We calculate the one-loop contributions to the polarization operator for scalar quantum electrodynamics in different external electromagnetic and gravitational fields. In the case of gravity, de Sitter space and its different patches were considered. It is shown that the Debye mass appears only in the case of alpha-vacuum in the Expanding Poincare Patch. It can be shown either by direct computations or by using analytical and causal properties of the de Sitter space. Also, the case of constant electric field is considered and the Debye mass is calculated.

Article funded by SCOAP3

0$ there can be some divergences and instabilities of the system. ]]>

\frac{D-1}{2}$.The harmonics in such a case are linear combinations of Hankel's functions \beq f_p(\eta) = \eta^{\frac{d-1}{2}} h_{i\mu}(p\eta) = \eta^{\frac{d-1}{2}}\left[\alpha H^{(1)}_{i\mu}(p\eta) + \beta H^{(2)}_{i\mu}(p\eta)\right],\quad \left|\alpha\right|^2 - \left|\beta\right|^2 = 1. \label{HarmDsEPP} \eeq where $\mu = \sqrt{m^2 - \left(\frac{D-1}{2}\right)^2}$. If $\beta=0$ then this vacuum is named after Bunch and Davies~\cite{Bunch-ml-1978yq}. This vacuum has some nice properties when the interactions are switched off. It can be thought as an analog of usual Minkowski vacuum in de Sitter space-time~\cite{Krotov-ml-2010ma,Akhmedov-ml-2013vka}. It means that in the distant past ($\eta=\infty$) modes are just plane waves $e^{i p \eta}$. If $\beta\neq 0$ we will name such a state as an alpha vacuum~\cite{Mottola-ml-1984ar}. The last states are usually considered to be unphysical due to the incorrect UV behavior, nevertheless they are function of a geodesic distance $Z$ on a de-Sitter space and therefore could respect de Sitter invariance (at least on a tree level). Therefore, it is interesting to consider alpha-vacuums and compare to the Bunch-Davies(BD) state. In terms of this modes the Debye mass is {\rdmathspace\beq m^2_{\rm Deb} = \lim \limits_{q\to 0} 2 e^2\! \int\limits^\infty_\eta\! d\eta' \eta^3 \eta' \int \frac{d^3 p}{(2\pi)^3} {\rm Im}\left[\left(h_{i\mu}(p\eta)\overleftrightarrow{\partial_\eta} h_{i\mu}(|p+q|\eta)\right) \left(h^*_{i\mu}(p\eta')\overleftrightarrow{\partial_{\eta'}} h^*_{i\mu}(|p+q|\eta')\right)\right]. \label{massEPP} \eeq}\relax Before taking the limit $\eta\to 0$ one can notice that the l.h.s.\ of the equation~\eqref{massEPP} depends only on the physical momentum, $q\eta$, $m^2 (q\eta), m^2_{\rm Deb} = \lim\limits_{q\to 0} m^2(q\eta)$. Indeed, one can rescale $p\to p\eta, \eta'\to \frac{\eta'}{\eta}$ and get that the only left parameter is $q\eta$. In this case the limit of the distant future $\eta \to 0$ and small momentum $q\to0$ coincide with the limit of small physical momentum $q\eta \to 0$. As we will see, in the case of CPP there could arise some problems with defining of the Debye mass. One can check that for Bunch-Davies vacuum there are no divergences as we integrate over $\eta'$, therefore there is no Debye mass $m_{\rm Deb,\, BD}^2 = 0$. In contrast, alpha-vacuums~\eqref{HarmDsEPP} in the limit of $\eta\to\infty$ are combinations of the positive and negative frequency harmonics (partially, due to this fact they do not have a proper UV behavior) that leads to some divergences \beq m^2_{\rm Deb} = -\frac{32 e^2}{\pi} \int \frac{d^3 p}{(2\pi)^3} {\rm Im}\left[\alpha^* \beta^* \left(h_{i\mu}(p) h'_{i\mu}(p) - p (h'_{i\mu}(p))^2 + p h_{i\mu}(p) h''_{i\mu}(p)\right)\right]. \eeq It can be seen that the mass is divergent at the large $p$ and also may be negative for specific chosen alpha-vacuums. This indicates, that the alpha-vacuum can effectively screen any charge. The same effect can happen for the gravitational mass. In such a case, large mass can drastically change the de Sitter solution or even make it unstable. The difference between alpha vacuums and the Bunch Davies vacuum can be shown in the following way. In the BD vacuum propagators have a good analytic properties \begin{eqnarray*} \hat{G}(X_1,X_2) &=& \begin{pmatrix} G_{--} & G_{-+}\\ G_{+-} & G_{++} \end{pmatrix} = \notag\\ &=&\begin{pmatrix} G\left(\frac{\left(\eta_1 - \eta_2 - i \eps(\eta_1 - \eta_2)\right)^2 - \vec{x}^2}{\eta_1 \eta_2}\right) & G\left(\frac{\left(\eta_1 - \eta_2 - i \eps\right)^2 - \vec{x}^2}{\eta_1 \eta_2}\right)\\ G\left(\frac{\left(\eta_1 - \eta_2 + i \eps\right)^2 - \vec{x}^2}{\eta_1 \eta_2}\right) & G\left(\frac{\left(\eta_1 - \eta_2 + i \eps(\eta_1 - \eta_2)\right)^2 - \vec{x}^2}{\eta_1 \eta_2}\right) \end{pmatrix}. \label{BDprop} \end{eqnarray*} Note the $i\eps$ prescription in this equation. All two point functions for BD vacuums such as polarization operators will have the same $i\eps$ prescriptions for time, even after Fourier transformation over the spatial coordinates. Now let us use this property in the formula for Debye mass in Kubo representation \begin{equation} m^2_{\rm Deb} = \lim\limits_{k\to 0}\int\limits^\eta_{\infty} \frac{d\eta'}{\eta'^2} \braket{[J_0(k,\eta), J_0(-k,\eta')]}. \label{massphotoncoorEPP} \end{equation} Because of the analytic properties discussed above this integral can be seen as the integral of the analytic function $\Pi_{00}(\eta') = \braket{J_0(k,\eta) J_0(-k,\eta')}$ of the variable $\eta'$, where the contour goes from $+\infty-i\eps$ to $\eta-i\eps$ turns up and goes from $\eta+i\eps$ to $+\infty+i\eps$. This allows us to make the integral~\eqref{massphotoncoorEPP} convergent by slightly changing the contour $\eta\to (1+i\delta) \eta$. Therefore we can interchange the integral and the limit to get \begin{equation} m^2_{\rm Deb, BD} = \int\limits^\eta_{\infty} \frac{d\eta'}{\eta'^2} \braket{[J_0(0,\eta), J_0(0,\eta')]} = 0. \end{equation} This proof will not work for other vacuums, because the functions will not be analytic on the whole Riemann surface, for example, $\eta=\infty$ will have an essential singularity. Namely, the product of two Green functions for an alpha-vacuum contains two cuts and contour lies between them. \begin{figure} \centering \includegraphics[scale=1]{3.pdf} ]]>

\frac{D-1}{2}$. These limits are drastically different, because of different mode behavior, like in the case of heavy fields only the Keldysh propagator gets secularly growing loop corrections and modes thermalize~\cite{Krotov-ml-2010ma}, while in the case of light fields even the higher correlators start growing and there is no thermalization of harmonics~\cite{Akhmedov-ml-2017ooy}. The case of vanishing mass corresponds to the light masses and makes the physics different from the heavy one that was considered in this paper, as it was indicated in the beginning of this section. That can cause a discrepancy between the results. ]]>

\mu, |p|< \mu e^{-2 T}\\ \frac{1}{e^{2\pi \mu} - 1}, \mu e^{-2 T} < |p| < \mu \end{matrix} \right. \end{equation} where $\mu = \sqrt{m^2 - \left(\frac{D-1}{2}\right)^2}$ and $m$ is a mass of the scalar field. In such a case we can use the above formula for Minkowski space-time that gives us \begin{equation} m^2_{\rm Deb} = \frac{e^2 \mu^2}{2\pi^2} \frac{1}{e^{2\pi \mu} - 1} \label{finalanswer} \end{equation} The formula is different from the Debye mass for a thermal state with Gibbons-Hawking temperature $T = \frac{1}{2\pi}$, it happens because of the infinite blue-shift during the expansion phase. One can expect then, that we should get the same answer for the case of the CPP with the contraction switched off after a while. However this means that we will get a contradiction with the above calculation of the Debye mass in CPP, because the same reasoning as in that case leads to the zero photon mass. This contradiction is easy to resolve. Indeed, we know that the limits of small $q$ and distant future do not commute. But in the case when we switch off the contraction there is no such a problem and we can take the limit of the distant future and get the non-zero photon mass. Let us consider the case when the EPP was turned on at some moment $\eta_0$. In this case after gluing modes will be described as follows \beq f_p(\eta) = \left\{ \begin{matrix} H^{(1)}_{i\mu}(p\eta),\quad p\eta_0 \gtrsim \mu\\ J_{i\mu}(p\eta), ~\quad p\eta_0 \lesssim \mu \end{matrix} \right. \eeq As one can notice, there is no singularity from the switching on the EPP, effectively if one changes all momentum to the physical ones in eq.~\eqref{massEPP}, he gets that the border between two states goes down $\mu' = \mu \frac{\eta}{\eta_0} \to 0, \eta \to 0$. So, we would expect that the state will be described by Bunch-Davies and that will not give any photon mass. ]]>