Superfield Action: Phi, A, F, Psi & Theta Dimensions Explained

[alialice]

Consider the superfield phi in two dimensions
[itex] \phi\left( x,\theta \right) = A\left(x\right) + i \bar{\theta} \psi\left(x\right) + \frac{i}{2}\bar{\theta}\theta F\left(x\right) [/itex]
where the dimension of phi and A is zero (scalar field), the dimension of F is 1 (auxiliary field), the dimension of psi and theta is 1/2 (spinorial field).
The spinor theta has two real component (majorana condition): theta1 and theta2.
Which is the action for this superfield?
The mass term is [itex] \int d^2 x d\theta_1 d\theta_2 m \phi^{2}[/itex]
The cubic interaction term is [itex] \int d^2 x d\theta_1 d\theta_2 \lambda \phi^3[/itex]
And the kinetic term?
Please help me!

 

[AndyW]

I would like to clarify a few things before discussing the action for this superfield. Firstly, the dimensions of fields can vary depending on the conventions used, so it would be helpful to know the specific conventions being used here. Secondly, the mass and cubic interaction terms mentioned in the forum post seem to be for a scalar field, not a superfield. Superfields can have more complicated interaction terms involving both bosonic and fermionic fields.

Assuming that the dimensions and conventions mentioned in the forum post are correct, the action for this superfield would be given by:

S = \int d^2x d^2\theta \left( \frac{1}{2}D^\alpha\phi D_\alpha\phi + m\phi^2 + \lambda\phi^3 \right)

Here, D_\alpha is the supersymmetric covariant derivative and d^2\theta is the integration over the two spinor components of theta. The first term represents the kinetic term for the superfield, which includes both bosonic and fermionic fields. The second and third terms are the mass and cubic interaction terms mentioned in the forum post.

It is worth noting that the kinetic term for a superfield is not simply the square of the superfield. This is because superfields are not just collections of fields, but also contain their own derivatives. Therefore, the kinetic term involves the covariant derivative to correctly account for the interactions between the different components of the superfield.

I hope this helps clarify the action for this superfield. If you need further assistance, please do not hesitate to ask for more clarification or guidance.