A Spinor product in Peskin-Schroeder problem 5.3

1. Mar 30, 2018

Manu_

Hello,

I am currently stuck on problem 5.3 (c) about spinor products in PS, where one needs to prove the Fierz identity:
$$\bar{u}_{L}(p_{1}) \gamma^{\mu} {u}_{L}(p_{2}) [\gamma_{\mu}]_{ab} = 2 [u_{L}(p_{2})\bar{u}_{L}(p_1) +u_{R}(p_{1})\bar{u}_{R}(p_2) ]_{ab}$$
They say that a Dirac matric M satisfies:
$$\gamma^{5} [M]=-[M]\gamma^{5}$$
hence should be of the form:
$$[M]= \left( \frac{1-\gamma^{5}}{2} \right) \gamma_{\mu} V^{\mu} + \left( \frac{1+\gamma^{5}}{2} \right) \gamma_{\mu} W^{\mu}$$

But then, to get the answer, I suppose that:
$$V^{\mu} = u_{L}(p_{2})\bar{u}_{L}(p_1)$$
$$W^{\mu} = u_{R}(p_{1})\bar{u}_{R}(p_2)$$
Honestly, I don't see exactly why. Can someone point me out the way to make this identification?

Next, in part (d), we should get an amplitude of the form:
$$i\mathcal{M} = (-ie)^{2} \bar{v}_{R} (k_{2}) \gamma^{\mu} u_{R} (k_{1}) \frac{-i}{s} \bar{u}_{R}(p_{1})\gamma_{\nu} v_{R}(p_2)$$
Thus, we have terms in u and v. However, all the spinor product formalism has been developed in terms of u. My question is: can one define a spinor product $$s(p_1,p_2)=\bar{v}_{R}(p_1) u_{L}(p_2)$$?

Thanks,
Emmanuel

2. Mar 30, 2018

MathematicalPhysicist

All these messy calculations involving Fierz transformations identities can be found in the books by Atkinson's and Radovanovic's.

3. Mar 31, 2018

Manu_

Thanks for the references, MathematicalPhysicist.
But I took a look on these books, and all I saw was the general Fierz identities, and these are already treated in PS.
However, I still don't see where does this result come from. I'm sure it's something basic, but I can't see it...

4. Apr 7, 2018

Manu_

Does anyone else have a suggestion?
Thanks!

5. Apr 8, 2018

6. Apr 9, 2018

Manu_

Hi!
Yes, I have seen this manual, but unfortunately, he states that this is all obvious...

7. Apr 16, 2018 at 3:24 PM

MathematicalPhysicist

Hi @Manu_ , have you tried the exercises textbook of Atkinson's?

8. Apr 19, 2018 at 12:19 PM

Manu_

Hi!
I took a look in this book and Radovanovic's, but they cover more or less what has been treated in P&S before problem 5.3. They do not mention Fierz identities used with spinor products.

9. Apr 19, 2018 at 1:32 PM