#  $\nu^{2}$-Flows: Fast and improved neutrino reconstruction in multi-
neutrino final states with conditional normalizing flows

John Andrew Raine<EMAIL_ADDRESS>Département de physique nucléaire et
corpusculaire, University of Geneva, Switzerland  Matthew Leigh
<EMAIL_ADDRESS>Département de physique nucléaire et corpusculaire,
University of Geneva, Switzerland  Knut Zoch  Département de physique
nucléaire et corpusculaire, University of Geneva, Switzerland  Laboratory for
Particle Physics and Cosmology, Harvard University, Cambridge, 02138 MA, USA
Tobias Golling  Département de physique nucléaire et corpusculaire, University
of Geneva, Switzerland

###### Abstract

In this work we introduce $\nu^{2}$-Flows, an extension of the $\nu$-Flows
method to final states containing multiple neutrinos. The architecture can
natively scale for all combinations of object types and multiplicities in the
final state for any desired neutrino multiplicities. In $t\bar{t}$ dilepton
events, the momenta of both neutrinos and correlations between them are
reconstructed more accurately than when using the most popular standard
analytical techniques, and solutions are found for all events. Inference time
is significantly faster than competing methods, and can be reduced further by
evaluating in parallel on graphics processing units. We apply $\nu^{2}$-Flows
to $t\bar{t}$ dilepton events and show that the per-bin uncertainties in
unfolded distributions is much closer to the limit of performance set by
perfect neutrino reconstruction than standard techniques. For the chosen
double differential observables $\nu^{2}$-Flows results in improved
statistical precision for each bin by a factor of 1.5 to 2 in comparison to
the Neutrino Weighting method and up to a factor of four in comparison to the
Ellipse approach.

##  I Introduction

At collider experiments in particle physics, such as those at the Large Hadron
Collider (LHC) [1], beams of hadrons or leptons are accelerated to high
energies and collided together. These collisions result in an array of
particles which are studied by experiments comprising complex detectors
surrounding the interaction points. General purpose detectors, such as ATLAS
[2] and CMS [3], are designed to record and reconstruct nearly all stable
particles predicted in the standard model of particle physics (SM). From these
reconstructed stable particles, both precision measurements of the SM as well
as searches for new phenomena beyond the SM (BSM) are performed.

Neutrinos, stable particles produced in many collisions, interact only through
the electroweak force and traverse the detectors without leaving a trace.
Their presence in collisions is inferred from a momentum imbalance in the
transverse plane perpendicular to the beam axis. This imbalance, known as the
missing transverse momentum $\vv{p}_{\mathrm{T}}^{\text{miss}}$, is calculated
from the negative vector sum of the transverse momenta of all reconstructed
objects in the transverse plane.

In order to reconstruct individual neutrinos, and thus fully reconstruct a
single event, underlying assumptions need to be made on their origin and
multiplicity. The $\vv{p}_{\mathrm{T}}^{\text{miss}}$ serves as a proxy for
all unobserved particles in the collision, but doesn’t indicate how many were
present, or how the momentum should be shared in the case of multiple
neutrinos. Furthermore, at hadron colliders there is no experimental
equivalent for the missing longitudinal momentum.

Several approaches are used to reconstruct neutrinos using
$\vv{p}_{\mathrm{T}}^{\text{miss}}$ and by setting constraints on the
invariant masses of intermediate particles. In the case of top quark pair
production ($t\bar{t}$) in the semileptonic decay channel, where there is one
lepton and one neutrino in the event, the neutrino momentum is typically
reconstructed by solving the longitudinal momentum component ${p}_{z}$ under
the assumption that the invariant mass of the lepton-neutrino pair is exactly
that of the $W$ boson [4, 5, 6, 7, 8, 9, 10, 11, 12]. For the case of events
with two neutrinos, such as $t\bar{t}$ production in the dileptonic channel,
more complicated methods [13, 14, 15] are employed in order to resolve the
share of momentum between the two neutrinos in the event [16, 17, 18, 19, 20,
21, 22, 21, 23, 24, 25, 26, 27, 28, 12]. These approaches still require the
invariant mass of the two lepton-neutrino pairs to match the $W$ boson mass,
and in addition require the invariant mass of the lepton-neutrino-jet triplets
to match the top-quark invariant mass.

In $\nu$-Flows [29] we use conditional normalizing flows [30, 31, 32] to learn
the probability distribution of the neutrino momentum vector given the
observed objects in an event for semileptonic $t\bar{t}$ events. From the
learned conditional probability distribution, solutions can be sampled per
event and the most probable solution can be determined from the learned
likelihood of the solution. In this work, we extend $\nu$-Flows to more
challenging final states with multiple neutrinos. We focus on $t\bar{t}$ final
states in which both top quarks decay semileptonically. This results in an
expected final state with exactly two oppositely charged leptons, at least two
jets, of which two should originate from $b$-quarks, and two neutrinos. In
comparison to the single neutrino case, where the main challenge arises in
recovering the longitudinal momentum component of the neutrino, events with
multiple neutrinos have the additional complexity of how to share the total
missing momentum vector between all neutrinos in the final state. We show that
$\nu$-Flows yields improved reconstruction performance in comparison to
standard approaches and demonstrate the direct benefit to the statistical
precision in a simplified double differential $t\bar{t}$ dilepton analysis in
observables such as the invariant mass of the $t\bar{t}$ system,
$m_{t\bar{t}}$, and the angular separation of the two leptons,
$\Delta\phi(\ell^{+}\ell^{-})$.

The repository111https://github.com/rodem-hep/nu2flows and
data222https://zenodo.org/record/8113516 [33] used in this work are both made
publicly available.

##  II Method

###  II.1 $\nu^{2}$-Flows

Figure 1: A schematic of the $\nu^{2}$-Flows network for learning the
conditional likelihood of multiple neutrinos in the event. The network uses a
transformer encoder (TE) with cross-attention (CA) with a learnable class
token (CT) to embed an event representation for any multiplicity of physics
objects. This operation is permutation invariant and can operate on any jet
and lepton multiplicity. Each physics object has its own dedicated embedding
network and additional event information (Misc) is used to condition the
transformer encoder blocks. The representation vector is used to condition the
transformation with the normalizing flow.

In $\nu$-Flows, conditional normalizing flows are used to capture the
distribution of possible solutions for the neutrino momenta given the
reconstructed momenta of observed objects in a collision. The overall model
comprises two components, the event feature extraction, and the conditional
normalizing flow. The event feature extraction learns a representative vector
of the event from the collections of reconstructed objects, namely the jets
and leptons, and $\vv{p}_{\mathrm{T}}^{\text{miss}}$. The resulting vector is
used as a conditional input to the normalizing flow, in order to learn the
conditional density of all possible neutrino solutions from the training data.

In this work we extend the initial $\nu$-Flows architecture to predict
multiple neutrinos and accommodate any number of leptons in addition to jets
by using attention transformers [34]. We label this architecture as
$\nu^{2}$-Flows to distinguish it from the general method. In order to handle
two neutrinos we double the dimensionality of the conditional normalizing flow
(from three to six). The neutrinos are also always predicted in the same order
for each event, with the momentum of the neutrino followed by the momentum of
the anti-neutrino. When increasing the neutrino multiplicity further, the same
procedure is used together with a predefined ordering for the neutrinos. The
architecture of the normalizing flow is otherwise kept largely the same.

The most substantial optimisation has been performed on the feature extraction
network. The feature extraction network attempts to produce a contextual
vector, specific to each event, to guide the transformations within the
normalizing flow. In the single lepton case, $\nu$-Flows uses an attention
pooled deep set [35] to process the jets, with
$\vv{p}_{\mathrm{x}}^{\text{miss}}$, $\vv{p}_{\mathrm{y}}^{\text{miss}}$, the
lepton four momentum, and some event level information as extra conditional
information. To extend $\nu^{2}$-Flows to multiple leptons, we require a
permutation invariant architecture that can accommodate a variable number of
both jets and leptons, motivating the move to attention transformers.

To train $\nu^{2}$-Flows all jets and leptons are represented by their four-
momentum vectors in the form $({p}_{x},{p}_{y},{p}_{z},\log E)$. Jets are
assigned an additional binary decision on whether they are tagged as
originating from a $b$-quark. Leptons are identified as being either an
electron or a muon, as well as whether they had positive or negative charge.
The target neutrino momenta are expressed as $({p}_{x},{p}_{y},{p}_{z})$ for
both the neutrino and anti-neutrino. The full set of inputs to the network are
provided in Table 1. The coordinates chosen to describe the input and target
object kinematics was the result of a grid search.

Table 1: The different input observables used as inputs to the feature
extraction network.

Category | Variables | Description
---|---|---
$\vv{p}_{\mathrm{T}}^{\text{miss}}$ |  $p_{x}^{\text{miss}}$, $p_{y}^{\text{miss}}$ | Missing transverse momentum 2-vector
Leptons |  $p_{x}^{\ell}$, $p_{y}^{\ell}$, $p_{z}^{\ell}$, $\log E^{\ell}$ | Lepton momentum 4-vector
$q^{\ell}$ | Lepton charge
$\ell^{flav}$ | Whether lepton is an electron or muon
Jets |  $p_{x}^{j}$, $p_{y}^{j}$, $p_{z}^{j}$,$\log E^{j}$ | Jet momentum 4-vector
$isB$ | Whether jet passes $b$-tagging criteria
Misc |  $N_{\text{jets}}$, $N_{\text{bjets}}$ | Jet and $b$-jet multiplicities in the event

A schematic of the new architecture for $\nu^{2}$-Flows is shown in Fig. 1,
which makes use of attention transformers and object specific embedding
networks. Initially, the jets, leptons and $\vv{p}_{\mathrm{T}}^{\text{miss}}$
are all independently embedded into higher dimensional space using object
specific multi-layer perceptrons (MLP).333Other final state objects such as
photons and tau leptons can also be accommodated by embedding them additional
MLPs for each particle type. The embedded objects subsequently interact
through a transformer encoder using several layers of multi-headed attention.
Additional event information (Misc) containing object multiplicities, is
injected into the network as conditional information by concatenating the
vector to each token within the transformer encoder blocks. To obtain a single
global vector from transformer as our conditioning vector for the normalizing
flow, we perform repeated cross-attention with a learnable class token (CT), a
common technique used in vision transformers [36].

The $\nu^{2}$-Flows model in this paper comprises three transformer encoder
blocks and two cross-attention blocks, each with an embedding dimension of 128
and 16 attention heads. All MLPs in the network have a single hidden layer
with 256 neurons and use the LeakyReLU activation [37] and Layer-Normalization
[38]. The output of the transformer is passed through an MLP to produce the
context tensor for the flow with a dimension of 128. The invertible neural
network employs 10 rational quadratic spline (RQS) coupling blocks [39]
interspersed with LU-decomposed linear layers, implemented with the nflows
package [40] and Pytorch v2.0 [41]. Each RQS has 10 bins with linear tail
bounds outside $\pm 4$. The conditional normalizing flow is trained with the
standard maximum likelihood estimation loss obtained through the change of
variables formula, and transforms the input neutrino momenta to a standard
multivariate normal distribution. The entire $\nu^{2}$-Flows model, including
the transformer, has around 600 000 trainable parameters.

We train the $\nu^{2}$-Flows model for 100 epochs using the AdamW optimizer
[42] with a learning rate cycling from ${10}^{-8}$ to ${10}^{-3}$ and back
every 50 epochs. We use weight decay with a strength of ${10}^{-4}$.

###  II.2 Reference methods

Several analytical techniques have been proposed to solve the reconstruction
of the two neutrinos in dilepton $t\bar{t}$ events. Amongst these are
_Neutrino weighting_ [13] ($\nu$-Weighting), an _algebraic_ solution [14], and
the _Ellipse_ method [15]. These have been successfully employed in a wide
range of measurements at the Tevatron and LHC [16, 17, 18, 19, 20, 21, 22, 21,
23, 24, 25, 26, 27, 28, 12], most notably $\nu$-Weighting. In this work we
compare $\nu^{2}$-Flows to the $\nu$-Weighting due to its common usage but
also the Ellipse method due to its reduced computation time.

####  $\nu$-Weighting

In $\nu$-Weighting the kinematic properties of the neutrino are extracted from
the identified leptons, jets and missing transverse momentum in the event. For
our implementation we follow the prescription described in Ref. [27].
Constraints on neutrino solutions using the invariant mass of the top quark
and $W$ boson

$\displaystyle(\ell_{1,2}+\nu_{1,2})^{2}=m_{w}^{2}=($80.38\text{\,}\mathrm{GeV}$)^{2},$
(1)
$\displaystyle(\ell_{1,2}+\nu_{1,2}+b_{1,2})^{2}=m_{t}^{2}=($172.5\text{\,}\mathrm{GeV}$)^{2},$
(2)

are applied, where $\ell_{1,2}$, $\nu_{1,2}$, and $b_{1,2}$ represent the
four-momenta of the charged leptons, neutrinos and $b$-tagged jets. However,
this is not enough to fully constrain the kinematics of the neutrinos.
Therefore, the neutrino and anti-neutrino rapidities ($\eta^{\nu}$ and
$\eta^{\bar{\nu}}$) are individually hypothesized and tested. For each pair of
values for $\eta^{\nu}$ and $\eta^{\bar{\nu}}$, we solve the mass equations in
Eq. 1, yielding two possible solutions for the full pair of neutrino
kinematics. Each solution produces an inferred missing transverse momentum
$\vv{p}_{\mathrm{T}}^{\nu\bar{\nu}}$ vector which can then be compared to the
observed $\vv{p}_{\mathrm{T}}^{\text{miss}}$. This comparison defines a weight

$w=\exp\left(-\frac{||\vv{p}_{\mathrm{T}}^{\text{miss}}-\vv{p}_{\mathrm{T}}^{\nu\bar{\nu}}||_{2}^{2}}{2\sigma^{2}}\right),$
(3)

where $\sigma$ is a fixed resolution scale related to the
$\vv{p}_{\mathrm{T}}^{\text{miss}}$ reconstruction in the detector. In
$\nu$-Weighting the hypothesis that maximizes $w$ is chosen as the correct
solution. To find all solutions we perform a grid search of $\eta^{\nu}$ and
$\eta^{\bar{\nu}}$ with values ranging from -5 to 5 with a step size of 0.2.
For each of these we also need to test all combinations of assigning
$b$-tagged jets to each of the $b$-quarks from the $t\bar{t}$ decay. This
represents a very costly computation scan that considers only discrete $\eta$
values.

Another significant drawback of this method is that despite the large number
of neutrino solutions being tested, it is still possible that the constraint
systems are not solvable. This can be due to mis-measurement or mis-assignment
of the leptons, jets, or $\vv{p}_{\mathrm{T}}^{\text{miss}}$. Alternatively,
this can arise from the masses of either the top quarks or $W$ bosons in the
event deviating from the nominal values. Therefore, to increase the success
rate of this method we also iterate over different values of $m_{t}$ from 171
to $174\text{\,}\mathrm{GeV}$ with a step size of $0.5\text{\,}\mathrm{GeV}$.
This further increases the computational requirements, but increases the
efficiency of finding a solution for each event.

After selecting the solution with the highest $w$, any solution which results
in $m_{t\bar{t}}<$300\text{\,}\mathrm{GeV}$$ or where either of the two
reconstructed top quarks have negative energy is rejected. The $\nu$-Weighting
method is unable to find a valid solution on the nominal dataset around 5% of
the time.

#### Ellipse method

The Ellipse method is derived from a geometric approach to analytically
constrain neutrino kinematics explicitly in processes where top quarks decay
into leptons and neutrinos [15]. For a single neutrino, its momentum can be
calculated as a function of the 4-momenta of the $b$-quark and the charged
lepton, the $W$ boson mass, and the top quark mass. The solution set for this
function defines the surface of an ellipse. By combining this information with
the observed $\vv{p}_{\mathrm{T}}^{\text{miss}}$, the solution set collapses
to a unique value. For events with two neutrinos in the final state, the
method is extended to calculate the solution for neutrino pairs which are most
likely to have produced the observed $\vv{p}_{\mathrm{T}}^{\text{miss}}$.

We use the implementation from the authors of the Ellipse
method444Implementation available at github.com/betchart/analytic-nu. To solve
the $b$-jet combinatorics we use a simple minimum $\Delta R$ matching between
the leptons and the $b$-jets in the event. To reduce the combinations we only
take the two leading jets in ${p}_{\mathrm{T}}$ passing $b$-tagging criteria.
If the $\Delta R$ matching yields no solutions for the neutrino kinematics
using the ellipse method, the opposite association is tested.

The drawbacks of this approach is that it requires accurate matching between
each lepton and the associated $b$-jet in the event. Furthermore, like
$\nu$-Weighting, it requires one to make hard assumptions on the mass of the
$W$ boson and top quark. Finally, it is possible that the method can yield no
solutions, just like $\nu$-Weighting. The implementation used in this work
fails to find solutions in 22% of $t\bar{t}$ dilepton events. In comparison to
$\nu$-Weighting, the Ellipse method requires much less computational resources
per event.

##  III Dataset

In this work, $\nu^{2}$-Flows is applied to simulated $t\bar{t}$ events where
both top quarks decay semileptonically, resulting in a final state with
exactly two leptons ($\ell$)555Leptons is used to designate either electrons
($e$) or muons ($\mu$)., two neutrinos ($\nu$) and two jets initiated by
$b$-hadrons ($b$-jets). Additional jets arise from initial and final state
radiation. All events are simulated in proton-proton collisions at a centre-
of-mass energy of $\sqrt{s}=13$ TeV. Two different samples are generated, each
using a different generator for the hard interactions in the matrix element.

In the nominal sample, hard interactions are simulated using MadGraph5_aMC@NLO
[43] (v3.1.0). The top-quark mass $m_{t}$ is set to 173 GeV for all events.
All events are interfaced to Pythia8 [44] (v8.243) for the parton shower and
hadronisation, using the NNPDF2.3LO PDF set [45] with
$\alpha_{S}(m_{Z})=0.130$ using the LHAPDF [46] framework.

In the alternative sample, both the hard interactions and parton shower are
simulated with Pythia8 (v8.307) with the Monash tuned set of parameters [47]
at leading order accuracy. The same PDF set is used as for the nominal sample.

The detector response is simulated using Delphes [48] (v3.4.2) with a
parametrisation similar to the response of the ATLAS detector [2]. Jets are
reconstructed using the anti-$k_{t}$ clustering algorithm [49] with a radius
parameter of $R=0.4$ using the FastJet package [50]. All jets are required to
have a transverse momentum ${p}_{\mathrm{T}}>25$ GeV and fall within
$|\eta|<2.5$. A $b$-tagging working point corresponding to 70% inclusive
signal efficiency is used to identify jets as originating from $b$-hadrons. Up
to 10 jets are selected per event, ordered in descending ${p}_{\mathrm{T}}$.

Events are required to have at least two $b$-tagged jets, and two oppositely
charged leptons, each with ${p}_{\mathrm{T}}>15$ GeV and $|\eta|<2.5$. Truth
association of jets to the $b$-quarks in the $t\bar{t}$ hard scatter is
performed using a $\Delta R$ matching, with partons matched to jets within
$\Delta R<0.4$. Events where multiple partons are matched to the same jet are
removed. Truth association of the lepton to the parent top quark is performed
assuming there is no charge misidentification, and the true neutrino momenta
are taken directly from simulation.

In total there are 1.02 million events in the nominal sample and 1.4 million
events in the alternative sample passing all selection requirements. 940,000
(970,000) events from the nominal (alternative) sample are used to train the
network, with 80,000 nominal samples used for evaluation.

##  IV Results

###  IV.1 Neutrino reconstruction

The first measure of performance is to validate that $\nu^{2}$-Flows is able
to correctly reconstruct the momenta and relative positions of the neutrino
pair. To establish a baseline for the impact of neutrino reconstruction in all
distributions, we define $\nu$-Truth to be the case where the truth neutrinos
are reconstructed perfectly, but all other objects in the event remain the
same. This defines the ground truth of the target distributions and is also
the upper limit in performance for any neutrino reconstruction approach. We
compare the reconstruction performance with $\nu^{2}$-Flows to the
$\nu$-Weighting and Ellipse methods.

Figure 2: The kinematics of the reconstructed (anti-)neutrinos for the three
reconstruction methods and $\nu$-Truth (shaded grey). The hashed areas
represent statistical uncertainties in the $\nu$-Truth prediction.

The individual neutrino kinematics and the angular separation between the two
neutrinos are shown in Figs. 2 and 3. Here we can see that both
$\nu$-Weighting and Ellipse overestimate the neutrino transverse momenta and
energy and tend to prefer central neutrinos with a visible peak at $\eta=0$.
The neutrino pair is also predominantly predicted to be back to back in
$\phi$, with the opening angle between them showing a large degree of tension
with the ground truth. In all neutrino kinematic distributions,
$\nu^{2}$-Flows is able to reproduce the ground truth accurately, with slight
discrepancies only visible in the low statistic tails.

Figure 3: The angular separation in $\eta$ and $\phi$ between the
reconstructed neutrino pair per event for the three reconstruction methods and
$\nu$-Truth (shaded grey). The hashed areas represent statistical
uncertainties in the $\nu$-Truth prediction.

From the reconstructed neutrinos it is also possible to reconstruct the $W$
bosons, top quarks and the full $t\bar{t}$ system in the event. Figure 4 shows
the reconstructed invariant mass of the $W$ boson and top quark. Here we
always use the perfect association of jets and leptons to the two top quarks.
The strong bias to the top quark and $W$ boson masses used in Eq. 1 for the
$\nu$-Weighting and Ellipse methods are clearly visible. In comparison,
$\nu^{2}$-Flows follows the full underlying target distribution for the $W$
boson mass and better captures the distribution in the reconstructed top-quark
mass, but with a lower resolution than $\nu$-Truth.

Figure 4: The reconstructed invariant mass of $W$ bosons (left) and top quarks
(middle), as well as the top quark ${p}_{\mathrm{T}}$ (right) when using the
three neutrino reconstruction methods in comparison to $\nu$-Truth (shaded
grey).

However, due to the dependence on the masses of the two particles for
$\nu$-Weighting and Ellipse it is more interesting to compare the
reconstructed kinematics of the individual top quarks and the $t\bar{t}$ pair.
The reconstructed top quark ${p}_{\mathrm{T}}$ is also compared for the three
methods in Fig. 4. The true distribution is reproduced with $\nu^{2}$-Flows
and up to approximately 200 GeV by $\nu$-Weighting. However, Ellipse tends to
reconstruct top quarks with a harder ${p}_{\mathrm{T}}$. In Fig. 5 the
invariant mass, ${p}_{\mathrm{T}}$ and the rapidity of the reconstructed
$t\bar{t}$ system $y_{t\bar{t}}$ are shown. Although discrepancies are
observed at low $m_{t\bar{t}}$ values, $\nu^{2}$-Flows is able to closely
reproduce the kinematics of the $t\bar{t}$ system much better than
$\nu$-Weighting and Ellipse. The $p_{\mathrm{T}}^{t\bar{t}}$ and
$y_{t\bar{t}}$ distributions are well reconstructed with $\nu$-Weighting but
there is an overestimate in the tail of the $m_{t\bar{t}}$ distribution,
whereas Ellipse shows poor modelling in all three observables.

Figure 5: The invariant mass, ${p}_{\mathrm{T}}$, and rapidity of the
reconstructed $t\bar{t}$ system when using the three neutrino reconstruction
methods in comparison to $\nu$-Truth (shaded grey).

As also performed in Ref. [29], we perform a cross check on the benefit of
using the normalizing flow in the $\nu^{2}$-Flows architecture. We train the
$\nu^{2}$-Flows architecture but without the flow and predict the two neutrino
momenta directly, establishing a simple machine learning baseline. The
performance achieved is substantially worse with strong biases in all neutrino
kinematics and resulting event level distributions.

One of the main drawbacks of $\nu$-Weighting, and why Ellipse is often
considered despite the reduced performance, is the computational resources
required. In comparison, $\nu$-Flows requires only a single forward pass for
each event. The typical inference times on a CPU for single event inference
are around 70 ms, with the computation time decreasing substantially with
parallelised execution on a GPU as summarised in Table 2.

Table 2: Required time for single event inference using $\nu^{2}$-Flows. Times representative of using a single core of an AMD EPYC 7742 2.25GHz CPU and an NVIDIA® RTX 3080 graphics card. Resource | Batch size | Time/event [ms]
---|---|---
CPU | 1 | 71.4
GPU | 1 | 33.3
1000 | 0.03

###  IV.2 Unfolding analysis

In order to evaluate the downstream impact of the improved neutrino
reconstruction from $\nu$-Flows, we follow the unfolding analysis performed in
Ref. [26], where a double differential cross section measurement is performed
to measure the spin correlation in $t\bar{t}$ events, by measuring the
invariant mass of the $t\bar{t}$ system $m_{t\bar{t}}$ and the angular
separation between the two leptons $\Delta\phi(\ell^{+}\ell^{-})$.
Reconstruction of the two neutrinos is crucial in order to fully reconstruct
the $t\bar{t}$ system, which in Ref. [26] is performed using $\nu$-Weighting.
To benchmark our model, we replace the neutrino reconstruction with the result
from $\nu^{2}$-Flows. In addition to $\nu$-Weighting we also compare the
performance to the Ellipse method due to its reduced computational complexity.

We focus on the reconstruction of individual observables dependent on the
neutrino kinematics as well as the statistical precision of the unfolded
distributions. In addition to $\Delta\phi(\ell^{+}\ell^{-})$, we look at other
observables in conjunction with $m_{t\bar{t}}$, motivated by the distributions
measured in Ref. [51]. These observables are described in Table 3 and the
corresponding bin edges for the double differential unfolding and
corresponding response matrices are shown in Table 4.

Table 3: Kinematic observables of the reconstructed $t\bar{t}$ system studied
for an unfolding analysis in dilepton events. Observables
---
$m_{t\bar{t}}$ | Invariant mass of $t\bar{t}$ system
$\Delta\phi(\ell^{+}\ell^{-})$ | Separation in $\phi$ between the two leptons
$p_{\mathrm{T}}^{t}$ | Transverse momentum of the top quark
$p_{\mathrm{T}}^{t\bar{t}}$ | Transverse momentum of the $t\bar{t}$ system
$y_{t\bar{t}}$ | Rapidity of the $t\bar{t}$ system
Table 4: Bin edges used for each of the kinematic observables of the reconstructed $t\bar{t}$ system studied for two dimensional unfolding analyses. Observable | Bin edges |
---|---|---
$m_{t\bar{t}}$ | [0, 400, 500, 800, inf] | GeV
$\Delta\phi(\ell^{+}\ell^{-})$ | [0.0, 0.25, 0.5, 0.75, 1.0] | rad/$\pi$
$p_{\mathrm{T}}^{t}$ | [0, 75, 125, 175, inf] | GeV
$p_{\mathrm{T}}^{t\bar{t}}$ | [0, 70, 140, 200, inf] | GeV
$y_{t\bar{t}}$ | [-inf, -1.0, 0.0, 1.0, inf] |

All distributions are compared to $\nu$-Truth, and the total uncertainty in
each bin after unfolding is calculated with respect to the optimal performance
achieved when using $\nu$-Truth. The correct jet and lepton association is
used for both top quarks in order to remove the effects arising from matching
inefficiencies. We perform the unfolding using the Singular Value
Decomposition (SVD) method [52] with a regularisation factor of 7, using the
implementation provided in RooUnfold [53]. The regularisation factor was
optimised for the $\nu$-Truth distributions for a reduced $\chi^{2}$ value
closest in agreement to one for the four double differential distributions. In
all cases we only consider the $t\bar{t}$ process and ignore the impact of
background estimation and subtraction in the methods.

Although more modern machine learning approaches for unfolding are an active
area of research [54, 55, 56, 57, 58, 59], we leave their study and
application to dilepton $t\bar{t}$ events to future studies, in particular the
study of unbinned multidimensional unfolding with the reconstructed neutrino
kinematics.

Figure 6: Binned response matrices for the double differential measurement of
$m_{t\bar{t}}$ and $\Delta\phi(\ell^{+}\ell^{-})$ when using each of the three
methods for neutrino reconstruction. The binning is symmetric for both the
parton and detector level observables, however the $m_{t\bar{t}}$ bins are
labelled on the $x$-axis with the $\Delta\phi(\ell^{+}\ell^{-})$ bins labelled
on the $y$-axis. The trace fraction is calculated for each method for a simple
quantitative comparison and is 0.73 when using $\nu$-Truth.

Figure 7: Binned response matrices for the double differential measurement of
$m_{t\bar{t}}$ and $p_{\mathrm{T}}^{t}$ when using each of the three methods
for neutrino reconstruction. The binning is symmetric for both the parton and
detector level observables, however the $m_{t\bar{t}}$ bins are labelled on
the $x$-axis with the $p_{\mathrm{T}}^{t}$ bins labelled on the $y$-axis. The
trace fraction is calculated for each method for a simple quantitative
comparison and is 0.62 when using $\nu$-Truth.

The response matrix using each of the neutrino reconstruction methods is shown
in Fig. 6 for the two dimensional binning in $m_{t\bar{t}}$ and
$\Delta\phi(\ell^{+}\ell^{-})$ and in Fig. 7 for $m_{t\bar{t}}$ and
$p_{\mathrm{T}}^{t}$. In the ideal case only the main diagonal would contain
entries, however due to inefficiencies in the neutrino reconstruction methods
as well as detector resolution effects, off diagonal elements are unavoidable.
In both cases it is clear that using $\nu^{2}$-Flows to reconstruct the
neutrino pair results in a more diagonal response matrix than the other two
approaches. This is quantified by the trace fraction of each matrix.

Although the trace fraction can give a good measure of which method is
performing best, the off diagonal elements still contribute to the unfolded
distributions. To quantitatively assess the true impact of using each method,
the response matrices are inverted using SVD and the overall uncertainties for
each bin at parton level are calculated.

Figure 8 shows the relative statistical uncertainty for each method with
respect to $\nu$-Truth. The values are obtained for each bin of the unfolded
distributions using SVD with the chosen level of regularisation. For the
individual bins of the four double differential distributions the
uncertainties are typically a factor of 1.5 to two times smaller when using
$\nu^{2}$-Flows compared to $\nu$-Weighting, and up to four times smaller in
comparison with Ellipse.

Figure 8: The relative statistical uncertainty in each unfolded bin of the two
dimensional distributions for $m_{t\bar{t}}$ and
$\Delta\phi(\ell^{+}\ell^{-})$ (top left), $p_{\mathrm{T}}^{t}$ (top right),
$p_{\mathrm{T}}^{t\bar{t}}$ (bottom left), and $y_{t\bar{t}}$ (bottom right).
Comparing the three reconstruction methods with respect to $\nu$-Truth (upper
pad), and the relative improvement of $\nu$-Flows with respect to
$\nu$-Weighting and Ellipse (lower pad).

###  IV.3 Robustness to training sample

In comparison to the standard analytical approaches, $\nu$-Flows is trained on
a specific sample of Monte Carlo simulated events. This could introduce a
performance dependence on the sample used for training, which may not be
optimal for all generators. It should be noted that the same model is used for
all events and, just like the analytical approaches, is independent of which
samples it is applied to. However, if $\nu$-Flows has learned sample specific
effects this can lead to a suboptimal performance or even unusable levels of
performance when applied to other samples.

To study the impact of this effect we train $\nu^{2}$-Flows using the
alternative $t\bar{t}$ dilepton sample ($\nu^{2}$-Flows (Pythia8)) and use it
to reconstruct the neutrinos for the nominal $t\bar{t}$ sample. We compare the
reconstructed kinematic distributions as well as the uncertainties in each bin
of the unfolded distributions.

Negligible differences are observed in the reconstructed neutrino kinematics,
though the difference can clearly be seen for the reconstructed $W$ boson mass
and a slight difference is also seen for the reconstructed top quark invariant
mass.

Figure 9: The invariant mass, ${p}_{\mathrm{T}}$, and rapidity of the
reconstructed $t\bar{t}$ system when using $\nu^{2}$-Flows trained on the
nominal or alternative sample, in comparison to the two baseline approaches
and $\nu$-Truth (shaded grey).

Some small differences are also observed in the tails of the reconstructed top
quark and $t\bar{t}$ properties in Fig. 9, however the performance is still
substantially improved in comparison to $\nu$-Weighting and Ellipse. The
response matrix for $\nu^{2}$-Flows (Pythia8) for the double differential
distribution in $m_{t\bar{t}}$ and $p_{\mathrm{T}}^{t}$ is shown in Fig. 10.
These differences translate to a very slight change in the statistical
precision in each bin after performing the unfolding.

Figure 10: Binned response matrices for the double differential measurement of
$m_{t\bar{t}}$ and $p_{\mathrm{T}}^{t}$ when using $\nu^{2}$-Flows (Pythia8)
but evaluating on the nominal $t\bar{t}$ sample. The binning is symmetric for
both the parton and detector level observables, however the $m_{t\bar{t}}$
bins are labelled on the $x$-axis with the $p_{\mathrm{T}}^{t}$ bins labelled
on the $y$-axis. The trace fraction is calculated for each method for a simple
quantitative comparison and is 0.62 when using $\nu$-Truth and 0.31 for the
nominal $\nu^{2}$-Flows.

###  IV.4 Measuring the top quark mass

In contrast to $\nu$-Weighting and Ellipse, which both assume values for the
top quark mass in the neutrino reconstruction, $\nu^{2}$-Flows only implicitly
learns this relation from the training data. To test whether $\nu^{2}$-Flows
has sensitivity to the underlying top quark mass, we use the default
$\nu^{2}$-Flows model to evaluate events in which the truth $m_{t}$ has been
changed to either 171 GeV or 175 GeV. These samples each have 160,000 events
and are otherwise the same as the nominal sample.

We compare the reconstructed top quark mass using the four momenta of the
lepton, $b$-quark and neutrino ($m_{t}$) to the invariant mass using only of
the lepton and $b$-quark from the same top decay ($m_{b\ell}$).

Figure 11: The reconstructed top quark mass using just from the lepton and
$b$-quark from the top quark decay ($m_{b\ell}$, left), the full invariant
mass using $\nu^{2}$-Flows to reconstruct the neutrinos from the top quarks
($m_{t}$, middle), and a smoothed template for $m_{t}$ obtained by sampling
256 solutions for each event with $\nu^{2}$-Flows (right).

The distributions are shown in Fig. 11. Despite training purely on events with
the nominal top quark mass, the reconstructed distribution using
$\nu^{2}$-Flows is sensitive to the difference in truth $m_{t}$. The
separation between the three templates is similar to using $m_{b\ell}$,
however for $\nu^{2}$-Flows the difference is more prominent in the bulk of
the distribution. Another benefit of $\nu^{2}$-Flows is also demonstrated,
with a smoother templates constructed by sampling multiple neutrino solutions
for each event.

##  V Conclusions

With $\nu^{2}$-Flows we have built upon the success of the $\nu$-Flows method
to employ conditional normalizing flows to reconstruct the momentum vectors of
multiple neutrinos in a single event. In comparison to other commonly used
approaches, $\nu^{2}$-Flows is able to reconstruct both neutrinos without
enforcing strong constraints on reconstructed particle masses in order to find
solutions in an under-constrained system. This translates to a reduced bias in
the reconstruction of neutrinos, without a preference for back-to-back
neutrinos, and with a more accurate reconstruction of the kinematics of
individual top quarks and the full $t\bar{t}$ system. The reconstructed
neutrinos can be used directly or potentially combined with other machine
learning approaches which aim to reconstruct the underlying hard scatter event
[60, 61, 62, 63, 64]. The generalised architecture introduced in
$\nu^{2}$-Flows has been designed to be easy to extend to any neutrino
multiplicity, and does not place restrictions on the multiplicities of
reconstructed objects, or how they are combined to extract information from
the event.

In applying $\nu^{2}$-Flows to dilepton $t\bar{t}$ events we achieve
significant improvements in the statistical precision of unfolded
distributions in comparison to the most commonly used analytical approaches
for a wide range of distributions of interest. These improvements can lead to
an increased sensitivity in the measurement of top quark spin correlation and
entanglement at the LHC. Furthermore, in contrast to other approaches,
$\nu^{2}$-Flows has solutions for all events and the fast single event
inference below 75 ms on a single computing core. As each sample is associated
with a probability from the transformation under the normalizing flow, this
could also provide opportunities for separating $t\bar{t}$ events from
background processes, similar to the weight in $\nu$-Weighting. This could be
done using the probability of the single solution per event, or, as multiple
solutions can be sampled for each event, the highest possible probability
value for an event.

## Acknowledgements

The authors would like to thank Chris Pollard for support producing the
Pythia8 standalone samples.

The authors acknowledge funding through the SNSF Sinergia grant “Robust Deep
Density Models for High-Energy Particle Physics and Solar Flare Analysis
(RODEM)” with funding number CRSII$5\\_193716$, and the SNSF project grant
200020_212127 “At the two upgrade frontiers: machine learning and the ITk
Pixel detector”. They would also like to acknowledge individual funding
acquired through the Swiss Government Excellence Scholarships for Foreign
Scholars and the Feodor Lynen Research Fellowship from the Alexander von
Humboldt foundation.

JAR and ML contributed equally to this work.

## References

  * Evans and Bryant [2008] L. R. Evans and P. Bryant, LHC Machine, JINST 3, S08001.
  * ATLAS Collaboration [2008] ATLAS Collaboration, The ATLAS experiment at the CERN Large Hadron Collider, JINST 3, S08003.
  * CMS Collaboration [2008] CMS Collaboration, The CMS experiment at the CERN LHC, JINST 3, S08004.
  * Grossi  _et al._ [2020] M. Grossi _et al._ , Comparing traditional and deep-learning techniques of kinematic reconstruction for polarization discrimination in vector boson scattering, Eur. Phys. J. C 80, 1144 (2020).
  * Kvita [2018] J. Kvita, Study of methods of resolved top quark reconstruction in semileptonic $t\bar{t}$ decay, Nucl. Instrum. Methods Phys. Res. A 900, 84 (2018).
  * ATLAS Collaboration [2019a] ATLAS Collaboration, Measurements of top-quark pair differential and double-differential cross-sections in the $\ell$+jets channel with $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector, Eur. Phys. J. C 79, 1028 (2019a), [Erratum: Eur. Phys. J. C 80, 1092 (2020)].
  * CMS Collaboration [2021] CMS Collaboration, Measurement of differential $t\bar{t}$ production cross sections in the full kinematic range using lepton+jets events from proton-proton collisions at $\sqrt{s}$ = 13 TeV, Phys. Rev. D 104, 092013 (2021).
  * ATLAS Collaboration [2015a] ATLAS Collaboration, Determination of the top-quark pole mass using $t\overline{t}$ \+ 1-jet events collected with the ATLAS experiment in 7 TeV pp collisions, JHEP 10, 121.
  * CMS Collaboration [2018] CMS Collaboration, Measurement of the top quark mass with lepton+jets final states using $pp$ collisions at $\sqrt{s}=13\,\text{TeV}$, Eur. Phys. J. C 78, 891 (2018), [Erratum: Eur. Phys. J. C 82, 323 (2022)].
  * ATLAS Collaboration [2019b] ATLAS Collaboration, Measurement of the top quark mass in the $t\bar{t}\rightarrow$ lepton+jets channel from $\sqrt{s}=8$ TeV ATLAS data and combination with previous results, Eur. Phys. J. C 79, 290 (2019b).
  * ATLAS Collaboration [2019c] ATLAS Collaboration, Measurement of the top-quark mass in $t\bar{t}+1$-jet events collected with the ATLAS detector in $pp$ collisions at $\sqrt{s}=8$ TeV, JHEP 11, 150.
  * ATLAS Collaboration [2022] ATLAS Collaboration, Evidence for the charge asymmetry in $pp\rightarrow t\bar{t}$ production at $\sqrt{s}=13$ TeV with the ATLAS detector (2022), arXiv:2208.12095 [hep-ex] .
  * Abbott  _et al._ [1998] B. Abbott _et al._ (D0), Measurement of the top quark mass using dilepton events, Phys. Rev. Lett. 80, 2063 (1998), arXiv:hep-ex/9706014 .
  * Sonnenschein [2005] L. Sonnenschein, Algebraic approach to solve $t\bar{t}$ dilepton equations, Phys. Rev. D 72, 095020 (2005), arXiv:hep-ph/0510100 .
  * Betchart  _et al._ [2014] B. A. Betchart, R. Demina, and A. Harel, Analytic solutions for neutrino momenta in decay of top quarks, Nucl. Instrum. Meth. A 736, 169 (2014), arXiv:1305.1878 [hep-ph] .
  * CMS Collaboration [2011] CMS Collaboration, Measurement of the $t\bar{t}$ production cross section and the top quark mass in the dilepton channel in $pp$ collisions at $\sqrt{s}=7$ TeV, JHEP 07, 049, arXiv:1105.5661 [hep-ex] .
  * CMS Collaboration [2012] CMS Collaboration, Measurement of the Top-Quark Mass in $t\bar{t}$ Events with Dilepton Final States in $pp$ Collisions at $\sqrt{s}=7$ TeV, Eur. Phys. J. C 72, 2202 (2012), arXiv:1209.2393 [hep-ex] .
  * ATLAS Collaboration [2013] ATLAS Collaboration, Measurement of Top Quark Polarization in Top-Antitop Events from Proton-Proton Collisions at $\sqrt{s}$ = 7 TeV Using the ATLAS Detector, Phys. Rev. Lett. 111, 232002 (2013), arXiv:1307.6511 [hep-ex] .
  * CMS Collaboration [2014] CMS Collaboration, Measurements of the $t\bar{t}$ charge asymmetry using the dilepton decay channel in pp collisions at $\sqrt{s}=$ 7 TeV, JHEP 04, 191, arXiv:1402.3803 [hep-ex] .
  * ATLAS Collaboration [2014] ATLAS Collaboration, Measurements of spin correlation in top-antitop quark events from proton-proton collisions at $\sqrt{s}=7$ TeV using the ATLAS detector, Phys. Rev. D 90, 112016 (2014), arXiv:1407.4314 [hep-ex] .
  * ATLAS Collaboration [2015b] ATLAS Collaboration, Measurement of the charge asymmetry in dileptonic decays of top quark pairs in $pp$ collisions at $\sqrt{s}=7$ TeV using the ATLAS detector, JHEP 05, 061, arXiv:1501.07383 [hep-ex] .
  * D0 Collaboration [2016] D0 Collaboration, Precise measurement of the top quark mass in dilepton decays using optimized neutrino weighting, Phys. Lett. B 752, 18 (2016), arXiv:1508.03322 [hep-ex] .
  * CMS Collaboration [2016a] CMS Collaboration, Measurements of t t-bar spin correlations and top quark polarization using dilepton final states in pp collisions at sqrt(s) = 8 TeV, Phys. Rev. D 93, 052007 (2016a), arXiv:1601.01107 [hep-ex] .
  * CMS Collaboration [2016b] CMS Collaboration, Measurements of $t\bar{t}$ charge asymmetry using dilepton final states in pp collisions at $\sqrt{s}=8$ TeV, Phys. Lett. B 760, 365 (2016b), arXiv:1603.06221 [hep-ex] .
  * ATLAS Collaboration [2017a] ATLAS Collaboration, Measurements of top quark spin observables in $t\overline{t}$ events using dilepton final states in $\sqrt{s}=8$ TeV pp collisions with the ATLAS detector, JHEP 03, 113, arXiv:1612.07004 [hep-ex] .
  * ATLAS Collaboration [2017b] ATLAS Collaboration, Measurements of top-quark pair differential cross-sections in the $e\mu$ channel in $pp$ collisions at $\sqrt{s}=13$ TeV using the ATLAS detector, Eur. Phys. J. C 77, 292 (2017b), arXiv:1612.05220 [hep-ex] .
  * ATLAS Collaboration [2020] ATLAS Collaboration, Measurements of top-quark pair spin correlations in the $e\mu$ channel at $\sqrt{s}=13$ TeV using $pp$ collisions in the ATLAS detector, Eur. Phys. J. C 80, 754 (2020), arXiv:1903.07570 [hep-ex] .
  * CMS Collaboration [2019] CMS Collaboration, Measurement of the top quark polarization and $\mathrm{t\bar{t}}$ spin correlations using dilepton final states in proton-proton collisions at $\sqrt{s}=$ 13 TeV, Phys. Rev. D 100, 072002 (2019), arXiv:1907.03729 [hep-ex] .
  * Leigh  _et al._ [2023] M. Leigh, J. A. Raine, K. Zoch, and T. Golling, $\nu$-Flows: Conditional Neutrino Regression, SciPost Phys. 14, 159 (2023), arXiv:2207.00664 [hep-ph] .
  * Tabak and Vanden-Eijnden [2010] E. G. Tabak and E. Vanden-Eijnden, Density estimation by dual ascent of the log-likelihood, Commun. Math. Sci. 8, 217 (2010).
  * Rezende and Mohamed [2015] D. Rezende and S. Mohamed, Variational inference with normalizing flows, in  _Proceedings of the 32nd International Conference on Machine Learning_ , PMLR, Vol. 37 (Lille, France, 2015) pp. 1530–1538, arXiv:1505.05770 [stat.ML] .
  * Ardizzone  _et al._ [2019] L. Ardizzone _et al._ , Guided image generation with conditional invertible neural networks (2019), arXiv:1907.02392 [cs.CV] .
  * Raine  _et al._ [2023] J. A. Raine, M. Leigh, K. Zoch, L. Ehrke, D. Sengupta, and T. Golling, Dileptonic ttbar neutrino regression dataset, 10.5281/zenodo.8113516 (2023).
  * Vaswani  _et al._ [2017] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, Attention is all you need, Advances in neural information processing systems 30 (2017).
  * Zaheer  _et al._ [2017] M. Zaheer _et al._ , Deep sets, in  _Proceedings of Advances in Neural Information Processing Systems_ , Vol. 30 (Long Beach, USA, 2017) arXiv:1703.06114 [cs.LG] .
  * Touvron  _et al._ [2021] H. Touvron, M. Cord, A. Sablayrolles, G. Synnaeve, and H. Jégou, Going deeper with image transformers, in  _Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)_ (2021) pp. 32–42.
  * Maas  _et al._ [2013] A. L. Maas, A. Y. Hannun, and A. Y. Ng, Rectifier nonlinearities improve neural network acoustic models, in  _Proceedings of The International Conference on Machine Learning_ , Vol. 28 (PMLR, Atlanta, USA, 2013) p. 3.
  * Ba  _et al._ [2016] J. L. Ba, J. R. Kiros, and G. E. Hinton, Layer normalization (2016), arXiv:1607.06450 [stat.ML] .
  * Durkan  _et al._ [2019] C. Durkan _et al._ , Neural spline flows, in  _Proceedings of Advances in Neural Information Processing Systems_ , Vol. 32 (Vancouver, Canada, 2019) arXiv:1906.04032 [stat.ML] .
  * Durkan  _et al._ [2020] C. Durkan _et al._ , nflows: normalizing flows in PyTorch (2020).
  * Paszke  _et al._ [2017] A. Paszke _et al._ , Pytorch: An imperative style, high-performance deep learning library, in  _Proceedings of Advances in Neural Information Processing Systems_ , Vol. 32 (Vancouver, Canada, 2017) pp. 8024–8035, arXiv:1912.01703 [cs.LG] .
  * Loshchilov and Hutter [2017] I. Loshchilov and F. Hutter, Decoupled weight decay regularization (2017), arXiv:1711.05101 [cs.LG] .
  * Alwall  _et al._ [2014] J. Alwall _et al._ , The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 07, 79.
  * Sjöstrand  _et al._ [2008] T. Sjöstrand, S. Mrenna, and P. Skands, A brief introduction to PYTHIA 8.1, Comput. Phys. Commun. 178, 852 (2008).
  * Ball  _et al._ [2013] R. D. Ball _et al._ , Parton distributions with LHC data, Nucl. Phys. B 867, 244 (2013).
  * Buckley  _et al._ [2015] A. Buckley _et al._ , LHAPDF6: parton density access in the LHC precision era, Eur. Phys. J. C 75, 132 (2015).
  * Skands  _et al._ [2014] P. Skands, S. Carrazza, and J. Rojo, Tuning PYTHIA 8.1: the Monash 2013 Tune, Eur. Phys. J. C 74, 3024 (2014), arXiv:1404.5630 [hep-ph] .
  * de Favereau  _et al._ [2014] J. de Favereau _et al._ , DELPHES 3, A modular framework for fast simulation of a generic collider experiment, JHEP 02, 057.
  * Cacciari  _et al._ [2008] M. Cacciari, G. P. Salam, and G. Soyez, The anti-kt jet clustering algorithm, JHEP 04, 063.
  * Cacciari  _et al._ [2012] M. Cacciari, G. P. Salam, and G. Soyez, Fastjet user manual, Eur. Phys. J. C 72, 1 (2012).
  * CMS Collaboration [2017] CMS Collaboration, Measurement of double-differential cross sections for top quark pair production in pp collisions at $\sqrt{s}=8$ TeV and impact on parton distribution functions, Eur. Phys. J. C 77, 459 (2017), arXiv:1703.01630 [hep-ex] .
  * Hocker and Kartvelishvili [1996] A. Hocker and V. Kartvelishvili, SVD approach to data unfolding, Nucl. Instrum. Meth. A 372, 469 (1996), arXiv:hep-ph/9509307 .
  * Brenner  _et al._ [2020] L. Brenner, R. Balasubramanian, C. Burgard, W. Verkerke, G. Cowan, P. Verschuuren, and V. Croft, Comparison of unfolding methods using RooFitUnfold, Int. J. Mod. Phys. A 35, 2050145 (2020), arXiv:1910.14654 [physics.data-an] .
  * Gagunashvili [2010] N. D. Gagunashvili, Machine learning approach to inverse problem and unfolding procedure, arXiv:1004.2006 [physics.data-an] (2010).
  * Datta  _et al._ [2018] K. Datta, D. Kar, and D. Roy, Unfolding with Generative Adversarial Networks, arXiv:1806.00433 [physics.data-an] (2018).
  * Andreassen  _et al._ [2020] A. Andreassen, P. T. Komiske, E. M. Metodiev, B. Nachman, and J. Thaler, OmniFold: A Method to Simultaneously Unfold All Observables, Phys. Rev. Lett. 124, 182001 (2020), arXiv:1911.09107 [hep-ph] .
  * Bellagente  _et al._ [2020] M. Bellagente, A. Butter, G. Kasieczka, T. Plehn, A. Rousselot, R. Winterhalder, L. Ardizzone, and U. Köthe, Invertible Networks or Partons to Detector and Back Again, SciPost Phys. 9, 074 (2020), arXiv:2006.06685 [hep-ph] .
  * Backes  _et al._ [2022] M. Backes, A. Butter, M. Dunford, and B. Malaescu, An unfolding method based on conditional Invertible Neural Networks (cINN) using iterative training, arXiv:2212.08674 [hep-ph] (2022).
  * Chan and Nachman [2023] J. Chan and B. Nachman, Unbinned Profiled Unfolding, arXiv:2302.05390 [hep-ph] (2023).
  * Fenton  _et al._ [2022] M. J. Fenton _et al._ , Permutationless many-jet event reconstruction with symmetry preserving attention networks, Phys. Rev. D 105, 112008 (2022).
  * Shmakov  _et al._ [2022] A. Shmakov _et al._ , Spanet: Generalized permutationless set assignment for particle physics using symmetry preserving attention, SciPost Phys. 12, 178 (2022).
  * Badea  _et al._ [2022] A. Badea, W. J. Fawcett, J. Huth, T. J. Khoo, R. Poggi, and L. Lee, Solving combinatorial problems at particle colliders using machine learning, Phys. Rev. D 106, 016001 (2022), arXiv:2201.02205 [hep-ph] .
  * Qiu  _et al._ [2023] S. Qiu, S. Han, X. Ju, B. Nachman, and H. Wang, Parton Labeling without Matching: Unveiling Emergent Labelling Capabilities in Regression Models, arXiv:2304.09208 [hep-ph] (2023).
  * Ehrke  _et al._ [2023] L. Ehrke, J. A. Raine, K. Zoch, M. Guth, and T. Golling, Topological reconstruction of particle physics processes using graph neural networks, Phys. Rev. D 107, 116019 (2023), arXiv:2303.13937 [hep-ph] .

##  Appendix A Additional material

Strong biases are introduced when replacing the normalizing flow with a naïve
feed-forward network to predict the neutrino momenta for each event directly.
This network, $\nu^{2}$-FF, has the same architecture as the feature
extraction network in $\nu^{2}$-Flows, but with a dense output layer for the
neutrino momenta, and is trained with an L2 loss. Figure 12 shows the
reconstructed invariant mass of the top quark and $t\bar{t}$ pair when using
$\nu^{2}$-FF to reconstruct the neutrinos in comparison to the other
reconstruction approaches. A strong negative bias can be seen in both
distributions.

Figure 12: The invariant masses of the reconstructed top quark and $t\bar{t}$
system when using the three neutrino reconstruction methods discussed in the
paper, as well as a feed-forward regression model $\nu^{2}$-FF, in comparison
to $\nu$-Truth (shaded grey). This plot highlights large negative bias induced
by the feed-forward model.

The relative uncertainty for each bin in the unfolded distributions is
compared in Appendix A for all methods with respect to $\nu$-Truth, including
both $\nu^{2}$-Flows and $\nu^{2}$-Flows (Pythia8). The reconstructed top
quark and $W$ boson invariant masses are compared in Fig. 13.

Table 5: Relative uncertainty in each bin of the respective unfolded double
differential distributions for each neutrino reconstruction method with
respect to the uncertainty when using $\nu$-Truth. The bins are ordered first
by increasing $m_{t\bar{t}}$ followed by the second variable, with vertical
dividers indicating the bin edges in $m_{t\bar{t}}$. The method with the
smallest relative increase in uncertainty in comparison to $\nu$-Truth is
highlighted in bold.

Observables | Method | Relative uncertainty to $\nu$-Truth (per bin)
---|---|---
$m_{t\bar{t}}$ | $y_{t\bar{t}}$ | $\nu$-Weighting |  2.9 |  3.2 |  3.0 |  2.6 |  2.9 |  3.4 |  3.1 |  2.6 |  2.7 |  2.8 |  2.9 |  2.6 |  2.4 |  2.5 |  2.6 |  2.7
Ellipse |  3.5 |  3.6 |  3.2 |  2.4 |  2.2 |  3.6 |  3.5 |  3.1 |  3.6 |  3.8 |  4.1 |  3.5 |  3.1 |  3.4 |  3.6 |  3.6
$\nu^{2}$-Flows |  1.8 |  1.9 |  1.9 |  1.8 |  1.9 |  2.1 |  1.9 |  1.8 |  1.8 |  1.8 |  1.8 |  1.7 |  1.6 |  1.6 |  1.6 |  1.7
$\nu^{2}$-Flows (Pythia8) |  1.9 |  1.9 |  1.9 |  1.8 |  1.9 |  2.1 |  2.0 |  1.8 |  1.8 |  1.8 |  1.8 |  1.7 |  1.7 |  1.6 |  1.7 |  1.8
$p_{\mathrm{T}}^{t\bar{t}}$ | $\nu$-Weighting |  3.5 |  2.6 |  1.6 |  2.1 |  3.4 |  3.0 |  2.4 |  2.3 |  2.5 |  2.7 |  2.5 |  2.2 |  2.1 |  2.2 |  2.2 |  2.3
Ellipse |  7.6 |  5.3 |  2.2 |  3.7 |  7.3 |  5.7 |  4.3 |  4.1 |  4.0 |  4.7 |  4.4 |  3.8 |  3.6 |  3.3 |  3.5 |  3.8
$\nu^{2}$-Flows |  1.9 |  1.6 |  1.2 |  1.4 |  2.0 |  1.7 |  1.5 |  1.5 |  1.6 |  1.6 |  1.6 |  1.5 |  1.5 |  1.5 |  1.5 |  1.5
$\nu^{2}$-Flows (Pythia8) |  2.0 |  1.6 |  1.2 |  1.4 |  2.0 |  1.7 |  1.5 |  1.5 |  1.7 |  1.6 |  1.6 |  1.5 |  1.5 |  1.5 |  1.5 |  1.5
$p_{\mathrm{T}}^{t}$ | $\nu$-Weighting |  3.1 |  2.3 |  1.7 |  2.5 |  3.2 |  3.2 |  2.8 |  2.9 |  2.8 |  2.9 |  2.9 |  2.3 |  2.2 |  2.2 |  2.3 |  2.4
Ellipse |  4.8 |  3.1 |  2.5 |  3.8 |  4.9 |  4.9 |  3.8 |  4.3 |  4.0 |  4.4 |  4.9 |  3.0 |  3.5 |  3.7 |  3.6 |  3.4
$\nu^{2}$-Flows |  2.2 |  1.9 |  1.8 |  2.1 |  2.5 |  2.4 |  2.0 |  2.2 |  2.1 |  2.0 |  2.0 |  1.7 |  1.9 |  1.9 |  1.7 |  1.6
$\nu^{2}$-Flows (Pythia8) |  2.3 |  1.9 |  1.9 |  2.2 |  2.5 |  2.4 |  2.0 |  2.3 |  2.2 |  2.1 |  2.0 |  1.7 |  2.0 |  2.0 |  1.7 |  1.6
$\Delta\phi(\ell^{+}\ell^{-})$ | $\nu$-Weighting |  2.0 |  2.0 |  1.5 |  1.4 |  1.9 |  2.1 |  2.0 |  1.9 |  1.9 |  2.0 |  2.2 |  2.0 |  1.5 |  1.6 |  1.9 |  2.0
Ellipse |  2.2 |  2.1 |  1.5 |  1.3 |  1.8 |  2.3 |  2.4 |  2.3 |  2.3 |  2.5 |  2.8 |  2.4 |  1.7 |  1.9 |  2.4 |  2.6
$\nu^{2}$-Flows |  1.5 |  1.5 |  1.4 |  1.4 |  1.6 |  1.6 |  1.5 |  1.5 |  1.5 |  1.5 |  1.6 |  1.5 |  1.3 |  1.4 |  1.5 |  1.5
$\nu^{2}$-Flows (Pythia8) |  1.5 |  1.5 |  1.4 |  1.4 |  1.6 |  1.6 |  1.5 |  1.5 |  1.5 |  1.5 |  1.6 |  1.5 |  1.3 |  1.4 |  1.5 |  1.5

Figure 13: The reconstructed invariant mass of the $W$ boson and top quarks
when using the three neutrino reconstruction methods in comparison to
$\nu$-Truth (shaded grey) as well as the alternative $\nu^{2}$-Flows (Pythia8)
model.

The response matrices for the double differential distributions in
$m_{t\bar{t}}$ and $p_{\mathrm{T}}^{t}$, and $m_{t\bar{t}}$ and
$y_{t\bar{t}}$, are shown in Figs. 14 and 15 respectively.

Figure 14: Binned response matrices for the double differential measurement of
$m_{t\bar{t}}$ and $p_{\mathrm{T}}^{t\bar{t}}$ when using each of the three
methods for neutrino reconstruction. The binning is symmetric for both the
parton and detector level observables, however the $m_{t\bar{t}}$ bins are
labelled on the $x$-axis with the $p_{\mathrm{T}}^{t\bar{t}}$ bins labelled on
the $y$-axis. The trace fraction is calculated for each method for a simple
quantitative comparison and is 0.63 when using $\nu$-Truth.

Figure 15: Binned response matrices for the double differential measurement of
$m_{t\bar{t}}$ and $y_{t\bar{t}}$ when using each of the three methods for
neutrino reconstruction. The binning is symmetric for both the parton and
detector level observables, however the $m_{t\bar{t}}$ bins are labelled on
the $x$-axis with the $y_{t\bar{t}}$ bins labelled on the $y$-axis. The trace
fraction is calculated for each method for a simple quantitative comparison
and is 0.70 when using $\nu$-Truth.

Detailed schema of the transformer encoder block and cross-attention block
used in $\nu^{2}$-Flows are detailed in Fig. 16.

Figure 16: Transformer encoder block (left) and cross-attention block (right)
comprising multi-headed attention, layer normalisation and simple linear
layers. Residual connections are used after the multi-headed attention and
linear operation. Conditional information is provided as context by
concatenating it to each token before the linear layer.