Search

Sterile neutrinos are well-motivated beyond the Standard Model (BSM) particles. The Standard Model Effective Field Theory (SMEFT) augmented with these new fields is known as the $\nu$SMEFT. We present the first code for solving the renormalization group equations (RGEs) of the $\nu$SMEFT in an automated way. For this purpose, we have implemented the $\nu$SMEFT as a new effective field theory (EFT) in the Wilson coefficient exchange format WCxf. Furthermore, we included anomalous dimensions depending on the gauge couplings and Yukawas in the python package wilson. This novel version of wilson allows a consistent inclusion of $\nu$SMEFT renormalization group (RG) running effects above the electroweak (EW) scale in phenomenological studies involving sterile neutrinos. Moreover, this new release allows us to study EW, strong, and Yukawa running effects separately within the SMEFT., Comment: 14 pages

This article presents a source list to support departments in creating more equitable, diverse, and inclusive mathematics classrooms. Included are relevant general-interest books, as well as articles and books presenting research results about how matters of inequity, exclusion, and homogeneity surface in educational contexts. We also present sources more specifically focused on research about the past and present of relationships between mathematics and gender, (dis)ability, race, and class, as well as articles proposing approaches and case studies in mathematics classrooms.

A novel method to treat effects from evanescent operators in next-to-leading order (NLO) computations is introduced. The approach allows, besides further simplifications, to discard evanescent-to-physical mixing contributions in NLO calculations. The method is independent of the treatments of $\gamma_5$ and can therefore be combined with different renormalization schemes. We illustrate the utility of this result by reproducing literature results of two-loop anomalous dimension matrices for both $|\Delta F| = 1$ and $|\Delta F| = 2$ transitions., Comment: 31 pages, 3 figures, explanations and appendix added, matches published version

One-loop Fierz identities are discussed, together with general basis transformations in Effective Field theories at the tree- and one-loop level. To this end, the notion of one-loop shifts is introduced, together with several examples that illustrate the virtues of this method., Comment: 4 pages, Proceedings of the International Workshop Beyond Standard Model: From Theory to Experiment (BSM- 2023), Hurghada, Egypt, 6-9 November 2023

In recent years, theoretical and phenomenological studies with effective field theories have become a trending and prolific line of research in the field of high-energy physics. In order to discuss present and future prospects concerning automated tools in this field, the SMEFT-Tools 2022 workshop was held at the University of Zurich from 14th-16th September 2022. The current document collects and summarizes the content of this workshop., Comment: 116 pages, 10 figures. Based on the contributions to SMEFT-Tools 2022 (https://indico.icc.ub.edu/event/128/). v2: version published in EPJC

We present a proof of the factorization of renormalization scheme in one-loop-corrected Fierz identities. This scheme factorization facilitates the simultaneous transformation of operator basis and renormalization scheme using only relations between physical operators; the evanescent operators in the respective bases may be chosen entirely independently of each other. The relations between evanescent operators in the two bases is automatically accounted for by the corrected Fierz identities. We illustrate the utility of this result with a two-loop anomalous dimension matrix computation using the Naive-Dimensional Regularization scheme, which is then transformed via one-loop Fierz identities to the known result in the literature given in a different basis and calculated in the Larin scheme. Additionally, we reproduce results from the literature of basis transformations involving the rotation of evanescent operators into the physical basis using our method, without the need to explicitly compute one-loop matrix elements of evanescent operators., Comment: 28 pages, 2 figures Updated to match published version

Despite the observation of significant suppression of $b\to s\mu^+\mu^-$ branching ratios no clear sign of New Physics (NP) has been identified in $\Delta F=2$ observables, in particular to $\varepsilon_K$. Assuming negligible NP contributions to these observables allows us to determine CKM parameters without being involved in the tensions between inclusive and exclusive determinations of $V_{cb}$ and $V_{ub}$. Furthermore, this method avoids the impact of NP on the determination of these parameters present likely in global fits. Simultaneously it provides SM predictions for numerous rare $K$ and $B$ branching ratios that are most accurate to date. Analyzing this scenario within $Z^\prime$ models we point out, following the 2009 observations of Monika Blanke and ours of 2020, that despite the absence of NP contributions to $\varepsilon_K$, significant NP contributions to several rare Kaon decays, $\varepsilon'/\varepsilon$ and $\Delta M_K$ can be present. In the simplest scenario, this is guaranteed by a single non-vanishing imaginary left-handed $Z^\prime$ coupling $g^L_{sd}$. This scenario implies very stringent correlations between the Kaon observables considered by us. In particular, the identification of NP in any of these observables implies automatically NP contributions to the remaining ones. A characteristic feature of this scenario is a strict correlation between $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$ branching ratios on a branch parallel to the Grossman-Nir bound. Moreover, $\Delta M_K$ is automatically suppressed as seems to be required by recent lattice QCD results., Comment: 14 Pages, 3 Figures

The present status of Kaon physics is summarized from a theory point of view. Focus is laid on $\Delta S=2$, $\Delta S=1$ as well as on rare Kaon decays, where progress has been made for instance in the case of $\varepsilon_K$ and $K_S\to \mu^+\mu^-$. Furthermore, several New Physics scenarios are discussed in the context of rare Kaon decays and other Kaon observables., Comment: 6 pages, Contribution to the Proceedings for the International Conference on Kaon Physics 2022 (KAON2022)

This note gives an overview of the tools for the precision matching of ultraviolet theories to the Standard Model effective field theory (SMEFT) at the tree level and one loop. Several semi- and fully automated codes are presented, as well as some supplementary codes for the basis conversion and the subsequent running and matching at low energies. A suggestion to collect information for cross-validations of current and future codes is made., Comment: 23 pages, 2 figures

We study the contribution from dipole operators to one-loop Fierz identities and provide the resulting QCD and QED shifts to the tree-level relations for all four-fermion operators. The results simplify one-loop basis changes as well as matching computations and allow one to consistently eliminate operators from an operator basis which give rise to complications, e.g. traces involving $\gamma_5$., Comment: 6 pages, 2 figures, 5 tables. Updated to match published version

In light of new data we present an updated phenomenological analysis of the simplified $U_1$-leptoquark model addressing charged-current $B$-meson anomalies. The analysis shows a good compatibility of low-energy data (dominated by the lepton flavor universality ratios $R_D$ and $R_{D^*}$) with the high-energy constraints posed by $pp\to \tau\bar\tau$ Drell-Yan data. We also show that present data are well compatible with a framework where the leptoquark couples with similar strength to both left- and right-handed third-generation fermions, a scenario that is well-motivated from a model building perspective. We find that the high-energy implications of this setup will be probed at the 95% confidence level in the high-luminosity phase of the LHC., Comment: 8 pages, 1 appendix, 5 figures. Fixes a bug with the old HFLAV average of RD and RD* & added new RD* data. Expanded appendix, results unchanged

The Standard Model prediction of the $B_c$ lifetime is discussed, together with the dominant uncertainties and strategies on how to improve them. Furthermore, a new method to compute the $B_c$ lifetime based on the operator product expansion is proposed. It relies on differences of $B,\,D$ and $B_c$ meson decay rates, in which the free-quark contributions cancel out, reducing the uncertainty of the theory prediction., Comment: 6 pages, 1 figure, Contribution to the proceedings of the 11th International Workshop on the CKM Unitarity Triangle (CKM2021), 22-26 November 2021, Based on arxiv:2203.16196 and arxiv:2209.12520. arXiv admin note: substantial text overlap with arXiv:2203.16196

The Standard Model decay rate of the $B_c$ meson is discussed together with a novel approach that uses experimental data in combination with an operator product expansion. In the new method differences of $B,\,D$ and $B_c$ meson decay rates are considered for which the free-quark contributions drop out, leading to a reduction of the theory prediction., Comment: 5 pages, contribution to the proceedings of "The Tenth Annual Conference on Large Hadron Collider Physics" (LHCP2022). arXiv admin note: substantial text overlap with arXiv:2204.06225

Fierz transformations for four-fermion operators are generalized to the one-loop level. A general renormalization scheme is used to compute QCD and QED corrections to the tree-level relations, which result from Fierz-evanescent operators. The results can be used to perform general one-loop basis transformations involving four-fermi and evanescent operators. We illustrate the usefulness of our results by discussing two examples from a matching calculation and a one-loop basis change., Comment: 40 pages, 3 figures

The Standard Model decay rate of the $B_c$ meson is discussed together with a novel approach based on the usage of experimental data in combination with an operator product expansion. In the new method differences of $B,\,D$ and $B_c$ meson decay rates are considered, for which the free-quark contributions drop out, leading to a reduction of the theory prediction., Comment: 5 pages, Contribution to the proceedings of the 30th International Symposium on Lepton Photon Interactions at High Energies, hosted by the University of Manchester, 10-14 January 2022

The Standard Model prediction of the $B_c$ lifetime is discussed, together with the dominant uncertainties and strategies on how to improve them. Furthermore, a new method to compute the $B_c$ lifetime based on the operator product expansion is proposed. It relies on differences of $B,\,D$ and $B_c$ meson decay rates, in which the free-quark contributions cancel out, reducing the uncertainty of the theory prediction., Comment: 7 pages, 1 figure, Contribution to the proceedings of the Corfu Summer Institute 2021 "School and Workshops on Elementary Particle Physics and Gravity", 29 August - 9 October 2021, Corfu, Greece

We present for the first time NLO QCD Renormalization Group (RG) evolution matrices for non-leptonic $\Delta F=2$ transitions in the Standard Model Effective Field Theory (SMEFT). To this end we transform first the known two-loop QCD anomalous dimension matrices (ADMs) of the BSM operators in the so-called BMU basis into the ones in the common Weak Effective Theory (WET) basis (the so-called JMS basis) for which tree-level and one-loop matching to the SMEFT are already known. This allows us subsequently to find the two-loop QCD ADMs for the SMEFT non-leptonic $\Delta F=2$ operators in the Warsaw basis. Having all these ingredients we investigate the impact of these NLO QCD effects on the QCD RG evolution of SMEFT Wilson coefficients for non-leptonic $\Delta F=2$ transitions from the new physics scale $\Lambda$ down to the electroweak scale $\mu_\text{ew}$. The main benefit of these new contributions is that they allow to remove renormalization scheme dependences present both in the one-loop matchings between the WET and SMEFT and also between SMEFT and a chosen UV completion. But the NLO QCD effects, calculated here in the NDR scheme, turn out to be small, in the ballpark of a few percent but larger than one-loop Yukawa top effects when only the $\Delta F=2$ operators are considered. The technology developed in our paper allows to obtain the ADMs in the SMEFT from the ones of the BMU basis also for non-leptonic $\Delta F=1$ decays and the results of this more involved analysis will be presented soon in another publication., Comment: 15 pages, slight change in presentation, conclusions unchanged, matches published version

We stress the importance of precise measurements of rare decays $K^+\rightarrow\pi^+\nu\bar\nu$, $K_L\rightarrow\pi^0\nu\bar\nu$, $K_{L,S}\to\mu^+\mu^-$ and $K_{L,S}\to\pi^0\ell^+\ell^-$ for the search of new physics (NP). This includes both branching ratios and the distributions in $q^2$, the invariant mass-squared of the neutrino system in the case of $K^+\rightarrow\pi^+\nu\bar\nu$ and $K_L\rightarrow\pi^0\nu\bar\nu$ and of the $\ell^+\ell^-$ system in the case of the remaining decays. In particular the correlations between these observables and their correlations with the ratio $\varepsilon'/\varepsilon$ in $K_L\to\pi\pi$ decays, the CP-violating parameter $\varepsilon_K$ and the $K^0-\bar K^0$ mass difference $\Delta M_K$, should help to disentangle the nature of possible NP. We stress the strong sensitivity of all observables with the exception of $\Delta M_K$ to the CKM parameter $|V_{cb}|$ and list a number of $|V_{cb}|$-independent ratios within the SM which exhibit rather different dependences on the angles $\beta$ and $\gamma$ of the unitarity triangle. The particular role of these decays in probing very short distance scales far beyond the ones explored at the LHC is emphasized. In this context the role of the Standard Model Effective Field Theory (SMEFT) is very important. We also address briefly the issue of the footprints of Majorana neutrinos in $K^+\rightarrow\pi^+\nu\bar\nu$ and $K_L\rightarrow\pi^0\nu\bar\nu$., Comment: 36 pp, 10 figures, contribution to Snowmass 2021

A model-independent anatomy of $\Delta F= 2$ transitions in the context of the Weak Effective Theory (WET) below the electroweak scale (EW) and the Standard Model Effective Field Theory (SMEFT) above the EW scale is discussed. Two master formulae for the BSM contribution of the mixing amplitude $M_{12}$, in terms of Wilson coefficients are presented. The coefficients entering these formulae contain all the information below the EW scale and the NP scale $\Lambda$, respectively. The renormalization group evolution from the top-quark Yukawa coupling has the largest impact on the result. The obtained expressions depend on whether the down-basis or the up-basis for SMEFT operators is considered., Comment: 7 pages, 2 figures, Contribution to the proceedings of the 11th International Workshop on the CKM Unitarity Triangle (CKM2021), 22-26 November 2021

The Standard Model Effective Field Theory (SMEFT) and the Low Energy Effective Field Theory (LEFT) can be extended by adding additional spin 0, 1/2 and 1 dark matter particles which are singlets under the Standard Model (SM) gauge group. We classify all gauge invariant interactions in the Lagrangian up to terms of dimension six, and present the tree-level matching conditions between the two theories at the electroweak scale. The most widely studied dark matter models, such as those based on the Higgs portal or on kinetic mixing between the photon and a dark photon, are based on dimension-four interactions with the SM sector. We consider a model with dark vector particles with a $\mathbb{Z}_2$ symmetry, so that the lightest dark matter particle is stable. The leading interaction with the SM is through dimension-six operators involving two dark vector field-strength tensors and the electromagnetic field-strength tensor. This model is a viable dark matter model in the freeze-in scenario for a wide range of parameters., Comment: 50 pages, 3 figures, matches published version

Basis transformations often involve Fierz and other relations which are only valid in $D=4$ dimensions. In general $D$ space-time dimensions however, evanescent operators have to be introduced, in order to preserve such identities. Such evanescent operators contribute to one-loop basis transformations as well as to two-loop renormalization group running. We present a simple procedure on how to systematically change basis at the one-loop level by obtaining shifts due to evanescent operators. As an example we apply this method to derive the one-loop basis transformation from the BMU (Buras, Misiak and Urban) basis useful for NLO QCD calculations, to the JMS (Jenkins, Manohar and Stoffer) basis used in the matching to the SMEFT., Comment: 25 pages, 2 figures, v2 matches with the published version

Using an operator product expansion (OPE) approach an updated Standard Model prediction of the $B_c$ lifetime is presented. The computation in three different mass schemes for the heavy quarks leads to three different values consistent with each other and with experiment. Furthermore a novel way to compute the $B_c$ lifetime is presented, taking differences of $B,D$ and $B_c$ meson decay rates. In this approach the leading contributions from free-quark decays cancel out, leading to a reduction of scale and scheme dependence., Comment: 5 pages, Contribution to the proceedings of the 22nd Particles and Nuclei International Conference (PANIC21), 5-10 September 2021

An Effective Field Theory (EFT) analysis of the magnetic moment of the muon is discussed. The expression for the dipole moment is given in terms of operator coefficients of the low-energy effective field theory (LEFT) and the Standard Model effective field theory (SMEFT), where one-loop renormalization group improved perturbation theory, the one-loop matching from SMEFT onto LEFT as well as one-loop lepton matrix elements of the effective-theory operators were taken into account. Interestingly only a very limited set of the SMEFT operators is able to explain the current deviation of the magnetic moment of the muon from its Standard Model expectation., Comment: 6 pages, 1 figure. Contribution to the proceedings of the European Physical Society Conference on High Energy Physics (EPS-HEP2021), 26-30 July 2021

To reduce the current theory uncertainties a novel way to determine the $B_c$ lifetime, $\tau_{B_c}$, is proposed. Taking the difference of the $B_c$, and $B$ and $D$ meson decay rates eliminates the leading contributions from the calculation, which exhibit large scale and scheme dependence. The uncertainties in the proposed determination of $\tau_{B_c}$ are analyzed and improvements are proposed. The method predicts a value of $\tau_{B_c}$ in tension with the experimental determination. Several explanations are considered, including underestimation of uncertainties, duality violation and new physics. We discuss quantitative evidence that our method is vitiated by duality violation in the large mass OPE used to calculate nonleptonic decay rates of $B$ and $D$ mesons., Comment: 8 pages, 3 figures, matches published version

As an important step towards a complete next-to-leading order (NLO) QCD analysis of the ratio $\varepsilon'/\varepsilon$ within the Standard Model Effective Field Theory (SMEFT), we present for the first time the NLO master formula for the BSM part of this ratio expressed in terms of the Wilson coefficients of all contributing operators evaluated at the electroweak scale. To this end we use the common Weak Effective Theory (WET) basis (the so-called JMS basis) for which tree-level and one-loop matching to the SMEFT are already known. The relevant hadronic matrix elements of BSM operators at the electroweak scale are taken from Dual QCD approach and the SM ones from lattice QCD. It includes the renormalization group evolution and quark-flavour threshold effects at NLO in QCD from hadronic scales, at which these matrix elements have been calculated, to the electroweak scale., Comment: 18 pages, no figures; v2: clarifying remarks and extended results

We reconsider the complete set of four-quark operators in the Weak Effective Theory (WET) for non-leptonic $\Delta F=1$ decays that govern $s\to d$ and $b\to d, s$ transitions in the Standard Model (SM) and beyond, at the Next-to-Leading Order (NLO) in QCD. We discuss cases with different numbers $N_f$ of active flavours, intermediate threshold corrections, as well as the issue of transformations between operator bases beyond leading order to facilitate the matching to high-energy completions or the Standard Model Effective Field Theory (SMEFT) at the electroweak scale. As a first step towards a SMEFT NLO analysis of $K\to\pi\pi$ and non-leptonic $B$-meson decays, we calculate the relevant WET Wilson coefficients including two-loop contributions to their renormalization group running, and express them in terms of the Wilson coefficients in a particular operator basis for which the one-loop matching to SMEFT is already known., Comment: 36 pages, match with the published version in JHEP

Applying an operator product expansion approach we update the Standard Model prediction of the $B_c$ lifetime from over 20 years ago. The non-perturbative velocity expansion is carried out up to third order in the relative velocity of the heavy quarks. The scheme dependence is studied using three different mass schemes for the $\bar b$ and $c$ quarks, resulting in three different values consistent with each other and with experiment. Special focus has been laid on renormalon cancellation in the computation. Uncertainties resulting from scale dependence, neglecting the strange quark mass, non-perturbative matrix elements and parametric uncertainties are discussed in detail. The resulting uncertainties are still rather large compared to the experimental ones, and therefore do not allow for clear-cut conclusions concerning New Physics effects in the $B_c$ decay., Comment: 42 pages, 1 figure

We perform a model-independent analysis of the magnetic and electric dipole moments of the muon and electron. We give expressions for the dipole moments in terms of operator coefficients of the low-energy effective field theory (LEFT) and the Standard Model effective field theory (SMEFT). We use one-loop renormalization group improved perturbation theory, including the one-loop matching from SMEFT onto LEFT, and one-loop lepton matrix elements of the effective-theory operators. Semileptonic four-fermion operators involving light quarks give sizable non-perturbative contributions to the dipole moments, which are included in our analysis. We find that only a very limited set of the SMEFT operators is able to generate the current deviation of the magnetic moment of the muon from its Standard Model expectation., Comment: 34 pages, 7 figures; v2: numbers updated to new experimental value for g-2; version published in JHEP

We present a model-independent anatomy of the $\Delta F=2$ transitions $K^0-\bar K^0$, $B_{s,d}-\bar B_{s,d}$ and $D^0-\bar D^0$ in the context of the Standard Model Effective Field Theory (SMEFT). We present two master formulae for the mixing amplitude $\big[M_{12} \big]_\text{BSM}$. One in terms of the Wilson coefficients (WCs) of the Low-Energy Effective Theory (LEFT) operators evaluated at the electroweak scale $\mu_\text{ew}$ and one in terms of the WCs of the SMEFT operators evaluated at the BSM scale $\Lambda$. The coefficients $P_a^{ij}$ entering these formulae contain all the information below the scales $\mu_\text{ew}$ and $\Lambda$, respectively. Renormalization group effects from the top-quark Yukawa coupling play the most important role. The collection of the individual contributions of the SMEFT operators to $\big[M_{12}\big]_\text{BSM}$ can be considered as the SMEFT ATLAS of $\Delta F=2$ transitions and constitutes a travel guide to such transitions far beyond the scales explored by the LHC. We emphasize that this ATLAS depends on whether the down-basis or the up-basis for SMEFT operators is considered. We illustrate this technology with tree-level exchanges of heavy gauge bosons ($Z^\prime$, $G^\prime$) and corresponding heavy scalars., Comment: 76 pages, 13 figures

Recently the RBC-UKQCD lattice collaboration presented new results for the hadronic matrix elements relevant for the ratio $\varepsilon'/\varepsilon$ in the Standard Model (SM). With the present knowledge of the Wilson coefficients and isospin breaking effects there is still much room for new physics (NP) contributions to $\varepsilon'/\varepsilon$ which could both enhance or suppress this ratio to agree with the data. The new SM value for the $K^0-\bar K^0$ mass difference $\Delta M_K$ from RBC-UKQCD is on the other hand by $2\sigma$ above the data hinting for NP required to suppress $\Delta M_K$. Simultaneously the most recent results for $K^+\rightarrow\pi^+\nu\bar\nu$ from NA62 and for $K_{L}\rightarrow\pi^0\nu\bar\nu$ from KOTO still allow for significant NP contributions. We point out that the suppression of $\Delta M_K$ by NP requires the presence of new CP-violating phases with interesting implications for $K\to\pi\nu\bar\nu$, $K_S\to\mu^+\mu^-$ and $K_L\to\pi^0\ell^+\ell^-$ decays. Considering a $Z^\prime$-scenario within the SMEFT we analyze the dependence of all these observables on the size of NP still allowed by the data on $\varepsilon'/\varepsilon$. The NP QCD penguin scenario for $\varepsilon'/\varepsilon$ is excluded by SMEFT renormalization group effects in $\varepsilon_K$ so that NP effects in $\varepsilon'/\varepsilon$ are governed by electroweak penguins. We also investigate for the first time whether the presence of a heavy $Z^\prime$ with flavour violating couplings could generate through top Yukawa renormalization group effects FCNCs mediated by the SM $Z$-boson. The outcome turns out to be very interesting., Comment: 35 pages, 12 figures; v2: added references, results unchanged

We study Yukawa Renormalization Group (RG) running effects in the context of the Standard Model Effective Theory (SMEFT).The Yukawa running being flavour dependent leads to RG-induced off-diagonal entries, so that initially diagonal Yukawa matrices at the high scale have to be rediagonalized at the electroweak (EW) scale. Performing such flavour rotations can lead to flavour violating operators which differ from the ones obtained through SMEFT RG evolution. We show, that these flavour rotations can have a large impact on low-energy phenomenology. In order to demonstrate this effect, we compare the two sources of flavour violation numerically as well as analytically and study their influence on several examples of down-type flavour transitions. For this purpose we consider $B_s-\bar B_s$ mixing, $b\to s\gamma$, $b\to s \ell \ell$ as well as electroweak precision observables. We show that the rotation effect can be comparable or even larger than the contribution from pure RGE evolution of the Wilson coefficients., Comment: 25 pages and 11 figures

We reanalyse the ratio $\varepsilon'/\varepsilon$ in the Standard Model (SM) using most recent hadronic matrix elements from the RBC-UKQCD collaboration in combination with most important NNLO QCD corrections to electroweak penguin contributions and the isospin-breaking corrections. We illustrate the importance of the latter by using their latest estimate from chiral perturbation theory (ChPT) based on the $octet$ approximation for lowest-lying mesons and a very recent estimate in the $nonet$ scheme that takes into account the contribution of $\eta_0$. We find $(\varepsilon'/\varepsilon)^{(8)}_\text{SM} = (17.4 \pm 6.1) \times 10^{-4}$ and $(\varepsilon'/\varepsilon)^{(9)}_\text{SM} = (13.9 \pm 5.2) \times 10^{-4}$, respectively. Despite a very good agreement with the measured value $(\varepsilon'/\varepsilon)_\text{exp} = (16.6 \pm 2.3) \times 10^{-4}$, the large error in $(\varepsilon'/\varepsilon)_\text{SM}$ still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for $(\varepsilon'/\varepsilon)_\text{BSM}$ valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters $B_6^{(1/2)}(\mu)$ and $B_8^{(3/2)}(\mu)$ at the $\mu$-scales used by the RBC-UKQCD collaboration and at lower scales $\mathcal{O}(1\,\text{GeV})$ used by ChPT and DQCD. We present semi-analytic formulae for $(\varepsilon'/\varepsilon)_\text{SM}$ in terms of these parameters and $\hat{\Omega}_\text{eff}$ that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the $\mathcal{O}(\alpha_\text{em})$ contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at $\mathcal{O}(\alpha_\text{em})$ in the present analyses of $\varepsilon'/\varepsilon$., Comment: 26 pages, 1 figure; v2: Analysis extended to the new value of Omega. More details on the numerical analysis given. Semi-analytic formula slightly modified

We consider scenarios with a heavy $Z^\prime$ gauge boson with flavour non-universal quark and lepton couplings with the goal to illustrate how the cancellation of gauge anomalies generated by the additional U(1)$^\prime$ gauge symmetry would imply correlations between FCNC processes in the quark sector, in the lepton sector and most interestingly between quark flavour and lepton flavour violating processes. We present scenarios with only left-handed flavour-violating $Z^\prime$ couplings and those with also right-handed flavour-violating couplings. These are characterized by a small number of free parameters but in contrast to gauge anomaly cancellation in the Standard Model, that takes place separately within each generation, in our scenarios anomaly cancellation involves simultaneously quarks and leptons of all three generations. Our models involve, beyond the ordinary quarks and leptons, three heavy right-handed neutrinos. The models with only left-handed FCNCs of $Z^\prime$ involve beyond $g_{Z^\prime}$ and $M_{Z^\prime}$ two real parameters characterizing the charges of all fermions under the U(1)$^\prime$ gauge symmetry and the CKM and PMNS ones in the quark and lepton sectors, respectively. The models with the right-handed FCNCs of $Z^\prime$ involve few additional parameters. Imposing constraints from well measured $\Delta F=2$ observables we identify interesting correlations that involve e.g. $\varepsilon'/\varepsilon$, $B_{s,d}\to\mu^+\mu^-$, $B\to K(K^*) \ell^+\ell^-$, $K^+\rightarrow\pi^+\nu\bar\nu$, $K_{L}\rightarrow\pi^0\nu\bar\nu$ and purely lepton flavour violating decays like $\mu\to e\gamma$, $\mu\to 3 e$, $\tau\to 3\mu$ and $\mu-e$ conversion} among others. Also $(g-2)_{\mu,e}$ are considered. The impact of the experimental $\mu\to e\gamma$, $\mu\to 3 e$ and in particular $\mu-e$ conversion bounds on rare $K$ and $B$ decays is emphasized., Comment: 37 pages, 18 figures