min}) \ ,
\label{Dp0n}$$ for all $n$. This identity is useful for the following discussions on the general features of the transverse momentum broadening. From Eq. (\[pilt\]), the transverse momentum broadening, $\Delta\langle
\ell_{T}^2\rangle_{1/3} (\ell_{\rm min})$, depends on the second moments $D^{(i)}(2,z_{\rm
min})$, which are plotted in Fig. \[fig3\].

Given the quark-to-pion fragmentation functions, we derive the following general relations between the transverse momentum broadening of $\pi^+$, $\pi^-$, and $\pi^0$ particles:

$$\Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^+}
- \Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^0} =
\Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^0}
- \Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^-} \ ,
\label{symmetry}$$

and $$\Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^+}
> \Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^0} 
> \Delta \langle \ell_{T}^2\rangle_{1/3}^{\pi^-} \ ,
\label{p0m}$$

independent of the details of the correlation functions $T_{qF}^A$. Eq. (\[symmetry\]) is a result of the isospin symmetry of the fragmentation functions given in Eq. (\[D\]) and the identity given in Eq. (\[Dp0n\]). The inequality given in Eq. (\[p0m\]) results from the fact that the contribution of each quark flavor is weighted by the square of the quark’s fractional charge, $e_q^2$, and the relation $D^+(z)>D^0(z)>D^-(z)$.

Since the correlation functions $T_{qF}^A$ are not known, in order to obtain numerical estimates of the transverse momentum broadening of pions, we adopt the following model for quark-gluon correlation functions [@LQS1; @LQS2]: 