How can we expand product of two currents into series?

[ndung200790]

Please teach me this:
In QFT book of Peskin&Schroeder write:
J$_{\mu}$(x)J$_{\nu}$(0)~C$_{\mu\nu}$$^{1}$(x).1+C$_{\mu\nu}$$^{q^{-}q}$q$^{-}$q(0)+C$_{\mu\nu}$$^{F^{2}}$(x)(F$_{\alpha\beta}$$^{a}$)$^{2}$(0)...
I do not know how to expand the product into this series.
Thank you very much for your kind helping.

 

[TriTertButoxy]

Oh dear... I would love to know the answer to this one.

The best answer I have been given to date is: think of all the operators that transforms in the same way as the product of two currents, and those will be the ones appearing in the expansion.

It would be so awesome if someone could give a constructive algorithm for generating the series expansion for the product of two currents (or for that matter, for the non-local product of any two local operators).

 

[ndung200790]

Given the series,they say that the coefficients in the series can be calculate by Feynman diagrams.Then I do not know how to give the Feynman diagrams for the coefficients.

 

[ndung200790]

Why is the coefficient of operator 1 being the sum of diagrams with no external legs other than the current insertions?

 

[ndung200790]

It seems that there is ''some arbitrary'' in OPE.We can ''adjust'' the coefficients in the series according with ''situation''.Is that correct?

 

[ndung200790]

It seem to me that we can expand product of operators by Wick theorem and Taylor series?