arXiv subjects

Jaroslav Trnka

Publications and source records attributed to Jaroslav Trnka.

At least 19 recordsLinked to original sources

arXiv2026

Multi-Loop Negative Geometries

Scattering amplitudes in planar ${\cal N}=4$ supersymmetric Yang-Mills theory are dual to expectation values of null polygonal Wilson loops. The Amplituhedron provides a geometric construction for the all-loop integrand as the canonical form on the geometric region in the Grassmannian defined by a certain set of inequalities. For a closely related object, the logarithm of the scattering amplitude, the integrand is reproduced in a similar way using negative geometries. When integrated over all loop momenta except one, the result is infrared (IR) finite and equal to the expectation value of a certain Wilson loop with a Lagrangian insertion. At four points, this quantity, ${\cal F}(g,z)$ only depends on a single cross ratio $z$ and the 't Hooft coupling $g$. At weak coupling, it is known up to three loops from perturbative Wilson loop computations and at strong coupling through the AdS/CFT correspondence at leading order. In this paper, we explore this object further through the lens of the Amplituhedron and negative geometries, which provide very natural IR finite building blocks. We perform an explicit three-loop computation of all negative geometries and show that the number of internal cycles in the diagram is closely linked to the depth of polylogarithms. We calculate the cusp anomalous dimension $\Gamma_{\rm cusp}$ by integrating ${\cal F}(g,z)$ over $z$. We show that the higher-cycle diagrams are suppressed if we consider separate odd and even zeta contributions. Furthermore, we focus on certain convergent infinite series of one-cycle diagrams, perform all-loop order resummations of such contributions, and discuss various features of the result.

hep-th↗

arXiv2026

A Surface Integrand for the Inverse KLT Kernel

We propose a loop-level generalization of the inverse string theory Kawai-Lewellen-Tye (KLT) kernel: the planar inverse KLT integrand. The integrand is defined constructively via a novel Berends-Giele-like recursion that exposes the inverse KLT kernel as the simplest toy model of a ``stringy amplitude''. We show that, to all loop orders, the inverse KLT integrand is structurally equivalent to integrands in the cubic scalar tr$ϕ^3$ theory. This simplicity is obscured in the conventional Feynman diagram approach, where the inverse KLT integrand receives contributions from an infinity of infinite towers of contact interactions. The inverse KLT integrand is a rational function of stringified kinematic variables and is naturally defined on the kinematic surface proposed by Arkani-Hamed et al.. It provides an elementary analogue of the surfacehedron integrand for the tr$ϕ^3$ theory involving only scalar resonances and unifies the scattering of cubic scalars and pions in the non-linear sigma model (NLSM) to all loop orders via kinematic $α'$-shifts.

hep-th↗

arXiv2025

Grassmannian Geometries for Non-Planar On-Shell Diagrams

On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the ${\cal N}=4$ Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $G(k,n)$. Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $N$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $G_+(k,n)$ which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $N$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $G(2,n)$. These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are \emph{internally planar} there always exists a strongly connected geometry, and for those that are \emph{irreducible}, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

hep-th↗

arXiv2025

Tree-Level Gravity Amplitudes at Infinity

In this note we study on-shell tree-level gravity amplitudes in the infinite momentum limit. In the case of the two-line BCFW shift, we have a famous improved behavior at infinity that allows for the amplitude to be reconstructed from the pole factorization. For other shifts, the poles at infinity are present and need to be considered, however general principles do not fix the residues of the amplitude on these poles. The web of all possible shifts is large, we focus primarily on a case of $(n{-}2)$-line anti-holomorphic shift, which also appears in the context of unitarity cuts of gravity loop integrands. We will find that for one class of shifts the gravity amplitudes at infinity exhibit a peculiar factorization property, quite different from the usual factorization on poles, while for other shifts, they evaluate to the same amplitude on shifted kinematics. We also discuss generalizations of our results to other anti-holomorphic shifts.

hep-th↗

arXiv2025

TrES-1 b: A Case Study in Detecting Secular Evolution of Exoplanet Orbits

We present a comprehensive analysis of transit, eclipse, and radial velocity data of the hot Jupiter TrES-1 b and confirm evidence of orbital variations on secular timescales. Apparent variations due to systemic motion and light travel time effects have been ruled out, indicating that the observed changes are dynamical in origin. Joint modeling of the TrES-1 b data favors an apsidal precession model, but the rapid precession rate of $4^\circ$ yr$^{-1}$ cannot be explained without invoking an undetected close-in planetary companion, which remains unseen in the data. While radial velocity measurements reveal a previously undetected companion candidate on a wide, eccentric orbit, it is unlikely to drive the observed evolution of TrES-1 b. However, an orbital decay model provides a plausible alternative if the loss of orbital energy is driven by planetary obliquity tides. We find that the best-fit orbital decay rate of $-7.1^{ +1.5}_{-1.6}$ ms yr$^{-1}$ is aligned with theoretical predictions for modified tidal quality factors of hot Jupiters if TrES-1 b has a planetary obliquity $\varepsilon_p > 30^\circ$. We encourage follow-up observations of this system, particularly of eclipse timing and radial velocities, to further constrain the nature of the observed evolution. This paper provides a practical framework for studying secular variations and aims to accelerate future research on similar systems.

astro-ph.EP↗

arXiv2025

Geometric Landau Analysis and Symbol Bootstrap

We investigate how the positive geometry framework for loop integrands in $\mathcal{N}{=}4$ super Yang-Mills theory constrains the structure of the integrated answers. This is done in the context of a geometric expansion of Wilson loops with a Lagrangian insertion, called negative geometries, extending ideas previously used for scattering amplitudes related to the Amplituhedron. The procedure we adopt combines the knowledge of all maximal codimension boundaries of the geometry, which characterize all possible leading singularities of the integral, with a geometrically informed Landau analysis. The interplay between geometry and Landau analysis arises from associating Landau diagrams to geometric boundaries. The boundary structure of the geometry then determines which solutions to the Landau equations are spurious and which ones are physical, that is, which singularities are actually present in the integral. This method allows us to efficiently determine the symbol alphabet of the associated integral, and serves as a starting point for the symbol bootstrap. We successfully implement this procedure and compute the six-point two-loop and five-point three-loop ladder negative geometries at the symbol level. We also present the conjectural alphabet for ladder negative geometries at two loops for all multiplicities. These are finite integrals that serve as building blocks for the Wilson loop with Lagrangian insertion, and therefore provide insights into the function space of the latter.

hep-th↗

arXiv2025

Universality of Colored Scalars from the Stringy KLT Kernel

A new perspective on the inverse string theory Kawai-Lewellen-Tye (KLT) kernel is provided which establishes the universality of scattering amplitudes in the bi-adjoint scalar (BAS) theory, pions in the Non-linear sigma model (NLSM), and mixed amplitudes (NLSM+$ϕ^3$) recently studied in the literature. We show that all these amplitudes can be viewed as equivalent, arising from a single function, the inverse string theory KLT kernel, evaluated at different kinematic points. In this way cubic colored scalars and pions become interchangeable through a procedure we call the $α'$-shift. The latter complements the $δ$-shift proposed by Arkani-Hamed et al., and demonstrates an inherent equivalence of scattering amplitudes in different quantum field theories by embedding them in a common stringy framework.

hep-th↗

arXiv2025

All-loop Leading Singularities of Wilson Loops

We study correlators of null, $n$-sided polygonal Wilson loops with a Lagrangian insertion in the planar limit of the ${\cal N}=4$ supersymmetric Yang-Mills theory. This finite observable is closely related to loop integrands of maximally-helicity-violating amplitudes in the same theory, and, conjecturally, to all-plus helicity amplitudes in pure Yang-Mills theory. The resulting function has been observed to have an expansion in terms of functions of uniform transcendental weight, multiplied by certain rational prefactors, called leading singularities. In this work we prove several conjectures about the leading singularities: we classify and compute them at any loop order and for any number of edges of the Wilson loop, and show that they have a hidden conformal symmetry. This is achieved by leveraging the geometric definition of the loop integrand via the Amplituhedron. The leading singularities can be seen as maximal codimension residues of the integrand, and the boundary structure of the Amplituhedron geometry restricts which iterative residues are accessible. Combining this idea with a further geometric decomposition of the Amplituhedron in terms of so-called negative geometries allows us to identify the complete set of leading singularities.

hep-th↗

arXiv2025

Two-loop four-point amplitudes on the Coulomb branch of ${\mathcal{N}}=4$ super Yang-Mills

We explore scattering amplitudes on the Coulomb branch of maximally supersymmetric Yang-Mills theory. We introduce a particular pattern of scalar vacuum expectation values that allow us to define amplitudes with a different mass pattern compared to what was studied previously. This is motivated by an extension of the Amplituhedron that leads to infrared-finite four-particle amplitudes involving massive particles. We work out the Feynman rules on the Coulomb branch and use them, together with generalized unitarity techniques, to perform consistency checks on the Amplituhedron expectations for the one- and two-loop integrands for the four-particle amplitude. We present details of the computation of the required two-loop four-point integrals via a four-dimensional version of the differential equations method. Finally, we study the Regge limit of the four-point amplitude, including the first power suppressed terms. We find that when organized in terms of a suitable expansion parameter, the subleading power term exponentiates, with the exponent matching the anomalous dimension of a cusped Wilson line with a local operator insertion. The latter is known from integrability, which leads to a prediction at higher loop orders in the Regge limit.

hep-th↗

arXiv2024

Uniqueness of MHV Gravity Amplitudes

We investigate MHV tree-level gravity amplitudes as defined on the spinor-helicity variety. Unlike their gluon counterparts, the gravity amplitudes do not have logarithmic singularities and do not admit Amplituhedron-like construction. Importantly, they are not determined just by their singularities, but rather their numerators have interesting zeroes. We make a conjecture about the uniqueness of the numerator and explore this feature from a more mathematical perspective. This leads us to a new approach for examining adjoints. We outline steps of our proposed proof and provide computational evidence for its validity in specific cases.

hep-th↗

arXiv2024

FaceDig: Automated tool for placing landmarks on facial portraits for geometric morphometrics users

Landmark digitization is essential in geometric morphometrics, enabling the quantification of biological shapes, such as facial structures, for in-depth morphological analysis. Traditional landmarking, which identifies specific anatomical points, can be complemented by semilandmarks when precise locations are challenging to define. However, manual placement of numerous landmarks is time-consuming and prone to human error, leading to inconsistencies across studies. To address this, we introduce FaceDig, an AI-powered tool designed to automate landmark placement with human-level precision, focusing on anatomically sound facial points. FaceDig is open-source and integrates seamlessly with analytical platforms like R and Python. It was trained using one of the largest and most ethnically diverse face datasets, applying a landmark configuration optimized for 2D enface photographs. Our results demonstrate that FaceDig provides reliable landmark coordinates, comparable to those placed manually by experts. The tool's output is compatible with the widely-used TpsDig2 software, facilitating adoption and ensuring consistency across studies. Users are advised to work with standardized facial images and visually inspect the results for potential corrections. Despite the growing preference for 3D morphometrics, 2D facial photographs remain valuable due to their cultural and practical significance. Future enhancements to FaceDig will include support for profile views, further expanding its utility. By offering a standardized approach to landmark placement, FaceDig promotes reproducibility in facial morphology research and provides a robust alternative to existing 2D tools.

cs.CV↗

arXiv2024

Positivity properties of five-point two-loop Wilson loops with Lagrangian insertion

In this paper we discuss the geometric integrand expansion of the five-point Wilson loop with one Lagrangian insertion in maximally supersymmetric Yang-Mills theory. We construct the integrand corresponding to an all-loop class of ladder-type geometries. We then investigate the known two-loop observable from this geometric viewpoint. To do so, we evaluate analytically the new two-loop integrals corresponding to the negative geometry contribution, using the canonical differential equations method. Inspecting the analytic result, we present numerical evidence that in this decomposition, each piece has uniform sign properties, when evaluated in the Amplituhedron region. Finally, we present an alternative bootstrap approach for the ladder-type geometries. We find that certain minimal bootstrap assumptions can be satisfied at two loops, but lead to a contradiction at three loops. This suggests to us that novel alphabet letters are required at this loop order. Indeed studying planar three-loop Feynman integrals, we do identify novel pentagon alphabet letters.

hep-th↗

arXiv2024

New Soft Theorems for Two-Scalar Sigma Models

In this paper, we study the scattering amplitudes and soft theorems for the sigma models with two scalars. We show that if the particles are Goldstone bosons, then you necessarily get Adler zero with no possibility for non-trivial soft theorems. For non-Goldstone bosons, the soft behavior is generically captured by the geometric soft theorem studied by Cheung et al., and the right-hand side contains derivatives of lower-point amplitudes. Inspired by the recent work on the 2D sigma models, we study one special two-scalar sigma model, where the presence of symmetries in the target space translates into a special but non-trivial soft theorem without derivatives. We further generalize the construction to two larger classes of such models and derive certain soft theorem sum rules, again avoiding the derivatives of amplitudes. Our analysis provides an interesting hierarchy of two-scalar sigma models and soft theorems, ranging from Goldstone boson case to a generic target space, and showing that there are interesting theories in between.

hep-th↗

arXiv2024

Hidden Amplitude Zeros From Double Copy

Recently, Arkani-Hamed et al. proposed the existence of zeros in scattering amplitudes in certain quantum field theories including the cubic adjoint scalar theory Tr($ϕ^3$), the $SU(N)$ non-linear sigma model (NLSM) and Yang-Mills (YM) theory. These hidden zeros are special kinematic points where the amplitude vanishes and factorizes into a product of lower-point amplitudes, similar to factorization near poles. In this letter, we show a close connection between the existence of such zeros and color-kinematics duality. In fact, all zeros can be derived from the Bern-Carrasco-Johansson (BCJ) relations. We also show that these zeros extend via the Kawai-Lewellen-Tye (KLT) relations to special Galileon amplitudes and their corrections, evincing that these hidden zeros are also present in permutation-invariant amplitudes.

hep-th↗

arXiv2024

An All-loop Soft Theorem for Pions

In this letter, we discuss a generalization of the Adler zero to loop integrands in the planar limit of the $SU(N)$ non-linear sigma model (NLSM). While possible to maintain at one-loop, the Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a non-zero soft theorem satisfied by loop integrands with any number of loops and legs. This requires a generalization of NLSM integrands to an off-shell framework with certain deformed kinematics. Defining an `algebraic soft limit', we identify a particularly simple non-vanishing soft behavior of integrands, which we call the `algebraic soft theorem'. We find that the proposed soft theorem is satisfied by the `surface' integrand of Arkani-Hamed, Cao, Dong, Figueiredo and He, which is obtained from the shifted ${\rm Tr}ϕ^3$ surfacehedron integrand. Finally, we derive an on-shell version of the algebraic soft theorem that takes an interesting form in terms of propagator renormalization factors and lower-loop integrands in a mixed theory of pions and scalars.

hep-th↗

arXiv2023

Goldstone bosons on celestial sphere and conformal soft theorems

In this paper, we study celestial amplitudes of Goldstone bosons and conformal soft theorems. Motivated by the success of soft bootstrap in momentum space and the important role of the soft limit behavior of tree-level amplitudes, our goal is to extend some of the methods to the celestial sphere. The crucial ingredient of the calculation is the Mellin transformation, which transforms four-dimensional scattering amplitudes to correlation functions of primary operators in the celestial CFT. The soft behavior of the amplitude is then translated to the singularities of the correlator. Only for amplitudes in "UV completed theories" (with sufficiently good high energy behavior) the Mellin integration can be properly performed. In all other cases, the celestial amplitude is only defined in a distributional sense with delta functions. We provide many examples of celestial amplitudes in UV-completed models, including linear sigma models and Z-theory, which is a certain completion of the SU(N) non-linear sigma model. We also comment on the BCFW-like and soft recursion relations for celestial amplitudes and the extension of soft bootstrap ideas.

hep-th↗

arXiv2023

Loops of Loops Expansion in the Amplituhedron

We study a novel geometric expansion for scattering amplitudes in the planar sector of N=4 super Yang-Mills theory, in the context of the Amplituhedron which reproduces the all-loop integrand as a canonical differential form on the positive geometry. In a paper by Arkani-Hamed, Henn and one of the authors, it was shown that this result can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momenta, represented by lines in momentum twistor space. One can then calculate an all-loop order result in the approximation where only tree graphs in the space of all loops are considered. Furthermore, using differential equation methods, it is possible to calculate and resum integrated expressions and obtain strong coupling results. In this paper, we provide a more general framework for the loops of loops expansion and outline a powerful method for the determination of differential forms for higher-order geometries. We solve the problem completely for graphs with one internal cycle, but the method can be used more generally for other geometries.

hep-th↗

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↗