$R_{D^{(*)}}$ and survival of the fittest scalar leptoquark
Damir Bečirević Svjetlana Fajfer Nejc Košnik Lovre Pavičić

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 and survival of the fittest scalar leptoquark
1.5cm.85Damir Bečirević^a, Svjetlana Fajfer^b,c, Nejc Košnik^b,c, Lovre Pavičić^c
1.3 cm
^a IJCLab, Pôle Théorie (Bat. 210), CNRS/IN2P3 and 
 
Université, Paris-Saclay, 91405 Orsay, France.

0.25 cm
^b Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia.


0.25 cm
^c 
J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia.


1.65cm



0.85cm

Motivated by the long-standing discrepancy in lepton flavor universality ratios R_D and R_D^∗ we assess the status of scalar leptoquark states R_2, R_2 and S_1 which can in principle provide a desired enhancement of ℬ(B→ D^(∗ )τν) in a minimal setup with two Yukawa couplings only. We consider unavoidable low-energy constraints, Z-pole measurements as well as high-p_T constraints. After setting mass of each leptoquark to 1.5TeV we find that of all considered states only S_1 leptoquark, coupled to both chiralities of leptons and quarks, is still a completely viable solution while the scenario with R_2 is in growing tension with Γ(Z →ττ) and with the LHC constraints on the di-tau tails at high-p_T. We comment on the future experimental tests of S_1 scenario. 





















§ INTRODUCTION


Experimental observations indicating the lepton flavor universality violation (LFUV) in the exclusive processes, based on transitions b→ cℓν and b→ sℓℓ, provided a huge boost in the high energy physics community to build a model of physics beyond the Standard Model (BSM) which would capture the effects of LFUV while remaining consistent with many experimental tests of the Standard Model (SM). In that respect the scenarios in which the SM is extended by one or more 𝒪(1 TeV)  leptoquarks proved to be the most attractive ones and a huge amount of work has been invested in (i) understanding the possible experimental signatures of such scenarios, and (ii) to figure out the ultraviolet (UV) completion of the proposed models. A practical advantage of the models which involved only scalar leptoquarks is that the resulting theory remains renormalizable and therefore the loop processes can be computed and results compared with experiments. However, no single scalar leptoquark could describe both types of LFUV and therefore the viable models necessarily involved at least two scalar leptoquarks <cit.>. An alternative to that situation was to use the simplest vector leptoquark, U_1, which could accommodate both types of LFUV, but in contrast to the models with scalar leptoquarks this scenario is not renormalizable and therefore the loop processes were explicitly dependent on the cutoff, which then necessitates to specify the UV completion, meaning new assumptions, new parameters (couplings and masses of additional states) <cit.>. 
 
Recently, however, the hints of LFUV in the b→ sℓℓ modes, which for almost a decade indicated that the measured R_K^(∗ ) = ℬ(B→ K^(∗ )μμ)/ℬ(B→ K^(∗ ) ee) < 1, were reexamined and the newly revised values established by LHCb were found to be, R_K = 0.949(47) and R_K^∗ = 1.027(75), thus fully consistent with lepton universality <cit.>. 
The world average values of R_D^(∗ ) = ℬ(B→ D^(∗ )τν)/ℬ(B→ D^(∗ ) l ν), with l∈{e,μ}, remain 
R_D^(∗ )^exp> R_D^(∗ )^SM and a model with one 𝒪(1 TeV) scalar leptoquark is still a viable option to accommodate that experimental deviation. In this paper we revisit such models in a minimalistic setup. One, however,  has to acknowledge the emergence of another observable that showed a departure from its value predicted in the SM, namely ℬ(B^±→ K^±νν̅) =  2.40(67)× 10^-5 <cit.>. Even though we do not intend to accommodate that deviation with respect to the SM prediction, we will monitor that our model does not lead to ℬ(B^±→ K^±νν̅) at odds with the experimental observations. 







§ LEPTON UNIVERSALITY VIOLATING EFFECTS IN B→ C ΤΝ



Before discussing the scalar leptoquark scenarios, we first consider the low-energy effective theory (LEET) of b→ c τν transitions. We extend the usual LEET Lagrangian by including a singlet fermion N_R (right-handed neutrino) because it is needed in the models involving the R_2 leptoquark which can couple to N_R at tree level. By focusing on the terms relevant to our paper we have:

    L_b→ cτν = 
    -2 √(2)  G_F V_cb   [(1+g_V_L)(c̅_Lγ^μ b_L)(τ̅_Lγ_μν_τ L)+g_V_R(c̅_Rγ^μ b_R)(τ̅_Lγ_μν_τ L) 
         
    +g_S_L(c̅_R b_L)(τ̅_Rν_τ L)+g_T(c̅_Rσ^μν b_L)(τ̅_Rσ_μνν_τ L) + 
          +g_S_R (c̅_L b_R)(τ̅_L N_R)+ g_T (c̅_Lσ^μν b_R)(τ̅_L σ_μνN_R)] +h.c. .

In the framework of the SM effective theory (SMEFT), the τ-specific g_V_R can only stem from dimension-8 operator in the linear realisation of SU(2)_L for the Higgs <cit.> and we will not pursue this option in the rest of the paper.[E.g. g_V_R can arise in a R_2-R_2 model via mixing term between the two leptoquarks.]








 §.§ Matching to SMEFT



In the following we specify the matching and the renormalization group running between the SMEFT Wilson coefficients and the low-energy coefficients g_i of Eq. (<ref>), defined at scale μ=m_b. Above the electroweak scale the SMEFT Lagrangian consists of operators invariant under the full SM gauge group <cit.>:

    ℒ_SMEFT = 1/Λ^2∑_i   C_i  O_i .

Since we consider models with a single leptoquark a natural choice for  normalization scale is Λ=m_LQ. For the semileptonic processes with the SM neutrino the following operators are relevant:

    O^(1)_lequ
    prst   = (l̅^a_p e_r) ϵ^ab (q̅^b_s  u_t) ,     O^(3)_lequ
     prst = (l̅^a_p σ^μν e_r) ϵ^ab (q̅^b_s  σ_μν u_t) ,
    O^(1)_lq
    prst   = (l̅^j_p γ^μ l_r) (q̅_s γ_μ  u_t) ,     O^(3)_lq
     prst = (l̅_p γ^μτ^I l_r)  (q̅_s  γ_μτ^I q_t) .

Here l and q denote the doublet lepton and quark fields, respectively, u and e are singlet up-quark and charged-lepton fields. Matrices τ^I are the Pauli-matrices (I=1,2,3) acting on SU(2)_L, while a,b=1,2 are the indices of SU(2)_L doublets. Finally, the flavor indices are denoted by prst and we employ the diagonal basis for the left handed down-type quarks and for charged leptons.
The presence of light N_R requires to consider a more general effective theory, often referred to as N_R-SMEFT, that besides a full set of SMEFT operators includes all possible operators containing a singlet N_R <cit.>. As far as the contribution to  goes, we will employ two such N_R-SMEFT operators:

    O^(1)_Nldq 
    Nrst = (N̅_R l^a_r) ϵ^ab(d̅_s q_t^b) ,    O^(3)_Nldq 
    N rst = (N̅_R σ^μνl^a_r) ϵ^ab(d̅_s  σ_μνq_t^b)   .

The literature on N_R-SMEFT considers operator O^(1)_Nldq and its version with exchanged roles of l and q, O^(1)_Nqdl. These two operators are related to those given in Eq. (<ref>) via the Fierz identities.

The relations between SMEFT Wilson coefficients and the low-energy Wilson coefficients of Lagrangian (<ref>) are obtained by the renormalization group running and matching of the two theories.
Vector current Wilson coefficients C^(1),(3)_lq do not run and the coefficient g_V_L is obtained as:

    g_V_L(m_b) = -v^2/m_LQ^2V_cs/V_cbC^(3)_lq 
    ττ s b (m_LQ) - v^2/m_LQ^2 C^(3)_lq 
    ττ b b (m_LQ) ,

where we have assumed both possibilities of quark flavors at high energy scales.
On the other hand, scalar and tensor operators C_lequ^(1) and C_lequ^(3) renormalize and mix among themselves.
The low energy Wilson coefficients present in (<ref>) are obtained by the renormalization group running and matching onto the low energy effective theory (<ref>)
 
    g_S_L (m_b)     =   -v^2/2 m_LQ^21/V_cb(α_s(m_b)/α_s(m_t))^12/23(α_s(m_t)/α_s(m_LQ))^4/7  C^(1)*_lequ
    ττ b c(m_LQ)
            =-0.56  C^(1)*_lequ
    ττ b c(m_LQ)  ,
    
            g_T (m_b)   =  -v^2/2 m_LQ^21/V_cb(α_s(m_b)/α_s(m_t))^-4/23(α_s(m_t)/α_s(m_LQ))^-4/21   C^(3)*_lequ
    ττ b c(m_LQ)= -0.28 C^(3)*_lequ
    ττ b c(m_LQ) ,

obtained to leading order in QCD. The off-diagonal mixings, i.e. mixing of C^(3)_lequ to g_S_L and C^(1)_lequ to g_T, are driven by electroweak radiative contributions and represent only small corrections with respect to the QCD driven multiplicative renormalization. 
The invariance of QCD under parity implies that the chirality-flipped operators involving right-handed N_R have exactly the same QCD running, i.e.:

    g_S_R (m_b)     =
        -0.56  C^(1)*_Nldq 
    τ bs(m_LQ)  , 
    g_T (m_b)   =   
          -0.28   C^(3)*_Nldq
    τ b s(m_LQ)  .

The above discussion is based on 1-loop running only, but in actual computations we included the higher corrections to 4-loops <cit.>. The net effect is that the right hand side of Eq.(<ref>) for  g_S_L (g_T) gets enhanced (suppressed) by about 6% (4%). In other words, the relation g_S_L = ± 4  g_T valid at the μ=m_LQ,  becomes  g_S_L = ± 8.8  g_T when the 4-loop running is used.[Notice that the effect of 1-loop running at μ=m_b is g_S_L = ± 8  g_T . ]
The same applies to Eq. (<ref>) regarding the couplings g_S_R and g_T.





 §.§ R_D and R_D^*

The goal of this study is to establish whether or not any of the scalar leptoquarks, with a minimalistic set of Yukawa couplings, fits the current experimental world average of R_D and R_D^*. 
The experimental averages are steadily updated in Ref. <cit.>. The most recent values are:

    R_D^exp = 0.344(26) ,     R_D^*^exp = 0.285(12) ,

with a correlation coefficient ρ=-0.39.

The SM predictions for these quantities are plagued by systematic uncertainties arising from hadronic matrix elements, i.e. from the relevant form factors. In the case of D-meson in the final state the problem is easier to handle because only the vector current contributes, i.e. only two form factors are present, f_+,0(q^2), namely,

    ⟨ D(k) | s̅γ^μ b | B(p) ⟩   = [(p+k)^μ- m_B^2-m_D^2q^2q^μ] f_+(q^2) +m_B^2-m_D^2q^2 q^μ f_0(q^2) ,

which are equal at q^2 = 0, f_+(0)=f_0(0), where q=p-k and q^2 ∈ [m_ℓ^2, (m_B-m_D)^2], with ℓ∈{e,μ,τ}. That condition is extremely helpful when extrapolating the form factors that are accessible through lattice QCD at large q^2's down to low q^2's. Two such lattice QCD analyses <cit.> agree and their results are combined in Ref. <cit.>, which is used to make the SM prediction,

    R_D^SM= 0.293(8) ,

which is a little less than 2σ smaller than measured, cf. Eq. (<ref>).



The situation with R_D^* is more complicated. Both the vector and the axial currents contribute to the B→ D^∗ℓν̅ decay amplitude, which then involves 4 independent form factors:

    ⟨ D^∗(k) | c̅γ_μ(1-γ_5) b    | B(p) ⟩ = ε_μνρσε^∗νp^ρ k^σ2 V(q^2)m_B+m_D^∗ -iε_μ^∗(m_B+m_D^∗)A_1(q^2) 
    
    +    i(p+k)_μ (ε^∗· q)A_2(q^2)m_B+m_D^∗
    +i q_μ (ε^∗· q) 2 m_D^∗q^2[A_3(q^2)-A_0(q^2)] ,

where ε_μ is the D^* polarization vector. Notice that the form factor A_3(q^2) is not independent but related to A_1,2(q^2), via 
2 m_D^∗ A_3(q^2) = (m_B+m_D^∗) A_1(q^2)-(m_B- m_D^∗) A_2(q^2). It is also related to A_0(q^2) at q^2=0 as A_3(0)=A_0(0). 
 We prefer the parametrization given in Eq. (<ref>) to the alternative one occasionally used in literature <cit.>, because the form factors are dimensionless by definition (<ref>). Physical results are obviously independent of the parametrization used. 
The fact that there are many more form factors and only one constraint [A_3(0)=A_0(0)] make the lattice QCD computation of the B→ D^∗ form factors particularly challenging when extrapolating the results obtained for a few small three-momenta given to the lighter meson down to lower end of q^2 = (p-k)^2 ∈ [m_ℓ^2, (m_B-m_D^∗)^2]. Very recently, the results of three detailed lattice QCD computations of these form factors have been reported in Refs. <cit.>. When converted to the same set of form factors, such as those defined in Eq. (<ref>), it appears that they are quite consistent when it comes to the dominant form factor, A_1(q^2), but they do not agree for the other form factors. That disagreement, when extrapolated to low q^2's can lead to very different predictions. To avoid such a situation, we will proceed as follows. Since both quarks participating in this process are heavy, m_c,b≫Λ_QCD, it is reasonable to use the decomposition of  hadronic matrix elements in terms of the form factors defined in heavy quark effective theory, namely, 

    1/√(m_B m_D^∗)D^∗(k) | c̅γ_μ(1-γ_5) b| B(p)    =  ε_μνρσε^∗νv^ρ v^'σ h_V(w) -iε_μ^∗(w+1)h_A_1(w) 
         + i (ε^∗· v) v_μ h_A_2(w) 
    +i (ε^∗· v) v^'_μ  h_A_3(w)  ,

which are used to fit the experimentally measured angular distributions of the B→ D^∗ (→ Dπ) lν̅ decay (l∈{e,μ}). In that way the normalization and shapes of the above form factors is reconstructed  from the measured angular distribution. In the above expressions, v_μ = p_μ/m_B, v^'_μ = k_μ/m_D^∗, and w=v· v^' = (m_B^2+m_D^∗^2 - q^2)/(2 m_B m_D^∗). More specifically, with the educated assumptions regarding the shapes, as proposed in Ref. <cit.>, the following expressions have been used:

    h_A_1(w)   =  h_A_1(1)  [
    1 - 8 ρ^2 z + (53 ρ^2 -15) z^2 - (231 ρ^2 - 91) z^3 ]
     , 
    
    R_1(w)    =h_V(w)/h_A_1(w) = R_1(1) - 0.12 (w-1) + 0.05 (w-1)^2 , 
    
    R_2(w)    =h_A_2(w)/h_A_1(w)  = R_2(1) + 0.11 (w-1) - 0.06 (w-1)^2 ,

where z= (√(w+1)-√(2))/(√(w+1)+√(2)).
Therefore, only four parameters are to be obtained from the fit with experimental data: h_A_1(1), ρ^2, R_1(1) and R_2(1). After combining the experimental results of several experiments, cf. Ref. <cit.>, the following values have been quoted: 

    ρ^2 = 1.121(24),    R_1(1)=1.269(26),   R_2(1)=0.853(17),

while the value of h_A_1(1), being an overall multiplicative factor, is immaterial for the discussion of R_D^∗. The correlation matrix for these parameters is given in Ref. <cit.> and we used it in our numerical estimates. Note that the Belle II results of the above parameters <cit.> appeared after the release of the HFLAV review <cit.> but the reported results are fully consistent with the numbers quoted in Eq. (<ref>). 

The relations between the form factors given in Eqs. (<ref>,<ref>) and those in (<ref>) are: 

    A_1(q^2)     = √(m_B m_D^*)/m_B + m_D^*( w +1)  h_A_1( w) , 
    V(q^2)/A_1(q^2)    =[ 1 - q^2/(m_B + m_D^*)^2]^-1 R_1( w) ,
    A_2(q^2)/A_1(q^2)    =[1 - q^2/(m_B + m_D^*)^2]^-1 R_2( w) ,

where, again, w≥ 1 is the recoil momentum, q^2=m_B^2+m_D^*^2 - 2 m_B m_D^* w. The values of parameters given in Eq. (<ref>) are obtained from experimental studies of the angular distribution of B→ D^∗ (→ Dπ) lν̅, with l∈{e,μ}, for which m_l^2/m_B,D^2 is negligibly small and therefore the pseudoscalar form factor A_0(q^2) cannot be accessed. That form factor, however, contributes significantly to 
B→ D^∗τν̅. To get that information we will define

    R_0(w) = [ 1 - q^2/(m_B + m_D^*)^2] A_0(q^2)/A_1(q^2) ,

which is known at w_max (i.e. q^2=0), due to 

    A_0(0)/A_1(0)  =
    1/2m_D^*[ m_B+ m_D^* - (m_B- m_D^*) A_2(0)/A_1(0)] .

We thus have 

    R_0(w_max) = m_B + m_D^*/2 m_D^* - m_B - m_D^*/2 m_D^* R_2(w_max) =1.087(14),

where we used information from Eqs (<ref>,<ref>) to get the
last number. We need at least one extra point to have the information on the slope of the form factor ratio  R_0(w). To that end we can use the lattice QCD results which are computed at small w values (or equivalently, large q^2's). From the information provided in their papers, at w=1, we find:  A_0/A_1 = 1.423(53), 1.387(49), 1.186(72) for MILC/FNAL <cit.>, JLQCD <cit.> and HPQCD <cit.>, respectively. After taking the average, 

    R_0(1) = 4 m_B  m_D^*/(m_B + m_D^*)^2A_0(q^2_max)/A_1(q^2_max) =1.087(26) ,

which, when compared with Eq. (<ref>), indicates a flat behavior of R_0(w). If, instead, we took separately the average of lattice values for A_0(q^2_max) and of A_1(q^2_max), and then combined them in the ratio, we would have obtained R_0(1) = 1.095(31),
indicating a small negative slope in w-1. To capture these effects we will take:

    R_0(w) = 1.09 - 0.16 (w-1)

to which we attribute an error of 10%, to account for the spread of lattice data at w=1. With the ingredients described above we obtain

    R_D^*^SM = 0.247(2) ,

hence over 3σ smaller than the current experimental average given in Eq. (<ref>).




We should stress once again that our basic assumption is that the BSM physics can modify the coupling to τ-lepton while leaving the couplings to lighter leptons intact. This is why we could use the experimental information on the form factors. If that assumption is relaxed then one ends up very large error bar on R_D^∗, reflecting the disagreement among various lattice QCD results <cit.>. To consider the BSM scenarios captured by the Lagrangian (<ref>) we need three more form factors T_1,2,3(q^2) defined as, 

    D^∗(k)|c̅σ_μν b|B(p)    =  ϵ_μναβ{ -ε^*α(p+k)^β T_1(q^2) + ε^*αq^βm_B^2-m_D^∗^2  q^2[T_1(q^2)-T_2(q^2)] 
    . 
         .      + (ε^* q)p^α k^β2  q^2[ T_1(q^2)-T_2(q^2) - q^2  m_B^2 - m_D^∗^2 T_3(q^2) ] } .

They have been very recently computed in Ref. <cit.>. From that paper we extract (at μ=m_b):

    T_1(q^2)/A_1(q^2)    = 0.96 + 0.041   q^2/GeV^2 ,
    T_2(q^2)/A_1(q^2)    = 0.96 - 0.025   q^2/GeV^2 - 0.002   q^4/GeV^4 ,
    T_3(q^2)/A_1(q^2)    = 0.35 + 0.040   q^2/GeV^2   .

with an overall error of 6%, 4% and 30%, respectively.


With the information given above it is convenient to write:

    R_D^(∗ )/R_D^(∗ )^SM  =    |1+g_V_L|^2 +a_S^D^(∗ )(|g_S_L|^2 +|g_S_R|^2)
    +a_T^D^(∗ )(|g_T|^2 + |g_T|^2)
          + a_S V^D^(∗ ) [(1+g_V_L)  g_S_L^*] + a_T V^D^(*) [ (1+g_V_L)   g_T^*]    ,

where in the case of D in the final state, thanks to the lattice QCD results <cit.>, we obtain <cit.>:

    a_S^D= 1.08(1),   
        a_T^D=0.83(5),   
        a_SV^D=1.54(2),  
        a_TV^D=1.09(3).



Instead, for the case of D^* in the final state, and based on the discussion presented above, we have:

    a_S^D^*= 0.037(4),   
        a_T^D^*=8.56(35),   
        a_SV^D^*=-0.107(11),  
        a_TV^D^*=-2.91(11) .

Note that the lattice QCD results of the form factors T_1,2,3  significantly reduced the values of the coefficients a_T^D^* and a_TV^D^*, when compared to the values given in Refs. <cit.>.





 §.§ Collider constraints on leptoquarks


As already mentioned above, we attribute the deviations of the observed R_D^( ∗) (<ref>) with respect to their values predicted in the SM (<ref>,<ref>) to a coupling of new physics particles to the third generation of leptons. In particular, we focus on the scenarios in which the scalar leptoquark couples to τ and either to c- or to b-quark. We refer to those coupling as to Yukawa couplings. One should monitor that such couplings do not become too large so that they could give significant modification of the high dilepton mass tails of pp →τν, ττ processes. Both ATLAS <cit.> and CMS <cit.> have presented results of their studies of such Drell-Yan processes at high dilepton masses. In this work we use the HighPT package <cit.> which provides us with a built-in likelihood function of the leptoquark couplings for each of the leptoquarks. They are obtained after recasting the results of Refs. <cit.> to the scenarios discussed in this paper. The exception is R_2 and its coupling to N_R which has not been considered in HighPT. In Sec. <ref> we will explain  how one can simulate Drell-Yan effects of R_2 by those of R_2 coupled to ordinary neutrinos.

If light enough leptoquarks can be produced in pairs via QCD interactions. Single leptoquark production, on the other hand, 
is completely determined by the Yukawa couplings. To satisfy the upper bounds on the leptoquark-mediated cross sections determined by ATLAS and CMS we set the leptoquark mass to m_LQ = 1.5TeV for all cases <cit.>.




 §.§ B → K^(*)νν̅

While our goal is not to accommodate the recently observed deviation of the measured ℬ(B → K νν̅) <cit.> with respect to the SM prediction <cit.>, we should monitor that our scenarios do not get in conflict with the experimental bounds on ℬ(B → K^(∗)νν̅) <cit.>. 
To do so we consider the following effective Lagrangian:
    ℒ_eff^b→ sνν= √(2)G_F α_emλ_tπ   [ C_L^ij (s̅_Lγ_μ b_L)(ν̅_L iγ^μν_L j) + C_R^ij (s̅_Rγ_μ b_R)(ν̅_L iγ^μν_L j). ,
       . + C_L^NN (s̅_Lγ_μ b_L)(N̅_Rγ^μ N_R) + C_LR^iN (s̅_L b_R)(ν̅_Li N_R).
       . + C_RL^Ni (s̅_R b_L)(N̅_Rν_Li)]+ h.c. ,

where i=e,μ,τ. The SM contribution is characterized by a unique and flavor diagonal left-handed interaction, C_L,SM^ij = C_L,SMδ^ij, with C_L,SM = -6.32(7). Leptoquarks, considered in this work can contribute to operators with different Lorentz structures or with N_R included in the interactions.
We will use the expressions and ingredients from Refs. <cit.> to check on the experimental bounds <cit.>.




 §.§ Loop-induced constraints: Z →ℓℓ, νν and τ→ l νν̅

Since accommodating ^exp>^SM requires significant coupling of the thrid family of leptons to leptoquarks, we should monitor that the resulting ℬ(Z→ττ) and ℬ(Z→νν̅) remain within the experimental error bars<cit.>. To do so we employ the
expressions derived in Ref. <cit.> for all of the scalar leptoquarks and confront them with measured effective couplings g_V,A^τ of Z boson. Similarly, the couplings to neutrinos should remain consistent with N_eff = 2.9840(82) <cit.>. 

Modifications of the on-shell fermionic couplings to W are less precisely known and we do not consider them here. On the other hand, if the quark loop is charged (e.g. t̅ b or c̅ b), it will induce, via W-exchange, leptonic decays of τ <cit.>, which we do consider in our analyses.














§ LEPTOQUARK MODELS



In this Section we focus on three specific scenarios of SM extended by a presence of a single scalar 𝒪(1 TeV) leptoquark that could provide us with a plausible explanation of ^exp>^SM through couplings to the third generation of leptons. In doing so we consider the models with a minimal number of Yukawa couplings. Three such scenarios allow for couplings to bτ and/or cτ and will be discussed one by one in the following.  




 §.§ R_2

In terms of the SM quantum numbers,[In the notation we employ the quantum numbers correspond to (SU(3),SU(2)_L)_U(1)_Y of the SM gauge group.] the R_2 doublet of scalar leptoquarks corresponds to (3,2)_7/6, so that the electric charge of its components is Q=2/3 and Q=5/3. 
The Yukawa interactions are described via

    ℒ_R_2 =  y_R^ij Q̅_i^a e_j R_2^a +  y_L^ij u̅_Ri R_2^T,aϵ^abL_j^b + h.c. ,

where Q and L are the quark and lepton doublets, i.e. Q_i = [(V^†u)_i,  d_i]^T, L = [ν_ℓ, ℓ]^T, with V being the Cabibbo–Kobayashi-Maskawa (CKM) matrix. Notice that the left-handed neutrinos are in the flavor basis.
The key element in building a model is to specify the relevant Yukawa couplings. We opt for minimal number of parameters and in the down-quark and charged-lepton mass basis we choose: 

    y_R =
      [      0      0      0;      0      0      0;      0      0 y_R^bτ ] ,      y_L = [      0      0      0;      0      0 y_L^cτ;      0      0      0 ] ,

which leads to the interaction Lagrangian:
  
    L_R_2   = y^bτ_R V^*_j b (u_j P_R τ) R^5/3_2 + y^bτ_R (b P_R τ) R^2/3_2 
      -y_L^cτ (c P_L τ) R^5/3_2 + y_L^cτ (c P_L ν_τ) R_2^2/3 + h.c. .

This R_2 model brings a tree level contribution to  which, when matched to SMEFT at the scale μ = m_R_2 = 1.5 TeV, is parametrized by two operators, O^(1)_lequ and O^(3)_lequ, the coefficients of which satisfy

    C^(1)_lequ
    ττ b c(m_R_2)     = 4  C^(3)_lequ
    ττ b c(m_R_2) = -y_R^bτy_L^c τ^* /2 .

In terms of the LEET (<ref>) the relevant couplings are g_S_L and g_T. Their relation at the matching scale (<ref>) g_S_L(m_R_2) = 4 g_T(m_R_2) is modified by the effects of running down to μ = m_b (<ref>), and reads:

    g_S_L(m_b) =  8.8× g_T(m_b) .





It is well known that accommodating ^exp>^SM in a scenario with R_2 necessitates introducing a complex coupling g_S_L(μ=m_b) <cit.>, in a way consistent with Eqs. (<ref>,<ref>). This is shown in the left panel of Fig. <ref>. That also means that the product of two Yukawa couplings has to be complex,[We choose to attribute the complex phase to y_R^bτ = |y_R^bτ| e^iφ. For alternative constraints on this CP-violating phase see Ref. <cit.>.]

    g_S_L(m_b) = 0.60×1/2  |y_R^bτ y_L^cτ|e^iφ  .

The best fit values to  are

    |g_S_L(m_b)|= 0.78 ,    φ=± 1.96 (=± 112^) .


In the right panel of Fig. <ref> we show the other important constraints, but this time in the plane span by the moduli of our Yukawa couplings (|y_R^bτ|,|y_L^cτ|). We see that the 2σ constraints arising from experimental studies of the di-tau and mono-tau high-p_T tails at the LHC are at odds with the values of Yukawa couplings preferred by R_D^(∗ )^exp. Otherwise the 2σ constraint stemming from consistency with the measured Γ(Z→ττ) [or, better, g_V,A^τ <cit.>] would select larger values of  |y_L^cτ| while keeping moderately small |y_R^bτ|. 
We reiterate that we use m_R_2=1.5 TeV, consistent with the lowest mass allowed for a leptoquark decaying mostly to cτ which is experimentally set to be 1.3 TeV <cit.>. Varying m_R_2 does not change our conclusion, which is that the constraints on Yukawa couplings deduced from experimental studies of pp →ττ, τν (+ soft jets) at high-p_T's are incompatible with those obtained from R_D^(∗ )^exp. Obviously, that statement is valid to 2 σ and not to 3σ or more. It is therefore difficult to make a strong statement on this issue because the 
 uncertainties on the high energy end, related to the reconstruction of τ leptons can be questioned <cit.>, and those on the low energy end related to the form factors used to compute R_D^*, may change once they are fully understood and the results of various lattice collaborations agree. Note, however, that the effect of propagation of R_2 has been properly taken into account. Notice also that dedicated experimental searches for a leptoquark signal in pp →ττ could definitively exclude this scenario.


Since we have aligned the couplings with the down-quark mass, the tree level flavor changing neutral semileptonic processes b → s or b → d are forbidden. We thus cannot expect significant effects contributing b → s νν or b → s ℓℓ processes. The corresponding b → d rare transitions are further CKM suppressed. 
 






 §.§ R_2


Another interesting scenario that could potentially describe the deviation ^exp>^SM is the one with a doublet of R_2 scalar  leptoquarks. In terms of the SM quantum numbers, R_2 is specified by (3,2)_1/6, and it is peculiar because besides its coupling to a lepton doublet, it can also couple to a lepton singlet state, N_R, namely:

    L=-y^ij_L d^iR_2^a ϵ^abL^j,b+y^iN_RQ^i,aR_2^a N_R+h.c. .

We again opt for a minimal setup and fix our model by choosing as non-zero Yukawa couplings:

    y_L =
      [      0      0      0;      0      0      0;      0      0 y_L^bτ ] ,     y_R = [      0; y_R^sN;      0 ] ,

needed to enhance ^SM
As before, the coupling to N_R is in the down-quark basis. For the sake of simplicity we introduce only one massive sterile state N_R but such that its mass is negligible with respect to all the other particles participating in B→ D^∗τ N_R. The interaction Lagrangian, in the mass basis of quarks, then reads:

    L=   -y^bτ_L (bP_L τ) R_2^2/3 +y^bτ_L (bP_L ν) R_2^-1/3+
       +y^sN_R (s P_R N_R) R_2^-1/3+
        y^sN_R V_js(u_j P_R N_R) R_2^2/3 +h.c. ,

where the superscript in R_2^Q denotes the leptoquark's electric charge. It is important to emphasize that N_R is not just a chirally flipped projection of the ordinary neutrino, but a completely different particle. As such, it entails the new physics contribution that does not interfere with the SM, nor with the BSM contribution involving left-handed neutrinos in the final state. In other words, the contribution involving N_R always increases the decay width with respect to the SM. Schematically, the branching fraction of a decay mode

    ℬ ∝ |A_SM+A_NP^ν_L|^2+|A_NP^N_R|^2 .

In this model there are two tree level contributions to  which are described by two N_R-SMEFT coefficients (<ref>). It is a simple matter to read them off at the matching scale μ = m_R_2 and get:

    C^(1)_Nldq 
    Nτ b s( m_R_2) = 4  C^(3)_Nldq 
    Nτ b s( m_R_2) = -1/2 y_R^sN y_L^bτ* .

Running from the matching scale down to μ = m_b is then made by means of Eq. (<ref>) so that one finally arrives at the low energy effective theory (<ref>) with a scenario g_S_R(m_b) = 8.8×g_T(m_b).


Besides , one should keep the width of B_c under control. This is usually enforced by requiring ℬ(B_c→τν)≤ 30% which is indicated in Fig. <ref>. Constraints from pp →ττ are obtained by the HighPT package in the leptoquark mediator mode. Conversely, N_R coupled to R̃_2 is not included within the HighPT package. Thus, in order to quantify the agreement of R̃_2-mediated pp → N_R τ with experimental data we first observe that in partonic processes pp →τ E_T,miss there is no interference between the SM and the amplitude with N_R. Thus, the signature of a R̃_2^2/3-mediated process, e.g. u_L b̅_R →τ_L N̅_R is experimentally indistinguishable from R_2^2/3-mediated process u_R b̅_L →τ_R ν_τ, provided the couplings and the masses of the two LQs are equal. In this way, by carefully adjusting R_2 couplings, we can fully assess the agreement of R̃_2 couplings with pp →τν_τ experimental searches. We show the combined pp →ττ,τν constraints in Fig. <ref>.

More problematic, however, is the fact that this model generates a huge contribution to ℬ(B→ K^(*)νν), so that accommodating the current ^exp would result in ℬ(B→ K^(*)νν) orders of magnitude larger than the current experimental bounds. For that reason this model should be discarded.

As a side remark, we can turn the above argument around and claim that 
this scenario can be used to describe the recently measured ℬ(B→ K νν) <cit.>, 3σ larger than predicted in the SM, should the improved measurement of ^exp lower the current average.








 §.§ S_1

The scalar singlet, often referred to as S_1, is the last of the three possible scalar leptoquarks that can accommodate the experimental hint of LFUV, ^exp >^SM, with a minimal number of Yukawa couplings. In terms of the SM quantum numbers this leptoquark is described by (3̅,1)_1/3. Its peculiarity is that it can couple to two fermion doublets or to two singlets, namely,

    L_S_1 = y_L^ij Q^C,a_i ϵ^ab  L_j^b S_1 +  y_R^ij u^C_i e_j S_1 + h.c. .

For the minimal setup of Yukawa couplings we choose,

    y_L =
      [      0      0      0;      0      0      0;      0      0 y_L^bτ ] ,      y_R = [      0      0      0;      0      0 y_R^cτ;      0      0      0 ] ,

defined in the mass basis of the down-type quarks, as before. In its more explicit form the above Lagrangian reads:

    L_S_1 = y_L^bτ  V^*_ib(u^C_i P_L τ) S_1 - y_L^bτ  (b^C P_L ν_τ) S_1 +y_R^cτ  (c^C P_R τ) S_1 + h.c. .

With the above choice of couplings one has, at the matching scale μ = m_S_1 = 1.5 TeV,

    C^(1)_lequ
    ττ b c(m_S_1)     = -4  C^(3)_lequ
    ττ b c(m_S_1) = -y_L^bτy_R^c τ^* /2 ,

i.e. the non-zero low energy effective couplings are g_S_L and g_T, with 
    g_S_L(m_S_1)= -
    v^2/4 V_cb y_L^bτy_R^c τ^* / m_S_1^2  .

Their relation at the matching scale g_S_L=-4 g_T, after running down to μ = m_b (<ref>), becomes:

    g_S_L(m_b) =  -8.8× g_T(m_b) .

In the S_1 case, and with the couplings chosen as in Eq. (<ref>), one also gets a non-zero g_V_L, namely

    g_V_L= 
    v^2/4 V_cb V_cb  | y_L^bτ|^2 / m_S_1^2  .


Contrary to the R_2 case, in this situation one can find a region in which all the constraints overlap for real values of couplings, which is shown in Fig. <ref>. As before, the measured Γ(Z→ττ) represents a powerful constraint on the Yukawa couplings but this time it is interesting to note that it is comparable to the constraint obtained from ℬ(τ→μν̅ν ) to which the leptoquark correction is also generated through a loop, cf. Ref. <cit.>. We have used Flavio package <cit.> for the leptonic τ decays.



Clearly, this is the only acceptable single scalar leptoquark solution to the problem of ^exp >^SM involving a minimal number of parameters. 
Since this model is viable, we can use the region of allowed parameters shown in Fig. <ref> and make several interesting predictions. 
 

  1. In the previous Section we made sure that ℬ(B_c→τν)≤ 30%. In our S_1 model such a requirement is not necessary since the correction to B_c→τν is generated through g_S_L and amounts to:

    ℬ(B_c→τν)^S_1/ℬ(B_c→τν)^SM∈ [1.13, 1.48] , where ℬ(B_c→τν)^SM = (2.24 ± 0.07)%×(V_cb/0.0417)^2,

which in fact is a prediction of this model, and it is well below ℬ(B_c→τν)≤ 30%.

  2. In the previous Section we also showed that the consistency with ^exp resulted in a huge enhancement of ℬ(B→ Kνν). In our S_1 model this is not the case because there is no tree-level coupling to sν. It can however generate, through the box or penguin diagrams involving one S_1 and one W-boson, a contribution to b → s ττ or b → s ν_τν̅_τ, as shown in Figs. <ref>,<ref>.

The leading contribution to b → s ττ is due to virtual top quark in the box, as shown in Fig. <ref> which then shifts the SM Wilson coefficients as

    δ C_9 = -δ C_10 = -|y_L^bτ|^2/8πα x_t log x_t/x_t - 1 = -0.31 |y_L^bτ|^2 ,

where x_q = m_q^2/m_S_1^2. 
The contribution from the box diagram is computed in the broken electroweak phase with massive quarks so that 
GIM actually annuls all the u_i-mass independent terms, leading to a finite result (even in the unitary gauge) and to vanishing of the whole diagram if the quarks were mass degenerate. 
Comparing this to the SM Wilson coefficients, C_9^SM = 4.2, C_10^SM = -4.1, we use δ C_9/C_9^SM≈δ C_10/C_10^SM = -0.075|y_L^bτ|^2 and find:

    ℬ(B_s→ττ)^S_1/ℬ(B_s→ττ)^SM∈ [0.73, 0.98] ,  ℬ(B→ K ττ)^S_1/ℬ(B→ K ττ)^SM∈ [0.73, 0.98]    (@ 2 σ) .

This is quite a remarkable result since the suppression occurs only due to y_L^bτ≠ 0. That means that even if R_D^(∗ )^exp were equal to R_D^(∗ )^SM, one could simply have y_R^cτ→ 0, and still have the above suppression of the b→ sττ rates. Knowing that y_L^bτ is very difficult to constrain either through the LHC studies of high-p_T tails of pp→ττ, or via the low energy constraints, such as ℬ(Υ(nS) →ττ), actually measuring ℬ(B_s→ττ) and/or ℬ(B→ K^(∗ )ττ) would be the only way to understand whether or not the above suppression indeed takes place. 

  3. 

Penguin contribution to b → s ν_τν̅_τ: 
We already stated that the models we consider in this paper are not meant to solve the 2.7σ discrepancy between the first measurement of ℬ(B→ Kνν)^Belle II and its SM prediction. Since in this case there is no tree-level contribution to b→ sνν there is no worry that ℬ(B→ K^(∗ )νν) in this model could then be in conflict with experimental upper bounds. This model in fact leads to a loop induced contribution to b → s ν_τν_τ which is again proportional to |y_L^bτ|^2. Main features of the computation of the diagrams shown in Fig. <ref> are described in Appendix <ref>. In terms of the relevant Wilson coefficient, 

    C_L^ij= C_L^SMδ^ij + δ C_L^S_1 δ^iτδ^jτ ,  δ C_L^S_1=  |y_L^bτ|^2/16 πα ∑_i=u,c,tλ_i/λ_t x_W  g(x_i,x_W) ,

where C_L^SM=-6.32(7) <cit.>. In the S_1 contribution we sum over all up-quarks in the loop that contribute to the renormalized y^L_sτ. Besides x_i = m_u_i^2/m_S_1^2, we also introduced x_W = m_W^2/m_S_1^2, and the CKM factors λ_i = V_ib V_is^*. The explicit expression for the loop function g(x_i,x_W) can be found in Eq. (<ref>). Here we just note an important feature that a mild dependence of the loop function on x_i
leads to an efficient GIM cancellation. For illustration, we observe that g(x_u,x_W) ≈ g(x_c,x_W) = 31.0 - 36.8 i is not much different from g(x_t,x_W) = 19.1 - 36.2 i. The imaginary part arises from the fact that all fermions propagating in the loop can be on their mass shell. 
Finally the S_1 contribution to the Wilson coefficient is

    C_L^S_1=  (-9.3 + 0.4 i)× 10^-2  |y_L^bτ|^2 .

The resulting shift of the physical decay rates is:

    ℬ(B→ K^(∗ )νν)^S_1/ℬ(B→ K^(∗ )νν)^SM = | 1 + δ C_L^S_1/3 C_L^SM|^2 ∈ [1.001, 1.02]   (@2σ) .



  4. Among other (semi-)leptonic decays, we may expect an appreciable contribution to t → b τν decay. Even though it is proportional to |y_L^bτ|^2 that contribution is too small to be distinguished experimentally.[The latest reported ℬ(t→ bτν)=0.105(1)(7) <cit.> is measured with 7% of systematic uncertainty, which is much larger than the leptoquark contribution to this decay.]

  5. A term proportional to V_ub |y_L^bτ|^2 can contribute to the b → u τν amplitude and thus it can modify ℬ(B^- →τν) and ℬ(B →πτν). The leading term interferes with SM and gives at most 3% enhancement of the SM branching fractions.

  6. Besides ℬ(B→ D^(∗ )τν), one can infer a number of observables from the angular distribution of this decay, cf. for example Refs. <cit.>. In the case of τ in the final state, the fraction of the decay rate to a longitudinally polarized D^⋆ has been measured and the two measurements do not agree, F_L^D^*,Belle = 0.60(9) and F_L^D^*,LHCb = 0.43(7), former being larger and latter consistent with the SM prediction, F_L^D^*,SM = 0.46(1). Belle also managed to measure the τ-polarization asymmetry in the B→ D^∗τν, and found P_τ^D^*,Belle = -0.37± 0.54 which, with increasing accuracy, may become an important observable to select among various BSM scenarios. Its SM value is known, P_τ^D^*,SM = -0.51(2). In our S_1, we obtain:

    F_L^D^* =  0.44(1) ,     P_τ^D^* = -0.53(3) .

Similarly, for the forward backward asymmetry we find

    A_fb^D^* =  -0.05(1) ,     A_fb^D =   0.33(1) ,

the values that are to be compared to the SM predictions A_fb^D,SM =   0.3600(4), and 
A_fb^D^∗ ,SM =   -0.06(1).







 §.§ S_1 alternative



In Eq. (<ref>) we chose both the left- and the right-handed Yukawa couplings. In that way we provided a viable solution to ^exp >^SM, as shown in Fig. <ref>. However, as it can be seen in Lagrangian (<ref>), one could also opt for left-handed Yukawa couplings only and generate a contribution to . One (minimal) possibility is to choose

    y_L =
      [      0      0      0;      0      0 y_L^sτ;      0      0 y_L^bτ ] ,      y_R = 0
       ,

so that the only non-zero coupling in LEET (<ref>) is g_V_L, which is expressed by the first term in Eq. (<ref>), i.e.

    g_V_L=- v^2/m_S_1^2V_cs/V_cb y_L^sτy_L^bτ.

It is the CKM enhancement that makes this model appealing but it nevertheless gets excluded by the B_s-B_s mixing (i.e. Δ m_B_s) which is incompatible with the constraint stemming from ℬ(τ→μνν̅), as it can be seen in Fig. <ref>. Note that in the numerical analysis we used the lattice QCD result for the hadronic parameters f_B_s√(B̂_B_s)=256(6) MeV, as obtained from the lattice QCD simulation with N_f=2+1+1 sea quark flavors <cit.>. The feature of this model that we emphasized above, namely that the constraints arising from Δ m_B_s^exp and from ℬ(τ→μνν̅)^exp are not mutually compatible, would be even more pronounced if we used the world average lattice QCD result with N_f=2+1 <cit.>, f_B_s√(B̂_B_s)=274(8) MeV.










§ SUMMARY


In this paper we revisited the possibilities of explaining the experimental hint of LFUV suggesting that ^exp >^SM to more than 3σ, by extending the SM by a single scalar leptoquark in the minimal setup, i.e. with the minimal number of Yukawa couplings. Of three scenarios that could potentially contribute to  we found that only S_1 leptoquark can provide the desired enhancement without being in conflict with other constraints, most notably those stemming from ℬ(Z→ττ)^exp and from the LHC studies of the tails of differential cross section of pp→ττ, τν (+ soft jets) at high p_T. 
While this latter constraint will be steadily improved with higher luminosity of the experimental data, it is important to emphasize that an important progress should be made on the low energy side too. Current lattice QCD estimates of the hadronic matrix elements relevant to B→ D^∗ℓν still suffer from important systematic uncertainties: the shapes of the form factors are not clear and, apart from the dominant form factor A_1(q^2) various collaborations do not agree among themselves. In such a situation, we decided to use the form factors extracted from the experimental studies regarding the angular distribution of B→ D^∗ (→ Dπ) lν, with l=(e,μ), and to evaluate R_D^∗^SM we needed only one information from lattice QCD, namely 
[A_0(q^2_max)]. This is also licit in the rest of our considerations since our main assumption is that the new physics (leptoquarks) can couple to the third generation of leptons only.[Note also that the tensor form factors were recently computed on the lattice <cit.>, which we use in form of the ratios T_1,2,3(q^2)/A_1(q^2). ] 

Our only scenario, that we deem as viable, is the one with S_1 leptoquark and with Yukawa couplings to both left- and right-handed quark/lepton doublets. We checked that the other possibility, in which only the left-handed couplings are allowed to be non-zero, is not simultaneously consistent with the constraints arising from Δ m_B_s and from ℬ(τ→ℓνν̅). We provided several predictions that can help supporting or invalidating the model we propose. 

As for the other models: R_2-model exhibits strong tension between the constraints arising from  and from the high-p_T tails mentioned above. That tension disappears, however, at 3σ and even though we deem the model unsatisfactory, one should monitor how the world average of ^exp will evolve and in what way will move the constraints coming from experimental studies of high-p_T tails with the next acquisition of data at the LHC.

On the other hand, the R_2 model cannot provide an explanation of ^exp >^SM because it necessarily results in a huge contribution to b→ sνν which then overshoots the experimental bounds on ℬ(B→ K^(∗ )νν) by orders of magnitude. 






§ ACKNOWLEDGMENTS

D.B. received a support of the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie Staff Exchange agreement No 101086085 - ASYMMETRY and -Curie grant agreement No 860881-HIDDeN.
S.F., N.K. and L.P. acknowledge financial support from the Slovenian Research Agency (research core funding No. P1-0035 and N1-0321). N.K. acknowledges support of the visitor programme at CERN Department of Theoretical Physics, where part of this work was done.






§ APPENDIX









§ B_S MIXING

The S_1 leptoquark with the choice of couplings as in Sec. <ref> contributes via a box-diagram to the B_s-B_s mixing amplitude. For convenience we match directly to the low-energy effective theory  and account for the QCD renormalization group running between the scales μ = m_S_1 and μ = m_b. The relevant Lagrangian reads:

    ℒ_bs = -4 G_F/√(2)λ_t^2 C^LL_bs (s̅_L
      γ^μ b_L) (s̅_L
      γ_μ b_L) + h.c. ,

in the notation in which λ_t = V_tb^* V_ts. 
At the matching scale, μ=m_S_1=1.5 TeV, for the Wilson coefficient we get

    C^LL(S_1)_bs(m_S_1) = v^2/256 π^2 m_S_1^2(y_L^sτ* y_L^bτ)^2/λ_t^2 ,

which, due to 2-loop running down to μ = m_b, is rescaled by a factor η_S_1=0.617, i.e.

    C^LL(S_1)_bs(m_b) = η_S_1  C^LL(S_1)_bs(m_S_1) .

This is to be added to the SM contribution to the same Wilson coefficient <cit.>:

    C_b s^L L(SM)(m_b)  =η_B m_W^2 S_0(m_t^2 / m_W^2)/16 π^2 v^2 
     =0.862(3) × 10^-3 ,

where η_B = 0.55.[Note that we use the renormalization group invariant definitions of η_S_1, η_B and of bag parameter B̂_B_s.] Finally, the frequency of oscillations of the B_s-B_s system is

    Δ M_B_s =Δ M_B_s^SM |1+C^LL(S_1)_bs(m_b)/C^LL(SM)_bs(m_b)| ,

where for the SM value, after using f_B_s√(B̂_B_s) = 0.256(6) GeV <cit.>, we obtain

    Δ M_B_s^SM = 4 m_B_s/3 f_B_s^2 B̂_B_s |λ_t|^2 C^LL(SM)_bs(m_b)/v^2 = 17.1(8) ps^-1(λ_t/0.04106)^2 .

Precise prediction depends on the values taken for the bag parameters and CKM elements, which is to be compared to 
Δ M_B_s^exp=17.765(6) ps^-1 <cit.>. Notice also that the above SM value of Δ M_B_s does not include the uncertainty in λ_t which is actually significant, cf. Ref. <cit.>. 
In our numerical analysis we used the above-mentioned values, including the only lattice QCD estimate of f_B_s√(B̂_B_s) obtained from simulations with N_f=2+1+1 flavors. If, instead, we used the world average value, f_B_s√(B̂_B_s) =0.274(8) GeV, as obtained from simulations with N_f=2+1 flavors <cit.>, we would have, Δ M_B_s^SM= (19.6± 1.1) ps^-1, modulo uncertainty on λ_t^2.




§ PENGUIN CONTRIBUTION TO B→ SΝΝ IN OUR S_1 MODEL 

In this Appendix we describe the weak-interaction renormalization of y_L^sτ, a coupling that is assumed to vanish at tree-level. In the process S_1^* → s ν_τ presented in Fig. <ref> we choose the kinematical point such that S_1 has an arbitrary time-like momentum q while the fermions are massless and on-shell. We calculate the right diagram iΔ y_L^sτ *(q^2) in Fig. <ref> in R_ξ gauge and find it is divergent and gauge-dependent.[Similar 1-particle-irreducible Goldstone-mediated diagrams are proportional to m_τ^2 and we accordingly neglect them.] Clearly, the amplitude also depends on momentum q^2. The renormalized coupling y_L^sτ(q^2) = y_L^sτ(0) + Δ y_L^sτ(q^2) is the sum of the loop- and tree-level contribution y^sτ(0)_L. We fix the latter by the on-shell renormalization condition:

    y_L^sτ(q^2 = m_S_1^2) = 0 .

Notice that once the weak radiative corrections are present, the S_1 Yukawa couplings run with q^2 and we can only set y_L^sτ =0 at a given momentum scale which we choose to be such that the on-shell amplitude S_1^* → s ν_τ vanishes. In Δ y_L^sτ(q^2) we have not considered 1-particle-reducible diagrams with u_i ↔ s flavor-changing loops on the quark leg since these are q^2-independent and are thus removed by the renormalization condition (<ref>).

The resulting y_L^sτ(q^2) is finite and gauge-independent:

    y_L^sτ *(q^2) =g^2   y_L^bτ */32 π^2 ∑_i = u,c,tλ_i m_S_1^2/q^2 h(q^2/m_S_1^2, x_i, x_W ) ,
  
where the loop function reads
    
    h(z,x_q,x_W)    =   -2 (x_q-1) z (1-x_W/x_q-1)-
        2 x_q (z-1)  (1-x_q/x_W)
          +2 (x_q-z) (1-x_W/x_q-z)-z(x_q-1) log^2(x_W/x_q-1)
          +(x_q-z) log^2(x_W/x_q-z) .

We have expressed the loop diagram in terms of Passarino-Veltman functions <cit.> with the help of FeynCalc package <cit.>. To obtain analytic expressions for certain C_0 functions we have used Package-X <cit.> and the results were numerically cross-checked against numerical values returned by program LoopTools <cit.>. At the scale of B meson the momentum transfer q^2 is well below  m_S_1^2 and we can safely take a limit z → 0, which leads us to

    g(x_i, x_W)    ≡lim_z→ 0h(z,x_i,x_W)/z
       =-2 x_i (1-x_q/x_W)-2  (1-x_W/x_i) +2 x_i log(x_W/x_i)/x_i-x_W
         -(x_i-1) [2  (1-x_W/x_i-1)+log^2(x_W/x_i-1)]-log^2(x_W/x_i) .



h-physrev



