1. Renormalizable Expansion for Nonrenormalizable Theories
D.I.Kazakov and G.S.Vartanov
based on hep-th/
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

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
# of loops # of φ fields # of σ fields
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
Vartanov G S

5. Degree of divergences No other 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
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7. Effective Lagrangian
No coupling -> we introduce dimensionless coupling h associated with the triple vertex h
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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 )
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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