Ultraviolet via dimensional regularization

[kassem84]

The integral:
$\int$d$^{3}k$$\frac{1}{k^{2}+m^{2}}$
is linearly divergent i.e. ultraviolet divergent.
However, If one performs dimensional regularization to the above integral:
$\frac{1}{(2\pi)^d}$$\int$d$^{d}k$$\frac{1}{k^{2}+m^{2}}$=$\frac{(m^{2})^{d/2-1}}{(4\pi)^{d/2}}$$\Gamma(1-d/2)$
As you can notice that the poles of the Gamma function are for even dimension i.e. d=2,4,6..etc and that the integral is convergent for d=3 for example!
What is the reason behind this convergence? Is it due to the Veltman's formula:
$\frac{1}{(2\pi)^d}$$\int$d$^{d}k$$(k^{2})^{n-1}$ = 0, for n=0,1,2,..
I am dealing with a divergent integral of power divergence (ultraviolet, quadratic, quartic,... and no logarithmic divergence). Do you advise me to use dimensional regularization or other methods?

 

[LightHero]

The reason why the integral is convergent in 3 dimensions when using dimensional regularization is because the Gamma function in the expression has a pole at d = 2, which cancels out the divergence of the integral. This is known as the 'dimensional regularization trick'. Dimensional regularization is a useful tool for dealing with divergent integrals, especially those of power divergence. It can also be used to regularize logarithmic divergences, though this requires some additional work. Other methods such as Pauli-Villars regularization and Zeta function regularization can also be used to regularize divergent integrals.