Form of Dilaton term in polarization tensor

Question

The vertex operator associated with massless state is

$$V(k,\epsilon) = -\frac{2}{\alpha}\epsilon_{\mu\nu}(k)\bar{\partial}X^\mu(\bar{z})\partial X^\nu(z)e^{ik\cdot X(z,\bar{z})}$$

The polarization tensor can be decomposed into symmetric (Graviton), antisymmetric and Trace bit (Dilaton) Lust,theisen 16.9

$$\epsilon^{(h)}_{\mu\nu} = \epsilon^{(h)}_{\nu\mu},\qquad\epsilon^{(h)}_{\mu\nu} \eta^{\mu\nu} = k^{\mu}\epsilon^{(h)}_{\mu\nu} = 0,$$

$$\epsilon^{(B)}_{\mu\nu} = -\epsilon^{(B)}_{\nu\mu},\qquad k^{\mu}\epsilon^{(B)}_{\mu\nu} = 0,$$

$$\epsilon^{(D)}_{\mu\nu} = \frac{1}{\sqrt{d-2}}(\eta_{\mu\nu}-k_{\mu}\bar{k_{\nu}}-k_{\nu}\bar{k_{\mu}})$$

$\bar{k}$ is an arbitrary light like vector orthogonal to $k$. Can you please tell me why we took that particular form for the Dilaton?

This post imported from StackExchange Physics at 2014-09-07 18:45 (UCT), posted by SE-user sol0invictus

Answer 1

The polarisation tensor is not unique and subject to the equivalence $\epsilon_{\mu\nu}(k) \sim \epsilon_{\mu\nu}(k) + k_\mu\xi_\nu+k_\nu\xi_\mu$ which is nothing but  diffeomorphisms in target space.   this gauge symmetry shows up in the form the existence of a null state  which can be constructed from the state given in Eq (16.11). One fixes this gauge symmetry by choosing $k^\mu \epsilon_{\mu\nu}=0$ and   $k^\nu \epsilon_{\mu\nu}=0$ (this is the analog of $\partial_\mu A^\mu=0$ in $U(1)$ gauge theories).  This can be viewed as a transversality condition on the polarisation tensor. On decomposing into a symmetric traceless, antisymmetric and trace part, this condition remains to be implemented. For the dilation, this is done by introducing an reference vector $\bar{k}$. $\bar{k}$ is chosen such that $\bar{k}^2=0$ and $\bar{k}\cdot k=1$ (not $0$ as the OP has stated). Then, $(\eta_{\mu\nu}-k_\mu\bar{k}_\nu-k_\nu\bar{k}_\mu)$ is the projector on to the space transverse to $k$. 

Answer 2

The graviton represents only the traceless symmetric part of the polarization tensor. The dilaton part belongs to the symmetric part too (the trace part). The symmetric dilaton   $\epsilon^{(D)}_{\mu\nu} (k)$represents only $1$degree of freedom, so it is necessarily expressed as a function of $k$, and it must verify the transversality condition :  $k^\mu\epsilon^{(D)}_{\mu\nu} (k)==0$

The general expression is then $\epsilon^{(D)}_{\mu\nu} (k) \sim \eta_{\mu\nu} -(k_\mu \bar k_\nu + k_\nu \bar k_\mu)$

The transversality condition implies $k_\nu - (k.\bar k) k_\nu =0$, and, so finally $(k.\bar k)=1$

One may add a supplementary  transversality condition : $\bar k^\mu\epsilon^{(D)}_{\mu\nu} (k)=0$, this clearly implies $\bar k^2=0$

This supplementary condition is compatible with the precedent conditions, in particular the dilaton and graviton degrees of freedom are still orthogonal: $\epsilon^{(D)}_{\mu\nu} (k) (\epsilon^{(G)}) ^{\mu\nu} (k)=0$