# Efficiency and time resolution of monolithic silicon pixel detectors in SiGe
BiCMOS technology

G. Iacobucci,11footnotetext: Corresponding author. L. Paolozzi  P. Valerio  T.
Moretti  F. Cadoux  R. Cardarelli22footnotetext: Also at INFN Section of Roma
Tor Vergata, Via della ricerca scientifica 1, Roma, Italy. R. Cardella  S.
Débieux  Y. Favre  D. Ferrere  S. Gonzalez-Sevilla  Y. Gurimskaya  R. Kotitsa
C. Magliocca  F. Martinelli  M. Milanesio  M. Münker  M. Nessi  A. Picardi  J.
Saidi  H. Rücker  M. Vicente Barreto Pinto  S. Zambito

###### Abstract

A monolithic silicon pixel detector prototype has been produced in the SiGe
BiCMOS SG13G2 130 nm node technology by IHP. The ASIC contains a matrix of
hexagonal pixels with pitch of approximately 100 $\mu$m. Three analog pixels
were calibrated in laboratory with radioactive sources and tested in a 180
GeV/c pion beamline at the CERN SPS. A detection efficiency of
(99.9$\begin{subarray}{c}+0.1\\\ -0.2\end{subarray}$)% was measured together
with a time resolution of $(36.4\pm 0.8)~{}$\mathrm{ps}$$ at the highest
preamplifier bias current working point of 150 $\mu$A and at a sensor bias
voltage of 160 V. The ASIC was also characterized at lower bias voltage and
preamplifier current.

##  1 Introduction

Since several years the particle-physics community has devoted an increasing
interest to the timing performance of silicon [1, 2] for the detection of
ionising radiation. In a pixelated hybrid detector, the NA62 GigaTracker [3]
achieved time resolution down to 150 ps. The ATLAS and CMS Collaborations are
upgrading their detectors for the High Luminosity LHC program with planes of
Low Gain Avalanche Detectors (LGAD) [4, 5]. LGADs are planar silicon sensors
that exploit the increase of the signal-to-noise ratio given by an internal
gain of 10-50 to provide time resolutions of approximately 30 ps. Those
employed in the ATLAS and CMS timing layer have an area of about 1.6 mm2.

An alternative approach to obtain the signal-to-noise ratio necessary to
achieve such time resolution comes from the exploitation of the unparalleled
analog performance of Silicon-Germanium Heterojunction Bipolar Transistors
(SiGe HBT) to produce a fast signal amplification with low noise at low
amplifier current density [6][7]. An R$\&$D effort started in recent years to
exploit commercial SiGe BiCMOS - a mainstream microelectronic technology with
Very Large Scale Integration (VLSI) capability - to produce monolithic pixel
detectors with small pixel size and time resolution comparable with that of
LGAD, without recurring to an avalanche gain mechanism [8][9].

Laboratory measurements with radioactive sources of the first monolithic
silicon pixel detector prototype in the 130 nm SG13G2 IHP process [10] with
100 $\mu$m pixel pitch demonstrated that time resolutions at the level of 50
ps can be achieved [11][12]. Such remarkable timing performance for a silicon
sensor without an internal gain layer was obtained at rather high
discrimination threshold, which was possibly affecting the sensor detection
efficiency. Furthermore, there were indications that the resolution on the
signal time-over-threshold (TOT), which was used to correct for signal time-
walk, was limiting the timing performance of the detector. These
considerations suggested that a study of the analog signals and an improved
time-walk correction method would be necessary to fully exploit the timing
capability of this type of silicon detector maintaining at the same time very
high efficiency.

For this purpose a second monolithic silicon pixel ASIC prototype was produced
in the same 130 nm IHP process. The ASIC contains four sub-matrices of pixels
with digital readout as well as four analog pixels with the preamplification
stage followed by an analog driver with dedicated output pads. Although a full
description of the ASIC is given in Section 2, the measurements presented in
this paper focus on the analog pixels, since the possibility to measure the
full output waveform allows for a better insight on the circuit performance,
decoupling the front-end characteristics from the digital logic, therefore
permitting a detailed study of the efficiency and timing response. The analog
channels were characterized in laboratory tests with radioactive sources at
the University of Geneva and with minimum ionizing particles (MIPs) at the
CERN SPS testbeam facility.

##  2 Description of the ASIC

###  2.1 Architecture and Floorplan

The ASIC prototype was produced in the IHP SG13G2 $130\text{\,}\mathrm{nm}$
BiCMOS technology with a wafer resistivity of
$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$. It features a
matrix of 12$\times$12 monolithic hexagonal pixels of 65 $\mu$m side with a
pixel capacitance of 80 fF333The sensor implemented in the pixel is
essentially the same sensor of [12].. The ASIC floorplan is shown in Figure 1.
Each pixel includes an HBT-based frontend, a fast discriminator and an 8-bit
threshold-tuning DAC. A common digital logic placed beside the matrix (from
now on called "periphery") contains the readout logic, a dedicated timing
digitization circuit, a number of DACs to provide the bias voltages, and
currents to the pixels and an SPI slow-control interface to program them.

The pixels are built using the n-well as sensor cathode, biased at a positive
bias voltage that can be changed via an external power supply, and the p-doped
substrate as the anode. The anode is biased at a negative high voltage ($\leq$
$-80\text{\,}\mathrm{V}$ in the data presented here). The presence of this
high-voltage bias forces all the electronics to be insulated by the substrate
it in order to avoid breakdown. Hence, every nMOS is placed in insulated
p-wells, divided in three power domains. A series of 18 guard-rings are placed
around the edge of the chip to insulate the substrate.

The output of the discriminator is sent to a Time-to-Digital Converter (TDC)
positioned in the periphery to calculate a timestamp for the hit and measure
the TOT of the signal. The ASIC integrates three separate TDCs, each wired to
a fast-or line connected to different pixels. The fast-or lines are
interleaved so that two adjacent pixels are always connected to separate TDCs:
this ensures that in the event of a particle hitting multiple adjacent pixels,
the timing information can be measured from all of them independently. The TDC
architecture is described in [13]. A 9-stage pseudo-nMOS feed-forward ring
oscillator is used with a set of latches to sample the oscillator state. The
TDC resolution is about $30\text{\,}\mathrm{ps}$. Each TDC features three
measurement channels: one for the event timestamp, one for ToT and one for
calibration purposes.

Figure 1:  Floorplan of the monolithic pixel detector prototype. The ASIC size
is 2.3$\times$2.5 mm2. It contains four sub-matrices of 6$\times$6 hexagonal
pixels with a pitch of approximately 100 $\mu$m.

###  2.2 Front-end flavors

The pixel matrix is divided into four 6$\times$6 sub-matrices, each with a
different amplifier design. The discriminator, calibration DACs and readout
interface are the same. The basic schematic of the preamplifier can be seen in
Figure 2. It consists of a SiGe HBT used as a charge sensitive amplifier, with
a pMOS load and an MOS-based floating resistor used as a feedback. The
architecture is similar to the one the authors developed for [11], which
yielded to excellent timing performance. This circuit is followed by a MOS
discriminator that compares the output of the preamplifier to a fixed
threshold to determine if the pixel was hit by a particle.

Figure 2:  Schematic layout of the BJT-based preamplifier of the prototype
ASIC. The left block is a common-emitter configuration capacitively coupled to
the sensor, while the right one implements a floating MOS-based feedback
resistor.

The four sub-matrices contain different variations of this circuit:

  * •

The circuit in Figure 2, with the preamplifier, discriminator and calibration
DAC placed outside of the active pixel area;

  * •

The same circuit, with the preamplifier integrated in the pixel n-well;

  * •

The same amplifier, but with an added HBT common collector driving stage
placed before the discriminator;

  * •

A version of the circuit with two discriminators with different thresholds, to
calculate the rising slope of the input signal directly and perform a more
accurate time-walk correction.

###  2.3 Analog pixels

One of the sub-matrices hosts four analog pixels (Figure 3). They contain the
same HBT amplifier represented in Figure 2, followed by an analog driver
directly connected to output pads to be measured with an oscilloscope. The
driver is composed by two consecutive HBTs in common collector configuration,
with the first stage AC coupled to the output of the amplifier. The first
stage of the driver has a configurable output impedance, while the second
stage is terminated on a $500\text{\,}\mathrm{\SIUnitSymbolOhm}$ resistor in
chip. The entire front-end is placed outside of the pixel area, in proximity
of the output pads, as shown in Figure 3. The short routing from the pixel to
the amplifier input is estimated to contribute a capacitance of less than 10
fF.

Figure 3:  Detail of the ASIC layout that shows the four pixels directly
connected to output pads. During the CERN SPS testbeam, pixels OA0, OA1 and
OA2 were read by an oscilloscope. This study is based on the data acquired
with these three analog pixels.

###  2.4 Slow-control and readout interface

The periphery of the chip contains, in addition to the I/O pads, thirteen
8-bit DACs to tune all the bias voltages and currents used by the analog
blocks in the pixels. Each DAC is programmable via a standard SPI interface
using 16-bit commands (8 to address a specific DAC, 8 for the DAC value). The
same SPI interface can be used to program a "mask" bit for individual pixels
(i.e. force their output to be ignored by the readout logic). This is useful
in case of malfunctioning pixels, or to test only some parts of the chip at a
time. A special SPI command is also used to initiate a frame readout. As soon
as this command is sent, the content of the TDCs and the address of the pixel
hit is read out via a low-voltage differential line. The state of the pixel is
then reset, to be able to acquire a new frame.

##  3 Amplifier response calibrations

Laboratory measurements with X-ray sources were performed to characterize the
analog response of the front-end electronics of two ASICs, that will be called
here Detector Under Study 0 (DUT0) and 1 (DUT1). For these measurements and
those described in Section 4, the outputs of the analog drivers were connected
to a $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ coaxial cable and AC coupled via a
$100\text{\,}\mathrm{nF}$ capacitor to a Lecroy WaveMaster 820zi oscilloscope.
The oscilloscope has a sampling rate of 40 Gs/s and analog bandwidth limited
to 4 GHz. A $\mathrm{{}^{109}Cd}$ source was used to measure the gain of the
electronic chain, which comprises the pixel, the preamplifier and the driver.
To study the response of the electronics as a function of the power
consumption, four different working points were adopted, with amplifier bias
current $I_{\textnormal{\it preamp}}=$ $7\text{\,}\mathrm{\SIUnitSymbolMicro
A}$, $20\text{\,}\mathrm{\SIUnitSymbolMicro A}$,
$50\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and
$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$.

The two main photons from the $\mathrm{{}^{109}Cd}$ source derive from the
${}^{109}Ag$ X-ray K lines of energy $E_{1}=$22.16\text{\,}\mathrm{keV}$$ and
$E_{2}=$24.94\text{\,}\mathrm{keV}$$. The spectrum of $\mathrm{{}^{109}Cd}$
was fitted with a double Gaussian function. Figure 4 shows one example of the
spectra from DUT0, while Figure 5 shows the amplitude values obtained from the
fit of the same spectrum as a function of the deposited charge, assuming an
energy of $3.6\text{\,}\mathrm{eV}$ to generate and electron-hole pair in
silicon. A linear fit of the data was used to estimate the charge gain
($A_{q}$) of the front-end electronics.

Figure 4: Spectrum of the $\mathrm{{}^{109}Cd}$ source measured at the working
point $I_{\textnormal{\it preamp}}=$$150\text{\,}\mathrm{\SIUnitSymbolMicro
A}$ and $HV=140~{}V$. The two peaks, corresponding to the X-ray emission of
${}^{109}Ag$, were fitted with a double Gaussian function.  Figure 5: Example
of the amplitude measurement as a function of the number of electrons obtained
with the $\mathrm{{}^{109}Cd}$ source for one of the working points for DUT0.
The superimposed linear fit is used to parameterise the response of the front-
end at this working point.

The Equivalent Noise Charge (ENC) of the front-end was then calculated as

$ENC=\frac{\sqrt{\sigma_{V}^{2}-\sigma_{\textnormal{\it scope}}^{2}}}{A_{q}}$

where $\sigma_{V}$ is the standard deviation of the electronics-noise
distribution measured on the oscilloscope when the front-end is connected, and
$\sigma_{\textnormal{\it scope}}=$420\text{\,}\mathrm{\SIUnitSymbolMicro V}$$
is the voltage noise of the oscilloscope with the open input connector. The
results of the gain and ENC measurements are summarized in Table 1. The two
DUTs show the same trend within approximately 10%. At $I_{\textnormal{\it
preamp}}=$50\text{\,}\mathrm{\SIUnitSymbolMicro A}$$ the ENC is as low as 50
electrons. As expected, it increases significantly for smaller
$I_{\textnormal{\it preamp}}$ values, which could produce a degradation of the
detection efficiency. The increase of the ENC at $I_{\textnormal{\it
preamp}}=$ 150 µA is due to the configuration of the amplifier used for this
working point.

|  | DUT0 | DUT1
---|---|---|---
| $I_{\textnormal{\it preamp}}$
---
[$\mu$A]
| Power density
---
[mW/cm2]
| $\sigma_{V}$
---
[mV]
| Aq
---
[mV/fC]
| ENC
---
[electrons]
| $\sigma_{V}$
---
[mV]
| Aq
---
[mV/fC]
| ENC
---
[electrons]
7 | 84 | 1.35 | 42.3 | 194 $\pm$ 1 | 1.41 | 49.2 | 171 $\pm$ 2
20 | 240 | 0.82 | 43.4 | 102 $\pm$ 1 | 0.83 | 46.0 |  97 $\pm$ 1
50 | 600 | 0.70 | 55.0 |  54 $\pm$ 1 | 0.58 | 50.5 |  50 $\pm$ 1
150 | 1800 | 0.81 | 52.4 |  76 $\pm$ 1 | 0.72 | 44.4 |  82 $\pm$ 1
Table 1: Power density, voltage noise, charge gain and ENC of the analog
front-end electronics at the four $I_{\textnormal{\it preamp}}$ working points
for HV = 120 V. The power density is calculated for a matrix of pixels with
100 µm pitch. The lower ENC value measured at $I_{\textnormal{\it
preamp}}=$50\text{\,}\mathrm{\SIUnitSymbolMicro A}$$ is due to a different
configuration of the amplifier and driver working point that acts as a filter
for noise, thus reducing the bandwidth.

##  4 Efficiency and time resolution measurement

###  4.1 Testbeam experiment setup and data set

The detection efficiency and time resolution of the prototypes were measured
at the CERN SPS testbeam facility with a pion beam with
$180\text{\,}\mathrm{GeV}$/c momentum. The experimental setup consisted of the
UniGe FEI4 telescope for particle tracking [14], with the two devices under
test (DUT0 upstream with respect to the beam, DUT1 downstream) placed after
three detection planes of the telescope as shown in Figure 6. The DUTs were
operated at room temperature. The DUTs were read by two oscilloscopes with
analog bandwidth of 4 GHz and a sampling rate of 40 GS/s and 20 GS/s,
respectively. The DUTs were positioned specularly to each other and pixels OA0
(shown in Figure 3) from the two chips were aligned and sent to the
oscilloscope with the largest sampling rate to make Time-Of-Flight (TOF)
measurements. The data from pixels OA1 and OA2 of the two DUTs, which were not
aligned among the two DUTs, were also acquired.

Figure 6:  Schematic view of the experimental setup, showing the five FEI4
telescope [14] planes that were operated and the two DUTs in green. Plane TEL3
was not operational during this testbeam measurement. The FEI4 readout chip
has a a matrix of $80\times 336$ pixels with a pixel size of $250\times
50~{}${\mathrm{\SIUnitSymbolMicro m}}^{2}$$. The telescope planes are
alternatively rotated by 90∘ to optimise the space resolution on the two
transversal directions. A region of interest, shown by the yellow area, was
imposed to the first plane of the telescope, and was put in coincidence with
the last telescope plane to generate the trigger.

The analysis of the data was performed using the full waveform information
acquired by the oscilloscopes. The signals from the DUTs were delayed to
guarantee that they were always in the second half of the waveform time window
acquired by the oscilloscope. This configuration allowed using the first half
of the waveform to determine the voltage noise $\sigma_{V}$ at the output of
the analog front-end and set a discrimination threshold $V_{th}$ as a multiple
of the voltage noise, independently for the two DUTs. Figure 7 shows a typical
waveform with a signal pulse from a MIP at the working point with
$I_{\textnormal{\it preamp}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. The
dashed line represents the discrimination threshold at
$V_{th}=6~{}\sigma_{V}$.

Figure 7:  Example of a waveform from working point $I_{\textnormal{\it
preamp}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. The shaded region below
250 ns is the portion of the waveform used to extract $\sigma_{V}$. The dashed
line shows the discrimination threshold used for this working point.

The FEI4 telescope provided the trigger to the oscilloscopes. A Region Of
Interest (ROI) of $250\times 600$ ${\mathrm{\SIUnitSymbolMicro m}}^{2}$ was
set on one of the trigger planes of the telescope, centered around the pixels
OA0 of the two DUTs that were aligned with respect to the beamline and used
for the TOF measurement.

Data were acquired with DUT0 and DUT1 at the same four front-end working
points used for the characterization of the DUTs with radioactive sources
(Section 3) for a bias voltage of 120 V. At this potential the substrate is
not fully depleted. The depletion depth is estimated to be
$23\text{\,}\mathrm{\SIUnitSymbolMicro m}$, which corresponds to a most
probable charge $Q_{MPV}\approx 1300$ electrons for a MIP.

A high voltage scan was also performed only for the working point
$I_{\textnormal{\it preamp}}=$ $150\text{\,}\mathrm{\SIUnitSymbolMicro A}$.

To evaluate the efficiency and time resolution of our DUTs in the cleanest
possible way, a selection was applied on the quality of the tracks
reconstructed by the FEI4 telescope. The selection consisted in discarding
events in which more than one track was reconstructed by the telescope, and in
accepting only those events with the reconstructed track having an associated
hit in each of the five telescope planes and a $\chi^{2}/NDF\leq 1$. About 30%
of the triggered events survived this stringent selection on the tracks from
the telescope. For this final sample the telescope pointing resolution on the
DUT planes was estimated to be approximately 10 $\mu$m [15].

###  4.2 Cross talk and robustness to induced noise

During laboratory and testbeam measurements, cross talk between the channels
was observed for events with large charge deposition, corresponding to
approximately five times the MIP most probable charge. The analysis of these
events showed that this cross talk was not influenced by the relative position
of the pixels within the matrix or the routing of the signal before and after
the amplification. Therefore the observed cross talk could be ascribed to two
possible causes. The first is a feedback path passing through the ground of
the board, which is then injected to the backside of the chip through the HV
decoupling capacitors and finally in the amplifier input. This hypothesis is
supported by the fact that the system becomes less stable at bias voltages
below $50\text{\,}\mathrm{V}$, when the pixel capacitance is significantly
larger. The second cause is noise induced by the driver pulse propagating in
the power supply, since in this prototype the analog and driver electronics
share the same supply lines.

As a consequence of this cross talk, only events in which the pixel under test
had the largest signal among the three acquired pixels were selected for the
time resolution measurement. No selection was applied for cross talk in the
efficiency measurement, since cross talk appears only for efficient events.

###  4.3 Efficiency measurement

For the calculation of the efficiency, all the selected pion tracks
reconstructed by the FEI4 telescope were extrapolated to the surface of each
of the two DUTs. Only the tracks crossing a DUT within the area of the three
pixels under study, or outside the external edges of the pixels by at most one
standard deviation of the telescope resolution, were retained. An event was
considered efficient if the signal in at least one of the pixels crossed the
discrimination threshold.

Figure 8 shows the efficiency map for the two DUTs at the $I_{\textnormal{\it
preamp}}=$ $150\text{\,}\mathrm{\SIUnitSymbolMicro A}$ working point, for a
threshold of $6\sigma_{V}$ and a sensor bias $HV=$ 120 V. The panels show the
efficiency map for the entire surface of the three pixels (the pixel edges are
represented by the black lines). The degraded efficiency measured nearby the
twelve external sides of the hexagonal pixels is produced by the FEI4
telescope resolution, as attested by the fact that such degradation of the
efficiency is not observed in the region of the three internal hexagon sides
between the pixels. To avoid biases stemming from the pointing resolution of
the FEI4 telescope, the measurement of the detection efficiency was restricted
to events with tracks extrapolated inside the triangular area444It should be
noted that this triangular area constitutes exactly one sixth of the total
area of the three pixels, which is representative of all the zones of a pixel
(central-pixel zone; zone in between two pixels; zone in between three pixels)
in the right geometrical proportions. Therefore, the efficiency measured
inside it provides a reliable estimation of the efficiency over the entire
hexagonal pixel area, unbiased by the telescope resolution. in between the
three pixels that is delimited by the red lines in Figure 8.

Figure 8:  Efficiency map measured for DUT0 (left) and DUT1 (right) at
$I_{\textnormal{\it preamp}}=$150\text{\,}\mu$$A, threshold
$V_{th}=6~{}\sigma_{V}$ and $HV=$120\text{\,}\mathrm{V}$$. The pixel edges are
shown by the black lines. The efficiency degradation around the external edges
of the three pixels is due to the FEI4 telescope resolution. The efficiency
measured inside the triangular area delimited by the red lines is unaffected
by the telescope pointing resolution and is used throughout this study.

Figure 9 and Table 2 show the efficiency obtained within the triangular area
for the four preamplifier working points at a discrimination threshold of 6
$\sigma_{V}$ and at a sensor bias voltage of 120 V. The efficiency follows the
trend expected from the ENC measurement of Table 1. All the preamplifier
working points show an efficiency well above 99% except for the one at
$7\text{\,}\mathrm{\SIUnitSymbolMicro A}$. At this current a larger depletion
depth would be required to operate the front-end at even higher efficiency. It
is also noted that DUT0 shows a lower efficiency, compatibly with the larger
ENC measured with the $\mathrm{{}^{109}Cd}$ source (Section 3). At higher
currents DUT0 is slightly more efficient than DUT1. To investigate this
difference, the efficiency as a function of the discrimination threshold was
inspected. The results are reported in Figure 10 for
$HV=$120\text{\,}\mathrm{V}$$. The threshold scan indicates a clear difference
in performance, in spite of the fact that both DUTs reach the efficiency
plateau at a threshold of six standard deviations of the noise.

Figure 9:  Efficiency vs. $I_{\textnormal{\it preamp}}$ for $HV=$120\text{\,}\mathrm{V}$$, evaluated in the triangular inter-pixel zone for DUT0 (in red) and DUT1 (in blue). Efficiency measured at HV = 120 V |
---|---
$I_{\textnormal{\it preamp}}$ [$\mu$A] | 7 | 20 | 50 | 150 |
Efficiency DUT0 [%] | $96.1_{-1.7}^{+1.4}$ | $99.75_{-0.17}^{+0.12}$ | $99.94_{-0.05}^{+0.03}$ | $99.91_{-0.08}^{+0.05}$ |
Efficiency DUT1 [%] | $98.4_{-0.4}^{+0.3}$ | $99.45_{-0.2}^{+0.2}$ | $99.86_{-0.07}^{+0.05}$ | $99.78_{-0.11}^{+0.08}$ |
Table 2: Efficiency of the two DUTs at different $I_{\textnormal{\it preamp}}$ for $HV=$120\text{\,}\mathrm{V}$$. The efficiency is measured according to the definition given in the text. The uncertainties are statistical only. Figure 10:  Efficiency vs. discrimination threshold measured within the triangular inter-pixel zone for DUT0 (in red) and DUT1 (in blue). Figure 11:  Efficiency vs. sensor bias voltage for the two DUTs at $I_{\textnormal{\it preamp}}$ = 150 $\mu$A and voltage threshold of 6 $\sigma_{V}$. The vertical error bars show the statistical uncertainties. Efficiency measured at $I_{\textnormal{\it preamp}}=150\mu A$ |
---|---
HV [V] | 80 | 100 | 120 | 140 | 160 |
Efficiency DUT0 [%] | $98.97_{-0.46}^{+0.35}$ | $99.78_{-0.26}^{+0.14}$ | $99.91_{-0.08}^{+0.05}$ | $99.81_{-0.71}^{+0.18}$ | $99.95_{-0.18}^{+0.04}$ |
Efficiency DUT1 [%] | $96.70_{-0.70}^{+0.61}$ | $99.21_{-0.37}^{+0.28}$ | $99.78_{-0.11}^{+0.08}$ | $100._{-0.42}^{+0.00}$ | $99.88_{-0.20}^{+0.09}$ |
Table 3: Efficiency at different HV values for the two DUTs for
$I_{\textnormal{\it preamp}}$ = 150 µA. The efficiency is measured according
to the definition given in the text. The uncertainties are statistical only.

The efficiency measurement of a sensor HV scan carried out for the working
point $I_{\textnormal{\it preamp}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro
A}$$ is reported in Figure 11 and Table 3. The scan shows that the two DUTs
are in the efficiency plateau at 120 V. At the
$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$ substrate
resistivity of this prototype, increasing the HV from $120\text{\,}\mathrm{V}$
to $160\text{\,}\mathrm{V}$ increases the depletion depth by 15%, which has
little or no contribution for the working point at the highest power
consumption, but could have been beneficial for the front-end operation at
$7\text{\,}\mathrm{\SIUnitSymbolMicro A}$ for which a small drop in efficiency
was observed (Figure 9).

###  4.4 Time resolution measurement

For the time resolution measurement, the pixels OA0 of the two DUTs were
carefully aligned with respect to the beamline and events in which the two
pixels registered signals with amplitudes above a discrimination threshold of
$6~{}\sigma_{V}$ in coincidence were selected. Furthermore, the telescope-
track quality selection described in Section 4.1 and the cross-talk selection
described in Section 4.2 were applied. To avoid biasing the sample with a
geometrical selection, no requirement on the telescope-track position was
imposed.

Figure 12:  Distributions of the difference in TOA between the two DUTs vs.
the inverse of the amplitude that was used for the time-walk correction of
DUT0 (left) and DUT1 (right). Both DUTs were operated at $I_{\textnormal{\it
preamp}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$$ and
$HV=$160\text{\,}\mathrm{V}$$. The time-walk correction points (in red) were
obtained by a Gaussian fit on each bin of the inverse of the amplitude. The
red segments show the linear interpolation between the time-walk correction
points used to correct the data. The TOA difference contains an arbitrary
offset that is irrelevant for the measurement of the time resolution.

Time-walk correction

Figure 12 shows the difference in the Time-Of-Arrival (TOA) measured in pixels
OA0 of DUT0 (TOA0) and DUT1 (TOA1) as a function of the inverse of the signal
amplitude in DUT0 (left) and DUT1 (right) for the working point at
$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and $HV=$160\text{\,}\mathrm{V}$$.
The data show a large variation of the average of the difference TOA0$-$TOA1
as a function of the signal amplitudes, of the order of a few hundreds ps,
that was corrected in the following way.

The data were divided in variable-size bins of the inverse of the amplitude
containing at least 200 entries. For each of these bins for DUT0 in Figure 12
left, the most probable value of TOA0-TOA1 was obtained by a Gaussian fit (red
points in the figure). That value was associated to the average value of the
inverse of the DUT0 amplitude distribution within that bin (instead than to
the center of the bin). An event-by-event correction was then applied to the
inverse of the amplitude of the signal in DUT0, using the value provided by
the linear interpolation (red segments in the figure) of the two adjacent
time-walk correction points. Once DUT0 was time-walk corrected in this way,
the entire procedure was repeated for DUT1 (shown in Figure 12 right) to
complete the time-walk correction555Given its importance for this measurement,
the time-walk correction was performed also with an unbinned maximum-
likelihood fit. This second method was used for a simultaneous extraction of
the resolution parameter $\sigma_{\textnormal{\it TOA0-TOA1}}~{}$and of the
two time-walk correction functions for DUT0 and DUT1. For all the data samples
analysed, the results were within few percent from those obtained by the
method described in the text..

Extraction of the time resolution

Once data were corrected for time walk, Gaussian fits were performed to the
TOA0-TOA1 distributions, including only bins containing more than 25% of the
entries at the maximum of the distribution. It was then assumed that the two
DUTs have the same resolution, so that the time resolution of each DUT can be
estimated as $\sigma_{t}=\sigma_{\textnormal{\it TOA0-TOA1}}/\sqrt{2}$.

As an example, Figure 13 shows the resulting TOA difference distribution after
time-walk correction for the data acquired at the working points
$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$ and $HV=$160\text{\,}\mathrm{V}$$
(left) and 7 $\mu$A and 120 V (right). In the case of the former, that is the
best working point for time resolution, the standard deviation obtained by the
Gaussian fit is measured to be $\sigma_{\textnormal{\it TOA0-TOA1}}=(51.4\pm
1.1)$ ps. Therefore the time resolution of each DUT is estimated to be

$\sigma_{t}=\frac{\sigma_{\textnormal{\it TOA0-TOA1}}}{\sqrt{2}}=(36.4\pm
0.8)$\mathrm{ps}$.$ (4.1)

Figure 13:  TOA difference between pixels OA0 of DUT0 and DUT1 after time-walk
correction for the two working points reported in the panels. A constant
arbitrary offset is present, which is irrelevant for the time-resolution
calculation. The red lines show the results of the Gaussian fit using only the
bins with more than 25% of the entries in the maximum of the distribution. The
full red lines show the ranges used for the fits, while the dashed red lines
allow the estimation of the non-Gaussian components in the tails.

The fraction of events exceeding the Gaussian fit in the tails of the
distribution of Figure 13 is approximately 5%. This fraction of events
represents the typical non-Gaussian component found in the tails of the time-
resolution distribution for all the data sets acquired at the testbeam at
different sensor and preamplifier bias. As a consequence the resolutions
quoted in the following refer to 95% of the signals acquired by the DUTs.

Figure 14 top shows the time resolution as a function of the HV for the
highest power consumption working point $I_{\textnormal{\it
preamp}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$$. The time resolution
varies between 60 and 36 ps with the HV between 80 and 160 V. At
$HV=$120\text{\,}\mathrm{V}$$ the timing performance is approximately 20%
worse than the one measured at 160 V.

Figure 14 bottom shows the time resolution measured at
$HV=$120\text{\,}\mathrm{V}$$ for the four $I_{\textnormal{\it preamp}}$
working points. As expected, the time resolution depends on the preamplifier
current. A significant degradation of the performance is observed for the
lowest power-consumption working point studied $I_{\textnormal{\it preamp}}=$
$7\text{\,}\mathrm{\SIUnitSymbolMicro A}$, for which the time resolution still
remains at the level of 200 ps.

Figure 14:  Top: time resolution as a function of sensor bias voltage at
$I_{\textnormal{\it preamp}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$$.
Bottom: time resolution as a function of $I_{\textnormal{\it preamp}}$ for
sensor bias voltage $HV=$ 120 $V$. The time resolution is defined as
$(\sigma_{\textnormal{\it TOA0-TOA1}})/\sqrt{2}$. It refers to the Gaussian
component of the data, which is approximately 95% of the total.

##  5 Conclusions

A monolithic silicon detector prototype with 100 µm pixel pitch was produced
in 130 nm SiGe BiCMOS technology on
$50\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$ wafers and tested
at a beamline with minimum-ionizing particles. The analysis of the data
acquired with analog channels using a threshold of $6~{}\sigma_{V}$ shows that
this detector technology can achieve efficiencies up to 99.9%. This result
proves that the low ENC reachable with SiGe HBTs allows operating at very high
efficiency even with a thin depletion layer of approximately 20 µm. The small
drop in efficiency observed at the lowest amplifier current studied, caused by
an increase of the ENC, could be compensated by increasing the sensor
depletion depth. Therefore, further optimization of this detector technology
is possible by adopting higher resistivity substrates to obtain full depletion
of the sensitive volume.

Results with a previous prototype [12] that provided uniquely the TOT
measurement to perform the time-walk correction, showed that a remarkable
timing performance at the level of 50 ps was possible only at large
discrimination threshold, therefore compromising the detection efficiency. The
data presented here demonstrate that a time-walk correction that uses the
signal amplitude instead of the TOT can provide an even better timing
performance at the lowest discrimination threshold permitted by the ENC of the
amplifier: when operated at a preamplifier current as low as
$20\text{\,}\mathrm{\SIUnitSymbolMicro A}$ the SiGe HBT amplifier implemented
in the ASIC provides a time resolution better than $100\text{\,}\mathrm{ps}$
and an efficiency of 99.6%; at a preamplifier current of
$150\text{\,}\mathrm{\SIUnitSymbolMicro A}$ a time resolution of
$36\text{\,}\mathrm{ps}$ and an efficiency of 99.9% are measured, even without
the support of an avalanche gain layer to boost the signal-to-noise ratio.
These results prove that this technology is suitable for applications that
require the combination of tracking capabilities with excellent time
resolution.

## Acknowledgments

The authors wish to thank Alexander Gebershagen and the CERN SPS H8-beamline
team for the support and coordination during the experimental activities, as
well as the technical staff of the DPNC of the University of Geneva. This
research would not have been possible without the support of the H2020
programme through the ERC Advanced MONOLITH project (grant agreement ID
884447) and the ATTRACT Phase1 MonPicoAD project, as well as of the Swiss
National Science Foundation grant number 200020_188489.

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††institutetext:  1Physics Department, Indiana University, Bloomington, IN
47405, USA

# 2$b$ or not 2$b$

Beni Pazar1 Enrico Lunghi1<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by new physics models which lead to final states containing a high
multiplicity of bottom and top quarks; we develop a tagging strategy to
suppress reducible and non-reducible multi-jet backgrounds. The idea takes
advantage of the properties of light parton showers and of the gluon
fragmentation into heavy quarks to reject jets that do not originate from a
bottom quark.

##  1 Introduction

Many extensions of the Standard Model (SM) involve heavy new particles with
couplings to the third generation of SM fermions and yield final states with
multiple bottom and top quarks. When feasible, experimental searches require
the presence of high transverse momentum leptons and/or missing energy in
order to suppress backgrounds or the use of data driven background estimation
techniques. In this paper, we focus on techniques to suppress backgrounds to
new physics signals with final states involving multiple third generation
quarks (i.e. combinations of $b$ and hadronic $t$ quarks).

For instance, in models with extra Higgs Bosons and vectorlike quarks, cascade
decays involving Higgses and vectorlike quarks can lead to final states in
which the two $b$-quarks do not form a resonance (e.g. $pp\to H\to
b_{4}\bar{b}\to Zb\bar{b}\to\mu^{+}\mu^{-}b\bar{b}$) Dermisek:2016via ;
Dermisek:2019heo  and to purely hadronic final states involving a large number
of $b$ and $t$-quarks (e.g. $pp\to b_{4}\bar{b}_{4}\to hb\;h\bar{b}\to
b\bar{b}b\bar{b}b\bar{b}$) Dermisek:2020gbr . In both cases the dominant
backgrounds are multi-jets in which low-mass light-flavored jets are mistagged
as originating from a $b$ in the hard interaction.

The ability to isolate jets containing $b$-quarks is important for studies of
QCD (e.g. heavy flavor production), of Higgs and top physics, and of new
physics models which often couple to the third generation only (to avoid the
strong constraints on couplings to the light quarks). More recently, there has
been an increased interest in heavy resonances produced at large transverse
momentum and whose decay products are, therefore, emitted in a narrow cone
(i.e. with low relative $\Delta R$111$\Delta R$ between two 4-momenta is
defined as $\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$, where $\eta$ and $\phi$
are pseudorapidity and azimuthal angle of the two 4-momenta, respectively.).
The identification of these fat jets with a quite complicated substructure has
spurred the development of dedicated tagging strategies. In particular, ATLAS
has developed taggers aimed precisely at the identification of $X\to b\bar{b}$
resonances ATLAS:2012xna ; ATLAS:2022qpg .

Our interest lies not much in resonances decaying to collimated $b\bar{b}$
pairs or $b+\text{jets}$ (like in decays of high-$p_{T}$ top quarks), but in
distinguishing pure b-jets originating from a $b$ quark produced directly in
the partonic interaction (which we refer to as 1b jets) from jets in which $b$
quarks appear in the parton shower of lighter partons (which we refer to as 2b
jets). Previous theoretical investigations of this issue have been presented
in ref. Goncalves:2015prv  (where the focus was on distinguishing $b$ vs $g\to
b\bar{b}$ jets) and ref. Bhattacherjee:2016bpy  (in the context of searches
for gluinos at LHC).

Our work builds on the analysis of ref. Goncalves:2015prv , discuss in detail
the impact of the parton shower in the properties b-tagged jet not originating
from a partonic $b$ quark while also including detector effects.

The paper is organized as follows. In section 2 we discuss the present status
of $b$-tagging and its implementation in Delphes (a simplified detector
simulation); in sections 3 and 4 we present an in-depth analysis of the
various jet sub-structure observables that we consider and proceed to the
construction of the 1b2b tagger; in section 5 we show an example of the
performance of the 1b2b tagger in multi-jet events; finally, in section 6 we
present our conclusions.

##  2 Preliminary considerations

In this section we discuss the current status of b-tagging at ATLAS and CMS.
We first review their standard approach and then describe recent developments
aimed at identifying boosted resonances that decay to pairs of $b$ quarks.

ATLAS approach is based on the MV2 and DL1 algorithms ATLAS:2019bwq . MV2 is a
boosted decision tree that combines the outputs of multiple lower level
tagging algorithms. The lower level algorithms include kinematics, impact
parameters, secondary vertex finders, and the topological multi-vertex finder
JETFITTER ATLAS:2018nnq . DL1, in contrast, is a deep feed-forward neural
network trained on the same lower level algorithms along with the JETFITTER
c-tagging variables. The two have relatively similar efficiencies when
distinguishing between $b$/$c$-jets and $b$/light-jets. At a working point of
$85\%$ signal efficiency they misidentify approximately $38\%$ of $c$-jets and
$3\%$ of light-jets. These algorithms define the flavor of jets in a
hierarchical manner: first looking to see if there are any b-hadrons among the
jet constituents, then $c$-hadrons, otherwise labeling the jet as “light”.
This method does not distinguish between b-hadrons originating from $b$-quarks
produced in the hard interaction and those which come from gluon splitting
ATLAS:2012xna ; ATLAS:2015dex ; ATLAS:2017bcq . This broad classification
leaves the b-taggers completely blind to the origin of the tagged $b$-quark.
JETFITTER uses the variable $f_{E}$ which is defined as the fraction of the
charged jet energy in all secondary vertices. With this definition, $f_{E}$
peaks near $1$ for both 1b and 2b jets (for the latter, this is only the case
as long as the splitting happens early enough).

CMS has multiple approaches to $b$-tagging CMS:2012feb ; CMS:2017wtu . In
particular jets containing $b$-hadrons are identified at CMS used the Combined
Secondary Vertex (CSV) algorithm (presented in ref. CMS:2012feb  for Run 1 and
updated in ref. CMS:2017wtu  for Run 2), which combines flight distance of the
secondary vertex with track based lifetime information (if no secondary
vertices are found). On top of this, CMS has a version of this tagger based on
a deep neural network (DeepCSV) which yields a slight increase in performance.

Early studies of jets containing multiple $b$-hadrons have been performed at
Tevatron (see, for instance, the CDF measurement ref. CDF:2004mmv ), were
based on direct reconstruction of secondary vertices and had limited success
to the difficulty of identifying and reconstructing two $b$-hadrons (double
$b$-tag). The ATLAS and CMS collaborations performed studies focused on
tagging $b\bar{b}$ jets both in the context of gluon splitting and of decays
of heavy Higgs like resonances to $b\bar{b}$. In both cases, the focus was to
identify $b\bar{b}$ at relatively low transverse momentum ($p_{T}\lesssim
200\;{\rm GeV}$).

A first approach adopted by ATLAS considers jet substructure variables and
used the Toolkit for Multivariate Data Analysis (TMVA) Hocker:2007ht  to
produce a discriminator based on track-jet width, jet width maximum track
$\Delta R$, charged track multiplicity, girth (see eq. (7)), and
n-subjettiness Thaler:2010tr . The jet $p_{T}$ analyzed lie in the $[40,480]$
GeV range. At a signal efficiency of $80\%$ this tagger keeps between
$20\%-30\%$ of 2b background ATLAS:2012xna . More recently, ATLAS developed
DeXTer (Deep set $X\to b\bar{b}$ Tagger) ATLAS:2022qpg  which uses
reconstructed tracks and vertices in a Deep Set Neural Network. The range is
$p_{T}\in[20,200]$GeV and, at a working point of $80\%$, keeps approximately
$20\%-25\%$ of background. Additionally, ATLAS developed tagging strategies
aimed exclusively at identifying heavy color neutral resonances which decay to
$b\bar{b}$, for which gluon splitting to $b\bar{b}$ provide the dominant
background. In ref. atlas:2023 , ATLAS proposed a novel algorithm, GN2X, that
supersedes previous incarnations of this strategy atlas:2020 ; atlas:2021  and
that is based on a transformer neural network.

In ref. cms:2015 ; CMS:2017wtu , CMS presented an analysis of b-tagging in
boosted jets based on kinematics and independent tagging of both $b$ quarks
and aimed at the identification of resonances decaying to $b\bar{b}$ (in
particular $H$ and $Z$).

Figure 1: Impact of $b$-tagging on 4-jet events originating from an inclusive $pp\to 4g$ hard interaction. The ATLAS vs CMS tagging efficiencies have been modeled within Delphes by rescaling the event weights by the efficiency factors given in Eqs. (1)–(3). The statistics of the tagged events is degraded with respect to the pre-$b$-tag distribution because the efficiencies for light jets are much smaller than those for $c$ and $b$ jets (the peaks correspond to the $O(100)$ events in which each of the jets contains an actual $b$ quark originating from gluon splitting). $p_{T}$ | Btag | f | Btagf | Btagf/Btag | prob jetf | Btag prob of jetf
---|---|---|---|---|---|---
|  | b | 0.0062 | 0.27 | 0.012 | 0.52
20-50 GeV | 0.023 | c | 0.0062 | 0.27 | 0.048 | 0.13
|  | l | 0.010 | 0.46 | 0.94 | 0.011
|  | b | 0.013 | 0.34 | 0.019 | 0.68
50-100 GeV | 0.037 | c | 0.013 | 0.34 | 0.061 | 0.21
|  | l | 0.012 | 0.32 | 0.92 | 0.013
|  | b | 0.020 | 0.37 | 0.035 | 0.57
100-200 GeV | 0.054 | c | 0.018 | 0.34 | 0.085 | 0.21
|  | l | 0.016 | 0.29 | 0.88 | 0.018
|  | b | 0.025 | 0.40 | 0.04 | 0.63
200-500 GeV | 0.062 | c | 0.019 | 0.31 | 0.10 | 0.19
|  | l | 0.018 | 0.29 | 0.86 | 0.021
|  | b | 0.025 | 0.35 | 0.052 | 0.48
0.5-1 TeV | 0.072 | c | 0.018 | 0.25 | 0.13 | 0.14
|  | l | 0.029 | 0.40 | 0.82 | 0.035
|  | b | 0.019 | 0.24 | 0.07 | 0.27
1-1.5 TeV | 0.081 | c | 0.017 | 0.21 | 0.16 | 0.11
|  | l | 0.045 | 0.55 | 0.77 | 0.058
|  | b | 0.015 | 0.19 | 0.077 | 0.19
1.5-2 TeV | 0.089 | c | 0.014 | 0.20 | 0.18 | 0.078
|  | l | 0.059 | 0.61 | 0.75 | 0.079
Table 1: Breakdown of CMS b-tagging for gluon initiated jets as a function of
$p_{T}$. The columns are: $p_{T}$ range of the jet($p_{T}$), total b-tagging
probability(Btag), identified flavor of the jet(f) ($l$ stands for jets not
containing bottoms or charms), probability of b-tagging a jet with flavor
f(Btagf), fraction of the total b-tagging probability coming from jets with
flavor f(Btagf/Btag), the probability of a jet having flavor f(prob jetf ),
and the b-tagging rate of jets with flavor f(Btag prob of jetf).

We now discuss the current status of implementation of the ATLAS and CMS
standard b-tagging algorithms in the detector simulator Delphes
deFavereau:2013fsa . For both experiments, the tagging efficiencies are
modeled following the results presented in refs ATLAS:2015dex  and CMS:2012feb
by ATLAS and CMS, respectively. The $p_{T}$ dependent efficiency functions for
jets tagged as bottom, charm and light are:

$\displaystyle\varepsilon_{\rm light}$
$\displaystyle=\begin{cases}0.01+3.8\times 10^{-5}\;p_{T}&\text{CMS}\\\
0.002+7.3\times 10^{-6}\;p_{T}&\text{ATLAS}\end{cases}$ (1)
$\displaystyle\varepsilon_{\rm charm}$
$\displaystyle=\begin{cases}0.25\;\frac{\tanh(0.018\;p_{T})}{1+0.0013\;p_{T}}&\text{CMS}\\\
0.20\;\frac{\tanh(0.02\;p_{T})}{1+0.0034\;p_{T}}&\text{ATLAS}\end{cases}$ (2)
$\displaystyle\varepsilon_{\rm bottom}$
$\displaystyle=\begin{cases}0.85\;\frac{25\;\tanh(0.0025\;p_{T})}{1+0.063\;p_{T}}&\text{CMS}\\\
0.80\;\frac{30\;\tanh(0.003\;p_{T})}{1+0.086\;p_{T}}&\text{ATLAS}\end{cases}$
(3)

As an example, in figure 1 we show the impact of requiring 4 $b$-tagged jets
on process originating from the inclusive $pp\to 4g$ hard interaction. We used
MadGraph5 Alwall:2011uj  for parton level generation, Pythia8 Sjostrand:2006za
; Sjostrand:2014zea  for parton shower and hadronization, and Delphes for
detector simulation. Jets are reconstructed using the anti-$k_{t}$ algorithm
with $\Delta R=0.5$. For both CMS and ATLAS the fraction of events that
survive is about $0.05\%$. In this simulation we do not actually impose the
stochastic $b$-tagging but rescale the weight of each event using the
efficiencies in Eqs. (1)–(3) for CMS and ATLAS.

In table 1 we present the $b$-tagging anatomy of a single jet produced with
$p_{T}\in[20\;{\rm GeV},2\;{\rm TeV}]$ and originating from a hard gluon. The
flavor ($f$) of the jet is obtained by looking at the event prior to
hadronization according to whether a $b$ or $c$ quark is found anywhere in the
shower. The $b$-tag probability given in the last column is obtained directly
from Eqs. (1)–(3) for CMS. The probability of observing a jet with a given
flavor depends exclusively on the details of the parton shower as handled by
Pythia8. An important feature of these rates is the increased
misidentification at high $p_{T}$, driven by mistagged light jets. Another
interesting aspect of these results is that, at moderate $p_{T}$, $b$-tagged
gluon jets are equally likely to originate from jets with showers containing
$b$-quarks, $c$-quarks or no heavy quarks at all: the small $b$-tagging
probabilities of charm and light jets are compensated by their significantly
larger production rates.

##  3 Observables

Following ref. Goncalves:2015prv  we focus on three observables: the ratio of
the energy of the $b$-tagged hadron to the total jet energy ($X_{E}$), the
transverse size of the jet (girth, $g$) and the number of charged tracks in
the jet ($N_{\rm ch}$).

$X_{E}$ is a key discriminator due to the significant difference in the
$b$-quark and gluon fragmentation functions. The mass of the $b$-quark greatly
suppresses collinear gluon radiation, thus also suppressing secondary partons.
This phenomenon is known as the “dead cone effect”. $b$-quarks tend to retain
the majority of their initial energy, leading to an average $X_{E}\sim 0.8$.
In contrast, gluons tend to radiate a considerable fraction of their energy
(larger at higher $p_{T}$) prior to the $b\bar{b}$ splitting. The latter tend
to be produced with similar energy (though asymmetric splitting can and does
occur). Both effects tend to lower the energy in the leading $b$-quark found
in a gluon shower and the resulting $X_{E}$ is considerably smaller than
typical values observed in $b$-quark initiated jets. Note that, in order to
distinguish between 1b and 2b jets, the variable $X_{E}$ is much more
appropriate than $f_{E}$ (which sums over all displaced vertices) because the
latter peaks near 1 for both types of jets.

In order to verify the above intuitive arguments, we can start from the
theoretically predicted distribution of $b\bar{b}$ pairs in a jet as a
function of the invariant masses of the jet ($m_{j}^{2}$) and of the gluon
($k^{2}$) which splits to $b\bar{b}$. This is given by Mueller:1985zz ;
Mangano:1992qq ; ellis_stirling_webber_1996 :

$\displaystyle R_{b\bar{b}}(m_{j}^{2},k^{2})$
$\displaystyle=\frac{1}{6\pi}\alpha_{s}(k^{2})\left[1+\frac{2m_{b}^{2}}{k^{2}}\right]\sqrt{1-\frac{4m_{b}^{2}}{k^{2}}}n_{g}(m_{j}^{2},k^{2})$
(4) $\displaystyle n_{g}(m_{j}^{2},k^{2})$
$\displaystyle=\left[\frac{\log(m_{j}^{2}/\Lambda^{2})}{\log(k^{2}/\Lambda^{2})}\right]^{a}\cosh\left\\{\sqrt{\frac{2C_{A}}{\pi
b}}\left[\sqrt{\log(m_{j}^{2}/\Lambda^{2})}-\sqrt{\log(k^{2}/\Lambda^{2})}\right]\right\\}$
(5)

where $a=-1/4-5n_{f}/(54\pi b)$, $b=(33-2n_{f})/(12\pi)$ and
$\alpha_{s}(k^{2})=1/[b\log(k^{2}/\Lambda^{2})]$. This distribution allows us
to build a quantity that should resemble $X_{E}$ by using it to weight the
average energy of the $b$-quarks produced in the $g\to b\bar{b}$ splitting:

$\displaystyle X_{E}(m_{j}^{2})$
$\displaystyle=\int_{4m_{b}^{2}}^{m_{j}^{2}}\frac{dk^{2}}{k^{2}}R_{b\bar{b}}(m_{j}^{2},k^{2})E_{b}^{\rm
av}(k^{2})\;.$ (6)

After extracting the relation between gluon virtuality and average $b$-quark
energy from Monte Carlo simulations, we plot the resulting $X_{E}$ in figure
2. The blue points in the figure are obtained by selecting jets in which a
$b\bar{b}$ pair appears in the shower and calculating the ratio of the average
energy of the $b$-quarks and of the jet energy. The agreement between the
rough theoretical calculation and the Monte Carlo results is quite
satisfactory. In the figure we also show that forcing the $g\to b\bar{b}$
split in the hard interaction (before the parton shower) yields values of
$X_{E}$ very close to 0.5 (black squares).

The construction of $X_{E}$ for a generic jet is done as follows. If a $b$ or
a $\bar{b}$-quark is found anywhere in the shower, we seek the highest energy
$b$-hadron in the generator level constituents of the jet and use its energy
to construct $X_{E}$. When a $c$-quark is found instead, we seek the highest
energy $c$-hadron, assume that the displaced vertex selected by the $b$-tagger
corresponds to this hadron decay and use its energy to calculate $X_{E}$. If
no heavy quarks are found, we simply construct $X_{E}$ by combining the
energies of five randomly chosen charged tracks selected among the jet
constituents.

Figure 2: Comparison between $X_{E}$ calculated using Eq. (6) (solid line) and
extracted from Monte Carlo simulations. Black squares are values of $X_{E}$
obtained from $g\to b\bar{b}$ prompt events (i.e. no gluon radiation is
allowed prior to splitting). The blue circles correspond to gluon initiated
jets. In order to compare directly to the theoretical estimate we require both
$b$ and $\bar{b}$ to be found in the shower and we calculate $X_{E}$ using the
average of the $b$ and $\bar{b}$ energies.

Figure 3: $X_{E}$ distributions for gluon jets in which a $b\bar{b}$ pair is
produced during the shower. In both cases, we require the decay products of
the two $b$-hadrons to be among the jet constituents. The distributions are
obtained by either imposing the $g\to b\bar{b}$ splitting directly in the hard
scattering (gray) or allowing the splitting to be controlled by the parton
shower (blue). These distributions correspond to blue and gray points in
figure 2.

In figures 3-6 we present various $X_{E}$ distributions which we obtain by
varying the hard interaction (i.e. whether the $g\to b\bar{b}$ splitting is
prompt or not) and the composition of the jet (whether we allow jets with
$n=0,1,2$ $b$-quarks). The purpose of this analysis is to highlight the
differences between the observables studied in Ref. Goncalves:2015prv  with
respect to the setup considered in this work.

In Figure 3 the gray and blue histograms correspond to the gray and blue
points in figure 2. Distributions for gluon jets in which the splitting to
$b\bar{b}$ is forced prior to the parton shower peak at $X_{E}\sim 0.5$ and
can be compared directly to those presented in ref. Goncalves:2015prv . When
the $g\to b\bar{b}$ is not imposed in the hard scattering and is controlled by
the parton shower, the $X_{E}$ distributions shift to lower values in quite
good agreement with the theoretical expectation shown in figure 2.

In figure 4 we look at gluon initiated jets in which the $b\bar{b}$ splitting
happens at some point in the shower (i.e. at least one bottom quark is found
in the cone of the jet but no restriction is imposed on $\Delta
R_{b\bar{b}}$). Blue and green histograms correspond to jets with exactly one
or two $b$ quarks in a $\Delta R<0.5$ cone around the jet direction. At
relatively low $p_{T}$ ($p_{T}\sim(50,100)\;{\rm GeV}$) we find the expected
peak at $X_{E}\sim 0.5$. At large $p_{T}$ (right panel) the large amount of
gluon radiation, described at leading order by the $n_{g}$ distribution in Eq.
(5), lowers $X_{E}$ and the peak almost disappears.

Figure 4: $X_{E}$ distributions for b-tagged flavor 5 gluon initiated jets
(i.e. jets that originate from a gluon, are b-tagged and contain an actual
$b$-quark at some point in the parton shower). Blue and green histograms
correspond to events in which only one and both $b$ quarks are found in the
shower. Figure 5: $X_{E}$ distribution in gluon initiated jets based on
flavor. The distributions are dependent on $p_{T}$, but for all energies the
events which pass b-tagging contain proportional amounts of flavors 5,4, and
0. See the text for further details.

In Figure 5 we illustrate the effects of $b$-tagged gluon jets which do not
contain $b$ quarks. We present $X_{E}$ distributions labeled according to
whether $b$’s (flavor 5), $c$’s (flavor 4) or no heavy quarks (flavor 0) are
found in the shower. Upper panels are prior to $b$-tagging and clearly show
the dominant production of flavor 0 jets. The lower panels, in contrast, are
after $b$-tagging. For every $p_{T}$ range, we see that each flavor
contributes roughly equal amounts to the final distribution, with flavor 5
jets yielding larger $X_{E}$ at low $p_{T}$ (as discussed above). These
distributions help explain why the $X_{E}$ distributions we find yield values
of $X_{E}$ that are much lower than the blue points in figure 2.

Finally, in figure 6 we show an example of the final distributions that we
obtain for bottom and gluon initiated jets. The former peak at $X_{E}\sim 0.8$
while the latter peak at $X_{E}\lesssim 0.2$ without any more trace of the
peak at $X_{E}\sim 0.5$.

Figure 6: $X_{E}$ distributions in gluon and bottom initiated jets for
$p_{T}>50$ GeV and $p_{T}>1$ TeV.

The two other discriminators we consider, girth ($g$) and charged track
multiplicity ($N_{\rm ch}$), are calculated in standard fashion. Girth is the
sum of the transverse momentum fractions carried by each jet constituent
($p_{T}^{(i)}/p_{T}$) weighted by its angular distance ($\Delta R_{iJ}$) from
the jet axis:

$\displaystyle g=\sum_{i\in{\rm jet}}\frac{p_{T}^{(i)}}{p_{T}}\Delta
R_{iJ}\;.$ (7)

Gluon initiated jets are expected to have girth distributions that are wider
than those for bottom initiated jets. In fact, gluons tend to radiate more
than $b$-quarks due to differences in color charges ($N_{A}=3$ vs $C_{F}=4/3$)
and collinear radiation for the latter is limited by the heavy quark mass. The
same argument leads to larger values of $N_{\rm ch}$ for gluon jets.

##  4 Construction of the 1b2b Tagger

We consider Monte Carlo samples containing gluon or $b$-quark initiated jets
with $p_{T}$ between 20 GeV and 2 TeV. Events have been generated as discussed
at the end of section 2. The three jet sub-structure observables that we
consider ($X_{E}$, $g$ and $N_{\rm ch}$) are calculated as detailed in section
3.

We build the effective tagging variable using a Logistic Regression, a
statistical model whose output is the probability of an event occurring
logistic-regression-hosmer . We chose this strategy over a straightforward
Principal Component Analysis (PCA), because the latter is highly dependent on
the variance of the input variables. This requires the data to be standardized
to a fixed mean and standard deviation. Standardization is especially
important since girth and charged track multiplicity differ dramatically in
scale. On the other hand, the logistic regression approach has the advantage
of being able to take discrete and continuous inputs which can posses very
different scales and variances. This is especially important in our case
because $N_{\rm ch}$, in particular, has no typical scale compared to $X_{E}$
and girth.

Logistic regression algorithms take as input triplets of $(X_{E},g,N_{\rm
ch})$ for both signal and background event files. The inputs are combined into
a single discriminant
$\beta=\beta_{0}+\beta_{X_{E}}X_{E}+\beta_{g}g+\beta_{N_{\rm ch}}N_{\rm ch}$
which is then fed to the sigmoid function $\sigma(x)=1/(1+e^{-x})$. An
optimization strategy is then used to find a set of coefficients
$(\beta_{0},\beta_{X_{E}},\beta_{g},\beta_{N_{\rm ch}})$ that achieve maximum
separation of background and signal events ($\sigma(\beta)$ near 0 and 1,
respectively). We used the logistic regression as implemented in the Python
package scikit-learn scikit-learn  and adopted the Limited-memory
Broyden–Fletcher–Goldfarb–Shanno algorithm. The fact that the log-odds
(logarithm of the inverse of the sigmoid) is simply a linear combination of
the input observables means that the regression is easy to train and
implement. And since the coefficients are directly tied to each parameter it’s
easy to interpret the importance of each parameter. Despite the simplicity of
the logistic regression, it is a strong classifier.

Since we expect the dominant variable to be $X_{E}$ we present the linear
combination of the input parameters selected by the logistic regression as:

$\displaystyle X_{E}^{\rm eff}=X_{E}+\alpha_{\rm ch}\;N_{\rm
ch}+\alpha_{g}\;g$ (8)

The cut-off values for this variable are constructed such that for $X_{E}^{\rm
eff}>X_{E}^{\rm cut}$ the jet is positively tagged (i.e. is identified as
originating from an initial $b$ quark). The efficiency rate for $b$-quark
initiated jets and the mistag rate for gluon initiated jets are indicated as
$\varepsilon_{b}$ and $\bar{\varepsilon}_{g}$, respectively.

In table 2 we present the coefficients of $X_{E}^{\rm eff}$ and the cut-off
values corresponding to various signal efficiencies $\varepsilon_{b}$ for
various jet $p_{T}$ ranges. In table 3, which is one of the main results of
this analysis, we present the corresponding gluon mistag rates
$\bar{\varepsilon}_{g}$. These results are presented graphically in figure 7.
The top two panels present the dependence of the signal tag and background
mistag rates as a function of $p_{T}$. The bottom panel shows the Receiver
Operating Characteristic (ROC) curves, namely the tag vs mistag rates of $b$
and $g$ initiated jets.

Details about the one and two dimensional distributions of the input variables
$X_{E}$, g, and $N_{\rm ch}$ for different jet transverse momenta are
presented in Appendix A (figures 10-16). From these distributions it is clear
that $X_{E}$ is the dominant discriminator and that the tagger becomes more
powerful at larger values of the jet $p_{T}$.

$p_{T}$ range | $p_{T}^{\rm avg}$ | $\varepsilon_{b}=0.6$ | $\varepsilon_{b}=0.7$ | $\varepsilon_{b}=0.8$ | $\varepsilon_{b}=0.9$ | $\alpha_{\rm ch}$ | $\alpha_{g}$
---|---|---|---|---|---|---|---
20–50 GeV | 33 GeV | 0.817 | 0.766 | 0.699 | 0.584 | 0.00581 | 0.416
50–100 GeV | 69 GeV | 0.747 | 0.692 | 0.615 | 0.488 | 0.00309 | -0.038
100–200 GeV | 134 GeV | 0.652 | 0.581 | 0.483 | 0.329 | -0.00061 | -0.463
200–500 GeV | 263 GeV | 0.551 | 0.469 | 0.356 | 0.182 | -0.00466 | -0.750
0.5–1 TeV | 0.61 TeV | 0.482 | 0.391 | 0.269 | 0.114 | -0.0059 | -0.084
1–1.5 TeV | 1.15 TeV | 0.490 | 0.404 | 0.287 | 0.126 | -0.00397 | 0.309
1.5–2 TeV | 1.66 TeV | 0.481 | 0.389 | 0.275 | 0.121 | -0.00363 | 0.609
Table 2: Coefficients $\alpha_{\rm ch}$ and $\alpha_{g}$ of the logistic regression output ($X_{E}^{\rm eff}$) and cut-off values $X_{E}^{\rm cut}$ for several signal efficiency working points ($\varepsilon_{b}\in[0.6,0.9]$). $p_{T}$ range | $\varepsilon_{b}=0.6$ | $\varepsilon_{b}=0.7$ | $\varepsilon_{b}=0.8$ | $\varepsilon_{b}=0.9$
---|---|---|---|---
20–50 GeV | 0.123 | 0.16 | 0.212 | 0.321
50–100 GeV | 0.117 | 0.15 | 0.206 | 0.332
100–200 GeV | 0.104 | 0.142 | 0.216 | 0.365
200–500 GeV | 0.08 | 0.119 | 0.195 | 0.348
0.5–1 TeV | 0.057 | 0.094 | 0.157 | 0.265
1–1.5 TeV | 0.038 | 0.058 | 0.102 | 0.193
1.5–2 TeV | 0.024 | 0.041 | 0.072 | 0.143
Table 3: Mistag rate for gluon initiated $b$-tagged jets
($\bar{\varepsilon}_{g}$), for fixed signal efficiencies ($\varepsilon_{b}$).

Figure 7: Tag and mistag rates as a function of $p_{T}$. Top left panel:
signal tag rate ($\varepsilon_{b}$) as function of $p_{T}$ for various values
of the background rejection rate ($0.01\leq\bar{\varepsilon}_{g}\leq 0.5$).
Top right panel: background mistag rate ($\bar{\varepsilon}_{g}$) as a
function of $p_{T}$ for various values of the background rejection rate
($0.6\leq\varepsilon_{g}\leq 0.9$). Lower panel: signal tag rate as a function
of the background mistag rate for various values of $p_{T}\in[20,2000]\;{\rm
GeV}$.

##  5 Application of the 1b2b Tagger to multi-jet final states

An important application of the tagger we are proposing is the suppression of
SM background to new physics signals with multiple (typically more than four)
prompt $b$-quarks. Under these circumstances requesting the presence of
multiple 1b2b tagged jets (which have all been previously b-tagged), yields
extremely tiny survival probabilities. For instance, starting with a
combination of four jets events originating from a combination of gluons and
light quarks with $\Delta R>0.5$ and $p_{T}>20$ GeV, the requirement of four
1b2b tagged jets reduces the cross section by a factor of about $10^{6}$.
Simulations under these conditions are clearly unmanageable. In this section
we explore how well a simple re-weighting of the events using the $p_{T}$
dependent efficiencies presented in table 3 reproduces the actual
implementation of the tagger on an event-by-event basis.

We generated a large sample ($5\times 10^{5}$ events) of
$pp\to(4b,4g,2g2q,4q)$ events at $\Delta R_{gg}>0.5$ and $p_{T}>20$ GeV
($q=u,d,s,c$). As a first step we checked whether the leading or subleading
jet 1b2b tagging is well reproduced by the efficiencies in table 3. The
results of this test are presented in the two panels of figure 8, where solid
lines are normalized distributions after b-tagging. The application of the
1b2b tagger is performed either exactly (dotted lines) or using the
efficiencies in table 3 (dashed lines). It is clear that both for the leading
and subleading jet 1b2b tagger is very accurately reproduced by use of the
efficiency table.

In the left panel of figure 9 we restrict the histograms to the $pp\to 4g$
case and impose the 1b2b requirement on leading two and four jets in each
event (left and right panels, respectively). The solid blue lines are the
normalized distributions prior to b-tagging, red lines show the impact of
b-tagging and green lines that of 1b2b-tagging. When considering the leading
two jets only (left panel), we were able to compare the exact implementations
(solid red and green lines) with the approximate one (dashed lines). It is
clear that the use of the efficiency tables perfectly simulates the 1b2b
tagger even when imposed simultaneously on two jets. In the right panel, only
the approximate tagging is displayed because the low probabilities would have
required several million events to obtain reasonable distributions.

Figure 8: Distributions of the leading(left) and subleading(right) jet
$p_{T}$. Initially with only the b-tagging from Delphes(solid). Then applying
the 1b2b tagger directly(dotted) or as a probability(dashed). All
distributions are normalized to the b-tag distribution to better show the
impact of the 1b2b tagger

The strength of the probabilistic application of the taggers is most evident
when requiring multiple jets to be tagged. It not only maintains the same
cross section as the exact taggers, but also the overall distribution of the
data. Whereas the exact tagger will no longer show the distribution when there
are too few events.

Figure 9: Update Caption 4-jets tagged 4 gluon background events with the
standard b-tagger and 1b2b tagger applied exactly or reweighted using the
probabilities of tagging. We started with close to half a million events but
see that when applying the taggers exactly we lose statistics along with the
distribution shape. We can maintain statistics and the distribution by
applying the taggers in the probabilistic manner.

##  6 Conclusions

In this paper we studied the properties of $b$-tagged jets originating from
prompt $b$ quarks, light quarks and gluons and proposed a strategy to build a
tagger able to isolate the former and reject the latter two. This work builds
on the approach of ref. Goncalves:2015prv  and is motivated by the need to
reject multi-jet backgrounds to new physics signals consisting, at the parton
level, of multiple $b$ and $t$ quarks. When top quarks are presents, search
strategies requiring leptons and missing energy are possible but suffer from
smaller branching ratios due to leptonic decays of the tops (see, for
instance, ref. Han:2018hcu ). On the other hand, for purely hadronic multi-$b$
decays (for instance, in ref. Dermisek:2020gbr , models with extra Higgses and
vectorlike fermions yield final states with up to 6 $b$ quarks), experimental
searches require a rejection of multi-jet QCD backgrounds beyond what
conventional $b$-taggers allow.

Both ATLAS atlas:2023  and CMS cms:2015  have studied tagging strategies
focused on jets with two $b$ quarks in connection to isolating heavy color
neutral resonances decaying to $b\bar{b}$ (e.g. heavy Higgses). One of the
main discriminants in this case is the fraction of the total energy found in
all secondary vertices: the propensity of gluons to radiate usually leads to
$b\bar{b}$ pairs carrying a small fraction of the total jet energy.

We focus on three jet substructure variables (girth, charged track
multiplicity and the fraction of the jet energy in the leading $b$-hadron
candidate) and show how it is possible to construct a 1b2b tagger that leads
to typical gluon rejection efficiencies in the 7–20% range (with the stronger
rejection corresponding to higher $p_{T}$ jets), while allowing roughly $80\%$
of prompt $b$-jets.

We show that this tagger works in multi-jet final states but that when
generating backgrounds for which more than two jets are required to be 1b2b
tagged, it is necessary to simulate the impact of the tagger by rescaling
event weights using the efficiencies we present in table 3. The impact of this
tagger on experimental studies of explicit new physics models with final
states involving multiple $b$ quarks will be presented in a forthcoming
publication.

##  Appendix A Additional figures

Figure 10: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[20,50]\;{\rm GeV}$.

Figure 11: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[50,100]\;{\rm GeV}$.

Figure 12: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[100,200]\;{\rm GeV}$.

Figure 13: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[200,500]\;{\rm GeV}$.

Figure 14: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[500,1000]\;{\rm GeV}$.

Figure 15: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[1,1.5]\;{\rm TeV}$.

Figure 16: Charged track multiplicity, girth and $x_{E}$ for jets with
$p_{T}\in[1.5,2]\;{\rm TeV}$.

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