# Phenomenology of doublet-triplet fermionic dark matter in nonstandard
cosmology and multicomponent dark sectors

Amalia Betancur1,2<EMAIL_ADDRESS>Óscar Zapata2
<EMAIL_ADDRESS>1 Grupo Física Teórica y Aplicada, Universidad
EIA, A.A. 7516, Medellín, Colombia,  2 Instituto de Física, Universidad de
Antioquia, Calle 70 No. 52-21, Medellín, Colombia.

###### Abstract

We consider the doublet-triplet fermionic model in the scheme of the custodial
limit when the dark matter (DM) candidate is pure doublet and lies at the
electroweak scale. This scheme, despite being an appealing and promising DM
model, is severely constrained by the interplay between the DM relic density
constraint and the LHC measurement of the Higgs diphoton decay rate. In this
work, the DM relic density is considered to arise from either a nonstandard
cosmology scenario or as a part of a larger sector encompassing other DM
particles, this in order to saturate the observed relic abundance. For these
scenarios we investigate the impact of the new sector in different collider
observables, and study constraints coming from direct detection and indirect
detection of gamma-rays both in the diffuse and linelike spectrum. As a
result, we find that in the nonstandard cosmology scenario most experiments
impose, up to a certain point, restrictions, though large portions of the
parameter space are still viable. For the multicomponent case, only direct
detection imposes constraints.

##  I Introduction

As early as 1930s, astronomical observations hinted the existence of an
unknown form of matter. In the last decades the evidence for the existence of
this type of matter, dubbed dark matter (DM) Zwicky:1933gu , has become
abundant and overwhelming Tanabashi:2018oca . We now know that it makes up
about 80$\%$ of the universe matter-content Aghanim:2018eyx  pointing out to
the existence of one or several new particles that are not part of the
Standard Model (SM) of particle physics. Among the DM canditates Feng:2010gw ,
weakly interacting massive particles (WIMPs) Steigman:1984ac  are some the
most well-motivated candidates since the thermal annihilation cross section
needed to account for the observed DM relic density is obtained for DM
particles with electroweak interactions and masses. That is, the WIMPs lie
naturally at what is expected to be the scale of physics beyond the Standard
Model (BSM). Moreover, their abundance Steigman:2012nb  is governed by the
generic mechanism of chemical frezee-out which has also played a role in the
abundance of light elements as well as the cosmic microwave background
radiation Kolb:1990vq , both of which are in stark agreement with current
observations.

It is worth noting, however, that the WIMP paradigm is not free of challenges,
both at theoretical and experimental levels Baer:2014eja . For instance, it is
not always a given that a WIMP candidate will automatically account the total
DM relic abundance, which, in some cases, implies the need for some degree of
fine-tuning in order for the models to still be viable111In particular, one of
the most well studied WIMPs, the neutralino, tends to yield too much relic
abundance if it is bino-like while its relic abundance tends to be suppressed
(as long as its mass is below 2.4 TeV) if it is mainly wino Jungman:1995df ;
Bertone:2004pz .. Additionally, despite large efforts to find evidence of
WIMPs through production at colliders, elastic scattering with heavy nuclei
while passing through the Earth or observation of the self-annihilation
byproducts in regions with high DM density, no concluding evidence has been
reported. The null results have lead to more and more constraints on the
parameter space of popular WIMP models Bertone:2010at ; Baer:2014eja ;
Escudero:2016gzx ; Arcadi:2017kky .

Among the different approaches to overcome the challenges on WIMP models, we
recall those that depart completely or partially from standard cosmology
scenario Kamionkowski:1990ni ; Giudice:2000ex ; Gelmini:2006pw ;
Acharya:2009zt  or consider WIMPs to make up only a fraction of the total DM
of the Universe Zurek:2008qg ; Profumo:2009tb . In standard cosmology, the
WIMP relic density is calculated assuming a set conditions during the time
before Big Bang Nucleosynthesis (BBN), but there are no data or indirect
evidence supporting such assumptions. For instance, the reheating temperature
is an unknown quantity which could be as low as $\sim 1$ MeV, the temperature
at which nucleosynthesis begins. If this temperature is small, it may have a
profound implication on the DM relic density (be suppressed or enhanced
compared to the standard scenario) Giudice:2000ex ; Gelmini:2006pw . More
recently, several works have addressed the problem of DM abundance under
nonstandard cosmologies Aparicio:2016qqb ; DEramo:2017gpl ; Hamdan:2017psw ;
Visinelli:2017qga ; DEramo:2017ecx ; Bernal:2018ins ; Bernal:2018kcw ;
Hardy:2018bph ; Drees:2017iod ; Arbey:2018uho . For instance, in Refs.
Bernal:2018kcw ; Hardy:2018bph  the case for scalar DM was addressed, while in
Ref. Drees:2017iod  a generic calculation for the DM abundance with a late
decaying scalar was considered, and in Ref. Arbey:2018uho  the relic abundance
is considered as a probe of the conditions of the Universe pre-BBN.
Interestingly enough, such deviations from the standard cosmology do not
affect the prospects regarding DM detection and BBN. On the other hand,
although most proposals contain one single WIMP, there is no reason to
consider that the DM of the Universe is composed by just one type of DM
particles. The total DM relic abundance could be a result of the presence of
several DM particles, a scenario referred to as multicomponent DM. In such
scenarios, the DM of the Universe is set by the WIMP and other DM candidate,
such as QCD axions Baer:2011hx ; Bae:2013hma ; Dasgupta:2013cwa ;
Alves:2016bib ; Ma:2017zyb  or even another WIMP Zurek:2008qg ; Profumo:2009tb
; Esch:2014jpa ; Arcadi:2016kmk ; Bernal:2018aon . In light of this, it makes
sense to study models where the relic abundance is not imposed as a
constraint, either because in nonstandard cosmology it is possible to fulfil
this requirement when the right combination of parameters is achieved, or
because is achieved by the interplay of two separate dark sectors.

For the aforementioned reasons, we consider the doublet-triplet fermionic
model (DTF) Dedes:2014hga ; Abe:2014gua ; Freitas:2015hsa ; Lopez-
Honorez:2017ora , which is one of the ultraviolet realizations of the
fermionic Higgs portal Patt:2006fw , and is part of the minimal setup expected
when the SM is extended by new physics which is, to some extent, related to
lepton and baryon number conservation Arbelaez:2015ila ; Arkani-Hamed:2015vfh
. In the DTF, the SM is enlarged with two colorless fermions, a vectorlike
$SU(2)_{L}$ doublet and a majorana $SU(2)_{L}$ triplet, both being odd an
exact $Z_{2}$ symmetry222This symmetry can be recognized as remnant symmetry
at the end of the symmetry breaking chain of the SO(10) group to the SM
Arbelaez:2015ila . which is imposed in order to render stable the lightest
$Z_{2}$-odd particle333This particle setup has been also considered in studies
associated to strengthening the electroweak phase transition Carena:2004ha ,
presicion test in future colliders Arkani-Hamed:2015vfh ; Bertuzzo:2017wam ;
Xiang:2017yfs , electroweak precision tests Cai:2016sjz , generation of
neutrino masses Betancur:2017dhy , fake split-supersymmetry Benakli:2013msa ,
and UV completion of doublet fermion DM Dedes:2016odh .. The DM candidate
turns out to be a mixture, generated by the interaction with the Higgs boson,
between the neutral component of the triplet and the neutral components of the
doublet vectorlike fermion. The viable dark matter regions comprise a DM mass
around the electroweak scale and above 1 TeV Dedes:2014hga ; Abe:2014gua ;
Freitas:2015hsa ; Betancur:2017dhy . The electroweak DM region arises in the
scheme of the custodial limit (the new Yukawa couplings are equal) when the DM
candidate is pure doublet. However, the contribution of new the charged
fermions to $h\rightarrow\gamma\gamma$ generates a considerable suppression on
the Higgs diphoton decay making such a scheme severely constrained by the
interplay between the DM relic density constraint and the LHC measurement of
the Higgs diphoton decay rate Dedes:2014hga ; Abe:2014gua ; Freitas:2015hsa ;
Betancur:2017dhy .

In this work we will study the custodial limit of the DTF within either a
nonstandard cosmology scenario, i.e., we depart from the standard relic
density calculation, or a multicomponent DM setup but assuming that the WIMP
relic density is obtained through the thermal freeze out. We will establish
the current constraints on the DFT coming from collider searches and Higgs
diphoton decay, without taking care of the constraint on the DM relic density.
Then, we will go on to determine the restrictions resulting from direct
detection (DD) experiments and indirect detection (ID) with gamma rays, both
in the framework of nonstandard cosmology and the DM candidate as part of a
multicomponent system.

The rest of the paper is organized as follows. In Sec. II we present the
doublet-triplet fermion model with its mass spectra and discuss the allowed
interactions. In Sect. III we present the model restrictions arising from
electroweak production at colliders of charginos and neutralinos and from
precision measurements of the Higgs diphoton decay rate. In Section IV we
study the constraints arising from DD and ID with gamma rays for the case of a
nonstandard cosmology, whereas in Section V, the same analysis is made for the
case where the DM candidate is part of a multicomponent system. Finally, we
conclude in Sec. VI.

##  II The Model

Doublet-triplet fermion DM consists on extending the SM with an $SU(2)_{L}$
vectorlike doublet with $Y=-1$ and a Majorana $SU(2)_{L}$ triplet, both odd
under an exact $Z_{2}$ symmetry. Expressing the new $SU(2)_{L}$ multiplets as

$\displaystyle\psi=\left(\begin{array}[]{ccc}\psi^{0}\\\
\psi^{-}\end{array}\right),\hskip
28.45274pt\Sigma_{L}=\left(\begin{array}[]{ccc}\Sigma_{L}^{0}/\sqrt{2}&\Sigma_{L}^{+}\\\
\Sigma_{L}^{-}&-\Sigma_{L}^{0}/\sqrt{2}\end{array}\right),$ (5)

the most general renormalizable Lagrangian, invariant under the
$SU(2)_{L}\times U(1)_{Y}\times Z_{2}$ symmetry can be written as

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\rm{SM}}+\mathcal{L}_{\rm{F}}+\mathcal{L}_{\rm{I}}-\mathcal{V}_{\rm{SM}}.$
(6)

Here $\mathcal{L}_{\rm{SM}}$ is the SM Lagrangian and $\mathcal{V}_{\rm{SM}}$
is the scalar potential of the Higgs doublet
$H=(0,(h+v)/\sqrt{2})^{\text{T}}$, with $h$ being the Higgs boson and $v=246$
GeV. $\mathcal{L}_{\rm{F}}$ refers to the kinetic and mass terms of the new
fermions,

$\displaystyle\mathcal{L}_{F}$ $\displaystyle=\bar{\psi}i\cancel{D}\psi-
M_{\psi}\bar{\psi}\psi+{\rm
Tr}[\bar{\Sigma}_{L}i\cancel{D}\Sigma_{L}]-\frac{1}{2}{\rm
Tr}(\bar{\Sigma}_{L}^{c}M_{\Sigma}\Sigma_{L}+\mbox{h.c.}),$ (7)

and $\mathcal{L}_{\rm{I}}$ contains the Yukawa interactions of the new
fermions with the Higgs doublet,

$\displaystyle\mathcal{L}_{\rm{I}}=-y_{1}H^{\dagger}\bar{\Sigma}_{L}^{c}\epsilon\psi_{R}^{c}+y_{2}\bar{\psi}_{L}^{c}\epsilon\Sigma_{L}H+{\rm
h.c.},$ (8)

where $y_{1}$ and $y_{2}$ are new Yukawa couplings that generate the mixing
between the new fermion fields. Note that the $Z_{2}$ symmetry not only
guarantees the stability of the lightest $Z_{2}$-odd particle (the DM
particle) but also avoids Higgs-mediated flavor changing neutral currents at
tree level through the mixing terms $\bar{\psi}He_{R}$ and
$\overline{\Sigma}^{c}_{L}\tilde{H}^{\dagger}L$. Therefore, lepton flavor
violating processes such as $\mu\to e\gamma$ are forbidden.

The particle spectrum contains three new Majorana mass eigenstates
$\chi_{\alpha}^{0}$ ($\alpha=a,b,c$) and two new charged fermions
$\chi_{a,b}^{\pm}$ . The mass matrix for the neutral fermions is Dedes:2014hga
; Betancur:2017dhy  (in the basis
$\Xi^{0}=(\Sigma_{L}^{0},\psi^{0}_{L},\psi^{0c}_{R})^{T}$),

$\displaystyle\mathbf{M}_{\Xi^{0}}=\begin{pmatrix}M_{\Sigma}&\frac{1}{\sqrt{2}}yv\cos\beta&\frac{1}{\sqrt{2}}yv\sin\beta\\\
\frac{1}{\sqrt{2}}yv\cos\beta&0&M_{\psi}\\\
\frac{1}{\sqrt{2}}yv\sin\beta&M_{\psi}&0\end{pmatrix},$ (9)

with $y=\sqrt{(y_{1}^{2}+y_{2}^{2})/2}$ and $\tan\beta=y_{2}/y_{1}$. On the
other hand, the charged fermions mass matrix reads

$\displaystyle\hskip
28.45274pt\mathbf{M}_{\Xi^{\pm}}=\begin{pmatrix}M_{\Sigma}&yv\cos\beta\\\
yv\sin\beta&M_{\psi}\\\ \end{pmatrix},$ (10)

which is diagonalized by a rotation of the gauge eigenstates into the physical
states defined via

$\displaystyle\left(\begin{array}[]{cc}\Sigma^{+}\\\
\psi^{+}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{cc}\chi_{a}^{+}\\\
\chi_{b}^{+}\end{array}\right),\hskip 56.9055pt\sin(2\theta)=\frac{\sqrt{2}\
y\ v}{m_{\chi_{a}^{+}}-m_{\chi_{b}^{+}}^{2}}.$ (17)

Either one of the mass eigenstates $\chi_{a}^{\pm}$ and $\chi_{b}^{\pm}$ could
be the lightest charged $Z_{2}$-odd fermion. From now on, we will assume that
$\chi_{1}^{\pm}$ is the lightest between $\chi_{a}^{\pm}$ and
$\chi_{b}^{\pm}$, while $\chi_{2}^{\pm}$ is the heaviest. Thus, for the masses
$m_{\chi_{1}^{\pm}}$ and $m_{\chi_{2}^{\pm}}$ a mass ordering is implied. On
the other hand, these mass matrices evoke the neutralino and chargino mass
matrices (in the wino-higgsino limit) in the minimal supersymmetric standard
model Martin:1997ns , which case is realized when $y=g/\sqrt{2}$ and has been
exploited in studies such as Carena:2004ha ; Arkani-Hamed:2015vfh ;
Benakli:2013msa .

###  II.1 The custodial limit

An interesting scheme arises when $\tan\beta=1$ (the custodial limit) since
several consequences arise. First, the interaction Lagrangian
$\mathcal{L}_{\rm{I}}$ becomes invariant under an $SU(2)_{R}$ symmetry which
protects the $T$ and $U$ parameters Cai:2016sjz ; Dedes:2014hga , thus making
the model free of the constraints coming from electroweak precision tests.
Second, all diagonal tree-level couplings of $\chi_{\alpha}^{0}$ to the $Z$
boson are zero. And third, the neutral mass matrix may be written as444 This
can be done via a similarity transformation
$\mathbf{M}^{\prime}_{\Xi^{0}}=O^{\dagger}\mathbf{M}_{\Xi^{0}}O$, with
$\displaystyle O=\begin{pmatrix}1&0&0\\\
0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\ \end{pmatrix}.$ (18) .

$\displaystyle\mathbf{M}^{\prime}_{\Xi^{0}}=\begin{pmatrix}M_{\Sigma}&yv&0\\\
yv&M_{\psi}&0\\\ 0&0&-M_{\psi}\\\ \end{pmatrix},$ (19)

which shows that there is a decoupled eigenvalue $M_{\psi}$ (the mass is not
obtained from the electroweak symmetry breaking), meaning that one of the mass
eigenstates (which will be labelled as $\chi_{1}^{0}$ with mass
$m_{\chi_{1}^{0}}=-M_{\psi}$) is an equal mixture of the doublets with no
triplet component. Moreover, the coupling of $\chi_{1}^{0}$ to the Higgs
$(g_{h\chi_{1}^{0}\chi_{1}^{0}})$ is also zero at tree level.

It follows that, for the custodial limit scheme with $\chi_{1}^{0}$ as the DM
candidate ($\chi_{1}^{0}$ is the lightest between $\chi_{a}^{0},\chi_{b}^{0}$
and $\chi_{c}^{0}$)555In this case we have that the heavier neutral fermions
are mass-degenerate with the charged fermions
($m_{\chi_{2}^{0}}=m_{\chi_{1}^{\pm}}$ and
$m_{\chi_{3}^{0}}=m_{\chi_{2}^{\pm}}$), so
$|m_{\chi_{1}^{0}}|<|m_{\chi_{2}^{0}}|<|m_{\chi_{3}^{0}}|$., the above
features lead to the following important implications for the phenomenology of
the model.

  * •

DM ($\chi_{1}^{0}$) annihilates into weak gauge bosons through $t$\- and $u$\-
channels via exchange of heavier $Z_{2}$ fermions, and so, the DM annihilation
cross section is suppressed. Furthermore, for large Yukawa couplings666This is
the reason for which such a scheme can not be obtained in the MSSM.
$(0.5\lesssim y\lesssim 3)$, the splitting between the DM candidate and the
heavier new fermions is large which further suppresses the annihilation cross
section, thus allowing the DM candidate to saturate the relic abundance for
masses as low as the electroweak scale ($80\lesssim m_{\chi_{1}^{0}}\lesssim
200$ GeV).

  * •

Within the region where the correct DM relic density is obtained, there are
two different allowed triplet mass regions for any value of the pair
$y$-$m_{\chi_{1}^{0}}$, one where $M_{\Sigma}$ is always negative
($M_{\Sigma}<-m_{\chi_{1}^{0}}$) and one where it can be either positive or
negative but larger than $-m_{\chi_{1}^{0}}$.

  * •

Since the interactions between $\chi_{1}^{0}$ and both $h$ and $Z$ are zero a
tree-level, there are not contributions to direct detection at tree-level, so
the leading contributions appear at one-loop level (more on this in Sec. IV.1)
for both spin-independent (SI) and spin-dependent cross sections. These blind
spots for the model has been studied in works such as Dedes:2014hga ;
Freitas:2015hsa ; Betancur:2017dhy

  * •

Though the custudial limit scheme presents interesting possibilities, it is
also severely constrained by the Higgs diphoton decay measurement. The origin
of this is due to the presence of new charged fermions that couple to the
Higgs boson and create an effective $h\gamma\gamma$ coupling through a
triangular diagram similar to the top quark contribution (though with a larger
electric charge). The decay ratio with respect to the SM rate can be
parametrized as

$\displaystyle
R_{\gamma\gamma}=\left|1+\frac{1}{A_{SM}}\left[\frac{y^{2}v^{2}}{m_{\chi_{2}^{\pm}}-m_{\chi_{1}^{\pm}}}\left(\frac{A_{F}(\tau_{\chi_{2}^{\pm}})}{m_{\chi_{2}^{\pm}}}-\frac{A_{F}(\tau_{\chi_{1}^{\pm}})}{m_{\chi_{1}^{\pm}}}\right)\right]\right|^{2},$
(20)

where $A_{{\rm SM}}=-6.5$ includes the contribution from all charged SM
particles such as gauge bosons and fermions, and the loop factor is
$A_{F}(\tau)=2\tau^{-2}[\tau+(\tau-1)\arcsin^{2}{\sqrt{\tau}}]$ for $\tau\leq
1$ where $\tau_{X}=m^{2}_{h}/(4m^{2}_{X})$. As can be seen from Eq. (20), the
new fermion contribution is always positive for real Yukawa couplings (which
is the case at hand) and because it is opposite in sign to SM contribution, it
may suppress its value causing large deviations from the results published by
the ATLAS Aaboud:2018xdt  and CMS Sirunyan:2018ouh  collaborations.

Indeed, that is what occurs when large values of $y$ as well as large
splitting between the $Z_{2}$ odd heavier fermions are required, the same
conditions that lead to the correct DM relic abundance. As a result, the new
contribution conspirates to create large discrepancies from the expected
value, up to the point of only allowing the DM mass to lie between the narrow
range of $70<m_{\chi_{1}^{0}}<80$ (GeV).

All in all, despite the custodial limit of the DTF model being an appealing
and promising scenario (thus being an excellent exponent of the WIMP
paradigm), it is severely constrained by the interplay between the DM relic
density constraint and the LHC measurement of the Higgs diphoton decay rate.

Nonetheless, if the DM abundance is generated within a nonstandard cosmology
or is part of multicomponent dark sector, the requirement of having large
values for $y$ as well as a large mass splitting between the charged fermions
would be substantially weaken. This in turn would result in a larger portion
of the parameter space that is still allowed777Other way of recovering part of
the parameter space is via introduction of additional charged scalars
Betancur:2017dhy ..

In the following sections, we will explore the DTF in the case of
$\tan\beta=1$ with $\chi_{1}^{0}$ as the DM candidate, our aim is to find
current as well as upcoming restrictions due to collider, direct detection and
indirect detection experiments. We choose the set of free parameters of the
model to be $M_{\psi},M_{\Sigma}$ and $y$, and due to the freedom to make
field redefinitions we assume $M_{\psi},y>0$ and $M_{\Sigma}$ to be real
Dedes:2014hga , implying CP conservation in the $Z_{2}$-odd sector and that
three intrinsic CP phases of the $Z_{2}$-odd fields (including $\chi_{1}^{0}$)
are fixed.

##  III Collider Bounds

The Large Hadron Collider (LHC) is currently running at an outstanding 13 TeV
energy and has collected more that 100 fb-1 of data. One of its current goals
is to probe BSM models either by direct production of new particles or by
measuring possible deviations from the SM. In that regard, the parameter space
of the DTF model may be constrained by the ATLAS and CMS experiments.

###  III.1 Higgs diphoton decay

Figure 1: Scan on the parameter space of the model against $R_{\gamma\gamma}$
for the region where $M_{\Sigma}<-m_{\chi^{0}_{1}}$ (left panels) and the
region where $M_{\Sigma}>-m_{\chi^{0}_{1}}$ (right panels). The solid and
dashed horizontal lines represent the lowest bound at a 2$\sigma$ deviation
from the central value reported by the ATLAS and CMS collaboration
respectively.

As explained before, the DTF may generate large deviations from the current
measurements of the Higgs diphoton decay rate. The ATLAS and CMS
collaborations have presented results for the decay at $\sqrt{s}=13$ TeV with
$\sim 36\,\mathrm{fb^{-1}}$ of data in Refs. Aaboud:2018xdt  and CMS
Sirunyan:2018ouh , respectively. We use these results to find the regions of
the parameter space that are in agreement with the Higgs diphoton decay rate.
For this we consider models that deviate at most $2\sigma$ from the central
value reported by each collaboration.

In order to obtain constraints from $R_{\gamma\gamma}$ without imposing the DM
relic density constraint, we performed a scan of the free parameters of the
model as follows:

$\displaystyle 0.1<$ $\displaystyle\;y<3.5,$ $\displaystyle-2000\
\mathrm{GeV}<$ $\displaystyle\;m_{\Sigma}<2000\,\mathrm{GeV},$ $\displaystyle
75.0\ \mathrm{GeV}<$ $\displaystyle\;m_{\psi}<500\ \mathrm{GeV}.$ (21)

Additionally, we only considered models where the lightest charged fermion is
heavier than 100 GeV in order to satisfy LEP limits Achard:2001qw .

The results are presented in Fig. 1 where the parameter space has been divided
into two regions, $M_{\Sigma}<-m_{\chi^{0}_{1}}$ (left panels) and
$M_{\Sigma}>-m_{\chi^{0}_{1}}$ (right panels). We will do this throughout the
paper because the phenomenology in these two regions tend to yield different
results. The scan shows that, considering the ATLAS results, for the region
where $M_{\Sigma}<-m_{\chi^{0}_{1}}$ there are no restrictions on the Yukawa
coupling $y$ or on $M_{\Sigma}$ whereas in the region
$M_{\Sigma}>-m_{\chi^{0}_{1}}$ the decay rate forbids $y$ values larger than
2.25 and $M_{\Sigma}\lesssim-60$ GeV. On the other hand, CMS results yield the
severe constraint $y<2.0$ for $M_{\Sigma}<-m_{\chi^{0}_{1}}$ and $y<1.0$ for
$M_{\Sigma}>-m_{\chi^{0}_{1}}$, whereas $M_{\Sigma}$ must be less than
$\sim-92$ GeV. Hence, positive triplet masses are no longer consistent with
$R_{\gamma\gamma}$ results.

We also consider the impact on the fermion mixing angle from
$R_{\gamma\gamma}$ (see Fig 2). For the region where
$M_{\Sigma}<-m_{\chi^{0}_{1}}$ (left panel) the mixing angle must be small
such that $|\cos\theta|\lesssim 0.3$ ($|\cos\theta|\lesssim 0.2$) in order to
be consistent with ATLAS (CMS) measurements. Accordingly, in that region the
lightest charged fermion is mostly doublet. On the other hand, the results for
the region where $M_{\Sigma}>-m_{\chi^{0}_{1}}$ (right panel) point out that
the lightest charged fermion is mostly triplet, with $|\cos\theta|\gtrsim
0.94$ ($|\cos\theta|\gtrsim 0.98$) for the ATLAS (CMS) data. A comment
regarding this region is in order: for $R_{\gamma\gamma}\sim 0.2$, the mixing
angle exhibits a rather complex behavior which is seen as large changes in
$\cos\theta$ (from -0.8 to 0.6) right next to a boundary where $\cos\theta\sim
0.8$. This stems from the fact that at this boundary the triplet mass is
changing sign, thus having an impact on the mixing angle behavior.

The results for the mixing angle in both region are important because they
will have a direct impact on the production cross section of the heavier
fermions at the LHC, as will be discussed below.

Figure 2: Impact of the cosine of the mixing angle $\theta$ on the Higgs
diphoton decay rate for the allowed values of the DM mass. The conventions are
the same as those of Fig. 1.

###  III.2 Constraints from electroweak production searches

Other LHC results that may potentially constraint the DTF model are those
searching for electroweak production of neutralinos and charginos in different
simplified SUSY models (with all other SUSY particles decoupled), where the
relevant detection channels are those with several leptons (and missing
energy) in the final state. In the DTF, $\chi_{1,2}^{\pm}$ and
$\chi_{2,3}^{0}$ play the role of charginos and heavier neutralinos,
respectively, with the same mass degeneracy that characterizes the simplified
supersymmetric scenarios.

The CMS collaboration has recently published results for such searches at
$\sqrt{s}=13$ TeV and 35.9 fb-1 Sirunyan:2018ubx . For the case of
$m_{\chi_{1}^{0}}\lesssim 500$ GeV and a nondegenerate spectrum, the most
sensitive channel is that with three final state leptons where at least two of
them have opposite sign and same flavor. Thus, DM production proceeds via the
following process

$\displaystyle q\bar{q}^{\prime}\rightarrow
W^{*\pm}\rightarrow\chi^{\pm}\chi_{2,3}^{0}:\begin{cases}\chi^{\pm}\rightarrow\chi^{0}_{1}W^{*\pm}\rightarrow\chi_{1}^{0}\ell^{\pm}\nu_{\ell},\\\
\chi_{2,3}^{0}\rightarrow\chi^{0}_{1}Z^{*}\rightarrow\chi_{1}^{0}\ell^{+}\ell^{-}.\end{cases}$
(22)

where the mediators $\chi^{\pm}$ and $\chi^{0}$ are considered to be winos and
thus mass degenerate, with the neutral fermion decaying 100$\%$ via $Z$ boson.
To recast the LHC constraints (and other experimental restrictions that will
be discussed below) for the DTF we implemented the model in SARAH-4.12.3
package Staub:2013tta  whose output was used with SPheno-4.0.3 Porod:2003um
in order to obtain the particle spectrum and with MadGraph5_aMC@NLO to obtain
the production cross sections Alwall:2014hca .

Figure 3: DM mass versus lightest chargino mass for the regions where
$M_{\Sigma}<-m_{\chi^{0}_{1}}$ (left panel) and $M_{\Sigma}>-m_{\chi^{0}_{1}}$
(right panel). The region below the blue dashed line is excluded from CMS
electroweak production while the region bounded by the blue solid line
represents the exclusion by the ATLAS collaboration using compressed spectra.
Points below the solid (dashed) black contour are excluded by the
$R_{\gamma\gamma}$ results reported by the ATLAS (CMS) collaboration.

Fig. 3 shows the constraints from electroweak production, where the excluded
region corresponds to the points below the blue dashed line. Moreover, the
points below the solid and dashed black lines yield a lower diphoton decay
ratio than the one allowed by ATLAS and CMS, respectively. In the region where
$M_{\Sigma}>-m_{\chi^{0}_{1}}$ (right panel) the diphoton decay ratio
restricts the lightest charged and the next-to-lightest neutral fermions to be
mostly doublet. A consequence of this is that the production cross section is
nearly the same even for all values of the allowed Yukawa coupling and the
triplet mass, which means that the boundary of the excluded region is nearly
independent of $y$ and $M_{\Sigma}$. Moreover, due to the mixing angle, the
production cross section resembles that of SUSY Higgsino with all scalars
decoupled. The figure shows that the strongest constraints come mostly from
$R_{\gamma\gamma}$ , except for a small area where electroweak production
cross section is more restrictive. Nonetheless, there are no additional
restrictions placed on the free parameters $M_{\psi}$, $y$ and $M_{\Sigma}$.

In the region where $M_{\Sigma}<-m_{\chi^{0}_{1}}$, the diphoton decay rate
restricts $\chi_{2}^{0}$ and $\chi_{1}^{\pm}$ to be mostly triplet. The
production cross section is large but again independent of $y$ and
$M_{\Sigma}$, and so, the region excluded by electroweak production is
presented with only one contour. In this region, due to the larger production
cross section, the curve is shifted to the left in the $m_{\chi_{1}^{+}}$ line
and so $R_{\gamma\gamma}$ places the strongest constrains for the whole plain.

###  III.3 Constraints from compressed spectra searches

The ATLAS collaboration has also published relevant results for the DTF for
the case of compressed spectra Aaboud:2017leg , i.e., the next-to-lightest
fermion is close in mass to the neutralino DM ($\leqslant$35 GeV) and a mass
degeneracy between the next-to-lightest neutralino and lightest chargino. In
that region, the DM production proceeds via:

$\displaystyle q\bar{q}^{\prime}\rightarrow
W^{*\pm}\rightarrow\chi^{\pm}\chi_{2,3}^{0}:\,\,\,\begin{cases}\chi^{\pm}\rightarrow\chi^{0}_{1}W^{*\pm}\rightarrow\chi^{0}_{1}q\bar{q}^{\prime},\\\
\chi_{2,3}^{0}\rightarrow\chi^{0}_{1}Z^{*}\rightarrow\chi_{1}^{0}\ell^{+}\ell^{-}.\end{cases}$
(23)

The search then focuses on two leptons with opposite sign and same flavor with
soft momentum and large $\not\mathrel{E}$ which is present due to the two DM
particles recoiling against initial state radiation. For this search, small
mass splittings are required, in order to ensure DM coannihilations. In the
DTF this low mass splitting is not needed, in fact $0\lesssim
m_{\chi_{1}^{\pm}}-m_{\chi_{1}^{0}}\lesssim 140$ GeV, however, we may still
use the constraints for small mass splittings between the next-to-lightest
$\chi^{\pm}$ and the DM. We find that, for the region
$M_{\Sigma}>-m_{\chi_{1}^{0}}$ where $\chi_{1}^{\pm}$ and $\chi_{2}^{0}$ are
mostly triplet, and so the restrictions resemble those of the ATLAS
collaboration which is shown with a solid blue contour. In terms of the free
parameters, we find that the triplet mass is now restricted to be smaller than
$\sim-165$ GeV, whereas the Yukawa coupling is not constrained. For the case
of $M_{\Sigma}<-m_{\chi_{1}^{0}}$, since $\chi_{1}^{\pm}\ \chi_{2}^{0}$ are
mostly doublet, there is a lower production cross section and so the
restriction is negligible.

##  IV DM detection in a nonstandard cosmology

In a nonstandard cosmology scenario, the late decay of a heavy scalar field
could either increase or decrease the DM relic abundance compared to the
standard calculation. For the DTF888The case of the SUSY neutralino was
studied in Ref. Gelmini:2006pw , whereas in Ref. Cheung:2012qy  the wino-
higgsino (similar to the DTF) case was also considered., it could be possible
to satisfy the relic abundance, for instance, due to the presence of a heavy
scalar particle which decays into heavier $Z_{2}$-odd particles that later on
decay into the DM candidate, thus increasing the relic abundance compared to
that of the model in the standard cosmology. Hence, we expect the DTF model to
saturate, in one way or another, the DM relic abundance. Therefore, we look
into current experimental constraints coming from direct searches and indirect
detection via gamma rays. Since the diphoton decay is by far more restrictive
than production at colliders, in this section we impose the $R_{\gamma\gamma}$
restriction coming from ATLAS and when relevant to the parameter space, we
present the restriction arising in this observable from the CMS experiment.

###  IV.1 Direct detection

Figure 4: Feynman diagrams contributing to the spin independent DD. The
diagrams arise from a loop correction to the $h\chi_{1}^{0}\chi_{1}^{0}$
coupling which is absent at tree-level when $\tan\beta=1$. The interaction
shown in the left diagram is independent on the Yukawa coupling $y$ while the
right one depends on it. The $Z_{2}$-odd $\chi$ fermion shown in both figures
represent the charged $\chi_{1,2}^{\pm}$ (neutral $\chi_{2,3}^{0}$) fermion
when the loop is mediated by $W$ ($Z$) boson.

Within the custodial limit scheme, the SI elastic scattering is only achieved
at the loop level since both $g_{\chi_{1}^{0}\chi_{1}^{0}Z}$ and
$g_{\chi_{1}^{0}\chi_{1}^{0}h}$ couplings vanish at tree-level. However, at
loop-level there are, in principle, several contributions that could be
relevant. First, there is an effective nonzero $\chi_{1}^{0}\chi_{1}^{0}h$
coupling originating from loops mediated by the new heavier fermions and weak
gauge bosons, thus allowing for spin-independent direct detection (see Fig.
4). Additionally, box diagrams mediated by gauge bosons and twist-2 operators
Hisano:2011cs ; Drees:1993bu  contribute to the SI cross section. In
principle, these two contributions should be taken into account to obtain a
reliable calculation. However, it has been shown that they are sub-leading
except when the two contributions arising from the Higgs vertex corrections
cancel each other out, which happens for low values of $\sigma_{SI}$
($\lesssim 10^{-47}$cm2) Drees:1993bu . Moreover, the authors of Ref.
Hisano:2011cs  have shown that when the cancellation happens, two-loop
contribution to an effective scalar interaction with external gluons are of
the same order as the box ones. Since these calculations (boxes from gauge,
twist-2 and two-loop) are quiet involved and only relevant for the case of
specific cancellations, we will not take them into account for the calculation
of the SI cross section. Moreover, they tend to create a larger suppression of
the cross section that is already out of reach of current experiments. As a
result, the restrictions that we will present below from DD are not strongly
affected by this assumption.

In order to obtain the most up-to-date limits from DD, we calculated the
effective $g_{h\chi_{1}^{0}\chi_{1}^{0}}$ coupling following Ref.
Freitas:2015hsa  and used that to compute the spin independent (SI) cross
sections. We then compared to the current upper limits on the DM-nucleon SI
scattering cross section, where the strongest ones (within the DM mass range
we are considering) are those reported by the XENON1T collaboration
Aprile:2018dbl . We also show the projected sensitivity of DARWIN
Aalbers:2016jon , the most sensitive DD experiment planned for DM at the
electroweak scale. However, the expected SI cross section around the DARWIN
limit must be taken with a grain of salt since sub-leading corrections might
change $\sigma_{SI}$ in that region.

Figure 5: Spin-independent cross section for the regions
$M_{\Sigma}<-m_{\chi_{1}^{0}}$ (left) and $M_{\Sigma}>-m_{\chi_{1}^{0}}$
(right). The blue curve represents the upper limit imposed by XENON1T
Aprile:2018dbl  whislt the green curve shows the projected sensitivity of
DARWIN Aalbers:2016jon . The black dashed line represents the limit when the
$R_{\gamma\gamma}$ restriction from CMS is considered.

In Fig. 5 we display the results for the spin-independent cross section as a
function of the DM mass, for the regions $M_{\Sigma}<-m_{\chi_{1}^{0}}$ (left)
and $M_{\Sigma}>-m_{\chi_{1}^{0}}$ (right). It follows that XENON1T restricts
the coupling to be less than 1.75 if the lower bound on $R_{\gamma\gamma}$
from ATLAS is imposed. The dashed black line in both panels shows the CMS
lower bound on $R_{\gamma\gamma}$, which excludes models even further and for
the region $M_{\Sigma}>-m_{\chi_{1}^{0}}$ imposes $y\leq 1.2$. We also checked
the impact of DD results on the other free parameter of the model,
$M_{\Sigma}$, but we found that they place no further restrictions on it. The
prospects coming from the DARWIN experiment correspond to the green solid
line, which show that couplings as small as 0.5 may be probed. It is worth
mentioning that the lower limit on the SI cross section is due to the
cancellation between the two one-loop corrections to the
$h\chi_{1}^{0}\chi_{1}^{0}$ vertex; hence, in order to have a precise value of
$\sigma_{SI}$ in this region, a more detailed calculation is necessary.

###  IV.2 Indirect detection from dwarf spheroidal galaxies

Figure 6: ID restrictions and prospects coming from the observation of dSphs
of the Fermi satellite applied to the regions $M_{\Sigma}<-m_{\chi_{1}^{0}}$
(left panel) and $M_{\Sigma}>-m_{\chi_{1}^{0}}$ (right panel). The blue and
green curve show current limits from the $W^{+}W^{-}$ channel for 6 years of
observation and 15 dSphs, and the projected sensitivity for 45 dSphs and 15
years of observation, respectively. Whereas points above the red dotted line
are excluded from CMB measurements by the Planck Collaboration. Points below
the black dashed line are excluded when the $R_{\gamma\gamma}$ restriction
from CMS is considered.

In regions of high dark matter density such as dwarf spheroidal galaxies
(dSphs) or the center of the Milky Way, DM particles may more easily find each
other and annihilate into SM particles. The dSphs are particularly interesting
because of their proximity to the Milky Way, their high DM to baryon mass-
ratio, and their low background, thus making the DM detection via gamma-rays
feasible. The Fermi satellite has searched for gamma rays in dSphs founding no
deviations from the expected spectrum, which has lead to upper limits on the
thermally averaged DM annihilation cross section Ackermann:2015zua . It is
worth noting that in this model DM annihilation can also affect the cosmic
microwave radiation (CMB). If DM annihilates during the time of recombination,
it will inject energy that will ionize Hydrogen. This will have a direct
effect on the anisotropies of the CMB, thus, altering what is currently
observed. Therefore, measurements of the CMB can constrain the parameter space
of DM models, with the advantage that for this observable astrophysical
uncertainties do not play a role Kawasaki:2015peu ; Lopez-Honorez:2013lcm .

For the DTF, the DM annihilation proceeds in the same channels as the ones in
the early Universe, i.e., via $t$\- and $u$-channel annihilation into
$W^{+}W^{-}$ and $ZZ$ bosons. The gauge bosons then decay and produce, for
instance, gamma rays that may be detected as an excess in the spectrum.To
obtain the constraints we calculated the thermally averaged cross section
using the public available package micrOMEGAS Belanger:2013oya  and used this
to compare with limits reported in Ref. Ackermann:2015zua . In the DTF this
cross section is, to a leading order approximation, independent of the DM
velocity. Thus, its value matches that of the early universe, which allow us
to compare our results with the limits reported in Ref. Aghanim:2018eyx .

The results are shown in Fig. 6 where all points shown satisfy the ATLAS
$R_{\gamma\gamma}$ constraint and DD bounds as explained in previous sections.
As can be seen, the Fermi satellite observation over 15 dSphs imposes
stringent limits on the model in a such a way that a large portion of the DM
mass range is ruled out. Moreover, stringent limits on the mass of the next-
to-lightest fermion also arise, since such particles act as the mediators in
the $t$\- and $u$-channels of the DM annihilation. On the other hand, though
CMB measurements do place constraints, they are far less restrictive than
those coming from dSphs. For the region where $M_{\Sigma}<-m_{\chi_{1}^{0}}$
we find that $86\,\text{GeV}<m_{\chi_{1}^{0}}<280$ GeV is already ruled out,
this also leads to a restriction on $m_{\chi^{+}}>340$ GeV for
$m_{\chi_{1}^{0}}>280$ GeV. On the other hand, for the region where
$M_{\Sigma}>-m_{\chi_{1}^{0}}$ we find that the diffuse spectrum requires that
$m_{\chi_{1}^{0}}>280$ GeV, $m_{\chi^{+}}>300$ GeV while
$M_{\Sigma}\lesssim-230$ GeV. The expected 15 years and 45 dSphs observation
will explore the whole region of the right panel and will leave a very narrow
range of $m_{\chi_{1}^{0}}$ of $\sim$ 80 GeV un-explored. We also note that
points that satisfy the $R_{\gamma\gamma}$ restriction of the CMS experiment
are those with the higher $\langle\sigma v\rangle$ since both observables
depende on the mixing angle and are maximal for large $y$999For
$R_{\gamma\gamma}$ the dependence was already shown in Sec. III, while for the
diffuse spectrum, the dependence on $\cos\theta$ enters through the vertices
of the annihilation channels. In Appendix A this dependence is shown for the
DM interaction with the $W^{\pm}$ gauge boson and a $Z_{2}$-odd charged
fermion..

###  IV.3 Indirect detection from gamma-ray lines

Another promising detection channel is DM annihilation into two photons within
regions with high DM density. In this case, the photon energies will be
closely related to the DM mass leading to a spectrum exhibiting a sharp peak
referred to as a linelike feature Funk:2013gxa . In this regard, the Fermi
Ackermann:2015lka  and H.E.S.S. Abdalla:2016olq ; Abdallah:2018qtu
collaborations have looked for gamma-ray lines coming from the center of the
Milky Way, with no evidence of DM so far. This in turn has lead to constraints
on the DM $\langle\sigma v\rangle_{\gamma\gamma}$ annihilation into photons.

In the DTF, the DM annihilation into two photons is mediated by heavier
$Z_{2}$-odd fermions interacting with vector and Goldstone bosons. Though the
annihilation cross section in this case is loop suppressed, it may be possible
to place constraints. In order to calculate the $\langle\sigma
v\rangle_{\gamma\gamma}$ we follow the procedure given in Ref. Garcia-
Cely:2016hsk  (the specific calculations along with the topologies that
contribute are given in Appendix A). After considering all the restrictions
coming from collider, DD and ID in the diffuse spectrum, our results show that
the Fermi and H.E.S.S. results do not place additional constraints on the
model for both $M_{\Sigma}<-m_{\chi_{1}^{0}}$ and
$M_{\Sigma}>-m_{\chi_{1}^{0}}$ regions since $\langle\sigma
v\rangle_{\gamma\gamma}\sim 10^{-29}\,\text{cm}^{3}/\text{s}$, which is nearly
an order of magnitude lower than the most sensitive results which are
presented by the H.E.S.S collaboration in Ref. Abdallah:2018qtu . As a result,
this observable does not restrict the parameter space of the model.

##  V DM detection in multicomponent dark sectors

An interesting possibility that has recently taken momentum is for the DM to
be composed of different sectors, which is a far more general setting than the
usual one DM candidate. For instance, the observed relic density could be the
result of WIMP and Axion particles. In this case, it is possible that the
sectors do not communicate, and so they behave as two completely independent
DM particles, without affecting each other’s relic density and experimental
bounds. For this section we will consider the WIMP DM candidate from the DTF
to be part of multicomponent DM, that is, we obtain experimental bounds for
models where the WIMP’s relic density is less than or equal to the central
value reported by the Planck collaboration, $\Omega_{\text{Planck}}$
Aghanim:2018eyx .

Figure 7 shows the ration
$\epsilon_{\chi_{1}^{0}}=\Omega_{\chi_{1}^{0}}/\Omega_{\text{Planck}}$ as a
function of $m_{\chi_{1}^{0}}$. For the region where
$M_{\Sigma}<-m_{\chi_{1}^{0}}$, the relic abundance is at most 40$\%$ of the
observed value except for the narrow region where $m_{\chi_{1}^{0}}\sim 80$
GeV (where annihilation into weak gauge bosons is kinematically suppressed).
On the other hand, in the region where $M_{\Sigma}>-m_{\chi_{1}^{0}}$ there
are no models that saturate relic density, and so, the DTF accounts at most
40$\%$ of the Universe DM content. We must add a comment here, unlike the
previous section, we are assuming that the DM arises from a standard cosmology
scenario, and in that sence the relic abundance of the WIMP DM is the one
calculated through the usual method of solving the Boltzman’s equation (the
one calculated via micrOMEGAS Belanger:2013oya ).

Figure 7:  $\epsilon_{\chi_{1}^{0}}$ vs. $m_{\chi_{1}^{0}}$ for
$M_{\Sigma}<-m_{\chi_{1}^{0}}$ (left panel) and $M_{\Sigma}>-m_{\chi_{1}^{0}}$
(right panel). All points satisfy collider bounds presented in Sec. III.

Now we set out to investigate experimental bounds on the model. For colliders,
the restrictions are the same as those presented in Sec. III since they are
independent of the DM abundance. On the other hand, DD and ID rates do depend
in the local DM density, and as a result the constraints presented in Sec. IV
will be different in this scenario. To quantify this, we used the parameter
$\epsilon_{\chi_{1}^{0}}$ Cao:2007fy ; Hur:2007ur  to re-scale the DD and ID
observables. For the case of DD, the expected scattering rate will be rescaled
by $\epsilon_{\chi_{1}^{0}}$ which means that the SI cross section is
effectively rescaled to be
$\sigma_{SI}=\epsilon_{\chi_{1}^{0}}\sigma_{SI-\chi_{1}^{0}}$; hence, DD
constrains are now relaxed. The results are shown in Fig. 8 for
$M_{\Sigma}<-m_{\chi_{1}^{0}}$ (left panel) and $M_{\Sigma}>-m_{\chi_{1}^{0}}$
(right panel). The left panel shows that for models that satisfy the lowest
ATLAS limit on $R_{\gamma\gamma}$, DD imposes $y\leqslant 2.1$ while for
models that satisfy lowest CMS limits $y\leqslant 1.9$. On the other hand, in
the right panel, for models that satisfy the lowest ATLAS limit on
$R_{\gamma\gamma}$, DD imposes $y\leqslant 2.2$ while for models that satisfy
lowest CMS limits $y\leqslant 0.95$ which means that in this case CMS diphoton
decay is more restrictive than DD (even considering DARWIN prospects).

For indirect detection, the situation is far less restrictive because the
thermally averaged cross section is rescaled by a factor of
$\epsilon_{\chi_{1}^{0}}^{2}$, thus suppressing it. As a result, ID does not
impose additional constraints on the model.

Figure 8: Direct detection results for $M_{\Sigma}<-m_{\chi_{1}^{0}}$ (left
panel) and $M_{\Sigma}>-m_{\chi_{1}^{0}}$ (right panel), the conventions are
the same as in Fig. 5.

##  VI Conclusions

In this work we have studied a simplified DM model, the doublet-triplet
fermionic model, where the SM is enlarged with an $SU(2)_{L}$ vectorlike
doublet and a Majorana triplet, both new fields are odd under a $Z_{2}$
symmetry while the SM fields are even. As a result the new fields Lagrangian
include two Yukawa type interactions with the Higgs field. It follows that
when the two allowed Yukawa couplings of the dark sector are equal, the model
exhibits a custodial symmetry and, when the DM particle is pure doublet, the
diagonal couplings of it with the Higgs and $Z$ bosons are forbidden at tree-
level. In this case, the model may saturate the relic abundance at the
electroweak scale, but that comes with the disadvantage of a strong
suppression of the Higgs diphoton decay rate.

For this reason, we have considered the model (in the aforementioned case)
framed within two different scenarios: one where the relic abundance is set by
nonstandard cosmology, i.e., we assumed the relic abundance is saturated due
to new physics before BBN, and another where the DM is made up of multiple
particles which do not affect each other’s abundance or DD limits. As a
result, the mass of the heavier charged and neutral fermions may lie close to
the DM mass, which lifts partly the $R_{\gamma\gamma}$ restriction.

Regarding DD and ID for the nonstandard cosmology scenario, we found that
Xenon1T results demand a Yukawa coupling $y<1.75$, whereas the Fermi results
imply that the DM mass is in general restricted to be $m_{\chi_{1}^{0}}<280$
GeV except for a narrow region of $m_{\chi_{1}^{0}}\sim 80$ GeV when
$M_{\Sigma}<m_{\chi_{1}^{0}}$. On the other hand, for the scenario of the DM
as part of the multicomponent dark sectors we found that DD impose a less
severe constraint on the Yukawa coupling ($y<2.2$) while current ID does not
impose additional constraints on the model.

## Acknowledgements

We are thankful to Diego Restrepo, Andrés Rivera and Guillermo Palacio for
enlightening discussions. We are also thankful to Florian Staub for help with
the SARAH package and Olivier Mattelaer for help with MadGraph. A. B. has been
supported by Colciencias, Universidad EIA and Fulbright Colombia. O.Z. has
been supported by the Sostenibilidad program of Universidad de Antioquia UdeA
and by COLCIENCIAS through the grants 111565842691 and 111577657253. O.Z.
acknowledges the ICTP Simons associates program and the kind hospitality of
the Abdus Salam ICTP where the final stage of this work was done.

##  Appendix A Annihilation into two photons

In this appendix, we give the explicit calculation to obtain the thermally
averaged cross section for the annihilation of DM in the DTF model into two
photons (e.g. gamma-ray lines). The procedure was obtained with the results
presented in Ref. Garcia-Cely:2015khw .

The thermally averaged cross section is given by

$\displaystyle\langle\sigma v\rangle=\frac{1}{4}\frac{|\mathcal{B}|^{2}}{32\pi
m_{\chi_{1}^{0}}^{2}},$ (24)

where $\mathcal{B}=\mathcal{B}_{W}+\mathcal{B}_{S}$. Here $\mathcal{B}_{W}$
and $\mathcal{B}_{S}$ denote the contributions coming from the charged gauge
bosons ($W$) and scalars (Goldstone bosons, $S$) running in the loop,
respectively (see Fig. 9). They read

$\displaystyle\mathcal{B}_{i}=$
$\displaystyle\frac{\alpha}{\pi}\bigg{(}x_{1}\frac{C_{0}(0,1,-1,r_{\mathrm{even}}^{2},r_{\mathrm{even}}^{2},r_{\mathrm{odd}}^{2})}{(r_{\mathrm{odd}}^{2}-r_{\mathrm{even}}^{2})(1+r_{\mathrm{odd}}^{2}-r_{\mathrm{even}}^{2})}+x_{2}\frac{C_{0}(0,1,-1,r_{\mathrm{odd}}^{2},r_{\mathrm{odd}}^{2},r_{\mathrm{even}}^{2})}{(r_{\mathrm{even}}^{2}-r_{\mathrm{odd}}^{2})(1-r_{\mathrm{odd}}^{2}+r_{\mathrm{even}}^{2})}$
(25)
$\displaystyle+x_{3}\frac{C_{0}(0,4,0,r_{\mathrm{even}}^{2},r_{\mathrm{even}}^{2},r_{\mathrm{even}}^{2})}{(1+r_{\mathrm{odd}}^{2}-r_{\mathrm{even}}^{2})}+x_{4}\frac{C_{0}(0,4,0,r_{\mathrm{odd}}^{2},r_{\mathrm{odd}}^{2},r_{\mathrm{odd}}^{2})}{(1-r_{\mathrm{odd}}^{2}+r_{\mathrm{even}}^{2})}\bigg{)}.$

Here $r_{\mathrm{even(odd)}}=m_{even(odd)}/m_{\chi_{1}^{0}}$ with the label
even (odd) indicating that the particle is $Z_{2}$ even (odd) and
$C_{0}(r_{1}^{2},r_{2}^{2},r_{3}^{2},r_{4}^{2},r_{5}^{2},r_{6}^{2})$ is the
usual Passarino-Veltman function Peskin:1995ev . In the case of the charged
Goldstone boson the mass $m_{even}=m_{W}$. On the other hand, the factors
$x_{i}$ are different depending if the mediator is a scalar or a vector boson:

Scalars

$\displaystyle x_{1}=$
$\displaystyle\sqrt{2}r_{\mathrm{even}}^{2}\big{(}r_{\mathrm{even}}^{2}-r_{\mathrm{odd}}^{2}-1\big{)}(g_{Ls}^{2}+g_{Rs}^{2}),$
(26) $\displaystyle x_{2}=$
$\displaystyle\sqrt{2}r_{\mathrm{even}}^{2}\big{(}r_{\mathrm{even}}^{2}-r_{\mathrm{odd}}^{2}-1\big{)}(g_{Ls}^{2}+g_{Rs}^{2})+4\sqrt{2}r_{\mathrm{odd}}\big{(}r_{\mathrm{even}}^{2}-r_{\mathrm{odd}}^{2}-1\big{)}(g_{Ls}\
g_{Rs}),$ $\displaystyle x_{3}=$ $\displaystyle 0,$ $\displaystyle x_{4}=$
$\displaystyle-2\sqrt{2}r_{\mathrm{odd}}(r_{\mathrm{odd}}(g_{Ls}^{2}+g_{Rs}^{2})+2g_{Ls}g_{Rs}),$

where $g_{Ls}=g_{Rs}=-y\cos\theta/\sqrt{2}$ for the lightest $Z_{2}$-odd
charged fermion and $g_{Ls}=g_{Rs}=y\sin\theta/\sqrt{2}$ for the heaviest
$Z_{2}$-odd charged fermion.

Vector Bosons

$\displaystyle x_{1}=$ $\displaystyle 2\sqrt{2}((r_{\mathrm{even}}^{4}+4\
r_{\mathrm{odd}}^{2}-r_{\mathrm{even}}^{2}(1+r_{\mathrm{odd}}^{2}))(g_{Lw}^{2}+g_{Rw}^{2}),$
(27) $\displaystyle-8\
r_{\mathrm{odd}}(1-r_{\mathrm{even}}^{2}+r_{\mathrm{odd}}^{2})g_{Lw}\ g_{Rw},$
$\displaystyle x_{2}=$
$\displaystyle-2\sqrt{2}(r_{\mathrm{odd}}^{2}(-3-r_{\mathrm{even}}^{2}+r_{\mathrm{odd}}^{2})(g_{Lw}^{2}+g_{Rw}^{2})+8\
r_{\mathrm{odd}}\ g_{Lw}\ g_{Rw}),$ $\displaystyle x_{3}=$ $\displaystyle 8\
\sqrt{2}(-1+r_{\mathrm{even}}^{2})(g_{Lw}^{2}+g_{Rw}^{2}),$ $\displaystyle
x_{4}=$ $\displaystyle 4\ \sqrt{2}\
r_{\mathrm{odd}}(r_{\mathrm{odd}}(g_{Lw}^{2}+g_{Rw}^{2})-4\ g_{Lw}\ g_{Rw}),$

where $g_{Lw}=g_{Rw}=-g_{L}\sin\theta/2$ for the lightest $Z_{2}$-odd charged
fermion and $g_{Lw}=g_{Rw}=-g_{L}\cos\theta/2$ for the heaviest $Z_{2}$-odd
charged fermion.

Figure 9: Topologies that lead to the annihilation of DM into two photons. The
external straight lines represent the DM particles, whereas the internal ones
represent a charged $Z_{2}$ odd fermion (shown with a cyan solid line), gauge
boson or Goldstone boson (shown with a black solid line). The external wavy
lines represent the photons (the gamma-rays).

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# High harmonic spectroscopy of disorder-induced Anderson localization

Adhip Pattanayak  Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai 400076, India  Á. Jiménez-Galán  Max-Born Straße 2A,
D-12489 Berlin, Germany  Misha Ivanov  Max-Born Straße 2A, D-12489 Berlin,
Germany  Department of Physics, Humboldt University, Newtonstraße 15, D-12489
Berlin, Germany.  Blackett Laboratory, Imperial College London, South
Kensington Campus, SW7 2AZ London, United Kingdom.  Gopal Dixit
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai 400076, India

###### Abstract

Exponential localization of wavefunctions in lattices, whether in real or
synthetic dimensions, is a fundamental wave interference phenomenon.
Localization of Bloch-type functions in space-periodic lattice, triggered by
spatial disorder, is known as Anderson localization and arrests diffusion of
classical particles in disordered potentials. In time-periodic Floquet
lattices, exponential localization in a periodically driven quantum system
similarly arrests diffusion of its classically chaotic counterpart in the
action-angle space. Here we demonstrate that nonlinear optical response allows
for clear detection of the disorder-induced phase transition between
delocalized and localized states. The optical signature of the transition is
the emergence of symmetry-forbidden even-order harmonics: these harmonics are
enabled by Anderson-type localization and arise for sufficiently strong
disorder even when the overall charge distribution in the field-free system
spatially symmetric. The ratio of even to odd harmonic intensities as a
function of disorder maps out the phase transition even when the associated
changes in the band structure are negligibly small.

Disorder is an ubiquitous effect in crystals Evers and Mirlin (2008). The
seminal work by Anderson Anderson (1958) predicted that above a critical
disorder value, the electronic wavefunction will change from being delocalized
across the lattice to exponentially localized (insulating state) due to the
interference of multiple quantum paths originating from the scattering with
random impurities and defects. Anderson localization is a fundamental wave
phenomenon and thus permeates many branches of physics; it has been observed
in matter waves Billy  _et al._ (2008), light waves Wiersma  _et al._ (1997),
and microwaves Dalichaouch  _et al._ (1991). Anderson localization also finds
direct analogues in periodically driven systems, with time-periodic dynamics
taking the role of space-periodic structure. While periodically driven
classical systems can develop chaotic behaviour for sufficiently strong
driving fields, leading to delocalization of the original ensemble across the
whole phase space, their quantum counterpart shows exponential localization of
the light-dressed states Casati  _et al._ (1987).

Dramatic changes in a wavefunction during transition from a delocalized to a
localized state may lead to changes in the nonlinear optical response of the
system. In this context, symmetry–forbidden harmonics of the driving field are
an appealing bellwether candidate. Indeed, while even-order harmonics are
known to be forbidden in systems with inversion symmetry Boyd (2008), they are
also known to arise in such systems if and when charges localize Silva  _et
al._ (2016); Bandrauk and Lu (2005). The required symmetry breaking can then
be triggered by even a small asymmetry in the oscillating electric field of
the driving laser pulse. Such asymmetry is natural in short laser pulses and
is controlled by the phase of the electric field oscillations under the pulse
envelope, i.e., the carrier-envelope phase (CEP) Krausz and Ivanov (2009).
Exponentially localized states in symmetric multiple well potentials appear to
be particularly sensitive to even small field asymmetries, leading to even
harmonics in the nonlinear response even for pulses encompassing tens of
cycles Silva  _et al._ (2016).

High harmonic generation (HHG) is a powerful tool for ultrafast spectroscopy
Smirnova and Ivanov (2013); Lein (2007). Extremely large coherent bandwidth of
harmonic spectra enables sub-femtosecond resolution. HHG is a sensitive probe
of Cooper minima Bertrand  _et al._ (2012), Auger decay Leeuwenburgh  _et al._
(2013), attosecond dynamics of optical tunnelling Shafir  _et al._ (2012);
Pedatzur  _et al._ (2015) and the dynamics of electron exchange Sukiasyan  _et
al._ (2009) in atoms, ultrafast hole dynamics Eckart  _et al._ (2018); Gaal
_et al._ (2007); Bruner  _et al._ (2016); Smirnova and Ivanov (2013), nuclear
motion Baykusheva  _et al._ (2014); Baker  _et al._ (2006); Lan  _et al._
(2017) in small molecules, and enantio-sensitive electronic response in more
complex chiral molecules Cireasa  _et al._ (2015); Ayuso  _et al._ (2018);
Neufeld  _et al._ (2019); Baykusheva  _et al._ (2019). In solids, high
harmonic spectroscopy has allowed observation of dynamical Bloch oscillations,
band structure tomography Vampa  _et al._ (2015), probing of defects in solids
Pattanayak  _et al._ (2020); Mrudul  _et al._ (2020a), sub-fs monitoring of
core excitons Luu  _et al._ (2015), optical measurement of the valley
pseudospin Langer  _et al._ (2018); Jimenez-Galan  _et al._ (2020); Mrudul
_et al._ (2020b), tracking of van Hove singularities Uzan  _et al._ (2020),
picometer resolution of valence band electrons Lakhotia  _et al._ (2020),
imaging internal structures of a unit cell Mrudul  _et al._ (2019), monitoring
of light-driven insulator-to-metal transitions Silva  _et al._ (2018), and
probing of topological effects Bauer and Hansen (2018); Silva  _et al._
(2019).

In this work, we employe HHG to track phase transition between delocalized and
localized states in the Aubry-André (AA) system Aubry and Andre (1980)
(similar to that proposed in Ref. Harper (1955)), where localization occurs
only above a critical value of disorder, already in one dimension. This model
captures the metal-to-insulator transition, is a workhorse to study non-
trivial topology, and has been realized in optical lattices and photonic
quasi-crystals Sanchez-Palencia and Lewenstein (2010); Roati  _et al._ (2008).

The model system is described by the following tight-binding Hamiltonian,

$\hat{H}=-t_{0}\sum_{j=1}^{L-1}\left(c_{j}^{\dagger}c_{j+1}+\textrm{h.c.}\right)+V\sum_{j=1}^{L}\cos(2\pi\sigma
j)c_{j}^{\dagger}c_{j},$ (1)

where $t_{0}$ is the nearest neighbour hopping term, $V$ is the strength of
the potential, $c_{j}^{\dagger}$ and $c_{j}$ are, respectively, the fermionic
creation and annihilation operators at site $j$, h.c. stands for the hermitian
conjugate, $L$ is the total number of lattice sites and $\sigma$ determines
the periodicity of the potential. A rational value of $\sigma$ corresponds to
a periodic potential and consequently to delocalized electronic wavefunctions.
If $\sigma$ is irrational, the potential becomes quasi-periodically disordered
(for finite systems, this may also happen for rational $\sigma$). For a
disordered potential, the system undergoes the localization phase transition
at $V/t_{0}=2$. For $V/t_{0}>2$ all states are exponentially localized on one
site, while for $V/t_{0}<2$ the states are delocalized.

Figures 1(a) and (b) show the eigenspectrum of our system, for
$\sigma=\frac{\sqrt{5}+1}{2}$, with 100 lattice sites, in the delocalized
($V/t_{0}=1.9$) and localized ($V/t_{0}=2.1$) phases, respectively. The
hopping term $t_{0}=0.26$ eV and the lattice constant $a_{0}=7.56$ atomic unit
of length ($\sim$0.4 nm) are used throughout. The Fermi energy is $E_{F}=-0.2$
eV, so that red-colored states correspond to fully occupied valence band
states, while the blue-colored states are unoccupied. Differences in the
eigenspectrum for both phases are indiscernible. In contrast, the individual
eigenstates of the system present a clear localization phase transition: Figs.
1(c) and (d) show the occupation number of the eigenstate with index=10 (other
eigenstates show similar behaviour). The eigenstate is delocalized for
$V/t_{0}=1.9$ but fully localized for $V/t_{0}=2.1$. These differences between
the charge distribution in individual eigenstates disappear completely when we
consider the fully-filled valence band, see Figs. 1(e) and (f): the
differences between both phases are barely visible.

Figure 1: Electronic structure of the model system, for
$\sigma=\frac{\sqrt{5}+1}{2}$. Left panels (a,c,e) show the delocalized phase
($V/t_{0}=1.9$), right panels (b,d,f) show the localized phase
($V/t_{0}=2.1$). (a,b) – eigenspectrum; (c,d) – electron density per site for
one eigenstate $(m=10)$, (e,f) lattice site occupation numbers in a fully-
filled valence band. Red and blue curves in (a) and (b) represent valence and
conduction states, respectively.

The question is: will the non-linear optical response be sensitive to the
phase transition? To address this question, we consider a non-resonant, low-
frequency field polarized along the 1D chain,

$F(t)=F_{0}f(t)\cos(\omega t+\phi),$ (2)

with $\phi$ the carrier-envelope phase and $f(t)$ the sine-squared envelope
with a full duration of 10 optical cycles. We include the laser-matter
interaction via the time-dependent Peierls phase: $t_{0}\to
t_{0}\,e^{ia_{0}eA(t)}$, where $A(t)$ is the field vector potential,
$F(t)=-{dA(t)}/{dt}$, and $e$ is the electron charge. The carrier
$\omega=0.136$ eV is set well below the bandgap $\Delta\simeq 0.4$ eV.

Two characteristic regimes describe laser-induced electron dynamics in such
low-frequency fields.

In the localized phase, efficient resonant tunnelling between localized states
at different sites, including transitions from the valence to the conduction
band, becomes possible when the peak voltage between the adjacent sites
$F_{0}a_{0}$ approaches and/or exceeds the characteristic energy gap $\Delta$,
$F_{0}a_{0}\simeq\Delta$. This regime enables rapid energy gain by the system
within a few laser cycles, allowing it to climb to the top of the energy scale
Ivanov  _et al._ (1996). In our case, its signature would be the emission of
harmonics up to the maximum transition frequency of the system (harmonic 10),
for all fields enabling resonant tunnelling (RT). The onset of this regime
corresponds to $F_{0,\text{RT}}\sim\Delta/a_{0}\simeq 0.4$ MV/cm in our
system. In the localized phase, exponential sensitivity of resonant tunnelling
to positions of individual states and to the field strength create ideal
conditions for symmetry breaking, leading to generation of even harmonics; the
direction in which the symmetry is broken is controlled by the pulse CEP.

In the delocalized case, the delocalized states adiabatically follow
oscillations of the low-frequency driving field, preventing symmetry breaking.
The latter requires light-induced electron localization, which occurs when
$F_{0}a_{0}\omega\geq\Delta^{2}$ Dietrich  _et al._ (1996), i.e., when
$F_{0}\geq 1$ MV/cm in our system. Even harmonic generation should therefore
only emerge around $F_{0}\simeq 1$ MV/cm.

Our numerical simulations below fully confirm all of these expectations. The
time-dependent Schrödinger equation is solved independently for the $m$
normalized eigenstates $|\psi_{m}(t=t_{i})\rangle$ of the field-free
Hamiltonian Eq. (1) that lie below $E_{F}$,
$|\psi_{m}(t)\rangle=e^{-i\int_{t_{i}}^{t}\hat{H}(t^{\prime})dt^{\prime}}|\psi_{m}(t=t_{i})\rangle$,
where $\hat{H}(t)$ is the Hamiltonian in Eq. (1) with the time-dependent
Peierls substitution. The current from a single eigenstate $m$ is calculated
as

$j_{m}(t)=\left\langle\psi_{m}(t)|\hat{J}(t)|\psi_{m}(t)\right\rangle$ (3)

where the current operator is defined as

$\hat{J}(t)=-iea_{0}t_{0}\sum_{j=1}^{L}\left(e^{-ia_{0}eA(t)}c_{j}^{\dagger}c_{j+1}-e^{ia_{0}eA(t)}c_{j+1}^{\dagger}c_{j}\right).$
(4)

The total current from all the valence band eigenstates $m$, i.e., the fully-
filled valence band, is,

$j(t)=\sum_{m}j_{m}(t).$ (5)

The harmonic spectra are then calculated from the Fourier transform of the
time derivative of the total current. Prior to the Fourier transform, we apply
an envelope to the current that coincides with the laser pulse envelope Wu
_et al._ (2015), to filter out emission after the end of the laser pulse. We
consider 100 lattice sites and 0.01 atomic unit of time-step ($\sim$0.25 as)
to get the converged spectra.

Figure 2: (a,b) High harmonic spectra for the system in the (a, c, e, g, i)
delocalized and (b, d, f, h, j) localized phase: (a, b) HHG from the 10th
eigenstate only for a field strength $F_{0}=0.6$ MV/cm. HHG from the fully-
filled valence band for a field strength (c, d) $F_{0}=0.6$ MV/cm, (e, f)
$F_{0}=2.2$ MV/cm, (e, f) $F_{0}=0.4$ MV/cm, and (i, j) $F_{0}=1.0$ MV/cm.

Figure 2 shows the HHG spectrum for different initial states and field
strengths in the localized and delocalized phases. First, in Figs. 2(a) and
(b), we consider a single valence band eigenstate as our initial state
($m=10$, shown in Fig. 1(c,d)). The charge distribution of the initial state
is strongly asymmetric in both phases, which breaks the left-right symmetry of
the chain in the HHG process and leads to the appearance of even harmonics,
both in the delocalized [Fig.2(a)] and localized [Fig.2(b)] phases.

Figure 3: High harmonic spectra in the delocalized (top panels, red curve) and
the localized (lower panels, green curve) phase calculated for different
carrier-envelope phases (CEP) of the field: (a,d) CEP=$+\pi/2$, (b,e)
CEP=$-\pi/2$ and (d,f) the coherent superposition of CEP=$+\pi/2$ and
CEP=$-\pi/2$.

However, the equilibrium initial state corresponds to the fully-filled valence
band, with the charge distributed relatively evenly between all sites; the
distribution is virtually identical between the two phases [Figs. 1(e) and
(f)]. With this initial condition, the HHG spectra of the two phases, shown in
Figs. 2(c) and (d) for a field strength of $F_{0}=0.6$ MV/cm, are now
strikingly different.

Destructive interference from different initial states completely suppresses
even harmonics in the delocalized phase [Fig. 2(c)], but they remain prominent
in the localized phase [Fig. 2(d)]. This result follows from our previous
discussion. The field strength $F_{0}=0.6$ MV/cm is above the threshold for
resonant tunneling between localized states in our system
($F_{0,\text{RT}}=0.4$ MV/cm), where the instantaneous field brings the energy
levels into resonance, generating coherence between all sites. Resonant
tunneling between the sites depends sensitively on the instantaneous field
strength, leading to CEP-dependent symmetry breaking and the appearance of
even harmonics in the localized phase. We find that even harmonics emerge
already at $F_{0}\simeq 0.4$ MV/cm. As resonant tunnellng induces coherences
between the localized sites, it enables population of the highest band and
generation of higher harmonics in the localized phase than in the delocalized
phase [Figs. 2(c) and (d)]. In the latter, the system follows adiabatically
the field oscillations and transitions to the highest states are suppressed.

The field strength used in Figs. 2(c) and (d) is below the threshold field for
laser-induced localization in the delocalized phase, which is $F_{0}\sim 1$
MV/cm for our system. Therefore, the harmonic spectrum in the delocalized
regime shows no sign of even harmonics and a smaller cut-off energy [Fig.
2(c)]. Even harmonics emerge as soon as the field amplitude crosses this
threshold. At $F_{0}=2.2$ MV/cm the harmonic spectra in both phases become
very similar [Figs. 2(e) and (f)]. For this field strength, the cut-off is the
same in both phases, and corresponds to the (Stark-shifted) maximum transition
frequency of the system.

To confirm the origin of symmetry breaking and even harmonics emission, Fig. 3
shows the harmonic spectra for two values of the pulse carrier-envelope phase
(CEP) shifted by $\pi$, and their coherent superposition in Figs. 3(c) and
(f). In the delocalized case, even harmonics are absent regardless of the CEP
[Figs. 3(a) and (b)]. In the localized case, they are identical in both cases,
Figs. 3(d) and (e), but with opposite phase: upon coherent addition even
harmonics are completely washed out [Fig. 3(f)]. The reason is that the laser-
induced asymmetry in the electron charge distribution at CEP=$+\pi/2$ is
exactly opposite to that at CEP=$-\pi/2$, as graphically illustrated by the
cancellation of the emission upon interference.

Figure 4: Even-to-odd harmonic peak ratio (R) for different $V/t_{0}$ ranging
from 1.7 to 2.3. The ratio R is calculated as
$\mathrm{R}=\mathrm{log_{10}}(\bar{H}_{even})/\mathrm{log_{10}}(\bar{H}_{odd})$,
where $\bar{H}=H^{peak}/H^{min}$. Blue tiangles (orange circles) represent the
R value correspond to 4th (6th) harmonic and 5th (7th) harmonic. The increment
of R value at $V/t_{0}>2$ indicates the phase transition phenomenon.

Figure 4 shows that the relative intensity of even harmonics does indeed track
the metal to insulator (delocalized – localized) phase transition in the
system. The Figure plots the ratio of even to odd harmonics for different
values of $V/t_{0}$, at a field strength of $F_{0}=0.6$ MV/cm, i.e., that
which generates resonant tunneling in the localized phase but not laser-
induced localization in the delocalized phase. The even–odd ratio increases
dramatically at the phase transition $V/t_{0}=2$, showing that HHG, driven by
a phase-stable CEP pulse, is able to map disorder-induced electron
localization in a solid.

In conclusion, we have used high harmonic spectroscopy to track delocalized to
localized phase transition in the Aubry-André system. For aperiodic potentials
($\sigma=\frac{\sqrt{5}+1}{2}$), the localized and delocalized phases show
almost identical eigenspectra and site occupation numbers for a state with
fully-filled valence band. Yet, high harmonic spectra between the two phases
show striking differences, especially in the appearance of forbidden (even)
harmonics. This effect is a consequence of dynamical symmetry breaking induced
by the field but enabled by initial electron localization, which promotes
resonant tunneling and leads to CEP-dependent symmetry breaking even in the
low-frequency regime. Our work shows that the localisation-delocalisation
phase transition, which can be driven by a very small modification of the on-
site energy, can be effectively traced by HHG spectroscopy.

A. P. acknowledges sandwich doctoral fellowship from Deutscher Akademischer
Austauschdienst (DAAD, reference no. 57440919). M.I. and A. J-G acknowledge
support of the FET-Open Optologic grant. M.I. acknowledges support from the
Deutsche Forschungsgemeinschaft (DFG) Quantum Dynamics in Tailored Intense
Fields (QUTIF) grant. G. D. acknowledges financial support from Science and
Engineering Research Board (SERB) India (Project No. ECR/2017/001460).

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