Relations for Einstein-Yang-Mills amplitudes from the CHY representation

We show that a recently discovered relation, which expresses tree-level single trace Einstein-Yang-Mills amplitudes with one graviton and $(n-1)$ gauge bosons as a linear combination of pure Yang-Mills tree amplitudes with $n$ gauge bosons, can be derived from the CHY representation. In addition we show that there is a generalisation, which expresses tree-level single trace Einstein-Yang-Mills amplitudes with $r$ gravitons and $(n-r)$ gauge bosons as a linear combination of pure Yang-Mills tree amplitudes with $n$ gauge bosons. We present a general formula for this case.Comment: 13 pages, version to be publishe

Properties of scattering forms and their relation to associahedra

We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms -- the scattering forms -- on the moduli space of a Riemann sphere with $n$ marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor $\overline{\mathcal M}_{0,n} \backslash {\mathcal M}_{0,n}$. Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.Comment: 40 pages, version to be publishe

Proof of the fundamental BCJ relations for QCD amplitudes

The fundamental BCJ-relation is a linear relation between primitive tree amplitudes with different cyclic orderings. The cyclic orderings differ by the insertion place of one gluon. The coefficients of the fundamental BCJ-relation are linear in the Lorentz invariants $2 p_i p_j$. The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in ${\mathcal N}=4$ super-Yang-Mills theory. Recently, it has been conjectured that the BCJ-relations hold also for QCD amplitudes. In this paper we give a proof of this conjecture. The proof is valid for massless and massive quarks.Comment: 24 pages, version to be publishe

Streuamplituden in Yang-Mills-Theorie und Gravitation auf Baumniveau

Neulich wurde eine neue Darstellung von S-Matrix-Elementen in Eichtheorien und Gravitation auf Baumniveau und in jeder Raumzeitdimension gefunden. Diese so genannte Cachazo-He-Yuan(CHY)-Darstellung basiert auf einem Satz von algebraischen Gleichungen, welche als Streugleichungen bekannt sind. Diese Gleichungen liefern eine Abbildung zwischen dem Modulraum der $n$-fach punktierten Riemannschen Sphäre $mathcal M_{0,n}$, wobei $n$ assoziiert ist mit der Anzahl der gestreuten Teilchen, und dem kinematischen Raum der Mandelstam-Invarianten. Wir werden diesen Formalismus verwenden, um Relationen zwischen Ein-Spur-Amplituden in Einstein-Yang-Mills-Theorie mit einem Graviton und farbgeordneten Yang-Mills-Amplituden, welche von Stieberger und Taylor gefunden wurden, zu zeigen. Weiterhin werden wir unter Benutzung einer Eigenschaft der Lösungen der Streugleichungen eine Verallgemeinerung dieser Relationen auf eine beliebige Anzahl an Gravitonen angeben. Die Bausteine im CHY-Formalismus für Yang-Mills-Amplituden, ein zyklisch invarianter Parke-Taylor-Faktor und ein Polarisationsfaktor, geben Anlass zu der Definition von Differenzialformen, welche als Streuformen bezeichnet werden, für die bi-adjungierte skalare Theorie und die Yang-Mills-Theorie auf der Kompaktifizierung von $mathcal M_{0,n}$. Es ist bekannt, dass die Streuform der bi-adjungierten skalaren Theorie assoziiert ist mit einer positiven Geometrie und dass der Pushforward dieser Form unter den Streugleichungen genau die CHY-Darstellung der farbgeordneten Amplitude in der bi-adjungierten skalaren Theorie liefert. Dies motiviert uns die Streuformen für die bi-adjungierte skalare Theorie und die Yang-Mills-Theorie genauer zu studieren. Wir werden zeigen, dass alle Singularitäten der Streuformen auf dem Rand von $mathcal M_{0,n}$ liegen, dass jede Singularität logarithmisch ist und dass das Residuum auf jeder Randkomponente in zwei Streuformen vom niedrigeren Grad faktorisiert.Recently a new formulation of S-matrix elements for gauge theories and gravity at tree-level and in any spacetime dimension has been found. This so called Cachazo-He-Yuan (CHY) formalism is based on a set of algebraic equations known as the scattering equations. These equations provide a map between the moduli space of Riemann spheres with $n$ marked points $mathcal M_{0,n}$, where $n$ is associated to the number of scattered particles and the kinematic space of Mandelstam invariants. We will utilize this formalism in order to prove relations between single-trace amplitudes in Einstein-Yang-Mills theory with one graviton and color-ordered Yang-Mills amplitudes found by Stieberger and Taylor. Furthermore, we will generalize these relations towards an arbitrary number of gravitons using a property of the solutions of the scattering equations. The building blocks in the CHY formalism for Yang-Mills amplitudes, a cyclically invariant Parke-Taylor factor and a polarization factor, give rise to the definition of differential forms called scattering forms for bi-adjoint scalar theory and Yang-Mills theory on the compactification of $mathcal M_{0,n}$. It is known that the scattering form for the bi-adjoint scalar theory is associated to a positive geometry and that the pushforward of this form under the scattering equations gives exactly the CHY representation of the double-ordered amplitude in bi-adjoint scalar theory. This motivates us to study scattering forms for bi-adjoint scalar theory and Yang-Mills theory in more detail. We will show that all singularities of the scattering forms are on the boundary of $mathcal M_{0,n}$, that each singularity is logarithmic and that the residue on each boundary component factorizes into two scattering forms of lower point

Structural and Functional analysis of the GABARAP interaction motif (GIM)

© 2017 The Authors. Published under the terms of the CC BY 4.0 license. Through the canonical LC3 interaction motif (LIR), [W/F/Y]-X 1 -X 2 -[I/L/V], protein complexes are recruited to autophagosomes to perform their functions as either autophagy adaptors or receptors. How these adaptors/receptors selectively interact with either LC3 or GABARAP families remains unclear. Herein, we determine the range of selectivity of 30 known core LIR motifs towards individual LC3s and GABARAPs. From these, we define a GABARAP Interaction Motif (GIM) sequence ([W/F] -[V/I]-X 2 -V) that the adaptor protein PLEKHM1 tightly conforms to. Using biophysical and structural approaches, we show that the PLEKHM1-LIR is indeed 11-fold more specific for GABARAP than LC3B. Selective mutation of the X 1 and X 2 positions either completely abolished the interaction with all LC3 and GABARAPs or increased PLEKHM1-GIM selectivity 20-fold towards LC3B. Finally, we show that conversion of p62/SQSTM1, FUNDC1 and FIP200 LIRs into our newly defined GIM, by introducing two valine residues, enhances their interaction with endogenous GABARAP over LC3B. The identification of a GABARAP-specific interaction motif will aid the identification and characterization of the expanding array of autophagy receptor and adaptor proteins and their in vivo functions