D^+(z)=D_u^{\pi ^+}=D_d^{\pi ^-}=D_{\bar{u}}^{\pi ^-}
=D_{\bar{d}}^{\pi ^+} &=&\frac{1-z^2}{4z} \ ;
\label{D+} \\
D^-(z)=D_u^{\pi ^-}=D_d^{\pi ^+}=D_{\bar{u}}^{\pi ^+}
=D_{\bar{d}}^{\pi ^-} &=& \frac{(1-z)^2}{4z} \ ;
\label{D-} \\
D^0(z)=D_u^{\pi ^0}=D_d^{\pi ^0}=D_{\bar{u}}^{\pi ^0}
=D_{\bar{d}}^{\pi ^0} &=& \frac{1-z}{4z} \ ;
\label{D01} \end{aligned}$$

with $D^0(z)=[D^+(z)+D^-(z)]/2$. For the strange quark fragmentation functions, we use $D_s^{\pi ^+}=D_s^{\pi ^-}=D_s^{\pi ^0}=D^-$. Notice that the fragmentation functions given in Eq. (\[D\]), as well as those given in Ref. [@newDD; @newD], violate Gribov-Lipatov reciprocity, which requires that the power of $(1-z)$ must be even. However, since it is not our purpose to invent better fragmentation functions in this paper, we will first use the fragmentation functions in Eq. (\[D\]) in our numerical calculation to illustrate the size and the general features of the transverse momentum broadening. Our predictions can be easily adjusted for other sets of fragmentation functions.

In order to evaluate the transverse momentum broadening, Eq. (\[pilt\]) requires the moments of the fragmentation functions. We therefore introduce $$D^{(i)}(n,z_{\rm min}) \equiv \int_{z_{\rm min}}^1 \,
dz\, z^n\, D^{(i)}(z) \ ,
\label{Dzn}$$ where $i=+, -$ and $0$ for $\pi^+$, $\pi^-$ and $\pi^0$, respectively. Using the fragmentation functions defined in Eq. (\[D\]), we obtain the following identity, $$D^{+}(n,z_{\rm min}) - D^{0}(n,z_{\rm min}) 
= D^{0}(n,z_{\rm min}) - D^{-}(n,z_{\rm 