One loop field strengths of charges and dipoles on a locally de Sitter background

Abstract

We use the one loop vacuum polarization induced by scalar quantum electrodynamics to compute the electric and magnetic fields of point charges and magnetic dipoles on a locally de Sitter background. Our results are consistent with the physical picture of an inflating universe filling with a vast sea of charged particles as more and more virtual infrared scalar are ripped out of the vacuum. One consequence is that vacuum polarization quickly becomes nonperturbatively strong. Our computation employs the Schwinger–Keldysh effective field equations and is done in flat, conformal coordinates. Results are also obtained for static coordinates.

Appendices

Appendix A: Integrals from Sect. 4.2

The following integrals (times $-\alpha/\pi\times F^{(0)}_{0i}$) sum to give our result (98) for $F^{1G}_{0i}$ in Sect. 4.2:

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

The integrands of (A.6), (A.7) and (A.9), (A.10) diverge at x′=−η, which results in the ill-defined factors of ±ln(1−aHx). However, these divergences cancel between (A.6), (A.7) and between (A.9), (A.10), as do the problematic logarithms.

Appendix B: Integrals from Sect. 4.3

We begin with some general comments. The expression (135) we must evaluate is a double integration over three regions which can be termed “I”, “II” and “III”,

(B.1)

Each of these double integrals takes the form

$$ \int_{B}^{A} d\eta' \int _{D(\eta')}^{C(\eta')} dx' , $$

(B.2)

where the various limits are

(B.3)

(B.4)

(B.5)

(B.6)

Of course the integrand is of crucial importance. From expressions (125) and (126) we see that xJ ^GK(η,x) can be broken up into seven components,

(B.7)

(B.8)

(B.9)

(B.10)

(B.11)

(B.12)

(B.13)

In each case the x′ integration can be expressed in terms of elementary functions. For $\mathcal{I}_{10}$ the η′ integration also results in elementary functions. $\mathcal{I}_{2}$ and $\mathcal{I}_{9}$ give polylogarithms, more about which later. The remaining integrands—$\mathcal{I}_{1}$, $\mathcal{I}_{34}$, $\mathcal{I}_{56}$ and $\mathcal{I}_{78}$—all take the form of a′² times the x′ derivative of an elementary function of η′ and x′. For these integrands the best strategy is partially integrate on η′ after having performed the x′ integration,

(B.14)

(B.15)

It turns out that the surface terms in (B.15) cancel between the three regions,

(B.16)

(B.17)

(B.18)

Integrands $\mathcal{I}_{2}$ and $\mathcal{I}_{9}$, and the volume terms (B.15) from $\mathcal{I}_{1}$, $\mathcal{I}_{34}$, $\mathcal{I}_{56}$ and $\mathcal{I}_{78}$, all give rise to η′ integrations of the form 1/η′ times logarithms. Integrations of this form result in polylogarithms, of which the two we require are

$$ \begin{aligned} \lefteqn{\mathrm{Li}_2(z) \equiv-\int_0^z dt \frac{\ln(1 - t)}{t} ,}\\\lefteqn{\mathrm{Li}_3(z) \equiv\int _0^z dt \frac{ \mathrm{Li}_2(t)}{t} .} \end{aligned} $$

(B.19)

Both are real for z≤1. Their expansions for small z and for large −z are

(B.20)

(B.21)

Care must be taken to arrange things so that the arguments of Li₂(z) and Li₃(z) lie in the range z≤1 for which the function is real. This is simple to accomplish for the dilogarithm Li₂(z) but it requires some effort for Li₃(z). Li₃(z) derives exclusively from $\mathcal{I}_{2}$, when changing to dimensionless variables converts the η′ integration to either ∫dt/t×ln²(1−t) or ∫dt/t×ln(t)ln(1−t). The final integral only makes sense for 0≤t≤1 so it involves no choices,

$$ \int dt \frac{\ln(t) \ln(1 - t)}{t} = \mathrm{Li}_3(t) - \ln(t) \mathrm{Li}_2(t) . $$

(B.22)

If the dimensionless parameter t lies in the range 0≤t≤1 we write the first integral as

(B.23)

However, if t<0 we must use Landen’s identity to re-express it as

(B.24)

It can happen that the appropriate choice to make between (B.23) and (B.24) depends upon whether x or r=ax is held fixed for large a, in which case we report the choice appropriate for holding x fixed at large a.

Our results for the seven integrals are

(B.25)

(B.26)

(B.27)

(B.28)

(B.29)

(B.30)

(B.31)

It remains just to give the expansions for large a at fixed x, and at fixed r=ax. In both cases the leading term goes like ln(a). For fixed x we find

(B.32)

(B.33)

(B.34)

(B.35)

(B.36)

(B.37)

(B.38)

At fixed r the results are

(B.39)

(B.40)

(B.41)

(B.42)

(B.43)

(B.44)

(B.45)