EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-325 LHCb-PAPER-2017-039

Search for the rare decay ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p\mu^{+}\mu^{-}$

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

A search for the flavor-changing neutral-current decay ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}}{\mu^{-}}$ is reported using a
data set corresponding to an integrated luminosity of $3.0\mbox{\,fb}^{-1}$
collected by the LHCb collaboration. No significant signal is observed outside
of the dimuon mass regions around the $\phi$ and $\omega$ resonances and an
upper limit is placed on the branching fraction of ${\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}})<7.7~{}(9.6)\times
10^{-8}~{}{\rm at}~{}90\%~{}(95\%)$ confidence level. A significant signal is
observed in the $\omega$ dimuon mass region for the first time.

Published in Phys. Rev. D97 (2018) 091101(R)

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

The flavor-changing neutral-current (FCNC) decay ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}}{\mu^{-}}$ (inclusion of the
charge-conjugate processes is implied throughout) is expected to be heavily
suppressed in the Standard Model (SM) by the Glashow-Iliopoulos-Maiani
mechanism [1]. The branching fractions for short-distance $c\rightarrow
u\ell^{+}\ell^{-}$ contributions to the transition are expected to be of
$\mathcal{O}(10^{-9})$ in the SM, but can be enhanced by effects beyond the
SM. However, long-distance contributions proceeding via a tree-level
amplitude, with an intermediate meson resonance decaying into a dimuon pair
[2, 3], can increase the branching fraction up to $\mathcal{O}(10^{-6})$ [4].
The short-distance and hadronic contributions can be separated by splitting
the data set into relevant regions of dimuon mass. The ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}}{\mu^{-}}$ decay has been
previously searched for by the BaBar collaboration [5] yielding $11.1\pm
5.0\pm 2.5$ events and an upper limit on the branching fraction of $4.4\times
10^{-5}$ at 90% CL.

Similar FCNC transitions for the ${b}$-quark system ($b\rightarrow
s\ell^{+}\ell^{-}$) exhibit a pattern of consistent deviations from the
current SM predictions both in branching fractions [6] and angular observables
[7], with the combined significance reaching 4 to 5 standard deviations[8, 9].
Processes involving $c\rightarrow u\ell^{+}\ell^{-}$ transitions are far less
explored at both the experimental and theoretical levels, which makes such
measurements desirable. Similar analyses of the $D$ system have reported
evidence for the long-distance contribution [10], however, the short-distance
contributions have not been established [11].

In this paper we report on the search for the ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ decay, using a data
set corresponding to an integrated luminosity of $3.0\mbox{\,fb}^{-1}$ of $pp$
collisions collected in 2011 and 2012 with the LHCb experiment. The branching
fraction is measured with respect to the branching fraction of the decay
${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi(1020)$ with
$\phi(1020)\rightarrow{\mu^{+}\mu^{-}}$ (here and after denoted as
${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi({\mu^{+}\mu^{-}})$)
decay, which has the benefit of having the same initial and final states and
consequently many sources of systematic uncertainty are expected to cancel.

The LHCb detector [12, 13] 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 includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region [14], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes [15] placed
downstream of the magnet. The tracking system provides a measurement of
momentum, of charged particles with a relative uncertainty that varies from
0.5% at low momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
minimum distance of a track to a primary vertex (PV), the impact parameter
(IP), is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[16]. Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers
[17]. The online event selection is performed by a trigger [18], which
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction.

Samples of simulated events are used to understand the properties of the
signal and normalization channels. The $pp$ collisions are generated using
Pythia [19, *Sjostrand:2007gs] with a specific LHCb configuration [21]. Decays
of hadronic particles are described by EvtGen [22], in which final-state
radiation is generated using Photos [23]. The decay of the ${\mathchar
28931\relax}^{+}_{c}$ baryon to $p$ $\mu^{+}\mu^{-}$ is simulated with a
three-body phase-space model. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [24,
*Agostinelli:2002hh] as described in Ref. [26]. The ${{\mathchar
28931\relax}^{+}_{c}}$ baryons are produced in two ways at a hadron collider;
as prompt ${{\mathchar 28931\relax}^{+}_{c}}$ or in $b$–hadron decays. The
simulation contains a mixture of these two production mechanisms, according to
the known ${{\mathchar 28931\relax}^{+}_{c}}$ and $b$–hadron production cross-
sections [27, 28].

The simulated samples are used to determine the selection criteria, in
particular to train a multivariate classifier that is aimed at distinguishing
signal signatures in the background-dominated data set. The simulated samples
are also used to calculate the efficiencies of several selection steps.

Candidate events of ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}}{\mu^{-}}$ decay are
reconstructed by combining a pair of charged tracks identified as muons with
one identified as a proton. Candidates that pass the trigger selections are
subject to further requirements consisting of kinematic and particle
identification criteria and based on the response of a multivariate
classifier. Each of the final-state tracks is required to be of good quality,
to have $\mbox{$p_{\mathrm{T}}$}>300{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
to be incompatible with originating from any of the PVs in the event. The
tracks are also required to form a good-quality secondary vertex with a
corresponding flight distance of at least $0.1~{}\rm mm$ from all of the PVs
in the event. The invariant mass of the dimuon system is required to be
smaller than $1400{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Three dimuon mass
regions are defined:

  * •

a region around the known $\phi$ mass,
$\left[985,1055\right]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, used as a
normalization channel;

  * •

a region around the known $\omega$ mass (the $\omega$ denotes hereafter the
$\omega(782)$ meson),
$\left[759,805\right]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, used to
isolate the ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}\omega$ decay;

  * •

a nonresonant region $({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}})$, with excluded ranges
$\pm 40{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known $\omega$ and
$\phi$ masses.

After the preselection, the normalization channel is still dominated by
combinatorial background, i.e. combinations of tracks that do not all
originate from a genuine ${\mathchar 28931\relax}^{+}_{c}$ baryon. A boosted
decision tree (BDT) is trained to reduce the combinatorial background to
manageable level. The BDT is trained using the kinematic and topological
variables of the ${\mathchar 28931\relax}^{+}_{c}$ candidate, related to its
flight distance, decay vertex quality, $p_{\mathrm{T}}$ and impact parameter
with respect to the primary vertex. In the BDT training, ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ simulated events are
used as a proxy for the signal and data outside the signal $p$
$\mu^{+}\mu^{-}$ invariant mass region extending up to $\pm
300{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known ${{\mathchar
28931\relax}^{+}_{c}}$ mass is used as a proxy for the background.

A k-folding technique is used to ensure the training is unbiased [29], while
keeping the full available data sets for further analysis. A loose BDT cut is
applied to reduce the background to the same level as the normalization
channel yield.

A fit to the $p{\mu^{+}\mu^{-}}$ invariant-mass distribution of ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi({\mu^{+}\mu^{-}})$ candidates after
the loose BDT requirement is shown in Fig. 1. The shape of the ${\mathchar
28931\relax}^{+}_{c}$ peak is parametrized by a Crystal Ball function [30]
with parameters determined from the simulation, while the background is
modeled with a first-order polynomial. The yield of the ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi({\mu^{+}\mu^{-}})$ decay is
determined to be $395\pm 45$ candidates. This sample is used for the final
optimization of selection requirements. It is checked at this stage that the
variables used in the signal selection are well described by simulation within
the available sample size.

Figure 1: Mass distribution of ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ candidates in the
$\phi$ region after the first BDT requirement. The solid line shows the result
of the fit described in the text, while the dashed line indicates the
background component.

For the final selection a second BDT is trained, which includes additional
variables related to ${\mathchar 28931\relax}^{+}_{c}$-baryon decay properties
and the isolation of the proton and muons in the detector. The final
discrimination is performed in three dimensions: the BDT variable and two
particle identification (PID) variables, the proton-identification
discriminant and the muon-identification discriminant. The optimal set of BDT
and PID requirements is determined by finding the best expected upper limit on
the branching fraction of the signal relative to the normalization channel
using the CLs method [31] by means of Monte Carlo methods.

Several sources of background have been considered. An irreducible background
due to long-distance contributions originates from ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}V({\mu^{+}\mu^{-}})$ decays, with
intermediate resonances indicated by $V$. The $\rho(770)^{0}$, $\omega$ and
$\eta$ resonances are studied, however, their contribution to the nonresonant
region is expected to be negligible, because the $V$ meson mass is well
separated from the nonresonant region and/or the ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}V({\mu^{+}\mu^{-}})$ branching fraction is
small. Another background source considered is due to misidentification of
final-state particles in hadronic ${D}^{+}$, ${D}^{+}_{s}$ and ${\mathchar
28931\relax}^{+}_{c}$ decays. The expected contribution from this source has
been estimated using large samples of simulated events. Given the tight PID
requirements obtained from the optimization, only $2.0\pm 1.1$ candidates are
expected to fall into the ${\mathchar 28931\relax}^{+}_{c}$ mass window in the
nonresonant region.

The ratio of branching fractions is measured using

$\frac{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}})}{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi){\cal
B}(\phi\\!\rightarrow{\mu^{+}\mu^{-}})}=\frac{\rm{\epsilon\mathstrut_{norm}}}{\rm{\epsilon\mathstrut_{sig}}}\times\frac{N_{\rm
sig}}{N_{\rm norm}},$ (1)

where $N_{\rm sig}$ ($N_{\rm norm}$) is the observed yield for the signal
(normalization) decay mode. The factors $\rm\epsilon\mathstrut_{sig}$ and
$\rm\epsilon\mathstrut_{norm}$ indicate the corresponding total efficiencies
for signal and normalization channels, respectively. The efficiencies are
determined from the simulation.

In the case of the observation of the decay ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}V$, the ratio of branching fractions is
determined by

$\frac{{\cal B}({{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}V){\cal
B}(V\rightarrow{\mu^{+}\mu^{-}})}{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi){\cal
B}(\phi\\!\rightarrow{\mu^{+}\mu^{-}})}=\frac{\rm{\epsilon\mathstrut_{norm}}}{\rm{\epsilon_{V}}}\times\frac{N_{V}}{N_{\rm
norm}},$ (2)

where $N_{V}$ ($N_{\rm norm}$) is the number of candidates observed for the
${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}V$ (normalization) decay
mode. The factors $\rm\epsilon\mathstrut_{V}$ and
$\rm\epsilon\mathstrut_{norm}$ indicate the corresponding total efficiencies
for ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}V$ and normalization
channel, respectively.

As the final states of the signal and normalization channels are identical,
many sources of systematic uncertainty cancel in the ratio of the
efficiencies. There are three significant sources of systematic uncertainty.
The first is related to the finite size of the simulation samples, which
limits the precision on the efficiency ratio. The second is linked to residual
differences between data and simulation of the BDT distribution. The third is
associated to the simulation of PID and is determined from the uncertainty on
the PID calibration samples. The values of the contributions are given in
Table 1.

Several other sources of systematic uncertainty were considered: the trigger
efficiency, the shapes used in the invariant mass fit for signal and
normalization channels, the shape of the combinatorial background, and the
fraction of prompt ${\mathchar 28931\relax}^{+}_{c}$ baryons and ${\mathchar
28931\relax}^{+}_{c}$ baryons from $b$–hadron decays. All of these, however,
are at negligible level when compared to three dominant sources of systematic
uncertainty.

Table 1:  Systematic uncertainties on the efficiency ratio used in the determination of the branching fraction in the nonresonant and $\omega$ regions.  Uncertainty source | Value [%] | Value [%]
---|---|---
| ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ | ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}V({\mu^{+}}{\mu^{-}})$
| nonresonant |  $\omega$ region
Size of simulation samples | 4.4 | 10.0
BDT cut | 4.8 |  4.8
PID cut | 0.7 |  0.7
Total | 6.5 | 11.1

The simulated ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ decays have been
generated according to a phase-space model for the decay products. As the
exact physics model for the decay is not known, no systematic uncertainty is
assigned. Instead, the weights needed to recast the result in terms of any
physics model are provided in Fig. 2. The weights are described by a function
of the dimuon invariant mass squared $m^{2}({\mu^{+}\mu^{-}})$ and the
invariant mass of the proton and the negatively charged muon squared
$m^{2}({p}\mu^{-})$. The weights are normalized to the average efficiency.

Figure 2: The efficiency weights for ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ as a function of the
dimuon invariant mass squared $m^{2}({\mu^{+}\mu^{-}})$ and by the invariant
mass of the proton and the negatively charged muon squared
$m^{2}({p}\mu^{-})$. The weights are normalized to the average efficiency.

The distributions of the $p$ $\mu^{+}\mu^{-}$ invariant mass for the
${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$
candidates after final selections in the three dimuon mass ranges are
presented in Fig. 3. The ${\mathchar 28931\relax}^{+}_{c}$ peak is
parametrized by a Crystal Ball [30] function with parameters determined from
the simulation and the background is described by a first-order polynomial.
The fits are used to determine the signal yields.

Figure 3: Mass distribution for selected $p$ $\mu^{+}\mu^{-}$ candidates in
the three regions of the dimuon invariant mass: a) nonresonant region, b)
$\phi$ region, c) $\omega$ region. The solid lines show the results of the fit
as described in the text. The dashed lines indicate the background component.

No significant signal is observed in the nonresonant region (Fig. 3a). The
yield for the normalization channel is determined to be $96\pm 11$ candidates
(Fig. 3b). An accumulation of $13.2\pm 4.3$ candidates at the ${\mathchar
28931\relax}^{+}_{c}$ mass is observed in the $\omega$ region (Fig. 3c). The
statistical significance of the excess is determined to be $5.0\sigma$ using
Wilks’ theorem [32].

The distribution of the dimuon invariant mass of the ${\mathchar
28931\relax}^{+}_{c}$ candidates is shown in Fig. 4. An excess is seen at the
known $\omega$ and $\phi$ resonance masses. The data is well described by a
simple model including these resonances and a background component. The
$\omega$ and $\phi$ peaks are parametrized as Breit-Wigner functions of
relevant decay width [33] convolved with a Gaussian function to take into
account the experimental resolution. The addition of a component for the
$\rho(770)^{0}$ resonance (and its interference with the $\omega$ meson) does
not improve the fit quality. It is therefore assumed that the observed
candidates in the $\omega$ region are dominated by decays via the $\omega$
resonance.

Figure 4: Invariant mass distribution $m({{\mu^{+}\mu^{-}}})$ for ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ candidates with mass
$\pm 25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the ${\mathchar
28931\relax}^{+}_{c}$ mass. The solid line shows the result of the fit, while
the dashed line indicates the background component.

As no evidence for nonresonant ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ decays is found, an
upper limit on the branching fractions is determined using the CLs method. The
systematic uncertainties are included in the construction of CLs. The
following upper limits are obtained at different confidence levels (CL)

$\displaystyle\frac{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}})}{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi){\cal
B}(\phi\\!\rightarrow{\mu^{+}\mu^{-}})}<0.24~{}(0.28)~{}~{}~{}~{}{\rm
at}~{}~{}90\%~{}(95\%)~{}{\rm CL}$

The corresponding distribution of CLs is shown in Fig. 5.

Figure 5: The CLs value as a function of the ${\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}})$ branching fraction.
The median expected value of an ensemble (assuming no signal component) is
shown by the dashed line, with the $\pm 1\sigma$ and $\pm 2\sigma$ regions
shaded. The observed distribution is shown by the solid line.

Using the values of the branching fractions for ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi$ and
$\phi\\!\rightarrow{\mu^{+}\mu^{-}}$ decays from Ref. [33] and including their
uncertainties in the CLs construction, an upper limit on the branching
fraction is determined to be

$\displaystyle{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}})<7.7~{}(9.6)\times
10^{-8}~{}~{}~{}~{}{\rm at}~{}~{}90\%~{}(95\%)~{}{\rm CL}.$

Under the above-mentioned assumption of the ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\omega$ dominance in the $\omega$
region, the relative branching fraction with respect to the normalization
channel is determined according to Eq. 2

$\displaystyle\frac{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\omega){\cal
B}(\omega\\!\rightarrow{\mu^{+}\mu^{-}})}{{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\phi){\cal
B}(\phi\\!\rightarrow{\mu^{+}\mu^{-}})}=0.23\pm 0.08~{}({\rm stat})\pm
0.03~{}({\rm syst}).~{}~{}~{}$

Using the relevant branching fractions from Ref. [33], the branching fraction
of ${{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow{p}\omega$ is determined to
be

$\displaystyle{\cal B}({{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}\omega)=(9.4\pm 3.2~{}({\rm stat})\pm
1.0~{}({\rm syst})\pm 2.0~{}({\rm ext}))\times 10^{-4},~{}~{}~{}$

where the first uncertainty is statistical, the second corresponds to the
above-mentioned systematic effects and the third is due to the limited
knowledge of the relevant branching fractions. Assuming lepton universality,
the branching fraction ${\cal B}(\omega\rightarrow e^{+}e^{-})$ is used
instead of ${\cal B}(\omega\\!\rightarrow{\mu^{+}\mu^{-}})$.

In summary, a search for the ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{\mu^{+}\mu^{-}}$ decay is reported,
using $pp$ data collected with the LHCb experiment. The analysis is performed
in three regions of dimuon mass: $\phi$, $\omega$ and nonresonant. The upper
limit on the nonresonant mode is improved by two orders of magnitude with
respect to the previous measurement [5]. For the first time the signal is seen
in the $\omega$ region with a statistical significance of 5 standard
deviations.

## 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 and OCEVU, and Région Auvergne-Rhône-Alpes (France), RFBR, RSF
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).

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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, M. Atzeni42, 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. Belyaev32,40, 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. Bizzeti18,u, M. Bjørn57, T.
Blake50, F. Blanc41, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31, I. Bordyuzhin32, S. Borghi56,40, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
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, M. Cattaneo40, G. Cavallero20,40,h, R. Cenci24,t, D.
Chamont7, M.G. Chapman48, M. Charles8, Ph. Charpentier40, G.
Chatzikonstantinidis47, M. Chefdeville4, S. Chen16, S.F. Cheung57, S.-G.
Chitic40, V. Chobanova39, M. Chrzaszcz42, A. Chubykin31, P. Ciambrone19, X.
Cid Vidal39, G. Ciezarek40, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco40, J. Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30,
P. Collins40, T. Colombo40, A. Comerma-Montells12, A. Contu16, 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, C.L. Da Silva73, E. Dall’Occo43, J.
Dalseno48, A. Davis3, O. De Aguiar Francisco40, K. De Bruyn40, 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, J.M. Durham73, D. Dutta56, R. Dzhelyadin37, M. Dziewiecki12, A.
Dziurda40, A. Dzyuba31, S. Easo51, 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, L. Ferreira Lopes41,
F. Ferreira Rodrigues2, M. Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M.
Fiorini17,g, M. Firlej28, C. Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, M.
Fontana16,40, F. Fontanelli20,h, 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. Gandini22, Y. Gao3, L.M. Garcia Martin71, J. García Pardiñas39, J. Garra
Tico49, L. Garrido38, D. Gascon38, C. Gaspar40, L. Gavardi10, G. Gazzoni5, D.
Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, Ph. Ghez4, S. Gianì41,
V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V. Gligorov8, D.
Golubkov32, A. Golutvin55,69, 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. Grillo56, 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, C.
Hasse40, M. Hatch40, J. He63, M. Hecker55, K. Heinicke10, A. Heister9, K.
Hennessy54, P. Henrard5, L. Henry71, E. van Herwijnen40, M. Heß67, A.
Hicheur2, D. Hill57, P.H. Hopchev41, W. Hu65, W. Huang63, Z.C. Huard59, W.
Hulsbergen43, T. Humair55, M. Hushchyn35, D. Hutchcroft54, P. Ibis10, M.
Idzik28, P. Ilten47, R. Jacobsson40, J. Jalocha57, E. Jans43, A. Jawahery60,
M. Jezabek27, 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, F. Keizer49, M. Kelsey61, M. Kenzie49, T.
Ketel44, E. Khairullin35, B. Khanji12, C. Khurewathanakul41, K.E. Kim61, T.
Kirn9, S. Klaver19, K. Klimaszewski29, T. Klimkovich11, S. Koliiev46, M.
Kolpin12, R. Kopecna12, P. Koppenburg43, A. Kosmyntseva32, S. Kotriakhova31,
M. Kozeiha5, L. Kravchuk34, M. Kreps50, F. Kress55, P. Krokovny36,w, W.
Krzemien29, W. Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K.
Kuonen41, 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, X.
Liang61, 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, B. Malecki27, A. Malinin68, T. Maltsev36,w, G.
Manca16,f, G. Mancinelli6, 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
Vidal71, 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. Mombächer10, 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. Müller40, 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. Onderwater72, A. Ossowska27, J.M. Otalora
Goicochea2, P. Owen42, A. Oyanguren71, P.R. Pais41, A. Palano14, M.
Palutan19,40, G. Panshin70, A. Papanestis51, M. Pappagallo52, L.L.
Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18,40, A. Pastore14,d,
M. Patel55, C. Patrignani15,e, A. Pearce40, A. Pellegrino43, G. Penso26, M.
Pepe Altarelli40, S. Perazzini40, D. Pereima32, P. Perret5, L. Pescatore41, K.
Petridis48, A. Petrolini20,h, A. Petrov68, M. Petruzzo22,q, E. Picatoste
Olloqui38, B. Pietrzyk4, G. Pietrzyk41, M. Pikies27, D. Pinci26, F. Pisani40,
A. Pistone20,h, A. Piucci12, V. Placinta30, S. Playfer52, M. Plo Casasus39, F.
Polci8, M. Poli Lener19, A. Poluektov50, 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, J. Qin63, R. Quagliani8, B.
Quintana5, B. Rachwal28, J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S.
Rangel2, I. Raniuk45,†, F. Ratnikov35,x, G. Raven44, M. Ravonel Salzgeber40,
M. Reboud4, F. Redi41, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz71, V.
Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, P. Robbe7,
A. Robert8, A.B. Rodrigues41, E. Rodrigues59, J.A. Rodriguez Lopez66, A.
Rogozhnikov35, S. Roiser40, A. Rollings57, V. Romanovskiy37, A. Romero
Vidal39,40, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz Valls71, J. Ruiz
Vidal71, J.J. Saborido Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f,
V. Salustino Guimaraes62, C. Sanchez Mayordomo71, B. Sanmartin Sedes39, R.
Santacesaria26, C. Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, G.
Sarpis56, A. Sarti19,k, C. Satriano26,s, A. Satta25, D.M. Saunders48, D.
Savrina32,33, S. Schael9, M. Schellenberg10, M. Schiller53, H. Schindler40, M.
Schmelling11, T. Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40,
H.F. Schreiner59, 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, Y. Shcheglov31, T.
Shears54, L. Shekhtman36,w, V. Shevchenko68, B.G. Siddi17, R. Silva
Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49,
N. Skidmore48, T. Skwarnicki61, 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, F. Stagni40, M. Stahl12, S. Stahl40, P. Stefko41, S.
Stefkova55, O. Steinkamp42, S. Stemmle12, O. Stenyakin37, M. Stepanova31, H.
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Straticiuc30, U. Straumann42, S. Strokov70, J. Sun3, L. Sun64, K. Swientek28,
V. Syropoulos44, 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. Tisserand5, M. Tobin41, S. Tolk49, L. Tomassetti17,g, D.
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S.A. Wotton49, K. Wyllie40, Y. Xie65, M. Xu65, Q. Xu63, Z. Xu3, Z. Xu4, Z.
Yang3, Z. Yang60, Y. Yao61, H. Yin65, J. Yu65, X. Yuan61, O. Yushchenko37,
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Zheng63, X. Zhu3, V. Zhukov9,33, 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
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 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
21Sezione INFN di 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 RAS),
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
39Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade 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 University of Science and Technology MISIS, Moscow, Russia,
associated to 32
70National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
71Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
72Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43
73Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

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, Vietnam
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
xNational Research University Higher School of Economics, Moscow, Russia
†Deceased


role on growth in finite time [40]. Our approach to growth considered a finite
number of species and reactions. Exploring chemical growth in CRNs where the
number of species can grow as the dynamics evolves (see e.g. [41]) is
certainly very important.

###### Acknowledgements.

The authors would like to thank Danilo Forastiere and Gianmaria Falasco for
helpful discussions. This research was supported by the Luxembourg National
Research Fund (FNR), via the research funding schemes PRIDE (Grant No.
19/14063202/ACTIVE), CORE project ChemComplex (Grant No. C21/MS/16356329), and
by project INTER/FNRS/20/15074473 funded by F.R.S.-FNRS (Belgium) and FNR
(Luxembourg).

##  Appendix A Eigenvalues of matrix $\mathbb{V}$

The rate equation of unimolecular and pseudo-unimolecular CRNs under mixed and
concentration control can be recast in the form of Eq. (34) (see Secs. III.2
and IV.2.1) where $\mathbb{V}=\mathbb{W}-\mathbb{D}$ with $\mathbb{W}$ being
the rate matrix and $\mathbb{D}$ being a nonegative diagonal matrix. In this
subsection, we prove that all the eigenvalues of $\mathbb{V}$ have negative
real parts under mixed and concentration control. The proof is split into two
parts.

First, we prove that $\mathbb{V}$ has no null eigenvalue. Since $\mathbb{W}$
is a rate matrix, namely, $\mathbb{W}_{\alpha,\beta}\geq 0$ and
$\mathbb{W}_{\alpha,\alpha}=-\sum_{\beta\neq\alpha}\mathbb{W}_{\beta,\alpha}<0$
(see Eqs. (29) and (30)) and $\mathbb{D}$ is a nonnegative diagonal matrix,
then $\mathbb{V}$ is irreducible as ${\mathbb{W}}$ is irreducible, and
$\sum_{\beta\neq\alpha}|\mathbb{V}_{\beta,\alpha}|\leq|\mathbb{V}_{\alpha,\alpha}|$
for all columns of $\mathbb{V}$ with a strict inequality for at least one
$\alpha$ (where $\mathbb{D}$ is nonzero). Because of these properties,
$\mathbb{V}$ is an irreducible weakly diagonally dominant matrix that is known
to be nonsingular [42, 43]. Hence, matrix $\mathbb{V}$ has no zero eigenvalues
unlike $\mathbb{W}$.

Second, we prove that $\mathbb{V}$ can only admit eigenvalues with non-
positive real parts. To do so, we use Gershgorin’s theorem [42] which states
that the eigenvalues of $\mathbb{V}$ can always be found in the union of discs
in the complex plane defined as follows. For every $\alpha$, there is a disc
centered on the negative real axis at $\mathbb{V}_{\alpha,\alpha}<0$ with
radius $\sum_{\beta\neq\alpha}|\mathbb{V}_{\beta,\alpha}|$. Since
$\sum_{\beta\neq\alpha}|\mathbb{V}_{\beta,\alpha}|\leq|\mathbb{V}_{\alpha,\alpha}|$,
the discs are either fully contained in the left half of the complex plane or
can be (at most) tangential to the origin. Therefore all eigenvalues of
$\mathbb{V}$ have nonpositive real parts.

The final implication of the two arguments is that all the eigenvalues of
$\mathbb{V}$ have negative real parts. Note that this proof did not need
$\mathbb{W}$ to be detailed balanced.

##  Appendix B Solutions of Eq. (34)

For unimolecular and pseudo-unimolecular CRNs, under any chemostatting
procedure, the rate equation Eq. (2) can be recast in the general form of Eq.
(34) (see Secs. III.2 and IV.2.1) where $\bm{a}$ is the vector of dynamical
variables, $\mathbb{V}$ is an appropriately chosen (real) matrix and
$\bar{\bm{I}}_{a}$ is a constant vector. Formally, the solution of Eq. (34)
with the initial condition $\bm{a}_{0}$ is given by

$\bm{a}(t)=e^{\mathbb{V}t}\bm{a}_{0}+\int_{0}^{t}ds\,e^{\mathbb{V}(t-s)}\bar{\bm{I}}_{a}\,,$
(156)

where the integral on the right-hand side of Eq. (156) satisfies

$\mathbb{V}\int_{0}^{t}ds\,e^{\mathbb{V}(t-s)}=e^{\mathbb{V}t}-\mathbb{1}\,.$
(157)

The specific dynamical behavior described by the formal solution (156) depends
on the spectral properties of $\mathbb{V}$. First, under mixed or
concentration control, $\mathbb{V}$ is invertible since all eigenvalues have
negative real part (see App. A). This implies that $\bm{a}(t)$ relaxes to a
steady state as we prove in App. B.1. Second, under flux control, $\mathbb{V}$
coincides with the rate matrix $\mathbb{W}$ and is therefore non-invertible
with a unique zero eigenvalue (see Secs. III.2 and IV.2.1). If $\mathbb{V}$ is
diagonalizable (which is always the case if $\mathbb{W}$ is detailed
balanced), $\bm{a}(t)$ will grow as shown in Secs. III.4 and IV.2.3. If
$\mathbb{V}$ is nondiagonalizable (which is possible when $\mathbb{W}$ is
nondetailed balanced [44, 45]), $\bm{a}(t)$ still grows as we prove in App.
B.2.

###  B.1 Invertible Case

We consider here the case of invertible $\mathbb{V}$ (corresponding to
unimolecular and pseudo-unimolecular CRNs under mixed or concentration
control). Since all the eigenvalues of $\mathbb{V}$ have negative real parts
(see App. A),

$\lim_{t\to\infty}e^{\mathbb{V}t}\bm{a}_{0}=\bm{0}\,,$ (158)

for any initial condition $\bm{a}_{0}$ [46]. Since $\mathbb{V}$ is invertible,
Eq. (157) implies that the general solution (156) reduces to

$\bm{a}(t)=e^{\mathbb{V}t}\bm{a}_{0}+\mathbb{V}^{-1}\left(e^{\mathbb{V}t}-\mathbb{1}\right)\bar{\bm{I}}_{a}\xrightarrow[t\to\infty]{}-\mathbb{V}^{-1}\bar{\bm{I}}_{a}\,,$
(159)

where the second limit uses Eq. (158). Hence, in the long time limit,
$\bm{a}(t)$ relaxes to the steady state $-\mathbb{V}^{-1}\bar{\bm{I}}_{a}$.

###  B.2 Noninvertible Case

We consider here the case of noninvertible nondigaonalizable $\mathbb{V}$
(corresponding to pseudo-unimolecular CRNs under flux control). From the
Perron Frobenius theorem, its largest eigenvalue is unique and equal to $0$.
The corresponding right and left null eigenvectors will be denoted as
$\bm{\pi}$ and $\ell^{m}=(1,1,\dots)^{\intercal}$, respectively (as done in
Sec. III.1). All other eigenvalues have negative real parts.

We start by converting $\mathbb{V}$ into its Jordan normal form by means of
the invertible matrix $\mathbb{P}$ of generalized eigenvectors [46]:

$\mathbb{\Lambda}=\mathbb{P}^{-1}\mathbb{V}\mathbb{P}\,,$ (160)

where the matrix $\mathbb{\Lambda}$ has the block diagonal form

$\Lambda=\left(\begin{array}[]{ccccc}0&&&&\\\ &\mathbb{B}_{2}&&&\\\
&&\ddots&&\\\ &&&\mathbb{B}_{r-1}&\\\ &&&&\mathbb{B}_{r}\\\
\end{array}\right)\,.$ (161)

The block matrices $\mathbb{B}_{k}$ take the form

$\mathbb{B}_{k}=\left(\begin{array}[]{ccccc}\lambda_{k}&1&&&\\\
&\lambda_{k}&1&&\\\ &&\ddots&&\\\ &&&\lambda_{k}&1\\\ &&&&\lambda_{k}\\\
\end{array}\right)\,,$ (162)

when the corresponding eigenvalue $\lambda_{k}$ is real, and the form

$\mathbb{B}_{k}=\left(\begin{array}[]{ccccc}\Omega_{k}&\mathbb{1}_{2}&&&\\\
&\Omega_{k}&\mathbb{1}_{2}&&\\\ &&\ddots&&\\\ &&&\Omega_{k}&\mathbb{1}_{2}\\\
&&&&\Omega_{k}\\\ \end{array}\right)\,,$ (163)

when the corresponding eigenvalue is complex
$\lambda_{k}=\kappa_{k}+\iota\omega_{k}$. Here, $\mathbb{1}_{2}$ is the 2x2
identity matrix,

$\Omega_{k}=\begin{pmatrix}\kappa_{k}&-\omega_{k}\\\
\omega_{k}&\kappa_{k}\end{pmatrix}\,,$ (164)

and $\lambda_{k}<0$ and $\kappa_{k}<0$. Note that the matrix $\mathbb{B}_{k}$
in Eq. (162) (resp. Eq. (163)) is invertible with all eigenvalues being
$\lambda_{k}$ (resp. $\kappa_{k}+\iota\omega_{k}$).

We now make a linear transformation into the space of generalized eigenvectors
$\bm{u}=\mathbb{P}^{-1}\bm{a}$. Correspondingly, Eq. (34) becomes

$d_{t}\bm{u}=\Lambda\bm{u}+\mathbb{P}^{-1}\bar{\bm{I}}_{a}\,.$ (165)

The first component $u_{1}(t)$ grows linearly in time as

$d_{t}u_{1}=(\mathbb{P}^{-1}\bar{\bm{I}}_{a})_{1}=\sum_{m}(\bar{{I}}_{a})_{m}\,,$
(166)

where the second equality used the fact that the left null eigenvector
$\ell^{m}=(1,1,\dots)^{\intercal}$ is the first row of $\mathbb{P}^{-1}$. The
remaining components of $\bm{u}$ can be grouped into (sub)vectors $\bm{u}_{k}$
such that each (sub)vector evolves as

$d_{t}\bm{u}_{k}=\mathbb{B}_{k}\bm{u}_{k}+(\mathbb{P}^{-1}\bar{\bm{I}}_{a})_{k}\,.$
(167)

Note the similarity between Eqs. (167) and (34). Furthermore, since each
$\mathbb{B}_{k}$ is invertible with all eigenvalues having negative real
parts, the results of Appendix B.1 apply. Thus, the components $\bm{u}_{k}$ in
Eq. (167) relax to the values

$\bm{u}_{k}(t)\xrightarrow[t\to\infty]{}-\mathbb{B}_{k}^{-1}(\mathbb{P}^{-1}\bar{\bm{I}}_{a})_{k}\,.$
(168)

By combining Eqs. (166) and (168) with $\bm{a}=\mathbb{P}\bm{u}(t)$, at
sufficiently long times,

$\bm{a}(t)=\bm{\pi}u_{1}(t)+\underbrace{\sum_{k>1}\mathbb{P}_{k}\left(-\mathbb{B}_{k}^{-1}\left((\mathbb{P}^{-1}\bar{\bm{I}}_{a})_{k}\right)\right)}_{=\bm{c}(\overline{\bm{I}})}\,,$
(169)

where we used the fact that the first column of $\mathbb{P}_{1}=\bm{\pi}$ and
$\mathbb{P}_{k}$ refers to the (sub)matrix formed from the columns of
$\mathbb{P}$ corresponding to the $k$-th Jordan Block. By comparing Eqs. (169)
and (67b), we observe that the only change due to the nondiagonalizability of
$\mathbb{V}$ is in the form of the offset $\bm{c}(\overline{\bm{I}})$. Hence,
in the long time limit, $\bm{a}(t)$ grows.

##  Appendix C Broken Conservation Laws in Fully Open Pseudo-unimolecular CRNs

When one of the $Z_{l}$ species is chemostatted the conservation law
$\bm{\ell}^{ml}$ is broken. Thus, the set of potential $Y_{p}$ (resp. force
$Y_{f}$) species includes both dynamically linear and hidden species, i.e.,
$Y_{p}={Y_{pl}}\cup Y_{ph}$ (resp. $Y_{f}={Y_{fl}}\cup Y_{fh}$). As for
unimolecular CRNs (see Subs. III.4.2) the set $Y_{pl}$ includes only one
species. Hence, matrix $\mathbb{L}$ in Eq. (59) is equivalent to the matrix
$\mathbb{L}^{b}$ (introduced in Sec. II.3) as it collects now only broken
conservation laws and can be written as

$\mathbb{L}^{b}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\bm{\ell}^{ml}_{X}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\ell^{ml}_{pl}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{\ell}^{ml}_{fl}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\mathbb{L}}_{X}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\bm{\ell}}_{{p{l}}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\mathbb{L}}_{{f{l}}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\mathbb{L}}_{{p{h}}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\mathbb{L}}_{{f{h}}}$\hfil\kern
5.0pt\crcr}}}}\right)$}}\,.$ (170)

From Eq. (170), the matrix $\mathbb{L}^{b}_{Y_{p}}$ and its inverse [47] read

$\mathbb{L}^{b}_{Y_{p}}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\ell^{ml}_{pl}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\bm{\ell}}_{{p{l}}}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\tilde{\mathbb{L}}_{{p{h}}}$\hfil\kern
5.0pt\crcr}}}}\right)$}}\,,\\\$ (171a)
$(\mathbb{L}^{b}_{Y_{p}})^{-1}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern
0.0pt\kern 2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle(\ell^{ml}_{pl})^{-1}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle-(\tilde{\mathbb{L}}_{{p{h}}})^{-1}\tilde{\bm{\ell}}_{{p{l}}}(\ell^{ml}_{pl})^{-1}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle(\tilde{\mathbb{L}}_{{p{h}}})^{-1}$\hfil\kern
5.0pt\crcr}}}}\right)$}}\,.$ (171b)

By plugging Eqs. (171b) and (170) in Eq. (21), the concentration vector of the
moieties reads

$\bm{m}=\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left(\kern
0.0pt\kern-2.5pt\kern-6.66669pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle\color[rgb]{0.50,0.50,0.50}\definecolor[named]{pgfstrokecolor}{rgb}{0.50,0.50,0.50}\pgfsys@color@gray@stroke{0.50}\pgfsys@color<EMAIL_ADDRESS>5.0pt&5.0pt\hfil$\scriptstyle(\ell^{ml}_{pl})^{-1}\bm{\ell}^{ml}_{X}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 1$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle(\ell^{ml}_{pl})^{-1}\bm{\ell}^{ml}_{fl}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}\\\$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\overline{\mathbb{L}}_{X}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\bm{0}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\overline{\mathbb{L}}_{fl}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\mathbb{I}$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\overline{\mathbb{L}}_{fh}$\hfil\kern
5.0pt\crcr}}}}\right)$}}\cdot\bm{z}\,,$ (172)

with

$\displaystyle\overline{\mathbb{L}}_{X}=(\tilde{\mathbb{L}}_{{p{h}}})^{-1}\left(\tilde{\mathbb{L}}_{X}-(\ell^{ml}_{pl})^{-1}\tilde{\bm{\ell}}_{{p{l}}}\bm{\ell}^{ml}_{X}\right)\,,$
(173)
$\displaystyle\overline{\mathbb{L}}_{fl}=(\tilde{\mathbb{L}}_{{p{h}}})^{-1}\left(\tilde{\mathbb{L}}_{fl}-(\ell^{ml}_{pl})^{-1}\tilde{\bm{\ell}}_{{p{l}}}\bm{\ell}^{ml}_{fl}\right)\,,$
(174)
$\displaystyle\overline{\mathbb{L}}_{fh}=(\tilde{\mathbb{L}}_{{p{h}}})^{-1}\tilde{\mathbb{L}}_{{fh}}\;.$
(175)

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