# Conductance asymmetry in proximitized magnetic
topological insulator junctions with Majorana modes

Daniele Di Miceli  Institute for Cross-Disciplinary Physics and Complex
Systems IFISC (CSIC-UIB), E-07122 Palma, Spain  Department of Physics and
Materials Science, University of Luxembourg, 1511 Luxembourg, Luxembourg
Eduárd Zsurka  Department of Physics and Materials Science, University of
Luxembourg, 1511 Luxembourg, Luxembourg  Peter Grünberg Institute (PGI-9),
Forschungszentrum Jülich, 52425 Jülich, Germany  JARA-Fundamentals of Future
Information Technology, Jülich-Aachen Research Alliance, Forschungszentrum
Jülich and RWTH Aachen University, Germany  Julian Legendre  Department of
Physics and Materials Science, University of Luxembourg, 1511 Luxembourg,
Luxembourg  Kristof Moors  Peter Grünberg Institute (PGI-9), Forschungszentrum
Jülich, 52425 Jülich, Germany  JARA-Fundamentals of Future Information
Technology, Jülich-Aachen Research Alliance, Forschungszentrum Jülich and RWTH
Aachen University, Germany  Thomas L. Schmidt  Department of Physics and
Materials Science, University of Luxembourg, 1511 Luxembourg, Luxembourg
School of Chemical and Physical Sciences, Victoria University of Wellington,
P.O. Box 600, Wellington 6140, New Zealand  Llorenç Serra  Institute for
Cross-Disciplinary Physics and Complex Systems IFISC (CSIC-UIB), E-07122
Palma, Spain  Department of Physics, University of the Balearic Islands,
E-07122 Palma, Spain

###### Abstract

We theoretically discuss electronic transport via Majorana states in magnetic
topological insulator-superconductor junctions with an asymmetric split of the
applied bias voltage. We study normal-superconductor-normal (NSN) junctions
made of narrow (wire-like) or wide (film-like) magnetic topological insulator
slabs with a central proximitized superconducting sector. The occurrence of
charge non-conserving Andreev processes entails a nonzero conductance related
to an electric current flowing to ground from the proximitized sector of the
NSN junction. We show that topologically-protected Majorana modes require an
antisymmetry of this conductance with respect to the point of equally split
bias voltage across the junction.

##  I Introduction

Majorana modes in solid-state physics are zero-energy quasiparticle
excitations with the unusual property of being their own antiparticles [1, 2],
which emerge in 1D and 2D topological superconductors (TSCs) near boundaries
and vortices [3]. These fascinating states can be distinguished into Majorana
chiral propagating states (MCPS) in two-dimensional superconducting phases [4,
5, 6], and zero-energy Majorana bound states (MBS) in spinless $p$-wave
superconducting wires [7]. The former are dispersive modes analogous to
quantum anomalous Hall and quantum spin Hall edge states in superconducting
materials [5, 6], while the latter are localized in-gap modes emerging at the
ends of gapped phases of 1D topological superconducting wires. Achieving
topological superconductivity is a crucial step toward the realization of non-
Abelian braiding statistics and fault-tolerant quantum computing [8, 9, 10].
Magnetic topological insulators [11], i.e., 3D topological insulators (TIs)
with topological surface states and ferromagnetic ordering, are promising
candidates for the realization of such robust platforms for quantum
computation, since in presence of proximity coupling to an ordinary $s$-wave
superconductor they are expected to realize different TSCs with either
propagating or localized Majorana modes [12, 13, 14, 15].

Despite the growing interest in proximitized MTIs [16], the experimental
detection of Majorana modes is still inconclusive [17, 18, 19, 20, 21]. In
this paper, we highlight a characteristic feature of Majorana states that can
be used in their detection. Through theoretical analysis and numerical
simulations, we find that both types of Majorana states lead to a peculiar
transport signature in NSN junctions between normal (N) and proximitized (S)
magnetic topological insulators, when the bias between the two N sections is
split asymmetrically with respect to the central S lead. Without Majorana
modes or trivial Andreev bound states (ABSs), which may be found in non-
topological 1D superconductors [22, 23], the electric currents flowing through
the N leads are equal and opposite, independently of how the bias is split
between left and right leads. In the presence of Andreev processes, instead,
the currents in the two N leads can be different, depending on the fraction of
bias applied to each side of the junction. When the electric currents in the N
leads have different intensities, charge conservation requires the existence
of a third current going to ground from the superconductor, defining a nonzero
differential conductance.

We show in this work that in presence of nontrivial MBSs or MCPSs, the NSN
conductance of an MTI slab must be _antisymmetric_ with respect to the
splitting of the bias. This reflects the existence of identical scattering
amplitudes at the two interfaces of the junction. Observing how the total
conductance varies with the bias splitting provides a selective criterion,
although not absolutely conclusive, to rule out electric signals coming from
trivial ABS in the proximitized MTI slabs, and constitutes a novel alternative
approach for transport measurements in NSN junctions. Monitoring the
conductance asymmetry with a continuous change of the bias split is more
selective than just observing zero-bias conductance peaks when the full bias
drop is applied in turn to each side of the junction. Furthermore, the
proposed symmetry analysis can be useful to discriminate between the two
different types of Majorana excitations which can be found in MTIs, and
provide an additional control parameter, i.e., the bias split, while
maintaining the correlation between the transport behaviour on the two
interfaces of the junction. Similar criteria to detect MBSs on the ends of
proximitized semiconducting wires have been discussed in recent works, with a
focus on multi-terminal transport measurements [24, 25, 26, 27, 28] and noise
correlations [29, 30]. Conductance matrix symmetries of particle-hole type
have also been investigated [31, 32, 33], discussing the inversion of a common
potential acting on all normal leads attached to the superconducting sector.
By contrast, the symmetry discussed in our work corresponds to the role of a
bias between the two normal leads of our three-terminal setup.

The paper is structured as follows. In Sec. II we discuss the model
Hamiltonian and the topological states in finite-size MTI slabs. In Sec. III
we compute the electric conductance in the NSN junction and discuss its
symmetry properties with bias splitting. In Sec. IV we show some numerical
results supporting our conclusions. Sec. V concludes the manuscript.

##  II Model Hamiltonian

To start, we consider the Hamiltonian of a 3D TI in presence of ferromagnetic
ordering. In the basis
$\phi_{k\sigma}^{\tau}=(c_{k\uparrow}^{+},c_{k\uparrow}^{-},c_{k\downarrow}^{+},c_{k\downarrow}^{-})^{T}$,
where $c_{k\sigma}^{\tau}\equiv c_{k\sigma}^{\tau}(y,z)$ annihilates an
electron with longitudinal wave number $k\equiv k_{x}$, spin
$\sigma=\uparrow,\downarrow$ and orbital index $\tau=\pm$, the effective 3D
Hamiltonian for magnetic TIs takes the following form [34, 35]

$\displaystyle\mathcal{H}_{0}(\mathbf{k})=\epsilon(\mathbf{k})+M(\mathbf{k})\tau_{z}+A(\mathbf{k})\tau_{x}+\Lambda\sigma_{z}\,,$
(1)

where

$\begin{split}\epsilon(\mathbf{k})&=\mu-
C_{\perp}\left(k_{x}^{2}+\hat{k}_{y}^{2}\right)-C_{z}\hat{k}_{z}^{2}\,,\\\
M(\mathbf{k})&=M_{0}-M_{\perp}\left(k_{x}^{2}+\hat{k}_{y}^{2}\right)-M_{z}\hat{k}_{z}^{2}\,,\\\
A(\mathbf{k})&=A_{\perp}\left(k_{x}\sigma_{x}+\sigma_{y}\hat{k}_{y}\right)+A_{z}\sigma_{z}\hat{k}_{z}\,.\end{split}$
(2)

Here, $\mathbf{k}=(k_{x},\hat{k}_{y},\hat{k}_{z})$ and the transverse momentum
operators are given by $\hat{k}_{y}=-i\hbar\partial_{y}$ and
$\hat{k}_{z}=-i\hbar\partial_{z}$. The Pauli matrices $\sigma_{i}$ and
$\tau_{i}$ ($i\in\\{x,y,z\\}$) act on the spin and orbital subspaces,
respectively, the magnetization along $z$ is represented by the Zeeman term
$\Lambda\sigma_{z}$, and $\mu$ is the chemical potential. This Hamiltonian is
suitable for describing TIs such as Bi2Se3, Bi2Te3 and Sb2Te3 through a proper
choice of parameters [34, 35]. In our simulations, we used the values given in
Ref. [36] for a topological insulator where the asymmetry between the
conduction and valence bands as well as the anisotropy of the Dirac cones have
been neglected. When placed in proximity to a superconductor, the system can
be described by the Bogoliubov-de Gennes (BdG) Hamiltonian [37]

$\mathcal{H}_{\text{BdG}}(\mathbf{k})=\begin{pmatrix}\mathcal{H}_{0}(\mathbf{k})&\Delta^{\star}\\\
\Delta&-\sigma_{y}\mathcal{H}_{0}^{\star}(-\mathbf{k})\sigma_{y}\end{pmatrix}\,,$
(3)

expressed in the basis [38]

$\Phi_{k\sigma}^{\tau}=\begin{pmatrix}c_{k\uparrow}^{+},c_{k\uparrow}^{-},c_{k\downarrow}^{+},c_{k\downarrow}^{-},-c_{-k\downarrow}^{+\dagger},-c_{-k\downarrow}^{-\dagger},c_{-k\uparrow}^{+\dagger},c_{-k\uparrow}^{-\dagger}\end{pmatrix}^{T}\,,$
(4)

where we assumed a local $s$-wave pairing with amplitude
$\Delta\equiv\Delta(y,z)$ induced by proximity. In the following, we fix the
thickness of the slab to $d=4$ nm and consider a wire-like geometry with width
$L_{y}=20$ nm and a film-like one with $L_{y}=160$ nm. For $d=4$ nm, the
surface states on opposite sides of the MTI slab are coupled [39, 40], and a
finite size gap opens up in the energy spectrum. The magnetization can induce
a gap inversion, yielding nontrivial topological states. We will assume a
constant pairing field along $y$, and model the proximity coupling on the
upper surface of the magnetic TI by

$\Delta(y,z)=\Delta\,\Theta(z-d/2)\,.$ (5)

where $\Theta$ is the Heaviside step function. An asymmetric pairing on the
top and bottom surfaces is indeed required to achieve topological
superconductivity in the MTI slab [13, 15].

All the numerical results below are obtained with $\Delta=5$ meV for the wire
and $\Delta=10$ meV for the film geometry. These values are unrealistically
large compared to experiments, but they are convenient for numerical
simulations and qualitatively similar results can be obtained for smaller
pairings and rescaled systems. Indeed, in a quasi-1D superconducting wire, the
decay length (along the longitudinal direction $x$) of Majorana end states is
inversely proportional to the pairing potential $\xi\propto 1/|\Delta|$. This
means that, in order to guarantee well-separated MBSs, a smaller pairing can
be compensated by a greater length $L_{x}$, as long as the ratio $\xi/L_{x}$
is unchanged. Similarly, in the effective 2D superconductor, the edge modes
localization length $l_{c}$ (along the transverse direction $y$) scales with
the inverse of the pairing amplitude: a smaller gap requires the thin film to
be wider to maintain a constant ratio $\l_{c}/L_{y}$ and ensure decoupled edge
modes. Therefore, a larger pairing $\Delta$ allows us to reduce the
computational effort by using smaller systems and, at the same time, gives us
the opportunity to enhance the energy gap for MBSs and increase the width of
the region with MCPSs. A similar scaling has already been proposed in graphene
[41].

Figure 1:  (a) Low-energy states at $k=0$ and (b)-(c) energy spectrum for an
infinitely long thin film with $\mu=0$. The black dashed line in (a) stands
for the $k=0$ gap, while the band structures are computed with (b)
$\Lambda=15$ meV and (c) $\Lambda=30$ meV. (d) Energy gap at $k=0$ and (e)-(f)
band structures for an infinite wire with $\mu=10$ meV. The band structures
are obtained with (e) $\Lambda=10$ meV and (f) $\Lambda=30$ meV. Red and blue
colours represent electron and hole modes, respectively, and purple denotes a
superposition.

When confined along the $z$ direction, the MTI Hamiltonian (1) can be used to
describe the physical properties of a 2D (thin film) or 1D (wire) geometry
[42, 43, 13, 15, 14]. The Hamiltonian for a 2D system with particle-hole
symmetry belongs to the D symmetry class; therefore, an integer invariant
$\mathcal{N}$ characterizes the topological state of the two-dimensional slab
[44, 45]. A chiral TSC with odd Chern invariant and unpaired Majorana modes
can be realized in a 2D thin film starting from the quantum anomalous Hall
(QAH) phase, which is routinely achieved in MTIs [42, 46, 47]. For $\mu=0$,
the proximity pairing induces a novel region between the $\mathcal{N}=0$
trivial superconductor and the $\mathcal{N}=2$ QAH state [48]. In this
intermediate region, the MTI thin film realizes an $\mathcal{N}=1$ TSC with
_unpaired_ chiral Majorana modes on the edges [12, 13]. The occurrence of this
chiral TSC region is shown in Fig. 1(a), which displays the $k=0$ low-energy
eigenvalues of Eq. (3) solved in the thin film geometry as a function of the
Zeeman field $\Lambda$. The black dashed line represents the $k=0$ bulk energy
gap, showing the existence of two distinct transition points where topological
phase transitions occur with the emergence of gapless edge modes within the
bulk gap. Figs. 1(b)-(c) display the full band structure of these phases: the
first shows a single crossing of unpaired MCPSs which characterizes the
$\mathcal{N}=1$ chiral topological superconductor, the second corresponds to
the BdG quasiparticle spectrum of an $\mathcal{N}=2$ proximitized QAH system.

While a large thin film can realize different 2D topological superconducting
states, a narrow MTI wire with $\mu\neq 0$ can be used to achieve a quasi-one-
dimensional TSC with end-localized MBSs. Since the effective BdG Hamiltonian
of a QAH/SC heterostructure in a 1D geometry fits in the BDI symmetry class
[15], the topological properties of the system are characterized by an integer
invariant [44, 45] which discriminates between trivial $N_{BDI}=0$ and
topological $N_{BDI}=1$ states with unpaired Majorana edge modes in finite-
length systems. In principle, even higher topological states can be realized,
with $N_{BDI}\geq 2$ MBSs at the same end of the ribbon, protected by chiral
symmetry. However, the latter is broken in presence of disorder, and a pair of
Majorana modes localized at the same end of the wire will fuse into a trivial
fermion [7]. Therefore, only the $(-1)^{N_{BDI}}=-1$ phases are topologically
nontrivial in realistic samples, as a single _unpaired_ Majorana mode can be
protected by particle-hole symmetry alone [15].

The spectral gap at $k=0$ for an infinitely long wire with $\mu=10$ meV is
shown in Fig. 1(d), where the closing and reopening of the energy gap signals
a phase transition between trivial and topological states. The full band
structures of the two distinct phases are depicted in Figs. 1(e)-(f). It can
be noted in Fig. 1(f) that the normal order of the energy bands around $k=0$
is inverted, indicating a nontrivial topology of the bulk and, as a
consequence of the bulk-boundary correspondence, the presence of topologically
protected MBSs at the ends of wires with finite length [49].

##  III Antisymmetric Conductance

S-Phase | $G_{1}$ | $G_{2}$ | $G_{t}$
---|---|---|---
$\mathcal{N}=0$ | 0 | 0 | 0
$\mathcal{N}=1$ | $\alpha e^{2}/h$ | $(\alpha-1)e^{2}/h$ | $(2\alpha-1)e^{2}/h$
$\mathcal{N}=2$ | $e^{2}/h$ | $-e^{2}/h$ | 0
$N_{BDI}=0$ | 0 | 0 | 0
$N_{BDI}=1$ | $2\alpha e^{2}/h$ | $2(\alpha-1)e^{2}/h$ | $2(2\alpha-1)e^{2}/h$
ABS | $2\alpha e^{2}/h$ | 0 | $2\alpha e^{2}/h$
Table 1:  Low-bias conductances $G_{1},G_{2}$ and $G_{t}=G_{1}+G_{2}$ computed
through Eqs. (10)-(11). The first column summarizes all the possible phases in
the central S lead of the junction. Here, we considered a trivial ABS with
perfect Andreev reflection on the left side of the junction. The conductances
are given for $\beta=1-\alpha$.  Figure 2:  (a) Experimental setup proposed
for the detection of topologically-protected Majorana modes. The potential
$V_{0}=-e\mu$ is set by the back-gade electrode. (b) Sketches of the
transmission processes at the interfaces of the junction for different
superconducting phases in the central sector. Red and blue colors stand for
electron and hole currents.

Next, we analyse the electric transport through an NSN junction consisting of
an MTI slab with central proximitized sector, exploring the regime in which
the bias voltage drops asymmetrically over the left and right leads. The
experimental setup is schematically shown in Fig. 2(a). The electric current
$I_{i}$ in the normal terminals $i=1,2$ of a double junction can be computed
as [50, 51, 52]

$I_{i}=\int_{0}^{+\infty}dE\sum_{a}s_{a}\,[J_{i}^{a}(E)-K_{i}^{a}(E)]\,,$ (6)

where $a\in\\{e,h\\}$ denotes electron and hole degrees of freedom,
$s_{e,h}=\pm 1$, and

$\displaystyle J_{i}^{a}(E)$ $\displaystyle=$
$\displaystyle\frac{e}{h}N_{i}^{a}(E)\,f_{i}^{a}(E)\,,$ (7) $\displaystyle
K_{i}^{a}(E)$ $\displaystyle=$
$\displaystyle\frac{e}{h}\sum_{jb}P_{ij}^{ab}(E)\,f_{j}^{b}(E)\,,$ (8)

are the incoming and outgoing fluxes of quasiparticles, respectively. The
electric current is expressed in terms of the number of propagating modes in
each terminal $N_{i}^{a}$ and the Fermi distribution functions $f_{i}^{a}$.
Moreover, $P_{ij}^{ab}$ denotes the transmission probability of a
quasiparticle of type $b\in\\{e,h\\}$ in lead $j$ to a quasiparticle of type
$a$ in lead $i$, such that both normal ($a=b$) and Andreev ($a\neq b$)
reflection ($i=j$) and transmission ($i\neq j$) are taken into account. We
define the differential conductance in the normal terminals of the double
junction as

$G_{i}=\frac{\partial I_{i}}{\partial V}\,,$ (9)

where $V=V_{1}-V_{2}$ is the _total_ bias across the junction and $V_{i}$ is
the voltage difference between the $i$-th lead and the central sector. Here,
we assume an asymmetric bias $V_{1}=\alpha V$ and $V_{2}=-\beta V$ with
$0\leq\alpha\leq 1$ and $\alpha+\beta=1$, such that the total bias between
left and right terminals is fixed. With this assumption, we can derive the
following expressions for the conductance in the normal leads

$\displaystyle G_{1}(V)$ $\displaystyle=$
$\displaystyle\alpha\frac{e^{2}}{h}\left[N_{1}^{e}(\alpha
V)-P_{11}^{ee}(\alpha V)+P_{11}^{he}(\alpha V)\right]+$ (10) $\displaystyle+$
$\displaystyle\beta\frac{e^{2}}{h}\left[P_{12}^{hh}(\beta V)-P_{12}^{eh}(\beta
V)\right]\,,$ $\displaystyle G_{2}(V)$ $\displaystyle=$
$\displaystyle\beta\frac{e^{2}}{h}\left[-N_{2}^{h}(\beta V)-P_{22}^{eh}(\beta
V)+P_{22}^{hh}(\beta V)\right]+$ (11) $\displaystyle+$
$\displaystyle\alpha\frac{e^{2}}{h}\left[P_{21}^{he}(\alpha
V)-P_{21}^{ee}(\alpha V)\right]\,.$

The different possible transport processes in the junction are sketched in
Fig. 2(b) and depend on the topological phase of the proximitized sector. In
the thin-film configuration, the scattering amplitudes can be merely inferred
from the connection between edge modes in different sectors of the junction.
If the normal leads are held into a QAH state, a pair of zero-energy chiral
modes run along the edges of the system, due to the particle-hole degeneracy
of the $\Delta=0$ BdG Hamiltonian. When $\mathcal{N}=2$, the superconducting
sector is topologically equivalent to the QAH insulator in the normal
terminals [12]. The chiral states run uninterruptedly through normal and
proximitized leads, and the edge current is perfectly transmitted [48, 53].
Conversely, the $\mathcal{N}=0$ trivial superconductor does not support edge
modes within the gap, and the boundary between the QAH phase and the
proximitized region requires the occurrence of a gapless chiral state along
the interface. This mode is responsible for the complete backscattering of the
edge current flowing toward the superconductor [48, 53]. Finally, when
$\mathcal{N}=1$, the proximitized sector supports a single unpaired Majorana
mode on each edge. The injected modes from the QAH regions separate into two
MCPSs at the interfaces of the junction, one is perfectly transmitted while
the other is totally reflected. Within the Blonder-Tinkham-Klapwijk formalism
for the electric conductance in NSN junctions [50, 51], this process
corresponds to equal probability of normal reflection, Andreev reflection,
normal transmission and Andreev transmission [13, 48].

A similar framework can also be obtained for the wire geometry. When the width
of the slab is smaller than the localization length of the edge modes, the QAH
edge states are coupled into a single conducting channel, resembling a
spinless metallic phase. Since the proximitized sector realizes an effective
1D $p$-wave superconductor [15], the interfaces between normal and
superconducting MTI reproduce the physics of a NS junction between a normal
metals and a $p$-wave superconductor [54]. In the $N_{BDI}=1$ topological
state with end-localized MBSs, perfect Andreev reflection occurs for a bias
lower than the energy gap [55]. Conversely, in the $N_{BDI}=0$ trivial phase,
the electric conductance is expected to vanish in the low-bias limit [54],
meaning that the scattering processes are dominated by normal reflection.
Choosing appropriate values for the transmission probabilities $P_{ij}^{ab}$
in order to recover the scenarios above, the conductance on the two terminals
of the junction can be easily computed from Eqs. (10)-(11). Their values are
summarized in Table 1 for the different phases in the proximitized MTI and for
a trivial ABS perfectly coupled to the left side of the junction in a wire
geometry. It can be noted that the total conductance $G_{t}=G_{1}+G_{2}\neq 0$
only in presence of Majorana modes or ABSs, meaning that the currents in the N
leads are different, being proportional to the fraction $\alpha,\beta$ of the
total bias applied on the two sides of the junction.

We claim that the analysis of the total conductance $G_{t}$ as a function of
bias splitting $\alpha$ can provide a useful criterion to rule out transport
signatures from trivial Andreev processes. While not being a conclusive proof,
an antisymmetric $G_{t}(\alpha)$ around $\alpha=0.5$ would point towards
Majorana modes because trivial Andreev levels are typically not constrained to
such an antisymmetric profile. The gapped superconductor and the proximitized
QAH state exhibit a constant $G_{t}=0$, but, more generally, the emergence of
trivial Andreev levels allow other $G_{t}(\alpha)$ trends, depending on how
the ABS couples with the two interfaces of the junction. For instance, in the
aforementioned case of a trivial ABS perfectly coupled to the left lead only,
the total conductance is antisymmetric around $\alpha=0$ (completely
unbalanced bias splitting). Quite remarkably, topological states with Majorana
modes require $G_{t}(\alpha)$ to be _antisymmetric_ around $\alpha=0.5$ (equal
bias splitting), because identical scattering amplitudes are expected at the
two interfaces of the junction. In spite of the same symmetry, a different
ratio $G_{t}/G_{0}$, where $G_{0}=e^{2}/h$ is the conductance quantum,
characterizes different types of Majorana modes in the superconductor: in the
case of a MCPS, $G_{t}/G_{0}=(2\alpha-1)$ due to all scattering probabilities
in Eqs. (10)-(11) being 0.25, while in presence of a MBS,
$G_{t}/G_{0}=2(2\alpha-1)$ indicates perfect Andreev reflection at the
extremities of the proximitized sector. Similar signals could be, in
principle, obtained due to trivial ABSs, but the antisymmetry around
$\alpha=0.5$ would in this case require fine-tuned equal conditions on the two
interfaces of the junction.

The total conductance $G_{t}\neq 0$ is related to the existence of an electric
current going to ground from the superconductor, which ensures charge
conservation when $G_{1}\neq G_{2}$ and the current injected on the left lead
is different from the one flowing out on the right one. This current can be
easily detected through electric measurements, providing a measure of the
electric conductance in the two normal leads, while maintaining the
correlation between transport on the two interfaces of the junction. We point
out that, within our simplified model, the only current flowing through the
$s$-wave superconductor is due to Cooper pairs originating in the proximitized
MTI. Indeed, for bias lower than the bulk gap, no quasiparticle modes can be
excited, preventing unintended transmissions between the terminals of the
junction. We also neglected scattering processes occurring between the normal
leads and the $s$-wave superconductor, since the presence of a physical
interface between the two distinct materials would make suppress them compared
to the scattering events which take place within the MTI slab.

### Conductance Matrix

The antisymmetric relation involving the total conductance $G_{t}$ can be
expressed in the equivalent language of the conductance matrix, where the
current-voltage relation reads

$\begin{pmatrix}I_{1}\\\ I_{2}\end{pmatrix}=\begin{pmatrix}g_{11}&g_{12}\\\
g_{21}&g_{22}\end{pmatrix}\begin{pmatrix}V_{1}\\\ V_{2}\end{pmatrix}\,.$ (12)

By considering 2-terminal transport between the N leads of our 3-terminal
device (see Fig. 2), we can extract information about the Andreev processes
taking place at the proximitized section of the MTI slab [32, 33]. The
conductance matrix elements are defined as $g_{ij}=\partial I_{i}/\partial
V_{j}$ and can be distinguished into local ($i=j$) and nonlocal ($i\neq j$)
components. The conductance $G_{i}$ for the current in the $i=1,2$ terminal
takes the form

$\begin{split}G_{1}&=\alpha\,g_{11}+(\alpha-1)\,g_{12}\,,\\\
G_{2}&=\alpha\,g_{21}+(\alpha-1)\,g_{22}\,,\end{split}$ (13)

and the total conductance can be written as

$G_{t}=-(g_{12}+g_{22})+\alpha(g_{11}+g_{12}+g_{21}+g_{22})\,.$ (14)

Therefore, in terms of local and nonlocal matrix elements the antisymmetric
condition around $\alpha=0.5$ can be written explicitly as

$g_{11}-g_{12}=g_{22}-g_{21}\,,$ (15)

meaning that the difference between local and nonlocal conductances must be
the same in both terminals.

The novelty of our approach compared with previous works is that we focus on
the symmetry of $G_{t}$ around the limit $\alpha=0.5$ of equally-split bias
voltage. For instance, in the case of a superconducting wire with end-
localized MBSs, the local conductances $g_{11}$ and $g_{22}$ are usually
measured separately, looking for correlations of $2e^{2}/h$ zero-bias peaks on
the two sides of the nanowire. In our framework of asymmetrically split bias,
these measurements are equivalent to $\alpha=1$ and $\alpha=0$, respectively.
Our work shows that measuring $G_{t}$ continuously as a function of $\alpha$
can provide a more robust transport signature. Similarly, in a proximitized
MTI thin film, a single transport measurement of $G_{t}$ in a junction with
equal bias split $\alpha=0.5$ has been already proved to be incapable of
detecting Majorana chiral propagating states [17, 19]. Determining $G_{t}$
with different bias configurations can provide more information on the phase
of the proximitized sector.

##  IV Numerical Results

Figure 3:  Conductance $G_{t}$ computed in the NSN junction as a function of
(a) magnetization and (b) bias split. In the left panel $\alpha=0.25$ and the
blue (green) line stands for the wire (thin film) geometry. The ABS is
modelled adding a barrier on the right side of the junction with the
proximitized sector in the $N_{BDI}=1$ phase. (c)-(d) Transmission amplitudes
$P_{ij}^{ab}$ for the left terminal of the junction as a function of the
length $L_{x}$ of the central proximitized sector. The probabilities are
computed for a (c) $N_{BDI}=1$ superconductor with MBSs and a (d)
$\mathcal{N}=1$ superconductor with MCPSs. In all the pictures, the total bias
$V=0.1$ meV is chosen within the bulk gap. The values of $\Lambda$ in panels
(b),(c) and (d) are chosen according to Fig. 1 to reproduce the different
TSCs.

We simulated a NSN junction between proximitized and normal MTI slabs using a
complex band structure approach. This numerical technique allows us to
describe not only propagating modes with $k\in\mathbb{R}$, but also evanescent
states (like end-localized MBSs) originating in the superconductor, which are
related to complex longitudinal wavenumbers $k\in\mathbb{C}$. Within this
framework, the electron wavefunction in a homogeneous sector of the junction
can be chosen as a superposition of transverse wavefunctions
$\Psi_{k}(y,z;\eta)$ with a proper wavenumber $k$. The full wavefunction takes
the generic form

$\Psi(x,y,z;\eta)=\sum_{k}{c_{k}}\,\Psi_{k}(y,z;\eta)\,e^{ikx}$ (16)

where $c_{k}$ are complex numbers and $\eta=\\{\sigma,\tau,\delta\\}$
represents the set of spin $\sigma$, orbital index $\tau$ and particle-hole
type $\delta$. Notably, the wavefunction for a finite-length sector of the
junction can be constructed from the set of bulk wavenumbers and coefficients
$\\{k,c_{k}\\}$ obtained in a full translational-invariant system.
Furthermore, while a grid discretization is required for the confined
dimensions $y$ and $z$, the dependence of $\Psi(x,y,z;\eta)$ along the
longitudinal axis $x$ is parametric, and enables to describe a junction of any
length $L_{x}$. Further details about the technique are given in Sec. A.

We computed numerically the conductances $G_{1},G_{2}$ and the sum
$G_{t}=G_{1}+G_{2}$ in the NSN junction with a magnetic TI in the wire and
thin film configurations, reproducing the properties of 1D and 2D topological
superconductors, respectively. Fig. 3(a) displays $G_{t}$ versus the
magnetization of the MTI for an asymmetric bias $\alpha=0.25$. In the thin
film geometry, a region with $G_{t}\neq 0$ distinguishes the $\mathcal{N}=1$
chiral TSC from the $\mathcal{N}=0$ trivial superconductor and the
$\mathcal{N}=2$ QAH phase, where $G_{t}=0$ denotes that the electric currents
in the two terminals are equal and opposite independently of the bias split.
For the chosen $\alpha$, the conductance for $\mathcal{N}=1$ was expected to
be quantized at $G_{t}=-e^{2}/2h$, which is roughly the value reached in the
nontrivial region with MCPSs. Similarly, in the wire geometry a plateau
$G_{t}=-e^{2}/h$ characterizes the $N_{BDI}=1$ nontrivial phase, while the
$N_{BDI}=0$ gapped superconductor exhibits $G_{t}=0$. Fig. 3(b) shows $G_{t}$
as a function of the split parameter $\alpha$ for all the nontrivial phases
realized by the proximitized MTI. A trivial ABS is also simulated in the wire
geometry as an $N_{BDI}=1$ superconductor with an insulating barrier on the
right side of the junction. Here, the values of the conductance are in perfect
agreement with our prediction in Table 1. We emphasize here that a
symmetrically distributed bias $\alpha=0.5$ is insufficient to discriminate a
TSC from the trivial state and the proximitized QAH phase, since this
particular configuration implies always $G_{t}=0$.

The lower panels in Fig. 3 display the amplitudes $P_{ij}^{ab}$ for all the
scattering processes occurring on the left interface of the junction, i.e.,
normal reflection $R_{N}$, Andreev reflection $R_{A}$, normal transmission
$T_{N}$ and Andreev transmission $T_{A}$. The figures correspond to a
$N_{BDI}=1$ topological superconducting wire with unpaired MBSs in (c) and a
$\mathcal{N}=1$ TSC thin film with MCPSs in (d). The former shows that, when
the junction is sufficiently large to prevent transmission by evanescent
modes, the injected electron undergoes perfect Andreev reflection $R_{A}=1$ in
presence of MBSs. The latter indicates that due to MCPSs, normal and Andreev
transmission and reflection occur with equal probability
$R_{N}=R_{A}=T_{N}=T_{A}=0.25$. Oscillations around the expected plateaus are
due to the interference between back-scattered chiral modes from the two
interfaces of the double junction, resulting in an interferometric behaviour
[56]. For all the above results, the total bias across the junction is $V=0.1$
meV, which is always lower than the bulk energy gap. Such low bias ensures
that no bulk modes are activated in the proximitized sector and that the
injected electrons and holes interact in the condensate only with
topologically-protected Majorana boundary states. Indeed, the proposed
framework does not hold for higher bias, which implies interaction with
multiple active modes in the superconductor.

All the numerical simulations are obtained without taking into account the
effect of disorder in the system. Even tough this may seem a crude
approximation, in the 2D thin film the electric current is transported through
the junction by topologically-protected chiral fermionic or Majorana edge
modes. These electronic states are known to be insensitive to weak disorder,
because no energy modes are available for backscattering [49]. Conversely, in
the regime of a narrow ribbon opposite edge states are strongly coupled and it
is difficult to maintain the ballistic nature of the chiral channels. Despite
the fragility of the QAH edge states, the MBSs arising in the quasi-1D
topological phase are expected to be robust against weak disorder, maintaining
well-quantized zero-bias peaks in tunneling spectroscopy [57]. Therefore, with
asymmetric bias splitting, we expect the low-bias conductance to keep its
antisymmetric behaviour in the presence of weak disorder.

##  V Conclusion

In summary, an asymmetric bias voltage drop applied across an NSN junction
provides a useful, but not conclusive, criterion to rule out conductance
signals produced by trivial Andreev levels in MTI slabs with a central
proximitized section. We showed that the antisymmetry of the conductance
$G_{t}$ with respect to the point of equal bias splitting ($\alpha=0.5$) is a
necessary condition for topologically-protected Majorana modes in normal-
superconductor junctions. Detailed model calculations for a narrow (wire-like)
and a wide (film-like) slab, hosting MBSs and MCPSs respectively, are shown to
support our conclusions. Our results will be useful for the experimental
detection of the elusive Majorana quasiparticles, contributing to the progress
towards a solid platform for quantum computing.

###### Acknowledgements.

This project is financially supported by the QuantERA grant MAGMA, by the
National Research Fund Luxembourg under the grant
INTER/QUANTERA21/16447820/MAGMA, by the German Research Foundation under grant
491798118, by MCIN/AEI/10.13039/501100011033 under project PCI2022-132927, and
by the European Union NextGenerationEU/PRTR. L.S. acknowledges support from
Grants No. PID2020-117347GB-I00 funded by MCIN/AEI/10.13039/501100011033 and
No. PDR2020-12 funded by GOIB. K.M. acknowledges the financial support by the
Bavarian Ministry of Economic Affairs, Regional Development and Energy within
Bavaria’s High-Tech Agenda Project ”Bausteine für das Quantencomputing auf
Basis topologischer Materialien mit experimentellen und theoretischen
Ansätzen” (grant allocation no. 07 02/686 58/1/21 1/22 2/23).

##  Appendix A Numerical Method

Our numerical results are obtained using a grid discretization of the
continuum Hamiltonian (3). States with real $k$, i.e., propagating modes like
those in Fig. 1, can be directly obtained by matrix diagonalization of the
corresponding BdG energy eigenvalue problem

$\mathcal{H}_{BdG}(k)\,\Psi=E\,\Psi\;.$ (17)

However, transport in non-translation-invariant systems like the NSN junction
requires also more general (evanescent) states described by a complex wave
number $k$. We modelled this case adapting the complex-band-structure approach
discussed in Refs. [58, 59].

We first rewrite the Hamiltonian by explicitly separating the $k$-dependent
terms as $\mathcal{H}_{BdG}={\cal A}+{\cal B}k+{\cal C}k^{2}$. Then, defining
an enlarged wave function $(\Psi_{1},\Psi_{2})^{T}=(\Psi,k\Psi)^{T}$ it is
possible to reformulate the energy eigenvalue problem in Eq. (17) into a
$k$-eigenvalue problem. After some straightforward algebra, this reads

$\left(\begin{array}[]{cc}0&1\\\ -{\cal C}^{-1}({\cal A}-E)&-{\cal
C}^{-1}{\cal B}\end{array}\right)\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)=k\left(\begin{array}[]{c}\Psi_{1}\\\
\Psi_{2}\end{array}\right)\;.$ (18)

In terms of the original parameters of the MTI Hamiltonian Eq. (3) it is

$\displaystyle{\cal A}$ $\displaystyle=$ $\displaystyle
C_{0}-C_{\perp}k_{y}^{2}-C_{z}k^{2}_{z}$ (19) $\displaystyle+$
$\displaystyle(M_{0}-M_{\perp}k^{2}_{y}-M_{z}k^{2}_{z})\tau_{z}$
$\displaystyle+$
$\displaystyle(A_{\perp}k_{y}\sigma_{y}+A_{z}k_{z}\sigma_{z})\tau_{x}\,,$
$\displaystyle{\cal B}$ $\displaystyle=$ $\displaystyle
A_{\perp}\sigma_{x}\tau_{x}\,,$ (20) $\displaystyle{\cal C}$ $\displaystyle=$
$\displaystyle-(C_{\perp}+M_{z}\tau_{z})\,.$ (21)

Note that the eigenvalue problem (18) requires non-Hermitian matrix solvers to
include the possibility of complex wave numbers $k$.

In the modelling of the NSN double junction, we first solve the matrix version
of Eq. (18) for a large set of modes $\\{k^{(a)},\Psi^{(a)}_{k}\\}$ in each
sector, where $a=L,C,R$, refers to left, centre and right, respectively. The
wave function is then represented by a collection of input (output) amplitudes
$a_{k}^{(a)}$ $(b_{k}^{(a)})$ in each part as

$\begin{split}\Psi^{(a)}(x,y,z;\eta)&=\sum_{k}{a_{k}^{(a)}}\,\Psi_{k}^{(a)}(y,z;\eta)\,e^{ik^{(a)}(x-x_{k}^{(a)})}\\\
&+\sum_{k}{b_{k}^{(a)}}\,\Psi_{k}^{(a)}(y,z;\eta)\,e^{ik^{(a)}(x-x_{k}^{(a)})}\;,\end{split}$
(22)

where $\eta=\\{\sigma,\tau,\delta\\}$ represent the set of spin $\sigma$,
orbital index $\tau$ and particle-hole type $\delta$, and the input/output
character of each mode $\Psi_{k}$ is determined according to the sign of its
probability flux

$I_{k}=\langle\Psi_{k}|\partial H/\partial k_{x}|\Psi_{k}\rangle\,.$ (23)

The $e^{ik^{(a)}x_{k}^{(a)}}$ factors in Eq. (22) are a gauge choice that
helps avoid numerical instabilities [59].

Due to truncation, the total number of unknowns
$\\{b_{k}^{(L)},b_{k}^{(C)},b_{k}^{(R)}\\}$ is finite and their values must be
fixed by imposing continuity of the wave function and its $x$-derivative at
the two interfaces $x=x_{1}$ and $x=x_{2}$. In practice, those equations are
projected onto the total discrete set of complex modes by means of the overlap
matrices

${\cal
M}_{k^{\prime}k}^{(ab)}=\sum_{\sigma\tau\delta}{}\int{dydz\,\Psi_{k^{\prime}}^{a*}(y,z;\eta)\,\Psi_{k}^{b}(y,z;\eta)}\;.$
(24)

In detail, the linear system reads

$\begin{array}[]{rrll}\displaystyle\sum_{k^{(L)}}{{\cal
M}_{k^{\prime}k}^{(aL)}\,{b_{k}^{(L)}}}&-\displaystyle\sum_{k^{(C)}}{{\cal
M}_{k^{\prime}k}^{(aC)}\,e^{ik^{(C)}(x_{1}-x_{k}^{(C)})}\,{b_{k}^{(C)}}}&=-\displaystyle\sum_{k^{(L)}}{{\cal
M}_{k^{\prime}k}^{(aL)}\,{a_{k}^{(L)}}}\;,&\;\;{\rm
if}\;\left\\{\begin{array}[]{c}a=L\\\
x_{k^{\prime}}^{(a)}=x_{1}\end{array}\right.\;,\\\
\rule{0.0pt}{22.76228pt}\displaystyle\sum_{k^{(L)}}{{\cal
M}_{k^{\prime}k}^{(aL)}\,k^{(L)}{b_{k}^{(L)}}}&-\displaystyle\sum_{k^{(C)}}{{\cal
M}_{k^{\prime}k}^{(aC)}\,e^{ik^{(C)}(x_{1}-x_{k}^{(C)})}\,k^{(C)}{b_{k}^{(C)}}}&=-\displaystyle\sum_{k^{(L)}}{{\cal
M}_{k^{\prime}k}^{(aL)}\,k^{(L)}{a_{k}^{(L)}}}\;,&\;\;{\rm
if}\;\left\\{\begin{array}[]{c}a=C\\\
x_{k^{\prime}}^{(a)}=x_{1}\end{array}\right.\;,\\\
\rule{0.0pt}{22.76228pt}\displaystyle\sum_{k^{(R)}}{{\cal
M}_{k^{\prime}k}^{(aR)}\,k^{(R)}{b_{k}^{(R)}}}&-\displaystyle\sum_{k^{(C)}}{{\cal
M}_{k^{\prime}k}^{(aC)}\,e^{ik^{(C)}(x_{2}-x_{k}^{(C)})}\,k^{(C)}{b_{k}^{(C)}}}&=-\displaystyle\sum_{k^{(R)}}{{\cal
M}_{k^{\prime}k}^{(aR)}\,k^{(R)}{a_{k}^{(R)}}}\;,&\;\;{\rm
if}\;\left\\{\begin{array}[]{c}a=C\\\
x_{k^{\prime}}^{(a)}=x_{2}\end{array}\right.\;,\\\
\rule{0.0pt}{22.76228pt}\displaystyle\sum_{k^{(R)}}{{\cal
M}_{k^{\prime}k}^{(aR)}\,{b_{k}^{(R)}}}&-\displaystyle\sum_{k^{(C)}}{{\cal
M}_{k^{\prime}k}^{(aC)}\,e^{ik^{(C)}(x_{2}-x_{k}^{(C)})}\,{b_{k}^{(C)}}}&=-\displaystyle\sum_{k^{(R)}}{{\cal
M}_{k^{\prime}k}^{(aR)}\,{a_{k}^{(R)}}}\;,&\;\;{\rm
if}\;\left\\{\begin{array}[]{c}a=R\\\
x_{k^{\prime}}^{(a)}=x_{2}\end{array}\right.\;.\end{array}$ (25)

By solving Eq. (25) with $a_{k}^{(a)}=1$ for a particular input propagating
mode, with all other inputs vanishing, we obtain a particular input/output
transmission probability
$p_{k^{\prime}k}^{a^{\prime}a}=|b_{k^{\prime}}^{a^{\prime}}|^{2}$. The sum of
all these individual probabilities discriminating their electron/hole
character in the normal leads finally yields the total probabilities defined
in Sec. III

$P_{ji}^{he}=\sum_{k_{h}k_{e}}{p_{k_{h}k_{e}}^{ji}}\,.$ (26)

A good control of the model truncations, regarding grid size and number of
complex modes, is given by the flux conservation, which we typically require
to be better than 1%.

##  Appendix B Derivation of the Differential Conductances

We derive here the equations for the conductances $G_{1}$ and $G_{2}$ given in
Eqs. (10) and (11). In our approach [51], the bias dependence is contained in
Fermi energy distribution functions for electrons and holes injected from far-
distant reservoirs into the N leads

$f_{i}^{a}(E)=\begin{cases}\frac{1}{1+e^{\left(E-eV_{i}\right)/k_{B}T}}&\text{if
}a=e\;,\\\ \frac{1}{1+e^{\left(E+eV_{i}\right)/k_{B}T}}&\text{if
}a=h\;,\end{cases}$ (27)

with $V_{i}$ being the voltage difference between the $i$-th reservoir and the
MTI slab. By making explicit the sum over the quasiparticle types of Eq. (6)
and using Eqs. (7) and (8), the electric current can be rewritten as

$\begin{split}I_{i}&=\frac{e}{h}\int_{0}^{+\infty}dE\left[J_{i}^{e}-K_{i}^{e}-J_{i}^{h}+K_{i}^{h}\right]=\frac{e}{h}\int_{0}^{+\infty}dE\left[N_{i}^{e}f_{i}^{e}-\sum_{jb}P_{ij}^{eb}f_{j}^{b}-N_{i}^{h}f_{i}^{h}+\sum_{jb}P_{ij}^{hb}f_{j}^{b}\right]\\\
&=\frac{e}{h}\int_{0}^{+\infty}dE\,\left[N_{i}^{e}f_{i}^{e}-\sum_{j}\left(P_{ij}^{ee}f_{j}^{e}+P_{ij}^{eh}f_{j}^{h}\right)-N_{i}^{h}f_{i}^{h}+\sum_{j}\left(P_{ij}^{he}f_{j}^{e}+P_{ij}^{hh}f_{j}^{h}\right)\right]\,,\end{split}$
(28)

where for simplicity we omitted the energy dependence. Expanding the sum over
the terminals $j=1,2$ we can write the electric current into the two leads of
the junction as

$\displaystyle I_{1}$
$\displaystyle=\frac{e}{h}\int_{0}^{+\infty}dE\Biggl{\\{}\left[N_{1}^{e}-P_{11}^{ee}+P_{11}^{he}\right]f_{1}^{e}+\left[-N_{1}^{h}-P_{11}^{eh}+P_{11}^{hh}\right]f_{1}^{h}+\left[P_{12}^{he}-P_{12}^{ee}\right]f_{2}^{e}+\left[P_{12}^{hh}-P_{12}^{eh}\right]f_{2}^{h}\Biggr{\\}}\,,$
$\displaystyle I_{2}$
$\displaystyle=\frac{e}{h}\int_{0}^{+\infty}dE\Biggl{\\{}\left[N_{2}^{e}-P_{22}^{ee}+P_{22}^{he}\right]f_{2}^{e}+\left[-N_{2}^{h}-P_{22}^{eh}+P_{22}^{hh}\right]f_{2}^{h}+\left[P_{21}^{he}-P_{21}^{ee}\right]f_{1}^{e}+\left[P_{21}^{hh}-P_{21}^{eh}\right]f_{1}^{h}\Biggr{\\}}\,.$
(29)

We assume that the bias is asymmetrically distributed as $V_{1}=\alpha V$ and
$V_{2}=-\beta V$ with $0\leq\alpha\leq 1$ and $\beta=1-\alpha$ such that the
total voltage drop across the NSN junction is fixed to $V_{1}-V_{2}=V$. In the
zero-temperature limit the Fermi functions take the form of step functions

$\displaystyle f_{1}^{e,h}$ $\displaystyle=\frac{1}{1+e^{\left(E\mp e\alpha
V\right)/k_{B}T}}\xrightarrow[T\to 0]{}\Theta(E\mp\alpha eV)$ (30)

for the left terminal of the junction and analogously for the right junction.
The expressions of the currents in the two terminals can thus be simplified as

$\displaystyle I_{1}$ $\displaystyle=\frac{e}{h}\int_{0}^{\alpha
eV}dE\left[N_{1}^{e}-P_{11}^{ee}+P_{11}^{he}\right]+\frac{e}{h}\int_{0}^{\beta
eV}dE\left[P_{12}^{hh}-P_{12}^{eh}\right]\,,$ (31) $\displaystyle I_{2}$
$\displaystyle=\frac{e}{h}\int_{0}^{\beta
eV}dE\left[-N_{2}^{h}-P_{22}^{eh}+P_{22}^{hh}\right]+\frac{e}{h}\int_{0}^{\alpha
eV}dE\left[P_{21}^{he}-P_{21}^{ee}\right]\,,$ (32)

and the differential conductance can be computed as the derivative of Eqs.
(31)-(32) with respect to the total bias $V$ across the junction, leading to

$\displaystyle G_{1}(V)=\frac{\partial I_{1}}{\partial V}$
$\displaystyle=\alpha\frac{e^{2}}{h}\left[N_{1}^{e}(\alpha
V)-P_{11}^{ee}(\alpha V)+P_{11}^{he}(\alpha
V)\right]+\beta\frac{e^{2}}{h}\left[P_{12}^{hh}(\beta V)-P_{12}^{eh}(\beta
V)\right]\,,$ (33) $\displaystyle G_{2}(V)=\frac{\partial I_{2}}{\partial V}$
$\displaystyle=\beta\frac{e^{2}}{h}\left[-N_{2}^{h}(\beta V)-P_{22}^{eh}(\beta
V)+P_{22}^{hh}(\beta V)\right]+\alpha\frac{e^{2}}{h}\left[P_{21}^{he}(\alpha
V)-P_{21}^{ee}(\alpha V)\right]\,.$ (34)

The left-terminal conductance (33) is given by the number of injected
electrons $N_{1}^{e}$, the normal $P_{11}^{ee}$ and Andreev $P_{11}^{he}$
reflection amplitudes for electrons injected in lead 1, and the normal
$P_{12}^{hh}$ and Andreev $P_{12}^{eh}$ transmission amplitudes for holes
injected in lead 2. Similarly, the right-terminal conductance Eq. (34) is
given by the number of injected holes $N_{2}^{h}$, normal $P_{22}^{hh}$ and
Andreev $P_{22}^{eh}$ reflection amplitudes for holes injected in lead 2, and
normal $P_{21}^{ee}$ and Andreev $P_{21}^{he}$ transmission amplitudes for
electrons injected in lead 1. The number of injected quasiparticles
$N_{i}^{a}$ and the values of the transmission amplitudes $P_{ij}^{ab}$ in the
low-bias scenario described in the main article are given in Tables 2-3 for
all the topological phases of the superconducting sector. Table 4 summarizes
the corresponding values of the conductances $G_{1},G_{2}$ and their sum
$G_{t}=G_{1}+G_{2}$.

S-Phase | $N_{1}^{e}$ | $P_{11}^{ee}$ | $P_{11}^{he}$ | $P_{12}^{hh}$ | $P_{12}^{eh}$ | $G_{1}$
---|---|---|---|---|---|---
$N_{BDI}=0$ | 1 | 1 | 0 | 0 | 0 | 0
$N_{BDI}=1$ (MBS) | 1 | 0 | 1 | 0 | 0 | $2\alpha e^{2}/h$
$\mathcal{N}=0$ | 1 | 1 | 0 | 0 | 0 | 0
$\mathcal{N}=1$ (MCPS) | 1 | 0.25 | 0.25 | 0.25 | 0.25 | $\alpha e^{2}/h$
$\mathcal{N}=2$ (QAH) | 1 | 0 | 0 | 1 | 0 | $e^{2}/h$
Table 2:  Transmission amplitudes and number of electronic modes required to compute the conductance $G_{1}$ through Eq. (33). The values are given for all the possible topological phases which can be found in the central superconducting sector of the NSN junction. The conductances are given taking into account that $\beta=\alpha-1$.  S-Phase | $N_{2}^{h}$ | $P_{22}^{eh}$ | $P_{22}^{hh}$ | $P_{21}^{he}$ | $P_{21}^{ee}$ | $G_{2}$
---|---|---|---|---|---|---
$N_{BDI}=0$ | 1 | 0 | 1 | 0 | 0 | 0
$N_{BDI}=1$ (MBS) | 1 | 1 | 0 | 0 | 0 | $2(\alpha-1)e^{2}/h$
$\mathcal{N}=0$ | 1 | 0 | 1 | 0 | 0 | 0
$\mathcal{N}=1$ (MCPS) | 1 | 0.25 | 0.25 | 0.25 | 0.25 | $(\alpha-1)e^{2}/h$
$\mathcal{N}=2$ (QAH) | 1 | 0 | 0 | 0 | 1 | $-e^{2}/h$
Table 3:  Transmission amplitudes and number of hole modes required to compute the conductance $G_{2}$ through Eq. (34). The values are given for all the possible topological phases which can be found in the central superconducting sector of the NSN junction. The conductances are given taking into account that $\beta=\alpha-1$.  S-Phase | $G_{1}$ | $G_{2}$ | $G_{t}$
---|---|---|---
$N_{BDI}=0$ | 0 | 0 | 0
$N_{BDI}=1$ (MBS) | $2\alpha e^{2}/h$ | $2(\alpha-1)e^{2}/h$ | $2(2\alpha-1)e^{2}/h$
$\mathcal{N}=0$ | 0 | 0 | 0
$\mathcal{N}=1$ (MCPS) | $\alpha e^{2}/h$ | $(\alpha-1)e^{2}/h$ | $(2\alpha-1)e^{2}/h$
$\mathcal{N}=2$ (QAH) | $e^{2}/h$ | $-e^{2}/h$ | 0
Table 4:  Conductances $G_{1},G_{2}$ and their sum $G_{t}=G_{1}+G_{2}$
computed through Eqs. (33)-(34) using the transmission probabilities given in
Tables 2-3. The sum of the conductance on the two terminal is non-zero only in
presence of topologically-protected Majorana modes. Furthermore, the value of
$G_{t}$ discriminates between end-localized MBSs and dispersive MCPSs.

##  Appendix C Role of an Interface Barrier

S-Phase | $N_{1}^{e}$ | $P_{11}^{ee}$ | $P_{11}^{he}$ | $P_{12}^{hh}$ | $P_{12}^{eh}$ | $G_{1}$ | $G_{2}$ | $G_{t}$
---|---|---|---|---|---|---|---|---
$N_{BDI}=1$ (MBS) | 1 | 0 | 1 | 0 | 0 | $2\alpha e^{2}/h$ | 0 | $2\alpha e^{2}/h$
$\mathcal{N}=1$ (MCPS) | 1 | 0.5 | 0.5 | 0 | 0 | $\alpha e^{2}/h$ | 0 | $\alpha e^{2}/h$
$\mathcal{N}=2$ (QAH) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0
Table 5:  Transmission amplitudes for the scattering processes on the left
interface and conductances $G_{1},G_{2}$ and $G_{t}$ assuming an insulating
barrier on the right side of the system. The values are given for all the
topologically nontrivial cases.

We consider in this section the role of an interface barrier between the
central proximitized (S) sector and the right (N) lead. That is, an NSN’N
structure where N’ represents a slab of a normal MTI material without any
propagating modes. The presence of N’ breaks the left-right symmetry with
respect to the central sector and, depending on the barrier transparency, it
will affect the electric connection to the right side. A small barrier length
mimics interface disorder, while a large barrier length corresponds to the
complete electric insulation.

We show here that the presence of a barrier does not change our conclusions
about the relevance of $G_{t}$. Indeed, despite the fact that the value of the
conductance $G_{1}$ and $G_{2}$ may change, the total conductance keeps its
meaning, with $G_{t}\neq 0$ as long as Andreev processes take place in the
junction. The different cases can be clearly understood when a completely
insulating barrier is introduced, for instance, on the right side of the
system: as the right lead is electrically disconnected from the proximitized
MTI, $G_{2}=0$ regardless of the topological phase realized in the
proximitized sector. In an $N_{BDI}=1$ superconducting wire, perfect Andreev
reflections occurs on the left interface of the junction due to interaction
with MBS. In an $\mathcal{N}=1$ TSC film, the electron is completely
reflected, since the transmission to the right side is prevented by the
barrier. Normal and Andreev processes take place with same probability. In an
analogous way, in an $\mathcal{N}=2$ TSC, the electrons are perfectly
reflected from the barrier, and no Andreev processes take place in the
junction. The conductances $G_{1}$ and $G_{2}$ can be easily computed through
Eqs. (10)-(11). Their values, together with the transmission amplitudes for
the left interface of the junction, are summarized in Table 5.

Figure 4:  Total conductance $G_{t}$ for a NSN’N junction with a proximitized
sector in the (a) $N_{BDI}=1$ phase and (b) $\mathcal{N}=1$ TSC with an
insulating barrier on the right side of the system. The conductance is
computed as a function of the length $L_{x}$ of the barrier. The plots are
obtained with $\alpha=0.25$.

A numerical simulation for the conductance in the NSN’N junction with
$\alpha=0.25$ is shown in Fig. 4 for (a) a wire geometry with a proximitized
sector in the $N_{BDI}=1$ state and (b) a film geometry with a $\mathcal{N}=1$
TSC in the central sector. We focus on the dependence on $L_{x}$, the length
of the intermediate barrier N’. For a completely transparent barrier
($L_{x}\approx 0$) the total conductance for $\alpha=0.25$ is $G_{t}=-e^{2}/h$
in presence of MBS and $G_{t}=-e^{2}/2h$ in presence of MCPS. An increasingly
opaque barrier ($L_{x}\to\infty$) changes these values, keeping $G_{t}\neq 0$
as long as Andreev processes occur in the proximitized MTI. For the case of
MBSs, a barrier with $L_{x}\gtrsim 50\,{\rm nm}$ is long enough to prevent the
electric transmission on the right side, leading to $G_{t}=e^{2}/2h$.
Remarkably, Fig. 4(b) shows a very different decay length along $x$ for MCPSs.
Indeed, a larger barrier $L_{x}\gtrsim 5\;\mu{\rm m}$ is required to prevent
completely the electric transmission in presence of MCPSs, changing the total
conductance to $G_{t}=e^{2}/4h$. Both limiting values are in agreement with
Table 5 for the selected bias split parameter $\alpha=0.25$. Focusing on the
short barrier limit, which may represent interface disorder effects, Fig. 4
suggests that, for the chosen set of parameters, the antisymmetry of
$G_{t}(\alpha)$ is robust for barriers with $L_{x}\lesssim 5$ nm for the MBS
and $L_{x}\lesssim 0.5$ $\mu$m for the MCPS, since $G_{t}$ is almost
unaffected by the barrier in these cases.

## References

  * Majorana [1937] E. Majorana, Teoria simmetrica dell’elettrone e del positrone, Il Nuovo Cimento 14, 171 (1937).
  * Wilczek [2009] F. Wilczek, Majorana returns, Nat. Phys. 5, 614 (2009).
  * Alicea [2012] J. Alicea, New directions in the pursuit of majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012).
  * Deng  _et al._ [2012] S. Deng, L. Viola, and G. Ortiz, Majorana modes in time-reversal invariant $s$-wave topological superconductors, Phys. Rev. Lett. 108, 036803 (2012).
  * Qi  _et al._ [2009] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Time-reversal-invariant topological superconductors and superfluids in two and three dimensions, Phys. Rev. Lett. 102, 187001 (2009).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Kitaev [2001] A. Y. Kitaev, Unpaired majorana fermions in quantum wires, Phys. Usp. 44, 131 (2001).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  * Kitaev [2003] A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
  * Beenakker  _et al._ [2019] C. W. J. Beenakker, P. Baireuther, Y. Herasymenko, I. Adagideli, L. Wang, and A. R. Akhmerov, Deterministic creation and braiding of chiral edge vortices, Phys. Rev. Lett. 122, 146803 (2019).
  * Tokura  _et al._ [2019] Y. Tokura, K. Yasuda, and A. Tsukazaki, Magnetic topological insulators, Nat. Rev. Phys. 1, 126 (2019).
  * Qi  _et al._ [2010] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Chiral topological superconductor from the quantum hall state, Phys. Rev. B 82, 184516 (2010).
  * Wang  _et al._ [2015] J. Wang, Q. Zhou, B. Lian, and S.-C. Zhang, Chiral topological superconductor and half-integer conductance plateau from quantum anomalous hall plateau transition, Phys. Rev. B 92, 064520 (2015).
  * Zeng  _et al._ [2018] Y. Zeng, C. Lei, G. Chaudhary, and A. H. MacDonald, Quantum anomalous hall majorana platform, Phys. Rev. B 97, 081102 (2018).
  * Chen  _et al._ [2018] C.-Z. Chen, Y.-M. Xie, J. Liu, P. A. Lee, and K. T. Law, Quasi-one-dimensional quantum anomalous hall systems as new platforms for scalable topological quantum computation, Phys. Rev. B 97, 104504 (2018).
  * Bernevig  _et al._ [2022] B. A. Bernevig, C. Felser, and H. Beidenkopf, Progress and prospects in magnetic topological materials, Nature 603, 41 (2022).
  * He  _et al._ [2017] Q. L. He, L. Pan, A. L. Stern, E. C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E. S. Choi, K. Murata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C. Zhang, K. Liu, J. Xia, and K. L. Wang, Retracted: Chiral majorana fermion modes in a quantum anomalous hall insulator-superconductor structure, Science 357, 294 (2017).
  * Thorp [2022] H. H. Thorp, Editorial retraction, Science 378, 718 (2022).
  * Kayyalha  _et al._ [2020] M. Kayyalha, D. Xiao, R. Zhang, J. Shin, J. Jiang, F. Wang, Y.-F. Zhao, R. Xiao, L. Zhang, K. M. Fijalkowski, P. Mandal, M. Winnerlein, C. Gould, Q. Li, L. W. Molenkamp, M. H. W. Chan, N. Samarth, and C.-Z. Chang, Absence of evidence for chiral majorana modes in quantum anomalous hall-superconductor devices, Science 367, 64 (2020).
  * Zhang  _et al._ [2018] H. Zhang, C.-X. Liu, S. Gazibegovic, D. Xu, J. A. Logan, G. Wang, N. Van Loo, J. D. Bommer, M. W. De Moor, D. Car, _et al._ , Retracted article: Quantized majorana conductance, Nature 556, 74 (2018).
  * Zhang  _et al._ [2021] H. Zhang, C.-X. Liu, S. Gazibegovic, D. Xu, J. A. Logan, G. Wang, N. van Loo, J. D. Bommer, M. W. de Moor, D. Car, _et al._ , Retraction note: Quantized majorana conductance (2021).
  * Prada  _et al._ [2020] E. Prada, P. San-Jose, M. W. de Moor, A. Geresdi, E. J. Lee, J. Klinovaja, D. Loss, J. Nygård, R. Aguado, and L. P. Kouwenhoven, From andreev to majorana bound states in hybrid superconductor–semiconductor nanowires, Nat. Rev. Phys. 2, 575 (2020).
  * Sauls [2018] J. Sauls, Andreev bound states and their signatures, Phil. Trans. R. Soc. A 376, 20180140 (2018).
  * Souto  _et al._ [2022] R. S. Souto, M. M. Wauters, K. Flensberg, M. Leijnse, and M. Burrello, Multiterminal transport spectroscopy of subgap states in coulomb-blockaded superconductors, Phys. Rev. B 106, 235425 (2022).
  * Pan  _et al._ [2021] H. Pan, J. D. Sau, and S. Das Sarma, Three-terminal nonlocal conductance in majorana nanowires: Distinguishing topological and trivial in realistic systems with disorder and inhomogeneous potential, Phys. Rev. B 103, 014513 (2021).
  * Zazunov  _et al._ [2016] A. Zazunov, R. Egger, and A. Levy Yeyati, Low-energy theory of transport in majorana wire junctions, Phys. Rev. B 94, 014502 (2016).
  * Anselmetti  _et al._ [2019] G. L. R. Anselmetti, E. A. Martinez, G. C. Ménard, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, C. M. Marcus, L. Casparis, and A. P. Higginbotham, End-to-end correlated subgap states in hybrid nanowires, Phys. Rev. B 100, 205412 (2019).
  * Pikulin  _et al._ [2021] D. I. Pikulin, B. van Heck, T. Karzig, E. A. Martinez, B. Nijholt, T. Laeven, G. W. Winkler, J. D. Watson, S. Heedt, M. Temurhan, _et al._ , Protocol to identify a topological superconducting phase in a three-terminal device, arXiv preprint arXiv:2103.12217 (2021).
  * Manousakis  _et al._ [2020] J. Manousakis, C. Wille, A. Altland, R. Egger, K. Flensberg, and F. Hassler, Weak measurement protocols for majorana bound state identification, Phys. Rev. Lett. 124, 096801 (2020).
  * Jonckheere  _et al._ [2019] T. Jonckheere, J. Rech, A. Zazunov, R. Egger, A. L. Yeyati, and T. Martin, Giant shot noise from majorana zero modes in topological trijunctions, Phys. Rev. Lett. 122, 097003 (2019).
  * Maiani  _et al._ [2022] A. Maiani, M. Geier, and K. Flensberg, Conductance matrix symmetries of multiterminal semiconductor-superconductor devices, Phys. Rev. B 106, 104516 (2022).
  * Danon  _et al._ [2020] J. Danon, A. B. Hellenes, E. B. Hansen, L. Casparis, A. P. Higginbotham, and K. Flensberg, Nonlocal conductance spectroscopy of andreev bound states: Symmetry relations and bcs charges, Phys. Rev. Lett. 124, 036801 (2020).
  * Ménard  _et al._ [2020] G. C. Ménard, G. L. R. Anselmetti, E. A. Martinez, D. Puglia, F. K. Malinowski, J. S. Lee, S. Choi, M. Pendharkar, C. J. Palmstrøm, K. Flensberg, C. M. Marcus, L. Casparis, and A. P. Higginbotham, Conductance-matrix symmetries of a three-terminal hybrid device, Phys. Rev. Lett. 124, 036802 (2020).
  * Zhang  _et al._ [2009] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in $\text{Bi}_{2}\text{Se}_{3}$, $\text{Bi}_{2}\text{Te}_{3}$ and $\text{Sb}_{2}\text{Te}_{3}$ with a single dirac cone on the surface, Nat. Phys. 5, 438 (2009).
  * Liu  _et al._ [2010] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Model hamiltonian for topological insulators, Phys. Rev. B 82, 045122 (2010).
  * Moors  _et al._ [2018] K. Moors, P. Schüffelgen, D. Rosenbach, T. Schmitt, T. Schäpers, and T. L. Schmidt, Magnetotransport signatures of three-dimensional topological insulator nanostructures, Phys. Rev. B 97, 245429 (2018).
  * De Gennes and Pincus [2018] P.-G. De Gennes and P. A. Pincus,  _Superconductivity of metals and alloys_ (CRC Press, 2018).
  * de Juan  _et al._ [2019] F. de Juan, J. H. Bardarson, and R. Ilan, Conditions for fully gapped topological superconductivity in topological insulator nanowires, Scipost Phys. 6, 060 (2019).
  * Zhang  _et al._ [2010] Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, _et al._ , Crossover of the three-dimensional topological insulator bi2se3 to the two-dimensional limit, Nature Physics 6, 584 (2010).
  * Linder  _et al._ [2009] J. Linder, T. Yokoyama, and A. Sudbø, Anomalous finite size effects on surface states in the topological insulator ${\text{bi}}_{2}{\text{se}}_{3}$, Phys. Rev. B 80, 205401 (2009).
  * Liu  _et al._ [2015] M.-H. Liu, P. Rickhaus, P. Makk, E. Tóvári, R. Maurand, F. Tkatschenko, M. Weiss, C. Schönenberger, and K. Richter, Scalable tight-binding model for graphene, Phys. Rev. Lett. 114, 036601 (2015).
  * Yu  _et al._ [2010] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Quantized anomalous hall effect in magnetic topological insulators, Science 329, 61 (2010).
  * Chang  _et al._ [2023] C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Colloquium: Quantum anomalous hall effect, Rev. Mod. Phys. 95, 011002 (2023).
  * Schnyder  _et al._ [2008] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78, 195125 (2008).
  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * Chang  _et al._ [2013] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator, Science 340, 167 (2013).
  * Deng  _et al._ [2020] Y. Deng, Y. Yu, M. Z. Shi, Z. Guo, Z. Xu, J. Wang, X. H. Chen, and Y. Zhang, Quantum anomalous hall effect in intrinsic magnetic topological insulator $\textrm{MnBi}_{2}\textrm{Te}_{4}$, Science 367, 895 (2020).
  * Chung  _et al._ [2011] S. B. Chung, X.-L. Qi, J. Maciejko, and S.-C. Zhang, Conductance and noise signatures of majorana backscattering, Phys. Rev. B 83, 100512 (2011).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Blonder  _et al._ [1982] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion, Phys. Rev. B 25, 4515 (1982).
  * Lambert  _et al._ [1993] C. Lambert, V. Hui, and S. Robinson, Multi-probe conductance formulae for mesoscopic superconductors, J. Phys.: Condens. Matter 5, 4187 (1993).
  * Lim  _et al._ [2012] J. S. Lim, R. Lopez, and L. Serra, Transport through majorana nanowires attached to normal leads, New J. Phys. 14, 083020 (2012).
  * Lian  _et al._ [2016] B. Lian, J. Wang, and S.-C. Zhang, Edge-state-induced andreev oscillation in quantum anomalous hall insulator-superconductor junctions, Phys. Rev. B 93, 161401 (2016).
  * Setiawan  _et al._ [2015] F. Setiawan, P. M. R. Brydon, J. D. Sau, and S. Das Sarma, Conductance spectroscopy of topological superconductor wire junctions, Phys. Rev. B 91, 214513 (2015).
  * Law  _et al._ [2009] K. T. Law, P. A. Lee, and T. K. Ng, Majorana fermion induced resonant andreev reflection, Phys. Rev. Lett. 103, 237001 (2009).
  * Osca and Serra [2018] J. Osca and L. Serra, Conductance oscillations and speed of chiral majorana mode in a quantum anomalous hall two-dimensional strip, Phys. Rev. B 98, 121407 (2018).
  * Burke  _et al._ [2023] D. Burke, D. Heffels, K. Moors, P. Schüffelgen, D. Grützmacher, and M. R. Connolly, Robust majorana bound states in magnetic topological insulator nanoribbons with fragile chiral edge channels, arXiv preprint arXiv:2302.10982 (2023).
  * Osca and Serra [2019] J. Osca and L. Serra, Complex band-structure analysis and topological physics of Majorana nanowires, Eur. Phys. J. B 92, 101 (2019).
  * Benchtaber  _et al._ [2021] N. Benchtaber, D. Sánchez, and L. Serra, Scattering of topological kink-antikink states in bilayer graphene structures, Phys. Rev. B 104, 155303 (2021).


# Review of the measurements of the strong coupling constant in CMS at 13 TeV

A. Potrebko on behalf of the CMS collaboration  Institute of Particle Physics
and Accelerator Technologies
Faculty of Natural Sciences and Technology
Riga Technical University, 7 Paula Valdena, Riga LV-1048, Latvia

###### Abstract

The strong coupling constant is the least known of the coupling constants in
the standard model. Nevertheless it appears in the calculations of cross
sections of all the processes at the LHC. We present a review of the strong
coupling constant measurements conducted at the CMS experiment, focusing on
those performed at a center-of-mass of 13 TeV.

##  1 Introduction

The strong coupling constant ($\alpha_{S}$) is the only free parameter in QCD
and is also the least known of the coupling constants in the standard model.
The world average value of $\alpha_{S}$ at the reference point of Z boson pole
mass is $\alpha_{S}(m_{Z})=0.1179\pm 0.0009$. Nevertheless, powers of
$\alpha_{S}$ appear in the calculations of perturbative quantum chromodynamics
(pQCD) for cross sections of virtually all processes measured in LHC. The
value of $\alpha_{S}$ decreases with the energy scale of the underlying
process, $Q$, as predicted by the renormalization group equations (RGE)
starting from the Landau pole value $\Lambda_{QCD}\approx 0.2$ GeV. The
consistency of the running of $\alpha_{S}$ with the RGE has been shown to be
consistent over three orders of magnitude (see, figure 1). In the CMS
experiment [2] $\alpha_{S}$ has been measured in vector boson, top, jet and
jet substructure measurements. In this review, we summarize the measurements
performed in the CMS at the center-of-mass energy $\sqrt{s}=13$ TeV, first
highlighting three new measurements.

##  2 New measurements of the strong coupling constant in CMS

A novel approach in HEP is to measure $\alpha_{S}$ using energy correlators
inside jets [1]. N-particle energy correlators (ENC) describe the correlations
of kinematic properties of particles inside jets. In the new measurement, two-
and three-particle energy correlators (E2C and E3C, respectively), were
measured. E2C (E3C) sum up all the combinations of pairs (triplets) of
particles inside jets, scaled with an energy weight. For example,

$\mathrm{E3C}=\sum_{i,j,k}\int
d\sigma\frac{E_{i}E_{j}E_{k}}{E^{3}}\delta\Bigl{(}x_{L}-\mathrm{max}(\Delta
R_{i,j},\Delta R_{i,k},\Delta R_{j,k})\Bigr{)},$ (1)

where $\Delta R_{i,j}=\sqrt{(\Delta\eta_{i,j})^{2}+(\Delta\phi_{i,j})^{2}}$
represents the angular distance between two constituents. The dependence of
E2C and E3C on $x_{L}$ reveals mappings of various stages of parton showers.
Short $x_{L}$ describe the final stages of fragmentation, that is free hadrons
that are uncorrelated. The scaling is linear in this case. Conversely, large
$x_{L}$ describe interacting partons, characterized by an inverse scaling.

In this measurement, dijet events with jet rapidity $|\eta^{jet}|<2.1$ and
transverse momentum $p_{T}^{jet}>97$ GeV were selected. In jets, all charged
and neutral particles with $p_{T}>1$ GeV were selected. The data used for the
extraction of $\alpha_{S}$ were corrected for the detector effects (unfolded).
The ratio of energy correlators $\mathrm{E2C/E3C}\leavevmode\nobreak\
\sim\alpha\mathrm{log}R$. In this way for the extraction of $\alpha_{S}$,
$\mathrm{E2C/E3C}$ was compared to analytical calculations available at an
approximate next-to-next-to-leading logarithm
($\mathrm{NNLL}_{\mathrm{approx}}$) matched to next-to-leading order (NLO)
perturbative QCD (pQCD) calculations. This yielded the worlds most precise
$\alpha_{S}$ measurement from jet substructure:
$\alpha_{S}=0.1229^{+0.0040}_{-0.0050}$. While the uncertainty of the
measurement is larger compared to measurements obtained from jet and top quark
cross section measurements, $\alpha_{S}$ obtained from jet substructure is
more sensitive to collinear effect. In this way, this measurement helps to
probe the consistency of $\alpha_{S}$ in different phase spaces.

Another new measurement of $\alpha_{S}$ at CMS was performed using azimuthal
correlations among jets [3, 4]. A ratio observable

$R_{\Delta\phi}(p_{T})=\frac{\sum_{i=0}^{N_{jet}(p_{T})}N_{nbr}^{(i)}(\Delta\phi,p_{T,min}^{nbr})}{N_{jet}(p_{T})}$
(2)

was defined, where the denominator counts the number of jets pet $p_{T}$ bin
while the numerator counts the number of neighboring jets for a given jet $i$.
Neighboring jets in this measurement were taken to fall within the interval
$\frac{2\pi}{3}<\Delta\phi<\frac{7\pi}{8}$, i.e., neighboring jets do not have
to be spatially close to each other. Such a choice of $\Delta\phi$ ensures
that in the dijet event, each jet has 0 neighbors, resulting in the numerator
of $R_{\Delta\phi}$ counting only the 3+ jet topologies while the denominator
counts all the jets.

Figure 1: Slope of the ratio of the three- and two-particle energy
correlators, E3C/E2C, with respect to the distance between the constituents,
$x_{L}$, for different jet energies [1] (left). Unfolded data are overlaid
with analytical calculations at $\mathrm{NNLL}_{\mathrm{approx}}$ accuracy.
The ratio section illustrates changes in the slope of E3C/E2C for values of
$\alpha_{S}$ higher ($\alpha_{S}=0.136$) and lower ($\alpha_{S}=0.100$) than
the default value of $\alpha_{S}=0.118$. Running of the strong coupling
constant using the world average (yellow band) compared to measurements
performed at different scales, $Q$ [3] (right). Four new extractions from the
$R\Delta_{\phi}$ measurement, reaching the highest scale probed to date, are
added.

For the extraction of $\alpha_{S}$, perturbative QCD calculation using
NLOJet++ within fastNLO framework was compared to the unfolded data. Non-
perturbative (NP) effects for data were estimated using Monte Carlo (MC)
predictions taking a ratio of the distributions with and without NP effects
$C^{NP}=(\sigma^{PS+MPI+HAD})/\sigma^{PS}$. NLO electroweak (EW) effects were
estimated from the SHERPA MC generator in interfaced to RECOLA. The fit
yielded $\alpha_{S}(m_{Z})=0.1177^{+0.0117}_{-0.0074}$. The main contribution
to the uncertainty is the scale uncertainty (${}^{+114}_{-68}$) which could be
reduced by around threefold if next-to-NLO (NNLO) predictions were available.
Experimental, NP and EW uncertainties are small due to cancellations when
taking the ratio. The measurement also probed running of $\alpha_{S}$,
splitting the whole $p_{T}$ range into four subregions. Notably, the
measurement reaches to the highest scale to date, $Q\approx 2081$ TeV showing
no deviations from RGE.

In the multi-differential dijet cross section measurement, a double-
differential (2D) cross section as a function of the invariant mass of the two
jets, $m_{1,2}$, and the largest absolute rapidity, $|y|_{max}$, was used to
extract $\alpha_{S}$ [5]. A simultaneous fit of parton distribution functions
(PDF) and $\alpha_{S}$ was performed by fitting the unfolded data to pQCD
predictions available up to NNLO. The predictions were obtained with the
NNLOJET program interfaced with fastNLO. The fit resulted in
$\alpha_{S}(m_{Z})=0.1179\pm 0.0019$. A triple-differential (3D) distribution
as a function of the rapidity separation, $y^{*}$, the total boost, $y_{b}$,
and $m_{1,2}$ was also used to extract $\alpha_{S}$ and yielded
$\alpha_{S}=0.1181\pm 0.022$ which is in a good agreement with the 2D result.
The main uncertainty in this measurement is the fit, i.e., the experimental
uncertainty mostly caused by the jet energy scale uncertainty and luminosity.

##  3 Summary of the measurements of the strong coupling constant in CMS

A summary of the measurements of $\alpha_{S}$ in CMS at $\sqrt{s}=$ 7, 8 and
13 TeV can be seen in figure 2. In this section, we will review the
measurements of $\alpha_{S}$ at $\sqrt{s}=13$ Tev, not mentioned in section 2.

\begin{overpic}[width=429.28616pt]{pics/alpha_S.pdf} \put(90.0,585.0){ \href
https://link-springer-com.ezproxy.cern.ch/article/10.1007/JHEP06(2020)018 }
\put(90.0,559.0){ \href
https://www.sciencedirect.com/science/article/pii/S0370269313009854 }
\put(90.0,520.0){ \href https://link-springer-
com.ezproxy.cern.ch/article/10.1140/epjc/s10052-019-6863-8 } \put(90.0,485.0){
\href https://link-springer-
com.ezproxy.cern.ch/article/10.1140/epjc/s10052-020-7917-7 } \put(90.0,450.0){
\href https://journals-aps-
org.ezproxy.cern.ch/prd/abstract/10.1103/PhysRevD.98.092014 }
\put(90.0,130.0){ \href https://cms-results.web.cern.ch/cms-results/public-
results/publications/SMP/ALPHAS.html } \end{overpic} Figure 2: Summary of the
$\alpha_{S}$ $(m_{Z})$ measurements in CMS and the world average (black maker
and the yellow band). The error bands illustrate the full uncertainty of the
measurement.

In the inclusive jet measurement [6], $\alpha_{S}$ was extracted from a fit to
a 2D cross section of jet $p_{T}$ and $|\eta|$. A simultaneous fit of PDF and
$\alpha_{S}$ was performed using NNLO predictions corrected for NP and EW
effects. The value obtained is $\alpha_{s}(m_{Z})=0.1166\pm 0.0017$. The
uncertainty is dominated by the fit uncertainty primarily stemming from the
jet energy scale uncertainty.

When fitting $\alpha_{S}$ using top quark datasets, the simultaneous fit
should also include top mass, $m_{t}$. However, in the inclusive cross section
measurement only one parameter can be used to extract $\alpha_{s}$. In the top
quark pair ($t\overline{t}$) inclusive cross section measurement in the two
lepton decay channel, instead of a simultaneous fit, $\alpha_{s}$ was
extracted several times for different PDF sets using $m_{t}$ default to each
PDF set [7]. The obtained value, $\alpha_{s}=0.1151^{+0.0040}_{0.0035}$, is
consistent with the world average. In addition, the fit stability was assessed
by repeating the fit for different $m_{t}$ values. The result for $\alpha_{S}$
was found to deviate by no more than one sigma if $m_{t}$ default to other PDF
sets was used instead.

In the multi-differential $t\overline{t}$ cross section measurement,
$\alpha_{S}$ was extracted from a 3D distribution of the number of jets in the
event, $N_{jet}$, mass and the absolute rapidity of the $t\overline{t}$
system, $M(t\overline{t})$ and $\eta(t\overline{t})|$ respectively [8]. A
simultaneous fit of $\alpha_{S}$, $m_{t}$ and PDF was performed compared to
NLO calculations. The value of $\alpha_{S}(m_{Z})=0.1135^{+0.0021}_{-0.0017}$
and $m_{t}=170.5\pm 0.8$ GeV was obtained. In the measurement possible effects
from Coulomb and soft-gluon resummation at the $t\overline{t}$ threshold were
neglected. This would cause an increase of $m_{t}$ by an order of 1 GeV and
would also pull $\alpha_{S}$ by an order of 0.001 to a higher value. Despite
the absence of this contribution, the competitive uncertainty suggests that
the data should be reexamined. While the given measurement only analyzed the
CMS data acquired in 2016, a new measurement extended to the full Run 2 data
is now available and can be included in the extraction of $\alpha_{S}$ [9].

Finally $\alpha_{S}$ was extracted from bottom jet substructure in
$t\overline{t}$ events [10]. In this measurement, the angle between the
groomed subjects, $\Delta R_{g}$, was used for $\alpha_{S}$ extraction. The
data were fitted with the predictions generated by the POWHEG generator and
showered using the PYTHIA 8 program. This only provides LO+LL precision for
distributions within jets, resulting in a large uncertainty in the final
measurement $\alpha_{S}=0.115^{+0.015}_{-0.013}$. Soft-gluon emissions in this
measurement were incorporated using the CMW scheme.

##  4 Summary

At $\sqrt{s}=13$ TeV, the CMS Collaboration has provided 7 different
$\alpha_{S}$ extractions spanning jet, top and jet substructure measurements.
Where NNLO calculation are available the experimental fit uncertainties are
typically the dominant ones. An option to mitigate this is to use ratio
observables like $R_{\Delta\phi}$ or $R_{32}$ where several uncertainties
cancel out. Conversely, measurements like $R_{\Delta\phi}$ and jet
substructure would greatly benefit from higher-order predictions.

## References

## References

  * [1] CMS Collaboration, submitted to Phys. Rev. Lett., arXiv:2402.13864
  * [2] CMS Collaboration, JINST 3 (2008) S08004
  * [3] CMS Collaboration, CMS-PAS-CMP-22-005, https://cds.cern.ch/record/2868568
  * [4] CMS Collaboration, submitted to EPJC, arXiv:2404.16082
  * [5] CMS Collaboration, submitted to EPJC arXiv:2312.16669
  * [6] CMS Collaboration, accepted to JHEP, arXiv:2111.10431
  * [7] CMS Collaboration, accepted to EPJC, arXiv:1812.10505
  * [8] CMS Collaboration, accepted to EPJC, arXiv:1904.05237
  * [9] CMS Collaboration, accepted to JHEP, arXiv:2402.08486
  * [10] CMS Collaboration, accepted to Phys. Rev. D, arXiv:1808.07340


Probing the nature of electroweak symmetry breaking with Higgs boson pairs in
ATLAS

Bartłomiej Żabiński
On behalf of the ATLAS Collaboration

The Henryk Niewodniczański
Institute of Nuclear Physics
Polish Academy of Sciences

> Constraints on the Higgs boson trilinear self-coupling modifier
> $\kappa_{\lambda}$ and non-SM HHVV coupling strength $\kappa_{2V}$ are set
> by combining di-Higgs boson analyses using $b\bar{b}b\bar{b}$,
> $b\bar{b}\tau^{+}\tau^{-}$ and $b\bar{b}\gamma\gamma$ decay channels. The
> data used in these analyses were recorded by the ATLAS detector at the Large
> Hadron Collider in proton–proton collisions at $\sqrt{s}$ = 13 TeV and
> corresponding to an integrated luminosity of 126–139 $fb^{-1}$. The
> combination of the di-Higgs analyses sets an upper limit of signal strength
> $\mu_{HH}<$ 2.4 at 95% confidence level on the di-Higgs production and
> constraints for $\kappa_{\lambda}$ between -0.6 and 6.6. The obtained
> confidence interval for $\kappa_{2V}$ coupling modifier are [0.1,2.0] . The
> High Luminosity Large Hadron Collider prospects have been considered as
> well. The expected signal strength is 0.55 and $\kappa_{\lambda}$ between
> 0.0 and 2.5 for the baseline scenario.

> PRESENTED AT
>
>
>
>
> DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related
> Subjects,
> Michigan State University, USA, 27-31 March 2023
>

111$\copyright$ [2023] CERN for the benefit of the ATLAS Collaboration.
Reproduction of this article or parts of it is allowed as specified in the CC-
BY-4.0 license

##  1 Introduction

The discovery of the Higgs boson by ATLAS [1] and CMS [2] experiments at the
CERN Large Hadron Collider [3] opened opportunities to measure the boson’s
properties and test new hypotheses. In the Standard Model [4, 5, 6] Higgs
boson potential [7, 8] provides spontaneous electroweak symmetry breaking that
gives rise to its self-interactions. The Higgs boson self-interactions can be
described among others by trilinear self-coupling $\lambda_{HHH}$ that can be
predicted at lowest order from Fermi constant $G_{F}$ [9] and the Higgs mass
$m_{H}$ expressed by equation (1).

$\lambda_{HHH}=\frac{m^{2}_{H}G_{F}}{\sqrt{2}}$ (1)

The validity of the SM in the Higgs sector can be tested using the ‘kappa
framework’ [10, 11], in which a coupling modifier $\kappa_{m}$ is defined as
the ratio of the coupling strength between the particle m and Higgs boson to
its SM value. The deviation of $\kappa_{m}$ from unity would indicate physics
processes beyond SM. For di-Higgs production $\kappa_{\lambda},\kappa_{t}$,
$\kappa_{V}$ and $\kappa_{2V}$ are considered. The $\kappa_{V}$ and
$\kappa_{t}$ describe modifying the SM Higgs boson coupling to W or Z bosons
and up-type quarks, respectively. The $\kappa_{2V}$ is related to the VVHH
interaction vertex. The $\kappa_{\lambda}$ coupling modifiers can be measured
in the gluon-gluon fusion process (ggF HH) and vector boson fusion (VBF). The
$\kappa_{V}$ and $\kappa_{2V}$ can be measured in the VBF processes,
$\kappa_{2V}$ = 0 in the SM. At the LHC, Higgs boson pair-production is
dominated by ggF processes and the overall cross-section in the SM is
$\sigma^{SM}_{ggF}(pp\rightarrow HH)$ = $31.05^{+6\%}_{-23\%}$ fb at
$\sqrt{s}$ = 13 TeV [12]. The second largest Higgs pair-production process at
LHC is VBF with $\sigma^{SM}_{VBF}$ = $1.72\pm 0.04$ fb at $\sqrt{s}$ = 13 TeV
[13, 14]. Figure 1 represents leading order diagrams of the ggF and VBF
production processes at the LHC featuring coupling modifiers described above.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 1: Examples of leading-order Feynman diagrams for Higgs boson pair
production: for ggF production, diagram (a) is proportional to the square of
the top-quark Yukawa coupling, while diagram (b) is proportional to the
product of the top-quark Yukawa coupling and the Higgs boson self-coupling.
For VBF production, diagram (c) and diagram (d) are proportional to the
interactions $\kappa_{V}$ and $\kappa_{2V}$ between two vectors bosons and two
Higgs bosons [18].

##  2 Higgs boson pair production in ATLAS

The Higgs pair-production is measured in different decay channels, but in this
note, only the most sensitive channels $b\bar{b}b\bar{b}$,
$b\bar{b}\gamma\gamma$ and $b\bar{b}\tau\tau$ [15, 16, 17] and their
combination [18] will be discussed. Analysis-specific details can be found in
the above-referenced publications. The first measured parameter of the
combined results is signal strength $\mu_{HH}$ defined as a ratio of di-Higgs
production cross-section including only ggF and VBF processes to its SM
prediction of 32.7 fb [19, 20, 21]. The expected upper limit of $\mu_{HH}$ is
2.9. The observed upper limit in the absence of HH production, at 95% CL, is
2.4. The best-fit value obtained from the fit to the data $\mu_{HH}$=-0.7
$\pm$ 1.3. is compatible with the SM prediction of unity, with $p$-value of
0.2. The obtained limits of $\mu_{HH}$ for each particular channel entering
the combination and the results of the combination itself are shown in Figure
2 Figure 1.

Figure 2: Observed and expected 95% CL upper limits on the signal strength for
di-Higgs production from the $b\bar{b}b\bar{b}$, $b\bar{b}\tau^{+}\tau^{-}$
and $b\bar{b}\gamma\gamma$ decay channels, and their statistical combination.
The value $m_{H}$ = 125.09 GeV is assumed when deriving the predicted SM
cross-section. The expected limit and the corresponding error bands are
derived assuming the absence of the HH process and with all nuisance
parameters profiled to the observed data [18].

The combination constraints for $\kappa_{\lambda}$ and $\kappa_{2V}$ coupling
modifiers, related to $VVHH$ interaction vertex, are shown in Figure 3.
Presented limits for $\kappa_{\lambda}$ and $\kappa_{2V}$ are obtained by
using the values of the test statistic as a function of $\kappa_{\lambda}$ in
the asymptotic approximation and including the theoretical uncertainty of the
cross-section predictions. The obtained confidence interval for
$\kappa_{\lambda}$ in each discussed analysis are collected in Table 1, and
the combination constrain at 95% CL are -0.6 $<\kappa_{\lambda}<$ 6.6
(observed) and -2.1 $<\kappa_{\lambda}<$7.8 (expected). The $\kappa_{2V}$
coupling modifier for each mentioned channel and their combination have been
obtained by fixing all other couplings modifiers to unity and with the
expected value derived from the SM hypothesis. The expected and observed at
95% CL combined constraint for $\kappa_{2V}$ are 0.0 $<\kappa_{2V}<$ 2.1 and
0.1 $<\kappa_{2V}<$ 2.0, respectively. Table 1 includes $\kappa_{2V}$
constraints for each channel under combination.

|
---|---
(a) | (b)
Figure 3: Observed and expected 95% CL exclusion limits on the production cross-sections of (a) the combined ggF HH and VBF HH processes as a function of $\kappa_{\lambda}$ and (b) the VBF HH process as a function of $\kappa_{2V}$, for the three di-Higgs search channels and their combination. The expected limits assume no HH production or no VBF HH production, respectively. The red line shows (a) the theory prediction for the combined ggF HH and VBF HH cross-section as a function of $\kappa_{\lambda}$ where all parameters and couplings are set to their SM values except for $\kappa_{\lambda}$, and (b) the predicted VBF HH cross-section as a function of $\kappa_{2V}$. The bands surrounding the red cross-section lines indicate the theoretical uncertainty band in (b) is smaller than the width of the plotted line [18]. | $b\bar{b}b\bar{b}$ | $b\bar{b}\tau\tau$ | $b\bar{b}\gamma\gamma$ | combination
---|---|---|---|---
Observed 95% CL |  -3.3 $<\kappa_{\lambda}<$ 11.4 |  -2.7 $<\kappa_{\lambda}<$ 9.5 |  -1.4 $<\kappa_{\lambda}<$ 6.5 |  -0.6 $<\kappa_{\lambda}<$ 6.6
Expected 95% CL |  -5.2 $<\kappa_{\lambda}<$ 11.6 |  -3.1 $<\kappa_{\lambda}<$ 10.2 |  -3.2 $<\kappa_{\lambda}<$ 8.1 |  -2.1 $<\kappa_{\lambda}<$ 7.8
Obs. value | $\kappa_{\lambda}=6.2^{+3.0}_{-5.2}$ | $\kappa_{\lambda}=1.5^{+5.9}_{-2.5}$ | $\kappa_{\lambda}=2.8^{+2.0}_{-2.2}$ | $\kappa_{\lambda}=3.1^{+1.9}_{-2.0}$
Observed 95% CL |  0.0 $<\kappa_{2V}<$ 2.1 |  -0.6 $<\kappa_{2V}<$ 2.7 |  -0.8 $<\kappa_{2V}<$ 3.0 |  0.1 $<\kappa_{2V}<$ 2.0
Expected 95% CL |  -0.0 $<\kappa_{2V}<$ 2.1 |  -0.5 $<\kappa_{2V}<$ 2.7 |  -1.6 $<\kappa_{2V}<$ 3.7 |  0.0 $<\kappa_{2V}<$ 2.1
Obs. value | $\kappa_{2V}=1.0^{+0.7}_{-0.6}$ | $\kappa_{2V}=1.5^{+0.7}_{-1.7}$ | $\kappa_{2V}=1.1^{+1.0}_{-1.0}$ | $\kappa_{2V}=1.1^{+0.6}_{-0.6}$
Table 1: Summary of $\kappa_{\lambda}$ and $\kappa_{2V}$ observed and expected
constraints and corresponding observed best-fit values with their
uncertainties for the $HH\rightarrow b\bar{b}b\bar{b}$, $HH\rightarrow
b\bar{b}\tau\tau$, $HH\rightarrow b\bar{b}\gamma\gamma$, analyses and for the
di-Higgs combination. Limits are obtained using the test statistic (-2
ln$\Lambda$) in the asymptotic approximation. The expected constraints are
derived under the SM assumption. All other coupling modifiers are fixed to the
SM value.

##  3 Prospects of double Higgs production at the HL-LHC

The High Luminosity LHC (HL-LHC) is expected to start in 2029. During the HL-
LHC operation it is expected to obtain a total integrated luminosity of 3000
$fb^{-1}$ at a center of mass energy $\sqrt[]{s}$ = 14 TeV. The expected total
number of the collected data is significantly larger than the collected data
during the LHC Run-2 offers great conditions to measure the Higgs trilinear
self-coupling. The prospects of the di-Higgs production at HL-LHC were
obtained by scaling full Run-2 distributions and considering multiple
scenarios of systematic uncertainties [22]. The scale factors are applied to
take into account the integrated luminosity and increase of center-of-mass
energy. Four cases of future uncertainties are considered:

  * •

Only statistical uncertainties are considered (No syst. unc).

  * •

A baseline scenario where the relevant systematic uncertainties were scaled
down with respect to expected HL-LHC data delivered [23]. The data-driven
background uncertainties are also reduced according to the integrated
luminosity. This relies on a 50% reduction of the bootstrap uncertainty at the
HL-LHC, while the shape uncertainty is assumed to be identical to Run-2.

  * •

A scenario where experimental uncertainties from Run-2 are unchanged and
theoretical uncertainties associated with HH signal are halved (Theoretical
unc. halved).

  * •

The experimental uncertainties from Run-2 are left unchanged (Run-2 syst.
unc).

Figure 4 represents the result of the prospect studies. The signal strength
for the baseline scenario at 95% CL is 0.55. The constraint on
$\kappa_{\lambda}$ in the same scenario is within the range from 0.0 to 2.5.
The results for the rest of the scenarios and more details about discussed
prospects are in the note [22].

##  4 Conclusions

The double Higgs production searches offer a unique opportunity to investigate
the mechanism of electroweak symmetry-breaking. The measurement of
$\lambda_{HHH}$ is challenging due to very small cross-sections of the di-
Higgs production processes. Constraints are set on the signal strength $\mu$
as well as on $\kappa_{\lambda}$ and $\kappa_{2V}$. Obtained limits scope for
each considered channel are wide, but by the combination of all channels, the
constraints have been limited to 0.1 $<\kappa_{2V}<$ 2.0 and -0.6
$<\kappa_{\lambda}<$ 6.6. The di-Higgs searches are also a good probe for
beyond Standard Model heavy resonance searches [24, 25, 26]. The ATLAS
experiment is in the Run-3 phase, where it is expected to collect two times
more data than during Run-2 and finally reach LHC runs with high luminosity.
That significant amount of data will offer spectacular results in the
mentioned channels and even more, so please stay tuned.

|
---|---
(a) | (b)
Figure 4:  Projected 95% CL upper limits on the signal strength (a) and
negative log-profile-likelihood as a function of $\kappa_{\lambda}$ evaluated
on Asimov datasets constructed under the SM hypothesis of $\kappa_{\lambda}$ =
1 (b), combining the $b\bar{b}b\bar{b}$, $b\bar{b}\gamma\gamma$ and
$b\bar{b}\tau\tau$ channels at 3000 $fb^{-1}$ and $\sqrt{s}$ = 14 TeV for four
uncertainty scenarios described in the text. The limits are derived assuming
no HH production.

Funding informaion This work is supported in part by the Polish Ministry of
Education and Science project no. 2022/WK/08.

## References

  * [1] ATLAS Collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Physics Letters B 716 (2012) 1, arXiv:1207.7214 [hep-ex]
  * [2] CMS Collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Physics Letters B 716 (2012) 30, arXiv:1207.7235
  * [3] L. Evans, P. Bryant (Eds.), LHC Machine, JINST 3 (2008) S08001.
  * [4] S.L. Glashow, Partial-symmetries of weak interactions, Nucl. Phys. 22 (1961) 579.
  * [5] A.Salam, Weak and Electromagnetic Interactions, Conf. Proc. C680519 (1968). Proceedings of the eight Nobel sumphosium.
  * [6] S.Weinberg, A model of leptons, Phys. Rev. Lett. 19 (1967) 1264.
  * [7] F.Englert and R.Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett 13 (1964) 321.
  * [8] P.W.Higgs, Broken symmetries, massles particles and gauge fields, Phys. Rev. Lett. 12 (1964) 132.
  * [9] Particle Data Group, R.L. Workman et al. Review of Particle Physics, PTEP 2022 (2022) 083C01.
  * [10] LHC Higgs Cross Section Working Group, S. Heinemeyer et al.,
Handbook of LHC Higgs Cross Sections:3. Higgs Properties, CERN-2013-004
(CERN,Geneva,2013), arXiv:1307.1347 [hep-ph].

  * [11] LHC Higgs Cross Section Working Group, D. de Florian et al.,
Handbook of LHC Higgs Cross Sections:4. Description of Nature of the Higgs
Sector, CERN-2017-002-M (CERN,Geneva,2017), arXiv:1610.07922 [hep-ph].

  * [12] J. Baglio, F. Campanario, S. Glaus, M. Mühlleitner, J. Ronca and M. Spira, $gg\to HH$ : Combined uncertainties, Phys. Rev. D 103 (2021) no.5, 056002 doi:10.1103/PhysRevD.103.056002, arXiv:2008.11626 [hep-ph]
  * [13] R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, P. Torrielli, E. Vryonidou, and M. Zaro, Higgs pair production at the LHC with NLO and parton-shower effects, Phys. Lett. B 732 (2014) 142, arXiv:1401.7340
  * [14] Frédéric A. Dreyer, Alexander Karlberg, Fully differential Vector-Boson Fusion Higgs Pair Production at Next-to-Next-to-Leading Order, Phys. Rev. D 99, 074028 (2019), arXiv:1811.07918 [hep-ph]
  * [15] ATLAS Collaboration, Search for nonresonant pair production of Higgs bosons in the $b\bar{b}b\bar{b}$ final state in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, CERN-EP-2022-235, arXiv:2301.03212 [hep-ex]
  * [16] ATLAS Collaboration, Search for Higgs boson pair production in the two bottom quarks plus two photons final state in $pp$ collisions at $\sqrt{s}=13\mathrm{TeV}$ with the ATLAS detector, Phys. Rev. D, 106 (2022) 052001, arXiv:2112.11876 [hep-ex]
  * [17] ATLAS Collaboration, Search for resonant and non-resonant Higgs boson pair production in the $b\bar{b}\tau^{+}\tau^{-}$ decay channel using 13 TeV $pp$ collision data from the ATLAS detector, CERN-EP-2022-109, arXiv:2209.10910 [hep-ex]
  * [18] ATLAS Collaboration, Constraints on the Higgs boson self-coupling from single- and double-Higgs production with the ATLAS detector using pp collisions at $\sqrt{=}13$ TeV, Phys. Lett. B, 843 (2023) 137745, arXiv:2211.01216 [hep-ex]
  * [19] M. Grazzini et al., Higgs boson pair production at NNLO with top quark mass effects, JHEP 05 (2018) 059, arXiv: 1803.02463 [hep-ph].
  * [20] G. Heinrich, S. P. Jones, M. Kerner, G. Luisoni and L. Scyboz, Probing the trilinear Higgs boson coupling in di-Higgs production at NLO QCD including parton shower effects, JHEP 06 (2019) 066, arXiv: 1903.08137 [hep-ph].
  * [21] D. de Florian and J. Mazzitelli, Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD, Phys. Rev. Lett. 111 (2013) 201801, arXiv: 1309.6594 [hep-ph].
  * [22] ATLAS Collaboration, HL-LHC prospects for the measurement of Higgs boson pair production in the $b\bar{b}b\bar{b}$ final state and combination with the $b\bar{b}\gamma\gamma$ and $b\bar{b}\tau\tau$ final states at the ATLAS experiment, ATL-PHYS-PUB-2022-053
  * [23] A. Dainese et al., Report on the Physics at the HL-LHC, and Perspectives for the HE-LHC, CERN Yellow Report CERN-2019-007, 2019, url: https://cds.cern.ch/record/2703572 (cit. on pp. 2, 6).
  * [24] ATLAS Collaboration, Search for Higgs boson pair production in the two bottom quarks plus two photons final state in $pp$ collisions at $\sqrt{(}s)$=13 TeV with the ATLAS detector, Phys. Rev. D 106 (2022) 052001, arXiv:2112.11876 [hep-ex].
  * [25] ATLAS Collaboration, Reconstruction and identification of boosted di$-\tau$ systems in a search for Higgs boson pairs using 13 TeV proton-proton collision data in ATLAS, JHEP 11 (2020) 163, arXiv:2007.14811 [hep-ex].
  * [26] ATLAS Collaboration, Search for resonant pair production of Higgs bosons in the $b\bar{b}b\bar{b}$ final state using pp collisions at $\sqrt{(}s)$ = 13 TeV with the ATLAS detector, Phys. Rev. D 105 (2022) 092002, arXiv:2202.07288 [hep-ex].