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5.2 Double Higgs Boson Production in G2HDM at the LHC

5.2.3 Numerical Results

(

f =q,q

Â

H

Â

2 i=1

kf f highih1h1vDhi(ˆs)FD(ˆs,tf i) +

Â

f =q,qH

k2f f h1F(ˆs,tf i) 2

+

Â

f =q,qH

k2f f h1G(ˆs,tf i) 2 )

, (5.28)

where GF andasare the Fermi constant and strong coupling constant respectively,

kqqhi=OH1i, (5.29)

kqHqHhi=OH2i v

vF, (5.30)

Dhi(ˆs) = 1

(ˆs m2hi+imhiGhi), (5.31) and FD(ˆs,tf i), F(ˆs,tf i), G(ˆs,tf i)withtf i=4m2f/m2hi are form factors that can be found in the Appendix A.1 of Ref. [136]. For later purpose, we also definelhih1h1=ghih1h1/gSMhhh where gSMhhh=6lSMv.

5.2.3 Numerical Results

In this section we will present the numerical analysis for double h1production in G2HDM at LHC and compare the results with SM predictions.

Before we proceed, let us present the set up of the parameter space in the model. We will adopt the allowed ranges for all thel-parameters, lH,F,D,lH0 ,lHF,HD,FD,lHF0 , which satisfy the theoretical constraints from (VS + PU) obtained in [125]. Recall that these theoretical constraints are only relevant for the quartic couplings in the scalar potential. For the Higgs phenomenology constraints, other parameters in the scalar potential are also involved and we will vary their ranges as follows

0.1GeV < vD< 4TeV , (5.32)

30TeV < vF < 100TeV , (5.33)

3TeV < MHD < 3TeV , (5.34)

0 < MFD < 15GeV . (5.35)

5.2 Double Higgs Boson Production in G2HDM at the LHC 105

The SM VEV v is fixed at 246 GeV.

First, we scan all the parameters with the set-up ranges defined above and require them to pass through all the theoretical and Higgs phenomenological constraints presented in [125] without worrying about the dark matter constraints. The constraints from direct Z0 resonance search at the latest ATLAS and CMS 13 TeV results have been taken into account in our scanning. Furthermore, as mentioned in [10], in order to avoid the invisible decay constraints of 125 GeV Higgs boson in the vector boson fusion production mode which yields BR(h1! invisible) < 0.28 at 95% CL [158, 159], we set mh1 <2mDsuch that the invisible mode of h1! DD is not kinematically allowed. As we have mentioned above, to ensure a stable DM candidate D, we impose the mD to be lightest among the masses of ˜D, W0(p,m), H± and all heavy fermions. The masses of heavy fermions are assumed to be degenerate and set to be 3 TeV. We also require mh2 >2mh1 to allow h2decays on shell into h1h1.

In Fig. 5.3, we show the scatter plots of the ratio of production cross sections for a pair of 125 GeV Higgs bosons between the G2HDM and SM on the planes of (lh1h1h1, BR(h2 ! h1h1)) (Fig. 5.3a), (lh1h1h1, mh2) (Fig. 5.3b), (lh1h1h1, kqqh1) (Fig. 5.3c) and (lh1h1h1,kqqh2) (Fig. 5.3d) without taking into account the DM constraints. The color palette on the right of each of the plots in Fig. 5.3 indicates the signal strength of the double h1Higgs boson production. From Fig. 5.3a, one can see that the trilinear self-coupling of Higgs boson in G2HDM can significantly deviate from SM value, it can even flips its sign to be negative.

From Fig. 5.3b, one can see the branching ratio of the heavier scalar h2decay into a pair of h1s can vary from 0 up to 100%. As expected, when |lh1h1h1| and BR(h2! h1h1)are getting larger, the triangle diagram will become the dominant channel and enhance the production cross section. Note that whenlh1h1h1 becomes negative, there is a constructive interference between the two types of box and triangle Feynman diagrams in Fig. 5.2. However when one of the channels, either the box or triangle Feynman diagram, becomes the dominant contribution to the total production cross section, the interference effect is not significant anymore. It is also shown in Fig. 5.3b that for a heavier h2mass the cross section of double h1 production will be much smaller. Furthermore, due to the constraints from the Higgs physics, the absolute value of quark Yukawa couplings with h1can not be deviated too much from its SM value which is demonstrated in Fig. 5.3c, while Fig. 5.3d shows the couplings between quark Yukawa and Higgs boson h2could be small due to the smallness of mixing between SU(2)L doublet scalar H and SU(2)H doublet scalarFH. The contributions from the heavy quarks in G2HDM are found to be small because the Yukawa couplingskqHqHhi

in (5.30) are scaled by the small VEV ratio v/vF.

Next, we impose further the dark matter constraints on the parameter space scan. We used the MadDM package [160] to calculate the relic density of the DM candidate and its elastic

106 Double Higgs Boson Production in G2HDM at the LHC

(a) (b)

(c) (d)

Fig. 5.3The scatter plots of relevant parameters to Higgs boson pair production without the experi-mental constraints from DM relic density and direct searches. The colour palette indicates the ratio of double Higgs boson production cross sections between G2HDM and SM.

5.2 Double Higgs Boson Production in G2HDM at the LHC 107

(a) (b)

Fig. 5.4 The scatter plots for the ratio of production cross sections for a pair of 125 GeV Higgs bosons between G2HDM and SM on the plane of dark matter mass and a) relic density of DM, b) spin-independent cross section of DM and nucleon. The green (yellow) band corresponds to 1s (3s) range of the PLANCK’s relic density measurement of DM [161]. The green and orange line represent the upper limit on spin-independent cross section of DM and nucleon from XENON1T [162] and PandaX-II Experiment [163], respectively.

scattering cross sections with nucleon. In Fig. 5.4, we present the scatter plots for the ratio of production cross sections for a pair of 125 GeV Higgs bosons between G2HDM and SM on plane of the dark matter mass and a) relic density of DM, b) spin-independent cross section of DM and nucleon. The green (yellow) band corresponds to 1s (3s) range of the PLANCK’s relic density measurement of DM [161]. The green and orange lines represent the upper limit on spin-independent cross section of DM and nucleon from XENON1T [162] and PandaX-II Experiment [163], respectively. Imposing the mass of dark matter candidate D to be the lightest among ˜D, W0(p,m), H±and heavy fermions implies mDto be less than ⇠ 2.7 TeV. In the region of mD>500 GeV, there are correlations between Higgs boson pair production cross section and DM relic density as well as DM-nucleon cross section. In particular, the cross section of gluon-gluon fusion to double h1tends to be larger when DM relic density becomes smaller or DM-nucleon cross section becomes larger. The first correlation, shown in Fig. 5.4a, is due to the fact that the |lh1h1h1| and BR(h2! h1h1)can control not only the Higgs boson pair production but also DM annihilation cross section. Indeed, when they both become bigger, the DM annihilation process will be dominated by DD ! hi! h1h1channel, implying the DM annihilation cross section becomes larger or DM relic density becomes smaller. The second correlation, shown in Fig. 5.4b, is due to the DM-nucleon cross section has about half its contributions coming from the one loop heavy quarks (mainly top quark) in the triangle diagram which also appear in the double Higgs boson production process.

108 Double Higgs Boson Production in G2HDM at the LHC

(a) (b)

(c) (d)

Fig. 5.5Same as Fig. 5.3 but after taking into account the experimental constraints from DM relic density from PLANCK [161] and direct searches from XENON1T [162] and PandaX-II [163].

5.2 Double Higgs Boson Production in G2HDM at the LHC 109

The DM relic density and direct searches put stringent constraints on the parameter space of G2HDM. On the one hand, the PLANCK’s relic density measurement constrains the parameter space in a small 3s band as shown in Fig. 5.4a. On the other hand, from Fig. 5.4b, one can also see that the DM direct search constraints cut off almost all the parameter space which significantly enhances the cross section of double Higgs boson production. Moreover, when both relic density and direct search constraints are imposed, only about 2% of the data points survived.

Same as Fig. 5.3, we show in Fig. 5.5 the scatter plots of relevant parameters to Higgs boson pair production after taking into account the constraints from DM relic density and direct searches. The allowed points in the parameter space are selected within 3s of the PLANCK’s relic density measurement of DM [161] and below the upper limits of the DM direct detection searches from XENON1T [162] and PandaX-II experiments [163]. Under these combined constraints of (VS+PU+HP+DM), the parameter space can be narrowed down further. For example, we now have the 1  lh1h1h1  1.3 and BR(h2! h1h1) is less than about 80%. The negative value oflh1h1h1 gives an enhancement of the production cross section because the constructive interference occur between box and triangle diagrams.

Overall, the production cross section of h1 pair is about one order of magnitude lower as compared with the one before imposing the DM constraints.

To do further analysis, we pick seven benchmark points from the final allowed parameter space satisfying the (VS+PU+HP+DM) constraints, at which the mass of heavier scalar h2 varying from 300 to 900 GeV. In Table 5.2, we show the fundamental parameters in the scalar potential, derived couplings, mixing parameters, mass spectra of the scalars, branching ratio BR(h2! h1h1)and the signal strength for Higgs boson pair production at each benchmark point. For benchmark point A, the trilinear self-couplinglh1h1h1 is about the same as SM value. For the points C and D, we have the negative values for lh1h1h1 which can lead to constructive interference between the box and triangle diagrams, while point B is chosen in which large branching ratio of h2! h1h1 can be achieved to see the enhancement effects of heavy scalar resonance on the production cross section. For the points E, F and G, the production cross section is about twice of its SM value. One can see that the Yukawa couplings between SM quarks and the h1Higgs boson are close to SM values in all benchmark points except benchmark point D.

In Table 5.3, we show the branching ratio of h2decays to all two body final states in our benchmark points. We observe that the heavy scalar h2mainly decays into a pair of SM-like Higgs boson h1, W and Z bosons, and top quark.

In our simulation, we first implement the G2HDM model into the FeynRules package [164] and pass the UFO model files into MadGraph5 aMC@NLO [165] to generate the events

110 Double Higgs Boson Production in G2HDM at the LHC

Table 5.2 Seven benchmark points allowed by the combined (VS+PU+HP+DM) constraints.

Benchmark point A B C D E F G

lH 0.35 0.60 0.80 1.79 0.66 2.49 2.22

lF 2.75 1.83 1.43 2.45 1.44 1.68 3.43

lD 0.84 0.37 2.43 0.05 0.67 1.97 0.08

lH0 -3.78 -0.75 -6.55 -2.52 -17.80 1.31 0.45

lHF -1.37 1.41 -0.05 -2.24 0.003 0.83 -2.15

lHD -0.75 1.30 0.034 -0.53 -0.316 0.86 -0.31

lFD 3.06 2.11 3.78 0.73 2.08 3.64 0.95

lHF0 6.04 6.94 7.59 7.41 1.46 0.40 6.16

vD(GeV) 1926 1793 3378 621 1520 3212 3458

vF(GeV) 36220 36274 41580 30800 51914 86229 33348

MHD(GeV) 199.6 2203 1625 1117 -2293 2214 -2986

MFD(GeV) 0.91 8.72 11.09 0.50 1.80 0.64 3.56

lh1h1h1 0.85 0.15 0.53 0.20 0.84 0.35 0.41

lh2h1h1 0.76 3.03 3.88 3.25 3.32 5.42 7.16

kqqh1 0.95 0.91 0.81 0.77 0.93 0.75 0.86

kqqh2 0.29 0.41 0.58 0.64 0.37 0.65 0.52

kqHqHh1 5 ⇥ 10 5 10 4 3.7 ⇥ 10 4 10 5 4 ⇥ 10 5 8 ⇥ 10 5 7 ⇥ 10 5 kqHqHh2 2 ⇥ 10 4 1.7 ⇥ 10 4 5.1 ⇥ 10 4 4 ⇥ 10 5 9 ⇥ 10 5 9 ⇥ 10 5 8 ⇥ 10 5

(OH11)2 0.91 0.83 0.65 0.59 0.86 0.56 0.73

(OH12)2 0.09 0.17 0.34 0.41 0.14 0.43 0.26

(OH13)2 ⇠ 0 10 5 ⇠ 0 10 5 ⇠ 0 ⇠ 0 10 5

(OD21)2 5 ⇥ 10 5 5 ⇥ 10 5 3 ⇥ 10 5 6 ⇥ 10 5 2 ⇥ 10 5 10 5 5 ⇥ 10 5

(OD22)2 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0

(OD23)2 0.999 0.999 0.999 0.999 0.999 0.999 0.999

mh2(GeV) 300 400 500 600 700 800 900

mh3(TeV) 85 69.49 70.77 68.22 88.35 158.2 87.39

mD(GeV) 398 1278 1210 467 883 619 553

m˜D(TeV) 62.94 67.61 81.03 59.29 44.38 38.87 58.45

mH±(TeV) 62.94 67.60 81.03 59.29 44.39 38.87 58.44

BR(h2! h1h1) 0.33 0.58 0.30 0.12 0.18 0.10 0.16

s(gg!h1h1)

sSM 8.2 27.3 16.7 4.6 2.1 2.1 2.1

Table 5.3 Branching ratios of the two body decays of h2in the seven benchmark points.

Benchmark point A B C D E F G

h2! h1h1 0.329 0.575 0.298 0.113 0.175 0.100 0.161 h2! W+W 0.462 0.255 0.391 0.496 0.471 0.529 0.500 h2! ZZ 0.206 0.119 0.186 0.240 0.230 0.260 0.247 h2! t¯t 0 0.049 0.123 0.150 0.122 0.114 0.091

h2! b¯b ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0 ⇠ 0

5.2 Double Higgs Boson Production in G2HDM at the LHC 111

of h1pair production. We use MadSpin [168] package to make h1decays into b¯b andgg. The Pythia8 package [166] is used for parton showering and hadronization the events, while the Delphes3 package [167] (with ATLAS setting) is used as a fast detector simulation. Finally, we use our own codes to analysis the events.

In the next two subsections, we will concentrate on the b¯bgg and b¯bb¯b final states.

According to the current searches of the Higgs boson pair production at LHC, the b¯bgg final state channel is a good channel search for the lower mass regime of the heavy scalar, while the b¯bb¯b search channel has better sensitivity for the search for the heavier mass regime of the heavy scalar. Thus we use our benchmark points A, B, C, D to study the b¯bgg final state channel, while E, F, G are used for b¯bb¯b final state channel.

The b¯bgg Final State Channel

In this section, we perform the kinematic distributions of the b¯bgg final state. We select 4 benchmark points A, B, C, D, at which the mass of heavy scalar h2 equals 300, 400, 500, 600 GeV respectively, to study this channel. Here, we follow the cuts used in the ATLAS for b¯bgg channel analysis [151]. In particular, we require to have at least 2 photons and exactly 2 b-jets in the final state. The leading b-jet must have transverse momentum PT >55GeV, while the sub-leading b-jet is required to have PT >35GeV. All other jets must have their pseudorapidities |h| < 2.5 and transverse momenta PT >25GeV. Furthermore, the diphoton invariant mass, mgg, is required to lie between 105 GeV and 160 GeV and the b-jet pair invariant mass, mb¯b, is required to fall into a mass window of 95 GeV to 135 GeV.

In Fig. 5.6, we present the kinematic distributions of the b¯bgg channel for a) the invariant mass of b¯bgg, b) the opening angle DR of two photons and c) of 2 b-jets in SM and in G2HDM with mh2=300, 400, 500 and 600 GeV at ps = 13TeV for ATLAS. In Fig. 5.6a, it is obvious to see that the invariant mass distributions peaked at the corresponding mass of heavy scalar h2, while the peak around 400 GeV is for the SM. The peaks are getting lower when the mass of h2becomes heavier, this is due to the fact that when the mass of h2

becomes heavier the non-resonant process is getting more relevant to the total production cross section. One can also observe that the opening angles of the two photons and of the pair of b-jets in Fig. 5.6b and Fig. 5.6c respectively, tend to be narrower when the mass of h2

becomes heavier. This is expected because when the parent decaying particle h2becomes heavier, the two daughter h1Higgs bosons will be more boosted implying the opening angles DRgg andDRbb would be smaller.

As mentioned above, the kinematic distribution of invariant mass of b¯bgg final state is peaked at about 400 GeV for the case of the SM. In Fig. 5.7, we perform ac2test at the high luminosity LHC for the SM and a fictitious benchmark point B0(pseudo data) with mh2 =400

112 Double Higgs Boson Production in G2HDM at the LHC

(a) (b)

(c)

Fig. 5.6The kinematic distributions of the b¯bgg channel for a) the invariant mass of b¯bgg, b) the opening angleDR of two photons and c) of 2 b-jets in SM and in G2HDM with mh2=300, 400, 500 and 600 GeV at ps = 13TeV for ATLAS detectors.

5.2 Double Higgs Boson Production in G2HDM at the LHC 113

200 250 300 350 400 450 500 550 600 650 700

)-1 = 3000.0 fbintNevent (L

0 0.5 1 1.5 2 2.5 3

3.5 SM

= 400GeV

h2

m @LHC 13TeV 1.31σ

20 40 60 80 100 120 140 160 180 200 220

(GeV)

bb γ

Mγ

200 250 300 350 400 450 500 550 600 650 700

PseudoData/SM

0 1 2 3 4

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4

)-1 = 3000.0 fb int pairs) (LγγN (

0 0.5 1 1.5 2 2.5 3

3.5 SM

= 400GeV

h2

m

@LHC 13TeV σ

0.03

20 40 60 80 100 120 140 160 180 200 220

γ

Rγ

0 0.5 1 1.5 2 2.5 3 3.5 4

PseudoData/SM

0 0.51 1.52

(b)

Fig. 5.7c2test for the benchmark point B0 (pseudo data) and SM at 13 TeV LHC with an integrated luminosity of Lint=3000 fb 1.

σ = σ��

@��� ��� ���

mh2=400GeV

0 200 400 600 800 1000

0 1 2 3 4 5

Integrated Luminosity (fb-1)

StandardDeviation

Fig. 5.8The integrated luminosity versus standard deviation withc2test for the benchmark point B0 and SM at 100 TeV LHC.s = sSM means the cross section for the process gg ! h1h1! gg ¯bb of benchmark point B0same as SM value.

114 Double Higgs Boson Production in G2HDM at the LHC

GeV and has a total production cross section same as the SM value. The error is naively taken as square root of the pseudo data bin content. It turns out that at 13 TeV LHC with Lint=3000fb 1, we can not distinguish the signals between the heavy scalar resonance with a mass of 400 GeV and the SM. In particular, it is about 1.31s deviated from the SM in the case of Mb¯bgg kinematic distribution, and only 0.03s in case of DRgg distribution. However, the signals can be distinguished at future collider machine which has higher center-of-mass energy and/or luminosity. For illustration, we present the corresponding integrated luminosity versus standard deviation withc2test for the benchmark point B0and SM at 100 TeV LHC.

The result shown in Fig. 5.8 is telling us in order to distinguish, says a signal for a 400 GeV scalar resonance with a standard deviation 3s larger than the SM Higgs, one would need Lint 500fb 1at 100 TeV LHC.

The b¯bb¯b Final State Channel

We select benchmark points E, F, G at which mh2 = 700, 800, 900 GeV respectively for studying the kinematic distributions of the b¯bb¯b final state channel. In this case, we follow the cuts used in the ATLAS resolved analysis for b¯bb¯b final state channel [153]. To be more specific, we first require the event contains at least four b-jets with PT >30GeV and |h| < 2.5.

Furthermore, we pair up these four b-jets to construct two 125 GeV Higgs boson candidates and then impose additional mass-dependent cuts for these two Higgs boson candidates as follows:

m4 j360/GeV 0.5 <DRj j,lead< m 655

4 j/GeV+0.475

m4 j235/GeV <DRj j,subl<m4 j875/GeV+0.35 )

if m4 j<1250GeV ,

0 <DRj j,lead<1 0 <DRj j,subl<1

)

if m4 j>1250GeV ,

whereDRj j,lead is the opening angle of two jets which the leading Higgs boson candidate decay into and DRj j,subl for the sub-leading candidate. Here, the leading Higgs boson candidate indicates the Higgs boson with the highest scalar sum of jet PT. We further apply a new algorithm to choose the best pairing of b-jets into Higgs boson candidates. In particular, we select the b-jet pairs that have minimum distance Dhh from the pairing’s (mlead2 j ,msubl2 j ) point to the line connecting the two points (0 GeV, 0 GeV) and (120 GeV, 115 GeV). One can write down Dhh as follows:

Dhh =r⇣

mlead2 j2

+⇣

msubl2 j2

sin tan 1 msubl2 j mlead2 j

!

tan 1✓115 120

◆!

. (5.36)

5.2 Double Higgs Boson Production in G2HDM at the LHC 115

Here, mlead/subl2 j is the mass of the leading/sub-leading Higgs boson candidate and the values of 120 GeV and 115 GeV are centre of signal regions in mlead2 j and msubl2 j respectively. In additional, two Higgs boson candidates are required to have the transverse momenta PTlead and PTsubl, the opening angleDR(h,h) and the pseudorapidity difference |Dhhh| satisfying the following mass-dependent cuts:

PTlead >0.5m4 j 90GeV , PTsubl >0.33m4 j 70GeV , DR(h,h) > 1.5 ,

and

|Dhhh| <

( 1.1 ifm4 j <850GeV ,

2 ⇥ 10 3 m4 j/GeV 0.6 ifm4 j>850GeV .

Finally, the mass of Higgs boson candidate must lie in the signal region Xhh defined by

Xhh= vu

ut mlead2 j 120GeV 0.1mlead2 j

!2

+ msubl2 j 115GeV 0.1msubl2 j

!2

<1.6 . (5.37)

In Fig. 5.9, we present the kinematic distributions of the b¯bb¯b channel for a) invariant mass of b¯bb¯b, b) opening angle of 2 b-jets associated with leading Higgs boson candidate DRleadbb and c) 2 b-jets associated with sub-leading Higgs boson candidateDRsublbb , with mh2 = 700, 800, 900 GeV at ps = 13TeV for ATLAS detectors. We note that these 3 benchmark points E, F, G selected to study this b¯bb¯b final state channel have the same production cross sections and are about twice the SM value. In Fig. 5.9a, one can observe that the non-resonant contributions, peaked around 400 GeV, become dominant in the benchmark points F and G with mh2=800 and 900 GeV respectively, while the benchmark point E with mh2=700 GeV represents a more dominant contribution from the resonant process. Next, in both Figs. 5.9b and 5.9c, theDR distributions tend to separate into two peaks, one is about DR ⇡ 1, while another is aboutDR ⇡ 3. This behaviour is expected because of the different contributions from non-resonant and resonant processes to the total production cross section. In particular the peak about 1 represents the resonant contribution, while the peak about 3 represents non-resonant contribution. Therefore, in this case, one can use opening angleDR to separate the non-resonant and resonant contributions. We note that theDRleadbb distribution is more preferable to have a peak located at ⇠ 1, due to its more energetic parent Higgs boson.

116 Double Higgs Boson Production in G2HDM at the LHC

(a) (b)

(c)

Fig. 5.9The kinematic distributions of the b¯bb¯b channel for a) invariant mass of b¯bb¯b, b) opening angleDR of 2 b-jets associated with leading Higgs boson candidate and c) of 2 b-jets associated with sub-leading Higgs boson candidate with mh2=700, 800, 900 GeV at ps = 13TeV for ATLAS detectors.

5.2 Double Higgs Boson Production in G2HDM at the LHC 117

5.2.4 Conclusion

Studying Higgs boson pair production is an important way to probe for the details of Higgs boson properties, especially for the self-coupling of the Higgs boson. It is important to know if the self-coupling of 125 GeV Higgs is SM-like or can be modified by BSM physics. We have studied this process in the G2HDM, a model that promotes the discrete Z2symmetry

Studying Higgs boson pair production is an important way to probe for the details of Higgs boson properties, especially for the self-coupling of the Higgs boson. It is important to know if the self-coupling of 125 GeV Higgs is SM-like or can be modified by BSM physics. We have studied this process in the G2HDM, a model that promotes the discrete Z2symmetry

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