Tuesday, 14 April 2015

quantum field theory - Formula for Symmetry Factor


In ϕ3 theory, are there any formula for determining the Symmetry factor as that is found for the ϕ4 theory in any standard book of Quantum Field Theory?



Answer



Peskin and Schroeder provide a sufficiently detailed explanation regarding the computation of symmetry factors for Feynman diagrams. The paper by Palmer et al. present a general formula,


S=1R(12)D1(12!)D2(13!)D3(14!)D4


where the constants are defined in their paper, which require an understanding of the derivation and its notation. The expression is applicable to QED, QCD and ϕ3,4 theory but generalizable to others. For a diagram such as (considered in their paper as figure 1),



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For this case, D1=D3=D4=0, R=1 and D2 = 1 which yield, S=1/2 as expected. In Dong's paper for real and complex scalar field theories, he presents the general formula,


S=g2β2dn(n!)αn


where (quoting from the paper): g is the number of of interchanges of vertices leaving the diagram topologically unchanged, β is the number of lines connecting a vertex to itself, d is the number of double bubbles, and αn is the number of vertex pairs connected by n identical lines.


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