Unitarity Constraints in the SM
with a singlet scalar
2013. 7. 30 @ KIAS
Jubin Park
collaborated with Prof. Sin Kyu Kang,
and based on arXiv:1306.6713 [hep-ph]
2013. 7. 30 @ KIAS, Jubin Park
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Contents
1. Motivation
2. Model
3. How to derive the unitarity condition ?
4. Unitarity of S-matrix and Numerical Results :
4.1 <S> ≠ 0 case
4.2 <S> = 0 case
5. Implications :
5.1 Unitarized Higgs inflation
5.2 TeV scale singlet dark matter
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1. Motivation
Why a (singlet) scalar field ?
1. A new discovery of a scalar particle at LHC.
Higg particle in the SM ~ 124 ~ 126 GeV
??
2. can modify the production and/or decay rates of the Higgs field.
B. Batell, D. McKeen and M. Pospelov, JHEP 1210, 104 (2012) [arXiv:1207.6252 [hep-ph]].
S. Baek, P. Ko, W. -I. Park and E. Senaha, arXiv:1209.4163 [hep-ph].
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3. can supply a dark matter candidate by using a discrete Z_2 symmetry
C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B 619, 709 (2001) [hep-ph/0011335].
E. Ponton and L. Randall, JHEP 0904, 080 (2009) [arXiv:0811.1029 [hep-ph]].
4. can give a solution of baryogenesis via the first of electroweak phase transition
S. Profumo, M. J. Ramsey-Musolf and G. Shaughnessy, JHEP 0708, 010 (2007)
[arXiv:0705.2425 [hep-ph]].
5. can solve the unitarity problem of the Higgs inflation.
G. F. Giudice and H. M. Lee, Phys. Lett. B 694, 294 (2011) [arXiv:1010.1417 [hep-ph]].
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Higgs mass implications on the stability
of the electroweak vacuum
Joan Elias-Miroa, Jose R. Espinosaa;b, Gian F. Giudicec,
Gino Isidoric;d, Antonio Riottoc;e, Alessandro Strumiaf
arXiv:1112.3022v1 [hep-ph]
The RG running of Higgs
quartic coupling can give
a useful hint about the
structure of given theory
at the very short distance
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Stabilization of the Electroweak Vacuum
by a Scalar Threshold Effect
Joan Elias-Miro, Jose R. Espinosa, Gian F. Giudicec, Hyun Min Lee,
Alessandro Strumia
arXiv:1203.0237v1 [hep-ph]
The RG running of Higgs
quartic coupling can give
a useful hint about the
structure of given theory
at the very short distance
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But, (my) real motivation is
In fact, we want to study 2HD + 1S case, where the potential is generated radiatively.
So we have to consider the unitarity condition in this case.
But, I could not find any paper about this. Note that there are many papers about 2HD.
So, I decided to attack this problem, and I tried to find a more easy case such as 1HD(SM) + 1S.
Frankly speaking I found one paper, but they just consider a limited case not a general case.
After all, I tried to study the unitarity constraints of the 1HD(SM) + 1S case first.
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2. Model
The potential form is given by
★
S is a singlet scalar and H is a Higgs particle in the SM.
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2. 1. <S> ≠ 0
<𝐻 >=
1
2
𝑣 , < 𝑆 >=η , v = ( 2 𝐺𝐹 )−1/2
Mixing angles
𝜼
𝐶β = 𝑣/ 𝑣 2 + η2
𝜷
𝑆β = η/ 𝑣 2 + η2
𝒗
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This is important !!!!
Three couplings can be rewritten in terms of physical masses, 𝑚ℎ1 and 𝑚ℎ2 .
★
ξ2 = 𝑣 2 + η2
𝑚ℎ1 < 𝑚ℎ2
𝐶β = 𝑣/ 𝑣 2 + η2
2013. 7. 30 @ KIAS, Jubin Park
𝑆β = η/ 𝑣 2 + η2
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★ Stability conditions ★
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2. 2. <S> = 0
2
𝑚𝑠 =
𝑣2
2
μ𝑠 +λ𝐻𝑆
2
Imposing Z_2 symmetry, this case can give a Z_2 odd singlet scalar as a dark matter candidate.
There is no bi-linear mixing term (~hs) in the potential.
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3. How to derive the unitarity constraints ?
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① The scattering amplitude
Differential cross section
② Optical theorem
Imaginary part
in the forward direction
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③ Identity
Finally,
★
Unitarity condition
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with vanishing external particle masses
General form of amplitude 𝑎0 (s)
s
t
u
Three point vertex.
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4. Unitarity of S-matrix and Numerical Results
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4. 1. <S> ≠ 0
s ≫ 𝑚ℎ2 ,𝑚𝑠2
Three vertex parts of 𝑎0 is negligible !!!
Only, four vertex part is important !!!
0
0
1
𝑎0 =
𝐴
16𝜋
2013. 7. 30 @ KIAS, Jubin Park
~
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Neutral states from
|𝑊 + 𝑊 − >, |
1
𝑍𝑍>,
2
|
1
ℎℎ>,
2
|
1
𝑠𝑠>,
2
|
1
ℎ𝑠>,
2
|ℎ𝑍>
For example,
𝑎0 (|𝑊 + 𝑊 − > → 𝑊 + 𝑊 − > = −
λ𝐻
4𝜋
Charged states just give
the diagonal elements of 𝑇0 ,
1
which is equal to .
2
A≡
λ𝑆
λ𝐻
, B≡
1 λ𝐻𝑆
6 λ𝐻
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×1
Perturbative unitarity can be given in terms of eigenvalues of 𝑇0 ,
λ𝐻
1
× 𝑐𝑖 <
4𝜋
2
eigenvalues of 𝑇_0
Therefore,
★
The maximal eigenvalue can give the most strong bound !!
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SM case
1
𝑍𝑍>,
2
|𝑊 + 𝑊 − >, |
|
1
ℎℎ>,
2
|
1
𝑠𝑠>,
2
|
1
→
SM limit
λ𝐻
−
4𝜋
1
8
1
8
0
Therefore,
1
ℎ𝑠>,
2
|ℎ𝑍>
1
1
8
3
4
1
4
8
1
0
4
3
0
4
1
0
2
0
0
★
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1
1
𝑇𝑆𝑀 =
8
1
8
0
1
1
8
3
4
1
4
8
1
0
4
3
0
4
1
0
2
0
0
3 3 3 1
, , , of eigenvalues of 𝑇0
2 2 2 2
𝟑
𝟐
λ𝐻 𝟑
1
× <
4𝜋 𝟐
2
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λ𝐻 𝟑
1
× <
4𝜋 𝟐
2
and from
𝑀𝐻2
= 2 λ𝐻 𝑣 = 2 λ𝐻 ×
2
𝐺𝐹 2
λ𝐻 = 𝑀𝐻
2
1
2𝐺𝐹
Lee-Quigg-Thacker bound
𝑀𝐻 ≤
8 2𝜋
3 𝐺𝐹
1
2
≡ 𝑀𝐿𝑄𝑇 ≈ 1 TeV
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Again, we go back to
s ≫ 𝑚ℎ2 ,𝑚𝑠2
2. 1. <S> ≠ 0
As we check the characteristic equation of 𝑇0 , we get this equation,
4
with two trivial eigenvalues 3 𝐵 and
1
2
of 𝑇0 , where A and B are given by
2013. 7. 30 @ KIAS, Jubin Park
A≡
λ𝑆
λ𝐻
, B≡
1 λ𝐻𝑆
6 λ𝐻
.
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Also, from the eigenvalue of 𝑇0
→
λ𝐻
1
× 𝑐𝑖 <
4𝜋
2
This bound on the coupling is translated into the bound on the mass given by,
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Now let us find the eigenvalues,
→
Λ = Λ (λ𝑆 , λ𝐻 , λ𝐻𝑆 )
A≡
First, we fix λ𝐻𝑆
and check
the allowed regions
from
λ𝑆
λ𝐻
, B≡
1 λ𝐻𝑆
6 λ𝐻
the stability conditions,
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Allowed regions from unit. and stab.
Unitarity
λ𝑯𝑺 = 9.8
λ𝑯𝑺 = 5
Stability
λ𝑯𝑺 = 1
λ𝑯𝑺 ≈ 𝟗. 𝟖
λ𝑯𝑺 = 0
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After all, we get the contour plots
Allowed region
λ𝑯𝑺 ~0
λ𝑯𝑺 ~9.8
Allowed region
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From the maximum eigenvalue 3 of 𝑇0
𝑎0 <
1
2
𝑎0 < 1
𝑎0 <
1
2
𝑎0 < 1
126 GeV
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4. 1. <S> ≠ 0
𝑠 ≫ 𝑚ℎ2 , 𝑠 ~ 4𝑚𝑠2 ~ (1 𝑇𝑒𝑉)
Neutral states from
|𝑊 + 𝑊 − >, |
1
𝑍𝑍>,
2
𝑇0 matrix is written by
2
|
1
ℎℎ>,
2
|
1
𝑠𝑠>,
2
|
1
ℎ𝑠>,
2
|ℎ𝑍>
where,
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Explicit form of scattering amplitudes
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The contour plots
λ𝑯𝑺 ~0
2013. 7. 30 @ KIAS, Jubin Park
λ𝑯𝑺 ~9.8
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4. 2. <S> = 0
𝑠 ≫ 𝑚ℎ2 ,𝑚𝑠2 limit
① No 𝑠 − ℎ − ℎ coupling !
② The odd parity of s forbids
following processes :
𝑊 + 𝑊 − → ℎ𝑠, 𝑍𝑍→ ℎ𝑠, ℎℎ→ ℎ𝑠, ℎℎ→ ss
𝑇0 matrix is written by
|𝑊 + 𝑊 −>, |
1
2
𝑍𝑍>, |
1
2
ℎℎ>, |
1
2
𝑠𝑠>, |
1
2
ℎ𝑠>, |ℎ𝑍>
0
0
0
2. 2. <S> ≠ 0 case
0
3
𝐵
4
2013. 7. 30 @ KIAS, Jubin Park
𝑆 ≫ 𝑚ℎ2 ,𝑚𝑠2 limit
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But because of no mixing between ℎ and 𝑠,
we can not constrain the mass bound of 𝑠.
→
α = 0 and 𝑐𝑚𝑎𝑥 = 3
The unitarity condition gives
𝑚ℎ ≤
1
2
𝑀𝐿𝑄𝑇
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4. 2. <S> ≠ 0
s ≫ 𝑚ℎ2 , 𝑠 ~ 4𝑚𝑠2 ~ (1 𝑇𝑒𝑉)
2
The characteristic equation is
with one trivial eigenvalue of
1
2
.
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The contour plots
𝟑
𝟐
𝑚ℎ ≤
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2 𝑀𝐿𝑄𝑇
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Let us summarize our results for a while.
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5. Implications
5.1 Unitarized Higgs inflation
Potential :
From unitarity :
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Imposing the COBE result for
normalization of the power spectrum,
≈ 10 GeV
13
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Mixing angle vs Mass of singlet scalar s
Very small mixing allowed
Allowed region
↑
𝑀𝑠 ≈ 1013 GeV
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5.2 TeV scale singlet dark matter
Dominant annihilation channel :
When 𝑚𝑠 ≈ 1 TeV
Relic density :
2
Ω
ℎ
From the 9-year WMAP result: 𝐷𝑀 = 0.1138 ± 0.0045
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λ𝐻𝑆 vs 𝑚𝑠
Unitarity
Ω𝐷𝑀 ℎ2 = 0.1138 ± 0.0045
★
𝑚𝑠 ≤ 30. 490 TeV
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Conclusion
1. Taking into account full contributions to the scattering amplitudes,
we have drived unitarity conditions that can be translated into
bounds on the masses of sclar fields.
2. While the upper mass bound of the singlet scalar becomes divergent
in the decoupling limit α → 0 , the bound becomes very strong,
𝜋
𝑚𝑠 ≤ 400 GeV in the maximal angle α → .
2
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Conclusion
1. In the unitarized Higgs inflation scenario, a tiny mixing angle α ~ 10−10
is required for the singlet scalar with around 1013 GeV mass.
2. In the TeV scale dark matter scenario, we have drived upper bound
on the singlet scalar mass, 𝑚𝑠 ≤ 30. 490 TeV , by combining the observed
relic abundance with the unitarity.
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