arXiv subjects

Matthias Staudacher

Publications and source records attributed to Matthias Staudacher.

At least 19 recordsLinked to original sources

arXiv2024

Supersymmetric brick wall diagrams and the dynamical fishnet

We consider the double scaling limit of $\beta$-deformed planar N = 4 supersymmetric Yang-Mills theory (SYM), which has been argued to be conformal and integrable. It is a special point in the three-parameter space of double-scaled $\gamma_i$-deformed N = 4 SYM, preserving N = 1 supersymmetry. The Feynman diagrams of the general three-parameter models form a "dynamical fishnet" that is much harder to analyze than the original one-parameter fishnet, where major progress in uncovering the model's integrable structure has been made in recent years. Here we show that by applying N = 1 superspace techniques to the $\beta$-deformed model the dynamical nature of its Feynman graph expansion disappears, and we recover a regular lattice structure of brick wall (honeycomb) type. As a first application, we compute the zero-mode-fixed thermodynamic free energy of this model by applying Zamolodchikov's method of inversion to the supersymmetric brick wall diagrams.

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arXiv2024

Brick wall diagrams as a completely integrable system

We study the free energy of an integrable, planar, chiral and non-unitary four-dimensional Yukawa theory, the bi-fermion fishnet theory discovered by Pittelli and Preti. The typical Feynman-diagrams of this model are of regular "brick-wall"-type, replacing the regular square lattices of standard fishnet theory. We adapt A. B. Zamolodchikov's powerful classic computation of the thermodynamic free energy of fishnet graphs to the brick-wall case in a transparent fashion, and find the result in closed form. Finally, we briefly discuss two further candidate integrable models in three and six dimensions related to the brick wall model.

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arXiv2022

Spectrum of the Hypereclectic Spin Chain and Pólya Counting

In earlier work we proposed a generating function that encodes the Jordan block spectrum of the integrable Hypereclectic spin chain, related to the one-loop dilatation operator of the dynamical fishnet quantum field theory. We significantly improve the expressions for these generating functions, rendering them much more explicit and elegant. In particular, we treat the case of the full spin chain without imposing any cyclicity constraints on the states, as well as the case of cyclic states. The latter involves the Pólya enumeration theorem in conjunction with q-binomial coefficients.

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arXiv2021

Combinatorial Solution of the Eclectic Spin Chain

The one-loop dilatation operator in the holomorphic 3-scalar sector of the dynamical fishnet theory is studied. Due to the non-unitary nature of the underlying field theory this operator, dubbed the eclectic spin chain Hamiltonian, is non-diagonalisable. The corresponding spectrum of Jordan blocks leads to logarithms in the two-point functions, which is characteristic of logarithmic conformal field theories. It was previously conjectured that for certain filling conditions and generic couplings the spectrum of the eclectic model is equivalent to the spectrum of a simpler model, the hypereclectic spin chain. We provide further evidence for this conjecture, and introduce a generating function which fully characterises the Jordan block spectrum of the simplified model. This function is found by purely combinatorial means and is simply related to the q-binomial coefficient.

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arXiv2020

Minkowski Box from Yangian Bootstrap

We extend the recently developed Yangian bootstrap for Feynman integrals to Minkowski space, focusing on the case of the one-loop box integral. The space of Yangian invariants is spanned by the Bloch-Wigner function and its discontinuities. Using only input from symmetries, we constrain the functional form of the box integral in all 64 kinematic regions up to twelve (out of a priori 256) undetermined constants. These need to be fixed by other means. We do this explicitly, employing two alternative methods. This results in a novel compact formula for the box integral valid in all kinematic regions of Minkowski space.

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arXiv2020

The Integrable (Hyper)eclectic Spin Chain

We refine the recently introduced notion of eclectic spin chains by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n>2), where their "spectrum" consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not "random", but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above "generic" spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of N=4 Super Yang-Mills Theory.

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arXiv2020

The Dual Conformal Box Integral in Minkowski Space

The dual conformal box integral in Minkowski space is not fully determined by the conformal invariants $z$ and $\bar{z}$. Depending on the kinematic region its value is on a 'branch' of the Bloch-Wigner function which occurs in the Euclidean case. Dual special conformal transformations in Minkowski space can change the kinematic region in such a way that the value of the integral jumps to another branch of this function, encoding a subtle breaking of dual conformal invariance for the integral. We classify conformally equivalent configurations of four points in compactified Minkowski space. We show that starting with any configuration, one can reach up to four branches of the integral using dual special conformal transformations. We also show that most configurations with real $z$ and $\bar{z}$ can be conformally mapped to a configuration in the same kinematic region with two points at infinity, where the box integral can be calculated directly in Minkowski space using only the residue theorem.

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arXiv2019

Grassmannian Integrals in Minkowski Signature, Amplitudes, and Integrability

We attempt to systematically derive tree-level scattering amplitudes in four-dimensional, planar, maximally supersymmetric Yang-Mills theory from integrability. We first review the connections between integrable spin chains, Yangian invariance, and the construction of such invariants in terms of Grassmannian contour integrals. Building upon these results, we equip a class of Grassmannian integrals for general symmetry algebras with unitary integration contours. These contours emerge naturally by paying special attention to the proper reality conditions of the algebras. Specializing to psu(2,2|4) and thus to maximal superconformal symmetry in Minkowski space, we find in a number of examples expressions similar to, but subtly different from the perturbative physical scattering amplitudes. Our results suggest a subtle breaking of Yangian invariance for the latter, with curious implications for their construction from integrability.

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arXiv2018

The One-Loop Spectral Problem of Strongly Twisted $\mathcal{N}$=4 Super Yang-Mills Theory

We investigate the one-loop spectral problem of $γ$-twisted, planar $\mathcal{N}$=4 Super Yang-Mills theory in the double-scaling limit of infinite, imaginary twist angle and vanishing Yang-Mills coupling constant. This non-unitary model has recently been argued to be a simpler version of full-fledged planar $\mathcal{N}$=4 SYM, while preserving the latter model's conformality and integrability. We are able to derive for a number of sectors one-loop Bethe equations that allow finding anomalous dimensions for various subsets of diagonalizable operators. However, the non-unitarity of these deformed models results in a large number of non-diagonalizable operators, whose mixing is described by a very complicated structure of non-diagonalizable Jordan blocks of arbitrarily large size and with a priori unknown generalized eigenvalues. The description of these blocks by methods of integrability remains unknown.

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arXiv2017

The Tetrahedron Zamolodchikov Algebra and the AdS5 x S5 S-matrix

The S-matrix of the $AdS_5 \times S^5$ string theory is a tensor product of two centrally extended su(2|2) S-matrices, each of which is related to the R-matrix of the Hubbard model. The R-matrix of the Hubbard model was first found by Shastry, who ingeniously exploited the fact that, for zero coupling, the Hubbard model can be decomposed into two XX models. In this article, we review and clarify this construction from the AdS/CFT perspective and investigate the implications this has for the $AdS_5 \times S^5$ S-matrix.

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arXiv2017

On Form Factors and Correlation Functions in Twistor Space

In this paper, we continue our study of form factors and correlation functions of gauge-invariant local composite operators in the twistor-space formulation of N=4 super Yang-Mills theory. Using the vertices for these operators obtained in our recent papers arXiv:1603.04471 and arXiv:1604.00012, we show how to calculate the twistor-space diagrams for general N^kMHV form factors via the inverse soft limit, in analogy to the amplitude case. For general operators without $\dotα$ indices, we then reexpress the NMHV form factors from the position-twistor calculation in terms of momentum twistors, deriving and expanding on a relation between the two twistor formalisms previously observed in the case of amplitudes. Furthermore, we discuss the calculation of generalized form factors and correlation functions as well as the extension to loop level, in particular providing an argument promised in arXiv:1410.6310.

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arXiv2016

All Tree-Level MHV Form Factors in $\mathcal{N}=4$ SYM from Twistor Space

We incorporate all gauge-invariant local composite operators into the twistor-space formulation of N=4 SYM theory, detailing and expanding on ideas we presented recently in arXiv:1603.04471. The vertices for these operators contain infinitely many terms and we show how they can be constructed by taking suitable derivatives of a light-like Wilson loop in twistor space and shrinking it down to a point. In particular, these vertices directly yield the tree-level MHV super form factors of all composite operators in N=4 SYM theory.

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arXiv2016

Composite Operators in the Twistor Formulation of $\mathcal{N}=4$ SYM Theory

We incorporate gauge-invariant local composite operators into the twistor-space formulation of $\mathcal{N}=4$ Super Yang-Mills theory. In this formulation, the interactions of the elementary fields are reorganized into infinitely many interaction vertices and we argue that the same applies to composite operators. To test our definition of the local composite operators in twistor space, we compute several corresponding form factors, thereby also initiating the study of form factors using the position twistor-space framework. Throughout this letter, we use the composite operator built from two identical complex scalars as a pedagogical example; we treat the general case in a follow-up paper.

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arXiv2015

Grassmannian Integrals as Matrix Models for Non-Compact Yangian Invariants

In the past years, there have been tremendous advances in the field of planar N=4 super Yang-Mills scattering amplitudes. At tree-level they were formulated as Grassmannian integrals and were shown to be invariant under the Yangian of the superconformal algebra psu(2,2|4). Recently, Yangian invariant deformations of these integrals were introduced as a step towards regulated loop-amplitudes. However, in most cases it is still unclear how to evaluate these deformed integrals. In this work, we propose that changing variables to oscillator representations of psu(2,2|4) turns the deformed Grassmannian integrals into certain matrix models. We exemplify our proposal by formulating Yangian invariants with oscillator representations of the non-compact algebra u(p,q) as Grassmannian integrals. These generalize the Brezin-Gross-Witten and Leutwyler-Smilga matrix models. This approach might make elaborate matrix model technology available for the evaluation of Grassmannian integrals. Our invariants also include a matrix model formulation of the u(p,q) R-matrix, which generates non-compact integrable spin chains.

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arXiv2015

A Twistorial Approach to Integrability in N=4 SYM

While the achievements in the study of N=4 Super Yang-Mills through the application of integrability are impressive, the precise origins of the exact solvability remain shrouded in mystery. In this note, we propose that viewing the problem through the lens of twistor theory should help to clarify the reasons for integrability. We illustrate the power of this approach by rederiving the model's one-loop spin chain dilatation operator in the SO(6) sector.

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arXiv2014

N=4 Scattering Amplitudes and the Deformed Grassmannian

Some time ago the general tree-level scattering amplitudes of N=4 Super Yang-Mills theory were expressed as certain Grassmannian contour integrals. These remarkable formulas allow to clearly expose the super-conformal, dual super-conformal, and Yangian symmetries of the amplitudes. Using ideas from integrability it was recently shown that the building blocks of the amplitudes permit a natural multi-parameter deformation. However, this approach had been criticized by the observation that it seemed impossible to reassemble the building blocks into Yangian-invariant deformed non-MHV amplitudes. In this note we demonstrate that the deformations may be succinctly summarized by a simple modification of the measure of the Grassmannian integrals, leading to a Yangian-invariant deformation of the general tree-level amplitudes. Interestingly, the deformed building-blocks appear as residues of poles in the spectral parameter planes. Given that the contour integrals also contain information on the amplitudes at loop-level, we expect the deformations to be useful there as well. In particular, applying meromorphicity arguments, they may be expected to regulate all notorious infrared divergences. We also point out relations to Gelfand hypergeometric functions and the quantum Knizhnik-Zamolodchikov equations.

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arXiv2014

Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes

We propose that Baxter's Z-invariant six-vertex model at the rational gl(2) point on a planar but in general not rectangular lattice provides a way to study Yangian invariants. These are identified with eigenfunctions of certain monodromies of an auxiliary inhomogeneous spin chain. As a consequence they are special solutions to the eigenvalue problem of the associated transfer matrix. Excitingly, this allows to construct them using Bethe ansatz techniques. Conceptually, our construction generalizes to general (super) Lie algebras and general representations. Here we present the explicit form of sample invariants for totally symmetric, finite-dimensional representations of gl(n) in terms of oscillator algebras. In particular, we discuss invariants of three- and four-site monodromies that can be understood respectively as intertwiners of the bootstrap and Yang-Baxter equation. We state a set of functional relations significant for these representations of the Yangian and discuss their solutions in terms of Bethe roots. They arrange themselves into exact strings in the complex plane. In addition, it is shown that the sample invariants can be expressed analogously to Grassmannian integrals. This aspect is closely related to a recent on-shell formulation of scattering amplitudes in planar N=4 super Yang-Mills theory.

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arXiv2014

A Shortcut to General Tree-level Scattering Amplitudes in N=4 SYM via Integrability

We combine recent applications of the two-dimensional quantum inverse scattering method to the scattering amplitude problem in four-dimensional N=4 Super Yang-Mills theory. Integrability allows us to obtain a general, explicit method for the derivation of the Yangian invariants relevant for tree-level scattering amplitudes in the N=4 model.

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