Mathematical aspects of scattering amplitudes

In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their Hopf algebra structure. We show how these mathematical concepts are useful in physics by illustrating on several examples how these algebraic structures are useful to perform analytic computations of loop integrals, in particular to derive functional equations among polylogarithms.Comment: 58 pages. Lectures presented at TASI 201

PolyLogTools - Polylogs for the masses

We review recent developments in the study of multiple polylogarithms, including the Hopf algebra of the multiple polylogarithms and the symbol map, as well as the construction of single valued multiple polylogarithms and discuss an algorithm for finding fibration bases. We document how these algorithms are implemented in the Mathematica package PolyLogTools and show how it can be used to study the coproduct structure of polylogarithmic expressions and how to compute iterated parametric integrals over polylogarithmic expressions that show up in Feynman integal computations at low loop orders.Comment: Package URL: https://gitlab.com/pltteam/pl

Higgs production in bottom-quark fusion to third order in the strong coupling

We present the inclusive cross section at next-to-next-to-next-to-leading order (N$^3$LO) in perturbative QCD for the production of a Higgs boson via bottom-quark fusion. We employ the five-flavour scheme, treating the bottom quark as a massless parton while retaining a non-vanishing Yukawa coupling to the Higgs boson. We find that the dependence of the hadronic cross section on the renormalisation and factorisation scales is substantially reduced. For judicious choices of the scales the perturbative expansion of the cross section shows a convergent behaviour. We present results for the N$^3$LO cross section at various collider energies. In comparison to the cross section obtained from the Santander-matching of the four and five-flavour schemes we predict a slightly higher cross section, though the two predictions are consistent within theoretical uncertainties.Comment: 3 pretty plots with pretty colours. Published versio

Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral

We introduce a class of iterated integrals that generalize multiple polylogarithms to elliptic curves. These elliptic multiple polylogarithms are closely related to similar functions defined in pure math- ematics and string theory. We then focus on the equal-mass and non-equal-mass sunrise integrals, and we develop a formalism that enables us to compute these Feynman integrals in terms of our iterated integrals on elliptic curves. The key idea is to use integration-by-parts identities to identify a set of integral kernels, whose precise form is determined by the branch points of the integral in question. These kernels allow us to express all iterated integrals on an elliptic curve in terms of them. The flexibility of our approach leads us to expect that it will be applicable to a large variety of integrals in high-energy physics.Comment: 22 page

The analytic structure and the transcendental weight of the BFKL ladder at NLL accuracy

We study some analytic properties of the BFKL ladder at next-to-leading logarithmic accuracy (NLLA). We use a procedure by Chirilli and Kovchegov to construct the NLO eigenfunctions, and we show that the BFKL ladder can be evaluated order by order in the coupling in terms of certain generalised single-valued multiple polylogarithms recently introduced by Schnetz. We develop techniques to evaluate the BFKL ladder at any loop order, and we present explicit results up to five loops. Using the freedom in defining the matter content of the NLO BFKL eigenvalue, we obtain conditions for the BFKL ladder in momentum space at NLLA to have maximal transcendental weight. We observe that, unlike in moment space, the result in momentum space in N = 4 SYM is not identical to the maximal weight part of QCD, and moreover that there is no gauge theory with this property. We classify the theories for which the BFKL ladder at NLLA has maximal weight in terms of their field content, and we find that these theories are highly constrained: there are precisely four classes of theories with this property involving only fundamental and adjoint matter, all of which have a vanishing one-loop beta function and a matter content that fits into supersymmetric multiplets. Our findings indicate that theories which have maximal weight are highly constrained and point to the possibility that there is a connection between maximal transcendental weight and superconformal symmetry.Comment: 45 pages, 1 figure, 1 table. v2: published versio

Modulation of growth and differentiation of mesenchymal cells for cartilage and bone tissue engineering

Tissue engineering is a highly promising technology for the treatment of challenging cartilage and bone lesions for which no adequate therapeutic options are available yet. However for their widespread use, engineered tissues will first have to prove a predictable clinical success. To reach this objective, a reproducibly high product quality will be required, which can be achieved by a better knowledge and a continuous control of the cell phenotype during all phases of the tissue engineering process. The aim of this thesis is therefore to demonstrate how two types of mesenchymal cells, chondrocytes and bone marrow-derived mesenchymal/stromal cells (BMSC), can be modulated during growth and differentiation in order to conserve and fully exploit their potential. The present work is divided into 4 chapters. Chapter I will reveal how different chondrocyte subpopulations change their phenotype during in vitro proliferation and how this can lead to the detection of cells with increased differentiation capacity. In chapter II, the parameters governing the maintenance of osteogenic potential of BMSC during expansion will be analyzed. Chapter III will demonstrate how the coculture of BMSC with macrophages can result in a better cartilage-forming capacity. Finally in chapter IV, the effect of ascorbic acid on chondrogenic differentiation will be established. Chapter I: The De-differentiation of Human Chondrocytes is Linked to Individual Cell Divisions The relationship between proliferation and de-differentiation of chondrocytes during in vitro culture remains poorly understood. It was hypothesized here that cell proliferation tracking could reveal differences in the progression of de-differentiation and chondrogenic potential among subpopulations proliferating at different rates. Results showed that changes in the expression of cell surface markers and extracellular matrix genes were linked to individual cell divisions. Different culture conditions influenced cell doubling rates but not the relationship between cell divisions and phenotypic alterations, which indicated a strong coupling between both phenomena. Interestingly the highest chondrogenic potential was measured for slowly growing chondrocytes, even after a same number of total doublings was reached for all subpopulations. The increased understanding of the link between proliferation, de-differentiation and re-differentiation capacity will lead to innovative ways to maintain chondrogenic differentiation potential during chondrocyte expansion. It will also facilitate the identification of progenitor populations with intrinsically superior capacity for the generation of enhanced engineered cartilage grafts. Chapter II: Perfused 3D Scaffolds and Hydroxyapatite Substrate Maintain the Osteogenic Potential of Human Bone Marrow-Derived Mesenchymal Stromal Cells during Expansion In previous studies it was repeatedly shown that the expansion of bone marrow-derived mesenchymal/stromal cells (BMSC) on 3D ceramic scaffolds resulted in increased maintenance of osteogenic potential as compared to culture on 2D polystyrene (PS). Since several culture parameters completely differ between 3D ceramic and 2D PS culture, the individual influences of the 3D scaffold and the ceramic material, as well as of the extracellular matrix deposition were investigated here. Results revealed that BMSC expanded on 2D PS only yielded bone matrix if the culture time was not longer than 2 weeks. Cells cultured for 3 weeks on both 3D PS and 3D ceramic scaffolds produced a dense bone matrix. The number of explants containing bone was higher with cells expanded on 3D ceramic compared to 3D PS. However there were no significant differences between cells extracted from 3D ceramic and directly implanted constructs. These findings suggest that the bone-forming capacity of BMSC can be maintained by a 3D environment and further improved by a ceramic substrate material, but that a preexisting 3D niche is not required for bone formation. The preservation of BMSC with osteogenic potential during 3D expansion in bioreactors opens the perspective for a streamlined production of large-scale bone grafts for clinical use. Chapter III: Anti-Inflammatory/Tissue Repair Macrophages Enhance the Cartilage-Forming Capacity of Human Bone Marrow-Derived Mesenchymal Stromal Cells Macrophages play a key role in healing processes, by regulating inflammation and stimulating tissue repair. However their influence on the tissue formation potential of BMSC is unknown. The effect of the coculture of macrophages with either pro-inflammatory or tissue-remodeling traits on the chondrogenic differentiation capacity of BMSC was therefore tested here. Results showed that the coculture of BMSC with tissue-repair but not with pro-inflammatory macrophages resulted in significantly higher glycosaminoglycan content and type II collagen expression, while type X collagen expression was unaffected. This chondro-inductive effect was found to be caused by an increased survival and higher clonogenic and chondrogenic capacity of BMSC that were cocultured with tissue-repair macrophages. No difference was detected however in the cartilage tissue maturation in nude mice, as evidenced by similar accumulation of type X collagen and calcified tissue. These results demonstrated that a coculture with tissue-repair macrophages can improve the chondrogenic differentiation capacity of BMSC. This increased knowledge can lead to new coculture strategies for the manufacturing of cartilage grafts with enhanced quality. Chapter IV: Chondrogenic Differentiation and Collagen Synthesis of Human Chondrocytes in the Absence of Ascorbic Acid Ascorbic acid is considered to be an important supplement for cartilage tissue engineering because of its role in collagen hydroxylation in vivo. Due to its instability, ascorbic acid requires specific liquid handling conditions, which poses significant challenges to the automation of cartilage graft manufacturing. The aim of this study was to investigate the effect of ascorbic acid on chondrogenesis in vitro, with special regard to collagen synthesis and hydroxylation. Results showed that cartilage gene expression, tissue formation, and production of glycosaminoglycans were indistinguishable whether chondrocyte micromass pellets were cultured with or without ascorbic acid. Not adding ascorbic acid caused a reduction of collagen deposition, but collagen hydroxylation was not significantly different. Collagen secretion was unaffected and collagens showed a similar fibril structure in the absence of ascorbic acid. In conclusion, ascorbic acid did not influence chondrogenesis except for a small effect on collagen quantity, and can thus be omitted to simplify automation for a more cost-efficient cartilage graft manufacturing. Conclusion: In this work, four different approaches to modulate the growth and differentiation of chondrocytes and BMSC were presented. With the gained knowledge the cell phenotype can be better controlled during manufacturing processes, which will be required for the production of engineered tissue grafts with reproducibly high quality for clinical translation