Dimensional regularization

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Consider an infinite charged line with charge density ${\displaystyle s}$ , and we calculate the potential of a point distance ${\displaystyle x}$  away from the line. The integral diverges:$${\displaystyle V(x)=A\int _{-\infty }^{\infty }{\frac {dy}{\sqrt {x^{2}+y^{2}}}}}$$ where ${\displaystyle A=s/(4\pi \epsilon _{0}).}$ 

Since the charged line has 1-dimensional "spherical symmetry" (which in 1-dimension is just mirror symmetry), we can rewrite the integral to exploit the spherical symmetry:$${\displaystyle \int _{-\infty }^{\infty }{\frac {dy}{\sqrt {x^{2}+y^{2}}}}=\int _{-\infty }^{\infty }{\frac {dt}{\sqrt {(x/x_{0})^{2}+t^{2}}}}=\int _{0}^{\infty }{\frac {\mathrm {vol} (S^{1})dr}{\sqrt {(x/x_{0})^{2}+r^{2}}}}}$$ where we first removed the dependence on length by dividing with a unit-length ${\displaystyle x_{0}}$ , then converted the integral over ${\displaystyle \mathbb {R} ^{1}}$  into an integral over the 1-sphere ${\displaystyle S^{1}}$ , followed by an integral over all radii of the 1-sphere.

Now we generalize this into dimension ${\displaystyle d}$ . The volume of a d-sphere is ${\displaystyle {\frac {2\pi ^{d/2}}{\Gamma (d/2)}}}$ , where ${\displaystyle \Gamma }$  is the gamma function. Now the integral becomes $${\displaystyle {\frac {2\pi ^{d/2}}{\Gamma (d/2)}}\int _{0}^{\infty }{\frac {r^{d-1}dr}{\sqrt {(x/x_{0})^{2}+r^{2}}}}}$$ When ${\displaystyle d=1-\epsilon }$ , the integral is dominated by its tail, that is, ${\displaystyle \int _{0}^{\infty }{\frac {r^{d-1}dr}{\sqrt {(x/x_{0})^{2}+r^{2}}}}\sim \int _{c}^{\infty }r^{d-2}dr={\frac {1}{d-1}}c^{d-1}=\epsilon ^{-1}c^{-\epsilon },}$  where ${\displaystyle c=\Theta (x/x_{0})}$  (in big theta notation). Thus ${\displaystyle V(x)\sim (x_{0}/x)^{\epsilon }/\epsilon }$ , and so the electric field is ${\displaystyle V'(x)\sim x^{-1}}$ , as it should.

Suppose one wishes to dimensionally regularize a loop integral which is logarithmically divergent in four dimensions, like

${\displaystyle I=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {1}{\left(p^{2}+m^{2}\right)^{2}}}.}$ 

First, write the integral in a general non-integer number of dimensions ${\displaystyle d=4-\varepsilon }$ , where ${\displaystyle \varepsilon }$  will later be taken to be small,$${\displaystyle I=\int {\frac {d^{d}p}{(2\pi )^{d}}}{\frac {1}{\left(p^{2}+m^{2}\right)^{2}}}.}$$  If the integrand only depends on ${\displaystyle p^{2}}$ , we can apply the formula^[7]$${\displaystyle \int d^{d}p\,f(p^{2})={\frac {2\pi ^{d/2}}{\Gamma (d/2)}}\int _{0}^{\infty }dp\,p^{d-1}f(p^{2}).}$$  For integer dimensions like ${\displaystyle d=3}$ , this formula reduces to familiar integrals over thin shells like ${\textstyle \int _{0}^{\infty }dp\,4\pi p^{2}f(p^{2})}$ . For non-integer dimensions, we define the value of the integral in this way by analytic continuation. This gives$${\displaystyle I=\int _{0}^{\infty }{\frac {dp}{(2\pi )^{4-\varepsilon }}}{\frac {2\pi ^{(4-\varepsilon )/2}}{\Gamma \left({\frac {4-\varepsilon }{2}}\right)}}{\frac {p^{3-\varepsilon }}{\left(p^{2}+m^{2}\right)^{2}}}={\frac {2^{\varepsilon -4}\pi ^{{\frac {\varepsilon }{2}}-1}}{\sin \left({\frac {\pi \varepsilon }{2}}\right)\Gamma \left(1-{\frac {\varepsilon }{2}}\right)}}m^{-\varepsilon }={\frac {1}{8\pi ^{2}\varepsilon }}-{\frac {1}{16\pi ^{2}}}\left(\ln {\frac {m^{2}}{4\pi }}+\gamma \right)+{\mathcal {O}}(\varepsilon ).}$$  Note that the integral again diverges as ${\displaystyle \varepsilon \rightarrow 0}$ , but is finite for arbitrary small values ${\displaystyle \varepsilon \neq 0}$ .

• Bollini, Carlos; Giambiagi, Juan Jose (1972), "Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter.", Il Nuovo Cimento B, 12 (1): 20–26, Bibcode:1972NCimB..12...20B, doi:10.1007/BF02895558, S2CID 123505054
• Etingof, Pavel (1999), "Note on dimensional regularization", Quantum fields and strings: a course for mathematicians, Vol. 1,(Princeton, NJ, 1996/1997), Providence, R.I.: Amer. Math. Soc., pp. 597–607, ISBN 978-0-8218-2012-4, MR 1701608
• Hooft, G. 't; Veltman, M. (1972), "Regularization and renormalization of gauge fields", Nuclear Physics B, 44 (1): 189–213, Bibcode:1972NuPhB..44..189T, doi:10.1016/0550-3213(72)90279-9, hdl:1874/4845, ISSN 0550-3213