Anatomy of $b \to c \, \tau \, \nu$ anomalies

In recent times, one of the strongest hints of Physics Beyond the Standard Model (BSM) has been the anomaly in the ratios $R_D$ and $R_{D^\ast}$ measured in the charged current decays of $B$-mesons. In this work, we perform a comprehensive analysis of these decay modes, first in a model independent way and subsequently, in the context of composite Higgs models. We discuss in depth as to how linearly realised $\rm SU(2)_L \times U(1)_Y$ symmetry imposes severe constraint on the various scenarios because of correlations with other $\Delta F =1$ processes and $Z\,\tau\,\tau$ and $Z\,\nu\,\nu$ couplings. In the composite Higgs paradigm with partial compositeness, we show that, irrespective of the flavour structure of the composite sector, constraints from $\Delta F =2$ processes bring the compositeness scale down to $\sim 650$ GeV which is in tension with electroweak precision observables. In the presence of composite leptoquarks, the situation improves only marginally (a factor of $\sqrt{2}$ in the compositeness scale), thus making the new states soon discoverable by direct searches at the LHC. We also comment on the possible explanation of the $R_{K,K^*}$ anomalies within the composite Higgs framework.


Introduction
The Standard Model (SM) of particle physics has been remarkably successful in explaining almost all the measurements made till date in accelerator-based experiments, ranging from a few GeV in centre-of-mass energy to a few hundred GeV. However, deviations from the SM expectations approximately at the 2σ − 4σ level have shown up in a number of recent measurements involving semi-leptonic B-meson decays, both in charged current and neutral current channels.
In this work, we focus mainly on the analysis of the charged current anomalies 1 , namely R D and R D * defined in the following way where the 0 stands for either e or µ. The experimental results as well as the SM predictions for R D and R D * are summarised in table 1. We also show two other relevant recent Observable SM prediction Measurement 0.300 ± 0.008 [1] R D 0.299 ± 0.011 [2] 0.407 ± 0.046 [3] 0.299 ± 0.003 [4] R D * 0.252 ± 0.003 [5] 0.304 ± 0.015 [3] 0.260 ± 0.008 [6] P τ (D * ) −0.47 ± 0.04 [6] −0.38 ± 0.51(stat.) +0.21 −0.16 (syst.) [7,8] R J/ψ 0.290 0.71 ± 0.25 [9] Table 1: Observables, their SM predictions and the experimentally measured values. The experimental averages for R D and R D * shown in the third column are based on [7,[10][11][12][13][14][15]. The SM prediction of R J/ψ is based on the form-factors given in [16], see appendix-B for more details. As the B c → J/ψ form-factors are not very reliably known, we do not show any uncertainty for R J/ψ . However, it is expected to be similar to that of R D * .
measurements, which are however rather imprecise at the moment -the τ polarisation, P τ (D * ), in the decay B → D ( * ) τ ν, and R J/ψ , a ratio similar to Eq. 1.1 for the decay B c → J/ψ τ ν. It can be seen from Table 1 that a successful explanation of the R D,D * anomalies requires a new physics (NP) contribution of the order of at least 10 -20% of the SM contribution to the branching ratio. As the SM contribution is generated at the tree level by W ± boson exchange, this is a rather large effect. Such a large effect puts any NP explanation under strong pressure arising from experimental measurements of other ∆F = 1 and ∆F = 2 processes, electroweak precision observables and direct searches at the LHC.
In the first part of this work (sections 2 -4), we provide a comprehensive analysis of the possible explanations of these anomalies in as model independent way as possible.
We discuss the various implications of (linearly realised) SU(2) × U(1) symmetry. In particular, we focus on the correlations among the observables which could be used in the future to decipher the physics behind these anomalies.
Some of the results which are presented in this part already exist in the literature in some form or another. While we try to perform the analysis in a comprehensive and systematic way, and present our results within a unified language, we will explicitly point out to the existing literature wherever appropriate.
In the second part of the paper (section 5), we apply these results to composite Higgs models with partial compositeness. An explanation of the flavour anomalies within this framework has received a lot of attention in the recent past [17][18][19][20][21][22][23][24][25][26][27][28]. However, this scenario, motivated by the Higgs mass naturalness problem, generically predicts flavour violating effects which are often too large to be compatible with experimetal measurements. This can be partially cured by introducing additional flavour symmetries suppressing the undesirable flavour violating effects [17,20,24]. In this work, instead of explicitly relying on flavour symmetries, we take an agnostic approach and rely only on the correlations among the various flavour violating observables coming from partial compositness. Interestingly, even without making any assumption on the flavour structure of the composite sector, we are able to find strong correlations between ∆F = 1 and ∆F = 2 observables leading to an upper bound on the scale of compositeness.

Operators for b → c ν decay
The relevant Lagrangian for the quark level process b → c ν can be written as, where the ellipses refer to terms which are suppressed by additional factors of ( ∂ Λ ) 2 . As ( ∂ Λ ) 2 ∼ ( M B Λ ) 2 for the processes we are interested in, these ellipses are completely negligible for new physics (NP) scales heavier than the weak scale. Here, (2.2) and the definition of the operators are Note that the operators in the right hand side of Eq. (2.3) (referred to as right-chiral operators below) involve right-chiral neutrinos. In this work, we assume that light rightchiral neutrinos do not exist in nature 2 . Moreover, even in their presence, the operators involving them do not interfere with those involving left-chiral neutrinos (and hence not to the SM operators). This means, by naive power counting, that in order to explain the experimental data by the right-chiral operators, the required NP scale of these operators have to be lower than that for the left-chiral operators.
For notational convenience, we will normalise the new WCs by (2.4) Thus, we have In the SM, C cb ν VL = 1, C cb ν AL = −1.
Although there are five operators with left-chiral neutrinos, not all of them contribute to both R D and R D * . This is because the following matrix elements vanish identically: (2.8) Thus, the operators O cb ν AL and O cb ν PL do not contribute to R D , and similarly, the operator O cb ν SL does not contribute to R D * .
We now provide semi-numerical formulae for the branching ratios in terms of the Wilson coefficients (WCs) (from now onwards, instead of using "cb ν" we will just use " " in the superscript of the operators and WCs) : Here, ∆C i correspond to the NP WCs g i of Eq. 2.1. The above formulas are based on the analytic expressions of the decay amplitudes given in [31], and are reproduced in apendix C for later convenience. In order to obtain the various numerical coefficients, central values of the form-factors (see [31] for more details) and other parameters have been used.
If NP is assumed to exist only in the decay to τ leptons, one gets from the above formulas We report similar formulas for B c → η c ν and B c → J/ψ ν in appendix C.3 and C.4 respectively.

Explaining R D and R D *
In this section, we systematically study the possible role of various dimension-6 operators in explaining the R D and R D * anomalies. As mentioned before, some of the results that will be shown in this section are not new, and already exist in the literature in some form or another [32][33][34][35][36][37][38][39][40][41][42][43]. Our aim would be to offer a coherent picture to the readers and add some important insights and aspects to the discussion.

Vector and Axial-vector operators
Here we consider only the operators O τ VL and O τ AL , and investigate whether they are capable of explaining R D and R D * anomalies simultaneously. We also comment on the compatibility with the recent measurement of R J/ψ . In Fig. 1, we show the regions in the C τ VL -C τ AL plane that satisfy the experimental data on R D , R D * and R J/ψ within 1σ. Note that the uncertainties in the form-factors have been carefully taken into account in obtaining the various allowed regions. However, the semi-numerical formulas given in the previous section can be used to qualitatively understand the results. It can be seen that there is an overlap region (the overlap between the red and green bands) that successfully explains both R D and R D * . This overlap region is outside the 1σ experimental measurement of R J/ψ , but consistent with R J/ψ at ≈ 1.5 σ.  Figure 1: The vertical red band corresponds to the values of C τ VL that satisfy the experimental measurement of R D within 1σ. Similarly, the green (blue) region corresponds to the values of C τ VL and C τ AL that satisfy the experimental measurement of R D * (R J/ψ ) within 1σ. All the WCs are defines at the m b scale. The oblique dashed line is the locus of the equation It is interesting that C τ VL = −C τ AL ≈ 1.1 falls in the overlap region mentioned above. As we will see in the next section, the relation C τ VL = −C τ AL is expected if SU(2) L × U(1) Y gauge invariance in linearly realised at the dimension-6 level. Note that the vector and axial-vector operators do not have anomalous dimensions if only QCD interactions are considered (see, for example, appendix-E of [31] and also [44]). Hence, we take C τ VL,AL (Λ) = C τ VL,AL (m b ).

Scalar and Pseudo-scalar operators
Here we consider the scalar and pseudo-scalar operators, O τ SL and O τ PL respectively. In the left panel of Fig. 2, we show the parameter space that satisfies the individual experimental data on R D , R D * and R J/ψ within 1σ. As discussed before, while the operator O τ SL contributes to R D only, the operator O τ PL contributes only to R D * . This explains the vertical and horizontal nature of the allowed regions for R D and R D * respectively.
Note that the operator O τ PL directly contributes to the decay B c → τ ν also (refer to appendix-A for more details). The regions below the two horizontal dashed lines cor- and C τ PL that satisfy the experimental measurement of R D and R D * (R J/ψ ) within 1σ respectively. The values of C τ PL that correspond to Br(B c → τ ν) < 30% and < 10% are also shown. Right panel : renormalisation group running of the WCs C τ SL and C τ PL .
respond to Br(B c → τ ν) < 30% and < 10%, which were claimed to be the indirect experimental upper bounds by the authors of [45] and [46] respectively. Thus, an explanation of R D * by the operator O τ PL is in serious tension with the upper bound on Br(B c → τ ν).
The right panel of Fig. 2 shows the renormalisation group (RG) running (considering only QCD interactions) of the WCs C τ SL and C τ PL from the m b scale to 5 TeV using the following equation [31], where, γ = −8. The values at the m b scale are taken from the allowed bands in the left panel.

Tensor operators
We now turn to the discussion of the tensor operator. In Fig. 3, we show the allowed values of C τ TL that are consistent with the 1σ experimental measurements of R D , R D * and R J/ψ . The values enclosed by the green vertical dashed lines correspond to simultaneous explanation of R D and R D * anomalies. Note however that the prediction for R J/ψ in this C τ TL region is ≈ 0.17 − 0.23, which is below the SM prediction and quite far from the current experimental central value. The RG running of C τ TL is shown in the right panel of Fig. 3 (using Eq. 3.1 with γ = 8/3 [31]) where the initial values of C τ TL at the m b scale correspond to the range enclosed by the two vertical dashed lines in the left panel. Note that the tensor operator does not contribute to the decay B c → τ ν because the matrix element 0|c σ µν b|B c identically vanishes. Hence, there is no constraint on C τ TL from the process B c → τ ν.

Combination of Tensor, Scalar and Pseudo-scalar operators
In this section, we consider the scenario in which the scalar, pseudo-scalar and tensor operators are present simultaneously. In the upper left panel of Fig. 4, we show the various allowed regions in the C τ TL -C τ SL plane assuming the relation C τ SL = −C τ PL . From the upper panel of Fig. 4, it can be seen that a simultaneous explanation of the R D and R D * anomalies requires C τ SL (m b ) = −C τ PL (m b ) ∈ [0.08, 0.23] and C τ TL (m b ) ∈ [−0.11, −0.06] (the small overlap of the red and green regions for positive values of C τ SL and negative values of C τ TL ). We are ignoring the overlap regions with C τ PL > 1 because of the bound from Br(B c → τ ν). There is also an overlap region enclosing C τ SL = −C τ PL = 0 and for non-zero C τ TL which corresponds to the tensor solution discussed in the previous section.   Figure 4: The red and green (blue) shaded regions in the left panel correspond to the values of C τ SL = −C τ PL and C τ TL that satisfy the experimental measurement of R D and R D * (R J/ψ ) within 1σ respectively. The small overlap of the red and green regions for positive (negative) values of C τ SL (C τ TL ) is magnified separately in the middle panel. The right panel shows the RG evolution of the coupling ratio C τ SL /C τ TL assuming C τ SL /C τ TL = 2 at 3 TeV. See text for more details.
We would like to comment in passing that there exist scalar leptoquark models that generate the operator (cP L ν) (τ P L b) at the matching scale 3 Λ, see e.g., [47]. This operator can be written in terms of the operators in Eq. (2.3) after performing the Fierz transformation 4 , Thus, at the matching scale one gets This was our motivation to consider C τ SL = −C τ PL in Fig. 4. The ratio C τ SL /C τ TL , however, increases with the decreasing RG scale as shown in the right panel of Fig. 4.
3 This operator arises from a SU(2) L × U(1) Y gauge invariant operator l k u jk q j e which, by using Fierz transformation, gives l k u jk q j e = − 1 8 4 l j e jk q k u + l j σ µν e jk q k σ µν u .
See section 4 below for the notations. 4 Note that vector leptoquarks, after Fierz transformation, generate vector operators only in the basis of section 2. A scenario with vector leptoquarks will be discussed in section 5.2.

Combination of vector and scalar operators
In this section, we briefly comment on the scenario where both vector and scalar operators are present (see, for example [48] for a model). In Fig. 5, we show the allowed regions in the C τ VL -C τ SL plane assuming C τ VL = −C τ AL and C τ SL = ±C τ PL .   It can be seen that the overlap of the red and green regions (that corresponds to the simultaneous solution of R D and R D * ) touches the C τ SL = ±C τ PL = 0 point. Thus, a combination of the vector and scalar operators extends the solution with only the vector operator discussed in section 3.1. Interestingly, if the red shaded region shrinks in the future due to more precise measurement of R D (without affecting the current central value much), the combination of scalar and vector operators may lead to a better fit than with only vector operators.

Distinguishing the various explanations
In the previous subsections we saw that simultaneous explanations of the R D and R D * anomalies are possible by 1. a combination of vector and axial-vector operators (the overlap of red and green regions in Fig. 1) 2. a combination of scalar and pseudo-scalar operators (the overlap of red and green regions in Fig. 2) 3. tensor operator only (the region between the two dashed vertical lines in Fig. 3) 4. a combination of scalar, pseudo-scalar and tensor operators (the overlap of red and green regions in Fig. 4, in particular, the region with positive values of C τ SL and negative values of C τ TL .) The second solution is quite strongly disfavoured by the existing indirect upper bound on the branching ratio of B c → τ ν. So we ignore it here. We also ignore the scenario with a combination of vector and scalar operators because of the reason mentioned in the last paragraph of of section 3.5.
We now very briefly comment on the possibility of distinguishing the three possible solutions 1), 3) and 4) by measuring the τ -polarisation (P τ (D * )), forward-backward asymmetry (A FB (D * )) and more interestingly, R J/ψ . In Fig. 6, we plot the predictions for P τ (D * ), A FB (D * ) and R J/ψ for values of the WCs that correspond to various simultaneous solutions of R D and R D * anomalies.   Figure 6: Predictions for P τ (D * ), A FB (D * ) and R J/ψ for values of the WCs that correspond to various simultaneous solutions of R D and R D * anomalies. See text for more details.
It can be seen that it is indeed possible to discriminate the three solutions by measuring P τ (D * ), A FB (D * ) and R J/ψ . In fact, as can be seen from the right panel of Fig. 6, R J/ψ can be a very good discriminating observable between the solutions 1) and 3). Of course, with more data, various kinematical distributions can also be used to discriminate the different Lorentz structures [32,40,43,[49][50][51][52][53].
4 Linearly realised SU(2) L × U(1) Y gauge invariance In the previous sections, we considered operators which were manifestly SU(3) × U(1) em invariant, but invariance under the full electroweak group was not demanded. We investigate the consequences of SU(2) L × U(1) Y invariance in this section.

List of operators
The SU(3) × SU(2) L × U(1) Y invariant dimension-6 operators that lead to b → c τ ν decay are given by (using the same notation as in [54]; the primes represent the fact that the operators and couplings are written in the gauge basis) where, ij is antisymmetric with 12 = +1, and (Note that, the operator structure l j p σ µν e r d s σ µν q j t vanishes algebraically.) The operators [O  are not relevant for the explanation of the R D and R D * anomalies, and we will ignore them in the rest of the paper.
It can be seen from Eq. (4.1) and Eq. (4.5) that the operators only involve left chiral fields. Consequently, these operators only lead to V-A interactions. We stress that it is not true in general that linearly realised SU(2) L × U(1) Y gauge invariance forbids V+A operators at the dimension-6 level. For example, the operator in Eq. (4.7) generates V+A operator, but, as mentioned before, it does not lead to lepton non-universality at the dimension-6 level. This is an important consequence of linearly realised SU(2) L × U(1) Y gauge invariance.
Note however that at the dimension-8 level, the operator (O τ VL +O τ AL ) with the possibility of lepton non-universality can be generated. For example, consider the operator where the objects inside each of the parenthesis are constructed as SU(2) singlets. After electroweak symmetry breaking, this operator generates the following interaction term with right handed current in the quark sector. We will however ignore dimension-8 operators in the rest of this paper.

Correspondence with section 2
We now expand the various SU(2) structures in order to relate the WCs of the SU(2) L × U(1) Y invariant operators to those in section 2: The scalar and tensor operators can be decomposed similarly. It is clear that, as a consequence of the manifest SU(2) L × U(1) Y gauge invariance, the operators relevant for the explanation of the R D and R D * anomalies get related to other operators, in particular, to operators that give rise to neutral current decays. However, in order to understand these correlations more concretely, we have to first rotate the fields from the gauge to the mass eigenstates.
From the gauge to the mass eigenstates: We introduce the following mixing matrices which relate the gauge and mass eigenstates 14) The CKM and PMNS matrices are defined as Using the above definition of the mixing matrices, we get (see appendix D) Similar relations can also be found for the scalar and tensor operators, see appendix D.
see appendix D for more details.
Note that the operator φ † i ← → D I µ φ l p σ I γ µ l r modifies the leptonic charged and neutral current vertices of W and Z bosons respectively (see Eq. 4.13). So in order to explain the R D ( * ) data by this operator, lepton non-universality has to be introduced at these vertices. However, a strong bound on such lepton non-universality exists from LEP [55]: This means that the branching ratio of W + → τ + ν can exceed that of W + → µ + ν or W + → e + ν at most by 10.3% at 1σ. Thus the correction to the W τ ν vertex can at most be 5% of the SM, assuming that the Wμν and Wēν vertices have no NP. This gives (using Eq. 4.13 and appendix D) 5 which, from the second line of Eq. 4.16, implies As we saw in section 3.1, ∆C τ VL = −∆C τ AL < 0.05 is not enough to explain the R D ( * ) data within their 1 σ experimental ranges. Moreover, as can be seen from Eq. (4.16), contribution of this operator to the WCs C τ VL = −C τ AL is suppressed by V cb compared to the other contribution. This operator also modifies the Z µτ γ µ P L τ coupling, and the modification is given by 6 Using the experimental constraint from LEP [56] , and assuming that there is no NP in the decays to light leptons, we get 7 Similarly, The SM couplings are defined by where g τ W = 1. Any deviation from the SM will be denoted by ∆g τ W . 6 We define the SM couplings to be where g τ L = −1/2 + sin 2 θ W ≈ −0.27 and g τ R = sin 2 θ W ≈ 0.23. The vector and axial-vector couplings are defined by g τ V,A = g τ L ± g τ R . 7 Here we have assumed (V PMNS ) 33 = 1. However, given the strong experimental constraints, our results will not change if correct values of (V PMNS ) 13 and (V PMNS ) 23 are used.
which should be compared with the experimental constraint [56] Given these constraints, it is clear that the operator O φl alone is unable to explain the R D,D * data. We will thus not consider this operator anymore in this work. Before closing this section, we would like to mention that a much stronger indirect constraint (compared to Eq. 4.21) on the W τ ν coupling can be obtained from measurements of leptonic tau decays assuming that no other four-fermion operator that can either contribute to τ → e νν or µ → e νν exists [57]. Assuming no NP in the W µ ν vertex, this gives, (4.29)

Correlations
The linearly realised SU(3) × SU(2) L × U(1) Y symmetry leads to various correlations among the flavour violating neutral and charged current observables. To start with, we assume that only the operator(s) that is (are) needed for the explanation of the R D,D * anomalies is (are) present. In particular, we consider the operator of Eq. 4.1 and investigate the correlations arising from it. We will not consider the scalar and tensor operators anymore because, as discussed in the previous sections, an explanation of R D,D * anomalies by scalar (or a combination of scalar and tensor) operators is strongly disfavoured by the upper bound on Br(B c → τ ν), and it is rather difficult to generate only the tensor operator.
Without loss of generality, we now go to a basis where the left-chiral down quarks and left-chiral charged leptons are in the mass basis. This amounts to setting It should be emphasised that we are not making any assumption here by going to a particular basis. This just means that our primed WCs of section 4.1 are defined in this basis. In this basis, we have Let us first consider the contribution to the operator (τ γ µ P L ν) (cγ µ P L b). From Eq. (D.1) and (D.2), one can read off the coefficient of this operator. For simplicity, we assume that the NP Wilson coefficients, [C lq ] p r s t , are diagonal in the Lepton flavours. We get, Note that ν Lτ = [V † PMNS ] 3r ν L r , ν Lτ being the τ -flavour neutrino. As we discussed in the previous sections, in order to explain the anomalies at the 1 σ level, the coefficient of the operator (τ γ µ P L ν τ ) (cγ µ P L b) in Eq. 4.33 should at least be ∼ 0.16 for a NP scale Λ = Λ SM . This gives, We would now like to understand whether this condition is consistent with the other measurements of B meson decays. We first consider the decay B → K * ν ν, or in other words, the operator (νγ µ P L ν) (sγ µ P L b). Contribution to this operator is given by Experimental bound on Br(B 0 → K * 0ν ν) [58] then requires the Wilson coefficients to satisfy (the SM prediction is taken from [59]), The first term on the left hand side of Eq. 4.34 contributes to b → dν ν processes. Using the experimental bound on the Br(B 0 → π 0ν ν) [58] and the corresponding SM prediction from [60], we obtain The same coupling can also be constrained by measurement of Br(B u → τ ν τ ). Assuming the maximum allowed value of Br(B u → τ ν τ ) to be twice that of the SM [61] and, in the absence of cancellations (see Eq. 4.51 below), we get See also [62] for a related discussion. We now investigate whether the coupling ([C (3) lq ] 3333 + [C (3) lq ] * 3333 ) is constrained by other measurements. First of all, notice that the coefficient of the operator (τ γ µ P L τ ) b γ µ P L b is given by (4.40) Direct searches of processes involving two τ leptons in the final state constrain this coupling weakly [63]: Moreover, as we show below, the same coupling also appears in the coefficient of the operator (τ γ µ P L τ ) (tγ µ P L t). From Eq. (D.1) and (D.2) we get, for the coefficient of this operator, where, The 2nd, 3rd, 5th and 6th terms in Eq. 4.43 are small because of Eqs. 4.36 and 4.38.
All the other terms which are of the form [C (3) lq ] 33ij , i, j = 1, 2 are constrained by direct searches of τ τ final state (taking into account the enhancement of the di-jet → τ τ crosssection compared to that ofb b → τ + τ − due to larger parton distribution functions, we get bounds which are stronger than Eq. 4.41 by a factor of ∼ 2 for ([C (3) lq ] 3322 + c.c) to a factor of ∼ 8 for ([C (3) lq ] 3311 + c.c)). Thus, the only term which remains is of the form This term, once the top quarks in the operator (τ γ µ P L τ ) (tγ µ P L t) form a loop and are attached to Z, contributes to ∆g τ L [64]. As ∆g τ L is very strongly constrained, see Eq. 4.25, this provides a stringent constraint on the coupling of Eq. 4.44. Indeed, from Eq. E.10 in appendix E, we find which clearly rules out the possibility of explaining R D,D * anomalies by this term, unless there are other contributions to the modifications of the Z couplings making it compatible with the experimental observations.
It is worth mentioning that in our discussion so far we have made the assumption that no other operator is present except the one required for the R D,D * anomaly. The presence of other operator(s) can however help evade some of these constraints [65]. For example, one possibility is to assume the presence of the singlet operator (4.47) This makes the explanation of the R D,D * anomalies by the third term of Eq. 4.34 impossible even in the presence of the operator of Eq. 4.46. This leaves us with two possibilities : I. The anomaly is explained by the second term in Eq. 4.34. The tension with the Br(B 0 → K * 0ν ν) in Eq. 4.36 is assumed to be cured by cancellation against the contribution of the operator in Eq. 4.46. However, in this case, the flavour structure of the BSM sector must be such that the last term of Eq. 4.34 is smaller than the second by at least factor of ∼ 3.
II. The other possibility is to assume the presence of appropriate UV contribution at the matching scale that takes care of the ∆g τ,ν L constraints. In this case, one can explain the anomalies by the third term of Eq. 4.34 alone.
As we will see in section 5.1 and 5.2, where we study the explanation of the R D,D * anomalies within the partial compositeness framework, elements of both the above mechanisms can in principle be present there.
Before concluding this section, we would like to comment on a few observables which do not provide relevant constraints at this moment, but may become important in the future.
Now we assume that we are in scenario (I) where B → K * ν ν transition is cancelled by the singlet operator, and Eq. 4.34 is saturated by the second term. Then, in the standard notation, we get for the WCs, ∆C τ 9 = −∆C τ 10 = −35, with the corresponding Lagrangian: giving rise to large enhancement in B s → τ + τ − (by a factor of ∼ 50 compared to the SM in the branching fraction) and B → K/K * τ + τ − decays (by a factor of ∼ 60 (for K), 75 (for K * ) compared to the SM in the branching fraction). It is interesting to note that large enhancement in Br(B s → τ + τ − ) was also proposed as a possible solution to the like-sign di-muon charge asymmetry observed in one of the experiments in Tevatron [67,68].
• b → uτ ν transition: in this case, the operator (τ γ µ P L ν) (ūγ µ P L b) is generated with the Wilson coefficient: where, we have again used ν Lτ = [V ν L ] 3r ν r , and assumed that the NP Wilson coefficients, [C (3) lq ] p r s t , are diagonal in the Lepton flavours. Assuming that we have both singlet and triplet operators, the b → s(d)νν transitions do not lead to any constraints. Then, if the second (third) term in Eq. 4.34 is responsible for R D,D * anomalies (i.e., saturates the inequality), we get which leads to approximately 45% (20%) increase in Br(B u → τ ν τ ). Instead, if one assumes that the first term in Eq. 4.34 is responsible for R D,D * anomalies and saturates the inequality, the corresponding NP coupling for b → uτ ν becomes, which is obviously ruled out by experiment. Thus, even in the presence of cancellation in b → dν ν, an explanation of R D,D * by the first term in Eq. 4.34 seems very unlikely.
Similar analysis can also be done for the scalar, pseudo-scalar and tensor operators. Since the scalar and pseudo-scalar operators alone cannot explain the anomalies because of the strong constraint from B c → τ ν (see section 3.2), we do not discuss them anymore. The tensor operator, [C lequ ] p r s t l j p σ µν e r jk q k s σ µν u t , on the other hand, is not affected by the process B c → τ ν, and generates, along with the charged current operator which is relevant for R D ( * ) , also neutral current operators involving up-type quarks.

Going beyond the dimension-6 analysis: partial compositeness
In the previous section, we illustrated that some other processes e.g., B → K * ν ν, Z τ τ and Z ν ν couplings can provide stringent restrictions on the possible explanations of R D and R D * anomalies. It would be interesting also to study the correlations with the various ∆F = 2 observables where the constraints on NP are particularly strong. Such an analysis requires specific assumptions on the underlying UV theory, or some powercounting rules. As we discussed before, explanations for the R D and R D * anomalies call for NP close to the TeV scale, which is also expected for the naturalness of the Higgs mass. This coincidence of scales advocates for the speculation of a common origin of these two seemingly unrelated phenomena. This motivates us to consider the Composite Higgs paradigm [69], and, in particular, the models where fermion masses are generated via the Partial Compositeness (PC) mechanism [70]. In fact, recently there has been a lot of effort invested in analysing the B-meson anomalies within this framework [19][20][21][24][25][26][27], all of which, however, focusses on specific models. A novel feature of our study would instead be to carry out the analysis in the EFT language, emphasising the correlations among the various observables. In particular, our aim would be to identify the key features that these models should possess in order to satisfy the experimental data. Our main results will be independent of the concrete realisation of PC, and are thus expected to be quite generic.

R D,D * from the composite electroweak resonances
In the PC paradigm, the SM fermion masses are generated via linear mixing to fermionic bound states (see, [71,72] for a review). We denote the elementary and composite fields byψ andÕ respectively. The SM fields, collectively denoted by ψ below 8 , are related to these fields by a suitable diagonalising matrix: where O is the combination which is linearly independent of the SM fields ψ , which are massless before electroweak symmetry breaking. Note that each entry of the rotation matrix is a matrix in flavour space.
We are interested in the dimension-6 four-fermion operators. These operators are generated by the exchange of the composite vector resonances. The flavour structure of these operators can be found by examining the coupling of the composite vectors which is given byψ where g(g * ) denotes the couplings of the SM (composite) sector collectively (see appendix F for definitions and more details). In the limit of g * g, the flavour nonuniversal part arises predominantly from the first term of Eq. 5.2. Thus, the four-fermion operators at the dimension-6 level schematically take the form Our aim would now be to understand the correlations among the flavour-changing ∆F = 2 operators and those that contribute to the R D ( * ) anomalies. The effective Lagrangian for the ∆F = 2 transitions can be written as where the constant in front is given by in which the first term inside the parenthesis corresponds to the contribution of the composite gluon, the second to the SU(2) L,R triplets and the third to U(1) X vector bosons (the number 4/9 is fixed by the U (1) X charge assignment of the up-like multipletÕ q 1 , see Eq.F.6). Experimental data onK-K,B d -B d andB s -B s mixings give the following constraints 9 , 10 −3 , fromK-K mixing, i.e., i = 1, j = 2 [73] 1.1 × 10 −3 , fromB d -B d mixing, i.e., i = 1, j = 3 [74] 4 × 10 −3 , fromB s -B s mixing, i.e., i = 2, j = 3 [74] , (5.6) where the numerical values are obtained by running the couplings to the scale M * . Keeping the above constraints from ∆F = 2 processes in mind, we now look at the b → cτ ν transitions. We assume that the NP contribution arises from the exchange of a composite vector field which is a triplet of SU(2) L . This generates the interaction Lagrangian where we have assumed g * 2 = g * , M * 2 = M * . If we decide to remain agnostic about the leptonic sector, we can still use the loose upper bound which is satisfied even for maximal possible τ compositeness. Thus, for the explanation of R D and R D * anomalies at the 1 σ level , we need where the numerical factor 0.2 corresponds to ∆C τ VL = −∆C τ AL = 0.08 (see Fig. 1). Expanding Eq. 5.10, we get Here, we have assumed that only one ∆F = 2 operator (the operator Q 1 in the basis of [73]) is generated. In principle, other operator(s) may also be generated at the matching scale, and cancel part of the contribution from Q 1 . However, barring large accidental cancellations, our results should always hold.
Using the upper bounds on V d † Lŝ q †ŝ from Eq. 5.6 and the trivial inequality V d †ŝ q †ŝ q V d 3 3 ≤ 1, we now get As the first two terms are negligibly small compared to the third term (for small (M * /TeV)/g * ) on the left hand side, we finally get Note that partial compositeness automatically selects the scenario (II) (see discussion after Eq. 4.47) for fitting the R D,D * anomalies. This solution, as mentioned in section 4.3, requires the presence of additional UV contributions to protect g τ,ν L couplings of the Z boson. In the appendix H, we explicitly show how this can be achieved. Interestingly, the generated operators automatically satisfy the condition of scenario (I) due to the SO(4) structure of the model.
We would like to make a few comments here regarding the robustness of this result and its applicability to the various models employing partial compositeness. The only assumption that we have made in deriving the above result is that the charged current operator (see Eq. 5.7) is generated by a vector field which is a triplet of electroweak SU(2) L . The rest of the discussion is completely model independent and applies to various embeddings of the SM fermions into the composite multiplets, choices of the off-diagonal elementary-composite mixing parametersŝ, and is practically independent of the mass of the composite gluon and the mass of the U (1) X vector. It should also be emphasised that we have been completely agnostic of the dynamics that allows the model under consideration to satisfy the constraints from ∆F = 2 processes namely, those given in Eq. 5.6. In fact, it is well known that models based on anarchic partial compositeness [75,76] can satisfy these bounds only for M * 10 − 20 TeV, which is too high a scale to explain the R D and R D * anomalies. The scale of the compositeness can however be lowered and made consistent with the R D ( * ) anomalies by invoking additional flavour symmetries, for example U(2) [17,77,78].
The constraint in Eq. 5.12, in general, can pose serious difficulties with the electroweak precision observables and measurement of Higgs's couplings to electroweak vector bosons. Indeed the constraints from electroweak precision tests [79][80][81] require the scale of compositeness to be 1.2 TeV in order to satisfy the data at 2σ level. At the same time the mass of the vector resonance is related to the scale of compositeness, f , as where a ρ is a number of O(1). In an explicit two-site construction, a ρ = 1/ √ 2 (see for example [72]) so that the compositeness scale is constrained to f 0.64 TeV. This is incompatible with the bound from electroweak precision measurements mentioned above. It may however be possible to accommodate the electroweak precision observables by additional UV contributions, see for example, [82][83][84].
The tension with meson mixing data makes it interesting to think of other possibilities of enhancing the contributions to R D and R D * without modifying the ∆F = 2 observables considerably. This can be partially achieved in scenarios with composite vector leptoquarks which we discuss in the next section.

Leptoquark contribution
The composite vector leptoquark scenario in connection to the B meson anomalies was first proposed in [17,20,24]. In this construction, the global symmetry of the composite sector is extended from SO (5

) × SU(3) (where SU(3) is weakly gauged later and becomes the SU(3) of QCD) to SO(5) × SU(4). The composite gluon, which is an octet of SU(3), lies inside the 15 dimensional adjoint of SU(4) and is accompanied by two SU(3) triplets (3 and3) and a singlet. The SU(4) global symmetry is broken to SU(3) both by the QCD gauging as well as by the elementary-composite mixing (see appendix G.)
Here we focus only on the relevant interaction term for R D,D * anomalies, 14) The effective Lagrangian for the b → c τ ν processes can now be written as, Without loss of generality, we now go to the basis of elementary fields and composite fields in which the lepton compositeness matrix has the following form: where * stands for non-zero entry. We now assume that only the third family of leptons has a strong mixing with the composite sector i.e. only (ŝ l ) 33 ∼ 1 and the rest of the elements are much smaller. In this case, the WC in Eq. 5.16 is controlled by, Our aim now is to understand how big can be, consistently with an almost diagonal V d † Lŝ q †ŝ q V d L (as the off-diagonal elements are constrained to be 10 −3 , see Eq. 5.6). Similar to the leptonic elementary-composite mixing matrixŝ l , we can also makeŝ q triangular by suitable field redefinitions of the elementary fields. Thus, without loss of generality, we can write, Let us now consider the special case where only the third generation quark mixes strongly with the composite sector so that, It can be noticed that these elements have an additional suppression of (s ij /s 33 ) compared to the naive expectation. This renders the contributions of the first two terms of Eq. 5.18 subdominant. Thus, adding the contribution of the electroweak triplet from Eq. 5.11, for the explanation of R D,D * anomalies at the 1 σ level, we must have where we have assumed that the electroweak triplet and the leptoquarks have the same mass and coupling. Hence, the role of leptoquarks is just to double the contribution to R D,D * without worsening the other low energy observables. This increase of the upper bound on the scale of compositeness by a factor of √ 2 helps ameliorate the constraints from S and T parameters which are now in agreement at almost 2 σ level 10 .
10 Note that if we assume g * 4 M * 4 > g * 2 M * 2 i.e., smaller masses of the SU(4) resonances than those of the SO(4) fields, we can be in the situation where the composite leptoquark contribution dominates in R D,D * and the tension with electroweak precision observables can be relaxed even further.
It is worth emphasising that the result of Eq. 5.22 was derived assuming the hierarchical nature (see Eq. 5.20) of the mixing matrixŝ q and the constraint of Eq. 5.22 can be relaxed if this assumption is not valid. For example, if we assume that the matrix s q is not hierarchical but unitary, then V d †

R K,K * anomalies
In this section, we investigate very briefly whether the R K,K * anomalies can be explained within the composite Higgs framework (see [18-22, 24-28, 86] for related discussion). It is known that the discrepancy of the experimental data on R K and R K * with the SM expectations can be alleviated by the following operator [87-92] 11 L b→sµµ = − 1 Λ 2 (sγ µ P L b)(μγ µ P L µ) , (6.1) with 1/Λ 2 1/(38 TeV) 2 at the 1 σ level. 11 Actually, the experimental value of R Low K * cannot be explained simultaneously with R K and R Central K * by this operator (see, for example, the upper left panel of Fig. 6 in [92]), and either additional light fields [92] or tensor operators [93] are required.
In models with partial compositeness, such an operator can be generated by the exchange of either a neutral Z vector boson or a vector leptoquark. We examine the flavour structures of these two cases and identify the features that can explain the data.
Z contribution: Following the analysis of section 5.1, neutral composite bosons ρ 3 L,R , ρ X will generate which, after implementing theB s -B s mixing constraint from Eq. 5.6, gives (assuming V e L = 1 and diagonalŝ l ), for the explanation of the R K,K * anomaly, where 'const' is defined in Eq.5.5 and g * ,2 = g * , M * ,2 = M * . This inevitably requires large muon compositeness. Let us compare our results with the discussion in the previous two sections 5. = ( being some small parameter), ∆F = 2 observables scale as 2 and R K as . It is precisely this extra power of suppression that can make the explanation for the two measurements consistent [20].
Leptoquark contribution: The flavour structure in this case is different from the Z contribution and the relevant operator in given by In this case correlations with the other low energy measurements are less strict and as an illustration we will consider two extreme scenarios (for simplicity we will assume diagonalŝ l,q ) • flavour trivial lepton sector: In this case we assume V e L = 1 and this obviously evades all the constraints from LFV processes like τ → 3µ. In such a scenario, the main constraint comes from B s mixing. Using the bound from Eq. 5.6, we get where s s,b are equal to the maximal possible values allowed by theB s − B s mixing. Interestingly, even in this case the bound becomes less strict compared to the one obtained for the Z contribution (see Eq. 6.3). However, the scale of the muon compositeness must still be quite high.
• flavour trivial down quarks [24]: If we assume V d L = 1, R K can be generated solely by the leptoquark contribution and Z mediated diagrams vanish. Interestingly, we can correlate the R K with the flavour violating τ decay τ → 3µ arising from the operator: where now const = 1 2 comes from the contributions of the ρ 3 L , ρ 3 R , ρ X and ρ T 15 and we have assumed that the λ l mixing is the dominant one (see appendix H). Assuming that the mixing with the first generation is small, we focus only on the µ − τ rotations with the mixing angle θ. The experimental bound on τ → 3µ [94] gives const 2 If the angle θ is small, say θ ∼ s µ /s τ 1, then the upper bound on the muon compositeness becomes On the other hand, the bound on R K implies s s s b s µ s τ sin 2θ ≥ 10 −3 M * /TeV g * 2 (6.10) If s b ∼ 1 and s τ ∼ 1 we see that there is no tension between the R K data and the τ → 3µ data; in fact Eq. (6.9) translates into a bound on the compositeness scale of the strange quark which is similar to the naive expectations for the left-handed strange quark compositeness s s ∼ V ts .
Thus we can conclude that it is possible to fit R K as well within the partial compositeness paradigm.

Summary and outlook
In this paper, we have studied various aspects of the R D and R D * anomalies in depth.
The main objective of our work has been to understand potential correlations of R D,D * to other ∆F = 1 and ∆F = 2 processes that give rise to constraints on the NP explanations, and thus allowing us to identify the desired properties of the underlying UV theory.
We then extend our analysis to composite Higgs paradigm with the partial compositeness mechanism to generate the fermion masses. In this case, because of an available power-counting rule, the ∆F = 2 processes, namely K, B d and B s mixing measurements turn out to be extremely constraining. We show that generically the models with partial compositeness can offer an explanation of these anomalies only if the scale of compositeness is below 0.90 (0.64) TeV for scenarios with (without) leptoquarks. While the requirement of such a low scale is favoured by the electroweak hierarchy problem, it is problematic from direct searches, and also indirect electroweak precision measurements unless some additional cancellations are involved.
Finally, in section 6 we also comment on the possibility of explaining the other neutral current B-meson anomalies R K and R K * in this framework.
As the charged current anomalies require a NP scale which is rather low (∼ TeV), they might as well be the harbingers of new physics at the TeV scale. It is thus important to critically examine the models that can provide simultaneous solutions to different problems at the TeV scale. At this point, it seems that the manifestation of New Physics, if any, in the dynamics of flavour transitions is likely to be quite non-generic and subtle. Thus the interpretation of any NP signal would require a large amount of data with a high precision. It is encouraging that such a large amount of data are expected to come from both the LHCb and Belle-II in the near future, and hopefully, we are not far from an exciting discovery.
DB was supported in part by the Israel Science Foundation (grant no. 780/17).

Appendix A Decay Width of the B c meson
The differential decay rate for the process B − c (p) → τ − (k 1 ) +ν τ (k 2 ) is given by where, k 1 is the 3-momentum of the τ in the rest frame of the B c meson, and The matrix element is given by, (The operators O cb ν VL,VR , O cb ν SL,SR and O cb ν TL,TR do not contribute because the corresponding matrix elements, 0|cγ µ b|B c , 0|cb|B c and 0|cσ µν b|B c identically vanish) In the above step, we have used 0|cγ µ γ 5 b|B c (p) = if Bc p µ , and Br(B c →τν τ )=10% The Br(B c → τ ν τ ) as a function of C τ AL and C τ PL . The upper bounds, 30% and 10% on this branching fraction from [45] and [46] respectively are also shown. The SM branching ratio is ≈ 2%. We have used f Bc = 0.434 ± 0.015 GeV [95] in our calculation.
This gives, Thus, the partial decay rate is given by, which gives, for the branching ratio, The B c → η c matrix elements are parametrised in the same way as the B → D matrix elements, see for example section-4 of [31]. Unfortunately, only preliminary lattice results are available for B c → η c matrix elements [96]. In Fig. 8, we show the pQCD estimates of the F + and F 0 form-factors from [16]. The preliminary lattice results from [96] are also overlaid.
Similarly, parametrisation of the different B c → J/ψ matrix elements are the same as those for B → D * matrix elements, see again [31] for the notations. The pQCD estimates [16] for the B c → J/ψ form-factors: V , A 0 , A 1 and A 2 , are shown in Fig. 9. Preliminary lattice results for V and A 1 from [96] are also shown.  Figure 8: Form factors F 0 (q 2 ) Bc→ηc and F + (q 2 ) Bc→ηc from pQCD [16] and lattice [96].

B.2 Tensor form-factors
As no estimate of the tensor form-factors exists in the literature, we use the quark level equations of motion to relate them to the other form-factors. We show this explicitly below.
The LHS of Eqn B.6, using the tensor form factor, is then Noting that q = p Bc − p ηc , we can equate the coefficients of q ν on either side of Eqn.
B.6. This gives us the relation between the tensor and the vector form factors to be The hadronic matrix elements forB → V transition are parametrised by Using, the LHS of Eqn. B.14 yields, The first term vanishes, and after using Eqn. B.10 and B.14, leaves Consider the term V (p V , , M V ) |c iσ µν q ν γ 5 b|B(p B , M B ) . Using σ µν γ 5 = i 2 ε µνρσ σ ρσ (we use 0123 = 1, which implied that 0123 = −1) we have Simplification of the RHS using Eqn. B.14, we get, Comparing the coefficients of * .q q µ /q 2 from either side, we get The equations B.17, B.20 and B.21 agree with those given in [97].
Appendix C Formulae for calculating branching ratios The double differential branching fractions for the decays B → D ν and B → D * ν can be written as The normalisation factor, N and the absolute value of the D ( * ) -meson momentum, |p D ( * ) | are given by, where λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca). The angle θ is defined as the angle between the lepton and D ( * ) -meson in the lepton-neutrino centre-of-mass frame, and q 2 is the invariant mass squared of the lepton-neutrino system.

C.1 B → D decay
The quantities a D , b D and c D for negative and positive helicity lepton are given by [31] : Negative helicity lepton: Positive helicity lepton: Negative helicity lepton : Positive helicity lepton: φl ] p r − 1 2 where p ,r ,s ,t [C (1) lequ ] p r s t l j p e r jk q k s u t : p ,r ,s ,t [C (1) lequ ] p r s t l j p e r jk q k s u t = p,r,s,t p ,r ,s ,t lequ ] p r s t l j p σ µν e r jk q k s σ µν u t : lequ ] p r s t l j p σ µν e r jk q k s σ µν u t = p,r,s,t p ,r ,s ,t ] prst (ē p σ µν P R e r ) (ū s σ µν P R u t ) , This gives (1 + 0.6 log(Λ/TeV)) (E.5) TeV Λ

Appendix F Two-site Lagrangian
In this appendix we will briefly sketch the minimal Composite Higgs construction (for the details see the orginal paper [99] and reviews [71,72]). The global symmetry breaking pattern is taken as follows: We will study the phenomenology of the model using an effective field theory, so called two site model [100]. Within this framework the theory consists of two sectors: the composite sector invariant under SO(5) × SU (3) × U (1) X and the elementary sector invariant under SU (2) × SU (3) × U (1) Y . The SM gauge symmetry is identified with the diagonal subgroup, where the "composite hypercharge" generator is defined as follows The Higgs boson appears as the Goldstone boson of the spontaneous symmetry breaking SO(5)/SO(4). We will use the CCWZ formalism [101,102] to parametrise the nonlinearly realised symmetry SO(5)/SO(4) for the composite sector (in our discussion we will follow closely the notations of [103]). Then the Higgs boson appears inside the usual Goldstone boson matrix U which in the unitary gauge is equal to: where h is the Higgs boson and the f is the scale of the global symmetry breaking. It is customary to define two covariant derivatives (Maurer-Cartan 1-form) decomposing it along the broken Tâ and unbroken T a generators. The Higgs kinetic term and the mass of the gauge terms come from the two derivative term of the chiral Lagrangian Elementary fermions obtain their masses by mixing with the composite fermionic states, and in this study we will focus only on the model where the composite fields appear as a fiveplet of SO(5) , i.e. MCHM5 model [99]. These fivepletes after the SO(5)/SO(4) breaking can be decomposed as a fourplet of SO(4) and a singlet. The fourplet of SO(4) has in its turn two SU (2) L doublets: one with the standard model quantum numbers denoted as O SM 12 and another one O EX , where the doublets are related by SU (2) R transformations. The singlet operators are denoted asÕ u,d,e and the full spectrum is 5-plet Ψ q 1 = Õ q 1 ,Õ u (F.6) 12 We intend that it has the same quantum numbers as the elementary doublet under the SU (3) ×  10) where the charges of the components of a SU (2) R doubletÕ = Õ 1 ,Õ 2 under T 3 R are equal to + 1 2 and − 1 2 respectively. The elementary fields are denoted asq L ,l L ,ũ R ,d R , e R . Each field is a 3-vector in the flavour generation space and the subscript of χ field indicates its electric charge.
The elementary SU (2) L doubletq L is embedded in the incomplete fiveplet of SO(5). Thus, the group representations and charges of the fermion states are depicted in table 2. Note that we have two composite doubletsÕ q 1 SM andÕ q 2 SM which have the same quantum numbers under the SM gauge group; similarly for the leptons. The symmetries of the composite sector are broken explicitly to the diagonal subgroup by the mixing with the elementary sector, which is given by: L f lavour = λ q M * q L U (h)Ψ q 1 +λ q M * q L U (h)Ψ q 2 + λ u M * ũR U (h)Ψ q 1 + λ d M * dR Ψ q 2 +λ l M * lL U (h)Ψ l 1 +λ l M * lL U (h)Ψ l 2 + λ e M * ẽR U (h)Ψ l 2 (F. 11) where the SM doublets where uplifted to incomplete 5-plets as follows: λ qqL ≡ λ q [(0, q L ), 0] λ qqL ≡λ q [(q L , 0), 0] , (F.12) where we put zeros in all the missing components and (q L , 0) singles out the SO(4) multiplet. Note also that symmetries of the model allow us to further split λ q mixing into two independent parameters where the sum over repeating indices is understood. Let us look at the fermion spectrum before EWSSB. Due to the mixing λ we will have one massless SM state and one heavy field with the mass M * (1 + λ)/ √ 1 + λ 2 , which becomes M * in the limit 1 λ . This leads to the mixing between the elementary and composite states which can be described by the mixing angles defined as follows: with sin θ ψ ≡ŝ = λ √ 1 + λ 2 , cos θ ψ ≡ĉ = 1 √ 1 + λ 2 , (F. 15) whereŝ,ĉ are the sine and cosine of the corresponding mixing angles. Then the SM Yukawa coupling will scale as y u,d ∼ s q 1,2 s u,d M * f . (F.16)

Vector fields
We are interested in the interactions between the SM fermions and the composite vector fields. We will follow the vector formalism [103] (see for example the Ref. [104] for the comparison of various formalisms)for a spin-1 fields where it is assumed that the vector fields transform non-homogeneouslỹ where H is unbroken subgroup (SO(4)) transformation. Then the following interactions are allowed by the CCWZ symmetries: where A µ a are the elementary vector fields. The interaction betweenρ and the composite fermions can be deduced from the symmetries L f erm =Ψγ µ (i∂ µ + g * ρµ ) Ψ, (F.20) whereρ µ =ρ a µ T co a and T co a are the generators of the global symmetry group of the composite sector 13 . In order to get mass eigenstates vectors, a diagonalisation of the matrix of masses and mixing is needed whereρ µ is an eigenstate with mass of M * 1 + g 2 el /g 2 * and the orthogonal A SM µ is the massless state, that is identified with the SM gauge boson. Rotating to the mass eigenstate basis we get where the first term comes from the mixing of the elementary and composite fermions and the second term corresponds to the mixing between composite and elementary vector bosons Eq.F.21. In this paper we are mainly interested in the flavour non universal and flavour violating effects, so the contribution of the last term will be subleading since g * g and the non-universalities inĉ ∼ 1 −ŝ 2 /2 have an extraŝ supression.
B (1,1) 0 that is associated to the T 15 generator of SU (4) and there are also the lepto- where g * is the strong coupling for the SU (4) of the composite sector. This interaction after integrating out the heavy fermions reduces to g * √ 2ψ q ŝ q †ŝ Since the bound on g τ Z is a bit stronger, it is natural to assume that λ l >λ l and the contribution to R D,D * is dominated by λ l . Note that this coupling does not enter the leading expression of the τ mass which scales as m τ ∝ λ eλl . (H.3) Then in order to pass the constraints from g ν Z we will have to tune additionally the parameter c 1 as was suggested in [24].