Light resonances and the low- q2 bin of $$ {R}_{K^{*}} $$

Journal of High Energy Physics, Mar 2018

Wolfgang Altmannshofer, Michael J. Baker, Stefania Gori, Roni Harnik, Maxim Pospelov, Emmanuel Stamou, Andrea Thamm

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Light resonances and the low- q2 bin of $$ {R}_{K^{*}} $$

JHE Light resonances and the low-q2 Wolfgang Altmannshofer 1 2 4 8 9 10 11 12 Michael J. Baker 1 2 4 6 9 10 11 12 Stefania Gori 1 2 4 8 9 10 11 12 Roni Harnik 1 2 4 7 9 10 11 12 Maxim Pospelov 1 2 3 4 5 9 10 11 12 g Emmanuel Stamou 0 1 2 4 9 10 11 12 Andrea Thammg 1 2 4 9 10 11 12 0 Enrico Fermi Institute, University of Chicago 1 Wilson Street and Kirk Road, Batavia, IL 60510 , U.S.A 2 Winterthurerstrasse 190 , 8057 Zurich , Switzerland 3 Department of Physics and Astronomy, University of Victoria 4 155 B McMicken Hall , Cincinnati, OH 45221 , U.S.A 5 Perimeter Institute for Theoretical Physics 6 Physik-Institut, Universitat Zurich 7 Theoretical Physics Department , Fermilab 8 Department of Physics, University of Cincinnati 9 1211 Geneva , Switzerland 10 5640 S Ellis Ave , Chicago, IL 60637 , U.S.A 11 31 Caroline St N , Waterloo, ON N2L 2Y5 , Canada 12 3800 Finnerty Road, Victoria, BC V8P 5C2 , Canada LHCb has reported hints of lepton- avor universality violation in the rare decays B ! K( )`+` , both in high- and low-q2 bins. Although the high-q2 hint may be explained by new short-ranged interactions, the low-q2 one cannot. We thus explore the possibility that the latter is explained by a new light resonance. We nd that LHCb's in the low-q2 bin is achievable in a restricted parameter space of new-physics scenarios in which the new, light resonance decays preferentially to electrons and has a mass within approximately 10 MeV of the di-muon threshold. Interestingly, such an explanation can have a kinematic origin and does not require a source of lepton- avor universality violation. A model-independent prediction is a narrow peak in the di erential B ! K e+e rate close to the di-muon threshold. If such a peak is observed, other observbetween models. However, if a low-mass resonance is not observed and the low-q2 anomaly increases in signi cance, then the case for an experimental origin of the lepton- avor universality violating anomalies would be strengthened. To further explore this, we also point Beyond Standard Model; Heavy Quark Physics - ables, such as the di erential B ! Ke+e central value of RK out that, in analogy to J= and + decays of mesons can be used as a cross check of lepton- avor universality by LHCb with 5 fb 1 of integrated luminosity. 1 Introduction 2 Model-independent analysis 3 4 5 2.1 2.2 3.1 3.2 Discussion and conclusions A Light o -shell V in b ! s`` B Form factors 1 Introduction Cross-checking lepton-universality violation current decays of B mesons. In the SM, these decays are induced at one-loop level and are additionally suppressed by the Glashow-Iliopoulos-Maiani (GIM) mechanism. For these decays, observables that are sensitive to lepton- avor universality (LFU) are ratios of decay rates to muons and electrons, i.e., RM = BR(B ! M BR(B ! M e+e ) ) ; + M = K; K ; Xs; : : : Recently, the LHCb collaboration determined [1, 2] 0:05 ; for q2 2 [1:1; 6] GeV2 ; { 1 { where q2 is the di-lepton invariant mass squared. The SM predictions for these observables have small, percent-level uncertainties. Away from the di-muon threshold, q 0:045 GeV2, RKSM and RKSM are 1 with high precision [3, 4]. RKSM in the low-q2 bin is slightly 2 = 4m2 ' below 1, mainly due to phase space e ects [4]: RKSM = 1:00 0:01 ; for q2 2 [1; 6] GeV2 ; RKSM = (0:91 1:00 2:5 tension in the two bins for RK . If the discrepancies between measurements and SM predictions are due to New Physics (NP) from four-fermion contact interactions, the ratio RK is expected to have a non-trivial q2 dependence. At low di-lepton invariant mass, the B ! K `+` rates are dominated by a 1=q2 enhanced photon contribution, which strongly dilutes NP e ects in the low-q2 bin. Model independent analyses [5{9] nd that a NP contact interaction that explains RK and RK in the high-q2 bins a ects RK in the low-q2 bin typically by at most 10%. We are, therefore, led to explore the possibility that the low-q2 discrepancy in RK may be a hint for new light degrees of freedom, which cannot be described by an e ective Lagrangian with only SM elds (see, however, also ref. [10]). The possible e ects of resonances below the electroweak scale on LFU in B ! K( )`+` have been previously considered in refs. [11{18]. In this work, we point out that a light, new resonance can a ect the low-q2 bin of RK only in a very restricted range of parameter space once all relevant constraints are taken into account. If the resonance has a mass signi cantly below the di-muon threshold, it a ects RK from an o -shell exchange. We nd, however, that the related two-body decays of B mesons into nal states containing the resonance on-shell typically oversaturate the total B width. We thus exclude such a scenario. If the resonance mass is close to or above the di-muon threshold, strong constraints exist from the existing measurements of the di erential B ! K e+e resonance searches in the B ! K + decay [20]. rate [19] and from di-muon Our main result is that a light new resonance can produce a suppression of RK in the low-q2 bin only if the resonance decays preferentially to electrons and its mass is within approximately 10 MeV of the di-muon threshold. Such a situation can occur either because the resonance couples non-universally to charged leptons or because its decay to muons is kinematically forbidden even if its coupling is universal, e.g., dark-photon models. This leads to testable consequences for other LHCb measurements. In particular, it implies that the di erential B ! K e+e should be searched for experimentally. Analogously, the Bs ! feature a peak close to the di-muon threshold of the same relative size. A peak should also and be present in the di erential B ! Ke+e rate close to the di-muon threshold. While K are vectors, K is a pseudoscalar. Therefore, the size of the peak in B ! Ke+e is e+e spectrum has to rate close to the di-muon threshold features a peak that { 2 { model dependent and allows us to distinguish between di erent avor violating interactions of the resonance to bottom and strange quarks. The connection between the deviation in the low-q2 bin of RK and the peaks in the B ! K e+e e+e spectra is robust. This allows us to further conclude that if the low-q2 deviation persists and becomes statistically signi cant, but no peak is observed, the case for a systematic experimental origin of the deviation would be strengthened. This will have implications for the interpretation of any anomaly in the high-q2 bin, if it persists. The paper is organized as follows: in section 2, we show how a light, new resonance can a ect the low-q2 bin of RK taking into account all relevant experimental constraints. independent implications for the B ! K e+e current and show the corresponding model dependent implications for the B ! Ke+e decay. In section 4, we propose additional LFU measurements for the LHCb experiment that could lead to further insights into the origin of the low-q2 anomaly. Finally, in section 5 we discuss and summarize our results. In section A we elaborate on the o -shell case, and in section B we report the form factors used in our analysis. 2 Model-independent analysis In this section, we discuss the impact of a light, new resonance, X, in RK , keeping the discussion as model independent as possible. 2.1 O -shell e ect of a light resonance The o -shell exchange of a resonance far below the di-muon threshold can in principle contribute to the B ! K `+` rate in the low-q2 bin. The propagator is approximately proportional to 1=q2, which enhances the o -shell contribution at low q2 (like the SM photon). We thus expect such o -shell exchanges to have a high impact on measurements at low q2, which could account for the anomaly in the low-q2 bin of RK . However, we show here that such a setup is unlikely to satisfy existing experimental constraints. To illustrate this point, we consider a very light resonance, X, with a mass far below the low-q2 bin of RK , i.e., m2X 0:045 GeV2, that couples to leptons (with coupling g`, ` = ; e) and o -diagonally to bottom and strange quarks. If the o -shell exchange of X produces a visible e ect in RK , then this would typically imply a two-body inclusive B ! XsX width that exceeds the total B width. For example, if we assume that X has a { 3 { avor changing dipole interaction,1 we estimate that (B ! XsX) SM B;tot e 2 4g`2 ( RK ) 2 BR(B ! Xs ) ' 800% low-q2 bin), and where we have used BR(B ! Xs ) = (3:32 0:15) 10 4 [21]. Given that RKSM RK (in the the coupling of light ( 10's of MeV) new degrees of freedom to electrons and muons are constrained to be . 10 3 (see gure 7 in section A), the B ! XsX decay width typically exceeds the experimentally determined total B width by a factor of a few, which excludes such a scenario. For the derivation of eq. (2.1) we assumed that the resonance couples only to one type of lepton. Barring cancellations, the same argument leads to even more stringent constraints if we assume couplings to both muons and electrons. We quantify our argument in detail for a vector resonance in section A. + + 2.2 On-shell production of a light resonance Having argued that the o -shell exchange of a light resonance cannot a ect the low-q2 bin of RK in an appreciable way, we now discuss scenarios in which on-shell production of the resonance (B ! K X with X ! `+` ) a ects the low-q2 bin. In the case of a narrow resonance, this is possible as long as the mass of the resonance is inside the [0:045; 1:1] GeV2 bin, up to experimental resolution e ects. In the on-shell approximation there is no interference with the SM b ! s`` amplitudes, so the resonance can only enhance the B ! K `+` rates. Therefore, in order to explain RK in this scenario, the resonance has to decay more often into electrons than into muons, i.e., BR(X ! e+e ) > BR(X ! + ). In general, the scenario can be model independently de ned by the following set of parameters: (i) the mass of the resonance, mX ; (ii) the B meson branching ratio BR(B ! K X); (iii) the leptonic branching ratios of the resonance, BR(X ! e+e ) and BR(X ! ); (iv) the total width of the resonance tXot. We will nd that the mass of X has to be close to the di-muon threshold. Far below the threshold, the e ect in RK constraints from the measured B ! K e+e becomes negligible, while far above the threshold the spectrum and searches for B ! K X(! ) are severe. In the following, we therefore focus on the case of X masses for which the decay to 's or to two or more hadrons is kinematically forbidden. The total X width is then the sum of the partial width into the visible nal states of electrons and muons, as well as the width into invisible nal states like neutrinos and any other kinematically accessible decay channel of the X to \dark", non-SM particles.2 We work in the limit of X narrow width, tot mX . The width of X is bounded from below, as the leptons and the K are observed to originate from the same vertex [2]. Demanding that the X decays 1The qualitative conclusions remain the same for di erent choices of the particle X and its interactions with fermions. 2We do not consider the decay X ! do not consider the decay X ! 0 , which has a tiny branching ratio in typical models. We also that is possible if X is a (pseudo)scalar. { 4 { . 2mm) and using a typical boost factor of 200,3 we nd This is compatible with the narrow-width assumption for the range of masses we consider. The new resonance then a ects the B ! K `+` di-lepton invariant mass [qm2in; qmax] in the following way 2 branching ratios in a given bin of hBR``i qmin qmax = hBR`S`Mi qmin qmax + BR(B ! K X) BR(X ! `+` ) G(r`)(qmin; qmax) : (2.2) The function G detector. We assume a Gaussian smearing such that (r`)(qmin; qmax) models the imperfect di-lepton mass resolution of the LHCb G(r`)(qmin; qmax) = p 1 Z qmax 2 r` qmin djqje (jqj mX )2 2r`2 : (2.3) For the resolutions we use re = 10 MeV for electrons [23] and r = 2 MeV for muons [24]. We neglect the dependence of the mass resolution on q2, as we always consider a very narrow range of masses for X. sponding modi ed branching ratios The NP prediction for RK in the bin [qm2in; qm2ax] is then determined by the correRK = hBR i qqmmianx .hBReei qmin qmax : (2.4) branching ratios hBR`S`Mi qmin qmax . We use flavio [25] to compute the SM predictions and uncertainties of RK and the As long as the mass of the new resonance is not more than O(re) = O(10 MeV) outside the lower edge of the [0:045; 1:1] GeV2 bin, the B ! K X and X ! `+` branching ratios can be adjusted to account for the RK value measured by LHCb. Various other measurements constrain the NP parameter space. The most stringent constraints are: The LHCb search for a resonance in the di-muon invariant mass spectrum in the B ! K X(! + the product BR(B ! K X) BR(X ! the X mass and the X width. If X is to explain the low-q2 bin of RK , the bounds for a promptly decaying X apply. No bound can be obtained from this search for ) decay [20]. This search places very stringent upper limits on + ), which are given as a function of mX < 2m , where BR(X ! + ) = 0. The di erential branching ratio of B ! K e+e the product of BR(B ! K X) measured by LHCb [19] constrains BR(X ! e+e ) for resonance masses below and above the di-muon threshold. LHCb presents measurements of six bins of q2, ranging from 0:0004 GeV2 to 1 GeV2 [19]. Interestingly, a small excess of B ! K e+e events is observed in the q2 bin below the di-muon threshold, leading to a slight preference for a non-zero BR(B ! K X) The bounds on BR(B ! K BR(X ! e+e ) for mX < 2m . ) obtained at the B factories [26, 27] are relevant for the case in which the resonance has a sizeable branching ratio into invisible nal BR(X ! invisible) < 5:5 10 5 at 90% Con dence Level (C.L.). states. The most stringent bound is obtained by Belle [26]; it reads BR(B ! K X) 3We estimate the boost factor using a mean energy of 80 GeV for the B mesons at LHCb [22]. { 5 { For the numerical analysis, we construct a 2 function based on a gaussian likelihood search, the B ! K e+e distribution, and the B ! K function that contains the low-q2 bin of RK , the limits from the B ! K X(! + ) bound. To account for the asymmetric error of RK we use the positive (negative) side of error if the RK prediction lies above (below) the experimental central value. From ref. [20] we extract the bound on prompt X decays BR(B ! K X) BR(X ! implement in the 2 for all masses mX > 2m + ) < 3 10 9 at 95% C.L., which we close to the di-muon threshold. We take into account all q2 bins measured in the LHCb analysis [19] of B ! K e+e . The theory uncertainties in this q2 region are mainly due to form factors and CKM elements. We, therefore, assume that these uncertainties are 90% correlated across the bins. E ciency e ects are estimated by comparing the SM prediction from flavio with the ones presented in the LHCb analysis [19]. To capture possible uncertainties of this procedure, we in ate the theory uncertainties from flavio by a factor of 1:5 to be conservative. Taking into account the correlation, we add the theory errors in quadrature with the experimental errors. We have checked that choosing a di erent level of correlation in the theory uncertainties does not lead to qualitative changes in our results. For a given set of NP parameters, we plot contours of 2 = 2 and B ! K the low-q2 bin of RK to the preferred regions at 68:27% and 95:45% C.L., and give both comparison. We also show separately the preferred 68:27% C.L. region for RK in the low-q2 bin ignoring all constraints. For the constraints from B ! K X(! + ), B ! K e+e , we shade the part of the parameter space that is excluded at 95% C.L. We nd that once the above mentioned constraints are taken into account, the discrepancy in can only be addressed in a very restricted range of NP parameter space. We rst illustrate this in a simple benchmark scenario, in which we identify the resonance with a dark photon, i.e., X ) A0. We then discuss the viable parameter space in the case of a generic resonance. 2min corresponding 2min and 2 SM for 2.2.1 Dark photon | LFU violation without LFU violation If the resonance is a dark photon, A0, its branching ratios to electrons and muons are xed by the dark-photon mass, mA0 , its total width, tAo0t, and either the kinetic-mixing parameter A0 or equivalently the dark-photon partial width to non-SM particles, other. In the mass range we consider, the total width is given by with tAo0t = eAe0 + A0 + other ; A0 `A`0 = 2 e 2 12 m2 ` m2A0 s m2 m2A0 mA0 1 + 2 1 4 ` (m2A0 4m`2) : (2.5) (2.6) We nd it convenient to parameterize other = ( eAe0 + A0 A0 ). In this parametrization, the dark-photon branching ratios to electrons and muons are independent of . A dark-photon { 6 { 4 3 green the preferred region from the measurement of the low-q2 bin of RK alone. In grey the 95% CL exclusion regions from the measurements listed in the legend. In the left plot, the dark photon is assumed to decay 100% to electrons and muons; the dark-photon mass and BR(B ! K A0) are varied. In the right plot, the dark-photon mass is xed to mA0 = 208 MeV; the BR(B ! K A0) and the invisible width (parameterized by , see text) are varied. The red cross at BR(B ! K A0) = 1:2 10 7 and mA0 = 208 MeV (left), and BR(B ! K A0) = 1:2 10 7 and = 0 (right) are the best- t values in each case. benchmark is then fully speci ed by choosing4 mA0 ; BR(B ! K A0); : (2.7) In the left panel of gure 1 we consider the case of = 0 and show the constraints and preferred region in the parameter space of mA0 and BR(B ! K A0). In green we show the preferred 68:27% C.L. region for the low-RK bin and in magenta the preferred 68.27% and 95.45% C.L. regions of the combined B ! K X(! from B ! K + ) exclude the shaded grey regions at 95% C.L. There is no constraint here since = 0. The best- t point of the joint 2 is at mA0 ' 208 MeV and BR(B ! K A0) ' 1:2 10 7 (red cross in gure 1).5 We see that the preferred region (magenta) is constrained to be below and close to the di-muon threshold. After pro ling away the BR(B ! K A0) direction we nd that mA0 2 [203; 211] MeV at 68:27% C.L. The comparison of the minimum of the joint 2 , 2 min = 5:2, to the SM one, 2SM = 15:8, shows 2 . LHCb's constraints on B ! K e+e and and hBRSMijlow-q2 ' 1:2 10 7. 4The kinetic mixing parameter determines the total width of the dark photon. As long as is large enough such that the dark photon decays promptly, the exact value of is not relevant for our discussion. 5For comparison, note that in the SM the branching ratios in the low-q2 bin are hBReSeMijlow-q2 ' 1:3 10 7 { 7 { that a dark photon in the preferred region describes low-q2 data signi cantly better than the SM alone. This is driven by an improved t to RK . Next we turn on the partial width of A0 to light non-SM particles, i.e., 6= 0. The presence of these additional decay channels reduces the branching ratios of A0 to electrons and muons. Correspondingly, a larger BR(B ! K A0) is required to explain the anomaly. This is illustrated in the right panel of gure 1 where we x the dark-photon mass to mA0 = 208 MeV and show the preferred region of parameter space in the vs. BR(B ! K A0) plane. We see that for large values of > O(103), the constraint from B ! K + invisible excludes an explanation of RK by the dark photon. However, the point with = 0 is slightly preferred. Interestingly, the dark-photon explanation of the low-q2 bin does not introduce any sources of LFU violation beyond the SM. In this attractive, minimal scenario, the modi cation of RK arises due to the di erence of electron and muon mass. Note that the value of RK does not depend on the kinetic-mixing parameter, , as long as the dark photon decays promptly. At a mass of 210 MeV the dark photon is constrained by the APEX, MAMI, and BaBar experiments to have a mixing . 10 3 [28{32]. A dark photon with a coupling that saturates this limit has a decay length of about 80 microns including a typical Lorentz boost factor of 200 (see footnote 3). This is fully compatible with the maximal displacement of 2 mm seen in the RK measurement [2]. 2.2.2 Generic resonance In the generic case, we treat the electron and muon branching ratios of the resonance as independent parameters. Instead of introducing a Lagrangian, for which we would have to specify the spin of the resonance and the chiral structure of its couplings, we introduce the parameter y 2 [0; 1] which interpolates between the case of BR(X ! and BR(X ! e+e ) = 0 for y = 1. We thus use the parameterization + ) = 0 for y = 0 The generic scenario is then fully speci ed by the parameter set BR(X ! e+e ) = BR(X ! + ) = BR(X ! other) = 1 1 1 + 1 + 1 + : (1 y ; y) ; mX ; BR(B ! K X); y; : (2.8) (2.9) (2.10) (2.11) HJEP03(218) For a resonance mass below the di-muon threshold, i.e., mX < 2m , the branching ratio to muons vanishes and, thus, at these masses this scenario is identical to the dark-photon model discussed in the previous section. In gure 2 we pick a mass for the resonance above the di-muon threshold, mX = 220 MeV. In the left panel, we show the preferred region in the space of BR(B ! K X) and y, xing = 0 corresponding to the case of no invisible decays. We observe that a resonance with a larger branching ratio to electrons than to muons, i.e., y < 0:5, is preferred. The dashed vertical line at y = 0:29 corresponds { 8 { RKlow 6 8 :2 7% C L all 2min( 2SM)=12:2(15:8) 0:1 0:2 0:3 y y = 0 (left), and BR(B ! K X) = 3:5 10 8 and bin of RK . In magenta the preferred region including all constraints (see main text) and in green the preferred region from the measurement of the low-q2 bin of RK alone. In grey the 95% CL exclusion regions from the measurements listed in the legend. The mass of the resonance is xed to 220 MeV. In the left plot the invisible branching ratio is set to zero ( = 0). In the right plot the branching ratio to muons is set to zero (y = 0). The red crosses at BR(B ! K X) = 3:5 10 8 and = 0 (right) correspond to the best- t values. to the case of the dark-photon scenario discussed above. In the right panel, we vary BR(B ! K X) and , xing y = 0 corresponding to BR(X ! of the dark photon, a large invisible branching ratio is allowed. We see that for mX = 220 MeV, the minimum of the total 2 is signi cantly larger than for the dark-photon case above ( 2min = 12:2 and 5:2, respectively) and corresponds to BR(B ! K X) = 3:5 10 8 and y = 0 in the case of 3:5 10 8 and = 0 in the case of y = 0 (red crosses in = 0, and to BR(B ! K X) = gure 2). This is predominantly due to the tension between the low-q2 bin in RK and the B ! K e+e constraint for this choice of mX . If we increase the X mass to values above 220 MeV, the constraint from the B ! K e+e spectrum becomes stronger excluding an explanation of the low-q2 anomaly + ) = 0. As in the case in RK . 2.2.3 Model-independent predictions As discussed above, any on-shell explanation of the low-q2 bin of RK requires a resonance close to the di-muon threshold decaying preferentially into electrons.6 A model-independent 6In the past, a new particle in a very similar mass range had been proposed in connection with avor physics. A light unspeci ed resonance was invoked as an explanation for the anomalous clustering of events with di-muon mass at 214:3 0:5 MeV in the decay by the HyperCP collaboration [33]. Recent LHCb measurements of the same decay mode do not lend further support to a hypothesis of a new 214 MeV mX = 208 MeV and BR(B ! K X) = 1:2 10 7 BR(X ! e+e ) = 100% HJEP03(218) 0:00 0:02 0:04 0:06 0:08 0:10 0:12 0:14 in the presence of a resonance with the best- t values for its mass and BR(B ! K X) (solid red). We also show the binned predictions together with the LHCb measurements from ref. [19]. The vertical line indicates the lower boundary of the low-q2 bin in the RK measurement. key prediction is therefore a peak in the di erential B ! K e+e rate at a q2 close to the di-muon threshold. For a resonance that decays only to electrons (y = 0; = 0), the 68:27% C.L. region for the mass is mX 2 [203; 212] MeV. If instead the resonance has a non-negligible decay mode into muons (like the dark photon) the 68:27% C.L. region is mX 2 [203; 211] MeV. For a resonance mass below the di-muon threshold, the size of the peak is completely xed. Above the di-muon threshold the size of the peak scales as BR(X ! e+e )=(BR(X ! e+e ) BR(X ! + )). In gure 3 we show the peak for the best t point below the di-muon threshold for = 0. We calculate the SM rate using flavio. We see that the SM rate rises as q2 ! 0, due to the contribution from the photon pole. We assume that the resonance is narrow and that the spread in the NP events comes from the experimental resolution in electron reconstruction. Even taking this into account, the peak still rises prominently above the background. Also shown in the plot are SM and NP predictions of BR(B ! K e+e ) for the q2 bins measured by LHCb [19] together with the experimental results. More data and a ner q2 binning should resolve the peak if it is present. An analogous peak with the same relative size is predicted in the Bs ! e+e decay. 3 Model-dependent implications We now consider possible operators that could induce the B ! K X transition for the case in which X is a generic vector resonance, i.e., X ) V . In addition to constraining the Wilson coe cients, this will allow us to make predictions for other observables, i.e., the di erential rate of B ! Ke+e and RK . We shall nd that a future precise measurement of the di erential B ! Ke+e rate and of RK at low q2 can distinguish the di erent operators if they are responsible for the anomalous measurement of RK in the low-q2 bin. We concentrate here on vector resonances with masses just below the di-muon threshto compensate for the reduced V ! e+e branching ratio. old, such that the branching ratio into di-muons is zero. We assume 100% branching ratio into prompt electrons, neglecting possible decays into a dark sector or neutrinos. As we have shown in gure 1, in the presence of a non-negligible invisible width of the resonance, a larger B ! K V branching ratio and, therefore, larger couplings to quarks are required We consider avor-violating couplings of the vector to bottom and strange quarks up to dimension six Le = Cd(d)4 Q(d) + Cd(0d)4 Q0(d) ! + h.c. ; Q(4) = (sL Q(5) = (sL Q(6) = (sL X d=4;5;6 bL)V ; bR)V ; ; Q0(4) = (sR Q0(5) = (sR Q0(6) = (sR bR)V ; bL)V ; ; where the operators are given by with V @ V . In eq. (3.1), we also included the primed operators with a coupling of opposite chirality with respect to the non-primed operators. The widths and the analysis presented here for the primed operators are equivalent to the ones of the corresponding non-primed operators. We thus refrain from explicitly showing the results for the primed operators. In what follows, we shall assume that only one Wilson coe cient contributes at a time. The presence of more than one operator may produce additional interference e ects. Note that if one restricts to processes involving on-shell V , even this minimal set of operators is over-complete. In particular, the free equation of motion for V relates Q(6) = m2V Q(4) and Q0(6) = m2V Q0(4). Nevertheless, these operators are not fully equivalent as the amplitudes with o -shell V exchange di er for Q(4) and Q(6). One can also wonder how a direct coupling of the vector to the bs current in Q(4) and Q0(4) is possible. Recent studies have shown that these interactions do indeed arise in models where a light V couples to quark currents that are not conserved when the SM mass terms and/or quantum anomaly e ects are taken into account [35, 36]. Models with direct avour-universal couplings of V to the axial-vector current of quarks tend to develop Q(4) at one loop, while models with a coupling of V to any linear combination of lepton and baryon currents other than B L induce Q(4) at the two-loop level. In concrete UV completions, the Wilson coe cients in eq. (3.1) will be suppressed by loop factors and couplings. To make a connection to such UV models we pick a set of assumptions motivated by concrete examples and de ne a rescaled Ce(d) for each C(d) as follows. For Q(4) we assume that the interaction is induced by the couplings of the vector to anomalous currents in which case the coupling is two-loop suppressed [35] and we have (3.1) (3.2) (3.3) (3.4) (3.5) The concrete model of ref. [35] is gauged baryon number with a gauge coupling gX and with a small kinetic mixing of the U(1)B and the photon. The translation from our coupling Ce4 to this model is Ce(4) = sin43 W F mt2 m2W or at tree-level [37, 38]. one de ned in [35]. In other classes of models, this coupling can be induced at one-loop [36], gX 102gX ; where F (x) is a loop function of order For Q(5) and Q(6) we assume that as in the SM the relevant couplings are one-loop suppressed and that Minimal Flavor Violation aligns the avor structure of the couplings with the corresponding photonic operators in the SM: C(5) = C(6) = 2 e2 mb 4GF VtbVts 16 2 e Ce(5) p2 e2 1 4GF VtbVts 16 2 e Ce(6) p2 m2W ; 2 e m2W : 2 e Setting ~ = mW gives a Lagrangian in the normalization most frequently employed for the e ective Lagrangian in the SM.7 3.1 Model interpretations of B ! K data The decay width (B ! K V ) induced by each of the operators in eq. (3.1) is (B ! K V ) Q(4) (B ! K V ) Q(5) mm2K2B ; m2V m2B ; mm2K2B ; m2V m2B ; ; where we have de ned the kinematical function 1 + m4K m4B + m4V m4B 2 m2K m2B 2 m2V m2B 2 m2K m2B mm2V2B ; (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) and used B ! K form factors from ref. [39] to compute F1 and F2 (see eqs. (B.3) and (B.4) in section B). Analogous expressions hold for the C(0d) coe cients. Notice that the same combination of form factors enter the decay induced by Q(4) and Q(6). We perform the 2 t outlined in the previous section, including the constraints from B ! K V (! + and from the measured value of RK in the low-q2 bin. In table 1, we list the best- t value of the Wilson coe cients, having xed = 1 TeV, and vice versa the value of having xed the Wilson coe cient to be one. This is shown both for C(d) and the rescaled Ce(d). of UV models, as well as in connection with the high-q2 bin of RK . We can now interpret the results of our best t for the Ce(d) in table 1 in the context 7To see this for Q(6), use the equation of motion for V to relate Q(6) to the semileptonic vector fourfermion operator appearing in the SM (O9). d 4 5 6 C(bde)st t =1 TeV best t | 4:1 106 TeV 16 TeV 1:0 10 2 3:3 mass of the new resonance is approximately 208 MeV. The normalization of the Wilson coe cients Ce(d) is de ned in eqs. (3.5){(3.7). The dimension-four operators that can account for the low-q2 anomaly are wellbehaved perturbative models, even if the coupling is suppressed by two loops as in eq. (3.5). It is notable that in the model of gauged baryon number [36] this is achieved without any avor violation neither in the quark nor the lepton sector, the former being generated by the CKM matrix and the latter by phase space. The dimension- ve, dipole-type operators can t the low-q2 deviation for NP at the TeV scale that is perturbative (coupling of order one) and respects Minimal Flavor Violation. Note that the same can be said for explanations of the high-q2 deviation of RK . Turning this statement around, a generic TeV-scale explanation of the high-q2 anomaly can be augmented by a light mediator, for instance a dark photon, with a judiciously chosen mass in order to explain the low-q2 anomaly as well. The dimension-six interpretation of the low-q2 anomaly appears to be disfavored with our UV assumptions as a non-perturbative coupling is required if the scale of NP is at the electroweak scale or higher. Existing data does not further constrain the parameter space of the models we discussed. Current experiments, however, can test and distinguish these models. 3.2 Predictions for B ! K data Equipped with speci c models we can now correlate the B ! K results of the previous subsection with currently possible B ! K measurements. We focus on the di erential spectrum of B ! Ke+e and RK . The relevant B ! KV partial width induced by each of the operators is 10 5 2 mV = 208 MeV and C(4), C(6)= 2 C(5)= C(5)= with rate enhanced by 100 SM 0:02 0:04 0:08 0:10 0:06 in the presence of a light vector resonance with mass 208 MeV produced via the operators Q(4) (dashed red), Q(5) (dotted green) and Q(6) (dashed red). In each case, we use the best- t value for the corresponding Wilson coe cients (see table 1). The predictions for Q(4) and Q(6) are identical. The Q(5) peak is much less prominent. For illustration we also show this case after enhancing the NP rate by a factor of 100 (dotted dashed green). where now mB denotes the B+ mass and mK the K+ mass. In the kinematical function de ned in eq. (3.11) mK should be replaced by mK . The B ! K form factors f+ and fT are taken from ref. [40] (see section B). From eqs. (3.8){(3.10) and eqs. (3.12){(3.14) we see that: (i) The B ! KV and B ! K V decay widths induced by Q(5) depend on di erent form factors than those induced by Q(4)/Q(6). (ii) The B ! KV and B ! K V decay widths induced by Q(5) have di erent scaling with the vector mass mV , while those induced by Q(4)/Q(6) do not. Therefore, the magnitude of the peak in the B ! K`+` spectra can be used as a way to disentangle the two production modes of the resonance. (iii) The fact that (B ! K V ) Q(4) = implies that the correlation of the B ! K `+` and B ! K`+` observables is identical in both production modes. Therefore, the two production modes cannot be distinguished via the B ! K`+` spectra or a measurement of RK at low q2. Due to the absence of the photon-pole contribution to B ! Ke+e , a peak in the B ! Ke+e spectrum from the new resonance is potentially even more prominent than in B ! K e+e . In gure 4 we show the di erential BR(B ! Ke+e ) as a function of q2. The solid black line depicts the predicted branching ratio in the SM, computed using flavio. The red and green lines show the SM plus NP contribution from Q(4)=Q(6) and Q(5), 0 200 coe cients C(4) (left) and C(6) (right), and the dark-photon mass, mA0 . Superimposed is the best- t region from the measurement of RK , B ! K A0(! + ) and B ! K e+e . We do not show the corresponding case for Q(5) because in the best- t region the e ects in RK are unobservably small. respectively, at the best- t points given in table 1. The bands correspond to the 68:27 C.L. regions of BR(B ! K V ) from the 2 for the case mV = 208 MeV. The prediction from Q(4) and Q(6) coincide due to eq. (3.15), which is why they are represented by the same line. While Q(4) and Q(6) yield a sizeable deviation from the SM, the contribution of Q(5) at the best- t point (dotted green line) is small compared to the SM theory uncertainties (grey band). The dotted dashed green line shows the Q(5) contribution if the NP rate is enhanced by a factor 100 with respect to the best- t rate. The reason for this large suppression of the dipole contribution is that the B decay into the pseudoscalar K and the vector V via Q(5) is suppressed by m2V =m2B compared to the decay into the two vectors K and V , due to angular-momentum conservation. In gure 5, we consider the case of a dark photon with = 0. We show contours of the predicted value of RK in a bin of q2 2 [0:045; 1] GeV2, along with the 68:27% and 95:45% C.L. regions of the 2 including the constraints from B ! K A0(! + and the measured value of RK in the low-q2 bin. We nd that if the new resonance is produced via the operators Q(4) (left panel) or Q(6) (right panel) then RK can be as low as 0:3 in the 95:45% C.L. preferred region. If the new resonance is instead coupled via Q(5) then RK is barely altered from its SM value and we do not show this case. In gure 6, we consider the case of an \electrophilic" vector resonance, i.e., BR(V ! e+e ) = 100%, and, analogously to gure 5, show RK contours and the preferred regions from the 2. In this case, the bounds from the B ! K V (! + not apply and even a resonance with a mass above the di-muon threshold can account for ) resonance search do the low-q2 bin of RK and signi cantly a ect RK . 0:2 0:0 as a function of the Wilson coe cients C(4) (left) and C(6) (right), and the resonance mass, mV . Superimposed is the best- t region from the measurement of RK and the electron distribution. We do not show the corresponding case for Q(5) because in the best- t region the e ects in RK are unobservably small. For both the dark photon and the electrophilic case we see that, if a future measurement of RK in such a low-q2 bin nds a value signi cantly smaller than the SM expectation, the Q(4) and Q(6) production modes would be favored, while the Q(5) mode would be disfavored. 4 Cross-checking lepton-universality violation The central issue looming over the subject of lepton universality in semileptonic B decays (and over NP speculations about its origin) is the question of experimental uncertainties and of possible unaccounted bias in the reconstruction of e+e pairs. If the deviation in the lowand Bs ! q2 bin of RK persists in the future, and at the same time no peaks in the B ! K( )e+e e+e spectra are observed, the case for a systematic experimental origin of the deviation would be strengthened. The LHCb collaboration has performed detailed analyses of the leptonic decays of J= . Those are known to be universal: BR(J= ! `+` ) are equal for muon and electron nal states to very good accuracy [41]. Therefore, LHCb uses these resonant sources of `+` as a normalization for the continuum contribution in RK and RK . The collaboration also tests the overall consistency of the e+e electrons in the K nal states of B0 decays. reconstruction using photon conversion to Here we would like to point out that additional tests can and should be made in other channels where one would not expect large deviations from lepton universality, namely in decays to hadronic nal states with the lowest resonance, m = 1020 MeV. The q2 value corresponding to ! `+` is 1:04 GeV2 and is, therefore, very close to the interesting D D0 ! ! Ds ! Ds ! K + meson are possible at LHCb. The individual branching ratios are extracted using PDG tables [44], while the leptonic branching to individual avors is obtained by multiplying with BR( ! `+` ), which we take to be 2:9 10 4 with 5 fb 1 is obtained by a simple rescaling of results from ref. [43]. . The estimates for the number of expected events values for q2. mesons are copiously produced in a hadronic environment and can be clearly seen as a peak in the di-muon invariant mass spectrum [42]. However, in order to have the maximum resemblance to the semileptonic B decays, one should explore the decay channels of charmed mesons that lead to charged hadrons and a , with decaying leptonically (e.g. D+ ! In table 2, we summarize the relevant decay modes of charmed mesons that can be investigated by the LHCb collaboration. Table 2 suggests that the studies of leptonic decays of generated by charmed mesons are entirely feasible given the number of expected events. ! + We take a previous study of D by LHCb, which recorded several thousand -mediated lepton pairs with 1 fb 1 of integrated luminosity, as an example [43] and make a simple rescaling to higher integrated luminosity to estimate the number of expected events with 5 fb 1 . Unlike the case of B decays where continuum contributions are comparable to the resonant one, the hadron + `+` decay modes of D mesons are dominated by resonances [43]. Therefore, if the suggested test would produce highly discrepant yields for lepton pairs from decays, e.g., by 30% as is the case for RK and RK , then this would likely indicate a potential problem with the LHCb reconstruction of electron pairs. If on the other hand, the results for the -mediated `+` e ects come out to be avor universal, then it would further strengthen the case for NP in RK and RK . As a note of potential curiosity, the comparison of the currently most precise results for the leptonic widths of from KLOE [45] and Novosibirsk [46] already produces a mildly non-universal answer at an approximately 10% level. In particular, taking the combination of p BR( BR( )=BR( ! e+e ) = 1:09 )=BR( ! + ! e+e ) = 0:97 by LHCb would be highly welcome to solidify the situation. !ee ! from Novosibirsk, we nd BR( measured by KLOE and combining it with the measurement of !ee ! + )=BR( ! e+e ) = 1:15 0:06. When we include all KLOE and Novosibirsk measurements of leptonic widths of the discrepancy is milder, decays. Also the PDG averages of branching ratios (which are partly based on older measurements of the branching ratio [47, 48]) are compatible with universality, 0:05. KLOE data by itself supports LFU in 0:06. A new measurement of ! `+` decays We explored possible NP explanations of the anomaly in the low-q2 bin of RK observed by LHCb. Heavy NP parameterized in terms of an e ective Lagrangian typically does not a ect the low-q2 bin appreciably. We found that e ects from new, light degrees of freedom can account for the observation, but are strongly constrained and an explanation of the excess is only possible in a very narrow range of parameter space. In particular, we argued that o -shell exchange of a light resonance, X, (signi cantly below the di-muon threshold) can be excluded as the origin of the discrepancy at low q2, as the implied two-body decay rate B ! XsX typically exceeds the measured total B width. An explanation in terms of one new resonance is possible if the resonance mass is close Bs ! e+e BR(A0 ! e+e ) to the di-muon threshold, mX ' 2m ' 211 MeV, and if the resonance decays predominantly into electrons. Notably, in this mass range the di erence of muon and electron mass are enough to trigger e ects in RK originating solely from kinematics, without requiring a lepton- avor non-universal coupling. A simple interesting example model is given by a dark photon. To explain the low-q2 discrepancy one needs BR(B ! K A0) 10 7. The light resonance near the di-muon threshold a ects mainly the low-q2 bin of RK , while its e ect at higher q2 is negligible. Additional NP is required to explain the high-q2 bin of RK and the anomaly in RK . Turning the statement around, a generic TeV-scale explanation of RK and the high-q2 bin of RK can be augmented by a light mediator to explain the low-q2 anomaly as well. A fairly robust model-independent implication of a light-NP origin of the low-q2 discrepancy is a prominent peak close to the di-muon threshold in the B ! K e+e and di-electron invariant-mass spectra. Within speci c models, we investigated all possible couplings of a vector resonance up to dimension six. We found that couplings from dimensions-four and ve operators can originate from plausible UV completions in the sense that their Wilson coe cients may be induced from perturbative NP at or above the electroweak scale. One possibility for a model for dimension-four couplings is gauged (anomalous) baryon number with a gauge coupling of order 10 3 . This model does not require any sources of avor violation beyond the SM, neither in the quark nor the lepton sector. It is also notable that the scale at which the dimension- ve couplings are induced in order to account for the low-q2 RK explain its high-q2 counterpart. We also investigated the implications of these models for B ! K data and found that the size of a corresponding peak in the B ! Ke+e di-electron invariant-mass spectrum dimension- ve interaction (dipole) leaves the B ! Ke+e depends on the nature of the avor violating b ! s coupling of the resonance. In particular, dimension-four and six interactions lead to a prominent peak in B ! Ke+e , while the decay SM-like to an excellent approximation. If the predicted peaks are not observed in future measurements, then this would suggest that the e ect is unlikely to originate from NP. In such a case, a persistent anomaly in the low-q2 bin could imply a systematic experimental origin of the deviation, which may also a ect the interpretation of other LFU violation hints, such as the high-q2 bins of RK( ) . anomaly is compatible with the scale needed to HJEP03(218) Possible exotic NP explanations that would not predict a peak may still be possible. These include unparticles, or a large discrete set of resonances that are so close in mass that they cannot be resolved as peaks experimentally. As an additional experimental cross check of LFU violation, we proposed measurements of the leptonic branching ratios at LHCb. To have the maximum resemblance to the semileptonic B decays, we suggested to explore the decay channels of charmed mesons to charged hadrons and a . We identi ed several D ; D0, and Ds meson decay modes, each of which lead to O(104) leptonically decaying 's with 5 fb 1 of data. This suggests excellent prospects for a precise measurement of the ratio of ! + and ! e+e HJEP03(218) branching ratios. Acknowledgments We thank Kaladi Babu and Pedro Machado for discussions. WA and SG thank the Mainz Institute for Theoretical Physics (MITP) for its hospitality and support during parts of this work. The work of WA and SG was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. The research of WA is supported by the National Science Foundation under Grant No. PHY-1720252. SG is supported by a National Science Foundation CAREER Grant No. PHY-1654502. MJB and AT would like to thank Fermilab for its kind hospitality and support during the early stages of this project. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. MJB was supported by the German Research Foundation (DFG) under Grant Nos. KO 4820/1-1 and FOR 2239, by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 637506, \ Directions"), by Horizon 2020 INVISIBLESPlus (H2020-MSCA-RISE-2015-690575) and by the Swiss National Science Foundation (SNF) under contract 200021-175940. The work of AT was supported under the International Cooperative Research and Development Agreement for Basic Science Cooperation (CRADA No. FRA-2016-0040) between Fermilab and Johannes Gutenberg University Mainz, and partially by the Advanced Grant EFT4LHC of the European Research Council (ERC) and the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA | EXC 1098). A Light o -shell V in b ! s`` In this appendix we demonstrate that the measured value of RK in the low-q2 bin cannot be explained by the o -shell exchange of a light vector boson, V , with vectorial couplings to leptons and a mass signi cantly below the di-muon threshold. Such a light vector could in principle lead to a NP e ect in the three-body decay B ! K `+` that is enhanced at low q2 by m2B=q2. In practice, however, we nd that such an e ect is severely constrained by limits on the partial width of the inclusive two-body decay B ! XsV and limits on the couplings of V to leptons. A robust limit on the B ! XsV partial width, which is completely independent of the possible V decay modes, is given by the measured total B width, (B ! XsV ) < 1= B. An equally robust and slightly stronger constraint can be obtained from measurements of the charm yield per B meson decay. The BaBar analysis [49] nds that the average number of charm quarks per B decay is Nc = 0:968+00::004453, where we added the statistical uncertainty, the systematic uncertainty, and the uncertainty from charm branching ratios in quadrature. The measured value of Nc implies that the branching ratios of non-standard charmless decay modes such as B ! XsV are bounded by 11:8% at the 2 level. It follows that (B ! XsV ) . 11:8% 1= B ' 4:7 10 14 GeV ; (A.1) derived from B ! K( )e+e where we used the lifetime of the charged B meson B = 1:638 0:004 ps [44]. We note that in many cases much stronger bounds on the B ! XsV branching ratio can be obtained depending on the V decay modes. If the V decays dominantly to invisible nal states or is stable on detector scales, constraints from B ! K( ) imply BR(B ! KV ) . 1:7 10 5 [27] and BR(B ! K V ) . 4:0 10 5 [26] at 90% C.L. Constraints at a similar level can be measurements [19, 50], if the V decays promptly into electrons and has a mass mV & 20 MeV. We will not consider these much stronger constraints in the following, as the model-independent constraint in eq. (A.1) turns out to be su ciently powerful to exclude observable e ects in the low-q2 bin of RK . In eq. (3.1) we introduced the possible avor violating interactions of a new vector to SM quarks up to dimension six. The contribution of the dimension-six interaction, Q(06), to B ! K `+` are not enhanced at low q2 by m2B=q2, and we, therefore, only consider Q(4) and Q(5) in the following. In the limit mV mB, we nd the following partial decay widths of the inclusive decay B ! XsV (B ! XsV ) Q(4) = jC(4)j2 m3 32 m2Vb ; (B ! XsV ) Q(5) = jC(5)j2 m3 b : 2 4 Using eq. (A.1) and the PDG value for the bottom pole mass mb = 4:78 0:06 GeV [44], we nd the following bounds on the couplings C(4) and C(5) jC(4)j . mV 100 MeV jC(5)j The o -shell corrections to the low-q2 bin of RK do not only depend on the quark avor violating couplings of the V , but also on the V couplings to muons and electrons. Here we focus on vector couplings8 Lleptons ge(e e)V + g ( )V : Two concrete setups that induce such vector couplings are: (i) kinetic mixing of V with the photon; (ii) the gauging of avor-speci c lepton number. In the case of kinetic mixing, 8Introducing simultaneously axial-vector couplings and/or couplings from higher-dimensional operators may open up the possibility of tuned cancellations in some of the constraints discussed below. We do not consider the possibility of such cancellations here. ( ) (A.2) (A.4) one has ge = g . In the case of gauged lepton number, one simultaneously also generates couplings to the corresponding neutrinos. In the latter case, strong constraints on the coupling to muons, g , can be derived from the measured rate of neutrino trident production [51, 52]. The bound is at the level of g . 10 3 and is shown in the left panel of gure 7. This bound is independent of the decay modes of the V . Also shown in the plot is the region of parameter space that would allow us to address the (g 2) anomaly at the 2 level, as well as the exclusion by (g 2) at the 5 level. The (g 2) bound is independent of both the V decay modes and the couplings to neutrinos. Strong constraints on g could be obtained at a muon beam dump experiment [53]. The anomalous magnetic moment of the electron, (g 2)e, leads to a bound on the V coupling to electrons that is independent of the V decay modes. The anomaly in the gyromagnetic ratio of the electron, ae = 12 (g 2)e, can be predicted in the SM with high precision using measurements of the ne-structure constant in atomic physics experiments. This results in the bound [54] j aej . 8:1 10 13 : The contribution to ae from a V loop implies a bound on the coupling ge g 1:9 10 3 ; (A.5) (A.6) where we assumed mV me. Additional strong constraints on the coupling to electrons can be obtained from the xed target experiment NA64 [55, 56] and from BaBar searches [31, 57]. In the relevant range of V masses and couplings, there are only two possible decay modes of V : (i) the coupling ge allows the V to decay promptly to electrons; (ii) V can decay invisibly into neutrinos or a light dark sector. If invisible decays are absent or negligibly small, the BaBar search for dark photons [31] leads to constraints on ge that are stronger than the constraints from (g 2)e for masses mV & 20 MeV. In the central panel of gure 7 we show both constraints. If the invisible decays dominate, the BaBar mono-photon search [57] and the NA64 search for dark photons [55, 56] lead to strong constraints on ge as summarized in the right panel of gure 7. Finally, if V also has couplings to quarks (as in the dark-photon case) additional constraints become relevant that restrict the parameter space further [29]. In gure 8 we show the maximal e ects in the low-q2 bin of RK that can be induced by the o -shell exchange of a light vector as function of the vector mass taking into account the constraints on the couplings to quarks and leptons discussed above. We consider separately the production from Q(4) and Q(5), taking their Wilson coe cients to saturate the bounds in eq. (A.3). To identify the maximal e ect in RK we vary the sign of the Wilson coe cients and the phase di erence between the SM and the NP contribution. In the left panel, we consider a muophilic case in which V couples solely to muon avor. For a given V mass we take the maximally allowed g coupling from the -trident bound (left panel of gure 7). In the central panel, we consider an electrophilic case in which V decays solely to electrons. For a given V mass we take the maximally allowed ge coupling from the combination of the bounds in the central panel of gure 7. In the right panel, we consider the case in Muons (g 2) > 5 CCFR (g 2) 2 Electrons ! invisible dom. BaBar BaBar N MV [MeV] 102 101 and right) as a function of the vector mass. The central plot assumes the absence of a relevant invisible decay rate of the vector. The right plot assumes that the invisible decays dominate. The constraints from the (g 2) of the muon and the electron are indirect constraints from having V in the loop, while the constraints from BaBar [31] and NA64 [55, 56] come from direct decays to electrons. On the left panel, the constraint from -trident production based on measurements by the CCFR collaboration [52] assumes that the V couples with equal strength to the left-handed muon and muon neutrino [51], i.e., to the SU(2) doublet L . This constraint does not apply to models in which V does not couple to neutrinos, i.e., dark-photon models. 10 3 Muons Q(4) Q(5) Electrons Q(4) V ! invisible dom. Q(5) Q(4) Q(5) 50 100 150 MV [MeV] 50 100 150 MV [MeV] 50 100 150 of a light vector, V , as function of the vector mass. In the left panel, V decays solely to muons with a coupling that saturates the -trident bound from the left panel of gure 7. In the central panel, V decays solely to electrons with a coupling saturating the bounds in the central panel of gure 7. In the right panel, V couples to electrons but primarily decays invisibly, the maximally allowed coupling to electrons is plotted in the right panel of gure 7. For each case, we show the maximal value of RK[0:045;1:1] SM RK[0:045;1:1] Q(d) much smaller than the current discrepancy (horizontal green band). as a function of the V mass. We see that the e ects are which V can decay to electrons but primarily decays invisibly. For a given V mass we take of gure 7. For each case, we show the maximal value of RK[0:045;1:1] the maximally allowed ge coupling from the combination of the bounds in the right panel SM as a function of the V mass. We nd that the e ects are much smaller than the current For the computation of the decay width (B ! K( )V ) we use the form factors given in refs. [39, 40]. In the limit of vanishing momentum transfer, q2 ! 0, the relevant B ! K and for the relevant B ! K form factors f+(0) = 0:335 None of these form factors change appreciably between q2 = 0 and q2 = m2V our numerics we thus use the zero-momentum transfer values above. The functions F1 and F2 in eqs. 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Wolfgang Altmannshofer, Michael J. Baker, Stefania Gori, Roni Harnik, Maxim Pospelov, Emmanuel Stamou, Andrea Thamm. Light resonances and the low- q2 bin of $$ {R}_{K^{*}} $$, Journal of High Energy Physics, 2018, 188, DOI: 10.1007/JHEP03(2018)188