Alternative approach

The purpose of this appendix is to provide an alternative derivation of the effective interaction vertex (3.9) by applying a method which works also for generic dilaton gravity theories and which has proven useful already for tree level vertices [15,16,17,25,26]. Since the Polyakov action contains the zweibeine at classical level it is sufficient to solve the classical equations of motion with matter replaced by a localized source $(\partial_0\phi)^2\to c_0\delta (x-y)$. The linear order terms in $c_0$ will produce the vertex of interest. In this manner one obtains by expanding the logarithm of (2.18)

$$\ln{\hat{E}_1^+}=\ln{E_1^+}-c_0\left(1-\frac{y^0}{x^0}\right)\theta(x^0-y^0)\delta (x^1-y^1)+\mathcal{O}(c_0^2)\,.$$ (34)

The differential operator $\Gamma$ does not receive any $c_0$ corrections.

As pointed out below eq. (3.8) there seem to be two ways to obtain the vertex: naively, one would just take the first order in $c_0$ of the whole Polyakov action (``symmetric variant''); alternatively, by taking the origin of the Polyakov action, namely the conformal anomaly, seriously one has to take the first order term in $c_0$ of $\ln{\hat{E}_1^+}$ and to multiply it with the zeroth order of the curvature term, $\Gamma \ln{\hat{E}_1^+}$ (``correct variant''). The result comprising both cases is

$$V=\int\limits_{-\infty}^{\infty} dy^1\int\limits_{x^0_h+\varepsilon }^{\infty} dy^0 (\partial_0 \phi)^2 \left[\frac{c}{y^0}+d\right]\,.$$ (35)

The symmetric variant yields $c=-M/(192\pi\lambda ^2)$ and $d=1/48\pi$ while otherwise $c=-M/(192\pi\lambda ^2)$ and $d=0$ is obtained. The lower integration limit is explained as follows: $x^0_h=M/\lambda$ is the horizon of the background geometry and $\varepsilon >0$ is a cutoff parameter to regularize the $y^0$ integration in (A.2).

For $d=0$ the result (A.2) is equivalent to (3.9). To show that $d$ must be zero it is sufficient to study propagation in the vacuum, i.e. a vanishing BH mass $M=0$ can be considered. Then $c=0$, but $d$ might be vanishing or nonvanishing, depending on whether the ``correct'' or the ``symmetric'' variant is applied. The scalar field can be represented as

$$\begin{multline} \phi=\frac{1}{\sqrt{2\pi}}\int\limits_0^\infty \frac{dk}{\sqrt{... ...k\left(x^1+\int^{x^0}\frac{dz}{\lambda z}\right)\right)}\Bigg]\,, \end{multline}$$

where $b^\pm$ create and annihilate the right movers and $a^\pm$ the left movers. The normalization is given by $[a_k^-,a_q^+]=\delta (k-q)=[b_k^-,b_q^+]$ (all other commutators vanish).

In the quantity $(\partial_0\phi)^2$ only the left movers survive; therefore, only left movers may acquire nontrivial quantum corrections. The relation between in and out vacua is trivial and the T-matrix corresponding to a propagator correction can be calculated straightforwardly:

$$T=_{\rm out}<0\vert a^-_k V a^+_q\vert>_{\rm in} \propto \delta (k-q) \frac{d}{\varepsilon }$$ (36)

Thus, in the correct variant no additional correction to the propagator arises, while the symmetric variant yields a cutoff dependent result which diverges when the cutoff approaches zero. This provides a physical way to see why the ``correct'' variant can be the only valid one: after all, for vanishing BH mass ($M=0$) the scalar field should propagate freely on a Minkowskian background.

In the massive case ($M\neq 0$) the standard route would be to introduce a mode decomposition for the scalar field (carefully distinguishing between in modes and out modes) and to calculate vacuum expectation values with the insertion of (3.9). Assuming that $V$ is a small perturbation standard methods can be applied and then the problem reduces to the determination of quantum corrected Bogoliubov coefficients between in states and out states. Once these coefficients are known also corrections to the Hawking temperature can be extracted. However, there are difficulties as the vertex introduced in (A.2) depends on the cutoff parameter $\varepsilon$ and diverges in the limit $\varepsilon \to 0$. Therefore, the method presented in the main part is more suitable if one wants to calculate just corrections to the Hawking temperature rather than generic scattering processes.