AJB-21-1

The $\varepsilon^{\prime}/\varepsilon$-Story: $1976-2021$
Andrzej J. Buras
TUM Institute for Advanced Study, Lichtenbergstr. 2a, D-85748 Garching,
Germany
Physik Department, TU München, James-Franck-Straße, D-85748 Garching, Germany
E-mail<EMAIL_ADDRESS>
###### Abstract

The ratio $\varepsilon^{\prime}/\varepsilon$ measures the size of the direct
CP violation in $K_{L}\to\pi\pi$ decays $(\varepsilon^{\prime})$ relative to
the indirect one described by $\varepsilon$ and is very sensitive to new
sources of CP violation. As such it played a prominent role in particle
physics already for 45 years. Due to the smallness of
$\varepsilon^{\prime}/\varepsilon$ its measurement required heroic efforts in
the 1980s and the 1990s on both sides of the Atlantic with final results
presented by NA48 and KTeV collaborations at the beginning of this millennium.
On the other hand, even 45 years after the first calculation of
$\varepsilon^{\prime}/\varepsilon$ we do not know to which degree the Standard
Model agrees with this data and how large is the room left for new physics
(NP) contributions to this ratio. This is due to significant non-perturbative
(hadronic) uncertainties accompanied by partial cancellation between the QCD
penguin contributions and electroweak penguin contributions. In addition to
the calculation of hadronic matrix elements of the relevant operators
including isospin breaking effects and QED corrections, it is crucial to
evaluate accurately the Wilson coefficients of the relevant operators. While
the significant control over the latter short distance effects has been
achieved already in the early 1990s, with several improvements since then,
different views on the non-perturbative contributions to
$\varepsilon^{\prime}/\varepsilon$ have been expressed by different authors
over last thirty years. In fact even at the dawn of the 2020s the uncertainty
in the room left for NP contributions to $\varepsilon^{\prime}/\varepsilon$ is
still very significant, which I find to be very exciting.

My own work on $\varepsilon^{\prime}/\varepsilon$ started in 1983 and involved
both perturbative and non-perturbative calculations. This writing is a non-
technical recollection of the steps which led to the present status of
$\varepsilon^{\prime}/\varepsilon$ including several historical remarks not
known to everybody. The present status of the $\Delta I=1/2$ rule is also
summarized.

This story is dedicated to my great
$\varepsilon^{\prime}/\varepsilon$-collaborator Jean-Marc Gérard on the
occasion of the 35th anniversary of our collaboration and his 64th birthday.

##  1 Overture

Let me open this writing with the 2020 formula for
$\varepsilon^{\prime}/\varepsilon$ within the SM presented in [1]. It reads

$\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)_{\text{SM}}={\rm
Im}\lambda_{\rm t}\cdot\left[\,\big{(}1-\hat{\Omega}_{\rm
eff}\big{)}\big{(}-2.9+15.4\,B_{6}^{(1/2)}(\mu^{*})\big{)}+2.0-8.0\,B_{8}^{(3/2)}(\mu^{*})\,\right].$
(1)

It includes NLO QCD corrections to the QCD penguin (QCDP) contributions and
NNLO contributions to electroweak penguins (EWP). The coefficients in this
formula and the parameters $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ are scale
dependent. Their values for different scales are collected in Table 1 of [1].
Here we will set $\mu^{*}=1\,{\rm GeV}$ because at this scale it is most
convenient to compare the values for $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$
obtained in various non-perturbative approaches. The four contributions in (1)
are dominated by the following operators:

  * •

The terms involving the non-perturbative parameters $B_{6}^{(1/2)}$ and
$B_{8}^{(3/2)}$ contain only the contributions from the dominant QCDP operator
$Q_{6}$ and the dominant EWP operator $Q_{8}$, respectively. There are two
main reasons why $Q_{8}$ can compete with $Q_{6}$ here despite the smallness
of the electroweak couplings relative to the QCD one. In the basic formula for
$\varepsilon^{\prime}/\varepsilon$ its contribution is enhanced relative to
the $Q_{6}$’s one by the factor ${{\rm Re}A_{0}/{\rm Re}A_{2}}=22.4$ with
$A_{0,2}$ being isospin amplitudes. In addition its Wilson coefficient is
enhanced for the large top-quark mass which is not the case of the $Q_{6}$’s
one. The expressions for these two operators and the remaining operators
mentioned below are given in Appendix A.

  * •

The term $-2.9$ is fully dominated by the QCDP operator $Q_{4}$.

  * •

The term $+2.0$ is fully dominated by EWP operators $Q_{9}$ and $Q_{10}$.

  * •

The quantity $\hat{\Omega}_{\rm eff}$ represents the isospin breaking
corrections and QED corrections beyond EWP contributions. It is not in the
ballpark of a few percent as one would naively expect, because in
$\varepsilon^{\prime}/\varepsilon$ it is enhanced by the factor ${{\rm
Re}A_{0}/{\rm Re}A_{2}}=22.4$.

  * •

${{\rm Im}\lambda_{\rm t}}$ is the CKM factor that within a few percent is in
the ballpark of ${1.45\cdot 10^{-4}}$.

On the other hand the experimental world average of
$\varepsilon^{\prime}/\varepsilon$ from NA48 [2] and KTeV [3, 4]
collaborations reads

$(\varepsilon^{\prime}/\varepsilon)_{\text{exp}}=(16.6\pm 2.3)\times
10^{-4}\,.$ (2)

In what follows we will describe the theoretical developments which led over
four decades to the present values of $B_{6}^{(1/2)}$, $B_{8}^{(3/2)}$ and
$\hat{\Omega}_{\rm eff}$ obtained in various non-perturbative approaches.
Inserting them into (1) will allow us to get various expectations for
$\varepsilon^{\prime}/\varepsilon$ in the SM and to compare them with the data
in (2).

##  2 The First Period: $1976-1989$

The story of $\varepsilon^{\prime}/\varepsilon$ begins on February 1 1976 when
John Ellis, Mary K. Gaillard and Dimitri Nanopoulos submitted a paper to
Nuclear Physics B in which the first calculation of
$\varepsilon^{\prime}/\varepsilon$ has been presented [5]. This pioneering
calculation does not resemble by any means the present calculations of
$\varepsilon^{\prime}/\varepsilon$ but this was the first one. In particular,
it does not include renormalization group effects and takes only QCDP
contributions into account. Moreover in 1976 one had no idea about the matrix
elements of QCDP operators contributing to $\varepsilon^{\prime}/\varepsilon$.

However, already in 1975, Shifman, Vainshtein and Zakharov [6], suggested that
the QCDPs could be responsible for the $\Delta I=1/2$ rule: a large
enhancement of the $K\to\pi\pi$ isospin amplitude $A_{0}$ over $A_{2}$ one as
seen already above. This was quite natural at that time because QCDPs
contribute only to $A_{0}$ and the attempts in 1974 to explain this rule
through current-current operators ($Q_{1}$ and $Q_{2}$) including leading
order QCD corrections in [7, 8] turned out to be unsuccessful. As found twelve
years later in the framework of the Dual QCD (DQCD) approach [9] and recently
confirmed by the RBC-UKQCD lattice QCD collaboration [10], this failure was
caused by the poor knowledge of hadronic matrix elements of these operators at
that time. We will be more explicit about it at the end of this writing.

In order to explain the $\Delta I=1/2$ rule with the help of QCDPs Shifman,
Vainshtein and Zakharov have simply chosen the values of their hadronic matrix
elements so that this rule could be reproduced. But the same matrix elements
enter $\varepsilon^{\prime}/\varepsilon$ and Gilman and Wise [11] using their
values from the Russian trio and performing renormalization group analysis for
$m_{t}$ much smaller than $M_{W}$, as thought in 1979, found
$\varepsilon^{\prime}/\varepsilon$ to be in the ballpark of $5\times 10^{-2}$.
Soon after other analyses appeared, in particular the one by Guberina and
Peccei in [12], finding values of $\varepsilon^{\prime}/\varepsilon$ in the
ballpark of $10^{-3}-10^{-2}$.

My first real encounter with $\varepsilon^{\prime}/\varepsilon$ goes back to
1981 when Bruce Winstein, the spokesman of future E731 and KTeV
collaborations, entered my office at Fermilab and asked me about the work of
Gilman and Wise. I told him that I never studied this ratio, but that one
should seriously consider their result after they wrote other important
papers, in particular the one in [13]. Bruce told me that if
$\varepsilon^{\prime}/\varepsilon$ was as large as claimed by Fred and Mark,
he can definitely measure it and I encouraged him to do it because of the
importance of CP violation.

My first paper on $\varepsilon^{\prime}/\varepsilon$, in collaboration with
Slominski and Steger, appeared in 1983 [14]. It was just a phenomenology of
this ratio as a function of the hadronic matrix element of $Q_{6}$, of the CKM
parameters and of $m_{t}$ in the ballpark of $30\,{\rm GeV}$ as expected at
that time.

Over the 1980s the calculations of $\varepsilon^{\prime}/\varepsilon$ were
refined through the inclusion of isospin breaking in the quark masses [15,
16], the inclusion of QED penguin effects for $m_{t}\leq M_{W}$ [17, 15, 16,
18, 19] and in particular through the first calculation of hadronic matrix
elements of QCDP and EWP operators in QCD by Bardeen, Gérard and myself in the
framework of the DQCD [20, 21, 9]. The latter calculations were done in the
strict large $N$ limit of colours, but this was sufficient to see that QCDPs
are not responsible for the $\Delta I=1/2$ rule 111More about it in Section 7.
and that $\varepsilon^{\prime}/\varepsilon$ was rather $\mathcal{O}(10^{-3})$
than $\mathcal{O}(10^{-2})$ as claimed by Gilman and Wise in [11]. Thus
already in 1986 the values of $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ were known:

$B_{6}^{(1/2)}=1.0,\qquad B_{8}^{(3/2)}=1.0,\qquad(N\to\infty)$ (3)

and the study of their renormalization scale dependence for $\mu\geq 1\,{\rm
GeV}$ in the leading logarithmic (LO) approximation indicated that it was
small, although as we will see later it is non-negligible when higher order
QCD corrections are taken into account and scales below $1\,{\rm GeV}$ in the
framework of DQCD are considered.

It is rather surprizing that we did not calculate these two parameters
including $1/N$ corrections in the 1980s. Such a calculation has been
performed by Jean-Marc Gérard and me 29 years later with interesting
consequences. I will report on it in Section 5.

While the numerical coefficients in (1) include higher order QCD corrections
that were unknown in the 1980s, it is tempting to use this formula for
$B_{6}^{(1/2)}=1.0$ and $\hat{\Omega}_{\rm eff}=0.29$ setting the last two
terms to zero because EWP are irrelevant for low values of $m_{t}$ expected at
that time. Setting ${{\rm Im}\lambda_{\rm t}}=1.45\cdot 10^{-4}$ we find

$\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)_{\text{SM}}=12.9\times
10^{-4}\,,$ (4)

in a good agreement with (2) which was first known fifteen years later.
Setting $\hat{\Omega}_{\rm eff}=0$ would increase this value to $18.1\times
10^{-4}$. This is also consistent with experiment but this change illustrates
the importance of isospin breaking corrections in the evaluation of
$\varepsilon^{\prime}/\varepsilon$. We will return to it in Section 5, where
we will explain the dynamics behind $\hat{\Omega}_{\rm eff}=0.29$. There is no
point in discussing the errors in these estimates already now. We will do it
in Section 6.

One of the last analyses of $\varepsilon^{\prime}/\varepsilon$ in the SM for
$m_{t}\leq M_{W}$, that included all effects known at that time, is the one in
[22] in which typical values for $\varepsilon^{\prime}/\varepsilon$ were found
in the ballpark of a few $10^{-3}$.

In December 1988 I attended the Kaon conference in Vancouver. Fred Gilman
during a lunch told me about his paper with Claudio Dib and Isard Dunietz
[23], in which they calculated electroweak contributions to
$K_{L}\to\pi^{0}e^{+}e^{-}$ for an arbitrary top quark mass finding for
$m_{t}>150\,{\rm GeV}$ a large contribution from the $Z^{0}$-penguin that
increased with $m_{t}$ roughly as $m_{t}^{2}$. A similar result has been
obtained by Jonathan Flynn and Lisa Randall [24]. This gave me the idea to
calculate $Z^{0}$-penguin contribution to $\varepsilon^{\prime}/\varepsilon$
for large $m_{t}$. The QCDP and photon penguin contributions were known
already at that time to have a very weak $m_{t}$ dependence.

I moved to TUM in November 1988, but already in January 1989 I got a number of
very good diploma students. Two of them, Gerhard Buchalla and Michala
Harlander were supposed to perform the calculation of
$\varepsilon^{\prime}/\varepsilon$ together with me for an arbitrary top quark
mass. But this was their first calculation of that type and I was
simultaneously busy preparing first lectures in my life as well starting the
project on NLO QCD corrections to all flavour violating processes in
collaboration with Peter Weisz [25]. Consequently my project with Gerhard and
Michaela took longer than I initially expected and in April 1989222The paper
appeared already in March but in 1989 the arXiv did not exist and we learned
about it several weeks later. we were surprized by a paper by Flynn and
Randall [26] in which $\varepsilon^{\prime}/\varepsilon$ including QCDP, EWP
($\gamma$ and $Z^{0}$ penguins) and the relevant box diagrams were calculated
for an arbitrary top quark mass. Significant suppression of
$\varepsilon^{\prime}/\varepsilon$ for $m_{t}>150\,{\rm GeV}$ by EWPs has been
found by them. While definitely Flynn and Randall should be given the credit
for pointing out the importance of $Z^{0}$ penguins in
$\varepsilon^{\prime}/\varepsilon$ in print, fortunately for us their
calculation of the QCDP contribution for large $m_{t}$ was incorrect so that
in fact the first correct calculation of $\varepsilon^{\prime}/\varepsilon$ in
the SM for arbitrary $m_{t}$ including LO QCD corrections has been presented
by us in [27]. In fact the suppression of $\varepsilon^{\prime}/\varepsilon$
for large $m_{t}$ turned out to be significantly stronger than reported by
Jonathan and Lisa in [26]. The strong cancellation between QCDP and EWP for
$m_{t}>150~{}\,{\rm GeV}$ found by us was soon confirmed in the erratum to
[26] and subsequently by other authors [28, 29]. In fact at that time, due to
the aforementioned cancellation between QCDP and EWP contributions, the
vanishing of $\varepsilon^{\prime}/\varepsilon$ in the SM and negative values
for it could not be excluded.

It is then tempting to include next the EWP terms in (1) valid for the
presently known $m_{t}$ and set $B_{6}^{(1/2)}=B_{8}^{(3/2)}=1.0$ and
$\hat{\Omega}_{\rm eff}=0.29$ to find this time

$\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)_{\text{SM}}=4.2\times
10^{-4},$ (5)

that is by a factor of $4$ below the experimental value in (2). Setting
$\hat{\Omega}_{\rm eff}=0$ would increase this value to $9.4\times 10^{-4}$,
roughly by a factor of two below the data and showing the importance of
isospin breaking effects dominating the parameter $\hat{\Omega}_{\rm eff}$.

As the upper bound on $m_{t}$ from electroweak precision tests was still
unknown in 1989, one can find in our paper plots of
$\varepsilon^{\prime}/\varepsilon$ as a function of $m_{t}$ with
$\varepsilon^{\prime}/\varepsilon$ vanishing for $m_{t}\approx 200\,{\rm
GeV}$. I was visiting CERN in November 1989, one month after our paper
appeared. In the CERN cafeteria I met Jack Steinberger and asked him the
standard question how he was doing. He told me that NA31 collaboration was
shocked by the theory. I asked him which theory. Your paper with Buchalla and
Harlander he replied. In fact at that time only the first result from NA31
collaboration was known [30] implying $\varepsilon^{\prime}/\varepsilon=(33\pm
11)\times 10^{-4}$. The second NA31 analysis in 1992 gave
$\varepsilon^{\prime}/\varepsilon=(23\pm 7)\times 10^{-4}$ [31] and the one
from the E731 experiment at Fermilab $\varepsilon^{\prime}/\varepsilon=(7.4\pm
5.9)\times 10^{-4}$ [32]. Therefore, the experimental situation of
$\varepsilon^{\prime}/\varepsilon$ was still unclear in the first half of the
1990s.

It should be emphasized that the calculations described above were done in the
leading logarithmic approximation (e.g. one-loop anomalous dimensions of the
relevant operators) with the exception of the $m_{t}$-dependence which in the
analyses [26, 27, 28] has been already included at the NLO level. While such a
procedure is not fully consistent, $m_{t}$ dependence enters
$\varepsilon^{\prime}/\varepsilon$ first at NLO in the renormalization group
(RG) improved perturbation theory, it allowed for the first time to exhibit
the strong $m_{t}$-dependence of the EWP contributions and to identify the
partial cancellation between QCDP and EWP contributions which is not seen in
the strict leading logarithmic approximation.

##  3 The Second Period: $1990-2000$

During the early 1990s considerable progress has been made by calculating
complete NLO corrections to $\varepsilon^{\prime}/\varepsilon$. This means
both QCD and electroweak corrections at NLO in the RG improved perturbation
theory. This rather heroic effort has been accomplished by the Munich and Rome
teams led by me [33, 34, 35, 36] and Guido Martinelli [37, 38], respectively.
I stress heroic because in that days the technology for two-loop calculations
in weak decays, involving issues of $\gamma_{5}$ and evanescent operators, was
at an early stage and moreover some of my colleagues told me that I was crazy
doing such calculations in view of poorly known hadronic matrix elements. But
while the main goal in these papers was to calculate Wilson coefficients of
QCDP and EWP operators at the NLO level in the context of
$\varepsilon^{\prime}/\varepsilon$, as a byproduct, many of our results could
also be soon used for non-leptonic $B$ and $D$ decays. Reviews of this period
of NLO calculations can be found in [39, 40] and in my recent book on weak
decays [41].

Thus by 1993 $\varepsilon^{\prime}/\varepsilon$ was known at the NLO level as
far as the Wilson coefficients of the contributing operators are concerned. On
the other hand, the matrix elements of the contributing operators were either
calculated in the large $N$ limit, represented for $Q_{6}$ and $Q_{8}$
operators by (3), or vacuum insertion method which for these two operators
gives basically the same result as the large $N$ limit333It should be stressed
that the vacuum insertion method has no QCD basis and gives even wrong signs
for the $1/N$ corrections to hadronic matrix elements as pointed out already
in [42, 20, 21, 9] and confirmed numerically by the RBC-UKQCD collaboration in
[43].. But as other operators also play some role, significant uncertainties
in the estimate $\varepsilon^{\prime}/\varepsilon$ have been found. In order
to reduce them it has been suggested in [36] to assume that the $\Delta I=1/2$
rule can completely be explained by the SM dynamics and to determine the
matrix elements of the current-current operators $Q_{1}$ and $Q_{2}$ by using
the experimental values of the $K\to\pi\pi$ isospin amplitudes $A_{0}$ and
$A_{2}$. This in turn allowed with the help of isospin symmetry to determine
the matrix elements of $(V-A)\times(V-A)$ operators like $Q_{4},Q_{9},Q_{10}$.
This method has been used in all analyses performed in Munich since then with
some refinements made recently in [44, 45]. In this manner the terms like
$-2.9$ and $+2.0$ in (1) could be more accurately determined than the
remaining dominant terms.

However, this procedure has one weak point. If NP is responsible for a
fraction of the $\Delta I=1/2$ rule this trick will not provide fully correct
results for hadronic matrix elements and it is safer to calculate them using
non-perturbative methods like LQCD or DQCD which avoids automatically any NP
contributions to hadronic operator matrix elements. NP can only affect the
Wilson coefficients of these operators. Therefore, in fact the formula (1)
does not assume that the SM explains fully the $\Delta I=1/2$ rule and to find
the terms $-2.9$ and $+2.0$ only hadronic matrix elements from RBC-UKQCD
collaboration [10] have been used, properly evolved to $\mu=1~{}\,{\rm GeV}$.
The formula (1) exhibits only central values of various contributions. We will
return to uncertainties in the final sections of this writeup.

Unfortunately in the 1990s there were no results from LQCD for any of the
terms in (1) and the relevant hadronic matrix elements of $Q_{6}$ and $Q_{8}$
operators, known only from DQCD at that time, and given by [42, 16]

$\displaystyle\langle Q_{6}(\mu)\rangle_{0}$
$\displaystyle=-\,4\left[\frac{m_{K}^{2}}{m_{s}(\mu)+m_{d}(\mu)}\right]^{2}(F_{K}-F_{\pi})\,B_{6}^{(1/2)}(\mu),$
(6) $\displaystyle\langle Q_{8}(\mu)\rangle_{2}$
$\displaystyle=\sqrt{2}\left[\frac{m_{K}^{2}}{m_{s}(\mu)+m_{d}(\mu)}\right]^{2}F_{\pi}\,B_{8}^{(3/2)}(\mu),$
(7)

with $B_{6}^{(1/2)}=B_{8}^{(3/2)}=1.0$ in the large N-limit, were subject to
the large uncertainty in the strange quark mass. We did not expose this
dependence in (1) because in 2021 this uncertainty is very small. It is
exhibited in the older Munich papers like [46, 47]. Moreover, in the 1990s it
was not clear what was the error on $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ so
that often in the phenomenological analyses of the Munich and also Rome group
[48] their values were varied within the range $0.8-1.2$. We will see later
that while presently the $B_{8}^{(3/2)}$ is already accurately known, this is
not the case for $B_{6}^{(1/2)}$.

In any case it turned out that NLO corrections led to additional suppression
of $\varepsilon^{\prime}/\varepsilon$ beyond the one by EWP so that typical
values of $\varepsilon^{\prime}/\varepsilon$ in Munich papers were in the
ballpark of $7\times 10^{-4}$ and the ones from Rome more like $3\times
10^{-4}$, both values being significantly below the experimental result in
(2).

The origin of the difference in $\varepsilon^{\prime}/\varepsilon$ resulting
from the phenomenological analyses performed in Munich and Rome has been
clarified only in 1998 in a paper written in collaboration with Paolo Gambino
and Ulrich Haisch [49]. We have pointed out that without NNLO QCD corrections
to EWP contribution the results for $\varepsilon^{\prime}/\varepsilon$ are
renormalization-scheme dependent and exhibit significant non-physical
dependences on the scale $\mu_{t}$ at which the top-quark mass
$m_{t}(\mu_{t})$ is evaluated as well as on the scale $\mu_{W}$, at which the
full SM is matched onto the low-energy effective theory without top,
$W^{\pm}$, $Z$ and the Higgs. It turned out then that the dominant difference
between Munich and Rome NLO results originated in the use of different schemes
for $\gamma_{5}$, NDR scheme in Munich and t’Hooft-Veltman scheme in Rome.
This difference, as well as scale uncertainties mentioned above, have been
significantly reduced in our paper, the first NNLO analysis of
$\varepsilon^{\prime}/\varepsilon$ that concentrated on the EWP contributions.
The corresponding NNLO analysis for QCDPs should hopefully be completed in
2021 [50, 51], although the first steps have already been made in 2004 by
Martin Gorbahn and Ulrich Haisch, who calculated the three-loop anomalous
dimension matrix of the relevant operators [52]. But we know already today
from preliminary results in [50, 51], that while the NNLO QCD corrections to
EWPs increase their importance in $\varepsilon^{\prime}/\varepsilon$ [49], the
ones to QCDPs suppress the latter. Consequently there is a visible suppression
of $\varepsilon^{\prime}/\varepsilon$ by the NNLO QCD corrections. But the
percentual effect of this suppression strongly depends on the values of the
relevant hadronic matrix elements which in the case of QCDPs are still poorly
known.

One of the last papers of this period was the one in [53] in which we
addressed the first messages from the new round of
$\varepsilon^{\prime}/\varepsilon$-experiments by NA48 and KTeV collaborations
that led eventually to the result in (2). We found that it was not possible to
explain the data within the SM which motivated a detailed analysis in the MSSM
[54].

##  4 The Third Period: $2000-2015$

Already in the second period, but with even increased energy in the third
period, the authors in [55, 56, 57, 58, 59, 60, 61] using the ideas from
Chiral Perturbation Theory (ChPT) made a strong claim that final state
interactions (FSI) enhance $B_{6}^{(1/2)}$ above unity and suppress
$B_{8}^{(3/2)}$ below it so that the SM value for
$\varepsilon^{\prime}/\varepsilon$ according to them is fully consistent with
experiment. Albeit only a very inaccurate value $(17\pm 9)\times 10^{-4}$ [61]
could be obtained at that time. More about it later. Critical remarks about
some of these papers appeared in [62] but they did not slow down the ChPT
experts.

The main theoretical activity on $\varepsilon^{\prime}/\varepsilon$ within the
SM in this period was another look at the parameter $\hat{\Omega}_{\rm eff}$
which includes both isospin breaking corrections and QED corrections beyond
EWP contributions. Intensive analyses of these corrections can be found in
[63, 64, 65] were references to earlier analyses can be found. The renaissance
of such analyses with ChPT updates and also including the update of my 1987
analysis with Jean-Marc Gérard in [16] will be described in the fourth period.
It modified significantly ChPT results, basically confirming with smaller
uncertainties our 1987 value for $\hat{\Omega}_{\rm eff}$ that was only
slightly lower than $\hat{\Omega}_{\rm eff}\approx 0.29$ used in our examples
above.

In the absence of LQCD results and the claim of ChPT experts that the SM can
reproduce the experimental value in (2), the interest in
$\varepsilon^{\prime}/\varepsilon$ among theorists decreased significantly in
this period. This was also due to the fact that the new experimental results
in other sectors of particle physics, in particular in $B$ physics and
neutrino physics, could be easier addressed theoretically than
$\varepsilon^{\prime}/\varepsilon$. Selected reviews of
$\varepsilon^{\prime}/\varepsilon$ in this period can be found in [66, 47, 67,
68]. On my part I have performed with my PhD students, postdocs and other
collaborators a number of analyses of $\varepsilon^{\prime}/\varepsilon$ in
various extensions of the SM that can easily be found in the hep-arXiv. I will
not review them here because the input parameters and often the scale of NP
changed in the last decade and it is better to summarize the status of NP
models in the last period presented here. But the formulae presented in these
papers could still turn out to be useful one day when the hadronic
uncertainties will be reduced by much.

##  5 The Fourth Period: $2015-2019$

The new era for $\varepsilon^{\prime}/\varepsilon$ began in 2015, when the
RBC-UKQCD collaboration [69, 70] presented their first results for
$K\to\pi\pi$ hadronic matrix elements. From these results one could extract
values of $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ for $\mu=1.30\,{\rm GeV}$. They
were [44]

$B_{6}^{(1/2)}=0.57\pm 0.19,\qquad B_{8}^{(3/2)}=0.76\pm
0.05\,,\qquad{\rm(RBC-UKQCD,~{}2015)}.$ (8)

It is evident that with such values there is a strong cancellation between
QCDP and EWP contributions. With $\hat{\Omega}_{\rm eff}\approx 0.15$, as used
in 2015, and NLO QCD corrections one finds values of
$\varepsilon^{\prime}/\varepsilon$ in the ballpark of $(1-2)\times 10^{-4}$.
This means one order of magnitude below the experimental value. But with an
error in the ballpark of $5\times 10^{-4}$ one could talk of an
$\varepsilon^{\prime}/\varepsilon$ anomaly of at most $3\sigma$. The relevant
analyses that extracted some matrix elements from data by assuming the $\Delta
I=1/2$ rule in the SM can be found in [44, 71]. The RBC-UKQCD lattice
collaboration [70, 69], calculating directly hadronic matrix elements of all
operators, but not including isospin breaking (IB) effects, found similar
result but with an error of $7\times 10^{-4}$.

Motivated by the RBC-UKQCD results in (8), Jean-Marc Gérard and me calculated
already in July 2015 $1/N$ corrections to the large $N$ limit in (3) [72].
These corrections, loop corrections in the meson theory with a physical cut-
off $\Lambda\approx 0.7\,{\rm GeV}$, are the leading non-factorizable
corrections to hadronic matrix elements of $Q_{6}$ and $Q_{8}$. Two main
results in this paper are:

  * •

Realization that the large-$N$ result in (3) is not valid at scales
$\mathcal{O}(1\,{\rm GeV}$), as assumed in all papers before, but at much
lower scales $\mathcal{O}(m_{\pi}^{2})$. In order to find it out one has to
calculate the non-factorizable $1/N$ contributions represented in DQCD by
meson loops with a physical cut-off $\Lambda$, which separates the long
distance and short distance contributions444For a detailed exposition of this
point and comparison with ChPT which uses dimensional regularization see
Section 3 in [73].. In the large $N$ limit one cannot determine the scale in
$B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ and as for $\mu\geq 1\,{\rm GeV}$ the
$\mu$ dependence of these parameters is weak [36], without knowing $1/N$
corrections it was useful to neglect this dependence.

  * •

Calculation of $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ at scales
$\mathcal{O}(1\,{\rm GeV})$ by performing the meson evolution from the low
factorization scale $\mathcal{O}(m_{\pi}^{2})$ to the physical cutoff
$\Lambda$ of DQCD with the result

$B_{6}^{(1/2)}\leq 0.6,\qquad B_{8}^{(3/2)}=0.80\pm
0.10,\qquad(\text{DQCD}-2015),$ (9)

in a very good agreement with the RBC-UKQCD results in (8).

As already mentioned, for scales above $1\,{\rm GeV}$ both parameters decrease
very slowly. This was already known from the 1993 analysis in [36], but as
seen in Figs. 11 and 12 of that paper $B_{6}^{(1/2)}$ decreases faster with
increasing scale than $B_{8}^{(3/2)}$ in accordance with the pattern at low
scales found in 2015 by Jean-Marc and myself. This can also be shown
analytically [72]. Unfortunately, not knowing $1/N$ corrections to
$B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ in 1993, both parameters have been set at
$\mu=m_{c}$ in [36] to unity, which is clearly wrong.

While we did not exclude the possibility that our bound on $B_{6}^{(1/2)}$
could be violated by $1/N^{2}$ corrections, vector meson contributions and
other effects like final state interactions (FSI) not taken by us into
account, one should notice that with only pseudoscalars included in the loops,
the cut-off $\Lambda$ has to be chosen below $1\,{\rm GeV}$ so that these
omitted effects, even if they would increase $B_{6}^{(1/2)}$, could still be
at least partially compensated by the running to higher scales that are
explored by lattice QCD. In any case we expected $B_{6}^{(1/2)}$ at scales
$\mathcal{O}(1\,{\rm GeV})$ to be below unity.

Therefore it appeared to us in 2015 that we could understand the QCD dynamics
behind the LQCD values which was important for the following reason. There is
no other lattice collaboration beyond RBC-UKQCD one calculating
$B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ at present, so that in the lattice world
the results of the RBC-UKQCD collaboration for
$\varepsilon^{\prime}/\varepsilon$ could not be tested in 2015 and this is
also the case now. As we will emphasize below, ChPT by itself, has no means to
verify or disprove the RBC-UKQCD results for $B_{6}^{(1/2)}$ and
$B_{8}^{(3/2)}$. As already stated above, according to our analysis in [72],
the main QCD dynamics behind the lattice values in (8) was the meson evolution
at long distances, analogous to the well known quark-gluon evolution at short
distance scales.

At a flavour workshop in Mainz in January 2016 two important ChPT experts,
Gilberto Colangelo and Antonio Pich, expressed serious doubts about the RBC-
UKQCD result in (8), because the $(\pi\pi)_{I=0}$ phase shift
$\delta_{0}\approx(24\pm 5)^{\circ}$ obtained by RBC-UKQCD disagreed with
$\delta_{0}\approx 34^{\circ}$ obtained by combining dispersion theory with
experimental input [74].

This criticism appeared in print in 2017 [75] and in two subsequent conference
proceedings [76, 77]. It is in line with the one expressed many years ago in
[55, 56, 57, 58, 59, 60, 61], but one should realize that with
$\delta_{0}\approx 24^{\circ}$ a big portion of FSI has been already taken
into account in (8). Therefore, from Jean-Marc’s and my point of view it
appeared rather unlikely that increasing $\delta_{0}$ up to its dispersive
value $\delta_{0}\approx 34^{\circ}$ would shift
$\varepsilon^{\prime}/\varepsilon$ upwards by one order of magnitude. In fact,
soon after the Mainz workshop we have expressed this view in [78]
demonstrating that possible enhancement of $\varepsilon^{\prime}/\varepsilon$
by FSI cannot be as large as its suppression through meson evolution absent in
the calculations in [75]. Subsequently in a Christmas story [79] I have
illustrated possible impact of the meson evolution on the result of the
authors of [76, 77], finding much lower values of
$\varepsilon^{\prime}/\varepsilon$ than claimed by them. Moreover, based on
the insight from DQCD and NNLO QCD corrections as well as isospin breaking
effects, I summarized my expectations for 2026, the 50th anniversary of the
first $\varepsilon^{\prime}/\varepsilon$ calculation and also my 80th
birthday, by

$\boxed{(\varepsilon^{\prime}/\varepsilon)_{\rm SM}=(5\pm 2)\cdot
10^{-4},\qquad(2026).}$ (10)

This expectation is in accord with our analysis [78] from which values of
$B_{6}^{(1/2)}$ above unity at $\mu=1\,{\rm GeV}$ are rather unlikely even
after the inclusion of FSI. But for $B_{6}^{(1/2)}=1$ we found already (5) and
the slight decrease of $B_{8}^{(3/2)}$ could increase the value of
$\varepsilon^{\prime}/\varepsilon$ by a bit. The error is just a guess
estimate but even if it is large, confirmation of this result by LQCD would
imply a significant anomaly and NP at work.

While waiting for the new RBC-UKQCD result, the calculation of isospin
breaking affects and QED corrections, represented by $\hat{\Omega}_{\rm eff}$,
has been updated within ChPT in [80] with the result

$\hat{\Omega}_{\rm eff}^{(8)}=(17\pm 9)\,10^{-2},\qquad(\text{ChPT}-2019),$
(11)

where the index “(8)” indicates that only contributions from the octet of
pseudoscalars have been taken explicitly into account. The large error of
$50\%$ in this estimate originates from the difficulties in the matching of
the long distance (LD) and short distance (SD) contributions in this framework
so that the effects of the flavour singlet $\eta_{0}$ cannot be explicitly
included in this approach. They are buried in a poorly known low-enery
constant $L_{7}$.

Using (11) the most recent estimate from ChPT [81, 82] reads

$\displaystyle(\varepsilon^{\prime}/\varepsilon)_{\text{SM}}$
$\displaystyle=(14\pm 5)\times 10^{-4}\,,\qquad(\text{ChPT}-2019).$ (12)

It is in contrast to (10) fully consistent with experiment but because of the
large error related to the problematic matching of LD and SD contributions in
this approach, it still allows for significant NP contributions. We will
return to this result in the next section.

Parallel to these analytic developments RBC-UKQCD presented already in 2018 a
new result for $\delta_{0}$ that with $\delta_{0}=(32.3\pm 2.1)^{\circ}$ is
within $1\sigma$ from its dispersive value555See talks by Ch.Kelly and T. Wang
at Lattice 2018.. But the most important recent result from this collaboration
is the new result for $\varepsilon^{\prime}/\varepsilon$ to be presented soon.

The hints for the possible $\varepsilon^{\prime}/\varepsilon$ anomaly
motivated several authors to perform BSM analyses of this ratio. We collected
a selection of these papers in Table 1 in Appendix A. Here we just mention
that if this anomaly will be confirmed one day by more precise calculations,
the leptoquark models, with the possible exception of the vector
$\text{U}_{1}$ model, will not be able to explain it because of the
constraints from rare Kaon decays [83]. This shows how crucial correlations of
$\varepsilon^{\prime}/\varepsilon$ with other observables in a given NP
scenario are. As indicated in Table 1, such correlations have been analyzed in
other NP scenarios.

Moreover, the lessons gained from the SMEFT analysis in [84] should be very
helpful in identifying NP behind possible $\varepsilon^{\prime}/\varepsilon$
anomaly. Such a general analysis allows to take the constraints from other
processes, in particular from electroweak precision tests and collider
processes, into account. To this end the master formula for
$\varepsilon^{\prime}/\varepsilon$ in [85], valid in any extension of the SM,
should facilitate the search for the dynamics behind the possible anomaly in
question. This formula is based on hadronic matrix elements of SM operators
from LQCD and the BSM four-quark operators from DQCD [86]. The ones of
chromomagnetic operators are known from LQCD [87] and DQCD [88]. They turned
out to be less important than expected in the past. This master formula has
been updated in [1] and will be soon generalized to include NLO QCD
corrections to the BSM contributions.

##  6 The Fifth Period: $2020$

The fifth period begins with a surprising new result from the RBC-UKQCD
collaboration [10]

$\displaystyle(\varepsilon^{\prime}/\varepsilon)_{\rm SM}$
$\displaystyle=(21.7\pm 8.4)\times 10^{-4}\,,$ (13)

where statistical, parametric and systematic uncertainties have been added in
quadrature. The central value is by an order of magnitude larger than the
central 2015 value presented by this collaboration but is subject to large
systematic uncertainties which dominate the quoted error. It is based on the
improved values of the hadronic matrix elements of QCDP, includes the Wilson
coefficients at the NLO level but does not include isospin breaking effects
and NNLO QCD effects

However, as already demonstrated in [45], the inclusion of the effects in
question, that are absent in (13) is important. Indeed after including
isospin-breaking effects from [80] in (11) and NNLO QCD corrections to EWP
contributions [49], one finds using the hadronic matrix elements of RBC-UKQCD
[1]

$\displaystyle(\varepsilon^{\prime}/\varepsilon)^{(8)}_{\text{SM}}=(17.4\pm
6.1)\times 10^{-4}\,$ (14)

instead of (13). The index “(8)” indicates that only the octet of
pseudoscalars has been included in the evaluation of isospin breaking effects
summarized by $\hat{\Omega}_{\rm eff}^{(8)}$ in (11).

Yet, already in 1987 Jean-Marc and me [16]666See also [15]. pointed out that
the contribution of $\eta_{0}$ and of the resulting $\eta-\eta^{\prime}$
mixing cannot be neglected in the evaluation of
$\varepsilon^{\prime}/\varepsilon$. Updating and significantly improving our
1987 analysis we presented last spring the improved estimate of
$\hat{\Omega}_{\rm eff}$ [89]

$\boxed{\hat{\Omega}_{\rm eff}^{(9)}=(29\pm
7)\,10^{-2},\qquad(\text{NIB}-2020),}$ (15)

where the index “(9)” indicates that the full nonet of pseudoscalars has been
taken into account. This is the value we have used in our examples. Note that
the percentual error, even if sizable, amounts to $24\%$, a factor of two
smaller than in (11). Most importantly the central value is by more than
$50\%$ higher than in (11).

At the FCPC 2020 conference Toni Pich referred to our result in (15) as the
naive IB. As a coauthor of the naive dimensional regularization scheme (NDR)
[25] I have no problem with this terminology and accepted this name in (15).
It is well known that the NDR scheme for $\gamma_{5}$ is used these days by
almost everybody even if it is not as sophisticated as the ’t Hooft-Veltman
scheme. I expect that the future of NIB will be similar. The point is that our
analysis in [89] is really not as naive as one would conclude from its name.
In the decoupling limit for $\eta_{0}$ our approach reproduces IB from ChPT
within $10\%$, but in contrast to ChPT we are able to include the effect of
$\eta_{0}$ that is very important. We should stress again that in our approach
this effect is included explicitly, while in the octet scheme, necessarily
used in ChPT, it is buried in a poorly known low-energy constant $L_{7}$. But
$L_{7}$ can only be extracted from the data in the large $N$-limit which in
decays like $K\to\pi\pi$ is a bad approximation. Therefore I expect that
including $\eta_{0}$ effect in $\varepsilon^{\prime}/\varepsilon$ will remain
a big challenge for ChPT for some time and whether ChPT will ever be able to
match the present NIB result is unclear to me at present.

Using the value of $\hat{\Omega}_{\rm eff}^{(9)}$ in (15) together with
hadronic matrix elements of RBC-UKQCD one finds [89, 1]

$\displaystyle\boxed{(\varepsilon^{\prime}/\varepsilon)^{(9)}_{\text{SM}}=(13.9\pm
5.2)\times 10^{-4}\,.}$ (16)

This is in my view the present best estimate of this ratio in the SM if one
accepts the the results from RBC-UKQCD on hadronic matrix elements. However, I
will stress below that I am not ready to do it at present and the main reason
for the analyses in [89, 1] was the service to the community by improving the
RBC-UKQCD analysis through the addition of isospin breaking effects and NNLO
QCD corrections to EWPs.

The result in (16) agrees well with experiment and with the ChPT expectations
(12) but in view of our comments on the ChPT analysis it is on a more solid
footing. Moreover, as we will demonstrate soon, its agreement with the ChPT
value in (12) is a pure numerical coincidence. We expect further reduction of
$\varepsilon^{\prime}/\varepsilon$ by roughly $(5-10)\%$ when NNLO QCD
corrections to QCDP contributions will be taken into account [50, 51]. We look
forward to the final results of these authors.

It is a good place to list the values of $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$
at $\mu=1\,{\rm GeV}$ that can be extracted from the most recent RBC-UKQCD
results for hadronic matrix elements for QCDPs [10] and from [69] for EWPs.
This collaboration calculated them respectively for $\mu=4\,{\rm GeV}$ and
$\mu=3\,{\rm GeV}$ and one extracts at these scales [1]

$B_{6}^{(1/2)}(4\,{\rm GeV})=1.11\pm 0.20,\qquad B_{8}^{(3/2)}(3\,{\rm
GeV})=0.70\pm 0.04.$ (17)

Performing the RG evolution down to $1\,{\rm GeV}$ one finds [1] instead

$B_{6}^{(1/2)}(1\,{\rm GeV})=1.49\pm 0.25,\qquad B_{8}^{(3/2)}(1\,{\rm
GeV})=0.85\pm 0.05\,,$ (18)

demonstrating the decrease of both parameters with increasing $\mu$ in
accordance with DQCD [72].

Let us begin with the good news. Comparing the LQCD value for $B_{8}^{(3/2)}$
with DQCD one in (9) we find a very good agreement between LQCD and DQCD as
far as EWP contribution to $\varepsilon^{\prime}/\varepsilon$ is concerned.
This implies that this contribution to $\varepsilon^{\prime}/\varepsilon$,
that is unaffected by leading IB corrections, is already known within the SM
with acceptable accuracy:

$\boxed{(\varepsilon^{\prime}/\varepsilon)^{\text{EWP}}_{\text{SM}}=-(7\pm
1)\times 10^{-4}\,,\qquad(\text{LQCD~{}and~{}DQCD}).}$ (19)

Because both LQCD and DQCD can perform much better in the case of EWPs than in
the case of QCDPs I expect that this result will remain with us for coming
years.

On the other hand ChPT expected $B_{8}^{(3/2)}\approx 0.55$ [81] with the
suppression below unity caused by FSI. Evidently this large suppression has
not been confirmed by the RBC-UKQCD collaboration. Including only the last two
terms in (1) we find the EWP contribution estimated by ChPT to be roughly by a
factor of 2 below the result in (19). This already signals that the agreement
of (16) with (12) is an accidental numerical coincidence. This is undermined
by the fact that the ChPT result was obtained with $\hat{\Omega}^{(8)}_{\rm
eff}$ in place of $\hat{\Omega}^{(9)}_{\rm eff}$ and ${\rm
Im}\lambda_{t}=(1.35)\times 10^{-4}$ instead of our ${\rm
Im}\lambda_{t}=(1.45)\times 10^{-4}$.

The case of QCDPs is a different story. Here the LQCD value overshoots the
DQCD one by more than a factor of two and consequently despite the agreement
on EWP contribution the result in (16) based on RBC-UKQCD hadronic matrix
elements differs by roughly a factor of three from my expectations for 2026 in
(10). The difference from the RBC-UKQCD result that does not include IB, QED
corrections and NNLO QCD effects in (13) is even by a factor of four. On the
other hand the ChPT estimate of $B_{6}^{(1/2)}$ being in the ballpark of
$1.35$ is at first sight in the ballpark of the LQCD value. However, such a
direct comparison is incorrect because the ChPT value of $B_{6}^{(1/2)}$
corresponds to much lower scales and as demonstrated in Fig. 2 of [79] the
inclusion of the meson evolution would make it significantly smaller, in the
ballpark of unity.

Let me next make a few additional critical remarks about ChPT estimate of
$\varepsilon^{\prime}/\varepsilon$:

  * •

First of all one should realize that strictly speaking
$\varepsilon^{\prime}/\varepsilon$ cannot be calculated in ChPT by itself
because several important contributions in this framework depend on low-energy
constants which have to be taken from LQCD calculations or low energy data.
While in the case of semi-leptonic decays this procedure is rather successful,
in the case of $K\to\pi\pi$ it encounters serious problems which I doubt will
be solved in the coming years.

  * •

In particular the parameters $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ are evaluated
in the strict large $N$ limit which as we stressed corresponds to the scales
much lower than the scales at which Wilson coefficients can be calculated. But
ChPT by itself does not have meson evolution and consequently the matching
with Wilson coefficients is practically impossible. The authors of [81] admit
this stating by themselves that the dominant error in their estimate of
$\varepsilon^{\prime}/\varepsilon$ originates from their ignorance about this
matching. The low-energy constant which they have to know to overcome this
difficulty is $L_{5}$. Its value obtained from LQCD is still rather uncertain
implying large error in (12). However, this error could be an underestimate
for the following reasons.

  * •

The expression for $\varepsilon^{\prime}/\varepsilon$ in terms of low-energy
constants presented by these authors is obtained in the large $N$ limit. While
the numerical value for $L_{5}$ from LQCD certainly includes some $1/N$
corrections, there could still be some missing $1/N$ contributions both in the
formula for $\varepsilon^{\prime}/\varepsilon$ used by them and also in the
extraction of $L_{5}$ in case they would try to find its value from some low-
energy data.

  * •

Equally problematic is the inclusion of the singlet $\eta_{0}$ and of the
$\eta-\eta^{\prime}$ mixing which has been known for more than 30 years to
suppress $\varepsilon^{\prime}/\varepsilon$ significantly [15, 16, 89]. This
effect is buried again in a poorly known low energy constant, this time
$L_{7}$. If the authors of [81] would use the NIB of (15) they would end up,
even without the meson evolution, with the value of
$\varepsilon^{\prime}/\varepsilon$ in the ballpark of $(9-11)\times 10^{-4}$.

  * •

There is one point where ChPT could be superior to DQCD. These are FSI.
However, at least in the case of EWP contribution to
$\varepsilon^{\prime}/\varepsilon$, its strong suppression through FSI
predicted by ChPT experts has not been confirmed by RBC-UKQCD which obtains
the result for $B_{8}^{(3/2)}$ in agreement with DQCD. Here presently these
effects are not included and the modest suppression of $B_{8}^{(3/2)}$ below
unity in DQCD is due to the meson evolution.

Next let me move to make several comments on the most recent analysis of the
RBC-UKQCD collaboration. I am doing it not only because all my recent analyses
of $\varepsilon^{\prime}/\varepsilon$ [79, 45, 90, 1, 89] have been simply
ignored by this collaboration. In particular I want to list the arguments
while I still expect the final result for $\varepsilon^{\prime}/\varepsilon$
in the SM to be close to my 2026 expectations and certainly by a factor of at
least two below the present RBC-UKQCD value in (13).

While one should admire the RBC-UKQCD collaboration for their heroic efforts
over at least one decade to calculate $\varepsilon^{\prime}/\varepsilon$, as
they emphasize from first principles, here are my main problems with accepting
their result in (13) despite the large error they admit.

  * •

The inclusion of isospin breaking corrections as a symmetric error to their
value without these corrections. This is like stating that these corrections
could in principle enhance $\varepsilon^{\prime}/\varepsilon$ implying thereby
an anomaly in this ratio that would require NP to suppress this ratio to agree
with data. Equivalently, it amounts to question all the work done over more
than thirty years by different authors, with different methods that imply
significant suppression of $\varepsilon^{\prime}/\varepsilon$ by these
corrections, in particular by the presence of the $\eta_{0}$ and related
$\eta-\eta^{\prime}$ mixing. This is evident by comparing the value in (13)
with (16) which use the same hadronic matrix elements from RBC-UKQCD but in
(16) also isospin breaking corrections are included. While it could be
legitimate to use LQCD for the calculation of all effects, talking to various
colleagues who are closer to LQCD than me it is likely that we will not see a
value for $\varepsilon^{\prime}/\varepsilon$ from LQCD including isospin
corrections, in particular those from $\eta-\eta^{\prime}$ mixing, before
2026. I do hope very much that these expectations are wrong and RBC-UKQCD
collaboration or other LQCD groups will surprize us again by calculating this
time these corrections with respectable precision.

  * •

The absence of GIM mechanism, the crucial property of the SM and of a great
relevance for kaon physics. RBC-UKQCD works at $4\,{\rm GeV}$ without the
inclusion of charm. While this omission is more important for the QCDP
contributions to the $\Delta I=1/2$ rule (see below), because in the case of
$\varepsilon^{\prime}/\varepsilon$ GIM mechanism is broken already at higher
scales by the disparity of top quark and charm quark masses, it is to be
expected that the inclusion of charm will play a role also for
$\varepsilon^{\prime}/\varepsilon$. I expect that this issue will be solved
before a satisfactory inclusion of isospin breaking effects by LQCD in
general.

  * •

Matching of the lattice renormalization scheme to the $\overline{\rm MS}$
scheme used for the calculation of Wilson coefficients. In order to reduce the
errors in the matching, that is known presently at the one-loop level, RBC-
UKQCD works at $4\,{\rm GeV}$, which without charm is problematic but indeed
could help in improving the accuracy of the matching because of the smaller
value of $\alpha_{s}$ at this scale than at $\mu=1.5\,{\rm GeV}$ used by them
in 2015. On the other hand the problem with the omission of charm at this
lower scale is smaller.

Finally, I am looking forward to more accurate LQCD results on
$\varepsilon^{\prime}/\varepsilon$ from Japan [91]. This could help in
resolving the controversy described above.

##  7 The Present Status of the $\Delta I=1/2$ Rule

I cannot resist to add a few lines about the $\Delta I=1/2$ rule [92, 93]
after RBC-UKQCD indicated in [10] that they should be credited for the
identification of the dynamics behind this rule. This is disappointing,
because in their first paper [43] they were more careful about the history of
this rule as are the authors of [94, 95]. This history is summarized in our
2014 paper [73] and in particular in Section 7.2.3 of my recent book [41].
From this it is evident that the credit for the identification of the basic
dynamics behind this rule should go to the authors of [9] who demonstrated
that the current-current operators and not QCDP operators777The fact that
QCDPs cannot be important for the $\Delta I=1/2$ rule has also been noticed by
the authors of [96]. as claimed by the Russian masters [6] are dominantly
responsible for this rule. While the short-distance contributions analyzed in
[7, 8] provided only a small enhancement of the $K\to\pi\pi$ isospin amplitude
$A_{0}$ over $A_{2}$ one, the continuation of this quark evolution by meson
evolution in the non-perturbative region down to very low energy scales within
DQCD approach allowed to obtain the enhancement of the ratio ${{\rm
Re}A_{0}}/{{\rm Re}A_{2}}$ up to $16\pm 2$ from $\sqrt{2}$, in the absence of
QCD dynamics, compared with the experimental value of $22.4$. While a number
of authors suggested different solutions that were published in the 1990s [97,
98, 55, 99, 100, 66, 101, 102, 103], the recent result from the RBC-UKQCD
collaboration seems to confirm our findings of 1986 although I have the
impression that they cannot see it doing purely numerical work. In itself it
is a very important result and the RBC-UKQCD collaboration should be
congratulated for it. Also recent dissection of the $\Delta I=1/2$ rule at
large $N$ by other LQCD experts [94, 95] shows that in this decade we should
know whether NP plays any role in this rule. Here also the studies from [91]
will be important.

As stated above it is dominantly the meson evolution responsible for the
enhancement of ${{\rm Re}A_{0}}$ and the suppression of ${{\rm Re}A_{2}}$ with
respect to the values in which QCD interactions are switched off by going to
the large $N$ limit888Only operator $Q_{2}$ contributes in this limit and the
mesons are non-interacting in this limit. See [73, 41] for a detailed
presentation.:

${\rm Re}A_{0}=3.59\times 10^{-8}\,{\rm GeV},\quad{\rm Re}A_{2}=2.54\times
10^{-8}\,{\rm GeV}~{},\quad\frac{{\rm Re}A_{0}}{{\rm Re}A_{2}}=\sqrt{2},$ (20)

in plain disagreement with the experimental value of 22.4. It should be
emphasized that the explanation of the missing enhancement factor of $15.8$
through some dynamics must simultaneously give the correct values for ${\rm
Re}A_{0}$ and ${\rm Re}A_{2}$ [104]:

${\rm Re}A_{0}=27.04(1)\times 10^{-8}~{}\,{\rm GeV}.\quad\quad{\rm
Re}A_{2}=1.210(2)\times 10^{-8}~{}\,{\rm GeV}.$ (21)

This means that these dynamics should suppress ${\rm Re}A_{2}$ by a factor of
$2.1$, not more, and enhance ${\rm Re}A_{0}$ by a factor of $7.5$. This tells
us that while the suppression of ${\rm Re}A_{2}$ is an important ingredient in
the $\Delta I=1/2$ rule, it is not the main origin of this rule. It is the
enhancement of ${\rm Re}A_{0}$, as already emphasized in [6], even if, in
contrast to this paper, as demonstrated in [9], the current-current operators
are responsible dominantly for this rule and not QCD penguins. More details
can be found in our papers and in my book.

I am making this point because the RBC-UKQCD collaboration in their papers and
talks stressed more the suppression of ${\rm Re}A_{2}$ and not the enhancement
of ${\rm Re}A_{0}$ as the major dynamics behind this rule. The simple
discussion above shows that this is simply not true. But working numerically
at $4\,{\rm GeV}$ and being not able to switch-off QCD interactions in LQCD
one simply cannot see properly what is really going on. Yet, as demonstrated
by Jean-Marc and myself in [105], in the context of BSM $K^{0}-\bar{K}^{0}$
matrix elements, meson evolution allows to explain analytically the values
obtained by various lattice collaborations [106, 107, 108, 109, 110]. Without
the meson evolution one would fail to explain some of these matrix elements by
factors of three. Therefore we are convinced that this is also the case of
$K\to\pi\pi$, that is of the $\Delta I=1/2$ rule and of
$\varepsilon^{\prime}/\varepsilon$. In other words meson evolution is bound to
be present in RBC-UKQCD calculations and its effects will hopefully be more
visible in $\varepsilon^{\prime}/\varepsilon$ in the next round of
calculations by this collaboration.

This is also supported by the comparison of LQCD with DQCD in this context in
Section 9 of [73] where we use the language of contractions used by LQCD
experts. As already noticed in [43] the dominant two contractions work
constructively to enhance ${\rm Re}A_{0}$ and destructively to suppress ${\rm
Re}A_{2}$, but at $4\,{\rm GeV}$ this only describes in the lattice language
the property of asymptotic freedom that has been found in corresponding Wilson
coefficients by Altarelli and Maiani [7] and Gaillard and Lee [8] by now 45
years ago. This in itself is an important result because it indicates that a
proper matching of hadronic matrix elements from LQCD with the Wilson
coefficients is possible.

However, in order to understand why the contribution of the sum of these
contractions $2C_{1}+C_{2}$ to ${\rm Re}A_{0}$999As seen in equation (122) of
[73] and also in [43] there is a second constructive contribution
$C_{1}+2C_{2}$ but with a much smaller coefficient multiplying it, it is
subleading. is by a factor of 22 larger than their difference ($C_{1}-C_{2}$)
entering ${\rm Re}A_{2}$, one has to understand physically the dynamics behind
the values of $C_{1}$ and $C_{2}$. But these dynamics must be below $1\,{\rm
GeV}$, not at $4\,{\rm GeV}$. Otherwise the authors of [7, 8] working above
$1\,{\rm GeV}$ would identify it 45 years ago, but they did not. Thus, while
DQCD working below $1\,{\rm GeV}$ can identify these dynamics as the meson
evolution, LQCD, working at $4\,{\rm GeV}$ can only see its implications in
their numerical values which summarize all the contributions below $4\,{\rm
GeV}$.

Yet, we agree with CHPT experts that a part of the ${{\rm Re}A_{0}}$
enhancement over ${{\rm Re}A_{2}}$ comes from FSI, although they were not the
first to point it out. To my knowledge these were nuclear physicists [111]
motivated by our first results in the 1980s.

Finally, let me emphasize the following problem in the present calculation of
the $\Delta I=1/2$ rule by the RBC-UKQCD collaboration despite the impressive
result they obtained for the ratio in question

$\frac{{\rm Re}A_{0}}{{\rm Re}A_{2}}=19.9(2.3)(4.4),\qquad{\rm RBC-
UKQCD~{}~{}(2020)}$ (22)

that is consistent with the DQCD value of $16\pm 2$ and is in agreement with
the experimental value $22.4$.

The inspection of various contributions to ${{\rm Re}A_{0}}$ in the RBC-UKQCD
result reveals the surprizing fact that at the scale of $4\,{\rm GeV}$ the
QCDP contribution, known to be governed due to GIM mechanism by the
$m_{c}-m_{u}$ difference, not only amounts to $10\%$ of ${{\rm Re}A_{0}}$ at
this scale but even suppresses this amplitude thereby working against the
$\Delta I=1/2$ rule. This contribution would be absent in the presence of
charm contributions due to GIM mechanism shifting the result in (22) towards
the experimental value. But in the presence of charm other contributions could
modify this result and we have to wait until next round of LQCD calculations
which hopefully will include GIM mechanism. I am well aware of the fact that
QCDP contribution to ${{\rm Re}A_{0}}$ is scale dependent and increases with
lowering the scale but at $4\,{\rm GeV}$ it should be negligible. From DQCD
estimates at $1\,{\rm GeV}$ it could amount to a $10-15\%$ effect but then it
would slightly enhance and not suppress ${{\rm Re}A_{0}}$.

I do hope very much that the RBC-UKQCD collaboration appreciates my detailed
analysis of their results101010Partly presented already in [1, 89]. despite my
reservations and that my critical remarks will motivate them to provide next
time much improved results for both $\varepsilon^{\prime}/\varepsilon$ and the
$\Delta I=1/2$ rule with all important effects taken into account and with
much smaller errors. There is no doubt that from present perspective LQCD will
one day give us most precise values for the $\Delta I=1/2$ rule and
$\varepsilon^{\prime}/\varepsilon$ within the SM, hopefully revealing some NP
contributions to both. Yet, this could still take a decade of tedious
calculations. I expect that other LQCD collaborations like the ones in [94,
91] will make similar efforts.

In summary, in my view the status of the $\Delta I=1/2$ rule as of 2020 is as
follows:

  * •

The dominant dynamics behind this rule is our beloved QCD. It is simply the
quark evolution from $M_{W}$ down to scale $\mathcal{O}(1\,{\rm GeV})$ as
analysed first by Altarelli and Maiani [7] and Gaillard and Lee [8], followed
by the meson evolution down to very low scales at which QCD becomes a theory
of weakly interacting mesons and free theory of mesons in the strict large $N$
limit [112, 113, 114, 115]. This non-perturbative evolution within the Dual
QCD approach dominates by far the enhancement of ${{\rm Re}A_{0}}$ over ${{\rm
Re}A_{2}}$ as demonstrated by Bardeen, Gérard and myself in [9, 73].

  * •

This picture appears to be confirmed in particular by the RBC-UKQCD
collaboration [43, 10] when one takes the insight from [105] into account, but
also other LQCD collaborations [94, 91] made significant progress here. Very
importantly, from present perspective only LQCD can provide satisfactory
estimate of the room left for new physics in this rule. Most likely it is at
most at the level of $20\%$. A detailed analysis in [116] shows that heavy
neutral coloured gauge bosons $G^{\prime}$ but not $Z^{\prime}$ could provide
such contributions while satisfying all existing constraints.

##  8 A Strategy for the Coming Years

Evidently, there is no question about that the situation with
$\varepsilon^{\prime}/\varepsilon$ is very unclear at present. Personally, I
am truly delighted that my expectations for $\varepsilon^{\prime}/\varepsilon$
in the SM, based on the collaboration with Jean-Marc, are very different from
the ones of RBC-UKQCD and of the ChPT experts. There are two reasons for this.
First of all if we all agreed that the SM agrees with data on
$\varepsilon^{\prime}/\varepsilon$, the future of
$\varepsilon^{\prime}/\varepsilon$ would be rather boring. Equally important,
if one day my expectations in (10) will be confirmed by several LQCD
collaborations, it will be evident who should be credited for the
identification of the $\varepsilon^{\prime}/\varepsilon$-anomaly.

However, even ChPT practitioners and RBC-UKQCD collaboration, who strongly
disagree with Jean-Marc’s and my claims about
$\varepsilon^{\prime}/\varepsilon$, cannot exclude that at a certain level NP
will be required to fit its experimental value. Yet, analyzing NP models
containing new parameters for various values of $B_{6}^{(1/2)}$ and
$B_{8}^{(3/2)}$ complicates the search for NP by much. Here comes one idea
which in my view could give us a clue which NP models could have a chance to
explain possible anomaly dependent on its size [117].

Instead, of varying $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ we can just write

$\frac{\varepsilon^{\prime}}{\varepsilon}=\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)^{\rm
SM}+\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)^{\rm BSM}\,$ (23)

and assume that NP provides a shift in $\varepsilon^{\prime}/\varepsilon$:

$\left(\frac{\varepsilon^{\prime}}{\varepsilon}\right)^{\rm
BSM}=\kappa_{\varepsilon^{\prime}}\cdot
10^{-3},\qquad-0.5\leq\kappa_{\varepsilon^{\prime}}\leq 1.0,$ (24)

with the range for $\kappa_{\varepsilon^{\prime}}$ indicating conservatively
the room left for BSM contributions. This range is dictated by the recent
analyses in [89, 1]. Personally, I would vary $\kappa_{\varepsilon^{\prime}}$
only in the range $0.5\leq\kappa_{\varepsilon^{\prime}}\leq 1.5$ but this
would mean ignoring the results from ChPT and RBC-UKQCD which I do not want to
do today. We are fortunate that there is no interference between these two
contributions although the Wilson coefficients of SM operators can be affected
by NP. The corresponding modifications are included in the BSM term.

Now, in a given NP model $\varepsilon^{\prime}/\varepsilon$ is correlated with
other observables, in particular those in the K-meson decays like
$K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$, $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$,
$K_{L}\rightarrow\pi^{0}\mu^{+}\mu^{-}$ and $K_{S}\rightarrow\mu^{+}\mu^{-}$.
As a consequence one can study the dependence of the corresponding branching
ratios as functions of $\kappa_{\varepsilon^{\prime}}$ which depends on the
model considered. We refer to numerous plots of such dependences in [117,
118].

One can now ask what is the uncertainty in BSM contributions due to hadronic
matrix elements. Here comes a good news. It turns out that in most NP models
considered until now the dominant shift in $\varepsilon^{\prime}/\varepsilon$
comes from the modifications of the EWP contributions because similar to
isospin breaking effects they are enhanced by a factor ${{\rm Re}A_{0}/{\rm
Re}A_{2}}$ relative to those of QCDPs. But $B_{8}^{(3/2)}$ is already well
known so that the main uncertainty in NP contributions comes in this case from
new parameters in BSM models which can be constrained by other processes.

Yet, there are also contributions from BSM operators, in particular scalar and
tensor operators. Their matrix elements have only been calculated within DQCD
[86]. But the master formula for all BSM scenarios presented in [85, 84]
demonstrates very clearly the dominance of the $\Delta I=3/2$ contributions
over the $\Delta I=1/2$ ones also in this case. While the computation of the
matrix elements of these new operators within LQCD has still to be done, it is
expected that the uncertainties in the dominant $\Delta I=3/2$ contributions
will be smaller than the present uncertainty in the matrix element of the QCDP
operator $Q_{6}$.

##  9 Final Remarks

Our $\varepsilon^{\prime}/\varepsilon$-story approaches the end. I have
concentrated here on the non-perturbative calculations because no consensus
has been reached among theorists until now. However, being privileged to be
one of two111111The second is Guido Martinelli in the context of the
contributions of chromomagnetic penguins to $\varepsilon^{\prime}/\varepsilon$
[87]. theorists who calculated both short distance and long distance
contributions to $\varepsilon^{\prime}/\varepsilon$, I want to emphasize that
without NLO calculations of Wilson coefficients performed in the early 1990s
in Munich [33, 34, 35, 36] and Rome [37, 38], the uncertainties in the
prediction for $\varepsilon^{\prime}/\varepsilon$ would be even larger. In
particular without these corrections the matching of Wilson coefficients to
hadronic matrix elements performed by LQCD would not be possible. The result
would be simply renormalization scheme dependent. The story of these
calculations is described in [40] and in this context I want to thank Guido
Martinelli and his strong team for a very friendly competition we had.

The uncertainties in various steps leading to (16) should still be
significantly decreased in the coming years and I do hope very much that by
2026 the picture of $\varepsilon^{\prime}/\varepsilon$ with respect to
possible NP contributions will be much clearer than it is today. In particular
the identification of new sources of CP violation in the data of NA48 and KTeV
collaborations would be very important because they could play in principle a
role in the explanation of our existence. I really have no idea whether NP in
$\varepsilon^{\prime}/\varepsilon$, if found, would be responsible for our
existence. Not only because this is presently beyond my skills but also
because we did not yet identify what this NP could be, although several ideas
have been put forward. They are listed in Table 1. My bet would be a heavy
$Z^{\prime}$ and/or vector-like quarks but not leptoquarks.

My recollection of the $\varepsilon^{\prime}/\varepsilon$ efforts dealt
dominantly with theory. Yet, without the measurements of
$\varepsilon^{\prime}/\varepsilon$ by NA48 and KTeV collaborations, all these
discussions between RBC-UKQCD, ChPT and DQCD experts would be much less
exciting and we should thank these two important experimental groups for the
result in (2).

##  10 Acknowledgements

Over 37 years I had 39 collaborators with whom I have written papers on
$\varepsilon^{\prime}/\varepsilon$ but Jean-Marc Gérard is the one with whom I
have written the highest number of $\varepsilon^{\prime}/\varepsilon$-papers
and in fact among the male physicists the one with whom I have written the
highest number of papers to date. In particular this includes papers on the
$\Delta I=1/2$ rule, and large $N$ calculations in the context of the Dual QCD
approach, 20 journal papers in total. Only Monika Blanke and Jennifer
Girrbach-Noe can compete with him in this respect. The important virtue of
this collaboration was that we had to struggle for 35 years against the
spanish matadors, of three physics generations by now, Eduardo de Rafael,
Antonio Pich and Hector Gisbert. In this context we were declared to be naive,
both as far as final state interactions and isospin breaking corrections to
$\varepsilon^{\prime}/\varepsilon$ are concerned, which clearly united us. But
fortunately the collected joint number of citations (2656) demonstrates that
our work has not been ignored by the community and there is no doubt that this
number will increase in the future. I want to thank Jean-Marc for this great
and most pleasant collaboration and in particular for his deep insight into
low-energy QCD from which I benefited in many ways. Thanks go also to those
with whom I performed many analyses of $\varepsilon^{\prime}/\varepsilon$
within the SM and in various NP scenarios. Several of their names appeared in
the reference list to this writing. The remaining ones can be found in
INSPIRE.

Finally, I would like to thank the Max-Planck Institute for Physics, the
Physics Department at TUM, the TUM Institute for Advanced Study, DFG (German
Research Foundation), BMFT (Federal Ministry of Education and Research), the
Alexander von Humboldt Foundation, European Research Council and two
Excellence Clusters (2006-2019), founded by the DFG under Germany’s Excellence
Strategy, for the financial support over 37 years in the research presented
here. For the coming years the main support for my research will come from the
TUM-IAS and the Alexander von Humboldt Foundation. But also support from the
Excellence Cluster ORIGINS – EXC-2094 – 390783311 is highly appreciated.

##  Appendix A Operators and New Physics Analyses

We list the operators mentioned in the text:

Current–Current:

$Q_{1}=(\bar{s}_{\alpha}u_{\beta})_{V-A}\;(\bar{u}_{\beta}d_{\alpha})_{V-A},~{}~{}~{}~{}~{}~{}Q_{2}=(\bar{s}u)_{V-A}\;(\bar{u}d)_{V-A},$
(25)

QCD Penguins:

$Q_{3}=(\bar{s}d)_{V-A}\\!\\!\sum_{q=u,d,s,c,b}(\bar{q}q)_{V-A},~{}~{}~{}~{}~{}Q_{4}=(\bar{s}_{\alpha}d_{\beta})_{V-A}\\!\\!\sum_{q=u,d,s,c,b}(\bar{q}_{\beta}q_{\alpha})_{V-A},$
(26)
$Q_{5}=(\bar{s}d)_{V-A}\\!\\!\sum_{q=u,d,s,c,b}(\bar{q}q)_{V+A},~{}~{}~{}~{}~{}Q_{6}=(\bar{s}_{\alpha}d_{\beta})_{V-A}\\!\\!\sum_{q=u,d,s,c,b}(\bar{q}_{\beta}q_{\alpha})_{V+A},$
(27)

Electroweak Penguins:

$Q_{7}=\frac{3}{2}\,(\bar{s}d)_{V-A}\\!\\!\sum_{q=u,d,s,c,b}e_{q}\,(\bar{q}q)_{V+A},~{}~{}~{}~{}~{}Q_{8}=\frac{3}{2}\,(\bar{s}_{\alpha}d_{\beta})_{V-A}\\!\\!\sum_{q=u,d,s,c,b}e_{q}\,(\bar{q}_{\beta}q_{\alpha})_{V+A},$
(28)
$Q_{9}=\frac{3}{2}\,(\bar{s}d)_{V-A}\\!\\!\sum_{q=u,d,s,c,b}e_{q}\,(\bar{q}q)_{V-A},~{}~{}~{}~{}~{}Q_{10}=\frac{3}{2}\,(\bar{s}_{\alpha}d_{\beta})_{V-A}\\!\\!\sum_{q=u,d,s,c,b}e_{q}\,(\bar{q}_{\beta}q_{\alpha})_{V-A}\,.$
(29)

Here, $\alpha,\beta$ denote colour indices and $e_{q}$ denotes the electric
quark charges reflecting the electroweak origin of $Q_{7},\ldots,Q_{10}$.
Finally,
$(\bar{s}d)_{V-A}\equiv\bar{s}_{\alpha}\gamma_{\mu}(1-\gamma_{5})d_{\alpha}$.

NP Scenario | References | Correlations with
---|---|---
LHT | [119] | $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
$Z$-FCNC | [117, 120, 121] |  $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
$Z^{\prime}$ | [117] |  $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ and $\Delta M_{K}$
Simplified Models | [122] | $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
331 Models | [123, 124] | $b\to s\ell^{+}\ell^{-}$
Vector-Like Quarks | [125] |  $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$ and $\Delta M_{K}$
Supersymmetry | [126, 127, 128, 129, 130] |  $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
2HDM | [131, 132] |  $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$ and $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
Right-handed Currents | [133, 134] | EDMs
Left-Right Symmetry | [135, 136] | EDMs
Leptoquarks | [83] | all rare Kaon decays
SMEFT | [84] | several processes
SU(8) | [137] |  $b\to s\ell^{+}\ell^{-}$, $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
Diquarks | [138, 139] |  $\varepsilon_{K}$, $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
3HDM + $\nu_{R}$ | [140] |  $R(K^{(*)})$, $R(D^{(*)})$
Vectorlike compositeness | [141] |  $R(K^{(*)})$, $R(D^{(*)})$, $\varepsilon_{K}$, $K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$, $K_{L}\rightarrow\pi^{0}\nu\bar{\nu}$
$\text{U(2)}^{3}$ flavour symmetry | [142] | hadronic $B\to K\pi$, $B_{s,d}\to(KK,\pi\pi)$, $B_{s}\to\phi(\rho^{0},\pi^{0})$

Table 1: Papers studying implications of a possible
$\varepsilon^{\prime}/\varepsilon$ anomaly.

## References

  * [1] J. Aebischer, C. Bobeth, and A. J. Buras, $\varepsilon^{\prime}/\varepsilon$ in the Standard Model at the Dawn of the 2020s, Eur. Phys. J. C 80 (2020), no. 8 705, [arXiv:2005.05978].
  * [2] NA48 Collaboration, J. Batley et al., A Precision measurement of direct CP violation in the decay of neutral kaons into two pions, Phys. Lett. B544 (2002) 97–112, [hep-ex/0208009].
  * [3] KTeV Collaboration, A. Alavi-Harati et al., Measurements of direct CP violation, CPT symmetry, and other parameters in the neutral kaon system, Phys. Rev. D67 (2003) 012005, [hep-ex/0208007].
  * [4] KTeV Collaboration, E. Abouzaid et al., Precise Measurements of Direct CP Violation, CPT Symmetry, and Other Parameters in the Neutral Kaon System, Phys. Rev. D83 (2011) 092001, [arXiv:1011.0127].
  * [5] J. R. Ellis, M. K. Gaillard, and D. V. Nanopoulos, Lefthanded Currents and CP Violation, Nucl. Phys. B109 (1976) 213–243.
  * [6] M. A. Shifman, A. Vainshtein, and V. I. Zakharov, Light Quarks and the Origin of the $\Delta I=1/2$ Rule in the Nonleptonic Decays of Strange Particles, Nucl. Phys. B120 (1977) 316.
  * [7] G. Altarelli and L. Maiani, Octet Enhancement of Nonleptonic Weak Interactions in Asymptotically Free Gauge Theories, Phys. Lett. B52 (1974) 351–354.
  * [8] M. Gaillard and B. W. Lee, $\Delta I=1/2$ Rule for Nonleptonic Decays in Asymptotically Free Field Theories, Phys. Rev. Lett. 33 (1974) 108.
  * [9] W. A. Bardeen, A. J. Buras, and J.-M. Gérard, A Consistent Analysis of the $\Delta I=1/2$ Rule for K Decays, Phys. Lett. B192 (1987) 138.
  * [10] RBC, UKQCD Collaboration, R. Abbott et al., Direct CP violation and the $\Delta I=1/2$ rule in $K\to\pi\pi$ decay from the standard model, Phys. Rev. D 102 (2020), no. 5 054509, [arXiv:2004.09440].
  * [11] F. J. Gilman and M. B. Wise, The $\Delta I=1/2$ Rule and Violation of CP in the Six Quark Model, Phys. Lett. 83B (1979) 83–86.
  * [12] B. Guberina and R. D. Peccei, Quantum Chromodynamic Effects and CP Violation in the Kobayashi-Maskawa Model, Nucl. Phys. B163 (1980) 289–311.
  * [13] F. J. Gilman and M. B. Wise, Effective Hamiltonian for $\Delta S=1$ Weak Nonleptonic Decays in the Six Quark Model, Phys. Rev. D20 (1979) 2392.
  * [14] A. J. Buras, W. Slominski, and H. Steger, B Meson Decay, CP Violation, Mixing Angles and the Top Quark Mass, Nucl. Phys. B238 (1984) 529–560.
  * [15] J. F. Donoghue, E. Golowich, B. R. Holstein, and J. Trampetic, Electromagnetic and Isospin Breaking Effects Decrease $\varepsilon^{\prime}/\varepsilon$, Phys. Lett. B179 (1986) 361. [Erratum: Phys. Lett.B188,511(1987)].
  * [16] A. J. Buras and J. M. Gérard, Isospin Breaking Contributions to $\varepsilon^{\prime}/\varepsilon$, Phys. Lett. B192 (1987) 156.
  * [17] J. Bijnens and M. B. Wise, Electromagnetic Contribution to $\varepsilon^{\prime}/\varepsilon$, Phys. Lett. B137 (1984) 245–250.
  * [18] S. R. Sharpe, On the Contribution of Electromagnetic Penguins to $\epsilon^{\prime}$, Phys. Lett. B 194 (1987) 551–556.
  * [19] M. Lusignoli, Electromagnetic Corrections to the Effective Hamiltonian for Strangeness Changing Decays and $\epsilon^{\prime}/\epsilon$, Nucl. Phys. B 325 (1989) 33–61.
  * [20] W. A. Bardeen, A. J. Buras, and J.-M. Gérard, The $\Delta I=1/2$ Rule in the Large $N$ Limit, Phys. Lett. B180 (1986) 133.
  * [21] W. A. Bardeen, A. J. Buras, and J.-M. Gérard, The $K\to\pi\pi$ Decays in the Large N Limit: Quark Evolution, Nucl. Phys. B293 (1987) 787\.
  * [22] A. J. Buras and J. Gérard, $\varepsilon^{\prime}/\varepsilon$ in the Standard Model, Phys. Lett. B203 (1988) 272.
  * [23] C. Dib, I. Dunietz, and F. J. Gilman, CP Violation in the $K_{L}\to\pi^{0}\ell^{+}\ell^{-}$ Decay Amplitude for Large $m_{t}$, Phys. Lett. B218 (1989) 487–492.
  * [24] J. Flynn and L. Randall, The CP Violating Contribution to the Decay $K_{L}\to\pi^{0}e^{+}e^{-}$, Nucl. Phys. B326 (1989) 31. [Erratum: Nucl. Phys.B334,580(1990)].
  * [25] A. J. Buras and P. H. Weisz, QCD Nonleading Corrections to Weak Decays in Dimensional Regularization and ’t Hooft-Veltman Schemes, Nucl. Phys. B333 (1990) 66–99.
  * [26] J. M. Flynn and L. Randall, The Electromagnetic Penguin Contribution to $\varepsilon^{\prime}/\varepsilon$ for Large Top Quark Mass, Phys. Lett. B224 (1989) 221.
  * [27] G. Buchalla, A. J. Buras, and M. K. Harlander, The Anatomy of $\varepsilon^{\prime}/\varepsilon$ in the Standard Model, Nucl. Phys. B337 (1990) 313–362.
  * [28] E. A. Paschos and Y. L. Wu, Correlations between $\varepsilon^{\prime}/\varepsilon$ and heavy top, Mod. Phys. Lett. A6 (1991) 93–106.
  * [29] M. Lusignoli, L. Maiani, G. Martinelli, and L. Reina, Mixing and CP violation in K and B mesons: A Lattice QCD point of view, Nucl. Phys. B369 (1992) 139–170.
  * [30] NA31 Collaboration, H. Burkhardt et al., First Evidence for Direct CP Violation, Phys. Lett. B206 (1988) 169–176.
  * [31] NA31 Collaboration, G. D. Barr et al., A New measurement of direct CP violation in the neutral kaon system, Phys. Lett. B317 (1993) 233–242.
  * [32] L. K. Gibbons et al., Measurement of the CP violation parameter Re($\epsilon^{\prime}/\epsilon$), Phys. Rev. Lett. 70 (1993) 1203–1206.
  * [33] A. J. Buras, M. Jamin, M. E. Lautenbacher, and P. H. Weisz, Effective Hamiltonians for $\Delta S=1$ and $\Delta B=1$ nonleptonic decays beyond the leading logarithmic approximation, Nucl. Phys. B370 (1992) 69–104. [Addendum: Nucl. Phys.B375,501(1992)].
  * [34] A. J. Buras, M. Jamin, M. E. Lautenbacher, and P. H. Weisz, Two loop anomalous dimension matrix for $\Delta S=1$ weak nonleptonic decays. 1. $\mathcal{O}(\alpha_{s}^{2})$, Nucl. Phys. B400 (1993) 37–74, [hep-ph/9211304].
  * [35] A. J. Buras, M. Jamin, and M. E. Lautenbacher, Two loop anomalous dimension matrix for $\Delta S=1$ weak nonleptonic decays. 2. $\mathcal{O}(\alpha\alpha_{s})$, Nucl. Phys. B400 (1993) 75–102, [hep-ph/9211321].
  * [36] A. J. Buras, M. Jamin, and M. E. Lautenbacher, The Anatomy of $\varepsilon^{\prime}/\varepsilon$ beyond leading logarithms with improved hadronic matrix elements, Nucl. Phys. B408 (1993) 209–285, [hep-ph/9303284].
  * [37] M. Ciuchini, E. Franco, G. Martinelli, and L. Reina, $\varepsilon^{\prime}/\varepsilon$ at the Next-to-leading order in QCD and QED, Phys. Lett. B301 (1993) 263–271, [hep-ph/9212203].
  * [38] M. Ciuchini, E. Franco, G. Martinelli, and L. Reina, The $\Delta S=1$ effective Hamiltonian including next-to-leading order QCD and QED corrections, Nucl. Phys. B415 (1994) 403–462, [hep-ph/9304257].
  * [39] G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125–1144, [hep-ph/9512380].
  * [40] A. J. Buras, Climbing NLO and NNLO Summits of Weak Decays, arXiv:1102.5650.
  * [41] A. J. Buras, Gauge Theory of Weak Decays.  Cambridge University Press, 6, 2020.
  * [42] A. J. Buras and J.-M. Gérard, $1/N$ Expansion for Kaons, Nucl. Phys. B264 (1986) 371.
  * [43] RBC, UKQCD Collaboration, P. Boyle et al., Emerging understanding of the $\Delta I=1/2$ Rule from Lattice QCD, Phys. Rev. Lett. 110 (2013), no. 15 152001, [arXiv:1212.1474].
  * [44] A. J. Buras, M. Gorbahn, S. Jäger, and M. Jamin, Improved anatomy of $\varepsilon^{\prime}/\varepsilon$ in the Standard Model, JHEP 11 (2015) 202, [arXiv:1507.06345].
  * [45] J. Aebischer, C. Bobeth, and A. J. Buras, On the importance of NNLO QCD and isospin-breaking corrections in $\varepsilon^{\prime}/\varepsilon$, Eur. Phys. J. C80 (2020), no. 1 1, [arXiv:1909.05610].
  * [46] A. J. Buras, M. Jamin, and M. E. Lautenbacher, A 1996 analysis of the CP violating ratio $\varepsilon^{\prime}/\varepsilon$, Phys. Lett. B389 (1996) 749–756, [hep-ph/9608365].
  * [47] A. J. Buras and M. Jamin, $\varepsilon^{\prime}/\varepsilon$ at the NLO: 10 years later, JHEP 01 (2004) 048, [hep-ph/0306217].
  * [48] M. Ciuchini, E. Franco, G. Martinelli, L. Reina, and L. Silvestrini, An Upgraded analysis of $\varepsilon^{\prime}/\varepsilon$ at the next-to-leading order, Z. Phys. C68 (1995) 239–256, [hep-ph/9501265].
  * [49] A. J. Buras, P. Gambino, and U. A. Haisch, Electroweak penguin contributions to non-leptonic $\Delta F=1$ decays at NNLO, Nucl. Phys. B570 (2000) 117–154, [hep-ph/9911250].
  * [50] M. Cerdá-Sevilla, M. Gorbahn, S. Jäger, and A. Kokulu, Towards NNLO accuracy for $\varepsilon^{\prime}/\varepsilon$, J. Phys. Conf. Ser. 800 (2017), no. 1 012008, [arXiv:1611.08276].
  * [51] M. Cerdà-Sevilla, NNLO QCD Contributions to $\varepsilon^{\prime}/\varepsilon$, Acta Phys. Polon. B49 (2018) 1087–1096.
  * [52] M. Gorbahn and U. Haisch, Effective Hamiltonian for non-leptonic $|\Delta F|=1$ decays at NNLO in QCD, Nucl. Phys. B713 (2005) 291–332, [hep-ph/0411071].
  * [53] S. Bosch et al., Standard model confronting new results for $\varepsilon^{\prime}/\varepsilon$, Nucl. Phys. B565 (2000) 3–37, [hep-ph/9904408].
  * [54] A. J. Buras, P. Gambino, M. Gorbahn, S. Jäger, and L. Silvestrini, $\varepsilon^{\prime}/\varepsilon$ and rare $K$ and $B$ decays in the MSSM, Nucl. Phys. B592 (2001) 55–91, [hep-ph/0007313].
  * [55] V. Antonelli, S. Bertolini, M. Fabbrichesi, and E. I. Lashin, The $\Delta I=1/2$ selection rule, Nucl. Phys. B469 (1996) 181–201, [hep-ph/9511341].
  * [56] S. Bertolini, J. O. Eeg, and M. Fabbrichesi, A New estimate of $\varepsilon^{\prime}/\varepsilon$, Nucl. Phys. B476 (1996) 225–254, [hep-ph/9512356].
  * [57] E. Pallante and A. Pich, Strong enhancement of $\varepsilon^{\prime}/\varepsilon$ through final state interactions, Phys. Rev. Lett. 84 (2000) 2568–2571, [hep-ph/9911233].
  * [58] E. Pallante and A. Pich, Final state interactions in kaon decays, Nucl. Phys. B592 (2001) 294–320, [hep-ph/0007208].
  * [59] M. Büchler, G. Colangelo, J. Kambor, and F. Orellana, A Note on the dispersive treatment of $K\to\pi\pi$ with the kaon off-shell, Phys. Lett. B521 (2001) 29–32, [hep-ph/0102289].
  * [60] M. Büchler, G. Colangelo, J. Kambor, and F. Orellana, Dispersion relations and soft pion theorems for $K\to\pi\pi$, Phys. Lett. B521 (2001) 22–28, [hep-ph/0102287].
  * [61] E. Pallante, A. Pich, and I. Scimemi, The Standard model prediction for $\varepsilon^{\prime}/\varepsilon$, Nucl. Phys. B617 (2001) 441–474, [hep-ph/0105011].
  * [62] A. J. Buras et al., Final state interactions and $\varepsilon^{\prime}/\varepsilon$: A critical look, Phys. Lett. B480 (2000) 80–86, [hep-ph/0002116].
  * [63] V. Cirigliano, A. Pich, G. Ecker, and H. Neufeld, Isospin violation in $\epsilon^{\prime}$, Phys. Rev. Lett. 91 (2003) 162001, [hep-ph/0307030].
  * [64] V. Cirigliano, G. Ecker, H. Neufeld, and A. Pich, Isospin breaking in $K\to\pi\pi$ decays, Eur. Phys. J. C33 (2004) 369–396, [hep-ph/0310351].
  * [65] J. Bijnens and F. Borg, Isospin breaking in $K\to 3\pi$ decays III: Bremsstrahlung and fit to experiment, Eur. Phys. J. C40 (2005) 383–394, [hep-ph/0501163].
  * [66] S. Bertolini, M. Fabbrichesi, and J. O. Eeg, Theory of the CP violating parameter $\epsilon^{\prime}/\epsilon$, Rev. Mod. Phys. 72 (2000) 65–93, [hep-ph/9802405].
  * [67] V. Cirigliano, G. Ecker, H. Neufeld, A. Pich, and J. Portoles, Kaon Decays in the Standard Model, Rev. Mod. Phys. 84 (2012) 399, [arXiv:1107.6001].
  * [68] S. Bertolini, J. O. Eeg, A. Maiezza, and F. Nesti, New physics in $\epsilon^{\prime}$ from gluomagnetic contributions and limits on Left-Right symmetry, Phys. Rev. D86 (2012) 095013, [arXiv:1206.0668].
  * [69] RBC, UKQCD Collaboration, Z. Bai et al., Standard Model Prediction for Direct CP Violation in $K\to\pi\pi$ Decay, Phys. Rev. Lett. 115 (2015), no. 21 212001, [arXiv:1505.07863].
  * [70] T. Blum et al., $K\rightarrow\pi\pi$ $\Delta I=3/2$ decay amplitude in the continuum limit, Phys. Rev. D91 (2015), no. 7 074502, [arXiv:1502.00263].
  * [71] T. Kitahara, U. Nierste, and P. Tremper, Singularity-free next-to-leading order $\Delta$S = 1 renormalization group evolution and $\epsilon_{K}^{\prime}/\epsilon_{K}$ in the Standard Model and beyond, JHEP 12 (2016) 078, [arXiv:1607.06727].
  * [72] A. J. Buras and J.-M. Gérard, Upper Bounds on $\varepsilon^{\prime}/\varepsilon$ Parameters $B_{6}^{(1/2)}$ and $B_{8}^{(3/2)}$ from Large N QCD and other News, JHEP 12 (2015) 008, [arXiv:1507.06326].
  * [73] A. J. Buras, J.-M. Gérard, and W. A. Bardeen, Large $N$ Approach to Kaon Decays and Mixing 28 Years Later: $\Delta I=1/2$ Rule, $\hat{B}_{K}$ and $\Delta M_{K}$, Eur. Phys. J. C74 (2014), no. 5 2871, [arXiv:1401.1385].
  * [74] G. Colangelo, J. Gasser, and H. Leutwyler, $\pi\pi$ scattering, Nucl. Phys. B603 (2001) 125–179, [hep-ph/0103088].
  * [75] H. Gisbert and A. Pich, Direct CP violation in $K^{0}\to\pi\pi$: Standard Model Status, Rept. Prog. Phys. 81 (2018), no. 7 076201, [arXiv:1712.06147].
  * [76] H. Gisbert and A. Pich, Updated Standard Model Prediction for $\varepsilon^{\prime}/\varepsilon$, Nucl. Part. Phys. Proc. 300-302 (2018) 137–144, [arXiv:1810.04904].
  * [77] H. Gisbert, Current status of $\varepsilon^{\prime}/\varepsilon$ in the Standard Model, PoS Confinement2018 (2018) 178, [arXiv:1811.12206].
  * [78] A. J. Buras and J.-M. Gérard, Final state interactions in $K\rightarrow\pi\pi$ decays: $\Delta I=1/2$ rule vs. $\varepsilon^{\prime}/\varepsilon$, Eur. Phys. J. C77 (2017), no. 1 10, [arXiv:1603.05686].
  * [79] A. J. Buras, $\epsilon^{\prime}/\epsilon$-2018: A Christmas Story, arXiv:1812.06102.
  * [80] V. Cirigliano, H. Gisbert, A. Pich, and A. Rodríguez-Sánchez, Isospin-violating contributions to $\epsilon^{\prime}/\epsilon$, JHEP 02 (2020) 032, [arXiv:1911.01359].
  * [81] V. Cirigliano, H. Gisbert, A. Pich, and A. Rodríguez-Sánchez, Theoretical status of $\varepsilon^{\prime}/\varepsilon$, J. Phys. Conf. Ser. 1526 (2020) 012011, [arXiv:1912.04736].
  * [82] H. Gisbert, Nonleptonic $K\to 2\,\pi$ decay dynamics, 12, 2020.  arXiv:2012.09483.
  * [83] C. Bobeth and A. J. Buras, Leptoquarks meet $\varepsilon^{\prime}/\varepsilon$ and rare Kaon processes, JHEP 02 (2018) 101, [arXiv:1712.01295].
  * [84] J. Aebischer, C. Bobeth, A. J. Buras, and D. M. Straub, Anatomy of $\varepsilon^{\prime}/\varepsilon$ beyond the Standard Model, Eur. Phys. J. C79 (2019), no. 3 219, [arXiv:1808.00466].
  * [85] J. Aebischer, C. Bobeth, A. J. Buras, J.-M. Gérard, and D. M. Straub, Master formula for $\varepsilon^{\prime}/\varepsilon$ beyond the Standard Model, Phys. Lett. B792 (2019) 465–469, [arXiv:1807.02520].
  * [86] J. Aebischer, A. J. Buras, and J.-M. Gérard, BSM Hadronic Matrix Elements for $\varepsilon^{\prime}/\varepsilon$ and $K\to\pi\pi$ Decays in the Dual QCD Approach, JHEP 02 (2019) 021, [arXiv:1807.01709].
  * [87] ETM Collaboration, M. Constantinou, M. Costa, R. Frezzotti, V. Lubicz, G. Martinelli, D. Meloni, H. Panagopoulos, and S. Simula, $K\to\pi$ matrix elements of the chromomagnetic operator on the lattice, Phys. Rev. D97 (2018), no. 7 074501, [arXiv:1712.09824].
  * [88] A. J. Buras and J.-M. Gérard, $K\to\pi\pi$ and $K-\pi$ Matrix Elements of the Chromomagnetic Operators from Dual QCD, JHEP 07 (2018) 126, [arXiv:1803.08052].
  * [89] A. J. Buras and J.-M. Gérard, Isospin-breaking in $\varepsilon^{\prime}/\varepsilon$: impact of $\eta_{0}$ at the dawn of the 2020s, Eur. Phys. J. C 80 (2020), no. 8 701, [arXiv:2005.08976].
  * [90] A. J. Buras, The Optimal Strategy for $\varepsilon^{\prime}/\varepsilon$ in the SM: 2019, J. Phys. Conf. Ser. 1526 (2020) 012019, [arXiv:1912.12306].
  * [91] N. Ishizuka, K.-I. Ishikawa, A. Ukawa, and T. Yoshié, Calculation of $K\to\pi\pi$ decay amplitudes with improved Wilson fermion action in non-zero momentum frame in lattice QCD, Phys. Rev. D 98 (2018), no. 11 114512, [arXiv:1809.03893].
  * [92] M. Gell-Mann and A. Pais, Behavior of neutral particles under charge conjugation, Phys. Rev. 97 (1955) 1387–1389.
  * [93] M. Gell-Mann and A. Rosenfeld, Hyperons and heavy mesons (systematics and decay), Ann.Rev.Nucl.Part.Sci. 7 (1957) 407–478.
  * [94] A. Donini, P. Hernández, C. Pena, and F. Romero-López, Dissecting the $\Delta I=1/2$ rule at large $N_{c}$, Eur. Phys. J. C 80 (2020), no. 7 638, [arXiv:2003.10293].
  * [95] P. Hernández and F. Romero-López, The Large $N_{c}$ limit of QCD on the lattice, 12, 2020.  arXiv:2012.03331.
  * [96] R. S. Chivukula, J. Flynn, and H. Georgi, Polychromatic Penguins Don’t Fly, Phys.Lett. B171 (1986) 453–458.
  * [97] A. Pich and E. de Rafael, Four quark operators and nonleptonic weak transitions, Nucl. Phys. B 358 (1991) 311–382.
  * [98] M. Jamin and A. Pich, QCD corrections to inclusive Delta S = 1,2 transitions at the next-to-leading order, Nucl. Phys. B 425 (1994) 15–38, [hep-ph/9402363].
  * [99] A. Pich and E. de Rafael, Weak K amplitudes in the chiral and $1/N$ expansions, Phys.Lett. B374 (1996) 186–192, [hep-ph/9511465].
  * [100] S. Bertolini, J. Eeg, M. Fabbrichesi, and E. Lashin, The $\Delta I=1/2$ rule and B(K) at $\mathcal{O}(p^{4})$ in the chiral expansion, Nucl. Phys. B 514 (1998) 63–92, [hep-ph/9705244].
  * [101] J. Bijnens and J. Prades, The $\Delta I=1/2$ rule in the chiral limit, JHEP 01 (1999) 023, [hep-ph/9811472].
  * [102] T. Hambye, G. Kohler, E. Paschos, P. Soldan, and W. A. Bardeen, $1/N$ corrections to the hadronic matrix elements of $Q_{6}$ and $Q_{8}$ in $K\to\pi\pi$ decays, Phys. Rev. D58 (1998) 014017, [hep-ph/9802300].
  * [103] R. Crewther and L. C. Tunstall, $\Delta I=1/2$ rule for kaon decays derived from QCD infrared fixed point, Phys. Rev. D 91 (2015), no. 3 034016, [arXiv:1312.3319].
  * [104] Particle Data Group Collaboration, M. Tanabashi et al., Review of Particle Physics, Phys. Rev. D98 (2018), no. 3 030001.
  * [105] A. J. Buras and J.-M. Gérard, Dual QCD Insight into BSM Hadronic Matrix Elements for $K^{0}-\bar{K}^{0}$ Mixing from Lattice QCD, Acta Phys. Polon. B50 (2019) 121, [arXiv:1804.02401].
  * [106] ETM Collaboration, N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz, G. C. Rossi, S. Simula, and C. Tarantino, $\Delta S=2$ and $\Delta C=2$ bag parameters in the standard model and beyond from Nf=2+1+1 twisted-mass lattice QCD, Phys. Rev. D92 (2015), no. 3 034516, [arXiv:1505.06639].
  * [107] SWME Collaboration, B. J. Choi et al., Kaon BSM B-parameters using improved staggered fermions from $N_{f}=2+1$ unquenched QCD, Phys. Rev. D93 (2016), no. 1 014511, [arXiv:1509.00592].
  * [108] RBC/UKQCD Collaboration, N. Garron, R. J. Hudspith, and A. T. Lytle, Neutral Kaon Mixing Beyond the Standard Model with $n_{f}=2+1$ Chiral Fermions Part 1: Bare Matrix Elements and Physical Results, JHEP 11 (2016) 001, [arXiv:1609.03334].
  * [109] RBC, UKQCD Collaboration, P. A. Boyle, N. Garron, R. J. Hudspith, C. Lehner, and A. T. Lytle, Neutral kaon mixing beyond the Standard Model with nf = 2 + 1 chiral fermions. Part 2: non perturbative renormalisation of the $\Delta F=2$ four-quark operators, JHEP 10 (2017) 054, [arXiv:1708.03552].
  * [110] P. Boyle, N. Garron, J. Kettle, A. Khamseh, and J. T. Tsang, BSM Kaon Mixing at the Physical Point, EPJ Web Conf. 175 (2018) 13010, [arXiv:1710.09176].
  * [111] G. Brown, J. Durso, M. Johnson, and J. Speth, Final state interactions in K meson decays, Phys. Lett. B 238 (1990) 20–24.
  * [112] G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B72 (1974) 461.
  * [113] G. ’t Hooft, A Two-Dimensional Model for Mesons, Nucl. Phys. B75 (1974) 461.
  * [114] E. Witten, Baryons in the $1/N$ Expansion, Nucl. Phys. B160 (1979) 57.
  * [115] S. B. Treiman, E. Witten, R. Jackiw, and B. Zumino, Current Algebra and Anomalies.  1986\.
  * [116] A. J. Buras, F. De Fazio, and J. Girrbach, $\Delta I=1/2$ rule, $\varepsilon^{\prime}/\varepsilon$ and $K\rightarrow\pi\nu\bar{\nu}$ in $Z^{\prime}(Z)$ and $G^{\prime}$ models with FCNC quark couplings, Eur. Phys. J. C74 (2014) 2950, [arXiv:1404.3824].
  * [117] A. J. Buras, New physics patterns in $\varepsilon^{\prime}/\varepsilon$ and $\varepsilon_{K}$ with implications for rare kaon decays and $\Delta M_{K}$, JHEP 04 (2016) 071, [arXiv:1601.00005].
  * [118] J. Aebischer, A. J. Buras, and J. Kumar, Another SMEFT Story: $Z^{\prime}$ Facing New Results on $\varepsilon^{\prime}/\varepsilon$, $\Delta M_{K}$ and $K\to\pi\nu\bar{\nu}$, JHEP 12 (2020) 097, [arXiv:2006.01138].
  * [119] M. Blanke, A. J. Buras, and S. Recksiegel, Quark flavour observables in the Littlest Higgs model with T-parity after LHC Run 1, Eur. Phys. J. C76 (2016), no. 4 182, [arXiv:1507.06316].
  * [120] C. Bobeth, A. J. Buras, A. Celis, and M. Jung, Yukawa enhancement of $Z$-mediated new physics in $\Delta S=2$ and $\Delta B=2$ processes, JHEP 07 (2017) 124, [arXiv:1703.04753].
  * [121] M. Endo, T. Kitahara, S. Mishima, and K. Yamamoto, Revisiting Kaon Physics in General $Z$ Scenario, Phys. Lett. B771 (2017) 37–44, [arXiv:1612.08839].
  * [122] A. J. Buras, D. Buttazzo, and R. Knegjens, $K\to\pi\nu\bar{\nu}$ and $\epsilon^{\prime}/\epsilon$ in Simplified New Physics Models, JHEP 11 (2015) 166, [arXiv:1507.08672].
  * [123] A. J. Buras and F. De Fazio, $\varepsilon^{\prime}/\varepsilon$ in 331 Models, JHEP 03 (2016) 010, [arXiv:1512.02869].
  * [124] A. J. Buras and F. De Fazio, 331 Models Facing the Tensions in $\Delta F=2$ Processes with the Impact on $\varepsilon^{\prime}/\varepsilon$, $B_{s}\to\mu^{+}\mu^{-}$ and $B\to K^{*}\mu^{+}\mu^{-}$, JHEP 08 (2016) 115, [arXiv:1604.02344].
  * [125] C. Bobeth, A. J. Buras, A. Celis, and M. Jung, Patterns of Flavour Violation in Models with Vector-Like Quarks, JHEP 04 (2017) 079, [arXiv:1609.04783].
  * [126] M. Tanimoto and K. Yamamoto, Probing SUSY with 10 TeV stop mass in rare decays and CP violation of kaon, PTEP 2016 (2016), no. 12 123B02, [arXiv:1603.07960].
  * [127] T. Kitahara, U. Nierste, and P. Tremper, Supersymmetric Explanation of CP Violation in $K\to\pi\pi$ Decays, Phys. Rev. Lett. 117 (2016), no. 9 091802, [arXiv:1604.07400].
  * [128] M. Endo, S. Mishima, D. Ueda, and K. Yamamoto, Chargino contributions in light of recent $\epsilon^{\prime}/\epsilon$, Phys. Lett. B762 (2016) 493–497, [arXiv:1608.01444].
  * [129] A. Crivellin, G. D’Ambrosio, T. Kitahara, and U. Nierste, $K\to\pi\nu\overline{\nu}$ in the MSSM in light of the $\epsilon^{\prime}_{K}/\epsilon_{K}$ anomaly, Phys. Rev. D96 (2017), no. 1 015023, [arXiv:1703.05786].
  * [130] M. Endo, T. Goto, T. Kitahara, S. Mishima, D. Ueda, and K. Yamamoto, Gluino-mediated electroweak penguin with flavor-violating trilinear couplings, JHEP 04 (2018) 019, [arXiv:1712.04959].
  * [131] C.-H. Chen and T. Nomura, $Re(\epsilon^{\prime}_{K}/\epsilon_{K}$) and $K\to\pi\nu\bar{\nu}$ in a two-Higgs doublet model, JHEP 08 (2018) 145, [arXiv:1804.06017].
  * [132] C.-H. Chen and T. Nomura, $\epsilon^{\prime}/\epsilon$ from charged-Higgs-induced gluonic dipole operators, Phys. Lett. B787 (2018) 182–187, [arXiv:1805.07522].
  * [133] V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, An $\varepsilon^{\prime}$ improvement from right-handed currents, Phys. Lett. B767 (2017) 1–9, [arXiv:1612.03914].
  * [134] S. Alioli, V. Cirigliano, W. Dekens, J. de Vries, and E. Mereghetti, Right-handed charged currents in the era of the Large Hadron Collider, JHEP 05 (2017) 086, [arXiv:1703.04751].
  * [135] N. Haba, H. Umeeda, and T. Yamada, $\epsilon^{\prime}/\epsilon$ Anomaly and Neutron EDM in $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ model with Charge Symmetry, JHEP 05 (2018) 052, [arXiv:1802.09903].
  * [136] N. Haba, H. Umeeda, and T. Yamada, Direct CP Violation in Cabibbo-Favored Charmed Meson Decays and $\epsilon^{\prime}/\epsilon$ in $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ Model, JHEP 10 (2018) 006, [arXiv:1806.03424].
  * [137] S. Matsuzaki, K. Nishiwaki, and K. Yamamoto, Simultaneous interpretation of $K$ and $B$ anomalies in terms of chiral-flavorful vectors, JHEP 11 (2018) 164, [arXiv:1806.02312].
  * [138] C.-H. Chen and T. Nomura, $\epsilon_{K}$ and $\epsilon^{\prime}/\epsilon$ in a diquark model, JHEP 03 (2019) 009, [arXiv:1808.04097].
  * [139] C.-H. Chen and T. Nomura, Left-handed color-sextet diquark in the Kaon system, Phys. Rev. D99 (2019), no. 11 115006, [arXiv:1811.02315].
  * [140] C. Marzo, L. Marzola, and M. Raidal, Common explanation to the $R_{K^{(*)}}$, $R_{D^{(*)}}$ and $\epsilon^{\prime}/\epsilon$ anomalies in a 3HDM+$\nu_{R}$ and connections to neutrino physics, Phys. Rev. D100 (2019), no. 5 055031, [arXiv:1901.08290].
  * [141] S. Matsuzaki, K. Nishiwaki, and K. Yamamoto, Simultaneous explanation of $K$ and $B$ anomalies in vectorlike compositeness, in 18th Hellenic School and Workshops on Elementary Particle Physics and Gravity (CORFU2018) Corfu, Corfu, Greece, August 31-September 28, 2018, 2019.  arXiv:1903.10823.
  * [142] A. Crivellin, C. Gross, S. Pokorski, and L. Vernazza, Correlating $\epsilon^{\prime}/\epsilon$ to hadronic $B$ decays via $U(2)^{3}$ flavour symmetry, Phys. Rev. D 101 (2020), no. 1 015022, [arXiv:1909.02101].