Using the approach based on conformal symmetry we calculate the three-loop (NNLO)
contribution to the evolution equation for flavor-nonsinglet leading twist operators in
the
Article funded by SCOAP3
0, m\geq 0\,, $$ which is required by the reciprocity symmetry under the $j_N\!\mapsto\! 1-j_N$ transformation~\mbox{\cite{Basso-ml-2006nk,Alday-ml-2015eya,Alday-ml-2015ewa}}. Under the same condition ($\widetilde \chi(\rho)$ only has poles at integer $\rho$ values in the right half-plane), the kernels $\chi(\tau)$ at small values of the conformal ratio have the expansion $$\chi(\tau)=\sum_{km}c_{km} \tau^k \ln^m\tau \,.$$ The Mellin transform of the one-loop kernel is very simple: \begin{align} \chi_{\rm inv}^{(1)} = - 4 C_F \quad\leftrightarrow\quad \widetilde \chi_{\rm inv}^{(1)} = - 4 C_F \Big[ 2\pi i \delta(\rho)\Big]\,. \end{align} More examples are collected in table~\ref{Tab:Examples}, see appendix~\ref{app:H2loop}. One finds that the Mellin space kernels have a rather simple form, while the corresponding anomalous dimensions can be quite involved. Also the two-loop evolution kernels in Mellin space are given by rather compact expressions, see eq.~\eqref{mellin-twoloop}. If the anomalous dimensions are written in terms of the splitting functions, eq.~\eqref{def:split}, \begin{align} \gamma_{\rm inv}^{(k\pm)}(N) = -\int_0^1 dx\, x^N H_{\rm inv}^{(k\pm)}(x)\,, \end{align} the corresponding Mellin-transformed invariant kernels can be calculated as \begin{align}\label{delta-rho} \widetilde{\chi}_{\rm inv}^{(k)}(\rho) &= -\frac{\Gamma(2\rho+2)}{\Gamma^2(1+\rho)}\int_0^1dx \, H_{\rm inv}^{(k+)}(x)\, \frac{1}{x\bar x} \left(\frac{1+x}{1-x}\right) \left(\frac x{\bar x^2}\right)^\rho\, \end{align} and similar for $ \widetilde{\chi}_{\rm inv}^{\mathbb{P}(k)}(\rho)$, with the replacement $H_{\rm inv}^{(k+)}(x)\to H_{\rm inv}^{(k-)}(x)$. The kernels in $\tau$ space can finally be obtained by the inverse Mellin transformation~\eqref{mellin-transform-h}. We found this two-step approach to be the most effective for the three-loop case. For the remainder function $\delta H^{(3\pm)}_{\rm inv}(x)$ written in the form~\eqref{ansatz1} one obtains in \linebreak Mellin~space \begin{align} \delta\widetilde{\chi}^{(3\pm)}_{\rm inv}(\rho)= -\frac{\Gamma(2\rho+2)}{\Gamma^2(1+\rho)}\int_0^\infty dt\, h_\pm(t) \, t^{\rho-1}\,, \end{align} and for the simple ansatz in eq.~\eqref{ansatz2} \begin{align} \delta\widetilde{\chi}^{(3\pm)}_{\rm inv}(\rho) &= -\frac{\Gamma(2\rho+2)}{\Gamma^2(1+\rho)} \frac{\pi}{\sin (\pi \rho ) } H^\pm_0(1 + b_\pm \rho)a_\pm^{\rho } \,. \end{align} Using eq.~\eqref{mellin-transform-h} we finally obtain the following expression for the corresponding contribution to the invariant kernel \begin{align} \delta {\chi}^{(3\pm)}_{\rm inv}(\tau) = \frac{H^\pm_0}{\left(1+4a_\pm \tau/\bar \tau\right)^{5/2}} \left[1 +a_\pm \frac{\tau}{\bar \tau} (4-6\,b_\pm) \right] - H^\pm_0 \,. \label{deltachi-tau} \end{align} We also need the expressions for the functions $\phi_k$~\eqref{f-k} in $\rho$- and $\tau$-space defined as \begin{align} \widetilde{\phi}_k(\rho) &\equiv -\frac{\Gamma(2\rho+2)}{\Gamma^2(1+\rho)}\int_0^1dx \, \phi_k(x) \, \frac{1}{x\bar x} \left(\frac{1+x}{1-x}\right) \left(\frac x{\bar x^2}\right)^\rho \equiv \int_0^1 \frac{d\tau}{\tau\bar \tau} \left(\bar\tau/\tau\right)^\rho \, \varphi_k(\tau)\,. \end{align} One obtains \begin{align} \widetilde{\phi}_1(\rho)&=-{\pi}/(\rho\sin\pi\rho), \notag\\ \widetilde{\phi}_2(\rho)&= \widetilde{\phi}_1(\rho)\, \frac{\bar\rho}{\rho}, \notag\\ \widetilde{\phi}_3(\rho)&= \widetilde{\phi}_1(\rho)\,\bigg( -2\,\frac{\bar\rho}{\rho}+\psi'(\rho)-\frac{\pi^2}{6}\bigg), \notag\\ \widetilde{\phi}_4(\rho)&= \widetilde{\phi}_1(\rho)\,\bigg( 5\,\frac{\bar\rho}{\rho}+\frac{(1-3\rho)}{\rho} \bigg[\psi'(\rho)-\frac{\pi^2}{6}\bigg]\bigg), \end{align} and \begin{align} \varphi_1(\tau) & = \ln\bar\tau=H_1\left(-\frac{\tau}{\bar \tau}\right)\,, \notag\\ \varphi_2(\tau) & = -\varphi_1(\tau) + H_{01}\left(-\frac{\tau}{\bar\tau}\right) = -\ln\bar\tau +\Li_2\left(-\tau/\bar\tau\right)\,, \notag\\ \varphi_3(\tau) & = -2\varphi_2(x)-H_{101}\left(-\frac{\tau}{\bar \tau}\right)= 2H_1\left(-\frac{\tau}{\bar \tau}\right)-2H_{01}\left(-\frac{\tau}{\bar \tau}\right)-H_{101}\left(-\frac{\tau}{\bar \tau}\right)\,, \notag\\ &= 2\ln\bar\tau -2\Li_2\left(-\tau/\bar\tau\right) - 2\Big(\Li_3(\bar \tau)-\Li_3(1)\Big)+\ln\bar\tau\left(\Li_2(\bar\tau) +\frac{\pi^2}6\right) +\frac16 \ln^3\bar\tau \notag\\ \varphi_4(\tau) & =-3\varphi_3(\tau)-\varphi_2(\tau) -H_{0101}\left(-\frac{\tau}{\bar \tau}\right) \notag\\ &= -5H_1\left(-\frac{\tau}{\bar \tau}\right) + 5H_{01}\left(-\frac{\tau}{\bar \tau}\right)+3H_{101}\left(-\frac{\tau}{\bar \tau}\right) -H_{0101}\left(-\frac{\tau}{\bar \tau}\right), \label{phi-tau} \end{align} where $H_{p\ldots q}(z)$ are harmonic polylogarithms, see ref.~\cite{Remiddi-ml-1999ew}. With these expressions at hand, our result for the invariant kernel~\eqref{Hinv3} is complete. We obtain \begin{align}\label{tau-x} \chi_{\rm inv}^{(3)}(\tau) &= \sum_{k=1}^{4} B^{(3+)}_k \varphi_k(\tau) - C^{(3+)}_0 + C^{(3+)}_1 \big[\ln\left(\tau/\bar\tau\right)+2\big] + \delta {\chi}^{(3+)}_{\rm inv}(\tau)\,, \notag\\ \chi_{\rm inv}^{\mathbb{P}(3)}(\tau) &= \sum_{k=1}^{4} B^{(3-)}_k \varphi_k(\tau) - C^{(3-)}_0 + \delta {\chi}^{(3-)}_{\rm inv}(\tau)\,, \end{align} where the functions $\varphi_k(\tau)$ are defined in eq.~\eqref{phi-tau}, the coefficients $B^{(3\pm)}_k$, $C^{(3\pm)}_k$ can be found in table~\ref{tab:resultcoeff} and the parameters for $\delta {\chi}^{(3\pm)}_{\rm inv}(\tau)$~\eqref{deltachi-tau} are collected in table~\ref{tab:fitcoeff}. \sloppy For illustration, in figure~\ref{fig:chi} we compare the full NNLO invariant functions \mbox{$\chi(a) = a \chi^{(1)} + a^2 \chi^{(2)} + a^3 \chi^{(3)}$} with the NLO, $\mathcal{O}(a^2)$, and the LO, $\mathcal{O}(a)$, results for a typical value of the coupling $\alpha_s/\pi = 0.1$ and, for definiteness, $n_f=4$. In the same plot the NNLO results using the exact three-loop functions obtained by the numerical integration of eq.~\eqref{Legendre} are shown by dots. One sees that the accuracy of our parametrization is rather good. The remaining entries in the invariant kernel are, for the same values $n_f=4$ and $\alpha_s/\pi = 0.1$, \begin{align} \Gamma_{\rm cusp} &= a \Gamma^{(1)} (1+8.019 a+ 80.53 a^2 +\ldots) = a \Gamma^{(1)} (1+ 0.2005 + 0.0503 +\ldots) \,, \notag\\ \chi_0 &= a \chi_0^{(1)} (1 -0.7935 a - 141.3 a^2+\ldots) = a \chi_0^{(1)} (1 - 0.0198 - 0.0883 +\ldots) \,. \end{align} \fussy \begin{figure}[t] \centering \includegraphics[width=\textwidth]{fig2.pdf} ]]>
n\,, \qquad \gamma_{nn} = \gamma_{n}\,. \end{equation} Since $\gamma_{nn'}$ does not depend on $k$, the second subscript $k$ is essentially redundant. In what follows we will use a ``hat'' for the anomalous dimensions and other quantities in matrix notation \begin{equation} \widehat{\boldsymbol{\gamma}} \equiv \gamma_{nn'}\,. \end{equation} The light-ray operator~\eqref{LRO} can be expanded in terms of the local operators defined in eq.~\eqref{Onk} \begin{align} \label{localExpand} \mathcal O(x; z_1,z_2 ) = \sum _{n=0}^\infty\sum_{k=n}^{\infty} \Phi_{nk}(z_1,z_2) \mathcal O_{nk}(x) \,, \end{align} where the coefficients $\Phi_{nk}(z_1,z_2)$ are homogeneous polynomials of two variables of degree $k$~\cite{Braun-ml-2011dg}: \begin{align} \Phi_{nk}(z_1,z_2) = \omega_{nk} (S_+^{(0)})^{k-n} z_{12}^n \,, && \omega_{nk} = 2 \frac{2 n + 3}{(k-n)!} \frac{\Gamma(n + 2) }{\Gamma(n + k + 4)}\,. \label{Phi-nk} \end{align} These polynomials are mutually orthogonal and form a complete set of functions\footnote{Our notation in this section is adapted to facilitate the comparison with ref.~\cite{Belitsky-ml-1998gc} and differs from the notation used in~\cite{Braun-ml-2011dg}. In particular the functions $\Phi_{nk}$ defined in~\eqref{Phi-nk} correspond to $\Phi_{n,k-n}$ in~\cite{Braun-ml-2011dg}.} w.r.t.\ the canonical $\SL(2)$ scalar product (see, e.g.,~\cite{Braun-ml-2011dg}) \begin{align} \label{ortho} \langle \Phi_{nk} | \Phi_{n'k'} \rangle = \delta_{kk'}\delta_{nn'} ||\Phi_{nk}||^2 =\delta_{kk'}\delta_{nn'} \,\omega_{nk} \rho_n^{-1} \,, && \rho_n = \frac12{(n+1)(n+2)!} \end{align} so that the local operators~\eqref{Onk} can be obtained by the projection on the corresponding ``state'' \begin{align} \mathcal O_{nk} &= \rho_n\omega_{nk}^{-1} \langle \Phi_{nk} |\mathcal O(z_1,z_2) \rangle = \rho_n \langle \left( S_+\right)^{k-n}z_{12}^n |\mathcal O(z_1,z_2) \rangle\,. \end{align} The (canonical) conformal spin generators $S^{(0)}_\pm$ act as rising and lowering operators on this space whereas $S^{(0}_0$ is diagonal \begin{align}\label{SPhi} S_0^{(0)} \Phi_{nk}(z_1,z_2) =&~ (k+2)\Phi_{nk}(z_1,z_2)\,, \notag\\ S_+^{(0)} \Phi_{nk}(z_1,z_2) =&~ (k-n+1)(n+k+4) \Phi_{nk+1}(z_1,z_2) \,, \notag\\ S_-^{(0)} \Phi_{nk}(z_1,z_2) =&~ - \Phi_{nk-1}(z_1,z_2) \,. \end{align} Thus the set of coefficient functions $\Phi_{n,k}(z_1,z_2)$ for $k=\{n\ldots\infty\} $ forms an irreducible representation of the $\SL(2)$ algebra, which is usually referred to as the conformal tower. Let~$\mathcal A$ be a certain operator~$\mathcal A$ acting on quantum fields. Its action can be realized by the expansion in terms of local operators with ``matrix elements'' serving as expansion coefficients \begin{align} \label{Alocal} [\mathcal A, \mathcal O_{nk}] = \sum _{n'k'} A_{nn'}^{kk'}\,\mathcal O_{n'k'} \,. \end{align} Alternatively one can represent $\mathcal A$ by some integro-differential operator~${\bf A}$ acting on the arguments $z_1,z_2$ of the light-ray operator $\mathcal O(z_1,z_2)$ and, by means of the expansion~\eqref{localExpand}, on the coefficient functions, \begin{equation} \label{Anonlocal1} [\mathcal A, O](z_1,z_2) = \sum_{nk} \Phi_{nk}(z_1,z_2)\sum _{n'k'} A_{nn'}^{kk'}\,\mathcal O_{n'k'} \equiv [{\bf A} O](z_1,z_2) = \sum_{nk} [{\bf A}\Phi_{nk}](z_1,z_2) O_{nk}\,. \end{equation} Comparing the representations in eqs.~\eqref{Alocal} and~\eqref{Anonlocal1} we see that the action of ${\bf A}$ on the coefficient functions of local operators is given by the transposed matrix \begin{align} [{\bf A}\Phi_{nk}](z_1,z_2) = \sum_{n'k'} A_{n'n}^{k'k} \Phi_{n'k'}(z_1,z_2) \,. \end{align} Using the orthogonality relation~\eqref{ortho} one obtains \begin{align} A_{nn'}^{kk'} = ||\Phi_{nk}||^{-2}\langle \Phi_{nk}(z_1,z_2) | [{\bf A}\Phi_{n'k'}](z_1,z_2) \rangle \equiv \langle {nk} | {\bf A}|{n'k'} \rangle\,, \end{align} which is the desired conversion, for a generic operator, between the light-ray and local operator representations. The ``matrix elements'' $A_{nn'}^{kk'}$ depend in general on four indices. However, if the operator $\bf A$ has a certain (canonical) dimension, i.e $[S^{(0)}_0, {\bf A}]= d_{\bf A} \bf A$, then its matrix elements are nonzero only if the indices satisfy the constraint $d_{\bf A}=k-k'$. This reduces the number of independent indices by one and allows one to write $A_{nn'}^{kk'} \equiv A_{nn'}(k)\delta_{k, k'+d_{\bf A}}$. If, in addition, the operator $\mathcal A$ is invariant under translations, i.e.\ $[S_-, {\bf A}]=0$, then it follows from~eqs.~\eqref{SPhi} that the matrix elements $A_{nn'}^{kk'} \equiv A_{nn'}$ do not depend on the upper indices~at~all. Constraints on the operator mixing in the light-ray operator representation that follow from conformal algebra take the form~\eqref{HDelta} \begin{align}\label{HDelta1} \big[S_+^{(0)},\mathbb{H}(a)\big] &= \big[\mathbb{H}(a),z_1+z_2\big]\left(\bar\beta(a)+ \frac12\mathbb{H}(a)\right)+\big[\mathbb{H}(a),z_{12} \Delta(a)\big]. \end{align} To translate this equation into the local operator representation, we define the matrices \begin{align} \mathbf{a}_{mn}(k) &= \langle m,k | S_+^{(0)}| n,k-1\rangle\,, \notag\\ \mathbf{b}_{mn}(k) &= \langle m,k | z_1+z_2 | n,k-1\rangle\,, \notag\\ \boldsymbol{\gamma}_{mn} &= \langle m,k | \mathbb{H} | n,k\rangle\,, \notag\\ \mathbf{w}_{mn} &= \langle m,k | z_{12}\Delta | n,k-1\rangle\,. \end{align} The first two matrix elements are easily computed, \begin{align} \mathbf{a}_{mn}(k) &= -(m-k)(m+k+3)\delta_{mn} \equiv - {\bf a}(m,k)\delta_{mn}\,, \notag\\ \mathbf{b}_{mn}(k) &= 2(k-n)\delta_{mn}-2(2n+3)\vartheta_{mn} \,, \label{ab} \end{align} where we introduced a discrete step function \begin{align*} \vartheta_{mn} = \begin{cases} 1 \text{ if } m - n > 0 \text{ and even} \\ 0 \text { else.} \end{cases} \end{align*} The remaining two are nontrivial and can be written as a perturbative expansion \begin{align} \widehat{\boldsymbol{\gamma}}(a) &= a \widehat{\boldsymbol{\gamma}}^{(1)} + a^2 \widehat{\boldsymbol{\gamma}}^{(2)} + a^3 \widehat{\boldsymbol{\gamma}}^{(3)} +\ldots\,, \qquad a= \frac{\alpha_s}{4\pi}\,. \notag\\ \widehat{\mathbf{w}}(a) &= a \widehat{\mathbf{w}}^{(1)} + a^2 \widehat{\mathbf{w}}^{(2)} +\ldots\,. \end{align} Eq.~\eqref{HDelta1} becomes in matrix notation \begin{align} \label{GammaW} [\widehat {\bf a},\widehat{\boldsymbol{\gamma}}(a) ] = [\widehat{\boldsymbol{\gamma}}(a),\widehat {\bf b}]\left(\bar \beta(a)+\frac12\widehat{\boldsymbol{\gamma}}(a)\right) + [\widehat{\boldsymbol{\gamma}},\widehat {\bf w}(a)]\,. \end{align} \looseness=-1 Note that the matrices $\widehat {\bf a}(k)$ and ${\widehat {\bf b}}(k)$~\eqref{ab} depend in principle on the total number of derivatives $k$. However, due to the fact that only diagonal elements depend on this parameter, the dependence on $k$ drops out in the commutator. Hence we can safely omit it. In complete analogy to the light-ray operator formulation, this equation fixes the non-diagonal (i.e.\ canonically non-invariant) part of the anomalous dimension matrix. Indeed, the commutator on the l.h.s.\ of eq.~\eqref{GammaW} takes the form \begin{align} [\widehat {\bf a},\widehat{\boldsymbol{\gamma}}(a) ]_{mn} &= (- \mathbf{a}(m,k) + \mathbf{a}(n,k)) \gamma_{mn} = - \mathbf{a}(m,n) \gamma_{mn}\,, \end{align} so that the non-diagonal elements of the mixing matrix are given by~\cite{Mueller-ml-1993hg} \begin{align} \widehat{\boldsymbol{\gamma}}^{\text{ND}}(a) = \mathcal G \left\{ [\widehat{\boldsymbol{\gamma}}(a),\widehat {\bf b}]\left(\frac12\widehat{\boldsymbol{\gamma}}(a)+\bar\beta(a) \right) + [\widehat{\boldsymbol{\gamma}}(a),\widehat {\bf w}(a)] \right\}, \label{mueller} \end{align} where \begin{align} \mathcal G \big\{\widehat M \big\}_{mn} = -\frac{M_{mn}}{ {\bf a }(m,n) } \,. \end{align} In particular to the two-loop accuracy \begin{align} \widehat{\boldsymbol{\gamma}}^{(2),\text{ND}} = \mathcal G \left\{ [\widehat{\boldsymbol{\gamma}}^{(1)},\widehat {\bf b}]\left(\frac12\widehat{\boldsymbol{\gamma}}^{(1)}+\beta_0 \right) + [\widehat{\boldsymbol{\gamma}}^{(1)},\widehat {\bf w}^{(1)}] \right\}. \end{align} Here $\widehat{\boldsymbol{\gamma}}^{(1)}$ is the well-known (diagonal) matrix of one-loop anomalous dimensions \begin{align} {\gamma}_{mn}^{(1)} & = \gamma^{(1)}_n \delta_{mn} = 2 \delta_{mn} C_F \bigg(4S_1(n+1)-\frac{2}{(n+1)(n+2)}-3\bigg) \end{align} and $\widehat {\bf w}^{(1)}$ is the one-loop conformal anomaly \begin{align} {\bf w}_{mn}^{(1)} &= 4 C_F (2n+3)\, \mathbf{a}(m,n) \left( \frac{A_{mn} - S_1(m+1)}{(n+1)(n+2)} + \frac{2A_{mn}}{\mathbf{a}(m,n)} \right) \vartheta_{mn}\,, \end{align} where \begin{align} A_{mn} = S_1\left(\frac{m+n+2}2\right) -S_1\left(\frac{m-n-2}2\right) +2S_1(m-n-1)-S_1(m+1). \end{align} Collecting everything one obtains the two-loop anomalous dimension matrix: \begin{align} \gamma^{(2)}_{mn} &= \delta_{mn}\gamma^{(2)}_n - \frac{\gamma^{(1)}_m-\gamma^{(1)}_n}{\mathbf{a}(m,n)} \biggl\{ - 2(2n+3) \left(\beta_0 + \frac12 \gamma^{(1)}_n\right)\vartheta_{mn}+ {\bf w}_{mn}^{(1)} \biggr\}. \end{align} The first few elements $(0\leq n\leq7,~0\leq m\leq7)$ for $N_c=3$ are \begin{align} {\gamma}_{mn}^{(2)} & = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{23488}{243} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{260}{9} & 0 & \frac{34450}{243} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{8668}{243} & 0 & \frac{5241914}{30375} & 0 & 0 & 0 & 0 \\ \frac{52}{9} & 0 & \frac{8512}{243} & 0 & \frac{662846}{3375} & 0 & 0 & 0 \\ 0 & \frac{120692}{8505} & 0 & \frac{261232}{7875} & 0 & \frac{83363254}{385875} & 0 & 0 \\ -\frac{2054}{14175} & 0 & \frac{34243}{2025} & 0 & \frac{2208998}{70875} & 0 & \frac{718751707}{3087000} & 0 \\ 0 & \frac{226526}{35721} & 0 & \frac{982399}{55125} & 0 & \frac{7320742}{250047} & 0 & \frac{557098751203}{2250423000} \end{pmatrix} \notag \\[5pt] &\quad -n_f \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{512}{81} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{40}{9} & 0 & \frac{830}{81} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{88}{27} & 0 & \frac{26542}{2025} & 0 & 0 & 0 & 0 \\ \frac{104}{45} & 0 & \frac{1064}{405} & 0 & \frac{31132}{2025} & 0 & 0 & 0 \\ 0 & \frac{1144}{567} & 0 & \frac{232}{105} & 0 & \frac{1712476}{99225} & 0 & 0 \\ \frac{4108}{2835} & 0 & \frac{242}{135} & 0 & \frac{1804}{945} & 0 & \frac{3745727}{198450} & 0 \\ 0 & \frac{2372}{1701} & 0 & \frac{506}{315} & 0 & \frac{2860}{1701} & 0 & \frac{36241943}{1786050} \\ \end{pmatrix}. \end{align} Our expressions for the one-loop conformal anomaly and the two-loop anomalous dimension matrix coincide identically with the results in~\cite{Belitsky-ml-1998gc}.\footnote{The explicit relation to the notations in~\cite{Belitsky-ml-1998gc} is as follows: $\widehat {\mathbf{a}}(k)|_{\text{\cite{Belitsky-ml-1998gc}}} = -\frac12 \widehat {\mathbf{a}}(k)$, $\widehat {\mathbf{b}}(k)|_{\text{\cite{Belitsky-ml-1998gc}}} = \widehat {\mathbf{b}}(k)$, $\widehat {\mathbf{w}}^{(1)}|_{\text{\cite{Belitsky-ml-1998gc}}} = - \widehat {\mathbf{w}}^{(1)}$, and $\widehat{\gamma}^{(i)}_{\text{\cite{Belitsky-ml-1998gc}}} = 2^i \widehat\gamma^{(i-1)}$. A perturbative expansion in~\cite{Belitsky-ml-1998gc} is done in powers of $\alpha_s/(2\pi)$, e.g., $\widehat{\boldsymbol{\gamma}}(a) = \sum_i (2a)^i\widehat{\boldsymbol{\gamma}}^{(i-1)} $.} Expanding eq.~\eqref{mueller} to the third order, we obtain the three-loop nondiagonal anomalous dimension matrix in the form \begin{equation} \widehat{\boldsymbol{\gamma}}^{(3),\text{ND}} = \mathcal G \bigg\{ [\widehat{\boldsymbol{\gamma}}^{(2)},\widehat{\bf b}] \left(\frac12\widehat{\boldsymbol{\gamma}}^{(1)}\!+\!\beta_0\right) + [\widehat{\boldsymbol{\gamma}}^{(2)},\widehat{\bf w}^{(1)}]+ [\widehat{\boldsymbol{\gamma}}^{(1)},\widehat{\bf b}] \left(\frac12\widehat{\boldsymbol{\gamma}}^{(2)}\!+\!\beta_1\right) + [\widehat{\boldsymbol{\gamma}}^{(1)},\widehat{\bf w}^{(2)}] \bigg\}\,. \end{equation} In addition to the already known quantities, this expression involves the matrix element of the two-loop conformal anomaly~\eqref{Delta2plus}, \begin{align} \mathbf{w}_{mn}^{(2)} = \langle m,k| z_{12} \Delta^{(2)} | n,k-1 \rangle = \frac14 [\widehat{\boldsymbol{\gamma}}^{(2)}, \widehat{\mathbf{b}}] + \Delta {\bf w}_{mn}^{(2)} \end{align} where \begin{align} \Delta {\bf w}_{mn}^{(2)} = \langle m,k| z_{12} \Delta_+^{(2)} | n,k-1 \rangle \,. \end{align} The explicit expression for the operator $\Delta_+^{(2)}$~\eqref{Delta+twoloops} can be found in~\cite{Braun-ml-2016qlg}. We have not found a closed analytic expression for the matrix $\Delta {\bf w}_{mn}^{(2)}$, but the values for given $m,n$ can be evaluated in a straightforward way. Splitting the result into different color structures, \begin{align*} \Delta \widehat{\bf w}^{(2)} = C_F^2\Delta \widehat{\bf w}^P + \frac{C_F}{N_c} \Delta \widehat{\bf w}^{FA} + \beta_0 C_F \Delta \widehat{\bf w}^{bF}, \end{align*} we get for the first few elements $(0\leq n \leq 5 ,~1 \leq m \leq 7)$ \begin{align} \Delta \widehat{\bf w}^{FA} \!&=\! \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0\\ -\frac{75}{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{5075}{108} & 0 & 0 & 0 & 0\\ -\frac{679}{15} & 0 & -\frac{58723}{720} & 0 & 0 & 0\\ 0 & -\frac{7399}{90} & 0 & -\frac{724339}{6000} & 0 & 0\\ -\frac{1070777}{16800} & 0 & -\frac{12001}{96} & 0 & -\frac{123357091}{756000} & 0 \\ 0 & -\frac{22974677}{211680} & 0 & -\frac{101507627}{588000} & 0 & -\frac{308384869}{1481760} \\ \end{pmatrix}, \notag\\ \Delta \widehat{\bf w}^{P} \!&=\! \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ \! -\frac{2965}{144} & 0 & 0 & 0 & 0 & 0 \\ 0 & \!-\frac{1176553}{10800} & 0 & 0 & 0 & 0 \\ \!-\frac{140959}{9000} & 0 & \!-\frac{7387709}{36000} & 0 & 0 & 0\\ 0 & \!-\frac{75208391}{617400} & 0 & \!-\frac{2111899581}{6860000} & 0 & 0\\ \!-\frac{68372343}{5488000} & 0 & \!-\frac{5045910661}{21168000} & 0 & \!-\frac{307457793929}{740880000} & 0\\ 0 & \!-\frac{99911324293}{800150400} & 0 & \!-\frac{808931234579}{2222640000} & 0 & \!-\frac{2942615103467}{5601052800} \\ \end{pmatrix}, \notag\\ \Delta \widehat{\bf w}^{bF} \!&=\! \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 35 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1855}{27} & 0 & 0 & 0 & 0\\ \frac{105}{2} & 0 & \frac{2555}{24} & 0 & 0 & 0 \\ 0 & \frac{2891}{30} & 0 & \frac{146839}{1000} & 0 & 0 \\ \frac{6459}{100} & 0 & \frac{390313}{2700} & 0 & \frac{2552407}{13500} & 0 \\ 0 & \frac{202829}{1764} & 0 & \frac{4798313}{24500} & 0 & \frac{14365013}{61740} \\ \end{pmatrix}. \end{align} Using these expressions and the diagonal matrix elements from~\cite{Moch-ml-2004pa} we obtain the full three-loop anomalous dimension matrix \begin{align} \widehat{\boldsymbol{\gamma}}^{(3)} = {\rm diag}\left\{\gamma_0^{(3)}, \gamma_1^{(3)},\ldots \right\} + \widehat{\boldsymbol{\gamma}}^{(3)}_{\langle 1\rangle}+ n_f\, \widehat{\boldsymbol{\gamma}}^{(3)}_{\langle n_f\rangle} + n_f^2\, \widehat{\boldsymbol{\gamma}}^{(3)}_{\langle n_f^2\rangle} \,, \end{align} where the off-diagonal matrices for $N_c=3$ and different powers of $n_f$ in the range $0\!\leq\! n \!\leq\! 7 $, $0 \leq m \leq 7$ are given by the following expressions: \par\vskip -.7\baselineskip {\small\begin{align} \widehat{\boldsymbol{\gamma}}^{(3)}_{\langle 1\rangle} \!&=\! \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{49024}{81} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{36623912}{54675} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{3911}{27} & 0 & \frac{23599891}{36450} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{8049304723}{31255875} & 0 & \frac{320657981731}{520931250} & 0 & 0 & 0 & 0 \\ \frac{281851388261}{7501410000} & 0 & \frac{208052194247}{714420000} & 0 & \frac{21898269506047}{37507050000} & 0 & 0 & 0 \\ 0 & \frac{7192640196053}{56710659600} & 0 & \frac{159898280729473}{525098700000} & 0 & \frac{220023775251709}{396974617200} & 0 & 0 \\ \end{pmatrix}, \notag\\ \widehat{\boldsymbol{\gamma}}^{(3)}_{\langle n_f\rangle} \!&=\! \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{28700}{243} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{5762188}{54675} & 0 & 0 & 0 & 0 & 0 & 0 \\ -\frac{1279108}{30375} & 0 & -\frac{26434828}{273375} & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{849255644}{18753525} & 0 & -\frac{516077668}{5788125} & 0 & 0 & 0 & 0 \\ -\frac{54942827}{2500470} & 0 & -\frac{636248861}{13395375} & 0 & -\frac{77507831071}{937676250} & 0 & 0 & 0 \\ 0 & -\frac{1660976917}{67512690} & 0 & -\frac{7496172461}{156279375} & 0 & -\frac{36406093529}{472588830} & 0 & 0 \\ \end{pmatrix}, \notag\\ \widehat{\boldsymbol{\gamma}}^{(3)}_{\langle n_f^2\rangle} \!&=\! \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{236}{81} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{3172}{1215} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1316}{2025} & 0 & \frac{41356}{18225} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{187412}{178605} & 0 & \frac{22012}{11025} & 0 & 0 & 0 & 0 \\ \frac{55169}{893025} & 0 & \frac{10793}{9450} & 0 & \frac{176539}{99225} & 0 & 0 & 0 \\ 0 & \frac{726193}{1607445} & 0 & \frac{681503}{595350} & 0 & \frac{515359}{321489} & 0 & 0 \end{pmatrix}. \end{align}}For completeness we also write the first few anomalous dimensions~\cite{Moch-ml-2004pa}: \par\vskip -.7\baselineskip {\small\begin{align} \gamma^{(3)}_{0} \!&= 0 \,, \notag\\ \gamma^{(3)}_1 \!&= \frac{2560}{81}\zeta_3+\frac{11028416}{6561} -\left(\frac{2560}{27} \zeta_3 +\frac{334400}{2187}\right)n_f-\frac{1792 }{729}n_f^2 \,, \notag\\ \gamma_2^{(3)} \!&= \frac{2200}{81}\zeta_3+\frac{64486199}{26244} -\left(\frac{4000}{27} \zeta _3 + \frac{967495}{4374}\right) n_f -\frac{2569 }{729}n_f^2 \,, \notag\\ \gamma_3^{(3)} \!&= \frac{11512 }{405}\zeta _3+ \frac{245787905651}{82012500} -\left(\frac{5024 }{27}\zeta _3+\frac{726591271}{2733750}\right) n_f -\frac{384277}{91125} n_f^2 \,, \notag\\ \gamma_4^{(3)} \!&= \frac{11312}{405} \zeta _3+\frac{559048023977}{164025000} -\left(\frac{5824}{27} \zeta _3+\frac{90842989}{303750}\right) n_f -\frac{431242 }{91125}n_f^2 \,, \notag\\ \gamma_5^{(3)} \!&= \frac{558896}{19845} \zeta _3+\frac{10337334685136687}{2756768175000} -\left(\frac{45376}{189} \zeta _3+\frac{713810332943}{2187911250}\right) n_f -\frac{160695142}{31255875} n_f^2 \,, \notag\\ \gamma_6^{(3)} \!&= \frac{185482}{6615} \zeta _3+\frac{59388575317957639}{14702763600000} -\left(\frac{16432}{63} \zeta _3+\frac{12225186887503}{35006580000}\right) n_f -\frac{1369936511}{250047000} n_f^2 \,, \notag\\ \gamma_7^{(3)} \!&= \frac{5020814}{178605} \zeta _3 \!+\frac{46028648192099544431}{10718314664400000} \!-\!\left(\!\frac{158128}{567}\zeta _3\!+\!\frac{349136571992501}{945177660000}\!\right) n_f \!-\!\frac{38920977797}{6751269000} n_f^2 \,. \end{align}}To visualize the size of the three-loop correction we consider the full NNLO nondiagonal part of the anomalous dimension matrix for $n_f=4$ in the same range $(0\leq n\leq7$, $0\leq m\leq7)$: \par\vskip -.7\baselineskip {\small\begin{equation} \widehat{\boldsymbol{\gamma}}^{\rm ND}= a^2 \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 11.1 + 179 a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 22.6 +290 a & 0 & 0 & 0 & 0 & 0 & 0 \\ -3.47-13.2 a & 0 & 24.5 +297 a & 0 & 0 & 0 & 0 & 0 \\ 0 & 6.12 + 93.2 a & 0 & 24.3+291 a & 0 & 0 & 0 & 0 \\ -5.94 - 49.3 a & 0 & 9.74 +120 a & 0 & 23.5+282 a & 0 & 0 & 0 \\ 0 & 0.0764+35.6 a & 0 & 11.4+131 a & 0 & 22.6+272 a & 0 & 0 \end{pmatrix}. \end{equation}}For realistic values of the strong coupling $a = \alpha_s/(4\pi) \sim 1/40$ the three-loop contribution is on the average about 30\% of the two-loop result. ]]>