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.

Phenomenology

Evaluation of the three-loop form factor for massive quarks [Uwer]. Very recently first results on the three-loop form factor have been presented by R. Lee, A. Smirnov, V. Smirnov, and M. Steinhauser. So far no independent cross check of this fundamental field theoretic quantity exists. Using the tools developed in the phenomenology group the PhD project aims in providing an independent cross check and if possible extend the calculation to so far unknown contributions. In the first year the required diagrams are generated and reduction to master integrals shall be performed. This step shall be performed for vector, axial-vector, scalar and pseudo-scalar currents. In the second year known results for the vector-current shall be reproduced using existing results for the master integrals. Furthermore, applying various approximations existing results for the remaining form factors shall be calculated. In the last year it should be explored whether the approximations made before can be relaxed.

NLO Matrix Element Method for tt ̄+Higgs production [Uwer, Schulze]. This project explores the capabilities of modern analysis methods using the full event information based on first-principle QFT matrix elements, elevated to higher order. In the first year, the student extends existing technology to study the tt ̄ process including top quark decays. Depending on the numerical performance, a FKS partitioning of the phase space can be considered. In the second year, the more complicated tt ̄+Higgs process is studied, which subsequently (third year) can be extended by anomalous interactions in an EFT framework including a phenomenological analysis.

Duality in linear dilaton background and gauge-Higgs unification models [Grojean]. The linear dilaton geometry in five dimensions has recently been understood as the continuum limit of the clock- work model aiming at solving the hierarchy problem via a chain of copies of the SM gauge group linked to each others according to a specific pattern. This geometry interpolates between warped space-time and large flat extra-dimension setups. This project will consider fermions and gauge symmetries em- bedded in the bulk (year 1). A model of gauge-Higgs unification in which the Higgs boson emerges as the component of a gauge field along the extra-dimension will be studied (year 2). The radiative scalar potential will be computed and interpreted in terms of dual strongly interacting theories (year 3).

Phenomenology of relaxion models [Grojean]. In the recently proposed relaxion models, the Higgs boson is coupled to an axion-like field that explores large phase space regions during a long period of inflation. The axion unchains a dynamical screening of the Higgs mass. In principle, no new degree of freedom around the TeV scale is needed anymore to screen the Higgs mass from large quantum corrections. This has profound implications for the physics agenda of the LHC and beyond. The goal of this project is to further elaborate on this approach by exploring different sources of friction to prevent the uncontrolled acceleration of the relaxion field during its cosmological evolution (year 1). The phenomenological and cosmological signatures at colliders and beyond will be studied (year 2). And theoretical constraints, e.g. the consistency with the weak gravity conjecture, will be examined (year 3).

Phenomenology and Lattice Simulations of Composite Higgsmodels [Gröber,Grojean,Patella]. The idea of the project is to study the phenomenology of non-minimal Composite Higgs Models with low-energy effective methods and non-perturbative methods. The first task is to identify QCD- like four- dimensional models and study their phenomenology with special focus on the Higgs phenomenology and new exotic resonances using effective field theory (year 1), while learning the basics of lattice field theory. In the next step, existing simulation codes need to be adapted to BSM theories and the parameter space of lattice-discretized theory needs to be studied in order to identify the physically- relevant region (year2) . The final step of the computation of the spectrum with lattice simulations will be performed in year 3 and confronted with the EFT predictions.

Trilinear Higgsself coupling determination[Gröber,Grojean]. One of the main goals of the high- luminosity LHC is the determination of the trilinear Higgs self-coupling, needed for a reconstruction of the Higgs potential. The potential in the SM is postulated ad-hoc and the questions remains if there is more profound origin of electroweak symmetry breaking. Experimentally the trilinear Higgs self-coupling is directly accessible in di-Higgs production, with the experimental measurement being extremely difficult due to small signal cross section and large backgrounds. In order to reach the interesting range in coupling modifications new ideas both on experimental side and theoretical side are needed. The goal of this project is two-fold: i.) a formal investigation of the possible deviations in the trilinear Higgs self-coupling, taking into account theoretical arguments for a consistent QFT such as positivity, unitarity and causality (year 1), and ii.) a phenomenological global analysis (in the presence of the full set of effective operators) of the capabilities to constrain the trilinear Higgs self-coupling using different di-Higgs production modes, single Higgs production measurements and electroweak precision data to constrain the trilinear Higgs self-coupling at the high-luminosity LHC (year 2) and future colliders (year 3).

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.

Isospin breaking corrections [Patella]. The goal of this project is to calculate isospin breaking cor- rections to hadronic masses, by coupling QCD to QED with C⋆ boundary conditions (which allow to preserve locality in finite volume). In year 1, joining to the ongoing effort of the RC⋆ collaboration, the doctoral researcher will contribute the generation of QCD+QED gauge configurations, and a code to measure for hadron correlators needs to be developed with state-of-the art techniques. In year 2, hadron correlators will be measured on available gauge configurations, and used to extract hadron masses, after developing a suitable analysis strategy. Finally, in year 3 the possibility to extract hadrons states with real photons by means of a variational approach is going to be investigated.

Renormalization of composite operators via gradient-flow [Patella, Sommer]. 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.

AdS/CFT Correspondence

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.