On evolution kernels of twist-two operators

The study of deeply-virtual Compton scattering (DVCS) gives one access to the generalized parton distributions [1, 2, 3] (GPDs) that encode the information on the transverse position of quarks and gluons in the proton in dependence on their longitudinal momentum. In order to extract the GPDs from experimental data one has to know, among other things, their scale dependence. The latter is governed by the renormalization group equations (RGEs) or, equivalently, evolution equations for the corresponding twist two operators. Essentially the same equations govern the scale dependence of the ordinary parton distribution functions (PDFs) in the Deep Inelastic Scattering (DIS) process. In DIS one is interested in the scale dependence of forward matrix elements of the local twist-2 operators and therefore can neglect the operator mixing problem between local operators from the operator product expansion (OPE). In the nonsinglet sector, there is only one operator for a given spin/dimension. The anomalous dimensions of such operators are known currently with the three-loop accuracy [4, 5] and first results at four loops are becoming available [6, 7]. In contrast, the DVCS process corresponds to non-zero momentum transfer from the initial to the final state and, as a consequence, the total derivatives of the local twist-two operators have to be taken into consideration. All these operators mix under renormalization and the RGE has a matrix form. The DIS anomalous dimensions appear as the diagonal entries of the anomalous dimensions matrix which, in general, has a triangular form for the latter.

It was shown by Dieter Müller [8, 9] that the off-diagonal part of the anomalous dimension matrix is completely determined by a special object, the so-called conformal anomaly. Moreover, in order to determine the off-diagonal part of the anomalous dimension matrix with $\ell$-loop accuracy it is enough to calculate the conformal anomaly at one loop less. This technique was used to reconstruct all relevant evolution kernels/anomalous dimension matrices in QCD at two loops [10, 11, 12].

A similar approach, but based on the analysis of QCD at the critical point in non-integer dimensions, was developed in refs. [13, 14, 15]. It was shown that the evolution kernels in $d=4$ in the $\overline{\mathrm{MS}}$-like renormalization scheme inherit the symmetries of the critical theory in $d=4-2\epsilon$ dimensions. As expected, the symmetry generators deviate from their canonical form. Corrections to the generators have a rather simple form if they are written in terms of the evolution kernel and the conformal anomaly. It was shown in ref. [16] that by changing a renormalization scheme one can get rid of the conformal anomaly term in the generators bringing them into the so-called “minimal” form. Beyond computing the evolution kernels, the conformal approach has also been employed to calculate the NNLO coefficient (hard) functions of vector and axial-vector contributions in DVCS [17, 18], the latter in agreement with a direct Feynman diagram calculation [19]. Moreover, the conformal technique is also applicable to computing kinematic higher-power corrections in two-photon processes as was recently shown in refs. [20, 21].

In this paper we construct a similarity transformation that brings the full quantum generators back to the canonical form. Correspondingly, the transformed evolution kernel is invariant under the canonical $\mathrm{SL}(2,\mathrm{R})$ transformation. Moreover, we will show that the eigenvalues of this kernel are given by the so-called parity respecting anomalous dimension, $f(N)$ [22, 23] which is related to the PDF anomalous dimension spectrum $\gamma(N)$ as

$\displaystyle\gamma(N)=f\left(N+{\bar{\beta}}(a)+\frac{1}{2}\gamma(N)\right),$

(1)

where $\bar{\beta}(a)=-\beta(a)/2a$ with $\beta(a)$ being the QCD beta function. The strong coupling $\alpha_{s}$ is normalized as $a=\alpha_{s}/(4\pi)$. We develop an effective approach to restore the canonically invariant kernel from its eigenvalues $\gamma(N)$. As an example, we present explicit expressions for three-loop invariant kernels in QCD and ${\cal N}=4$ supersymmetric Yang-Mills (SYM) theory. The answers are given by linear combinations of harmonic polylogarithms [24], up to weight four in QCD and up to weight three in ${\cal N}=4$ SYM. We also compare our exact result with the approximate expression for the three-loop kernels in QCD given in ref. [16].

The paper is organized as follows: in section II we describe the general structure of the evolution kernels of twist-two operators. In section III we explain how to effectively recover the evolution kernel from the known anomalous dimensions and present our results for the invariant kernels in QCD and ${\cal N}=4$ SYM. Sect. IV contains the concluding remarks. Some technical details are given in the Appendices.

We are interested in the scale dependence of the twist-two light-ray flavor nonsinglet operator [25]

$\displaystyle\mathcal{O}(z_{1},z_{2})$

$\displaystyle=[\bar{q}(z_{1}n)\gamma_{+}[z_{1}n,z_{2}n]q(z_{2}n)]_{\overline{\text{MS}}},$

(2)

where $n^{\mu}$ is an auxiliary light-like vector, $n^{2}=0$, $z_{1,2}$ are real numbers, $\gamma_{+}=n^{\mu}\gamma_{\mu}$ and $[z_{1}n,z_{2}n]$ stands for the Wilson line ensuring gauge invariance, and the subscript $\overline{\rm MS}$ denotes the renormalization scheme. This operator can be viewed as the generating function for local operators, ${\mathcal{O}}^{\mu_{1}\dots\mu_{N}}$ that are symmetric and traceless in all Lorentz indices $\mu_{1}\dots\mu_{N}$.

The renormalized light-ray operator (2) satisfies the RGE

$\displaystyle\Big{(}\mu\partial_{\mu}+\beta(a)\partial_{a}+\mathbb{H}(a)\Big{)}{\mathcal{O}}(z_{1},z_{2})=0,$

(3)

where $\beta(a)$ is $d$-dimensional beta function

$\displaystyle\beta(a)=-2a\big{(}\epsilon+\beta_{0}a+\beta_{1}a^{2}+O(a^{3})\big{)},$

(4)

$\beta_{0}=11/3N_{c}-2/3n_{f}$, etc., and $\mathbb{H}(a)=a\mathbb{H}_{1}+a^{2}\mathbb{H}_{2}+\ldots$ is an integral operators in $z_{1},z_{2}$.

It follows from the invariance of the classical QCD Lagrangian under conformal transformations that the one-loop kernel $\mathbb{H}_{1}$ commutes with the canonical generators of the collinear conformal subgroup, $S_{0},S_{\pm}$,

	$\displaystyle S_{-}$	$\displaystyle=-\partial_{z_{1}}-\partial_{z_{2}}\,,$
	$\displaystyle S_{0}$	$\displaystyle=z_{1}\partial_{z_{1}}+z_{2}\partial_{z_{2}}+2\,,$
	$\displaystyle S_{+}$	$\displaystyle=z_{1}^{2}\partial_{z_{1}}+z_{2}^{2}\partial_{z_{2}}+2z_{1}+2z_{2}\,.$		(5)

This symmetry is preserved beyond one loop albeit two of the generators, $S_{0},S_{+}$ receive quantum corrections, $S_{\alpha}\mapsto\widetilde{S}_{\alpha}(a)=S_{\alpha}+\Delta S_{\alpha}(a)$. The explicit form of these corrections can be found in ref. [15].

It is quite useful to bring the generators to the following form using the similarity transformation [16],

	$\displaystyle\mathbb{H}(a)=e^{-X(a)}\mathrm{H}(a)e^{X(a)}\,,$
	$\displaystyle\widetilde{S}_{\alpha}(a)=e^{-X(a)}\mathrm{S}_{\alpha}(a)e^{X(a)}\,,$		(6)

where $X(a)=aX_{1}+a^{2}X_{2}+\ldots$ is an integral operator known up to terms of $O(a^{3})$ [11, 16]. This transformation can be thought of as a change in a renormalization scheme.

The shift operator $\mathrm{S}_{-}$ is not modified and hence identical to $S_{-}$ in Eq. (II), and the quantum corrections to $\rm S_{0}$ and $\rm S_{+}$ come only through the evolution kernel

	$\displaystyle\mathrm{S}_{0}(a)$	$\displaystyle=S_{0}+\bar{\beta}(a)+\frac{1}{2}\mathrm{H}(a)\,,$		(7a)
	$\displaystyle\mathrm{S}_{+}(a)$	$\displaystyle=S_{+}+(z_{1}+z_{2})\left(\bar{\beta}(a)+\frac{1}{2}\mathrm{H}(a)\right)\,,$		(7b)

where $\bar{\beta}(a)=\beta_{0}a+\beta_{1}a^{2}+\cdots$ is the beta function in four dimensions, cf. Eq. (1). The form of the generator $\mathrm{S}_{0}(a)$ is completely fixed by the scale invariance of the theory, while Eq. (7b) is the “minimal” ansatz consistent with the commutation relation $[\mathrm{S}_{+},\mathrm{S}_{-}]=2\mathrm{S}_{0}$. Since the operator $\mathrm{H}(a)$ commutes with the generators, $[\mathrm{H}(a),\mathrm{S}_{\alpha}(a)]=0$ its form is completely determined by its spectrum (anomalous dimensions). However, since the generators do not have the simple form as in Eq. (II), it is yet necessary to find a way to recover the operator from its spectrum.

To this end we construct a transformation which brings the generators $\mathrm{S}_{\alpha}(a)$ to the canonical form $S_{\alpha}$, Eq. (II). Let us define an operator $\mathrm{T}(\mathrm{H})$:

$\displaystyle\mathrm{T}(\mathrm{H})=\sum_{n=0}^{\infty}\frac{1}{n!}\mathrm{L}^{n}\left(\bar{\beta}(a)+\frac{1}{2}\mathrm{H}(a)\right)^{n}\,,$

(8)

where $\mathrm{L}=\ln z_{12}$, $z_{12}\equiv z_{1}-z_{2}$. Recall that $z_{1},z_{2}$ are real variables, so for $z_{12}<0$ it is necessary to choose a specific branch of the logarithm function. Although this choice is irrelevant for further analysis we chose the $+i0$ recipe for concreteness, i.e., $\mathrm{L}=\ln(z_{12}+i0)$. It can be shown that the operator $\mathrm{T}(\mathrm{H})$ intertwines the symmetry generators $\mathrm{S}_{\alpha}(a)$ and the canonical generators, $S_{\alpha}$. Namely,

$\displaystyle\mathrm{T}(\mathrm{H})\,\mathrm{S}_{\alpha}(a)=S_{\alpha}\,\mathrm{T}(\mathrm{H}),$

(9)

see Appendix A for details. Let us also define a new kernel $\widehat{\mathrm{H}}$ as

$\displaystyle\mathrm{T}(\mathrm{H})\,\mathrm{H}(a)=\widehat{\mathrm{H}}(a)\,\mathrm{T}(\mathrm{H}).$

(10)

It follows from Eqs. (9), (10) that the operator $\widehat{\mathrm{H}}$ commutes with the canonical generators in Eq. (II)

$\displaystyle[S_{\alpha},\widehat{\mathrm{H}}(a)]=0.$

(11)

The problem of restoring a canonically invariant operator $\widehat{\rm H}(a)$ from its spectrum is much easier than that for the operator $\mathrm{H}(a)$ and will be discussed in the next section. It can be shown that the inverse of $\mathrm{T}(\mathrm{H})$ takes the form

$\displaystyle\mathrm{T}^{-1}(\mathrm{H})=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\mathrm{L}^{n}\left(\bar{\beta}(a)+\frac{1}{2}\widehat{\mathrm{H}}(a)\right)^{n}\,,$

(12)

see Appendix A. Further, it follows from Eq. (10) that

	$\displaystyle\mathrm{H}(a)$	$\displaystyle=\mathrm{T}^{-1}(\mathrm{H})\,\widehat{\mathrm{H}}(a)\,\mathrm{T}(\mathrm{H})$
		$\displaystyle=\widehat{\mathrm{H}}(a)+\sum_{n=1}^{\infty}\frac{1}{n!}\mathrm{T}_{n}(a)\left(\bar{\beta}(a)+\frac{1}{2}{\mathrm{H}}(a)\right)^{n}\,.$		(13)

The operators $\mathrm{T}_{n}(a)$ are defined by recursion

$\displaystyle\mathrm{T}_{n}(a)=[\mathrm{T}_{n-1}(a),\mathrm{L}]$

(14)

with the boundary condition $\mathrm{T}_{0}(a)=\widehat{\mathrm{H}}(a)$. The $n$-th term in the sum in Eq. (II) is of order $\mathcal{O}(a^{n+1})$ so that one can easily work out an approximation for $\mathrm{H}(a)$ with arbitrary precision, e.g.,

	$\displaystyle\mathrm{H}(a)$	$\displaystyle=\widehat{\mathrm{H}}(a)+\mathrm{T}_{1}(a)\left(1+\frac{1}{2}\mathrm{T}_{1}(a)\right)\left(\bar{\beta}(a)+\frac{1}{2}\widehat{\mathrm{H}}(a)\right)$
		$\displaystyle+\frac{1}{2}\mathrm{T}_{2}(a)\left(\bar{\beta}(a)+\frac{1}{2}\widehat{\mathrm{H}}(a)\right)^{2}+\mathcal{O}(a^{4})\,.$		(15)

It can be checked that this expression coincides with that obtained in ref. [16, Eq. (3.9)] ^***The notations adopted here and in ref. [16] differ slightly. To facilitate a comparison we note that the operators $\mathrm{T}_{n}$ defined here satisfy the equation $[S_{+},\mathrm{T}_{n}]=n[\mathrm{T}_{n-1},z_{1}+z_{2}]$..

The evolution kernel $\widehat{\mathrm{H}}(a)$ can be realized as an integral operator. It acts on a function of two real variables as follows

$\displaystyle\widehat{\mathrm{H}}(a)\,f(z_{1},z_{2})$

$\displaystyle=\!Af(z_{1},z_{2})\!+\!\!\int_{+}h(\tau)f(z_{12}^{\alpha},z_{21}^{\beta}),$

(16)

where $A$ is a constant, $z_{12}^{\alpha}\equiv z_{1}\bar{\alpha}+z_{2}\alpha$, $\bar{\alpha}\equiv 1-\alpha$, and

$\displaystyle\int_{+}\equiv\int_{0}^{1}d\alpha\int_{0}^{\bar{\alpha}}d\beta.$

(17)

$\tau=\alpha\beta/\bar{\alpha}\bar{\beta}$ is called conformal ratio. The weight function $h(\tau)$ in Eq. (16) only depends on this particular combination of the variables $\alpha,\beta$ as a consequence of invariance properties of $\widehat{\mathrm{H}}$, Eq. (11).

It is easy to find that the operators $\mathrm{T}_{n}$ take the form

$\displaystyle\mathrm{T}_{n}(a)f(z_{1},z_{2})$

$\displaystyle=\!\int_{+}\ln^{n}(1\!-\!\alpha\!-\!\beta)h(\tau)f(z_{12}^{\alpha},z_{21}^{\beta})\,,$

(18)

that again agrees with the results of ref. [16]. Note, that this expression does not depend on the choice of the branch of the logarithm defining the function $\mathrm{L}=\ln z_{12}$ in Eq. (8), see Appendix A for more discussion.

First of all let us establish a connection between the eigenvalues of the operators $\mathrm{H}$ and $\widehat{\mathrm{H}}$. Since both of them are integral operators of the functional form in Eqs. (16), (18), both operators are diagonalized by functions of the form $\psi_{N}(z_{1},z_{2})=(z_{1}-z_{2})^{N-1}$, where $N$ is an arbitrary complex number. One may worry that the continuation of the function $\psi_{N}$ for negative $z_{12}$ is not unique and requires special care. But it does not matter for our analysis. Indeed, $z_{12}^{\alpha}-z_{21}^{\beta}=(1-\alpha-\beta)z_{12}$ with $\alpha+\beta<1$, therefore the operators do not mix the regions $z_{12}\gtrless 0$. For definiteness let us suppose that

$\displaystyle\psi_{N}(z_{1},z_{2})=\theta(z_{12})z_{12}^{N-1}.$

(19)

Let $\gamma(N),\widehat{\gamma}(N)$ be eigenvalues (anomalous dimensions) of the operators $\mathrm{H}$, $\widehat{\mathrm{H}}$ corresponding to the function $\psi_{N}$, respectively,

	$\displaystyle\mathrm{H}(a)\psi_{N}$	$\displaystyle=\gamma(N)\psi_{N},$		(20)
	$\displaystyle\widehat{\mathrm{H}}(a)\psi_{N}$	$\displaystyle=\widehat{\gamma}(N)\psi_{N}.$		(21)

The anomalous dimensions $\gamma(N),\widehat{\gamma}(N)$ are analytic functions of $N$ in the right complex half-plane, $\mathrm{Re}(N)>0$. For integer even (odd) $N$, $\gamma(N)$ gives the anomalous dimensions of the local (axial)vector operators ^†††As usual one has to consider the operators of certain parity, $\mathcal{O}_{\pm}(z_{1},z_{2})=\mathcal{O}(z_{1},z_{2})\mp\mathcal{O}(z_{2},z_{1})$, then the functions $\gamma_{\pm}(N)$ give the anomalous dimensions of local operators, for even and odd $N$ respectively..

Now let us note that the operator $\mathrm{T}(\mathrm{H})$ acts on $\psi_{N}$ as follows

	$\displaystyle\mathrm{T}(\mathrm{H})\psi_{N}(z_{1},z_{2})$	$\displaystyle=\sum_{n=0}^{\infty}\frac{\mathrm{L}^{n}}{n!}\left(\bar{\beta}(a)+\frac{1}{2}\gamma(N)\right)^{n}\!\psi_{N}(z_{1},z_{2})$
		$\displaystyle=z_{12}^{\bar{\beta}(a)+\frac{1}{2}\gamma(N)}\psi_{N}(z_{1},z_{2})$
		$\displaystyle=\psi_{N+\bar{\beta}+\frac{1}{2}\gamma(N)}(z_{1},z_{2}).$		(22)

Thus, it follows from Eq. (II) that the anomalous dimensions $\gamma(N)$ and $\widehat{\gamma}(N)$ satisfy the relation (cf. also Eq. (1))

$\displaystyle\gamma(N)=\widehat{\gamma}\left(N+\bar{\beta}(a)+\frac{1}{2}\gamma(N)\right).$

(23)

This relation appeared first in refs. [23, 22] as an generalization of the Gribov-Lipatov reciprocity relation [26, 27]. It was shown that the asymptotic expansion of the function $\widehat{\gamma}(N)$ for large $N$ is invariant under the reflection $N\to-N-1$, see e.g., refs. [22, 28, 29, 30]. This property strongly restricts harmonics sums which can appear in the perturbative expansion of the anomalous dimension $\widehat{\gamma}(N)$ [29]. Explicit expressions for $\widehat{\gamma}(N)$ are known at four loops in QCD [6] and at seven loops in the ${\cal N}=4$ SYM, see refs. [31, 29, 32, 33, 34].