Erratum to “Two-loop QCD anomalous dimensions of flavour-changing four-quark operators within and beyond the standard model” [Nucl. Phys. B 586 (2000) 397] (Nuclear Physics, Section B (2000) 586(1–2) (397–426), (S0550321300004375), (10.1016/S0550-3213(00)00437-5))

Andrzej J. Buras, Mikołaj Misiak, Jörg Urban

Research output: Contribution to journalComment/debate


The matrix in eq. (4.9) of our article should read [Formula presented] Corrections occur in its first column only. The mistake was pointed out in ref. [1]. Our original article contained no details on the determination of this matrix. Below, we provide an extra appendix where our (corrected) procedure is outlined. Appendix F Let us begin with the SM QCD-penguin operators [Formula presented], …[Formula presented] that are listed in eq. (4.1). Their [Formula presented] ADM can be split into contributions from the current-current and penguin diagrams: [Formula presented]. The structure of [Formula presented], …[Formula presented] implies that [Formula presented] where [Formula presented] and [Formula presented] up to the two-loop level are given in eqs. (3.2)–(3.5). The matrix [Formula presented] is most easily found by subtracting [Formula presented] from the full [Formula presented]. One- and two-loop contributions to the latter matrix in the NDR–[Formula presented] are listed in table 5 of ref. [2].1 This way one finds [Formula presented] Let us now extend the considered operator set to [Formula presented], where the extra operators have been defined in eqs. (4.2) and (4.6). At this point, we ignore the fact that [Formula presented] and [Formula presented] are related by a Fierz relation in [Formula presented], i.e. we treat both of them as independent normal (non-evanescent) operators. The four extra operators [Formula presented] can be obtained from [Formula presented] by skipping the u-, c- and b-quarks in the sum over flavours. Consequently, the full [Formula presented] ADM up to two loops takes the form [Formula presented] We have tacitly assumed here that each of our operators is accompanied by a corresponding one-loop evanescent operator containing triple products of Dirac matrices inside the quark currents, defined in full analogy to [Formula presented], [Formula presented], [Formula presented] and [Formula presented] in appendix B, including the coefficients at the [Formula presented] terms there. In the next step, we transform our [Formula presented] ADM to the basis [Formula presented], where [Formula presented] is still treated as a normal (non-evanescent) operator. The transformed ADM reads [Formula presented] Finally, we depart from the MS scheme for [Formula presented] (still thinking of it as of a normal operator though) by introducing finite terms in the one-loop renormalization constants that correspond to its mixing via penguin diagrams into [Formula presented], …[Formula presented]. This amounts to replacing [Formula presented] in eq. (2.15) by [Formula presented] when the index i corresponds to [Formula presented], and the index k corresponds to [Formula presented], …[Formula presented]. We adjust [Formula presented] to make the renormalized one-loop penguin matrix element of [Formula presented] vanish. The one-loop ADM remains intact, while the resulting transformation of the two-loop ADM reads (cf. eq. (6.10)) [Formula presented] At this point, our non-MS scheme with [Formula presented] treated as a normal operator becomes equivalent to the MS scheme with [Formula presented] treated as an evanescent operator.2 We explicitly verify that the 8th row of [Formula presented] up to two loops has only a single non-vanishing entry that corresponds to the mixing of [Formula presented] with itself. Consequently, the Wilson coefficient of [Formula presented] has no effect on the RG evolution of the Wilson coefficients of normal operators, as it should be for any evanescent operator in the MS scheme. Let us note that it would not be the case if the transformation (F.14) was not performed. To find the actual ADM for the normal operators only (now with [Formula presented] treated as an evanescent operator, and in the MS scheme), we remove the 8th row and 8th column from the matrix [Formula presented]. The resulting [Formula presented] matrix reads [Formula presented] where the one- and two-loop contributions to [Formula presented], [Formula presented] and [Formula presented] coincide with what has been already given in eqs. (F.9)-(F.10), (4.7) and (4.8)-(4.9), respectively. Our final results for [Formula presented] in eqs. (4.8)-(4.9) have actually been extracted from eq. (F.15). As far as [Formula presented] in eq. (4.7) is concerned, eq. (F.15) gives us a nice confirmation of the result that has in practice been determined using a much simpler method. The transformation (F.14) was missed in the original version of our paper. The mistake was pointed out in ref. [1]. The reader might wonder why our finite subtraction in eq. (F.13) was restricted to the penguin matrix elements only, i.e. why no similar operation was necessary for the one-loop current-current matrix element of [Formula presented]. It was the case because such a matrix element turns to vanish after subtracting the evanescent counterterms only, and passing to [Formula presented], as already discussed in section 6, below eq. (6.18).

Original languageEnglish
Article number116529
JournalNuclear Physics, Section B
StatePublished - May 2024


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