Unitarity Constraints in the SM with a singlet scalar KIAS Jubin Park collaborated with Prof. Sin Kyu Kang, and based on arXiv: [hep-ph]arXiv: KIAS, Jubin Park1

Contents KIAS, Jubin Park2 1.Motivation 2.Model 3.How to derive the unitarity condition ? 4.Unitarity of S-matrix and Numerical Results : 4.1 ≠ 0 case 4.2 = 0 case 5.Implications : 5.1 Unitarized Higgs inflation 5.2 TeV scale singlet dark matter 5.Implications : 5.1 Unitarized Higgs inflation 5.2 TeV scale singlet dark matter

1. Motivation KIAS, Jubin Park3 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: [hep-ph]]. S. Baek, P. Ko, W. -I. Park and E. Senaha, arXiv: [hep-ph].

KIAS, Jubin Park4 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/ ]. E. Ponton and L. Randall, JHEP 0904, 080 (2009) [arXiv: [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: [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: [hep-ph]].

Higgs mass implications on the stability of the electroweak vacuum KIAS, Jubin Park5 Joan Elias-Miroa, Jose R. Espinosaa;b, Gian F. Giudicec, Gino Isidoric;d, Antonio Riottoc;e, Alessandro Strumiaf arXiv: v1 [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

Stabilization of the Electroweak Vacuum by a Scalar Threshold Effect KIAS, Jubin Park6 Joan Elias-Miro, Jose R. Espinosa, Gian F. Giudicec, Hyun Min Lee, Alessandro Strumia arXiv: v1 [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

But, (my) real motivation is KIAS, Jubin Park7 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.

2. Model KIAS, Jubin Park The potential form is given by ★ S is a singlet scalar and H is a Higgs particle in the SM. 8

KIAS, Jubin Park ≠ 0 Mixing angles

KIAS, Jubin Park10 This is important !!!! ★

KIAS, Jubin Park11 ★ Stability conditions ★

KIAS, Jubin Park = 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.

3. How to derive the unitarity constraints ? KIAS, Jubin Park13

① The scattering amplitude KIAS, Jubin Park14 Differential cross section ② Optical theorem

KIAS, Jubin Park15 ③ Identity Finally, Unitarity condition ★

16 with vanishing external particle masses Three point vertex. s t u KIAS, Jubin Park

4. Unitarity of S-matrix and Numerical Results KIAS, Jubin Park17

KIAS, Jubin Park ≠ 0 0 0

Neutral states from KIAS, Jubin Park19 For example,

KIAS, Jubin Park20 eigenvalues of _0 The maximal eigenvalue can give the most strong bound !! Therefore, ★

SM case KIAS, Jubin Park21 → SM limit Therefore, ★

KIAS, Jubin Park22

KIAS, Jubin Park23 Lee-Quigg-Thacker bound

Again, we go back to KIAS, Jubin Park ≠ 0

KIAS, Jubin Park25 → This bound on the coupling is translated into the bound on the mass given by,

Now let us find the eigenvalues, KIAS, Jubin Park26 →

Allowed regions from unit. and stab KIAS, Jubin Park27 Unitarity Stability

After all, we get the contour plots KIAS, Jubin Park28 Allowed region

KIAS, Jubin Park GeV

Neutral states from KIAS, Jubin Park ≠ 0 where,

Explicit form of scattering amplitudes KIAS, Jubin Park31

The contour plots KIAS, Jubin Park32

KIAS, Jubin Park = 0

KIAS, Jubin Park34 → The unitarity condition gives

KIAS, Jubin Park ≠ 0 The characteristic equation is

The contour plots KIAS, Jubin Park36

Let us summarize our results for a while KIAS, Jubin Park37

5. Implications KIAS, Jubin Park Unitarized Higgs inflation Potential : From unitarity :

Imposing the COBE result for normalization of the power spectrum, KIAS, Jubin Park39

Mixing angle vs Mass of singlet scalar s KIAS, Jubin Park40 Allowed region ↑ Very small mixing allowed

5.2 TeV scale singlet dark matter KIAS, Jubin Park41 Dominant annihilation channel : Relic density : From the 9-year WMAP result:

KIAS, Jubin Park42 Unitarity ★

Conclusion KIAS, Jubin Park43

KIAS, Jubin Park44 Conclusion