SU(3)-breaking ratios for D (s) and B (s) mesons
Peter A Boyle, Luigi Del Debbio, Nicolas Garron, Andreas Juttner, Amarjit Soni, Justus Tobias Tsang, Oliver Witzel
SSU(3)-breaking ratios for D ( s ) and B ( s ) mesons RBC and UKQCD Collaborations
P. A. Boyle, a,d
L. Del Debbio, a N. Garron, b A. Jüttner, c A. Soni, d J. T. Tsang a,e andO. Witzel a,f a Higgs Centre for Theoretical Physics, School of Physics & Astronomy, University of Edinburgh,EH9 3FD, United Kingdom b Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liv-erpool L69 3BX, United Kingdom c School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UnitedKingdom d Physics Department, Brookhaven National Laboratory, Upton, NY 11973, United States e CP3-Origins and IMADA, University of Southern Denmark, Campusvej 55, 5230 Odense M,Denmark f Department of Physics, University of Colorado Boulder, Boulder, CO 80309, United States
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We present results for the SU (3) breaking ratios of decay constants f D s /f D and f B s /f B and - for the first time with physical pion masses - the ratio of bag parameters B B s /B B d , as well as the ratio ξ , forming the ratio of the non-perturbative contributionsto neutral B ( s ) meson mixing. Our results are based on Lattice QCD simulations withchirally symmetric 2+1 dynamical flavours of domain wall fermions. Eight ensembles atthree different lattice spacings in the range a = 0 . − .
07 fm enter the analysis, twoof which feature physical light quark masses. Multiple heavy-quark masses are simulatedranging from below the charm quark mass to half the bottom-quark mass. The SU (3) breaking ratios display a very benign heavy-mass behaviour allowing for extrapolation tothe physical bottom-quark mass.The results in the continuum limit including all sources of systematic errors are f D s /f D =1 . stat (cid:0) +68 − (cid:1) sys , f B s /f B = 1 . stat (cid:0) +95 − (cid:1) sys , B B s /B B d = 0 . stat (cid:0) +80 − (cid:1) sys and ξ = 1 . stat (cid:0) +95 − (cid:1) sys . Combining these with experimentally measured valueswe extract the ratios of CKM matrix elements | V cd /V cs | = 0 . exp (cid:0) +12 − (cid:1) lat and | V td /V ts | = 0 . exp (cid:0) +162 − (cid:1) lat . Corresponding author. a r X i v : . [ h e p - l a t ] J un ontents In the Standard Model (SM) one can parameterise the QCD contribution to weak decaysof charged pseudoscalar mesons (e.g. B ± , D ± and D ± s ) into a lepton and a neutrino viathe leptonic decay constants f B ± and f D ± ( s ) . Similarly the mass difference between the twomass eigenstates of neutral mesons, which mix under the weak interaction, (e.g. B − ¯ B and B s − ¯ B s mixing) can be parametrised in terms of Standard Model free parameters andexperimentally known quantities. Both these parametrisations involve elements of the CKM– 1 –atrix [1, 2], which are not known a priori. However, the structure of the SM constrainsthis matrix to be unitary, so by independent precise determinations of the elements of thismatrix, its unitarity can be tested and hence tests of the SM performed.For charged pseudoscalar mesons P with quark content ¯ q q experiments measure thedecay rates Γ( P → lν l ) which can be expressed as Γ( P → lν l ) = | V q q | f P K + O ( α EM ) . (1.1)Here K are perturbatively known expressions, V q q is the relevant CKM matrix elementand f P is the decay constant. When electromagnetic effects are neglected (c.f. equation(1.1)), the decay rate factorises and hence precise knowledge of the non-perturbative quan-tity f P allows for an extraction of the CKM matrix element under consideration.These decay rates have been measured for P ± = D ± ( s ) and B ± by CLEO-c [3–9],BaBar [10, 11], Belle [12, 13] and BESIII [14, 15]. After accounting for the perturbativecontributions K , we can identify the product of the relevant CKM matrix element andthe charged decay constants ( f P ) as summarised by the Particle Data Group (PDG) [16]leading to the following global averages: | V cd | f D + = 45 . .
05) MeV [3, 4, 14] | V cs | f D + s = 250 . .
0) MeV [5–10, 12] | V ub | f B + = 0 . [11, 13] . (1.2)We note that the very recent result by BESIII [15] quoting | V cs | f D + s = 246 . . stat (3 . sys is not included in this average yet. Adding this into the average, by treating [15] and theaverage presented in [16] as uncorrelated, we obtain | V cs | f D + s = 249 . .
2) MeV [5–10, 12, 15] , (1.3)in full agreement with the PDG value, but with a slightly reduced error.Similarly, the mass differences between the mass eigenstates of the B − ¯ B and B s − ¯ B s systems can be measured to great precision as oscillation frequencies. When considering themixing of B s ) mesons in the SM, the right diagram in Figure 1 is dominated by top loops(i.e. q = q (cid:48) = t ) and therefore by short distance contributions. The SM prediction of themass differences ∆ m d and ∆ m s (for P = B , B s , respectively) can again be expressed as afunction of known perturbative factors ( K ), CKM matrix elements and non-perturbativequantities such as decay constants f P and renormalisation group invariant bag parameters ˆ B P , i.e. ∆ m q = (cid:12)(cid:12) V ∗ tq V tb (cid:12)(cid:12) K f P m P ˆ B P . (1.4)The mass difference ∆ m d has been measured by ALEPH [17], BaBar [18–22], Belle [23–25],CDF [26–30], D0 [31], DELPHI [32, 33], OPAL [34], L3 [35], LHCb [36–39], whilst ∆ m s hasonly been measured by CDF [40] and LHCb [38, 39, 41, 42]. The values for both observableshave been summarised and averaged in Ref. [16] leading to the global averages ∆ m d = 0 . − [17–39] , ∆ m s = 17 . − [38–42] , (1.5)– 2 –here the first error is statistical and the second systematical. Note that the perturbativefactor K in (1.4) cancels in the ratio ∆ m s / ∆ m d leading to ∆ m s ∆ m d = (cid:12)(cid:12)(cid:12)(cid:12) V ts V td (cid:12)(cid:12)(cid:12)(cid:12) m B s m B f B s ˆ B B s f B ˆ B B . (1.6)Similar to the case of leptonic decays, precise predictions of the non-perturbative quantities f P and ˆ B P (for P = B s ) ) enables the extraction of | V ts /V td | .The current central values and one σ error band for the CKM matrix elements asdetermined by the CKMfitter group [43, 44] (left) and the UTfit [45] group (right) are . (cid:0) +254 − (cid:1) = | V cd | = 0 . . (cid:0) +50 − (cid:1) = | V cs | = 0 . . (cid:0) + 86 − (cid:1) = | V td | = 0 . . (cid:0) + 28 − (cid:1) = | V ts | = 0 . . (1.7)The CKMfitter [43, 44] (left) and UTfit [45] (right) groups quotes their current best estimatefor the ratios | V cd /V cs | and | V td /V ts | to be . (cid:0) +280 − (cid:1) = | V cd /V cs | . (cid:0) +16 − (cid:1) = | V td /V ts | = 0 .
211 (3) . (1.8)Further detail on how the numbers in equations (1.7) and (1.8) are obtained are given inRef. [43–45].The non-perturbative quantities f P and ˆ B P can be calculated in lattice QCD. The baredecay constants and bare bag parameters are defined as (cid:10) (cid:12)(cid:12) A µq q (cid:12)(cid:12) P ( p ) (cid:11) = if P p µP (1.9)and B P = (cid:10) ¯ P (cid:12)(cid:12) O V V + AA (cid:12)(cid:12) P (cid:11) / f P m P , (1.10)where P is the pseudoscalar meson under consideration with four-momentum p µ and mass m P . In particular we will consider P = D ( s ) , B ( s ) , i.e. q = c, b and q = u/d, s . A µq q isthe axial vector current defined by A µq q = ¯ q γ µ γ q and the four-quark operator O V V + AA is given by (¯ q γ µ (1 − γ ) q ) (¯ q γ µ (1 − γ ) q ) . Quark flow diagrams that describe theseprocesses are shown in Figure 1.In this paper we consider the leptonic weak decays of charged mesons ( D ± , D ± s and B ± ) as well as the mixing of the neutral B s ) -meson with its antiparticle ¯ B s ) . Morespecifically, we will consider ratios which are typically more precise since common factors We thank Sébastien Descotes-Genon, Jérôme Charles and Marcella Bona for private communication ofthese results. We use the notation B s ) to simultaneously refer to B ≡ B d and B s . – 3 –nd parts of the systematic errors and of the statistical noise cancel. In particular we willconsider the SU (3) breaking ratios f D s /f D , f B s /f B , ˆ B B s / ˆ B B d ≡ B B s /B B d and ξ ≡ f B s (cid:112) B B s f B (cid:112) B B d . (1.11)As was first pointed out in Ref. [46], precise knowledge of SU(3) breaking ratios, such as B B s /B B d , f B s /f B and ξ , can be combined with the measured mass differences to extractthe ratio | V td /V ts | from (cid:12)(cid:12)(cid:12)(cid:12) V td V ts (cid:12)(cid:12)(cid:12)(cid:12) = (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) ∆ m d ∆ m s m B s m B d (cid:33) exp (cid:0) ξ (cid:1) lat . (1.12)As a result, we present constraints for the ratios | V cd /V cs | and | V td /V ts | .A summary of relevant lattice results for f D s /f D , f B s /f B , ξ and B B s /B B d was presentedby the Flavour Lattice Averaging Group (FLAG) [47]. Whilst lattice computations ofheavy-light decay constants have become more mature over the last few years, there arestill only few results for direct simulations at the physical pion mass [49–51]. For the caseof neutral meson mixing ( B B s /B B d and ξ ) this is the first result that is obtained fromsimulations including physical pion masses. For the ratio f D s /f D FLAG averaged the results presented in Refs. [49–51, 53–60].Similarly, the ratio of decay constants f B s /f B have also been computed by various lat-tice groups [50, 59–66]. For ξ and B B s /B B d only a few collaborations have publishedresults [52, 59, 66–70]. For results in the b -sector, the lattice formulations of the heavyquark vary widely, leading to differing systematic errors. The results presented in this pa-per are obtained from a chirally symmetric action which renormalises multiplicatively andtherefore is free of renormalisation uncertainties. A more detailed discussion of these resultsis presented in Section 5.The remainder of this paper is organised as follows. In Section 2 we describe ourensembles, our choice of heavy quark discretisation and our strategy to obtain correlationfunctions. In Section 3 we describe our correlation function analysis to deduce the requiredenergies and matrix elements, before addressing the global fit and the full error budgetin Section 4. Section 5 provides a comparison of our results with the known literature.Section 6 assesses the phenomenological implications, such as the determination of ratiosof CKM matrix elements before we conclude in Section 7. The status of this calculationwas previously reported in [71, 72]. We are performing this calculation in isospin symmetric lattice QCD with N f = 2 + 1 flavours, thereby capturing the dynamical effects of light (degenerate up and down) and In version 1 of this work we referred to the previous version of FLAG, i.e. FLAG16 [48]. Since the first version of this work a new lattice computation which also uses physical pion masses hasappeared [52]. We note that the lattice result [52] and the QCD sum rule result [70] have appeared after the firstversion of this paper has been posted. – 4 – + u, d, s ¯ c (¯ b ) ¯ lν l ¯ P q ¯ q ′ P b l, s ¯ b ¯ l, ¯ s Figure 1 . Left : Quark flow diagrams for the decay of a charged pseudoscalar meson.
Right : Quarkflow diagram for neutral meson mixing. For the shown diagram (i.e. for P = B ( s ) ), the quarks q and q (cid:48) in the loop have charge +2/3 (i.e. u,c,t). For the case D − ¯ D mixing the bottom quarkwould be replaced by a charm quark and q, q (cid:48) would have charge -1/3. strange quarks in the sea. We utilise RBC/UKQCD’s N f = 2 + 1 ensembles with physicallight quarks at a − ∼ . , . [73] and one ensemble with a finer lattice spacing of a − ∼ . and m π ≈
230 MeV [49]. We complement our dataset with RBC/UKQCD’sheavier pion mass ensembles [74–76], to guide the small correction of the fine ensembletowards the physical pion mass.For the heavy quarks we adopt a similar strategy to our previous work [49] by sim-ulating a range of heavy quark masses m h from slightly below the charm quark mass toapproximately half the b quark mass. For the neutral meson mixing computation, we onlyconsider the charge assignment suitable for B ( s ) meson mixing (cf. Figure 1), so that in thelimit m h → m b , we recover the correct quantities for B ( s ) -meson mixing.As we will lay out in Sections 2.1 and 2.2, our discretisation differs between thelight/strange and the heavy quark sector, resulting in a mixed action. In this work wesolely focus on results for observables where the renormalisation constants cancel. Workis in process to calculate the required mixed-action renormalisation factors (as laid out in[49]) in order to also obtain results for the individual decay constants and bag parameters,rather than their ratios. All ensembles use the Iwasaki gauge action [77] and the domain wall fermion action [78–81].The ensembles with heavier pion masses (C1-2, M1-3) use the Shamir action approximationto the sign function [81, 82], the remaining ensembles (C0, M0, F1M) the Möbius actionapproximation with the same H T kernel [83]. In the convention of Ref. [83], the Shamiraction is simulated with b = 1 , c = 0 , whereas for the Möbius action b = 1 . , c = 0 . is used. The parameters of both of these actions are chosen such that they lie on thesame scaling trajectory, allowing for a combined continuum limit [73]. Details of the mainparameters of these ensembles are summarised in Tables 1 and 2.Table 2 also describes the light and strange valence parameters. All light quarks aresimulated at their unitary value am sea l = am val l whilst the valence strange quark masseswere tuned to their physical values as determined in Refs. [49, 73] with the exception of theF1M ensemble where we simulated at the unitary strange quark mass. All propagators were For more detail on the F1M ensemble, please refer to Appendix A. – 5 –ame
L/a T /a a − [GeV] m π [MeV ] m π L hits × N conf C0 48 96 1.7295(38) 139.17(0.35) 3.86 × C1 24 64 1.7848(50) 339.76(1.22) 4.57 × C2 24 64 1.7848(50) 430.63(1.38) 5.79 × M0 64 128 2.3586(70) 139.34(0.46) 3.78 × M1 32 64 2.3833(86) 303.56(1.38) 4.08 × M2 32 64 2.3833(86) 360.71(1.58) 4.84 × M3 32 64 2.3833(86) 410.76(1.74) 5.51 × F1M 48 96 2.708(10) 232.01(1.01) 4.11 × Table 1 . This table summarises the main parameters of the ensembles used for the presentedcalculation. All ensembles have N f = 2 + 1 flavours in the sea. C stands for coarse, M formedium and F for fine. The columns hits and N conf give the number of measurements on a givenconfiguration and the total number of configurations used, respectively. Name DWF M L s am sea , val l am seas am phys s σ N σ C0 M 1.8 24 0.00078 0.0362 0.03580(16) 4.5 400C1 S 1.8 16 0.005 0.04 0.03224(18) 4.5 400C2 S 1.8 16 0.01 0.04 0.03224(18) 4.5 100M0 M 1.8 12 0.000678 0.02661 0.02539(17) 6.5 400M1 S 1.8 16 0.004 0.03 0.02477(18) 6.5 400M2 S 1.8 16 0.006 0.03 0.02477(18) 6.5 100M3 S 1.8 16 0.008 0.03 0.02477(18) 6.5 100F1M M 1.8 12 0.002144 0.02144 0.02217(16) - -
Table 2 . Domain wall parameters for the light and strange quarks. All quoted values for am l,s are bare quark masses in lattice units. The column DWF corresponds to the chosen domain wallfermion formulation, i.e. M(öbius) or S(hamir) domain wall fermions. generated using Z -wall sources [84–86]. For the light and strange quark propagators onthe coarse and medium ensembles, we used Gaussian smearing [87–89] to achieve a betteroverlap with the ground state. The smearing parameters σ and N σ are listed in Table 2. In our previous work [49, 90] the limitations of our formalism prohibited the direct simu-lation of the physical charm quark mass on the coarse ensembles. We therefore required aslight extrapolation in the heavy quark mass to reach the physical charm quark mass on ourcoarsest ensembles. We found that it is possible to increase the heavy-quark mass reach bystout smearing [91] the gauge fields prior to performing the charm quark inversions [71, 92].A comparison of the effect on the residual chiral symmetry breaking parameter m res waspresented in Ref. [71]. We found that three hits of stout smearing with the standard pa-– 6 –ame DWF L s M am h C0 M 12 1.0 0.51, 0.57, 0.63, 0.69C1 M 12 1.0 0.50, 0.58, 0.64, 0.69C2 M 12 1.0 0.51, 0.59, 0.64, 0.68M0 M 12 1.0 0.41, 0.50, 0.59, 0.68M1 M 12 1.0 0.41, 0.50, 0.59, 0.68M2 M 12 1.0 0.41, 0.50, 0.59, 0.68M3 M 12 1.0 0.41, 0.50, 0.59, 0.68F1M M 12 1.0 0.32, 0.41, 0.50, 0.59, 0.68
Table 3 . Bare heavy quark masses in lattice units. Γ Γ hl, st src = 0 tL LS S/L ¯ P l, ¯ sc ( b ) P l, s ¯ c (¯ b ) t = 0 t = ∆ TtO
V V + AA LS SL
Figure 2 . Schematic description of the set-up of our two-point (left) and three-point (right)correlation functions. rameter ρ = 0 . extends the reach in the heavy quark mass compared to our previouswork [71, 92]. Table 3 lists the domain wall parameters as well as the quark masses thatwere used on the various ensembles. Since the charm quark is quenched in our calculationsthis has no additional unitarity implications which are not already present. The left panel of Figure 2 shows our set-up for the computation of two-point functions.These take the form C s ,s Γ , Γ ( t ) ≡ (cid:88) x (cid:28)(cid:16) O s Γ ( x , t ) (cid:17) (cid:16) O s Γ ( , (cid:17) † (cid:29) = N = ∞ (cid:88) n =0 (cid:16) M s Γ (cid:17) n (cid:16) M s Γ (cid:17) ∗ n E n (cid:16) e − E n t ± e − E n ( T − t ) (cid:17) , (2.1)where the interpolation operators O s Γ i define the quantum numbers of the meson underconsideration and are given by O s Γ i ( t, x ) = (cid:32) ¯ q ( t, x ) (cid:88) y ω s ( x , y )Γ i q ( t, y ) (cid:33) . (2.2)– 7 –ere q and q give the quark content of the meson and we consider the cases Γ i = γ ≡ P (pseudoscalar) and Γ i = γ γ ≡ A (axial vector). ω s denotes that each propagator can besmeared (S) or local (L) at both the source and the sink. For the local case, ω reduces to aKronecker-delta (i.e. ω L = δ x , y ). In principle we consider the cases s ∈ { LL, SL, LS, SS } for each of the two operators (where the first entry corresponds to the smearing of thesource and the second entry to that of the sink). For the smeared case, ω s is obtainedby Gaussian smearing via Jacobi iteration [87, 89, 93], the parameters of which are givenin Table 2. In practice, we never smear the heavy quark propagator. On the coarse andmedium ensembles we always smear the source of the light and strange quark propagatorsand alow both options for the sink. On the fine ensemble (F1M), both source and sinkof all quark propagators are kept local. For the heavy-light systems under considerationwe thus consider SL and SS only for the coarse and medium ensembles and LL on thefine ensemble, where in both cases we have dropped the indices corresponding to the heavypropagators. The overlap coefficients M s i Γ i for state n are given by (cid:16) M s i Γ i (cid:17) n = (cid:68) X n (cid:12)(cid:12)(cid:12) O s i Γ i (cid:12)(cid:12)(cid:12) (cid:69) , (2.3)where X n is the n th excited meson state X with the correct quantum numbers. In theremainder of this paper we will omit the label for the state if only one state is considered.The right panel of Figure 2 shows how we obtain the three point functions from whichthe bag parameters are determined. We create a state with the quantum numbers of ¯ P at t = 0 , let it propagate to the operator insertion t , where it is transformed to the state P and then annihilate this state at ∆ T . Noting that for the coarse and medium (fine)ensembles the external states are always build from a smeared (local) light or strangepropagator there is no need to label the smearing combination for the three point function C ( t, ∆ T ) . Considering the zero momentum projected three point function, we can rewritethe correlation functions as C ( t, ∆ T ) ≡ (cid:68) P (∆ T ) O V V + AA ( t ) ¯ P † (0) (cid:69) = (cid:88) n,n (cid:48) m n m n (cid:48) (cid:0) M iP (cid:1) n (cid:10) n (cid:12)(cid:12) O V V + AA ( t ) (cid:12)(cid:12) n (cid:48) (cid:11) (cid:0) M iP (cid:1) ∗ n (cid:48) × (cid:16) e − (∆ T − t ) m n + e − ( T − ∆ T + t ) m n (cid:17) (cid:16) e − tm n (cid:48) + e − ( T − t ) m n (cid:48) (cid:17) ≈ m (cid:0) M iP (cid:1) e − (∆ T − t ) m (cid:104) P | O V V + AA ( t ) | P (cid:105) (cid:0) M iP (cid:1) ∗ e − tm , (2.4)where i = S ( i = L ) for the coarse and medium (fine) ensembles. In the final line, weassumed that only the ground state contributes and that “around-the-world” contributionsare negligible.The signal-to-noise ratio quickly deteriorates for large times so obtaining a signal inthe low t region is favourable. Hence a trade off between choosing ∆ T as small as possiblewithout pollution from excited states is required, which will be discussed in Section 3.2.We place a Z -wall source on every second time slice across the lattice, hence produce allrequired correlation functions ( T /a ) / times per configuration (cf. column hits in Table 1).– 8 –hese correlation functions are translated in time and binned into one effective measurementper configuration prior to any statistical analysis. In addition to improving the statisticalsignal, this allows us to compute the bag parameters for many source-sink separations ∆ T (compare Figure 2) without needing to invert additional propagators. For a given ∆ T thesethree point functions are obtained by contracting the propagators originating from differentwall source positions with the four-quark operator. Finally, this multi-source approachallows us to make efficient use of the HDCG algorithm [94], rendering this computationaffordable. We bin all measurements on a given configuration into one effective measurement. Prior toany analysis step, we make use of the last lines of equations (2.1) and (2.4) and symmetriseall two and three point correlation functions with respect to
T / and ∆ T / , respectivelyand before restricting the data to the temporal extent from t ∈ [0 , T / and t ∈ [0 , ∆ T / ,respectively.We conservatively choose to illustrate all correlator fits for the heaviest mass point onthe M0 ensemble, since this is a worst case scenario given the large difference between thephysical light quark mass and the heavier-than-charm quark mass. The error propagation iscarried out by using bootstrap resampling using 2000 bootstrap samples. We use differentseeds for the random number generator for different ensembles, to avoid the introductionof any spurious correlations. For the coarse and medium ensembles we extract values for the masses and matrix elementsby performing a simultaneous double-exponential fit (i.e. n = 0 , in (2.1)) to six correlationfunctions in the interval t ∈ [ t min , t max ) . In particular we simultaneously fit the correlationfunctions C SLAA , C SSAA , C SLAP , C SSAP , C SLP P and C SSP P . From this we obtain the mass m i as well asthe bare matrix elements M LP,n , M SP,n , M LA,n and M SA,n , where n = 0 , refers to the groundstate and the first excited state, respectively. The result of such a fit is shown in the lefthand panel of Figure 3. The coloured data points (circles and squares) show the effectivemass of the correlation functions that enter the fit, the grey horizontal band depicts theground state mass that is obtained from a fit to the data in the range [ t min , t max ) (indicatedby the vertical dotted lines). The coloured shaded bands show the effective mass obtainedby reconstructing the respective correlation functions from the fit-results. We can see thatthe data is well described by these fits. In the case of the F1M ensemble this situationsimplifies due to the absence of source and sink smearing and the above reduces to a jointfit of C LLAA , C LLAP and C LLP P to extract the matrix elements M LP,n and M LA,n . The results toall correlation functions fits are tabulated in the appendix in Table 6.Whilst for a pure ground state fit, the effective mass provides a visual cross-check of a plateau range in which one can approximate the correlation function as a single state, thisis more difficult for fits including excited states. We circumvent this in the following: As-suming we are in a range where only the ground state and the first excited states contribute– 9 – t/a a m e ff C SLAA C SSAA C SLAP C SSAP C SLPP C SSPP t/a a m e ff C AP ( t ) C PP ( t ) D SL ( t ) D SL ( t ) Figure 3 . Example correlation function fit for heaviest D -like meson on the M0 ensemble. Thedata points in the left panel show the effective masses of the correlation functions that enter thefit. On top of this, the effective mass of the fit results is superimposed. The grey horizontal bandshows the ground state fit result obtained in this way. The right panel shows the effective mass ofthe linear combinations of correlation functions that are mentioned in the text. The dashed verticallines correspond to the values of t min and t max . (and for simplicity restricting ourselves to t (cid:28) T / ), we can write C pqab ( t ) ≡ ∞ (cid:88) n =0 e − E n t E n ( M pa ) n (cid:0) M qb (cid:1) n ≈ e − E t E ( M pa ) (cid:0) M qb (cid:1) + e − E t E ( M pa ) (cid:0) M qb (cid:1) . (3.1)We now consider linear combinations of the form E X,Y ( t ) = C ( t ) X − D ( t ) Y , (3.2)where C and D are two of the original correlation functions and X, Y are some constants.Assuming we have carried out a fit to determine the matrix elements (cid:0) M LA (cid:1) n , (cid:0) M SA (cid:1) n , (cid:0) M LP (cid:1) n and (cid:0) M SP (cid:1) n for n = 0 , , we can now choose C ( t ) and D ( t ) , such that they have oneof the two matrix element factors in common. Furthermore we identify the factors X and Y with the excited state matrix element of the respective other correlation function whichthey do not have in common. More precisely, we construct the linear combinations C AP ( t ) ≡ C SSAP ( t ) (cid:0) M LA (cid:1) | fit − C LSAP ( t ) (cid:0) M SA (cid:1) | fit C P P ( t ) ≡ C SSP P ( t ) (cid:0) M LP (cid:1) | fit − C LSP P ( t ) (cid:0) M SP (cid:1) | fit D SL ( t ) ≡ C SLAP ( t ) (cid:0) M LA (cid:1) | fit − C SLAA ( t ) (cid:0) M LP (cid:1) | fit D SL ( t ) ≡ C SLP P ( t ) (cid:0) M LA (cid:1) | fit − C LSAP ( t ) (cid:0) M LP (cid:1) | fit , (3.3)where (cid:0) M SP (cid:1) | fit and (cid:0) M LP (cid:1) | fit refer to the central values of the fit. We stress that this isapplicable to any pair of two-point correlation functions that have the same spectrum andone matrix element in common. We note that if the backwards travelling contribution comes with the opposite sign between the twocorrelation functions in this difference, this only holds for values of t where temporal “around-the-world”effects are negligible. However for heavy-light quantities this contribution is suppressed by a factor smallerthan e − ET/ where the smallest simulated values are ET / ∼ . This is therefore negligible. – 10 – .0010.0020.0030.0040.0050.0060.007 a m +1.016 chosen t max =18 at max =20 at max =22 at max =24 a t min /a a f b a r e a m +1.0588 chosen t max =20 at max =22 at max =24 at max =26 a t min /a a f b a r e +1.128e 1 Figure 4 . Impact of the choice of fit range on the observables of interest, i.e. the mass m and the(bare) decay constant f . Results are shown for the heaviest heavy-light (left) and heavy-strange(right) mesons on the M0 ensemble. If the fit describes the data well, the excited state contribution cancels in this differenceand such an effective mass plot should show a plateau in the region of the fit. Furthermorethis plateau needs to coincide with the fit result for the ground state energy. This proceduretherefore serves as a strong a posteriori check.The right panel of Figure 3 shows the effective masses of some of the linear combinationswhich can be obtained from the correlation functions. The grey horizontal band shows theground state mass which is obtained from the fit. We note that in between the two verticallines, the effective mass of the reconstructed data points lie within the grey band. Inaddition to this strong visual check, we also varied t min and t max to investigate stabilityunder these changes. This is presented for the case of the heaviest heavy quark mass onthe M0 ensemble in Figure 4. All variations of the fit range are well within the quotedstatistical uncertainty, particularly for the heavy-light case which dominates the error onall the presented ratios. Since we are interested in B P , we construct ratios in which the matrix elements M SP cancel(c.f. equation (2.4)). More precisely we construct ratios R ( t, ∆ T ) which, in the limit oflarge t and ∆ T , plateau to the value of the bag parameter B P R ( t, ∆ T ) = C ( t, ∆ T )8 / C SLP A (∆ T − t ) C LSAP ( t ) → (cid:104) P | O V V + AA ( t ) | P (cid:105) / m P f P ≡ B P for t, ∆ T (cid:29) . (3.4)Figures 5 and 6 show example fits of such plateaux and the fits to them for the case ofthe heaviest heavy-light and heavy-strange mass points on M0, respectively.– 11 – t/a R l h ( t , ∆ T ) fitdata for ∆ T = 22 a
20 22 24 26 28 30 32 ∆ T/a
Figure 5 . Example fits for the heavy-light bag parameters on the M0 ensemble for the heaviestheavy quark mass for the respective choices of ∆ T . The left panel shows the fit to a constant forthe chosen value of ∆ T /a = 22 . The right panels show the stability as a function of the source-sinkseparation ∆ T . The magenta star illustrates our chosen value for ∆ T and the obtained result. t/a R s h ( t , ∆ T ) fitdata for ∆ T = 32 a
20 22 24 26 28 30 32 34 ∆ T/a
Figure 6 . Same plot as Figure 5 but for the heavy-strange bag parameter. Note that the heavy-strange quantity is nearly an order of magnitude more precise than the heavy-light one.
From the fits to the correlation function data we have obtained decay constants and bagparameters for a range of charm quark and pion masses and lattice spacings. Due to the useof chiral fermions all of these observables renormalise multiplicatively. So by constructingthe ratios f D s /f D and ξ (see (1.11)) all renormalisation constants cancel, such that wecan replace ˆ B B s / ˆ B B by B B s /B B in the same equation. Some of the statistical noise anddiscretisation effects also cancel, making these observables cleaner. Figure 7 shows the ratioof decay constants (left) and the ratio of bag parameters (right) as a function of the inverseheavy-strange pseudoscalar meson mass. The behaviour of these ratios as a function of theheavy quark mass (set via a meson mass containing a heavy quark) is very benign, lendingconfidence to the use of inter/extrapolations in the heavy meson mass. By comparing the– 12 – .1 0.2 0.3 0.4 0.5 0.6 m − D s [GeV − ] f s h / f l h bcC0C1C2M0M1M2M3F1 m − D s [GeV − ] B s h / B l h bcC0C1C2M0M1M2M3F1 Figure 7 . Summary of the ratio of decay constants (left) and the ratio of bag parameters B B s /B B (right) as a function of the inverse D s mass. C0 and M0 ensembles (which are at the same pion mass, but differ in lattice spacing), wenote that the discretisation effects appear to also be mild. We notice a stronger dependenceon the pion mass, as is expected for SU (3) -breaking ratios, since in the limit of m π → m K they are identically unity. This is the first computation of the ratio of bag parametersand the SU (3) breaking ratio ξ using ensembles at the physical pion mass, so that themain reason for this extrapolation is to guide the small extrapolation of the F1M ensembletowards the physical pion mass.Recalling how ξ is constructed from the ratio of decay constants and the square rootof the ratio of bag parameters (compare (1.11)) and noting that (cid:112) B hs /B hl is very closeto unity, we expect f B s /f B and ξ to be very similar in magnitude, in agreement with acomment made in Ref. [95]. This in turn implies that with a high degree of accuracy thecalculation of the SU(3) breaking ratio ξ can be approximated by just studying the ratioof two-point functions required for determining pseudoscalar decay constants. From our simulation data we determine observables O as a function of the lattice spacing a , the finite volume V and the quark masses. To combine this with the experimental data,we need to extra/interpolate our data to the physical values of the quark masses as wellas to the continuum ( a = 0 ) and infinite volume. Since quark masses are experimentallynot directly accessible quantities, we set the heavy quark mass by inter/extrapolating theresults to the physical value of appropriate meson masses. We set the light quark massby extrapolating to the neutral pion mass of
135 MeV [16]. The charm (bottom) quarkmass is fixed by the heavy-light ( m D -like), heavy-strange ( m D s -like) or heavy-heavy ( η c -like) pseudoscalar meson mass. From our previous experience [72], we find that the chiralslope in our data and the continuum limit artifacts are well described by terms linear in ∆ m π ≡ m π − (cid:16) m phys π (cid:17) and a , respectively. In the past, we further found that the heavyquark behaviour is captured well, by expanding in ∆ m − H ≡ /m H − /m expand H where m H is the meson chosen to set the heavy quark mass, and m expand H is the point around whichthe expansion is performed. A small caveat arises from the fact that the physical strange– 13 –uark mass on the F1M ensemble was not known prior to the data production, so thesimulation was carried out at the unitary value am uni s (c.f. Table 2 and appendix A). Wecompensate for this slight mistuning by including a term proportional to the mistuning ∆ m s ≡ (cid:16) am phys s − am uni s (cid:17) /am phys s . This term is only non-zero for the F1M ensemble.We therefore describe the data O ( a, m π , m H ) at given lattice spacing a , pion mass ( m π )and heavy meson mass ( m H ), by the fit ansatz f ( a, m π , m H ) = O (0 , m phys π , m phys H ) + C χ ∆ m π + C CL a + C H ∆ m − H + C s ∆ m s . (4.1)To check the validity and to estimate any systematic errors induced by this ansatz, wesystematically vary this ansatz and the data that enters the fit (cuts). For example, weconsider the impact of various pion mass cuts, the exclusion of the heaviest data points etc.Finally, we will also estimate higher order effects that are not captured by this fit form. In addition to the number of data points N obs of the observable under consideration ( f sh /f lh , B B s /B d or ξ ), also the parameters that the expression in (4.1) depends on, enter the fit.These are the N obs values of the heavy meson mass m H (there is a corresponding mesonmass for each value of the observable), the N ens values of the pion masses (one per ensemble)and the N a values of distinct lattice spacings (i.e. C1/2 and M1/2/3 share the same latticespacings respectively). We will collectively refer to these N x ≡ N obs + N ens + N a values as x i and note that their uncertainties have to be taken into account correctly. For the mesonmasses m π and m H these arise from correlator fits and are therefore fully correlated betweenthe observables and each other. However, this is not the case for the lattice spacing a , sincethis was determined from a different analysis including a larger set of gauge ensembles asdescribed in Refs. [49, 73]. To propagate this uncertainty, we generate a Gaussian bootstrapdistribution with the correct central value and match its width to the error.The fit is then carried out via χ minimisation, where χ is defined as χ = N tot (cid:88) i =1 N tot (cid:88) j =1 [ y i − F ( x i )] C − ij [ y j − F ( x j )] , (4.2)with N tot = N obs + N x , (i.e. N obs values for the observable, N obs values for the correspondingheavy meson mass, N ens values of the pion mass and N a values of the lattice spacing). The y i in (4.2) are given by y i = (cid:40) O i ( a i , m πi , m H i ) for i ≤ N obs x i otherwise. (4.3)The appropriate values of f ( x i ) are given by F i ( x i ) = (cid:40) f ( a i , m πi , m H i ) for i ≤ N obs x i otherwise. (4.4)– 14 –bservable C CL [GeV ] C χ [GeV − ] C H [GeV] C s d . o . f . χ / d . o . f . p -value f hs /f hl B B s /B B d ξ Table 4 . Results of the base fit (i.e. m max π = 350 MeV and M H = m sh ). We list the determinedcoefficients for the continuum limit slope ( C CL ), pion mass dependence ( C χ ), heavy mass depen-dence ( C H ) and strange mistuning ( C s ), the number of degrees of freedom in the fit as well as thegoodness of fit measures χ / d . o . f . and the corresponding p -values. Since C ij is the full covariance matrix (i.e. of size N tot × N tot ), this procedure takes all corre-lations between the various data points (pion masses, heavy meson masses and observables)into account.In summary, the fit determines not only the parameters O (0 , m phys π , m phys H ) , C χ , C CL and C H but also re-determines the x values. We note that this does not add any degrees offreedom, since the same number of additional parameters that are added to the fit are alsore-determined by it. For the observables considered in this work, we find that the relativeerror on the arguments of (4.1) are sufficiently small that the inclusion of the x -errors onlyhas a negligible effect (i.e. the effect is far smaller than the statistical error). We check thatthe output values ( x i ) are within errors of the input values ( x i ). We now present the results of the global fits described in the previous sections. We chooseas our central value the results obtained from a fit to the data according to (4.1) with apion mass cut of
350 MeV and the heavy mass being set by the heavy-strange pseudoscalarmass. The central values and statistical errors of these fits are f D s /f D = 1 . stat f B s /f B = 1 . stat B B s /B B d = 0 . stat . (4.5)The coefficients obtained from these fits together with the goodness-of-fit measure χ / d . o . f . and the associated p -values are listed in Table 4. The coefficient C s is small and compatiblewith zero in all cases, indicating that the strange quark mass mistuning has no significanteffect and we confirmed that performing the same fit without such a term leads to fullycompatible results. We note that the χ / d . o . f . values of all three fits are excellent, producinggood p -values. This is remarkable, given the small number of fit parameters (5) and thelarge number of degrees of freedom (16).The left panel of Figure 8 shows the data for the ratio of decay constants entering ourpreferred fit together with the fit result (magenta band) for the case of the ratio of decayconstants. The coloured bands and dashed lines show the fit function (c.f. equation (4.1))evaluated at the physical strange quark mass and the respective pion masses and latticespacings for each ensemble. The closed and open green diamonds show the F1M data beforeand after the adjustement for the strange quark mass mistuning, respectively.– 15 – .1 0.2 0.3 0.4 0.5 0.6 m − D s [GeV − ] f s h / f l h m phys B s m phys D s C0C1M0M1F1M f D s /f D f B s /f B m − D s [GeV − ] d a t a / r e s u l t o ff i t f o r f s h / f l h All datapoints shifted to m π = m phys π and a =0 m phys B s m phys D s C0C1M0M1F1M
Figure 8 . The global fit result for the ratio of decay constants as a function of the inverse heavymeson mass (left) and between the data and the fit results (right) as described in equation (4.6). Thered circles (blue squares, green diamonds) show the data for the coarse (medium, fine) ensemblesthat enter the fit. The open green diamonds show the correction due to the strange quark massmistuning. The magenta line shows the fit function evaluated at physical pion masses in thecontinuum. The magenta band illustrates the statistical error. The black and magenta stars showthe result (statistical error only) for f D s /f D and f B s /f B , respectively. We stress that due to the high degree of correlation of the data points on a givenensemble, care needs to be taken when trying to consider the contribution to the value of χ from a given data point. In the right panel of Figure 8 we present the data correctedto the physical pion mass and vanishing lattice spacing, normalised by the heavy massbehaviour. More precisely we show O ( a, m π , m H ) − C CL a − C χ ∆ m π − C s ∆ m s f (0 , m phys π , m H ) . (4.6)This illustrates that all data points are compatible with the fit at the ∼ σ level. Weobserve the above mentioned correlations by noting that data points on a given ensembleremain at a roughly constant distance from the fit. This lends further confidence in ourdescription of the behaviour as the heavy mass is varied. Whilst the goodness-of-fit forthe presented fit is excellent, we note that the largest contribution to the χ / d . o . f . arisesfrom the ensemble M0. This is conservatively addressed in our systematic error analysis byinvestigating different choices of pion mass cuts, leading to one of our dominant systematicerrors.To expose the functional behaviour with respect to each of the three parameters ( a , m π and m − H ) expression (4.1) depends on, we shift the data points to their physical valuesalong two of these three directions, to validate the behaviour in the third. Figure 9 showsthe data points shifted to the physical pion mass and after discretisation effects have beenremoved. Note the change in the y -axis between Figures 8, 9 and 10. The data pointsdisplay a very linear behaviour all the way from the lightest data point (below D s whichis shown by the vertical dotted line) up to heaviest data point (at approximately half the B s mass, which is indicated by the vertical dash-dotted line). This linear behaviour allowsus to extrapolate our results to obtain results at the b -quark mass. The difference betweenthe data at the charm mass and at the bottom mass is only of the order of ∼ , making– 16 – .1 0.2 0.3 0.4 0.5 0.6 m − D s [GeV − ] f s h / f l h All datapoints shifted to m π = m phys π and a = 0 m phys B s m phys D s C0C1M0M1F1M f D s /f D f B s /f B Figure 9 . Chosen global fit for the ratio of decay constants. All data points are shifted to thephysical pion mass and zero lattice spacing, so the plot shows the behaviour as a function of theinverse heavy mass. this extrapolation very benign. We note that this is largely due to the fact that the heavyquark behaviour cancels in the ratio of decay constants. Figure 10 shows the projections ofthe data points shown in the left panel of Figure 8 to the physical charm quark mass, setby the D s mass. In the left (right) panel the data points are also shifted to the physicalpion mass (zero lattice spacing), so that we can compare the continuum limit (pion mass)behaviour with the data. We see that the continuum limit is rather flat (cf. coefficients C CL in Table 4), with discretisation effects of around one percent for the coarsest ensemble. Thebehaviour with m π is stronger, as expected for an SU (3) breaking ratio, with the ensembleat m π ∼
340 MeV differing by ∼ − compared to the physical pion mass. This is very welldescribed by the linear ansatz in ∆ m π . We emphasise that since our simulation includestwo ensembles at the physical pion mass, the main impact of this slope is to guide thesmall extrapolation on the fine ensemble to the physical pion mass. The same behaviour isobserved for the projection to the physical b -quark mass, and we refer to these very similarlooking plots (cf. Figure 23) in appendix C.For the ratio of bag parameters B B s /B B d , the discretisation effects are very similar tothe above. The chiral behaviour is suppressed with a coefficient that is roughly an orderof magnitude smaller. So a pion mass of
340 MeV only leads to a difference of ∼ compared to the physical value. The behaviour with the heavy mass, is very benign andsimilar in magnitude to the ratio of decay constants, but opposite in sign. These resultsare summarised in Figures 11 and 12.We can obtain the observable ξ in two ways: We can construct ξ ( a, m π , m H ) ensembleby ensemble and perform the global fit (see equation (4.1)) on this quantity. Alternativelywe can take the output of the global fit for f B s /f B and B B s /B B d and then construct ξ from these outputs (via equation (1.11)) in the continuum limit and after the extrapolation– 17 – .00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 a [GeV − ] f s h / f l h ( m ph y s D s , m ph y s π ) m π [GeV ] f s h / f l h ( m ph y s D s , a = ) Figure 10 . Global fit result for the ratio of decay constants. All data points are projected to thephysical charm quark mass, set via m D s . The data points on the left (right) panels are also shiftedto the physical pion mass (zero lattice spacing) and hence illustrates the scaling (chiral) behaviourof our data. We slightly shift data points along the horizontal axis for better visualisation of thedifferent data points. m − D s [GeV − ] B s h / B l h All datapoints shifted to m π = m phys π and a = 0 m phys B s m phys D s C0C1M0M1F1M B B s /B B d Figure 11 . Global fit result for B B s /B B . All data points are shifted to the physical pion massand zero lattice spacing. to physical masses but including all correlations . We will refer to the former as direct andlatter as indirect determinations. The results with statistical error of these two are ξ = 1 . direct ξ = 1 . indirect . (4.7)The central value remains within the statistical uncertainty, but the statistical error of the– 18 – .00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 a [GeV − ] B s h / B l h ( m ph y s B s , m π = m ph y s π ) m π [GeV ] B s h / B l h ( m ph y s B s , a = ) Figure 12 . Continuum limit and pion mass dependence obtained from the chosen global fit for B B s /B B . indirect determination is reduced by roughly 30%. This occurs due to stronger cancellationsof statistical errors in the individual ratios f hs /f hl and B sh /B hl as opposed to the directconstruction of ξ where some of these correlations appear to be washed out. We willtherefore take the indirect determination as our preferred value.The results of the direct determination are presented in Table 4 and Figures 13 and 14.Figure 13 again shows the data points shifted to the physical pion mass and zero latticespacing. The heavy mass behaviour displayed is very similar to the case of the ratio ofdecay constants, with the data being well described by a linear term in the inverse heavymeson mass that is chosen (here m D s -like mesons). The two panels in Figure 14 showthe projections of the data points shown in Figure 13 to the physical B s mass. The sameobservations as in the case of the ratio of decay constants hold true for this case. However,the approach to the continuum is slightly steeper with discretisation effects on the coarsestensemble being ∼ (compare C CL in Table 4). We will now estimate the various systematic errors. These are tabulated in Table 5. Westart by considering the systematic errors due to the global fitting procedure. To this end,we consider variations in the fit ansatz. First we compare the results obtained from theglobal fit for different pion mass cuts. The results for all of these fits are listed in Tables 8,9, 10 and 11. We note some of the fits including the heavier pion mass ensembles displaya poorer fit quality (some with unacceptable values of χ / d . o . f . ). This occurs in particularfor m max π = 450 MeV and m max π = 430 MeV . For this reason we only consider the pion masscuts of m max π = 400 , , and
250 MeV (the red, blue, green and yellow data points inFigures 15 - 18, respectively) for our systematic error estimation. These cuts successivelyeliminate ensembles from the fit. For the pion mass cut of
250 MeV the fit in the form of(4.1) becomes insufficiently constrained. Noting that the C s parameter is compatible with– 19 – .1 0.2 0.3 0.4 0.5 0.6 m − D s [GeV − ] ξ All datapoints shifted to m π = m phys π and a = 0 m phys B s m phys D s C0C1M0M1F1M ξ Figure 13 . Global fit result for ξ . Similar to Figure 9, all data points are shifted to the physicalpion mass and zero lattice spacing. a [GeV − ] ξ ( m ph y s B s , m π = m ph y s π ) m π [GeV ] ξ ( m ph y s B s , a = ) Figure 14 . Continuum limit and pion mass dependence obtained from the chosen global fit for ξ . All data points are shifted to the physical B s meson mass and the physical pion mass (left) orvanishing lattice spacing (right). zero (cf. Table 4) and C s as well as ∆ m s are small numbers, we drop this term from thefits with the
250 MeV pion mass cut.Furthermore we consider including (“inc”) all data points or excluding the heaviestdata point on all coarse (“exc h/C”) or all (“exc h/all”) ensembles. This choice is justified,since we expect the heaviest mass points and the coarse ensembles to be most stronglyafflicted by discretisation effects. For the case of f D s /f D , for which we have data bracketingthe physical value on all ensembles, we also consider a fit where a physical mass cut of– 20 – .
76 GeV ≤ m PDG D s ≤ .
15 GeV is applied to the heavy-strange meson mass (labelled as“phys mh cut”). The left hand panels of Figures 15, 16, 17 and 18 list the outcomes of thesevariations for f D s /f D , f B s /f B , B B s /B B d and ξ , respectively. We conservatively assign asystematic error due to the chiral-continuum limit part of the fit as the maximum spreadof the central values from the chosen fit. This is labelled “fit chiral-CL” in Table 5. All ofthese variations remain within the quoted statistical error. We use the pion mass cut of
350 MeV as our central value, since this better constrains the coefficients (therefore fullyexploiting the third lattice spacing) whilst giving an excellent goodness-of-fit. This choicemight change if additional ensembles at light quark masses became available. One desirablechoice for such an ensemble would be a physical pion mass ensemble at a − ∼ . (F0).Recalling that we have two ways to determine ξ , which have different statistical proper-ties, we choose the indirect determination of ξ as our central value, as discussed in equation(4.7). For this determination, we take both, the ratio of decay constants and the ratio ofbag parameters from fits with the specified cuts. For comparison we also show the resultsof the direct determinations as open symbols in Figure 18.To assess the systematic errors due to the heavy mass dependence we compare settingthe heavy quark mass via a heavy-light ( D and B ), heavy-strange ( D s and B s ) or heavy-heavy ( η c and η b ) pseudoscalar meson mass. These are respectively shown as diamonds,circles and squares in Figures 15-18. The physical masses we use are given by the PDGaverages given in Ref. [16] m D = 1 . m ± D s = 1 . m η c = 2 . m B = 5 . m B s = 5 . m η b = 9 . . (4.8)We note that the η c contains a small quark-disconnected contribution which we neglectin our simulation. In addition to the smallness of this contribution, its effect is furthersuppressed due to the very benign behaviour with the heavy quark mass, displayed in theSU(3) breaking ratios under consideration. Since in the base fit we choose to fix the heavyquark mass with the heavy-strange meson mass, this small quark-disconnected contributiondoes not affect the final result. For f D s /f D , we expand around m expand H = m PDG H . We alsocompare fits where we additionally include a term C h (cid:0) /m H − /m PDG H (cid:1) in (4.1). Wenote that we cannot resolve this additional coefficient from zero, since the data does notdisplay significant curvature. The variations of the results are shown in the right panels ofFigure 15. We note that for f D s /f D , there is no significant variation due to these choices,due to the presence of precise data in and around the charm region.For quantities involving a b quark, we require to extrapolate from the region where wehave data to the B ( s ) or η b mass. Motivated by heavy quark effective theory (HQET) [96]we take the expansion point to be the static limit, i.e. /m expand H = 0 . We again test thestability of our fit result by setting the heavy quark mass using the PDG values B , B s and η b as well as systematically applying cuts to the data that enters the fit. For each choiceto set the heavy mass, we carry out the following variations:1. baseline fit (inc) For a linear fit, this amounts simply to a re-definition of the constant f (0 , m phys π , m expand H ) . – 21 – n c e x c h / C e x c h / a ll p h y s m h c u t i n c e x c h / C e x c h / a ll p h y s m h c u t i n c e x c h / C e x c h / a ll p h y s m h c u t i n c e x c h / C e x c h / a ll p h y s m h c u t f D s / f D m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV chiral-continuum stability i n c q u a d i n c q u a d i n c q u a d m H = D m H = D s m H = η c Heavy mass stability
Figure 15 . Stability of fit results for f D s /f D . The left plot shows variations of the pion mass cuts(separated by dotted vertical lines, from left to right) between m max π = 400 , , and
250 MeV .The right hand panel compares different approaches for the heavy quark interpolation. More detailcan be found in the text.
2. excluding the heaviest mass point of each coarse ensemble (exc h/C)3. excluding the heaviest mass point of all ensembles (exc h/all)4. excluding the lightest mass point of each coarse ensemble (exc l/C)5. excluding the lightest mass point of all ensembles (exc l/all)We again take the full spread of the central values as our systematic error for the linearpart of the heavy quark extrapolation, which is slightly larger than one statistical standarddeviation. The right hand panels of Figures 16, 17 and 18 show the corresponding resultsfor f B s /f B , B B s /B B d and ξ , respectively.We estimate neglected higher order terms to be of the form O| static (cid:34) α Λ m B s + β (cid:18) Λ m B s (cid:19) (cid:35) = O| static (cid:20) α Λ m B s (cid:18) βα Λ m B s (cid:19)(cid:21) (4.9)for some scale Λ . Assuming that the coefficients of this expansion are of similar order,we approximate the missing higher order contributions to be the difference between ourbaseline fit result and the observable evaluated at the physical heavy meson mass. At thephysical pion mass, zero lattice spacing and β ≡ , equation (4.9) reproduces equation(4.1) if we identify α ≡ C h / (Λ C ) . Taking Λ = 500 MeV and conservatively allowingfor a large coefficient (i.e. β/α = 5 ), we can substitute C H , C from the fit. We obtain– 22 – n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll f B s / f B m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV chiral-continuum stability i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll m H = B m H = B s m H = η b Heavy mass stability
Figure 16 . Stability of fit results for f B s /f B . The left plot shows variations of the pion mass cuts(separated by dotted vertical lines, from left to right) between m max π = 400 , , and
250 MeV .The right plot shows variations of the fit for different choices of the heavy pseudoscalar meson massused to set the bottom quark mass (triangles for H = B , circles for H = B s , squares for H = η b )as well as different different cuts to the data which are described in more detail in the text. i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll B B s / B B d m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV chiral-continuum stability i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll m H = B m H = B s m H = η b Heavy mass stability
Figure 17 . Variations of the fit result for the ratio of bag parameters B B s /B B d , analogous toFigure 16 ∆ f B s /f B = 0 . and ∆ ξ = 0 . , which we assign as a (sub-leading) systematic errorfor higher order extrapolation terms (labelled “H.O. heavy” in Table 5).– 23 – n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll i n c e x c h / C e x c h / a ll ξ m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV m max π =400MeV m max π =350MeV m max π =330MeV m max π =250MeV chiral-continuum stability i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll i n c e x c h / C e x c h / a ll e x c l / C e x c l / a ll m H = B m H = B s m H = η b Heavy mass stability
Figure 18 . Stability of fit results for ξ analogous to Figures 16 and 17. The closed data pointsdisplay the indirect determinations of ξ , whilst the open symbols show the direct determinations(cf. equation (4.7)). Our strategy to asses the systematic errors due to strong isospin breaking and toestimate higher order discretisation errors which are not included in our fit form closelyfollows Ref. [49]. Since our simulations are done with degenerate light quark masses ( m u = m d = m l ), we need to account for the missing strong isospin corrections in our error budget.We estimate this, by considering the difference between using the charged or neutral pion, D and B meson masses. The corrections due to the pion mass are given by ∼ C χ (cid:0) m π ± − m π (cid:1) .Using C χ from Table 4, this amounts to . for the ratio of decay constants and to . for ξ . Similarly, using the slope C h with the inverse D ( B ) meson mass and applying itto the difference between the charged and the neutral one gives an error of × − for f D s /f D , × − for f B s /f B and × − for ξ . We add the relevant terms in quadratureand list them in Table 5 as “ m u (cid:54) = m d ”. Assuming O ( a ) discretisation effects to be present,would lead to terms of the form C CL a + D CL a = C CL a (cid:18) D CL C CL a (cid:19) (4.10)in the fit ansatz. Since the leading order discretisation effects are accounted for in our fit, itonly remains to quantify the correction to them. Assuming that discretisation effects growas a/ Λ with Λ = 500 MeV , i.e. D CL /C CL = (0 . , we can simply substitute the valuesfor a and C CL (compare Table 4) to obtain the corrections such a term would cause. Fromthis, we find the O ( a ) corrections on the finest (coarsest) ensemble to be 0.0001 (0.0009)for the ratios of decay constants and 0.0003 (0.0021) for ξ . We conservatively take the erroron the estimated corrections on the coarse ensembles and list these errors as “H.O. disc.” inTable 5. Finally, for the finite size effects, we evaluate the one-loop finite-volume HM χ PT– 24 – D s /f D f B s /f B ξ B B s /B B d absolute relative absolute relative absolute relative absolute relativecentral 1.1740 1.1949 1.1939 0.9984stat 0.0051 0.43% 0.0060 0.50% 0.0067 0.56% 0.0045 0.45%fit chiral-CL +0 . − . . − . % +0 . − . . − . % +0 . − . . − . % +0 . − . . − . %fit heavy mass +0 . − . . − . % +0 . − . . − . % +0 . − . . − . % +0 . − . . − . %H.O. heavy . . % . . % . . % . . %H.O. disc. . . % . . % . . % . . % m u (cid:54) = m d . . % . . % . . % . . %finite size . . % . . % . . % . . %total systematic +0 . − . . − . % +0 . − . . − . % +0 . − . . − . % +0 . − . . − . %total sys+stat +0 . − . . − . % +0 . − . . − . % +0 . − . . − . % +0 . − . . − . % Table 5 . Summary of central values, statistical errors and all sources of systematic errors. Thetotal systematic is found by adding the respective errors in quadrature. For ease we separately listthe absolute and the relative errors, where the latter are presented in %. expressions given in Ref. [95] for our choice of pion masses and volumes. For a reasonablechoice of parameters we find the maximal deviation to be less than 0.18%, which we assignas the finite size error as listed in Table 5.
The results of our analysis are summarised in Table 5. We will now compare our valueswith those published in the literature.
Figure 19 shows a comparison of our results with the literature for the ratios f D s /f D (left)and f B s /f B (right). The result obtained in this work is shown as the magenta star and thevertical magenta band.For f D s /f D we find excellent agreement with our previous result [49] which was obtainedon the same ensembles but with a different choice of discretisation for the charm quarks.There is also no significant tension with the published literature [49–51, 53–60] or theaveraged values presented by FLAG [47]. We note that other than in this work, thereare still only very few computations including data directly calculated at the physical pionmass [49–51].For the ratio f B s /f B there are a variety of different results using different methods inthe literature [50, 57, 59–66, 68, 97]. We note that some of the results in Refs. [62, 66]have been carried out on a subset of the ensembles (C1/2 and M1/2/3) used in this study,however using different choices for the heavy quark discretisation. Besides the use of a fullyrelativistic formulation, our results improve upon these by the inclusion of physical pionmass ensembles and a third lattice spacing, leading to a more than three-fold reduction inerror. – 25 – .10 1.15 1.20 1.25 1.30 1.35FNAL/MILC 17 +FLAG19 N f =2+1+1 FNAL/MILC 14ETM 14
THIS WORK
RBC/UKQCD 17FLAG19 N f =2+1 HPQCD 12FNAL/MILC 11HPQCD 07FNAL/MILC 05FLAG19 N f =2 ETM 13ETM 11ETM 09 f D s /f D N f =2+1+1 HPQCD 13
THIS WORK
FLAG19 N f =2+1 RBC/UKQCD 14ARBC/UKQCD 14HPQCD 12FNAL/MILC 11RBC/UKQCD 10FLAG19 N f =2 ALPHA 14ETM 13ETM 11 f B s /f B Figure 19 . Comparison of our result (magenta star and band) with results from the literature forthe (isospin symmetric) ratios of decay constants f D s /f D (left) and f B s /f B (right). The squares,circles and diamonds correspond to N f = 2+1+1 , N f = 2+1 and N f = 2 flavour calculations. Theblack triangles show the averages published in the 2019 FLAG report [47] for the given number of seaquark flavours with the results entering this average shown below these black triangles. Referencesfor all the displayed data points are given in the text. We note that in FNAL/MILC 17 [50] and RBC/UKQCD 14A [62], no isospin symmetricresult for the ratio f B s /f B is quoted. For the comparison in Figure 19 we instead take thecorrelated average of the results quoted for f B s /f B ± and f B s /f B which are plotted as thered square [50] and the blue circle [62] in the right panel of Figure 19. Prior to this work,only two fully relativistic fermion actions have been employed as heavy quark discretisation,namely the HISQ action in Ref. [50] and the twisted mass action in Refs. [57, 59, 63, 64].Other than the result presented here, only one computation [50] with physical pion massesis currently available for f B s /f B . Figure 20 summarises the current status of the literature for the mixing parameter ξ andthe ratios of bag parameters B B s /B B d [52, 59, 66–70]. We note that compared to theratio of decay constants, there are far fewer computations for these observables. This is thefirst calculation for ξ and B B s /B B d which includes ensembles with physical pion masses. The only other result [59] that employs a fully relativistic set-up is presented in the N f = 2 calculation using twisted mass fermions. We obtain a similar error with a somewhat smallercentral value for the quantity ξ compared to Ref. [67]. Ref. [67] used the PDG [98] averageof the decay constants f B s /f B to obtain the ratio of bag parameters, resulting in a largererror for this quantity. For the ratio B B s /B B d , our result is two times more precise thanthe previously most precise lattice QCD value obtained by [59]. Refs. [52, 70] appeared after version 1 of this paper was posted. Ref. [52] appeared after the first version of this paper. – 26 – .10 1.15 1.20 1.25 1.30 1.35HPQCD 19
THIS WORK
FNAL/MILC 16FLAG19 N f =2+1 RBC/UKQCD 14FNAL/MILC 12HPQCD 09FLAG19 N f =2 ETM 13King et al 19 ξ THIS WORK
FNAL/MILC 16 [PDG f B s /f B used]FLAG19 N f =2+1 RBC/UKQCD 14FNAL/MILC 12HPQCD 09FLAG19 N f =2 ETM 13King et al 19 B B s /B B d Figure 20 . Comparison of our result (magenta star and band) with results from the literaturefor the SU (3) breaking ratios ξ (left) and the ratio of bag parameters B B s /B B (right) analogousto Figure 19. Closed symbols refer to lattice QCD computations, whilst the open symbol shows arecent QCD sum-rule result. References for the displayed data points are given in the text. We stress that our systematic errors differ from most other lattice computation sincedue to the use of a chiral action the decay constants and bag parameter renormalise multi-plicatively and therefore cancel in the considered ratios. As a consequence, our computationis free from lattice renormalisation uncertainties.
Having obtained SM predictions for the ratios of decay constants f D s /f D and ξ , we arenow in a position to combine these with experimentally observed quantities to obtain ratiosof CKM matrix elements. Treating the experimental averages from the first two lines of(1.2) as uncorrelated we obtain | V cd /V cs | = 0 . × f D s /f D . If we choose the averageincluding the new BESIII results [15] (see equation (1.3)), this changes to | V cd /V cs | =0 . × f D s /f D . Similarly, combining the experimental averages ∆ m d and ∆ m s (see(1.5)) with the PDG values for m B s and m B (see (4.8)) yields | V td /V ts | = 0 . × ξ .Were we to consider the decay rates of the individual charged decays or the individualmass differences ∆ m q for q = s, d , we would have to correct for electromagnetic effectsbefore extracting V cd , V cs , V ub , V td or V ts from the pure QCD entities f D ( s ) and f B ( s ) (cid:113) ˆ B B ( s ) .However, given that D + s and D + and respectively B s and B are identical when replacingthe s by the d quark, and both of these have the same charge, we assume that these effectsare highly suppressed in the ratios we consider. At the time version 1 of this work was posted, [15] only existed as a pre-print. – 27 –nserting the lattice results, propagating the errors and assuming that there are no newphysics contributions in the experimental measurements leads to the ratios | V cd /V cs | = 0 . exp (cid:0) +12 − (cid:1) lat , | V td /V ts | = 0 . exp (cid:0) +162 − (cid:1) lat . (6.1)where we included the new BESIII result [15] in our determination. Had we not includedthe new BESIII result we would obtain | V cd /V cs | = 0 . exp (cid:0) +12 − (cid:1) lat . These resultsare slightly lower than those currently reported (compare equation (1.8)). We anticipate,that the global fit values will change as a result of this work.The error on the ratio | V cd /V cs | is currently dominated by the experimental uncertainty.For | V td /V ts | , the situation is reversed and the theoretical uncertainty dominates the error.This work improves on this by providing a first computation based on chiral fermions withphysical pion mass ensembles. We have, for the first time, predicted the SU (3) breaking ratios B B s /B B d and ξ in acalculation based on ensembles with physical pion masses, therefore eliminating any largechiral extrapolations. Furthermore, we present for the first time, results for SU (3) breakingratios in the B ( s ) mesons systems obtained from an all-domain wall calculation. We haveillustrated that such ratios display a very benign behaviour from below the charm mass to ∼ half the bottom quark mass and that lattice artefacts in our choice of discretisation aresmall for these observables. We found that nearly all of the SU (3) breaking effects observedin the difference of ξ from unity, arise from the ratio of decay constants f B s /f B . This yieldsthe to-date most precise computation of the ratio of CKM matrix elements | V td /V ts | .Looking forwards, we anticipate the generation of a third ensemble with physical pionmasses, at the same lattice spacing as our currently finest ensemble (F1M). This will addressour leading systematic error, namely the chiral-continuum limit and heavy quark massextrapolation. It will allow to lower the pion mass cut to ∼
250 MeV whilst still constrainingthe continuum limit with three lattice spacings.Based on the presented dataset we are also working on the mixed action renormalisa-tion, to deduce the decay constants f D ( s ) , f B ( s ) and the standard model bag parameters B B ( s ) . We will also address the full set of beyond the SM four-quark operators for B ( s ) -mixing and the short distance contribution to D − ¯ D mixing. This will be analogous to thecomputation presented in [99] for the Kaon sector. Acknowledgments
The authors would like to thank the members of the RBC and UKQCD collaborations formany valuable discussions with special thanks to Felix Erben. The authors further thankthe CKMfitter and the UTfit groups for providing their current estimates for | V cd /V cs | and | V td /V ts | Ensemble properties of F1M
We noticed that the Shamir action approximation was used for the ensemble generationof the original F1 ensemble (presented in Ref. [49]), whilst valence measurements werecarried out with the Möbius action approximation. To avoid any ambiguity, this ensem-ble will henceforth be referred to as F1S. At fixed ensemble parameters this results in alarger residual chiral symmetry breaking (residual mass) compared the the Möbius actionapproximation which in turn results in slightly heavier pion masses. Previously, the valencemeasurements were carried out with the Möbius action approximation on the F1S ensemblewhich is a partial quenching effect in the quark masses. The pion mass and residual massthat were determined in Ref. [49] were am π = 0 . and am l res = 0 . ,respectively. The scale setting fit gave a − = 2 . and am phys s = 0 . .To resolve this, we performed valence measurements with the Shamir action approx-imation on the original ensemble and additionally generated an ensemble with identicalparameters but using the Möbius approximation to the sign function (referred to as F1M).The results for the unitary pion and residual masses are F1S : am π = 0 . , am l res = 0 . , F1M : am π = 0 . , am l res = 0 . . (A.1)By repeating the scale setting analysis of Refs. [49, 73] under inclusion of both of theseensembles we obtained determinations of the inverse lattice spacing and the physical strangequark mass F1S : a − = 2 . , am phys s = 0 . , F1M : a − = 2 . , am phys s = 0 . . (A.2)We note that the shift in lattice spacing at fixed value of β when changing from Shamirto Möbius is in agreement with previous work [73]. Combining the above we obtain pionmasses of
232 MeV (F1M) and
267 MeV (F1S). In this paper only the F1M ensemble isused.
B Results of correlation function fits
In this section we list the relevant results of the correlation function fits for the two pointfunctions (Table 6) and the three point functions (Table 7). The fit strategy is discussed inmore detail in the text. – 30 – ame am h range am hl af hl χ / dof range am hs af hs χ / dof am hh f sh /f lh C0 0.51 [ 7,17) 0.90759(67) 0.14135(57) 0.041 [ 7,24) 0.96797(13) 0.16485(13) 0.033 1.413572(53) 1.1663(47)C0 0.57 [ 7,17) 0.97440(79) 0.14208(69) 0.053 [ 7,24) 1.03298(15) 0.16599(16) 0.058 1.525400(50) 1.1682(56)C0 0.63 [ 7,17) 1.03899(94) 0.14172(84) 0.069 [ 8,24) 1.09592(18) 0.16573(21) 0.051 1.632449(49) 1.1694(69)C0 0.69 [ 7,17) 1.1006(12) 0.1397(10) 0.094 [ 8,24) 1.15600(21) 0.16337(27) 0.098 1.732940(49) 1.1692(88)C1 0.50 [ 6,17) 0.90826(78) 0.14842(64) 0.029 [ 6,24) 0.95316(40) 0.16324(35) 0.029 1.39276(20) 1.0998(38)C1 0.58 [ 6,17) 0.99656(95) 0.14935(84) 0.044 [ 6,24) 1.03965(44) 0.16454(43) 0.037 1.54135(19) 1.1018(53)C1 0.64 [ 6,17) 1.0601(11) 0.1486(11) 0.065 [ 7,24) 1.10187(52) 0.16372(58) 0.034 1.64697(18) 1.1018(75)C1 0.69 [ 6,17) 1.1106(14) 0.1465(13) 0.094 [ 7,24) 1.15137(59) 0.16136(70) 0.047 1.72972(18) 1.1012(98)C2 0.51 [ 7,18) 0.93015(72) 0.15441(64) 0.016 [ 7,20) 0.96602(43) 0.16537(39) 0.012 1.41243(20) 1.0710(27)C2 0.59 [ 7,18) 1.01801(88) 0.15548(85) 0.021 [ 8,20) 1.05216(56) 0.16646(60) 0.013 1.55999(18) 1.0706(37)C2 0.64 [ 8,18) 1.0708(13) 0.1549(15) 0.022 [ 8,20) 1.10400(64) 0.16581(74) 0.017 1.64761(18) 1.0705(70)C2 0.68 [ 8,18) 1.1115(15) 0.1534(18) 0.028 [ 9,20) 1.14377(85) 0.1639(11) 0.013 1.71429(18) 1.0684(94)M0 0.41 [ 9,22) 0.74184(60) 0.10269(52) 0.056 [10,24) 0.78458(11) 0.120553(99) 0.018 1.170921(44) 1.1739(57)M0 0.50 [ 9,22) 0.84158(80) 0.10247(72) 0.061 [10,24) 0.88275(12) 0.12096(12) 0.021 1.344482(41) 1.1804(80)M0 0.59 [ 9,22) 0.9350(11) 0.10017(97) 0.070 [10,24) 0.97514(14) 0.11896(15) 0.034 1.507094(39) 1.188(11)M0 0.68 [ 9,22) 1.0198(14) 0.0948(13) 0.090 [10,24) 1.05931(18) 0.11338(19) 0.085 1.654201(37) 1.195(15)M1 0.41 [ 8,22) 0.74931(91) 0.10867(65) 0.030 [ 8,24) 0.78445(36) 0.12142(27) 0.016 1.17063(15) 1.1174(54)M1 0.50 [ 8,22) 0.8489(11) 0.10873(82) 0.053 [ 8,24) 0.88259(39) 0.12183(31) 0.028 1.34397(14) 1.1206(70)M1 0.59 [ 8,22) 0.9423(13) 0.1067(10) 0.086 [10,24) 0.97496(47) 0.11990(41) 0.030 1.50613(13) 1.1241(94)M1 0.68 [ 8,22) 1.0272(17) 0.1014(13) 0.124 [11,24) 1.05894(59) 0.11425(59) 0.041 1.65226(13) 1.126(13)M2 0.41 [ 9,22) 0.7531(10) 0.11027(96) 0.031 [10,24) 0.78463(48) 0.12139(43) 0.023 1.17078(19) 1.1009(77)M2 0.50 [ 9,22) 0.8524(14) 0.1102(14) 0.041 [10,24) 0.88280(56) 0.12179(54) 0.031 1.34407(17) 1.105(11)M2 0.59 [ 9,22) 0.9456(18) 0.1079(19) 0.050 [11,24) 0.97500(85) 0.1196(10) 0.035 1.50625(16) 1.109(17)M2 0.68 [ 9,22) 1.0299(25) 0.1021(26) 0.061 [11,24) 1.0588(11) 0.1137(13) 0.046 1.65232(15) 1.114(25)M3 0.41 [ 9,22) 0.75830(78) 0.11308(71) 0.016 [ 9,24) 0.78609(41) 0.12240(33) 0.017 1.17153(17) 1.0824(50)M3 0.50 [ 9,22) 0.85767(100) 0.1131(10) 0.017 [ 9,24) 0.88421(48) 0.12270(45) 0.020 1.34477(16) 1.0848(73)M3 0.59 [ 9,22) 0.9509(13) 0.1109(14) 0.022 [10,24) 0.97626(74) 0.12019(92) 0.022 1.50691(15) 1.084(10)M3 0.68 [ 9,22) 1.0357(17) 0.1053(19) 0.032 [10,24) 1.0599(10) 0.1139(13) 0.029 1.65305(15) 1.082(15)F1M 0.32 [ 9,24) 0.61852(51) 0.09136(39) 0.054 [10,24) 0.65291(17) 0.10409(16) 0.025 0.967402(95) 1.1394(51)F1M 0.41 [ 9,24) 0.72222(64) 0.09205(52) 0.067 [10,24) 0.75503(19) 0.10532(19) 0.034 1.149482(87) 1.1442(68)F1M 0.50 [ 9,24) 0.81987(79) 0.09117(64) 0.075 [10,24) 0.85160(22) 0.10468(24) 0.046 1.321898(82) 1.1482(87)F1M 0.59 [ 9,24) 0.91076(96) 0.08835(77) 0.090 [10,24) 0.94167(27) 0.10172(29) 0.059 1.482450(77) 1.151(11)F1M 0.68 [ 9,24) 0.9922(12) 0.08272(87) 0.141 [10,24) 1.02245(33) 0.09534(35) 0.090 1.625929(74) 1.153(13)
Table 6 . Results of the correlation function fits. – 31 – ame am h ∆ T t min B hl χ / dof ∆ T t min B hs χ / dof B sh /B lh C0 0.51 20 6 0.7875(32) 0.059 32 9 0.81071(31) 0.026 1.0295(42)C0 0.57 20 6 0.7971(39) 0.073 32 9 0.82014(39) 0.033 1.0289(50)C0 0.63 20 5 0.8066(44) 0.085 32 7 0.82935(46) 0.066 1.0282(56)C0 0.69 20 4 0.8161(50) 0.091 32 6 0.83880(58) 0.079 1.0278(63)C1 0.50 22 5 0.7916(25) 0.204 26 8 0.80895(74) 0.198 1.0219(32)C1 0.58 20 6 0.8054(32) 0.343 26 7 0.82186(90) 0.220 1.0204(42)C1 0.64 20 6 0.8144(39) 0.285 26 6 0.8312(11) 0.147 1.0206(50)C1 0.69 20 5 0.8228(43) 0.283 26 5 0.8392(13) 0.193 1.0200(56)C2 0.51 22 7 0.7978(19) 0.103 24 8 0.81038(85) 0.028 1.0158(23)C2 0.59 22 6 0.8107(23) 0.121 24 7 0.8229(10) 0.046 1.0150(28)C2 0.64 22 5 0.8185(26) 0.115 24 7 0.8304(12) 0.025 1.0145(32)C2 0.68 22 5 0.8245(30) 0.121 24 7 0.8364(14) 0.025 1.0145(37)M0 0.41 22 8 0.7905(21) 0.146 32 12 0.80410(30) 0.276 1.0172(27)M0 0.50 22 8 0.8091(29) 0.074 32 12 0.82039(39) 0.173 1.0140(36)M0 0.59 22 6 0.8275(36) 0.098 32 11 0.83506(49) 0.130 1.0091(43)M0 0.68 22 5 0.8456(46) 0.017 32 9 0.84962(61) 0.044 1.0047(54)M1 0.41 20 7 0.7932(22) 0.147 28 9 0.80395(85) 0.053 1.0135(30)M1 0.50 18 7 0.8105(26) 0.198 28 9 0.8203(11) 0.050 1.0121(33)M1 0.59 18 7 0.8246(35) 0.154 28 8 0.8350(14) 0.128 1.0126(44)M1 0.68 18 7 0.8377(47) 0.122 28 8 0.8496(20) 0.179 1.0143(58)M2 0.41 20 8 0.7914(21) 0.272 28 8 0.80290(87) 0.429 1.0145(28)M2 0.50 18 7 0.8097(24) 0.144 28 8 0.8190(11) 0.297 1.0115(30)M2 0.59 18 7 0.8241(30) 0.087 28 8 0.8335(14) 0.178 1.0114(39)M2 0.68 18 6 0.8387(35) 0.146 28 8 0.8480(19) 0.082 1.0112(48)M3 0.41 20 8 0.7940(18) 0.093 22 9 0.80460(87) 0.000 1.0134(23)M3 0.50 20 8 0.8110(22) 0.031 22 7 0.82136(96) 0.073 1.0128(29)M3 0.59 18 7 0.8253(24) 0.033 22 7 0.8357(12) 0.095 1.0126(33)M3 0.68 18 7 0.8398(30) 0.001 22 6 0.8498(15) 0.310 1.0120(41)F1M 0.32 28 12 0.7777(20) 0.023 34 15 0.78754(40) 0.010 1.0127(25)F1M 0.41 28 11 0.7989(24) 0.016 34 15 0.80743(48) 0.010 1.0107(30)F1M 0.50 28 10 0.8164(30) 0.009 34 15 0.82342(61) 0.009 1.0085(36)F1M 0.59 28 10 0.8327(38) 0.063 34 15 0.83771(82) 0.006 1.0060(46)F1M 0.68 28 8 0.8493(40) 0.190 34 14 0.8519(10) 0.006 1.0031(48)
Table 7 . Results of the bag parameter fits. – 32 –
Results of the global fit
Here we list the results of the global fits for the SU (3) breaking ratios f D s /f D (Table 8), f B s /f B (Table 9), B B s /B B d (Table 10) and ξ (Table 11). cut name m max π / MeV m H f D s /f D C CL / GeV C χ / GeV − C H / GeV C s d.o.f χ / dof p inc 450 D s D s D s D s D s D s D s D s D s D s D s D s D s D s D s D s η c η c η c η c D D D D D s D s D s D s D s D s D s D s Table 8 . Results of the global fit for f D s /f D . – 33 – ut name m max π / MeV m H f B s /f B C CL / GeV C χ / GeV − C H / GeV C s d.o.f χ / dof p inc 450 B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s η b η b η b η b η b B B B B B B s B s B s B s B s B s B s B s B s B s Table 9 . Results of the global fit for f B s /f B . – 34 – ut name m max π / MeV m H B B s /B B d C CL / GeV C χ / GeV − C H / GeV C s d.o.f χ / dof p inc 450 B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s η b η b η b η b η b B B B B B B s B s B s B s B s B s B s B s B s B s Table 10 . Results of the global fit for B B s /B B d . – 35 – ut name m max π / MeV m H ξ C CL / GeV C χ / GeV − C H / GeV C s d.o.f χ / dof p inc 450 B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s B s η b η b η b η b η b B B B B B B s B s B s B s B s B s B s B s B s B s Table 11 . Results of the global fit for ξ . – 36 – .1 0.2 0.3 0.4 0.5 0.6 m − D s [GeV − ] ξ m phys B s m phys D s C0C1M0M1F1M ξ m − D s [GeV − ] d a t a / r e s u l t o ff i t f o r ξ All datapoints shifted to m π = m phys π and a =0 m phys B s m phys D s C0C1M0M1F1M
Figure 21 . Analogous plots to Figure 8 for ξ . m − D s [GeV − ] B s h / B l h m phys B s m phys D s C0C1M0M1F1M B B s /B B d m − D s [GeV − ] d a t a / r e s u l t o ff i t f o r B s h / B l h All datapoints shifted to m π = m phys π and a =0 m phys B s m phys D s C0C1M0M1F1M
Figure 22 . Analogous plots to Figure 8 for B B s /B B d . a [GeV − ] f s h / f l h ( m ph y s B s , m π = m ph y s π ) m π [GeV ] f s h / f l h ( m ph y s B s , a = ) Figure 23 . The analogous plots to Figure 10, displaying the chiral and continuum limit behaviourfor the ratio f B s /f B . – 37 – eferences [1] N. Cabibbo, Unitary Symmetry and Leptonic Decays , Phys. Rev. Lett. (1963) 531–533.[2] M. Kobayashi and T. Maskawa, CP Violation in the Renormalizable Theory of WeakInteraction , Prog. Theor. Phys. (1973) 652–657.[3] CLEO collaboration, M. Artuso et al.,
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