## Irreversibility of the RG flow

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 review my explanation of the irreversibility of the renormalization-group flow in even dimensions greater than two and address new investigations and tests.

I derive a procedure to generate sum rules for the trace anomalies $a$ and $a’$. Linear combinations of $\Delta a = a_{UV}-a_{IR}$ and $\Delta a’ = a’_{UV}-a’_{IR}$ are expressed as multiple flow integrals of the two-, three- and four-point functions of the trace of the stress tensor. Eliminating $\Delta a’$, universal flow invariants are obtained, in particular sum rules for $\Delta a$. The formulas hold in the most general renormalizable quantum field theory (unitary or not), interpolating between UV and IR conformal fixed points. I discuss the relevance of these sum rules for the issue of the irreversibility of the RG flow. The procedure can be generalized to derive sum rules for the trace anomaly $c$.

JHEP 0111:033 (2001) | DOI: 10.1088/1126-6708/2001/11/033

A flow invariant is a quantity depending only on the UV and IR conformal fixed points and not on the flow connecting them. Typically, its value is related to the central charges a and c. In classically-conformal field theories, scale invariance is broken by quantum effects and the flow invariant $a_{UV}-a_{IR}$ is measured by the area of the graph of the beta function between the fixed points. There exists a theoretical explanation of this fact. On the other hand, when scale invariance is broken at the classical level, it is empirically known that the flow invariant equals $c_{UV}-c_{IR}$ in massive free-field theories, but a theoretical argument explaining why it is so is still missing. A number of related open questions are answered here. A general formula of the flow invariant is found, which holds also when the stress tensor has improvement terms. The conditions under which the flow invariant equals $c_{UV}-c_{IR}$ are identified. Several non-unitary theories are used as a laboratory, but the conclusions are general and an application to the Standard Model is addressed. The analysis of the results suggests some new minimum principles, which might point towards a better understanding of quantum field theory.

Class.Quant.Grav. 18 (2001) 4417-4442 | DOI: 10.1088/0264-9381/18/21/304

I discuss the properties of the central charges $c$ and $a$ for higher-derivative and higher-spin theories (spin 2 included). Ordinary gravity does not admit a straightforward identification of c and a in the trace anomaly, because it is not conformal. On the other hand, higher-derivative theories can be conformal, but have negative $c$ and $a$. A third possibility is to consider higher-spin conformal field theories. They are not unitary, but have a variety of interesting properties. Bosonic conformal tensors have a positive-definite action, equal to the square of a field strength, and a higher-derivative gauge invariance. There exists a conserved spin-2 current (not the canonical stress tensor) defining positive central charges $c$ and $a$. I calculate the values of $c$ and $a$ and study the operator-product structure. Higher-spin conformal spinors have no gauge invariance, admit a standard definition of $c$ and $a$ and can be coupled to Abelian and non-Abelian gauge fields in a renormalizable way. At the quantum level, they contribute to the one-loop beta function with the same sign as ordinary matter, admit a conformal window and non-trivial interacting fixed points. There are composite operators of high spin and low dimension, which violate the Ferrara-Gatto-Grillo theorem. Finally, other theories, such as conformal antisymmetric tensors, exhibit more severe internal problems. This research is motivated by the idea that fundamental quantum field theories should be renormalization-group (RG) interpolations between ultraviolet and infrared conformal fixed points, and quantum irreversibility should be a general principle of nature.

Class.Quant.Grav. 17 (2000) 2847-2866 | DOI: 10.1088/0264-9381/17/15/301

I review recent results on conformal field theories in four dimensions and quantum field theories interpolating between conformal fixed points, supersymmetric and non-supersymmetric. The talk is structured in three parts: $i$) central charges, $ii$) anomalous dimensions and $iii$) quantum irreversibility.

I identify the class of even-dimensional conformal field theories that is most similar to two-dimensional conformal field theory. In this class the formula, elaborated recently, for the irreversibility of the renormalization-group flow applies also to massive flows. This implies a prediction for the ratio between the coefficient of the Euler density in the trace anomaly (charge $a$) and the stress-tensor two-point function (charge $c$). More precisely, the trace anomaly in external gravity is quadratic in the Ricci tensor and the Ricci scalar and contains a unique central charge. I check the prediction in detail in four, six and eight dimensions, and then in arbitrary even dimension.

Phys.Lett. B476 (2000) 182-187 | DOI: 10.1016/S0370-2693(00)00135-0

Some recent ideas are generalized from four dimensions to the general dimension $n$. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density $G_n$ and $\Box^{n/2-1}R$, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, $G_n$ can be extended to a new notion of Euler density, linear in the conformal factor. We call it pondered Euler density. This notion relates the trace-anomaly coefficients $a$ and $a’$ of $G_n$ and $\Box^{n/2-1}R$ in a universal way ($a=a’$) and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for $n=6$ and check the prediction to the fourth-loop order in the $\phi^3$-theory. The formula of quantum irreversibility for general n even can be extended to $n$ odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content.

Nucl.Phys. B567 (2000) 331-359 | DOI: 10.1016/S0550-3213(99)00479-4

The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and $\Box R$, with coefficients, properly normalized, called $c$, $a$ and $a’$, the latter being ambiguously defined by an additive constant. Considerations about unitarity and positivity properties of the induced actions allow us to show that the total RG flows of $a$ and $a’$ are equal and therefore the $a’$-ambiguity can be consistently removed through the identification $a’=a$. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and renormalization implies that the flux of the renormalization group is irreversible. A monotonically decreasing $a$-function interpolating between the appropriate values is naturally provided by $a’$. The total $a$-flow is expressed non-perturbatively as the invariant (i.e. scheme-independent) area of the graph of the beta function between the fixed points. We test this prediction to the fourth loop order in perturbation theory, in QCD with $N_f \lesssim 11/2 N_c$ and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and $\phi^4$-theory). Arguments for the positivity of $a$ are also discussed.

Annals Phys. 276 (1999) 361-390 | DOI: 10.1006/aphy.1999.5949