Analytic renormalization via continuous space dimension

The third-order QCD corrections to deep-inelastic scattering by photon exchange

“…(2.11) a regularization procedure and a renormalization scheme need to be selected. We choose dimensional regularization [46][47][48][49] and the modified [50] minimal subtraction [51] scheme, MS, the standard choice for modern higher-order calculations in QCD. For this choice the running coupling in D = 4 − 2ε dimensions evolves according to where β n denote the usual four-dimensional expansion coefficients of the beta function in QCD [52][53][54][55][56][57], β 0 = 11 − 2/3 n f etc, with n f representing the number of active quark flavours.…”

“…For brevity suppressing the function arguments for the rest of this section, the expansion of the 'master formula' (2.19) to the third order in the strong coupling α s can by written as The factor S ε = exp(ε{ln(4π) − γ e }), where γ e denotes the Euler-Mascheroni constant, is an artefact of dimensional regularization [46][47][48][49] kept out of the coefficient functions and anomalous dimensions in the MS scheme [50]. δ k collects the quark charge factors,…”

The iterated structure of the all-order result for the two-loop sunrise integral

“…Due to the presence of ultraviolet or infrared divergences these integrals may require regularisation. It is common practice to use dimensional regularisation [1][2][3] with regularisation parameter ε and to present the result for a Feynman integral as a Laurent series in ε. It is a natural question to ask, what transcendental functions appear in the ε j -term.…”

Charged current deep-inelastic scattering at three loops

“…In [24][25][26][27] in D = 4 − 2ε dimensions and the modified minimal subtraction [28,29] scheme, MS. The running coupling evolves according to…”