††thanks: Email: pedro.delatorreluque@fysik.su.se,ORCID:
0000-0002-4150-2539††thanks: Email<EMAIL_ADDRESS>ORCID:
0000-0002-3082-0929††thanks: Email: linden@fysik.su.se,ORCID:
0000-0001-9888-0971

# Gamma-Ray Lines in 15 Years of Fermi-LAT Data:
New Constraints on Higgs Portal Dark Matter

Pedro De La Torre Luque  Stockholm University and The Oskar Klein Centre for
Cosmoparticle Physics, Alba Nova, 10691 Stockholm, Sweden  Polytechnic School,
Universidad San Pablo-CEU, CEU Universities. 28668 Boadilla del Monte, Spain
Juri Smirnov  Department of Mathematical Sciences, University of Liverpool,
Liverpool, L69 7ZL, United Kingdom  Stockholm University and The Oskar Klein
Centre for Cosmoparticle Physics, Alba Nova, 10691 Stockholm, Sweden  Tim
Linden  Stockholm University and The Oskar Klein Centre for Cosmoparticle
Physics, Alba Nova, 10691 Stockholm, Sweden

###### Abstract

Monoenergetic $\gamma$-ray spectral lines are among the cleanest signatures of
dark matter annihilation. We analyze 15 years of Fermi-LAT data, find no
spectral lines, and place strong constraints on dark matter annihilation to
monoenergetic $\gamma$-rays. Additionally, we produce the first double-line
analysis of the coupled signals from $\gamma\gamma$ and $Z\gamma$ lines, which
proves particularly powerful for dark matter masses above $\sim 150$ GeV. From
our constraints on a double-line feature, we investigate and constrain some
minimal models where the Galactic Center Excess (GCE) can be fit by dark
matter annihilation through the Higgs boson into Standard Model particles.

††preprint: LTH-1335

##  I Introduction

Models that generate the observed dark matter (DM) abundance through thermal
freezeout provide one of the most compelling explanations for the cosmological
evolution of our universe Zeldovich (1965); Lee and Weinberg (1977); Steigman
and Turner (1985); D’Eramo and Thaler (2010); Hochberg  _et al._ (2014, 2015);
Farina  _et al._ (2016); Bertone and Hooper (2018); Dey  _et al._ (2017);
Cline  _et al._ (2017); Dolgov (2017); Arcadi  _et al._ (2018); Roszkowski
_et al._ (2018); Dey  _et al._ (2019); Maity and Ray (2019); Kim and Kuflik
(2019); Smirnov and Beacom (2020); Asadi  _et al._ (2022); Parikh  _et al._
(2023). Fortunately, scenarios dominated by a $2\rightarrow 2$ process (e.g.
$\chi\chi\rightarrow\text{SM SM}$) also provide us with a precise, testable
target for DM annihilation searches Steigman  _et al._ (2012); Bringmann  _et
al._ (2021). The thermal freeze-out mechanism does not generically predict the
Standard Model (SM) final states or branching ratios, and thus it is common to
examine DM models dominated by tree-level annihilations to different standard-
model particle states, such as $b\bar{b}$, $\tau^{+}\tau^{-}$, $W^{+}W^{-}$ or
other leptonic and hadronic pairs.

However, these models are simplified for two reasons. First, DM annihilation
may include tree-level couplings to a number of final states, with branching
ratios that depend on the decay widths of the intermediate particles. Second,
in addition to tree-level processes, there are guaranteed loop-level
processes. Some of these final states, like those that produce $\gamma\gamma$
or Z$\gamma$ lines, may be more detectable than tree-level annihilation
processes despite their subdominant branching fractions.

While the two-photon channel leads to a mono-energetic line at
$E_{\gamma}=m_{\rm DM}$. The $Z\gamma$ channel is kinematically accessible at
DM masses of $m_{\rm DM}>m_{Z}/2$, and leads to final state photons with
energies centered around

$E_{Z\gamma}=E_{\gamma\gamma}\left(1-\left(\frac{m_{Z}}{2E_{\gamma\gamma}}\right)^{2}\right)\,.$
(1)

The spectrum has an intrinsic width due to the finite life-time of the $Z$
boson, which is given by Beenakker and Hollik (1988); Abrams  _et al._ (1989)

Figure 1: The results of our model-independent single-line (blue solid) and
Higgs-portal double-line (red solid) analysis, given an NFW profile in ROI41.
Single-line analyses by the Fermi-LAT collaboration (2015) Ackermann  _et al._
(2015)(gray solid) and a recent result by Ref. Foster  _et al._ (2022a)
(magenta dashed) are shown for comparison.

$\displaystyle\Gamma_{Z\gamma}\approx\frac{\Gamma_{Z}m_{Z}}{2\sqrt{3}E_{\gamma\gamma}}<\frac{\Gamma_{Z}}{\sqrt{3}}\approx
1.2\text{ GeV}\,.$ (2)

This width leads to an effectively monochromatic signal at the current Fermi-
LAT energy resolution. Making use of this fact, we propose a double-line
search that further increases our experimental sensitivity.

The relationship between the branching ratios to all final states depends on
the mediator choice. The simplest, and most predictive scenario is DM coupling
though the Higgs portal, a singlet operator $H^{\dagger}H$ Silveira and Zee
(1985); McDonald (1994); Burgess  _et al._ (2001); O’Connell  _et al._ (2007);
Cline  _et al._ (2013); Duerr  _et al._ (2015, 2016a, 2016b). In this case,
the branching ratios to all SM final states are entirely fixed by the well-
known properties of the Higgs boson. Thus, combining collider-grade accuracy
with the freeze-out condition entirely fixes our signal expectation for a
given DM mass.

In this work, we reanalyse existing Fermi-LAT data, choosing CLEAN events from
180 months of the PASS8 data (Atwood  _et al._ , 2009), and perform both
single- and double-line searches for annihilating DM. We find that double-line
analyses have a superior constraining power, especially at large DM masses.
Furthermore, we show that our limits on Higgs-mediated DM annihilation are in
tension with Higgs-portal interpretations of the GCE at masses near the
$m_{H}/2$ resonance Duerr  _et al._ (2016b); Fraser  _et al._ (2021).

Fig. 1 shows the limits of our single- and double-line analyses on the
$\langle\sigma v\rangle_{\gamma\gamma}$ annihilation cross section, and
compares our results with previous work Ackermann  _et al._ (2015); Foster
_et al._ (2022a). Our analysis provides stronger constraints, particularly at
large $\gamma$-ray energies.

##  II Methodology and analysis

###  II.1 Gamma-ray datasets

The Fermi-LAT is a pair-conversion telescope that measures $\gamma$-rays with
energies between $\sim$20 MeV and $\sim$1 TeV (Atwood  _et al._ , 2009). In
this paper, we use $\sim$180 months of data spanning from 2008-08-04 to
2023-07-20 selecting CLEAN events from the PASS8 data. We include events from
good quality time intervals and remove periods when the LAT was operating at
rocking angles $\theta_{r}>52\deg$ ((DATA_QUAL$>0$) && (LAT_CONFIG==1) &&
ABS(ROCK_ANGLE)$<52$). We also apply the zenith-angle cut $\theta_{z}<90\deg$,
to avoid contamination from the Earth limb. We limit our analysis to EDISP3
events evtype $=512$), which have the best energy reconstruction (hence, best
energy resolution), to minimize uncertainties relating to instrumental energy
dispersion. We employ the P8R3_CLEAN_V3 version of the instrument response
functions. The extraction of Fermi-LAT data and calculation of exposure maps
is performed using the most up-to-date version of the ScienceToolsFer (2021)
(2.0.8; released on 01/20/2021). In addition, we performed different
consistency checks of our analyses using PASS7 and PASS8 front- and back-
converted events, which allows us to compare our results with those obtained
by the Fermi-LAT Collaboration Ackermann  _et al._ (2013).

We extract data spanning from $10$ GeV to $300$ GeV in $130$ logarithmically
uniform energy bins, which constitute an energy resolution $\Delta E/E\sim
3\%$ (which is better by at least a factor of two than the intrinsic energy
resolution of EDISP3 events). Closely following previous Fermi analyses, we
divide our data into three regions of interest (ROIs), which are spherical
regions of $3$, $16$, and $41$ degrees around the Galactic center, focusing on
the inner regions studied by the Fermi collaboration in Refs. Ackermann  _et
al._ (2013, 2015) (see also Weniger (2012) for a similar approach). For each
ROI, the regions of $68\%$ containment around known sources are subtracted in
every dataset following the 4FGL_DR2 catalog, except for the ROI3 region,
where no source subtraction is done. We also mask the galactic plane in
regions with $|b|<5^{\circ}$ and $|l|>6^{\circ}$ as in Refs. Ackermann  _et
al._ (2013, 2015).

###  II.2 Single and double line analyses

We closely follow the strategy used by the Fermi collaboration in Ref.
Ackermann  _et al._ (2013) and search for spectral lines by performing maximum
likelihood fits in each of our ROIs and in $88$ sliding energy intervals (the
sliding window technique from Ref. Vertongen and Weniger (2011)) from $10$ GeV
to $300$ GeV. For each interval, we fit the count spectrum in an energy window
that surrounds the central energy with a width of $\Delta E\geq
3\times\sigma_{E}(E)$111We repeated the analysis for window sizes between $2$
and $6\times\sigma_{E}(E)$ and found that the chosen width has almost no
effect on our results. (where $\sigma_{E}(E)$ is the half width of the $68\%$
exposure-weighted energy resolution of each dataset (See Per (2021)), assuming
that the background is described by a power-law and adding a line-like signal
of free amplitude, which is smeared due to the energy resolution of the Fermi-
LAT (Following EDI (2022)). We use a likelihood function described by a
Poisson distribution in the number of events at every energy window:

$\mathcal{L}^{ROI}=\prod_{i}\frac{e^{-n(E_{i},E^{\prime}_{i})}\times
n(E_{i},E^{\prime}_{i})^{N_{i}(E_{i})}}{N(E_{i})!},$ (3)

where $N_{i}$ is the observed number of events at energy $E_{i}$ in each
dataset (ROI), and $n_{i}$ is the expected number of counts at energy
$E^{\prime}$ that are reconstructed at energy $E$ by the instrument. Under the
null hypothesis, there is no line-like signal and the expected number of
counts is found by fitting the data to a power-law describing the background
emission ($n_{i}=n_{bkg}$). In the alternative hypothesis, the number of
counts is described as $n_{i}=n_{sig}D_{eff}+n_{bkg}$, where $D_{eff}$ is the
energy dispersion matrix that allow us to account for the energy
reconstruction of the events by the LAT, and which was obtained using the
gtdrm Fermitool.

It is important to remark that, within the energy range where we perform this
analysis, the systematic uncertainties in the spectral reconstruction are
expected to be negligible compared to the statistical uncertainties in the
photon count, as reported by Refs. Ackermann  _et al._ (2013, 2015). This
assumption does not hold at much lower energies, which would require more
complex modeling of Fermi-LAT responses, as in Ref. Ackermann  _et al._
(2015). The inclusion of systematic uncertainties would only slightly weaken
our bounds at low energies, leaving the main conclusions of this manuscript
unchanged.

To perform the fits we rely on the Markov Chain Monte Carlo (MCMC) package
Emcee Foreman-Mackey  _et al._ (2013), since this technique is more robust
than conventional optimizers and less prone to finding false local minima.
This analysis produces probability distribution functions for every parameter
in the fit, which are used to estimate the credible intervals of each
parameter and the DM limits. Since the best-fit number of signal events that
we obtain is very low, we use the Feldman-Cousins (FC) method Feldman and
Cousins (1998)222We acknowledge the use of the package from
https://github.com/usnistgov/FCpy/tree/main. to ensure that we are not mis-
evaluating the confidence intervals and, hence, the limits. Using the FC
method produces roughly the same upper-limits as the MCMC algorithm, except
when the best-fit number of source counts is very small. Concretely, we take
the best-fit values for the number of background events and number of signal
events obtained from the MCMC procedure, and apply the FC method with the
likelihood functions defined in Eq. 3. In this way, we reject unrealistically
strong upper-limits, particularly for the downward fluctuations.

For the double-line analysis, we repeat the same procedure as the single-line
analysis, but add a correlated second line signal that accounts for DM
annihilation through the Higgs into a second $Z\gamma$ line
($\chi+\chi\rightarrow H\rightarrow Z+\gamma$). The relative amplitude of the
$\gamma\gamma$ and Z$\gamma$ signals is correlated by the branching ratio of
the Higgs boson to each channel. Moreover, the energy of the line signal for
$Z\gamma$ production is connected to the energy of the $\gamma\gamma$ signal
as described by Eq. 1, and we set $E_{\gamma\gamma}=m_{\chi}$ for the
annihilation process that we are considering.

To account for the fact that the energy window in our double line analysis
must accommodate both the $\gamma\gamma$ and Z$\gamma$ line energies, the
lower edge of the energy window considered in this analysis is set to be the
minimum value between $E_{Z\gamma}$ and $E_{\gamma\gamma}-3\sigma_{E}$, to
cover the double-line feature. This results in a larger energy window for the
double-line analysis from $50$ to $\sim 80$ GeV, above which the lower limit
of the sliding energy window coincides with the one used in the single-line
analysis (i.e. the lower limit is always the $3\sigma_{E}$ above $80$ GeV).

##  III DM bounds from the line search

The expected $\gamma$-ray flux from the annihilation of DM particle through
the process $\chi+\chi\rightarrow\gamma+\gamma$ in a region of the sky with
angular size $\Delta\Omega$ is

$\frac{d\Phi}{dE}=\frac{1}{8\pi}\frac{\left<\sigma
v\right>_{\gamma\gamma}}{m^{2}_{\chi}}\left(\frac{dN}{dE}\right)_{\gamma\gamma}\times\mathcal{J^{ROI}}(\Delta\Omega)\,\,\,\,\,,$
(4)

where $\mathcal{J^{ROI}}(\Delta\Omega)$ is the astrophysical J-factor that
describes the expected annihilation rate given a specific choice of ROI and a
DM distribution,
$\left(\frac{dN}{dE}\right)_{\gamma\gamma}=2\times\delta(E-E_{\gamma\gamma})$
is the $\gamma$-ray yield per annihilation, $m_{\chi}=E_{\gamma\gamma}$ is the
mass of the WIMP, and $\left<\sigma v\right>_{\gamma\gamma}$ is the
annihilation rate to the $\gamma\gamma$ channel and is related to the total
annihilation rate via the mediator-dependent branching fraction $\left<\sigma
v\right>_{\gamma\gamma}=\text{BR}_{\gamma\gamma}\times\left<\sigma
v\right>_{\rm ann}$, where $\left<\sigma v\right>_{\rm ann}$ is the total DM
annihilation rate.

For the $Z\gamma$ process, this same formula holds, but with $\left<\sigma
v\right>_{Z\gamma}=\text{BR}_{Z\gamma}\times\left<\sigma v\right>_{\rm ann}$
and $\left(\frac{dN}{dE}\right)_{Z\gamma}=\delta(E-E_{Z\gamma})$, where the
photon is produced at the energy given by Eq. 1. The main uncertainty in
deriving limits on the annihilation rate is the J-factor,
$\mathcal{J^{ROI}}(\Delta\Omega)$, which directly depends on the Milky Way DM
distribution. Here, we assume a local DM density of $0.4$ GeV cm-3 Iocco  _et
al._ (2011) and a distance from the Solar System to the GC of $8.5$ kpc. We
characterize two DM distributions, the NFW and a contracted-NFW profile with
an index $\gamma=1.3$ (motivated by studies of the GCE Ackermann and others
(The Fermi LAT Collaboration); Di Mauro (2021)), both with a scale radius of
$r_{s}=20$ kpc.

###  III.1 Significance for lines in the gamma-ray spectrum

Figure 2 shows the test-statistic (TS) computed for the single-line and
double-line analyses as a function of the DM mass. This produces an accurate
calculation (assuming Wilks’ theorem holds) for the local significance of any
line signal ($\sigma_{local}\sim\frac{n_{line}}{\sqrt{n_{bck}}}$), which can
be calculated as

${\rm TS}=2\frac{\mathcal{L}(n_{sig}=n_{sig,{\rm
Best}})}{\mathcal{L}(n_{sig}=0)}.$ (5)

Although, the J-factor constitutes the largest uncertainty on the expected
annihilation signal from the GC region, we remind the reader that the TS is
independent of the J-factor employed. We note no statistically significant
peaks (exceeding a 3$\sigma$ local significance) in any dataset. The most
statistically significant peaks hardly exceed $2\sigma$ and are not repeatedly
present in the different ROIs (i.e. a fluctuation not present in all the
datasets). We have performed an analogous analysis, without fixing the
branching ratio between the $Z\gamma$ and $\gamma\gamma$ processes to the
values predicted by the Higgs portal. Also in this general case no significant
excess signals were observed and the local significance is roughly identical
to the one obtained in the double-line analysis.

Figure 2: Local significance obtained in the $10$ to $300$ GeV range. We
compare the result of the single-line analysis (solid lines) and double-line
analysis (dashed lines) for each ROI: $3^{\circ}$ (upper), $16^{\circ}$
(middle) and $41^{\circ}$ (lower) around the center of the Galaxy. For those
signals where there is no preference for a positive number of counts over the
background the significance ($\sigma_{local}$) is set to 0.

###  III.2 DM bounds from the single-line and double-line analyses

Given the lack of significant excesses in the $\gamma$-ray spectrum, we derive
the confidence limits for both the single- and double-line analyses. We
produced the bands depicting the $68\%$ and $95\%$ confidence intervals of the
observed limits by generating mock data following a power-law distribution
with spectral index of $-2$ and Poissonian noise. We repeat the analysis for
$750$ iterations (we find that the bands remain stable above $\sim 400$
iterations), similar to the approach of Refs. Ackermann  _et al._ (2013,
2015).

Figure 3 shows the derived limits for ROI41, which provide slightly stronger
(but similar) limits than the other ROIs. We show results for other ROIs in
the Figure 7 in Appendix C. We also include the observed limits for the
double-line analysis as a dashed line, finding that these constraints become
stronger at higher energies when the $Z\gamma$ cross-section becomes larger
than the $\gamma\gamma$ cross section. The ratio BRZγ/BRγγ is derived from
Dittmaier  _et al._ (2011) and is discussed in the supplemental material.

In Figure 1, we compare our single-line limits with those obtained by the
Fermi-LAT collaboration (using $5.8$ yr of data) Ackermann  _et al._ (2015)
and those obtained by Foster et al. Foster  _et al._ (2022b) (using $\sim$14
years of Fermi-LAT data). The main differences between our analysis and Ref.
Foster  _et al._ (2022b) include: (1) their analysis employs a constant
energy-window size (of $\Delta E/E\sim 0.64$), while our analysis utilizes a
variable window size based on the local Fermi-LAT energy resolution, (2) their
analysis employs a single ROI focused on the inner $30^{\circ}$ around the GC
and utilizes a different Galactic plane cut, (3) they utilize SOURCE class
photon events with energy reconstructions spanning EDISP 1–3 (the top 75% of
well-reconstructed energy events) while we use only CLEAN class photon events
from EDISP3 (the $25\%$ of reconstructed events with the best energy
resolution) (4) they do not subtract regions surrounding bright $\gamma$-ray
point sources, while we eliminate these background-dominated regions from our
analysis in all ROIs except for ROI3. Despite these differences, our single-
line results are in good agreement, as seen in Fig. 1. The DAMPE collaboration
recently published the results of their single-line analysis using $5$ yr of
data collected by the DAMPE instrument Alemanno  _et al._ (2022), obtaining
similar bounds to those found in Ref. Ackermann  _et al._ (2015) (see also
Ref. Cheng  _et al._ (2023)).

Figure 3: DM bounds for the ROI41 region assuming a NFW Galactic DM profile,
including the $68\%$ and $95\%$ confidence intervals obtained from the single-
line analysis. We show the single-line and double-line limits as a solid and a
dashed line, respectively, and compare to the limits from Refs. Ackermann  _et
al._ (2015).

##  IV Implication for Higgs-Mediated Annihilation

Higgs mediated annihilation is unique from the perspective that collider level
precision can be used in a DM framework. Since SM processes govern the
branching ratios for the annihilation final states, and the DM coupling to the
Higgs is fixed by the freeze-out condition, the only unknown parameter is the
DM mass.

However, there are several model choices that affect the relationship between
the annihilation cross-section and the expected event rates in direct
detection and collider experiments Fraser  _et al._ (2021); Abdallah  _et al._
(2015). Here we mention two well-motivated models:

  * •

A singlet scalar model $S$, with mass $m_{S}$, which after the electroweak
symmetry breaking has the following relevant coupling to the Higgs boson, $h$

$\displaystyle-\mathcal{L}\supset\frac{1}{2}m_{S}^{2}S^{2}+\frac{\lambda_{p}v_{H}}{2}hS^{2}\,,$
(6)

where $\lambda_{p}$ is a dimensionless coupling, and $v_{H}$ is the Higgs
field vacuum expectation value Silveira and Zee (1985); McDonald (1994);
Burgess  _et al._ (2001). In this case the spin-independent direct detection
cross section is not suppressed, and only $\lambda_{p}<10^{-3}$ values are
compatible with the current limits Duerr  _et al._ (2016a); Aalbers  _et al._
(2022). The GCE signal can thus only be explained in a very narrow mass range
around the Higgs resonance Duerr  _et al._ (2016b).

  * •

A Majorana fermion model $\chi$, with mass $m_{\chi}$ and the following low-
energy couplings

$\displaystyle-\mathcal{L}\supset\frac{1}{2}m_{\chi}\bar{\chi}\chi+i\frac{y_{p}}{2}h\bar{\chi}\gamma_{5}\chi\,$
(7)

where $y_{p}$ is a dimensionless coupling parameter. In this case the direct
detection cross section is suppressed for real values of $y_{p}$ Fraser  _et
al._ (2021); Foster  _et al._ (2022a), which means that we would not expect a
signal in direct detection searches. Therefore, this model is not constrained
by direct detection experiments, and only visible in collider and indirect
detection searches.

We note that both scenarios are subject to invisible Higgs decay constraints,
which limit $\text{BR}(h\rightarrow\text{inv.})<0.11$ Wang (2022), and
difficult to avoid.

The total annihilation cross section in both scenarios is given by

$\displaystyle\sigma_{\rm ann.}(s)=\frac{f(m_{\rm DM},\lambda_{\rm
DM})}{\left(s-m_{h}^{2}+\Gamma_{h}^{2}m_{h}^{2}\right)^{2}}\,$ (8)

where $s$ is the total s-channel four-momentum, $m_{\rm DM}$ and $\lambda_{\rm
DM}$ are the DM mass and coupling strength to the Higgs, and $\Gamma_{h}$ is
the total Higgs boson decay width. As discussed in Ref. Gondolo and Gelmini
(1991) the thermally averaged cross-section at a resonance needs to be
performed without the non-relativistic expansion of the annihilation cross
section. The fact that the thermal average of the annihilation cross section
in the early universe is significantly different from the average at late
times leads to a strong late-time mass dependence of the annihilation cross
section around the resonance, as discussed in detail in Ref. Duerr  _et al._
(2016a).

Figure 4 shows the Higgs portal parameter space with the predicted total
annihilation rate as a function of the DM mass. It is intriguing that
$\gamma$-ray line searches have the strongest sensitivity around the Higgs
resonance, a region in parameter space that is typically challenging to test.
We make use of the fact that, given the Fermi-LAT energy resolution, the line
signatures can be well distinguished from the continuum emission, such as
final state radiation, up to $\gamma$-ray energies of $E_{\gamma\gamma}\sim
300$ GeV, as discussed in Refs. Duerr  _et al._ (2016b, 2015).

Figure 4:  The Higgs portal total annihilation rate as a function of the DM
mass. We superimpose our constraint from the double-line analysis on the total
annihilation rate (red solid). Additionally we show the dwarf spheroidal
limits from the Fermi-LAT collaboration Ahnen  _et al._ (2016) (green solid),
as well as the constraints from invisible Higgs decay searches Wang (2022)
(gray dashed). The rate factor predicted by the relic density constraint is
shown as a function of DM mass (black dashed).

##  V Implications for the Galactic Center Excess

Observations of $\gamma$-ray emission from the Milky Way galactic center have
long observed a $\gamma$-ray excess that has been named the GCE Goodenough and
Hooper (2009); Hooper and Linden (2011); Abazajian and Kaplinghat (2012);
Daylan  _et al._ (2016); Calore  _et al._ (2015); Ackermann  _et al._ (2017).
While the origin of the GCE is disputed, the two most compelling explanations
involve DM annihilation or the combined emission from a population of
millisecond pulsars (MSPs) Hooper and Goodenough (2011); Abazajian (2011);
Hooper and Linden (2011); Hooper  _et al._ (2013); Cholis  _et al._ (2015);
Lee  _et al._ (2016); Bartels  _et al._ (2016); Leane and Slatyer (2019,
2020a, 2020b); Buschmann  _et al._ (2020); Macias  _et al._ (2018, 2019);
Cholis  _et al._ (2019, 2022). Within the context of DM models, a DM candidate
that annihilates predominantly via $\chi\chi\rightarrow b\bar{b}$ with a mass
$m_{\chi}$ between $40$ GeV and $70$ GeV is highly consistent with the data
Daylan  _et al._ (2016); Calore  _et al._ (2015).

Since this tree-level final state involves charged particles, there is
unavoidably a loop process leading to monoenergetic $\gamma\gamma$ and, as
dictated by electroweak symmetry, narrow $Z\gamma$ photon lines. However, the
simple $b-$quark loop produces a branching ratio of the order of
$\text{BR}_{\gamma\gamma}\sim\alpha_{\rm EM}^{2}/(4\pi)\sim 10^{-6}$, which
falls far below current experimental sensitivities.

On the other hand, a dominant branching fraction into $b-$quark states in this
mass range is hard to explain unless the interaction is related to the quark
Yukawa couplings. Thus, we are naturally led to the Higgs-boson mediated
scenario. Notably, a Higgs-motivated $b\bar{b}-$annihilation rate that fits
the GCE unambiguously predicts bright $\gamma\gamma$ and $Z\gamma$ signals.

Figure 5 shows the $95\%$ confidence interval for the GCE signal-predictions
for the annihilation rates $\langle\sigma v\rangle_{\gamma\gamma}$ and
$\langle\sigma v\rangle_{Z\gamma}$, as a red and magenta ellipse,
respectively. Those regions are derived from the best-fit region of Ref.
Calore  _et al._ (2015). We compare the predicted rates with our best limits
from the double-line analysis of 15 years of Fermi-LAT data, and find that are
results are beginning to be in tension with minimal Higgs portal mediated
scenarios as an explanation for the GCE. By minimal we mean models based on
the interactions in Eqs. 6, and 7, in which the relic density is determined by
the s-channel Higgs annihilation (shown as black dashed line), and given the
searches for invisible Higgs decays apply. Note that other approaches find
best-fit parameter regions Di Mauro and Winkler (2021), that are only
consistent with the relic density predictions within the excluded parameter
range.

Figure 5:  The predictions for the $\gamma\gamma$ (dark red) and $Z\gamma$
(magenta) annihilation rates in the Higgs portal model, given the GCE $95\%$
confidence interval found in Ref. Calore  _et al._ (2015). We superimpose our
double-line search limits from a gNFW profile in ROI41 (red solid), and the
$\langle\sigma v\rangle_{\gamma\gamma}$ values predicted by the freeze-out
(black dashed). Note that in contrast to constraints from dwarf galaxies this
search for $\gamma-$lines in the GC does not have an intrinsic J-factor
uncertainty, when the comparison to the continuum excess is made.

##  VI Discussion and Conclusion

In this _letter_ , we have reanalyzed 15 years of Fermi-LAT data and studied
spectral signatures stemming from DM annihliation to monoenergetic lines. We
find no evidence for any statistically significant excesses and set strong
limits on DM annihilation to mono-energetic photons. Our results can be
applied to a broad range of DM scenarios, constraining DM masses up to (or
beyond) the electroweak scale in many well-motivated DM models. If non-
perturbative effects, such as bound-state formation are taken into account
Mitridate  _et al._ (2017); Smirnov and Beacom (2019); Becker  _et al._
(2022), the reach extends to even higher DM masses.

Additionally, we performed the first double-line analysis of the full Fermi-
LAT data-set, using 15 years of data. We placed strong limits on thermal DM
that is coupled to the Standard Model through the Higgs portal, in a largely
model independent way. Our results are in moderate tension (but do not
entirely rule out), Higgs Portal models of the GCE with dark matter masses
that sit near the mH/2 resonance. This parameter space is of interest due to
the fact that it is the only portion of the Higgs Portal parameter space that
is consistent with the GCE and constraints from the branching ratios to
invisible particles. Moreover, we note that our constraints are roughly
independent of the dark matter density profile near the galactic center, as
the cross-sections for both the GCE continuum and the line search shift in the
same way.

We emphasise, that the double-line technique can be applied to other datasets,
and is particularly promising at energies near the Higgs resonance.
Furthermore, it has an enhanced discovery potential for $\gamma$-ray signals
that have a limited sensitivity due to low photon counts, as it makes use of
additional information from photons in correlated energy bins.

## Acknowledgements

We would like to thank Ben Safdi, Josh Foster, Rebecca Leane, and Linda Xu for
helpful comments and discussions. PD and TL are supported in part by the
European Research Council under grant 742104 and the Swedish National Space
Agency under contract 117/19. JS was also supported by the European Research
Council under grant 742104 during the initial stage of the project. TL is also
supported by the Swedish Research Council under contracts 2019-05135 and
2022-04283. This project used computing resources from the Swedish National
Infrastructure for Computing (SNIC) under project Nos. 2021/3-42, 2021/6-326,
2021-1-24 and 2022/3-27 partially funded by the Swedish Research Council
through grant no. 2018-05973.

##  Appendix A Fermi counts spectra and signal fit

In this appendix we show examples of the derived count spectra, and the
template functions used in the single- and double-line analyses in the first
figure panel. We furthermore show the limits obtained from the single-line
analysis for all ROIs studied in this work in the second figure panel, and for
the double-line analysis in the third figure panel. Finally, we report the
ratio BR${}_{Z\gamma}/$BRγγ as function of the DM mass, as derived from
Dittmaier  _et al._ (2011) (consistent with analytic calculations from Djouadi
(2008)), which motivates the application of the double-line analysis for
$\mathcal{O}$(TeV) gamma-ray data

Figure 6 shows the count spectrum for the ROI16 region at two different
energies: at around $106$ GeV (upper row), where fluctuations at the level of
$1-2\sigma$ are observed for all ROIs (see Fig. 2) and at $155$ GeV, where the
signals from the photons produced in the $\gamma\gamma$ and $Z\gamma$ decay
start to merge into the same energy bin (leading to a significantly stronger
constraint compared to the one from the single-line analysis above this
energy). Here, we include the fitted count spectra assuming only background
(null fit) and assuming background+signal (signal fit), which allows us to see
a comparison of the signals searched in the single-line (right panels) and
double-line (left panels) analysis.

Figure 6: Count spectra for two different energy windows comparing the double-
line (left column) and single-line (right column) best-fit signals. In the
upper panels, the energy window is centered at E${}_{\gamma\gamma}\sim 106$
GeV (E${}_{Z\gamma}\sim 87$ GeV) and in the lower panels at
E${}_{\gamma\gamma}\sim 155$ GeV (E${}_{Z\gamma}\sim 142$ GeV). At energies
greater than E${}_{\gamma\gamma}\sim 180$ GeV both signals expected in the
double-line analysis merge.

##  Appendix B Single-line limits

Figure 7 shows the limits obtained from the single-line analysis compared to
those from Refs. Alemanno  _et al._ (2022); Ackermann  _et al._ (2015); Foster
_et al._ (2022b). The ROI16 region constitutes the region that leads to better
bounds. Thus, in the upper panels, we show the limits derived assuming an NFW
profile, on the left, and a contracted-NFW profile (with index $\gamma=1.3$),
on the right. The contracted-NFW profile has been found in several papers to
be the DM profile most compatible with the GCE. In this case, the limits from
other works have been rescaled accordingly. Then, in the lower panels we show
the limits obtained for the ROI3 (left) and ROI41 (right) regions. As we see,
the ROI3 region offers the most conservative limits, which is due to the
smaller ROI, which produces many fewer counts and larger Poissonian
uncertainties, while the ROI41 region leads to similar but slightly higher
bounds than in the ROI16 region. Larger ROIs can have large systematic
uncertainties associated to the analysis and a lower expected signal-to-noise
ratio for a DM signal.

Figure 7: DM bounds and confidence bands for different ROIs obtained from the
single-line analysis, compared to those from Refs. Alemanno  _et al._ (2022);
Ackermann  _et al._ (2015); Foster  _et al._ (2022b). The upper panels show
the limits for the ROI16 region, assuming an NFW and a contracted-NFW (with
index $\gamma=1.3$) DM profile, for the upper-left and upper-right panels,
respectively. The limits from other works have been rescaled in the case where
we show the limits assuming a c-NFW profile. The lower panels show the limits
for the ROI3 (left) and ROI41 (right) regions.

##  Appendix C Double-line limits

Figure 8 shows the DM bounds and confidence bands obtained from the double-
line analysis for the ROI16 (left panel) and ROI41 (right panel), compared to
the limits derived from the single-line analyses, drawn assuming a c-NFW
profile (with index $\gamma=1.3$). This show that the double-line analysis
would constitute a more sensitive way to look for line-like gamma-ray signals
at very high energies. The reason is that at high energies the photons from
the $\gamma\gamma$ and $Z\gamma$ decays are produced at roughly the same
energy and the branching ratio for the $Z\gamma$ decay becomes much higher
than for the $\gamma\gamma$ process, leading to a more constraining limit of
$\langle\sigma v\rangle_{\gamma\gamma}$. This means that a search for lines at
the energies where the $Z\gamma$ decay dominates lead to a much stronger
constraint due to the sum of the signals coming from the $\gamma\gamma$ and
$Z\gamma$ decays and the branching ratio for the $Z\gamma$ process.

Figure 8: DM bounds and confidence bands for different ROIs obtained from the
double-line analysis, compared to the limits derived from the single-line
analyses, for the ROI16 (left panel) and ROI41 (right panel), assuming a c-NFW
profile (with index $\gamma=1.3$).

Figure 9 displays the signal strength ratio between the $Z\gamma$ and
$\gamma\gamma$ mono-energetic photon signals for Higgs mediated annihilation.
At low energies the massive Z-boson emission suppresses the signal, in the
intermediate mass range the ratio is determined by the coupling ratio between
the weak and electromagnetic couplings $\alpha_{\rm EW}\sim 1/29$ and
$\alpha_{\rm EM}\sim 1/137$, as well as the photon multiplicity. Finally, at
larger masses the loop factor of the $\gamma\gamma$ signal is suppressed due
to the destructive interference of virtual particles, which enhances the
relative $Z\gamma$ signal strength.

Figure 9: The signal-strength ratio of the $Z\gamma$ and $\gamma\gamma$ mono-
energetic photon signals for Higgs mediated annihilation Dittmaier  _et al._
(2011).

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# Separating the Edges of a Graph by a Linear Number of Paths This research
has been partially supported by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior – Brasil – CAPES – Finance Code 001. M. Bonamy is supported by
ANR project DISTANCIA (Metric Graph Theory, ANR-17-CE40-0015). F. Botler is
supported by CNPq (304315/2022-2), FAPERJ (211.305/2019 and 201.334/2022) and
CAPES-PRINT (88887.695773/2022-00). T. Naia is supported by CNPq
(201114/2014-3) and FAPESP (2019/04375-5, 2019/13364-7, 2020/16570-4). FAPERJ
and FAPESP are, respectively, Research Foundations of Rio de Janeiro and São
Paulo. CNPq is the National Council for Scientific and Technological
Development of Brazil.

Marthe Bonamy  Fábio Botler  François Dross  Tássio Naia  Jozef Skokan

###### Abstract

Recently, Letzter proved that any graph of order $n$ contains a collection
$\mathcal{P}$ of $\bigoh(n\log^{\star}n)$ paths with the following property:
for all distinct edges $e$ and $f$ there exists a path in $\mathcal{P}$ which
contains $e$ but not $f$. We improve this upper bound to $19n$, thus answering
a question of G.O.H. Katona and confirming a conjecture independently posed by
Balogh, Csaba, Martin, and Pluhár and by Falgas-Ravry, Kittipassorn, Korándi,
Letzter, and Narayanan. Our proof is elementary and self-contained.

O psn title = Separating the Edges of a Graph by a Linear Number of Paths,
author = Marthe Bonamy, Fábio Botler, François Dross, Tássio Naia, and Jozef
Skokan, plaintextauthor = Marthe Bonamy, Fabio Botler, François Dross, Tassio
Naia, and Jozef Skokan, plaintexttitle = Separating the Edges of a Graph by a
Linear Number of Paths, runningtitle = Separating the Edges of a Graph by a
Linear Number of Paths, runningauthor = M. Bonamy, F. Botler, F. Dross, T.
Naia, and J. Skokan, copyrightauthor = M. Bonamy, F. Botler, F. Dross, T.
Naia, and J. Skokan, year=2023, number=6, received=20 February 2023,
published=27 October 2023, doi=10.19086/aic.2023.6,

[classification=text]

A path _separates_ an edge $e$ from an edge $f$ if it contains the former and
not the latter. How many paths do we need to separate any edge from any other?

This question was first asked by G.O.H. Katona in 2013 (see [6]), in line with
the general study of separating systems initiated by Rényi in 1961 [10]. Note
that the roles of $e$ and $f$ are not symmetric. There are in fact two
variants of the problem, which we clarify now. A collection $\mathcal{P}$ of
paths in a graph $G$ is a _strongly-separating path system_ of $G$ if, for any
two distinct edges $e$ and $f$ of $G$, there exist paths $P_{e}$ and $P_{f}$
in $\mathcal{P}$ such that $e\in P_{e}$, $e\not\in P_{f}$ and $f\in P_{f}$,
$f\not\in P_{e}$. The collection $\mathcal{P}$ is a _weakly-separating path
system_ of $G$ if for any two distinct edges $e$ and $f$, there is a path
$P\in\mathcal{P}$ that contains one and not the other. The _size_ of a (weakly
or strongly) separating path system is the number of paths in the collection.
We recommend the excellent introduction in [7] for a more complete description
of the history of the problem. We are interested in the following conjecture.

######  Conjecture 1 ([1, 6])

Every graph on $n$ vertices admits a strongly-separating path system of size
$\bigoh(n)$.

This conjecture was formulated independently by Falgas-Ravry, Kittipassorn,
Korándi, Letzter, and Narayanan [6] for weakly-separating path systems, and
later strengthened by Balogh, Csaba, Martin, and Pluhár [1] to strongly-
separating path systems. Very recently, Letzter [7] made major progress toward
Conjecture 1 by proving that every graph on $n$ vertices admits a strongly-
separating system with $\bigoh(n\cdot\log^{\star}n)$ paths. A key component in
[7] is the use of sublinear expanders, motivated by the recent work of Bucić
and Montgomery [3]. In this note we obtain a linear bound, thus confirming
Conjecture 1.

###### Theorem 2

Every graph on $n$ vertices admits a strongly-separating path system of size
$19n$.

Our proof uses Pósa rotation–extension tools [9] as presented by Brandt,
Broersma, Diestel, and Kriesell [2], and is inspired by Lemma 2.2 in Conlon,
Fox, and Sudakov [4]. In essence, we reduce the general problem to _traceable
graphs_ , i.e., graphs with a path that spans all vertices, known as a
Hamiltonian path.

Before delving into technical arguments, we point out that the upper bound of
$19n$ is likely far from optimal. Balogh et al. [1, Theorem 10] showed that
the complete bipartite graph $K_{\varepsilon n,(1-\varepsilon)n}$ cannot be
strongly separated by fewer than $2(1-2\varepsilon)n$ paths. However, we are
not aware of any graph with a larger lower bound. Therefore, the following
might be true.

###### Problem 3

Does every graph on $n$ vertices admit a strongly-separating path system of
size $2n$ ?

Determining the exact constant is challenging even for cliques [11]. In our
initial attempts to solve Conjecture 1, we stumbled upon this question:

###### Problem 4

Does every properly edge-colored graph $G$ admit $\bigoh(|V(G)|)$ rainbow
paths that cover $E(G)$ ?

In this context, a rainbow path is a path that does not contain two edges of
the same color. To the best of our knowledge, this question has not been
stated anywhere else. We pose this question here, since we believe it to be
natural and interesting in its own right. Problem 4, if answered
affirmatively, would yield an alternative proof of Conjecture 1: start with a
linear path decomposition (e.g., with Theorem 0.7), decompose each path into
two matchings, and define two graphs each containing precisely one of the
matchings of each path. The matchings form a proper edge coloring of each such
graph, and the two linear rainbow path decompositions of these graph together
with the original path decomposition form a strongly separating path system.
We remark that we can verify a version of Problem 4 where “paths” is replaced
by “trails”.

### Proof of Theorem 2

From now on, every separating path system is a strongly-separating path
system. We use the following standard notation. Given a graph $G$, and a set
$S\subseteq V(G)$, we denote by $N_{G}(S)$ the set of vertices _not_ in $S$
adjacent in $G$ to some vertex in $S$. We omit subscripts when clear from the
context.

Pósa rotation-extension. Given a graph $G$ and vertices $u,v$ in $G$, let
$P=u\cdots v$ be a path from $u$ to $v$. If $x\in V(P)$ is a neighbor of $u$
in $G$ and $x^{-}$ is the vertex preceding $x$ in $P$, then
$P^{\prime}=P-xx^{-}+ux$ is a path in $G$ for which $V(P^{\prime})=V(P)$. We
say that $P^{\prime}$ _has been obtained from_ $P$ by an _elementary exchange_
fixing $v$ (see Figure 1). A path obtained from $P$ by a (possibly empty)
sequence of elementary exchanges fixing $v$ is said to be a path _derived_
from $P$. The set of endvertices of paths derived from $P$ that are distinct
from $v$ is denoted by $S_{v}(P)$. Since all paths derived from $P$ have the
same vertex set as $P$, we have $S_{v}(P)\subseteq V(P)$. When $P$ is a
longest path ending at $v$, we obtain the following.

$u\vphantom{x^{-}}$$x^{-}$$x\vphantom{x^{-}}$$v\vphantom{x^{-}}$ Figure 1: a
path (highlighted) obtained by an elementary exchange fixing $v$.

######  Lemma 5 ([2])

Let $P=u\cdots v$ be a longest path of a graph $G$ and let $S=S_{v}(P)$. Then
$|N_{G}(S)|\leq 2|S|$.

######  Proof 0.6 (Proof rephrased from [2, Lemma 2.6]).

Since the maximum degree of $P$ is $2$, it suffices to show that
$N_{G}(S)\subseteq N_{P}(S)$. Let $x\in S$ and $y\in N_{G}(x)\setminus S$ be
given. As $x\in S$ there is a longest path $Q=x\cdots v$ derived from $P$.
Note that, by the maximality of $Q$, we have $y\in V(Q)$. Let $z$ denote the
predecessor of $y$ in $Q$, and note that $Q+xy-yz$ is obtained from $Q$ by an
elementary exchange. Now, consider the first edge $e$ incident to $y$ that is
removed by an elementary exchange in the “derivation sequence” from $P$ to
$Q+xy-yz$, and note that $e\in E(P)$. Note that when an elementary exchange
removes an edge $ab$ of a path derived from $P$, then $a$ or $b$ belongs to
$S$. Therefore one of the vertices of $e$ is in $S$. Since $y\not\in S$ and
$e\in E(P)$, we obtain $y\in N_{P}(S)$, as desired.

Our argument requires a lemma stating that every $n$-vertex graph admits an
edge-cover by $\bigoh(n)$ paths. While elementary arguments yield a cover by
at most $6n$ paths (as we discuss at the end of the paper), we optimize the
multiplicative constant in Theorem 2 using a result of Dean and Kouider [5]
that strengthens a well-known result of Lovász [8].

######  Theorem 0.7 ([5]).

Every graph $G$ contains at most $2|V(G)|/3$ edge-disjoint paths covering
$E(G)$.

Let $H$ and $G$ be (not necessarily disjoint) graphs. We say that a set of
paths $\mathcal{P}$ in $H\cup G$ _separates_ $H$ from $G$ if for each pair
$(e,f)\in E(H)\times E(G)$ with $e\neq f$ there is a path in $\mathcal{P}$
that contains $e$ and does not contain $f$. Now, we are able to prove our
result.

######  Proof 0.8 (Proof of Theorem 2).

We proceed by induction on $n$. Let $G$ be a graph with $n$ vertices. If $G$
is empty, the result trivially holds. If not, we consider $P$ and $S$ as in
Lemma 5. Let $H$ be the subgraph of $G$ induced by the edges with at least one
vertex in $S$, set $n^{\prime}=|V(H)|$, and note that $n^{\prime}\leq 3|S|$.
Let $G^{\prime}=G\setminus S$ and note that $G$ is the (edge) disjoint union
of $G^{\prime}$ and $H$. Note that, since $G$ is not empty, $S$ is not empty.
By the induction hypothesis, $G^{\prime}$ admits a separating path system
$\mathcal{Q}^{\prime}$ of size $19(n-|S|)$. Since $\mathcal{Q}^{\prime}$
covers $G^{\prime}$, the family $\mathcal{Q}^{\prime}$ separates $G^{\prime}$
from $H$. In what follows, we construct a set of paths $\mathcal{P}$ that
separates $H$ from $G$. Moreover, we obtain such $\mathcal{P}$ so that
$|\mathcal{P}|\leq 19|S|$, and hence $\mathcal{Q}^{\prime}\cup\mathcal{P}$ is
a separating path system of $G$ with cardinality at most $19(n-|S|)+19|S|=19n$
as desired.

First, let $P_{S}=E(P)\cap E(H)$ be the set of edges of $P$ having at least
one vertex in $S$ (see Figure 2), and let $\mathcal{P}_{S}$ be the set of
paths each consisting of a single edge in $P_{S}$, i.e.,
$\mathcal{P}_{S}=\\{(e,\\{e\\}):e\in P_{S}\\}$. Now, let $\mathcal{D}$ be the
path decomposition of $H^{\prime}=H\setminus P_{S}$ given by Theorem 0.7. Note
that $\mathcal{P}_{S}\cup\mathcal{D}$ has size at most $2|S|+2n^{\prime}/3\leq
4|S|$ and separates (i) $H$ from $G^{\prime}$; (ii) $H^{\prime}$ from $P$; and
(iii) $P_{S}$ from $G$. It remains to create a set of at most $5n^{\prime}\leq
15|S|$ paths that separates $H^{\prime}$ from itself (see Figure 2).

$S$$N(S)$

$H^{\prime}$$G^{\prime}$$P_{S}$$P_{k}$’s and
$Q_{k}$’s$\mathcal{Q}^{\prime}$$\mathcal{P}_{S}$$\mathcal{D}$$\mathcal{Q}^{\prime}$$\mathcal{D}$$\mathcal{P}_{S}$$\mathcal{Q}^{\prime}$$\mathcal{P}_{S}$

Figure 2: Left: a set $S$ in a traceable graph and its neighborhood $N(S)$; a
Hamiltonian path is highlighted, dashed red edges are the edges in $P_{S}$.
Right: paths that separate subgraphs of $G$, where
$A\stackrel{{\scriptstyle\alpha}}{{\to}}B$ indicates that $\alpha$ separates
$A$ from $B$ (for instance, $\mathcal{Q^{\prime}}$ separates $G^{\prime}$ from
$H^{\prime}$).

Let us write $P=u_{1}\cdots u_{m}$, and let $v_{1},\ldots,v_{n^{\prime}}$ be
the vertices of $H$ in the order that they appear in $P$, i.e., so that if
$v_{i}=u_{s_{i}}$ and $v_{j}=u_{s_{j}}$, then $i<j$ if and only if
$s_{i}<s_{j}$. In what follows, each edge $v_{i}v_{j}$ is written so that
$i<j$. We define $5n^{\prime}$ sets of edges as follows: for
$k\in[2n^{\prime}]$ let $M_{k}=\\{v_{i}v_{j}\in E(H^{\prime}):i+j=k\\}$, and
for $k\in[3n^{\prime}]$ let $N_{k}=\\{v_{i}v_{j}\in E(H^{\prime}):i+2j=k\\}$.
We shall build a family of $5n^{\prime}$ paths, covering each of the $M_{k}$
and $N_{k}$. (Strictly speaking, we need fewer than $5n^{\prime}$ paths, since
some $M_{k}$ and $N_{k}$ are empty. However, as far as we can tell, this only
leads to marginal improvements.) Note that
$\mathcal{M}=\\{M_{1},\ldots,M_{2n}\\}$ (resp.
$\mathcal{N}=\\{N_{1},\ldots,N_{3n}\\}$) partitions $E(H^{\prime})$.

Figure 3: Set of comparable edges $M_{k}$ (dashed) and a highlighted path
$P_{k}$ joining them.

We observe that for any $k$, the edges in $M_{k}$ are pairwise comparable
(similarly for $N_{k}$), in the sense that given
$v_{i}v_{j},v_{i^{\prime}}v_{j^{\prime}}\in M_{k}$ with $i>i^{\prime}$ we have
$j^{\prime}>j$, and in fact $j^{\prime}>j>i>i^{\prime}$, i.e., each $M_{k}$
and $N_{k}$ is a set of “nested” edges with respect to $P$. Moreover, the set
$M_{k}$ is a subset of
$\\{v_{1}v_{k-1},\ldots,v_{\frac{k}{2}-1}v_{\frac{k}{2}+1}\\}$, with the last
indices tweaked to fit the parity of $k$. In the same manner
$N_{k}\subseteq\\{v_{1}v_{\frac{k-1}{2}},\ldots,v_{\frac{k}{3}-2}v_{\frac{k}{3}+1}\\}$,
again with tweaks depending on the value of $k\bmod 6$. Now, for each
$k\in[2n^{\prime}]$, we find a path $P_{k}$ using all edges of $M_{k}$ and
(possibly) some edges in $P$ (see Figure 3). Let
$M_{k}=\\{x_{1}y_{1},\ldots,x_{s}y_{s}\\}$ where the $x_{i}$’s are chosen in
increasing order on $P$ (hence the $y_{i}$’s in decreasing order). By
induction on $i$ from $s$ down to $1$, there is a path $R_{i}$ in
$P_{i}\cup\\{x_{i}y_{i},\ldots,x_{s}y_{s}\\}$ that contains
$x_{i}y_{i},\ldots,x_{s}y_{s}$ and ends in either $x_{i}$ or $y_{i}$, where
$P_{i}$ denotes the subpath of $P$ between $x_{i}$ and $y_{i}$. Therefore,
$P_{k}=R_{1}$ is the desired path. Analogously, for $k\in[3n^{\prime}]$, we
obtain a path $Q_{k}$ that contains $N_{k}$ and some edges of $P$.

We claim that the collection of $P_{k}$’s and $Q_{k}$’s together separates
each pair of edges in $H^{\prime}$. Indeed, let $v_{i}v_{j}$ and
$v_{i^{\prime}}v_{j^{\prime}}$ be two edges of $H^{\prime}$. Each such edge
belongs to exactly one $M_{k}$ and exactly one $N_{k}$. If they belong to
different $M_{k}$’s, they belong to different $P_{k}$’s and are separated.
Similarly, if they belong to different $N_{k}$’s. Therefore, we may assume
that $i+j=i^{\prime}+j^{\prime}$ and $i+2j=i^{\prime}+2j^{\prime}$. This
immediately yields $j=j^{\prime}$ and $i=i^{\prime}$, as desired. Therefore,
any two distinct edges in $H^{\prime}$ are separated by this collection, which
involves $5n^{\prime}$ paths. This concludes the proof.

### Discussion and further work

As mentioned in the introduction, the best possible constant factor in Theorem
2 might well be $2$. Here we obtain $19$, which can still be reduced a little
with some care111For example, only applying Theorem 0.7 once for the whole
graph, for all edges not in any $P$, would give an improvement of $1.3333$. We
decided against implementing this and other small improvements, as they do not
seem worth the loss of clarity and simplicity. . While partial progress would
only be of limited interest, we highlight here straightforward candidates for
improvement. First, any improvement of the ratio $\frac{n^{\prime}}{|S|}$
(currently $3$) would immediately significantly improve the factor. Second,
with the arguments in this paper we can prove that the edges of any traceable
graph with $n$ vertices can be separated by $(6+2/3)n$ paths. In fact, any
improvement of that could be plugged in the proof and used to improve the
bound. In contrast, we observe that Theorem 0.7 is tight for disjoint
triangles.

We also note that by using the $M_{k}$’s together with Lemma 5, we can prove
that the edges of any graph can be covered by $6n$ paths without using Theorem
0.7. This allows us to verify Conjecture 1 in a self-contained manner.

As a final remark, we observe that robust sublinear expanders, which play a
key role in [7], have also been applied in [3] to investigate the well-known
Erdős–Gallai conjecture, that says that the edges of any $n$-vertex graph can
be partitioned into $\bigoh(n)$ edges and cycles. It is natural to wonder if
our approach might be useful in tackling this conjecture as well.

## Acknowledgments

The first author expresses gratitude to Bhargav Narayanan for sharing the
question with her at the Oberwolfach Workshop id:2201, and to the organizers
and participants of that workshop for creating a stimulating environment. We
also thank Matija Bucić, Victor Falgas-Ravry and Lyuben Lichev for insightful
comments which enriched this manuscript.

## References

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[bonamy] Marthe Bonamy
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400
Talence, France
marthe.bonamyu-bordeauxfr
https://www.labri.fr/perso/mbonamy/ [botler] Fábio Botler
Professor
Programa de Engenharia de Sistemas e Computação
Instituto Alberto Luiz Coimbra de Pós-Graduação e Pesquisa em Engenharia
Universidade Federal do Rio de Janeiro
Rio de Janeiro, Brasil
fbotlercosufrjbr
http://www.cos.ufrj.br/ fbotler [dross] François Dross
Associate Professor
Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400
Talence, France
francois.drossu-bordeauxfr
https://sites.google.com/view/francoisdross/ [naia] Tássio Naia
María de Maetzu fellow
Centre de Reserca Matemática
tnaiamemberfsforg
https://tassio.gitlab.io/site/ [skokan] Jozef Skokan
Professor
Department of Mathematics
London School of Economics and Political Science (LSE)
Houghton Street, London, WC2A 2AE, United Kingdom
j.skokanlseacuk
http://www.maths.lse.ac.uk/personal/jozef/