arXiv subjects

John M. Cornwall

Publications and source records attributed to John M. Cornwall.

At least 19 recordsLinked to original sources

arXiv2015

Exploring dynamical gluon mass generation in three dimensions

In the d=3 gluon mass problem in pure-glue non-Abelian $SU(N)$ gauge theory we pay particular attention to the observed (in Landau gauge) violation of positivity for the spectral function of the gluon propagator. This causes a large bulge in the propagator at small momentum. Mass is defined through $m^{-2}=Δ(p=0)$, where $Δ(p)$ is the scalar function for the gluon propagator in some chosen gauge, it is not a pole mass and is generally gauge-dependent, except in the gauge-invariant Pinch Technique (PT). We truncate the PT equations with a new method called the vertex paradigm that automatically satisfies the QED-like Ward identity relating the 3-gluon PT vertex function with the PT propagator. The mass is determined by a homogeneous Bethe-Salpeter equation involving this vertex and propagator. This gap equation also encapsulates the Bethe-Salpeter equation for the massless scalar excitations, essentially Nambu-Goldstone fields, that necessarily accompany gauge-invariant gluon mass. The problem is to find a good approximate value for $m$ and at the same time explain the bulge, which by itself leads, in the gap equation for the gluon mass, to excessively large values for the mass. Our point is not to give a high-accuracy determination of $m$ but to clarify the way in which the propagator bulge and a fairly accurate estimate of $m$ can co-exist, and we use various approximations that illustrate the underlying mechanisms. The most critical point is to satisfy the Ward identity. In the PT we estimate a gauge-invariant dynamical gluon mass of $m \approx Ng^2/(2.48 π)$. We translate these results to the Landau gauge using a background-quantum identity involving a dynamical quantity $κ$ such that $m=κm_L$, where $m_L^{-2}\equiv Δ_L(p=0)$. Given our estimates for $m,κ$ the relation is fortuitously well-satisfied for $SU(2)$ lattice data.

hep-ph↗

arXiv2014

Resolving infrared singularities of QCD through the vertex paradigm

We furnish details and extensions for the vertex paradigm and related ideas. The vertex paradigm is a method for dealing non-perturbatively with the Schwinger-Dyson equations (SDE) of asymptotically-free (AF) gauge theories such as QCD, even in the face of necessary approximations. It provides a useful truncation for the infinitely-many SDE of the gauge- and renormalization-group invariant Pinch Technique (PT-RGI). We implement the vertex paradigm by successive approximations at the one-dressed-loop level, postulating input tree-level gluon and ghost propagators and a 3-gluon vertex that are well-behaved in the infrared and also satisfy several crucial PT-RGI Ward identities that are QED-like and ghost-free. Good IR behavior is assured by including a (non-running) gauge-invariant dynamical gluon and ghost mass as part of the input. The non-trivial part of the vertex paradigm is that, with our inputs, the one-loop output vertex then satisfies the correct Ward identity from which we can construct the output gluon propagator, by taking account of Nambu-Goldstone-like massless scalars and related technical problems that arise whenever there is dynamical gluon mass generation. The one-loop outputs show a number of desirable features: They are PT-RGI; free of any reference to a coupling (dimensional transmutation); satisfy the ghost-free Ward identities connecting them; give exactly the known one-loop UV behavior; and are free of IR singularities. We give a much simpler illustration of the main principles of the vertex paradigm in a modified $ϕ^3_6$ model that is AF. Our successive-approximation scheme is not designed to estimate the gluon dynamical mass, but it shows that there is a lower limit to the mass below which the AF theory breaks down.

hep-ph↗

arXiv2013

Three easy exercises in off-shell string-inspired methods

Off-shell string-inspired methods (OSSIM) calculate off-shell QCD Green's functions using Schwinger-Feynman proper-time techniques, always in the background field method (BFM) Feynman gauge for technical convenience, and so far only at one loop. We already know that these results are gauge-invariant, because this gauge realizes the prescriptions of the Pinch Technique (PT), a Feynman-graph formulation for any gauge, but the idea of the first exercise is to show this directly in OSSIM. In this exercise we extend proper-time OSSIM beyond the BFM Feynman gauge so that one can apply PT algorithms, and show that the intrinsic PT is equivalent to resolving ambiguities in OSSIM in other gauges. In the second exercise we use forty-year-old rules of the author and Tiktopoulos for expressing loop integrals with numerator momenta directly in terms of Feynman parameters after momentum integration (the goal of OSSIM) and show that these rules elegantly and with economy of effort give rise, at least at one loop, to standard OSSIM algorithms. In the third exercise we apply world-line techniques to the problem of the breaking of adjoint strings, requiring a non-perturbative treatment that in the end reduces to a variant of the Schwinger result for production of electron-positron pairs in an electric field. This generalizes OSSIM to non-perturbative processes.

hep-ph↗

arXiv2013

Positivity violations in QCD

Both lattice simulations and theoretical studies show that the spectral function of the gluon propagator of QCD (in various gauges, as well as for the gauge-invariant Pinch Technique, or PT, propagator) is not non-negative everywhere, although it should be if it has a physical interpretation as in QED. Theory says moreover that the non-positive spectral function of the Landau-gauge or of the PT gluon propagator is further constrained to obey a superconvergence relation (the integral of the spectral function vanishes). We review the theoretical and lattice evidence for violation of positivity as well as various interpretations of this violation, and consider methods for checking superconvergence on the lattice (so far undone). The most common interpretation is that positivity violation implies confinement of gluons, so the gluon propagator does not describe processes with physical gluons. Another more direct and gauge-invariant interpretation arises from the PT: Asymptotic freedom alone demands non-positivity and superconvergence.

hep-ph↗

arXiv2012

Can supersymmetry breaking lead to electroweak symmetry breaking via formation of scalar bound states?

The recent discovery of the putative 125-GeV Higgs boson has motivated a number of attempts to reconcile its relatively large mass with the predictions of the minimal supersymmetric standard model (MSSM). Some approaches invoked large trilinear supersymmetry-breaking terms A_t between stops and one of the elementary Higgs fields. We consider the possibility that electroweak symmetry breaking may be triggered by supersymmetry breaking with a large A_t, large enough to generate a composite field with the same quantum numbers as the Higgs boson and with a non-vanishing vacuum expectation value. In the resulting vacuum, the usual relation between the gauge couplings and the Higgs self-coupling does not apply, and there is no reason to expect the same upper bound on the mass of the lightest Higgs boson. In a simple model where the bound state is assumed to have no mixing with the other fields, we calculate the critical coupling A_t necessary for symmetry breaking using the lowest-order Bethe-Salpeter (BS) equation. Study of the BS equation is complicated by the structure of its lowest-order kernel, which is a crossed box graph, but we find an accurate approximation to its solution. In a realistic model, the mixing of the bound state with the fundamental Higgs boson creates a symmetry-breaking seesaw. We outline the steps toward a realistic model.

hep-ph↗

arXiv2012

The QCD running charge and its RGI three-gluon vertex parent in the Pinch Technique

We give a brief review of an elementary extension of the Pinch Technique (PT) that yields renormalization-group invariant (RGI) Green's functions, called PT-RGI. These are also gauge- and process-independent, show dimensional transmutation, and are the natural ingredients of skeleton expansions of physical processes. Because of a dynamically-generated gluon mass all PT-RGI Green's functions are IR-finite. Next we show from the ghost-free Ward identities of the PT how the conventional running charge is recovered from the full PT-RGI three-gluon vertex, which depends on three momenta. The usual running charge, depending on only one momentum, is not necessarily a good substitute for this PT-RGI three-gluon vertex. We show that at one dressed loop a good approximation to the full dressed loop PT-RGI three-gluon vertex, both in the UV and in the IR, comes from input propagators and vertices that are free, except that the propagator is modified by introducing a (constant) mass term. Finally,we illustrate these ideas in the much simpler context of a scalar theory with cubic interactions in d=6, which is asymptotically free.

hep-ph↗

arXiv2012

Entropy in quantum chromodynamics

We review the role of zero-temperature entropy in several closely-related contexts in QCD. The first is entropy associated with disordered condensates, including $< G_{μν}^2>$. The second is vacuum entropy arising from QCD solitons such as center vortices, yielding confinement and chiral symmetry breaking. The third is entanglement entropy, which is entropy associated with a pure state, such as the QCD vacuum, when the state is partially unobserved and unknown. Typically, entanglement entropy of an unobserved three-volume scales not with the volume but with the area of its bounding surface. The fourth manifestation of entropy in QCD is the configurational entropy of light-particle world-lines and flux tubes; we argue that this entropy is critical for understanding how confinement produces chiral symmetry breakdown, as manifested by a dynamically-massive quark, a massless pion, and a $< \bar{q}q>$ condensate.

hep-ph↗

arXiv2011

Renormalization group invariance in the Pinch Technique

We show how to construct, using an elementary extension of the Pinch Technique, all off-shell Green's functions of a non-Abelian gauge theory so that they are locally gauge-invariant and renormalization-group invariant (RGI), as the S-matrix is, as well as being process-independent, coupling-constant independent (dimensional transmutation), and satisfying QED-like Ward identities. We call these PT-RGI Green's functions and outline how to construct an approximate three-gluon PT-RGI vertex with three physical scales and no dependence on the renormalization point $μ$. Properties of the PT-RGI Schwinger-Dyson equations are discussed, mostly in the context of a modified form of $ϕ^3_6$. The PT-RGI property of all off-shell Green's functions, plus other work of long ago, leads to a near-realization of the old dreams of S-matrix theorists.

hep-th↗

arXiv2010

Entropy, confinement, and chiral symmetry breaking

This paper studies the way in which confinement leads to chiral symmetry breaking (CSB) through a gap equation. We argue that entropic effects cut off infrared singularities in the standard confining effective propagator $1/p^4$, which should be replaced by $1/(p^2+m^2)^2$ for a finite mass $m\sim K_F/M(0)$ [$M(0)$ is the zero-momentum value of the running quark mass]. Extension of an old calculation of the author yields a specific estimate for $m$. This cutoff propagator shows semi-quantitatively two critical properties of confinement: 1) a negative contribution to the confining potential coming from entropic forces; 2) an infrared cutoff required by gauge invariance and CSB itself. Entropic effects lead to a proliferation of pion branches and a $\bar{q}q$ condensate, and contribute a negative term $\sim -K_F/M(0)$ to the effective pion Hamiltonian allowing for a massless pion in the presence of positive kinetic energy and string energy. The resulting gap equation leads to a well-behaved running constituent quark mass $M(p^2)$ with $M^2(0)\approx K_F/π$. We include one-gluon terms to get the correct renormalization-group ultraviolet behavior, with the improvement that the prefactor (related to $<\bar{q}q>$) can be calculated from the confining solution. We discuss an integrability condition that guarantees the absence of IR singularities at $m=0$ in Minkowski space through use of a principal-part propagator.

hep-ph↗

arXiv2009

Open issues in confinement, for the lattice and for center vortices

Topological confinement by center vortices does not immediately explain either a minimum-area law for non-planar Wilson loops or the Lüscher term. I conjecture that both a minimal-area law and a Lüscher term arise in a confinement model of random ensembles of vortices with no propagating gluons (a polymer model), and propose their test by polymer-like lattice simulations. I also consider the role of dynamically-massive gluons propagating from one point to another on a Wilson loop, and conjecture an approximate duality between the gluon-chain model and a condensate of center vortices with nexuses (magnetic monopoles) propagating on the vortex surfaces. I explore the old fishnet model, updated to deal with propagating massive QCD gluons, and argue that it leads to a surface tension and therefore a Lüscher term, as expressed through an effective action of the Dirichlet form that describes tension. I propose various lattice studies of non-planar Wilson loops to investigate such issues. Finally, in a different vein I urge the lattice community to study gauge-fixing to the background-field Feynman gauge, which will yield the gauge-invariant off-shell Greens functions of the pinch technique.

hep-ph↗

arXiv2009

Positivity issues for the pinch-technique gluon propagator and their resolution

Although gauge-boson propagators in asymptotically-free gauge theories satisfy a dispersion relation, they do not satisfy the Källen-Lehmann (KL) representation because the spectral function changes sign. We argue that this is a simple consequence of asymptotic freedom. On the basis of the QED-like Ward identities of the pinch technique (PT) we claim that the product of the coupling $g^2$ and the scalar part $\hat{d}(q^2)$ of the PT propagator, which is both gauge invariant and renormalization-group invariant, can be factored into the product of the running charge $\bar{g}^2(q^2)$ and a term $\hat{H}(q^2)$ both of which satisfy the KL representation although their product does not. We show that this behavior is consistent with some simple analytic models that mimic the gauge-invariant PT Schwinger-Dyson equations (SDE) provided that the dynamic gauge boson mass is sufficiently large. The PT SDEs do not depend directly on the PT propagator through $\hat{D}$ but only through $\hat{H}$.

hep-ph↗

arXiv2008

Center vortices, the functional Schrodinger equation, and CSB

The functional Schrodinger equation (FSE) for QCD gives a unique perspective on generation of a gluon mass m, as required for center vortices. The FSE, which yields a special d=3 gauge action, combined with lattice calculations strictly in d=3 give a value for the dimensionless ratio of d=3 coupling to mass g_3^2/m. From this we infer a reasonably accurate value for the d=4 running coupling g^2(0) in the region of low momentum where it is nearly constant. The result, consistent with other estimates, is too low to drive chiral symmetry breaking (CSB) for quarks in a standard gap equation that has no explicit confinement effects. We recall and improve on old work showing that confinement implies CSB for quarks, and consider CSB for test (that is, quenched) Dirac fermions in the adjoint representation. Here the previously-found value of $g^2(0)$ is large enough to drive CSB in a gap equation, which we relate to the presence of center vortices (non-confining, for the adjoint) and nexuses that drive fermionic zero modes. We discuss the extension of adjoint CSB to finite temperature.

hep-ph↗

arXiv2007

A conjecture on the infrared structure of the vacuum Schrodinger wave functional of QCD

The Schrodinger wave functional for the d=3+1 SU(N) vacuum is a partition function constructed in d=4; the exponent 2S in the square of the wave functional plays the role of a d=3 Euclidean action. We start from a gauge-invariant conjecture for the infrared-dominant part of S, based on dynamical generation of a gluon mass M in d=4. We argue that the exact leading term, of O(M), in an expansion of S in inverse powers of M is a d=3 gauge-invariant mass term (gauged non-linear sigma model); the next leading term, of O(1/M), is a conventional Yang-Mills action. The d=3 action that is the sum of these two terms has center vortices as classical solutions. The d=3 gluon mass, which we constrain to be the same as M, and d=3 coupling are related through the conjecture to the d=4 coupling strength, but at the same time the dimensionless ratio in d=3 of mass to coupling squared can be estimated from d=3 dynamics. This allows us to estimate the QCD coupling $α_s(M^2)$ in terms of this strictly d=3 ratio; we find a value of about 0.4, in good agreement with an earlier theoretical value but a little low compared to QCD phenomenology. The wave functional for d=2+1 QCD has an exponent that is a d=2 infrared-effective action having both the gauge-invariant mass term and the field strength squared term, and so differs from the conventional QCD action in two dimensions, which has no mass term. This conventional d=2 QCD would lead in d=3 to confinement of all color-group representations. But with the mass term (again leading to center vortices), N-ality = 0 mod N representations are not confined.

hep-th↗

arXiv2006

Probing the center-vortex area law in d=3: The role of inert vortices

In center vortex theory, beyond the simplest picture of confinement several conceptual problems arise that are the subject of this paper. Confinement arises through averaging of phase factors which are gauge-group center elements raised to the power of the Gauss linking numbers of vortices. The simplest approach to confinement counts this link number by counting the number of vortices, considered in d=3 as infinitely-long closed self-avoiding random walks on a cubical lattice, piercing any surface spanning the Wilson loop. A given vortex, however, may pierce the spanning surface multiply with a link number smaller than the number of piercings. We call such vortices inert (although they may be only partially-inert). We estimate the dilution factor from inert vortices that reduces the ratio of fundamental string tension to vortex areal piercing density as roughly 0.6. Next we show how inert vortices resolve the old problem that the link number of a given vortex configuration is the same for any choice of spanning surface, yet only one such surface appears in the Wilson loop expectation value. Third, we discuss semi- quantitatively a configuration of two distinct Wilson loops separated by a variable distance, and show how inert vortices govern the transition between two allowed forms of the area law, one at small separation and one at large. The result is a finite-range Van der Waals force between the loops. Finally, in a problem related to the double-loop problem, we argue that inert vortices do not affect the fact that in the SU(3) baryonic area law, the mesonic string tension appears.

hep-th↗

arXiv2004

Baryonic hybrids: Gluons as beads on strings between quarks

We analyze the ground state of the heavy-quark hybrid system composed of three quarks and a gluon. The known string tension K and approximately-known gluon mass M lead to a precise specification of the long-range non-relativistic part of the potential binding the gluon to the quarks with no undetermined phenomenological parameters, in the limit of large interquark separation R. Our major tool (also used earlier by Simonov) is the use of proper-time methods to describe gluon propagation within the quark system, which reveals the gluon Wilson line as a composite of co-located quark and antiquark lines. We show that (aside from color-Coulomb and similar terms) the gluon potential energy in the presence of quarks is accurately described via attaching these three strings to the gluon, which in equilibrium sits at the middle of the Y-shaped string network joining the three quarks. The gluon undergoes small harmonic fluctuations that slightly stretch these strings and quasi-confine the gluon to the neighborhood of the middle. In the non-relativistic limit (large R) we use the Schrodinger equation, ignoring mixing with l=2 states. Relativistic corrections (smaller R) are applied with a variational principle for the relativistic harmonic oscillator. We also consider the role of color-Coulomb contributions. We find leading non-relativistic large-R terms in the gluon string energy which behave like the square root of K/(MR). The relativistic energy goes like the cube root of K/R. We get an acceptable fit to lattice data with M = 500 MeV. We show that in the quark-antiquark hybrid the gluon is a bead that can slide without friction on a string joining the quark and anti-quark. We comment briefly on the significance of our findings to fluctuations of the minimal surface.

hep-ph↗

arXiv2004

A three-dimensional scalar field theory model of center vortices and its relation to k-string tensions

In d=3 SU(N) gauge theory, we study a scalar field theory model of center vortices that furnishes an approach to the determination of so-called k-string tensions. This model is constructed from string-like quantum solitons introduced previously, and exploits the well-known relation between string partition functions and scalar field theories in d=3. Center vortices corresponding to magnetic flux J (in units of 2π/N) are composites of J elementary J=1 constituent vortices that come in N-1 types, with repulsion between like constituents and attraction between unlike constituents. The scalar field theory involves N scalar fields ϕ_i (one of which is eliminated) that can merge, dissociate, and recombine while conserving flux mod N. The properties of these fields are deduced directly from the corresponding gauge-theory quantum solitons. Every vacuum Feynman graph of the theory corresponds to a real-space configuration of center vortices. We study qualitatively the problem of k-string tensions at large N, whose solution is far from obvious in center-vortex language. We construct a simplified dynamical picture of constituent-vortex merging, dissociation, and recombination, which allows in principle for the determination of vortex areal densities and k-string tensions. This picture involves point-like "molecules" (cross-sections of center vortices) made of constituent "atoms" that combine and disassociate dynamically in a d=2 test plane . The vortices evolve in a Euclidean "time" which is the location of the test plane along an axis perpendicular to the plane. A simple approximation to the molecular dynamics is compatible with k-string tensions that are linear in k for k<< N, as naively expected.

hep-th↗

arXiv2003

On the center-vortex baryonic area law

We correct an unfortunate error in an earlier work of the author, and show that in center-vortex QCD (gauge group SU(3)) the baryonic area law is the so-called $Y$ law, described by a minimal area with three surfaces spanning the three quark world lines and meeting at a central Steiner line joining the two common meeting points of the world lines. (The earlier claim was that this area law was a so-called $Δ$ law, involving three extremal areas spanning the three pairs of quark world lines.) We give a preliminary discussion of the extension of these results to $SU(N), N>3$. These results are based on the (correct) baryonic Stokes' theorem given in the earlier work claiming a $Δ$ law. The $Y$-form area law for SU(3) is in agreement with the most recent lattice calculations.

hep-th↗

arXiv2003

Relativistic center-vortex dynamics of a confining area law

We offer a physicists' proof that center-vortex theory requires the area in the Wilson-loop area law to involve an extremal area. Area-law dynamics is determined by integrating over Wilson loops only, not over surface fluctuations for a fixed loop. Fluctuations leading to to perimeter-law corrections come from loop fluctuations as well as integration over finite -thickness center-vortex collective coordinates. In d=3 (or d=2+1) we exploit a contour form of the extremal area in isothermal which is similar to d=2 (or d=1+1) QCD in many respects, except that there are both quartic and quadratic terms in the action. One major result is that at large angular momentum \ell in d=3+1 the center-vortex extremal-area picture yields a linear Regge trajectory with Regge slope--string tension product α'(0)K_F of 1/(2π), which is the canonical Veneziano/string value. In a curious effect traceable to retardation, the quark kinetic terms in the action vanish relative to area-law terms in the large-\ell limit, in which light-quark masses \sim K_F^{1/2} are negligible. This corresponds to string-theoretic expectations, even though we emphasize that the extremal-area law is not a string theory quantum-mechanically. We show how some quantum trajectory fluctuations as well as non-leading classical terms for finite mass yield corrections scaling with \ell^{-1/2}. We compare to old semiclassical calculations of relativistic q\bar{q} bound states at large \ell, which also yield asymptotically-linear Regge trajectories, finding agreement with a naive string picture (classically, not quantum-mechanically) and disagreement with an effective-propagator model. We show that contour forms of the area law can be expressed in terms of Abelian gauge potentials, and relate this to old work of Comtet.

hep-th↗