# Sample PhD Projects

## General Interests

Quantum chromodynamics, dimensional regularization, conformal field theory in d-dimensions, curved spacetime, supersymmetry; quantum mechanics, quantum computing, statistical mechanics, proton structure and Mellin moments, high energy physics, lattice gauge theory

### Dr Pavel Buividovich

### Anomalous transport in quark-gluon plasma

Due to the presence of nearly massless u- and d-quarks, quark-gluon plasma is expected to exhibit nontrivial transport phenomena related to quantum anomalies (violations of certain classical symmetries in quantum theory, such as the Adler-Bell-Jackiw axial anomaly). Our aim is to measure the strength of anomalous transport responses in strongly interacting quark gluon plasma. One of the nontrivial parts of the project is the lattice implementation of Noether currents such as the axial current and energy-momentum tensor, for which the corresponding symmetries of continuum QCD are broken by lattice regularization.

### Fractional quantum Hall effect in graphene from Monte-Carlo simulations

So far Monte-Carlo simulations could not be directly applied to study the microscopic mechanism of the fractional quantum Hall effect due to inevitable sign problem in finite-density fermionic systems subject to magnetic field. We will attempt to tackle this problem using the combination of LLR method and Hybrid Monte-Carlo simulation techniques, which are being actively developed at Liverpool University and by collaborators in Germany.

### Diagrammatic Monte-Carlo for non-Abelian quantum field theories

Diagrammatic Monte-Carlo is an alternative to conventional Monte-Carlo operating in field space, and is based on importance sampling in the space of Feynman diagrams (or any other diagrams, e.g. for the strong-coupling expansion). Our aim is to apply DiagMC to non-Abelian gauge theories with finite-density fermions, with the hope to milden the fermionic sign problem.

### Dr Martin Gorbahn

### Precision flavour physics

Flavour changing meson decays probe new physics at the TeV scale. An improved standard model prediction of these decays will increase the new physics sensitivity. To this end, we will use a combination of perturbative and non-perturbative methods and obtain high precision predictions for flavour physics.

### New physics contributions to precision observables

New particles predicted in extensions of the standard model modify properties of flavour, Higgs and electroweak precision physics. We will derive these modified properties for the class of perturbative unitary models. We will then discriminate concrete model scenarios of new physics by comparing our theory predictions with the upcoming experimental data.

### Professor John Gracey

### Infrared structure of Yang-Mills theory and the Gribov problem

The dynamics of gluon and quark fields at low energy is related to the fundamental problem of confinement. This project is aimed at understanding the Gribov-Zwanziger Lagrangian which is a model of confinement. It has the intriguing property that in the zero momentum limit the propagator of a bosonic field has a dipole behaviour. This is believed to be necessary for quark confinement. However the implications of this for quarks at low energy has yet to be determined. The aim of the project is explore various avenues of current thought including operator condensation.

### Multiloop renormalization in QCD

Understanding the structure of Quantum Chromodynamics at orders of perturbation theory beyond the first is required to have precise information for experiments. This project is geared to examining QCD in a variety of renormalization schemes and gauges to understand the deeper structure.

This will include the renormalization of local operators which appear in the operator product expansion and hence QCD sum rules. Understanding how these operators evolve with the renormalization scale will aid the extraction of more precise estimates for a variety of vacuum expectation values such as the gluon condensate. The computations are carried out by the use of symbolic manipulation computer programs.

### Lattice matching via loop calculations

Lattice gauge theory methods provide a numerical way of measuring the properties of the fundamental operators underlying physical particles such as the proton. These are necessarily non-perturbative in nature. However in order to aid precision the lattice results have to agree with the known high energy behaviour of the same quantity. These are computed in perturbation theory at high loop order. The aim is to provide this perturbative information to as high a loop order as possible in schemes which are appropriate for the lattice. On practical grounds it will require the extension of existing computer algebra routines and the development of numerical evaluation code.

### Professor Ian Jack

### Supersymmetric Standard Model

Even if the Higgs boson is discovered at the LHC, there are reasons to think that the Standard Model of particle physics is still incomplete since the masses of the particles are much smaller than might be expected theoretically. This is the so-called “Hierarchy Problem”. Currently the most plausible solution is that the full theory has a property called supersymmetry. One consequence is that every known particle should have a partner particle called a superpartner. These particles will be sought at the LHC and future particle accelerators. We are engaged in the effort to provide highly precise predictions for the properties of these new particles (such as their masses) so that they can be clearly identified if they are eventually seen and so that different variants of the supersymmetric theory can be distinguished.

### The Lee-Wick model

There have been alternative suggestions to the solution of the hierarchy problem which avoid supersymmetry; one of them is the Lee-Wick model. This has some oddities from the theoretical point of view and it is still not completely clear whether it is a fully consistent theory. A possible project would be to investigate in detail the properties of this model to confirm its viability and to obtain testable predictions.

### Dr Paul Rakow

Lattice Gauge Theory is a technique to study strongly interacting systems, in particular Quantum Chromodynamics (QCD) by large scale computer simulations.

Researchers in the Liverpool lattice group are also members of the UKQCD collaboration and have played a leading role in exploiting the resources of UKQCD to maintain and increase the collaboration's high international research profile.

Currently Dr Rakow is using lattice QCD to investigate the following topics:

### Flavour symmetry breaking

Understanding how the difference between the masses of the strange, up and down quarks effects the masses and internal structure of baryons and mesons.

### The magnetic moment of the muon

Calculating the contribution of QCD to the anomalous magnetic moment of the muon, a quantity which is known experimentally to an astonishing accuracy.

### Hadronic decays

As well as calculating hadron masses, we are starting to use lattice QCD to calculate the decay rates of resonances (excited hadron states) such as the Delta and rho.

### Stochastic Perturbation Theory

Using lattice gauge theory techniques to generate extremely long perturbative series (20 terms or more), and then compare perturbation theory with the full non-perturbative calculation. The goal is to understand the asymptotic behaviour of perturbation theory, and to better describe non-perturbative effects.

### Dr David Schaich

Lattice field theory provides non-perturbative methods to investigate strongly interacting quantum field theories (QFTs) through high-performance computing. PhD projects in the following areas will develop novel applications of this approach, and can involve collaboration with researchers in the US, UK, Europe and India.

### Lattice supersymmetry

Supersymmetry plays prominent roles in modern theoretical physics, as a tool to better understand quantum field theory, as an ingredient in many proposed extensions of the standard model of particle physics, and as a means to study quantum gravity via holographic duality. Recent advances in lattice formulations of supersymmetric QFTs enable several new projects suitable for PhD students.

### Electroweak symmetry breaking

The Higgs boson may be a composite particle arising from new strong dynamics beyond the standard model, which would explain how it is protected against large quantum corrections. This possibility will be probed by experiments at the Large Hadron Collider and future particle colliders. Theoretical investigations of such new strong dynamics rely on lattice field theory for quantitative predictions from first principles, and involve more general analyses of QFTs with strongly coupled near-conformal behaviour qualitatively different from quantum chromodynamics.

### Approaches to finite-density sign problems

A non-zero net density of fermions often causes a 'sign problem' in lattice field theory calculations, with traditional importance sampling algorithms encountering negative or complex numbers where they expect probabilities. This project will investigate methods to solve or at least ameliorate sign problems in simple QFTs, such as a density-of-states approach or a canonical reformulation combined with a cluster algorithm.

### Quantum simulation

A more radical approach to solving sign problems in lattice calculations is to develop the emerging technology of quantum simulation and quantum computation. This project will contribute to this new and rapidly evolving field by reformulating simple systems in terms of entangled qubits and designing algorithms to simulate them using existing and near-future quantum devices.

### Professor Thomas Teubner

### Hadronic effects in g-2

The anomalous magnetic moment of the muon, g-2, is at odds with the Standard Model. This hints at New Physics but the case is not yet conclusive. To settle this, new experiments are planned at Fermilab and in Japan, which will require a better theoretical determination of g-2. The project will be concerned with improving the hadronic contributions to g-2 by both perturbative and non-perturbative methods, including the use of experimental data for hadronic cross sections and computations within lattice gauge field theory.

### High energy scattering at the LHC

To understand the data collected at the LHC, particle production in the high energy regime needs to be understood with better accuracy. For this, dedicated calculations are needed in higher order of perturbative QCD, which will allow us to predict so-called exclusive and also multi-jet production in the high-energy regime.

### Heavy quark and supersymmetric particle production at the next electron-positron collider

The next particle accelerator after the LHC will most probably be an electron-positron linear collider, which will be ideally suited to measure the production of light Higgs bosons and supersymmetric particles, if they exist. For this, detailed calculations are required to predict production cross sections and subsequent decays, especially close to production thresholds. This project will build on similar work done for top quark pair production, where large corrections have been calculated in higher orders in the framework of Effective Field Theories.