Neutral $B$-meson mixing from full lattice QCD at the physical point

We calculate the bag parameters for neutral $B$-meson mixing in and beyond the Standard Model, in full four-flavor lattice QCD for the first time. We work on gluon field configurations that include the effect of $u$, $d$, $s$, and $c$ sea quarks with the highly improved staggered quark (HISQ) action at three values of the lattice spacing and with three $u/d$ quark masses going down to the physical value. The valence $b$ quarks use the improved nonrelativistic QCD action and the valence light quarks, the HISQ action. Our analysis was blinded. Our results for the bag parameters for all five operators are the most accurate to date. For the Standard Model operator between $B_s$ and $B_d$ mesons we find ${\widehat{B}}_{B_s}=1.232(53)$ and ${\widehat{B}}_{B_d}=1.222(61)$. Combining our results with lattice QCD calculations of the decay constants using HISQ quarks from the Fermilab/MILC Collaboration and with experimental values for $B_s$ and $B_d$ oscillation frequencies allows determination of the Cabibbo-Kobayashi-Maskawa (CKM) elements $V_{ts}$ and $V_{td}$. We find $|V_{ts}|=0.04189(93)$, $|V_{td}|=0.00867(23)$, and $|V_{ts}|/|V_{td}|=0.2071(27)$. Our results agree well (within $2\sigma$) with values determined from CKM unitarity constraints based on tree-level processes (only). Using a ratio to $\mathrm{\Delta }{M}_{s,d}$ in which CKM elements cancel in the Standard Model, we determine the branching fractions $\mathrm{Br}(B_s\to {\mu }^{+}{\mu }^{-})=3.81(18)\times {10}^{-9}$ and $\mathrm{Br}(B_d\to {\mu }^{+}{\mu }^{-})=1.031(54)\times {10}^{-10}$. We also give results for matrix elements of the operators $R_0$, $R_1$, and ${\tilde{R}}_1$ that contribute to neutral $B$-meson width differences.

Figure 1

An example from the Standard Model of a mechanism that mixes the neutral $B_q$ and ${\overline{B}}_q$. The amplitude is well approximated by a contact term for matrix elements between $B_q$-meson states.

Figure 2

Coefficients $z_{ij}$ of $\mathcal{O}({\alpha }_s)$ terms in the matching of lattice NRQCD-HISQ 4-quark operators to the $\overline{\mathrm{MS}}$ scheme plotted as a function of the bare NRQCD $b$ quark mass in lattice units. The top plot shows the diagonal coefficients ($i=j$) that enter the renormalization of a given operator; the lower plot shows the off-diagonal coefficients ($i\ne j$) corresponding to the mixing of different operators. See Eq. (5) for the definition of $z_{ij}$ and Table 2 for the values. The $ij$ values are indicated in the key. “$A_0$” refers to the $\mathcal{O}({\alpha }_s)$ coefficient for the renormalization of the temporal axial current ($z_{A_0}$ in Table 2).

Figure 3

Comparison of the $⟨B_s|O_n|{\overline{B}}_s{⟩}_{\overline{\mathrm{MS}}}^{(m_b)}/(f_{B_s}{M}_{B_s}{)}^2$ values from individual configurations sets (colored data points) with the final extrapolated values (gray bands and dotted lines) for each 4-quark operator. Errors shown include correlated uncertainties from operator normalization and lattice spacing effects as discussed in the text. The data are plotted versus $a{m}_b$, falling into three groups corresponding to lattice spacings of 0.09, 0.12, and 0.15 fm. Results are shown for three different values of light-quark mass $m_l\equiv (m_u+m_d)/2$ corresponding to $m_l/m_s=1/5$ (green, $\times \mathrm{s}$), $m_l/m_s=1/10$ (blue, boxes), and the physical mass (red, circles). The dotted lines show the extrapolated values, while the gray bands show the $\pm 1\sigma$ uncertainty in those values. The analogous figure for $B_d$ mesons is very similar.

Figure 4

Comparison of the ratio of bag parameters $B_{B_s}^{(n)}/B_{B_d}^{(n)}$ from individual configuration sets (colored data points) with the final extrapolated values (gray bands and dotted lines) for each 4-quark operator. The data are plotted versus $m_{\pi }^2$, falling into three groups corresponding approximately to the physical value, 2.7 times the physical value, and 5.4 times the physical value. Results are shown for three different lattice spacings corresponding approximately to 0.15 fm (green, $\times \mathrm{s}$), 0.12 fm (blue, boxes), and 0.09 fm (red, circles). The dotted lines show the extrapolated values, while the gray bands show the $\pm 1\sigma$ uncertainty in those values.

Figure 5

A comparison of our results (red filled circles at $n_f=4$) to previous lattice QCD values for the $B_s$ bag parameters $B_{B_s}(m_b)$ in the $\overline{\mathrm{MS}}$ scheme for all five SM and BSM operators. Previous results come from the Fermilab/MILC Collaboration on $n_f=3$ gluon field configurations (blue crosses) [8] and the ETM Collaboration on $n_f=2$ gluon field configurations (purple filled diamonds) [7]. Note that the ETM results for $O_4$ and $O_5$ have been converted to the definition of the bag parameter given in Eq. (4). The filled green square at $n_f=3$ for the $O_1$ operator comes from an earlier HPQCD calculation using NRQCD $b$ quarks [4]. The $n_f=2$ results are missing $s$ sea quarks, whose impact cannot be estimated perturbatively (and no uncertainty is included for this in the error bars). It is therefore unclear what level of agreement to expect between these results and those for $n_f=3$ and 4. Since we do not expect missing $c$ in the sea to have a significant impact on the bag parameters [8] we can meaningfully compare $n_f=3$ and $n_f=4$. The grey bands are the weighted average of our new results with those of [8], and the average value of the bag parameter $B_{B_s}^{(n)}(m_b)$ for each operator $O_n$ is indicated in that panel. We include a vertical line at value 1.0 for comparison to the vacuum saturation approximation.

Figure 6

A comparison of our results (red filled circles at $n_f=4$) to previous lattice QCD values for the ratio of $B_s$ to $B_d$ bag parameters for all five SM and BSM operators. Previous results come from the Fermilab/MILC Collaboration on $n_f=3$ gluon field configurations (blue crosses) [8] using their quoted correlations to reconstruct the ratio. Since we do not expect missing $c$ in the sea to have a significant impact on the bag parameters [8] we can meaningfully compare $n_f=3$ and $n_f=4$. The grey bands are the weighted average of these two sets of results and the average value for each operator is indicated in that panel. For $O_1$ at $n_f=3$ we also show previous results from HPQCD (green filled square) using NRQCD $b$ quarks [4] and RBC/UKQCD (purple filled diamond) using domain-wall quarks with masses of $m_c$ and above and extrapolating results to the $b$ quark mass [9]. We include a vertical line at value 1.0 to make clear which ratios are above, and which below, this value.

Figure 7

A comparison of our results (red filled circles at $n_f=4$) to previous lattice QCD values for the combination of decay constant and square root of bag parameter $f_{B_s}\sqrt{{\widehat{B}}_{B_s}^{(1)}}$. Previous results (blue filled squares) come from the Fermilab/MILC Collaboration [8] and from HPQCD [4] on $n_f=3$ gluon field configurations. The Fermilab/MILC results include a 1% uncertainty for missing $c$ in the sea. The grey band is the weighted average of our new results and those of [8] and the new lattice QCD average value is quoted at the top.

Figure 8

A comparison of our results (red filled circles at $n_f=4$) for $\xi$, defined in Eq. (13), to previous lattice QCD values for $n_f=3$ (filled blue squares). Previous results come from the Fermilab/MILC Collaboration [8] and from HPQCD [4] using calculations at the physical $b$ quark mass. Results are also shown from RBC/UKQCD using domain-wall quarks and extrapolating to the $b$ from the $c$ quark region and above [9] and using static (infinitely massive) $b$ quarks [6]. The grey band is the weighted average of our new results and those of [8] with the result for the average quoted above it.

Figure 9

A comparison of $\pm 1\sigma$ constraints on $V_{ts}$ and $V_{td}$ from experimental results on $B_s$ and $B_d$ oscillation frequencies compared to SM calculations. This is an update of Fig. 7 in [39] to include the results presented here. The lattice QCD constraints shown come from the following: this paper, dark grey; [8], red; [9], light blue, $|V_{ts}|/|V_{td}|$ ratio only. The light blue lozenge is from sum rules [39]. The lozenges with dashed boundaries include a full unitarity triangle fit: light pink is from CKMfitter [59, 61] and orange from UTFit [60, 62]. The green lozenge with dotted boundary is the result of a unitarity triangle fit for tree-level processes only from CKMfitter.

Figure 10

A comparison of the SM branching fractions for $B_s$ and $B_d$ to decay to ${\mu }^{+}{\mu }^{-}$ from lattice QCD results with the current experimental measurements. The grey lozenge shows results from our calculation here of the bag parameters and a ratio to experimental results for $\mathrm{\Delta }{M}_q$ [Eqs. (20) and (22)]. The red lozenge shows results from a Fermilab/MILC calculation of $B_d$ and $B_s$ decay constants, combined with input CKM elements [13]. The blue band shows the current experimental average for $\mathrm{Br}(B_s\to {\mu }^{+}{\mu }^{-})$ [11]; only an upper bound exists for $\mathrm{Br}(B_d\to {\mu }^{+}{\mu }^{-})$.

Figure 11

A schematic diagram of the 3-point function for $B-\overline{B}$ mixing. ${\mathcal{O}}_n$ marks the insertion at time $t$ of a 4-quark operator, when the meson and antimeson operators are located at times 0 and $T$.

Figure 12

A comparison of chiral logarithm terms in $m_{\pi }$ that appear in the continuum chiral perturbation theory for $R_d$ [Eq. (b5)] with those in staggered chiral perturbation theory [Eq. (b4)] for HISQ quarks. The solid black line gives the continuum form [Eq. (b5)], and the dashed blue and red curves give the staggered form in Eq. (b4) for HISQ quarks on very coarse, coarse, and fine lattices, respectively.

Figure 13

A comparison of the function $J(m_{\pi },\mathrm{\Delta })/{\mathrm{\Lambda }}_{\chi }^2$ to the chiral logarithm to which it is equal at $\mathrm{\Delta }=0$ ($(m_{\pi }^2/{\mathrm{\Lambda }}_{\chi }^2)\log (m_{\pi }^2/{\mu }_{\chi }^2)$) as a function of $m_{\pi }$.

Figure 14

The function $\tilde{X}$, defined in Eq. (b8), plotted for HISQ light quarks against the square of the pion mass. We give three curves, for lattice spacing values corresponding to our very coarse, coarse, and fine lattices. This function appears in the “wrong-sign” tadpole terms in heavy meson staggered chiral perturbation theory.

Figure 15

The ratio of approximate to exact eigenvalues of the correlation matrix for $N_G=512$ correlated data points is plotted versus the size of the exact eigenvalues divided by the maximum eigenvalue. The approximate eigenvalues are determined [Eq. (d2)] from random samples of different sizes ranging (by powers of 2) from $N_s=N_G/4$ to $N_s=32N_G$. The red (dashed) line corresponds to $N_s=N_G$. Both sets of eigenvalues (approximate and exact) are ordered from smallest to largest.

Figure 16

Correlation-matrix eigenvalues computed from $N_s$ random samples of $N_G$ correlated data points are compared with eigenvalues computed from bootstrapped copies of the random sample. Results are shown for $N_G=50$ (top) and $N_G=500$ (bottom), with $N_s=4N_G$ in each case. The blue data points are ratios of bootstrapped eigenvalues to eigenvalues from the random sample itself; the error bars show the spread across different bootstrapped copies. The solid red line (mostly hidden in the bottom panel) shows ratios of eigenvalues from the random sample to those from the underlying distribution used to generate the random sample. The locations of the SVD cuts $\kappa$ are shown by the vertical dashed red lines.