# Observations of Thomson scattering from a loop-prominence system

Juan Carlos Martínez Oliveros Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  Juan Camilo Guevara Gómez Rosseland
Centre for Solar Physics, University of Oslo, Postboks 1029 Blindern, 0315
Oslo, Norway  Institute of Theoretical Astrophysics, University of Oslo,
Postboks 1029 Blindern, 0315 Oslo, Norway  Pascal Saint-Hilaire Space Sciences
Laboratory, University of California, Berkeley, CA 94720-7450, USA  Hugh
Hudson Space Sciences Laboratory, University of California, Berkeley, CA
94720-7450, USA  SUPA Department of Physics and Astronomy, University of
Glasgow, G12 8QQ, UK  Säm Krucker Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  Institute for Data Science, School
of Engineering, University of Applied Sciences and Arts Northwestern
Switzerland, 5210 Windisch, Switzerland

(Received April 1, 2019; Revised March 5, 2021)

###### Abstract

We describe observations of the white-light structures in the low corona
following the X8.2 flare SOL2017-09-10, as observed in full Stokes parameters
by the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics
Observatory. These data show both bright loops and a diffuse emission region
above them. We interpret the loops as the white-light counterpart of a
classical loop-prominence system, intermediate between the hot X-ray loops and
coronal rain. The diffuse emission external to the loops is linearly polarized
and has a natural interpretation in terms of Thomson scattering from the hot
plasma seen prior to its cooling and recombination. The polarimetric data from
HMI enable us to distinguish this contribution of scattering from the HMI
pseudo-continuum measurement, and to make a direct estimation of the coronal
mass in the polarized source. For a snapshot at 16:19 UT, we estimate a mass
$8\times 10^{14}$ g. We further conclude that the volumetric filling factor of
this source is near unity.

Sun: flares, Sun: corona, Sun: X-rays

††journal: ApJ

##  1 Introduction

Classical loop-prominence systems associated with powerful solar flares have
been observed since the 1930’s (e.g. Bruzek, 1964; Švestka, 1968). These are
commonly observed in H$\alpha$ and other chromospheric lines in the form of
loop arcades. Since their initial observations, loop prominence systems (LPS)
were a topic of particular interest for solar physicists as they seemed to
have a clear relationship with solar flares. Moreover, LPS display filamentary
structures with episodes of plasma condensation (Waldmeier, 1963), leading to
the somewhat misleading term “sporadic coronal condensation” for the flare-
associated LPS. Švestka (1968) found that nearly 70% of all LPS are observed
in active regions, an in particular active regions that also hosted solar
flares and were often associated with the coronal yellow line of Ca xv,
implying higher than normal active-region temperatures. The early observations
further suggested that the LPS are highly correlated with active regions, and
in particular with active regions that can produce highly energetic flares
(e.g. proton flares) (Waldmeier, 1963). Since their discovery loop-prominence
systems have been identified differently with “loop-prominences”, “flare
arcades”, “eruptive loops”, “post-flare loops”, or “post-eruptive arcades”
(for an in-depth discussion see Švestka, 2007). Although similar in nature,
loop-prominence systems should not be confused with ordinary prominences.
According to Švestka (2007) the key difference between these two phenomena is
their evolution. Prominences grow by expanding the individual loops that
compose them, while loop-prominence systems grow by the illumination of new
higher loops while the lower fade. The time evolution of loop prominence
systems is related to the thermodynamic and radiative properties of the
chromospheric-temperature plasma. From observations (e.g., Bruzek, 1964) it is
clear that these systems do not grow from lower to higher altitudes (see
Figure 2 in Waldmeier, 1963), instead their generation is the product of
higher loops getting more intense due to cooling to low temperatures, while
the lower ones decay. This evolutionary process can take several tens of
minutes to several hours (Švestka, 2007). We now understand this in terms of
the cooling of arcade loop systems from X-ray temperatures, down through Ca xv
and then abrupt condensation to produce emission in cool lines like H$\alpha$.

Although LPS commonly occur in flares near the limb, their observation in
white light or pseudo-white light above the solar limb is extremely rare, even
more so without the usage of coronagraphs since the signatures are faint
relative to the intensity of the emission from the disk. Only a few cases
exist in the literature, to our knowledge, describing visual observations of
such events: SOL1980-06-21 (X2.6), an event visually observed by J. Harvey and
T. Duvall (Hudson & Willson, 1983; Chupp, 1990), SOL1989-08-14 ($\sim$X20;
Hiei et al., 1992), and SOL2003-11-04 ($>$X17; Leibacher et al., 2004) may
comprise the whole list of such visual observations. Recently, though,
Martínez Oliveros et al. (2014) and Saint-Hilaire et al. (2014) found two
excellent examples in ordinary optical imaging, but from space:
SOL2013-05-13T02:17 and SOL2013-05-13T16:00. These observations made use of
the Helioseismic and Magnetic Imager (HMI; Schou et al., 2012; Scherrer et
al., 2012) of the Solar Dynamics Observatory (SDO; Pesnell et al., 2012).
Jejčič et al. (2018) have now also analyzed SOL2017-09-10 using HMI data, and
in this paper we analyze its polarization properties.

Our study of SOL2017-09-10 focuses on the development of the thermal sources
in the flare. Many earlier papers describe this fascinating event in great
detail, but mainly bypassing the basic thermal emissions in favor of studying
more complicated aspects of the flare development, such as current-sheet
formation (e.g. Warren et al., 2018) and magnetic reconnection, especially via
the new capabilities of microwave imaging spectroscopy (e.g. Gary et al.,
2018; Yu et al., 2020; Chen et al., 2021).

The HMI observations describe off-limb emission at the solar limb and
relatively low in altitude (below 1.03 R$\odot$ elongation). Both of the 2013
flares, the first reported from the HMI database, also showed white-light (WL)
footpoint sources at the level of the photosphere. The gradual coronal
emissions can be identified as visual counterparts of the classical loop-
prominence system, but were brighter than expected and have some ambiguity in
their emission mechanisms; a continuum could come from scattering or from the
free-bound extension of the hot plasma seen in soft X-rays, or from direct
emission (lines and continua), depending mainly upon the density (Jejčič et
al., 2018). In their interpretation, the coronal sources detected by HMI in
these flares represent flare loops, initially heated to X-ray temperatures,
and detected in the process of cooling. The authors found the HMI flux to
exceed the long-wavelength extrapolation of the bremsstrahlung of the flare
soft X-ray sources by at least one order of magnitude, implying the
contribution of cooler material that could produce free–bound continua and
possibly line emission detectable by HMI. Their analysis suggested electron
densities as high as 1013 cm-3, and under these conditions optically-thin
Paschen continuum is not important.

Saint-Hilaire et al. (2014) reported the detection of linearly polarized
scattered light from an LPS (SOL2013-05-13T02:17 and SOL2013-05-13T16:00) via
the HMI white-light polarimetric data. This revealed Thomson scattering and
therefore enabled a direct mass estimate and inferences about the source
density, using the techniques described in Minnaert (1930). Saint-Hilaire et
al. (2014) concluded that only a fraction of white-light emission in LPS was
due to Thomson scattering and inferred a lower limit of the free electron
density of about 3.5$\times$1011 cm-3, assuming a line-of-sight depth of 2.2
Mm.

In this article, we describe similar polarization measurements and analysis of
the well-studied SOL2017-09-10 limb flare. The HMI white-light intensity
(i.e., Stokes I) of this event were reported by Jejčič et al. (2018), whose
study took into account all continuum emission processes in the flare loops,
as well as optical-depth effects; see also Hiei et al. (1992). The LPS appear
in an image annulus extending about 44′′ above the limb, evolving for several
hours in a close relationship with the EUV structures observed by AIA (e.g.
Warren et al., 2018; Heinzel et al., 2020). For reference, we show the spatial
relationships of these sources in Figure 1. The diffuse HMI source above the
LPS was bright enough, in this case, for us to make a first estimate of the
density/filling factor for the hot plasma routinely seen as recombination
radiation in the soft X-ray and microwave bands in major flares. We note that
French et al. (2019, 2020), Zhao et al. (2021) and Cheng et al. (2018) have
also analyzed this event with the complementary Mauna Loa K-Coronameter
observations.

Figure 1: A snapshot of the multiwavelength sources observed in SOL2017-09-10
at 16:16:41 UT, with the background image an AIA 94/193/211 composite. The
blue contours show the bright HMI loops at this moment, and the RHESSI image
at 12 – 18 keV is shown by red contours (10%, 30%, 50%, 70%, 90%).

##  2 Observations and Instrumentation

On 2017 September 10, a loop prominence system developed during the gradual
phase of an X8.2 flare (SOL2017-09-10). This LPS was well observed by the
SDO/AIA (e.g. Warren et al., 2018) and various other aspects of this classical
eruptive flare event have been widely reported in the literature. In
particular, Jejčič et al. (2018) have previously reported on the HMI
observations, but not using its polarimetric capability.

The HMI instrument contains two cameras (front and side) designed for on-disk
observations of the whole Sun, primarily for helioseismology and the
characterization of the photospheric magnetic field, and for these purposes it
makes high-resolution filtergrams of the photospheric Fe i absorption line at
6173.3 Å. HMI observes the line via six passbands spanning a range of about
345 mÅ around the target line. It also obtains full Stokes profiles (Schou et
al., 2012). The “front camera” standard data are created from a combination of
sequences of 12 distinct images (six filters and two polarizations each called
a filtergram), while the “side camera” data are reconstructed from
observations with the same filters and the following polarizations: I+/–U,
I+/–Q, and I+/–V. The individual wavelengths (and polarization settings) for
these line profiles are not observed simultaneously, but in a programmed
sequence extending over 45 s and 135 s frames for the HMI front and side
cameras, respectively, and therefore have limitations when rapid transients
occur (e.g., Martínez Oliveros et al., 2011, 2014). The resulting images are
cropped and later during the process, truncated at about 65′′ above the limb
to generate the standard HMI observables (intensity, velocity and magnetic
field). The observations we report here, though unambiguous photometrically,
thus occur in a parameter space not optimized by the design of the telescope.

Figure 2 shows the evolution of the LPS at two different and characteristic
times, for HMI/STOKES I (top row) and AIA 94/193/211 Å composite images
(bottom row). The RHESSI 10%, 30%, 50%, 70%, 90% intensity contours are shown
in red and plotted for the same times. The RHESSI images were reconstructed
using the CLEAN algorithm with a period of integration that encompasses the
HMI image cadence. Figure 2 (bottom row) shows the HMI/STOKES I contours
(blue) for each evolutionary time at 75%, 85%, 95% over the composite AIA
images. On the top row three regions (white boxes) are shown and designated as
A, B and C. Regions A and B denote the areas where the HMI/Stokes Q, U and V
were analyzed. Region C is our control region.

Figure 2: Top: HMI/Stokes I at two different times during the evolution of the
LPS. White boxes A, B and C show the regions of analysis. Bottom: AIA
94/193/211 combination at same times, showing presence of cool plasma; blue
contours are for the LPS feature from HMI/Stokes I at 75%, 85%, 95%. The
RHESSI 12 – 18 keV intensity contours at 10%, 30%, 50%, 70%, 90% are shown in
all the images in red color for two evolutionary times close to the HMI image
time. The lower-left panel matches Figure 1. The orange solid line represent
the SDO/HMI limb

The data provided by HMI team through the Joint Science Operations Center
(JSOC) has a cadence of 90 seconds and a roll angle or rotation angle close to
180∘. The analysis to be described was made before applying the roll angle
rotation in order to avoid mathematical artifacts introduced by the
interpolation. The data also have a small jitter in every frame, so, the first
operation was to recenter all frames in order to have all the images properly
aligned in time and later the six filters were average for each Stokes
component. In addition, to increase the signal, images were summed over four
consecutive frames to produce a single image every 6 minutes, which gives us
sufficient temporal resolution for the purpose of this research. Finally, Q
and U were rotated to increase the signal to noise ratio for the Stokes
parameters throughout by aligning the plane of reference of the measurement
with the location of the LPS (see Schutgens et al., 2004). These new Stokes
are named Q′ and U′ on the expectation that most of the polarized signal, in
the case of Thomson scattering above a uniform photosphere, will appear in Q′
and little in U′ (Saint-Hilaire et al., 2014, 2021).

Figure 3 shows the measured SDO/HMI Stokes parameters in the three regions of
interest (A, B, C). The top panel shows the intensity values as a function of
time. The GOES 1-8 Å X-ray flux is shown as the gray shadow area. It is clear
that the highest value is reached in region A which during the event contains
the visible part of the LPS. Region B is not as intense as region A and its
peak occurred several tens minutes later than the peak in region A. Our
control region C shows no appreciable changes in intensity that can be
attribute to the LPS. The second panel from the top shows the Stokes parameter
Q′, U′ and V. Regions A and B show a change in the Stokes parameter (Q′)
during the event of about 20% with a maximum close to 44%. The control region
C shows no apparent change. The U′ and V Stokes parameters show values close
to zero. The third panel shows the $\Delta Q’$/$\Delta I$ values for regions A
and B, region C is not shown since the values are undetermined in the time
interval. Here we note a 19% increase of this ratio in region A and about 21%
in region B in the time intervals selected to match the RHESSI observations.
The bottom panel shows the spatial-temporal evolution of the LPS intensity
within a 10′′ slit centered at the LPS brightest feature. We calculated the
average propagation velocity of the LPS during two stages of its evolution.
The blue line indicates the apparent motion of the LPS with an average
velocity of $7.88~{}\text{km\,s}^{-1}$ whereas the green line indicates the
final phase of the event with an average velocity of
$1.75~{}\text{km\,s}^{-1}$, these values are in agreement with those reported
for this type of structures (e.g. Bruzek, 1964; Gallagher et al., 2002).

Figure 3: SDO/HMI Stokes parameters in the regions A, B, C. The top panel
shows the intensity values as a function of time with GOES 1-8Å in gray. The
second panel from the top shows the Q′, U′ and V Stokes parameters. The third
panel shows the ratio $\Delta Q^{\prime}/\Delta I$ for the regions A and B.
The bottom panel shows the spatial and temporal evolution of the LPS intensity
within a 10′′ slit centered at the LPS brightest feature, the blue line
indicates the apparent motion of the LPS with an average velocity of
$7.88~{}\text{km\,s}^{-1}$ whereas the green line indicates the final phase of
the event with an average velocity of $1.75~{}\text{km\,s}^{-1}$.

Table 1 summarizes the measurements of the excess fluxes for Boxes A, B and C
at the two reference times indicated in Figure 2, and interpreted as
successive loops cooling and draining sequentially.

Table 1: Observed Fluxes Box | Time (UT) | Position | I (DN/s) | Q′ (DN/s) | Area (arcsec2)
---|---|---|---|---|---
A | 16:19 | [982,-140] | 48.7$\pm 1.5$ |  $\sim$ -0.44$\pm$0.05 | 196
B | 16:19 | [996,-142] | 39.9$\pm 1.5$ |  $\sim$ -0.27$\pm$0.03 | 196
C | 16:19 | [996,-170] | 37.6$\pm 1.5$ |  $\sim$ -0.08$\pm$0.009 | 196
A | 16:43 | [982,-140] | 56.1$\pm 1.4$ |  $\sim$ -0.38$\pm$0.04 | 196
B | 16:43 | [996,-142] | 39.9$\pm 1.4$ |  $\sim$ -0.24$\pm$0.03 | 196
C | 16:43 | [996,-170] | 37.5$\pm 1.4$ |  $\sim$ -0.03$\pm$0.004 | 196

##  3 Mass, density and filling factor from Thomson scattering

In the following sections we discuss the region above the bright loops. This
faint and apparently diffuse volume corresponds to the hot plasma of the soft
X-ray source, an almost definitive flare component and one that is not well
imaged by EUV observations such as those of AIA.

###  3.1 Mass

The K-component of the white-light continuum observed from coronal structures
consists of Thomson-scattered photospheric continuum, with the resulting
linear polarization (e.g. Billings, 1966). This component of coronal
brightness scales directly with $N_{e}$ (the column electon density) and hence
the electron density $n_{e}$, while collisional atomic emissions (collisional
bremstrahlung) scale as $n_{e}n_{i}$. The total free electron number
${\mathcal{N}}_{e}$ then determines the mass of the scattering material, if
the abundances are known and if the location of the source justifies the
“plane-of-the-sky” approximation. At high densities the atomic emissions
processes may dominate and even produce optically-thick continuum (Hiei et
al., 1992; Jejčič et al., 2018). This may obscure a part of the polarization
source; accordingly the mass estimate represents a lower limit. Such an effect
is likely to be a small one because of the efficiency of the thermal collapse
(Field, 1965). The dust-scattered F-corona is not important in the lower
corona. Thus where Thomson scattering dominates, its brightness directly
determines the mass of the scattering material, if the photospheric radiation
field is known (Saint-Hilaire et al., 2021):

$M=\mu m_{p}\mathcal{N}_{e}=\frac{F_{s}}{F_{\lambda}}\cdot\frac{\mu m_{p}(\pi
R_{\odot}^{2})}{\sigma_{T}W}\ \ ,$ (1)

where $\mu$ is the mean molecular weight, $m_{p}$ the mass of the proton,
$\mathcal{N}_{e}$ the number of scattering electrons, $F_{s}/F_{\lambda}$ the
ratio of scattered irradiance to the total solar spectral irradiance at the
HMI wavelength, $W$ a geometrical factor as estimated by Saint-Hilaire et al.
(2014), and $\mathrm{\sigma}_{T}=6.65\times 10^{-25}~{}\text{cm}^{-2}$ the
Thomson cross section. This estimate assumes fully ionized plasma and does not
depend upon the source density or filling factor $\eta$, which we discuss in
the following sections. Note that we use $\mathcal{N}_{e}$ for the total
number of scattering electrons, $N_{e}$ for the column density (cm-2), and
$n_{e}$ for the number density (cm-3) here and below.

###  3.2 Density

With further assumptions we can estimate the density of the scattering source
and its filling factor. The basic idea of this technique is to derive the
Thomson scattering contribution of the measured flux using the ratio between
the observed linear polarization and the total intensity data. Following
Rybicki & Hummer (1994) and Jejčič & Heinzel (2009) the density $n_{e}$ of an
off-limb source emitting by Thomson scattering is:

$n_{e}=\frac{j(\nu)}{I_{0}(\nu)}\frac{1}{W}\frac{1}{\sigma_{T}}=\frac{\varepsilon_{I}}{I_{0}}\frac{1}{\sigma_{T}}\frac{1}{G_{I}}\,\text{\hskip
8.53581pt(eq.14in{\cite[citet]{\@@bibref{Authors
Phrase1YearPhrase2}{2021ApJ...923..276S}{\@@citephrase{}}{\@@citephrase{}}}})}$
(2)

where $j(\nu)$ is the volume emissivity at a given prominence location,
I${}_{0}(\nu)$ is the emitted intensity from the solar disk center, $W=G_{I}$
is a geometrical dilution factor dependent on the height over the limb which
represents our knowledge of the source structure. Saint-Hilaire et al. (2014)
described how to use the theoretical values from Minnaert (1930) in order to
rewrite Equation 2 in terms of the effective Thomson scattering cross section
at a given height $H$ over the solar limb $\sigma_{T}(H)=\sigma_{T}\cdot
G_{I}(H)$111$G_{I}(H)$ is $\approx$ 0.3 for the observed source heights and
the known limb darkening at 6173Å, the line-of-sight thickness $D$ and the
ratio between the scattered intensity from the source at the given height
$\Delta I(H)$, understood as the intensity excess value above the baseline,
and the intensity from the solar disk center such that:

$n_{e}(H)=\frac{\Delta I(H)}{I_{0}}\frac{1}{\sigma_{T}(H)}\frac{1}{D}\ \ .$
(3)

This allows us to estimate the electron density of a source, with $\Delta I$
being the intensity produced by only the scattered-light component. The
observed $\Delta I$, however, also may have a component due to optically-thick
sources seen in emission. As these latter sources are unpolarized and the
expected degree of polarization due to Thomson scattering is known as a
function of height (e.g. Saint-Hilaire et al., 2014), the two components can
be separated using the actual measured degree of polarization $P_{m}$ and the
expected one $P_{T}$. Taking this factor into account in Equation 3, it is
possible to calculate the density of the material responsible for the Thomson-
scattered signal:

$n_{e}=\frac{\Delta
I}{I_{0}}\frac{1}{\sigma_{T}(H)}\frac{P_{m}}{P_{T}}\frac{1}{D}\ \ .$ (4)

The column density $N$ (electrons cm-2) is implicit in Equation 4; it is the
integral of the electron density along the line of sight through the emitting
volume. Here we estimate the column depth by taking the width of the 70% image
contour of the RHESSI source. This is shown as the box dimension in Figure 2.
Thus, the column density, with the polarization-factor correction, becomes:

$N_{e}=\frac{\Delta I}{I_{0}}\frac{1}{\sigma_{T}(H)}\frac{P_{m}}{P_{T}}\ \ .$
(5)

Figure 4: Top left and right panels show the column densities for regions A
and B respectively, the shadow refers to the errors in each case. In both
cases, the dashed curve is the column density without correction by the
polarization factor whereas the solid and shadowed region have the correction
done. The bottom panels are the densities for the regions A and B derived from
the column densities assuming a line-of-sight thickness between
$2.9~{}\text{Mm}$ and $40.7~{}\text{Mm}$ (shadowed regions).

The upper panels of Figure 4 show the column densities for regions A and B
respectively, as calculated using Equation 5. If the depolarization factor is
not applied, i.e., if the flux originated entirely from Thomson scattering,
the column densities would follow the dashed curves. The correction factor in
each box increases with time, as expected for sources cooling and recombining.
For region A, which is the closest to the photosphere and therefore is reached
first by the white-light structure seen in HMI/Stokes $I$ (see Figure 2), the
uncorrected column density is as large as $10^{21}~{}\text{cm}^{-2}$. However,
when the column density is corrected by the polarization, gray curve in top
left panel, it drops by almost an order of magnitude as this represents the
actual Thomson scattered component only. For region B, there is no significant
variation between the uncorrected and corrected column densities initially,
meaning that the source is mainly dominated by Thomson scattering until the
observed structure reaches it at a later stage. Finally, the bottom panels
shows the expected electron densities derived from the corrected column
densities using Equation 4 and assuming the line-of-sight thickness $D$
between $2.9~{}\text{Mm}$ and $40.7~{}\text{Mm}$ for regions A and B. Although
the electron densities for the two regions are quite similar during the
observation, region A shows a slightly higher density, which is expected as it
is closest to the photosphere and consequently influenced by the material just
above the bright loops. We performed an error propagation analysis to
determine the uncertainty in our calculation of the column density and hence
the electron density, finding this relative error to be about 11%, and to be
dominated by the observational error of Q’. An additional and ill-defined
uncertainty is due to the assumption of a value for the column depth.

This geometrical uncertainty does not affect the mass $M$ estimated via
Equation 1, using the excess flux $F_{S}$ observed from each source region.
From the first entry in Table 1, we find that $F_{s}/F_{\odot}=6.7\times
10^{-9}$ for Box A at 16:19 UT. This corresponds to an electron number
$4\times 10^{38}$ and a (total) mass $8\times 10^{14}$ g. This mass estimate
is interesting as it is a simple to measure the gravitational potential energy
of the scattering source, here roughly $5\times 10^{28}$ erg. For further
discussion see Section 4.3.

###  3.3 X-ray and microwave observations

The polarized continuum observed by HMI relates directly to the free-free and
free-bound continua observed in the microwave and X-ray bands. The same
electrons contribute both to the emission measure ($\propto n_{e}^{2}$) and to
the polarized flux ($\propto n_{e}$). The X-ray continuum also has significant
free-bound and bound-bound contributions, whereas the microwaves do not. Each
can serve to estimate the value of $n_{e}n_{i}\eta V$, and all three
wavelength ranges should agree for a hot isothermal source according to
Equation 6 (e.g. Hudson & Ohki, 1972):

$f_{\nu}\propto(n_{e}n_{i}V)\Lambda(n_{e},T)\cdot\frac{e^{-h\nu/kT}}{\sqrt{T}}\
\ ,$ (6)

where $f_{\nu}$ is the spectral flux in erg (cm2 s Hz)-1, including the slowly
varying Coulomb logarithm $\Lambda(n_{e},T)$. Note that the microwave flux has
only a weak dependence upon the source temperature, but that it may have low-
frequency cutoffs due to opacity or the Razin-Tsytovich efffect (e.g. Krucker
et al., 2013); furthermore the relatively weak thermal microwave continuum
also may have to compete with other emission mechanisms such as
gyrosynchrotron or plasma radiations.

Figure 5 shows single-frequency data from the San Vito site of the Radio Solar
Telescope Network (RSTN). Based on Equation 6, we expect to see a long-
wavelength extension of the X-ray and white light continua through the IR, mm-
wave, and longer radio wavelengths. This extension should have only the weak
frequency dependence of the Gaunt factor, and so
$f_{\nu}\approx{\mathrm{c}onst.}$ – a flat spectrum. What we see in the Figure
is different from this expectation, mainly because of gyrosynchrotron
emission. The major time-series peak prior to 16:20 UT is likely to be from
this mechanism, and the emission after about 16:25 UT may also come from a
competing source. We interpret the minimum at around 16:20-25 UT in terms of
the thermal signature of the hot plasma detected by RHESSI, but only as an
upper limit. The spectral character of the minimum-flux epoch is consistent
with optically thin free-free emission (Equation 6, which could reflect the
long-wavelength extension of the RHESSI hot plasma and the polarized white-
light continuum. The microwave flux does indeed have the right magnitude for
the free-free interpretation, but this is just a consistency check.

Figure 5: Single-frequency observations of SOL2017-09-10 from RSTN/San Vito
(4.995, 8.8, and 15.4 GHz), with pre-event background levels subtracted. The
minimum coincides approximately with the Box A snapshot time. The radio
spectral irradiances are measured in standard solar flux units, 10-22 W (m2
Hz)-1.

The two boxes A and B introduced in Figure 2 have been chosen to coincide with
the main X-ray sources seen by RHESSI. We defer the details of the RHESSI
measurements to Section 3.4 but report the result for the thermal emission
measure of the source in Box A here. From the thermal component of the
spectral fits we can derive a volumetric emission measure $n_{e}n_{i}\eta V$
cm-3 for the hot component in terms of the volumetric filling factor $\eta$.
We compare this with the same quantity derived from the whole-Sun fluxes
observed by GOES/XRS. Finally we can also interpret the
RSTN222http://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-
features/solar-radio/rstn-1-second/ whole-Sun microwave observations for this
parameter, noting that the lack of temperature senstivity for the microwave
thermal component makes it possible to check for low-temperature flare sources
not visible in soft X-rays (e.g., Penn et al., 2016). Table 2 lists all three
of these estimates.

Table 2: Volumetric Emission Measures Source | Value (cgs)
---|---
RHESSI | $(3.0\pm 0.3)\times 10^{50}$
GOES-15/XRS | $(3.0\pm 0.1)\times 10^{50}$
RSTN/San Vito | $<(3.3\pm 0.5)\times 10^{50}$

This agreement between microwave and X-ray emission measures confirms the
identification of the sources and precludes the presence of substantial flare
emission at low temperatures, according to Equation 6, this would destroy the
agreement between microwave and X-ray emission-measure estimates displayed in
the “loci plot” of Figure 6.

Figure 6: Left: the “loci plot” comparing the microwave emission measure with
X-rays. The GOES and RHESSI exclusion lines agree well with the RHESSI 12-18
keV flux, and match the short-wavelength extrapolation of the microwave
thermal flux. Right (Section 4): constraints on the mean source density for
the polarization source, combining HMI observations with X-ray and microwave
data. The colored lines at top are limits imposed by the Razin-Tsytovich
effect (red) and by free-free opacity (blue), in the absence of a long-
wavelength spectral break (e.g., Belkora, 1996). The heavy black line
corresponds to filling factory unity, with the thin lines for values of 0.3
and 0.1. The vertical dashed lines are the Box A width times 0.5, 1, and 2.

The agreement of the HMI estimate of column density $N_{e}$ with the radio and
X-ray values also allows us, uniquely for this flare, to estimate the line-of-
sight thickness of the source $\eta D$ in terms of the observed volumetric
filling factor EM and the total electron number ${\mathcal{N}}$ as derived in
Section 3. The emission measure implies an electron content
${\mathcal{N}}^{\prime}=n_{e}^{\prime}V^{\prime}=\sqrt{EM\cdot\eta DA}$, which
we can combine with the true electron number to give the estimate for the
source thickness

$\eta D=\frac{{\mathcal{N}}^{2}}{A\cdot EM}\ ,$ (7)

based on simple estimates of the source area $A$ and depending on the
volumetric filling factor $\eta$.

###  3.4 RHESSI imaging

The final information needed for characterizing the polarization sources comes
from the RHESSI imaging, from which we can obtain estimates of the projected
area of the joint X-ray/Thomson-scattering/thermal-microwave source identified
by the agreement in Table 2.

At the time of the observations of the September 10 flare, the RHESSI
spectrometer had been operating in space for more than 16 years and only 4 of
the 9 detectors were functioning. Even for these four detectors (detectors 1,
3, 6, and 8), radiation damage had substantially affected the spectral
resolution and effective area. Because radiation damage affects each detector
differently, we performed spectral fitting for each detector separately and
then compare the derived parameters to judge their consistency.

The RHESSI observations ended at about 16:17:40 UT when the RHESSI spacecraft
entered the South Atlantic Anomaly. We selected the interval from 16:16:30 to
16:17:20 UT for our comparison with the HMI observations taken at 16:19 UT.
The orbit-night background from 15:49 to 15:52 UT has been subtracted before
the standard RHESSI spectral fitting of a thermal component plus a non-thermal
component was done. The thermal fits dominate the total count spectrum for
energies below 25 keV, so we take the 12-18 keV images shown in Figure 1 to
represent the thermal sources. The derived temperatures from the four
different detectors range from 18.2 to 19.3 MK in good agreement. The
corresponding emission measures vary from 2.6 to 3.5 $\times 10^{50}$ cm-3,
with the average value of 3.3$\pm$0.5$\times 10^{50}$ cm-3 as quoted above in
Table 2. The temperature and emission measure derived from GOES-15 are 19.1 MK
and 3.0$\pm$0.1$\times 10^{50}$ cm-3, respectively, in surprisingly good
agreement with RHESSI, considering the effects of radiation damage to the
effective area of the RHESSI detectors.

The RHESSI parameters derived above refer to the total counting rates. We can
also use imaging spectroscopy to estimate temperature and emission measure for
the RHESSI source contained within Box A (e.g. Krucker & Lin, 2002). The
fluxes from within Box A are derived from CLEAN images at 1 keV resolution,
which were reconstructed in the same way as the images shown in Figure 1.
Using the energy range 13–18 keV gave parameters [21.0$\pm$1.1 MK,
1.2$\pm$0.4$\times 10^{50}$ cm-3]. Here the errors have been derived by adding
Gaussian random noise at 10% rms in a Monte Carlo procedure; this gave
parameters [19.2–22.6 MK, and 0.6–2.0 $\times 10^{50}$ cm-3]. The derived
temperature of the main source is thus slightly hotter than the flare-averaged
value. Considering the uncertainties the difference is of minor significance.
The emission measure of the bright source is about a factor of 3 smaller,
indicating that the fainter sources outside Box A cannot be neglected and
indeed affect the flare-averaged fit parameters.

In the final step of the RHESSI analysis, we estimate the source area of the
thermal hard X-ray source. We use CLEAN images to estimate the source size of
the main compact RHESSI source (source A). From the RHESSI clean image shown
in Figure 1 (top), we derive a FWHM source size of the compact part of the
source of 15.8′′. The image is reconstructed with a CLEAN beam of 12.2′′.
Hence, we get a deconvolved source size of 10.0′′.

###  3.5 The Box B interval

RHESSI exited the South Atlantic Anomaly (SAA) around 16:39 UT; the second
interval (Box B) was fully observed by RHESSI with both attenuators still
inserted. The same analysis is applied to the later interval as has been done
for the earlier interval using the orbit-night background from 16:55 to 16:58
UT. The non-thermal fit is marginal at this time, with very steep spectral
slopes ($\approx 8$). The temperature ranges from 16.0 to 17.7 MK with an
average value of 17.0$\pm$0.6 MK; and the emission measures vary between 0.7
to 1.3 $\times 10^{50}$cm-3, with an average value of $1.0\pm 0.2\times
10^{50}$cm-3. GOES at this time gives [T, EM] = [15.5 MK, 1.3$\times
10^{50}$cm-3], again in good agreement with RHESSI. Imaging-spectroscopy fits
for the main compact source give higher temperatures (22.8$\pm$1.1 MK). As the
temperature of the main source is higher than the flare-averaged temperature,
a significantly lower emission measure of $0.10\pm 0.04\times 10^{50}$cm-3 of
the compact source already produces a good fit with the observed hard X-ray
flux in the box B. As extreme values for the emission measure we found
$0.04\times 10^{50}$cm-3 and $0.3\times 10^{50}$cm-3.

The CLEAN image shown in Figure 1 has an area with the 50% contour level of
18.5′′ $\times$18.5′′, giving a deconvolved source diameter of 14.0′′.
Compared to the earlier interval around 16:19 UT, the source area apparently
had roughly doubled in area.

##  4 Results

With these data we can now incorporate the polarization observations into a
description of the RHESSI soft X-ray source. The essential augmentation or our
knowledge is twofold: we can directly estimate the number of electrons, and
hence the mass, of the scattering source; in addition that we can determine
the product $\eta D$ (Equation 7).

###  4.1 Electron number, mass, and energy

As described above, the measurement of scattered flux directly determines the
total number of free electrons, with the plausible assumptions of plane-of-
the-sky location and nominal limb-darkened solar radiance (Saint-Hilaire et
al., 2021). We find $4\ \times\ 10^{38}$ for the total electron number. This
in turn determines the coronal mass at the time of observation, plus its
gravitational potential energy. The values we have derived for the Box A
snapshot are $8\times 10^{14}$ g and $5\times 10^{28}$ erg, respectively.

###  4.2 Constraints on density and filling factor

The right panel of Figure 6 summarizes the constraints imposed by the observed
value of $\eta D=4\times 10^{9}$ cm. To this we have added constraints from
the Razin-Tsytovich effect and from free-free opacity (Belkora, 1996). These
limits are based on the assumption that the RSTN three-point spectrum at the
Box A epoch (Figure 5) does not show a spectral break. The free-free limit
definitely restricts the density estimate for unity filling factor, shown in
the Figure as the thick solid line (the depth-dependence of mean density for
$\eta$ = 1). For some part of the range of possible source thicknesses, we can
exclude this limit. We would expect a filling factor below unity in any case
from the fine structure in the developing arcade, but there is no real
constraint from the microwave or X-ray imaging. From the diagram we conclude
that the probable value for the mean source density does not exceed $2\times
10^{10}$ cm-3 for line-of-sight depths comparable to the dimension of the
integration box.

###  4.3 Interpretation

This remarkable solar flare (SOL2017-09-10) has allowed us to consider in
detail the information about hot coronal emission sources in a major flare.
These sources consist of hot plasma, and we have shown for the first time that
observations of linear polarization in the optical continuum match the
extrapolated soft X-ray thermal emission spectrum right down to the microwave
range. This convincingly establishes that flares do not require the injection
into the corona of cool material (below about 1 MK), which would readily be
detectable in the Paschen continuum and in the microwave free-free emission in
the Rayleigh-Jeans spectrum (e.g. Penn et al., 2016). The absence of few-MK
temperatures is also consistent with the AIA-derived emission measure
distributions for this event, as reported by Warren et al. (2018). The
polarization brightness is directly proportional to the mass of the hot
coronal cloud; this information cannot be obtained otherwise except in model-
dependent ways. The coronal mass, $8\times 10^{14}$ g, estimated some 30
minutes after the onset of the flare’s impulsive phase, suggest that the
magnetically trapped mass can rival that of the most massive CME, often quoted
as about 1016 g. The potential energy estimate, however, is less than a
percent of total energy estimates for such an event, either flare or CME (e.g.
Emslie et al., 2012).

The standard picture of the development of a major eruptive flare envisions a
“magnetic explosion” (eg Moore et al., 2001) creating a CME, followed by an
extended period of magnetic reconnection that forms the loop prominence system
finally seen in chromospheric emission lines. The plasma that fills these
loops, according to this well-accepted picture, originated via chromospheric
“evaporation” and appear as the hot cloud our observations show and as
depicted in many cartoon descriptions. This hot mass has been driven into the
corona and heated, although the details of how this happens remain unclear.
Our data are consistent with the idea that the heating can be identified with
particle acceleration leading to plasma $\beta$ of order unity (Krucker et
al., 2010). The standard picture, dating back at least to Bruzek (1964),
involves hot mass in larger and larger loops cooling sequentially, implying
that the instantaneous mass we infer at the epoch of Box A is a lower limit to
the total mass for the entire flare process. After cooling, the ionized plasma
recombines and may become optically thick, radiating strongly via collisional
processes (Jejčič et al., 2018) at densities much higher than we infer for the
hot plasma.

##  5 Summary

The successful measurement of linear polarization in the optical continuum has
allowed us to make direct estimates of the electron content of the flare’s
coronal plasma, and hence its mass. The information provided by this new tool
is in agreement with previous observations of classical loop prominence
systems (Bruzek, 1964; Švestka, 2007; Hiei et al., 1992). Essentially we have
extended the familiar methods of K-corona studies to the bottom of the corona
(Saint-Hilaire et al., 2021).

The work reported here made serendipitous use of HMI data, which were never
intended for studying off-limb sources and are limited in performance, for
this purpose, by design factors in the optics and the spectral bandwidth. In
particular, we have not attempted to use these data to analyze either the
image structure of the polarization or its development in time. Both of these
would contain novel information about the evolution of the excess coronal mass
created by the flare. Our simple mass estimate is of special interest now
because of the remarkable EOVSA microwave observations of this flare
(SOL2017-09-10), which reveal important processes outside the regions of the
EUV loop/cusp/current-sheet geometry (Chen et al., 2020; Fleishman et al.,
2022). Our multi-wavelength analysis does suggest that the polarization source
can be identified with its counterparts in the soft X-rays and microwaves at
high electron temperatures, and can disambiguate these observations in terms
of electron density and filling factor. This just gives us a taste of what may
be learned by better-optimized observations in the future. Better imaging
polarimetry would allow for a far deeper investigation of the coronal
structure of a suitable flare and of its evolution (e.g. DKIST, Rast et al.,
2021).

J.C.G.G. wish to acknowledge the SolarALMA project, which has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement No. 682462), and by
the Research Council of Norway through its Centres of Excellence scheme,
project number 262622. H.S.H. thanks the University of Glasgow for hospitality
during the completion of this work.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 LHCb-DP-2021-004

Performance of the LHCb RICH detectors during LHC Run 2

R. Calabresea,b, M. Fiorinia,b, E. Luppia,b, L. Minzonia,b, I. Slazyka,b, L.
Tomassettia,b, M. Bartolinic,d, R. Cardinalec,d, F. Fontanellic,d, A.
Petrolinic,d, A. Pistonec,d,1, M. Calvie,f, C. Matteuzzie, A. Lupatog,2, G.
Simig, M. Kucharczykh, B. Maleckih, M. Witekh, S. Bensoni, M. Blagoi, G.
Cavalleroi, A. Contui, C. D’Ambrosioi, C. Freii, T. Gysi, J. Hei, D.
Piedigrossii, K. Wylliei, M. Adinolfij, M.G. Chapmanj, J. Dalsenoj,3, S.
Harnewj, S. Maddrell-Manderj, P. Naikj, C. Prouvej,3, J. Rademackerj, A.
Solominj, J. Garra Ticók, V. Gibsonk, C.R. Jonesk, S. Tolkk, S.A. Wottonk, S.
Easol, A. Papanestisl, F. F. Wilsonl, L. Carsonm, S. Eisenhardtm, S.
Gambettam, K. Gizdovm, H. Luom, A. Morrism, F. Muheimm, M. Pappagallom, I.
Smithm, J. Websterm, J. Zonneveldm, S. Karodian, S. Ogilvyn, P. Sailn, F.J.P.
Solern, P. Spradlinn, M. Trailln, S. Bakero, D. Clarko, I. Clarko, K.
Ladhamso, M. McCann222Corresponding author.o, R.D. Moiseo,4, D. Nardinio, M.
Patelo, T. Savidgeo, E. Smitho,5, S. Stefkovao,6, D. Websdaleo, M. Bjornp, F.
Cheungp, R. Gaop, N. Harnewp, D. Hillp, S. Maldep, A. Nandip, S. Topp-
Jorgensenp.

a INFN Sezione di Ferrara, Ferrara, Italy
b Università di Ferrara, Ferrara, Italy
c INFN Sezione di Genova, Genova, Italy
d Università di Genova, Genova, Italy
e INFN Sezione di Milano-Bicocca, Milano, Italy
f Università di Milano Bicocca, Milano, Italy
g Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
h Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
i European Organization for Nuclear Research (CERN), Geneva, Switzerland
j H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
k Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
l STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
m School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
n School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
o Imperial College London, London, United Kingdom
p Department of Physics, University of Oxford, Oxford, United Kingdom

1 Now at Italian institute of Technology
2 Now at Department of Physics and Astronomy, University of Manchester,
Manchester, United Kingdom
3 Now at Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
4 Now at I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
5 Now at Physik-Institut, Universität Zürich, Zürich, Switzerland
6 Now at Deutsches Elektronen-Synchrotron, Hamburg, Germany

The performance of the ring-imaging Cherenkov detectors at the LHCb experiment
is determined during the LHC Run 2 period between 2015 and 2018. The stability
of the Cherenkov angle resolution and number of detected photons with time and
running conditions is measured. The particle identification performance is
evaluated with data and found to satisfy the requirements of the physics
programme.

Submitted to JInst.

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

##  1 Introduction

The LHCb experiment is designed for the precision study of decays of hadrons
containing $b$ and $c$ quarks, with the aim of discovering new physics beyond
the Standard Model of particle physics and measuring the parameters of the CKM
matrix. A key requirement for isolating specific decays is the identification
of long-lived charged hadrons, i.e. ${\pi}^{\pm}$, ${K}^{\pm}$,
$p$/$\overline{p}$ and $d$/$\overline{d}$. LHCb uses a system of two ring-
imaging Cherenkov (RICH) detectors to discriminate between these particles.

The performance of the RICH detectors was studied during the LHC Run 1
(2011–2012) [1]. During Run 2 (2015–2018), the LHCb experiment took data with
a 25 ns $pp$-collision spacing, half that of Run 1, and an increased centre-
of-mass energy, $\sqrt{s}$ =13 TeV, creating a more challenging environment
with higher track multiplicities and larger detector occupancies. A system of
automated real-time alignment and calibration measurements of the LHCb
detector was implemented for Run 2, allowing hadron identification to be used
in the software trigger for the first time.

The RICH detectors are currently undergoing a major upgrade in preparation for
the significantly increased data rates expected during LHC Run 3, with
completely new sensors and readout chain, and substantially different optics
allowed by removal of an aerogel radiator at the end of Run 1. This paper
addresses the ultimate performance of the LHCb RICH detectors in their
original configuration.

This paper describes the performance of the RICH detectors in LHC Run 2. In
section 2 the LHCb spectrometer and the RICH detectors are described. The
calibrations and alignments of the RICH detectors are given in section 3.
Measurements of the Cherenkov angle resolution are presented in section 4, and
the number of detected photons per charged track in section 5. The particle
identification (PID) performance measured with several calibration channels is
presented in section 6, along with its implications on representative LHCb
physics analyses demonstrating the utility of the excellent hadron PID.

##  2 Detector and reconstruction

Figure 1:  A view of the LHCb detector [2]. The proton-proton interaction
point is on the extreme left at $z=0$ m.

The LHCb detector [3, 4] is a single-arm forward spectrometer covering the
pseudorapidity range ${2<\eta<5}$, designed for the study of particles
containing $b$ or $c$ quarks. A schematic view of the detector is given in
figure 1. The detector elements include: 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 the momentum, $p$, of charged particles
as they traverse a dipole magnetic field with a bending power of about
$4{\mathrm{\,Tm}}$; and two RICH detectors that are able to discriminate
between different species of charged particles. Photons, electrons and hadrons
are additionally identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers. The magnetic field is
periodically flipped between vertically up and down to reduce systematic
uncertainties associated with opposing charges experiencing different parts of
the detector, and left-right detector asymmetries.

(a)(b)

Figure 2:  The optical system of (a) RICH1 and (b) RICH2 [2].

The two RICH detectors are each configured with a single gas radiator for Run
2. Upstream of the dipole magnet is RICH1, optimised for low-momentum
particles using C4F10 as a radiator with refractive index 1.0014, and covering
the full angular acceptance. Downstream of the magnet is RICH2, optimised for
a higher momentum range, using CF4 as a radiator with refractive index 1.0005,
and covering approximately a quarter of the solid-angular acceptance of the
spectrometer, $3<\eta<5$.

Both RICH detectors use spherical mirrors to focus the cones of Cherenkov
light emitted from a charged particle as rings onto an array of hybrid photon
detectors (HPDs). An intermediate flat mirror between the spherical mirror and
HPD array allows the HPDs to be located outside of the spectrometer
acceptance. A schematic of the two RICH detectors is shown in figure 2.

Figure 3:  The focusing optics of an HPD [2]. A photoelectron generated at the
photocathode is accelerated to the silicon sensor chip.

The HPDs [5] consist of a photocathode deposited on the inside of the quartz
window of a vacuum tube containing acceleration optics that guide the electron
produced at the photocathode to a silicon sensor where it is read out as a
hit. A schematic of an HPD is shown in figure 3. The tube operates with a high
voltage range of 16 to 20 kV applied between the anode and cathode. The high
energy of the electron entering the silicon sensor allows the signal to be
well isolated from the electronic noise pedestal, making a very efficient,
almost noise free photon detector. In total, across the two RICH detectors
there are approximately 500 000 active channels and in a typical LHC event
readout window of 25ns only 10 dark noise counts are observed. The trajectory
of the photoelectron through the focusing optics of the HPD is affected by
external electric and magnetic fields. While most of these effects are
minimised by ample magnetic shielding placed around the HPDs, residual
distortions are corrected in software.

Charged particles are probabilistically identified by the construction of a
log-likelihood value from the measured hits on the detector plane and the
expected hit patterns calculated from all the tracks measured through the
detector [6]. Each particle-type hypothesis is tried iteratively to find the
configuration of particle hypotheses that minimises the global likelihood.
With all other tracks under the hypothesis that minimises the global
likelihood, a delta-log-likelihood ($\Delta$LL) value is computed for each
track as the difference between the log-likelihoods when the track is given a
specific hypothesis and that of the pion hypothesis. This value is used to
discriminate between different hypotheses further along the analysis chain.

##  3 Detector calibration

To achieve optimal performance several detector effects need to be accounted
for through various calibration schemes. These include the timing of the
readout electronics relative to the transit of charged particles, the
alignment of the mirrors, changes in the refractive index of the radiators,
and the effects of electric and magnetic fields on the images of the HPD’s
photocathode on the silicon sensor. Each of these requires a careful
calibration process, which is described and evaluated below. Further details
of the calibration procedures, and their implementation can be found in ref.
[7].

###  3.1 Time alignment

The difference between the arrival time of photons at an HPD’s photocathode
and the peak of the output from the silicon sensor depends on the sensor
characteristics, particularly the leakage current. In addition, the arrival
time of photons relative to a $p$ $p$ collision is dependent on the photon
path length through the optical system to each HPD. These effects are
compensated for by applying an offset of the trigger signal to each of the
HPD’s readout boards, L0 boards [8], relative to the global LHCb clock. Each
L0 board reads out two HPDs, so care is taken to select HPD pairs with similar
characteristics.

To time align the HPDs, dedicated calibration data is taken when there is a
very large bunch spacing in the LHC, typically with a single colliding bunch
per orbit. The number of photons detected by each HPD is recorded with
different values of time offset relative to the LHCb clock. The offset is
adjusted in steps of 1 ns in the range $\pm 25$ ns from a point seen to be in
the correct 25 ns window. Typically 10 000 events are taken per step. An
example of such a scan for one pair of HPDs is shown in figure 4. The scan is
performed without input from the tracking system, so the total photon count
cannot be normalised to the number of tracks. Instead, the instantaneous
luminosity as measured by the calorimeters is monitored during the scan to
ensure it remains stable. For each pair of photon detectors a distribution of
detected photons as a function of time is produced. This distribution is
typically a top hat, and the centre point of the high region (defined as 90%
of maximum) is taken as the optimal timing. To further account for variation
in the beam conditions, several scans are performed and the required time
adjustment for each board computed from the average.

Figure 4:  The number of hits from two HPDs (blue and red points) read out by
the same L0 board and the combination (black points) for data taken with
differing trigger offsets in steps of 1 ns. Each step integrates the number of
photons detected over 10 000 bunch crossings.

The distribution of the residual time differences between the trigger and the
centre point of the photon signal is shown in figure 5. The mean and RMS of
the L0 board timing distributions are well centred, giving an alignment better
than 1 ns, comparable with the resolution of the timing data taken. The
combined HPD response for each L0 board is typically a top-hat function with a
width of 10 ns, making a 1 ns alignment sufficiently precise.

(a)(b)

Figure 5:  Distribution of the residual time differences between the trigger
to each L0 board and the central point of the photon signal from that board
for (a) RICH1 and (b) RICH2. Both detectors are aligned to better than 1 ns.

###  3.2 Magnetic distortion correction

The HPDs are sensitive to magnetic fields, which alter the trajectory the
photo-electrons take between the cathode and anode sensor. The LHCb dipole
creates a large magnetic field that extends over the region including RICH1
with a peak value of about $60{\mathrm{\,mT}}$ at the location of the photon
detectors. To attenuate this field, the HPDs are contained within a soft-iron
box, and are wrapped in individual mu-metal shields. However they still
experience a residual, predominantly axial, field of up to 2.5 mT. The
distortion caused by this field is the largest effect that must be corrected
in order to achieve the optimal resolution in RICH1. A dedicated calibration
system, which scans a grid of point light sources across the detector plane to
map the magnetic distortions, is periodically run (typically once per field
polarity per year) to derive corrections that are applied to the hit
positions. A detailed description of the operation and performance of the
correction system can be found in ref. [9]. The magnetic distortion maps are
one of the largest sources of differences in reconstruction performance
between the magnetic field polarities in RICH1.

RICH2 is in a region of much lower magnetic field, and the HPDs are aligned
perpendicular to the predominant direction of the field. This leads to much
smaller distortions, the time dependence of which can be ignored. Corrections
for RICH2 were computed during the early detector commissioning and remain
unchanged [10].

(a)(b)

Figure 6:  The distribution of the difference of reconstructed and expected
Cherenkov angles in (a) RICH1 and (b) RICH2. The data is shown for
reconstruction with corrections for the distortions caused by magnetic field
applied (black) and ignored (red). A significant improvement is seen in the
width of the distributions in RICH1, and hence the Cherenkov angle
resolutions, when applying the corrections.

Figure 6 shows the change in angular resolution for both RICH1 and RICH2
without corrections for the magnetic field distortions, and when applying
corrections. A large degradation of approximately a factor of five in the
resolution is observed in RICH1 when no corrections are applied. Almost no
degradation is seen in RICH2.

###  3.3 Mirror alignment

RICH1 contains 4 spherical mirrors and 16 flat mirrors whilst RICH2 contains
56 spherical mirrors and 40 flat mirrors. The relative orientation of the
mirrors with respect to the tracking system and/or the HPDs could change with
time through movements in the tracking stations, the detector support
structures, or genuine movements of the mirrors. Such effects can be corrected
using a data driven determination of the orientation of the mirrors. The
alignment of the mirrors is monitored once per fill of the LHC and if found to
change by more than 0.03  mrad, the alignment used by the reconstruction is
recomputed and updated.

The alignment of the mirrors is computed from the distribution of the photon
hits as a function of the angular position around the tracks. If the
orientation of the mirrors is perfectly known, the mean reconstructed
Cherenkov angle would be constant as a function of the reconstructed azimuthal
angle around the Cherenkov ring. Misalignments from the orientation used in
the reconstruction introduce a sinusoidal dependency to the mean. The extent
of this sinusoidal component is measured for spherical/flat mirror pairs and
is used to correct mirror alignment. The process is performed iteratively
until the corrections fall below a predefined threshold. A full description of
the alignment procedure and its performance can be found in ref. [11].
Throughout Run 2 the mirror alignments have been stable with very few updates
required.

###  3.4 Refractive index

The refractive index, $n$, of the radiators used in the RICH detectors is a
function of the pressure, temperature, and purity of the gases. The
temperature and pressure are constantly monitored, but the purity of the gases
is only measured periodically. A parameterisation of the refractive index as a
function of pressure and temperature is used to approximate the actual
refractive index, to which a correction factor, measured using data, is
applied.

Figure 7 shows the difference in the Cherenkov angle reconstructed from hits
around a track and the expected angle calculated using only the temperature
and pressure parameterisation of the refractive index. The refractive index
must be monitored frequently and early in the reconstruction chain, so a high-
statistics sample requiring little additional processing is used. The
distribution is formed from a sample of high momentum tracks, for which the
velocity approaches $c$ and the Cherenkov angle is approximately a function of
$n$ only, collected over a one hour period. The peak representing the true
Cherenkov photons is not centred at zero, indicating that the refractive index
used in the calculation of the expected angle is different from the true
value. This displacement is used to compute a corrective scale factor applied
to the estimated refractive index in the reconstruction.

(a)(b)

Figure 7:  Example distribution of photons detected in (a) RICH1 and (b) RICH2
from online monitoring data, from which the refractive index correction is
computed. The correction is derived from the displacement of the mean of the
Gaussian from zero.

After correction, there is a residual difference between the refractive index
used in the construction and the true refractive index. This difference is
evaluated using the offset from zero of the signal peak of low-multiplicity
dimuon data, selected only using information from the tracking and muon
systems to avoid any bias from selecting tracks well identified in the RICH.
The estimated residual difference as a function of time is shown in figure 8.
A small bias is observed in the refractive index after correction, but this
results in angular deviations an order of magnitude smaller than those due to
the HPD pixel size, and therefore is negligible. The stability of the
correction is excellent over a period of several years. Figure 9 shows the
residual difference split by magnet polarity, and track charge for each year.
All differences are very small. The differences seen between magnet polarities
follows the same pattern as seen in the resolutions determined in section 4,
where the resolution can affect the determination of the calibration. The
differences between years follows the same pattern as the photon yields
determined in section 5, which again can affect the ability to extract the
calibration.

(a)(b)

Figure 8:  The relative difference in the refractive index used in the
reconstruction after the calibration, and the true refractive index of the
radiators in (a) RICH1 and (b) RICH2 evaluated using the peak position of low-
multiplicity dimuon data as a function of time. Each point integrates data
taken over a 24 hour period. The stability of the correction is excellent over
several years.

(a)(b)
(c)(d)

Figure 9:  The relative difference in the refractive index used in the
reconstruction after the calibration, and the true refractive index of the
radiators in (left) RICH1 and (right) RICH2 evaluated using the peak position
of low-multiplicity dimuon data split by magnet state (a and b), and track
charge (c and d).

The gas system in RICH1 is known to have a slowly increasing contamination
with air. The gas is regularly cleaned to keep the contamination below 2 %.
The level of contamination and the associated correction factor are shown in
figure 10, along with the corresponding effect on the residual of the
refractive index determination for different levels of contamination. No
significant bias is seen as a function of this contamination. The bias is not
expected to be constant as a function of the contamination with air. The
refractive index correction procedure was tuned using data taken over a short
time period, therefore not exhibiting the full range of contamination. The air
contamination in the gas radiator of RICH2 has been kept below 2 % over 10
years without the need to purify the gas.

(a)(b)

Figure 10:  (a) The air contamination in RICH1 as a function of time (black
circles), and the correction factor applied to the refractive index (blue
crosses). The axis for the correction factor is reversed. (b) The relative
difference in the refractive index used in the reconstruction after the
calibration, and the true refractive index of the radiator in RICH1 for
differing levels of contamination by air.

###  3.5 HPD photocathode image position

Charging and temperature effects in the HPD can cause the image of the
photocathode projected on to the silicon sensor to move slowly with time. The
image the circular photocathode makes on the square pixel array from an
accumulated set of events is sufficiently circular that it is well described
by a centre point and radius. This image drift is compensated for by fitting a
circle to the distribution of hits in an HPD, integrated over approximately
one hour, to locate the projected centre point of the photocathode. The
difference between this and the centre of the silicon sensor is applied as a
correction to individual photon hits. Before fitting, the distribution is
cleaned and smoothed to remove artefacts and to compensate for the non-uniform
illumination across an HPD. A Sobel filter [12] is applied to enhance the edge
of the photocathode image.

The position of the centre point of the images of nine HPDs’ photocathodes as
a function of time is shown separately for four fills of the LHC in figure 11.
There is a range of behaviour, with some HPDs showing significant movements
compared to the size of a pixel (0.5 mm $\times$ 0.5 mm).

Figure 11:  The paths taken by the centre point of the image of the
photocathode of nine typical HPDs measured as displacement from the centre of
the silicon chip, with four paths per HPD taken from four day-long fills of
the LHC. The transitioning of the colour of the paths indicates the passage of
time, with blue closest to the HPDs being switched on, and yellow the end of
data taking. Dashed lines delimit pixels. A large range of behaviours is
observed.

(a)(b)

Figure 12:  The distribution of the difference of reconstructed and expected
Cherenkov angles in (a) RICH1 and (b) RICH2. The data is shown for
reconstruction with corrections for the HPD photocathode image shifts applied
(black) and ignored (red). A significant improvement is seen in the width of
the distributions, and hence the Cherenkov angle resolutions, when applying
the corrections.

Figure 12 shows the change in angular resolution for both RICH1 and RICH2
without corrections for the HPD image shifts, and when applying corrections. A
significant degradation of approximately 20% in the resolution is observed in
both RICH detectors when no corrections are applied.

##  4 Cherenkov angle resolution

The single photon resolution of both RICH detectors is measured on high
momentum tracks. Photons emitted from the tracks used to calculated the
prediction form a peak around zero in the distribution of the difference of
reconstructed and expected Cherenkov angles. Photons from other tracks or
random noise form a background below the peak, where in a typical $pp$
collision there is an occupancy of 1% averaged across the detector plane in
RICH1 and 0.6% in RICH2. To obtain the resolutions a fit to this distribution
is performed, modelling the correctly matched photons with a Gaussian
distribution and the background as a second order polynomial. The resolution
is taken to be the width of the Gaussian distribution.

The shape of the background peaks under the signal and tends to bias the
resolutions obtained towards larger values. To get a reliable estimate of the
absolute resolution, events with very-low charged-track multiplicity are used.
This both reduces the relative amount of background and flattens the
background shape under the signal peak. An example of the photon angle
distribution from low-multiplicity events is shown in figure 13. The
systematic uncertainty from the background shape is estimated by taking the
difference in resolution obtained from the nominal fit and one where the
background is modelled as linear, as this has been shown to cover for a wide
range of potential background shapes using simulation. The resolutions as a
function of time are show in figure 14. The resolution for each year in Run 2
and in each magnet polarity are shown in figure 15. The uncertainty is a
combination of the uncertainty obtained from the fit and the background
systematic, dominated by the latter. In RICH1 there does appear to be a small
dependence on magnet polarity, with the resolution of the Up polarity being
slightly better than the Down. There are several potential sources of
differences between polarities in the detector, with the largest effects
thought to come from differences in the magnetic distortion correction and
tracking resolution.

(a)(b)

Figure 13:  Example distribution of photons detected in (a) RICH1 and (b)
RICH2 from low track-multiplicity events. A fit to this distribution is
overlaid, with a Gaussian signal component (red), polynomial background
component (blue), and the total distribution (black)

(a)(b)

Figure 14:  Resolutions for (a) RICH1 and (b) RICH2 measured between 2016 and
2018. Each data point is from low-multiplicity dimuon data accumulated over 24
hours. The uncertainties show individual bars for the statistical and
systematic uncertainty, with the statistical uncertainty dominating.

(a)(b)
(c)(d)

Figure 15:  Resolutions for (a) RICH1 and (b) RICH2 measured per year and per
magnet polarity, and for (c) RICH1 and (d) RICH2 measured per year and per
track charge integrated over data taken with each magnet polarity. The
uncertainties are dominated by the systematic uncertainty arising from the
background shape.

When both magnet polarities are averaged there should be very little
difference in the performance between positively and negatively charge
particles. The resolutions for each charge in each Run 2 year, integrated over
magnet polarities, is shown in figure 15, where the resolutions determined for
both charges are in excellent agreement.

Combining all run periods the Cherenkov angle resolution in RICH1 is $1.662\pm
0.023\text{\,mrad}$ and in RICH2 is $0.621\pm 0.012\text{\,mrad}$, where, now,
the statistical uncertainty is negligible and the systematic uncertainty is
dominated by the shape of the background distribution. The measurements for
both detectors are within 10% of the expectation from simulation.

##  5 Detected photon yield

The number of detected Cherenkov photons emitted per track is determined from
low track-multiplicity events. The determination is performed per track, where
a simple linear background subtraction is applied to the Cherenkov angle
distribution of reconstructed photons. The selected tracks are positively
identified as muons by the muon system, with a high momentum that produces a
ring with Cherenkov angle close to its saturation value, and falls within the
geometric acceptance of the detector, the same selection used to study the
refractive index. An example of the distribution of the difference between the
reconstructed and expected Cherenkov angles for photons around a typical track
is shown in figure 16.

The determination of the photon yield is highly dependent on the shape of the
background, which can peak close to the signal. Even with the low backgrounds,
simulations show that the absolute value of the number of detected photons
could be biased by up to 7 % depending on the exact source of the background.
However this bias is not a strong function of yield, and is highly correlated
between measurements, so is ignored for the following relative comparisons.

(a)(b)

Figure 16:  Example distribution of the difference between reconstructed and
expected Cherenkov angle for photons detected around a typical track in a low-
multiplicity event for (a) RICH 1 and (b) RICH 2.

The background-subtracted mean number of detected Cherenkov photons for both
RICH1 and RICH2 is shown in figure 17. The number of detected photons slowly
decreases each year as may be expected by ageing of the photon detectors.

The number of detected photons per track for each charge, integrated over both
magnet polarities, is shown in figure 17. In general there is very good
agreement between the number of detected photons for both charges, although a
small charge asymmetry is seen for RICH2 in 2018 and a smaller asymmetry is
seen in 2017. The cause of this asymmetry is not fully understood, but it is
small enough to not significantly affect the performance.

(a)(b)
(c)(d)

Figure 17:  Number of Cherenkov photons emitted per track for (a) RICH1 and
(b) RICH2 measured per year and per magnet polarity, and (c) RICH1 and (d)
RICH2 measured per year and per track charge. The uncertainty is statistical
only.

The average detected photon yield for the period 2016–2018, determined from a
fit to the integrated data set, in RICH1 is $30\pm 2$ and in RICH2 is $18.5\pm
1.2$, where the uncertainty is dominated by the background shape bias. This is
in good agreement with expectations from simulation.

##  6 PID performance

The quality of the detector calibration, and the values of the Cherenkov angle
resolution and photon yields all impact the performance of physics selections
that utilise the RICH particle identification $\Delta\text{LL}$ information to
identify the signal of interest and to reject backgrounds. The ultimate
performance of the RICH system can be determined in terms of the ability to
discriminate between the species of charged hadrons and also in the impact of
those selections on the physics output of the experiment.

###  6.1 Efficiency versus purity

The efficiency of the system for discriminating between different species of
hadrons can be determined from control samples of well-identified particles
obtained purely through kinematic selections. These calibration samples are
selected without using the RICH information in order to not bias the results.
High-purity samples of charged kaons and pions are obtained from the decay
products of the $D^{0}$ meson identified in the decay chain $D^{*+}\rightarrow
D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ and the charge-conjugated decays.
Samples of protons are obtained from $\Lambda^{0}\rightarrow p\pi^{-}$ decays,
and the charge-conjugate decay. Details of the selection of the calibrations
samples are given in ref. [13].

The efficiency of a particular $\Delta\text{LL}$-based PID selection on a
probe hadron (pion, kaon or proton) is computed by performing an unbinned
maximum likelihood fit to the invariant mass distributions of the parent
particle in the appropriate samples, allowing the separation of the decay
candidates from any residual background. The fit is performed with and without
the associated PID selection applied to the particle of interest, and the
efficiency computed from the ratio of yields of candidate decays between the
two fits. The PID selection is scanned over a range of loose to tight
$\Delta\text{LL}$ selections, and a profile of selection efficiencies (e.g.
kaons identified as kaons) and associated misidentified particle leakage (e.g.
pions misidentified as kaons) obtained. The separation between kaons and
pions, proton and pions, protons and kaons, is obtained by using the
information provided by the $\Delta\text{LL}(K,\pi)$,
$\Delta\text{LL}(p,\pi)$, $\Delta\text{LL}(p,K)$ distributions, respectively.

Trigger selections vary across the different data-taking periods, and the
kinematics of the $D^{0}$ and $\Lambda^{0}$ decay products are significantly
different. Since the RICH performance is a function of the momentum,
pseudorapidity and activity in the detectors, the latter being parametrised by
the number of reconstructed tracks in the LHCb acceptance, a weighting
procedure is used to equalise the distributions of these variables between
different species of hadrons and across data-taking periods. The momentum
range is chosen such that under at least one of the particle hypotheses
compared the particle would be above the velocity threshold for generating
Cherenkov radiation. Pseudorapidity ranges are chosen to match the RICH
detectors’ acceptances. The weighting in the number of tracks cancels most of
the differences in data-taking conditions and trigger thresholds.

The performance curves for separating kaons and pions, protons and pions, and
protons and kaons, split by charge, are shown in figure 18 for the 2018 data-
taking year and integrated over the two magnet polarities. Each datapoint
corresponds to a $\Delta\text{LL}$ value, with tighter selections when moving
towards the bottom-left corner of each plot. The agreement between the
performance curves of positive and negative hadrons, of particular interest
for $C\\!P$ violation measurements, is excellent. The same behaviour is
observed for all the data-taking years.

(a)(b)
(c)(d)
(e)(f)

Figure 18:  The efficiency of selecting kaons (a and b), protons (c, d, e and
f), with the associated leakage from misidentifying pions (a, b, c d) and
kaons (e and f) in data taken during 2018. (a, c and e) The efficiency curves
are shown for positively (black, solid) and negatively (red, dotted) charged
particles. (b, d and f) The efficiency curves are shown for Up (red, dotted)
and Down (black, solid) magnetic field polarities. Uncertainties are
statistical only, and are highly correlated between points on the same curve.

The same performance curves, but split by magnet polarity, are shown in figure
18 for the 2018 data-taking year and integrated over the hadron charges. Also
here the agreement is excellent and the same behaviour is observed for all the
data-taking years.

The curves showing the stability of the PID performances across Run 2 are
shown for the years 2015–2018 in figure 19. Within the corresponding
kinematics ranges, the probability to correctly identify each charged hadron
is consistently above the 90% for a 5% misidentification probability.

(a)(b)
(c)

Figure 19:  The efficiency of selecting kaons (a), protons (b and c), with the
associate leakage from misidentifying pions (a and b) and kaons (c). The
efficiency curves are shown for 2015 (blue, dashed), 2016 (green, dash-
dotted), 2017 (red, dotted), 2018 (black solid). Uncertainties are statistical
only, and are highly correlated between points on the same curve.

The PID efficiency and the misidentification rate are also determined as a
function of the track momentum, as shown in figure 20. In this case two
selections are considered: a loose selection on the relevant $\Delta\text{LL}$
distribution, resulting in a high signal efficiency, and a tight selection on
the same variable, resulting in a good background rejection.

(a)(b)
(c)

Figure 20:  The efficiency of selecting kaons (a), protons (b and c), with the
associate leakage from misidentifying pions (a and b) and kaons (c) as a
function of momentum. Two selections are made, a loose selection (hollow
circles) and a tight selection (solid circles).

###  6.2 Physics impact

The performance curves shown in section 6.1 are highly illustrative, but do
not tell the whole story of the performance of the hadronic particle ID within
LHCb. The benefits of the application of PID on particular physics observables
is dependent on the kinematics of the signal channels, and the nature of the
background channels. In this section a more qualitative demonstration of the
performance of the hadron PID is given through several examples of its
importance for the LHCb physics programs.

An example of the hadron PID used in rare decay analyses is shown in figure 21
[14]. Separate samples of $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ and
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ decay candidates are obtained, where
the pion mode is suppressed by the ratio of CKM elements
$\left|\frac{V_{td}}{V_{ts}}\right|^{2}$, approximately a factor of 25. The
hadronic PID selection suppresses the kaon mode relative to the pion mode by a
factor of 80, while retaining 82 % of pion candidates. A clean peak from the
pion mode is seen, which would be impossible to obtain without effective
hadron PID.

(a)(b)

Figure 21:  The effect of applying hadron PID selections to suppress the
dominant kaon mode (a) to measure the CKM suppressed pion mode (b), in rare
$B$ decays [14]. Here 82 % of pion events are retained, suppressing the kaons
by a factor of 80 relative to the pions, leaving a clear peak from the pion
mode.

Measurements of the CKM angle $\gamma$ rely on distinguishing purely hadronic
final states from $B$ decays. An example of isolating $B^{\pm}\rightarrow
D^{0}K^{\pm}$ candidates from the CKM favoured $B^{\pm}\rightarrow
D^{0}\pi^{\pm}$ is show in figure 22.

(a)(b)

Figure 22:  The effect of applying hadron PID selections to suppress the
dominant pion mode (a) to measure the CKM suppressed kaon mode (b), in CP
violation analyses [15].

Hadron PID is also crucial in the spectroscopy program, where the capability
to efficiently discriminate between long-lived hadrons is of fundamental
importance in both precision measurements [16] and new resonance searches
[17]. An application of the excellent proton identification performance is
shown in figure 23, where the $B_{c}^{+}$ meson invariant mass peak can be
reconstructed in two decay modes only differing for the presence of a
$p\overline{p}$ pair instead of a $\pi^{+}\pi^{-}$ pair in the final state.
The $B_{c}^{+}\rightarrow J/\psi p\overline{p}\pi^{+}$ decay mode is
suppressed by a factor $\sim 15$ with respect to the $B_{c}^{+}\rightarrow
J/\psi\pi^{+}\pi^{-}\pi^{+}$ decay channel [18], but the proton selection
efficiency and pion rejection provided by the RICH detectors allow clear
identification of the $B_{c}^{+}\rightarrow J/\psi p\overline{p}\pi^{+}$
signal.

(a)(b)

Figure 23:  The effect of applying hadron PID selections to distinguish
between $B_{c}^{+}\rightarrow J/\psi\pi^{+}\pi^{-}\pi^{+}$ (a) and
$B_{c}^{+}\rightarrow J/\psi p\bar{p}\pi^{+}$ (b) decay modes [19].

##  7 Conclusions

The performance of the LHCb RICH detectors has been investigated. The utility
of the calibration and alignment procedure has been demonstrated to give
remarkable stability.

The angular resolution of the detectors has been measured to be $1.662\pm
0.023\text{\,mrad}$ and $0.621\pm 0.012\text{\,mrad}$ for RICH1 and RICH2,
respectively. The resolutions in RICH1 show small differences between magnetic
field polarities due to limitations of mapping out distortions caused by the
magnetic field, but otherwise the resolutions are excellent, and close to
expectations from simulation.

The number of detected photons per track has been measured to be $30\pm 2$ and
$18.5\pm 1.2$ for RICH1 and RICH2, respectively. A slight decline in the
photon yield is seen in both detectors as a function of time, as expected from
the ageing of the HPD photocathodes.

The purity and efficiency of particle selections made using the information
from both RICH detectors has been shown, yielding very pure samples with high
efficiency. The performance has been shown to be very stable over time and
independent of the track charge. The benefits of the excellent charged hadron
identification have been shown in a number of physics channels demonstrating
the access to a unique physics program from the LHC.

The RICH detectors are undergoing a substantial upgrade program, which will
improve the performance for the much larger data sets that will be taken
during Run 3 of the LHC. This will ensure the excellence of the LHCb charged
hadron identification for years to come.

## 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: INFN (Italy); MNiSW and NCN (Poland); MEN/IFA
(Romania); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), 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.

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