ε ' / ε in the Standard Model at the Dawn of the 2020s.

We reanalyse the ratio ε ' / ε in the Standard Model (SM) using most recent hadronic matrix elements from the RBC-UKQCD collaboration in combination with most important NNLO QCD corrections to electroweak penguin contributions and the isospin-breaking corrections. We illustrate the importance of the latter by using their latest estimate from chiral perturbation theory (ChPT) based on the octet approximation for lowest-lying mesons and a very recent estimate in the nonet scheme that takes into account the contribution of η 0 . We find ( ε ' / ε ) SM ( 8 ) = ( 17.4 ± 6.1 ) × 10 - 4 and ( ε ' / ε ) SM ( 9 ) = ( 13.9 ± 5.2 ) × 10 - 4 , respectively. Despite a very good agreement with the measured value ( ε ' / ε ) exp = ( 16.6 ± 2.3 ) × 10 - 4 , the large error in ( ε ' / ε ) SM still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for ( ε ' / ε ) BSM valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters B 6 ( 1 / 2 ) ( μ ) and B 8 ( 3 / 2 ) ( μ ) at the μ -scales used by the RBC-UKQCD collaboration and at lower scales O ( 1 GeV ) used by ChPT and Dual QCD (DQCD). We present semi-analytic formulae for ( ε ' / ε ) SM in terms of these parameters and Ω ^ eff that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the O ( α em ) contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at O ( α em ) in the present analyses of ε ' / ε .

Introduction

The direct CP-violation in decays, represented by the ratio , plays a very important role in the tests of the Standard Model (SM) and more recently in constraining its possible extensions [1]. In the SM is governed by QCD penguins (QCDP) but receives also an important contribution from the electroweak penguins (EWP), pointed out already in 1989 [2, 3], that entering with the opposite sign to QCDP suppress this ratio significantly. The partial cancellation of these two contributions in addition to the evaluation of the hadronic matrix elements of QCDP and EWP operators is the reason why even today a precise prediction for in the SM is not available. Yet, significant progress has been made during the last years and the goal of our paper is to update the SM value of taking into account all available informations both from lattice QCD (LQCD) and analytic approaches most relevant for the evaluation of the Wilson coefficients but presently also for the estimate of the isospin-breaking corrections to the isospin amplitudes.

The situation of in the SM before April 20, 2020 has been summarized by us in [4]. In short there are presently three approaches to calculate hadronic matrix elements entering :In view of the fact that LQCD calculations contain both the meson evolution2 and FSI, while the estimate of in the other two approaches does not include one of them, we have recently proposed the optimal strategy for the evaluation of as of 2020 [4, 19] Recently significant progress in the estimate of in the SM has been made through the improved values of the hadronic matrix elements presented by the RBC-UKQCD collaboration [21]. Not only statistical errors have been significantly decreased but also a better agreement with the experimental values of the strong interaction phases has been obtained. The RBC-UKQCD collaboration, using their new results for the hadronic matrix elements and known Wilson coefficients at the NLO level [20, 27–31] but not accounting for isospin-breaking corrections, finds [21]to be compared with the experimental world average from NA48 [32] and KTeV [33, 34] collaborations,While the result in (1) is in full agreement with the experimental value in (2) the theoretical error of does not allow for clear cut conclusions whether some amount of new physics contributions is present in or not. The same is the case of the earlier updated ChPT analysis [15], which resulted inwith an error of 36%, very close to the LQCD one. But it should be remarked that with the present best values of the CKM parameters as used by us the central value in (3) would be raised to .

Lattice QCD, lead by the RBC-UKQCD LQCD collaboration. Using their results from 2015 for matrix elements [5, 6] and including isospin-breaking corrections from [7, 8] as done in [9, 10], leads to a value for in the ballpark of . Although exhibiting a large error of the result lies one order of magnitude below the data. Taking these analyses at face value one could talk about an anomaly of at most .

DQCD approach [11-13], which gave a support to these values and moreover provided an upper bound on in the ballpark of . The main QCD dynamics suppressing in this approach is represented by the meson evolution, which is necessary to match long-distance contributions to short-distance ones. On the other hand it has been argued in [13] that final state interactions (FSI) should have only a minor impact on and the quoted bound does not include them.

Chiral Perturbation theory (ChPT) [14-16] where, using ideas from ChPT, the authors found attributing an important role to FSI in this result. While in agreement with the measurement, the large uncertainty, that expresses the difficulties in matching long-distance and short-distance contributions in this framework, does not allow for any clear-cut conclusions.1

Use LQCD results for hadronic matrix elements of the dominant QCDP and EWP operators and , respectively. They are represented by the parameters and defined in Sect. 2. On the other hand the hadronic matrix elements of operators can be determined from the experimental data on the real parts of the amplitudes as performed in [9, 20]. In fact this procedure has been recently adopted with slight modifications by the RBC-UKQCD collaboration [21] with the goal to decrease their errors. This procedure is clearly legitimate when testing the consistency of the SM with the data, if the reduction of the uncertainties is significant. But the RBC-UKQCD analysis shows only a little gain and therefore we will not use it in the present work, but rather base the full analysis on all hadronic matrix elements from RBC-UKQCD that are absolutely free from new physics. We will only use the experimental values of in the basic formula for because they automatically take possible NP contributions into account.

Include isospin-breaking corrections from ChPT [15] that are compatible due to large uncertainties with the results obtained already 33 years ago in [22]. Very recently the latter analysis has been updated [23] and we will include these new findings as well.

Include NNLO QCD contributions to EWP in [24] thereby reducing the unphysical scale and renormalization scheme dependences in the matching at , with the largest part due to the top-quark mass. The removal of the dependence on at NNLO has still to be done, see also the next point.

Take into account NNLO QCD contributions to QCDP [25, 26]. This reduces the left-over renormalization scale uncertainties present at the NLO level, in particular those due to the matching scale .

Despite large errors both results deviate significantly from the DQCD values of in the ballpark of stressed in particular in [17]. While there is no question about that meson evolution necessary for a proper matching between Wilson coefficients and hadronic matrix elements at scales must play a role in the evaluation of it appears from present RBC-UKQCD results that precisely in the case of the matrix element of the operator its suppression is overcompensated by other QCD dynamics which was hidden due to the contamination of the excited states present in their 2015 analysis. It has been removed in the latest analysis. In fact as we will see soon the value of obtained using the optimal procedure with hadronic matrix elements from [21], agrees very well with the one advocated in [15] and given in (3). Yet, it is not evident at present that FSI, as claimed by ChPT experts, are responsible for this agreement. Possibly other dynamical QCD effects apparently not taken into account both in the ChPT and DQCD approaches are responsible for the enhancement of relative to DQCD expectations.3 However, a clear-cut conclusion on this issue is difficult because of rather different techniques that are used in these three approaches. The fact that the central value in (3) differs significantly from the central LQCD value in (1) is dominantly due to the omission of isospin-breaking effects in the RBC-UKQCD prediction that are included in (3).

Even if the new improved calculation of hadronic matrix elements in [21] is an important advance towards the accurate calculation of , the result in (1) does not represent the present SM value of properly. Indeed, as we emphasized in [4] the hadronic matrix elements in question are only a part of the story. The three additional advances, listed in the context of the optimal strategy, that are not taken into account in the result in (1) are also important, in particular because they all lower the value of . As we will demonstrate below, the final result for differs significantly from the one obtained by the RBC-UKQCD collaboration. Indeed after including isospin-breaking effects from [15] that include the effects from the octet of lowest-lying mesons and NNLO QCD corrections to EWP contributions, we find using the hadronic matrix elements of RBC-UKQCDOn the other hand including the singlet in this estimate one arrives at [23]Both results agree very well with experiment and with the ChPT expectations but in view of our comments on the ChPT analysis are on a more solid footing. We expect further reduction of by roughly (5–10)% when NNLO QCD corrections to QCD penguin contributions will be taken into account [25, 26]. We look forward to the final results of these authors.

Our paper is organized as follows. In Sect. 2, after recalling a number of basic formulae, we determine the parameters and using the recent RBC-UKQCD results and compare them with the expectations from ChPT [15] and DQCD [12, 13]. It turns out that while there is a good agreement on the value of between LQCD and ChPT, the rather precise value of from RBC-UKQCD is by a factor of 1.5 larger than the ChPT one when both are evaluated at . On the contrary, while there is a good agreement on the value of between LQCD and DQCD [12], the most recent value of from RBC-UKQCD is by a factor of two larger than the values quoted in [12, 13]. We close this section with an updated formula for in terms of and . In Sect. 3 we derive the results in (4) and (5) which take into account the updated isospin-breaking effects [15, 23] and also NNLO QCD corrections to EWP contributions [24]. We also perform a detailed anatomy of various contributions. In Sect. 4 we update the BSM master formula for [35, 36] in view of the new RBC-UKQCD results. A brief summary and an outlook are given in Sect. 5. Some additional information on the numerical analysis are given in appendices. This includes the values of the hadronic matrix elements from RBC-UKQCD and the Wilson coefficients at various scales. We discuss in detail the effect of isospin-breaking corrections present in the renormalization group (RG) flow on in Appendix C.

Basic formulae

Preliminaries

The amplitudes for , with denoting strong isospin of the final state, are given as where and are the Wilson coefficients and the hadronic matrix elements of the operators , both in the scheme at the low-energy factorization scale in the flavour theory [20]. By convention the strong phase shifts are not included in , and therefore the are real-valued. FurtherThe real parts are given entirely by the , because the are strongly suppressed by , on the other hand the imaginary parts and depend only on . The Wilson coefficients of the QCD penguin (QCDP) operators are usually larger compared to those of the electroweak penguin (EWP) operators , as can be seen in Table 7.

Table 7

The Wilson coefficients at various scales in the NDR- scheme for the renormalization scales , and using NLO and partial NNLO matching results for . Further

	μ=1.3GeV		μ=3.0GeV		μ=4.006GeV
	NLO	NNLO’	NLO	NNLO’	NLO	NNLO’
z_1	-0.3938	←	-0.2368	←	-0.1984	←
z_2	1.2020	←	1.1096	←	1.0892	←
z_3×10^2	0.4231	←	-0.3540	←	-0.4679	←
z_4×10^2	-1.2693	←	1.5289	←	2.1423	←
z_5×10^2	0.4231	←	-0.3142	←	-0.5236	←
z_6×10^2	-1.2693	←	1.0955	←	1.5460	←
z_7×10^4	0.4780	←	0.8969	←	1.2560	←
z_8×10^4	0	0	-0.9518	←	-1.0783	←
z_9×10^4	0.4780	←	0.2914	←	0.5490	←
z_10×10^4	0	0	0.7362	←	0.8552	←
y_3×10^2	2.6958	←	2.0441	←	1.8743	←
y_4×10^2	-5.4542	←	-5.3848	←	-5.3689	←
y_5×10^2	0.5579	←	1.1474	←	1.2634	←
y_6×10^2	-8.2572	←	-5.9125	←	-5.4750	←
y_7×10^2	-0.0180	-0.0192	-0.0137	-0.0146	-0.0119	-0.0128
y_8×10^2	0.0981	0.1050	0.0622	0.0666	0.0547	0.0585
y_9×10^2	-1.1167	-0.9939	-1.0184	-0.9063	-0.9975	-0.8878
y_10×10^2	0.3981	0.3025	0.2366	0.1798	0.1997	0.1518

The scheme and scale dependences cancel between the Wilson coefficients and the matrix elements individually in and . We will take advantage of this freedom to use different scales and in the evaluation of and , respectively. In particular we choose the values at which the RBC-UKQCD lattice collaboration presents their results of the  [21] and [6] matrix elements. There are only seven linearly independent and three linearly independent in the flavour theory [9, 20], since RBC-UKQCD work in the isospin-symmetric limit, where also QED corrections are not included yet.

We remind that the amplitudes and the strong phase shifts are related to the decay amplitudes relevant for as followswith the experimental values of for given in Table 2, whereas RBC-UKQCD works with the convention . These relations are valid also in the presence of finite QED corrections, as long as virtual infrared-divergent contributions, and also Coulomb corrections, are properly subtracted and combined with real photon radiation [8] when determining the amplitudes and phases from data.

Table 2

Numerical input: The CKM elements and combinations thereof and the uncertainties are derived from Wolfenstein parameters from PDG 2019. The experimental results for amplitudes are for normalization . The quark masses are FLAG averages for from [42–47].

Parameter	Value	References	Parameter	Value	References
G_F	1.166379×10^-5GeV^-2	[39]
λ	0.22453(44)	[39]	A	0.836(15)	[39]
ρ¯	0.122(-17+18)	[39]	η¯	0.355(-11+12)	[39]
V_ud	0.97446(10)		VtdVts∗	[-3.40(15)+i1.45(8)]×10^-4
V_us	0.22453(45)		τ	[15.58(67)-i6.62(35)]×10^-4
ReA_0|_exp	27.04(1)×10^-8 GeV	[40]	ε_K	0.002228(11)	[39]
ReA_2|_exp	1.210(2)×10^-8 GeV	[40]	m_K	497.614 MeV	[39]
F_π	130.41(20) MeV	[39]	m_d(2GeV)	4.67(9) MeV	[41]
F_K/F_π	1.194(5)	[41]	m_s(2GeV)	92.0(1.1) MeV	[41]

Basic formula for

As in [9], our starting expression is the formulawhere [8, 15]Here a and summarise isospin-breaking corrections. The latter include strong isospin violation , the correction to the isospin limit coming from transitions and electromagnetic corrections as first summarized in [7, 8] and recently updated in [15]These analyses are based on the so-called octet scheme which includes only the octet of the lowest-lying pseudoscalar mesons. The inclusion of the singlet in the nonet scheme has been known already for 33 years [22, 37] to give stronger suppression of through the mixing, but only very recently this estimate has been updated and put on a more solid basis than it was possible in 1987. Withthe role of isospin-breaking effects is enhanced relative to the ChPT estimate in (12).

The inclusion of the isospin-breaking corrections requires a modification in the evaluation of the part in as follows [9]such that only leading isospin-breaking corrections are included.

A strong reduction of the uncertainty of can be achieved firstly [20] by the use of the experimental values of in the denominators of (10). Secondly, the real parts of the relations (6) and (7) allow to eliminate one and one , respectively, in favour of the measured values of and , respectively. These can then be used in the numerators and , as proposed in [9]. The particular choice of j and k is subject to optimisation. However, as already announced previously we will not use this procedure here.

The real parts of the isospin amplitudes in (10) are then extracted from the branching ratios on decays in the isospin limit. In the limit and the formula in (10) reduces to the one used by RBC-UKQCD [21], where all isospin breaking-corrections except for EWP contributions at the NLO level have been set to zero.

Extracting and from LQCD

In the past the so-called bag factors have been frequently used in phenomenological analyses and it is interesting to provide their values in view of the updated matrix elements. The and parameters, that enter the formula (20), are defined as follows with [22, 38]in the large-N limit. We have introduced the factor h in order to emphasize different normalizations of these matrix elements present in the literature.

We find from the latest RBC-UKQCD results for [21] matrix elements at the scales , and and for from [6] for , and to be compared with the 2015 values and from RBC-UKQCD [5, 6]. In principle only4 the central value of has been changed by a factor of more than two, but with slightly larger uncertainty, which would correspond to a discrepancy. However, in view that the systematic uncertainty of the 2015 results for the matrix elements has been underestimated [21], the uncertainty quoted for the 2015 result of must not be taken at face value anymore.

The new value of is in the ballpark of values advocated in [15], but it is unclear to us at present whether this is a numerical coincidence or due to FSI dynamics. Moreover, the large uncertainty in the value of does not yet rule out the values of as expected from the DQCD approach [12]. Similar, the decrease of both parameters with increased , pointed out already in [20] and seen above, is also present below within the DQCD allowing smooth matching between hadronic matrix elements and Wilson coefficients. On the other hand it turns out that while there is a good agreement on the value of between LQCD and DQCD [12], its rather precise value from RBC-UKQCD is by a factor of 1.5 larger than the ChPT one, in the ballpark of 0.55, when both are evaluated at .

An analytic formula for

As is well-known and shown also in the full analysis later, is strongly dominated by the two terms and . For convenience we provide a semi-analytic result of in terms of these two parameters. Contrary to [4, 9], we evaluate and at the two different scales and and use now for the remaining matrix elements the RBC-UKQCD results. Thenwith the coefficients given in Table 1 for various choices of . The numerical input of the various parameters entering (20) is given in Table 2 and details on the Wilson coefficients at scales are collected in Appendix B. The quark masses in (15) and (16) have been calculated as well at the two scales and , respectively. The coefficients are comparable to [4], but differ because of the updated values for the remaining matrix elements and changed values of the down- and strange-quark masses. Note that contains the and contributions of the EWPs at the scales , where the QCDP matrix elements for are zero because the lattice calculation is done in the isospin limit. In general, when using the RG equations to evolve these matrix elements to different scales , the isospin-breaking in quark charges in the RG flow lead to nonvanishing QCDP matrix elements that would also contribute to . As explained in more detail in Appendix C, we evolve the matrix elements of the operators from the initial scales to the scales and only with NLO QCD RG equations instead of NLO QCDQED, which maintains isospin relations for these matrix elements.

Table 1

Coefficients entering the semi-analytic formula (20), when the amplitudes and are evaluated at the scales and , respectively, for different choices of . The central values for with these approximate formulas are given in the last two lines.

(μ_0,μ_2)[GeV]	(1.0,1.0)	(1.3,1.3)	(4.006,3.0)
B6(1/2)(μ_0)	1.49±0.25	1.36±0.23	1.11±0.20
B8(3/2)(μ_2)	0.85±0.05	0.79±0.05	0.70±0.04
m_d[MeV]	6.37	5.52	(3.88,4.16)
m_s[MeV]	125.48	108.81	(76.50,81.89)
a^QCDP(μ_0)	-2.86	-3.37	-5.64
a6(1/2)(μ_0)	15.15	16.98	22.77
a^EWP(μ_0,μ_2)	-2.02	-2.12	-2.27
a8(3/2)(μ_2)	8.00	8.79	9.85
10^4×(ε^′/ε)^(8)	17.2	17.1	17.3
10^4×(ε^′/ε)^(9)	13.7	13.7	13.9

in the Standard Model

The new results for the matrix elements from RBC-UKQCD imply a modification of in the SM relative to those values presented in 2015 in [5, 9, 10], taking into account additional advances listed in Sect. 1. Here we include the isospin-breaking corrections and NNLO QCD corrections to EWPs calculated in [24]. Both contributions lead to a considerable reduction of , as discussed previously [4]. Note that the RBC-UKQCD collaboration [21] prefers to use the magnitude of the isospin-breaking corrections from ChPT in the octet scheme (12) exclusively as an estimate of their size, thereby introducing an additional large uncertainty in . In contrast to previous predictions [4, 9], here we use in obtaining the final result for directly the LQCD values of matrix elements and . For the interested readers, we provided the updated values of the two most important bag factors and in Sect. 2.3.

We find for the amplitudes () and where NNLO QCD corrections have been included in EWP parts [4]. The statistical uncertainties due to the matrix elements (ME, stat) were determined including the available correlations for , whereas the systematic ones (ME, syst) are based on the overall for and for , as estimated by RBC-UKQCD in [6, 21], respectively. For comparison, these values are very close to the RBC-UKQCD predictions , , , , from Eqs.  (77a, 85, 90) [21] and Eq.  (64) [6], respectively.5 The scale uncertainties are obtained by varying and for the NLO expressions, shown in Fig. 1. Note that we use , and hence the variation includes the top-mass scheme dependence. We emphasize that the uncertainty for , and , is very conservative, because we actually include here partial NNLO QCD corrections to EWPs [24], which remove the implicit dependence associated with the top-quark mass and some of the explicit dependence as well, see also [4] for more details. The parametric uncertainty due to the input value for the top-quark mass in Table 6 is denoted by “”.

Fig. 1

The dependence [left] and dependence [right] at NLO accuracy of the various quantities normalized to their value at and , respectively

Table 6

Numerical input for Wilson coefficients

Parameter	Value	Ref.	Parameter	Value	Ref.
αs(5)(m_Z)	0.1181(11)	[39]	m_Z	91.1876 GeV	[39]
αem(5)(m_Z)	1/127.955(10)	[39]	m_W	80.385 GeV	[39]
sW2=sin^2(θ_W)	0.23126	[39]	mtpole	173.1(6)(5) GeV	[39]

The various relative contributions of the operators to and are listed in Table 3 when using . These numbers show that are dominated by the current–current operators. In the dominates with almost 96%, whereas the and contributions of about 12% cancel each other and there are subleading and contributions from and . In the of and the of enter with opposite signs and there is a subleading contribution from of %. On the other hand the is dominated by QCDP operators, where the 121% contribution of is mainly reduced by and . The is dominated by EWPs, in particular by due to , which is partially cancelled by . The corrections from and cancel each other.

Table 3

The contribution in % of each operator to and at .

	Q_1	Q_2	Q_3	Q_4	Q_5	Q_6	Q_7	Q_8	Q_9	Q_10
ReA_0	12.7	95.8	0.2	2.4	1.1	-12.0	0.1	-0.2	0.0	0.0
ReA_2	-27.4	128.6	0.0	0.0	0.0	0.0	0.3	-1.5	0.0	0.0
ImA_0	0.0	0.0	-2.7	-16.9	-7.5	121.8	-0.2	3.4	1.8	0.4
ImA_2	0.0	0.0	0.0	0.0	0.0	0.0	-5.8	120.2	-18.0	3.6

In the SM receives contributions from QCDP and EWP via the matrix elements and from EWP via the matrix elements, that exhibit quite some hierarchies as can be seen in (36) and (38), respectively. These hierarchies are strongly counteracted by those present in the Wilson coefficients at the two scales and , where we evaluate and , respectively. This is illustrated by the following semi-analytic results of that include the NNLO QCD corrections to EWPs [24]Here the experimental values of have been used only in the denominator of (10). As a remark on the side, we note that at the scale the relations hold because the RBC-UKQCD calculations so far do not include isospin-breaking corrections, neither due to quark masses nor quark charges. In consequence no such contributions appear in (25). Moreover isospin relations (37) have been used to substitute . As mentioned before in Sect. 2.4, a straight-forward application of the NLO QCDQED RG equations to the matrix elements to evolve them to some different scale will generate nonvanishing , because the RG flow includes isospin-breaking effects from quark charges. The effect on is discussed in further detail in Appendix C, where we provide the analogous result to (25) at .

The hierarchy of the Wilson coefficients signaled for instance by large coefficients in front of is strongly counteracted by a hierarchy in the hadronic matrix elements modifying the pattern of the various contributions:where the “” indicate the origin of the contribution. This shows much clearer the relevance of and for and to some extend . Eventuallyshows the contributions of QCDP in and the partial cancellation of EWP contributions from and . Note that this statement is scale dependent, i.e. at some other scales the composition changes slightly due to RG flow.

The final result for , using in the octet scheme (12), with NNLO QCD in EWP and other parameters as collected in Tables 2 and  6 isConsidering the new value from the nonet scheme (13) we find:There is a statistical error due to the matrix elements from the lattice, based on covariance matrices for , propagated with Monte Carlo methods as well as individually available statistical errors for matrix elements. The systematic uncertainty due to various sources related to the lattice approach is entirely dominated by the systematic error of in . The isospin-breaking corrections to QCDP from ChPT, summarized in in (11), contribute a relative uncertainty of . There is an overall relative uncertainty of from due to the CKM input.

The NNLO QCD corrections to EWPs lead to a decrease of [4] and without them the central value would be . Since our numerical input and the treatment of short-distance contributions differs slightly from RBC-UKQCD our central value does not agree exactly with their prediction [21], such that after setting , , and using only NLO QCD EWP we obtain slightly higher , but well within the uncertainties. The inclusion of NNLO QCD EWP reduces this to .

In our prediction we made only use of the experimental values in the denominator of (10). As proposed in [9], in addition also in the numerator one of the and one of the matrix elements could be eliminated in favour of to improve the accuracy in the framework of the SM. Here we did not adapt this strategy, because in agreement with the RBC-UKQCD collaboration [21], we did not find evidence for a substantial improvement when employing it to the amplitude. It must be also noted that this strategy leads to a slightly reduced value of compared to the result without the additional information from .

The “ rule” is given by the ratioand agrees with the experimental result . Our value almost coincides with the RBC-UKQCD prediction. The RBC-UKQCD lattice results show that QCD dynamics, present dominantly in current–current operators, is responsible for this large ratio thereby confirming the findings within DQCD obtained many years ago [11, 48]. This is also seen in another recent LQCD analysis [49].

BSM master formula

In this section we report the updated master formula coefficients describing the new physics effects beyond the SM (BSM) in ,which were first presented in [35, 36]. The BSM contribution to is given by the weight factors for each Wilson coefficient of the operators and their chirality-flipped counterparts listed in Table 4. The contain the information of the RG evolution from the low-energy scale to the electroweak (EW) scale and are linearly dependent on the hadronic matrix elements of the operators at the scale , such that the -dependence cancels. The master formula takes the simple formwith the effective Hamiltonianleading to dimensionless Wilson coefficients and factors. The sum runs over all Wilson coefficients of the operators in Table 4. These operators are a complete basis for non-leptonic transitions in the absence of any other light degrees of freedom [36]. The Wilson coefficients and their weight factors are evaluated at the particular value of the EW scale. For more details we refer to [35, 36].

Table 4

Updated coefficients entering the master formula for NP effects in

Class	Q_i	P_i	ΛTeV
A)	QVLLu=(s¯^iγ_μP_Ld^i)(u¯^jγ^μP_Lu^j)	-3.3±0.8	57
	QVLRu=(s¯^iγ_μP_Ld^i)(u¯^jγ^μP_Ru^j)	-124±11	351
	Q~VLLu=(s¯^iγ_μP_Ld^j)(u¯^jγ^μP_Lu^i)	1.1±1.2	32
	Q~VLRu=(s¯^iγ_μP_Ld^j)(u¯^jγ^μP_Ru^i)	-430±40	656
	QVLLd=(s¯^iγ_μP_Ld^i)(d¯^jγ^μP_Ld^j)	1.8±0.5	42
	QVLRd=(s¯^iγ_μP_Ld^i)(d¯^jγ^μP_Rd^j)	117±11	342
	QSLRd=(s¯^iP_Ld^i)(d¯^jP_Rd^j)	204±20	451
	QVLLs=(s¯^iγ_μP_Ld^i)(s¯^jγ^μP_Ls^j)	0.1±0.1	7
	QVLRs=(s¯^iγ_μP_Ld^i)(s¯^jγ^μP_Rs^j)	-0.17±0.04	12
	QSLRs=(s¯^iP_Ld^i)(s¯^jP_Rs^j)	-0.4±0.1	19
	QVLLc=(s¯^iγ_μP_Ld^i)(c¯^jγ^μP_Lc^j)	0.5±0.1	22
	QVLRc=(s¯^iγ_μP_Ld^i)(c¯^jγ^μP_Rc^j)	0.8±0.1	28
	Q~VLLc=(s¯^iγ_μP_Ld^j)(c¯^jγ^μP_Lc^i)	0.7±0.1	26
	Q~VLRc=(s¯^iγ_μP_Ld^j)(c¯^jγ^μP_Rc^i)	1.3±0.2	35
	QVLLb=(s¯^iγ_μP_Ld^i)(b¯^jγ^μP_Lb^j)	-0.33±0.03	18
	QVLRb=(s¯^iγ_μP_Ld^i)(b¯^jγ^μP_Rb^j)	-0.22±0.03	14
	Q~VLLb=(s¯^iγ_μP_Ld^j)(b¯^jγ^μP_Lb^i)	0.3±0.1	17
	Q~VLRb=(s¯^iγ_μP_Ld^j)(b¯^jγ^μP_Rb^i)	0.4±0.1	19
B)	Q_8g=m_s(s¯σ^μνT^aP_Ld)Gμνa	-0.35±0.12	18
	QSLLs=(s¯^iP_Ld^i)(s¯^jP_Ls^j)	0.05±0.02	7
	QTLLs=(s¯^iσ_μνP_Ld^i)(s¯^jσ^μνP_Ls^j)	-0.14±0.05	12
	QSLLc=(s¯^iP_Ld^i)(c¯^jP_Lc^j)	-0.26±0.09	16
	QTLLc=(s¯^iσ_μνP_Ld^i)(c¯^jσ^μνP_Lc^j)	-0.15±0.05	12
	Q~SLLc=(s¯^iP_Ld^j)(c¯^jP_Lc^i)	-0.23±0.07	15
	Q~TLLc=(s¯^iσ_μνP_Ld^j)(c¯^jσ^μνP_Lc^i)	-5.9±1.9	76
	QSLLb=(s¯^iP_Ld^i)(b¯^jP_Lb^j)	-0.35±0.12	18
	QTLLb=(s¯^iσ_μνP_Ld^i)(b¯^jσ^μνP_Lb^j)	-0.11±0.03	10
	Q~SLLb=(s¯^iP_Ld^j)(b¯^jP_Lb^i)	-0.34±0.11	18
	Q~TLLb=(s¯^iσ_μνP_Ld^j)(b¯^jσ^μνP_Lb^i)	-13.4±4.5	115
C)	QSLLu=(s¯^iP_Ld^i)(u¯^jP_Lu^j)	74±16	272
	QTLLu=(s¯^iσ_μνP_Ld^i)(u¯^jσ^μνP_Lu^j)	-162±36	402
	Q~SLLu=(s¯^iP_Ld^j)(u¯^jP_Lu^i)	-15.6±3.3	124
	Q~TLLu=(s¯^iσ_μνP_Ld^j)(u¯^jσ^μνP_Lu^i)	-509±108	713
D)	QSLLd=(s¯^iP_Ld^i)(d¯^jP_Ld^j)	-87±16	295
	QTLLd=(s¯^iσ_μνP_Ld^i)(d¯^jσ^μνP_Ld^j)	191±35	436
E)	QSLRu=(s¯^iP_Ld^i)(u¯^jP_Ru^j)	-266±21	515
	Q~SLRu=(s¯^iP_Ld^j)(u¯^jP_Ru^i)	-60±5	244

In Table 4 we summarize the updated factors after taking into account the most recent matrix elements reported by RBC-UKQCD [21]. Table 4 has been obtained by taking into account the tree-level matching [50] and one-loop running [51] below the EW scale using the public codes wilson [52] and WCxf [53]. Only the factors of operators in Class A are affected by this change, since they depend exclusively on matrix elements of the SM operators. In all other classes the ’s depend on matrix elements of BSM operators or the chromomagnetic operator and remain unchanged. The central values as well as statistical and systematic uncertainties of the matrix elements of all operators are listed in Table 5 at the common scale .

Table 5

Numerical values of hadronic matrix elements from the literature in units of in the scheme at the scale , with statistical and systematic uncertainties. For the SM four-quark operators we provide the values obtained by RG equations with NLO QCDQED and only NLO QCD. The normalization convention is chosen to be for all operators.

Q_i	⟨Q_i⟩_0		⟨Q_i⟩_2
SM operators
	QCD×QED	QCD	QCD×QED	QCD
Q_1	-0.065(17)(10)	-0.065(17)(10)	0.0087(2)(5)	0.0085(2)(5)
Q_2	0.087(13)(14)	0.087(13)(14)	0.0085(2)(5)	0.0085(2)(5)
Q_3	-0.075(57)(12)	-0.075(57)(12)	0.0000	0
Q_4	0.093(51)(15)	0.093(52)(15)	-0.0003	0
Q_5	-0.120(53)(19)	-0.121(53)(19)	0.0002	0
Q_6	-0.641(46)(101)	-0.644(46)(101)	0.0011	0
Q_7	0.217(16)(34)	0.216(16)(34)	0.0996(68)(30)	0.0989(68)(30)
Q_8	1.583(30)(249)	1.581(30)(249)	0.684(19)(41)	0.683(19)(41)
Q_9	-0.059(17)(9)	-0.061(17)(9)	0.0132(3)(8)	0.0128(3)(8)
Q_10	0.092(18)(14)	0.092(18)(14)	0.0130(3)(8)	0.0128(3)(8)
Q_8g	-0.013(4)		0	0
Beyond the SM operators
Q1SLL,u	-0.005(1)		-0.0030(6)
Q2SLL,u	-0.044(9)		-0.031(6)
Q3SLL,u	-0.371(74)		-0.262(52)
Q4SLL,u	-0.214(43)		-0.151(30)
Q1SLL,d	0.0070(14)		-0.002(4)
Q2SLL,d	-0.088(18)		0.031(6)
Q1SLR,u	-0.015(3)		0.0030(6)
Q2SLR,u	-0.141(28)		0.050(10)

The changes are moderate of not more than 30% for operators that contribute directly to , whereas changes can be larger for those operators (with s, c, b-quarks) that enter via RG running from the EW scale down to the low-energy scale and have smaller coefficients. The last column of Table 4 shows the suppression scale that would generate for , assuming the presence of only this particular operator. For comparison, the theory uncertainty of the SM prediction (28) is about . The scale is strongly dependent on the uncertainties of the matrix elements, which did not all decrease in the latest RBC-UKQCD predictions. A comparison to the previous values [35] shows a slight increase of for the first seven operators, which contribute directly to . In general also increases for the remaining Class A operators, with a few exceptions, pushing the NP scale also in these cases up, even though they are entering only via RG mixing. This shows that the new results for the matrix elements from RBC-UKQCD will lead to stronger bounds on CP violation beyond the SM.

Eventually we point out that the large increase of the central value of in the SM from with the 2015 RBC-UKQCD results to with the 2020 results constitutes one order of magnitude and hence has significant impact on excluded regions of parameter spaces of BSM scenarios. The 2015 SM predictions [5, 9, 10] suggested a strong anomaly with a constructive to reach agreement with the experimental value . Contrary, the predictions based on 2020 results do not show anymore an anomaly, but allow now for both, a constructive and destructive interference, that can be still sizable in view of the large theory uncertaintiesas a rough range. The complete error propagation can be obtained properly for general BSM scenarios with the master formula, which is implemented in the public code flavio [54, 55]. Despite the large uncertainties, was and remains one of the strongest constraints on CP violation in the quark-flavour sector, as has been shown for different BSM scenarios in the past. The BSM studies based on the 2015 SM predictions of used mostly the working hypothesis of a constructive of similar size, see references in [4], and the obtained conclusions for are still mostly valid.

Summary and outlook

Our final result for in (5) differs significantly from the one of the RBC-UKQCD collaboration but in view of large uncertainties in both results they are in agreement with each other and with experiment. But as emphasized in [23] the perfect agreement of (5) with the ChPT result in (3) is a pure numerical coincidence because the latter was obtained with in place of and with the values of and , which differ from ours. Still it would be important to clarify whether the QCD dynamics enhancing the parameter over unity in LQCD and in ChPT is the same.

The recent advances in LQCD allow us to hope that in the coming years we should be able to have a value of within the SM with a comparable error to the experimental one. In order to reach this goal and thereby to obtain an assessment on the allowed room for NP contributions to it is important to perform a number of steps:Several BSM analyses of have been performed, which are collected in [4]. A recent example of a model with explicit gauge anomaly cancellation has been discussed in [58]. Furthermore leptoquark models, except the model, would not be able to explain large deviations of the SM value from the data due to constraints coming from rare K decays [59]. This underlines the importance of correlations of with other observables in NP scenarios. The new SM value in (5) removes the difficulties of leptoquark models pointed out in [59], but these problems could return with an improved analyses of within the SM.

A more precise determination of or . At least a second LQCD collaboration should calculate , in order to confirm the large enhancement of found by RBC-UKQCD that has not been identified in DQCD. Also the errors in other matrix elements should be decreased.

A more precise determination of . In particular in LQCD calculations isospin-breaking corrections and corrections in hadronic matrix elements required for the removal of renormalization scheme dependence at this order should be taken into account. The present status is summarized in [56].

A more precise determination of the short distance contributions, especially in the QCD penguin sector, which in the context of the RBC-UKQCD analysis will decrease the sensitivity to the matching scale . Despite the fact that the NNLO analysis of QCD corrections to EWP contributions practically removed the sensitivity of to the renormalization scheme of the top-quark mass and , our analysis shows that the significant uncertainty in the EWP sector still has to be removed through the matching of to effective theory at the NNLO level.

The computation of the BSM hadronic matrix elements of four-quark operators by lattice QCD, which are presently known only from the DQCD approach [57].

Furthermore the lessons from the SMEFT analysis in [36] should be useful in this respect. Such general analyses allow to take into account constraints from other processes such as collider processes, electroweak precision tests, neutral meson mixing as well as electric dipole moments. Finally the master formula for presented in [35] valid for any BSM scenario should facilitate the derivation of constraints on CP-violating phases beyond the SM imposed by . In this respect we point out that also has a very precise SM prediction and can be predicted rather precisely also in BSM scenarios, providing thus a second observable besides to constrain also real parts of the Wilson coefficients of non-leptonic operators.