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Jaroslav Trnka

Publications and source records attributed to Jaroslav Trnka.

At least 37 records · Page 2Linked to original sources
arXiv2023

Coulomb Branch Amplitudes from a Deformed Amplituhedron Geometry

The Amplituhedron provides, via geometric means, the all-loop integrand of scattering amplitudes in maximally supersymmetric Yang-Mills theory. Unfortunately, dimensional regularization, used conventionally for integration, breaks the beautiful geometric picture. This motivates us to propose a 'deformed' Amplituhedron. Focusing on the four-particle amplitude, we introduce two deformation parameters, which can be interpreted as particle masses. We provide evidence that the mass pattern corresponds to a specific choice of vacuum expectation values on the Coulomb branch. The deformed amplitude is infrared finite, making the answer well-defined in four dimensions. Leveraging four-dimensional integration techniques based on differential equations, we compute the amplitude up to two loops. In the limit where the deformation parameters are taken to zero, we recover the known Bern-Dixon-Smirnov amplitude. In the limit where only one deformation parameter is taken to zero, we find a connection to the angle-dependent cusp anomalous dimension.

hep-th↗
arXiv2023

Gravity Amplitudes From Double Bonus Relations

In this letter we derive new expressions for tree-level graviton amplitudes in $\mathcal{N}=8$ supergravity from BCFW recursion relations combined with new types of bonus relations. These bonus relations go beyond the famous $1/z^2$ behavior under a large BCFW shift, and use knowledge about certain zeroes of graviton amplitudes in collinear kinematics. This extra knowledge can be used in the context of global residue theorems by writing the amplitude in a special form using canonical building blocks. In the NMHV case these building blocks are dressed one-loop leading singularities, the same objects that appear in the expansion of Yang-Mills amplitudes, where each term corresponds to an $R$-invariant. Unlike other approaches, our formula is not an expansion in terms of cyclic objects and does not manifest color-kinematics duality, but rather preserves the permutational symmetry of its building blocks. We also comment on the possible connection to Grassmannian geometry and give some non-trivial evidence of such structure for graviton amplitudes.

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arXiv2023

Scalar BCJ Bootstrap

In this letter, we study tree-level scattering amplitudes of scalar particles in the context of effective field theories. We use tools similar to the soft bootstrap to build an ansatz for cyclically ordered amplitudes and impose the Bern-Carrasco-Johansson (BCJ) relations as a constraint. We obtain a set of BCJ-satisfying amplitudes as solutions to our procedure, which can be thought of as special higher-derivative corrections to SU(N) non-linear sigma model amplitudes satisfying BCJ relations to arbitrary multiplicity at leading order. The surprising outcome of our analysis is that BCJ conditions on higher-point amplitudes impose constraints on lower-point amplitudes, and they relate coefficients at different orders in the derivative expansion. This shows that BCJ conditions are much more restrictive than soft limit behavior, allowing only for a very small set of solutions.

hep-th↗
arXiv2023

The SAGEX Review on Scattering Amplitudes, Chapter 7: Positive Geometry of Scattering Amplitudes

Scattering amplitudes are both a wonderful playground to discover novel ideas in Quantum Field Theory and simultaneously of immense phenomenological importance to make precision predictions for e.g.~particle collider observables and more recently also for gravitational wave signals. In this review chapter, we give an overview of some of the exciting recent progress on reformulating QFT in terms of mathematical, geometric quantities, such as polytopes, associahedra, Grassmanians, and the amplituhedron. In this novel approach, standard notions of locality and unitarity are derived concepts rather than fundamental ingredients in the construction which might give us a handle on a number of open questions in QFT that have evaded an answer for decades. We first give a basic summary of positive geometry, before discussing the associahedron -- one of the simplest physically relevant geometric examples -- and its relation to tree-level scattering amplitudes in bi-adjoint $ϕ^3$ theory. Our second example is the amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory.

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arXiv2023

The SAGEX Review on Scattering Amplitudes

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments.

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arXiv2022

Poles At Infinity in On-shell Diagrams

In this paper we study on-shell diagrams in ${\cal N}{<}4$ supersymmetric Yang-Mills (SYM) theory. These are on-shell gauge invariant objects which appear as cuts of loop integrands in the context of generalized unitarity and serve as building blocks for amplitudes in recursion relations. In the dual formulation, they are associated with cells of the positive Grassmannian $G_+(k,n)$ and the on-shell functions can be reproduced as canonical differential forms. While for the case of the ${\cal N}{=}4$ maximally supersymmetric Yang-Mills theory all poles in on-shell diagrams correspond to IR poles when the momentum flows in edges are zero, for ${\cal N}{<}4$ SYM theories there are new UV poles when the loop momenta go to infinity. These poles originate from the prefactor of the canonical dlog form and do not correspond to erasing edges in on-shell diagrams. We show that they can be interpreted as a diagrammatic operation which involves pinching a loop and performing a ``non-planar twist'' on external legs, which gives rise to a non-planar on-shell diagram. Our result provides an important clue on the role of poles at infinite momenta in on-shell scattering amplitudes, and the relation to non-planar on-shell functions.

hep-th↗
arXiv2022

Snowmass TF04 Report: Scattering Amplitudes and their Applications

The field of scattering amplitudes plays a central role in elementary-particle physics. This includes various problems of broader interest for collider physics, gravitational physics, and fundamental principles underlying quantum field theory. We describe various applications and theoretical advances pointing towards novel descriptions of quantum field theories. Comments on future prospects are included.

hep-th↗
arXiv2022

Non-planar BCFW Grassmannian Geometries

In this paper, we study non-adjacent BCFW recursion relations and their connection to positive geometry. For an adjacent BCFW shift, the $n$-point N$^k$MHV tree-level amplitude in ${\cal N}=4$ SYM theory is expressed as a sum over planar on-shell diagrams, corresponding to canonical dlog forms on the cells in the positive Grassmannian $G_+(k,n)$. Non-adjacent BCFW shifts naturally lead to an expansion of the amplitude in terms of a different set of objects, which do not manifest the cyclic ordering and the hidden Yangian symmetry of the amplitude. We show that these terms can be interpreted as dlog forms on the non-planar Grassmannian geometries, generalizing the cells of the positive Grassmannian $G_+(k,n)$ to a larger class of objects which live in $G(k,n)$. We focus mainly on the case of NMHV amplitudes and discuss in detail the Grassmannian geometries. We also propose an alternative way to calculate the associated on-shell functions and dlog forms using an intriguing connection between Grassmannian configurations and the geometry in the kinematical space.

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arXiv2022

Solving Scattering in N=4 Super-Yang-Mills Theory

As part of the Snowmass community planning exercise, we highlight an ongoing program of research into the structure of scattering amplitudes in N=4 super-Yang-Mills theory, particularly in the planar limit of a large number of colors. This theory sits at the nexus of a number of exciting topics in high-energy particle physics, including the AdS/CFT correspondence, conformal field theory, integrability, and string theory, and is believed to be exactly solvable in four dimensions. In many ways, planar N=4 super-Yang-Mills theory is the "hydrogen atom" of relativistic scattering: It has proven indispensable for learning about new geometrical formulations of quantum field theory, for exploring mathematical properties at high perturbative orders, and for developing powerful new computational methods that have found applicability in precision collider physics.

hep-th↗
arXiv2022

Recursion Relations for One-Loop Goldstone Boson Amplitudes

In this letter, we construct the recursion relations for one-loop planar integrands in the SU(N) non-linear sigma model. This generalizes the soft recursions for tree-level amplitudes in a variety of quantum field theories with special soft limits. The main ingredient is the definition of the one-loop planar integrand, which is fixed by cuts in the sense of generalized unitarity and by requiring the Adler zero on all external legs. We show that this does not uniquely fix the integrand, so additional constraints on the soft behavior of the loop momentum have to be imposed. Our work is the first step in extending modern amplitudes methods for loop amplitudes to effective field theories with special soft limits.

hep-th↗
arXiv2021

Nonperturbative Negative Geometries: Amplitudes at Strong Coupling and the Amplituhedron

The amplituhedron determines scattering amplitudes in planar ${\cal N}=4$ super Yang-Mills by a single "positive geometry" in the space of kinematic and loop variables. We study a closely related definition of the amplituhedron for the simplest case of four-particle scattering, given as a sum over complementary "negative geometries", which provides a natural geometric understanding of the exponentiation of infrared (IR) divergences, as well as a new geometric definition of an IR finite observable ${\cal F}(g,z)$ - dually interpreted as the expectation value of the null polygonal Wilson loop with a single Lagrangian insertion - which is directly determined by these negative geometries. This provides a long-sought direct link between canonical forms for positive (negative) geometries, and a completely IR finite post-loop-integration observable depending on a single kinematical variable $z$, from which the cusp anomalous dimension $Γ_{\rm cusp}(g)$ can also be straightforwardly obtained. We study an especially simple class of negative geometries at all loop orders, associated with a "tree" structure in the negativity conditions, for which the contributions to ${\cal F}(g,z)$ and $Γ_{\rm cusp}$ can easily be determined by an interesting non-linear differential equation immediately following from the combinatorics of negative geometries. This lets us compute these "tree" contributions to ${\cal F}(g,z)$ and $Γ_{\rm cusp}$ for all values of the 't Hooft coupling. The result for $Γ_{\rm cusp}$ remarkably shares all main qualitative characteristics of the known exact results obtained using integrability.

hep-th↗
arXiv2021

Integrands of Less-Supersymmetric Yang-Mills at One Loop

We construct a prescriptive, bubble power-counting basis of one-loop integrands suitable for representing amplitude integrands in less-supersymmetric Yang-Mills theory. With the exception of massless bubbles, all integrands have unambiguous, leading singularities as coefficients defined in field theory; for the massless bubbles on external legs, we find two natural choices which lead to different integrands that highlight distinct aspects of field theory. For concreteness, we give the all-multiplicity integrands for MHV amplitudes, and the split-helicity amplitude integrand for six-particle NMHV. The basis we construct is mostly pure and is divided into to separately UV- and IR-finite sectors of fixed transcendental weight, resulting in UV- and IR-finite ratio functions of n-particle helicity amplitudes.

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arXiv2021

Multi-spin soft bootstrap and scalar-vector Galileon

We use the amplitude soft bootstrap method to explore the space of effective field theories (EFT) of massless vectors and scalars. It is known that demanding vanishing soft limits fixes uniquely a special class of EFTs: non-linear sigma model, scalar Galileon and Born-Infeld theories. Based on the amplitudes analysis, we conjecture no-go theorems for higher-derivative vector theories and theories with coupled vectors and scalars. We then allow for more general soft theorems where the non-trivial part of the soft limit of the (n+1)-pt amplitude is equal to a linear combination of n-pt amplitudes. We derive the form of these soft theorems for general power-counting and spins of particles and use it as an input into the soft bootstrap method in the case of Galileon power-counting and coupled scalar-vector theories. We show that this unifies the description of existing Galileon theories and leads us to the discovery of a new exceptional theory: Special scalar-vector Galileon.

hep-th↗
arXiv2021

Learning from the Pandemic: the Future of Meetings in HEP and Beyond

The COVID-19 pandemic has by-and-large prevented in-person meetings since March 2020. While the increasing deployment of effective vaccines around the world is a very positive development, the timeline and pathway to "normality" is uncertain and the "new normal" we will settle into is anyone's guess. Particle physics, like many other scientific fields, has more than a year of experience in holding virtual meetings, workshops, and conferences. A great deal of experimentation and innovation to explore how to execute these meetings effectively has occurred. Therefore, it is an appropriate time to take stock of what we as a community learned from running virtual meetings and discuss possible strategies for the future. Continuing to develop effective strategies for meetings with a virtual component is likely to be important for reducing the carbon footprint of our research activities, while also enabling greater diversity and inclusion for participation. This report summarizes a virtual two-day workshop on Virtual Meetings held May 5-6, 2021 which brought together experts from both inside and outside of high-energy physics to share their experiences and practices with organizing and executing virtual workshops, and to develop possible strategies for future meetings as we begin to emerge from the COVID-19 pandemic. This report outlines some of the practices and tools that have worked well which we hope will serve as a valuable resource for future virtual meeting organizers in all scientific fields.

physics.soc-ph↗
arXiv2021

Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops

We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all soft-collinear singularities in both theories. As a consequence, this integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as ancillary files. This work has been updated to take into account the results of [arXiv:1911.09106], which leads to a simpler and more uniform representation of these amplitudes.

hep-th↗
arXiv2021

Gravity loop integrands from the ultraviolet

We demonstrate that loop integrands of (super-)gravity scattering amplitudes possess surprising properties in the ultraviolet (UV) region. In particular, we study the scaling of multi-particle unitarity cuts for asymptotically large momenta and expose an improved UV behavior of four-dimensional cuts through seven loops as compared to standard expectations. For N=8 supergravity, we show that the improved large momentum scaling combined with the behavior of the integrand under BCFW deformations of external kinematics uniquely fixes the loop integrands in a number of non-trivial cases. In the integrand construction, all scaling conditions are homogeneous. Therefore, the only required information about the amplitude is its vanishing at particular points in momentum space. This homogeneous construction gives indirect evidence for a new geometric picture for graviton amplitudes similar to the one found for planar N=4 super Yang-Mills theory. We also show how the behavior at infinity is related to the scaling of tree-level amplitudes under certain multi-line chiral shifts which can be used to construct new recursion relations.

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arXiv2020

Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes

In this paper, we present new expressions for n-point NMHV tree-level gravity amplitudes. We introduce a method of factorization diagrams which is a simple graphical representation of R-invariants in Yang-Mills theory. We define the gravity analogues which we call G-invariants, and expand the NMHV gravity amplitudes in terms of these objects. We provide explicit formulas of NMHV gravity amplitudes up to eight points in terms of G-invariants, and give the general definition for any number of points. We discuss the connection to BCFW representation, special behavior under large momentum shift, the role of momentum twistors and the intricate web of spurious poles cancelation. Because of the close connection between R-invariants and the (tree-level) Amplituhedron for Yang-Mills amplitudes, we speculate that the new expansion for gravity amplitudes should correspond to the triangulation of the putative Gravituhedron geometry.

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