Anti-commutation of Dirac Spinor and Gamma-5

[Dewgale]

Homework Statement

Given an interaction Lagrangian $$ \mathcal{L}_{int} = \lambda \phi \bar{\psi} \gamma^5 \psi,$$ where $\psi$ are Dirac spinors, and $\phi$ is a bosonic pseudoscalar, I've been asked to find the second order scattering amplitude for $\psi\psi \to \psi\psi$ scattering. I've been able to get extremely close, but my main sticking point is whether I can anti-commute $\gamma^5$ and $\bar{\psi}$. Details are below.

Homework Equations

$$S = T( e^{-i \int_{-\infty}^\infty dt H})$$

The Attempt at a Solution

The Hamiltonian density for this Lagrangian density is $$\mathcal{H} = - \mathcal{L} = - \lambda \phi \bar{\psi} \gamma^5 \psi.$$

We therefore have $$ \left< f \right| S \left| i \right> = \left< f \right| T( e^{i \lambda \int d^4x \phi \bar{\psi} \gamma^5 \psi}) \left| i \right>.$$

At second order, this gives
$$\left< f \right| S \left| i \right> \sim \left< f \right|T(\int d^4x \int d^4y \phi(x) \bar{\psi}(x) \gamma^5 \psi(x) \phi(y) \bar{\psi}(y) \gamma^5 \psi(y)) \left| i \right>$$

Let's identify the momenta of the inbound states as $p,q$, and the outbound states as $p',q'$. Following Peskin and Schroeder's example for the Yukawa interaction, we can contract the two $\phi$s to get their Feynman propagator. This leaves us with two cases: where $\bar{\psi}(x)$ and $\psi(x)$ are contracted to the "same" momentum (i.e. q' and q, respectively), and when they are contracted to "opposite" momenta (i.e. p' and q, respectively). These correspond the to t-channel and the u-channel, respectively.

I'm not sure how to write Wick contractions on Physics Forums version of latex, but if we don't write the contractions we have

$$ \left< 0 \right| \hat{a}_{q'} \hat{a}_{p'}\, \bar{\psi}(x) \gamma^5 \psi(x)\, \bar{\psi}(y) \gamma^5 \psi(y) \,\hat{a}_p^\dagger \hat{a}_q^\dagger \left| 0 \right>$$

Recall that the two cases are
(1) $\bar{\psi}(x)$ contracted with $\hat{a}_{q'}$ and $\psi(x)$ contracted with $\hat{a}_q^\dagger$ (the other two are appropriately contracted)
(2) $\bar{\psi}(x)$ contracted with $\hat{a}_{p'}$ and $\psi(x)$ contracted with $\hat{a}_p^\dagger$ (the other two are again appropriately contracted)

We can see that, if we assume $[\gamma^5, \bar{\psi}]=0$ (which isn't true), then the first situation requires two interchanges to "untangle" the contractions, while the second case only requires one. This is the case of the Yukawa interaction, and as each interchange provides a factor of $(-1)$, it gives the appropriate statistics in that case.

In the second case, no spinors need to cross a $\gamma^5$, so it's not an issue. However, in the first case, we do need a spinor ($\bar{\psi}(y)$) to cross a $\gamma^5$.

My question, then, is if $\{\gamma^5, \bar{\psi}\} = 0$ is true? My rational is that
$$
\gamma^5 \bar{\psi} =\gamma^5 \psi^\dagger \gamma^0 \stackrel{?}{=} \psi^\dagger \gamma^5 \gamma^0 = - \psi^\dagger \gamma^0 \gamma^5 = - \bar{\psi} \gamma^5
$$
If it's not true, then I'm not sure how to figure out the statistics for this. It's pretty obvious the net effect will be to put a $\gamma^5$ between each spinor in the numerator of each term, but the relative minus sign between the t-channel and u-channel is important.

Thanks in advance for the help!

 

[king vitamin]

In these situations where you may get confused by strings of products of spinors, it is usually useful to write the indices out. First:
$$\bar{\psi}(x) \gamma^5 \psi(x)\, \bar{\psi}(y) \gamma^5 \psi(y) = \bar{\psi}_a(x) \gamma^5_{ab} \psi_b(x)\, \bar{\psi}_c(y) \gamma^5_{cd} \psi_d(y)$$
From here, you may temporarily move the $\gamma^5$ factors out of the expectation value (but keep track of their indices!). Then, order the fermionic degrees of freedom using the canonical anticommutator:
$$\{ \psi_a(x) , \bar{\psi}_b(y) \} = \delta^{d}(x - y) \gamma^0_{ab}$$
(you can move any factors of $\gamma^0$ out of the expectation value as well). Once you've moved things to their desired place for the given Wick ordering, then you can replace the expectation value with a pair of Dirac propagators, and then realign your propagators/gamma matrices by their indices to get the spinor multiplication which you've generated.