2022 Update on $\varepsilon_K$ with lattice QCD inputs

We present recent updates for $\varepsilon_K$ determined directly from the standard model (SM) with lattice QCD inputs such as $\hat{B}_K$, $|V_{cb}|$, $|V_{us}|$, $\xi_0$, $\xi_2$, $\xi_\text{LD}$, $f_K$, and $m_c$. We find that the standard model with exclusive $|V_{cb}|$ and other lattice QCD inputs describes only 65% of the experimental value of $|\varepsilon_K|$ and does not explain its remaining 35%, which leads to a strong tension in $|\varepsilon_K|$ at the $5.1\sigma \sim 3.9\sigma$ level between the SM theory and experiment. We also find that this tension disappears when we use the inclusive value of $|V_{cb}|$ obtained using the heavy quark expansion based on the QCD sum rule approach, although this inclusive tension is small ($\approx 1.4\sigma$) but keeps increasing as time goes on.

Here, we follow the color convention of our previous papers [1][2][3][4][5][6][7] in Tables 1-8. We use the red color for the new input data which is used to evaluate . We use the blue color for the new input data which is not used for some obvious reason.

Input parameters: Wolfenstein parameters
In Table 1 (a), we present the most updated Wolfenstein parameters available in the market. As explained in Ref. [3,7], we use the results of angle-only-fit (AOF) in Table 1 (a) in order to avoid unwanted correlation between ( , | |), and (¯,¯). We determine from | | which is obtained from the ℓ2 and ℓ3 decays using lattice QCD inputs for form factors and decay constants as explained in Ref. [8]. We determine the parameter from | |.

Input parameters: | |
In Table 2 (a) and (b), we present recently updated results for exclusive | | and inclusive | | respectively. In Table 2 (a), we summarize results for exclusive | | obtained by various groups: HFLAV, BELLE, BABAR, FNAL/MILC, LHCb, and FLAG. Results from LHCb comes from analysis on → * ℓ¯decays which are not available in the -factories. Since results for decay channels have poor statistics, we drop out them here without loss of fairness. The rest of results for exclusive | | have comparable size of errors and are consistent with one another within 1.0 . In addition, we find that the results are consistent between the CLN and BGL analysis, after the clamorous debates [3,14].
In Table 2 (b), we present recent results for inclusive | |. The Gambino group has reported updated results for inclusive | | in 2021. There are a number of attempts to calculate inclusive | | in lattice QCD, but they belong to a category of exploratory study rather than that of precision measurement yet [15].

Input parameter 0
The absorptive part of long distance effects on is parametrized into 0 .
There are two independent methods to determine 0 in lattice QCD: the indirect and direct methods. The indirect method is to determine 0 using Eq. (1) with lattice QCD results for 2 combined with experimental results for / , , and . The direct method is to determine 0 directly using the lattice QCD results for Im 0 , combined with experimental results for Re 0 .
In Table 3 (a), we summarize experimental results for Re 0 and Re 2 . In Table 3 (b), we summarize lattice results for Im 0 and Im 2 calculated by RBC-UKQCD. In Table 3 (c), we summarize results for 0 which is obtained using results in Table 3 (a) and (b).
Here, we use results of the indirect method for 0 to evaluate , since its systematic and statistical errors are much smaller than those of the direct method.

Input parameters:ˆ, LD , and others
In FLAG 2021 [8], they report lattice QCD results forˆwith = 2, = 2 + 1, and = 2 + 1 + 1. Here, we use the results forˆwith = 2 + 1, which is obtained by taking an  Table 3: Results for 0 . Here, we use the same notation as in Table 2. average over the four data points from BMW 11, Laiho 11, RBC-UKQCD 14, and SWME 15 in  The dispersive long distance (LD) effect is defined as As explained in Refs. [3], there are two independent methods to estimate LD : one is the BGI estimate [31], and the other is the RBC-UKQCD estimate [32,33]. The BGI method is to estimate  the size of LD using chiral perturbation theory as follows, The RBC-UKQCD method is to estimate the size of LD as follows, Here, we use both methods to estimate the size of LD .
In Table 1 (b), we present higher order QCD corrections: with , = , . A new approach using − unitarity instead of − unitarity appeared in Ref. [34], which is supposed to have a better convergence with respect to the charm quark mass. But we have not incorporated this into our analysis yet, which we will do in near future.
In Table 4 (b), we present other input parameters needed to evaluate .

Quark masses
In Table 5, we present the charm quark mass ( ) and top quark mass ( ). From FLAG 2021 [8], we take the results for ( ) with = 2 + 1, since there is some inconsistency among the lattice results of various groups with = 2 + 1 + 1. For the top quark mass, we use the PDG 2022 results for the pole mass to obtain ( ).    In Table 6 (a), we plot top pole mass as a function of time. Here we find that the average value drifts downward a little bit and the error shrinks fast as time goes on, thanks to accumulation of high statistics in the LHC experiments. The data for 2020 is dropped out intentionally to reflect on the absence of Lattice 2020 due to COVID-19.

boson mass
In Fig. 7 (a), we plot ( boson mass) as a function of time. The corresponding results for are summarized in Table 7 (b). In Fig. 7 (a), the light-green band represents the standard model (SM) prediction, the red circles represents the PDG results, and the brown cross represents the CDF-2022 result. The upside is that the CDF-2022 result is the most precise and latest experimental result for . The downside, however, is that it has a 6.9 tension from that of SM-2022 (the standard model prediction). Here, we use the SM-2022 result for to evaluate .

Results for
In Fig. 1, we show results for | | evaluated directly from the standard model (SM) with lattice QCD inputs given in the previous sections. In Fig. 1 (a), the blue curve represents the theoretical evaluation of | | obtained using the FLAG-2021 results forˆ, AOF for Wolfenstein parameters, the [FNAL/MILC 2022, BGL] results for exclusive | |, results for 0 with the indirect method, and the RBC-UKQCD estimate for LD . The red curve in Fig. 1 represents the experimental results for | |. In Fig. 1 (b), the blue curve represents the same as in Fig. 1 (a) except for using the 1S scheme results for the inclusive | |. Our results for | | SM and Δ are summarized in Table 8. Here, the superscript SM represents the theoretical expectation value of | | obtained directly from the SM. The superscript Exp represents the experimental value of | | = 2.228(11) × 10 −3 . Results in Table 8 (a) are obtained using the RBC-UKQCD estimate for LD , and those in Table 8 (b) are obtained using the BGI estimate for LD . In Table 8 (a), we find that the theoretical expectation values of | | SM with lattice QCD inputs (with exclusive | |) has 5.12 ∼ 3.93 tension with the experimental value of | | Exp , while there is no tension with inclusive | | (obtained using heavy quark expansion and QCD sum rules). We also find that the tension with inclusive | | is small but keeps increasing with respect to time.
In Fig. 2 (a), we show the time evolution of Δ starting from 2012 till 2022. In 2012, Δ was 2.5 , but now it is 5.05 with exclusive | | (FNAL/MILC-2022, BGL). In Fig. 2    show the time evolution of the average Δ and the error Δ during the period of 2012-2022. At present, we find that the largest error (≈ 50%) in | | SM comes from | |. Hence, it is essential to reduce the errors in | | as much as possible. To achieve this goal, there is an on-going project to extract exclusive | | using the Oktay-Kronfeld (OK) action for the heavy quarks to calculate the form factors for¯→ ( * ) ℓ¯decays [38][39][40][41][42][43][44].
A large portion of interesting results for | | SM and Δ could not be presented in Table 8 and in Fig. 2 due to lack of space: for example, results for | | SM obtained using exclusive | | (FLAG 2021), results for | | SM obtained using 0 determined by the direct method, and so on. We plan to report them collectively in Ref. [45].