Renormalization Group Treatment of Non-renormalizable Interactions

1. Renormalization Group Treatment of Non-renormalizable Interactions Dmitri Kazakov in collaboration with G.Vartanov JINR / ITEP Questions: • Can one treat non-renormalizable interactions • in a consistent way at least at low energies? • Is it possible to apply RG to sum up the leading • terms of perturbative expansion? • Power law running: is it reliable ? Approaches: • Renormalization Operation in local QFT • Explicit calculation of Feynman diagrams • QFT Renormalization Group • - Analytical continuation from critical dimension • - Pole equations and True RG

2. Non-Renormalizable Interactions What is the problem ? Uncontrollable UV behaviour Operator of higher dimension Operator of dimension > D Does not repeat L Example One loop • Each loop creates new higher dim operators • The number of structures is infinite • They are all relevant in loop expansion

3. R-operation for Renormalizable Interactions Lagrangian Repeates L Coupling Dim Reg This procedure removes all UV divergences

4. QFT RG for Renormalizable Interactions does not depend on the scale Pole Eqs Simple pole PT Leadingorder This leads to summation of the leading logs

5. Pole Equations in Arbitrary QFT (True RG) Local operators Lagrangian Pole Eqs Leading term One-loop only

6. Explicit form of Counter Terms (Interactions without derivatives) Local ! D=4 D>4 Conjecture ! Local ! One should take local part of this expression and cut the rest

7. Example: QFT in D=6 Non-renormalizable intearfction One-loop counter term Only two terms survive These counter terms do not repeat the original Lagrangian but are the higher order operators

8. Higher Order Terms General form Explicit form Variational derivative Contains local part

9. Evaluation of Variational Derivatives 3 + 3 + 4 + +

10. Leading Higher Order Terms

11. Summation of Leading Terms Counter terms in momentum space 4-point function 6-point fun 4-point function 6-point fun 8-point fun 4-point function

12. Four-point Green Function Leading Divergences ?! Checked: Explicitly – 4 loops RG recursion – 5 loops 4-point Green Function (sym point)

13. Conclusions Questions: • Can one treat non-renormalizable interactions • in a consistent way at least at low energies? • Is it possible to apply RG to sum up the leading • terms of perturbative expansion? • Power law running: is it reliable ? Answers: • Absorb UV divergences into the renormalization • of higher dim operators (infinite #) and ignore these • operators at low energies • Sum up the leading terms (contain no arbitrariness !!) • using one-loop conjecture and RG recursion • Summation results do not resemble those for renormalizable • interactions with logs replaced by power law.