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Ultraviolet via dimensional regularization

  1. Jul 16, 2011 #1
    The integral:
    [itex]\int[/itex]d[itex]^{3}k[/itex][itex]\frac{1}{k^{2}+m^{2}}[/itex]
    is linearly divergent i.e. ultraviolet divergent.
    However, If one performs dimensional regularization to the above integral:
    [itex]\frac{1}{(2\pi)^d}[/itex][itex]\int[/itex]d[itex]^{d}k[/itex][itex]\frac{1}{k^{2}+m^{2}}[/itex]=[itex]\frac{(m^{2})^{d/2-1}}{(4\pi)^{d/2}}[/itex][itex]\Gamma(1-d/2)[/itex]
    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:
    [itex]\frac{1}{(2\pi)^d}[/itex][itex]\int[/itex]d[itex]^{d}k[/itex][itex] (k^{2})^{n-1}[/itex] = 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?
     
  2. jcsd
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