Matter loop effects

The term $j_3\hat E_1^+$ in (2.17) provides the only nontrivial interaction of the scalars with geometry in the CGHS model. The basic idea is to perform the path integration for the quadratic part of $\phi$ in (2.16) and to consider higher powers in $\phi$ perturbatively. Without interaction and without source term the integral is of the type

$$\int(\mathcal{D}\phi f^{1/2})\exp{i\int d^2xf\left[\frac{1}{2}g^{\mu\nu}(\partial_\mu\phi)(\partial_\nu\phi)\right]} = e^{iL_P}\,,$$ (21)

where Polyakov's effective action is given by [27]

$$L_P=\frac{1}{96\pi}\int_x\int_y f R_x \square^{-1}_{xy}R_y$$ (22)

with $\square:=g^{\mu\nu}\nabla_{\mu}\partial_\nu$ and $R$ being the curvature scalar of the background geometry $g_{\mu\nu}$. In the gauge (2.5) the components of the inverse metric are given by

$$g^{01}=f^{-1}=g^{10}\,,\quad g^{00}=-2f^{-1}E_1^-\,,\quad g^{11}=0\,,$$ (23)

with $E_1^-$ as defined in (2.13) and the wave operator becomes

$$\Gamma := \frac{1}{2}f\square = \left(\partial_1-\partial_0(\lambda x^0-\frac{M}{2\lambda })\right)\partial_0\,.$$ (24)

Now the 1-loop effective action (3.2) in the gauge (2.5),

$$L_P=\frac{1}{48\pi}\int_x\int_y (\partial_0^2E_1^--\Gamma \ln{f})_x\Gamma _{xy}^{-1}(\partial_0^2E_1^--\Gamma \ln{f})_y\,,$$ (25)

for the CGHS reduces to a local quantity because of (2.13)

$$L_P=\frac{1}{48\pi}\int \ln f \left(\Gamma \ln f\right)\,.$$ (26)

Performing the integration $\mathcal{D}f$ to leading order just implies the replacement $f\to E_1^+$ in (3.1)-(3.6), i.e. the result of quantization on a fixed background. Clearly our intention is to proceed beyond that level.

There are three steps to be performed: A) integrate out the scalars thus obtaining the 1-loop effective action; B) integrate out the auxiliary field, thus replacing $f$ by $\hat{E}_1^+$; C) expand $\hat{E}_1^+$ in powers of $\phi$ (or in powers of $-i\delta /\delta \sigma$). The order is essential since, for instance, integration over the scalars can hardly be performed after integration over the auxiliary field because the measure would contain nonpolynomial factors in $\phi$. Thus A) must be performed before B). To perform A) before C) also is advantageous in order to impose perturbation theory at the last possible instant.

Reinserting the interaction term $j_3 \hat{E}_1^+$ and the source term $\sigma \phi$ and performing the path-integration over $\tilde{\phi}$ for the generating functional (2.16) yields

$$\widetilde{W} [f,j_3,\sigma ] = \exp \left[i\int j_3\hat{E}_1^+\l... ...xp \left[i \left(L_P+\frac{1}{4}\int\sigma \Gamma ^{-1}\sigma \right)\right]\,.$$ (27)

The focus will be on the next to leading order term which effectively produces an interaction of a pair of scalars with the Polyakov loop, because the $\mathcal{D}f$ integration essentially replaces $f$ in (3.6) by $\hat{E}_1^+(\phi\rightarrow-i\delta /\delta \sigma )$ (expanded in powers of $\phi$ up to the investigated order, i.e. $\phi^2$).

cghs1loop.epsiPropagator plus correction term In fig. 1 the relevant Feynman diagrams are depicted. The first term contains just the propagator $\Gamma ^{-1}$ related to (3.4), while the second one encodes the Polyakov-loop induced correction (the double line marks the part of the scalars which has been integrated out) between two such propagators. The formula corresponding to these diagrams is the usual one, $-\frac1W\frac{\partial W}{\partial\sigma _x\partial\sigma _y}\vert _{\sigma =0... ... \Gamma ^{-1}\sigma )(\int\sigma \Gamma ^{-1}\sigma )+\dots)]\vert _{\sigma =0}$, where $V$ is the vertex calculated below (cf. eqs. (3.8) and (3.9)). It contains two derivatives with respect to the source $\sigma$. Moreover, it contains two $\partial_0$ derivatives . The interpretation is as follows: a scalar field interacts via the Polyakov loop with itself. Since the external legs are actually of the form $(\partial_0\phi)$ rather than just $\phi$ it is to be expected that the Killing norm effectively acquires a nontrivial correction which is determined to a large extent by the renormalization prescription implicit in the Polyakov action (3.6).

In the appendix a more straightforward but also more lengthy derivation for the simpler case of $M=0$ is presented. That method is suitable for generic dilaton gravity. However, the CGHS allows for substantial simplifications already from the very beginning.

By virtue of (2.19) the interaction term to order $\phi^2$ is

$$V=-\frac{1}{24\pi}\int_x\int_y \left(\frac{1}{2\lambda x^0}\Gamma \ln{E_1^+}\right)_x(\partial_0^{-2})_{xy}(\partial_0\phi)_y^2\,,$$ (28)

where strictly speaking $\phi$ should be replaced by $-i\delta /\delta \sigma$. Naively, one might expect a symmetric expression of the type (first order) $\times(\Gamma$ zeroth order)$+$(zeroth order) $\times(\Gamma$ first order), rather than just twice (first order) $\times(\Gamma$ zeroth order) as in (3.8). However, this is another instance where special care must be taken when an argument is based upon an effective action of Polyakov type the reliable basis of which being the conformal anomaly. More precisely, we use that to first order the Polyakov action should read $\int (\delta E_1^+)(\delta L_P/\delta E_1^+)$, where $(\delta L_P/\delta E_1^+)$ is given by the conformal anomaly. This property actually defines the Polyakov action and must be considered as more fundamental than its non-local expression (3.2) (cf. [28] for a discussion on this point). Indeed, the symmetric form would not produce the correct conformal anomaly $\Gamma \ln{E_1^+}$ and as a consequence the vacuum expectation value of the propagator would receive a nontrivial correction. This feature is demonstrated more explicitly in the appendix.

There are several ways to deal with the integral kernel in (3.8). The simplest seems to be to act with the double integral on the $x^0$-dependent term and to fix the integration constants to zero on the basis that no change of the asymptotics should occur. This yields immediately

$$V=-\frac{1}{192\pi} \int_y \frac{M}{\lambda ^2 y^0}(\partial_0\phi)^2_y\,.$$ (29)

Thus the interaction of a pair of scalar fields with the Polyakov loop produces a (local) vertex which is proportional to the BH mass and couples only to the $(\partial_0\phi)^2$ part of the kinetic term.

But this means effectively a shift of $E_1^-\to E_1^-+M/(192\pi\lambda ^2 x^0)$ in (2.13). Together with (2.19) and a coordinate redefinition as in (2.20) this modifies the classical Killing norm in the line element (2.20) to

$$\hat{\xi}=1-\frac{M}{\lambda }e^{-2\lambda r}+\frac{M}{48\pi\lambda }e^{-4\lambda r}\,,$$ (30)

and (to this order) matter fields just propagate on this shifted ``background'' geometry. This somewhat resembles the situation in semiclassical approaches where the quantum effects provide corrections to an otherwise classical calculation. That something similar happens here too - despite of our background independent quantization - is due to the fact that in the CGHS model only a local self interaction is produced (cf. fig. 1). The correction term becomes dominant only close to the singularity where our approximation breaks down. The Killing norm (3.10) exhibits now two horizons: one close to the classical one ($r=r_h$) and the other close to the singularity; however, the latter should be only an artifact of our approximation - it anyhow has no effect upon the asymptotic Hawking flux.

Now well-known methods can be applied to extract the corrected Hawking temperature

$$T_H=T_H^0\left(1-\frac{\lambda }{48\pi M}\right)\,,$$ (31)

where $T_H^0=\lambda /(2\pi)$ is the ``classical'' Hawking radiation. This formula holds as long as $\lambda \ll M$, i.e. the 1-loop approximation is valid. The specific heat is then given by (1.1) which concludes the proof of our main statement.

It should be noted that in principle one could use also the background field formalism to calculate perturbative corrections to the static BH, starting from the nonlocal zweibein (2.19). However, our method seems to be simpler as it allows to avoid complicated problems of treating the nonlocalities of the Polyakov action. The relation of the present approach to the background field formalism has been discussed also in ref. [15].

Finally, we would like to address the issue of $M$ in (2.13). Like the other ``constants'' in (2.9)-(2.11) it is related to a residual gauge freedom left by the condition (2.5). So far it has been fixed to a constant, but in principle it could be any function of $x^1=u$ as well. In the absence of matter it essentially coincides with the ADM mass (whenever this notion makes sense - for the CGHS model it does) and it has to be constant even at quantum level [15]. In the presence of matter it can be identified with the (outgoing) Bondi mass [30] and thus becomes in general a function monotonically decreasing with $u$. Within the present approach to leading order the Bondi mass coincides with the ADM mass because of the imposed large BH approximation $\lambda \ll M$. The same condition ensures the validity of the 1-loop approximation. Since the leading order Hawking temperature is mass independent any correction coming from a varying Bondi mass will not be relevant until next-to-next-to leading order.

Nevertheless, it is amusing that all steps leading to the result (3.11) remain valid even for non-constant $M=M(u)$. Thus, one can calculate the leading order mass decrease by virtue of the $2D$ Stefan-Boltzmann law:

$$\frac{dM}{du}= -\frac{\pi}{6} (T_H^0)^2\left(1-\frac{\lambda }{24\pi M}\right) + {\cal O}\left(\frac{\lambda ^2}{M^2}\right)$$ (32)

Supposing that at $u=u_0$ the initial mass of the BH was $M=M_0$ this equation can be integrated straightforwardly

$$M(u)\approx M_0-\frac{\pi}{6} (T_H^0)^2(u-u_0)+\frac{\lambda }{24\pi}\ln{\frac{M(u)-\lambda /(24\pi)}{M_0-\lambda /(24\pi)}}\,.$$ (33)

The first term is the ADM mass, the second term corresponds to a linear decrease due to the (in leading order) constant Hawking flux and the third term provides the first nontrivial correction. Up to the considered order in $\lambda /M$ from (3.13) one can express $M(u)$ in terms of the Lambert $W$-function [31].