# Exploration of superconducting multi-mode cavity architectures for quantum
computing
††thanks: This material is based upon work supported by the U.S. Department of
Energy, Office of Science, National Quantum Information Science Research
Centers, Superconducting Quantum Materials and Systems Center (SQMS) under
contract number DE-AC02-07CH11359.

Alessandro Reineri  Fermi National Accelerator Laboratory
Batavia, USA
<EMAIL_ADDRESS>Silvia Zorzetti  Fermi National Accelerator Laboratory
Batavia, USA
<EMAIL_ADDRESS>Tanay Roy  Fermi National Accelerator Laboratory
Batavia, USA
<EMAIL_ADDRESS>Xinyuan You  Fermi National Accelerator Laboratory
Batavia, USA
<EMAIL_ADDRESS>
###### Abstract

Superconducting radio-frequency (SRF) cavities coupled to transmon circuits
have proven to be a promising platform for building high-coherence quantum
information processors. An essential aspect of this realization involves
designing high quality factor three-dimensional superconducting cavities to
extend the lifetime of quantum systems. To increase the computational
capability of this architecture, we are exploring a multi-mode approach. This
paper presents the design optimization process of a multi-cell SRF cavity to
perform quantum computation based on an existing design developed in the scope
of particle accelerator technology. We perform parametric electromagnetic
simulations to evaluate and optimize the design. In particular, we focus on
the analysis of the interaction between a nonlinear superconducting circuit
known as the transmon and the cavity. This parametric design optimization is
structured to serve as a blueprint for future studies on similar systems.

###### Index Terms:

Quantum, Multi-mode, Cavity, Qubit, Simulation, Energy-participation Ratio

##  I Introduction

Superconducting radio-frequency (SRF) cavity and transmon coupled systems have
the potential to be a primary architecture for the realization of high-
coherence quantum information processors [1]. Moreover, such systems show good
scalability toward the metric of quantum volume [2, 3]. The information is
typically encoded in the lowermost $N$ states of a cavity mode, forming a more
complex object called “qudit”. The control of the cavity states is performed
by the transmon [4], a superconducting nonlinear oscillator. One way of
scaling up the amount of information encoded in these systems is to use higher
order modes of the cavity, which usually show a lower coherence time than the
fundamental mode, because of the difference in the modes’ quality factor. A
better approach is to use a multi-cell cavity containing several modes with
nearly the same quality factor, where the mentioned modes are found within a
bandwidth. Multi-cell cavities were originally developed in high-energy
physics for particle acceleration purposes. Consisting of a repetition of the
single-cell cavity geometry, the multi-cell cavity is an intrinsic multi-modal
resonator with a number of high quality factor modes equal to the number of
cells. Additionally, such a resonator geometry allows for a bigger Hilbert
space to implement more complex quantum algorithms.

At Fermilab, the superconducting multi-cell architecture has been studied and
manufactured since the year 2000 [6]. Among the most widespread multi-cell
designs, the TESLA-shape cavity has shown astonishing $Q_{0}$ values for all
its fundamental modes, up to $10^{10}$ [5], where $Q_{0}$ stands for the
cavity’s internal quality factor, i.e. the number of oscillations a cavity
mode’s electric or magnetic field undergoes before being dissipated. However,
since the mentioned cavity geometry has been developed for particle
acceleration purposes, the strength of interaction between transmon and
different cavity modes varies by orders of magnitude. This ultimately poses a
major challenge in controlling the transmon-cavity coupled system with the
available driving protocols [8, 9]. Therefore, the multi-cell cavity shape has
to be modified to decrease the difference in transmon-mode interactions among
all the cavity fundamental $\mathrm{TM}_{010}$ modes. Yet, given the novelty
of the architecture, there is no established cavity design optimization
process for multi-mode resonator shapes to be used in the field of quantum
computation.

In this paper, we present an example of multi-mode cavity design along with
the optimization workflow. The goal is to find the most suitable design for
quantum computation purposes. The optimization, based on finite-element
electromagnetic simulations implemented with the software CST Studio Suite®
and Ansys® High-Frequency Structure Solver (HFSS™), aims to determine the
cavity geometric parameters of interest and their influence on the cavity-
transmon interaction strength for each fundamental eigenmode. At first, by
carefully varying the identified parameters, the workflow allows us to obtain
a multi-cell resonator shape that meets the desired requirement. Then, we
evaluate the interaction between the fields and the transmon by inserting a
representative sample of a transmon chip inside the cavity and performing a
second series of electromagnetic simulations.

For a better explanation of the design optimization workflow we also include
in the paper a preliminary example of multi-cell cavity design developed using
the shown process. Specifically, the presented design is found starting from
the aforementioned, already-established TESLA-shape structure. We chose to
limit our investigation to a relatively small 3-cell cavity to keep the number
of modifiable geometric parameters low, though the process can be generalized
to resonators with more complex geometry.

The cavity-transmon interaction parameters we find with this second set of
simulations are compatible with some of the available control protocols [8,
9]. Moreover, the mentioned parameters’ ranges also cover the values recently
obtained in an experimental study using the same transmon positioning method,
though with a different cavity design [14].

##  II The TESLA-shape geometry

The starting cavity geometry we considered is the multi-cell superconducting
TESLA shape. It is based on the single-cell structure originally developed in
the field of particle accelerator technology [6]. The single-cell outline is
comprised of two ellipses’ arcs joined together by their common tangent. The
cell is then realized by rotating the contour around the beam axis and adding
the resulting half-cell to another half-cell (Fig. 1(a) \- 1(b)). This
particular cell shape allows for extremely high quality factor values, up to
$10^{10}$ for the fundamental $\mathrm{TM_{010}}$ mode of a niobium single-
cell cavity [7]. The multi-cell cavity is then realized by combining together
several single-cell structures through geometric interfaces called irises
(Fig. 1(c)).

From the electromagnetic point of view, the multi-cell cavity can be modeled
as a series of LC circuits linked with coupling capacitances (Fig. 1(d)). This
way, the eigenvalues problem is analytically solved yielding the following
expression for the $n$-th mode

$\left(\frac{\nu_{n}}{\nu_{0}}\right)^{2}=1+2k_{cc}\left[1-\cos\left(\frac{n\pi}{N}\right)\right],$
(1)

where $\nu_{0}$ is the resonant frequency of the LC circuit representing a
single cell, $N$ is the number of single LC elements and the constant
$k_{cc}$, called cell-to-cell coupling, is equal to the ratio of each LC
circuit’s capacitance over the coupling one $k_{cc}=\frac{C}{C_{k}}$. Equation
$\left(\ref{eq:EigvalRelation}\right)$ well describes the actual cavity’s
fundamental $\mathrm{TM_{010}}$ band: the single, high quality factor mode of
the single-cell architecture splits into $N$ high quality factor modes, as
many as the number of cells, which still show the same electric and magnetic
field orientations of $\mathrm{TM_{010}}$ modes [10].

At the cavity level, the frequencies are determined by acting on the half-cell
geometric parameters. In particular, the last mode’s frequency, often referred
to as $\pi$-mode, is related to the half-cell length by

$l=\frac{c}{2\nu_{\pi}},$ (2)

$c$ being the speed of light in vacuum. The $\pi$-mode frequency also
determines the half-cell equatorial radius through the empirical relation

$r_{0}=\frac{c}{\nu_{\pi}}.$ (3)

On the other hand, the fundamental modes’ bandwidth is related to the cell-to-
cell coupling factor $k_{cc}$ via

$k_{cc}=\frac{\nu_{\pi}^{2}-\nu_{0}^{2}}{\nu_{0}^{2}}.$ (4)

This parameter, at the cavity level, gauges the amount of electromagnetic
energy exchanged between each cell. Its magnitude depends on the geometric
shape of the iris connecting two subsequent cells, i.e. on its radius $r_{i}$.
For iris radius values of $0.4\ r_{0}$, characterizing the TESLA-shape design,
$k_{cc}$ is very small, around $0.02$. Consequently, the spectral width of the
fundamental band of cavities operating in the GHz regime is limited to few
tens of MHz.

(a)

(b)

(c) $\mathrm{\hat{u}}_{z}$iriscouplingholecell $n-1$cell $n$cell $n+1$

(d) $L$$C$$L$$C$$L$$C$$C_{k}$$C_{k}$$C_{b}$$C_{b}$

Figure 1: TESLA-shaped SRF cavities. (a) A single-cell cavity. (b) The shape
is determined by a set of geometric parameters defining the half-cell outline.
(c) Individual cells are joined using irises to make a multi-cell cavity. (d)
Electrical circuit model of the cavity, consisting of capacitively-coupled LC
circuits.

Moreover, the bandwidth remains constant as the number of cells is increased,
resulting in more fundamental modes being inserted in the same frequency
range, further reducing the modes’ spacing. A very small frequency separation
between the modes is not ideal: when resonantly driving one of the cavity
modes, the same drive could also off-resonantly drive other modes, causing
unwanted classical cross-talk [8].

By inserting a single transmon inside a multi-cell TESLA-shape cavity it is
possible to realize a multi-qudit processor. However, the coupling strengths
between the transmon and the cavity modes show sizable differences in
magnitude. This is due to the variation of the electric field magnitude among
the modes at the ends of the cavity, where the transmon is usually placed. As
the transmon interacts with the cavity through an electric dipole term,
differences in the electric field component parallel to the transmon dipole
moment are reflected and amplified into variations among the parameters
characterizing each transmon-cavity mode interaction. The differences in the
interaction strengths can even span orders of magnitude, resulting in the
inability to drive all the eligible modes of the transmon-cavity coupled
system with the available control protocols.

Contrarily to the case of the bandwidth and modes’ separation, there is no
direct connection, in literature, between a target mode’s electric field
distribution within a portion of the cavity volume and a certain cavity
geometric parameter. Nonetheless, through several simulation iterations, it
has been noticed a slight dependence of the electric field variation toward
the cell-to-cell coupling, as will be discussed in the next section.

##  III Cavity design optimization process

In order to meet the aforementioned goals for the multi-cell cavity design
improvement we started a design optimization process. The process is based on
finite-element electromagnetic simulations involving the cavity alone and it
is divided into two parts. In the first one, through the use of a CAD model of
the cavity with all the half-cells parameterized independently, the
simulations are set up with an eigenmode solver and prompted to compute the
cavity’s first $N$ eigenmodes for the considered geometry. Then, to study the
effect of the iris radius modification on the bandwidth and modes’ separation,
the $r_{i}$ of all irises, all equal, are changed by the same amount and the
eigenmode solver is rerun. In the second part of the optimization process,
another series of eigenmode simulations are performed to evaluate the
distribution of the electric field component parallel to the transmon dipole
moment for all the $\mathrm{TM_{010}}$ modes. The analysis focuses on the
neighborhood of the cavity entrances, along a line in which the transmon is
typically inserted (Fig. 3(a) \- 3(b)). Then, by modifying the two ellipses’
semi-axis oriented in the $\mathrm{\hat{u}}_{z}$ direction by the same
quantity for every half-cell, the electric field component is re-evaluated and
compared to the former TESLA-shape design to assess eventual improvements in
its variation among the modes.

All these simulations are performed by considering the cavity walls made up of
a perfect conductor, effectively not taking into account any electromagnetic
loss. Additionally, the choice of modifying the geometric parameters of
interest by the same quantity for all the half-cells is not a mandatory
requirement. Indeed, it is possible to change each variable differently, all
the half-cells being parameterized independently. However, having done so
allows us to keep the cavity geometry simple and maintain the symmetry of the
electric field distribution at both ends of the cavity, yet achieving the
desired improvements.

We start the optimization process considering a three-cell TESLA-shape design
sized to have a $\mathrm{TM_{010}}$ $\pi$-mode frequency of $\nu_{\pi}=6\
\mathrm{GHz}$, i.e. with a half-cell length of $l=12.5\ \mathrm{mm}$, an
equatorial radius $r_{0}=23.55\ \mathrm{mm}$, an iris radius $r_{i}=10\
\mathrm{mm}$ and a radius ratio of $\frac{r_{i}}{r_{0}}=0.42$. With this
choice of parameters, the frequencies of the fundamental modes become 5.77
GHz, 5.88 GHz and 6.08 GHz, resulting in a 200 MHz wide $\mathrm{TM_{010}}$
passband with a maximum modes’ separation of 110 MHz. By increasing the iris
radius value relatively to $r_{0}$ and evaluating the fundamental band
bandwidth, we notice that $\Delta\nu_{\mathrm{TM_{010}}}$ follows a parabolic
relation towards $\frac{r_{i}}{r_{0}}$ (Fig. 2(a)). Consequently, with the
number of modes remaining constant, the modes’ separation increases as the
iris is enlarged. In addition to that, for every iris radius value analyzed,
all the fundamental $\mathrm{TM_{010}}$ modes keep their electromagnetic
features, always showing the electric field oriented parallel to
$\mathrm{\hat{u}}_{z}$. To not end up with a too-sharp and pointy cavity
design that would affect its quality factor, we choose a radius ratio value of
0.63 which still provides a sizable improvement in the bandwidth, becoming 900
MHz large, and in the modes’ separation, respectively of 270 MHz between the
$\frac{\pi}{3}$ and the $\frac{2}{3}\pi$ modes and of 630 MHz between the
$\frac{2}{3}\pi$ and the $\pi$-mode (Table I).

(a)

(b)

Figure 2: $\mathrm{TM_{010}}$ bandwidth and inter-mode spacing as a function
of the $\frac{r_{i}}{r_{0}}$ ratio. (a) The bandwidth shows a parabolic
dependence. (b) Minimum and maximum modes’ separation as a function of
$\frac{r_{i}}{r_{0}}$.

For the second step of the optimization process, as mentioned before, we act
on the parameters $r_{0}$ and $a_{2}$, slightly modifying them from the
original values of the TESLA-shape design to see how the electric field’s
$z$-component distribution among the modes is affected. With the chosen radius
ratio value from the first optimization step, prompting $r_{0}=25.5\
\mathrm{mm}$ for all the half-cells, $a_{2}=2.7\ \mathrm{mm}$ for the half-
cells next to the beam stubs and $a_{2}=3.8\ \mathrm{mm}$ for the other half-
cells, we notice a reduction of the 35% in the $E_{z}$ variation among all the
$\mathrm{TM_{010}}$ modes in the designated neighborhoods of the cavity
entrances along the line $l=\left(-4,0,z\right)$ compared to the initial
TESLA-shape design (Fig. 3). We refer to the optimized design with TESLA-like
for its resemblance to the original TESLA-shape one.

(a)

(b)

(c)

(d)

Figure 3: (a) TESLA-shape and (b) TESLA-like CAD models. The black lines with
the red segments show the axis $l=\left(-4,0,z\right)$ along which the
comparison of $E_{z}$ variation among the $\mathrm{TM}_{010}$ modes has been
made. The plots (c) and (d) show that, although the $E_{z}$ mean value stays
almost the same, there is a sizable reduction of the component variation in
the optimized design case at both ends of the cavity, around 35%.

In modifying the cavity geometry, not only the $\mathrm{TM_{010}}$ band and
modes are affected but also the higher order $\mathrm{TE_{111}}$ ones. In
particular, the two bands, while being fairly apart in the case of the initial
TESLA-shape design, intersect one another in the case of the TESLA-like
design, resulting in the $\mathrm{TE_{111}}$ $\frac{2}{3}\pi$-mode to be found
between the $\mathrm{TM_{010}}$ $\frac{2}{3}\pi$-mode and the $\pi$-mode (Fig.
4). Despite that, the unwanted $\mathrm{TE_{111}}$ modes are not expected to
interfere with the correct functioning of the cavity when the transmon is
inserted. That is both due to a still fairly large separation between the
$\mathrm{TE_{111}}$ $\frac{2}{3}\pi$ mode and the nearest $\mathrm{TM_{010}}$
mode, around 100 MHz (Table I), and to the fact that, being transverse-
electric modes, their electric field is oriented almost-perpendicularly to the
transmon dipole. There might be a small $E_{z}$ component that couples with
the transmon, for the cavity geometry makes the fields bend slightly in the
vicinity of the cavity walls, though it is not expected to have a sizable
effect on the system.

Figure 4: Band diagram comparison between TESLA-shape and TESLA-like cavity designs. The $\mathrm{TM_{010}}$ and $\mathrm{TE_{111}}$ bands, well separated in the first case, intersect one the other in the second case. TABLE I: Band data comparison between TESLA-like and TESLA-shape designs. [5pt]Mode |  $\boldsymbol{\nu}$ (TESLA-like) [GHz] |  $\boldsymbol{\nu}$ (TESLA-shape) [GHz]
---|---|---
[5pt]$\mathrm{TM_{010}}$ $\frac{\pi}{3}$ | 5.1813 | 5.7666
[5pt]$\mathrm{TM_{010}}$ $\frac{2}{3}\pi$ | 5.4597 | 5.8780
[5pt]$\mathrm{TM_{010}}$ $\pi$ | 6.0780 | 5.9519
[5pt]$\mathrm{TE_{111}}$ $\frac{\pi}{3}$ | 4.8957 | 6.6127
[5pt]$\mathrm{TE_{111}}$ $\frac{2}{3}\pi$ | 5.9663 | 6.8589
[5pt]$\mathrm{TE_{111}}$ $\pi$ | 7.6470 | 7.4106

The found design meets the initial requirements of larger bandwidth and larger
modes’ separation. Moreover, we also obtain some improvement in the $E_{z}$
component variation among the modes. All the achievements did not modify the
nature of the $\mathrm{TM_{010}}$ modes which still show their electric field
original orientation. Figure 5 shows that the electric field orientation is
the same along the $x=0$ cutting plane and that each mode keeps the number of
field intensity maxima unchanged between the two designs.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Electric field distribution for the $\mathrm{TM_{010}}$ modes along
x=0 plane. (a), (b), (c) show the three fundamental modes for the TESLA-shape
cavity and (d), (e), (f) show the same modes for the TESLA-like design. All
the $\mathrm{TM_{010}}$ modes maintain their electromagnetic properties, i.e.
their electric and magnetic field orientations, and the number of antinodes
remain the same.

##  IV Transmon-cavity interaction evaluation

The last step of the multi-cell cavity design optimization process involves
assessing the interaction between the optimized TESLA-like design and the
transmon. In particular, we want to evaluate all the parameters defining the
Hamiltonian in the dispersive regime which, for an intrinsic multi-modal
electromagnetic environment, reads [11]

$\begin{split}\hat{H}_{\mathrm{disp}}\simeq&\sum_{m}\hbar\left(\omega_{m}+\gamma_{m}\right)\hat{a}_{m}^{{\dagger}}\hat{a}_{m}\\\
&+\frac{1}{2}\sum_{m}\hbar
K_{m}\left(\hat{a}_{m}^{{\dagger}}\right)^{2}\hat{a}_{m}^{2}\\\
&+\sum_{m>n}\hbar\chi_{m,n}\hat{a}_{m}^{{\dagger}}\hat{a}_{m}\hat{a}_{n}^{{\dagger}}\hat{a}_{n},\end{split}$
(5)

where $\left\\{\hat{a}_{m},\hat{a}_{m}^{{\dagger}}\right\\}_{m\in\mathbb{N}}$
are the $m$-th mode’s annihilation and creation operators (including the
transmon ones) and $\left\\{\omega_{m}\right\\}_{m\in\mathbb{N}}$ are the
uncoupled system’s eigenfrequencies (including the transmon one). The
knowledge of the dispersive regime parameters
$\left\\{\gamma_{m}\right\\}_{m\in\mathbb{N}}$,
$\left\\{K_{m}\right\\}_{m\in\mathbb{N}}$ and
$\left\\{\chi_{m,n}\right\\}_{m,n\in\mathbb{N},m\neq n}$, named respectively
linear corrections, self-Kerrs and cross-Kerr allows for a complete assessment
of the transmon-cavity interaction in the dispersive regime working region
[4]. The parameters are not all independent one another; instead, the cross-
Kerr interaction can be used to express the other quantities with the
following

$\displaystyle\gamma_{m}$ $\displaystyle=\frac{1}{2}\sum_{n}\chi_{m,n},$ (6)
$\displaystyle K_{m}$ $\displaystyle=\frac{\chi_{m,m}}{2}.$ (7)

In addition to that, the cross-Kerr interactions evaluation is fundamental to
implementing quantum computation algorithms on a transmon-cavity coupled
system [8, 9]. A last, yet important parameter that does not pertain directly
to the dispersive regime approximation is called Rabi coupling. It gives a
measure of the dipole interaction strength between each cavity
$\mathrm{TM_{010}}$ mode and the transmon mode and can be calculated from the
parameters in the equation (5) as [11]

$g_{m}=\sqrt{\frac{-\chi_{0,m}\Delta_{m}\left(\Delta_{m}-\frac{E_{C}}{\hbar}\right)}{\frac{E_{C}}{\hbar}}}.$
(8)

where $\Delta_{m}$ is the frequency difference between the transmon and the
$m$-th cavity mode, $\chi_{0,m}$ is the cross-Kerr interaction strength
between the transmon and the $m$-th mode and $E_{C}$ is the transmon
capacitive energy, related to the transmon self-Kerr interaction which, in
literature, is called anharmonicity [4].

To obtain all the mentioned parameters we use the energy-participation ratio
analysis method [12]. This protocol is based on finite-element eigenmode
simulations and requires inserting a transmon element inside the cavity
volume. It evaluates the fraction of electromagnetic energy stored in the
transmon for each system’s eigenmode from the $\mathbf{E}$ ad $\mathbf{H}$
distributions in the volume and, through that, returns all the parameters
characterizing equation (5). The simulations are built by inserting a transmon
at one end of the cavity on a loss-free silicon rod. The transmon geometry is
similar to the ones already in literature and consists of a pair of antenna
pads, a pair of qubit pads and a rectangle for the actual transmon junction,
all modeled as perfect conductors (Fig. 6). An additional boundary condition
is set up on the junction rectangle by defining an integration line across it
and assigning it a lumped inductance value that corresponds to the transmon
linear Josephson inductance $L_{J_{0}}$. The chosen value $L_{J_{0}}=11.57\
\mathrm{nH}$ for this series of simulations corresponds to a transmon
frequency of $\nu_{0}=4.5\ \mathrm{GHz}$.

Figure 6: CAD model of the transmon inserted at one side of the cavity for EPR
simulations.

Afterwards, we initialize the eigenmode solver with the number of desired
eigenmodes we want the electric and magnetic fields to be calculated. At this
stage, it is of the utmost importance to seed a mesh refinement both on the
entire CAD model and locally around the transmon junction, for its small
physical dimensions can make the solver not detect its impedance and result in
failing the simulations. Finally, we set up a parametric sweep over the
relative position between the cavity end and the transmon to see the behavior
of the parameters around the neighborhood on which $E_{z}$ variation reduction
among the modes is observed from the TESLA-shape to the TESLA-like design.

With the parameter values obtained from the cavity-transmon relative position
sweep we plot Fig. 7 showing each mode’s parameter behavior toward the
transmon position. Starting from the transmon-modes cross-Kerr interactions,
we notice that the parameters $\chi_{01}$ and $\chi_{02}$ corresponding to the
$\mathrm{TM_{010}}$ $\frac{\pi}{3}$-mode and $\frac{2}{3}\pi$-mode, are very
close with each other for the considered transmon positions. The cross-Kerr
$\chi_{03}$, instead, is much lower than the other two, about one order of
magnitude less for each transmon position. As expected, all the cross-Kerr
magnitudes decrease in absolute value as the transmon is moved out of the
cavity, due to a smaller electric field. Moreover, all the modes’ cross-Kerr
remain negative within the entire position sweep. The performed sweep also
shows that the cavity-transmon interaction can be tuned by repositioning the
nonlinear circuit element with respect to the cavity entrance. This way, the
coupled system can be rather flexible toward the applicable driving protocol
by simply moving the transmon chip and retaining the multi-cell cavity design:
the cross-Kerr interactions go from the order of MHz to tens of kHz, modifying
the transmon-cavity interaction from strong dispersive regime to weak
dispersive regime. This allows the system to be driven by either control
protocols developed for strongly coupled [8] or weakly coupled systems [9].

Figure 7: Cross-Kerr interaction between transmon and cavity modes as a
function of transmon position for all cavity $\mathrm{TM_{010}}$ modes. The
transmon position is given considering the system of coordinate origin at the
beginning of the cavity beam stub so that larger values of $d$ imply the
transmon being deeper inside the cavity.

The second set of parameters we evaluate as a function of the transmon
position is the self-Kerr set, including the transmon anharmonicity. Since, as
expected, the anharmonicity is considerably larger than the cavity modes’
self-Kerrs, we report their absolute values in a logarithmic plot (Fig. 8). As
highlighted before for the cross-Kerr interactions, the self-Kerr parameters
of the first two $\mathrm{TM_{010}}$ modes are very similar for every transmon
position, whereas the third mode’s self-Kerr is roughly one order of magnitude
less than the other two throughout the whole position sweep. All the self-Kerr
magnitudes and the transmon anharmonicity, although reported in absolute value
in the plots, are negative for any transmon position and decrease in absolute
value as the transmon is moved outward. The cavity modes’ self-Kerr magnitudes
are considerably less than the corresponding modes’ cross-Kerr values.
Consequently, their effect on the coupled system behavior is very weak per
electromagnetic excitation of a cavity mode, as stated by equation
$\left(\ref{eq:MultiModeDispHam}\right)$.

(a)

(b)

Figure 8:  Absolute values of the self-Kerr interactions as a function of
transmon position for (a) cavity’s $\mathrm{TM_{010}}$ modes and (b) transmon.

Proceeding with the energy-participation ratio analysis, the third ensemble of
dispersive regime parameters we examine towards the transmon position is the
linear corrections set. As for the self- and cross-Kerr cases, the first two
$\mathrm{TM_{010}}$ modes’ linear corrections are fairly similar throughout
the entire position sweep (Fig. 9). This supports the validity of Eq. (6)
linking together all the dispersive regime parameters. Moreover, from the plot
it is possible to notice that the corrections are almost equal to half the sum
of self-Kerr and transmon-mode cross-Kerr magnitudes for each mode, implying
that the cross-Kerr interactions between the cavity modes are negligible.

Figure 9: Linear correction magnitude to cavity modes as a function of
transmon position for all cavity $\mathrm{TM_{010}}$ modes.

Finally, the cross-Kerr interaction values are used to calculate the Rabi
couplings through equation (8) and their behavior towards the transmon
position is plotted. Contrarily to all the previous parameters, all the Rabi
couplings are different from each other throughout the entire transmon
position sweeps (Fig. 10). However, this does not go against the similarities
between the first two modes’ parameters, as from equation (8) it can be seen
that the difference in frequency $\Delta_{m}$ can compensate the $g_{m}$
variation, yielding an almost equal cross-Kerr magnitude for the two mentioned
modes. This fact also justifies the difference in the parameters’ magnitude
characterizing the $\pi$-mode: its Rabi coupling is not enough to compensate
for the frequency difference towards the transmon. Consequently, the
$\pi$-mode parameters of cross-Kerr, self-Kerr and linear correction are
smaller than the ones characterizing the other $\mathrm{TM_{010}}$ modes’
interactions.

Figure 10: Rabi coupling magnitude as a function of transmon position for all
$\mathrm{TM_{010}}$ modes.

##  V Conclusions

To summarize, we optimize the geometry of a high-coherence multi-mode cavity
to improve its performance as a quantum processor. First of all, we
systematically broaden the spectral width of the fundamental
$\mathrm{TM_{010}}$ band, consequently increasing the spacing between modes
and potentially resolving the issue of frequency crowding. Afterward, we
establish a qualitative connection between some of the cavity geometric
parameters and the electric field distribution of the fundamental modes. The
results allow for more efficient optimization of the cavity design. The
parameters found with the transmon position sweep in the second set of
simulations allow a certain flexibility in the driving protocol choice,
modifying the transmon-cavity interaction from strongly dispersive to weakly
dispersive while retaining the similar cavity design. Future work will focus
on the experiments to test the developed geometry with a prototype TESLA-like
3-cell cavity, which will provide guidance to further scale up the quantum
processor.

## References

  * [1] Reinhold, P., et al., “Error-corrected gates on an encoded qubit”, Nature Physics 16, 822–826 (2020).
  * [2] Cross, A. W., et al., “Validating quantum computers using randomized model circuits”, arXiv preprint arXiv:1811.12926v2 (2019).
  * [3] Jurcevic, P., et al., “Demonstration of quantum volume 64 on a superconducting quantum computing system”, arXiv preprint arXiv:2008.08571v2 (2020).
  * [4] Koch, J., et al. “Charge-insensitive qubit design derived from the Cooper pair box”, Phys. Rev. A 76, 042319 (2007).
  * [5] Alam, M. Sohaib, et al. “Quantum computing hardware for HEP algorithms and sensing”, arXiv preprint arXiv:2204.08605 (2022).
  * [6] Aune, B., Bandelmann, R., Bloess, D. et al. “Superconducting TESLA cavities”, Phys. Rev. ST Accel. Beams 3, 092001 (2000).
  * [7] Romanenko, A., et al. “Three-dimensional superconducting resonators at $T<20$ mK with photon lifetimes up to $\tau$=2 s”, Phys. Rev. Applied 13, 034032 (2020).
  * [8] Heeres, R. W., Sivak, Vlastakis B., Holland E. et al. “Cavity State Manipulation Using Photon-Number Selective Phase Gates”, Phys. Rev. Lett. 115, 137002, (2015).
  * [9] Eickbusch, A., Sivak, V., Ding, A.Z. et al. “Fast universal control of an oscillator with weak dispersive coupling to a qubit”, Nat. Phys. 18, 1464–1469 (2022).
  * [10] Padamsee, H., Knobloch, J., Hays, T., “RF superconductivity for accelerators”, John Wiley & Sons (2009).
  * [11] Blais, A., Grimsmo, A. L., Girvin, S. M. and Wallraff, A., “Circuit quantum electrodynamics”, Rev. Mod. Phys. 93, 025005 (2021).
  * [12] Minev, Z. K., et al. “Energy-participation quantization of Josephson circuits”, npj Quantum Information 7, 131 (2021).
  * [13] Krantz, P., et al. “A quantum engineer’s guide to superconducting qubits”, Applied Physics Reviews 6, 021318 (2019).
  * [14] Milul O., et al., “A superconducting quantum memory with tens of milliseconds coherence time” arXiv preprint arXiv:2302.06442v2 (2023).


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-189 LHCb-PAPER-2017-023 November 4, 2017

Search for baryon-number-violating ${\mathchar 28932\relax}^{0}_{b}$
oscillations

LHCb collaboration†††Authors are listed at the end of this Letter.

A search for baryon-number-violating ${\mathchar 28932\relax}^{0}_{b}$
oscillations is performed with a sample of ${p}{p}$ collision data recorded by
the LHCb experiment, corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$. The baryon number at the moment of production is
identified by requiring that the ${\mathchar 28932\relax}^{0}_{b}$ come from
the decay of a resonance ${{{\mathchar
28932\relax}^{*-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}}$ or ${{{\mathchar
28932\relax}^{\prime-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}}$, and the baryon number at the moment of
decay is identified from the final state using the decays ${{\mathchar
28932\relax}^{0}_{b}}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{{\pi}^{-}},~{}{{\mathchar
28932\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$. No evidence of baryon
number violation is found, and an upper limit at the 95% confidence level is
set on the oscillation rate of $\omega<0.08\,{\mathrm{\,ps}}^{-1}$, where
$\omega$ is the associated angular frequency.

Published as Phys. Rev. Lett. 119, 181807 (2017)

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

Three conditions are necessary for the formation of a matter-dominated
universe: $C$ and $C\\!P$ violation, baryon number violation (BNV), and the
absence of thermal equilibrium [1]. The existence of $C$ and $C\\!P$ violation
has been established experimentally for decades [2, 3, 4], although the amount
of $C\\!P$ violation present in the Standard Model (SM) is known to be
insufficient to generate the matter-antimatter asymmetry observed in the
universe today [5, 6], prompting numerous searches for sources of $C\\!P$
violation beyond the SM. By contrast, despite baryon number conservation being
an accidental low-temperature symmetry of the SM, BNV has never been observed
experimentally, and stringent lower limits have been placed on the mean
lifetimes of protons and of bound neutrons [7]. These limits impose
constraints on generic models of physics beyond the SM. In particular, in
supersymmetric extensions of the SM, a mechanism such as $R$-parity
conservation is required to naturally suppress baryon number violation [8, 9,
10, 11]. An alternative is that the new physics has nongeneric flavour
interactions, such that only certain BNV processes are allowed and the
experimental constraints are respected. One possibility would be for new BNV
couplings to be entirely flavour diagonal [12, 13], such as a six-fermion
operator that couples two fermions from each generation. This would couple two
from each of $\\{u,d,e,\nu_{e}\\}$, $\\{c,s,\mu,\nu_{\mu}\\}$, and
$\\{t,b,\tau,\nu_{\tau}\\}$, with duplication allowed within a generation,
e.g. a $usb\overline{usb}$ vertex would be permitted. Such an operator could
arise in models with leptoquarks or $R$-parity-violating supersymmetric
extensions of the SM [13, 14]. The six-fermion operator could allow BNV while
being consistent with the experimental limit on the proton lifetime, since the
proton initial state contains only first-generation fermions and, therefore,
its coupling to the operator would require two flavour-changing neutral
processes and would be heavily suppressed [13].

Most experimental processes involving such an operator are difficult to
observe, since they include multiple third-generation fermions. For example,
the signatures proposed in Ref. [13] require performing asymmetry measurements
of same-sign dilepton pairs produced in association with a top-quark jet.
However, there is a process that could give rise to a clean, unambiguous
experimental signature: baryon-antibaryon oscillations of hadrons that contain
a valence quark from each generation. The only such baryon observed to date
that decays weakly is the ${\mathchar 28932\relax}^{0}_{b}$ ($bsu$). The
interest of searching for ${\mathchar 28932\relax}^{0}_{b}$ oscillations was
noted in Refs. [15, 16], with an oscillation period potentially as short as
$\mathcal{O}(0.1{\mathrm{\,ps}})$ suggested. More recently, heavy baryon
oscillations have been proposed as a possible mechanism for baryogenesis [17,
14].

The signature for a BNV process is that a ${\mathchar 28932\relax}^{0}_{b}$
baryon is produced and decays weakly as an antibaryon to a final state such as
${{\overline{\mathchar 28932\relax}}{}_{c}^{-}}\pi^{+}$ (or, vice versa, that
an antibaryon is produced and decays as a baryon). The strong decays111 The
inclusion of charge-conjugate processes is implied throughout.  ${{\mathchar
28932\relax}^{\prime-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$ and ${{\mathchar
28932\relax}^{*-}_{b}}\rightarrow{{\mathchar 28932\relax}^{0}_{b}}{{\pi}^{-}}$
(denoted ${{\mathchar 28932\relax}^{\prime,*\,-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$), where ${\mathchar
28932\relax}^{\prime-}_{b}$ and ${\mathchar 28932\relax}^{*-}_{b}$ are the
narrow resonances ${{\mathchar 28932\relax}^{\prime}_{b}}(5935)^{-}$ and
${{\mathchar 28932\relax}^{*}_{b}}(5955)^{-}$ recently observed by the LHCb
collaboration [18], allow the baryon number at the time of production to be
determined from the charge of the pion. Figure 1 shows quark-level diagrams of
example (a) non-BNV and (b) BNV processes.

Figure 1:  (a) A non-BNV quark diagram for a ${{\mathchar
28932\relax}^{\prime,*\,-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$ strong decay followed by a ${{\mathchar
28932\relax}^{0}_{b}}\rightarrow{{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}}$
weak decay. (b) The corresponding BNV diagram with a ${\mathchar
28932\relax}^{0}_{b}$ to ${\overline{\mathchar 28932\relax}}{}_{b}^{0}$
oscillation followed by a decay to the final state ${{\overline{\mathchar
28932\relax}}{}_{c}^{-}}{{\pi}^{+}}$.

For baryon states propagating in free space, the formalism for oscillations is
similar to that of neutral mesons [7, 19, 20], which has been studied
extensively in the context of ${K}^{0}$, ${D}^{0}$, ${B}^{0}$, and
${B}^{0}_{s}$ mixing [21]. However, a difference arises in the presence of a
magnetic field, $\vec{B}$, due to the nonzero magnetic moment, $\mu$,
possessed by the baryons, resulting in a splitting of the baryon and
antibaryon energy levels of $\Delta E=2\vec{\mu}\cdot\vec{B}$. This splitting
leads to a damping of the oscillations over time. For the case of neutron
oscillations, even a modest ambient magnetic field would greatly suppress the
oscillation probability on the timescale of the neutron lifetime [22]. The
criterion for the effect of the magnetic field to be negligible is $|\Delta
E|t/2\ll 1$, where $t$ is the time of propagation of the baryon. Taking the
${\mathchar 28932\relax}^{0}_{b}$ magnetic moment to be comparable to the
nuclear magneton [7], the energy splitting associated with the magnetic field
in the interaction region of the LHCb detector, which is $\lesssim 10$ mT, may
be computed. For a typical time of propagation equal to the known ${\mathchar
28932\relax}^{0}_{b}$ lifetime [23] of $1.477\pm 0.032$${\mathrm{\,ps}}$,
$|\Delta E|t/2\lesssim 10^{-4}$. This effect can therefore be neglected and,
in the limit of small mixing, the ratio of the rate of oscillated decays,
$P_{X\rightarrow\overline{X}}(t)$, to the rate of nonoscillated decays,
$P_{X\rightarrow X}(t)$, varies over time as

$R(t)\equiv\frac{P_{X\rightarrow\overline{X}}(t)}{P_{X\rightarrow
X}(t)}=\tan^{2}(t/\tau_{\mathrm{mix}})\simeq\frac{t^{2}}{\tau_{\mathrm{mix}}^{2}}\equiv(\omega
t)^{2},$ (1)

where $2\pi\tau_{\mathrm{mix}}$ is the oscillation period, and
$\omega=1/\tau_{\mathrm{mix}}$ gives the corresponding angular frequency and
is zero in the absence of oscillations. This angular frequency is related to
the mass difference $\Delta M$ and the width difference $\Delta\Gamma$ between
the eigenstates of the Hamiltonian by $\omega^{2}=(\Delta
M/2)^{2}+(\Delta\Gamma/4)^{2}$, and in the limit that BNV in the decay itself
is negligible, $\omega=\Delta M/2$.

This Letter presents a search for baryon-number-violating ${\mathchar
28932\relax}^{0}_{b}$ oscillations, performed with a sample of ${p}{p}$
collision data recorded by the LHCb experiment, corresponding to an integrated
luminosity of 3$\mbox{\,fb}^{-1}$ collected at centre-of-mass energies
$\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$ and
8$\mathrm{\,Te\kern-1.00006ptV}$. This is the first such search for
oscillations in heavy baryons. The LHCb detector [24, 25] is a single-arm
forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed
for the study of particles containing $b$ or $c$ quarks. The detector elements
that are particularly relevant to this analysis are: a silicon-strip vertex
detector surrounding the $pp$ interaction region that allows $c$ and $b$
hadrons to be identified from their characteristically long flight distance; a
tracking system that provides a measurement of momentum, $p$, of charged
particles; and two ring-imaging Cherenkov detectors that are able to
discriminate between different species of charged hadrons. Samples of
simulated events are used to study the detector response and its effect on the
measurement. In the simulation, $pp$ collisions are generated using Pythia
[26, *Sjostrand:2006za] with a specific LHCb configuration [28]. Decays of
hadronic particles are described by EvtGen [29], in which final-state
radiation is generated using Photos [30]. The interaction of the generated
particles with the detector, and its response, are implemented using the
Geant4 toolkit [31, *Agostinelli:2002hh] as described in Ref. [33].

Two classes of ${\mathchar 28932\relax}^{\prime,*\,-}_{b}$ candidates are
defined. Baryon-number-conserving decays, in which a strong decay ${{\mathchar
28932\relax}^{\prime,*\,-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$ is followed by weak decays ${{\mathchar
28932\relax}^{0}_{b}}\rightarrow{{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}}$
and ${{\mathchar 28932\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$, are
referred to as opposite-sign (OS) candidates, since the ${\pi}^{-}$ emitted in
the strong decay and the $p$ have charges of opposite sign. Conversely, in
same-sign (SS) candidates the first decay ${{\mathchar
28932\relax}^{\prime,*\,-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$ is followed by weak decays to a final state
of different baryon number, ${{\overline{\mathchar
28932\relax}}{}_{b}^{0}}\rightarrow{{\overline{\mathchar
28932\relax}}{}_{c}^{-}}{{\pi}^{+}}$ and ${{\overline{\mathchar
28932\relax}}{}_{c}^{-}}\rightarrow{\overline{p}}{{K}^{+}}{{\pi}^{-}}$.

The reconstruction and selection procedures are the same as those described in
Ref. [18], except for one additional requirement on the track quality of the
pion produced in the ${{\mathchar 28932\relax}^{\prime,*\,-}_{b}}$ decay. This
requirement rejects a source of peaking background that can arise when a
genuine ${{\mathchar 28932\relax}^{\prime,*\,-}_{b}}\rightarrow{{\mathchar
28932\relax}^{0}_{b}}{{\pi}^{-}}$ decay occurs but the ${{\pi}^{-}}$ track is
misreconstructed such that its charge is incorrect and the candidate migrates
from the OS to the SS class [34]. In studies of simulated events, the fitted
SS yield of this contribution is found to be smaller than that of correctly
reconstructed OS signal by a factor of $(1.3\pm 0.3)\times 10^{-3}$, where the
uncertainty is statistical. Applying the additional track quality requirement
reduces the SS contribution in simulation by an order of magnitude, such that
it becomes smaller than the OS yield by a factor of $(1.6\pm 2.0)\times
10^{-4}$, corresponding to an expected SS peaking background yield of less
than 0.1, which is negligible. The track quality requirement also reduces the
OS signal yield in data by approximately 10% and the combinatorial background
by approximately 20% compared to Ref. [18]. Figure 2 shows the spectra of the
mass difference, $\delta m$, for the selected OS and SS candidates, defining
$\delta m\equiv m({{\mathchar 28932\relax}^{0}_{b}}\pi)-m({{\mathchar
28932\relax}^{0}_{b}})-m_{\pi}$, where $m_{\pi}$ is the known $\pi^{\pm}$ mass
[7], and $m({{\mathchar 28932\relax}^{0}_{b}}\pi)$ and $m({{\mathchar
28932\relax}^{0}_{b}})$ are the reconstructed invariant masses of the
${{\mathchar 28932\relax}^{0}_{b}}\pi$ and ${\mathchar 28932\relax}^{0}_{b}$
candidates. The figure also shows an unbinned extended maximum likelihood fit
to the OS candidates, performed following the same procedure as described
below and in Ref. [18], as a blue curve.

Figure 2:  Spectra of the mass difference $\delta m\equiv m({{\mathchar
28932\relax}^{0}_{b}}\pi)-m({{\mathchar 28932\relax}^{0}_{b}})-m_{\pi}$ in
data after the full selection, for the opposite-sign (OS) sample (black points
with error bars) and same-sign (SS) decays (red, hatched histogram). The blue
curve is a fit to the OS data. The ${\mathchar 28932\relax}^{\prime-}_{b}$ and
${\mathchar 28932\relax}^{*-}_{b}$ peaks are at $\delta m\approx 3.7$ and
$24$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and the $\delta m$ resolution
at these points is approximately $0.2$ and
$0.5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively; the ${\mathchar
28932\relax}^{*-}_{b}$ also has a non-negligible natural width of
$\Gamma\approx 1.7$$\mathrm{\,Me\kern-1.00006ptV}$ [23]. Inset: detail of the
region $2.0<\delta m<5.5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The data are divided into seven bins of decay time (illustrated in Fig. 3)
that have approximately equal OS signal yields and cover the range
$0<t<8$${\mathrm{\,ps}}$, corresponding to approximately $5.4$ times the mean
${\mathchar 28932\relax}^{0}_{b}$ lifetime. The OS resonance yields in the
$i^{\mathrm{th}}$ bin are determined from a fit to the $\delta m$ distribution
of the OS data in that bin, with the resonance masses and the ${\mathchar
28932\relax}^{*-}_{b}$ width fixed to values obtained in a fit to the whole OS
data sample. In each bin of decay time, the shape and normalisation of the SS
combinatorial background are obtained from a fit to the $\delta m$ sideband
regions of the SS data in that bin (the sidebands being 0–2, 6–15, and
32–45${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). For a given value of the
angular frequency $\omega$ of the oscillations, the expected ratio of SS to OS
decays in the bin may be computed. In combination with the OS yield and the
shape and normalisation of the SS background obtained as described above, this
fully determines the probability density function for the SS data in bin $i$,
and the corresponding likelihood $L_{i}(\omega)$ is evaluated. The overall
likelihood is obtained by combining all bins as
$L(\omega)=\prod_{i}L_{i}(\omega)$.

A test statistic $\Delta$ is defined based on the likelihood ratio approach as
${\Delta\equiv 2\ln L(\hat{\omega})-2\ln L(0)}$, where $\hat{\omega}$ is the
best-fit value of $\omega$ and is estimated from a likelihood scan. Only the
physical domain $\omega\geq 0$ is considered, and consequently $\hat{\omega}$
is expected to be zero approximately half of the time under the null
hypothesis. The best-fit value for the data is found to be $\hat{\omega}=0$
and the test statistic is therefore $\Delta=0$.

Since no evidence of BNV oscillations is found, an upper limit at the 95%
confidence level is placed on $\omega$ following the $\mathrm{CL}_{s}$ method
[35, 36]. Ensembles of parameterised simulations, referred to as
pseudoexperiments, are generated for a range of different oscillation angular
frequencies $\omega$. The pseudoexperiments include variation of efficiency
with decay time, decay time and mass resolution, combinatorial background, and
misclassification of OS candidates as SS via the misreconstruction described
earlier. To incorporate the associated systematic uncertainties, the input
parameters used to define the distributions (the masses and yields of the
resonances, the natural width of the ${\mathchar 28932\relax}^{*-}_{b}$, the
background yield and shape parameters, and the signal misclassification rate)
are varied randomly within their uncertainties for each pseudoexperiment. Each
pseudoexperiment is analysed in the same way as data, and its test statistic
$\Delta$ computed. Coverage tests with pseudoexperiments indicate that the
procedure overcovers for small values of $\omega$, with 100% coverage at
$\omega=0$, and that the coverage converges asymptotically to 95% as the true
value of $\omega$ increases.

An upper limit of $\omega<0.08\,{\mathrm{\,ps}}^{-1}$ at the 95% confidence
level is obtained, which corresponds to
$\tau_{\mathrm{mix}}>13$${\mathrm{\,ps}}$. This result can also be expressed
in terms of the time-integrated mixing rate $\chi$, defined as the fraction of
particles produced as ${\mathchar 28932\relax}^{0}_{b}$ that decay as
${\overline{\mathchar 28932\relax}}{}_{b}^{0}$, or vice versa. Under the
assumption of quadratic time dependence for $R(t)$,
$\chi=2\,\omega^{2}\,\tau^{2}<2.7\%$ at the 95% confidence level, where $\tau$
is the known ${\mathchar 28932\relax}^{0}_{b}$ lifetime [23]. For the purposes
of illustration, the evolution of the expected SS yield with decay time for
$\omega=0.08\,{\mathrm{\,ps}}^{-1}$ and $\omega=0.16\,{\mathrm{\,ps}}^{-1}$ is
shown in Fig. 4, and compared to the SS yield in data as obtained with a
simplified statistical procedure.

Figure 3:  Distribution of $\delta m$ vs decay time. The opposite-sign (OS)
data are shown as grey points, and the same-sign (SS) data as larger red
triangles. The vertical lines indicate the decay time bins. The horizontal
lines are intended to guide the eye and indicate the ${\mathchar
28932\relax}^{\prime-}_{b}$ and ${\mathchar 28932\relax}^{*-}_{b}$ regions.
Due to selection requirements, few candidates are present at short decay
times.

Figure 4:  For illustration, plots of the decay time distribution of signal
candidates. In these plots, the two resonance regions are combined, and
background is subtracted with a simplified statistical procedure that, unlike
the likelihood described in the text, allows negative SS yields. (upper) The
OS and SS data. (lower) Detail of the SS data, along with the expected SS
yields under various hypotheses for the angular frequency $\omega$ of the
oscillations.

In summary, a search is performed for baryon-antibaryon oscillations in the
${\mathchar 28932\relax}^{0}_{b}$ system. This is the first such search in the
heavy-flavour sector, and is of particular interest since ${\mathchar
28932\relax}^{0}_{b}$ baryons may couple directly to flavour-diagonal six-
fermion operators that violate baryon number [13]. No evidence of baryon-
number-violating oscillations is found. In the limit of a small oscillation
rate, the ratio of same-sign to opposite-sign decays is expected to increase
quadratically with decay time. A limit on the oscillation angular frequency
$\omega<0.08\,{\mathrm{\,ps}}^{-1}$ at the 95% confidence level is obtained,
equivalent to $\tau_{\mathrm{mix}}>13$${\mathrm{\,ps}}$. This rules out
oscillations with a period comparable to the ${\mathchar 28932\relax}^{0}_{b}$
lifetime, as proposed in Ref. [15].

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom),
RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF
(Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities
behind the multiple open-source software packages on which we depend.
Individual groups or members have received support from AvH Foundation
(Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union),
ANR, Labex P2IO, ENIGMASS and OCEVU, and Région Auvergne-Rhône-Alpes (France),
RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith
Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust
(United Kingdom).

## References

  * [1] A. D. Sakharov, _Violation of $C\\!P$ invariance, $C$ asymmetry, and baryon asymmetry of the universe_, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32, [JETP Lett. 5 (1967) 24]
  * [2] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, _Evidence for the $2\pi$ decay of the $K_{2}^{0}$ meson_, Phys. Rev. Lett. 13 (1964) 138
  * [3] BaBar collaboration, B. Aubert et al., _Observation of $C\\!P$ violation in the $B^{0}$ meson system_, Phys. Rev. Lett. 87 (2001) 091801, arXiv:hep-ex/0107013
  * [4] Belle collaboration, K. Abe et al., _Observation of large $C\\!P$ violation in the neutral $B$ meson system_, Phys. Rev. Lett. 87 (2001) 091802, arXiv:hep-ex/0107061
  * [5] M. B. Gavela, P. Hernández, J. Orloff, and O. Pène, _Standard model CP violation and baryon asymmetry_ , Mod. Phys. Lett. A9 (1994) 795, arXiv:hep-ph/9312215
  * [6] M. B. Gavela et al., _Standard model CP violation and baryon asymmetry. Part 2: Finite temperature_ , Nucl. Phys. B430 (1994) 382, arXiv:hep-ph/9406289
  * [7] Particle Data Group, C. Patrignani et al., _Review of particle physics_ , Chin. Phys. C40 (2016) 100001, and 2017 update
  * [8] A. Salam and J. A. Strathdee, _Supersymmetry and fermion number conservation_ , Nucl. Phys. B87 (1975) 85
  * [9] P. Fayet, _Supergauge invariant extension of the Higgs mechanism and a model for the electron and its neutrino_ , Nucl. Phys. B90 (1975) 104
  * [10] S. Weinberg, _Supersymmetry at ordinary energies. Masses and conservation laws_ , Phys. Rev. D26 (1982) 287
  * [11] G. R. Farrar and S. Weinberg, _Supersymmetry at ordinary energies. II. $R$ invariance, Goldstone bosons, and gauge fermion masses_, Phys. Rev. D27 (1983) 2732
  * [12] C. Smith, _Proton stability from a fourth family_ , Phys. Rev. D85 (2012) 036005, arXiv:1105.1723
  * [13] G. Durieux, J.-M. Gérard, F. Maltoni, and C. Smith, _Three-generation baryon and lepton number violation at the LHC_ , Phys. Lett. B721 (2013) 82, arXiv:1210.6598
  * [14] K. Aitken, D. McKeen, A. E. Nelson, and T. Neder, _Baryogenesis from oscillations of charmed or beautiful baryons_ , Phys. Rev. D96 (2017) 075009, arXiv:1708.01259
  * [15] V. A. Kuzmin, _Might fast $B$-violating transitions be found soon?_, in Future prospects of baryon instability search in ${p}$ decay and ${n}\rightarrow{\overline{n}}$ oscillation experiments: Proceedings, international workshop, Oak Ridge, US, 28-30 Mar, 1996, p. 89, 1996.  arXiv:hep-ph/9609253
  * [16] J. D. Bjorken, _What lies ahead?_ , in The project, the progress, the physics: Proceedings of the 1991 symposium on the Superconducting Supercollider Laboratory 13-17 Oct. 1991 Corpus Christi, Texas, p. 651, 1991.  SLAC-PUB-5673
  * [17] D. McKeen and A. E. Nelson, _$C\\!P$ violating baryon oscillations_, Phys. Rev. D94 (2016) 076002, arXiv:1512.05359
  * [18] LHCb collaboration, R. Aaij et al., _Observation of two new ${{\mathchar 28932\relax}^{-}_{b}}$ baryon resonances_, Phys. Rev. Lett. 114 (2015) 062004, arXiv:1411.4849
  * [19] P. F. Harrison and H. R. Quinn, _The BABAR physics book: Physics at an asymmetric $B$ factory_, SLAC-R-504, 1998
  * [20] BaBar and Belle collaborations, A. J. Bevan et al., _The physics of the $B$ factories_, Eur. Phys. J. C74 (2014) 3026, arXiv:1406.6311
  * [21] Heavy Flavor Averaging Group, Y. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016_, arXiv:1612.07233, updated results and plots available at http://www.slac.stanford.edu/xorg/hflav/
  * [22] D. G. Phillips II et al., _Neutron-antineutron oscillations: Theoretical status and experimental prospects_ , Phys. Rept. 612 (2016) 1, arXiv:1410.1100
  * [23] LHCb collaboration, R. Aaij et al., _Precision measurement of the mass and lifetime of the ${{\mathchar 28932\relax}^{0}_{b}}$ baryon_, Phys. Rev. Lett. 113 (2014) 032001, arXiv:1405.7223
  * [24] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [25] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [26] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [27] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [28] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [29] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [30] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [31] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [32] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [33] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [34] LHCb collaboration, R. Aaij et al., _First observation of ${{D}^{0}}-{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ oscillations in ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays and a measurement of the associated coherence parameters_, Phys. Rev. Lett. 116 (2016) 241801, arXiv:1602.07224
  * [35] A. L. Read, _Presentation of search results: The $CL_{s}$ technique_, J. Phys. G28 (2002) 2693
  * [36] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, _Asymptotic formulae for likelihood-based tests of new physics_ , Eur. Phys. J. C71 (2011) 1554, arXiv:1007.1727

LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B.
Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S.
Bachmann12, J.J. Back50, A. Badalov38,m, C. Baesso62, S. Baker55, V.
Balagura7,b, W. Baldini17, A. Baranov35, R.J. Barlow56, C. Barschel40, S.
Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V. Battista41, A.
Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1, A. Beiter61,
L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K. Belous37, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S. Beranek9, A.
Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt42, F. Betti15, M.-O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42,
S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bitadze56, A. Bizzeti18,u, M.
Bjørn57, T. Blake50, F. Blanc41, J. Blouw11,†, S. Blusk61, V. Bocci26, T.
Boettcher58, A. Bondar36,w, N. Bondar31, W. Bonivento16, I. Bordyuzhin32, A.
Borgheresi21,i, S. Borghi56, M. Borisyak35, M. Borsato39, F. Bossu7, M.
Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12, T.
Britton61, J. Brodzicka27, D. Brundu16, E. Buchanan48, C. Burr56, A.
Bursche16,f, J. Buytaert40, W. Byczynski40, S. Cadeddu16, H. Cai64, R.
Calabrese17,g, R. Calladine47, M. Calvi21,i, M. Calvo Gomez38,m, A.
Camboni38,m, P. Campana19, D.H. Campora Perez40, L. Capriotti56, A.
Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i,
L. Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina21, L. Castillo
Garcia41, M. Cattaneo40, G. Cavallero20,40,h, R. Cenci24,t, D. Chamont7, M.
Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S.
Chen56, S.F. Cheung57, S.-G. Chitic40, V. Chobanova39, M. Chrzaszcz42,27, A.
Chubykin31, P. Ciambrone19, X. Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52, M.
Clemencic40, H.V. Cliff49, J. Closier40, J. Cogan6, E. Cogneras5, V.
Cogoni16,f, L. Cojocariu30, P. Collins40, T. Colombo40, A. Comerma-Montells12,
A. Contu40, A. Cook48, G. Coombs40, S. Coquereau38, G. Corti40, M. Corvo17,g,
C.M. Costa Sobral50, B. Couturier40, G.A. Cowan52, D.C. Craik58, A.
Crocombe50, M. Cruz Torres1, R. Currie52, C. D’Ambrosio40, F. Da Cunha
Marinho2, E. Dall’Occo43, J. Dalseno48, A. Davis3, O. De Aguiar Francisco54,
S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d,
P. De Simone19, C.T. Dean53, D. Decamp4, L. Del Buono8, H.-P. Dembinski11, M.
Demmer10, A. Dendek28, D. Derkach35, O. Deschamps5, F. Dettori54, B. Dey65, A.
Di Canto40, P. Di Nezza19, H. Dijkstra40, F. Dordei40, M. Dorigo40, A. Dosil
Suárez39, L. Douglas53, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany8,
P. Durante40, R. Dzhelyadin37, M. Dziewiecki12, A. Dziurda40, A. Dzyuba31, S.
Easo51, M. Ebert52, U. Egede55, V. Egorychev32, S. Eidelman36,w, S.
Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53, S. Ely61, S.
Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S. Farry54, D.
Fazzini21,i, L. Federici25, D. Ferguson52, G. Fernandez38, P. Fernandez
Declara40, A. Fernandez Prieto39, F. Ferrari15, F. Ferreira Rodrigues2, M.
Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M. Fiore17,g, M. Fiorini17,g, M.
Firlej28, C. Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, K. Fohl40, M.
Fontana16,40, F. Fontanelli20,h, D.C. Forshaw61, R. Forty40, V. Franco Lima54,
M. Frank40, C. Frei40, J. Fu22,q, W. Funk40, E. Furfaro25,j, C. Färber40, E.
Gabriel52, A. Gallas Torreira39, D. Galli15,e, S. Gallorini23, S. Gambetta52,
M. Gandelman2, P. Gandini57, Y. Gao3, L.M. Garcia Martin70, J. García
Pardiñas39, J. Garra Tico49, L. Garrido38, P.J. Garsed49, D. Gascon38, C.
Gaspar40, L. Gavardi10, G. Gazzoni5, D. Gerick12, E. Gersabeck12, M.
Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gianì41, V. Gibson49, O.G. Girard41,
L. Giubega30, K. Gizdov52, V.V. Gligorov8, D. Golubkov32, A. Golutvin55,40, A.
Gomes1,a, I.V. Gorelov33, C. Gotti21,i, E. Govorkova43, J.P. Grabowski12, R.
Graciani Diaz38, L.A. Granado Cardoso40, E. Graugés38, E. Graverini42, G.
Graziani18, A. Grecu30, R. Greim9, P. Griffith16, L. Grillo21,40,i, L.
Gruber40, B.R. Gruberg Cazon57, O. Grünberg67, E. Gushchin34, Yu. Guz37, T.
Gys40, C. Göbel62, T. Hadavizadeh57, C. Hadjivasiliou5, G. Haefeli41, C.
Haen40, S.C. Haines49, B. Hamilton60, X. Han12, T.H. Hancock57, S. Hansmann-
Menzemer12, N. Harnew57, S.T. Harnew48, J. Harrison56, C. Hasse40, M. Hatch40,
J. He63, M. Hecker55, K. Heinicke10, A. Heister9, K. Hennessy54, P. Henrard5,
L. Henry70, E. van Herwijnen40, M. Heß67, A. Hicheur2, D. Hill57, C.
Hombach56, P.H. Hopchev41, Z.C. Huard59, W. Hulsbergen43, T. Humair55, M.
Hushchyn35, D. Hutchcroft54, P. Ibis10, M. Idzik28, P. Ilten58, R.
Jacobsson40, J. Jalocha57, E. Jans43, A. Jawahery60, F. Jiang3, M. John57, D.
Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N. Jurik57, S. Kandybei45, M.
Karacson40, J.M. Kariuki48, S. Karodia53, N. Kazeev35, M. Kecke12, M.
Kelsey61, M. Kenzie49, T. Ketel44, E. Khairullin35, B. Khanji12, C.
Khurewathanakul41, T. Kirn9, S. Klaver56, K. Klimaszewski29, T. Klimkovich11,
S. Koliiev46, M. Kolpin12, I. Komarov41, R. Kopecna12, P. Koppenburg43, A.
Kosmyntseva32, S. Kotriakhova31, M. Kozeiha5, L. Kravchuk34, M. Kreps50, P.
Krokovny36,w, F. Kruse10, W. Krzemien29, W. Kucewicz27,l, M. Kucharczyk27, V.
Kudryavtsev36,w, A.K. Kuonen41, K. Kurek29, T. Kvaratskheliya32,40, D.
Lacarrere40, G. Lafferty56, A. Lai16, G. Lanfranchi19, C. Langenbruch9, T.
Latham50, C. Lazzeroni47, R. Le Gac6, A. Leflat33,40, J. Lefrançois7, R.
Lefèvre5, F. Lemaitre40, E. Lemos Cid39, O. Leroy6, T. Lesiak27, B.
Leverington12, P.-R. Li63, T. Li3, Y. Li7, Z. Li61, T. Likhomanenko68, R.
Lindner40, F. Lionetto42, V. Lisovskyi7, X. Liu3, D. Loh50, A. Loi16, I.
Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio Martinez39, H. Luo52, A.
Lupato23, E. Luppi17,g, O. Lupton40, A. Lusiani24, X. Lyu63, F. Machefert7, F.
Maciuc30, V. Macko41, P. Mackowiak10, S. Maddrell-Mander48, O. Maev31,40, K.
Maguire56, D. Maisuzenko31, M.W. Majewski28, S. Malde57, A. Malinin68, T.
Maltsev36,w, G. Manca16,f, G. Mancinelli6, P. Manning61, D. Marangotto22,q, J.
Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito38, M. Marinangeli41,
P. Marino41, J. Marks12, G. Martellotti26, M. Martin6, M. Martinelli41, D.
Martinez Santos39, F. Martinez Vidal70, D. Martins Tostes2, L.M. Massacrier7,
A. Massafferri1, R. Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21, A.
Mauri42, E. Maurice7,b, B. Maurin41, A. Mazurov47, M. McCann55,40, A. McNab56,
R. McNulty13, J.V. Mead54, B. Meadows59, C. Meaux6, F. Meier10, N. Meinert67,
D. Melnychuk29, M. Merk43, A. Merli22,40,q, E. Michielin23, D.A. Milanes66, E.
Millard50, M.-N. Minard4, L. Minzoni17, D.S. Mitzel12, A. Mogini8, J. Molina
Rodriguez1, T. Mombacher10, I.A. Monroy66, S. Monteil5, M. Morandin23, M.J.
Morello24,t, O. Morgunova68, J. Moron28, A.B. Morris52, R. Mountain61, F.
Muheim52, M. Mulder43, D. Müller56, J. Müller10, K. Müller42, V. Müller10, P.
Naik48, T. Nakada41, R. Nandakumar51, A. Nandi57, I. Nasteva2, M. Needham52,
N. Neri22,40, S. Neubert12, N. Neufeld40, M. Neuner12, T.D. Nguyen41, C.
Nguyen-Mau41,n, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A.
Nogay68, D.P. O’Hanlon50, A. Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19,
R. Oldeman16,f, C.J.G. Onderwater71, A. Ossowska27, J.M. Otalora Goicochea2,
P. Owen42, A. Oyanguren70, P.R. Pais41, A. Palano14,d, M. Palutan19,40, A.
Papanestis51, M. Pappagallo14,d, L.L. Pappalardo17,g, W. Parker60, C.
Parkes56, G. Passaleva18, A. Pastore14,d, M. Patel55, C. Patrignani15,e, A.
Pearce40, A. Pellegrino43, G. Penso26, M. Pepe Altarelli40, S. Perazzini40, P.
Perret5, L. Pescatore41, K. Petridis48, A. Petrolini20,h, A. Petrov68, M.
Petruzzo22,q, E. Picatoste Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26,
A. Pistone20,h, A. Piucci12, V. Placinta30, S. Playfer52, M. Plo Casasus39, F.
Polci8, M. Poli Lener19, A. Poluektov50,36, I. Polyakov61, E. Polycarpo2, G.J.
Pomery48, S. Ponce40, A. Popov37, D. Popov11,40, S. Poslavskii37, C.
Potterat2, E. Price48, J. Prisciandaro39, C. Prouve48, V. Pugatch46, A. Puig
Navarro42, H. Pullen57, G. Punzi24,p, W. Qian50, R. Quagliani7,48, B.
Quintana5, B. Rachwal28, J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S.
Rangel2, I. Raniuk45,†, F. Ratnikov35, G. Raven44, M. Ravonel Salzgeber40, M.
Reboud4, F. Redi55, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz70, V.
Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, V. Rives
Molina38, P. Robbe7, A. Robert8, A.B. Rodrigues1, E. Rodrigues59, J.A.
Rodriguez Lopez66, P. Rodriguez Perez56,†, A. Rogozhnikov35, S. Roiser40, A.
Rollings57, V. Romanovskiy37, A. Romero Vidal39, J.W. Ronayne13, M. Rotondo19,
M.S. Rudolph61, T. Ruf40, P. Ruiz Valls70, J. Ruiz Vidal70, J.J. Saborido
Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino Guimaraes1,
C. Sanchez Mayordomo70, B. Sanmartin Sedes39, R. Santacesaria26, C.
Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, G. Sarpis56, A.
Sarti26, C. Satriano26,s, A. Satta25, D.M. Saunders48, D. Savrina32,33, S.
Schael9, M. Schellenberg10, M. Schiller53, H. Schindler40, M. Schlupp10, M.
Schmelling11, T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40,
H.F. Schreiner59, K. Schubert10, M. Schubiger41, M.-H. Schune7, R.
Schwemmer40, B. Sciascia19, A. Sciubba26,k, A. Semennikov32, E.S. Sepulveda8,
A. Sergi47, N. Serra42, J. Serrano6, L. Sestini23, P. Seyfert40, M. Shapkin37,
I. Shapoval45, Y. Shcheglov31, T. Shears54, L. Shekhtman36,w, V. Shevchenko68,
B.G. Siddi17,40, R. Silva Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S.
Simone14,d, M. Sirendi49, N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T.
Smith52, J. Smith49, M. Smith55, l. Soares Lavra1, M.D. Sokoloff59, F.J.P.
Soler53, B. Souza De Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F.
Stagni40, M. Stahl12, S. Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42,
S. Stemmle12, O. Stenyakin37, M. Stepanova31, H. Stevens10, S. Stone61, B.
Storaci42, S. Stracka24,p, M.E. Stramaglia41, M. Straticiuc30, U. Straumann42,
J. Sun3, L. Sun64, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, M.
Szczekowski29, T. Szumlak28, M. Szymanski63, S. T’Jampens4, A. Tayduganov6, T.
Tekampe10, G. Tellarini17,g, F. Teubert40, E. Thomas40, J. van Tilburg43, M.J.
Tilley55, V. Tisserand4, M. Tobin41, S. Tolk49, L. Tomassetti17,g, D.
Tonelli24, F. Toriello61, R. Tourinho Jadallah Aoude1, E. Tournefier4, M.
Traill53, M.T. Tran41, M. Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P.
Tsopelas43, A. Tully49, N. Tuning43,40, A. Ukleja29, A. Usachov7, A.
Ustyuzhanin35, U. Uwer12, C. Vacca16,f, A. Vagner69, V. Vagnoni15,40, A.
Valassi40, S. Valat40, G. Valenti15, R. Vazquez Gomez19, P. Vazquez
Regueiro39, S. Vecchi17, M. van Veghel43, J.J. Velthuis48, M. Veltri18,r, G.
Veneziano57, A. Venkateswaran61, T.A. Verlage9, M. Vernet5, M. Vesterinen57,
J.V. Viana Barbosa40, B. Viaud7, D. Vieira63, M. Vieites Diaz39, H. Viemann67,
X. Vilasis-Cardona38,m, M. Vitti49, V. Volkov33, A. Vollhardt42, B. Voneki40,
A. Vorobyev31, V. Vorobyev36,w, C. Voß9, J.A. de Vries43, C. Vázquez Sierra39,
R. Waldi67, C. Wallace50, R. Wallace13, J. Walsh24, J. Wang61, D.R. Ward49,
H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden42, M. Whitehead40, J.
Wicht50, G. Wilkinson57,40, M. Wilkinson61, M. Williams56, M.P. Williams47, M.
Williams58, T. Williams47, F.F. Wilson51, J. Wimberley60, M. Winn7, J.
Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K.
Wraight53, K. Wyllie40, Y. Xie65, Z. Xu4, Z. Yang3, Z. Yang60, Y. Yao61, H.
Yin65, J. Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V.
Zhukov33, J.B. Zonneveld52, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Universita & INFN, Milano-Bicocca, Milano, Italy
22Sezione di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
71Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased