F-\frac{C_A}{2}\)\ln\(\frac{q^+}{\hat p^+_\xi}\)
+C_A\ln\(\frac{q^+}{\hat p^+_A}\)\Big]\Bigg\}+O(a_s^2).
\end{eqnarray}
The factor $Z_{U1}^{\text{sub}}$ is a half of the TMD renormalization constant and coincides with it \cite{Echevarria:2016scs}. The factor $Z^{\text{sub}}_{U2}$ is a new one, to our best knowledge.

The rapidity divergent factor $R$ cancels between the soft factor and TMD operators. The leftover of the rapidity divergences are the scaling parameters $\nu^\pm$. In the final definition of the TMD operator in eq.~(\ref{def:TMDop-sub}), we replace $\nu^\pm$ by $\zeta$ and $\bar \zeta$ using that
\begin{eqnarray}
\nu^+=q^+\sqrt{\frac{\nu^2}{\zeta}},\qquad \nu^-=q^-\sqrt{\frac{\nu^2}{\bar \zeta}}.
\end{eqnarray}
The $\nu^2=2\nu^+\nu^-$ is a low-energy parameter related to the definition of soft modes, and it can be hidden in the definition of TMD distribution. The scaling variables $\zeta$ and $\bar \zeta$ satisfy
\begin{eqnarray}\label{def:zeta*zeta=q+q-}
\zeta \bar \zeta=(2q^+q^-)^2.
\end{eqnarray}

The definition of the subtracted TMD operator incorporates the remnant of the soft factor $S_{0}$, which is a finite number that depends on $b^2$. We recall that the function $S_0$ is nonperturbative. The absorption of $S_0$ into TMD distributions is a part of a scheme definition for the rapidity renormalization. Effectively it adds finite terms to the ``minimal subtraction scheme'' factor $R$. The equation (\ref{def:OO/S=OO}) serves as the definition of the scheme, i.e. the final expression for the effective operator in DY, SIDIS, and SIA does not contain any extra factors. This statement defines the commonly used physical TMD distributions. It also implies that the TMD factorization formula for processes, which contain a soft factor different from $\mathcal{S}_{\text{LP}}$, would have an extra nonperturbative function composed from the remnants of the soft factors. For example, such function occurs in the factorization theorem for quasi-TMD distribution \cite{Vladimirov:2020ofp}.

\subsection*{Evolution equations for TMD operators}

The TMD operator $\mathcal{O}_{NM,\bar n}$ has two scaling parameters $\mu$ and $\zeta$. The evolution equations with respect to these parameters follow from the scaling invariance of bare TMD operator eq.~(\ref{def:ONM-renorm}).

The evolution equation with respect to $\mu$ is
\begin{eqnarray}\label{evol:mu}
\mu^2 \frac{d}{d\mu^2}\mathcal{O}^{\text{sub}}_{NM,\bar n}(\zeta,\mu)
=\(\gamma_N(\mu,\zeta)+\gamma_M(\mu,\zeta)\)\otimes \mathcal{O}^{\text{sub}}_{NM,\bar n}(\zeta,\mu),
\end{eqnarray}
where
\begin{eqnarray}
\gamma_N(\mu,\zeta)=-\mu^2\frac{dZ^{\text{sub}}_{N}(\zeta)}{d\mu^2},
\end{eqnarray}
The symbol $\otimes$ indicate a possible integral convolution. Each anomalous dimension acts on corresponding arguments. 

The evolution equation (\ref{evol:mu}) is general in the sense that a TMD operator of any twist-(N+M) satisfies it. The value of twist is preserved by the evolution, which is a part of geometrical twist definition. However, if there are several operators $U_N$ of the same twist and other quantum numbers they can mix with each other.

In the present case, we have deal on with twist-1 and twist-2 operators. The corresponding anomalous dimensions are
\begin{eqnarray}
\gamma_1(\mu,\zeta)&=&a_sC_F\(\frac{3}{2}+\ln\(\frac{\mu^2}{\zeta}\)\)+O(a_s^2),
\\
\gamma_2(\mu,\zeta)&=&a_s\Bigg\{
2\mathbb{H}_1
+\Big[C_F\(\frac{3}{2}+\ln\(\frac{\mu^2}{\zeta}\)\)
\\\nonumber &&\qquad
+2\(C_F-\frac{C_A}{2}\)\ln\(\frac{q^+}{\hat p^+_\xi}\)
+C_A\ln\(\frac{q^+}{\hat p^+_A}\)\Big]\Bigg\}+O(a_s^2).
\end{eqnarray}
In momentum space, the expression for $\mathbb{H}$ is given in app.~\ref{app:evol-momentum} and $\hat p^+/q^+$ should be replaced by corresponding momentum fractions. E.g. for $\gamma_2\otimes \mathcal{O}_{21,\bar n}(x_{1,2,3},b)$ one replaces
\begin{eqnarray}
\ln\(\frac{q^+}{\hat p^+_\xi}\)\to -\ln x_1,\qquad \ln\(\frac{q^+}{\hat p^+_A}\)\to -\ln x_2.
\end{eqnarray}

These anomalous dimensions can be written in a more general form
\begin{eqnarray}\label{def:gamma1}
\gamma_1(\mu,\zeta)&=&\frac{\Gamma_{\text{cusp}}}{4}\ln\(\frac{\mu^2}{\zeta}\)-\frac{\gamma_V}{4},
\\\label{def:gamma2}
\gamma_2(\mu,\zeta)&=&\mathbb{H}_{A\xi}+
\frac{\Gamma_{\text{cusp}}}{4}\ln\(\frac{\mu^2}{\zeta}\)
+\frac{\Gamma_{\text{cusp}}-\Gamma_g}{2} \ln\(\frac{q^+}{\hat p^+_\xi}\)
+\frac{\Gamma_g}{2}\ln\(\frac{q^+}{\hat p^+_A}\),
\end{eqnarray}
where $\Gamma_{\text{cusp}}$ is the (quark) light-like cusp anomalous dimension (known up to N$^3$LO \cite{Henn:2019rmi,vonManteuffel:2020vjv}), $\gamma_V$ is the anomalous dimension of the quark vector form factor (known up to NNLO, see e.g. \cite{Gehrmann:2010ue}). The integral kernel $\mathbb{H}_{A\xi}$ is the evolution kernel for the quark-gluon pair. And $\Gamma_g$ is some constant $\Gamma_g=2a_sC_A+O(a_s^2)$. The sum of the last two terms in eq.~(\ref{def:gamma2}) must be $\Gamma_{\text{cusp}}$, since the sum of $\ln(q^+)$ generates $\ln(\zeta)$ whose coefficient is fixed by the integrability condition eq.~(\ref{def:integrability}).

The evolution with respect to the rapidity parameter follows from the independence of bare TMD operators eq.~(\ref{def:O11-renorm}, \ref{def:O12-renorm}) on the parameter $\nu^+$. It gives
\begin{eqnarray}\label{evol:zeta}
\zeta \frac{d\mathcal{O}^{\text{sub}}_{NM,\bar n}(\zeta,\mu)}{d\zeta}=-2\nu^+\frac{d\mathcal{O}^{\text{sub}}_{NM,\bar n}(\zeta,\mu)}{d\nu^+}=-\mathcal{D}(b,\mu)\mathcal{O}^{\text{sub}}_{NM,\bar n}(\zeta,\mu),
\end{eqnarray}
with
\begin{eqnarray}
\mathcal{D}(b,\mu)=-\frac{\nu^+}{2}\frac{d}{d\nu^+}R\(b^2,\frac{\delta^+}{\nu^+}\).
\end{eqnarray}
The function $\mathcal{D}(b,\mu)$ is the rapidity anomalous dimension. It is also known as the Collins-Soper kernel ($\tilde K=-2\mathcal{D}$), originally introduced in ref.~\cite{Collins:1981uk}. For a relation among different definitions of the CS-kernel, see ref.~\cite{Scimemi:2018xaf}. The LO expression at small $b$ is
\begin{eqnarray}\label{RAD-LO}
\mathcal{D}(b,\mu)&=&-2a_s C_F\Bigg[
\Gamma(-\epsilon)\(-\frac{b^2 \mu^2}{4e^{-\gamma_E}}\)^\epsilon+\frac{1}{\epsilon}\Bigg]+O(a_s^2)
\\\nonumber &=&2a_SC_F \ln\