with B arbitrary. Employing the relation [33] where γ = 0 , 5772 ... is the Euler-Mascheroni con- stant and ζ is the zeta Riemann function, and ex- panding ( − B $^{2}$x$^{2}$) a as a power series one can compare both sides of (52). After some algebra one gets Expanding the exponential in power series and equating the coefficients of equal powers of a one obtains This work was supported by CAPES and CNPq, by grants of FCT, FEDER, OE, POCI 2010, CERN/FP/83510/2008, and by HadronPhysics2, Grant Agreement n. 227431 under the Seventh Framework Programme of EU. with ¯ B 2 ≡ 4 e 2 $_{γ}$B $^{2}$. ∫ d $^{4}$x e ipx x 2 ln $^{2}$( − x $^{2}$B$^{2}$) = − i 4 π 2 p 2 ln 2 ( − p 2 ¯ B 2 ) . (58) ∫ d $^{4}$x e ipx x 2 ln $^{3}$( − x $^{2}$B$^{2}$) = i 4 π 2 p 2 ( ln 3 ( − p 2 ¯ B 2 ) +4 ζ (3) ) . (59) ∫ d $^{4}$x e ipx x 2 ln $^{4}$( − x $^{2}$B$^{2}$) = − i 4 π 2 p 2 ( ln 4 ( − p 2 ¯ B 2 ) + 16 ζ (3) ln ( − p 2 ¯ B 2 ) . (60) ( − B $^{2}$x$^{2}$) a = ∞ ∑ n =0 a n n ! ln $^{n}$( − x $^{2}$B$^{2}$) (54) Γ(1 + a ) Γ(1 − a ) = e − 2 $^{γa}$exp [ 2 ∞ ∑ n =1 ζ (2 n + 1) 2 n + 1 a 2 n +1 ] , (53) ∞ ∑ n =0 a n n ! ∫ d $^{4}$x e ipx 1 x 2 ln $^{n}$( − x $^{2}$B$^{2}$) = − i 4 π 2 p 2 × exp [ − a ln ( $_{−}$p $^{2}$e2 γ 4 B 2 ) − 2 ∞ ∑ n =1 ζ (2 n + 1) 2 n + 1 a 2 n +1 ] . (55) ∫ d $^{4}$x e ipx x 2 = − i 4 π 2 p 2 . (56) ∫ d $^{4}$x e ipx x 2 ln( − x $^{2}$B$^{2}$) = i 4 π 2 p 2 ln ( − p 2 ¯ B 2 ) . (57) ACKNOWLEDGMENTS 8 [20] G. Dunne, N. Rius, Phys. Lett. B 293 , 367 (1992). [10] J. Mas, M. Perez-Victoria, C. Seijas, JHEP 0203 [9] N. A-Hamed and H. Murayama, JHEP 0006 030 (2000). [19] P. E. Haagensen, J. I. Latorre, Ann. Phys. 221 , 77 (1993). [18] D. Z Freedman, G. Grignani, K. Johnson and N. Rius, Ann. Phys. 218 , 75 (1992). [17] R. Mu˜ noz-Tapia, Phys. Lett. B 295 , 95 (1992). [8] N. A-Hamed and H. Murayama, Phys. Rev. D 57 (1998) 6638. [7] V. Novikov, M. Shifman, A. Vainstein, V. Za- kharov, Phys. Lett. B 166 329 (1985). [16] P. E. Haagensen, Mod. Phys. Lett. A 7 , 893 (1992). [15] D. Z. Freedman, K. Johnson and J. I. Latorre, Nucl. Phys. B 371 , 353 (1992). [14] D. Stockinger, Nucl. Phys. Suppl. 160 (2006) 250; JHEP 0503 076 (2005). [6] L. A. M. Souza, Marcos Sampaio, M. C. Nemes, Phys. Lett. B 632 717 (2006). [5] A. P. Baˆeta Scarpelli, M. Sampaio, M. C. Nemes and B. Hiller, Phys. Rev. D 64 , 046013 (2001). [4] A. P. Baˆeta Scarpelli, M. Sampaio and M. C. Nemes, Phys. Rev. D 63 , 046004 (2001). [3] O. A. Battistel and M. C. Nemes, Phys. Rev. D 59 , 055010 (1999). [2] A. Brizola, O. A. Battistel, M. Sampaio and M. C. Nemes, Mod. Phys. Lett. A 14 , 1509 (1999). [1] O. A. Battistel, A. L. Mota and M. C. Nemes, Mod. Phys. Lett. A 13 , 1597 (1998). [13] L. C. Ferreira, M. C. Nemes, M. Sampaio, Work in progress . [12] D. Carneiro, A. P. Baˆeta Scarpelli, M. Sampaio and M. C. Nemes, JHEP 12 044 (2003). [11] A. Zee, Quantum Field Theory in a Nutshell (2003). 049 (2002).