# Constraints on the trilinear and quartic Higgs couplings from triple Higgs
production
at the LHC and beyond

Panagiotis Stylianou<EMAIL_ADDRESS>Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
Georg Weiglein<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany
II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany

###### Abstract

Experimental information on the trilinear Higgs boson self-coupling
$\kappa_{3}$ and the quartic self-coupling $\kappa_{4}$ will be crucial for
gaining insight into the shape of the Higgs potential and the nature of the
electroweak phase transition. While Higgs pair production processes provide
access to $\kappa_{3}$, triple Higgs production processes, despite their small
cross sections, will provide valuable complementary information on
$\kappa_{3}$ and first experimental constraints on $\kappa_{4}$. We
investigate triple Higgs boson production at the HL-LHC, employing efficient
Graph Neural Network methodologies to maximise the statistical yield. We show
that it will be possible to establish bounds on the variation of both
couplings from the HL-LHC analyses that significantly go beyond the
constraints from perturbative unitarity. We also discuss the prospects for the
analysis of triple Higgs production at future high-energy lepton colliders
operating at the TeV scale.

††preprint: DESY-23-203

##  I Introduction

Since the discovery of a Higgs boson with a mass of about $125\,\text{GeV}$ in
2012 Aad et al. (2012); Chatrchyan et al. (2012), a tremendous and ongoing
effort has been enacted in order to gain insights into the properties and
interactions of the detected state. Its couplings with third generation
fermions and weak gauge bosons, as well as the loop-induced couplings with
gluons and photons, have been investigated in detail, indicating agreement
with the predictions of the Standard Model (SM) within the present
experimental and theoretical uncertainties. In view of the plethora of
possible connections of the detected Higgs boson to sectors of physics beyond
the SM (BSM), probing the Higgs interactions with respect to possible effects
of BSM physics will be of central importance at the present and future runs at
the LHC and at any future collider.

In this context the Higgs boson self-couplings are of particular relevance,
while experimentally these couplings are very difficult to access.
Experimental information about the trilinear and quartic Higgs couplings is
needed to gain insights about the shape of the Higgs potential, which will
have implications for a better understanding of the electroweak phase
transition in the early universe and may be instrumental for explaining the
observed asymmetry between matter and anti-matter in the universe. In the SM
the Higgs potential is given by

$V(\Phi)=\lambda(\Phi^{\dagger}\Phi)^{2}-\mu^{2}\Phi^{\dagger}\Phi$ (1)

in terms of the single Higgs doublet field $\Phi$. In extended scalar sectors
the potential can have a much richer structure. While the cubic and quartic
Higgs couplings arising from Eq. (1) are correlated in the SM and can be
predicted in terms of the known experimental values of the mass of the
detected Higgs boson and the vacuum expectation value, large deviations from
the SM predictions for the Higgs self-couplings are possible even in scenarios
where the other couplings of the Higgs boson at $125\,\text{GeV}$ are very
close to the SM predictions (see e.g. Ref. Bahl et al. (2022) for a recent
discussion of this point for the case of the trilinear Higgs coupling).
Experimental constraints on the trilinear and quartic Higgs self-couplings can
be expressed in terms of the so-called $\kappa$-framework, where $\kappa_{3}$
($\kappa_{4}$) denotes the coupling modifier of the cubic (quartic) coupling
from its SM value at lowest order, i.e. $\kappa_{i}=g_{i}/g_{i}^{\text{SM}}$,
where $g_{i}$ denotes the value of the coupling and $g_{i}^{\text{SM}}$ its
lowest-order SM prediction, and $i=3,4$.

The most direct probe of the trilinear Higgs coupling at the LHC is the
production of a pair of Higgs bosons, where $\kappa_{3}$ enters at leading
order (LO). Both the ATLAS Aad et al. (2023a) and CMS Tumasyan et al. (2022a)
collaborations determine the limits on $\kappa_{3}$ from both gluon fusion and
weak boson fusion (WBF) from different decay channels of the Higgs boson. At
next-to-leading order (NLO), the trilinear Higgs coupling contributes to the
Higgs-boson self-energy and also enters in additional one-loop and two-loop
diagrams in WBF and gluon fusion, respectively, enabling the possibility of an
indirect measurement through single-Higgs production Degrassi et al. (2016);
Maltoni et al. (2017); Di Vita et al. (2017); Gorbahn and Haisch (2016); Bizon
et al. (2017). The inclusion of single-Higgs information by the ATLAS
collaboration results in the most stringent bound to date on $\kappa_{3}$:
$\left[-0.4,6.3\right]$. Triple-Higgs production is known to suffer from very
small cross sections, but yields additional information on $\kappa_{3}$ which
could be used in combination with the aforementioned searches. Furthermore, it
can provide the first experimental constraints on the quartic Higgs coupling
$\kappa_{4}$.

The paper is structured as follows. In Sect. II we discuss the allowed values
of $\kappa_{3}$ and $\kappa_{4}$ from the perspective of perturbative
unitarity and show that sizeable contributions to $\kappa_{4}$ can occur,
especially if $\kappa_{3}$ deviates from the SM value. We explore in Sect. III
how well the HL-LHC will be able to constrain $\kappa_{3}$ and $\kappa_{4}$
from the $6b$ and $4b2\tau$ channels. Lepton colliders are additionally
explored in Sect. IV before conclusions are presented in Sect. V.

##  II Current bounds, unitarity and theoretical motivation

Besides the experimental constraints from Higgs pair and triple Higgs
production processes, which will be discussed below, theoretical bounds can be
placed on the Higgs self-couplings from the requirement of perturbative
unitarity. In our analysis we employ the unitarity constraints obtained at
tree level.

A general matrix element for $2\rightarrow 2$ scattering with initial and
final states $\ket{i}$ and $\ket{f}$, respectively, can be decomposed in terms
of partial waves through the Jacob-Wick expansion Jacob and Wick (1959)

${\cal{M}}_{if}=16\pi\sum_{J}(2J+1)a_{fi}^{J}({\cal{D}}_{\lambda_{i},\lambda_{f}}^{J}(\theta,\phi))^{*}\;,$
(2)

where $J$ indicates the total angular momentum of the corresponding amplitude,
and the $\lambda_{i,f}$ denote the helicities of the initial and final states.
The most relevant channel at tree level for constraining $\kappa_{3}$ and
$\kappa_{4}$ is $HH\rightarrow HH$ scattering, where the Wigner-D functions
${\cal{D}}_{\lambda_{i},\lambda_{f}}^{J}$ reduce to unity for the zeroth
partial wave. Conservation of probability leads to the requirement that the
perturbative expansion must satisfy the optical theorem, which can be used to
obtain an upper bound on the zeroth partial wave of

$\absolutevalue{\text{Re}\left(a_{ii}^{0}\right)}\leq\frac{1}{2}\;,$ (3)

which if violated indicates inconsistencies in the perturbative calculation.

The zeroth partial wave at tree level can be calculated as111We project out
the zeroth partial waves from the matrix element computed through FeynArts
Kublbeck et al. (1990); Hahn (2001) and FormCalc Hahn and Schappacher (2002),
which is in agreement with the result of Ref. Liu et al. (2018).

$\begin{split}a_{ii}^{0}=\frac{3M_{H}^{2}\sqrt{s^{2}-4M_{H}^{2}s}}{32\pi
s(s-M_{H}^{2})v^{2}}\left[\kappa_{4}(s-M_{H}^{2})-3\kappa_{3}^{2}M_{H}^{2}\right.\\\
\left.+\frac{6\kappa_{3}^{2}M_{H}^{2}(s-M_{H}^{2})}{s-4M_{H}^{2}}\log\left(\frac{s}{M_{H}^{2}}-3\right)\right].\end{split}$
(4)

In the limit where the centre of mass energy is high, $a_{ii}^{0}$ solely
depends on $\kappa_{4}$, while at lower energies a sizeable contribution from
$\kappa_{3}$ can yield a peak in $a_{ii}^{0}$ that surpasses the allowed
limit. We have calculated the zeroth partial wave for different values of
$\kappa_{3}$ and $\kappa_{4}$ for a large range of energies in order to
identify the parameter regions that are allowed by tree-level perturbative
unitarity. Fig. 1 shows the bounds from perturbative unitarity along with the
current experimental bounds on the trilinear coupling from ATLAS,
$\kappa_{3}\in[-0.4,6.3]$ at the 95% C.L. Aad et al. (2023a), and the $95$%
combined ATLAS and CMS HL-LHC projection under the SM hypothesis,
$\kappa_{3}\in[0.1,2.3]$ Cepeda et al. (2019).

Figure 1: Bounds from perturbative unitarity on $\kappa_{3}$ and $\kappa_{4}$
as obtained from $HH\rightarrow HH$ scattering. In addition, the current
experimental bounds on $\kappa_{3}$ are shown (black dashed lines), as well as
the expected projections from the HL-LHC (black solid lines). The shaded light
blue region indicates where the dimension-eight contributions to $\kappa_{4}$
are smaller than the dimension-six ones, while the dotted blue line
corresponds to the case where the dimension-eight contributions vanish,
$\kappa_{4}-1\simeq 6(\kappa_{3}-1)$.

The unitarity bounds on $\kappa_{4}$ are significantly weaker than the ones on
$\kappa_{3}$. This feature can be understood from the Effective Field Theory
(EFT) perspective, where effects from higher-dimensional operators to the
potential are incorporated as an expansion in terms of inverse powers of a UV-
scale $\Lambda$ Boudjema and Chopin (1996) (see also the discussion in Ref.
Maltoni et al. (2018)),

$V_{\text{BSM}}=\frac{C_{6}}{\Lambda^{2}}\left(\Phi^{\dagger}\Phi-\frac{v^{2}}{2}\right)^{3}+\frac{C_{8}}{\Lambda^{4}}\left(\Phi^{\dagger}\Phi-\frac{v^{2}}{2}\right)^{4}+{\cal{O}}({\frac{1}{\Lambda^{6}})}\;.$
(5)

We use the convention where in the unitary gauge
$\Phi=\left(0,(v+H)/\sqrt{2}\right)$, where $v$ denotes the electroweak vacuum
expectation value (VEV), and $H$ is the $125$ GeV Higgs boson. The benefit of
this parameterisation of the higher-dimensional operators is that $\kappa_{3}$
receives corrections purely from dimension-six operators, while $\kappa_{4}$
only from dimension-six and dimension-eight operators (interaction vertices
with more Higgs legs would additionally receive corrections from
${\cal{O}}(1/\Lambda^{6})$ terms and so on). With the definitions of
$\kappa_{i}$ as before, the coupling modifiers receive corrections

$\begin{split}(\kappa_{3}-1)&=\frac{C_{6}v^{2}}{\lambda\Lambda^{2}}\;,\\\
(\kappa_{4}-1)&=\frac{6C_{6}v^{2}}{\lambda\Lambda^{2}}+\frac{4C_{8}v^{4}}{\lambda\Lambda^{4}}\\\
&\simeq
6(\kappa_{3}-1)+{\cal{O}}\left(\frac{1}{\Lambda^{4}}\right)\;.\end{split}$ (6)

Thus, if a small correction is induced in $\kappa_{3}$, one should expect that
in an EFT theory with high scale cutoff where the dimension-eight terms are
negligible, the deviation in $\kappa_{4}$ from the SM expectation would be six
times larger. Even if the higher-dimensional contributions are relevant,
$\lvert(\kappa_{4}-1)-6(\kappa_{3}-1)\rvert<6\lvert\kappa_{3}-1\rvert$ needs
to be satisfied in order to maintain a well-behaved expansion in powers of
$\Lambda$. Although in this work we choose to work in all generality without
any EFT assumptions on the $\kappa_{3}$ and $\kappa_{4}$ modifiers, we
indicate the region where this condition is fulfilled in Fig. 1.

Figure 2: The left plot shows the impact of an increasing splitting between
the masses of the BSM Higgs bosons on the one-loop prediction for the
trilinear Higgs coupling $\kappa_{3}$, where $M=m_{H}=600$ GeV and
$m_{A}=M_{H^{\pm}}$ is varied, in agreement with the results of Ref. Bahl et
al. (2022) for the quoted benchmark point. On the right, the respective plot
for $\kappa_{4}$ is shown. The shorthand notations $s_{\beta}$, $c_{\beta}$
and $t_{\beta}$ denote $\sin\beta,\cos\beta$ and $\tan\beta$, respectively.
Figure 3:  Correlation of $\kappa_{3}$ and $\kappa_{4}$ for different values
of $M=m_{H}$ and $m_{A}=M_{H^{\pm}}$. The light purple area indicates the
range of 2HDM predictions for the correlation between $\kappa_{3}$ and
$\kappa_{4}$ for $m_{A},M\in[0.3,10]$ TeV. We show specific choices of $M$ as
solid lines with varying $m_{A}$, in order to demonstrate the effect on
$\kappa_{3}$ and $\kappa_{4}$. The gray (dark) shaded regions are the areas
excluded by tree-level perturbative unitarity according to the analysis of
Fig. 1, and the shaded light blue region is defined in the same way as in Fig.
1.

In order to present an example where Eq. (6) can be realised, we consider the
Two-Higgs Doublet Model (2HDM), where beyond tree level, the cubic and quartic
self-couplings can receive significant contributions, as shown in Ref. Bahl et
al. (2022) (see also Ref. Bahl et al. (2023a)). A review of the 2HDM can be
found in Ref. Branco et al. (2012). We work in the alignment limit with the
lightest scalar identified as the $125$ GeV Higgs boson (after Electroweak
Symmetry Breaking (EWSB) and rotation to the Higgs basis) and perform a one-
loop calculation222We use FeynArts Kublbeck et al. (1990); Hahn (2001),
FormCalc Hahn and Schappacher (2002) and LoopTools Hahn and Perez-Victoria
(1999), using the model of Ref. Wu (2023). of the trilinear and quartic
couplings employing the on-shell renormalisation scheme. As a motivated
example we pick a benchmark point from Refs. Bahl et al. (2022, 2023b) which
is compatible with the latest experimental results while also receiving
sizeable trilinear-coupling corrections. We reproduce the one-loop result of
Refs. Bahl et al. (2022, 2023b) and also show the quartic coupling in Fig. 2.
As expected, the prediction for the quartic coupling quickly rises to values
even beyond what is allowed by tree-level perturbative unitarity in the
$\kappa$-framework if the splitting between the mass of the CP-odd Higgs boson
in the 2HDM, $m_{A}$, and the BSM Higgs scale $M=m_{12}/(c_{\beta}s_{\beta})$
increases. In the displayed example the unitarity bound is violated if $m_{A}$
surpasses $\sim 1100$ GeV, and further two-loop contributions would tighten
the bound on $m_{A}$.

In Fig. 3 the linear relation between $\kappa_{3}$ and $\kappa_{4}$ in the
2HDM is shown for variations of the scale $M$ and for masses $m_{A}\leq 1.5$
TeV. Varying the values of $m_{A}$ and $M$ shifts the relation between the
self-couplings while maintaining a linear correlation between them. For
$\kappa_{3}=6$ the corresponding results for $\kappa_{4}$ vary between
$\kappa_{4}\approx 22$ and $\kappa_{4}\approx 31$ for the displayed scenarios.
Thus, the largest allowed values for $\kappa_{3}$ according to the present
bounds are correlated in the 2HDM with very large shifts in $\kappa_{4}$. As
indicated by the shaded light blue region in the plot these predictions for
$\kappa_{3}$ and $\kappa_{4}$ are associated with a well-behaved power
expansion within an EFT framework. While it would also be of interest to
explore which models can induce an even larger deviation of $\kappa_{4}$ for
relatively small values of $\kappa_{3}$, potentially resulting in regions that
require a non-linear effective prescription (for instance the Electroweak
Chiral Lagrangian), we leave such an investigation for future work.

##  III Triple Higgs production at the HL-LHC

The production of three Higgs scalars at the LHC and future colliders is
highly suppressed compared to single and double Higgs production, severely
limiting the available final states that can be explored at the LHC. In order
to obtain the highest values for the product of cross section and branching
ratios, one needs to consider the dominant production mode through gluon
fusion, but also the main decay channel to a $b$-quark pair. The latter is
difficult in hadron collisions due to the sizeable multi-jet background from
QCD processes. It can be problematic for typical cut-and-count analyses to
sufficiently suppress the background while at the same time avoiding a large
reduction of signal events in order to maximise significance. In this work we
resort to Machine Learning (ML) techniques for appropriately selecting the
signal region of the considered channels.

In order to identify which of the decay channels of the on-shell Higgs bosons
can be utilised for the analysis at the LHC, we start with an optimistic
estimate of the number of events for the $6b$, $4b2\tau$, $2b4\tau$ and
$4b2\gamma$ final states.333The different final states from triple-Higgs
production have been discussed in Ref. Papaefstathiou and Sakurai (2016).
While in principle channels with two $W$ bosons can also be of relevance, we
choose not to explore them in view of the difficulty of the final states.
Within the SM the involved branching ratios are given as

$\displaystyle\text{BR}(H\to b\bar{b})=0.584,$ (7)
$\displaystyle\text{BR}(H\to\tau^{+}\tau^{-})=6.627\times 10^{-2}$
$\displaystyle\text{BR}(H\to\gamma\gamma)=2.26\times 10^{-3}\;.$

We note that the $4b2\gamma$ and $2b4\tau$ final states only produce a few
events at $3$/ab, even at relatively large coupling modifiers
$\kappa_{3}\gtrsim 4.5$, $\kappa_{4}\gtrsim 30$ (taking into account K-factors
of 1.7 de Florian et al. (2020) and tagging efficiencies of all taus and all-
but-one $b$-quarks). It is therefore unlikely that these channels will be
statistically significant at the HL-LHC, even though they can be highly
relevant for colliders utilising higher energies, as shown in Refs. Fuks et
al. (2016); Papaefstathiou and Sakurai (2016); Chen et al. (2016). We
therefore will not consider these channels further, and instead focus on the
$6b$ and $4b2\tau$ channels.

The background processes for the $6b$ final state have been thoroughly
discussed in Ref. Papaefstathiou et al. (2019) (see also Ref. Papaefstathiou
et al. (2021)), and it is expected that the dominant contribution arises from
multi-jet QCD $6b$ events. This is the only background that is taken into
consideration for this final state in this work, and we neglect subdominant
channels.

In the $4b2\tau$ channel444See Ref. Fuks et al. (2017) for an analysis at FCC
energies., the dominant backgrounds arise from the production of four
$b$-quarks along with two $W$ bosons ($WWbbbb$) or one $Z$ boson ($Zbbbb$).
The former includes the production of a top and bottom pair ($ttbb$) with
subsequent decays $t\rightarrow Wb$. The production of a top pair associated
with a Higgs ($ttH$) or a $Z$ boson ($ttZ$) also yields noteworthy
contributions. Here the $ttH$ channel is particularly problematic if a
reconstructed resonance close to the $125$ GeV mass is required during an
analysis to isolate the triple-Higgs signal. The final background included in
our analysis is the four top production ($tttt$).

###  III.1 Analysis

####  III.1.1 Event generation and pre-selection

We use Madgraph Alwall et al. (2014); Hirschi and Mattelaer (2015) for event
generation and modify the provided SM model file in the UFO Degrande et al.
(2012) format to introduce the modifications of the trilinear and quartic
Higgs couplings $\kappa_{3}$ and $\kappa_{4}$, respectively.555We checked that
our (loop-induced) leading order cross sections for the production of three
(undecayed) Higgs bosons is in good agreement with Refs. Plehn and Rauch
(2005); Binoth et al. (2006).  Signal events are generated for $pp\rightarrow
hhh$ and are subsequently decayed on-shell with Madspin Artoisenet et al.
(2013) in order to obtain the cross section rates. Due to the complexity of
the multi-particle final states we generate events with a minimum transverse
momentum for the $b$-quarks of $p_{T}(b)>28$ GeV and within the pseudorapidity
region $\absolutevalue{\eta}<2.5$, while we will later impose stricter cuts
during the analysis. Additionally, since the signal consists of three on-shell
Higgs bosons, we impose a cut on the invariant mass of the process of
$\sqrt{\hat{s}}>350$ GeV at generation level.

While in principle one could explore different cuts in order to efficiently
identify the signal region, the complexity of the final states would render
this a cumbersome and difficult procedure, possibly requiring the use of
complicated observables. Instead, we resort to Graph Neural Networks (GNNs)
for an efficient discrimination between signal and background events. This
requires an appropriate embedding of particle events to graphs. Before we
address the ML aspects of the analysis it is appropriate to define pre-
selection conditions required to be satisfied by each event that gets passed
to the network.

Showering and hadronisation is performed with Pythia8 Sjöstrand et al. (2015)
saving the resulting events as HepMC files Dobbs and Hansen (2001). FastJet
Cacciari et al. (2012); Cacciari and Salam (2006) is interfaced through Rivet
Buckley et al. (2020); Bierlich et al. (2020), and jets are clustered using
the anti-kT algorithm Cacciari et al. (2008) of radius $0.4$ and requiring a
transverse momentum of $p_{T}(j)>30$ GeV. We use Rivet to calculate the events
that will pass the pre-selection using a $b$-tagging efficiency (independent
of $p_{T}$) of $0.8$. For the $6b$ ($4b2\tau$) channel, at least five (three)
$b$-quarks are required, satisfying the conditions $p_{T}(b)>30$ GeV and
$\absolutevalue{\eta(b)}<2.5$. For the $4b2\tau$ channel two $\tau$ particles
must also be identified in the central part of the detector,
$\absolutevalue{\eta(\tau)}<2.5$, with $p_{T}(\tau)>10$ GeV. The $\tau$
particles are identified with the TauFinder class of Rivet, and at least one
$\tau$ particle must decay hadronically.666We take inspiration from the
$bb\tau\tau$ analysis of Ref. Aad et al. (2023b). We apply an efficiency of
$0.8$ for both leptonic and hadronic taus.777It should be noted that such
efficiencies have already shown to be achievable, see e.g. Ref. Tumasyan et
al. (2022b). The invariant mass of the sum of the four-momenta of the above
final states should exceed $350$ GeV, otherwise the event is vetoed. Finally,
we form combinations of $b$-quark pairs, and at least one pair is required to
have an invariant mass close to the mass of the Higgs boson,
$m_{b\bar{b}}\in\left[110,140\right]$ (in GeV). In the case of the $4b2\tau$
channel the event passes the pre-selection also if the invariant mass
criterion is satisfied by the invariant mass of the di-tau state,
$m_{\tau\tau}$.

####  III.1.2 Graph Embedding and Neural Network Architecture

GNNs, stemming from the idea that certain types of data can be efficiently
represented as graphs, have been increasingly utilised in particle physics.
Various works have indicated their applicability for BSM-relevant tasks such
as event classification Blance and Spannowsky (2020), jet-tagging Qu and
Gouskos (2020); Dreyer and Qu (2021), particle reconstruction Pata et al.
(2021), identifying anomalies in data arising from BSM interactions Atkinson
et al. (2022a); Atkinson et al. (2021) and obtaining constraints on parameters
in SMEFT or the $\kappa$-framework Atkinson et al. (2022b); Anisha et al.
(2022).888For a detailed list of references in particle physics using GNNs,
see Ref. Feickert and Nachman (2021). The latter is what we aim to achieve by
performing a fit within the $\kappa_{3}$–$\kappa_{4}$ plane after the
efficient selection of a signal region from the GNN. Similar architectures
using graphs have also been recently utilised by experiments at the LHC, see
e.g. Ref. Aad et al. (2023c).

The generated events need to be embedded in graphs before they are passed to
the neural network. We explore two different paths999All considered graphs are
bidirectional.:

  1. 1.

Fully Connected (FC): Add nodes for all the considered final states (i.e. $b$
quarks and $\tau$ leptons, denoted as $b_{i}$ and $\tau_{i}$ according to
their $p_{T}$ values) and edges connecting all the nodes. We use the
transverse momentum, pseudorapidity, azimuthal angle, energy, mass and PDG
identification number as node features,
$\left[p_{T},\eta,\phi,E,m,\text{PDGID}\right]$, while no edge features are
introduced. A node is also added for the missing momentum of the event.

  2. 2.

Reconstructed Nodes (RN): Add fully connected nodes for $b$ quarks (and $\tau$
leptons for the $4b2\tau$ final state) as before, but additionally add nodes
$H_{i}$ for reconstructed pairs of particles $i$, $j$ that are (relatively
well) compatible with the Higgs-boson mass, $m_{ij}=125\pm 25$ GeV. This is
achieved by forming combinations between all the $b$-quarks and (if
applicable) the $\tau$-pair. The $H_{i}$ nodes correspond to the four-momentum
and mass of the reconstructed pair, ordered according to which is closest to
the Higgs-boson mass of $125$ GeV. All the nodes have
$\left[p_{T},\eta,\phi,E,m,\text{PDGID}\right]$ as associated features, where
the PDGID for $H_{i}$ is zero.

Such physics-inspired approaches according to the expected chain of the event
have been shown to improve results in semi-leptonic top decays Atkinson et al.
(2022b) and are actively being explored Ehrke et al. (2023).

GNNs operate by calculating messages using node features (and edge features if
these exist) and iteratively updating the node features for each message
passing layer. We rely on the EdgeConv Wang et al. (2018) operation for
message passing, where the message of node $i$ at the $l$-th message passing
layer is calculated with

$m_{ij}^{(l)}=\text{\sc{ReLU}}\left(\Theta(\vec{x}_{j}^{\,(l)}-\vec{x}_{i}^{\,(l)})+\Phi(\vec{x}_{i}^{\,(l)})\right)\;,$
(8)

where $\Theta$ and $\Phi$ indicate linear layers. The node features for $l=0$
are the kinematical quantities we have defined as inputs, and the updated node
features are obtained from the messages by averaging over the neighbouring
nodes,

$\vec{x}_{i}^{(l+1)}=\frac{1}{\absolutevalue{{\cal{N}}(i)}}\sum_{j\in{\cal{N}}(i)}m_{ij}^{(l)}\;.$
(9)

The final node features after all EdgeConv operations are aggregated to a
single vector using a ‘mean’ graph readout operation. In principle, it is
possible to additionally include further (non-graph related) layers at this
stage. The final network score is obtained with a linear layer with the
SoftMax activation Bridle (1989) that reduces the resulting features to a two-
dimensional vector, with each entry representing the probability that the
event was signal or background. The amount of EdgeConv and the following
linear layers need to be optimised to achieve high performance while at the
same time avoiding overfitting. After experimenting with different setups, we
settled on using two EdgeConv layers with hidden features of size $96$ before
the output layer for both channels.

####  III.1.3 Training and comparison of graph embeddings

The data is split into subsamples of $\sim 56\%$, $19\%$ and $25\%$ for
training, validation and testing, respectively101010In total we include 22000
events for each class for the $5b$ case and 35000 events for the $3b2\tau$
case., and we minimise the cross-entropy loss function in order to train the
network using the Adam optimiser Kingma and Ba (2014). The learning rate is
one of the hyperparameters requiring tuning, and for our case the value of
$0.001$ ($0.01$) performs best for $3b2\tau$ ($5b$). If for three epochs in a
row the loss has not decreased, then the learning rate is reduced by a factor
of $0.1$. In principle the training can run for up to $200$ epochs, although
we impose early stopping conditions if the loss has not improved for ten
consecutive epochs. A batch size of $128$ is used for every update of the loss
function.

The GNN for the $5b$ analysis is trained on two classes (signal and
background). The situation is more involved for the $3b2\tau$ case where the
analysis benefits significantly from a multi-class classification which allows
identifying different thresholds for the different background scores. In
particular we choose to train on the $WWbbbb$, $Zbbbb$ and
$t\bar{t}(H\rightarrow\tau^{+}\tau^{-})$ contributions. The signal events used
for training are always for the $(\kappa_{3},\kappa_{4})=(1,1)$ point (using
different values does not significantly alter the performance of the network).

We use the EdgeConv implementation from the Deep Graph Library Wang et al.
(2020) with PyTorch Paszke et al. (2019) as backend. The graph embedding
relies on PyLHE Heinrich et al.  to extract events from the Les Houches Events
(LHE) files Alwall et al. (2007). In order to compare the different graph
embeddings, we use functionality from scikit-learn Pedregosa et al. (2011) to
calculate the true and false positive rates at different thresholds111111As we
perform multi-class classification for the $3b2\tau$ analysis, we binarise the
output of the network for the purpose of this comparison., and we show the
corresponding Receiver Operating Characterestic (ROC) curves for both channels
in Fig. 4.

The ROC curves and the distributions allow one to conclude that the RN
embedding utilising the reconstructed Higgs boson mass can lead to significant
improvements. This is not unexpected as additional information (available at
detector level) is passed to the network to aid classification. While in
principle a sufficiently deep neural network with fully-connected graphs could
also eventually learn to map the input features of the $b$-jets (and taus) to
the masses of the reconstructed Higgs bosons, including the information in the
graph embedding allows easier optimisation and quick convergence with a
shallow network. We therefore utilise only the RN embedding for performing the
final signal region selection.

Figure 4: ROC curves for the $5b$ and $3b2\tau$ analyses are displayed on the
left and right, respectively, showing the performance of the two embedding
cases. For the $3b2\tau$ case we binarise in a one-vs-rest scheme, grouping
together the background classes. The areas under the ‘RN’ and ‘FC’ curves for
the $5b$ case are 0.862 and 0.823, respectively. For the $3b2\tau$ analysis
the ‘RN’ area is 0.909 and the ‘FC’ area is 0.833.

####  III.1.4 HL-LHC Results

For simplicity we will use the signal efficiency of the network for the
$(\kappa_{3},\kappa_{4})=(1,1)$ point and assume that it will be mostly the
same irrespective of the coupling modifier values. This is not unreasonable as
a change of $\kappa_{3}$, $\kappa_{4}$ mostly affects the overall cross
section which is not used as input for the GNN. For the $5b$ analysis we
optimise the signal selection to reduce the false positive rate to $\sim
0.6\%$. In the $3b2\tau$ channel we require the following conditions to be
satisfied on the background network scores:

  * •

$\text{P}[WWbbbb]<3\%$ ,

  * •

$\text{P}[Zbbbb]<10\%$ ,

  * •

$\text{P}[t\bar{t}(H\rightarrow\tau^{+}\tau^{-})]<30\%$ .

It should be noted that even though the network was trained only on a subset
of possible background contributions, it still performs well, as discussed
below, and manages to remove background contributions from other sources as
well. We calculate the efficiencies for each contribution and show the
reduction of cross sections in Tab. 1. Our results for both channels include a
K-factor for the signal of $1.7$ de Florian et al. (2020) and a conservative
estimate of the higher-order contributions to the background processes in
terms of a K-factor of $2$.

We define the significance for our analysis according to Cowan et al. (2011)

$Z=\sqrt{2\bigg{(}(S+B)\ln{(1+\frac{S}{B})}-S\bigg{)}}\;,$ (10)

where $S$ and $B$ denote the signal and background events, respectively. This
allows us to obtain $1\sigma$ and $2\sigma$ bounds within the
$\kappa_{3}$–$\kappa_{4}$ plane (which roughly correspond to $68\%$ and $95\%$
CL, respectively), as shown in Fig. 5 for the $5b$ and $3b2\tau$ analyses. We
assume an integrated luminosity at the HL-LHC of $3$/ab and a combined ATLAS
and CMS luminosity of $6$/ab.

Overall, we observe that the $3b2\tau$ analysis is more sensitive than the
$5b$ analysis, and the latter will additionally suffer from further
subdominant electroweak contributions to the background that have not been
included. However both channels should be utilised in combinations to maximise
the significance. Assuming for simplicity zero correlations between the
channels and using the Stouffer method Stouffer et al. (1949), we combine the
significances as $Z_{\text{comb}}=(Z_{5b}+Z_{3b2\tau})/\sqrt{2}$, giving rise
to the contours shown in Fig. 6. While the projected bounds of about $\pm 20$
times the predicted value for the quartic Higgs self-coupling in the SM may
appear to be quite weak, in view of our discussion above we emphasise that
such bounds go much beyond the existing theoretical bounds. Furthermore,
deviations of this size in $\kappa_{4}$ are well compatible with the existing
experimental bounds on $\kappa_{3}$ according to the correlations between
$\kappa_{3}$ and $\kappa_{4}$ that are present in the BSM scenarios analysed
above. Regarding the sensitivity to $\kappa_{3}$ from triple Higgs boson
production at the HL-LHC, Fig. 6 shows that the expected sensitivity in this
channel at the HL-LHC is weaker than the present experimental limits that have
been derived from di-Higgs production. Combining this independent set of
experimental information on $\kappa_{3}$ with the experimental results from
di-Higgs production may nevertheless turn out to be useful. While our analysis
may be optimistic in some respects (e.g. we neglect fake taus), on the other
hand we note that further developments of the triggers, tagging and
reconstruction algorithms of final states could result in higher efficiencies
than the values that we have adopted in our analysis, enhancing the
significance. The ability to discriminate between jet flavours is highly
important for $HHH$ studies (as well as $HH$ studies) and could also allow
experiments to study fully hadronic final states where $H$ decays to $W$
bosons.

| $\sigma(\text{gen.})(\text{fb})\;\;$ | $\sigma(\text{sel.})(\text{fb})\;\;$ | $\sigma(\text{NN})(\text{fb})$
---|---|---|---
$tt(H\rightarrow\tau\tau)$ | $3.8$ | $0.17$ | $0.011$
$WWbbbb$ | $31$ | $4.6$ | $8.1\times 10^{-3}$
$tt(H\rightarrow bb)$ | $3.5$ | $0.89$ | $3.8\times 10^{-3}$
$Zbbbb$ | $4.3$ | $0.45$ | $3.3\times 10^{-4}$
$tt(Z\rightarrow bb)$ | $0.77$ | $0.15$ | $3.1\times 10^{-4}$
$tt(Z\rightarrow\tau\tau)$ | $4.7$ | $0.080$ | $2.2\times 10^{-4}$
$tttt$ | $0.38$ | $0.091$ | $2.1\times 10^{-4}$
Table 1: Background contributions included in the $3b2\tau$ analysis and
reduction of the generated cross sections (labelled as “gen.”) after pre-
selection cuts (“sel.”) and GNN selection (“NN”).

Figure 5: Projected contours indicating the $1\sigma$ and $2\sigma$ bounds in
the $\kappa_{3}$–$\kappa_{4}$ plane from the $5b$ (left) and the $3b2\tau$
(right) analysis, including effects from showering, hadronisation and
reconstruction.  Figure 6: Projected contours indicating the $1\sigma$ and
$2\sigma$ bounds in the $\kappa_{3}$–$\kappa_{4}$ plane obtained from a
combination of the $5b$ and $3b2\tau$ channels under the assumption that there
are no correlations.

###  III.2 Interpretability of NN scores

Understandably, NN techniques are often viewed as “black boxes”, due to their
inability to indicate the input features that are most important for
determining their predicted scores. In order to address this shortcoming,
various approaches have been explored in the recent years with the goal to
yield interpretability, allow efficient debugging of the network, better
understand the mapping between input and output, and ultimately allow the
identification of ways to improve it. These methods gained traction in
particle physics in the recent years to obtain a better insight for various
different tasks such as jet- and top-tagging and detector triggers de Oliveira
et al. (2016); Chang et al. (2018); Agarwal et al. (2021); Chakraborty et al.
(2019); Andreassen et al. (2019); Mahesh et al. (2021); Khot et al. (2023).

Figure 7: Attributes for the features of the $b$ jets, $\tau$ leptons and the
reconstructed invariant Higgs-boson mass that is closest to the $125$ GeV
resonance. The height of the attribution value indicates to what extent the
network is using the particular feature in order to discriminate between
signal and background. In the figure we denote collectively all the background
classes as ‘B’ and the signal as ‘S’.

There are various techniques for gaining interpretability in ML, but in
general they can be separated into two categories: intrinsically interpretable
models that are specifically designed to increase transparency providing
intuition and post-hoc explanation methods that were developed to enhance our
understanding of generic ML models. The latter is what applies to the case of
this work. However, many post-hoc techniques lack certain properties that are
beneficial to maintain; for example one could directly use the product of the
gradients computed during backpropagation and the input in order to attribute
the most relevant features Baehrens et al. (2010); Simonyan et al. (2013). As
the gradients of the network hold information regarding variations of the
inputs, it should be possible to use them to quantify the dependence of the
score on features. It is known, though, that gradient methods can yield the
same attribution for an input and a baseline that differ from each other and
have different outputs (for an example see Ref. Sundararajan et al. (2017)),
due to the gradient becoming flat (this is often the case as NNs are trained
until the loss is saturated).

Shapley Shapley (1951) values (originating from Game Theory), are formulated
based on certain axioms to distribute the attributions amongst the
participating variables in a ML approach and have been applied for obtaining
interpretations Lundberg and Lee (2016) (for an application in particle
physics, see Refs. Grojean et al. (2021); Alasfar et al. (2022); Grojean et
al. (2022)). Their attractiveness stems from the fact that they follow
axiomatic principles unlike earlier methods (e.g. DeepLift Shrikumar et al.
(2017) or Layer-wise relevance propagation Bach et al. (2015)). However, their
evaluation is often computationally expensive and requires multiple calls of
the neural network.

Integrated Gradients (IGs) is an alternative approach, designed in Ref.
Sundararajan et al. (2017) using axiomatic considerations, which requires
significantly fewer calls to the network function. The trade-off is the
requirement that the ML technique must be differentiable, which is the case
for NNs optimised through gradient descent approaches, and the application of
IGs also requires access to the gradient of the model121212Often techniques
such as Shapley values are called “black-box” approaches as they have no
access to anything other than the output of the ML approach, while IGs and
similar techniques are refered to as “white-box” approaches.. Let a generic
classification NN denoted as $F:\mathbb{R}^{n}\rightarrow\left[0,1\right]$ for
input features $x\in\mathbb{R}^{n}$ and $x^{\prime}\in\mathbb{R}^{n}$ denote
an appropriate baseline (e.g. a zero vector). Integrating over all the
gradients of $F$ in a straight path from $x^{\prime}$ to $x$ defines IGs as

${\cal{I}}_{i}(x)=(x_{i}-x_{i}^{\prime})\int_{0}^{1}d\alpha\frac{\partial
F(x^{\prime}+\alpha(x-x^{\prime}))}{\partial x_{i}}\;.$ (11)

Figure 8: Histograms showing the attribution for different events against the
value of the reconstructed mass for the (true) signal and backgrounds. The
plot on the left (right) shows the reconstructed mass of the Higgs boson that
is closest (second-closest) to the $125$ GeV resonance. A positive attribution
close to 1 indicates events with a high output score (i.e. identified as
signal), while lower values of the attribution imply a low output score.

We thus utilise IGs, implemented in the Captum Kokhlikyan et al. (2020)
library, in order to obtain attributions for our predictions and identify the
most relevant inputs for our processes.

The attributions obtained from IGs allow us to interpret the results of the
network in terms of the input parameters for each node, as shown in Fig. 7,
although some care is necessary when interpreting such results. Quite
intuitively the transverse momenta and the energy of the b-jets are relevant
parameters that receive high attributions. This is expected since restricting
to higher values of $p_{T}$ can help in the discrimination between signal and
background (this was also the reason for applying a pre-selection momentum
cut). Angular momenta are not so helpful for discrimination; this is not
unexpected as we are dealing with scalars. The network additionally utilises
the PID of the tau leptons more than the identification of the $b$ quarks;
this is likely due to the fact that the di-tau state is correlated with the
highly discriminative reconstructed invariant mass of the Higgs boson. We
clearly see that the introduction of the reconstructed masses sigificantly
boosts the performance of the network, being the most important observable for
the signal events. We note that as a reconstructed particle, the Higgs node
has been assigned a PID of zero which as required by the ‘dummy’
axiom131313The ‘dummy’ axiom states that a variable that is not contributing
to the output of the network should have no attribution, ensuring that the
attribution is insensitive to irrelevant inputs. It is a standard axiom
imposed by interpretation methods (see e.g. Refs. Sundararajan et al. (2017);
Shapley (1951)). has no attribution and thus zero contribution to the network
results.

Taking a closer look at the reconstructed masses and their attributions, we
see in Fig. 8 that the node with a reconstructed mass that is closest to 125
GeV receives a sizeable attribution.141414We note that the reconstructed
quantity that is closer to the actual mass of the observed Higgs boson can be
the di-tau state, which is less affected by showering the events than for the
case where the Higgs boson decays to $b$ quarks and can yield more events
closer to the actual mass of the Higgs boson. The attributions from the mass
of the $H_{1}^{\text{reco}}$ node indicate that due to the similarity of the
$t\bar{t}(H\rightarrow\tau^{+}\tau^{-})$ background with the signal, the
network is unable to clearly discriminate the two classes based on this
feature alone. The inclusion of the mass of the second reconstructed Higgs
boson, however, helps the network as indicated by the higher attributions
assigned to the signal events as compared to the other sources of backgrounds.
This implies that the inclusion of reconstructed observables can enhance the
performance of GNNs in certain analyses, as also expected from the discussion
in Sec. III.1.3.

We stress that while the IG attributions provide an indication of the most
important variables, our approach does not yield detailed information on how
the specific correlations between the input features can impact the network
score. While in many cases this would be desired, this is beyond the scope of
our work where we use IGs as a method to verify that the introduction of the
reconstructed Higgs-boson mass is indeed the most relevant variable. We leave
explorations of alternative techniques (also specific to GNNs) pinpointing to
important connections between input features and nodes for future work.

In our work we utilised interpretation methods mostly to ensure that the GNN
works as expected and in order to identify potential issues during the
implementation of the network. However, the usefulness of such techniques
extends well beyond this. For example, in the case of limited computing
resources one could check which features are irrelevant and remove them from
the analysis before scaling the network up. Indeed, in our analysis we checked
that if we remove the seemingly unused angular information, we obtain similar
results as before (resulting in no visible changes in Fig. 5 for $3b2\tau$).
Additionally for analyses with multiple final states the most practical
observable that can be exploited is not always straightforward to identify.
Interpretation techniques could therefore be used as a first step to
identifying the most relevant observables before optimising the analysis to
enhance its significance.

##  IV Reach assessment for lepton colliders and comparison with the HL-LHC

For comparison with the prospects of the HL-LHC, we finally consider the
expected upper limits on $\kappa_{3}$ and $\kappa_{4}$ from possible future
lepton colliders.151515This topic has previously been explored in Refs.
Maltoni et al. (2018); Gonzalez-Lopez et al. (2021); Chiesa et al. (2020). We
consider an inclusive analysis of $\ell\ell\rightarrow HHH+\text{other}$ which
includes both the associated $ZHHH$ production and the production through WBF.
In principle one could consider dedicated analyses for each channel,
optimising the selection of final states; however, we choose to perform an
inclusive analysis to avoid further assumptions on the identification of other
states which could vary depending on the collider concept and the detector. We
will consider the decay $H\rightarrow b\bar{b}$ of the Higgs boson, which
yields the largest possible cross section for the signal, and assume
throughout that the $b$-tagging efficiencies will be $0.8$. Our analysis
relies solely on identifying $5b$ jets in the clean environment provided by
lepton collisions. We apply an additional $\sim 0.83$ efficiency which arises
from requiring the $p_{T}$ of the $b$ jets to be larger than $30$ GeV. We note
that in practice the results for an electron or muon collider would be
similar, i.e. the obtained contours for the limits in the
$\kappa_{3}$–$\kappa_{4}$ plane for a given collider c.m. energy and
integrated luminosity would not be expected to significantly differ for the
two collider types. Therefore we will refer to generic lepton colliders in the
following, although we use the centre-of-mass energies of $1$ and $3$ TeV
envisaged for the ILC and CLIC, as well as $10$ TeV collisions that could be
realised at a muon collider. We scan over different values of $\kappa_{3}$ and
$\kappa_{4}$ for the aforementioned energies and subsequently apply the
relevant tagging efficiencies.

An important limitation of high-energy lepton collisions in this case,
however, arises from the region where the detectors can tag $b$ jets. While
for energies $\sim 1$ TeV the $b$ quarks are in the central part of the
detector, the situation is significantly different for $10$ TeV collisions, as
shown in Fig. 9. It is thus necessary to explore possibilities for extending
the tagging capabilities of future detectors to even $\absolutevalue{\eta}\sim
4$ in order to avoid a significant loss of events.

Figure 9: Pseudorapidity distribution for the leading $b$ quark for different
collision energies.

Figure 10: On the left, the projected 95% CL contours for lepton colliders at
different energies and integrated luminosities are shown, mainly focusing on
the energies of ILC, CLIC and a possible muon collider. The SM value is shown
as a black dot. The plot on the right shows a zoomed-in version.  Figure 11:
Comparison of the projected 95% CL contours for the $5b$ and $3b2\tau$
analyses at the HL-LHC as well as their combination with the projected 95% CL
sensitivities at lepton colliders with different energies (indicated by the
different coloured regions). The shaded gray area indicates the region that is
excluded by the bound from tree-level perturbative unitarity.

The leptonic collisions deliver clean signals avoiding the large background
contamination from QCD that is present at hadron colliders. We checked the
backgrounds of the signature $(5b+\text{other})$ from SM processes and found
that the cross sections of these background processes are small. Assuming that
the selection of the signal region will enforce $p_{T}(b)>30$ GeV, the
requirement that one di-bottom pair should be compatible with the mass of the
observed Higgs boson and a cut ensuring that the total invariant mass of the
final state particles produced in the process is at least $350$ GeV would
result in no remaining background events (even with more relaxed cuts, the
number of events is negligible when taking $b$-tagging efficiencies into
account). We therefore turn to a Poissonian analysis as described in Ref.
Workman et al. (2022), where $n$ corresponds to the number of events expected
from the SM, i.e. for $(\kappa_{3},\kappa_{4})=(1,1)$. Upper limits on the
mean value of the Poisson distribution $\mu$ are then calculated with

$\mu_{\text{up}}=\frac{1}{2}F^{-1}_{\chi^{2}}\bigg{[}2(n+1);\text{CL}\bigg{]}\;,$
(12)

where $F^{-1}_{\chi^{2}}$ denotes the inverse of the cumulative distribution
of the $\chi^{2}$ distribution, and CL is the confidence level (e.g. 95%).

The resulting bounds at $95\%$ CL are shown in Fig 10 for different centre-of-
mass energies and integrated luminosities. The plots show that the large
luminosities expected to be utilised by colliders at $3$ TeV and $10$ TeV (as
envisaged for CLIC and muon-colliders, see Refs. de Blas et al. (2018);
Accettura et al. (2023)) provide significant constraining power via the triple
Higgs production process for $\kappa_{4}$ and $\kappa_{3}$.

The lepton collider projections are compared with our results for the HL-LHC
in Fig. 11. We find that the HL-LHC sensitivity for $\kappa_{4}$ is
competitive with the one achievable at a $1$ TeV lepton collider such as the
ILC. In particular the comparison shows that for negative $\kappa_{4}$ the HL-
LHC is expected to have a better sensitivity than a $1$ TeV lepton collider,
while a $1$ TeV lepton collider has a higher sensitivity in the large and
positive $\kappa_{3}$ and $\kappa_{4}$ region.

As discussed above further developments in ML could increase both the tagging
and selection efficiencies beyond our assumptions, and additional channels
will provide additional information.

##  V Conclusions

Our investigation for the prospects at the HL-LHC shows that even though
triple-Higgs production is limited by low rates at the LHC, its exploration
provides interesting information even if it does not receive additional
contributions from new scalar resonances. Bounds can be placed on $\kappa_{4}$
significantly beyond the theoretical constraints from perturbative unitarity.

While as expected the bounds on $\kappa_{3}$ will be much weaker than the ones
from double-Higgs production, they should be useful for improving the
sensitivity through combinations. Additionally, if deviations from the SM are
found, the correlation between the Higgs self-couplings can shed light on the
possible scenarios of physics beyond the SM.

If an excess in the triple Higgs production process is observed, the
correlation with the result for double-Higgs production will be immensely
informative. On the one hand, if no deviation from the SM value is identified
in $\kappa_{3}$ from other channels, an indication for a large deviation in
$\kappa_{4}$ would likely imply the presence of non-linear effects that cannot
be described consistently within an effective field theory approach via the
expansion in terms of a heavy scale. On the other hand, a deviation in both
coupling modifiers could indicate a correlation between $\kappa_{3}$ and
$\kappa_{4}$ that can be confronted with predictions of specific models such
as the 2HDM and of effective field theories.

The physics gain that can be achieved via the statistically limited channel of
triple Higgs production at the HL-LHC crucially depends on an efficient
signal–background discrimination. For this purpose we have employed in our
analysis the use of GNNs. It is already evident from current experimental
searches that such ML techniques will be the centerpiece of future studies.
However, it is especially important in particle physics to be able to identify
the relevant kinematical features that contribute to the identification of the
signal. An unintuitive behaviour (e.g. a high-attribute quantity that is
already known to be irrelevant) could indicate a possible issue in the
learning framework. Alternatively, potentially interesting quantities could be
identified that could provide discriminative power even in simpler analyses
that do not use ML algorithms. We have explored interpretability within GNNs
using IGs which satisfy necessary axioms. We have shown that, as expected, the
invariant mass of bottom and tau pairs is the most important feature in the
data that is utilised for discrimination. We expect that such techniques will
play an important role not only for the development of analyses for BSM
searches but also for further applications in particle physics.

Our comparison of the prospects at the HL-LHC with future lepton colliders
shows that the sensitivity to $\kappa_{4}$ at the HL-LHC should be competitive
with a $1$ TeV lepton collider such as ILC. While the sensitivities of lepton
colliders at $3$ and $10$ TeV (e.g. CLIC or a possible muon-collider) are
expected to be considerably higher, these results will presumably become
available only on a longer time scale, such as the one for a future higher-
energetic hadron collider. Thus, it can be expected that the HL-LHC will be
able to establish the first bounds on $\kappa_{4}$ beyond theoretical
considerations.

Acknowledgements — We thank Henning Bahl, Akanksha Bhardwaj, Johannes
Braathen, Philipp Gadow, Greg Landsberg and Jürgen Reuter for useful comments
and discussions. This work is supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2121
“Quantum Universe” - 390833306 and has been partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - 491245950.

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# Mod $p$ homology of unordered configuration spaces of surfaces

Matthew Chen  Wayzata High Shcool, Plymouth, MN 55446<EMAIL_ADDRESS>and  Adela YiYu Zhang  Department of Mathematics, Massachusetts Institute of
Technology, Cambridge MA 02139<EMAIL_ADDRESS>
###### Abstract.

We provide a short proof that the dimensions of the mod $p$ homology groups of
the unordered configuration space $B_{k}(T)$ of $k$ points in a torus are the
same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology
has no $p$-power torsion. The same argument works for the punctured genus $g$
surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via
Lubin-Tate theory.

##  1\. Introduction

The unordered configuration space of $k$ points in a manifold $M$ is the orbit
space

$B_{k}(M):=\mathrm{Conf}_{k}(M)_{\Sigma_{k}}=\\{(x_{1},\ldots,x_{k})\in
M^{\times k},x_{i}\neq x_{j}\mathrm{\ for\ }i\neq j\\}/\Sigma_{k}.$

The main new result of this paper concerns the odd primary homology of
$B_{k}(T)$, where $T$ is a closed torus.

######  Theorem 1.1.

Let $p$ be an odd prime. The dimension of $H_{i}(B_{k}(T);\mathbb{F}_{p})$
over $\mathbb{F}_{p}$ is given by the $i$th Betti number $\beta_{i}(B_{k}(T))$
for all $i$ and $k\leq p$. Hence the integral homology of $B_{k}(T)$ has no
$p$-power torsion for $k\leq p$.

The study of unordered configuration spaces dates back to as early as Segal
[Seg73] and McDuff [McD75]. The rational homology groups of these objects are
relatively well understood in cases of interests via classical methods, see
for instance [BC88][BCT89][Kri94][Tot96][FT00]. In contrast, the odd primary
homology groups of unordered configuration spaces have remained mostly
intractable. Classically, the only known cases are the following: when
$M=\mathbb{R}^{n}$ for $1\leq n\leq\infty$ where $\bigoplus_{k\geq
0}H_{*}(B_{k}(M);\mathbb{F}_{p})$ is the mod $p$ homology of the free
$\mathbb{E}_{n}$-algebra on $\mathbb{S}$ [May72] [CLM76][BMMS88], and when the
dimension of $M$ is odd where $\bigoplus_{k\geq
0}H_{*}(B_{k}(M);\mathbb{F}_{p})$ depends only on the $\mathbb{F}_{p}$-module
$H_{*}(M;\mathbb{F}_{p})$ [BCT89][ML88][BCM93].

More recently, advances in the computation of the homology of unordered
configuration spaces are made possible by a result of Knudsen [Knu18]. For any
manifold $M$ and spectrum $X$, we can consider the labeled configuration
spectrum

$B_{k}(M;X):=\Sigma^{\infty}_{+}\mathrm{Conf}_{k}(M)\mathbin{\mathop{\otimes}\displaylimits_{\Sigma_{k}}}X^{\mathbin{\mathop{\otimes}\displaylimits}k}.$

In particular $\Sigma^{\infty}_{+}B_{k}(M)=B_{k}(M;\mathbb{S})$. Denote by
$s\mathscr{L}$ the monad associated to the free spectral Lie algebra functor
$\mathrm{Free}^{s\mathscr{L}}$. The $\infty$-category of spectral Lie algebras
is cotensored in Spaces, and $(-)^{M^{+}}$ denotes the cotensor with the one-
point compactification of $M$ in this category. Using the machinery of
factorization homology, Knudsen established the following equivalence.

######  Theorem 1.2.

[Knu18, Section 3.4] Let $M$ be a parallelizable $n$-manifold and $X$ a
spectrum of weight one. Then there is an equivalence of weighted spectra

(1) $\bigoplus_{k\geq
1}B_{k}(M;X)\simeq\mid\mathrm{Bar}_{\bullet}(\mathrm{id},s\mathscr{L},\mathrm{Free}^{s\mathscr{L}}(\Sigma^{n}X)^{M^{+}})\mid.$

The left hand side is weighted by the index $k$ and right hand side induced by
the weight on $X$.

Using the bar spectral sequence with rational coefficients associated to the
right hand side of (1), Knudsen [Knu17] provided a general formula for the
Betti numbers of unordered configuration spaces. Building on Knudsen’s work,
Drummond-Cole and Knudsen [DCK17] produced explicit formulae of the Betti
numbers of unordered configuration spaces of surfaces. In [BHK19], Brantner,
Hahn, and Knudsen studied Knudsen’s spectral sequence with coefficients in
Morava $E$-theory at an odd prime. They computed the weight $p$ part of the
labeled configuration spaces in $\mathbb{R}^{n}$ and punctured genus $g$
surfaces $\Sigma_{g,1}$ for $g\geq 1$ with coefficient in a sphere. By letting
the height go to infinity, they deduced that:

######  Theorem 1.3.

[BHK19, Theorem 1.10] Let $p$ be an odd prime. The integral homology of
$B_{p}(\Sigma_{g,1})$, $g\geq 1$ has no $p$-power torsion.

The computation via Morava $E$-theory becomes rather convoluted if one would
like to apply it to the closed genus one surface $T$, which is the only
parallelizable closed surface. Following a similar approach, the second author
studied Knudsen’s spectral sequence with odd primary coefficients in [Zha21]
and showed that $H_{*}(B_{k}(M);\mathbb{F}_{p}),\ k=2,3$ depends on the
cohomology ring $H^{*}(M^{+};\mathbb{F}_{p})$ when $M$ is an even dimensional
parallelizable manifold, which is in contrast to the case when $M$ is odd
dimensional.

In this paper, we build on the work of the second author and attack the
computation of $H_{*}(B_{k}(T);\mathbb{F}_{p})$ directly. We prove Theorem 1.1
by identifying the higher differentials in the odd primary Knudsen’s spectral
sequence

$E^{2}_{s,t}(k)=\pi_{s}\pi_{t}\big{(}\mathrm{Bar}_{\bullet}\big{(}\mathrm{id},s\mathscr{L},\mathrm{Free}^{s\mathscr{L}}(\Sigma^{n}\mathbb{S})^{T^{+}}\big{)}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p}\big{)}(k)\Rightarrow
H_{s+t}(B_{k}(T);\mathbb{F}_{p})$

for $k\leq p$. The argument we use involves a comparison with the spectral
sequence computing $H_{*}(B_{k}(T);\mathbb{Q})$ studied by Knudsen [Knu17] and
simple dimension counting. The exact same argument works for the punctured
genus $g$ surface for $g>0$, thereby providing a more direct proof of Theorem
1.3.

###  1.1. Outline

In Section 2, we review the construction of Chevalley-Eilenberg complex of a
shifted graded Lie algebra over a field and the structure of the odd primary
homology of spectral Lie algebras. Then we recall previous computations of the
$E^{2}$-page of the odd primary Knudsen’s spectral sequence. Comparing with
the computation of the rational homology of $B_{k}(M)$ by Knudsen where $M=T$
or $\Sigma_{g,1}$, we deduce the main result of the paper Theorem 1.1 in
Section 3.

###  1.2. Acknowledgements.

The first author wishes to thank the Research Science Institute program and
the Department of Mathematics at MIT for the opportunity. The second author
would like to thank Jeremy Hahn, Haynes Miller, and Andrew Senger for helpful
conversations.

###  1.3. Conventions

Let $k$ be a field. A weighted graded $k$-module $M$ is an
$\mathbb{N}$-indexed collection of $\mathbb{Z}$-graded $k$-modules
$\\{M(w)\\}_{w\in\mathbb{N}}$. The weight grading of an element $x\in M(w)$ is
$w$. Morphisms are weight preserving morphisms of graded $k$-modules. Denote
by $\mathrm{Mod}_{k}$ the category of weighted graded $k$-modules. We omit the
adjectives weighted, graded from here on. The Day convolution
$\mathbin{\mathop{\otimes}\displaylimits}$ makes $\mathrm{Mod}_{k}$ a
symmetric monoidal category and the Koszul sign rule
$x\mathbin{\mathop{\otimes}\displaylimits}y=(-1)^{|x||y|}y\mathbin{\mathop{\otimes}\displaylimits}x$
depends only on the internal grading. Denote by $\mathrm{wt}_{i}(M)$ the
weight $i$ part of the $k$-module $M$.

##  2\. Preliminaries

###  2.1. The Chevalley-Eilenberg complex

A shifted Lie algebra $L$ over $k$ is a $k$-module equipped with a shifted Lie
bracket

$[-,-]:L_{m}\mathbin{\mathop{\otimes}\displaylimits}L_{n}\rightarrow
L_{m+n-1}$

that satisfies graded commutativity $[x,y]=(-1)^{|x||y|}[y,x]$, the graded
Jacobi identity

$(-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,$

and adds weight. When $p=3$ we further require that $[[x,x],x]=0$ for all
$x\in L$. Denote by $\mathrm{Lie}^{s}_{k}$ the category of shifted weighted
graded Lie algebras over $k$.

######  Definition 2.1.

[CE48][May66A] Suppose that $k$ has characteristics away from two. For a
$\mathrm{Lie}^{s}_{k}$-algebra $L$ , let $L_{\text{even}}$ and
$L_{\text{odd}}$ denote the elements in $L$ with even and odd degree,
respectively. The Chevalley-Eilenberg complex of $L$ is the chain complex

$\mathrm{CE}(L;k)=(\Gamma^{\bullet}(L_{\text{even}})\otimes\Lambda^{\bullet}(L_{\text{odd}}),\partial),$

where $\Gamma^{\bullet}$ and $\Lambda^{\bullet}$ are respectively the graded
shifted divided power and exterior algebra functor over $k$, and the
differential $\partial$ on an element
$\gamma_{k_{1}}(x_{1})\cdots\gamma_{k_{m}}(x_{m})\langle
y_{1},\dots,y_{n}\rangle\in\Gamma^{\bullet}(L_{\text{even}})\otimes\Lambda^{\bullet}(L_{\text{odd}})$
is

$\displaystyle\sum_{1\leq i<j\leq
m}\gamma_{k_{1}}(x_{1})\cdots\gamma_{k_{i}-1}(x_{i})\cdots\gamma_{k_{j}-1}(x_{j})\cdots\gamma_{k_{m}}(x_{m})\langle[x_{i},x_{j}],y_{1},\dots
y_{n}\rangle$ $\displaystyle+$ $\displaystyle\sum_{1\leq i<j\leq
n}(-1)^{i+j-1}\gamma_{k_{1}}(x_{1})\cdots\gamma_{k_{m}}(x_{m})\langle[y_{i},y_{j}],y_{1},\dots,\widehat{y_{i}},\dots\widehat{y_{j}},\dots
y_{n}\rangle$ $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i=1}^{m}\gamma_{k_{1}}(x_{1})\cdots\gamma_{k_{i}-2}(x_{i})\cdots\gamma_{k_{m}}(x_{m})\langle[x_{i},x_{i}],y_{1},\dots,y_{n}\rangle$
$\displaystyle+$
$\displaystyle\sum_{i=1}^{m}\sum_{j=1}^{n}(-1)^{j-1}\gamma_{1}([x_{i},y_{j}])\gamma_{k_{1}}(x_{1})\cdots\gamma_{k_{i}-1}(x_{i})\cdots\gamma_{k_{m}}(x_{m})\langle
y_{1},\dots,\widehat{y_{j}},\dots,y_{n}\rangle.$

Note that the differential $\partial$ preserves weights, so
$H_{*,*}(\mathrm{wt}_{k}(\mathrm{CE}(L;k)))=\mathrm{wt}_{k}(H_{*,*}(\mathrm{CE}(L;k)))$.
The Chevellay-Eilenberg complex is useful in computing the
$\mathrm{Lie}^{s}_{k}$-algebra homology.

######  Theorem 2.2.

[May66A][Pri70] For $L$ a $\mathrm{Lie}^{s}_{k}$-algebra, its
$\mathrm{Lie}^{s}_{k}$-algebra homology is given by

$H^{\mathrm{Lie}^{s}_{k}}_{*,*}(L):=\pi_{*,*}(\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Lie}^{s}_{k},L)\oplus
k)\cong H_{*,*}(\mathrm{CE}(L;k)).$

###  2.2. Operations on the odd primary homology of spectral Lie algebras

Building on the work of Arone-Mahowald [AM99] and Johnson [Joh95], Ching
[Chi05] and Salvatore showed that the Goodwillie derivatives
$\\{\partial_{n}(\mathrm{Id})\\}_{n}$ of the identity functor
$\mathrm{Id}:\mathrm{Top}_{*}\rightarrow\mathrm{Top}_{*}$ form an operad in
Spectra that is Koszul dual to the non-unital $\mathbb{E}_{\infty}$-operad.
This is called the spectral Lie operad and we denote it by $s\mathscr{L}$.
Algebras over the operad $s\mathscr{L}$ are called spectral Lie algebras, with
structure maps
$\partial_{k}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits_{h\Sigma_{k}}}L^{\mathbin{\mathop{\otimes}\displaylimits}k}\rightarrow
L$ for all $k\geq 1$.

In [Kja18], Kjaer studied the structure of the mod $p$ homology of spectral
Lie algebras for $p>2$ following the approach of Behrens [Beh12] and Antolín-
Camarena [AC20] when $p=2$.

######  Proposition 2.3.

[Kja18, Definition 3.2] Let $L$ be a spectral Lie algebra. Then
$H_{*}(L;\mathbb{F}_{p})$ admits unary operations of weight $p$

$\overline{\beta^{\epsilon}Q^{j}}:H_{*}(L;\mathbb{F}_{p})\rightarrow
H_{*+2(p-1)i-\epsilon-1}(L;\mathbb{F}_{p}),\
x\mapsto\xi_{*}(\sigma^{-1}\beta^{\epsilon}Q^{j}(x))$

for $\epsilon\in\\{0,1\\},j\in\mathbb{Z}$. Here
$\xi:\partial_{p}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits_{h\Sigma_{p}}}L^{\mathbin{\mathop{\otimes}\displaylimits}p}\rightarrow
L$ is the $p$th structure map of the spectral Lie algebra $L$, $\sigma^{-1}$
the desuspension isomorphism, and $\beta^{\epsilon}\bar{Q}^{j}$ a mod $p$
Dyer-Lashof operation.

It follows from the unstability of Dyer-Lashof operations that
$\overline{\beta^{\epsilon}Q^{j}}(x)=0$ if $j<\frac{|x|}{2}$. There is also a
$\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-algebra structure on
$H_{*}(L;\mathbb{F}_{p})$, induced by the second structure map

$\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits_{h\Sigma_{2}}}L^{\mathbin{\mathop{\otimes}\displaylimits}2}\simeq\partial_{2}(\mathrm{Id})\mathbin{\mathop{\otimes}\displaylimits}L^{\mathbin{\mathop{\otimes}\displaylimits}2}_{h\Sigma_{2}}\simeq\mathbb{S}^{-1}\mathbin{\mathop{\otimes}\displaylimits}L^{\mathbin{\mathop{\otimes}\displaylimits}2}_{h\Sigma_{2}}\rightarrow
L.$

######  Proposition 2.4.

[Kja18, Proposition 3.7] For $L$ a spectral Lie algebra,
$[\overline{\beta^{\epsilon}Q^{j}}(x),y]=0$ for any $\epsilon,j$ and $x,y\in
H_{*}(L;\mathbb{F}_{p})$ .

Define a functor
$\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}:\mathrm{Mod}_{\mathbb{F}_{p}}\rightarrow\mathrm{Mod}_{\mathbb{F}_{p}}$
as follows. For $M\in\mathrm{Mod}_{\mathbb{F}_{p}}$, let $A$ be an
$\mathbb{F}_{p}$-basis for the free shifted Lie algebra
$\mathrm{Free}^{\mathrm{Lie}^{s}_{\mathbb{F}_{p}}}(M)$. The graded
$\mathbb{F}_{p}$-module $\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(M)$ has basis

$\\{\overline{\beta_{1}^{\epsilon_{1}}Q^{j_{1}}}\cdots\overline{\beta_{k}^{\epsilon_{k}}Q^{j_{k}}}|x,\
\ x\in A,j_{k}\geq\frac{|x|}{2},j_{i}\geq pj_{i+1}-\epsilon_{i+1}\forall
i\\}.$

The $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-structure on
$\mathrm{Free}^{\mathrm{Lie}^{s}_{\mathbb{F}_{p}}}$ can be extended to that on
$\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(M)$ via Proposition 2.4.

######  Theorem 2.5.

[Kja18, Theorem 5.2] For $X$ a spectrum. there is an isomorphism of
$\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-algebras

$\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(H_{*}(X;\mathbb{F}_{p}))\rightarrow
H_{*}(\mathrm{Free}^{s\mathscr{L}}(X);\mathbb{F}_{p}).$

###  2.3. Odd primary Knudsen’s spectral sequence

The odd primary Knudsen’s spectral sequence was first investigated by the
second author in [Zha21]. Using the skeletal filtration of the geometric
realization of the bar construction in Theorem 1.2, we obtain Knudsen’s
spectral sequence with mod $p$ coefficients

(2)
$E^{2}_{s,t}(k)=\pi_{s}\pi_{t}\big{(}\mathrm{Bar}_{\bullet}\big{(}\mathrm{id},s\mathscr{L},\mathrm{Free}^{s\mathscr{L}}(\Sigma^{n}X)^{M^{+}}\big{)}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p}\big{)}(k)\Rightarrow
H_{s+t}(B_{k}(M;X);\mathbb{F}_{p}).$

By repeatedly applying Theorem 2.5, we see that the $E^{2}$-page is the
homotopy group of a simplicial $\mathbb{F}_{p}$-module $V_{\bullet}$ with
$(\mathrm{Lie}^{s}_{\bar{\pazocal{R}}})^{\circ s}(L)$ as the $s$th simplicial
level, where

$L=H_{*}(\mathrm{Free}^{s\mathscr{L}}(\Sigma^{n}X)^{M^{+}};\mathbb{F}_{p})\cong\widetilde{H}^{*}(M^{+};\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(\Sigma^{n}H_{*}(X;\mathbb{F}_{p})).$

Then $L$ has a $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-structure given by

$[y_{1}\mathbin{\mathop{\otimes}\displaylimits}x_{1},y_{2}\mathbin{\mathop{\otimes}\displaylimits}x_{2}]=(y_{1}\cup
y_{2})\mathbin{\mathop{\otimes}\displaylimits}[x_{1},x_{2}]$

[BHK19, Proposition 5.9] and the action of the unary operations is given by

$\overline{\beta^{\epsilon}Q^{j}}(y\mathbin{\mathop{\otimes}\displaylimits}x)=y\mathbin{\mathop{\otimes}\displaylimits}\overline{\beta^{\epsilon}Q^{j}}(x)$

if there is no nonzero Steenrod operation on $H^{*}(M;\mathbb{F}_{p})$ other
than $Sq^{0}$ [Zha21, Proposition 4.4].

At the time of this work, there is no published result on the relations among
the unary operations $\overline{\beta^{\epsilon}Q^{j}}$.111Through private
communication, we were informed that Nikolay Konovalov has forthcoming work
computing the odd primary relations via Goodwillie calculus. Nonetheless, at
weight $k<p^{2}$ it suffices to know how unary operations and
$\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-brackets commute, which is established by
Proposition 2.4. The face maps on $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-brackets
are simply $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-algebra structure maps. In
[Zha21], the second author computed the $E^{2}$-page of the spectral sequence
(2) in weight $k\leq p$ in terms of
$\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-algebra homology.

######  Proposition 2.6.

[Zha21, Proposition 6.5] Let $M$ be a parallelizable manifold of dimension $n$
and $X$ any spectrum. Set

$\mathfrak{g}=\widetilde{H}^{*}(M^{+};\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}\mathrm{Lie}^{s}_{\mathbb{F}_{p}}(\Sigma^{n}H_{*}(X;\mathbb{F}_{p}))$

with $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-structure given by
$[y_{1}\mathbin{\mathop{\otimes}\displaylimits}x_{1},y_{2}\mathbin{\mathop{\otimes}\displaylimits}x_{2}]=(y_{1}\cup
y_{2})\mathbin{\mathop{\otimes}\displaylimits}[x_{1},x_{2}].$

  1. (1)

For $k<p$, the weight $k$ part of the spectral sequence (2) has $E^{2}$-page
given by

$E^{2}_{s,t}(k)\cong\mathrm{wt}_{k}H_{s,t}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p})).$

  2. (2)

For $p\geq 5$, the weight $p$ part of the spectral sequence (2) has
$E^{2}$-page given by

$E^{2}(p)_{*,*}\cong\mathrm{wt}_{p}H_{*,*}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p}))\oplus\bigoplus_{y\in
H,x\in
B}\mathbb{F}_{p}\Big{\\{}\overline{\beta^{\epsilon}Q^{j}}|y\mathbin{\mathop{\otimes}\displaylimits}x,\
\frac{|x|-|y|}{2}\leq j<\frac{|x|}{2}\Big{\\}},$

where $H$ is an $\mathbb{F}_{p}$-basis of
$\widetilde{H}^{*}(M^{+};\mathbb{F}_{p})$ and $B$ an $\mathbb{F}_{p}$-basis of
$H_{*}(X;\mathbb{F}_{p})$.

######  Remark 2.7.

[Zha21, Remark 6.6] When $p=3$, there has to be an identity
$\overline{\beta^{\epsilon}Q^{j}}(x)=[[x,x],x]$ when $x$ is a class of degree
$2j$ in the mod 3 homology of a spectral Lie algebra $L$. This can be seen by
computing the weight 3 part of the spectral sequence (2) for
$M=\mathbb{R}^{n}$, $X=\mathbb{S}^{2j}$ and comparing with the weight 3 part
of the free $\mathbb{E}_{n}$-algebra on the generator $x$ given in [CLM76,
III].

In other words, the mod 3 homology of a spectral Lie algebra should have the
structure of an operadic $\mathrm{Lie}^{s}_{\mathbb{F}_{3}}$-algebra, denoted
by $\mathrm{Lie}^{s,\mathrm{op}}_{\mathbb{F}_{3}}$, which does not require
$[[x,x],x]=0$. The underlying module of
$\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(M)$ is thus given as follows: let $B$ be
an $\mathbb{F}_{3}$-basis for the free
$\mathrm{Lie}^{s,\mathrm{op}}_{\mathbb{F}_{3}}$-algebra on $M$. The graded
$\mathbb{F}_{3}$-module $\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(M)$ has basis
the quotient of

$\\{\overline{\beta_{1}^{\epsilon_{1}}Q^{j_{1}}}\cdots\overline{\beta_{k}^{\epsilon_{k}}Q^{j_{k}}}|x,\
\ x\in B,j_{k}\geq\frac{|x|}{2},j_{i}\geq 3j_{i+1}-\epsilon_{i+1}\forall i\\}$

by the relation $\overline{\beta^{\epsilon}Q^{j}}(x)=[[x,x],x]$ for all $x\in
M$ with even degree.

##  3\. Mod $p$ homology of $B_{k}(T)$

Let $M=\Sigma_{g}$ be a genus $g$ surface with $g\geq 1$. Its integral
cohomology ring is

$H^{*}(\Sigma_{g};\mathbb{Z})=\left\\{\begin{array}[]{lcl}\mathbb{Z}\\{d\\}&*=0\\\
\mathbb{Z}\\{a_{i}\oplus b_{i},\ i=1,\ldots,g\\}&*=1\\\
\mathbb{Z}\\{c\\}&*=2\\\ 0&\mbox{otherwise}\end{array}\right.$

with cup product given by $a_{i}\cup b_{i}=c$ for all $i$, $d\cup y=y$ for all
$y\in H^{*}(\Sigma_{g};\mathbb{Z})$, and zero otherwise. Hence
$H^{*}(\Sigma_{g,1}^{+};\mathbb{Z})$ has product structure given by $a_{i}\cup
b_{i}=c$ for all $i$ and zero otherwise, where $\Sigma_{g,1}$ is the punctured
genus $g$ surface.

Using the fact that Knudsen’s spectral sequence with rational coefficients
always collapses on the $E^{2}$-page [Knu18], Drummond-Cole and Knudsen
produced explicit formulae for the Betti numbers of $B_{k}(\Sigma_{g})$ and
$B_{k}(\Sigma_{g,1})$ for all $k$ and $g$.

######  Notation 3.1.

Denote by $\mathfrak{g}$ the $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-algebra

$\mathfrak{g}=H^{*}(\Sigma_{g};\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}\mathrm{Lie}^{s}_{\mathbb{F}_{p}}(\mathbb{F}_{p}\\{x_{2}\\})$

with $x_{2}$ in internal degree 2 and weight 1, with the
$\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-structure is given by
$[y\mathbin{\mathop{\otimes}\displaylimits}x,y^{\prime}\mathbin{\mathop{\otimes}\displaylimits}x^{\prime}]=(y\cup
y^{\prime})\mathbin{\mathop{\otimes}\displaylimits}[x,x^{\prime}].$ An
$\mathbb{F}_{p}$-basis for $\mathfrak{g}$ is
$B=\\{y\mathbin{\mathop{\otimes}\displaylimits}x_{2},y\mathbin{\mathop{\otimes}\displaylimits}[x_{2},x_{2}],\
y=a_{1},b_{1},\ldots,a_{g},b_{g},c,d\\}$.

Let $\mathfrak{g}^{\prime}$ be $\mathrm{Lie}^{s}_{\mathbb{Q}}$-algebra

$H^{*}(\Sigma_{g};\mathbb{Q})\otimes\mathrm{Free}^{\mathrm{Lie}^{s}_{\mathbb{Q}}}(\mathbb{Q}\\{x_{2}\\})$

with brackets given by the same formula as above. A $\mathbb{Q}$-basis for
$\mathfrak{g}^{\prime}$ is also $B$.

######  Notation 3.2.

Denote by $\mathfrak{g}_{1}$ the $\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-algebra

$\mathfrak{g}_{1}=\widetilde{H}^{*}(\Sigma_{g,1}^{+};\mathbb{F}_{p})\mathbin{\mathop{\otimes}\displaylimits}\mathrm{Lie}^{s}_{\mathbb{F}_{p}}(\mathbb{F}_{p}\\{x_{2}\\})$

with $x_{2}$ in internal degree 2 and weight 1, with the
$\mathrm{Lie}^{s}_{\mathbb{F}_{p}}$-structure is given by
$[y\mathbin{\mathop{\otimes}\displaylimits}x,y^{\prime}\mathbin{\mathop{\otimes}\displaylimits}x^{\prime}]=(y\cup
y^{\prime})\mathbin{\mathop{\otimes}\displaylimits}[x,x^{\prime}].$ An
$\mathbb{F}_{p}$-basis for $\mathfrak{g}$ is
$B_{1}=\\{y\mathbin{\mathop{\otimes}\displaylimits}x_{2},y\mathbin{\mathop{\otimes}\displaylimits}[x_{2},x_{2}],\
y=a_{1},b_{1},\ldots,a_{g},b_{g},c\\}$.

Let $\mathfrak{g}^{\prime}_{1}$ be $\mathrm{Lie}^{s}_{\mathbb{Q}}$-algebra

$\widetilde{H}^{*}(\Sigma_{g}^{+};\mathbb{Q})\otimes\mathrm{Free}^{\mathrm{Lie}^{s}_{\mathbb{Q}}}(\mathbb{Q}\\{x_{2}\\})$

with brackets given by the same formula as above. A $\mathbb{Q}$-basis for
$\mathfrak{g}^{\prime}_{1}$ is also $B_{1}$.

######  Theorem 3.3.

[Knu17] The $i$th Betti number of $B_{k}(\Sigma_{g})$ and
$B_{k}(\Sigma_{g,1})$ are respectively equal to the dimension over
$\mathbb{Q}$ of
$\bigoplus_{s+t=i}H_{s,t}\big{(}\mathrm{wt}_{k}(\mathrm{CE}(\mathfrak{g}^{\prime};\mathbb{Q}))\big{)}$
and
$\bigoplus_{s+t=i}H_{s,t}\big{(}\mathrm{wt}_{k}(\mathrm{CE}(\mathfrak{g}^{\prime}_{1};\mathbb{Q}))\big{)}$
for all $i$.

Explicit formulae for the Betti numbers $\beta_{i}(B_{k}(T))$ were obtained by
Drummond-Cole and Knudsen in [DCK17, Corollary 4.5-4.7] and
$\beta_{i}(B_{k}(\Sigma_{g,1}))$ in [DCK17, Proposition 3.5].

We will deduce the higher differentials in the spectral sequence

(3)
$E^{2}_{s,t}(k)=\pi_{s}\pi_{t}\big{(}\mathrm{Bar}_{\bullet}\big{(}\mathrm{id},s\mathscr{L},\mathrm{Free}^{s\mathscr{L}}(\Sigma^{n}\mathbb{S})^{M^{+}}\big{)}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p}\big{)}(k)\Rightarrow
H_{s+t}(B_{k}(M);\mathbb{F}_{p})$

where $M=T$ or $\Sigma_{g,1}$ by combining Proposition 2.6 with Theorem 3.3.

######  Theorem 3.4 (Theorem 1.1).

For $k\leq p$, the dimension of $H_{i}(B_{k}(T);\mathbb{F}_{p})$ over
$\mathbb{F}_{p}$ is equal to the Betti number $\beta_{i}(B_{k}(T))$ for all
$i$. Hence the integral homology of $B_{k}(T)$ has no $p$-power torsion for
$k\leq p$.

###### Proof.

The isomorphism between the $\mathbb{F}_{p}$-basis of $\mathfrak{g}$ and the
$\mathbb{Q}$-basis of $\mathfrak{g}^{\prime}$ (cf. 3.1) induces compatible
isomorphisms between the basis

$\big{\\{}\gamma_{k_{1}}(u_{1})\cdots\gamma_{k_{m}}(u_{m})\langle
v_{1},\dots,v_{n}\rangle,\ u_{1},\ldots,u_{m},v_{1},\ldots v_{m}\in
B\big{\\}}$

on each simplicial level of $\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p})$ and
$\mathrm{CE}(\mathfrak{g}^{\prime};\mathbb{Q})$. Since the integral cohomology
of $T$ is torsion-free and
$\gamma_{p}(y\mathbin{\mathop{\otimes}\displaylimits}x_{2})$ in weight $p$
does not receive differentials, the differentials in the two $\mathrm{CE}$
complexes preserve the isomorphism in weight $k\leq p$. Furthermore, when
$p=3$ the simplicial objects
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Lie}^{s}_{\mathbb{F}_{3}},\mathfrak{g})$
and
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Lie}^{s,\mathrm{op}}_{\mathbb{F}_{3}},\mathfrak{g})$
are isomorphic in weight $k<p$. Hence

(4)
$\dim_{\mathbb{F}_{p}}\bigoplus_{s+t=i}\mathrm{wt}_{k}H_{s,t}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p}))=\dim_{\mathbb{Q}}\bigoplus_{s+t=i}\mathrm{wt}_{k}H_{s,t}(\mathrm{CE}(\mathfrak{g}^{\prime};\mathbb{Q}))=\beta_{i}(B_{k}(T))$

for all $i$ and $k\leq p$, with the right equality given by Theorem 3.3.

By Proposition 2.6.(1), the $E^{2}$-page of the weight $k<p$ part of the
spectral sequence (3) is

$E^{2}_{s,t}(k)\cong\mathrm{wt}_{k}H_{s,t}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p})).$

Since $B_{k}(T)$ is of finite type, the dimension of
$H_{i}(B_{k}(T);\mathbb{F}_{p})$ over $\mathbb{F}_{p}$ is at least
$\beta_{i}(B_{k}(T))$ for all $i$. Therefore no higher differential can happen
by Equation 4.

Now we tackle the case $k=p$. First we consider $p>3$. By Proposition 2.6.(2),
the $E^{2}$-page of the weight $k$ part of the spectral sequence (3) is given
by

$E^{2}_{*,*}(p)\cong\mathrm{wt}_{p}H_{*,*}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p}))\oplus\mathbb{F}_{p}\\{\overline{Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2},\overline{\beta
Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2}\\}.$

Note that the class $\overline{\beta
Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2}\in E^{2}_{1,-2}$ has
total degree $-1$, so it has to killed by a class with total degree $0$ and
simplicial degree at least 3. There is exactly one class of total degree $0$
in $\mathrm{wt}_{p}H_{*,*}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p}))$ since
$\beta_{0}(B_{p}(T))=1$, which is the class
$\gamma_{p}(c\mathbin{\mathop{\otimes}\displaylimits}x_{2})\in
E^{2}_{p-1,1-p}$. Therefore there has to be a $d_{p-2}$-differential
$\gamma_{p}(c\mathbin{\mathop{\otimes}\displaylimits}x_{2})\mapsto\overline{\beta
Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2}.$ Appealing to the
inequality

$\sum_{i}\dim_{\mathbb{F}_{p}}H_{i}(B_{k}(T);\mathbb{F}_{p})\geq\sum_{i}\beta_{i}(B_{k}(T))\stackrel{{\scriptstyle(\ref{CEsamedim})}}{{=}}\dim_{\mathbb{F}_{p}}\bigoplus_{s,t}\mathrm{wt}_{p}H_{s,t}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{p}))=\sum_{s,t}\dim_{\mathbb{F}_{p}}E^{2}_{s,t}(p)-2,$

we deduce that no other higher differential can happen.

For $p=3$, we deduce from Remark 2.7 that the normalized complexes of
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Lie}^{s}_{\mathbb{F}_{3}},\mathfrak{g})$
and
$\mathrm{Bar}_{\bullet}(\mathrm{id},\mathrm{Lie}^{s,\mathrm{op}}_{\mathbb{F}_{3}},\mathfrak{g})$
differ in the weight 3 part in that there is a differential
$\gamma_{3}(c\mathbin{\mathop{\otimes}\displaylimits}x_{2})\mapsto\overline{\beta
Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2}$ in the latter but not
in the former. Hence in the spectral sequence (3), the element

$\gamma_{3}(c\mathbin{\mathop{\otimes}\displaylimits}x_{2})=[[c\mathbin{\mathop{\otimes}\displaylimits}x_{2},c\mathbin{\mathop{\otimes}\displaylimits}x_{2}],c\mathbin{\mathop{\otimes}\displaylimits}x_{2}]\in\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}\circ\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}\Big{(}H^{*}(T;\mathbb{F}_{3})\mathbin{\mathop{\otimes}\displaylimits}\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(\mathbb{F}_{3}\\{x_{2}\\})\Big{)}$

with the two brackets come from different iterations of
$\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}$ is mapped by the $d_{1}$-differential
to

$[[c\mathbin{\mathop{\otimes}\displaylimits}x_{2},c\mathbin{\mathop{\otimes}\displaylimits}x_{2}],c\mathbin{\mathop{\otimes}\displaylimits}x_{2}]\in\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}\Big{(}H^{*}(T;\mathbb{F}_{3})\mathbin{\mathop{\otimes}\displaylimits}\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}(\mathbb{F}_{3}\\{x_{2}\\})\Big{)}$

with both brackets coming from the same iteration of
$\mathrm{Lie}^{s}_{\bar{\pazocal{R}}}$. At weight 3, there are no more
differentials between elements containing a unary operation and brackets. It
follows that the $E^{2}$-page of the weight 3 part of the spectral sequence
(3) has a basis given by the union of an $\mathbb{F}_{3}$-basis of
$\mathrm{wt}_{3}H_{*,*}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{3}))/\mathbb{F}_{3}\\{\gamma_{3}(c\mathbin{\mathop{\otimes}\displaylimits}x_{2})\\}$
and $\\{\overline{Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2}\\}$.
Combining with Equation 4, we see that

$\sum_{i}\beta_{i}(B_{3}(T))=\dim_{\mathbb{F}_{3}}\bigoplus_{s,t}\mathrm{wt}_{3}H_{s,t}(\mathrm{CE}(\mathfrak{g};\mathbb{F}_{3}))=\sum_{s,t}\dim_{\mathbb{F}_{3}}E^{2}_{s,t}(3)\geq\sum_{i}\dim_{\mathbb{F}_{3}}H_{i}(B_{3}(T);\mathbb{F}_{3}).$

Hence no higher differential can happen and equality is achieved.

Therefore the dimensions of $H_{*}(B_{k}(T);\mathbb{F}_{p})$ over
$\mathbb{F}_{p}$ agree with the Betti numbers for $k\leq p$. Since $B_{k}(T)$
is a finite complex, we further deduce that its integral homology has no
$p$-power torsion for $k\leq p$. ∎

######  Remark 3.5.

The exact same argument works for the punctured genus $g$ surface by comparing
the $\mathrm{Lie}^{s}$-algebra homology groups of $\mathfrak{g}_{1}$ and
$\mathfrak{g}^{\prime}_{1}$ (3.2) weight $k\leq p$, thereby providing an
elementary proof for Theorem 1.3.

######  Remark 3.6.

The $d_{p-2}$-differential
$\gamma_{p}(c\mathbin{\mathop{\otimes}\displaylimits}x_{2})\mapsto\overline{\beta
Q^{0}}|c\mathbin{\mathop{\otimes}\displaylimits}x_{2}$ should be viewed as a
universal differential, in the sense that it occurs in the universal case
$M=\lim_{n\rightarrow\infty}\mathbb{R}^{n}$, cf. [Zha21, Proposition 6.6].
Then
$\lim_{n\rightarrow\infty}\Omega^{n}\mathrm{Free}^{s\mathscr{L}}(\Sigma^{n}X)\simeq
X$ and the spectral sequence (2) becomes

$E^{2}_{s,t}=\pi_{s}\pi_{t}\mathrm{Bar}_{\bullet}(\mathrm{id},s\mathscr{L},\mathbb{S}^{r}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p})\Rightarrow\pi_{s+t}|\mathrm{Bar}_{\bullet}(\mathrm{id},s\mathscr{L},\mathbb{S}^{r}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p})|\cong\pi_{s,t}(\mathrm{Free}^{\mathbb{E}^{\mathrm{nu}}_{\infty}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p}}(\mathbb{S}^{r})),$

where
$\mathbb{E}^{\mathrm{nu}}_{\infty}\mathbin{\mathop{\otimes}\displaylimits}\mathbb{F}_{p}$
is the non-unital $\mathbb{E}_{\infty}$-operad in the category of
$\mathbb{F}_{p}$-module spectra. Heuristically, this is because the bottom
non-vanishing mod $p$ Dyer-Lashof operation on a class $x$ of degree $2j$ in
an $\mathbb{E}^{\mathrm{nu}}_{\infty}$-$\mathbb{F}_{p}$-algebra is given by
$\bar{Q}^{j}(x)=x^{\mathbin{\mathop{\otimes}\displaylimits}p}$, so
$\gamma_{p}(x)$ is redundant.

######  Remark 3.7.

We expect the first $p$-power-torsion classes in $H_{*}(B_{k}(M);\mathbb{Z})$
for $M=T,\Sigma_{g,1}$ to show up at weight $k=2p$ in the form of
$\overline{\beta^{\epsilon}Q^{1}}|c\mathbin{\mathop{\otimes}\displaylimits}[x_{2},x_{2}]\in
H_{*}(B_{2p}(M);\mathbb{F}_{p})$ with $\epsilon=0,1$.

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