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

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

7. Solution to the RG Equation Nonperturbative solution The fixed points: is unknown is known exactly ! Non-Gaussian fixed point Gaussian fixed point IR freedom Asymptotic freedom

8. Dimensional Counting background gauge 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!

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

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

11. 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

12. 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

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

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

15. Leading Higher Order Terms

16. 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

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

18. 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.