broadening, $\Delta\langle \ell_{T}^2\rangle_{1/3} (\ell_{\rm min})$, depend on the explicit functional form of the quark-to-pion fragmentation functions and the quark-gluon correlation functions. In this section, without assuming any specific form for the quark-gluon correlation functions, we derive relations for the transverse momentum broadening between $\pi^{\pm}$ and $\pi^0$ based on isospin symmetry and the charge conjugation invariance. Furthermore, by using the simple model proposed in Ref. [@LQS1] for the quark-gluon correlation functions, we explore both the normalization and functional dependence of the momentum broadening.

Although new parameterizations of the quark-to-pion fragmentation functions were obtained recently [@newDD], we will use the simple parameterizations of Ref.[@Dpi] for the following analytical derivations and discussions on the general features of the momentum broadening. Later, when we present our figures for the numerical values of the momentum broadening, we will use the parameterizations from both Refs. [@Dpi] and [@newD], and demonstrate the similarities and differences.

Like the parton distributions, the quark-to-pion fragmentation functions have scaling violation (or $Q^2$-dependence). To simplify our discussion on the general features of transverse momentum broadening, we ignore the scaling violation of the fragmentation functions, and take for the fragmentation functions the input distributions of Ref. [@Dpi], which are given as:

\[D\] $$\begin{aligned}
