arXiv subjects

John M. Cornwall

Publications and source records attributed to John M. Cornwall.

33 records · Page 2Linked to original sources

arXiv2002

Sphalerons, knots, and dynamical compactification in Yang-Mills-Chern-Simon theories

Euclidean d=3 SU(2) Yang-Mills-Chern-Simons (YMCS) theory, including Georgi-Glashow (GGCS) theory, may have solitons in the presence of appropriate mass terms. For integral CS level k and for solitons carrying integral CS number, YMCS is gauge-invariant and consistent. However, individual solitons such as sphalerons and linked center vortices with CS number of 1/2 and writhing center vortices with arbitrary CS number are non-compact; a condensate of them threatens compactness of the theory. We study various forms of the non-compact theory in the dilute-gas approximation, treating the parameters of non-compact large gauge transformations as collective coordinates. We conclude: 1) YMCS theory dynamically compactifies; non-compact YMCS have infinitely higher vacuum energy than compact YMCS. 2) An odd number of sphalerons is associated with a domain- wall sphaleron, a pure-gauge configuration on a closed surface enclosing them and with a half-integral CS number. 3) We interpret the domain-wall sphaleron in terms of fictitious closed Abelian magnetic field lines that express the links of the Hopf fibration. Sphalerons are over- and under-crossings of knots in the field lines; the domain-wall sphaleron is a superconducting wall confining these knots to a compact domain. 4) Analogous results hold for center vortices and nexuses. 5) For a CS term induced with an odd number of fermion doublets, domain-wall sphalerons are related to non-normalizable fermion modes. 6) GGCS with monopoles is compactified with center-vortex-like strings.

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arXiv2001

On topological charge carried by nexuses and center vortices

In this paper we further explore the question of topological charge in the center vortex-nexus picture of gauge theories. Generally, this charge is locally fractionalized in units of 1/N for gauge group SU(N), but globally quantized in integral units. We show explicitly that in d=4 global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy $Π_3(S^3)\simeq Z$; this homotopy insures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy $Π_2(S^2)\simeq Z$, a homotopy usually associated with 't Hooft-Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We find a new type of nexus which can carry topological charge by linking to vortices or carry d=3 Chern-Simons number without center vortices present; the Chern-Simons number is connected with twisting and writhing of field lines, as the author had suggested earlier. In general, no topological charge in d=4 arises from these specific static configurations, since the charge is the difference of two (equal) Chern-Simons number, but it can arise through dynamic reconnection processes. We complete earlier vortex-nexus work to show explicitly how to express globally-integral topological charge as composed of essentially independent units of charge 1/N.

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arXiv2001

Alternative Dimensional Reduction via the Density Matrix

We give graphical rules, based on earlier work for the functional Schrodinger equation, for constructing the density matrix for scalar and gauge fields at finite temperature T. More useful is a dimensionally-reduced effective action (DREA) constructed from the density matrix by further functional integration over the arguments of the density matrix coupled to a source. The DREA is an effective action in one less dimension which may be computed order by order in perturbation theory or by dressed-loop expansions; it encodes all thermal matrix elements. The DREA is useful because it gives a dimensionally-reduced field theory usable at any T including infinity, where it yields the usual dimensionally-reduced field theory (DRFT). However, it cannot and does not have spurious infinities which sometimes occur in the density matrix or the DRFT; these come from ln T factors at infinite temperature. An example of spurious divergences in the DRFT occurs in d 2+1 $ϕ^4$ theory dimensionally reduced to d=2. We show that the rules for the DREA replace these "wrong" divergences in physical parameters by calculable powers of ln T; we also compute the phase transition temperature of this theory at one loop order.

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arXiv2000

Baryon number non-conservation and phase transitions at preheating

Certain inflation models undergo pre-heating, in which inflaton oscillations can drive parametric resonance instabilities. We discuss several phenomena stemming from such instabilities, especially in weak-scale models; generically, these involve energizing a resonant system so that it can evade tunneling by crossing barriers classically. One possibility is a spontaneous change of phase from a lower-energy vacuum state to one of higher energy, as exemplified by an asymmetric double-well potential with different masses in each well. If the lower well is in resonance with oscillations of the potential, a system can be driven resonantly to the upper well and stay there (except for tunneling) if the upper well is not resonant. Another example occurs in hybrid inflation models where the Higgs field is resonant; the Higgs oscillations can be transferred to electroweak (EW) gauge potentials, leading to rapid transitions over sphaleron barriers and consequent B+L violation. Given an appropriate CP-violating seed, we find that preheating can drive a time-varying condensate of Chern-Simons number over large spatial scales; this condensate evolves by oscillation as well as decay into modes with shorter spatial gradients, eventually ending up as a condensate of sphalerons. We study these examples numerically and to some extent analytically. The emphasis in the present paper is on the generic mechanisms, and not on specific preheating models; these will be discussed in a later paper.

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arXiv1999

Center Vortices, Nexuses, and Fractional Topological Charge

It has been remarked in several previous works that the combination of center vortices and nexuses (a nexus is a monopole-like soliton whose world line mediates certain allowed changes of field strengths on vortex surfaces) carry topological charge quantized in units of 1/N for gauge group SU(N). These fractional charges arise from the interpretation of the standard topological charge integral as a sum of (integral) intersection numbers weighted by certain (fractional) traces. We show that without nexuses the sum of intersection numbers gives vanishing topological charge (since vortex surfaces are closed and compact). With nexuses living as world lines on vortices, the contributions to the total intersection number are weighted by different trace factors, and yield a picture of the total topological charge as a linking of a closed nexus world line with a vortex surface; this linking gives rise to a non-vanishing but integral topological charge. This reflects the standard 2πperiodicity of the theta angle. We argue that the Witten-Veneziano relation, naively violating 2πperiodicity, scales properly with N at large N without requiring 2πN periodicity. This reflects the underlying composition of localized fractional topological charge, which are in general widely separated. Some simple models are given of this behavior. Nexuses lead to non-standard vortex surfaces for all SU(N) and to surfaces which are not manifolds for N>2. We generalize previously-introduced nexuses to all SU(N) in terms of a set of fundamental nexuses, which can be distorted into a configuration resembling the 't Hooft-Polyakov monopole with no strings. The existence of localized but widely-separated fractional topological charges, adding to integers only on long distance scales, has implications for chiral symmetry breakdown.

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arXiv1999

Center Vortices, Nexuses, and the Georgi-Glashow Model

In a gauge theory with no Higgs fields the mechanism for confinement is by center vortices, but in theories with adjoint Higgs fields and generic symmetry breaking, such as the Georgi-Glashow model, Polyakov showed that in d=3 confinement arises via a condensate of 't Hooft-Polyakov monopoles. We study the connection in d=3 between pure-gauge theory and the theory with adjoint Higgs by varying the Higgs VEV v. As one lowers v from the Polyakov semi- classical regime v>>g (g is the gauge coupling) toward zero, where the unbroken theory lies, one encounters effects associated with the unbroken theory at a finite value v\sim g, where dynamical mass generation of a gauge-symmetric gauge- boson mass m\sim g^2 takes place, in addition to the Higgs-generated non-symmetric mass M\sim vg. This dynamical mass generation is forced by the infrared instability (in both 3 and 4 dimensions) of the pure-gauge theory. We construct solitonic configurations of the theory with both m,M non-zero which are generically closed loops consisting of nexuses (a class of soliton recently studied for the pure-gauge theory), each paired with an antinexus, sitting like beads on a string of center vortices with vortex fields always pointing into (out of) a nexus (antinexus); the vortex magnetic fields extend a transverse distance 1/m. An isolated nexus with vortices is continuously deformable from the 't Hooft-Polyakov (m=0) monopole to the pure-gauge nexus-vortex complex (M=0). In the pure-gauge M=0 limit the homotopy $Π_2(SU(2)/U(1))=Z_2$ (or its analog for SU(N)) of the 't Hooft monopoles is no longer applicable, and is replaced by the center-vortex homotopy $Π_1(SU)N)/Z_N)=Z_N$.

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arXiv1998

Nexus solitons in the center vortex picture of QCD

It is very plausible that confinement in QCD comes from linking of Wilson loops to finite-thickness vortices with magnetic fluxes corresponding to the center of the gauge group. The vortices are solitons of a gauge-invariant QCD action representing the generation of gluon mass. There are a number of other solitonic states of this action. We discuss here what we call nexus solitons, in which for gauge group SU(N), up to N vortices meet a a center, or nexus, provided that the total flux of the vortices adds to zero (mod N). There are fundamentally two kinds of nexuses: Quasi-Abelian, which can be described as composites of Abelian imbedded monopoles, whose Dirac strings are cancelled by the flux condition; and fully non-Abelian, resembling a deformed sphaleron. Analytic solutions are available for the quasi-Abelian case, and we discuss variational estimates of the action of the fully non-Abelian nexus solitons in SU(2). The non-Abelian nexuses carry Chern-Simons number (or topological charge in four dimensions). Their presence does not change the fundamentals of confinement in the center-vortex picture, but they may lead to a modified picture of the QCD vacuum.

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arXiv1998

Center vortices and confinement vs. screening

We study adjoint and fundamental Wilson loops in the center-vortex picture of confinement, for gauge group SU(N) with general N. There are N-1 distinct vortices, whose properties, including collective coordinates and actions, we study. In d=2 we construct a center-vortex model by hand so that it has a smooth large-N limit of fundamental-representation Wilson loops and find, as expected, confinement. Extending an earlier work by the author, we construct the adjoint Wilson-loop potential in this d=2 model for all N, as an expansion in powers of $ρ/M^2$, where $ρ$ is the vortex density per unit area and M is the vortex inverse size, and find, as expected, screening. The leading term of the adjoint potential shows a roughly linear regime followed by string breaking when the potential energy is about 2M. This leading potential is a universal (N-independent at fixed fundamental string tension $K_F$) of the form $(K_F/M)U(MR)$, where R is the spacelike dimension of a rectangular Wilson loop. The linear-regime slope is not necessarily related to $K_F$ by Casimir scaling. We show that in d=2 the dilute vortex model is essentially equivalent to true d=2 QCD, but that this is not so for adjoint representations; arguments to the contrary are based on illegal cumulant expansions which fail to represent the necessary periodicity of the Wilson loop in the vortex flux. Most of our arguments are expected to hold in d=3,4 also.

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arXiv1997

On One-Loop Gap Equations for the Magnetic Mass in d=3 Gauge Theory

Recently several workers have attempted determinations of the so-called magnetic mass of d=3 non-Abelian gauge theories through a one-loop gap equation, using a free massive propagator as input. Self-consistency is attained only on-shell, because the usual Feynman-graph construction is gauge-dependent off-shell. We examine two previous studies of the pinch technique proper self-energy, which is gauge-invariant at all momenta, using a free propagator as input, and show that it leads to inconsistent and unphysical result. In one case the residue of the pole has the wrong sign (necessarily implying the presence of a tachyonic pole); in the second case the residue is positive, but two orders of magnitude larger than the input residue, which shows that the residue is on the verge of becoming ghostlike. This happens because of the infrared instability of d=3 gauge theory. A possible alternative one-loop determination via the effective action also fails. The lesson is that gap equations must be considered at least at two-loop level.

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arXiv1997

Speculations on Primordial Magnetic Helicity

We speculate that above or just below the electroweak phase transition magnetic fields are generated which have a net helicity (otherwise said, a Chern-Simons term) of order of magnitude $N_B + N_L$, where $N_{B,L}$ is the baryon or lepton number today. (To be more precise requires much more knowledge of B,L-generating mechanisms than we currently have.) Electromagnetic helicity generation is associated (indirectly) with the generation of electroweak Chern-Simons number through B+L anomalies. This helicity, which in the early universe is some 30 orders of magnitude greater than what would be expected from fluctuations alone in the absence of B+L violation, should be reasonably well-conserved through the evolution of the universe to around the times of matter dominance and decoupling, because the early universe is an excellent conductor. Possible consequences include early structure formation; macroscopic manifestations of CP violation in the cosmic magnetic field (measurable at least in principle, if not in practice); and an inverse-cascade dynamo mechanism in which magnetic fields and helicity are unstable to transfer to larger and larger spatial scales. We give a quasi-linear treatment of the general-relativistic MHD inverse cascade instability, finding substantial growth for helicity of the assumed magnitude out to scales $\sim l_Mε^{-1}$, where $ε$ is roughly the B+L to photon ratio and $l_M$ is the magnetic correlation length. We also elaborate further on an earlier proposal of the author for generation of magnetic fields above the EW phase transition.

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arXiv1997

Large-Order Perturbation Theory in Infrared-Unstable Superrenormalizable Field Theories

We study the factorial divergences of Euclidean $ϕ^3_5$, a problem with connections both to high-energy multiparticle scattering in d=4 and to d=3 (or high-temperature) gauge theory, which like $ϕ^3_5$ is infrared-unstable and superrenormalizable. At large external momentum p (or small mass M) and large order N one might expect perturbative bare skeleton graphs to behave roughly like $N!(ag^2/p)^N$ with a>0, so that no matter how large p is there is an $N\sim g^2/p$ giving rise to strong perturbative amplitudes. The semi- classical Lipatov technique (which works only in the presence of a mass) is blind to this momentum dependence, so we proceed by direct summation of bare skeleton graphs. We find that the various limits of large N, large p, and small M do not commute, and that when $N\gg p^2/M^2$ there is a Borel singularity associated with $g^2/M$, not $g^2/p$. This is described by the zero-momentum Lipatov technique, and we find the necessary soliton for $ϕ^3_5$; the corresponding sphaleron-like solution for unbroken Yang-Mills theory has long been known. We also show that the massless theories have no classical solitons. We discuss non-perturbative effects based partly on known physical arguments concerning the cancellation by solitons of imaginary parts due to the pert- urbative Borel singularity, and partly on the dressing of bare skeleton graphs by dressed propagators showing non-perturbative mass generation, as happens in d=3 gauge theory.

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arXiv1996

The Baryon Wilson Loop Area Law in QCD

There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL) of QCD. Strong-coupling (i.e., finite lattice spacing in lattice gauge theory) approximations suggest the form $\exp [-KA_Y]$, where $K$ is the $q\bar{q}$ string tension and $A_Y$ is the global minimum area, generically a three-bladed area with the blades joined along a Steiner line ($Y$ configuration). However, the correct answer is $\exp[-(K/2)(A_{12}+A_{13}+A_{23})]$, where, e.g., $A_{12}$ is the minimal area between quark lines 1 and 2 ($Δ$ configuration). This second answer was given long ago, based on certain approximations, and is also strongly favored in lattice computations. In the present work, we derive the $Δ$ law from the usual vortex-monopole picture of confine- ment, and show that in any case because of the 1/2 in the $Δ$ law, this law leads to a larger value for the BWL (smaller exponent) than does the $Y$ law. We show that the three-bladed strong-coupling surfaces, which are infinitesimally thick in the limit of zero lattice spacing, survive as surfaces to be used in the non-Abelian Stokes' theorem for the BWL, which we derive, and lead via this Stokes' theorem to the correct $Δ$ law. Finally, we extend these considerations, including perturbative contributions, to gauge groups $SU(N)$, with $N>3$.

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arXiv1996

On The Phase Transition in D=3 Yang-Mills Chern-Simons Gauge Theory

$SU(N)$ Yang-Mills theory in three dimensions, with a Chern-Simons term of level $k$ (an integer) added, has two dimensionful coupling constants, $g^2 k$ and $g^2 N$; its possible phases depend on the size of $k$ relative to $N$. For $k \gg N$, this theory approaches topological Chern-Simons theory with no Yang-Mills term, and expectation values of multiple Wilson loops yield Jones polynomials, as Witten has shown; it can be treated semiclassically. For $k=0$, the theory is badly infrared singular in perturbation theory, a non-perturbative mass and subsequent quantum solitons are generated, and Wilson loops show an area law. We argue that there is a phase transition between these two behaviors at a critical value of $k$, called $k_c$, with $k_c/N \approx 2 \pm .7$. Three lines of evidence are given: First, a gauge-invariant one-loop calculation shows that the perturbative theory has tachyonic problems if $k \leq 29N/12$.The theory becomes sensible only if there is an additional dynamic source of gauge-boson mass, just as in the $k=0$ case. Second, we study in a rough approximation the free energy and show that for $k \leq k_c$ there is a non-trivial vacuum condensate driven by soliton entropy and driving a gauge-boson dynamical mass $M$, while both the condensate and $M$ vanish for $k \geq k_c$. Third, we study possible quantum solitons stemming from an effective action having both a Chern-Simons mass $m$ and a (gauge-invariant) dynamical mass $M$. We show that if $M \gsim 0.5 m$, there are finite-action quantum sphalerons, while none survive in the classical limit $M=0$, as shown earlier by D'Hoker and Vinet. There are also quantum topological vortices smoothly vanishing as $M \rightarrow 0$.

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arXiv1995

String Tension and Chern-Simons Fluctuations in the Vortex Vacuum of d=3 Gauge Theory

Based on a model of the d=3 SU(2) pure gauge theory vacuum as a monopole-vortex condensate, we give a quantitative (if model-dependent) estimate of the relation between the string tension and a gauge-invariant measure of the Chern-Simons susceptibility, due to vortex linkages, in the absence of a Chern-Simons term in the action. We also give relations among these quantities and the vacuum energy and gauge-boson mass. Both the susceptibility and the string tension come from the same physics: The topology of linking, twisting, and writhing of closed vortex strings. The closed-vortex string is described via a complex scalar field theory whose action has a precisely-specified functional form, inferred from previous work giving the exact form of a gauge-theory effective potential at low momentum. Applications to high-T phenomena, including B+L anomalous violation, are mentioned.

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arXiv1995

Toy Models of Non-perturbative Asymptotic Freedom in $ϕ^3_6$

We study idealizations of the full nonlinear Schwinger-Dyson equations for the asymptotically free theory of $ϕ^3$ in six dimensions in its meta-stable vacuum. We begin with the cubic nonlinearity and go on to all-order nonlinearities which contain instanton effects. We show how our toy models of the cubic Schwinger-Dyson equations contain the usual diseases of perturbation theory in the massless limit (e.g., factorially divergent $β-$functions, singular Borel-transform kernels associated with infrared renormalons) and show how these models yield specific mechanisms for removing such singularities when there is a mass gap. In the all-order nonlinear equation we show how to recover the usual renormalization-group-improved instanton effects and associated factorial divergences.

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