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I study the problem of renormalizing a non-renormalizable theory with a reduced, eventually finite, set of independent couplings. The idea is to look for special relations that express the coefficients of the irrelevant terms as unique functions of a reduced set of independent couplings $\lambda$, such that the divergences are removed by means of field redefinitions plus renormalization constants for the $\lambda$s. I consider non-renormalizable theories whose renormalizable subsector $R$ is interacting and does not contain relevant parameters. The “infinite” reduction is determined by $i$) perturbative meromorphy around the free-field limit of $R$, or $ii$) analyticity around the interacting fixed point of $R$. In general, prescriptions $i$) and $ii$) mutually exclude each other. When the reduction is formulated using $i$), the number of independent couplings remains finite or slowly grows together with the order of the expansion. The growth is slow in the sense that a reasonably small set of parameters is sufficient to make predictions up to very high orders. Instead, in case $ii$) the number of couplings generically remains finite. The infinite reduction is a tool to classify the irrelevant interactions and address the problem of their physical selection.


JHEP 0508 (2005) 029 | DOI: 10.1088/1126-6708/2005/08/029


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14B1 D. Anselmi

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Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)