Universal Unitarity Triangle 2016 and the Tension
Between and in CMFV Models
Monika Blanke and Andrzej J. Buras Institut fur Kernphysik, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany
Institut fur Theoretische Teilchenphysik, Karlsruhe Institute of Technology, Engesserstraße 7, D-76128 Karlsruhe, Germany
TUM-IAS, Lichtenbergstr. 2a, D-85748 Garching, Germany
Physik Department, TUM, D-85748 Garching, Germany
[10pt] Motivated by the recently improved results from the Fermilab Lattice and MILC Collaborations on the hadronic matrix elements entering in 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 and of the angle . They follow in this framework from the experimental values of and of the CP-asymmetry . As in CMFV models the new contributions to meson mixings can be described by a single flavour-universal variable , we next determine the CKM matrix elements , , and as functions of using the experimental value of as input. The lower bound on in these models, derived by us in 2006, implies then upper bounds on these four CKM elements and on the CP-violating parameter , which turns out to be significantly below its experimental value. This strategy avoids the use of tree-level determinations of and that are presently subject to considerable uncertainties. On the other hand if is used instead of as input, are found significantly above the data. In this manner we point out that the new lattice data have significantly sharpened the tension between and 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 . We also present the implications of these results for , and within the Standard Model.
Already for decades the transitions in the down-quark sector, that is and 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 being the mass differences in mixings and and the corresponding mixing induced CP-asymmetries. describes the size of the indirect CP violation in mixing. and are already known with impressive precision. The asymmetries and are less precisely measured but have the advantage of being subject to only very small hadronic uncertainties. We do not include in (1) as it is subject to much larger theoretical uncertainties than the five observables in question.
The hadronic uncertainties in and 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 and , an impressive progress has recently been made by the Fermilab Lattice and MILC Collaborations (Fermilab-MILC) that find 
with uncertainties of and , respectively. An even higher precision is achieved for the ratio
This value is significantly lower than the central value in the previous lattice estimates  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 mixing . 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 , or , 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 . However we note that the result for obtained by the ETM collaboration supports a rather low value of from the universal unitarity triangle (UUT). An extensive list of references to other lattice determinations of these parameters can be found in .
Lattice QCD also made an impressive progress in the determination of the parameter which enters the evaluation of [6, 7, 8, 9, 10, 11]. The most recent preliminary world average from FLAG reads , very close to its large value [13, 14]. Moreover the analyses in [15, 16] show that cannot be larger than but close to it. Taking the present results and precision of lattice QCD into account it is then a good approximation to set . In the evaluation of we also take into account long distance contributions parametrised by . Note that at present the theoretical uncertainty in is dominated by the parameter  summarising NLO and NNLO QCD corrections to the charm quark contribution. We take these uncertainties into account.
With determined already very precisely, the main uncertainties in the CKM parameters reside in
with 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 and to a lesser extent of do not yet allow for clear cut conclusions on their values. Moreover, the current direct measurement of is not precise 
This is consistent with from the U-spin analysis of and decays () . The U-spin analysis by LHCb  on the other hand finds a lower value in good agreement with the result from the UUT analysis in (25).
The present uncertainties in and from tree-level decays preclude then a precise determination of the so-called reference unitarity triangle (RUT)  which is expected to be practically independent of the presence of NP. In addition the uncertainty in prevents precise predictions for and 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)  for which the knowledge of and is not required. The UUT can be constructed from
and this in turn allows to determine and .
The important virtue of this determination is its universality within CMFV models. In the case of 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 , with collectively denoting the parameters of a given CMFV model. This function enters universally , and 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 is bounded from below by its SM value 
with 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 and . This in turn implies a precise knowledge of the ratio and the angle , both to be compared with their tree-level determinations. Also the side of the UUT can be determined precisely in view of the result for in (4).
In order to complete the determination of the full CKM matrix without the use of any tree-level determinations, except for , we will use two strategies:
strategy in which the experimental value of is used to determine as a function of , and is then a derived quantity.
strategy in which the experimental value of is used, while is then a derived quantity and 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 , , and as functions of . Yet their outcome is very different, which signals the tension between and 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 upper bounds on , , and which are saturated in the SM, and in turn allows to derive an upper bound on 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 also upper bounds on , , and which are saturated in the SM. However the dependence of these elements determined in this manner differs from the one obtained in , which in turn allows to derive lower bounds on 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 and [29, 30, 31, 32, 33]. It should be emphasized that in CMFV models only the version of this tension in , i. e. NP in , is possible as in these models there are no new CP-violating phases. Therefore has to be used to determine the sole phase in these models, the angle in the UT, or equivalently the CKM phase, through the unitarity of the CKM matrix. The resulting low value of can be naturally raised in CMFV models by enhancing the value of or/and increasing the value of . However, as pointed out in [27, 28], this spoils the agreement of the SM with the data on , signalling the tension between and in CMFV models. The 2013 analysis of this tension in  found that the situation of CMFV with respect to transitions would improve if more precise results for and turned out to be lower than the values known in the spring of 2013. The recent results from  in (3) show the opposite. Both and increased. Moreover the more precise and significantly smaller value of 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 , and than in 2013, our strategy outlined above differs from the one in . In particular we take 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  and also to  at no place in our paper tree-level determinations of , and 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 mass differences and the tree-level determination of the CKM matrix within the SM. Complementary to their findings, we identify a significant tension within processes, that is between and in the whole class of CMFV models. Moreover, we determine very precisely the UUT, in particular the angle in this triangle and the ratio , 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  and in particular in 2000, when the UUT was first suggested . Subsequently we execute the strategies and defined above. The values of , , and , resulting from these two strategies, differ significantly from each other which is the consequence of the tension between and in question. In Section 3 we present the implications of these results for , and within the SM, obtaining again rather different results in and . In Section 4 we briefly discuss how the models face the new lattice data and comment briefly on other models. We conclude in Section 5.
2 Deriving the UUT and the CKM
2.1 Determination of the UUT
We begin with the determination of the UUT. For the mass differences in the systems we have the very accurate expressions
The value in the normalization of is its SM value for obtained from
|= ||= |
It is significantly higher than the value in (12). It should be emphasized that the values of and to a very good approximation do not enter this ratio. Therefore this discrepancy is not a consequence of the tree-level determinations of and . As we will demonstrate below it is the consequence of the value of the angle , which due to the small value of found in  turns out to be significantly smaller than its tree-level value in (6).
with being one of the sides of the unitarity triangle (see Fig. 1) and
where we have used
Having determined and we can construct the UUT shown in Fig. 1, from which we find
Subsequently, using the relation
allows a very precise determination of the ratio
This implies, as shown in Fig. 2, a strict correlation between and 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  and are given as follows
We note that after the recent Belle data on , the exclusive and inclusive values of are closer to each other
than in the past. On the other hand in the case of there is a very significant difference. But the inclusive value for
implies new CP phases in order to accomodate the data on and
consequently the CMFV framework selects the exclusive value of as we
will see below.
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 and , respectively, as explained below and summarized in Table 2. We observe significant tensions both between the results in and and also between them and the inclusive tree-level determination of . On the other hand the exclusive determination of accompanied by the inclusive one for gives , very close to the result in (21). However the separate values of and in (22) and (23) used to obtain this result are not compatible with our findings in , implying problems with as we will see below.
With the very precise value of and consequently we can precisely determine the angle independently of the values of , and . In Fig. 3 we show as a function of 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 is very small. In order to appreciate this result one can read off the plot in Fig. 3 that the old range of corresponds to .
2.2 : Upper Bounds on , , , and
Returning to (9) and (10), we note that the overall factors on the r.h.s. equal the central experimental values of and , respectively. We can therefore read off from these formulae the central values of and 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 and in the CMFV models that are consistent with the data on and
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 . 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 and their tree-level determinations found in  within the SM is larger in any other CMFV model. Interestingly the values of and extracted from the rare semi-leptonic decays and agree with the ones in (28) and (12), respectively :
For , 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  for a more detailed comparison of rare semileptonic -decays with mixing results and the relevant references.
This dependence is represented by the red band in Fig. 4 with defined by
For illustrative purposes we also show the tree-level values in (22) and (23). Evidently the exclusive determinations of are favoured in . Furthermore with increasing , quickly drops significantly below the value in (22).
Having the full CKM matrix as a function of , we can calculate the CP-violating parameter . We use the usual formulae which can be found in . It should be noted that depends directly on
with . Consequently, the value of is not needed for this evaluation.
Now, the dominant contribution to is proportional to
where we have used (28). Thus with and determined through , the parameter decreases with increasing , in contrast to the analysis in which the CKM parameters are taken from tree-level decays. In that case increases with increasing .
Consequently using we find the upper bound on in CMFV models to be
We conclude that the imposition of the constraints within CMFV models implies an upper bound on , 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 with the data by . Once more we stress that this shift cannot be obtained within CMFV models without violating the constraints from .
In Table 2 we collect the values of the most relevant CKM parameters as well as the real and imaginary parts of . In particular the value of is important for the ratio . Its value found in is lower than what has been used in the recent papers [51, 52, 53, 54], thereby further decreasing the value of in the SM.
2.3 : Lower Bounds on
The strategy uses the construction of the UUT as outlined above, but then instead of using for the complete extraction of the CKM elements, the experimental value of is used as input. Taking the lower bound in (8) into account, this strategy again implies upper bounds on , , and . However this time their dependence differs from the one in (28), as seen in the case of in Fig. 4, where is represented by the blue band. The weaker dependence in , together with the higher values, is another proof that the tension between and cannot be removed within the CMFV framework and is in fact smallest in the SM limit.
In order to understand this weaker dependence of on we use the formula for extracted from
that has been derived in . We recall
it here for convenience
where for the central values of the QCD corrections and in Table 1 one finds
Values of and in the full range of and can be found in Table 3 of .
Therefore, with (8), we find lower bounds on and that are significantly larger than the data
Consequently, our results for and in the SM differ from their experimental values by and , respectively. This difference increases for other CMFV models. On the other hand, as seen in Fig. 4, the value of in is fully compatible with its tree-level determination from inclusive decays, but for small larger than its exclusive determination.
The ratio of the central values of obtained by us
perfectly agrees with the data as this ratio is used in and 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 and with extracted from , 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  find in the SM
Compared with (41), this shows the inconsistency between the tree-level determination of the CKM matrix and processes in CMFV models.
In Table 2 we compare the results for the CKM elements obtained in with the ones found using . In both cases we use the SM value for , as it allows to obtain values of in and of in closest to the data. But as we can see, the values of the CKM elements obtained in differ by much from the corresponding ones in , and in particular favour the inclusive determination of . Also the value of is larger, however it differs only by a few percent from the one used in recent calculations of [51, 52, 53, 54].
We conclude therefore, as already indicated by the analysis in , that it is impossible within CMFV models to obtain a simultaneous agreement of and with the data. The improved lattice results in (3) and (4) allow to exhibit this difficulty stronger. In the context of the strategies and , the tension between and is summarized by the plots of vs. in Fig. 5. Note that these plots differ from the known plots of vs. in CMFV models (see e.g. Fig. 5 in ). In the latter plot the CKM parameters were taken from tree-level decays, and varying increased both and 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 and 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 , 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 and . As the first three columns correspond to and , 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 and are significantly below their recent values from  in (3). Moreover, with increasing 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 , where the correlations between and and between and 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  are listed in Section 3 of that paper.
3 Implications for Rare and Decays in the SM
In the previous section we have determined the full CKM matrix using in turn the strategies and . It is interesting to determine the impact of these new determinations on the branching ratios of the rare decays , and within the SM. To this end we use for and the parametric formulae derived in  which we recall here for completeness
For one finds 
In Table 4 we collect the results for the four branching ratios in the SM obtained using the strategies and 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 and in the SM, holding more generally in CMFV models.