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.