Exact Beta Function Research Articles

Non-perturbative SQED beta function using the functional renormalization group approach and the NSVZ exact beta function

Abstract The renormalization group equations of massive $\mathcal {N}=1$ supersymmetric quantum electrodynamics are studied using the functional renormalization group approach. A non-perturbative form of the beta function has been computed via a derivative expansion of the effective action. In the local potential approximation, the functional form of the non-perturbative beta function is closely related to the form of the Novikov–Shifman–Vainshtein–Zakharov (NSVZ) exact beta function; this relationship is exact if an effective fine-structure constant is defined. The non-massive limit of the same is also analyzed. Furthermore, the calculation of the beta function has been improved by incorporating the influence of momentum modes on the propagation of the superfields in the non-perturbative running of the electric charge, applying a second-order truncation for the derivative expansion, which we use to find the momentum contributions to the β function. Again, we find the NSVZ relation for an effective fine-structure constant. It is with sadness that I say goodbye to my professor, Iván Schmidt Andrade, who left us during the course of this work. His passion for research and his special vision of physics work will remain with us. Thank you for everything.

Renormalization group flow of the Aharonov–Bohm scattering amplitude

The Aharonov–Bohm elastic scattering with incident particles described by plane waves is revisited by using the phase-shifts method. The formal equivalence between the cylindrical Schrödinger equation and the one-dimensional Calogero problem allows us to show that up to two scattering phase-shifts modes in the cylindrical waves expansion must be renormalized. The renormalization procedure introduces new length scales giving rise to spontaneous breaking of the conformal symmetry. The knowledge of the exact beta function permits us to describe the renormalization group flows within the two-parametric family of renormalized Aharonov–Bohm scattering amplitudes.

Renormalization group in super-renormalizable quantum gravity

One of the main advantages of super-renormalizable higher derivative quantum gravity models is the possibility to derive exact beta functions, by making perturbative one-loop calculations. We perform such a calculation for the Newton constant by using the Barvinsky–Vilkovisky trace technology. The result is well-defined in a large class of models of gravity in the sense that the renormalization group beta functions do not depend on the gauge-fixing condition. Finally, we discuss the possibility to apply the results to a large class of nonlocal gravitational theories which are free of massive ghost-like states at the tree-level.

The scalar glueball operator, the a-theorem, and the onset of conformality

We show that the anomalous dimension γG of the scalar glueball operator contains information on the mechanism that leads to the onset of conformality at the lower edge of the conformal window in a non-Abelian gauge theory. In particular, it distinguishes whether the merging of an UV and an IR fixed point – the simplest mechanism associated to a conformal phase transition and preconformal scaling – does or does not occur. At the same time, we shed light on new analogies between QCD and its supersymmetric version. In SQCD, we derive an exact relation between γG and the mass anomalous dimension γm, and we prove that the SQCD exact beta function is incompatible with merging as a consequence of the a-theorem; we also derive the general conditions that the latter imposes on the existence of fixed points, and prove the absence of an UV fixed point at nonzero coupling above the conformal window of SQCD. Perhaps not surprisingly, we then show that an exact relation between γG and γm, fully analogous to SQCD, holds for the massless Veneziano limit of large-N QCD. We argue, based on the latter relation, the a-theorem, perturbation theory and physical arguments, that the incompatibility with merging may extend to QCD.

Integrable flows between exact CFTs

We explicitly construct families of integrable σ-model actions smoothly inter-polating between exact CFTs. In the ultraviolet the theory is the direct product of two current algebras at levels k1 and k2. In the infrared and for the case of two deformation matrices the CFT involves a coset CFT, whereas for a single matrix deformation it is given by the ultraviolet direct product theories but at levels k1 and k2 − k1. For isotropic deformations we demonstrate integrability. In this case we also compute the exact beta-function for the deformation parameters using gravitational methods. This is shown to coincide with previous results obtained using perturbation theory and non-perturbative symmetries.

Quantum aspects of doubly deformed CFTs

We study quantum aspects of the recently constructed doubly λ-deformed σ-models representing the effective action of two WZW models interacting via current bilinears. We show that although the exact beta-functions and current anomalous dimensions are identical to those of the λ-deformed models, this is not true for the anomalous dimensions of generic primary field operators in accordance with the fact that the two models differ drastically. Our proofs involve CFT arguments, as well as effective σ-model action and gravity calculations.

Finite quantum gauge theories

We explicitly compute the one-loop exact beta function for a nonlocal extension of the standard gauge theory, in particular Yang-Mills and QED. The theory, made of a weakly nonlocal kinetic term and a local potential of the gauge field, is unitary (ghost-free) and perturbatively super-renormalizable. Moreover, in the action we can always choose the potential (consisting of one "killer operator") to make zero the beta function of running gauge coupling constant. The outcome is "a UV finite theory for any gauge interaction". Our calculations are done in D=4, but the results can be generalized to even or odd spacetime dimensions. We compute the contribution to the beta function from two different killer operators by using two independent techniques, namely the Feynman diagrams and the Barvinsky-Vilkovisky traces. By making the theories finite we are able to solve also the Landau pole problems, in particular in QED. Without any potential the beta function of the one-loop super-renormalizable theory shows a universal Landau pole in the running coupling constant in the ultraviolet regime (UV), regardless of the specific higher-derivative structure. However, the dressed propagator shows neither the Landau pole in the UV, nor the singularities in the infrared regime (IR).

Making supersymmetric connectedN=(0,2)sigma models

We construct "connected" (0,2) sigma models starting from n copies of (2,2) CP(N-1) models. General aspects of models of this type (known as T+O deformations) had been previously studied in the context of heterotic string theories. Our construction presents a natural generalization of the nonminimally deformed (2,2) model with an extra (0,2) fermion superfield on tangent bundle T CP(N-1) x C^1. We had thoroughly analyzed the latter model previously, found the exact beta function and a spontaneous breaking of supersymmetry. In contrast, in certain connected sigma models the spontaneous breaking of supersymmetry disappears. We study the connected sigma models in the large-N limit finding supersymmetric vacua and determining the particle spectrum. While the Witten index vanishes in all the models under consideration, in these special cases of connected models one can use a permutation symmetry to define a modification of the Witten index which does not vanish. This eliminates the spontaneous breaking of supersymmetry. We then examine the exact beta functions of our connected (0,2) sigma models.

Supersymmetric QCD: exact results and strong coupling

We revisit two longstanding puzzles in supersymmetric gauge theories. The first concerns the question of the holomorphy of the coupling, and related to this the possible definition of an exact (NSVZ) beta function. The second concerns instantons in pure gluodynamics, which appear to give sensible, exact results for certain correlation functions, which nonetheless differ from those obtained using systematic weak coupling expansions. For the first question, we extend an earlier proposal of Arkani-Hamed and Murayama, showing that if their regulated action is written suitably, the holomorphy of the couplings is manifest, and it is easy to determine the renormalization scheme for which the NSVZ formula holds. This scheme, however, is seen to be one of an infinite class of schemes, each leading to an exact beta function; the NSVZ scheme, while simple, is not selected by any compelling physical consideration. For the second question, we explain why the instanton computation in the pure supersymmetric gauge theory is not reliable, even at short distances. The semiclassical expansion about the instanton is purely formal; if infrared divergences appear, they spoil arguments based on holomorphy. We demonstrate that infrared divergences do not occur in the perturbation expansion about the instanton, but explain that there is no reason to think this captures all contributions from the sector with unit topological charge. That one expects additional contributions is illustrated by dilute gas corrections. These are infrared divergent, and so difficult to define, but if non-zero give order one, holomorphic, corrections to the leading result. Exploiting an earlier analysis of Davies et al, we demonstrate that in the theory compactified on a circle of radius beta, due to infrared effects, finite contributions indeed arise which are not visible in the formal limit that beta goes to infinity.

QuasiBPSWilson loops, localization of loop equation by homology and exact beta function in the large-Nlimit of SU(N) Yang-Mills theory

We localize the loop equation of large-N YM theory in the anti-self-dual variables on a critical equation for an effective action by means of homological methods as opposed to the cohomological localization of equivariantly closed forms in local field theory. Our localization occurs for some special simple quasi BPS Wilson loops, that have no perimeter divergence and no cusp anomaly for backtracking cusps, in a partial Eguchi-Kawai reduction from four to two dimensions of the non-commutative theory in the limit of infinite non-commutativity and in a lattice regularization in which the anti-self-dual integration variables live at the points of the lattice, thus implying an embedding of parabolic Higgs bundles in the YM functional integral. We find that the beta function of the effective action is saturated by the non-commutative anti-self-dual vortices of the Eguchi-Kawai reduction. An exact canonical beta function of Novikov-Shifman-Vainshtein-Zakharov type, that reproduces the universal first and second perturbative coefficients follows by the localization on vortices. Finally we argue that a scheme can be found in which the canonical coupling coincides with the physical charge between static quark sources in the large-N limit and we compare our theoretical calculation with some numerical lattice result.

Linear Models for Flux Vacua

We construct worldsheet descriptions of heterotic flux vacua as the IR limits of N=2 gauge theories. Spacetime torsion is incorporated via a 2d Green-Schwarz mechanism in which a doublet of axions cancels a one-loop gauge anomaly. Manifest (0, 2) supersymmetry and the compactness of the gauge theory instanton moduli space suggest that these models, which include Fu-Yau models, are stable against worldsheet instantons, implying that they, like Calabi-Yaus, may be smoothly extended to solutions of the exact beta functions. Since Fu-Yau compactifications are dual to KST-type flux compactifications, this provides a microscopic description of these IIB RR-flux vacua.

Exact beta function from the holographic loop equation of large-NQCD4

We construct and study a previously defined quantum holographic effective action whose critical equation implies the holographic loop equation of large-N QCD_4 for planar self-avoiding loops in a certain regularization scheme. We extract from the effective action the exact beta function in the given scheme. For the Wilsonean coupling constant the beta function is exacly one loop and the first coefficient agrees with its value in perturbation theory. For the canonical coupling constant the exact beta function has a NSVZ form and the first two coefficients agree with their value in perturbation theory.

Exactly Marginal Deformations of Quiver Gauge Theories as Seen from Brane Tilings

We study the relation between exactly marginal deformations in a large class of N=1 superconformal quiver gauge theories described by brane tilings and the degrees of freedom in the corresponding 5-brane systems. We show, with the help of NSVZ exact beta functions, that there are generically d-1 complex exactly marginal deformations of a gauge theory, where d is the perimeter of the corresponding n it, and the other two, the diagonal gauge coupling and a beta-like deformation, as background supergravity fields.toric diagram. We identify d-3 complex marginal deformations as deformations of the brane system and the Wilson lines on it, and the other two, the diagonal gauge coupling and a beta-like deformation, as background supergravity fields.

NONCOMMUTATIVE U(1) GAUGE THEORY AS A NONLINEAR SIGMA MODEL

Noncommutative U (1) gauge theory in four dimensions is shown to be equivalent in some scaling limit to an ordinary nonlinear sigma model in two dimensions. The model in this regime is solvable and the corresponding exact beta function is found. We also show that classical U (n) gauge theory on [Formula: see text] can be approximated by a sequence of ordinary (d-2)-dimensional Georgi–Glashow models with gauge groups U (n(L+1)), where L+1 is the matrix size of the regularized noncommutative plane [Formula: see text].

The Kramers–;Wannier Symmetry and S-Duality in the Two-Dimensional g03A6;4 Theory

We show that the exact beta function of the two-dimensional gΦ4 theory possesses two dual symmetries. These are the Kramers–Wannier symmetry d(g) and the strong–weak-coupling symmetry, or the S-duality f(g), connecting the strong- and weak-coupling domains lying above and below the fixed point g*. We obtain explicit representations for the functions d(g) and f(g). The S-duality transformation f(g) allows using the high-temperature expansions to approximate the contributions of the higher-order Feynman diagrams. From the mathematical standpoint, the proposed scheme is highly unstable. Nevertheless, the approximate values of the renormalized coupling constant g* obtained from the duality equations agree well with the available numerical results.

Gauge singlet renormalisation in softly-broken supersymmetric theories

We consider the renormalisation of a softly-broken supersymmetric theory with singlet fields and a superpotential with a linear term. We show that there exist exact beta-functions for both the linear term in the superpotential and the associated linear term in the Lagrangian. We also construct exact renormalisation group invariant trajectories for these terms, corresponding to the conformal anomaly solution for the soft masses and couplings.

Holomorphy, rescaling anomalies and exact β functions in supersymmetric gauge theories

There have been known "exact" beta functions for the gauge coupling in N=1 supersymmetric gauge theories, the so-called NSVZ beta functions. Shifman and Vainshtein (SV) further related these beta functions to the exact 1-loop running of the "Wilsonian" gauge coupling. All these results, however, remain somewhat mysterious. We attempt to clarify these issues by presenting new perspectives on the NSVZ beta function. Our interpretation of the results is somewhat different than the one given by SV, having nothing to do with the distinction between "Wilsonian" and "1PI" effective actions. Throughout we work in the context of the Wilsonian Renormalization Group; namely, as the cutoff of the theory is changed from M to M', we determine the appropriate changes in the bare couplings needed to keep the low energy physics fixed. The entire analysis is therefore free of infrared subtleties. When the bare Lagrangian given at the cutoff is manifestly holomorphic in the gauge coupling, we show that the required change in the holomorphic gauge coupling is exhausted at 1-loop to all orders of perturbation theory, and even non-perturbatively in some cases. On the other hand, when the bare Lagrangian has canonically normalized kinetic terms, we find that the required change in the gauge coupling is given by the NSVZ beta function. The higher order contributions in the NSVZ beta function are due to anomalous Jacobians under the rescaling of the fields done in passing from holomorphic to canonical normalization. We also give prescriptions for regularizing certain N=1 theories with an ultraviolet cutoff M preserving manifest holomorphy, starting from finite N=4 and N=2 theories. It is then in principle possible to check the validity of the exact beta function by higher order calculations in these theories.

Duality symmetry, strong-coupling expansion and universal critical amplitudes in two-dimensional Phi4field models

We show that the exact beta-function (g) in the continuous two-dimensional g4 model possesses the Kramers-Wannier duality symmetry. The duality symmetry transformation = d(g) such that (d(g)) = d´(g)(g) is constructed and the approximate values of g* computed from the duality equation d(g*) = g* are shown to agree with the available numerical results. The calculation of the beta-function (g) for the 2D scalar g4 field theory based on the strong-coupling expansion is developed and the expansion of (g) in powers of g-1 is obtained up to order g-8. The numerical values calculated for the renormalized coupling constant g+* are in reasonably good agreement with the best modern estimates recently obtained from the high-temperature series expansion and with those known from the perturbative four-loop renormalization-group calculations. The application of Cardy's theorem for calculating the renormalized isothermal coupling constant gc of the 2D Ising model and the related universal critical amplitudes is also discussed.

Exact renormalization group equation in presence of rescaling anomaly

Wilson's approach to renormalization group is reanalyzed for supersymmetric Yang-Mills theory. Usual demonstration of exact renormalization group equation must be modified due to the presence of the so called Konishi anomaly under the rescaling of superfields. We carry out the explicit computation for N=1 SUSY Yang-Mills theory with the simpler, gauge invariant regularization method, recently proposed by Arkani-Hamed and Murayama. The result is that the Wilsonian action S_M consists of two terms, i.e. the non anomalous term, which obeys Polchinski's flow equation and Fujikawa-Konishi determinant contribution. This latter is responsible for Shifman-Vainshtein relation of exact beta-function.

Branes and electric-magnetic duality in supersymmetric QCD

We study the type IIA limit of the M-theory fivebrane configuration corresponding to N = 1 supersymmetric QCD with massless quarks. We identify the effective gauge coupling constant that fits with the Novikov-Shifman-Veinshtein-Zakharov exact beta function. We find two different type IIA limits that correspond to the electric and magnetic descriptions of SQCD, as observed in the massive case by Schmaltz and Sundrum. The analysis is extended to the case of symplectic and orthogonal gauge groups. In any of the cases considered in this paper, the electric and magnetic configurations are smoothly interpolated via M-theory. This is in sharp contrast with the proposed derivation of N = 1 duality within the weakly coupled type IIA string theory where a singularity is inevitable unless one turns on a parameter that takes the theory away from an interesting point.