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## XII. APPENDIX A: THE BETA FUNCTION 

Consider a single free boson with action eq. (2). Setting the external energy and momentum to zero, the 4-point vertex at tree-level plus 1-loop is 

<img src='content_image/337760.jpg'>

The ω integral can be done by closing the contour in the upper half plane picking up the pole at ω = − k $^{2}$/ 2 m + iϵ : 

Introducing an ultraviolet cut-off Λ: 

<img src='content_image/337761.jpg'>

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