On page 388 in section 11.6 of Peskin and Shroeder.
There appears an equation of the inverse propagator(the second functional derivative of the effective action) for a theory that contains several scalar fields: K2ij=:∫d4xeip⋅(x−y)δ2Γδϕiδϕj(x,y)=0
When diagonalizing
K2ij=PikPjl˜K2kl=(P˜KPt)ij,P:an orthogonal matrix
the property that K2ij: real is needed. After diagonalizing ˜K2ii=p2−m2i0−M2i(p2)i : no sum,
where mi0 is the bare mass of the ith scalar field and M2i(p2) is the sum of one-particle-irreducivle two-point diagrams.
Is M2i(p2) always a real number?
That's simply because
˜K2ii=0⇔m2i=m2i0−M2i(m2i)?
,where mi is the physical mass.
Is that whole the story? Is there some other reason?
Thanks.
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