2013. 7. 30 @ KIAS, Jubin Park 1

Unitarity Constraints in the SM

with a singlet scalar2013. 7. 30 @ KIAS

Jubin Parkcollaborated with Prof. Sin Kyu Kang,

and based on arXiv:1306.6713 [hep-ph]

2013. 7. 30 @ KIAS, Jubin Park 2

Contents1. 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. MotivationWhy 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 symmetryC. 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 transi-tionS. 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 sta-bility 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 isIn fact, we want to study 2HD + 1S case, where the potential is generated radia-tively.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. ModelThe 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 =η , v

Mixing angles

𝐶 β=𝑣 /√𝑣2+η2 𝑆β=η /√𝑣2+η2

𝒗

𝜼

𝜷

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This is important !!!!

Three couplings can be rewritten in terms of physical masses, and

★

𝐶 β=𝑣 /√𝑣2+η2 𝑆β=η /√𝑣2+η2

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★ Stability conditions ★

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2. 2. <S> = 0 +

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 con-straints ?

14

① The scattering amplitude

2013. 7. 30 @ KIAS, Jubin Park

Spin J partial wave

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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General form of amplitude (s)with vanishing external particle masses

Four point vertex Three point vertex.

s t u

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4. Unitarity of S-matrix and Numerical Results

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4. 1. <S> ≠ 0 s , → Three vertex parts of is negligible !!!

Only, four vertex part is important !!!

0 0

→𝑎0=

116𝜋 𝐴

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Neutral states from >, >, >, >, >, >

A , B

For example,

Charged states just give the diagonal elements of ,which is equal to .

Note

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Perturbative unitarity can be given in terms of eigenvalues of ,

|λ𝐻4𝜋 ×𝑐 𝑖|< 12eigenvalues of

_0𝑇

The maximal eigenvalue can give the most strong bound !!

Therefore,★

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SM case >, >, >, >, >, >

(− λ𝐻4 𝜋 )(1 1

√81√8

0

1√8

34

14

0

1√8

14

34

0

0 0 0 12

)→SM limit

It is important to check the eigenvalue of →Therefore,★

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𝑇 𝑆𝑀=(1 1

√81

√80

1√8

34

14

0

1√8

14

34

0

0 0 0 12

) →The maximal eigenvalue of is

|λ𝐻4𝜋 ×𝟑𝟐|<12

of eigenvalues of

→→

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|  λ𝐻4 𝜋 ×𝟑𝟐|< 12 and from =

TeVLee-Quigg-Thacker bound

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Again, we go back to 2. 1. <S> ≠ 0 s ,

As we check the characteristic equation of , we get this equa-tion,

with two trivial eigenvalues and of , where A and B are given by A , B .

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Also, from the eigenvalue of

|λ𝐻4𝜋 ×𝑐 𝑖|< 12 →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,

→

A , B

First, we fix and check the allowed regions from the stability condi-tions,

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Allowed regions from unit. and stab. Unitarity

Stabilityλ𝑯𝑺≈𝟗 .𝟖 λ𝑯𝑺=1

λ𝑯𝑺=5

λ𝑯𝑺=9.8

λ𝑯𝑺=0

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After all, we get the contour plots

λ𝑯𝑺 0 λ𝑯𝑺 9.8

Allowed region

Allowed region

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From the maximum eigenvalue 3 of

126 GeV

|𝑎0|<12 |𝑎0|<

12|𝑎0|<1

|𝑎0|<1

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Neutral states from

4. 1. <S> ≠ 0 , >, >, >, >, >, >

matrix is written by where,

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Explicit form of scattering ampli-tudes

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The contour plots

λ𝑯𝑺 0 λ𝑯𝑺 9.8

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4. 2. <S> = 0 ,limit Note

matrix is written by >, >, >, >, >, >

00

00 → 2. 2. <S> 0 case

,limit

① No coupling !② The odd parity of s forbids following processes :→ , →

34 𝐵

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But because of no mixing between and ,we can not constrain the mass bound of .

and →The unitarity condition gives

𝑚h≤1√2

𝑀𝐿𝑄𝑇

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4. 2. <S> ≠ 0 s ,

The characteristic equa-tion is

with one trivial eigenvalue of .

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The contour plots

NoteThe maximal eigenvalue of is → →

𝑚h≤√2𝑀𝐿𝑄𝑇

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Let us summarize our results for a while.

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5. Implications5.1 Unitarized Higgs inflation

Potential :

From unitarity :

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Imposing the COBE result for normalization of the power spectrum, →

→GeV

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Mixing angle vs Mass of singlet scalar s

Allowed region

GeV↑

NoteVery small mixing allowed

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5.2 TeV scale singlet dark matter

Dominant annihilation channel :When TeV

Relic density :

Ω𝐷𝑀 h2=0.1138 ±0.0045From the 9-year WMAP result:

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vs Ω𝐷𝑀 h

2=0.1138 ±0.0045Unitarity

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 , the bound becomes very strong,GeV in the maximal angle .

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Conclusion

1. In the unitarized Higgs inflation scenario, a tiny mixing angle is required for the singlet scalar with around GeV mass.

2. In the TeV scale dark matter scenario, we have drived upper bound on the singlet scalar mass, TeV , by combining the observed relic abundance with the unitarity.