Conformal Invariance = Finiteness and Beta Deformed N=4 SYM Theory

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Conformal Invariance Finitenessand Beta
Deformed N4 SYM Theory

• D.I. Kazakov , L.V. Bork
• Bogoliubov Lab. JINR (Dubna) ITEP (Moscow)
• ITEP MEPHI (Moscow)
• arXiv0706.4245v1 hep-th

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N4 SYM Theory

• N4 SYM is magnificent playground to test
  nonperturbative features of QFT. One of them is
  AdS/CFT correspondence (Maldacena98
  Polyakov,Gubser,Klebanov98 E.Witten98).
• AdS/CFT requires from the field theory to be
  conformally invariant and not necessarily obtain
  the full N4 supersymmetry.

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Marginally deformed N4 SYM
Beta-deformation (LeighStrassler95)
A supergravity dual to this deformed theory in
AdS/CFT context was found (LuninMaldacena05).
Will Beta-deformed N4 SYM remain conformal
invariant on quantum level ( Will loop
corrections preserve conformal sym. ) ?
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Conformal invariance of Beta-deformed N4 SYM
One can reparametrise couplings
and try to find manifold of fixed points
order by order in P.T.(Kazakov,Tarasov,Ermushev85
Jones86 Oehme86).
One can write (Novikov,Zakharov,ShifmanVainshtain
83)
where
So one can see that
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Conformal invariancefiniteness 1-loop
Choosing N1 superspace formalism and background
field gauge in dimensional regularization
(reduction) and MS-bar scheme one can see that
(Zanon and et al.05 Rossi,Sokatchev,Stanev05 F
reedmanGursoy05)
1-loop
So at one loop level theory is conformal
invariant and finite.
What will happen in higher orders of P.T. (how
1-loop condition will be modified) ?
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Conformal invariance and finiteness at higher
loops
At higher orders of P.T. condition
will receive corrections at 4 loops in planar
limit (Rossi,Sokatchev,Stanev06 Zanon and et
al.06). If one use one fold expansion (in
context of dimensional regularization
(reduction))
4-loop
1,2,3-loop
1,2,3-loop
4-loop
One may think that there is contradiction between
conformal invariance and finiteness (Zanon and et
al.07) !
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Conformal invariancefiniteness
If previous assumptions are satisfied
(dimensional regularization (reduction) and
MS-bar scheme) then one have two fold expansion
(Kazakov85)
Or alternatively for bare couplings

Coefficients are defined order by order
in P.T. and one can obtain simultaneously
finiteness and conformal invariantness in any
given order of P.T.
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in beta-deformed N4 SYM up to
4 loops in planar limit
Straightforward calculations lead to the
following family of solutions
And the theory is finite and conformal invariant
up to 4-loops. Using algorithm described here one
can make the theory finite and conformal
invariant at any given order of P.T.
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in beta-deformed N4 SYM up to 4
loops in planar limit
Lets demonstrate how our two fold expansion
exactly works
4-loop
1-loop
1-loop
4-loop
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Conclusions

• Beta-deformed N4 SYM can be made simultaneously
  conformal invariant and finite.
• In a framework of dimensional regularization (
  reduction ) one has to use double series of
  Yukawa couplings over g and parameter of
  dimensional regularization.
• Deformation with complex beta is also allowed.