1. 1 Renormalization Group Treatment of Non-renormalizable Interactions Dmitri Kazakov 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 in collaboration with G.Vartanov

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

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

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

5. 5 Wilsonian RG near Critical Dimension Pure gauge theory Dimensions D=4 critical dimension Renormalization (ignoring all higher order operators ) Background gauge The gauge field anomalous dimension

6. 6 Solution to the RG Equation Perturbative solution 1 loop Power law running Log running When applied to GUTs: 1)the unification point does not depend on ε 2)can provide unification at any scale ?!

7. 7 Power law Running Couplings Susy ThresholdExtra Dim Threshold Dienes, Dudas, Ghergetta However, This picture ignores all higher dimensional operators which are relevant in the UV The correct meaning: IR running ONLY!!!!

8. 8 Solution to the RG Equation Nonperturbative solutionThe fixed points: Asymptotic freedomIR freedom is unknown is known exactly ! Gaussian fixed point Non-Gaussian fixed point

9. 9 Dimensional Counting At the fixed point the coupling is dimensionless in any D ! This statement does not depend on the gauge The theory at the fixed point is perturbatively nonrenormalizable, but nonperturbatively renormalizable! background gauge

10. 10 An Effective Theory at the Fixed Point D=6 No scale, the coupling is dimesionless Scale (conformal) invariance The exact anomalous dimensions are included Vanish on shell ( ) Can one guarantee that this fixed point exist? Is it perturbatively reachable? Can one calculate anything at the FP? Properties Questions

11. 11 Pole Equations in Arbitrary QFT (True RG) Lagrangian Local operators Pole Eqs Leading term One-loop only

12. 12 Explicit form of Counter Terms D=4 D>4 Conjecture ! Local ! One should take local part of this expression and cut the rest (Interactions without derivatives)

13. 13 Example: QFT in D=6 Only two terms survive These counter terms do not repeat the original Lagrangian but are the higher order operators Non-renormalizable intearfction One-loop counter term

14. 14 Higher Order Terms General form Explicit form Contains local part Variational derivative

15. 15 Evaluation of Variational Derivatives 3 + 3+ 4++

16. 16 Leading Higher Order Terms

17. 17 Summation of Leading Terms Counter terms in momentum space 4-point function6-point fun 4-point function6-point fun8-point fun 4-point function

18. 18 Four-point Green Function ?! Leading Divergences 4-point Green Function (sym point) Checked: Explicitly – 4 loops RG recursion – 5 loops

19. 19 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. Conclusions