C..3 Result for the regularized integral

From the results (47), (60) of the two previous subsections we obtain what we are interested in,

$$J ( r = \Lambda , \rho = \Lambda ; \kappa , m _{\mathrm{A}} ) - \lim _{(r , \rho) \to (0 , 0)} J ( r , \rho ; \kappa , m _{\mathrm{A}} ) \sim$$

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

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

$$\hphantom{\qquad \sim} -\, \frac{( \kappa + m _{\mathrm{A}} ^{2} ... ..._{\mathrm{A}} ^{2} ) ^{3}}{24 \kappa} \ln \left( m _{\mathrm{A}} ^{2} \right) .$$

$$\qquad \sim \left( I _{\Lambda} ^{(4)} - \ln 2 + \frac{3}{4} \rig... ... + I _{\ln \Lambda} ^{(2)} \Lambda ^{2} + I ^{(0)} + I ^{(0)} _{\ln \Lambda}\,,$$ (61)

where for convenience in the last line we have used the following definitions:

$$I _{\Lambda} ^{(4)} = 2 \ln 2 - \frac{13}{8}$$

$$I _{\ln \Lambda} ^{(4)} = 4$$

$$I _{\Lambda} ^{(2)} = - \frac{2 \kappa + 3 m _{\mathrm{A}} ^{2}}{24} - \frac{\kappa + 2 m _{\mathrm{A}} ^{2}}{4} \ln 2$$

$$I _{\ln \Lambda} ^{(2)} = - \frac{\kappa + 2 m _{\mathrm{A}} ^{2}}{2}$$

$$I ^{(0)} = \frac{49 \kappa ^{2} + 132 \kappa m _{\mathrm{A}} ^{2} + 132 m _{... ...appa ^{2} + 3 \kappa m _{\mathrm{A}} ^{2} + 3 m _{\mathrm{A}} ^{4}}{48} \ln 2 -$$

$$- \, \frac{( \kappa + m _{\mathrm{A}} ^{2} ) ^{3}}{24 \kappa} \ln... ...m _{\mathrm{A}} ^{2} ) ^{3}}{24 \kappa} \ln \left( m _{\mathrm{A}} ^{2} \right)$$

$$I ^{(0)} _{\ln \Lambda} = \frac{\kappa ^{2} + 3 \kappa m _{\mathrm{A}} ^{2} + 3 m _{\mathrm{A}} ^{4}}{24} \,.$$ (62)