Universal Unitarity Triangle 2016 and the Tension Between $\Delta M_{s,d}$ and $\varepsilon_K$ in CMFV Models

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 $|V_{ub}|/|V_{cb}|=0.0864\pm0.0025$ and of the angle $\gamma=(63.0\pm 2.1)^\circ$. They follow in this framework from the experimental values of $\Delta M_{d}/\Delta M_s$ and of the CP-asymmetry $S_{\psi K_S}$. As in CMFV models the new contributions to meson mixings can be described by a single flavour-universal 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 CP-violating parameter $\varepsilon_K$, which turns out to be significantly below its experimental value. This strategy avoids the use of tree-level determinations of $|V_{ub}|$ and $|V_{cb}|$ that 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 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 $K^+\to\pi^+\nu\bar\nu$, $K_L\to\pi^0\nu\bar\nu$ and $B_{s,d}\to\mu^+\mu^-$ within the Standard Model.


Introduction
Already for decades the ∆F = 2 transitions in the down-quark sector, that is B 0 s,d −B 0 s,d and K 0 −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).
The five observables of interest are with ∆M s,d being the mass differences in B 0 s,d −B 0 s,d mixings and S ψK S and S ψφ the corresponding mixing induced CP-asymmetries. ε K describes the size of the indirect CP violation in K 0 −K 0 mixing. ∆M s,d and ε K are already known with impressive precision. The asymmetries S ψK S and S ψφ are less precisely measured but have the advantage of being subject to only very small hadronic uncertainties. We do not include ∆M K in (1) as it is subject to much larger theoretical uncertainties than the five observables in question.
The hadronic uncertainties in ∆M s,d and ε K within the SM and CMFV models reside within a good approximation in the parameters Fortunately, during the last years these uncertainties decreased significantly. In particular, concerning F Bs B Bs and F B d B B d , an impressive progress has recently been made by the Fermilab Lattice and MILC Collaborations (Fermilab-MILC) that find [3] F Bs B Bs = (274.6 ± 8.8) MeV, with uncertainties of 3% and 4%, respectively. An even higher precision is achieved for the ratio This value is significantly lower than the central value 1.27 in the previous lattice estimates [4] and its reduced uncertainty by a factor of three plays an important role in our analysis.
The ETM Collaboration has also presented results for matrix elements of all five operators entering B d,s −B d,s mixing [5]. This work however only employs two flavours of sea quarks and does not estimate the uncertainty from quenching the strange quark. The ETM and Fermilab-MILC results for matrix elements differ by ∼ 5%, or ∼ 1σ, which could arise from the omitted strange sea. We think it is safer to avoid this issue and use only the Fermilab-MILC results with N f = 2 + 1. However we note that the result for ξ obtained by the ETM collaboration supports a rather low value of γ from the universal unitarity triangle (UUT). An extensive list of references to other lattice determinations of these parameters can be found in [3].
Lattice QCD also made an impressive progress in the determination of the parameterB K which enters the evaluation of ε K [6][7][8][9][10][11]. The most recent preliminary world average from FLAG readsB K = 0.7627(97) [12], very close to its large N valueB K = 0.75 [13,14]. Moreover the analyses in [15,16] show thatB K cannot be larger than 0.75 but close to it. Taking the present results and precision of lattice QCD into account it is then a good approximation to setB K = 0.750±0.015. In the evaluation of ε K we also take into account long distance contributions parametrised by κ ε = 0.94 ± 0.02 [17]. Note that at present the theoretical uncertainty in ε K is dominated by the parameter η cc = 1.87 ± 0.76 [18] summarising NLO and NNLO QCD corrections to the charm quark contribution. We take these uncertainties into account.
With |V us | determined already very precisely, the main uncertainties in the CKM parameters reside in with γ being one of the angles of the unitarity triangle (UT). These three parameters can be determined from tree-level decays that are subject to only very small NP contributions. However the tensions between inclusive and exclusive determinations of |V ub | and to a lesser extent of |V cb | do not yet allow for clear cut conclusions on their values. Moreover, the current direct measurement of γ is not precise [19] γ = (70.9 +7.1 −8.5 ) • .
This is consistent with γ from the U-spin analysis of B s → K + K − and B d → π + π − decays (γ = (68.2 ± 7.1) • ) [20]. The U-spin analysis by LHCb [21] on the other hand finds a lower value γ = (63.5 +7.2 −6.7 ) • in good agreement with the result from the UUT analysis in (25). The present uncertainties in |V ub |/|V cb | and γ from tree-level decays preclude then a precise determination of the so-called reference unitarity triangle (RUT) [22] which is expected to be practically independent of the presence of NP. In addition the uncertainty in |V cb | prevents precise predictions for ε K and ∆M s,d in the SM. However in the SM and more generally models with constrained minimal flavour violation (CMFV) [23][24][25] it is possible to construct the so-called universal unitarity triangle (UUT) [23] for which the knowledge of |V ub |/|V cb | and γ is not required. The UUT can be constructed from and this in turn allows to determine |V ub |/|V cb | and γ.
The important virtue of this determination is its universality within CMFV models. In the case of ∆F = 2 transitions in the down-quark sector various CMFV models can only be distinguished by the value of a single flavour universal real one-loop function, the box diagram function S(v), with v collectively denoting the parameters of a given CMFV model. This function enters universally ε K , ∆M s and ∆M d and cancels out in the ratio in (7). Therefore the resulting UUT is the same in all CMFV models. Moreover it can be shown that in these models S(v) is bounded from below by its SM value [26] S(v) ≥ S 0 (x t ) = 2.32 (8) with S 0 (x t ) given in (11).
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 ∆M d /∆M s and S ψK S . This in turn implies a precise knowledge of the ratio |V ub |/|V cb | and the angle γ, both to be compared with their tree-level determinations. Also the side R t of the UUT can be determined precisely in view of the result for ξ in (4).
In order to complete the determination of the full CKM matrix without the use of any tree-level determinations, except for |V us |, we will use two strategies: S 1 : ∆M s strategy in which the experimental value of ∆M s is used to determine |V cb | as a function of S(v), and ε K is then a derived quantity.
S 2 : ε K strategy in which the experimental value of ε K is used, while ∆M s is then a derived quantity and ∆M d follows from the determined UUT.
Both strategies use the determination of the UUT by means of (7) and allow to determine the whole CKM matrix, in particular |V ts |, |V td |, |V ub | and |V cb | as functions of S(v). Yet their outcome is very different, which signals the tension between ∆M s,d and ε K in this framework. As we will demonstrate below, this tension, known already from previous studies [27,28], has been sharpened significantly through the results in (3) and (4). Using these two strategies separately allows to exhibit this tension transparently. Indeed • 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 allows to derive an upper bound on ε 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 ∆M s,d in CMFV models that are reached in the SM but turn out to be significantly above the data.
It has been known since 2008 that the SM experiences some tension in the correlation between S ψK S and ε K [29][30][31][32][33]. It should be emphasized that in CMFV models only the version of this tension in [30], i. e. NP in ε K , is possible as in these models there are no new CP-violating phases. Therefore S ψK S has to be used to determine the sole phase in these models, the angle β in the UT, or equivalently the CKM phase, through the unitarity of the CKM matrix. The resulting low value of ε K can be naturally raised in CMFV models by enhancing the value of S(v) or/and increasing the value of |V cb |. However, as pointed out in [27,28], this spoils the agreement of the SM with the data on ∆M s,d , signalling the tension between ∆M s,d and ε K in CMFV models. The 2013 analysis of this tension in [34] found that the situation of CMFV with respect to ∆F = 2 transitions would improve if more precise results for F Bs B Bs and F B d B B d turned out to be lower than the values known in the spring of 2013. The recent results from [3] in (3) show the opposite. Both Moreover the more precise and significantly smaller value of ξ enlarges the tension in question.
In view of the new lattice results, in this paper we take another look at CMFV models.
Having more precise values for F Bs B Bs , F B d B B d and ξ than in 2013, our strategy outlined above differs from the one in [34]. In particular we take γ 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 tree-level determinations of |V ub |, |V cb | and γ are used. However we compare our results with them.
It should be mentioned that Fermilab-MILC identified a significant tension between their results for the B 0 s,d −B 0 s,d mass differences and the tree-level determination of the CKM matrix within the SM. Complementary to their findings, we identify a significant tension within ∆F = 2 processes, that is between ε K and ∆M s,d in the whole class of CMFV models. Moreover, we determine very precisely the UUT, in particular the angle γ in this triangle and the ratio |V ub |/|V cb |, both valid also in the SM.
Our paper is organized as follows. In Section 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 ε K and ∆M s,d in question. In Section 3 we present the implications of these results for K L → π 0 νν, K + → π + νν and B s,d → µ + µ − within the SM, obtaining again rather different results in S 1 and S 2 . In Section 4 we briefly discuss how the U (2) 3 models face the new lattice data and comment briefly on other models. We conclude in Section 5.  [37] S ψK S = 0.691 (17) [37] S ψφ = 0.015 (35) [37] The value 2.322 in the normalization of S(v) is its SM value for m t (m t ) = 163.5 GeV obtained from and η B is the perturbative QCD correction [35]. Our input parameters, equal to the ones used in [3], are collected in Table 1.
From (9) and (10) we find using (4) which perfectly agrees with [3]. The tree-level determination of this ratio, quoted in the latter paper and obtained from CKMfitter [41], reads It is significantly higher than the value in (12). It should be emphasized that the values of |V cb | and |V ub | to a very good approximation do not enter this ratio. Therefore this discrepancy is not a consequence of the tree-level determinations of |V cb | and |V ub |. As we will demonstrate below it is the consequence of the value of the angle γ, which due to the small value of ξ found in [3] turns out to be significantly smaller than its tree-level value in (6).
with R t being one of the sides of the unitarity triangle (see Fig. 1) and where we have used obtained from S ψK S = sin 2β = 0.691 ± 0.017 .
Thus using (12) and (14) we determine very precisely Having determined β and R t we can construct the UUT shown in Fig. 1, from which we findρ = 0.170 ± 0.013 ,η = 0.333 ± 0.011 . We observe that the UUT in Fig. 1 differs significantly from the UT obtained in global fits [41,42], with the latter exhibiting smallerρ and largerη values.
Subsequently, using the relation allows a very precise determination of the ratio This implies, as shown in Fig. 2, a strict correlation between |V ub | and |V cb | that can be compared with the tree-level determinations of both CKM elements, also shown in this plot. The exclusive determinations have been summarized in [43] and are given as follows They are based on [3,[44][45][46][47]. The inclusive ones are summarized well in [48,49].
We note that after the recent Belle data on B → D ν l [46], the exclusive and inclusive values of |V cb | are closer to each other than in the past. On the other hand in the case of |V ub | there is a very significant difference. But the inclusive value for |V ub | implies new CP phases in order to accomodate the data on S ψK S and consequently the CMFV framework selects the exclusive value of |V ub | as we will see below.  (4), and the yellow range displays the tree-level determination of γ (6).
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 summarized in Table 2.
We observe significant tensions both between the results in S 1 and S 2 and also between them and the inclusive tree-level 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 | = 0.0850± 0.0026, 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 ∆M s,d as we will see below.
Returning to the issue of the origin of the difference between (12) and (13), the new lattice results [3] have important implications on the angle γ in the UUT that can be determined by means of With the very precise value of ξ and consequently R t we can precisely determine the angle γ independently of the values of S(v), |V ub | and |V cb |. In Fig. 3 we show γ as a function of ξ from which we extract below its central value from tree-level decays in (6), and with an uncertainty that is by more than a factor of three smaller. We will use this value in what follows. We note that the uncertainty due to S ψK S is very small. In order to appreciate this result one can read off the plot in Fig. 3 that the old range of ξ = 1.27 ± 0.06 corresponds to γ = (70 ± 6) • .
Finally, from (16) and (25) we determine the angle α in the unitarity triangle It should be emphasized that the results in (16), (18), (21), (25) and (26) are independent of S(v) and therefore valid for all CMFV models.
2.2 S 1 : Upper Bounds on |V ts |, |V td |, |V cb |, |V ub | and ε K Returning to (9) and (10), we note that the overall factors on the r.h.s. equal the central experimental values of ∆M s and ∆M d , respectively. We can therefore read off from these formulae the central values of |V ts | and |V td | corresponding to the lattice results in (3). Including the uncertainties in the latter formula and taking into account the inequality (8) we find the maximal values of |V ts | and |V td | in the CMFV models that are consistent with the data on ∆M s and ∆M d It should be noted that where we suppressed the errors given in (27). Thus the bounds in (27) are saturated in the SM. The results within the SM are in excellent agreement with those obtained in [3]. Yet, here we also stress that these are upper bounds in CMFV models. Therefore, the tension between the values of these CKM elements extracted from ∆M s,d and their tree-level determinations found in [3] within the SM is larger in any other CMFV model.
Interestingly the values of |V td | and |V td |/|V ts | extracted from the rare semi-leptonic decays B → πµ + µ − and B → Kµ + µ − agree with the ones in (28) and (12), respectively [50]: For |V ts |, the values are found to be even smaller than in (28). However this determination of CKM parameters still suffers from large uncertainties. We refer to [3] for a more detailed comparison of rare semileptonic B-decays with B s,d mixing results and the relevant references.
With the knowledge of |V us |, |V ts |, |V td | and β we can determine |V ub | and |V cb | as functions of S(v) so that they can directly be compared with their determinations from semi-leptonic decays summarized in (22) and (23). We find This dependence is represented by the red band in Fig. 4 with ∆S(v) defined by For illustrative purposes we also show the tree-level values in (22) and (23). Evidently the exclusive determinations of |V cb | are favoured in S 1 . Furthermore with increasing ∆S(v), |V cb | quickly drops significantly below the value in (22).
Having the full CKM matrix as a function of S(v), we can calculate the CP-violating parameter ε K . We use the usual formulae which can be found in [34]. It should be noted that ε K depends directly on with β s = −1 • . Consequently, the value of |V cb | is not needed for this evaluation. Now, the dominant contribution to ε K is proportional to where we have used (28). Thus with |V ts | and |V td | determined through ∆M s,d , the parameter ε K decreases with increasing S(v), in contrast to the analysis in which the CKM parameters are taken from tree-level decays. In that case ε K increases with increasing S(v).
Consequently using S 1 we find the upper bound on ε K in CMFV models to be  Table 2: Upper bounds on CKM elements in units of 10 −3 and of λ 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 ).
We conclude that the imposition of the ∆M s,d constraints within CMFV models implies an upper bound on ε K , saturated in the SM, which is significantly below its experimental value given in Table 1. Therefore a non-CMFV contribution is required, implying a discrepancy of the SM and CMFV value of ε K with the data by 2.3 σ. Once more we stress that this shift cannot be obtained within CMFV models without violating the constraints from ∆M s,d .
In Table 2 we collect the values of the most relevant CKM parameters as well as the real and imaginary parts of λ t = V td V * ts . In particular the value of Imλ t is important for the ratio ε /ε. Its value found in S 1 is lower than what has been used in the recent papers [51][52][53][54], thereby further decreasing the value of ε /ε in the SM.

S 2 : Lower Bounds on ∆M s,d
The strategy S 2 uses the construction of the UUT as outlined above, but then instead of using ∆M s for the complete extraction of the CKM elements, the experimental value of ε 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 ε K and ∆M s,d cannot be removed within the CMFV framework and is in fact smallest in the SM limit.
In order to understand this weaker dependence of |V cb | on S(v) we use the formula for |V cb | extracted from ε K that has been derived in [34]. We recall it here for convenience 2 where for the central values of the QCD corrections η cc and η ct in Table 1 one finds v(η cc , η ct ) = 0.0282, h(η cc , η ct ) = 24.83 .
Values ofṽ(η cc , η ct ) and h(η cc , η ct ) in the full range of η cc and η ct can be found in Table 3 of [34].
Inserting (36) into (14) we find and consequently from (9) and (10) Therefore, with (8), we find lower bounds on ∆M s and ∆M d that are significantly larger than the data Consequently, our results for ∆M s and ∆M d in the SM differ from their experimental values by 1.8σ and 1.5σ, respectively. This difference increases for other CMFV models.
On the other hand, as seen in Fig. 4, the value of |V cb | in S 2 is fully compatible with its tree-level determination from inclusive decays, but for small ∆S(v) larger than its exclusive determination. perfectly agrees with the data as this ratio is used in S 1 and S 2 as experimental input in our analysis. The error on this ratio calculated directly from (40) is spurious as we impose this ratio from experiment and the true error is negligible. Only when one individually calculates ∆M s and ∆M d with |V cb | extracted from ε K , the errors in (40) are found. However they are correlated and cancel in the ratio.
On the other hand, using the tree-level determination of the CKM matrix, the authors of [3] Compared with (41), this shows the inconsistency between the tree-level determination of the CKM matrix and ∆F = 2 processes in CMFV models.
In Table 2 we compare the results for the CKM elements obtained in S 2 with the ones found using S 1 . In both cases we use the SM value for S(v), as it allows to obtain values of  Figure 5: ∆M s,d and ε K obtained from the strategies S 1 and S 2 for S(v) = S 0 (x t ), at which the upper bound on ε K in S 1 and lower bound on ∆M s,d in S 2 are obtained. The arrows show how the red and blue regions move with increasing S(v). The black dot represents the experimental values.
ε K in S 1 and of ∆M s,d in S 2 closest to the data. But as we can see, the values of the CKM elements obtained in S 2 differ by much from the corresponding ones in S 1 , and in particular favour the inclusive determination of |V cb |. Also the value of Imλ t is larger, however it differs only by a few percent from the one used in recent calculations of ε /ε [51][52][53][54].
We conclude therefore, as already indicated by the analysis in [34], that it is impossible within CMFV models to obtain a simultaneous agreement of ∆M s,d and ε K with the data. The improved lattice results in (3) and (4) allow to exhibit this difficulty stronger. In the context of the strategies S 1 and S 2 , the tension between ∆M d,s and ε K is summarized by the plots of ∆M s,d vs. ε K in Fig. 5. Note that these plots differ from the known plots of ∆M s,d vs. ε K in CMFV models (see e.g. Fig. 5 in [2]). In the latter plot the CKM parameters were taken from tree-level decays, and varying S(v) increased both ∆M s,d and ε 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 γ. As the first three columns correspond to γ = 63 • and ξ = 1.204, very close to the values of these quantities found in the present paper, there is a clear message from  (3). We also refer to the plots in Fig. 4 of [34], where the correlations between |V cb | and    (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 Section 3 of that paper.
The "bar" in (46) indicates that ∆Γ s effects [57][58][59] have been taken into account through where In Table 4 we collect the results for the four branching ratios in the SM obtained using the strategies S 1 and S 2 for the determination of the CKM parameters and other updated parameters collected in Table 1. We observe significant differences in these two determinations, which gives another support for the tension between ∆M s,d and ε K in the SM, holding more generally in CMFV models.
We observe that in S 1 the SM prediction for B s → µ + µ − is rather close to the data, while in the case of S 2 it is visibly larger. The most recent result from ATLAS [61], while not accurate, also favours a low value for B(B s → µ + µ − ).
Finally, in view of the improved lattice determinations of the parametersB Bs andB B d [3] B Bs = 1.44 ± 0.10,B B d = 1.38 ± 0.13 (53) it is tempting to calculate the B s,d → µ + µ branching ratios by normalizing them to ∆M s,d [62]. This eliminates not only the dependence on the CKM parameters and weak decay constants, but also reduces the dependence on m t . Neglecting the tiny uncertainties in η B , α s and τ Bq we find the very accurate expressions These expressions apply only to the SM and S 1 , where the experimental values of ∆M s,d are used to determine the CKM matrix. We then find These results agree perfectly with the ones in Table 4. This is not surprising because in S 1 the constraint from ∆M s,d has been imposed and the authors of [3] extracted the values ofB Bq from their results in (3) and F Bq in Table 1. The outcome of this exercise will be more illuminating once independent and more precise lattice determinations of theB B s,d parameters become available. In addition, the derived formulae (54) and (55) are much simpler than the ones in (46) and (49) For simplicity we will here restrict ourselves to solutions in which only SM operators are present.
A fully general and very convenient solution in this case is just to consider instead of the flavour universal function S(v) three functions It is evident that with two free parameters in each meson system it is possible to obtain an agreement with the data on ∆F = 2 observables. The simplest models of this type are models with tree-level Z and Z exchanges analysed in detail in [64]. The flavour violating couplings in these models are complex numbers (two free parameters) and can be chosen in such a manner that any problems of CMFV models in ∆F = 2 processes are removed by properly choosing these couplings. Effectively the observables in (1) are simply used to find these parameters or equivalently S i . The test of these scenarios is only offered through the correlations with ∆F = 1 processes, that is rare K or B s,d decays, which in these simple models involve the same couplings. The analysis in [64] then shows that when constraints from ∆F = 1 processes are taken into account it is easier to obtain an agreement with the data for ∆F = 2 processes in the case of Z models than models with tree-level Z exchanges.
Here we would like to discuss only the models with a minimally broken U (2) 3 flavour symmetry [65,66] which are more constrained. In these models, as discussed in detail in [67], in addition to the unitary CKM matrix one has and with r B being a real parameter which could be larger or smaller than unity. The important difference from the CMFV scenario is that it cannot be tested without invoking tree-level determinations of at least some elements of the CKM matrix. The main features of this scenario are: • No correlation between the K and B s,d systems, so that the tension between ε K and ∆M s,d is absent in these models.
• However as r K ≥ 1, finding one day ε K in the SM to be larger than the data would exclude this scenario. Presently such a situation seems rather unlikely.
• S d ≡ S s are complex functions and r B can be larger or smaller than unity. Consequently, through interference with the SM contributions, ∆M s,d can be suppressed or enhanced as needed.
• With the new phase ϕ new and r B not bounded from below there is more freedom than in the CMFV scenario.
However, due to the equality S d = S s there are two important implications that can be tested.
The first one is the CMFV relation [67] from which one can obtain the ratio |V td |/|V ts | as done already in section 2, see (12), which can be compared with its tree-level determination. As stated before, the tree-level determination of this ratio, quoted in (13), is significantly larger, and consequently MU (2) 3 models have the same difficulty here as CMFV models. Yet, a firm conclusion will only be reached after the result in (13) will be superseded by a more precise tree-level determination of the angle γ.
The second one is the correlation between the two CP asymmetries that results from the equality of NP phases in As β s is very small in the SM, a precise measurement of S ψφ determines ϕ new . From the measured value of S ψK S we then obtain β. The latter value can be compared with the one obtained from the tree-level determination of |V ub |/|V cb | and either R t or the treelevel determination of γ. However β is strongly correlated with |V ub |/|V cb |, with very weak dependence on γ and R t . Therefore eventually (61) implies a triple correlation between [67] S ψK S , S ψφ , which provides another important test of the MU (2) 3 scenario once the three observables will be known precisely.
In summary, MU (2) 3 models face the new lattice data better than CMFV, but similar to the latter models have difficulties with the value of γ and of the ratio |V td |/|V ts | being significantly below their tree-level determinations.
Concerning more complicated models like the Littlest Higgs model with T-parity [68,69] or 331 models [70], it is clear that the new lattice data has an impact on the allowed ranges of new parameters. However such a study is beyond the scope of our paper 4 .

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 ∆M s or of |ε 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 tree-level determinations of |V ub | and |V cb |.
The main messages from this analysis are as follows: • The extracted angle γ in the UUT is already known precisely and is somewhat smaller than its tree-level determination. This is a direct consequence of the small value of ξ in (4). In turn the ratio |V td |/|V ts | also turns out to be smaller than its tree-level 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 ∆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 |ε K | that is significantly below the data.
• Requiring CMFV to reproduce the data for ε 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 ∆M s,d are then significantly above the data.
• The tension between ε K and ∆M s,d in CMFV models with either |ε K | being too small or ∆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 ξ 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 ∆F = 2 data.
• The inconsistency of ∆M d,s and ε 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 Section 3 and Table 4.
• As the correlation between ε K and ∆M s,d is broken in models with U (2) 3 flavour symmetry, these models perform better than CMFV models. Still the correlation between ∆M s and ∆M d , that is of CMFV type, predicted by these models is in conflict with the tree-level determinations already pointed out in [3] within the SM. See (12) and (13).
Our analysis of CMFV models shows that they fail to properly describe the existing data on ∆F = 2 observables simultaneously and implies thereby the presence of either new sources of flavour violation and/or new operators. Several models analysed in the literature like Z models, 331 models, or the Littlest Higgs model with T-parity (LHT) could help in bringing the theory to agree with the data. In fact in the case of 331 models it was demonstrated recently in [71] that the tensions pointed by us here could be removed in these models. We expect that this is also the case in the LHT model, firm conclusions would however require a dedicated study.
Certainly, further improvements on the hadronic matrix elements from lattice QCD and on the tree-level determinations of |V ub |, |V cb | and γ will sharpen the prediction for the size of required NP contributions to ∆F = 2 observables, thereby selecting models which could bring the theory to agree with experimental data. In particular finding the value of γ from tree-level determinations in the ballpark of 70 • would imply the violation of the CMFV relation (60). On the other hand resolving the discrepancy between exclusive and inclusive tree-level determinations of |V ub | in favour of the latter, would indicate the presence of new CP-violating phases affecting S ψK S . Moreover, the correlations of ∆F = 2 transitions with rare K and B s,d decays and ε /ε 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 [54].