Revisiting the vector leptoquark explanation of the B-physics anomalies
Claudia Cornella Javier Fuentes-Martin Gino Isidori






 ZU-TH-15/19  









     Revisiting the vector leptoquark explanation
 of the B-physics anomalies
       
 [1.0 cm] 
   Claudia Cornella,  Javier Fuentes-Martín, Gino Isidori    

  Physik-Institut, Universität Zürich, CH-8057 Zürich, Switzerland  






Abstract

We present a thorough investigation of the vector leptoquark hypothesis for 
a combined explanation of  the B-physics anomalies. We analyze this hypothesis 
from a twofold perspective, taking into account recent results from B-physics observables 
and high-p_T searches. 
 First, using a simplified model, 
we determine the general conditions for a  successful low-energy fit 
in presence of right-handed  leptoquark couplings (neglected in previous analyses).
Second, we show how  these conditions, in particular a sizable 2-3 family mixing, 
can be achieved in a  motivated ultraviolet completion.
Our analysis reinforces the phenomenological success of the vector leptoquark hypothesis 
in addressing the anomalies, and its compatibility with
motivated extensions of the Standard Model based on the idea of flavor non-universal gauge interactions.
The implications of right-handed leptoquark couplings for a series of key low-energy observables, 
namely B_s →ττ and τ→μ 
lepton flavor violating processes, both in τ and in B decays, 
are discussed in detail. The role of the  ultraviolet completion in precisely 
estimating other low-energy observables, most notably Δ F=2 amplitudes,
is also addressed.


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§ INTRODUCTION



The hints of Lepton Flavor Universality (LFU) violation in charged-current semi-leptonic b→ c ℓν
decays <cit.>, as well as in  b→ s ℓℓ transitions <cit.>, represent  a very intriguing 
phenomenon and a fascinating challenge. 
Recent data confirm numerous discrepancies from the Standard Model (SM) predictions in both these sectors.
Despite the fact that there is not a single measurement with a high statistical significance, and that recent 
data have slightly decreased the overall significance of the anomalies, the global picture is still extremely interesting:
the internal consistency of available data is remarkable and, once combined, the significance of the LFU violating observables
exceeds 3.7σ in b→ s ℓℓ and  3.1σ  in b→ c ℓν. 
A  common origin of the two sets of anomalies is not obvious, but it is a very appealing  possibility 
from the theoretical point of view. If confirmed as clear signals of physics beyond the Standard Model (SM),
the two anomalies combined would point to non-trivial dynamics at the TeV scale,
possibly linked to a solution of the SM flavor puzzle.

The initial efforts to address both sets of anomalies in terms of beyond the SM (BSM) physics  
have been focused on Effective Field Theory (EFT)  approaches 
(see <cit.> for the early attempts).
However, the importance of complementing EFT 
approaches with appropriate simplified models with new heavy mediators was soon realized <cit.>,
and in this context leptoquark models played a key role <cit.>.
Explicit heavy mediators are essential to address the compatibility of this class of SM extensions with other low-energy constraints and with high-p_T data.
Due to the relatively low scale of new physics hinted at by the charged-current anomalies, high-p_T
constraints are indeed quite relevant <cit.>.
Given the success of some EFT approaches and simplified models in describing available data, 
the attention has shifted recently towards the development of 
 more complete (and more complex) models with a consistent  ultraviolet (UV)  behavior
 (see in particular <cit.>).


Already in early attempts <cit.>, 
the U_1∼(3,1)_2/3 vector leptoquark, 
coupled mainly to third-generation fermions, emerged as an excellent
mediator for the explanation of both sets of anomalies. The effectiveness of this state as 
single-mediator accounting for all available low-energy data has been established in <cit.>. 
However, the analysis of Ref. <cit.>, as most other phenomenological analyses of the U_1 leptoquark
in B physics (see in particular <cit.>), 
is based on the simplifying hypothesis of vanishing U_1 couplings to right-handed (RH) SM fermions. 
This hypothesis is motivated by the absence of clear indications of non-standard RH currents in present data, and by 
the sake of minimality, but it does not have a strong theoretical justification. Indeed 
the quantum numbers of the U_1 allow for RH couplings at the renormalizable level, 
and in motivated UV completions such couplings naturally appear <cit.>.

The first goal of the present paper is the generalization of existing EFT/simplified-model 
studies on the U_1 impact in low-energy observables,
taking into account non-vanishing RH couplings (mainly to the third generation). 
As pointed out  first in <cit.>
in the context of a specific UV completion, and as we show in more general terms below, such couplings lead to 
a series of interesting modifications in the low-energy phenomenology compared to the pure left-handed case.
We also update the analysis taking into account recent results on semileptonic B-meson
decays. New data by both LHCb <cit.> and 
Belle <cit.> have not changed the overall 
significance of the anomalies in b→ s ℓℓ <cit.>, while preliminary 
data from Belle <cit.> have slightly decreased the significance in b→ c ℓν. However, as anticipated, 
the overall significance of the anomalies remains very high and the possibility of a combined 
explanation has become even  more consistent from a theoretical point of view,
due to the reduced tension with high-p_T data and other low-energy observables.

The second goal of this paper is to assess whether the conditions necessary for a successful low-energy fit to present data,
compatible with high-p_T constraints <cit.>, 
can be achieved in the context of a consistent UV completion of the simplified model.
Here the main difficulty is to achieve a sizable 2-3 family mixing for the U_1 without introducing excessively large contributions 
to Δ F=2 observables from other mediators required by the consistency of the theory.
As we show, this can be achieved by means of a simple extension of the scalar sector of the model 
originally proposed in <cit.> (see also <cit.>).

We provide a detailed implementation of the U_1 leptoquark in a renormalizable model based on the (flavor non-universal) 
gauge group SU(4)_3× SU(3)_1+2× SU(2)_L × U(1)^', which in turn 
can be embedded in PS^3 <cit.>. In this context, we complement the simplified-model analysis
by including one-loop contributions to low-energy observables (most notably  Δ F=2 amplitudes and dipole operators) 
which can be reliably computed only
within a UV-complete framework.


The paper is organized as follows. In Section <ref> we present the simplified-model analysis: we introduce the  
Lagrangian describing the U_1 couplings to SM fermions, and analyze its low-energy limit.
We discuss all the observables insensitive to the UV completion (Section <ref>), which are later used to fit  
low-energy data (Section <ref>). We finally comment 
 on the high-p_T constraints  (Section <ref>). 
 The UV-complete model is presented and discussed in Section <ref>: on the model-building side 
 we pay particular attention to the flavor structure of the model (Section <ref>); 
 on the phenomenological side we present 
 complete expressions for the UV-dependent (loop-induced) 
 observables, which were omitted in the low-energy fit (Section <ref>).
 The results are summarized in Section <ref>. 

 




§ THE SIMPLIFIED U_1 MODEL AND ITS PHENOMENOLOGY





 §.§ Effective interactions of the U_1 to SM fields

We consider the most general Lagrangian for the SU(2)_L-singlet vector leptoquark, U_1^μ∼ (3,1)_2/3, coupled to both left- and right-handed SM fields

    ℒ_U=   -1/2 U_1 μν^†  U_1^μν+M_U^2 U_1 μ^†  U_1^μ -ig_c(1-κ_c) U_1 μ^† T^a U_1 ν G^a μν
       -i 2/3 g_Y  (1-κ_Y) U_1 μ^†  U_1 ν B^μν+(U_1^μ J_μ + h.c.) ,

where U_1 μν = D_μ U_1 ν - D_ν U_1 μ , with D_μ = ∂_μ - i g_c  G_μ^aT^a - i 2/3 g_Y  B_μ. Here G_μ^a (a=1,…,8) and B_μ denote the SM SU(3)_c and U(1)_Y gauge bosons, with g_c and g_Y gauge couplings respectively, and T^a are the SU(3)_c generators. In models in which the vector leptoquark has a gauge origin, κ_c=κ_Y=0, while this is not necessarily the case for models in which the U_1 arises as a bound state from a strongly-coupled sector. The fermion current reads[We ignore possible couplings to right-handed neutrinos. Such particles, if present, are assumed to be heavy enough such that they do not to play any role in low-energy observables. Vector leptoquark solutions of the B-physics anomalies involving right-handed neutrinos, light enough to fake the SM ones, have been discussed in <cit.>.]

    J_μ=g_U/√(2)[ β_L^iα (q̅_L^ iγ_μℓ_L^α)  +     β_R^i α (d̅_R^ iγ_μ   e_R^α)]  .
 
Here the couplings β_L and β_R are complex 3 × 3 matrices in flavor space. Without loss of generality, we adopt the down-quark and charged-lepton mass eigenstate basis for the SU(2)_L multiplets, i.e.

    q_L^i=
    [ V_ji^* u^j_L;        d_L^i ] ,        ℓ_L^i=
    [ ν^i_L; e_L^i ] .

In this flavor basis, we assume the following structure for the β_L and β_R couplings

    β_L = 
    [       0       0  β_L^dτ;       0 β_L^s μ β_L^s τ;       0 β_L^b μ       1 ] ,        β_R = 
    [       0       0       0;       0       0       0;       0       0 β_R^b τ ] ,

where the normalization of g_U is chosen such that β_L^b τ =1. 
The assumed structure contains the minimal set of couplings directly connected to a combined explanation of the 
B-physics anomalies. The null entries in Eq. (<ref>) should be understood as small terms which have a negligible 
impact in the observables we analyze. We discuss later on the implications of this requirement on the values of β_L^dμ and β_R^b μ. 
Under the assumption of a natural (CKM-like) flavor structure, the entries in (<ref>) are expected to follow the relations
 β_L^d τ, β_L^s μ≪β_L^s τ, β_L^b μ≪ 1, and   β_R^b τ = 𝒪(1). 
 As we shall see, 
the hierarchy of the β's is well compatible and, at least in some cases, it emerges from the fit to low-energy data. 
The only parameter we force to be small a priori in the phenomenological analysis is β_L^sτ, which 
is largely unconstrained in the simplified model (using only low-energy data) and 
plays a key role in setting the overall mass scale for the U_1. We set the upper limit  
|β_L^sτ | ≤  0.25 (see Section <ref>). 
 In Section <ref> we show how this hierarchical structure of the β's is naturally enforced 
 in the proposed UV completion. 


By integrating out the vector leptoquark at tree level, we obtain the following high-scale (μ∼ M_U) 
effective Lagrangian:

    ℒ_eff  = - 2  C_U/v^2   [   -2   (β_L^i α)^*  β_R ^ l β (ℓ̅_L^αe_R^β  ) (d̅_R^l  q_L^i ) + h.c. + β_R^i α  (β_R^ l β)^* (e̅_R^βγ_μ e_R^α) (d̅_R^iγ^μ  d_R^l) 
     .
       .  +1/2β_L^i α  (β_L^ l β)^* (ℓ̅_L^βγ_μℓ^α_L)(q̅^i_Lγ^μq_L^l )  +   1/2β_L^i α  (β_L^ l β)^*  (ℓ̅_L^βσ^aγ_μℓ^α_L)(q̅^i_Lσ^aγ^μq_L^l )  ]  ,

where C_U≡ g_U^2 v^2/(4 M_U^2) and v=(√(2)  G_F)^-1/2≈246 GeV is the SM Higgs vacuum expectation value (vev).





 §.§ The relevant low-energy observables



Since the Lagrangian in (<ref>) is not renormalizable, only a limited set of low-energy observables can be reliably estimated 
in this setup. These are observables where the four-fermion interactions in (<ref>) contribute at the tree level,
or where they induce (log- or Yukawa-enhanced) loop contributions which are largely insensitive to the UV completion.
In this section and in the corresponding low-energy fit we consider only such class of observables. 
The discussion of the UV-dependent one-loop contributions is postponed to Section <ref>.

Taking into account these considerations,  the most relevant low-energy observables 
can be classified as follows: 


  i) b→ c(u)τν. Sizable tree-level contributions arise in the LFU ratios R_D and R_D^*. 
Due to the presence of the right-handed coupling, β_R^bτ, the  usual V-A contribution is supplemented by a (large) scalar contribution, yielding the following approximate expressions for R_D and R_D^* (see <cit.> for the hadronic matrix elements of the scalar contribution)

    R_D   ≈ R_D^SM[1+ 2 C_U Re{( 1-1.5 η_S  (β_R^b τ)^* )(1+ V_cs/V_cbβ_L^s τ + V_cd/V_cbβ_L^d τ) }] , 
    
     R_D^*   ≈  R_D^*^SM[1+ 2 C_U Re{(  1 -0.14  η_S (β_R^b τ)^* )( 1  + V_cs/V_cbβ_L^s τ + V_cd/V_cbβ_L^d τ) }] ,

where η_S≈1.8 accounts for the running of the scalar operator from M_U=4 TeV to m_b <cit.>. 
Interestingly, due to the scalar contribution, we obtain a significantly different scaling of the NP effect in R_D and in R_D^*, depending on the value of β_R^bτ (see Figure <ref>). 
 
Additionally, because of the chiral enhancement of the scalar contribution, large NP effects are expected in ℬ( B_c →τν) 
 
    ℬ( B_c →τν)    =τ_B_c m_B_cf_B_c^2 G_F^2|V_cb|^2/8π m_τ^2(1-m_τ^2/m_B_c^2)^2 
         ×|1+ C_U (1 -(β_R^b τ)^*2 η_S m_B_c^2/m_τ(m_b + m_c))(1+V_cs/V_cbβ_L^s τ+V_cd/V_cbβ_L^d τ) |^2  .

The most stringent bounds on this observable are obtained from LEP data from which the authors of <cit.> obtain ℬ( B_c →τν)≲10% (see also <cit.>). 

Concerning the b→ u transitions, the only measured observable in this category is ℬ( B →τν), for which we obtain the following expression

    ℬ( B →τν)    = ℬ( B →τν)_ SM|1+ C_U (1 -(β_R^b τ)^*2 η_S m_B^2/m_τ(m_b + m_u))(1+V_us/V_ubβ_L^s τ+V_ud/V_ubβ_L^d τ) |^2  .

Also here, we expect large NP effects due to the chirally enhanced scalar contribution. However, in this case the connection with R_D^(*) is less robust due to the possible sizable contribution from β_L^dτ, which now receives a larger CKM-enhancement than the one from β_L^sτ. 





  ii) b→ sℓℓ. The vector leptoquark  yields potentially large contributions to b→ sℓℓ transitions, both at tree level and at one loop. Given our assumption of  vanishing couplings to electrons (β_L,R^qe=0, for any q), tree-level contributions affect only b→ s μμ and b→ s ττ transitions along the direction Δ𝒞_9^ℓℓ =  - Δ𝒞_10^ℓℓ with ℓ=μ,τ (see Appendix <ref> for the Wilson coefficient definitions)

    Δ𝒞_9^ℓℓ   =  - Δ𝒞_10^ℓℓ  = - 2 π/α V_tb V_ts^∗  C_Uβ_L^s ℓ (β_L^b ℓ)^* .
 

Being lepton non-universal, these Wilson coefficient modifications affect the R_K^(*) ratios in the following way <cit.>

    Δ R_K   ≡ R_K^[1,6] GeV^2-1≈ 0.46   Δ𝒞_9^μμ ,
    Δ R_K^*   ≡ R_K^*^[1.1,6] GeV^2-1≈ 0.47 Δ𝒞_9^μμ .


As discussed in <cit.>,  a large β_L^sτ coupling can also yield a sizable lepton-universal contribution to b → s ℓℓ transitions in the Δ𝒞_9 direction via a (log-enhanced) photon penguin. Since the 
dominant contribution is given by the log-enhanced piece, it can be unambiguously recovered from the corresponding 
EFT computation which gives

    Δ𝒞_9^U    ≈ -1/V_tbV_ts^* 2/3 C_U∑_ℓ=e, μ, τβ_L^sℓ(β_L^bℓ)^* log(m^2_b/M_U^2)  
       ≈ -1/V_tbV_ts^* 2/3 C_U 
    β_L^sτ(β_L^bτ)^* log(m^2_b/M_U^2) .

For non-zero β_R^bμ, scalar-current contributions are generated in b→ sμμ transitions.
As shown in <cit.>, a stringent bound on β_R^bμ follows from B_s →μ^+μ^-, since the scalar-current contribution is chirally enhanced. 
Fixing the other parameters to fit the current central value of R_K^(*), present data imply |β_R^bμ|  0.02 |β_L^bμ|. Once this condition is imposed, the effect of 
β_R^bμ on other b→ sμμ observables is negligible. We can thus directly compare the 
corrections to 𝒞_9, 10^μμ in (<ref>) and 𝒞_9^U in (<ref>)
with the global fits of these Wilson coefficients reported in <cit.> (see also <cit.> for details on the fit methodology followed in <cit.>). 

On the other hand, scalar currents are necessarily present in b→ sττ transitions if β_R^bτ =𝒪(1).
The most interesting observable in this respect is 

    ℬ(B_s→τ^+τ^-)    = ℬ(B_s→τ^+τ^-)_ SM[|1+2π/α V_tbV_ts^*C_U/C_10^ SM β_L^s τ(1  - η_S  m_B_s^2/m_τ(m_s + m_b)   (β_R^b τ)^* )|^2. 
         .+(1-4m_τ^2/m_B_s^2)|2π/α V_tbV_ts^*C_U/C_10^ SMη_S  m_B_s^2/m_τ(m_s + m_b) β_L^s τ (β_R^b τ)^* |^2]  .

A milder, but still sizable, chiral enhancement occurs in ℬ(B → K τ^+τ^-), which can be large and within the reach of future experiments, especially for β_R^bτ =𝒪(1). Using the hadronic form factors in <cit.> we find

    ℬ(B → K τ^+τ^-)     ≈ 1.5·10^-7+10^-3  C_U (1.4 Re{β_L^sτ}-3.3 Re{β_L^sτβ_R^bτ *}) 
         + C_U^2 |β_L^sτ|^2  (3.5 - 16.4 Re{β_R^bτ}+ 95.0  |β_R^bτ|^2) .


An interesting feature of the vector-leptoquark solution is the absence of tree-level contributions to b→ sνν observables, letting this setup easily pass the current constraints from B→ K^(*)νν.



  iii) Dipoles. For β_R^b τ≠ 0, the presence of both left- and right-handed leptoquark couplings gives rise to contributions to the radiative LFV decay τ→μγ that are m_b-enhanced. Taking κ_Y=0, the m_b-enhanced piece is finite and can be unambiguously computed already in the dynamical model. We find

    ℬ(τ→μγ)    ≈1/Γ_τα/64 π^4m_τ^3m_b^2 /v^4  C_U^2 | β_R^b τ (β_L^b μ)^*|^2 .

Analogous loop effects in the b→ sγ(g) transitions are more sensitive to the specifics of the UV completion.
Indeed the contribution proportional to the internal mass in the U_1-mediated amplitude leads to a 𝒪(m_τ^2/m_b^2)
suppression, rather than an enhancement, compared to the one proportional to the external mass. This latter contribution is sensitive to the details of the UV completion and cannot be reliably estimated. 
We thus postpone their discussion to Section <ref>.


  iv) LFV observables. The vector leptoquark can also yield sizable tree-level contributions to semileptonic LFV transitions. The most interesting observables are those involving the b → s τμ transition. One of the observables in this category for which experimental limits are available is B^+ → K^+ τμ. The simplified expressions are given by <cit.>

    ℬ(B^+→ K^+τ^+μ^-)     ≈ C_U^2  |β_L^sμ|^2(8.3 + 155.2  |β_R^b τ|^2 - 42.3 Re{β_R^b τ}) ,
    ℬ(B^+→ K^+τ^-μ^+)     ≈ 8.3 C_U^2  |β_L^bμ(β_L^sτ)^*|^2 .

Note that, for large values of β_R^b τ, the τ^+μ^- channel is expected to yield the dominant NP contribution, provided the other couplings follow the natural flavor scaling discussed in Section <ref>.

As in B_s→ττ, the NP effect in B_s→τμ is chirally enhanced for β_R^b τ≠ 0, making this observable of particular interest. Its expression reads

    ℬ(B_s→τ^-μ^+) = τ_B_s m_B_s  f_B_s^2 G_F^2/8 π  m_τ^2(1 - m_τ^2/m_B_s^2)^2  C_U^2| β_L ^s μ (β_L^b τ)^* - 2 η_S  m_B_s^2/m_τ(m_s + m_b) β_L^s μ (β_R^b τ)^*|^2 .

The LHCb Collaboration has recently performed the first measurement of this observable, setting the upper limit 
ℬ(B_s→τ^±μ^±) < 4.2× 10^-5 at 95% C.L. <cit.>, whose 
 implications  are discussed in the next section.

Another interesting LFV observable, relevant in the limit of large β_L^s τ, is τ→μϕ (see e.g. <cit.>).
Here we find

    ℬ( τ→μϕ)    = 1/Γ_τ f_ϕ^2  G_F^2/16 π m_τ^3(1-  m_ϕ^2/m_τ^2)^2(1 + 2 m_ϕ^2/m_τ^2)  C_U^2|β_L^s τ(β_L^s μ)^*|^2 .
 


  v) LFU in τ decays.
At the one-loop level,  the effective Lagrangian in (<ref>) leads to modifications of the Z and W couplings to fermions
and, more generally,  to LFU breaking effects in purely leptonic charged-current transitions, 
as extensively discussed in  <cit.>. The most constraining bounds arise from LFU tests in τ decays, in particular from the ratio  g_τ/g_μ. Using the results in <cit.>,
we can describe these effects via the following simplified expression

    ( g_τ/g_μ)_ℓ, π, K   ≈ 1 -  0.08   C_U ,

where we have set M_U = 4 TeV in the evaluation of the leptoquark loop.



  vi) Δ F=2 observables. Though important, loop contributions to Δ F=2 transitions mediated by the vector leptoquark are divergent and cannot be reliably estimated without a UV completion. The discussion of these effects is therefore postponed to Section <ref>.



