Intheoretical physics,dimensional regularization is a method introduced byJuan José Giambiagi andCarlos Guido Bollini [es]^[1] as well as – independently and more comprehensively^[2] – byGerard 't Hooft andMartinus J. G. Veltman^[3] forregularizingintegrals in the evaluation ofFeynman diagrams; in other words, assigning values to them that aremeromorphic functions of a complex parameterd, the analytic continuation of the number of spacetime dimensions.

Dimensional regularization writes aFeynman integral as an integral depending on the spacetime dimensiond and the squared distances (x_i−x_j)² of the spacetime pointsx_i, ... appearing in it. InEuclidean space, the integral often converges for −Re(d) sufficiently large, and can beanalytically continued from this region to a meromorphic function defined for all complexd. In general, there will be a pole at the physical value (usually 4) ofd, which needs to be canceled byrenormalization to obtain physical quantities.Pavel Etingof showed that dimensional regularization is mathematically well defined, at least in the case of massive Euclidean fields, by using theBernstein–Sato polynomial to carry out the analytic continuation.^[4]

Although the method is most well understood when poles are subtracted andd is once again replaced by 4, it has also led to some successes whend is taken to approach another integer value where the theory appears to be strongly coupled as in the case of theWilson–Fisher fixed point. A further leap is to take the interpolation through fractional dimensions seriously. This has led some authors to suggest that dimensional regularization can be used to study the physics of crystals that macroscopically appear to befractals.^[5]

It has been argued thatzeta function regularization and dimensional regularization are equivalent since they use the same principle of using analytic continuation in order for a series or integral to converge.^[6]

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.^[7] The integral diverges:$${\displaystyle V(x)=A{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dy}{\sqrt{x^2+y^2}}}$$where${\displaystyle A=s/(4\pi {ϵ}_0).}$

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:$${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dy}{\sqrt{x^2+y^2}}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{dt}{\sqrt{(x/x_0{)}^2+t^2}}={\int }_0^{\mathrm{\infty }}\frac{\mathrm{v}\mathrm{o}\mathrm{l}(S^1)dr}{\sqrt{(x/x_0{)}^2+r^2}}}$$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${\displaystyle \frac{2{\pi }^{d/2}}{\mathrm{\Gamma }(d/2)}}$, where${\displaystyle \mathrm{\Gamma }}$ is thegamma function. Now the integral becomes$${\displaystyle \frac{2{\pi }^{d/2}}{\mathrm{\Gamma }(d/2)}{\int }_0^{\mathrm{\infty }}\frac{r^{d-1}dr}{\sqrt{(x/x_0{)}^2+r^2}}}$$When${\displaystyle d=1-ϵ}$, the integral is dominated by its tail, that is,${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{r^{d-1}dr}{\sqrt{(x/x_0{)}^2+r^2}}\sim {\int }_c^{\mathrm{\infty }}{r}^{d-2}dr=\frac{1}{d-1}{c}^{d-1}={ϵ}^{-1}{c}^{-ϵ},}$ where${\displaystyle c=\mathrm{\Theta }(x/x_0)}$ (inbig Theta notation). Thus${\displaystyle V(x)\sim (x_0/x{)}^{ϵ}/ϵ}$, and so the electric field is${\displaystyle V^{\prime }(x)\sim x^{-1}}$, 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${\displaystyle d=4-\epsilon }$, where${\displaystyle \epsilon }$ 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^[8]$${\displaystyle \int d^{d}pf(p^2)=\frac{2{\pi }^{d/2}}{\mathrm{\Gamma }(d/2)}{\int }_0^{\mathrm{\infty }}dp{p}^{d-1}f(p^2).}$$ For integer dimensions like${\displaystyle d=3}$, this formula reduces to familiar integrals over thin shells like${\int }_0^{\mathrm{\infty }}dp4\pi p^{2}f(p^2)$. For non-integer dimensions, wedefine the value of the integral in this way by analytic continuation. This gives$${\displaystyle I={\int }_0^{\mathrm{\infty }}\frac{dp}{(2\pi {)}^{4-\epsilon }}\frac{2{\pi }^{(4-\epsilon )/2}}{\mathrm{\Gamma }\left(\frac{4-\epsilon }{2}\right)}\frac{p^{3-\epsilon }}{{\left(p^2+m^2\right)}^2}=\frac{2^{\epsilon -4}{\pi }^{\frac{\epsilon }{2}-1}}{\sin \left(\frac{\pi \epsilon }{2}\right)\mathrm{\Gamma }\left(1-\frac{\epsilon }{2}\right)}{m}^{-\epsilon }=\frac{1}{8{\pi }^2\epsilon }-\frac{1}{16{\pi }^2}\left(\ln \frac{m^2}{4\pi }+\gamma \right)+\mathcal{O}(\epsilon ).}$$ Note that the integral again diverges as${\displaystyle \epsilon \to 0}$, but is finite for arbitrary small values${\displaystyle \epsilon \ne 0}$.

• 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

• Quantum field theory
• Summability methods

• CS1 maint: location missing publisher
• Articles with short description
• Short description is different from Wikidata