# Neutralino Interactions

1. Jul 30, 2014

### ChrisVer

Do you have any source where I can check for the neutralino (higgsino or chargino/bino -like) interaction processes?
In general I'm trying to find the amplitudes in the Appendix A of:
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.47.376
But without seeing a Lagrangian, I can't understand the possible contributing channels I think...

Last edited: Jul 30, 2014
2. Aug 4, 2014

### Staff: Admin

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

3. Aug 5, 2014

### ChrisVer

Yes I am able to clear a lot of difficulties I've been having with it. But it's still a little bit complicated.
For example one can have the figures I attached for $χχ \rightarrow ZZ$... which are 6 in number (n=1,2,3,4 for the neutralinos), and H,h are the two neutral scalar higgs bosons.
So in general I can write the amplitude for each one, right?
However I am not sure how is this kind of amplitudes written... Could someone check if the formula I'm using is correct for the small higgs?

$M= [ \bar{u}_{χ} \gamma^{\mu} u_{χ}] \frac{1}{k^{2} -m_{h}^{2} +i m_{h} \Gamma_{h}} j_{\mu}^{ZZ}$

where u's are the spinors for the χ neutralinos... k is the momentum of the scalar higgs, $m_{h}$ its mass, $\Gamma_{h}$ its width and $j_{\mu}^{ZZ}$ the current of ZZ bosons (I don't know its form- any help?).
Thanks

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4. Aug 6, 2014

### ofirg

I think that the $\gamma^{\mu}$ in the $\bar{u}_{\chi}\gamma^{\mu}{u}_{\chi}$ should be removed since the neutralinos couple to a scalar, not a vector. For the coupling of the higgs to the Z see feynman rules references, peskin for example. I think it it something like $\frac{m_{Z}^{2}}{v}$ .

5. Aug 6, 2014

### ChrisVer

I think in general the $M$ is the coupling of the one current with the other through the propagator.
$M= j_{1}^{\mu} [prop]_{\mu \nu} j_{2}^{\nu}$
No?
A current then is supposed to have an index.

6. Aug 7, 2014

### ChrisVer

Well I tried to think of someway to do it, can someone check the amplitude please?
it's for: $χχ \rightarrow h \rightarrow W^{+} W^{-}$
Can someone help me with how I can use the Feynman rules I've found?
For the coupling of $χχh$ I have:
$-ig_{2} (c_{L} P_{L} + c_{R} P_{R} )$
So for this it's better to work with the left and right neutralinos separately and then add the amplitudes ($M= M(χ_{L}χ_{L} \rightarrow W^{+}W^{-}) +M(χ_{R}χ_{R} \rightarrow W^{+}W^{-})$ )

For the $h W^{\pm}$ vertex I found:
$ig_{2} m_{W} n^{\mu \nu} \cos(\beta-\alpha)$

And the propagator is as given:
$\frac{i}{k^{2}-m_{h}^{2} + i m_{h} \Gamma_{h}}$

What am I missing to get the $M$ is how to represent the outgoing particles....
is it fine to write for the fermionic neutralinos the $\bar{u}_{χ} \gamma^{\mu} u_{χ'}$ ?
I am not sure...
in any case it's like:

$i M(χ_{j}χ_{j} \rightarrow W^{+}W^{-})= (-ig_{2} c_{j}) \frac{i }{k^{2}-m_{h}^{2} + i m_{h} \Gamma_{h}} ( ig_{2} m_{W} \cos(\beta-\alpha)) n^{\mu \nu} \epsilon_{\mu} \epsilon^{*}_{\nu}$

is that right?

Last edited: Aug 7, 2014
7. Aug 8, 2014

### Orodruin

Staff Emeritus
The vertex for the $\chi\chi h$ interaction is what should be within the fermion bilinear. Basically you should have (given your Feynman rules)
$$-ig_{2} \bar u_\chi (c_L P_L + c_R P_R) u_{\chi'} \frac{i }{k^{2}-m_{h}^{2} + i m_{h} \Gamma_{h}} ( ig_{2} m_{W} \cos(\beta-\alpha)) n^{\mu \nu} \epsilon_{\mu} \epsilon^{*}_{\nu}.$$
Since the higgs is a scalar, it cannot interact with the vector current of the form $\bar u \gamma^\mu u$. There is simply no way to contract the free Lorentz index.

8. Aug 8, 2014

### ChrisVer

Aha... so it's more like I'm having a RL and LR helicities.

9. Aug 9, 2014

### ChrisVer

do the self couplings between gauge bosons change from SM to SUSY?
eg the coupling of $Z^0 _{\lambda} (q), W^+_{\mu}(k_+), W^-_{\nu}(k_-)$ is it still
$i g \cos(\theta_{w}) [g^{\mu \nu} (k_{-}-k_{+})^{\lambda}+ g^{\nu \lambda} (-q-k_-)^{\mu} + g^{\mu \lambda} (q+k_+)^{\nu}]$
as given in Peskin Fig 21.9, or is it changed?

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