How do I know if this field has a mass term?

[Lecticia]

1. Special Relativity

2. Homework Statement
Consider this Lagrangian:
L=(1/2) (\partial_{\mu} \Psi)(\partial^{mu} \Psi) + \exp(-(a\times \Psi)^2)

Have this field a mass term?

Homework Equations

The Attempt at a Solution

 

[Astronuc]

Is this what one intended to write, or is this given in some text?

L= \frac{1}{2} (\partial_{\mu} \Psi)(\partial^{\mu} \Psi) + e^{-{(a \Psi)}^2}

 

[Lecticia]

Is this what one intended to write, or is this given in some text?

L= \frac{1}{2} (\partial_{\mu} \Psi)(\partial^{\mu} \Psi) + e^{-{(a \Psi)}^2}

Yes, exactly this Lagrangian, where \Psi is a scalar field.

 

[nrqed]

Yes, exactly this Lagrangian, where \Psi is a scalar field.

This is a nonlinear field theory so, strictly speaking, there is no clear meaning for a mass term.
But I am guessing that they want you to treat the parameter "a" as small and to do a Taylor expansion of the exponential. If you do that, you will generate a mass term.

That's my guess.

 

[dextercioby]

Well, just one question, you have the lagrangian, what are the field eqn's ?

 

[Lecticia]

Well, just one question, you have the lagrangian, what are the field eqn's ?

Do you mean the motion equations?

 

[Lecticia]

This is a nonlinear field theory so, strictly speaking, there is no clear meaning for a mass term.
But I am guessing that they want you to treat the parameter "a" as small and to do a Taylor expansion of the exponential. If you do that, you will generate a mass term.

That's my guess.

Thanks, I'll think about this...