Flavor Phenomenology of the Leptoquark Singlet-Triplet Model

In recent years, experiments revealed intriguing hints for new physics (NP) in semi-leptonic $B$ decays. Both in charged current processes, involving $b \to c \tau \nu$ transitions, and in the neutral currents $b\to s\ell^+\ell^-$, a preference for NP compared to the standard model (SM) of more that $3\sigma$ and $5\sigma$ was found, respectively. In addition, there is the long-standing tension between the theory prediction and the measurement of the anomalous magnetic moment (AMM) of the muon ($a_\mu$) of more than $3\sigma$. Since all these observables are related to the violation of lepton flavor universality (LFU), a common NP explanation seems not only plausible but is even desirable. In this context, leptoquarks (LQs) are especially promising since they give tree-level effects in semi-leptonic $B$ decays, but only loop-suppressed effects in other flavor observables that agree well with their SM predictions. Furthermore, LQs can lead to a $m_t/m_\mu$ enhanced effect in $a_{\mu}$, allowing for an explanation even with (multi) TeV particles. However, a single scalar LQ representation cannot provide a common solution to all three anomalies. In this article we therefore consider a model in which we combine two scalar LQs: the $SU(2)_L$ singlet and the $SU(2)_L$ triplet. Within this model we compute all relevant one-loop effects and perform a comprehensive phenomenological analysis, pointing out various interesting correlations among the observables. Furthermore, we identify benchmark points which are in fact able to explain all three anomalies ($b \to c \tau \nu$, $b\to s\ell^+\ell^-$ and $a_\mu$), without violating bounds from other observables, and study their predictions for future measurements.


Introduction
While the Large Hadron Collider (LHC) at CERN has not directly observed any particles beyond the ones of the SM (see e.g. Refs. [1,2] for an overview) intriguing indirect hints for NP have been acquired in flavor observables. In particular, measurements of semi-leptonic B meson decays, involving the charged current b → cτ ν or the flavor changing neutral current b → s + − , point towards the violation of LFU. Furthermore, also the AMM of the muon, which measures LFU violation as it vanishes in the massless limit, points convincingly towards physics beyond the SM. In order to explain these deviations from the SM predictions -also called anomalies -one thus needs NP that couples differently to tau leptons, muons and electrons. As we will see, LQs are prime candidates for such an explanation in terms of physics beyond the SM.
In tauonic B decays, BaBar measured in 2012 the ratios significantly above the SM predictions [48]. This is in agreement with the later LHCb measurements [49][50][51] of R(D * ), while BELLE found values closer to the SM in its latest analysis [52]. In combination, these deviations from the SM amount to 3.1σ [53] 2 . Interestingly, also the ratio While this is not significant on its own, the central value is very well in agreement with the expectation from b → s + − assuming a V td /V ts -like scaling [125] of the NP effect as obtained in models possessing an U (2) flavor symmetry in the quark sector (see e.g. Refs. [73,[126][127][128] for accounts in the context of the flavor anomalies). This means that an effect of the same order and sign as in b → s + − , relative to the SM, is preferred. Once more, LQs are prime candidates for an explanation. In particular the U 1 vector LQ SU (2) L singlet [66, 67, 69-74, 76, 77, 79-82, 129, 130], the U 3 vector LQ SU (2) L triplet [67-70, 72, 77, 80, 130, 131] and the Φ 3 scalar LQ SU (2) L triplet [68, 70-72, 77, 80, 130, 132] can explain data very well via a purely left-handed current.
From the discussion above it is clear that there are several options for a combined explanation of the flavor anomalies with LQs. Here we will consider the singlet-triplet model introduced in Refs. [42,107] which was also studied in the context of Dark Matter [108]. Within this model, a combined explanation can be possible since Φ 1 can account for the anomaly in a µ and affects b → cτ ν transitions while Φ 3 can explain b → s + − data and enters b → cτ ν processes. Furthermore, their combined effects in b → sνν processes can be destructive, relieving the bounds. However, in order to perform a complete phenomenological analysis, an inclusion of all relevant loop effects is necessary. We will compute these effects and extend the analysis of Ref. [42], allowing for couplings of Φ 1 to right-handed fermions.
The outline of the article is as follows: In the next section we will define our setup. The conventions for the various observables as well as the results of the matching, taking into account the relevant loop effects, are given in Sec. 3 before we perform our phenomenological analysis in Sec. 4 and conclude in Sec. 5.

Setup
The scalar LQ singlet-triplet model is obtained by adding a scalar LQ SU (2) L singlet (Φ 1 ) and an SU (2) L triplet (Φ 3 ), each carrying hypercharge −2/3, to the SM particle content. While the couplings to gauge bosons are completely determined by the representations of the LQs under the SM gauge symmetry, their couplings to the SM fermions and the SM Higgs 5 are free parameters of the Lagrangian After electroweak symmetry breaking the Higgs acquires its vacuum expectation value v ≈ 174 GeV. The last term in Eq. (2.1) then leads to a shift in the bi-linear mass terms of the LQs. However, this shift can be absorbed by defining Thus, the terms ξ KK have (at leading order in perturbation theory) no impact on the low energy flavor phenomenology of the singlet-triplet model but would only enter processes with an external Higgs (or at higher loop level). Furthermore, by unitary rotations of the LQ fields, we can now diagonalize their bi-linear mass terms via unitary rotations U 1,2 : In turn, these rotations lead to an effect in the couplings to the Higgs which can however be absorbed by the definition The LQ field rotations in Eq. (2.3) have to be applied to their fermionic interactions as well. Here, they can again be absorbed by a redefinition of the couplings Couplings to the Higgs lead to mixing among different LQ representations. Via this mixing LQs are able to generate Majorana masses for neutrinos [33,[133][134][135][136][137]. 6 In the R-parity violating MSSM this would correspond to the number of generations for the singlet. However, in general N and M do not need to be equal.
Hence, we are left with diagonal bi-linear mass terms with entries m 2 Φ 1 II and m 2 Φ 3 JJ and off-diagonal Φ 1 − Φ 3 mixing governed byρ IJ . While the LQs with Q em = {2/3, −4/3} are already in their mass eigenstates, we have to diagonalize the resulting full matrix of the Φ 1 − Φ 3 system with Q em = −1/3  Concerning the QCD evolution of these coefficients, O 8 mixes into O 7 at O(α s ), yielding the relation [138,139] (3.6) Here, f denotes the number of active quark flavors, µ h(l) refers to the high (low) energy scale and where α s needs to be evaluated with the number of active flavors at a given scale as well.
Even though b → s + − can be induced at tree level in our model, there are still scenarios in which loop effects are phenomenologically important. As pointed out in Ref. [79], the large couplings to tau leptons, needed to explain b → cτ ν data, also lead to huge Wilson coefficients C τ τ 9,sb = −C τ τ 10,sb . In turn, O τ τ 9,sb mixes into O 9,sb via the off-shell photon penguin [140], shown in the right diagram of Fig. 1. In our UV complete model, we cannot only calculate this mixing, but also the finite part of the effect, contained in the matching contribution This means that even if couplings to light leptons are absent at tree level, they are generated via loop effects in the presence of tau couplings. Since we will mainly focus on b → s transitions, we shorten our notation in the following and write C sb 7(8) ≡ C 7 (8) , C f i 9 (10),sb ≡ C f i 9 (10) . The logarithm involving µ LQ in Eq. (3.8) originates from the fact that the right-diagram in Fig. 1 is divergent. To get rid of this dependence one has to solve the RGE governing the mixing between O ii 9 with different lepton flavors: with γ = 2α 3π . Here, we do not take into account the running of α and do not consider the running of C ii 9 (i.e. just the mixing of O ii 9 into C jj 9 with i = j). This then has the solution For B meson decays, this amounts to replacing the high scale µ LQ in Eq. (3.8) by the low scale of the processes µ b . In addition, at the B meson scale, O τ τ 9 gives a q 2 dependent contribution to C 9,eff , which however is numerically small [140] and not accessible with the SM independent fit. QCD corrections to the matching of scalar LQs for semi-leptonic processes (both charged and neutral current) can be taken into account by applying the following shifts to the Wilson coefficients of vector (V ), scalar (S) and tensor (T ) operators [141] with l µ = log (µ 2 /M 2 ) (where M can be either M 1 or M 3 ) and C F = 4/3 as the color factor. Since QCD is insensitive to flavor, electric charge and chirality, these corrections can be applied in a straightforward way to all other semi-leptonic processes, particularly to b → sνν and b → cτ ν.

Observables
As mentioned in the introduction, a main motivation for this anlysis is the explanation of the hints for NP in b → s + − data. In order to resolve this discrepancy between SM and experiment, an O(20%) effect to C 9,10 is required compared to the SM contribution which is given by [142,143] C SM 9 (4.8 GeV) = 4.07 , C SM 10 (4.8 GeV) = −4.31 . (3.12) In a global fit one finds preference for scenarios like C µµ 9 = −C µµ 10 , as generated in our model at tree level. However, a C µµ 9 = −C µµ 10 effect complemented by a LFU one in C 9 gives an even better fit to data [118,144]. As we will see, this is exactly the pattern that arises in our model, taking into account the loop effects discussed above.
For b → sτ + τ − transitions we have on the experimental side [145] Br [146] and an upper limit of LHCb [145] Br (3.14) These measurements are compatible at the 2σ level. The SM predictions read [147,148] Br B s → τ + τ − SM = (7.73 ± 0.49) × 10 −7 , In our model we find 16) and the analogous expression for b → d transitions. Also the branching ratios of semileptonic b → sτ + τ − processes can be expressed in terms of NP Wilson coefficients [149] Br These branching ratios together with the corresponding coefficients are shown in Tab. 2.

ddνν Processes
To describe d k → d j νν processes we use the Hamiltonian (3.24) At tree level we find contributions from Φ 1 and Φ 3 resulting in Since these processes are generated at tree level, we do not need to calculate loop effects, which would only amount to numerically small corrections. Again, we simplify the notation for b → s transitions, writing C f i L,sb ≡ C f i L . The QCD matching corrections are given in Eq. (3.11) and there is no QCD evolution of these operators.

Observables
For B → K ( * ) νν we follow Ref. [154] and use C SM L ≈ −1.47/s 2 w . The branching ratios normalized to the SM read This has to be compared to the current experimental limits R νν K < 3.9 and R νν K * < 2.7 [155] (both at 90% C.L.). The expected BELLE II sensitivity for B → K ( * ) νν is 30% of the SM branching ratio [156].

du ν Processes
For the charged current semi-leptonic processes we define the effective Hamiltonian as where in the SM C SM V L = 1. The contribution of our model to the SM Wilson coefficient from Φ 1 and Φ 3 is given by while scalar and tensor operators are generated by Φ 1 only Since we are mainly interested in b → c transitions, we abbreviate Again, the QCD matching corrections are given in Eq. (3.11). We also include the 2-loop QCD and the 1-loop EW RGE. Using the results of Ref. [157], we have . (3.31)

Observables
With these conventions, the ratios R(D ( * ) ) are given by [158] R(D) in terms of the Wilson coefficients given at the B meson scale. Furthermore, the branching ratio of B c → τ ν reads [158,159] Br In this work we use the most stringent limit of Ref. [160] Br even though this bound might be too restrictive (see Refs. [60,160] for theoretical discussions). However, we will see that even this limit does not constrain our model significantly.
A further constraint comes from the determination of the CKM element V cb when comparing electron and muon final states. Here Ref. [161] finds that For observables including first and second generation quarks such as τ → πν, K → µν/K → eν or D decays, the Wilson coefficients can be applied using appropriate indices. The corresponding formulas and analyses can be found e.g. in Refs. [45,162].

∆F = 2 processes
Dealing with ∆F = 2 processes, concretely B s −B s mixing, we use the effective Hamiltonian In our model we obtain at the high scale µ LQ . Here the first term originates only from Φ 1 and the second one only from Φ 3 . The last term originates from a box diagram where both LQ representations contribute. One of the corresponding Feynman diagram is shown in Fig. 2.
The formula for B d and Kaon mixing follow trivially. We can write the mass difference ∆m Bs (including NP) normalized to the SM one as ∆m Bs ∆m SM given at the high scale. Since both the SM and LQ contribute to C 1 , the QCD running down to µ b is the same for both and therefore cancels in Eq. (3.39), neglecting the evolution from µ LQ to the EW scale.
Observables B s −B s mixing has been measured to very good precision [164] and the current world average reads [165] ∆m exp Bs = (17.757 ± 0.021) × 10 12 s −1 . (3.41) The theoretical prediction suffers strongly from the uncertainties in QCD effects. While Ref. [166] and Ref. [167] fit well to the measurement (with rather large errors) Ref. [168] obtains a larger SM value The bounds on the imaginary part of the Wilson coefficient is even more stringent. In our phenomenological anlysis we will assume real couplings and allow for NP effects of up to 20% with respect to the SM prediction.

γ Processes
In case of charged lepton interactions with on-shell photons we define Here we obtain the following matching contribution in case of a top quark in the loop from the Feynman diagram given in Fig. 2 with N c = 3 already included. Note that we have C R f i = C L * i f due to the hermiticity of the Hamiltonian. Here we quoted explicitly the formula for the top quark, which we integrated out together with the LQ at the scale M ≈ M 1 ≈ M 3 . In case of light quarks, some comments concerning the use of Eq. (3.46) are in order: In principle, one has to integrate out only the LQ at the scale M but keep the quark as a dynamical degree of freedom. In this way, the matching contribution to C L f i acquires an infrared divergence, which is cancelled by the corresponding UV divergence of the contribution of the tensor operator 7 , obtained by integrating out the LQ at tree level. This amounts to a replacement of m t by µ LQ in the logarithm in Eq. (3.46). Now, at the low scale, the solution to the RGE (disregarding QED effects) leads to a replacement of µ LQ by the scale of the processes, or by the quark mass in case this mass is bigger than the scale. Therefore, in the case of light quarks, Eq. (3.46) can be considered as an effective Wilson coefficient at the low scale, which includes the effect of 4-fermion operators (up to QED corrections) and can therefore be used for the numerical evaluation.
Considering i → f transition with an off-shell photon, we define the amplitude where (3.49)

Observables
We can now express the branching ratios of flavor changing radiative lepton decays in terms of the Wilson coefficients as where τ i is the life time of the initial state lepton. The AMM of a charged lepton i is given by The expression for the electric dipole moment of the lepton is quite similar to the one for the AMM, namely In case of the AMM of the muon we already discussed the experimental situation in the introduction. In summary, the difference between the experiment and the SM prediction is corresponding to a 3.5σ deviation. Note that in our case the Wilson coefficient is in general complex and could therefore lead to sizable EDMs [28]. The current limits for radiative LFV decays are [169,170] representing relevant constraints for our analysis. The off-shell photon penguins contribute to processes like τ → 3µ which we will consider later. and Z → ν fνi processes.

Z and Zνν Processes
In this subsection we compute the amplitudes for Z → − i + f and Z → ν fνi processes for massless leptons. At zero momentum transfer (or equivalently vanishing Z mass), these amplitudes are directly related to effective Z i f and Zν i ν f couplings, which will enter flavor observables like for example in τ → 3µ. We write the amplitude in an analogous way to the case with the off-shell photon where ε µ is the polarization vector of the Z and (3.55) At tree-level the SM couplings read with s w being the Weinberg angle. Beyond tree-level, the SM coefficients receive momentum dependent corrections which are included in the predictions for EW observables. The corresponding Feynman diagrams generating these amplitudes in our model are depicted in Fig. 3. For the calculation we include the up-type quark masses (which become relevant in case of the top) and the Z mass up to the order m 2 u /M 2 LQ and m 2 Z /M 2 LQ , respectively. In this setup we obtain The corresponding loop functions F L,R , G L and H 1,2 are given in Eq. (A.17) and Eq. (A.22). In case of Z decays we have q 2 = m 2 Z . For the effective Z i f and Zν fνi couplings (at zero momentum transfer), we define (3.58) In this case, only the top contribution is relevant and the effective couplings become (3.59)

Observables
The branching ratio of a Z decaying into a charged lepton pair reads with Γ tot ≈ 2.5 GeV. The case for a pair of neutrinos in the final state follows trivially. The effective number of active neutrinos, including the corrections in our model, are given by (3.61) At LEP [171] the lepton flavor conserving Z boson couplings were measured precisely. We give the experimental results for each flavor separately Figure 4: Feynman diagrams contributing to W − → − fν i . The right diagram is only present for the triplet since the singlet does not couple to the W boson (at tree-level).
The SM predictions at the Z pole are Concerning lepton flavor violating Z decays the limits from LEP are [172][173][174] Br Z → e ± µ ∓ = 7.5 × 10 −7 , (3.64) From Z → νν one can determine the number of active neutrinos to be [171] N ν = 2.9840 ± 0.0082 . (3.65) As mentioned before, Z couplings (at zero momentum transfer) contribute to processes like τ → 3µ. Furthermore, Z couplings in Z decays can be measured much more precisely at an FCC-ee which could produce more than 10 11 Z bosons [175].

W ν Processes
Computing the amplitude of this process, we obtain where with the loop functions F 1,2 given in Eq. (A.29). Again, we set all down-type quark masses to zero but included the up-type quark masses, which are relevant for the top. At the level of effective couplings, we define the Lagraigian The LQ contribution then reads Out of this formula one deduces a destructive interference between the contribution of the singlet and the triplet in case of lepton flavor conservation.

Observables
Experimentally, the modification of the W τ ν coupling extracted from τ → µνν and τ → eνν decays reads [165,176] |Λ W τ ν (0)| exp ≈ 1.002 ± 0.0015 (3.71) and provides a better constraint than data of W decays. Here we averaged the central values of the muon and tau mode, but did not add the errors in quadrature in order to be conservative. We see that a positive NP effect is preferred which means that the triplet contribution should exceed the one of the singlet.

4 Processes
We define the effective Hamiltonian as with Here we sum over flavor indices. In this way, no distinction for the cases of equal flavors are necessary in the matching and tensor and scalar LR operators do not need to be included since they follow from Fierz identities.
The photon contribution reads W q c Figure 6: Penguin diagrams that contribute to b → a ν iνf transitions. In case of the Z boson, lepton flavor is conserved at tree-level vertex (f = i). For the W penguins we applied Fierz identities in order to match on the effective operators. The box diagrams look similar to the one in Fig. 3 but turn out to be numerically insignificant.

νν Processes
We define the effective Hamiltonian as Figure 7: Tree-level Feynman diagrams contributing to gg → ΦΦ. Figure 8: Tree-level diagrams contributing to qq → ΦΦ and gq → LΦ. Except for the left diagram, the cross-sections depend on the couplings of the LQ to SM fermions. L can be either a neutrino or a charged lepton, depending on the specific LQ representation. with (3.81) At the one-loop level, LQs can contribute to these processes through three types of Feynman diagrams: W -penguins, Z-penquins and pure LQ box diagrams. Again, the boxes are numerically not relevant due to the small couplings to muons. Therefore, we only present these results with full generality in the appendix.
The W penguin given in terms of the modified W ν couplings of Eq. (3.70) gives Finally we also have the Z-penguins, yielding where we used the effective Z and Zνν couplings given in Eq. (3.59).

Phenomenology
Now we turn to the phenomenological analysis of our singlet-triplet model. We consider the processes discussed above and include the loop effects calculated in the previous section. Our strategy is as follows: First we will discus the LHC bounds on thirdgeneration LQs. Then we will consider how one can explain b → cτ ν data taking into account these limits and then study the impacts on other observables like B s → τ + τ − and W → τ ν. For this purpose, only couplings to tau leptons (but not to muons or electrons) are necessary. In a next step we will include b → s + − data in our analysis and thus allow for non-zero couplings to left-handed muons, while disregarding couplings to electrons due to the strong constraints from µ → eγ [180]. In a final step, we search for benchmark points which can explain b → cτ ν, b → s + − and a µ simultaneously. For this purpose we also include couplings to right-handed muons in our analysis.

LHC bounds
Both Φ 1 and Φ 3 could obviously be produced at the LHC. Since LQs are charged under SU (3) c they can be pair produced via gluons (depicted in Fig. 7), which in general gives the best bound. However, for a third generation LQ, which is the case for our model to a good approximation, also t-channel production from bottom fusion is possible as well as single production via bottom-gluon fusion (see Fig. 8). ATLAS and CMS performed searches in these channels. In particular, in Ref. [181] CMS analyzed data taken at a center-of-mass energy of 13 TeV with an integrated luminosity of 35.9 fb −1 for the scalar singlet Φ 1 . Assuming Br Φ 1 → tτ = 100%, LQ masses up to 900 GeV are excluded. ATLAS searched for typical signals of the scalar triplet Φ 3 , using 36.1 fb −1 of data at √ s = 13 TeV [182]. Focusing on NP effects in third generation quarks and leptons, i.e. Φ 3 → tν/bτ and Φ 3 → tτ /bν, they find a lower limit on the LQ mass of 800 GeV. This limit can be raised up to 1 TeV if one of the aforementioned decay channels is dominating. Therefore, a third generation scalar LQ with mass above 1 TeV is consistent with LHC searches. We will assume this as a lower limit in the following phenomenological analysis of flavor observables. For more extensive analyses of LQ searches in combination with the flavor anomalies we refer e.g. to Refs. [103,[183][184][185][186][187][188].

b → cτ ν
Concerning b → cτ ν processes one can address the anomalies with couplings to third generation leptons, i.e. the tau lepton and the tau neutrino, while disregarding couplings to muons and electrons. In a first step we consider the simplified case of lefthanded couplings only, i.e.λ = 0. Furthermore, we can safely neglect CKM suppressed effects from first-generation quark couplings and are therefore left with the couplings λ 23,33 and κ 23,33 , involving second and third generation quarks (i.e. bottom and strange Here we scanned over λ 23 , κ 23 ∈ − 1, 1 for all points and λ 33 , κ 33 ∈ − 1, 1 (blue) or λ 33 , κ 33 ∈ − 2, 2 (red), respectively. The blue points are displayed on top of the red ones, showing only points that are allowed by R νν K * . The dark gray points are in agreement with R νν K * , but excluded by B s −B s mixing. The horizontal contour lines depict the LFU contribution to C 9 while the green band represents the region for R(D ( * ) ) preferred by data at the 1σ level. quark in the down-basis). In this case the box contributions to B s −B s in Eq. (3.38), together with the tree-level effect in b → sνν in Eq. (3.25) put an upper limit on the possible contribution to b → cτ ν processes (see Fig. 9). While the relative effect in b → sνν compared to b → cτ ν is independent of the LQ mass, the relative effect in B s −B s mixing compared to b → cτ ν amplitudes turns out to have a quadratic scaling with the mass. In fact, assuming real couplings and an exact cancellation in R νν K ( * ) , ∆m Bs can be expressed in terms of the NP effect in R(D ( * ) ) as ∆m Bs ∆m SM This relation holds once small CKM rotations are neglected which is possible in the case of an anarchic flavor structure, i.e. V cb λ 33 λ 23 and V cb κ 33 κ 23 . The tau loops also generate an effect in C 7 as well as a LFU contribution to C 9 . Both these effects are directly correlated to b → sτ + τ − processes, induced by the tree-level Figure 10: Correlations between the NP effect in ∆m Bs and the corrections to the effective W τ ν τ coupling Λ W τ ντ (0), constrained from τ → µνν and τ → eνν. Like in Fig. 9 we only considered the couplings λ 23,33 and κ 23,33 , i.e. only couplings to lefthanded taus, scanning over λ 23 and κ 23 (λ 33 and κ 33 ) between ±1 (±2) and setting M 1 = M 3 = M = 1 TeV. The blue region is preferred by τ → µνν and τ → eνν data at the 1σ level.
coefficients C τ τ 9 = −C τ τ 10 . We find neglecting the different running of C 7 from µ LQ down to m t . One can also relate these two coefficients, yielding This situation is illustrated in Fig. 9, where we show the correlations between B s → τ + τ − and R(D ( * ) ).
Note that for left-handed couplings R(D)/R(D) SM = R(D * )/R(D * ) SM is predicted. The bound from B s −B s mixing limits the possible effect, both in B s → τ + τ − and R(D ( * ) ), depending on the LQ mass. Heavier LQs lead to larger effects in B s −B s with respect to B s → τ + τ − and R(D ( * ) ) than lighter LQs. For the same scenario, i.e. only left-handed couplings to tau leptons, we  also show corrections to the W τ ν coupling in Fig. 10. Note that effect of Φ 1 has opposite sign than the one of Φ 3 . Furthermore, if one aims at increasing R(D ( * ) ), the effect of Φ 1 (Φ 3 ) in W → τ ν is destructive (constructive) such that it increases (decreases) the slight tension in τ → µνν data.
Next, let us allow for non-zero right-handed couplingsλ 23,33 of Φ 1 to quarks and leptons. In this case the left-handed vector current encoded in C τ τ V L (originating from Φ 1 and Φ 3 via λ 23,33 and κ 23,33 only) is now complemented by a C τ τ SL = −4C τ τ T L effect from Φ 1 . This breaks the common rescaling of R(D)/R(D) SM and R(D * )/R(D * ) SM , depicted by the green line in Fig. 11. The constraint from B s −B s only limits C V L but not C SL = −4C T L . The resulting correlations between R(D) and R(D * ) are shown in Fig. 11. One can see that for deviations of R(D ( * ) )/R(D ( * ) ) SM from unity of more than ≈ 10%, our model predicts R(D)/R(D) SM > R(D * )/R(D * ) SM . The size and correlation between C 7 and a LFU effect in C 9 , induced by the tau loop,  Figure 12: Correlations between C 7 and C 9 , both given at the B meson scale. Here we imposed that the points satisfy B s −B s mixing (i.e. yield a maximal effect of 20%) and lie within the 1σ (yellow) or 2σ (blue) region preferred by the global fit to b → cτ ν data. Note that non-zero effects in C 7 (µ b ) and C 9 (µ b ) are mandatory in order to explain b → cτ ν data at 1σ and that C 9 (µ b ) has the sign preferred by the fit if this is required. Both coefficients include O(α s ) corrections. Again we scanned over the couplings λ 23,33 , κ 23,33 andλ 23 ∈ [−1.5, 1.5] and the LQ masses is shown in Fig. 12. Interestingly, to account for b → cτ ν data within 1σ, we predict −0.5 < C 9 < −0.2 (inclduing right-handed couplings) which is in very good agreement with the global fit on b → s + − data, especially if it is complemented by a C µµ 9 = −C µµ 10 LFUV effect [118,144].
In the same way, b → dτ ν data can be addressed. Here, it was shown in Ref. [105] that already a 10% effect with respect to the SM could lead to a neutron EDM observables in the near future.

b → cτ ν and b → s + −
Let us now turn to the case where we allow for couplings to left-handed muons as well.
Here, it is clear that, disregarding for the moment R(D ( * ) ) and thus tau couplings, one can explain b → s + − data with a tree-level C µµ 9 = −C µµ 10 effect from Φ 3 without running into the danger of violating bounds from other flavor observables. However, the situation gets more interesting if one aims at explaining b → s + − and b → cτ ν data simultaneously. In this case LFV τ − µ effects necessarily arise e.g. in B → Kτ µ, τ → φµ, Z → τ µ and τ → 3µ. Note that our model does not posses scalar currents in the down sector, therefore B s → τ µ does not receive a chiral enhancement. The correlations between B → Kτ µ and τ → φµ are shown in Fig. 13, finding that they are in general anti-correlated despite fine-tuned points.
4.4 b → cτ ν, b → s + − and a µ Finally, we aim at explaining the anomaly in the AMM of the muon in addition to b → cτ ν and b → s + − data. Accounting for δa µ alone is possible and the only unavoidable effect occurs in Z → µ + µ − , which can however only be tested at the FCC-ee [189]. Furthermore, explaining δa µ together with b → s + − data does not pose a problem either since Φ 1 can account for δa µ while Φ 3 can explain b → s + − . However, once one wants to account for b → cτ ν data the situation becomes nontrivial. Scanning over 10 million points 8 we found approximately 350 points which can explain all three anomalies at the same time. The corresponding range for the couplings of these 350 points is shown in Fig. 14. Only allowing for an effect of 20% in B s −B s mixing, the number of points is reduced to 40, where an effect as low as 10% is possible. In addition, we choose (out of these 350 points) four benchmark points, shown in color in Fig. 14 points are given in Tab. 3. Interestingly, even though in general τ → µγ represents the most restrictive constraint on our model in case one aims at an explanation of all three anomalies, we still find points that give a relatively small contribution of roughly one order of magnitude below the current experimental bound. The branching ratio of B s → τ τ is enhanced by a factor of roughly 100 with respect to the SM, which also is below the current experimental bound. While the effects in Λ W τ ν are small, they are always positive, reducing the slight tension in the effective W τ ν coupling. The effects in B → Kτ µ and τ → φµ range from being negligible to close to the current experimental bounds while effects in τ → µee and τ → 3µ lie roughly two orders of magnitude below the current experimental limit. Furthermore, the effects in Z → τ τ would clearly be measurable at an FCC-ee [175].

Conclusions
Motivated by the intriguing hints for LFU violating NP in R(D ( * ) ), b → s + − processes and a µ , we studied the flavor phenomenology of the LQ singlet-triplet model. We first defined the most general setup for the model, including an arbitrary number of LQ "generations" as well as mixing among them. With this at hand, we performed the matching of the model on the effective low energy theory and related the Wilson coefficients to flavor observables. Here, we included the potentially relevant loop effects, e.g. in B s −B s mixing, b → sγ, LFU contributions to C 9 and a µ as well as in modified Z and W couplings.
Our phenomenological analysis proceeded in three steps: First, we disregarded the anomalies related to muons and considered the possibility of explaining R(D ( * ) ) and the resulting implication for other observables. We found that, including only couplings to left-handed fermions, the size of the possible effect depends crucially on the mass of the LQ: the larger (smaller) the mass (couplings) the bigger the relative effect in B s −B s . Together with b → sνν, this is the limiting factor here. For M = 1 TeV and values of κ 33 up to ±2, a 20% effect in R(D ( * ) ) is possible, while for M = 1.5 TeV and |κ 33 | < 1 only a 10% effect with respect to the SM can be generated (see Fig. 9). At the same time, an enhancement of B s → τ + τ − of the order of 10 2 is predicted, which, via loop effects, leads to a LFU C 9 ≈ −0.3. Once couplings to right-handed leptons are included, larger effects in b → cτ ν processes are possible and R(D)/R(D) SM > R(D * )/R(D) * SM is predicted, see Figs. 11 and 12.
In a second step, we aimed at a simultaneous explanation of b → s + − data together with R(D ( * ) ). In this case, effects in lepton flavor violating processes like B → Kτ µ and τ → φµ are predicted as shown in Fig. 13. These effects are still compatible with current data but can be tested soon by LHCb and BELLE II.
Finally, including in addition the AMM of the muon in the analysis is challenging since then right-handed couplings to muons are required which, together with the couplings needed to explain R(D ( * ) ), lead to chirally enhanced effects in τ → µγ. It is still possible to find a common solution to all three anomalies but only a small region of the parameter space can do this. Nonetheless, we identified four benchmark points which can achieve such a simultaneous explanation to all three anomalies (see Fig. 14).
In summary, the LQ singlet-triplet model is a prime candidate for explaining the flavor anomalies and we would like to emphasize that there is no renormalizable model on the market which is more minimal (only two new particles are needed here) and capable to address all three prominent flavor anomalies together.

A.3 uuγ and EDM
We define the effective Hamiltonian as and obtain in the case of one generation of LQs and no mixing among them The contributing diagram is depicted in Fig. 1. For the neutron EDM we set j = k = 1 and reproduce (setting m u = 0) our result from [105], where also the relevant RGE can be found. In case of LQ mixing, we have (A.10)

A.4 du ν
For the effective Hamiltonian defined in Eq. (3.27) we find and for B s −B s mixing A.6 γ, Z and Zνν In case of i → f γ transitions and the effective Hamiltonian given by Eq. (3.44) we have with N c = 3 already included. For the off-shell photon, as given by the amplitude in Eq. (3.47), we obtaiñ where the loop functions F (y) and G(y) are defined in Eq. (3.49).
For Z decays, where the amplitude is given by Eq. (3.54) and the ∆ The amplitude for Z → νν is again given by Eq. (3.54). For the Σ LQ f i q 2 , introduced in Eq. (3.55), we obtain with H 1 (q 2 , M 2 ) = N c q 2 864π 2 M 2 3(3 − 2s 2 w ) log with F W m 2 u , q 2 , M 2 = 1 9M 2 6(2q 2 − 3m 2 u ) log (A. 26) Additionally, there are terms that do not trivially decouple, however, they vanish in the decoupling limit. They read A.8 τ → 3µ, τ → µe + e − and µ → 3e The relevant effective Hamiltonian is given in Eq. (3.72). The contributions of the photon and Z penguin diagrams are given by Eq. Γ L,P * u k a Γ L,K u k b Γ L,K * u j f Γ L,P u j i + Γ L,K * u j a Γ L,K u k b Γ L,P * u k f Γ L,P u j i D 2 m 2 u k , m 2 u j , m 2 K , m 2 A.9 τ → νν and µ → eνν As it was the case for the previous results, we consider the top as the only non-zero quark mass and in cases where the result is proportional to the quark mass (squared), we directly write the result in terms of the top. The effective Hamiltonian for the process is given in Eq. (3.80). The box diagrams read