Archive for September 2003
I show that under certain conditions it is possible to define consistent irrelevant deformations of interacting conformal field theories. The deformations are finite or have a unique running scale (“quasi-finite”). They are made of an infinite number of lagrangian terms and a finite number of independent parameters that renormalize coherently. The coefficients of the irrelevant terms are determined imposing that the beta functions of the dimensionless combinations of couplings vanish (“quasi-finiteness equations”). The expansion in powers of the energy is meaningful for energies much smaller than an effective Planck mass. Multiple deformations can be considered also. I study the general conditions to have non-trivial solutions. As an example, I construct the Pauli deformation of the IR fixed point of massless non-Abelian Yang-Mills theory with $N_c$ colors and $N_f \lesssim 11N_c/2$ flavors and compute the couplings of the term $F^3$ and the four-fermion vertices. Another interesting application is the construction of finite chiral irrelevant deformations of $N=2$ and $N=4$ superconformal field theories. The results of this paper suggest that power-counting non-renormalizable theories might play a role in the description of fundamental physics.
JHEP 0310 (2003) 045 | DOI: 10.1088/1126-6708/2003/10/045
arXiv:hep-th/0309251
As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory $C$. The Newton constant and the marginal parameters of $C$ are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop- and energy-expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern-Simons $U(1)$ gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity.
Nucl.Phys. B687 (2004) 124-142 | DOI: 10.1016/j.nuclphysb.2004.03.024
arXiv:hep-th/0309250
In three spacetime dimensions, where no graviton propagates, pure gravity is known to be finite. It is natural to inquire whether finiteness survives the coupling with matter. Standard arguments ensure that there exists a subtraction scheme where no Lorentz-Chern-Simons term is generated by radiative corrections, but are not sufficiently powerful to ensure finiteness. Therefore, it is necessary to perform an explicit (two-loop) computation in a specific model. I consider quantum gravity coupled with Chern-Simons U(1) gauge theory and massless fermions and show that renormalization originates four-fermion divergent vertices at the second loop order. I conclude that quantum gravity coupled with matter, as it stands, is not finite in three spacetime dimensions.
Nucl.Phys. B687 (2004) 143-160 | DOI: 10.1016/j.nuclphysb.2004.03.023
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Quantum Gravity 
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14B1 D. Anselmi
Renormalization
Course on renormalization, taught in 2015.
Last update: September 15th 2023, 242 pages
The final (2023) edition is vaibable 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
The pdf file of the 2015 Edition is available here: PDF
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