Spectroscopic analysis of the phase diagram of Yang-Mills-Higgs theory

Axel Maas and Tajdar Mufti

Phys. Rev. D 91, 113011 – Published 18 June 2015

Abstract

Yang-Mills-Higgs theory, being the standard-model Higgs sector for a suitable choice of a gauge and custodial group, offers a rich set of physics. In particular, in some regions of its parameter space it has QCD-like behavior, while in some other regions it is Higgs-like. Therefore, it is possible to study a plethora of phenomena within a single theory. Here, the physics of the standard-model version is studied using lattice gauge theory at a qualitative level. To this end, the low-lying spectrum in several different channels is obtained for more than 160 different sets of bare parameters throughout the phase diagram. The theory shows quite different behaviors in the different regions, from almost Yang-Mills-like to the one of an essentially free gas of massive photons. Especially, not always is the behavior as naively expected.

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Received 25 February 2015

DOI:https://doi.org/10.1103/PhysRevD.91.113011

Authors & Affiliations

Axel Maas¹ and Tajdar Mufti²

¹Institute of Physics, University of Graz, Universitätsplatz 5, A-8010 Graz, Austria
²Institute for Theoretical Physics, University of Jena, Max-Wien-Platz 1, D-07743 Jena, Germany

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Issue

Vol. 91, Iss. 11 — 1 June 2015

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Images

Figure 1
The effective mass of the operators $O_{ρ}^{n}$ (left panel) and $O_{1^{-}}^{n}$ as a function of time for different numbers $n$ of APE sweeps. Results are at $β = 2.4728$ , $κ = 0.2939$ , and $λ = 1.036$ , which is in the Higgs-like domain with a physical $0_{1}^{+}$ to $1_{3}^{-}$ ratio and 88192 configurations.
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Figure 2
The effective energy levels in the $1_{3}^{-}$ channel, together with fits of type (8). Different styles correspond to different eigenvalues. Points with too large error bars have been omitted for visibility. See text for details. Here and hereafter, the full line is the fit to the average values, while the dashed one is the fit to the $1 σ$ -shifted data, and thus gives an error band.
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Figure 3
The effective energy levels in the $0_{1}^{+}$ channel, together with fits of type (8). Different styles correspond to different eigenvalues. Points with too large error bars have been omitted for visibility. See text for details.
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Figure 4
The effective energy levels in the $0_{1}^{-}$ channel (top panel) and $2_{1}^{+}$ channel (bottom panel), together with fits of type (8). Different styles correspond to different eigenvalues. Points with too large error bars have been omitted for visibility. See text for details.
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Figure 5
A two-dimensional hyperplane of the phase diagram. The axes are some functions of the parameters. For a quantitative version, see [8, 9].
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Figure 6
The top plot shows the phase diagram of Yang-Mills-Higgs theory as a function of bare gauge coupling, Higgs 4-point coupling, and the bare Higgs mass in units of the $0_{1}^{+}$ mass. Green points are confinementlike, and purple points are Higgs-like. The lighter the points, the smaller is the lattice spacing. The bottom plot shows the same in terms of the lattice bare parameters of inverse gauge coupling, hopping parameter, and four-Higgs coupling; see (1) for their relation to the continuum parameters.
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Figure 7
The left-hand plot shows the bare continuum parameters qualifying as to be in the QLD, while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 8
The ground states in the different quantum number channels in the QLD. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 9
The ground states in the different quantum number channels in the QLD. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 10
The next two states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels) in the QLD. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 11
The left-hand plot shows the bare continuum parameters qualifying as to be in the crossover region, while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 12
The ground states in the different quantum number channels in the crossover region. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 13
The ground states in the different quantum number channels in the crossover region. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 14
The two next states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels) in the crossover region. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 15
The left-hand plot shows the bare continuum parameters exhibiting a light $0_{1}^{+}$ in the HLD, while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 16
The ground states in the different quantum number channels. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 17
The ground states in the different quantum number channels. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 18
The two next states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels). Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 19
The left-hand plot shows the bare continuum parameters exhibiting a $0_{1}^{+}$ ground state in the HLD within the physical mass window, while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 20
The ground states in the different quantum number channels. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 21
The ground states in the different quantum number channels. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 22
The two next states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels). Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 23
The left-hand plot shows the bare continuum parameters exhibiting a $0_{1}^{+}$ ground state in the HLD above the physical mass window, but still stable, while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 24
The ground states in the different quantum number channels. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 25
The ground states in the different quantum number channels. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 26
The two next states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels). Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 27
The left-hand plot shows the bare continuum parameters exhibiting a $0_{1}^{+}$ ground state in the HLD with the $0_{1}^{+}$ at the decay threshold into two $1_{3}^{-}$ , while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 28
The ground states in the different quantum number channels, lattice spacing decreasing from right to left. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 29
The ground states in the different quantum number channels, lattice spacing decreasing from right to left. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 30
The two next states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels), lattice spacing decreasing from right to left. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 31
The left-hand plot shows the bare continuum parameters exhibiting a $0_{1}^{+}$ ground state for anomalous points, while the right-hand plot shows the same in terms of the lattice bare parameters.
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Figure 32
The ground states in the different quantum number channels, lattice spacing decreasing from right to left. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to the lightest mass.
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Figure 33
The ground states in the different quantum number channels, lattice spacing decreasing from right to left. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. For the normalization, see text.
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Figure 34
The next two states after the ground state in the $0_{1}^{+}$ channel (upper panels) and the $1_{3}^{-}$ channel (bottom panels), lattice spacing decreasing from right to left. Open symbols have an energy greater than 1, but below 1.5, in lattice units, while closed symbols are below 1. All levels are normalized to scattering states, as detailed in the text.
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Figure 35
The volume dependence of the extracted states in the QLD. Note that here and hereafter the different channels have been slightly displaced horizontally for better visibility.
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Figure 36
The volume dependence of the extracted states for a light Higgs. Symbols and lines have the same meaning as in Fig. 35.
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Figure 37
The volume dependence of the extracted states for a physical Higgs. Symbols and lines have the same meaning as in Fig. 35.
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Figure 38
The volume dependence of the extracted states for the threshold case. Symbols and lines have the same meaning as in Fig. 35.
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Figure 39
The cutoff dependence of the extracted states for a physical Higgs. Different channels have been slightly displaced for better visibility. Symbols and lines have the same meaning as in Fig. 35. Especially, the black solid lines are in the $0^{+}$ channel made from constituents without relative momenta, and the red dashed lines are in the $1^{-}$ channel made from constituents without relative momenta.
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