Accessing directly the properties of fundamental scalars in the confinement and Higgs phase
Axel Maas
备注：
11 pages, 6 figures, 1 table; v2: one minor error corrected; v3: one appendix on systematic uncertainties added and some minor changes, version to appear in EPJC
The properties of elementary particles are encoded in their respective propagators and interaction vertices. For a SU(2) gauge theory coupled to a doublet of fundamental complex scalars these propagators are determined in both the Higgs phase and the confinement phase and compared to the Yang-Mills case, using lattice gauge theory. Since the propagators are gauge-dependent, this is done in the Landau limit of 't Hooft gauge, permitting to also determine the ghost propagator. It is found that neither the gauge boson nor the scalar differ qualitatively in the different cases. In particular, the gauge boson acquires a screening mass, and the scalar's screening mass is larger than the renormalized mass. Only the ghost propagator shows a significant change. Furthermore, indications are found that the consequences of the residual non-perturbative gauge freedom due to Gribov copies could be different in the confinement and the Higgs phase.
We propose a novel description for the Higgs mechanism by which a gauge boson acquires the mass. We do not assume spontaneous breakdown of gauge symmetry signaled by a non-vanishing vacuum expectation value of the scalar field. In fact, we give a manifestly gauge-invariant description of the Higgs mechanism in the operator level, which does not rely on spontaneous symmetry breaking. This enables us to discuss the confinement-Higgs complementarity from a new perspective. The "Abelian" dominance in quark confinement of the Yang-Mills theory is understood as a consequence of the gauge-invariant H…
We show that the Higgs and confinement phases of a gauge Higgs theory, with the Higgs field in the fundamental representation of the gauge group, are distinguished both by a broken or unbroken realization of the global center subgroup of the gauge group, and by the type of confinement in each phase. This is color confinement in the Higgs phase, and a stronger property, which we call "separation-of-charge" confinement, in the confining phase.
The property of color confinement ("C confinement"), meaning that all asymptotic particle states are color neutral, holds not only in QCD, but also in gauge-Higgs theories deep in the Higgs regime. In this talk we describe a new and stronger confinement criterion, separation-of-charge confinement or "S$_c$ confinement," which is an extension of the Wilson area-law criterion to gauge + matter theories. We will show that there is a transition between S$_c$ and C confinement in the phase plane of gauge-Higgs theories, and we will also explain what symmetry is actually broken in the Higgs phase of…
We propose that the Higgs phase of a gauge Higgs theory is the phase of spontaneously broken custodial symmetry, and present a new gauge invariant order parameter for custodial symmetry breaking which is very closely analogous to the Edwards-Anderson order parameter for spin glasses. Custodial symmetry is a global symmetry acting on the Higgs field alone, and we show here that the spin glass transition in gauge Higgs theories, from a QCD-like phase to a Higgs phase of broken custodial symmetry, coincides with the transition between two distinct types of confinement. These are color confinement…
We propose to use the different kinds of vacua of the gauge theories coupled to matter as a laboratory to test confinement ideas of pure Yang-Mills theories. In particular, the very poor overlap of the Wilson loop with the broken string states supports the 't Hooft and Mandelstam confinement criteria. However in the Z(2) gauge-Higgs model we use as a guide we find that the condensation of monopoles and center vortices is a necessary, but not sufficient condition for confinement.
Gauge-independent transition separating confinement and Higgs phases in lattice SU(2) gauge theory with a scalar field in the fundamental representation
High density YM-theory is addressed by means of an effective theory where a Higgs type field corresponds to the diquark order parameter. The effective Higgs theory mimics the Cooper instability at the Fermi surface. After Landau gauge fixing, the theory possesses a global color degree of freedom. Color superconductivity corresponds to a spontaneous breakdown of the residual global color symmetry. Numerical simulations illustrate this mechanism of spontaneously broken global color for the case of the standard electro-weak phase transition and for the case of the transition between the confineme…
In SU($N$) gauge-Higgs theories, with a single Higgs field in the fundamental representation, there exists in addition to the local gauge symmetry a global SU(2) symmetry, at $N=2$, and a global U(1) symmetry, for $N \ne 2$. We construct a gauge-invariant order parameter for the breaking of these global symmetries in the Higgs sector, and calculate numerically the transition lines, in coupling-constant space, for SU(2) and SU(3) gauge theories with unimodular Higgs fields. The order parameter is non-local, and therefore its non-analyticity does not violate the theorem proved by Osterwalder and…
We re-examine by numerical simulation the phase structure of the three-dimensional Abelian lattice gauge theory (LGT) with $Z(2)$ gauge fields coupled to $Z(2)$-valued Higgs fields. Concretely, we explore two different order parameters which are able to distinguish the three phases of the theory: (i) the Fredenhagen-Marcu operator used to discriminate between deconfinement and confinement/Higgs phases and (ii) the Greensite-Matsuyama overlap operator proposed recently to distinguish confinement and Higgs phases. The latter operator is an analog of the overlap Edwards-Anderson order parameter f…
According to recent work of Greensite and Matsuyama, the Higgs phase of a gauge Higgs theory is distinguished from the confinement and massless phases by the spontaneous breaking of a global center subgroup of the gauge group, and by confinement type. This is contrary to the notion that there is no essential distinction between the Higgs and confinement phases when the Higgs field is in the fundamental representation of the gauge group. Although this new symmetry breaking order parameter has been investigated in $D=4$ dimensions, there is so far no check in a non-abelian gauge theory containin…
It is argued that the phase diagram of the 3D Compact U(1) Lattice Higgs Model is more refined than generally thought. The confined and Higgs phases are separated by a well-defined phase boundary, marked by proliferating vortices. It is shown that the confinement mechanism at work is precisely the dual superconductor scenario.
We study the confinement-deconfinement transition in $SU(2)$ gauge theory in the presence of massless bosons using lattice Monte Carlo simulations. The nature of this transition depends on the temporal extent ($N_τ$) of the Euclidean lattice. We find that the transition is a cross-over for $N_τ=2,4$ and second order with $3D$ Ising universality class for $N_τ=8$. Our results show that the second order transition is accompanied by realization of the $Z_2$ symmetry.
We determine the phase structure of an SU(2) gauge theory with an adjoint scalar on $R^{3}\times S^{1}$ using semiclassical methods. There are two global symmetries: a $Z(2)_{H}$ symmetry associated with the Higgs field and a $Z(2)_{C}$ center symmetry. We analyze the order of the deconfining phase transition when different deformation terms are used. After finding order parameters for the global symmetries, we show that there are four distinct phases: a deconfined phase, a confined phase, a Higgs phase, and a mixed confined phase. The mixed confined phase occurs where one might expect a phase…
We present a method for finding lines of constant physics in the confinement phase of the SU(2) Higgs model on the lattice. The model is considered at finite values of the cut-off where it behaves like an effective field theory with three independent couplings. In particular, a renormalised quantity sensitive to a variation of the bare Higgs quartic self-coupling is constructed from generalised Binder cumulants. Numerical results for the non-perturbative matching of the bare parameters of the model between beta=2.2 and beta=2.4 are presented.
It has been noticed long ago that in Higgs models with `complete symmetry breaking' one can move from the confinement to the Higgs regime without crossing a phase boundary, a fact sometimes called referred to as `complementarity'. In a recent paper some doubt was raised about the correctness of the mathematics underlying this fact and it was claimed that the supposed `flaw' would resolve the `paradox' seen in this complementarity. Here we briefly revisit the facts both from a mathematical and a physical point of view and point out that (a) there is no paradox and (b) there is no flaw in the ma…
We further investigate center vortex percolation and Coulomb gauge remnant symmetry breaking in the SU(2) gauge-Higgs model. We show that string breaking is visible in Polyakov line correlators on the center projected lattice, that our usual numerical tests successfully relate P-vortices to center vortices, and that vortex removal removes the linear potential, as in the pure gauge theory. This data suggests that global center symmetry is not essential to the vortex confinement mechanism. But we also find that the line of vortex percolation-depercolation transitions, and the line of remnant sym…
In the previous works, we proposed atomic quantum simulations of the U(1) gauge-Higgs model by ultra-cold Bose gases. By studying extended Bose-Hubbard models (EBHMs) including long-range repulsions, we clarified the locations of the confinement, Coulomb and Higgs phases. In this paper, we study the EBHM with nearest-neighbor repulsions in one and two dimensions at large fillings by the Gutzwiller variational method. We obtain phase diagrams and investigate dynamical behavior of electric flux from the gauge-theoretical point of view. We also study if the system exhibits glassy quantum dynamics…
Yang-Mills-Higgs theory offers a rich set of physics. In particular, in some region of its parameter space it has QCD-like behavior, while in some other range it is Higgs-like. Furthermore, for the choice of the gauge group SU(2) and an SU(2) Higgs flavor symmetry it is the Higgs sector of the standard model. Therefore, it is possible to study a plethora of phenomena within a single theory. Here the standard-model version is studied using lattice gauge theory. Choosing non-aligned minimal Landau gauge, its propagators and three-point vertices will be determined in both the QCD-like and Higgs-l…
We present results for the masses of the low-lying states with quantum numbers $0^{++}$, $2^{++}$ and $1^{--}$ as well as Polyakov line correlations in the Higgs and confinement regions of the 3d SU(2) Higgs model. In the confinement phase we find a dense spectrum of bound states approximately split into two disjoint sectors. One consists of W-balls nearly identical to the glueball spectrum of the pure gauge theory, the other of bound states of scalars.
Odd Toric Code in a tilted field: Higgs-confinement multicriticality, spontaneous self-duality symmetry breaking, and valence bond solids
Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）；Max Planck Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学弗里茨·哈伯分子动力学研究中心）
We examine the phase structure of massive Yang-Mills theory in 1+1 dimensions. This theory is equivalent to a gauged principal chiral sigma model. It has been previously shown that the gauged theory has only a confined phase, and no Higgs phase in the continuum, and at infinite volume. There are no massive gluons, but only hadron-like bound states of sigma-model particles. The reason is that the gluon mass diverges, being proportional to the two-point correlation function of the renormalized field of the sigma model at $x=0$. We use exact large-$N$ results to show that after introducing a latt…
The real-time dynamics of topological defects and turbulent configurations of gauge fields for electric and magnetic confinement are studied numerically within a 2+1D Abelian Higgs model. It is shown that confinement is appearing in such systems equilibrating after a strong initial quench such as the overpopulation of the infrared modes. While the final equilibrium state does not support confinement, metastable vortex defect configurations appear in the gauge field which are found to be closely related to the appearance of physically observable confined electric and magnetic charges. These phe…
In this review, we provide a short outlook of some of the currently most popular pictures and promising approaches to non-perturbative physics and confinement in gauge theories. A qualitative and by no means exhaustive discussion presented here covers such key topics as the phases of QCD matter, the order parameters for confinement, the central vortex and monopole pictures of the QCD vacuum structure, fundamental properties of the string tension, confinement realisations in gauge-Higgs and Yang-Mills theories, magnetic order/disorder phase transition among others.
Further numerical evidences for the gauge-independent separation between Confinement and Higgs phases in lattice SU(2) gauge theory with a scalar field in the fundamental representation
A generalization of the Wilson loop area-law criterion is proposed, which is applicable to gauge theories with matter in the fundamental representation of the gauge group. This new criterion, like the area law, is stronger than the statement that asymptotic particle states are massive color singlets, which holds even for theories described by the Brout-Englert-Higgs mechanism.
We study the Landau gauge photon propagator in the three--dimensional Abelian Higgs model with compact gauge field and fundamentally charged matter in the London limit. The total gauge field is split into singular and regular parts. On the confinement side of the string breaking crossover the momentum dependence of the total propagator is characterized by an anomalous dimension similarly to 3D compact QED. At the crossover and throughout the Higgs region the anomalous dimension disappears. This result perfectly agrees with recent observations that the monopole--antimonopole plasma leads to non…
We study the Z_N symmetry in SU(N)+Higgs theories with the Higgs field in the fundamental representation. The distributions of the Polyakov loop show that the Z_N symmetry is explicitly broken in the Higgs phase. On the other hand, inside the Higgs symmetric phase the Polyakov loop distributions and other physical observables exhibit the Z_N symmetry. This effective restoration of the Z_N symmetry changes the nature of the confinement-deconfinenement transition. We argue that the Z_N symmetry will lead to time independent topological defect solutions in the Higgs symmetric deconfined phase whi…
We present arguments suggesting that large size overlapping instantons are the driving mechanism of the confinement-deconfinement phase transition at nonzero chemical potential mu. The arguments are based on the picture that instantons at very large chemical potential in the weak coupling regime are localized configurations with finite size ρ\simμ^{-1}. At the same time, the same instantons at smaller chemical potential in the strong coupling regime are well represented by the so-called instanton-quarks with fractional topological charge 1/N_c. We estimate the critical chemical potential mu_c(…
We study theoretical bases of the dual Higgs theory for confinement physics in QCD in terms of monopoles and the gluon configuration in the maximally abelian (MA) gauge. Abelian dominance for the confinement force can be analytically proved by regarding the off-diagonal angle variable as a random variable in the lattice formalism. In the long-distance scale, the contribution of off-diagonal gluons to the Wilson loop cancels each other and exhibits a perimeter law behavior, which leads to exact abelian dominance on the string tension if the finite size effect of the Wilson loop is removed. We i…
We discuss theories with the exceptional centerless gauge group G(2), paying attention to confinement and the pattern of chiral symmetry breaking. Exploiting the Higgs mechanism to break the symmetry down to SU(3), we also present how the familiar features of confinement and chiral symmetry breaking of SU(3) gauge theories reemerge. G(2) gauge theories show up as an unusual theoretical framework to study SU(3) gauge theories without the ``luxury'' of a center.
We study the phase structure of SU(2) gauge-Higgs model in two dimensions using lattice simulations. We show the result for the plaquette expectation value, static potential, and W propagator. Our results suggest that a confinement-like region and a Higgs-like region appear even in two dimensions. The behavior of the plaquette expectation value is consistent with a smooth cross-over in accordance with the Fradkin-Shenker-Osterwalder-Seiler theorem. In the confinement-like region, the static potential seems to rise linearly with string breaking at large distances, while in the Higgs-like region…
The Fradkin-Shenker theorem proves analyticity in a region that connects Higgs to confinement regimes, precluding a phase transition. This conflicts with a simpler analyticity argument applicable to any symmetry-breaking phase transition that requires the phase diagram to be bifurcated. A flaw in the Fradkin-Shenker and related Osterwalder-Seiler proofs is found which removes this paradox. Higgs and Confinement regions are everywhere separated by a phase boundary. A new order parameter allowing this transition to be traced with Monte-Carlo simulations without gauge fixing is introduced.
We present an efficient local hybrid Monte-Carlo algorithm to investigate G(2) gluodynamics with and without Higgs field in 3 and 4 dimensions. Additionaly we implemented a modified version of the multi-level Lüscher-Weisz algorithm with exponential error reduction to measure expectation values of Wilson and Polyakov loops. In three dimensions we show that at intermediate scales the potential between static charges in the eight lowest-dimensional representations of G(2) scale with the eigenvalues of the quadratic Casimir operator. For the fundamental representations we detect string breaking f…
The SU(2)-Higgs model, with a single Higgs field in the fundamental representation and a quartic self-interaction, has a Higgs region and a confinement region which are analytically connected in the parameter space of the theory; these regions thus represent a single phase. The effect of multiple Higgs fields on this phase structure is examined via Monte Carlo lattice simulations. For the case of N>=2 identical Higgs fields, there is no remaining analytic connection between the Higgs and confinement regions, at least when Lagrangian terms that directly couple different Higgs flavours are omitt…
The confining and topological properties of the compact Abelian Higgs model with doubly-charged Higgs field in three space-time dimensions are studied. We consider the London limit of the model. We show that the monopoles are forming chain-like structures (kept together by ANO vortices) the presence of which is essential for getting simultaneously permanent confinement of singly-charged particles and breaking of the string spanned between doubly-charged particles. In the confinement phase the chains are forming percolating clusters while in the deconfinement (Higgs) phase the chains are of fin…
We study the properties of topological defects in the lattice compact Abelian Higgs Model with charge Q=2 matter field. We find that monopoles and antimonopoles form chain-like structures which are dense in the confinement/symmetric phase. In this phase the mentioned structures explain both the confinement of single-charged and the breaking of strings spanned between doubly-charged test particles. This observation helps to understand how the non-diagonal gluons, once taken into consideration in the Abelian projection of gluodynamics, could reproduce in this framework the string breaking for ad…
The one-dimensional lattice Schwinger model has recently been realized by using bosons in optical lattices. This model contains both confinement and deconfinement phases, whose phase diagram is controlled by the mass of the matter field and the topological angle. Since varying the mass of matter field is straightforward experimentally, we propose how to tune the topological angle, allowing accessing the entire phase diagram. We propose that direct experimental evidence of confinement and deconfinement can be obtained by measuring whether a physical charge is localized around a fixed gauge char…
In the present paper we investigate the phase transition "Coulomb--confinement" in the Higgs model of abelian scalar dyons -- particles having both, electric $e$ and magnetic $g$, charges. It is shown that by dual symmetry this theory is equivalent to scalar fields with the effective squared electric charge e^{*2}=e^2+g^2. But the Dirac relation distinguishes the electric and magnetic charges of dyons. The following phase transition couplings are obtained in the one--loop approximation: α_{crit}=e^2_{crit}/4π\approx 0.19, \tildeα_{crit}=g^2_{crit}/4π\approx 1.29 and α^*_{crit}\approx 1.48.
The confinement mechanism in the nonperturbative QCD is studied in terms of topological excitation as QCD-monopoles and instantons. In the 't Hooft abelian gauge, QCD is reduced into an abelian gauge theory with monopoles, and the QCD vacuum can be regarded as the dual superconductor with monopole condensation, which leads to the dual Higgs mechanism. The monopole-current theory extracted from QCD is found to have essential features of confinement. We find also close relation between monopoles and instantons using the lattice QCD. In this framework, the lowest $0^{++}$ glueball (1.5 $\sim$ 1.7…
Based on the dual superconductor picture, we study the confinement phenomena systematically, using the lattice QCD, the monopole-current dynamics and the dual Ginzburg-Landau (DGL) theory. (1) We study the origin of abelian dominance for the confinement force in the maximally abelian (MA) gauge in terms of the gluon-field properties using the lattice QCD. In the MA gauge, the off-diagonal gluon amplitude is strongly suppressed, and the off-diagonal gluon phase tends to be random, according to the weakness of the constraint from the QCD action. Within the random-variable approximation for the o…
We study the +/- J random-plaquette Z_2 gauge model (RPGM) in three spatial dimensions, a three-dimensional analog of the two-dimensional +/- J random-bond Ising model (RBIM). The model is a pure Z_2 gauge theory in which randomly chosen plaquettes (occuring with concentration p) have couplings with the ``wrong sign'' so that magnetic flux is energetically favored on these plaquettes. Excitations of the model are one-dimensional ``flux tubes'' that terminate at ``magnetic monopoles.'' Electric confinement can be driven by thermal fluctuations of the flux tubes, by the quenched background of ma…
