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D.I.Kazakov and G.S.Vartanov based on hep-th/0607177

Renormalizable Expansion for Nonrenormalizable Theories. D.I.Kazakov and G.S.Vartanov based on hep-th/0607177. Museo Storico della Fisica e Centro Studi e Ricerche "Enrico Fermi", Rome, Italy Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia

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D.I.Kazakov and G.S.Vartanov based on hep-th/0607177

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  1. Renormalizable Expansion for Nonrenormalizable Theories D.I.Kazakov and G.S.Vartanov based on hep-th/0607177 Museo Storico della Fisica e Centro Studi e Ricerche "Enrico Fermi", Rome, Italy Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia e-mail: Vartanov@theor.jinr.ru

  2. 1/N expansion Let us introduce the Lagrange multiplier At the tree level the σ propagator : “i”; after corrections - ? Vartanov G S

  3. Resumming of the σ propagator when D is odd no divergences where Vartanov G S

  4. Degree of divergences • φ propagator: w(G)=L*D-(2L-1)*2-L(D-4)=2 -> gives us the logarithmic divergence! in any D • φ2σ vertex: w(G)= L*D-2L*2-L(D-4)=0 -> again logarithmic divergence! in any D # of loops # of φ fields # of σ fields Vartanov G S

  5. Degree of divergences σ propagator: w(G)=D-4 -> gives us no global divergence, after subtracting divergent subgraphs we don’t have any divergences No other divergences!!! Vartanov G S

  6. Leading order of 1/N expansion Vartanov G S

  7. Effective Lagrangian • No coupling -> we introduce dimensionless coupling h associated with the triple vertex h Vartanov G S

  8. Renormalization group for h Solution of the RG equation for the small coupling will be What will give us in the leading order the usual leading logarithmic behavior of the effective coupling constant with IR or UV asymptotic behavior depending on the dimension D. Vartanov G S

  9. Checking the renormalization group equations (example of the second order contribution to the φ propagator ) Vartanov G S

  10. Checking the renormalization group equations (example of the second order contribution to the φ propagator ) Vartanov G S

  11. Conclusions • In nonrenormalizable theory we constructed renormalizable 1/N expansion • The parameter expansion is dimensionless and the coupling constant is running logarithmically • Properties of the 1/N expansion doesn’t depend on the space-time dimension if it is odd. Vartanov G S

  12. Acknowledgements • I want to thank the organizers of the ISSP school for the opportunity to give here a talk • I want to thank “Enrico Fermi Center” Vartanov G S

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