Wednesday, September 20, 2017

Complete renormalization in phi4-theory?


In the one-loop renormalization of ϕ4-theory, only 1PI vertex functions Γ(2) and Γ(4) are regularized and renormalized. But they do not exhaust all the irreducible connected diagrams at one loop. One can have a diagram, for example, with one-loop, 3 vertices and 6 external lines, or with one-loop, 4 vertices and 8 external lines and so on. What about these diagrams? They respectively correspond to Γ(6) and Γ(8). What about these 1PI diagrams with one-loop? Shouldn't they require renormalization as well? In fact these diagrams contribute to the effective potential.


EDIT : arxiv.org/abs/hep-ph/9901312 This might be an useful reference. Please look at the one-loop diagrams in the calculation of the effective potential in ϕ4-theory.



Answer



The naive power counting approach for a d-dimensional theory with coupling constant λ tells us that the amplitude of diagrams with E external lines and V vertices behaves with the cutoff Λ as ΛD with D=d[λ]Vd22E where [˙] is the mass dimension. Since ϕ4 in four dimension has a dimensionless coupling, D=4E and since only diagrams with D0 need renormalization, the only diagrams needing it in 4D ϕ4 are those with E4. All diagrams with an odd number of external lines vanish due to the ϕϕ symmetry, so what's left to renormalize is E=0,2,4, which are the vacuum energy, the propagator, and the 4-vertex, respectively.


The diagrams you ask about exist, but have D<0, and do not need to be renormalized, since they are not diverging when we take the cutoff to infinity.


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