Show that current conservation at the hadronic vertex requires qμW^μv=qνW^μv=0 . The proof may b

Step 1: **Understanding Current Conservation at the Hadronic Vertex** Current conservation at th

Show that current conservation at the hadronic vertex...

Show that current conservation at the hadronic vertex requires $$ q_\mu W^{\mu v}=q_\nu W^{\mu v}=0 . $$ 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 independent; 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
$$
q_\mu W^{\mu v}=q_\nu W^{\mu v}=0 .
$$

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 independent; 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

Lorentz Invariance and Tensor Decomposition

Lorentz invariance requires the physical observables to be independent of the choice of reference frame. This principle constrains the form of the hadronic tensor, which must be constructed from the available four-momenta and the metric tensor in a Lorentz-covariant manner. The tensor decomposition uses these invariant combinations to express the hadronic tensor in a form that automatically respects the symmetry properties of spacetime, hence ensuring the correct transformation behavior under Lorentz transformations.

Kinematic Variables in Inelastic Scattering

In inelastic scattering processes, two independent Lorentz scalar variables naturally arise: the squared four-momentum transfer, q², and the energy transfer scaled by the hadron mass, ?. These variables are used to describe the energy and momentum exchanged in the interaction and are directly related to measurable quantities in scattering experiments. The invariant mass of the final hadronic state is then expressed in terms of these variables, providing a complete kinematic description of the reaction.

Current Conservation

Current conservation is a fundamental principle arising from gauge invariance. In the context of hadronic interactions, it implies that the divergence of the current, ??J??, is zero. This results in the requirement that the four-momentum q? contracted with the hadronic tensor W?? (i.e. q?W?? and q?W??) vanishes. This property ensures that only physical degrees of freedom, consistent with the conservation law, contribute to the scattering process.

Hadronic Tensor and Structure Functions

The hadronic tensor, W??, encapsulates all the information about the hadronic structure in inelastic scattering and is expressed in terms of several structure functions. These structure functions, such as W1 and W2, are scalar functions that depend on the Lorentz invariants constructed from the momenta at the vertex. They are crucial for describing how the internal structure of the hadron influences the scattering process. The relations among the structure functions, derived from current conservation, reduce the number of independent functions, thus simplifying the analysis.

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