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Register a new userRenormalization programme for effective theories
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V.Vereshagin
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We summarize our latest developments in perturbative treating the effective theories of strong interactions. We discuss the principles of constructing the mathematically correct expressions for the S-matrix elements at a given loop order and briefly review the renormalization procedure. This talk shall provide the philosophical basement as well as serve as an introduction for the material presented at this conference by A. Vereshagin and K. Semenov-Tian-Shansky.
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We discuss the peculiar features of the renormalization procedure in the case of infinite-component effective theory. It is shown that in the case of physically interesting theories (namely, those leading to the amplitudes with asymptotic behavior governed by known Regge intercepts) the full system of required renormalization prescriptions only contains those needed to fix the minimal counterterm vertices with one, two, three and (in some cases) four lines. There is no necessity in imposing the prescriptions for n-leg counterterm vertices with (n > 4). Moreover, the prescriptions for n =< 4 cannot be taken arbitrary: an infinite system of bootstrap constraints must be taken into account. The general method allowing one to write down the explicit form of this system is explained and illustrated by the example of elastic scattering process.
We develop the idea that renormalization, decoupling of heavy particle effects from low energy physics and the construction of effective field theories are intimately linked to the momentum space entanglement of disparate modes of an interacting quantum field theory. Using unitary transformations to decouple these modes at the perturbative level, we show in a scalar field theoretical model with light and heavy fields, how renormalization may be consistently implemented and how the low energy effective field theory can be constructed. We also obtain a renormalization group equation in this framework and apply it to the scalar field theoretical model.
These lectures review the formalism of renormalization in quantum field theories with special regard to effective quantum field theories. While renormalization theory is part of every advanced course on quantum field theory, for effective theories some more advanced topics become particularly important. This includes the renormalization of composite operators, operator mixing under scale evolution, and the resummation of large logarithms of scale ratios. This course thus sets the basis for many of the more specialized lecture courses delivered at the 2017 Les Houches Summer School.
The off-shell one-loop renormalization of a Higgs effective field theory possessing a scalar potential $simleft(Phi^daggerPhi-frac{v^2}2right)^N$ with $N$ arbitrary is presented. This is achieved by renormalizing the theory once reformulated in terms of two auxiliary fields $X_{1,2}$, which, due to the invariance under an extended Becchi-Rouet-Stora-Tyutin symmetry, are tightly constrained by functional identities. The latter allow in turn the explicit derivation of the mapping onto the original theory, through which the (divergent) multi-Higgs amplitude are generated in a purely algebraic fashion. We show that, contrary to naive expectations based on the loss of power counting renormalizability, the Higgs field undergoes a linear Standard Model like redefinition, and evaluate the renormalization of the complete set of Higgs self-coupling in the $Ntoinfty$ case.
This paper has been withdrawn to address an omission. It will be resubmitted in the near future.
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