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.
JHEP 0507 (2005) 077 | DOI: 10.1088/1126-6708/2005/07/077
arXiv:hep-th/0502237
I formulate a deformation of the dimensional-regularization technique that is useful for theories where the common dimensional regularization does not apply. The Dirac algebra is not dimensionally continued, to avoid inconsistencies with the trace of an odd product of gamma matrices in odd dimensions. The regularization is completed with an evanescent higher-derivative deformation, which proves to be efficient in practical computations. This technique is particularly convenient in three dimensions for Chern-Simons gauge fields, two-component fermions and four-fermion models in the large N limit, eventually coupled with quantum gravity. Differently from even dimensions, in odd dimensions it is not always possible to have propagators with fully Lorentz invariant denominators. The main features of the deformed technique are illustrated in a set of sample calculations. The regularization is universal, local, manifestly gauge-invariant and Lorentz invariant in the physical sector of spacetime. In flat space power-like divergences are set to zero by default. Infinitely many evanescent operators are automatically dropped.
Int.J.Mod.Phys. A20 (2005) 1389-1418 | DOI: 10.1142/S0217751X0501983X
arXiv:hep-th/0404053
In flat space, $\gamma_5$ and the epsilon tensor break the dimensionally continued Lorentz symmetry, but propagators have fully Lorentz invariant denominators. When the Standard Model is coupled with quantum gravity $\gamma_5$ breaks the continued local Lorentz symmetry. I show how to deform the Einstein lagrangian and gauge-fix the residual local Lorentz symmetry so that the propagators of the graviton, the ghosts and the BRST auxiliary fields have fully Lorentz invariant denominators. This makes the calculation of Feynman diagrams more efficient.
Phys. Lett. B 596 (2004) 90 | DOI: 10.1016/j.physletb.2004.06.089
arXiv:hep-th/0404032
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
I study some aspects of the renormalization of quantum field theories with infinitely many couplings in arbitrary space-time dimensions. I prove that when the space-time manifold admits a metric of constant curvature the propagator is not affected by terms with higher derivatives. More generally, certain lagrangian terms are not turned on by renormalization, if they are absent at the tree level. This restricts the form of the action of a non-renormalizable theory, and has applications to quantum gravity. The new action contains infinitely many couplings, but not all of the ones that might have been expected. In quantum gravity, the metric of constant curvature is an extremal, but not a minimum, of the complete action. Nonetheless, it appears to be the right perturbative vacuum, at least when the curvature is negative, suggesting that the quantum vacuum has a negative asymptotically constant curvature. The results of this paper give also a set of rules for a more economical use of effective quantum field theories and suggest that it might be possible to give mathematical sense to theories with infinitely many couplings at high energies, to search for physical predictions.
Class.Quant.Grav. 20 (2003) 2355-2378 | DOI: 10.1088/0264-9381/20/11/326
I discuss several issues about the irreversibility of the RG flow and the trace anomalies $c$, $a$ and $a’$. First I argue that in quantum field theory: $i$) the scheme-invariant area $\Delta a’$ of the graph of the effective beta function between the fixed points defines the length of the RG flow; $ii$) the minimum of $\Delta a’$ in the space of flows connecting the same UV and IR fixed points defines the (oriented) distance between the fixed points; $iii$) in even dimensions, the distance between the fixed points is equal to $\Delta a =a_{UV}-a_{IR}$. In even dimensions, these statements imply the inequalities $0 \leq \Delta a \leq \Delta a’$ and therefore the irreversibility of the RG flow. Another consequence is the inequality $a \leq c$ for free scalars and fermions (but not vectors), which can be checked explicitly. Secondly, I elaborate a more general axiomatic set-up where irreversibility is defined as the statement that there exist no pairs of non-trivial flows connecting interchanged UV and IR fixed points. The axioms, based on the notions of length of the flow, oriented distance between the fixed points and certain “oriented-triangle inequalities”, imply the irreversibility of the RG flow without a global a function. I conjecture that the RG flow is irreversible also in odd dimensions (without a global a function). In support of this, I check the axioms of irreversibility in a class of $d=3$ theories where the RG flow is integrable at each order of the large $N$ expansion.
Class.Quant.Grav. 21 (2004) 29-50 | DOI: 10.1088/0264-9381/21/1/003
I study some classes of RG flows in three dimensions that are classically conformal and have manifest $g \rightarrow 1/g$ dualities. The RG flow interpolates between known (four-fermion, Wilson-Fischer, $\phi_3^6$) and new interacting fixed points. These models have two remarkable properties: $i$) the RG flow can be integrated for arbitrarily large values of the couplings g at each order of the $1/N$ expansion; $ii$) the duality symmetries are exact at each order of the $1/N$ expansion. I integrate the RG flow explicitly to the order ${\cal O}(1/N)$, write correlators at the leading-log level and study the interpolation between the fixed points. I examine how duality is implemented in the regularized theory and verified in the results of this paper.
Nucl.Phys. B658 (2003) 440 | DOI: 10.1016/S0550-3213(03)00174-3
I review my explanation of the irreversibility of the renormalization-group flow in even dimensions greater than two and address new investigations and tests.
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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
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