$CP$ violation in the $B_s^0$ system

Experimental and theoretical studies of $CP$ violation in the $B_s^0$ system are reviewed. Updated predictions for the mixing parameters of the $B_s^0$ mesons expected in the standard model (SM) are given, namely, the mass difference $\mathrm{\Delta }{M}_s^{\mathrm{SM}}=18.3\pm 2.7{\mathrm{ps}}^{-1}$, the decay rate difference $\mathrm{\Delta }{\mathrm{\Gamma }}_s^{\mathrm{SM}}=0.085\pm 0.015{\mathrm{ps}}^{-1}$, and the flavor-specific $CP$ asymmetry $a_{\mathrm{fs}}^{s,\mathrm{SM}}=(2.22\pm 0.27)\times {10}^{-5}$ and the equivalent quantities in the $B^0$ sector. Current experimental values of $\mathrm{\Delta }{M}_s$ and $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ agree with remarkable precision with theoretical expectations. This agreement supports the applicability of theoretical tools such as the heavy quark expansion to these decays. $CP$-violating studies in the $B_s^0$ system provide essential information to test the SM expectations and to unveil a possible contribution of the new physics (NP). NP effects on $\mathrm{\Delta }{M}_s$ of the order of 15% are still possible. The $CP$ phase ${\varphi }_s$ due to $CP$ violation in interference of decays and mixing can accommodate effects of the order of $\mathcal{O}(100\%)$. The semileptonic $CP$ asymmetry $a_{\mathrm{sl}}^s$ due to $CP$ violation in mixing could still be a factor of 130 larger than its robust SM expectation and thus provides a very clean observable for NP searches. Theoretical improvements that are necessary to make full use of the experimental precision are discussed.

Figure 1

Standard model diagrams for the transition between $B_s^0$ and ${\overline{B}}_s^0$ mesons. The contribution of internal on-shell particles (only the charm and the up quark can contribute) is denoted by ${\mathrm{\Gamma }}_{12}^s$; the contribution of internal off-shell particles (all depicted particles can contribute) is denoted by $M_{12}^s$.

Figure 2

To ${\mathrm{\Gamma }}_{12}^s$ only the box diagrams with internal up and charm quarks are contributing in the standard model; see Fig. 1. Integrating out the heavy $W$ boson, we are left with a bilocal object, which is shown here for internal charm and anticharm quarks.

Figure 3

Comparison of HQE predictions for lifetime ratios of heavy hadrons with experimental values. The theory values are taken from [209]. Experimental numbers are taken from (fall 2014) HFAG ([58]).

Figure 4

Standard model diagrams contributing to ${\mathrm{\Gamma }}_{12}^s$ at NLO QCD, i.e., ${\mathrm{\Gamma }}_3^{s,(1)}$. For obtaining the NLO QCD Wilson coefficients one has to calculate one-loop corrections to the $\mathrm{\Delta }B=2$ operators ($E_1–E_4$) and also two-loop corrections to the double insertion of $\mathrm{\Delta }B=1$ operators ($D_1–D_{12}$). An explicit cancellation of all infrared singularities in the matching was shown by [79] and later by [84] and [128]. Such an IR safety is crucial for the consistency of the HQE. The next future steps will be the determination of ${\mathrm{\Gamma }}_4^{s,(1)}$ and ${\mathrm{\Gamma }}_3^{s,(2)}$. For that one has to take into account in these diagrams a nonvanishing strange quark momentum and one has to add a further gluon in these diagrams.

Figure 5

The proper decay time resolution measured by the CDF Collaboration. From [52].

Figure 6

Proper decay time distribution for the selected $B_s^0$ decay candidates tagged as mixed (different flavor at decay and production; red, solid line) or unmixed (same flavor at decay and production; blue, dotted line). The data and the fit projections are plotted in a signal window around the reconstructed $B_s^0$ mass of $5.32–5.55\mathrm{GeV}/c^2$. From [16].

Figure 7

The average of all the $B_s^0\to J/\psi \varphi$ and $B_s^0\to J/\psi K^{+}{K}^{-}$ results is shown as the red contour, and the constraints given by the effective lifetime measurements of $B_s^0$ to flavor-specific [see Eq. (31)], pure $CP$-odd and pure $CP$-even final states are shown as the blue, green, and purple bands, respectively. The average taking all constraints into account is shown as the gray-filled contour. The yellow band is the theory prediction given in Eq. (89) that assumes no new physics in $B_s^0$ mixing. From [58].

Figure 8

The possible enhancement factors of $\mathrm{\Delta }{\mathrm{\Gamma }}_d$ by new scalar (lhs) and vector (rhs) $bd\tau \tau$ operators are indicated by the shaded (yellow) regions. In the case of a scalar operator $\mathrm{\Delta }{\mathrm{\Gamma }}_d$ can still be enhanced to about 1.6 times of the SM values. In the case of a vector operator $\mathrm{\Delta }{\mathrm{\Gamma }}_d$ can even be enhanced to about 3.5 times of the SM values. More precise bounds on $B^0\to {\tau }^{+}{\tau }^{-}$, $B^0\to X_d{\tau }^{+}{\tau }^{-}$, and $B^{+}\to {\pi }^{+}{\tau }^{+}{\tau }^{-}$ could further shrink the allowed enhancement factor. The relation between the bounds $B^0\to {\tau }^{+}{\tau }^{-}$, $B^0\to X_d{\tau }^{+}{\tau }^{-}$, and $B^{+}\to {\pi }^{+}{\tau }^{+}{\tau }^{-}$ and the possible enhancement factor of $\mathrm{\Delta }{\mathrm{\Gamma }}_d$ is given by the solid (red), long-dashed (blue) and dashed (green) lines.

Figure 9

The weighted $K^{+}{K}^{-}{\pi }^{\mp }$ invariant mass distribution for the $\mu \varphi {\pi }^{\mp }$ sample. The solid line represents the result of the fit and the dashed line shows the background parametrization. The lower mass peak is due to the decay $D^{\mp }\to \varphi {\pi }^{\mp }$ and the second peak is due to the $D_s^{\mp }$ meson decay. Note the suppressed zero on the vertical axis. From [49].

Figure 10

Invariant mass distributions of $K^{+}{K}^{-}{\pi }^{+}$ and $K^{+}{K}^{-}{\pi }^{-}$ in the three Dalitz regions studied in the LHCb analysis, summed over both magnet polarities and data taking periods. Overlaid are the results of the fits, with signal and combinatorial background components as indicated in the legend. From [37].

Figure 11

Relative muon efficiency as a function of the muon momentum. From [20].

Figure 12

Overview of measurements in the $a_{\mathrm{sl}}^d\text{−}{a}_{\mathrm{sl}}^s$ plane. Direct measurements of $a_{\mathrm{sl}}^s$ and $a_{\mathrm{sl}}^d$ listed in Table 4 ($B^0$ average as the vertical band, $B_s^0$ average as the horizontal band, D0 dimuon result as the yellow ellipse). The black point close to (0; 0) is the standard model prediction reported in this paper with error bars multiplied by 10. From [37].

Figure 13

Different decay topologies contributing to the decays $B_s^0\to J/\psi \varphi$ (lhs) and $B_s^0\to K^{-}{\pi }^{+}$ (rhs). The top row shows the color suppressed topologies (in the case of $B_s^0\to J/\psi \varphi$ this is the tree-level contribution and in the case of $B_s^0\to K^{-}{\pi }^{+}$ this is the penguin contribution) and the lower row shows the color allowed topologies (in the case of $B_s^0\to J/\psi \varphi$ this is now the penguin contribution and in the case of $B_s^0\to K^{-}{\pi }^{+}$ this is the tree-level contribution). Since the $J/\psi$ meson is color neutral, we have to add additional gluons for the penguin contribution to $B_s^0\to J/\psi \varphi$.

Figure 14

(a) Tree level and (b) penguin diagrams for ${\overline{B}}^0$ decays into $J/\psi {\pi }^{+}{\pi }^{-}$, which also contains ${\overline{B}}^0\to J/\psi {\rho }^0$.

Figure 15

Definition of the helicity angles. For details see text. The angle ${\theta }_{\mu }$ is denoted as ${\theta }_l$. From [246].

Figure 16

Decay time and helicity-angle distributions for $B_s^0\to J/\psi K^{+}{K}^{-}$ decays (data points) with the one-dimensional fit projections overlaid. The solid blue lines show the total signal contributions, which are composed of $CP$-even (long-dashed, red), $CP$-odd (short-dashed, green) and $S$-wave (dot-dashed, purple) contributions. From [36].

Figure 17

The angular distributions ($\cos {\theta }_T$, $\cos {\psi }_T$, and ${\phi }_T$) of the $B_s^0$ candidates from data (solid markers). The angles ${\theta }_T$ and ${\psi }_T$ are the polar and azimuthal angles, respectively, of the ${\mu }^{+}$ in the rest frame of the $J/\psi$, where the $x$ axis is defined by the direction of the $\varphi$ meson in the $J/\psi$ rest frame, and the $x\text{−}y$ plane is defined by the decay plane of the $\varphi \to K^{+}{K}^{-}$ decay. The helicity angle ${\psi }_T$ is the angle of the $K^{+}$ in the $\varphi$ rest frame with respect to the negative $J/\psi$ momentum direction. The solid line is the result of the fit, the dashed line is the signal fit, and the dot-dashed line is the background fit. From [188].

Figure 18

Projections of (a) $m({\pi }^{+}{\pi }^{-})$, (b) $\cos {\theta }_{{\pi }^{+}{\pi }^{-}}$, (c) $\cos {\theta }_{J/\psi }$, and (d) $\chi$ for the solution with the five resonance discussed in the text. The points with error bars represent data. The dashed (red) lines represent the signals, the dotted (black) lines represent the backgrounds, and the solid (blue) lines represent the total fit. From [23].

Figure 19

68% C.L. regions in $B_s^0$ width difference $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ and weak phase ${\varphi }_s$ obtained from individual and combined CDF, D0, ATLAS, CMS, and LHCb likelihoods of $B_s^0\to J/\psi \varphi$, $B_s^0\to J/\psi K^{+}{K}^{-}$, and $B_s^0\to J/\psi {\pi }^{+}{\pi }^{-}$. The expectation within the standard model is shown as the black rectangle. From [58].

Figure 20

Contours corresponding to 68% (dashed) and 95% (solid) confidence levels for the number of degrees of freedom (ndf) of 2, respectively, for the penguin amplitude parameters $a^{\prime }$ and ${\theta }^{\prime }$. From [33].

Figure 21

Limits on the penguin parameters $a_i$ and ${\theta }_i$ obtained from intersecting contours derived from the $CP$ asymmetries and branching fraction information in $B_s^0\to J/\psi {\overline{K}}^{\star 0}$ and $B^0\to J/\psi {\rho }^0$. Superimposed are the confidence level contours obtained from a ${\chi }^2$ fit to the data. The longitudinal (top), parallel (middle), and perpendicular (bottom) polarizations are shown. From [32].

Figure 22

One-dimensional projections of the $B_s^0\to \varphi \varphi$ fit for (a) decay time, (b) helicity angle $\varphi$, and (c), (d) cosines of the helicity angles ${\theta }_1$ and ${\theta }_2$, respectively. The data are represented as points, with the one-dimensional fit projections overlaid. The solid blue line shows the total signal contribution, which is composed of $CP$-even (long-dashed, red), $CP$-odd (short-dashed, green) and $S$-wave (dotted, blue) contributions. From [10].

Figure 23

Tree-level contribution to the decays $B_s^0\to D_s^{+}{K}^{-}$ and $B_s^0\to D_s^{-}{K}^{+}$. Both diagrams are color allowed and their CKM structure is similar in size, although the difference of the CKM phases is given by the CKM angle $\gamma$.

Figure 24

(a), (b) The invariant mass spectra obtained using the event selection adopted for the best sensitivity to $A_{CP}(B^0\to K^{+}{\pi }^{-})$; (c), (d) the invariant mass spectra obtained using the event selection adopted for the best sensitivity to $A_{CP}(B_s^0\to K^{+}{\pi }^{-})$. (a), (c) The $K^{+}{\pi }^{-}$ invariant mass is shown, while (b), (d) show the $K^{-}{\pi }^{+}$ invariant mass. From [12].

Figure 25

Result of the decay time (top) $s$ fit and (bottom) $c$ fit to the $B_s^0\to D_{s}K$ candidates. From [21].

Figure 26

Current bounds (Summer 2014) on the complex parameters ${\mathrm{\Delta }}_d$ (left) and ${\mathrm{\Delta }}_s$ (right) from different mixing observables. The point ${\mathrm{\Delta }}_q=1+0i$ corresponds to the SM—no deviation from the SM is visible. From [122].

Figure 27

Bounds on the complex parameters ${\mathrm{\Delta }}_d$ (left) and ${\mathrm{\Delta }}_s$ (right) from different mixing observables with data available till 2010. The point ${\mathrm{\Delta }}_q=1+0i$ corresponds to the SM. Unfortunately this quite clear hint for new physics effects has completely vanished by now. From [204].