On the Superiority of the $|V_{cb}|-\gamma$ Plots over the Unitarity Triangle Plots in the 2020s

The UT plots played already for three decades an important role in the tests of the SM and the determination of the CKM parameters. As of 2022, among the four CKM parameters, $V_{us}$ and $\beta$ are already measured with respectable precision, while this is not the case of $|V_{cb}|$ and $\gamma$. In the case of $|V_{cb}|$ the main obstacle are the significant tensions between its inclusive and exclusive determinations from tree-level decays. The present uncertainty in $\gamma$ of $4^\circ$ from tree-level decays will be reduced to $1^\circ$ by the LHCb and Belle II collaborations in the coming years. Unfortunately in the UT plots $|V_{cb}|$ is not seen and the experimental improvements in the determination of $\gamma$ from tree-level decays at the level of a few degrees are difficult to appreciate. In view of these deficiencies of the UT plots with respect to $|V_{cb}|$ and $\gamma$ and the central role these two CKM parameters will play in this decade, the recently proposed plots of $|V_{cb}|$ versus $\gamma$ extracted from various processes appear to be superior to the UT plots in the flavour phenomenology. We illustrate this idea with $\Delta M_s$, $\Delta M_d$, $\epsilon_K$ and with rare decays $B_s\to\mu^+\mu^-$, $B_d\to\mu^+\mu^-$, $K^+\to \pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$. The power of $\epsilon_K$, $K^+\to\pi^+\nu\bar\nu)$ and $K_{L}\to\pi^0\nu\bar\nu)$ in the determination of $|V_{cb}|$, due to their strong dependence on $|V_{cb}|$, is transparently exhibited in this manner. Combined with future reduced errors on $\gamma$ and $|V_{cb}|$ from tree-level decays such plots can better exhibit possible inconsistenices between various determinations of these two parameters, caused by new physics, than it is possible with the UT plots.


Introduction
The unitary CKM matrix [1,2] can be conveniently parametrized by the four parameters with β and γ being two angles in the UT, shown in Fig. 1. By now |V us |, β and γ, as measured in tree-level decays, are found to be [3].
|V us | = 0.2253 (8), Moreover, in the coming years the determination of γ by the LHCb [4,5] and Belle II [6] collaborations should be significantly improved with the uncertainty brought down to 1 • . Also some reduction of the error on β from tree-level decays is expected. A review of various methods can be found in Chapter 8 in [7]. The situation with |V cb | is very different. There is a persistent tension between its inclusive and exclusive determinations [8,9] which is clearly disturbing. This is the case when one makes SM predictions for rare K and B decays branching ratios and for quark mixing observables like ∆M s , ∆M d and ε K . In particular rare K decay branching ratios, like the ones for K + → π + νν and K L → π 0 νν, exhibit |V cb | 2.8 and |V cb | 4 dependence, respectively, while |ε K | the |V cb | 3.4 one [10]. B physics observables exhibit typically |V cb | 2 dependence. This implies large modifications in the SM predictions for the observables in question when the inclusive values of |V cb | are replaced by the exclusive ones [11]. This problematic is not seen directly in the usual UT plots [12,13], simply because |V cb | does not enter Fig. 1 explicitly. Moreover the bands resulting from rare K decays and ε K are bound to be broad because these observables being very sensitive to |V cb | suffer from this large parametric uncertainty. However, already in 1994, it has been pointed out in [14] that the branching ratio for K L → π 0 νν growing like |V cb | 4 is a powerful tool to determine |V cb | in the absence of NP provided its branching ratio could be measured precisely. But even a measurement of this branching ratio with 10% accuracy would determine |V cb | with an error of 2.5% because in this decay the hadronic uncertainties are negligible. A similar comment applies to K + → π + νν up to long distance charm quark contribution and to ε K , for which the theoretical uncertainties have been recently reduced [15].
It is then evident that the usual UT plots are not the arena where this nice property of Kaon processes can be used properly and in fact it becomes their Achilles tendon there. Moreover, the |V cb | problematic cannot be properly monitored with the help of the UT plots as |V cb | is hidden in computer codes 2 . These appears not to be a problem for computer code experts but in my view it is a deficiency. In particular, why should we give the pleasure to the computer to see directly the impact of loop induced processes on the value of |V cb | without being able to monitor directly what is going on.
As the removal of the |V cb | obstacle could still take some years before the experts agree on a unique value of this parameter, it is time to develop a new strategy for the coming years. In fact the |V cb | − γ plots proposed recently in collaboration with Elena Venturini in [10,11,16] appear to be superior to the usual UT plots in this context in three ways: • They exhibit |V cb | and its correlation with γ determined through a given observable in the SM, allowing thereby monitoring the progress on both parameters expected in the coming years. Violation of this correlation in experiment will clearly indicate new physics (NP) at work.
• They utilize the strong sensitivity of rare K decay processes to |V cb | thereby providing precise determination of |V cb | even with modest experimental precision on their branching ratios.
• They exhibit the action of ∆M s and of B s decays together with other processes. In particular the recently found anomaly in B s → µ + µ − decay [11,17] can be exhibited in this manner which is not possible in a UT-plot.
It appears then that the |V cb | − γ plots for ∆M s , ∆M d and ε K presented in [10,11] are more useful in this context than the usual UT plots that exhibit the impact of quark mixing and rare decays in the (¯ ,η) plane [18,19] 3 . The goal of the present paper is to demonstrate the usefulness of |V cb | − γ plots also for K + → π + νν, K L → π 0 νν, B s → µ + µ − and B d → µ + µ − decays. Clearly, other decays can be considered as well, in particular the short distance contribution to K S → µ + µ − which has the same CKM dependence as K L → π 0 νν. Despite these comments we do not claim by no means that during the RUN 3 of the LHC and the Belle II era the UT plots should be abandoned. Indeed with improved measurements of various observables they will exhibit the CKM unitarity tests with γ and β measured in tree level non-leptonic B-decays. In this sense they offer complementary tests of the SM. It should be remarked that the authors of [20] performed recently a determination of |V cb | and |V ub | from loop processes, rare decays and quark mixing, by assuming no NP contributions to these observables. To this end they used only well measured observables in the B system and ε K . This strategy has already been explored in [21] where ε K , ∆M d and ∆M s and S ψK S have been considered. This was also the case of the analyses in [10,11].
Our present analysis extends the |V cb |−γ strategy, developed in [10,11], to rare K decays, not considered in [20], resurrecting some old ideas from the 1990s [14,22,23] and improving significantly on them. In this context it should be noted that in [22] the possibility of the determination of β from K + → π + νν and K L → π 0 νν, basically independently of |V cb | and γ, has been pointed out. As pointed out recently in [10] β can also be determined practically independently of |V cb | and γ either from K + → π + νν and |ε K | or K L → π 0 νν and |ε K |.
The determination of |V cb | from K L → π 0 νν alone proposed in [14] requires still the value of γ as we will see explicitly in Section 3. In [23] the determination of the full CKM matrix has been achieved with the help of the measurement of K L → π 0 νν and of the angle α in the UT in tree-level decays. However, from the present perspective, the use of γ is favoured over α because of smaller hadronic uncertainties in its tree-level determinations. The outline of our paper is as follows. In Section 2 we recall the basic formulae for the observables in question. In Section 3 we list useful formulae for |V cb | as a function of γ and β resulting from the seven magnificant observables considered by us. We illustrate the application of these formulae by presenting a few examples of the |V cb | − γ plots. A brief summary and an outlook are given in Section 4.

Basic Formulae
Explicit expressions for various observables in terms of the CKM parameters in (1), used in our paper, are the ones from [10] modified only by adjusting some reference input as stated below.
For ∆M d and ∆M s they are The value 2.307 in the normalization of S 0 (x t ) is its SM value for m t (m t ) = 162.83 GeV.
The central values of |V td | and |V ts | exposed here are chosen to make the overall factors in these formulae to be equal to the experimental values of the two observables. The reference values for B B d F B d and B Bs F Bs are those from the HPQCD collaboration [24] as used by us already in [11]. Correspondingly also the resulting reference values for |V td | and |V ts | agree perfectly with those quoted in [24]. Similar results for ∆M d and ∆M s hadronic matrix elements have been obtained within the HQET sum rules in [25] and [26], respectively. Of importance is also the mixing induced CP-asymmetry [3] S ψK S = sin(2β) = 0.699(17), which implies the value of β in (2). Next [10], The expression above provides an approximation of the exact formula of [15] with an accuracy of 1.5%, in the ranges 38 < |V cb | × 10 3 < 43, 60 • < γ < 75 • , 20 • < β < 24 • . For the rare decays K + → π + νν and K L → π 0 νν we have [10] where we do not show explicitly the parametric dependence on λ = |V us | and set λ = 0.225. The 3.5% uncertainty in K + → π + νν is dominated by the long distance effects in the charm contribution [27], fully negligible in Here r(y s ) summarizes ∆Γ s effects with r(y s ) = 0.935 ± 0.007 within the SM [30][31][32].
As to an excellent accuracy r(y d ) = 1, one has this timē Next where with the expression for |V ts | being an excellent approximation. Therefore, even if β can be determined precisely by measuring S ψK S , the strong dependence of all rare decay branching ratios on |V cb | precludes in the presence of the tensions mentioned above, a useful measurement of γ with the help of a given rare decay within the SM and its confrontation with its tree-level measurements. Similarly, sin γ determined through ε K suffers from the |V cb | 2 dependence. Two messages follow from these observations: • Unless the |V cb | tension will be removed, only direct measurement of γ using tree-level decays will provide a useful determination of γ.
• The same comment applies to the angle β, but here the |V ub | tension is even more important.
As no explicit information on |V cb |, beyond the one resulting from computer codes, is provided by the UT plots, let us turn our attention to the |V cb | − γ plots. They illustrate that considering simultaneously |ε K |, ∆M d and ∆M s and imposing the constraint on β in (7) one is able already now to obtain respectable determination on |V cb | and γ within the SM [11]. In turn future precise measurements of γ in tree-level decays and also experimental improvements on rare decay branching ratios will offer very powerful tests of the SM. In particular when the experts also agree on the unique value of |V cb |.

|V cb | − γ Plots
Using the formulae just listed we can now find |V cb | as a function of γ and β resulting separately from each of the seven magnificant observables considered by us.
∆M d In [10,11] only |V cb | − γ plots for ∆M s , ∆M d and ε K have been presented. One of such plots from [11] is shown in Fig. 2.
An important observation should be made in this plot. The ε K -band is thiner than the one coming from ∆M d . This is dominantly related to the stronger dependence of ε K on |V cb |. This fact makes the action of ε K less useful in the (¯ ,η) plane than that of ∆M d while in the |V cb | − γ plane it is reversed. The formulae above demonstrate this in explicit terms. Moreover, while the action of ∆M s is invisible in a UT-plot it is clearly exhibited in Fig. 2.
It updates the previous prediction of [17], based on different hadronic matrix elements, that exhibited a 2.1σ anomaly. These are the most precise SM predictions for these decays to date. For K + → π + νν and K L → π 0 νν they were obtained by using the experimental values of ε K and S ψK S . The ones for B s,d → µ + µ − using the strategy of [36] and experimental values of ∆M s,d 4 . No information on |V cb | was required to obtain these results and the left-over γ dependence in rare K decay branching ratios, once ε K constraint was imposed, turned out to be negligible in the full range 60 • ≤ γ ≤ 75 • investigated by us. One can verify all these nice properties using the formulae above or inspect the plots in Fig. 15 of [10].
In this context I am astonished by statements made by some computer code practitioners that setting in this strategy these four ∆F = 2 observables to their experimental values one has to assume the absence of new physics (NP). The goal of this strategy is not to make an overall SM fit but to predict the SM branching ratios in question. In the SM there are no NP contributions to ∆F = 2 transitions and no assumption on the absence of NP is needed. More on this in [37].
The 2.7σ anomaly in B s → µ + µ − can also be seen in the |V cb |-independent ratio  Figure 2: The values of |V cb | extracted from ε K , ∆M d and ∆M s as functions of γ with the hadronic matrix elements for ∆M s,d obtained with 2 + 1 + 1 flavours [24]. The green band represents experimental S ψK S constraint on β. From [11].
Yet, it also turns out that the simultaneous description of the data for ∆M d , ∆M s , ε K and S ψK S can be made without any participation of NP which gives an additional support for the SM predictions in (24) and (25). Indeed, as seen in Fig. 2, the SM predictions for ε K , ∆M d and ∆M s turn out to be consistent with each other and with the data for the following values of the CKM parameters [11] |V cb | = 42.6(7) × 10 −3 , γ = 64.6(4.0) • , β = 22.2(7) • , |V ub | = 3.72(13) × 10 −3 . (28) As emphasized in [11] this agreement is only obtained with the hadronic matrix elements with 2+1+1 flavours from the lattice HPQCD collaboration [24]. For 2+1 flavours significant inconsistencies within the SM were found. See Fig. 8 in [11]. All the input parameters used by us are collected in Table 1.
Our value for |V cb | is consistent with the inclusive one from [8] and |V ub | value with the exclusive one from FLAG [9]. The value for γ agrees well with the one from the LHCb in (2).
In order to illustrate the action of the seven observables in the |V cb | − γ plane we show in Fig. 3 the results in the SM setting all uncertainties for transparency reasons to zero. We make the following observations.
• For fixed β = 22.2 • , ε K , K + → π + νν and K L → π 0 νν are represented to an excellent approximation by the same line which is already a very good test of the SM. This is simply because as seen in (19), (20) and (21) the γ dependence in the three observables is practically the same, the fact pointed out first in [22] and strongly emphasized in [10]. The dependence on β is different and this allows to determine within the SM the angle β from any pair of these observables independently of the value of γ. For the pair of the rare K branching ratios this was pointed out in [22]. For the other two pairs in [10]. But the determination of β with the help of the plot in Fig. 3 is not useful and  it is better to use the |V cb |-independent ratios R 0 , R 11 , R 12 of [10] with [16]  (29) and with explicit expressions for R 11 and R 12 given in (90) and (91) of [10], respectively. They all can be derived from (8), (9) and (10) given in the previous section.
• ∆M d and B d → µ + µ − are represented by a single line and a different line represents ∆M s and B s → µ + µ − . This is precisely the illustration of the SM relations pointed out long time ago in [36].
While, as seen in Fig. 2, SM describes ε K , ∆M d , ∆M s simultaneously very well, this not need be the case for the four rare decays in question. This is illustrated in Fig. 4. To obtain these results we have set the branching ratio for B s → µ + µ − to the experimental world average from LHCb, CMS and ATLAS [33][34][35] but decreased its error from 11% down to 5%. For the remaining branching ratios we have chosen values resulting from hypothetical future measurements that differ from the SM predictions in (24) and (25). We kept the errors at 5% as in the case of B s → µ + µ − to exhibit the superiority of rare K decays over rare B decays as far as the determination of |V cb | is concerned. We use then  Figure 4: The impact of hypothetical future measurements of the branching ratios for K + → π + νν, K L → π 0 νν, B d → µ + µ − and B s → µ + µ − as given in (31) and (32)  While the experimental errors are futuristic, we expect that the theoretical errors will go down with time so that the bands in Fig. 4 could apply one day with less accurate meassurements. This plot confirms all the statements made above. The superiority of K L → π 0 νν over the remaining decays is clearly seen. The blue band will be narrowed once the long distance charm contributions to K + → π + νν will be known with higher precision from lattice QCD calculations [38] than they are known now [27].

Conclusions and Outlook
In the present paper we have emphasized, resurrecting by now almost thirty years old ideas of [14,22,23], that the rare K and B decay branching ratios, being subject to small hadronic uncertainties, could soon give us a powerful tool to determine the CKM parameters, in particular the controvertial parameter |V cb |. They could also provide a useful insight in the value of γ beyond its tree-level determinations. In this context we have proposed to monitor future progress on the determination on |V cb | and γ in the |V cb | − γ plane rather then in the (¯ ,η) plane used in the context of the common UT-fits. We also reemphasized in this context the important role of ∆M s , ∆M d and ε K . To this end we derived seven expressions m Bs = 5366.8(2)MeV [3] m B d = 5279.58 (17) [9], PDG [3] and HFLAV [39].
for |V cb | by means of which this CKM element can be determined. First in the case of with the relevant expression for |V cb | as a function of γ, β and the observable involved given in (17), (18) and (19), respectively. The corresponding four formulae for |V cb | from rare decays are given in (20), (21), (22) and (23), respectively. The very good consistency of the three observables (33) with each other within the SM allowed, after the imposition of the S ψK S constraint, a satisfactory determination of the four CKM parameters in [11] and given in (28). As emphasized in [11], this agreement is only obtained with the hadronic matrix elements with 2 + 1 + 1 flavours from the lattice HPQCD collaboration [24]. For 2 + 1 flavours significant inconsistencies within the SM are found. See Fig. 8 in [11].
In the present paper this agreement is seen in the plot in Fig. 2). The 2.7σ anomaly in B s → µ + µ − found in [11] would increase with the improved measurement as assumed in (32) to 5.2σ. In the |V cb | − γ plane it will be signalled by the inconsistency with the SM yellow disc on which the SM prediction for this decay in (25) is based.
The data on the remaining rare decay branching ratios allow still for significant NP contributions and inconsistencies between various determinations of |V cb | as a function of γ from different decays could be found one day. We illustrated it in Fig. 4. In this context a precise measurement of γ by the LHCb and Belle II collaborations and the improvements on β will allow a very precise determination of |V cb | within the SM, first with the help of precisely measured |ε K | and later K + → π + νν and K L → π 0 νν.
Simultaneously the very accurate |V cb | independent SM predictions for rare decay branching ratios found in [10,11] and recalled here in (24) and (25) will play, in case of inconsistencies in the |V cb | − γ plane, an important role in the identification of a particular NP at work. The 16 |V cb |-independent ratios of various flavour observables derived in [10] will also be useful in this context. We are looking forward to new data from LHCb, NA62, KOTO and Belle II collaborations as well as to improved hadronic matrix elements from LQCD which will allow one to use this strategy and the ones outlined in [10,11] more efficiently than it is possible now.