PLB34568S03702693(19)30248510.1016/j.physletb.2019.04.016The Author(s)PhenomenologyTable 1Table of ΔS = 1 operators contributing to (ε′/ε)BSM with coefficients Pi(μew) for μew = 160GeV, and corresponding suppression scales. The Hamiltonian is normalized as HΔS=1(5)=−∑iCi(μew)(1TeV)2Oi.Table 1ClassOiPiΛTeVsmeft
A)OVLLu=(s¯iγμPLdi)(u¯jγμPLuj)−4.3 ± 1.065✓
OVLRu=(s¯iγμPLdi)(u¯jγμPRuj)−126 ± 9354✓
O˜VLLu=(s¯iγμPLdj)(u¯jγμPLui)1.5 ± 1.738✓
O˜VLRu=(s¯iγμPLdj)(u¯jγμPRui)−436 ± 34659✓
OVLLd=(s¯iγμPLdi)(d¯jγμPLdj)2.3 ± 0.648✓
OVLRd=(s¯iγμPLdi)(d¯jγμPRdj)123 ± 10350✓
OSLRd=(s¯iPLdi)(d¯jPRdj)−214 ± 19462✓
OVLLs=(s¯iγμPLdi)(s¯jγμPLsj)−0.4 ± 0.118✓
OVLRs=(s¯iγμPLdi)(s¯jγμPRsj)−0.32 ± 0.0517✓
OSLRs=(s¯iPLdi)(s¯jPRsj)−0.0 ± 0.16✓
OVLLc=(s¯iγμPLdi)(c¯jγμPLcj)0.7 ± 0.125✓
OVLRc=(s¯iγμPLdi)(c¯jγμPRcj)0.7 ± 0.126✓
O˜VLLc=(s¯iγμPLdj)(c¯jγμPLci)0.2 ± 0.113✓
O˜VLRc=(s¯iγμPLdj)(c¯jγμPRci)0.4 ± 0.220✓
OVLLb=(s¯iγμPLdi)(b¯jγμPLbj)−0.30 ± 0.0317✓
OVLRb=(s¯iγμPLdi)(b¯jγμPRbj)−0.28 ± 0.0316✓
O˜VLLb=(s¯iγμPLdj)(b¯jγμPLbi)−0.0 ± 0.14✓
O˜VLRb=(s¯iγμPLdj)(b¯jγμPRbi)−0.1 ± 0.18✓

B)O8g=ms(s¯σμνTaPLd)Gμνa−0.3 ± 0.118✓
OSLLs=(s¯iPLdi)(s¯jPLsj)−0.05 ± 0.027
OTLLs=(s¯iσμνPLdi)(s¯jσμνPLsj)0.15 ± 0.0512
OSLLc=(s¯iPLdi)(c¯jPLcj)0.2 ± 0.115✓
OTLLc=(s¯iσμνPLdi)(c¯jσμνPLcj)0.14 ± 0.0511✓
O˜SLLc=(s¯iPLdj)(c¯jPLci)0.2 ± 0.114✓
O˜TLLc=(s¯iσμνPLdj)(c¯jσμνPLci)5.4 ± 1.873✓
OSLLb=(s¯iPLdi)(b¯jPLbj)0.4 ± 0.118
OTLLb=(s¯iσμνPLdi)(b¯jσμνPLbj)0.11 ± 0.0310
O˜SLLb=(s¯iPLdj)(b¯jPLbi)0.3 ± 0.118
O˜TLLb=(s¯iσμνPLdj)(b¯jσμνPLbi)13.5 ± 4.3116

C)OSLLu=(s¯iPLdi)(u¯jPLuj)−74 ± 16272✓
OTLLu=(s¯iσμνPLdi)(u¯jσμνPLuj)162 ± 36402✓
O˜SLLu=(s¯iPLdj)(u¯jPLui)15.6 ± 3.3124✓
O˜TLLu=(s¯iσμνPLdj)(u¯jσμνPLui)509 ± 108713✓

D)OSLLd=(s¯iPLdi)(d¯jPLdj)87 ± 16295
OTLLd=(s¯iσμνPLdi)(d¯jσμνPLdj)−191 ± 35436

E)OSLRu=(s¯iPLdi)(u¯jPRuj)266 ± 21515
O˜SLRu=(s¯iPLdj)(u¯jPRui)60 ± 5244✓
Master formula for ε′/ε beyond the Standard ModelJasonAebischeraChristophBobethba⁎christoph.bobeth@ph.tum.deAndrzej J.BurascJeanMarcGérarddDavid M.StraubaaExcellence Cluster Universe, Technische Universität München, Boltzmannstr. 2, 85748 Garching, GermanyExcellence Cluster UniverseTechnische Universität MünchenBoltzmannstr. 2Garching85748GermanybPhysik Department, TU München, JamesFranckStraße, 85748 Garching, GermanyPhysik DepartmentTU MünchenJamesFranckStraßeGarching85748GermanycTUM Institute for Advanced Study, Lichtenbergstr. 2a, 85748 Garching, GermanyTUM Institute for Advanced StudyLichtenbergstr. 2aGarching85748GermanydCentre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de Louvain, Chemin du Cyclotron 2, 1348 LouvainlaNeuve, BelgiumCentre for Cosmology, Particle Physics and Phenomenology (CP3)Université catholique de LouvainChemin du Cyclotron 2LouvainlaNeuve1348Belgium⁎Corresponding author.Editor: B. GrinsteinAbstractWe present for the first time a master formula for ε′/ε, the ratio probing direct CP violation in K→ππ decays, valid in any ultraviolet extension of the Standard Model (BSM). The formula makes use of hadronic matrix elements of BSM operators calculated recently in the Dual QCD approach and the ones of the SM operators from lattice QCD. We emphasize the large impact of several scalar and tensor BSM operators in the context of the emerging ε′/ε anomaly. We have implemented the results in the open source code flavio.The nonconservation of the product of parity (P) and chargeconjugation (C) symmetries in nature, known under the name of CP violation, was established experimentally for the first time in 1964 via K→ππ decays [1]. Since then, this fundamental phenomenon has been confirmed also in other processes in the quark sector and is rather consistently described by the socalled CabibboKobayashiMaskawa (CKM) mixing matrix [2,3] within the Standard Model (SM) of elementary particle physics. Currently there are experimental efforts to establish analogous CP violation in the lepton sector.CP violation proves to be a prerequisite [4] for our present understanding of matter dominance over antimatter in the universe. However, the CPviolating contribution from the CKM matrix in the SM fails to account for this observation and it remains to be seen whether the CPviolating contributions in the lepton sector will be able to do so. As direct collider searches have not yet revealed any presence of new physics, rare processes in the quark sector remain a good territory to search for new sources of CP violation. This is especially the case for the kaon physics observables ε and ε′, which measure indirect and direct CP violation in K0K¯0 mixing and K0 decay into ππ, respectively.Recently, there has been a renewed interest in the ratio ε′/ε [5–25], due to hints for a significant tension between measurements and the SM prediction from the RBCUKQCD lattice collaboration [26,27] and the Dual QCD approach (DQCD) [28,29]. While on the experimental side the world average from the NA48 [30] and KTeV [31,32] collaborations reads(1)(ε′/ε)exp=(16.6±2.3)×10−4, the lattice collaboration [26,27] and the NLO analyses in [33,34] based on their results find (ε′/ε)SM in the ballpark of (1−2)×10−4, that is by one order of magnitude below the data, but with an error in the ballpark of 5×10−4. An independent analysis based on hadronic matrix elements from DQCD [28,29] gives a strong support to these values and moreover provides an upper bound on (ε′/ε)SM in the ballpark of 6×10−4. A different view has been expressed in [35] where, using ideas from chiral perturbation theory but going beyond it, the authors find (ε′/ε)SM=(15±7)×10−4 in agreement with the data, albeit with a large uncertainty.The results from RBCUKQCD and DQCD motivated several authors to look for various extensions of the SM which could bring the theory to agree with data. For a recent review see [36]. In all the models studied to date, the rescue comes from the modification of the Wilson coefficient of the dominant electroweak leftright (LR) penguin operator Q8, but also solutions through a modified contribution of the dominant QCD LR penguin operator Q6 could be considered [10]. However, in generic BSM scenarios, also operators not present in the SM could play an important role. The very recent calculation of the K→ππ hadronic matrix elements of all BSM fourquark operators, in particular scalar and tensor operators in DQCD [37] and the one of the chromomagnetic operator by the ETM lattice collaboration [38] and in DQCD [39], allow for the first time the study of ε′/ε in an arbitrary extension of the SM. While the matrix element of the chromomagnetic operator has been found to be much smaller than previously expected, the values of the BSM matrix elements of scalar and tensor operators are found to be in the ballpark of the ones of Q8, the dominant electroweak penguin operator in the SM. Consequently, these operators could help in the explanation of the emerging ε′/ε anomaly.As far as shortdistance contributions encoded in the Wilson coefficients are concerned, they have been known for the SM operators already for 25 years at the NLO level [40–45] and for the BSM operators twoloop anomalous dimensions have been known [46,47] for almost two decades. First steps towards the NNLO predictions for ε′/ε have been made in [48–51] and the complete NNLO result should be available soon [52].Having all these ingredients from longdistance and shortdistance contributions at hand, we are in the position to present for the first time a master formula for ε′/ε that can be applied to any ultraviolet extension of the SM. Neglecting isospin breaking corrections, ε′/ε can be written as(2)(ε′ε)th=−ω2εK[ImA0ReA0−ImA2ReA2], where ω=ReA2/ReA0 and A0,2 are the K→ππ isospin amplitudes,(3)A0,2=〈(ππ)I=0,2HΔS=1(3)K0〉. Isospin breaking corrections have been considered in [53,54]. These corrections will affect only the A0 contributions that are suppressed by ω∼1/22. They can only be relevant in NP scenarios in which, similar to the case of the SM, the Wilson coefficients of the operators contributing to A0 are by more than one order of magnitude larger than those relevant for the A2 amplitude. Here HΔS=1(3) denotes the ΔS=1 effective Hamiltonian with only the three lightest quarks (q=u,d,s) being dynamical, obtained by decoupling the heavy W±, Z0, and h0 bosons and the top quark at the electroweak scale μew∼mW and the bottom and charm quarks at their respective mass thresholds [55].Assuming that no particles beyond the SM ones with mass below the electroweak scale exist, any BSM effect is encoded in the Wilson coefficients of the most general ΔS=1 dimensionsix effective Hamiltonian. The values of the Wilson coefficients Ci(μew) in this effective Hamiltonian at the electroweak scale with Nf=5 active quark flavours,(4)HΔS=1(5)=−NΔS=1∑iCiOi, are connected to those of HΔS=1(3), entering ε′/ε, by the usual QCD and QED renormalization group (RG) evolution. In full generality, three classes of operators can contribute, directly or via RG mixing, to K→ππ decays:a.fourquark operators:(5)OXABq=(s¯iΓXPAdi)(q¯jΓXPBqj),(6)O˜XABq=(s¯iΓXPAdj)(q¯jΓXPBqi),b.electro and chromomagnetic dipole operators:(7)O7γ(′)=ms(s¯σμνPL(R)d)Fμν,(8)O8g(′)=ms(s¯σμνTaPL(R)d)Gμνa,c.semileptonic operators:(9)OXABℓ=(s¯ΓXPAd)(ℓ¯ΓXPBℓ). Here i,j are colour indices, A,B=L,R, and X=S,V,T with ΓS=1, ΓV=γμ, ΓT=σμν.11For ΓT there is only PA=PB in four dimensions but not PA≠PB. Throughout it is sufficient to consider the case A=L, whereas results for the case A=R follow analogously due to parity conservation of QCD and QED. We will choose the overall normalization factor NΔS=1 below such that the coefficients Ci are dimensionless.In the following, we will neglect the electromagnetic dipole and semileptonic operators, which only enter through small QED effects. This leaves 40 fourquark operators for Nf=5 and one chromomagnetic dipole operator of a given chirality which have to be considered at the electroweak scale. A detailed renormalization group analysis of these operators, model independently and in the context of the Standard Model effective field theory (SMEFT), is performed in [56]. The goal of the present letter is to provide the central result of [56] and [37], the master formula for ε′/ε, in a form that could be used by any model builder or phenomenologist right away without getting involved with the technical intricacies of these analyses.Writing(10)(ε′ε)=(ε′ε)SM+(ε′ε)BSM, our formula allows to calculate automatically (ε′/ε)BSM once the Wilson coefficients of all contributing operators are known at the electroweak scale μew. It reads as follows:(11)(ε′ε)BSM=∑iPi(μew)Im[Ci(μew)−Ci′(μew)], where(12)Pi(μew)=∑j∑I=0,2pij(I)(μew,μ)[〈Qj(μ)〉IGeV3].The sum in (11) extends over the Wilson coefficients Ci of the linearly independent fourquark and chromomagnetic dipole operators listed in Table 1. The Ci′ are the Wilson coefficients of the corresponding chiralityflipped operators obtained by replacing PL↔PR. 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 Wilson coefficients in (11) include only BSM contributions.The dimensionless coefficients pij(I)(μew,μ) include the QCD and QED RG evolution from μew to μ∼O(1GeV) for each Wilson coefficient as well as the relative suppression of the contributions to the I=0 amplitude due to ReA2/ReA0≪1 for the matrix elements 〈Qj(μ)〉I of all the operators Qj present at the lowenergy scale. The index j includes also i so that the effect of selfmixing is included. We refer the reader to [56] for the numerical values of the pij(I)(μew,μ) and 〈Qj(μ)〉I for our choice of the set of Qj. The details given there allow to easily account for future updates of the matrix elements. The Pi(μew) do not depend on μ to the considered order, because the μdependence cancels between matrix elements and the RG evolution operator. Moreover, it should be emphasized that their values are modelindependent and depend only on the SM dynamics below the electroweak scale, which includes short distance contributions down to μ and the long distance contributions represented by the hadronic matrix elements. The BSM dependence enters our master formula in (11) only through the Wilson coefficients Ci(μew) and Ci′(μew). That is, even if a given Pi is nonzero, the fate of its contribution depends on the difference of these two coefficients. In particular, in models with exact leftright symmetry this contribution vanishes as first pointed out in [57].The numerical values of the Pi(μew) are collected in Table 1 for(13)μew=160GeV,NΔS=1=(1TeV)−2. They have been calculated with the flavio package [58], where we have implemented general BSM contributions to ε′/ε. As seen in (12), the Pi depend on the hadronic matrix elements 〈Qj(μ)〉I and the RG evolution factors pij(I)(μew,μ). The numerical values of the hadronic matrix elements rely on lattice QCD in the case of SM operators [26,27] and on results for scalar and tensor operators obtained in DQCD [37]. The matrix element of the chromomagnetic dipole operator is from [38] and [39] which agree with each other.The operators in Table 1 have been grouped into five distinct classes.Class A: All hadronic matrix elements can be expressed in terms of the ones of SM operators calculated by lattice QCD [26,27].Class B: All operators contribute only through RG mixing into the chromomagnetic operator O8g so that only one hadronic matrix element is involved and taken from [38,39].Class C: RLRL type operators with flavour (s¯d)(u¯u) that contribute via BSM matrix elements [37] or by generating the chromomagnetic dipole matrix element [38,39] through mixing.Class D: RLRL type operators with flavour (s¯d)(d¯d) that contribute via BSM matrix elements [37] or the chromomagnetic dipole matrix element [38,39].Class E: RLLR type operators with flavour (s¯d)(u¯u) that contribute exclusively via BSM matrix elements [37].Besides the Pi, we provide in the nexttolast column of Table 1 the suppression scale Λ that would generate (ε′/ε)BSM=10−3 for Ci=(1TeV)2/Λ2. It gives an indication of the maximal scale probed by ε′/ε for any given operator.Among the 40 fourquark operators present in HΔS=1(5), four have been omitted in Table 1, namely OSLRb,c and O˜SLRb,c, since they neither contribute directly nor via RG mixing at the level considered, i.e. they have Pi=0.In models with a mass gap above the electroweak scale, v≪Λ, where v is the Higgs vacuum expectation value and Λ the BSM scale, some of the operators in Table 1 are not generated at leading order in an expansion in v/Λ. As discussed in more detail in [56], these operators violate hypercharge, that is conserved in the SMEFT above the electroweak scale.22As an exception, the hypercharge constraint can be avoided for the operator O˜SLRu, if in the intermediate SMEFT the dimensionsix operator with righthanded modified W± couplings (OHud in the basis of [70]) is generated, as for example in a leftright symmetric UV completion of the SM due to treelevel WL–WR mixing. The O˜SLRu is then generated at the electroweak scale by the treelevel W± exchange of a single insertion of OHud with a dimensionfour SM coupling of W± and quarks. In the rightmost column of Table 1, we have indicated whether the operator can arise from a treelevel matching of SMEFT at dimension six onto the ΔS=1 effective Hamiltonian (cf. [59,60]).Inspecting the results in Table 1, the following comments are in order.•The largest Pi values in Class A can be traced back to the large values of the matrix elements 〈Q7,8〉2, the dominant electroweak penguin operators in the SM, and the enhancement by 1/ω≈22 of the I=2 contributions.•The small Pi values in Class B are the consequence of the fact that each one is proportional to 〈O8g〉0, which has recently been found to be much smaller than previously expected [38,39]. Moreover, as 〈O8g〉2=0, all contributions in this class are suppressed by the factor 1/ω relative to contributions from other classes.•The large Pi values in Classes C and D can be traced back to the large hadronic matrix elements of scalar and tensor operators calculated recently in [37]. Due to the smallness of 〈O8g〉0, the contribution of the chromomagnetic dipole operator is negligible.•While the operators in Classes D have sizable Pi, they violate hypercharge as discussed above, so they do not arise in a treelevel matching from SMEFT at dimension six.•While the I=0 matrix elements of the operators in Class E cannot be expressed in terms of SM ones, the I=2 matrix elements can, and the large Pi values can be traced back to the large SM matrix elements 〈Q7,8〉2.Almost all existing BSM analyses of ε′/ε in the literature are based on the contributions of operators from Class A or the chromomagnetic dipole operator. Table 1 shows that also other operators, in particular the ones in Class C, could be promising to explain the emerging ε′/ε anomaly and can play an important role in constraining BSM scenarios.However, in a concrete BSM scenario, the Wilson coefficients with the highest values of Pi could vanish or be suppressed by small couplings. Moreover, additional constraints on Wilson coefficients can come in SMEFT and from correlations with other observables as discussed in more detail in [56].Next, we would like to comment on the accuracy of the values of the Pi listed in Table 1. As far as short distance contributions to the Pi are concerned, they have been calculated in the leading logarithmic approximation to RG improved perturbation theory using the results of [61–65]. Although the inclusion of nexttoleading corrections is possible already now, such contributions are renormalization scheme dependent and can only be cancelled by the one of the hadronic matrix elements. While in the case of SM operators this dependence has been included in the DQCD calculations in [66], much more work still has to be done in the case of BSM operators.The uncertainties from the matrix elements depend on the operator classes in Table 1. In the Pi column, we have given the uncertainties obtained from varying the individual matrix elements, assuming them to be uncorrelated. In Class A, they stem from the lattice matrix elements. Here we point out that due to the enhancement of the I=2 contributions by the factor 1/ω≈22, the largest Pi are dominated by the I=2 matrix elements, which are known to 5–7% accuracy from lattice QCD [27]. For the small Pi in Class A, in some cases there are cancellations between contributions from different matrix elements, leading to larger relative uncertainties. The matrix elements of fourquark BSM operators entering Classes C–E have only been calculated recently in DQCD approach [37] and it will still take some time before lattice QCD will be able to provide results for them. Previous results of DQCD imply that it is a successful approximation of lowenergy QCD and that the uncertainties in the largest Pi are at most at the level of 20%. While not as precise as ultimate lattice QCD calculations, DQCD offered over many years an insight in the lattice results and often, like was the case of the ΔI=1/2 rule [67] and the parameter BˆK [68], provided results almost three decades before this was possible with lattice QCD. The agreement between results from DQCD and lattice QCD is remarkable, in particular considering the simplicity of the former approach compared to the sophisticated and computationally demanding numerical lattice QCD one. The most recent example of this agreement was an explanation by DQCD of the pattern of values of B6(1/2) and B8(3/2) entering ε′/ε obtained by lattice QCD [28,29] and of the pattern of lattice values for BSM parameters Bi in K0K¯0 mixing [69]. This should be sufficient for the exploration of new phenomena responsible for the hinted ε′/ε anomaly. Similar comments apply to the hadronic matrix element of the chromomagnetic dipole operator, entering mainly the Pi in Class B, that was recently calculated in DQCD in [39] and found to be in agreement with the lattice QCD result from [38]. Since the Pi in Class B only receive a single contribution, their relative uncertainties mirror the relative uncertainty of the chromomagnetic matrix element, that was estimated at 30% in [39].The usefulness of our master formula is twofold. 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