We present a systematic study of higher-twist distribution
amplitudes (DAs) of the B-meson which give rise to power-suppressed
Article funded by SCOAP3
0$ and $-\infty < x < \infty$. It turns out that the corresponding anomalous dimensions can be written as a sum of terms depending on $s$ and $x$ separately. The $s$-dependent part can be absorbed in the same universal factor $R(s;\mu,\mu_0)$ as for the leading twist so that one obtains~\cite{Braun-ml-2015pha} \begin{align} \eta_3(s,x,\mu) &= L^{\gamma_3(x)/\beta_0}R(s;\mu,\mu_0)\,\eta_3(s,x,\mu_0)\,, \notag\\ \eta_3^{(0)}(s,\mu) &= L^{N_c/\beta_0}R(s;\mu,\mu_0)\eta_3^{(0)}(s,\mu_0)\,, \label{scale:twist23} \end{align} with the anomalous dimension~\cite{Braun-ml-2015pha}\footnote{To be precise, $\gamma_3(x)$ and $\gamma_4(x)$ defined below in eq.~\eqref{gamma4} correspond to the \emph{difference} of anomalous dimensions between the higher- and leading-twist operators. The scale dependence for the leading-twist DA is included in the $R$-factor.} \begin{equation} \gamma_3(x) = N_c\big[\psi\big(3/2+ix\big)+ \psi\big(3/2-ix\big) +2\gamma_E\big]\,,\qquad \gamma_3^{(0)}=\gamma_3(x= {i}/{2}) = N_c\,. \label{gamma3} \end{equation} Note that in addition to the integral over all real values of $x$ the DA $\Phi_3(\underline{z},\mu)$ contains an extra contribution, the first term in eq.~\eqref{Phi3}, corresponding to a particular imaginary value $x=i/2$. This special term has a lower anomalous dimension separated by a finite number from the rest, continuum spectrum, and can be interpreted as the asymptotic DA. This interpretation fails, however, for large quark and/or gluon momenta $\omega_1,\omega_2 \gtrsim \mu$ (alias small coordinates $z_1,z_2 \lesssim 1/\mu$) in which case contributions with all anomalous dimensions have to be included, see ref.~\cite{Braun-ml-2015pha} for a detailed discussion. Note also that the twist-three contribution to the two-particle DA $\Phi_-$ in eq.~\eqref{Phi+Phi-} is determined entirely by this special term, $\eta_3^{(0)}(s,\mu)$. The ``genuine'' three-particle twist-three contributions to $\Phi_3$ encoded in $\eta_3(s,x,\mu)$ decouple from $\Phi_-$ to the stated $1/N_c^2$ accuracy. \begin{figure} \centerline{ \begin{picture}(210,140)(0,0) \put(-5,0){\epsfxsize7.8cm\epsffile{anomalousdimensions.eps}} \put(105,-8){$x$} \put(55,100){$\gamma_3(x)$} \put(30,60){\textcolor{blue}{$\gamma_4(x)$}} \end{picture} } ]]>
0$ and $x\in \mathbb{R}$ related to the eigenvalues of the conserved charges, \begin{equation} \mathbb{Q}_1^{(3)} Y_{3}(s,x|\,\underline{z})=s\,Y_{3}(s,x|\,\underline{z})\,, \qquad \mathbb{Q}_2^{(3)} Y_{3}(s,x|\,\underline{z})=-s\, x^2\,Y_{3}(s,x|\,\underline{z})\,. \end{equation} The charges $\mathbb{Q}_i^{(3)}$ are self-adjoint operators w.r.t.\ the $\SL(2)$-invariant scalar product so that the eigenfunctions $Y_{3}(s,x\,|\,\underline{z})$ are mutually orthogonal, see appendix~\ref{app:scalarproduct}. For the twist-four case, in addition to the two conserved charges $\mathbb{Q}_1$ and $\mathbb{Q}_2$, one can construct an extra, ``supplementary" charge $\mathbb{Q}_3$, such that $[\mathbb{Q}_{{1,2}},\mathbb{Q}_3]=0$. The ``supplementary" charges have a rather simple form \begin{equation} \mathbb{Q}_3=i\begin{pmatrix} -S_{g,3/2}^+&z_1/z_2S^+_{g,1} \\ z_2/z_1S^+_{q,1/2}& - S^+_{q,1} \end{pmatrix}\,, \qquad \widebar{\mathbb{Q}}_3=i\begin{pmatrix} S^+_{q,3/2}\quad &(S^0_{q,1}+1)(S^0_{g,1}+1)\\ z_1z_2\quad&S^+_{g,1} \end{pmatrix}\,, \end{equation} and are quite helpful for constructing of the eigenfunctions. To this end we start with the following ansatz \begin{equation} \Phi_k(\underline{z})= {z_1^{-2j_1} z_2^{-2j_2}} \int_0^1 du e^{is(u/z_1+\bar u/z_2)} \varphi_k(u)\,, \end{equation} where $j_1$ and $j_2$ are the conformal spins of the light fields in the corresponding operator. Such functions are, by construction, eigenfunctions of the first charge $\mathbb{Q}_1$ ($\widebar{\mathbb{Q}}_1$): \begin{equation} \mathbb{Q}_1\begin{pmatrix}\Phi_1(\underline{z})\\\Phi_2(\underline{z})\end{pmatrix} =s \begin{pmatrix}\Phi_1(\underline{z})\\\Phi_2(\underline{z})\end{pmatrix}. \end{equation} Requiring that the supplementary charge $\mathbb{Q}_3$ ($\widebar{\mathbb{Q}}_3$) is diagonalized yields a simple relation between the ``upper'' and ``lower'' components, $\varphi_1(u)$ and $\varphi_2(u)$, after which the eigenvalue problem for $\mathbb{Q}_2$ ($\widebar{\mathbb{Q}}_2$) reduces to a second order differential equation for, e.g., $\varphi_1(u)$. Selecting the solutions with the required analytic properties one obtains the following set of eigenfunctions: \paragraph{\boldmath $\bullet$} Chiral operators (eigenfunctions of $\mathbb{Q}_i$-charges): \begin{align} \label{eq:Y-states-app} \begin{pmatrix} Y_{4;1}^{(-)}\\ Y_{4;2}^{(-)}\end{pmatrix}(s,x\,|\,\underline{z}) &= \frac{s^{3/2}}{z_1^2z_2^3}\int^1_0du\,\e^{is(u/z_1+\bar u/z_2)}\, {}_2F_1\left(\genfrac{}{}{0pt}{}{\frac12-ix,\frac12+ix}{2}\Big|-\frac{u}{\bar u}\right) \begin{pmatrix}u z_1\\ u z_2\end{pmatrix}, \notag\\ \begin{pmatrix} Y_{4;1}^{(+)}\\ Y_{4;2}^{(+)}\end{pmatrix}(s,x\,|\,\underline{z}) &= \frac{s^{3/2}}{z_1^2z_2^3}\int^1_0du\,\e^{is(u/z_1+\bar u/z_2)}\, {}_2F_1\left(\genfrac{}{}{0pt}{}{ -ix, +ix}{1}\Big|-\frac{u}{\bar u}\right) \begin{pmatrix}\bar u z_1 \\ -u z_2\end{pmatrix}. \end{align} \paragraph{\boldmath $\bullet$} Operators of different chirality (eigenfunctions of $\widebar{\mathbb{Q}}_i$ charges): \begin{align} \label{app:Zfunctions} \begin{pmatrix} Z_{4;1}^{(-)}\\ Z_{4;2}^{(-)}\end{pmatrix}(s,x\,|\,\underline{z}) &= \frac{s^{3/2}}{z_1^3z_2^3}\int^1_0du\, {u^2}\,\e^{is(u/z_1+\bar u/z_2)}\, {}_2F_1\left(\genfrac{}{}{0pt}{}{-ix, ix}{3}\Big|-\frac{u}{\bar u}\right) \begin{pmatrix}is\,\bar u\\ z_1z_2\end{pmatrix}, \\ \begin{pmatrix} Z_{4;1}^{(+)}\\ Z_{4;2}^{(+)}\end{pmatrix}(s,x\,|\,\underline{z}) &= \frac{s^{3/2}}{z_1^3z_2^3}\int^1_0du\,{u}{\bar u}\,\e^{is(u/z_1+\bar u/z_2)}\, {}_2F_1\left(\genfrac{}{}{0pt}{}{-\frac12-ix,-\frac12+ix}{2}\Big|-\frac{u}{\bar u}\right) \begin{pmatrix}-is\, u\\ z_1z_2\end{pmatrix} \notag\\ \begin{pmatrix} Z_{4;1}^{(0)}\\ Z_{4;2}^{(0)}\end{pmatrix}(s,x\,|\,\underline{z}) &= \begin{pmatrix} Z_{4;1}^{(+)}\\ Z_{4;2}^{(+)}\end{pmatrix}(s,x=i/2\,|\,\underline{z}) ~=~ \frac{s^{3/2}}{z_1^3z_2^3}\int^1_0du \,u\bar u\,\e^{is(u/z_1+\bar u/z_2)}\, \begin{pmatrix}-is\, u\\ z_1z_2\end{pmatrix}.\nonumber \end{align} Both sets form a complete system with the respect of the $\SL(2)$-invariant scalar product, see appendix~\ref{app:scalarproduct}. Note that $Z_4^{(+)}$-functions are related to the twist-three eigenfunctions $Y_3$ as follows: \begin{equation} Z_{4;1}^{+}(s,x|\underline{z}) = -\frac{i}{\sqrt{s}} \big(z_1\partial_{z_1}+2\big)\, Y_3(s,x|\underline{z})\,, \qquad Z_{4;2}^{+}(s,x|\underline{z}) = -\frac{iz_2}{\sqrt{s}}\, Y_3(s,x|\underline{z})\,, \label{ZYidentity} \end{equation} and the same relation is valid between $Z_4^{(0)}$ and $Y_3^{(0)}$. The last step is to calculate the eigenvalues of the Hamiltonians (anomalous dimensions): \begin{equation} N_c\mathbb{H}_4\begin{pmatrix} Y_{4;1}^{(\pm)}\\ Y_{4;2}^{(\pm)}\end{pmatrix} = \gamma_4^{\pm} \begin{pmatrix} Y_{4;1}^{(\pm)}\\ Y_{4;2}^{(\pm)}\end{pmatrix}, \qquad N_c \mathbb{\widebar H}_4\begin{pmatrix} Z_{4;1}^{(\pm)}\\ Z_{4;2}^{(\pm)}\end{pmatrix} = \widebar{\gamma}_4^{\pm} \begin{pmatrix} Z_{4;1}^{(\pm)}\\ Z_{4;2}^{(\pm)}\end{pmatrix}. \end{equation} This can most easily be done by comparing the large-$z_1,z_2$ asymptotic behavior on the both sides. In this way one obtains \begin{align} \gamma_4^{(+)} &= \widebar \gamma_4^{(-)} = N_c[\ln (\mu s) +\gamma_E -5/4] + \gamma_4(x)\,, \notag\\ \gamma_4^{(-)} &= \widebar \gamma_4^{(+)} = N_c[\ln (\mu s) +\gamma_E -5/4] + \gamma_3(x)\,, \end{align} where $\gamma_3(x)$ and $\gamma_4(x)$ are defined in eqs.~\eqref{gamma3} and~\eqref{gamma4}, respectively. Expansion of the B-meson DAs over the eigenfunctions of the evolution equations reads \begin{equation} \Phi_3(\underline{z},\mu) = \int_0^\infty ds \bigg[ \eta_3^{(0)}(s,\mu)\,{Y}_3^{(0)}(s\,| \underline{z}) + \frac12 \int_{-\infty}^\infty dx\,\eta_3(s,x,\mu)\,{Y}_{3}(s,x\,|\,\underline{z})\bigg], \label{Phi3-1} \end{equation} and \begin{align} \begin{pmatrix}\! -2\Phi_4(\underline{z},\mu) \\ [\Psi_4 + \widetilde\Psi_4](\underline{z},\mu) \!\end{pmatrix} \!& =\!- \int\limits^\infty_0{ds}\int\limits^\infty_{-\infty}dx\, \biggl\{\eta_4^{(-)}(s,x,\mu)\,{Y}_{4}^{(-)}(s,x\,|\underline{z}) + \eta_4^{(+)}(s,x,\mu)\,{Y}_{4}^{(+)}(s,x\,|\underline{z})\biggr\}\, , \notag\\ \begin{pmatrix}\! -2\Xi_4(\underline{z},\mu) \\ [\Psi_4 - \widetilde\Psi_4](\underline{z},\mu) \!\end{pmatrix} \!&=\! - \int\limits^\infty_0{ds}\int\limits^\infty_{-\infty}dx\, \biggl\{\varkappa_4^{(-)}(s,x,\mu)\, {Z}_{4}^{(-)}(s,x\,|\underline{z}) + \varkappa_4^{(+)}(s,x,\mu)\,{Z}_{4}^{(+)}(s,x\,|\underline{z})\biggr\}\notag\\ &\quad - {2}\int\limits^\infty_0{ds}\,\varkappa_4^{(0)}(s,\mu)\,{Z}_{4}^{(0)}(s\,|\underline{z})\,, \label{eq:t4general} \end{align} where \begin{equation} {Y}_{4}^{(\pm)} = \begin{pmatrix} {Y}_{4;1}^{(\pm)}\\ {Y}_{4;2}^{(\pm)}\end{pmatrix}, \qquad\qquad {Z}_{4}^{(\pm)} = \begin{pmatrix} {Z}_{4;1}^{(\pm)}\\ {Z}_{4;2}^{(\pm)}\end{pmatrix}. \end{equation} The coefficient functions $\eta_3$, $\eta_4^{(\pm)}$ and $\varkappa_4^{(0,\pm)}$ have autonomous scale dependence (up to $1/N_c^2$ corrections): \begin{align} \eta_3(s,x,\mu) &= L^{\gamma_3(x)/\beta_0}R(s;\mu,\mu_0)\,\eta_3(s,x,\mu_0)\,, \notag\\ \eta_3^{(0)}(s,\mu) &= L^{N_c/\beta_0}R(s;\mu,\mu_0)\eta_3^{(0)}(s,\mu_0)\,, \notag\\ \eta_4^{(+)} (s,x,\mu) &= L^{\gamma_4(x)/\beta_0}R(s;\mu,\mu_0)\, \eta_4^{(+)}(s,x,\mu_0)\,, \notag\\ \eta_4^{(-)} (s,x,\mu) &= L^{\gamma_3(x)/\beta_0}R(s;\mu,\mu_0)\, \eta_4^{(-)}(s,x,\mu_0)\,, \notag\\ \varkappa_4^{(+)} (s,x,\mu) &= L^{\gamma_3(x)/\beta_0}R(s;\mu,\mu_0)\, \varkappa_4^{(+)}(s,x,\mu_0)\,, \notag\\ \varkappa_4^{(-)} (s,x,\mu) &= L^{\gamma_4(x)/\beta_0}R(s;\mu,\mu_0)\, \varkappa_4^{(-)}(s,x,\mu_0)\,, \notag\\ \varkappa_4^{(0)} (s,\mu) &= L^{N_c/\beta_0}R(s;\mu,\mu_0) \varkappa_4^{(0)}(s,\mu_0)\,, \label{scaledep:t34} \end{align} where $L = {\alpha_s(\mu)}/{\alpha_s(\mu_0)}$ and $R(s;\mu,\mu_0)$ is defined in eq.~\eqref{Rfactor}. The expressions in~\eqref{eq:t4general} are valid for the most general case. We will show in appendix~\ref{app:WW} that neglecting contributions of four-particle (quasipartonic)~\cite{Bukhvostov-ml-1985rn} operators of the type $\bar q GG h_v$ and $\bar q q\bar q h_v$, the following relations hold: \begin{align} 2\big(z_1\partial_{z_1}+1\big)\Phi_4(\underline{z}) &= \big(z_2\partial_{z_2}+2\big)\left[\Psi_4 (\underline{z})+\widetilde\Psi_4 (\underline{z})\right], \notag\\ 2z_1\,{\Xi_4}(\underline{z})&= \big(z_2\partial_{z_2}+2\big)\left[\Psi_4 (\underline{z})-\widetilde\Psi_4 (\underline{z})\right]-2\Phi_3(\underline{z})\,. \label{WWidentities} \end{align} Using the representation in eq.~\eqref{eq:t4general} and taking into account eq.~\eqref{ZYidentity} it is easy to show that the relations between the DAs in~\eqref{WWidentities} imply the following relations between the coefficient functions (to the same accuracy): \begin{align} \eta_4^{(-)}(s,x,\mu_) &= 0\,, \notag\\ \sqrt{s} \varkappa_4(s,x,\mu) &= \eta_3(s,x,\mu)\,, \notag\\ \sqrt{s} \varkappa^{(0)}_4(s,\mu) &= \eta^{(0)}_3(s,\mu)\,. \end{align} The resulting simplified expressions for the DAs are given in eq.~\eqref{eq:t4final} in the main text. We remind that quasipartonic four-particle operators do not mix with three-particle operators to the one-loop accuracy~\cite{Bukhvostov-ml-1985rn}. For this reason neglecting such contributions is consistent with the scale dependence. ]]>
0$, imply that they are analytic functions of the light parton coordinates $z$ in the lower half plane, $\text{Im}\,z<0$ and, second, the evolution kernels at one loop order are functions of the $\SL(2,\mathbb{R})$ generators. These properties invite for using a formalism where $z$ is treated as a complex number and conformal symmetry of the equations is implemented explicitly. Such a formalism is well known in mathematical literature. One defines the $\SL(2,\mathbb{R})$ invariant scalar product for functions holomorphic in the lower complex half-plane~\cite{Gelfand} \begin{equation} \label{eq:SL2scalarproduct} \vev{\Phi_1\big|\Phi_2}_j=\int_{\mathbb{C}_-}{\cal D}_jz\,\Phi^*_1(z)\,\Phi_2(z)\, , \end{equation} where the integration goes over the lower half-plane $\mathbb{C}_-$ of the complex plane, $\text{Im}\,z<0$, and the integration measure for spin $j$ is defined as $${\cal D}_jz=\frac{2j-1}\pi d^2z\,[i(z-\bar z)]^{2j-2}.$$ The scalar product~\eqref{eq:SL2scalarproduct} is invariant w.r.t.\ to the $\SL(2,\mathbb{R})$ transformations~\cite{Gelfand} \begin{equation} \Phi(z) \mapsto \frac{1}{(cz+d)^{2j}} \Phi\left(\frac{az+b}{cz+d}\right)\,, \qquad ad-bc =1\,. \end{equation} The generator of special conformal transformation $i S^{(j)}_+$ is self-adjoint w.r.t.\ this scalar product and its eigenfunctions \begin{equation} iS^{(j)}_+\, Q_s^{(j)}(z)=s\,Q_s^{(j)}(z)\,, \qquad Q_s^{(j)}(z)=\frac{e^{-i\pi j}}{z^{2j}} e^{is/z}\,, \end{equation} are orthogonal and form a complete set of functions in the Hilbert space\footnote{This is the so-called Hilbert space of holomorphic functions, see ref.~\cite{Hall} for a review.} defined by eq.~\eqref{eq:SL2scalarproduct}: \begin{align} \langle{Q_s^{(j)}|Q_{s'}^{(j)}\rangle}_j&=\frac{\Gamma(2j)}{s^{2j-1}}\,\delta(s-s')\,, \\ \frac1{\Gamma(2j)}\int_0^\infty ds \, s^{2j-1}\,Q_s^{(j)}(z)\,\overline{Q_s^{(j)}(z')} &= \frac{e^{-i\pi j}}{(z-\bar z')^{2j}}\,. \label{RK} \end{align} The expression on the r.h.s.\ of eq.~\eqref{RK} is the reproducing kernel (unit operator)~\cite{Hall}, i.e.\ for arbitrary function (holomorphic in the lower half-plane) \begin{equation} \Psi(z)=\frac{2j-1}\pi\int_{\mathbb{C}_-} \mathcal{D}_j z'\, \frac{e^{-i\pi j}}{(z-\bar z')^{2j}}\,\Psi(z')\,. \end{equation} Exponential functions $e^{-i\omega z}$, $\omega >0$ form another complete orthogonal set w.r.t.\ the same scalar product, \begin{equation} \langle{e^{-i\omega z}|e^{-i\omega' z}\rangle}_j ={\Gamma(2j)}\,{\omega^{1-2j}}\,\delta(\omega-\omega'). \label{ortho} \end{equation} The momentum space DAs defined by the Fourier transform \begin{equation} \Phi_j(z,\mu)=\int_0^\infty d\omega \, e^{-i\omega z }\,\phi_j(\omega,\mu) \end{equation} can be found making use of~\eqref{ortho} and the following relation: \begin{equation} \langle e^{-i\omega z}|Q_s^{(j)}\rangle_j = \Gamma(2j)\,(\omega s)^{1/2-j}\, J_{2j-1}(2\sqrt{\omega s})\,. \end{equation} In this way one obtains \begin{align} \Phi_j(z,\mu) &= \frac{1}{\Gamma(2j)} \int_0^\infty ds\, s^{2j-1} Q_s^{(j)}(z)\, \eta_j(s,\mu)\,, \notag\\ \phi_j(\omega,\mu) &=\int_0^\infty ds\,\eta_j(s,\mu)\, (s \omega)^{j-1/2}\,J_{2j-1}(2\sqrt{\omega s})\,. \end{align} In particular for $j=1/2$ corresponding to the $B$-meson DA $\phi_-(\omega,\mu)$ the conformal expansion goes over Bessel functions $J_{0}(2\sqrt{\omega s})$ as compared to $J_{1}(2\sqrt{\omega s})$ for the leading twist, in which case $j=1$. The coefficient functions of DAs in the $s$-representation can be written as the scalar products as well, e.g.\ for the leading twist \begin{equation} \eta_+(s,\mu) = \langle Q_s^{(1)} | \Phi_+\rangle\,. \end{equation} The $\SL(2)$ invariant scalar product for the functions of two variables reads, \begin{equation}\label{scalar2} \vev{\Psi \big|\Phi}_{(j_1,j_2)}= \int_{\mathbb{C}_-}\!\!{\cal D}_{j_1}z_1\int_{\mathbb{C}_-}\!\!{\cal D}_{j_2}z_2\,\Psi^*(\underline{z})\,\Phi(\underline{z}) = \Gamma(2j_1)\Gamma(2j_2) \int^\infty_0\!\frac{d\omega_1}{\omega_1^{2j_1-1}}\frac{d\omega_2}{\omega_2^{2j_2-1}}\,\psi^*(\underline\omega)\phi(\underline\omega) \end{equation} in position and momentum space representations, respectively. The scalar product for the twist-$4$ doublets, $\vec{\Phi}(\underline{z})=(\Phi_1(\underline{z}),\Phi_2(\underline{z}))$, can be written in the form \begin{equation} \vev{\overrightarrow{{\Phi}}|\overrightarrow{{\Psi}}} = \vev{\Phi_1|\Psi_1} + c \vev{\Phi_2|\Psi_2}\,, \end{equation} where the two terms on the r.h.s.\ are defined by the scalar product~\eqref{scalar2} with the conformal spins matching those of the corresponding --- up or down --- component of the doublet. The coefficient $c$ is fixed by the requirement that the Hamiltonian and conserved charges are self-adjoint operators. It is easiest to impose this condition on the ``supplementary'' charges, $\mathbb{Q}_3 (\widebar{\mathbb{Q}}_3)$. For example, for the same-chirality doublet one has to require \begin{equation} \langle \Phi_1|\mathbb{Q}_3^{12}\Psi_2\rangle_{(\frac12\frac32)}=c\,\langle \mathbb{Q}_3^{21}\Phi_1| \Psi_2\rangle_{(11)}\,, \end{equation} where from it follows that $c=2$. In this way we obtain (for all possible combinations of superscripts $\pm,0$ in the bra- and ket-states) \begin{align} \vev{Y_4 (s,x)\big|Y'_4 (s',x')}&=\vev{Y_{4;1}(s,x)\big|Y'_{4;1}(s',x')}_{(\frac12,\frac32)} +2\,\vev{Y_{4;2}(s,x)\big|Y'_{4;2}(s',x')}_{(1,1)}\,, \notag\\ \vev{Z_4 (s,x)\big|Z'_4 (s',x')}&=\vev{Z_{4;1}(s,x)\big|Z'_{4;1}(s',x')}_{(\frac32,\frac32)} +4\,\vev{Z_{4;2}(s,x)\big|Z'_{4;2}(s',x')}_{(1,1)}\,. \end{align} The normalization conditions of the twist-three and four eigenstates read: \begin{align} \langle {Y}^{(0)}_{3}(s),{Y}^{(0)}_{3}(s') \rangle_{(1,\frac32)}&=\delta(s-s')\,, \notag\\ \langle {Y}_{3}(s,x),{Y}_{3}(s',x') \rangle_{(1,\frac32)}&=\delta(s-s') \delta(x-x') \frac{\coth\pi x}{x(x^2+9/4)}\,, \notag\\ \vev{Y^{(-)}_4 (s,x)\big|Y^{(-)}_4 (s',x')}&=\delta(s-s')\delta(x-x')\frac{\coth\pi x}{x(x^2+1/4)}\, , \notag\\ \vev{Y^{(+)}_4 (s,x)\big|Y^{(+)}_4 (s',x')}&=\delta(s-s')\delta(x-x')\frac{\tanh\pi x}{x}\, , \notag\\ \vev{Z^{(-)}_4 (s,x)\big|Z^{(-)}_4 (s',x')}&=\delta(s-s')\delta(x-x')\frac{8\tanh\pi x}{x(x^2+1)(x^2+4)}\, , \notag\\ \vev{Z^{(+)}_4 (s,x)\big|Z^{(+)}_4 (s',x')}&=\delta(s-s')\delta(x-x')\frac{2\coth\pi x}{x(x^2+9/4)}\, , \notag\\ \vev{Z^{(0)}_4 (s)\big|Z^{(0)}_4 (s')}&=2\delta(s-s')\,. \end{align} Here it is assumed that $x>0$ and $x'>0$. This is sufficient because all eigenfunctions are symmetric under reflection $x\to -x$. Scalar products for the pairs of the eigenfunctions with different superscripts vanish. The coefficient functions appearing in the expansion of the DAs in the eigenstates of the evolution equation in~\eqref{Phi3-1},~\eqref{eq:t4general} can be calculated as, for twist three, \begin{equation} \eta_3(s,x,\mu) = \frac{x(x^2+9/4)}{\coth\pi x} \vev{Y_3(s,x)\big| \Phi_3}\,, \qquad\qquad \eta^{(0)}_3(s,\mu) = \vev{Y^{(0)}_3(s)\big| \Phi_3}\,, \end{equation} and for twist four, \begin{align} \eta_4^{(-)}(s,x,\mu)&= \frac{x(x^2+1/4)}{\coth\pi x} \left[\vev{Y^{(-)}_{4;1}(s,x)\big|\Phi_4}_{(\frac12,\frac32)}-\vev{Y^{(-)}_{4;2}(s,x)\big|(\Psi_4+\widetilde\Psi_4)}_{(1,1)}\right], \notag \\ \eta_4^{(+)}(s,x,\mu)&= \frac{x}{\tanh\pi x} \left[\vev{Y^{(+)}_{4;1}(s,x)\big|\Phi_4}_{(\frac12,\frac32)} -\vev{Y^{(+)}_{4;2}(s,x)\big|(\Psi_4+\widetilde\Psi_4)}_{(1,1)}\right], \notag\\ \varkappa_4^{(-)}(s,x,\mu)&=\frac{x(x^2\!+\!1)(x^2\!+\!4)}{8\tanh\pi x} \left[\vev{Z^{(-)}_{4;1}(s,x)\big|{\Xi}_4}_{(\frac32,\frac32)}-2\vev{Z^{(-)}_{4;2}(s,x)\big|(\Psi_4-\widetilde\Psi_4)}_{(1,1)}\right], \notag\\ \varkappa_4^{(+)}(s,x,\mu)&=\frac{x(x^2+9/4))}{2\coth\pi x} \left[\vev{Z^{(+)}_{4;1}(s,x)\big|{\Xi}_4}_{(\frac32,\frac32)}-2\vev{Z^{(+)}_{4;2}(s,x)\big|(\Psi_4-\widetilde\Psi_4)}_{(1,1)}\right]. \end{align} Note that $\eta_3^{(0)}(s,\mu)$ is related to the residue of $\eta_3(s,x,\mu)$ at imaginary $x=i/2$: \begin{equation} \eta_3(s,x,\mu)\Big|_{x\to i/2} = \frac{x (x^2+9/4)}{\coth \pi x} \eta_3^{(0)}(s,\mu)\Big|_{x\to i/2} = \frac{1}{x-i/2} \frac{i}{\pi} \eta_3^{(0)}(s,\mu) + \ldots \end{equation} ]]>
2\right)\, , \\ [\widebar H^{21}_{qg}\varphi](z_1,z_2)&=\frac{N_cz_{12}}9\frac{\Gamma\Big(\frac13J^{gg}_{(\frac32,\frac32)}-2/3\Big)} {\Gamma\Big(\frac13J^{gg}_{(\frac32,\frac32)}+4/3\Big)}-\frac{2z_{12}(-1)^{J^{qg}_{(\frac32,\frac32)}}}{N_c}\frac{\Gamma\Big(J^{qg}_{(\frac32,\frac32)}-2\Big)} {\Gamma\Big(J^{qg}_{(\frac32,\frac32)}+2\Big)}\, ,\nonumber\\* [\widebar H^{22}_{qg}\varphi](z_1,z_2)&=N_c\Big[\psi\Big(J^{qg}_{(1,1)}+1\Big)+\psi\Big(J^{qg}_{(1,1)}-1\Big)-2\psi(1)-\frac34\Big] -\frac{\delta_{J^{qg}_{(1,1)},2}}{3N_c}\, , \end{align}}where operator $J^{qg}$ is defined in terms of their corresponding quadratic Casimir operators $J^{qg}_{(j_1,j_2)}(J^{qg}_{(j_1,j_2)}-1)=(\vec S_{(q,j_1)}+\vec S_{(g,j_2)})^2$ and we have labeled the conformal spins of each $J^{qg}$ accordingly. $\Gamma(z)$ stands for the Euler-Gamma function while $\psi(z)$ is the $0$-th order polygamma function. ]]>