arXiv subjects

Till Bargheer

Publications and source records attributed to Till Bargheer.

At least 19 recordsLinked to original sources

arXiv2026

Structure Constants from Q-Systems and Separation of Variables

We introduce a novel method to compute structure constants from Q-functions in the scalar sector of planar N=4 super Yang-Mills (SYM) and related theories. The method derives from operatorial as well as functional separation of variables, and the structure constants are expressed as determinants of matrices whose entries are integrals over products of Q-functions. In this framework, each operator is twisted by an external angle, mirroring the cusped Maldacena-Wilson loop. The structure constants of local single-trace operators in N=4 SYM are recovered in the untwisting limit, where we obtain a one-to-one correspondence between our key building blocks and those of the Hexagon formalism. Retaining appropriate twists, our structure constants also perfectly match those of the orbifold points of N=4 SYM. Our results thus far are valid at leading order in the weak-coupling expansion, but their formulation in terms of Q-functions provides a natural starting point for including loop corrections. Many of the methods we develop in this work apply more generally to the computation of correlation functions in integrable models.

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arXiv2025

Higher-Point Correlators in N=4 SYM: Ten-Dimensional Null Polygons

We consider higher-point generalizations of the "octagon" large-charge four-point function in planar N=4 super Yang-Mills theory. These n-point polygon correlators are defined as ten-dimensional null limits of generating functions of n-point correlators of protected scalar operators with arbitrary R-charge, and are conjecturally dual to massive scattering amplitudes of W-bosons on the Coulomb branch of the theory. We present a twistor-based direct computation of the polygon loop integrands, and determine the two-loop integrands for any number of external operators. The result is expressed as a linear combination of conformal integrals, with coefficients that are rational functions of the spacetime distances and operator polarizations. We obtain an exact match with a one-loop integrability computation, and study the near-massless limit of the pentagon function.

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arXiv2025

The Non-Planar Four-Point Integrand and Konishi Dimension in N=4 Super Yang-Mills Theory at Five Loops

We compute the complete non-planar integrand for the correlation function of four lightest scalar operators in N=4 super Yang-Mills theory at five-loop order. This is equivalent to the super-correlator of nine stress-tensor multiplets in the self-dual theory. Starting with an ansatz of f-graphs, we impose constraints from light-cone limits, and fix the remaining freedom by using the reformulation of the theory in twistor space. We develop an efficient GPU-based algorithm for the numerical evaluation of the twistor rules. As an application, we extract the five-loop non-planar anomalous dimension of the Konishi operator. Our code and result are provided in ancillary files.

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arXiv2025

Higher-Point Correlators in N=4 SYM: Generating Functions

We construct generating functions of five- and six-point correlators up to two loops at weak 't Hooft coupling in planar N=4 SYM. These generating functions unify the correlators of the lightest scalar operator in the stress-tensor multiplet with those of all higher R-charge single-trace half-BPS scalar operators, thereby extending previous results for four-point loop integrands. At the integrated level, they are represented as sums of conformal integrals with coefficients exhibiting ten-dimensional poles that combine spacetime and R-charge distances. Our results show that higher-order poles are captured by products of lower-point generating functions. We also extract new OPE data on spinning structure constants, and compare these to integrability-based computations, finding good agreement.

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arXiv2025

Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory

We construct orthogonality relations in the Separation of Variables framework for the sl(2) sector of planar N=4 supersymmetric Yang-Mills theory. Specifically, we find simple universal measures that make Q-functions of operators with different spins vanish at all orders in perturbation theory, prior to wrapping corrections. To analyze this rank-one sector, we relax some of the assumptions thus far considered in the Separation of Variables framework. Our findings may serve as guidelines for extending this formalism to other sectors of the theory as well as other integrable models

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arXiv2024

Wilson Loops with Lagrangians: Large Spin OPE and Cusp Anomalous Dimension Dictionary

In the context of planar conformal gauge theory, we study five-point correlation functions between the interaction Lagrangian and four of the lightest single-trace, gauge-invariant scalar primaries. After performing two light-cone OPEs, we express this correlator in terms of the three-point functions between two leading-twist spinning operators and the Lagrangian. For finite values of spin, we compute these structure constants in perturbation theory up to two loops in N=4 super Yang-Mills theory. Large values of spin are captured by null polygon kinematics, where we use dualities with null polygon Wilson loops as well as factorization properties to bootstrap the universal behavior of the structure constants at all loops. We find explicit maps that relate the Lagrangian structure constants with the leading-twist anomalous dimension. From the large-spin map, we recover the cusp anomalous dimension at strong and weak coupling, including genus-one terms.

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arXiv2023

Higher-Point Integrands in N=4 super Yang-Mills Theory

We compute the integrands of five-, six-, and seven-point correlation functions of twenty-prime operators with general polarizations at the two-loop order in N=4 super Yang-Mills theory. In addition, we compute the integrand of the five-point function at three-loop order. Using the operator product expansion, we extract the two-loop four-point function of one Konishi operator and three twenty-prime operators. Two methods were used for computing the integrands. The first method is based on constructing an ansatz, and then numerically fitting for the coefficients using the twistor-space reformulation of N=4 super Yang-Mills theory. The second method is based on the OPE decomposition. Only very few correlator integrands for more than four points were known before. Our results can be used to test conjectures, and to make progresses on the integrability-based hexagonalization approach for correlation functions.

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arXiv2019

Octagons I: Combinatorics and Non-Planar Resummations

We explain how the 't Hooft expansion of correlators of half-BPS operators can be resummed in a large-charge limit in N=4 super Yang-Mills theory. The full correlator in the limit is given by a non-trivial function of two variables: One variable is the charge of the BPS operators divided by the square root of the number Nc of colors; the other variable is the octagon that contains all the 't Hooft coupling and spacetime dependence. At each genus g in the large Nc expansion, this function is a polynomial of degree 2g+2 in the octagon. We find several dual matrix model representations of the correlators in the large-charge limit. Amusingly, the number of colors in these matrix models is formally taken to zero in the relevant limit.

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arXiv2019

Octagons II: Strong Coupling

The octagon function is the fundamental building block yielding correlation functions of four large BPS operators in N=4 super Yang-Mills theory at any value of the 't Hooft coupling and at any genus order. Here we compute the octagon at strong coupling, and discuss various interesting limits and implications, both at the planar and non-planar level.

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arXiv2019

Handling Handles. Part II. Stratification and Data Analysis

In a previous work, we proposed an integrability setup for computing non-planar corrections to correlation functions in $\mathcal{N}=4$ super Yang-Mills theory at any value of the coupling constant. The procedure consists of drawing all possible tree-level graphs on a Riemann surface of given genus, completing each graph to a triangulation, inserting a hexagon form factor into each face, and summing over a complete set of states on each edge of the triangulation. The summation over graphs can be interpreted as a quantization of the string moduli space integration. The quantization requires a careful treatment of the moduli space boundaries, which is realized by subtracting degenerate Riemann surfaces; this procedure is called stratification. In this work, we precisely formulate our proposal and perform several perturbative checks. These checks require hitherto unknown multi-particle mirror contributions at one loop, which we also compute.

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arXiv2019

Handling Handles: Nonplanar Integrability in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory

We propose an integrability setup for the computation of correlation functions of gauge-invariant operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory at higher orders in the large $N_{\text{c}}$ genus expansion and at any order in the 't Hooft coupling $g_{\text{YM}}^2N_{\text{c}}$. In this multi-step proposal, one polygonizes the string worldsheet in all possible ways, hexagonalizes all resulting polygons, and sprinkles mirror particles over all hexagon junctions to obtain the full correlator. We test our integrability-based conjecture against a non-planar four-point correlator of large half-BPS operators at one and two loops.

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arXiv2019

The Multi-Regge Limit from the Wilson Loop OPE

The finite remainder function for planar, color-ordered, maximally helicity violating scattering processes in N=4 super Yang-Mills theory possesses a non-vanishing multi-Regge limit that depends on the choice of a Mandelstam region. We analyze the combined multi-Regge collinear limit in all Mandelstam regions through an analytic continuation of the Wilson loop OPE. At leading order, the former is determined by the gluon excitation of the Gubser-Klebanov-Polyakov string. We illustrate the general procedure at the example of the heptagon remainder function at two loops. In this case, the continuation of the leading order terms in the Wilson loop OPE suffices to determine the two-loop multi-Regge heptagon functions in all Mandelstam regions from their symbols. The expressions we obtain are fully consistent with recent results by Del Duca et al.

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arXiv2018

Four-Point Functions with a Twist

We study the OPE of correlation functions of local operators in planar N=4 super Yang-Mills theory. The considered operators have an explicit spacetime dependence that is defined by twisting the translation generators with certain R-symmetry generators. We restrict to operators that carry a small number of excitations above the twisted BMN vacuum. The OPE limit of the four-point correlator is dominated by internal states with few magnons on top of the vacuum. The twisting directly couples all spacetime dependence of the correlator to these magnons. We analyze the OPE in detail, and single out the extremal states that have to cancel all double-trace contributions.

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arXiv2017

Systematics of the Multi-Regge Three-Loop Symbol

We review the systematics of Mandelstam cut contributions to planar scattering amplitudes in the multi-Regge limit. Isolating the relevant cut terms, we explain how the BFKL expansion can be used to construct the perturbative n-point multi-Regge limit amplitude in certain kinematic regions from a finite number of basic building blocks. At three loops and at leading logarithmic order, two building blocks are required. Their symbols are extracted from the known three-loop six-point and seven-point symbols for general kinematics. The new seven-point building block is constructed in terms of single-valued multiple polylogarithms to the extent it can be determined using the symbol as well as further symmetry and consistency constraints. Beyond the leading logarithmic order, the subleading and sub-subleading terms require two and one further building block, respectively. The latter could either be reconstructed from further perturbative data, or from BFKL integrals involving yet-unknown corrections to the central emission block.

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arXiv2016

The Two-Loop Symbol of all Multi-Regge Regions

We study the symbol of the two-loop n-gluon MHV amplitude for all Mandelstam regions in multi-Regge kinematics in N=4 super Yang-Mills theory. While the number of distinct Mandelstam regions grows exponentially with n, the increase of independent symbols turns out to be merely quadratic. We uncover how to construct the symbols for any number of external gluons from just two building blocks which are naturally associated with the six- and seven-gluon amplitude, respectively. The second building block is entirely new, and in addition to its symbol, we also construct a prototype function that correctly reproduces all terms of maximal functional transcendentality.

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arXiv2015

Integrable Amplitude Deformations for N=4 Super Yang-Mills and ABJM Theory

We study Yangian-invariant deformations of scattering amplitudes in 4d N=4 supersymmetric Yang-Mills theory and 3d N=6 Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. In particular, we obtain the deformed Grassmannian integral for 4d N=4 super Yang-Mills theory, both in momentum and momentum-twistor space. For 3d ABJM theory, we initiate the study of deformed scattering amplitudes. We investigate general deformations of on-shell diagrams, and find the deformed Grassmannian integral for this theory. We furthermore introduce the algebraic R-matrix construction of deformed Yangian invariants for ABJM theory.

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arXiv2014

Computing Three-Point Functions for Short Operators

We compute the three-point structure constants for short primary operators of N=4 super Yang-Mills theory to leading order in the inverse coupling by mapping the problem to a flat-space string theory calculation. We check the validity of our procedure by comparing to known results for three chiral primaries. We then compute the three-point functions for any combination of chiral and non-chiral primaries, with the non-chiral primaries all dual to string states at the first massive level. Along the way we find many cancellations that leave us with simple expressions, suggesting that integrability is playing an important role.

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arXiv2013

Long-Range Deformations for Integrable Spin Chains

We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest-neighbor) integrable spin chains, it allows to construct an infinite set of conserved long-range charges. We explain the moduli space of deformation parameters by different classes of generating operators. The rapidity map and dressing phase in the long-range Bethe equations are a result of these deformations. The closed chain asymptotic Bethe equations for long-range spin chains transforming under a generic symmetry algebra are derived. Notably, our construction applies to generalizations of standard nearest-neighbor chains such as alternating spin chains. We also discuss relevant properties for its application to planar D=4, N=4 and D=3, N=6 supersymmetric gauge theories. Finally, we present a map between long-range and inhomogeneous spin chains delivering more insight into the structures of these models as well as their limitations at wrapping order.

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