Universal Unitarity Triangle 2016 and the tension between \(\Delta M_{s,d}\) and \(\varepsilon _K\) in CMFV models
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Abstract
Motivated by the recently improved results from the Fermilab Lattice and MILC Collaborations on the hadronic matrix elements entering \(\Delta M_{s,d}\) in \(B_{s,d}^0\)–\(\bar{B}_{s,d}^0\) mixing, we determine the universal unitarity triangle (UUT) in models with constrained minimal flavour violation (CMFV). Of particular importance are the very precise determinations of the ratio \(V_{ub}/V_{cb}=0.0864\pm 0.0025\) and of the angle \(\gamma =(62.7\pm 2.1)^\circ \). They follow in this framework from the experimental values of \(\Delta M_{d}/\Delta M_s\) and of the CPasymmetry \(S_{\psi K_S}\). As in CMFV models the new contributions to meson mixings can be described by a single flavouruniversal variable S(v), we next determine the CKM matrix elements \(V_{ts}\), \(V_{td}\), \(V_{cb}\) and \(V_{ub}\) as functions of S(v) using the experimental value of \(\Delta M_s\) as input. The lower bound on S(v) in these models, derived by us in 2006, implies then upper bounds on these four CKM elements and on the CPviolating parameter \(\varepsilon _K\), which turns out to be significantly below its experimental value. This strategy avoids the use of treelevel determinations of \(V_{ub}\) and \(V_{cb}\), which are presently subject to considerable uncertainties. On the other hand, if \(\varepsilon _K\) is used instead of \(\Delta M_s\) as input, \(\Delta M_{s,d}\) are found to be significantly above the data. In this manner we point out that the new lattice data have significantly sharpened the tension between \(\Delta M_{s,d}\) and \(\varepsilon _K\) within the CMFV framework. This implies the presence of new physics contributions beyond this framework that are responsible for the breakdown of the flavour universality of the function S(v). We also present the implications of these results for \(K^+\rightarrow \pi ^+\nu \bar{\nu }\), \(K_{L}\rightarrow \pi ^0\nu \bar{\nu }\) and \(B_{s,d}\rightarrow \mu ^+\mu ^\) within the Standard Model.
1 Introduction
Already for decades 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) [1, 2]. However, theoretical uncertainties related to the hadronic matrix elements entering these transitions and their large sensitivity to the CKM parameters so far precluded clear cut conclusions about the presence of new physics (NP).
Lattice QCD also made impressive progress in the determination of the parameter \(\hat{B}_K\), which enters the evaluation of \(\varepsilon _K\) [6, 7, 8, 9, 10, 11]. The most recent preliminary world average from FLAG reads \(\hat{B}_K=0.7627 (97)\) [12], very close to its large N value \(\hat{B}_K=0.75\) [13, 14]. Moreover, the analyses in [15, 16] show that \(\hat{B}_K\) cannot be larger than 0.75 but must be close to it. Taking the present results and precision of lattice QCD into account it is then a good approximation to set \(\hat{B}_K=0.750 \pm 0.015\). In the evaluation of \(\varepsilon _K\) we also take into account long distance contributions parametrised by \(\kappa _\varepsilon = 0.94\pm 0.02\) [17]. Note that at present the theoretical uncertainty in \(\varepsilon _K\) is dominated by the parameter \(\eta _{cc} = 1.87 \pm 0.76\) [18] summarising NLO and NNLO QCD corrections to the charm quark contribution. We take these uncertainties into account.
The recent results in (3) and (4) have a profound impact on the determination of the UUT. The UUT can be determined very precisely from the measured values of \(\Delta M_d/\Delta M_s\) and \(S_{\psi K_S}\). This in turn implies a precise knowledge of the ratio \(V_{ub}/V_{cb}\) and the angle \(\gamma \), both to be compared with their treelevel determinations. Also the side \(R_t\) of the UUT can be determined precisely in view of the result for \(\xi \) in (4).

\({\varvec{S}_1}\): \({{\varvec{\Delta }}M_s}\) strategy in which the experimental value of \(\Delta M_s\) is used to determine \(V_{cb}\) as a function of S(v), and \(\varepsilon _K\) is then a derived quantity.

\({\varvec{S}_2}\): \({\varvec{\varepsilon }_K}\) strategy in which the experimental value of \(\varepsilon _K\) is used, while \(\Delta M_{s}\) is then a derived quantity and \(\Delta M_d\) follows from the determined UUT.

The lower bound in (8) implies in \(S_1\) upper bounds on \(V_{ts}\), \(V_{td}\), \(V_{ub}\) and \(V_{cb}\) which are saturated in the SM, and in turn it allows to derive an upper bound on \(\varepsilon _K\) in CMFV models that is saturated in the SM but turns out to be significantly below the data.

The lower bound in (8) implies in \(S_2\) also upper bounds on \(V_{ts}\), \(V_{td}\), \(V_{ub}\) and \(V_{cb}\) which are saturated in the SM. However the S(v) dependence of these elements determined in this manner differs from the one obtained in \(S_1\), which in turn allows to derive lower bounds on \(\Delta M_{s,d}\) in CMFV models that are reached in the SM but turn out to be significantly above the data.
In view of the new lattice results, in this paper we take another look at CMFV models. Having more precise values for \(F_{B_s}\sqrt{\hat{B}_{B_s}}\), \(F_{B_d} \sqrt{\hat{B}_{B_d}}\) and \(\xi \) than in 2013, our strategy outlined above differs from the one in [34]. In particular we take \(\gamma \) to be a derived quantity and not an input as done in the latter paper. Moreover, we will be able to reach much firmer conclusions than it was possible in 2013. In particular, in contrast to [34] and also to [3] at no place in our paper treelevel determinations of \(V_{ub}\), \(V_{cb}\) and \(\gamma \) are used. However, we compare our results with them.
It should be mentioned that FermilabMILC identified a significant tension between their results for the \(B^0_{s,d}\bar{B}^0_{s,d}\) mass differences and the treelevel determination of the CKM matrix within the SM. Complementary to their findings, we identify a significant tension within \(\Delta F = 2\) processes, that is between \(\varepsilon _K\) and \(\Delta M_{s,d}\) in the whole class of CMFV models. Moreover, we determine very precisely the UUT, in particular the angle \(\gamma \) in this triangle and the ratio \(V_{ub}/V_{cb}\), both valid also in the SM.
Our paper is organised as follows. In Sect. 2 we determine first the UUT as outlined above, that in 2016 is significantly better known than in 2006 [25] and in particular in 2000, when the UUT was first suggested [23]. Subsequently we execute the strategies \(S_1\) and \(S_2\) defined above. The values of \(V_{ts}\), \(V_{td}\), \(V_{cb}\) and \(V_{ub}\), resulting from these two strategies, differ significantly from each other which is the consequence of the tension between \(\varepsilon _K\) and \(\Delta M_{s,d}\) in question. In Sect. 3 we present the implications of these results for \(K_{L}\rightarrow \pi ^0\nu \bar{\nu }\), \(K^+\rightarrow \pi ^+\nu \bar{\nu }\) and \(B_{s,d}\rightarrow \mu ^+\mu ^\) within the SM, obtaining again rather different results in \(S_1\) and \(S_2\). In Sect. 4 we briefly discuss how the \(U(2)^3\) models match the new lattice data and comment briefly on other models. We conclude in Sect. 5.
2 Deriving the UUT and the CKM
2.1 Determination of the UUT
\(m_{B_s} = 5366.8 (2)\, \mathrm{MeV}\) [36]  \(m_{B_d}=5279.58 (17)\, \mathrm{MeV}\) [36] 
\(\Delta M_s = 17.757 (21) \,\text {ps}^{1}\) [37]  \(\Delta M_d = 0.5055 (20) \,\text {ps}^{1}\) [37] 
\(S_{\psi K_S}= 0.691 (17)\) [37]  \(S_{\psi \phi }= 0.015 (35)\) [37] 
\(V_{us}=0.2253 (8)\) [36]  \(\varepsilon _K= 2.228 (11)\cdot 10^{3}\) [36] 
\(F_{B_s}\) = \(226.0 (22)\, \mathrm{MeV}\) [38]  \(F_{B_d}\) = \(188 (4)\, \mathrm{MeV}\) [39] 
\(m_t(m_t)=163.53 (85)\, \mathrm{GeV}\)  \(S_0(x_t)=2.322 (18)\) 
\(\eta _{cc}=1.87 (76)\) [18]  \(\eta _{ct}= 0.496 (47)\) [40] 
\(\eta _{tt}=0.5765 (65)\) [35]  
\(\tau _{B_s}= 1.510 (5)\,\text {ps}\) [37]  \(\Delta \Gamma _s/\Gamma _s=0.124 (9)\) [37] 
\(\tau _{B_d}= 1.520 (4)\,\text {ps}\) [37]  \(\kappa _\varepsilon = 0.94 (2)\) [17] 
Upper bounds on CKM elements in units of \(10^{3}\) and of \(\lambda _t\) in units of \(10^{4}\) obtained using strategies \(S_1\) and \(S_2\) as explained in the text. We set \(S(v)=S_0(x_t)\)
\(S_i\)  \(V_{ts}\)  \(V_{td}\)  \(V_{cb}\)  \(V_{ub}\)  \(\mathrm{Im}\lambda _t\)  \(\mathrm{Re}\lambda _t\) 

\(S_1\)  \(38.9\,(13)\)  \(7.95\,(29)\)  \(39.5\,(1.3)\)  \(3.41\,(15)\)  \( 1.20\,(8)\)  \(2.85\,(19)\) 
\(S_2\)  \(42.7\,(12)\)  \(8.74\,(27) \)  \(43.4\,(1.2) \)  \(3.75\,(15)\)  \(1.44\,(8)\)  \(3.44\,(19)\) 
We observe that within the CMFV framework only special combinations of these two CKM elements are allowed. The red and blue squares represent the ranges obtained in the strategies \(S_1\) and \(S_2\), respectively, as explained below and summarised in Table 2. We observe significant tensions both between the results in \(S_1\) and \(S_2\) and also between them and the inclusive treelevel determination of \(V_{ub}\). On the other hand, the exclusive determination of \(V_{ub}\) accompanied by the inclusive one for \(V_{cb}\) gives \(V_{ub}/V_{cb}=0.0881\pm 0.0041\), very close to the result in (21). However, the separate values of \(V_{ub}\) and \(V_{cb}\) in (22) and (23) used to obtain this result are not compatible with our findings in \(S_1\), implying problems with \(\Delta M_{s,d}\) as we will see below.
2.2 \(S_1\): upper bounds on \(V_{ts}\), \(V_{td}\), \(V_{cb}\), \(V_{ub}\) and \(\varepsilon _K\)
In Table 2 we collect the values of the most relevant CKM parameters as well as the real and imaginary parts of \(\lambda _t=V_{td} V^*_{ts}\). In particular the value of \(\mathrm{Im}\lambda _t\) is important for the ratio \(\varepsilon '/\varepsilon \). Its value found in \(S_1\) is lower than what has been used in the recent papers [50, 51, 52, 53], thereby further decreasing the value of \(\varepsilon '/\varepsilon \) in the SM.
2.3 \(S_2\): lower bounds on \(\Delta M_{s,d}\)
The strategy \(S_2\) uses the construction of the UUT as outlined above, but then instead of using \(\Delta M_s\) for the complete extraction of the CKM elements, the experimental value of \(\varepsilon _K\) is used as input. Taking the lower bound in (8) into account, this strategy again implies upper bounds on \(V_{ts}\), \(V_{td}\), \(V_{cb}\) and \(V_{ub}\). However, this time their S(v) dependence differs from the one in (28), as seen in the case of \(V_{cb}\) in Fig. 4, where \(S_2\) is represented by the blue band. The weaker S(v) dependence in \(S_2\), together with the higher \(V_{cb}\) values, is another proof that the tension between \(\varepsilon _K\) and \(\Delta M_{s,d}\) cannot be removed within the CMFV framework and is in fact smallest in the SM limit.
CMFV predictions for various quantities as functions of S(v) and \(\gamma \). The four elements of the CKM matrix are in units of \(10^{3}\), \(F_{B_s} \sqrt{\hat{B}_{B_s}}\) and \(F_{B_d} \sqrt{\hat{B}_{B_d}}\) in MeV and \(\mathcal {B}(B^+\rightarrow \tau ^+\nu )\) in units of \(10^{4}\). From [34]
S(v)  \(\gamma \)  \(V_{cb}\)  \(V_{ub}\)  \(V_{td}\)  \(V_{ts}\)  \(F_{B_s}\sqrt{\hat{B}_{B_s}}\)  \(F_{B_d} \sqrt{\hat{B}_{B_d}}\)  \(\xi \)  \(\mathcal {B}(B^+\rightarrow \tau ^+\nu )\) 

2.31  \(63^\circ \)  43.6  3.69  8.79  42.8  252.7  210.0  1.204  0.822 
2.5  \(63^\circ \)  42.8  3.63  8.64  42.1  247.1  205.3  1.204  0.794 
2.7  \(63^\circ \)  42.1  3.56  8.49  41.4  241.8  200.9  1.204  0.768 
We conclude therefore, as already indicated by the analysis in [34], that it is impossible within CMFV models to obtain a simultaneous agreement of \(\Delta M_{s,d}\) and \(\varepsilon _K\) with the data. The improved lattice results in (3) and (4) allow one to exhibit this difficulty stronger. In the context of the strategies \(S_1\) and \(S_2\), the tension between \(\Delta M_{d,s}\) and \(\varepsilon _K\) is summarised by the plots of \(\Delta M_{s,d}\) vs. \(\varepsilon _K\) in Fig. 5. Note that these plots differ from the known plots of \(\Delta M_{s,d}\) vs. \(\varepsilon _K\) in CMFV models (see e.g. Fig. 5 in [2]). In the latter plot the CKM parameters were taken from treelevel decays, and varying S(v) increased both \(\Delta M_{s,d}\) and \(\varepsilon _K\) in a correlated manner. Even if the physics in those plots and in the plots in Fig. 5 is the same, presently the accuracy of the outcome of strategies \(S_1\) and \(S_2\) shown in Fig. 5 is higher.
The problems with CMFV models encountered here could be anticipated on the basis of the first three rows of Table 2 from [34], which we recall in Table 3. In that paper a different strategy has been used and various quantities have been predicted in CMFV models as functions of S(v) and \(\gamma \). As the first three columns correspond to \(\gamma =63^\circ \) and \(\xi =1.204\), very close to the values of these quantities found in the present paper, there is a clear message from Table 3. The predicted values of \(F_{B_s}\sqrt{\hat{B}_{B_s}}\) and \(F_{B_d} \sqrt{\hat{B}_{B_d}}\) are significantly below their recent values from [3] in (3). Moreover, with increasing S(v) there is a clear disagreement between the values of these parameters favoured by CMFV and the values in (3). We also refer to the plots in Fig. 4 of [34], where the correlations between \(V_{cb}\) and \(F_{B_d} \sqrt{\hat{B}_{B_d}}\) and between \(V_{cb}\) and \(F_{B_s} \sqrt{\hat{B}_{B_s}}\) implied by CMFV have been shown. Already in 2013 there was some tension between the grey regions in that figure representing the 2013 lattice values and the CMFV predictions. With the 2016 lattice values in (3), the grey areas shrunk and moved away from the values favoured by CMFV. Other problems of CMFV seen from the point of view of the strategy in [34] are listed in Sect. 3 of that paper.
3 Implications for rare K and B decays in the SM
SM predictions for rare decay branching ratios using the strategies \(S_1\) and \(S_2\), as explained in the text
\(S_i\)  \( {\mathcal {B}}(K^+\rightarrow \pi ^+\nu \bar{\nu })\)  \( {\mathcal {B}}(K_{L}\rightarrow \pi ^0\nu \bar{\nu })\)  \(\overline{\mathcal {B}}(B_{s}\rightarrow \mu ^+\mu ^)\)  \(\mathcal {B}(B_{d}\rightarrow \mu ^+\mu ^)\) 

\(S_1\)  \(6.88\,(70)\cdot 10^{11}\)  \(2.11\,(25)\cdot 10^{11}\)  \(3.14\,(22)\cdot 10^{9}\)  \(0.84\,(7)\cdot 10^{10}\) 
\(S_2\)  \(8.96\,(79)\cdot 10^{11}\)  \(3.08\,(32)\cdot 10^{11}\)  \(3.78\,(23)\cdot 10^{9}\)  \(1.02\,(8)\cdot 10^{10}\) 
4 Beyond CMFV
Our analysis of CMFV models signals the violation of flavour universality in the function S(v), signalling 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.^{2} For simplicity we will here restrict ourselves to solutions in which only SM operators are present.

No correlation between the K and \(B_{s,d}\) systems, so that the tension between \(\varepsilon _K\) and \(\Delta M_{s,d}\) is absent in these models.

However, as \(r_K\ge 1\), finding one day \(\varepsilon _K\) in the SM to be larger than the data would exclude this scenario. Presently such a situation seems rather unlikely.

\(S_d \equiv S_s\) are complex functions and \(r_B\) can be larger or smaller than unity. Consequently, through interference with the SM contributions, \(\Delta M_{s,d}\) can be suppressed or enhanced as needed.

With the new phase \(\varphi _\mathrm{new}\) and \(r_B\) not bounded from below there is more freedom than in the CMFV scenario.
In summary, M\(U(2)^3\) models match the new lattice data better than CMFV, but similar to the latter models they have difficulties with the value of \(\gamma \) and of the ratio \(V_{td}/V_{ts}\) being significantly below their treelevel determinations.
Concerning more complicated models like the Littlest Higgs model with Tparity [66, 67] or 331 models [68], it is clear that the new lattice data have an impact on the allowed ranges of new parameters. However, such a study is beyond the scope of our paper.
5 Conclusions
In this paper we have determined the Universal Unitarity Triangle (UUT) of constrained minimal flavour violation (CMFV) models. We then derived the full CKM matrix, using either the experimental value of \(\Delta M_s\) or of \(\varepsilon _K\) as input. The recently improved values of the hadronic matrix elements in (3) and (4) [3] have been crucial for this study. In contrast to many analyses in the literature, we avoided treelevel determinations of \(V_{ub}\) and \(V_{cb}\).

The extracted angle \(\gamma \) in the UUT is already known precisely and is significantly smaller than its treelevel determination. This is a direct consequence of the small value of \(\xi \) in (4). In turn the ratio \(V_{td}/V_{ts}\) also turns out to be smaller than its treelevel determination, as already pointed out in [3].

The precise relation between \(V_{ub}\) and \(V_{cb}\) obtained by us in (21) provides another test of CMFV. See Fig. 2.

Requiring CMFV to reproduce the data for \(\Delta M_{s,d}\) (strategy \(S_1\)), we find that low values of \(V_{ub}\) and \(V_{cb}\) are favoured, in agreement with their exclusive determinations. More importantly we derived an upper bound on \(\varepsilon _K\) that is significantly below the data.

Requiring CMFV to reproduce the data for \(\varepsilon _K\) (strategy \(S_2\)), we find a higher value of \(V_{ub}\), still consistent with exclusive determinations, but \(V_{cb}\) significantly higher than in \(S_1\) and in agreement with its inclusive determination. The derived lower bounds on \(\Delta M_{s,d}\) are then significantly above the data.

The tension between \(\varepsilon _K\) and \(\Delta M_{s,d}\) in CMFV models with either \(\varepsilon _K\) being too small or \(\Delta M_{s,d}\) being too large cannot be removed by varying S(v). This would only be possible, as stressed in [34], if the values in (3) turned out to be significantly smaller and \(\xi \) larger than in (4). With the present values of these parameters, the SM performs best among all CMFV models, even if, as seen in Fig. 5, it falls short in properly describing the \(\Delta F=2\) data.

The inconsistency of \(\Delta M_{d,s}\) and \(\varepsilon _K\) in the SM and CMFV is also signalled by rather different predictions for rare decay branching ratios obtained using strategies \(S_1\) and \(S_2\). See Sect. 3 and Table 4.

As the correlation between \(\varepsilon _K\) and \(\Delta M_{s,d}\) is broken in models with \(U(2)^3\) flavour symmetry, these models perform better than CMFV models. Still the correlation between \(\Delta M_s\) and \(\Delta M_d\), that is of CMFV type, predicted by these models is in conflict with the treelevel determinations already pointed out in [3] within the SM. See (12) and (13).
Certainly, further improvements on the hadronic matrix elements from lattice QCD and on the treelevel determinations of \(V_{ub}\), \(V_{cb}\) and \(\gamma \) will sharpen the prediction for the size of required NP contributions to \(\Delta F=2\) observables, thereby selecting models which could bring the theory to agree with experimental data. In particular finding the value of \(\gamma \) from treelevel determinations in the ballpark of \(70^\circ \) would imply the violation of the CMFV relation (59). On the other hand, resolving the discrepancy between exclusive and inclusive treelevel determinations of \(V_{ub}\) in favour of the latter, would indicate the presence of new CPviolating phases affecting \(S_{\psi K_S}\). Moreover, the correlations of \(\Delta F=2\) transitions with rare K and \(B_{s,d}\) decays and \(\varepsilon '/\varepsilon \) could eventually give us a deeper insight into the NP at short distance scales that is responsible for the anomalies indicated by the new lattice data, as reviewed in [2] and recently stressed in [53].
Footnotes
Notes
Acknowledgments
We thank Aida ElKhadra and Andreas Kronfeld for many illuminating discussions and information on the progress in their lattice calculations and in particular for sharing their new results with us prior to publication. The research of AJB was fully financed and done in the context of the ERC Advanced Grant project “FLAVOUR” (267104) and was partially supported by the DFG cluster of excellence “Origin and Structure of the Universe”.
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