MSSM Higgs Potential Homework: Get Equ. (1.70)

[Safinaz]

Homework Statement

Hi,

I study the Higss sector of the MSSM from this review " arxiv:0503173v2", "The Higgs bosons in the Minimal Supersymmetric Model",

In Sec.: 1.2, it gives the Higgs potential by Equ. (1.60), then after acquiring the vevs and minimizing the potential to get the masses of the Higgs bosons , it yields two minimization conditions (1.70)

Homework Equations

I can not get Equ. (1.70)

The Attempt at a Solution

First I wrote the potential of the neutral components of the Higgs doublets: ## H_1,~ H_2##, as following:

$$ V_{H^0} = \bar{m}^2_1 |H^0_1|^2 + \bar{m}^2_2 |H^0_2|^2 + 2 B \mu H^0_1 H^0_2 + \frac{g_1^2+ g_2^2}{8} ( |H^0_1|^2 - |H^0_2|^2)^2.$$

Then minimized the potential
$$ \frac{\partial v }{\partial H_1^0} = \bar{m}^2_1 H^0_1 + 2 B \mu H^0_2 + \frac{g_1^2+ g_2^2}{4} ( |H^0_1|^2 - |H^0_2|^2) H^0_1=0, $$
$$ \frac{\partial v }{\partial H_2^0} = \bar{m}^2_2 H^0_2 + 2 B \mu H^0_1 -\frac{g_1^2+ g_2^2}{4} ( |H^0_1|^2 - |H^0_2|^2) H^0_2=0, $$

taking the minima Equ. (1.67), and using Equ. (1.68) and (1.61) in the reference, I got,
$$ ( \mu^2 +m_{H_1}^2) v_1 + 2 B \mu v_2 + \frac{M_z^2}{v^2} ( v_1^2 - v_2^2) v_1=0, $$
$$ ( \mu^2 +m_{H_2}^2) v_2 + 2 B \mu v_1 - \frac{M_z^2}{v^2} ( v_1^2 - v_2^2) v_2=0, $$
But then what I do to reach (1.70) ?

Thanks

 

[fzero]

Obviously you can multiply the first equation by ##v_2## and subtract the 2nd multiplied by ##v_1## to solve for ##B\mu##. The opposite difference can be solved for ##\mu^2##. Then we have to eliminate ##v,v_1,v_2## in favor of expressions involving ##\beta##. It might be convenient to use (1.68) and (1.69) to express ##v_2 = v_1 \tan\beta##, ##v^2=v_1^2 (1+\tan^2\beta)##. Then the ##v_1## dependence cancels in the expressions for ##B\mu## ##\mu^2##, which are functions of ##\beta## and the masses.

 

[Safinaz]

Thanks, I got it..

 

[Safinaz]

Hi,

I preferred to continue in the same thread because I have a question in the same section of the referred reference:

Homework Equations

I can't get the mass matrices of the cp even Higgs scalars nor the cp odd, as Equ. (1.75) and (1.76)

The Attempt at a Solution

[/B]
I got the first element in the mass matrix of the cp even Higgs by:

## \frac{\partial V_H}{\partial H_1^0 \partial H_1^0} = ( \mu^2 + m_{H_1}^2 ) + \frac{3}{2} M_z^2 \cos^2 \beta ## ,

Now to get (1.75 ) matrix , at which this first term contains ##\bar{m_3}^2 ## or ## B\mu ## - Equ. (1.61)- , I tried to use the constrains (1.70 ), but did not reach it..

The mixed terms like ## \frac{\partial V_H}{\partial H_1^0 \partial H_2^0} ## are fine with me..

Bests.

 

[fzero]

Hi,

I got the first element in the mass matrix of the cp even Higgs by:

## \frac{\partial V_H}{\partial H_1^0 \partial H_1^0} = ( \mu^2 + m_{H_1}^2 ) + \frac{3}{2} M_z^2 \cos^2 \beta ## ,

Now to get (1.75 ) matrix , at which this first term contains ##\bar{m_3}^2 ## or ## B\mu ## - Equ. (1.61)- , I tried to use the constrains (1.70 ), but did not reach it..

The mixed terms like ## \frac{\partial V_H}{\partial H_1^0 \partial H_2^0} ## are fine with me..

You should be able to use the equations that you have in post 1 for ##\mu^2 + m_{H_{1,2}}^2## to write that in terms of ##B\mu## and ##M_Z## alone.