Contraction of Leptonic Tensors in ElectromagneticInteractions

For the scattering discussed in Chapter 9, the transition matrix elementsquared is written as

where is the momentum transfer and the factor of is for the averaging overthe initial electron and muon spins.

The leptonic current is given by

In Eq. (D.2),

• • Adjoint Dirac spinor is a 1 × 4matrix,

• • 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 andthe Hermitian conjugate are the same thing.

Therefore, instead of, we may write

we can rewrite the aforementioned expression in the component form for anelectronic 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 mediatedby a W boson, the transition matrix element squared is expressed as

where the factor of is for the averaging over the initial muon spin.

Leptonic tensor,

Therefore,

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 obtainedas

Contraction of Weak Leptonic Tensor with Hadronic Tensor

Contracting the various terms of hadronic tensor with the leptonic tensor, weget

Where