Exploiting the gauge freedom associated with the Volkov description of a charge propagating in a ... more Exploiting the gauge freedom associated with the Volkov description of a charge propagating in a plane wave background, we identify a new type of gauge choice which significantly simplifies the theory. This allows us to develop a compact description of the propagator for both scalar and fermionic matter, in a circularly polarised background. It is shown that many of the usually observed structures are gauge artefacts. We then analyse the full ultraviolet behaviour of the one-loop corrections for such charges. This enables us to identify and contrast the different renormalisation prescriptions needed for both types of matter.
In this paper we study the infra-red behaviour of a gauge invariant and physically motivated desc... more In this paper we study the infra-red behaviour of a gauge invariant and physically motivated description of a charged particle in 2+1 dimensions. We show that both the mass shift and the wave function renormalisation are infra-red finite on-shell.
There is a widespread belief in particle physics that there is no relativistic description of a c... more There is a widespread belief in particle physics that there is no relativistic description of a charged particle. This is claimed to be due to persistent, long range interactions which distort the in and out going plane waves and generate infra-red divergences. In this paper we will show that this is not the case in QED. We construct locally gauge invariant charged fields which do create in and out Fock states. In a companion paper we demonstrate that the Green's functions of these fields have a good pole structure describing particle propagation.
Gauge theories are characterised by long range interactions. Neglecting these interactions at lar... more Gauge theories are characterised by long range interactions. Neglecting these interactions at large times, and identifying the Lagrangian matter fields with the asymptotic physical fields, leads to the infra-red problem. In this paper we study the perturbative applications of a construction of physical charges in QED, where the matter fields are combined with the associated electromagnetic clouds. This has been formally shown, in a companion paper, to include these asymptotic interactions. It is explicitly demonstrated that the on-shell Green's functions and S-matrix elements describing these charged fields have, to all orders in the coupling, the pole structure associated with particle propagation and scattering. We show in detail that the renormalisation procedure may be carried out straightforwardly. It is shown that standard infra-red finite predictions of QED are not altered and it is speculated that the good infra-red properties of our construction may open the way to the ...
The particle Fock space of the matter fields in QED can be constructed using the free creation an... more The particle Fock space of the matter fields in QED can be constructed using the free creation and annihilation operators. However, these particle operators are not, even at asymptotically large times, the modes of the matter fields that enter the QED Lagrangian. In this letter we construct the fields which do recover such particle modes at large times. We are thus able to demonstrate for the first time that, contrary to statements found in the literature, a relativistic description of charged particles in QED exists.
We calculate the screening and anti-screening contributions to the inter-quark potential in 2+1 d... more We calculate the screening and anti-screening contributions to the inter-quark potential in 2+1 dimensions, which is relevant to the high temperature limit of QCD. We demonstrate that the relative strength of screening to anti-screening agrees with the 3+1 dimensional theory to better than one percent accuracy.
We consider the Nielsen identities for the two-point functions of full QCD and QED in the class o... more We consider the Nielsen identities for the two-point functions of full QCD and QED in the class of Lorentz gauges. For pedagogical reasons the identities are first derived in QED to demonstrate the gauge independence of the photon self-energy, and of the electron mass shell. In QCD we derive the general identity and hence the identities for the quark, gluon and ghost propagators. The explicit contributions to the gluon and ghost identities are calculated to one-loop order, and then we show that the quark identity requires that in on-shell schemes the quark mass renormalisation must be gauge independent. Furthermore, we obtain formal solutions for the gluon self-energy and ghost propagator in terms of the gauge dependence of other, independent Green functions.
A crucial element of scattering theory and the LSZ reduction formula is the assumption that the c... more A crucial element of scattering theory and the LSZ reduction formula is the assumption that the coupling vanishes at large times. This is known not to hold for the theories of the Standard Model and in general such asymptotic dynamics is not well understood. We give a description of asymptotic dynamics in field theories which incorporates the important features of weak convergence and physical boundary conditions. Applications to theories with three and four point interactions are presented and the results are shown to be completely consistent with the results of perturbation theory.
The Lee-Nauenberg theorem is a fundamental quantum mechanical result which provides the standard ... more The Lee-Nauenberg theorem is a fundamental quantum mechanical result which provides the standard theoretical response to the problem of collinear and infrared divergences. Its argument, that the divergences due to massless charged particles can be removed by summing over degenerate states, has been successfully applied to systems with final state degeneracies such as LEP processes. If there are massless particles in both the initial and final states, as will be the case at the LHC, the theorem requires the incorporation of disconnected diagrams which produce connected interference effects at the level of the cross-section. However, this aspect of the theory has never been fully tested in the calculation of a cross-section. We show through explicit examples that in such cases the theorem introduces a divergent series of diagrams and hence fails to cancel the infrared divergences. It is also demonstrated that the widespread practice of treating soft infrared divergences by the Bloch-N...
We show how to construct physical, minimal energy states for systems of static and moving charges... more We show how to construct physical, minimal energy states for systems of static and moving charges. These states are manifestly gauge invariant. For charge-anticharge systems we also construct states in which the gauge fields are restricted to a finite volume around the location of the matter fields. Although this is an excited state, it is not singular, unlike all previous finite volume descriptions. We use our states to model the processes of pair creation and annihilation.
Asymptotic freedom arises from the dominance of anti-screening over screening in non-abelian gaug... more Asymptotic freedom arises from the dominance of anti-screening over screening in non-abelian gauge theories. In this paper we will present a simple and physically appealing derivation of the anti-screening contribution to the interquark potential. Our method allows us to identify the dominant gluonic distribution around static quarks. Extensions are discussed.
We consider an electron which is electromagnetically dressed in such a way that it is both gauge ... more We consider an electron which is electromagnetically dressed in such a way that it is both gauge invariant and that it has the associated electric and magnetic fields expected of a moving charge. We study the propagator of this dressed electron and, for small velocities, show explicitly at one loop that at the natural (on-shell) renormalisation point, p_0=m, p= m v, one can renormalise the propagator multiplicatively. Furthermore the renormalisation constants are infra-red finite. This shows that the dressing we use corresponds to a slowly moving, physical asymptotic field.
Starting from the observation that colour charge is only well defined on gauge invariant states, ... more Starting from the observation that colour charge is only well defined on gauge invariant states, we construct perturbatively gauge invariant, dynamical dressings for individual quarks. Explicit calculations show that an infra-red finite mass-shell renormalisation of the gauge invariant, dressed propagator is possible and, further, that operator product effects, which generate a running mass, may be included in a gauge invariant way in the propagator. We explain how these fields may be combined to form hadrons and show how the interquark potential can now be directly calculated. The onset of confinement is identified with an obstruction to building a non-perturbative dressing. We propose several methods to extract the hadronic scale from the interquark potential. Various extensions are discussed.
In this paper we construct a wide class of Gribov copies in Coulomb gauge SU(2) gauge theory. Inf... more In this paper we construct a wide class of Gribov copies in Coulomb gauge SU(2) gauge theory. Infinitesimal copies are studied in some detail and their non-perturbative nature is made manifest. As an application it is shown that the copies prevent a non-perturbative definition of colour charge.
We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory... more We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory. A state describing the flux tube between quarks as a thin string of glue is found to be a poor description of the continuum ground state; the infinitesimal thickness of the string leads to UV artifacts which suppress the overlap with the ground state. Contrastingly, a state which surrounds the quarks with non-abelian Coulomb fields is found to have a good overlap with the ground state for all charge separations. In fact, the overlap increases as the lattice regulator is removed. This opens up the possibility that the Coulomb state is the true ground state in the continuum limit.
The possibility of constructing charged particles in gauge theories has long been the subject of ... more The possibility of constructing charged particles in gauge theories has long been the subject of debate. In the context of QED we have shown how to construct operators which have a particle description. In this paper we further support this programme by showing how the screening interactions arise between these charges. Unexpectedly we see that there are two different gauge invariant contributions with opposite signs. Their difference gives the expected result.
In this article we investigate charged particles in gauge theories. After reviewing the physical ... more In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the Abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green's functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions.
We investigate the properties of a dressed electron which reduces, in a particular class of gauge... more We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We show explicitly that an infra-red finite, multiplicative, mass shell renormalisation is possible for this dressed electron, or, equivalently, for the usual fermion in the abovementioned gauges. The results are in complete accord with previous conjectures.
It is sometimes claimed that one cannot describe charged particles in gauge theories. We identify... more It is sometimes claimed that one cannot describe charged particles in gauge theories. We identify the root of the problem and present an explicit construction of charged particles. This is shown to have good perturbative properties and, asymptotically before and after scattering, to recover particle modes.
We show that the `dressing' approach, which describes physical charges as gauge invariant com... more We show that the `dressing' approach, which describes physical charges as gauge invariant composites of matter and clouds of gauge bosons, arises naturally in gauge theories. We give perturbative examples of dressings for both asymptotic charges and for states in which the fields are confined to a compact volume as is required, for example, by causality in pair creation. In QCD, we use dressed states to demonstrate explicitly how Gribov copies obstruct the non-perturbative construction of colour charges.
Exploiting the gauge freedom associated with the Volkov description of a charge propagating in a ... more Exploiting the gauge freedom associated with the Volkov description of a charge propagating in a plane wave background, we identify a new type of gauge choice which significantly simplifies the theory. This allows us to develop a compact description of the propagator for both scalar and fermionic matter, in a circularly polarised background. It is shown that many of the usually observed structures are gauge artefacts. We then analyse the full ultraviolet behaviour of the one-loop corrections for such charges. This enables us to identify and contrast the different renormalisation prescriptions needed for both types of matter.
In this paper we study the infra-red behaviour of a gauge invariant and physically motivated desc... more In this paper we study the infra-red behaviour of a gauge invariant and physically motivated description of a charged particle in 2+1 dimensions. We show that both the mass shift and the wave function renormalisation are infra-red finite on-shell.
There is a widespread belief in particle physics that there is no relativistic description of a c... more There is a widespread belief in particle physics that there is no relativistic description of a charged particle. This is claimed to be due to persistent, long range interactions which distort the in and out going plane waves and generate infra-red divergences. In this paper we will show that this is not the case in QED. We construct locally gauge invariant charged fields which do create in and out Fock states. In a companion paper we demonstrate that the Green's functions of these fields have a good pole structure describing particle propagation.
Gauge theories are characterised by long range interactions. Neglecting these interactions at lar... more Gauge theories are characterised by long range interactions. Neglecting these interactions at large times, and identifying the Lagrangian matter fields with the asymptotic physical fields, leads to the infra-red problem. In this paper we study the perturbative applications of a construction of physical charges in QED, where the matter fields are combined with the associated electromagnetic clouds. This has been formally shown, in a companion paper, to include these asymptotic interactions. It is explicitly demonstrated that the on-shell Green's functions and S-matrix elements describing these charged fields have, to all orders in the coupling, the pole structure associated with particle propagation and scattering. We show in detail that the renormalisation procedure may be carried out straightforwardly. It is shown that standard infra-red finite predictions of QED are not altered and it is speculated that the good infra-red properties of our construction may open the way to the ...
The particle Fock space of the matter fields in QED can be constructed using the free creation an... more The particle Fock space of the matter fields in QED can be constructed using the free creation and annihilation operators. However, these particle operators are not, even at asymptotically large times, the modes of the matter fields that enter the QED Lagrangian. In this letter we construct the fields which do recover such particle modes at large times. We are thus able to demonstrate for the first time that, contrary to statements found in the literature, a relativistic description of charged particles in QED exists.
We calculate the screening and anti-screening contributions to the inter-quark potential in 2+1 d... more We calculate the screening and anti-screening contributions to the inter-quark potential in 2+1 dimensions, which is relevant to the high temperature limit of QCD. We demonstrate that the relative strength of screening to anti-screening agrees with the 3+1 dimensional theory to better than one percent accuracy.
We consider the Nielsen identities for the two-point functions of full QCD and QED in the class o... more We consider the Nielsen identities for the two-point functions of full QCD and QED in the class of Lorentz gauges. For pedagogical reasons the identities are first derived in QED to demonstrate the gauge independence of the photon self-energy, and of the electron mass shell. In QCD we derive the general identity and hence the identities for the quark, gluon and ghost propagators. The explicit contributions to the gluon and ghost identities are calculated to one-loop order, and then we show that the quark identity requires that in on-shell schemes the quark mass renormalisation must be gauge independent. Furthermore, we obtain formal solutions for the gluon self-energy and ghost propagator in terms of the gauge dependence of other, independent Green functions.
A crucial element of scattering theory and the LSZ reduction formula is the assumption that the c... more A crucial element of scattering theory and the LSZ reduction formula is the assumption that the coupling vanishes at large times. This is known not to hold for the theories of the Standard Model and in general such asymptotic dynamics is not well understood. We give a description of asymptotic dynamics in field theories which incorporates the important features of weak convergence and physical boundary conditions. Applications to theories with three and four point interactions are presented and the results are shown to be completely consistent with the results of perturbation theory.
The Lee-Nauenberg theorem is a fundamental quantum mechanical result which provides the standard ... more The Lee-Nauenberg theorem is a fundamental quantum mechanical result which provides the standard theoretical response to the problem of collinear and infrared divergences. Its argument, that the divergences due to massless charged particles can be removed by summing over degenerate states, has been successfully applied to systems with final state degeneracies such as LEP processes. If there are massless particles in both the initial and final states, as will be the case at the LHC, the theorem requires the incorporation of disconnected diagrams which produce connected interference effects at the level of the cross-section. However, this aspect of the theory has never been fully tested in the calculation of a cross-section. We show through explicit examples that in such cases the theorem introduces a divergent series of diagrams and hence fails to cancel the infrared divergences. It is also demonstrated that the widespread practice of treating soft infrared divergences by the Bloch-N...
We show how to construct physical, minimal energy states for systems of static and moving charges... more We show how to construct physical, minimal energy states for systems of static and moving charges. These states are manifestly gauge invariant. For charge-anticharge systems we also construct states in which the gauge fields are restricted to a finite volume around the location of the matter fields. Although this is an excited state, it is not singular, unlike all previous finite volume descriptions. We use our states to model the processes of pair creation and annihilation.
Asymptotic freedom arises from the dominance of anti-screening over screening in non-abelian gaug... more Asymptotic freedom arises from the dominance of anti-screening over screening in non-abelian gauge theories. In this paper we will present a simple and physically appealing derivation of the anti-screening contribution to the interquark potential. Our method allows us to identify the dominant gluonic distribution around static quarks. Extensions are discussed.
We consider an electron which is electromagnetically dressed in such a way that it is both gauge ... more We consider an electron which is electromagnetically dressed in such a way that it is both gauge invariant and that it has the associated electric and magnetic fields expected of a moving charge. We study the propagator of this dressed electron and, for small velocities, show explicitly at one loop that at the natural (on-shell) renormalisation point, p_0=m, p= m v, one can renormalise the propagator multiplicatively. Furthermore the renormalisation constants are infra-red finite. This shows that the dressing we use corresponds to a slowly moving, physical asymptotic field.
Starting from the observation that colour charge is only well defined on gauge invariant states, ... more Starting from the observation that colour charge is only well defined on gauge invariant states, we construct perturbatively gauge invariant, dynamical dressings for individual quarks. Explicit calculations show that an infra-red finite mass-shell renormalisation of the gauge invariant, dressed propagator is possible and, further, that operator product effects, which generate a running mass, may be included in a gauge invariant way in the propagator. We explain how these fields may be combined to form hadrons and show how the interquark potential can now be directly calculated. The onset of confinement is identified with an obstruction to building a non-perturbative dressing. We propose several methods to extract the hadronic scale from the interquark potential. Various extensions are discussed.
In this paper we construct a wide class of Gribov copies in Coulomb gauge SU(2) gauge theory. Inf... more In this paper we construct a wide class of Gribov copies in Coulomb gauge SU(2) gauge theory. Infinitesimal copies are studied in some detail and their non-perturbative nature is made manifest. As an application it is shown that the copies prevent a non-perturbative definition of colour charge.
We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory... more We study trial states modelling the heavy quark-antiquark ground state in SU(2) Yang-Mills theory. A state describing the flux tube between quarks as a thin string of glue is found to be a poor description of the continuum ground state; the infinitesimal thickness of the string leads to UV artifacts which suppress the overlap with the ground state. Contrastingly, a state which surrounds the quarks with non-abelian Coulomb fields is found to have a good overlap with the ground state for all charge separations. In fact, the overlap increases as the lattice regulator is removed. This opens up the possibility that the Coulomb state is the true ground state in the continuum limit.
The possibility of constructing charged particles in gauge theories has long been the subject of ... more The possibility of constructing charged particles in gauge theories has long been the subject of debate. In the context of QED we have shown how to construct operators which have a particle description. In this paper we further support this programme by showing how the screening interactions arise between these charges. Unexpectedly we see that there are two different gauge invariant contributions with opposite signs. Their difference gives the expected result.
In this article we investigate charged particles in gauge theories. After reviewing the physical ... more In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the Abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green's functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions.
We investigate the properties of a dressed electron which reduces, in a particular class of gauge... more We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We show explicitly that an infra-red finite, multiplicative, mass shell renormalisation is possible for this dressed electron, or, equivalently, for the usual fermion in the abovementioned gauges. The results are in complete accord with previous conjectures.
It is sometimes claimed that one cannot describe charged particles in gauge theories. We identify... more It is sometimes claimed that one cannot describe charged particles in gauge theories. We identify the root of the problem and present an explicit construction of charged particles. This is shown to have good perturbative properties and, asymptotically before and after scattering, to recover particle modes.
We show that the `dressing' approach, which describes physical charges as gauge invariant com... more We show that the `dressing' approach, which describes physical charges as gauge invariant composites of matter and clouds of gauge bosons, arises naturally in gauge theories. We give perturbative examples of dressings for both asymptotic charges and for states in which the fields are confined to a compact volume as is required, for example, by causality in pair creation. In QCD, we use dressed states to demonstrate explicitly how Gribov copies obstruct the non-perturbative construction of colour charges.
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Papers by Martin Lavelle