New decay modes of heavy Higgs bosons in a two Higgs doublet model with vectorlike leptons
Accepted: May
New decay modes of heavy Higgs bosons in a two Higgs doublet model with vectorlike leptons
Radovan Derm sek 0 1 2 3
Enrico Lunghi 0 1 3
Seodong Shin 0 1 3
0 Seoul National University , 1 Gwanakro, Gwanakgu, Seoul, 151747 Korea
1 727 E. Third St. , Bloomington, IN, 47405 U.S.A
2 Department of Physics and Astronomy and Center for Theoretical Physics
3 Physics Department, Indiana University , USA
In models with extended Higgs sector and additional matter elds, the decay modes of heavy Higgs bosons can be dominated by cascade decays through the new fermions rendering present search strategies ine ective. heavy neutral Higgses in two Higgs doublet model with vectorlike leptons. We also discuss constraints from existing searches and discovery prospects. Among the most interesting signatures are monojet, mono Z, mono Higgs, and Z and Higgs bosons produced with a pair of charged leptons.
Beyond Standard Model; Higgs Physics

2.1
2.2
3.1
3.2
3.3
3.4
3.5
1 Introduction
3 Signatures
H ! W
H ! Z
H ! Z
H ! h
H ! h
4 Conclusions
1
Introduction
2 Two Higgs doublet model with vectorlike leptons
Branching ratios of the heavy Higgs boson
Branching ratios of the lightest neutral and charged leptons
and H ! e4 , where e4 and 4 are the lightest new charged and neutral leptons. These
decay modes can be very large when the mass of the heavy Higgs boson is below the tt
threshold and the light Higgs boson (h) is SMlike so that H ! ZZ; W W are suppressed
or not present. In this case, avor changing decays H ! 4
or H ! e4 compete only
with H ! bb and for su ciently heavy H also with H ! hh. Subsequent decay modes of
e4 and 4: e4 ! W
, e4 ! Z , e4 ! h
and 4 ! W , 4 ! Z , 4 ! h
lead to the
following 6 decay chains of the heavy Higgs boson:
H !
H ! e4
4
ν4
νμ
νμ
μ
e4
W
H
(b)
H
(e)
νμ
ν4
νμ
e4
μ
μ
Z
Z
H
(c)
H
(f)
νμ
ν4
νμ
e4
μ
μ
h
h
are kinematically open. Moreover, the
nal states are the same as in pair production of
vectorlike leptons. We will not consider these possibilities here. Finally, although we focus
on the second family of SM leptons in nal states, the modi cation for a di erent family
of leptons or quarks is straightforward.
We show that in a large range of the parameter space branching ratios for the decay
modes (1.1) and (1.2) can be sizable or even dominant while satisfying constraints from
searches for heavy Higgs bosons, pair production of vectorlike leptons [2] obtained from
searches for anomalous production of multilepton events and constraints from precision
electroweak (EW) observables [
3
]. Since the Higgs production cross section can be very
large, for example the cross section for a 200 GeV Higgs boson at 8 TeV (13 TeV) LHC
for tan
= 1 is 7pb (18pb) [4], the
nal states above can be produced in large numbers.
Thus searching for these processes could lead to the simultaneous discovery of a new Higgs
boson and a new lepton if they exist. Some of the decay modes in
gure 1 also allow for
full reconstruction of the masses of both new particles in the decay chain.
The nal states of the processes (1.1–) a2n–d (1.2) are the same as nal states of pp !
W W; ZZ; Zh production or H ! W W; ZZ decays with one of the gauge bosons decaying
into second generation of leptons. Since searching for leptons in
nal states is typically
advantageous, our processes contribute to a variety of existing searches. Even searches for
processes with fairly large cross sections can be signi cantly a ected. For example, the
contribution of pp ! H
!
4
to pp ! W W can be close to current limits
while satisfying the constraints from H ! W W . This has been recently studied in ref. [1]
{ 2 {
in the two Higgs doublet model we consider here, and also in a more model independent
way in ref. [5]. However, the processes with tiny SM rates would be the best place to look
for this scenario and here we will focus on such signatures. Examples of almost background
free processes include H
! h
and H
! h
with h !
.
Vectorlike quarks and leptons near the electroweak scale provide a very rich
phenomenology. For example, similar processes to (1.1) and (1.2) involving SMlike Higgs
boson decaying into 2`2 or 4` through a new lepton were previously studied in ref. [6] and
the 4` case also in ref. [7]. An explanation of the muon g 2 anomaly with vectorlike leptons
was studied in [8, 9]. Vectorlike quarks and possibly Z0 o er possibilities to explain
anomalies in Zpole observables [10{13]. Extensions of the SM with complete vectorlike families
HJEP05(216)48
were considered to provide an understanding of values of gauge couplings from IR
xed
point behavior and threshold e ects of vectorlike fermions [14, 15]. Vectorlike quarks and
leptons were also considered in supersymmetric framework, see for example refs. [16{21].
Further discussion and references can be found in a recent review [22].
This paper is organized as follows. In section 2 we brie y summarize the model,
discuss constraints and present result for branching ratios of the heavy Higgs boson and
new leptons. In section 3 we discuss relevant existing searches and the most promising
search strategies for each of the six processes.
We summarize and present concluding remarks in section 4.
2
Two Higgs doublet model with vectorlike leptons
In ref. [1] we introduced an explicit model consisting of a typeII two Higgs doublet model
augmented by vectorlike pairs of new leptons: SU(2) doublets LL;R, SU(2) singlets EL;R
and SM singlets NL;R, where the LL and ER have the same hypercharges as leptons in the
SM. In order to avoid dangerous rates of lepton avor changing transitions between the light
leptons we further assume that the new leptons mix only with one family of SM leptons
and we consider the mixing with the second family as an example. This can be achieved
by requiring that the individual lepton number is an approximate symmetry (violated
only by light neutrino masses). With these assumptions, one can write the most general
renormalizable Lagrangian containing Yukawa and mass terms for the second generation
of SM leptons and new vectorlike leptons.
After spontaneous symmetry breaking, Hu0
= vu and
H0
d
= vd with
righthanded elds ( R; LR; ER)T on the right [9],
v = 174 GeV (we also de ne tan
vu=vd), the model can be summarized by mass
matrices in the charged lepton sector, with lefthanded elds ( L; LL ; EL) on the left and
qvu2 + vd2 =
0y vd 0
E vd1
vd CA ;
0
vd ME
(2.1)
and in the neutral lepton sector, with lefthanded elds ( ; L0L; NL) on the left and
right{ 3 {
handed elds ( R = 0; L0R; NR)T on the right [1],
vu CA :
0 vu MN
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
The superscripts on vectorlike elds represent the charged and the neutral components (we inserted
R = 0 for the righthanded neutrino which is absent in our framework in
order to keep the mass matrix 3
3 in complete analogy with the charged sector). The
usual SM Yukawa coupling of the muon is denoted by y , the Yukawa couplings to Hd are
denoted by various s, the Yukawa couplings to Hu are denoted by various s and
nally
the explicit mass terms for vectorlike leptons are given by ML;E;N . Note that explicit mass
terms between SM and vectorlike elds (i.e. LLR and EL R) can be rotated away. These
mass matrices can be diagonalized by biunitary transformations and we label the two new
charged and neutral mass eigenstates by e4; e5 and 4
; 5 respectively:
ULy MeUR = diag (m ; me4 ; me5 ) ;
VLyM VR = diag (0; m 4 ; m 5 ) :
Since SU(2) singlets mix with SU(2) doublets, the couplings of all involved particles to
the Z, W and Higgs bosons are in general modi ed. The avor conserving couplings receive
corrections and avor changing couplings between the muon (or muon neutrino) and heavy
leptons are generated. The relevant formulas for these couplings in terms of diagonalization
matrices de ned above can be found in refs. [1, 9]. In the limit of small mixing, approximate
analytic expressions for diagonalization matrices can be obtained which are often useful for
understanding of numerical results. These are also given in refs. [1, 9].
2.1
Branching ratios of the heavy Higgs boson
In order to focus on decay modes (1.1) or (1.2) we either allow mixing only in the neutral
sector,
couplings, or only in the charged sector,
couplings. We further assume that the
relevant lighter new lepton, 4 or e4, is heavier that mH =2 to avoid decays into pairs of
new leptons and the heavier new lepton, 5 or e5, is heavier than H. Finally, we work in
the limit with the light Higgs boson being fully SMlike and thus the heavy CP even Higgs
H has no direct couplings to gauge bosons. We apply these requirements on randomly
generated points in the parameter space speci ed by following ranges:
N ; ;
or
L; E ; ;
tan
mH 2 [130; 340] GeV ;
where we focus on mH < 2mt in order to avoid H ! tt and on the small tan
region
where the heavy Higgs production cross section is the largest.
{ 4 {
We impose constraints from precision EW data related to the muon and muon neutrino:
muon lifetime, Zpole observables (Z partial width to
, the invisible width,
forwardbackward asymmetry, leftright asymmetry) and the W partial width; constraints from
oblique corrections, namely from S and T parameters; and the LEP limit on the mass
of a new charged lepton, 105 GeV. These constraints are obtained from ref. [
3
]. We also
impose constraints on pair production of vectorlike leptons [2] obtained from searches for
+
anomalous production of multilepton events.1
In addition to constraints on new leptons we also impose constraints from searches
for new Higgses: H ! W W [24, 25] and H
!
[26]. Although the processes in (1.1)
or (1.2) do not contribute to H ! W W directly, they contribute to the same nal states as
obtained from decays of W bosons. In applying the constraints from H ! W W we follow
the analyses presented in refs. [5, 27]. We implement the cutbased analysis using the data
for e
e
nal state in [24, 25] but the results using
would be similar. Since our H
has no direct coupling to the W boson, it is only the top quark and new charged leptons in
loops that contribute to H !
. The top quark contribution to H !
and thus this process is highly constraining the parameter space at very small tan . In the
scales as cot2
usual two Higgs doublet model this was studied in ref. [28]. However, new charged leptons
can enhance or partially cancel the contribution from the top quark depending on the signs
of the new Yukawa couplings. Thus the allowed range of tan
is expected to extend to
lower values compared to the usual two Higgs doublet model. Since new leptons also couple
to the SMlike Higgs boson, the constraints from h !
at present still allow for a sizable new physics contribution [29].
have to be also satis ed. These
Results of the scan in the case of mixing in the neutral lepton sector, allowing for
and h !
H ! 4 , are depicted in gure 2 in various planes of relevant branching ratios and tan .
The dark colored points satisfy all the limits summarized above. The gray points satisfy
all the limits on new leptons but are excluded by H !
; W W and h !
light colors represent the points which are phenomenologically viable and satisfy H !
. Finally, the
limits only if mixing in the charged lepton sector is simultaneously allowed
which partially cancels the contribution from the top quark. In this case, for simplicity, we
allow L and E to mix but not with the muon and we conservatively extend the ranges for
and
to [ 1; 1]. We clearly see the lower bound on tan
' 0:55. Without the mixing
in the charged sector the lower limit on tan
moves to about 0.7.
Similarly, results of the scan in the case of mixing only in the charged lepton sector,
allowing for H ! e4 , are given in
gure 3 in the same planes and color scheme. In this
case, the tan
dependence is not so signi cant because the new Yukawa coupling inducing
in the same way as the bb coupling. The lowest possible value
H ! e4 scales with tan
of tan
is 0.7 as it was in the H ! 4
[ 1; 1], as indicated by light colors, only allows tan
& 0:6.
case. Even extending the ranges for
and
to
For completeness we plot the same points in the plane of Higgs branching ratios for
H ! 4
and H ! e4
versus the Higgs mass in
gure 4. Finally, although we focussed
on the CP even Higgs, the results for CP odd Higgs would be qualitatively similar.
1For the discussion of prospects of multilepton searches for vectorlike leptons in the case of mixing with
the 3rd generation of SM leptons, see ref. [23].
{ 5 {
points) additionally satisfy the bounds from H !
allowed. Various colors indicate di erent ranges of mH and BR(H ! hh) speci ed in the legend.
2.2
Branching ratios of the lightest neutral and charged leptons
The branching ratios of new lightest neutral and charged leptons decaying through Z and
W bosons are shown in
gure 5. Although not explicitly shown, the remaining branching
ratio of the decay through the SM Higgs boson can be easily read out since only three
decay modes are possible. In the left panels only constraints from EW precision data and
direct searches are imposed. In the right panels we include the impact of constraints from
searches for anomalous production of multilepton events [2]. In these plots, the colors
indicate the doublet fractions of 4 or e4. The blue, cyan, magenta, and red points have
doublet fractions in the ranges [95,100]%, [50,95]%, [5, 50]%, and [0, 5]% respectively. The
doublet fraction the a (a = 4; 5) is de ned as
1 h
2
(VLy)a2
i2
h
+ (VLy)a4
i2
h
+ (VRy)a4
i2
(2.9)
and the doublet fraction of ea (a = 4; 5) is obtained by replacing the VL;R matrices by UL;R
matrices in the formula above. Singlet fractions are given by 1  (doublet fraction).
{ 6 {
decay. Color coding is the same as in gure 2. We inserted the sub gure extending tan
and do not even allow a doubletlike charged lepton in the mass range considered. Note
however that in the case of charged leptons with large BR(e4 ! W ) the constraints come
from the pair production of 4 that accompanies doubletlike e4 or from e4 4 production; the
e4e4 pair production is not directly constrained by multilepton searches. Without mixing
in the neutral sector, BR( 4 ! W ) = 1 and this decay mode is highly constrained [2].
Allowing simultaneously full mixing in both charged and neutral sectors would relax this
constraint somewhat.
The main features of plots in gure 5 can be understood from analytic formulas for
couplings that can be obtained in the limit of small mixing [1]. For singletlike e4 or 4
the avor changing couplings to W and Z have, in the leading order, the same dependence
on parameters controlling the mixing. Thus this dependence disappears in the ratio and
we nd
this slope.
(e4 ! W
)= (e4 ! Z ) to be approximately 2:1 leading to the red bands with
For doubletlike 4, the
avor changing couplings to A = W; Z; h have the form
N ( A
+
A ) where
A and
A are functions of tan , MN and ML. This implies
immediately that, for xed tan , MN and ML, a scan over the three couplings
N ,
{ 7 {
axis of these ellipses is almost horizontal for ML & v; for smaller ML the ellipses collapse
onto the diagonal, which corresponds to BR( 4 ! h )
gures 3, 4 and 7 of ref. [1] for various choices of m 4 .
Finally, for the doubletlike e4, there are two couplings,
L and
E , connecting the
muon to vectorlike fermions and thus controlling the overall strengths of avor changing
couplings to W , Z, and h. In result, there is signi cantly more freedom and generating
couplings to W , Z, and h are less correlated.
In order to illuminate more subtle features of the scenario, we plot the same points in
the planes of branching ratios versus the masses of 4 and e4 in gure 6.
0. This behavior can be seen in
3
Signatures
In this section we discuss each of the novel heavy Higgs decay modes with details of the
main features of each channel, of existing experimental searches to which these new process
contribute, and of possible new searches. Considering the variety of possible
nal states,
detailed Monte Carlo studies of these signatures are beyond the scope of this paper.
For estimates of rates of various processes we will use, as a reference point, the
production cross section of a 200 GeV Higgs boson at 8 TeV LHC for tan
= 1 which is 7pb.
The corresponding cross section at 13 TeV LHC is 18pb, and for di erent values of tan
these numbers should be divided by tan2 . Furthermore, we will assume BR(H !
4 ) or
BR(H ! e4 ) to be 50% and branching ratios of 4 or e4 relevant for a given process to be
100%. These branching ratios are close to the upper possible values allowed as we saw in
the previous section. Note however that after imposing all constraints e4 ! Z
branching
ratios above 40%, while possible, are di cult to achieve.
3.1
H
The diagrams in gures 1a and 1d yield the W
nal state. A detailed study of these
topologies with W ! ` (` = e; ) has been presented in refs. [1, 5], where with focus on
contributions to the existing pp ! W W and pp ! H ! W W measurements.
{ 8 {
straints from precision EW data and in the right plots also by searches for anomalous production of
multilepton events. The blue, cyan, magenta, and red points have doublet fractions in the ranges
[95,100]%, [50,95]%, [5, 50]%, and [0, 5]% respectively.
A crucial feature of the process in gure 1a is that the intermediate 4 ! W
decay,
with a hadronically decaying W , allows the kinematic reconstruction of the neutral fermion
4. Experimental studies of this mode can be a ected by searches for semileptonic decays
of a heavy Higgs (H ! W W ! ` jj) which have been presented in refs. [30{34]. In these
searches, the assumption that the observed missing transverse momentum is caused by a
neutrino emitted in the W decay, is used to reconstruct the complete fourmomentum of the
neutrino; thus allowing the reconstruction of the Higgs mass (m2H = (p` + p + pjj )2). The
e ciency loss due to this reconstruction procedure is about 50%. However, in our case, the
neutrino does not originate from a W and this procedure does not reconstruct the correct
Higgs mass. An alternative approach is to consider the transverse mass mT = (E= 2T
that is expected to have an edge at mH . Moreover the pT distribution of the neutrino is
T
p=2 )1=2
expected to be di erent because the latter does not originate from a W . Finally, for Higgs
masses above 200 GeV our signal is potentially enhanced, with respect to the one studied
in ref. [30], by the ratio of BR(H ! 4
! W
maximal H and 4 branching ratios are taken from
! jj
) . 0:5
1
0:7
0:35 (where the
gures 4 and 6) to BR(H ! W W !
jj
) = 2
0:74
0:7
0:1 ' 0:1 (where we included a combinatoric factor of 2 for the
{ 9 {
new lepton mass.
H !
3.2
(with a leptonic tau) and H !
and is constrained by searches for
The diagram in gure 1b leads to the Z
nal state. For Z !
our process contributes
to the jet plus missing ET signature, that is common to monojet searches for dark matter
pair production or for invisible Higgs boson decays. For instance, in refs. [35, 36] the
ATLAS and CMS collaborations placed upper limits on the visible cross section de ned
as the product of cross section, Monte Carlo acceptances and detector e ciencies, i.e. the
observed number of events divided by the integrated luminosity. Requiring E= T > 150 GeV,
the ATLAS limit is 726 fb which is larger than the typical values that we can obtain. For
our reference point, even before requiring an extra highpT jet and including acceptances
and detector e ciencies, the total cross section is (pp ! H)
BR(H ! 4
! Z
` `) . (7 pb)
0:5
1
0:2
! ``, our signal contributes to the pp ! ZZ
! ``
measurement [37] and to searches for H ! ZZ ! ``
[38], monoZ with missing energy,
Z + E= T [39, 40] and Zh(H) ! `` + E= T [41]. For instance, the CMS search for Z + E= T [40]
nds a limit on the visible cross section (for various E= T cuts) at the 1{2 fb level; in our case
the mH = 200 GeV total cross section is up to
(pp ! H)
210 fb, before acceptances and e ciencies. Therefore,
this channel is likely to o er the strongest constraint.
The hadronic Z ! jj mode is less clean and has been studied in the context of the
!
!
H ! ZZ ! jj
the Higgs, PH = pZ + p 1 + p 2 , and charged vectorlike lepton, Pe4 = pZ + p 1 (up to a
dilution due to picking the wrong muon).
The most promising channel is H
! Z
! ``
. A recent ATLAS search for
e4 ! Z` ! 3` with the e4 produced with another vectorlike lepton in a DrellYan process
is presented in ref. [42] where the visible cross section into 4`, 3` + jj and 3` nal states is
found to be smaller than 1 fb. For our reference point, we obtain
underlying process and kinematics with respect to vectorlike leptons pair production, we
expect our acceptance for the ATLAS search to be somewhat smaller than the quoted
(20{50)% acceptance. In addition, one could additionally search for a fourth lepton and
require one lepton pair to form a Z while imposing a Z veto on the second pair thus
suppressing ZZ backgrounds; the four leptons invariant mass is also expected to peak at
the heavy Higgs mass.
In ref. [43], CMS presented a study of a heavy SMlike Higgs decaying to ZZ and
found that for Higgs masses smaller than about 500 GeV the 95% upper limit on the total
cross section into four leptons is 0:1
this corresponds to about 0:1
takes into account that only the 4
and 2e2
nal states should be included), which has
to be compared to our reference cross section of about 210 fb discussed above. Since these
searches require two onshell Z bosons compared to the single Z in our signal, we expect
the Monte Carlo level acceptances for this process to be very small. A similar argument
applies to pp ! ZZ ! 4` [44]. However, one can again impose a Z veto on two of the
charged leptons, almost completely removing the SM background.
The Z ! jj mode is also interesting but it yields experimental bounds that are roughly
an order of magnitude weaker than in the dilepton case [43]. The invisible Z channel is
problematic because it leads to a dimuon plus missing energy
nal state in which the two
muons do not reconstruct a Z. This nal state would also contribute to a W W search.
This mode stems from the diagram in gure 1f and leads to large contributions to several
very promising
nal states: (
)h
, (ZZ )h
! 6`, (W W )h
! 4` + MET and
(bb)h
(the subscript indicates the particles whose invariant mass reconstruct the SM
Higgs). Standard Model backgrounds to the rst three modes are essentially absent making
them golden channels to discover this process. In fact, the dominant SM backgrounds are
pp ! h( ; Z); moreover, the latter can be further suppressed by requiring the Z to be
virtual by vetoing dileptons with invariant mass close to mZ .
For our reference point we
nd
almost zero background) could be already present in the existing data set.
BR(H
! e4
BR(H ! e4
! h
! h
!
! ZZ
) .
!
0:4 fb. A large number of expected events (over
3.5
The SM Higgs plus missing energy signal, depicted in gure 1c, is also very interesting
because it overlaps with dark matter searches.
The clear golden mode is the (
)h + E= T
nal state. A search for this signal has been
performed in ref. [45] where a small excess of 3 events has been observed over (essentially)
no background. In fact, the pp ! Zh !
0.2 fb and, with a 10% acceptance, leads to about 0.5 events at 20 fb 1. The cross section
for our reference point is about
0:5
1
10:5 fb. Assuming that our signal acceptance is identical to the
acceptance used in this search (10%), we expect about 21 events with 20 fb 1 of integrated
luminosity. Future updates of this search will certainly place interesting constraints on our
process has a total cross section of about
model.
In ref. [46] the pp ! (Z ! bb)(h ! invisible) ! bb + E= T was studied. Interestingly
a small excess of 20 events is observed for mbb
125 GeV that would be compatible with
our signal with h ! bb. For our reference point and considering the h ! bb decay of
the SM Higgs we get a cross section of up to
2:0 pb. This is further reduced by the b tagging
e ciency (0:72 = 0:5) and by the acceptance. In ref. [47], CMS presents a similar study
but the dijet invariant mass distribution after btagging is not presented; hence we do not
know whether an excess at mbb = 125 GeV is seen.
In refs. [48] ATLAS performed a dedicated search for pp ! (h ! bb)Z=W . Events
are classi ed according to the number of leptons in the
nal state: Z !
(0leptons),
W ! ` (1lepton), Z ! `` (2leptons). The 0lepton channel shares the nal state with
the search presented in ref. [46] (that we discussed above) and a small excess compatible
with the one observed in that search has also been observed. A similar search has been
performed by CMS in ref. [49].
A search for h ! bb produced in association with dark matter has been presented in
ref. [50] where, unfortunately, the bb invariant mass distribution is not shown.
Other interesting nal states are pp ! h + E= T ! (
modes are similar to the corresponding SM pp ! h ! (
; ; W W ; ZZ ) + E= T . These
; ; W W ; ZZ ) ones, albeit
with sizable extra missing energy that further reduces all backgrounds.
4
Conclusions
In two Higgs doublet model type II with vectorlike leptons, the decay modes of heavy Higgs
bosons can be dominated by cascade decays through the new leptons into W , Z and Higgs
bosons and SM leptons. These processes are listed in eqs. (1.1) and (1.2) and corresponding
Feynman diagrams are in gure 1. After applying constraints from precision electroweak observables, searches for heavy Higgs bosons and constraints on pair production of vectorlike leptons obtained from searches for anomalous production of multilepton events we found that branching ratios of
H ! 4
and H ! e4 , where e4 and 4 are the lightest new charged and neutral leptons
can be as large as 50%. These decay modes are especially relevant below the tt threshold
and when the light Higgs boson (h) is SMlike so that H ! ZZ; W W are suppressed or
not present, competing only with H ! bb and for su ciently heavy H also with H ! hh.
Furthermore, we found that each of the subsequent decay modes of e4 and 4: e4 !
W
providing many possible search opportunities. Among the most interesting signatures
are monojet, mono Z, mono Higgs, and Z and Higgs bosons produced with a pair of
charged leptons. Some of these signatures are almost background free. Combining this
with potentially large production cross section for these processes presents great discover
prospects at the LHC.
Acknowledgments
The work of RD and EL was supported in part by the U.S. Department of Energy under
grant number DESC0010120. RD was supported in part by the Ministry of Science, ICT
and Planning (MSIP), South Korea, through the Brain Pool Program. RD also thanks the
Galileo Galilei Institute for Theoretical Physics for hospitality and support during part of
this work.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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