We analyze the behavior of unstable modes in the glasma produced in high energy heavy ion collisions, using a simple model with effective homogeneous longitudinal color electric and magnetic fields. The unstable modes are approximately described as Nielsen-Olesen unstable modes under the homogeneous longitudinal gauge fields. We find that the Nielsen-Olesen unstable modes show properties very similar to those of the exponentially increasing unstable modes in the glasma recently shown by Romatschke and Venugopalan. Although initial gauge fields in the glasma are much stronger than the ones in our model, they decay with the production of the Nielsen-Olesen unstable modes. We discuss why we can reproduce the features of the glasma by using the effective homogeneous weak magnetic fields. Our analysis supports an idea that the decay of the gauge fields in the glasma is caused by Nielsen-Olesen instability.
Vortex configurations in the electroweak gauge theory are investigated. Two gauge-inequivalent solutions of the field equations, the Z and W vortices, have previously been found. They correspond to embeddings of the abelian Nielsen-Olesen vortex solution into a U(1) subgroup of SU(2)xU(1). It is shown here that any electroweak vortex solution can be mapped into a solution of the same energy with a vanishing upper component of the Higgs field. The correspondence is a gauge equivalence for all vortex solutions except those for which the winding numbers of the upper and lower Higgs components add…
We obtain an effective string theory of the Abrikosov-Nielsen-Olesen vortices of the Abelian Higgs model. The theory has an anomaly free effective string action which, when the extrinsic curvature is set equal to zero, yields the Nambu-Goto action. This generalizes previous work in which a string representation was obtained in the London limit, where the magnitude of the Higgs field is fixed. Viewed as a model for long distance QCD, it provides a concrete picture of the QCD string as a fluctuating Abrikosov-Nielsen-Olesen vortex of a dual superconductor on the border between type I and type II…
We consider a generalization of abelian Chern-Simons-Higgs model by introducing a nonstandard kinetic term. In particular we show that the Bogomolnyi equations of the abelian Higgs theory may be obtained, being its solutions Nielsen-Olesen vortices with electric charge. In addition we study the self-duality equations for a generalized non-relativistic Maxwell-Chern-Simons model.
The stability of an abelian (Nielsen-Olesen) vortex embedded in the electroweak theory against W production is investigated in a gauge defined by the condition of a single-component Higgs field. The model is characterized by the parameters $β=({M_H\over M_Z})^2$ and $γ=\cos^2θ_{\rm w}$ where $θ_{\rm w}$ is the weak mixing angle. It is shown that the equations for W's in the background of the Nielsen-Olesen vortex have no solutions in the linear approximation. A necessary condition for the nonlinear equations to have a solution in the region of parameter space where the abelian vortex is classi…
We investigate analytically and numerically the asymptotic behavior of the Nielsen-Olesen vortex solutions and show that they approach their asymptotic values exponentially but with exponents that differ from the ones quoted in the literature. In particular, it is shown that for values of the Higgs self-coupling that are larger than a critical value, both the Higgs field and the gauge field approach exponentially their asymptotic values with {\it equal} exponents that are {\it independent} of the Higgs self-coupling.
We revisit the classical instability of closed strings in the Abelian Higgs model. The instability is expressed by a vanishing energy as the torus-like configuration shrinks to zero size. We treat the radius of the torus as a collective coordinate and demonstrate that a quantum mechanical treatment of this coordinate leads to a stabilization of the closed string at small radii.
We consider soliton solutions to Bekenstein's theory, for which the fine structure constant $α=e^2/(4π\hbar c)$ is allowed to vary due to the presence of a dielectric field pervading the vacuum. More specifically we investigate the effects of a varying $α$ upon a complex scalar field with a U(1) electromagnetic gauge symmetry subject to spontaneous symmetry breaking. We find vortex solutions to this theory, similar to the Nielsen-Olesen vortex. Near the vortex core the electric charge is typically much larger than far away from the string, lending these strings a superconducting flavour. In ge…
We discuss gluon production by the Schwinger mechanism in collinear color-electric and magnetic fields which may be realized in pre-equilibrium stages of ultra-relativistic heavy-ion collisions. Fluctuations of non-Abelian gauge fields around a purely color-magnetic field contain exponentially growing unstable modes in a longitudinally soft momentum region, which is known as the Nielsen--Olesen instability. With a color-electric field imposed parallelly to the color-magnetic field, we can formulate this instability as the Schwinger mechanism. This is because soft unstable modes are accelerated…
It is shown that there is a second properly normalized KNO scaling function, $nP_n(n/\bar n)=φ(z)$, which has certain advantages in the analysis of KNO scaling. First, the $nP_n$ are not influenced by the statistical and systematic uncertainties of $\bar n$ hence $φ(z)$ provides more selective power than the original KNO scaling function $\bar nP_n(n/\bar n)=ψ(z)$. Second, the new scaling function generates scale parameter $σ=1$ since it depends only on the combination of $z$ and the scale parameter of $ψ(z)$. An analysis of $φ(z)$ is given using $e^+e^-$ annihilation data for charged particle…
In the large flux limit vortices become flux tubes with almost constant magnetic field in the interior region. This occurs in the case of non-Abelian vortices as well, and the study of such configurations allows us to reveal a close relationship between vortex zero modes and the gyromagnetic instabilities of vector bosons in a strong background magnetic field discovered by Nielsen, Olesen and Ambjørn. The BPS vortices are exactly at the onset of this instability, and the dimension of their moduli space is precisely reproduced in this way. We present a unifying picture in which, through the stu…
We investigate the behaviour of a model with two oppositely charged scalar fields. In the Bogomol'nyi limit this may be seen as the scalar sector of N=1 supersymmetric QED, and it has been shown that cosmic strings form. We examine numerically the model out of the Bogomol'nyi limit, and show that this remains the case. We then add supersymmetry-breaking mass terms to the supersymmetric model, and show that strings still survive. Finally we consider the extension to N=2 supersymmetry with supersymmetry-breaking mass terms, and show that this leads to the formation of stable cosmic strings, unli…
We study properties of Abrikosov-Nielsen-Olesen (ANO) strings with the Coleman-Weinberg (CW) potential, which we call CW-ANO strings. While the scale-invariant scalar potential has a topologically trivial vacuum admitting no strings at the classical level, quantum correction allows topologically nontrivial vacua and stable string solutions. We find that the system of the CW potential exhibits significant difference from that of the conventional Abelian-Higgs model with the quadratic-quartic potential. While a single-winding string is qualitatively similar in both systems, and the static interv…
Zero modes of strings in the abelian Higgs model are analyzed. In spite of the fact that the gauge symmetry is not broken in the string center, the corresponding zero mode is shown to exist and to see it one has to analyze carefully the dependence on transverse coordinates for the excitations. The analysis of this kind is also important for the Witten model of superconducting string. Unusual properties of the zero modes connected with the broken gauge symmetry in the string background are investigated. One of the modes carries the current quite similar to that in the Witten model and gives bac…
Using Mueller's dipole formalism for deep inelastic scattering in QCD, we formulate and solve the evolution for the generating function for the multiplicities of the produced particles, in hadronic processes at high energy. The solution for the multiplicities satisfies Koba-Nielsen-Olesen (KNO) scaling, with good agreement with the recently re-analyzed data from the H1 experiment at HERA (DESY), and the old ALEPH data for hadronic $Z$ decay at LEP (CERN). The same scaling function with KNO scaling, carries to the hadronic multiplicities from jets in electron-positron annihilation. This agreeme…
New solutions to the abelian U(1) Higgs model, corresponding to vortices of integer and half-integer winding number bound onto the edges of domain walls and possibly surrounded by annular current flows, are described, based on a fine-grained analysis of the topology of such configurations in spacetime. The existence of these states, which saturate BPS bounds in specific limits and are quite reminiscent of D-branes and membranes in general, could have interesting and some important consequences in a wide range of physical contexts. For instance, they raise the possibility that for some regimes …
We modify the standard Abelian-Higgs model by introducing spatially-dependent couplings for the scalar and vector fields. We investigate static, non-cylindrically symmetric solutions of the resulting field equations and propose a pinch solution which interpolates between degenerate vacua along the string, labelled by $\pm |n|$. This configuration corresponds to a vortex which shrinks to Planck scale before re-emerging as an anti-vortex, resulting in the formation of a bead pair with one bead at either side of the intersection. The solution is then topologically stable. A key assumption is that…
We study out-of-equilibrium dynamics of intense non-abelian gauge fields. Generalizing the well-known Nielsen-Olesen instabilities for constant initial color-magnetic fields, we investigate the impact of temporal modulations and fluctuations in the initial conditions. This leads to a remarkable coexistence of the original Nielsen-Olesen instability and the subdominant phenomenon of parametric resonance. Taking into account that the fields may be correlated only over a limited transverse size, we model characteristic aspects of the dynamics of color flux tubes relevant in the context of heavy-i…
A homogeneous color magnetic field is known to be unstable for the fluctuations perpendicular to the field in the color space (the Nielsen-Olesen instability). We argue that these unstable modes, exponentially growing, generate an azimuthal magnetic field with the original field being in the z-direction, which causes the Nielsen-Olesen instability for another type of fluctuations. The growth rate of the latter unstable mode increases with the momentum p_z and can become larger than the former's growth rate which decreases with increasing p_z. These features may explain the interplay between th…
We find evidence for the existence of solutions of the Einstein and Abelian Higgs field equations describing a black hole pierced by a Nielsen-Olesen vortex. This situation falls outside the scope of the usual no-hair arguments due to the non-trivial topology of the vortex configuration and the special properties of its energy-momentum tensor. By a combination of numerical and perturbative techniques we conclude that the black hole horizon has no difficulty in supporting the long range fields of the Nielsen Olesen string. Moreover, the effect of the vortex can in principle be measured from inf…
We study static cosmic string solutions in the context of Bekenstein-type models. We show that there is a class of models of this type for which the classical Nielsen-Olesen vortex is still a valid solution. However, in general static string solutions in Bekenstein-type models strongly depart from the standard Nielsen-Olesen solution with the electromagnetic energy concentrated along the string core seeding spatial variations of the fine structure constant, $α$. We consider models with a generic gauge kinetic function and show that equivalence principle constraints impose tight limits on the a…
Within the non-Abelian SU(2) Proca-Higgs theory, we study localised axially symmetric solutions possessing a finite field energy. It is shown that in a certain sense such solutions are analogues of the Nielsen-Olesen tube, since they have a longitudinal magnetic field creating a flux of this field over the central cross-section of the Proca tube. The main difference between the Proca tube and the Nielsen-Olesen tube is that the Proca tube is described by a topologically trivial solution and has finite size, since its energy density decreases exponentially with distance. The dependence of the t…
We classify the spectrum, family structure and stability of Nielsen-Olesen vortices embedded in a larger gauge group when the vacuum manifold is related to a symmetric space.
We study the fate of U(1) strings embedded in a non-Abelian gauge theory with a hierarchical pattern of the symmetry breaking: G->U(1) at V->nothing at v, V>>v. While in the low-energy limit the Abrikosov-Nielsen-Olesen string (flux tube) is perfectly stable, being considered in the full theory it is metastable. We consider the simplest example: the magnetic flux tubes in the SU(2) gauge theory with adjoint and fundamental scalars. First, the adjoint scalar develops a vacuum expectation value V breaking SU(2) down to U(1). Then, at a much lower scale, the fundamental scalar (quark) develops a …
We discuss quantum mechanical decay of color magnetic field generated initially at high energy heavy ion collisions. It is caused by Nielsen-Olesen unstable modes and is accomplished possibly in a period $<1$fm/c. We show that the decay products, i.e. incoherent gluons may be thermalized in a sufficiently short period ($< 1$fm/c). The precise determination of the period is made by calculating two point function of the color magnetic field in a model of color glass condensate.
It has been argued that the rapid decay of coherent color gauge fields generated immediately after high energy heavy ion collisions produces quark gluon plasma. But there are no convincing mechanisms for the rapid decay of the gauge fields which satisfy phenomenological constraints on their life time. We show by using classical statistical field theory that the production of magnetic monopoles cause the rapid decay of the gauge fields. The monopoles are unstable modes just like Nielsen-Olesen modes and are enormously produced by dual Schwinger mechanism.
It has been discussed that quark gluon plasma is produced following the rapid decay of coherent color gauge fields generated immediately after high energy heavy ion collisions. But there are no convincing mechanisms known for the rapid decay of the gauge fields which satisfy phenomenological constraints on their lifetime. We show by using classical statistical field theory that the gauge fields rapidly decay into magnetic monopoles. For comparison, we show how fast they decay into Nielsen-Olesen unstable modes. We find that the rapid decay of the glasma is caused by the monopole production.
It is argued that the phenomenon of a flux tube in quantum chromodynamics is closely connected with a spontaneously symmetry breakdown of gauge theory. It is shown that in the presence of a mass term in the SU(2) gauge theory the Nielsen-Olesen equations describe the flux tube surrounded by an external field.
The Nielsen - Olesen flux tube in the SU(2) Yang - Mills - Higgs theory dressed with the color electric $E^a_{ρ,ϕ}$ and magnetic $H^a_{ρ,ϕ}$ fields is derived. On the next step it is argued that this flux tube can be considered as a result of the nonperturbatuve analytical calculations in the SU(3) quantum theory.
We study the Abrikosov-Nielsen-Olesen multi-vortices. Using a numerical code we are able to solve the vortex equations with winding number up to n=25,000. We can thus check the wall vortex conjecture previously made in hep-th/0507273 and hep-th/0507286. The numerical results show a remarkable agreement with the theoretical predictions.
We discuss vacuum stability in Froggatt-Nielsen (FN) models. One concern in FN models is that for large flavon VEVs the running of the quartic Higgs coupling is enhanced what might lead to a more severe instability compared to the Standard Model (SM). We study this issue using the renormalization-group improved scalar potential. Another issue is that the mixing between the Higgs and the flavon can potentially destabilize the potential. However, taking current bounds on the flavon phenomenology into account, we find that both effects do not lead to an instability that is more severe than in the…
We extend work done previously concerning the formation of vortices in a U(1)->1 gauge theory at a first order phase transition to the symmetry breaking SU(2)xU(1)->U(1). It is shown that the collision of bubbles, appearing at the phase transition, allow the possibility for forming non-topological strings associated with the gauge group. The method used also shows clearly how these vortices are related to the Nielsen-Olesen vortices.
Six-dimensional Nielsen-Olesen vortices are analyzed in the context of a quadratic gravity theory containing Euler-Gauss-Bonnet self-interactions. The relations among the string tensions can be tuned in such a way that the obtained solutions lead to warped compactification on the vortex. New regular solutions are possible in comparison with the case where the gravity action only consists of the Einstein-Hilbert term. The parameter space of the model is discussed
The collinear color electric and magnetic fields have been discussed to be produced immediately after high energy heavy ion collisions. We discuss anomalous gluon production under the background gauge fields. The gluons are Nielsen-Olesen unstable modes. The production rate of the modes by Schwinger mechanism has recently been found to be anomalously larger than the rate of quarks or other stable gluons. Analyzing classical evolutions of the modes with initial conditions given by vacuum fluctuations, we find that their production makes the color electric field decay very rapidly. The life time…
We analyze the problem of the Nielsen-Olesen unstable modes in the SU(2) lattice gauge theory by means of a recently introduced gauge-invariant effective action. We perform numerical simulations in the case of a constant Abelian chromomagnetic field. We find that for lattice sizes above a certain critical length the density of effective action shows a behaviour compatible with the presence of the unstable modes. We put out a possible relation between the dynamics of the unstable modes and the confinement.
Resonance phenomena in vortex-antivortex collisions
Arnold Sommerfeld Center, Ludwig-Maximilians-Universitat（阿诺德·索末菲中心，慕尼黑大学）；Max-Planck-Institut fur Physik（马克斯·普朗克物理研究所）；Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理研究所）；Hiroshima University（广岛大学）
Motivated by the electroweak hierarchy problem, we consider theories with two extra dimensions in which the four-dimensional scalar fields are components of gauge boson in full space. We explore the Nielsen-Olesen instability for SU(N) on a torus, in the presence of a magnetic background. A field theory approach is developed, computing explicitly the minimum of the complete effective potential, including tri-linear and quartic couplings and determining the symmetries of the stable vacua. We also develop appropriate gauge-fixing terms when both Kaluza-Klein and Landau levels are present and int…
We investigate instabilities of classical Yang-Mills fields in a time-dependent spatially homogeneous color magnetic background field in a non-expanding geometry for elucidating the earliest stage dynamics of ultra-relativistic heavy-ion collisions. The background gauge field configuration considered in this article is spatially homogeneous and temporally periodic, and is alluded by Berges-Scheffler-Schlichting-Sexty (BSSS). We discuss the whole structure of instability bands of fluctuations around the BSSS background gauge field on the basis of Floquet theory, which enables us to discuss the …
Multiplicity fluctuations at midrapidity in p+p collisions at high energies are described by a negative binomial distribution and exhibit approximate Koba-Nielsen-Olesen (KNO) scaling. We find that these KNO fluctuations are important also for reproducing the multiplicity distribution in d+Au collisions observed at RHIC, adding to the Glauber fluctuations of the number of binary collisions or participants. We predict that the multiplicity distribution in p+Pb collisions at the LHC also deviates little from the KNO scaling function. Finally, we analyze various moments of the eccentricity of the…
The dual superconductor picture of the QCD vacuum is thought to describe various aspects of the strong interaction including confinement. Ordinary superconductivity is described by the Ginzburg-Landau (GL) equation. In the present work we show that it is possible to arrive at a GL-like equation from pure SU(2) gauge theory. This is accomplished by using Abelian projection to split the SU(2) gauge fields into an Abelian subgroup and its coset. The two gauge field components of the coset part act as the effective, complex, scalar field of the GL equation. The Abelian part of the SU(2) gauge fiel…
