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 1 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 2013. 7. 30 @ KIAS, Jubin Park 2 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 3 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]]. 2013. 7. 30 @ KIAS, Jubin Park 4 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 5 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 2013. 7. 30 @ KIAS, Jubin Park 6 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 7 2. Model The potential form is given by β S is a singlet scalar and H is a Higgs particle in the SM. 2013. 7. 30 @ KIAS, Jubin Park 8 2. 1. <S> ≠ 0 <π» >= 1 2 π£ , < π >=η , v = ( 2 πΊπΉ )−1/2 Mixing angles πΌ πΆβ = π£/ π£ 2 + η2 π· πβ = η/ π£ 2 + η2 π 2013. 7. 30 @ KIAS, Jubin Park 9 This is important !!!! Three couplings can be rewritten in terms of physical masses, πβ1 and πβ2 . β ξ2 = π£ 2 + η2 πβ1 < πβ2 πΆβ = π£/ π£ 2 + η2 2013. 7. 30 @ KIAS, Jubin Park πβ = η/ π£ 2 + η2 10 β Stability conditions β 2013. 7. 30 @ KIAS, Jubin Park 11 2. 2. <S> = 0 2 ππ = π£2 2 μπ +λπ»π 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. 2013. 7. 30 @ KIAS, Jubin Park 12 3. How to derive the unitarity constraints ? 2013. 7. 30 @ KIAS, Jubin Park 13 β The scattering amplitude Differential cross section β‘ Optical theorem Imaginary part in the forward direction 2013. 7. 30 @ KIAS, Jubin Park 14 β’ Identity Finally, β Unitarity condition 2013. 7. 30 @ KIAS, Jubin Park 15 with vanishing external particle masses General form of amplitude π0 (s) s t u Three point vertex. 2013. 7. 30 @ KIAS, Jubin Park 16 4. Unitarity of S-matrix and Numerical Results 2013. 7. 30 @ KIAS, Jubin Park 17 4. 1. <S> ≠ 0 s β« πβ2 ,ππ 2 Three vertex parts of π0 is negligible !!! Only, four vertex part is important !!! 0 0 1 π0 = π΄ 16π 2013. 7. 30 @ KIAS, Jubin Park ~ 18 Neutral states from |π + π − >, | 1 ππ>, 2 | 1 ββ>, 2 | 1 π π >, 2 | 1 βπ >, 2 |βπ> For example, π0 (|π + π − > → π + π − > = − λπ» 4π Charged states just give the diagonal elements of π0 , 1 which is equal to . 2 A≡ λπ λπ» , B≡ 1 λπ»π 6 λπ» 2013. 7. 30 @ KIAS, Jubin Park 19 ×1 Perturbative unitarity can be given in terms of eigenvalues of π0 , λπ» 1 × ππ < 4π 2 eigenvalues of π_0 Therefore, β The maximal eigenvalue can give the most strong bound !! 2013. 7. 30 @ KIAS, Jubin Park 20 SM case 1 ππ>, 2 |π + π − >, | | 1 ββ>, 2 | 1 π π >, 2 | 1 → SM limit λπ» − 4π 1 8 1 8 0 Therefore, 1 βπ >, 2 |βπ> 1 1 8 3 4 1 4 8 1 0 4 3 0 4 1 0 2 0 0 β 2013. 7. 30 @ KIAS, Jubin Park 21 1 1 πππ = 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 π0 2 2 2 2 π π λπ» π 1 × < 4π π 2 2013. 7. 30 @ KIAS, Jubin Park 22 λπ» π 1 × < 4π π 2 and from ππ»2 = 2 λπ» π£ = 2 λπ» × 2 πΊπΉ 2 λπ» = ππ» 2 1 2πΊπΉ Lee-Quigg-Thacker bound ππ» ≤ 8 2π 3 πΊπΉ 1 2 ≡ ππΏππ ≈ 1 TeV 2013. 7. 30 @ KIAS, Jubin Park 23 Again, we go back to s β« πβ2 ,ππ 2 2. 1. <S> ≠ 0 As we check the characteristic equation of π0 , we get this equation, 4 with two trivial eigenvalues 3 π΅ and 1 2 of π0 , where A and B are given by 2013. 7. 30 @ KIAS, Jubin Park A≡ λπ λπ» , B≡ 1 λπ»π 6 λπ» . 24 Also, from the eigenvalue of π0 → λπ» 1 × ππ < 4π 2 This bound on the coupling is translated into the bound on the mass given by, 2013. 7. 30 @ KIAS, Jubin Park 25 Now let us find the eigenvalues, → Λ = Λ (λπ , λπ» , λπ»π ) A≡ First, we fix λπ»π and check the allowed regions from λπ λπ» , B≡ 1 λπ»π 6 λπ» the stability conditions, 2013. 7. 30 @ KIAS, Jubin Park 26 Allowed regions from unit. and stab. Unitarity λπ―πΊ = 9.8 λπ―πΊ = 5 Stability λπ―πΊ = 1 λπ―πΊ ≈ π. π λπ―πΊ = 0 2013. 7. 30 @ KIAS, Jubin Park 27 After all, we get the contour plots Allowed region λπ―πΊ ~0 λπ―πΊ ~9.8 Allowed region 2013. 7. 30 @ KIAS, Jubin Park 28 From the maximum eigenvalue 3 of π0 π0 < 1 2 π0 < 1 π0 < 1 2 π0 < 1 126 GeV 2013. 7. 30 @ KIAS, Jubin Park 29 4. 1. <S> ≠ 0 π β« πβ2 , π ~ 4ππ 2 ~ (1 πππ) Neutral states from |π + π − >, | 1 ππ>, 2 π0 matrix is written by 2 | 1 ββ>, 2 | 1 π π >, 2 | 1 βπ >, 2 |βπ> where, 2013. 7. 30 @ KIAS, Jubin Park 30 Explicit form of scattering amplitudes 2013. 7. 30 @ KIAS, Jubin Park 31 The contour plots λπ―πΊ ~0 2013. 7. 30 @ KIAS, Jubin Park λπ―πΊ ~9.8 32 4. 2. <S> = 0 π β« πβ2 ,ππ 2 limit β No π − β − β coupling ! β‘ The odd parity of s forbids following processes : π + π − → βπ , ππ→ βπ , ββ→ βπ , ββ→ ss π0 matrix is written by |π + π −>, | 1 2 ππ>, | 1 2 ββ>, | 1 2 π π >, | 1 2 βπ >, |βπ> 0 0 0 2. 2. <S> ≠ 0 case 0 3 π΅ 4 2013. 7. 30 @ KIAS, Jubin Park π β« πβ2 ,ππ 2 limit 33 But because of no mixing between β and π , we can not constrain the mass bound of π . → α = 0 and ππππ₯ = 3 The unitarity condition gives πβ ≤ 1 2 ππΏππ 2013. 7. 30 @ KIAS, Jubin Park 34 4. 2. <S> ≠ 0 s β« πβ2 , π ~ 4ππ 2 ~ (1 πππ) 2 The characteristic equation is with one trivial eigenvalue of 1 2 . 2013. 7. 30 @ KIAS, Jubin Park 35 The contour plots π π πβ ≤ 2013. 7. 30 @ KIAS, Jubin Park 2 ππΏππ 36 Let us summarize our results for a while. 2013. 7. 30 @ KIAS, Jubin Park 37 5. Implications 5.1 Unitarized Higgs inflation Potential : From unitarity : 2013. 7. 30 @ KIAS, Jubin Park 38 Imposing the COBE result for normalization of the power spectrum, ≈ 10 GeV 13 2013. 7. 30 @ KIAS, Jubin Park 39 Mixing angle vs Mass of singlet scalar s Very small mixing allowed Allowed region ↑ ππ ≈ 1013 GeV 2013. 7. 30 @ KIAS, Jubin Park 40 5.2 TeV scale singlet dark matter Dominant annihilation channel : When ππ ≈ 1 TeV Relic density : 2 Ω β From the 9-year WMAP result: π·π = 0.1138 ± 0.0045 2013. 7. 30 @ KIAS, Jubin Park 41 λπ»π vs ππ Unitarity Ωπ·π β2 = 0.1138 ± 0.0045 β ππ ≤ 30. 490 TeV 2013. 7. 30 @ KIAS, Jubin Park 42 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 ππ ≤ 400 GeV in the maximal angle α → . 2 2013. 7. 30 @ KIAS, Jubin Park 43 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, ππ ≤ 30. 490 TeV , by combining the observed relic abundance with the unitarity. 2013. 7. 30 @ KIAS, Jubin Park 44