Contraction of Leptonic Tensors in Electromagnetic Interactions
For the scattering discussed in Chapter 9, the transition matrix element squared is written as
where is the momentum transfer and the factor of is for the averaging over the initial electron and muon spins.
The leptonic current is given by
In Eq. (D.2),
• Adjoint Dirac spinor is a 1 × 4 matrix,
• Dirac spinor (u) is a 4 × 1 matrix,
• γμ is a 4 × 4 matrix,
• Ultimately, we have (1 × 4)(4 × 4)(4 × 1) = A, a number,
• For any number A, the complex conjugate and the Hermitian conjugate are the same thing.
Therefore, instead of, we may write
we can rewrite the aforementioned expression in the component form for an electronic tensor as:
where we have used the trace properties,
The trace of an odd number of gamma matrices is zero. Similarly,
Using Eqs. (D.3) and (D.4), we get
Contraction of Leptonic Tensors in the Case of Weak Interactions
For the scattering discussed in Chapter 9, where the interaction is mediated by a W boson, the transition matrix element squared is expressed as
where the factor of is for the averaging over the initial muon spin.
We can rewrite this expression in the component form as:
Similarly, for the muonic tensor,
Using Eqs (D.8) and (D.9), the transition matrix element squared is obtained as
Contraction of Weak Leptonic Tensor with Hadronic Tensor
Contracting the various terms of hadronic tensor with the leptonic tensor, we get