I'm having some trouble understanding the kinematics in SIDIS.
For example take the reaction $l+N\to l'+h+X$, where $l$ is a lepton, $N$ is the target Nucleon and $h$ is a hadron.
The kinematic variables used to describe this proccess are $$q=(l-l')$$ $$\nu=(E_l-E_{l'})$$ $$Q^2=-q^2$$ $$x_{Bj}=Q^2/2P\cdot q=Q^2/2M\nu$$ $$z=P\cdot p/ P\cdot q$$
Where q is the 4-momentum transfer and P and p are the 4-momentum of N and h.
The differential multiplicity of the measured hadron is
$$\frac{dN}{dzdp_t^2}$$
So the final state hadron is specified from the measurements of $(z,p_t)$
Now what I'm having trouble to understand is:
The energy of the hadron is $E_h=\sqrt{m_h^2 + p_t^2 + p_L^2}$, where $p_t$ and $p_L$ are the transverse and longitudinal momentum
On the other hand, in a fixed target experiment where the 3-momentum of the nucleon is $0$, we have $z=E_h/\nu \Rightarrow E_h=z\nu$. But the problem here is $E_h$ doesn't depend on the hadron's $p_t$. So where did that dependence go?
In reactions such as those in proton-proton collisions, we usually have the rapidity from the measured hadron $\eta$ which is a function of $p_L$, then we can recover $E_h$ using the usual energy-momentum relation. Can we relate $z$ to the longitudinal momentum?
