arXiv subjects

Matthias Staudacher

Publications and source records attributed to Matthias Staudacher.

At least 37 records · Page 2Linked to original sources

arXiv2013

Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory

Planar N=4 Super Yang-Mills theory appears to be a quantum integrable four-dimensional conformal theory. This has been used to find equations believed to describe its exact spectrum of anomalous dimensions. Integrability seemingly also extends to the planar space-time scattering amplitudes of the N=4 model, which show strong signs of Yangian invariance. However, in contradistinction to the spectral problem, this has not yet led to equations determining the exact amplitudes. We propose that the missing element is the spectral parameter, ubiquitous in integrable models. We show that it may indeed be included into recent on-shell approaches to scattering amplitude integrands, providing a natural deformation of the latter. Under some constraints, Yangian symmetry is preserved. Finally we speculate that the spectral parameter might also be the regulator of choice for controlling the infrared divergences appearing when integrating the integrands in exactly four dimensions.

hep-th↗

arXiv2013

Harmonic R-matrices for Scattering Amplitudes and Spectral Regularization

Planar N=4 super Yang-Mills appears to be integrable. While this allows to find this theory's exact spectrum, integrability has hitherto been of no direct use for scattering amplitudes. To remedy this, we deform all scattering amplitudes by a spectral parameter. The deformed tree-level four-point function turns out to be essentially the one-loop R-matrix of the integrable N=4 spin chain satisfying the Yang-Baxter equation. Deformed on-shell three-point functions yield novel three-leg R-matrices satisfying bootstrap equations. Finally, we supply initial evidence that the spectral parameter might find its use as a novel symmetry-respecting regulator replacing dimensional regularization. Its physical meaning is a local deformation of particle helicity, a fact which might be useful for a much larger class of non-integrable four-dimensional field theories.

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arXiv2012

Review of AdS/CFT Integrability: An Overview

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010.

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arXiv2011

Baxter Operators and Hamiltonians for "nearly all" Integrable Closed gl(n) Spin Chains

We continue our systematic construction of Baxter Q-operators for spin chains, which is based on certain degenerate solutions of the Yang-Baxter equation. Here we generalize our approach from the fundamental representation of gl(n) to generic finite-dimensional representations in quantum space. The results equally apply to non-compact representations of highest or lowest weight type. We furthermore fill an apparent gap in the literature, and provide the nearest-neighbor Hamiltonians of the spin chains in question for all cases where the gl(n) representations are described by rectangular Young diagrams, as well as for their infinite-dimensional generalizations. They take the form of digamma functions depending on operator-valued shifted weights.

math-ph↗

arXiv2011

Review of AdS/CFT Integrability, Chapter III.1: Bethe Ansätze and the R-Matrix Formalism

The one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe's ansatz.

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arXiv2011

A Shortcut to the Q-Operator

Baxter's Q-operator is generally believed to be the most powerful tool for the exact diagonalization of integrable models. Curiously, it has hitherto not yet been properly constructed in the simplest such system, the compact spin-1/2 Heisenberg-Bethe XXX spin chain. Here we attempt to fill this gap and show how two linearly independent operatorial solutions to Baxter's TQ equation may be constructed as commuting transfer matrices if a twist field is present. The latter are obtained by tracing over infinitely many oscillator states living in the auxiliary channel of an associated monodromy matrix. We furthermore compare and differentiate our approach to earlier articles addressing the problem of the construction of the Q-operator for the XXX chain. Finally we speculate on the importance of Q-operators for the physical interpretation of recent proposals for the Y-system of AdS/CFT.

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arXiv2011

Baxter Q-Operators and Representations of Yangians

We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the Yang-Baxter equation connected with harmonic oscillator algebras. These infinite-state solutions of the Yang-Baxter equation serve as elementary, "partonic" building blocks for other solutions via the standard fusion procedure. As a first example of the method we consider sl(n) compact spin chains and derive the full hierarchy of operatorial functional equations for all related commuting transfer matrices and Q-operators. This leads to a systematic and transparent solution of these chains, where the nested Bethe equations are derived in an entirely algebraic fashion, without any reference to the traditional Bethe ansatz techniques.

math-ph↗

arXiv2011

Oscillator Construction of su(n|m) Q-Operators

We generalize our recent explicit construction of the full hierarchy of Baxter Q-operators of compact spin chains with su(n) symmetry to the supersymmetric case su(n|m). The method is based on novel degenerate solutions of the graded Yang-Baxter equation, leading to an amalgam of bosonic and fermionic oscillator algebras. Our approach is fully algebraic, and leads to the exact solution of the associated compact spin chains while avoiding Bethe ansatz techniques. It furthermore elucidates the algebraic and combinatorial structures underlying the system of nested Bethe equations. Finally, our construction naturally reproduces the representation, due to Z. Tsuboi, of the hierarchy of Baxter Q-operators in terms of hypercubic Hasse diagrams.

math-ph↗

arXiv2008

A Generalized Scaling Function for AdS/CFT

We study a refined large spin limit for twist operators in the sl(2) sector of AdS/CFT. We derive a novel non-perturbative equation for the generalized two-parameter scaling function associated to this limit, and analyze it at weak coupling. It is expected to smoothly interpolate between weakly coupled gauge theory and string theory at strong coupling.

hep-th↗

arXiv2007

Nesting and Dressing

We compute the anomalous dimensions of field strength operators Tr F^L in N=4 SYM from an asymptotic nested Bethe ansatz to all-loop order. Starting from the exact solution of the one-loop problem at arbitrary L, we derive a single effective integral equation for the thermodynamic limit of these dimensions. We also include the recently proposed phase factor for the S-matrix of the planar AdS/CFT system. The terms in the effective equation corresponding to, respectively, the nesting and the dressing are structurally very similar. This hints at the physical origin of the dressing phase, which we conjecture to arise from the hidden presence of infinitely many auxiliary Bethe roots describing a non-trivial "filled" structure of the theory's BPS vacuum. We finally show that the mechanism for creating effective nesting/dressing kernels is quite generic by also deriving the integral equation for the all-loop dimension of a certain one-loop so(6) singlet state.

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arXiv2006

Transcendentality and Crossing

We discuss possible phase factors for the S-matrix of planar N=4 gauge theory, leading to modifications at four-loop order as compared to an earlier proposal. While these result in a four-loop breakdown of perturbative BMN-scaling, Kotikov-Lipatov transcendentality in the universal scaling function for large-spin twist operators may be preserved. One particularly natural choice, unique up to one constant, modifies the overall contribution of all terms containing odd zeta functions in the earlier proposed scaling function based on a trivial phase. Excitingly, we present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on AdS_5xS^5 once the constant is fixed to a particular value. Our proposal, if true, might therefore resolve the long-standing AdS/CFT discrepancies between gauge and string theory.

hep-th↗

arXiv2006

Planar N=4 Gauge Theory and the Hubbard Model

Recently it was established that a certain integrable long-range spin chain describes the dilatation operator of N=4 gauge theory in the su(2) sector to at least three-loop order, while exhibiting BMN scaling to all orders in perturbation theory. Here we identify this spin chain as an approximation to an integrable short-ranged model of strongly correlated electrons: The Hubbard model.

hep-th↗

arXiv2005

Long-Range PSU(2,2|4) Bethe Ansaetze for Gauge Theory and Strings

We generalize various existing higher-loop Bethe ansaetze for simple sectors of the integrable long-range dynamic spin chain describing planar N=4 Super Yang-Mills Theory to the full psu(2,2|4) symmetry and, asymptotically, to arbitrary loop order. We perform a large number of tests of our conjectured equations, such as internal consistency, comparison to direct three-loop diagonalization and expected thermodynamic behavior. In the special case of the su(1|2) subsector, corresponding to a long-range t-J model, we are able to derive, up to three loops, the S-matrix and the associated nested Bethe ansatz from the gauge theory dilatation operator. We conjecture novel all-order S-matrices for the su(1|2) and su(1,1|2) subsectors, and show that they satisfy the Yang-Baxter equation. Throughout the paper, we muse about the idea that quantum string theory on AdS_5xS^5 is also described by a psu(2,2|4) spin chain. We propose asymptotic all-order Bethe equations for this putative "string chain", which differ in a systematic fashion from the gauge theory equations.

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arXiv2004

The Factorized S-Matrix of CFT/AdS

We argue that the recently discovered integrability in the large-N CFT/AdS system is equivalent to diffractionless scattering of the corresponding hidden elementary excitations. This suggests that, perhaps, the key tool for finding the spectrum of this system is neither the gauge theory's dilatation operator nor the string sigma model's quantum Hamiltonian, but instead the respective factorized S-matrix. To illustrate the idea, we focus on the closed fermionic su(1|1) sector of the N=4 gauge theory. We introduce a new technique, the perturbative asymptotic Bethe ansatz, and use it to extract this sector's three-loop S-matrix from Beisert's involved algebraic work on the three-loop su(2|3) sector. We then show that the current knowledge about semiclassical and near-plane-wave quantum strings in the su(2), su(1|1) and sl(2) sectors of AdS_5 x S^5 is fully consistent with the existence of a factorized S-matrix. Analyzing the available information, we find an intriguing relation between the three associated S-matrices. Assuming that the relation also holds in gauge theory, we derive the three-loop S-matrix of the sl(2) sector even though this sector's dilatation operator is not yet known beyond one loop. The resulting Bethe ansatz reproduces the three-loop anomalous dimensions of twist-two operators recently conjectured by Kotikov, Lipatov, Onishchenko and Velizhanin, whose work is based on a highly complex QCD computation of Moch, Vermaseren and Vogt.

hep-th↗

arXiv2004

Bethe Ansatz for Quantum Strings

We propose Bethe equations for the diagonalization of the Hamiltonian of quantum strings on AdS_5 x S^5 at large string tension and restricted to certain large charge states from a closed su(2) subsector. The ansatz differs from the recently proposed all-loop gauge theory asymptotic Bethe ansatz by additional factorized scattering terms for the local excitations. We also show that our ansatz quantitatively reproduces everything that is currently known about the string spectrum of these states. Firstly, by construction, we recover the integral Bethe equations describing semiclassical spinning strings. Secondly, we explain how to derive the 1/J energy corrections of arbitrary M-impurity BMN states, provide explicit, general formulae for both distinct and confluent mode numbers, and compare to asymptotic gauge theory. In the special cases M=2,3 we reproduce the results of direct quantization of Callan et al. Lastly, at large string tension and relatively small charge we recover the famous 2 (n^2 lambda)^(1/4) asymptotics of massive string modes at level n. Remarkably, this behavior is entirely determined by the novel scattering terms. This is qualitatively consistent with the conjecture that these terms occur due to wrapping effects in gauge theory. Our finding does not in itself cure the disagreements between gauge and string theory, but leads us to speculate about the structure of an interpolating Bethe ansatz for the AdS/CFT system at finite coupling and charge.

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arXiv2003

The N=4 SYM Integrable Super Spin Chain

Recently it was established that the one-loop planar dilatation generator of N=4 Super Yang-Mills theory may be identified, in some restricted cases, with the Hamiltonians of various integrable quantum spin chains. In particular Minahan and Zarembo established that the restriction to scalar operators leads to an integrable vector so(6) chain, while recent work in QCD suggested restricting to twist operators, containing mostly covariant derivatives, yields certain integrable Heisenberg XXX chains with non-compact spin symmetry sl(2). Here we unify and generalize these insights and argue that the complete one-loop planar dilatation generator of N=4 is described by an integrable su(2,2|4) super spin chain. We also write down various forms of the associated Bethe ansatz equations, whose solutions are in one-to-one correspondence with the set of all one-loop planar anomalous dimensions in the N=4 gauge theory. We finally speculate on the non-perturbative extension of these integrable structures, which appears to involve non-local deformations of the conserved charges.

hep-th↗

arXiv2001

Limiting Geometries of Two Circular Maldacena-Wilson Loop Operators

We further analyze a recent perturbative two-loop calculation of the expectation value of two axi-symmetric circular Maldacena-Wilson loops in N=4 gauge theory. Firstly, it is demonstrated how to adapt the previous calculation of anti-symmetrically oriented circles to the symmetric case. By shrinking one of the circles to zero size we then explicitly work out the first few terms of the local operator expansion of the loop. Our calculations explicitly demonstrate that circular Maldacena-Wilson loops are non-BPS observables precisely due to the appearance of unprotected local operators. The latter receive anomalous scaling dimensions from non-ladder diagrams. Finally, we present new insights into a recent conjecture claiming that coincident circular Maldacena-Wilson loops are described by a Gaussian matrix model. We report on a novel, supporting two-loop test, but also explain and illustrate why the existing arguments in favor of the conjecture are flawed.

hep-th↗

arXiv2001

Two Loops to Two Loops in N=4 Supersymmetric Yang-Mills Theory

We present a full two-loop O(g^6) perturbative field theoretic calculation of the expectation value of two circular Maldacena-Wilson loops in D=4 N=4 supersymmetric U(N) gauge theory. It is demonstrated that, after taking into account very subtle cancellations of bulk and boundary divergences, the result is completely finite without any renormalization. As opposed to previous lower order calculations existing in the literature, internal vertex diagrams no longer cancel identically and lead to subleading corrections to the dominant ladder diagrams. Taking limits, we proceed to extract the two-loop static potential corresponding to two infinite anti-parallel lines. Our result gives some evidence that the existing strong-coupling calculations using the AdS/CFT conjecture might sum up the full set of large N planar Feynman diagrams.

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