$c \to u \nu {\bar \nu}$ transitions of $B_c$ mesons: 331 model facing Standard Model null tests

The Glashow-Iliopoulos-Maiani mechanism is extremely efficient to suppress the flavour-changing neutral current decays of charmed hadrons induced by the $c \to u$ transitions, making such processes particularly sensitive to phenomena beyond the Standard Model. In particular, $c \to u$ decays with a neutrino pair in the final state are theoretically appealing due to the small long-distance contributions. Moreover, in the framework of the Standard Model Effective Field Theory (SMEFT), the $SU(2)_L$ invariance allows to relate the Wilson coefficients in the effective Hamiltonian governing the $c \to u \nu {\bar \nu}$ decays to the coefficients in the $c \to u \ell^+ \ell^-$ Hamiltonian. We analyze the $B_c \to B^{(*)+} \nu {\bar \nu}$ decays, for which branching fractions of at most ${\cal O}(10^{-16})$ are predicted in the Standard Model including short- and long-distance contributions, so small that they can be considered as null tests. Using SMEFT and the relation to the $c \to u \ell^+ \ell^-$ processes we study the largest enhancement achievable in generic new physics scenarios. Then we focus on a particular extension of the Standard Model, the 331 model. SMEFT relations and the connection with $c \to u \ell^+ \ell^-$ imply that ${\cal B}(B_c \to B^{(*)+} \nu {\bar \nu})$ could even reach ${\cal O}(10^{-6})$, an extremely large enhancement. A less pronounced effect is found in the 331 model, with ${\cal O}(10^{-11})$ predicted branching fractions. Within the 331 model correlations exist among the $B_c \to B^{(*)+}\nu \bar \nu$ and $K\to \pi \nu \bar \nu$, $B\to (X_s, K, K^*) \nu \bar \nu$ channels.


I. INTRODUCTION
In the Standard Model (SM) the flavor-changing neutral current (FCNC) transitions occur at loop level and are generally characterized by Cabibbo-Kobayashi-Maskawa (CKM) and loop suppressions. The CKM cancellation mechanism is particularly efficient in the processes involving up-type quarks which take place through penguin and box diagrams with internal down-type quark exchanges. This is the case of the charmed hadron decays induced by the c → u + − and c → uνν transitions, for which tiny branching fractions are predicted in SM considering the short-distance amplitude [1]. The modes with charged dileptons are polluted by long-distance (LD) hadronic contributions, and the phase-space regions where such terms are large must be cut to pin down the effects of the short-distance term [2]. In the c → u dineutrino modes long-distance effects are smaller than in the charged dilepton modes. Therefore, such processes represent genuine null tests of the SM: their observation would be an indication of phenomena beyond the Standard Model (BSM). Among all hadrons, the decays of B c induced by the c → u transitions are particularly interesting, since in this case the main long-distance contributions affect a region of the phase-space near the end point, differently, e.g., from D, D s and Λ c . Hence, B c plays an important role in testing the Standard Model [3]. On general grounds, searching for new physics (NP) effects requires the analysis of several modes induced by the same underlying transition: the correlations among the various observables are important to identify the possible NP contributions and to relate them to the structure of the SM extensions.
The short-distance low-energy Hamiltonian governing the c → uνν transition has a simple structure. For left-handed neutrinos it consists of two operators, with Q L = (ūγ µ (1 − γ 5 )c)(νγ µ (1 − γ 5 )ν) In the SM the Hamiltonian comprises only Q L . The Wilson coefficient C SM L is obtained from loop diagrams with down-type quark exchanges, In (3) G F is the Fermi constant, α the fine structure constant, θ W the Weinberg angle, λ q = V * cq V uq with V ij the CKM matrix elements. The Inami-Lim function X(x q ), depending on x q = m 2 q /M 2 W , can be found in [4]. The dominant contribution from the intermediate b quark provides |C SM L | O(10 −13 ). Analogously, the transitions s → dνν and b → (s, d)νν are governed by a low-energy Hamiltonian with the structure of (1) and intermediate up-type quarks. BSM phenomena can manifest themselves through the enhancement of C L and through the effects of the operator Q R .
As for the charm sector, a few studies have analyzed the FCNC dineutrino modes in the SM and BSM frameworks [24][25][26][27]. Here, we focus on B c → B ( * )+ νν decays, for which lattice QCD results for the hadronic form factors can be used [28], with a control of the theoretical uncertainty related to nonperturbative QCD quantities. From the experimental point of view, these modes will be accessible at high energy e + e − colliders, namely the planned future circular collider FCC-ee machine running at the Z 0 peak.
We proceed both in a model-independent way and in a defined BSM framework. In the next section we apply the Standard Model effective field theory (SMEFT) to relate the Wilson coefficients in the low-energy c → uνν Hamiltonian (1) to the coefficients in the c → u + − Hamiltonian, as done in [24,25]. This allows us to establish the largest enhancement for the B c branching fractions achievable in a generic NP scenario, with the numerical results discussed in Sec. III. In Sec. IV we focus on a definite NP model, the 331 model in four variants. We observe that in this framework it is possible to relate the charm to the strange and beauty quark sectors, and that the c → u processes can be constrained using bounds from ∆S = 2 and ∆B = 2 observables. The correlations among B c and kaon and B meson dineutrino decays are described in Sec. VI. In the last section we draw our conclusions.

II. RELATING THE c → u DINEUTRINO AND CHARGED DILEPTON MODES USING SMEFT
A relation between the c → u dineutrino and the c → u charged dilepton modes can be established on the basis of SU (2) L invariance using the Standard Model effective theory [24,25]. Considering the possibility of lepton flavor violation, one focuses on c → uν iνj transitions, with the indices i, j denoting the neutrino flavors. The coefficients C L and C R in the low-energy Hamiltonian (1) become lepton-flavor dependent C i,j L,R and can be combined giving and The combinations (4) and (5) account for the contributions of both the operators Q L and Q R .
The relation of C L,R to the Wilson coefficients in the c → u + − low-energy Hamiltonian has been proposed in [24,25]. For two generic quarks q 1 and q 2 the q 1 → q 2 + − general Hamiltonian reads [29]: with the operators Q T = (q 2 σ µν q 1 )(¯ σ µν ) and P R,L = 1 ± γ 5 2 . The relations are obtained using the SMEFT operators classified in Ref. [30]. The treelevel matching of the dimension-6 four-fermion operators invariant under the SM SU (3) C × SU (2) L × U (1) Y gauge group with the Hamiltonian (1) gives the relations: The operators corresponding to the coefficients in the rhs of Eq. (8) are expressed in the Warsaw basis [30]. In this equation v is the electroweak vacuum expectation value and Λ the matching scale of NP with the SMEFT.
The relations between the coefficients in the c → u + − Hamiltonian (6) and the coefficients of the SMEFT operators can also be worked out: . The SMEFT operators have generation indices. The coefficients in the rhs of Eqs. (8) and (9) read C = C ij12 , with i and j the lepton generation indices and 1, 2 indicating the u and c quark in the first and second generation. 1 The coefficients of the SMEFT operators appearing in the rhs of Eq. (8) are also comprised in the rhs of Eq. (9). This allows us to translate the experimental bounds on the c → u + − modes, together with data on the s → d + − modes, into an upper bound for the combination x U in Eq. (5) [24,25]. Indeed, the SU (2) L symmetry links the c → u + − with c → uνν modes, and the s → d + − with c → uνν modes. The bound on x U is obtained assuming conservatively that the experimental limits on the charged dilepton branching fractions are saturated by the short-distance Hamiltonian (6). The limit depends on additional assumptions on the structure of the transitions; the most stringent one is obtained assuming lepton universality (LU) and charged lepton flavor conservation (cLFC) [24,25], with the results The bounds (10) and (11) have been considered in the analysis of D ± , D 0 , D s and charmed baryon decays induced by c → uνν [24,25]. Here we focus on B + c → B ( * )+ νν. We use the lattice QCD B c → B d form factors in [28], and the B c → B * d form factors derived in [3] applying the heavy quark spin symmetry [32,33]. The B + c → B ( * )+ form factors are obtained invoking the isospin symmetry.
the particle momenta are p, p , k 1 , k 2 and is the B * polarization vector. Denoting by E miss the energy of the neutrino pair in the B c rest frame, the dimensionless variable The hadronic matrix elements in the decay amplitudes are parametrized in terms of form factors: and The B + c → B + νν missing energy distribution obtained from (1) involves the form factor f + (q 2 ): with q = p−p and λ the Källén function. For B + c → B * + νν the missing energy distributions for longitudinally and transversely polarized B * read with p and E the B * three-momentum and energy in the B c rest frame. In Eqs. (14) and (15) the relation q 2 = m 2 Bc (2x − 1) + m 2 B ( * ) is used; the factor 3 is due to the sum over the three neutrino flavors. As inferred from (14) and (15), depends on both combinations. Using the parameters in Table I and The largest values of B(B + c → B + νν) and B(B + c → B * + νν) correspond to the largest x + U and x − U , respectively. We scan the branching fractions using for the two cases (10) (LU bound). We extend the computation up to the cLFC bound (11), which has been established by the analysis of the charged lepton modes, to investigate the size of the enhancement in this case. In Fig. 1 we show the largest enhancement for the dB(B + c → B + νν)/dx distribution obtained for x + U = 2 x U . In Fig. 2 we depict the maximum enhancement for the missing energy distribution and for the distributions of longitudinally and transversely polarized Integrating over x we have:  (11) (cFLC bound -red curve). The widths of the curves are obtained varying the form factor parameters [3,28].  (10) and (11) and varying x + U ∈ [0, 2 x U ] the branching fractions can be read in the plot in Fig. 3.
The enhancements in Eqs. (18) and (19), achievable in generic NP scenarios, must be taken with caution, since they would be the manifestation of BSM phenomena affecting other processes to a level that is necessary to control. For this reason it is worth considering a well defined extension of the Standard Model, as discussed in the next section. The electric charge generator Q is defined by Equation (21) provides the bound |β| ≤ 1 tan θ W (M Z ) which corresponds to |β| < 1.737 for the sine of the Weinberg angle sin θ W (M Z = 1 TeV) = 0.249. In all the 331 variants there is a neutral gauge boson Z mediating tree-level FCNC in the quark sector, with universal and diagonal Z couplings to leptons. The extended Higgs sector involves three SU (3) L triplets and one sextet. New heavy fermions are also present in the spectrum.
As in the SM, quark mass eigenstates are defined upon rotation of flavor eigenstates through two unitary matrices, U L (for up-type quarks) and V L (for down-type quarks). The relation V CKM = U † L V L holds. However, while in the SM V CKM only enters in charged current interactions and the two rotation matrices do not appear individually. In 331 model only one matrix, either U L or V L , can be expressed in terms of V CKM and of the other one. The remaining rotation matrix affects the Z couplings to the quarks. Choosing V L as the surviving rotation matrix, it can be parametrized as withc i = cos θ i ,s i = sin θ i , and phases δ 1,2,3 . With this parametrization, considering the Z couplings to the quarks, one finds that the B d system involves the parameterss 13 and δ 1 , the B s systems 23 and δ 2 , and the kaon systems 13 ,s 23 and δ 2 − δ 1 . This provides remarkable correlations among observables in kaon, B d, and B s systems [42][43][44][45][46]. It is interesting to observe that the relation allows to bound the Z mediated FCNC transitions of up-type quarks using the constraints established in the down-type quark sector [47]. Such a relation connecting the down-type and up-type quark FCNC processes is a peculiar feature of the 331 model. The Z coupling to ordinary fermions, for a generic value of the β parameter, is encoded in the 331 Lagrangian density: where s W = sin θ W , c W = cos θ W , q u (q d ) denotes an up (down)-type quark (i, j are generation indices), and v ij and u ij are the elements of the V L and U L matrices, respectively. The models corresponding to β = ± 2 √ 3 and β = ± 1 √ 3 , together with the choice of the fermions in the third generation as transforming as SU (3) L antitriplets, satisfy a number of phenomenological constraints [43]. In particular, it is possible to select a region of the parameter space compatible with the constrains from ∆F = 2 observables in the B d , B s and K systems and from the electroweak precision observables, provided that the Z mass is not lighter than 1 TeV.
As shown in [44], the Z − Z mixing can be neglected in ∆F = 2 transitions, while it must be taken into account in decays with neutrinos in the final state. The Z − Z mixing angle is written as [44] sin ξ = c 2 where and v 2 ± are given in terms of the vacuum expectation values of two Higgs triplets ρ and η: v 2 The parameter a is expressed in terms of tanβ = v ρ v η as in two Higgs doublet models (we useβ to distinguish this parameter from β defining the 331 model in (20)) [44]: We consider the four variants scrutinized in [43]. For the modes with a neutrino-antineutrino pair in the final state, the Z − Z mixing is included replacing R L νν (a) is defined as with B(β, a) in (25) and ∆ νν (Z) the SM Z coupling to neutrinos. In the 331 model the B c → B ( * )+ νν modes present several features. The structure of the 331 model allows us to use data from B and K decays to constrain c → u modes. Moreover, Z mediates FCNC at tree level only in the case of left-handed quarks; hence, the coefficient C R in the Hamiltonian (1) vanishes in all the model variants.
Considering the contribution from the tree-level diagram in Fig. 4 and using the coupling of Z to quarks and neutrinos derived from Eq. (24), the coefficient C L in (1) reads: where The elements u ij are obtained from Eqs. (22) and (23). As a consequence, C L depends on the parameters s 13 , δ 1 ,s 23 , δ 2 that in pair control the B d and B s decays, respectively, and altogether govern the K decays. C L also depends on the Z − Z mixing parameter a. The B + c → B ( * )+ νν processes must be studied in such a parameter space. In the numerical analysis of B + c → B ( * )+ νν in the 331 model we follow the method described in [45]. We select the model parameters imposing that ∆M B d , S J/ψK S and ∆M Bs , S J/ψφ , whose measurements are quoted in Table I, lie in their experimental ranges within 2σ. In the kaon sector we require that ε K is in the range [1.6, 2.5] × 10 −3 and ∆M K varies between [0.75, 1.25] × (∆M K ) SM , i.e., (∆M K ) SM = 0.0047 GeV using V ub in Table I. The formulas for such observables in the SM and in 331 models can be found in [42]. For ε K we use the updated result in [48]. The other input quantities are also collected in Table I. For the CKM matrix elements the Table displays the four entries chosen as the independent ones, the others are derived.
The obtained allowed regions in the parameter spaces 13 , δ 1 ,s 23 , δ 2 are in Fig. 5 for M Z ∈ [1, 5] TeV. The regions (s 13 , δ 1 ) are obtained imposing the constraints on ∆M B d , S J/ψK S , the regions (s 23 , δ 2 ) using ∆M Bs , S J/ψφ . In our computation of the observables in the 331 model, we varys 13 , δ 1 ,s 23 , δ 2 in their allowed ranges and select the values for which the constraints from ∆F = 2 processes in the kaon sector are also satisfied. For each value of β Fig. 5 shows the presence of two ranges for the phases δ 1,2 which are independent of M Z . Choosing the 331 parameters in the allowed ranges, the coefficient C 331 L can be computed and the B c → B ( * )+ νν branching fractions can be predicted.
In Fig. 6 we plot the missing energy distributions for the set ofs 13 , δ 1 ,s 23 , δ 2 and a maximizing the dineutrino B c branching fractions for each M Z up to 5 TeV. In all cases the choice a = 1 provides the largest enhancement. The central values of the branching fractions are: B(B + c → B * + νν) = 6.08 The enhancement with respect to the SM is large, even though the branching fractions do not exceed O(10 −11 ). We have remarked that a peculiar feature of the 331 model is the possibility of constraining FCNC up-type quark processes using information on FCNC down-type quark transitions. On this basis we can establish the correlations between B c → B ( * )+ νν and the s → dνν induced transitions K + → π + νν and K L → π 0 νν, and between B c → B ( * )+ νν and B → {X s , K, K * } νν induced by b → sνν. In the SM such transitions proceed through box and Z 0 penguin diagrams. The low-energy s → dνν Hamiltonian reads in the SM: Hs →dνν takes into account the internal charm contribution [49][50][51][52][53][54], the function describes the internal top contribution. η X = 0.994 is a QCD correction computed for m t = m t (m t ) [50,55].
In the charged K + → π + νν mode both contributions must be taken into account, in K L → π 0 νν the top quark contribution dominates. The top quark contribution also dominates in the b → sνν modes governed by the effective Hamiltonian The 331 contribution from the tree-level Z exchange can be included in the Hamiltonian replacing X(x t ) → X(M ) with and The Z − Z mixing is included multiplying the rhs of Eqs. (40) and (41) by (1 + R L νν (a)), with R L νν in (31).

VII. CONCLUSIONS
Null tests, like rare FCNC charm decays, are useful to investigate the existence of phenomena beyond the Standard Model. The B c → B ( * )+ νν modes are predicted within the SM with branching ratios not exceeding O(10 −16 ) and can be used for null tests. Exploiting the relations between the Wilson coefficients of the c → uνν and c → u + − low-energy Hamiltonian obtained from the SMEFT, together with the experimental bounds on the charged dilepton c → u processes, we have derived the largest enhancement for the dineutrino modes in generic NP scenarios, finding branching fractions up to O(10 −6 ). Specific NP scenarios predict smaller effects. To investigate this point we have predicted the branching fractions for these processes in the 331 model. The reason to consider this framework is that in this model the NP parameters entering in FCNC charm decays are the same that govern the B, B s , K FCNC transitions. This provides nontrivial correlations among the observables in the various systems. In the 331 models the effective c → uνν Hamiltonian comprises only one operator, as in the SM, with a modification of the Wilson coefficient enhancing B(B c → B ( * )+ νν) up to O(10 −11 ). Moreover, in this model a correlation with down-type quark dineutrino processes can be established. We have found that the largest branching fractions correspond to β = −2/ √ 3 and are anticorrelated with K + → π + νν, K L → π 0 νν and B → {X s , K, K * }νν. We estimate the main long-distance contributions to B c → B ( * )+ νν represented by the processes B c → B ( * )+ V 0 → B ( * )+ νν, with V 0 = ρ 0 , ω, φ [2]. The nonleptonic color suppressed B c → B ( * )+ V 0 amplitude can be estimated using naive factorization in terms of the B c → B ( * ) form factors. The V 0 → ν ν amplitude involves the hadronic matrix elements 0|qγ µ (g q V − g q A γ 5 )q|V 0 (q, ) (A.1) with g q V,A the vector and axial-vector couplings constants of the neutral current for quarks. Actually, (A.1) takes contribution only from the vector quark current. These matrix elements can be obtained from the V 0 matrix element of the em current J µ em = q e qq γ µ q, with e q the quark charges, parametrized as 0|J µ em |V 0 (q, ) = Using the V 0 masses, widths and V 0 → e + e − (µ + µ − ) branching fractions [34], we have f ρ 0 = 4.99 ± 0.03 (5.08 ± 0. 16