Show that current conservation at the hadronic vertex requires qμ W^μν=qν W^μν=0 The proof may b

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Show that current conservation at the hadronic vertex...

Show that current conservation at the hadronic vertex requires $$ q_{\mu} W^{\mu \nu}=q_{\nu} W^{\mu \nu}=0 $$ The proof may be left until after (8.39); it follows from $\partial_{\mu} \tilde{J}^{\mu}=0$. As a result of (8.26), verify that $$ \begin{aligned} &W_{5}=-\frac{p \cdot q}{q^{2}} W_{2} \\ &W_{4}=\left(\frac{p \cdot q}{q^{2}}\right)^{2} W_{2}+\frac{M^{2}}{q^{2}} W_{1} \end{aligned} $$ Thus, only two of the four inelastic structure functions of (8.24) are indepenlent; so we may write $$ W^{\mu \nu}=W_{1}\left(-g^{\mu \nu}+\frac{q^{\mu} q^{\nu}}{q^{2}}\right)+W_{2} \frac{1}{M^{2}}\left(p^{\mu}-\frac{p \cdot q}{q^{2}} q^{\mu}\right)\left(p^{\nu}-\frac{p \cdot q}{q^{2}} q^{\nu}\right), $$ where the $W_{i}$ 's are functions of the Lorentz scalar variables that can be constructed from the four-momenta at the hadronic vertex. Unlike elastic scattering, there are two independent variables, and we choose $$ q^{2} \quad \text { and } \quad \nu \equiv \frac{p \cdot q}{M} . $$ The invariant mass $W$ of the final hadronic system is related to $\nu$ and $q^{2}$ by $$ W^{2}=(p+q)^{2}=M^{2}+2 M \nu+q^{2} . $$

Show that current conservation at the hadronic vertex requires
$$
q_{\mu} W^{\mu \nu}=q_{\nu} W^{\mu \nu}=0
$$
The proof may be left until after (8.39); it follows from $\partial_{\mu} \tilde{J}^{\mu}=0$. As a result of (8.26), verify that
$$
\begin{aligned}
&W_{5}=-\frac{p \cdot q}{q^{2}} W_{2} \\
&W_{4}=\left(\frac{p \cdot q}{q^{2}}\right)^{2} W_{2}+\frac{M^{2}}{q^{2}} W_{1}
\end{aligned}
$$
Thus, only two of the four inelastic structure functions of (8.24) are indepenlent; so we may write
$$
W^{\mu \nu}=W_{1}\left(-g^{\mu \nu}+\frac{q^{\mu} q^{\nu}}{q^{2}}\right)+W_{2} \frac{1}{M^{2}}\left(p^{\mu}-\frac{p \cdot q}{q^{2}} q^{\mu}\right)\left(p^{\nu}-\frac{p \cdot q}{q^{2}} q^{\nu}\right),
$$
where the $W_{i}$ 's are functions of the Lorentz scalar variables that can be constructed from the four-momenta at the hadronic vertex. Unlike elastic scattering, there are two independent variables, and we choose
$$
q^{2} \quad \text { and } \quad \nu \equiv \frac{p \cdot q}{M} .
$$
The invariant mass $W$ of the final hadronic system is related to $\nu$ and $q^{2}$ by
$$
W^{2}=(p+q)^{2}=M^{2}+2 M \nu+q^{2} .
$$

Key Concepts

Current Conservation

Current conservation is a fundamental principle in quantum field theory that follows from gauge invariance. It dictates that the divergence of the current, ??J^?, must vanish. In scattering processes, this leads to important constraints such as q_?W^?? = 0, ensuring that only physical, transverse degrees of freedom contribute and that the theory remains consistent and renormalizable.

Hadronic Tensor

The hadronic tensor encapsulates the complex internal dynamics of hadronic matter when probed by an external current. It is constructed to respect Lorentz invariance and current conservation, and its structure is parameterized by a set of scalar functions. These functions, or structure functions, summarize the response of the hadronic system and incorporate details of the quarks and gluons within the nucleon.

Inelastic Structure Functions

Inelastic structure functions are scalar functions that characterize the distribution of momentum and spin among the constituents of a nucleon or hadron during inelastic scattering processes. Due to the constraints imposed by current conservation, only a subset of the initially defined structure functions are independent. This reduction simplifies the description of the hadronic response and is crucial for understanding the underlying physics in deep inelastic scattering experiments.

Lorentz Invariance

Lorentz invariance is a symmetry principle that requires the equations governing physical processes to remain form-invariant under Lorentz transformations. This principle ensures that physical observables in scattering processes depend only on scalar quantities constructed from the four-momenta, leading to a simplified and frame-independent description of the dynamics.

Invariant Kinematic Variables

Invariant kinematic variables, such as q² and ?, are constructed from the four-momenta of the interacting particles to ensure a frame-independent description of the scattering process. These variables, along with derived quantities like the invariant mass of the final system, play a crucial role in parameterizing the physical observables and in the analysis of both elastic and inelastic scattering phenomena.

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