Anatomy of new physics in $B$-$\overline{B}$ mixing

We analyze three different new physics scenarios for $\Delta F=2$ flavor-changing neutral currents in the quark sector in the light of recent data on neutral-meson mixing. We parametrize generic new physics contributions to $B_q\mathrm{\text{−}}{\overline{B}}_q$ mixing, $q=d$, $s$, in terms of one complex quantity ${\Delta }_q$, while three parameters ${\Delta }_K^{tt}$, ${\Delta }_K^{ct}$, and ${\Delta }_K^{cc}$ are needed to describe $K\mathrm{\text{−}}\overline{K}$ mixing. In scenario I, we consider uncorrelated new physics contributions in the $B_d$, $B_s$, and $K$ sectors. In this scenario, it is only possible to constrain the parameters ${\Delta }_d$ and ${\Delta }_s$ whereas there are no nontrivial constraints on the kaon parameters. In scenario II, we study the case of minimal flavor violation (MFV) and small bottom Yukawa coupling, where $\Delta \equiv {\Delta }_d={\Delta }_s={\Delta }_K^{tt}$. We show that $\Delta$ must then be real, so that no new $CP$ phases can be accommodated, and express the remaining parameters ${\Delta }_K^{cc}$ and ${\Delta }_K^{ct}$ in terms of $\Delta$ in this scenario. Scenario III is the generic MFV case with large bottom Yukawa couplings. In this case, the kaon sector is uncorrelated to the $B_d$ and $B_s$ sectors. As in the second scenario one has ${\Delta }_d={\Delta }_s\equiv \Delta$, however, now with a complex parameter $\Delta$. Our quantitative analyses consist of global Cabibbo-Kobayashi-Maskawa (CKM) fits within the Rfit frequentist statistical approach, determining the standard model parameters and the new physics parameters of the studied scenarios simultaneously. We find that the recent measurements indicating discrepancies with the standard model are well accommodated in Scenarios I and III with new mixing phases, with a slight preference for Scenario I that permits different new $CP$ phases in the $B_d$ and $B_s$ systems. Within our statistical framework, we find evidence of new physics in both $B_d$ and $B_s$ systems. The standard model hypothesis ${\Delta }_d={\Delta }_s=1$ is disfavored with $p$-values of $3.6\sigma$ and $3.3\sigma$ in Scenarios I and III, respectively. We also present an exhaustive list of numerical predictions in each scenario. In particular, we predict the $CP$ phase in $B_s\to J/\psi \varphi$ and the difference between the $B_s$ and $B_d$ semileptonic asymmetries, which will be both measured by the LHCb experiment.

Figure 1

Constraint on the angle $\gamma$ from a combined analysis of $B\to D^{(*)}{K}^{(*)}$ decays.

Figure 2

Constraint on the angle $\alpha$ from the isospin analyses of $B\to \pi \pi$ and $B\to \rho \rho$ decays and a Dalitz plot analysis of $B\to \rho \pi$ decays. In the presence of new physics in $B_d$ mixing, the quantity shown is $\pi -\gamma -\beta -{\varphi }_d^{\Delta }/2$.

Figure 3

Constraint on the CKM $(\overline{\rho },\overline{\eta })$ coordinates from the global standard model CKM fit. Regions outside the shaded (colored) areas have $\mathrm{C}.\mathrm{L}.>95.45\%$. For the combined fit the (yellow) area inscribed by the contour line at the apex of the triangle represents points with $\mathrm{C}.\mathrm{L}.<95.45\%$. The shaded area inside this region represents points with $\mathrm{C}.\mathrm{L}.<68.3\%$.

Figure 4

Constraint in the [$\sin 2\beta$, $\mathcal{B}(B\to \tau \nu )$] plane. The colored (shaded) constraint represents the prediction for these quantities from the global fit when these inputs are removed while the cross represents the measurements with a $1\sigma$ uncertainty.

Figure 5

Comparison between the ${\widehat{\mathcal{B}}}_{B_d}$ parameter used in the global fit and its prediction. The colored (shaded) constraint represents the prediction of ${\widehat{\mathcal{B}}}_{B_d}$ quantities from the global fit while the blue point represents the input value with the corresponding $1\sigma$ uncertainty as used in the global fit.

Figure 6

Constraint on $f_{B_d}$ and $f_{B_d}\sqrt{{\widehat{\mathcal{B}}}_{B_d}}$. The horizontal (green) band shows the 95.45% C.L. constraint on $f_{B_d}$ when $\mathcal{B}(B\to \tau \nu )$ is included in the fit. The vertical (orange) band represents the 95.45% C.L. constraint on $f_{B_d}\sqrt{{\widehat{\mathcal{B}}}_{B_d}}$ thanks to the $\Delta m_d$ measurement. The combined 68.3% C.L. and 95.45% C.L. constraint is shown as oval (red and yellow) regions, respectively. The black point shows the $1\sigma$ uncertainty on $f_{B_d}$ and $f_{B_d}\sqrt{{\widehat{\mathcal{B}}}_{B_d}}$ as used in the fit.

Figure 7

Constraint on the CKM $(\overline{\rho },\overline{\eta })$ coordinates, using only the inputs which are not affected by new physics in mixing: $|V_{ud}|$, $|V_{us}|$, $|V_{cb}|$, $|V_{ub}|$ from semileptonic decays and from $B\to \tau \nu$, $\gamma$ [directly and from the combination $\gamma (\alpha )$ of $\alpha -{\varphi }_d^{\Delta }/2$ and $\beta +{\varphi }_d^{\Delta }/2$]. Regions outside the shaded areas have $\mathrm{C}.\mathrm{L}.>95.45\%$. For the combined fit, two solutions are available: the usual solution corresponds to the (yellow) area at the apex of the triangle (points with $\mathrm{C}.\mathrm{L}.<95.45\%$, the shaded region corresponding to points with $\mathrm{C}.\mathrm{L}.<68.3\%$), and the second solution corresponds to the (purple) dark region for $\overline{\eta }<0$.

Figure 8

Constraint on the complex parameter ${\Delta }_d$ from the fit in scenario I. For the individual constraints the shaded areas represent regions with $\mathrm{C}.\mathrm{L}.<68.3\%$. For the combined fit the shaded (red) area shows the region with $\mathrm{C}.\mathrm{L}.<68.3\%$ while the two additional contour lines inscribe the regions with $\mathrm{C}.\mathrm{L}.<95.45\%$, and $\mathrm{C}.\mathrm{L}.<99.73\%$, respectively.

Figure 9

Constraint on the complex parameter ${\Delta }_s$ from the fit in scenario . For the individual constraints the shaded areas represent regions with $\mathrm{C}.\mathrm{L}.<68.3\%$. For the combined fit the shaded (red) area shows the region with $\mathrm{C}.\mathrm{L}.<68.3\%$ while the two additional contour lines inscribe the regions with $\mathrm{C}.\mathrm{L}.<95.45\%$, and $\mathrm{C}.\mathrm{L}.<99.73\%$, respectively.

Figure 10

Fit result for the complex parameter ${\Delta }_s$ in scenario I when excluding $\mathcal{B}(B\to \tau \nu )$ from the list of inputs. For the combined fit the shaded (red) area shows the region with $\mathrm{C}.\mathrm{L}.<68.3\%$ while the two additional contour lines inscribe the regions with $\mathrm{C}.\mathrm{L}.<95.45\%$, and $\mathrm{C}.\mathrm{L}.<99.73\%$, respectively.

Figure 11

Fit result for the complex parameter ${\Delta }_d$ in scenario I when excluding $\mathcal{B}(B\to \tau \nu )$ from the list of inputs. For the combined fit the shaded (red) area shows the region with $\mathrm{C}.\mathrm{L}.<68.3\%$ while the two additional contour lines inscribe the regions with $\mathrm{C}.\mathrm{L}.<95.45\%$, and $\mathrm{C}.\mathrm{L}.<99.73\%$, respectively.

Figure 12

New physics scenario I. Indirect constraint on the $CP$ phase ${\varphi }_s^{\psi \varphi }$, compared with the direct measurement by the Tevatron: previous world average [83] (in black) and the summer 2010 CDF and D0 updates [31, 32] (in blue). The dotted line represents the full scenario I fit with the previous world average for $a_{\mathrm{fs}}$, while the solid (green) curve is the update after the D0 evidence for a nonzero dimuonic asymmetry.

Figure 13

Constraint on the real parameter $\Delta$ from the fit in scenario II. The red dashed line represents the constraint if only $B_d$ and $B_s$ observables are used excluding $\mathcal{B}(B\to \tau \nu )$ and ${ϵ}_K$. When adding ${ϵ}_K$ the constraint is essentially unchanged (blue dotted-dashed line). A significantly stronger constraint is obtained when adding $\mathcal{B}(B\to \tau \nu )$ (dotted green). The full constraint when adding both, $\mathcal{B}(B\to \tau \nu )$ and ${ϵ}_K$, is shown as a solid (green) curve.

Figure 14

Constraint on the complex parameter $\Delta \equiv {\Delta }_d={\Delta }_s$ from the fit in scenario III. For the individual constraints the colored areas represent regions with $\mathrm{CL}<68.3\%$. In addition, a $\mathrm{C}.\mathrm{L}.<95.45\%$ contour is shown for the individual constraints obtained from $\Delta m_d$ and $\Delta m_s$, from $A_{SL}$, $a_{SL}^d$, and $a_{SL}^s$, and from ${\varphi }_d^{\psi K}$ and ${\varphi }_s^{\psi \varphi }$. For the combined fit the shaded (red) area shows the region with $\mathrm{C}.\mathrm{L}.<68.3\%$ while the two additional contour lines inscribe the regions with $\mathrm{CL}<95.45\%$ and $\mathrm{CL}<99.73\%$, respectively.

Figure 15

Indirect fit prediction for the difference $a_{sl}^s-a_{sl}^d$ as a function of ${\varphi }_s^{\Delta }-2{\beta }_s$, in the standard model and the new physics scenarios I and III. The allowed regions correspond to $\mathrm{C}.\mathrm{L}.<95.45\%$. The prediction from scenario II is not shown since it is very close to the SM. Note that for this plot, the direct measurement of ${\varphi }_s^{\Delta }-2{\beta }_s$ in $B_s\to J/\psi \varphi$ is removed from the inputs.

Figure 16

Constraint on $|{ϵ}_K|$ predicted from the global fit, compared to the experimental value. The green (shaded) curve is obtained with our Rfit treatment of the uncertainties for the parameters entering Eq. (26). The dashed curve is obtained by treating all the errors as Gaussian.

Figure 17

Constraint on ${\kappa }_{ϵ}$ predicted from the global fit using Eq. (26), compared to three different theoretical inputs ${\kappa }_{ϵ}=0.94\pm 0.02$ with either a Rfit or a Gaussian treatment of the error, or ${\kappa }_{ϵ}=0.94\pm 0.013\pm 0.023$ corresponding to our own treatment of the uncertainties involved in its evaluation.