Vector Leptoquarks Beyond Tree Level III: Vector-like Fermions and Flavor-Changing Transitions

Extending previous work on this subject, we evaluate the impact of vector-like fermions at next-to-leading order accuracy in models with a massive vector leptoquark embedded in the $SU(4)\times SU(3)^\prime\times SU(2)_L\times U(1)_X$ gauge group. Vector-like fermions induce new sources of flavor symmetry breaking, resulting in tree-level flavor-changing couplings for the leptoquark not present in the minimal version of the model. These, in turn, lead to a series of non-vanishing flavor-changing neutral-current amplitudes at the loop level. We systematically analyze these effects in semileptonic, dipole and $\Delta F=2$ operators. The impact of these corrections in $b\to s\nu\nu$ and $b\to c\tau\nu$ observables are discussed in detail. In particular, we show that, in the parameter region providing a good fit to the $B$-physics anomalies, the model predicts a $10\%$ to $50\%$ enhancement of $\mathcal{B}(B\to K^{(*)}\nu\nu)$.

As pointed out in [26,27], in order to investigate the interplay between precision measurements and collider searches in this class of models, it is important to explore the relation between low-and high-energy observables beyond the tree level. In [26,27] we have presented a systematic analysis of the next-to-leading-order (NLO) corrections induced by the two largest gauge couplings, namely α 4 and α s . Such NLO effects lead to a sizable en- * fuentes@physik.uzh.ch † isidori@physik.uzh.ch ‡ matthias.koenig@uzh.ch § nudzeim@physik.uzh.ch hancement of the LQ contribution in low-energy semileptonic observables, at fixed on-shell coupling, that could reach up to 40% in specific amplitudes [26]. The analysis of [26,27], being focused on NLO effects related to the gauge sector, has been performed in a simplified version of 4321 models characterized by the minimal fermion and scalar field content. The purpose of this paper is to go beyond this limitation by analyzing the impact of one-loop corrections due to the exchange of massive vector-like fermions. The latter are a key ingredient for a successful description of the anomalies, and also a necessary ingredient to describe the subleading entries in the effective Yukawa couplings of the SM-like chiral fermions [16][17][18][19][20].
More precisely, the purpose of the paper is twofold. On the one hand, extending the model with the inclusion of vector-like fermions, we evaluate the modifications of the leading O(α 4 ) corrections to the matching conditions to the semileptonic operators computed in [26]. As an application of this result, we present a detailed discussion of the relative weight of vector and scalar contributions to the b → cτ ν decay amplitude. On the other hand, since vector-like fermions introduce a new source of flavor violation with respect to the minimal version of the model, we present a systematic analysis of all the flavorchanging neutral-current (FCNC) amplitudes generated beyond the tree level at O(α 4 ). The latter effects turn out to be particularly relevant for processes such as b → sνν or B-B mixing, which do not receive a tree-level contribution in this class of models. Combing the NLO amplitudes computed in this paper with those analyzed in [26], we present the first complete analysis of the U 1 impact in b → sνν decays, which is of great phenomenological interest.
The structure of the paper is as follows: In Section II, we introduce the minimal version of the model and the relevant interactions for the loop computations, and discuss in detail the effect of including vector-like fermions. In Section III we present our results of the loop-induced FCNCs. The phenomenological implications in b → sνν and b → cτ ν transitions are discussed in Section IV. The results are summarized in Section V. Appendices A and B provide further details on the vector-like fermion implementation and on the loop computations, respectively.

II. THE MODEL
A. Minimal field content The 4321 models are based on the SU (4) × SU (3) × SU (2) L × U (1) X gauge symmetry. We denote the corresponding gauge fields by H A µ , C a µ , W I µ and B µ , with indices A = 1, . . . , 15, a = 1, . . . , 8 and I = 1, 2, 3, and the gauge couplings by g 4 , g 3 , g 2 and g 1 . The SM gauge group corresponds to the 4321 subgroup SU (3) c × U (1) Y ≡ [SU (4) × SU (3) × U (1) X ] diag , with SU (2) L being the SM one. The hypercharge, Y , is defined in terms of the U (1) X charge, X, and the SU (4) generator T 15 4 = 1 2 √ 6 diag(1, 1, 1, −3) by Y = X + 2/3 T 15 4 . As in the SM case, it is useful to define the mixing angles θ 1,3 , relating the 4321 gauge couplings to the SM ones with g s and g Y denoting the SU (3) c and U (1) Y gauge couplings, and where we used a shorthand notation for the sine (s 1,3 ) and cosine (c 1,3 ) of the mixing angles. The SM gluon, G a µ , and hypercharge gauge boson, B µ , written in terms of 4321 gauge bosons and mixing angles, read The additional gauge bosons transform under the SM gauge group as U 1 ∼ (3, 1, 2/3), G ∼ (8, 1, 0) and Z ∼ (1, 1, 0). In terms of the 4321 gauge eigenstates, they are given by These gauge bosons become massive after the sponta- where ω 1,3 are the Ω 1,3 vacuum expectation values. In the limit ω 1 = ω 3 and g 3,1 = 0 the massive vectors are degenerate: this is the result of an unbroken global symmetry, that we denote as SU (4) V custodial symmetry. The latter is defined by the diagonal combination of the SU (4) × SU (4) groups, with SU (4) being the global group that contains SU (3) and (part of) U (1) X as subgroup. Electroweak symmetry breaking proceeds as in the SM through the vev of a SM-like Higgs, which could either be fundamental or composite [21].
The minimal matter content of the model and their 4321 representations are described in Table I. The Ω 1,3 scalar fields decompose under the SM subgroup as the corresponding gauge fields, S 1 and S 3 are SM singlet physical scalars, and tan β ≡ ω 1 /ω 3 . In the limit of heavy radial modes (m hi , m S1,3 m U,Z ,G ), we are left with a non-linear realization of the SU (4) × SU (3) × U (1) X → SU (3) c × U (1) Y symmetry breaking, as in the composite model in [21]. As we show in Section III, most NLO corrections can be evaluated also in the non-linear case . The prime in the left-handed fields indicate that these are not mass eigenstates (see II C).
with marginal ambiguities on the size of the effects.
We consider a version of the 4321 model were the would-be SM fields (in the absence of fermion mixing) are charged non-universally under the 4321 gauge group, see Table I. The fermion content charged under SU (4) consists of three fields transforming as Pati-Salam representations under SU (4) × SU (2) L × U (1) X : one SU (2) L doublet, ψ L , and two SU (2) L singlets, ψ ± R . In addition, we have two identical SM-like families, singlets under SU (4) and transforming as the SM fermions under SU (3) × SU (2) L × U (1). In the absence of fermion mixing (see II C), the SU (4)-charged fermions would correspond to the SM third generation (plus a right-handed neutrino), and the SU (4)-singlets to the light-generation SM fermions.

B. Relevant interactions
We describe only those interactions that are relevant for the loop computations below. The U 1 interactions with SM gauge bosons are given by where If we neglect terms of O(g 2 SM ), with g SM being any of the SM couplings, the triple gauge interactions of two U 1 with Z (G ) are the same as with B (G) with the replacement g Y → g 4 3/2 (g s → g 4 ). The relevant interactions of Goldstone and radials to gauge bosons read TABLE II. Additional fermion content. Here χL = (Q L L L ) and χR = (QR LR) . The prime in the χL components indicates that these are not mass eigenstates.
with T a being the SU (3) generators. In the absence of fermion mixing (see section below), and neglecting once more terms of O(g 2 SM ), the interactions between the massive vectors and fermions read where . Finally, the couplings of Goldstones and radials to fermions depend on the specific vector-like implementation (see section below) and are described in Appendix A.

C. Vector-like fermions
We now discuss the inclusion of massive fermions, vector-like under the SM gauge group, to the minimal model discussed in the previous section. In realistic 4321 models, these are introduced to induce couplings between the SU (4) vectors and the light SM families. For simplicity, here we focus on the mixing with a single SM-like family, and therefore introduce only one vector-like family. More precisely, we add to the minimal model one family of left-handed fermions, transforming in the fundamental representations of SU (4) and SU (2) L , and one family of right-handed partners. The massive fermions are vector-like under the SM gauge group, therefore, the right-handed partners should transform in the fundamental of SU (2) L , but there is freedom in the SU (4)×SU (4) transformations. As shown in Table II, we consider two possible implementations: They are either SU (4) singlets and transform in the fundamental of SU (4) (model I), as in [21]; or they are SU (4) singlets and transform in the fundamental of SU (4) (model II), as in [20].
Having two SU (4) charged fields with the same Lorentz and gauge transformation properties, ψ L and χ L , leads to a new flavor symmetry that we denote U (2) ξ , where This symmetry is broken by the fermion masses, giving rise to a possible mixing among ψ L and χ L , and possibly also the SU (4)-singlet fermions, q L and L , after the breaking of the SU (4) symmetry. The SU (4) breaking in the fermion masses could either be due to the vevs of Ω 1,3 or via new sources. We discuss the details for each implementation in Appendix A.
In either case, the mass terms after SU (4) breaking read with the left-handed fermions arranged into the the flavor vectors and where M q, are 3-dimensional mass vectors. Without loss of generality, these mass vectors can be written as where m Q,L are the vector-like fermion masses. Here, the 3 × 3 orthogonal matrices O q, parametrize the mixing among different SU (4) representations, and take the explicit form with s Q,L (c Q,L ) being the sine (cosine) of the θ Q,L mixing angles.
On the other hand, the 3 × 3 unitary matricesW q, parametrize the mixing among SU (4) states, and can be decomposed asW with W q, being unitary 2 × 2 matrices. To better understand the origin of the flavor mixing matrices, it is convenient to decompose the mass vector in (10) into two components where M 1 q, is real and M 4 q, is a 2-vector. Their combined presence encodes two different flavor symmetry breakings: i) The U (2) ξ alignment of M 4 q and M 4 is at the origin of the W q, matrices. Indeed, we have with M 3,1 being real parameters with mass dimension, and W q, as before. The SU (4) breaking from M 4 q, is analogous to the SU (2) L breaking in the SM from the up-and down-type fermion masses. As we show below, only the misalignment of quarks and leptons in U (2) ξ space, encoded in W = W † q W , is physical.
ii) The ratio between M 1 q, and M 1,3 determines the breaking of the U (2) q, flavor symmetry of the light fermions. Such breaking appears in the form of the O q, mixing matrices, with tan θ Q,L = M 1 q, /M 3,1 . In the mass basis, the SU (4) vector interactions with left-handed fermions (in the limit g 3,1 = 0) take the form with P 23 ≡ diag(0, 1, 1) projecting into the SU (4) components of Ψ q, L . SinceW q, and P 23 commute, the in-dividualW q, matrices are not observable, but only the combinationW It is convenient to rewrite these interactions in an SU (4) basis, or in the quark (Q i L ) and lepton (L i L ) components of ξ i L , that in the mass-eigenstate basis are given by In this basis, the interactions in (17) take the simple form (i = 1, 2) The unitary matrix W can be regarded as a generalization of the CKM matrix to SU (4) or quark-lepton space.
Similarly to the CKM case, the W matrix is the only source of flavor-changing transitions among SU (4) states, and it appears only in interactions involving both quarks and leptons. In this sense, the vector LQ, U µ , is analogous to the SM W µ . Similarly, the Z µ , G µ are analogous to the SM Z µ and their interactions are SU (4) flavor-conserving at tree-level. In analogy to the SM, we will denote U µ transitions as charged current and Z µ , G µ transitions as neutral currents. As in the SM, flavorchanging neutral currents (FCNCs) proportional to the W matrix are generated at the loop level. We compute these contributions in Section III.
Finally, note that the structure in (10) holds in the limit of unbroken SU (2) L symmetry, with a single family of SU (4)-singlet fermions (corresponding to the SM with 2 generations). Its generalization to a 3 generation case, and the inclusion of SU (2) L -breaking effects from the SM Yukawa couplings is straightforward, as long as we neglect light-quark mass effects. Note in particular that the 2-3 mixing form the SM Yukawa couplings (corresponding to a 1-2 mixing in the Ψ q, L space) can effectively be encoded via the replacement O q, → O q, L q, , where L q, are rotation matrices in 1-2 space resulting from the diagonalization of the SM Yukawa couplings.

A. Generalities
Before presenting the results, it is illustrative to show explicitly the unitarity cancellations taking place in the FCNC loops, analogous to the so-called Glashow-Iliopoulos-Maiani (GIM) mechanism in the SM. For instance, for the fermion self-energy graph shown in diagram (i) in Figure 3 we have where f ψ 2 is the loop function, we took i L massless, and P 3 = diag(0, 0, 1). In the second line, we used the property P 23W =W P 23 ,W (O ) unitarity (orthogonality), and the following relation Similar unitarity cancellations also take place in vertices and boxes. It is worth stressing some features that are common to all the FCNC loops presented here: i. Since we are dealing with SU (4) interactions only, the external states can always be written in the SU (4) basis defined in (19).
ii. Similarly to the SM, the SU (4) flavor-changing contribution is proportional toW P 3W † = W i2 W * j2 . The effect of O is seen in the factor c 2 L , which gives the projection of the massive component in the SU (4) state.
iii. The FCNC part of the amplitude is proportional to the flavor-and SU (4) V -breaking component of the vector-like mass in (10): in the limit of small breaking c 2 L W i2 W * j2 ≈ W i =j and, by means of Eq. (A16), we can interpret the flavor-violating amplitude as the result of inserting the symmetry-breaking mass term on the vector-like fermion propagator.
iv. While our computations present many similarities with those in the SM, one should not be tempted to simply rescale the SM contributions. Indeed, the presence of both W and O q, mixing matrices, instead of just the CKM matrix, yield loop functions that are different from their SM analog. In the limit of small breaking, this can be understood from the fact that symmetry breaking terms and fermion masses (controlling the loop functions) can be varied independently in our case, while they are in one-to-one correspondence in the SM.
In the next subsection, we present the result of the effective flavor-changing vertices of the Z and G massive vectors to fermions, using the SU (4) basis in (19). These (gauge dependent) vertices, which are evaluated in the Feynman gauge, are then combined with the box amplitudes in order to obtain the (gauge independent) contributions to the Wilson coefficients (WC) of the semileptonic FCNC operators (written in the SMEFT basis). In subsection III D, we present the results of dipole-type effective operators, and in subsection III E of the ∆F = 2 hadronic operators.

B. Z and G flavor-changing vertices
We define the following effective vertices that encode the FCNCs where where The gauge and Goldstone contributions to the vertex func- , and M = I, II denotes the different vector-like models in Table II. As expected, the unitarity cancellation discussed in the previous section ensures that the flavor-changing vertices vanish in the limit x i → 0. The loop functions F V are given in Appendix B 1 c. For reference, we give the value of F V in the limit x V = x i = 1 and neglecting terms of O(θ 2 i ): The functions multiplying the divergent piece read In a renormalizable model, we expect d V = 0, since the FCNC vertices are not present at tree level. Indeed, this is the case also in our models, but only after the introduction of the radial contributions. These depend on the different implementations of the scalar sector, and thus should be discussed for each model separately.

Model I
The scalar content of this model is the same as the one described in Section II. The only radial mode that can mediate flavor-changing transitions proportional to the W matrix is the LQ radial h U (see (5)). Similarly to what we did with the gauge and Goldstone contributions, we decompose the contribution from the scalar LQ as As expected, the LQ radial contribution cancel exactly the divergence from the Goldstone sector. The corresponding expres- loop functions are given in Appendix B 1 c. In the limit of heavy radials, i.e.x i → 0 and x R → ∞, these reduce to Therefore, the net effect of the LQ radials in the heavy radial limit is to replace the divergence in (25) by namely by a logarithm of the mass ratio plus an O(1) constant that is the same for all three effective vertices.

Model II
Apart from the Ω 1,3 fields introduced in Section II, this model requires an additional scalar with non-zero vev, Ω 15 , to generate a non-trivial W matrix (see Appendix A for details). This field transforms in the adjoint of SU (4) and therefore it contains a scalar LQ. In the limit ω 1,3 = 0 (or equivalently x Z ,G = 0), this LQ is identified with the would-be Goldstone boson and the gauge and Goldstone contributions become finite (see (27)). However, in the general case where ω 1,3 = 0, as needed to have θ L,Q = 0 (see Appendix A), the Goldstone contribution is divergent and all LQ radials have to be considered. Since the scalar sector is more involved in this case, we do not compute the LQ radial contributions here. However, we note that, as in the case above, their effect in the heavy and degenerate radial mass limit is to replace the divergence in (25) by where, similarly to the case above, x R = m 2 R /m 2 U , with m R being the mass of the LQ radials, and f R is a (universal) constant, expected to be of O(1).

C. ∆F = 1 semileptonic operators
We define the following effective Lagrangian for the semileptonic operators involving ∆F = 1 flavor-changing transitions that are absent at tree level with G U = √ 2 g 2 4 /8m 2 U and the effective operators Neglecting contributions of O(g 2 SM /g 2 4 ), with g SM being the SM couplings, the corresponding Wilson coefficients at the matching scale are given by (M = I, II) where we omitted the arguments in the loop functions to simplify the notation. The expressions for the B ijkl f g loop functions, whose flavor indices refer to the SU (4) basis, are given in Appendix B 2. If we neglect terms of O(|W 12 | 2 ), we find the following expression for the box functions We also provide the corresponding amplitudes for the hadronic and leptonic boxes in Appendix B 2. The hadronic and leptonic ∆F = 1 EFT contributions can thus be obtained with trivial replacements in the expressions given here.

D. ∆F = 1 dipole operators
For dipole-type operators, we define the following effective Lagrangians with the dipole operators defined as whereH = iσ 2 H * , the Higgs vev is normalized such that We compute the Wilson coefficients to first order in the (third-generation) SM Yukawa couplings and, consistently with the semileptonic amplitudes discussed above, we neglect flavor-violating effects from the CKM matrix. In this limit, the Wilson coefficients for the ∆Q = 1 down-type operators can be written in the following general form where The loop functions vanish for x → 0 and approach G 1 (1) → −11/24 and G 2 (1) → −5/24 in the x → 1 limit. The separate contributions from each loop diagram are reported in Appendix B 3. The expression of the (phenomenologically less interesting) coefficient C u A is obtained from C d A replacing y b with y t , and y τ with the third generation neutrino Yukawa coupling y ν . In the lepton case we find where Q B q L = 1 6 , Q W q L = 1, and N c = 3 is the number of colors of the particles in the loop. Due to the colorless nature of the leptons, there is no gluon-dipole operator.
For completeness, we note that the coefficients of the photon-dipole operators which are particularly interesting from the phenomenological point of view, can be obtained by the coefficients above as C W /2 or, equivalently, by using (39) and (42), with Q A f being the electric charges of the corresponding states.

E. ∆F = 2 hadronic operators
We write the effective Lagrangian for ∆Q = 2 transitions as The loop function is such that F ∆F =2 (1) = 5/48 and, for small x, The expressions for the individual contributions from each diagram are reported in Appendix B 2. Note that the loop function in (45) does not agree with the expression in [16,20], which was obtained by rescaling the W box contribution. As we already mentioned in Section III A, the different fermion mixing structure of the model compared to the SM does not allow for a naive rescaling of the SM amplitudes. Adopting the same normalization, the loop function in [16,20] has the same x → 0 behavior as F ∆F =2 (x) in (46), but a steeper raise for larger x values, reaching 3/16 for x = 1. As a result, we deduce that the bound on x L derived in [16,20] from B s mixing is slightly overestimated. We finally note that an analogous expression for ∆L = 2 transitions is found by replacing (W * 12 Before electroweak (EW) symmetry breaking, the part of the effective Lagrangian relevant for b → sνν decays is where β 23 = W * 12 W 22 s Q c 2 L . At the matching scale, C 3333 q can be decomposed as 2 with δC NLO q (1) ≈ 8 [26]. We have checked explicitly using DsixTools (based on the Renormalization Group Evolution (RGE) equations in [28][29][30]) that RGE effects in C 3333 q (µ = m Z ), including the mixing with the flavorconserving leptoquark mediated operator, are below 20%.
After EW symmetry breaking, we can project the contributions of O 3333 q and O 3323 q onto the coefficients of the the operators (s L γ µ b L )(ν γ µ ν ), that we normalize as in the SM, where α w = α/s 2 w = g 2 2 /(4π) and V ij are CKM matrix elements. Neglecting the tiny contributions suppressed by light quark masses, the lepton-universal SM contribution read where X t = 1.48 ± 0.01 [31]. Taking into account the contribution of (47) after diagonalizing the Yukawa couplings, we get C ≈ C SM for = e, µ and where Here, s b is the 2-3 mixing from the left-handed diagonalization of Y d , defined as in [20,32], and we have neglected terms of O(s 2 b ). Employing the notation of [20], we further identified C U = G U /G F and β 23 = β * bτ β sτ +O(s 2 L,d ), 2 In the notation of [26], C 3333 (53) With this notation, we can write Of the two contributions to ∆C τ in (52), the one proportional to s b can induce at most a ±3% correction to B(B → K ( * ) νν): the value of s b is indeed severely constrained by the tree-level Z and G contributions to B s mixing, which imply |s b | < ∼ 0.1 × |V ts | [15][16][17]20]. The contribution proportional to β 23 ≈ β sτ β * bτ can be larger, yielding up to O(60%) corrections to the B(B → K ( * ) νν) SM value. Moreover, the sign of the correction is unambiguously connected to the sign of the new physics contributions to R D ( * ) . More precisely, an enhancement of the R D ( * ) ratios requires a positive β 23 that, in turn, implies an enhancement also in B(B → K ( * ) νν).
In Fig. 1 we plot as a function of x L setting β 23 = 3|V ts |, C U = 0.01 and g 4 = 3, which are natural benchmark values to fit R D ( * ) while avoiding direct searches [32]. Changing β 23 and C U leads to uniform linear re-scaling of the plot The value of ∆ B→K ( * ) νν at x L = 0 corresponds to the contribution of C RGE U in (52), whereas the growth with x L is due to C 3323 q . As a result, a change of g 4 would rescale only the latter contribution. It is worth noting that the Belle II Collaboration should be able to measure B(B → K ( * ) νν) with a 10% error, assuming the SM value [34], and thus should be able to probe most of the parameter space of the model relevant to fit the B-physics anomalies.

B. b → cτ ν transitions
In this section we evaluate the modifications of b → cτ ν decay amplitudes, and their impact in R D and R D * , with respect to the NLO effects estimated in [26,27] in the limit of minimal field content. In the rest of the section we refer to these previous works as Ref. I [26] and II [27].
Before EW symmetry breaking, the effective Lagrangian relevant to charged-current transitions can be decomposed as where we have left the flavor indices implicit, and the operators are defined as Restricting the attention to b → cτ ν decays, quark flavor indices assume the values 3 and 2, whereas the lepton fla-vor indices are always third generation (in close analogy to the b → sνν case discussed above where φ LR parametrizes the arbitrary relative phase between left-and right-handed currents, related to the embedding of SM quark and leptons in SU (4) multiplets [17]. The first term in all the expressions above corresponds to the tree-level contribution that, compared to Ref. I and II, is modulated by a combination of W entries also in the flavor-conserving case. The NLO corrections can be further decomposed into a factorizable contribution due to the renormalization of g 4 (under both α 4 and α s corrections) and non-factorizable finite contributions due to box amplitudes and nonuniversal vertex corrections. In order to follow the approach adopted in Ref. I and II as closely as possible, we renormalize g 4 from the on-shell inclusive decay width of the LQ into a τ lepton and any quark species, that we denote as Γ U τ . In the absence of high-energy observables sensitive to W ij , we treat W 21 (and correspondingly W 11 ) as an effective low-energy parameter that we do not need to normalize.
By construction, the α s corrections are flavor blind and can be directly extracted from the result in Ref. II. Summing factorizable and non-factorizable contributions, and assuming the custodial limit for the vector masses, yields Here the subscript I refers to the flavor-blind result obtained in Ref. I that, in the custodial limit for the vector masses, yields where the subscript F (NF) denotes the factorizable (nonfactorizable) contributions. The factorizable contribution due to vector-like quarks, δC (4) | VL F , corresponds to the two-point function corrections that can be found in Appendix C 1. This is the only effect due to these additional degrees of freedom that does not vanish in the x Q,L → 0 limit. We find that this contribution yields an O(5% − 10%) reduction of the WCs, for fixed onshell coupling g 4 = 3. As far as non-factorizable corrections are concerned, δC (4),ij LL(R) | VL NF , we neglect the contributions generated from the vertex, consistently with what we did with the g 4 renormalization, and consider only the box contributions. Given the results in Ref. I, the vertex contributions are expected to be numerically subleading compared to the box amplitudes. The complete expressions for the box amplitudes can be found in Appendix C 2. In the custodial limit for the vector masses, we have δC (4),33 The ratio between LL and LR effective operators is of phenomenological relevance, since it affects the relative weight of scalar and vector contributions to R D and R D * . At the tree level, this ratio is completely determined by W 11 and the phase φ LR . At NLO accuracy it gets modified by the non-factorizable corrections and becomes flavor dependent. To parametrize this effect, we define the WC ratios at the matching scale At fixed g 4 = 3, we have ρ 33 LR ≈ 1.29 e iφ LR /W 11 and ρ 32 LR /ρ 33 LR ∈ [1.0, 0.92] for c Q = 1 and x Q ∈ [0, 1]. We have now collected all the ingredients to provide a description of the LQ contributions to the R D ( * ) ratios at NLO accuracy. Expressing the quark fields in terms of mass eigenstates (after EW symmetry breaking), and evolving the effective operators down to µ = m b , we obtain where X = D, D * . Here η S is the factor encoding the RGE evolution of O U LR , which for m U = 4 TeV assumes the value η S ≈ 1.8 [33]. The coefficients c X encode the ratios of the hadronic matrix elements of scalar and vector operators in the two modes. According to [35,36], they are given by c D ≈ 1.5 and c D * ≈ 0.14.

V. CONCLUSIONS
In this paper we have presented a systematic analysis of the impact of vector-like fermions, beyond the tree level, in models based on the (flavor non-universal) The inclusion of such heavy fields in this class of models is necessary for a successful phenomenological description of the SM spectrum at low-energies, in particular to describe masses and mixing angles for the light generations [17,18,20]. Vector-like fermions are also a key ingredient to enhance the 3 − 2 flavor mixing in the effective coupling of the TeV-scale LQ field to SM fermions, providing a better fit to the charged-current B anomalies [16]. We have considered two possible embeddings of the vector-like fermions into the model, both satisfying these phenomenological requirements. Interestingly, most of the conclusions we have derived are, to a large extent, independent of the specific embedding.
The new sources of flavor symmetry breaking due to the additional mass terms associated to the vector-like fermions lead to non-vanishing FCNC amplitudes that are not present in the minimal version of the model. We have elucidated the origin of this phenomenon in general terms, and we have systematically analyzed the matching conditions for FCNC semileptonic, dipole and ∆F = 2 operators. Using these results, combined with previous NLO results in [26], we present the first complete analysis of the impact of the U 1 leptoquark in B → K ( * ) νν decays beyond the tree level. As shown in Fig. 1, the branching ratios of these rare modes are unambiguously predicted to be enhanced by 10% to 50% in the parameter region of the model providing a good fit to the B-physics anomalies.
The inclusion of vector-like fermions leads also to sizable NLO effects in amplitudes which are non-vanishing already at the tree-level, such as charged-current semileptonic transitions. Extending our previous works [26,27], we have analyzed these additional NLO effects. Using these results, we have derived phenomenological expressions of the R D ( * ) ratios, in terms of the model parameters, which include all the relevant corrections at O(α 4 ) and O(α s ). These results will allow us to perform precise compatibility tests of the B-physics anomalies, if confirmed as clear signals of physics beyond the SM, with the predictions of 4321 models. As discussed in II C, there are several possible implementations for the massive fermions. In this appendix, we discuss in more detail the two realizations corresponding to model I and II in Table II. We also complete the discussion in II C by including Goldstone boson and (or) physical scalar interactions with fermions for each implementation.

Model I
This model consists of a simplified version of the composite model in [21], where a single vector-like family is included. In this implementation, the vector-like mass and fermion mixing terms are given by (i = 2, 3) where Ψ i L = (Ψ q i L Ψ i L ) with Ψ q, L defined as in (11), and with Ω 1,3 as in (5). In the composite model in [21], one has ω 1 = ω 3 , so the vevs of Ω 1,3 preserve the custodial SU (4) V symmetry. Moreover, only the Goldstone and vev part of Ω 1,3 is the same as in (5), while the physical scalars, together with other composite resonances, are expected to have masses around the compositeness scale, Λ ≈ 4πω 1,3 , much larger than the heavy gauge boson masses. However, to illustrate the effect of the radial modes in the computation of the FCNCs, we leave this model general by treating ω 1 and ω 3 as independent parameters and we keep the leptoquark radial in Ω 1,3 .
After Ω 1,3 acquires a vev, it is straightforward to write the fermion mass terms in the form of (10) (see also (15)), with Moving to the SU (4) basis defined in (19), the Lagrangian in (A1) can be rewritten as and where, in the radial interactions, we included only the leptoquark interactions.

Model II
This model consists of a simplified version of the one in [20], with only one vector-like family. Since in this implementation χ R is an SU (4) multiplet, a new source of SU (4) breaking beyond the Ω 1,3 vevs is needed to generate a mixing between SU (4) flavor states. This can be obtained from the vev of a new scalar field, Ω 15 , transforming in the adjoint of SU (4) and singlet under the rest of the 4321 group. Once this new field is introduced, the vector-like mass and fermion mixing terms for this model read (i = 2, 3) defined as in (11). Since the scalar sector of this model is more complicated, we do not discuss the radial modes here. The Goldstone and vev part of Ω 1,3 retain the same form as in (5) (with m U as in (A8)), while the Goldstone and vev part of Ω 15 decompose under the SM group as where the dots represent radial excitations that we do not consider. The presence of a vev for Ω 15 introduces an explicit breaking of the custodial SU (4) symmetry in the gauge boson masses. Indeed, this vev does not affect the Z and G masses, but it does change the U 1 mass compared to the one given in (4). More precisely, we now have Once more, it is possible to write the fermion mass terms in the form of (10) after Ω 1,3,15 acquire a vev. Namely, where M 15 ≡ λ 15 ω 15 /(2 √ 6). Note that, in the limit ω 15 = 0, the mass vectors are aligned and the W = W † q W matrix becomes the identity. Using the same decomposition as in (A3), we now find for the Goldstone boson interactions with q L = c Q q 2 L +s Q Q L , and analogously for L (see (13) for the definition of the mixing angles). Note that the first term in the φ U interactions coincides with the one in the previous model, c.f. (A4). The second term is new and is related to the fact that χ R is now charged under SU (4). Also note that, contrary to the previous case, there are no Goldstone couplings to φ Z ,G in the limit s q, → 0, making manifest the custodial symmetry breaking. The interactions involving the SM fields are however the same in both models. 3. SU (4)V structure of W and Oq, A complementary (model-independent) view about the mixing matrices W and O q, is obtained by looking at their transformation properties under the SU (4) V custodial symmetry. To do so, we rewrite (10) using a SU (4)invariant notation, with ξ L defined in (9). Since the two M a mix different SU (4) representations with the same right-handed field, one of them necessarily break the SU (4) gauge symmetry (M 4 in model I, and M 1 in model II). We can further decompose the M a in the SU (4) V space as where T Q−L = 3 2 (T B−L + 1 3 ), such that the M a q, defined in (15) are and let us consider the limiting case where all the other contributions to M L are small relative to M χ . Then from the perturbative diagonalization of M L M † L we obtain Form this we deduce that The flavor-changing G and Z one-loop vertices are given in Figure 2, with the internal curvy (dashed) line denoting the U 1 LQ (Goldstone). Using the same normalization as in (24), the contribution from each diagram at s = 0 and in the Feynman gauge reads where i = L (Q) for the quark (lepton) vertices, x Q,L = m 2 Q,L /m 2 U , N L = 3 N Q = 3, and with ∆ U = 1 − γ E + ln 4π + ln µ 2 m 2 U . Note that we have applied the unitarity relations discussed in III A in diagrams (a.i), (b), (c) and (d). Due to this unitarity cancellations, the contributions from these diagrams are finite. Diagrams (f ) and (g) require a fermion mass insertion and are also finite. On the other hand, diagrams (a.ii), (e) and (h) are divergent.
The couplings are given by The right-handed fermion couplings are different in model I and II. For model I, these couplings are zero, while in model II g L Z q ,Z = g R Z q ,Z .

b. Contributions from radial modes
We discuss here the contributions from the radial modes, which we compute only for model I. The diagrams to be computed are the same as in Figure 2 replacing the Goldstone by a radial leptoquark, except for (h) which has two contributions: one with two radials, and one with a radial and a Goldstone. We find , and the loop functions are defined as and the radial couplings are given by In the limit of very heavy radial mass compared to gauge boson and vector-like fermion masses, wherex i → 0 and x R → ∞, the loop functions above reduce to Here we compile the results from the previous sections, using the same notation as in Section III B. For the gauge and Goldstone contributions, the regular functions in model I are while the regular functions in model II read with x Q,L = m 2 Q,L /m 2 U and c Q,L as in (13), and the function F 1 defined as The regular functions for the radial contributions, which we computed only for model I, are given by , , where we used the mass relations x Z = 1 2 + sin 2 β and x G = 2 cos 2 β, and the function F 2 defined as .

Box diagrams
Here, we provide further details on the calculation of the box amplitudes. We present the results in the SU (4) basis (see (19)), but we focus on the cases where only SM particles are present in the external states. There are four possible topologies contributing to these amplitudes. These are shown in Fig. 4, with the curvy line denoting the U 1 exchange, and the dashed line denoting a φ U Goldstone exchange. Note that mixed diagrams with Goldstone and gauge leptoquarks necessarily contain a vector-like fermion as external state, which we do not consider here. Furthermore, we do not consider radial box contributions. These can be easily obtained from the Goldstone-box contribution by appropriately replacing the couplings, and are power-suppressed in the limit of heavy radial masses.

a. Semileptonic amplitudes
The amplitudes for the semileptonic box contribution with two left-handed currents read 3 Only the non-planar diagrams contribute to this amplitude, i.e. B q = B c q + B d q . The contributions of each diagram read In the Feynman gauge, the loop functions are given by which, as expected, are finite in this gauge. On the other hand, the box amplitudes for the case with one left-handed and one right-handed current are The amplitude to the hadronic box contribution with two left-handed currents reads in which only planar diagrams contribute, i.e. B qq = B a qq + B b qq . The contribution of each diagram is given by with the same loop functions as in the semileptonic case, c.f. (B15).

c. Leptonic amplitude
The corresponding box amplitude with two left-handed currents is given by As in the hadronic case, only planar diagrams contribute. Each of them yields the following contribution [B a ] αβγδ = 3 2 δ αβ δ γδ B(0, 0) with the loop functions in (B15).

SM dipole diagrams
We provide here further details on the computation of the SM dipoles. The diagrams to be computed are (c), (d), (e) and (h) of Figure 2 with the external gauge boson being a SM gauge boson, and with appropriate Higgs insertions in the fermion lines. The only cases where all diagrams are not vanishing are the B µν -dipoles, since both internal fermions and the LQ have non-vanishing U (1) Y charges. Considering the case of the O d B operator as representative example and normalizing the Wilson coefficient as in (36), we get where Y ,e,U denote the corresponding hypercharges, and D L,R k are the contributions from each of the diagrams in Figure 2 evaluated in the Feynman gauge. In the limit of vanishing external momenta, these diagrams are given by We further need to subtract the corresponding contributions from the EFT matrix elements. This is only nonvanishing for the diagrams associated to the D R c contribution, shown in Figure 5. We find which exactly cancels the contribution from the corresponding UV diagram. This curious cancellation can swiftly be reproduced from computing the hard region of the corresponding loop graph in the full theory [37][38][39][40] and seeing that it vanishes exactly.

(C9)
As indicated in (C7), summing all contributions we can decompose the result into a term independent from the vector-like mass, which is equivalent to the loop function appearing in the flavor-conserving amplitude, and a term which vanishes in the limit x Q → 0. The former coincides with the loop function obtained in [26]. Using the notation of the latter paper, we have with f V = ln x V /(x V − 1) and x V = m 2 V /m 2 U . In the custodial limit for the massive vectors, the terms that vanish at x Q → 0 read