Emerging \(\Delta M_{d}\)anomaly from treelevel determinations of \(V_{cb}\) and the angle \(\gamma \)
Abstract
We point out that the recently increased value of the angle \(\gamma \) in the unitarity triangle (UT), determined in treelevel decays to be \(\gamma =(74.0^{+5.0}_{5.8})^\circ \) by the LHCb collaboration, combined with the most recent value of \(V_{cb}\) implies an enhancement of \(\Delta M_{d}\) over the data in the ballpark of \(30\%\). This is larger by roughly a factor of two than the enhancement of \(\Delta M_{s}\) that is independent of \(\gamma \). This disparity of enhancements is problematic for models with constrained minimal flavour violation (CMFV) and also for \(U(2)^3\) models. In view of the prospects of measuring \(\gamma \) with the precision of \(\pm 1^\circ \) by Belle II and LHCb in the coming years, we propose to use the angles \(\gamma \) and \(\beta \) together with \(V_{cb}\) and \(V_{us}\) as the fundamental parameters of the CKM matrix until \(V_{ub}\) from treelevel decays will be known precisely. Displaying \(\Delta M_{s,d}\) as functions of \(\gamma \) clearly demonstrates the tension between the value of \(\gamma \) from treelevel decays, free from new physics (NP) contributions, and \(\Delta M_{s,d}\) calculated in CMFV and \(U(2)^3\) models and thus exhibits the presence of NP contributions to \(\Delta M_{s,d}\) beyond these frameworks. We calculate the values of \(V_{ub}\) and \(V_{td}\) as functions of \(\gamma \) and \(V_{cb}\) and discuss the implications of our results for \(\varepsilon _K\) and rare K and B decays. We also briefly discuss a future strategy in which \(\beta \), possibly affected by NP, is replaced by \(V_{ub}\).
1 Introduction
The \(\Delta F=2\) transitions in the downquark sector, that is \(B^0_{s,d}{\bar{B}}^0_{s,d}\) and \(K^0{\bar{K}}^0\) mixings, have been vital in constraining the standard model (SM) and in the search for new physics (NP) for several decades [1, 2]. However, theoretical uncertainties related to the hadronic matrix elements entering these transitions and their large sensitivity to the CKM parameters made clear cut conclusions about the presence of NP impossible. As we demonstrate in this paper, this could change in the near future.
Ideally, one would like to use the second pair which allows to construct the socalled reference unitarity triangle (RUT) [4] that is supposed to be free of NP contributions. Unfortunately, the persistent discrepancy between inclusive and exclusive determinations of \(V_{ub}\) from treelevel decays precludes a satisfactory determination of the RUT at present.

In [14], we have considered two strategies. One in which \(\varepsilon _K\) has been used to determine \(V_{cb}\), implying a value consistent with the inclusive determination as well as \(\Delta M_{s,d}\) values well above the data. In the second strategy, \(V_{cb}\) has been determined from \(\Delta M_s\) resulting in a low value of \(V_{cb}\) consistent with the exclusive determination at that time. The predicted \(\varepsilon _K\) then turned out to be well below its experimental value. The recent improvements in the determinations of \(V_{cb}\) [16, 17] disfavours the second strategy and also the recent claim in [18] that there is a \(4\sigma \) anomaly in \(\varepsilon _K\).

More importantly, in view of the improved value of \(\gamma \), we decided to use it as an input in the present analysis, instead of the usual determination of the UT in CMFV models through \(S_{\psi K_S}~(\beta )\) and the side \(R_t\) of the UT determined from the ratio \(\Delta M_{d}/\Delta M_{s}\) and \(\xi \) in (9).

The most recent discussions, see in particular [19, 20], dealt exclusively with the implications of the enhanced value of \(\Delta M_s\) and not \(\Delta M_d\), for which in addition to the increased value of \(V_{cb}\) also the increased value of \(\gamma \) matters.
One could wonder why the emerging \(\Delta M_d\) anomaly pointed out by us has not been noticed in the global fits performed by the CKMfitter and UTfit collaborations. In our view such global fits, involving simultaneously many quantities, are likely to miss NP effects present in only a subset of observables, in particular when the significance has not reached the discovery level. We are optimistic that the findings of this paper pointing towards NP in the \(B_d\) system will motivate both theorists and experimentalists to intensify the search for NP in \(b\rightarrow d\) transitions, after the last five years being dominated by the study of \(b\rightarrow s\) and \(b\rightarrow c\) transitions.
Our paper is organized as follows. In Sect. 2 we present the determination of the UT and of the CKM matrix using the \((\beta ,\gamma )\) strategy. In Sect. 3 we evaluate \(\Delta M_{d}\) and \(\Delta M_{s}\) as functions of \(\gamma \), finding their values to disagree with the data. The new result relative to [14] and other recent papers [19, 20] is the disagreement of \(\Delta M_{d}\) and the ratio \(\Delta M_{d}/\Delta M_{s}\) with the data, a direct consequence of the increased value of \(\gamma \). On the other hand \(\varepsilon _K\) agrees well with the data. We therefore provide the SM predictions for the branching ratios of \(K^+\rightarrow \pi ^+\nu {\bar{\nu }}\) and \(K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}\) for different values of \(\gamma \), \(\beta \) and \(V_{cb}\). In Sect. 4 we have a look at the \((R_b,\gamma )\) strategy, which could become favourable in the next decade, once the treelevel determination of \(V_{ub}\) is settled. In Sect. 5 we briefly investigate what kind of NP could be responsible for the \(\Delta M_{s,d}\) anomalies found in Sect. 3 and what are the implications for NP in \(\Delta F =1\) transitions. We conclude in Sect. 6.
2 Deriving the UT and the CKM matrix
Our determination of the UT and of the CKM matrix proceeds in two steps:
Step 1:
In Fig. 2 we show the constraints on the UT from the treelevel measurement of \(\gamma \), from \(\beta \) extracted from \(S_{\psi K_S}\), and \(R_t\) from \(\Delta M_d/\Delta M_s\). The advantage of the \((\gamma ,\beta )\) strategy over the \((R_t,\beta )\) strategy is not seen yet because of a significant error in \(\gamma \). With the future uncertainty on \(\gamma \) of \(\pm 1^\circ \) represented by the black area, the power of the \((\gamma ,\beta )\) strategy in determining the UT is clearly visible. However, already now we observe that the apex of the UT obtained from the \((\gamma ,\beta )\) strategy disagrees with the one from the \((R_t,\beta )\) one. This tension indicates the presence of some NP contributions.
In Fig. 3 we show \(V_{td}\) as a function of \(\gamma \) and \(V_{ub}\) as a function of \(\beta \) for different values of \(V_{cb}\). The dependences of \(V_{td}\) on \(\beta \) and of \(V_{ub}\) on \(\gamma \) are very small. These plots will allow to monitor the values of \(V_{td}\) and \(V_{ub}\) that enter various observables as the uncertainties of \(\gamma \), \(\beta \) and \(V_{cb}\) will shrink with time.
3 Calculating observables
\(m_{B_s} = 5366.8(2)\, \mathrm{MeV}\) [22]  \(m_{B_d}=5279.58(17)\, \mathrm{MeV}\) [22] 
\(\Delta M_s = 17.757(21) \text {ps}^{1}\) [6]  \(\Delta M_d = 0.5055(20) \text {ps}^{1}\) [6] 
\(S_{\psi K_S}= 0.691(17)\) [6]  \(S_{\psi \phi }= 0.015(35)\) [6] 
\(V_{us}=0.2253(8)\) [22]  \(\varepsilon _K= 2.228(11) \times 10^{3}\) [22] 
\(F_{B_s} = 228.6(3.8)\, \mathrm{MeV}\) [23]  \(F_{B_d} = 193.6(4.2)\, \mathrm{MeV}\) [23] 
\(m_t(m_t)=163.53(85)\, \mathrm{GeV}\)  \(S_0(x_t)=2.322(18)\) 
\(\eta _{cc}=1.87(76)\) [24]  \(\eta _{ct}= 0.496(47)\) [25] 
\(\eta _{tt}=0.5765(65)\) [21]  
\(\tau _{B_s}= 1.510(5) \text {ps}\) [6]  \(\Delta \Gamma _s/\Gamma _s=0.124(9)\) [6] 
\(\tau _{B_d}= 1.520(4) \text {ps}\) [6]  \(\kappa _\varepsilon = 0.94(2)\) [27] 
In Fig. 4 we show in the left panel \(\Delta M_d\) and \(\Delta M_s\) normalized to their experimental values. Evidently, for central values of all parameters, \(\Delta M_d\) differs by roughly \(30\%\) from the data while in the case of \(\Delta M_s\) the corresponding difference amounts only to \(12\%\). But the uncertainties in other parameters like \(V_{cb}\) and the hadronic parameters in (8) are still significant. However, we expect that in the coming years these uncertainties will significantly be reduced.
In the right panel of Fig. 4 we show the ratio \(\Delta M_s/\Delta M_d\) as a function of \(\gamma \). The dependence on \(V_{cb}\) cancels in this ratio and the error on \(\xi \) in (9) is much smaller than the errors in (8). Consequently the disagreement of the ratio in question with the data, shown as a horizontal line at 35.1, is clearly visible and expresses the problem of CMFV models and those based on the \(U(2)^3\) symmetry.
Left: Central values for the branching ratio \({\mathcal {B}}(K^+\rightarrow \pi ^+\nu {\bar{\nu }})\) for various values of \(\gamma \) and \(V_{cb}\). The angle \(\beta \) is fixed to \(\beta =21.85^\circ \) determined from \(S_{\psi K_S}\). Right: Central values for the branching ratio \( {\mathcal {B}}(K_L\rightarrow \pi ^0\nu {\bar{\nu }})\) for various values of \(\beta \), \(\gamma \) and \(V_{cb}\)
\(\gamma [^\circ ]\)  \(10^3\cdot V_{cb}\)  \(\beta [^\circ ]\)  \(\gamma [^\circ ]\)  \(10^3\cdot V_{cb}\)  

39  40  41  42  43  39  40  41  42  43  
\(10^{11}\cdot {\mathcal {B}}(K^+\rightarrow \pi ^+\nu {\bar{\nu }})\)  \(10^{11}\cdot {\mathcal {B}}(K_L\rightarrow \pi ^0\nu {\bar{\nu }})\)  
64  6.7  7.2  7.8  8.3  8.9  21.85  65  2.1  2.3  2.5  2.8  3.1 
66  6.9  7.4  7.9  8.5  9.1  69  2.2  2.4  2.7  3.0  3.2  
68  7.1  7.6  8.1  8.7  9.3  73  2.3  2.6  2.8  3.1  3.4  
70  7.2  7.7  8.3  8.9  9.5  77  2.4  2.7  3.0  3.3  3.6  
72  7.4  7.9  8.5  9.1  9.7  24.0  65  2.5  2.7  3.0  3.3  3.7 
74  7.5  8.1  8.6  9.2  9.9  69  2.6  2.9  3.2  3.5  3.9  
76  7.7  8.2  8.8  9.4  10.0  73  2.8  3.1  3.4  3.8  4.1  
78  7.8  8.4  9.0  9.6  10.3  77  3.0  3.3  3.6  4.0  4.4 
4 \((R_b,\gamma )\) strategy
It is likely that in the next decade the \((\beta ,\gamma )\) strategy will be replaced by the \((R_b,\gamma )\) strategy. This could turn out to be even necessary if the value of \(V_{ub}\) determined from treelevel processes turned out to be very different from the one determined in the previous section. Therefore for completeness we want to give the relevant formulae for this strategy.
5 Going beyond CMFV
Our analysis signals the violation of flavour universality in the function S(v), characteristic for CMFV models. It hints for the presence of new sources of flavour and CPviolation and/or new operators contributing to \(\Delta F=2\) transitions beyond the SM \((VA)\otimes (VA)\) ones.^{3} For simplicity we restrict first our discussion of NP scenarios to the ones in which only SM operators are present.
 A clear breakdown of the universality of S(v) with$$\begin{aligned} \Delta S_s < \Delta S_d . \end{aligned}$$(31)
 The new phasesin order not to spoil the good agreement of the SM with the experimental values of \(S_{\psi K_S}\) and \(S_{\psi \phi }\).$$\begin{aligned} \delta _s\approx \delta _d\approx 0 \end{aligned}$$(32)

In the case of \(K^0{\bar{K}}^0\) mixing, the good agreement of \(\varepsilon _K\) with its measured value implies a small imaginary part of the NP contribution. This can either be achieved by a small value of \(\Delta S_K\), or by an appropriately chosen value of the new phase \(\delta _K\).
The simplest models beyond the CMFV and \(U(2)^3\) frameworks one could consider are models with treelevel \(Z^\prime \) and Z exchanges. While in [34, 35, 36, 37] general studies of such scenarios have been considered, specific examples are models with vectorlike quarks [38] and 331 models [39]. These models have sufficient numbers of parameters to obtain an agreement with the data for \(\Delta F=2\) processes. This is explicitly shown for the case of 331 models in [39].
The minus sign in (30) has been introduced by us by hand. Strictly speaking, as already discussed in the context of \(\Delta M_s\) in [19], in the presence of only lefthanded currents the minus sign in (30) truly requires the NP phases to be \(\pi +\delta _{d,s}\). Following the reasoning in [40], this implies the CPviolating phases in the corresponding \(\Delta F =1 \) \(b\rightarrow d,s\) transitions to be close to \(\pi /2\), i.e. maximal. We hence conclude that, within models with only lefthanded currents, the observed suppression of \(\Delta M_{d}\) and to a lesser extent \(\Delta M_s\) implies significant deviations from the SM in CPasymmetries of radiative and rare \(b\rightarrow d\) and \(b\rightarrow s\) decays. As quantitative predictions for these observables are modeldependent, we leave their thorough analysis for future work.
In the presence of both left and righthanded couplings, on the other hand, the suppression of \(\Delta M_d\) is much easier to achieve without introducing large CPviolating phases. In this context probably most interesting are models in which the SMEFT operator \({\mathcal {O}}_{Hd}\) involving righthanded flavour violating couplings to downquarks is generated at the NP scale. As demonstrated in [37], the renormalisation group evolution to lowenergy scales involving also lefthanded currents present already within the SM generates leftright \(\Delta F=2\) operators representing FCNCs mediated by the Z boson. At NLO this effect has also been discussed in [36]. An explicit realization of such a NP scenario is provided by models with vectorlike quarks with an additional U(1) gauge symmetry so that both treelevel Z and \(Z^\prime \) exchanges are present, and in some models of this type also box diagram contributions with vectorlike quarks, Higgs and other scalar and pseudoscalar exchanges are important [38]. The test of these scenarios is then mainly offered through the correlations of \(\Delta M_{d,s}\) with \(\Delta F=1\) processes, that is rare K or \(B_{s,d}\) decays, the ratio \(\varepsilon '/\varepsilon \) and other observables. This is evident from the analyses in [37, 38] and once the data on \(\gamma \), \(V_{cb}\) and \(V_{ub}\) improve, could be an arena for further investigation of the implications of the \(\Delta M_d\) anomaly pointed out here.
6 Summary
The main message of our paper is the emerging \(\Delta M_d\) anomaly which is significantly larger than the \(\Delta M_s\) one discussed in [14, 19, 20]. Its fate will depend strongly on the improved values of \(\gamma \) and \(V_{cb}\) from treelevel decays and, to a lesser extent, on \(V_{ub}\), which is more relevant for the prediction of \(\sin 2\beta \) in the SM. This anomaly, if confirmed, will have implications for observables sensitive to \(b\rightarrow d\) transitions like \(b\rightarrow d \ell ^+\ell ^\) and \(b\rightarrow d \nu {\bar{\nu }}\) which will be explored by Belle II. It will open a new oasis of NP, analogous to the one related to the recent anomalies in \(b\rightarrow s \ell ^+\ell ^\) and their implications for \(b\rightarrow s \nu {\bar{\nu }}\) transitions. Depending on the NP flavour structure, it could also have implications for \(K^+\rightarrow \pi ^+\nu {\bar{\nu }}\) and \(K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}\).
Footnotes
Notes
Acknowledgements
The research of AJB was fully supported by the DFG cluster of excellence “Origin and Structure of the Universe”.
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