|Vcb| and γ from B-mixing

In this addendum to “Bs mixing observables and |Vtd/Vts| from sum rules” [1] we study the impact of the recent improvements in the theoretical precision of B meson mixing onto CKM unitarity fits. Our key results are the most precise determination of the angle γ = (63.4± 0.9)◦ in the unitarity triangle and a new value for the CKM element |Vcb| = (41.6± 0.7) · 10−3.

In our recent works we have determined the hadronic matrix elements for B-mixing with HQET sum rules [1,2] (cf. also [3]) and combined the results with lattice determinations [4][5][6] to obtain updated the predictions [7] for the mass differences ∆M d and ∆M s . Here we use the weighted averages for the matrix elements presented in [7] to determine the following combinations of CKM elements from the experimental measurements of the mass differences, updating the results in [1]. Motivated by the well-known discrepancy between the direct determination of the CKM elements V cb and V ub from semi-leptonic b-hadron decays (see [8] for some recent discussion) and the prospect of a measurement of the CKM angle γ with an uncertainty of 1.5 • by 2023 from the LHCb collaboration [9] we study the impact of these values on CKM unitarity fits.
The effects of B-mixing on CKM unitarity fits can be illustrated with the unitarity triangle shown in Figure 1. The combinations of CKM elements (1) and (2) we determined from ∆M s and ∆M d appear in the lengths of the two non-trivial sides of the triangle if we expand to leading order in the Wolfenstein parameter λ = |V us |. Up to reflection with respect to theρ axis the apex of the triangle is exactly fixed with the addition of |V ub | and the precisely measured |V us |. Here, we use this information to determine the angle γ. Furthermore, we can extract |V cb | = |V ts V tb | × [1 + O(λ 2 )] with a precision that is competitive with direct measurements.
We perform a minimalistic CKM unitarity fit, first taking only the direct measurements of the CKM element |V us | = 0.2243 ± 0.0005 [12] and the mass differences ∆M d and ∆M s into account. This strongly constrains the length of the side R t . Figure 2 shows our results in the |V ub | − γ and |V cb | − γ planes where the shaded blue regions indicate the parameter space satisfying the inputs within one and two standard deviations. For values of γ larger than about 65 • the unitarity triangle does not close within the two-sigma region 1 . This 1 Similar observations were made in e.g. [13].    Figure 2: Our results for a minimalistic CKM unitarity fit based on direct measurements of |V us | and the mass differences ∆M d and ∆M s are given as shaded blue regions. Including the exclusive or inclusive measurements of |V ub | yields the orange and red regions, respectively. See text for details. behaviour is illustrated in Figure 3 and allows us to derive a stringent upper limit on γ.
At the level of five standard deviations we obtain which is indicated by the horizontal dashed line in Figure 2 and quite a bit smaller than the direct measurements of γ [10-12, 14, 15] summarised there. We note that the indirect determinations of γ from the CKMfitter [14] and UTfit [15] collaborations also yield smaller values than direct measurements, albeit larger ones than our analysis. We used the CKMlive [16] tool to perform the standard CKMfitter analysis without direct measurements of γ or the mass differences and obtained the result which is in good agreement with the direct measurements of γ and has a significantly larger uncertainty than the indirect fit results. This demonstrates that the smaller indirect values in the CKMfitter and UTfit studies are solely driven by ∆M s and ∆M d and implies that the confrontation of the planned improvements by LHCb and Belle II for the experimental determination of γ with constraints from the mass differences is a very promising indicator for BSM physics. Assuming the central value of the direct measurement remains the expected precision of ±1.5 • by 2023 will lead to a significant tension as indicated in Figure 3.
For smaller values of γ there are two intersections between the circle of length R t around the point (1,0) and the line crossing the origin at angle γ, leading to two degenerate perfect-fit results for |V ub | and |V cb | at a fixed value of γ. This degeneracy can be broken by constraining the length of the side R u by including the measurements of |V ub | in the fit. Due to the well-known puzzle about different results in exclusive and inclusive measurements (shown by the orange and red horizontal error bars in Figure 2, values from HFLAV [11]) this step would normally have to be taken with a grain of salt. However, due to a lucky numerical coincidence the values of |V ub | are very close to the region where the intersection point of the circles of length R t and R u lies at the maximal value of γ allowed by R t as shown by the orange and red ellipses in Figure 2 which are the results of the fit when the exclusive or inclusive measurements of |V ub | are included. Thus, the dependence of γ on the exact value of |V ub | is rather small. Indeed we find We take the envelope of both values as our final result to be sufficiently conservative about the uncertainty associated with the direct measurements of |V ub |. Eq. (7) represents the most precise determination of γ to date. The result is fairly insensitive to the input value for |V us |. If we inflate the error in |V us | by a factor of three we obtain γ = (63.4 ± 1.3) • and the upper five-sigma bound (3) becomes 68.9 • , which still poses a very stringent constraint. The effect of the exclusive or inclusive |V ub | measurements on the fit is also indicated in the |V cb | − γ plane by the orange and red ellipses, respectively. The difference in the extracted values of |V cb | is negligible and we again adopt the envelope as our final result |V cb | = (41.6 ± 0.7) · 10 −3 .
We also show the exclusive and inclusive HFLAV averages [11] and the result of a recent reanalysis BJvD [17] of exclusive determinations in Figure 2. Our result yields a competitive uncertainty and the one-sigma region overlaps with the inclusive and the BJvD results, while there is a 1.7 and 2.9 σ tension with respect to the B → D ν and B → D * ν values quoted by HFLAV. The result (8) remains unaffected when we inflate the |V us |-uncertainty by a factor of three.
In summary, we have performed a minimal χ 2 fit of the CKM parameters based on the mass differences in the B system and direct measurements of |V us | and |V ub |. We found competitive results for |V cb | which are in good agreement with the inclusive determinations and obtained the currently most precise value for the angle γ in the unitarity triangle. Our analysis clearly shows that more precise measurements of γ are a sensitive probe of new physics effects in the flavour sector. We are looking forward to updates of the complete CKM unitarity fits by the CKMfitter and UTfit collaborations where the latest theoretical developments [1][2][3][4][5][6][7] in B-mixing are taken into account.