Calculating Cross Section: Is (1-\gamma^5) Same as (1-\gamma_5)?

[indigojoker]

Is there a difference between $$(1-\gamma^5)$$ and $$(1-\gamma_5)$$ ? I see the two used interchangeably when calculating cross section.

 

[Norman]

Try using:

$$\gamma^5 \equiv i \gamma^0 \gamma^1 \gamma^2 \gamma^3$$
and
$$\gamma_{\mu} = \eta_{\mu \nu} \gamma^{\nu}$$

where $\eta_{\mu \nu}$ is the Minkowski metric.

 

[indigojoker]

Well, Perkins 3rd edition page 383 gives the amplitude using $$\gamma_5$$ while Halzen and Martin calculates the amplitude using $$\gamma^5$$ on equation 12.56
I'm not sure why they could be interchanged.

 

[humanino]

Well, Perkins 3rd edition page 383 gives the amplitude using $$\gamma_5$$ while Halzen and Martin calculates the amplitude using $$\gamma^5$$ on equation 12.56
I'm not sure why they could be interchanged.

You will not be able to find all answers to all questions in books. Try to do the calculation by yourself as indicated earlier, it is much more rewarding.

 

[Vanadium 50]

The "try the calculation" advice is good. You will see that every time that a covariant index occurs, a contravariant index also occurs, so when you contract them you get a scalar. Exactly which indices go up and which go down is a matter of convention.

Picking pieces out of different books - which may use different conventions - is a recipe for making errors.

 

[nrqed]

Is there a difference between $$(1-\gamma^5)$$ and $$(1-\gamma_5)$$ ? I see the two used interchangeably when calculating cross section.

I have no idea what the other posters have in mind...

As far as I know, $$\gamma^5$$ and $$\gamma_5$$ are exactly the same thing. The 5 here is not a Lorentz index so there is no meaning to having it upstairs or downstairs.

for example, nachtmann (Elementary particle physics) defines

$$\gamma_5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3$$

Peskin defines $$\gamma^5$$ exactly the same way.

(But Donoghue et al have a minus sign in the definition)



An important point is that one may write gamma_5 as

$$\gamma_5 = \frac{i}{4!} ~\epsilon_{\mu \nu \rho \sigma} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma$$
which shows clearly that gamma_5 is a scalar. (well, a pseudoscalar to be more precise since it reverses sign under a reflection in space).

 

[Mr.Slava]

An important point is that one may write gamma_5 as

$$\gamma_5 = \frac{i}{4!} ~\epsilon_{\mu \nu \rho \sigma} \gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma$$

It is really important definition if you use dimensional regularization (dimensionality of space-time is $$d$$) where

$$\eta_{\mu \nu} \gamma^{ \mu } \gamma^{ \nu } = d$$