aainstitutetext: Rudolf Peierls Centre for Theoretical Physics, Clarendon
Laboratory, Parks Road, Oxford OX1 3PU, UK bbinstitutetext: CERN, EP
Department, CH-1211 Geneva 23, Switzerland

# Jet tagging in the Lund plane with graph networks

Frédéric A. Dreyer b Huilin Qu

###### Abstract

The identification of boosted heavy particles such as top quarks or vector
bosons is one of the key problems arising in experimental studies at the Large
Hadron Collider. In this article, we introduce LundNet, a novel jet tagging
method which relies on graph neural networks and an efficient description of
the radiation patterns within a jet to optimally disentangle signatures of
boosted objects from background events. We apply this framework to a number of
different benchmarks, showing significantly improved performance for top
tagging compared to existing state-of-the-art algorithms. We study the
robustness of the LundNet taggers to non-perturbative and detector effects,
and show how kinematic cuts in the Lund plane can mitigate overfitting of the
neural network to model-dependent contributions. Finally, we consider the
computational complexity of this method and its scaling as a function of
kinematic Lund plane cuts, showing an order of magnitude improvement in speed
over previous graph-based taggers.

††preprint: OUTP-20-15P

##  1 Introduction

As the Large Hadron Collider continues to explore proton collisions at the
energy frontier, an important task to search for signals of new physics beyond
the Standard Model (SM) is the identification of heavy particles at the
electroweak scale, which might emerge from decays of yet unknown heavier
particles.

An entity of particular interest in this quest is the jet, which is
essentially defined as a collimated bunch of particles with a certain energy
and direction, typically determined using a sequential recombination algorithm
Salam:2009jx . One important problem is that electroweak scale particles such
as vector bosons or top quarks produced from yet heavier new states can become
sufficiently boosted such that their hadronic decays are reconstructed as
single jets. It is therefore crucial to have efficient tools to probe the
radiation patterns within jets and determine their physical origin. This topic
has been the focus of much attention over the past decade, with a range of
approaches being developed to extract information from a jet’s substructure
Abdesselam:2010pt ; Altheimer:2012mn ; Altheimer:2013yza ; Adams:2015hiv ;
Marzani:2019hun . In recent years, a new generation of tools based on deep
learning models have emerged, which can achieve very high performance on
specific benchmarks deOliveira:2015xxd ; Komiske:2016rsd ; Louppe:2017ipp ;
Kasieczka:2017nvn ; Butter:2017cot ; Larkoski:2017jix ; Cheng:2017rdo ;
Macaluso:2018tck ; Moreno:2019bmu ; Kasieczka:2019dbj ; Qu:2019gqs ;
Lim:2020igi  and provide some insights into what kinematic variables drive the
discrimination performance Datta:2017rhs ; Datta:2017lxt ; Lim:2018toa ;
Komiske:2017aww ; Komiske:2018cqr ; Chakraborty:2019imr ; Kasieczka:2020nyd ;
Agarwal:2020fpt ; Chakraborty:2020yfc ; Dolan:2020qkr . A limitation of such
deep learning-based methods is the difficulty to estimate their uncertainties,
as well as their proneness to rely on unphysical features present in the
training data to achieve their high performance, as this data is generally
derived from Monte Carlo simulations of proton collisions.

In this article, we introduce a novel method to identify jets using graph
networks. To this end, we represent jets through their so-called Lund plane,
associating each Lund declustering with a node on the graph. Compared with
other state-of-the-art tools, our new method shows improved performance,
notably for processes with complicated topologies such as top decays, while
requiring substantially less training time. We will also investigate the
robustness of our new tagger, and show how through kinematic cuts on the Lund
variables one can mitigate overfitting to the model-dependent effects of Monte
Carlo simulations, reducing the reliance of the neural network on non-
perturbative contributions.

We provide a brief review of the Lund plane for jet physics in section 2, and
describe the LundNet model in section 3. Results for a range of benchmarks are
described in section 4, of which a summary is given in table 1. The robustness
and computational complexity of the models is explored in section 5. Finally,
we offer our conclusions in section 6.

|  |  | AUC |  |  $1/\epsilon_{B}$ at $\epsilon_{S}\\!=\\!0.5$ |  $1/\epsilon_{B}$ at $\epsilon_{S}\\!=\\!0.7$
---|---|---|---|---|---|---
| LundNet-5 |  | 0.938 |  | 609.8 | 70.4
| LundNet-3 |  | 0.935 |  | 500.0 | 61.5
$W$ tagging $p_{t}$>500 GeV | ParticleNet |  | 0.936 |  | 480.8 | 60.4
| LundNet-5 |  | 0.958 |  | 12500.0 | 813.2
| LundNet-3 |  | 0.956 |  | 8333.3 | 641.0
$W$ tagging $p_{t}$>2 TeV | ParticleNet |  | 0.958 |  | 8333.3 | 626.4
| LundNet-5 |  | 0.987 |  | 5000.0 | 1315.8
| LundNet-3 |  | 0.982 |  | 1785.7 | 333.3
top tagging $p_{t}$>500 GeV | ParticleNet |  | 0.983 |  | 2000.0 | 382.8
| LundNet-5 |  | 0.902 |  | 31.6 | 12.3
| LundNet-3 |  | 0.893 |  | 27.7 | 10.7
$q/g$ discrim. $p_{t}$>500 GeV | ParticleNet |  | 0.904 |  | 34.1 | 12.9
Table 1: Benchmarks of several jet tagging algorithms for a range of
processes. The first column gives the area under the ROC curve, and the later
two show the background rejection at two different signal efficiencies, 50%
and 70% respectively. In each case, larger values indicate better performance.

##  2 Jets in the Lund plane

The models we introduce in this article rely on the Lund plane
Andersson:1988gp . This representation provides a useful mapping of the
emission phase-space to a two dimensional plane representing the angle and
transverse momentum of a given emission with respect to its emitter, and which
is often used in discussions of resummations of large logarithms in
perturbation theory or of Monte Carlo parton showers. Each emission then
creates an additional triangular leaf corresponding to the phase space for
further emissions. It was shown in recent work that the Lund plane provides a
useful basis to achieve an efficient description of the clustering sequence of
a jet, containing a rich set of information about its substructure, with
notable potential for jet tagging Dreyer:2018nbf . The Lund jet plane allows
for a visual representation of the clustering history of a jet. This
systematic encoding of a jet’s radiation patterns can be measured
experimentally Aad:2020zcn , allowing for comparisons between theoretical
predictions and experimental data Lifson:2020gua  and with potential for
constraining general purpose Monte Carlo event generators Dasgupta:2020fwr .

Figure 1: The Lund plane representation of a jet (left) where each emission is
positioned according to its $\Delta$ and $k_{t}$ coordinates, and the
corresponding mapping to a binary Lund tree of tuples (right). The thick blue
line represents the primary sequence of tuples $\mathcal{L}_{\text{primary}}$.

The Lund plane is obtained by first reclustering a jet’s constituents with the
Cambridge/Aachen (CA) algorithm Dokshitzer:1997in ; Wobisch:1998wt , which
sequentially identifies and combines the pair of particles $a$ and $b$ closest
in rapidity $y$, a measure of relativistic velocity along the beam axis, and
azimuthal angle $\phi$ around the same axis, i.e. minimising
$\Delta^{2}=(y_{a}-y_{b})^{2}+(\phi_{a}-\phi_{b})^{2}$. We then iterate over
this clustering sequence, starting from the full jet and proceeding by:

  1. 1.

Declustering the current (pseudo)jet into two transverse momentum ordered
pseudojets $a$ and $b$ such that $p_{t,a}>p_{t,b}$, and where we consider $b$
to be the emission of the $(a+b)$ emitter.

  2. 2.

Determining a number of kinematic variables associated with the declustering
step $i$, which we denote as a tuple $\mathcal{T}^{(i)}$

$\mathcal{T}^{(i)}=\\{k_{t},\Delta,z,m,\psi\\}\,.$ (1)

Here $k_{t}=p_{t,b}\Delta$ is the transverse momentum of emission $b$ with
respect to its emitter in the limit where $p_{t,b}\ll p_{t,a}$, $\Delta$ is
the previously defined rapidity-azimuth distance,
$z=p_{t,b}/(p_{t,a}+p_{t,b})$ is the momentum fraction of the softer subjet
$b$, $m$ is the invariant mass of the $(a+b)$ pair, and
$\psi=\tan^{-1}\big{(}\tfrac{y_{b}-y_{a}}{\phi_{b}-\phi_{a}}\big{)}$ is the
azimuthal angle around subjet $a$’s axis.

  3. 3.

Repeating this procedure for pseudojets $a$ and $b$ if they contain more than
one particle.

This procedure produces a binary Lund tree with a tuple of variables
$\mathcal{T}^{(i)}$ for each node $i$ of the Lund tree, as shown in figure 1.
The first two elements of the tuple provide the coordinates in the Lund plane
of the corresponding splitting, and the remaining ones provide complementary
kinematic information. A subset of this tree of particular significance is the
primary list of tuples $\mathcal{L}_{\text{primary}}$ containing the kinematic
variables of each splitting along the primary branch of the tree, i.e.
following only the pseudojet with larger transverse momentum in step 3. of the
algorithm above, corresponding to points on the blue primary plane in figure
1. The primary Lund sequence can be used notably for two-dimensional visual
representations of the radiation patterns in a jet Dreyer:2018nbf ;
Aad:2020zcn ; Andrews:2018jcm .

Figure 2: Left: Distribution of the number of Lund declusterings per jet, for
different choices of $k_{t}$ cuts in the Lund plane. The dashed lines indicate
the mean of each distribution. Right: Average of the number of nodes per jet
as a function of the $k_{t}$ cut.

Corrections to the Lund plane originating from non-perturbative hadronisation
effects affect the low $k_{t}$ region of the plane. One can therefore limit
the dependence on the non-perturbative region of any model trained on Lund
declusterings by removing emissions that fall below a certain transverse
momentum $k_{t}$ threshold. In figure 2 (left), we show the distribution of
the number of Lund declusterings per jet for several choices of $k_{t}$ cut in
2 TeV QCD jets simulated using Pythia 8.223 Sjostrand:2014zea . The mean of
each distribution is indicated as a dashed line. An additional benefit of a
$k_{t}$ threshold is that even for small cut values the number of nodes per
jet is significantly reduced, and therefore correspondingly so the
computational cost of training a machine learning model on these inputs. The
right-hand side of figure 2 shows the average number of nodes per jet as a
function of the $k_{t}$ cut, which decreases quadratically as the cut is
increased.

##  3 LundNet Models

Figure 3: (a) Illustration of the EdgeConv operation on a node of the Lund
tree. (b) Architecture of the EdgeConv block used in the LundNet model. (c)
Architecture of the LundNet model.

The Lund plane encodes a rich set of information of the substructure and
radiation patterns of a jet, therefore serving as a natural input to machine
learning models for jet physics. The use of Lund planes for jet tagging was
first proposed in Ref. Dreyer:2018nbf  where log-likelihood and deep learning
models are applied, and good performance was observed for tagging boosted
electroweak bosons. However, the main focus of Ref. Dreyer:2018nbf  was the
primary Lund plane, which inevitably leads to some loss of information due to
the omission of the secondary and tertiary splittings. In this article, we
propose LundNet, a new deep learning model capable of digesting the full Lund
plane. Graph neural networks are used in this model to better exploit the
structural information associated with the Lund plane representation of a jet,
leading to significantly improved performance on a range of jet tagging
benchmarks.

The LundNet model starts with transforming the Lund tree into a graph, where
each node corresponds to a Lund declustering and carries the tuple of
kinematic variables $\mathcal{T}^{(i)}$ as its input features, and
bidirectional edges are formed following the structure of the Lund
declustering tree. The graph network architecture is adapted from the
ParticleNet Qu:2019gqs  model, with the EdgeConv operation proposed in Ref.
DGCNN  as a core step. Figure 3(a) illustrates how EdgeConv operates for one
node (the highlighted one) in the Lund tree. It consists of two steps: First,
a shared multi-layer perceptron (MLP) is applied to each of its incoming
edges, using features of the node pair connected by the edge as inputs, and
produces a learned “edge feature”. As the Lund tree is a binary tree, there
are only up to three edges for each node, which do not require a nearest-
neighbour search, therefore the computational cost is much lower than for the
ParticleNet model. As shown in figure 3(b), we use two layers for this shared
MLP, each consisting of a linear layer followed by a batch normalization (BN)
DBLP:journals/corr/IoffeS15  and a ReLU activation glorot2011deep . Then, an
aggregation step is performed for the node by taking an element-wise average
of the learned edge features of all the incoming edges. A shortcut connection
he2016deep  is also added to take the original node features into account
directly, and the node feature is then updated to the new value. This
operation is performed for all the nodes using the same shared MLPs, therefore
updating all the node features but keeping the graph structure unchanged.

The architecture of the LundNet model is shown in figure 3(c). We stack six
such EdgeConv blocks to form a deep graph network. The number of channels of
the MLPs are $(32,32)$, $(32,32)$, $(64,64)$, $(64,64)$, $(128,128)$ and
$(128,128)$ for the six EdgeConv blocks, respectively. Outputs from these
EdgeConv blocks are concatenated per node and further processed by another MLP
with 384 channels to better aggregate features learned at different stages. A
global average pooling is applied afterwards to read out information from all
nodes in the graph. This is followed by a fully connected layer with 256 units
and a dropout layer with a drop probability of 0.1, before the final
classification output.

The LundNet model uses the Lund kinematic variables defined in equation (1) as
the input node features. Two variants of the LundNet models are investigated
in this article. The first one uses all five Lund variables,

$(\ln k_{t},\ln\Delta,\ln z,\ln m,\psi)\,$ (2)

as input features to extract as much information as possible from the Lund
plane to maximize the jet tagging performance and is referred to as LundNet-5.
The second one uses only three Lund variables,

$(\ln k_{t},\ln\Delta,\ln z)\,$ (3)

and is referred to as LundNet-3. The removal of the $\ln m$ and $\psi$
variables significantly increases the resilience of the model to non-
perturbative effects at only a small cost of the performance, as will be
discussed in Section 5.

We implement the LundNet model with the Deep Graph Library 0.4.3 wang2019dgl
using the PyTorch 1.7 NEURIPS2019_9015  backend. The training is performed on
a Nvidia GTX 1080 Ti graphics card with a minibatch size of 256. The Adam
optimizer DBLP:journals/corr/KingmaB14  is used to minimize the cross entropy
loss. The training is performed for 30 epochs, with an initial learning rate
of 0.001, and subsequently lowered by a factor of 10 after the 10th and the
20th epochs. A snapshot of the model is saved at the end of each epoch, and
the model snapshot showing the best accuracy on the validation dataset is
selected for the final evaluation.

##  4 Jet tagging in the Lund plane

Let us now turn to a detailed evaluation of our models for the identification
of several hallmark signals at the LHC. We will look at four different
benchmarks: the tagging of boosted electroweak $W$ for two different
transverse momentum cuts, the tagging of top quarks, and the discrimination
between quark and gluon jets. The data samples consist in each benchmark of
1.2m signal and background jets simulated through the corresponding process
with Pythia 8.223 Sjostrand:2014zea , with an equal split between signal and
background events. Events are generated at hadron-level with underlying event
turned on, but without including detector effects or the presence of
additional pile-up collisions. A subset of 100k jets each are used as
validation and test data, with the same number of signal and background events
in both samples. All data sets are taken from Ref. Carrazza:2019efs ;
groomRL_data . Jets are clustered using the anti-$k_{t}$ algorithm
Cacciari:2008gp ; Cacciari:2011ma  with a radius $R=1.0$ using FastJet 3.3.2,
and are required to pass a selection cut, with transverse momentum $p_{t}>500$
GeV or $p_{t}>2$ TeV as indicated, and rapidity $|y|<2.5$. In each event, only
the two jets with the highest transverse momentum are considered, and are
saved as training data if they pass the selection cuts. A summary of these
benchmarks is given in table 1.

###  4.1 $W$ tagging

Figure 4: Background rejection $1/\epsilon_{\text{QCD}}$ versus signal
efficiency $\epsilon_{W}$ for $W$ jet tagging with transverse momentum
$p_{t}>500$ GeV.

We start by considering the identification of hadronically decaying $W$
bosons, one of the key objects commonly appearing in high energy proton
collisions. The signal data is obtained from 600k jets passing the selection
cuts and simulated using the $pp\rightarrow WW$ process, where the $W$ bosons
are decayed hadronically. The background consists of the same number of QCD
jets simulated through a sample of $pp\rightarrow jj$ events. Training of the
neural network weights for every model is performed using 500k of the $W$ and
background samples each. At the end of each epoch, the performance is
monitored on a separate validation sample consisting of 100k jets. The final
performance of each model is then evaluated using a further independent sample
of 100k jets with an equal number of signal and background events.

In fig. 4 we show for each model the background rejection
$1/\epsilon_{\mathrm{QCD}}$ against the signal efficiency $\epsilon_{W}$ for
$W$ bosons, for jets passing a transverse momentum cut of $p_{t}>500$ GeV.
Better performance translates to curves that achieve a higher background
rejection for a given signal efficiency, i.e. which are closer to the top
right corner of the figure. We compare the LundNet-3 and LundNet-5 models with
three recent benchmarks: the ParticleNet model introduced in Qu:2019gqs , the
RecNN model from Louppe:2017ipp  and the Lund+LSTM model from the original
Lund plane paper Dreyer:2018nbf , which uses an LSTM network on the primary
Lund sequence. Both the RecNN and the Lund+LSTM models, while superior to
heuristic substructure algorithms, are vastly outperformed by all of the graph
based methods considered. The LundNet-3 model is able to achieve about the
same signal purity as ParticleNet, but can be trained in substantially less
time, as will be discussed in more detail in section 5.3, and takes only a
small 3-dimensional input for each declustering node in the Lund plane. By
including more kinematic information, the LundNet-5 model is able to provide a
slightly higher performance, but as we will see in section 5, this comes at
the price of being less robust to non-perturbative effects than its lower-
dimensional counterpart.

Figure 5: Background rejection $1/\epsilon_{\text{QCD}}$ versus signal
efficiency $\epsilon_{W}$ for $W$ jet tagging with transverse momentum
$p_{t}>2$ TeV.

In figure 5, we show the same process but with a transverse momentum selection
cut of $p_{t}>2$ TeV for the jets. Here we can observe roughly the same
qualitative behaviour as at lower transverse momentum, but with the LundNet-5
model now clearly outperforming the remaining taggers even at high signal
efficiencies. At higher transverse momentum, the peak in the Lund plane
associated with the $W$ splitting, and the corresponding depletion associated
with the colour-singlet nature of the $W$, become more distinguishable. The
Lund+LSTM model, which relies purely on the primary Lund sequence, also shows
a strong performance, although it is still lags significantly behind all the
graph-based approaches.

###  4.2 Top tagging

Figure 6: Background rejection $1/\epsilon_{\text{QCD}}$ versus signal
efficiency $\epsilon_{\text{Top}}$ for top jet tagging with transverse
momentum $p_{t}>500$ GeV.

We now turn to the identification of jets originating from top quark decays.
Top quarks are of particular interest at the LHC, interacting strongly with
the Higgs boson and providing a valuable avenue in searches for new physics,
as well as being the only quarks to decay before hadronising. Here the signal
data is obtained from the $pp\rightarrow t\bar{t}$ process in Pythia 8.223,
where the top quarks decay to hadrons and the jets are required to pass a 500
GeV transverse momentum cut. The background QCD jets are identical to the ones
used in figure 4. Each model is again trained using 500k signal and 500k
background jets, with further validation and testing samples that are both one
tenth the size of the training data.

In fig. 6, we show the QCD background rejection as a function of the top
efficiency. In this case, the Lund+LSTM model does not perform as well as
RecNN. This is to be expected, as it was designed for one or two-pronged jet
identification and uses only information from the primary Lund declustering
sequence. It therefore contains information about the structure of only one of
the initial decay products of the original top quark, limiting the performance
that can be achieved without input from secondary planes. It is however
interesting to see that in this process with more complex topology, the
LundNet-5 model provides a substantial performance gain over existing state-
of-the-art methods such as ParticleNet. This is due to the nature of its
input, which contains already high-level kinematic information about the
radiation patterns of the jet, making it much simpler for the neural network
to learn how to distinguish signals with more involved signatures. Thus the
LundNet-3 model achieves almost the same signal purity as the ParticleNet
algorithm, despite having as input only a reduced 3-tuple of kinematic
variables per node and taking about an order of magnitude less time to train.
Interestingly, the performance gap between the two LundNet taggers is entirely
due to the addition of the subjet mass and azimuthal angle $\psi$ to the input
features of each declustering for the LundNet-5 model.

###  4.3 Quark/Gluon discrimination

Figure 7: Background rejection $1/\epsilon_{\text{Gluon}}$ versus signal
efficiency $\epsilon_{\text{Quark}}$ for quark/gluon discrimination between
$R=0.4$ anti-$k_{t}$ jets with transverse momentum $p_{t}>500$ GeV.

Our final benchmark considers the discrimination between quark and gluon
initiated jets, a core challenge in collider physics which has seen much
research in recent years Gallicchio:2011xq ; Larkoski:2013eya ;
Larkoski:2014pca ; Bhattacherjee:2015psa ; Komiske:2016rsd ; Frye:2017yrw ;
Metodiev:2018ftz ; Larkoski:2019nwj . For this study, we consider a signal
sample of 500k quark-initiated jets obtained through the $q\bar{q}\rightarrow
q\bar{q}$ process in Pythia 8.223, while the background is obtained from
$gg\rightarrow gg$ events. The jets are clustered with an anti-$k_{t}$
algorithm with radius $R=0.4$ and are again required to pass a transverse
momentum $p_{t}>500$ GeV and rapidity $|y|<2.5$ selection cut.

The gluon-jet rejection as a function of the quark-jet efficiency is shown in
figure 7. In this case there is not as large a hierarchy between models, with
the Lund+LSTM model performing somewhat below the competing approaches.
ParticleNet has a slight edge over the other algorithms at small quark
efficiencies, but is indistinguishable from the LundNet-5 tagger at high
efficiency. The LundNet-3 and RecNN models show similar performance at high
efficiency, with RecNN providing slightly higher gluon rejection at lower
quark efficiencies.

##  5 Robustness study

We will now investigate the robustness of the different models we considered
in our benchmarks. To this end we will consider three axes: their resilience
to non-perturbative effects, their resilience to detector effects, and the
complexity and computational cost of each tagger.

###  5.1 Non-perturbative effects

Figure 8: Performance $\tfrac{\epsilon_{W}}{\sqrt{\epsilon_{\text{QCD}}}}$
versus resilience to non-perturbative effects.

Beyond its raw performance, it is important for practical applications that a
tagger be relatively robust to model-dependent non-perturbative effects. To
carry out studies of sensitivity to non-perturbative effects, we compare
performance between a data sample of both 50k signal and background jets
produced at parton level, and a sample obtained with hadronisation and
underlying event models turned on in the event generator. The same model,
trained on hadron-level data, is evaluated on both samples for the comparison.
For this study, we use the same 2 TeV $W$ jet sample as was used in section
4.1 as well as the corresponding models shown in figure 5, which are now used
to label jets from both parton and hadron-level data.

Figure 9: Background rejection as a function of $W$ tagging efficiency. Dotted
lines indicate a $W$ tagger applied on parton-level data.

Figure 8 shows the robustness of the tagger in conjunction with its
performance. This robustness is measured through the resilience
$\zeta_{\text{NP}}$ Bendavid:2018nar , calculated using both the efficiency on
the hadron-level sample, $\epsilon$, and that on the parton-level sample,
$\epsilon^{\prime}$

$\zeta_{\text{NP}}=\left(\frac{\Delta\epsilon_{W}^{2}}{\langle\epsilon\rangle^{2}_{W}}+\frac{\Delta\epsilon_{\mathrm{QCD}}^{2}}{\langle\epsilon\rangle^{2}_{\mathrm{QCD}}}\right)^{-1/2}\,,$
(4)

where $\Delta\epsilon=\epsilon-\epsilon^{\prime}$ and
$\langle\epsilon\rangle=1/2\left(\epsilon+\epsilon^{\prime}\right)$. The
efficiencies are obtained with a fixed cut corresponding to a signal
efficiency $\epsilon_{W}=70\%$ on the hadron-level sample. The curves in
figure 8 are obtained by increasing a transverse momentum cut on the $k_{t}$
variable of the Lund plane, progressively removing declustering nodes that
fall below the cut. Each curve starts on the upper left of figure 8, with a
model trained without any cuts on the Lund plane, and ends in the lower right
part of the figure with a model trained with a transverse momentum cut $\ln
k_{t}/\text{GeV}>2$ that has higher resilience but lower performance due to
the removal of parts of the Lund tree. We can observe that despite their good
performance, the ParticleNet and RecNN models have very little resilience to
non-perturbative effects, and have no handles through which such robustness
can be consistently imposed. Somewhat surprisingly, the LundNet-5 also offers
relatively poor robustness to non-perturbative effects. This is due to its
higher dimensional input state, which allows the neural network to extrapolate
some information on emissions in the non-perturbative regime despite the
presence of a transverse momentum cut. In contrast, the LundNet-3 model
becomes very resilient to non-perturbative effects as the transverse momentum
cut is increased, outperforming the Lund+LSTM model by a factor two for the
same resilience value.

In figure 9 (left), we show the ROC curve for each model trained on the
hadron-level $W$ data, with the ROC curve obtained on the parton-level data
shown as a dotted line. The lower panel provides the ratio between the parton-
level ROC curve and the hadron-level one. The right-hand side of figure 9
gives the ROC curve of the LundNet-3 models obtained for several choices of
the $\ln k_{t}$ transverse momentum cut applied on the Lund tree. Here we can
observe the improved resilience as the $k_{t}$ cut is increased, with the $\ln
k_{t}/\text{GeV}>1$ model providing almost the same performance at parton and
hadron level, albeit at the cost of a factor 20 in background rejection when
compared to the unconstrained model.

###  5.2 Detector effects

Figure 10: Performance $\tfrac{\epsilon_{W}}{\sqrt{\epsilon_{\text{QCD}}}}$
versus resilience to detector effects.

Let us now turn to the impact of detector effects on the model robustness. To
this end, we create a sample of 100k 2 TeV $W$ and QCD jets using Pythia
8.223, including fast detector simulation with Delphes v3.4.1 using the
delphes_card_CMS_NoFastJet.tcl card to simulate both detector effects and
particle flow reconstruction deFavereau:2013fsa . The effects of detector
granularity are then partially mitigated by applying a subjet-particle
rescaling algorithm Dreyer:2018nbf ; Son:2012mb , where the Delphes particle-
flow objects in a jet are reclustered into subjets using a CA algorithm with
$R_{h}=0.12$ and rescaling the particle flow charged-particle and photons by a
factor

$f=\frac{\sum_{i\in\text{subjet}}p_{t,i}}{\sum_{i\in\text{subjet}(h^{\pm},\gamma)}p_{t,i}}.$
(5)

before discarding neutral hadron candidates. The resulting particles of all
subjets are then reclustered into a single jet on which the Lund tree can be
measured.

Figure 11: Background rejection as a function of $W$ tagging efficiency.
Dotted lines indicate a $W$ tagger applied on detector-level data.

Applying the 2 TeV $W$ taggers trained in section 4.1 on this sample, we can
now compute an index of resilience to detector effects $\zeta_{\text{D}}$ in
the same way as was done for non-perturbative effects, but taking now
$\epsilon^{\prime}$ in equation (4) to be the detector-level efficiency. We
show in figure 10 the resilience as a function of performance
$\epsilon_{W}/\sqrt{\epsilon_{\text{QCD}}}$ for a signal efficiency of
$\epsilon_{W}=70\%$ on the hadron-level data. One can observe here that while
for high performance, good resilience can be achieved, a transverse momentum
cut in the Lund plane does not result models that are particularly insensitive
to detector effects. Adding a further Lund-plane angular cut $\ln 1/\Delta<4$
to remove unmitigated effects due to electromagnetic calorimeter granularity
did not provide any noticeable improvement, as is shown in dashed lines in the
figure for the Lund+LSTM model.

The limitations in achieving higher resilience values for any of the
considered models are due to the consistently enhanced performance of taggers
at hadron-level. We show the ROC curves for each model in figure 11 (left),
with the dotted lines showing the hadron-level model applied on detector-level
data. Here one can note the performance of the Lund+LSTM model on the
detector-level sample, achieving performance quite close to the LundNet and
ParticleNet models. The LundNet-5 model in particular, is performing slightly
worse than LundNet-3 when applied on the detector-level sample, despite having
a substantial edge over it on the hadron-level data that both were trained on.
The lower panel gives the ratio between both curves, with the background
rejection ratio of the Lund+LSTM tagger with an angular cut $\ln 1/\Delta<4$
shown in dashed lines. In the right-hand side of figure 11, one can see the
ROC curve of the LundNet-3 model trained for increasing $k_{t}$ cuts, showing
somewhat improved robustness at larger cut values.

###  5.3 Complexity of models

An important quality for a deep learning-based jet tagger is the simplicity of
the model, and the speed of its training and inference on new samples. To
quantify these considerations we measure three different metrics for the
models:111We do not include a comparison with RecNN as this model was trained
on a CPU.

  * •

the number of trainable parameters of the model,

  * •

the training time of each model per data sample and per epoch, which provides
a measure of the time needed to train a full tagger on a given data set,

  * •

and finally the inference time per sample of the trained model on new data
points, which provides a measure of how efficiently an existing model can be
deployed to label a given sample of jets.

|  | Number of |  | Training time | Inference time
---|---|---|---|---|---
|  | parameters |  | [ms/sample/epoch] | [ms/sample]
LundNet |  | 395k |  | 0.472 | 0.117
ParticleNet |  | 369k |  | 3.488 | 1.036
Lund+LSTM |  | 67k |  | 0.424 | 0.131
Table 2: Summary for each model of the number of parameters, training time per
sample and epoch, and inference time per sample. The time is measured in
milliseconds as obtained when running the models on an Nvidia GTX 1080 Ti
card.

We evaluate the training time on the 2 TeV $W$ and QCD training sample used
previously in section 4.1, and the inference time on the corresponding test
data of 100k jets. The results are shown in table 2 for the graph-based models
and the LSTM tagger. As LundNet-3 and LundNet-5 only differs in the dimension
of the input features, the number of parameters and the computational cost are
essentially the same, therefore we do not distinguish between them in this
section and provide numbers derived from the LundNet-3 tagger. The Lund+LSTM
model has a much simpler architecture, resulting in only 67k trainable
parameters, significantly less than any of the graph-based models. It is
however not substantially faster than these larger models, and even
underperforms the LundNet models in inference time. The relatively long
training time is partly due to the smaller learning rate used when training
the LSTM network, and the smaller number of epochs needed for the Lund+LSTM
model to converge. Due to its increased number of EdgeConv blocks, the LundNet
model has 26k more parameters than ParticleNet. However, the direct use of the
Lund tree as the graph structure removes the need for a costly nearest-
neighbour search and also significantly reduces the number of edges for each
node, therefore increasing both the training and inference speed by almost an
order of magnitude. This is compounded by the fact that due to their higher-
level kinematic inputs, the LundNet models take significantly less epochs to
converge to a good solution.222We note that in this benchmark the time needed
to pre-process jets from list of particles to input data to each model is not
included. Due to its reliance on recursion, our python implementation takes
about 4.3 ms per jet to recluster a jet and transform the clustering tree into
a graph of Lund nodes. This is however completely dependent on the data format
used when saving Pythia events and can be therefore significantly reduced
through a more efficient processing pipeline implementation.

Figure 12: Inference time per jet of the LundNet model as a function of the
mean number of Lund declusterings per 2 TeV QCD jet. Each circle corresponds
to a separate LundNet model trained for a different $k_{t}$ cut, as indicated
in the figure text.

An interesting side-effect of the $k_{t}$ cut applied in the Lund plane to
improve the robustness of the model as described in previous sections is that
it also reduces the number of nodes present in the graph. As such, both the
training and the inference time of the model are expected to be reduced as the
transverse momentum cut is increased and more nodes are removed from the input
graph. To demonstrate this we show in figure 12 the inference time per sample
as a function of the average number of Lund declusterings per QCD jet,
obtained through models trained with different Lund plane $k_{t}$ cuts, each
of which is shown as blue circle. As expected, the inference time scales
linearly with the number of nodes in the graph, such that computing time
increases quadratically as the $\ln k_{t}$ cut is reduced.

##  6 Conclusions

In this article, we have introduced LundNet, a novel algorithm used to detect
signals at the LHC. We showed that this method provides substantial
improvements over existing methods on the identification of key benchmark
processes, as well as in training speed and robustness to non-perturbative
effects.

The LundNet model combines the power of graph convolutional networks with an
efficient representation of the radiation patterns within a jet to optimally
extract information from its substructure. Jets are represented through a Lund
tree constructed from the CA clustering tree of each jet. Each node of the
Lund tree contains a tuple of kinematic information for the corresponding
pairwise splitting, used as input to the graph network. By using the
clustering tree structure to aggregate information in the graph convolution,
the weights of the LundNet model can be trained ten times faster than previous
graph-based methods such as ParticleNet, with a similar gain on the inference
time when applying the trained model to identify new jets.

We introduced two taggers, LundNet-3 and LundNet-5 which rely on a three- and
five-dimensional emission feature space respectively. The LundNet-5 tagger
outperforms current state-of-the-art methods on several pivotal jet tagging
benchmarks, most notably for the identification of jets originating from top
decays, while the lower-dimensional LundNet-3 tagger matches the performance
of current jet taggers despite its reduced kinematic input size.

We provided a concrete study of its resilience to model-dependent non-
perturbative and detector effects. Through the use of an appropriate
transverse momentum cut in the Lund plane, we showed how one can establish an
algorithm that retains high performance while maintaining a handle on
robustness. Due to its limited kinematic input, the LundNet-3 tagger is best
positioned to provide jet identification that is relatively insensitive to
non-perturbative effects and detector smearing, while substantially
outperforming previous methods based on the primary Lund plane.

These results offer a concrete avenue to implementing effective machine-
learning based taggers that can be robust to model-dependent effects present
in the training data, a key feature for real-life applications of artificial
intelligence at the LHC. In this context, the work presented in this article
provides a key step towards a new generation of efficient, robust and
tractable jet substructure tools for LHC physics.

## Acknowledgments

We are grateful to Gregory Soyez and Gavin Salam for useful discussions and
comments on the manuscript. F.D. is supported by the Science and Technology
Facilities Council (STFC) under grant ST/P000770/1.

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# A tentative 114-minute orbital period challenges the ultra-compact nature of
the X-ray binary 4U 1812–12

Montserrat Armas Padilla Instituto de Astrofísica de Canarias, E-38205 La
Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La
Laguna, E-38206 La Laguna, Tenerife, Spain<EMAIL_ADDRESS>Pablo
Rodríguez-Gil Instituto de Astrofísica de Canarias, E-38205 La Laguna,
Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna,
E-38206 La Laguna, Tenerife, Spain  Teo Muñoz-Darias Instituto de Astrofísica
de Canarias, E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica,
Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain  Manuel
A.P.Torres Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife,
Spain  Departamento de Astrofísica, Universidad de La Laguna, E-38206 La
Laguna, Tenerife, Spain  Jorge Casares Instituto de Astrofísica de Canarias,
E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad
de La Laguna, E-38206 La Laguna, Tenerife, Spain  Nathalie Degenaar Anton
Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands  Vik S. Dhillon Department of Physics and
Astronomy, University of Sheffield, Sheffield S3 7RH, UK  Instituto de
Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain  Craig O. Heinke
Department of Physics and Astronomy, University of Alberta, Edmonton T6G 2E9,
Alberta, Canada  Stuart P. Littlefair Department of Physics and Astronomy,
University of Sheffield, Sheffield S3 7RH, UK  Thomas R. Marsh Department of
Physics, University of Warwick, Coventry CV4 7AL, UK

###### Abstract

We present a detailed time-resolved photometric study of the ultra-compact
X-ray binary candidate 4U 1812–12. The multicolor light curves obtained with
HiPERCAM on the 10.4-m Gran Telescopio Canarias show a $\simeq 114$ min
modulation similar to a superhump. Under this interpretation, this period
should lie very close to the orbital period of the system. Contrary to what
its other observational properties suggest (namely, persistent dim luminosity,
low optical-to-X-ray flux ratio and lack of hydrogen features in the optical
spectrum), this implies that 4U 1812–12 is most likely not an ultra-compact
X-ray binary, which are usually defined as systems with orbital periods lower
than 80 min. We discuss the nature of the system, showing that a scenario in
which 4U 1812–12 is the progenitor of an ultra-compact X-ray binary may
reconcile all the observables.

Stellar accretion disks (1579) — Low-mass x-ray binary stars (939) — Neutron
stars (1108)

††facilities: GTC(HiPERCAM)††software: ANOVA (Schwarzenberg-Czerny, 1996),
HiPERCAM pipeline (https://github.com/HiPERCAM/), Peranso (www.peranso.com)

##  1 Introduction

The ultra-compact family of low mass X-ray binaries (LMXBs) is composed of
those systems in which a compact object, a neutron star or a black hole,
accretes material from an evolved, hydrogen-deficient companion star in a
tight orbit with a period $P_{\mathrm{orb}}$ $<80$ min (Rappaport et al.,
1982; Verbunt & van den Heuvel, 1995). Three evolutionary channels, depending
on the nature of the donor star, have been proposed to explain the origin of
such compact systems: the white dwarf channel, the helium star channel and the
evolved main-sequence star channel (see e.g., Nelemans et al., 2010; Heinke et
al., 2013). Distinguishing between these formation paths is not always
straightforward, since the final state of the donor is similar for the three
scenarios. In this regard, the study of the progenitors of ultra-compact
systems (i.e., before the orbital period becomes shorter than 80 min) may be
key to shed light on the origin of this family.

The neutron star LMXB 4U 1812–12 is a strong ultra-compact candidate. Its
persistently low X-ray luminosity ($\sim 4\times 10^{35}$
$\mathrm{erg~{}s}^{-1}$) and low optical–to–X-ray flux ratio suggest that it
harbors a small accretion disk (van Paradijs & McClintock, 1994; Bassa et al.,
2006; in ’t Zand et al., 2007). In addition, its optical spectrum lacks
hydrogen spectral features, which suggest a hydrogen-exhausted donor star
(Armas Padilla et al., 2020).

In order to confirm the ultra-compact nature of 4U 1812–12, we performed a
detailed time-resolved photometric study using data taken with the HiPERCAM
imager on the 10.4-m Gran Telescopio Canarias (GTC).

##  2 Observations and reduction

We obtained images of 4U 1812–12 with HiPERCAM (Dhillon et al., 2021) on the
GTC in La Palma. This high-speed camera uses four dichroic beamsplitters and
five frame-transfer CCDs to simultaneously image the $u_{\mathrm{s}}$,
$g_{\mathrm{s}}$, $r_{\mathrm{s}}$, $i_{\mathrm{s}}$ and $z_{\mathrm{s}}$
optical bands111HiPERCAM is equipped with ”Super” SDSS filters, high-
throughput versions of the SDSS filters, hence the ”s” subscript.. The CCD
detectors were used in full-frame mode with slow readout and no binning. The
observations presented here were taken on 2021 June 15 with an exposure time
of 12.9 s (with only 8 ms dead time between exposures) for 1129 frames ($\sim
4$-hour total coverage). The cadence in the $u_{\mathrm{s}}$ and
$g_{\mathrm{s}}$ bands was slower by a factor of 15 in an attempt to increase
the signal-to-noise ratio in these bands. The data were reduced using the
HiPERCAM pipeline222https://github.com/HiPERCAM/. We block averaged the
$r_{\mathrm{s}}$, $i_{\mathrm{s}}$ and $z_{\mathrm{s}}$ images and obtained 60
images per band with improved signal-to-noise ratio. We then extracted the
count rates of 4U 1812–12 and a comparison star333Gaia DR2 4153779729434598528
via variable aperture photometry by direct summing of the sky subtracted flux
over the aperture, i.e. with no profile weighting. The signal-to-noise ratios
of the $u_{\mathrm{s}}$ and $g_{\mathrm{s}}$ images were insufficient to
extract reliable fluxes. In fact, the source is not detected in either the
$u_{\mathrm{s}}$ or $g_{\mathrm{s}}$ stacked images, so these bands are not
discussed further in this paper. The measured seeing gradually degraded during
the last third of the observation from a median value in the $z_{\mathrm{s}}$
band of 1.1 to 1.8 arcsec.

Figure 1: GTC/HiPERCAM $r_{\mathrm{s}}$$i_{\mathrm{s}}$$z_{\mathrm{s}}$ light
curves of 4U 1812–12. The individual images were block averaged into 60 bins
prior to flux extraction to enhance the signal-to-noise ratio.

##  3 Period analysis and results

The light curves of 4U 1812–12 are shown in Fig. 1. The average magnitudes of
the system are $r_{\mathrm{s}}=23.17\pm 0.02$, $i_{\mathrm{s}}=21.93\pm 0.01$
and $z_{\mathrm{s}}=21.28\pm 0.01$ mag. To compute the light curves we used
the Pan-STARRS Data Release 1 magnitudes of the comparison star $r=20.33\pm
0.02$, $i=18.58\pm 0.01$ and $z=17.42\pm 0.01$ mag.

We analyzed the $z_{\mathrm{s}}$-band light curve of 4U 1812–12 using the
analysis of variance (ANOVA) method (Schwarzenberg-Czerny, 1996). This uses
the ANOVA statistic, $\Theta$, to assess the goodness of fits to the data with
periodic orthogonal polynomials. The periodogram is displayed in Fig. 2 and
favors a period of $114.5\pm 3.5$ min ($0.0795\pm 0.0024$ d), where the
1-$\sigma$ confidence interval was calculated using a postmortem analysis
(Schwarzenberg-Czerny, 1991) and is defined as the width of the periodogram
peak at the mean noise power level (MNPL; $\Theta_{\mathrm{MNPL}}=2.52$ in
this case) in its vicinity. We also used the ANOVA method with the
$r_{\mathrm{s}}$ and $i_{\mathrm{s}}$ light curves and obtained consistent
results. In Fig. 3 we show the average-subtracted light curves phase binned on
this period. A peak-to-peak amplitude of $\approx 0.4$ mag is found for the
three bands.

Figure 2: Analysis of variance (ANOVA) periodogram of the
$z_{\mathrm{s}}$-band light curve of 4U 1812–12. A period of $114.5\pm 3.5$
min ($0.0795\pm 0.0024$ d) is favored.

The observed photometric modulation can be a reflection of the orbital period
and may be produced by either X-ray irradiation of the donor star or a
superhump with a period a few per cent longer than the orbital period, caused
by the precession of an eccentric accretion disk (van Paradijs et al., 1988;
Whitehurst & King, 1991). While X-ray heating of the donor produces sinusoidal
light curves (e.g., van Paradijs & McClintock, 1995), changes in the disk size
and shape and resonance between the Keplerian orbits in the disk and the
orbital motion of the donor star can produce a more complex morphology in the
superhump modulations (O’Donoghue & Charles, 1996; Haswell et al., 2001;
Zurita et al., 2008).

The sawtooth-like modulation that we detected (Fig. 1) supports the superhump
scenario, and the phase-binned light curves presented in Fig. 3 are similar to
the superhumps observed in cataclysmic variables (CVs; Patterson et al. 2005)
and X-ray binaries (O’Donoghue & Charles, 1996; Zurita et al., 2002, 2008).
Furthermore, the modulation is found to be color independent, as is also the
case for superhumps (Zurita et al. 2008 and references therein). Persistent
CVs show permanent superhumps with typical amplitudes (peak-to-peak) of
$\approx 0.1$ mag (Smak, 2010). These are smaller than the $\approx 0.4$ mag
amplitude in 4U 1812–12 (Fig. 3). However, superhumps with larger amplitudes
($\approx 0.25-0.6$) have been observed during CV superoutbursts (Smak, 2010).
These are also consistent with those found in superhumps detected during LMXB
outbursts ($\approx 0.1$ to 0.6 mag; O’Donoghue & Charles 1996; Zurita et al.
2008; Thomas et al. 2022) and permanent superhumps in persistent LMXBs. For
instance, a $\approx 0.6$ mag amplitude permanent superhump has been detected
in the ultra-compact X-ray binary 4U 1915$-$05 (Callanan et al., 1995; Chou et
al., 2001; Haswell et al., 2001; Retter et al., 2002), which has a low
persistent luminosity, similar to 4U 1812–12.

Figure 3: GTC/HiPERCAM $r_{\mathrm{s}}$$i_{\mathrm{s}}$$z_{\mathrm{s}}$ light
curves phase binned (20 bins) on the $114.5\pm 3.5$ min period, in which the
respective average magnitudes have been subtracted. The phases are computed
relative to the time of the first data point. Colors are the same as in Fig.
1. The whole cycle has been plotted twice for continuity.

It is important to bear in mind, however, that our observation spans two and a
half periods, and therefore we can not fully discard a non-coherent origin for
the modulation, although we deem it unlikely. Rapid aperiodic variability with
a suggested origin in the accretion disk is commonly observed in X-ray
binaries (Shahbaz et al., 2003; Zurita et al., 2003; Hynes et al., 2004;
Shahbaz et al., 2013; Casares & Jonker, 2014). However, this flickering
activity results in erratic variations of typically very short time scales
(from seconds to minutes). Longer flares with time scales of hours have also
been observed, particularly in systems with long orbital periods (e.g., the
6-h flares in the 6.5-d orbital period LMXB V404 Cyg, Casares & Jonker 2014).
However, such a long orbital period would require a much higher mass-accretion
rate in 4U 1812–12 to sustain its persistent nature, which would not be
consistent with its observed X-ray luminosity. Flaring events produced by the
compact jet are also unlikely, since the jet is not expected to make a
dominant contribution to the optical emission in neutron star X-ray binaries
(e.g., Russell et al., 2006; Migliari et al., 2010). Further, they should be
more prominent in the red bands (e.g., Gandhi et al., 2016). All considered,
although we cannot fully discard that the modulation may result from the
detection of two $\approx 114$ min long, color-independent and remarkably
similar (in shape and amplitude) flares, a superhump origin for the observed
modulation seems to be the most plausible scenario. We therefore tentatively
propose an orbital period $P_{\mathrm{orb}}$ $\approx 114$ min for 4U
1812–12.444Negative superhumps, with a period slightly shorter than the
$P_{\rm orb}$, have also been detected in some systems, and are suggested to
be produced by retrograde precession of a tilted disk.

##  4 Discussion

4U 1812–12 is a strong ultra-compact X-ray binary candidate. It shows several
of the distinctive characteristics of the class, related to their compact
geometry (small accretion disk) and nature (hydrogen-exhausted donor; see more
details in in ’t Zand et al. 2007; Armas Padilla & López-Navas 2019). In
particular, since its discovery 50 years ago by the Uhuru mission (Forman et
al., 1976), the system has been repeatedly detected in the X-rays at $\sim$
(3–10) $\times 10^{-10}~{}\mathrm{erg~{}cm}^{-2}~{}\mathrm{s}^{-1}$ (depending
on the X-ray band), which together with the suggested distance of $2.3-4.6$
kpc (Cocchi et al., 2000; Jonker & Nelemans, 2004; Galloway et al., 2020),
translates to a persistent X-ray luminosity of $\sim$ 0.1–1 $\times
10^{36}~{}\mathrm{erg~{}s}^{-1}$ (e.g. Warwick et al., 1981; Barret et al.,
2003; Wilson et al., 2003; Muno et al., 2005; Tarana et al., 2006). This
persistent activity at such low X-ray luminosity prompted in ’t Zand et al.
(2007) to propose an ultra-compact orbit for the source, since only small
disks can be entirely ionized at such low accretion rates (Lasota, 2001). In
the same way, Bassa et al. (2006) suggested a short orbital period based on
the very dim optical counterpart ($g\simeq 25$, $r\simeq 23$ mag; Armas
Padilla et al. 2020), since the X-ray (to optical) reprocessing scales with
the size of the accretion disk (van Paradijs & McClintock, 1994). Yet,
possibly the most compelling evidence for the ultra-compact nature of 4U
1812–12 is the absence of hydrogen features in its optical spectrum (Armas
Padilla et al., 2020), in particular H$\alpha$, which is typically the most
prominent optical emission line in LMXBs with hydrogen-rich donor stars
(Charles & Coe, 2006).

Figure 4: Stability limits for non-irradiated pure-helium disks (red solid
line), for irradiated pure-helium disks (red dashed line), for non-irradiated
C/O disks (blue solid line), for irradiated mixed-composition disks (green
dashed line), for non-irradiated solar-composition disks (magenta solid line)
and for irradiated solar-composition disks (magenta dashed line) according to
Menou et al. (2002) and Lasota et al. (2008). The black dot corresponds to 4U
1812–12 assuming the tentative $\approx 114$-min orbital period and the mass
transfer rate reported in this work (see Section 4).

If we assume that 4U 1812–12 is indeed an ultra-compact system (i.e., hydrogen
poor), the numerous short thermonuclear bursts displayed by the source would
need to be fuelled by pure He, pointing to a He-rich donor (see Armas Padilla
et al. 2020). Considering evolutionary tracks for donor stars in ultra-compact
binaries, the derived mass transfer rate555We derived the mass transfer rate
following Coriat et al. (2012), assuming an average 2–10 keV X-ray unabsorbed
continuum flux of $3.8\times
10^{-10}~{}\mathrm{erg~{}cm}^{-2}~{}\mathrm{s}^{-1}$ (Cocchi et al., 2000), a
bolometric correction of 2.9 (for which we account a 25 per cent uncertainty;
in ’t Zand et al., 2007) and a distance of $2.3-4.6$ kpc (Jonker & Nelemans,
2004; Galloway et al., 2020). of $\sim(3.4\pm 2.5)\times
10^{-10}\,\mathrm{M}_{\odot}$ yr-1 translates into an orbital period of
$\simeq 20$ min and $\gtrsim 35$ min for a He white dwarf and a He-star donor,
respectively. However, according to disk instability models for irradiated He
accretion disks, the orbital period should be $\lesssim 25$ min in order to
sustain a stable disk ($\lesssim 40$ min in the case of a C/O disk; Menou et
al. 2002; Lasota et al. 2008).

###  4.1 An ultra-compact X-ray binary progenitor?

Our tentative $P_{\mathrm{orb}}$ $\approx 114$ min lies outside the ultra-
compact zone of $P_{\mathrm{orb}}$ $<80$ min, contrary to what was expected
from the aforementioned observational properties. The binary orbital
separation is still large enough to fit a main sequence star. In fact, a
114-min orbit would require a very late M-type main sequence star or a brown
dwarf to fill its Roche lobe (Faulkner et al., 1972; Cox, 2000; Rappaport et
al., 2021). In such cases, the system mass ratio would be $q=M_{2}/M_{1}<0.04$
(assuming $M_{1}=1.4\,\mathrm{M}_{\odot}$), which is in agreement with the low
mass ratio required to produce superhump modulations (Whitehurst & King,
1991). Further, according to evolutionary sequences for LMXBs, systems with an
initial companion star mass $\approx 0.6-3\,\mathrm{M}_{\odot}$ and an orbital
period below the bifurcation period evolve by shrinking the orbit to a minimum
period that can be as low as $P_{\rm orb}$ $\simeq 80$ min (see e.g.,
Rappaport et al., 1982; Podsiadlowski et al., 2002). Both scenarios could be
valid for 4U 1812–12. However, the lack of hydrogen features in its optical
spectrum is difficult to reconcile with these binary solutions. Nevertheless,
we note that some transient LMXBs with hydrogen-rich companions did not show
H$\alpha$ emission during some phases of the outburst (see Jiménez-Ibarra et
al. 2019; Stoop et al. 2021 and references therein).

A very appealing alternative for the nature of the system is that 4U 1812–12
is a progenitor of an ultra-compact X-ray binary. In this case, the donor star
would be an evolved main-sequence star that started mass transfer near or just
after the point of central hydrogen exhaustion, and is progressively getting
closer to the compact object on its path to the ultra-compact period regime
(i.e., the system is in the so-called evolved main-sequence star channel; see
Nelemans et al. 2010). In this scenario, traces of hydrogen can still be
present in the donor star photosphere ($X_{\rm s}\sim 0.1$), and therefore in
the accretion disk (Nelemans et al., 2010; Nelson & Rappaport, 2003;
Podsiadlowski et al., 2002). This scenario would reconcile all the observables
of 4U 1812–12. First, this solution would be consistent with it being a
persistent system, since the mass transfer rate is of the same order as the
critical mass transfer rate in an accretion disk with mixed composition
($X=0.1$ and $Y=0.9$, Lasota et al. 2008, see Fig. 4). Second, such a low
fraction of hydrogen would not affect the duration of thermonuclear bursts
(Cumming, 2003), which is in agreement with the numerous short thermonuclear
events displayed by the source (Cocchi et al., 2000; Galloway et al., 2020).
Finally, this low fraction of hydrogen would not be detectable in the optical
spectrum (Werner et al., 2006), in agreement with the observations of Armas
Padilla et al. (2020).

###  4.2 An ultra-compact X-ray binary with a 114-min orbital period?

According to evolutionary models, ultra-compact X-ray binaries can evolve
towards orbital periods of $100-110$ min or longer if the donor is heated and
inflated, or if the donor’s mass loss via winds is taken into account (van
Haaften et al., 2012; Van Haaften et al., 2012). However, the predicted mass
transfer rate at these longer periods is below $\sim
10^{-12}~{}\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}$, which is two orders of
magnitude lower than that of 4U 1812–12. We note that these computed tracks
are for a helium or carbon-oxygen white dwarf companion that fully fills its
Roche lobe. The evolutionary tracks for He-star donors presented in Heinke et
al. (2013) provide higher mass transfer rates, which may explain the group of
persistent ultra-compact systems with orbital periods longer than $\simeq 40$
min and high mass accretion rates. Still, these tracks were calculated only
for orbital periods up to 60 min. Thus, the evolution beyond this value is
unclear, as is the stage where the He-star core stops expanding.

Setting aside the uncertainties that surround the above He-star evolutionary
paths, this tantalizing scenario (i.e., an evolved 114-min ultra-compact
system) might be still plausible from a pure stability point of view. Although
the mass transfer rate of 4U 1812–12 sits below the disk instability lines for
both pure He and non-irradiated C/O disks, an irradiated C/O disk would
maintain stability if the critical mass transfer rate drops by a similar
amount than He and Solar-abundance disks when irradiation is taken into
account (i.e., by a factor of $6-10$, see Fig. 4, Menou et al. 2002; Lasota et
al. 2008). As a matter of fact, evolutionary calculations show that He-star
donors can be C/O rich stars with some traces of helium left (Nelemans et al.,
2010). Furthermore, we note that our calculated mass transfer rate is an upper
limit, since it does not account for possible mass loss via outflows (see
e.g., Fender & Muñoz-Darias 2016; Hernández Santisteban et al. 2019; Marino et
al. 2019), and therefore the actual mass-transfer rate might still sit above
the instability thresholds.

##  5 Conclusions

We have presented time-resolved HiPERCAM/GTC photometry of the ultra-compact
X-ray binary candidate 4U 1812–12. The
$r_{\mathrm{s}}$$i_{\mathrm{s}}$$z_{\mathrm{s}}$ light curves show a clear
sawtooth-like periodic modulation that resembles a superhump. An ANOVA
periodogram of the $z_{\mathrm{s}}$-band light curve favors a period of
$\simeq 114$ min. We tentatively propose this as the orbital period of 4U
1812–12, challenging the previously proposed ultra-compact nature of the
binary.

We discuss possible scenarios for the nature of the system. Based on its
properties, namely its persistently dim luminosity, optical spectrum and short
thermonuclear type I bursts, we suggest that 4U 1812–12 is an ultra-compact
X-ray binary progenitor whose orbit is shrinking towards the ultra-compact
regime and has an evolved main-sequence star with a low fraction of
photospheric hydrogen as donor star.

Additional high-quality observations of 4U 1812–12 are desirable in order to
verify the persistence of the periodic modulation and confirm the orbital
period of the system.

This work is supported by the Spanish Ministry of Science under grants
PID2020–120323GB–I00 and EUR2021–122010. We acknowledge support from the
Consejería de Economía, Conocimiento y Empleo del Gobierno de Canarias and the
European Regional Development Fund (ERDF) under grant with reference
ProID2021010132 and ProID2020010104. The design and construction of HiPERCAM
was funded by the European Research Council under the European Union’s Seventh
Framework Programme (FP/2007–2013) under ERC–2013–ADG Grant Agreement no.
340040 (HiPERCAM). HiPERCAM operations and VSD are supported by STFC grant
ST/V000853/1. Based on observations made with the Gran Telescopio Canarias
(GTC), installed at the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofísica de Canarias, on the island of La Palma. This research
made use of Peranso (www.peranso.com), a light curve and period analysis
software.

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