Dimensional regularization

In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini^[1] for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of an auxiliary complex parameter d, called (somewhat confusingly) the dimension.

Dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and the squared distances (x_i−x_j)² of the spacetime points x_i, ... appearing in it. In Euclidean space, the integral often converges for −Re(d) sufficiently large, and can be analytically continued from this region to a meromorphic function defined for all complex d. In general, there will be a pole at the physical value (usually 4) of d, which needs to be canceled by renormalization to obtain physical quantities. Etingof (1999) showed that dimensional regularization is mathematically well defined, at least in the case of massive Euclidean fields, by using the Bernstein–Sato polynomial to carry out the analytic continuation.

There is a tradition of confusing the parameter d appearing in dimensional regularization, which is a complex number, with the dimension of spacetime, which is a fixed positive integer (such as 4). The reason is that if d happens to be a positive integer, then the formula for the dimensionally regularized integral happens to be correct for spacetime of dimension d. For example, the surface area of a unit (d − 1)-sphere is $\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}$ where Γ is the gamma function when d is a positive integer, so in dimensional regularization it is common to say that this is the surface area of a sphere in d dimensions even when d is not an integer. Whereas there is no such thing as a sphere in non-integral dimensions, the formulas such as this are nonetheless a useful mnemonics in dimensional regularization. This failure to distinguish between the dimension of spacetime and the formal parameter d has led to speculation about spacetimes of non-integral dimension.

If one wishes to evaluate a loop integral which is logarithmically divergent in four dimensions, like

$\int\frac{d^dp}{(2\pi)^d}\frac{1}{\left(p^2+m^2\right)^2},$

one first rewrites the integral in some way so that the number of variables integrated over does not depend on d, and then we formally vary the parameter d, to include non-integral values like d = 4 − ε.

This gives

$\int_0^\infty \frac{dp}{(2\pi)^{4-\varepsilon}} \frac{2\pi^{(4-\varepsilon)/2}}{\Gamma\left(\frac{4-\varepsilon}{2}\right)} \frac{p^{3-\varepsilon}}{\left(p^2+m^2\right)^2}=\frac{2^{\varepsilon -4}\pi^{\frac{\varepsilon}{2}-1}}{\sin(\frac{\pi\varepsilon}{2}) \Gamma(1-\frac{\varepsilon}{2})}m^{-\varepsilon}=\frac{1}{8\pi^2\varepsilon}-\frac{1}{16\pi^2}\left(\ln \frac{m^2}{4\pi}+\gamma\right)+ \mathcal{O}(\varepsilon).$

Emilio Elizalde has shown that Zeta 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.

Notes

1. Bollini 1972, p. 20.

References

• Bollini, Carlos; Giambiagi, Juan Jose (1972), Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter. 12, Il Nuovo Cimento B, pp. 20–26 line feed character in |title= at position 54 (help)
• 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, ISSN 0550-3213
• Kleinert, H.; Schulte-Frohlinde, V. (2001), Critical Properties of φ⁴-Theories, pp. 1–474, ISBN 978-981-02-4659-4, Paperpack ISBN 978-981-02-4659-4 (also available online). Read Chapter 8 for Dimensional Regularization.