new gauged two higgs doublet model: theoretical and...

Gauged two Higgs doublet model: theoreticaland phenomenological constraints

Raymundo Ramos1

Institute of Physics, Academia Sinica

Windows on the UniverseICISE, �y Nhon, Vietnam, August 8, 2018

1A collaboration with: Tzu-Chiang Yuan, Yue-Lin Sming Tsai, Wei-Chih Huangand Abdesslam Arhrib.

Overview

Motivation

The Gauged 2 Higgs Doublet Model

Ma�er content of the G2HDM

The Scalar Potential

Theoretical constraints

Higgs phenomenology

Final Remarks

Motivation

A few years a�er the discovery of the Higgs boson, we can still asksome questionsI is it "the" Higgs or is it "a" Higgs?I What about dark ma�er candidates?I And neutrino masses (oscillation)?I Are there any extra symmetries?

A popular class of models that aims to solve some of these problemsis the two Higgs doublet model.

Motivation: Two Higgs Doublet models

Simple extensions that explain BSM physicsI MSSM: 2 Higgs doublets due to superpotential.I 2HDM: the prototype, extra CP phases.I IHDM: dark ma�er candidate, no FCNC at tree level.I G2HDM: more details ahead...

The G2HDM

Main feature: the two Higgs doublets are embedded in a doublet ofan extra SU(2)HX New gauge group SU(2)H×U(1)XX This approach simplifies the potential for the doublets.

X The additional complex vector fields are neutral

X Anomaly free through the addition of heavy fermions.

X One of the Higgs can be inert→ DM candidate.

! The cost is the introduction of new scalars: An SU(2)H tripletand a doublet.

Ma�er content of the G2HDM

Ma�er Fields SU(3)C SU(2)L SU(2)H U(1)Y U(1)XQL = (uL dL)

T 3 2 1 1/6 0

UR =(uR uHR

)T3 1 2 2/3 1

DR =(dHR dR

)T3 1 2 −1/3 −1

uHL 3 1 1 2/3 0dHL 3 1 1 −1/3 0

LL = (νL eL)T 1 2 1 −1/2 0

NR =(νR νHR

)T1 1 2 0 1

ER =(eHR eR

)T1 1 2 −1 −1

νHL 1 1 1 0 0eHL 1 1 1 −1 0

NOT relevant for today! (for more information check 1512.00229)

Scalar content of the G2HDM

Relevant for today:

Ma�er Fields SU(3)C SU(2)L SU(2)H U(1)Y U(1)XH = (H1 H2)

T 1 2 2 1/2 1

∆H =©«∆32

∆p√

2∆m√

2−∆32

ª®¬ 1 1 3 0 0

ΦH = (Φ1 Φ2)T 1 1 2 0 1


V (H,∆H,ΦH) = V (H) + V (ΦH) + V (∆H) + Vmix (H,∆H,ΦH) ,

where

V (H) = µ2H

(H†αiHαi

)+ λH

(H†αiHαi

)2

+12λ′Hεαβε

γδ(H†αiHγi

) (H†βjHδj

),

V (ΦH) = µ2ΦΦ†

HΦH + λΦ

(Φ†

HΦH

)2,

V (∆H) = − µ2∆tr

(∆†

H∆H

)+ λ∆

[tr

(∆†

H∆H

)]2

,


Vmix (H,∆H,ΦH) = +MH∆

(H†∆HH

)−MΦ∆

(Φ†

H∆HΦH

)+ λHΦ

(H†H

) (Φ†

HΦH

)+ λ′HΦ

(H†ΦH

) (Φ†

HH)

+ λH∆

(H†H

)tr

(∆†

H∆H

)+ λΦ∆

(Φ†

HΦH

)tr

(∆†

H∆H

),

Theoretical constraints: Vacuum stability (VS)

Consider all the quartic terms in the scalar potential

V4 = +λH

(H†H

)2+ λ′H

(−H†1H1H

†2H2 + H

†1H2H

†2H1

)+λΦ

(Φ†

HΦH

)2+ λ∆

(Tr

(∆†

H∆H

))2

+λHΦ

(H†H

) (Φ†

HΦH

)+ λ′HΦ

(H†ΦH

) (Φ†

HH)

+λH∆

(H†H

)Tr

(∆†

H∆H

)+ λΦ∆

(Φ†

HΦH

)Tr

(∆†

H∆H

).

The challenge: Write V4 as a matrix


We introduce the following basis x,y, z and two ratios ξ and η,

x ≡ H†H ,

y ≡ Φ†

HΦH ,

z ≡ Tr(∆†

H∆H

),

ξ ≡

(H†ΦH

) (Φ†

HH)

(H†H)(Φ†

HΦH

) , 0 < ξ < 1

η ≡

(−H†1H1H

†2H2+H

†1H2H

†2H1

)(H†H)

2 , −1 < η < 0.


Write V4 as a quadratic form of x,y, z with parameters ξ and η

V4 =(x y z

)·Q(ξ, η) ·

©«xyz

ª®®¬ ,with

Q(ξ, η) =©«

λ̃H(η)12 λ̃HΦ(ξ)

12λH∆

12 λ̃HΦ(ξ) λΦ

12λΦ∆

12λH∆

12λΦ∆ λ∆

ª®®¬ ,where λ̃H(η) ≡ λH + ηλ

′H, λ̃HΦ(ξ) ≡ λHΦ + ξλ

′HΦ.


x , y and z are always positive. From copositive criteria (K. Kannike,1205.3781) the resulting conditions are

λ̃H(η) ≥ 0 , λΦ ≥ 0 , λ∆ ≥ 0 ,

ΛHΦ(ξ, η) ≡ λ̃HΦ(ξ) + 2√λ̃H(η)λΦ ≥ 0 ,

ΛH∆(η) ≡ λH∆ + 2√λ̃H(η)λ∆ ≥ 0 ,

ΛΦ∆ ≡ λΦ∆ + 2√λΦλ∆ ≥ 0 ,

√λ̃H(η)λΦλ∆ +

12

(λ̃HΦ(ξ)

√λ∆ + λH∆

√λΦ + λΦ∆

√λ̃H(η)

)+

12

√ΛHΦ(ξ, η)ΛH∆(η)ΛΦ∆ ≥ 0 .

Theor. constraints: Perturbative Unitarity (PU)

The sca�ering amplitudes for the 2→ 2 processes have initial andfinal states belonging to{

hh√

2,G0G0

√2,G+G−,H0∗

2 H02,H

+H−,φ2φ2√

2,G0HG

0H

√2,Gp

HGmH ,δ3δ3√

2,∆p∆m

}The matrix of sca�ering amplitudesM1 is

©«

3λH λH2√2λH

2√2λH

2√2λ̃H

12λHΦ

12λHΦ

1√2λ̃HΦ

12λH∆

1√2λH∆

− 3λH 2√2λH

2√2λH

2√2λ̃H

12λHΦ

12λHΦ

1√2λ̃HΦ

12λH∆

1√2λH∆

− − 4λH 2λ̃H 2λH 1√2λHΦ

1√2λHΦ λ̃HΦ

1√2λH∆ λH∆

− − − 4λH 2λH 1√2λ̃HΦ

1√2λ̃HΦ λHΦ

1√2λH∆ λH∆

− − − − 4λH 1√2λ̃HΦ

1√2λ̃HΦ λHΦ

1√2λH∆ λH∆

− − − − − 3λΦ λΦ2√2λΦ

12λΦ∆

1√2λΦ∆

− − − − − − 3λΦ 2√2λΦ

12λΦ∆

1√2λΦ∆

− − − − − − − 4λΦ 1√2λΦ∆ λΦ∆

− − − − − − − − 3λ∆ 2√2λ∆

− − − − − − − − − 4λ∆

ª®®®®®®®®®®®®®®®®®®®®®¬


In both basis{hH0∗

2 ,G0H0∗

2 ,G+H−

}and

{hG+,H0∗

2 H+,G0G+}

we findthe matrix

M2 =

©«2λH 0

√2

2 λ′H

0 2λH −i√

22 λ′H√

22 λ′H + i

√2

2 λ′H 2λH

ª®®®¬with eigenvalues 2λH , 2λH ± λ′H .


And we have a series of individual amplitudes:

2λH : hG0 ←→ hG0 , G+H+ ←→ G+H+ ,

2λ̃H :hH+ ←→ hH+

G0H+ ←→ G0H+

H0∗2 G+ ←→ H0∗

2 G+

M

(G+H0

2 ←→ hH+)= 1√

2λ′H

M

(G+H0

2 ←→ G0H+)= − i√

2λ′H

2λΦ :φ2G0

H ←→ φ2G0H , φ2G

pH ←→ φ2G

pH ,

G0HG

pH ←→ G0

HGpH ,

...


In summary, perturbative unitarity limits the parameters to thefollowing ranges:

1. Between −4π and +4π:I the parameters λH , λΦ and λ∆

2. Between −8π and +8π:I the eigenvalues λi(M1), for i = 1, . . . ,10I the combinations 2λH ± λ′H and λHΦ + λ′HΦI the parameters λHΦ, λH∆ and λΦ∆

3. Between −8√

2π and +8√

2π:I the parameters λ′H and λ′HΦ

Theoretical constraints: Results

10−4 10−3 10−2 0.1 1 10λΦ

−24

−18

−12

−6

0

6

12

18

24

λH

Φ

VS

PU

VS + PU

−25 −20 −15 −10 −5 0 5 10

λ′H

−24

−18

−12

−6

0

6

12

18

24

λH

Φ

VS

PU

VS + PU

Green lower bound controlled by: λ̃HΦ(ξ) + 2√λ̃H(η)λΦ ≥ 0


10−4 10−3 10−2 0.1 1 10λ∆

−24

−18

−12

−6

0

6

12

18

24

λH

∆

VS

PU

VS + PU

−25 −20 −15 −10 −5 0 5 10

λ′H

−24

−18

−12

−6

0

6

12

18

24

λH

∆

VS

PU

VS + PU

Green lower bound controlled by: λH∆ + 2√λ̃H(η)λ∆ ≥ 0


−24 −18 −12 −6 0 6 12 18 24λHΦ

−24

−18

−12

−6

0

6

12

18

24

λH

∆

VS

PU

VS + PU

−24 −18 −12 −6 0 6 12 18 24λHΦ

−24

−18

−12

−6

0

6

12

18

24

λΦ

∆

VS

PU

VS + PU

Non-diagonal couplings limited mostly by perturbative unitarity


−24 −18 −12 −6 0 6 12 18 24

λ′HΦ

−24

−18

−12

−6

0

6

12

18

24

λH

Φ

VS

PU

VS + PU

−24 −18 −12 −6 0 6 12 18 24

λ′HΦ

−24

−18

−12

−6

0

6

12

18

24

λH

∆

VS

PU

VS + PU

Non-diagonal couplings limited mostly by perturbative unitarity

Higgs phenomenology

Signal S trength

0 1 2 3 4 5 6 7

Run-1μ

Run-2μ

ggHμ

VBFμ

VHμ

topμ

ATLAS Preliminary-1 = 13 TeV, 36.1 fbs

Total Stat

Total Stat Syst Theo

0.08−

0.12+ 0.08−

0.10+ 0.23−

0.23+ 0.26−

0.28+ = 1 .1 7 Run-1

μggH @ NNLO

0.05−

0.06+ 0.05−

0.06+ 0.11−

0.12+ 0.14−

0.14+ = 0.99 Run-2

μLO3

ggH @ N

0.05−

0.06+ 0.06−

0.07+ 0.16−

0.16+ 0.18−

0.19+ = 0.80 ggH

μ

0.2 −

0.3 + 0.2 −

0.3 + 0.5 −

0.5 + 0.6 −

0.6 + = 2.1 VBF

μ

0.1 −

0.2 + 0.2 −

0.2 + 0.8 −

0.8 + 0.8 −

0.9 + = 0.7 VH

μ

0.0 −

0.1 + 0.1 −

0.1 + 0.5 −

0.6 + 0.6 −

0.6 + = 0.5 top

μ

Figure 11: Summary of the signal strengths measured for the different production processes (ggH, VBF, VH andtt̄H + tH) and globally (µRun−2), compared to the global signal strength measured at 7 and 8 TeV (µRun−1) [76].The black (red) error bar shows the total (statistical) uncertainty. The µRun−1 was derived assuming the Higgsproduction cross section based on Ref. [19, 100]. In the more recent theoretical predictions used in this analysis [10,34], the gluon fusion production cross section is larger by approximately 10%. In this measurement, the bb̄Hcontributions are scaled with ggH (µbbH = µggH), and the sum of tH and tt̄H productions are measured together(µtop = µttH+tH). Associated production with Z or W bosons is assumed to be affected by a single signal strengthparameter (µVH = µZH = µWH).

Table 7: Expected and observed local significances of the VBF, VH and top-quark associated production modesignal strengths.

Measurement Exp. Z0 Obs. Z0

µVBF 2.6 σ 4.9 σµVH 1.4 σ 0.8 σµtop 1.8 σ 1.0 σ

37

Taken from: arXiv:1802.04146, ATLAS Collaboration

Higgs phenomenology

The 125 GeV Higgs boson h1 is a linear combination of h, φ2 and δ3:

h1 = O11h + O21φ2 + O31δ3,

Narrow Higgs decay width→ Higgs production dominated by theresonance region→ Approximate σ(pp→ h1 → γγ) by

σ(gg → h1 → γγ

)=

π2

8s mh1 Γh1

fgg

(mh1√s

)Γ

(h1 → gg

)Γ

(h1 → γγ

)

Higgs phenomenology

Γ (h1 → γγ) =GF α

2 m3h1O2

11

128√

2 π3

��A1(τW ) +∑f

NcQ2f A1/2(τf )

+ ChλHv2

m2H±

A0(τH±) +O21

O11

vvΦ

∑F

NcQ2FA1/2(τF )

��2

,

Ch = 1 +O21

O11

λHΦvΦ2λHv

−O31

O11

2λH∆v∆ +MH∆

4λHv.

Γ(h1 → gg) =α2s m

3h1O2

11

72 v2 π3

��∑f 34A1/2(τf ) +

O21

O11

vvΦ

∑F

34A1/2(τF )

��2

.

Higgs phenomenology results

v = 246 TeV, vΦ = 10 TeV, v∆ = [0.5,20] TeV

0.1 1 10λH

−24

−18

−12

−6

0

6

12

18

24

λH

Φ

HP

VS + PU + HP

−25 −20 −15 −10 −5 0 5 10

λ′H

−24

−18

−12

−6

0

6

12

18

24

λH

ΦVS + PU

VS + PU + HP

λH lower bound corresponds to the SM limit.


v = 246 TeV, vΦ = 10 TeV, v∆ = [0.5,20] TeV

0.1 1 10λH

−24

−18

−12

−6

0

6

12

18

24

λH

∆

HP

VS + PU + HP

−25 −20 −15 −10 −5 0 5 10

λ′H

−24

−18

−12

−6

0

6

12

18

24

λH

∆

VS + PU

VS + PU + HP

Higgs phenomenology constrains non-diagonal couplingssymmetrically around the zero.


v = 246 TeV, vΦ = 10 TeV, v∆ = [0.5,20] TeV

103 104

mH± (GeV)

10−9

10−7

10−5

10−3

10−1

H±

/SM

for

loop

cont

rib

uti

ons

Destructive

Constructive

0.5 0.6 0.7 0.8 0.9 1.0

O211

0.6

0.8

1.0

µγγggH

HP

VS + PU + HP

Light charged Higgs has the most (new physics) relevantcontribution


v = 246 TeV, vΦ = 10 TeV, v∆ = [0.5,20] TeV

103 104

mH± (GeV)

10−9

10−7

10−5

10−3

10−1

HF

/SM

for

loop

cont

rib

uti

ons

Destructive

Constructive

−0.5 0.0 0.5 1.0

O21/O11

0.6

0.8

1.0

µγγggH

HP

VS + PU + HP

Small heavy fermion contribution. Shape controlled by overall O211

and O21/O11 factor from gluon side.

10−4

10−2

1λ

Φ

10−4

10−2

1

λ∆

−20

0

20

λH

Φ

−20

0

20

λH

∆

−20

0

20

λΦ

∆

−20

0

20

λ′ H

Φ

10−4 10−2 1

−20

0

20

λ′ H

10−4 10−2 1 10−4 10−2 1 −20 0 20−20 0 20−20 0 20−20 0 20

10−4 10−2 1λΦ

10−4 10−2 1λ∆

−20 0 20λHΦ

−20 0 20λH∆

−20 0 20λΦ∆

−20 0 20

λ′HΦ−20 0 20

λ′H

10−4

10−2

1

10−4

10−2

1

10−4

10−2

1

−20

0

20

−20

0

20

−20

0

20

−20

0

20

10−4 10−2 1λH

10−4

10−2

1

λH

λH0.13,2.70

λΦ < 4.19

λ∆ < 5.03

λHΦ

-5.69,3.52

λH∆

-3.90,9.25

λΦ∆

-5.41,13.25

λ′HΦ

-0.38,17.11

−20 0 20

−20

0

20

λ′H-23.82,2.54

Final Remarks

I 2HDM represent a popular extension to the SM.I Each variant has its own merits and failures.I The G2HDM represents a new setup still requiring exploration.I We found scalar parameter regions of this model that follows

standard constraints.I In the future we plan to extend this analysis to dark ma�er, the

muon g−2, electroweak precision test and so on.

BACKUP

Symmetry Breaking

If −µ2∆< 0, then SU(2)H is spontaneously broken by the vev

〈∆3〉 = −v∆ , 0, 〈∆p,m〉 = 0

Shi� the triplet diagonal components

∆H =©«−v∆+δ3

21√2∆p

1√2∆m

v∆−δ32

ª®¬ .

Symmetry Breaking

The quadratic terms for H1 and H2 have the following coe�icients

|H1 |2 : µ2

H −12MH∆ · v∆ +

12λH∆ · v2

∆+

12λHΦ · v2

Φ ,

|H2 |2 : µ2

H +12MH∆ · v∆ +

12λH∆ · v2

∆+

12(λHΦ + λ

′HΦ) · v

2Φ ,

Similarly, for Φ1 and Φ2

|Φ1 |2 : µ2

Φ +12MΦ∆ · v∆ +

12λΦ∆ · v2

∆+

12(λHΦ + λ

′HΦ) · v

2 ,

|Φ2 |2 : µ2

Φ −12MΦ∆ · v∆ +

12λΦ∆ · v2

∆+

12λHΦ · v2 ,

Symmetry Breaking

Shi� the rest of the scalars

H1 =

(G+

v+h√2+ i G

0√

2

), H2 =

(H+

H02

),

ΦH =©«

GpH

vΦ+φ2√2+ i

G0H√2

ª®¬ .

Symmetry Breaking

Substituting the VEVs in the potential V leads to

V (v, v∆, vΦ) =14

[λHv4 + λΦv4

Φ + λ∆v4∆+ 2

(µ2Hv

2 + µ2Φv

2Φ − µ

2∆v2∆

)−

(MH∆v2 +MΦ∆v2

Φ

)v∆ + λHΦv2v2

Φ + λH∆v2v2∆+ λΦ∆v2

Φv2∆

].

Minimization of the potential leads to the conditions:

v ·(2λHv2 + 2µ2

H −MH∆v∆ + λHΦv2Φ + λH∆v

2∆

)= 0 ,

vΦ ·(2λΦv2

Φ + 2µ2Φ −MΦ∆v∆ + λHΦv

2 + λΦ∆v2∆

)= 0 ,

4λ∆v3∆− 4µ2

∆v∆ −MH∆v2 −MΦ∆v2

Φ + 2v∆(λH∆v2 + λΦ∆v2

Φ

)= 0 .

Scalar Mass Spectrum

In the basis of S = {h, φ2, δ3} it is given by

M20 =

©«2λHv2 λHΦvvΦ v

2 (MH∆ − 2λH∆v∆)− 2λΦv2

Φ

vΦ2 (MΦ∆ − 2λΦ∆v∆)

− − 14v∆

(8λ∆v3

∆+MH∆v2 +MΦ∆v2

Φ

)ª®®®¬ .

This matrix can be diagonalized by a rotation with an orthogonalmatrix O

OT · M20 · O = Diag(m2

h1,m2

h2,m2

h3) ,


The second block is also 3 × 3. In the basis of G = {GpH,H

0∗2 ,∆p}, it is

given by

M ′20 =

©«MΦ∆v∆ + 1

2λ′HΦv

2 12λ′HΦvvΦ − 1

2MΦ∆vΦ12λ′HΦvvΦ MH∆v∆ + 1

2λ′HΦv

2Φ

12MH∆v

− 12MΦ∆vΦ

12MH∆v 1

4v∆

(MH∆v2 +MΦ∆v2

Φ

)ª®®®¬ .

One eigenvalue vanishes, corresponding to the physical Goldstoneboson G̃p

H


The other two eigenvalues are the masses of two physical fields ∆̃and D. They are given by

M2∆̃,D

=B ±√B2 − 4AC2A

,

with

A = 8v∆ ,

B = 2[MH∆

(v2 + 4v2

∆

)+MΦ∆

(4v2∆+ v2Φ

)+ 2λ′HΦv∆

(v2 + v2

Φ

)],

C =(v2 + v2

Φ + 4v2∆

) [MH∆

(λ′HΦv

2 + 2MΦ∆v∆)+ λ′HΦMΦ∆v

2Φ

].


The final block is 4 × 4 diagonal, giving

m2H±2= MH∆v∆ −

12λ′Hv

2 +12λ′HΦv

2Φ ,

for the physical charged Higgs H±2 , and

m2G± = m2

G0 = m2G0H= 0 ,

for the three Goldstone boson fields G±, G0 and G0H .


Remember that this analysis uses an 8-dim parameter space(x,y, {z1}

)VS OK on green,(

x,y, {z2})

PU OK on blue,(x,y

)on red only if there is {z} that is both VS+PU OK.

For example:

(λ∆ = 1, λH∆ = −4, λH = 4π) is VS OK but no PU,

(λ∆ = 1, λH∆ = −4, λH = 0.13) is PU OK but no VS,

new gauged two higgs doublet model: theoretical and...

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