$\mathrm{\Delta }S=2$ and $\mathrm{\Delta }C=2$ bag parameters in the standard model and beyond from $N_f=2+1+1$ twisted-mass lattice QCD

We present unquenched lattice QCD results for the matrix elements of four-fermion operators relevant to the description of the neutral $K$ and $D$ mixing in the standard model and its extensions. We have employed simulations with $N_f=2+1+1$ dynamical sea quarks at three values of the lattice spacings in the interval 0.06–0.09 fm and pseudoscalar meson masses in the range 210–450 MeV. Our results are extrapolated to the continuum limit and to the physical pion mass. Renormalization constants have been determined nonperturbatively in the RI-MOM scheme. In particular, for the kaon bag parameter, which is relevant for the ${\overline{K}}^0-K^0$ mixing in the standard model, we obtain $B_K^{\mathrm{RGI}}=0.717(24)$.

Figure 1

A compilation of unquenched lattice results for the RGI value of the $B_K$ parameter. From top to bottom results are taken from Refs. [12, 32, 34, 35, 36, 37, 38, 39]. Circle, squares, and triangle correspond to $N_f=2$, $N_f=2+1$, and $N_f=2+1+1$ dynamical quark computations, respectively. The full blue square indicates the FLAG average [32] over $N_f=2+1$ data. For reader’s convenience some information on the basic features of each computation is also given.

Figure 2

A compilation of $K^0$ meson bag parameters $B_i$, $i=2,\dots ,5$. From top to bottom data have been taken from Refs. [12, 40, 41]. The work reported in Ref. [40] is an updated computation of Ref. [37]. The label “CL” stands for continuum limit computation. Work in progress by the RBC/UKQCD has been reported in [42].

Figure 3

Results for the $D^0$ meson bag parameters $B_i$, $i=1,\dots ,5$ obtained by the ETMC with $N_f=2+1+1$ (this paper) and $N_f=2$ [43] dynamical flavor lattice simulations.

Figure 4

${\mathcal{R}}_i^{(b)}(x_0)$ $(i=1,\dots ,5)$ simulation points plotted against $x_0/T_{\text{sep}}$ for the ${\overline{K}}^0-K^0$ case: (a) $\beta =1.90$, $(a{\mu }_{\ell },a{\mu }_s)=(0.0030,0.0185)$, $\text{volume}={32}^3\times 64$; (b) $\beta =1.95$, $(a{\mu }_{\ell },a{\mu }_s)=(0.0025,0.0180)$, $\text{volume}={32}^3\times 64$; (c) $\beta =2.10$, $(a{\mu }_{\ell },a{\mu }_s)=(0.0015,0.0151)$, volume $={48}^3\times 96$. The dotted lines delimit the plateau region. Points for ${\mathcal{R}}_1^{(b)},\dots ,{\mathcal{R}}_4^{(b)}$ at $\beta =2.10$ have been slightly shifted upward by $+0.05$ for accommodating data from all three $\beta$’s in the same plotting scale.

Figure 5

${\mathcal{R}}_i^{(b)}(x_0)$ $(i=1,\dots ,5)$ simulation points plotted against $x_0/T_{\text{sep}}$ for the ${\overline{D}}^0-D^0$ case: (a) $\beta =1.90$, $(a{\mu }_{\ell },a{\mu }_c)=(0.0030,0.25)$, $\text{volume}={32}^3\times 64$; (b) $\beta =1.95$, $(a{\mu }_{\ell },a{\mu }_c)=(0.0025,0.22)$, volume $={32}^3\times 64$; (c) $\beta =2.10$, $(a{\mu }_{\ell },a{\mu }_c)=(0.0015,0.17)$, $\text{volume}={48}^3\times 96$. The dotted lines delimit the plateau region. Points for ${\mathcal{R}}_1^{(b)}$ and ${\mathcal{R}}_2^{(b)}$ at $\beta =2.10$ have been slightly shifted upward by $+0.1$ for accommodating data from the three $\beta$’s in the same plotting scale.

Figure 6

Combined chiral and continuum extrapolation for the five $B_i$’s of the ${\overline{K}}^0-K^0$ case. Bag parameters are renormalized in the $\overline{\mathrm{MS}}$ scheme of [31] at the scale of 3 GeV. Left and right panels correspond to M1-type and M2-type four-fermion RCs, respectively, following the nomenclature of Ref. [73]. In each panel open circles and stars represent the value at the physical point corresponding to the linear and NLO ChPT fits, respectively.

Figure 7

Combined chiral and continuum extrapolation for the five ${\overline{D}}^0-D^0$ bag parameters, $B_i$, renormalized in the $\overline{\mathrm{MS}}$ scheme of [31] at the scale of 3 GeV. Left and right panels correspond to M1-type and M2-type four-fermion RCs, respectively, following the nomenclature of Ref. [73]. In each panel open circles and stars represent the value at the physical point corresponding to the linear and NLO HMChPT fit, respectively. For $B_1,B_2$, and $B_3$ the polynomial (linear) and the HMChPT fits are practically indistinguishable.

Figure 8

Combined chiral and continuum extrapolation in the ${\overline{K}}^0-K^0$ case of the combinations defined in Eq. (3.2). We use M1-type RCs in the $\overline{\mathrm{MS}}$ scheme of [31] at 3 GeV. For the combinations shown in the right panels the polynomial (linear) and the NLO ChPT fit Ansätze coincide.

Figure 9

Distribution of $B_{1,\dots ,5}$ results for the neutral $K$-mixing renormalized in the $\overline{\mathrm{MS}}$ scheme of Ref. [31] at the scale of 3 GeV. The solid vertical line marks the central value (average) while the gray band indicates the systematic error determined from Eq. (3.7).

Figure 10

Same as in Fig. 9 but for the neutral $D$-mixing.

Figure 11

Graphical representation of the error budget owing to the lattice computation (i.e., without including the estimate for the systematic uncertainty due to the perturbative matching between the ${\mathrm{RI}}^{\prime }$ and $\overline{\mathrm{MS}}$ schemes) for the $K$ bag parameters, as reported in Table 4.

Figure 12

Same as in Fig. 11 for the $D$ bag parameters.

Figure 13

Fitting procedure for the GB pole subtraction on the block-diagonal elements of the dynamic matrix. We show two examples from the coarsest lattice, in particular for the ensembles A2m (left) and A2p (right), at a relatively small value of momentum, namely, $(a\tilde{p}{)}^2\simeq 1.57$.

Figure 14

Same as in Fig. 13 but at a larger value of momentum, namely, $(a\tilde{p}{)}^2\simeq 2.19$.

Figure 15

Same as in Fig. 13 at $\beta =2.10$, for the ensembles C2m (left) and C2p (right) and $(ap{)}^2\simeq 1.57$.

Figure 16

For each of block-diagonal matrix element we form the product of $(a^2{\tilde{p}}^2{)}^2$ with the fit parameter $\mathcal{D}(a^2{\tilde{p}}^2)$ that is associated to the term of the double GB pole [cf. Eq. (b5)] and we plot it against $(a^2{\tilde{p}}^2)$ for the ensembles A1p and A1m of $\beta =1.90$.

Figure 17

Example of the sea chiral extrapolation of the RC estimators at three values of $\beta$ computed at $(a\tilde{p}{)}^2\simeq 1.57$.

Figure 18

Best linear fits [cf. Eq. (b6)] in the momentum interval $(a\tilde{p}{)}^2\in [1.8:2.2]$ for the block-diagonal RCs, $Z_{ij}$, at each value of $\beta$, renormalized in the $\overline{\mathrm{MS}}$ scheme at the scale of 3 GeV.

Figure 19

$B_{1,\dots ,5}$ estimates computed at the physical strange quark value and at a fixed reference quark mass ${\mu }_{\ell }^{\text{ref}}=12.0\mathrm{MeV}$ plotted against $a^2$ for the three values of the lattice spacing. We compare the scaling behavior of the bag-parameter estimates computed with the four possible combinations of M1 and M2 type for the four- and two-fermion RCs. The best linear fit in $a^2$ and the corresponding C.L. value for each RC combination is also shown.

Figure 20

The behavior of the off-diagonal mixing coefficients ${\mathrm{\Delta }}_{ij}$ as a function of $(a\tilde{p}{)}^2$ for $\beta =2.10$.

Figure 21

${\mathrm{\Delta }}_{ij}$ computed with the M2 method against $a^2$.