\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\}\
