## Trace anomalies

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 study various properties of the critical limits of correlators containing insertions of conserved and anomalous currents. In particular, I show that the improvement term of the stress tensor can be fixed unambiguously, studying the RG interpolation between the UV and IR limits. The removal of the improvement ambiguity is encoded in a variational principle, which makes use of sum rules for the trace anomalies $a$ and $a’$. Compatible results follow from the analysis of the RG equations. I perform a number of self-consistency checks and discuss the issues in a large set of theories.

J.Math.Phys. 43 (2002) 2965-2977 | DOI: 10.1063/1.1475766

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

Various formulas for currents with arbitrary spin are worked out in general space-time dimension, in the free field limit and, at the bare level, in presence of interactions. As the n-dimensional generalization of the (conformal) vector field, the $(n/2-1)$-form is used. The two-point functions and the higher-spin central charges are evaluated at one loop. As an application, the higher-spin hierarchies generated by the stress-tensor operator-product expansion are computed in supersymmetric theories. The results exhibit an interesting universality.

Class.Quant.Grav. 17 (2000) 1383-1400 | DOI: 10.1088/0264-9381/17/6/305

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