Update of global TwoHiggsDoublet model fits
Accepted: May
Update of global TwoHiggsDoublet model ts
Debtosh Chowdhury 0 1 2 3 4 5 6
Otto Eberhardt 0 1 2 6
Email: 0 1 2 6
0 F91405 Orsay Cedex , France
1 F91128 Palaiseau Cedex , France
2 Piazzale Aldo Moro 2 , I00185 Roma , Italy
3 Centre de Physique Theorique, Ecole Polytechnique
4 Laboratoire de Physique Theorique, Universite ParisSud
5 Istituto Nazionale di Fisica Nucleare , Sezione di Roma
6 Parque Cient co, C/Catedratico Jose Beltran , 2, E46980 Paterna , Spain
We perform global ts of TwoHiggsDoublet models with a softly broken strengths and direct search limits obtained at p
Beyond Standard Model; Higgs Physics

s = 8 TeV and p
Z2 symmetry to recent results from the LHC detectors CMS and ATLAS, that is signal
s = 13 TeV. We combine
all available ATLAS and CMS constraints with the other relevant theoretical and
experimental bounds and present the latest limits on the model parameters. We obtain that
deviations from the socalled alignment limit
=
=2 cannot be larger than 0:03 in
type I and have to be smaller than 0:02 in the remaining three types. For the latter we also
observe lower limits on the heavy Higgs masses in the global t. The splittings between
these masses cannot exceed 200 GeV in the types I and X and 130 GeV in the types II and
nd that the decay widths of the heavy Higgs particles cannot be larger
than 7% of their masses if they are lighter than 1:5 TeV.
1 Introduction 2
Model
3 Constraints and tting setup
4 h signal strengths
5 Heavy Higgs searches
6 Combination of all constraints
7 Conclusions
A.1 H and A decays to tt
A.2 H and A decays to bb
A.3 H and A decays to
A.4 H and A decays to
A.5 H and A decays to Z
A.6 H decays to ZZ or W W
A.7 H decays to hh
A.8 A decays to hZ
A.9 H+ decays
A Detailed description of the direct Higgs searches
B Prior dependence of the massive parameters
1
Introduction
The discovery of a new scalar resonance with a mass around 125 GeV [1, 2] in the Run 1
phase of the Large Hadron Collider (LHC) has paved the way for new directions in
highenergy particle physics. Analyzing the properties of this particle has suggested strong
evidence that it is the Higgs boson of the Standard Model (SM), i.e. a scalar CPeven state
which has SMlike couplings to the other particles. Currently the combined analysis based
on the Run 1 (7 and 8 TeV) LHC data shows that its couplings with the vector bosons
are found to be compatible with those expected from the SM within a
10% uncertainty,
whereas the coupling to the third generation fermions (top, bottom quarks and the
lepton) is compatible within an uncertainty of
15
20% [3]. Thus the current status of
the Higgs properties still allows to explore new interpretations of the observation coming
from new physics of di erent underlying structures.
{ 1 {
2HDM has two Higgs doublets in contrast to the single Higgs doublet in the SM. This
extension of the Higgs sector leads to the existence of ve scalar bosons, namely a heavy
and light CPeven Higgs boson, H and h, a CPodd Higgs boson, A, and a pair of charged
Higgs bosons, H . Whether the scalar boson observed in the Run 1 of LHC is a part of
an extended Higgs sector is an outstanding question and is at the cynosure of attention of
the current Run 2 (13 TeV) phase of the LHC.
The questions we ask is: which parts of the 2HDM parameter space are favoured after
possible nal states at the LHC: besides the fermionic nal states tt, bb, +
, tb and +
they include gauge bosons (
, Z , ZZ, W +W ) and Higgs particles (hh, hZ, HZ, AZ)
as the decay products of a heavy resonance. So far, these searches for heavy resonances
have remained elusive in the ATLAS and CMS data, and thus the measurements put
modelindependent 95% C.L. upper limits on the production cross section times branching
ratios for di erent production processes and decay modes. In the present work, we assess
the status of all four types of softly broken Z2 symmetric 2HDM with natural
avour
conservation when all the experimental constraints coming from the latest LHC data are
taken into account. We confront these with the theoretical constraints on these models
(positivity, stability and nexttoleading order unitarity). Furthermore, we perform global
Bayesian ts to all relevant constraints on these models, which also include electroweak
precision and avour observables, and highlight the complementarity between them.
This paper is organized as follows: the 2HDM is de ned in section 2. In section 3 we
list all relevant constraints and explain the tting setup. The results are presented in the
subsequent sections, rst taking into account only the Higgs signal strengths in section 4
and the direct searches in section 5, before combining them with the other constraints in
section 6. We conclude in section 7. In appendix A we show the details of our
implementation of the direct Higgs searches and in appendix B we explain how we treat the prior
dependence of the massive parameters.
2
Model
The TwoHiggsDoublet model with a softly broken Z2 symmetry is characterized by the
following scalar potential:
V = m121 y1 1 + m222 y2 2
m122( y1 2 +
y2 1) + 12 1( y1 1
)2 + 12 2( y2 2
)
2
+ 3( y1 1)( y2 2) + 4( y1 2)( y2 1) + 12 5 ( y1 2
)2 + ( y2 1
h
)2i ;
(2.1)
{ 2 {
Yb;1 = Yl;1 = 0
Yb;2 = YbSM= sin
Yl;2 = YlSM= sin
Yb;2 = Yl;2 = 0
Yb;1 = YbSM= cos
Yl;1 = YlSM= cos
Yb;1 = Yl;2 = 0
Yb;2 = YbSM= sin
Yl;1 = YlSM= cos
Yb;2 = Yl;1 = 0
Yb;1 = YbSM= cos
Yl;2 = YlSM= sin
Type X (\lepton speci c") Type Y (\ ipped")
2 are the two Higgs doublets. While writing the potential we have assumed
that the scalar potential is CP conserving. Instead of the eight potential parameters from
eq. (2.1) we will use the physical parameters in the rest of this article. They consist of the
vacuum expectation value v, the CPeven Higgs masses mh and mH , the CPodd Higgs
mass mA, the mass of the charged Higgs, mH+, the two diagonalization angles
and the soft Z2 breaking parameter m212. Assuming the observed scalar of mass
and ,
1.6 TeV, that is beyond the region where the 125 GeV scalar was found. Moreover, we trade
the angles
and
with
and tan , since these combinations can be directly related
to physical observables. All SM parameters were xed to their best t values [34, 35].
Neglecting the rst two generations of quarks and the rst generation of the leptons,
the Yukawa part of the 2HDM Lagrangian reads as follows:
LY =
YtQLi 2 2tR
Yb;1QL 1bR
Yb;2QL 2bR
Yl;1LL 1lR
Yl;2LL 2lR + h:c:
Here, l stands for or . In the above Lagrangian, by convention the top quark only couples to
2; its Yukawa coupling is related to the SM value YtSM by Yt = YtSM= sin .
With an unbroken Z2 symmetry in the Yukawa sector, there are only four possibilities
through which the Higgs elds couple to the bottom quark and the leptons at treelevel, if
we assume that the leptons couple to the same doublet. They are commonly called type I,
type II, type X or \lepton speci c" and type Y or \ ipped". In table 1 we categorize the
corresponding Yukawa coupling assignments.
3
Constraints and tting setup
Our statistical analysis of the 2HDM is a Bayesian t, in which the following priors are
used for the previously de ned parameters:
1:1
0
;
log(tan )
1:7 (equivalent to 0:08
tan
50);
strength
Value
tth
ZZ
ggF
ZZ
VBF
W W
ggF
W W
VBF
ggF
VBF
W W
Wh
Wh
W W
Zh
Zh
W W
tth
tth
bb
Wh
bb
Zh
bb
tth
pp
0.25
0.14
0.26
0.16
0.37
1
0
0
1
1
1
0
0
1
0.21
0.25
0.26
0.16
0.37
1
0
0
0
1
1
1
0
0
0
0
1
0.47
1:4
1:4
0.25
1:9
3:5
0.47
0.1. The colours in the rst column indicate the decay category in gures 1 and 2.
Extreme tan
values outside the chosen prior are expected to be excluded due to the
absence of strong 2HDM e ects in certain
avour observables (see e.g. reference [36]); the
aforementioned interval is a very conservative estimate. The only implicit assumption we
make is that the 125 GeV scalar is the light CPeven Higgs particle of the 2HDM and that
the other scalars should be heavier, yet in LHC reach.
The focus of this article is on LHC Higgs observables, that is h signal strengths and
searches for H, A and H+.
Most details of the implementation of the corresponding observables can be found in our last article [31]. The modi cations to this will be explained in the following. { 4 {
For the signal strengths, we de ne production, where \production" stands for the ggF,
VBF, Vh, Zh, Wh, tth or pp production channels of the h, while \decay" denotes the
subsequent h decay products
, ZZ, W W ,
1 For the last one, only
upper limits are available; we assign to this signal strength a central value of 0 and adjust
the Gaussian error such that the likelihood distribution has the 95% limit at the value
provided by the experimental collaborations. All h couplings are calculated at leading
order: while the fermionic decays and the bosonic decays to W W and ZZ are possible at
treelevel, we apply oneloop expressions for the decays into nal states including massless
bosons (that is gg,
and Z ) [37].
A list of the available experimental signal strength values from LHC Run 1 and 2 can
HJEP05(218)6
be found in the tables 2 (ATLAS and CMS combination for Run 1), 3 (ATLAS numbers
for Run 2) and 4 (CMS measurements for Run 2). For the Run 2 data, we also list
the corresponding integrated luminosities L. The numbers for the correlations in table 2
can be found in the mentioned document. For Run 2, ATLAS provides correlations only
for the combination of the
and ZZ decays; observing very similar numbers in the
corresponding Run 1 data, we assume identical correlations for the
and ZZ
nal states.
The correlation between
tth
tth was extracted from
gure 17 in [42]. (We assume
that the V V
nal state therein is dominated by W W .) Also the CMS correlations in table 4
were reconstructed from the signal strength contours (or cross section times branching ratio
contours) in the plane of VBF vs. ggF production. In section 4 we discuss the individual
impact of the signal strengths on the 2HDM parameters, ordered by the decay products.
Concerning the direct searches for the heavy CPeven, the CPodd and the charged
Higgs, we have updated the number of used LHC analyses from 16 in [31] to 50 Run 1 and
2 measurements in the present article. We calculate the product of the production cross
section [105{114] and the branching ratio [115, 116] of a speci c decay,
compare it with the experimental bounds, we assign Gaussian likelihoods with a central
value of 0 to the ratio of the theoretical value and the observed upper limit of
method agrees with the treatment of the upper limit of the Z
signal strength mentioned
above and coincides with our approach in [31] under the assumption that the observed
upper limit does not deviate from the expected one. With no evidence for such a deviation
in any of the searches, this approximation seems to be justi ed. The experimental input
from Run 1 and 2 can be found in tables 5, 6 and 7. These analyses comprise a large variety
of searches for heavy resonances decaying into fermionic or bosonic states: bb,
and
Z limits can be applied to both, the CPeven and CPodd Higgs bosons; signatures with
a pair of massive bosons or AZ in the
nal state can exclusively stem from H decays at
treelevel in the 2HDM, while a CPodd resonance decaying to one h or H and one Z is
interpreted as A; nally, the searches for charged Higgs particles were performed looking
B. In order to
B
. This
for the
instance (
nal states tb or
. If the branching ratio into a speci c
nal state  like for
)(bb)  is included in the upper limit, we list it in the table. If the nal state
is not included in the
B limits, but its information is needed to distinguish it from other
searches, we write it in square brackets. The secondary decay products of one particle are
1In order to improve readability, we drop charge or conjugation labels when there is no ambiguity.
{ 5 {
strength
ggF
VBF
Vh
tth
ZZ
ggF
ZZ
VBF
ZZ
Vh
ZZ
tth
W W
VBF
W W
Wh
W W
tth
tth
bb
Vh
bb
tth
pp
Z
pp
0.29
0.22
0.29
0.22
1
0
1
0
1
0.31
amount of data. For instance, the latest CMS results of the searches for an H decaying via
ZZ into two leptons and two neutrinos are available only for mH > 600 GeV. Lighter mH
scenarios will be constrained using an older publication based on an integrated luminosity
of 2:3 fb 1
. Also for the upper limits on H ! hh ! (bb)(bb) by CMS and on H+ ! tb by
ATLAS we apply di erent searches depending on the masses. For gg ! X !
, CMS
combined their 8 and 13 TeV data; the limits are given for the 13 TeV production cross
section. A detailed discussion of how the di erent searches constrain the 2HDM can be
found in section 5, where we show the results ordered by the decay products.
Apart from the discussed treelevel Higgs observables the 2HDM scalars can also
contribute to the quantum corrections of other observables, the most important ones being
the electroweak precision observables, the b ! s branching ratio and the mass di erence
{ 6 {
strength
ggF
VBF
Vh
tth
ZZ
ggF
ZZ
VBF
ZZ
Vh,h
ZZ
Vh,l
ZZ
tth
W W
ggF
pp
ggF
VBF
tth
bb
VBF
bb
Vh
bb
tth
pp
W W
VBF+Vh
1:11
0:5
2:3
2:2
1:20
0:06
0
0
0
1:02
0:89
1:17
0:84
1:11
0:72
3:7
1:2
0:19
0:7
0:19
0:6
1:1
0:9
0:22
1:03
2:85
2:78
1:19
0:27
0:67
0:44
0:89
0:35
0:58
2:5
0:4
0:81
1:0
1
0.32
1
0.43
1
0.24
0.32
1
0.43
1
0.24
1
in the Bs meson system. While the implementation into HEPfit was already explained
in [31], we updated the experimental values [35, 117, 118]. Also for the treatment of
theoretical constraints we refer to [31], with two exceptions: we do not apply any constraints
arising from the renormalization group evolution and de ne our model at the electroweak
scale. And for the nexttoleading order unitarity bounds we chose the most conservative
approach that appeared reasonable to us, namely requiring that the real and imaginary
parts of the Smatrix eigenvalues should be between
0:5 and 0:5 and between 0 and
1, respectively. Moreover we impose perturbativity by discarding scenarios for which the
oneloop contribution to these eigenvalues exceeds the treelevel term in magnitude.
{ 7 {
Channel Experiment
2 2b
[100;900]
[90;1000]
[90;1000]
[90;1000]
[90;1000]
[65;600]
[200;1600]
[200;1200]
[140;1000]
[140;1000]
[300;1500]
[300;1500]
[145;1000]
[270;1100]
[260;1100]
[260;1000]
[300;1000]
[220;1000]
[220;1000]
[225;600]
[220;350]
[130;1000]
[130;1000]
[180;1000]
[180;600]
[200;600]
[180;600]
rst column, we assign a label and colour to each search, which correspond to the ones in
gures 6
to 12. Details of production and decay modes are given in the second column. The third column
contains the corresponding reference. The mass ranges, for which the corresponding limits on
are given, and the integrated luminosity the searches are based on, can be found in the fourth and
fth column. The CMS Run 1 limits of the diphoton channel are included in their Run 2 bounds.
V V refers to either W W or ZZ. C8V V provides signal strength limits. A8hh contains information
B
about the decays of hh to 4b, 2 2b, 2 2b and 2 2W .
{ 8 {
A13
C13
2HDM. For an explanation, see the description below table 5. In the last column, we additionally
highlight an underlying integrated luminosity of around 3, 13 or 36 fb 1 in red, yellow or green,
respectively.
{ 9 {
HJEP05(218)6
Channel Experiment Mass range [TeV]
2HDM. For an explanation, see the description below table 5.
As numerical setup we use the opensource package HEPfit [119], interfaced with the
HJEP05(218)6
release candidate of the Bayesian Analysis Toolkit (BAT) [120]. The former calculates all
mentioned 2HDM observables and feeds them into the parallelized BAT, which applies the
Bayesian t with Markov chain Monte Carlo simulations.
4
h signal strengths
In this section we show the impact of the h signal strengths on the 2HDM parameters. The
ts were done with the most uptodate experimental inputs; for a comparison with the
status before EPSHEP 2017, see [
121
]. The di erences in the Z2 symmetry assignment
to the fermions result in a type dependent treatment of their couplings to the light Higgs
boson. The signal strength of the process with a given initial state i producing an h which
decays to the nal state f can be written as
if = ri
rf
f0
P rf0 BSM(h ! f 0)
;
(4.1)
where rx is the ratio of the 2HDM and the SM partial width of an h decaying into x and
BSM(h ! x) is the corresponding SM branching ratio. From this equation one can see that
every signal strength depends on the 2HDM h couplings of all decay products.
In gure 1 we show the individual impact of the signal strengths with a speci c nal
state on the
plane as well as their combination in all four types of Z2
symmetry. We have tried to adopt the colouring scheme from
gure 14 of the Run 1
combination [3]. All contours delimit the regions allowed with a probability of 95.4%. The
upper limit on pZp is included in the combination, but not shown separately as its e ect
In type I all fermions have the same relative coupling to h: rtt = rbb = r
= r
)]2. This can only deviate signi cantly from 1 if tan
is not close to the alignment limit =2. In these regions, the
di=
is
photon signal strengths are the most constraining ones, see the upper left panel of gure 1.
For larger tan
values, the ZZ and W W signal strengths become the most important
constraints. In the combined t to all signal strengths, the largest possible deviation of
from
=2 is 0:26 at 95.4% if we marginalize over all other parameters.
The upper right panel of gure 1 shows t results with the same inputs for type II.
Here, the relative downtype fermion and lepton couplings to h are di erent from the
vs. tan
plane in all four 2HDM types. We show the 95.4% posterior probability contours for individual
ts to data from h decays to
, bb,
,
, W W and ZZ in red, cyan, purple, orange, blue and
green, respectively. The resulting 95.4% regions of the combined ts to all signal strengths are the
blackbordered grey areas.
top coupling, and thus the fermionic signal strengths yield more powerful constraints.
But also the signal strengths with a bosonic
nal state become stronger because of the
modi cations of the loop coupling rgg and the fermionic couplings in the denominator of
eq. (4.1). Especially the W W and ZZ signal strengths constrain
to be very close
to
=2; the largest deviation from the alignment limit in the onedimensional t to all
signal strengths is 0:055 at 95.4%. The socalled \wrongsign" solution for the fermionic
couplings, which is represented by the lower \branches" of the individual W W , ZZ, bb
and
signal strength ts for tan
> 3 and
< 1:5, can (cannot) be excluded in the
combined t to all signal strengths with a probability of 95:4% (99:7%). These scenarios
have also been shown to be incompatible with the assumption that the 2HDM of type II
is stable under a renormalization group evolution up to O(1) TeV [31, 122].
In type X, the h couplings of the downtype quarks agree with the ones of the top
quark, but the leptonic couplings are like in type II. Consequently, the contour of the bb
decays in the lower left panel of gure 1 has a similar shape as the one of type I, while the
and
decays behave more like in type II for large tan for
< =2. For tan
the latter two are the dominant signal strengths. For very large tan , the wrongsign
solution of the fermion couplings is allowed at 95:4%. However, no larger deviations of
from
=2 than 0:069 are allowed at the 95:4% level if we combine all signal strength
information and marginalize over all other parameters.
Finally, the type Y t can be found in the lower right panel of gure 1. Like in type II,
has to be very close to the alignment limit with the bosonic signal strengths being the
strongest constraints. But like in type X, the wrongsign coupling of the fermions cannot be
completely excluded at 95:4% in the t combining all signal strengths, although it is only
possible for very large tan . In this type's combined t and marginalizing over the other
As compared to the status before EPSHEP 2017 [121], the W W ,
and bb signal
strengths have become more constraining; the latter changed drastically due to additionally
released data. In type II, a small spot of the \wrong sign" branch around tan
= 3 and
= 1 was allowed at the 95:4% before summer 2017 and has disappeared now.
The two angles
and
de ne all treelevel couplings of fermions and bosons to the
light Higgs h, but the loop couplings to gluons and photons are more complicated. In order
to analyse their allowed ranges, we show the rgg vs. r
plane in
gure 2. Apart from the
individual ts to the di erent
nal states and their combination like in
gure 1, we also
add the contours from a t to only Run 1 and only Run 2 data, respectively.
The relative oneloop couplings read like this:
8
rgg = <rtt = rbb in type I and X,
h
: rtt gtgF + rbb gbgF + prttrbb( ggF
t
ggF
i
gbgF ) = ggF
in type II and Y,
r
=
prV V F1( W ) + P
f
prff Ncf ef2 F1=2( f ) + 23mm2H2h+ chH+H F0( H+ )
2
F1( W ) + P
f Ncf ef2 F1h=2( f )
2
Here, the SM gluonfusion production cross section is ggF whereas its pure top and bottom
loop contribution are denoted as gtgF and gbgF , respectively. Their values were calculated
with HIGLU [106]. For the diphoton coupling, we follow the notation of [37], in which the
loop functions F1, F1=2 and F0 are de ned in eq. (2.17) and i = 4mi2=m2h. Ncf and ef
denote the colour factor and the electric charge of the fermion f , respectively. The triple
Higgs coupling chH+H
can be found after eq. (5) of [18].
In all types, the combined
t to all signal strengths is dominated by the bosonic
decays. While the
mainly delimit r , rgg is constrained also by the
measurements. The maximal deviation of r
(rgg) from its SM value is roughly 30%
(20%). The wrongsign solution for the fermion couplings can be seen in type II, X and
Y: the regions for rgg > 1 in gure 2 contain the lower branches of gure 1. For type II, it
has been shown that the wrongsign couplings feature increased rgg and reduced r
[12].
In the lower right panel of gure 2, this \second solution" is visible between 1:1 and 1:2
for rgg as spikes in the W W , ZZ and
contours for large r
as well as in the combined
in the plane of the relative oneloop couplings of the h to gluons and photons. While the colours
of the shaded contours with solid borders correspond to the ones in
gure 1, the Run 1 and Run 2
combinations are bounded by the dark green and brown dashed contours, respectively.
signal strength t in both r
directions. Comparing all Run 1 signal strengths with all
Run 2 signal strengths, one can see that generally the Run 2 data is more constraining in
all types. However, the Run 1 signal strengths prefer a smaller rgg and thus determine the
upper limit of the gluon coupling ratio in the combined t to all signal strengths. This can
be seen in the types II and Y.
The run time for ts with 120 million iterations was 230 80 CPU hours for the diphoton nal state and 70 20 CPU hours for the other single channels.
5
Heavy Higgs searches
The detailed analysis of the individual direct searches can be found in A. Here we only
want to discuss the impact of the relevant combined decay categories. The narrow width
approximation will be applied throughout this section; we will comment on its validity at
the end of the next section.
In gure 3 we show the available parameter space for 2HDM masses and angles from the
t where the heavy Higgs searches are taken into account. The region inside the various
coloured patches are disfavoured by the corresponding search category denoted in the
legend. The central areas inside the solid grey line mark the 95.4% allowed regions when all
heavy Higgs searches are considered in the t, including also CAZ and CHZ . In the panels
8 8
in the rst row the black dashed lines mark the limit from the t to theory constraints only.
The combination of all H/A/H+ searches is represented by the orange/blue/green dashed
contours. The channels described in the previous subsections which are not constraining
or very weakly constraining in this mass vs. angles plane are not shown in the gure.
From the rst row of gure 3 we can see that the region around = =2 remains
unconstrained in all four types of 2HDMs when all the heavy Higgs searches are taken
in account. From moderate to high masses, the diHiggs channels dominate the excluded
regions in all four types whereas H !
, H !
and H ! V V are the most important
constraints below the hh threshold. In type I and X, the nal exclusion region is mainly
dominated by the heavy Higgs to two light Higgs channel, while in type II and Y this
is only true if
>
=2; for
<
=2 the
nal constraint is weak except for
250 GeV. Although the diphoton searches alone disfavour mH . 600 GeV for regions
'
=2 this region is allowed when considering all the heavy Higgs searches
in the t. The H
! V V decays only are susceptible to the tan
. 1 region and for
mH & 800 GeV in all four types; this again is an e ect of netuning. In the displayed mH
vs. tan
ranges, all the H searches become ine ective for mH & 1000 GeV in type II and
Y, whereas in type I and X the tan
. 1 regions remain inaccessible even if the heavy
Higgs mass is as large as 1500 GeV. This is an e ect of the combination of all heavy Higgs
searches, in which scenarios with small tan
are sampled less by the tter. This feature is
not worrisome, because these regions are also suppressed by the avour observabels as we
will show in the next section.
In the pseudoscalar mass vs. tan
planes we see that the diphoton channel constrains
low tan
and mA up to 600 GeV for all four types. The A ! hZ channel can exclude tan
values up to 10 and mA almost as heavy as 1000 GeV in all four types. The exclusion also
applies for the large tan
regions in the types II and Y. The next most important channel
for the pseudoscalar searches in type II is the A !
channel, which e ciently excludes
high tan
regions for mA as large as 1000 GeV. In type X this channel is susceptible to
tan
. 20 and mA . 400 GeV. The exclusion from this channel is weaker for type I and Y.
The contour for all pseudoscalar searches is mainly dominated by the hZ channel in type I
and Y, and a combination of hZ and
in type II and X. Combining all the pseudoscalar
Higgs searches the present data constrain certain regions where mA . 1000 GeV. For type
X, the mA < 400 GeV region remains available only if tan
& 10. In the same mass regions
a very narrow range of intermediate tan
remains accessible for type II when all constrains
are taken into account. The bound on large pseudoscalar masses is similar to the heavy
Higgs case when all constraints are taken into account.
The main channel which constrains the charged Higgs mass is H+ ! tb. The exclusion
region of this channel is shown in red in the last row of gure 3. From the gures we see
that the present searches for the charged Higgs mass can only constrain the regions with
searches with a probability of 95.4% by the central area inside the grey solid line. We compare them
with the areas excluded by searches in various nal states represented by the coloured patches. The
areas inside the coloured dashed lines correspond to the exclusion at 95.4% when all H searches
(orange), all A searches (dark blue) and all H+ searches (dark green) are considered. In the rst
row the limits from theory constraints are shown by black dashed lines.
tan
. 1 and mH+ . 1000 GeV in all four types. The inclusion of all the searches in
the t yields stronger constraints in the mH+ vs. tan
plane than the t to all charged
Higgs searches only. In type II and Y this is due to the H ! hh ! 4b searches, which are
particularly sensitive to large tan
and disfavour certain regions featuring tan
type I and X it is more di cult to pinpoint one particular channel for the seeming exclusion
of tan
values up to 2 for charged Higgs masses above 1 TeV, but in the next section we
see that these bounds can be relaxed once we take into account also other constraints.
6
Combination of all constraints
After discussing the individual e ects of the h signal strengths and the searches for H,
A and H+ on the 2HDM, we want to confront these constraints with the other bounds
on the parameters. In
gure 4 we copy the information about all heavy Higgs searches
gure 3 in the mass vs. angle planes, and add the bounds from signal strengths
and theoretical constraints to the mH vs.
planes and the impact of the
avour
observables to the planes with tan . Finally, the global t to all constraints is represented
by the grey regions.
The signal strength bounds do not depend on the masses of the heavy Higgs particles;
their limits on
di er for each type, see section 4. The theory conditions force
the 2HDM's into the alignment limit for mH > 600 GeV, decoupling the heavy Higgs
particle from physics around the electroweak scale. The type dependence of this e ect is
negligible as it only enters via subleading Yukawa terms in the beta function parts of the
NLO unitarity conditions. Besides the obvious consequences for certain constellations of
mH and
, the combination of the signal strengths with the theory constraints also
disfavours large values of tan . This is because extreme values for the latter result in a
destabilization of the unitarity conditions [31]. Speci c combinations of the 2HDM angles
can still ful l the theoretical constraints, but these solutions are highly
netuned and
thus have a low posterior probability. The avour constraints have been discussed many
times in the literature; the summary is that in all types the Bs mass di erence sets lower
limits on tan
(at least for masses within the reach of the LHC), while the branching
ratio of b ! s processes enforces mH+ & 580 GeV at 95% C.L. in type II and Y [124].
In combination with theory and electroweak precision bounds, these limits on the charged
Higgs mass can be translated to lower limits on the neutral masses. (The individual impact
of ST U will be explained below as it is not visible in these twodimensional projections of
the parameter space.)
The combination of all constraints is more intricate than the naive superposition of
all individual bounds. First of all, we should mention that it also depends on the prior
we choose for the masses: while the direct experimental observables depend on the masses
of the heavy Higgs bosons, the theoretical bounds and the loopinduced e ects are only
sensitive to the mass squares. Since we want to combine both, we need to decide whether
we want to use a at prior for the masses or the mass squares. A detailed discussion can
be found in appendix B. The contours we show here are a superposition of a t with
at
mass priors and a t with at mass square priors in order to be as conservative as possible.
In the mH vs.
planes of type I and X, the combination of signal strengths and theory
only leave a very small strip around the alignment limit of
=
=2. Heavy Higgs
searches additionally exclude mH < 380 GeV in type X. The reason for this are not only
with a probability of 95.4% in light grey and compare them with the areas excluded by various sets
of bounds: the 95.4% contours of heavy Higgs searches (dark grey), avour observables (yellow)
and h signal strengths (pink) as well as the 99.7% limits from theory constraints (purple).
the H !
searches as observed in gure 3, as they are similar to the bounds in type I,
but mainly it is an interplay of the strong bounds of the A searches for low tan
and the
abovementioned exclusion of large tan
due to signal strengths and theory, which together
disallow mA < 400 GeV. This bound translates to a limit on mH using the unitarity and
electroweak precision constraints, since these bounds delimit the mass splittings, see below.
Also in the type II and Y planes we see a lower limit of mH > 550 GeV, which in this case
derives from the lower bound on mH+ from the b ! s
measurements and the fact that
the mass di erence mH
mH+ cannot be too large. The absolute maximal deviation of
from
=2 is 0.03 in type I and 0.02 in the types II, X and Y. (This corresponds to
1
sin(
) < 5 10 4 and < 2 10 4, respectively.) In the mH vs. tan
planes one can
see that the netuning for large tan
scenarios disfavours these regions and pushes the
allowed contours towards smaller values of tan . Only in type I and for mH < 350 GeV,
the heavy Higgs searches have a visible impact on this plane, excluding tan
. 2:5. More
or less the same holds for mA vs. tan , where in type I tan
< 3 is excluded by direct
A searches if mA < 350 GeV. We already mentioned above that in type X, the interplay
between netuning and A searches sets a lower limit of 400 GeV on mA. Having a look at
the mH+ vs. tan
planes, one can see the lower bounds on the charged Higgs mass in type II
and Y from b ! s , which we quantify to be 600 GeV in our t. Also here, we observe that
large tan
values are disfavoured and the posterior regions are shifted towards small tan .
The t only to electroweak precision data does not exclude any region in the
twodimensional mass vs. angle projections in
gure 4. What it does constrain are the mass
di erences between H, A and H+. That is why in gure 5 we show the di erence between
the pseudoscalar and charged Higgs mass, once depending on the H mass (left column)
and once against the mH
mH+ di erence. In the mA
mH+ vs. mH planes the dominant
constraints come from the theory bounds, at least in the decoupling limit mH > 600 GeV.
The ST U pseudoobservables are stronger if mH < 600 GeV and mH+ > mA. In type I,
they yield a lower bound on mA
mH+ in the global t for mH < 250. If we look at the
mA
mH+ planes, we observe that here the oblique parameters are the
strongest constraint on mA
mH+ if the charged Higgs mass is larger than mH . Combining
all constraints and marginalizing over all other parameters, we obtain the following ranges
for the mass di erences allowed with a probability of 95%:
mH
mA [GeV]
mH
mH+ [GeV]
mA
mH+ [GeV]
We can thus exclude the decays H ! H+H , H ! AA, H ! H+W
as well as H ! AZ
in all four types with a probability of 95%.
For all heavy Higgs search limits, we implicitly assumed the narrow width
approximation. In a simultaneous t to all constraints except for these direct search limits we nd
that with a probability of 95% the decay widths of H, A and H+ never exceed 5:5% of
the mass of the particle in the types II and Y. For masses below 1 TeV the maximal decay
widths are less than 3:5% in these two types. In type I and X and for
yield
=m
< 3:5% (< 5%) if m
< 1 TeV (< 1:5 TeV). Only the ratio
= H; H+ the ts
A=mA can reach
7% for mA
550 GeV, but in the decoupling limit mA > 600 GeV similar bounds apply
as for H and H+. We would like to stress here that all these 95% limits are maximally
Figure 5. In the mA
mH+ vs. mH (left panels) and mA
mH+ (right panels)
planes we show the allowed regions by various sets of constraints: the heavy Higgs searches, the
oblique parameters and the avour observables determine the 95.4% allowed contours in dark grey,
light blue and yellow, respectively. For the theoretical constraints, the 99.7% regions are given by
the purple shaded areas. We superimpose the 95.4% probability combination from the global t to
all observables in light grey.
allowed values and that in a typical 2HDM scenario the widths are signi cantly smaller.
Therefore we conclude that the narrow width approximation is a reasonable choice for
2HDM scenarios. { 19 {
Finally, addressing the last variable of our chosen parametrization, we also observe
limits on the soft Z2 breaking parameter m212. The upper limits strongly depend on the
maximally allowed physical Higgs masses and are around (1 TeV)2 in all types. Due to
the lower mass limits on the physical Higgs particles in the types II, X and Y, we also
observe that m212 is limited from below, the respective minimal values being (280 GeV)2,
(170 GeV)2 and (240 GeV)2. Only in type I an unbroken Z2 symmetry is still compatible
with all constraints.
The run time for the global ts to all constraints with 240 million iterations was
130 CPU hours.
Conclusions
HJEP05(218)6
550
7
j
In all four 2HDM types with a softly broken Z2 symmetry we have presented global ts to
the most recent data.
Focussing on the latest measurements from LHC, we have showed explicitly how the
individual signal strengths a ect the leading order h couplings at treelevel and at
oneloop level. Combining all information about the signal strengths, we nd that the quantity
=2j cannot exceed 0.26, 0.055, 0.069 and 0.056 in the types I, II, X and Y. The
oneloop couplings of the h to gluons and photons cannot di er by more than 20% and
30%, respectively, relative to their SM values.
In order to systematically discuss the searches for H, A and H+, we have categorized
them according to their decay products and have compared the exclusion strength of the
single available ATLAS and CMS analyses on the production cross section times branching
ratio, depending on the masses. We have then combined all decay categories and have
showed their impact on the 2HDM masses and mixing angles. For mH below 1 TeV we
observe strong bounds on \extreme" values for the angles, that is if
is very di erent
from the alignment limit
=2 or if tan
is smaller than 1 or larger than 10. The exact
limits depend on the model type and mH . Also the LHC searches for pseudoscalars severely
constrain the 2HDM parameters: for mA < 1 TeV, the lower limits on tan
reach values
of around 10 in the types I and X. In the types II and Y, these limits are weaker, but there
are also mass dependent upper limits. The bounds from charged Higgs searches are less
constraining in comparison; nevertheless, they also start to be stronger than the indirect
constraints in the regions with low mH+ and low tan .
Finally, we have confronted the LHC h signal strengths and heavy Higgs searches
with all other relevant indirect constraints from theory and experiment. In detail, we
have showed how stability and unitarity constraints and B physics observables set mass
dependent limits on the 2HDM angles and on the di erences between the heavy Higgs
masses, while electroweak precision data only a ect the latter.
We have compared all
di erent sets of constraints and have showed the results in the mass vs. angle planes as
well as in the mA
mH+ vs. mH ( mH+) planes for all four types of Z2 symmetric 2HDM's
together with the simultaneous t to all constraints. In this global t we nd the following
95% probability limits on the 2HDM parameters marginalizing over all other parameters:
j
II and Y, mH > 700 GeV, mA > 750 GeV, mH+ > 740 GeV and m212 > (240 GeV)2,
=2j cannot be larger than 0:03 in type I and 0:02 in the other types. In type
while we observe lower mass limits of mH > 450 GeV, mA > 500 GeV, mH+ > 460 GeV
and m212 > (170 GeV)2 for type X. For the latter, it is the rst time that a statistically
signi cant lower limit on the massive parameters has been observed in a global t for
the analyzed mass ranges. Also, if we discard particularly
netuned scenarios, only the
following ranges for tan
are allowed for masses below 1:6 TeV: [0.93; 10.5] in type I, [0.93;
are no strict bounds and have to be taken with a grain of salt. Moreover, we can put type
dependent upper limits of order of 100 GeV on the di erences between mH , mA and mH+,
and thus kinematically exclude all decays of H or A into another heavy Higgs particle.
As a consequence, the decay widths of H and H+ cannot exceed 5:5% of their mass in
all types, at least as long as we consider masses below 1:5 TeV. While in the types II and
Y we see a similar limit for the A decay width, it can amount to up to 7% of mA in the
types I and X.
Acknowledgments
We thank Enrico Franco, Ayan Paul, Maurizio Pierini and Luca Silvestrini for useful
discussions. The ts were run on the Roma Tre Cluster. We are grateful for the availability
of these resources and especially want to thank Antonio Budano for the support. We also
thank Jose Miguel No and Ken Mimasu for giving us a hand with the implementation of
the H
! AZ and A ! HZ search limits. This work was supported by the European
Research Council under the European Union's Seventh Framework Programme
(FP/20072013) / ERC Grant Agreement n. 279972, by the IndoFrench Center for Promotion of
Advanced Research/CEFIPRA (Project no. 54042) and by the Spanish Government and
ERDF funds from the European Commission (Grants No. FPA201453631C21P and
SEV20140398).
A
Detailed description of the direct Higgs searches
In the following, we will scrutinize the impact of the searches for heavy Higgs particles
in all four types of a 2HDM with a softly broken Z2 symmetry, ordered by their decay
products. First we will address the fermionic decays to tt, bb and
and the loop induced
decays with
and Z
in the nal state. The searches for signals in these channels apply
to both, H and A bosons. After that, we will turn towards the H speci c decays into
two massive vector bosons or two h bosons and the A speci c channel with an h and a Z
nal state. Finally, the decays of a charged Higgs to
and tb will be discussed.
The limits CAZ and CHZ on the decays H ! AZ and A ! HZ were only used in the
8 8
combination of all heavy searches; their impact on the mA vs. mH plane is found to be
weak if we marginalize over all other parameters.
The grey shaded regions in the plots in this section depict the prediction of the 13 TeV
B for the corresponding channel without applying any theoretical or experimental
constraints on the model; in other words they correspond to our full priors. The black dashed
lines delimit the full available posterior ranges of the
B for the corresponding channel
HJEP05(218)6
when only the theoretical constraints de ned in the section 3 have been used in the t.
The areas within the various coloured solid lines depict the 95:4% probability posterior
ranges of
B after imposing the experimental constraints from the LHC for a particular
measurement. The legend of each plot refers to the channels described in tables 5, 6 and 7.
The horizontal coloured lines on the top of the panels mark the mass ranges analyzed at
the LHC for each of the searches denoted in the legend. In the following plots the posterior
prediction of
B after considering a particular direct search replicates the prior behaviour
unless it deviates from rest of the posteriors for the same channel. In other words, the
direct searches only have a visible impact on the 2HDM parameters if their contour is
lower than the other coloured contours. Otherwise, they represent the 95.4% probability
contours of the prior distributions. Also the lower lines of the searches mostly show the
priors and do not pose any signi cant lower limit on
B
.
Since we want to combine all information about a speci c heavy Higgs decay into one
gure, we have to \project" the actual measurement onto a
B value which we consider
searches, the contours of the A13
as common denominator of the decay. For instance, the searches H ! hh comprise a large
spectrum of nal state signatures (like 4b or
W W ). So we decided to predict the possible
values of (pp ! H) B(H ! hh) from the information of the search limits involving further
h decays. In other words, we divide the actually tted quantity by the branching ratios of
the h decays if necessary, in order to obtain the bare (pp ! H) B(H ! hh). This is why
the same model independent search limit can have varying e ects and the exclusion curves
are not necessarily the same for the di erent 2HDM types. In our example of the H ! hh
2 2W channel will be the same in all four types, because the
information about the h decays was removed by the ATLAS collaboration, who provide
(pp ! H) B(H ! hh) limits. On the other hand, the impact of C123b2 will look di erent
in all types, because the h ! bb and h !
decays are included in the CMS limits.
The ts to the single experiments in this section took 60
7 CPU hours with 120 million iterations.
A.1
H and A decays to tt
For H and A masses heavier than two top quarks the decay to tt is the dominant in the
2HDM, at least for moderate values of tan . Unfortunately, a possible signal strongly
interferes with the treelevel background process gg ! tt. The only available experimental
limits of an analysis which takes into account this interference is [123] for the 2HDM of type
II. Its limits, however, constrain only a small region for tan
1 and mH=A
500 GeV,
which has been excluded by indirect constraints. Therefore, we do not take into account
this direct measurement.
A.2
H and A decays to bb
The direct search for H
! bb decay does not put any constraint on
B in all four
2HDM types considered in the analysis. In gure 6 one can see that theoretical constraints
provide a suppression of
B by roughly an order of magnitude compared to the t without
any constraint in type I and X. Similar to the previous case, pseudoscalar decaying to bb
searches do not provide any stronger constraint on
B than the t with theory constraint
alone in all four types. For this search, theory constraints alone restrict
B by at least one
order in magnitude with respect to the t without any constraints in the parameter space
analyzed for all the types. This suppression is more dominant in type II and Y compared
to the other two cases. In type II and Y, in the regime mA . 600 GeV the A ! bb search
from Run 1 suppresses
B compared to the t without any constraints but it remains
subdominant or at most of similar strength to the t with theory constraint alone.
A.3
H and A decays to
In the upper panel of gure 7 we show that the H !
searches suppress the
B limit
by at least one order of magnitude compared to the t with theory constraints alone in
the regime where the heavy Higgs mass is below 250 GeV. In this regime the strongest
constraint comes from Run 1 data and the suppression of
types I and Y. Theory constraints alone restrict
B is more pronounced in the
B by roughly an order of magnitude
compared to the t without any constraints for type I and Y, whereas in type II and X,
theory constraints raise the lower limit of
B for mH > 500 GeV. This can be understood
as a sensitivity of the t to
netuned scenarios with extreme tan
values, which are
disfavoured in a t to the theoretical bounds.
From the t without any constraints in the lower panel of gure 7 we see that the
predicted ranges of
B for the pseudoscalar decaying to
are quite narrow for type II
and X compared to the other two 2HDM types. The theory constraints yield a suppression
by at least one order of magnitude for
B compared to the t without any constraints
10 5 .
in the types I and Y whereas for type II and X the theory constraints push up the lower
limit on
B by one order of magnitude compared to the t without any constraints for
mA > 350 GeV. The direct search limits for A !
suppress
B by roughly one to two
orders of magnitude compared to the t with theory constraints alone for all the types
except for type Y as long as mA . 300 GeV. In type Y, the experimental upper limits
on
B are stronger than the theoretical ones for pseudoscalar masses between 200 and
400 GeV, in type II even up to 1 TeV. Scenarios with very light A and
B values between
B . 10 pb seem also to be excluded at 95% by the prior. However, since we
have no experimental data on this region, this prior dependence is not an issue here.
A.4
H and A decays to
The theory constraints on H=A !
suppress
B by one to three orders of magnitude
compared to the t without any constraints in all four 2HDM types, see gure 8. Direct
searches for a heavy CPeven Higgs decaying to two photons constrain
order of magnitude compared the t with theory constraints for mH . 250 GeV in all types.
The searches in the diphoton decay channel of a pseudoscalar Higgs yield a suppression of
B by one to three orders of magnitude compared to the t with theory constraints for
mA . 600 GeV for all four types considered. In the types II and Y, we observe again that
certain intermediate
B regions for low mA are disfavoured by the prior.
B by roughly one
four 2HDM types (top: H, bottom: A). For details, see text.
B vs. mH=A planes for the
four 2HDM types (top: H, bottom: A). For details, see text.
in the
B vs. mH=A planes for the
four 2HDM types (top: H, bottom: A). For details, see text.
in the
B vs. mH=A planes for the
H and A decays to Z
We see from the top panel in
gure 9 that for H ! Z , theory constraints suppress
by one to four orders of magnitude compared to the t without any constraints for all
types. In all four types, direct search for this channel does not provide any constraint on
B except for a very small window below mH ' 250 GeV, but it remains subdominant
compared to the t with theory bounds.
Similar to the heavy CPeven Higgs case, theory constraints yield a suppression of
by one to two orders of magnitude for the decay A ! Z
compared to the t without any
constraints in all four types. Direct searches for this channel provide a suppression of
by an order of magnitude compared to the t with theory constraints in the mass window
B
B
B
250 . mA . 350 GeV in all four types. Again, some parts of the
masses are disfavoured by the prior in the types II and Y.
B region for light A
A.6
H decays to ZZ or W W
The heavy Higgs decays to massive gauge bosons can be divided into searches for ZZ and
W W , but the relative coupling of H to two vector bosons V V = ZZ; W W is universal and
type independent. However, the production of the H di ers between the types. We show
the H ! ZZ channels in the upper panel of gure 10 and the searches for H ! W W as well
as the combined searches for H ! V V in its lower panel. The
B are constrained by the
theoretical bounds in the decoupling limit, where mH > 600 GeV. The direct LHC searches
for this channel yield a strong suppression of
B by one to three orders of magnitude
compared to the t with theory constraint in the mass regime 150 . mH . 800 GeV
(150 . mH . 700) for the ZZ (W W ) channel. For the ZZ searches the mH . 250 GeV
region is constrained by Run 1 data whereas Run 2 data determine the dominant limits
for the rest of the region. For the W W searches, Run 1 data dictate the limit until
mH ' 600 GeV and the high mass range is dominated by Run 2 data. Additionally the
CV V search for H ! V V severely constrains
8
B in the 200 . mH . 250 GeV region for
type II and Y.
A.7
H decays to hh
In the upper panel of gure 11 we show that for the H decaying to two h bosons, theory
constraints already yield a strong suppression of
constraints in all four types. Direct searches suppress
B compared to the t without any
B by at most one order of magnitude
compared to the t with the theory constraints in the mass range 200 . mH . 600 (500)
GeV in type I and X (type II and Y). The main constraints stem from Run 1 data and
the Run 2 searches for hh resonances decaying to two photons and two bottom quarks.
Although di erent searches at Run 1 and Run 2 continue to constrain the
channel up to mH ' 1200 GeV, they remain subdominant to the limit from the t to the
B for this
theory constraints.
A.8
A decays to hZ
The searches for A decaying into hZ are shown in the
of gure 11; more precisely, they are projected onto the
B vs. mA planes in the lower panel
B of the decay to (bb)Z. As
B vs. mH=A planes for the
(bottom) in the
B vs. mH planes for the four 2HDM types. For details, see text.
B vs. mH=A planes for the four 2HDM types. For details, see text.
compared to the t without any constraints, theory constraints e ectively are important
in the decoupling limit in all types as well as for mA . 300 GeV in type I and X. Direct
searches for this channel yield a strong suppression (one to three orders of magnitude) of
B compared to the t with theory constraints for all four types as long as mA . 800 GeV.
In all types, the strongest bounds come from searches in the A ! hZ ! (bb)Z channel; in
type X we additionally observe that the A ! hZ ! ( )Z limits yield the most important
constraints for mA between 200 GeV and 300 GeV.
A.9
H+ decays
At the present, charged Higgs decaying into does not provide any constraint on the 2HDM's under consideration, see the upper panel of gure 12. The theory constraints yield a suppression of
B by roughly one order of magnitude as compared to the t
without any constraint for all types but type X as well as for all types if the charged Higgs
decays into tb (lower panel of gure 12). In the latter case, the direct searches from Run 1
B in
(Run 2) provide even stronger limits on
only small di erences between the 2HDM types.
B between 180 GeV and 600 GeV (1 TeV), with
B
Prior dependence of the massive parameters
In this appendix we discuss the prior dependence of our analysis when all the constraints
have been taken into consideration. In gure 13 we compare the allowed parameter space
in the mass versus angles planes and the mass di erence versus mass (di erence) planes
from the t with at mass priors (blue solid and dashed curves) with the t with at mass
square priors (red solid and dashed curves). At a rst glance one can see that the t with
at mass square priors prefers high mass regions and raises the lower limit on the masses
by O(100) GeV compared to the t with
at mass priors.
It is well known that Bayesian statistics do not provide a unique rule to determine the
prior distribution and in general the posterior distribution is a prior dependent quantity.
A thumb rule would be to choose a at prior for the parameter on which the observables
depend linearly. For example, if an observable quadratically depends on a particle mass,
one would choose a
at mass square prior. Unfortunately, in the 2HDM the theoretical
and indirect experimental constraints depend on the mass squares, whereas the direct
experimental observables dependent on the masses. Not having su ciently constraining
data makes assigning the mass priors in the
t with all constraints is a delicate task.
This would not be problematic if the observables were measured with a high precision;
in fact, we can see that in the types II and Y the di erence between the two priors are
considerably smaller as there are strong lower mass limits. In order to be as conservative
and prior independent as possible, we decided to combine the 95.4% regions for both priors:
the light grey contours for the ts with all constraints in
gures 4, 5 and 13 are obtained
by superimposing a t with
at mass priors and a t with
at mass square priors. The
corresponding numerical results mentioned in section 6 are based on the more conservative
t; for instance, the limits for masses and mass di erences were extracted from the t with
at mass priors, while the upper limits on the decay widths were larger in the ts using
at mass square priors.
(top) and H+
! tb (bottom) in the
B vs. mH+ planes for the four 2HDM types. For details, see text.
planes, in the mH vs.
planes (corresponding to
gure 4) and in the mA
vs. mH and mA
mH+ vs. mH
mH+ planes (corresponding to gure 5), from top to bottom.
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. { 33 {
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