维度正规化

Wilson–Fisher固定点、分形维数

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

若d = 4 − ε，上文的積分是

${\displaystyle \int _{0}^{\infty }{\frac {dp}{(2\pi )^{4-\varepsilon }}}{\frac {2\pi ^{(4-\varepsilon )/2}}{\Gamma \left({\frac {4-\varepsilon }{2}}\right)}}{\frac {p^{3-\varepsilon }}{\left(p^{2}+m^{2}\right)^{2}}}={\frac {2^{\varepsilon -4}\pi ^{{\frac {\varepsilon }{2}}-1}}{\sin \left({\frac {\pi \varepsilon }{2}}\right)\Gamma \left(1-{\frac {\varepsilon }{2}}\right)}}m^{-\varepsilon }={\frac {1}{8\pi ^{2}\varepsilon }}-{\frac {1}{16\pi ^{2}}}\left(\ln {\frac {m^{2}}{4\pi }}+\gamma \right)+{\mathcal {O}}(\varepsilon ).}$

有人認為Ζ函數正規化和維度正規化是等效的等同因為解析延拓。

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• Etingof, Pavel, , , Providence, R.I.: Amer. Math. Soc.: 597–607, 1999 , ISBN 978-0-8218-2012-4, MR 1701608, （原始内容存档于2019-05-29）
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