A simple dirac prescription for two-loop anomalous dimension matrices

A novel method to treat effects from evanescent operators in next-to-leading order (NLO) computations is introduced. The approach allows, besides further simplifications, to discard evanescent-to-physical mixing contributions in NLO calculations. The method is independent of the treatments of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _5$$\end{document}γ5 and can therefore be combined with different renormalization schemes. We illustrate the utility of this result by reproducing literature results of two-loop anomalous dimension matrices for both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Delta F| = 1$$\end{document}|ΔF|=1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Delta F| = 2$$\end{document}|ΔF|=2 transitions.


Introduction
When performing loop-level computations in perturbative quantum field theories, one frequently encounters the issue of unphysical ultraviolet (UV) divergences.These UV divergences are handled in a systematic way using renormalization, where one absorbs the UV divergences into the parameters of the theory, and reinserts them in physical processes, where the divergences cancel order-by-order.However, in order to extract the UV divergences, one must first render them finite through the use of a regularization method, the most commonly used being dimensional regularization where the dimensionality of spacetime is continued from d = 4 to d = 4 − 2ϵ, and the limit of ϵ → 0 is taken to obtain a physical result.One issue arises when including fermions into the theory, since the Dirac algebra cannot always be trivially continued to arbitrary dimensionality.In the majority of the literature, this issue is handled by using evanescent operators: operators which vanish in the physical limit, but account for the fact that operator relations in this limit may be altered when working in an arbitrary number of dimensions.The standard story of evanescent operators is as follows: In a computation, physical matrix elements may project onto a set of Dirac structures, {D}.In four dimensions, these structures are not independent of the set of physical operators, {Q}, i.e.
where we have specified that this relation is only true in four-dimensional spacetime.When continuing to d = 4 − 2ϵ, the relation in Eq. (1.1) will no longer hold in general, meaning that the operators D and Q are no longer necessarily linearly dependent.
To compensate for this, one adds an additional operator to the basis, known as an evanescent operator (EV) [1,2] which by definition vanishes from the operator basis when we take the physical limit ϵ → 0. Note that the matrix F needs not be exactly equal to F 4 due to the fact that the only requirement is that E vanishes when ϵ → 0. Therefore, we may include an arbitrary dependence on positive powers of ϵ so that F = F 4 + σ where where the arbitrary constants σ n fix the renormalization scheme [3].We may therefore write the interaction Hamiltonian in d-dimensions as where C Q and C E correspond to the physical and evanescent Wilson coefficients (WCs), respectively.The choice of evanescent operator basis is not unique.In fact, to write down such a basis, one must first pick a scheme for how to handle Dirac matrix manipulations in d-dimensions (hereby referred to as a "prescription"), e.g.naive dimensional regularization (NDR), 't Hooft-Veltmann (HV), or dimensional reduction (DRED).Once a prescription is chosen, it allows one to reduce certain Dirac structures, but not all.The different prescriptions then lead to different irreducible structures, and for each such structure, an evanescent operator must be introduced.However, not even the evanescent operators for a given prescription are unique.Since the only requirement for an evanescent operator is that it vanishes when we take d → 4, we may introduce a general evanescent-scheme dependence (hereby just called the scheme dependence) like we have done in Eq. (1.3).The key point is the following: any physical observable must be independent of the choice of both the prescription and scheme as alternative choices differ by only O(ϵ) contributions.However, the set of structures which are reducible or irreducible in each prescription are different.This means that, in one prescription, a structure may be reducible, but in another prescription, the same structure will require an evanescent operator.The issue is that the treatment of the structure can be substantially different in the two prescriptions: in the second prescription, we must introduce finite subtractions for this structure to avoid the issue of specifying an infinite number of initial conditions for evanescent operators.This finite subtraction does not correspond to a choice of renormalization scheme and cannot be neglected in order to obtain the correct result.In the first prescription, we simply treat the structure as any other and move on.In this article, we examine the notion of different prescriptions and suggest a novel approach for treating structures that can not be trivially reduced to the physical basis.The method is recursive and independent of the chosen renormalization scheme and is hence well suited for automation.
The rest of the article is structured as follows: In Sec. 2 an example in scalar QED is considered in order to illustrate different prescriptions and their relations among each other.Furthermore, in Sec. 3 a general prescription at the NLO is presented and applied to several examples in the literature.Finally, we conclude in Sec. 4. Details of the two-loop computation from Sec. 2 are collected in App. A.

An Example: Muon decay in scalar QED
As a simple example, we will examine NLO QED corrections to charged lepton decay, mediated by a charged scalar particle with interaction Lagrangian where P L and P R denote the left-and right-handed projection operators, respectively.We will only study the left-left decays, as this sector will run independently of the right-right and mixed sectors under only vector-coupled, parity conserving QED.
We will consider matching this theory onto an effective Hamiltonian with two physical operators which mix under QED renormalization group running with and C s and C t the respective WCs.The purpose of this procedure is to fully compute this process featuring non-trivial off-diagonal running in three different ways: first, we will use the standard NDR prescription with scheme-dependent evanescent operators.Second, we will consider the case where we give no prescription for the treatment of γ 5 and instead only define evanescent operators for all Dirac structures which do not immediately reduce with d-dimensional Dirac algebra.We will call this prescription the "no-prescription" (NP) method.Finally, we will consider an extension to NDR featuring the so-called "Greek trick" [4] to define a prescription which covers all nontrivial Dirac structures and gives no evanescent operators.We note however, that our way of employing the Greek method differs from the conventional one used in the literature.Traditionally, the Greek identities are used in combination with introducing corresponding EVs, as further explained in App.B. To simplify the discussion and comparison with our method we refrain from this approach and simply apply the greek identities, without introducing EVs.This method will be referred to as the "Greek prescription" (GP) method.Hence, in the following the matching and running computations will be performed in these three prescriptions in order to compare the different results.

Matching
We begin by computing the leading-order (LO) matching onto the effective theory.The tree-level amplitude in the UV theory is given by where M S denotes the mass of the heavy scalar.In the EFT, we expand the WCs as which gives the LO matching conditions At NLO, the MS-renormalized amplitude reads in the hard region (2.7) The 1/ϵ pole arises from the infrared divergence (ϵ < 0) and matches the corresponding UV pole in the EFT up to a sign, following the method of regions.Using this method, we can immediately extract the matching conditions from this expression, neglecting the 1/ϵ pole, once we reduce the Dirac algebra to project onto the operators in the EFT.

NDR Prescription
In NDR, the structure appearing in the matching can be immediately re-written in terms of the physical operators Making this replacement in Eq. (2.7) gives the NDR matching conditions (2.9)

NP Prescription
Since in NP, we do not specify any way of treating γ 5 , we require an evanescent operator in order to reduce the Dirac algebra in the matching.We choose the definition with arbitrary coefficients σ s1 , σ t1 .Note that this implies that the NLO amplitude in the UV theory projects onto the evanescent operator E 1 .This projection is irrelevant in the matching since we will end up subtracting any evanescent-to-physical mixing, and therefore we need not specify initial conditions for the WCs corresponding to evanescent operators. 1Neglecting this projection, we find Clearly, this reproduces the same matching conditions as in NDR if we set σ s1 = 2 and σ t1 = 0.

GP Prescription
In the case of GP, we can again directly evaluate the structure appearing in the matching, giving the same result as in NDR.Using this, we find the same UV and EFT amplitudes as in NDR and therefore the same matching conditions (2.12) Here, one may already expect an inconsistency to arise: in NDR, we will have evanescent operators which will introduce additional terms corresponding to the finite subtractions for evanescent-to-physical mixing.Therefore, it seems that the two prescriptions could produce different anomalous dimension matrices (ADMs), yet have the same matching conditions, as discussed in Ref. [5].

One-Loop EFT Renormalization
In order to perform the NLL running from the matching scale to the scale where we evaluate the matrix element, we must perform the one-and two-loop renormalization in the EFT.We will use MS augmented by finite subtractions for evanescent-to-physical mixing.The UV poles can be extracted using the methods described in Ref. [6].At one loop, we only have a single diagram to compute for each operator insertion, which is depicted in Fig. 1.For an operator insertion with general Dirac structure (Γ 1 ) ⊗ (Γ 2 ), we find the result of the diagram in a general R ξ gauge where m is the IR regulator mass (see Ref. [6] for details).The ξ-dependence in Eq. (2.13) will drop out after the renormalization of the external legs.We have kept the finite logarithmic term as a check for the cancellation of non-local divergences when inserting operator counterterms.At this point, we must specify how to treat the Dirac algebra in order to project back onto the physical basis and complete the physical-tophysical and physical-to-evanescent renormalization.In particular, we must know how to treat the two structures which arise from the insertions of physical operators.The former structure has already been addressed in the matching computation.The latter one will now be discussed again in the different prescriptions.

NDR Prescription
In NDR, to reduce the second structure, we must introduce an evanescent operator which we choose to take the form We then find Using Eq. (2.13) along with the one-loop lepton field renormalization in the general R ξ -gauge gives the one-loop renormalization constants where Q = {Q s , Q t } and we organize the renormalization constants according to which relates the bare and renormalized operators in the Lagrangian via (2.20) The operator renormalization constant in eq. ( 2.18) agrees with the findings in [7].Only the ϵ-independent piece of the treatment of Dirac algebra enters the one-loop poles, so the physical-to-physical counterterms will be the same in all prescriptions.

NP Prescription
When we do not introduce a prescription for treating γ 5 , we choose a second evanescent operator to treat the second Dirac structure arising from an insertion of This gives, in addition to the physical-to-physical counterterm Z QQ in Eq. (2.18), the physical-to-evanescent mixing (ordering 1 2×2 . (2.22)

GP Prescription
In the case of GP, we can immediately evaluate the two structures appearing in the amplitudes.We find for the second structure (2.23) Since we do not have any evanescent operators in this prescription, we cannot specify any renormalization constant Z Q Ẽ corresponding to physical-to-evanescent mixing.

Evanescent-To-Physical Mixing
In both NDR and NP, we have introduced unphysical evanescent operators whose mixing into the physical sector must be taken into account.To avoid the need of specifying an infinite number of initial conditions, we must subtract off the finite evanescentto-physical mixing to decouple the two sectors.This is done by inserting evanescent operators into Eq.(2.13) and keeping the finite pieces.

NDR Prescription
In the case of NDR, we must introduce one additional evanescent operator which appears when inserting the four-gamma structure into Eq.(2.13).We choose this operator to be With this definition we find the relation: (2.25) Also, After inserting E NDR into the one-loop diagram and projecting onto the physical basis, we find the finite counterterm . (2.27) Note that we do not need to insert E ′ NDR since the physical operators do not mix into this evanescent operator at the considered order.

NP Prescription
Upon inserting our evanescent operators in NP, we will require two additional evanescent operators, which we choose to be With these, we can insert E 1 and E 2 into the one-loop diagram to find the finite subtractions (2.29) Here, we point out the (somewhat obvious) observation that these counterterms are purely scheme-dependent.This is due to the fact that the scheme-independent pieces, i.e. the purely four-dimensional parts, cancel trivially due to the requirement that the structures appearing in loop diagrams must reduce to the four-dimensional counterparts when d → 4. We are therefore only left with pieces where the 1/ϵ poles of the loop integrals hit the ϵ terms in the definition of the evanescent operators.This pure scheme dependence is much more clear in NP than in NDR due to the fact that, in NDR, the Dirac matrices appearing in the amplitudes are re-ordered to project onto the chosen evanescent operators, thereby introducing additional O(ϵ) terms from ddimensional Dirac algebra.In NP, on the other hand, no such re-arranging occurs, making the pure scheme-dependence more explicit.

Anomalous Dimension Matrix
The next step is the computation of the one-and two-loop anomalous dimension matrix corresponding to physical-to-physical mixing.The one-loop ADM can immediately be found from the one-loop counterterms.Expanding the ADM as we find in all prescriptions.The two-loop ADM is found in the MS-scheme by computing only the UV-divergent pieces of all two-loop diagrams with physical operator insertions.The UV poles can be extracted using the methods presented in Ref. [6].We must additionally compute all one-loop diagrams with single counterterm insertions in order to subtract subdivergences in the two-loop diagrams.In total, this amounts to the computation of thirteen diagrams, up to Dirac algebra: seven true two-loop diagrams, five dimension-four operator counterterm insertions, and one dimension-six operator counterterm insertion.The necessary counterterms and the pole structure of the corresponding diagrams are collected in App. A.

NDR Prescription
In NDR, using the previously-defined evanescent operators, we find the (local and gauge-invariant) counterterms QQ and we also find that the ADM is independent of σ ′ s and σ ′ t , as must be the case in order for the scheme-dependence to properly cancel with the one-loop matching and matrix elements.

NP Prescription
Performing the same computation using NP, we find the counterterms We again see the independence of the two-loop evanescent scheme, indicating that scheme-dependence will properly cancel with the one-loop matching and matrix elements.

Discussion of the Example
In this example, we have performed the computation of heavy-scalar-mediated muon decay to NLO in QED using three different prescriptions for the treatment of Dirac algebra in d-dimensions.Our goal was to gain further insight into the relevance of evanescent operators in such computations.All three prescriptions not only generated different forms of the evanescent operators, but also different numbers of evanescent operators as well as free parameters used to fix the scheme.Despite this, all three prescriptions produce the same result when a consistent evanescent scheme is chosen.This is particularly surprising when considering the Greek prescription, which did not introduce any evanescent operators.Upon closer inspection, it is clear why the GP reproduces the same result as the other two prescriptions: for the particular choices of σ (′) i and σ (′) i which give the same Dirac algebra replacements as those found using the GP, the evanescent-to-physical counterterms vanish.So, in GP, the problematic mixing of evanescent structures into the physical sector is automatically handled simply by the replacement we use for d-dimensional Dirac algebra.It seems that one can circumvent the arguments of Ref. [5] and ignore the effects of evanescent operators in the calculation of ADMs if one can ensure a priori the vanishing of evanescent-to-physical mixing.
This becomes even clearer by a simple re-interpretation of evanescent operators in general.Consider a set of operators, D i which reduce to the set of physical operators, Q i in four dimensions via When upgrading to d-dimensions, we will assert that the following relation, which reduces to the four-dimensional one for d → 4, holds where (2.41) We next consider inserting Eq. ( 2.40) into a one-loop amplitude and we find where ∆ ij ̸ = 0 and moreover is finite.The latter point is very important since we can immediately take the limit of Eq. (2.42) as d → 4, and we find a result which clearly violates Eq. (2.39).We see that, when using a general prescription with dimensional regularization, loop-effects explicitly spoil four-dimensional Dirac algebra, reminiscent of loop effects inducing gauge anomalies.However, unlike gauge anomalies, the violation of fourdimensional Dirac relations is entirely local, arising only from UV poles.We can therefore erase these problematic shifts by introducing additional, local operators into our basis and defining a subtraction scheme to exactly cancel these terms.These are precisely the evanescent operators.

A Prescription free of Evanescent-to-Physical mixing
In this section, we introduce a general methodology for choosing evanescent operator definitions to guarantee the cancellation of evanescent mixing into the physical sector for four-fermion operators.We then apply this technique to two non-trivial examples in the Weak Effective Theory (WET) and reproduce known results from the literature.

General discussion
As exemplified by the results of Sec. 2, it is clear that, if we can ensure that no evanescent-to-physical mixing arises when inserting evanescent operators into loop diagrams, then the effect of evanescent operators on physical ADMs is equivalent to simple "replacement rules" like those used in the Greek prescription.More importantly, loop diagrams with evanescent operator insertions need not be computed, as no additional finite subtractions are necessary, thereby sequestering the evanescent sector to only mix with itself.To begin, we consider a set of physical four-fermion operators defined as where a i are flavor/color indices.Inserting Eq. (3.1) into a one-loop diagram results in where we accounted for the fact that the one-loop insertions can change the color and flavor structure of the external legs.In d-dimensions, this structure does not reduce to any appearing in the physical basis, and we must therefore introduce an evanescent operator.As previously discussed, the choice of evanescent operators is not unique, so we make the choice where C kℓ;c α 1 iα 2 ;α 3 jα 4 ;a→b contains not only the coefficients corresponding to the reduction of Eq. (3.2) in four dimensions, but also arbitrary O(ϵ n ) terms for n ≥ 1.The super/subscript a → b refers to the fact that Eq. (3.3) arises from inserting operators with physical flavor/color structure {a ℓ } and projecting onto structure {b ℓ }.With this, we now must account for the fact that the evanescent operators introduced in Eq. (3.3) can mix into the physical sector, thereby requiring finite subtractions.Inserting Eq. ( 3.3) into one-loop diagrams will give the new Dirac structure Note that, since we have explicitly separated the Dirac structures arising from the loop insertions, i.e.Γ β i , from the operator Dirac structures (Γ α k Γ i Γ α ℓ and Γ i , respectively for the two terms), the two terms in Eq. (3.4) receive the same proportionality constant from e.g.loop integration and reduction of Γ β i in the chosen scheme for the mass dimension-four sector.
For the first term in Eq. (3.4), we must again introduce an additional evanescent operator to treat the irreducible Dirac structure.We then choose
This procedure simplifies in the case that the reduction of structures onto the physical basis in four-dimensions does not require a rearrangement of the external fields, i.e.
C kℓ;d α 1 iα 2 ;α 3 jα 4 ;a→b = C kℓ α 1 iα 2 ;α 3 jα 4 ;a→b δ bd . (3.9) In this case (which we will refer to as the non-Fierz case), the four-dimensional reduction only changes the Dirac structure and is completely independent of the particular choice of external legs.We can then define universal evanescent tensor products without reference to external states (3.10) Similar to before, we can require that the evanescent mixing into the physical sector vanishes, giving the requirement As we have already discussed, once we have guaranteed that this mixing vanishes, the effect of the evanescent operators on physical ADMs is equivalent to that of replacement rules taking (3.12) However, the second line of Eq. (3.12) simply corresponds to the recursive application of the first line, as is required for such a prescription to be self-consistent.
In our example of scalar-mediated muon decay, choosing this particular scheme corresponds to fixing the two relations generated by one-loop physical operator insertions These relations, along with the Lorentz-and gauge-invariant condition {γ µ , γ ν } = 2η µν , completely fixes the Dirac reductions for the problem at hand.For example, using these relations, we find Notice that this is equivalent to fixing the σ ′ i in Eq. (2.28) to the values in the curly brackets.As expected, the evanescent-to-physical mixing exactly vanishes with this choice of scheme.
It is worth emphasizing that one must be extremely cautious when using the NDR prescription due to the fact that pushing factors of γ 5 across the fermion line, or mixing the Dirac structures from the mass-dimension four and operator insertions changes the Dirac structures in each fermion line, thus obscuring the necessary cancellation in Eq. (3.7).For example, if we fix the NDR relation then the one-loop insertions of the left-hand side of this relation will generate structures like If we immediately use Eq.(3.15), then we will see the explicit cancellation of the shifts to the four-dimensional Dirac algebra, but if we first use the naive anticommutation of γ 5 and re-order the Dirac matrices in both lines, we will now obtain structures like However, this spoils the overall proportionality seen in Eq. (3.7) since the re-ordering will be different between the two terms, thus violating four-dimensional Dirac algebra relations and requiring finite subtractions.Indeed, if one tries to fix σ ′ i in E ′ NDR according to the reduction of (3.17) via Eq.(3.15), one no longer sees the explicit cancellation of the evanescent-to-physical mixing.
At this point, we wish to stress that this prescription only applies to non-Fierz cases.It is unclear to the authors whether such a recursive method can be extended to the more general "Fierz-like" case in Eq. (3.8), and for the time being must be treated on a case-by-case basis2 .
In the following two subsections we will apply this method to two examples to reproduce known results from the literature, namely for |∆F | = 1 and |∆F | = 2 two-loop ADMs.

Example I: Two-Loop QCD ADM for |∆F | = 1
As a first example we begin by considering the charged-current weak decay governed by the effective Hamiltonian where r = s, b for |∆S| = 1 and |∆B| = 1 processes, respectively and Inserting these operators into one-loop diagrams with a single gluon generates four non-trivial structures to which we assign the replacements Furthermore, for future simplicity, we choose σ F = σ V 1 .In this basis, the one-loop QCD ADM is diagonal.It can be found, for example, in Ref.s [10,11], and since it is scheme-independent, we will trivially reproduce the same result.
At two-loops, all 1/ϵ 2 poles reduce using only the relations given in Eq. (3.20), as they must for the cancellation of subdivergences.At O(1/ϵ), additional irreducible structures arise.These structures can be treated in an identical way, but for an NLO ADM computation, only the four-dimensional piece is relevant, so we do not give the explicit relations.
We separate the results by their scheme-dependence as and we find (fixing the number of colors N c = 3 and number of active flavors f = 5) In Ref. [10], the same ADM is computed in the NDR scheme using the evanescent operator which reduces to that given in [10] for the standard choice of a = 4.After performing a change of renormalization scheme (see e.g.Ref.s [9,12,13]) from Eq.s (3.20) and (3.23) and applying the resulting transformation to Eq. (3.22), we exactly reproduce Eq. (3.24).
Here, we wish to emphasize a few critical points.First, we see that in our scheme, the |∆F | = 1 QCD mixing remains diagonal for any choice of scheme constants.This is in contrast to the case of NDR which is only diagonal for a particular choice of scheme.
To understand why this occurs, we first note that the Q ± operators are self-Fierz and anti-self-Fierz conjugate operators, respectively.Consider the Fierz-evanescent operator corresponding to a V − A operator insertion where i, j, k, ℓ are Dirac indices.After insertion into a one-loop diagram, we find both terms in the one-loop insertion of E F reduce the same way using the third relation in Eq. (3.20) and this contribution vanishes for σ F = 0.The same is true for the two terms reduce using the first and second lines of Eq. (3.20), respectively, and the one-loop Fierz-evanescent insertion only vanishes for the specific choice σ F = σ V 1 .Hence, using this scheme choice preserves tree-level Fierz identities and consequently the diagonality of the ADM.
Next, we reiterate that we only needed to consider physical operators and we worked in a pure MS scheme, requiring no finite subtractions.Furthermore, we did not need to fix our physical basis in order to eliminate evanescent operators; the relations appearing in Eq. (3.20) guarantee the preservation of four-dimensional Dirac relations when inserted into loop diagrams and recursively reduced.Finally, since we reproduce the NDR result using only a standard transformation of renormalization scheme, our method is equivalent to the computation in NDR up to the unphysical choice of scheme.

Example II: Two-Loop QCD ADM for |∆S| = 2
In this section, we show that the proposed scheme can also be applied to other nontrivial scenarios aside from four-quark operators with gauge boson loops.In this case, we consider the |∆S| = 2 Hamiltonian where and λ i = V * is V id .Here, we have used the u − t unitary basis [14] instead of the c − t basis used in Ref. [10].Due to the large suppression from λ 2 t , the dimension-eight operator Q7 gives a similar contribution as that from Q S2 despite the additional m 2 c /M 2 W suppression.The WC C7 obtains no matching conditions, but is instead generated by RG running from double-insertions of the |∆S| = 1 operators in Eq. (3.18) (we neglect penguin operator insertions and only focus on the charged-current subspace).
In our scheme, we use the same relations as given in Eq. (3.20), and the one-loop double insertion gives one additional relation (3.29) In principle, we also need the Fierz-evanescent operators however, the (N c -and f -independent) choice of σ F = σ V 1 along with F 1 = F1 = 0 leads to vanishing evanescent-to-physical mixing at this order as discussed in the previous section, so the evanescent operators may be ignored with this choice and we used simple tree-level Fierz relations.Again, at two-loop the 1/ϵ 2 poles reduce using only the one-loop relations, and more relations must be introduced for the 1/ϵ poles.The single-insertion mixing can be immediately extracted from the "++" component of Eq. (3.22) since Q S2 is equal to its Fierz conjugate in this scheme.
The two-loop anomalous dimension tensor computed from double insertions of |∆S| = 1 operators with gluon loops is given by γ(1),S.I. (3.31) The NDR results of the |∆S| = 2 mixing can, in principle, be extracted from Ref. [10] using the basis change in Ref. [14] along with a partial conversion from the simplified considerably compared to conventional methods, since evanescent insertions together with the renormalization of EVs can be neglected.
We have also shown in an explicit case that a careful treatment of the proposed scheme can in some cases also ensure the vanishing of the mixing of Fierz-evanescent operators into the physical basis, but it is at this point unclear whether this is a general feature or only applicable to the considered cases.In the case where this method does not generalize, one must still consider loop-insertions of Fierz-evanescent operators, or use one-loop Fierz identities [7][8][9].Furthermore, a matching computation onto this scheme might involve rather complicated scheme transformations.Finally, a generalization of this approach to higher orders or to include also non-internal structures would be desirable.We leave such studies for the future.
Finally, we note that the presented prescription method for performing NLO calculations is well suited for automation, due to its simple and recursive character.It would therefore be very interesting to implement this approach in order to make it accessible for the community.
The results are presented immediately after loop integration with no reduction of Dirac algebra.All results are multiplied by −iC 12 (α/(4π)) 2 and non-local divergences are neglected as they are equal to twice the 1/ϵ 2 poles.Furthermore, we set Q µ = Q e = −1.

B Greek method
In this appendix we review the greek method to reduce Dirac structures to the physical basis.Traditionally, this method is used in the literature to define EVs in higher-order calculations [1,4,24].As was shown in Sec. 3, when choosing the EV basis such that the evanescent-to-physical mixing vanishes, the greek identities can simply be used to reduce Dirac structures.

B.1 Greek identities
In this subsection we will review how to obtain the greek identities, which can be used to either directly reduce Dirac structures to the physical basis, or to define EVs.
Starting from a given Dirac structure, in order to find the corresponding greek identity, an Ansatz has to be made.In this Ansatz one assumes that the Dirac structure is proportional to its four-dimensional counterpart.The proportionality constants will contain besides the four-dimensional part also O(ϵ n ) terms.To fully determine the proportionality constants the "greek trick" is used, which consists of replacing the tensor product by appropriate Dirac structures and therefore collapsing it to a single Dirac string.The constants can then be determined by reducing the Dirac strings on both sides and solving the resulting system of equations.In order to illustrate this method, let us consider the example of the following operator: This operator reduces in four dimensions to the standard vector operator O V,LR = (f 1 γ µ P L f 2 )(f 3 γ µ P R f 4 ) , (B.2) using the Chisolm identity, which fixes the four-dimensional part of the greek identity.In four dimensions one finds: In general d-dimensions however, four-dimensional Dirac relations can not be used anymore, since they are only valid up to O(ϵ).Following the greek prescription we therefore make the Ansatz (γ µ γ ν γ ρ P L ) ij ⊗ (γ µ γ ν γ ρ P R ) kl = A (γ µ P L ) ij ⊗ (γ µ P R ) kl , (B.4) where the constant A needs to be fixed.This is achieved in the Greek method by replacing the tensor product by the identity matrix, therefore collapsing the tensor product of two Dirac currents to a single one. 3One finds: (γ µ γ ν γ ρ P L γ µ γ ν γ ρ P R ) ij = A (γ µ P L γ µ P R ) ij , , (B.5) Both sides can now be reduced by using for example the NDR scheme.This leads to Therefore, the greek identity for the operator structure in eq.(B.1) reads: (γ µ γ ν γ ρ P L ) ij ⊗ (γ µ γ ν γ ρ P R ) kl = 4(1 + ϵ) (γ µ P L ) ij ⊗ (γ µ P R ) kl , (B.8) which agrees for instance with the findings in [25].
In case the Dirac structure to reduce is proportional to two Dirac structures in four dimensions, two proportionality constants have to be used.This is exemplified for instance by tensor structures.Considering for instance a TLL structure, the general Ansatz dictated by the four-dimensional case is given by (σ µν P L γ α γ β ) ij ⊗(σ µν P L γ α γ β ) kl = A T (P L ) ij ⊗(P L ) kl +B T (σ µν P L ) ij ⊗(σ µν P L ) kl .(B.9) The constants A T , B T are then fixed by replacing the tensor product once by the identity and once by γ τ 1 γ τ 2 .Reducing the resulting structures and solving the system of equations leads to A T = −48 + 80ϵ , B T = 12 − 6ϵ . (B.10) Finally, for structures that do not generate chirality flips the tensor product has to be replaced by an odd number of gamma matrices for the resulting equations to be non-trivial.
Hence, with the greek trick all Dirac structures can be mapped back to the physical basis, by fixing the scheme-dependent constants in a particular way.In our approach the factors proportional to ϵ are left general in order to be agnostic about the used γ 5 -scheme.

Figure 1 :
Figure 1: One-loop QED vertex corrections to the four-point interaction in the EFT.

Figure 2 :
Figure 2: Genuine two-loop diagrams relevant for the muon decay in the EFT, discussed in Sec. 2.

Figure 3 :
Figure 3: Counterterm diagrams for the muon decay in the EFT, discussed in Sec. 2.

Table 2 :
One-loop results with an insertion of operator with structure (Γ 1 ) ⊗ (Γ 2 ) and a single standard model counterterm insertion.The results are presented immediately after loop integration and replacement of explicit counterterms with no reduction of Dirac algebra.All results are multiplied by −iC 12 (α/(4π)) 2 and non-local divergences are neglected as they are equal to the 1/ϵ 2 poles.Furthermore, we set Q µ = Q e = −1.