Inclusive semileptonic $\Lambda_b$ decays in the Standard Model and beyond

Inclusive semileptonic decays of beauty baryons are studied using the heavy quark expansion to ${\cal O}(1/m_b^3)$, at leading order in $\alpha_s$. The case of a polarized decaying baryon is examined, with reference to $\Lambda_b$. An extension of the Standard Model effective Hamiltonian inducing $b \to U \ell {\bar \nu}_ \ell$ transitions ($U=u,\,c$ and $\ell=e,\,\mu,\,\tau$) is considered, which comprises the full set of D=6 semileptonic operators with left-handed neutrinos. The effects of the new operators in several observables are described.


Introduction
The observations of anomalies in b → c semileptonic exclusive decays of B mesons, with hints toward possible violation of lepton flavour universality (LFU) 1 , require new analyses of related processes involving heavy hadrons with a single b quark, to enlarge the set of observables suitable to test the Standard Model (SM) predictions. The inclusive semileptonic modes are theoretically appealing, since the nonperturbative effects of strong interactions, which necessarily must be taken into account, can be systematically considered by an expansion in the inverse heavy quark mass [2,3]. The expansion involves a set of long-distance hadronic matrix elements of operators of increasing dimension, which can be classified and parametrized. For each term in the heavy quark expansion perturbative QCD corrections can also be computed at increasing order in α s , therefore a double expansion in 1/m Q and α s is obtained. Improving the control of QCD effects, in the inclusive as well as in the exclusive processes, is the premise to disentangle the origin of the observed anomalies.
The present study is devoted to the inclusive b → c, u semileptonic modes of b-flavoured baryons, in particular Λ b → X c,u −ν . The formalism is developed for a generic baryon, therefore it can also be applied to Ξ b and Ω b . In our study the heavy quark mass expansion is considered at O(1/m 3 Q ), and the parametrization of the baryon matrix elements relevant at this order is provided. Moreover, the case of polarized baryon decays is considered at this order, the unpolarized case being recovered averaging over the initial baryon polarizations. The semileptonic transitions are analyzed in the Standard Model and in an extension of the SM effective weak Hamiltonian comprising vector, scalar, pseudoscalar, tensor and axial operators. Such Hamiltonian densities have been scrutinized in connection with the flavour anomaly problem, considering B meson exclusive modes, see, e.g., [4][5][6], but less information is available about their impact on inclusive observables [7][8][9][10].
For Λ b the decay rates and the angular distributions can be considered, although the latter measurements are experimentally challenging. Moreover, the systematic study of New Physics (NP) effects for polarized and unpolarized Λ b would provide a wealth of information. The prime purpose of such investigations is to identify the correlations between different processes induced by the same short-distance transition. If the B anomalies are due to new interactions, correlated pattern of same size effects must be observed in B and Λ b decays: we aim at describing such patterns, starting from the same extended low-energy Hamiltonian scrutinized for B and choosing the same benchmark points studied in that case, so that the size of the possible deviations from SM in the meson and baryon case can be compared. We shall provide the formulae for various observables, they can be used in experimental simulations to assess the sensitivity to the NP operators.
We have to say that measurements of the Λ b polarization at LHC give results compatible with zero [31][32][33][34], which means that the b quarks hadronizing in Λ b are mainly produced by QCD processes. However, a sizable longitudinal Λ b polarization is expected for b quarks produced in Z and top quark decays, as shown by the measurements at LEP [35][36][37]. For this reason, the effects beyond the Standard Model (BSM) in the polarized case have been scrutinized for the exclusive Λ b → Λ c −ν modes [38][39][40][41], in addition to the case of unpolarized baryon [42][43][44][45][46][47].
The plan of our study is as follows. In Sec. 2 we introduce the semileptonic b → c, u effective Hamiltonian, which generalizes the SM one by the inclusion of the set of D = 6 fourfermion semileptonic operators weighted by complex coefficients. The heavy quark expansion (HQE) to describe the inclusive process H b (p, s) → X c,u −ν is discussed in Sec. 3 considering the terms up to O(1/m 3 b ). In Sec. 4 we construct the fully differential Λ b → X c,u −ν decay distributions in the case of polarized Λ b . In Sec. 5 we analyze several observables in SM and with the extended Hamiltonian density, at a benchmark point in the parameter space of the effective couplings to investigate the sensitivity to the new operators. The last Section contains the conclusions.
The appendices contain the ingredients developed in our analysis. In Appendix A we collect the baryon matrix elements relevant for the OPE at order 1/m 3 b , considering the spin of the baryon. In Appendix B we write the expressions of the structure functions for the hadronic tensor in SM and in the case of the extended Hamiltonian. Appendix C contains the coefficients appearing in the 1/m b expansion the full semileptonic decay widths, for the SM and for the generalized Hamiltonian.

Effective Weak Hamiltonian
We consider the inclusive semileptonic decay of a baryon H b comprising a single b quark with s the spin of the decaying baryon. We assume that the process is induced by the lowenergy effective Hamiltonian density which extends the SM one, H eff consists of D=6 four-fermion operators with complex lepton-flavour dependent coefficients V,S,P,T,R . Only left-handed neutrinos are considered. U can be either the u or the c quark, V U b is the corresponding Cabibbo-Kobayashi-Maskawa (CKM) matrix element. V U b and V are independent parameters: the product V U b (1 + V ) is not a mere redefinition of the SM V U b , due to the lepton-flavour dependence of V . A comment is in order, concerning the operator with right-handed (RH) quark vector current O R = (Ū γ µ (1 + γ 5 )b)(¯ γ µ (1 − γ 5 )ν). This operator is in the set of D = 6 operators constituting the effective Hamiltonian (2), and has been previously considered [48][49][50][51][52][53][54]. We shall give analytic formulae that comprise its contribution, providing general expressions for the Λ b fully differential decay distributions and for the integrated distributions. However, in the Standard Model Effective Field Theory the only D=6 operator with a RH quark current, invariant under the SM gauge group, is nonlinear in the Higgs field: i Ū R γ µ b R (H † D µ H), with D µ the electroweak (ew) covariant derivative, H the SU(2) Higgs doublet,H i = ij H j * , and ij the totally antisymmetric tensor [55][56][57]. At the ew symmetry breaking scale this operator modifies the W U b coupling, but the resulting low energy four-fermion O R operator in the effective b → c, u semileptonic Hamiltonian does not violate LFU. Hence, in the framework of this effective field theory it is not involved in B flavour anomalies, and it has been omitted in several analyses [58][59][60]. Modifications of the W U b vertex are connected to modifications of the quark Z vertices, which are tightly constrained by the electroweak precision observables [57,61]. Stringent bounds can also be obtained from different processes [62,63]. For these reasons we shall not include O R in our phenomenological analysis, since this would require a dedicated study beyond the purposes of the present work. The Hamiltonian (2) can be written as with C 1 = (1 + V ) and C 2,3,4,5 = S,P,T,R . J (i) M indicates the hadronic and L (i)M the leptonic current in each operator, M generically denotes the set of Lorentz indices contracted between J and L. The SM Hamiltonian corresponds to i = 1 and V,S,P,T,R = 0. We keep the mass of the charged lepton = e, µ, τ different from zero.

Inclusive decay width
The decay width of the processes (1) is given by where G F is the Fermi constant, q = p + p ν the lepton-pair momentum, and dΣ the phase The hadronic tensor (W ij ) M N is obtained from the discontinuity of the forward amplitude across the cut corresponding to the process (1): T ij and W ij can be computed exploiting an operator product expansion (OPE) with expansion parameter the inverse b quark mass [2,3]. To construct the OPE, the hadron momentum p = m H v, with four-velocity v, is expressed in terms of the heavy quark mass m b and of a residual momentum k: still defined in QCD and satisfying the equation of motion: where P + is the velocity projector P + = 1+ v 2 . In terms of b v (x) one has: withĴ (i) containing the field b v . The heavy quark expansion is obtained from with p X = m b v + k − q and S U (p X ) the U quark propagator. Replacing k → iD, with D the QCD covariant derivative, the U quark propagator can be expanded: where S This expression involves H b matrix elements of QCD operators of increasing dimensions, written as = Tr (Γ with a, b Dirac indices. The hadron matrix elements can be expressed in terms of nonperturbative parameters, the number of which increases with the dimension of the operators. The expansion to order O(1/m 3 b ) requires A method to compute M µ 1 ...µn is exploited in [64] for B meson, and more parameters than those listed in (14)(15)(16)(17) are needed for n = 4. The order n = 5 has also been analyzed [65]. For a heavy baryon, the dependence on the spin four-vector s µ must be kept in (13). This is important since, for hadrons with spin, considering the hadron polarization leads to interesting observables to analyze.
In Appendix A we collect the expressions of the matrix elements needed for the expansion at O(1/m 3 b ) keeping the s µ dependence. The computation procedure is described in [64]. One starts from the highest dimension operator, which in our case n = 3 has dimension 6, and determines it in the static heavy quark limit, replacing b v (x) → h v (x), the heavy quark field defined in the heavy quark effective theory (HQET).
can be expanded over the set of 16 independent Dirac matrices. However, in HQET it is given in terms of only two Dirac structures, P + andŜ µ = P + γ µ γ 5 P + , an observation which simplifies the parametrization [66]. On the other hand, the matrix elements of lower dimension operators are computed in QCD expanding over the full set of Dirac matrices. The coefficients of Dirac structures in the d dimension matrix element are recursively computed from the d + 1 terms, and Eqs.(14)-(17) are used.
The parameters in Eqs. (14)- (17) are denoted by a hat to distinguish them from the corresponding parameters defined in HQET, with b v replaced by h v . Forμ 2 π ,ρ 3 D ,ρ 3 LS the difference between the two definitions involves terms appearing at O(1/m 4 b ), hence in our casê LS , a combination often present in our expressions. The formalism is suitable for the analysis in the Standard Model and in NP with the Hamiltonian (2). Our results are obtained for non-vanishing charged lepton mass, at order 1/m 3 b in the HQE, in the case of a polarized baryon and with all operators in (2) taken into account. In the existing literature one or more of the above points are relaxed. For the inclusive semileptonic B decays and non vanishing lepton masses, NP operators have been considered at order 1/m 2 b in [10], and we agree with those results at that order after taking the spin-average in our expressions. V + A and S − P operators have been studied at the leading order in the 1/m b expansion in [8], while a V + A operator has been considered for the mode B → Xτν τ performing the HQE at O(1/m 2 b ) in [67]. The hadronic tensor has been computed by an OPE in terms of operators comprising the HQET field h v in [68]. In this analysis the polarized Λ b inclusive semileptonic decay is considered at order 1/m 2 b in SM for massless leptons, the case of massive leptons at the same order in 1/m b is studied in [69]. We agree with such results at that order in the b mass expansion. As a last remark, in b → u semileptonic transition we neglect weak annihilation contributions, which mainly affect the endpoint region of the charged lepton energy spectrum [70].
Using the matrix elements M µ 1 ...µn collected in Appendix A the hadronic tensor can be computed. It is expanded in Lorentz structures depending on v, q and s. The related invariant functions are given in Appendix B for the Standard Model and for the effective Hamiltonian Eq. (2).

Decay distributions
For the H b (v, s) → X(p X ) − (p )ν (p ν ) transition the four-fold differential decay distribution is given by with p = (E , p ) and θ P the angle between p and s in the H b rest frame. The structure functions in which the hadronic tensor is expanded depend on q 2 and v · q. The various decay distributions are obtained integrating (18) over the phase space [71]. To compute the spectrum in q 2 and in the charged lepton energy E the order in the integration must be specified. Integrating first in E , the integration limits are The replacement 1 π in the hadronic tensor can be used to integrate over v · q. The last q 2 integration is for To compute the charged lepton energy spectrum one integrates in a different order [72]. The first integration over v · q is in the range where E ± = E ± E 2 − m 2 . Then one integrates over q 2 with integration limits The range for the last integration in E is Keeping the dependence on cos θ P , the corresponding double decay distributions are obtained. Notice that the kinematics involves the quark masses, the dependence on the decaying hadron is contained in the matrix elements of the OPE operators. However, the OPE breaks down in the endpoint region of the spectra, as signaled by singularities as the derivatives of the δ function. Such terms must be resummed in a H b shape function. Convolution of the distributions with such a function smears the spectra at the endpoint and transforms the phase space boundaries from the partonic to the hadronic kinematics: , with m X the mass of the lightest hadron containing the U quark produced in the decay. We do not include the effects of the shape function, the profile of which is not known in the baryon case, keeping in mind that the OPE results loose reliability in the endpoint region.
Expanding the tensor T ij in invariant functions, as provided in Appendix B, the fully differential distribution is obtained upon contraction with the leptonic tensor. We express the distribution as In this expression the first term is ImT a and G a = 1 π ImS a with the index a = 1, 2, . . Let us consider the other terms in Eq. (25) for the NP contributions. Considering the scalar and pseudoscalar operators, we have: with W S(P ),1 and G SP,1 obtained from the imaginary parts of the functions T and S in (B.14)-(B.16). From the interference terms, we have: Continuing with the distributions, we have: with W T i and G T i from the imaginary parts of the functions T and S in (B.28)-(B.60); with W SM T,i and G SM T,i from the imaginary parts of the functions T and S in (B.62)-(B.70); have the same form of Eq. (26) with suitable substitutions: d 4 Γ 55 is obtained from Eq. (26) replacing ImS Ra with the functions T R and S R collected in (B.77). In the case of d 4 Γ 15+51 the replacements are: with the functions T SM R and S SM R collected in (B.78)-(B.86). Analogously, the distributions have the same form of Eq.
have the same form of Eq. (31) with the substitutions: The above expressions can be used to compute all double and single decay distributions. We do not present the lengthy formulae here, but only give the full decay width, which can be cast in the form: The index i indicates the contribution of the various operators and of the interferences: i = SM, S, P, T, R, SP, SM S, SM P , SP, ST, SM T , and SM R, SR, P R, T R. The coefficients C (i) are collected in Appendix C and contain the couplings V,S,P,T,R in the effective Hamiltonian. All C For the SM terms in Eq. (36) various perturbative corrections are known. The leading electroweak correction A ew = 1.014 is a multiplying factor. QCD corrections are known at O(α 2 s ) for the leading term and at O(α s ) for the 1/m 2 b terms in (36), and can be included following, e.g., [73][74][75][76][77][78]. In ratios of decay widths involving different lepton species such corrections largely cancel out. We do not include QCD corrections in the decay distributions analyzed in the following.

Numerical results
In our numerical study we use the heavy quark masses in the kinetic scheme m kin b (µ = 0.75 GeV) = 4.62 GeV, m kin c (µ = 0.75 GeV) = 1.20 GeV, and the up quark mass in the MS scheme m u (2 GeV) = 2.16± 0.49 0.26 MeV [79]. In the case of B mesons the HQE parameters are constrained fitting the measured lepton energy and the hadronic mass distributions and their moments in B → X c ν decay [78]. For Λ b only few theoretical estimates of µ 2 π (Λ b ) exist [80]. A relation between µ 2 π (Λ b ) and µ 2 π (B) in terms of the measured mass differences between beauty and charmed mesons and baryons can be exploited [81]: (m B,D is the spin-averaged B ( * ) and D ( * ) mass), to obtain µ 2 π (Λ b ) from the value of µ 2 π (B). Moreover, the approximation ρ 3 can be adopted, increasing the uncertainty on ρ 3 D (Λ b ) with respect to the value for B. In our analysis we use: The HQE parameters µ 2 G and ρ 3 LS are sensitive to the total angular momentum of the light degrees of freedom in the hadron. For Λ b they vanish since the light degrees of freedom have spin zero, but for other baryons they are different from zero. For this reason we include the contributions involving such parameters in the various expressions in the Appendices, so that they can be used for different heavy hadrons.
The description of NP effects requires input values for the couplings V,S,P,T,R in the Hamiltonian (2). As anticipated, in the phenomenological analysis we do not consider the contribution of the operator O R , hence we set R = 0. For U = u, allowed regions for the other couplings have been determined from the analysis of purely leptonic B decays and of semileptonic B transitions to π and ρ(770) [82]. Accordingly, for b → u µν µ we set the benchmark  [59]. At odds with the B case, at present there is not enough experimental information on Λ b decays to restrict the ranges of the effective couplings. For b → c modes, semileptonic Λ b transitions to Λ + c , Λ + c π + π − , Λ c (2595), Λ c (2625), Σ c (2455) 0 π + and Σ c (2455) ++ π − have been observed. The branching fractions are measured for the modes into Λ c baryons, and the result Using |V cb | = 0.042 and |V ub | = 0.0037, together with , we obtain the inclusive Λ b branching fractions for the two quark transitions and for final τ and µ lepton. The results for the Standard Model, for the central value of the parameters and neglecting QCD corrections, are collected in Table 1.
A remark about the various sources of uncertainties is in order. In a first-principle computation, such as the one we have described here, the theoretical uncertainties are connected to the quark masses, to the hadronic parameters, to the perturbative corrections and to the size of next-order terms in the heavy quark expansion. All such uncertainties can be reduced in a systematic way. This is the case, in particular, of the values of the hadronic parameters, the knowledge of which can be improved using nonperturbative QCD methods, such as QCD sum rules and lattice QCD. For example, for µ 2 π (Λ b ) and ρ 3 D (Λ b ) we have quoted an uncertainty of 20% and 50%, respectively, which could be reduced by dedicated QCD analyses. For other baryons, such parameters are even less known and deserve new studies. On the other hand, the sensitivity to NP effects of the observables we have described needs to be assessed in the actual experimental conditions. In this respect, the analytic formulae we have provided can be used, e.g., to scrutinize by appropriate simulations the individual effects of the various low-energy operators in (2), when the experimental analyses are planned.

Observables in the
for the SM and NP at the chosen benchmark point is shown in Fig. 1 for the muon and for the τ final state. In the case of NP there is an enhancement of the distribution, more pronounced in the τ case, for charged lepton energy E µ 1.7 GeV and E τ 2.1 GeV. The charged lepton energy spectrum is useful to assess the role of the various terms in the 1/m b expansion. In Fig. 2 we show the result for the muon and the τ case. The impact of the next-to-leading and next-to-next-to-leading corrections in the HQE is higher for large values of E ( = µ, τ ), excluding the end-point region where the expansion breaks down. In the case of τ the corrections affects a wider energy range. The parametric hierarchy between 1/m 2 b and 1/m 3 b corrections is numerically confirmed. Comparison of the SM prediction (at O(1/m 3 b )) to NP at the benchmark point is provided in Fig. 3, where the NP enhancement already observed in the double distribution is evident, in particular for the τ mode. The enhancement due to NP can also be observed in the q 2 spectrum, Fig. 4: in the muon mode the impact is larger for smaller values of q 2 , while in the τ modes the spectrum displays an enhancement in almost all the q 2 range.
A significant sensitivity to NP is found in the cos θ P distribution displayed in Fig. 5. The dependence of dΓ d cos θ P on cos θ P is linear, and NP contributions modify both the slope and the intercept of the curve. In principle, a measurement of few points in the distribution would allow to access NP. This is confirmed by the comparison of different scenarios, corresponding to different benchmark points. Fig. 6 shows how the various operators have a different impact on the intercept and slope of the distribution. In particular, the tensor operator produces a large deviation from SM.
The top and bottom panels refer to = µ and = τ , respectively, the left and right plots to the Standard Model and to NP at the benchmark point.  Figure 3: Charged lepton energy spectrum for Λ b → X c ν , with = µ (left) and = τ (right).
The solid line is the SM result, the dashed line the result for NP at the benchmark point.
line is the SM result, the dashed line the NP result at the benchmark point.  Figure 6: of the effective couplings. The line NP(BP1) corresponds to τ T chosen in this paper. The lines NP(BP2), "only τ V " and "only τ S and τ P " correspond to the values of the couplings fixed in ref. [59], see the text.

5.2
Λ b → X u ν mode b → u transition displays similar features. The enhancement due to NP appears in the double differential spectra in Fig. 7, although in this case it is similar in the µ and τ mode. The various terms in the HQE alter the lepton energy spectrum for large energy, as shown in Fig. 8. NP affects a wide E range, with a similar impact for the muon and τ modes, Fig. 9. The enhancement in the q 2 spectrum, displayed in Fig. 10, is lower than in the decay to charm. The distribution in cos θ P is sensitive to NP also in this mode.

Ratio
For inclusive semileptonic Λ b decays it is interesting to consider a ratio analogous to R(D ( * ) ) for B meson, to compare the τ and the muon mode using a quantity in which several theoretical uncertainties are canceled: For this ratio we obtain: As with the other quantities in this study, the ratios (39)- (40) are obtained at leading order in α s . O(α 2 s ) corrections have been included in the ratio R b (X c ) [83], showing that they are small and supporting the expectation that perturbative corrections cancel in the ratios to a large extent. Our results suggest a higher sensitivity of the charm mode to NP. It would be important to observe the correlation of this measurement with the results for B mesons.
As a last observable, we define another ratio sensitive to lepton flavour universality violating NP effects. It can be constructed from the distribution

The intercept of the distribution is
has a definite value in SM, and can deviate from it due to NP.
In SM we find: R c S = 0.1 and R u S = 0.08. A correlation between R Λ b (X U ) and R U S can be costructed. As an example, for Λ b → X c τν τ with the effective Hamiltonian extended including a tensor operator, we vary the couplings (Re( µ T ), Im( µ T )) and (Re( τ T ), Im( τ T )) in the regions determined in [5]. The correlation plot in Fig. 12 shows that the (challenging) measurement of the two ratios would separate SM from NP.

Conclusions
We have presented a reappraisal of the calculation of the inclusive semileptonic decay width of a heavy hadron, focusing on polarized Λ b . We present the expressions for the full differential    Our study improves the SM result, previously known at order O(1/m 2 b ) for a polarized hadron, providing the expressions of the hadronic matrix elements. This allows to analyze other b-flavoured baryon modes, as well as other inclusive processes. Moreover, the study supplies the elements for analyzing different operators in the effective weak Hamiltonian density.
Possible NP effects are systematically investigated in various distributions. In particular, in view of the tension in the ratios R(D ( * ) ) for B mesons, we have studied the analogous ratios R Λ b (X c,u ). Among other results, we have found that the dΓ d cos θ P distribution, linear in cos θ P , is sensitive to NP. The slope of the distribution, for a hadronic final state X c or X u , depends on the final lepton species, hence the ratio R c,u S of the slopes, for = τ vs = µ, is sensitive to possible lepton flavour universality violation. For Λ b → X c ν when the effective Hamiltonian includes a tensor operator, we have shown that a deviation from SM in R Λ b (X c ) is related to a deviation in R c S , an interesting, although challenging, correlation to investigate.

A Hadronic Matrix Elements
In this appendix we collect the hadronic matrix elements involved in the heavy quark expansion to O(1/m 3 b ). The relations are employed: The terms independent of the spin four-vector s µ agree with Ref. [64].

Dimension 6 operator
The matrix element is computed in HQET: with a, b Dirac indices. A (D6) and B (D6) are parity-even and parity-odd, respectively. Using the expansion we find: This gives the expression of the matrix element:

Dimension 5 operator
The matrix element, computed in QCD, can be expressed as: with A (D5) , C (D5) , E (D5) parity-even and B (D5) , D (D5) parity-odd. They can be expanded as: We obtain: The previous expressions allow us to write the matrix element:

Dimension 4 operator
The procedure for computing the matrix element in QCD, using the expansion in Dirac matrices, is analogous to the D = 5 case. The results is:

Dimension 3 operator
The matrix element computed in QCD reads: (A.10) The matrix elements can be related using the equation of motion for b v : for a generic Dirac matrix Γ. This allows to relate the coefficients of matrix elements of operators of different dimensions, providing a check of the results [84].

B Hadronic tensor for the Standard Model and for the extended Hamiltonian
We provide the tensor T ij for the b → U modes (U = u, c) for the Standard Model and for the effective Hamiltonian in Eq. (2), expanded in invariant functions. We provide their expressions for the single operators (Standard Model, S, P, T) and for the interferences.

• Standard Model
This case amounts to choosing i = j = 1 in Eq. (5) and J For a polarized baryon the corresponding tensor T µν SM can be expanded in terms of the functions T 1,...,5 and S 1,...,13 [68]: The 1/m b expansion of the these functions reads: with S 4 = 0 and S 5 = S 3 , and S 10,11,12,13 = 0.

• Scalar operator in H ef f
This case amounts to choosing i = j = 2 in Eq. (5). T S is expanded as T S = T S1 + (q · s)S S1 (B.14) with and S S1 = 0.

• Pseudoscalar operator in H ef f
The tensor is obtained choosing i = j = 3 in Eq. (5), and is expanded as The two functions in (B.16) are given by the corresponding ones in (B.14) replacing m U → −m U .
• Interference between the SM and the scalar operator in H ef f The tensor is obtained when (i, j) = (1, 2) and (2, 1) in Eq. (5). We denote the two contributions as T SM S and T SSM , respectively. Using the expansion and the analogous one for T SSM , we find: • Interference between the SM and the pseudoscalar operators in H ef f The tensor is obtained for (i, j) = (1, 3) and (i, j) = (3, 1) in Eq. (5). We denote the two contributions as T SM P and T P SM , respectively, with the expansion and analogously for T P S , we have T SP,1 = T P S,1 = 0 and

• Tensor operator in H ef f
This case amounts to choosing i = j = 4 in Eq. (5). The corresponding tensor T T can be expanded as: The various functions are given by: with S T 4 = 0, with T T 13 = S T 9 = S T 10 = S T 13 = 0, • Interference between the SM and the tensor operators in H ef f The tensor is obtained for (i, j) = (1, 4) and (i, j) = (4, 1) in Eq. (5). We denote the two contributions with T SM T and T T SM , respectively. We write: The results for the various functions read: with T SMT,6 = T SMT,8 = 0, Several functions vanish: S SMT, (2,4,5,6,7,8) = 0 and S SMT, (15,19,20,21,22,23,24,26)  • Interference between the scalar and tensor operators in H ef f The tensor is obtained for (i, j) = (2, 4) and (i, j) = (4, 2) in Eq. (5). We denote the two contributions as T µν ST and T µν T S , respectively. Writing we obtain:  3,4,8,9,10,11,12,13) S Ri = −S i (i = 1, 2, 5, 6, 7) (B.77) • Interference between the SM and the O R operators in H ef f and S SM R,(1,2,3,4,5,6,7,12,13) = 0. In addition we have: • Interference between the right-handed and the scalar operators in H ef f The tensor is obtained when (i, j) = (5, 2) and (2,5) in Eq. (5). We denote the two contributions as T RS and T SR , respectively. Using the expansion as in Eq. (B.17) changing T SM Si → T RSi and S SM Si → S RSi we find: and analogous relations in the case of the structures in T SR .
• Interference between the right-handed and the pseudoscalar operators in H ef f The tensor is obtained when (i, j) = (5, 3) and (3,5) in Eq. (5). We denote the two contributions as T RP and T P R , respectively. Using an expansion analogous to (B.24), substituting T SM P i → T RP i and S SM P i → S RP i , we find: • Interference between the right handed and the tensor operators in H ef f The tensor is obtained for (i, j) = (5, 4) and (i, j) = (4, 5) in Eq. (5), with the two contributions denoted as T RT and T T R , respectively. Using an expansion as in Eq. (B.61), with T SM T i → T RT i and S SM T i → S RT i , we find: In the formulae √ λ stays for λ(1, ρ, ρ ). Factorizing the effective couplings in the Hamiltonian Eq. (3), we define for A = 0, µ 2 π , µ 2 G , ρ 3 D : C A for i=S, P, T, R, and C Our results for SM agree with [8] in the case of massless leptons and differ for the coefficient C The result for C (SM ) ρ 3 D coincides with that recently reported in [85,86].