Neutral B Oscillations

neutral-b-osc

# $B^0$ Oscillations

## Introduction

The $B^0_d(d\bar{b})$ and $B^0_s(s\bar{b})$ mesons undergo matter-antimatter oscillations due to second-order weak interactions.[^osc]

![](http://www.ppd.stfc.ac.uk/PPD/resources/image/jpg/mixing.jpg.jpg)

The asymmetry of the $B^0$ meson decay tends to favor matter production over antimatter production, but the Standard Model predicts that the $B^0$ decay asymmetry alone cannot account for the preponderance of matter over antimatter in the universe.[^LHCb-trouble] Flavor eigenstates do not share an eigenbasis with physical particle states, and the mass difference between the physical particle states give rise to a phase difference in the physical state time evolutions, causing pure flavor states to mix after some time. Experiments exploring this phenomenon are crucial tests of the Standard Model and place restrictions on viable physical theories.

## B-factories and $\Delta m$

Exploring B meson physics can most practically be achieved in experiment through the use of a B-factory. At the $\Upsilon(4S)$ resonance (10579.4±1.2 MeV), there is enough energy to produce B meson pairs. Typically, factories are implemented with particle-antiparticle pairs tuned to the $\Upsilon(4S)$ resonance.[^upsilon] Then, detectors are used to track decay products of the B mesons, and software such as neural networks trained on Monte-Carlo simulations of B decay are used to reconstruct events from the data. The ability to distinguish decay products is crucial for event reconstruction, and experiments typically use either dE/dx measurements or a ring-imaging Cherenkov (RICH) detector for this purpose.

The Collider Detector at Fermilab (CDF), the Large Electron–Positron Collider (LEP), the semileptonic decay group at the Stanford Linear Accelerator (SLD at SLAC), and the Large Hadron Collider beauty (LHCb) collaborations are all experimental groups that have measured neutral B oscillations in the aforementioned fashion. The mass difference of the physical eigenstates of the $B^0_d$ meson $\Delta m_d$, which is the same as the speed of the oscillation, is a crucial parameter obtained from these experiments:[^LHCb-measure] [^CDF-measure] [^LEP-measure]

$$
$$\begin{array}{c|c|} 
 & \Delta m_d ({\rm  ps^{-1}}) \\ \hline
\text{LHCb} & 0.5036 ± 0.0020 \text{(stat)} ± 0.0013 \text{(sys)}  \\ \hline
\text{CDF} & 0.509 ± 0.010 \text{(stat)} ± 0.016 \text{(sys)}  \\ \hline
\text{LEP, SLD, and CDF} & 0.477 ± 0.017  \\ \hline
\end{array}$$
$$

Modern experiments have probed the energy scales where the heavier B mesons can be produced and analyzed. These mesons require particle collisions in the TeV range. The analogous oscillation for $B^0_s$ has been measured: [^LHCb-Deltams] [^CDF-Deltams]

$$
$$\begin{array}{c|c|} 
 & \Delta m_s ({\rm  ps^{-1}}) \\ \hline
\text{LHCb} & 17.93 ± 0.22 \text{(stat)} ± 0.15 \text{(sys)}  \\ \hline
\text{CDF} & 17.31 ^{+0.33}_{-0.18} \text{(stat)} ± 0.07 \text{(sys)}  \\ \hline
\end{array}$$
$$

## Electroweak flavor-changing

Particle-antiparticle oscillations arise due to a flavor changing term in the Lagrangian:

$$
\mathcal{L} = {g \over \sqrt{2}} \begin{pmatrix} \bar{u} & \bar{c} & \bar{t} \end{pmatrix} V_\text{CKM} \gamma_\mu \begin{pmatrix} d \\ s \\ b \end{pmatrix}_L W^\mu + \text{h.c.}
$$

where $V_\text{CKM}$ is the Cabibbo-Kobayashi-Maskawa matrix. The oscillation diagrams are dominated by the top quark tree level diagrams. Considering only these diagrams and the fact that $\Delta m$ is proportional to the amplitude give rise to the approximate relation:

$$
{\Delta m_s\over\Delta m_d} = {m_{B_s} f^2_{B_s} B_{B_s}\over m_{B_d} f^2_{B_d} B_{B_d}}\left|{V_{ts}\over V_{td}}\right|^2
$$

The $f_B$ parameters are decay constants, and the $B_B$ parameters are the bag factors. The decay constants can be obtained through experiment, but only crudely, and they can also be gotten from QCD calculations. The bag factors are corrections to an approximation of the effective matrix element using vacuum insertion, and can be calculated with lattice techniques. We can use this relation to find the magnitude of the ratio of the CKM matrix elements from the ratio of the oscillation speeds.[^b-mixing]

The CDF collaboration found the value of this ratio to be

$$
\left|{V_{td}\over V_{ts}}\right|=0.2060±0.0007\text{(exp)}^{+0.0081}_{-0.0060}\text{(theor)}
$$ 

## Conclusions

Neutral B oscillations are well documented and provide confirmation for the Standard Model. Furthermore, better measurements of this effect constrain the collection of viable theories. There are still experimental difficulties in measuring the $f_B$ decay constants, and the CKM matrix cannot be fully specified with just these measurements, but due to unitarity, the CKM matrix only has 4 degrees of freedom, so these measurements actually provide a lot of information about the Standard Model electroweak theory. This thread has been a huge collaborative effort in particle physics.

  [^osc]: J. Boucrot, [*Oscillations of neutral B mesons systems*][1]

  [^LHCb-trouble]: John Timmer, [*LHCb detector causes trouble for supersymmetry theory*][2]

  [^b-mixing]: Colin Gay, [*B Mixing*][3]

  [^CDF-Deltams]: CDF Collaboration, [*Observation of $B^0_s$-$\bar{B^0_s}$ Oscillations*][4]

  [^upsilon]:(Cornell U.), [*The Upsilon System*][5]

  [^LHCb-measure]: Vecchi, Stefania (Ferrara U.); Khanji, Basem (Universita & INFN, Milano-Bicocca (IT)), [*A precise measurement of the $B^0$ meson oscillation frequency*][6]

  [^CDF-measure]: CDF Collaboration, [*Mixing and Tagger Performance in Semileptonic B Decays with $\text{~}1\,\rm{fb}^{-1}$*][7]

  [^LEP-measure]: Achille Stocchi, [*$B^0$-$\bar{B^0}$ oscillations and measurements of $|V_{ub}|/|V_{db}|$ at LEP*][8]

  [^LHCb-Deltams]: LHC Collaboration, [*Observation of $B^0_s$-$\bar{B^0_s}$ mixing and measurement of mixing frequencies using semileptonic B decays*][9]

[1]: http://arxiv.org/pdf/hep-ex/9901035v1.pdf
[2]: http://arstechnica.com/science/2011/08/lhcb-detector-causes-trouble-for-supersymmetry/
[3]: http://arxiv.org/pdf/hep-ex/0103016v1.pdf
[4]: http://arxiv.org/pdf/hep-ex/0609040v1.pdf
[5]: http://www.lns.cornell.edu/public/lab-info/upsilon.html
[6]: http://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/LHCb-CONF-2015-003.html
[7]: http://www-cdf.fnal.gov/physics/new/bottom/060406.blessed-semi_B0mix/
[8]: http://arxiv.org/pdf/hep-ex/9902004.pdf
[9]: http://arxiv.org/pdf/1308.1302v3.pdf