IR-conformal dynamics in SU(2) gauge theories with adjoint fermions

An urgent question in Particle Physics is related to the possible existence of new interactions beyond the Standard Model. The existence of those interactions is strongly suggested by currently open problems such as the nature of Dark Matter and the naturalness of the measured value of the Higgs mass.  A framework that could provide a solution to a significant number of open problems is near-conformal dynamics beyond the Standard Model. This framework is based on the existence of a novel strong force whose dynamics, rather than being confining, is conformal in the infrared (i.e., at asymptotically large distances).

Figure 1: The ratio of the mass of the vector meson over the mass of the scalar meson as a function of the constituent fermion mass for the SU(2) gauge theory with one adjoint fermion flavour. Different colours indicate different values of the gauge coupling, and larger couplings correspond to smaller lattice spacings. As the constituent mass goes to zero, this ratio is expected to diverge if the theory is confining and to be approximately constant if the theory is infrared conformal.

Currently, no example of a model displaying near-conformality in the infrared is known in nature. Therefore, as a part of the investigation effort, a set of signatures for near-conformality needs to be identified. The problem being a typically strong coupling one, Monte Carlo simulations of the model discretised on a spacetime lattice provide the best quantitative first principles approach. Previous  simulations pioneered by our collaboration and conducted on earlier DiRAC hardware have shown that a robust signal of near-conformality is provided by the behaviour of ratios of spectral masses: in the near-conformal case, these ratios are very mild functions of the mass of the constituent fermions, while in the better understood confining case there is a state (the Goldstone boson associated with the breaking of a global internal symmetry) whose mass goes to zero as the constituent mass is decreased towards zero, with all other masses staying instead finite in the same zero constituent mass limit. Taking the continuum limit of spectral ratios in nearly conformal theories is a remarkable computational challenge that can be addressed only with the latest hardware technologies.

Using the capabilities of the latest DiRAC3 Extreme Scaling facility, we have obtained preliminary results that, if confirmed by more extended simulations, will provide the first evidence of the persistence of near-conformal behaviour in the SU(2) gauge theory with one Dirac fermion flavour in the adjoint representation as the lattice spacing is reduced (see Fig. 1). This finding enables us to take the studied model as a robust template of near-conformal behaviour that will be used in successive investigations to inform phenomenologically relevant studies. 

The Occam’s razor equation

This equation is science in a nutshell. On the left-hand-side it has the Bayesian Evidence. This is the quantity which is responsible for updating our belief in a scientific model in light of data (for example, how much we believe in the concordance model of cosmology in light of new supernovae measurements). On the right hand side it has the posterior averaged loglikelihood, and the Kullback Liebler divergence.

The first of these is a Bayesian “goodness of fit” measure, and the second of these is an information theory “model complexity”. The equation therefore quantifies Occam’s razor, telling us that the degree to which a model is scientifically convincing is composed in equal part how well it describes the data and in the simplicity with which it does so.

This equation and philosophical interpretation was discovered as part of a DiRAC supported Bayesian study in cosmology [2102.11511, equation 9].

The paper showcases some of the subtle but important details relevant when applying Bayesian model comparison in a modern cosmological setting. As concrete examples it considers the challenge of using measurement of the Universe to weigh and determine the hierarchy of neutrino masses, as well as detecting primordial gravitational waves generated near to the big bang during cosmic inflation. Such tasks are theoretically, observationally and computationally extremely demanding, requiring high performance computing to extract robust inferences from cosmological data. The paper emphasises that as we move to more and more stringent constraints on the tensor-to-scalar ratio using BICEP and LiteBIRD these considerations will become increasingly vital for interpreting the scientific content of increasingly powerful experiments.

Nested sampling beyond Bayesian statistics

Nested sampling is a numerical tool widely used in cosmology for performing Bayesian data analysis: using astrophysical data and models of the universe to extract parameters such as its age and its size, as well as to numerically determine which model is preferred by the data.

This paper [2105.13923] showcases that the fundamental nested sampling technique is actually applicable in a far wider set of physical and statistical contexts, by applying it to a frequentist analysis commonly used in Particle Physics: that of computing the ‘p value’ for detection of a new particle.

The plot shows that while the current state-of-the-art Monte Carlo approach performs well for computing significances of 1 and 2 sigma, it becomes exponentially expensive for computing higher significances. Indeed, in order to reach the gold standard of ‘five sigma’ (as for example reached in the discovery of new particles such as the Higg’s Boson), it is shown that nested sampling is thousands of times more efficient, with the nested sampling algorithm MultiNest performing well in low dimensions d<30, but at higher dimensionalities as demanded by modern cosmological analyses, PolyChord becomes more efficient.

This thorough analysis was supported by a DiRAC grant for applying nested sampling to cosmology and particle physics analyses. The wide applicability and general Physics interest of this research was recognised by its publication in the high-impact flagship journal Physical Review Letters.

Composite Higgs Phenomenology at the LHC

The figure from [1] shows p-wave phase shift as a function of the energy in the center of mass frame of the scattering of two Goldstone bosons. The red solid line shows the Breit-Wigner resonance fit. The results allow to predict the effective coupling constant between the vector resonance and the two Goldstone bosons.

Understanding the nature of the Standard Model Higgs boson is still an open problem. An appealing possibility is that the Higgs boson be a composite particle resulting from a novel strong interaction. The lack of observation of otherwise unexplained particles interacting through this conjectured novel force and with mass comparable to that of the Higgs boson would then require a mechanism that keeps the Higgs parametrically light. Such a mechanism can be provided by the spontaneous breaking of the global flavour symmetry in the new interaction. In this framework, the Higgs boson is interpreted as a Pseudo-Nambu-Goldstone Boson (PNGB) of the novel interaction. Among candidate realisations of PNGB compositeness are Sp(2N) gauge theories. The two simplest theories in this class are Sp(2), coinciding with SU(2), and Sp(4). Given the strong nature of the novel force, first-principle calculations allow to determine quantitative predictions that are used to test the experimental viability of these theories.

The figure from [2] shows a representative meson spectrum in lattice units for the Sp(4) model, at selected lattice parameters for constituent fermions in the fundamental (blue) and in the antisymmetric representation (red). The purple point indicates the mass of the chimera baryon – a fermion bound state with admixture of valence fermions in different representations.

In our project, we performed the first calculation of the scattering of two Goldstone bosons in the vector resonance channel for an SU(2) gauge theory with two fundamental fermions[1].  The calculation allows to obtain the first prediction of the vector resonance effective coupling to Goldstone bosons in a phenomenologically viable model. The coupling is used by phenomenologists to constrain Composite Higgs model at the LHC and is therefore relevant to shed light on the role played by the vector resonance in a particular realisation of a Composite Higgs model.

In the Sp(4) model, we performed the first calculation of the spectrum in a phenomenologically relevant model in which the matter content is provided by two Dirac fermions in the fundamental representation and three Dirac fermions in the antisymmetric representation [2]. In particular, we have determined the mass of the chimera baryon, a bound state whose constituents are two fermions in the fundamental representation and one in the antisymmetric representation and whose importance is related to the partial top compositeness mechanism, which has been advocated to account for the large mass of the top quark.

[1] V. Drach et al., Scattering of Goldstone Bosons and resonance production in a Composite Higgs model on the lattice, JHEP 04 (2021), 117 [arXiv:2012.09761]
[2] E. Bennett et al., Lattice studies of the Sp(4) gauge theory with two fundamental and three antisymmetric Dirac fermions, arXiv:2202.05516. 

Performing the First Resolution Study for Primordial Star Formation Simulations

The Population III initial mass function (IMF) is currently unknown, but recent studies agree that fragmentation of primordial gas gives a broader IMF than the initially suggested singular star per halo. In this study we introduce sink particle mergers into the moving mesh code Arepo, to perform the first resolution study for primordial star formation simulations and present the first Population III simulations to run up to densities of 10-6 g cm-3 for hundreds of years after the formation of sink particles.

The maximum resolution of a simulation is set by the maximum density it is allowed to run to. When the gas density increases during gravitationally collapse, the Jeans length (the scale that structures can support themselves thermally against gravitational collapse) shrinks, and so the simulation mesh must refine to resolve these scales and prevent artificial instability. This process continues until the maximum density is reached and a sink particle is introduced to represent the densest gas on the smallest scales. In this resolution study, we vary the sink particle creation density from 10-10 – 10-6 g cm-3 and track the fragmentation behaviour of the gas and the resulting IMF.

The image shows how the structure of the star forming cloud changes as the maximum resolution is increased, and how they evolve in time. Sink particles are displayed as purple dots.

We find that the total number of sink particles formed continues to increase with increasing sink particle creation density, without achieving numerical convergence. However, the total mass in sinks remains invariant to the maximum resolution and is safe to estimate using low resolution studies. This results in an IMF that shifts towards lower masses with increasing resolution. Greater numbers of sinks cause increased fragmentation-induced starvation of the most massive sink, yielding lower accretion rates, masses and ionising photons emitted per second. The lack of convergence up to densities 2 orders of magnitudes higher than all relevant chemical reactions suggests that the number of sinks will continue to grow with increasing resolution until H2 is fully dissociated and the collapse becomes almost adiabatic at 10-4 g cm-3. We also note that in the highest resolution runs, sinks with stellar masses capable of surviving until the present day had an ejection fraction of 0.21.

These results imply that many Population III studies utilising sink particles have produced IMFs which have overestimated the masses of primordial stars and underestimated the number of stars formed.

Likelihood-free inference with Dark Energy Survey Y3 data

Project summary:

Using optimised “active learning’’ to efficiently sample the cosmological model space, we will generate a suite of simulated catalogues of mock Dark Energy Survey (DES) data. These mock data are the primary input (along with the actual observed data) to the DES likelihood-free inference pipeline. This pipeline uses state-of-the-art statistical methods to avoid many current assumptions that are known to be likely sources of error, and which can lead to incorrect cosmological inference or the creation of artificial model tensions. This project uses a new simulation-based inference approach creating a new frontier for DES to understand dark energy, dark matter, neutrinos, and the initial conditions of the Universe.

The result of this proposed research will significantly improve the robustness of new discoveries and of cosmological model inference from the DES Year 3 Weak Lensing analysis (one of the two main DES probes). The resulting methodology, code and simulation products will then be further incorporated in the final (Year 6) DES analysis in the following year (2022+). The DiRAC hardware provides the only machine available in the UK to carry out this full analysis.

Project Science Highlight:  We cannot currently share the cosmology “science” results (including inference for dark energy models) due to DES policy regarding unblinding at this stage. We can, however, share a visualisation of a typical simulation that has been run on a GPU as part of the suite generated in the project.

Image description: This is a visualisation of the output of the simulation – a dark matter map. This is a “view from Earth” looking out into the simulation over a certain radial distance. The structures you can see in the image are bright spots representing high density of matter and darker spots showing lower densities of matter (cosmic voids). Overall you can see the cosmic web.

The structures in this image are expected to change depending on the cosmological model. For example, if we have a different value for the Hubble parameter or a different equation-of-state for Dark Energy, these structures would appear different. The structures would also appear to be at different distances from us. 

By using the DiRAC GPUs, we are able to run multiple realisations of different Universes that can be compared to the observed dark matter map (from the Dark Energy Survey). In particular, as we can run fast independent simulations in parallel (with one simulation per GPU to avoid message passing overheads) we are able to sample many cosmological parameters. We can then use deep learning techniques and methods of “likelihood-free inference” to estimate our uncertainty in those cosmological parameters given the observed DES data – this has never been done  yet with a full galaxy survey.

Solar Energetic Particle propagation in the 3D heliosphere

diffusion-based model (Laitinen et al 2016) of SEP propagation from the Sun (yellow dot in the middle) as guided by Parker spiral (black curve).

During solar eruptions, charged particles are accelerated to relativistic energies. These solar energetic particles (SEPs) propagate through the interplanetary space to Earth and cause a serious Space Weather hazard to humans and technology in space and for high latitude air travellers. To understand and mitigate these risks, it is crucial to be able to understand the processes behind both the acceleration of these particles, and how they propagate from the Sun to our Near-Earth environment.

Understanding the propagation of SEPs in the interplanetary space is particularly challenging. The medium is turbulent plasma, and the charged particles thus experience a stochastic force field due to fluctuating magnetic fields. The medium is also very inhomogeneous: the magnetic field and turbulence typically vary as powers of radial distance and as a result the transport parameters can vary orders of magnitude during the SEP propagation. A particular problem is caused by the spiral shape of the mean field, the Parker spiral, which is due to the rotating Sun. The large-scale curvature and gradients cause the SEPs to drift, and the coupling between the drifts and the stochastic motion are currently only approximated using diffusive transport models. These are not satisfactory, as we observe the particles typically at times when the diffusion approximation is not yet applicable. We use 3D particle orbit simulations to get a full picture of this link.

depiction of locations where SEPs cross a 1-au sphere in modelled turbulence, with 3D particle simulations. The white square depicts the source region at the Sun (preliminary data).

In this project, we are investigating how the large scale drifts affect the stochastic motion of the particles affect one another. In one branch of the project, we investigate how large-scale drifts influence the stochastic motion of the particles, to ascertain if the asymmetry caused by the drifts.  

The second part of the project uses a novel model to address a major complication which has prevented full investigation of SEP propagation in turbulence that is set about the Parker spiral. A preliminary result in the top right figure shows latitudinal asymmetry, as well as local features.

Due to the complications during the Covid pandemic, the running of simulations and publication of the first results were delayed, with necessity to perform the planned development and analysis of the second part in a shorter timeframe than desired. Both datasets are now available, and will result in several publications in near future.

Tracing the small-scale structure of the cosmic web over 12 billion years

The intergalactic medium (IGM) is the rarefied material that spans the vast distances between galaxies in the Universe.   In our DiRAC thematic project, we model the ionisation and thermal structure of the IGM using a combination of hydrodynamical structure formation simulations and GPU accelerated radiative transfer.  The results form part of the “Sherwood” and “Sherwood-Relics” simulation projects.

The image shows the temperature of the IGM predicted by one of the Sherwood-Relics models, performed with the DiRAC Memory Intensive and Data Intensive Cambridge facilities (credit: E. Puchwein).  It shows a region of the Universe 24 million light years across, at a time when the Universe was only 6 per cent of its current age.  This corresponds to period known as the epoch of reionisation, when the first stars and galaxies were forming and emitting the ultraviolet (UV) radiation that ionised and heated the IGM.  The colour scale shows IGM temperatures ranging from cold (dark blue at a few hundred Kelvin), to very hot (red at >100,000 Kelvin).  The sharp boundaries between the dark blue and yellow regions mark the transition from gas that is ionised and heated by the UV radiation emitted by the first stars, to cold, neutral gas that has yet to be illuminated.   

Selected science highlights for 2021include new measurements of the IGM temperature at redshifts z~2-4 from the Ly-a forest.  The results are consistent with the epoch of helium reionisation occurring rapidly at z~3 (Gaikwad et al. 2021, MNRAS, 506, 4389).   We have also completed the first simulations of the 21-cm forest in late reionisation models, showcasing the potential of SKA1-low observations to place informative lower limits on the soft X-ray background and neutral hydrogen spin temperature at z~6 with 21-cm absorbers in the diffuse IGM (Šoltinský et al. 2021, MNRAS, 506, 5818).   Finally, we have recently presented a new measurement of the mean free path for the ionising photons through the IGM at z~6 (Becker et al. 2021, MNRAS, 508, 1853).   This has yielded a surprisingly low value of ~3.6 cMpc/h, which is a factor of ~2 smaller than expected even for a very late end to reionisation at z~5.3.  This suggests that the ionising photon production from galaxies at z~6 may have to be dramatically boosted to ensure reionisation ends by z~5.3.

Transport properties of finite-density SU(2) lattice gauge theory

Quantum Chromodynamics (QCD) is a fundamental theory of nuclear interactions, describing nucleons as composite states of quarks and gluons. Heavy-ion collision experiments create a very hot and dense phase of nuclear matter, in which nucleons melt into a quark-gluon plasma. With more than 100 nucleons participating in each collision, the large numbers of produced quarks and gluons allow us to use hydrodynamics to describe the dynamics of QGP. Since nucleons and quarks are electrically charged, a hot QCD medium has nonzero electric conductivity and is also subject to strong magnetic field generated during the collision, which calls for a magneto-hydrodynamic description. Magneto-hydrodynamic equations require electric conductivity as a first-principle input. In particular, it is of direct importance for the lifetime of magnetic field.

We carried out the first numerical study of the density dependence of electric conductivity. While this study is hardly possible in real QCD because of the fermionic sign problem which makes the path integral weight non-positive and does not allow for Monte-Carlo simulations, we circumvented the sign problem by replacing the SU(3) gauge group of QCD with the SU(2) gauge group. The resulting SU(2) gauge theory is similar to QCD for small densities. Using SU(2) gauge theory, we found that the density dependence of electric conductivity is mostly close to that of free quarks, and becomes somewhat more pronounced in the vicinity of the chiral crossover which separates the hadronic and the quark-gluon plasma regimes (see Figure). Furthermore, quantum anomalies of gauge theory (violations of classical chiral symmetry at the quantum level) generate new terms in magneto-hydrodynamic equations, known as anomalous transport phenomena. We used SU(2) gauge theory with finite fermion density to study one of these phenomena, the generation of fermionic chirality flow along the magnetic field in dense plasma, dubbed the Chiral Separation Effect. Justifying some of the popular phenomenological approaches, we found that the strength of the Chiral Separation Effect is well described by the free quark gas approximation everywhere in the phase diagram except for the low-temperature, low-density regime.

Spontaneous Symmetry Breaking in the 3d Thirring Model

PI: Simon Hands, University of Liverpool

The Thirring Model describes relativistic fermions moving in a two-dimensional plane and interacting via a contact term between covariantly conserved currents. The physical system it most resembles is that of low-energy electronic excitations in graphene. For free electrons at half-filling on a honeycomb lattice, conduction and valance bands form cones just touching at their vertices at two “Dirac points” lying within the first Brillouin zone. Since the density of states vanishes, and the effective fine structure constant is boosted by a factor vF/c≈1/300, where the pitch of the cones vF is the Fermi velocity, the resulting physics is described by a strongly-interacting relativistic quantum field theory, with equal likelihood of exciting electrons or holes.

Besides possible applications in layered condensed matter systems, the Thirring model is of interest in its own right, as possibly the simplest relativistic theory of fermions requiring a computational solution. A major question is whether or not for sufficient interaction strength a bilinear condensate <ψψ> forms, restructuring the ground state and causing a gap to form between the conduction and valence bands, resulting in a phase transition from semimetal to insulator – this process is precisely analogous to chiral symmetry breaking in QCD. It is anticipated that this can occur if the number of fermion species N lies below some critical Nc: accurate determination of Nrequires control over non-perturbative quantum field theory.

In lattice field theory, the problem requires a precise rendering of the correct U(2N) fermion global symmetries. We use Domain Wall Fermion formulation, in which U(2N) GInsparg-Wilson symmetries are recovered in the limit that the wall separation Ls→∞. Simulations have been performed using the RHMC algorithm on 163 systems with Ls ranging from 8 to 80. At strong couplings g2 and light fermion masses m recovery of GW symmetry is slow – nonetheless our data for <ψψ> as a function of g2, m are well-fitted by an equation of state which assumes a continuous symmetry-breaking transition with non-mean field theory critical exponents at a critical βc≡agc-2≈0.28 (see figure), characteristic of an interacting conformal field theory.

A major outcome of the project is the confirmation that the critical number of flavors for symmetry breaking in the Thirring model Nc >1. This is a significant step towards understanding strongly-interacting fermions from first principles. We are now calculating propagators to understand the quasiparticle and bound-state excitation spectra.