Flavor violating Higgs and Z decays at FCC-ee

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Introduction.Flavor Changing Neutral Currents (FCNCs) are forbidden at tree level in the Standard Model (SM), and are as such ideal to search for effects of beyond the SM (BSM) physics.Most of the FCNC observables are accessible at experiments that are done at relatively low energies, but with large statistics.The list of such observables is very long, and involves both quarks and leptons.The classic examples are B(µ → eγ), µ → e conversion rate, B (s) − B(s) , or K − K mixing, B(B s → µ + µ − ), and many more (for reviews see, e.g., [1][2][3][4][5]).
In this Letter we show that, somewhat surprisingly, the on-shell FCNC decays of the Higgs, B(h → bs) ≡ B(h → bs + bs) and B(h → cu) ≡ B(h → cu + cū), can be added to the list of high energy FCNC observables, since they can be probed at a phenomenologically interesting level at a future lepton collider, such as the FCC-ee [14].Over the full running period of FCC-ee, the collider is expected to produce N h = 6.7×10 5 h's [15] and N Z = 5 × 10 12 Z's [16,17].As we show in the following, FCC-ee is projected to have a sensitivity to B(h → bs) and B(h → cu) below the indirect bounds from B s − Bs and D − D mixing, cf.Table I, and we expect similar sensitivities to apply also to CEPC [18].For a recent analysis of the h → bs reach at ILC, but using b-and c−taggers, see [19], where the leptonic channel reach is consistent with our results 1 .The main reasons for these significant improvements are: i) the recent advances in b-, c− and s-jet tagging, ii) the analysis technique that we advocate for below, which results in excellent sensitivity to these FCNC transitions, and iii) the relatively clean environment of e + e − collisions.The same approach can also be applied to B(Z → bs) and B(Z → cu), however, the phenomenologically interesting branching ratios are still below the floor set by the systematic uncertainties of taggers.
Accessing flavor violating transitions.An analysis strategy that has been successfully applied to h → cc decays [20], as well as to suppressed t → (s, d)W transitions [21,22], is to distribute events into different event types according to how many flavor tagged (and antitagged) jets they contain.In particular, the inclusion of information about events with light jets was shown in Ref. [21] to lead to significant improvement in sensitivity to V ts,td .
Here, we modify the approach of Ref. [21] and apply it to the case of h → bs, cu and Z → bs, cu decays.For notational expediency we focus first on just the bs final state, and then extend these results to the analysis of cu decays.In both h → bs and Z → bs decays there are two jets in the final state; in e + e − → hZ(h → bs, Z → ee, µµ) there are also two isolated leptons, while the e + e − → Z → bs events only have two jets.Applying the band s-taggers to the two jets, the events are distributed in (n b , n s ) ∈ {(0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2)} bins, where n b(s) denotes the number of b(s)-tagged jets in the authors find a slight improvement by going to higher energies, where the relative importance of the Zh contribution decreases, and W -fusion production of the h is the main signal channel.Because the backgrounds are lower, the power of the analysis therefore increases.We focus instead exclusively on the leptonic channel were Zh can be more easily separated from the backgrounds and our Monte-Carlo-free analysis is more trustworthy.We do note that the increase in performance is still qualitatively consistent with our results.A quantitative comparison would require a more detailed study of the systematic uncertainties, which is beyond the scope of present work.
event.The b-and s-taggers need to be orthogonal to ensure no event populates two different (n b , n s ) bins and is double-counted 2 .We denote the tagger efficiencies as ϵ b β and ϵ s β , where β = {l, s, c, b} denotes the flavor of the initial parton (l = g for h and l = u, d for Z).
The expected number of events in the bin (n b , n s ) is given by where the summation is over the relevant (signal and background) decay channels, f = {gg, ss, cc, b b, bs} for the h and f = {uū + d d, ss, cc, b b, bs} for the Z.The expect number of events in each decay channel is given by where B(Z/h → f ) are the corresponding branching fractions, N Z/h are the number of Z and h bosons expected to be produced during the FCC-ee run, while A is the detector acceptance including reconstruction efficiency, which we assume for simplicity to be the same for all the relevant decay channels.
In writing down Eq. ( 1) we have neglected the backgrounds: the τ + τ − for Z → bs and the Drell-Yan, W W, ZZ for h → bs.We expect that the inclusion of these backgrounds will not qualitatively change our results, since for most part they are small enough to constitute only a subleading effect.Perhaps the most worrisome is the ZZ background for h → bs.Even this we expect in the actual experimental analysis to be either reduced enough through optimized selection to be ignored (e.g., through use of a multivariate classifier trained on other kinematic observables such as the invariant masses and angular correlations), or alternatively it can, in the proposed analysis strategy, be treated as an appropriate small re-scaling of the predicted Nf .
The probability distribution p(n b , n s |f, ν) for a given event to end up in the (n b , n s ) bin depends on a number of nuisance parameters, ν = {B(h → f ), B(Z → f ′ ), ϵ α β , N Z/h , A}, which are varied within the uncertainties in the numerical analysis 3 .We build a probabilistic model for p(n b , n s |f, ν), with a graphical representation given in Fig. 1 4 .The probability p(n b , n s |f, ν) depends Decay SM prediction exp.bound indir.constr.B(h → bs) (8.9 ± 1.5) • 10 −8 0.16 2 × 10 −3 B(h → bd) (3.8 ± 0.6) • 10 −9 0.16 on the flavor of the initial Z/h → f parton decay, where f = {uū + d d(gg), ss, cc, b b, bs} for Z(h), since the tagging efficiencies ϵ α β , α = b, s, depend on the flavor of the initial parton.
Experimentally, the value of B(Z/h → bs) would be determined by comparing the measured number of events in each (n b , n s ) bin, N (n b ,ns) , with the expected value N(n b ,ns) .The highest sensitivity to B(Z/h → bs) is expected from the (n b , n s ) = (1, 1) bin, however, keeping also the (2, 0) and (0, 2) bins increases the overall statistical power.In order to estimate the sensitivity of FCC-ee to B(Z/h → bs), as a proof of concept, we can bypass the need for Monte Carlo simulations and work within the Asimov approximation [23] 5 , both because of the simplicity of the study and especially due to the high statistics environment.That is, we consider an ideal dataset where the observed number of events equals N A (n b ,ns) = N(n b ,ns) (B(Z/h → bs) 0 , ν = ν 0 ), that is, it equals to the expected number of events for the nominal values of nuisance parameters and the input value of B(Z/h → bs) 0 .The expected upper bound on B(Z/h → bs) 0 is then obtained from a maximum likelihood, allowing nuisance parameters to float 6 .
Expected reach at FCC-ee.We first focus on the simplified case where only the b-tagger is used, and obtain the expected exclusion limits on FCNC decays summed over light quark flavors, However, the inclusion of strangeness tagging can result in further appreciable improvements in the expected sensitivity.
Fig. 2 (bottom) shows the expected 95% CL bounds on B(h → bs) obtained from the comparison of all possible (n b , n s ) bins with the predictions.Here, the possible bins are (n b , n s ) = {(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (0, 2)}, where the signal mostly populates the (n b , n s ) = (1, 1) bin, while the remaining bins constrain the backgrounds.To scan over possible taggers we assume in Fig. 2 (bottom) for the purpose of presentation a common TPR for b-and s−tagging, ϵ b b = ϵ s s , and similarly a common FPR, ϵ b lsc = ϵ s lcb .This assumption is not crucial, and is for instance relaxed in the analysis in Sec.S3 of the supplementary material.Nevertheless, we anticipate it to give a reasonable guidance on the expected reach at FCC-ee, if the common FPR is identified as FPR=max(ϵ b s , ϵ s b ), where ϵ b s , ϵ s b are the actual tagger working point mis-identification rates.The reason is that the backgrounds with two misidentified jets are highly suppressed relative to the backgrounds with one misidentified jet, and this is more often than not dominated by the larger mis-identification rate.For instance, the performance of the common medium working point (TPR, FPR) = (0.80, 0.004), denoted with a star in Fig. 2 (bottom), is very close to the expected 95% upperlimit B(h → bs) < 9.6 × 10 −4 , obtained when considering all the different efficiencies in the medium working point of the b-and s-taggers introduced in Refs.[25,26], and assuming a 1% systematic uncertainty (the taggers still need to be calibrated).This limit, which does not consider other backgrounds such as Drell-Yan, W W, ZZ, q q, 0. which we expect to not affect significantly the projected reach, is competitive with indirect measurements and represents a complementary direct probe.We use this as a benchmark expected exclusion in our exploration of the impact on new physics (NP) searches.Note that the SM prediction is orders of magnitude smaller, see Table I, so that any positive signal would mean discovery of NP.
In Fig. 2 (bottom) the relative uncertainties on the eight tagger parameters ϵ α β are taken to be 1% (the uncertainties are treated as independent, while the central values are common TPR, FPR).The 1% uncertainty is currently below the calibrated scale factors in the LHC analyses [27,28].However, given the high statistics environment at the FCC-ee, it is reasonable to expect that a dedicated calibration for high precision taggers could reach such relatively low uncertainties.For 1% systematic uncertainties the expected upper bounds on B(h → bs) are statistics limited, except for very large FPR.Incidentally, this also justifies the neglect of systematics in Fig. 2

(top).
A similar analysis can be performed to arrive at the expected FCC-ee sensitivity to B(h → cu).The main difference is that the sensitivity is determined just by the performance of the c-tagger (there is currently no well established "u-tagger").Using the loose (medium) working point for the c-tagger [25,26] leads to the 95% CL expected bound for B(h → cu) < 2.9(2.5)× 10 −3 . 7e move next to the case of Z → bs decays.As before, we perform a scan over tagger efficiencies, taking the same TPR for b-and s-taggers, ϵ b b = ϵ s s , and similarly for the FPR, ϵ b udsc = ϵ s udcb .The resulting expected 95% CL upper limits are shown in Fig. 3, where the solid (dashed, dotted) lines correspond to the default 1% (0.1%, no) systematic uncertainties.The FPR of 10 −4 for ϵ b s and few×10 −3 for ϵ s b were estimated to be achievable at FCC-ee in Ref. [25,26].Obtaining the ϵ s b well below 10 −3 level will be hard, since this is roughly the fraction of b-quarks that decay effectively promptly, within the projected vertexing resolution of FCC-ee detectors [29].To further improve on ϵ s b one would thus need to rely on jet shape variables to distinguish between s-and b-jets.For rather optimistic FPR of 10 −4 the expected reach on B(Z → bs) is O(10 −6 ) (O(10 −7 )) when assuming systematics of 1% (rather aggressive 0.1%), which is still well above the SM value (see Table I).Given existing indirect constraints on effective Zbs couplings coming from b → sℓ + ℓ − transitions, which have already been determined at SM rates, we conclude that it will be challenging to reach bounds on B(Z → bs) that probe parameter space sensitive to NP.Similarly, the expected reach for Z → cu is B(Z → cu) ∼ 2 × 10 −38 , and thus well above the sensitivity of indirect probes, e.g., B(D 0 → µ + µ − ).We further quantify these statements below.
Sensitivity to NP.We define the effective FCNC couplings of the h and Z bosons to b and s quarks as and similarly for couplings to c and u (or b and d) quarks, with obvious changes in the notation.Eq. ( 3) can be obtained as the effective low energy realization of various extensions of the SM, e.g., the addition of vectorlike quarks [19,30], or in the Two-Higgs-Doublet Model (2HDM) [31,32].We provide details on these models in Sec.S5 of the supplementary material, while here we focus on the relevant phenomenology.
Existing direct limits on the non-standard hadronic decays of the Z follow from the agreement of the measurement and the SM prediction for the Z hadronic width [33], giving B(Z → qq ′ ) < 2.9 × 10 −3 at 95 % CL, cf.Table I.Similarly, existing Higgs boson studies at the LHC already impose limits on its undetermined decays B(h → undet.)< 0.16 at 95 % CL [34,35].Assuming this bound is saturated by h → bs or h → cu decays, we obtain |y ij , y ji | ≲ 7 × 10 −3 , where ij = {cu, bs} (shown as purple contours in Fig. 4).
At energies below the h and Z masses, the effective couplings in Eq. (3) give rise to additional contributions in numerous observables, such as the B s − Bs mass splitting and the branching ratio for leptonic decay B s → µ + µ − .Starting from Eq. (3), we perform the matching to the Weak Effective Theory (WET) operators and employ the package wilson [36] to compute the RGE running down to the scale µ ∼ m b , where we use flavio [37] and smelli [38] to compute contributions to the relevant flavour observables and construct the resulting likelihoods.
The Z −bs couplings generate the effective C (′)  [39], which are probed by the B s meson mixing observables.The resulting bounds on flavor changing couplings read |y bs , y sb | ≲ 10 −3 (baring large cancellations), as shown by the red regions in the upper panel in Fig. 4. Similarly, the D − D mixing constraints lead to the indirect constraints on |y cu , y uc | ≲ few×10 −3 , shown in the lower panel in Fig. 4. Excluding the regions with large cancellations, this leads to the approximate indirect bounds on B(h → q i q j ) quoted in Tab.I10 .This is to be compared with the projected upper limits of FCCee on B(h → bs) and B(h → cu) shown with black lines in Fig. 4. Taking the medium working point for jet-flavor taggers, the expected reach B(h → bs) < 9.6 × 10 −4 translates to the bound |y bs , y sb | ≲ 5 × 10 −4 , whereas B(h → cu) < 2.5×10 −3 translates to |y cu , y uc | ≲ 8×10 −4 , as shown by the black solid lines.The latter thus improves the strongest indirect constraints on flavorchanging Higgs couplings by a factor of a few.For completeness, we show with lighter lines the expected bounds obtained employing less performative taggers.Details about h → bd can be found in Sec.S5, as well as more examples of constraints on 2HDM parameter space away from the limit of light Higgs being the dominant contribution.
Conclusions.The FCC-ee, running at the center of mass energies between the Z boson mass and the t t threshold, will allow to measure flavor, electroweak and Higgs processes with an unprecedented level of precision.In this Letter we demonstrated the potential of FCC-ee to explore flavor changing decays of the Higgs and Z bosons (with similar expectations for CEPC).The projected sensitivities to B(h → bs, cu), in particular, go well beyond the current constraints from indirect probes, such as the B s and D meson oscillations.The expected reach does strongly depend on the performance of the flavor taggers, for which we explored a range of achievable efficiencies and uncertainties, based on existing measurements and ongoing studies.Auspiciously, even with rather conservative assumptions, where only the b-tagger is used in the analysis, the projected reach is already such that it will be able to probe significant portions of unconstrained NP parameter space as demonstrated in Fig. 4 (and on the example of a type III 2HDM in S5 C).Finally, as a sideresult we have also updated the SM predictions for the h → bs, cu, and Z → bs, cu branching ratios.These are orders of magnitude smaller, so that any signal in these channels would unambiguously imply existence of New Physics.
Flavor violating Higgs and Z decays at the FCC-ee Supplementary Material Jernej F. Kamenik, Arman Korajac, Manuel Szewc, Michele Tammaro, and Jure Zupan In this supplementary material we give further details on the probabilistic model implemented in the analysis of the projected FCC-ee reach, in Sec.S1, and define the relevant statistical estimates implemented in this work, in Sec.S2.We expand on the results for h → bs, h → cu, Z → bs and Z → cu in Sec.S3.Updated theoretical calculations for the SM FCNC branching ratios are given in Sec.S4, while Sec.S5 contains additional details about the two BSM examples: type III 2HDM and a model with vectorlike quarks.

S1. THE PROBABILISTIC MODEL
In this section we provide further details on the probabilistic model implemented to obtain the FCC-ee reach.The probability distribution functions p(n b , n s |f, ν) give the probability for an event with the initial parton flavor configuration f to end up in the (n b , n s ) bin.By convoluting these distributions with the expected number of events in each f decay channel and summing over all possible f , we can obtain the total number of expected events in the (n b , n s ) bin, see Eq. ( 1).
Given the parton configuration of the two final state jets, indicated as j 1 and j 2 respectively, we denote the flavor tagging of the two final state jets 11 with n q;i = 0, 1, where q = b, s, and i = 1, 2. The probability density distribution then is (see also Fig. 1 for a representation of the model) where we sum over all the possible jet flavor tagging allowed in the (n b , n s ) bin, with the constraint n b;i + n s;i = 1.
Here p(n b;1 |j 1 ) is the (anti)-b−tagging probability of jet j 1 when n b;1 = (0)1.Assuming no kinematic dependence of the taggers, this is simply the probability to get n b;1 b-tags, with tagging probability ϵ b 1 , where ϵ b 1 ≡ ϵ b j1 is the b-tagger efficiency for the initial parton configuration of jet j 1 .Thus, we can model such probability as a Binomial distribution, where Similarly, p(n s;1 |j 1 , n b;1 ) is the (anti)-s-tagging probability of jet j 1 , conditioned over the n b;1 flavor tagging.Namely, where the probability ϵ s 1 is weigthed by a factor (1 − ϵ b 1 ) −1 to account for the fact that we are s-tagging the non b-tagged jets.
The same reasoning applies to the flavor-tagging of jet j 2 , thus the full probability distribution reads (S5) 11 The jet flavor tagging labels are not independent, but rather satisfy n q;2 = nq − n q;1 .The (n b , ns) = (0, 2) bin thus has events with the For flavour-conserving decays, i.e., for all f parton configurations apart from bs, this expression simplifies to Note that the ordering of the tagging that we assumed in the probabilistic model is irrelevant.By imposing the constraint n b;i + n s;i = 1, one can easily verify that Finally, note that all the efficiencies are implicit functions of the nuisance parameters.

S2. THE DETAILS ON STATISTICAL ESTIMATES
Here we give further details on how we arrived at the statistical estimates of the FCC-ee sensitivities, quoted in the main text.As the parameter of interest we use the ratio of the observed FCNC branching ratio normalized to the SM prediction To relate the observed number of events in the (n b , n s ) bin N (n b ,ns) to the number of expected events in the same bin we can construct the extended likelihood where P is the Poisson likelihood and p(ν) is the appropriate distribution for the nuisance parameters.Note that N(n b ,ns) (µ, ν) depends on the parameter of interest through the number of expected decays to the bs partonic final state, N bs , cf.Eq. ( 1).The probability density functions p(n b , n s |f, ν), which also enter Eq. ( 1), are given explicitly in Eq. (S5).Given this probabilistic model for N(n b ,ns) (µ, ν), we follow Ref. [23] and define the profile likelihood ratio with the associated test statistic Here, ν(µ) are the maximum likelihood estimates (MLE) of the nuisance parameters, obtained by maximizing L(µ, ν), varying ν, but keeping µ fixed.The maximum likelihood estimates, μ, ν, are instead obtained by finding the global maximum of L(µ, ν), varying both ν and µ.
Since the expected statistics of events in each (n b , n s ) bin is large, we work in the Asimov approximation [23] to obtain the expected confidence level interval.In the Asimov approximation, one sets the observed values of , where µ true is the input value of µ, and we have defined the nuisance parameters such that ν is the difference to their nominal input values.For Asimov dataset the MLE are thus by definition μ = µ true and ν = 0. Using the Asimov dataset in conjunction with the test statistic t µ , we can make three statements regarding µ: • We set confidence intervals on µ assuming µ true = 1.In particular, the 68% confidence interval [µ low , µ up ] is obtained by solving for t µ = 1.For simplicity, we report as the result the upper relative uncertainty (µ up − μ)/μ = (µ up − 1).If we modify Eq. (S11) to account for the fact that µ ≥ 0, the lower relative uncertainty will be the minimum between (1 − µ low ) and 1.
• We obtain the discovery significance as the significance of ruling out µ = 0, given that µ true = 1.This is done by computing q 0 = t 0 , with the median expected significance Z 0,A ≡ √ q 0 .
• We set upper limits on B(h/Z → bs) assuming µ true = 0.In particular, the 95% CL upper bound is obtained by solving for t µ = (Φ −1 (1 − 0.05)) 2 , where Φ −1 is the inverse of the Gaussian error function.Note that here the limit is one-sided and thus it will not coincide with the 95% µ up which is instead obtained from a two-sided test.
In all three instances, the minimization over the relevant nuisance parameters is assumed.This is achieved with the iminuit python package [41].The nuisance parameters for the h → bs and Z → bs searches are listed in Tables S1  and S4, respectively.In our simplified model we know the functional dependency of N(n b ,ns) on all these parameters are given explicitly by the probabilistic model, Eqs. ( 1), ( 2), (S5).The ν i are taken to be constrained by additional measurements which have yielded the nominal values of nuisance parameters, which we subtract, so that ν i,0 = 0.The additional likelihood term will thus be where σ i is the estimated uncertainty for the i-th nuisance parameter.

S3. FURTHER DETAILS ON THE FCC-EE REACH
In this section we collect further details regarding our estimates of the FCC-ee sensitivity to the Z/h → bs, bd, cu decays.

A. The h → bs decays
The h → bs study has a relatively low statistic, cf.N h in Table S1.Because of this, and the smallness of the SM B(h → bs) branching ratio, the expected sensitivity is well above the SM values.We therefore focus mainly on setting the upper limits on B(h → bs) assuming no h → bs decays in the data.The nominal values and uncertainties of several nuisance parameters can be found on Table S1.
The N h and A are taken from Ref. [15] (note that their relative uncertainties are higher than for the equivalent parameters for Z → bs, discussed in the next subsection).The nominal values and the uncertainties on the flavor conserving branching ratios are taken to be the quoted central values and statistical uncertainties on the signal strength from the preliminary projections of the FCC-ee fits in the Z(→ ν ν)h mode [42], which are consistent with Ref. [15].We verified that increasing the uncertainties on the branching ratios by one order of magnitude will not impact significantly the performance of the analysis.The leading uncertainties are the systematic uncertainties on the tagger efficiencies.We set these to 1% relative uncertainty across all the taggers, which we anticipate to be realistically achievable for the b-and s-taggers at the FCC-ee.
In the main part of the manuscript we showed the results for the true positive rate (TPR) ϵ b b = ϵ s s and the false positive rate (FPR), ϵ b gsc = ϵ s gcb .This parameterization assumes that the two sets of probabilities are the same for both taggers as well as, perhaps more importantly, that the mistag probabilities are the same for every type of jet.This is almost certainly not the case for the actual taggers that will be used, since most taggers have quite different behaviors in the case of heavy and light jets.However, we can interpret any (TPR, FPR) choice as a conservative choice where all mistags are the same as the least stringent of the ones achievable for the individual cases, Another factor that encourages us to take this approximation is that because gg (or uu + dd for Z) and cc final states require two misidentifications in order to populate the signal dominated region, their contributions will be strongly suppressed both with respect to the signal and with respect to the two main backgrounds, ss and bb.In this sense, FPR can be thought of as representing mostly the ϵ b s and ϵ s b values (taken in most of the analyses to be also the same).In Table S2 we list instead the 95% CL upper limits on B(h → bs), which were obtained for two realistic working points for the b-and s-taggers, introduced in Refs.[25,26] where the label runs over β = {g, s, c, b}.The best performance is achieved by combining the Loose b-tagger and Medium s-tagger, although all performances are very similar.This projected limit on B(h → bs) does not take into account other backgrounds such as Drell-Yan, W W, ZZ, q q, which we expect to be subleading, and should not affect significantly the projected reach.The projected FCC-ee sensitivity to h → bs decays is competitive with indirect measurements and represents a complementary direct probe, as we discuss in the main text.

B. The h → cu decays
A similar analysis can be performed for the FCC-ee sensitivity to B(h → cu).Due to the difficulties in implementing a "u-tagger", the main text considers just the case where only the c-tagger is applied (similar to the B(h → bq) case shown in the top row in Fig. 2).We parameterize the c-tagger in terms of the TPR, ϵ c c , and a common FPR for all the other initial partons.We consider a c-tagger with four parameter ϵ c g , ϵ c uds , ϵ c c , ϵ c b , so that the number of nuisance parameters is thus four, one for each efficiency.The systematic uncertainties considered are the same as those listed in Table S1, with the caveat that the efficiencies now refer to the four ϵ c β .The resulting 95% CL upper limits on B(h → cu) as a function of FPR and TPR are shown in Fig. S1 (left).For a given FPR, TPR values the expected upper bounds on B(h → cu), assuming just statistical errors, are stronger than the ones for B(h → bq), Fig. 2 (top), because the dominant background for h → cu, due to h → cc, is smaller than the h → bb background for h → bq.
Besides the two dimensional scan, we also list in Table S3 the 95% CL upper limits on B(h → cu) obtained for two realistic working points for the c-tagger introduced in Refs.[25,26], Loose : ϵ c β;Loose = {0.07,0.07, 0.90, 0.04}, (S15) where the labels run over β = {g, uds, c, b}.The best performance is achieved with the medium working point, although the two performances are again very similar.As for B(h → bs), this projected limit on B(h → cu) does not take into account other backgrounds such as Drell-Yan, W W, ZZ, q q, which we expect to not affect significantly the projected reach.The projected FCC-ee sensitivity to h → cu decays is competitive with indirect measurements and represents a complementary direct probe, as we discuss in the main text.
Regarding the u-tagger, there were very recently developments based on the jet charge [43].Fig. S1 (right) shows the expected 95% CL upper limits on B(h → cu) as a function of FPR and TPR, obtained from the same two tagger approach that we used for h → bs in the main text, but now using the c and the u-taggers.We use the same medium working point as we did for the case of just the c-tagger (TPR,FPR) = (0.80, 0.02), shown in Fig. S1 (left).Note that this is consistent with the tagger performance curves reported in Ref. [43].Note that the u/d separation is rather weak with the u-tagger, but it is not needed in our analysis, since none of the backgrounds contain a d quark.Both the c-tagger and the u-tagger were assumed to distinguish between u/d and s-quarks, ϵ c,u g , ϵ c,u ud , ϵ c,u s , ϵ u,c c , ϵ u,c b with a resulting total of ten nuisance parameters, one for each efficiency (we lump together u and d quarks, but this is mostly for completeness: neither the backgrounds nor the signal posses a d-quark).The systematic uncertainties considered are the same as those listed in Table S1, with the caveat that the efficiencies now refer to the ten ϵ c,u β .If such a u-tagger were to be implemented in practice, the power of the analysis would increase considerably, with the medium working point yielding a 95% CL expected bound B(h → cu) < 6.6 × 10 −4 , compared to 2.5 × 10 −3 for the c-tagger only analysis.The increase in the statistical power is due to both the additional observables, as well as  due to the lack of a background with u-jets that could populate the (n c , n u ) = (1, 1) bin (this is different from the h → bs case, where the h → ss decays can end up in the (n b , n s ) = (1, 1) with a still relatively high probability).
Although very encouraging, one should keep in mind that the u-taggers are still in the early stages of development.The performance one will be able to achieve in practice may thus well differ from the medium working point we assumed above.We also comment in passing that in principle the s-tagger could be applied as a light-jet tagger instead of the u-tagger.The hope would be that this would increase the sensitivity much in the same way as the s-tagger increases the sensitivity of the h → bs analysis compared to the one with the b-tagger alone.However, we find that the performance of the analysis does not noticeably increase, because the s-tagger acts mostly just as the background rejector.In conclusion, while the preliminary results from a two-tagger analysis are encouraging, a more detailed analysis is called for.

C. The Z → bs decays
As for the Higgs decays, we first consider the statistical reach on B(Z → bq) = B(Z → bd) + B(Z → bs), i.e., summing over the Z → bd and Z → bs decay modes, thus only using the b-tagger.The resulting bounds as functions of TPR and FPR are shown in Fig. S2 (left).We see that even ignoring systematic uncertainties, the projected sensitivity is well above the SM B(Z → bq) ratio.
Once we introduce the s-tagger, we can explore the discovery potential of Z → bs decays and the respective confidence interval.We first consider a simplified set-up which allows for a two-dimensional scan, and parameterize the taggers as a set of true positive rates (TPR) ϵ b b = ϵ s s and a set of false positive rates (FPR) ϵ b udsc = ϵ s udcb .The uncertainties on the flavour conserving branching ratios were taken from the Particle Data Group [44], while the uncertainties on the number of Z bosons and the acceptances were taken from Ref. [17].These are shown in Table S4.
Table S5 shows the upper 1σ error, σ + µ , on the measurement of B(Z → bs) SM (2nd column), the discovery significance (3d column), and the expected 95% upper limits (4th column) for two working points, each with either the default 1% systematic and the significantly reduced 0.1% tagger uncertainties.The two working points are chosen as (TPR,FPR)=(0.4,10 −4 ) and (TPR,FPR)=(0.2,10 −5 ).While the first is a reasonable choice for a future tagger performace, the latter represents a very aggressive projection which is clearly hard to achieve.As shown in Table S5, the aggressive choice of tagger efficiencies, coupled with a low systematic uncertainty, is the only scenario where FCC-ee can reach the SM value of B(Z → bs), although with a low discovery significance and with systematic dominated uncertainty.The results discussed above are generalized in Fig. S2 (middle), where we show the expected discovery significance as a function of FPR and TPR, with the solid (dashed) lines denoting the case of 1% (0.1%) systematic errors on the taggers, while the dotted lines assume only statistical errors.Fig. S2 (right) shows, similarly, the expected upper error on µ for each of these cases.Neglecting systematics we observe that an uncertainty on the SM Z → bs branching ratio below 30% is achieved already with the conservative choice of efficiencies, (TPR,FPR)=(0.5,10−4 ), which is not altogether impossible to implement with current state of the art taggers.However, this performance is highly degraded by the introduction of systematic uncertainties, i.e., the measurement is expected to be completely systematic dominated.
From this preliminary study, we can conclude that one will not be able to measure the Z → bs decay rate with enough precision to impact the searches for beyond standard model physics.

D. The Z → cu decays
Similarly to the h → cu decays, we can obtain the expected 95% CL upper limits on B(Z → cu) when applying just the c-tagger.In Fig. S3 we show the projected upper limits as functions of charm tagger TPR and FPR, assuming again the three cases of 1%, 0.1% and no systematic uncertainties on the tagger efficiency.For a given value of (TPR, FPR) the upper bounds on B(Z → cu), when assuming negligible systematics, are almost identical to the ones for B(Z → bq), cf.Fig. S2 (top), obtaining an upper limit of 2.3 × 10 −7 for the medium WP (TPR,FPR) = (0.80, 0.02).This is not surprising, given that in both cases only a single tagger is used as the discriminator, while the Z → bb and Z → cc backgrounds are almost identical in size.Once the systematic effects are taken into account, these completely dominate the performance, saturating the achievable reach (note that for this reason the FPR range displayed in the left and right panels in Fig. S3 differ significantly).For the medium WP (TPR,FPR) = (0.80, 0.02) and 1% (0.1%) systematic uncertainties we obtain an expected 95% CL upper limit on B(Z → cu) of 2.3 × 10 −3 (4.0 × 10 −4 ).
We reiterate that the above results rely on just the c-tagger.As done for h → cu, we could implement a u-tagger to study its impact.However, in this case the background is larger as both the Z → uu, dd decays are important (even if better than a random tagger, the reported tagger in Ref. [43] yields similar ϵ u u and ϵ u d ).This implies that the background from Z → uu + dd would be essentially twice the one given by Z → ss in the Z → bs case.If already   for Z → bs the background contamination is a limiting factor that forces us to consider extreme efficiencies, and systematic uncertainties render a precise measurement very difficult, in this case we are searching for a smaller signal with a larger background and more experimental taggers.We thus deem the application of both a c-and u-tagger more unrealistic than for h → cu.As mentioned for h → cu, in principle one may also try to apply instead an s-tagger as an additional discriminator between signal and background.As in h → cu, preliminary studies of applying an s-tagger as a light-jet tagger have shown no increase in the statistical power.We leave a more complete study of applying a two-tagger analysis for Z → cu for future work.

S4. UPDATED CALCULATIONS OF THE Z/h FLAVOR CHANGING DECAY WIDTHS IN SM
The SM predictions for B(h → bs, bd, cu) were presented in [45,46], and for B(Z → bs, bd) in [47][48][49].Here, we repeat the calculations and update the numerical predictions for the SM value of B(Z/h → qq ′ ) ≡ B(Z/h → q q′ ) + B(Z/h → qq ′ ), with the final results collected in Table I.The numerical inputs are summarized in Table S6.The SM RG evolution of the parameters is performed using the three-loop β-functions [50,51], in order to obtain the t and b quark masses and α, α s at µ = m h , m Z .For the values of the CKM matrix elements we use the results of the global fit from the CKMfitter collaboration [52] (the Moriond 2021 update).
The partial decay widths are, to the order we are working, given by where m h(Z) is the mass of the Higgs (Z) boson, N C = 3 the number of colors, and | M(h/Z → q q′ )| 2 the spin-averaged squared decay amplitude.In the SM, the h/Z → q q′ transitions occur at one-loop, through an up-type quark and W boson exchange for qq ′ = bs, bd, while qq ′ = cu requires a down-type quark.A representative diagram for each decay is shown in Fig. S4, where u k = u, c, t and We first focus on the Z → bq decays.Due to the GIM mechanism, the decay amplitude is proportional to u k and is thus dominated by the top quark contribution.Counting top mass as m t ∼ m Z , gives the following naive dimensional analysis estimate for the decay amplitude, where g is the SU (2) L gauge coupling, s W = sin(θ W ) and c W = cos(θ W ) are the sine and cosine of the weak mixing angle respectively, while V ij are the CKM matrix elements entering the u i − d j − W vertex.The corresponding r 8 1 t 5 7 j b 6 r x v d b 9 0 N / c / V L q t O C + d V 0 7 T 6 T g 7 z r 7 z y T l w j h x a e 1 3 7 X P t W O 6 x / r Z / V f 9 Z / n U O X l 6 q a F 8 4 N q / / + D 3 G 0 a T I = < / l a t e x i t > h b, c Representative one-loop diagrams for the Z → q q′ (left) and h → q q′ (right) decays.The crosses indicate mass insertions.
estimates for the partial decay widths are These turn out to be good approximations to the exact result, given below.We perform the calculation of the one-loop decay amplitude by first generating the one loop Z → bq diagrams using FeynArts [53], and then evaluating the amplitudes using FeynCalc [54][55][56], including the reduction of the loop integrals to the Passarino-Veltmann (PaVe) functions.Finally, we use LoopTools [57] to perform the numerical evaluation of the full amplitude.As an additional check, we have also evaluated the PaVe functions analytically, using the Package-X [58].Isolating the 1/ϵ divergent term, this is of the form where P L = (1 − γ 5 )/2, while u b , u q and ϵ Z are the b-and q-quark spinors, and the Z boson polarization vector, respectively.From the unitarity of the CKM matrix we then obtain M div = 0, giving an independent check on our calculation.In general, all the m k -independent terms in the amplitude give vanishing contributions due to the CKM unitarity; we check independently that each of these pieces cancel.Using the inputs from Table S6, gives for the Z → bq partial decay widths, The quoted theoretical uncertainties reflect only the uncertainties on the inputs, and are dominated by the errors on |V ts | and |V td | CKM elements, which amount in both cases to a relative uncertainty, at the 1σ level, of ∼ 1%, where for simplicity we have symmetrized the uncertainty interval reported in Table S6.The latter translates to a ∼ 2% uncertainty on the Z → bs and Z → b d widths.Other inputs are known with a sub-percent precision and thus we neglect their contribution to the total error budget.
The second source of theoretical uncertainty are the higher order QCD corrections, which we estimate by using the partial two-loop calculation of mixed QCD-EW diagrams from [59][60][61], and include them in the total uncertainty budget.In general, one can write where the ellipses denote the O(α 2 s ) corrections that we have neglected.The vector and axial couplings, g V (A)bq , can be extracted from the full one-loop calculation.The radiators R V (A) represent the corrections from the sum of virtual < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 s 2 f / v C S 0 P R 9 T 3 1 H 0 4 e b l m x K t q o 1 T 5 7 9 y M R n G q S I S L j f w 0 B I o D 8 8 0 D j w q C V T j V E 4 g F 1 b 0 C f A I F x E r / G S p O m L M Y q o H d 0 c / u 5 m G 1 5 / n p R H V c 9 u V w V i e H v Y 7 9 p v P q c 2 9 3 7 0 M Z 3 I b 1 x H p q N S z b e m v t W Z + s f e v A w j V c + 1 E 7 q / 2 s n 9 V / 1 X / X / x T o + l q 5 5 r F V G f W / / w H A W H X 8 < / l a t e x i t > u g 5 j W w 4 d p q l a 6 1 l v r A R e P B p N Z 5 c q 2 P a r P D J T M W U l e H p P G / O C + T Q g M g U p l 0 t v h x g l m 2 h R T 9 o 1 5 8 N n S z x x i a D B 2 G 2 h g c F D u 4 W 8 P R g 8 T w c F n m L 0 1 x c r Z z 4 W m 2 4 W + + 0 O 8 V C 6 x u 7 3 N S t c h 0 N d / 8 5 n q S Z Y J G m I U 7 T g d 2 J t Z t j p T k N Q d L J U h Z j O s Y B G 8 A 2 w o K l b l 6 8 s l P 0 H C I e 8 q W C X 6 R R E V 2 u y L F I 0 4 k g Q A q s R + n V n A l u y g 0 y 7 b 9 z c x 7 F m W Y R n T X y s x B p i c z 7 j z y u G N X h B D a Y K g 6 z I j r C C l M N X 4 k V J S p F j H X P b s O 9 u 4 V b + 3 M D w V K w y 7 5 q z v r m p N u 2 3 7 R f f e r W D 9 + X x u 1 Y T 6 1 n V s O y r b f W o f X R O r K O L V q R l e + V i 8 q P 6 k X 1 Z / V X 9 f c M 3 d 4 q a 5 5 Y K 6 v 6 5 z 9 3 t H q a < / l a t e x i t > D y 7 a T f t d 8 + 2 X 9 v 7 p x 9 K 4 b e u l 9 c q q W 7 b 1 3 j q 1 P l t n 1 r n F K p u V e s W u t K t 2 t V v 9 V v 0 + h W 5 u l D 0 v r J V V 5 f 8 A 4 p S l K A = = < / l a t e x i t > and real (emitted) gluons attached to the external quark legs, see the middle and right diagrams in Fig. S5.To the order we are working these corrections are, in the MS scheme, equal to R V = R A ≃ α s (m Z )/π ∼ 3.7 × 10 −2 [59,60].The coefficients c V (A) in (S22) indicate the numerically leading two-loop gluon correction, see the left diagram in Fig. S5.Using the results from Ref. [61] we get (again in the MS scheme), c V = c A = −α s (m t ) ∼ −0.1.The factor of 2 in (S22) takes into account that this is a correction on the coupling, while the decay width is proportional to The sum of the two contributions gives (R A,V + 2c A,V ) ≃ 0.17.This shift in the predicted decay width is much larger than the uncertainty on the CKM elements.Our estimate for the relative error on the theoretical prediction is therefore ∼ 17%, leading to Using the value of the total Z width [62], Γ Z = 2.4952±0.0023GeV, this then translates to the following predictions for the B(Z → bq) ≡ B(Z → bq) + B(Z → bq) branching ratios, We can repeat the above procedure for the calculation of the remaining decay widths.A representative one loop diagram generating the h → bq transition is shown in Fig. S4 (right).To obtain the correct chirality, we need at least two mass insertions, indicated with a cross in Fig. S4, one of which will be on the external quark legs.Thus we expect the dominant contribution to the h → bq amplitude to be suppressed by an additional factor of bottom Yukawa, y b , compared to the result for the Z → bq decay, Eq. (S18).The NDA estimate for the h → bq decay amplitude is thus and the corresponding NDA estimates for the partial decay widths Using the same FeynArts+ FeynCalc+LoopTools pipeline as for the Z → bq decays above, along with the numerical inputs at µ = m h in Table S6, gives where as a rough guidance we assigned the same ∼ 17% uncertainty due to the missing higher order QCD corrections, which we assigned above for the Z → bq decay.Dividing by the SM prediction for the Higgs width, Γ h = 4.12 ± 0.06 MeV where in the last expression we neglected terms proportional to y u .The GIM mechanism now leads to numerically even more suppressed decays, with the Z/h → cū decay amplitudes about O(10 6 ) times smaller than the Z/h → bs decay amplitudes in Eqs.(S18) and (S26).The full numerical evaluation leads to where the errors reflect our rough estimate of higher order QCD corrections, due to which we assign a ∼ 17% uncertainty on the result.Note that the B(h → cu) prediction quoted above differs from the one reported in Ref. [45] by orders of magnitude, most likely due to an incorrect normalization of the Passarino-Veltmann integrals, while it is consistent with the prediction in Ref. [46].As a consistency check we obtained our result in two partially independent ways; after generating the diagrams and the corresponding amplitudes, the loop integrals were either computed numerically using LoopTools or first computed analytically with Package-X and then evaluated numerically, with the two results agreeing with each other.

S5. ADDITIONAL DETAILS ON THE BSM MODELS
Here we give further details on the indirect constraints on the FCNC couplings of the Z boson and the Higgs to quarks, Eq. (3).We show three different examples of NP effects.In section S5 A we show the constraints on the Z − bs, Z − bd and Z − cu couplings from low energy observables, assuming that these are the dominant NP effects.The same results for the h − bs and h − cu couplings, for which the only low energy constraints are due to B s − Bs and D − D mixing, respectively, were already shown in the main text, cf.Fig. 4.Here we complete this list with constraints on h − bd couplings, dominated by B d − Bd mixing observables.We also show constraints for two UV complete NP models, in section S5 B for the SM extended by a set of vector-like quarks, and in section S5 C for the type III two Higgs doublet model (2HDM) with a particular flavor violating structure of Yukawas.

A. Indirect bounds on FCNC Z couplings
The bs couplings of the Z boson, g L,R sb , Eq. ( 3), result in a shift in a number of low-energy observables, such as the branching ratio of B s → µ + µ − , as well as angular observables in B → K ( * ) ℓ + ℓ − decays, etc. Integrating out the Z S7.The 1σ, 2σ, 3σ limits (from dark to light blue) on g L uc , g R uc couplings of the Z, Eq. ( 3), from rare D meson decay data.The lines denote the 3σ limits that were obtained from various constraints, as indicated in the legend.The two projected 95% C.L. upper limits at the FCC-ee for the medium WP with 1% (0.1%) systematic uncertainties are shown as dotted (dashed) lines, cf.Sec.S3 D.

B. Vector-like quarks
Next, we turn to a concrete NP model, where we add to the SM a single generation of vector-like singlet down-type quarks, (D L , D R ), singlets under SU (2) L and with hypercharge −1/3.These have Yukawa couplings to the SM quarks, see, e.g., [30], where y ij d,u are the SM Yukawa couplings, y i D are the Yukawa couplings to the vector-like quarks, and M D is the vector-like quark mass.After EWSB, the down-type SM quarks and the vectorlike-quarks mix.Diagonalization of the mass matrix leads to flavor-changing couplings of down quarks with the Z and the Higgs boson, where X d ij in general has both nonzero diagonal and off-diagonal entries.Focusing on the bs couplings, this gives for the couplings in the effective Lagrangian (3), Alternatively, one could extend the SM by a single generation of doublet vector-like quarks (Q L , Q R ) with hypercharge 1/6.These can have Yukawa couplings of Q L with both the right-handed SM up-and down-quarks.The Lagrangian reads: After mass diagonalization these then generate right-handed flavor-changing neutral currents in both up and down sectors.Focusing on the down quark sector, the Lagrangian (S41) is replaced by while the couplings in the effective Lagrangian (3) are now given by, The combination X Q sb + X Q * bs is tightly constrained from the fits to the low energy b → sℓℓ data.Extending the SM by a doublet VLQ leads to flavor-violating right-handed currents in the direction of g R sb , whose values are closer to the SM expectation than the left-handed g L sb , explaining the relative position of allowed regions in the vertical direction in the left and right panels in Fig. S8.The orthogonal linear combination X Q sb − X Q * bs is limited by the B s − Bs mixing.Both X Q sb and X Q * bs generate effective contributions to h → bs; X Q sb (X * Q bs ) induces the effective operator C 2 (C ′ 2 ), and these both enter in C 4 .Since X Q sb contribution is enhanced by an m b /m s factor compared X Q bs , the bounds imposed by the B s − Bs mixing on tree level Higgs exchanges are relevant mostly for X Q sb and are saturated at approximately ∼ O(0.1), cf.Fig. S8 (similar discussion applies to X d sb,bs ).

C. Type III two Higgs doublet model
The second example we consider is the type-III two-Higgs-doublet model [31].Denoting the two Higgs fields as H 1,2 , the most general form of the quark Yukawa couplings is given by where q i L (q ′i L ) are the left-handed quark doublet fields written in the down (up) quark mass basis.We work in the Higgs basis, where only H 1 has a nonzero vev, so that the two Higgs doublets are given by, Here, G 0 and G + are the Goldstone bosons, and A the CP-odd heavy Higgs.The CP-even Higgs mass eigen-states h, H are an admixture of h 1 and h 2 , Here h denotes the SM-like Higgs, and we abbreviated c α ≡ cos α, s α ≡ sin α.
We work in the limit where H and A are much heavier than the h.Diagonalizing the mass matrix gives for the light Higgs coupling to quarks, after electroweak symmetry breaking, FIG. 1. Graphical representation of the probabilistic model for determining B(Z/h → bs).Starting with the Z/h → f partonic decay, where f = {uū + d d(gg), ss, cc, b b, bs} for Z(h), the tagged flavours of the two final state jets, Z/h → j1j2, are determined by the corresponding s− and b−tagger efficiencies, ϵ α β .The arrows denote the probabilities for each event to end up in the (n b , ns) bin.

3 95%FIG. 2 .
FIG. 2. Top: Expected 95% CL upper bounds on B(h → bq) as a function of the b-tagger efficiencies, neglecting systematic uncertainties.Bottom: Expected 95% CL upper bounds on B(h → bs) as a function of TPR and FPR.Solid (dashed) lines and colors are with default (no) systematic uncertainties.The Medium Working Point is based on the taggers introduced in Refs.[25, 26].See main text for details.

FIG. 4 .
FIG.4.Top: Current and projected limits on y sb and y bs .Bottom: Current and projected limits on yuc and ycu.The 1σ, 2σ, 3σ regions are depicted from darker to lighter red.

3 × 10 − 3 5 × 10 − 3 7 × 10 − 3 9 × 10 − 3
FIG. S1.Left: Expected 95% CL upper bounds on B(h → cu) as a function of the c-tagger efficiencies Right: Expected 95% CL upper bounds on B(h → cu) as a function of the TPR and FPR of the c-and u-taggers.Solid (dashed) lines and colors are with default (no) systematic uncertainties.The Medium Working Point is based on the c-tagger introduced in Refs.[25,26] and the u-tagger introduced in Ref.[43].

3 TABLE 3 FPR
FIG. S2.Left: The expected 95% CL upper bounds on B(Z → bq) as a function of the b-tagger efficiencies, assuming no systematic uncertainties.Middle: The expected discovery significance for the SM B(Z → bs) as a function of TPR and FPR.Solid (dashed, dotted) lines and colors are for the default systematic uncertainties of 1% (0.1%, 0%).Right: The expected uncertainty σ + µ as a function of TPR and FPR.Solid lines and colors are with default systematic uncertainties while dashed lines correspond to reduced tagging uncertainties and dotted lines to no systematic uncertainties, see text for details.
FIG.S3.The expected 95% CL upper bounds on B(Z → cu) as a function of the c-tagger efficiencies.Left: Solid lines and colors are for no systematic uncertainties.Right: Solid (dashed, dotted) lines and colors are for the default systematic uncertainties of 1% (0.1%, 0%), while the temperature map assumes 1% systematics.
g Q y F J I h m n w 0 9 e E p E Z L y 6 I u a x s R l M I i o T z F U W h r v r J 8 7 i A Q 0 y h S d f I 8 p V q k g s 0 0 A S t U n 0 4 j B y C h a 0 0 a K f A O N 4 y b I n M a x 0 5 z 1 l w s j Q Y M T N e h h B r g P j lv A 4 8 r V t F K a v x Z V h h s y 4 q 2 C E P F T M g c N M r c d o p t s N a E 5 l D e L V p p F J O R f r z R O b j R O D J f k x g 6 C A i Q L 9 2 p Q g d 0 E U I H G y x b o v H / d 7 F + q 9 h b V t t 3 p z u s F 5 F E Y C M h e g C y / B y A b E S 8 g c u A T w f S r c 0 v Z j C L i d h v k C A i h f r b n A 6 e R j i d O s 4 y q t c z n 2 W l e h 1 P V k 2 t 9 z D Y V P i l c z L J S n s 3 r 5 p D o G h g h L n W 3 1 c W L B o p q C 1 z p 7 c b F k 0 n S 7 w J 0 u B B m K 7 g Q Y 7r M H N 4 1 i 8 T N I e Y R N 7 F E c 7 v K s d 8 c 7 y 9 2 + 1 0 8 w F W J 3 Y 5 2 b X K s T / e P n c 8 j l N G I o V D K O X I 7 s b K z a B Q F I f a 0 U k l i S G e w I C M 9 D S C j E g 3 y t e x i t s h a 1 _ b a s e 6 4 = " s x 8 g Q f z i d y4 5 Y 6 9 Z O G h q + f U O K k I = " > A A A F H H i c h Z R N b 9 N A E I b d J k A J X y 0 c u a y I E A l K o z j i S 6 o i V X D h W C T a V I 2 t a H e 9 d l b x e u 3 1 u h C s / A y u 5 c 9 w Q 1 y R + C 8 c m H W c J m m i d q V E 6 5 l n 3 h m / X p v E I U 9 1 p / N 3 a 7 t S v X X 7 z s 7 d 2 r 3 7 D x 4 + 2 t 1 7 f J L K T F F 2 T G U o 1 S n B K Q t 5 x I 4 1 1 y E 7 j R X D g o S s T 8 Y f T L 5 / z l T K Z f R Z T 2 L m C h x E 3 O c U a w g N 9 y o 7 D m E B j 3 L N x 9 9 i T n W m 2 L S G U B n 1 2 S Q S O D I R i I G Q Z l 9 R4 6 y J c q d x 5 j S n B 8 u J g e L B S P e 6 V C D p o 7 M W 8 q R 2 g d Y a + G t R b b i + Y N 4 6 i I k 8 Z 3 P Q I H P Z P r l J F g j g S D E s FIG. S5.Leading two-loop QCD corrections to the Z → bq process.

[ 63 ] 4 FIG
FIG. S6.Left:The 1σ, 2σ, 3σ limits (from dark to light colors) on left-handed and right-handed FCNC couplings of Z to b and s quarks, g L bs , g R bs , from current low-energy experiments, see Eq. (3) in the main text.Right: The 1σ, 2σ, 3σ limits on couplings of Z to bd.
cu) = 4.0 × 10 −4 B(Z → cu) = 2.3 × 10 −3 Combined D decays fit B(D + → π + µ + µ − ), full region FigureS8(left) displays the results of the global fit to the low energy observables using flavio and smelli, and assuming real values of the couplings for simplicity.The negative values of the sum X d sb + X d bs are favored by the fit, since this particular combination enters δC 9,ℓℓ and δC 10,ℓℓ Wilson coefficients.That is, the NP contributions to these two Wilson coefficients are in the direction set by g L bs , for which negative values are preferred, see Fig.S6.The orthogonal direction of the parameter space is constrained by the B s − Bs mixing, which receives contributions from both the tree level Higgs and Z exchanges.Alternatively, one could extend the SM by a single generation of doublet vector-like quarks (Q L , Q R ) with hypercharge 1/6.These can have Yukawa couplings of Q L with both the right-handed SM up-and down-quarks.The Lagrangian reads:

5 FIG
FIG.S8.The parameter space, Eqs.(S41) and (S44), of the SM extended by either a down-type singlet vector-like quark (left) or a doublet vector-like quark (right), which is allowed at the 1σ, 2σ, 3σ level (from dark to light blue).For simplicity, we assume X d,Q bs and X d,Q sb to be real.
FIG. S9.Constraints on the parameter space of the off-diagonal Yukawas Y sb and Y bs in the full 2HDM model, for values of the mixing angle sin α = 1 × 10 −2 (left) and sin α = 1 × 10 −1 (right).The masses of the heavier Higgses are assumed to be mH = mA = 1 TeV.The legend in the left panel holds also for the right panel.

TABLE I .
The SM predictions and current experimental upper bounds on hadronic FCNC decays of h and Z, either from direct searches (3rd column) or indirect constraints (4th column), where the indirect bounds on B(h → qq ′ ) assume no large cancellations, see main text for details.For details on the SM calculations see supplementary material, Sec.S4. ,

TABLE S1 .
Nuisance parameters and their relative uncertainties, entering the h → bs sensitivity estimation.

TABLE S5 .
Examples of possible tagger choices with their uncertainties and the corresponding measurement uncertainty, discovery significance and 95% upper limits for Z → bs.

TABLE S6 .
GeV mt(m h ) 167.036 ± 0.315 GeV m b (mZ ) 2.871 ± 0.024 GeV m b (m h ) 2.796 ± 0.024 GeV The numerical inputs used for the SM prediction of Z/h → bq decay widths.The mt and m b masses are given in the MS scheme for two values of µ.