# SM neutrinos coupling to Z boson

1. Dec 16, 2015

### majorana2015

Hi All

1. The problem statement, all variables and given/known data

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

2. Dec 16, 2015

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

3. Dec 17, 2015

### majorana2015

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

4. Dec 17, 2015

### Staff: Mentor

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

5. Dec 17, 2015

### 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?

6. Dec 17, 2015

Staff Emeritus
Can you simplify that?

7. Dec 17, 2015

### majorana2015

what could I use ?

8. Dec 17, 2015

### Staff: Mentor

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

9. Dec 17, 2015

### 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

$$\frac{1}{4} Z_{\mu} { \bar {\nu}_{Dirac}} { \gamma}^{ \mu} (1- { \gamma}_{ 5} ) \nu_{Dirac}$$

10. Dec 18, 2015

### Staff: Mentor

Can you?
Can you check the individual steps?

11. Dec 19, 2015

### majorana2015

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

12. Dec 19, 2015

### Staff: Mentor

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.

13. Dec 19, 2015

### Orodruin

Staff Emeritus
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.