From B-Meson Anomalies to Kaon Physics with Scalar Leptoquarks

TUM-HEP-1344-21

From B-meson anomalies to Kaon physics with scalar leptoquarks

David Marzoccaa, Sokratis Triﬁnopoulosa, Elena Venturinib

(a) INFN, Sezione di Trieste, SISSA, Via Bonomea 265, 34136, Trieste, Italy (b) Technische Universit¨atM¨unchen,Physik-Department, James-Franck-Straße 1, 85748 Garching, Germany

Abstract In this work we study possible connections between B-meson anomalies and Kaon physics observables in the context of combined solutions with the singlet and triplet scalar leptoquarks S1 and S3. By assuming a flavor structure for the leptoquark couplings dictated by a minimally broken U(2)5 flavor symmetry we can make a sharp connection between these two classes of observables. We find that bound on B(K+ → π+νν) from NA62 puts + − already some tension in the model, while the present limits on B(KL → µ µ ) and µ → e conversion in nuclei can be saturated. Relaxing instead the flavor assumption we study + + 0 + − what values for B(K → π νν), as well as for B(KL → π νν) and B(KL,S → µ µ ), are viable compatibly with all other phenomenological constraints. arXiv:2106.15630v1 [hep-ph] 29 Jun 2021

1 Contents

1 Introduction2

2 Setup 4

3 Scalar leptoquarks and U(2)5 ﬂavor symmetry5 3.1 Analysis and discussion ...... 7

4 General case with right-handed couplings 11

5 Conclusions 12

A Analysis of observables and pseudo-observables 17 0 + + A.1 KL → π νν and K → π νν ...... 17 0 A.2 KL(S) → µµ and KL → π `` ...... 19 A.3 LFV in Kaon decays ...... 21 A.4 Rare D-meson decays ...... 22 A.5 b → dµµ and b → dee decays ...... 24 A.6 µ–e conversion in nuclei ...... 25 A.7 Z boson couplings ...... 26 0 A.8 K /K ...... 28 A.9 Neutron EDM ...... 29

1 Introduction

The observed deviations from Standard Model (SM) predictions in semileptonic B-meson decays persist as some of the most significant experimental hints for the presence of possible New Physics (NP) beyond the SM at the TeV scale. One set of deviations regards Lepton Flavor Universality (LFU) ratios of charged-current semileptonic B decays between the third and lighter lepton families, R(D(∗)) = B(B → D(∗)τν)/B(B → D(∗)`ν) [1–11], with a combined significance of approximately 3σ [12]. Another set of deviations from the SM are observed in LFU ratios between second and first lepton families in (∗) + − (∗) + − neutral-current B decays, RK(∗) = B(B → K µ µ )/B(B → K e e )[13–17], as well + − ∗ + − as in Bs → µ µ , in angular observables of the B → K µ µ process as well as in branching ratios of other decay processes which involve the b → sµ+µ− transition [18–23]. The global significance of these deviations, obtained with very conservative estimates of SM uncertainties, is 3.9σ [24]. On the other hand, global fits show that along the preferred directions in effective field theory (EFT) space the pulls from the SM can be even up to 5 − 7σ [25–29]. When attempting to address at the same time both sets of anomalies, leptoquark (LQ) mediators are by far the preferred candidates. This is mainly due to the fact that, while semileptonic operators required for the B-anomalies can be induced at the tree-level, four- quark and four-lepton operators, that are strongly constrained by meson mixing or LFV, are induced only at one loop and thus automatically suppressed. An interesting scenario for a combined explanation of the anomalies involves the two scalar LQs S1 = (3¯, 1, 1/3) and S3 = (3¯, 3, 1/3) [30–39]. Interestingly enough, the S1

2 couplings to right-handed fermions allow also an explanation of the observed deviation in the muon anomalous magnetic moment (g − 2)µ [40–42]. As a possible ultraviolet (UV) completion for this model, the two scalars could arise, together with the Higgs boson, as pseudo-Nambu-Goldstone bosons from a new strongly coupled sector at the multi- TeV scale, which would also address the hierarchy problem of the electroweak scale in a composite Higgs framework [32,38] (see also [43–45] for related works). The solution to B-anomalies involves NP couplings to second and third generations of quarks and leptons, while the couplings to first generation could in principle be very small, which is also required by strong experimental constraints [46–49]. Nevertheless, the natural expectation is that NP should couple to all generations, possibly with some flavor structure dictated by a dynamical mechanism or a symmetry. In light of this, a question one can pose is: what are the expected effects in Kaon physics and electron observables for models that address the B-anomalies? This is the main question we aim at addressing with this paper, in the context of S1 + S3 solutions. Specifically, the golden channels of + + 0 rare Kaon decays, K → π νν and KL → π νν, are now actively being investigated by the NA62 [50] and KOTO [51] experiments, respectively, and substantial improvements are expected in the future. Furthermore, an improvement by a few orders of magnitude in sensitivity is expected for µ → e conversion experiments COMET and Mu2e [52–55] as well as in µ → 3e from the Mu3e experiment at PSI [56]. Can these experiments expect to observe a signal, possible related to the B-anomalies? Our starting point is the analysis of how this setup can address the B-anomalies (as well as the (g − 2)µ) performed in Ref. [37]. In that work several scenarios were considered and, for each of them, a global analysis was performed including all the relevant observables computed at one-loop accuracy using the complete one-loop matching between the two mediators and the SMEFT obtained in Ref. [57]. In particular, two scenarios able to address both sets of B-anomalies were found:

• LH couplings. If the S1 couplings to right-handed fermions are zero, then both charged and neutral-current B-anomalies can be addressed, while the muon magnetic moment deviation cannot. It was also observed that the preferred values of the couplings to second generation was compatible with the structure hinted to by an approximate U(2)5 ﬂavor symmetry [37] (see also [31,32]).

• All couplings. If, instead, all couplings are allowed, then both the B-anomalies and the (g −2)µ can be addressed, but the coupling structure is not compatible with 5 the U(2) flavor symmetry since a large coupling to cRτR is required, with a small coupling to tRτR instead [37]. We extend our previous work by considering also Kaon and D decays, as well as all processes sensitive to the µ → e lepton flavor violating (LFV) transition. To find possible connections between B-anomalies and Kaon physics we take two different approaches for the two scenarios listed above. For the first scenario we impose from the beginning a concrete assumption on the flavor structure, in particular the U(2)5 flavor symmetry [58–60] and perform the first complete study of B-anomalies and Kaon physics with S1 and S3 within this context. The main feature of this approximate symmetry is that strict relations between the LQ coupling to first and second generations are predicted, implying that Kaon decays become strictly connected with B decays. Our choice of the flavor symmetry is motivated by the observation that the approximate U(2)q × U(2)` flavor symmetry, that acts on the light-generations of SM quarks and

3 leptons and is a subset of U(2)5, appears to provide a consistent picture of all low-energy data. In fact, it was shown in the general EFT context [31, 61–64], as well as also in concrete UV realizations [32,37,65–70], that not only it can reproduce the observed hierarchies in the SM Yukawa sector, but also successfully control the strength of NP couplings allowing for sufficiently large effects in processes involving third-generation fermions. We mention that the flavor symmetry does not need to be a fundamental property of the UV theory, but it could arise as an accidental low-energy symmetry. For the second scenario, since no evident flavor structure emerges from the B-anomalies fit when also right-handed couplings are included, we let vary the couplings relevant for K → πνν, while keeping the other couplings fixed to the best-fit values required by the B-anomalies and muon magnetic moment. In this way we can find the allowed values for K → πνν that are compatible with the B-anomalies, in this general setup. The paper is structured as follows. In Section2 we briefly introduce the model, the setup, and the statistical tool used for the analysis. In Section3 we discuss the structure of the LQ couplings predicted by the minimally broken U(2)5 flavor symmetry and study the results of the global fit in this framework. Section4 is instead devoted to the study of rare Kaon decays and electron LFV processes in the general case where the flavor assumption is lifted. We conclude in Section5. Details on the observables not discussed in [37] are collected in AppendixA.

2 Setup

We consider in this work the two scalar LQs S1 = (3¯, 1, 1/3) and S3 = (3¯, 3, 1/3), where the quantum numbers under the SM gauge group SU(3)c × SU(2)L × U(1)Y are indi- cated. The Lagrangian to be added to the SM one, assuming baryon and lepton number conservation, is

2 2 2 2 2 2 LLQ =|DµS1| + |DµS3| − M |S1| − M |S3| − VLQ(S1,S3,H)+ 1 3 (2.1) 1L c 1R c 3L c I I + (λ )iα q i `α + (λ )iα u ieα S1 + (λ )iα q i σ `αS3 + h.c. ,

1L 1R 3L where = iσ2,(λ )iα, (λ )iα, (λ )iα ∈ C, and VLQ includes LQ self-couplings and interactions with the Higgs boson, which are omitted since they are not relevant for the phenomenology studied here. We denote SM quark and lepton fields by qi, ui, di, `α, and eα, while the Higgs doublet is H. We adopt latin letters (i, j, k, . . . ) for quark flavor indices and greek letters (α, β, γ, . . . ) for lepton flavor indices. We work in the down-quark and charged-lepton mass eigenstate basis, where

∗ j α VjiuL νL qi = i , `α = α , (2.2) dL eL

and V is the CKM matrix. We use the same conventions as Refs. [37, 57], to which we refer for further details. Our goal is to study the phenomenology of the S1 + S3 model extending the list of observables already accounted for in Ref. [37] to all the other relevant Kaon, D-meson, and electron LFV ones. We do this by employing the same multistep procedure: the LQ model is matched at one-loop level into the SMEFT [57], which is then matched into the Low Energy EFT (LEFT) [71,72]; within any EFT the renormalization group evolution (RGE)

4 is taken into account, as well as the one-loop rational contributions within the LEFT, in terms of whose coefficients the observables and pseudo-observables are expressed (barring the case of observables that are measured at the electroweak scale). In our global analysis for the two LQs we add the following observables to those 0 observables already studied in Ref. [37]: rare and LFV Kaon decays, K /K , rare D-meson decays, b → d`` decays, µ → e conversion in nuclei, and the neutron EDM. In Tables2, 3, and4, we show the complete list of observables that we analyze, together with their SM predictions and experimental bounds. In App.A we collect details on the low-energy observables that were not considered in Ref. [37], together with others which turn out to set relevant constraints in the fit, such as limits from Z couplings measurements and K → πνν, that we repeat for sake of completeness. For all the observables, the full set of one-loop corrections is considered in the numerical analysis. We perform a χ2 fit, thus defining the likelihood as

2 X (Oi(λx,Mx) − µi) −2 log L ≡ χ2(λ ,M ) = , (2.3) x x σ2 i i

where Oi(λx,Mx) is the expression of the observable as function of the model parameters, µi its central measured value, and σi the associated standard deviation. In the analysis presented in this paper, 73 observables are taken into account, for which, within the SM, 2 2 the χ is χSM = 104.0. In each scenario we first find the best-fit point by minimizing the χ2. We then perform a numerical scan over the parameter space, with samples of size O(104), using a Markov Chain Monte Carlo algorithm, to select points that are within the 68 or 95% CL from the best-fit point. These scans are used to obtain preferred regions in parameter space or for selected pairs of interesting observables.

3 Scalar leptoquarks and U(2)5 ﬂavor symmetry

In the limit where only third generaton fermions are massive, the SM enjoys the global ﬂavor symmetry [58–60]

GF = U(2)q × U(2)` × U(2)u × U(2)d × U(2)e . (3.1) Masses of the ﬁrst two generations of fermions and their mixing break this symmetry. In the quark sector the largest breaking is of size ≈ yt|Vts| ≈ 0.04 [64]. Formally, the symmetry breaking terms in the Yukawa matrices can be described in terms of spurions transforming under representations of GF . The minimal set of spurions that can reproduce the observed masses and mixing angles is 1 V ∼ (2, 1, 1, 1, 1) , V ∼ (1, 2, 1, 1, 1) , q ` (3.2) ∆u ∼ (2, 1, 2¯, 1, 1) , ∆d ∼ (2, 1, 1, 2¯, 1) , ∆e ∼ (1, 2, 1, 1, 2¯) . In terms of these spurions the SM Yukawa matrices can be written as ∆ x V ∆ x V Y = y u(d) t(b) q ,Y = y e τ ` , (3.3) u(d) t(b) 0 1 e τ 0 1

1 Strictly speaking V` is not required in the SM, since in absence of neutrino masses lepton mixing is unphysical. It is however usually added for symmetry with the quark sector and, in our case, because it (∗) is needed in order to address the R(K ) anomalies, which requires |V`| ∼ O(0.1) [31, 61].

5 with xt,b,τ are O(1) complex numbers, ∆’s are 2 × 2 matrices, and Vq,` are 2-component vectors. In the context of the B-anomalies, this flavor symmetry was introduced as a possible explanation for the LFU breaking hints, that point to largest effects for τ leptons, smaller for muons, and even smaller for electrons. Furthermore, it was observed in Refs. [31, 32, 37, 61–64] that the LQ couplings to second and third generations, required to fit the anomalies, were consistent with the expectations given by this symmetry. In this Section 5 we study if, indeed, a complete implementation of U(2) flavor symmetry for the S1 and S3 scalar LQs, including the couplings to first generation fermions, is consistent with the observed anomalies. In the same flavor basis used to write the Yukawa couplings of Eq. (3.3), the S1 and S3 LQ couplings have the following structure:

1(3)L ∗ † 1(3)L ∗ ! 1(3)L 1(3) x˜q` Vq × V` x˜q Vq λ = λ 1(3)L † 1(3)L , (3.4) x˜` V` x˜bτ

1R † ∗ 1R 1 O(∆uVq∆eV`)x ˜u ∆uVq 1 0 0 λ = λR 1R † ∗ 1R ≈ λR 1R , (3.5) x˜e V`∆e x˜tτ 0x ˜tτ 1(3) 1 where λ and λR are overall couplings, allx ˜ are O(1) parameters, and in the last step in λ1R we neglected all the terms that give too small couplings to have a significant influence to our observables. In the following we can thus neglect the presence of the λ1R 5 couplings in the U(2) scenario since the tRτR coupling does not affect in a relevant way the phenomenology. By diagonalizing the SM lepton and down-quark Yukawa matrices one can put in relation some of the parameters in Eq. (3.3) with observed masses and CKM elements, we refer to Ref. [64] for a detailed discussion on this procedure. For our purposes, the main result is that in this basis the quark doublet spurion is fixed by the CKM up to an overall ∗ ∗ T O(1) factor, Vq = κq(Vtd,Vts) , while the size of the leptonic doublet spurion V` ≡ |V`| as well as the angle that rotates left-handed electrons and muons, se ≡ sin θe, are free. The same rotations that diagonalize the (lepton and down quark) Yukawas also apply to the LQ couplings. The final result of this procedure is the following structure for the LQ couplings in the mass basis:

 1(3) 1(3) 1(3)  xq` seV`Vtd xq` V`Vtd xq Vtd λ1(3)L = λ1(3)  1(3) 1(3) 1(3)  . (3.6)  xq` seV`Vts xq` V`Vts xq Vts  1(3) 1(3) x` seV` x` V` 1 All x1(3) parameters are expected to be O(1) complex numbers and we absorbed the O(1) coefficient in the 3-3 component inside the definition of λ1(3). It directly follows from Eq. (3.6) that the flavor symmetry imposes some strict relations between families:

1(3)L 1(3)L Vtd 1(3)L 1(3)L λ1α = λ2α , λi1 = λi2 se . (3.7) Vts For the two LQs we thus remain with the two overall couplings λ1(3), that we can always 1(3) 1(3) 1(3) take to be positive, six O(1) complex parameters (xq` , xq , x` ), one small angle se that regulates the couplings to electrons compared to the muon ones, and ﬁnally V` that sets the size of muon couplings compared to tau ones.

6 ¯ Since S1 does not mediate di → dj`α`β at tree level, its contributions proportional to 1 1 x` and xq` are only very weakly constrained. For this reason, to simplify the numerical scan we fix them to be equal to 1.2 We provide here some simplified expressions for the most relevant NP effects in this setup, deferring for details to App.A:

(∗) 1 2 1∗ ∗ 3 2 3∗ ∗ ∆R(D ) 2 |λ | (1 − xq Vtb) |λ | (1 − xq Vtb) (∗) ≈ v 1.09 2 − 1.02 2 , (3.8) R(D )SM 2M1 2M3 3 2 2 3 3∗ sbµµ sbµµ π |λ | |V`| x` xq` ∆C9 = −∆C10 ≈ √ 2 , (3.9) 2GF αVtb M3 ∗ 3 2 2 3 2 sdµµ sdµµ πVtsVtd |λ | |V`| |xq`| ∆C9 = −∆C10 ≈ √ 2 , (3.10) 2GF α M3 |λ1|2x1∗ |λ3|2x3∗ LV LL ≈ V ∗ q + q , (3.11) νd ντ ντ sb ts 2 2 2M1 2M3 1 2 1 2 3 2 3 2 V LL ∗ |λ | |xq| |λ | |xq| L ≈ V Vts + , (3.12) νd ντ ντ ds td 2 2 2M1 2M3 |λ1|2 |λ1|2 103δgZ ≈ 0.59 + 0.80 , (3.13) τL 2 2 2 2 M1 / TeV M1 / TeV

∗ 1 4 1 4 3 4 3 4 1 2 3 2 1 2 3 2 2 2 1 VtsVtd |λ | |xq| |λ | |xq| |λ | |λ | |xq| |xq| log M3 /M1 CK ≈ 2 2 + 5 2 + 2 2 , (3.14) 128π M3 M3 M3 − M1 where LV LL are the Wilson coefficients (WCs) of the low-energy operators νd ντ ντ didj (¯ν γ ν )(d¯i γµdj ), δgZ describes the deviation in the Z couplings to τ , and C1 is the τ µ τ L L τL L K 2 coefficient of the (¯sγµPLd) operator. The leading contribution to s → dµµ transitions has a phase fixed to be equal to the SM one, so no large effect in KS → µµ can be expected. Analogously, also in s → dνν the NP coefficients have the same phase as in the SM, since the x coefficients enter with the absolute value squared. This implies that no cancellation between the two LQs can take place in this channel and that we expect a 0 + + tight correlation between KL → π νν and K → π νν, independently on the phases of the couplings. Similar considerations apply for all s → d transitions. On the other hand, non-trivial phases can appear in b → s transitions and a mild cancellation can alleviate the B → K(∗)νν bound [31]. Since real couplings are favored by the B-anomalies and since in any case the phases in Kaon physics observables are fixed by the U(2)5 flavor structure, in our numerical analysis we only consider real values for all parameters.

3.1 Analysis and discussion Using the global likelihood we ﬁnd the best-ﬁt point in parameter space, allowing the x’s 1(3) to vary in the range |x| < 5, while λ ,V` > 0. Fixing M1 = M3 = 1.1 TeV we get

2We checked that, as expected, if left free these parameters have an almost uniform distribution in the whole interval of interest [−5, 5].

7 Figure 1: Results of a parameter scan in the U(2)5 scenario. The green (yellow) points are within the 68% (95%) CL from the best-fit point (shown in black). We also overlay 2σ constraints from single observables, where other parameters are fixed to the best-fit point (3.15).

2 2 3 χSM − χbest−ﬁt = 47.6, for:

1 3 5 λ ≈ 0.79 , λ ≈ 0.72 ,V` ≈ 0.071 , se ≈ 0 , best-ﬁt U(2) : 1 3 3 3 (3.15) xq ≈ −0.97 , xq ≈ 1.6 , x` ≈ 3.6 , xq` ≈ −2.0 .

We then perform a numerical scan on all the parameters in Eq. (3.15), selecting only 2 2 2 points with a ∆χ = χ − χbest−fit corresponding to a 68% (green points) or 95% (yellow) confidence level. The results are shown in Fig.1, where we project the points to several 2D planes. We also show 2σ constraints from single observables, obtained by fixing the parameters not in the plot to the corresponding best-fit values, Eq. (3.15).4 We observe from Fig.1 (bottom-left) that values V` ≈ 0.1 and |se| . 0.02 are preferred. We can also see that all the x’s can be of O(1), with no tendency towards parametrically smaller or larger values, hence the structure of the U(2)5 symmetry is respected.

3This puts the LQs above present LHC limits, see e.g. Ref. [35] for a recent review. We note that the ﬁt slightly worsens when the masses are increased. 4This procedure misses the correlations between the plotted parameters and those that are ﬁxed, that is however fully kept in the parameter scan. For this reason the bounds shown in this way are useful mainly to gain an understanding of the relevant observables in each plane.

8 Figure 2: Results, for the B-anomalies, of the parameter scan in the U(2)5 scenario shown in Fig.1. The green (yellow) points are within the 68% (95%) CL from the best-fit point (3.15). The gray lines describe different CL regions in the fit of the anomalies (see App.A for details).

In Figs.2 and3 we show the values of particularly interesting pairs of observables obtained with the same sets of parameter-space points. From Fig.2 we observe that, while neutral-current B-anomalies can be addressed entirely, this setup can enhance R(D(∗)) only by . 7% of the SM size, i.e. at the 2σ level of the present combination. This situation should be compared with the result of the analogous similar fit with S1 and S3 with only couplings to left-handed fermions shown in [37] (c.f. Fig. 5), where both anomalies can be satisfied in a scenario where couplings to the second generation quarks were compatible 5 1(3)L 1(3)L with a U(2) flavor structure, |λs` | ∼ |Vts/Vtb||λb` |, but couplings to first generation were set to zero. Therefore, the reason for the inability of the U(2)5-symmetric scenario to fully address charged-current anomalies must be found in first-generation constraints, specifically Kaon physics. Indeed, this can be seen in the first row of Fig.1, where we observe that the + + 1 bounds from K → π νν¯ , Eq. (3.12), and K (i.e. Im CK ), Eq. (3.14), in combination with the constraints on λ1,3 from Z → ττ¯ , Eq. (3.13), don’t allow the fit to enter the region preferred by R(D(∗)), due to the precise relations between couplings to the first and the second generation, derived from the flavor structure, i.e. Eq. (3.7). Regarding Kaon physics observables, from Fig.3 we see that B(K+ → π+νν¯ ) can take all values currently allowed by the NA62 bound [50] (we show with vertical lines the best-fit and the ±1σ intervals) and therefore any future update on this observable will put further strong constraints on this scenario. Furthermore, since the phase in s → dνν is fixed by the corresponding CKM phase, Eq. (3.12), a correlation between this mode 0 −10 and B(KL → π νν¯ ) is obtained, with values ∼ 10 also for the latter. Therefore, even by the end of stage-I the KOTO experiment won’t be able to reach the sensitivity to test this model (brown horizontal dotted line). However, the future sensitivity goals by NA62 (10% [74]) and KOTO at stage-II, or KLEVER, (20% [75]) would be able to completely test this scenario (purple ellipse). The model also predicts short-distance contributions to B(KL → µµ) of the order of the present bound, although it remains challenging to improve this constraint in the future due to non-perturbative contributions to the long-distance component. Also in this

9 Figure 3: Results, for rare Kaon decays and µ → e conversion processes, of the parameter scan in the U(2)5 scenario shown in Fig.1. The green (yellow) points are within the 68% (95%) CL from the best-ﬁt point. In the upper plot, the gray region is excluded by the Grossman-Nir bound [73], the red solid and dashed lines represent the present measurement from NA62, the dotted brown line the sensitivity prospect for KOTO after stage-I, while the dotted purple one the ﬁnal sensitivity expected from NA62 and KOTO (stage-II). In the lower plot the red lines describe the present 95% CL bound (see App.A for details).

10 ∗ channel the phase of the NP WC is fixed to the one of VtsVtd, see Eq. (3.10). The KS → µµ mode is thus completely correlated with the KL decay and the New Physics effect adds constructively to the SM short-distance amplitude, the expected size is however below the SM long-distance contribution of ≈ 5 × 10−12. The expected relative effect in the −7 −9 tree-level transitions s → uµν¯µ and d → uµν¯µ is of order 10 and 10 , respectively, excluding possible signatures from these decays. Finally, the rotation angle between electrons and muons, se, is constrained mainly by LFV µ → e processes, the strongest bound presently given by µ → e conversion in gold atoms, while the predicted effect in titanium is strictly correlated in our model, with an approximate relation B(µ → e)Au ≈ 1.3B(µ → e)Ti. In Fig.3 we show the correlation with µ → 3e. These measurements are expected to improve substantially in the future, reaching limits of the order of 10−16 in µ → e conversion in nuclei from COMET and Mu2e [52–55] or even 10−18 by the PRISM proposal, and also a level of 10−16 in µ → 3e from the Mu3e experiment [56]. These will put further constraints on the se parameter, or a signal could be observed if this mixing angle is large enough. For what regards LFV in Kaon decays, given the largest allowed values for se and V` −15 + + −18 from present limits, we obtain at most B(KL → µe) . 10 and B(K → π µe) . 10 .

4 General case with right-handed couplings

In this section we depart from the flavor symmetry assumption and examine the fit of the S1 +S3 model in the general case where all the couplings, including the right-handed ones, are allowed. As investigated in Ref. [37], if the couplings are a priori uncorrelated, there is enough freedom to accommodate both B-physics anomalies as well as the discrepancy in the anomalous magnetic moment of the muon. Due to the high dimensionality of the parameter space, it is computationally too expensive to perform a χ2 minimization by random search techniques. For the purposes of our discussion it suffices then to use the best-fit point of Eq. (3.11) in Ref. [37]5 in order to fix the relevant couplings and let only the additional couplings vary. A further simplification constitutes in switching off the couplings to electrons. This choice is justified by the fact that the necessary suppression required to pass the stringent bounds from LFV µ → e processes is of the same order in both the flavor-symmetry motivated case and the general one. Using the best-fit point from Ref. [37] assures that all the anomalies are addressed 0 within 1σ (c.f. Fig. 5 therein). In order to assess the viable values of B(KL → π νν) and B(K+ → π+νν) in this setup we perform a likelihood scan of the complex couplings 1(3)L 1(3)L λsτ and λdτ (see Eq. (A.10)) and keep only values that have a likelihood within the 68% or 95% CL from the best-fit point. The corresponding branching ratios are reported in the left plot of Fig.4, from which we observe that, compared to the U(2)5 case, the viable values are significantly expanded. We also notice that the two decays are induced by the same short distance operators and are trivially related through isospin, leading to the so-called Grossman-Nir bound [73]. The LQ interactions fall under a category of models that may saturate the bound [78]. Yet this is not entirely possible due to the constraints on the magnitude of the couplings to muons and electrons. Nevertheless, we

5A ﬁt with the updated values for the various ﬂavor constraints including the new measurement of RK does not result in any substantial variations.

11 Out[ ]=

1(3)L 1(3)L Figure 4: Allowed region obtained by varying the couplings λsτ and λdτ ,relevant for 3L 3L the B → Kνν decays (left), and the couplings λsµ and λdµ relevant for the KL,S → µµ decays (right). The rest of the couplings are fixed to the best-fit point couplings in Eq. (3.11) of Ref. [37] and compatibility with the global fit is retained at 68% (green) and 95% (yellow) CL. On the left plot, the gray region is excluded by the Grossman-Nir bound, the red solid and dashed lines represent the present measurement from NA62, the dotted brown line the sensitivity prospect for KOTO after stage-I, while the dotted purple one the final sensitivity expected from NA62 and KOTO (stage-II). On the right plot, the red solid line represents the bound set in Ref. [76] and the red dashed line the future prospects by LHCb [77].

0 see that B(KL → π νν) could potentially take values that can be probed by the end of stage 1 of the KOTO experiment. 3L 3L By letting vary the λsµ and λdµ complex couplings we find that the short distance contribution to KS → µµ can reach values of the order of the long-distance one, while that to the KL mode saturate the theory-derived constraint [76]. In the right plot of Fig.4 we present the 68% and 95% CL regions from the best-fit point in these two observables. Considering the ≈ 30% uncertainty on the SM prediction for the KS mode, we notice that a significant part of the preferred region features NP effects that are distinguishable from the SM ones. As a matter of fact, the future prospects look promising, since the LHCb Upgrade II plans to exclude branching fractions down to near the SM prediction [77] (see dashed line in Fig.4). Regarding µ → e transition in nuclei we expect a very similar result to the one shown in Fig.3, since even in that case the observables saturate the present bounds and there is additional freedom if the flavor symmetry assumption is removed.

5 Conclusions

The observed anomalies in B decays can potentially be addressed in LQ scenarios. While couplings to ﬁrst generation of fermions are not required to describe these deviations, and could therefore be set to zero in a bottom-up approach, the typical expectation from UV

12 models is that all couplings should be generated. In this work we study the impact that LQ couplings to first generation fermions can have on Kaon and electron LFV observables, in scenarios that aim at addressing the B-anomalies. Correlations between couplings to different generations, and therefore between B and Kaon decays, can be obtained only if the flavor structure is specified. In this work we consider the approximate U(2)5 flavor symmetry, that is motivated from the observed SM fermions mass hierarchies and also predicts a pattern of deviations in B decays consistent with the observed one. With this assumption, the LQ coupling to left-handed d quarks are fixed to be equal to those to left-handed s quarks times the small CKM factor Vtd/Vts (see Eq. (3.7)). This structure correlates strongly Kaon physics with B decays. After performing a global likelihood analysis of a large set of observables in this scenario, we (∗) find that the S1 + S3 LQs can address the R(D ) anomalies only at the 2σ level. The + + most important observables preventing a successful fit are B(K → π νν), K , and the Z couplings to τ. We thus expect future improved measurements of K+ → π+νν from NA62 to further test this scenario. The LQ couplings to electrons, instead, are constrained mainly by the limits from µ → e transition in nuclei and µ → 3e decay, that will be improved by several orders of magnitude in the near future. Going beyond the flavor symmetric scenario, we also studied the allowed values for + + 0 + − B(K → π νν), B(KL → π νν), and KL/S → µ µ in the general case where no 0 flavor structure is imposed. We find that, in this setup, values of B(KL → π νν) and + − B(KS → µ µ ) that could be potentially probed by KOTO stage-1 or LHCb, respectively, are allowed. The B-anomalies are expected to receive further experimental inputs in the next few years by LHCb, Belle-II, as well as CMS and ATLAS experiments. However, even assuming these will be confirmed, in order to understand the flavor structure of the underlying New Physics the connection to Kaon physics and to observables sensitive to electron couplings will be crucial.

Acknowledgements DM and ST acknowledge support by MIUR grant PRIN 2017L5W2PT. DM is also partially supported by the INFN grant SESAMO and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement 833280 (FLAY). EV has been partially funded by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311, by the Collaborative Research Center SFB1258 and the BMBF grant 05H18WOCA1 and thanks the Munich Institute for Astro- and Particle Physics (MIAPP) for hospitality.

13 Observable SM prediction Experimental bounds b → s`` observables [37] sbµµ ∆C9 0 −0.43 ± 0.09 [79] univ C9 0 −0.48 ± 0.24 [79] b → cτ(`)ν observables [37] RD 0.299 ± 0.003 [12] 0.34 ± 0.027 ± 0.013 [12] ∗ RD 0.258 ± 0.005 [12] 0.295 ± 0.011 ± 0.008 [12] D∗ Pτ −0.488 ± 0.018 [80] −0.38 ± 0.51 ± 0.2 ± 0.018 [7] FL 0.470 ± 0.012 [80] 0.60 ± 0.08 ± 0.038 ± 0.012 [81] + + B(Bc → τ ν) 2.3% < 10% (95% CL) [82] µ/e RD 1 0.978 ± 0.035 [83,84] b → sνν and s → dνν [37] ν RK 1 [85] < 4.7 [86] ν RK∗ 1 [85] < 3.2 [86] b → dµµ and b → dee App. A.5 B(B0 → µµ) (1.06 ± 0.09) × 10−10 [87,88] (1.1 ± 1.4) × 10−10 [89,90] B(B+ → π+µµ) (2.04 ± 0.21) × 10−8 [87,88] (1.83 ± 0.24) × 10−8 [89,90] B(B0 → ee) (2.48 ± 0.21) × 10−15 [87,88] < 8.3 × 10−8 [51] B(B+ → π+ee) (2.04 ± 0.24) × 10−8 [87,88] < 8 × 10−8 [51] B LFV decays [37] ± ∓ −5 B(Bd → τ µ ) 0 < 1.4 × 10 [91] ± ∓ −5 B(Bs → τ µ ) 0 < 4.2 × 10 [91] B(B+ → K+τ −µ+) 0 < 5.4 × 10−5 [92] < 3.3 × 10−5 [92] B(B+ → K+τ +µ−) 0 < 4.5 × 10−5 [93]

Table 1: Observables from B and D meson decays. Upper limits correspond to 95%CL.

14 Observable SM prediction Experimental bounds D leptonic decay [37] and App. A.4 −2 −2 B(Ds → τν) (5.169 ± 0.004) × 10 [94] (5.48 ± 0.23) × 10 [51] B(D0 → µµ) ≈ 10−11 [95] < 7.6 × 10−9 [96] B(D+ → π+µµ) O(10−12)[97] < 7.4 × 10−8 [98] Rare Kaon decays (νν) App. A.1 B(K+ → π+νν) 8.64 × 10−11 [99] (11.0 ± 4.0) × 10−11 [100] 0 −11 −9 B(KL → π νν) 3.4 × 10 [99] < 3.6 × 10 [101] Rare Kaon decays (``) App. A.3 and A.2 −10 −9 B(KL → µµ)SD 8.4 × 10 [102] < 2.5 × 10 [76] −12 −10 B(KS → µµ) (5.18 ± 1.5) × 10 [76,103,104] < 2.5 × 10 [105] 0 −11 −10 B(KL → π µµ) (1.5 ± 0.3) × 10 [106] < 4.5 × 10 [107] 0 +1.2 −11 −10 B(KL → π ee) (3.2−0.8) × 10 [108] < 2.8 × 10 [109] LFV in Kaon decays App. A.3 and A.2 −12 B(KL → µe) 0 < 4.7 × 10 [110] B(K+ → π+µ−e+) 0 < 7.9 × 10−11 [111] B(K+ → π+e−µ+) 0 < 1.5 × 10−11 [112] CP-violation App. A.8 0 −4 −4 K /K (15 ± 7) × 10 [113] (16.6 ± 2.3) × 10 [51] Table 2: D meson and Kaon physics observables with the corresponding SM predictions and experimental bounds. Upper limits correspond to 95% CL.

15 Observable SM prediction Experimental bounds ∆F = 2 processes [37] 0 B0 − B : |C1 | 0 < 9.1 × 10−7 TeV−2 [114,115] Bd 0 0 1 −5 −2 Bs − Bs: |CBs | 0 < 2.0 × 10 TeV [114,115] 0 0 1 −7 −2 K − K : Re[CK ] 0 < 8.0 × 10 TeV [114,115] 0 0 1 −9 −2 K − K : Im[CK ] 0 < 3.0 × 10 TeV [114,115] 0 0 1 −7 −2 D − D : Re[CD] 0 < 3.6 × 10 TeV [114,115] 0 0 1 −8 −2 D − D : Im[CD] 0 < 2.2 × 10 TeV [114,115] 0 0 4 −8 −2 D − D : Re[CD] 0 < 3.2 × 10 TeV [114,115] 0 0 4 −9 −2 D − D : Im[CD] 0 < 1.2 × 10 TeV [114,115] 0 0 5 −7 −2 D − D : Re[CD] 0 < 2.7 × 10 TeV [114,115] 0 0 5 −8 −2 D − D : Im[CD] 0 < 1.1 × 10 TeV [114,115] LFU in τ decays [37] 2 |gµ/ge| 1 1.0036 ± 0.0028 [116] 2 |gτ /gµ| 1 1.0022 ± 0.0030 [116] 2 |gτ /ge| 1 1.0058 ± 0.0030 [116] LFV observables [37] B(τ → µφ) 0 < 1.00 × 10−7 [117] B(τ → 3µ) 0 < 2.5 × 10−8 [118] B(τ → µγ) 0 < 5.2 × 10−8 [119] B(τ → eγ) 0 < 3.9 × 10−8 [119] B(µ → eγ) 0 < 5.0 × 10−13 [120] B(µ → 3e) 0 < 1.2 × 10−12 [121] (Ti) −12 Bµe 0 < 5.1 × 10 [122] (Au) −13 Bµe 0 < 8.3 × 10 [123] EDMs [37] −44 −29 |de| < 10 e · cm [124,125] < 1.3 × 10 e · cm [126] −42 −19 |dµ| < 10 e · cm [125] < 1.9 × 10 e · cm [127] −41 −17 dτ < 10 e · cm [125] (1.15 ± 1.70) × 10 e · cm [37] −33 −26 dn < 10 e · cm [128] < 2.1 × 10 e · cm [129] Anomalous [37] Magnetic Moments SM −13 −13 ae − ae ±2.3 × 10 [130,131] (−8.9 ± 3.6) × 10 [132] SM −11 −11 aµ − aµ ±43 × 10 [42] (279 ± 76) × 10 [40,42] SM −8 −7 aτ − aτ ±3.9 × 10 [130] (−2.1 ± 1.7) × 10 [133] Table 3: Meson-mixing and leptonic observables and EDMs, with their SM predictions and experimental bounds. Upper limits correspond to 95%CL.

16 A Analysis of observables and pseudo-observables

0 + + A.1 KL → π νν and K → π νν

The relevant parton level processes s → dνανβ are described by the eﬀective four-fermion Lagrangian 4GF α X Ldsνν ⊃ √ V ∗ V C O , (A.1) eﬀ 4π td ts i i 2 i with the following ∆F = 1 operators

dsαβ ¯ µ dsαβ ¯ µ OL = dγµPLs (¯ναγ (1 − γ5)νβ) , OR = dγµPRs (¯ναγ (1 − γ5)νβ) . (A.2)

The relations between the WCs of the operators above and the LEFT ones (in the basis of Ref. [71]) are given by

X V,LL X V,LR [CL]dsαβ = Nds [Lνd ]αβds , [CR]dsαβ = Nds [Lνd ]αβds , (A.3) αβ αβ √ ∗ −1 where Nds = ( 2GF αVtdVts/π) . Within the SM, the K → πνν decays are among the cleanest observables to determine the mixing between the top quark and the light generations as generated at loop level, as well as the magnitude of CP-violating (CPV) eﬀects in the quark sector. The coeﬃcients of the relevant four-fermion interaction are [99]

∗ SM 1 VcdVcs α SM [CL]dsαβ = − 2 Xt + ∗ Xc δαβ , [CR]dsαβ = 0 , (A.4) sW VtdVts e µ −3 τ −3 with Xt = 1.47, Xc = Xc = 1.053 × 10 , and Xc = 0.711 × 10 . In scenarios beyond the SM, these observables are highly sensitive to new sources of both CPC and CPV ﬂavor mixing. In our case, at tree-level only CL is not vanishing. While at loop level a contribution to CR is generated (and included in the numerical analysis), it is suppressed by small down-quarks Yukawa couplings, and thus it is negligible. The leading contributions to the WCs at mb scale, in terms of the UV parameters, are

1L∗ 1L 3L∗ 3L 2 3L∗ 3L ! hλdα λsβ λdα λsβ 1 1 mt h ∗ 2 λbα λbβ [CL]dsαβ = Nds 2 + 2 + 2 2 24VtdVts|Vtb| 2 + 2M1 2M3 16π 12 v M3 1L∗ 1L 3L∗ 3L ! 2 2 λdα λkβ λdα λkβ ∗ − 3(3 + 2 log(M /mt )) 2 + 2 VtsVtk+ 2M1 2M3 1L∗ 1L 3L∗ 3L ! ! λkα λsβ λkα λjβ ∗ ii + 2 + 2 VtkVtd + ..., 2M1 2M3

[CR]dsαβ ≈ 0 . (A.5)

+ + 0 In the CR ≈ 0 approximation, the branching ratios for the K → π νν and KL → π νν decays can be expressed in terms of the SM branching ratios, via a rescaling of the SM

17 Observable Experimental bounds Z boson couplings App. A.7 δgZ (0.3 ± 1.1)10−3 [134] µL δgZ (0.2 ± 1.3)10−3 [134] µR δgZ (−0.11 ± 0.61)10−3 [134] τL δgZ (0.66 ± 0.65)10−3 [134] τR δgZ (2.9 ± 1.6)10−3 [134] bL δgZ (−3.3 ± 5.1)10−3 [134] cR Nν 2.9963 ± 0.0074 [135] Drell-Yan σ(pp → µ+µ−) [136,137] σ(pp → τ +τ −) [137,138]

Table 4: Limits on the deviations in Z boson couplings to fermions from LEP I and from Drell-Yan high-energy tails at LHC.

CL coeﬃcient, in the following way [C ] 2 + + + + X L dsαβ B(K → π νν) = B(K → π νeνe)SM δαβ + SM + (A.6) [CL] α,β=1,2 ds11 " # 2 2 2 X [CL]dsα3 [CL]ds3α [CL]ds33 + B(K+ → π+ν ν ) + + 1 + , τ τ SM [C ]SM [C ]SM [C ]SM α=1,2 L ds33 L ds33 L ds33 " −1 2 1 X Im[N [CL]dsαβ] B(K → π0νν) = B(K → π0νν) δ + ds + L L SM 3 αβ Im[N −1[C ]SM ] α,β=1,2 ds L ds11 −1 2 −1 2! −1 2 # X Im[N [CL]dsα3] Im[N [CL]ds3α] Im[N [CL]ds33] + ds + ds + 1 + ds , Im[N −1[C ]SM ] Im[N −1[C ]SM ] Im[N −1[C ]SM ] α=1,2 ds L ds33 ds L ds33 ds L ds33 (A.7) where + + −11 B(K → π νeνe)SM = 3.06 × 10 , + + −11 B(K → π ντ ντ )SM = 2.52 × 10 , (A.8) 0 −11 B(KL → π νν)SM = 3.4 × 10 . The experimental bounds from NA62 [50] and KOTO [51] are + + +4.0 −11 B(K → π νν) = (10.6−3.5) × 10 , 0 −9 (A.9) B(KL → π νν) < 3.57 × 10 (95%CL) . NA62 expects to reach a ﬁnal sensitivity of about 10% of the SM K+ → π+νν breaching 0 ratio [74], while KOTO expects a future 95%CL upper limit for B(KL → π νν) of about 1.8 × 10−10 at the end of stage-I. A proposed KOTO upgrade (stage-II)6 or the KLEVER

6From a CERN EP seminar given in February 2019 (slides at https://indico.cern.ch/event/ 799787/attachments/1797668/2939627/EPSeminar_YuChen.pdf) and of a talk presented at the Ren- contres de Moriond 2019 (slides at http://moriond.in2p3.fr/2019/EW/slides/1_Sunday/1_morning/ 5_Nanjo.pdf).

18 proposal at CERN [75] would be able to reach a ﬁnal 20% sensitivity of the SM rate. Assuming that the contribution in ντ is the dominant one we get the following approximate numerical expressions

1L∗ 1L 10 + + λdτ λsτ 10 B(K → π νν) ≈ 0.61 + 0.25 1 + (1.58 − i0.51) 2 + |Vtd||Vts|m1 3L∗ 3L 2 λdτ λsτ +(1.45 − i0.47) 2 + ... , (A.10) |Vtd||Vts|m3 1L∗ 1L 3L∗ 3L 2 10 0 λdτ λsτ λdτ λsτ 10 B(KL → π νν) ≈ 0.23 + 0.11 1 − Im 5.4 2 + 4.9 2 + ... , |Vtd||Vts|m1 |Vtd||Vts|m3

where mi = Mi/TeV.

0 A.2 KL(S) → µµ and KL → π `` The standard notation for the eﬀective operators relevant to the s → d`+`− rare processes is 4GF α X Hsd`` ⊃ − √ C O , (A.11) eﬀ 4πs2 i i 2 W i

where the Ci are evaluated at ms scale and the operators Oi are deﬁned as

O(0)sd`` = (¯sγ P d)(`γ¯ α`) , O(0)sd`` = (¯sγ P d)(`γ¯ αγ `) , 9 α L(R) 10 α L(R) 5 (A.12) (0)sd`` ¯ (0)sd`` ¯ OS = (¯sPR(L)d)(``) , OP = (¯sPR(L)d)(`γ5`) . The expressions for the WCs of the above operators in terms of the LEFT ones are

−1 −1 sd`` Nsd V,LR V,LL 0sd`` Nsd V,RR V,LR C9(10) = 2 [Lde ]sd`` ± [Led ]``sd , C9(10) = 2 [Led ]``sd ± [Led ]``sd , −1 −1 sd`` Nsd S,RR S,RL ∗ 0sd`` Nsd S,RL S,RR ∗ CS(P ) = 2 [Led ]``sd ± [Led ]``sd , CS(P ) = 2 [Led ]``sd ± [Led ]``sd . (A.13) 4√GF α where Nsd = 2 . In the Standard Model, the only non vanishing coeﬃcients are [102] 2 4πsW

sd`` sd`` ∗ ∗ C9,SM = −C10,SM = (VcsVcdYNL + VtsVtdY (xt)) . (A.14) −4 with YNL = 3 × 10 and Y (xt) = 0.94. In our analysis, the above coeﬃcients are computed at one-loop level. The short-distance contributions to the branching ratios for

19 the leptonic KL → µµ and KS → µµ decays are given by s 2 2 2 4m2 + − τKL fK GF α µ B(KL → µ µ )SD = 3 4 1 − 2 32π mK sW mK 2 2 h 2 2 sdµµ 0sdµµ mK (mK − 4mµ) Re CS − CS + ms + md m2 2 i +m2 Re Csdµµ + Csdµµ − C0sdµµ 2m + Re Csdµµ − C0sdµµ K , K 10,SM 10 10 µ P P m + m s s d 2 2 2 4m2 + − τKS fK GF α µ B(KS → µ µ )SD = 3 4 1 − 2 32π mK sW mK 2 2 h 2 2 sdµµ 0sdµµ mK (mK − 4mµ) Im CS − CS + ms + md 2 2 2 sdµµ sdµµ 0sdµµ sdµµ 0sdµµ mK i +mK Im C10,SM + C10 − C10 2mµ + Im CP − CP . ms + md (A.15) To these one should add the SM long-distance contribution, dominated by the intermedi- ∗ ∗ + − ate KL,S → γ γ → µ µ process. Calculating this in the SM proves to be very challenging (see e.g. Refs. [76,104]) and while for the KS mode the long-distance component can + − −12 be estimated to about ≈ 30% accuracy, BLD = B(KS → µ µ )LD = (5.18 ± 1.50) × 10 [104], for the KL mode the situation is worsened by the unknown sign of the amplitude A(KL → γγ). For the rare semileptonic KL decays, we neglect the contributions from operators that are not generated at tree-level, which are also the ones not interfering with the SM interaction. Therefore, we apply a rescaling, with respect to SM, of the left-handed sd`` sd`` coeﬃcient C9 − C10

2 sdµ+µ− sdµ+µ− ! 0 + − 0 + − Im[C9 − C10 ] B(KL → π µ µ ) = B(KL → π µ µ )SM 1 + + − + − , Im[Csdµ µ − Csdµ µ ] 9,SM 10,SM (A.16) + − + − !2 Im[Csde e − Csde e ] B(K → π0e+e−) = B(K → π0e+e−) 1 + 9 10 , L L SM sde+e− sde+e− Im[C9,SM − C10,SM ] where the SM WCs are as in Eq. A.14 and

0 + − −11 B(KL → π µ µ )SM = (1.5 ± 0.3) × 10 [106] , 0 + − +1.2 −11 (A.17) B(KL → π e e )SM = (3.2−0.8) × 10 [108] . We evaluate these observables at one-loop level; however, in most of the regions of the parameter space, the tree-level contribution is dominant. At the tree-level, the branching

20 fractions as a function of the model parameters are

2 " 3L∗ 3L #! + − −10 4 λ sµλ dµ B(KL → µ µ )SD ≈ 8.4 × 10 1 − 1.50 × 10 Re 2 2 , M3 / TeV 2 " 3L∗ 3L #! λ λ dµ B(K → µ+µ−) ≈ B + 1.86 × 10−13 1 + 4.23 × 104 Im sµ , S LD M 2/ TeV2 3 (A.18) 2 " 3L∗ 3L #! 0 + − −11 4 λ sµλ dµ B(KL → π µ µ ) ≈ (1.5 ± 0.3) × 10 1 + 4.23 × 10 Im 2 2 , M3 / TeV 3L∗ 3L 2 0 + − +1.2 −11 4 λ seλ de B(KL → π e e ) ≈ (3.2−0.8) × 10 1 + 4.23 × 10 Im 2 2 . M3 / TeV The experimental bounds, at 95% CL, on the studied Kaon branching fractions, whose expressions in the model are shown in the previous equations, are the following

+ − −9 B(KL → µ µ )SD . 2.5 × 10 [76] , + − −10 B(KS → µ µ ) < 2.5 × 10 [105] , 0 + − −10 (A.19) B(KL → π µ µ ) < 4.5 × 10 [107] , 0 + − −10 B(KL → π e e ) < 2.8 × 10 [109] .

A.3 LFV in Kaon decays

The LFV Kaon decays are absent in SM; thus any non zero value for the KL → µe branching ratio would be a signal of NP. In terms of LEFT coeﬃcients in the San Diego − − + basis [71], the branching ratios for the two leptonic channels KL → µ e+ and KL → e µ decays are given by (see also Ref. [139]):

2 2 2 !2 − + τKL fK mK mµ mµ B(KL → µ e ) = 1 − 2 × 128π mK " 2 2 ∗ V,LR µesd V,LL µesd mK S,RR eµds S,RL eµds [Led ] − [Led ] + (s ↔ d) − [Led ] − [Led ] + (s ↔ d) + mµ(ms + md) 2 2# V,RR µesd V,LR sdµe mK S,RR µesd S,RL µesd + [Led ] − [Lde ] + (s ↔ d) − [Led ] − [Led ] + (s ↔ d) , mµ(ms + md)

2 2 2 !2 + − τKL fK mK mµ mµ B(KL → µ e ) = 1 − 2 × (A.20) 128π mK " 2 2 V,LR µesd V,LL µesd mK S,RR µesd S,RL µesd [Led ] − [Led ] + (s ↔ d) + [Led ] − [Led ] + (s ↔ d) + mµ(ms + md) 2# 2 ∗ V,RR µesd V,LR sdµe mK S,RR eµds S,RL eµds + [Led ] − [Lde ] + (s ↔ d) + [Led ] − [Led ] + (s ↔ d) , mµ(ms + md)

where we approximated mµ me. In our analysis, the LEFT coeﬃcients are computed at one-loop level. However, in the relevant parameter space the tree-level contribution

21 from S3 exchange is dominant. In this approximation we have

2 λ3L∗λ3L + λ3L∗λ3L B(K → µ−e+) ≈ B(K → µ+e−) ≈ 2.5 × 10−2 dµ se sµ de , (A.21) L L 2 m3

where m3 = M3/TeV. The experimental bound, at 95% CL, on this LFV Kaon branching fraction is [110] ± ∓ −12 B(KL → µ e ) < 4.7 × 10 . (A.22) The contributions to LFV Kaon decays K+ → π+µ−e+ and K+ → π+e−µ+ can be described following Refs. [139, 140]. For simplicity, and since these will not be the most constraining observables in our model, we keep only the tree-level contribution from the S3 exchange:

∗ 2 λ3L λ3L B(K+ → π+µ−e+) ≈ 1.1 × 10−3 dµ se , 2 m3 (A.23) ∗ 2 λ3L λ3L B(K+ → π+e−µ+) ≈ 1.1 × 10−3 de sµ . 2 m3

The recent NA62 constraint on the ﬁrst decay is B(K+ → π+µ−e+) < 6.6 × 10−11 at 90% CL [111]. The best bound on the second decay mode is instead still from KTeV at BNL: B(K+ → π+e−µ+) < 1.3 × 10−11 at 90% CL [112].

A.4 Rare D-meson decays While in general charm physics suffers from large uncertainties in the theory predictions, some channels offer good sensitivity to NP [95,97,141]. Particularly relevant to our setup are the D0 → `+`− and D+ → π+`+`− decays, which probe the FCNC c → u`+`− transition. These can be described by the following effective Hamiltonian at the µc = mc scale: " # 4GF α X X Hcu`` ⊃ − √ (C O + C0O0) + C O , (A.24) eff 4π i i i i i i 2 i6=T,T 5 i=T,T 5 with

`(0) ¯ α `(0) ¯ α O9 = (¯uγαPL(R)c)(`γ `) , O10 = (¯uγαPL(R)c)(`γ γ5`) , O`(0) = (¯uP c)(``¯ ) , O`(0) = (¯uP c)(`γ¯ `) . S R(L) P R(L) 5 (A.25) (0) mc µν (0) mcgs a µν O7 = e (¯uσµνPR(L)c)F , O8 = e2 (¯uσµνT PR(L)c)Ga , ` 1 ¯ µν OT (T 5) = 2 (¯uσµνc)(`σ (γ5)`) .

` In the SM the only short-distance contributions arise along the C7 and C9 coeﬃcients (see e.g. [95] and references therein), while long-distance contributions are very challenging to evaluate due to mc ∼ ΛQCD and can reliably be computed only on the lattice. The purely leptonic decays are dominated by the long-distance contribution D0 → γ∗γ∗ → `+`−, which gives a branching ratio estimated to be of order ∼ 10−11. On the other hand the

22 present experimental constraints at 95%CL are still larger by several orders of magnitude:

B(D0 → e+e−) < 7.9×10−8 , B(D0 → µ+µ−) < 6.2×10−9 , B(D0 → µ±e∓) < 1.3×10−8 . (A.26) For this reason, when setting limits on NP one can thus safely neglect the SM contribution, writing the branching ratio as:

2 2 5 2 s 2 " 2 0 + − GF α mD0 fD0 4mµ 4mµ µ µ0 2 B(D → µ µ ) = 3 2 1 − 2 1 − 2 CS − CS + 64π mc ΓD m 0 mD D (A.27) 2 # µ µ0 2mµmc µ µ0 + CP − CP + 2 C10 − C10 , mD0

−13 where fD0 ≈ 206.7MeV, mD0 ≈ 1864.83MeV, τD0 ≈ 4.101 × 10 s. Also in the semileptonic decay D+ → π+`+`− one can neglect the SM contribution, given the present experimental accuracy. The 95%CL limit on the muonic channel gives the constraint [95,97]:

+ 2 µ+ 2 µ+ 2 µ+ 2 µ+ 2 µ 2 µ 2 1.3|C7 | + 1.3|C9 | + |C10 | + 2.6|CS | + 2.7|CP | + 0.4|CT | + 0.4|CT 5| + h µ+ µ∗i h µ+ µ+∗i h + µ+∗i h + µ∗i + 0.3 Re C9 CT + 1.1 Re C10 CP + 2.6 Re C7 C9 + 0.6 Re C7 CT < 1 , (A.28)

+ 0 where Ci = Ci + Ci. In terms of the LEFT coeﬃcients at the mc scale, the Ci’s are given by:

C` = 1 [LV,LR] + [LV,LL] , C` = 1 [LV,LR] − [LV,LL] , 9 2Ncu ue uc`` eu `ùc 10 2Ncu ue uc`` eu `ùc C`0 = 1 [LV,RR] + [LV,LR] , C`0 = 1 [LV,RR] − [LV,LR] , 9 2Ncu eu `ùc eu `ùc 10 2Ncu eu `ùc eu `ùc C = 1 e [L ] , C0 = 1 e [L ]∗ , (A.29) 7 Ncu mc uγ uc 7 Ncu mc uγ cu C = C = 1 [LS,RR] , C0 = −C0 = 1 [LS,RR]∗ , S P 2Ncu eu `ùc S P 2Ncu eu ``cu C`0 = 1 [LT,RR] + [LT,RR]∗ , C`0 = 1 [LT,RR] − [LT,RR]∗ , T Ncu eu `ùc eu ``cu T 5 Ncu eu `ùc eu ``cu

where N = 4√GF α . While in our numerical analysis we keep all the one-loop contri- cu 2 4π butions to these coeﬃcients arising from the two LQs (including one-loop matching and RGE), for illustrative purposes we report here only the tree-level contributions, which are expected to give the dominant terms. In this limit, the only non-vanishing coeﬃcients are:

1L∗ 1L 3L∗ 3L ! V,LL (0) ∗ λi` λj` λi` λj` [Leu ]`ùc = VuiVcj 2 + 2 , 2M1 2M3 1R∗ 1R V,RR (0) λu` λc` (A.30) [Leu ]`ùc = 2 , 2M1 1L∗ 1R S,RR (0) T,RR (0) λi` λc` [Leu ]`ùc = −4[Leu ]`ùc = −Vui 2 . 2M1

23 A.5 b → dµµ and b → dee decays We consider the leptonic branching ratios

B(B0 → µ+µ−) , B(B0 → e+e−) , (A.31) and the semileptonic ones

B(B+ → π+µ+µ−) , B(B+ → π+e+e−) . (A.32)

At partonic level, these processes are induced by the same low-energy operators as b → dµµ and b → dee, respectively. Among these operators, the ones that in our framework can arise at tree-level are

db`` ¯ ¯ α db`` ¯ ¯ α O9 = (dγαPL(R)b)(`γ `) , O10 = (dγαPL(R)b)(`γ γ5`) . (A.33) where ` = µ, e. The expressions for the WCs of the above operators in terms of the LEFT ones are −1 db`` Ndb V,LR V,LL C9(10) = 2 [Lde ]db`` ± [Led ]``db , (A.34) where N = 4√GF α V ∗ V . db 2 4π td tb

In our model, the tree-level generated effective vertices lead to left-handed four-fermion 9 10 interactions with Cdb`` = −Cdb``, which are the ones that interfere with the SM processes. (0)db`` The effective Lagrangian for these processes also contains scalar (OS ) and pseudoscalar (0)db`` (0)db (OP ) semileptonic operators, as well as dipoles (O7 ) and other two vector four- 0db`` fermion operators (O9(10)). However, their contributions to the b → dµµ(ee) transitions is negligible in the S1 + S3 model with respect to the one coming from the operators 9 10 in Eq. (A.33). Therefore, we take into account only the Cdb`` and Cdb`` contributions to the considered B-mesons decays, including however, in the numerical analysis, also the one-loop corrections to these coefficients. The branching ratios are given then by,

10 2 0 − + − + Cdb`` B(B → ` ` ) = B(Bd → ` ` )SM 1 + , C10 SM db`` (A.35) 9 10 2 + + + − + + + − Cdb`` − Cdb`` B(B → π ` ` ) = B(B → π ` ` )SM 1 + 9 SM 10 SM , Cdb`` − Cdb`` where [142] 9,SM 10,SM Cdb`` = −Cdb`` = 4.2 . (A.36) The SM predictions are [87,88],

0 −10 B(B → µµ)SM = (1.06 ± 0.09) × 10 , + + − + −8 B(B → π µ µ )SM = (2.04 ± 0.21) × 10 , 0 −15 (A.37) B(B → ee)SM = (2.48 ± 0.21) × 10 , + + − + −8 B(B → π e e )SM = (2.04 ± 0.24) × 10 . The branching ratios are measured for the case of the muons [89,90],

0 −10 B(B → µµ)exp = (1.1 ± 1.4) × 10 , + + − + −8 (A.38) B(B → π µ µ )exp = (1.83 ± 0.24) × 10 ,

24 while for the electrons only upper bounds exists so far [51],

0 −8 B(B → ee)exp < 8.3 × 10 , + + − + −8 (A.39) B(B → π e e )exp < 8 × 10 .

Numerically, at tree-level, we get:

0 − + + + − + 3L 3L∗ 2 B(B → ` ` ) B(B → π ` ` ) λb` λd` 0 − + = + + − + ≈ 1 + 139.23 2 2 . (A.40) B(B → ` ` )SM B(B → π ` ` )SM M3 / TeV

A.6 µ–e conversion in nuclei Searches for µ–e conversion in nuclei yield competitive LFV constraints on LQ interactions coupling light quarks to electrons and muons. The relevant observable is the branching ratio Γconversion BRµe = , (A.41) Γcapture

where Γconversion denotes the rate of conversion of a muon captured by the 1s nucleus A to an electron µA → e + A and Γcapture the normalising rate µA → νµ + A. The current upper bounds are

(Ti) −12 (Au) −13 BRµe < 4.3 × 10 and BRµe < 7 × 10 (90% CL) , (A.42)

as set by experiments on titanium [122] and gold targets [123], respectively. The COMET (JPARC) [52] and Mu2e (Fermilab) [53–55] experiments are expected to reach the 10−16 sensitivity, while the PRISM proposal at JPARC aims at the 10−18 level. Numerical calculations of the relevant transition matrix elements in the diﬀerent nuclei [143] yield the following formula for the conversion rates

2 ∗ (u) (d) (p) (u) (d) (n) Γconversion = 2GF ARD + (2gLV + gLV )V + (gLV + 2gLV )V +(G(u,p)g(u) + G(d,p)g(d) + G(s,p)g(s))S(p) S LS S LS S LS (A.43) 2 (u,n) (u) (d,n) (d) (s,n) (s) (n) +(GS gLS + GS gLS + GS gLS)S + (L ↔ R) .

(u,p) (d,n) (d,p) The numerical factors that are the same for both nuclei are GS = GS = 5.1, GS = (u,n) (s,p) (s,n) GS = 4.3, and GS = GS = 2.5 [144] and the ones that are diﬀerent are presented in Table5. The various expressions containing WCs are given in the LEFT basis by √ √ 2 2 g(q) = [LV,LL] + [LV,LR] , g(q) = [LV,RR] + [LV,LR] , LV G eq eµqq qe qqeµ RV G eq eµqq eq eµqq √F √F 2 2 g(q) = [LS,RR] + [LS,RL] , g(q) = [LS,RR]∗ + [LS,RL]∗ , (A.44) LS G eq eµqq eq eµqq RS G eq µeqq eq µeqq F √ √F 2e 2e ∗ AL = [Leγ]µe ,AR = [Leγ]µe . 4GF mµ 4GF mµ

25 Nucleus D[10−9GeV] V (p)[10−9GeV] V (n)[10−9GeV] 48 Ti22 9.91 4.54 5.37 197 Au79 21.68 11.17 16.75

(p) −9 (n) −9 6 −1 S [10 GeV] S [10 GeV] Γcapture[10 s ] 48 Ti22 4.22 4.99 2.59 197 Au79 7.04 10.53 13.07

Table 5: Data relevant for the µ → e conversion in nuclei [143].

In the model these receive both tree-level and loop contributions and, since the tree-level ones are typically suppressed by small couplings to 1st generation, loop eﬀects are important and the expressions are quite cumbersome. Expanding numerically these coeﬃcients, the terms with largest numerical pre-factor are (mi ≡ Mi/ TeV):

∗ 1L∗ 1L ∗ 3L∗ 3L λ3L∗ λ3L u VuiVuj λie λiµ VuiVuj λie λiµ d(s) d(s)e d(s)µ gLV ≈ 0.065 2 + 0.060 2 + . . . , gLV ≈ 0.12 2 + ..., m1 m3 m3 1R∗ 1R u λue λuµ d(s) gRV ≈ 0.062 2 + . . . , gRV ≈ 0 , m1 ∗ 1L∗ 1R u Vuiλie λuµ d(s) gLS ≈ −0.11 2 + . . . , gLS ≈ 0 , m1 1L 1R∗ u Vuiλiµ λue d(s) gRS ≈ −0.11 2 + . . . , gRS ≈ 0 , m1 1L∗ 1R λbµ λte 2 AL ≈ 0.055 2 (1 + log m1) + ..., m1 1L 1R∗ λbe λtµ 2 AR ≈ 0.055 2 (1 + log m1) + .... m1 (A.45)

A.7 Z boson couplings The couplings of the Z boson have been measured very precisely at LEP 1. At one-loop, the LQ model generates contributions to these very well measured quantities, which pose strong constraints on the model. The RGE-induced contributions in models aimed at addressing the B-anomalies have first been studied in Ref. [145–147]. Here we include the effect of finite corrections from the one-loop matching. The pseudo-observables corresponding to the pole couplings of the Z boson to fermions correspond to these effective Lagrangian couplings:

g Z SM Z ¯ µ Leff = − ([gψ ]ij + [δgψ ]ij)Zµ(ψiγ ψj) , (A.46) cθ

Z SM ψ 2 where [gψ ]ij = δij(T3L − Qψsθ). The measurements of these pseudo-observables and the predictions for the SM contributions can be found in [134]. Also often used is the eﬀective number of neutrino species at the Z peak, which depends on the couplings to neutrinos as Z 2 X [δgν ]αβ Nν = δαβ + , (A.47) [gZ ]SM αβ ν Z SM where [gν ] ≈ 0.502. The latest update on the extraction of Nν from LEP data is given in [135]. We collect in Table4 the limits used in our ﬁt.

26 Working at one-loop accuracy, there are two possible contributions to these pseudo- observables in our setup: tree-level contributions from the SMEFT operators7, one-loop matrix elements from the tree-level generated semileptonic operators. The result is: Z 2 1 (1+3) (1) Nc (1−3) (0) MS 2 [δg ]αβ ≈ − v [C ] (µ) + [C ] I (ui, m , µ) , eL 2 H` αβ 16π2 `q αβii L Z Z 2 1 (1) Nc (0) MS 2 [δg ]αβ ≈ − v [CHe] (µ) + [Ceu] I (ui, m , µ) , eR 2 αβ 16π2 αβii R Z 1 N [δgZ ] ≈ − v2 [C(1−3)](1)(µ) + c [C(1+3)](0) IMS(u , m2 , µ) , ν αβ 2 H` αβ 16π2 `q αβii L i Z 2 Z v (1+3) (1) (A.48) [δg ]ij ≈ − [C ] (µ) , dL 2 Hq ij 2 Z v ∗ (1−3) (1) [δg ]ij ≈ − VikV [C ] (µ) , uL 2 jl Hq kl 2 Z v (1) [δg ]ij ≈ − [CHu] (µ) , uR 2 ij 2 Z v (1) [δg ]ij ≈ − [CHd] (µ) , dR 2 ij

(1±3) (1) (3) where µ is the scale at which the operators are evaluated, CX = CX ± CX , and we included only one-loop matrix elements from up-type quark loops, since the top quark gives the dominant eﬀect. The expressions for the one-loop matching of the SMEFT operators to the LQ model can be found in [57]. The loop functions are given by 1h v2IMS(u , q2, µ) = 5q2(1 − 3s2) − 6m2 (1 + 6s2)+ L i 9 θ ui θ − 3 m2 − q2 + 3(2m2 + q2)s2 DiscB(q2, m , m )+ ui ui θ ui ui µ2 i + 3(−3m2 + q2 − 3q2s2) log , ui θ m2 ui 2 MS 2 5 2 2 2 2 v IR (ui, q , µ) = − q sθ + mu (2 − 4sθ)+ (A.49) 3 i + m2 − (2m2 + q2)s2 DiscB(q2, m , m )+ ui ui θ ui ui µ2 i + (m2 − q2s2) log , ui θ m2 ui pq2(q2 − 4m2) 2m2 − q2 + pq2(q2 − 4m2) DiscB(q2, m, m) = log . q2 2m2

MS 2 MS 2 Note that IL (c, mZ , µ = 1.5 TeV) = 0.102 + i0.044, IR (c, mZ , µ = 1.5 TeV) = −0.23 − MS 2 MS 2 i0.10 while IL (t, mZ , µ = 1.5 TeV) = −1.91 and IR (t, mZ , µ = 1.5 TeV) = 1.83. Furthermore, in the limit q2 → 0,

2 2 2 MS mt µ µ IR,L (t, 0, µ) ≈ ± 2 log 2 = ±1.95 log 2 . (A.50) v mZ 1.5 TeV

7 There might be some indirect contributions via modiﬁcations to GF , but are negligible in our model.

27 When all contributions are included, numerical expressions are: (λ3L)2 (λ1L)2 103δgZ ≈0.80 bα (1 + 0.35 log m2) + 0.59 bα (1 + 0.39 log m2)+ eαL 2 3 2 1 m3 m1 3L 2 1L 2 (λsα ) (λsα ) + 0.11 2 − 0.10 2 + ..., (A.51) m3 m1 (λ3L)2 (λ1L)2 (λ1L)2 λ3Lλ3L 103δgZ ≈1.31 bα (1 + 0.37 log m2) + 0.11 bα + 0.11 sα − 0.16 bα sα + ..., να 2 3 2 2 2 m3 m1 m1 m3 (A.52) (λ1R)2 (λ1R)2 (λ1R)2 103δgZ ≈ − 0.67 tα (1 + 0.37 log m2) + 0.059 cα + 0.030 uα + ..., (A.53) eαR 2 1 2 2 m1 m1 m1 (λ1L)2 + (λ1L)2 103δgZ ≈ − 0.044 bτ bµ + ..., (A.54) bL 2 m1 (λ1R)2 + (λ1R)2 103δgZ ≈ − 0.014 cτ cµ + ..., (A.55) cR 2 m1

where mi ≡ Mi/ TeV and the dots represent smaller, thus negligible, contributions. The present measurements are reported in Table4.

0 A.8 K/K 0 The observable K /K is strongly suppressed within the SM, not only due to the CKM hierarchy but also to an accidental SM low-energy property (the so-called ∆I = 1/2 rule). Consequently this observable is a particularly sensitive probe of non-standard sources of both CP and ﬂavor symmetry breaking. The experimental world average [51] reads

0 −4 (K /K )exp = (16.6 ± 2.3) × 10 . (A.56) 0 The main diﬃculty in making predictions for K /K resides with the theory side. The uncertainty on the SM calculation is plagued by long-distance eﬀects on the hadronic matrix elements of the relevant operators. The value we use in the present work is taken from [113] 0 −4 (K /K )SM ≈ (15 ± 7) × 10 , (A.57) and it is also in good overall agreement with other recent results [148,149]. The master formula [150] for the NP contribution is given by

0 X 0 (K /K )NP = Pi(µew)Im [Ci(µew) − Ci(µew)] . (A.58) i

In Table6 we list all the relevant operators Qi that are non-vanishing in our framework, together with their WCs Ci and the respective Pi values. The relations between these WCs and the LEFT ones are given by

ui V 1,LL ui V 1,LR ui 1 V 8,LL C = [L ] i i ,C = [L ] i i , C˜ = [L ] i i , V LL ud u u sd V LR du sdu u V LL 2 ud u u sd ui 1 V 8,LR di V,LL di V 1,LR C˜ = [L ] i i ,C = [L ] i i ,C = [L ] i i , V LR 2 du sdu u V LL dd sdd d V LR dd sdd d di V 8,LR CSLR = −[Ldd ]didsdi . (A.59)

28 Qi Pi u i i j µ j QV LL = (¯s γµPLd )(¯u γ PLu ) −3.3 ± 0.8 u i i j µ j QV LR = (¯s γµPLd )(¯u γ PRu ) −124 ± 11 ˜u i j j µ i QV LL = (¯s γµPLd )(¯u γ PLu ) 1.1 ± 1.2 ˜u i j j µ i QV LR = (¯s γµPLd )(¯u γ PRu ) −430 ± 40 d i i ¯j µ j QV LL = (¯s γµPLd )(d γ PLd ) 1.8 ± 0.5 d i i ¯j µ j QV LR = (¯s γµPLd )(d γ PRd ) 117 ± 11 d i i ¯j j QSLR = (¯s PLd )(d PRd ) 204 ± 20 s i i j µ j QV LL = (¯s γµPLd )(¯s γ PLs ) 0.1 ± 0.1 s i i j µ j QV LR = (¯s γµPLd )(¯s γ PRs ) −0.17 ± 0.04 s i i j j QSLR = (¯s PLd )(¯s PRs ) −0.4 ± 0.1 c i i j µ j QV LL = (¯s γµPLd )(¯c γ PLc ) 0.5 ± 0.1 c i i j µ j QV LR = (¯s γµPLd )(¯c γ PRc ) 0.8 ± 0.1 ˜c i j j µ i QV LL = (¯s γµPLd )(¯c γ PLc ) 0.7 ± 0.1 ˜c i j j µ i QV LR = (¯s γµPLd )(¯c γ PRc ) 1.3 ± 0.2 b i i ¯j µ j QV LL = (¯s γµPLd )(b γ PLb ) −0.33 ± 0.03 b i i ¯j µ j QV LR = (¯s γµPLd )(b γ PRb ) −0.22 ± 0.03 ˜b i j ¯j µ i QV LL = (¯s γµPLd )(b γ PLb ) 0.3 ± 0.1 ˜b i j ¯j µ i QV LR = (¯s γµPLd )(b γ PRb ) 0.4 ± 0.1 Table 6: The operators corresponding to the WCs entering the master formula for NP eﬀects in Eq. (A.58) (and are non-vanishing in the S1 + S3 framework) and the respective Pi values.

i i 0 where u can be the u or c quark and d the d, s or b quark. The Ci are the WCs of the corresponding chirality-ﬂipped operators obtained by interchanging PL ↔ PR. Keeping only the terms with the largest numerical pre-factors in our model, we ﬁnd the approximate expression (mi ≡ Mi/ TeV):

1R 2 1R 2 1R 1R 1R 1R 0 −6 2.4((λtµ ) + (λtτ ) ) − 0.3(λcµ λtµ + λcτ λtτ ) (K /K )NP ≈ 10 2 m1 1L 2 1L 2 1L 1L 2 3L 2 3L 2 2 −8 ((λsµ ) + (λsτ ) − 2λsτ λuτ ) log m1 −8 1.1((λsµ ) + (λsτ ) )(1 + log m3) + 10 2 + 10 2 . m1 m3 (A.60)

A.9 Neutron EDM The neutron electric-dipole moment (EDM) is a CP-odd observable whose measurement can set bounds on the imaginary parts of the model parameters. It receives contributions from the short range QCD interactions involving the EDMs dq and chromo-EDMs ˆ (cEDMs) dq of the light quarks (u, d and s) that are valence quarks of the nucleon. These (c)EDMs are deﬁned by the following Lagrangian

d dˆ L(c)EDM = −i q qσ¯ µνγ q F − i q qσ¯ µνT aγ q Ga . (A.61) 2 5 µν 2 5 µν

29 The quark dipole moments can be expressed in terms of the LEFT WCs ˆ du = − Im (2[Luγ]uu(1GeV)) , du = − Im (2[LuG]uu(1GeV)) , ˆ dd = − Im (2[Ldγ]dd(1GeV)) , dd = − Im (2[LdG]dd(1GeV)) , (A.62) ˆ ds = − Im (2[Ldγ]ss(1GeV)) , ds = − Im (2[LdG]ss(1GeV)) , where 1GeV is the neutron mass scale. One should notice, however, that in our model there is not any imaginary part for the Ldγ(dG) coefficients, as a consequence of the fact that the S1 LQ is not coupled to right-handed down quarks. In principle, the neutron EDM receives a contribution from the CP-odd gluon Weinberg operator (O ) as well. 3Ge However, in our model this operator arises at two-loops level. The neutron EDM can be expressed in terms of the quark (c)EMDs as follows v uG ˆ dG ˆ uγ dγ sγ dn = − √ β du + β dd + β du + β dd + β ds = 2 n n n n n (A.63) v uG ˆ uγ = −√ β du + β du , 2 n n qV where the βn are the hadronic matrix elements and in the second line we have taken into account the fact that in our model the down quark dipoles are vanishing. For the cEDMs qG uG +1.5 −4 the βn factors are estimated using QCD sum rules and they are βn = −2−0.8 ×10 e fm dG +3.1 −4 and βn = −4−1.7 × 10 e fm [151, 152]; one can notice that there are currently affected by a large uncertainty, bigger than 50%. The contributions from the strange quark cEDM are cancelled if a Peccei-Quinn mechanism is used to remove the contribution from the qγ QCD θ-term [153]. The βn matrix elements are evaluated with lattice QCD [154–157] and their most recent estimates are − √v βuγ = −0.204+0.011, − √v βdγ = +0.784 ± 0.028 2 n −0.010 2 n and − √v βsγ = −0.0027 ± 0.0016. 2 n The experimental bound, on the absolute value of the neutron EDM, at 90% CL is [129] −26 |dn| < 1.8 × 10 e · cm . (A.64) In our analysis, for simplicity we take into account the central values for the hadronic matrix elements. We find that EDMs give negligible contributions in constraining the model once the other observables entering the fit are taken into account.

References

(∗) − [1] BaBar Collaboration, J. P. Lees et al., Evidence for an excess of B¯ → D τ ν¯τ decays, Phys. Rev. Lett. 109 (2012) 101802, [arXiv:1205.5442].

(∗) − [2] BaBar Collaboration, J. P. Lees et al., Measurement of an Excess of B¯ → D τ ν¯τ Decays and Implications for Charged Higgs Bosons, Phys. Rev. D 88 (2013), no. 7 072012, [arXiv:1303.0571]. [3] LHCb Collaboration, R. Aaij et al., Measurement of the ratio of branching fractions 0 ∗+ − 0 ∗+ − B(B¯ → D τ ν¯τ )/B(B¯ → D µ ν¯µ), Phys. Rev. Lett. 115 (2015), no. 11 111803, [arXiv:1506.08614]. [Erratum: Phys.Rev.Lett. 115, 159901 (2015)]. [4] Belle Collaboration, M. Huschle et al., Measurement of the branching ratio of (∗) − (∗) − B¯ → D τ ν¯τ relative to B¯ → D ` ν¯` decays with hadronic tagging at Belle, Phys. Rev. D 92 (2015), no. 7 072014, [arXiv:1507.03233].

30 [5] Belle Collaboration, Y. Sato et al., Measurement of the branching ratio of 0 ∗+ − 0 ∗+ − B¯ → D τ ν¯τ relative to B¯ → D ` ν¯` decays with a semileptonic tagging method, Phys. Rev. D 94 (2016), no. 7 072007, [arXiv:1607.07923]. [6] Belle Collaboration, S. Hirose et al., Measurement of the τ lepton polarization and ∗ ∗ − R(D ) in the decay B¯ → D τ ν¯τ , Phys. Rev. Lett. 118 (2017), no. 21 211801, [arXiv:1612.00529]. [7] Belle Collaboration, S. Hirose et al., Measurement of the τ lepton polarization and ∗ ∗ − R(D ) in the decay B¯ → D τ ν¯τ with one-prong hadronic τ decays at Belle, Phys. Rev. D 97 (2018), no. 1 012004, [arXiv:1709.00129].

0 ∗− + [8] LHCb Collaboration, R. Aaij et al., Measurement of the ratio of the B → D τ ντ 0 ∗− + and B → D µ νµ branching fractions using three-prong τ-lepton decays, Phys. Rev. Lett. 120 (2018), no. 17 171802, [arXiv:1708.08856]. [9] LHCb Collaboration, R. Aaij et al., Test of Lepton Flavor Universality by the 0 ∗− + measurement of the B → D τ ντ branching fraction using three-prong τ decays, Phys. Rev. D 97 (2018), no. 7 072013, [arXiv:1711.02505]. [10] LHCb Collaboration, B. G. Siddi, Measurement of R(D∗) with hadronic τ decays, J. Phys. Conf. Ser. 956 (2018), no. 1 012015.

[11] Belle Collaboration, G. Caria et al., Measurement of R(D) and R(D∗) with a semileptonic tagging method, Phys. Rev. Lett. 124 (2020), no. 16 161803, [arXiv:1910.05864]. [12] HFLAV Collaboration, Y. Amhis et al., Averages of b-hadron, c-hadron, and τ-lepton properties as of summer 2016, Eur. Phys. J. C 77 (2017), no. 12 895, [arXiv:1612.07233]. [13] LHCb Collaboration, R. Aaij et al., Test of lepton universality using B+ → K+`+`− decays, Phys. Rev. Lett. 113 (2014) 151601, [arXiv:1406.6482]. [14] LHCb Collaboration, R. Aaij et al., Test of lepton universality with B0 → K∗0`+`− decays, JHEP 08 (2017) 055, [arXiv:1705.05802]. [15] LHCb Collaboration, R. Aaij et al., Search for lepton-universality violation in B+ → K+`+`− decays, Phys. Rev. Lett. 122 (2019), no. 19 191801, [arXiv:1903.09252]. [16] Belle Collaboration, A. Abdesselam et al., Test of Lepton-Flavor Universality in B → K∗`+`− Decays at Belle, Phys. Rev. Lett. 126 (2021), no. 16 161801, [arXiv:1904.02440]. [17] LHCb Collaboration, R. Aaij et al., Test of lepton universality in beauty-quark decays, arXiv:2103.11769. [18] LHCb Collaboration, R. Aaij et al., Measurement of Form-Factor-Independent Observables in the Decay B0 → K∗0µ+µ−, Phys. Rev. Lett. 111 (2013) 191801, [arXiv:1308.1707]. [19] LHCb Collaboration, R. Aaij et al., Angular analysis of the B0 → K∗0µ+µ− decay using 3 fb−1 of integrated luminosity, JHEP 02 (2016) 104, [arXiv:1512.04442]. [20] LHCb Collaboration, R. Aaij et al., Angular analysis and diﬀerential branching fraction 0 + − of the decay Bs → φµ µ , JHEP 09 (2015) 179, [arXiv:1506.08777].

31 0 + − [21] LHCb Collaboration, R. Aaij et al., Measurement of the Bs → µ µ branching fraction and eﬀective lifetime and search for B0 → µ+µ− decays, Phys. Rev. Lett. 118 (2017), no. 19 191801, [arXiv:1703.05747].

0 0 [22] ATLAS Collaboration, M. Aaboud et al., Study of the rare decays of Bs and B mesons into muon pairs using data collected during 2015 and 2016 with the ATLAS detector, JHEP 04 (2019) 098, [arXiv:1812.03017].

[23] LHCb Collaboration, R. Aaij et al., Measurement of CP -Averaged Observables in the B0 → K∗0µ+µ− Decay, Phys. Rev. Lett. 125 (2020), no. 1 011802, [arXiv:2003.04831].

[24] D. Lancierini, G. Isidori, P. Owen, and N. Serra, On the signiﬁcance of new physics in b → s`+`− decays, arXiv:2104.05631.

[25] W. Altmannshofer and P. Stangl, New Physics in Rare B Decays after Moriond 2021, arXiv:2103.13370.

[26] L.-S. Geng, B. Grinstein, S. J¨ager,S.-Y. Li, J. Martin Camalich, and R.-X. Shi, Implications of new evidence for lepton-universality violation in b → s`+`− decays, arXiv:2103.12738.

[27] M. Algueró,B. Capdevila, S. Descotes-Genon, J. Matias, and M. Novoa-Brunet, b → s`` global fits after Moriond 2021 results, in 55th Rencontres de Moriond on Electroweak Interactions and Unified Theories, 4, 2021. arXiv:2104.08921.

[28] M. Alguer´o,B. Capdevila, A. Crivellin, S. Descotes-Genon, P. Masjuan, J. Matias, M. Novoa Brunet, and J. Virto, Emerging patterns of New Physics with and without Lepton Flavour Universal contributions, Eur. Phys. J. C 79 (2019), no. 8 714, [arXiv:1903.09578]. [Addendum: Eur.Phys.J.C 80, 511 (2020)].

[29] M. Ciuchini, A. M. Coutinho, M. Fedele, E. Franco, A. Paul, L. Silvestrini, and M. Valli, New Physics in b → s`+`− confronts new data on Lepton Universality, Eur. Phys. J. C 79 (2019), no. 8 719, [arXiv:1903.09632].

[30] A. Crivellin, D. M¨uller,and T. Ota, Simultaneous explanation of R(D(∗)) and b → sµ+µ−: the last scalar leptoquarks standing, JHEP 09 (2017) 040, [arXiv:1703.09226].

[31] D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, B-physics anomalies: a guide to combined explanations, JHEP 11 (2017) 044, [arXiv:1706.07808].

[32] D. Marzocca, Addressing the B-physics anomalies in a fundamental Composite Higgs Model, JHEP 07 (2018) 121, [arXiv:1803.10972].

[33] P. Arnan, D. Becirevic, F. Mescia, and O. Sumensari, Probing low energy scalar leptoquarks by the leptonic W and Z couplings, JHEP 02 (2019) 109, [arXiv:1901.06315].

[34] A. Crivellin, D. M¨uller,and F. Saturnino, Flavor Phenomenology of the Leptoquark Singlet-Triplet Model, JHEP 06 (2020) 020, [arXiv:1912.04224].

[35] S. Saad, Combined explanations of (g − 2)µ, RD(∗) , RK(∗) anomalies in a two-loop radiative neutrino mass model, Phys. Rev. D 102 (2020), no. 1 015019, [arXiv:2005.04352].

32 [36] A. Crivellin, D. Müller,and F. Saturnino, Leptoquarks in oblique corrections and Higgs signal strength: status and prospects, JHEP 11 (2020) 094, [arXiv:2006.10758]. [37] V. Gherardi, D. Marzocca, and E. Venturini, Low-energy phenomenology of scalar leptoquarks at one-loop accuracy, JHEP 01 (2021) 138, [arXiv:2008.09548]. [38] L. Da Rold and F. Lamagna, A model for the Singlet-Triplet Leptoquarks, arXiv:2011.10061. [39] M. Bordone, O. Catà,T. Feldmann, and R. Mandal, Constraining flavour patterns of scalar leptoquarks in the effective field theory, JHEP 03 (2021) 122, [arXiv:2010.03297]. [40] Muon g-2 Collaboration, G. W. Bennett et al., Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL, Phys. Rev. D 73 (2006) 072003, [hep-ex/0602035]. [41] Muon g-2 Collaboration, B. Abi et al., Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm, Phys. Rev. Lett. 126 (2021), no. 14 141801, [arXiv:2104.03281]. [42] T. Aoyama et al., The anomalous magnetic moment of the muon in the Standard Model, Phys. Rept. 887 (2020) 1–166, [arXiv:2006.04822]. [43] B. Gripaios, Composite Leptoquarks at the LHC, JHEP 02 (2010) 045, [arXiv:0910.1789]. [44] B. Gripaios, M. Nardecchia, and S. A. Renner, Composite leptoquarks and anomalies in B-meson decays, JHEP 05 (2015) 006, [arXiv:1412.1791]. [45] E. Alvarez, L. Da Rold, A. Juste, M. Szewc, and T. Vazquez Schroeder, A composite pNGB leptoquark at the LHC, JHEP 12 (2018) 027, [arXiv:1808.02063]. [46] S. Davidson, D. C. Bailey, and B. A. Campbell, Model independent constraints on leptoquarks from rare processes, Z. Phys. C 61 (1994) 613–644, [hep-ph/9309310]. [47] I. Dorˇsner,S. Fajfer, A. Greljo, J. F. Kamenik, and N. Koˇsnik, Physics of leptoquarks in precision experiments and at particle colliders, Phys. Rept. 641 (2016) 1–68, [arXiv:1603.04993]. [48] V. Gherardi, D. Marzocca, M. Nardecchia, and A. Romanino, Rank-One Flavor Violation and B-meson anomalies, JHEP 10 (2019) 112, [arXiv:1903.10954]. [49] A. Crivellin and L. Schnell, Combined Constraints on First Generation Leptoquarks, Phys. Rev. D 103 (2021) 115023, [arXiv:2104.06417]. [50] NA62 Collaboration, E. Cortina Gil et al., Measurement of the very rare K+ → π+νν¯ decay, arXiv:2103.15389. [51] Particle Data Group Collaboration, M. Tanabashi et al., Review of Particle Physics, Phys. Rev. D 98 (2018), no. 3 030001. [52] COMET Collaboration, Y. Kuno, A search for muon-to-electron conversion at J-PARC: The COMET experiment, PTEP 2013 (2013) 022C01. [53] C. Ankenbrandt, D. Bogert, F. DeJongh, S. Geer, D. McGinnis, D. Neuffer, M. Popovic, and E. Prebys, Using the Fermilab proton source for a muon to electron conversion experiment, physics/0611124.

33 [54] Mu2e Collaboration, K. Knoepfel et al., Feasibility Study for a Next-Generation Mu2e Experiment, in Community Summer Study 2013: Snowmass on the Mississippi, 7, 2013. arXiv:1307.1168.

[55] Mu2e Collaboration, L. Bartoszek et al., Mu2e Technical Design Report, arXiv:1501.05241.

[56] Mu3e Collaboration, K. Arndt et al., Technical design of the phase I Mu3e experiment, arXiv:2009.11690.

[57] V. Gherardi, D. Marzocca, and E. Venturini, Matching scalar leptoquarks to the SMEFT at one loop, JHEP 07 (2020) 225, [arXiv:2003.12525]. [Erratum: JHEP 01, 006 (2021)].

[58] R. Barbieri, G. Isidori, J. Jones-Perez, P. Lodone, and D. M. Straub, U(2) and Minimal Flavour Violation in Supersymmetry, Eur. Phys. J. C 71 (2011) 1725, [arXiv:1105.2296].

[59] R. Barbieri, D. Buttazzo, F. Sala, and D. M. Straub, Flavour physics from an approximate U(2)3 symmetry, JHEP 07 (2012) 181, [arXiv:1203.4218].

[60] R. Barbieri, D. Buttazzo, F. Sala, D. M. Straub, and A. Tesi, A 125 GeV composite Higgs boson versus ﬂavour and electroweak precision tests, JHEP 05 (2013) 069, [arXiv:1211.5085].

[61] A. Greljo, G. Isidori, and D. Marzocca, On the breaking of Lepton Flavor Universality in B decays, JHEP 07 (2015) 142, [arXiv:1506.01705].

[62] R. Barbieri, G. Isidori, A. Pattori, and F. Senia, Anomalies in B-decays and U(2) ﬂavour symmetry, Eur. Phys. J. C 76 (2016), no. 2 67, [arXiv:1512.01560].

[63] M. Bordone, G. Isidori, and S. Triﬁnopoulos, Semileptonic B-physics anomalies: A general EFT analysis within U(2)n ﬂavor symmetry, Phys. Rev. D 96 (2017), no. 1 015038, [arXiv:1702.07238].

[64] J. Fuentes-Mart´ın,G. Isidori, J. Pag`es,and K. Yamamoto, With or without U(2)? Probing non-standard ﬂavor and helicity structures in semileptonic B decays, Phys. Lett. B 800 (2020) 135080, [arXiv:1909.02519].

[65] R. Barbieri, C. W. Murphy, and F. Senia, B-decay Anomalies in a Composite Leptoquark Model, Eur. Phys. J. C 77 (2017), no. 1 8, [arXiv:1611.04930].

[66] A. Greljo and B. A. Stefanek, Third family quark–lepton uniﬁcation at the TeV scale, Phys. Lett. B 782 (2018) 131–138, [arXiv:1802.04274].

[67] M. Bordone, C. Cornella, J. Fuentes-Mart´ın, and G. Isidori, Low-energy signatures of the PS3 model: from B-physics anomalies to LFV, JHEP 10 (2018) 148, [arXiv:1805.09328].

[68] S. Triﬁnopoulos, Revisiting R-parity violating interactions as an explanation of the B-physics anomalies, Eur. Phys. J. C 78 (2018), no. 10 803, [arXiv:1807.01638].

[69] L. Di Luzio, J. Fuentes-Martin, A. Greljo, M. Nardecchia, and S. Renner, Maximal Flavour Violation: a Cabibbo mechanism for leptoquarks, JHEP 11 (2018) 081, [arXiv:1808.00942].

34 [70] S. Triﬁnopoulos, B -physics anomalies: The bridge between R -parity violating supersymmetry and ﬂavored dark matter, Phys. Rev. D 100 (2019), no. 11 115022, [arXiv:1904.12940].

[71] E. E. Jenkins, A. V. Manohar, and P. Stoﬀer, Low-Energy Eﬀective Field Theory below the Electroweak Scale: Operators and Matching, JHEP 03 (2018) 016, [arXiv:1709.04486].

[72] W. Dekens and P. Stoffer, Low-energy effective field theory below the electroweak scale: matching at one loop, JHEP 10 (2019) 197, [arXiv:1908.05295].

[73] Y. Grossman and Y. Nir, K(L) —> pi0 neutrino anti-neutrino beyond the standard model, Phys. Lett. B 398 (1997) 163–168, [hep-ph/9701313].

[74] NA62 Collaboration, G. Ruggiero, Status of the CERN NA62 Experiment, J. Phys. Conf. Ser. 800 (2017), no. 1 012023.

[75] KLEVER Project Collaboration, F. Ambrosino et al., KLEVER: An experiment to 0 measure BR(KL → π νν¯) at the CERN SPS, arXiv:1901.03099. [76] G. Isidori and R. Unterdorfer, On the short distance constraints from K(L,S) —> mu+ mu-, JHEP 01 (2004) 009, [hep-ph/0311084].

[77] LHCb Collaboration, R. Aaij et al., Physics case for an LHCb Upgrade II - Opportunities in ﬂavour physics, and beyond, in the HL-LHC era, arXiv:1808.08865.

[78] Y. Grossman, G. Isidori, and H. Murayama, Lepton ﬂavor mixing and K —> pi nu anti-nu decays, Phys. Lett. B 588 (2004) 74–80, [hep-ph/0311353].

[79] J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl, and D. M. Straub, B-decay discrepancies after Moriond 2019, Eur. Phys. J. C 80 (2020), no. 3 252, [arXiv:1903.10434].

[80] M. Bordone, M. Jung, and D. van Dyk, Theory determination of B¯ → D(∗)`−ν¯ form 2 factors at O(1/mc ), Eur. Phys. J. C 80 (2020), no. 2 74, [arXiv:1908.09398]. [81] Belle Collaboration, A. Abdesselam et al., Measurement of the D∗− polarization in the 0 ∗− + decay B → D τ ντ , in 10th International Workshop on the CKM Unitarity Triangle, 3, 2019. arXiv:1903.03102.

[82] A. G. Akeroyd and C.-H. Chen, Constraint on the branching ratio of Bc → τν¯ from LEP1 and consequences for R(D(∗)) anomaly, Phys. Rev. D 96 (2017), no. 7 075011, [arXiv:1708.04072].

[83] BaBar Collaboration, B. Aubert et al., Measurements of the Semileptonic Decays anti-B —> D l anti-nu and anti-B —> D* l anti-nu Using a Global Fit to D X l anti-nu Final States, Phys. Rev. D 79 (2009) 012002, [arXiv:0809.0828].

[84] Belle Collaboration, R. Glattauer et al., Measurement of the decay B → D`ν` in fully reconstructed events and determination of the Cabibbo-Kobayashi-Maskawa matrix element |Vcb|, Phys. Rev. D 93 (2016), no. 3 032006, [arXiv:1510.03657]. [85] A. J. Buras, J. Girrbach-Noe, C. Niehoﬀ, and D. M. Straub, B → K(∗)νν decays in the Standard Model and beyond, JHEP 02 (2015) 184, [arXiv:1409.4557].

35 [86] Belle Collaboration, J. Grygier et al., Search for B → hνν¯ decays with semileptonic tagging at Belle, Phys. Rev. D 96 (2017), no. 9 091101, [arXiv:1702.03224]. [Addendum: Phys.Rev.D 97, 099902 (2018)].

[87] C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou, and M. Steinhauser, + − Bs,d → l l in the Standard Model with Reduced Theoretical Uncertainty, Phys. Rev. Lett. 112 (2014) 101801, [arXiv:1311.0903].

[88] Fermilab Lattice, MILC Collaboration, J. A. Bailey et al., B → π`` form factors for new-physics searches from lattice QCD, Phys. Rev. Lett. 115 (2015), no. 15 152002, [arXiv:1507.01618].

0 + − [89] CMS Collaboration, S. Chatrchyan et al., Measurement of the Bs → µ µ Branching Fraction and Search for B0 → µ+µ− with the CMS Experiment, Phys. Rev. Lett. 111 (2013) 101804, [arXiv:1307.5025].

[90] LHCb Collaboration, R. Aaij et al., First measurement of the diﬀerential branching fraction and CP asymmetry of the B± → π±µ+µ− decay, JHEP 10 (2015) 034, [arXiv:1509.00414].

[91] LHCb Collaboration, R. Aaij et al., Search for the lepton-ﬂavour-violating decays 0 ± ∓ 0 ± ∓ Bs → τ µ and B → τ µ , Phys. Rev. Lett. 123 (2019), no. 21 211801, [arXiv:1905.06614].

[92] BaBar Collaboration, J. P. Lees et al., A search for the decay modes B+− → h+−τ +−l, Phys. Rev. D 86 (2012) 012004, [arXiv:1204.2852].

[93] LHCb Collaboration, R. Aaij et al., Search for the lepton ﬂavour violating decay + + − + ∗0 B → K µ τ using Bs2 decays, JHEP 06 (2020) 129, [arXiv:2003.04352]. [94] S. Aoki et al., Review of lattice results concerning low-energy particle physics, Eur. Phys. J. C 77 (2017), no. 2 112, [arXiv:1607.00299].

[95] H. Gisbert, M. Golz, and D. S. Mitzel, Theoretical and experimental status of rare charm decays, Mod. Phys. Lett. A 36 (2021), no. 04 2130002, [arXiv:2011.09478].

[96] LHCb Collaboration, R. Aaij et al., Search for the rare decay D0 → µ+µ−, Phys. Lett. B 725 (2013) 15–24, [arXiv:1305.5059].

[97] S. de Boer and G. Hiller, Flavor and new physics opportunities with rare charm decays into leptons, Phys. Rev. D 93 (2016), no. 7 074001, [arXiv:1510.00311].

[98] LHCb Collaboration, R. Aaij et al., Searches for 25 rare and forbidden decays of D+ + and Ds mesons, arXiv:2011.00217. [99] A. J. Buras, Weak Hamiltonian, CP violation and rare decays, in Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, 6, 1998. hep-ph/9806471.

[100] NA62 Collaboration, E. Cortina Gil et al., An investigation of the very rare K+ → π+νν decay, JHEP 11 (2020) 042, [arXiv:2007.08218].

0 0 0 [101] KOTO Collaboration, J. K. Ahn et al., Search for the KL →π νν and KL →π X decays at the J-PARC KOTO experiment, Phys. Rev. Lett. 122 (2019), no. 2 021802, [arXiv:1810.09655].

36 + + + − [102] G. Buchalla and A. J. Buras, The rare decays K → π νν¯ and KL → µ µ beyond leading logarithms, Nucl. Phys. B 412 (1994) 106–142, [hep-ph/9308272].

[103] G. Ecker and A. Pich, The Longitudinal muon polarization in K(L) —> mu+ mu-, Nucl. Phys. B 366 (1991) 189–205.

[104] G. D’Ambrosio and T. Kitahara, Direct CP Violation in K → µ+µ−, Phys. Rev. Lett. 119 (2017), no. 20 201802, [arXiv:1707.06999].

0 + − [105] LHCb Collaboration, R. Aaij et al., Constraints on the KS → µ µ Branching Fraction, Phys. Rev. Lett. 125 (2020), no. 23 231801, [arXiv:2001.10354].

0 + − [106] G. Isidori, C. Smith, and R. Unterdorfer, The Rare decay KL → π µ µ within the SM, Eur. Phys. J. C 36 (2004) 57–66, [hep-ph/0404127].

0 + − [107] KTEV Collaboration, A. Alavi-Harati et al., Search for the Decay KL → π µ µ , Phys. Rev. Lett. 84 (2000) 5279–5282, [hep-ex/0001006].

[108] G. Buchalla, G. D’Ambrosio, and G. Isidori, Extracting short distance physics from K(L,S) —> pi0 e+ e- decays, Nucl. Phys. B 672 (2003) 387–408, [hep-ph/0308008].

[109] KTeV Collaboration, A. Alavi-Harati et al., Search for the rare decay K(L) —> pi0 e+ e-, Phys. Rev. Lett. 93 (2004) 021805, [hep-ex/0309072].

[110] BNL Collaboration, D. Ambrose et al., New limit on muon and electron lepton number violation from K0(L) —> mu+- e-+ decay, Phys. Rev. Lett. 81 (1998) 5734–5737, [hep-ex/9811038].

[111] NA62 Collaboration, E. Cortina Gil et al., Search for lepton number and ﬂavour violation in K+ and π0 decays, arXiv:2105.06759.

[112] KTeV Collaboration, E. Abouzaid et al., Search for lepton ﬂavor violating decays of the neutral kaon, Phys. Rev. Lett. 100 (2008) 131803, [arXiv:0711.3472].

[113] H. Gisbert and A. Pich, Updated Standard Model Prediction for ε0/ε, Nucl. Part. Phys. Proc. 300-302 (2018) 137–144, [arXiv:1810.04904].

[114] UTﬁt Collaboration, M. Bona et al., Model-independent constraints on ∆F = 2 operators and the scale of new physics, JHEP 03 (2008) 049, [arXiv:0707.0636].

[115] UTﬁt Collaboration, Latest results from UTﬁt , http://www.utfit.org/UTfit/ 2016.

[116] A. Pich, Precision Tau Physics, Prog. Part. Nucl. Phys. 75 (2014) 41–85, [arXiv:1310.7922].

[117] Belle Collaboration, Y. Miyazaki et al., Search for Lepton-Flavor-Violating tau Decays into a Lepton and a Vector Meson, Phys. Lett. B 699 (2011) 251–257, [arXiv:1101.0755].

[118] K. Hayasaka et al., Search for Lepton Flavor Violating Tau Decays into Three Leptons with 719 Million Produced Tau+Tau- Pairs, Phys. Lett. B 687 (2010) 139–143, [arXiv:1001.3221].

[119] BaBar Collaboration, B. Aubert et al., Searches for Lepton Flavor Violation in the Decays tau+- —> e+- gamma and tau+- —> mu+- gamma, Phys. Rev. Lett. 104 (2010) 021802, [arXiv:0908.2381].

37 [120] MEG Collaboration, A. M. Baldini et al., Search for the lepton ﬂavour violating decay µ+ → e+γ with the full dataset of the MEG experiment, Eur. Phys. J. C 76 (2016), no. 8 434, [arXiv:1605.05081].

[121] SINDRUM Collaboration, U. Bellgardt et al., Search for the Decay mu+ —> e+ e+ e-, Nucl. Phys. B 299 (1988) 1–6.

[122] SINDRUM II Collaboration, C. Dohmen et al., Test of lepton ﬂavor conservation in mu —> e conversion on titanium, Phys. Lett. B 317 (1993) 631–636.

[123] SINDRUM II Collaboration, W. H. Bertl et al., A Search for muon to electron conversion in muonic gold, Eur. Phys. J. C 47 (2006) 337–346.

[124] M. Pospelov and A. Ritz, CKM benchmarks for electron electric dipole moment experiments, Phys. Rev. D 89 (2014), no. 5 056006, [arXiv:1311.5537].

[125] C. Smith and S. Touati, Electric dipole moments with and beyond ﬂavor invariants, Nucl. Phys. B 924 (2017) 417–452, [arXiv:1707.06805].

[126] ACME Collaboration, V. Andreev et al., Improved limit on the electric dipole moment of the electron, Nature 562 (2018), no. 7727 355–360.

[127] Muon (g-2) Collaboration, G. W. Bennett et al., An Improved Limit on the Muon Electric Dipole Moment, Phys. Rev. D 80 (2009) 052008, [arXiv:0811.1207].

[128] I. B. Khriplovich and A. R. Zhitnitsky, What Is the Value of the Neutron Electric Dipole Moment in the Kobayashi-Maskawa Model?, Phys. Lett. B 109 (1982) 490–492.

[129] nEDM Collaboration, C. Abel et al., Measurement of the permanent electric dipole moment of the neutron, Phys. Rev. Lett. 124 (2020), no. 8 081803, [arXiv:2001.11966].

2 [130] A. Keshavarzi, D. Nomura, and T. Teubner, g − 2 of charged leptons, α(MZ ) , and the hyperﬁne splitting of muonium, Phys. Rev. D 101 (2020), no. 1 014029, [arXiv:1911.00367].

[131] R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. M¨uller, Measurement of the ﬁne-structure constant as a test of the Standard Model, Science 360 (2018) 191, [arXiv:1812.04130].

[132] D. Hanneke, S. Fogwell, and G. Gabrielse, New Measurement of the Electron Magnetic Moment and the Fine Structure Constant, Phys. Rev. Lett. 100 (2008) 120801, [arXiv:0801.1134].

[133] DELPHI Collaboration, J. Abdallah et al., Study of tau-pair production in photon-photon collisions at LEP and limits on the anomalous electromagnetic moments of the tau lepton, Eur. Phys. J. C 35 (2004) 159–170, [hep-ex/0406010].

[134] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group, SLD Heavy Flavour Group Collaboration, S. Schael et al., Precision electroweak measurements on the Z resonance, Phys. Rept. 427 (2006) 257–454, [hep-ex/0509008].

[135] P. Janot and S. Jadach, Improved Bhabha cross section at LEP and the number of light neutrino species, Phys. Lett. B 803 (2020) 135319, [arXiv:1912.02067].

38 [136] ATLAS Collaboration, M. Aaboud et al., Search for new high-mass phenomena in the √ dilepton final state using 36 fb−1 of proton-proton collision data at s = 13 TeV with the ATLAS detector, JHEP 10 (2017) 182, [arXiv:1707.02424]. [137] A. Angelescu, D. Beˇcirević,D. A. Faroughy, and O. Sumensari, Closing the window on single leptoquark solutions to the B-physics anomalies, JHEP 10 (2018) 183, [arXiv:1808.08179]. [138] ATLAS Collaboration, M. Aaboud et al., Search for additional heavy neutral Higgs and √ gauge bosons in the ditau final state produced in 36 fb−1 of pp collisions at s = 13 TeV with the ATLAS detector, JHEP 01 (2018) 055, [arXiv:1709.07242]. [139] A. Angelescu, D. A. Faroughy, and O. Sumensari, Lepton Flavor Violation and Dilepton Tails at the LHC, Eur. Phys. J. C 80 (2020), no. 7 641, [arXiv:2002.05684]. [140] N. Carrasco, P. Lami, V. Lubicz, L. Riggio, S. Simula, and C. Tarantino, K → π semileptonic form factors with Nf = 2 + 1 + 1 twisted mass fermions, Phys. Rev. D 93 (2016), no. 11 114512, [arXiv:1602.04113]. [141] S. Fajfer and N. Koˇsnik, Prospects of discovering new physics in rare charm decays, Eur. Phys. J. C 75 (2015), no. 12 567, [arXiv:1510.00965].

+ − [142] A. Khodjamirian and A. V. Rusov, Bs → K`ν` and B(s) → π(K)` ` decays at large recoil and CKM matrix elements, JHEP 08 (2017) 112, [arXiv:1703.04765]. [143] R. Kitano, M. Koike, and Y. Okada, Detailed calculation of lepton flavor violating muon electron conversion rate for various nuclei, Phys. Rev. D 66 (2002) 096002, [hep-ph/0203110]. [Erratum: Phys.Rev.D 76, 059902 (2007)]. [144] T. S. Kosmas, S. Kovalenko, and I. Schmidt, Nuclear muon- e- conversion in strange quark sea, Phys. Lett. B 511 (2001) 203, [hep-ph/0102101]. [145] F. Feruglio, P. Paradisi, and A. Pattori, Revisiting Lepton Flavor Universality in B Decays, Phys. Rev. Lett. 118 (2017), no. 1 011801, [arXiv:1606.00524]. [146] F. Feruglio, P. Paradisi, and A. Pattori, On the Importance of Electroweak Corrections for B Anomalies, JHEP 09 (2017) 061, [arXiv:1705.00929]. [147] C. Cornella, F. Feruglio, and P. Paradisi, Low-energy Effects of Lepton Flavour Universality Violation, JHEP 11 (2018) 012, [arXiv:1803.00945]. [148] V. Cirigliano, H. Gisbert, A. Pich, and A. Rodr´ıguez-Sánchez, A complete update of ε0/ε in the Standard Model, PoS EPS-HEP2019 (2020) 238, [arXiv:1911.06554]. [149] J. Aebischer, C. Bobeth, and A. J. Buras, On the importance of NNLO QCD and isospin-breaking corrections in ε0/ε, Eur. Phys. J. C 80 (2020), no. 1 1, [arXiv:1909.05610]. [150] J. Aebischer, C. Bobeth, and A. J. Buras, ε0/ε in the Standard Model at the Dawn of the 2020s, Eur. Phys. J. C 80 (2020), no. 8 705, [arXiv:2005.05978]. [151] J. Hisano, J. Y. Lee, N. Nagata, and Y. Shimizu, Reevaluation of Neutron Electric Dipole Moment with QCD Sum Rules, Phys. Rev. D 85 (2012) 114044, [arXiv:1204.2653]. [152] J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Electric Dipole Moments of Nucleons, Nuclei, and Atoms: The Standard Model and Beyond, Prog. Part. Nucl. Phys. 71 (2013) 21–74, [arXiv:1303.2371].

39 [153] M. Pospelov and A. Ritz, Neutron EDM from electric and chromoelectric dipole moments of quarks, Phys. Rev. D 63 (2001) 073015, [hep-ph/0010037].

[154] T. Bhattacharya, V. Cirigliano, R. Gupta, H.-W. Lin, and B. Yoon, Neutron Electric Dipole Moment and Tensor Charges from Lattice QCD, Phys. Rev. Lett. 115 (2015), no. 21 212002, [arXiv:1506.04196].

[155] PNDME Collaboration, T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, A. Joseph, H.-W. Lin, and B. Yoon, Iso-vector and Iso-scalar Tensor Charges of the Nucleon from Lattice QCD, Phys. Rev. D 92 (2015), no. 9 094511, [arXiv:1506.06411].

[156] T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, H.-W. Lin, and B. Yoon, Axial, Scalar and Tensor Charges of the Nucleon from 2+1+1-ﬂavor Lattice QCD, Phys. Rev. D 94 (2016), no. 5 054508, [arXiv:1606.07049].

[157] R. Gupta, B. Yoon, T. Bhattacharya, V. Cirigliano, Y.-C. Jang, and H.-W. Lin, Flavor diagonal tensor charges of the nucleon from (2+1+1)-ﬂavor lattice QCD, Phys. Rev. D 98 (2018), no. 9 091501, [arXiv:1808.07597].