The notation below, is consistent with Wess and Bagger's https://www.amazon.com/Supersymmetry-Supergravity-Julius-Wess/dp/0691025304".(adsbygoogle = window.adsbygoogle || []).push({});

Given a Majorana spinor field in 4D, written in 2-component notation as

[tex] \Psi(x) = \begin{pmatrix} \psi(x) \\\\ \bar\psi(x) \end{pmatrix} ,

\quad (\psi_\alpha)^* = \bar\psi_{\dot\alpha} \ ,

[/tex]

how many linearly independent Lorentz invariants can be formed using 2n spinors and n space-time derivatives, tied together with the sigma/Pauli matrices and various metrics?

And, more importantly for my application, how many are there modulo total derivatives?

Is there a general (eg representation theory) approach to this type of problem?

Notes:

- I am mainly (at the moment) concerned with the n=2 case.
- Due to anticommutativity, there are no such terms with n>4.
- This can obviously be rewritten using 4-component Majorana spinors and Dirac matrices.

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For example with n=1 the only invariant is the standard kinetic term

[tex] \psi \sigma^a \partial_a \bar\psi

= \psi^\alpha \sigma^a_{\alpha\dot\alpha} \partial_a \bar\psi^{\dot\alpha}

= - (\partial_a\psi) \sigma^a \bar\psi + \text{total derivative} \ .

[/tex]

For [tex] n=2 [/tex], I believe (and want to prove) that there are only 6 invariants up to total derivatives.

Defining the matrix

[tex] v_a{}^b = i \psi\sigma^b\partial_a\bar\psi [/tex]

and its complex conjugate

[tex] \bar v_a{}^b = -i (\partial_a\psi)\sigma^b\bar\psi [/tex],

I chose the basis(?)

[tex] (\partial^a\psi^2)(\partial_a\bar\psi^2) \ ,\; tr(v)tr(\bar v) \ , \;

tr(v)^2\ ,\; tr(v^2)\ ,\; tr(\bar v)^2 \,\; tr(\bar v^2)\ .

[/tex]

Other terms being related by (for example)

[tex] tr(v\bar v) = tr(v)tr(\bar v)

+\tfrac12\Big(tr(v^2)-tr(v)^2+tr(\bar v^2)-tr(\bar v)^2\Big)

+ \text{total derivative}

[/tex]

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# Number of 4-fermion, 2-derivative Lorentz invariants

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