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The $O(N)$ model with scalar quartic interactions at its ultraviolet fixed point, and the $O(N)$ model with scalar cubic interactions at its infra-red fixed point are conjectured to be equivalent. This has been checked by comparing various features of the two models at their respective fixed points. Recently, the scaling dimensions of a family of operators of fixed charge $Q$ have been shown to match at the FPs up to $\cal{O}\left(\frac{1}{N^2}\right)$at leading order (LO) and next-to-leading order (NLO) in $Q$ using a semiclassical computation which is valid to all orders in the coupling. Here we perform a complementary but overlapping comparison using a perturbative calculation in six dimensions, up to three-loop order in the coupling, to compare these critical scaling dimensions beyond NLO in $Q$, in fact to all relevant orders in $Q$. We also obtain the corresponding results at $\cal{O}\left(\frac{1}{N^3}\right)$ for the cubic theory., Comment: 14 pages, 2 figures, reference added, typos corrected

Recently it was shown that the scaling dimension of the operator $\phi^n$ in $\lambda(\bar\phi\phi)^2$ theory may be computed semiclassically at the Wilson-Fisher fixed point in $d=4-\epsilon$, for generic values of $\lambda n$, and this was verified to two loop order in perturbation theory at leading and subleading $n$. This result was subsequently generalised to operators of fixed charge $Q$ in $O(N)$ theory and verified up to four loops in perturbation theory at leading and subleading $Q$. More recently, similar semiclassical calculations have been performed for the classically scale-invariant $U(N)\times U(N)$ theory in four dimensions, and verified up to two loops, once again at leading and subleading $Q$. Here we extend this verification to four loops. We also consider the corresponding classically scale-invariant theory in three dimensions, similarly verifying the leading and subleading semiclassical results up to four loops in perturbation theory., Comment: 8 figures, 5 tables, 24 pages. arXiv admin note: text overlap with arXiv:2101.09820

Recently it was shown that the scaling dimension of the operator $\phi^n$ in $\lambda(\phi^*\phi)^2$ theory may be computed semi-classically at the Wilson-Fisher fixed point in $d=4-\epsilon$, for generic values of $\lambda n$ and this was verified to two loop order in perturbation theory at leading and sub-leading $n$. In subsequent work, this result was generalised to operators of fixed charge $Q$ in $O(N)$ theory and verified up to three loops in perturbation theory at leading and sub-leading order. Here we extend this verification to four loops in $O(N)$ theory, once again at leading and sub-leading order. We also investigate the strong-coupling regime., Comment: 16 pages. 6 Figures. Typos corrected; references and an acknowledgement added. Some minor errors corrected in a figure and equations

Recently it was shown that the scaling dimension of the operator $\phi^n$ in scale-invariant $d=3$ theory may be computed semiclassically, and this was verified to leading order (two loops) in perturbation theory at leading and subleading $n$. Here we extend this verification to six loops, once again at leading and subleading $n$. We then perform a similar exercise for a theory with a multiplet of real scalars and an $O(N)$ invariant hexic interaction. We also investigate the strong-coupling regime for this example., Comment: 25 pages, 8 figures, 4 tables. Uses tikz. Additional references and acknowledgements added. Minor corrections and additions made

We analyze $SO(N)$ and $SU(N)$ gauge theories with scalars in adjoint and fundamental representations, coupled to renormalisable, classically scale invariant gravity. In the specific case of $SO(12),$ we show that the quantum field theory can be can be asymptotically free in all couplings (hence ultra-violet complete). For a region of parameter space, Dimensional Transmutation occurs, with the adjoint vacuum expectation value breaking $SO(12) \to SU(6)\otimes U(1)$ and producing a Low Energy Effective Theory having Einstein-Hilbert gravity. We verify that certain minima are locally stable and lie within the catchment basin of the ultraviolet fixed points., Comment: 27 pages, 2 attachments (v3 includes brief comparision with perturbative asymptotic safety, new references, and minor typos corrected.)

We analyse scale invariant quadratic quantum gravity incorporating non-minimal coupling to a multiplet of scalar fields in a gauge theory, with particular emphasis on the consequences for its interpretation resulting from a transformation from the Jordan frame to the Einstein frame. The result is the natural emergence of a de Sitter space solution which, depending the gauge theory and region of parameter space chosen, can be free of ghosts and tachyons, and completely asymptotically free. In the case of an SO(10) model, we present a detailed account of the spontaneous symmetry breaking, and we calculate the leading (two-loop) contribution to the dilaton mass., Comment: 22 pages, no figures (Except for the additional "Post-publication note added" and associated reference, v3 is essentially as published.)

We calculate the $\beta$-functions for $SO(N)$ and $SU(N)$ gauge theories coupled to adjoint and fundamental scalar representations, correcting long-standing, previous results. We explore the constraints on $N$ resulting from requiring asymptotic freedom for all couplings. When we take into account the actual allowed behavior of the gauge coupling, the minimum value of $N$ in both cases turns out to be larger than realized in earlier treatments. We also show that in the large $N$ limit, both models have large regions of parameter space corresponding to total asymptotic freedom., Comment: 8 pages, no figures. v6: Minor nonsubstantive changes in text and formatting

There are five well-known zero modes among the fluctuations of the metric of de~Sitter (dS) spacetime. For Euclidean signature, they can be associated with certain spherical harmonics on the $S^4$ sphere, viz., the vector representation $\bf5$ of the global $SO(5)$ isometry. They appear, for example, in the perturbative calculation of the on-shell effective action of dS space, as well as in models containing matter fields. These modes are shown to be associated with collective modes of $S^4$ corresponding to certain coherent fluctuations. When dS space is embedded in flat five dimensions $E^5,$ they may be seen as a legacy of translation of the center of the $S^4$ sphere. Rigid translations of the $S^4$-sphere on $E^5$ leave the classical action invariant but are unobservable displacements from the point of view of gravitational dynamics on $S^4.$ Thus, unlike similar moduli, the center of the sphere is not promoted to a dynamical degree of freedom. As a result, these zero modes do not signify the possibility of physically realizable fluctuations or flat directions for the metric of dS space. They are not associated with Killing vectors on $S^4$ but can be with certain non-isometric, conformal Killing forms that locally correspond to a rescaling of the volume element $dV_4.$ For convenience, we frame our discussion in the context of renormalizable gravity, but the conclusions apply equally to the corresponding zero modes in Einstein gravity. We expect that these zero modes will be present to all orders in perturbation theory. They will occur for Lorentzian signature as well, so long as the hyperboloid $H^4$ is locally stable, but there remain certain infrared issues that need to be clarified. We conjecture that they will appear in any gravitational theory having dS background as a locally stable solution of the effective action, regardless of whether additional matter is included., Comment: v4, 28pages, no figures; final journal form, minor changes in text and refs from v3

As an illustration of a renormalizable, asymptotically-free model of induced gravity, we consider an $SO(10)$ gauge theory interacting with a real scalar multiplet in the adjoint representation. We show that dimensional transmutation can occur, spontaneously breaking $SO(10)$ to $SU(5){\otimes}U(1),$ while inducing the Planck mass and a positive cosmological constant, all proportional to the same scale $v$. All mass ratios are functions of the values of coupling constants at that scale. Below this scale (at which the Big Bang may occur), the model takes the usual form of Einstein-Hilbert gravity in de Sitter space plus calculable corrections. We show that there exist regions of parameter space in which the breaking results in a local minimum of the effective action, and a {\bf positive} dilaton $(\hbox{mass})^2$ from two-loop corrections associated with the conformal anomaly. Furthermore, unlike the singlet case we considered previously, some minima lie within the basin of attraction of the ultraviolet fixed point. Moreover, the asymptotic behavior of the coupling constants also lie within the range of convergence of the Euclidean path integral, so there is hope that there will be candidates for sensible vacua. Although open questions remain concerning unitarity of all such renormalizable models of gravity, it is not obvious that, in curved backgrounds such as those considered here, unitarity is violated. In any case, any violation that may remain will be suppressed by inverse powers of the reduced Planck mass., Comment: 44 pages, 5 figures, 2 tables. v2 has new discussion concerning stability of SSB plus related appendix. Additional references added. v3 is version to be published; contains minor revisions

We show that classically scale invariant gravity coupled to a single scalar field can undergo dimensional transmutation and generate an effective Einstein-Hilbert action for gravity, coupled to a massive dilaton. The same theory has an ultraviolet fixed point for coupling constant ratios such that all couplings are asymptotically free. However the catchment basin of this fixed point does not include regions of coupling constant parameter space compatible with locally stable dimensional transmutation. In a companion paper we will explore whether this more desirable outcome does obtain in more complicated theories with non-Abelian gauge interactions., Comment: 17 pages, 3 figures. v2, additional references; slight changes in wordng; no changes in results. v3, additional references

The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the beta-functions via a gradient flow equation involving a positive definite metric. We demonstrate the existence of a candidate a-function for renormalisable Chern-Simons theories in three dimensions, involving scalar and fermion fields, in both non-supersymmetric and supersymmetric cases., Comment: 15 pages, 2 figures, uses axodraw. Revised with minor corrections; also additional clarifications and references included

We discuss the renormalization of higher-derivative gravity, both without and with matter fields, in terms of two primary coupling constants rather than three. A technique for determining the dependence of the Gauss-Bonnet coupling constant on the remaining couplings is explained, and consistency with the local form of the Gauss-Bonnet relation in four dimensions is demonstrated to all orders in perturbation theory. A similar argument is outlined for the Hirzebruch signature and its coupling. We speculate upon the potential implications of instantons on the associated nonperturbative coupling constants., Comment: 11 pages, no figures, revtex-4.1. v2 includes additional references and minor changes in text. v3 has minor changes in text in Section II.B. Results unchanged

We discuss the nature of quantum field theories involving gravity that are classically scale-invariant. We show that gravitational radiative corrections are crucial in the determination of the nature of the vacuum state in such theories, which are renormalisable, technically natural, and can be asymptotically free in all dimensionless couplings. In the pure gravity case, we discuss the role of the Gauss-Bonnet term, and we find that Dimensional Transmutation (DT) \`a la Coleman-Weinberg leads to extrema of the effective action corresponding to nonzero values of the curvature, but such that these extrema are local maxima. In even the simplest extension of the theory to include scalar fields, we show that the same phenomenon can lead to extrema that are local minima of the effective action, with both non-zero curvature and non-zero scalar vacuum expectation values, leading to spontaneous generation of the Planck mass. Although we find an asymptotically free (AF) fixed point exists, unfortunately, no running of the couplings connect the region of DT to the basin of attraction of the AF fixed point. We also find there remains a flat direction for one of the conformal modes. We suggest that in more realistic models AF and DT could be compatible, and that the same scalar vacuum expectation values could be responsible both for DT and for spontaneous breaking of a Grand Unified gauge group., Comment: 49 pages, 4 figures, JHEP3; v2 includes additional references, one equation replaced, note added, but no substantive changes. v3 corrects beta-functions (below eq. E.1) & Table 2, fixed typos and format, acknowledgements. v4 contains additional references and slight revisions in text. v5 is the published JHEP version, with no changes in results or conclusions

We have investigated a number of GUT models for the possibility that their Higgs fields might be responsible for inflation in the early universe. In addition to models having an intrinsic Planck mass parameter, we have entertained classically scale invariant models in which the Planck scale itself as well as the GUT scale is induced by spontaneous gauge symmetry breaking. We found that in a non-supersymmetric SU(5) with the usual adjoint Higgs, but with large non-minimal coupling to the curvature, there are flat directions that might lead to inflation. Interestingly, the one of lowest energy breaks into SU(3) x SU(2) x U(1), as suggested by gauge coupling unification. Further, we show that this flat direction is stable against small fluctuations in other directions. We attempted to extend this to supersymmetric GUTS, both global and supergravity, but did not succeed in finding an acceptable model of this type. As is often the case, such models suffered either from a negative vacuum energy or from tachyonic modes. We also considered a variant of an "inverted hierarchy" model in which the GUT scale is set by dimensional transmutation, but were unable to find a phenomenologically acceptable model., Comment: 32 pages, no figures; minor corrections, improved discussion and added references

We explore in the supergravity context the possibility that a Higgs scalar may drive inflation via a non-minimal coupling to gravity characterised by a large dimensionless coupling constant. We find that this scenario is not compatible with the MSSM, but that adding a singlet field (NMSSM, or a variant thereof) can very naturally give rise to slow-roll inflation. The inflaton is necessarily contained in the doublet Higgs sector and occurs in the D-flat direction of the two Higgs doublets., Comment: 13 pages, 1 figure

We discuss the non-anticommutative (N=1/2) supersymmetric SU(N)\otimes U(1) gauge theory including a superpotential. We show how recent proposals for obtaining a renormalisable version of the theory may be implemented in the component formalism at the one-loop level., Comment: 25pp, 14 figures, LaTeX; minor typo fixed, reference added, notation clarified

We discuss the non-anticommutative (N=1/2) supersymmetric U(1) gauge theory in four dimensions, including a superpotential. We perform the one-loop renormalisation of the model, including the complete set of terms necessary for renormalisability, showing in detail how the eliminated and uneliminated forms of the theory lead to equivalent results., Comment: 26pp, 5 figures

We discuss the non-anticommutative (N=1/2) supersymmetric Wess-Zumino model in four dimensions. Firstly we introduce differential operators which implement the non-anticommutative supersymmetry algebra acting on the component fields and action. Then we perform the renormalisation of the model up to two-loop order, including the complete set of terms necessary for renormalisability. We show that (at least up to this order) the results obtained when we eliminate the auxiliary field after renormalisation are equivalent to those obtained when we eliminate the auxiliary fields before quantisation., Comment: 28 pages, 7 figures, uses LaTeX

We present four-loop results for the gauge beta-function and the fermion mass anomalous dimension for a gauge theory with a general gauge group and a multiplet of fermions transforming according to an arbitrary representation, calculated using the dimensional reduction scheme. In the special case of a supersymmetric theory we confirm previous calculations of both the gauge beta-function and the gaugino mass beta-function., Comment: 44 pages, added references (v2) minor changes (v3)

We review the calculation of the the effective potential with particular emphasis on cases when the tree potential or the renormalisation-group-improved, radiatively corrected potential exhibits non-convex behaviour. We illustrate this in a simple Yukawa model which exhibits a novel kind of dimensional transmutation. We also review briefly earlier work on the Standard Model. We conclude that, despite some recent claims to the contrary, it can be possible to infer reliably that the tree vacuum does not represent the true ground state of the theory., Comment: 23 pages; 5 figures; v2 includes minor changes in text and additional references

We construct a superpotential for the general N=1/2 supersymmetric gauge theory coupled to chiral matter in the fundamental and adjoint representations, and investigate the one-loop renormalisability of the theories., Comment: 67 pages, including 17 figures. Plain TeX. Uses Harvmac and epsf. Combined and condensed version of hep-th/0607194 and hep-th/0607195 with some added material including in particular a generalisation of the Lunin-Rey classification of potentially divergent operators

We construct the general N=1/2 supersymmetric gauge theory coupled to massive chiral matter, and show that it is renormalisable at one loop., Comment: 23 pages, including 6 figures. Plain TeX. Uses Harvmac and epsf. References added and typos corrected

We construct a superpotential for the general N=1/2 supersymmetric gauge theory coupled to chiral matter in the adjoint representation, and investigate the one-loop renormalisability of the theory., Comment: 44 pages, including 15 figures. Plain TeX. Uses Harvmac and epsf. References added and typos corrected

We show that the renormalisation of the N=1 supersymmetric gauge theory when working in the component formalism, without eliminating auxiliary fields and using a standard covariant gauge, requires a non-linear renormalisation of the auxiliary fields., Comment: 9 pages, including 4 figures. Plain TeX. Uses Harvmac and epsf; reference added and minor changes to Introduction

We investigate the one-loop renormalisability of a general N=1/2 supersymmetric gauge theory coupled to chiral matter, and show the existence of an N=1/2 supersymmetric SU(N)xU(1) theory which is renormalisable at one loop., Comment: 30 pages, including 8 figures. Plain TeX. Uses Harvmac and epsf

We show that N=1/2 supersymmetric gauge theory is renormalisable at one loop, but only after gauge invariance is restored in a non-trivial fashion., Comment: 11 pages, including 3 figures. Plain TeX. Uses Harvmac and epsf. Minor changes and reference added

We give an explicit construction of general classical solutions for the noncommutative CP^(N-1) model in two dimensions, showing that they correspond to integer values for the action and topological charge. We also give explicit solutions for the Dirac equation in the background of these general solutions and show that the index theorem is satisfied., Comment: 11 pages, Harvmac. Typos corrected, references added

We consider the ultra-violet divergence structure of general noncommutative supersymmetric $U(N_c)$ gauge theories, and seek theories which are all-orders finite., Comment: 11 pages, Tex, one figure. Uses harvmac (big) and epsf

We discuss the ultra-violet properties of bosonic and supersymmetric noncommutative non-linear sigma-models in two dimensions, both with and without a Wess-Zumino-Witten term., Comment: 10 pages, including 1 figure. Plain TeX. Uses Harvmac and epsf. Minor revisions and additional reference. Version to be published in Phys. Lett. B

We establish the ultra-violet finiteness of various classes of noncommutative gauge theories., Comment: 10 pages, 4 figures, TeX. Uses Harvmac (big) and epsf. Some minor typos corrected and added references in v2

We calculate the three loop contribution to the beta-function of the gauge coupling constant in a general, anomaly-free, renormalisable gauge field theory involving a single gauge coupling using the background field method in the MSbar scheme., Comment: 10 latex pages

We present an elegant exact formula for the gaugino $\beta$-function in a softly-broken supersymmetric gauge theory, of the form $\beta_M = {\cal O}(\beta_g/g)$, where $\beta_g$ is the gauge $\beta$ function and ${\cal O}$ is a simple differential operator acting on the gauge coupling $g$ and the Yukawa coupling., Comment: 11 pages, tex, no figures. Uses harvmac. Minor error in derivation of Eq. (14) corrected

We present calculations of the leading and O(1/N) terms in a large-N expansion of the \beta-functions for various supersymmetric theories: a Wess-Zumino model, supersymmetric QED and a non-abelian supersymmetric gauge theory. In all cases N is the number of a class of the chiral superfields in the theory., Comment: 9 pages, tex, five figures. Uses harvmac and epsf. Revised to include a discussion of higher loop DRED ambiguities. Some references added, and notation clarified

We calculate the three loop anomalous dimension for a general $N=1$ supersymmetric gauge theory. The result is used to probe the possible existence of renormalisation invariant relationships between the Yukawa and gauge couplings., Comment: 18 pages. Uses Harvmac. Revised version includes discussion of the special case of the Wess-Zumino model

The two-dimensional supersymmetric $\s$-model on a K\"ahler manifold has a non-vanishing $\b$-function at four loops, but the $\b$-function at five loops can be made to vanish by a specific choice of renormalisation scheme. We investigate whether this phenomenon persists at six loops, and conclude that it does not; there is a non-vanishing six-loop $\b$-function irrespective of renormalisation scheme ambiguities., Comment: 13pp, uses phyzzx, LTH 287 (published in Int. J. Mod. Phys. A)

We derive an exact expression for the tachyon $\beta$-function for the Wess-Zumino-Witten model. We check our result up to three loops by calculating the three-loop tachyon $\beta$-function for a general non-linear $\sigma$-model with torsion, and then specialising to the case of the WZW model., Comment: 13pp, uses harvmac (with figures appended as a a standard PostScript file), LTH 317

We derive an explicit, exactly conformally invariant form for the action for the non-abelian Toda field theory. We demonstrate that the conformal invariance conditions, expressed in terms of the $\beta$-functions of the theory, are satisfied to all orders, and we use our results to obtain a value for the central charge agreeing with previous calculations., Comment: 28pp, harvmac (choose "b" option)

We show that Witten's two-dimensional string black hole metric is exactly conformally invariant in the supersymmetric case. We also demonstrate that this metric, together with a recently proposed exact metric for the bosonic case, are respectively consistent with the supersymmetric and bosonic $\sigma$-model conformal invariance conditions up to four-loop order., Comment: 14p