1. 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]arXiv:1306.6713 2013. 7. 30 @ KIAS, Jubin Park1

2. Contents 2013. 7. 30 @ 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

3. 1. Motivation 2013. 7. 30 @ 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:1207.6252 [hep-ph]]. S. Baek, P. Ko, W. -I. Park and E. Senaha, arXiv:1209.4163 [hep-ph].

4. 2013. 7. 30 @ 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/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]].

5. Higgs mass implications on the stability of the electroweak vacuum 2013. 7. 30 @ KIAS, Jubin Park5 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

6. Stabilization of the Electroweak Vacuum by a Scalar Threshold Effect 2013. 7. 30 @ KIAS, Jubin Park6 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

7. But, (my) real motivation is 2013. 7. 30 @ 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.

8. 2. Model 2013. 7. 30 @ KIAS, Jubin Park The potential form is given by ★ S is a singlet scalar and H is a Higgs particle in the SM. 8

9. 2013. 7. 30 @ KIAS, Jubin Park9 2. 1. ≠ 0 Mixing angles

10. 2013. 7. 30 @ KIAS, Jubin Park10 This is important !!!! ★

11. 2013. 7. 30 @ KIAS, Jubin Park11 ★ Stability conditions ★

12. 2013. 7. 30 @ KIAS, Jubin Park12 2. 2. = 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.

13. 3. How to derive the unitarity constraints ? 2013. 7. 30 @ KIAS, Jubin Park13

14. ① The scattering amplitude 2013. 7. 30 @ KIAS, Jubin Park14 Differential cross section ② Optical theorem

15. 2013. 7. 30 @ KIAS, Jubin Park15 ③ Identity Finally, Unitarity condition ★

16. 16 with vanishing external particle masses Three point vertex. s t u 2013. 7. 30 @ KIAS, Jubin Park

17. 4. Unitarity of S-matrix and Numerical Results 2013. 7. 30 @ KIAS, Jubin Park17

18. 2013. 7. 30 @ KIAS, Jubin Park18 4. 1. ≠ 0 0 0

19. Neutral states from 2013. 7. 30 @ KIAS, Jubin Park19 For example,

20. 2013. 7. 30 @ KIAS, Jubin Park20 eigenvalues of _0 The maximal eigenvalue can give the most strong bound !! Therefore, ★

21. SM case 2013. 7. 30 @ KIAS, Jubin Park21 → SM limit Therefore, ★

22. 2013. 7. 30 @ KIAS, Jubin Park22

23. 2013. 7. 30 @ KIAS, Jubin Park23 Lee-Quigg-Thacker bound

24. Again, we go back to 2013. 7. 30 @ KIAS, Jubin Park24 2. 1. ≠ 0

25. 2013. 7. 30 @ KIAS, Jubin Park25 → This bound on the coupling is translated into the bound on the mass given by,

26. Now let us find the eigenvalues, 2013. 7. 30 @ KIAS, Jubin Park26 →

27. Allowed regions from unit. and stab. 2013. 7. 30 @ KIAS, Jubin Park27 Unitarity Stability

28. After all, we get the contour plots 2013. 7. 30 @ KIAS, Jubin Park28 Allowed region

29. 2013. 7. 30 @ KIAS, Jubin Park29 126 GeV

30. Neutral states from 2013. 7. 30 @ KIAS, Jubin Park30 4. 1. ≠ 0 where,

31. Explicit form of scattering amplitudes 2013. 7. 30 @ KIAS, Jubin Park31

32. The contour plots 2013. 7. 30 @ KIAS, Jubin Park32

33. 2013. 7. 30 @ KIAS, Jubin Park33 4. 2. = 0

34. 2013. 7. 30 @ KIAS, Jubin Park34 → The unitarity condition gives

35. 2013. 7. 30 @ KIAS, Jubin Park35 4. 2. ≠ 0 The characteristic equation is

36. The contour plots 2013. 7. 30 @ KIAS, Jubin Park36

37. Let us summarize our results for a while. 2013. 7. 30 @ KIAS, Jubin Park37

38. 5. Implications 2013. 7. 30 @ KIAS, Jubin Park38 5.1 Unitarized Higgs inflation Potential : From unitarity :

39. Imposing the COBE result for normalization of the power spectrum, 2013. 7. 30 @ KIAS, Jubin Park39

40. Mixing angle vs Mass of singlet scalar s 2013. 7. 30 @ KIAS, Jubin Park40 Allowed region ↑ Very small mixing allowed

41. 5.2 TeV scale singlet dark matter 2013. 7. 30 @ KIAS, Jubin Park41 Dominant annihilation channel : Relic density : From the 9-year WMAP result:

42. 2013. 7. 30 @ KIAS, Jubin Park42 Unitarity ★

43. Conclusion 2013. 7. 30 @ KIAS, Jubin Park43

44. 2013. 7. 30 @ KIAS, Jubin Park44 Conclusion