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Register a new userRenormalization for critical orders close to 2N
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We study the dynamics of the renormalization operator acting on the space of pairs (v,t), where v is a diffeomorphism and t belongs to [0,1], interpreted as unimodal maps x-->v(q_t(x)), where q_t(x)=-2t|x|^a+2t-1. We prove the so called complex bounds for sufficiently renormalizable pairs with bounded combinatorics. This allows us to show that if the critical exponent a is close to an even number then the renormalization operator has a unique fixed point. Furthermore this fixed point is hyperbolic and its codimension one stable manifold contains all infinitely renormalizable pairs.
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We present a systematic implementation of differential renormalization to all orders in perturbation theory. The method is applied to individual Feynamn graphs written in coordinate space. After isolating every singularity. which appears in a bare diagram, we define a subtraction procedure which consists in replacing the core of the singularity by its renormalized form givenby a differential formula. The organizationof subtractions in subgraphs relies in Bogoliubovs formula, fulfilling the requirements of locality, unitarity and Lorentz invariance. Our method bypasses the use of an intermediate regularization andautomatically delivers renormalized amplitudes which obey renormalization group equations.
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