PHYSICAL REVIEW D 100, 035028 (2019)
R2 as a single leptoquark solution to RDð Þ and RKð Þ
† ‡ Oleg Popov,1,* Michael A. Schmidt,2, and Graham White3, 1Physics and Astronomy Department, University of California, Riverside, California 92521, USA Institute of Convergence Fundamental Studies, Seoul National University of Science and Technology, Seoul 139-743, Korea 2School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia 3TRIUMF Theory Group, 4004 Wesbrook Mall, Vancouver, British Columbia V6T2A3, Canada
(Received 27 May 2019; published 26 August 2019)
We show that, up to plausible uncertainties in BRðBc → τνÞ, the R2 leptoquark can simultaneously explain the observation of anomalies in RKð Þ and RDð Þ without requiring large couplings. The former is achieved via a small coupling to first generation leptons, which boosts the decay rate ΓðB¯ → K¯ ð Þeþe−Þ. Finally we motivate a neutrino mass model that includes the S3 leptoquark, which can alleviate a mild tension with the most conservative limits on BRðBc → τνÞ.
DOI: 10.1103/PhysRevD.100.035028
I. INTRODUCTION level [8]. The SM calculation is reliable as it is largely insensitive to hadronic uncertainties, which cancel out in There have recently been multiple independent anoma- the ratios R ð Þ . lous measurements of semileptonic B decays that depart D LHCb has similarly found an intriguing deviation from from standard model (SM) predictions. Rare decays into ð Þ ð Þ LFU in the semileptonic B meson decays to K mesons. D mesons show a discrepancy from SM predictions in The LFU ratios BABAR [1,2], Belle [3–5], and LHCb [6,7] measurements of the lepton flavor universality (LFU) ratios. New results Γ ¯ → ¯ ð Þμþμ− from Belle combined with measurements from BABAR and ðB K Þ R ð Þ 1 K ¼ ¯ ¯ − ð3Þ LHCb give [8] ΓðB → Kð Þeþe Þ Γ ¯ → τν¯ 0.299 0.003 SM ½9 R ðB D Þ provide a clean probe of new physics effects because D ¼ ¯ ¼ ΓðB → De=μν¯Þ 0.335 0.031 observed ½8 hadronic uncertainties cancel out in the ratios as long as new physics effects are small [11–13]. LHCb measured the ð1Þ 2 ratios for the dilepton invariant mass range 1.1 GeV < q2 < 6 2 and GeV . A combination of run I and run II from LHCb gives ¯ ΓðB → D τν¯Þ 0.258 0.005 SM ½10 R ¼ ¼ : 1 0003 0 0001 14 D Γ ¯ → μν¯ 0 298 0 015 8 . . SM ½ ðB D e= Þ . . observed ½ R ¼ ð4Þ K 0.846þ0.06 stat þ0.016 sys observed 15 ð2Þ −0.054ð Þ−0.014 ð Þ ½
When the correlation between the two observables is taken and 3 1σ into account the significance of the anomaly is at the . 1.00 0.01 SM ½16 R ¼ ð5Þ *[email protected] K 0 716þ0.070 17 18 † . −0.057 observed ½ ; [email protected] ‡ [email protected] 1The best-fit value and error bars have been extracted from the where we combined the LHCb measurement [18] of RK figure on slide 9. with the new Belle measurement [17] using the methods described in Ref. [19]. Experimental sensitivity to both of Published by the American Physical Society under the terms of these anomalies is expected to improve by orders of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to magnitude over the next few years and make a potential the author(s) and the published article’s title, journal citation, confirmation of a departure from the SM imminent. The and DOI. Funded by SCOAP3. measurements are not just quantitatively different from the
2470-0010=2019=100(3)=035028(11) 035028-1 Published by the American Physical Society POPOV, SCHMIDT, and WHITE PHYS. REV. D 100, 035028 (2019)
SM but qualitatively so as well, because the SM has no leptoquark. We then explain the most relevant constraints in notable violation of lepton flavor universality. Sec. III and show that R2 can explain RKð Þ and RDð Þ .In The most common explanation for these anomalies is Sec. IV we introduce a minimal model for neutrino masses to extend the SM by leptoquarks (see Refs. [20–38] for based on the R2 and S3 leptoquarks. Finally we conclude a leptoquark solution to the RKð Þ anomalies, Refs. [34, in Sec. V. – – 39 45] for the RDð Þ anomalies and Refs. [34,46 69] for simultaneous explanations). Vector leptoquarks have issues II. EFFECTIVE FIELD THEORY ANALYSIS with ultraviolet (UV) completion and their tendency is to be FOR THE R2 LEPTOQUARK heavy in UV complete models. Therefore it is attractive to ∼ 3 2 7 6 consider scalar leptoquark solutions to these anomalies. To The R2 ð ; ; = Þ leptoquark is an electroweak dou- date the only known candidate that simultaneously explains blet and couples to both left-handed and right-handed SM quarks and leptons. Its Yukawa couplings with SM fer- both sets of anomalies is the S1 leptoquark [46,57,70],butit 2 − σ mions are only satisfies RKð Þ at [57]. In this work we show that the R2 leptoquark can provide a simultaneous solution at L − ¯ αϵ β − ¯ † 1 − σ consistent with all known constraints. R2 ¼ ðY2ÞabuaR2 αβPLLb ðY4ÞabeaR2PLQb þ H:c: In addition to the anomalous measurements of RDð Þ and ð6Þ RKð Þ , two other anomalies have generated interest: On the 0 one hand, the value of the angular observable P5 [71,72] We work in the basis, where the flavor eigenstates of down- and more generally the data of b → sμμ¯ point to a deviation type quarks and charged leptons coincide with their mass from the SM [73]. While these anomalies are intriguing eigenstates. In particular, the component of R2 with electric they are currently less clean signals of new physics due charge 2=3 couples right-handed charged leptons to left- to large hadronic uncertainties and the difficulty in esti- handed down-type quarks and right-handed up-type quarks 0 mating a signal for the P5 anomalies [72]. On the other to neutrinos and thus contributes to both b → s and the → hand, similar to the LFU ratios RDð Þ the LFU ratio b c processes. þ þ RJ=ψ ¼ ΓðBc → J=ψτνÞ=ΓðBc → J=ψμνÞ points to a For energies below the mass of the leptoquark, it is larger branching fraction to τ leptons compared to muons, convenient to write an effective Lagrangian to capture the but it is still consistent with the SM at 2 − σ due to the large relevant contributions beyond the SM. Using the Warsaw error bars [74]. We therefore leave the consideration of such basis [84] of the SM effective field theory (SMEFT), the anomalies to future work. relevant terms in the effective Lagrangian are The R2 leptoquark has quantum numbers ð3; 2; 7=6Þ with L qe ¯ γ ¯ γμ lu ¯ γ ¯ γμ respect to the SM gauge group SUð3Þ ×SUð2Þ ×Uð1Þ and ¼ CabcdðQa μQbÞðec edÞþCabcdðLa μLbÞðuc udÞ lequ1 j has been proposed as a cause of the RDð Þ anomalies ¯ ϵ ¯ k þ Cabcd ðLaebÞ jkðQcudÞ [10,75,76] with O(1) couplings as well as the RKð Þ lequ3 ¯ j σ ϵ kσμν anomalies with very large couplings through a new con- þ Cabcd ðLa μνebÞ jkðQc udÞ; ð7Þ tribution to the decay b → sμμ¯ [77–81]. These operators σ i γ γ are induced at the one-loop level and thus require undesir- where μν ¼ 2 ½ μ; ν with Wilson coefficients ably large couplings with at least one coupling needing to ðY4Þ ðY4Þ be a lot larger than 1. We reopen the case of this leptoquark Cqe ðm Þ¼− ab cd bdca R2 2 2 and find that a more promising route to what has previously mR2 been studied is to boost the denominator in Eq. (3),by Y2 Y2 2 lu − ð Þabð Þcd allowing the leptoquark to couple to electrons. As the C ðm 2 Þ¼ ð8Þ dbac R 2m2 relevant operator is generated at tree level, the required R2 couplings are quite small. The deviations in the LFU ratios lequ1 lequ3 ðY2ÞabðY4Þcd R ð Þ can be explained at the same time by introducing a C ðm Þ¼4C ðm Þ¼− ; ð9Þ D bcda R2 bcda R2 2m2 coupling of the R2 leptoquark to τ leptons. A mild tension R2 with the theoretically inferred constraint on BRðBc → τνÞ which are defined at the renormalization scale μ m , the S3 ¼ R2 [83] can be resolved by the introduction of the qe leptoquark, which can be motivated within a radiative mass of leptoquark R2. The vector Wilson coefficient Csbee neutrino mass model. contributes to b → see and thus modifies the LFU ratios The structure of this paper is as follows. In Sec. II we RKð Þ . This is illustrated in the left panel of Fig. 1. The blue- 1 − σ perform an effective field theory (EFT) analysis of the R2 shaded region indicates the -allowed region for RKð Þ. 1 For a fixed leptoquark mass mR2 ¼ TeV, the LFU ratio 2A model independent discussion of couplings to electrons and RK decreases when increasing the magnitude of the muons using effective field theory has been performed in Yukawa couplings jðY4ÞesðY4Þebj, thus increasing the qe Ref. [82]. magnitude of the Wilson coefficient Csbee.
035028-2 R2 AS A SINGLE LEPTOQUARK SOLUTION TO … PHYS. REV. D 100, 035028 (2019)
1 FIG. 1. Dependence of RDð Þ and RKð Þ on the relevant Yukawa couplings for fixed leptoquark mass mR2 ¼ TeV.
Similarly the scalar and tensor Wilson coefficients couplings ðY4Þes and ðY4Þeb differs, when attempting to lequ1;3 → τν explain both R ð Þ and R ð Þ simultaneously. Cντbc contribute to b c and thus modify the LFU K D A minimal set of Yukawa couplings to accommodate a ratios RDð Þ . As the final state neutrino is not measured, there is a contribution from all three flavors. The coupling simultaneous solution to RKð Þ and RDð Þ is ν μ to e;μ is accompanied by couplings to e and respectively 0 1 0 1 and hence there are additional constraints from lepton 00 0 0 ðY4Þ ðY4Þ B C B es eb C flavor violating processes. In order to avoid these additional Y2 ¼ @ 00ðY2Þ ν A;Y4 ¼ @ 00 0A; ν c τ constraints, we only consider couplings to τ. The depend- 00 0 00 ðY4Þτb ence of RD to the magnitude of the Yukawa couplings jðY2Þcντ ðY4Þτbj is illustrated in the right panel of Fig. 1. The ð10Þ blue-shaded region indicates the 1 − σ-allowed region for R ð Þ . The Yukawa couplings jðY2Þ ν j and jðY4Þτ j are D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic τ b which we focus on in the following. Before discussing the phenomenology of the R2 lepto- generally of order 1 with jðY2Þ ν ðY4Þτ j ∼ 1 and thus c τ b quark we briefly make a connection to the operators in the typically larger than the Yukawa couplings required to commonly used operator basis in B-physics. We limit our explain RKð Þ . As there is generally operator mixing, when discussion to the operators induced after integrating out the evolving the Wilson coefficients from the scale of the R2 leptoquark. In the weak effective theory, after integrat- leptoquark to the scale of the b-quark, the large Wilson ing out the Higgs, Z- and W-bosons and the top quark, the lequ1;3 ð Þ coefficients Cνττbc typically modify the result for RK and relevant operators in the effective Lagrangians governing thus the interesting parameter range for the Yukawa b → sll and b → clν decays are
4G α X L ffiffiffiF em l ¯γ l¯γμl l ¯γ l¯γμγ l sbll ¼ p VtbVts 4π ½C9 ðs μPLbÞð ÞþC10ðs μPLbÞð 5 Þ 2 l 4 X GF ij μ ¯ ij ¯ ij μν ¯ L lν ¼ − pffiffiffi V ½C ðc¯γ P bÞðl γμP ν ÞþC ðcP¯ bÞðl P ν ÞþC ðc¯σ P bÞðl σμνP ν Þ ; ð11Þ cb 2 cb V L i L j S L i L j T L i L j i;j
respectively, with the CKM mixing matrix elements Vij. coefficients and the calculation of most processes. We vary The Wilson coefficients in weak effective theory are related the magnitude of the four Yukawa couplings over the to the ones in SMEFT by range consistent with perturbativity and the explanation of
the R ð Þ and R ð Þ anomalies at 1 − σ and their phases over π qe D K e e Cbsee the whole allowed range ½0; 2π while fixing the mass C9 ¼ C10 ¼ pffiffiffi 2 α 1 VtbVtsGF em mR2 ¼ TeV. lequ1 τν τν Cν τ C τ ¼ 4C τ ¼ − pffiffiffi τ bc : ð12Þ III. EXPERIMENTAL CONSTRAINTS AND THE S T 2 2 VcbGF VIABLE PARAMETER SPACE In our numerical analysis we use the flavio package [85] In this section we first summarize the most significant for the renormalization group evolution of the Wilson constraints on the couplings of the R2 leptoquark in
035028-3 POPOV, SCHMIDT, and WHITE PHYS. REV. D 100, 035028 (2019)
1 − σ τ− → −γ → τν − → FIG. 2. Points that simultaneously explain RDð Þ and RKð Þ at with observables BRð e Þ,BRðBc Þ,BRðB K−eþτ−Þ shown in the top left, top right, and bottom left panels, respectively. The bottom right panel shows the relative size of the largest Yukawa couplings. The orange dashed (dotted) gridlines indicate the contributions to the Zττ coupling at the level of the 1 − σ 2 − σ 1 − σ ( ) experimental error. All points explain RKð Þ and RDð Þ at the level. Dark blue points satisfy all constraints. Light blue points satisfy strict limits on BRðτ− → e−γÞ but are excluded by other constraints. In the top right we show the three most stringent bounds on BRðBc → τνÞ inferred from different theoretical considerations [83,86,87] and note that only the most stringent one rules out the scenario.
Sec. III A, followed by a discussion of the viable parameter in the limit of vanishing final state electron mass and to space in Sec. III G. leading order in the quark masses in the loop. The branching ratio for the SM purely leptonic τ decay is τ → ν ν¯ 0 178 A. Constraints BRð e τ eÞ¼ . . In the numerical scan, we use the exact expression and impose the current limit on the We show the impact of the most relevant constraints on τ− → −γ 5 4 10−8 the parameter space satisfying a 1 − σ simultaneous sol- branching ratio BRð e Þ < . × obtained by HFLAV [89]. The HFLAV limit is less stringent than the ution for RKð Þ and RDð Þ in Fig. 2. The four most stringent constraints are posited by the decays τ → eγ, Bþ → limit quoted in the PDG, because it combines the BABAR Kþτþe−, Z → ττ, and B → τν. result [90] with the less stringent Belle result [91] while the c PDG [92] relies only on the former. The combined limit is τ − → e − γ less aggressive as Belle saw a small excess of this process B. (see Table 319 in Ref. [89]). Irrespective of whether the − − The radiative lepton flavor violating decay τ → e γ Belle result is included or not, the simultaneous explanation occurs at loop level. Its branching ratio takes the form [88] of both RKð Þ and RDð Þ is viable.
− − In the top left panel of Fig. 2 we show the branching ratio BRðτ → e γÞ 27α 1=2 ≃ em Y YT vs ðjðY4ÞebðY4ÞτbjÞ . For large couplings jðY4ÞebðY4Þτbj, τ− → −ν ν¯ 256π 2 4 ð 4 4 Þτe BRð e τ eÞ GFmR2 the branching ratio is dominated by the first term and thus X increases for increasing Yukawa couplings. For small 4 † mq − Y2 Y4V ð Þqτð Þeq ðY4Þτb, the Yukawa coupling ðY2Þcντ becomes large in 3 q¼u;c;t mτ order to explain R ð Þ as shown in the bottom right plot m2 2 D 1 − q and thus the second term in Eq. (13) dominates, which × ln 2 ð13Þ mR2 explains the increasing branching ratio for small
035028-4 R2 AS A SINGLE LEPTOQUARK SOLUTION TO … PHYS. REV. D 100, 035028 (2019) jðY4ÞebðY4Þτbj. The Belle II experiment [93] is expected to right panel of Fig. 2 as orange dashed (dotted) lines. The improve the sensitivity to τ → eγ by more than 1 order of dark blue points in the numerical scan do not lead to any magnitude to 3 × 10−9 (indicated by a dotted red line) and correction larger than the 1 − σ experimental uncertainties. thus probe a large part of the remaining parameter space. In reality, a full global fit to all electroweak observables is needed to impose a reliable constraint, and it is probable C. B + → K + τ + e − that significantly larger deviations to effective Z couplings can be accommodated. We leave such a work to the future Another constraint on the τ − e flavor violating proc- and comment here that even our pessimistic approach does esses originates from the semileptonic lepton flavor violat- − not rule out our model. ing B decay Bþ → Kþτþe . Its branching ratio satisfies þ þ þ − −5 BRðB → K τ e Þ < 1.5 × 10 [92]. The R2 leptoquark E. Bc → τν induces the vector operator The R2 leptoquark contributes to Bc → τν via the same couplings relevant to R ð Þ , since the same scalar operator qe − ðY4ÞesðY4Þτb D Csbτe ¼ 2 ; ð14Þ → τν 2 contributes to both RDð Þ and Bc . In the top right panel mR2 of Fig. 2 we show the prediction for BRðBc → τνÞ. We find which contributes to Bþ → Kþτþe− and thus constrains the branching ratios between 15% and 23% for the region of parameter space that explains both R ð Þ and R ð Þ at 1 − σ. simultaneous explanation of RKð Þ and RDð Þ . As we dem- D K onstrate in the bottom left panel of Fig. 2, it provides a Thus limits on this process pose a direct constraint on the moderate constraint on the parameter space that simulta- explanation of RDð Þ . Several groups inferred limits on BRðB → τνÞ < neously explains RKð Þ and RDð Þ . The region excluded by c − 0 1 0 6 Bþ → Kþτþe is also excluded by τ → eγ. ½ . ; . [52,83,86,96,97] via different theoretical argu- ments. In the top right panel of Fig. 2 we indicate the three D. Z decays most stringent theoretically argued upper limits: 10% [83], 20% [87], and 30% [86], which are indicated by a dotted, The R2 leptoquark also contributes to several Z-boson dash-dotted, and dashed red line, respectively. In particular, → ττ decay processes. In particular, its contribution to Z is Ref. [83] found that the branching ratio can be at most 10%, τ significant due to the large couplings to leptons. which is in tension with the viable parameter space of the Approximate expressions for the left-handed and right- R2 leptoquark explanation of R ð Þ . However, there is some τ D handed couplings of the Z-boson to leptons controversy over this constraint that relies on the proba- 2 bility that a bottom quark hadronizes with a charmed quark jðY4Þτ j 3 δgτ ≃ b − 1 that has not yet been measured at the LHC. There is Reð LÞ 16π2 2 xt½ þ ln xt currently an ongoing discussion on how the probability at 23 128 1 the LHC can be inferred from LEP measurements and 8 − 2θ þ xZ 12 þ 9 þ ln xt 3 ln xZ sin W Monte Carlo simulations, where the authors of the first paper [83,87] advocate values in the range 10%–20% ð15Þ compared to some recent papers that advocate liberal – 2 constraints in the range 39% 60% [96,97]. The large range jðY2Þ ν j 1 1 Reðδgτ Þ ≃ c τ x − ln x of values comes from the fact that the limit is quite sensitive R 16π2 Z 12 2 Z to the ratio of hadronization probabilities f =f , that is, the c u 1 2 probability of hadronizing with a charm or up quark, 2θ þ 18 þ 3 ln xZ sin W ð16Þ respectively. Figure 5 in Ref. [97], in particular, shows this dependency where a bound of 10% corresponds to −3 fc=fu ∼ 4 × 10 and the most liberal bound of 60% in terms of the Weinberg angle θW and the ratios xt ¼ 2 2 corresponds to a ratio that is a factor of 5 smaller, ðmt=mR2 Þ and xZ ¼ðmZ=mR2 Þ are obtained by expand- −4 f =f ∼ 8 × 10 . Reference [97] use PYTHIA8 [98,99], ing the expressions given in Ref. [94] to leading order in x , c u t which relies on experimental data to tune hadronization, to x and the quark mixing angles by taking V ≃ 1. Z tb estimate the hadronization fraction and derive a fairly The LEP experiments measured the Z-boson couplings 1 − σ δ τ liberal lower bound of 39%. However, they and others precisely [95] with uncertainties of jReð gLÞj < [87] express skepticism of the true lower bound. We remain 5 8 10−4 τ . × for couplings to left-handed leptons and agnostic with regards to this constraint and show our δ τ ≤ 6 2 10−4 τ jReð gRÞj . × for right-handed leptons. This prediction for this branching ratio in the top right panel translates to a constraint on the magnitude of the Yukawa of Fig. 2. ≲ 1 0 1 4 couplings jðY4Þτbj and jðY2Þcντ j of jðY4Þτbj . ð . Þ and Even the most stringent constraint of 10% may be ≲ 2 6 3 7 1 − σ 2 − σ jðY2Þcντ j . ð . Þ using ( ) experimental avoided by extending the model with a S3 leptoquark, as uncertainties, respectively. This is indicated in the bottom we discuss in Sec. IV.
035028-5 POPOV, SCHMIDT, and WHITE PHYS. REV. D 100, 035028 (2019)
ð Þ F. Other constraints Finally, as the deviation in the ratios RK originates Apart from the discussed constraints we also studied entirely from a modification to the branching fraction of → ð Þ þ − possible constraints from several other processes and we B K e e one may wonder whether this deviation is briefly summarize the results. The limits obtained from consistent with measurements of the branching ratio itself. 1σ lepton flavor violating semileptonic τ decays, τ → eP with Under the given assumptions taking the boundaries (central value) the experimental value implies an increase a pseudoscalar meson P K, π, are always substantially − ¼ dBrðBþ→Kþeþe Þ weaker than the limit from τ → eγ and thus we do not of the binned differential branching ratio h dq2 i × report them here. Furthermore, we considered leptonic ð1.1 GeV2 x3 − 11x2 þ 4x 3x3 S0ðxÞ¼ − ln x: ð20Þ G. Viable parameter space 4 x − 1 2 2 x − 1 3 ð Þ ð Þ We show the viable parameter space in Fig. 3 and the bsbs bottom right panel of Fig. 2. For a fixed leptoquark mass of The contribution of R2 to CVLL can be expressed in terms of e m ¼ 1 TeV, the bottom right panel of Fig. 2 shows that the Wilson coefficient C9. A simple order of magnitude R2 estimate shows that Cbsbs is several orders of magnitude an aggressive constraint from Z decays restricts two of the VLL;R2 ∈ 0 44 1 0 Yukawa couplings to the range jðY4Þτbj ½ . ; . and smaller as the SM contribution for the interesting R2 jðY2Þ ν j ∈ ½1.0; 2.6 , respectively. All quoted ranges are leptoquark mass range c τ approximate and are only intended to give an indication. bsbs 2 C α The product is also constrained by the need to explain R ð Þ VLL;R2 em mR2 e D ≃ C9 : ð21Þ 1 − σ ∈ 0 88 1 3 bsbs 2π at the level to the range jðY2Þcντ ðY4Þτbj ½ . ; . . CVLL;SM mW The product is almost purely imaginary with argððY2Þcντ × − ¯ ∈ 0 45 0 54 π We independently checked the contribution to Bs Bs ðY4ÞτbÞ ½ . ; . , irrespective of the experimental mixing using flavio with the same result. constraints, which confirms previous findings [10,75,76]. 035028-6 R2 AS A SINGLE LEPTOQUARK SOLUTION TO … PHYS. REV. D 100, 035028 (2019) FIG. 3. Relevant parameter range of the Yukawa couplings. The top left panel shows the phase vs the magnitude for ðY4ÞebðY4Þes and the top right panel the phase vs the magnitude of ðY2Þcντ ðY4Þτb. In the bottom panel we plot the absolute values of the Yukawa couplings 1 − σ entering RKð Þ against each other. All points explain RKð Þ and RDð Þ at the level. Dark blue points satisfy all experimental constraints. Light blue points satisfy strict limits on τ− → e−γ but are excluded by other constraints. The other two Yukawa couplings are generally smaller The R2 leptoquark especially generates the Wilson ∈ 0 11 0 37 ∈ 0 015 0 055 with jðY4Þebj ½ . ; . , jðY4Þesj ½ . ; . . Their coefficients ∈ product constrained to the narrow range jðY4ÞebðY4Þesj RL ððY4ÞesVcs þðY4ÞebVcbÞðY4ÞτbVtb ½0.0047; 0.0070 after imposing all experimental con- ϵτ ¼ pffiffiffi ; e 4 2 2 straints. This is shown in the bottom panel of Fig. 3. GFmR2 For the points that satisfy all experimental constraints, the Y V Y4 V Y4 V ϵRL ð 4Þτb cbðð Þffiffiffies ts þð Þeb tbÞ real part of the product is generally negative; the argument eτ ¼ p 2 : ð23Þ 4 2G m ∈ F R2 is weakly constrained to the range argððY4ÞebðY4ÞesÞ 0 47 1 0 qe → τ ∓ ½ . ; . . This implies that the Wilson coefficient Csbee We find that the decays t c e are generally tiny with is generally positive. The absolute valuep offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the product is by branching ratios contrast confined to a narrow range Y4 Y ∈ − −9 − −11 jð Þebð 4Þesj BRðt → cτþe Þ≲2×10 BRðt → cτ eþÞ≲8×10 4 7 7 0 10−3 ≪ ½ . ; . × . The hierarchy jðY4Þesj jðY4Þebj can be understood as follows. The constraint from Z-boson decays ð24Þ ≲ 1 0 constrains the coupling ðY4Þτb . and thus the coupling for the parameter space, which explains both RKð Þ and RDð Þ ðY2Þcντ has to be larger than 1 in order to explain RDð Þ . This 1 − σ 1 at the level with a leptoquark mass mR2 ¼ TeV and Y2 in turn leads to a stronger constraint on jð Þcντ j from thus we do not expect any signal at the LHC experiments. τ → eγ, because the suppression from the ratio ms=mτ is 2 2 IV. ALLEVIATING THE (POSSIBLE) TENSION compensated by the large logarithm, lnðms =mR2 Þ. Finally, we obtain a prediction for the branching ratios WITH BRðBc → τνÞ VIA A NEUTRINO → lþl− → lþl− ≃ MASS MODEL of the decays t c i j [111],BRðt i j þ cÞ 1.3 ϵRL 2 ϵRL ¯ 3 48π2 j ij j , where ij is the Wilson coefficient of the It is well known that the S3 ∼ ð3; 3; 1=3Þ leptoquark effective operator with the Yukawa interaction pffiffiffi 3 L −2 2 ϵRL ¯ γ ¯γμ The collider phenomenology of an S3 leptoquark has been ¼ GF ij ½ei μPRej ½c PLt : ð22Þ studied in Ref. [112]. 035028-7 POPOV, SCHMIDT, and WHITE PHYS. REV. D 100, 035028 (2019) q q FIG. 4. Neutrino mass from leptoquarks in the loop. The superscripts q of R2 and S3 denote the electromagnetic charge of the different components of R2 and S3, respectively. L − c ϵijϵkl 1 μ Σ0 2 S3 ¼ ðYSÞabQaiPLLbkS3jl þ H:c: ð25Þ h i mR2 ðmνÞ ≃ ðY2Þ m F ij 16π2 m2 − m2 ki k m2 R2 S k m2 and mass mS contributes to the Wilson coefficients − S ↔ F 2 Vkl ðYSÞlj þði jÞð28Þ mk 1 3ðY Þ ðY Þ 3 ðY Þ ðY Þ in the limit of small mixing between R2 and S3 leptoquarks, Clq ¼ S ab S cd Clq ¼ S ab S cd ð26Þ dbca 8m2 dbca 8m2 which is generated by the trilinear potential term S S ijk μΣ R2iS3jk. The Yukawa coupling matrices are defined in the basis where charged lepton and down-type quark mass of the vector operators [113–115] matrices are diagonal. Thus the loop diagram with up-type quarks in the loop is proportional to the CKM mixing matrix element Vkl. Roman indices i, j, k, l indicate lq1 μ L ¯ γ ¯ γ flavor, mk the up-type quark mass, mR2 ;mS are R2 and S3 ¼ CabcdðLa μLbÞðQc QdÞ leptoquark masses, respectively, and hΣ0i the vacuum lq3 ¯ γ τI ¯ γμτI þ CabcdðLa μ LbÞðQc QdÞ; ð27Þ expectation value of the neutral component of Σ. The loop function FðxÞ is defined as which can help alleviate the possible tension with x ln x FðxÞ¼1 − : ð29Þ BRðBc → τνÞ at the cost of a contribution to the decay x → νν B K . Such a model involving two leptoquarks can be The more general expression for a general mixing angle motivated by a neutrino mass model. In this section we between the R2 and S3 leptoquarks is given in Appendix. sketch out how this is possible leaving a detailed analysis to The two-loop contribution features a similar flavor structure. future work. V. CONCLUSION A. Neutrino masses We demonstrate that the R2 ∼ ð3; 2; 7=6Þ leptoquark is a Just extending the SM with R2 and S3 leptoquarks is new single particle candidate for explaining the anomalous not sufficient to generate nonzero neutrino masses.4 To lepton-flavor-universality ratios RKð Þ and RDð Þ . There is keep our model minimal we extend our two leptoquark possibly a mild tension with the theoretically derived limit 2 extension of the SM by a single particle, which is a SUð ÞL on the branching ratio BRðBc → τνÞ. Since we require the Σ ∼ 1 4 3 quadruplet with quantum numbers ð ; ; 2Þ. Then branching ratio of Bc → τν to be within a relatively narrow neutrino masses are generated at the one-loop level as range, the viability of the R2 leptoquark as a single particle shown in the left panel of Fig. 4. There is also a two-loop solution to these anomalies is a directly falsifiable scenario. contribution that is shown on the right. Another promising probe of the viable parameter space of If the two-loop diagram can be neglected, neutrino masses our model is BRðτ → eγÞ, where the projected sensitivity are approximately given by their one-loop contribution5 for Belle II is expected to improve by an order of magnitude [93]. Another intriguing possibility is whether the large 4See Ref. [116] for a discussion of different possibilities in the imaginary couplings needed to explain RDð Þ leave an context of a grand unified theory. permanent neutron electric dipole moment that is detectable 5Further details we relegate to the Appendix. in future experiments [117]. 035028-8 R2 AS A SINGLE LEPTOQUARK SOLUTION TO … PHYS. REV. D 100, 035028 (2019) The tension with the disputed aggressive limit on where ð−1Þqx is −1 for H and Σ and þ1 for the other scalar BRðBc → τνÞ can be alleviated through the introduction fields. Thus the general form for neutrino masses at one- of a S3 leptoquark, which can be motivated by a neutrino loop order is given by mass model as discussed in Sec. A. This suggests that even 1 m2 kl if future analysis indeed rules out the R2 leptoquark as a † Si ðmνÞ ¼ ðUsÞ ðY2Þ m F V single leptoquark solution to anomalous B decays, it still mn 16π2 R2si km k 2 kl mk can play a substantial role in an extended model. × Y U m ↔ n : A3 ð SÞlnð sÞsiS3 þð Þ ð Þ ACKNOWLEDGMENTS 2=3 2=3 0 The mixing between R2 and S3 is generated by hΣ i We thank John Gargalionis and David Straub and is obtained by diagonalizing the charge 2=3 leptoquark for useful discussions. We thank Andrew Akeroyd and 2 3 2 3 mass matrix, which is given in the ðR = ;S = Þ basis Nejc Kosnik for useful comments to the first version and 2 3 Monika Blanke for pointing out a typo in the first version. 2 2 ! 2 vH vΣ vΣffiffi μ þ λ þ λ Σ μ p O. P. is supported by the National Research Foundation 2 R HR 2 R 2 2 MS ¼ 2 2 of Korea Grants No. 2017K1A3A7A09016430 and 2 v v μ pvΣffiffi μ λ H λ Σ No. 2017R1A2B4006338. TRIUMF receives federal fund- 2 S þ HS 2 þ SΣ 2 ing via a contribution agreement with the National Research 2 2 ¼ UTDiagðm ;m ÞU ðA4Þ Council of Canada and the Natural Science and Engineering S S1 S2 S ffiffiffi ffiffiffi Research Council of Canada. This research includes compu- 0 p 0 p with v H = 2 and vΣ Σ = 2, the masses m tations using the computational cluster Katana supported by H ¼h i ¼h i Si 2 2 Research Technology Services at UNSW Sydney. and the × unitary mixing matrix US which defines the mass eigenstates Si in terms of the flavor eigenstates APPENDIX: LEPTOQUARK MIXING AND ! 2=3 NEUTRINO MASSES S1 R2 cos θ sin θ ¼ U U ¼ : ðA5Þ S 2=3 S − θ θ The relevant terms in the scalar potential V ¼ V0 þV1 are S2 S3 sin cos X λ X 2 2 x 4 2 2 −1 qx μ λ A straightforward calculation results in the following V0 ¼ ð Þ xjxj þ 2 jxj þ xyjxj jyj H;R2;S3; H;R2;S3; expressions for the rotation angle and the masses Σ∈x Σ∈fx 2 2 2 2 v vΣ μ þ μ þ H ðλ þ λ Þþ ðλ Σ þ λ ΣÞ m2 ¼ R S 2 HR HS 2 R S S1;2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 μ2 − μ2 vH λ − λ vΣ λ − λ 4μ2 vΣ 2 R S þ 2 ð HR HSÞþ 2 ð RΣ SΣÞ þ 2 : ðA7Þ μ For small and thus small mixing, the square of the masses mR2 and mS in the main part of the text can be identified with the 2 2 μ2 λ 2 2 λ 2 2 2 μ2 λ 2 2 λ 2 2 diagonal elements of the scalar mass matrix MS, mR2 ¼ R þ HRvH= þ RΣvΣ= and mS ¼ S þ HSvH= þ SΣvΣ= . [1] J. P. Lees et al. (BABAR Collaboration), Phys. Rev. Lett. [3] M. Huschle et al. (Belle Collaboration), Phys. Rev. D 92, 109, 101802 (2012). 072014 (2015). [2] J. P. Lees et al. (BABAR Collaboration), Phys. Rev. D 88, [4] Y. Sato et al. (Belle Collaboration), Phys. Rev. D 94, 072012 (2013). 072007 (2016). 035028-9 POPOV, SCHMIDT, and WHITE PHYS. REV. D 100, 035028 (2019) [5] S. Hirose et al. (Belle Collaboration), Phys. Rev. Lett. 118, [36] I. de Medeiros Varzielas and J. Talbert, Eur. Phys. J. C 79, 211801 (2017). 536 (2019). [6] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 115, [37] J.-H. Sheng, R.-M. Wang, and Y.-D. Yang, Int. J. Theor. 111803 (2015); 115, 159901(E) (2015). Phys. 58, 480 (2019). [7] R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 97, [38] S. Balaji, R. Foot, and M. A. Schmidt, Phys. Rev. D 99, 072013 (2018). 015029 (2019). [8] G. Caria (Belle Collaboration), talk given at Moriond [39] M. Freytsis, Z. Ligeti, and J. T. Ruderman, Phys. Rev. D (2019). 92, 054018 (2015). [9] J. A. Bailey et al. (MILC Collaboration), Phys. Rev. D 92, [40] A. Biswas, D. K. Ghosh, A. Shaw, and S. K. Patra, Int. J. 034506 (2015). Mod. Phys. A 34, 1950112 (2019). [10] M. Tanaka and R. Watanabe, Phys. Rev. D 87, 034028 [41] J. Zhang, Y. Zhang, Q. Zeng, and R. Sun, Eur. Phys. J. C (2013). 79, 164 (2019). [11] G. Hiller and F. Kruger, Phys. Rev. D 69, 074020 (2004). [42] U. Aydemir, T. Mandal, and S. Mitra, arXiv:1902.08108. [12] B. Capdevila, S. Descotes-Genon, J. Matias, and J. Virto, J. [43] T. Mandal, S. Mitra, and S. Raz, Phys. Rev. D 99, 055028 High Energy Phys. 10 (2016) 075. (2019). [13] B. Capdevila, A. Crivellin, S. Descotes-Genon, J. Matias, [44] S. Bansal, R. M. Capdevilla, and C. Kolda, Phys. Rev. D and J. Virto, J. High Energy Phys. 01 (2018) 093. 99, 035047 (2019). [14] C. Bobeth, G. Hiller, and G. Piranishvili, J. High Energy [45] S. Iguro, T. Kitahara, Y. Omura, R. Watanabe, and K. Phys. 12 (2007) 040. Yamamoto, J. High Energy Phys. 02 (2019) 194. [15] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 122, [46] M. Bauer and M. Neubert, Phys. Rev. Lett. 116, 141802 191801 (2019). (2016). [16] M. Bordone, G. Isidori, and A. Pattori, Eur. Phys. J. C 76, [47] S. Fajfer and N. Košnik, Phys. Lett. B 755, 270 440 (2016). (2016). [17] M. Prim (Belle Collaboration), talk given at Moriond [48] B. Bhattacharya, A. Datta, J.-P. Gu´evin, D. London, and R. (2019). Watanabe, J. High Energy Phys. 01 (2017) 015. [18] R. Aaij et al. (LHCb Collaboration), J. High Energy Phys. [49] S. Sahoo, R. Mohanta, and A. K. Giri, Phys. Rev. D 95, 08 (2017) 055. 035027 (2017). [19] R. Barlow, arXiv:physics/0306138. [50] D. Bečirević, S. Fajfer, N. Košnik, and O. Sumensari, [20] A. J. Buras, J. Girrbach-Noe, C. Niehoff, and D. M. Straub, Phys. Rev. D 94, 115021 (2016). J. High Energy Phys. 02 (2015) 184. [51] D. Bečirević,N.Košnik, O. Sumensari, and R. Z. Funchal, [21] B. Gripaios, M. Nardecchia, and S. A. Renner, J. High J. High Energy Phys. 11 (2016) 035. Energy Phys. 05 (2015) 006. [52] X.-Q. Li, Y.-D. Yang, and X. Zhang, J. High Energy Phys. [22] H. Päs and E. Schumacher, Phys. Rev. D 92, 114025 08 (2016) 054. (2015). [53] B. Chauhan and B. Kindra, arXiv:1709.09989. [23] R. Barbieri, C. W. Murphy, and F. Senia, Eur. Phys. J. C [54] L. Calibbi, A. Crivellin, and T. Li, Phys. Rev. D 98, 77, 8 (2017). 115002 (2018). [24] M. Duraisamy, S. Sahoo, and R. Mohanta, Phys. Rev. D [55] L. Di Luzio, A. Greljo, and M. Nardecchia, Phys. Rev. D 95, 035022 (2017). 96, 115011 (2017). [25] O. Sumensari, Proc. Sci., EPS-HEP2017 (2017) 245. [56] D. Buttazzo, A. Greljo, G. Isidori, and D. Marzocca, J. [26] D. Aloni, A. Dery, C. Frugiuele, and Y. Nir, J. High Energy High Energy Phys. 11 (2017) 044. Phys. 11 (2017) 109. [57] Y. Cai, J. Gargalionis, M. A. Schmidt, and R. R. Volkas, J. [27] O. Sumensari, in Proceedings, 52nd Rencontres de Mor- High Energy Phys. 10 (2017) 047. iond on Electroweak Interactions and Unified Theories: La [58] A. Crivellin, D. Müller, and T. Ota, J. High Energy Phys. Thuile, Italy (2017), pp. 445–448, ISBN: 9791096879021. 09 (2017) 040. [28] G. Hiller and I. Nisandzic, Phys. Rev. D 96, 035003 [59] D. Müller, EPJ Web Conf. 179, 01015 (2018). (2017). [60] C. Cornella, J. Fuentes-Martin, and G. Isidori, J. High [29] G. D’Amico, M. Nardecchia, P. Panci, F. Sannino, A. Energy Phys. 07 (2019) 168. Strumia, R. Torre, and A. Urbano, J. High Energy Phys. 09 [61] L. Da Rold and F. Lamagna, J. High Energy Phys. 03 (2017) 010. (2019) 135. [30] J. M. Cline, Phys. Rev. D 97, 015013 (2018). [62] M. Schmaltz and Y.-M. Zhong, J. High Energy Phys. 01 [31] S.-Y. Guo, Z.-L. Han, B. Li, Y. Liao, and X.-D. Ma, Nucl. (2019) 132. Phys. B928, 435 (2018). [63] B. Fornal, S. A. Gadam, and B. Grinstein, Phys. Rev. D 99, [32] A. Crivellin, D. Müller, A. Signer, and Y. Ulrich, Phys. 055025 (2019). Rev. D 97, 015019 (2018). [64] N. Assad, B. Fornal, and B. Grinstein, Phys. Lett. B 777, [33] A. K. Alok, B. Bhattacharya, A. Datta, D. Kumar, J. 324 (2018). Kumar, and D. London, Phys. Rev. D 96, 095009 (2017). [65] M. Blanke and A. Crivellin, Phys. Rev. Lett. 121, 011801 [34] A. Angelescu, D. Bečirević, D. A. Faroughy, and O. (2018). Sumensari, J. High Energy Phys. 10 (2018) 183. [66] D. Bečirević, I. Doršner, S. Fajfer, N. Košnik, D. A. [35] I. De Medeiros Varzielas and S. F. King, Phys. Rev. D 99, Faroughy, and O. Sumensari, Phys. Rev. D 98, 055003 095029 (2019). (2018). 035028-10 R2 AS A SINGLE LEPTOQUARK SOLUTION TO … PHYS. REV. D 100, 035028 (2019) [67] A. Azatov, D. Bardhan, D. Ghosh, F. Sgarlata, and E. Electroweak Group, SLD Heavy flavor Group), Phys. Rep. Venturini, J. High Energy Phys. 11 (2018) 187. 427, 257 (2006). [68] A. Azatov, D. Barducci, D. Ghosh, D. Marzocca, and L. [96] M. Blanke, A. Crivellin, S. de Boer, M. Moscati, U. Ubaldi, J. High Energy Phys. 10 (2018) 092. Nierste, I. Nišandžić, and T. Kitahara, Phys. Rev. D 99, [69] Z.-R. Huang, Y. Li, C.-D. Lu, M. A. Paracha, and C. Wang, 075006 (2019). Phys. Rev. D 98, 095018 (2018). [97] D. Bardhan and D. Ghosh, Phys. Rev. D 100, 011701 [70] O. Popov and G. A. White, Nucl. Phys. B923, 324 (2017). (2019). [71] S. Descotes-Genon, J. Matias, and J. Virto, Phys. Rev. D [98] T. Sjostrand, S. Mrenna, and P. Z. Skands, J. High Energy 88, 074002 (2013). Phys. 05 (2006) 026. [72] D. Guadagnoli, Mod. Phys. Lett. A 32, 1730006 (2017). [99] T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. [73] S. Descotes-Genon, L. Hofer, J. Matias, and J. Virto, J. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and High Energy Phys. 06 (2016) 092. P. Z. Skands, Comput. Phys. Commun. 191, 159 (2015). [74] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 120, [100] R. Fleischer, in High-Energy Physics. Proceedings, 4th 121801 (2018). Latin American CERN-CLAF School, Vina del Mar, Chile, [75] I. Doršner, S. Fajfer, N. Košnik, and I. Nišandžić, J. High 2007 (2008), pp. 105–157, https://doi.org/10.5170/CERN- Energy Phys. 11 (2013) 084. 2008-004. [76] Y. Sakaki, M. Tanaka, A. Tayduganov, and R. Watanabe, [101] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); Phys. Rev. D 88, 094012 (2013). 65, 1772(E) (1981). [77] S. Sahoo and R. Mohanta, Phys. Rev. D 91, 094019 [102] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. (2015). 107, 201802 (2011). [78] C.-H. Chen, T. Nomura, and H. Okada, Phys. Rev. D 94, [103] J. P. Lees et al. (BABAR), Phys. Rev. D 86, 032012 115005 (2016). (2012). [79] U. K. Dey, D. Kar, M. Mitra, M. Spannowsky, and A. C. [104] R. Aaij et al. (LHCb Collaboration), J. High Energy Phys. Vincent, Phys. Rev. D 98, 035014 (2018). 06 (2014) 133. [80] D. Bečirević and O. Sumensari, J. High Energy Phys. 08 [105] R. Aaij et al. (LHCb 133), J. High Energy Phys. 10 (2015) (2017) 104. 034. [81] B. Chauhan, B. Kindra, and A. Narang, Phys. Rev. D 97, [106] J. A. Bailey et al. (Fermilab Lattice and MILC Collabo- 095007 (2018). rations), Phys. Rev. Lett. 115, 152002 (2015). [82] G. Hiller and M. Schmaltz, J. High Energy Phys. 02 (2015) [107] D. Du, A. X. El-Khadra, S. Gottlieb, A. S. Kronfeld, J. 055. Laiho, E. Lunghi, R. S. Van de Water, and R. Zhou, Phys. [83] A. G. Akeroyd and C.-H. Chen, Phys. Rev. D 96, 075011 Rev. D 93, 034005 (2016). (2017). [108] C. Bouchard, G. P. Lepage, C. Monahan, H. Na, and J. [84] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Shigemitsu (HPQCD Collaboration), Phys. Rev. Lett. 111, J. High Energy Phys. 10 (2010) 085. 162002 (2013); 112, 149902(E) (2014). [85] D. M. Straub, arXiv:1810.08132. [109] A. Khodjamirian and A. V. Rusov, J. High Energy Phys. 08 [86] R. Alonso, B. Grinstein, and J. M. Camalich, Phys. Rev. (2017) 112. Lett. 118, 081802 (2017). [110] S. Jäger and J. M. Camalich, Phys. Rev. D 93, 014028 [87] A. Akeroyd (private communication). (2016). [88] L. Lavoura, Eur. Phys. J. C 29, 191 (2003). [111] S. Davidson, M. L. Mangano, S. Perries, and V. Sordini, [89] Y. Amhis et al. (HFLAV Collaboration), Eur. Phys. J. C 77, Eur. Phys. J. C 75, 450 (2015). 895 (2017). [112] E. Alvarez, L. Da Rold, A. Juste, M. Szewc, and T. V. [90] B. Aubert et al. (BABAR), Phys. Rev. Lett. 104, 021802 Schroeder, J. High Energy Phys. 12 (2018) 027. (2010). [113] I. Doršner, S. Fajfer, D. A. Faroughy, and N. Košnik, J. [91] K. Hayasaka et al. (Belle), Phys. Lett. B 666, 16 (2008). High Energy Phys. 10 (2017) 188. [92] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, [114] C.-H. Chen, T. Nomura, and H. Okada, Phys. Lett. B 774, 030001 (2018). 456 (2017). [93] W. Altmannshofer et al. (Belle-II Collaboration), [115] D. Das, C. Hati, G. Kumar, and N. Mahajan, Phys. Rev. D arXiv:1808.10567. 94, 055034 (2016). [94] P. Arnan, D. Becirevic, F. Mescia, and O. Sumensari, J. [116] I. Doršner, S. Fajfer, and N. Košnik, Eur. Phys. J. C 77, 417 High Energy Phys. 02 (2019) 109. (2017). [95] S. Schael et al. (ALEPH, DELPHI, L3, OPAL, SLD [117] W. Dekens, J. de Vries, M. Jung, and K. K. Vos, J. High Collaborations, LEP Electroweak Working Group, SLD Energy Phys. 01 (2019) 069. 035028-11