PRLPRLTAOPhysical Review LettersPhys. Rev. Lett.0031-90071079-7114American Physical Society10.1103/PhysRevLett.122.011805LETTERSElementary Particles and FieldsImportance of Loop Effects in Explaining the Accumulated Evidence for New Physics in B Decays with a Vector LeptoquarkCrivellinAndreas^{*}Paul Scherrer Institut, CH-5232 Villigen PSI, SwitzerlandGreubChristoph^{†}SaturninoFrancesco^{‡}Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, CH-3012 Bern, SwitzerlandMüllerDario^{§}Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland and Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

andreas.crivellin@cern.ch

greub@itp.unibe.ch

saturnino@itp.unibe.ch

dario.mueller@psi.ch

11January201911January2019122101180520July2018Published by the American Physical Society2019authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP^{3}.

In recent years experiments revealed intriguing hints for new physics (NP) in B decays involving b→cτν and b→sℓ+ℓ- transitions at the 4σ and 5σ level, respectively. In addition, there are slight disagreements in b→uτν and b→dμ+μ- observables. While not significant on their own, they point in the same direction. Furthermore, Vus extracted from τ decays shows a slight tension (≈2.5σ) with its value determined from Cabibbo-Kobayashi-Maskawa unitarity, and an analysis of BELLE data found an excess in Bd→τ+τ-. Concerning NP explanations, the vector leptoquark SU(2) singlet is of special interest since it is the only single particle extension of the standard model which can (in principle) address all the anomalies described above. For this purpose, large couplings to τ leptons are necessary and loop effects, which we calculate herein, become important. Including them in our phenomenological analysis, we find that neither the tension in Vus nor the excess in Bd→τ+τ- can be fully explained without violating bounds from K→πνν¯. However, one can account for b→cτν and b→uτν data finding intriguing correlations with Bq→τ+τ- and K→πνν¯. Furthermore, the explanation of b→cτν predicts a positive shift in C7 and a negative one in C9, being nicely in agreement with the global fit to b→sℓ+ℓ- data. Finally, we point out that one can fully account for b→cτν and b→sℓ+ℓ- without violating bounds from τ→ϕμ, ϒ→τμ, or b→sτμ processes.

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung10.13039/501100001711PZ00P2_154834200020_175449/1Introduction.—

So far, the LHC has not directly observed any particles beyond the standard model (SM). However, intriguing hints for lepton flavor universality (LFU) violating new physics (NP) have been acquired.

b→s(d)ℓ+ℓ-.—The ratios R(K(*))=Br[B→K(*)μ+μ-]Br[B→K(*)e+e-][1,2] indicate LFU violation with a combined significance of ≈4σ[3–8]. Taking also into account all other b→sμ+μ- observables, like the angular observable P5′[9] in the decay B→K*μ+μ-, the global fit of the Wilson coefficients to all available data even shows compelling evidence [10] for NP (>5σ).

Concerning b→dℓ+ℓ- transitions, the theoretical analysis of Ref. [11] shows that the LHCb measurement of B→πμ+μ-[12] 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→sℓ+ℓ- under the assumption of a Vtd/Vts-like scaling of the NP effect [13]. In other words, an effect of the same order and sign as in b→sℓ+ℓ-, relative to the SM, is preferred. Furthermore, an (unpublished) analysis of BELLE data found an excess in Bd→τ+τ-[14].

b→c(u)τν.—The ratios R(D(*))=Br[B→D(*)τν]Br[B→D(*)ℓν]withℓ={e,μ},which measure LFU violation in the charged current by comparing τ modes with light leptons (ℓ=e, μ), differ in combination from their SM predictions by ≈4σ[15]. Also, the ratio R(J/ψ)=Br[Bc→J/ψτν]Br[Bc→J/ψμν][16] exceeds the SM prediction in agreement with the expectations from R(D(*))[17,18].

Concerning b→uτν transitions, the theory prediction for B→τν crucially depends on Vub. While previous lattice calculations resulted in rather small values of Vub, recent calculations give a larger value (see Ref. [19] for an overview). However, the measurement is still above the SM prediction by more than 1σ, as can be seen from the global fit [20]. In R(π)=Br[B→πτν]Br[B→πℓν]there is also a small disagreement between theory [21] and experiment [22] which does not depend on Vub. These results are not significant on their own but lie again above the SM predictions like in the case of b→cτν.

Vusτ.—Vus extracted from τ lepton decays (Vusτ) shows a tension of 2.5σ compared to the value of Vus determined from Cabibbo-Kobayashi-Maskawa (CKM) unitarity (Vusuni) [15,23].

The only possible single particle explanation, which can (at least in principle) address all these anomalies is the vector leptoquark (LQ) SU(2)L singlet V1 with hypercharge [24]-4/3[25–31] arising in the famous Pati-Salam model [32]: This LQ can explain b→cτν data without violating bounds from b→sνν¯ and/or direct searches, provides (at tree level) a left-handed solution to b→sℓ+ℓ- data, and does not lead to proton decay. Therefore, a sizable effect in b→uτν and b→dℓ+ℓ- is straightforward, and also an explanation of Vusτ could be possible. A huge enhancement of b→sτ+τ- rates is predicted as well [33], making an amplification of Bd→τ+τ- possible.

Several attempts to construct a UV completion for this LQ to address the anomalies have been made [34–44]. In order to fully account for the b→cτν data (while respecting perturbativity), one needs sizable couplings to third generation leptons and V1 generates, via SU(2)L invariance, also large contributions to the operators didjττ and uiujντντ at tree level. These operators give rise to couplings of down quarks to neutrinos or light charged leptons at loop level (see Fig. 1).

110.1103/PhysRevLett.122.011805.f1

Feynman diagrams depicting the one-loop contributions of the vector LQ singlet to C7/8sb, b→sℓ+ℓ-, τ→μνν¯, and b→sνν¯ (from left to right).

In this Letter we will calculate these loop effects [45], which turn out to be not only numerically important but also give 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).

Model and one-loop effects.—

We work in a simplified model extending the SM by a vector LQ SU(2)L singlet with hypercharge -4/3, mass M, and interactions with fermions determined by LVμ=(κfiLQf¯γμLi+κfiRdf¯γμei)V1μ†+H.c.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 right-handed couplings, which are not necessary to explain the anomalies. This then generates the effective four-fermion interactions encoded in Leff=-κilLκjkL*M2Q¯iαγμQjβL¯kβγμLlα,where α and β label the SU(2) components. After electroweak symmetry breaking, we work in the down basis; i.e., no CKM elements appear in flavor changing neutral currents of down quarks. We recall our definitions and the tree-level results in the Supplemental Material [49], which includes Refs. [50–66].

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)τν 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 calculated reliably since they are gauge independent and finite (also in unitary gauge) [67]. 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 always absent at tree level (like b→sνν¯ processes) or are not present at tree level due to a specific coupling structure (like b→sμ+μ- processes in the absence of muon couplings). Furthermore, we neglect tiny dimension-8 effects of the SM Higgs particle. In these cases the loop effects are the leading contributions. We calculate all diagrams at leading order in the external momenta using asymptotic expansion [68].

<inline-formula><mml:math display="inline"><mml:mi>W</mml:mi></mml:math></inline-formula> boxes contributing to <inline-formula><mml:math display="inline"><mml:msub><mml:mi>d</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mi>ν</mml:mi><mml:mover accent="true"><mml:mi>ν</mml:mi><mml:mo stretchy="false">¯</mml:mo></mml:mover></mml:math></inline-formula>.—

We use the effective Hamiltonian, Heffνν=-4GF2VtdkVtdj*(CL,jkfiOL,jkfi+CR,jkfiOR,jkfi),OL,jkfi=α4π[d¯jγμPLdk][ν¯fγμ(1-γ5)νi],OR,jkfi=α4π[d¯jγμPRdk][ν¯fγμ(1-γ5)νi],with PR(L)=[1+(-)γ5]/2 and GF (α) being the Fermi (electromagnetic fine structure) constant. The result of the box contributions involving a W to di→dfνν¯ (an example diagram is shown on the right-hand side of Fig. 1) is gauge invariant in Rξ gauge and the same finite result is obtained in unitary gauge [with e=4πα and mt (mW) the top quark (W boson) mass]: CL,faij=-mW22e2V3aV3f*M2(6κfjLκaiL*log(mW2M2)+3(V3aV3k*κkiL*κfjL+V3f*V3kκkjLκaiL*)log(mt2mW2)1-mW2mt2+V3f*V3kκkjLV3aV3l*κliL*mt2mW2).

Here (see third diagram in Fig. 1) we obtain again a finite and gauge independent result for the Wilson coefficient; following the analysis of Ref. [69], we use Heffτμνfνi=4GF2DL,fiτμ[ν¯fγσPLνi][μ¯γσPLτ],with DL,fiτμ=Ncδi2V3k*κkfL*κ33L32π2mt2M2[1+2log(mt2M2)].We find, in agreement with Ref. [30], that the effect is small.

Photon and gluon penguins.—

We use the standard Hamiltonian (see, for example, Ref. [70]) also defined in the Supplemental Material [49]. For on-shell photons and gluons the result of the left-hand diagram in Fig. 1 is finite in unitary gauge and the same result is obtained in Rξ gauge: C7(8)sb=-2GFVtbVts*M21172(548)κ2iLκ3iL*.Taking into account the running from the LQ scale μLQ=M=1TeV down to μb=5GeV (see, e.g., Refs. [71,72]), we obtain C7sb(μb)≈0.29κ2iLκ3iL*.

For off-shell photons 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 C9,sbττ=-C10,sbττ into the four-fermion operators O9,sbℓℓ (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 effective field theory (below the LQ scale), generating the following mixing into the four-fermion operators with light leptons: C9,sbℓℓ=2GFVtbVts*M216log(M2μb2)κ2iLκ3iL*.Note that this result is model independent (at leading-log accuracy) in the sense that it does not depend on the model which generates C9,sbττ=-C10,sbττ. In principle, there are also Z penguins generating C9,sbℓℓ and C10,sbℓℓ. However, this effect is suppressed by light lepton masses (or small momenta) and is therefore of dimension 8. Further, note that there are no box diagram contributions which generate s¯bμ¯μ(s¯be¯e) operators if the couplings of the LQ to muons (electrons) are zero at tree level.

Box diagrams with LQs.—

What cannot be calculated consistently are box diagrams involving only LQs [35]. Here, the results are divergent in unitary gauge which corresponds to a gauge dependence in Rξ gauge. However, these effects are suppressed if |κL|<g2 and can be further suppressed in the presence of vectorlike fermions by a GIM-like mechanism [38] which, in analogy to the SM, would render the result finite.

Phenomenology.—

Assuming κ33LVcb≪κ23L, one is safe from LHC bounds, and the effects in Bs→τ+τ-, C7sb(μb) [Eq. (12)], and C9,sbℓℓ [Eq. (13)] directly depend on R(X)/R(X)SM (with X=D(*), J/ψ). In Fig. 2 we show these dependences. Intriguingly, the effect generated in C7sb(μb) and C9,sbℓℓ, within the preferred region from b→cτν data, exactly overlaps with the 1σ ranges of the model independent fit to b→sμ+μ- data excluding LFU violating observables [70,73] [therefore, only lepton flavor universality conserving observables like P5′ but not R(K(*)) can be explained].

210.1103/PhysRevLett.122.011805.f2

C9,sbℓℓ and C7sb(μb) as functions of R(X)/R(X)SM with X={D,D*,J/ψ}. The solid lines correspond to M=1TeV and the dashed ones to M=5TeV while the (dark) blue region is preferred by b→cτν data at the 1σ (2σ) level. From the global fit, taking into account only lepton flavor conserving observables we have -1.29<C9,sbℓℓ<-0.87[70] and -0.01<C7sb(μb)<0.05[10] at the 1σ level. Therefore, our model predicts just the right sign and size of the effect in C9,sbℓℓ and C7sb(μb) necessary to explain b→sℓ+ℓ- data, assuming an explanation of b→cτν.

Let us now include the effect of κ13L. Here, many correlations arise. First of all, b→c(u)τν is already at tree level correlated to b→s(d)τ+τ-. In addition, the W boxes in Eq. (8) 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 Bq→τ+τ-, there are striking correlations with K→πνν¯, as can be seen from Fig. 3. Furthermore, we get an effect δVusτ=Vusτ-Vusτ(0)Vusuni≈-Cusττ,where Vusτ(0) is the CKM matrix element extracted from τ decays without NP. However, Eq. (8) generates K→πνν¯, and respecting these bounds, the relative effect in Vusτ can only be at the per-mill level, |δVusτ|≈0.05%, excluding the possibility to account for the discrepancy of |Vusuni|=0.22547±0.00095 versus |Vusτ|=0.2212±0.0014[15,23]. The same is true about Bd→τ+τ-, where the currently preferred region of analysis using BELLE data [14] of Br[Bd→τ+τ-]exp=(4.39-0.83+0.80±0.45) lies outside the plot range.

310.1103/PhysRevLett.122.011805.f3

Predictions for Bq→τ+τ- and K→πνν¯ (contour lines) in the κ13L-κ23L plane for M=1TeV and κ33L=1. The colored regions are preferred by b→c(u)τν data, where we naively averaged (i.e., we computed the weighted average of the observables and added their errors in quadrature, disregarding correlations) R(D(*)) and R(J/ψ) or R(π) and B→τν, respectively. The gray region is excluded by K+→π+νν¯. Here we assumed all couplings κijL to be real.

Now, in addition to the couplings κ33L and κ23L, we allow nonvanishing κ32L and κ22L. These couplings give rise to tree-level effects in b→sμ+μ-. In Fig. 4 we show the allowed (colored) regions from b→sμ+μ- and b→cτν as well as the exclusions from b→sτμ and τ→ϕμ. Note that a simultaneous explanation of the anomalies is perfectly possible since the colored regions overlap and do not extend to the parameter space excluded by b→sτμ and τ→ϕμ. Interestingly, due to the loop effects originating from the b→cτν explanation, we predict a flavor universal effect in C9,sbℓℓ and C7sb which is supplemented by a tree-level effect of the form C9,sbμμ=-C10,sbμμ with muons only. This means that the relative NP effect compared to the SM in lepton flavor conserving observables (like P5′) should be larger than in R(K(*)), which is in perfect agreement with the global fit [74].

410.1103/PhysRevLett.122.011805.f4

Allowed (colored) regions in the C9,sbμμ=-C10,sbμμ(=^640κ22Lκ32L*)—R(X)/R(X)SM plane for M=1TeV and X=D, D*, J/ψ at the 1σ and 2σ level for κ33LVcb≪κ23L. The region above the black dashed (solid) line is excluded by τ→ϕμ (B→Kτμ) for κ33L=0.5=25κ32L (κ33L=0.5=2.5κ32L). The bound from τ→ϕμ (B→Kτμ) depends on κ33L and κ32L and gets stronger if κ32L gets smaller (larger). That is, for κ33L=0.5 and 2.7⪅κ33L/κ32L⪅27, the whole 2σ region preferred by b→cτν and b→sℓ+ℓ- data is consistent with B→Kτμ and τ→ϕμ.

Conclusions.—

The vector leptoquark SU(2) singlet is a prime NP candidate to explain the current hints for LFU violation. In this Letter, we calculated and studied the important loop effects arising within such a model and performed a phenomenological analysis.

We find that an explanation of b→cτν data generates lepton flavor universal effects in b→sℓ+ℓ- transitions which nicely agree with the model independent fit (see Fig. 2). Therefore, the C9=-C10-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. That is, with more precise measurements of lepton flavour universality violating and lepton flavor universality conserving effects, one can test if in fact there is a lepton flavor universality conserving contribution in addition to the lepton flavor universality violating ones [75]. Similar conclusions hold for the correlations between b→uτν data generating lepton flavor universal effects in b→dℓ+ℓ- processes.

We also find that NP in b→c(u)τν generates important effects in Bs(d)→τ+τ- which are even correlated to b→s(d)νν¯ processes and K→πνν¯ via W box contributions (see right-hand diagram in Fig. 1). The Vusτ puzzle (like the CP asymmetry in τ→KSπν[76]) cannot be solved due to the stringent constraints from K→πνν¯, and because of b→uτν bounds one cannot fully account for the BELLE excess in Bd→τ+τ- (see Fig. 3).

Additionally, b→cτν and b→sℓ+ℓ- data can be simultaneously explained without violating other bounds like τ→ϕμ (see Fig. 4). Furthermore, one could at the same time also account for NP effects in b→dμ+μ- without violating KL→μ+μ- bounds.

We are very grateful to Joaquim Matias and Bernat Capdevilla for providing us with the fit necessary for the b→sℓ+ℓ- region in Fig. 4 whose work is supported by an explora grant (FPA2014-61478-EXP) and to Aleksey Rusov for providing us with the fit for C9,bdμμ. The work of A. C. and D. M. is supported by an Ambizione Grant of the Swiss National Science Foundation (PZ00P2_154834). The work of C. G. and F. S. is supported by the Swiss National Foundation under Grant No. 200020_175449/1.

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Here, V refers to to the Cabibbo-Kobayashi-Maskawa (CKM) matrix.

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In our conventions, the left-handed lepton doublet has hypercharge -1.

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Similar loop effects for scalar LQs have been calculated in Refs. [46–48].

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In this Letter, we followed two approaches to check the results. First, we calculated the results in unitary gauge. Then, we derived the couplings of the LQ Goldstones to SM fermions by requiring the tree-level amplitude to be gauge independent. Finally, we calculated its contribution in Rξ gauge.

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See Ref. [75] for a recent analysis of such scenarios.

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