On the ultimate precision of meson mixing observables

Meson mixing is considered to be an ideal candidate for new physics searches. Current experimental precision has greatly increased over the recent years, excelling in several cases the theoretical precision. A possible limit in the theoretical accuracy could be a hypothetical breakdown of quark-hadron duality. We propose a simple model for duality violations and give stringent bounds on such effects for mixing observables, indicating regions, where future measurements of $\Delta \Gamma_d$, $a_{sl}^d$ and $a_{sl}^s$ would be clear signals of new physics. Finally, we turn our attention to the charm sector, and reveal that already a modest duality violation of about $20 \%$ could explain the huge difference between HQE predictions for D-mixing and experimental data.


Introduction
Despite having passed numerous tests, the standard model of particle physics, leaves many fundamental questions unanswered, which might be resolved by extensions of this model. Flavour physics is an ideal candidate for general indirect new physics searches, as well as for dedicated CP-violating studies, which might shed light on the unsolved problem of the baryon asymmetry in the Universe. For this purpose hadronic uncertainties on flavour observables have to be under control. Various flavour experiments have achieved a high precision in many observables, in many cases challenging the precision of theory calculations. LHCb in particular, as an experiment designed to study beauty and charm physics, contributes to the currently impressive status of experimental precision. As we attempt to test the SM to the highest level of precision, the question of how sure we can be about any deviations from the current theoretical predictions being evidence of new physics comes to the fore. Such a question is the subject we tackle in this paper. Many current theory predictions rely on the Heavy Quark Expansion (HQE), and we will examine how the idea of quark-hadron duality -which is assumed by the HQE -can be tested. We will use current data from B-mixing, the dimuon asymmetry, and B-meson lifetimes to constrain violations of quark-hadron duality, and then see how this affects the predicted values of other observables. We also investigate how the current trouble with inclusive predictions of mixing in the charm sector can be explained through quark-hadron duality violation. We discuss what improvements could be made in both theory and experiment in order to further constrain duality violating effects, and what level of precision would be necessary to properly distinguish between genuine new physics and merely a non-perturbative contribution to the SM calculation. In this spirit, we also provide a first attempt at improving the theory prediction, using the latest results and aggressive error estimates to see how theory uncertainties could reduce in the near future. Our paper is organised as follows: in Sec. 2 we explain the basic ideas of duality violation in the HQE. We introduce in Sec. 2.1 a simple parameterisation for duality violation in B-mixing and we derive bounds on its possible size. The dimuon asymmetry and the lifetime ratio τ (B 0 s )/τ (B 0 d ) can provide complementary bounds on duality violation, which is discussed in Sect. 2.2 and Sect. 2.3. The bounds in the B-system depend strongly on the theory uncertainties, hence we present in Sect. 3 a numerical update of the mixing observables with an aggressive error estimate for the input parameters. In Sec. 4 we study possible effects of duality violation in D-mixing. We conclude in Sec. 5 with a short summary and outlook. The appendices contain more details to the studies in the main text.

Duality violation
In 1979 the notion of duality was introduced by Poggio, Quinn and Weinberg [1] for the process e + +e − → hadrons. 1 The basic assumption is that this process can be well approximated by a quark level calculation of e + + e − → q +q. In this work we will investigate duality in the case of decays of heavy hadrons, which are described by the heavy quark expansion (see e.g. [4][5][6][7][8][9][10][11] for pioneering papers and [12] for a recent review). The HQE is a systematic 1 The concept of duality was already used in 1970 for electron proton scattering by Bloom and Gilman [2,3]. expansion of the decay rates of b-hadrons in inverse powers of the heavy quark mass.
Γ 4 + ... , (2.1) with Λ being of the order of the hadronic scale. One finds that there are no corrections of order 1/m b and that some corrections from the order 1/m 3 b onwards are enhanced by an additional phase space factor of 16π 2 . The HQE assumes quark hadron duality, i.e. that the hadron decays can be described at the quark level. A violation of duality could correspond to nonperturbative terms like exp[−m b /Λ], which give vanishing contributions, when being Taylor expanded around Λ/m b = 0 (see e.g. [13] and also [14] for a detailed discussion of duality, its violations and some possible models for duality violations). To estimate the possible size of these non-perturbative terms we note first that the actual expansion parameter of the HQE is the hadronic scale Λ normalised to the momentum release From this simple numerical exercise one finds that duality violating terms could easily be of a similar size as the expansion parameter of the HQE. Moreover decay channels like b → ccs might be more strongly affected by duality violations compared to e.g. b → uūs. This agrees with the naive expectation that decays with a smaller final state phase space might be more sensitive to duality violation. Obviously duality cannot be proved directly, because this would require a complete solution of QCD and a subsequent comparison with the HQE expectations, which is clearly not possible.
In this work we will follow strategy b) and use a simple parameterisation for duality violation in mixing observables and lifetime ratios, which will be most pronounced for the b → ccs channel. At this stage it is interesting to note that for many years there have been problems related to application of the HQE for inclusive b-hadron decays and most of them seemed to be related to the b → ccs channel: • The experimental Λ b lifetime was considerably lower than the theory prediction, see e.g. the discussion in [19], where also a simple model for a modification of the HQE was suggested in order to explain experiment, see also [20] and [21]. The dominant contribution to the Λ b lifetime is given by the b → cūd and b → ccs transitions. To a large extent the Λ b -lifetime problem has now been solved experimentally, see the detailed discussion in [12], mostly by new measurements from LHCb [22][23][24]. However, there is still a large theory uncertainty remaining due to unknown non-perturbative matrix elements that could be calculated with current lattice-QCD techniques.
• For quite some time the values of the inclusive semi-leptonic branching ratio of Bmesons as well as the average number of charm quarks per b-decay (missing charm puzzle) disagreed between experiment and theory, see e.g. [25][26][27][28]. Modifications of the decay b → ccs were considered as a potential candidate for solving this problem. This issue has been improved considerably by new data and and new calculations [29]. Again, there still a considerable uncertainty remains due to unknown NNLO-QCD corrections. First estimates suggest that such corrections could be large [30].
• Because of a cancellation of weak annihilation contributions it is theoretically expected (based on the HQE) that the B 0 s -lifetime is more or less equal to the B 0 d -lifetime, see e.g. [12]. For quite some time experiment found deviations of τ (B 0 s )/τ (B 0 d ) from onewe have plotted the experimental averages from HFAG [31] from 2003 onwards in Fig.  1. Currently there is still a small difference between data and the HQE prediction, which will be discussed further Section 2.3. Here again a modification of the b → cūd and/or the b → ccs transitions might solve the problem. Year • The large observed value of the dimuon asymmetry [32][33][34][35] could not only have hinted towards new physics but also to large values of Γ s 12 , which is dominated by b → ccs.
Thus it was suggested to investigate the dimuon asymmetry without making use of the theory prediction of Γ s 12 [36], which was criticised in [37]. The issue of the dimuon asymmetry is still not settled and we will discuss it further below.
All of these problems are currently considerably softened and huge duality violations are now ruled out by experiment [38], in particular by the measurement of ∆Γ s , which is to a good approximation a b → ccs transition. But there is still space for a small amount of duality violation -which will be quantised in this work. We will thus investigate the decay rate difference ∆Γ s in more detail. According to the HQE we get the following expansion for ∆Γ s = 2|Γ s 12 | cos φ s The leading term Γ s,(0) 3 has been calculated quite some time ago by [39][40][41][42][43][44], NLO-QCD corrections Γ s,(1) 3 have been determined in [45][46][47] and sub-leading mass corrections were done in [48][49][50]. Corresponding lattice values were determined by [51][52][53][54]. The most recent numerical update for the mixing quantities is given in [55] (superseding the numerical predictions in [56,57]) and can be compared to the experimental values from e.g. HFAG [31]. The theory prediction uses conservative ranges for the input parameters -we will present a more aggressive estimate in Sec. 3  This is currently the most precise prediction for the decay rate difference; in Section 3 we will give a less conservative estimate of the SM prediction for ∆Γ s , with an even smaller uncertainty.

B-mixing
The off-diagonal elements Γ s 12 and M s 12 of the mixing matrix for B 0 s -mesons can be expressed as (2.12) Eq.(2.12) introduces the a, b and c notation of [47]. The way of writing Γ s 12 /M s 12 in Eq.(2.11) and Eq.(2.12) can be viewed as a Taylor expansion in the small CKM parameter λ u /λ t , for 2 In the ratio ∆Γ s /∆M s theoretical uncertainties are cancelling and thus the corresponding theory error is smaller than for ∆Γ s alone. We would reintroduce this uncertainty by multiplying with the theory value of ∆M s . Using instead the experimental value of ∆M s , which has in comparison a negligible error we get a more precise prediction of ∆Γ s , which, however, only holds under the assumption that ∆M Exp s is given by its standard model value.
which we get (we use the same the CKM input as [55]; the values were taken in 2015 from CKMfitter [58], similar values can be obtained from UTfit [59].) In addition to the CKM suppression a pronounced GIM-cancellation [60] is arising in the coefficients a and b in Eq. (2.12). With the input parameters described in [55] we get for the numerical values of a, b and c: From this hierarchy we see, that ∆Γ q /∆M q is given to a very good approximation by −0.0001c and a q sl by 0.0001a (λ u /λ t ). Next we introduce a simple model for duality violation. Such effects are typically expected to be larger, if the phase-space of a B 0 s decay becomes smaller. Thus b-quark decays into two charm quarks are expected to be more strongly affected by duality violating effects compared to b-quark decays into two up quarks. Motivated by the observations in Section 2 we write to a first approximation 3 : Studying this expression, we find that the decay rate difference is mostly given by the first term on the r.h.s., so we expect ∆Γ s /∆M s ≈ −c(1 + 4δ) · 10 −4 , which is equivalent to our naive starting point of comparing experiment and theory prediction for ∆Γ s . The semileptonic CP asymmetries will be dominantly given by the second term on the r.h.s., a s sl ≈ (λ u /λ t ) [a + δ(6c + a)] · 10 −4 . Now the duality violating coefficient δ is GIM enhanced by (6c + a) compared to the leading term a. Having an agreement of experiment and theory for semi-leptonic CP asymmetries could thus provide very strong constraints on duality violation.
Using the values of a, b and c from Eq.(2.14) and the CKM elements from Eq.(2.13) we get for the observables ∆M q , ∆Γ q and a q sl the following dependence on the duality violating parameter δ: As expected we find that the duality violating parameter δ has a decent leverage on ∆Γ q and a sizeable one on a q sl . The expressions for ∆Γ q were obtained by simply multiplying the theory ratio ∆Γ q /∆M q with the theoretical values of the mass difference, as given in Eq.(2.4). Comparing experiment and theory for the ratio of the decay rate difference ∆Γ s and the mass difference ∆M s we found (see Eq.(2.5)) an agreement with a deviation of at most 19%. Thus the duality violation -i.e. the factor 1 + 3.94516δ in Table 2.19 -has to be smaller than this uncertainty: Equivalently this bound tells us that the duality violation in the cc-channel is at most +15.2% or −23.3%, if the effect turns out to be negative. If there would also be an 19% agreement of experiment and theory for the semi-leptonic asymmetry a s sl , then we could shrink the bound to δ down to 0.00851. Unfortunately experiment is still far away from the standard model prediction, see Eq.(2.4). However, we can turn around the argument: even in the most pessimistic scenario -i.e. having a duality violation that lifts GIM suppression -the theory prediction of a s sl can be enhanced/diminished at most to a s sl = [0.336, 5.12] · 10 −5 . (2.21) In the B 0 d -system a comparison of experiment and theory for the ratio of decay rate difference and mass difference turns out to be tricky, since ∆Γ d is not yet measured, see Eq.(2.4). If we would use the current experimental bound on the decay rate difference ∆Γ d , we would get artificially large bounds on δ. Looking at the structure of the loop contributions necessary to calculate Γ d 12 and Γ s 12 , one finds very similar cc-, ucc-, cūand uū-contributions. Our duality violation model is based on the phase space differences of decays like B 0 s → D s D s (cc), B 0 s → D s K (uc) and B 0 s → πK (uu), which are very pronounced. On the other hand we find that the phase space differences of B 0 s -and B 0 d -decays are not very pronounced, i.e. the difference between e.g. B 0 s → D s D s vs. B 0 d → D s D is small -compared to the above differences due to different internal quarks. Hence we conclude that the duality violation bounds from the B 0 ssystem can also be applied to a good approximation to the B 0 d -system. With the B 0 s -bound we get that the theory prediction of a d sl and ∆Γ d can be enhanced/diminished due to duality violations at most to  [32][33][34][35]. Currently a sizeable enhancement of ∆Γ d is not excluded by theoretical or experimental bounds [65]. Thus it clearly important to distinguish hypothetical duality violating effects in ∆Γ d from new physics effects. Since our conclusions (new physics or unknown hadronic effects) are quite far-reaching, we try to be as conservative as possible and we will firstly use a more profound statistical method, a likelihood ratio test 4 . Our more conservative bound for δ is now supposed to be given by with a 90% confidence level (1.6 standard deviations). This more conservative statistical method almost doubles the allowed region for δ. Inserting these values into the predictions for a d,s sl and ∆Γ d we see that duality violation can give at most the following ranges for the mixing observables: The second modification to ensure that our estimates are conservative concerns our ad-hoc ansatz in Eqs.(2.15), (2.16), (2.17), where we assumed that the cc-part is affected by duality violations four times as much as the cu-part and the uu-part is not affected at all; we can obtain more general results with the following modification with δ cc ≥ δ uc ≥ δ uu and the requirement that all δs must have the same sign. Now we get for the observables ∆Γ s ∆M s = 48.1(1 + 0.982δ cc + 0.0187δ uc − 0.000326δ uu ) · 10 −4 , (2.31) For our likelihood function we use a Gaussian function where, to take into account the uncertainty on both theory and experiment, we take for our error the quadrature sum, i.e.
The test we apply is −2 ln L/ L ≤ 2.71, where our choice of 2.71 gives a 90% confidence limit on our parameters and in principle we normalise by the maximum likelihood L. However, in our model the maximum likelihood of L = 1 is always achievable, and so our test reduces to simply −2 ln L ≤ 2.71.
In the case of ∆Γ s , which will be used to determine the size of the duality violating δs, the coefficients of the uu component are suppressed by more than three orders of magnitude compared to the rest and therefore neglected. For the semi-leptonic CP asymmetries the uu duality violating component is about two orders of magnitude lower than the rest, thus we neglect the uu component in the following. This might lead to an uncertainty of about 20% in the duality bounds for ∆Γ d , which we will keep in mind. Considering only δ cc and δ uc we get with the likelihood ratio test the bounds depicted in Fig. 2 at a 90% confidence level. Fig. 2 shows that a duality violation of no more than 35% is allowed in either Γ s cc or in Γ s uc . We also see that it is in principle possible to see duality violation in ∆Γ s but not in a s sl and vice versa. Moreover we find from the functional form of a s sl , that this quantity achieves a maximum (minimum) when δ uc = 0 and δ cc < 0 or (> 0). Our generalised parameterisation of duality violation gives now the most conservative bounds We are now in a position to make a strong statement: any measurement outside this range, cannot be due to duality violation and it will be an unambiguous signal for new physics.
Since the ranges in Eq.(2.35), Eq.(2.36) and Eq.(2.37) are considerably larger than the uncertainties of the corresponding standard model prediction given in Eq.(2.4) the question of how to further shrink the duality bounds is arising. Currently the bound on the duality violating parameters δ come entirely from ∆Γ s , where the current experimental and theoretical uncertainty adds up to ±19%. Any improvement on this uncertainty will shrink the allowed regions on δ. In Section 3 we will discuss a more aggressive estimate of the theory predictions for the mixing observable, indicating that a theory uncertainty of about ±10% or even ±5% in ∆Γ s /∆M s might come into sight. Including also possible improvements in experiment this would indicate a region for δ that is considerably smaller than the ones given in Eq.  Figure 3: Comparison of SM prediction (green), SM + duality violation (brown), SM + duality violation in future (orange) and current experimental (blue) bound for ∆Γ d (l.h.s.). One the r.h.s. the experimental bounds on a d sl (brown) and a s sl (blue) are shown in comparison to their theory values. Any measurement outside the allowed theory regions will be a clear indication for new physics. The theory uncertainties for a s sl are so small, that they cannot be resolved, they are depicted by the black line. For a d sl the duality allowed region (green) has quite some overlap with the experimental one, in future this region can be shrinked to the red region.
by the blue region, which can be compared to the standard model prediction (green). As we have seen above, because of still sizeable uncertainties in ∆Γ s duality violation of up to 35% can currently not be excluded -this would lead to an extended region (brown) for the standard model prediction including duality violation. If in future ∆Γ s will be known with a precision of about 5% both in theory and experiment, than the brown region will shrink to the orange one -here also the intrinsic precision of the SM value will be reduced. In other words, currently any measurement of ∆Γ d outside the brown region will be a clear signal of new physics; in future any measurement outside the orange region can be a signal of new physics. The same logic is applied for the r.h.s. of Fig. 3, where a d sl and a s sl are investigated simultaneously. For a s sl still any measurement outside the bounds in Eq.(2.35) would be clear indication of new physics. This bound is in Fig. 3 so small, compared to the current blue experimental region, that it can only be resolved as a single line (black). For a d sl the current experimental region is given by the orange region, which is only slightly larger, than the green region, which is indicating the theory prediction including duality violation. Future improvement in experiment and theory for ∆Γ s will reduce the green region to the red one and then any measurement outside the red region will be a clear signal of new physics. In addition we can ask if there are more observables that will be affected by the above discussed duality violations. An obvious candidate is the dimuon asymmetry, which depends on a d sl , a s sl and ∆Γ d . This will be discussed in Sec. 2.2. Another candidate is the the lifetime ratio τ (B 0 s )/τ (B 0 d ), where the dominant diagrams are very similar to the mixing ones, this observable will be studied further in Sec. 2.3.

Duality bounds from the dimuon asymmetry
The D0 collaboration has measured the like-sign dimuon asymmetry finding consistently deviations with the expected value from the Standard Model [32][33][34][35]. The most recent experimental determination found a discrepancy of 3.0 σ when interpreted as the result of CP violation in mixing and interference given in terms of the semileptonic asymmetries a s sl , a d sl and the life time difference ∆Γ d respectively, as suggested by [64] and further improved by [66]. Thus we want to investigate the possibility of explaining the discrepancy between theory and experiment as an effect of duality violation. The residual like-sign dimuon charge asymmetry A CP reads with coefficients that can be determined using the information provided in [35]. We also include a further correction factor in the interference contribution C int , as suggested by [66].  q can decay. Independent of this observation, our initial argument that the phase space for intermediate cc-states is smaller than the one for intermediate uc-states, which is again smaller that the uū-case, still holds. Hence we assume that the xȳ-loop for the lifetime ratio, has the same duality violating factor δ xy as the xȳ-loop for Γ q 12 . It turns out that the largest weak annihilation contribution to the B 0 s -lifetime is given by a cc-loop, while for the B 0 d -lifetime a uc-loop is dominating. This observation tells us that duality will not not drop out in the lifetime ratio, because the dominating contributions for B 0 s and B 0 d are affected differently. Using our above model and modifying the cc-loop with a factor 1 + 4δ and the uc-loop with a factor 1 + δ, we get with the expressions in [12,20,21,67]

Duality bounds from lifetime ratios
A detailed estimate of the theoretical error is given in the appendix. Unfortunately the standard model prediction relies strongly on lattice calculations that are already 15 years old [68] and no update has been performed since then. For a more detailed discussion of the status of lifetime predictions, see [12]. Nevertheless, one finds a big impact of the duality violating factor δ on the final result. A value of δ = 1 would have huge effects, compared to the central value within the standard model and its uncertainty. Our theory prediction can be compared to the current experimental value for the lifetime ratio [31] τ If the tiny deviation between theory and experiment is attributed to duality violation, then we get an allowed range for δ of There is currently a discrepancy of about 2.5σ between experiment (Eq.(2.46)) and theory (Eq.(2.45)) and this difference could stem from new physics or a sizeable duality violation of δ ≈ 0.5 in lifetimes. The allowed region of the duality violating parameter δ can be read off Fig. 5, where the current experimental bound from Eq.(2.46) is given by the blue region and theory prediction including hypothetical duality violation by the red region. It goes without saying that 2.5 standard deviations is much too little to justify profound statements, thus we consider next future scenarios where the experimental uncertainty of the lifetime ratio will be reduced to ±0.001.
• Scenario I: the central value will stay at the current slight deviation from one: This scenario corresponds to a clear sign of duality violation or new physics in the lifetime ratio. Assuming the first one, we get a range of δ of (see the violet region in Fig. 5 Thus the lifetime ratio requires large values of δ. Our final conclusions depend now on the future developments of ∆Γ s . Currently ∆Γ s requires small values of δ, which is in contrast to scenario I. Thus we have to assume additional new physics effects -either in mixing or in lifetimes -that might solve the discrepancy. If in future the theory value of ∆Γ s will go up sizeable or the experimental value will go down considerably, then mixing might also require a big value of δ and we then would have duality violation as a simple solution for explaining discrepancies in both lifetimes and B 0 s -mixing. • Scenario II: the central value will go up to the standard model expectation:  In that case we will find only a small allowed region for δ around zero, see the green region in Fig. 5 δ ∈ [−0.0708, 0.116] naive , (2.53) δ ∈ [−0.0853, 0.130] likelihood ratio 90% . (2.54) The above region is, however, still larger than the one obtained from ∆Γ s . New lattice determinations of lifetime matrix elements might change this picture and in the end the lifetime ratio might also lead to slightly stronger duality violating bounds than ∆Γ s . Again our final conclusion depends on future developments related to ∆Γ s . If both experiment and theory for mixing stay at their current central values, we simply get very strong bounds on δ. If theory or experiment will change in future, when we could indications for deviations in mixing, which have to be compared to the agreement of experiment and theory for lifetimes in Scenario II.
In Section 3 we will discuss a possible future development of future theory predictions for mixing observables. Before we proceed let us make a comment about our duality model. In principle we also could generalise our duality ansatz, and modify the cc-loop with a factor 1 + δ cc and the uc-loop with a factor 1 + δ uc , as we did in the mixing case. We get the following expression Here one sees a pronounced cancellation of the cc and the uc contribution, if one allows δ cc to be of similar size as δ uc . This is, however, not what we expect from our phase space estimates for duality violation. Thus we use for the lifetime ratio only our model given in Eq.(2.45).

Numerical Updates of Standard Model Predictions
We have already pointed out that more precise values of ∆Γ s are needed to derive more stringent bounds on duality violation in the B-system. Very recently the Fermilab MILC collaboration presented a comprehensive study of the non-perturbative parameters that enter B-mixing [79]. 5 . A brief summary of their results reads: • Improved numerical values for the non-perturbative matrix elements Q , Q S , Q S , R 0 , R 1 and R 1 that are necessary for ∆Γ q and ∆M q . Hence we have numerical values for all operators that are arising up to dimension seven in the HQE, up to R 2 and R 3 , which are still unknown and can only be estimated by assuming vacuum insertion approximation.
• The results provide a very strong confirmation of vacuum insertion approximation. All their bag parameters turn out to be in the range of 0.8 to 1.2. Sometimes in the literature different normalisations of the matrix elements are used, that lead to values of the bag parameters which differ from one in vacuum insertion approximation, see e.g. the discussion in [55]. The definitions in [79] are all consistent with B = 1 ± 0.2 in vacuum insertion approximation.
• The numerical values of f 2 Bq B are larger than most previous lattice calculations.
Based on these new results we perform a more aggressive -compared to the recent study in [55] -numerical analysis of the SM predictions, where we try to push the current theory uncertainties to the limits. In particular we will modify the predictions in [55] by using • Most recent values of the CKM parameter from CKMfitter [58] (similar values can be obtained from UTfit [59]).
• New Fermilab MILC results for the bag parameters of Q,Q S , R 0 , R 1 andR 1 . We do not try to average with other lattice results, e.g. the values given by FLAG [81].
• Assume vacuum insertion approximation for R 2 and R 3 with a small uncertainty of B = 1 ± 0.2. We note that this is not clearly justified yet and it has to be confirmed by independent determinations of the corresponding bag parameters.
• Use results derived from equations of motionB All inputs are listed in Appendix C. We first note that the overall normalisation due to f 2 Bq B seems to be considerably enhanced now, so we expect enhancements in ∆M q and ∆Γ q that will cancel in the ratio. Moreover the uncertainty in the bag parameter ratioB S /B is larger than e.g. in [55]. On the other hand the dominant uncertainty due to R 2 and R 3 will now be dramatically be reduced. Putting everything together we get with the new parameters the following predictions for the two neutral B-systems, which are compared with the more conservative theory predictions [55] and the experimental values from HFAG [31], that were already given in Eq. (2.4 13.0147 (1 ± 10) · 10 −3 (3.1) The new theory values for ∆M q and ∆Γ q are larger than the ones presented in [55] and they are further from experiment. For the ratios ∆Γ q /∆M q and a q sl the central values are only slightly enhanced. The overall error shrinks by about a factor of two for ∆M s and also sizeably for ∆M d , ∆Γ q and the ratios ∆Γ q /∆M q . For the semi-leptonic asymmetries the effect is less pronounced. In Appendix C a detailed analysis of the errors is given. Here one clearly sees the enhancements of the mass differences, which are up to 20% or more than two standard deviations above the experimental value. The decay rate difference ∆Γ s is also enhanced by about 20% above the measured value; due to larger uncertainties, this is statistically less significant. The dominant source for this enhancement is the new value of Q . The ratio ∆Γ s /∆M s is slightly lower than before, but still consistent with the corresponding experimental number. Taking the deviations above seriously, we can think about several possible interpretations: 1. Statistical fluctuations in the experimental results of the order of three standard deviations might explain the deviation in ∆Γ s , while the deviation in ∆M s cannot be explained by a fluctuation in the experiment.
2. Duality violations alone cannot explain these deviations, because they have no visible effects on ∆M q .
3. The lattice normalisation for f 2 B B is simply too high, future investigations will bring down the value and there is no NP in mixing. Currently there is no foundation for this possibility, but we try to leave no stone unturned. Since f 2 B B cancels in the ratio of mass and decay rate difference, we can use the new values to give the most precise SM prediction of ∆Γ s via ∆Γ s ∆M s · 17.757 ps −1 (≡ ∆M exp s ) = 0.0875 ± 0.0102 ps −1 . (3.10) Now the theory error is very close to the experimental one and it would be desirable to have more precise values in theory and in experiment. In that case we also get an indication of the short-term perspectives for duality violating bounds. The above numbers indicate an uncertainty of ±0.138 for the ratio ∆Γ s /∆M s , which corresponds -in the case of a perfect agreement of experiment and theory -to a bound on δ of ±0.035. This would already be a considerable improvement compared to the current situation.
4. Finally the slight deviation might be a first hint for NP effects.
(a) To explain the deviation in the decay rate difference one needs new physics effects in tree level decays, while deviation in M 12 might be solved by new physics effects in loop contributions.
(b) In principle one can think of the possibility of new tree-level effects that modify both ∆Γ s and ∆M s , but which cancels in the ratio. ∆M s is affected by a double insertion of the new tree-level operators. Following the strategy described in e.g.
[65], we found, however, that the possible effects on the mass difference are much too small.
(c) Finally there is also the possibility of having a duality violation of about 20% in ∆Γ s , while the effect in ∆M s is due to new physics in loops. This possibility can be tested in future by more precise investigations of the lifetime ratio τ (B 0 s )/τ (B 0 d ).
In order to draw any definite conclusions about these interesting possibilities, we need improvements in several sectors: from experiment we need more precise values for ∆Γ s and τ (B 0 s )/τ (B 0 d ). A first measurement of ∆Γ d will also be very helpful. A measurement of the semi-leptonic asymmetries outside the duality-allowed regions would already be a clear manifestation of new physics in the mixing system. From the theory side we need (in ranked order) 1. A first principle determination of the dimension 7 operators B R 2,3 and the corresponding colour-rearranged ones.
2. Independent lattice determinations of the matrix elements of Q, Q S ,Q S , R 0 , R 1 and R 1 .
3. NNLO QCD calculations for the perturbative part of Γ 12 .
These improvements seem possible in the next few years and they might lead the path to a detection of new physics effects in meson mixing. Using τ (D 0 ) = 0.4101 ps [69], this can be translated into

D-mixing
When trying to compare these numbers with theory predictions, we face the problem that it is not obvious if our theory tools are also working in the D-system. Till now the mixing quantities have been estimated via exclusive and inclusive approaches. The exclusive approach is mostly based on phase space and SU (3) F -symmetry arguments, see e.g. [70,71]. Within this approach values for x and y of the order of 1% can be obtained. Thus, even if it is not a real first principle approach, this method seems to be our best currently available tool to describe D-mixing. Inclusive HQE calculations worked very well in the B-system, but their naive application to the D-system gives results that are several orders of magnitude lower than the experimental result [72,73]. Hence it seems we are left with some of the following options: • The HQE is not valid in the charm system. This obvious solution might however, be challenged by the fact that the tiny theoretical D-mixing result is solely triggered by an extremely effective GIM cancellation [60], see e.g. the discussion in [74], and not by the smallness of the first terms of the HQE expansion. A breakdown of the HQE in the charm system could best be tested by investigating the lifetime ratio of D-mesons. From the theory side, the NLO QCD corrections have been determined for the lifetime ratio in [75] and it seems that the experimental measured values can be reproduced. To draw a definite conclusion about the agreement of experiment and theory for lifetimes and thus about the convergence of the HQE in the charm system, lattice evaluations of the unknown charm lifetime matrix elements are urgently needed. So this issue is currently unsettled.
• Bigi and Uraltsev pointed out in 2000 [76] that the extreme GIM cancellation in Dmixing might be lifted by higher terms in HQE, i.e. the 1/m c -suppression of higher terms in the HQE is overcompensated by a lifting of the GIM cancellation in higher order terms. There are indications for such an effect, see [74,77], but it is not clear whether the effect is large enough to explain the experimental mixing values. To make further progress in that direction we need the perturbative calculation of the dimension 9 and 12 terms of the OPE and an idea of how to estimate the matrix elements of the arising D=9 and D=12 operators. Hence this possibility is not ruled out yet.
• The deviation of theory and experiment could of course also be due to new physics effects. Bounds on new physics models from determining their contributions to Dmixing, while more or less neglecting the standard model contributions were studied e.g. in [78].
In this work we will investigate the related question, whether relatively small duality violating effects in inclusive charm decays could explain the deviation between experiment and the inclusive approach. We consider the decay rate difference ∆Γ D for this task. According to the relation (see the derivation in the appendix) we will only study |Γ 12 | and test whether it can be enhanced close to the experimental value of the decay rate difference. This is of course only a necessary, but not a sufficient condition for an agreement of experiment and theory. A complete answer would also require a calculation of |M 12 |, which is beyond the scope of this work. Γ 12 consists again of three CKM contributions with the CKM elements λ d = V cd V * ud and λ s = V cs V * us . Using again the unitarity of the CKM matrix (λ d + λ s + λ b = 0) we get The CKM-factor have now a very pronounced hierarchy, they read λ 2 s = 4.81733 · 10 −2 − 3.00433 · 10 −6 I , Looking at the expressions in Eq.(4.7) we see an extreme GIM cancellation in the CKMleading term, while the last term without any GIM cancellation is strongly CKM suppressed. We get Γ ss 12 − 2Γ sd 12 + Γ dd 12 = 1.17z 2 − 59.5z 3 + ... , (4.14) Γ   Eq. (4.20) shows that our duality violating model completely lifts the GIM cancellation and that even tiny values of δ will lead to an overall result that is much bigger than the usual standard model predictions within the inclusive approach. For our final conclusions we will use the generalised duality violating model with δ ss ≥ δ sd ≥ δ dd . Next we test for what values of δ the inclusive approach can reproduce the experimental results for ∆Γ D . The corresponding allowed regions for δ ss,sd,dd are given as shaded areas in Fig. 6. As expected, very small values of δ cannot give an agreement between HQE and experiment, surprisingly, however, values as low as δ ss ≈ 0.18 can explain the current difference. So a duality violation of the order of 20% in the HQE for the charm system is sufficient to explain the huge discrepancy between a naive application of the HQE and the measured value for ∆Γ D .

Summary and Conclusions
In this paper we have explored the possibility of duality violations in heavy meson decays.
The study of such effects has a long tradition in flavour physics. If the semi-leptonic asymmetries would agree with a similar precision between experiment and theory then the bound on δ would go down to ±0.009. Unfortunately, the semi-leptonic asymmetries are not observed yet, and we have only experimental bounds. The same is true for the decay rate difference ∆Γ d . Thus we use our bounds on δ from ∆Γ s to determine the maximal possible size of a q sl and ∆Γ d , if duality is violated. These regions are compared with current experimental ranges in Fig.3. Any measurement outside the region allowed by duality violation is a clear signal for new physics. We also show a future scenario in which the duality violation is further constrained by more precise values of ∆Γ s both in experiment and theory.
Duality violations would also affect the still unsolved problem of the dimuon asymmetry measured by the D0 collaboration, since it depends on a d sl , a s sl and ∆Γ d . We found, however, that an agreement between experiment and theory for the dimuon asymmetry would require values of δ in the region of −0.4 to −1.9, which is considerably outside the allowed region found above. Taking only allowed values of δ we find that the theory prediction including duality violation is still an order of magnitude smaller than experiment. Hence duality violation cannot explain the value of the dimuon asymmetry. We have shown that the duality violating parameter δ will also affect the lifetime ratio τ (B 0 s )/τ (B 0 d ), where we currently have a deviation of about 2.5 standard deviations between experiment and theory. Looking at the historical development of this ratio depicted in Fig.  1 one might be tempted to assume a statistical fluctuation. Taking the current deviation seriously, it is either a hint for new physics or for a sizeable duality violations of the order of δ ∼ 0.5, which is inconsistent with our bounds on δ derived from ∆Γ s . Here a future reduction of the experimental error of τ (B 0 s )/τ (B 0 d ) will give us valuable insight. We have studied two future scenarios in Fig. 5, which would either point towards new physics and duality violations or stronger bounds on duality violation. It is very important to note here that the theory prediction has a very strong dependence on almost unknown lattice parameters. In particular, we can see from our error budget for the lifetime ratio in Appendix A that any new calculation of the bag parameters 1,2 would bring large improvements in the theory prediction for τ (B 0 s )/τ (B 0 d ). By now we already mentioned several times necessary improvements in both experiment and theory for mixing observables and in particular for ∆Γ s . Therefore we presented an update of the SM predictions for the observables ∆Γ, ∆M, and a sl in both the B 0 s and B 0 d systems, based on the recent Fermilab-MILC lattice results [79] for non-perturbative matrix elements, the latest CKM parameters from CKMfitter [58], and an aggressive error estimate on the unknown bag parameters of dimension seven operators. With this input the current theory error in the mixing observables could be reduced by a factor of two for ∆M s or 1/3 for ∆M d , ∆Γ s , and ∆M s /∆Γ s . Thus we get for our fundamental relation to establish the possible size of duality violation ∆Γs ∆Ms Exp ∆Γs ∆Ms SM agr. = 0.94 ± 0.14 .
As expected, the overall uncertainty drops considerably, with a theory uncertainty almost compatible with the experimental one -thus demanding more precise values of ∆Γ s . On the other hand, we found in this new analysis that the central values of the mass differences and decay rate differences are enhanced to values of about 20% above the measurements with a significance of around 2 standard deviations. To find out whether this enhancement is real, we need several ingredients: 1) an independent confirmation of the larger values of the matrix element Q found by [79]. 2) a first principle calculation of R 2,3 -triggered by the results of [79] we simply assumed small deviations from vacuum insertion approximation. If the new central values turn out to be correct, we will get profound implications for new physics effects and duality violation in the B-system. For a further improvement of the theory uncertainties NNLO-QCD corrections for mixing have to be calculated. We finally focus on the charm-system, where a naive application of the HQE gives results that are several orders of magnitude below the experimental values. We found the unexpected result that duality violating effects as low as 20% could solve this discrepancy. Such a result might have profound consequences on the applicability of the HQE. As a decisive test we suggest a lattice calculation of the matrix elements arising in the ratio of charm lifetimes. This ratio is free of any GIM cancellation, which are very severe in mixing.
B Proof of ∆Γ ≤ 2|Γ 12 | In the B-system we get very simple expression for the mixing observables in terms of M 12 and Γ 12 because one can make of the pronounced hierarchy Γ 12 /M 12 and perform a Taylor expansion. In the D-system ∆Γ and ∆M are of the same order and one has to use the exact expression. One finds however, ∆Γ ≤ 2|Γ 12 |, which gives us the opportunity to calculate only Γ 12 and to give an upper bound on ∆Γ.
We start with the two fundamental equations for the mixing observables: where φ = arg(−M 12 /Γ 12 ). Next we eliminate ∆M by substituting Eq. B.2 into Eq. B.1, and then solve for |M 12 |.
Since M 12 ≥ 0, we can say that the numerator and denominator on the r.h.s. of Eq. B.5 must have the same sign. First, assume both terms are ≥ 0.

C Numerical update with new lattice inputs
In this appendix we give details of the new numerical analysis done in Sec. 3.

C.3 Error estimates
We get now the following error estimates, compared to [55]: The