Possible PhD Theses Projects

List of Thesis Projects February 2023

Scattering amplitudes

Amplitudes via homotopy transfer [Hohm]. There is a close relation between QFT techniques, say based on Feynman diagrams, and results in pure mathematics obtained in the realm of topology and homotopy algebras. Specifically, passing over from a classical field theory (as described by a Lagrangian) to the scattering amplitudes corresponds to a homotopy transfer from a homotopy algebra (like L-infinity or BV-algebras) to algebraic objects encoding the amplitudes. In the first year, these observations are confirmed and generalized by formulating them in a rigorous fashion at tree-level, with applications to gauge theories like Yang-Mills and gravity theories. In the second year, this will be generalized to loop order, using BV or Loop-L-infinity algebras. Finally, in the third year, this should be generalized to non-perturbative problems using the cohomology of BV algebras.

Amplitudes and effective field theories [Grojean, Plefka]. On-shell methods will be applied to the SM considered as an effective theory supplemented by higher dimensional operators. In the first year recursive techniques will be developed to implement efficient computations of tree-amplitudes involving dimension five and dimension six operators. Together with generalized unitarity these will be used to generate one-loop results (year 2). In the final year the structure of the renormalisation group and non-renormalisation theorems among certain classes of operators will be interpreted in the on-shell formalism.



Exploring the landscape of Effective Field Theories [Grojean]
Effective field theories are a convenient way to systematically explore quantum field theory in the presence of a large separation of physical scales, be there the weak scale and the scale of new physics. EFTs can be matched to any specific UV physics scenario but they can also be used to parametrise unknown physics in terms of few coefficients capturing local interactions of know particles. Locality and unitarity actually offer powerful constraints on the values of these coefficients. Consistency with the laws of gravity at the quantum level also impose some non-trivial constraints on these IR coefficients to separate a landscape of consistent EFTs from the swampland of EFTs that cannot be completed into a theory of quantum gravity. The aim of this project will be to investigate the geometrical structure of these constraints in terms of an amplituhedron and to study to which extent they can be probed at future colliders, like CEPC or FCC. In particular, the interactions violating CP and/or CP symmetry will be scrutinised. Special attention will also be devoted to scenarios with light and very weakly coupled degrees of freedom such as axion-like particles that can source local operators in the Standard Model Effective Field Theory.

Lattice Field Theory

QED corrections to hadronic observable [Patella] The only know way to extract quantitative properties of hadrons from QCD is by means of lattice simulations. When a subpercent precision is needed, one must also take into account QED effect on hadron physics. This poses substantial challenges from the theoretical and numerical point of view. The successful candidate will join the effort of the RC* collaboration, with a special focus on exploratory calculations of radiative corrections to hadronic decay rates. Numerical simulations and analysis will be accompanied by a theoretical study with effective-field theory techniques.

Renormalization of composite operators via gradient-flow [Patella]. Composite opera- tors (e.g. energy-momentum tensor, electroweak effective Hamiltonian) can be defined on the lattice by means of the gradient flow. The small-flow time expansion allows to recover the properly-defined local operators. The goal of this project is to explore the numerical feasibility of this program. In year 1, the doctoral researcher will investigate possible non-perturbative definitions of the Wilson coefficients of the small flowtime expansions of fermionic bilinears, and develop the code to calculate these Wilson coefficients. In year 2, after generating small-volume gauge configurations, and the Wilson coefficients will be calculated on these configurations. In year 3, the developed techniques are going to be tested against conventional methods for scalar, vector and axial vector bilinears, concerning agreement as well as statistical precision.

Baryon-baryon interactions and SU(3) symmetry [Green, Patella] Interactions between baryons containing strange quarks are difficult to measure experimentally. Thus, although an ab initio study of the physical deuteron may still be very challenging, there is an opportunity for lattice calculations to have an impact on our understanding of hypernuclear physics. The goals of this project will be to understand the interactions between two octet baryons, based on calculations with both exact and broken SU(3) symmetry, and to understand how the SU(3)-breaking effects can be described using effective field theories or models.

Energy-momentum tensor n-point functions in 2d sigma models [Patella, Forini]

Two-dimensional nonlinear sigma models on supersymmetric target spaces play a role in a variety of models in statistical mechanics and, notably, in string theory and the AdS/CFT correspondence. For their nonperturbative definition one possible starting point is the study we propose here, which considers the renormalization of the energy-momentum tensor in two-dimensional O(N) sigma model in perturbation theory and non-perturbatively by means of lattice simulations.  A detailed comparison of different methods for non- perturbative renormalization (shifted boundary conditions, Ward identities with probes at positive flowtime, small flowtime expansion) will be carried out, together with the investigation of the trace anomaly and calculation of the central charge in the conformal case (N=2). We will generalization to supergroups and compare with integrability predictions.

AdS/CFT Correspondence

Witten diagrammatics for effective string worldsheets [Forini]

The computation of Witten diagrams at tree and loop level has received in the last few years a very good deal of attention. The complicated interactions and regularization subtleties makes it, however, far from trivial to translate this progress into tools for the effective worldsheet string theories in AdS2 which holographically describe superconformal (Wilson) line defects in higher-dimensional CFTs. We will investigate the higher-order analysis in sigma-model perturbation theory of correlators and their regularization, developing unitarity techniques for this case as well as building a consistent basis of integrals at higher perturbative orders. We will also consider the Euclidean AdS version of these diagrams  as a simple laboratory for realizing explicitly the mapping to de Sitter space, the simplest cosmological spacetime, and investigate implications of this analysis in the area of the so-called cosmological bootstrap.

String compactifications in the AdS/CFT correspondence. [Malek] The AdS/CFT correspondence relates string theory in asymptotically anti-de Sitter (AdS) backgrounds times a compact space to strongly-coupled conformal field theories (CFTs) living on the boundary of AdS. The string compactification plays an important role in the correspondence: the spectrum of Kaluza-Klein modes encodes the spectrum of single-trace primary operators of the CFT, while the interactions of Kaluza-Klein modes encode n-point functions of these dual CFT operators. However, computing the masses and interactions of the Kaluza-Klein modes used to be technically impossible beyond the simplest examples. Building on our recent work of computing the Kaluza-Klein spectrum for backgrounds related to consistent truncations to maximal gauged supergravity, this project will push this technology further to study AdS compactifications outside the maximal gauged supergravity setting, such as Sasaki-Einstein manifolds, and extract general features of the Kaluza-Klein modes. The first half of year 1 will consist of gaining the relevant expertise in Exceptional Field Theory/Exceptional Generalised Geometry, which is a powerful formalism for describing string flux compactifications and their consistent truncations, and understanding the Kaluza-Klein spectroscopy techniques developed so far. The second half will study important string compactifications which do not admit a consistent truncation to maximal gauged supergravity, such as Sasaki-Einstein spaces, AdS backgrounds arising from wrapping D-branes on Riemann surfaces and AdS vacua in more than 5 dimensions. In years 2, the project will develop a formalism for computing the Kaluza-Klein masses around these more interesting string backgrounds and matching these with the superconformal index of the dual CFTs. Year 3 will push the technology further, for example to capture 3-point functions of the CFT.

Holography = Homotopy [Hohm] The “holographic” AdS/CFT correspondence is closely related to the notion of homotopy retract from topology. Here a bulk gravity theory is encoded in a L-infinity algebra and then mapped via homotopy to an L-infinity algebra on the boundary of AdS. This algebra encodes the correlation functions of the putative boundary CFT. In the first year, these general observations will be confirmed and applied to specific theories like maximal 5D supergravity and suitable subsectors of N=4 super Yang-Mills theory. In the second year, these techniques will be generalized to the complete Kaluza-Klein towers of type IIB supergravity, using techniques from exceptional field theory. If possible, in the third year an attempt will be made to formulate precise criteria under which the boundary L-infinity algebra can be viewed as coming from a local boundary theory, as would be needed in order to prove AdS/CFT.

Conformality and integrability of Feynman, on-shell, and twistor graphs [Staudacher]. An open question for integrable, conformal four-dimensional quantum field theories, such as N = 4 SYM and its deformations, is how exactly integrability interplays with weak-coupling, perturbative expansions. The latter can take many forms from the conventional Feynman graphs to on-shell diagrams and the graphs of twistor theory approaches. The aim of this thesis is to explain how integrable diagrams for various quantities of interest, such as scattering amplitudes, correlation functions and form factors, sum up to non-perturbative expressions. The first part of year 1 will be devoted to learning AdS/CFT integrability, the relevant parts of conformal quantum field theory techniques, and on-shell as well as twistor approaches. The rest of year one will be spent on doing easy calculations “to learn the trade”. In year 2 the bulk of new, cutting-edge results is to be obtained. We will start from clearly defined calculations that definitely can be done, gradually moving to more risky but also more rewarding explorations. The last year will be a mix of writing up the results, as well as exploring some further “high risk” directions.

Exact solution of integrable fishnet models [Staudacher]. Integrable models of “fishnet-type” have recently come into the focus of the studies on integrable four-dimensional quantum field theories. They are obtained from γi-deformed N=4 Super Yang-Mills in the limit of infinitely large, imaginary twist an- gles. In these strongly twisted quantum field theories only a very limited number of Feynman diagrams survives, while integrability is preserved. Even though this promises to lead to a complete understand- ing of these models, a number of intriguing mathematical and physical subtleties arises due to their non-unitarity. The goal of this thesis is to gain a complete understanding of these subtleties and to ex- tract lessons for the significantly more involved unitary cases. The first half of the first year will be spent on learning the relevant parts of the quantum inverse scattering method, AdS/CFT integrability, as well as the double-scaled, strongly twisted “fishnet” models. For the rest of this year some exploratory cal- culations, partly reproducing recent results from the literature, partly new, will be performed. Year 2 will then aim at obtaining entirely new results at the forefront of the subject. The third year will be spent on writing up the thesis, and on exploring promising but more risky directions.

Gravitational Waves

Color-kinematic duality for gravitational waves [Plefka]. Building upon our recent work the double copy of a classical color charged particles in Yang-Mills to a massive gravitationally interacting particle in worldline quantum field theory, this project will construct the classcical observables of dilaton-gravity using the double copy method further. In order to validate the method it will be pushed to next-to-next-to-leading order (3PM) (year 1). For this a comparison to the scalar- tensor theory results in the literature is available. If the color-kinematic construction works out correctly spin degrees of freedom will be taken into account starting in the later part of the first year. In the second half of the PhD project techniques known from scattering amplitudes to decouple the dilaton degrees of freedom will be adopted to perform this decoupling also in the case of the worldline quantum field theory computation.

Mathematical Physics

Double Copy and Homotopy Algebras [Hohm]
Recent work indicates that homotopy algebras are the right framework to provide a first-principe off-shell derivation of the “double copy” that relates gravity scattering amplitudes to gauge theory amplitudes. So far this has been achieved (partially) for N=0 supergravity. In the first year, these result will be generalized to N=4 super Yang-Mills theory whose double copy should be a (double field theory version of) N=8 supergravity, at first to cubic and quartic order. In the second year, this should be generalized to all orders, using operadic techniques from homotopy algebras. In the third year, the implications for the question of UV finiteness of N=8 supergravity, both in its original and double field theory form, will be explored, with possible cross-fertilizations to scattering amplitudes.