Question about the matrix of a pseudoscalar meson

[Mason20]

Hi everyone,
I have a question about the $q_{1}\bar{q_{2}}$ to vacuum :
$$ \langle 0 |\bar{q_{2}}\gamma_{\mu}\gamma_{5}q_{1}| q_{1}\bar{q_{2}}\rangle$$
That is the first time I try to solve the question like this.
How do we calculate the matrix about this question ?
Thank you so much!

 

[AndyW]

Hi there,

To calculate the matrix, we can use the Feynman diagram approach. First, we can represent the initial state of the system as a quark-antiquark pair, with one quark (q1) and one antiquark (q2). Then, we can use the Feynman rules to draw the diagram for the process described in the equation.

The Feynman rules tell us that the quark and antiquark in the initial state can interact by exchanging a virtual gluon, represented by a wavy line. The interaction is described by the gamma matrices, with the gamma mu representing the gluon's polarization and the gamma 5 representing the chirality of the quarks.

Once we have the diagram, we can use the Feynman rules to calculate the amplitude for the process. This involves multiplying the appropriate propagators for the quark and gluon, as well as the gamma matrices, and integrating over all possible momenta for the virtual particles.

Finally, the matrix element can be obtained by squaring the amplitude and summing over all possible final states. This will give us the probability for the initial state to transition into the final state, which in this case is the vacuum state.

I hope this helps! Let me know if you have any further questions.