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...
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?
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 [itex]χχ \rightarrow ZZ [/itex]... 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? [itex] M= [ \bar{u}_{χ} \gamma^{\mu} u_{χ}] \frac{1}{k^{2} -m_{h}^{2} +i m_{h} \Gamma_{h}} j_{\mu}^{ZZ}[/itex] where u's are the spinors for the χ neutralinos... k is the momentum of the scalar higgs, [itex]m_{h}[/itex] its mass, [itex]\Gamma_{h}[/itex] its width and [itex]j_{\mu}^{ZZ}[/itex] the current of ZZ bosons (I don't know its form- any help?). Thanks
I think that the [itex]\gamma^{\mu}[/itex] in the [itex]\bar{u}_{\chi}\gamma^{\mu}{u}_{\chi}[/itex] 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 [itex]\frac{m_{Z}^{2}}{v}[/itex] .
I think in general the [itex]M[/itex] is the coupling of the one current with the other through the propagator. [itex]M= j_{1}^{\mu} [prop]_{\mu \nu} j_{2}^{\nu}[/itex] No? A current then is supposed to have an index.
Well I tried to think of someway to do it, can someone check the amplitude please? it's for: [itex] χχ \rightarrow h \rightarrow W^{+} W^{-} [/itex] Can someone help me with how I can use the Feynman rules I've found? For the coupling of [itex]χχh[/itex] I have: [itex] -ig_{2} (c_{L} P_{L} + c_{R} P_{R} ) [/itex] So for this it's better to work with the left and right neutralinos separately and then add the amplitudes ([itex]M= M(χ_{L}χ_{L} \rightarrow W^{+}W^{-}) +M(χ_{R}χ_{R} \rightarrow W^{+}W^{-}) [/itex] ) For the [itex]h W^{\pm}[/itex] vertex I found: [itex] ig_{2} m_{W} n^{\mu \nu} \cos(\beta-\alpha) [/itex] And the propagator is as given: [itex]\frac{i}{k^{2}-m_{h}^{2} + i m_{h} \Gamma_{h}} [/itex] What am I missing to get the [itex]M[/itex] is how to represent the outgoing particles.... is it fine to write for the fermionic neutralinos the [itex] \bar{u}_{χ} \gamma^{\mu} u_{χ'}[/itex] ? I am not sure... in any case it's like: [itex] 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}[/itex] is that right?
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.
do the self couplings between gauge bosons change from SM to SUSY? eg the coupling of [itex]Z^0 _{\lambda} (q), W^+_{\mu}(k_+), W^-_{\nu}(k_-)[/itex] is it still [itex] i g \cos(\theta_{w}) [g^{\mu \nu} (k_{-}-k_{+})^{\lambda}+ g^{\nu \lambda} (-q-k_-)^{\mu} + g^{\mu \lambda} (q+k_+)^{\nu}][/itex] as given in Peskin Fig 21.9, or is it changed?