Theoretical challenges for flavor physics

Among the three types of $CP$ violation, CPV in decay and CPV in the interference of decay with and without mixing have been well established in many $b$ hadron decays. However, CPV in mixing has only been observed in $K^{0}$ mesons so far. Due to the many more decay channels available in $B_{d,s}^{0}$ mesons, CPV in $B$ mixing provides a complementary probe of BSM Laplace:2002ik . It can be measured via the $CP$ asymmetry in semileptonic $B$ decays, $A_{\rm SL}$.

The current world averages for the $B_{d}$ and $B_{s}$ mesons are $A_{\rm SL}^{d}=-(2.1\pm 1.7)\times 10^{-3}$ and $A_{\rm SL}^{s}=-(0.6\pm 2.8)\times 10^{-3}$ Amhis:2019ckw , respectively. These uncertainties are well above the SM expectations, $A_{\rm SL}^{d}=-(4.7\pm 0.6)\times 10^{-4}$ and $A_{\rm SL}^{s}=(2.22\pm 0.27)\times 10^{-5}$ Jubb:2016mvq . (The theory uncertainties are subject to some recent discussions Lenz:2020efu .) At the FCC-$ee$, the achievable experimental uncertainty has been estimated to be about $2.5\times 10^{-5}$ for both quantities Monteil ; Charles:2020dfl , which would allow a measurement of $A_{\rm SL}^{d}$ even at the SM level. These measurements would help probe BSM physics, as well as help discriminate between models, should BSM physics be discovered in other processes.

In a large class of models, the dominant BSM physics effects in the flavor sector may be those that modify neutral meson mixing amplitudes, while impacts on decay rates may be smaller. In addition to exploring specific models, this is justified, or maybe even expected, from an effective field theory viewpoint. The scale of dimension-6 operators contributing to neutral meson mixing is typically constrained to be higher by the data than the scales of operators affecting decays. A recent analysis of future sensitivities observed that beyond the Belle II and LHCb data taking in this decade, future improvements are hindered by the expected uncertainty of $|V_{cb}|$ Charles:2020dfl . To go beyond current expectations and reach the per mille level, not yet known theoretical progress would be needed.

Measurements of $CP$ asymmetries in $b$ decays shaped our understanding that the breaking of $CP$ symmetry observed in hadron decays is due to a single complex parameter of the CKM matrix. Currently, there is a lot of effort at the LHC and Belle II to test the CKM picture of CPV to much higher precision. The drive for this program is the fact that some observables are theoretically extremely clean, and thus any experimental progress will improve the sensitivity to BSM physics. At the same time, measurements of observables which can currently only be computed with large theoretical uncertainties due to hadronic physics, provide us with information about QCD.

The FCC-$ee$ is expected to deliver a very large and clean sample of $b$ hadrons. It is anticipated that the uncertainties of the CKM angle $\gamma$ will reach about 0.004 rad, of $\beta$ about 0.005, and of $\phi_{s}$ about 0.002 (see Table 7.3 of Ref. Abada:2019lih ). While these are only modest improvements over what is expected from LHCb (with 300/fb), these observables relate to measurements which can be done well at the LHC. To interpret measurements of $\beta$ and $\phi_{s}$ with such precisions, improvements in the theory are also needed to relate the results to parameters in the Lagrangian. The FCC-$ee$ should have greater advantages in modes containing neutrals, but few dedicated sensitivity studies exist so far. It will also allow combining results from experiments in different environments, and thus with different systematic uncertainties.

While most probes of CPV use rate asymmetries (with or without time-dependence) one other way to probe $CP$ violation is to use angular distribution asymmetries. They are usually called “triple products”, which is an idea that can be applied to many different angular distributions Durieux:2015zwa . The FCC-$ee$ can be used to measure many such modes. A theoretical question is how clean information can be extracted from them. The result is sensitive to hadronic matrix elements in a way similar to rate asymmetries. While it is interesting to measure them as a way to get insights into QCD, it would be even more significant to find a way to cleanly probe also the underlying weak interactions in such cases. The hope is that some progress in this direction will take place by the time the FCC-$ee$ starts collecting data.

The FCC-$ee$ will have unique capabilities to measure decay modes with large missing energy. These include decays with neutrinos (or $\tau$ leptons) in the final state. The FCC-$ee$ should also be able to measure electrons better than LHCb. Both exclusive and inclusive measurements are interesting, as the experimental and theoretical uncertainties are distinct, so they provide complementary probes of the underlying physics.

Prime examples are decays mediated by $b\to s\nu\bar{\nu}$ or $b\to s\tau^{+}\tau^{-}$ transitions, as well as their $b\to d$ counterparts. While LHCb is well suited to measure $B\to K^{(*0)}\mu^{+}\mu^{-}$ with high precision, its $b\to d$ analogs, $B\to\rho\mu^{+}\mu^{-}$ or $B_{s}\to K^{(*0)}\mu^{+}\mu^{-}$ are much more challenging Aaij:2018jhg ; MHS . The FCC-$ee$ is expected to be able to tackle these, as well as $B\to K^{(*0)}\tau^{+}\tau^{-}$, $\Lambda_{b}\to\Lambda\tau^{+}\tau^{-}$ Abada:2019lih , $B\to K^{(*)}\nu\bar{\nu}$, $B_{s}\to\phi\nu\bar{\nu}$, and $\Lambda_{b}\to\Lambda\nu\bar{\nu}$ decays, and maybe even $B\to\pi(\rho)\nu\bar{\nu}$.

The two-body decays $B\to\ell^{+}\ell^{-}$ are sensitive to particularly high scales among $B$ decays, especially if BSM physics alleviates the helicity suppression in the SM. From an effective theory point of view, these decays have some of the highest mass-scale sensitivities, comparable to $K\to\pi\nu\bar{\nu}$. The FCC-$ee$ is expected to be comparably sensitive to the HL-LHC for the decays $B_{s,d}\to\mu^{+}\mu^{-}$, but should be a lot more sensitive for $B_{s,d}\to e^{+}e^{-}$, and measure $B_{s}\to\tau^{+}\tau^{-}$ even at the SM level, ${\cal B}(B_{s}\to\tau^{+}\tau^{-})=(7.7\pm 0.5)\times 10^{-7}$ Bobeth:2013uxa (with 80 events/exp/year DonalHill ).

In many models inspired by the $R_{K^{(*)}}$ and $R(D^{(*)})$ anomalies hinting at lepton universality violation, there are correlated deviations from the SM predictions in transitions mediated by operators with flavor structures $b\bar{s}\ell^{+}\ell^{-}$ and $b\bar{s}\nu\bar{\nu}$, which makes these modes particularly interesting. If any of the anomalies become established, then searches for $b\to s\tau\mu$, $b\to s\tau e$, and similar modes would gain a lot in importance, since lepton universality violation in most models also leads to lepton flavor violation. The not yet observed $B_{c}\to\tau\bar{\nu}$ decay, which the FCC-$ee$ should be able to measure Amhis:2021cfy , is impacted by many models that attempt to explain the $R(D^{(*)})$ anomaly. The challenge to precision physics using $B_{c}$ decays may be the knowledge of production rates. These decay modes are important even if the current anomalies become less significant, as they are generic probes of many BSM scenarios Cohen:1996vb which treat the 3rd generation differently from the 1st and 2nd.

In order to fully benefit from these measurements, precise SM calculations of these rates are needed. Regarding the inclusive calculations, the operator product expansion will probably remain the main tool. The calculations for exclusive decays will likely rely on lattice QCD, and new developments are needed to extend the calculations for the full ranges of $q^{2}$. Another significant challenge for lattice QCD calculations is to account for QED corrections and isospin violation. These have been started to be addressed in limited contexts, and they will have to be included more comprehensively Sachrajda . Fully addressing electromagnetic corrections will necessitate dedicated interactions between theorists and experimentalists, for example to refine Monte Carlo tools. The hope is that by the time data from FCC-$ee$ is available, the theoretical uncertainties may be below the anticipated experimental ones.

Many probes of flavor that can be done with mesons can also be done with baryons. In many cases, adding baryons provide more statistics as well as a different set of systematics. There are, however, some ways baryons can probe short-distance physics that cannot be done with mesons. In particular, baryons can be polarized, which is not the case for the pseudoscalar mesons that are most often used to probe the weak interaction. The fact that $b$ and $c$ quarks in $Z$ decays are highly polarized, make polarization related physics well suited for the FCC-$ee$.

There are two aspects of polarization that can be exploited. First, by determining the polarization of the baryons, we can learn about the polarization of the underlying quark, which can teach us about the production mechanism Falk:1993rf ; Galanti:2015pqa . In particular, it can teach us about the Dirac structure of the operator that creates them. That information is washed out by hadronization into pseuduscalar mesons. (For example, if squarks are discovered, we would like to know if they are the left- or the right-handed ones. One way to probe this is to measure the polarization of the quarks that emerge from their decays.) In order to be able to use baryon polarization as a measure of the quark polarization, we need to know how much of the quark polarizations is retained by the baryons. It was shown that this can be obtained using top decays Galanti:2015pqa . A generalization of that idea to $Z$ decay is needed in order to fully exploit this method at the FCC-$ee$. Moreover, comparing the two determinations is another interesting check of the SM.

Polarization is also required in order to explore the full structure of the weak decays of the quarks. The point is that $\Lambda_{b}$ baryons produced at the FCC-$ee$ are highly polarized. The reason is that in $Z\to b\bar{b}$ the quarks are highly polarized and based on data from LEP Buskulic:1995mf ; Abbiendi:1998uz ; Abreu:1999gf and theoretical estimates Falk:1993rf ; Galanti:2015pqa , we expect an ${\cal O}(1)$ fraction of that polarization to be retained by the $\Lambda_{b}$. Thus, we can have a very large sample of polarized $\Lambda_{b}$ decays. For example, looking for semileptonic $\Lambda_{b}\to\Lambda_{c}\ell\nu$ decays with polarized $\Lambda_{b}$ can test the handedness of the weak interaction in similar ways that it is done with the Michel parameters in muon decays Manohar:1993qn . It can also be used to test the structure of FCNC decays, like $\Lambda_{b}\to\Lambda\ell^{+}\ell^{-}$ Das:2020qws . Similar studied can be done for $\Lambda_{c}$ decays deBoer:2017que .

One way to learn about some nonperturbative aspects of QCD is to study meson productions in exclusive $Z$ decays. For example, the rate of $Z\to X\gamma$ is sensitive to the light-cone distribution amplitude of the meson $X$ Grossmann:2015lea . This avenue to probe these amplitudes is very promising theoretically, as the expansion parameter is $1/Q$, where $Q$ is the energy carried by the hadron. Thus, using $Z$ decays, we can get theoretical sensitivity that is absent in $B$ decays. A promising candidate for such decay is $Z\to J/\psi\gamma$, where the branching ratio is expected to be of ${\cal O}(10^{-7})$. Thus, at the FCC-$ee$, we can hope to measure the rate with high precision.

The interest in these decays goes beyond QCD as they are closely related to decays of the form $H\to X\gamma$. Such decays can probe the couplings of the Higgs to light quarks Perez:2015aoa . These couplings are very hard to measure, and exclusive decays may be the best option. Thus, $Z$ decays provide important calibration for these calculations.

Another unique opportunity of the FCC-$ee$ is to look for FCNC $Z$ decays, e.g., $Z\to B_{s}\gamma$ or $Z\to B_{s}\mu^{+}\mu^{-}$. In the SM such decay rates are expected to be suppressed compared to $Z\to J/\psi\gamma$ by roughly $[|V_{cb}|/(16\pi^{2})]^{2}\sim 10^{-7}$, resulting in branching ratios smaller than about $10^{-14}$, and thus too small to measure. Yet, one can envision a situation where such rates are enhanced by some BSM physics. In many cases, such an enhancement is ruled out by rare $B_{s}$ and $B$ decays. Yet, there could be cancellations between several contributions in $B$ decays that is absent in $Z$ decays. While we are not aware of any model that predicts such an enhancement, the FCC-$ee$ is unique in its ability to probe directly such FCNC couplings of the $Z$ boson that is not available from $B$ decays. Theoretically, it would be interesting to study decays like $Z\to B^{+}K^{-}$ or $Z\to B^{+}K^{-}\gamma$ to check if they can provide sensitivity to FCNC $Z$ couplings.

$CP$ violation in both charm mixing and decay are probes of QCD dynamics and BSM physics. In 2011, the LHCb hint of a nearly 1% $CP$ violation in $D$ decays generated intense attention, since the central value was larger than what was previously considered to be allowed in the SM Isidori:2011qw ; Brod:2011re ; Pirtskhalava:2011va ; Brod:2012ud . In 2019 a smaller central value was established with more than $5\sigma$ significance, $A_{CP}(K^{-}K^{+})-A_{CP}(\pi^{-}\pi^{+})=-(1.54\pm 0.29)\times 10^{-3}$ Aaij:2019kcg . It is possible to accommodate this result in the SM, but it requires hadronic enhancements compared to the available (model dependent) calculations. It is expected that the FCC-$ee$ will be able to measure many individual $CP$ asymmetries with good precision, without having to construct differences, like the one recently measured by LHCb.

The FCC-$ee$ can probe many more observables that can teach us about CPV in the charm system. In particular, we should be able to have many tests of the SM, as we briefly explain below. These tests can either confirm the picture that there are large hadronic enhancements in certain channels, or will show that there is some BSM physics that is significant in the $D$ system.

The FCC-$ee$ can provide many such measurements. In particular, the anticipated superb time resolution can make them more precise than we can hope to get at the LHC and Belle II. Moreover, one problem for the LHC when it attempts to probe very small $CP$ asymmetries has to do with the production asymmetry between $c$ and $\bar{c}$, which can only be controlled using other measurements. This issue is not there in the FCC-$ee$, as the production is symmetric.

While we cannot make precision calculations for charm CPV because of hadronic uncertainties, we are still able to make rough predictions that can be tested. In particular, one can explore consequences of flavor $SU(3)$ symmetry. The point is that different observables arise at different orders of $SU(3)$ breaking, and while we cannot calculate how large this breaking is in specific matrix elements, it is typically of order $20\%$. Thus, by exploiting relations that hold to higher order in $SU(3)$ breaking, we can derive a pattern that can be used to probe the SM Kagan:2020vri ; Grossmann:2021aaa .

In particular, the SM predicts that to leading order in $SU(3)$ breaking all the time-dependent $CP$ asymmetries are identical. Moreover, there are subsets of them that are identical to second order in the breaking. To test these predictions, we need to have many different measurements, with uncertainties that are at the level of the anticipated $SU(3)$ symmetry breaking effects.

The situation can dramatically change if there is a theoretical breakthrough that enables the calculation of some of the hadronic input needed for these observables. The prime candidate is lattice QCD, and while some preliminary studies for the matrix elements relevant for $A_{CP}(K^{-}K^{+})-A_{CP}(\pi^{-}\pi^{+})$ have started, it is yet unknown what can be achieved for these observables.

We thank Stephane Monteil, Dean Robinson, and Marie-Helene Schune for helpful discussions. The work of YG is supported in part by the NSF grant PHY1316222. ZL was supported in part by the Office of High Energy Physics of the U.S. Department of Energy under contract DE-AC02-05CH11231.