### Course

19S1 D. Anselmi
Theories of gravitation

Program

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### Book

D. Anselmi
From Physics To Life

A journey to the infinitesimally small and back

In English and Italian

Available on Amazon:
US: book | ebook  (in EN)
IT: book | ebook  (in IT)

## Archive for February 2005

The maximum pole of a diagram with $V$ vertices and $L$ loops is at most $1/\varepsilon^{m(V,L)}$, where $m(V,L)=\min (V-1,L).$ The result holds in dimensional regularization, where $\varepsilon = d-D$, $d$ is the physical dimension and $D$ the continued one. Moreover, vertices are counted treating mass terms and the other non-dominant quadratic terms as “two-leg vertices”.

Certain power-counting non-renormalizable theories, including the most general self-interacting scalar fields in four and three dimensions and fermions in two dimensions, have a simplified renormalization structure. For example, in four-dimensional scalar theories, $2n$ derivatives of the fields, $n>1$, do not appear before the nth loop. A new kind of expansion can be defined to treat functions of the fields (but not of their derivatives) non-perturbatively. I study the conditions under which these theories can be consistently renormalized with a reduced, eventually finite, set of independent couplings. I find that in common models the number of couplings sporadically grows together with the order of the expansion, but the growth is slow and a reasonably small number of couplings is sufficient to make predictions up to very high orders. Various examples are solved explicitly at one and two loops.

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JHEP 0507 (2005) 077 | DOI: 10.1088/1126-6708/2005/07/077

arXiv:hep-th/0502237

Quantum Gravity

### Book

14B1 D. Anselmi
Renormalization

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

Last update: May 9th 2015, 230 pages

Avaibable on Amazon:

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

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