A Common Origin of Neutrino Masses and $R_{D^{(\ast)}}$, $R_{K^{(\ast)}}$ Anomalies

In this work, we present a solution to the persistent tensions in the decay observables $R_{D^{(\ast)}}$ and $R_{K^{(\ast)}}$ by introducing a $SU(2)_L$ doublet and a $SU(2)_L$ triplet scalar leptoquarks (LQs) that reside at the TeV energy scale. Neutrinos that remain massless in the Standard Model receive naturally small masses at one-loop level via the propagation of the same LQs inside the loop. Such a common origin of apparently disjoint phenomenological observations is appealing, and we perform a comprehensive analysis of this set-up. We identify the minimal Yukawa textures required to accommodate these flavor anomalies and to successfully incorporate neutrino oscillation data while being consistent with all experimental constraints. This scenario has the potential to be tested at the experiments by the future improved measurements in the lepton flavor violating processes. Furthermore, proper explanations of these flavor anomalies predict TeV scale LQs that are directly accessible at the LHC.


Introduction
In the Standard Model (SM) of particle physics, all the charged fermions receive their masses as a result of the electroweak (EW) symmetry breaking, whereas neutrinos remain massless.
Over the decades, neutrino oscillations have been firmly confirmed in the experiments [1][2][3][4][5][6][7] breaking the mold of the SM. However, the theoretical origin behind the neutrino mass generation is yet to be discovered. By extending the particle content of the SM, neutrino masses can be incorporated either at the tree-level via the seesaw mechanism [8][9][10][11][12][13][14] or at the loop-level by quantum corrections [15][16][17][18][19]. We consider a scenario where neutrinos are Majorana fermions and their masses that violate lepton number by two units ∆L = 2 are generated radiatively at the one-loop order. In our set-up, due to loop suppression, neutrino masses are naturally small relative to the charged fermion masses. Scenarios, where the beyond SM states running inside the loop have masses at the TeV scale are particularly interesting, as they can simultaneously solve other problems such as the recently observed flavor anomalies.
Observation of neutrino oscillations provides direct evidence for flavor violation in the neutral lepton sector, which is indeed a phenomenon beyond the SM. Consequently, the origin of neutrino masses may be directly linked with flavor physics. Recently, several measurements associated with some of the flavor observables appear to indicate violations of lepton flavor universality (LFU). Four of such flavor observables, the ratios R D ( * ) and R K ( * ) , are of great importance. It is intrigued to find a common origin of the neutrino masses and R D ( * ) , R K ( * ) anomalies, which is indeed the purpose of this work.
These results show 2.5σ deviation from SM predictions in both q 2 bins. Theory predictions of both R K and R K * are completely trustworthy since hadronic uncertainties cancel out in these ratios.
These experimental results have drawn much attention to the theory community over the last several years. Accurate measurements of these observables have the great potential to hunt for new physics (NP) since the corresponding ratios are reliably computed within the SM. In short, experimental data of these flavor observable ratios beg for extensions of the SM for their proper explanations. To incorporate these significant deviations from the SM values, leptoquark (LQ) solutions are extensively exploited in the literature (see for example Refs. ).
In this work, we propose a solution to the R D ( * ) and R K ( * ) anomalies by introducing a SU (2) L doublet and a SU (2) L triplet scalar LQs, respectively. Under the SM gauge group We perform a detailed investigation of this model to identify the minimal Yukawa parameters that are required to satisfy the above-mentioned semi-leptonic B meson decay anomalies and neutrino oscillation data. In our phenomenological analysis, we take into account all essential experimental constraints applicable to our model. We show that such a non-trivial task can be successfully achieved with only a limited number of parameters. The scenario presented in this work is particularly attractive since the same parameter set that generates neutrino masses and mixings, as well as accommodates semi-leptonic B meson decay anomalies, also links many other beyond SM processes. Among them the charged lepton flavor violating processes that receive large contributions provide a way to probe this scenario in the experiments. This model can also be tested at the LHC since existence of TeV scale LQs are required for consistent explanations of flavor anomalies.
There are few more flavor observables that depart from their theoretical values. Associated to charged current processes, the ratio R J/ψ shows enhancement [107] in the tauoniuc mode to the muonic mode for B → J/ψ ν, the longitudinal polarization of D * meson in the B → D * τ ν decay known as f D * L deviates from the SM value by ∼ 1.5σ [108], and P * τ the polarization asymmetry in the longitudinal direction of the tau in the D * mode is also found to show some tension with theory [24]. Furthermore, two important quantities linked with the neutral current processes are the angular observable P 5 [109][110][111], and combined fit to the anomalous b → s data in operators contributing to b → sµµ [60,61,[112][113][114][115][116][117][118][119]. These anomalies can also play prominent roles in finding NP, however, currently they have either large error bars or suffer from large hadronic uncertainties. This is why in this work we focus only on the anomalies associated with R D ( * ) and R K ( * ) ratios that provide clean signals of NP. Implementations of all these aforementioned anomalies will be presented elsewhere.
In Sec. 2, we introduce the proposed model. We discuss how to accommodate B physics anomalies and neutrino oscillation data in in Sec. 3 and Sec. 4, respectively. Then we discuss at length all the experimental constraints that are relevant to our model in Sec. 5.
Furthermore, we present our results in Sec. 6, and finally conclude in Sec. 7.

The Model
In addition to the SM particle content, our proposed model consists of two scalar LQs R K ( * ) anomalies, respectively. The corresponding Yukawa part of the Lagrangian is given by [120]: where Q L and L L are the left-handed quark and lepton doublets under SU (2) L , while d R , u R , and R are the right-handed down-type quark, up-type quark, and charged lepton, respectively that are singlets of SU (2) L . Furthermore, σ a , a = 1, 2, 3, are Pauli matrices, {i,j} are flavor indices, and S a 3 are the components of the LQ triplet. It is to be noted that S 3 couples to di-quarks, and we choose these couplings to be zero to ensure the stability of the proton. By expanding Eq. (7) one can write: In writing the above Lagrangian terms, we have made a transition from flavor to mass eigenstates of the fermions with the following transformations: such that V and U are the Cabibbo-Kobayashi-Maskawa (CKM) and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrices, respectively. We have chosen a basis where all the relative rotations are assigned to up-type quarks and neutrinos. From hereafter we will use the notationŷ L ij = (y L U ) ij . The main objective in this work is to construct a minimal model that can simultaneously satisfy both the neutrino masses and the B meson decay anomalies. However, with only R 2 and S 3 LQs, neutrino masses cannot be generated. For simplicity, instead of fermionic extensions, we prefer to enlarge the scalar sector. One such simple choice is to extend the particle content by a color triplet scalar χ 1 (3, 1, 2/3) = χ 2/3 [121], which is a singlet under SU (2) L . Then in our model, neutrinos are Majorana type particles, and their masses are generated at the one-loop level, as shown in Fig. 1. In general, χ 1 can have only di-quark coupling, which we assume to be sufficiently small. The completion of the one-loop diagram for neutrino mass requires the following two terms in the scalar potential: Here H is the SM Higgs doublet, v = H = 246 GeV, and is the rank-2 Levi-Civita tensor.
The one-loop neutrino mass matrix in this set-up is then given by: here m u 1,2,3 = {m u , m c , m t }, and M 1,2 are the two largest masses among R 2/3 , S −2/3 , χ 2/3 states. We denote the mass of the field χ by m χ , and the components of R 2 (S 3 ) LQ are assumed to be degenerate with a common mass m R (m S ). Note that within this model, neutrino masses also receive contributions from a two-loop diagram, as shown in Fig. 1, which is expected to be much smaller than the one-loop contributions due to a further loop suppression; hence we neglect such corrections. Inclusions of such terms do not change the results presented in this work.

Explaining the B Meson Decay Anomalies
In this section, we discuss how to incorporate B meson decay anomalies observed in the R D ( * ) and R K ( * ) ratios within our framework.

R D -R D * Anomalies
Deviations that have been observed in several experiments in the R D and R D * ratios can be accommodated by the scalar LQ R 2 (3, 2, 7/6). The charged current process b → cτ ν contributing to B → Dτ ν and B → D * τ ν decays after integrating out the heavy LQ state R 2 can be explained in terms of the following effective Hamiltonian: where C j S,T are the Wilson coefficients induced by the LQ state mediating the semileptonic decays via a tree-level contribution, as shown in Fig. 2 (left diagram). The first term in the above Hamiltonian is the SM contribution, which strictly corresponds to ν τ . On the other hand, the BSM contributions need not conserve lepton flavor, and the neutrino index j can in principle run over all three flavors [41,42]. Note that experiments do not identify the flavor state of the neutrinos in the R D -R D * measurements. After integrating out R 2 field, the expressions of the Wilson coefficients C j S,T at the matching scale µ = m R reads: The scale dependence of the scalar and tensor Wilson coefficients can be straightforwardly calculated following Ref. [41], and at the bottom-quark mass scale at which the matrix elements of hadronic currents are calculated, we find the relation C j GeV and m R = 1 TeV). Furthermore, the approximate relations [122] between the Wilson coefficients at the two different scales are given by: . Throughout this work, we will quote the values of these (C S , C T ) Wilson coefficients at the scale of the heavy NP particle From the discussion above, it is clear that a successful explanation of the R D -R D * anomalies requires the coupling of the b-quark to τ -lepton. Moreover, c-quark must couple to neutrinos ν j , implicating the following needed Yukawa textures [41]: We remind the readers that theŷ L matrix is related to the original y L matrix via PMNS The left plot corresponds to the case with real C e,µ S (C τ S = 0), and the right plot represents the scenario with complex C τ S (C e,µ S = 0). We utilize the Flavio package [123] to get these plots. We find that scenarios, where j = τ contribution is non-negligible toward R D ( * ) , it always needs to be very close to purely imaginary to get a good fit. For scenarios of this type, we choose y R 33 to be purely imaginary. On the other hand, all parameters can be taken to be real for cases when j = τ contribution is negligible (as can be seen from the left plot in Fig. 3).
As well known, S 3 LQ can also contribute to b → cτ ν process with a wrong sign that is at odds with the experimental observation. These contributions are proportional to the product of Yukawa couplings y 3j (V y * ) 23 , which within our set-up is too small to provide any significant contribution.
The purple (pink) band represents R D (R D * ) within experimental 1σ values. We are interested in the values of the corresponding Wilson coefficients in the overlapping shaded regions between these two bands.

R K -R K * Anomalies
The B meson decay anomalies associated with the neutral current processes correspond to an excess of B → K ( * ) µ + µ − over the B → K ( * ) e + e − channels. The effective Hamiltonian that describes these processes can be written as follows: Here the effective operators correspond to: Within our framework, these types of effective operators are generated at the tree-level via S 3 LQ, as presented in Fig. 2 (right diagram). After integrating out the heavy degrees of freedom, one gets the following Wilson coefficients: Wilson coefficients C 9,10 get additional contributions from the one-loop mediated processes lead by R 2 LQ, which can be safely ignored in the present scenario. Interestingly, S 3 is the only LQ that can explain both R K < R SM K and R K * < R SM K * at the tree-level. In this work, we assume the NP dominantly couples to the muon sector (hence = = µ), and assume the following texture of the corresponding Yukawa matrix: to alleviate discrepancies in the R K ( * ) ratios. The scenario with C µµ 9,10 is shown to provide an excellent fit to the R K -R K * anomalies, and the required values of the Wilson coefficients corresponding to best fit is given by [116]: The allowed values within the one-sigma and two-sigma correspond to

Incorporating Neutrino Oscillation Data
In our proposed model, neutrinos are Majorana fermions, and as aforementioned, neutrino mass is generated radiatively at the one-loop level. The computed neutrino mass matrix is given in Eq. (13) which can be decomposed into a from: Whereas from experimental results, we already know the definite sign of ∆m 2 21 > 0; however, the sign of ∆m 2 31 is yet to be measured. The case with ∆m 2 31 > 0 corresponds to the normal ordering (NO). In contrast, the case with ∆m 2 31 < 0 is known as the inverted ordering (IO). Since inverted mass ordering is less preferred by fits to data, this work only  Table I, which we use for numerical analysis. Since the Dirac phase in the neutrino sector is inadequately measured in the experiments, and the Majorana phases have not been measured, we do not include these quantities in our fits.
By combining the Yukawa textures Eqs. (16) and (20), along with the neutrino mass matrix given in Eq. (13), it is trivial to verify that neutrino oscillation data cannot be explained. Hence to properly incorporate neutrino masses and mixings, we must enlarge the Yukawa parameters. Since in y L matrix, which enters the neutrino mass matrix formula and contains non-zero entries (entry) in the second row that are (is) required for the explanation of R D ( * ) , it is natural to populate along the second row of y matrix as well, for the sake of minimality. Following this argument, our detailed numerical study performed in Sec. 6 reveals two minimal textures for the Yukawa coupling matrices to accommodate anomalies in the R D * , R K * ratios in conjunction with neutrino observables, which are: Here the entries in blue enter the explanation of R D , R D * ratios, whereas entries in red play a role in explaining R K , R K * ratios. Hence, we always keep these terms to be non-zero.
Additional entries shown in black allow fitting the neutrino observables as well. Moreover, in the above Yukawa coupling matrices, no entries for the first generation quarks are introduced, since they are highly constrained by experimental data. As will be shown in Sec. 6, with only a limited number of parameters as given in Eqs. (22) and (23)  It is worthwhile to mention that further reduction of the number of parameters in Eqs. (22) and (23) do not return the aforementioned observables within 1σ measured values. Hence we do not list or consider such textures here.

Radiative Leptonic Decays: → γ
LQ mediated one-loop contributions to the dangerous lepton flavor violating (LVF) processes → γ can severely constrain the couplings of the LQ with the charged leptons. In our model, some of the parameters are highly constrained by these radiative decay processes.
Following the general method described in Ref. [125] (see also Ref. [126]) in computing such processes, we derive the following contributions to LVF decays within our scenario: Here τ is the lifetime of the initial state lepton, and x q = (m q /m LQ ) 2 . In the above formula, the first (last) two terms are the contributions from the R 2 (S 3 ) LQ and the expression is in agreement with Ref. [62] ( [83]). The current experimental bounds on these processes are given by [127,128]: which can be easily translated into the upper bounds on the Yukawa couplings and the lower bound on the mass of the corresponding LQ. We find that, as long as these flavor violating processes are under control, processes like → are well below their current experimental bounds. Even though in our numerical analysis, we take into account all such processes, here for brevity, we do not write down the associated formulas.

µ − e Conversion
Neutrinoless µ − e conversion in nuclei can provide strong bounds on some of the Yukawa couplings in our model. The LQ S 3 mediates this dangerous process at the tree-level, and the associated Yukawa couplings are directly linked to neutrino observables and anomalies in the R K ( * ) ratios. This conversion rate can be written as follows: Here Γ capture (Z) represents the capture rate for a nucleus with atomic number Z, and the expression for the width Γ µ−e in our scenario is given by [126,129]: In Eq. (29) the nuclear form factors (in units of m 5/2 µ ) are given by [129] V (p) = 0.0974, V (n) = 0.146 for gold and the total capture rate for gold is Γ capture = 13.07 × 10 6 s −1 . The current sensitivity imposes strong bounds on the µ−e transition rate: CR(µ−e) < 7×10 −13 [130]. On the other hand, the future projected sensitivity that will provide an improvement on the current limit by almost four orders of magnitude CR(µ → e) < 10 −16 [131][132][133][134][135][136][137] is capable of ruling out most part of the parameter space of this theory.

Z-Boson Decays: Z →
Leptoquarks' contributions to Z decays to pair of leptons via one-loop processes provide substantial restrictions on the size of the Yukawa couplings. In the following, we derive such constraints on the relevant Yukawa couplings of the LQs. The effective Lagrangian of the following form can describe these processes [138]: with and θ W is the Weinberg angle. These dimensionless coupling parameters g ij are responsible for Zdecay processes Z → . LEP [139] measurements provide stringent constraints on the contributions to these rates for both flavor conserving and flavor violating Z decays. By following the computations done in Ref. [138], we derive the contributions by the LQs to Z decays, which are given by: For the same LQ, while calculating δg R we have: On the other hand, for S 3 LQ, w u ij = −(V * y) ij , w d ij = − √ 2y ij replacements must be made, and this LQ does not give any contribution to δg R . Some of the Yukawa couplings are strongly constrained by the precise measurements of these effective couplings by the LEP collaboration [140]: These experimental results can be translated into upper bounds on the Yukawa couplings of our theory for a fixed value of LQ masses. Moreover, LHC data [141] imposes an upper limit on the branching fraction, B(Z → eµ) < 7.5 × 10 −7 at the 95% confidence level. Within our scenario, contributions from S 3 can be safely neglected. Moreover, one of the essential bounds on the Yukawa couplings we get from the above LEP data is (with m R = 1 TeV): For our numerical study, we demand its value to lie within the 2σ range, which is crucial to achieving a good fit in our model.
Additionally, the effective number of neutrinos is another accurately measured quantity, and the observed value corresponds to [140]: which will constraint the LQ couplings to the neutrinos by the following equation: While performing a fit, we also incorporate these bounds on the Yukawa couplings. This has a significant impact on physics BSM that leads to B meson decay and constraints the yet unmeasured branching ratio Br(B c → τ ν). In our scenario, this branching ratio has the form [142]: In deriving this formula, we have used f Bc = 0.43 GeV and m Bc = 6.2749 GeV.
Since the branching ratio Br(B c → τ ν) has not been measured yet in the experiments, B c lifetime needs to be compared with the theoretical calculations [133,[143][144][145][146]. By doing so, the works in Ref. [147] and Ref. [148] advocated that the NP contributions to this decay must be Br(B c → τ ν) ≤ 10% and Br(B c → τ ν) ≤ 30%, respectively. However, as claimed in Ref. [149] (see also Ref. [150]), these calculations suffer from theoretical uncertainties, and suggested a conservative limit of Br(B c → τ ν) ≤ 60%. As will be shown, a somewhat larger branching ratio is required to be compatible with R D ( * ) measurements in our set-up.

Rare Meson Decays: B → K ( * ) ν ν
NP explanations of B-anomalies impose strong constraints on the couplings due to rare meson decays associated with processes like s → dνν and b → sνν. Note that according to the specific choice of the Yukawa coupling given in Eqs. (22) and (23), our set-up does not lead to the former mode; however the latter mode is open. This will correspond to B → K ( * ) ν ν that can be computed from the effective Hamiltonian given in Eq. (17).

Semileptonic K Decays: K → ν
Lepton flavor universality in kaon decays has also be measured with great accuracy. One such relevant observable in our scenario is the following ratio [50]: The SM prediction and the experimental measured values of this ratios are given as follows [155]: For our NP scenario, this constraint can be recast into an upper bound on the associated Yukawa coupling [50,156]: |y bµ | 0.5(m S /T eV ). This bound also plays a crucial role, since it is directly correlated with the explanation of R K ( * ) ratios. can be written as [157]:

Pseudoscalar Meson
here, q refers to up-type quarks and the function f is defined as: Whereas the leptonic decays B 0 s → ± ∓ are well predicted in the SM, at present, only the B s → µ + µ − decay mode has been observed, which is in good agreement with the SM prediction [158]: Br(B s → µ + µ − ) SM = (3.65 ± 0.23) × 10 −9 . The observed value for the corresponding branching fraction is Br(B s → µ + µ − ) exp = (3.0 ± 0.6) × 10 −9 , as reported by the LHC collaborations [159]. Furthermore, we also take into account the current limit of the lepton flavor violating B meson decay with eµ final state [139]: Br(B s → µ ± e ∓ ) < 5.4×10 −9 .

Muon Anomalous Magnetic Moment: (g − 2) µ
Anomalous magnetic moment (AMM) of the muon is one of the most preciously measured quantities in particle physics. Its value measured at the Brookhaven National Laboratory [160] corresponds to a deviation of 3.7σ compared to the SM prediction. Their measured value is given as follows: Both R 2 and S 3 LQs can in principle contribute to the AMM of the muon, which we derive to be [126,161]: Both of these provide wrong contributions to the muon AMM, hence the observed value quoted in Eq. (46) cannot be incorporated within our set-up. Moreover, due to large masses of the LQs, this observation provides very weak bounds on the above Yukawa couplings.

Bounds from LHC Searches
In this section, we investigate the LHC constraints on the leptoquark (LQ) masses at Large Hadron Collider (LHC) for the relevance of B-decays in this model. At hadron colliders, LQ is pair produced via gg or qq fusion. qq → LQ LQ † with a t-channel is highly suppressed compared to QCD induced production. Note, the coupling is assumed to be small in the production cross-section, gg(qq) → LQ LQ † , and it is a QCD driven process, entirely determined by the LQ mass and strong coupling constant. In both ATLAS and CMS, there are searches for the LQ pair production in different decay modes, LQ † LQ → qq ¯ , qqνν associated with first [162,163], second [162,[164][165][166], and third [165,167,168] generations of quarks and leptons.
In Table II,  with Yukawa coupling (y L U ) ij (i = c, j = ν e , ν µ , ν τ ) and (y R ) 33 lead to jjνν and bbττ final states, respectively. It is worth mentioning that S 3 LQ, relevant for R K − R K * , has all the above-mentioned decay modes.
However, we take the mass of the S 3 to be 2.5 TeV, which automatically satisfies all current LHC bounds.
To obtain the pair-produced LQ constraints at large Yukawa couplings, we quote Ref. [79] for various j and b type LQs. Here, t-channel diagram interferes with gg and qq fusion diagrams. The bounds indicated by Ref. [79] shows that t-channel starts to dominate for a LQ that couples to first generation quarks with coupling of y 1. However, LQ coupling to heavier quarks has main contribution from t-channel only for y 1. It is also to be pointed that Drell-Yan-like production with the + − final state puts bounds on LQ masses and the associated Yukawa couplings. Such as, LQ coupling to ce, cµ, cτ , and bτ has a bound on the Yukawa couplings of 1.8, 1.5, 2.3, and 2.3, respectively for 1 TeV LQ mass. The constraints on the Yukawa couplings and the bounds on the LQ masses discussed here will be used in the next section for our numerical analysis.

Results and Discussions
In this section, we present the numerical results of the model introduced in Sec. 2. As already aforementioned, in search of finding the minimal parameters of the theory, our detailed numerical analysis results in two separate textures that are summarized in Eqs.   We remind the readers that the overall factor m 0 in front of the neutrino mass matrix Eq. (13) is a free parameter of the theory, which is required to be tiny m 0 ∼ 10 −8 GeV to satisfy the neutrino oscillation data. Then Eq. (13) provides µλ ∼ 0.7 GeV, which clearly shows that for the phenomenological analysis, the mixing among the scalars can be ignored entirely within our scenario. It is because the corresponding mixings are parametrized by the cubic coupling µ and the quartic coupling λ present in the scalar potential.
As discussed in Sec. 3, the solution to the R D − R D * observables in scenarios with significant contributions from Wilson coefficient C τ L must be complex, which is always the case in our model. Consequently, we choose the parameter y R 33 to be purely imaginary, and the rest of the parameters are all taken to be real. A fit to the R K − R K * observables within 1σ measured values demands y bµ y sµ to lie in between [4.49 × 10 −3 , 6.09 × 10 −3 ]. In this search strategy, we appropriately include all the relevant constraints listed in Sec. 5 coming from various experimental results. It is beyond the scope of this work to explore the entire parameter space; instead we employ an optimization technique that helps us efficiently explore different Yukawa textures and verify their viabilities as discussed below.
Our numerical method is based on a constrained minimization where five of the neutrino observables (∆m 2 21 , ∆m 2 31 , sin 2 θ 12 , sin 2 θ 23 , sin 2 θ 13 ), along with the four flavor ratios (R D , R D * , R K , R K * ), are forced to stay close to their experimental measured values. Deviations from these values are parametrized by pulls defined by T i is the theory prediction, O i is the experimental central value, and E i is the associated 1σ error corresponding to an observable i. A χ 2 -function is then formed out of these pulls that enters the minimization. The theory parameters x k are further constrained by the other experimental measurements discussed in the previous section. Utilizing the set of formulas derived in Sec. 5, we impose those conditions to restrict the theory parameters x k , such as Br(µ → eγ)[x k ] < 4.2 × 10 −13 . By performing this constrained minimization for several different textures of Yukawa coupling matrices, we filter out scenarios that fail to give a reasonable fit.
Corresponding to Eqs. (22) and (23), the minimal and viable Yukawa textures that resulted from our numerical search, we provide one benchmark point for each in Eqs. (49) and (50), respectively. A list of observables associated with these parameter sets is tabulated in Table III. Theŷ L = y L U matrices associated to these benchmark points are listed in Appendix 1.
For each of these Yukawa textures, we iterate over the aforementioned constrained minimization procedure ten thousand times to explore the correlation among different physical quantities. These results are presented in Figs   As can be seen from these plots, experimental results of charged lepton flavor violating processes namely µ → e conversion in nuclei, and → γ decays have strong implications in our scenario. Future improved sensitivity of these experiments will probe most part of the parameter space. For example, Fig. 4 demonstrates the correlation of R K ( * ) ratios with µ → e in gold. This shows that for both the textures, specially for TX-I, only a small parameter space will remain viable, since µ → e conversion rate cannot be made arbitrarily small and still be consistent with observed R K * ratio. has decay modes to jjττ and ttττ with corresponding The solid red horizontal line corresponds to the present bound of CR(µ → e) < 7 × 10 −13 from SINDRUM II [130], and the gray solid horizontal line is for the most stringent bound of CR(µ → e) < 10 −16 from future projected sensitivity [131][132][133][134][135][136][137]. The vertical red dashed line corresponds to present experimental limit Br(τ → µγ) < 4.4 × 10 −8 [128]. The color code is the same as that of Fig. 4. LQs decaying to bbττ . However, we have set the LQ mass for R 2 at 1 TeV for demonstration, which automatically satisfies the mass limit from LHC, as quoted in Table II. Since, LQs in both of these textures only decay to heavier quarks (c, b, t), LQ pair production t-channel diagram and Drell-Yan-like production with + − final state do not put important constraints with the choice of Yukawa couplings presented in Eqs. (49) and (50). From the above analysis it can be inferred that consistent explanation of R D * ratios requires the mass of the R 2 LQ in the range ∼ 800-1000 GeV without making the corresponding Yukawa couplings much larger than unity. A scalar LQ of this type can be directly probed at the LHC by the upcoming high luminosity run. A detailed collider study is beyond the scope of this work.

Conclusion
In conclusion, we have proposed an economic model where three of the problems are inter- is achieved with only a limited number of Yukawa parameters. This is a highly non-trivial task since the same parameters that explain neutrino oscillation data, and R D ( * ) , R K ( * ) flavor ratios, also lead to many other flavor violating processes that are strongly constrained by experimental observations. Low energy experiments that are searching for rare lepton flavor violating decays provide tight constraints, and future improved sensitivity of these measurements will probe substantial part of the parameter space of the model. For the explanation of R D ( * ) , since complex Yukawa coupling of order unity is required, such a scenario can be directly probed in the future experiments by precise measurements of electric dipole moments [169]. In our set-up, the explanations of both R D and R D * ratios within their 1σ allowed range require the branching ratio Br(B c → τ ν) much larger than 10%, which in principle can be ruled out in future by reducing the uncertainties associated with QCD calculations. Furthermore, our description of flavor anomalies, particularly for R D ( * ) ratios predict LQ states around the TeV scale that are directly accessible at the LHC.

Appendix 1
In this appendix we present the numerical values of theŷ L matrices for the two benchmark points considered in the main text.  The red (gray) solid horizontal line represents current (future projected sensitivity) bound