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I have a pretty interesting Lagrangian setup:

[tex]\mathcal{L}=-\frac{1}{4}(\nabla_a u_b -\nabla_b u_a)(\nabla^a u^b -\nabla^b u^a)-\lambda(u_au^a-v^2)-\frac{1}{2}(\partial\phi^2)-\frac{1}{2}m^2\phi^2-\frac{1}{2\mu^2}u^au^b\partial_a\phi\partial_b\phi[/tex]

My term [itex]u^a[/itex] is a spacelike 5 vector that violates Lorentz Invariance in the 5th dimension only. The indices a,b run from 0 to 4.

My question:

In the paper I'm reading (http://arxiv.org/PS_cache/arxiv/pdf/...802.0521v1.pdf [Broken]), the above lagrangian is referred to have "the lowest order coupling". I'm guessing this comes from the last interaction term between [itex]u[/itex] and [itex]\phi[/itex]. Can anyone explain why the term has to be of the form [tex]\frac{1}{2\mu^2}u^au^b\partial_a\phi\partial_b\phi[/tex]. Whats wrong with just [tex]\frac{1}{2\mu^2}u^au_a\phi^2[/tex] for example??

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# Interaction terms

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