Proceedings of the CTD/WIT 2019 CLICdp-Conf-2019-005

New developments in conformal tracking for the CLIC detector

Erica Brondolin

On behalf of the CLICdp collaboration,
Experimental Physics Department at CERN
Geneva, Switzerland

> ABSTRACT
>
>
>
>
> Conformal tracking is an innovative track finding strategy adopted for the
> detector at the Compact Linear Collider (CLIC), a proposed future
> electron–positron collider. It features a pattern recognition in a
> conformal-mapped plane using the cellular automaton algorithm to reconstruct
> the trajectory of charged particles in a magnetic field. The efficiency and
> robustness of the algorithm are validated using full-simulation studies in
> the challenging beam-induced background conditions expected for the 3 TeV
> stage of the CLIC collider. The tracking performance requirements, set by
> the ambitious CLIC physics programme, have been shown to be met. Moreover,
> thanks to its flexibility and geometry-agnostic nature, this algorithm was
> also shown to be easily adaptable to different detector designs and beam
> conditions.

> PRESENTED AT
>
>
>
>
> Connecting the Dots and Workshop on Intelligent Trackers (CTD/WIT 2019)
>
> Instituto de Física Corpuscular (IFIC), Valencia, Spain
>
> April 2-5, 2019

##  1 Introduction

The next generation of tracking detectors foreseen for future
electron–positron colliders, such as the Compact Linear Collider (CLIC), will
feature ultra-low-mass systems and provide extremely high position accuracy to
fulfil the requirements imposed by the physics programme and the challenging
beam-induced background conditions. In order to fully exploit these features
in track reconstruction, a new track finding technique was developed by the
CLICdp collaboration that merges the two concepts of conformal mapping and
cellular automaton. We refer to this technique as _conformal tracking_. Thanks
to its flexibility and robustness, the new algorithm can also be adapted
easily to different detector designs and geometries.

The goal of this paper is to describe the conformal tracking algorithm, its
implementation as well as its performance in the track reconstruction used in
the CLIC detector (CLICdet). Section 2 introduces the future CLIC collider
with particular emphasis on the physics requirements and the beam-induced
background conditions including the resulting challenges in terms of track
performance. It concludes with a brief description of the tracking system for
CLICdet. In Section 3 the event simulation and software framework are
presented. Section 4 presents the track reconstruction process at CLIC. The
tracking performance in terms of efficiency, fake rate, track resolution and
CPU time is shown in Section 5, while conclusions are presented in Section 6.

##  2 Tracking challenges at CLIC

CLIC is a proposed
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
collider operating in three stages at centre-of-mass energies of
$380\text{\,}\mathrm{GeV}$, $1.5\text{\,}\mathrm{TeV}$ and
$3\text{\,}\mathrm{TeV}$, each stage lasting $7-8$ years [1, 2, 3]. CLIC’s
main goals are to measure the properties of the top quark and the Higgs boson
with high precision and to search for physics beyond the Standard Model.

In order to reach its design luminosity of $1.5-6\times 10^{34}$cm-2s-1, CLIC
will operate with very small bunch sizes [1, 2, 4]. CLIC’s bunch sizes lead to
strong beamstrahlung radiation from the electron and positron bunches in the
field of the opposite beam. The interactions involving beamstrahlung photons
result in two main types of background, incoherent
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
pairs and $\mathup{}\mathup{}\to\mathrm{hadron}$ events. Both backgrounds are
expected to affect the tracking performance in terms of detector occupancy.
The incoherent
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
pairs are mostly emitted in the forward direction and constrain the inner
detector radius to $\geq$31\text{\,}\mathrm{mm}$$ in the central region of the
detector. The $\mathup{}\mathup{}\to\mathrm{hadron}$ events extend to larger
angles with respect to the beam line, thus impacting the physics measurements.
At the $3\text{\,}\mathrm{TeV}$ stage, each bunch train contains 312 bunches,
separated by $0.5\text{\,}\mathrm{ns}$. The bunch trains are separated by
$20\text{\,}\mathrm{ms}$.

An accurate vertex reconstruction is needed to obtain an accurate and
efficient identification of secondary vertices from heavy-quark flavour
hadrons and tau-leptons. This requirement translates to a transverse impact-
parameter resolution at the level of
$\sigma_{d_{0}}^{2}=($5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$)^{2}+($15\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}\mathrm{GeV}$)^{2}/(p^{2}\sin^{3}\theta)$ and thus to a single
point resolution of $3~{}\upmu\mathrm{m}$ for the vertex detector, the
innermost detector of CLICdet [4, 5]. The vertex detector is made of $25\times
25~{}$\upmu\mathrm{m}$^{2}$ pixels which are arranged in three cylindrical
double-layers in the central region of CLICdet and in double-layer petals in a
spiral arrangement in the forward direction allowing for efficient air cooling
of the entire vertex detector.

In terms of track reconstruction, the scientific goals of CLIC place also a
demanding requirement on the transverse momentum resolution for high-momentum
tracks in the central detector region, which must be at the level of
$\sigma_{p_{\text{T}}}/p_{\text{T}}^{2}\leq$2\text{\times}{10}^{-5}\text{\,}{\mathrm{GeV}}^{-1}$$.
Optimization studies show that a tracker with a radius of
$1.5\text{\,}\mathrm{m}$ immersed in a magnetic field of 4 T with a single
point resolution smaller than $7~{}\upmu\mathrm{m}$ will achieve the required
resolution [4]. Therefore, a tracking detector made of silicon elongated pixel
layers is placed around the vertex detector.

To fulfil the stringent requirements from the physics programme, the material
budget of the CLICdet tracker must be limited to $0.2\%$ radiation length
($X_{0}$) per single layer in the vertex detector and $1-2\%X_{0}$ per layer
in the tracker. This ultra-light tracker with an extremely precise hit-
position resolution is placed in the innermost part of CLICdet as shown in
Figure 1(a). Its total amount of material expressed in term of radiation
length $X_{0}$ is shown in 1(b) and corresponds to about $4\%$ nuclear
interaction lengths in the central detector region.

(a)

(b)

Figure 1: (a) Tracking system layout in CLICdet [4]. The vertex-detector
layers are shown in orange, while the main tracker layers are depicted in
yellow. The tracker is surrounded by the electromagnetic calorimeter, depicted
in bright green. (b) The total thickness $x$ of the tracker material as a
function of the polar angle and averaged over azimuthal angles, expressed in
units of radiation length $X_{0}$ [6].

##  3 Event simulation and reconstruction

The performance of the track reconstruction algorithm used in CLICdet is
estimated with full simulation studies. The full tracker geometry including
the support material, cables and cooling is described with the DD4hep software
[7] and simulated in Geant4 [8] and includes a homogeneous solenoid field of
$4\text{\,}\mathrm{T}$.

To assess the tracking performance, two types of events are simulated and
reconstructed. The first sample contains single muons and possible secondary
particles produced by interaction with material. The second sample represents
a more complex scenario where a jet topology event, an
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
$\rightarrow$ $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ event at
$3\text{\,}\mathrm{TeV}$ centre-of-mass energy, is generated with WHIZARD [9]
and the $\mathup{}\mathup{}\to\mathrm{hadron}$ background expected at the
$3\text{\,}\mathrm{TeV}$ CLIC energy stage is superimposed [10]. The
$\mathup{}\mathup{}\to\mathrm{hadron}$ background overlaid corresponds to a
total of 30 bunch crossings (BX), 10 bunch crossings before and 20 bunch
crossings after the physics event. The
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ event at
$3\text{\,}\mathrm{TeV}$ produces on average about $90$ tracks, which
increases to about $550$ tracks when the
$\mathup{}\mathup{}\to\mathrm{hadron}$ background is overlaid.

The CLIC track reconstruction software is implemented in the linear collider
Marlin framework [11] and interfaced with the geometry using DD4hep [12].
Large simulation and reconstruction samples were produced with the iLCDirac
grid production system [13].

##  4 Track reconstruction

The main goal of the track reconstruction process is to estimate the momentum
and position parameters of charged particles crossing the detector in a
magnetic field. The first step of the track reconstruction is to create
_reconstructed_ hits. In CLICdet, the local position of the _simulated_ hits
from charged particles crossing the sensitive layers of the detector are
smeared with a Gaussian distribution with a width equal to the single point
resolution of the subdetector. This procedure is done for all simulated hits.
The reconstructed hits are used as input for the subsequent steps of the track
reconstruction.

The CLIC track reconstruction can be divided in three main blocks: the track
finding with the conformal tracking algorithm, the track fitting using a
Kalman filter and smoother, and the track selection. The final output of the
track reconstruction are tracks which include not only the hits they are made
of, but also the information of the estimated track parameters.

(a)  (b)

Figure 2: Hits produced by three muon tracks: prompt (blue), prompt with low
momentum (green) and non-prompt (red). Tracks are shown in the $(x,y)$ global
coordinate system (a) and in the $(u,v)$ conformal coordinate system (b).

###  4.1 Track finding with the conformal tracking method

In the conformal algorithm, point coordinates in global space $(x,y)$ are
translated into the conformal space $(u,v)$. The idea behind this coordinate
transformation is that circles passing through the origin of a cartesian
coordinate system $(x,y)$, can be transformed into straight lines in a
different coordinate system where the coordinates are defined as:

$u=\frac{x}{x^{2}+y^{2}},\qquad v=\frac{y}{x^{2}+y^{2}}.$ (1)

This is demonstrated in [14] and is valid if the circle is passing through the
origin. In the conformal space $(u,v)$, the circle equation
$(x-a)^{2}+(y-b)^{2}=r^{2}$ is translated into a straight line

$v=-\frac{a}{b}u+\frac{1}{2b}.$ (2)

Through the application of the conformal algorithm, curved tracks left by
charged particles bent by a magnetic field can be reduced to a straight lines
search. The radial order of the hit positions is inverted in the conformal
space with respect to the global space: track hits on the vertex detector
layers correspond to larger $(u,v)$ coordinates than hits on the tracker
layers.

In real measurements not all particle trajectories can be approximated by a
straight line due to possible deviations caused by particles undergoing
multiple scattering or simply not originated at the origin of the $(x,y)$
plane, also known as _displaced_ or _non-prompt_ particles. An example of each
of these three cases is given in Figure 2 where hits belonging to three muon
tracks are simulated using CLICdet. To take these deviations systematically
into account, pattern recognition in conformal space is performed via a
cellular automaton (CA) [15].

In the CA, cells are defined as segments connecting two hits and the algorithm
is based on the creation and extension of these cells. The role of the CA in
the conformal tracking is two-fold: firstly, it builds the so-called cellular
track candidates and then, these cellular track candidates are extended to
create full tracks and only those with the highest quality are selected.

####  4.1.1 Building of cellular track candidates

The hit collection given as input to the building step is referred to as the
_seeding collection_ and each hit in the collection is considered as a _seed
hit_ , the starting point of a cellular track candidate. From each seed hit,
one or more _seed cells_ are created connecting the seed hit with its nearest
neighbours. The nearest neighbour is defined as a hit which is within a
certain angular region and distance from the seed hit, does not lie on the
same detector layer, is located at a smaller conformal radius and is not
already part of a track. Each seed cell is then extended to _virtual hits_ ,
i.e. hits sitting on the prolongation in the radial direction of the cell,
from which the search for nearest neighbours is repeated. New cells are then
created and attached to the seed cell if the neighbouring hits do not lie on
the same detector layer as the end point of the seed cell, have smaller
conformal radius and had not been already included in a track. The extension
of the new cell is then repeated and more cells are created. Each cell
contains information about its start and end point as well as a _weight_
information. The weight of a cell indicates how many other cells are further
connected to it. Each subsequent link increments the cell weight by one unit,
such that the higher the weight, the higher the potential of the cell to make
a track.

Cellular track candidates are chains of cells, created by all cells compatible
within a certain angle window starting from the highest weighted seed cell. A
minimum number of hits for each candidate is required. Among all the cellular
track candidates only those with the best quality are kept. This step is
called _clone treatement_ and it assesses the candidate quality using the
length of the track and the two $\chi^{2}$ independently computed using a
linear regression fit on the conformal space and on the $(s,z)$ plane, where
$s$ is the coordinate along the helix arc segment. Longer tracks are
preferred, while the one with the best $\chi^{2}$ is chosen in case of equal
length. A further attempt to recover rejected cellular tracks is made by
removing, one by one, each hit on the track, refitting and recomputing the
normalised $\chi^{2}$. This allows one to keep good tracks that contain a
spurious hit. At the end of the process, all hits belonging to the created
cellular tracks are marked as _used_.

####  4.1.2 Extension of cellular track candidates

The hit collection given as input of this CA step is used to extend the
cellular track candidates created at the end of the building algorithm.
Cellular track candidates are extended in different ways according to their
particle transverse momentum estimated using the parameters extracted from the
linear regression fit in the $(s,z)$ space. All candidates with an estimated
$p_{\text{T}}$ above a threshold defined as external parameter are extended
first, as they are easier to reconstruct.

In this case, the extension proceeds in a similar manner as described for the
building step: end points of the previously formed cellular tracks are used as
seed hits and a search for nearest neighbours in polar angle is performed. The
search is performed limiting the compatible hits in the successive detector
layers. New cellular tracks are created including every valid neighbour. Among
those, the best track candidate is chosen based on the $\chi^{2}$ values from
the linear regression fit in $(u,v)$ and $(s,z)$. If no hit is found on one
layer, the cellular automaton proceeds on the subsequent layer. The extension
process is repeated until the last detector layer.

The extension algorithm then is performed for cellular track candidates whose
estimated $p_{\text{T}}$ is lower than the threshold. To allow the
reconstruction of very low-$p_{\text{T}}$ tracks, such as particles looping in
the tracker, no specific nearest neighbour search is performed but all unused
hits which are not located on the other side of the detector in $z$ are
considered for track extension. In this part, the search for compatible hits
is done including a quadratic term in the regression formula, in order to take
into account deviations from a straight line. At the end of the extension, all
CA track candidates are sorted by $p_{\text{T}}$ and the hits included in the
conformal tracks are marked as _used_.

####  4.1.3 Full conformal tracking chain

The two main CA algorithms are run on all reconstructed hits in several
iterations with different sets of parameters. The full chain used in the CLIC
track reconstruction is shown in Table 1. The first part aims to reconstruct
the prompt tracks in the vertex detector: step 0 builds cellular track
candidates using the hits in the vertex barrel as input, and they are extended
in step 1 using the hits in the vertex endcap. The remaining hits in the
combined vertex barrel and endcap detectors are then used to recover prompt
tracks more difficult to find. This is done in two steps using tighter cuts
first (step 2) and looser cuts afterwards (step 3). All cellular track
candidates reconstructed in steps 2 and 3 are then extended to the hits
belonging to the tracker detectors in step 4. The last part of the full chain
focuses on the reconstruction of non-prompt particles (step 5) using the whole
tracking system. During this step the CA is run from the outermost layer of
the tracker back to the innermost layer of the vertex detector. Moreover, a
quadratic term is included in the regression formula used to fit the cellular
track candidates in conformal space. An example of all cellular track
candidates reconstructed using the conformal tracking algorithm is given in
Figure 3.

Table 1: Overview of the configuration for the different steps of the pattern recognition chain. The last column shows some of the parameters of relevance for the cellular automaton as used for CLICdet: the maximum angle between cells, the maximum cell length, the minimum number of hits on a track, the maximum $\chi^{2}$ for valid track candidates, and the $p_{\text{T}}$ threshold used to discriminate between the two variations of the algorithm of track extension. Step | Algorithm | Hit collection | Parameters
---|---|---|---
max cell | max cell | min | $\chi^{2}$ |  $p_{\text{T}}$ threshold
angle [$\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$] | length [$\text{\,}\mathrm{mm}$-1] | $N_{\textnormal{hits}}$ | cut | [$\text{\,}\mathrm{GeV}$]
0 | Building | Vertex Barrel | 0.005 | 0.02 | 4 | 100 | -
1 | Extension | Vertex Endcap | 0.005 | 0.02 | 4 | 100 | 10
2 | Building | Vertex | 0.025 | 0.02 | 4 | 100 | -
3 | Building | Vertex | 0.05 | 0.02 | 4 | 2000 | -
4 | Extension | Tracker | 0.05 | 0.02 | 4 | 2000 | 1
5 | Building | Vertex & Tracker | 0.05 | 0.015 | 5 | 1000 | -

###  4.2 Track fitting and selection

The track fit in CLICdet consists of a Kalman filter and smoother in the
global $(x,y)$ coordinate space [16]. The fit is initialised with the
parameters obtained by a simple helix fit to three hits of the track –
typically first, middle, and last hit. The Kalman filter and smoother is then
run as implemented in the KalLib package described in [17]. If the fit fails,
the Kalman filter and smoothing procedure is tried again backwards.

In the track selection, some cuts are applied to the track candidates to
filter out the tracks that contain many spurious hits or share a number of
hits with other tracks. In order to do that, the clone treatment described in
Section 4.1.1 is performed on the fitted track collection. The $\chi^{2}$ used
to determine the highest quality track is the one calculated at the end of the
Kalman filter and smoother procedure. Moreover, only tracks with a number of
hits larger or equal to three is kept to maximise the track efficiency.

(a)  (b)

Figure 3: Cellular track candidates reconstructed for a single
$\mathup{{{b}}}\mathup{{\overline{{\mathup{{{b}}}}}}}$ event at
$500\text{\,}\mathrm{GeV}$ centre-of-mass energy (a) and for a single
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ event at
$3\text{\,}\mathrm{TeV}$ (b).

##  5 Performance

In this section, the performance of the track reconstruction for the CLICdet
experiment is investigated for simulated isolated muons and realistic
topologies in a high occupancy environment.

The tracking efficiency is defined as the ratio of _pure_ tracks to the number
of reconstructable particles, i.e. simulated particles which are stable, and
have $p_{\text{T}}>$100\text{\,}\mathrm{MeV}$$, $|\cos(\theta)|<0.99$, and at
least 4 hits on different layers. Pure tracks are defined as reconstructed
tracks which have at least 75% of their hits associated to a single
reconstructable particle. The efficiency is presented as a function of various
parameters of the simulated particles.

The fake rate is defined as the ratio of _fake_ tracks to the total number of
reconstructed tracks. Fake tracks are defined as reconstructed tracks which
have less then 75% of their hits associated to one reconstructable particle.
The fake rate is presented as a function of various parameters of the
reconstructed track.

The resolution for a specific track parameter is estimated from the track
parameter residuals, defined as the difference between the reconstructed-track
parameter and the simulated-particle parameter. The distribution of residuals
is fitted with a Gaussian function, whose standard deviation defines the
resolution.

###  5.1 Results for isolated muons

The tracking efficiency for single isolated muons is shown in Figure 4(a) as a
function of the polar angle of the simulated particle. The tracking is fully
efficient in the entire tracker acceptance and for any transverse momentum
[6], while the fake rate is negligible.

The resolutions of the following track parameters were studied using simulated
isolated muons [6]: transverse and longitudinal impact parameter, azimuthal
and polar angle, and transverse momentum. High-momentum tracks fulfil the
requirements on the impact parameter and transverse momentum set by the
physics programme at CLIC as described in Section 2. As expected, all track
parameter resolutions deteriorate in the forward region and for low-momentum
tracks due to multiple scattering. Moreover, for high-momentum tracks
resolutions of the impact parameters achieve the limit foreseen by the
assumption on the single point resolution in the vertex detector. As an
example, Figure 4(b) shows the resolution of the transverse momentum parameter
as a function of the $p_{\text{T}}$ of the simulated muon produced at $\theta$
= $None$, $None$, $None$, $None$ and $None$.

(a)  (b)

Figure 4: (a) Track reconstruction efficiency as a function of polar angle
$\theta$ for $p_{\text{T}}$ = $1\text{\,}\mathrm{GeV}$,
$10\text{\,}\mathrm{GeV}$, and $100\text{\,}\mathrm{GeV}$ for isolated muons
reconstructed with the CLICdet tracker. (b) Transverse momentum resolution for
isolated muons with polar angle $\theta$ = $None$, $None$, $None$, $None$ and
$None$ as a function of $p_{\text{T}}$.

###  5.2 Results for complex events

The robustness of the track reconstruction at CLIC is estimated using more
complex events in a realistic CLIC environment. Simulated
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
$\rightarrow$ $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events at
$3\text{\,}\mathrm{TeV}$ centre-of-mass energy with and without the
$\mathup{}\mathup{}\to\mathrm{hadron}$ background expected at the
$3\text{\,}\mathrm{TeV}$ energy stage of CLIC are used for this study.

In Figure 5 the efficiency and fake rate are shown as a function of the
$p_{\text{T}}$ of the simulated particle and of the reconstructed track,
respectively. The tracking is fully efficient for simulated particles with
$p_{\text{T}}>$1\text{\,}\mathrm{GeV}$$ and the efficiency is still well above
90% down to a $p_{\text{T}}$ approximately of $200\text{\,}\mathrm{MeV}$. Both
efficiency and fake rate are almost unaffected by the
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background expected at the
$3\text{\,}\mathrm{TeV}$ energy stage of CLIC.

(a)  (b)

Figure 5: Tracking efficiency (a) and fake rate (b) as a function of the
$p_{\text{T}}$ of the simulated particle and of the reconstructed track,
respectively.
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
$\rightarrow$ $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events
simulated at $3\text{\,}\mathrm{TeV}$ centre-of-mass energy are used with and
without the corresponding overlay of 30 BX of
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background expected at the
$3\text{\,}\mathrm{TeV}$ energy stage of CLIC.

Results were also studied as a function of the smallest distance between the
particle associated to the track and any other particle and as a function of
particle production vertex radius. The efficiency plots are shown in Figure 6.
The maximum efficiency loss for tracks produced in the vertex detector amounts
to only 1% for $\Delta_{\textnormal{MC}}<$0.02\text{\,}\mathrm{rad}$$. The
tracking is fully efficient provided that the particles are produced within
the vertex detector.

Thanks to the conformal tracking modularity, the track reconstruction in CLIC
was successfully adapted also to the CLD detector at FCC-ee which not only has
a different sub-detector design and magnetic field magnitude but also includes
different background conditions [18, 19].

(a)  (b)

Figure 6: Tracking efficiency as a function of particle proximity (a) and of
the particle production vertex radius (b).
$\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{+}}}\mathup{{{\mathup{{{e}}}}}^{\scriptstyle{-}}}$
$\rightarrow$ $\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events
simulated at $3\text{\,}\mathrm{TeV}$ centre-of-mass energy are used with and
without the corresponding overlay of 30 BX of
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background expected at the
$3\text{\,}\mathrm{TeV}$ energy stage of CLIC.

###  5.3 CPU execution time

Like in many other particle physics experiments, track reconstruction is one
of the most time consuming parts of the event reconstruction. A first
assessment was performed using the mean CPU time of ten
$\mathup{{{t}}}\mathup{{\overline{{\mathup{{{t}}}}}}}$ events at
$3\text{\,}\mathrm{TeV}$ centre-of-mass energy without and with the overlay of
$\mathup{}\mathup{}\to\mathrm{hadrons}$ background expected at the
$3\text{\,}\mathrm{TeV}$ energy stage of CLIC. If the background is not
included, most of the CPU time is spent in the fitting procedure, otherwise
the last CA building step (step 5) is the most time consuming part.

##  6 Conclusions

The conformal tracking presented in this paper is a new pattern recognition
technique for track finding in the context of future electron–positron
colliders. It uses a cellular automaton algorithm to find efficiently prompt
and non-prompt tracks in conformal space and it has been demonstrated that it
reconstructs tracks successfully also in the most challenging
low-$p_{\text{T}}$ regime. Its robustness against the beam-induced backgrounds
was proven using more complex events for the highest energy stage of CLIC.
Moreover, some preliminary studies in terms of CPU computing time were also
presented. Its use in the Particle Flow Analysis foreseen for particle
reconstruction at future electron–positron colliders is essential [6] and will
trigger further improvements in the near future.

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Variation of the fine structure constant
in the light of recent helium abundance measurement

Osamu Seto$\,{}^{1}$, Tomo Takahashi$\,{}^{2}$ and Yo Toda$\,{}^{1}$

1Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
2Department of Physics, Saga University, Saga 840-8502, Japan

We point out that the recent result of primordial helium-4 (4He) abundance
measurement by EMPRESS, which has reported a smaller 4He abundance than other
measurements, can be well fitted by assuming a time-variation of the fine
structure constant $\alpha$ which is slightly smaller than the present value
during big bang nucleosynthesis (BBN). We find that the EMPRESS result in
combination with deuterium abundance measurement indicates
$-2.6\%<\Delta\alpha/\alpha<-1.4\%$ (68% C.L.) where $\Delta\alpha$ is the
difference between the values of $\alpha$ at the BBN and present epochs, while
$-1.2\%<\Delta\alpha/\alpha<0.4\%$ (68% C.L.) is obtained from other previous
4He abundance data. We also investigate its effects in the framework where the
effective number of neutrino species and the lepton asymmetry, which are other
typical interpretations of the EMPRESS result, are allowed to vary. Once a
smaller $\alpha$ is adopted, the EMPRESS result can be explained without
assuming any non-standard values for the effective number of neutrino species
and lepton asymmetry.

EPHOU-23-010

##  1 Introduction

Time variation of the fundamental constants such as the fine structure
constant $\alpha$, the electron mass, the gravitational constant can arise in
various theories such as superstring theories, scalar-tensor theories, models
with extra dimension (for reviews, see, e.g., [1, 2, 3]). The variation of
such fundamental constants at various cosmic epochs have been probed with a
wide range of observations such as quasar absorption spectra [4, 5, 6, 7],
cosmic microwave background (CMB) [8, 9, 10], big bang nucleosynethesis (BBN)
[11, 12, 13, 14, 15, 16], the dating of meteorites [17, 18, 19], most of which
put severe constraints on them [1, 2, 3]. Some studies have obtained non-null
results for the variation of some constants in the past from the natural
reactor such as at Oklo [20] and quasar absorption spectra [5, 21], those have
been debated for example in Refs. [22, 23] nevertheless. Interestingly, the
time variation of some fundamental constants has recently been suggested as a
possible solution for the so-called Hubble tension ($H_{0}$ tension), which is
the issue of inconsistency (at almost $5\sigma$ level) between the values of
the Hubble constant $H_{0}$ obtained from local direct measurements such as
the Cepheid calibrated supernova distance ladder [24] and from indirect
measurements such as CMB from Planck satellite [25] (see reviews [26, 27] for
the current status of the tension). For instance, a model with time-varying
electron mass has been proposed to reduce the tension significantly [28] (see
also Refs. [29, 30, 31, 32, 33])#1#1#1 It may be worth noting that other
cosmological aspects can also be affected such as cosmological constraints on
neutrino masses [34] in time-varying electron mass model.  and a rapid
transition of the gravitational constant also has been suggested to resolve
the $H_{0}$ tension [35].

Recently, another cosmological tension seems to appear in the primordial
helium-4 abundance $Y_{p}$, in which the value of $Y_{p}$ reported by EMPRESS,
$Y_{p}=0.2370^{+0.0033}_{-0.0034}$ [36], is somewhat smaller than that
obtained in other recent observations [37, 38, 39]. Intriguingly, the combined
analysis of the value of $Y_{p}$ by EMPRESS and the deuterium (D) abundance
$D_{p}$ [40] indicates the effective number of neutrino species $N_{\rm eff}$
deviated from the standard value of $N_{\rm eff}=3.046$#2#2#2 For recent
precise calculations of $N_{\rm eff}$ for the standard case, see [41, 42, 43,
44, 45].  and also a non-zero lepton asymmetry, characterized by the electron
neutrino chemical potential or the degeneracy parameter $\xi_{e}$ [36]. The
EMPRESS result has stimulated several works for its interpretations and
implications of lepton asymmetry [46, 47, 48, 49], modified gravity [50],
early dark energy [51], and so on.

In this paper, we argue that the time variation of $\alpha$ can explain the
EMPRESS $Y_{p}$ result without resorting to non-standard values of $N_{\rm
eff}$ and $\xi_{e}$. To this end, we investigate the light element abundances
such as 4He and D and perform the statistical analysis by using the EMPRESS
$Y_{p}$ result along with the data of D abundance in Ref. [40]#3#3#3 The
compatibility of the time-varying $\alpha$ model with other recent BBN
observations are discussed [52, 53]. . Indeed, as we will show, $\alpha$ a few
percent smaller than the present value is preferred by the EMPRESS result and
in such a case, the standard values of $N_{\rm eff}=3.046$ and $\xi_{e}=0$ are
well within the allowed region.

The structure of this paper is as follows. In the next section, we summarize
how the time variation of $\alpha$ is implemented in the BBN calculation to
predict the light abundances and the method of our statistical analysis. In
the Section 3, we present our results: the time variation of $\alpha$ is
indeed preferred by the recent EMPRESS $Y_{p}$ result and also we do not need
to assume any other non-standard cosmological scenario. We conclude this paper
in the final section.

##  2 Setup and analysis method

In this section, we summarize the methodology of our analysis. First, we
describe how we implement the time-varying fine structure constant to
calculate the abundances of light elements. The description of our statistical
analysis methods follows.

###  2.1 Effects of varying fine structure constants

We apply the following modifications to implement the effects of varying fine
structure constant. For the details of what aspects of BBN calculation are
modified due to the change of $\alpha$, see e.g., Refs. [11, 15, 12, 16].

#### Weak reaction rate correction due to the mass difference variation

Increasing $\alpha$ decreases the mass difference between neutrons and protons
($Q\equiv m_{n}-m_{p}$). The response of the variation of $\alpha$ on $Q$ is
given by [54]

$Q=\left(-0.76\frac{\alpha}{\alpha_{0}}+2.05\right)\,\mathrm{MeV},$ ( 2.1)

with the present value of the fine structure constant $\alpha_{0}\simeq
1/137.036$.

The change of $Q$ is implemented in the weak reaction rate in the BBN era as

$\Gamma(n\rightarrow
p^{+}e^{-}\nu_{e})_{\mathrm{BBN}}=\Gamma_{n0}\frac{\lambda_{n\rightarrow
p^{+}}(Q)}{\lambda_{n\rightarrow p^{+}}(Q_{0})},$ ( 2.2)

where $\Gamma_{n0}=1/\tau_{n0}\simeq 1/(879.4~{}{\rm s})$ is the present
values of the neutron decay rate, $Q_{0}\simeq 1.29~{}\mathrm{MeV}$ is the
present value of the mass difference and $\lambda$ is the phase space integral
[55]

$\lambda_{n\rightarrow
p^{+}}(Q)\propto\left(\int_{-\infty}^{-m_{e}-Q}\frac{x^{2}(Q+x)^{2}\sqrt{1-\frac{m_{e}^{2}}{(Q+x)^{2}}}}{\left(e^{\frac{x}{T_{\nu}}}+1\right)\left(e^{-\frac{Q+x}{T}}+1\right)}dx+\int_{m_{e}-Q}^{\infty}\frac{x^{2}(Q+x)^{2}\sqrt{1-\frac{m_{e}^{2}}{(Q+x)^{2}}}}{\left(e^{\frac{x}{T_{\nu}}}+1\right)\left(e^{-\frac{Q+x}{T}}+1\right)}dx\right),$
( 2.3)

with $T_{\nu}$ being the temperature of neutrinos, and a similar expression
holds for the $p^{+}\rightarrow n$ weak interaction rate.

Although we assume that $\alpha$ does not change during the BBN epoch, it
should eventually evolve to the present value at some point in the course of
the evolution of the Universe. We have implemented the above change in
PArthENoPE [56, 57, 58], which we use to calculate the abundances of light
elements such as D and 4He.

#### Reaction rates corrections

Effects of the time-variation of $\alpha$ also appears in the Coulomb barrier
penetrability. For charged-particle reactions, the expression of the cross
section $\sigma(E)$ is given by

$\sigma(E)=\frac{S(E)}{E}\exp(-2\pi\alpha Z_{i}Z_{j}\sqrt{\mu/2E}),$ ( 2.4)

where $S(E)$ is the astrophysical $S$-factor, $\mu$ is the reduced mass, and
$Z_{i,j}$ are the atomic number of the nuclei. Since a smaller $\alpha$
increases the cross section, the nuclear reactions proceed more quickly.
Following Ref. [11], the $\alpha$ dependence of the reaction rates are
included by modifying PArthENoPE3.0.

#### Other corrections

We have also taken account of other minor effects [15]: final-state Coulomb
interaction corrections, radiative capture corrections, mass corrections due
to the electromagnetic contributions $\Delta M_{EM}$, and radiative capture
corrections.

#### Results

In Fig. 1, we show density plots of $D_{p}$ (left) and $Y_{p}$ (right) in the
$\eta$ – $\Delta\alpha/\alpha_{0}$ plane where $\eta$ is the baryon-to-photon
ratio and $\Delta\alpha=\alpha-\alpha_{0}$. Regions between red lines in the
left and right panels correspond to 1$\sigma$ allowed region from measurements
of $D_{p}$ [40] and the EMPRESS $Y_{p}$ [36], respectively. For comparison, in
the right panel, another recent helium abundance measurement from Aver et al.
[39] is also shown as region between green dashed lines. As is shown in the
figure, as the structure constant $\alpha$ decreases, the abundance of 4He
become smaller. Therefore the value of $Y_{p}$ reported by EMPRESS can be
explained by a smaller structure constant. The abundance of D is also affected
by the change of $\alpha$, however, its effect is canceled by the change of
$\eta$ as we explain below.

Figure 1:  Density plots of $D_{p}$ (left) and $Y_{p}$ (right) in the $\eta$ –
$\Delta\alpha/\alpha_{0}$ plane. In this figure, other parameters are fixed as
$\Delta N_{{\rm eff}}=0$ (the deviation from the standard value of $N_{\rm
eff}$) and $\xi_{e}=0$. Regions between red lines in the left and right panels
correspond to 1$\sigma$ allowed region from measurements of $D_{p}$ [40] and
the EMPRESS $Y_{p}$ [36], respectively. Regions between green dashed lines in
the right panel correspond to 1$\sigma$ allowed region from another $Y_{p}$
abundance measurement by Aver et al. [39].

The response of $Y_{p}$ against the change of $\alpha$ can be understood by
noticing that the mass difference $Q$ decreases by increasing $\alpha$ and the
neutron decay rate becomes larger (the neutron lifetime becomes shorter),
which makes the abundances of 4He larger. These behaviors in response to the
variation of the parameters, denoted as $\Delta\alpha$ and $\Delta\eta$, can
be understood by the following scaling relation [59]:

$\displaystyle\frac{\Delta D_{p}}{D_{p}}$ $\displaystyle=$ $\displaystyle
3.6\frac{\Delta\alpha}{\alpha}-1.6\frac{\Delta\eta}{\eta},$ ( 2.5)
$\displaystyle\frac{\Delta Y_{p}}{Y_{p}}$ $\displaystyle=$ $\displaystyle
1.9\frac{\Delta\alpha}{\alpha}+0.04\frac{\Delta\eta}{\eta}.$ ( 2.6)

The $\alpha$ dependence of $D_{p}$ can be somewhat canceled by the change of
$\eta$. This behavior indeed can be seen in the left panel of Fig. 1. On the
other hand, the 4He abundance is not so sensitive with respect to the
variation of $\eta$, and hence $\alpha$ almost controls the value of $Y_{p}$.
This also can be observed in the right panel of Fig. 1.

###  2.2 Analysis method

In the following, we investigate the impact of the EMPRESS result of $Y_{p}$
on models beyond the standard assumption, particularly focusing on the time-
varying $\alpha$. Since it has been discussed that the EMPRESS result may
suggest non-standard values of the effective number of neutrino $N_{{\rm
eff}}$ and non-zero lepton asymmetry, characterized by the chemical potential
(the degeneracy parameter) of the electron neutrino $\xi_{e}$ [36], we also
vary $N_{{\rm eff}}$ and $\xi_{e}$ in models with varying $\alpha$ in some
cases.

To study the preferred ranges for $\alpha,N_{{\rm eff}}$ and $\xi_{e}$, we
calculate the value of $\chi^{2}$ fitted to the observed values of $Y_{p}$ by
EMPRESS [36] and $D_{p}$ [40]. We also adopt the prior for the baryon density
derived from the CMB measurement of Planck [25]. The total value of $\chi^{2}$
is evaluated as

$\chi^{2}(\alpha,\eta,\xi_{e},N_{{\rm
eff}})=\chi_{Y_{p}}^{2}(\alpha,\eta,\xi_{e},N_{{\rm
eff}})+\chi_{D_{p}}^{2}(\alpha,\eta,\xi_{e},N_{{\rm
eff}})+\chi_{\eta}^{2}(\eta)\,,$ ( 2.7)

where $\chi_{Y_{p}}^{2}$ and $\chi_{D_{p}}^{2}$ are the one calculated from
the fit to the observed $Y_{p}$ by EMPRESS [36] and the measurement of $D_{p}$
[40], which are respectively given as

$\displaystyle\chi_{Y_{p}}^{2}(\alpha,\eta,\xi_{e},N_{{\rm
eff}})=\frac{(Y_{p,{\rm obs}}-Y_{p}^{{\rm(th)}}(\alpha,\eta,\xi_{e},N_{{\rm
eff}}))^{2}}{\sigma_{Y_{p},{\rm obs}}^{2}+\sigma_{Y_{p},{\rm th}}^{2}}\,,$ (
2.8) $\displaystyle\chi_{D_{p}}^{2}(\alpha,\eta,\xi_{e},N_{{\rm
eff}})=\frac{(D_{p,{\rm obs}}-D_{p}^{{\rm(th)}}(\alpha,\eta,\xi_{e},N_{{\rm
eff}}))^{2}}{\sigma_{D_{p},{\rm obs}}^{2}+\sigma_{D_{p},{\rm th}}^{2}}\,,$ (
2.9)

where $Y_{p,{\rm obs}}=0.2370$ and $\sigma_{Y_{P},{\rm obs}}=0.00335$ are the
mean value and 1$\sigma$ error from the observation of $Y_{p}$ [36]. For
comparison, we also make an analysis using the measurement of $Y_{p}$ from
Aver et al. [39] $Y_{p}=0.2449\pm 0.040$ in some cases. For deuterium
abundance, we take $D_{p,{\rm obs}}=2.527\times 10^{-5}$ and
$\sigma_{D_{p,{\rm obs}}}=0.030\times 10^{-5}$ which are from Cooke et al.
[40]. $\sigma_{Y_{p},{\rm th}}=\sqrt{0.00003^{2}+0.00012^{2}}$ is the
theoretical uncertainty originating from the nuclear rates and the neutron
lifetime, respectively corresponding to the first and second terms in the
square root, and $\sigma_{D_{p},{\rm th}}=\sqrt{0.06^{2}\times 10^{-10}}$ is
the one for $D_{p}$ from the nuclear reaction rates [58].

$Y_{p}^{{\rm(th)}}(\alpha,\eta,\xi_{e},N_{{\rm eff}})$ and
$D_{p}^{{\rm(th)}}(\alpha,\eta,\xi_{e},N_{{\rm eff}})$ are theoretically
predicted values of $Y_{p}$ and $D_{p}$ calculated by using PArthENoPE [56,
57, 58] which we have modified to implement the variation of $\alpha$.
$\chi_{\eta}^{2}$ is to take account of the prior on the baryon density from
the CMB measurement of Planck, which is written as

$\chi_{\eta}^{2}(\eta)=\frac{(\eta_{{\rm
Planck}}-\eta)^{2}}{\sigma_{\eta_{{\rm Planck}}}^{2}},$ ( 2.10)

where $\eta_{{\rm Planck}}=6.082\times 10^{-10}$ and $\sigma_{\eta_{\rm
Planck}}=0.060\times 10^{-10}$, which is taken from the analysis for the case
of $\Lambda$CDM$+N_{\rm eff}+Y_{p}$ model [25]. In the following analysis, we
also vary the value of $N_{\rm eff}$ and $\xi_{e}$, and hence we take the
value of baryon density from such an analysis#4#4#4 Since the analysis varying
$\xi_{e}$ is not available in Ref. [25], we instead adopt the constraint on
the baryon density in the case of $\Lambda$CDM$+N_{\rm eff}+Y_{p}$ model, as
has also been done in Ref. [36]. . Since $N_{\rm eff}$ and $Y_{p}$ are
correlated with the baryon density in CMB (see, e.g., [25, 60, 61, 62]), the
uncertainty of $\eta$ is somewhat larger than that for the $\Lambda$CDM case.

We calculate $\chi^{2}$ given in Eq. (2.7) to obtain constraints on
$\alpha,N_{{\rm eff}},\xi_{e}$ and $\eta$. We mainly show our results in a two
dimensional plane of two of these parameters. We present constraints on a two
dimensional plane by marginalizing over the other parameter(s), which in
practice is done by determining (fixing) the redundant parameter to the one
minimizing the value of $\chi^{2}$. In some analyses, we use $\eta$ as one of
variables to show a two dimensional plane and we do not include
$\chi_{\eta}^{2}$ in the total $\chi^{2}$.

##  3 Results

Figure 2:  1$\sigma$ and 2$\sigma$ constraints in the
$\eta$–$\Delta\alpha/\alpha_{0}$ plane from the EMPRESS measurements of
$Y_{p}$ and $D_{p}$ (red shaded). We also show the case where we adopt the
$Y_{p}$ measurement by Aver et al. and $D_{p}$ (unshaded black contours). In
this figure, other parameters are fixed as $\Delta N_{{\rm eff}}=0$ and
$\xi_{e}=0$. In this figure, $\chi_{\eta}^{2}$ is not included in the total
$\chi^{2}$. Figure 3:  1$\sigma$ and 2$\sigma$ constraints in the
$\eta$–$N_{{\rm eff}}$ plane from the measurements of $Y_{p}$ and $D_{p}$ with
the electron mass being fixed as $\alpha/\alpha_{0}=0.98$ (red shaded). The
neutrino chemical potential is set to $\xi_{e}=0$. For reference, the
constraint for the standard case (i.e., $\alpha/\alpha_{0}=1$) is shown with
blue lines. In this figure, $\chi_{\eta}^{2}$ is not included in the total
$\chi^{2}$.

Now in this section, we present our results by showing constraints on the
parameters obtained by fitting to the data of $Y_{p}$ from EMPRESS [36] and
$D_{p}$ from Cooke et al. [40] using $\chi^{2}$ explained in the previous
section. First, we show the constraint shaded with red in the
$\eta$–$\Delta\alpha/\alpha_{0}$ plane in Fig. 2, in which we fix $\Delta
N_{{\rm eff}}$ and $\xi_{e}$ to be the standard values, i.e., $\Delta N_{{\rm
eff}}=0$ and $\xi_{e}=0$. For comparison, we also show the constraints for the
case of the $Y_{p}$ measurement by Aver et al. [39] with the same $D_{p}$ data
with unshaded black contours. The figure indicates that the best-fit value for
the case with EMPRESS+Cooke et al. is $\alpha/\alpha_{0}\simeq 0.98$, and
$\alpha/\alpha_{0}\simeq 1$ is excluded more than 2$\sigma$, which shows that
the fine structure constant should be decreased by 2 % during BBN epoch
compared to the present value when the EMPRESS $Y_{p}$ is adopted, on the
other hand, $Y_{p}$ from Aver et al. shows no such deviation. In any case, the
varying $\alpha$ scenario seems to be favored from the recent EMPRESS result
of $Y_{p}$. After marginalizing over $\eta$, the bound for $\alpha$ is
obtained as $-2.6\%<\Delta\alpha/\alpha_{0}<-1.4\%\,(1\sigma)$ (EMPRESS+Cooke
et al.) and $-1.2\%<\Delta\alpha/\alpha_{0}<0.4\%\,(1\sigma)$ (Aver et
al.+Cooke et al.).

Next, we show the constraint in the $\eta$–$N_{{\rm eff}}$ plane in Fig. 3.
Here we fix $\alpha/\alpha_{0}=0.98$ (red shaded region), which is motivated
from the result presented in Fig. 2, and $\alpha/\alpha_{0}=1$ (blue
contours), which is the standard case, for comparison. Indeed it has already
been pointed out that the EMPRESS $Y_{p}$ result favors $N_{{\rm eff}}<3.046$
when there is no lepton asymmetry (i.e., $\xi_{e}=0$), and the baryon density
is also deviated from that obtained by Planck [36], although the discrepancy
is not so significant. On the other hand, as seen from Fig. 3, when we assume
$\alpha/\alpha_{0}=0.98$, the standard value of $\Delta N_{{\rm eff}}=0$ is
well within 1$\sigma$ allowed region, and moreover, the baryon density can
also be consistent with that obtained from Planck ($\eta_{{\rm Planck}}\simeq
6.1\times 10^{-10}$). Therefore, the time-varying fine structure constant can
remove the tensions between the EMPRESS $Y_{p}$ result and the standard
cosmological model.

Figure 4:  1$\sigma$ and 2$\sigma$ constraints on the $\eta$–$N_{{\rm eff}}$
plane (top), the $N_{{\rm eff}}$–$\xi_{e}$ plane (bottom right) and the
$\eta$–$\xi_{e}$ plane (bottom left). Here we fix the structure constant as
$\alpha/\alpha_{0}=0.98$ (red shaded region) and $1$ (blue contours), but vary
3 other parameters: $N_{{\rm eff}},\xi_{e}$ and $\eta$. To project the
constraints on 2 dimensional plane, we have eliminated the redundant parameter
which is not shown in the plane by fixing it so that the value of $\chi^{2}$
is minimized for the parameter.

Figure 5:  Density plots of $\xi_{e}$ which minimizes $\chi^{2}$ in
$\eta$–$N_{{\rm eff}}$ plane for the cases with $\alpha/\alpha_{0}=1$ (left
panel) and $\alpha/\alpha_{0}=0.98$ (right panel). Ellipse surrounded by red
lines correspond to 1$\sigma$ allowed region in Fig. 4.

To see how the preference for small $\alpha$ by the EMPRESS value of $Y_{p}$
is affected by the existence of the lepton asymmetry characterized by
$\xi_{e}$, we also analyze the cases where $\xi_{e}$ is also allowed to vary
in addition to $N_{{\rm eff}}$ and $\eta$. In Fig. 4, we show constraints on
the $\eta$–$N_{{\rm eff}}$ plane (top panel), the $N_{{\rm eff}}$–$\xi_{e}$
plane (bottom right panel) and the $\eta$–$\xi_{e}$ plane (bottom left panel).
In the figure, the fine structure constant is fixed as
$\alpha/\alpha_{0}=0.98$ (red shaded region) and $1$ (blue contours), however
the other 3 parameters, $N_{{\rm eff}},\xi_{e}$ and $\eta$ are varied. Since
we show the constraint in a two dimensional plane, we need to project a three
dimensional constraint to the two dimensional one, which is done by fixing the
redundant parameter so that the value of $\chi^{2}$ is minimized for the
parameter. In the top panel of Fig. 4, the best-fit value of $N_{\rm eff}$ for
the case with $\alpha/\alpha_{0}=0.98$ is shifted slightly towards a higher
ones than the standard value compared to that for $\alpha/\alpha_{0}=1$.
However, notice that this does not indicate that a smaller $\alpha$ prefers a
slightly larger $N_{\rm eff}$. This is attributed to the fact that, in the
analysis with $\xi_{e}$ being varied as performed in the top panel, a
positively large value of $\xi_{e}$ is favored, which drives $N_{\rm eff}$ to
be close to the standard value in the case of $\alpha/\alpha_{0}=1$. To see
this in a clear manner, we show the density plot of $\xi_{e}$ which minimizes
the value of $\chi^{2}$ for a given $\eta$ and $N_{\rm eff}$ with
$\alpha/\alpha_{0}=1$ (left) and $0.98$ (right) in Fig. 5. For the case of
$\alpha/\alpha_{0}=1$, the allowed region lies around $N_{\rm eff}=3.046$ and
$\xi_{e}$ takes a positively non-zero value. On the other hand, when
$\alpha/\alpha_{0}=0.98$, the slightly larger value of $N_{\rm eff}$ is
preferred with vanishing $\xi_{e}$. In any case, when $\xi_{e}$ is varied,
$N_{\rm eff}=3.046$ is well allowed in either case of $\alpha/\alpha_{0}=0.98$
or $1$.

From the constraint in the $N_{{\rm eff}}$–$\xi_{e}$ plane shown in the bottom
right panel of Fig. 4, although a non-zero value of $\xi_{e}$ and a slightly
higher value of $N_{{\rm eff}}$ are preferred from the EMPRESS $Y_{p}$ for
$\alpha/\alpha_{0}=1$ [36], when $\alpha/\alpha_{0}=0.98$ is assumed, we do
not need extra radiation component nor lepton asymmetry: $N_{{\rm eff}}=3.046$
and $\xi_{e}=0$ are well within 1$\sigma$ allowed region.

This can also be seen from the bottom left panel in Fig. 4 where the
constraint on the $\eta$–$\xi_{e}$ plane is shown with $N_{{\rm eff}}$ being
marginalized. One can see that $\xi_{e}$ can be zero for the case of
$\alpha/\alpha_{0}=0.98$ (on the other hand, as already known [36], a non-zero
positive value of $\xi_{e}$ is preferred in the case of
$\alpha/\alpha_{0}=1$). Therefore we can conclude that the time-varying fine
structure constant which is 2% smaller than the present value can explain the
recent EMPRESS result on $Y_{p}$ without assuming non-standard $N_{\rm eff}$
and $\xi_{e}$.

##  4 Conclusion and discussion

In this paper, we have investigated the impact of the time variation of
$\alpha$ to the recent measurement of the primordial helium abundance by
EMPRESS [36]. In the standard $\Lambda$CDM+$N_{\rm eff}$ model, the
combination of the measurements of $Y_{p}$ by EMPRESS and $D_{p}$ [40] may
indicate that $N_{\rm eff}$ is smaller than 3.046 and the baryon density is in
a slight tension with the one obtained by Planck. However, as shown in Fig. 2,
by assuming the value of $\alpha$ to be $2\%$ smaller than the present value,
the standard assumption of $N_{\rm eff}=3.046$ can well fit the data and the
baryon density can also be consistent with the one obtained from Planck.

Actually, if one allows the lepton asymmetry characterized by $\xi_{e}$ to
vary, the EMPRESS result may indicate a non-zero lepton asymmetry and a
slightly larger value of $N_{\rm eff}$ than the standard one [36]. However, as
we have shown, just by assuming the time variation of $\alpha$ in the BBN
epoch, one can have a good fit to the EMPRESS result, along with the
measurement of D abundance [40], without any extra assumptions such as lepton
asymmetry and non-standard value for $N_{\rm eff}$.

Our analysis suggests that the recent measurement of $Y_{p}$ by EMPRESS may
indicate the time variation of the fine structure constant, which would gives
a significant implication on the fundamental physics, since it means that some
fundamental constant could change in time. Although we have assumed that only
the fine structure constant changes with time in this paper, in some theories,
other constants can simultaneously be altered. It would be interesting to
pursue such possibilities in the light of the EMPRESS $Y_{p}$ result, which is
left for the future work.

## Acknowledgements

We would like to thank Kazuhide Ichikawa for sharing his BBN code with the
variation of the fine structure constant. We are also grateful to Akinori
Matsumoto for the correspondence regarding the analysis in Ref [36]. We would
also thank Masahiro Kawasaki for a useful conversation. This work was
supported by JSPS KAKENHI Grant Number 19K03874 (TT), 19K03860, 19K03865,
23K03402 (OS), MEXT KAKENHI 23H04515 (TT), and JST SPRING, Grant Number
JPMJSP2119 (YT).

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