arXiv subjects

Jacob L. Bourjaily

Publications and source records attributed to Jacob L. Bourjaily.

69 records · Page 4Linked to original sources

arXiv2010

The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM

We give an explicit recursive formula for the all L-loop integrand for scattering amplitudes in N=4 SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This generalizes the BCFW recursion relation for tree amplitudes to all loop orders, and extends the Grassmannian duality for leading singularities to the full amplitude. It also provides a new physical picture for the meaning of loops, associated with canonical operations for removing particles in a Yangian-invariant way. Loop amplitudes arise from the "entangled" removal of pairs of particles, and are naturally presented as an integral over lines in momentum-twistor space. As expected from manifest Yangian-invariance, the integrand is given as a sum over non-local terms, rather than the familiar decomposition in terms of local scalar integrals with rational coefficients. Knowing the integrands explicitly, it is straightforward to express them in local forms if desired; this turns out to be done most naturally using a novel basis of chiral, tensor integrals written in momentum-twistor space, each of which has unit leading singularities. As simple illustrative examples, we present a number of new multi-loop results written in local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point 3-loop MHV amplitude. The structure of the loop integrand strongly suggests that the integrals yielding the physical amplitudes are "simple", and determined by IR-anomalies. We briefly comment on extending these ideas to more general planar theories.

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arXiv2010

A Note on Polytopes for Scattering Amplitudes

In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in CP^2, we interpret the 1-loop MHV integrand as the volume of a polytope in CP^3x CP^3, which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each CP^3 of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical "square" of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into CP^4. In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.

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arXiv2010

Local Integrals for Planar Scattering Amplitudes

Recently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in N=4 SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the integrand can be expressed in an amazingly simple and manifestly local form when represented in momentum-twistor space using a set of chiral integrals with unit leading singularities. As examples, we present very-concise expressions for all 2- and 3-loop MHV integrands, as well as all 2-loop NMHV integrands. We also describe a natural set of manifestly IR-finite integrals that can be used to express IR-safe objects such as the ratio function. Along the way we give a pedagogical introduction to the foundations of the subject. The new local forms of the integrand are closely connected to leading singularities---matching only a small subset of all leading singularities remarkably suffices to determine the full integrand. These results strongly suggest the existence of a theory for the integrand directly yielding these local expressions, allowing for a more direct understanding of the emergence of local spacetime physics.

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arXiv2010

Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion in Mathematica

We describe an efficient implementation of the BCFW recursion relations for tree-amplitudes in N=4 super Yang-Mills, which can generate analytic formulae for general N^kMHV colour-ordered helicity-amplitudes-which, in particular, includes all those of non-supersymmetric Yang-Mills. This note accompanies the public release of the Mathematica package "bcfw", which can quickly (and automatically) generate these amplitudes in a form that should be easy to export to any computational framework of interest, or which can be evaluated directly within Mathematica given external states specified by four-momenta, spinor-helicity variables or momentum-twistors. Moreover, bcfw is able to solve the BCFW recursion relations using any one of a three-parameter family of recursive `schemes,' leading to an extremely wide variety of distinct analytic representations of any particular amplitude. This flexibility is made possible by bcfw's use of the momentum-twistor Grassmannian integral to describe all tree amplitudes; and this flexibility is accompanied by a remarkable increase in efficiency, leading to formulae that can be evaluated much faster-often by several orders of magnitude-than those previously derived using BCFW.

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arXiv2010

The Grassmannian and the Twistor String: Connecting All Trees in N=4 SYM

We present a new, explicit formula for all tree-level amplitudes in N=4 super Yang-Mills. The formula is written as a certain contour integral of the connected prescription of Witten's twistor string, expressed in link variables. A very simple deformation of the integrand gives directly the Grassmannian integrand proposed by Arkani-Hamed et al. together with the explicit contour of integration. The integral is derived by iteratively adding particles to the Grassmannian integral, one particle at a time, and makes manifest both parity and soft limits. The formula is shown to be related to those given by Dolan and Goddard, and generalizes the results of earlier work for NMHV and N^2MHV to all N^(k-2)MHV tree amplitudes in N=4 super Yang-Mills.

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arXiv2009

Effective Field Theories for Local Models in F-Theory and M-Theory

Requiring a strictly local origin of visible sector phenomenology is perhaps the strongest, most falsifiable condition that one can impose on string theory at the high scale: it at once excludes a vast majority of the string landscape, and yet leads naturally to constructions that can be surprisingly realistic (and familiar). Yet only for local models can gravity be made parametrically weak while keeping the strength of gauge- and Yukawa-couplings fixed--a limit which is well-motivated by low-energy experiments. Conveniently, the entire class of high-scale effective field theories that can arise from such local models in F-theory and M-theory can be classified according to simple, purely group-theoretic rules. In this note, we describe these rules from the viewpoint of an effective field theorist with little interest in the underlying geometry or high-scale physics, and we discuss the general predictions these models have for low-energy phenomenology.

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arXiv2009

Local Models in F-Theory and M-Theory with Three Generations

We describe a general framework that can be used to geometrically engineer local, phenomenological models in F-theory and M-theory based on ALE-fibrations, and we present several concrete examples of such models that feature three generations of matter with semi-realistic phenomenology. We show that the geometric structures required for generating interactions--triple-intersections of matter-curves in F-theory and supersymmetric three-cycles supporting multiple conical singularities in M-theory--are generic in such ALE-fibred manifolds, and that they can be understood in correspondence with one another. The models we can construct in this way are strictly limited in complexity by the maximality of the E8-ALE space, but turn out to be just complex enough to accommodate some of the most realistic string models to date.

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arXiv2008

Geometrically Engineerable Chiral Matter in M-Theory

We present a classification of the massless chiral matter representations that can arise locally in M-theory on G(2) through geometrically engineered singularities. We will find that several of the more exceptional singularities could have applications for model building.

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arXiv2007

Geometrically Engineering the Standard Model: Locally Unfolding Three Families out of E8

This paper extends and builds upon the results of an earlier paper, in which we described how to use the tools of geometrical engineering to deform geometrically-engineered grand unified models into ones with lower symmetry. This top-down unfolding has the advantage that the relative positions of singularities giving rise to the many `low energy' matter fields are related by only a few parameters which deform the geometry of the unified model. And because the relative positions of singularities are necessary to compute the superpotential, for example, this is a framework in which the arbitrariness of geometrically engineered models can be greatly reduced. In our earlier paper, this picture was made concrete for the case of deforming the representations of an SU(5) model into their Standard Model content. In this paper we continue that discussion to show how a geometrically engineered 16 of SO(10) can be unfolded into the Standard Model, and how the three families of the Standard Model uniquely emerge from the unfolding of a single, isolated E8 singularity.

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arXiv2007

Unfolding Geometric Unification in M-Theory

By reinterpreting the familiar tools and ideas of M-theory model building, we show how a G2-manifold locally engineered to give rise to massless matter representations of an SU(5) grand unified model can be smoothly unfolded into a G2-manifold giving rise to SU(3)xSU(2) gauge theory with the corresponding matter representations. These ideas could lead to new insights in string phenomenology because much of the arbitrariness of M-theory model building can be removed by supposing, for example, that the singularities giving rise to Standard Model particles could arise from unfolding a more singular, grand unified geometry.

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arXiv2007

Multiple Unfoldings of Orbifold Singularities: Engineering Geometric Analogies to Unification

Katz and Vafa showed how charged matter can arise geometrically by the deformation of ADE-type orbifold singularities in type IIa, M-theory, and F-theory compactifications. In this paper we use those same basic ingredients, used there to geometrically engineer specific matter representations, here to deform the compactification manifold itself in a way which naturally compliments many features of unified model building. We realize this idea explicitly by deforming a manifold engineered to give rise to an $SU_5$ grand unified model into a one giving rise to the Standard Model. In this framework, the relative local positions of the singularities giving rise to Standard Model fields are specified in terms of the values of a small number of complex structure moduli which deform the original manifold, greatly reducing the arbitrariness of their relative positions.

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arXiv2005

Outside the mSUGRA Box

Most studies of the potential for discovery of superpartners at the Fermilab Tevatron or CERN LHC have focused on the so-called mSUGRA (minimal supergravity mediated supersymmetry breaking) model, not because it is well motivated but because it has a minimal number of parameters. If signals are seen that could be superpartners, most analyses will attempt to interpret them in the mSUGRA framework since the needed software and computational tools exist. With only a few signal channels, and initially large statistical and systematic errors, it is very likely that an mSUGRA interpretation will look all right even if it is not. We present an approach to studying any potential signals of new physics in ``inclusive signature space'' that sensitively tests any proposed interpretation, and apply it to the mSUGRA case. The approach also has significant experimental advantages, reducing the sensitivity to jet energy corrections, dependence on beam luminosity, and other systematics. Basically, if one (or more) instances of reported data lies outside certain bounded regions of inclusive signature space characteristic of the physics being tested, that physics is excluded for any parameters. The approach can be used to study supersymmetry breaking and to point to the form of the underlying theory even without detailed measurements of a number of important parameters which will be difficult or impossible to measure at hadron colliders.

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arXiv2005

What is the Cosmological Significance of a Discovery of Wimps at Colliders or in Direct Experiments?

Although a discovery of wimps either at colliders or in direct experiments would have enormous implications for our understanding of particle physics, it would imply less than one would like about our understanding of the dark matter in the universe or in the galactic halo: it surely is possible that the discovered particles account for only a little of the total dark matter. To establish the cosmological significance of a wimp discovery, their density must be determined. We show that data from neither hadron colliders nor direct detection experiments alone can be sufficient to determine the local or relic density of discovered wimps, even allowing all needed assumptions about cosmology and astrophysics. We provide examples of dark matter models where nearly identical detector or collider signals correspond to very different densities. We show, however, that it may be possible to determine the density of wimps by combining data from both experiments; we present a general method to do this in the case of supersymmetric dark matter, and describe how similar studies could be made for other wimp candidates.

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arXiv2004

Weighing the Dark Matter Halo

The dark matter problem will be solved only when all of the dark matter is accounted for. Although wimps may be discovered in direct detection experiments soon, we will not know what fraction of the dark matter halo they compose until we measure their local density. In this talk, I will offer a novel method to determine the mass of a wimp from direct detection experiments alone using kinematical consistency constraints. I will then describe a general method to estimate the local density of wimps using both dark matter detection and hadron collider data when it becomes available. These results were obtained in collaboration with Gordon Kane at the University of Michigan.

astro-ph↗

arXiv2004

Determining the Actual Local Density of Dark Matter Particles

Even if dark matter particles are unambiguously discovered in experiments, there is no clear reason to expect that the dark matter problem has been solved. It is very easy to provide examples of dark matter scenarios (e.g. in supersymmetric models) where nearly identical detector signals correspond to extremely different relic densities. Therefore, the density of the particles discovered must be determined before their cosmological relevance is established. In this talk, I will present a general method to estimate the local density of dark matter particles using both dark matter and hadron collider experimental data when it becomes available. These results were obtained in collaboration with Gordon Kane at the University of Michigan.

astro-ph↗