R(D(∗)) from W′ and right-handed neutrinos

We provide an ultraviolet (UV) complete model for the R(D(∗)) anomalies, in which the additional contribution to semi-tauonic b → c transitions arises from decay to a right-handed sterile neutrino via exchange of a TeV-scale SU(2)L singlet W′. The model is based on an extension of the Standard Model (SM) hypercharge group, U(1)Y , to the SU(2)V × U(1)′ gauge group, containing several pairs of heavy vector-like fermions. We present a comprehensive phenomenological survey of the model, ranging from the low-energy flavor physics, direct searches at the LHC, to neutrino physics and cosmology. We show that, while the W′ and Z′-induced constraints are important, it is possible to find parameter space naturally consistent with all the available data. The sterile neutrino sector also offers rich phenomenology, including possibilities for measurable dark radiation, gamma ray signals, and displaced decays at colliders.


Introduction
Measurements of |V cb |-independent ratios have been performed by the Babar [1,2], Belle [3][4][5], and LHCb [6] collaborations. The results exhibit a tension with the Standard Model (SM) expectations at the 4σ level when data from both D and D * measurements are combined [7] (see also refs. [8][9][10][11][12]). The b → cτν τ decays occur at tree-level in the SM. New Physics (NP) explanations of the R(D ( * ) ) anomaly are therefore nontrivial, since they require new states close to the TeV scale. The NP contributions could, in principle, be due to tree level exchange of a -1 -

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Such a UV completion is rather minimal in its NP field content and can naturally lead to the largest NP effects in the b → cτν transitions.
Further advantages of an SU(2) L singlet W interaction can be understood by comparision to, e.g., the W model of ref. [16] (see also ref. [17]), which requires the W to be part of an SU(2) L triplet vector with the nearly degenerate Z , as dictated by the Z-pole observables. Gauge invariance further requires that the flavor structures of W and Z couplings are related through the SM CKM mixing matrix. In our '3221' model, by contrast, these requirements are lifted, so that the observable effects of Z can be suppressed below the present experimental sensitivity, while at the same time one can still explain the R(D ( * ) ) anomaly through the tree level exchange of the W . Our '3221' is much closer in spirit to the left-right model in refs. [47,48], where the third generation right-handed SM fermions are charged under the new SU(2) R . The addition of vector-like fermions in the '3221' model, however, allows for a flavor structure that is completely safe from the low energy constraints and at the same time avoids the LHC searches.
The paper is structured as follows. In section 2 we first present an EFT analysis of the W interaction with respect to the B → D ( * ) τν data, followed by presentation of the UV complete 3221 model in section 3. Relevant collider and flavor constraints are explored in section 4. Verification of compatibility of the right-handed neutrino with neutrino phenomenology, including its cosmological history, is presented in section 5. Section 6 contains our conclusions. In appendix A we describe the details on the flavor locking mechanism, while in appendix B we discuss the phenomenological implications, if the symmetry breaking of the new gauge interactions is non-minimal.

Operators and effective scale
We assume the SM field content is supplemented by a single new state, the right-handed sterile neutrino transforming as N R ∼ (1, 1, 0) under SU(3) c × SU(2) L × U(1) Y . This state may couple to the SM quarks via any of the four dimension-6 operators suppressing for now the generational indices (an operator containing a vector current with left-handed quarks is also possible, but requires two Higgs insertions and is thus dimension-8). We focus on the operator Q VR . This is generated in a simplified model by a tree level exchange of the W ∼ (1, 1, +1) mediator, with the interaction Lagrangian, with g V an overall coupling constant, while c ij q , c i N coefficients encode the flavor dependence of W interactions. Restoring the flavor structure to Q VR , the b → c N R decay then arises from TeV , (2.4) choosing a phase convention in which V cb is real. We work in the mass basis, such that setting i = 2, j = 3, k = 3 in eq. (2.3) generates the operator c R γ µ b R τ R γ µ N R . The definition for Λ eff in (2.4) is chosen such that the rate for the B → D ( * ) τN R decay is normalized to the SM rate for the B → D ( * ) τν process at C 23,3 = 1. The B → D ( * ) τν decays become an incoherent sum of two contributions: from the SM decay, b → cτν τ , as well as from the new decay channel, b → cτN R . The NP contributions therefore necessarily increase both of the B → D ( * ) τν branching ratios above the SM expectation, in agreement with the direction of the experimental observations for R(D ( * ) ).

Fit to the
The addition of b → cτN R transitions to the SM b → cτν τ process does not change significantly the differential distributions (see figure 2 and discussion in section 2.3 below). Computation of the B → (D * → DY )(τ → νX)ν differential distributions and corresponding R(D ( * ) ) predictions are obtained from the expressions in ref. [49], making use of the form factor fit 'L w≥1 +SR' of ref. [8]. This fit was performed at next-to-leading order in the heavy quark expansion, utilizing the recently published unfolded Belle B → D ( * ) lν data [50] and state-of-the-art lattice calculations beyond zero recoil [51,52]. Fitting the R(D ( * ) ) predictions to the current experimental world averages [7], gives the ∆χ 2 as a function of C 23,3 shown in figure 1 (dof = 2). The best fit value is obtained for C 23,3 0.46 ± 0.06 (68% CL, ∆χ 2 = 2.3) with minimum ∆χ 2 1.0, to be compared with ∆χ 2 20 at the SM point, C 23,3 = 0. This best fit corresponds to TeV , (2.6) and in the W simplified model to the W mass of in which we normalized, for illustration, g V to the approximate value of the SM weak coupling constant, g 2 . The additional W current also incoherently modifies the B c → τν decay rate with respect to the SM contribution, such that with f Bc 0.43 GeV [53] and τ Bc 0.507 ps [54]. We apply an aggressive upper bound Br(B c → τν) < 5% [32,33]. In figure 1 we show the corresponding exclusion region for |C 23,3 | (orange shaded regions), which is far from the best fit region. 3), to the current world average [7]. Also shown (shaded orange) are exclusion regions for Br(B c → τ ν) 5%.

Differential distributions
Crucial to the reliability of the above fit results is the underlying assumption that the differential distributions, and hence experimental acceptances, of the B → D ( * ) τν decays are not significantly modified in the presence of the W current. Experimental extraction of R(D ( * ) ) relies on a simultaneous float of background and signal data, and can be significantly model dependent (cf., e.g., the measured values of R(D ( * ) ) for the SM versus Type II 2HDM in ref. [3]). In figure 2 we show normalized differential distributions for the detector observables q 2 , E , m 2 miss and cos θ D arising from the cascades B → (D * → Dπ)(τ → ν ν τ )ν and B → D(τ → νν)ν, for the SM versus SM+W theories, taking N R to be massless, and applying the phase space cuts, as an approximate simulation of the measurements performed in refs. [2,3]. These distributions are generated as in ref. [49], using a preliminary version of the Hammer library [55].
In each plot, we show the variation in shape over the range C 23,3 = 0.46 ± 0.2, corresponding to a range greater than the 99%CL, and for the SM (C 23,3 = 0). One sees that the variation in shape is small over this range. The variations in other observables, such as cos θ Dπ , are not shown, since they are even smaller. This gives us good confidence that the measured R(D ( * ) ) in eq. (2.5) well-approximate the values that would be measured for a SM+W model template, though a fully forward-folded analysis is required to verify this.

Explicit UV completion: the '3221' gauge model
A massive vector requires a UV completion. We consider a '3221'-type gauge theory, with a gauge group G = SU(3) c × SU(2) L × SU(2) V × U(1) . The U(1) together with the SU(2) V symmetry will generate heavy vectors under spontaneous symmetry breaking SU(2) V × U(1) → U(1) Y . Our notation for the gauge fields in the G-symmetric phase is G a µ , W i µ , W j µ , and B µ , respectively, with g s , g L , g V , and g the corresponding gauge couplings. The content of the model is shown in table 1: three generations of SM-like chiral field content, denoted by primes, is extended by a right-handed neutrino ν R . Also included are one or more generations of vector-like quarks and leptons, Q i L,R and L i L,R that transform as doublets under SU(2) V . We will consider the phenomenological implications for the cases where either one or two sets of vector-like fermions are introduced, i.e., i = 1 or 2. In both cases, SU(2) V is asymptotically free; with a third set of vector-like fermions, SU(2) V has a Landau pole in the far ultraviolet. In the remainder of this section, we give a detailed account of this '3221' UV completion, while the related phenomenology is discussed in section 4.

Gauge symmetry and the spontaneous symmetry breaking pattern
The gauge group G is spontaneously broken in two steps, first G → G SM ≡ SU(3) c ×SU(2) L × U(1) Y , and then G SM → U(1) em . The first step of spontaneous symmetry breaking, G → G SM , occurs when the scalar, H V obtains a nonzero vacuum expectation value (vev),

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This results in three Goldstone modes being eaten by the W ± and Z gauge bosons, giving where tan θ V = g /g V . In the following, we will use the notation The V is the diagonal generator of SU(2) V . The hypercharge gauge coupling is This relation fixes the mixing angle to t V = g Y /(g 2 V − g 2 Y ) 1/2 , so that g V needs to be larger than g Y 0.36. For large values of g V we have t V g Y /g V . The second step in spontaneous symmetry breaking is the usual electroweak symmetry breaking within the SM, due to the Higgs vev, H = (0, v EW / √ 2) T , with the SM Higgs spanning the H ∼ (1, 2, 1, 1/2) representation of G. For simplicity we assume here and in the remainder of the manuscript that the mixed quartic, As a consequence, we can neglect the mixing between the two real scalar excitations, the SM Higgs, h, and the heavy Higgs, h V , simplifying the discussion.
The NP contributions to R(D ( * ) ) scale as g 4 V /m 4 W ∼ 1/v 4 V . In order to explain R(D ( * ) ) the vev v V should not be too large, while direct searches require m W ∼ g V v V to be large. A phenomenologically viable solution is obtained for g V 1, which then requires g g Y ≈ 0.36 (for a detailed numerical analysis see section 4). In this case the θ V mixing angle is small, sin θ V 0.3, and thus Z and W masses are almost degenerate, m Z − m W 0.05m W in the minimal G → G SM breaking scenario. Because of stringent constraints from Z searches, an extra source of mass splitting between the Z and W might be required in some cases: a possibility that we partially explore in appendix B.

Matter content and new Yukawa interactions
In order to make the phenomenology of the model more tractable and the notation more streamlined, we make the simplifying assumption that only one generation of the vectorlike fermions, Q i L,R and L i L,R , have appreciable couplings to the SM quarks and leptons. We denote the corresponding fields by just Q L,R , L L,R . They decompose under the SM gauge group as under G SM . This is the minimal field content required to generate the b → cτ N R transitions.

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An additional pair of vector-like fermions,Q L,R ,L L,R , are assumed to have very small couplings to the SM fields and will thus be relevant only in the discussion of collider searches.
The mixing of the SM-like chiral fermions and the vector-like fermions, Q L,R , L L,R , occurs through the following Yukawa interactions in the Lagrangian, where the Yukawa interactions between the SM fields are, as usual, Without loss of generality we can take M Q,L to be real positive, and set, using the flavour group u,d,e meaning a diagonal 3 × 3 matrix with real positive entries. (The neutrino sector is discussed separately below, in section 3.4.) To simplify the discussion we take all the λ i f to be real. In this basis, the couplings λ 3 d ≡ λ b , λ 2 u ≡ λ c , λ 3 e ≡ λ τ , and λ 3 ν , need to be large in order to explain the R(D ( * ) ) anomaly.
Before H V and H obtain vevs, the vector-like fermions Q L,R , L L,R have masses M Q,L , while the SM fermions are massless. The H V vev induces the mixing between the heavy vector-like fermions and the right-handed SM fermions. After the electroweak symmetry is broken by the Higgs vev, the SM fermions become massive, inducing mixing with the left-handed SM fermions. We first investigate the mixing between the vector-like fermions and the SM fermions for the simplified case of a 2 × 2 system, taking as an illustration the limit of only the bottom quark, b R , coupling to the vectorlike fermion Q L,R .
For H V = 0, but still keeping H = 0, the mass eigenstates are the D R , b R fermions with (left-handed components are not mixed so that D L = D L ) where the mixing angle satisfies, The heavy quark D has mass while b R remains massless. After electroweak symmetry breaking due to the Higgs vev, v EW = 0, also the lefthanded fields, i.e., the down component of q L , and the D L mix. The corresponding lefthanded mixing angle is in which the mass of the light quark is while the mass of the heavy state, D, remains ≈ M D . The above analysis extends straightforwardly to the three generations of SM quarks. In the first step now a linear combination of SM quarks mixes with D R when H V obtains a vev, H V = 0. The expressions for the second step, the electroweak symmetry breaking, can be found in section IV of ref. [56], where a general phenomenology of mixings with a singlet down-like vector-like quark has been worked out. The left-handed mixing (3.13) results in a tree-level modification of the effective Z boson couplings, which were precisely measured at LEP. In the limit of large tan θ b R 1, i.e., in the limit λ b v V M Q , the left-handed mixing is given by sin θ b L ≈ m b /M Q , implying a lower limit M Q 100 GeV [56]. The ratio of the light Higgs yukawa coupling to the light quark mass eigenstate v EW y b /m b cos θ b L 1 in this limit. Repeating the same analysis for charm, we find a comparable, yet somewhat less stringent, bound on M Q . Similar bounds apply also on M L from Z → τ + τ − and lepton flavor universality measurements in τ decays.
As we will discuss later on, the explanation of R(D ( * ) ) anomaly requires sin θ b R , sin θ c R , and sin θ τ R to be O(1). The analysis above the implies that one can take as a realistic benchmark tan θ b R ,c R ,τ R ≈ 10, i.e., the case where right-handed bottom and charm quarks, as well as the right-handed tau are mostly composed from the corresponding vector-like states, so that sin

Gauge boson interactions
For later convenience we also give the couplings of W ± and Z to fermions, all of which come from the covariant derivatives in the kinetic terms of the fermions. In the interaction basis we have, In the absence of fermion mass mixing the W only couples to the vector-like fermions. The Z , however, also couples to the f fermions. A phenomenologically viable scenario requires t V 1 in order to suppress pp → Z production from the valence quarks (see figure 3 and discussion in section 4).
We are now ready to map the above results to the notation we used for the EFT analysis of R(D ( * ) ), in section 2, eq. (2.2). Rotating to the fermion mass basis, the relevant JHEP09(2018)169 W boson couplings are, up to small corrections due to EW symmetry breaking, given by The corrections to R(D ( * ) ) are maximised in the limit c 23

Neutrino masses
The neutrino mass matrix, for a simplified case of a single SM-like neutrino flavor, has the following form in the basis (ν L , ν c R , N L , N c R ), where we have included a Majorana mass term µ for ν R , which is a singlet under G. For v EW = 0, the SM neutrino ν L decouples from the system and remains massless. In the remaining system of three Weyl fermions, the µ = 0 limit produces a massless Majorana , while the other two Weyl fermions combine into a Dirac fermion with mass As with the charged fermions (discussed above), for λ ν v V M L the massless right-handed neutrino has a large admixture of N c R , which is charged under SU(2) V ; this large mixing is necessary to induce a large coupling of the massless state to W in order to explain the R(D ( * ) ) anomaly. Introducing a nonzero but small µ M L , λ ν v V results in the lightest right-handed neutrino N R obtaining a mass M N R ≈ µ (M L /M N ) 2 and a small admixture of N L . The heavy Dirac fermion becomes a pseudo-Dirac state, composed of two O(M N ) mass states split by O(µ).
The above features persist for y ν v EW = 0, i.e., when the SM ν L state is coupled to this system, in the phenomenologically interesting limit y ν v EW µ. This also leads to a Type-I seesaw step that generates light Majorana neutrino masses ≈ y 2 ν v 2 EW /(2µ), with the smallness of the neutrino masses arising due to the smallness of the Yukawa couplings, y ν 10 −7 for µ ∼ GeV. It is straightforward to extend the above discussion to three generations of neutrinos, thereby accounting for the observed neutrino oscillation phenomena. In addition to the tree level neutrino masses discussed here, a Dirac mass term analogous to y ν v EW is also generated at two loops. The size of this contribution depends on the flavor structure of the theory, which will be discussed in the next section. Hence we postpone a discussion of the two loop Dirac mass term, along with the discussion of the phenomenology of the additional neutrino states, until section 5.

Constraints
In this section we derive the phenomenological constraints on the '3221' model. In addition to the SM states, the minimal model contains a light right-handed neutrino, N R , and several heavy states: the vectorlike quarks, U and D, with charges 2/3 and −1/3, the charged lepton, E, and a heavy pseudo-Dirac neutrino, N H , a heavy Higgs scalar h V from the SU(2) V Higgs doublet, H V ; and the W and Z gauge bosons. We also extend this minimal set-up by including up to two additional copies of vector-like fermions, requiring that mixings of the additional vector-like quarks with the SM model fermions are negligible (but large enough that they decay promptly and do not lead to displaced vertices). In appendix B we then also discuss the implications of non-minimal SU(2) V breaking sectors.

LHC constraints
In general, we expect the most important LHC constraints to arise from the resonant production of W and Z gauge bosons, and from the pair production of the heavy vectorlike quarks, U and D. The cross sections for pp → W , Z production depend crucially on the assumed flavor structure of the couplings. The couplings of W to SM quarks are induced from mixing of the light right-handed fermions, d i R , u i R with Q L , which is a doublet of SU(2) V . The couplings of SM quarks to Z arise from d i R , u i R and Q L gauge quantum numbers under SU(2) V × U(1) . Similarly, e i R and ν i R mix with L L , see eq. (3.7). The resulting interaction Lagrangian is For couplings to right-handed charged leptons we take for the Yukawa in eq. (3.7) λ i e ∼ (0, 0, 1), (4.2) so that there are no FCNCs induced among SM leptons at tree level, and the mixing is only among E R and τ R . We take v V M L , so that the heavy mass eigenstate has a mass O(v V ). The light eigenstate has a mass m τ = y τ v EW / √ 2, where y τ = y τ / 1 + v 2 V /(2M 2 L ), is the SM τ Yukawa, with y τ the coupling in (3.9). The mixing angle between right-handed τ and E R is sin θ τ R ∼ O(1), while the mixing among the left-handed τ and E L is highly suppressed, sin θ τ L ∼ O(m τ /v V ). One can allow for O(1) factors in eq. (4.2) which we absorb in the definition of c i N and write in the numerical analysis Note that the couplings of Z and W that involve the SM neutrinos are small and can be ignored. If one were able to expand in v V /M Q , the couplings in eq. (4.1) would bẽ

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This illustrates how the hierarchy in λ i d,u translates to a hierarchical structure of the couplings of SM quarks to Z and W gauge bosons. In the numerical analysis we work in a different limit, v V M Q ∼ O(v EW ). This introduces a new dimensionless ratio M Q /v V , that needs to be taken into account. We show first the results for the minimal set of nonzero Yukawa couplings, λ i d , λ i u , eq. (3.7), in order to explain the R(D ( * ) ) anomaly. We then modify this minimal assumption and show the relevant constraints from FCNCs.
We first fix the flavor structure to a particular realization of the flavor-locking mechanism, see appendix A, giving us the "flavor-locked 23 model" (FL-23). The new states only couple to c R and b R in the mass eigenstate basis, so that λ i d ∼ (0, 0, 1), and λ i u ∼ (0, 1, 0). (4.5) As stated before, we are interested in the limit v V m Q . For concreteness, we take m Q /v V ∼ λ, the usual Wolfenstein CKM parameter, and m Q ∼ v EW . In this case one obtains for the couplings in eq. (4.1) (4.6) Note that in (4.6) theb / W c coupling is c 23 q ∼ 1, which is parametrically larger than the corresponding CKM matrix element in the SM, V cb ∼ λ 2 . Similarly, the Z couples most strongly to charm and bottom quarks, withc 22 . For the FL-23 flavor structure the most severe LHC constraints are due W productions, decaying through W → τ N R , and from the Z production, decaying through the Z → τ τ . Following the FL-23 setup we assume in the numerical analysis of the LHC constraints that: i) there are sizeable mixings of vector-like fermions with b R , c R , s R and ν R , i.e., that sin θ b R ,c R ,τ R ,N 1, and ii) the mixings with the other SM fermions are negligible, as in eq. (4.6). The LHC constraints from pp → Z → τ τ, and pp → W → τ N R also depend crucially on how many other channels besides the one containing τ leptons are open. If only the decay channels to SM quarks are open, the W branching ratios in the FL-23 model are Br(W → τ N R ) : Br(W → cb) 1 : 3. In this case the LHC bounds from pp → W → τ +MET are already strong enough, that the model is pushed close to the perturbative limit. The situation somewhat changes if vector-like fermions are light enough that Z and W can decay into them.
In figure 3 figure 3 for τ + τ − (brown) and + − (gray), respectively. The parameter space consistent with the LHC data has g V g , or t V 1. This is required to suppress Z couplings to valence quarks and light charged leptons. In this regime, the dominant decay modes are to bb, cc, τ + τ − and N R N R , and the main production mechanism is from the charm fusion. Comparing instead the σ(pp → W ) × B(W → τ ν) to the upper limits from the ATLAS analysis [59] (see also [60]), leads to constraints shown with light blue. Introducing another vector-like fermion family helps reduce these constraints as shown in the right plot. Here we set the masses of vector-like fermions to 0.8 TeV, which is above the limits from the quark partner pair production [61]. We checked that in the interesting region of parameter space the W , Z induced production is subleading compared to the QCD pair production. We have also checked that V → jj searches are less sensitive compared to the dilepton. Nevertheless, should the signal show-up in a future dilepton search, both dijet and vector-like fermion signatures would provide a useful cross-check.
The dilepton exclusion limits in figure 3 assume the validity of the narrow width approximation. In fact, this approximation is expected to fail in the interesting region of parameter space for the future searches, Γ/M 20%, and this requires a dedicated analysis strategy (see e.g. figure 4 of [35]). The main point of the present analysis is to illustrate a parameter space consistent even with these (aggressive) exclusion limits.

Flavor constraints
We next turn our attention to the flavor constraints. In FL-23 model all the tree-level FCNCs are strongly suppressed, and are phenomenologically negligible. The one-loop induced FCNCs are also negligible, suppressed by both m W m W and the extreme smallness of the flavor-changing couplings c ij q , for ij = 23. Other flavor models, beside flavor-locking, may lead to a flavor structure similar to eq. (4.6), but with vanishing entries modified to some nonzero value. In fact, when writing eq. (4.6) we assumed that the two SM Yukawa structures are aligned for the right-handed fields with the FL-23 spurions, i.e., that no right-handed rotations are needed to diagonalize them. If we assume instead that the SM flavor structure comes from a Froggatt-Nielsen (FN) flavor model with a single horizontal U(1) [62,63], while the couplings to vector-like fermions are due to FL-23, some of the vanishing entries become nonzero. The largest correction to the vanishing entries in this case is inc 23 u ∼ O(λ 4 ), with λ = 0.23 the CKM parameter, and is O(λ 8 ) or less in all the other cases, all of which can still be safely ignored.
The off-diagonal Z couplings induce tree-level FCNCs which are stringently constrained by the bounds on the B → K ( * ) νν branching ratios and by the measurements of B d,s −B d,s , D 0 −D 0 and K 0 −K 0 mixing amplitudes. The constraints from B c → τ ν were already discussed in eq. (2.8), and were shown to be satisfied in these types of models.
The branching ratio for B → K ( * )N R N R normalized to the SM value for B → K ( * )ν ν , is given by where G F = 1.1663787(6) × 10 −5 GeV −2 is the Fermi constant, |V tb | 1, |V ts | = 40.0(2.7) × 10 −3 are the CKM elements, s 2 W 0.231 is the square of the sine of the weak mixing angle, α = 1/137 the fine-structure constant, and X(x t ) 1.31 the loop function. The present experimental bound is R K ( * ) νν < 5.2 at 95% C.L. [54], which signifies that for m Z ∼ 2 TeV one requiresc 23 dc 3 N λ 2 . A suppression of this size is usually not a challenge for flavor models that have suppressed FCNCs.
The branching ratio for D s → τ N R normalized to the SM prediction for D s → τ ν τ is given by The correction is well below the present experimental precision on this branching ratio, Br(D s → τ ν) = 5.48(23) × 10 −2 , even for c 22 q c 3 N ∼ O(1).

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The most severe bounds arise from the absence of any deviations seen in the meson mixing measurements. The contributions to the meson mixing from tree-level Z exchanges can be parametrized by the effective Hamiltonian [64]. The bounds on the Wilson coefficientC ij 1 , are [65,66] 1/|C sd where the bounds are due to the allowed size of the real and imaginary parts of the NP Wilson in K , the bound on weak phase in D −D mixing and on the size of NP matrix elements in B d(s) −B d(s) mixing, in all cases factoring out the SM weak phase. The tree level Z exchange gives for the Wilson coefficients of the NP operators The meson mixing bounds translate to where the CKM parameter λ = 0.23. The required suppressions ofc ij q are highly nontrivial, and would, e.g., be violated in most realizations of, otherwise phenomenologically viable, FN models.
Finally, the corrections to electroweak observables from heavy vectorlike fermions and due to W − W mixing are well below present experimental sensitivity. Since v V v EW the vectorlike fermions are heavy O(1 TeV), see eq. (3.12). The corrections to T parameter from W −W mixing arise effectively at 2-loops and are further suppressed by the m W mass.

Neutrino phenomenology
In this section, we study the phenomenology associated with the sterile neutrinos that are part of our framework. To simplify the discussion we assume that, as for the charged states, only one pair of vector-like fermions mixes appreciably with the SM fermions through Yukawa interactions. In addition to the three generations of SM neutrinos the relevant fields are thus the vector-like fermion pair N L , N R and the three singlet right handed neutrinos ν i R . These give rise to the following mass eigenstates: (3.18). It couples appreciably to the W and is responsible for the R(D ( * ) ) signal. For simplicity we take i = 1, i.e., µ 1 = µ, and treat the mass M N R as a free parameter.
• ν 2,3 R ≈ ν 2,3 R are the remaining two singlets. We assume that they couple negligibly to N L , N R and are approximately degenerate, so that they have masses M ν 2,3 R ≈ µ 2,3 ≈ µ.
These states are therefore expected to be heavier than N R by a factor ∼ (M N /M L ) 2 . We will use these states to generate the observed neutrino masses via the type-I seesaw mechanism, giving m ν L ≈ y 2 ν v 2 EW /(2µ). 1 • The remaining two degrees of freedom make up a pseudo-Dirac state composed of two states with masses of O(M N ) split by O(µ). These are heavy and decay rapidly, hence do not directly influence the low energy neutrino phenomenology or cosmology, and thus we do not discuss them further.
In our setup a Dirac mass term is generated at two loops, and is sensitive to the flavor structure of the theory, see figure 4. This contribution has been estimated in, e.g., [67][68][69].
Ignoring O(1) pre-factors and integration functions, the Dirac mass in the FL-23 scenario is approximately given by This is much smaller than the active neutrino mass scale, and therefore does not modify the discussions of neutrino masses and mixings above.

Cosmology
The same W mediated interaction that gives the R(D * ) signal will also produce N R in the early Universe, e.g., through the processes bc → τ N R or τ τ → N R N R . These interactions thermalize the N R population with the SM bath at high temperatures. Once the temperature drops below the masses of the SM fermions involved in these interactions, the 1 This imposes requirements on the Yukawa couplings of ν 2,3 R and mixing angles with SM neutrinos. Since solar and atmospheric neutrino oscillation data only fix two mass differences, while the absolute mass scale for active neutrinos is only bounded from above, only two sterile neutrinos are required to participate in the seesaw relation. The remaining sterile neutrinos, including NR, can in principle be decoupled from the seesaw constraint.   N R abundance freezes out. Since we have assumed m N R O(100 MeV), N R freezes out at temperature above m N R , so that its abundance is not Boltzmann suppressed, and it survives as an additional neutrino species in the early Universe.
It then becomes crucial to determine the fate of this N R population. The N R can decay either through N R → νγ via a two loop radiative process induced by its W couplings, or via a small mixing with the SM neutrinos, see figure 5. Since the N R mixing angle with the SM neutrinos can be arbitrarily small, the radiative decay process is generally the dominant decay channel. The decay rate for this process is approximately [70][71][72] where we have again ignored some O(1) pre-factors and integration functions. It should be emphasized that this decay rate is completely fixed by the fit to R(D ( * ) ), as there are no other free parameters that enter the above decay rate. For comparison, the decay rate for the tree level process, figure 5 right, is

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The mixing angle is bounded from above, sin 2 θ m ν /m N R , in order to remain consistent with the seesaw mechanism, but is typically much smaller, rendering this mode subdominant.
The radiative decay channel N R → νγ, if dominant, corresponds to a lifetime of ∼ 10 25 (m N R /keV) −3 s. For m N R < O(100) keV, the N R sterile neutrino therefore has a lifetime greater than the age of the Universe and could in principle form a component of dark matter. Such a dark matter interpretation, however, faces several challenges.
Sterile neutrino dark matter is generally studied in frameworks where its abundance freezes in rather than freezes out [73][74][75][76][77]; however, this is not true in the current scenario. It is well known that without other additional modifications of the standard cosmology, a species that undergoes relativistic freezeout around T ∼ m τ overcloses the Universe if its mass is greater than O(keV). Its relic abundance can be made to match the observed dark matter abundance through appropriate entropy dilution. For instance, species that grow to dominate the energy density in the early Universe and decay late, after dark matter has frozen out, release significant entropy into the SM thermal bath and dilute the abundance of dark matter. Such long-lived particles are present in our framework in the form of ν 2,3 R . If their masses lie at the GeV scale, 2 they can thermalize, undergo relativistic freezeout, and decay just before BBN, diluting the abundance of dark matter by a factor of 30 [72,80,81]. Significantly larger dilution factors can be achieved with late decaying sterile neutrinos that are not part of the seesaw mechanism (see e.g. [72]), although these are not as well motivated in general. It should be noted that a large entropy dilution also helps to make the dark matter colder, making the light dark matter candidate more compatible with warm dark matter constraints.
Even with the correct relic abundance, dark matter in this mass range is severely constrained by γ-ray bounds from various observations [82], which rule out dark matter lifetimes of O(10 26−28 )s in the keV-MeV window. These observations therefore rule out N R , which has a lifetime ∼ 10 25 (m N R /keV) −3 s, as constituting all of dark matter. It could still constitute a small fraction (sub-percent level) of dark matter, in which case future γ-ray observations could discover a line signal from its decay.
If N R is light, with a mass below keV, it can act as dark radiation and contribute to the effective number of relativistic degrees of freedom, N eff , at BBN and/or CMB decoupling. This is potentially problematic since a light sterile neutrino that undergoes relativistic freezeout and is long-lived effectively acts as an additional neutrino species, contributing ∆N eff ∼ 1, which is inconsistent with current observations. In our setup, the sterile neutrino N R freezes out at T ∼ GeV. Its abundance, relative to the SM neutrinos, gets diluted by a factor 10.75/g * (GeV) due to the entropy injected into the thermal bath by the subsequent freeze-out and decay of SM degrees of freedom. Furthermore, its energy relative to the SM neutrinos also gets diluted by (10.75/g * (GeV)) 1/3 . The net result is that N R contributes ∆N eff ∼ 0.1, which is consistent with current observations and at the same 2 Such a low scale seesaw, with sterile neutrinos at the GeV scale, is the central concept of the neutrino Minimal Standard Model (νMSM) [78][79][80], and the associated cosmology, phenomenology, and constraints have been extensively discussed in the literature. time possibly within reach of future instruments such as CMB-S4 [83]. It should be noted that further entropy dilution, for instance from the decays of GeV scale ν 2,3 R as discussed above, could further suppress ∆N eff , pushing it beyond the reach of planned experiments.
Alternatively, when N R is heavy enough that its lifetime is shorter than the age of the Universe, N R → νγ as the dominant decay channel results in a late injection of photons into the Universe, which can distort the CMB or contribute to the diffuse photon background. This problem can be avoided by enhancing the N R mixing with active neutrinos, to the extent allowed by the seesaw mechanism, so that N R primarily decays via this mixing (into channels such as N R → 3ν, see figure 5 right). For m N R > MeV, this introduces dominant decays channels into electrons or pions, which can also distort the CMB or contribute to the diffuse photon background. For masses below an MeV, such decays into charged states are not kinematically open; however, in addition to N R → 3ν, which might be compatible with all existing constraints, the active-sterile mixing also gives rise to the decay N R → γν at one loop with a significant branching fraction. Such considerations indicate that N R lifetimes shorter than the age of the Universe are incompatible with current observational constraints.
In summary, cosmology excludes the the case of N R lifetime shorter than the age of the Universe, but allows for longer lifetimes. If N R is light (< keV), it can contribute ∆N eff ∼ 0.1 to dark radiation, which is consistent with current constraints and within reach of future experiments. For heavy N R ( keV), gamma ray constraints rule out the possibility of it making up all of dark matter; a small (sub-percent level) fraction of dark matter is allowed, but requires significant entropy dilution to obtain the desired relic abundance, which can be achieved, e.g., with late decays of GeV scale ν 2,3 R .

Direct production of additional sterile neutrinos
The above discussion suggests that the sterile neutrinos ν 2,3 R might be light, at the GeV scale, such that their late decays dilute the abundance of N R in order to evade various cosmological constraints. This gives rise to the fascinating possibility that ν 2,3 R can be directly produced. Since they decay with lifetimes 1 s, their decays can lead to observable direct signatures. Note that production of ν 2,3 R requires them to carry small admixtures of ν L , which couples them to electroweak gauge bosons, or of N L , N R , which couples them to W , Z gauge bosons, as ν 2,3 R are singlets under G. If they carry small admixtures of N L , N R , the ν 2,3 R states can be produced in place of N R in B decays if kinematically allowed. The branching ratio is suppressed by the mixing angle with N L , N R as well as by the phase space. Although these states still appear as missing energy, as for the b → cτ N R decay, the distribution of visible final states will be affected by the relatively heavy masses of ν 2,3 R . Finally, ν 2,3 R can also be produced from the decays of W and Z at the LHC. Their relative long lifetimes 1 s could then lead to displaced decay signals at the LHC as well as at proposed detectors such as SHiP [84], MATHUSLA [85], FASER [86] or CODEX-b [87]. In the present manuscript we discussed the possibility that the R(D ( * ) ) anomaly is due to an additional right-handed neutrino, giving rise to the b → cτ N R decay through an exchange of W that couples to right-handed currents. Since such a decay does not interfere with the SM b → cτ ν τ transition, it coherently adds to the B → D ( * ) τ ν branching ratios, in agreement with the observed experimental trend. Assuming N R to have mass below O(100) MeV, this additional channel leads to only negligibly small deviations in the kinematic distributions of the B → D ( * ) τ ν decays.
The right-handed nature of the W interaction allows construction of a UV complete renormalizable model, based on extending the SM gauge group to SU(3) c × SU(2) L × SU(2) V × U(1) . The flavor and collider searches constrain the model to have a definite flavor structure -achievable in a flavor-locked framework -and to also contain additional copies of vector-like fermions, to which the W and Z bosons can decay. In this way the model becomes very predictive. The additional vector-like fermions cannot be too heavy. For W and Z with the mass of about 3 TeV the model becomes non-perturbative, since the two resonances become very wide. A clear prediction is therefore that there should be vector-like fermions with a mass below about 1.5 TeV.
Another set of predictions is related to neutrino phenomenology. The sterile neutrino N R is light and long-lived, and has significant relic abundance. Hence it can contribute measurably to N eff at both BBN and CMB, while its decay N R → νγ could also constitute an observable signal for current and future experiments. Likewise, the model also contains heavier sterile neutrinos, potentially in the GeV mass range; which could lead to additional signals either in B decays or in searches for displaced decays at colliders.
Irrespective of what the future of R(D ( * ) ) anomaly will be, we encourage the experimental collaborations to explore possible distortions of the kinematical distributions in semileptonic B meson decays due to the heavy right-handed neutrino in the final statean option which goes beyond the short-distance new physics effects typically considered.
Note added. Similar results to ours are found in the contemporaneous analysis by ref. [88], also based on the 3221 model.

JHEP09(2018)169
A Flavor-locked couplings Although the flavor structure of the Yukawa couplings in eq. (4.5) can be treated as an ansatz, they may also arise dynamically in a flavor-locking (FL) context [89,90], hence the terminology used in the main text. In the general setup of the FL mechanism for the SM, one posits the existence of three up and down type flavons y α=u,c,t ∼ 3 ×3 × 1 and yα =d,s,b ∼ 3 × 1 ×3, with respect to the flavor symmetry U(3) Q ⊗ U(3) U × U(3) D . Each flavon carries typically also a unique U(1) α,α (or a discrete symmetry), which is broken by 'hierarchon' operators, that gain vevs. The vacuum of the flavon potential ensures that the up and down type flavon vevs y α,α are aligned, rank-1 and disjoint. That is, one has a 3-way portalQ in which dynamically y u = diag{r y , 0, 0}, y c = diag{0, r y , 0}, y t = diag{0, 0, r y }, and similarly for yα =d,s,b , and the fermion mass hierarchies are controlled by s α,α . The CKM is a flat direction of this potential, but may be lifted to a realistic flavor structure by the introduction of additional physics in the Higgs sector [90]. Here we assume that the dynamics of the y α,α flavons is fixed to SM structure at a relatively high scale, and explore the dynamical generation of associated flavor violating couplings involving the W and Z . The case of interest for the b → cτ ν anomaly is when there are two additional flavons, λ u ∼ 1 ×3 × 1 and λ d ∼ 1 × 1 ×3, which can then appear in the W and Z couplings to SM quarks, that is in the operators where Λ is the scale connected with the dynamics of λ u,d flavons. The renormalizable potential for λ u,d has the general form noting that λ † u λ u ∼ 3 ×3 of U(3) U and similarly for λ d . All the couplings, µ u,d , ν qa and ν α,α , are real and positive.
The µ u,d and ν qa terms enforce | λ u | = r u and | λ d | = r d . Defining the diagonal matrix D 1 = diag{1, 0, 0}, then in the quark mass basis the operator (A.2) can be rewritten in matrix form, without loss of generality,

B Symmetry breaking beyond the minimal model
It is possible to break the relation between W and Z masses in eq. (3.3) by introducing additional sources of SU(2) V × U(1) → U(1) Y breaking. As an example consider that in addition to H V another complex scalar, Φ, obtains a vev. We take Φ to be in a (2j + 1)-dimensional representation of SU(2) V and to carry a U(1) charge Y = j. A phenomenologically viable possibility is that only the component of the Φ multiplet that has zero hypercharge, the Φ −j , acquires the vacuum expectation value (we use the notation The extra contributions to the W and Z masses are then, respectively, For large enough j it is therefore possible to keep W relatively light, as dictated by the R(D ( * ) ) anomaly, and at the same time increase the Z mass above the experimental bounds. In section 4 we will see that a large enough splitting is obtained already for j = 1, i.e., for Φ that is an SU(2) V triplet. We parametrize the ratio of the two vevs, v j and v V , so that a is a continuous parameter that can take values a ∈ [0, ∞). With this parametrization