BSM Master Formula for $\varepsilon'/\varepsilon$ in the WET Basis at NLO in QCD

As an important step towards a complete next-to-leading order (NLO) QCD analysis of the ratio $\varepsilon'/\varepsilon$ within the Standard Model Effective Field Theory (SMEFT), we present for the first time the NLO master formula for the BSM part of this ratio expressed in terms of the Wilson coefficients of all contributing operators evaluated at the electroweak scale. To this end we use the common Weak Effective Theory (WET) basis (the so-called JMS basis) for which tree-level and one-loop matching to the SMEFT are already known. The relevant hadronic matrix elements of BSM operators at the electroweak scale are taken from Dual QCD approach and the SM ones from lattice QCD. It includes the renormalization group evolution and quark-flavour threshold effects at NLO in QCD from hadronic scales, at which these matrix elements have been calculated, to the electroweak scale.


Introduction
The ratio ε ′ ε measures the size of the direct CP violation in K L → ππ decays (ε ′ ) relative to the indirect one (ε). It is very sensitive to new sources of CP violation. As such it played a prominent role in particle physics already for 45 years [1]. However, due to large hadronic uncertainties in the evaluation of ε ′ ε no consensus between theorists has been reached on its value within the Standard Model (SM). Even 20 years after its measurements from NA48 [2] and KTeV [3,4] collaborations implying the world average we do not know how much room is left for new physics (NP) contributions to this ratio. Within the SM ε ′ ε is governed by the difference between QCD penguin (QCDP) and electroweak penguin (EWP) contributions. Interestingly last year a consensus between the lattice QCD (LQCD) RBC-UKQCD collaboration [5] and the Dual QCD (DQCD) estimate [6] of the EWP contribution to ε ′ ε in the SM has been reached with the result (ε ′ ε) EWP SM = −(7 ± 1) × 10 −4 , (LQCD and DQCD).
The estimate of this contribution from Chiral Perturbation Theory (ChPT), having the same sign, is by roughly a factor of two lower in magnitude [7], but in view of large uncertainties consistent with (2). On the other hand there is no consensus on the size of the QCDP contribution, which originates on the one hand in different estimates of the hadronic matrix elements of QCDP operators and on the other hand in the estimate of isospin-breaking effects that suppress the QCDP contributions. While the RBC-UKQCD collaboration did not provide an estimate of the isospin-breaking effects until now, the suppression of QCDP by them provided by ChPT [8] amounts to roughly 15%, while the advocates of DQCD find the suppression as large as 30% [9]. This difference can be traced to the η − η ′ mixing, which is included in [9] but not in [8].
Adding isospin-breaking effects from [9] to the RBC-UKQCD results for QCDP hadronic matrix elements [5] one finds [9,10] (ε ′ ε) SM = (13.9 ± 5.2) × 10 −4 . ( A similar result is found using ChPT [7], but as discussed in [1] it is a numerical coincidence in view of the smaller estimate of EWP contributions by these authors relative to LQCD, and a different estimate of isospin-breaking corrections. In any case, while this estimate is fully compatible with the experimental value in (1), the large uncertainty related primarily to QCDP contributions does not exclude the modifications of ε ′ ε by NP as large as 10 −3 .
In fact as argued on the basis of the DQCD approach [6] in [1,11] the values of ε ′ ε in the SM in the ballpark of 5 × 10 −4 are still possible. This situation motivated various authors already for many years to perform analyses of ε ′ ε in various extensions of the SM with the goal to identify which models could allow for significant contributions to ε ′ ε taking existing constraints from other processes into account. They are listed in [12]. However, one could question the usefulness of these studies in view of very large uncertainties in the estimate of the SM value of ε ′ ε and thereby of the room left for NP contributions. In particular one could question the usefulness of sophisticated NLO calculations of Wilson coefficients of various operators within extensions of the SM at present that are multiplied by very uncertain hadronic matrix elements.
Here comes a good news. We have seen that the EWP contribution could be determined within the SM with a respectable precision already now. Being dominated by the ∆I = 3 2 transitions and contributing dominantly to the isospin amplitude A 2 , its contribution to ε ′ ε turns out to be automatically enhanced by a factor of 22.4 relative to the QCDP contribution that dominates the A 0 amplitude. The only reason why in the SM the QCDP contribution is, despite of this strong suppression relative to A 2 , larger in magnitude than the EWP contribution is that at scales relevant to the K L → ππ decays the value of α s is by roughly a factor of 50 larger than the one of α em . Consequently the Wilson coefficients of QCDP operators are much larger than those of EWP operators and this implies a positive sign of ε ′ ε as opposed to the result in (2).
While one cannot exclude that such a pattern of different contributions is present in some of the NP scenarios, in most of the BSM analyses to date the contributions of NP to the imaginary part of A 2 were by far dominant. It could be then that BSM contributions to ε ′ ε, as far as hadronic uncertainties are concerned, can be more precisely calculated than the SM contribution. Therefore, as proposed in [13], it is useful to write ε ′ ε as a sum of the SM and BSM contributions, and concentrate on the BSM one, until an improved estimate of the SM contribution will be available. Now, the novelty of the BSM contributions to ε ′ ε is the presence of new operators, absent or strongly suppressed in the SM. The hadronic matrix elements of all BSM fourquark operators have been calculated to date only within DQCD in [14] and it will still take some time before we know LQCD estimates for them. But already this estimate allowed to derive a master formula for ε ′ ε in [15]. Knowing the Wilson coefficients in any BSM scenario at the electroweak scale, this formula provides the BSM contribution to ε ′ ε. This then allowed to perform for the first time a SMEFT anatomy of ε ′ ε [16], demonstrating the possible interplay of NP contributions to ε ′ ε with BSM effects in other observables.
So far the master formula for ε ′ ε within WET in [15] is based on LO RG evolution from the low-energy scale µ had to the electroweak scale µ ew , at which the Wilson coefficients (WCs) are obtained from a UV completion or SMEFT. However, beyond the LO, the RG evolution from µ had to µ ew is known only in the so-called BMU basis [17], which is less suitable for the matching to the SMEFT. Moreover, for this matching the most convenient WET basis is the one introduced in [18], the so-called JMS basis. 1 In particular, for this basis the tree-level matching of SMEFT on to WET [18] and the corresponding one-loop matching [20,21] are known. Therefore it is evident that a master formula for ε ′ ε written in terms of WCs in the JMS basis would be more useful for BSM analyses than the one that uses BMU basis at the electroweak scale. Moreover, for future combination of WCs with hadronic matrix elements from LQCD it is mandatory to calculate the RG evolution at the NLO level.
With this goal in mind we have recently calculated, including NLO QCD corrections, the RG evolution matrixÛ BMU (µ had , µ ew ) [22] that allows to express the WCs ⃗ C BMU (µ had ) at the hadronic scale in the BMU basis in terms of the ⃗ C JMS (µ ew ) in the JMS basis at the electroweak scale as follows To this end the RG evolution from µ had to µ ew has been performed in the BMU basis, including quark-flavour threshold crossings at NLO in QCD, and subsequently the resulting ⃗ C BMU (µ ew ) in the BMU basis have been transformed to ⃗ C JMS (µ ew ) in the JMS basis at the one-loop level. This required a careful treatment of the evanescent operators, encoded in the matrixM JMS (µ ew ).
Having the matricesÛ BMU andM JMS at our disposal, the main result of our work is a new master formula for (ε ′ ε) BSM , which is superior to the one in [10,15,16] in two ways • It includes NLO QCD RG evolution below the electroweak scale. Further, also the LO QED and NLO QED×QCD evolution is known, and can be applied if needed [22].
• It is expressed in terms of the WCs evaluated at the electroweak scale in the JMS basis, thereby allowing with the help of the results in [20] to express it eventually in terms of the SMEFT WCs including one-loop corrections in the matching of the SMEFT on to WET. As such it can be readily used for all UV completions that reduce to SMEFT in their infra-red regime.
Optimally for model building it would be useful to have a master formula for ε ′ ε expressed in terms of WCs in the Warsaw SMEFT basis that are evaluated at the NP scale as we have done for ∆F = 2 transitions in [23]. We will present such a formula in a future publication.
As the two items above have been accomplished in [22], the main goal of the present paper is the evaluation of all hadronic matrix elements for I = 0 and I = 2 in the BMU basis at µ had using the results of [14] obtained in the DQCD approach. Combining them with (5), we will be able to present a new master formula for ε ′ ε with the virtues listed above.
Our paper is organized as follows. In Section 2 we present the general expression for the updated master formula for ε ′ ε. In Section 3 we present the numerical values of the relevant hadronic matrix elements and in Section 4 the values of the coefficients P b (µ ew ) for the 40 WCs in the JMS basis will be presented. The summary of our results and an outlook are presented in Section 5. The definitions of the JMS and BMU operator bases are given in Appendix A and Appendix B, respectively.
In the SM Im A 0 receives dominantly contributions from QCDPs, but also from EWP. On the other hand, Im A 2 receives contributions exclusively from EWPs. However, in BSM scenarios they are both affected by contributions of BSM operators and also the WCs of SM operators are modified. The amplitudes Re A 0,2 are extracted from the branching ratios of K → ππ decays in the isospin limit. Their values are given by The main reason for taking these amplitudes from the data is that we do not know presently the BSM contributions to them and proceeding in this manner one automatically includes both SM and BSM contributions. For the discussion of the BSM contributions with the WCs normalized at the electroweak scale we define with the operators Q b and Q ′ b given in the JMS basis as defined in Appendix A and the ordering of operators in VLL, VLR and SRR sectors as given in [22]. The chirality-flipped operators Q ′ b in the sectors VRR, VRL and SLL are obtained from Q b by interchanging the chirality-projectors P L ↔ P R . The relation of C ′ a (µ had ) in the BMU basis to C ′ b (µ ew ) in the JMS basis is given in (5).
The updated master formula [15] for the BSM part in (4) reads where with the sum over b running over the Wilson coefficients C b of all operators in the JMS basis and their chirality-flipped counterparts denoted by C ′ b . The relative minus sign accounts for the fact that their K → ππ matrix elements differ by a sign. Among the contributing operators are also operators present already in the SM, but their WCs in (10) are meant to include only BSM contributions. The list of all contributing operators in the JMS basis is given in Appendix A and the one for the BMU basis in Appendix B. We mention further that the choice of the JMS basis in [22] contains hermitian conjugated operators (9), such that actually their complex-conjugated Wilson coefficients enter ε ′ ε, but using Im C b (µ ew ) * = − Im C b (µ ew ), in (10) the minus sign is absorbed into P b .
It should be emphasized that the BMU operators can not be generated in the tree-level matching from SMEFT operators, because they do not conserve hypercharge. Comparing (10) with (6) and taking the minus sign in (9) into account we have 2 leading to and therefore implying a strong suppression of I = 0 coefficients relative to the I = 2 ones by On the other hand, this suppression is partially lifted because in most cases ⟨Q a ⟩ 2 < ⟨Q a ⟩ 0 . So far only four-quark operators have been considered, however also the chromo-and electromagnetic dipole operators (Q 8g and Q 7γ respectively) contribute to ε ′ ε in principle. Whereas the electromagnetic dipole operator has been neglected, the Q 8g was included in the previous LO master formula [10,15,16]. Also in this case an improvement towards NLO QCD evolution would be desirable, but here the corresponding anomalous dimension matrices for the mixing of the four-quark operators into Q 8g are not yet available, except the NLO mixing of Q 8g and Q 7γ [28]. In view of this, we include contributions that are proportional to ⟨Q 8g ⟩ I at LO only, based on the results of [16]. The involved operators belong to the three sectors SRR,s, c, b (called class B in [15]) and the sectors SRR, u (class C) and SRR, d (class D), which are part of the BMU basis. Eventually it remains to transform their Wilson coefficients at µ ew to the JMS basis, where we use the LO transformations given in [22], see also Appendix D. In the JMS and BMU bases the chromomagnetic dipole operator are defined as respectively, which differ by a factor of the strange quark mass (in the MS scheme). The corresponding chirality-flipped operator in the JMS basis is [Q dG ] † 12 . Class B operators contribute only via mixing into Q (′) 8g operators, hence in (11) The class C and D operators have non-vanishing K → ππ matrix elements of their own, resulting in a contribution due to four-quark operators and due to mixing a contribution from the dipole operator     can be found at LO for classes C and D from tables 8 and 9 [16], respectively. As can be seen there, in practice P dipole b ≪ P four-quark b , such that within theory uncertainties one can neglect for class C and D operators the P dipole b due to mixing into Q 8g .

Hadronic matrix elements
The calculation of the coefficients P b (µ ew ) requires the knowledge of the hadronic matrix elements The number of K → ππ matrix elements of four-quark operators in the N f = 3 quark-flavour theory has been discussed in full generality in [16]. Here we recall the most important points: • The matrix elements of the chirality-flipped operators are related to the non-flipped ones due to parity conservation as ⟨Q ′ i ⟩ I = −⟨Q i ⟩ I , thereby reducing the number of independent matrix elements by a factor of two.
• We assume that operators with quark-flavour content (sd)(ss) are strongly suppressed and neglect them, see explanations in [16]. In practice this concerns only the BSM operators, because for matrix elements of SM operators the LQCD calculations include these contributions.
• For the operators with quark-flavour content (sd)(uu) and (sd)(dd) there are 15 0 and 8 2 matrix elements for I = 0 and I = 2, respectively, when using exact or nearly exact symmetries of parity and isospin. Out of these, 7 0 and 3 2 are known from the SM operators from LQCD calculations [5,29]. The remaining 8 0 and 2 2 matrix elements of the BSM operators are available from the DQCD calculation [14].
As already stressed in the previous section it is useful to perform this calculation in the BMU operator basis because in this basis the RG evolution is available at NLO in QCD, contrary to the JMS basis. The SM operators are part of the BMU basis and their matrix elements from the LQCD calculations [5] for I = 0 and [29] for I = 2 can be used after RG evolution from the scales µ 0 = 4.006 GeV and µ 2 = 3.0 GeV, respectively, to the scale µ = 1.3 GeV that we will use here. The results and details of the RG evolution are given in [10]. 3 The matrix elements of the BSM operators from the DQCD calculation [14] are given in the BMU basis, with the exception of the SRR,d sector shown in Appendix C, at the scale µ had = 1.0 GeV. The results after the RG evolution to µ = 1.3 GeV are provided in [10]. In the case of the SRR,d sector Fierz identities have to be used in order to relate BMU operators to the ones used in [14]. To this end we will use tree-level relations and thereby neglect possible effects of evanescent operators required for a consistent treatment at NLO in QCD, simply because the present accuracy of the matrix elements obtained in the DQCD approach is at best at the level of 10%, such that effects at NLO in QCD can be neglected. However, we kept in [22] the NLO corrections in the evaluation of the RG evolution contained in Op.  ba (µ ew , µ had ). Indeed, anticipating more accurate future results from LQCD that should be performed directly in the BMU basis, such effects can be included consistently one day.
Working then for N f = 3, but neglecting (ss) contributions, most BSM operators in the BMU basis coincide with the ones in the basis of [14] and hence have the same matrix elements. 4 In consequence they can be directly obtained from the DQCD results in [14] at µ = 1.3 GeV with appropriate RG evolution. These BSM operators in the BMU basis are The operators Q 14,15,16 become linearly dependent on the SM operators when neglecting the (ss) contributions and we can obtain their matrix elements in this approximation as follows: from known LQCD results. 5 This leaves us with only 8 BSM operators (Q 19,20 , Q 25,26 , Q 29−32 ), that can be mapped onto the SD basis in [14]. Among these, the only BSM operator in the BMU basis for which a Fierz transformation is required is Q 26 , see (41). The complete basis transformation of the BMU operators to the SD basis of [14] is given in Appendix C and allows to determine the I = 0, 2 matrix elements of these operators at the input scale µ had = 1.0 GeV. We collect the values of hadronic matrix elements in Table 2. We notice in particular that the matrix elements of the operators Q i with i = 6, 8,15,20,26,31,32 have the largest matrix elements. These are the VLR, scalar and tensor operators. In comparing the values  Table 3: The numerical values for the quantities P b (µ ew ) that enter in the master formula (10) are shown for all JMS WCs contributing to ε ′ ε for µ ew = 160 GeV. The upper part is due to operators with non-vanishing K → ππ matrix elements at NLO in QCD with LO results for comparison. The lower part is due to operators that contribute via mixing into Q 8g at LO in QCD.
of the matrix elements of the SM operators with those present in the literature one should take into account that our SM operators are defined with P L,R instead of traditional V ∓ A, which brings in a factor of four: The chromomagnetic dipole operator has the nonvanishing matrix element ⟨Q ′ 8g ⟩ 0 = +0.013(4) GeV 3 [30,31].

Results for P b (µ ew ) in the JMS basis
In Table 3 we give the values of P b (µ ew ) entering in the master formula (10) at the electroweak scale. These values are self-explanatory and confirm what one would expect on the basis of the values of hadronic matrix elements and the size of the anomalous dimensions: the coefficients P b that multiply the WCs of left-right and scalar operators are typically by one order of magnitude and in a few cases by two orders of magnitude larger than of vector left-left and vector right-right operators. However, the importance of a given contribution to ε ′ ε will eventually depend on the size of WCs at the electroweak scale and this will depend on NP scenario considered. We recall that there are four out of the 40 four-quark operators that do not contribute to ε ′ ε [15,16], neither directly via a non-vanishing K → ππ matrix element nor via RG mixing. They correspond to the operators ] 2331 in the JMS basis. We departed from the previous treatment of certain BSM matrix elements in the master formula [10,15,16], by using here directly DQCD results [14] without employing isospin relations for I = 2 matrix elements. Using instead the previous treatment, following entries change P [C S1,RR uddu ] 1121 = −58.11, P [C S8,RR uddu ] 1121 = 24.21. The changes are in the ballpark of 10%, which is in general smaller compared to the uncertainties of about 20% of the matrix elements in the DQCD approach.
It is however clear that this table will be modified with time when LQCD will improve the values of the SM hadronic matrix elements and calculate the corresponding matrix elements for BSM operators. Whether also in this case there will be consensus between LQCD and DQCD on the I = 2 matrix elements remains to be seen. Such contributions are expected to dominate NP contributions to ε ′ ε because of the factor Re A 2 Re A 0 ∼ 1 22. However, to obtain the full picture some consensus has to be reached on I = 0 matrix elements, in particular within the SM because this determines the room left for NP contributions.
In using the master formula (10) it is crucial to take into account the definition of Wilson coefficients as given in (9). That is they enter the Hamiltonian with the minus sign and have the dimension energy −2 . While it is convenient to use the units 1 TeV 2 , it is not essential because at the end the final results for P b are dimensionless.
The theoretical uncertainties of the coefficients P b are determined firstly by the uncertainties of the hadronic matrix elements ⟨Q i ⟩ 0,2 , listed in Table 2. They enter the P b linearly and result in 10% − 50% relative uncertainty, depending on the P b [15]. Secondly, scheme dependences due to higher order corrections in the RG evolution are reduced here due to the inclusion of the NLO QCD corrections in the RG evolution and the threshold corrections. As discussed in [22], such effects are considerably reduced now. The comparison of the LO results that do not include NLO QCD corrections in the RG evolution of the Wilson coefficients in Table 3 shows sizeable effects. They are up to 60% for left-right vector operators, but also for left-left vector operators these corrections can become larger than 30%, whereas for the scalar right-right operators they are up to 25%. The NLO QCD effects of the RG evolution are sizeable, but unfortunately these improvements can not be fully appreciated in view of the remaining large hadronic uncertainties of the matrix elements.

Conclusions
The main result of our paper is the master formula (10) with the numerical values of the coefficients P b (µ ew ) given in Table 3. The advantage of this formula over the one presented in [15] is the inclusion of next-to-leading order (NLO) QCD corrections in the RG evolution in the Weak Effective Theory (WET) from the hadronic scale µ had to the electroweak scale µ ew and the use of the JMS basis [18]. NLO QCD evolution allows a correct matching of WCs to the matrix elements calculated by lattice QCD (LQCD) or other non-perturbative methods sensitive to renormalization scheme dependences. The use of the JMS basis on the other hand allows to generalize this formula to the Standard Model Effective Field Theory (SMEFT) one because in this basis the tree-level matching of SMEFT on to WET [18] and the one-loop matching [20,21] are known.
Our master formula can already be used for approximate estimates of BSM contributions to ε ′ ε using the JMS basis in the SMEFT. It constitutes an important step towards a complete NLO QCD analysis of the ratio ε ′ ε within the SMEFT. However, in order to finish this very ambitious project the following steps have still to be performed: • The K → ππ matrix elements of all operators of the BMU basis, in particular of the BSM ones, should be calculated by LQCD and also the ones already found using DQCD [14] could be in principle improved.
• The contributions due to the chromomagnetic dipole operator and mixing of four-quark operators into the former, including threshold corrections, need to be extended to the NLO level.
• The tree-level and one-loop matchings of SMEFT on WET that are already known [18,20,21] should be included. We have not done it because consistently also the next step should be made simultaneously.
• The RG QCD evolution within the SMEFT up to the NP scale should be generalized to the NLO level, i.e. using two-loop anomalous dimensions.
Only then one will be able to cancel all unphysical renormalization scheme dependences that we discussed in [22] at length. While the second step can be performed already now and the third one could be accomplished relatively soon, the first one is most difficult and could still take several years of intensive calculation by several LQCD collaborations. We are looking forward to the improvements of our formula and hope to report on the progress in a not too distant future.
tors that contain at least one pair of down-type quarks are given as Some of the listed operators are not equal to their hermitian conjugates with permuted generation indices. Although we do not list them explicitly here, they have to be treated as independent operators. Note that also Q V 1,LR uddu and Q V 8,LR uddu have Hermitian conjugates. The same holds for the operators (LR)(LR). This choice of basis eliminates all operators with Dirac structures σ µν . The colour structure is expressed in terms of the generators T A of SU(3) c . Above p, r, s, t denote quark flavour indices in the mass-eigenstate basis. These operators include both the "non-flipped" and the chirality-flipped ones.

B BMU operator basis for K → ππ
The BMU basis [17] contains several sectors of operators, depending on their quark-flavour content and Lorentz structures. They form separate blocks under RG evolution, thereby minimizing the operator mixing. General description of this basis valid also for the B-meson system is discussed in detail in the Appendix A.2 in [22]. Here we specify this basis to the case relevant for K → ππ decays. This amounts to setting d j = s and d i = d in the expressions listed in [22]. Moreover in some operators d k = b.
The SM basis consists of two current-current-type operators where the naming used in [17] is given as well. The SU(3) c colour indices are denoted by α and β. The choice of u k = u implies that the corresponding operators for u k = c are eliminated in favour of QCD-and QED-penguin operators, see below, which is most suited for Kaon physics.
There are eight penguin-type operators with left-handed (sγ µ P L d) vector currents: the P QCD and P QED The sum over q = u, d, s, c, b includes all five active quark flavours and Q q their electromagnetic charges: Q u = +2 3 and Q d = −1 3. Their chirality-flipped counterparts are not present in the SM. Then the SM basis contains ten operators only because the chirality-flipped operators are absent. No Wilson coefficients are generated for them due to the left-handed nature of electroweak interactions of the SM.
The definition of chirality-flipped operators is not explicitly shown here. The numbering of these operators is given as i.e. they are found by interchange of P L ↔ P R from the "non-flipped" operators.

C Basis transformation: BMU to SD
In this appendix we report the basis change from the BMU basis used in the RG evolution to the SD basis defined in [14], in which the matrix elements for the BSM operators are given. Both bases almost coincide, except for a few exceptions. The relations for the BSM operators Q i in the BMU basis and O i in the SD basis, respectively, read: