C..2 Ultraviolet limit

To tackle the problem of the ultraviolet behavior of the energy density, we analyze the limit in which $r \to \infty$, $\rho \to \infty$. As already discussed we will first set $r = \rho = \Lambda$ and then approximate the various quantities as $\Lambda \to \infty$. For the relevant expressions, already encountered above, we get⁶

$$\mbox{\boldmath$A$} = - 4 \kappa \Lambda ^{2} \left( 1 - \frac{( \kappa + m _{\mathrm{A... ...ppa + m _{\mathrm{A}} ^{2} ) ^{2}}{2 \kappa \Lambda ^{2}} \right) , \quad \dots$$ (49)

$$\mbox{\boldmath$B$} = 4 \kappa \Lambda ^{2} \left( 1 + \frac{m _{\mathrm{A}} ^{4}}{4 \k... ...t( 1 + \frac{m _{\mathrm{A}} ^{4}}{2 \kappa \Lambda ^{2}} \right) , \quad \dots$$ (50)

$$\mbox{\boldmath$C$} = - 4 \Lambda ^{2} \left( 1 - \frac{ m _{\mathrm{A}} ^{2}}{4 \Lambd... ...^{2} \left( 1 - \frac{ \kappa + m _{\mathrm{A}} ^{2}}{4 \Lambda ^{2}} \right) ,$$

$$\qquad \qquad ( \kappa + \mbox{\boldmath$C$}) ^{-1} \sim - \frac{... ...1 + \frac{ \kappa + m _{\mathrm{A}} ^{2}}{4 \Lambda ^{2}} \right) , \quad \dots$$ (51)

$$\mbox{\boldmath$D$} = 4 \Lambda ^{4} \left( 1 - \frac{m _{\mathrm{A}} ^{2}}{2 \Lambda ^{2}} \right) ,$$

$$\qquad \qquad \log ( - \kappa \ \rho ^{2} + \mbox{\boldmath$D$}) ... ...da ^{2}} - \frac{( \kappa + 2 m _{\mathrm{A}} ^{2} ) ^{4}}{32 \Lambda ^{2}} \,.$$ (52)

Moreover we also have the well known expansions

$${\mathrm{Arctanh}} \left( x \right) = x + \frac{x ^{3}}{3} + \frac{x ^{5}}{5} + {\mathcal{O}} (x ^{7})$$ (53)

$$\arctan \left( x \right) = x - \frac{x ^{3}}{3} + \frac{x ^{5}}{5} + {\mathcal{O}} (x ^{7}) \,,$$ (54)

which we are going to use in the following. In particular we can consider in generality the expansion of the following expression

$$- \frac{w ^{3/2}}{2} \ln \left( \frac{z - w ^{1/2}}{z + w ^{1/2}} \right) = w ^{3/2} {\mathrm{Arctanh}} \left( \frac{w ^{1/2}}{z} \right)$$

$$= \cases{ \vert w\vert ^{3/2} {\mathrm{Arctanh}} \displaystyle\left... ...w\vert ^{1/2} \quad \mbox{and$\quad w ^{3/2} = - \imath \vert w\vert ^{3/2}$} }$$

$$= \cases{ \vert w\vert ^{3/2} {\mathrm{Arctanh}} \displaystyle\left... ...playstyle\left( \frac{\vert w\vert ^{1/2}}{z} \right) \quad \mbox{if $w < 0$} }$$

$$= \cases{ \vert w\vert ^{3/2} {\mathrm{Arctanh}} \displaystyle\left... ...playstyle\left( \frac{\vert w\vert ^{1/2}}{z} \right) \quad \mbox{if $w < 0$} }$$

$$= \vert w\vert ^{3/2} \Biggl [ \left( \frac{\vert w\vert ^{1/2}}{z}... ...thrm{sign}} (w) \frac{1}{3} \left( \frac{\vert w\vert ^{1/2}}{z} \right) ^{3} +$$

$$\hphantom{\vert w\vert ^{3/2} \Biggl [} + \frac{1}{5} \left( \fra... ...sign}} (w) \left( \frac{\vert w\vert ^{1/2}}{z} \right) ^{7} + \cdots \Biggr] ,$$

so that

$$- \frac{w ^{3/2}}{2} \ln \left( \frac{z - w ^{1/2}}{z + w ^... ...^{0 , \infty} \frac{w ^{n}}{(2 n + 1) z ^{2 n + 1}} \right] .$$ (55)

From the above result, if we identify

$$w \longleftrightarrow \mbox{\boldmath$B$}\,,$$

$$z \longleftrightarrow \mbox{\boldmath$C$}$$

and we consider an overall $1/(24 \kappa)$ factor, using properly (50) and the first equation of (51), we get

$$\frac{\mbox{\boldmath$B$}^{3/2}}{48 k} \ln \left( \frac{\mb... ...{2}}{6} + \frac{\kappa ^{2} + 3 \kappa m ^{2} + 6 m ^{4}}{72}$$ (56)

for the second term in (38). In the same way, starting again from result (55), together with the identifications

$$w \longleftrightarrow \mbox{\boldmath$A$}\,,$$

$$z \longleftrightarrow \kappa + \mbox{\boldmath$C$}$$

and taking into account an overall $-1/(24 \kappa)$, (49) and (51), we obtain for the third term in (38)

$$- \frac{A ^{3/2}}{48 \kappa} \ln \left( \frac{\kappa + \mbo... ...} + \frac{4 \kappa ^{2} + 9 \kappa m ^{2} + 6 m ^{4}}{72} \,.$$ (57)

The last term in (38) has also a logarithmic part and, using (52), can be approximated as

$$\frac{1}{48} \left[ \kappa ^{2} + 3 \kappa m _{\mathrm{A}} ^{2} -... ...math$D$}\right] \ln \left( - \kappa \rho ^{2} + \mbox{\boldmath$D$}\right) \sim$$

$$\qquad \qquad \sim \frac{\kappa ^{2} + 3 \kappa m _{\mathrm{A}} ^... ...ambda ^{4} \right) + \frac{\Lambda ^{4}}{2} \ln \left( 4 \Lambda ^{4} \right) +$$

$$\hphantom{\qquad \qquad \sim} + \, \frac{( \kappa + 2 m _{\mathrm{A}} ^{2}) ^{2}}{64} - \frac{\kappa + 2 m _{\mathrm{A}} ^{2}}{8} \Lambda ^{2} \,.$$ (58)

We are now concerned with the easiest term in (38), namely the first, for which we get

$$\frac{ \rho ^{2} \left( \kappa + 3 \left( m _{\mathrm{A}} ^... ...athrm{A}} ^{2}}{24} \Lambda ^{2} - \frac{7}{8} \Lambda ^{4}\,.$$ (59)

Summing up equations (56), (57), (58), (59), we obtain

$$\Lambda ^{4} \left [ 2 \ln \Lambda + \ln 2 - \frac{7}{8} \right ]... ...ppa + 2 m _{\mathrm{A}} ^{2}}{4} \left( 2 \ln \Lambda + \ln 2 \right) \right] +$$ (60)

$$+ \, \frac{5 \kappa ^{2} + 12 \kappa m _{\mathrm{A}} ^{2} + 12 m ... ...m{A}} ^{2} + 3 m _{\mathrm{A}} ^{4}}{48} \left( 2 \ln \Lambda + \ln 2 \right) .$$