PHYSICAL REVIEW D 99, 075017 (2019) Are we overlooking lepton-flavor-universal new physics in b → sll ? Marcel Algueró,1,2 Bernat Capdevila,1,2 Sébastien Descotes-Genon,3 Pere Masjuan,1,2 and Joaquim Matias1,2 1 Grup de Física Teòrica (Departament de Física), Universitat Autònoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain 2 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, E-08193 Bellaterra (Barcelona), Spain 3 Laboratoire de Physique Théorique, UMR 8627, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay Cedex, France (Received 2 October 2018; published 15 April 2019) The deviations with respect to the Standard Model that are currently observed in b → sll transitions (the so-called flavor anomalies) can be interpreted in terms of different new physics (NP) scenarios within a model-independent effective approach. We reconsider the determination of NP in global fits from a different perspective by removing one implicit hypothesis of current analyses, namely, that NP is only lepton-flavor-universality violating (LFUV). We examine the roles played by LFUV NP and lepton-flavor- universal NP altogether, providing new directions to identify the possible theory beyond the SM responsible for the anomalies observed. New patterns of NP emerge due to the possibility of allowing at the same time large LFUV and lepton-flavor-universal NP contributions to C10μ , which provides a different mechanism to obey the constraint from the Bs → μþ μ− branching ratio. In this landscape of NP, we discuss how to discriminate among these scenarios in the short term, thanks to current and forthcoming observables. While the update of RK will be a major milestone to confirm the NP origin of the flavor anomalies, additional observables, in particular the LFUV angular observable Q5 , turn out to be central to assessing the precise NP scenario responsible for the observed anomalies. DOI: 10.1103/PhysRevD.99.075017 I. INTRODUCTION combines the experimental data from LHC experiments (LHCb [5–9] but also ATLAS [10] and CMS [11]) as well Besides the fundamental discovery of the Standard as the data from B factories (in particular, Belle [12,13]) Model (SM) Higgs, the first run at the LHC had two clear together with theoretical input concerning long-distance outcomes. On one side, no signals of new physics (NP) have been found in direct searches. On the other side, hadronic contributions [14–17]. They aim at extracting indirect searches have led to a large set of deviations with the value of the short-distance Wilson coefficients under respect to the SM (or anomalies) in both b → clν and in given NP hypotheses and at comparing them with the SM b → sll decays [1–3]. We can classify the latter (which we expectations. Even though different global analyses in the focus on) in two sets: b → sμμ anomalies related to literature use different approaches (statistical treatment, observables testing only muonic transitions, which we observables, hadronic inputs, etc.), they agree on the call lepton-flavor dependent (LFD), and lepton-flavor- emerging global picture [18–24]. For instance, in universality violating (LFUV) anomalies that correspond Ref. [19], a global fit including both LFD and LFUV to deviations in observables comparing muonic and elec- observables finds pulls (comparing the statistical tronic transitions. significance of the SM against that of a NP hypothesis) The b → sll anomalies have been analyzed in the between 5.0 and 5.8σ, depending on the particular NP effective Hamiltonian approach, which separate short- hypothesis used. The LFUV NP hypotheses involving and long-distance contributions in a model-independent either CNP9μ or C 9μ ¼ −C10μ are among those with the NP NP way (see, for instance, Ref. [4]). The analysis now highest significances. In this paper, we consider the possibility that short- distance Wilson coefficients will receive contributions from NP that are not only LFUV but also lepton-flavor universal. Published by the American Physical Society under the terms of Indeed, whereas LFUV NP contributions are mandatory the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to to explain RK and RK , b → sll processes are not the author(s) and the published article’s title, journal citation, restricted to such NP contributions alone. While several and DOI. Funded by SCOAP3. articles [19–24] allowed the presence of NP in electrons in 2470-0010=2019=99(7)=075017(9) 075017-1 Published by the American Physical Society MARCEL ALGUERÓ et al. PHYS. REV. D 99, 075017 (2019) global fits to b → sll, in the present paper, we go one step sum rule computation with one soft-gluon exchange [14] to beyond, and we impose different types of LFU structures get an order of magnitude estimate of this contribution, between all leptons. We show that a universal LFU NP without making any assumption about its sign and thus contribution, together with a LFUV NP contribution, gives allowing for constructive or destructive interference with rise to scenarios with a statistical significance at least as the other contributions to C9μ . relevant as the ones identified in Ref. [19], against a This effective approach is the basis for global fits to the common belief that the presence of such terms is not data in order to constrain the NP contributions CNP il under justified from the statistical point of view and should be various NP assumptions [19–22,24]. It turns out that the dropped. This may help motivate the construction of new combination of anomalies in some LFD (b → sμμ) angular models including not only LFUV but also LFU NP observables and in LFUV ratios RK and RK selects contributions. Thus, we reconsider the results of the fit, hypotheses with a large NP contribution to the Wilson allowing for the presence of two different types of NP coefficient C9μ (of order 25% of the SM) or NP contribu- that may lead to a new paradigm concerning the nature of tions to both C9μ and C10μ . the underlying theory beyond the SM. We then discuss the Following this perspective, we ought to be more precise next steps, to identify the NP scenario that is realized in on what goes under the “new physics” landscape. In this nature among the ones already favored, complementing paper, we consider that the short-distance Wilson coeffi- Refs. [16,19]. Following our findings, the UV completion cients Ciμ can contain two types of NP contribution, of the SM may require significant contributions from two different sectors (LFU and LFUV) instead of a single one, CNP il ¼ Cil þ Ci ; V U ð2Þ as often assumed. with l ¼ e, μ (the extension to τ is trivial, assuming true II. TWO TYPES OF NP CONTRIBUTIONS universality among e, μ, and τ), where CV il stands for lepton- flavor-universality violating NP and CU i stands for lepton- The b → sll processes can be analyzed within the flavor-universal NP contributions. These short-distance effective Hamiltonian framework [25,26]. The observables contributions are all independent of the external hadronic for exclusive decays can be written as interference terms states and their kinematics; they differ therefore from long- between helicity amplitudes which are given as (short- distance hadronic contributions, which are LFU but de- distance) Wilson coefficients multiplying (long-distance) pendent on the nature and kinematics of the hadronic states. hadronic matrix elements [27–29], with a separation We will define the separation between the two types of between short and long distances given by the factorization contributions by imposing that LFUV contributions affect scale μb ¼ Oðmb Þ. One can use the fact that mb is only muons significantly larger than the typical QCD scale in order to isolate perturbatively computable contributions to the CV ie ¼ 0: ð3Þ hadronic matrix elements (using effective approaches like QCD factorization). These perturbative contributions of There is no loss of generality here, since this term can hadronic origin can be lumped together with the purely always be absorbed in such a way that CV iμ can be short-distance contribution into effective Wilson coeffi- interpreted as the difference of NP contributions to muons cients (that will multiply nonperturbative hadronic form and electrons. factors) with the structure in the case of B → Kð Þll [30] C9l ðq2 Þ ¼ CSM 2 cc̄ 2 III. GLOBAL FITS IN THE PRESENCE OF LFU NP 9 pert ðq Þ þ C9 ðq Þ þ C9l NP C10l ¼ CSM LFUV NP contributions are mandatory to explain LFUV 10 þ C10l ; ð1Þ NP anomalies. The b → see measurements (in limited number, where l ¼ e, μ. The short-distance SM values [31] at this without significant deviations [13,35]) are compatible to no scale μb ¼ 4.8 GeV are CSM9 ¼ 4.07 and C 10 ¼ −4.31. We SM LFU-NP contributions (as often assumed), but they do not have C9 pert ¼ C9 þ Yðq Þ, where the function Yðq2 Þ SM SM 2 prevent these contributions from occurring. Assuming stems from one-loop matrix elements of four-quark oper- hadronic contributions properly assessed [16,24], we con- ators O1–6 , corresponding to the cc̄ continuum. It can be sider for the first time that LFU NP contributions can exist evaluated within perturbation theory at leading order, and for both C9 and C10 , together with LFUV NP contributions. corrections at Oðαs Þ to C9l to this function are known It is important to remark that this is not the same as simply [30,32,33]. In addition to this continuum, there is a long- allowing for NP in electrons; we impose as a constraint in distance contribution, which corresponds in particular to the fit that this contribution is the same for all leptons charmonium resonances Ccc̄ 9 and depends on the external and work out the consequences of this identity. The key hadron state. Several approaches are available to estimate point to lift the degeneracy between the various contribu- this contribution [17,22,34], all with similar outcomes tions through the fit consists in considering together [16,24]. Here, we follow Refs. [4,16], using the light-cone LFUV and LFD observables. The LFUV observables will 075017-2 ARE WE OVERLOOKING LEPTON-FLAVOR-UNIVERSAL NEW … PHYS. REV. D 99, 075017 (2019) TABLE I. Scenario 3: 4D hypothesis with CV 9μ and C 10μ and with V TABLE IV. Scenario 6: 2D hypothesis with CV 9μ ¼ −C 10μ and V C9 and C10 . U U 9 ¼ C 10 . CU U Best-fit point 1σ CI 2σ CI Best-fit point 1σ CI 2σ CI CV 9μ 0.08 ½−0.72; 0.80 ½−1.69; 1.49 CV 9μ ¼ −C 10μ V −0.64 ½−0.77; −0.51 ½−0.90; −0.39 CV 10μ 1.14 [0.66, 1.59] [0.12, 2.03] CU 9 ¼ C 10 U −0.44 ½−0.58; −0.29 ½−0.71; −0.14 CU 9 −1.26 ½−1.92; −0.25 ½−2.43; 1.62 CU 10 −0.91 ½−1.40; −0.40 ½−1.89; 0.16 (including Ref. [19]) and should be explained in more detail. The key observation is that RKðÞ -like TABLE II. Scenario 4: 3D hypothesis with CV observables may be also accommodated by CV 9μ ¼ −C 10μ and V 10μ with C9 and C10 . U U alone with a negligible CV 9μ , cf. the Appendix A. This result was not obtained in the two-dimensional (2D) Best-fit point 1σ CI 2σ CI fits with only LFUV NP contributions (setting CV ¼ −CV −0.68 ½−0.96; −0.45 ½−1.28; −0.26 CU 9μ 10μ i ¼ 0), since LFD observables led then to the CU −0.37 ½−0.68; −0.03 ½−0.95; 0.35 favored scenarios with b.f.p. CV 9μ ≃ −1 and C9μ ¼ V 9 CU 10 −0.51 ½−0.86; −0.18 ½−1.24; 0.13 −C10μ ≃ −0.7. Adding LFU contributions provides V complementary mechanisms to explain LFUV and LFD anomalies. On one side, the LFD anomalies constrain LFUV NP contributions (CVil ), whereas LFD are accommodated by CV 9μ þ C9 ≃ −1.18 and U observables will be sensitive to the sum of LFUV NP and LFU NP contributions (CU V CV10μ þ C10 ≃ þ0.23. On the other side, the LFUV U i þ Cil ). As we increase the number of parameters, we have more flexibility to describe observables are accommodated by CV 10μ ≃ 1.14. It is the data, which could lead to an improvement compared to thus not a surprise that the summed LFU and LFUV our earlier fits restricted to LFUV NP contributions only contributions for both C9;10 yield a result close to and opens the possibility of new NP models. the fit to all observables under the NP hypothesis 9μ , C10μ ) shown in Table III of Ref. [19]. Under (CNP We start from the results presented in the Table II of NP Ref. [19], for the global fits under (favored) one- this hypothesis, CV9μ changes sign with respect to fits dimensional (1D) hypotheses of NP in b → sμμ. The 1D without LFU in order to resolve the inner tensions hypothesis with CNP 9μ (scenario 1) led to a 68% confidence between LFUV and LFD observables. Moreover, the interval of ½−1.28; −0.94 with a pull with respect to the SM constraint from BðBs → μþ μ− Þ is obeyed by the sum of 5.8σ, whereas the hypothesis CNP 9μ ¼ −C 10μ (scenario 2) NP CV10μ þ C10 with opposite signs, thus allowing a large U had a 68% confidence interval of ½−0.75; −0.49 with a pull CV10μ . This important feature is observed for the first of 5.3σ. We consider now a set of nested fits named time here and opens new possibilities for models scenarios 3 to 8 and presented in Tables I–V in decreasing beyond the SM. order of complexity to better understand the interplay (ii) The hypothesis fCV 9μ ¼ −C10μ ; C 9 ; C10 g (Table II) is V U U between LFUV and LFU NP (more information and results, model-building motivated for theories with a sig- including the correlations among the parameters, are given nificant scale gap between SM and NP [36–40], as in the Appendix A and B): the additional NP contributions should be invariant (i) The general hypothesis fCV 9μ ; C 10μ ; C9 ; C10 g (Table I) V U U under SUð2ÞL. Remarkably, this three-dimensional has a pull of 5.6σ with respect to the SM. The result (3D) fit has a pull of 5.7σ to SM. The b.f.p. is in is remarkable: considering the best-fit point (b.f.p.), good agreement with the result found in Table II of CV9μ almost vanishes, C9;10 are far away from zero, U Ref. [19] but with LFU contributions differing from and CV 10μ is larger than 1. At first glance, this result zero at the 1σ level. The increase in the SM pull with seems to contradict the previous global analyses respect to the case without LFU (5.7σ vs 5.3σ [19], with two more parameters) hints at a slight prefer- TABLE III. Scenario 5: 3D hypothesis with CV ence for LFU NP in both C9;10 . 9μ and C10μ and V with C9 ¼ C10 . Confidence Intervals (CI) are also provided. U U (iii) The hypothesis fCV 9μ ; C10μ ; C9 ¼ C10 g (Table III) is V U U inspired by the fit in Table I, which suggests 1σ CI 2σ CI CU9 ≃ C10 . We find Table III with a pull of 5.8σ with Best-fit point U CV 9μ −0.16 ½−0.94; 0.46 ½−2.05; 0.98 respect to the SM, slightly larger than the four- CV 1.00 [0.18, 1.59] ½−1.35; 2.06 dimensional (4D) hypothesis. 10μ CU (iv) The hypothesis fCV 9μ ¼ −C10μ ; C9 ¼ C 10 g (Table IV) V U U 9 ¼ CU 10 −0.87 ½−1.43; −0.14 ½−1.91; 0.98 combines the suggestive results from both Tables I 075017-3 MARCEL ALGUERÓ et al. PHYS. REV. D 99, 075017 (2019) TABLE V. 2D hypotheses. Top: Scenario 7: LFUVand LFU NP previous results focused on LFUV NP contributions in CNP 9μ 9l only. Bottom: Scenario 8: C9μ ¼ −C 10μ and C 9 only. in CNP V V U without LFU NP. We consider two 2D fits: Best-fit point 1σ CI 2σ CI (i) The hypothesis fCV 9μ ; C9 g (Table V, top) has a pull U with respect to the SM of 5.7σ. The LFD observ- CV −1.57 ½−2.14; −1.06 ½−2.75; −0.58 ables are governed by the sum CV 9μ þ C9 ≃ −1.01 for 9μ U CU 0.56 [0.01, 1.15] ½−0.51; 1.78 9 the b.f.p., whereas the b.f.p. CV 9μ ¼ −1.57 is the CV 9μ ¼ −CV 10μ −0.42 ½−0.57; −0.27 ½−0.72; −0.15 dominant contribution to LFUV observables. Inter- CU 9 −0.67 ½−0.90; −0.42 ½−1.11; −0.16 estingly, these results can be linked to the results of Ref. [19] (without LFU NP), with the former in agreement with the b.f.p. of the fit to all data (−1.11) in Table II of Ref. [19] and the latter closer to the and III and yields a fit with a pull of 6.0σ with b.f.p. of the fit to LFUV observables (−1.76). respect to the SM. The b.f.p. CV 9μ ¼ −C10μ ¼ −0.64 V Therefore, the internal tension between LFD and obtained now is very similar to the one from Table II, LFUV observables in the global fit of Ref. [19] is and CU 9 ¼ C10 ¼ −0.44 is exactly the average of U resolved here due to the additional freedom allowed LFU contributions found in Table II. This particular by CV9μ, which enters the LFUV observables (always 2D correlation is shown in Fig. 1 (see the Appen- with a subleading contribution from CU 9μ ), whereas dix B for correlations under the other hypotheses). It is interesting that a C10 contribution gives rise to a the combination CV 9μ þ C U 9 is constrained by the LFD rather tight 1σ confidence interval, mainly due to observables. BðBs → μþ μ− Þ. Let us add that the hypothesis (ii) The hypothesis fCV 9μ ¼ −C10μ ; C9 g (Table V, bot- V U fCV tom) has a pull of 5.8σ and follows a similar pattern. 9μ ¼ −C10μ ; C9 ¼ −C10 g [once again of interest V U U for models based on SUð2ÞL invariance] has a pull The LFUV contribution CV 9μ ¼ −C10μ ¼ −0.42 (for V with respect to the SM lower by almost 1σ and is not the b.f.p.) accommodates the LFUV observables favored by the data, as can already be seen in Table I. well, while the sum CV 9μ þ C 9 ¼ −1.09 takes care U Two additional 2D hypotheses provide a bridge between of the LFD observables (recovering approximately the above hypotheses with LFU NP in both C9;10 and the fits to all data and to LFUV observables only from Ref. [19]). The similar pulls with respect to the SM of the various scenarios indicate that the current measurements cannot lift the degeneracy among the hypotheses, and a different strategy should be envisaged in order to distinguish them. IV. ROLE OF LFUV OBSERVABLES One of the most relevant outcomes of this work is the unexpected preference for a NP solution with a prominent CNP 10l signature, both LFUV and LFU. This may represent a shift of paradigm, since until now the vast majority of global analyses performed were signaling a single NP contribution to CV9μ as the most favored solution. The LFUV observables are natural candidates in order to identify the contributions from LFUV NP conclusively. While new and more precise measurements of RK and RK will certainly be useful, Refs. [19,41] pointed out the relevance of the Qi observables (difference of optimized angular observables in muon and electron modes) and the more exotic B5;6s observables. Indeed, these observables are not only very clean and stringent tests against the SM, similarly to RKðÞ , but they also contain additional infor- mation about the Wilson coefficients from a full angular FIG. 1. Confidence regions for scenario 6 in the plane analysis. In particular, while RK involves crossed LFUV 9μ ¼ −C10μ , C9 ¼ C 10 ). The regions for different experimental (CV V U U subsets correspond to a confidence level of 3σ, whereas the 1, 2, LFU terms such as CV 9μ C10 and C 10μ C9 , Q5 contains a C 9μ C9 U V U V U 3σ confidence regions are shown for the region associated with term, introducing complementary information to RK ; see the global fit to all data. the Appendix A. 075017-4 ARE WE OVERLOOKING LEPTON-FLAVOR-UNIVERSAL NEW … PHYS. REV. D 99, 075017 (2019) A natural candidate to disentangle LFU and LFUV NP is column in Fig. 2). hP05 i½4;6 can confirm this result by then hQ5 i½1.1;6 because of its high sensitivity to CV 9μ and its disentangling fCU 9 ; C10μ g from fC9 ¼ C10μ g, see the U U U 0 ties to the P5 anomaly sensitive to both types of NP Appendix C, if the experimental uncertainty on contributions. hRK i½1;6 and in second place hRK i½1;6 , hP05 i½4;6 is reduced by half [4]. despite the large theoretical uncertainties of the latter in (iii) If hQ5 i½1.1;6 ≲ 0.1, scenarios in which CV 9μ and C10μ V the presence of NP, should also play a role due to their are left free to vary independently (scenarios 3 0 sensitivity to CV 10μ . Finally, the very same hP5 i½4;6 should and 5) are preferred. Distinguishing among these help discern between the LFU contributions CU two scenarios is practically impossible, since only 9 and C10 . U We show the most interesting LFUV observables for the hB6s i½1.1;2.5 shows a very mild discrimination power b.f.p. of the above scenarios in Fig. 2 (from left to right, the (third column in Fig. 2) if measured at a very high SM and scenarios 1 to 8). Explicit expressions of these precision. þ − observables are given in the Appendix A. The value of CU 10 can be probed by BðBs → l l Þ, A decision tree can be built from the experimental assuming no significant scalar or pseudoscalar contribu- measurement of hQ5 i½1.1;6 , which exhibits a good discrimi- tions: nating power against the various scenarios considered above: (i) If hQ5 i½1.1;6 ≳ 0.3 (first column in Fig. 2), the 1D BðBs → eþ e− Þ m2e jCSM U 2 10 þ C10 j ¼ × : ð4Þ hypothesis CV 9μ is able to explain all anomalies. A BðBs → μþ μ− Þ m2μ jCSM V U 2 10 þ C 10μ þ C10 j confirmation can come from an updated measure- ment of hRK i½1;6 − 1 ≳ −0.2 (last column in Fig. 2). The inclusion of CU10 in this equation leads to and enhance- (ii) If 0.1 ≲ hQ5 i½1.1;6 ≲ 0.2, the hypotheses with only a ment of it between 30% and 60% with respect to the SM large CNP prediction, but with strong lepton-mass suppression for this 9μ are disfavored, while hypotheses with C10μ V observable to be available in the near future, and similarly, are favored. Actually, this range of values corresponds assuming no large LFUV-NP contributions in b → sττ, for to solutions involving CV 9μ ¼ −C10μ (scenarios 2, 4, 6, V the challenging measurement of BðBs → τþ τ− Þ. and 8). Knowing hRK i½1;6 with an uncertainty around 5% would help discriminate between the hypotheses V. CONCLUSIONS CV9μ ¼ −C10μ with and without LFU contributions in V CNP We have considered the current anomalies observed in 9μ (scenarios 2 and 8, respectively; see the fourth b → sll transitions and discussed the consequences of removing one hypothesis frequently made (and over- looked) in the global model-independent analyses, namely, that the anomalies are explained only by NP violating lepton–flavor universality. Instead, we explore the impli- cations of allowing both LFU and LFUV NP contributions in the Wilson coefficients C9 and C10 , providing more flexibility to describe the data. The LFUV observables will constrain LFUV NP contributions, whereas LFD observ- ables will be sensitive to the sum of LFUV NP and LFU NP contributions. We found a different mechanism with a large contribution to C10 to explain the data without transgressing the BðBs → μþ μ− Þ constraint, leading to an improvement compared to our earlier fits restricted to LFUV NP contributions only. The 4D hypothesis with both kinds of contributions to CNP 9l and C10l leads to two scenarios with high significances NP and well-constrained parameters (equivalent to scenarios with only LFUV NP contributions and thus a more limited set of parameters). Indeed, the fits favor either a large and positive CV 10μ together with large and negative LFU con- tributions in both CU9;10 (scenarios 3 and 5, Tables I and III) FIG. 2. Predictions for LFUV observables of interest under or a negative C9μ ¼ −CV V 10μ together with smaller (in various hypotheses of LFU and LFUV NP contributions currently favored by the global b → sll. From left to right: the SM is absolute value) but still negative LFU contributions in followed by scenarios 1 to 8 as described in the main text. We plot both CU9;10 (scenarios 4 and 6, Tables II and IV). If LFUV RK;K  − 1 to keep the figure compact. lepton interactions with V − A are favored, suggesting that 075017-5 MARCEL ALGUERÓ et al. PHYS. REV. D 99, 075017 (2019) SUð2ÞL invariance might be a guide for models for NP in Oi ¼ α0 þ α1 CU U 2 9 þ α2 C10 þ α3 C9μ þ α4 C10μ þ α5 ðC9 Þ U V V b → sll, LFU lepton interactions with a V þ A structure 2 V 2 V 2 are preferred. The size and structure of these LFU lepton þ α6 ðCU 10 Þ þ α7 ðC9μ Þ þ α8 ðC10μ Þ þ α9 C9 C10 U U interactions do not agree with a generation by radiative þ α10 CU 9 C9μ þ α11 C9 C10μ þ α12 C9μ C10 þ α13 C9μ C 10μ V U V V U V V effects from LFUV NP contributions, which leads to much smaller and purely vector LFU lepton interactions [42]. The þ α14 CU 10 C10μ : V ðA1Þ scenarios that we discuss would also require a deviation from popular model-building ideas relying on a strong hierarchy of NP contributions according to the generations The first block in Table VI (second column) contains the involved in order to provide a connection with b → cτν Standard Model prediction. In the second block (columns 3 anomalies [2,36,43–50]. to 6), one can find the coefficients of the linear terms To separate the various scenarios explaining the b → (CU9 , C10 , C9μ , and C10μ ), and the third block shows the U V V sll anomalies, a decision tree is proposed. Although the coefficients of the quadratic terms. Since in the four update of RK will be a major milestone, the measurement observables the terms α6;8 are zero, we have not included of Q5 (and the improvement of RK and P05 ) remains them in Table VI. essential to disentangle the possible scenarios of NP and hP05 i½4;6 , being the only LFD observable in Table VI, is to interpret the effective description in terms of a full- fledged UV-complete model of physics beyond the obviously the only observable with nonzero linear LFU 0 Standard Model. terms. The combination CU 9 þ C9μ dominates hP5 i½4;6 with V α1 ¼ α3 ¼ −0.207, which is approximately 25% of the SM ACKNOWLEDGMENTS value. The coefficients in front of CU;V 10;μ verify This work received financial support from European α2;4 ∼ 1=3α2;3 , while the coefficients of the quadratic terms 2 0 Regional Development Funds under the Spanish Ministry ðCU;V 9;μ Þ are α5;7 ∼ α2;4 . Moreover, hP5 i½4;6 also has crossed of Science, Innovation and Universities (projects terms mixing Ci and Cjμ (α9…14 ), even though they are U V No. FPA2014-55613-P and No. FPA2017-86989-P) and subleading with respect to α1;3 . from the Agency for Management of University and hQ5 i½1.1;6 is strongly sensitive to CV 9μ , with α3 ¼ −0.246 Research Grants of the Government of Catalonia (project being an order of magnitude larger than the rest of the SGR 1069) [M. A., B. C., P. M., J. M.]; and from European coefficients. hRK i½1;6 and hRK i½1;6 are linearly sensitive to Commission (Grant Agreements No. 690575, No. 674896 and No. 69219) [S. D. G.]. The work of P. M. is supported by both CV 9;10μ , but the former only contains crossed terms the Beatriu de Pinós postdoctoral program co-funded by the mixing universal and violating contributions of the type Agency for Management of University and Research Grants CU9ð10Þ C 10ð9Þμ . Contrarily, the latter hRK i½1;6 has also quad- V  of the Government of Catalonia and by the COFUND ratic terms such as C9ð10Þ C9ð10Þμ . This implies that if one sets U V program of the Marie Sklodowska-Curie actions under the either C9 or C10 to zero this kind of term remains in framework program Horizon 2020 of the European hRK i½1;6 while they vanish in hRK i½1;6 . This difference in Commission. J. M. gratefully acknowledges the financial structure can prove useful in disentangling different support by ICREA under the ICREA Academia programme. scenarios. The coefficients αi of these parametrizations have been APPENDIX A: POLYNOMIAL obtained by fitting the calculated expressions of the PARAMETRIZATION FOR SOME observables with the second-order polynomial in the OBSERVABLES OF INTEREST Wilson coefficients in Eq. (A1). We generated the central The observables hP05 i½4;6 , hQ5 i½1.1;6 , hRK i½1;6 , and values over a grid of values of CNP 9μ , C10μ , C9e , and C10e . The NP NP NP hRK i½1;6 can be parametrized as follows, with the coef- grid range of the grid varied from ½−1; 1 for the coefficients ficients αi for each observable collected in Table VI: CNP 10μ , C9e , and C10e to ½−2; 2 for the coefficient C9μ (with a NP NP NP TABLE VI. Coefficients of the polynomial parametrization of observables in Eq. (A1). α0 α1 α2 α3 α4 α5 α7 α9 α10 α11 α12 α13 α14 hP05 i½4;6 −0.814 −0.207 −0.066 −0.207 −0.066 0.058 0.058 0.011 0.116 0.011 0.011 0.011 0.008 hQ5 i½1.1;6 0 0 0 −0.246 −0.019 0 0 0.033 0.066 0 0 0 0.013 hRK i½1;6 1.001 0 0 0.230 −0.264 0 0 0 0 0.061 0.061 0 0 hRK i½1;6 1.000 0 0 0.157 −0.287 0 0 0 0.042 0.045 0.045 0 −0.016 075017-6 ARE WE OVERLOOKING LEPTON-FLAVOR-UNIVERSAL NEW … PHYS. REV. D 99, 075017 (2019) 9μ ; C10μ ; C9 ¼ C 10 g is (iii) Scenario 5-fCV V U U spacing of the grid of sampled points of 0.1). We stress that the above formulas correspond to central values only (the 0 1 associated uncertainties could be parametrized in a similar 1.00 0.88 −0.93 B C way), but they already help identify the main sensitivities of Corr5 ¼ @ 0.88 1.00 −0.92 A: these observables. −0.93 −0.92 1.00 9μ ¼ −C10μ ; C9 ¼ C10 g is (iv) Scenario 6-fCV V U U APPENDIX B: CORRELATIONS AMONG PARAMETERS OF THE FITS   1.00 −0.01 Corr6 ¼ : Figure 3 shows the 1, 2, and 3σ confidence regions of the −0.01 1.00 2D fits. We also provide information about the correlations 9μ ; C9 g is (v) Scenario 7-fCV between the different parameters of each of the fits U performed.   The correlations between the parameters of each fit are 1.00 −0.93 the following (in the order of the parameters given to Corr7 ¼ : −0.93 1.00 describe each scenario): (i) Scenario 3-fCV9μ ; C10μ ; C9 ; C10 g is V U U 9μ ¼ −C10μ ; C9 g is (vi) Scenario 8-fCV V U 0 1   1.00 0.59 −0.96 −0.52 1.00 −0.47 Corr8 ¼ : B 0.59 1.00 −0.56 −0.91 C −0.47 1.00 B C Corr3 ¼ B C: @ −0.96 −0.56 1.00 0.48 A −0.52 −0.91 0.48 1.00 The 4D fit (scenario 3) exhibits very strong anticorre- lations between CV 9μ and C9 and between C10μ and C10 . U V U This is logical since b → sμþ μ− constrains CV iμ þ Ci while U 9μ ¼ −C10μ ; C9 ; C10 g is (ii) Scenario 4-fCV V U U þ − b → se e constrains Ci and the LFUV observables U 0 1 constrain CV 1.00 −0.76 0.77 iμ . In fact, without LFUVobservables, we would B C find a correlation of −1 between CV iμ and C i because the U Corr4 ¼ @ −0.76 1.00 −0.64 A: LFD observables only see the sum of both types of 0.77 −0.64 1.00 contributions. The same pattern can be observed in the FIG. 3. Left: Correlation between CV 9μ and C 9 from the scenario 7 fit. Right: Correlation between the parameters C9μ ¼ −C 10μ and C 9 U V V U from the scenario 8 fit. The regions for different experimental subsets correspond to a confidence level of 3σ, whereas the 1, 2, 3σ confidence regions are shown for the region associated with the global fit to all data. 075017-7 MARCEL ALGUERÓ et al. PHYS. REV. D 99, 075017 (2019) other fits, although correlations are nominally less strong due to the fact that different and more involved structures, like CV9μ ¼ −C 10μ , are explored. V One should also stress that the correlation between the parameters CV 9μ ¼ −C10μ and C9 ¼ C10 in scenario 6 is V U U negligible, signaling its statistical independence. This means that the underlying structure of most of the LFD observables is such that, when imposing CV 9μ ¼ −C10μ , once V its value is fitted to the LFUV observables, the parameter CU9 ¼ C10 can be independently determined by the LFD U observables. APPENDIX C: FURTHER TESTS Figure 4 is a visual account of the decision tree discussed in the main text: in the case of an experimental determi- nation of hQ5 i½1.1;6 finding a value close to 0.4 with enough precision (green band), only a solution involving CNP 9μ (blue lines) can explain both hQ5 i½1.1;6 and hRK i½1;6 . However, this test has no discriminating power if hQ5 i½1.1;6 is FIG. 4. hRK i½1;6 as a function of hQ5 i½1.1;6 in the four of the measured to be around 0.2 (blue band), since both C9μ scenarios analyzed. 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