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I was confused as to how the equation 10 on page 7 or equation 21 on page 8 of this paper http://arxiv.org/abs/1211.1866 was derived. Can someone explain from where does this come and what do the "weird" arrows in Figure 1 on page 7 mean (..it doesn't look like a normal Feynman diagram..)

Especially how does this follow from the Schwinger-Dyson equation?

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It is supposed to be normal Feynman diagrams! They just use some additional vertices... The fermion propagators have arrows, to indicate the sign of charge and momentum, it's always the case. On the first diagram, they also indicate the extra 4-momentum carried by the current. Please take it as a shortened external line. But otherwise it's supposed to follow Feynman rules. –  Luboš Motl Dec 20 '12 at 11:30
    
@Lubos Motl Can you give a reference from where one can learn how to calculate such multi-loop 2 and 3 point functions of classically conserved currents? Only thing like this i have ever seen is calculating the free Fermion current correlators to see Bjorken scaling. And there it was just a two point function which was calculated by using Wick's theorem to reduce it to sum of products of free Fermion 2-point functions. (..that i guess would be called a tree level calculation..) I guess there are more sophisticated techniques for doing more loops. –  user6818 Jan 1 '13 at 5:58

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