Search for $W^{\prime}\to tb$ decays in the fully hadronic final state with
the ATLAS experiment

Kuan-Yu Lin on behalf of the ATLAS Collaboration111© 2021 CERN for the benefit
of the ATLAS Collaboration. Reproduction of this article or parts of it is
allowed as specified in the CC-BY-4.0 license

Department of Physics and Astronomy
Michigan State University

> A search for a new heavy boson $W^{\prime}$ in proton-proton collisions at
> $\sqrt{s}$ = 13 TeV is presented. The search focuses on the decay of the
> $W^{\prime}$ to a hadronic top quark and a bottom quark, using the full Run
> 2 dataset of the ATLAS detector. The hadronic decay of the top quark is
> identified using DNN-based boosted-object techniques. The dominant
> background is obtained by a data-driven method with small systematic
> uncertainties. The results are presented as upper limits on the production
> cross-section times decay branching ratio for the $W^{\prime}$ boson with
> right-handed couplings that decays to a top quark and a bottom quark, for
> several $W^{\prime}$ masses between 1.5 to 5 TeV.

> PRESENTED AT
>
>
>
>
> $14^{\mathrm{th}}$ International Workshop on Top Quark Physics
> (videoconference), 13–17 September, 2021

##  1 Introduction

A new heavy charged vector boson $W^{\prime}$ is found in various BSM
scenarios such as the Little Higgs models [1] and extra-dimensional models
[2]. In this analysis, the ATLAS collaboration [3] conducts a resonance search
for the $W^{\prime}$ in the all-hadronic $tb$ final state ($t\overline{b}$ and
$\overline{t}b$). The $tb$ channel is sensitive to $W^{\prime}$ candidates
with larger couplings with the third generation [4, 5, 6, 7] or quarks [8].
Hadronically decaying top-quarks are reconstructed as $R$ = 1 jets and
identifiable via jet substructure techniques [9].

##  2 The jet definitions and the invariant mass $m_{tb}$

Using the Run-2 ATLAS data equivalent to 139 $fb^{-1}$, this search considers
proton collision events with no leptons, at least one large-$R$ ($R$ = 1) jet
with $p_{\text{T}}$ $>$ 500 GeV, and a small-$R$ jet ($R$ = 0.4) with
$p_{\text{T}}$ $>$ 500 GeV in the opposite direction, as viewed on the plane
perpendicular to the proton beams. The two jets corresponds to the top quark
and bottom quark decaying from the $W^{\prime}$. These two jets are summed,
resulting in the invariant mass $m_{tb}$, which is expected to assume a
smoothly falling distribution for the SM backgrounds. By identifying jets
associated with the top and bottom quarks, a peaking structure around the
$W^{\prime}$ mass becomes significant.

##  3 The identification of jets associated with a top quark or a bottom quark

The energy pattern of the calorimeter cluster energy [10] in a large-$R$ jet
facilitates the top-quark tagging. Jet substructure variables such as the
N-subjettiness [11, 12] distinguish the energy deposits due to the three
quarks of top-decay from a light parton’s QCD radiation. Several substructure
variables, the jet mass, and $p_{\text{T}}$ are combined via a Deep Neural
Network (DNN) into a top-tagging score [13]. A large-$R$ jet with
$p_{\text{T}}$ $>$ 500 GeV and passing the 80% Working Point (80% WP, the
threshold corresponding to an 80% probability) is designated as a top-
candidate jet. Events with more than one top-candidate jet are rejected for
the suppression of the top-quark pair ($t\overline{t}$) background. In
contrast, events without a top-candidate jet but with large-$R$ jets
($p_{\text{T}}$ $>$ 500 GeV) passing the DNN score above $e^{-7}$ are reserved
for separate analysis regions. Such large-$R$ jets are called the top-proxy
jets.

The highest $p_{\text{T}}$ small-$R$ jet among those with $p_{\text{T}}$ $>$
500 GeV and $\Delta\phi$ $>$ 2 from the top-candidate jet is declared the
$b$-candidate jet. Each top-proxy jet is also paired with a $b$-candidate jet.
Despite bearing the letter $b$ in its name, the $b$-quark identification is
employed later for region assignment. The ATLAS $b$-tagging algorithm called
the DL1r [14] reconstructs the $b$-hadron decay vertices using charged
particle tracks around the small-$R$ jet axis. The $b$-tagging requirement
applied to the $b$-candidate jet is the 85% WP. Since the top decay products
include a $b$-quark, small-$R$ jets with $p_{\text{T}}$ $>$ 25 GeV inside the
top-candidate (top-proxy) jet are checked by the same $b$-tagging requirement.

##  4 Data-driven background estimation and region assignment

Figure 1: The Signal Regions (SR1 to SR3), the Validation Region (VR), the
Template Regions (TR1 to TR4), and the Control Regions (upper: CR1a to 4a and
lower: CR1b to CR4b) are shown with the cuts and selections. Taken from Ref.
[15].

The dominant background to this analysis is the QCD production of multi-jet
consisting of mainly light parton jets. A data-driven estimation for this
background is adopted under the assumption that the $b$-tagging probability
for $b$-candidate jets is unchanged for either top-proxy or top-candidate
jets. The tables in Figure 1 show the regions defined by the top-tagging and
$b$-tagging configuration, allowing the QCD multi-jet events in each of the
Signal Regions (SR1 to SR3) to be estimated using data events from the
neighboring Template Regions (TR1 to TR3) and the Control Regions (CR1a to 4a)
below. These Control Regions require a top-tagging score $>$ $e^{-4}$ for the
top-proxy jets such that the parton flavor compositions approach the Signal
and Template Regions. The two tables differ by the number of $b$-tagged
small-$R$ jets inside the top-candidate (top-proxy) jets: none for the 0 b-tag
in category and more than one for the 1 b-tag in category. This $b$-tag
requirement increases the contribution from heavy-flavor, impacting the
$b$-tagging rates of the $b$-candidate jet.

For the Template Regions (TRs), the only significant background other than
that from QCD multi-jets is the background from $t\overline{t}$ production.
The $t\overline{t}$ events estimated by event generators have to be subtracted
from the TRs as the Control Regions cannot account for them. All other
backgrounds are included in the data-driven background, dominated by the QCD
multi-jet. The DNN cut of $e^{-7}$ for the lower Control Regions – CR1b to
CR4b – corresponds to parton flavor variations between top-proxy and top-
candidate jets observed in multi-jet simulation. The double ratios between the
two rows of Control Regions are calculated in data to estimate the systematic
uncertainties of the data-driven estimate. The tighter 50% WP of top-tagging
is applied to top-candidate jets, leaving a signal-sensitive region SR1 and a
signal depleted Validation Region for cross-checking the data-driven method
before unblinding data in the SRs.

##  5 Post-fit invariant mass distribution

Figure 2: Post-fit invariant mass distribution in region SR1. The blue hatched
band shows the systematic plus statistical uncertainty of the total background
post-fit. The pre-fit total background is overlayed by blue dashed lines. The
$W^{\prime}_{R}$ shown in the red dashed histograms assume the expected cross-
section. Taken from Ref. [15].

The data-driven background and the MC-estimated $t\overline{t}$ background are
fit to the $m_{tb}$ distribution in data with a profile-likelihood function to
constrain systematic uncertainties. The post-fit distribution is compared with
data in Figure 2 for the region SR1. The data-driven background (pale red) is
plotted above the all-hadronic $t\overline{t}$ (green) and then the
semileptonic plus dileptonic $t\overline{t}$ (yellow). Their sum is consistent
with data adding the uncertainty bands, meaning that no significant presence
of the $W^{\prime}$ signal is observed. The overlayed right-handed
$W^{\prime}$ signal is not included in this background-only fit.

##  6 Exclusion upper limit

Figure 3: The exclusion upper limit on the $W^{\prime}_{R}$ production times
the $tb$ decay branching ratio at the 95% confidence level as a function of
the $W^{\prime}_{R}$ mass. The red band includes the theory uncertainties from
Parton Distribution Functions, the strong coupling constant, renormalization
and factorization scale, and the top quark mass. Taken from Ref. [15].

We compute the upper exclusion limit for several $W^{\prime}$ masses from 1.5
to 5 TeV at the 95% Confidence Level, as depicted in Figure 3. The black line
is the observed limit calculated with data; the blue dashed line is the
expected limit obtained by treating the pre-fit background as expected data.
The green (yellow) band corresponds to the one-sigma (two-sigma) uncertainties
for the expected limit. The red line follows the theoretical cross-section for
a right-handed $W^{\prime}$ with SM-like electroweak coupling, computed by the
ZTOP framework [16, 17]. As the observed limit excludes cross-sections above
the curve, the $W^{\prime}_{R}$ signal is excluded at the 95% Confidence Level
up to 4.4 TeV. For more details about this analysis, please refer to the
conference note [15].

## References

  * [1] M. Perelstein.  Little Higgs models and their phenomenology.  Prog. Part. Nucl. Phys., 58:247–291, 2007.
  * [2] H.-C. Cheng, C. T. Hill, S. Pokorski, and J. Wang.  The Standard Model in the Latticized Bulk.  Phys. Rev. D, 64:065007, 2001.
  * [3] ATLAS Collaboration.  The ATLAS experiment at the CERN large hadron collider.  Journal of Instrumentation, 3(08):S08003–S08003, 2008.
  * [4] X.-y. Li and E. Ma.  Gauge model of generation nonuniversality reexamined.  J. Phys. G, 19:1265–1278, 1993.
  * [5] E. Malkawi, T. M. P. Tait, and C. P. Yuan.  A Model of strong flavor dynamics for the top quark.  Phys. Lett. B, 385:304–310, 1996.
  * [6] D. J. Muller and S. Nandi.  Top flavor: A Separate SU(2) for the third family.  Phys. Lett. B, 383:345–350, 1996.
  * [7] R. Calabrese, A. De Iorio, D. F. G. Fiorillo, A. O. M. Iorio, S. Morisi, and G. Miele.  Top-flavor scheme in the context of $W^{\prime}$ searches at LHC.  Phys. Rev. D, 104(5):055006, 2021.
  * [8] H. Georgi, E. E. Jenkins, and E. H. Simmons.  The un-unified standard model.  Nuclear Physics B, 331(3):541–555, 1990.
  * [9] R. Kogler et al.  Jet Substructure at the Large Hadron Collider: Experimental Review.  Rev. Mod. Phys., 91(4):045003, 2019.
  * [10] ATLAS Collaboration.  Topological cell clustering in the ATLAS calorimeters and its performance in LHC Run 1.  Eur. Phys. J. C, 77:490, 2017.
  * [11] K. Thaler, J.and Van Tilburg.  Identifying Boosted Objects with N-subjettiness.  JHEP, 03:015, 2011.
  * [12] K. Thaler, J.and Van Tilburg.  Maximizing Boosted Top Identification by Minimizing N-subjettiness.  JHEP, 02:093, 2012.
  * [13] ATLAS Collaboration.  Performance of top-quark and $W$-boson tagging with ATLAS in Run 2 of the LHC.  Eur. Phys. J. C, 79(5):375, 2019.
  * [14] ATLAS Collaboration.  ATLAS $b$-jet identification performance and efficiency measurement with $t{\overline{t}}$ events in $pp$ collisions at $\sqrt{s}=13$ TeV.  Eur. Phys. J. C, 79(11):970, 2019.
  * [15] ATLAS Collaboration.  Search for vector boson resonances decaying to a top quark and a bottom quark in the hadronic final state using $pp$ collisions at $\sqrt{s}~{}=~{}13$ TeV with the ATLAS detector.  ATLAS-CONF-2021-043, 2021.  https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CONFNOTES/ATLAS-CONF-2021-043.
  * [16] Z. Sullivan.  Fully Differential $W^{\prime}$ Production and Decay at Next-to-Leading Order in QCD.  Phys. Rev. D, 66:075011, 2002.
  * [17] D. Duffty and Z. Sullivan.  Model independent reach for $W^{\prime}$ bosons at the LHC.  Phys. Rev. D, 86:075018, 2012.


# Using Limited Neural Networks to Assess Relative Mechanistic Influence on
Shock Heating in Granular Solids

Brenden W. Hamilton  [<EMAIL_ADDRESS>Timothy C. Germann

###### Abstract

The rapid compaction of granular media results in localized heating that can
induce chemical reactions, phase transformations, and melting. However, there
are numerous mechanisms in play that can be dependent on a variety of
microstructural features. Machine learning techniques such as neural networks
offer a ubiquitous method to develop models for physical processes. Limiting
what kinds of microstructural information is used as input and assessing
normalized changes in network error, the relative importance of different
mechanisms can be inferred. Here we utilize binned, initial density
information as network inputs to predict local shock heating in a granular
high explosive trained from large scale, molecular dynamics simulations. The
spatial extend of the density field used in the network is altered to assess
the importance and relevant length scales of the physical mechanisms in play,
where different microstructural features result in different predictive
capability.

Los Alamos National Laboratory] Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA

The rapid compaction of granular solids can lead to a wide variety of
microstructural 1, 2, 3, 4 and chemical 5, 6, 7, 8 responses that are highly
dictated by the initial local packing and structure of the material. While the
use of molecular dynamics (MD) and continuum mechanics simulations have helped
to elucidate the governing processes 9, 10, 11, 12, 13, 14, 15, 16, the wide
range of mechanisms in play have prevented a unified understanding of events,
especially the weighted relevance of various mechanisms.

A key example is the shock compression of energetic materials, in which the
materials are typically utilized as neat or polymer bonded granular solids
with a bimodal grain size distribution of larger grains and smaller “fills”
17, 18, 19. The shock initiation of chemistry, which can lead to a run to
detonation, is governed by the formation of localizations of excess energy
known as hotspots, which are typically defined by their temperature and size
20, 21. These hotspots form through shock induced processes such as intra- and
inter-granular void collapse, shear band formation, jetting of material, and
inter-granular friction 22, 23, 24, 25, 26. From system to system, these
individual processes can be influenced by material orientations, crystal
defect formation, surface properties, and void shapes and sizes 27, 28, 29,
30, 31, 32, 33. Void collapse is typically the dominant process, with the
energy localization increasing with increasing P-V work done 34. Hence,
broadly understanding the overall shock compression and initiation involves a
wide range of materials models and highly detailed structural information.

Additionally, shock compaction not only localizes thermal energy, but can also
deform individual molecules, causing them to exist in strained states 35.
These intra-molecular strains can alter reaction kinetics and pathways through
mechanochemistry 36, 37, 38. Interestingly, these strain energies are thought
to occur through fundamentally different processes than temperature
localization 33, 39. Being able to predict the localization of both
temperature and strain energy in a hotspot remains a grand challenge for the
energetic materials community and is highly relevant to general materials
compaction problems. Being able to predict these processes without running
computationally expensive molecular dynamics and hydro-code simulations, as
well as to better evaluate the key/necessary mechanisms, is crucial to the
materials physics and condensed matter chemistry communities.

Materials science has recently experienced a rapid increase in the use of
machine learning (ML) to extract and understand physical processes that can
occur 40, 41, 42. ML has played a key role in the development of computational
models 43, 44, 45, 46, 47, 48, 49, predicting properties, 50, 51, 52, 53, 54
and characterizing materials 55, 56, 57, 58. While predictions from large and
non-linear neural networks typically function as a black box, limiting and
altering the physical information that informs the network can help to tease
out which properties and mechanisms are critical to a processes by the
network’s ability to make predictions given its limited subset of input
information. This process, employed here, is similar to a ”leave one feature
out” scheme.

Here, we utilize non-reactive, all-atom MD simulations to model the shock
response of the granular high explosive 1,3,5-triamino-2,4,6-trinitrobenzene
(TATB). Simulation details are provided in the Methods section and
Supplemental Materials section SM-1. A neural network is used to predict the
final temperature and intra-molecular strain energy fields given just the
initial density field of the unshocked system. The level of coarsening and
total extent of the density information given is varied to assess the amount
of local information needed to properly predict energy localization from
granular compaction. Increasing density resolution provides more information
pertaining to pore shape and local curvature, but using only density restricts
potentially critical information such as local orientation and crystalline
defects. It should be noted that the purpose of this work is not to minimize
the error of the networks and make the best model possible. It is to
systematically change the inputs given to the network such that physical
trends and important physico-chemical mechanisms can be inferred from the
relative change in each network’s predictive ability. However, this
methodology does necessitate the network predictions be reasonable accurate as
a baseline for extracting materials physics, such that the model is presumable
learning some description of the physics.

Predictions of temperature ($T$) and intra-molecular strain energy
($U_{Latent}$) fields are done with a simple, non-linear neural network. The
system is binned in a Lagrangian fashion on the initial frame prior to shock
to provide the density descriptors. From the simulation, each molecule has a
defined time is is ”shocked”, and a ”composite frame” is constructed by taking
the position and thermodynamics at each molecule at 5ps after its shocked
time, i.e. molecules are taken from different simulation frames such that they
are at the same relative time compared to being compressed.

The input layer consists of the density of a bin and its $N$ sets of nearest
neighbors (initial frame), as shown in Figure 1 for the $N=1$ case (a second
shell of bins would be $N=2$ and just the central bin, no neighbors, is
$N=0$). Bins with zero molecules are considered as neighbor bins but not as
center bins as they will not have a final temperature/energy for an output.
The output layer is either the $T$ or $U_{Latent}$ of the center bin
(composite frame); each of these two values is trained with separate networks
of the same architecture. All networks consist of an input layer of size
$(2N+1)^{2}$, an output layer of size 1, and a single hidden layer of size
$\lfloor 0.5(2N+1)^{2}\rfloor$. Figure 1 exemplifies how the all-atom
structure is encoded into the neural network input layer, where the output
layer corresponds to a region of the all-atom results, the same center
Lagrangian bin, which has mean $T$ and $U_{Latent}$ values. Additional details
on network architecture are available in the Methods section and Supplemental
Materials section SM-2.

Figure 1: Initial configuration and composite temperature map for the testing
set system with example binning for network input and output.

Different square bin sizes of 2.5 nm, 3.0 nm, 4.0 nm and 5.0 nm are used. To
compare different bin size results, we define the spatial extent $(SE)$ of the
input layer, where $SE=L(N+\frac{1}{2})$ where $L$ is the bin length and $N$
is the number of sets of nearest neighbor bins included. This is the
equivalent to the radius of an inscribed circle for the total square of bins
used. Figure 2 shows parity plots of $T$ and $U_{Latent}$ for $L$ = 5.0 nm and
2.5 nm with nearly equivalent spatial extents of 27.5 and 26.25 nm,
respectively. For the temperature (left column plots), both show decent
correlation with the parity line, with RMS errors of 84.4 K and 108.9 K for
5.0 and 2.5 nm bins, respectively, where peak (individual) molecular
temperatures in the hotspots are near 2000 K and 600-700 K in the bulk shocked
materials.

When comparing the two sets at different bin sizes, it is crucial to consider
that while the 2.5 nm bins results in four times as many training and testing
points, it also samples a much wider range of values, as the smaller bins
provide less smoothing of extreme temperatures. The 2.5 nm cases have
considerably more values above 1000 K and its peak hotspot temperatures are
much higher than the bulk temperature, relative to the 5.0 nm bins. Both cases
mostly have slight over-predictions of values around 800 K and under-
predictions for those above 1000 K.

Figure 2: Parity plots of $T$ and $U_{Latent}$ for the 5.0 nm bins and 2.5 nm
bins for spatial extents of 27.5 and 26.25 nm, respectively. Data consists of
predictions of the smaller MD cell used exclusively for testing.

For the $U_{Latent}$ predictions shown in Figure 2, there is considerably less
correlation with the parity line. The predictions even appear to be less
correlated overall at the smaller bin sizes. While the $U_{Latent}$ fields,
shown in Supplemental Materials section SM-4, are more disperse that the $T$
fields, there are still notable regions of higher and lower strain energy. The
predicted fields are considerably more homogeneous and noisy, showing that the
density field alone is not enough to predict the mechanisms that drive
molecules to bent and distorted shapes, and that more complicated
microstructural or thermodynamic information is likely needed to make these
predictions. This additionally helps to verify previous results that concluded
that the $T$ and $U_{Latent}$ forming mechanisms are different, as their
localizations do not occur on a one-to-one basis 35. $U_{Latent}$ also has a
strong influence from pore size at larger pores 33, and the system sizes here
may not be large enough to provide a wide enough range of examples in the
training set. Compared to temperature, the density (and therefore P-V work)
alone is not enough to make decent quantitative predictions of the
$U_{Latent}$ field.

Figure 3 shows heat maps of the actual and predicted temperatures, as well as
the difference, for a 2.5 nm bin case. Other bin size and nearest neighbor
cases are available in Supplemental Materials section SM-3. From this, it can
be assessed that hotspots that are longer in the shock direction, like points
(a) and (b) in Figure 3, are under-predicted. However, hotspots that are
longer in the cross direction, like points (c) and (d) in Figure 3, are over-
predicted. The initial microstructure of this case is shown in Figure 1.
Vertical pores and high aspect curvature pores can often result in molecular
jetting, leading to high levels of expansion before getting re-compressed 59,
23, 31, however, wider or more circular pores often result in a more
hydrodynamic or plastic flow type responses that incurs much less P-V work
during re-compression relative to the jetted material, especially as material
will have less physical space to expand into the pore before getting re-
compressed. It should be noted that, while there is under predictions of the
hottest hotspots and over prediction of the colder ones, the model still
generally predicts vertical pores to be hotter than the horizontal ones.

Figure 3: Heat maps of the actual and predicted temperatures, as well as the
difference between the two, for the 2.5 nm bin case with a 26.25 nm spatial
extent. Points a-d on the actual temperature map correspond to specific
hotspots where a and b are under-predicted, and c and d are over-predicted.

The two key factors in qualitatively predicting hotspot temperatures from pore
collapse are the pore size and shape 9, 23, 33. Based on these results, the
predictions here appear to be fully considering size with the largest pores
(points (a) and (c)) predicting high temperatures and the smaller pores
typically predicting lower temperatures. Molecular jetting, which greatly
influences material expansion into pores, is a much more complex mechanical
process where things like curvature of the pore come into play 23, 31. Even
with small initial bins, this information is partially coarsened out,
preventing the network from making these predictions. Information on the shock
wave structure/shape, which will change as it progresses over the material, is
also unknown to the network and can cause different amounts of shock focusing
that leads to jetting. Additionally, the network does not have information
related to crystal orientation, which results in changes in hotspot
temperature on the order of the errors shown here 28, however, the grain
orientations chosen here minimize anisotropy between grains. Hence, while P-V
work from pore size is enough to make decently quantitative predictions of
hotspot temperature, the finer microstructural details are likely needed to
correct for errors on the order of several hundred Kelvin. It is most likely a
combination of these omitted features that drives the error in predictions,
especially the under-prediction of the highest temperature hotspots.

Figure 4 shows root mean square (RMS) errors and Linear Normalized RMS errors
(LN-RMSE) for all networks trained. The LN-RMSE are RMSE values normalized by
the RMSE of the linear regression between the density of a bin (no nearest
neighbor information) and the temperature. For the top row in Figure 4, which
is temperature, we interestingly see, for small to intermediate spatial
extent, similar RMSE from all bin sizes, followed by a divergence of values.
The slight uptick in errors for large spatial extent are attributed to a
static stopping criteria based on error reduction over the previous 200
epochs. For larger SE, the network itself is proportionally larger and
potentially learns at a slower rate, especially for smaller bin sizes where
$N$ is much larger for a given SE and the network size grows as $N^{2}$.

For an equal spatial extent between two different bin sizes, the smaller bin
will result in more training/testing data for a given simulation size.
However, the smaller bin system is also less coarse grained, and will have a
wider range of peak temperatures and more pronounced temperature gradients.
Hence, with more training data and more resolved microstructural data from the
density input, the smaller bins are able to reach roughly the same level of
predictive power as for the much smoother fields of the larger bins. For the
LN-RMSE, as smaller bins will have more variability in temperature with
density and therefore a larger normalizing constant, the smallest bin cases
perform the best. This normalized case shows the predictive power of adding
more microstructural information with a finer density field. As there is
significantly less accuracy in the $U_{Latent}$ predictions, the trends of
decreasing errors with SE and various bin size effects are less physically
meaningful. Plots of RMSE and LN-RMSE for $U_{Latent}$ are available in
Supplemental Materials section SM-5. These show similar trends with SE to
temperature, yet opposite trends with respect to bin size, which is most
likely an effect of coarsening significant noise and the ease of predicting a
more uniform field.

Figure 4: Root mean square errors for all temperature networks run, as a
function of spatial extent. Colored lines represent different bin sizes. LN-
RMSE (Linear Normalized RMSE) are RMS errors that are normalized by the RMS
error of a linear network for zero nearest neighbors ($N=0$).

In summary, MD simulations of shock compaction of a granular material resulted
in heterogeneous localization of both temperature and intra-molecular strain
energy, where the latter is known to be responsible for mechanochemical
effects. A Lagrangian binning of initial microstructures was used to embed
local density information, but intentionally excludes information pertaining
to local crystal structure, defects, orientation, and incident wave structure.
These density bins were utilized as an input layer to a neural network to
predict the shock induced $T$ and $U_{Latent}$ fields.

From trends in RMS errors for different bin sizes and spatial extents of the
input layer, we find that the predictability of a network increases with more
spatial extent, as well as with smaller bins which would carry more precise
microstructural information. These improvements are despite the smaller bins
leading to much larger variability and fluctuation in the temperature fields,
where larger bins smooth and coarsen that information.

While density information did not allow for accurate prediction of
$U_{Latent}$, the networks readily predicted the temperature fields with some
of the hotspots being over-predicted, while others are under-predicted. While
the network predicts hotter temperatures for larger pores, where more P-V work
can occur during compaction, it fails to fully capture other mechanisms such
as jetting and molecular ejecta, which can lead to extreme temperatures. This
failure manifests in pores that nucleate these jetting mechanisms, those
elongated in the shock direction, to have under-predicted temperatures.
Additionally, equi-axed or wider pores that do not jet are slightly over-
predicted. This leads the network to give less variation in prediction from
hotspot to hotspot, but still captrues the general trend with pore size.

Hence, we are able to utilize neural network predictions to show that, while
P-V work is the dominant mechanism in temperature localization, not having
finer microstructural details such as those on the nanometer to sub-nanometer
length scale leads to errors with the hotspot on the order of a few hundred
Kelvin. Additionally, P-V work and pore shape are shown to be much less
important mechanisms for the $U_{Latent}$ field, which may have considerable
influence from plasticity levels and material flow rate 25, 33.

This work shows promise in the use of these limited neural networks to assess
physical mechanisms in play for complex, physico-chemical processes in
condensed matter systems. In future work, a wider variety of input descriptors
can be utilize that, in addition to density, map features such as crystal
orientation, pre-existing crystalline defects, surface roughness in pores, and
the structure/shape of the incident shock wave which will be altered from
upstream microstructural features and shock instabilities. By coupling these
features with a ’leave one feature out’ type scheme and the varying spatial
extend scheme used here, a relative importance of key hotspot mechanisms and
their necessary descriptors can be deduced.

##  1 Methods

All simulations were run with all-atom MD using the LAMMPS software package
60, 61. Interatomic interactions were calculated using the nonreactive,
nonpolarizable forcefield from Bedrov et al. 62, with tailored harmonic bond
stretch and angle bend terms 63 and an intramolecular O-H repulsion term 64.
Electrostatics were solved for in real space with the Wolf potential 65. Van
der Waals interactions were modeled using the Buckingham potential. All
simulations are conducted with a 0.25 fs timestep.

Simulations cells of granular TATB were built with the PBXGen algorithm 17
using columnar grains with a bimodal grain size distribution with peaks at 40
nm and 8 nm. The smaller ’fill’ grains make up roughly 2/3 of the total grain
count. All grains are crystallographically oriented so that the TATB [001]
direction is in the periodic $Z$ direction with thickness 4.1 nm (into the
page in the Figures 1 and 3), which minimizes grain-to-grain anisotropy. Two
cells were constructed with different sizes of 100 x 400 nm (12440592 atoms),
and 100 x 200 nm (6183984 atoms), where the larger system is used to make up
the training set and smaller system is the testing set

Shock simulations are conducted using the reverse ballistic method with a
particle velocity of 2.0 km/s, with the resulting shock traveling in the $+Y$
direction (upwards in the Figures 1 and 3). The $Y$ direction is a free
boundary, whereas the other directions are periodic. Analysis is conducted
using a per molecule basis, using the molecular center of mass as its
position. Temperature ($T$) is the molecular roto-vibrational kinetic energy
in units of Kelvin. Intra-molecular strain energy ($U_{Latent}$) is defined as
the excess intra-molecular potential energy with respect to the equi-partition
theorem 28. Supplemental Materials section SM-1 provides additional MD and
PBXGen methods.

For the neural networks, a sigmoid function is used into the hidden layer, and
a linear function into the output layer. A bias is allowed for both layers.
Supplemental Materials section SM-2 provides additional machine learning
method details.

##  2 Acknowledgments

Funding for this project was provided by the Director’s Postdoctoral
Fellowship program at Los Alamos National Laboratory, project LDRD
20220705PRD1. Partial funding was provided by the Advanced Simulation and
Computing Physics and Engineering Models project (ASC-PEM). This research used
resources provided by the Los Alamos National Laboratory (LANL) Institutional
Computing Program. This work was supported by the U.S. Department of Energy
(DOE) through LANL, which is operated by Triad National Security, LLC, for the
National Nuclear Security Administration of the U.S. Department of Energy
(Contract No. 89233218CNA000001). Approved for Unlimited Release LA-
UR-23-24466.

##  3 Supplemental Material

## References

  * Cherukara et al. 2014 Cherukara, M. J.; Germann, T. C.; Kober, E. M.; Strachan, A. Shock Loading of Granular Ni/Al Composites. Part 1: Mechanics of Loading. _Journal of Physical Chemistry C_ 2014, _118_ , 26377–26386
  * Brujić et al. 2005 Brujić, J.; Wang, P.; Song, C.; Johnson, D. L.; Sindt, O.; Makse, H. A. Granular dynamics in compaction and stress relaxation. _Physical review letters_ 2005, _95_ , 128001
  * Lumay and Vandewalle 2005 Lumay, G.; Vandewalle, N. Experimental study of granular compaction dynamics at different scales: grain mobility, hexagonal domains, and packing fraction. _Physical review letters_ 2005, _95_ , 028002
  * Busignies et al. 2004 Busignies, V.; Tchoreloff, P.; Leclerc, B.; Hersen, C.; Keller, G.; Couarraze, G. Compaction of crystallographic forms of pharmaceutical granular lactoses. II. Compacts mechanical properties. _European journal of pharmaceutics and biopharmaceutics_ 2004, _58_ , 577–586
  * Cherukara et al. 2016 Cherukara, M. J.; Germann, T. C.; Kober, E. M.; Strachan, A. Shock Loading of Granular Ni/Al Composites. Part 2: Shock-Induced Chemistry. _Journal of Physical Chemistry C_ 2016, _120_ , 6804–6813
  * Strachan et al. 2003 Strachan, A.; van Duin, A. C.; Chakraborty, D.; Dasgupta, S.; Goddard, W. A. Shock Waves in High-Energy Materials: The Initial Chemical Events in Nitramine RDX. _Physical Review Letters_ 2003, _91_ , 7–10
  * Yang et al. 1997 Yang, Y.; Gould, R.; Horie, Y.; Iyer, K. Shock-induced chemical reactions in a Ni/Al powder mixture. _Applied physics letters_ 1997, _70_ , 3365–3367
  * Zhou et al. 2019 Zhou, X.; Miao, Y. R.; Shaw, W. L.; Suslick, K. S.; Dlott, D. D. Shock Wave Energy Absorption in Metal-Organic Framework. _Journal of the American Chemical Society_ 2019, _141_ , 2220–2223
  * Wood et al. 2018 Wood, M. A.; Kittell, D. E.; Yarrington, C. D.; Thompson, A. P. Multiscale modeling of shock wave localization in porous energetic material. _Physical Review B_ 2018, _97_ , 014109
  * Herring et al. 2010 Herring, S. D.; Germann, T. C.; Grønbech-Jensen, N. Effects of void size, density, and arrangement on deflagration and detonation sensitivity of a reactive empirical bond order high explosive. _Physical Review B_ 2010, _82_ , 214108
  * Rai et al. 2020 Rai, N. K.; Sen, O.; Udaykumar, H. S. Macro-scale sensitivity through meso-scale hotspot dynamics in porous energetic materials: Comparing the shock response of 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) and 1,3,5,7-tetranitro-1,3,5,7-tetrazoctane (HMX). _Journal of Applied Physics_ 2020, _128_ , 1–24
  * Bidault and Pineau 2018 Bidault, X.; Pineau, N. Granularity impact on hotspot formation and local chemistry in shocked nanostructured RDX. _Journal of Chemical Physics_ 2018, _149_
  * Xiong et al. 2019 Xiong, Y.; Li, X.; Xiao, S.; Deng, H.; Huang, B.; Zhu, W.; Hu, W. Molecular dynamics simulations of shock loading of nearly fully dense granular Ni-Al composites. _Physical Chemistry Chemical Physics_ 2019, _21_ , 20252–20261
  * Pouliquen et al. 2003 Pouliquen, O.; Belzons, M.; Nicolas, M. Fluctuating particle motion during shear induced granular compaction. _Physical review letters_ 2003, _91_ , 014301
  * Kadau et al. 2007 Kadau, K.; Germann, T. C.; Lomdahl, P. S.; Albers, R. C.; Wark, J. S.; Higginbotham, A.; Holian, B. L. Shock waves in polycrystalline iron. _Physical Review Letters_ 2007, _98_ , 135701
  * Rutherford et al. 2017 Rutherford, M. E.; Chapman, D. J.; Derrick, J. G.; Patten, J. R.; Bland, P. A.; Rack, A.; Collins, G. S.; Eakins, D. E. Probing the early stages of shock-induced chondritic meteorite formation at the mesoscale. _Scientific reports_ 2017, _7_ , 1–10
  * Li et al. 2022 Li, C.; Hamilton, B. W.; Shen, T.; Alzate, L.; Strachan, A. Systematic Builder for All-Atom Simulations of Plastically Bonded Explosives. _Propellants, Explosives, Pyrotechnics_ 2022, _47_ , e202200003
  * Skidmore et al. 1998 Skidmore, C.; Phililips, D.; Howe, P.; Mang, J.; Romero, J. The Evolution of Microstructural Changes in Pressed HMX Explosives. Eleventh International Detonation Symposium. Snowmass, Colorado, 1998
  * Willey et al. 2006 Willey, T. M.; Van Buuren, T.; Lee, J. R.; Overturf, G. E.; Kinney, J. H.; Handly, J.; Weeks, B. L.; Ilavsky, J. Changes in pore size distribution upon thermal cycling of TATB-based explosives measured by ultra-small angle X-ray scattering. _Propellants, Explosives, Pyrotechnics_ 2006, _31_ , 466–471
  * Hamilton et al. 2021 Hamilton, B. W.; Sakano, M. N.; Li, C.; Strachan, A. Chemistry Under Shock Conditions. _Annual Review of Materials Research_ 2021, _51_ , 101–130
  * Handley et al. 2018 Handley, C. A.; Lambourn, B. D.; Whitworth, N. J.; James, H. R.; Belfield, W. J. Understanding the shock and detonation response of high explosives at the continuum and meso scales. _Applied Physics Reviews_ 2018, _5_ , 011303
  * Davis 1981 Davis, W. C. High Explosives The Interaction of Chemistry and Mechanics. _Los Alamos Science_ 1981, _2_ , 48–75
  * Li et al. 2020 Li, C.; Hamilton, B. W.; Strachan, A. Hotspot formation due to shock-induced pore collapse in 1,3,5,7-tetranitro-1,3,5,7-tetrazoctane (HMX): Role of pore shape and shock strength in collapse mechanism and temperature. _Journal of Applied Physics_ 2020, _127_ , 175902
  * Cawkwell et al. 2008 Cawkwell, M. J.; Sewell, T. D.; Zheng, L.; Thompson, D. L. Shock-induced shear bands in an energetic molecular crystal: Application of shock-front absorbing boundary conditions to molecular dynamics simulations. _Physical Review B_ 2008, _78_ , 014107
  * Kroonblawd and Fried 2020 Kroonblawd, M. P.; Fried, L. E. High Explosive Ignition through Chemically Activated Nanoscale Shear Bands. _Physical Review Letters_ 2020, _124_ , 206002
  * Dienes 1985 Dienes, J. K. Discussion of a statistical theory of fragmentation processes. _Mechanics of Materials_ 1985, _4_ , 337–341
  * Zhao et al. 2020 Zhao, P.; Lee, S.; Sewell, T.; Udaykumar, H. S. Tandem Molecular Dynamics and Continuum Studies of Shock-Induced Pore Collapse in TATB. _Propellants, Explosives, Pyrotechnics_ 2020, _45_ , 196–222
  * Hamilton et al. 2022 Hamilton, B. W.; Kroonblawd, M. P.; Strachan, A. The Potential Energy Hotspot: Effects from Impact Velocity, Defect Geometry, and Crystallographic Orientation. _Journal of Physical Chemistry C_ 2022, _126_ , 3743–3755
  * Lafourcade et al. 2018 Lafourcade, P.; Denoual, C.; Maillet, J. B. Irreversible Deformation Mechanisms for 1,3,5-Triamino-2,4,6-Trinitrobenzene Single Crystal through Molecular Dynamics Simulations. _Journal of Physical Chemistry C_ 2018, _122_ , 14954–14964
  * Grilli and Koslowski 2018 Grilli, N.; Koslowski, M. The effect of crystal orientation on shock loading of single crystal energetic materials. _Computational Materials Science_ 2018, _155_ , 235–245
  * Li and Strachan 2022 Li, C.; Strachan, A. Shock-induced collapse of porosity, mapping pore size and geometry, collapse mechanism, and hotspot temperature. _Journal of Applied Physics_ 2022, _132_ , 065901
  * Bidault and Pineau 2018 Bidault, X.; Pineau, N. Impact of surface energy on the shock properties of granular explosives. _Journal of Chemical Physics_ 2018, _148_
  * Hamilton and Germann 2023 Hamilton, B. W.; Germann, T. C. Energy localization efficiency in 1, 3, 5-trinitro-2, 4, 6-triaminobenzene pore collapse mechanisms. _Journal of Applied Physics_ 2023, _133_ , 035901
  * Holian et al. 2002 Holian, B. L.; Germann, T. C.; Maillet, J. B.; White, C. T. Atomistic Mechanism for Hot Spot Initiation. _Physical Review Letters_ 2002, _89_ , 285501
  * Hamilton et al. 2021 Hamilton, B. W.; Kroonblawd, M. P.; Li, C.; Strachan, A. A Hotspot’s Better Half: Non-Equilibrium Intra-Molecular Strain in Shock Physics. _Journal of Physical Chemistry Letters_ 2021, _12_ , 2756–2762
  * Hamilton et al. 2022 Hamilton, B. W.; Kroonblawd, M. P.; Strachan, A. Extemporaneous Mechanochemistry: Shockwave Induced Ultrafast Chemical Reactions Due to Intramolecular Strain Energy. _Journal of Physical Chemistry Letters_ 2022, _13_ , 6657–6663
  * Hamilton and Strachan 2022 Hamilton, B. W.; Strachan, A. Many-Body Mechanochemistry : Intra-molecular Strain in Condensed Matter Chemistry. _chemRxiv_ 2022,
  * Hamilton and Germann 2023 Hamilton, B. W.; Germann, T. C. Interplay of Mechanochemistry and Material Processes in the Graphite to Diamond Phase Transformation. _arXiv preprint arXiv:2302.04684_ 2023,
  * Hamilton et al. 2023 Hamilton, B. W.; Kroonblawd, M. P.; Macatangay, J.; Springer, H. K.; Strachan, A. Intergranular Hotspots: A Molecular Dynamics Study on the Influence of Compressive and Shear Work. _arXiv preprint arXiv:2302.10728_ 2023,
  * Iten et al. 2020 Iten, R.; Metger, T.; Wilming, H.; Del Rio, L.; Renner, R. Discovering Physical Concepts with Neural Networks. _Physical Review Letters_ 2020, _124_ , 10508
  * Butler et al. 2018 Butler, K. T.; Davies, D. W.; Cartwright, H.; Isayev, O.; Walsh, A. Machine learning for molecular and materials science. _Nature_ 2018, _559_ , 547–555
  * Zhong et al. 2022 Zhong, X.; Gallagher, B.; Liu, S.; Kailkhura, B.; Hiszpanski, A.; Han, T. Explainable machine learning in materials science. _npj Computational Materials_ 2022, _8_ , 204
  * Huan et al. 2017 Huan, T. D.; Batra, R.; Chapman, J.; Krishnan, S.; Chen, L.; Ramprasad, R. A universal strategy for the creation of machine learning-based atomistic force fields. _NPJ Computational Materials_ 2017, _3_ , 37
  * Chan et al. 2019 Chan, H.; Cherukara, M. J.; Narayanan, B.; Loeffler, T. D.; Benmore, C.; Gray, S. K.; Sankaranarayanan, S. K. Machine learning coarse grained models for water. _Nature Communications_ 2019, _10_ , 379
  * Zuo et al. 2020 Zuo, Y.; Chen, C.; Li, X.; Deng, Z.; Chen, Y.; Behler, J.; Csányi, G.; Shapeev, A. V.; Thompson, A. P.; Wood, M. A., et al. Performance and cost assessment of machine learning interatomic potentials. _The Journal of Physical Chemistry A_ 2020, _124_ , 731–745
  * Yoo et al. 2021 Yoo, P.; Sakano, M.; Desai, S.; Islam, M. M.; Liao, P.; Strachan, A. Neural network reactive force field for C, H, N, and O systems. _npj Computational Materials_ 2021, _7_ , 1–10
  * Lindsey et al. 2017 Lindsey, R. K.; Fried, L. E.; Goldman, N. ChIMES: A Force Matched Potential with Explicit Three-Body Interactions for Molten Carbon. _Journal of Chemical Theory and Computation_ 2017, _13_ , 6222–6229
  * Hamilton et al. 2023 Hamilton, B. W.; Yoo, P.; Sakano, M. N.; Islam, M. M.; Strachan, A. High Pressure and Temperature Neural Network Reactive Force Field for Energetic Materials. _arXiv preprint arXiv:2302.04906_ 2023,
  * Zhang et al. 2019 Zhang, Y.; Hu, C.; Jiang, B. Embedded atom neural network potentials: Efficient and accurate machine learning with a physically inspired representation. _The Journal of Physical Chemistry Letters_ 2019, _10_ , 4962–4967
  * Fernandez et al. 2014 Fernandez, M.; Boyd, P. G.; Daff, T. D.; Aghaji, M. Z.; Woo, T. K. Rapid and accurate machine learning recognition of high performing metal organic frameworks for CO2 capture. _The Journal of Physical Chemistry Letters_ 2014, _5_ , 3056–3060
  * Xie and Grossman 2018 Xie, T.; Grossman, J. C. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. _Physical review letters_ 2018, _120_ , 145301
  * Sakano et al. 2020 Sakano, M. N.; Hamed, A.; Kober, E. M.; Grilli, N.; Hamilton, B. W.; Islam, M. M.; Koslowski, M.; Strachan, A. Unsupervised Learning-Based Multiscale Model of Thermochemistry in 1,3,5-Trinitro-1,3,5-triazinane (RDX). _Journal of Physical Chemistry A_ 2020, _124_ , 9141–9155
  * Ward et al. 2016 Ward, L.; Agrawal, A.; Choudhary, A.; Wolverton, C. A general-purpose machine learning framework for predicting properties of inorganic materials. _npj Computational Materials_ 2016, _2_ , 1–7
  * Zhuo et al. 2018 Zhuo, Y.; Mansouri Tehrani, A.; Brgoch, J. Predicting the band gaps of inorganic solids by machine learning. _The Journal of Physical Chemistry Letters_ 2018, _9_ , 1668–1673
  * Carbone et al. 2020 Carbone, M. R.; Topsakal, M.; Lu, D.; Yoo, S. Machine-learning X-ray absorption spectra to quantitative accuracy. _Physical Review Letters_ 2020, _124_ , 156401
  * Timoshenko et al. 2017 Timoshenko, J.; Lu, D.; Lin, Y.; Frenkel, A. I. Supervised machine-learning-based determination of three-dimensional structure of metallic nanoparticles. _The Journal of Physical Chemistry Letters_ 2017, _8_ , 5091–5098
  * Yao et al. 2022 Yao, Y.; Chan, H.; Sankaranarayanan, S.; Balaprakash, P.; Harder, R. J.; Cherukara, M. J. AutoPhaseNN: unsupervised physics-aware deep learning of 3D nanoscale Bragg coherent diffraction imaging. _npj Computational Materials_ 2022, _8_ , 1–8
  * Hu et al. 2019 Hu, W.; Ye, S.; Zhang, Y.; Li, T.; Zhang, G.; Luo, Y.; Mukamel, S.; Jiang, J. Machine learning protocol for surface-enhanced raman spectroscopy. _The journal of physical chemistry letters_ 2019, _10_ , 6026–6031
  * Germann et al. 2002 Germann, T. C.; Holian, B. L.; Lomdahl, P. S.; Heim, A. J.; Grønbech-Jensen, N.; Maillet, J.-B. Molecular dynamics simulations of detonation in defective explosive crystals. Proceedings of the Twelfth International Detonation Symposium. 2002; pp 711–717
  * Plimpton 1995 Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. _Journal of Computational Physics_ 1995, _117_ , 1–19
  * Thompson et al. 2022 Thompson, A. P.; Aktulga, H. M.; Berger, R.; Bolintineanu, D. S.; Brown, W. M.; Crozier, P. S.; in ’t Veld, P. J.; Kohlmeyer, A.; Moore, S. G.; Nguyen, T. D.; Shan, R.; Stevens, M. J.; Tranchida, J.; Trott, C.; Plimpton, S. J. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. _Computer Physics Communications_ 2022, _271_ , 108171
  * Bedrov et al. 2009 Bedrov, D.; Borodin, O.; Smith, G. D.; Sewell, T. D.; Dattelbaum, D. M.; Stevens, L. L. A molecular dynamics simulation study of crystalline 1,3,5-triamino-2,4,6- trinitrobenzene as a function of pressure and temperature. _Journal of Chemical Physics_ 2009, _131_
  * Kroonblawd and Sewell 2013 Kroonblawd, M. P.; Sewell, T. D. Theoretical determination of anisotropic thermal conductivity for crystalline 1,3,5-triamino-2,4,6-trinitrobenzene (TATB). _Journal of Chemical Physics_ 2013, _139_
  * Mathew et al. 2015 Mathew, N.; Sewell, T. D.; Thompson, D. L. Anisotropy in surface-initiated melting of the triclinic molecular crystal 1,3,5-triamino-2,4,6-trinitrobenzene: A molecular dynamics study. _Journal of Chemical Physics_ 2015, _143_
  * Wolf et al. 1999 Wolf, D.; Keblinski, P.; Phillpot, S. R.; Eggebrecht, J. Exact method for the simulation of Coulombic systems by spherically truncated, pairwise r-1 summation. _Journal of Chemical Physics_ 1999, _110_ , 8254–8282


# Charm lifetime measurements at Belle II

J. V. Bennett  University of Mississippi, University, MS, 38677  on behalf of
the Belle II collaboration

###### Abstract

Upgrades at the Belle II experiment and the SuperKEKB asymmetric-energy
electron-positron collider enable precise measurements of particle decays.
Even with early data, Belle II has made several world-leading measurements of
particle lifetimes, which are useful as tests of effective models used in
searches for physics beyond the standard model. Especially for charm hadrons,
these effective models depend on careful consideration of non-perturbative
effects to give an adequate description of lifetimes. Recent measurements at
other experiments have stimulated particular interest in charm baryons,
including several measurements that indicate the hierarchy of lifetimes is
different than once believed. The measurements of the $D^{+}$, $D^{0}$, and
$\Lambda_{c}^{+}$ lifetimes at Belle II are consistent with previous
measurements, but with improved precision. These measurements will improve the
world-average values, provide refined tests for effective models, and serve as
benchmarks for future measurements.

##  I Introduction

Searches for physics beyond the standard model (SM) of particle physics are
complicated by the need for accurate theoretical descriptions of strong
interactions at low energy. This is typically achieved with predictive tools
like the heavy quark expansion (HQE) Neubert ; Uraltsev ; Lenz ; Lenz15 ; Kirk
; Cheng18 ; Gratex . In this framework, decay widths of hadrons containing a
heavy quark, $Q$, are calculated with an expansion in terms of the heavy quark
mass, $m_{Q}$. Particle lifetimes, which are the inverse of the particle decay
widths, are sensitive to higher-order contributions Lenz ; Cheng15  and
therefore provide excellent tests for effective models. Charm hadrons, in
particular, require careful consideration of model-dependent effects. Non-
perturbative effects are relatively small for bottom hadrons, but are
significant for charm hadrons, for which the HQE to 1/$m_{c}^{3}$, where
$m_{c}$ is the mass of the charm quark, does not give a good description of
lifetimes Cheng18 ; Gratex . Precise measurements of charm hadron lifetimes
therefore provide stringent tests of effective models.

The world average values for charm hadron lifetimes are dominated by
measurements that were made two decades ago. The $D^{0}$ and $D^{+}$ lifetimes
were measured at the per-mille level by the photon-beam experiment FOCUS PDG ;
Link , the hyperon-beam experiment SELEX Kush , and the electron-positron
($e^{+}e^{-}$) experiment CLEO Bonv . These experiments also made percent-
level measurements of the lifetime of the $\Lambda_{c}^{+}$ Bonv ; Link02 ;
Mahm . More recently, the LHCb collaboration made precise measurements of the
lifetimes of several charm baryons, relative to the $D^{+}$ lifetime Aaij .
Indeed, these relative measurements, which benefit from reduced uncertainties
related to event selection that may bias the decay time at hadron colliders,
are systematics dominated due to the uncertainty on the lifetime of the
$D^{+}$. Additionally, the recent measurements by LHCb result in a hierarchy
of charm baryon lifetimes that is in conflict with that from previous
measurements Aaij18 ; Aaij22 . Precise, absolute measurements of charm hadrons
by the Belle II experiment help to clarify this picture and substantially
improve the world average values.

##  II The Belle II Experiment

Experiments at $e^{+}e^{-}$ colliders have an excellent potential for absolute
lifetime measurements due to the ability to reconstruct high charm-hadron
yields without biasing the measured decay time. The Belle II experiment at the
SuperKEKB asymmetric energy $e^{+}e^{-}$ collider BelleII ; SuperKEKB
benefits from excellent vertexing capabilities that provide a better decay-
time resolution than the previous $e^{+}e^{-}$ experiments at Belle and BaBar.
The Belle II detector BelleII  includes an upgraded tracking system comprised
of a two-layer silicon pixel detector, a four-layer double-sided silicon strip
detector, and a 56-layer central drift chamber. This tracking system enables
very good vertex resolution, which when coupled with a small beam size and
excellent alignment and calibration, enable precise, absolute measurements of
particle lifetimes. Lifetimes are calculated from the distance between the
production and decay vertices. In effect, lifetime measurements are therefore
an excellent probe of the beam spot and alignment calibrations using early
Belle II data.

The SuperKEKB collider provides world-leading instantaneous luminosity
according to the nano-beam scheme, in which the size of the electron and
positron beams in the plane transverse to their momentum is reduced by an
order of magnitude compared to those at the KEKB collider SuperKEKB . This
smaller beam size better constrains the event kinematics and allows for an
improved decay-time resolution for particles. The interaction region is
defined by the overlap of the beams, called the beam spot, which is calibrated
continuously using $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ events.

##  III Lifetime measurements

Precision measurements of the lifetimes of the $D^{0}$, $D^{+}$, and
$\Lambda_{c}^{+}$ are made at Belle II using unbinned, two-dimensional,
maximum-likelihood fits to the distributions of the decay-time t and decay-
time uncertainty $\sigma_{t}$ D ; Lc . Backgrounds are neglected for the
$D^{0}$ lifetime measurement, but the $D^{+}$ and $\Lambda_{c}^{+}$ lifetime
measurements account for backgrounds that remain after event selection by
applying a simultaneous fit to the events in the signal and sideband
invariant-mass regions. Fits to the mass distributions, as shown in Fig. 1 and
Fig. 2 for the $D^{0,+}$ and $\Lambda_{c}^{+}$ measurements, respectively, are
used to determine the rate of background contamination, which is applied as a
constraint in the lifetime fit.

Figure 1:  Invariant mass distributions for $D^{0}\rightarrow K^{-}\pi^{+}$
(top) and $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ (bottom) candidates with fit
projections overlaid D . The vertical dashed lines enclose the signal region.
The vertical dotted lines in the bottom figure enclose the sideband regions.
Figure 2:  Invariant mass distributions for $\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+}$ with fit projections overlaid Lc . The vertical dashed lines
enclose the signal region and the dotted lines enclose the sideband regions.

Using a data sample corresponding to 72 fb-1, the lifetime of the $D^{0}$ and
$D^{+}$ are measured to be $410.5\pm 1.1\pm 0.8$ fs and $1030.4\pm 4.7\pm 3.1$
fs, respectively, where the first uncertainty is statistical and the second is
systematic D . The decay-time distributions with lifetime fit projections
overlaid are shown in Fig. 3. These measurements are consistent with the
current world-averages, but with better precision. The sub-percent accuracy
establishes the excellent detector performance at Belle II and paves the way
for additional lifetime measurements.

Figure 3:  Decay-time distributions for $D^{0}\rightarrow K^{-}\pi^{+}$ (top)
and $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ (bottom) candidates in the
respective signal regions with fit projections overlaid D .

The lifetime of the $\Lambda_{c}^{+}$ is measured with a data sample
corresponding to 207.2 fb-1 using a relatively clean sample of
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ events. This data sample includes
additional data and an improved alignment calibration relative to the sample
used for the $D^{0}$ and $D^{+}$ lifetimes. Since the decay-time measurement
assumes prompt production of $\Lambda_{c}^{+}$ candidates at the interaction
region of the $e^{+}e^{-}$ beams, backgrounds due to
$\Xi_{c}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ and
$\Xi_{c}^{+}\rightarrow\Lambda_{c}^{+}\pi^{0}$ have the potential to bias the
lifetime measurement. These backgrounds were not accounted in the previous
measurement by CLEO Mahm  and are not relevant to the measurement made at LHCb
Aaij , which used $\Lambda_{c}^{+}$ candidates from semileptonic
$\Lambda_{b}^{0}$ decays. While the branching fraction for
$\Xi_{c}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ has been measured to be
$0.55\pm 0.20$% Xic0 , no measurement has been made for the branching fraction
for $\Xi_{c}^{+}\rightarrow\Lambda_{c}^{+}\pi^{0}$, though it was predicted to
be about twice that of $\Xi_{c}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ Xicp .
The lifetimes of the $\Xi_{c}^{0}$ and $\Xi_{c}^{+}$ are $153\pm 6$ fs and
$456\pm 5$ fs, respectively PDG . To reduce backgrounds from $\Xi_{c}$ decays,
events with an invariant mass difference between the $\Xi_{c}$ and
$\Lambda_{c}^{+}$ candidates within 2$\sigma$ of the expected value are
rejected. A conservative estimate of the remaining backgrounds is made by
fitting the impact parameter of the $\Lambda_{c}^{+}$ candidate in the plane
transverse to the beam direction with a signal and background shape. Since the
background shape includes both the $\Xi_{c}$ and combinatorial backgrounds, it
provides an estimated maximum contamination from $\Xi_{c}$ decays. Simulated
$\Xi_{c}^{0}\rightarrow\Lambda_{c}^{+}\pi^{-}$ and
$\Xi_{c}^{+}\rightarrow\Lambda_{c}^{+}\pi^{0}$ events are then added to a
simulated sample corresponding to five times the experimental data sample. The
difference in the $\Lambda_{c}^{+}$ lifetime extracted from a fit to the
simulated sample with and without the $\Xi_{c}$ events is 0.68 fs. Half of
this shift is subtracted from the $\Lambda_{c}^{+}$ lifetime measured in the
experimental data as a correction. The full amount of the correction is taken
as a systematic uncertainty associated with $\Xi_{c}$ backgrounds.

After the correction, the $\Lambda_{c}^{+}$ lifetime is determined to be
$203.20\pm 0.89\pm 0.77$ fs, where the first uncertainty is statistical and
the second is systematic Lc . This measurement is consistent with the current
world average, but with better precision than previous experiments. The slight
tension between the $\Lambda_{c}^{+}$ lifetime measurement by CLEO and other
measurements remains. The Belle II measurement provides a benchmark for future
baryon lifetime measurements.

Figure 4:  Decay-time distributions for $\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+}$ candidates in the signal region with fit projections overlaid
Lc .

##  IV Conclusions

The Belle II detector at the SuperKEKB collider benefit from upgrades in many
detector and accelerator components, in addition to improvements in
electronics, software, and physics analysis tools, and mark the beginning of
the next-generation super-B-factory. Even with early data, Belle II has shown
the capability to produce world-leading, high-precision results, including the
world’s best measurements of the lifetimes of the $D^{+}$, $D^{0}$, and
$\Lambda_{c}^{+}$. These measurements establish the excellent vertexing
capabilities of Belle II and provide benchmarks for future lifetime
measurements.

## References

  * (1) M. Neubert, Adv. Ser. Direct. High Energy Phys. 15, 239 (1998), arXiv:hep-ph/9702375.
  * (2) N. Uraltsev, in At the frontier of Particle Physics, edited by M. Shifman and B. Ioffe (2001) arXiv:hep-ph/0010328.
  * (3) A. Lenz and T. Rauh, Phys. Rev. D 88, 034004 (2013), arXiv:hep-ph/1305.3588.
  * (4) A. Lenz, Int. J. Mod. Phys. A 30, 1543005 (2015), arXiv:hep-ph/1405.3601.
  * (5) M. Kirk, A. Lenz, and T. Rauh, JHEP 12, 068 (2017), [Erratum: JHEP 06 (2020) 162], arXiv:hep-ph/1711.02100.
  * (6) H.-Y. Cheng, JHEP 11, 014 (2018), arXiv:hep-ph/1807.00916.
  * (7) J. Gratrex, B. Melić, and I. Nišandžić, (2022), arXiv:hep-ph/2204.11935.
  * (8) H.-Y. Cheng, Front. Phys. (Beijing) 10 (2015) 101406, arXiv:1508.07233.
  * (9) P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * (10) J. M. Link et al. (FOCUS Collaboration), Phys. Lett. B 537, 192 (2002).
  * (11) A. Kushnirenko et al. (SELEX Collaboration), Phys. Rev. Lett. 86, 5243 (2001), arXiv:hep-ex/0010014.
  * (12) G. Bonvicini et al. (CLEO Collaboration), Phys. Rev. Lett. 82, 4586 (1999), arXiv:hep-ex/9902011.
  * (13) J. M. Link et al. (FOCUS Collaboration), Phys. Rev. Lett. 88, 161801 (2002), arXiv:hep-ex/0202001.
  * (14) A. H. Mahmood et al. (CLEO Collaboration), Phys. Rev. Lett. 86, 2232 (2001), arXiv:hep-ex/0011049.
  * (15) R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 100, 032001 (2019), arXiv:hep-ex/1906.08350.
  * (16) R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 121, 092003 (2018), arXiv:hep-ex/1807.02024.
  * (17) R. Aaij et al. (LHCb Collaboration), Sci. Bull. 67, 479 (2022).
  * (18) T. Abe et al. (Belle II Collaboration), arXiv:ins-det/1011.0352.
  * (19) K. Akai, K. Furukawa, and H. Koiso, Nucl. Instrum. Meth. A 907, 188 (2018), arXiv:acc-ph/1809.01958.
  * (20) F. Abudinén et al. (Belle II Collaboration), Phys. Rev. Lett. 127, 211801 (2021), arXiv:hep-ex/2108.03216.
  * (21) F. Abudinén et al. (Belle II Collaboration), arXiv:hep-ex/2206.15227.
  * (22) R. Aaij et al. (LHCb Collaboration), Phys. Rev. D 102, 071101(R) (2020).
  * (23) P.-Y. Niu, Q. Wang, and Q. Zhao, Phys. Lett. B 826, 136916 (2022).