Importance of Loop Effects in Explaining the Accumulated Evidence for New Physics in B Decays with a Vector Leptoquark

In recent years experiments revealed intriguing hints for new physics (NP) in $B$ decays involving \bctaunu and $b\to s\ell^+\ell^-$ transitions at the $4\,\sigma$ and $5\,\sigma$ level, respectively. In addition, there are slight disagreements in $b\to u \tau\nu$ and $b\to d\mu^+\mu^-$ observables. While not significant on their own, they point in the same direction. Furthermore, $V_{us}$ extracted from $\tau$ decays shows a slight tension ($\approx2.5\,\sigma$) with its value determined from CKM unitarity. Additionally, BELLE found hints for an excess in $B_d\to\tau^+\tau^-$. Concerning NP explanations, the vector leptoquark $SU(2)$ singlet is of special interest since it is the only single particle extension of the SM which can (in principle) address all the anomalies described above. For this purpose, large couplings to $\tau$ leptons are necessary and loop effects, which we calculate in this article, become important. Including them in our phenomenological analysis, we find that neither the tension in $V_{us}$ nor the excess in $B_d\to\tau^+\tau^-$ can be fully explained without violating bounds from $K\to\pi\nu\bar\nu$. However, one can account for $b\to c\tau\nu$ and $b\to u\tau\nu$ data finding intriguing correlations with $B_{q}\to\tau^+\tau^-$ and $K\to \pi\nu\bar\nu$. Furthermore, the explanation of $b\to c\tau\nu$ predicts a positive shift in $C_7$ and a negative one in $C_9$, being nicely in agreement with the global fit to $b\to s\ell^+\ell^-$ data. Finally, we point out that one can fully account for \bctaunu and $b\to s\ell^+\ell^-$ without violating bounds from $\tau\to \phi\mu$, $\Upsilon\to\tau\mu$ or $b\to s\tau\mu$ processes.


I. INTRODUCTION
So far, the LHC did not directly observe any particles beyond the Standard Model (SM). However, intriguing hints for lepton flavor universality (LFU) violating NP were acquired: b → s(d) + − : The ratios R(K ( * ) ) = B→K ( * ) µ + µ − B→K ( * ) e + e − [1]( [2]) indicate LFU violation with a combined significance of ≈ 4 σ [3][4][5][6][7]. Taking also into account all other b → sµ + µ − observables (like P 5 [8]), the global fit even shows compelling evidence [9] for NP (> 5 σ). Concerning b → d + − transitions, the theoretical analysis of Ref. [10] shows that the LHCb measurement of B → πµ + µ − [11] slightly differs from the theory expectation. Even though this is not significant on its own, the central value is very well in agreement with the expectation from b → s + − under the assumption of a V td /V ts -like scaling of the NP effect. In other words an effect of the same order and sign as in b → s + − , rela-tive to the SM, is preferred. Furthermore, an analysis of BELLE data found an excess in B d → τ + τ − [12].
b → c(u)τ ν: The ratios R(D ( * ) ), which measure LFU violation in the charged current by comparing τ modes with light leptons (e, µ), differ in combination from their SM predictions by ≈ 4 σ [13]. Also, the ratio R(J/Ψ) = Bc→J/Ψτ ν Bc→J/Ψµν [14] exceeds the SM prediction in agreement with the expectations from R(D ( * ) ) [15,16]. Concerning b → uτ ν transitions, the theory prediction for B → τ ν crucially depends on V ub . While previous lattice calculations resulted in rather small values of V ub , recent calculations give a larger value (see Ref. [17] for an overview). However, the measurement is still above the SM prediction by more than 1 σ, as can be seen from the global fit [18]. In R(π) = B→πτ ν B→π ν there is also a small disagreement between theory [19] and experiment [20] which does not depend on V ub . These results are not significant on their own but lie again above the SM predictions like in the case of b → cτ ν. V τ us : V us extracted from τ lepton decays (V τ us ) shows a tension of 2.5 σ compared to the value of V us determined from CKM unitarity (V uni us ) [13,21]. charge −4/3 [22][23][24][25][26][27][28] arising in the famous Pati-Salam model [29]: This LQ can explain b → cτ ν data without violating bounds from b → sνν and/or direct searches, provides (at tree level) a C 9 = −C 10 solution to b → s + − data and does not lead to proton decay. Therefore, a sizable effect in b → uτ ν and b → d + − is straightforward and also an explanation of V τ us could be possible. A huge enhancement of b → sτ + τ − rates is predicted as well [30], making an amplification of B d → τ + τ − possible.
Several attempts to construct a UV completion for this LQ to address the anomalies have been made [31][32][33][34][35][36][37][38][39][40][41]. In order to fully account for the b → cτ ν data (while respecting perturbativity), one needs sizable couplings to third generation leptons and V 1 generates, via SU (2) L invariance, also large d i d j τ τ and u i u j ν τ ν τ currents at tree level. These currents can generate couplings of down quarks to neutrinos or light charged leptons at loop level (see Fig. 1).
In this article we will calculate these loop effects [71] which turn out to be not only numerically important but give also rise to additional correlations among observables. Even though a theory with a massive vector boson without an explicit Higgs sector is not renormalizable, we still identify several phenomenologically important loop effects which are gauge independent and finite and can therefore be calculated reliably (in analogy to flavor observables within the SM).
This article is structured as follows: In the next section we will establish our conventions for the various observables. In Sec. III we present our results for the loop effects. Sec. IV is devoted to a phenomenological analysis before we conclude in Sec. V.

II. MODEL, OBSERVABLES AND TREE LEVEL EFFECTS
We work in a simplified model obtained by extending the SM by a vector LQ SU (2) L singlet with hypercharge −4/3, mass M and interactions with fermions determined by Here, Q (L) are quark (lepton) SU (2) L doublets, d (e) are down quark (charged lepton) singlets and f, i are flavor indices. In the following, we will neglect the righthanded couplings which are not necessary to explain the anomalies. This then generates the effective 4-fermion interactions encoded in where α and β label the SU (2) components. After EW symmetry breaking, we work in the down basis, i.e. no CKM elements appear in flavor changing neutral currents of down quarks. Let us now define the various observables and recall the corresponding contributions of our leptoquark.
We define the effective Hamiltonian as and obtain at tree level at the (1 σ) 2 σ level, assuming a vanishing effect in electrons. In b → dµ + µ − transitions one finds for the Wilson coefficients assuming them to be real [10]. For τ leptons we have experimentally [42] Br and for B d → τ + τ − there is a (unpublished) measurement of BELLE [12] and an upper limit of LHCb [42] Br Both are compatible at the 2 σ level. The SM predictions are given by [43,44] Br In our model we have with q = s, d and C SM 10,qb ≈ −4.3 [45,46]. For the analysis of B → K ( * ) τ µ we will use the results of Ref. [47].
The blue regions are preferred by b → cτ ν data, the (light) grey region is excluded by (Bs → τ + τ − ) B → K * νν and the hatched region indicates the (approximate) excluded region by LHC, obtained by a naive rescaling of the PDFs using the bounds of Ref. [53]. Within the light red region Br[B → K * νν] is smaller than the expected BELLE II sensitivity.
Note that the LQ does not contribute at tree level. For K → πνν we use Ref [54] with the updated numerical values given in Ref. [55] resulting in with This has to be compared to the current experimental limits R νν K < 3.9 and R νν K * < 2.7 [57] (both at 90% C.L.). The future BELLE II sensitivity for B → K ( * ) νν is 30% of the SM branching ratio [58].
We define the effective Hamiltonian as where in the SM C f i jk,SM = δ f i . The contribution of our model is given by With these conventions we have for b → cτ ν transitions assuming vanishing contributions to the muon and electron channels with X = D, D * , J/Ψ. We obtain the analogous expression for b → uτ ν.
Concerning τ → K(π)ν we find that the CKM element V τ us extracted from these decays is given in terms of the one determined in the absence of NP contributions (V where we neglected LFV effects. This has to be compared to [13,21]

III. ONE-LOOP EFFECTS
In our setup, one-loop effects involving the LQ and third generation leptons (τ 's and τ neutrinos) can be very important, since we aim for large effects in b → c(u)τ ν and b → s(d)τ + τ − processes. In principle, a massive vector boson, like our LQ, without a Higgs sector is not renormalizable. However, in flavor physics most effects can still be reliably calculated since they are gauge independent and finite (also in unitary gauge) [72]. This is in analogy to the SM where the contribution of the W to flavor observables can be correctly calculated in unitary gauge without taking into account the Higgs sector.
We are only interested in effects which are absent at tree level. In these cases the loop effects are the leading contributions. We calculate all diagrams at leading order in the external momenta using asymptotic expansion [61].
A. W boxes contributing to di → df νν The result (an example diagram is shown on the right side of Fig. 1) is gauge invariant in R ξ gauge and the same finite result is obtained in unitary gauge:  ) and R(J/Ψ) or R(π) and B → τ ν, respectively). The gray region is excluded by K + → π + νν. Here we assumed all couplings κ L ij to be real.
B. W off-shell penguins contributing to τ → µνν Here (see third diagram in Fig. 1) we obtain again a finite and gauge independent result for the Wilson coefficient, following the conventions of [62] The ratio can be expressed in terms of the Wilson coefficients [62] and can be compared to the experimental values [63] R τ µ = 1.0060 ± 0.0030 , R τ e = 1.0022 ± 0.0030 . (27) We find, in agreement with Ref. [27], that the effect is small.

C. On-shell photon and gluon penguins
Again the result of the left diagram in Fig. 1 is finite in unitary gauge and the same result is obtained in R ξ gauge Taking into account the running from the LQ scale µ LQ = M = 1 TeV down to µ b = 5 GeV (see e.g. [64,65]), we obtain D. Off-shell γ penguins The full result (second diagram in Fig. 1) for the amplitude is gauge dependent and in general divergent. However, one can calculate the mixing of C τ τ 9,sb = −C τ τ 10,sb into the 4-fermion operators O 9,sb (containing light leptons as well) within the effective theory (i.e. after integrating out the LQ at tree level). In this way, a gauge independent result is obtained and the leading logarithm of the (unknown) full result is recovered. For off-shell photons we thus calculate the effect in the EFT (below the LQ scale),  generating the following mixing into the 4-fermion operators with light leptons Note that this result is model independent (at leadinglog accuracy) in the sense that it does not depend on the model which generates C τ τ 9,sb = −C τ τ 10,sb . In principle, there are also Z penguins generating C 9,sb and C 10,sb . However, this effect is suppressed by light lepton masses (or small momenta) and is therefore of dim-8. Further, note that there are no box-diagram contributions which generatesbμµ (sbēe) operators if the couplings of the LQ to muons (electrons) are zero at tree level.

E. Box diagrams with LQs
What cannot be calculated consistently are box diagrams involving only LQs [32]. Here, the results are divergent in unitary gauge which corresponds to a gauge dependence in R ξ gauge. However, these effects are suppressed if |κ L | < g 2 and can be further suppressed in the presence of vector-like fermions by a GIM-like mechanism [35] which, in analogy to the SM, would render the result finite.

IV. PHENOMENOLOGY
Let us start by assuming that κ L 23 and κ L 33 , which are necessary to explain b → cτ ν data, are the only non-zero  . b → sµ + µ − and b → dµ + µ − data can be simultaneously explained without violating bounds from KL → µ + µ − and the relative shift δV µ cb is below the percent level and thus compatible with data [67].
couplings. In Fig. 2 we see that in fact b → cτ ν data can be explained for a LQ mass of 1 TeV without violating bounds from B s → τ + τ − , B → K ( * ) νν or direct LHC searches if κ L 23 is of the same order as κ L 33 . (29)) and C 9,sb (Eq. (30)) directly depend on R(X)/R(X) SM (with X = D ( * ) , J/Ψ). In Fig. 3 we show these dependences. Intriguingly, the effect generated in C sb 7 (µ b ) and C 9,sb , within the preferred region from b → cτ ν data, exactly overlaps with the 1 σ range of the model independent fit to b → sµ + µ − data excluding LFU violating observables [59,66] (therefore only P 5 etc. but not R(K ( * ) ) can be explained).
Let us now include the effect of κ L 13 . Here, many correlations arise. First of all, b → c(u)τ ν is already at tree level correlated to b → s(d)τ + τ − . In addition, the W boxes in Eq. (24) generate effects in B → K ( * ) (π)νν and K → πνν. While the bounds from B → K ( * ) (π)νν turn out to be weaker than the ones from B q → τ + τ − , there are striking correlations with K → πνν as can be seen from Fig. 4. Furthermore, we get an effect where V τ (0) us is the CKM matrix element extracted from τ decays without NP. However, Eq. (24) generates K → πνν and respecting these bounds, the relative effect in V τ us can only be at the per-mill level, |δV τ us | ≈ 0.05%, excluding the possibility to account for this discrepancy. The same is true about B d → τ + τ − where the currently preferred region lies outside the plot. Now we allow in addition non-vanishing couplings κ L 32 and κ L 22 , obtaining tree level effects in b → sµ + µ − . We first correlate b → sµ + µ − with b → cτ ν, b → sτ µ and τ → φµ. Note that a simultaneous explanation of the anomalies is perfectly possible (see Fig. 5). Interestingly, due to the loop effects originating from the b → cτ ν explanation, we predict a flavor universal effect in C 9,sb and C sb 7 which is supplemented by the the tree-level effect in C µµ 9,sb = −C µµ 10,sb with muons only. This means that the relative NP effect compared to the SM in P 5 should be larger than in R(K ( * ) ) which is in perfect agreement with the global fit.
Finally, taking into account in addition couplings to the first generation, b → dµ + µ − and s → dµ + µ − effects arise. Furthermore, via CKM rotations, also b → c(u)µν is affected and we can correlate the effect to K L → µ + µ − (see Fig. 6). Note that the short distance contribution in K L → µ + µ − is compatible with a common explanation of b → dµ + µ − and s → dµ + µ − data. Also a shift in δV µ cb occurs which is however well within the allowed range [67].

V. CONCLUSIONS
The vector leptoquark SU (2) singlet is a prime NP candidate to explain the current hints for LFU violation. In this article we calculated and studied the important loop effects arising within such a model and performed a phenomenological analysis. We find: Explaining b → cτ ν data generates lepton flavor universal effects in b → s + − transitions which nicely agree with the model independent fit (see Fig. 3). Therefore, the C 9 = −C 10 -like tree level effect, which is in general LFU violating, is supplemented by these effects generating a new pattern for the Wilson coefficients. This can be tested with future data and similar conclusions hod for the correlations between b → uτ ν data generates lepton flavor universal effects in b → d + − processes.
NP in b → c(u)τ ν generates important effects in B s(d) → τ + τ − which are even correlated to b → s(d)νν processes and K → πνν via W box contributions (see right diagram in Fig. 1). The V τ us puzzle (like the CP asymmetry in τ → K S πν [68]) cannot be solved due to the stringent constraints from K → πνν and because of b → uτ ν bounds one cannot fully account for the BELLE excess in B d → τ + τ − (see Fig. 4).
b → cτ ν and b → s + − data can be simultaneously explained without violating other bounds like τ → φµ (see Fig. 5). Furthermore, one could at the same time also account for NP effects in b → dµ + µ − without violating K L → µ + µ − bounds.