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We report on a study of axion string networks using fixed-grid simulations of up to $16384$ points per side. The length of string can be characterised in terms of standard dimensionless parameters $\zeta_\text{w}$ and $\zeta_\text{r}$, the length density measured in the cosmic rest frame and the string rest frame, scaled with the cosmic time. The motion of the string can be characterised by the root-mean-square (RMS) velocity of the string. Starting from a range of initial length densities and velocities, we analyse the string network in the standard scaling framework and find evolution towards a fixed point with estimated values $\hat{\zeta}_{\text{w},*} = 1.220(57)$ and $\hat{\zeta}_{\text{r},*} = 1.491(93)$. The two measures are related by the RMS velocity, which we estimate to be $\hat{v}_{*} = 0.5705(93)$. The length density is consistent with previous measurements, while the velocity is about 5% lower. For simulations starting from low enough density, the length density parameters $\zeta_\text{w}$ and $\zeta_\text{r}$ remain below their fixed point values throughout, while growing slowly, giving rise to an impression of approximately logarithmic increase with time. This has been proposed as the true long-term behaviour. We find that the growth tends to slow down as the values of $\zeta_\text{w}$ and $\zeta_\text{r}$ identified as fixed points are approached. In the case of $\zeta_\text{r}$, the growth stops for simulations which started close to the fixed point length density. The difference between $\zeta_\text{w}$ and $\zeta_\text{r}$ can be understood to result from the continuing velocity evolution. Our results indicate that the growth of $\zeta_\text{w}$ is a transient appearing at low densities and while the velocity is converging. This highlights the importance of studying the string density and the velocity together, and the preparation of initial conditions., Comment: 19 pp, 18 figures

Gravitational waves (GWs) generated by a first-order phase transition at the electroweak scale are detectable by future space-based detectors like LISA. The lifetime of the resulting shock waves plays an important role in determining the intensity of the generated GWs. We have simulated decaying primordial acoustic turbulence in three dimensions and make a prediction for the universal shape of the energy spectrum by using its self-similar decay properties and the shape of individual shock waves. The shape for the spectrum is used to determine the time dependence of the fluid kinetic energy and the energy containing length scale at late times. The inertial range power law is found to be close to the classically predicted $k^{-2}$ and approaches it with increasing Reynolds number. The resulting model for the velocity spectrum and its decay in time is combined with the sound shell model assumptions about the correlations of the velocity field to compute the GW power spectrum for flows that decay in less than the Hubble time. The decay is found to bring about a convergence in the spectral amplitude and the peak power law that leads to a power law shallower than the $k^9$ of the stationary case., Comment: 32 pages, 18 figures, 9 tables

First order phase transitions in the very early universe are a prediction of many extensions of the Standard Model of particle physics and could provide the departure from equilibrium needed for a dynamical explanation of the baryon asymmetry of the Universe. They could also produce gravitational waves of a frequency observable by future space-based detectors such as the Laser Interferometer Space Antenna (LISA). All calculations of the gravitational wave power spectrum rely on a relativistic version of the classical nucleation theory of Cahn-Hilliard and Langer, due to Coleman and Linde. The high purity and precise control of pressure and temperature achievable in the laboratory made the first-order A to B transition of superfluid $^3$He an ideal for test of classical nucleation theory. As Leggett and others have noted the theory fails dramatically. The lifetime of the metastable A phase is measurable, typically of order minutes to hours, far faster than classical nucleation theory predicts. If the nucleation of B phase from the supercooled A phase is due to a new, rapid intrinsic mechanism that would have implications for first-order cosmological phase transitions as well as predictions for gravitational wave (GW) production in the early universe. Here we discuss studies of the AB phase transition dynamics in $^3$He, both experimental and theoretical, and show how the computational technology for cosmological phase transition can be used to simulate the dynamics of the A-B transition, support the experimental investigations of the A-B transition in the QUEST-DMC collaboration with the goal of identifying and quantifying the mechanism(s) responsible for nucleation of stable phases in ultra-pure metastable quantum phases., Comment: 34 pp, 4 figures. Based on a talk given at Quantum Fluids and Solids 2023, Manchester, U.K, on behalf of the QUEST-DMC collaboration

Hybrid Monte Carlo (HMC) simulations of lattice gauge theories with fermionic matter rely on the invertibility of the lattice Dirac operator. Near-zero modes of the latter can therefore significantly slow down the update algorithm and cause instabilities. This is in particular a problem when dealing with massless fermions. Homogeneous temporal Dirichlet boundary conditions can be used to remove zero modes from massless lattice Dirac operators, but the standard implementation of these boundary conditions can cause severe finite-volume cutoff effects in regions of parameter space where the physics at the ultraviolet (UV) cutoff scale is dominated by the fermionic instead of the gauge action. In lattice quantum chromodynamics (QCD) this is usually not an issue, as the gauge action dominates the UV physics and the problem does not show up. In studies of beyond standard model (BSM) theories, on the other hand, the finite-volume artifacts can be severe. We have identified the origin of these IR cutoff effects and propose a simple improvement on the homogeneous temporal Dirichlet boundary conditions to prevent them. We demonstrate the benefits of using our improved boundary conditions at the example of SU(2) lattice gauge theory with $N_f=24$ massless Wilson-clover flavors. Due to the large number of fermions in this theory, the boundary-related finite volume artifacts are particularly strong, and the effect from switching from the normal to our improved homogeneous Dirichlet boundary conditions is therefore distinct., Comment: 9 pages, 11 figures

Lattice gauge field theories may suffer from unphysical "bulk" phase transitions at strong lattice gauge coupling. We introduce a one-parameter family of lattice SU(N) gauge actions which, when used in combination with an HMC update algorithm, prevents the appearance of the bulk phase transition. We briefly discuss the (presumed) mechanism behind the prevention of the bulk transition and present test results for different SU(N) gauge groups., Comment: 14 pages, 8 figures, contains a refined discussion of the working principle of bulk-prevention in SU(N) gauge theories and a new appendix on the relation between bulk configurations and magnetic monopoles

A construction of a gravity dual to a physical gauge theory requires confronting data. We establish a proof-of-concept for precision holography, i.e., the explicit reconstruction of the dual background metric functions directly from the entanglement entropy (EE) of strip subregions that we extract from pure glue Yang-Mills theory discretized on a lattice. Our main focus is on a three-dimensional Euclidean SU(2) theory in the deconfining phase. Holographic EE suggests, and we find evidence for, that the scaling of the thermal entropy with temperature is to power 7/3 and that it approaches smoothly the critical point, consistent with black hole thermodynamics. In addition, we provide frugal results on the potential between quenched quarks by the computation of the Polyakov loop correlators on the lattice. Holographic arguments pique curiosity in the substratum of Debye screening at strong coupling., Comment: 59 pages, 28 figures, to be published in JHEP under the title "Progress in the lattice evaluation of entanglement entropy of three-dimensional Yang-Mills theories and holographic bulk reconstruction"

Using lattice simulations we calculate the rate of baryon number violating processes, the sphaleron rate, in the Standard Model with an external (hyper)magnetic field for temperatures across the electroweak cross-over, focusing on the broken phase. Additionally, we compute the Higgs expectation value and the pseudocritical temperature. The electroweak cross-over shifts to lower temperatures with increasing external magnetic field, bringing the onset of the suppression of the baryon number violation with it. When the hypermagnetic field reaches the magitude $B_Y \approx 2 T^2$ the cross-over temperature is reduced from $160$ GeV to $145$ GeV. In the broken phase for small magnetic fields the rate behaves quadratically as a function of the magnetic flux. For stronger magnetic fields the rate reaches a linear regime which lasts until the field gets strong enough to restore the electroweak symmetry where the symmetric phase rate is reached., Comment: 15 pages, 16 figures, corrected typos, published version

Entanglement entropy is a notoriously difficult quantity to compute in strongly interacting gauge theories. Existing lattice replica methods have suffered from a severe signal-to-noise ratio problem, making high-precision studies prohibitively expensive. Our improved lattice method mitigates this situation and allows us to probe holographic predictions for the behavior of entanglement entropies in three- and four-dimensional Yang-Mills theories. We use this data for the numerical reconstruction of holographic bulk metrics., Comment: 6 pages, 4 figures; contribution to the proceedings of the XVth Quark Confinement and the Hadron Spectrum conference at the University of Stavanger

The determination of entanglement measures in SU(N) gauge theories is a non-trivial task. With the so-called "replica trick", a family of entanglement measures, known as "R\'enyi entropies", can be determined with lattice Monte Carlo. Unfortunately, the standard implementation of the replica method for SU(N) lattice gauge theories suffers from a severe signal-to-noise ratio problem, rendering high-precision studies of R\'enyi entropies prohibitively expensive. In this work, we propose a method to overcome the signal-to-noise ratio problem and show some first results for SU(N) in 4 dimensions., Comment: 9 pages, 5 figures, contribution to the proceedings of the 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Bonn, Germany

We study the crossover phase transition of the SU(2) Georgi-Glashow model in three dimensions. In this model, a confining condensate of topological 't Hooft-Polyakov monopoles exists in the Higgs regime. We use lattice Monte Carlo simulations to study the monopole gas across a crossover transition, and demonstrate that gradient flow can be used to renormalize the otherwise divergent monopole number density. The condensation of the monopoles means that the theory admits also a massive photon-like excitation. We show that the renormalized monopole number density is approximately proportional to the square of the photon mass, in agreement with semiclassical results. Our results give insight into behaviour of the Higgs regime near crossover, which has boarder implications for beyond the Standard Model theories containing adjoint scalar fields., Comment: 16 pages, 6 figures

We study first-order electroweak phase transitions nonperturbatively, assuming any particles beyond the Standard Model are sufficiently heavy to be integrated out at the phase transition. Utilising high temperature dimensional reduction, we perform lattice Monte-Carlo simulations to calculate the main quantities characterising the transition: the critical temperature, the latent heat, the surface tension and the bubble nucleation rate, updating and extending previous lattice studies. We focus on the region where the theory gives first-order phase transitions due to an effective reduction in the Higgs self-coupling and give a detailed comparison with perturbation theory., Comment: 25 pages, 20 figures, for videos of bubble nucleation see https://doi.org/10.5281/zenodo.6548608, and for the dataset see https://doi.org/10.5281/zenodo.6539259; v3 typos corrected

We study the stochastic gravitational wave background (SGWB) produced by freely decaying vortical turbulence in the early Universe. We thoroughly investigate the time correlation of the velocity field, and hence of the anisotropic stresses producing the gravitational waves. With hydrodynamical simulations, we show that the unequal time correlation function (UETC) of the Fourier components of the velocity field is Gaussian in the time difference, as predicted by the "sweeping" decorrelation model. We introduce a decorrelation model that can be extended to wavelengths around the integral scale of the flow. Supplemented with the evolution laws of the kinetic energy and of the integral scale, this provides a new model UETC of the turbulent velocity field consistent with the simulations. We discuss the UETC as a positive definite kernel, and propose to use the Gibbs kernel for the velocity UETC as a natural way to ensure positive definiteness of the SGWB. The SGWB is given by a 4-dimensional integration of the resulting anisotropic stress UETC with the gravitational wave Green's function. We perform this integration using a Monte Carlo algorithm based on importance sampling, and find that the result matches that of the simulations. Furthermore, the SGWB obtained from the numerical integration and from the simulations show close agreement with a model in which the source is constant in time and abruptly turns off after a few eddy turnover times. Based on this assumption, we provide an approximate analytical form for the SGWB spectrum and its scaling with the initial kinetic energy and integral scale. Finally, we use our model and numerical integration algorithm to show that including an initial growth phase for the turbulent flow heavily influences the spectral shape of the SGWB. This highlights the importance of a complete understanding of the turbulence generation mechanism., Comment: 54 pages, 30 figures

The Laser Interferometer Space Antenna (LISA) has two scientific objectives of cosmological focus: to probe the expansion rate of the universe, and to understand stochastic gravitational-wave backgrounds and their implications for early universe and particle physics, from the MeV to the Planck scale. However, the range of potential cosmological applications of gravitational wave observations extends well beyond these two objectives. This publication presents a summary of the state of the art in LISA cosmology, theory and methods, and identifies new opportunities to use gravitational wave observations by LISA to probe the universe.

Gravitational waves from a phase transition associated with the generation of the masses of elementary particles are within the reach of future space-based detectors such as LISA. A key determinant of the resulting power spectrum, not previously studied, is the lifetime of the acoustic turbulence which follows. We study decaying acoustic turbulence using numerical simulations of a relativistic fluid in two dimensions. Working in the limit of non-relativistic bulk velocities, with an ultra-relativistic equation of state, we find that the energy spectrum evolves towards a self-similar broken power law, with a high-wavenumber behaviour of $k^{-2.08 \pm 0.08}$, cut off at very high $k$ by the inverse width of the shock waves. Our model for the decay of acoustic turbulence can be extended to three dimensions using the universality of the high-$k$ power law and the evolution laws for the kinetic energy and the integral length scale. It is used to build an estimate for the gravitational wave power spectrum resulting from a collection of shock waves, as might be found in the aftermath of a strong first order phase transition in the early universe. The power spectrum has a peak wavenumber set by the initial length scale of the acoustic waves, and a new secondary scale at a lower wavenumber set by the integral scale after a Hubble time. Between these scales a distinctive new power law appears. Our results allow more accurate predictions of the gravitational wave power spectrum for a wide range of early universe phase transition scenarios., Comment: 23 pages, 22 figures

We perform a lattice study of the phase transition in the $\mathrm{SU}(2)$ Georgi-Glashow model in three dimensions, where the gauge symmetry is broken to $\mathrm{U}(1)$ by the Higgs mechanism and a photon-like state appears. Due to condensation of monopoles the photon acquires a mass, which depends on the number density of the monopoles. We show that the monopole density can be renormalised on the lattice using gradient flow. Our preliminary results suggest that Polyakov's semiclassical relation between the photon mass and the renormalised monopole density is valid also at the nonperturbative level., Comment: 7 pages, 5 figures. The 38th International Symposium on Lattice Field Theory, LATTICE2021, 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We study SU(2) gauge field theory with $N_f=24$ quarks. The theory is asymptotically non-free and, at vanishing quark mass, governed by a Gaussian fixed point at long distances. On the other hand, at non-zero quark mass the quarks are expected to decouple at long distances and the system behaves like confining pure gauge SU(2) theory. We study the mass spectrum of the theory as the quark mass is varied and obtain scaling laws for meson masses and string tension. We also measure the evolution of the coupling constant at non-zero quark mass with gradient flow method. We observe unambiguously the decoupling of the quarks with the associated change of evolution of the coupling constant., Comment: 9 pages, 4 figures, contribution to the 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

We introduce a one-parameter family of SU(N) gauge actions which, when used in combination with an HMC update algorithm, prevent the gauge system from entering an artificial bulk-"phase". We briefly discuss the mechanism behind the bulk-prevention and present test results for different SU(N) gauge groups., Comment: 9 pages, 5 figures, contribution to the 38th International Symposium on Lattice Field Theory, LATTICE2021 26th-30th July, 2021, Zoom/Gather@Massachusetts Institute of Technology

SU(2) gauge theory with Nf=24 massless fermions is non-interacting at long distances, i.e. it has an infrared fixed point at vanishing coupling. With massive fermions the fermions are expected to decouple at energy scales below the fermion mass, and the infrared behaviour is that of confining SU(2) pure gauge theory. We demonstrate this behaviour non-perturbatively with lattice Monte Carlo simulations by measuring the gradient flow running coupling., Comment: 12 pages, 10 figures, appendix added

Building on the experience of [1], we develop a formalism to construct operators for higher derivatives of the pressure in hot QCD with respect to the quark chemical potential $\mu$. We provide formulae for the operators up to the sixth derivative, and obtain continuum-extrapolated results from lattice EQCD at zero and finite $\mu$ and at six different pairs of temperature T and number of massless quark flavors $n_f$. Our data is benchmarked against full-QCD lattice and perturbative results, allowing to judge the quality of the perturbative series expansion in EQCD and the dimensional reduction procedure as a whole., Comment: HIP-2021-28/TH

We present a numerical study of the spectrum of an asymptotically non-free $SU(2)$ gauge theory with $N_f=24$ massive fermion flavors. For such large number of flavors, asymptotic freedom is lost and the massless theory is governed by a gaussian fixed point at long distances. If fermions are massive they decouple at low energy scales and the theory is confining. We present a scaling law for the masses of the hadrons, glueballs and string tension as functions of fermion mass. The hadrons become effectively heavy quark systems, with masses approximately twice the fermion mass, whereas the energy scale of the confinement, probed by e.g. the string tension, is much smaller and vanishes asymptotically as $m_\text{fermion}^{2.18}$. Our results from lattice simulations are compatible with this behaviour.

We investigate the potential for observing gravitational waves from cosmological phase transitions with LISA in light of recent theoretical and experimental developments. Our analysis is based on current state-of-the-art simulations of sound waves in the cosmic fluid after the phase transition completes. We discuss the various sources of gravitational radiation, the underlying parameters describing the phase transition and a variety of viable particle physics models in this context, clarifying common misconceptions that appear in the literature and identifying open questions requiring future study. We also present a web-based tool, PTPlot, that allows users to obtain up-to-date detection prospects for a given set of phase transition parameters at LISA., Comment: 68 pages, 16 figures, some comments added, typos fixed

We present the first numerical study of the ultraviolet dynamics of non-asymptotically free gauge-fermion theories at large number of matter fields. As testbed theories we consider non-abelian SU(2) gauge theories with 24 and 48 Dirac fermions on the lattice. For these number of flavors asymptotic freedom is lost and the theories are governed by a gaussian fixed point at low energies. In the ultraviolet they can develop a physical cutoff and therefore be trivial, or achieve an interacting safe fixed point and therefore be fundamental at all energy scales. We demonstrate that the gradient flow method can be successfully implemented and applied to determine the renormalized running coupling when asymptotic freedom is lost. Additionally, we prove that our analysis is connected to the gaussian fixed point as our results nicely match with the perturbative beta function. Intriguingly, we observe that it is hard to achieve large values of the renormalized coupling on the lattice. This might be an early sign of the existence of a physical cutoff and imply that a larger number of flavors is needed to achieve the safe fixed point. A more conservative interpretation of the results is that the current lattice action is unable to explore the deep ultraviolet region where safety might emerge. Our work constitutes an essential step towards determining the ultraviolet fate of non asymptotically free gauge theories., Comment: 14 pages, 12 figures, references added

Making use of a dimensionally-reduced effective theory at high temperature, we perform a nonperturbative study of the electroweak phase transition in the Two Higgs Doublet model. We focus on two phenomenologically allowed points in the parameter space, carrying out dynamical lattice simulations to determine the equilibrium properties of the transition. We discuss the shortcomings of conventional perturbative approaches based on the resummed effective potential - regarding the insufficient handling of infrared resummation but also the need to account for corrections beyond 1-loop order in the presence of large scalar couplings - and demonstrate that greater accuracy can be achieved with perturbative methods within the effective theory. We find that in the presence of very large scalar couplings, strong phase transitions cannot be reliably studied with any of the methods.

We review our recent results on the infrared behaviour of the SU(2) gauge theory with $N_f$ massless fundamental flavour fermions. We have analyzed the running of the coupling in SU(2) gauge theories with six and eight fermionic flavours using the gradient flow step scaling method. From the running of the coupling, we see a clear indication of an infrared stable fixed point in theories with six and eight flavours. These results are confirmed by our mass spectrum study, where we varied the number of flavours from two to six. We also compute the anomalous dimensions of mass and coupling., Comment: 9 pages + 4 figures, Contribution to the proceedings of Quark Confinement and the Hadron Spectrum XIII conference (Confinement 2018), Maynooth, Ireland, August 2018

We compute the spectrum of SU(2) gauge theory with two, four or six Dirac fermions in the fundamental representation of the gauge group. We investigate the scaling of the meson masses as a function of the quark mass when approaching the chiral limit. We find behavior compatible with the usual pattern of chiral symmetry breaking at $N_f\le 4$ and with the existence of an infrared fixed point at $N_f=6$. In the six flavor case we determine the anomalous dimension of the quark mass operator from the spectrum and find results fully consistent with earlier analyses., Comment: 6 pages, 10 figures

We consider measurement of the leading irrelevant scaling exponent $\gamma_g^\ast$, given by the slope of the beta function, at the fixed point of SU(2) gauge theory with six or eight flavors. We use the running coupling measured using the gradient flow method and perform the continuum extrapolation by interpolating the measured beta function. We study also the dependence of the results on different discretization of the flow. For the eight flavor theory we find $\gamma_g^\ast=0.19(8)_{-0.09}^{+0.21}$. Applying the same analysis also for the six flavor theory, we find $\gamma_g^\ast=0.648(97)_{-0.1}^{+0.16}$ consistently with the earlier analysis., Comment: 11 pages, 15 figures, updated to match published version

We present the updated results of the infrared behavior of the SU(2) 6 and 8 fundamental representation fermions. We use the gradient flow method with the Schr\"odinger functional boundary conditions to measure the running of the coupling in these theories and find fixed points on both. We also measure the mass anomalous dimension from these configurations., Comment: 8 pages, 7 figures. the 35th International Symposium on Lattice Field Theory at Granada, Spain (18-24 June 2017)

We compute the running of the coupling in SU(2) gauge theory with six fermions in fundamental representation of the gauge group. We establish an infrared stable fixed point at strong coupling and measure also the anomalous dimension of the fermion mass operator at the fixed point. This theory therefore likely lies close to the boundary of the conformal window and will display novel infrared dynamics if coupled with the electroweak sector of the Standard Model., Comment: 7 pages + 6 pages of tables, 5 figures, updated to match the published version

We present results from large-scale numerical simulations of a first order thermal phase transition in the early universe, in order to explore the shape of the acoustic gravitational wave and the velocity power spectra. We compare the results with the predictions of the recently proposed sound shell model. For the gravitational wave power spectrum, we find that the predicted $k^{-3}$ behaviour, where $k$ is the wavenumber, emerges clearly for detonations. The power spectra from deflagrations show similar features, but exhibit a steeper high-$k$ decay and an extra feature not accounted for in the model. There are two independent length scales: the mean bubble separation and the thickness of the sound shell around the expanding bubble of the low temperature phase. It is the sound shell thickness which sets the position of the peak of the power spectrum. The low wavenumber behaviour of the velocity power spectrum is consistent with a causal $k^{3}$, except for the thinnest sound shell, where it is steeper. We present parameters for a simple broken power law fit to the gravitational wave power spectrum for wall speeds well away from the speed of sound where this form can be usefully applied. We examine the prospects for the detection, showing that a LISA-like mission has the sensitivity to detect a gravitational wave signal from sound waves with an RMS fluid velocity of about $0.05c$, produced from bubbles with a mean separation of about $10^{-2}$ of the Hubble radius. The shape of the gravitational wave power spectrum depends on the bubble wall speed, and it may be possible to estimate the wall speed, and constrain other phase transition parameters, with an accurate measurement of a stochastic gravitational wave background., Comment: 16+1 pages, 9+1 figures; v2: added a few paragraphs about the importance of turbulence, published in PRD; v3: added erratum correcting two equations and the SNR plot, methods and conclusions unchanged

We study the evolution of the coupling in SU(2) gauge field theory with $N_f=8$ fundamental fermion flavors on the lattice. This model is expected to have an infrared fixed point at high coupling. We use HEX-smeared Wilson-clover action, and measure the gradient flow running coupling with Dirichlet boundary conditions. Extrapolating our results to continuum, we find an infrared fixed point at $g^2=8.24(59)^{+0.97}_{-1.64}$, with statistical and systematic error estimates. We also measure the anomalous dimension of the quark mass operator, and find its value at the fixed point $\gamma \approx 0.15 \pm 0.02$., Comment: 24 pages, 11 figures, updated to match the published version

We perform the first numerical simulations of necklaces in a non-Abelian gauge theory. Necklaces are composite classical solutions which can be interpreted as monopoles trapped on strings, rather generic structures in a Grand Unified Theory. We generate necklaces from random initial conditions, modelling a phase transition in the early Universe, and study the evolution. For all cases, we find that the necklace system shows scaling behaviour similar to that of a network of ordinary cosmic strings. Furthermore, our simulations indicate that comoving distance between the monopoles or semipoles along the string asymptotes to a constant value at late times. This means that while the monopole-to-string energy density ratio decreases as the inverse of the scale factor, a horizon-size length of string has a large number of monopoles, significantly affecting the dynamics of string loops. We argue that gravitational wave bounds from millisecond pulsar timing on the string tension in the Nambu-Goto scenario are greatly relaxed., Comment: 15 pages, 8 figures; v2: Figs. 2 (bottom) and 4 (top) corrected, some discussion added, results unchanged, accepted by PRD

SU(2) with N_f = 6 and N_f = 8 are believed to have an infrared conformal fixed point. We use the spectral density method cross referenced with the mass step scaling method to evaluate the coupling constant dependence of the mass anomalous dimension for massless HEX smeared, clover improved Wilson fermions with Schr\"odinger functional boundary conditions., Comment: 7 pages, 4 figures, contribution to the proceedings of the 34rd International Symposium on Lattice Field Theory (Lattice 2016), 24-30 July 2016, Southampton, UK. arXiv admin note: substantial text overlap with arXiv:1511.03468

We present preliminary results of the running of the coupling in SU(2) gauge theory with 6 massless fundamental representation fermion flavors. We measure the coupling using the gradient flow method with Schr\"odinger functional boundary conditions. The results are consistent with the perturbation theory in the weak coupling and we see an indication of an infrared fixed point at strong coupling., Comment: 7 pages, 3 figures, Contribution to proceedings of the 34rd International Symposium on Lattice Field Theory (Lattice 2016), 24-30 July 2016, Southampton, UK

We study the properties of classical vortex solutions in a non-Abelian gauge theory. A system of two adjoint Higgs fields breaks the SU(2) gauge symmetry to $Z_2$, producing 't Hooft-Polyakov monopoles trapped on cosmic strings, termed beads; there are two charges of monopole and two degenerate string solutions. The strings break an accidental discrete $Z_2$ symmetry of the theory, explaining the degeneracy of the ground state. Further symmetries of the model, not previously appreciated, emerge when the masses of the two adjoint Higgs fields are degenerate. The breaking of the enlarged discrete symmetry gives rise to additional string solutions and splits the monopoles into four types of `semipole': kink solutions that interpolate between the string solutions, classified by a complex gauge invariant magnetic flux and a $Z_4$ charge. At special values of the Higgs self-couplings, the accidental symmetry broken by the string is continuous, giving rise to supercurrents on the strings. The SU(2) theory can be embedded in a wide class of Grand Unified Theories, including SO(10). We argue that semipoles and supercurrents are generic on GUT strings., Comment: 6 pages, 2 figures. We dedicate this paper to Tom Kibble; v2: some references added, typos fixed, phrasing changed in places; accepted by PRL

It is expected that SU(2) gauge theory with $N_f$ fundamental fermions has an infrared fixed point when $N_f$ is between $\sim 6$ and 10. We study the hadron spectrum and scale setting in SU(2) gauge field theory with $N_f=2,4,6$ using hypercubic stout smeared Wilson-clover (HEX) action. The case $N_f=2$ is QCD-like, whereas $N_f=6$ is close to the lower edge of the conformal window. In our study the length scales are determined by using the gradient flow approach., Comment: 7 pages, 7 figures, presented at the 33rd International Symposium on Lattice Field Theory, 14 - 18 July 2015, Kobe, Japan

First order phase transitions in the early Universe generate gravitational waves, which may be observable in future space-based gravitational wave observatiories, e.g. the European eLISA satellite constellation. The gravitational waves provide an unprecedented direct view of the Universe at the time of their creation. We study the generation of the gravitational waves during a first order phase transition using large-scale simulations of a model consisting of relativistic fluid and an order parameter field. We observe that the dominant source of gravitational waves is the sound generated by the transition, resulting in considerably stronger radiation than earlier calculations have indicated., Comment: Presented at the 33rd International Symposium on Lattice Field Theory, 14-18 July 2015, Kobe, Japan

We study the running of the coupling in SU(2) gauge theory with 8 massless fundamental representation fermion flavours, using the gradient flow method with the Schr\"odinger functional boundary conditions. Gradient flow allows us to measure robust continuum limit for the step scaling function. The results show a clear indication of infrared fixed point consistent with perturbation theory., Comment: 7 pages, 4 figures, Contribution to proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe, Japan. arXiv admin note: text overlap with arXiv:1411.4879

SU(2) with Nf=8 is believed to have an infrared conformal fixed point. We use the spectral density method to evaluate the coupling constant dependence of the mass anomalous dimension for massless HEX smeared, clover improved Wilson fermions with Schr\"odinger functional boundary conditions., Comment: 7 pages, 4 figures, presented at the 33rd International Symposium on Lattice Field Theory, 14 - 18 July 2015, Kobe, Japan

We study SU(2) gauge theory with two Dirac fermions in the adjoint representation of the gauge group on the lattice. Using clover improved Wilson fermion action with hypercubic truncated stout smearing we perform simulations at larger coupling than earlier. We measure the evolution of the coupling constant using the step scaling method with Schr\"odinger functional and study the remaining discretization effects. At weak coupling we observe significant discretization effects, which make it difficult to obtain a fully controlled continuum limit. Nevertheless, the data remains consistent with the existence of a fixed point in the interval $2.2\lesssim g^{\ast 2}\lesssim 3$. We also measure the anomalous dimension and find its value at the fixed point is $\gamma^\ast\simeq 0.2\pm 0.03$., Comment: 9 pages, 6 figures. Revisions in the abstract and introduction, as well as the handling of the continuum limit

With the physical Higgs mass the Standard Model symmetry restoration phase transition is a smooth cross-over. We study the thermodynamics of the cross-over using numerical lattice Monte Carlo simulations of an effective SU(2) X U(1) gauge + Higgs theory, significantly improving on previously published results. We measure the Higgs field expectation value, thermodynamic quantities like pressure, energy density, speed of sound and heat capacity, and screening masses associated with the Higgs and Z fields. While the cross-over is smooth, it is very well defined with a width of only approximately 5 GeV. We measure the cross-over temperature from the maximum of the susceptibility of the Higgs condensate, with the result $T_c = 159.5 \pm 1.5$ GeV. Outside of the narrow cross-over region the perturbative results agree well with non-perturbative ones., Comment: 10 pages

We present details of numerical simulations of the gravitational radiation produced by a first order thermal phase transition in the early universe. We confirm that the dominant source of gravitational waves is sound waves generated by the expanding bubbles of the low-temperature phase. We demonstrate that the sound waves have a power spectrum with a power-law form between the scales set by the average bubble separation (which sets the length scale of the fluid flow $L_\text{f}$) and the bubble wall width. The sound waves generate gravitational waves whose power spectrum also has a power-law form, at a rate proportional to $L_\text{f}$ and the square of the fluid kinetic energy density. We identify a dimensionless parameter $\tilde\Omega_\text{GW}$ characterising the efficiency of this "acoustic" gravitational wave production whose value is $8\pi\tilde\Omega_\text{GW} \simeq 0.8 \pm 0.1$ across all our simulations. We compare the acoustic gravitational waves with the standard prediction from the envelope approximation. Not only is the power spectrum steeper (apart from an initial transient) but the gravitational wave energy density is generically larger by the ratio of the Hubble time to the phase transition duration, which can be 2 orders of magnitude or more in a typical first order electroweak phase transition., Comment: 22 pages, 13 figures; v2: typos in section IIIE (Source lifetime) corrected, new appendix on GW production in an expanding universe, other minor corrections, version accepted for publication

We present preliminary results of the gradient flow running coupling with Dirichlet boundary condition in the SU(2) gauge theory with 8 fermion flavours. Improvements to the gradient flow measurement allow us to obtain a robust continuum limit. The results are consistent with perturbative running in the weak coupling region., Comment: 7 pages, 6 figures, Contribution to the 32nd International Symposium on Lattice Field Theory (Lattice 2014), 23-28 June 2014, Columbia University, New York, NY, USA

We present a lattice study of the momentum broadening experienced by a hard parton in the quark-gluon plasma. In particular, the contributions to this real-time phenomenon from soft modes are extracted from a set of gauge-invariant operators in a dimensionally reduced effective theory (electrostatic QCD), which can be simulated on a Euclidean lattice. At the temperatures accessible to present experiments, the soft contributions to the jet quenching parameter are found to be quite large. We compare our results to phenomenological models and to holographic computations., Comment: 4 pages, 2 figures, talk presented at the XXIV International Conference on Ultrarelativistic Nucleus-Nucleus Collisions "Quark Matter 2014" (19-24 May 2014, Darmstadt, Germany)