Constructing conserved current from lagrangian

[CAF123]

Homework Statement

Consider the following Lagrangian for a massive vector field $A_{\mu}$ in Euclidean space time: $$\mathcal L = \frac{1}{4} F^{\alpha \beta}F_{\alpha \beta} + \frac{1}{2}m^2 A^{\alpha}A_{\alpha}$$ where ##F_{\alpha \beta} = \partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}## which means $$\mathcal L = \frac{1}{4} (\partial^{\alpha}A^{\beta} - \partial^{\beta}A^{\alpha})(\partial_{\alpha}A_{\beta} - \partial_{\beta}A_{\alpha}) + \frac{1}{2}m^2A^{\alpha}_{\alpha} \,\,\,\,(1)$$ The canonical energy-momentum tensor is, using the relation

$$T^{\mu \nu}_c = -\eta^{\mu \nu} \mathcal L + \frac{\partial \mathcal L}{\partial (\partial_{\mu}\Phi)}\partial^{\nu}\Phi,\,\,\,\,(2)$$

$$T^{\mu \nu}_c = F^{\mu \alpha}\partial^{\nu}A_{\alpha} - \eta^{\mu \nu}\mathcal L$$

Then from ##T^{\mu \nu}_B = T^{\mu \nu}_c + \partial_{\rho}B^{\rho \mu \nu}##, it is found that $$B^{\alpha \mu \nu} = F^{\alpha \mu}A^{\nu}\,\,\,\,(3)$$ using the formula $$B^{\mu \rho \nu} = \frac{1}{2}i \left\{\frac{\partial \mathcal L}{\partial (\partial_{\mu}A_{\gamma})} S^{\nu \rho}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\rho}A_{\gamma})} S^{\mu \nu}A_{\gamma} + \frac{\partial \mathcal L}{\partial (\partial_{\nu}A_{\gamma})} S^{\mu \rho}A_{\gamma}\right\}$$ My question is how is this equation obtained and how did they obtain ##(3)##? Did they make use of the explicit form of the spin matrix for a vector field?

Homework Equations

(My question is from Di Francesco et al 'Conformal Field Theory' P.46-47).

The Attempt at a Solution

$$B^{\alpha \mu \nu} = \frac{i}{2}\left\{F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} + F^{\mu \gamma}S^{\alpha \nu}A_{\gamma} + F^{\nu \gamma}S^{\alpha \mu}A_{\gamma}\right\}$$ from simplifying the above. Concentrate on the first term. Then $$F^{\alpha \gamma}S^{\nu \mu}A_{\gamma} = F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b} (S_{ab})^c_dA^d$$ Inputting the form of $S$ for a vector field, I get $$F^{\alpha \gamma}\eta_{\gamma c}\eta^{\nu a}\eta^{\mu b}(\delta^c_a \eta_{bd} - \delta^c_b \eta_{da})A^d$$ But simplifying this and writing the other terms does not yield the result. Did I make a mistake upon insertion of the spin matrix?

 

[CAF123]

Can anybody help with this at all? Thanks.