In this paper, the authors say that the trace of the energy-momentum tensor has to vanish to allow for the existence of conserved dilatation or scaling currents, as defined on p 10, Eq(22)

$$ \Theta^{\mu} = x_{\nu} \Theta^{\mu\nu} + \Sigma^\mu.$$

$\Theta^{\mu\nu}$ is the energy-momentum tensor and $\Sigma^\mu = d_{\phi}\pi_i^{\mu}\phi^i$ is the internal part.

This fact is just mentioned and I dont understand why this has to be the case, so can somebody explain it to me?


Without the internal part:

The divergence $$\nabla_\mu (x_\nu \Theta^{\mu\nu}) = x_\nu \nabla_\nu \Theta^{\mu\nu} + \frac12(\nabla_\mu x_\nu + \nabla_\nu x_\mu) \Theta^{\mu\nu} $$ where I used that $\Theta^{\mu\nu}$ is symmetric. Recalling that the energy-momentum tensor is divergence free, the first term drops out. Assuming that $x^\nu$ generates a dilation/scaling symmetry (and not a bona fide symmetry), we know that its deformation $$ \nabla_\mu x_\nu + \nabla_\nu x_\mu \propto \mathcal{L}_x g_{\mu\nu} \propto g_{\mu\nu} $$ where $\mathcal{L}$ is the Lie derivative. (In the case $x^\nu$ generates a symmetry the term vanishes from Killing's equation.)

Hence in this case for the current to be conserved (that is, divergence free), we need that $g_{\mu\nu} \Theta^{\mu\nu} = 0$; that is, the energy momentum tensor is tracefree.

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