How to get these Equations?

[Monaliza Smile]

Hi all,

Do anyone know how from the mass eigenvalues:

$M_{H,h}^2 = \frac{1}{2} [\lambda_1 v_1^2 + \lambda_2 v_2^2 \mp \sqrt{(\lambda_1 v_1^2 - \lambda_2 v_2^2)^2 + 4 \lambda^2 v_1^2 v_2^2} ],$

and $\tan 2\alpha =\frac{2 \lambda v_1 v_2}{\lambda_1 v_1^2 - \lambda_2 v_2^2}$

To get the couplings $\lambda_{1,2}$ as in Equs. (6) in [arXiv:1508.00702v2[hep-ph]]
or Equs. (9) in [arXiv:1303.5098v1 [hep-ph]], Note that I'd like to put $m_{12} =0$.

PS. The angle $\alpha$ is the angle which diagnolize the mass matrix of the two cp- even Higgs scalars of the two Higgs doublets $\Phi_1$ and $\Phi_2$ in the 2HD potential (2), [arXiv:1508.00702v2[hep-ph]] to get the physical states: h, H, so it's Eigenvalues problem. But now after getting mh and mH, how to get the couplings $\lambda_{1,2,3}$ in terms of them ?

You can see in [arXiv:1508.00702v2[hep-ph]] that $\lambda_{4,5}$ can be driven easily from the charged and the cp- odd Higgs masses (5) .

Any help ?

 

[malawi_glenn]

You just have to do algebra

Mathematica or Maple helps for sure