Bs mixing observables and Vtd/Vts from sum rules

We consider the effects of a non-vanishing strange-quark mass in the determination of the full basis of dimension six matrix elements for $B_{s}$ mixing, in particular we get for the ratio of the $V-A$ Bag parameter in the $B_s$ and $B_d$ system: $\overline{B}^s_{Q_1} / \overline{B}^d_{Q_1} = 0.987^{+0.007}_{-0.009}$. Combining these results with the most recent lattice values for the ratio of decay constants $f_{B_s} / f_{B_d}$ we obtain the most precise determination of the ratio $\xi = f_{B_s} \sqrt{\overline{B}^s_{Q_1}}/ f_{B_d} \sqrt{\overline{B}^d_{Q_1}} = 1.2014^{+0.0065}_{-0.0072}$ in agreement with recent lattice determinations. We find $\Delta M_s=(18.5_{-1.5}^{+1.2})\text{ps}^{-1}$ and $\Delta M_d=(0.547_{-0.046}^{+0.035})\text{ps}^{-1}$ to be consistent with experiments at below one sigma. Assuming the validity of the SM, our calculation can be used to directly determine the ratio of CKM elements $|V_{td} / V_{ts} | = 0.2045^{+0.0012}_{-0.0013}$, which is compatible with the results from the CKM fitting groups, but again more precise.


Introduction
Mixing of B s mesons is experimentally well studied [1] and the mass difference ∆M s = 2|M s 12 | is known with a high precision [2] (based on the individual measurements [3][4][5][6][7]): ∆M Exp. s = (17.757 ± 0.021) ps −1 . (1.1) The corresponding theory expression for M s 12 reads with the CKM element λ t = V * ts V tb and the Inami-Lim function S 0 [8] describing the result of the 1-loop box diagrams in the standard model (SM). Perturbative 2-loop QCD corrections are compressed in the factorη B [9]. Since this observable is loopsuppressed in the SM, it is expected to be very sensitive to BSM effects. The bag parameter B ≡ B s Q 1 and the decay constant f Bs quantify the hadronic contribution to B-mixing; the uncertainties of their numerical values make up the biggest uncertainty by far in the SM prediction of the mass difference. These parameters have been determined by lattice simulations [10][11][12] and for the case of B d mesons with HQET sum rules [13][14][15][16]. There is also a recent lattice determination of the SU(3) breaking ratios [17]. Taking the most recent lattice average from the Flavour Lattice Averaging Group (FLAG) [18], which is more or less equivalent to the result in [12], one gets [19] a SM prediction for the mass difference, which is larger than the measurement: Such a value has dramatic consequences for some of the BSM models that are currently investigated in order to explain the flavour anomalies. In particular the parameter space of certain Z models is almost completely excluded [19].
In this work we extend the analysis of [15] with effects of a finite strange-quark mass, thus getting for the first time a HQET sum rule prediction for the mixing Bag parameter of B s mesons. Lattice simulations typically achieve a much higher precision than sum rule calculations, but in our case a sum rule for B − 1 can be written down.
Since the value of the Bag parameter B is close to 1, even a moderate precision of the sum rule of the order of 20 % for B − 1, turns into a precision of the order of 2% for the whole Bag parameter, which is highly competitive. Thus our determination constitutes an independent cross-check of the large lattice value found in [12]. In combination with a precise lattice determination of the decay constant f Bs our result for the Bag parameter can also be used for a direct determination of |V * ts V tb | from the measured mass difference ∆M Exp. s . Taking instead a ratio of the mass differences in the B d and the B s system one can get a clean handle on |V td /V ts |. Taking further a ratio of ∆M s and the rare branching ratio Br(B s → µ + µ − ) the decay constant and the CKM dependence cancel and the Bag parameter will be the only relevant input parameter. Our paper is organised as follows: in Section 2 we set up the sum rule for the Bag parameter and determine the m s corrections, in Section 3 we present a numerical study of the sum rules and we perform a phenomenological analysis. Finally, we conclude in Section 4.

Operator basis and definition of bag parameters
In this work we use the full dimension-six ∆B = 2 operator basis required for a calculation of ∆M s in the SM 1 and BSM theories and for a SM prediction of ∆Γ s . The QCD operators involved are while our HQET basis is defined as where h (+/−) (x) is the HQET bottom/anti-bottom field and we use the notation The matching condition is given by for which the NLO HQET-QCD matching coefficients C QQ were presented in [15]. We also use the same basis of evanescent operators. As mentioned in [15], the HQET evanescent operators are defined up to 3 constants a i with i = 1, 2, 3 in order to gauge the scheme dependence. We also note that in all of the following we work within the NDR scheme in dimensional regularisation with d = 4 − 2 .
The QCD bag parameters B s Q are defined through [20] with the coefficients A Q given by where M Bs denotes the B s meson mass, m q corresponds to quark pole masses and the B s meson decay constant f Bs is defined by The barred terms in the far right expression of (2.5) indicate that the quark masses used there are in the MS scheme. For the reasons discussed in [15] we prefer to use the pole masses for our analysis and then convert to this form at the end. Similarly, the HQET bag parameters are defined through with the coefficients AQ given by and where the matrix elements are taken between non-relativistically normalised The HQET decay constant F s (µ), appearing in (2.8) is defined by which is then related to the QCD decay constant f Bs through with [21] From our sum rule analysis we determine the HQET bag parameters B s Q . Using (2.4), (2.5), (2.8), and (2.12) we arrive at the relation which allows us to then match the values of B s Q onto their QCD counterparts.

Finite m s effects in the HQET decay constant
To illustrate our strategy for the treatment of finite m s effects we first consider the Borel sum rule for the HQET decay constant F s which has been derived in [22][23][24].
In the B s system it takes the form where ρ Π is the discontinuity of the two-point correlator with ω = p · v and the interpolating currentj + =sγ 5 h (+) . The leading perturbative part of the discontinuity is given by In the remainder of this subsection we consider the finite-energy (FESR) version of the sum rule (2.15) which is given by the limit t → ∞ to be able to present compact analytic results. We obtain Figure 1: Sample diagram involving a soft light-quark propagator (red thick line).
k ∼ ω or soft with momentum k ∼ m s whereas the heavy quark field is always hard. Up to and including the order m 2 s there are however only contributions from diagrams where all lines are hard. An example diagram involving a soft line is shown in Figure 1 s . Therefore, we only need to consider the fully hard momentum region where the integrand can be naively Taylor expanded in m s . We obtain

Finite m s effects in the Bag parameters
The sum rule for the Bag parameters is based on the three-point correlator where ω 1,2 = p 1,2 · v and the interpolating currents for the B s and B s mesons read .
where the second equation makes use of (2.15). The quantity ∆ρQ i is the nonfactorizable part of the double discontinuity In [15] we derived a simple analytical result for the HQET bag parameters by comparing (2.24) to the square of the sum rule for the decay constant (2.15) with an appropriately chosen weight function . (2.27) The generalization of this approach to the m s corrections is straightforward. Expanding the double discontinuity in m s , we obtain ∆ρ pert (2.28) where x = ω 2 /ω 1 and L ω = ln(µ 2 ρ /(4ω 1 ω 2 )). With this parametrization, the symmetry of the three-point correlator under exchange of ω 1 and ω 2 manifests as a symmetry under x ↔ 1/x of the r (j) Q i . The result for the deviation of the Bag parameters from the VSA reads (2.29) where L Λ+ms = ln(µ 2 ρ /(4(Λ + m s ) 2 )). We find that the result only depends on the value of the double discontinuity at ω 1 = ω 2 = Λ + m s . Thus, the knowledge of the m s -expanded double discontinuity is sufficient to determine the m s effects for the Bag parameters in B s mixing. However, the use of this weight function approach relies on the expanded version of the sum rule (2.15) for the decay constant. As discussed in the previous subsection, this approach gives an incorrect result at the order m 3 s and the result (2.29) is therefore limited to the quadratic order in m s .

Non-zero m s corrections to the non-factorizable part
We compute the m s -expanded result for the leading non-factorizable part of the three-point correlators using the expansion by regions [25,26]. As in the case of the two-point correlator, contributions involving soft propagators like the ones shown in Figure 3 first contribute at order m 3 s . Thus, we only have to consider the fully hard momentum region where all loop momenta admit the scaling l ∼ ω i m s and the loop integrands can be naively Taylor expanded in m s . We have performed two independent calculations. The amplitudes are either generated using QGRAF [29] with further processing in Mathematica or with a manual approach. The Dirac algebra is performed either with TRACER [30] or a private implementation. We employ FIRE [31] to generate IBP relations [32] between the loop integrals and to reduce them to a set of Master integrals with the Laporta algorithm [33]. The required master integrals have been computed to all orders in in [34]. We have expanded them up to the required order in using HypExp [35]. For completeness we state the results r For the linear terms r Last but not least, our results for the quadratic terms r (2.34)

Results and phenomenology
We determine the Bag parameters in Section 3.1, give our predictions for the B s mixing observables in Section 3.2 and use the results to determine the CKM elements |V td | and |V ts | in Section 3.3 and the top-quark MS mass in Section 3.4. We then present an alternative prediction of the branching ratios B(B q → µ + µ − ) from the ratios B(B q → µ + µ − )/∆M q in Section 3.5. Our analysis strategy closely follows the one we used in [15] in the limit m s = 0 and we only comment on where they differ due to the non-zero strange mass while referring to [15] for more details.

Bag parameters
We determine the HQET Bag parameters at the scale µ ρ = 1.5 GeV using the weight function approach (2.29). The strange-quark mass scheme in (2.29) is undetermined since any scheme change would only affect the expressions at higher orders which are not taken into account. We use the value in the MS scheme at the scale µ ρ which is determined from the central value of the average m s (2 GeV) = (95 +9 −3 ) MeV [36]. To account for the uncertainties related to the scheme choice and the truncation of the expansion in m s we increase the parametric uncertainty and use m s (2 GeV) = (95±30) MeV. To the perturbative part we add the condensate contributions [37,38]. The lattice simulation [39] shows that light and strange quark condensates agree within uncertainties and their result for the strange-quark condensate has since been confirmed with a different method [40]. With the factorization hypothesis qGq = m 2 0 qq the same holds for the quark-gluon condensate. We therefore assume the condensate corrections to be the same in the B 0 and B 0 s systems. We obtain where we have indicated the orders in m s with subscripts and find good convergence of the expansion. The differences in the leading terms with respect to the results for B d mixing obtained in [15] arise because the logarithms L Λ are replaced by L Λ+ms which we do not expand in m s /Λ. The results (3.1) are then evolved to the matching scale µ m = m b (m b ) where they are converted to QCD Bag parameters B s Q using (2.14). We do not consider the effects of a non-zero strange-quark mass in the QCD-HQET matching. The matching corrections are of the order α s (m b (m b ))/π × m s (m b )/m b (m b ) ∼ 0.001 and therefore subleading compared to the linear terms α s (µ ρ )/π×m s (µ ρ )/(Λ+m s (µ ρ )) ∼ 0.019 and even the quadratic terms α s (µ ρ )/π ×[m s (µ ρ )/(Λ+m s (µ ρ ))] 2 ∼ 0.003 in the sum rule. We do not include this uncertainty as a separate contribution in our error analysis since it is covered by the conservative variation of the input value for m s . Lastly, we convert the QCD Bag parameters to the usual convention which we denoted as B s Q in (2.5). We find where we have included the uncertainty from variation of m s in the sum rule (SR) error and M denotes the uncertainty from the QCD-HQET matching. We compare our results to other determinations from lattice simulations [10][11][12] and sum rules [13] and the FLAG averages [18] in Figure 4 and find very good agreement overall with similar uncertainties. We observe that the FNAL/MILC'16 value for B Q 1 is larger than all the other results -with respect to our value the difference corresponds to 1.1 sigma. We note that FNAL/MILC'16 determined the combination f 2 Bs B Q 1 and extracted the Bag parameter using the 2016 PDG average for the decay constant. They are currently working on a direct determination and, since their recent result [41] for f Bs is larger than the PDG value used in [12], we expect the Bag parameter to go down. On the other hand our Bag parameters for Q 4,5 are in good agreement with FNAL/MILC'16, while there is a tension of more than two sigmas with respect to the results of ETM'14. Similar tensions have been observed in the Kaon system [42] where it was conjectured that a difference in intermediate renormalization schemes might be responsible. We also consider the ratios B  The leading terms in the m s -expansion differ from unity because we do not expand the logarithms L Λ+ms in m s /Λ. Compared to the absolute Bag parameters we reduce the intrinsic sum rule error to 0.005, the condensate error to 0.002 and the uncertainty due to power corrections to 0.002 since the respective uncertainties cancel to a large extend in the ratios. However, we enhance the intrinsic sum rule and condensate error estimates for the operator Q 3 by a factor of five since the sum rule uncertainties for this operator are enhanced by large ratios of color factors A Q 1,2 /A Q 3 as discussed in [15]. A detailed overview of the uncertainties is given in Appendix A. Taking the where the ratio of decay constants and Bag parameters contributes equally to the error budget. A comparison with previous results is shown in Figure 5. There we also show how the result changes when the FLAG N f = 2 + 1 average is used for the ratio of the decay constants. Unfortunately FNAL/MILC and ETM do not provide values for B s/d Q i for i = 2, 3, 4, 5 so we cannot easily compare our results for these ratios.

B s mixing observables
In this section we present the results of our B mixing analysis. We consider the mass differences ∆M s and ∆M d , the decay rate differences ∆Γ s and ∆Γ d , and the ratio ∆M s /∆M d , of which the latter benefits from a reduced uncertainty due to the cancellation of CKM factors and hadronic effects. For the bottom-quark mass we studied the MS, PS [45], 1S [46] and the kinetic [47] mass schemes and found good agreement (see [15] for a more detailed discussion) -below we just quote the result in the PS scheme. We choose as our CKM parameter inputs the results of CKMfitter2018 [48] and collect these along with our other numerical inputs in Appendix A. For the non-perturbative input we use our SR determination of the Bag parameters (Eq.(3.2) and Eq. (3.3)) together with the lattice decay constants (N f = 2 + 1 + 1) from [18] (dominated by HPQCD'17 [43] and FNAL/MILC'17 [41]). Comparing our findings for ∆M s we see an excellent agreement with the experimental measurement We note that the update to our CKM input gives rise to an increase in ∆M SR s from the value presented in [15], despite the inclusion of m s -corrections which reduce the size of our hadronic input. Using instead the non-perturbative input purely from lattice determinations (FLAG 2019 [18], which is almost identical to the result in [12]), we get a considerably higher SM prediction for ∆M s : ∆M Lat. Recent measurements [49,50] that are not yet contained in the average [2] yield significantly smaller values for ∆Γ s which are however still in the one-sigma range of our prediction. The theoretical prediction for the decay rate difference includes NLO QCD [51][52][53][54] and 1/m b [55,56] corrections. The latter depend on matrix elements of dimension-seven operators which are currently only known in the vacuum saturation approximation, which results in uncertainties of approximately 25-30%. The sizable scale uncertainty can be reduced with a NNLO computation of the HQE matching coefficients -first steps towards this have recently been performed in [57]. Using instead the non-perturbative input from lattice [18], we again get higher values ∆Γ Lat.     [12] results for f 2 Bq B q Q i with i = 1 and R 0 . The weighted average over the sum rule and lattice results is shown in black. We indicate the updated Run 1 and Run 2 combinations for ∆Γ s presented by LHCb [49] and ATLAS [50] at Moriond EW 2019 by shaded gray regions.
Due to our new value for ξ we get a theoretical precision of about 3% for the ratio of mass differences in the B d and B s systems, which poses severe constraints on BSM models, that modify neutral B meson mixing. The uncertainty is now dominated by the CKM factors. Using lattice inputs one gets a slightly less precise value (∆M d /∆M s ) Lat. = 0.0295 +0.0010 −0.0012 = 0.0295 +0.0008 −0.0008 (had.) +0.0005 −0.0008 (exp.), which can be combined with the sum rule result to obtain (∆M d /∆M s ) Av. = 0.0297 +0.0006 −0.0009 = 0.0297 +0.0003 −0.0003 (had.) +0.0005 −0.0008 (exp.).

(3.12)
While there is good agreement for |V ts | the value of |V td | differs from our result by about 1.4 sigma. The value of the ratio |V td /V ts | can be determined more precisely based on the exact relation These values are all somewhat smaller than the expectation from CKM unitarity taken from CKMfitter [48] and UTfit [58] |V td /V ts | = 0.2088 +0.0016  Figure 7: Comparison of our constraints on the CKM parameters |V td | and |V ts | with other works based on B mixing [12,17] (solid boundaries) and unitarity [48,58] (dashed boundaries). Since the full CKM fit includes the mass differences we also show the tree-level fit from CKMfitter [48] (dotted boundaries).
but the effect is still only at the level of 1.3 (CKMfitter) and 2.0 (UTfit) standard deviations. Larger tensions are expected with respect to the CKM fit result based only on tree-level processes which is not publicly available. Thus, an improved determination of |V td | and |V td /V ts | from tree-level processes might provide an interesting hint towards new physics in the B d system. Similar considerations have recently led to claims about an emerging ∆M d anomaly [59]. An overview of the various results is presented in Figure 7, where the overlap of the one-sigma regions for |V td |, |V ts | and |V td /V ts | is indicated by the shaded regions. Our results provide an important input for future CKM unitarity fits and can be used to extract the angle γ in the unitarity triangle from the linear dependency between ξ and the CKM angle γ observed in [60].

Determination of the top-quark MS mass
The parametric error from the top-quark mass currently dominates the uncertainty in the determination of the stability or meta-stability of the electroweak vacuum [61]. Direct measurements quote very precise values m MC t = (173.0 ± 0.4) GeV for the top quark mass [36], but these results correspond to so-called Monte-Carlo (MC) masses and not the top-quark pole mass. One therefore needs to account for additional uncertainties from the scheme conversion [62] when these values are used for phenomenological predictions. Alternatively one can determine the top-quark mass by fitting observables like the total top-pair production cross section which can be predicted in terms of the top-quark mass in a well-defined scheme like MS. Similarly, we can use the mass differences ∆M q for a theoretically clean determination of m t (m t ). Using the CKMfitter values for V td and V ts as input we obtain where we have averaged over the hadronic and scale uncertainties, which are correlated, and treated the parametric uncertainties, which are dominated either by V td or V ts , as independent. This is in good agreement with the PDG average [36] m t (m t ) = (160 +5 −4 ) GeV, (3.19) of MS mass determinations from cross section measurements with our uncertainty being about 50% larger. A very precise measurement of the top-quark PS or MS mass with a total uncertainty of about 50 MeV is possible at a future lepton collider running at the top threshold [63][64][65].
The branching ratio Br(B q → l + l − ) is strongly suppressed in the SM and theoretically clean. Thus, it provides a very sensitive probe for new physics. At present it has been computed at NNLO QCD plus NLO EW [66] and the dominant uncertainties are parametric, stemming from the decay constant and the CKM parameters. Both uncertainties cancel out of the ratio [67] Br , (3.20) which in turn receives its dominant uncertainty from the Bag parameter B q Q 1 . Using our result (3.3) and including the power-enhanced QED corrections determined in [68] we predict the branching ratio by multiplying (3.20) with the measured mass differences where the uncertainties for the branching ratios are completely dominated by the error from B q Q 1 . The result for B 0 s → µ + µ − is in good agreement with the current experimental average [2] Br(B 0 while the latest measurements only provide upper bounds at 95% confidence level (ATLAS [71]) .

(3.23)
We compare our prediction (3.21) to the direct predictions from [41,66,68] which depend on the decay constants and CKM elements |V tq |, the prediction [12] from the ratios Br(B q → l + l − )/∆M q and the experimental average (3.22) in Figure 8. The shaded regions correspond to the overlap of the one-sigma regions for Br(B 0 s → µ + µ − ), Br(B 0 d → µ + µ − ) and Br(B 0 d → µ + µ − )/Br(B 0 s → µ + µ − ) where they were provided. We find good consistency among the various predictions with similar uncertainties for both approaches and good agreement with experiment whose uncertainty currently exceeds the theoretical one by a factor of about 3-4 in Br(B 0 s → µ + µ − ). For completeness we provide our predictions for the branching ratios to electrons which represents a reduction of the uncertainty by more than a factor of two compared to the latest lattice results [12,17]. Our results enable a rich phenomenology: we get updated SM predictions for the mixing observables ∆M s and ∆Γ s , which are in agreement with the experimental values. In particular we do not confirm the large values for ∆M s obtained with the non-perturbative values from FNAL/MILC [12], which led to severe bounds on BSM models. If V tb and ∆M q are used as inputs, we can precisely determine the CKM elements |V td | and |V ts | and we obtain the world's most precise determination of the ratio |V td /V ts |. Using all CKM elements as inputs we get constraints on the values of the top quark MS mass which are compatible with direct collider determinations. Finally our results lead also to precise SM predictions for the branching ratios of the rare decays B q → ll.
In future a still higher precision of our HEQT sum rule results can be obtained by the calculation of the HQET-QCD matching at NNLO (first steps in that direction have been performed in [16]). Another line of improvement could be the determination of 1/m b -corrections to the HQET limit. The computation of m s corrections to the Bag parameters of ∆F = 0 four-quark operators would enable an update of the predictions for the lifetime ratios τ (B s )/τ (B 0 ) [15] and τ (D + s )/τ (D 0 ) [72]. Finally a cross-check of our HQET sum results for mixing and lifetimes with modern lattice techniques would be very desirable. f Bs /f B 1.209 ± 0.005 [18] τ (B 0,H s ) (1.615 ± 0.009) ps −1 [36] τ (B 0 d ) (1.520 ± 0.004) ps −1 [36]