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Hi guys,
I have a very basic question about the WZ model. I want to show that it is invariant under SUSY transformations.
The action is \int{d^4 x} \partial^\mu \phi* \partial_\mu \phi +i\psi^† \bar{\sigma}^\mu \partial_\mu \psi
The SUSY transformations are \delta\phi = \epsilon \psi, \delta\phi* = \epsilon^† \psi^†, \delta\psi_\alpha = -i(\sigma^\nu \epsilon^†)_\alpha \partial_\nu \phi, \delta\psi^†_\alpha =i(\epsilon \sigma^\nu)_\alpha\partial_\nu \phi*
I know that I should get a total derivative that vanishes but there is one minus sign that is preventing me from completing the proof. It comes from the term i\psi^† \bar{\sigma}^\mu \partial_\mu \delta\psi in the variation. For this term I get the following:
\psi^† \bar{\sigma}^\mu \sigma^\nu \epsilon^† \partial_\mu \partial_\nu \phi
Then by using \bar{\sigma}^\mu \sigma^\nu + \bar{\sigma}^\nu \sigma^\mu = 2\eta^{\mu\nu} I have
\psi^† \epsilon^† \partial^\mu \partial_\mu \phi
I rewrite this as a total derivative minus another term
\partial^\mu(\psi^† \epsilon^† \partial_\mu \phi)-\partial^\mu\psi^† \epsilon^† \partial_\mu \phi
This is where the problem is. Now, in order for the second term to cancel a term from the variation of φ*, it should have exactly this form but with ψ† and ε† in reverse order. However, doing this introduces a minus sign as the spinors are grassmann odd. What have I done wrong in the manipulation, and how do I get rid of this extra minus sign?
I have a very basic question about the WZ model. I want to show that it is invariant under SUSY transformations.
The action is \int{d^4 x} \partial^\mu \phi* \partial_\mu \phi +i\psi^† \bar{\sigma}^\mu \partial_\mu \psi
The SUSY transformations are \delta\phi = \epsilon \psi, \delta\phi* = \epsilon^† \psi^†, \delta\psi_\alpha = -i(\sigma^\nu \epsilon^†)_\alpha \partial_\nu \phi, \delta\psi^†_\alpha =i(\epsilon \sigma^\nu)_\alpha\partial_\nu \phi*
I know that I should get a total derivative that vanishes but there is one minus sign that is preventing me from completing the proof. It comes from the term i\psi^† \bar{\sigma}^\mu \partial_\mu \delta\psi in the variation. For this term I get the following:
\psi^† \bar{\sigma}^\mu \sigma^\nu \epsilon^† \partial_\mu \partial_\nu \phi
Then by using \bar{\sigma}^\mu \sigma^\nu + \bar{\sigma}^\nu \sigma^\mu = 2\eta^{\mu\nu} I have
\psi^† \epsilon^† \partial^\mu \partial_\mu \phi
I rewrite this as a total derivative minus another term
\partial^\mu(\psi^† \epsilon^† \partial_\mu \phi)-\partial^\mu\psi^† \epsilon^† \partial_\mu \phi
This is where the problem is. Now, in order for the second term to cancel a term from the variation of φ*, it should have exactly this form but with ψ† and ε† in reverse order. However, doing this introduces a minus sign as the spinors are grassmann odd. What have I done wrong in the manipulation, and how do I get rid of this extra minus sign?
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