Renormalization of the
Higgs Triplet Model
Kei Yagyu (National Central U)
S. Kanemura, KY, arXiv: 1201.6287 [hep-ph], accepted
M. Aoki, S. Kanemura, M. Kikuchi. KY, arXiv: 1204.1951 [hep-ph]
National Taiwan University, 14th May 2012
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Phenomenology of the Higgs Triplet Model
at the LHC
Kei YAGYU (Univ. of Toyama)
柳生 慶
富山大学
M. Aoki, S. Kanemura, KY, arXiv: 1110.4625 [hep-ph]
2nd Jan. 2012 National Taiwan University
2/35
Plan of the talk
• Introduction
- The Higgs triplet model
• Renormalization of the Higgs Triplet Model
- Radiative corrections to the EW parameters
- Radiative corrections to the Higgs potential
• Phenomenology
- Higgs → 2 photon decay
- LHC physics
• Summary
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Introduction
・ The Higgs sector has not been confirmed.
- Minimal? or Non-minimal?
- The Higgs boson search is underway at the LHC.
The Higgs boson mass is constrained to be
115 GeV < mh < 127 GeV or mh > 600 GeV at 95% CL.
- There is a “SM-like” Higgs boson at around 125 GeV.
・ There are phenomena which cannot be explained in the SM.
- Tiny neutrino masses, dark matter and Baryon asymmetry of the Universe
・ New physics may explain these phenomena above the TeV scale.
100 GeV
TeV or 100 TeV or GUT scale or …
Energy scale
New physics model, where mν, DM, BAU are explained.
SM
100 GeV
TeV
Energy scale
SM with extended Higgs sectors, where mν, DM, BAU are explained.
4/35
Explanation by extended Higgs sectors
• Tiny neutrino masses
- The type II seesaw model
- Radiative seesaw models
(e.g. Zee model)
• Dark matter
- Higgs sector with an unbroken discrete symmetry
• Baryon asymmetry of the Universe
- Electroweak baryogenesis
Introduced extended Higgs sectors
SU(2) doublet Higgs + Singlet [U(1)B-L model]
+ Doublet [Inert doublet model]
+ Triplet [Type II seesaw model], etc…
Studying extended Higgs sectors is important to understand
the phenomena beyond the SM.
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How we can constrain various Higgs sectors?
An Extended Higgs Sector
Affects
Yukawa Interaction
・Flavor changing
neutral current
EW Parameters
・Rho parameter
・Gauge boson masses
Higgs Potential
・Higgs boson mass
・Higgs self coupling
Compare
Experimental Data
LEP, LHC, ILC, …
It is necessary to prepare precise calculations of observables in order to
compare the experimental data and to distinguish various Higgs sectors.
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How we can constrain various Higgs sectors?
An Extended Higgs Sector
Affects
Yukawa Interaction
・Flavor changing
neutral current
EW Parameters
・Rho parameter
・Gauge boson masses
Higgs Potential
・Higgs boson mass
・Higgs self coupling
Compare
Experimental Data
LEP, LHC, ILC, …
In this talk, we focus on the Higgs Triplet Model (HTM) and
its renormalization of electroweak parameters and the Higgs potential.
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The Higgs Triplet Model
The Higgs triplet field Δ is added to the SM.
・Important new interaction terms:
Cheng, Li (1980);
Schechter, Valle, (1980);
Magg, Wetterich, (1980);
Mohapatra, Senjanovic, (1981).
SU(2)I
U(1)Y
U(1)L
Φ
2
1/2
0
Δ
3
1
-2
Lepton number breaking parameter
・Neutrino mass matrix
MΔ : Mass of triplet scalar boson.
vΔ : VEV of the triplet Higgs
When the lepton number breaking parameter μ is taken to be of O(eV)
scale, we can take MΔ ~ O(100-1000) GeV.
In this case, the HTM can be tested at colliders.
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Important predictions (Tree-Level)
1. Rho parameter deviates from unity.
Renormalization of the electroweak parameters
2. Characteristic relations among the masses of the Δ-like Higgs bosons are predicted.
Renormalization of the Higgs potential
・Mass eigenstates:
Case I
(SM-like) h,
Under vΔ << vΦ (Required by ρexp ~ 1)
(Triplet-like) H±±, H±, H, A
Case II
Mass2
Mass2
A, H
H++
H+
H+
H++
A, H
mH++2 － mH+2 ≃ mH+2 － mA2
mA2 ≃ mH2
These relations are useful to test the model.
How these predictions are modified by the radiative corrections?
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Strategy
Renormalization of the EW parameters.
We search allowed parameter regions by the EW precision data.
Renormalization of the Higgs potential
By using the constrained parameter space,
we calculate one-loop corrected predictions of observables
in the Higgs potential (e.g., the mass spectrum, hhh coupling).
Phenomenology at the LHC
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Renormalization of the EW
parameters
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The electroweak rho parameter
★ The experimental value of the rho parameter is quite close to unity.
ρexp ~ 1
Tree-level expression for the rho parameter （Kinetic term of Higgs fields）
ρtree = 1
・ Standard Model
・Multi-doublet
(with singlets) model
There is the custodial SU(2) sym.
in the kinetic term
Higgs Potential
ρtree ≠ 1
・ Models with Higgs fields
whose isospin is larger than ½
e.g., the HTM.
The custodial SU(2) sym. is
broken in the kinetic term.
These sector affects
the rho parameter by the
loop effects.
Yukawa interaction
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Custodial Symmetry
The SM Lagrangian can be written by the 2×2 matrix form of the Higgs doublet:
When we take g’ and yA → 0,
Lagrangian is invariant under SU(2)L×SU(2)R
★ Kinetic term
,
After the Higgs field gets the VEV:
★ Higgs potential
this symmetry is reduced to
SU(2)L＝SU(2)R =SU(2)V (custodial symmetry).
★ Yukawa interaction (top-bottom sector)
SU(2)V breaking by g’ is included in
the definition of the rho parameter,
while that by yA is not.
There is a significant contribution to
the deviation of rho = 1 from the
top-bottom sector by the loop effect.
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Models with ρ = 1 at the tree level
There are 3 parameters: g, g’ and v in the kinetic term of Higgs fields.
The electroweak observables are described by the 3 input parameters.
We can choose αem, GF and mZ as the 3 input parameters.
αem(mz) = 128.903 ± 0.0015, GF = 1.16637 ± 0.00001 GeV-2,
mz = 91.1876 ± 0.0021 GeV
The weak angle sin2θW = sW2 and mW can be written in terms of the above 3 inputs.
The counter term of δsW2 is derived as
~ δρ = ρ-1 = αem T
The quantity δρ (or T) measures the violation of the custodial symmetry.
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Scalar boson and fermion loop contributions to δρ
The two Higgs doublet model case
αem T =
Custodial sym. breaking
in the top-bottom sector
Custodial sym. breaking
in the scalar sector
Dependence of the quadratic mass splitting
among particles in the same isospin multiplet
appears in the T (rho) parameter.
Peskin, Wells (2001);
Grimus, Lavoura, Ogreid, Osland (2008);
Kanemura, Okada, Taniguchi, Tsumura (2011).
mH+ = 300 GeV
mA = 200 – 300 GeV
500 GeV
mh = 117 GeV
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Models without ρ = 1 at the tree level
There are 4 parameters (instead of 3 in the SM):
g, g’, v and ρ0 in the kinetic term of Higgs fields.
This parameter causes ρtree ≠ 1 at the tree level.
Ex. the VEV of the triplet Higgs in the HTM.
The electroweak observables are described by the 4 input parameters.
^
We can choose αem, GF, mZ and sW2 as the input parameters.
^s
Blank, Hollik (1997)
e
W
2
is defined by the effective Zee vertex:
Z
e
^s 2 = 0.23146 ± 0.00012
W
In order to calculate the EW parameters at one-loop order,
we shift the bare parameters and impose reno. conditions.
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On-shell renormalization scheme
Parameters shift: g → g + δg, g’ → g’ + δg’,
v → v + δv,
s^W2 → s^W2 + δ ^sW2,
Wμ → ZW1/2 Wμ , Bμ → ZB1/2 Bμ
2-point
function
= IPI diagram
+
Counter term
3-point
function = IPI diagram + Counter term
On-shell renormalization conditions
From these 5 conditions, 5 counter terms (δg, δg’, δv, δZW and δZB) are determined.
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On-shell renormalization scheme
Parameters shift: g → g + δg, g’ → g’ + δg’,
v → v + δv,
s^W2 → s^W2 + δ ^sW2,
Wμ → ZW1/2 Wμ , Bμ → ZB1/2 Bμ
2-point
function
= IPI diagram
+
Counter term
3-point
function = IPI diagram + Counter term
On-shell renormalization conditions
^
How we can determine the counter term δ sW2 ?
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Radiative corrections to ρ in models w/o ρ = 1 at the tree level
^s
2
is an independent parameter.
→ Additional renormalization condition is necessary.
W
Blank, Hollik (1997)
=0
1PI diagram
Counter term
Model w/ ρtree = 1
Model w/o ρtree = 1
By the renormalization of δs^W2, quadratic dependence of mass splitting disappear.
Effect of the custodial symmetry breaking
In models without ρ = 1 at the tree level, one-loop contribution to
ρ dose not measure the violation of the custodial symmetry.
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Radiative corrections to ρ in models w/o ρ = 1 at the tree level
^s
2
is an independent parameter.
→ Additional renormalization condition is necessary.
W
Blank, Hollik (1997)
=0
1PI diagram
Counter term
Model w/ ρtree = 1
Model w/o ρtree = 1
By the renormalization of δs^W2, quadratic dependence of mass splitting disappear.
Effect of the custodial symmetry breaking
We apply this renormalization scheme to the HTM.
19/35
Renormalized ρ and mW in model w/o ρtree = 1
One-loop corrected ρ and mW are given by:
,
X
Renormalization conditions
=
Y
p
+ Vertex corrections and box diagram corrections
20/35
Kanemura, KY, arXiv: 1201.6287 [hep-ph]
Prediction of the W boson mass at the 1-loop level
mH++ = 150 GeV
mH++ = 300 GeV
Case I
Δm = mH++ - mH+
A, H
H+
H++
Case II
We fixed
mh=125 GeV
and
mt=173 GeV.
H++
H+
A, H
mA = 150 GeV
mA = 300 GeV
In Case I, by the effect of the mass splitting, there are allowed regions .
Case II is highly constrained by the data.
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One-loop corrected rho parameter in the HTM
Kanemura, KY, arXiv: 1201.6287 [hep-ph]
mH++ = 150 GeV, mh = 125 GeV
Case I
mA= 150 GeV, mh = 125 GeV
Case II
A, H
H++
H+
H+
H++
A, H
Δm = mH++ - mH+
vΔ is calculated by the tree level formula:
In Case I, mH++ = 150 GeV, 100 GeV <|Δm| < 400 GeV , 3 GeV < vΔ < 8 GeV is favored, while
Case II is highly constrained by the data.
22/35
Kanemura, KY, arXiv: 1201.6287 [hep-ph]
Heavy mass limit
ξ = mH++2 – mH+2
Case I
Case II
SM
prediction
When we take heavy mass limit, loop effects of the triplet-like scalar bosons disappear.
Even in such a case, the prediction does not coincide with the SM prediction.
23/35
Decoupling property of the HTM
HTM with L# (μ ≠ 0)
HTM with L# (μ = 0)
・ ρ ≠ 1 at the tree level
(vΔ ≠ 0)
・ 4 input parameters
(αem, GF, mZ and sW2)
・ ρ = 1 at the tree level (vΔ = 0)
・ 3 input parameters (αem, GF and mZ )
with sW2 = 1-mW2/mZ2
Heavy Δ-like fields limit
Heavy Δ-like fields limit
SM
・ L# is conserved.
・ ρ = 1 at the tree level
・ 3 input parameters (αem, GF and mZ )
with sW2 = 1-mW2/mZ2
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Renormalization of the Higgs potential
25/35
Higgs Potential
Direct evidence
Non-minimal
Higgs sectors
Discovery of extra Higgs bosons;
i.e., Charged Higgs bosons H±, …
★Measuring the mass spectrum, decay branching ratios
of these extra Higgs bosons.
Indirect signature
Deviation of (SM-like) Higgs couplings
(hVV, hff, hhh)
from those predicted values in the SM.
★ Once “SM-like” Higgs boson (h) is discovered, precise measurements
for the Higgs mass and the triple Higgs coupling turns to be very important.
Precise calculation of these physics quantities is very important to
discriminate various Higgs sectors.
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The Higgs potential in the HTM
・The most general form under the SU(2)×U(1) gauge sym.
There are 10 degrees of freedom of scalar states.
10 scalar states can be translated to 3 NG bosons, 1 SM-like scalar boson
and 6 Δ-like scalar bosons.
.
±±
±
6 Δ-like scalar bosons → H
,
Doubly-charged
H ,
Singly-charged
A and H
CP-odd
CP-even
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The Higgs potential in the HTM
・The most general form under the SU(2)×U(1) gauge sym.
8 physical parameters
8 parameters in the potential
μ , m , M , λ1 , λ2 , λ3 , λ4 , λ5
v , vΔ , mH++ , mH+ , mA , mH , mh , α
If the U(1) lepton number sym. is approximately restored (μ → 0, vΔ → 0 ),
5 parameters contribute to the masses
m , M , λ1 , λ4 , λ5
5 physical parameters
v , M , mh , + 2 triplet Higgs masses
2 of 4 triplet-like Higgs masses are not independent. → There must be 2 relations:
mH++2 － mH+2 = mH+2 － mA2 = -v2λ5 /4
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Renormalization of the Higgs potential
Aoki, Kanemura, Kikuchi, KY, arXiv: 1204.1951
8 physical parameters
8 parameters in the potential
μ , m , M , λ1 , λ2 , λ3 , λ4 , λ5
Counter terms
v , vΔ , mH++ , mH+ , mA , mh , mH , α
δv , δvΔ , δmH++2 , δmH+2 , δmA2, δmh2 , δmH2 , δα
Tadpole： δTφ , δTΔ ,
Wave function renormalization：δZH++, δZH+, δZA, δZH, δZh
Renormalization of GF and ^sw2
δv , δvΔ
δTφ , δTΔ
Vanishing 1-point function
On-shell condition
δmH++2 , δmH+2 , δmA2, δmh2 , δmH2
δZH++, δZH+, δZA, δZH, δZh
No-mixing
condition
where 2-point function is defined by
δα
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Radiative corrections to the mass spectrum
Aoki, Kanemura, Kikuchi, KY, arXiv: 1204.1951
Ratio of the squared mass difference R
Case I
Tree level:
Less than 10-3
Loop level:
Loop correction
(mA2)treeis determined by mH++2 and mH+2 :
In favored parameter sets by EW precision data: mH++ = O(100)GeV,
|Δm|～100GeV, ΔR can be as large as O(10)%.
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One-loop correctedAoki,hhh
coupling
Kanemura, Kikuchi, KY, arXiv: 1204.1951
On-shell renormalization of the hhh coupling:
For α = 0 and vΔ2 << v2
Quartic mass dependence of Δ-like Higgs bosons appears to the hhh coupling ⇒
Non-decoupling property of the Higgs sector.
Results for the renormalization of the EW
parameter suggests mH++ = O(100)GeV,
|Δm|～100GeV.
In this set, deviation of hhh is predicted
more than 25%
By measuring the mass spectrum
as well as the hhh coupling,
the HTM can be tested.
Case I
Unitarity is
broken 31/35
Phenomenology
32/35
Kanemura, KY, arXiv: 1201.6287 [hep-ph]
Higgs → two photon decay
+
SM contribution
mH++ = 150 GeV, mh = 125 GeV
＋
Triplet-like scalar loop contribution
mH++ = 300 GeV, mh = 125 GeV
Case I
A, H
H+
H++
The decay rate of h → γγ is around half in the HTM compared with that in the SM.
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Phenomenology at the LHC
★From the results of the renormalization of the EW parameters
Favored scenario by the precision data:
1. Case I (mH++ > mH+ > mA, mH)
2. Δm (=mH++ - mH+ ) = O(100) GeV
3. mH++ < 300-400 GeV
4. vΔ = O(1) GeV
★From the results of the renormalization of the Higgs potential
ΔR ~ - 10% → (mH++ 2 - mH+ 2)/(mH+ 2 – mA 2) ~ 0.9
★ Decays of triplet-like Higgs bosons (Multi-W boson event may be expected):
H++ → W+W+
H+
C.-W. Chiang, T. Nomura, K. Tsumura (2012)
→ Studied multi-W signature assuming Δm = 0.
→ W-H++
H, A → W±H∓ → W±W±H ∓ ∓
Cascade decay
M. Aoki, S. Kanemura, KY (2011)
→ Studied the pheno. of the HTM with Δm ≠ 0.
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Summary
• Precise calculations of EW parameters as well as Higgs couplings (hhh, hVV,
hff) are important to discriminate various Higgs sectors.
• The important predictions in the Higgs Triplet Model:
mH++2 － mH+2 ≃ mH+2 － mA2
• Renormalization of the EW parameters
- 4 input parameters (not 3 as the SM) are necessary in to describe
the EW parameters.
⇒ mA > mH+ > mH++ with Δm=O(100) GeV is favored.
• Renormalization of the Higgs potential
- One-loop corrected mass spectrum: ΔR = O(10)%
- One-loop corrected hhh coupling : deviation from the SM
prediction can be as large as O(100) %
• The decay rate of h→γγ is half compared to that of the SM prediction.
At the LHC, multi-W signatures can be expected.
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Back up slides
Transverse mass distribution
Ex.) Measurement for W boson mass
+ −
+
pp → 𝑊 𝑊 → (𝑙 𝜈)(𝑗𝑗)
Definition of the transverse mass
𝑀𝑇 𝑙 + 𝐸𝑇 =
𝑃𝑇𝑙+ + 𝑝𝑇
2
≃
Smith,Neerven,Vermaseren,PRL(1983)
𝑚𝑊
Counter terms δv and δvΔ
Measurements of hhh coupling at colliders
Baur, Plehn, Rainwater, PRL89 (2002)
LHC
gg → hh → W+W-W+W- → l+l+4j
- mh < 140 GeV : It is difficult to detect the hhh
coupling at the LHC.
ILC
e+e- → Zhh
e+e- → hhνν
Yasui, Kanemura, Kiyoura, Odagiri, Okada, Senaha,
Yamashita, hep-ph/0211047
It can be expected around 10-30% accuracy.
Detector level simulation is underway.
mh (GeV)
Mass reconstruction
qq’ → H++H- → (l+l+ννbb)(jjbb)
H++
140 GeV
H+
130 GeV
119 GeV
114 GeV
H, A
h
qq’ → H+H → (l+νbb)(bb)
vΔ = 10-2 GeV
qq → HA → (bb)(bb)
MT, Minv
MT
MT
Signal only
mH++
130 fb
42 fb
33 fb
12 fb
8.0 fb (14 TeV)
2.8 fb (7 TeV)
mH+
mH, mA
All the masses of the Δ-like scalar bosons may be reconstructed.
Large mh and mH++ case (Case I)
mh = 125 GeV and mH++ = 150 GeV
mh = 700 GeV and mH++ = 150 GeV
mh = 125 GeV and mH++ = 300 GeV
mh = 700 GeV and mH++ = 300 GeV
Approximately formulae of Δr
Case I:
mA > mH+ > mH++
Case II:
mH++ > mH+ > mA
By using
Case I:
with mlightest = mH++
Case II:
with mlightest = mA
Y=1 Higgs Triplet Model:Kinetic term
Covariant
Derivative
VEVs
Masses for
Gauge bosons
ρ parameter
Branching ratio of H++
Δm = 0
Δm = 10 GeV
mH++ = 150 GeV
mH++ = 500 GeV
mH++ = 300 GeV
mH++ = 150 GeV
mH++ = 300 GeV
mH++ = 500 GeV
vΔ = 0.1 MeV, mH++ = 200 GeV
Chakrabarti, Choudhury, Godbole, Mukhopadhyaya, (1998);
Chun, Lee, Park, (2003);
Perez, Han, Huang, Li, Wang, (2008);
Melfo, Nemevsek, Nesti, Senjanovic, (2011)
Phenomenology of Δm ≠ 0 is drastically
different from that of Δm = 0.
Branching ratios of H+, H and A
★ The H+ → f0 W+ mode can be dominant
in the case of Δm ≠ 0.
★ The f0 → bb mode can be dominant
when vΔ > MeV.
Renormalized hhh coupling in the 2HDM
Kanemura, Okada, Senaha, Yuan (2004)