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Unitarity Constraints in the SM with a singlet scalar

Unitarity Constraints in the SM with a singlet scalar. 2013. 7. 30 @ KIAS Ju b in Park collaborated with Prof. Sin Kyu Kang, and based on arXiv:1306.6713  [ hep-ph ]. Contents. Motivation Model How to derive the unitarity condition ?

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Unitarity Constraints in the SM with a singlet scalar

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  1. Unitarity Constraints in the SMwith a singlet scalar 2013. 7. 30 @ KIASJubin Parkcollaborated with Prof. Sin Kyu Kang,and based on arXiv:1306.6713 [hep-ph] 2013. 7. 30 @ KIAS, Jubin Park

  2. Contents Motivation Model How to derive the unitarity condition ? Unitarity of S-matrix and Numerical Results : 4.1 <S> ≠ 0 case 4.2 <S> = 0 case Implications : 5.1 Unitarized Higgs inflation 5.2 TeV scale singlet dark matter 2013. 7. 30 @ KIAS, Jubin Park

  3. 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]. 2013. 7. 30 @ KIAS, Jubin Park

  4. 3. can supply a dark matter candidate by using a discrete Z_2 symmetry C. P. Burgess, M. Pospelov and T. terVeldhuis, 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]]. 2013. 7. 30 @ KIAS, Jubin Park

  5. 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 2013. 7. 30 @ KIAS, Jubin Park

  6. Stabilization of the Electroweak Vacuum by a Scalar Threshold Effect Joan Elias-Miro, Jose R. Espinosa, Gian F. Giudicec, Hyun Min Lee, Alessandro StrumiaarXiv: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 2013. 7. 30 @ KIAS, Jubin Park

  7. 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. 2013. 7. 30 @ KIAS, Jubin Park

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

  9. 2. 1. <S> ≠ 0 =η , v Mixing angles 2013. 7. 30 @ KIAS, Jubin Park

  10. This is important !!!! Three couplings can be rewritten in terms of physical masses, and ★ 2013. 7. 30 @ KIAS, Jubin Park

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

  12. 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. 2013. 7. 30 @ KIAS, Jubin Park

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

  14. ① The scattering amplitude Spin J partial wave Differential cross section ② Optical theorem + Imaginary part in the forward direction 2013. 7. 30 @ KIAS, Jubin Park

  15. → ③ Identity Finally, + ★ Unitarity condition 2013. 7. 30 @ KIAS, Jubin Park

  16. with vanishing external particle masses General form of amplitude (s) u t s Four point vertex Three point vertex. 2013. 7. 30 @ KIAS, Jubin Park

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

  18. → Three vertex parts of is negligible !!! Only, four vertex part is important !!! s, 4. 1. <S> ≠ 0 0 0 → 2013. 7. 30 @ KIAS, Jubin Park

  19. Neutral states from >,>,>, >,>, > For example, Note Charged states just give the diagonal elements of ,which is equal to . A , B 2013. 7. 30 @ KIAS, Jubin Park

  20. Perturbative unitarity can be given in terms of eigenvalues of , eigenvalues of 𝑇_0 Therefore, ★ The maximal eigenvalue can give the most strong bound !! 2013. 7. 30 @ KIAS, Jubin Park

  21. SM case >,>,>, >,>, > → SM limit → Therefore, ★ It is important to check the eigenvalue of 2013. 7. 30 @ KIAS, Jubin Park

  22. → of eigenvalues of → The maximal eigenvalue of is → 2013. 7. 30 @ KIAS, Jubin Park

  23. and from = Lee-Quigg-Thacker bound TeV 2013. 7. 30 @ KIAS, Jubin Park

  24. Again, we go back to s , 2. 1. <S> ≠ 0 As we check the characteristic equation of , we get this equation, with two trivial eigenvalues and of , where A and B are given by A , B . 2013. 7. 30 @ KIAS, Jubin Park

  25. Also, from the eigenvalue of → This bound on the coupling is translated into the bound on the mass given by, 2013. 7. 30 @ KIAS, Jubin Park

  26. Now let us find the eigenvalues, → A , B First, we fix and check the allowed regions from the stability conditions, 2013. 7. 30 @ KIAS, Jubin Park

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

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

  29. From the maximum eigenvalue 3 of 126 GeV 2013. 7. 30 @ KIAS, Jubin Park

  30. Neutral states from , 4. 1. <S> ≠ 0 >,>,>, >,>, > matrix is written by where, 2013. 7. 30 @ KIAS, Jubin Park

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

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

  33. Note 4. 2. <S> = 0 ,limit ① No coupling !② The odd parity of s forbids following processes :→, → matrix is written by >,>,>, >,>, > → 2. 2. <S> 0 case ,limit 2013. 7. 30 @ KIAS, Jubin Park

  34. But because of no mixing between and,we can not constrain the mass bound of . → and The unitarity condition gives 2013. 7. 30 @ KIAS, Jubin Park

  35. s , 4. 2. <S> ≠ 0 The characteristic equation is with one trivial eigenvalue of . 2013. 7. 30 @ KIAS, Jubin Park

  36. The contour plots Note → The maximal eigenvalue of is → 2013. 7. 30 @ KIAS, Jubin Park

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

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

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

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

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

  42. vs Unitarity ★ TeV 2013. 7. 30 @ KIAS, Jubin Park

  43. Conclusion Taking into account full contributions to the scattering amplitudes, we have drivedunitarity conditions that can be translated into bounds on the masses of sclar fields. 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 . 2013. 7. 30 @ KIAS, Jubin Park

  44. Conclusion • In the unitarized Higgs inflation scenario, a tiny mixing angle is required for the singlet scalar with around GeV mass. • 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. 2013. 7. 30 @ KIAS, Jubin Park

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