dimensional regularization

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[edit] Formulae

[edit] Schwinger-Feynman parameters

\frac{1}{A_1A_2...A_n} = \int^1_0 dx_1dx_2...dx_n \delta(\sum x_i - 1) \frac{(n-1)!}{(x_1A_1+x_2A_2 + ... x_nA_n)^{n}}\,

[edit] Integration

\int \frac{d^Nl}{(2\pi)^N} \frac{(l^2)^R}{(l^2-m)^M} = \frac{i(-1)^{(R-M)}}{(16\pi^2)^{N/4}} m^{R-M+N/2} \frac{\Gamma(R+\frac{1}{2}N) \Gamma(M-R-\frac{1}{2}N)}{\Gamma(\frac{1}{2}N)\Gamma(M)}

Editor's note: What signature is this?

[edit] Symmetry substitution

l^\mu l^\nu \to \frac{1}{d} l^2 g^{\mu\nu}\,
l^\mu l^\nu l^\rho l^\sigma \to \frac{1}{d(d+2)} (l^2)^2 ( g^{\mu\nu}g^{\rho\sigma} + g^{\mu\rho}g^{\nu\sigma} + g^{\mu\sigma}g^{\nu\rho})\,

[edit] Surface-area of a d-dimensional sphere

\int\!d\Omega_d = \frac{2\pi^{\frac{d}{2}}} {\Gamma\left(\frac{d}{2}\right) }\,.

[edit] References

Further reading: [1]

  1. ↑ M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, Massachusetts, 1995
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