#### A new avenue to charged Higgs discovery in multi-Higgs models

Radovan Dermsek
1
Jonathan P. Hall
1
Enrico Lunghi
1
Seodong Shin
0
1
0
CTP and Department of Physics and Astronomy, Seoul National University
, Seoul 151-747,
Korea
1
Physics Department, Indiana University
, Bloomington,
IN 47405, U.S.A
Current searches for the charged Higgs at the LHC focus only on the , cs, and tb final states. Instead, we consider the process pp W H W +W A where is a heavy neutral Higgs boson, H is a charged Higgs boson, and A is a light Higgs boson, with mass either below or above the bb threshold. The cross-section for this process is typically large when kinematically open since H W A can be the dominant decay mode of the charged Higgs. The final state we consider has two leptons and missing energy from the doubly leptonic decay of the W +W and possibly additional jets; it is therefore constrained by existing SM Higgs searches in the W +W channel. We extract these constraints on the cross-section for this process as a function of the masses of the particles involved. We also apply our results specifically to a type-II two Higgs doublet model with an extra Standard-Model-singlet and obtain new and powerful constraints on mH and tan . We point out that a slightly modified version of this search, with more dedicated cuts, could be used to possibly discover the charged Higgs, either with existing data or in the future.
Contents
1 Introduction
2 Charged Higgs production and decay 2.1 Our example reference scenario: the type-II two Higgs doublet model with an additional SM singlet
2.2 Charged Higgs decays
2.3 Heavy neutral Higgs decays
2.4 Heavy neutral Higgs production
2.5 Total cross-sections
3 The constraint from Standard Model h W +W searches
3.1 Model independent study
3.2 The type-II 2HDM plus singlet case
4 Conclusions
A Decay rates
A.1 W
A.2 Other AH decays
A.3 Other H decays
A.4 Other H decays
A.5 Off-shell H
B Branching ratios
D CLs limits
1 Introduction
The quest to unveil the mechanism responsible for the breaking of the electroweak
symmetry made a huge leap forward with the recent discovery of a scalar particle whose quantum
numbers and interactions appear to be compatible, albeit with large uncertainties, with
those of the Standard Model (SM) Higgs boson [1, 2]. The presence of a fundamental scalar
particle renders electroweak physics sensitive to arbitrarily large scales possibly present in a
full theory of electroweak, strong, and gravitational interactions. Solutions to this problem
usually entail the introduction of new physics just above the electroweak scale. Amongst
others, hints that point to the incomplete nature of the SM are the strong empirical evidence
for particle dark matter, the baryon-antibaryon asymmetry of the universe, and the pattern
of neutrino masses and mixing. Even before addressing these problems it is important to
realize that while the structure of currently observed gauge interactions is completely
dictated by the SM gauge groups alone the pattern of electroweak symmetry breaking is not.
In particular, within the context of a perturbative (Higgs) mechanism there are absolutely
no symmetry reasons for introducing a single doublet (besides the empirical observation
that such a choice leads directly to the rather successful Cabibbo-Kobayashi-Maskawa
pattern of flavor changing and CP violation). Moreover, it is well known that supersymmetry,
one of the most popular extensions of the SM that actually addresses some of the above
mentioned problems, requires the introduction of a second Higgs doublet. In view of these
observations it is clear that understanding how many fundamental scalars are involved in
the electroweak spontaneous symmetry breaking mechanism is one of the most pressing
questions we currently face. In particular, any model with at least two doublets contain at
least two charged Higgs boson (H) and at least two extra neutral Higgses. In this paper
we investigate a previously overlooked technique that could uncover a charged Higgs from
a multi-Higgs scenario.
Direct charged Higgs production in the top-bottom fusion channel typically has
crosssections O(1 pb) [3] and discovery would be fairly difficult in this channel [4, 5]. If the
charged Higgs mass is lower than the top mass, it is possible to bypass this problem by
looking for charged Higgs bosons in top decays (t H+b), taking advantage of the very
large tt production cross-section. Moreover, most current experimental studies consider
only charged Higgs decays to pairs of fermions (H+ +, H+ cs, and H+ tb).
Under these assumptions ATLAS and CMS were able to place bounds on BR(t H+b)
at the 15 % level [69] for mH < mt.1 It is well known that the presence of a light
neutral Higgs can significantly modify these conclusions. In fact, the H+ W +A decay
(A being a neutral CP -even or -odd Higgs boson) can easily dominate the charged Higgs
decay width if it is kinematically allowed and the A has non-vanishing mixing with one
of the neutral components of a Higgs doublet. Such a light neutral pseudoscalar Higgs
(A = a1) has been looked for by BaBar [11, 12] in a1 ( , ) decays and by
ATLAS [13] and CMS [14] in pp a1 direct production. These bounds are easily
evaded by assuming that the lightest neutral Higgs a1 has a singlet component. Under this
condition, in the context of a type-II two Higgs doublet model (2HDM) with an additional
singlet, the BR(t bH+) can be as large as O(10 %) for tan < 6 (tan being the ratio
of the vacuum expectation values of the neutral components of the two Higgs doublets)
even for a1 as light as 8 GeV [15]. Trilepton events in tt production can be used to discover
at the LHC a charged Higgs produced in top decays and decaying to W A with as little
as 20 fb1 integrated luminosity at 8 TeV center of mass energy.
At the LHC the charged Higgs can be alternatively produced in the decay of a heavier
neutral Higgs (). Heavy neutral Higgs bosons are dominantly produced in gluon-gluon
fusion (ggF) with a significant cross-section, leading to sizable charged Higgs production
rates. For our somewhat model independent analysis, we ignore possible mass relations
amongst the various Higgs bosons as they depend on the exact Lagrangian of the model. In
1A preliminary result of ATLAS reduces this to O(0.1%) [10].
the presence of a light Higgs A the decay H+ W +A is mostly dominant for mH+ < mt
and remains comparable to H+ tb otherwise, depending on the values of the various
parameters. Note that the H+ W +h1 decay (we take h1 be the particle recently discovered
at the LHC) vanishes in the limit that h1 is completely SM-like.
In this study we consider the process pp HW W +W A as shown in
figure 1. The constraints we derive are valid for mA not too far above the bb threshold,
where the decay A bb should be dominant (they are also approximately valid below
this threshold, as discussed in section 3.1). At large transverse momentum of the bb pair
(transverse momentum relevant for the event selection), the angular separation of the two
bottom quarks is small and they are combined into a single jet.2 The final state we consider
is, therefore, constrained by the standard h W W searches by CMS [21] (with 19.5 fb1
at 8 TeV and 4.9 fb1 at 7 TeV) and ATLAS [22] (with 20.7 fb1 at 8 TeV and 4.6 fb1 at
7 TeV). We use the data provided in the CMS analysis to place bounds.
The impact of the experimental cuts depends on the kinematics and is controlled by
the masses of the three intermediate Higgs bosons only. We therefore derive constraints
on the LHC cross-section for the considered process that depend only on the masses of
the relevant particles and not on other model-dependent parameters or the CP nature of
the neutral Higgs bosons and A. We also apply our results to a CP conserving type-II
2HDM with an additional singlet [2326]. In this framework the lightest neutral Higgs
(A) is identified with the lightest CP -odd eigenstate a1 and the heavy Higgs () with
the heavy CP -odd Higgs a2. To the extent that the a2 H+W decay dominates over
other decays involving Higgs bosons (and this can easily be the case) and decays to other
beyond-the-Standard-Model particles our bounds depend on only ma2 , mH , ma1 , tan(),
and A (the mixing angle in the CP -odd sector). A novelty in our analysis is the exclusion
of parameter space regions at low tan . The 8 TeV LHC data analyzed so far allow one,
using our approach, to probe only a relatively light charged Higgs (roughly below the
tb threshold); in the future, regions in parameter space with a heavy charged Higgs will
be accessible as well. We also consider the same scenario but with one of the CP -even
states (h2) as the heavy neutral state . The types of scenario we consider and constrain
can easily be consistent with constraints on the custodial symmetry breaking parameter
= M W2 /(MZ2 cos2 W ).
The paper is organized as follows. In section 2 we discuss the production and decay
cross-section for our signal. In particular, after introducing the type-II 2HDM + singlet
scenario in section 2.1 we discuss charged (H) and neutral () Higgs decays in sections 2.2
and 2.3, the gg production cross-section in section 2.4, and the total cross-section
(production times branching ratios) in section 2.5. In section 3.1 we show the upper bound
on the total cross-section that we extract from SM Higgs to W W searches. In section 3.2 we
specialize the previous results to our reference scenario (type-II 2HDM with an additional
singlet, = a2 and A = a1) and present the new exclusion bounds at low tan that we
extract. Finally, in section 4, we present our conclusions.
2The ATLAS collaboration recently announced the results of a search for a similar process, where the
light state A is identified with the 125 GeV CP -even Higgs, dominantly decaying into two separable
bjets [16]. They consider the semileptonic decay of the W W . This was based on the suggestion put forward
in ref. [17]. See also ref. [1820] which includes the non-resonant production of HW .
W
W
W
W
Charged Higgs production and decay
In the multi-Higgs models containing at least two SU(2) doublets, there can exist a heavy
neutral Higgs () which decays into HW . The process is shown in figure 1 with the
charged Higgs decaying to a light neutral Higgs A and another W boson. Looking for this
process could be the first way the charged Higgs is discovered and its properties measured.
This is due to the large value of (gg W H W W A) when all particles
can be on-shell. In this section, we focus on showing how large such a production
crosssection times the branching ratios can be, especially in the context of the type-II 2HDM
+ singlet scenario. In the following subsections, we show that the branching ratios of
H W A and HW can be sizable when kinematics allow and the production
cross-section of is roughly as large as that of the SM Higgs. Our general cross-section
constraints depend only on the masses of the particles involved and will be discussed in the
next section. For the specific type-II 2HDM + singlet reference scenario we can constrain
physical parameters (the masses; tan ; and A, the mixing angle in the CP -odd sector)
without specifying the Lagrangian in the Higgs sector and we assume no mass relations
among the Higgs bosons states.
Our example reference scenario: the type-II two Higgs doublet model
with an additional SM singlet
Considering the type-II 2HDM with one extra complex singlet scalar we define the
fieldspace basis by
AN = 2ImS,
h cos() sin() 0 2ReHd0 vd
NH = si0n() cos0() 01 2 R2Re HeSu0 svu ,
AH = 2 cos()ImHu0 sin()ImHd0 ,
where S is the SM-singlet and s is its possibly non-zero VEV and tan = vu/vd. In this
convention, h interacts exactly as a SM Higgs in both gauge and Yukawa interactions; H
has no coupling to the gauge boson pairs and interacts with the up-type quarks (down-type
quarks and charged leptons) with couplings multiplied by cot (tan ) relative to the SM
Higgs couplings. The orthogonal state to AH and AN is the Z-boson Goldstone mode.
We define an orthogonal matrix U that transforms the CP -even field-space basis states
into the CP -even mass eigenstates
h1 U1h U1H U1N h
h2 = U2h U2H U2N H
h3 U3h U3H U3N N
We define h1 to be the particle recently discovered at the LHC and do not demand that
hi are ordered by mass. The overlap of h1 with the SM-like state h appears to be large.
The mass eigenstates h2 and h3 are then approximately superpositions of H and N only.
When we consider h2 to be the heavy state produced from pp collisions U2H , the overlap
of h2 and H, becomes an important parameter.
We define a mixing angle between the CP -odd mass eigenstates A by
where a1 is defined to be the lighter state.
The state a1 is identified with A in our process pp W H W W A. We
mainly consider to be the other CP -odd state a2 but also consider the case where it is
one of the CP -even states, defined to be h2.
When the mass of a1 is below the bb threshold the constraints from the decay
a1 ( , ) at BaBar and the light scalar search at the LHC (pp a1 ) lead to
an upper bound on cos A tan of about 0.5 [13, 14, 27]. We concentrate on two benchmark
a1 masses: 8 and 15 GeV. Our results depend weakly on this mass; therefore, the 8 GeV
threshold is representative of masses just below and just above the bb threshold, where the
constraint cos A tan . 0.5 does and does not apply respectively.
In the parameter region where one of the CP -even Higgses h2,3 is lighter than 150 GeV,
the direct search bounds for light neutral Higgses in associated production hia1 (i = 2, 3) at
LEP-II can be considered [28]. The final states can be, for example, 4b or 2b2 . However,
even for h2,3 light enough for this associated production to be possible, the cross-section
is proportional to the doublet component of a1 and is usually small in our scenario. The
upper bounds in [28] constrain cos2(A) Ui2H (i = 2, 3) times branching ratios as a function
of the masses, but this can easily be small enough to be consistent with the bounds. We
therefore ignore the LEP-II constraint throughout this paper.
The masses of the extra neutral and charged Higgs bosons can affect the custodial
symmetry breaking parameter = M W2 /(MZ2 cos2 W ), where W is the weak mixing angle.
Since we are considering extensions of the Higgs sector involving only SU(2) doublets and
singlets, contributions to 1 appear only at loop level. In our type-II 2HDM +
singlet reference scenario with = a2 (mostly doublet), A = a1 (mostly singlet) and the
SM-like Higgs boson discovered at the LHC identified with h1, depends also on the
two remaining CP -even states h2,3. For a simple demonstration of the constraint, we
assume that one of these two states is completely doublet (the field-space basis state H
defined above). Then the main contributions to the vacuum polarization of the W by
H a2 and H H loops need to be cancelled by that of the Z by H a2 loop. Therefore
one can roughly expect the contribution due to the mass difference between H and a2
can be cancelled by that between H and a2, while making that of the H H loop to
the W boson small. (In the 2HDM, complete contributions to the oblique parameters are
well depicted in the appendix D of the reference [29].) In figure 2, we show the H mass
range allowed at 95 % C.L. by the present determination of [30] for given masses of a2
and H. The solid (blue) contours give the maximum value of mH required to satisfy the
experimental constraint; the dashed (green) contours show the difference between the
maximum and minimum mH required and are therefore a measure of the (low) fine tuning
between mH and mH that we require. We find that in the parameter space where our
process is dominant (m mH & MW ) the contributions to can easily be compensated
by the contributions of other Higgs states, although the fine tuning between mH+ and mH
increases as ma2 does. It is quite possible for the H state to remain unconstrained by LHC
Higgs searches. Based on this result, we simply ignore the constraint throughout this
paper. We also ignore possible mass relations amongst the various Higgs bosons which
depend on the exact details of the Higgs sector Lagrangian.
Charged Higgs decays
When the charged Higgs is lighter than the top quark (light charged Higgs), investigating
only the usual or cs final states from its decay may not be enough for discovery. This
is because the process H+ W +A, whose decay rate is proportional to m3H+, can easily
dominate over the + and cs final states. The detailed analysis of the light charged
Higgs from the top quark decay in the context of the type-II 2HDM + singlet is shown in
ref. [15], where the lightest CP odd neutral Higgs a1 is the particle A. The main factors
determining the BR(H+ W +a1) are the SU(2) doublet fraction (at the amplitude level)
in a1 (cos A) and tan . According to that analysis, BR(H+ W +a1) rapidly approaches
unity for mH+ > MW + ma1 even when the light Higgs a1 is highly singlet-like, as long as
tan is small.
For a charged Higgs heavier than the top quark (heavy charged Higgs), the channel
H+ tb opens to compete with the process H+ W +A. In the context of the
typeII 2HDM + singlet, we show the dependence of the BR(H+ W +a1) on cos A and
tan in figure 3. For low tan . 5, the value of (H+ tb) is dominantly determined
by the (mt/v)2 cot2 term, so the BR(H+ W +a1) increases for larger tan . (See
appendix A for the detailed formulae.) Above threshold the ratio of the H+ W +a1
and H+ tb decay rates is proportional to cos2 A tan2 m2H+. For ma1 = 8 GeV the
constraint cos A tan . 0.5 applies and hence BR(H+ W +a1) is at most around 30 %
for mH+ < 400 GeV, increasing for larger charged Higgs masses. On the other hand, we do
not need to consider this bound when a1 is heavier than about 9 GeV, so in this case the
mmHax-mmHin
95% CL D allowed region, cos2A=0.1
150 200 250 300
BR(H+ W +a1) can be larger than 0.5, corresponding to larger values of cos A tan
when ma1 is set to 15 GeV in figure 3. Far above thresholds and at low tan we have
Consequently, BR(H W a1) can be still larger than 0.5 even after the on-shell H+
tb decay opens, as long as a1 is heavier than about 9 GeV. For large values of tan ( 7)
the tan dependence of BR(H+ W +a1) is reversed since the (mb/v)2 tan2 term in
(H+ tb) is dominant.
Heavy neutral Higgs decays
The W H decay can easily dominate over decays into SM fermions, including top
quarks. In the type II 2HDM + singlet scenario we set A = a1 and = a2 (which is the
heavy CP -odd Higgs); then figure 4 shows how BR(a2 HW ) varies with tan and
the various masses. For small tan , the branching ratio is affected by the partial width
a2 tt, whose rate depends on cot2 . Since we only consider this decay and decays
into SM fermions, all taking place via the doublet (AH ) component of a2, the sin2(A)
dependence cancels out of all of the branching ratios of a2. For our reference type-II
2HDM + singlet scenario we assume the possible decays a2 hiZ and a2 hia1 to be
subdominant compared to a2 HW , where hi is a CP -even neutral Higgs. This is
tan() = 1, cos() = 0.5
tan() = 1, cos() = 0.1
tan() = 2, cos() = 0.1
tan() = 5, cos() = 0.1
tan() = 1, cos() = 0.5
tan() = 1, cos() = 0.1
tan() = 2, cos() = 0.5
tan() = 2, cos() = 0.1
tan() = 5, cos() = 0.5
tan() = 5, cos() = 0.1
tan() = 10, cos() = 0.5
tan() = 10, cos() = 0.1
tan() = 20, cos() = 0.5
tan() = 20, cos() = 0.1
tan() = 50, cos() = 0.5
tan() = 50, cos() = 0.1
1
0.9
0.8
)
0.7
W
0.6
H
0.5
20.4
a
(R0.3
B
0.2
0.1
in order to reduce the number of parameters relevant for determining cross-section times
branching ratios in this reference scenario (to be compared to the general bounds on this
cross-section times branching ratios that we derive). The processes a2 hia1 are model
dependent even within the type II 2HDM + singlet scenario. As for the possible decay
modes a2 hiZ: the more SM-like the 125 GeV particle discovered at the LHC (h1) is (the
more h1 h, see section 2.1), the more suppressed the decay to h1Z will be. On the other
hand the other final states hi>1Z can reduce the relevant BR(a2 HW ) by up to about
1/3, if we consider that the constraint requires very approximate mass degeneracy of
H and any state significantly overlapping with H (The width to ZH is equal to the
width to H+W if one ignores the phase-space factor). The results we present (e.g. the
new bound in the (tan , mH ) plane for ma2 2mt) are not much affected by the presence
of this decay mode and we will neglect it altogether in the following. Far above thresholds
we have
(a2 H+W ) + (a2 HW +)
(a2 tt)
Heavy neutral Higgs production
The dominant production mechanism for hSM at the LHC is ggF mediated by quark loops,
mainly dominated by the top quark loop due to its large Yukawa coupling. The production
cross-section of depends on its modified couplings to up- and down-type quarks. The
AH and H interaction states, defined in section 2.1, have couplings to up-type quarks
suppressed by 1/ tan and couplings to down-type quarks enhanced by tan . The production
of a2 is also modified at leading order since there are different form factors for the scalar
and pseudoscalar couplings; CP -even Higgs bosons couple to fermions via scalar couplings
and CP -odd couple via pseudoscalar.
At leading order the ggF production cross-section for a scalar or pseudoscalar is
proportional to
3 X gqA1/2
A1H/2( ) = 2[ + ( 1)f ( )]/ 2
where g is the relative coupling to the quark q (relative to that of the SM Higgs) and mq
is the quark pole mass. The form factors A1/2 are equal to
for scalar and pseudoscalar couplings respectively. The universal scaling function f can
be found, for example, in ref. [31, 32]. In the limit 0 the functions A1H/2( ) and
8 TeV
102100 150 200 250 300 350 400 450 500 550 600
mh2/GeV
14 TeV & inc. bbF
14 TeV & inc. bbF
102100 150 200 250 300 350 400 450 500 550 600
mh2/GeV
A1A/2( ) tend to 4/3 and 2 respectively, so the ratio squared tends to 2.25. The K-factors
(the ratios of cross-sections to their leading order approximations) are typically around 1.8
and cannot be neglected. In this work, to calculate the CP -odd (AH ) and CP -even (H)
doublet production we take the 8 and 14 TeV ggF production cross-sections recommended
by the CERN Higgs Working Group [33] (calculated at NNLL QCD and NLO EW) for a
SM Higgs of the same mass M and multiply by the ratio
where gq = {tan(), cot()} for {down-, up-} type quarks q. (This is also the approach
taken in ref. [34].) We checked the consistency of this approach using the Fortran code
HIGLU [35, 36] at NNLO QCD and NLO EW level with the CTEQ6L parton distribution
functions. For the cases of a2 and h2 the cross-section will have an additional suppression
of approximately sin2 A and U2H respectively, since only the doublet admixture couples
2
to quarks. These production cross-sections at 8 and 14 TeV are shown in figure 5. Note
that for AH there is a sharp peak around the tt threshold region for small tan (where
the top loop dominates) due to the pseudoscalar form factor. Below the tt threshold the
shapes of the curves are highly dependent on whether the top or bottom loop dominates.
This is because the form factor looks quite different depending on whether one is above
threshold (bottom loop case) or below threshold (top loop case).
At moderate and large tan (i.e. tan & 5) heavy neutral Higgs production in bottom
fusion (bbF, upper right plot in figure 1) can be larger than in gluon fusion (ggF, upper left
plot in figure 1). In fact, although the probability to find a bottom quark in a proton is
small (whereas gluons have the largest parton distribution function at LHC center-of-mass
energies), this is compensated by the fact that bbF is an electroweak tree-level process
(whereas the ggF is one-loop suppressed). In the lower plots of figure 5 we show the
impact of adding the bbF cross-section (calculated using FeynHiggs [3740]) to the ggF one
for s = 14 TeV; clearly the effect is sizable only for large values of tan & 10. Note that
at small tan ggF is large and dominant and that at large tan bbF controls the
crosssection; at intermediate values of tan 5 the ggF suppression is not yet compensated by
the bbF enhancement and we find relatively small cross-sections.
Total cross-sections
Combining the previous results, we can obtain the complete cross-section times branching
ratios (gg a2 W +W a1) at 8 TeV in figure 6 for various masses and values of tan
and cos A. For small tan , we can easily obtain a total cross-section times branching ratios
O(pb), which is comparable to the SM Higgs production times BR(hSM W +W ). Hence
the LHC Higgs search result can constrain the maximum total cross-section of our process,
as will be discussed in the next section. For very large tan (& 20) our study is not very
sensitive because the tan dependence of the a2 production cross-section (responsible for
the enhancement of the latter at large tan ) is compensated by the tan suppression of the
branching ratio BR(a2 W +W a1). The complete branching ratios BR(a2 W +W a1)
are calculated as outlined in appendix B and are shown in figure 7.
For comparison, we also show the expected total cross-section at 14 TeV for both
= a2 and = h2 in figures 8 and 9 respectively. Note that in these plots we add the
ggF and bbF production cross-sections. The most important effect of adding the latter is
bmH = 160 GeV,
ma1 = 8 GeV
8 TeV
bmH = 160 GeV,
ma1 = 15 GeV
8 TeV
ma2 = 360 GeV,
ma1 = 8 GeV
8 TeV
ma2 = 360 GeV,
ma1 = 15 GeV
8 TeV
bmH = 110 GeV,
a2101
(
R
B102
)
a2103
a2101
(
R
B102
)
a2103
a2101
(
R
B102
)
a2103
g104100
g
(
a2101
(
R
B102
)
a2103
a2101
(
R
B102
)
a2103
a2101
(
R
B102
)
a2103
g104
g
(
to tan 5 we expect to be sensitive to all values of tan (depending on cos A). When
= h2, the complete cross-section (gg h2 W +W a1) divided by the
mixing1
0.9
)10.8
a0.7
W0.6
W0.5
20.4
(a0.3
R
B0.2
0.1
tan() = 1, cos() = 0.5
tan() = 1, cos() = 0.1
tan() = 2, cos() = 0.1
tan() = 5, cos() = 0.1
1
0.9
)10.8
a0.7
W0.6
W0.5
20.4
(a0.3
R
B0.2
0.1
tan() = 1, cos() = 0.5
tan() = 1, cos() = 0.1
tan() = 2, cos() = 0.5
tan() = 2, cos() = 0.1
tan() = 5, cos() = 0.5
tan() = 5, cos() = 0.1
tan() = 10, cos() = 0.5
tan() = 10, cos() = 0.1
tan() = 20, cos() = 0.5
tan() = 20, cos() = 0.1
tan() = 50, cos() = 0.5
tan() = 50, cos() = 0.1
element-squared U22H is shown. We show that it is possible to have total cross-sections
O(10 pb) in some regions of the parameter space. In these figures we include also a heavier
charged Higgs masses, above the tb threshold.
In this method of estimating the total cross-section, multiplying the production
crosssection by the branching ratios, the non-zero width of the heavy state is neglected. We
check that for m above the HW threshold, going beyond the zero-width approximation
for is a numerically small effect in the parameter space we consider. Below the HW
threshold the finite width effects can be important if the width of is already comparable
to the widths of H and W (the dominant contribution can come from going off-shell
rather than H or W ). We find that this can only occur at extreme values of tan ( 20
or 1 if m > 2mt). In these cases our method can underestimate the below threshold
(off-shell) total cross-section. See appendix C for more details of the width. Our
zerowidth approximation for the heavy state does not affect the limits that we derive. (For
the kinematics the finite width effects are included.)
The constraint from Standard Model h
W +W searches
Model independent study
The CMS collaboration observed a SM Higgs signal in the W +W `+` channel (final
states with zero jets or one jet were included) with a mass of approximatively 125 GeV [21]
2
(a101
R
B
)a2102
b
,bg103
(g
104
2
(a101
R
B
)a2102
b
,bg103
(g
104
2
(a101
R
B
)a2102
b
,bg103
(g
104
a2101
(
R
B
)102
a2
gg103
(
104
a2101
(
R
B
)102
a2
gg103
(
104
a2101
(
R
B
)102
a2
gg103
(
104
at a significance of 4. CMS also provides an exclusion bound for a SM Higgs bosons in
the mass range 128600 GeV at 95 % confidence level (C.L.). The process that we are
considering (pp W +W A) leads to a very similar final state, the only difference being
2101
h
(
R
B102
)
h2
b103
b
,
g
g
(104
2101
h
(
R
B102
)
h2
b103
b
,
g
g
(104
2101
h
(
R
B102
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h2
b103
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,
g
g
(104
the light Higgs A decay products that lead to extra jets or leptons. We, therefore, expect
this search to provide strong constraints on the charged Higgs production mechanism we
consider and potentially to offer an avenue to discover a charged Higgs. However, due to
the presence of the light Higgs A, the distributions of kinematic variables that we obtain
are different from those expected in the SM Higgs search. In order to apply the results in
ref. [21] we need to calculate how the efficiency of the various cuts adopted in that analysis
are affected by the presence of the light Higgs A.
The constraints that we derive are valid for a light Higgs A whose mass is just above
the bb threshold and which decays dominantly to a pair of bottom quarks. In the CMS
analysis the number of jets (for the purposes of separating the events into channels; 0, 1,
or more) is defined as the number of reconstructed jets with pT > 30 GeV (and || < 4.7),
reconstructed using the anti-kT clustering algorithm with distance parameter R = 0.5.
For purely kinematic reasons, when the pT of the A is as large as 30 GeV the angular
separation (R) between the two b quarks is going to be small (compared to 0.5) for the
A masses that we consider and therefore any A final state with high enough pT to count as
a jet will in fact have its final state b quarks cluster into a single jet most of the time. This
has been explicitly checked in ref. [15] (for A +, + ) and in ref. [41] (for A bb
see figure 6 therein). Using MadGraph we checked that for mA up to around 15 GeV,
the R angular opening of the two b quarks is small enough to treat the bb system as a
single fat jet (obviously for a low enough pT cut and/or a large enough mA, the two final
state b quarks can look like two distinct jets).
For mA below the bb threshold A will decay mainly to lepton pairs or maybe to
charm quark pairs. For example, for A = a1, decaying via its AH admixture, decays to
pairs will dominate until very low tan 1.3, where decays to charm pairs begin to
overtake [27]. For such decays into charm quarks the opening angle cannot exceed 0.5 and
the decay products will mostly be clustered into a single jet. For the decays into leptons,
the decay products will also mostly be clustered into a single jet and give no additional
isolated leptons; the exception is when both leptons decay leptonically (about 13 % of
the time). In this case there will be no jet and quite possibly extra isolated leptons that
would lead to the event not passing the selection criteria in the CMS analysis. This small
effect should not much affect our results.
The CMS collaboration presented exclusion bounds obtained using two different
techniques to isolate the signal from the background. The first is a cut-based analysis in which
separate sets of kinematic cuts are applied for each different Higgs mass hypothesis. The
second is a shape-based analysis applied to the distribution of events in the two-dimensional
(mT , m``) plane. In this paper, we apply the cut based analysis of ref. [21] to our signal; at
this time, we cannot proceed with the shape-based analysis since the CMS note does not
provide enough detail.
All of the CMS data are split into four channels depending on whether the two leptons
have different or the same flavor (DF, SF) and whether there is zero or one high pT
(> 30 GeV) jet (0j, 1j). In each channel the expected background, expected signal, and
observed data are given for several SM Higgs mass hypotheses. For each of these hypotheses
a different set of cuts is applied. The cuts used for SM Higgs searches with mass hypotheses
120, 125, 130, 160, 200, and 400 GeV are presented in table 1 of ref. [21]. Extra cuts are
also applied for the SF channels in order to suppress background from Drell-Yan processes.
In this paper, we analyze the 19.5 fb1 of data collected at s = 8 TeV and presented
in table 4 of ref. [21]. To obtain the observed upper limit from applying the cuts in
each channel and corresponding to each SM Higgs mass hypothesis we adopt a modified
frequentist construction [42, 43]. (A brief summary of the CLs method is presented in
appendix D.) The 95 % C.L. upper limit (on the number of events) that we obtain from
our analysis is indicated with `FHJ , where H refers to each of the SM Higgs mass hypotheses
and F J to the channel considered (F {DF, SF} and J {0j, 1j}). The value of `FHJ
has to be compared to the expected signal EFHJP , where P stands for the considered theory
and point in parameter space. (For the type-II 2HDM + singlet scenario P stands for the
relevant Higgs boson masses, cos A, and tan .)
In the type-II 2HDM + singlet reference scenario the expected signal in the 0j channel
is then
AP cos2 A
sin2 AP BaP2 AP cos2 A + BP aFH,Prel ,
|(gg{za2)}
where the exact A dependence has been factored out. Here sFH0 is the number of expected
events for each of the six SM Higgs mass hypotheses H in each channel F 0 in table 4 of
the CMS note [21]. BHH is the production cross-section times branching ratio for that
SM Higgs. The production cross-section times branching ratio for gg a2 HW is
given by sin2 AP BaP2. In the branching ratio for H a1W we factor out the cos2 A
dependence and define AP = (H A1H W ) and BP = (H / a1W ), where AiH is
the pure AH interaction state with the mass of ai. xHP is the fraction of events that have
F
one more jet (in addition to those from initial or final state QCD radiation) passing the
jet selection due to the decay of a1. Here these events are therefore removed from the 0j
channel and appear in the 1j channel. aFH,Prel is the relative acceptance for our signal and, for
each Higgs mass hypothesis H, is defined as the ratio of the fraction of pp a2 W W a1
events that survive a given cut H to the fraction of SM Higgs events that survive the same
cut. Both of these numbers depend on F since extra cuts are applied in the SF channels.
The exact definition of this relative acceptance is
For the expected signal in the 1j channels, we obtain
AP cos2 A
sin2 AP BaP2 AP cos2 A + BP aFH,Prel .
mA = 8 GeV
8 TeV
EFHJP < `FHJ .
We apply whichever of these conditions leads to the best upper limit on the production
cross-section times branching ratios for our signal. These limits on cross-section times
branching ratios are model independent in the sense that they apply to any model
containing , H, and A particles and depend only on the masses of these particles. Moreover,
they do not depend on the CP nature of the and A Higgs bosons because the is
produced on-shell and the structure of the 0V decay (where (0) are spin-0) does
not depend on the CP nature of the (0) (see appendix A). These cross-section limits are
shown in the upper plots in figure 10 and they are superimposed on our reference scenario
in figure 6. When deriving these limits l we assume a fractional systematic error for the
expected signal appearing in each channel of 30 %, which we consider to be conservative
(see appendix D). We find that the limits hardly vary with mA at all for the range that we
consider. The peaks that appear in the left plot are due to us only having data for discrete
values of the SM Higgs mass hypothesis. For instance, the most prominent peak
corresponds to the mass at which the 400 GeV cuts take over the 200 GeV cuts in providing
the best upper limit. Currently only very low values of tan (. 2) can be constrained in
our reference scenario. The strongest constraint is obtained near the tt threshold region,
for this reason we choose ma2 = 360 GeV as a reference point in the detailed parameter
space study presented in the next subsection.
If the analysis were to be performed again using a more appropriate set of cuts for each
set of masses the suppression due to the relative acceptance (see eq. (3.2)) could certainly
be reduced. In fact, since the SM Higgs to W W signal and our signal are very similar, it
is reasonable to presume that optimized cuts would lead to relative acceptances closer to
unity. This would remove the peaks and slightly lower the baseline in the plot in figure 10,
leading to an order of magnitude improvement on the upper limit in some parts of the
parameter space. Existing 8 TeV data could, therefore, be used to probe more moderate
values of tan . Estimating the possible sensitivity of a dedicated search at s = 14 TeV is
not simple, nonetheless the problem is one of distinguishing a signal over the uncertainty of
the background. Assuming that with more data the background determination continues
to be statistics limited and assuming that going from 8 to 14 TeV the background
crosssection roughly doubles we can very roughly predict that at 14 TeV with 100 fb1 (500 fb1)
of data a dedicated analysis could be sensitive to cross-sections of order 0.6 pb (0.3 pb),
to be compared with the kinds of signals predicted in figures 8 and 9. A proper analysis
would need to be carried out by the experimental groups after collecting more data.
It is also worth pointing out that our xHP parameter is almost always closer to unity
F
than to zero. In the SM search the limits coming from the 0j and 1j channels are
comparable. In our case, however, the best limit almost always comes from the 1j channels,
with the 0j channels setting much weaker limits. Almost as many events are moved out of
the 1j channels due to the non-zero xHP than are moved from the 0j into the 1j channels,
F
so the large xHP does not significantly increase the limits coming from the 1j channels;
F
it just weakens the limits coming from the 0j channels. However, if one were to look at
a 2j channel, with the same cuts as in the 0j and 1j channels, but requiring exactly two
high pT (> 30 GeV) jets, the situation could be different. Such a channel would not be
useful for the SM Higgs to W W search (the 2j channel discussed in the CMS analysis [21]
has completely different cuts and is designed to single out vector boson fusion production)
and is therefore not considered in SM searches. However, for our process the probability
to have two high pT jets even in the ggF production, one coming from initial or final state
radiation and another coming from the A decay, is significant. Such a 2j channel would
also likely have a smaller background and could lead to better limits than the 1j channels
for which we have data.
If we replace the a2 with one of the CP -even states, = h2, in our type-II 2HDM +
singlet scenario the analysis is similar. In this case there is, however, another independent
parameter, the H fraction in h2, U22H . This affects the production of but not the decays of
h2 under the assumptions outlined in subsection 2.3.
The type-II 2HDM plus singlet case
As explained in the previous section, SM Higgs W W searches allow one to place model
independent constraints on a charged Higgs produced in the decay of a heavy neutral Higgs
and decaying to W A, where A is a generic light neutral Higgs. In this section we apply
the results presented in section 3.1 to the special case of a type-II 2HDM with an extra SM
singlet. In the context of this model the limits worked out in section 3.1 apply at relatively
low tan (. 2).
In figure 11 we show the limits we obtain for ma2 = 360 GeV. As explained in the
previous section we choose ma2 = 360 GeV as a reference point because the constraints we
obtain are the strongest around the resonance region ma2 2mt. The figure shows the
excluded regions in the (mH , tan ) plane for various values of ma1 {8, 15} GeV and
cos2 A {0.1, 0.01}. The grey region is excluded by direct searches at LEP [5761]. The
blue and green regions are excluded by Tevatron and LHC searches in the [6, 7, 62] and
cs [63] final states, respectively. The pink region is excluded by a combination of searches
at BaBar [11, 12] (3s a1 channel) and at the LHC [13, 14] (direct gg a1
production); this pink exclusion only applies for ma1 just below the bb threshold and not
for ma1 just above. The red area is excluded by a dedicated t bH+ bW +a1
bW + + search at CDF [56]. The purple area surrounded by the thick black solid line
is the additional region of parameter space excluded by our study in the gg a2
W +W a1 channel.
At lower values of cos2 A the exclusion region narrows due to the cos2 A dependence
of BR(H W a1) (see the discussion in section 2.2). In particular, for (ma1 , cos2 A) =
(8 GeV, 0.1), the light charged Higgs parameter region analyzed in ref. [15] is completely
excluded (if a heavy Higgs with mass ma2 = 360 GeV is present). On one hand, at low
values of tan . 0.03 we lose sensitivity because the a2 width becomes dominated by
a2 tt. On the other hand, at large tan 10 either the a2 production cross-section or
BR(a2 W +W a1) are suppressed and our search loses sensitivity.
Our study extends also to charged Higgs masses above the tb threshold. Unfortunately,
sensitivity in this region is not currently very strong for the following two reasons. First,
in this region the H W a1 branching ratio is suppressed at low tan . 2 and very
large tan 10 unless the charged Higgs mass is fairly large (see figure 3). Second, as the
charged Higgs mass increases, the phase space for the a2 HW decay shrinks; this can
be compensated by raising the a2 mass at the price of a reduced production cross-section.
In conclusion, we do not currently find appreciable constraints for mH & 180 GeV. This
heavy charged Higgs parameter space could be constrained in the future with more data.
In figure 12 we show regions that we exclude in the (ma2 , cos2 A) plane at fixed values
of mH {110, 160} GeV, ma1 {8, 15} GeV, and tan . The region above the dotted line
is excluded by direct a1 searches at BaBar and at the LHC (tan cos A . 0.5 [15, 27])
mH = 110 GeV, ma1 = 8 GeV
mH = 160 GeV, ma1 = 8 GeV
mH = 160 GeV, ma1 = 15 GeV
103
103
when ma1 is just below the bb threshold. The reason for the weakening of the limits for
intermediate a2 masses in figure 12 is purely due to the fact that we have data for the
cuts corresponding to SM Higgs mass hypotheses of 200 GeV and 400 GeV, but nothing in
between. This then causes the peaks of weakening limits in figure 10 and the effects can
be seen in figure 12. (See also figure 6.)
The experimental discovery at the LHC of a particle compatible with the SM Higgs
boson is the first step towards a full understanding of the electroweak symmetry breaking
mechanism. Assuming that the particle discovered at the LHC is a fundamental scalar, it
becomes imperative to figure out what exactly the Higgs sector is. Many beyond-the-SM
scenarios contain a second Higgs doublet and predict the existence of at least one charged
Higgs and several neutral CP -even and -odd Higgs bosons. Most experimental searches
have been conducted under the rather traditional assumption that the charged Higgs
dominantly decays into or cs pairs at low-mass (mH . mt) and into tb otherwise. The
existence of a light neutral Higgs A opens the decay channel H+ W +A and offers new
discovery venues.
In this paper, we study a charged Higgs whose production mechanism relies on a heavy
neutral Higgs () and whose dominant decay is into a light neutral Higgs (A)
pp W H W +W A.
For mA & 2mb, this particle decays dominantly to pairs of b quarks that are detected,
at sufficiently high pT , as a single jet. Under these conditions, the final state is simply
W +W plus jets and is, therefore, constrained by SM Higgs searches in the W W channel
(this is also mostly true for mA below the bb threshold). For mA . 2mb, the A dominantly
decays into pairs, whose decay products will also mostly be clustered into a singlet jet
unless both s decay leptonically. (The latter case provides no extra jets and extra isolated
leptons that would lead to the event not passing the selection criteria in the CMS analysis.
This may however be another useful signal to search for.) Using existing data on searches
for a SM Higgs in the range 128600 GeV we are able to place constraints on this new
physics process. In particular, we find that the upper limit on the production cross-section
times branching ratios for the process in (4.1) are in the O(110 pb) range for a wide range
of , H and A masses. The results (presented at the top of figure 10) depend very loosely
on the details of a given model and will be useful to constrain a vast array of theories that
contain three such particles. In particular the limits depend only on the masses of the three
particles and not on the CP nature of and A. For the sake of definiteness we specialize
our results to an explicit type-II 2HDM plus singlet reference scenario and show that our
results are able, at low tan , to exclude previously open regions of parameter space.
The constraints we derive are shown in figures 11 and 12. They are limited both
because we only have partial access to the relevant data and because the cuts used for the
SM Higgs search are not quite optimized for the process we consider. We point out that
a slight modification of the search strategy, using more appropriate cuts that depend on
the hypothesized masses of and H, would lead to better limits and would be sensitive
to more moderate values of tan . Our analysis extends, in principle, to arbitrarily large
charged Higgs masses. In practice, the parametric dependence of the production
crosssection and branching ratios on the charged Higgs mass limits our present sensitivity to
mH . 180 GeV. However, the parameter space with a heavier charged Higgs could be
constrained in the future at the 14 TeV LHC. We point out that once the contribution to
production from bb fusion is taken into account alongside gg fusion, sensitivity to all values
of tan in our reference scenario should be achieved at the 14 TeV LHC. With 100 fb1
of data we very roughly estimate that sensitivity to cross-sections of order 0.6 pb would
be achieved, to be compared to the kinds of cross-sections predicted in figures 8 and 9. A
search for the process where the charged Higgs is produced in the same way but goes to tb
is also being considered [64].
Finally, let us comment on the possibility that our process might contribute sizably
to the total pp W +W cross-section. A recent CMS measurement with 3.54 fb1 of
integrated luminosity at 8 TeV, found a slight excess in this channel: 69.9 2.8 5.6
3.1 pb against a SM expectation of 57.3+21..46 pb without the inclusion of the SM Higgs
contribution [65]. Even after accounting for this an additional contribution of several pb
seems to be required (see for instance ref. [66] for a possible explanation of this tension in
a supersymmetric framework). If this discrepancy survives, the process discussed in this
paper could potentially offer a contribution of the correct order of magnitude.
Acknowledgments
R.D. is supported in part by the Department of Energy under grant number
DE-FG0291ER40661. S.S. is supported by the TJ Park POSCO Postdoc fellowship and NRF of
Korea No. 2011-0012630. E.L. would like to thank Frank Siegert for substantial help with
the Sherpa Monte Carlo.
Decay rates3
12 = (1 k1 k2)2 4k1k2,
1 = p11 = p1 4k1.
Allowing the W to be off-shell and assuming it can decay to all light fermions (excluding
tops), which we take to be massless, we can write
0 1x2k
(1 x1)(1 x2) k
(1 x1 x2 k + kW )2 + kW W
Here 1 and 2 label the fermions from the W decay.4 This formula is valid for AH
HW , H HW , and H AH W . For a2 HW and H a1W , with
the conventions defined in section 2.1, there is a suppression by sin2(A) and cos2(A)
respectively. Writing the integral in this way, the inner x1 integration can be performed
analytically and the remaining integrand behaves well for numerical integration and the
3A more complete list of two- and three-body tree-level decays relevant in Higgs sector extensions
containing doublets and singlets, along with accompanying C++ code, will be presented in ref. [68].
4This is for one particular charge of W . The equivalent formula for a Z boson is obtained by replacing
W Z everywhere. The formulae in ref. [32] (2.20) and ref. [67] (41,58,59) are a factor of 2 too large for
the W boson case, whereas the formula for the Z boson case are correct. This is because Z (as defined
in ref. [32]), rather than being the ratio of the Z and W widths times cos3 W , contains an extra factor of
1/2. This is the symmetry factor relevant for the V V decays, but not the V decays. There is also a typo
in the sin4 W term in Z in ref. [32].
outer integration over x2 can evaluated numerically very quickly. For completeness, above
threshold in the zero-width on-shell approximation we can write5
In this massless fermion approximation we can write
Other AH decays
For light quarks q
QCD(M ) + (35.94 1.36 nf ) QCD(2M )2 ,
mq is the running mass at the scale M = mAH , and nf is the QCD number of flavours at
M . Further QCD corrections for the scalar and pseudoscalar decays to quarks are derived
in refs. [69] and [70] and summarised in ref. [32], but these are only valid in the heavy top
mass limit, i.e. when the boson is light compared to the top quark.
For charged leptons l
+2kW ((1 xt)(1 xb) kW )
Here mb has been neglected in the integrand.The leading QCD correction can be included
by using the running mass for the mt2 factor that appears out front, which comes directly
from the Yukawa coupling in the Feynman rule. In the integrand and in the integration
limits the running mass is not used (for kt and kb) so that the threshold appears in the
correct place.6 For three-body decays written in terms of the xs (energies) of two (1 and 2)
5This is also for one particular charge of W and the equivalent formula for a Z boson is again obtained
by replacing W Z. This on-shell formula in ref. [32] (2.18) contains a typo that makes it dimensionally
inconsistent. The formulae in ref. [67] (38,39,51,57) are correct, except that (57) contains an erroneous
factor of cos2 W .
6This formula is correct in ref. [67] (55,56), but the expression in ref. [32] (2.8) is a factor of 2 larger.
The formula (2.8) as written is correct below threshold after one takes into account that either top can go
off-shell, but is then a factor of 2 too large above threshold. Our approach is given in the text.
of the three final states particles (1, 2, and 3) the kinematic limits are, without neglecting
any masses,
8GF 2 mt3(1 kW )(1 + 2kW )b1W/2.
A formula for (AH tt bW +bW ) that is valid both above and below threshold
can be obtained by doubling (AH tt tbW ) and using 4t in place of t. Above
threshold in the zero-width on-shell approximation we can write
(AH tt) = 43GF2 taMn2m( t2 ) t.
Other H decays
(H qq) = 3GF (gqAH )2M mq2(1 + qq) ,
4 2
1.57 32 ln
Again, a formula for (H tt) that is valid both above and below threshold can be
obtained by doubling (H tt tbW ) and using 4t in place of t. Above threshold
in the zero-width on-shell approximation we can write
(H tt) = 43GF2 taMn2m( t2 ) t3.
Other H decays
For light up- and down-type quarks u and d, assuming ku, kd
One of the (1 kl) factors comes from the matrix-element-squared and the other is the
phase-space factor l .
3GF M ptb
42
h(1 kt kb)(mb2 tan2() + mt2 cot2()) 4mbmtpktkbi . (A.21)
Off-shell H
In the off-shell decay W H W W A, the decay widths W and H roughly
decide which one is preferred to be off-shell. The full decay width of H in our type-II
2HDM + singlet reference scenario is shown in comparison with W in figure 13. For an
H with a mass much above the tb threshold the possible three-body decay of through
an off-shell H needs to be considered.
( W +H W +tb)
8The mt2 term given in ref. [67] (63) seems to be incorrect, producing a different shape to our formula
below threshold and not agreeing with the on-shell formula above threshold. All our formulae are checked to
make sure that they reproduce the on-shell zero-width approximation formulae sufficiently above threshold,
up to finite width effects.
eV101
G
/
H102
103
104
( W +H W +W )
eV101
G
/
H102
103
104
50 100 150 200 250 300 350 400 450 500
mH/GeV
this determines which one preferably goes off-shell.
4mbmtpkbkt
(x1 2kW ) 4kW (1 x1 + kW ) (1 x1 k)2 4kW k
2
(1 x1 + kW kH )2 + kH H
suppressed by mixing angles for other mass eigenstates that are not completely doublet.
B Branching ratios
Only two and three-body decay rates are calculated to allow for fast numerical integration.
product of the two branching ratios
These individual branching ratios can be calculated using off-shell W s and tops.
101
101
100 150 200 250 300 350 400 450 500 550 600
ma2/GeV
Alternatively, allowing the H to go off-shell, we can write
2( W +H W +W A)y + 2( W +H)yBR(H W A)
2( W +H W +X)y + 2( W +H)y + ( 9 W H) ,
should be used in place of the actual off-shell particle width in the integrand
denominator ( y2/M 2). Here tops and W s coming from the H are on-shell. This formula
is really only needed for H masses above the tb threshold anyway, as can be seen by
looking at figure 13. This formula provides a very good approximation to real answer
calculated using four-body decay widths (allowing both the H and W to be off shell)
much better than just allowing the particle with the largest width to be off-shell but is
built out of three-body decay widths and can therefore be quickly evaluated using single
numerical integration.
Figure 14 shows the width of a2 (divided by its doublet fraction) in the type-II 2HDM +
singlet scenario. The contribution to the total cross-section from a2 going off-shell (rather
than H or W ) can be important at high tan or at very low tan if mH + MW > 2mt.
For large a2 masses the width of a2 can become very large.
CLs limits
= P (D |H0)
P (D |H1)
B + S = b + s qb2 + s + s22,
where s is the expected signal, its statistical error is taken to be s, its fractional systematic
error is taken to be . In this paper, we set = 30% as a conservative bound.
We approximate everything as Gaussian. We therefore take
Where is the cumulative distribution function. For a given b, b, D, , and the 1
confidence level limit on s can therefore be found. We call this solution s = l. For our
calculation, the parameters are obtained from ref. [21].
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