y, \zeta)
\right\}
\otimes
\frac{1}{N_j (\nu) \zeta^{j + 1}}
\frac{y}{\zeta} \left( 1 - \frac{y}{\zeta} \right)
C_j^{\nu} \left( 2 \frac{y}{\zeta} - 1 \right),$$ where $\nu = \nu (k)$ and we introduced the shorthand notation $\otimes \equiv \int dy$. Here ${^{QQ}{\mit\Phi}}$ is known from Ref. [@Mue94], while ${^{QG}{\mit\Phi}}$ can be found with the help of the special conformal anomaly matrix in the quark-gluon channel. Fortunately, in this sector no subtle conformal symmetry breaking effects occur as it happens in the quark-quark transition amplitude. Thus, the special conformal anomaly can be obtained immediately and reads: $$\label{QGCAM}
{^{QG}\hat\gamma^{(c)}}
= - \frac{\alpha_s}{2 \pi} \hat{b}(l) \ {^{QG}\hat\gamma}.$$

Now we have all necessary results to solve the problem in question. First we outline the procedure for the evaluation of the coefficient function for the polarized non-forward gluon contribution and then we just list the results for other cases since the machinery we have developed is quite general and can be easily extended to the quark coefficient functions as well.

Polarized case.
---------------

For the $QG$-sector we find from Eq. (\[QGCAM\]) that the correction to the eigenfunctions are completely expressed in terms of the shift operator: $$\label{Phi}
{^{QG}{\mit\Phi}} (x, y, \zeta)
= ({\cal I} - {\cal D}) S (x, z) \otimes {^{QG}\!K^A} ( z, y, \zeta ),$$ which generates the shift of the Gegenbauer polynomials index: $$S(x, y) \otimes [y \bar y]^{\nu - \frac{1}{2}}
C^\nu_j (2y - 1)
= \left. \frac{d}{d\rho} \right|_{\rho = 0}
[x \bar x]^{\nu - \frac{1}{2} + \rho} C^{\nu + \rho}_j (2x - 1).$$ In Eq. (\[Phi\]) ${\cal I}$ is an identity operator and ${\cal D}$ extracts the diagonal part in the expansion of any test function $\tau (x, y)$ with respect to $C^\nu_j$, i.e. $$\begin{aligned}
{\cal D}\tau (x, y) = \sum_{ j= 0 }^{\infty}
[x \bar x]^{ \nu - \frac{1}{2}}/ N_j (\nu) C^\nu_j(2x - 1)
\tau_{jj} C^\nu_j (2y - 1).\end{aligned}$$ The generalized evolution kernel[^5] $${^{QG}\!K^A} (x , x ', \zeta)
= 2 N_f T_F \Theta_{112}^1 ( x , x - \zeta, x - x ' )$$ can be diagonalized with the following identity $$\label{QGmixing}
\zeta
\int dy C_{j}^{\frac{3}{2}} \left( 2 \frac{y}{\zeta} - 1 \right)
{^{QG}\!K^A} ( y, x, \zeta )
= {^{QG}\gamma^A_j}
C_{j - 1}^{\frac{5}{2}}
\left( 2 \frac{x}{\zeta} - 1 \right) \ \mbox{with} \
{^{QG}\gamma^A_j} = - \frac{12 N_f T_F}{(j + 1)(j + 2)}.$$ The coefficient ${^{QG}\gamma^A_j}$ in front of the Gegenbauer polynomial coincides with the DGLAP moments of the kernel ${^{QG}\!K^A} ( y, x, 0 )$ up to the common factor $6/j$, which arises as a result of the conventional definition of the Gegenbauer polynomials, namely, the coefficient of $x^j$ in $C_{j}^{\frac{3}{2}}
(x)$ is $3/j$ times the coefficient of $x^{j - 1}$ in $C_{j - 1}^{\frac{5}{2}} (x)$; an additional factor of 2 comes from the argument of the polynomial.

For the off-diagonal part of the amplitude we get from the conformal expansion (\[COPE\]) the following expression $$\begin{aligned}
\label{nondiagonal}
&&({\cal I} - {\cal D})
\left[
{^Q T_{(0)}} ( \omega, y ) \otimes S ( y, z )
-
{^Q T_{(0)}} ( \omega, z ) \ln ( 1 - z \omega )
\right]
\otimes
{^{QG}\!K^A} ( z, x, \zeta )
\otimes
{^G {\cal O}^A} ( x, \zeta ) \\
&& \hspace{0.5cm} = - 2 \sum_{j = 0}^{\infty}
{^{QG}\gamma^A_j}
B( j + 1 , j + 2 ) \omega^{j + 1}
\left. \frac{d}{d\rho} \right|_{\rho = 0}
{_2F_1} \left( \left.
{
1 + j + \rho,\
2 + j + \rho
\atop
4 + 2j + 2 \rho
}
\right| \omega \zeta \right)
\langle h' | {^G{\cal O}^A}_{jj} | h \rangle . \nonumber\end{aligned}$$ Its normalization is related to the LO prediction of the conformal OPE for the DVCS scattering: $${\cal F}^A_{LO} ( \omega, \zeta )
= - 2
\sum_{j = 0}^{\infty}
B( j + 1 , j + 2 )
\omega^{j + 1}
{_2F_1} \left( \left.
{
1 + j ,\
2 + j
\atop
4 + 2j
}
\right| \omega \zeta \right)
\langle h' | {^Q{\cal O}^A}_{jj} | h \rangle ,$$

To fix the diagonal part of the hard scattering amplitude we have to put ${^i\gamma^{\mit\Gamma}_j} = 0$ in Eq. (\[COPE\]), but keep the NLO result for the forward coefficient function of the polarized gluon distribution $\frac{\alpha_s}{2\pi}{^GC^A_j} (Q^2/\mu^2)$ [@PGCF; @AR88]. We can write the result as a convolution $$\begin{aligned}
\label{diagonal}
&&{^Q T_{(0)}} ( \omega, z ) \otimes F ( z, x, \zeta )
\otimes {^G {\cal O}^A} ( x, \zeta ) \\
&& \hspace{0.5cm} = - 2 \sum_{j = 0}^{\infty}
{^GC^A_j} (Q^2/\mu^2)
B( j + 1 , j + 2 ) \omega^{j + 1}
{_2F_1} \left( \left.
{
1 + j ,\
2 + j
\atop
4 + 2j
}
\right| \omega \zeta \right)
\langle h' | {^G{\cal O}^A}_{jj} | h \rangle , \nonumber\end{aligned}$$ with the following explicit expression for the function $F$, which corresponds to the DIS coefficient function in the ${\overline{\rm MS}}$ scheme [@AR88]: $$\begin{aligned}
\label{nondiagonal2}
{^GC^A}_j (Q^2/\mu^2)
&=& 2 N_f T_f \frac{6}{j}
\left\{ ( 2x - 1 )\ln \frac{Q^2\bar x}{\mu^2 x} + 3 - 4x \right\}
\otimes x^j \\
&\Rightarrow& F( z, x, \zeta )
= - {^{QG}\!K^A} (z , x , \zeta) \ln \frac{Q^2}{\mu^2}
+ \left[ 1  - {\cal D} \ln (1 - z \omega) \right]
{^{QG}\!K^A} (z , x , \zeta). \nonumber\end{aligned}$$

By adding the results of Eqs. (\[nondiagonal\]) and (\[nondiagonal2\]) together and subtracting the solution (\[Phi\]) of the evolution equation for $\beta = 0$, we finally come to the coefficient function in the ${\overline{\rm MS}}$ scheme: $${^G T_{(1)}^A}( \omega, x, \zeta, Q^2 ) =
{^Q T_{(0)}} ( \omega, y ) \otimes
\biggl\{
{^{QG}\!K^A} ( y, x, \zeta )\, \ln \frac{Q^2}{\mu^2}
+ \left[ \ln \left( 1 - y \omega \right) - 1 \right]
{^{QG}\!K^A} ( y, x, \zeta )
\biggr\}.$$

As an outcome of the same but a little bit more involved analysis for the quark scattering amplitude we get: $$\begin{aligned}
\label{unpolQCF}
{^Q T_{(1)}^A}( \omega, x, \zeta, Q^2 ) \!\!\!&=&\!\!\!
{^Q T_{(0)}} ( \omega, y ) \otimes
\biggl\{
{^{QQ}\!K^A} ( y, x, \zeta )\,
\ln \frac{Q^2}{\mu^2}
- \frac{3}{2} \left[ {^{Q}\!K^b} ( y, x, \zeta ) \right]_+
+ \frac{3}{2} \delta (y - x) \nonumber \\
&+& \ln \left( 1 - y \omega \right)
{^{QQ}\!K^A} ( y, x, \zeta )
- \left[ {^{Q}\!G} ( y, x, \zeta ) \right]_+
\biggr\} .\end{aligned}$$ Here $$\begin{aligned}
{^{QQ}\!K^A} (x , x ', \zeta)
\!\!\!&=&\!\!\! K^T (x , x ', \zeta)
+ C_F \, \Theta_{111}^0 ( x , x - \zeta, x - x ' ), \\
{^{Q}\!G} ( x, x', \zeta )
\!\!\!&=&\!\!\! \frac{x' - \zeta}{x - x'}
\ln \left( \frac{x' - x}{x' - \zeta} \right)
\Theta_{11}^0 (x - \zeta, x - x ')
+
\frac{x'}{x - x'}
\ln \left( \frac{x' - x}{x'} \right)
\Theta_{11}^0 (x, x - x '), \\
{^{Q}\!K^b} ( x, x', \zeta )
\!\!\!&=&\!\!\! \frac{x - \zeta}{x - x'}
\Theta_{11}^0 (x - \zeta, x - x ')
+
\frac{x}{x - x'}
\Theta_{11}^0 (x, x - x '),\end{aligned}$$ and a new kernel ${^{Q}\!G} ( x, x', \zeta )$ comes from the expression for ${^{QQ}{\mit\Phi}}$, which is of the form for $\beta = 0$: $${^{QQ}{\mit\Phi}} (x, y, \zeta)
= ({\cal I} - {\cal D})
\left\{
S (x, z) \otimes {^{QQ}\!K^A} ( z, y, \zeta )
- \left[ {^{Q}\!G} ( y, x, \zeta ) \right]_+
\right\}.$$ In the limit $\omega = 1$ and $\zeta = 1$ we obtain from Eq. (\[unpolQCF\]) the results of Refs. [@Cha81; @Bra83; @Rad86] for the pion transition form factor $F_{\pi\gamma\gamma}$.

Unpolarized case.
-----------------

Since the $QQ$-evolution kernels are the same in the polarized and non-polarized cases, the only difference will arise from the forward coefficient functions [@ZVN94]. Straightforward calculation gives for the quark as well as for gluon non-polarized coefficient functions: $$\begin{aligned}
{^Q T_{(1)}^V}( \omega, x, \zeta, Q^2 ) \!\!\!&=&\!\!\!
{^Q T_{(0)}} ( \omega, y ) \otimes
\biggl\{
{^{QQ}\!K^V} ( y, x, \zeta )\,
\ln \frac{Q^2}{\mu^2}
- \frac{5}{2} \left[ {^{Q}\!K^b} ( y, x, \zeta ) \right]_+
+ \delta (y - x) \nonumber \\
&+& \left[
\ln \left( 1 - y \omega \right) - 1
\right]
{^{QQ}\!K^V} ( y, x, \zeta )
- \left[ {^{Q}\!G} ( y, x, \zeta ) \right]_+
\biggr\} , \\
{^G T_{(1)}^V}( \omega, x, \zeta, Q^2 ) \!\!\!&=&\!\!\!
{^Q T_{(0)}} ( \omega, y ) \otimes
\biggl\{
{^{QG}\!K^V} ( y, x, \zeta )\,
\ln \frac{Q^2}{\mu^2}
- \frac{1}{2}
\left[
{^{QG}\!K^V} ( y, x, \zeta ) + {^{QG}\!K^A} ( y, x, \zeta )
\right]
\nonumber \\
&+& \ln \left( 1 - y \omega \right) {^{QG}\!K^V} ( y, x, \zeta )
\biggr\} ,\end{aligned}$$ the momentum space evolution kernels ${^{Q(Q,G)}\!K^V}$ read $$\begin{aligned}
{^{QQ}\!K^V} (x , x ', \zeta) &=& {^{QQ}\!K^A} (x , x ', \zeta),\\
{^{QG}\!K^V} (x , x ', \zeta)
&=& {^{QG}\!K^A} (x , x ', \zeta)
- 4 N_f T_F \frac{( x - x' )}{x'( x' - \zeta )}
\Theta_{111}^0 ( x , x - \zeta, x - x ' ). \nonumber\end{aligned}$$ These expressions coincide with the ones obtained in Ref. [@JiOs97] provided we have made some obvious redefinition of their conventions. Taking the limit intrinsic to the exclusive kinematics we obtain the well know result [@Cha81; @Bra83] for the radiative corrections to the coefficient functions of the parity-even vector meson transition form factor.

Renormalon chains in the non-singlet evolution kernel.
======================================================

The last issue of this paper that can be studied by the presently developed methods is the renormalon chains resummation of the ER-BL evolution kernel. Recently, this problem has been considered in the ${\overline{\rm MS}}$ scheme for the non-singlet channel [@Mik97; @GosKiv97] (see also the work for the DGLAP splitting kernels [@Gra97; @Mankiewicz97]). It has been found that the resulting series possesses a nonzero convergent radius, so that the infrared renormalons are absent in the kernels. From our point of view this result can be achieved by a partial resummation of conformal anomalies. The $\hat{B}$-matrix defined in Eq. (\[blefdt-1\]) diagonalizes the anomalous dimension matrix for the case $\beta=0$. From the conformal constraints (\[conf-constr-KD-full\]) we know that in the ${\overline{\rm MS}}$ scheme its off-diagonal part is induced by $\hat{\gamma}^c + 2\frac{\beta}{g}\hat{b}$. Therefore, the operator $$\begin{aligned}
\hat{B} =
{\hat{1} \over \hat{1} + {\cal J}
\left(\hat{\gamma}^c + 2 \frac{\beta}{g}\hat{b}\right)}\end{aligned}$$ diagonalize the anomalous dimension matrix[^6]. This formula can be understood as a resummation prescription for the conformal anomalies that induce the off-diagonal part. Knowing this matrix one can construct the eigenfunctions $\phi_l (x | \alpha_s) \equiv {^{QQ}\phi}_l (x, 1 | \alpha_s)$ of the ER-BL kernel [@Mue94] $$\phi_l(x | \alpha_s)=
\sum_{j=l}^\infty \frac{ x \bar x }{N_j(\frac{3}{2})}
C_j^{\frac{3}{2}}( 2 x - 1 ) B_{jl}(\alpha_s).$$ In Ref. [@Mue94] a differential equation for the eigenfunctions was derived in the conformal limit. Repeating the steps, it is easy to restore the neglected $\beta$ term. Employing the relation (see Appendix in [@Mue97]) $$\begin{aligned}
a(j, l) B_{jl}(\alpha_s)
= - \left\{\left(\hat{\gamma}^c(l)
+ 2 \frac{\beta}{g}\hat{b}(l)\right)\hat{B}\right\}_{jl}\end{aligned}$$ and the eigenvalue equation for the Gegenbauer polynomials [@BE53_2] we are able to derive an integro-differential equation for the eigenfunctions in the full theory, valid for the ${\overline{\rm MS}}$ scheme: $$\label{MSequation}
x \bar x \frac{d^2}{dx^2} \phi_l(x | \alpha_s)
+ ( l + 1 )( l + 2 ) \phi_l(x | \alpha_s)=
\frac{1}{2} \int_0^1 dy\,
\left(
\gamma_l^c(x,y,\alpha_s) + 2 \frac{\beta}{g} b_l(x,y)
\right)
\phi_l(y | \alpha_s),$$ where the kernels can be obtained from their non-forward analogues introduced in the preceding sections and read $$\begin{aligned}
\gamma_l^c(x,y,\alpha_s) &=& \frac{\alpha_s}{2\pi}
\left\{
2 \int_0^1 dz\, b_l(x,z) \left[ {^{QQ}V}(z,y) \right]_+
+ [g(x,y)]_+
\right\} +O(\alpha_s^2), \\
b_l(x,y)&=&  2 \left( l + 1 + \frac{ 2x - 1}{2} \frac{d}{dx} \right)
\delta( x - y ), \\
g(x,y)&=& C_F \theta( y - x ) \frac{\ln(1 - \frac{x}{y})}{ y - x }
+ \left\{{ x \to \bar x \atop y \to \bar y }\right\}, \\
{^{QQ}V} (x,y)&=&
C_F \theta( y - x )
\frac{x}{y} \left( 1 + \frac{1}{ y - x } \right)
+ \left\{ {x \to \bar x \atop y \to \bar y } \right\}.\end{aligned}$$ The non-perturbative solution of Eq. (\[MSequation\]) is unknown. However, in the large $N_f$ limit only the term proportional to $\beta$ survives. Taking into account also the non-leading $N_f$ terms included in the $\beta$-function, the equation (\[MSequation\]) can be reduced to $$\begin{aligned}
&&\hspace{-1cm}x \bar x \frac{d^2}{dx^2}\
\phi_l(x | \alpha_s)
+ ( l + 1 )( l + 2 ) \phi_l(x | \alpha_s) \\
&&\hspace{3cm} = \frac{\beta}{g} \int_0^1 dy\,
\left\{
2 \left( l + 1 + \frac{ 2 x - 1 }{2} \frac{d}{dx} \right)\delta(x-y) +
O(\alpha_s)
\right\}\
\phi_l (y | \alpha_s).
\nonumber\end{aligned}$$ Taking the known LO contribution on the right hand side yields a differential equation which has the solution: $$\begin{aligned}
\label{eigfunc-beta}
\phi_j(x | \alpha_s)
\sim
\left[ x \bar x \right]^{\frac{3}{2} - \frac{\beta}{g}}
C_j^{\frac{3}{2}-\frac{\beta}{g}}( 2 x - 1 ).\end{aligned}$$ This result coincide in the large $N_f$ limit with the resummation of the vacuum polarization bubbles in Refs. [@Mik97; @GosKiv97] and verifies the ‘naive non-abelianization’ hypothesis employed there [@GosKiv97]. Moreover, it is very interesting to see that this hypothesis can be extended in our case to the [*whole*]{} $\beta$-function. Unfortunately, the eigenfunctions (\[eigfunc-beta\]) have no practical importance for the solution of the evolution equation. Obviously, in this treatment the special anomaly matrix contribution is completely discarded, however, it was demonstrated by numerical studies in Ref. [@Mue95] that a considerable cancellation occurs between the former and the symmetry breaking term due to the $\beta$-function for the real case $N_f=3$.

Summary.
========

In the present study we employ the powerful restrictions from the conformal algebra on the amplitudes in the massless theory. In the leading order of perturbation theory it enables us to give the solution of the evolution equations for the off-forward distribution functions in terms of the conformal partial wave expansion. Beyond the leading order the conformal symmetry breaking term induced by nonvanishing $\beta$-function does not appear in the coefficient function so that we can rely on the hypothetical conformal limit of the theory in order to make the predictions for the two-photon processes in the generalized Bjorken kinematics. To this end the knowledge of the forward Wilson coefficients in the OPE and the diagonal anomalous dimensions of the corresponding operators which were determined earlier from the study of deep inelastic scattering is required .

The development of the present method is far from being finished since up to now all the results were derived in the theory with $U(1)$ symmetry group. The extension of the conformal constraints to purely Yang-Mills theory [@BelMul98] will allow to get much deeper insight into the conformal symmetry breaking by the non-abelian fields.

We wish to thank B. Geyer, S.V. Mikhailov, D. Robaschik, A. Schäfer for valuable discussions. D.M. would like to thank the Theory Group at the Joint Institute for Nuclear Research for the kind hospitality during his stay, where the initial part of this work has been done. A.B. was supported by the Russian Foundation for Fundamental Research, grant N 96-02-17631 and Deutsche Physikalische Gesellschaft. D.M. was financially supported by the Deutsche Forschungsgemeinschaft (DFG).

[99]{} D. Müller, Z. Phys. C 49 (1991) 293. D. Müller, Phys. Rev. D 49 (1994) 2525. A.V. Belitsky, D. Müller, [*Broken conformal invariance in Yang-Mills theory and NLO gluonic anomalous dimensions*]{}, (in preparation). D. Müller, D. Robaschik, B. Geyer, F. M. Dittes, J. Hořejši, Fortschr. Physik, 42 (1994) 101. X. Ji, Phys. Rev. Lett. 78 (1997) 610; Phys. Rev. D 55 (1997) 7114. A.V. Radyushkin, Phys. Lett. B 380 (1996) 417; Phys. Lett. B 385 (1996) 333;\
I.I. Balitsky, A.V. Radyushkin, [*Light-ray evolution equations and leading-twist parton helicity-dependent nonforward distributions*]{}, hep-ph/9706410. A.V. Radyushkin, Phys. Rev. D 56 (1997) (in print), hep-ph/9704207. T. Braunschweig, B. Geyer, D. Robaschik, Ann. Phys. (Leipzig) 44 (1987) 407;\
J. Blümlein, B. Geyer, D. Robaschik, Phys. Lett. B 406 (1997) 161. I.I. Balitsky, V.M. Braun, Nucl. Phys. B 311 (1988/89) 541. J.C. Collins, L.L. Frankfurt, M. Strikman, Phys. Rev. D 56 (1997) 2982. A.V. Belitsky, D. Müller, Nucl. Phys. B 503 (1997) 279, hep-ph/9702354;\
A.V. Belitsky, [*Leading order analysis of the twist-3 space-like and time-like cut vertices in QCD*]{}, XXXI PNPI Winter School (1997), hep-ph/9703432. H. Bateman, A. Erdélyi, [*Higher transcendental functions*]{}, V. 2, Mc Graw-Hill (1953). M.K. Chase, Nucl. Phys. B 174 (1980) 109;\
Th. Ohrndorf, Nucl. Phys. B 186 (1981) 153;\
M.A. Shifman, M.I. Vysotsky, Nucl. Phys. B 186 (1981) 475. For review, see S. Ferrara, R. Gatto, A.F. Grillo, [*Conformal algebra in space-time and operator product expansion*]{}, Springer tracts in modern physics, v. 67 (1973). D. Müller, [*Restricted conformal invariance in QCD and its predictive power for virtual two-photon processes*]{}, hep-ph/9704406. X. Ji, J. Osborne, [*One-loop QCD corrections to the deeply-virtual Compton scattering: the parton helicity-independent case*]{}, hep-ph/9707254. H. Bateman, A. Erdélyi, [*Higher transcendental functions*]{}, V. 1, Mc Graw-Hill (1953). J. Kodaira, Nucl. Phys. B 165 (1980) 129;\
R.D. Carlitz, J.C. Collins, A.H. Mueller, Phys. Lett. B 214 (1988) 229. G. Altarelli, G.G. Ross, Phys. Lett. B 212 (1988) 391;\
G.T. Bodwin, J. Qiu, Phys. Rev. D 41 (1990) 2755;\
A.V. Manohar, Phys. Rev. Lett. 66 (1991) 289. F. Del Aguila, M.K. Chase, Nucl. Phys. B 193 (1981) 517. E. Braaten, Phys. Rev. D 28 (1983) 524. E.P. Kadantseva, S.V. Mikhailov, A.V. Radyushkin, Sov. J. Nucl. Phys. 44 (1986) 326. E.B. Zijlstra, W.L. van Neerven, Nucl. Phys. B 383 (1992) 525; Nucl. Phys. B 417 (1994) 61; [*erratum*]{} Nucl. Phys. B 426 (1994) 245. S.V. Mikhailov, [*Renormalon chains contributions to the nonsinglet evolution kernels in $[\varphi^3]_6$ and QCD*]{}, hep-ph/9706326. P. Gosdzinsky, N. Kivel, [*Ressumation of $(-b_0\alpha_s)^n$ corrections to the photon-meson transition form-factor $\gamma^*
\gamma \to \pi^0$* ]{}, hep-ph/9707367. J.A. Gracey, Phys. Lett. B 322 (1994) 141; Nucl. Phys. B 480 (1996) 73. L. Mankiewicz, M. Maul, E. Stein, Phys. Lett. B 404 (1997) 345. D. Müller, Phys. Rev. D 51 (1995) 3855.

[^1]: We use the common definition of the Kronecker symbol $\delta_{jj'}$ and the following definition of the (discrete) step-function: $\theta_{jj'} =
    \left\{ \begin{array}{c} 1, \ j-j' \geq 0 \\ 0, \ j-j'<0 \end{array}
    \right.$.

[^2]: The explicit expressions as well as integral and algebraic properties of the $\Theta$-functions introduced in the main text: $\Theta^{m}_{i_1 i_2 ... i_n}
    (x_1,x_2,...,x_n)=\int_{-\infty}^{\infty}\frac{d\alpha}{2\pi i}
    \alpha^m \prod_{k=1}^{n}\left(\alpha x_k -1 +i0 \right)^{-i_k}$ can be found in Ref. [@BM97].

[^3]: For the $QG$ and $GQ$ sectors this is true up to some common factors (see Ref. [@Cha80] and Eq.(\[QGmixing\]) below).

[^4]: We omit everywhere the crossed $u$-channel contribution.

[^5]: These generalized kernels posses the following limits: for ER-BL kinematics $x \leq \zeta $: $K(x , x ',
    \zeta) = - V (x/\zeta , x'/\zeta)/\zeta$, for froward DGLAP evolution $\zeta = 0$: $K(x , x ', 0) = - P(x/x')$.

[^6]: Since the $\hat{B}$-matrix depends on the running coupling, the diagonalization of the anomalous dimension matrix provide the solution of the renormalization group equation only in the fixed coupling regime.
