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One of the obstacles to reconciling quantum theory with general relativity, is constructing a theory which is both consistent with observation, and and gives finite answers at high energy, so that the theory holds at arbitrarily short distances. Quantum field theory achieves this through the process of renormalisation, but famously, perturbative quantum gravity fails to be renormalisable, even without coupling to matter. Recently, an alternative to quantum gravity has been proposed, in which the geometry of spacetime is taken to be classical rather than quantum, while still being coupled to quantum matter fields [1, 2]. This can be done consistently, provided the dynamics is fundamentally stochastic. Here, we find that the pure gravity theory is formally renormalisable. We do so via the path integral formulation by relating the classical-quantum action to that of quadratic gravity which is renormalisable. Because the action induces stochastic dynamics of space-time, rather than deterministic evolution of a quantum field, the classical-quantum theory is free of tachyons and negative norm ghosts. The key remaining question is whether the renormalisation prescription retains completely positive (CP) dynamics. This consideration appears to single out the scale invariant and asymptotically free theory. We give further evidence that the theory is CP, by showing that the two-point function of the scalar mode is positive. To support the use of precision accelerometers in testing the quantum nature of spacetime, we also compute the power spectral density of the acceleration. The results presented here have a number of implications for inflation, CMB data, and experiments to test the quantum nature of spacetime. They may also provide a way to compute probabilities in the regime of quantum gravity where spacetime can be treated as effectively classical., Comment: v2: PSD pole prescription fand power spectral density for the acceleration added

If the metric is chosen to depend exponentially on the conformal factor, and if one works in a gauge where the conformal factor has the wrong sign propagator, perturbative quantum gravity corrections can be partially resummed into a series of terms each of which is ultraviolet finite. These new terms however are not perturbative in some small parameter, and are not individually BRST invariant, or background diffeomorphism invariant. With appropriate parametrisation, the finiteness property holds true also for a full phenomenologically relevant theory of quantum gravity coupled to (beyond the standard model) matter fields, provided massive tadpole corrections are set to zero by a trivial renormalisation., Comment: 39 pages. Further comments and clarifications added. Version accepted for publication in Jour Class and Quant Gravity

We investigate off-shell perturbative renormalisation of pure quantum gravity for both background metric and quantum fluctuations. We show that at each new loop order, the divergences that do not vanish on-shell are constructed from only the total metric, whilst those that vanish on-shell are renormalised by canonical transformations involving the quantum fields. Purely background metric divergences do not separately appear, and the background metric does not get renormalised. We highlight that renormalisation group identities play a crucial role ensuring consistency in the renormalisation of BRST transformations beyond one loop order. We verify these assertions by computing leading off-shell divergences to two loops, exploiting off-shell BRST invariance and the renormalisation group equations. Although some divergences can be absorbed by field redefinitions, we explain why this does not lead to finite beta-functions for the corresponding field., Comment: 45 pages

We study $d$-dimensional scalar field theory in the Local Potential Approximation of the functional renormalization group. Sturm-Liouville methods allow the eigenoperator equation to be cast as a Schrodinger-type equation. Combining solutions in the large field limit with the Wentzel-Kramers-Brillouin approximation, we solve analytically for the scaling dimension $d_n$ of high dimension potential-type operators $\mathcal{O}_n(\varphi)$ around a non-trivial fixed point. We find that $d_n = n(d-d_\varphi)$ to leading order in $n$ as $n \to \infty$, where $d_\varphi=\frac{1}{2}(d-2+\eta)$ is the scaling dimension of the field, $\varphi$, and determine the power-law growth of the subleading correction. For $O(N)$ invariant scalar field theory, the scaling dimension is just double this, for all fixed $N\geq0$ and additionally for $N=-2,-4,\ldots \,.$ These results are universal, independent of the choice of cutoff function which we keep general throughout, subject only to some weak constraints., Comment: 23 pages, no figures. Clarifications added. Version published in PRD

This article is a review of functional $f(R)$ approximations in the asymptotic safety approach to quantum gravity. It mostly focusses on a formulation that uses a non-adaptive cutoff, resulting in a second order differential equation. This formulation is used as an example to give a detailed explanation for how asymptotic analysis and Sturm-Liouville analysis can be used to uncover some of its most important properties. In particular, if defined appropriately for all values $-\infty

We explore the properties of a recently proposed background independent exact renormalization group approach to gauge theories and gravity. In the process we also develop the machinery needed to study it rigorously. The proposal comes with some advantages. It preserves gauge invariance manifestly, avoids introducing unphysical fields, such as ghosts and Pauli-Villars fields, and does not require gauge-fixing. However, we show that in the simple case of SU(N) Yang-Mills it does not completely regularise the longitudinal part of vertex functions already at one loop, invalidating certain methods for extracting universal components. Moreover we demonstrate a kind of no-go theorem: within the proposed structure, whatever choice is made for covariantisation and cutoff profiles, the two-point vertex flow equation at one loop cannot be both transverse, as required by gauge invariance, and fully regularised., Comment: 51 pages, no figures. Minor corrections. Version published in PRD

Interactions growing slower than a certain exponential of the square of a scalar field, are well behaved when evolved under the functional renormalization group linearised around the Gaussian fixed point. They satisfy properties usually taken for granted, and reproduce standard perturbative quantisation. However, ever more challenging effects appear the more interactions grow faster than this. We show explicitly that firstly the flow no longer splits uniquely into operators of definite scaling dimension; then (linearised) flows to the infrared can end prematurely in a singularity; and finally new interactions can spontaneously appear at any scale., Comment: 19 pages

We study an $f(R)$ approximation to asymptotic safety, using a family of non-adaptive cutoffs, kept general to test for universality. Matching solutions on the four-dimensional sphere and hyperboloid, we prove properties of any such global fixed point solution and its eigenoperators. For this family of cutoffs, the scaling dimension at large $n$ of the $n^\text{th}$ eigenoperator, is $\lambda_n\propto b\, n\ln n$. The coefficient $b$ is non-universal, a consequence of the single-metric approximation. The large $R$ limit is universal on the hyperboloid, but not on the sphere where cutoff dependence results from certain zero modes. For right-sign conformal mode cutoff, the fixed points form at most a discrete set. The eigenoperator spectrum is quantised. They are square integrable under the Sturm-Liouville weight. For wrong sign cutoff, the fixed points form a continuum, and so do the eigenoperators unless we impose square-integrability. If we do this, we get a discrete tower of operators, infinitely many of which are relevant. These are $f(R)$ analogues of novel operators in the conformal sector which were used recently to furnish an alternative quantisation of gravity., Comment: 35 pages, no figures; minor amendments. version published in JHEP

We show that perturbative quantum gravity based on the Einstein-Hilbert action, has a novel continuum limit. The renormalized trajectory emanates from the Gaussian fixed point along (marginally) relevant directions but enters the diffeomorphism invariant subspace only well below a dynamically generated scale. We show that for pure quantum gravity to second order in perturbation theory, and with vanishing cosmological constant, the result is the same as computed in the standard quantisation. Although this case is renormalizable at second order for kinematic reasons, the structure we uncover works in general. One possibility is that gravity has a genuine consistent continuum limit even though it has an infinite number couplings. However we also suggest a possible non-perturbative mechanism, based on the parabolic properties of these flow equations, which would fix all higher order couplings in terms of Newton's constant and the cosmological constant., Comment: 48 pages. 1 figure. 8 more pages with clarifications and new section on gauge parameter (in)dependence. To be publ in CQG

Recently a novel perturbative continuum limit for quantum gravity has been proposed and demonstrated to work at first order. Every interaction monomial $\sigma$ is dressed with a coefficient function $f^\sigma_\Lambda(\varphi)$ of the conformal factor field, $\varphi$. Each coefficient function is parametrised by an infinite number of underlying couplings, and decays at large $\varphi$ with a characteristic amplitude suppression scale which can be chosen to be at a common value, $\Lambda_\text{p}$. Although the theory is perturbative in couplings it is non-perturbative in $\hbar$. At second order in perturbation theory, one must sum over all melonic Feynman diagrams to obtain the particular integral. We show that it leads to a well defined renormalized trajectory and thus continuum limit, provided it is solved by starting at an arbitrary cutoff scale $\Lambda=\mu$ which lies in the range $0<\mu

A manifestly diffeomorphism invariant exact renormalization group requires extra diffeomorphism invariant ultraviolet regularisation at some effective cutoff scale $\Lambda$. This motivates construction of a `Parisi-Sourlas' supergravity, in analogy with the gauge theory case, where the superpartner fields have the wrong spin-statistics such that they can become Pauli-Villars regulator fields after spontaneous symmetry breaking. We show that in contrast to gauge theory, the free theory around flat space is already non-trivial and in a sense already displays some spontaneous symmetry breaking. We show that the fluctuating fields form multiplets whose mass matrices imply that the fields propagate into each other not only with the expected $1/p^2$ but also through propagators with improved ultraviolet properties, namely $1/p^4$ and $1/p^6$, despite the fact that the action contains a maximum of two space-time derivatives., Comment: 30 pages. Corrected typos. Further explained our strategy and relation to standard supergravity. Version accepted for Classical & Quantum Gravity

The Wilsonian renormalization group (RG) properties of the conformal factor of the metric are profoundly altered by the fact that it has a wrong-sign kinetic term. The result is a novel perturbative continuum limit for quantum gravity, which is however non-perturbative in $\hbar$. The ultraviolet part of the renormalized trajectory lies outside the diffeomorphism invariant subspace, entering this subspace only in the infrared, below a dynamically generated amplitude suppression scale. Interactions are dressed with coefficient functions of the conformal factor, their form being determined by the RG. In the ultraviolet, the coefficient functions are parametrised by an infinite number of underlying couplings. Choosing these couplings appropriately, the coefficient functions trivialise on entering the diffeomorphism invariant subspace. Here, dynamically generated effective diffeomorphism couplings emerge, including Newton's constant. In terms of the Legendre effective action, we establish the continuum limit to first order, characterising the most general form of such coefficient functions so as to verify universality., Comment: 47 pages, 2 figures; Expanded version including discussion section. Version to be publ in JHEP

We show that the Quantum Master Equation and the Wilsonian renormalization group (RG) flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naively break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov-Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov-Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to choice of renormalization scheme., Comment: 58 pages, one figure; Added comments to make clear where the work is (non)perturbative in coupling and Planck's constant. Version to be published in PTEP

The Effective Average Action is a form of effective action which depends on a cutoff scale suppressing the contribution of low momentum modes in the functional integral. It reduces to the ordinary effective action when the cutoff scale goes to zero. We derive the modifications of the scale Ward identity due to this cutoff and show how the resulting identity then intimately relates the trace anomaly to the Wilsonian realisation of the renormalization group., Comment: 23 pages, 2 figures; references added + extra discussion. Version to appear in PRD

We show that the Wilsonian renormalization group (RG) provides a natural regularisation of the Quantum Master Equation such that to first order the BRST algebra closes on local functionals spanned by the eigenoperators with constant couplings. We then apply this to quantum gravity. Around the Gaussian fixed point, RG properties of the conformal factor of the metric allow the construction of a Hilbert space $\mathfrak{L}$ of renormalizable interactions, non-perturbative in $\hbar$, and involving arbitrarily high powers of the gravitational fluctuations. We show that diffeomorphism invariance is violated for interactions that lie inside $\mathfrak{L}$, in the sense that only a trivial quantum BRST cohomology exists for interactions at first order in the couplings. However by taking a limit to the boundary of $\mathfrak{L}$, the couplings can be constrained to recover Newton's constant, and standard realisations of diffeomorphism invariance, whilst retaining renormalizability. The limits are sufficiently flexible to allow this also at higher orders. This leaves open a number of questions that should find their answer at second order. We develop much of the framework that will allow these calculations to be performed., Comment: 64 pages

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. In particular around the Gaussian fixed point, it supports a Hilbert space of renormalizable interactions involving arbitrarily high powers of the gravitational fluctuations. These interactions are characterised by being exponentially suppressed for large field amplitude, perturbative in Newton's constant but non-perturbative in Planck's constant. By taking a limit to the boundary of the Hilbert space, diffeomorphism invariance is recovered whilst retaining renormalizability. Thus the so-called conformal factor instability points the way to constructing a perturbatively renormalizable theory of quantum gravity., Comment: 5 pages; Received Honourable mention in Gravity Research Foundation essay competition-2018

The Wilsonian renormalization group properties of the conformal factor of the metric are profoundly altered by the fact that it has a wrong-sign kinetic term. If couplings are chosen so that the quantum field theory exists on $\mathbb{R}^4$, it fails to exist on manifolds below a certain size, if a certain universal shape function turns negative. We demonstrate that this is triggered by inhomogeneity in the cases of $\mathbb{T}^4$ and $\mathbb{T}^3\times\mathbb{R}$, including twisted versions. Varying the moduli, we uncover a rich phenomenology., Comment: 25 pages, 5 figures; version to be published in Class. Quantum Grav

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. Generically for the conformal sector, complete flows exist only in the reverse direction (i.e. from the infrared to the ultraviolet). The Gaussian fixed point supports infinite sequences of composite eigenoperators of increasing infrared relevancy (increasingly negative mass dimension), which are orthonormal and complete for bare interactions that are square integrable under the appropriate measure. These eigenoperators are non-perturbative in $\hbar$ and evanescent. For $\mathbb{R}^4$ spacetime, each renormalised physical operator exists but only has support at vanishing field amplitude. In the generic case of infinitely many non-vanishing couplings, if a complete RG flow exists, it is characterised in the infrared by a scale $\Lambda_\mathrm{p}>0$, beyond which the field amplitude is exponentially suppressed. On other spacetimes, of length scale $L$, the flow ceases to exist once a certain universal measure of inhomogeneity exceeds $O(1)+2\pi L^2\Lambda^2_\mathrm{p}$. Importantly for cosmology, the minimum size of the universe is thus tied to the degree of inhomogeneity, with spacetimes of vanishing size being required to be almost homogeneous. We initiate a study of this exotic quantum field theory at the interacting level, and discuss what the full theory of quantum gravity should look like, one which must thus be perturbatively renormalizable in Newton's constant but non-perturbative in $\hbar$., Comment: 52 pages, 4 figures; fixed typos; improved explanation of the sign of V, and the use of Sturm-Liouville theory. To be publ in JHEP

We show how the Weyl anomaly generated by gauge fields, can be computed from manifestly gauge invariant and diffeomorphism invariant exact renormalization group equations, without having to fix the gauge at any stage. Regularisation is provided by covariant higher derivatives and by embedding the Maxwell field into a spontaneously broken $U(1|1)$ supergauge theory. We first provide a realisation that leaves behind two versions of the original $U(1)$ gauge field, and then construct a manifestly $U(1|1)$ supergauge invariant flow equation which leaves behind only the original Maxwell field in the spontaneously broken regime., Comment: 24 pages

We explain how to find the asymptotic form of fixed point solutions in functional truncations, in particular $f(R)$ approximations. We find that quantum fluctuations do not decouple at large $R$, typically leading to elaborate asymptotic solutions containing several free parameters. By a counting argument, these can be used to map out the dimension of the fixed point solution spaces. They are also necessary to validate the numerical solution, and provide the physical part in the limit that the cutoff is removed: the fixed point equation of state. As an example we apply the techniques to a recent $f(R)$ approximation by Demmel et al, finding asymptotic matches to their numerical solution. Depending on the value of the endomorphism parameter, we find many other asymptotic solutions and fixed point solution spaces of differing dimensions, yielding several alternative scenarios for the equation of state. Asymptotic studies of other $f(R)$ approximations are needed to clarify the picture., Comment: Minor corrections. Version to be published in PRD

In single-metric approximations to the exact renormalization group (RG) for quantum gravity, it has been not been clear how to treat the large curvature domain beyond the point where the effective cutoff scale $k$ is less than the lowest eigenvalue of the appropriate modified Laplacian. We explain why this puzzle arises from background dependence, resulting in Wilsonian RG concepts being inapplicable. We show that when properly formulated over an ensemble of backgrounds, the Wilsonian RG can be restored. This in turn implies that solutions should be smooth and well defined no matter how large the curvature is taken. Even for the standard single-metric type approximation schemes, this construction can be rigorously derived by imposing a modified Ward identity (mWI) corresponding to rescaling the background metric by a constant factor. However compatibility in this approximation requires the space-time dimension to be six. Solving the mWI and flow equation simultaneously, new variables are then derived that are independent of overall background scale., Comment: 34 pages, 2 figures; various small improvements

We study the quantum behaviour of a particle moving in a one-dimensional double well potential. This double well is obtained by gluing together, at the origin, two shifted harmonic oscillator potentials. The Schr\"odinger equation is exactly solvable. The requirement that discontinuities, in the wavefunction and its first derivative, are absent at the origin, leads to the quantisation of the energy eigenvalues. We also show that oscillations in time take place between two nearby single harmonic oscillator ground states. Finally, the double well potential is augmented by a Dirac delta-function potentials at the origin and the corresponding Schr\"odinger equation is solved., Comment: This paper is withdrawn because the model is already treated in a substantially similar way in Merzbacher's text-book

The $\mathcal{O}(\partial^2)$ background independent flow equations for conformally reduced gravity are shown to be equivalent to flow equations naturally adapted to scalar field theory with a wrong sign kinetic term. This sign change is shown to have a profound effect on the renormalization group properties, broadly resulting in a continuum of fixed points supporting both a discrete and a continuous eigenoperator spectrum, the latter always including relevant directions. The properties at the Gaussian fixed point are understood in particular depth, but also detailed studies of the local potential approximation, and the full $\mathcal{O}(\partial^2)$ approximation are given. These results are related to evidence for asymptotic safety found by other authors., Comment: 58 pages, 15 figures. Added section that shows that for any sensible cutoff profile, space-time and field dimension, the LPA for standard scalar field theory can only have a discrete set of fixed points, while the sign change of the kinetic term results instead in a continuum of fixed points

We present an exact RG (renormalization group) analysis of $O(N)$-invariant scalar field theory about the Gaussian fixed point. We prove a series of statements that taken together show that the non-polynomial eigen-perturbations found in the LPA (local potential approximation) at the linearised level, do not lead to new interactions, \textit{i.e.} enlarge the universality class, neither in the LPA or treated exactly. Non-perturbatively, their RG flow does not emanate from the fixed point. For the equivalent Wilsonian effective action they can be re-expressed in terms of the usual couplings to polynomial interactions, which can furthermore be tuned to be as small as desired for all finite RG time. For the infrared cutoff Legendre effective action, this can also be done for the infrared evolution. We explain why this is nevertheless consistent with the fact that the large field behaviour is fixed by these perturbations., Comment: 32 pages, 1 figure; extended to exact treatment

Within the derivative expansion of conformally reduced gravity, the modified split Ward identities are shown to be compatible with the flow equations if and only if either the anomalous dimension vanishes or the cutoff profile is chosen to be power law. No solutions exist if the Ward identities are incompatible. In the compatible case, a clear reason is found for why Ward identities can still forbid the existence of fixed points; however, for any cutoff profile, a background independent (and parametrisation independent) flow equation is uncovered. Finally, expanding in vertices, the combined equations are shown generically to become either over-constrained or highly redundant beyond the six-point level., Comment: 31 pages, 4 figures

We construct a manifestly diffeomorphism invariant Wilsonian (Exact) Renormalization Group for classical gravity, and begin the construction for quantum gravity. We demonstrate that the effective action can be computed without gauge fixing the diffeomorphism invariance, and also without introducing a background space-time. We compute classical contributions both within a background-independent framework and by perturbing around a fixed background, and verify that the results are equivalent. We derive the exact Ward identities for actions and kernels and verify consistency. We formulate two forms of the flow equation corresponding to the two choices of classical fixed-point: the Gaussian fixed point, and the scale invariant interacting fixed point using curvature-squared terms. We suggest how this programme may completed to a fully quantum construction., Comment: 50 pages, 2 figures, minor typos fixed, minor additions to the Discussion, one reference added

Starting from a full renormalised trajectory for the effective average action (a.k.a. infrared cutoff Legendre effective action) $\Gamma_k$, we explicitly reconstruct corresponding bare actions, formulated in one of two ways. The first step is to construct the corresponding Wilsonian effective action $S^k$ through a tree-level expansion in terms of the vertices provided by $\Gamma_k$. It forms a perfect bare action giving the same renormalised trajectory. A bare action with some ultraviolet cutoff scale $\Lambda$ and infrared cutoff $k$ necessarily produces an effective average action $\Gamma^\Lambda_k$ that depends on both cutoffs, but if the already computed $S^\Lambda$ is used, we show how $\Gamma^\Lambda_k$ can also be computed from $\Gamma_k$ by a tree-level expansion, and that $\Gamma^\Lambda_k\to\Gamma_k$ as $\Lambda\to\infty$. Along the way we show that Legendre effective actions with different UV cutoff profiles, but which correspond to the same Wilsonian effective action, are related through tree-level expansions. All these expansions follow from Legendre transform relationships that can be derived from the original one between $\Gamma^\Lambda_k$ and $S^k$., Comment: 32 pages

We argue, at a very basic effective field theory level, that higher dimension operators in scalar theories that break symmetries at scales close to their ultraviolet completion cutoff, include terms that favour the breaking of translation (Lorentz) invariance, potentially resulting in striped, chequered board or general crystal-like phases. Such descriptions can be thought of as the effective low energy description of QCD-like gauge theories near their strong coupling scale where terms involving higher dimension operators are generated. Our low energy theory consists of scalar fields describing operators such as $\bar{q} q$ and $\bar{q} F^{(2n)} q$. Such scalars can have kinetic mixing terms that generate effective momentum dependent contributions to the mass matrix. We show that these can destabilize the translationally invariant vacuum. It is possible that in some real gauge theory such operators could become sufficiently dominant to realize such phases and it would be interesting to look for them in lattice simulations. We present a holographic model of the same phenomena which includes RG running. A key phenomenological motive to look at such states is recent work that shows that the non-linear response in $R^2$ gravity to such short range fluctuations can mimic a cosmological constant. Intriguingly in a cosmology with such a Starobinsky inflation term, to generate the observed value of the present day acceleration would require stripes at the electroweak scale. Unfortunately, low energy phenomenological constraints on Lorentz violation in the electron-photon system appear to strongly rule out any such possibility outside of a disconnected dark sector., Comment: 9 pages, 1 figure; minor changes. Version to be published in PRD

Within the conformally reduced gravity model, where the metric is parametrised by a function $f(\phi)$ of the conformal factor $\phi$, we keep dependence on both the background and fluctuation fields, to local potential approximation and $\mathcal{O}(\partial^2)$ respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale $k$ is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to $k$. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale, $\hat k$. The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of $f$. In general $f(\phi)$ is forced to be scale dependent and needs to be renormalised, but if this is avoided then $k$-fixed points are allowed and furthermore they coincide with $\hat k$-fixed points., Comment: 53 pages, broken reference corrected

We study the effective stress-energy tensor induced by cosmological inhomogeneity in $f(R)=R+cR^2$ and equivalent scalar-tensor theories, motivated both by models of early universe inflation and by phenomenological alternative cosmologies to the standard $\Lambda$-CDM. We use Green and Wald's framework for averaging over classical fluctuations of short-wavelength $\lambda$. By ensuring that the leading non-linear terms from the fluctuations of the Einstein terms and the corrections both contribute in the formal limit as $\lambda\to0$, we derive a diffeomorphism invariant effective stress-energy tensor whose trace is non-vanishing and of the right sign to potentially account for the current acceleration of the universe. However a more phenomenologically acceptable dark energy model would be required if this effect were to fully account for the current acceleration., Comment: 23 pages, references updated, some arguments clarified

Working within the familiar local potential approximation, and concentrating on the example of a single scalar field in three dimensions, we show that the commonly used approximation method of identifying the total and background fields, leads to pathologies in the resulting fixed point structure and the associated spaces of eigenoperators. We then show how a consistent treatment of the background field through the corresponding modified shift Ward identity, can cure these pathologies, restoring universality of physical quantities with respect to the choice of dependence on the background field, even within the local potential approximation. Along the way we point out similarities to what has been previously found in the f(R) approximation in asymptotic safety for gravity., Comment: 40 pages, version accepted by JHEP

In this paper we review the definition and properties of redundant operators in the exact renormalisation group. We explain why it is important to require them to be eigenoperators and why generically they appear only as a consequence of symmetries of the particular choice of renormalisation group equations. This clarifies when Newton's constant and or the cosmological constant can be considered inessential. We then apply these ideas to the Local Potential Approximation and approximations of a similar spirit such as the f(R) approximation in the asymptotic safety programme in quantum gravity. We show that these approximations can break down if the fixed point does not support a `vacuum' solution in the appropriate domain: all eigenoperators become redundant and the physical space of perturbations collapses to a point. We show that this is the case for the recently discovered lines of fixed points in the f(R) flow equations., Comment: 29 pages; improvements in presentation & correction of a subsidiary result, as reflected in the new abstract; typos corrected; references added

In the asymptotic safety programme for quantum gravity, it is important to go beyond polynomial truncations. Three such approximations have been derived where the restriction is only to a general function f(R) of the curvature R>0. We confront these with the requirement that a fixed point solution be smooth and exist for all non-negative R. Singularities induced by cutoff choices force the earlier versions to have no such solutions. However, we show that the most recent version has a number of lines of fixed points, each supporting a continuous spectrum of eigen-perturbations. We uncover and analyse the first five such lines. Sensible fixed point behaviour may be achieved if one consistently incorporates geometry/topology change. As an exploratory example, we analyse the equations analytically continued to R<0, however we now find only partial solutions.We show how these results are always consistent with, and to some extent can be predicted from, a straightforward analysis of the constraints inherent in the equations., Comment: Latex, 66 pages, published version, typos corrected

The canonical transformation that maps light-cone Yang-Mills theory to a Lagrangian description of the MHV rules is non-local, consequently the two sets of fields do not necessarily generate the same S-matrix. By deriving a new recursion relation for the canonical transformation expansion coefficients, we find a direct map between these coefficients and tree level light-cone diagrams. We use this to show that, at least up to one-loop with dimensionally regularised MHV vertices, the only difference is the omission of the one-loop amplitudes in which all gluons have positive helicity., Comment: 36 pages, JHEP3 style, published version, discussions on rational parts and supersymmetry added

The similarity of massive CSW scalar vertices and quark vertices can be understood using a kind of light-cone SUSY transformation presented in this paper. We also show that the canonical transformation generating the MHV-SQCD lagrangian, can be fixed by applying this light-cone SUSY transformation to the canonical transformation for MHV-QCD obtained in paper arxiv:0805.0239. Most of the massive CSW vertices for SQCD can also be pinned down in this way., Comment: 24 pages, JHEP3 style; minor typoes corrected; appendix A is moved into the main text; published version

We perform a canonical change of the field variables of light-cone gauge massless QCD to obtain a lagrangian whose terms are proportional up to polarisation factors to MHV amplitudes and continued off shell by the CSW prescription. We solve for this transformation as a series expansion to all orders in the new fields, and use this to prove that the resulting vertices are indeed MHV vertices as claimed. We also demonstrate how this works explicitly for the vertices with: two quarks and two gluons, four quarks, and a particular helicity configuration of two quarks and three gluons. Finally, we generalise the construction to massive QCD., Comment: **Important New Material:** Proof that vertices are MHV amps off-shell, **Plus** results for all MHV verts with massive quarks. Clarifications made, esp. on the link between massless quark-gluon vertices and known results, and the choice of massive poln. spinors. Citations revised. 42 pages, 2 figures, 4 tables. (This version published in JHEP.)

We demonstrate that the canonical change of variables that yields the MHV lagrangian, also provides contributions to scattering amplitudes that evade the equivalence theorem. This `ET evasion' in particular provides the tree-level (-++) amplitude, which is non-vanishing off shell, or on shell with complex momenta or in (2,2) signature, and is missing from the MHV (aka CSW) rules. At one loop there are ET-evading diagrammatic contributions to the amplitudes with all positive helicities. We supply the necessary regularisation in order to define these contributions (and quantum MHV methods in general) by starting from the light-cone Yang-Mills lagrangian in D dimensions and making a canonical change of variables for all D-2 transverse degrees of freedom of the gauge field. In this way, we obtain dimensionally regularised three- and four-point MHV amplitudes. Returning to the one-loop (++++) amplitude, we demonstrate that its quadruple cut coincides with the known result, and show how the original light-cone Yang-Mills contributions can in fact be algebraically recovered from the ET-evading contributions. We conclude that the canonical MHV lagrangian, supplemented with the extra terms brought to correlation functions by the non-linear field transformation, provide contributions which are just a rearrangement of those from light-cone Yang-Mills and thus coincide with them both on and off shell., Comment: 32 pages, 9 figures. v2: corrects typos and Fig. 8; adds further citations and acknowledgements

Building on recent work in SU(N) Yang-Mills theory, we construct a manifestly gauge invariant exact renormalization group for QCD. A gauge invariant cutoff is constructed by embedding the physical gauge theory in a spontaneously broken SU(N|N) gauge theory, regularized by covariant higher derivatives. Intriguingly, the construction is most efficient if the number of flavours is a multiple of the number of colours. The formalism is illustrated with a very compact calculation of the one-loop beta function, achieving a manifestly universal result and without fixing the gauge., Comment: 30 pages, 6 figures

We construct a manifestly gauge invariant Exact Renormalization Group for SU(N) Yang-Mills theory, in a form suitable for calculations without gauge fixing at any order of perturbation theory. The effective cutoff is incorporated via a manifestly realised spontaneously broken SU(N|N) gauge invariance. Diagrammatic methods are developed which allow the calculations to proceed without specifying the precise form of the cutoff structure. We confirm consistency by computing for the first time both the one and two loop beta function coefficients without fixing the gauge or specifying the details of the cutoff. We sketch how to incorporate quarks and thus compute in QCD. Finally we analyse the renormalization group behaviour as the renormalized coupling becomes large, and show that confinement is a consequence if and only if the coupling diverges in the limit that all modes are integrated out. We also investigate an expansion in the inverse square renormalized coupling, and show that under general assumptions it yields a new non-perturbative approximation scheme corresponding to expanding in 1/\Lambda_{QCD}., Comment: 24 pages, 14 figures, Fields Inst style file; Talk presented by TRM at RG2005, Helsinki, Finland, September 2005 and Renormalization and Universality in Mathematical Physics Workshop, Fields Institute, Toronto, Canada, October 2005, extended to include more details on the strong renormalized coupling expansion. To be publ. as proceedings by the Fields Institute

Recently, a canonical change of field variables was proposed that converts the Yang-Mills Lagrangian into an MHV-rules Lagrangian, i.e. one whose tree level Feynman diagram expansion generates CSW rules. We solve the relations defining the canonical transformation, to all orders of expansion in the new fields, yielding simple explicit holomorphic expressions for the expansion coefficients. We use these to confirm explicitly that the three, four and five point vertices are proportional to MHV amplitudes with the correct coefficient, as expected. We point out several consequences of this framework, and initiate a study of its implications for MHV rules at the quantum level. In particular, we investigate the wavefunction matching factors implied by the Equivalence Theorem at one loop, and show that they may be taken to vanish in dimensional regularisation., Comment: 23 pages, 1 figure. Makes major improvements on withdrawn paper hep-th/0605024

This paper has been withdrawn by the authors. A substantially altered paper will be submitted incorporating major corrections, additions and improvements., Comment: This paper has been withdrawn by the authors. A substantially altered paper will be submitted incorporating major corrections, additions and improvements

A field theoretic understanding of how the radial direction in the AdS/CFT Correspondence plays the role of a gauge invariant measure of energy scale has long been missing. In SU(N) Yang-Mills, a realization of a gauge invariant cutoff has been achieved by embedding the theory in spontaneously broken SU(N|N) gauge theory. With the recent discovery of ghost D-branes an AdS/CFT Correspondence version of this scheme is now possible. We show that a very simple construction precisely ties the two pictures together providing a concrete understanding of the radial RG flow on the field theory side., Comment: 6 pages, 1 figure. v2: refs. added, typos corrected

The manifestly gauge invariant Exact Renormalisation Group provides a framework for performing continuum computations in SU(N) Yang-Mills theory, without fixing the gauge. We use this formalism to compute the two-loop beta function in a manifestly gauge invariant way, and without specifying the details of the regularisation scheme., Comment: 69 pages, 29 figures. v2: published in prd; methodology refined, typos corrected

We take the manifestly gauge invariant exact renormalisation group previously used to compute the one-loop beta function in SU(N) Yang-Mills without gauge fixing, and generalise it so that it can be renormalised straightforwardly at any loop order. The diagrammatic computational method is developed to cope with general group theory structures, and new methods are introduced to increase its power, so that much more can be done simply by manipulating diagrams. The new methods allow the standard two-loop beta function coefficient for SU(N) Yang-Mills to be computed, for the first time without fixing the gauge or specifying the details of the regularisation scheme., Comment: 91 pages, 58 figures

We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. Calculations can be performed almost entirely diagrammatically. The method is formulated within the framework of an exact renormalisation group for QED. We demonstrate the technique with a calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing., Comment: 32 pages, Diagrams Fixed; version published in JHEP

In recent papers it has been noted that the local potential approximation of the Legendre and Wilson-Polchinski flow equations give, within numerical error, identical results for a range of exponents and Wilson-Fisher fixed points in three dimensions, providing a certain ``optimised'' cutoff is used for the Legendre flow equation. Here we point out that this is a consequence of an exact map between the two equations, which is nothing other than the exact reduction of the functional map that exists between the two exact renormalization groups. We note also that the optimised cutoff does not allow a derivative expansion beyond second order., Comment: 7 pages (improved statement of "central issue", 2 refs added)

Non-Abelian gauge theories may have continuum limits in more than four dimensions, supported by non-trivial ultra-violet fixed points. Moreover, such theories can be expected to be accessible to Wilson's epsilon expansion. We investigate this series for SU(N) Yang-Mills, in particular for the fixed point coupling and critical exponent nu, up to four loops. From the model-building point of view, such theories would be effectively perturbatively renormalizable in the normal way. A particularly attractive possibility is the construction of renormalizable extra-dimensional models of the weak interactions, which have the potential to address the full hierarchy problem. The simplest such gauge-Higgs unification model is however ruled out by a combination of theoretical and phenomenological constraints., Comment: 29 pages, 2 eps figures; extra refs added

We sketch the construction of a gauge invariant Exact Renormalization Group (ERG). Starting from Polchinski's equation, the emphasis is on how a series of ideas have combined to yield the gauge invariant formalism. A novel symmetry of the ERG allows the flow equation to be modified, in such a way that it is suitable for the computation of the (universal) two-loop beta-function. This computation has now been performed, within the framework of the ERG and, as such, in a manifestly gauge invariant way for the very first time., Comment: 10 pages, 1 figure. Proceeding of Quarks-2004, Pushkinskie Gory, Russia, May 24-30. v2: minor corrections compared to published version

We further develop an algorithmic and diagrammatic computational framework for very general exact renormalization groups, where the embedded regularisation scheme, parametrised by a general cutoff function and infinitely many higher point vertices, is left unspecified. Calculations proceed iteratively,by integrating by parts with respect to the effective cutoff, thus introducing effective propagators, and differentials of vertices that can be expanded using the flow equations; many cancellations occur on using the fact that the effective propagator is the inverse of the classical Wilsonian two-point vertex. We demonstrate the power of these methods by computing the beta function up to two loops in massless four dimensional scalar field theory, obtaining the expected universal coefficients, independent of the details of the regularisation scheme., Comment: 16 pages, 20 figures, uses revtex4. One- and two-loop seed action contributions added, typos corrected, version to appear in PRD

We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. The initial stages can even be computed diagrammatically. The method is formulated within the framework of an exact renormalization group for SU(N) Yang-Mills gauge theory, incorporating an effective cutoff through a manifest spontaneously broken SU(N|N) gauge invariance. We demonstrate the technique with a compact calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing, for the first time at finite N., Comment: 54 pages, 24 figures, uses JHEP3.cls