An Analytic Approximation for Researching Tunneling Rate from Black Hole in Proca Field

Abstract

The analytic solution of a static spherical symmetrical Proca black hole is discussed in this paper. As in the massive vector field, Proca black hole can be considered as the analogy of RN background plus a perturbation with the same order as μ ² due to the mass of vector particle μ satisfying μ ² ≪ 1. Through the action of Proca field, we find the analytic form with the first and arbitrary order approximation. Furthermore, we divide the results into 3 groups according to the real zero solutions of the background (i.e., spacetime in massless vector field). Finally we analyze the Hawking radiation of such black hole, which is significant for constructing black hole thermodynamic.

Appendix

In the appendix, for the case 1, case 2, case 3 mentioned in Section 2, we list the perturbation functions Ñ (r) and à ₀ (r) with corresponding subscripts respectively. Here we only retain the coefficients of δ.

Case 1

$1+\frac {Q^{2}}{r^{2}}-\frac {2M}{r}+\frac {r^{2}\Lambda }{3}=\frac {\Lambda }{3r^{2}}(r-r_{1})(r-r_{2})(r-r_{3})(r-r_{4})$:

$$\begin{array}{@{}rcl@{}} &{}\tilde{N}(r)&{}={}\frac{9 \mu^{2}Q^{2}}{\Lambda^{2}} \left\{\frac{\frac{{r_{1}^{3}}}{(r{}-{}r_{1}) (r_{1}{}-{}r_{3})^{2} (r_{1}{}-{}r_{3})^{2}}{}+{}\frac{{r_{2}^{3}}}{(r{}-{}r_{2}) (r_{2}{}-{}r_{3})^{2} (r_{2}{}-{}r_{4})^{2}}}{(r_{1}{}-{}r_{2})^{2}} {}+{}\frac{\frac{{r_{3}^{3}}}{(r{}-{}r_{3}) (r_{1}{}-{}r_{3})^{2} (r_{2}{}-{}r_{3})^{2}}{}+{}\frac{{r_{4}^{3}}}{(r{}-{}r_{4}) (r_{1}{}-{}r_{4})^{2} (r_{2}{}-{}r_{4})^{2}}}{(r_{3}{}-{}r_{4})^{2}} \right. \\ && \left. \frac{{r_{1}^{2}} \log (r-r_{1}) \left(3 {r_{1}^{3}}-{r_{1}^{2}} (r_{2}+r_{3}+r_{4})-r_{1} (r_{2} (r_{3}+r_{4})+r_{3} r_{4})+3 r_{2} r_{3} r_{4}\right)}{(r_{1}-r_{2})^{3} (r_{1}-r_{3})^{3} (r_{1}-r_{4})^{3}} \right. \\ && \left. +\frac{{r_{2}^{2}} \log (r-r_{2}) \left(r_{1} \left({r_{2}^{2}}+r_{2} (r_{3}+r_{4})-3 r_{3} r_{4}\right)+r_{2} \left(-3 {r_{2}^{2}}+r_{2} (r_{3}+r_{4})+r_{3} r_{4}\right)\right)}{(r_{1}-r_{2})^{3} (r_{2}-r_{3})^{3} (r_{2}-r_{4})^{3}} \right. \\ && \left. +\frac{{r_{3}^{2}} \log (r-r_{3}) (r_{1} (r_{2} (r_{3}-3 r_{4})+r_{3} (r_{3}+r_{4}))+r_{3} (r_{2} (r_{3}+r_{4})+r_{3} (r_{4}-3 r_{3})))}{(r_{1}-r_{3})^{3} (r_{3}-r_{2})^{3} (r_{3}-r_{4})^{3}} \right. \\ && \left. +\frac{{r_{4}^{2}} \log (r-r_{4}) (r_{1} (r_{2} (r_{4}-3 r_{3})+r_{4} (r_{3}+r_{4}))+r_{4} (r_{2} (r_{3}+r_{4})+r_{4} (r_{3}-3 r_{4})))}{(r_{1}-r_{4})^{3} (r_{4}-r_{2})^{3} (r_{4}-r_{3})^{3}}{\vphantom{\frac{\frac{{r_{1}^{3}}}{(r{}-{}r_{1}) (r_{1}{}-{}r_{3})^{2} (r_{1}{}-{}r_{3})^{2}}{}+{}\frac{{r_{2}^{3}}}{(r{}-{}r_{2}) (r_{2}{}-{}r_{3})^{2} (r_{2}{}-{}r_{4})^{2}}}{(r_{1}{}-{}r_{2})^{2}} {}+{}\frac{\frac{{r_{3}^{3}}}{(r{}-{}r_{3}) (r_{1}{}-{}r_{3})^{2} (r_{2}{}-{}r_{3})^{2}}{}+{}\frac{{r_{4}^{3}}}{(r{}-{}r_{4}) (r_{1}{}-{}r_{4})^{2} (r_{2}{}-{}r_{4})^{2}}}{(r_{3}{}-{}r_{4})^{2}}}}\right\}; \end{array}$$

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$$\begin{array}{@{}rcl@{}} &\tilde{A}_{0}(r)&{}={} \frac{18 \mu^{2} Q^{3}}{\Lambda^{2} r^{2}} \left\{\frac{{r_{1}^{2}} \log (r-r_{1})}{(r_{1}-r_{2})^{3} (r_{1}-r_{3})^{3} (r_{1}-r_{4})^{3}} \left[Q^{2} \left(9 {r_{1}^{3}}-3 {r_{1}^{2}} (r_{2}+r_{3}+r_{4})-3 r_{1} (r_{2} (r_{3}+r_{4})+r_{3} r_{4}) \right. \right. \right. \\ && \left.\left.\left. +9 r_{2} r_{3} r_{4}{\vphantom{9 {r_{1}^{3}}-3 {r_{1}^{2}} (r_{2}+r_{3}+r_{4})-3 r_{1} (r_{2} (r_{3}+r_{4})+r_{3} r_{4}}}\right)+\Lambda r_{1} (r_{1}-r_{2})^{2} (r_{1}-r_{3})^{2} (r_{1}-r_{4})^{2}\right]-\frac{{r_{2}^{2}} \log (r-r_{2})}{(r_{1}-r_{2})^{3} (r_{2}-r_{3})^{3} (r_{2}-r_{4})^{3}}\left(\Lambda r_{2} (r_{1}-r_{2})^{2} \right. \right. \\ && \left.\left. (r_{2}-r_{3})^{2} (r_{2}-r_{4})^{2} -3 Q^{2} (r_{4} (r_{1} (r_{2}-3 r_{3})+r_{2} (r_{2}+r_{3}))+r_{2} (r_{1} (r_{2}+r_{3})+r_{2} (r_{3}-3 r_{2})))\right) \right. \\ && -\frac{{r_{3}^{2}} \log (r-r_{3})}{(r_{1}-r_{3})^{3} (r_{3}-r_{2})^{3} (r_{3}-r_{4})^{3}} \left(\Lambda r_{3} (r_{1}-r_{3})^{2} (r_{2}-r_{3})^{2} (r_{3}-r_{4})^{2}-3 Q^{2} \left(r_{1} \left(r_{2} (r_{3}-3 r_{4}) \right.\right.\right.\\ && \left.\left.\left. +r_{3} (r_{3}+r_{4})\right)+r_{3} (r_{2} (r_{3}+r_{4})+r_{3} (r_{4}-3 r_{3}))\right){\vphantom{9 {r_{1}^{3}}-3 {r_{1}^{2}} (r_{2}+r_{3}+r_{4})-3 r_{1} (r_{2} (r_{3}+r_{4})+r_{3} r_{4}}}\right) -\frac{{r_{4}^{2}} \log (r-r_{4})}{(r_{1}-r_{4})^{3} (r_{4}-r_{2})^{3} (r_{4}-r_{3})^{3}}\\ && \left(Q^{2} \left(-3 {r_{4}^{2}} (r_{1}+r_{2}+r_{3})-3 r_{4} (r_{1} (r_{2}+r_{3})+r_{2} r_{3})+9 r_{1} r_{2} r_{3}+9 {r_{4}^{3}}\right) \right.\\ && +\frac{\frac{{r_{1}^{3}}}{(r-r_{1}) (r_{1}-r_{3})^{2} (r_{1}-r_{3})^{2}} +\frac{{r_{2}^{3}}}{(r-r_{2}) (r_{2}-r_{3})^{2} (r_{2}-r_{4})^{2}}}{(r_{1}-r_{2})^{2}}\\ && +\frac{\frac{{r_{3}^{3}}}{(r-r_{3}) (r_{1}-r_{3})^{2} (r_{2}-r_{3})^{2}}+\frac{{r_{4}^{3}}}{(r-r_{4}) (r_{1}-r_{4})^{2}(r_{2}-r_{4})^{2}}}{(r_{3}-r_{4})^{2}}\\ && \left. \left. +\Lambda r_{4}(r_{1}-r_{4})^{2} (r_{2}-r_{4})^{2} (r_{3}-r_{4})^{2}\right){\vphantom{\frac{{r_{1}^{2}} \log (r-r_{1})}{(r_{1}-r_{2})^{3} (r_{1}-r_{3})^{3} (r_{1}-r_{4})^{3}}}}\right\}; \end{array}$$

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Case 2

$1+\frac {Q^{2}}{r^{2}}-\frac {2M}{r}+\frac {r^{2}\Lambda }{3}=\frac {\Lambda }{3r^{2}}(r-r_{1})(r-r_{2})(r^{2}+ar+b)$:

$$\begin{array}{@{}rcl@{}} &\tilde{N}(r)&=\frac{9 \mu^{2} Q^{2} }{\Lambda^{2}}\left\{-\frac{(b-r_{1} r_{2}) \log (r (a+r)+b)}{2 (r_{1} (a+r_{1})+b)^{3} (r_{2} (a+r_{2})+b)^{3}} \left[-b r_{1} r_{2} \left(3 a^{2}+10 a (r_{1}+r_{2})+3 {r_{1}^{2}}+7 r_{1} r_{2}+3 {r_{2}^{2}}\right) \right. \right. \\ && \left.\left. +{r_{1}^{2}} {r_{2}^{2}} \left(-3 a^{2}-a (r_{1}+r_{2})+r_{1} r_{2}\right)-b^{2} \left(a (r_{1}+r_{2})+3 {r_{1}^{2}}+7 r_{1} r_{2}+3 {r_{2}^{2}}\right)+b^{3}\right] \right. \\ && \left. +\frac{1}{(r_{1}-r_{2})^{2}}\left[\frac{{r_{1}^{3}}}{(r-r_{1}) (r_{1} (a+r_{1})+b)^{2}}+\frac{{r_{2}^{3}}}{(r-r_{2}) (r_{2} (a+r_{2})+b)^{2}}\right] \right. \\ && -\frac{1}{\left(a^{2}-4 b\right) (r (a+r)+b) (r_{1} (a+r_{1})+b)^{2} (r_{2} (a+r_{2})+b)^{2}}\left[a^{3} r {r_{1}^{2}} {r_{2}^{2}}+a^{2} b \left(b^{2}+r_{1} r_{2} \left(2 r(r_{1}+r_{2}) \right.\right. \right.\\ &&\left.\left. +r_{1} r_{2}\right)\right)+a b \left(b^{2} (r+2 (r_{1}+r_{2}))+b \left(r \left({r_{1}^{2}}+4 r_{1} r_{2}+{r_{2}^{2}}\right)+2 r_{1} r_{2} (r_{1}+r_{2})\right)-3 r {r_{1}^{2}} {r_{2}^{2}}\right)\\ &&\left. +2 b^{2} \left(-b^{2}+b\left(2 r (r_{1}+r_{2})+{r_{1}^{2}}+4 r_{1} r_{2}+{r_{2}^{2}}\right) -r_{1} r_{2} (2 r(r_{1}+r_{2})+r_{1} r_{2})\right)\right]\\[-2pt]&&-\frac{\tan^{-1}\left(\frac{a+2 r}{\sqrt{4 b-a^{2}}}\right)}{\left(4 b-a^{2}\right)^{3/2} (r_{1}(a+r_{1})+b)^{3} (r_{2} (a+r_{2})+b)^{3}} \left[3 a^{5} r_{1} r_{2} \left(b^{2}+{r_{1}^{2}} {r_{2}^{2}}\right)+a^{4} (r_{1}+r_{2}) (b+r_{1} r_{2})\right.\\ &&\times\left(b^{2}+8 b r_{1} r_{2}+{r_{1}^{2}} {r_{2}^{2}}\right)+a^{3} \left(-b^{4} +b^{3} (r_{1}-3 r_{2}) (3 r_{1}-r_{2})+18 b^{2} r_{1} r_{2} \left({r_{1}^{2}}+3 r_{1} r_{2}+{r_{2}^{2}}\right)\right.\\&&\left.+b {r_{1}^{2}} {r_{2}^{2}} (r_{1}-3 r_{2}) (3 r_{1}-r_{2})-{r_{1}^{4}} {r_{2}^{4}}\right) -6 a^{2} b \left(b-{r_{1}^{2}}\right)\left(b-{r_{2}^{2}}\right) (r_{1}+r_{2}) (b+r_{1} r_{2})\\ && +2 a b \left(3 b^{4} +12 b^{3} r_{1}r_{2}+b^{2} \left({r_{1}^{4}}-20 {r_{1}^{3}} r_{2}-72 {r_{1}^{2}} {r_{2}^{2}}-20 r_{1} {r_{2}^{3}}+{r_{2}^{4}}\right)+12 b {r_{1}^{3}} {r_{2}^{3}}+3 {r_{1}^{4}} {r_{2}^{4}}\right) \\ &&\left. +8 b^{2} (r_{1}+r_{2}) (b+r_{1} r_{2})\left(3 b^{2}-b \left({r_{1}^{2}}+8 r_{1} r_{2}+{r_{2}^{2}}\right)+3 {r_{1}^{2}} {r_{2}^{2}}\right)\right] \\ &&-\frac{{r_{1}^{2}} \log (r-r_{1}) (r_{1} (r_{2} (a-r_{1})+r_{1} (a+3 r_{1}))-b (r_{1}-3 r_{2}))}{(r_{1} (a+r_{1})+b)^{3}(r_{1}-r_{2})^{3}}\\ && \left. -\frac{{r_{2}^{2}} \log (r-r_{2}) (r_{2} (a (r_{1}+r_{2})+r_{2} (3 r_{2}-r_{1}))+b (3 r_{1}-r_{2}))}{(r_{2} (a+r_{2})+b)^{3}(r_{1}-r_{2})^{3}}\right\}; \end{array}$$

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$$\begin{array}{@{}rcl@{}} &{}\tilde{A}_{0}(r)&=\frac{6 q^{3} \mu^{2} }{r^{2} \Lambda^{2}}\left\{\frac{{r_{2}^{2}} \left(-3 (b (3 r_{1}-r_{2})+r_{2} (a (r_{1}+r_{2})+r_{2} (3 r_{2}-r_{1}))) q^{2}-(r_{1}-r_{2})^{2} r_{2} (b+r_{2} (a+r_{2}))^{2} \Lambda \right)}{(r_{1}-r_{2})^{3} \left({r_{2}^{2}}+a r_{2}+b\right)^{3}}\right.\\[-4pt] &&\left. \log (r-r_{2}) +\frac{{r_{1}^{2}}\log (r-r_{1})}{\left({r_{1}^{2}}+a r_{1}+b\right)^{3} (r_{1}-r_{2})^{3}} \left(3 (r_{1} (r_{1} (3 r_{1}-r_{2})+a (r_{1}+r_{2}))-b (r_{1}-3 r_{2})) q^{2} \right. \right.\\[-4pt] && \left.\left. +r_{1} (b+r_{1} (a+r_{1}))^{2} (r_{1}-r_{2})^{2} \Lambda \right) +\frac{3}{(r_{1}-r_{2})^{2}} \left(\frac{{r_{1}^{3}}}{(r-r_{1}) (b+r_{1} (a+r_{1}))^{2}}+\frac{{r_{2}^{3}}}{(r-r_{2}) (b+r_{2} (a+r_{2}))^{2}}\right) \right. \\ && \left. -\frac{3}{\left(a^{2}-4 b\right) (b+r (a+r)) (b+r_{1} (a+r_{1}))^{2} (b+r_{2} (a+r_{2}))^{2}}\left(r {r_{1}^{2}} {r_{2}^{2}} a^{3}+b \left(b^{2}+r_{1} r_{2} (r_{1} r_{2}+2 r (r_{1}+r_{2}))\right) a^{2} \right.\right. \\[-2pt] && \left.\left. +b \left((r+2 (r_{1}+r_{2})) b^{2}+\left(2 r_{1} r_{2} (r_{1}+r_{2})+r \left({r_{1}^{2}}+4 r_{2} r_{1}+{r_{2}^{2}}\right)\right) b-3 r {r_{1}^{2}} {r_{2}^{2}}\right) a+2 b^{2} \left(-b^{2}+\left({r_{1}^{2}}+4 r_{2} r_{1} \right.\right.\right. \right.\\ && \left.\left.\left.\left. +{r_{2}^{2}} +2 r (r_{1}+r_{2})\right) b-r_{1} r_{2} (r_{1} r_{2}+2 r (r_{1}+r_{2}))\right)\right)+\frac{3 }{\left(4 b-a^{2}\right)^{3/2} (b+r_{1} (a+r_{1}))^{3} (b+r_{2} (a+r_{2}))^{3}} \right.\\ && \left. \left[3 r_{1} r_{2} \left(b^{2}+{r_{1}^{2}} {r_{2}^{2}}\right) a^{5}+(r_{1}+r_{2}) (b+r_{1} r_{2}) \left(b^{2}+8 r_{1} r_{2} b+{r_{1}^{2}} {r_{2}^{2}}\right) a^{4}+\left(-b^{4}+(r_{1}-3 r_{2}) (3 r_{1}-r_{2}) b^{3} \right.\right.\right. \\ && \left.\left.\left. +18 r_{1} r_{2} \left({r_{1}^{2}}+3 r_{2} r_{1}+{r_{2}^{2}}\right) b^{2} +{r_{1}^{2}} (r_{1}-3 r_{2}) (3 r_{1}-r_{2}) {r_{2}^{2}} b-{r_{1}^{4}} {r_{2}^{4}}\right) a^{3}-6 b \left(b-{r_{1}^{2}}\right) (r_{1}+r_{2}) \right.\right.\\ && \left.\left. (b+r_{1} r_{2}) \left(b-{r_{2}^{2}}\right) a^{2}+2 b \left(3 b^{4}+12 r_{1} r_{2} b^{3}+\left({r_{1}^{4}}-20 r_{2} {r_{1}^{3}}-72 {r_{2}^{2}} {r_{1}^{2}}-20 {r_{2}^{3}} r_{1}+{r_{2}^{4}}\right) b^{2} \right.\right.\right.\\ && \left.\left.\left. +12 {r_{1}^{3}} {r_{2}^{3}} b+3 {r_{1}^{4}} {r_{2}^{4}}\right) a+8 b^{2} (r_{1}+r_{2}) (b+r_{1} r_{2}) \left(3 b^{2}-\left({r_{1}^{2}}+8 r_{2} r_{1}+{r_{2}^{2}}\right) b+3 {r_{1}^{2}} {r_{2}^{2}}\right)\right] \right.\\ && \left. +\left(a^{2}-4 b\right) (b+r_{1} (a+r_{1}))^{2} (b+r_{2} (a+r_{2}))^{2} \left(r_{1} r_{2} a^{3}+b (r_{1}+r_{2}) a^{2}+b (b-3 r_{1} r_{2}) a \right.\right.\\[-2pt] && \left.\left. -2 b^{2} (r_{1}+r_{2})\right)\Lambda \tan^{-1}\left(\frac{a+2 r}{\sqrt{4 b-a^{2}}}\right)+\frac{1}{2 \left({r_{1}^{2}}+a r_{1}+b\right)^{3} \left({r_{2}^{2}}+a r_{2}+b\right)^{3}}\left(3 (b-r_{1} r_{2}) \right.\right.\\ && \left.\left. \left(b^{3}-\left(3 {r_{1}^{2}}+7 r_{2} r_{1}+3 {r_{2}^{2}}+a (r_{1}+r_{2})\right) b^{2}-r_{1} r_{2} \left(3 a^{2}+10 (r_{1}+r_{2}) a+3 {r_{1}^{2}}+3 {r_{2}^{2}}+7 r_{1} r_{2}\right)\right.\right.\right.\\ && \left.\left. b+{r_{1}^{2}} {r_{2}^{2}} \left(-3 a^{2}-(r_{1}+r_{2}) a+r_{1} r_{2}\right)\right) +(b+r_{1} (a+r_{1}))^{2}(b+r_{2} (a+r_{2}))^{2} \right.\\ && \left.\left. \left(r_{1} r_{2} a^{2}+b^{2}+b (a (r_{1}+r_{2})-r_{1} r_{2})\right) \Lambda \right) \log {\vphantom{\left\{\frac{{r_{2}^{2}} \left(-3 (b (3 r_{1}-r_{2})+r_{2} (a (r_{1}+r_{2})+r_{2} (3 r_{2}-r_{1}))) q^{2}-(r_{1}-r_{2})^{2} r_{2} (b+r_{2} (a+r_{2}))^{2} \Lambda \right)}{(r_{1}-r_{2})^{3} \left({r_{2}^{2}}+a r_{2}+b\right)^{3}}\right.}}\left(r^{2}+a r+b\right)\right\} \end{array}$$

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Case 3

$1+\frac {Q^{2}}{r^{2}}-\frac {2M}{r}+\frac {r^{2}\Lambda }{3}=\frac {\Lambda }{3r^{2}}(r^{2}+ar+b)(r^{2}+cr+d)$:

$$\begin{array}{@{}rcl@{}} &{}\tilde{N}(r)&=-\frac{9 q^{2} \mu^{2} }{\Lambda^{2}}\left\{\frac{1}{\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{2}}\left(\frac{1}{\left(a^{2}-4 b\right) (b+r (a+r))}\right. \right.\\[-3pt] && \left.\left.\left. \times\left(d^{2} r a^{3} +b \left(b^{2} +d(d-2 c r)\right) a^{2}+b \left(\left(-3d^{2}+2 b d+b \left(c^{2}+b\right)\right) r-2 b c (b+d)\right) \right. \right. \right.\right.\\[-6pt]&&\left. \times a-2 b^{2} \left(b^{2}-\left(c^{2}-2 r c+2 d\right) b+d (d-2 c r)\right)\right)+\frac{1}{\left(c^{2}-4 d\right) (d+r (c+r))}\left(\left(c^{2}-2 d\right) d+c\right.\\ &&\left.\times\left(c^{2}-3 d\right) r\right) b^{2}+2 d \left(d (2 d+c r) -a\left(r c^{2}+d c-2 d r\right)\right) b+d^{2} \left((2 d+c r) a^{2}-2 d (c+2 r) a\right.\\[-6pt]&&\left.\left.+d (c (c+r)-2 d)\right){\vphantom{\left(c^{2}-3 d\right)}}\right) -\frac{1}{\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{3}}-\frac{1}{\left(4 b-a^{2}\right)^{3/2}}\\[-4pt]&& \times\left(-3 d \left(b^{2}+d^{2}\right) a^{5}+c (b+d) \left(b^{2}+8 d b+d^{2}\right) a^{4}+\left(b^{4}+ \left(16 d-3 c^{2}\right) b^{3}-18 d \left(c^{2}+d\right) b^{2}\right.\right. \\ &&\left.+d^{2} \left(16 d-3 c^{2}\right) b+d^{4}\right) a^{3}-6 b c (b+d) \left(b^{2}-c^{2} b+2 d b+d^{2}\right) a^{2}-2 b \left(3 b^{4}+12 d b^{3}\right.\\ &&\left.+\left(c^{4}-24 d c^{2}-30 d^{2}\right) b^{2}+12 d^{3} b+3 d^{4}\right) a+8 b^{2} c (b+d) \left(3 b^{2}-\left(c^{2}+6 d\right) b\right.\\[-4pt]&&\left.\left.+3 d^{2}\right)\right) \tan ^{-1}\left(\frac{a+2 r}{\sqrt{4 b-a^{2}}}\right)-\frac{1}{\left(4 d-c^{2}\right)^{3/2}}\left(-6 (c-4 a) d^{5}+\left(-8 a^{3}-6 \left(c^{2}+4 b\right) a \right.\right.\\[-4pt]&& \left. +c^{3}-24 b c\right) d^{4}+\left(a c^{4}+\left(16 b-3 a^{2}\right) c^{3}+6 a \left(a^{2}-3 b\right) c^{2}-2 \left(a^{4}-24 b a^{2}-30 b^{2}\right) c\right.\\&& \left.-8 a b \left(a^{2}+3 b\right)\right) d^{3}-3 b \left(c^{5}-3 a c^{4}+6 \left(a^{2}+b\right) c^{3}-2 a \left(a^{2}-3 b\right) c^{2}+8 b^{2} c-8 a b^{2}\right)d^{2} \\[-4pt] && +b^{2} c \left(-6 b^{2}+2 c (8 c-3 a) b\left.\left.-3 a (a-3 c) c^{2}\right) d+b^{3} c^{3} (b+(a-3 c) c)\right) \tan^{-1}\left(\frac{c+2 r}{\sqrt{4 d-c^{2}}}\right) \right.\\&& -\frac{(b-d)}{2 \left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{3}}\left(b^{3}+((a-3 c) c-d) b^{2}-d ((a-3 c) (3a-c)+d) b\right.\\[-4pt] && \left.\left.+d^{2} (a (c-3 a)+d)\right) (\log (b+r (a+r))-{\vphantom{\left\{\frac{1}{\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{2}}\left(\frac{1}{\left(a^{2}-4 b\right) (b+r (a+r))}\right. \right.}}\log (d+r (c+r))\left.{\vphantom{\frac{1}{\left(a^{2}-4 b\right) (b+r (a+r))}}}\right)\right\}; \end{array}$$

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$$\begin{array}{@{}rcl@{}} &\tilde{A}_{0}(r)&=\frac{6 q^{3} \mu^{2} }{r^{2} \Lambda^{2}}\left\{\frac{3 }{\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{2}}\left(\frac{1}{\left(a^{2}-4 b\right) (b+r (a+r))}\left(-d^{2} r a^{3}-b \right.\right.\right.\\ && \left.\left.\left.\left. \times\left(b^{2}+d (d-2 c r)\right) a^{2}+b \left(2 b c (b+d)-\left(-3 d^{2}+2 b d+b \left(c^{2}+b\right)\right) r\right) a+2 b^{2} \left(b^{2}-\left(c^{2}-2 r c+2 d\right) b \right. \right. \right. \right.\right.\\ && \left.\left.\left.\left. +d (d-2 c r)\right)\right)-\frac{1}{\left(c^{2}-4 d\right) (d+r (c+r))}\left(\left(c^{2}-2 d\right) d+c \left(c^{2}-3 d\right) r\right) b^{2}+2 d \left(d (2 d+c r) \right. \right. \right.\\ && \left.\left.\left. -a \left(r c^{2}+d c-2 d r\right)\right) b+d^{2} \left((2 d+c r) a^{2}-2 d (c+2 r) a+d (c (c+r)-2 d)\right)\right) \right. \\ &&\left. +\frac{\left(4 b-a^{2}\right)^{3/2}}{\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-ca+d\right)\right)^{3}}\left(\left(a^{2}-4 b\right) \left(d a^{3}-b c a^{2}+b (b-3 d) a+2 b^{2}c\right) \Lambda \right.\right. \\ &&\left.\left.\left. \times\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{2}+3 \left(3 d \left(b^{2}+d^{2}\right) a^{5}-c (b+d) \left(b^{2}+8 d b+d^{2}\right) a^{4} \right. \right.\right.\right. \\ &&\left.\left.\left. -\left(b^{4}+\left(16 d-3 c^{2}\right) b^{3}-18 d \left(c^{2}+d\right) b^{2}+d^{2} \left(16 d-3 c^{2}\right) b+d^{4}\right) a^{3}+6 b c (b+d) \left(b^{2}-c^{2} b \right.\right.\right.\right. \\ &&\left.\left.\left.\left. +2 d b+d^{2}\right) a^{2}+2 b \left(3 b^{4}+12 d b^{3}+\left(c^{4}-24 d c^{2}-30 d^{2}\right) b^{2}+12 d^{3} b+3 d^{4}\right) a-8 b^{2} c (b+d) \right.\right.\right. \\ &&\left.\left.\left. \times\left(3 b^{2}-\left(c^{2}+6 d\right) b+3 d^{2}\right)\right) \right) \tan^{-1}\left(\frac{a+2 r}{\sqrt{4 b-a^{2}}}\right)+\frac{1}{\left(4 d-c^{2}\right)^{3/2}}\left(\left(c^{2}-4 d\right)\left(b c^{3}-(3 b+a c) d c \right.\right.\right.\\ &&\left.\left.\left. +(2a+c) d^{2}\right) \Lambda \left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{2}+3 \left(6 (c-4 a) d^{5}+\left(8 a^{3}+6 \left(c^{2}+4 b\right) a \right.\right.\right.\right.\\ &&\left.\left.\left.\left. -c^{3}+24 b c\right) d^{4}+\left(-a c^{4}+\left(3 a^{2}-16 b\right) c^{3}-6 a \left(a^{2}-3 b\right) c^{2}+2 \left(a^{4}-24 b a^{2}-30 b^{2}\right) c \right.\right.\right.\right.\\ &&\left.\left.\left.\left. +8 a b \left(a^{2}+3 b\right)\right) d^{3}+3 b \left(c^{5}-3 a c^{4}+6 \left(a^{2}+b\right) c^{3}-2 a \left(a^{2}-3 b\right) c^{2}+8 b^{2} c-8 a b^{2}\right) d^{2} \right.\right.\right.\\ &&\left.\left.\left. +b^{2} c \left(6 b^{2}+2 (3 a-8 c) c b+3 a (a-3 c) c^{2}\right) d-b^{3} c^{3} (b+(a-3 c) c)\right) \right) \tan ^{-1}\left(\frac{c+2 r}{\sqrt{4 d-c^{2}}}\right) \right. \\ && \left. +\frac{1}{2 \left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{3}}\left(\left(d a^{2}+b^{2}-b (a c+d)\right) \Lambda \left(b^{2}+\left(c^{2}-a c-2 d\right) b \right.\right.\right.\\ && \left.+d \left(a^{2}-c a+d\right)\right)^{2}+3 (b-d) \left(b^{3}+((a-3 c) c-d) b^{2}-d ((a-3 c) (3 a-c)+d) b\right.\\&&\left.\left.+d^{2} (a (c-3 a)+d)\right)\right) \log (b+r (a+r))+\left(\left(b c^{2}+d^{2}-(b+a c) d\right)\left(b^{2}+\left(c^{2}-a c-2 d\right) b\right.\right.\\&&\left.+d \left(a^{2}-c a+d\right)\right)^{2} \Lambda-3 (b-d) \left(b^{3}+((a-3 c)c-d) b^{2}-d ((a-3 c) (3 a-c)+d) b\right.\\&&\left.\left.\left.+d^{2} (a (c-3 a)+d)\right) \right){\vphantom{\left\{\frac{3 }{\left(b^{2}+\left(c^{2}-a c-2 d\right) b+d \left(a^{2}-c a+d\right)\right)^{2}}\left(\frac{1}{\left(a^{2}-4 b\right) (b+r (a+r))}\left(-d^{2} r a^{3}-b \left(b^{2} \right. \right. \right.\right.}}\log \left(d+cr+r^{2}\right)\right\}. \end{array}$$

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