Coupling of vector gauge to a tensor field problem

1. Jul 22, 2012

aries0152

1. The problem statement, all variables and given/known data
Hi
I was reviewing this paper.
But I stuck in the last part. (equation 25)

According to the paper the Lagrangian is -
$$2L=(\dot{V^{T})^{2}}-(\nabla\times V^{T})^{2}+[\dot{B}^{T}-\nabla\times A^{T}]^{2}+2mV^{T}.(\nabla\times A^{T})+2mB^{T}.\dot{V}^{T}+2\mu^{2}A^{T}.B^{T}------(1)$$
And after imposing the conditions below-
$$B^{L}=A^{L}=0=\dot{V^{L}}-\nabla U------(2)$$
$$A^{T}.B^{T}=-(\nabla\times A^{T}).(\nabla^{2})^{-1}(\nabla\times B^{T})------(3)$$
$$(\nabla\times A^{T})=\dot{B^{T}}-mV^{T}+\mu^{2}(\nabla^{2})^{-1}(\nabla\times B^{T})------(4)$$

There would be only two terms (according to the paper)-
$$2L= (\dot{V^{T})^{2}}+\mathbf{V}^{T}\nabla^{2}\mathbf{V}^{T}$$

My Calculation:
But after I use the condition (2), (3) and (4) and put them into (1). But I have found the equation-
$$2L= (\dot{V^{T})^{2}}+\mathbf{V}^{T}\nabla^{2}\mathbf{V}^{T}+2mB^{T}.\dot{V}^{T}-m^{2}(V^{T})^{2}-\mu^{4}(\nabla^{2})^{-2}(\nabla\times B^{T})^{2}$$
$$+2\dot{B}^{T}mV^{T}+2mV^{T}\mu^{2}(\nabla^{2})^{-1}(\nabla\times B^{T})-2\mu^{2}(\nabla^{2})^{-1}(\nabla\times B^{T})\dot{B}^{T}$$
So there are some extra terms that need to be eliminated.

How can I do it? Is there any other condition that needed to be applied? Please help.