Angular and polarization observables for Majorana-mediated B decays with effective interactions

We probe the effective field theory extending the Standard Model with a sterile neutrino in B meson decays at B factories and lepton colliders, using angular and polarization observables. We put bounds on different effective operators characterized by their distinct Dirac-Lorentz structure, and probe the $N$-mediated B decays sensitivity to these interactions. We define a Forward-Backward asymmetry $A_{FB}^{\ell \gamma}$ between the muon and photon directions for the $B \to \mu \nu \gamma$ decay, which allows us to separate the SM contribution from the effective lepton number conserving and violating processes, mediated by a near on-shell $N$. Using the most stringent constraints on the effective parameter space from Belle and BaBar we find that a measurement of the final polarization $P_{\tau}$ in the rare $B^-\rightarrow \ell^-_{1}\ell^-_{2} \pi^+$ decays can help us infer the scalar or vector interaction content in the $N$ production or decay vertices. We find that the B meson decays are more sensitive to scalar operators.


I. INTRODUCTION.
Besides the remarkable performance of the standard model (SM) of particle physics in describing nature, neutrino oscillations are currently the most compelling experimental evidence of the need to extend the SM in order to include mechanisms for neutrino mass generation. In the recent years, the LHC experiments have put stringent constraints on the existence of new physics involving colored states, but still the possible extensions of the electroweak (EW) sector are far less restricted. A variety of new physics scenarios may be hidden at energies well above the EW scale, and their study is being consistently tackled by the use of the standard model effective field theory (SMEFT) [1,2].
But also new physics might involve weakly coupled fields at the EW scale. Many models leading to neutrino masses predict the existence of sterile right-handed neutrinos with Majorana masses, as the Type I [3][4][5][6][7] and also the linear and inverse seesaw mechanisms. This possibly not-that-heavy degrees of freedom can be described by an EFT including them (SM-NEFT) [8][9][10][11], and here we concentrate on a simplified scenario with only one right-handed neutrino added [8]. The phenomenology of models extending the Type I seesaw renormalizable Lagrangian with effective interactions of higher dimension for the right-handed neutrinos are being studied [12,13], and recently complemented with the implications of the Minimal Flavor Violation ansatz on the new heavy neutrino interactions [14,15].
The simplified scenario considering only one sterile state N has started to get attention in connection to the novel dimension five Higgs-neutrino interactions [16] and the heavy neutrino magnetic moment dipole portal [17]. In particular, some studies have set constraints on the effective operators in this simplified scenario using existing LHC searches [18] and recent work explores the matching between the off-shell EW-scale operator basis and a low-energy on-shell basis [19,20].
On the other hand, works on approaches as the so called neutrino non standard interactions (NSI) and general neutrino interactions (GNI) are incorporating right-handed neutrinos to the SMEFT [21,22] considering their Majorana and/or Dirac nature. Also effective neutrino long-range interactions are being studied [23] in an EFT approach.
The presence of a Majorana mass term for the right-handed neutrinos is the source of lepton number violation (LNV) in these models. While the phenomenology of LNV has been thoroughly studied in the past in the context of seesaw scenarios, for recent reviews see [24,25], the study of lepton number violating and conserving (LNC) interference effects in final states with light neutrinos is usually discarded. Here we assess the chances of disentangling possible contributions from effective operators with distinct Dirac-Lorentz structure to B leptonic decays where the final neutrinos do not allow for the identification of LNV or LNC interactions.
In this article we focus on the simplified scenario with only one heavy Majorana neutrino N , and neglect the effect of the renormalizable Yukawa term N Lφ giving the heavy-active neutrino mixings U lN , given that it is strongly constrained not only by the naive seesaw relation U 2 lN ∼ m ν /M N ∼ 10 −14 − 10 −10 required to account for the light ν masses [25,26], but also by the experimental constraints on a toy-like model in which the SM is extended with a massive Majorana neutral fermion, assumed to have non-negligible mixings with the active states, without making any hypothesis on the neutrino mass generation mechanism [27,28]. Such a minimal SM extension leads to contributions to LNV observables which are already close, or even in conflict, with current data from meson and tau decays, for masses M N below 10 GeV (see [27,29] and the references therein).
Our group has studied the Majorana N decays [30,31] and phenomenology mostly in colliders for masses m N above the EW-scale [32][33][34][35][36] focusing on LNV processes, and in the order 1 − 10 GeV scale, where its decay is dominated by the N → νγ channel [37], which was also considered in [38].
The pure leptonic and radiative decays of pseudoscalar mesons mediated by a sterile N in this effective scenario were initially studied in [39], where the authors only took into account vectorial operators contributions. The study of N -mediated lepton number violation in rare B meson decays has been pursued, for example, in [26,27,[40][41][42][43][44][45][46][47][48] and the references therein.
Also, the tension with the SM values of the ratios of branching fractions R(D) and R(D * ) in semileptonic B decays [49] measured at Belle, BaBar and LHCb, has led to proposals involving sterile neutrinos as solutions, in seesaw scenarios [50] and also including EFTs with right-handed neutrinos [51], with sterile N interactions mediated by leptoquarks [52] or a W SU (2) L singlet vector boson [53,54].
In this work we go on studying the effect on B meson decays of the presence of a Majorana N with effective interactions. In a previous paper [55] we studied the constraints imposed by the bound on the B − → µ ± µ ∓ π + decay by LHCb [56] and in the radiative leptonic B → µνγ decay by Belle [57] on the effective operators. Here we propose to use final tau polarization and angular observables to disentangle the possible contributions of a Majorana neutrino N with effective interactions to the B − → τ − ν, the radiative B → µνγ and the LNV B − → − 1 − 2 π + decays in future experiments. The paper is organized as follows. In Sec. I A we introduce the effective Lagrangian formalism for the Majorana N , discuss the existing bounds on the couplings weighting the different effective operators contributions I B, and review the experimental prospects for B meson leptonic decays I C. In Sec.II we study the B tauonic decay. The radiative B → νγ decay is assessed in Sec.III and the LNV B − → − 1 − 2 π + in Sec.IV. We summarize our results in Sec.V.

A. Effective interactions formalism.
We extend the SM Lagrangian including only one right-handed neutrino N R with a Majorana mass term, giving a relatively light massive state N as an observable degree of freedom. The new physics effects are parameterized by a set of effective operators O J constructed with the SM and the Majorana neutrino fields and satisfying the SU (2) L ⊗U (1) Y gauge symmetry [8,10,58].
The effect of these operators is suppressed by inverse powers of the new physics scale Λ.
The total Lagrangian is organized as follows: where n is the mass dimension of the operator O (n) J 1 .
The dimension 5 operators were studied in detail in [9]. These include the Weinberg gives Majorana masses and couplings of the heavy neutrinos to the Higgs (its LHC phenomenology has been studied in [12][13][14]), and the operator O In the following, as the dimension 5 operators do not contribute to the studied processes 1 Note that we do not include the Type-I seesaw Lagrangian terms giving the Majorana and Yukawa terms for the sterile neutrinos.

Operator Notation Type Coupling
Operator Notation Type Coupling -discarding the heavy-light neutrino mixings-we will only consider the contributions of the dimension 6 operators, following the treatment presented in [8,10], and shown in Tab.I.
The effective operators above can be classified by their Dirac-Lorentz structure into scalar, vectorial and tensorial. The couplings of the tensorial operators are naturally suppressed by a loop factor 1/(16π 2 ), as they are generated at one-loop level in the UV complete theory [8,60].
In this paper we will consider the B decays B → τ N in Sec.II, B − → µ − νγ in Sec.III and B − → − 1 − 2 π + in Sec.IV, mediated by a Majorana neutrino N . We can thus take into account the following effective Lagrangian terms involved in those processes: and The one-loop generated effective Lagrangian contributes to the N → νγ decay. Here −P (A) is the 4-momentum of the outgoing photon and s W and c W are the sine and cosine of the weak mixing angle.
In the effective four-fermion terms in (2) the quark fields are flavor eigenstates with family j = 1, 2, 3. In order to find the contribution of the effective Lagrangian to the B − decays we are studying, we must write it in terms of the massive quark fields. Thus, we consider that the contributions of the dimension 6 effective operators to the Yukawa Lagrangian are suppressed by the new physics scale with a factor 1 Λ 2 , and neglect them, so that the matrices that diagonalize the quark mass matrices are the same as in the pure SM.
Writing with a prime symbol the flavor fields, we take the matrices U R , U L , D R and D L to diagonalize the SM quark mass matrix in the Yukawa Lagrangian. Thus the left-and righthanded quark flavor fields (subscript j) are written in terms of the massive fields (subscript β) as: With this notation, the SM V CKM mixing matrix corresponds to the term V ββ = 3 j=1 (U jβ L ) † D jβ L appearing in the charged SM current J µ CC = u β V ββ γ µ P L d β . Thus in the mass basis the tree-level generated Lagrangian in (2) can be written in terms of the massive quarks as For the sake of simplicity, we will rename the new quark mixing matrix element products as These new quark flavor-mixing matrices combinations Y ββ are unknown, and in principle their entries may be found by independent measurements, as is done in the case of the SM V CKM matrix. In this occasion we will make an ansatz and consider that all the Y ββ values in (6) shall be of the order of the SM V ββ CKM value, taking it as a measure of the strength of the coupling between the respective quarks: This will be done in the numerical calculations, while we leave the explicit expressions in our analytical interactions N LW when neutrino mixing is taken into account [26]. Inspired in this interaction we consider the combination which is derived from the Lagrangian term coming from the operator O (i) LN φ in Tab.I and allows a direct comparison with the mixing angles in the Type I seesaw scenarios [26]. Updated reviews of the existing bounds on the Type I seesaw mixings U N can be found in refs. [40,61].
Some of the operators involving the first fermion family (with indices i = 1) are strongly constrained by the neutrinoless double beta decay bounds, currently obtained by the KamLAND-Zen collaboration [62]. Following the treatment already made in [31], the values of the 0νββ-decay constrained couplings α 1/2 for Λ = 1 T eV . These operators, although not directly involved in the processes we consider in this work, appear as contributions to the Γ N total 2 The matrices in (4) could also be reabsorbed into the definition of the effective couplings α J , as is done for instance in [23].

Operators
Couplings  T for i = 2, 3. In the different sections of the paper, we will take complementary approaches to assign these values, that will be explained in a timely manner. However, we will use the sets defined in Tab. II to see which operators are taken into account in each case.
For the calculation of the total decay width of the Majorana neutrino Γ N we include all the kinematically allowed channels for m N in the range 1 GeV < m N < 5 GeV . The details of the calculation are described in our ref. [31]. The sets 1, 2 and 3 in Tab. II take into account the contributions of the tensorial one-loop-generated effective couplings in (3) to the N decay width. In particular these sets allow for the existence of the N → νγ decay channel. As we found in [31], this channel gives the dominant contribution to the N decay width for the low mass m N range considered in this work. The sets 4, 5 and 6 discard this contribution. The total Γ N width is around three orders of magnitude higher in the sets 1, 2 and 3 than in the sets 4, 5 and 6 [55].

C. B meson leptonic and semileptonic decays
Many ongoing and future experiments will copiously produce B mesons, enabling the study of its leptonic and semileptonic decays. The number of B mesons is expected to be over 10 10 at Belle II, over 10 13 at the 300f b −1 luminosity LHCb upgrade, and a similar number is expected at SHiP. The estimated number of B mesons to decay in the MATHUSLA detector volume is over 10 14 , and the expected number of B meson pairs to be produced at the Z peak at the FCC-ee is above 10 11 [40].
Measuring pure leptonic B decays is challenging. The B → τ ν predicted branching ratio is of order 10 −5 in the SM. For muonic and electronic B decays it is much smaller due to the m 2 factor coming from helicity suppression. The spinless B meson -like the pion-prefers to decay to the heaviest possible charged lepton because balancing the spins of the outgoing leptons requires them to have the same handedness, and the neutrino forces its charged partner into the unfavored helicity. The SM decay B − → τ −ν τ (or its conjugate) has not yet been measured by one single experiment with 5σ significance, and in Sec. II we use the value of the combined measurement [63] to put bounds on the effective couplings α J involved in the B → τ N decay. However, the prospects are for Belle II to achieve a measurement with approximately 2 ab −1 (assuming the SM branching ratio) [64]. This would be a first step towards disentangling the interference with possible new physics, as B → τ N decays, for instance with the aid of the final tau lepton polarization information.
Final taus are the only fermions whose polarization is accessible by means of the energy and angular distribution of its decay products. These measurements rely on the dependence of kinematic distributions of the observed tau decay products on the helicity of the parent tau. Tau lepton polarization measurements at Belle have focused on the R (D * ) anomaly [65,66]. In Sec. IV we find that measuring the final tau polarization in eventual LNV 1 and/or 2 = τ − could help to discern between the contributions of different types of effective operators. The Belle II prospects for measuring SM rare semitauonic modes as B + → τ + τ − π + are discouraging, because it is very difficult to reconstruct both final taus, and the lepton flavor violating B + → τ + µ − π + decays with final taus and muons are expected to be easier to measure [64]. One can hope that the LNV semi-tauonic decays could be discovered in experiments with more B meson events, as MATHUSLA and SHiP [40]. However, these would not allow for the reconstruction of the taus polarization, leaving us with the eventual measurements at the FCC-ee [67,68] to test our proposal.
B factories as Belle II produce B mesons at the center of mass (CM) energy of the Υ(4S) resonance, which decays in 50% to B + B − pairs. The decay of the Υ(4S) produces B mesons in pairs: if one (the tag B tag ) is reconstructed, the rest of the event must be a B meson (the signal B sig ). In hadronic B tags all the charged and neutral particles are identified, and used to reconstruct the tag-side B meson. The efficiency of this method is low -of order 10 −3 -but a very pure sample of B mesons is obtained. In semileptonic B tags a charmed meson is reconstructed together with a high momentum lepton. The efficiency of this method is higher, of order 10 −2 , though the obtained sample is not so pure. Inclusive tagging combines the four-momenta of all particles in the rest of the event of the signal-side B candidate. The achieved tagging efficiency is usually one order of magnitude above the hadronic and semileptonic tagging approaches [64].
The SM radiative leptonic B decays have been extensively studied in the literature [69][70][71][72][73][74], as they are a means of probing the strong and weak SM interactions in a heavy meson system. While the measurement of pure leptonic B decays is very difficult due to helicity suppression and the fact of having only one detected final state particle, the radiative modes can be larger as they escape helicity suppression, and are easier to reconstruct because of the extra real final photon. In Sec. IV we use the bounds imposed on the effective couplings by the last Belle measurements on the B → µνγ partial branching fraction [57]. Now, as the signal yields for the B → µνγ decay are expected to be three times higher in Belle II compared to Belle [64], in Sec. III we explore the possibility to, if the signal is found, use angular information from the hard photon and the muon flight directions to study the interference of N mediation in this decay. We are particularly interested in the inclusive tagging technique mentioned above, which allows for the reconstruction of the B sig meson flight direction, and the use of its rest frame, without a high loss in efficiency, as has been recently done in [75]. This technique could be used for implementing the angular Forward-Backward asymmetry A γ F B between the final charged lepton and photon flight directions we propose in Sec. III 1.

II. B TAUONIC DECAY: FINAL TAU POLARIZATION
In this section we aim to study the possibility to measure the effects of the presence of a Majorana neutrino N , considering the final tau polarization in the SM decay B − → τ −ν τ and the effective B → τ N decay. The latter mode can leave the same final tau plus missing energy / E T signal as the SM process when the N escapes the detector before decaying into observable particles.
The expression for the effective decay width Γ B→τ N is given below. The details of the calculation 3 can be found in our ref. [55]. The result is where Here f B is the B meson decay constant.
The effective couplings in A V,S -as the subscript indicates-correspond to vectorial and scalar interactions. We find that, due to the pseudo-scalar nature of the B meson the scalar operators contribution to the B → τ N decay width is enhanced with respect to the vectorial contribution. This leads to the presence of the quark masses m u and m b in the denominator of the scalar term in (8). In turn, this will enable us to put more stringent bounds on the scalar operators contributions to this process [55].
In order to find the allowed values for the effective couplings in this context, we compare the experimental value Br exp (B → τ ν) = (1.09 ± 0.24) × 10 −4 to the theoretical prediction subject to the constraint τ N > 10 3 ps on the N lifetime, enforcing it to escape undetected [56].
The SM result for the branching ratio is The and τ N > 10 3 ps.
In the numerical calculation we consider all the values of the effective couplings to be equal: α We now turn to calculate the polarized B → τ ν and B → τ N decays. We will consider the final tau polarization observable P τ in terms of the decay widths Γ(h), with h = ±1 for the positive and negative tau helicity states. Thus we write We will also assume that tau decays preserve CP invariance and therefore the distributions for h = ± anti-taus τ + follow those of h = ∓ taus τ − .
1/2 and we use the notation in (8). We find that the pure scalar contribution in the second term is divided by the squared sum of the quark mass quotients x u and x b . This enhances its contribution (and also the contribution of the second mixed term) compared to the pure vectorial contribution of the first term.  In the case of a specific final lepton flavor, take = µ, we consider the amplitude in the SM, which conserves lepton number: B − → µ −ν µ γ (M SM νµ ) and the one mediated by the Majorana neutrino represented in Fig.2. This effective process gives a LNC contribution to the same final state (M N νµ ) together with LNC contributions to final states with the other two anti-neutrino flavors (M N ν (e,τ ) ) and the LNV B − → µ − ν j γ (M N LN V ) with final neutrinos. The first two must be added coherently 4 in order to calculate the squared matrix element, while the last must be added incoherently. Thus we write: The SM amplitude is written following the treatment in [69] with the B meson momentum q ρ = m B v ρ , p and p ν the final lepton and anti-neutrino momenta, and k the photon momentum. The constants f V and f A are the vector and axial-vector form factors in [69,73], e is the electron charge and G F is the Fermi constant.
The LNC Majorana contribution to the amplitude is and the LNV part is Here p N is the intermediate Majorana momentum and P N ( The indices (j) in C correspond to the final (anti-)neutrino flavor, and (i) in C V and C B S to the final charged lepton flavor. A sum over the corresponding indices j = 1, 2, 3 in (16) is understood. The details of the calculation can be found in the Appendix C in our ref. [55].
With the amplitudes above, we perform a phase space Monte Carlo integration, and use it to calculate a Forward-Backward asymmetry between the final charged lepton and photon flight directions in the B rest frame, which can be used as an observable to disentangle the effective Majorana from the SM contribution to the leptonic radiative B decay.

Forward-Backward lepton-photon asymmetry
The Forward-Backward asymmetry A γ F B between the final charged lepton and photon flight directions is defined as where N ± is the number of events with a positive (negative) value of cos(θ), the angle between the final lepton and the photon flight directions in the B meson rest frame.
In Fig.3 we show our results for the value of the A γ F B asymmetry for the SM process B − → µ −ν µ γ considering a final muon, for each photon energy bin E γ (black dots with error bars). The SM asymmetry value is negative in all the allowed photon energy range. The photon has maximum energy when the muon and neutrino have parallel momenta, opposed to p γ . In this case, the asymmetry value is strictly A γ F B = −1, as can be seen in the plot. In the massless muon limit, the energy allowed for the photon in the B rest frame is where ϕ is the angle between the muon and neutrino momenta p µ and p ν . Thus the photon has maximum energy when the muon and neutrino have the same flight directions in the B rest frame, meaning the photon and muon momenta are back-to-back.  (14), which are taken from [73].

Sensitivity to effective couplings
The numerical value of the effective contribution to the Forward-Backward asymmetry The different curves in both panels in Fig.3 show that the effective contribution to the asymmetry is moderate, except for an interval in the photon energy E γ range, where it steps towards less negative values. This means that for this energy interval, some of the effective events contribute with a positive value of cos(θ) to the asymmetry. This interval is different for different m N values.
The effect can be explained by kinematical reasons. In the effective B → µN → µνγ decay, for each intermediate m N mass, the photon energy E γ in the B meson rest frame can take the values shown in Fig.4a. This is due in part to the cut value E cut = 1 GeV introduced to ensure the validity of the QCD treatment in the calculation [73], and to the minimal allowed energy E min γ = E N (1 + β N )/2 the photon can have in the B rest frame, according to the energy E N and boost velocity β N of the intermediate Majorana neutrino.
The maximal energy is given by the formula corresponding to (18) for m µ = 0. Thus, the effective contribution to the asymmetry only exists in the region between the curves in Fig.4a. The minimum value of E γ in Fig.4a for each m N is the value where the effective contribution starts [55]. This contribution stops to rise the asymmetry value when the scalar product p µ . p γ = | p µ || p γ | cos(θ) starts to be negative again: In Fig.4b, we plot eq. (19) for the m N values in Fig.3 as an aid to visualize the E γ value where the sign of cos(θ) starts to be negative.
Let us consider the curves for m N = 4 GeV in Fig.3. The step in both panels starts at E γ = 1.5 GeV , which is the minimum value we obtain from the curve in Fig.4a. At this energy cos(θ) is positive (it is entering the allowed range | cos(θ)| ≤ 1), and changes sign when E γ = 1.95 GeV , as can be seen in Fig.4b. Here the step in the curves in Fig.3 comes to an end. We also note that when we consider only vectorial operators, in Fig.3b, the curve for m N = 4 GeV stands between the SM value error bars, while the scalar contribution in Fig.3a increases above them. This can be studied by considering a ∆χ 2 distribution to see which values of the effective couplings α J would be compatible (or not) with the SM prediction. We construct a ∆χ 2 function as where E i are the photon energy bins shown in Fig.3, and δ i are the errors for the SM prediction, and A T ot F B is the sum of the SM and the effective value for A γ F B in each photon energy bin. In Fig.5 we show the contours in the (α S , α V ) plane corresponding to o a coverage probability (1 − α) of 68.27% (1σ), 90% and 95% from the SM prediction for A γ F B , for different m N values.
Let us consider the panel for m N = 4 GeV in Fig.5c. The point α S = 0, α V = 1 lies in the region inside the 1σ contour, and thus the effective contribution to A T ot F B lies between the error bars in Fig.3b, while the point α S = 1, α V = 0 is far outside the outer contour, and the effective contribution can be well distinguished from the pure SM curve in Fig.3a. The same effect occurs for other m N values: the asymmetry is more sensitive to the scalar interactions contribution. This can be seen easily by inspection of eqs. (16) and (15). The scalar contribution term C B S is weighted by a factor of order m B /(m u + m b ) with respect to the vectorial contribution, and this is again due to the pseudoscalar nature of the B meson.

IV. N MEDIATED LNV B DECAYS WITH FINAL TAUS
In this section we study the LNV decay B − → − 1 − 2 π + , mediated by a Majorana neutrino, as depicted in Fig.6. In the case where one or two of the final leptons are tau leptons, we would like to see if a measurement of their polarization could give a hint on the kind of new physics involved in the production or decay of the intermediate N . As we will see, the polarization of the final taus in these decays could be used to distinguish the vectorial and scalar operators contributions to the N production and decay vertices, complementing those proposed for a higher m N scale in [36].
As we want to keep track of the final taus polarization, we perform a calculation of the decay amplitude considering the intermediate N to be near the mass shell, but without recurring to the narrow width approximation. The details of the calculation are summarized in the Appendix A 2.
The amplitude for the LNV decay B − → − 1 − 2 π + can be written as with C (I)  In the case of only one final tau, we define the final-state polarization as and the two-taus final state polarization in B − → τ − τ − π + as Here the number of events (N ± , N ±± ) with subscripts + and − correspond respectively to the number of h = + and h = − polarization states of each final tau 1 or 2 in Fig.6 measured in the experiment, as considered in (A2). The defined final state polarizations, being a quotient, are independent of the total number of B decay events considered.
In our previous paper [55], we studied the bounds that can be set on the different couplings value in (7). While this constraint is obtained from interactions involving the = µ, i = 2 family, we will use it here to fix also the α J couplings in (21), for the sake of simplicity 5 .
To appreciate the ability of the final taus polarization to determine the kind of effective operators involved in the studied interaction, we define a parameter λ ∈ [0, 1] to measure the proportion of vectorial and scalar operators contributing to the processes. Thus we multiply the vector operators by λ and the scalars by (1 − λ), so that λ = 1 means pure vectorial and λ = 0 means pure scalar interactions.
In our numerical code, given a value of the scalar -vector interaction content λ, we let the value of the intermediate m N mass vary in the allowed interval for each region in Tab. III. After fixing the numerical values of the effective couplings (α) and mixing matrices (Y ) in (21), this gives us as output an interval of values for the final-state polarization P τ as defined in eqs. (22) and (23) for one or two tau processes, consisting of a band in the (λ, In Fig.7 we show the curves of maximum and minimum values giving the allowed values for the final tau polarization P τ in (22) depending on the λ parameter for Majorana neutrino masses m N in the kinematic regions I (Fig.7a), II (Fig.7b) and III (Fig.7c) defined in Table   III. For the case of two final taus (23), the curves are shown in Fig.7d.
In the experiment, such as FCC-ee as we discussed in Sec.I C, the first and second leptons can in principle be distinguished, as 1 should have the higher momentum. Then, the m N mass range can be reconstructed with a measurement of the invariant mass M ( 2 , π). This would allow us to distinguish the three regions I, II and III.
If only one tau lepton is found in the final state, we can distinguish between regions I and III by comparing its momentum with the muon momentum. If p τ > p µ , we should be in region I, and the invariant mass M (µ, π) value should confirm this. In this case, the final tau is produced together with the N in the primary vertex (see Fig.6). This means in this case we are probing the C (I) V and C B S coefficients in (21). As we find in Fig.7a, this tau leptons are expected to be mostly negatively-polarized if we only consider scalar interactions (λ = 0), and an unpolarized final tau state could be reached in the case of purely vector interactions (λ = 1). If p µ > p τ , we should be in region III, which can be checked with the invariant mass M (τ, π). Here, by measuring the tau polarization we access the N decay vertex, and test the C (II) V and C π S coefficients in (21). Fig.7c shows that finding a negative P τ would point towards a dominant scalar interaction, while a positive value would signal a vector-dominated N decay.
The case of region II shows that we expect mostly a negative final state tau polarization, indicating the dominance of the scalar interactions in both the N production and decay vertices. This is consistent with the fact that the scalar interactions are weighted in both cases by the quark masses in the denominators in (21), which as we mentioned in the discussion of (8), enhances the scalar operators contribution, due to the pseudoscalar nature of both the B and π mesons.

V. SUMMARY
The effective field theory extending the standard model with right-handed neutrinos (SMNEFT) parameterizes new high-scale weakly coupled physics in a model independent manner. We consider massive Majorana neutrinos coupled to ordinary matter by dimension 6 effective operators, focusing on a simplified scenario with only one right-handed neutrino added, which provides us with a manageable parameter space to probe. These massive neutrinos would be produced in leptonic and semileptonic decays of B mesons, a theoretically clean system to probe new physics effects in lepton colliders and B factories. We exploit the known bounds on B meson decays to constrain the effective parameter space, and propose to use final taus polarization and angular observables to disentangle vectorial or scalar type effective contributions.
The B meson couples preferably to the scalar operators in the left column of Tab.I in comparison to the vectorial ones. Thus, as we already found in [55], probing its decay to a Majorana neutrino N we can either put more stringent bounds or give more precise predictions on the scalar operators effective couplings.
In Sec.II we considered the bounds on the tauonic B decay [76][77][78][79] to constrain the allowed regions in the (m N , α) plane (Fig.1), and find that a measure of the final tau polarization could help to separate the pure SM decay, which gives P τ = 1, from the effective B → τ N , where the N escapes undetected. This contribution can lower the P τ value to 0.8 in the allowed region (with m N < 3.5 GeV ) for heavy neutrinos coupling only to the third lepton family. In Tab.II we define six different numerical coupling sets which turn on/off the scalar, vector or tensorial interactions. We find that the sets 4 and 6, which involve scalar N interactions, are the ones giving the more interesting results.
In Sec.III we define a Forward-Backward asymmetry A γ F B between the muon and photon directions for the B → µνγ decay, which allows us to separate the SM lepton number conserving contribution from the effective lepton number conserving and violating processes, mediated by a near on-shell N . We find that the effective contributions tend to deviate A γ F B from the SM value in photon energy intervals depending on the mass m N , according to the boost of the N in the B meson rest frame (Fig.3). This measure could be done in Belle II with the aid of inclusive tagging techniques, which allow for the reconstruction of the signal B frame [75]. Again, the asymmetry is more sensitive to the scalar interactions, as is found in Fig.5.
In Sec.IV we study the final taus polarization in the lepton number violating B − → − 1 − 2 π + decay when − 1 and/or 2 = τ − . Using the bounds on the effective couplings obtained in [55] we probe the chances to disentangle different effective operators contributions to these decays. Weighting the vectorial and scalar operators by a factor λ ∈ [0, 1] with λ = 1 (purely vectorial) and λ = 0 (purely scalar) contributions. In Fig.7 we find that a measurement of the final polarization P τ can help us infer the scalar or vector content in the N production or decay vertices, provided that the invariant mass M ( 2 , π) can be reconstructed.
The B meson provides a rich environment to search for and constrain possible high-energy physics effects in the case a GeV-mass heavy N is the only new accessible state. This study complements probes of lepton number violating same-sign dilepton signals, Higgs-neutrino interactions and other collider searches to characterize the Standard Model Neutrino effective field theory parameter space. In section II we showed the B → τν τ polarized decay result in the SM. For reference purposes, we summarize the calculation here. The SM amplitude can be written as with q m W the B meson momentum and q = p + k, the tau and anti-neutrino momenta, respectively.
We take the squared amplitude The leptons spin vectors can be written in terms of their helicity h = ± as Thus we have, in the B meson rest frame which gives us a polarized decay width with x τ = m τ /m B , which shows that we only have taus with h = 1 in the final state, giving a polarization P τ = 1 in the SM.

Final taus polarization in B → π decays
In section IV we presented the decay B − → − The amplitude is finally written as in (21) Here p 1,2 are the 1,2 momenta, f B and f π are the B and π mesons constants and C (I) are the terms corresponding to the hadronic B current in the N production vertex. On the N decay side we have the terms corresponding to the hadronic π current, We then take the squared amplitude, and as we did in A 1, we use the expressions in eq.