- #1
ice109
- 1,714
- 6
i'm doing an integral for my advisor that is way beyond me but i have pages from a textbook that tell me how to do it so here goes
[tex]\int\frac{d^4\ell}{(2\pi)^4}\frac{1}{(\ell^2+A^2)^2} = \frac{1}{2}B(0,2)[/tex]
which is divergent
but in arbitrary dimensions
you get
[tex]\int\frac{d^4\ell}{(2\pi)^4}\frac{1}{(\ell^2+A^2)^2} = \frac{1}{(4\pi)^\frac{D}{2}}\frac{\Gamma (2-\frac{D}{2})}{\Gamma (2)}(\frac{1}{A})^{2-\frac{D}{2}}[/tex]
and setting [tex] \epsilon = 4-D[/tex]
and then letting D approach 4 we get
[tex]\frac{1}{(4\pi)^{2}}(\frac{2}{\epsilon}-log\Delta - \gamma +log(4\pi) + O(\epsilon))[/tex]
where
[tex]\gamma \approx .5772[/tex]
the thing is i only understand why this works on a very very superficial level. something along the lines of taking a limit and studying behavior as the limit approaches a "pole." but i really have no intuitive clue as to what an integral in arbitrary space even means. can anyone give me some sense of what's going on.
[tex]\int\frac{d^4\ell}{(2\pi)^4}\frac{1}{(\ell^2+A^2)^2} = \frac{1}{2}B(0,2)[/tex]
which is divergent
but in arbitrary dimensions
you get
[tex]\int\frac{d^4\ell}{(2\pi)^4}\frac{1}{(\ell^2+A^2)^2} = \frac{1}{(4\pi)^\frac{D}{2}}\frac{\Gamma (2-\frac{D}{2})}{\Gamma (2)}(\frac{1}{A})^{2-\frac{D}{2}}[/tex]
and setting [tex] \epsilon = 4-D[/tex]
and then letting D approach 4 we get
[tex]\frac{1}{(4\pi)^{2}}(\frac{2}{\epsilon}-log\Delta - \gamma +log(4\pi) + O(\epsilon))[/tex]
where
[tex]\gamma \approx .5772[/tex]
the thing is i only understand why this works on a very very superficial level. something along the lines of taking a limit and studying behavior as the limit approaches a "pole." but i really have no intuitive clue as to what an integral in arbitrary space even means. can anyone give me some sense of what's going on.