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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 present a design improvement for a recently proposed type of Shack-Hartmann wavefront sensor that uses a cylindrical (lenticular) lenslet array. The improved sensor design uses optical binning and requires significantly fewer detector pixels than the corresponding conventional or cylindrical Shack-Hartmann sensor, and so detector readout noise causes less signal degradation. Additionally, detector readout time is significantly reduced, which reduces the latency for closed loop systems, and data processing requirements. We provide simple analytical noise considerations and Monte-Carlo simulations, and show that the optically binned Shack-Hartmann sensor can offer better performance than the conventional counterpart in most practical situations, and our design is particularly suited for use with astronomical adaptive optics systems., Comment: Accepted for publication in Applied Optics

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

A manifestly gauge invariant exact renormalization group for pure SU(N) Yang-Mills theory is proposed, along with the necessary gauge invariant regularisation which implements the effective cutoff. The latter is naturally incorporated by embedding the theory into a spontaneously broken SU(N|N) super-gauge theory, which guarantees finiteness to all orders in perturbation theory. The effective action, from which one extracts the physics, can be computed whilst manifestly preserving gauge invariance at each and every step. As an example, we give an elegant computation of the one-loop SU(N) Yang-Mills beta function, for the first time at finite N without any gauge fixing or ghosts. It is also completely independent of the details put in by hand, e.g. the choice of covariantisation and the cutoff profile, and, therefore, guides us to a procedure for streamlined calculations., Comment: LaTeX, 14 pages, 12 figs, merged from talks hep-th/0207154 and hep-th/0207153, to be published in Acta Physica Slavonica

Using a gauge invariant exact renormalization group, we show how to compute the effective action, and extract the physics, whilst manifestly preserving gauge invariance at each and every step. As an example we give an elegant computation of the one-loop SU(N) Yang-Mills beta function, for the first time at finite N without any gauge fixing or ghosts. It is also completely independent of the details put in by hand, e.g. the choice of covariantisation and the cutoff profile, and, therefore, guides us to a procedure for streamlined calculations., Comment: 8 pages, 11 figures, talk presented at the fifth international Conference "RG2002", Strba, Slovakia

A manifestly gauge invariant exact renormalization group for pure SU(N) Yang-Mills theory is proposed, allowing gauge invariant calculations, without any gauge fixing or ghosts. The necessary gauge invariant regularisation which implements the effective cutoff, is naturally incorporated by embedding the theory into a spontaneously broken SU(N|N) super-gauge theory. This guarantees finiteness to all orders in perturbation theory., Comment: 8 pages, 2 figures, talk presented at the fifth international Conference "RG 2002", Strba, Slovakia

The standard demand for the quantum partition function to be invariant under the renormalization group transformation results in a general class of exact renormalization group equations, different in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. Such scheme independence is elegantly illustrated in the scalar case by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and in this way the universal result is recovered., Comment: 6 pages, no figures, talk presented at the fifth international Conference "RG 2002", Strba, Slovakia

The requirement that the quantum partition function be invariant under a renormalization group transformation results in a wide class of exact renormalization group equations, differing in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. We demonstrate this scheme independence in four dimensional scalar field theory by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and thus, under very general conditions, the universal result is recovered., Comment: 13 pages, no figures, version to appear in JHEP, includes some minor corrections

We construct a gauge invariant regularisation scheme for pure SU(N) Yang-Mills theory in fixed dimension four or less (for N = infinity in all dimensions), with a physical cutoff scale Lambda, by using covariant higher derivatives and spontaneously broken SU(N|N) supergauge invariance. Providing their powers are within certain ranges, the covariant higher derivatives cure the superficial divergence of all but a set of one-loop graphs. The finiteness of these latter graphs is ensured by properties of the supergroup and gauge invariance. In the limit Lambda tends to infinity, all the regulator fields decouple and unitarity is recovered in the renormalized pure SU(N) Yang-Mills theory. By demonstrating these properties, we prove that the regularisation works to all orders in perturbation theory., Comment: Latex, 43 pages, extended to explain ERG context, preregularisation and why it is unecessary in less than 4 dimensions or at infinite N, and explain issues in early attempts at gauge invariant Pauli Villars regularisation

A gauge invariant Wilsonian effective action is constructed for pure SU(N) Yang-Mills theory by formulating the corresponding flow equation. Manifestly gauge invariant calculations can be performed i.e. without gauge fixing or ghosts. Regularisation is implemented in a novel way which realises a spontaneously broken SU(N|N) supergauge theory. As an example we sketch the computation of the one-loop beta function, performed for the first time without any gauge fixing., Comment: 13 pages, 5 figs, ws-p9-75x6-50.cls (included), epsf. Talk presented at the 2nd Conference on Exact RG, Rome 2000

The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relations. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions., Comment: latex, 4pages, to appear in the proceedings of the Rome workshop on Exact REnormalization Group

The convergence of the derivative expansion of the exact renormalisation group is investigated via the computation of the beta function of massless scalar lambda phi^4 theory. The derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. Convergence of the derivative expansion of the Legendre flow equation is trivial at one loop, but also can occur at two loops and in particular converges for an exponential cutoff., Comment: 6 pages, 3 figures, presented at the 2nd Conference on the Exact RG, Rome 2000

Scheme independence of exact renormalization group equations, including independence of the choice of cutoff function, is shown to follow from general field redefinitions, which remains an inherent redundancy in quantum field theories. Renormalization group equations and their solutions are amenable to a simple formulation which is manifestly covariant under such a symmetry group. Notably, the kernel of the exact equations which controls the integration of modes acts as a field connection along the flow., Comment: TeX, harvmac; 15 pages; ref added + last 2 sentences improved. version accepted for jhep

A manifestly gauge invariant and regularized renormalization group flow equation is constructed for pure SU(N) gauge theory in the large N limit. In this way we make precise and concrete the notion of a non-perturbative gauge invariant continuum Wilsonian effective action. Manifestly gauge invariant calculations may be performed, without gauge fixing, and receive a natural interpretation in terms of fluctuating Wilson loops. Regularization is achieved by covariant higher derivatives and by embedding in a spontaneously broken SU(N|N) supergauge theory; the resulting heavy fermionic vectors are Pauli-Villars fields. We prove the finiteness of this method to one loop and any number of external gauge fields. A duality is uncovered that changes the sign of the squared coupling constant. As a test of the basic formalism we compute the one loop beta function, for the first time without any gauge fixing, and prove its universality with respect to cutoff function., Comment: TeX, harvmac, epsf; 73 pages, 25 figs; several clarifying sentences added

A manifestly gauge invariant continuous renormalization group flow equation is constructed for pure SU(N) gauge theory. The formulation makes sense without gauge fixing and manifestly gauge invariant calculations may thus be carried out. The flow equation is naturally expressed in terms of fluctuating Wilson loops, with the effective action appearing as an integral over a `gas' of Wilson loops. At infinite N, the effective action collapses to a path integral over the trajectory of a single particle describing one Wilson loop. We show that further regularization of these flow equations is needed. (This is introduced in part II.), Comment: TeX, harvmac, epsf; 35 pages, 15 figs; a few typos corrected

We investigate the convergence of the derivative expansion of the exact renormalization group, by using it to compute the beta function of scalar field theory. We show that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. Finally, we show that the two loop contributions to certain higher derivative operators (not involved in beta) have divergent momentum-scale expansions for sharp cutoff, but the smooth exponential cutoff gives convergent derivative expansions for all such operators with any number of derivatives., Comment: Latex inc axodraw. 20 pages

In these lectures we describe the construction of a gauge invariant renormalization group equation for pure non-Abelian gauge theory. In the process, a non-perturbative gauge invariant continuum Wilsonian effective action is precisely defined. The formulation makes sense without gauge fixing and thus manifest gauge invariance may be preserved at all stages. In the large N limit (of SU(N) gauge theory) the effective action simplifies: it may be expressed through a path integral for a single particle whose trajectory describes a Wilson loop. Regularization is achieved with the help of a set of Pauli-Villars fields whose formulation follows naturally in this picture. Finally, we show how the one loop beta function was computed, for the first time without any gauge fixing., Comment: Invited lectures at the Workshop on the Exact Renormalization Group, Faro, Portugal, Sept. 10-12, 1998. 32 pages including 24 figs. Conventions and notation changed to correspond to imminent papers, plus minor corrections

These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the role of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons., Comment: Invited lectures at the Yukawa International Seminar '97. 20 pages including 6 eps figs. LaTeX. PTPTeX style files included