Cluster adjacency properties of scattering amplitudes

We conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalise the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture.Comment: 6 pages, 3 figures. Symbols of integrals presented in four ancillary files, as detailed in README.tx

Cluster adjacency beyond MHV

We explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the $\bar{Q}$-equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.Comment: 51 pages, 25 figures, 4 table

Form factors in superconformal theories in four and three dimensions

PhDThis thesis focuses on form factors in superconformal theories, in particular maximally supersymmetric Yang-Mills (MSYM) and ABJM. Scattering amplitudes in these theories have a wealth of special properties and significant amount of insight has been developed for these along with the modern techniques to calculate them. In this thesis, it is presented that form factors have very similar properties to scattering amplitudes and the techniques for scattering amplitudes can be successfully applied to form factors. After a review of the methods employed, the results for tree-level and multi-loop form factors of protected operators are derived. In four dimensions, it is shown that the tree-level form factors can be computed using MHV diagrams BCFWrelations by augmenting the set of vertices with elementary form factors. Tree and loop-level MHV and non-MHV form factors of protected operators in the stress-tensor multiplet of MSYM are computed as examples. A solution to the BCFW recursion relations for form factors is derived in terms of a diagrammatic representation. Supersymmetric multiplets of form factors of protected operators are constructed. In three dimensions, Sudakov form factor of a protected biscalar operator is computed in ABJM theory. This form factor captures the IR divergences of the scattering amplitudes. It is found that this form factor can be written in terms of a single, non-planar Feynman integral which is maximally transcendental. Additionally, the sub-leading colour corrections to the one-loop four-particle amplitude in ABJM is derived using unitarity cuts. Finally a basis of two-loo pure master integrals for the Sudakov form factor topology is constructed from a principle that relies on certain unitarity cuts

Modular Graph Functions

In earlier work we studied features of non-holomorphic modular functions associated with Feynman graphs for a conformal scalar field theory on a two-dimensional torus with zero external momenta at all vertices. Such functions, which we will refer to as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of single-valued elliptic multiple polylogarithms, which are defined as elliptic functions associated with Feynman graphs with vanishing external momenta at all but two vertices. These functions depend on a coordinate, $\zeta$, on the elliptic curve and reduce to modular graph functions when $\zeta$ is set equal to $1$. We demonstrate that these single-valued elliptic multiple polylogarithms are linear combinations of multiple polylogarithms, and that modular graph functions are sums of single-valued elliptic multiple polylogarithms evaluated at the identity of the elliptic curve, in both cases with rational coefficients. This insight suggests the many interrelations between modular graph functions (a few of which were established in earlier papers) may be obtained as a consequence of identities involving multiple polylogarithms, and explains an earlier observation that the coefficients of the Laurent polynomial at the cusp are given by rational numbers times single-valued multiple zeta values.Comment: 42 pages, significant clarifications added in section 5, minor typos corrected, and references added in version

Δεύτερο ιξώδες σε ολογραφικά μοντέλα με ασυμπτωτικά εκθετικά δυναμικά

Αριθμητικά και ολογραφικά μοντέλα για ισχυρά συζευγμένα συστήματα υποδεικνύουν μία αύξηση του δεύτερου ιξώδους στην περιοχή της αλλαγής φάσης από παγιδευμένες καταστάσεις χρώματος σε ελεύθερες του πλάσματος κουάρκ γλοιονίων. Μελετάμε την συμπεριφορά του δεύτερου ιξώδους σε ολογραφικά μοντέλα τα οποία είναι Einstein-dilaton βαρυτικές θεωρίες σε 4+1 διαστάσεις με δυναμικά τα οποία είναι ασυμπτωτικά εκθετικά στο IR. Βρίσκουμε ότι σε αυτά τα μοντέλα η αλλαγή φάσης είναι συνεχής, ο λόγος του δεύτερου ιξώδους προς την πυκνότητα εντροπίας έχει ένα μέγιστο λίγο πάνω από την κρίσιμη θερμοκρασία και παραμένει πεπεραμένος.Numerical and holographic models for strongly coupled systems indicated a rise in the vicinity of the confinement – deconfinement phase transition of the quark gluon plasma. We investigate the behaviour of bulk viscosity in holographic models that are 4+1 dimensional Einstein – dilaton gravity theories with potentials that are asymptotically exponential in the IR. We find that in such models where the phase transition is continuous, the bulk viscosity / entropy density ratio has a maximum just above a critical temperature and remains finite

Bootstrapping a stress-tensor form factor through eight loops

We bootstrap the three-point form factor of the chiral stress-tensor multiplet in planar N = 4 supersymmetric Yang-Mills theory at six, seven, and eight loops, using boundary data from the form factor operator product expansion. This may represent the highest perturbative order to which multi-variate quantities in a unitary four-dimensional quantum field theory have been computed. In computing this form factor, we observe and employ new restrictions on pairs and triples of adjacent letters in the symbol. We provide details about the function space required to describe the form factor through eight loops. Plotting the results on various lines provides striking numerical evidence for a finite radius of convergence of perturbation theory. By the principle of maximal transcendentality, our results are expected to give the highest weight part of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD through eight loops. These results were also recently used to discover a new antipodal duality between this form factor and a six-point amplitude in the same theory.</p

Folding Amplitudes into Form Factors:An Antipodal Duality

We observe that the three-gluon form factor of the chiral part of the stress-tensor multiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory is dual to the six-gluon MHV amplitude on its parity-preserving surface. Up to a simple variable substitution, the map between these two quantities is given by the antipode operation defined on polylogarithms (as part of their Hopf algebra structure), which acts at symbol level by reversing the order of letters in each term. We provide evidence for this duality through seven loops.Comment: 5+2 pages, 2 figures, 2 tables. v2: added further checks at kinematic points, some clarifications and references; version to appear in PR

Folding Amplitudes into Form Factors: An Antipodal Duality

We observe that the three-gluon form factor of the chiral part of the stress-tensor multiplet in planar $\mathcal{N}=4$ super-Yang-Mills theory is dual to the six-gluon MHV amplitude on its parity-preserving surface. Up to a simple variable substitution, the map between these two quantities is given by the antipode operation defined on polylogarithms (as part of their Hopf algebra structure), which acts at symbol level by reversing the order of letters in each term. We provide evidence for this duality through seven loops