How to Determine the Coupling Strength for Dirac Neutrinos with the Z Boson?

[majorana2015]

Hi All

1. Homework Statement
with the known coupling strenght between SM neutrinos and Z boson

$$Z_{\mu} \bar{\nu}_{L} { \gamma}^{ \mu} \nu_{L}$$

how can I get the coupling strength for Dirac neutrinos
$$Z_{\mu} \bar{\nu}_{Dirac} { \gamma}^{ \mu} (1- { \gamma}_{ 5} ) \nu_{Dirac}$$ ?

thanks

 

[Vanadium 50]

Just clipped off the part of the template where you show your attempt, did you?

Please show your attempt.

 

[majorana2015]

Yes I did, mainly because I have no idea how to start. Pointless to keep that part .

 

[DrClaude]

Sorry, but you will have to provide an attempt if you want help.

 

[majorana2015]

we have $$Z_{\mu} \bar{\nu}_{L} { \gamma}^{ \mu} \nu_{L}$$
given that
$${\nu}_{L}= \frac {(1- { \gamma}_{ 5} )}{2} \nu_{Dirac}$$
and
$$\bar{\nu}_{L}= { \bar {\nu}_{Dirac}} \frac {(1- { \gamma}_{ 5} )}{2}$$ then

$$Z_{\mu} \bar{\nu}_{L} { \gamma}^{ \mu} \nu_{L} = Z_{\mu} { \bar {\nu}_{Dirac}} \frac {(1- { \gamma}_{ 5} )}{2} { \gamma}^{ \mu} \frac {(1- { \gamma}_{ 5} )}{2} \nu_{Dirac}$$babysteps... but is it at least kind of right?

 

[Vanadium 50]

Can you simplify that?

 

[majorana2015]

what could I use ?

 

[mfb]

What do you know about products of gamma matrices?
This is the part where "relevant equations" from the homework template are important.

 

[majorana2015]

thanks!. Do you mean?
$$\frac{1}{4} (1- { \gamma}_{ 5} ) { \gamma}^{ \mu} (1- { \gamma}_{ 5} ) = \frac{1}{4} ( { \gamma}^{ \mu} (1- { \gamma}_{ 5})^{2}$$ can I just do that with 1 − γ_5 term? so I'dhave $$(1- { \gamma}_{ 5})^{2} = (1- { \gamma}_{ 5})$$ and then $$\frac{1}{4} <br /> Z_{\mu} { \bar {\nu}_{Dirac}}<br /> <br /> { \gamma}^{ \mu} (1- { \gamma}_{ 5} ) <br /> \nu_{Dirac}$$

 

[mfb]

can I just do that with 1 − γ_5 term?

Can you?
Can you check the individual steps?

 

[majorana2015]

is this a joke? i come here for help and all I'm getting are just simpleton questions

 

[mfb]

It is not a joke. Your question can be answered with a series of simple steps, and you should be able to do all those steps (or look them up). You try to do too many steps at the same time, which does not lead to the right result, so I encourage you to make smaller steps, and to show them, so we can see what goes wrong.

 

[Orodruin]

is this a joke? i come here for help and all I'm getting are just simpleton questions

Based on the work you have shown, you are making several trivial errors which mess up your computation, which is why you are being asked to check your individual steps. You will not get the answer handed to you here, this is not how PF works. Instead, you can get guidance and help to figure things out on your own, which is generally much better from a learning perspective. You can either chose to scream about this, which will get you nowhere, or accept the help and guidance you are given. One of the two will benefit you in te long run.