arXiv subjects

David Simmons-Duffin

Publications and source records attributed to David Simmons-Duffin.

At least 19 recordsLinked to original sources

arXiv2026

Bootstrapping currents and stress tensors in 3d CFTs

We perform a numerical conformal bootstrap study of the mixed system of correlation functions involving a spin-1 $\mathrm{U}(1)$ current $J$ and the stress-energy tensor $T$ in parity-preserving 3d CFTs. We find universal bounds on the stress-tensor two-point function $c_T$, which numerically reproduce the conformal collider bounds on $\langle JJT\rangle$ and $\langle TTT\rangle$, as well as bounds on the leading parity-even and parity-odd scalar operator dimensions. Under mild assumptions, we determine the values of the $\langle JJT\rangle$ and $\langle TTT\rangle$ three-point functions in the $\mathrm{O}(2)$ vector model. The $\langle TTT\rangle$ result is new and $\langle JJT\rangle$ is consistent with a previous result in the literature.

hep-th↗

arXiv2026

Accurate bootstrap bounds from optimal interpolation

We develop new methods for approximating conformal blocks as positive functions times polynomials, with applications to the numerical bootstrap. We argue that to obtain accurate bootstrap bounds, conformal block approximations should minimize a certain error norm related to the asymptotics of dispersive functionals. This error norm can be made small using interpolation nodes with an appropriate optimal density. The optimal density turns out to satisfy a kind of force-balance equation for charges in one dimension, which can be solved using standard techniques from large-N matrix models. We also describe how to use optimal density interpolation nodes to improve condition numbers inside the semidefinite program solver SDPB. Altogether, our new approximation scheme and improvements to condition numbers lead to more accurate bootstrap bounds with fewer computational resources. They were crucial in the recent bootstrap study of stress tensors in the 3d Ising CFT.

hep-th↗

arXiv2025

Seeing through the confinement screen: DGLAP/BFKL mixing and light-ray matching in QCD

We argue that collider observables such as hadron number flux can be matched onto a linear combination of detectors/light-ray operators in perturbative QCD. The spectrum of detectors in QCD is subtle, due to recombination between the DGLAP and BFKL trajectories. We explain how to define and renormalize these trajectories at one-loop, systematically incorporating their recombination. The leading and subleading soft gluon theorems play an important role, and our analysis suggests the presence of an infinite series of further subleading soft theorems for squared-amplitudes/form factors. Combined with our light-ray matching hypothesis, the anomalous dimensions of recombined DGLAP/BFKL detectors yield a prediction for the energy dependence of the number of particles in a jet, as well as other predictions for more general energy-weighted hadron measurements. We compare these predictions to Monte-Carlo simulations, finding good agreement.

hep-th↗

arXiv2025

A genus-2 crossing equation in $d\geq 2$

We explore a "genus-2" crossing equation obeyed by CFTs in general dimensions $d\geq 2$. This crossing equation relates two different decompositions of the "genus-2 partition function" -- namely the partition function on the connected sum $M_2=(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})$. The "sunrise" channel decomposition expresses $M_2$ as a pair of three-punctured spheres glued together with cylinders, while the "dumbbell" channel decomposition expresses $M_2$ as a gluing of two one-point functions on $S^1\times S^{d-1}$. We introduce coordinates to describe each channel, and write down Casimir equations obeyed by the corresponding blocks. We also explain why equality between the two channels guarantees mapping class group invariance of the genus-2 partition function in 3d CFTs. As an application of the genus-2 crossing equation, we derive a novel relation between asymptotics of "heavy-heavy-heavier" OPE coefficients and squares of thermal one-point coefficients in 3d CFTs. Along the way, we demonstrate how expectation values of conformal generators can help locate saddle points in large quantum number limits.

hep-th↗

arXiv2025

Weight Shifting Operators and Conformal Blocks

We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for "seed conformal blocks" in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including derivation of recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/"crossing kernels") of the conformal group.

hep-th↗

arXiv2025

Quantum Scaling in Energy Correlators Beyond the Confinement Transition

We study the QCD scaling behavior of the small-angle Energy-Energy Correlator (EEC), focusing on the transition between its perturbative pre-confinement and non-perturbative post-confinement regimes. Applying the light-ray Operator Product Expansion (OPE), we develop a formalism that describes the scaling of the EEC with the input energy $Q$ in the transition and the post-confinement region, where the latter quantum scaling is determined by the $J=5$ DGLAP anomalous dimension. A key result of our work is a novel connection between the light-ray OPE and the dihadron fragmentation function (DFF), where we show that the non-perturbative OPE coefficients correspond to moments of the DFF. This finding establishes a new paradigm for studying hadronization. Our theoretical predictions are validated against Monte Carlo simulations for both $e^+e^-$ and $pp$ collisions, showing excellent agreement. The potential role of the quantum scaling in the precision determination of $α_s$ is also discussed.

hep-ph↗

arXiv2025

Regge trajectories, detectors, and distributions in the critical ${\rm O}(N)$ model

We explore light-ray operators in the critical O$(N)$ model in the large-$N$ limit, focusing on leading-twist and leading ``horizontal" trajectories. We distinguish between light-ray operators in two conformal frames: detector operators, which characterize event shapes of final states, and distribution operators, which probe initial-state distributions. In particular, we identify parton distribution functions (PDFs) and collinear functions as matrix elements of appropriate distribution operators. We renormalize some simple detector operators at leading order in $1/N$, allowing us to extract the Regge intercept and the anomalous spin of the leading horizontal trajectory. We furthermore renormalize distribution versions of these operators, obtaining the leading-twist splitting function and a BFKL-type kernel, which match results from the detector frame. Finally, we show how these results can be read off from OPE data encoded in the Bethe-Salpeter resummation of conformal four-point functions.

hep-th↗

arXiv2025

Bootstrapping the 3d Ising Stress Tensor

We compute observables of the critical 3d Ising model to high precision by applying the numerical conformal bootstrap to mixed correlators of the leading scalar operators $σ$ and $ε$, and the stress tensor $T_{μν}$. We obtain new precise determinations of scaling dimensions $(Δ_σ, Δ_ε) = (0.518148806(24), 1.41262528(29))$ as well as OPE coefficients involving $σ$, $ε$, and $T_{μν}$. We also describe several improvements made along the way to algorithms and software tools for the numerical bootstrap.

hep-th↗

arXiv2025

Spinning dispersive CFT sum rules and bulk scattering

We use commutativity of null-integrated operators on the same null plane to construct dispersive CFT sum rules for spinning operators. The contribution of heavy blocks to these sum rules is dominated by a saddle configuration that we call the "scattering crystal." Correlators in this configuration have a natural flat-space interpretation, which allows us to build a dictionary between dispersive CFT sum rules for stress-tensors and flat-space dispersion relations for gravitons. This dictionary is a crucial step for establishing the HPPS conjecture for stress tensor correlators.

hep-th↗

arXiv2024

Light-Ray Wave Functions and Integrability

Using integrability, we construct (to leading order in perturbation theory) the explicit form of twist-three light-ray operators in planar $\mathcal{N}=4$ SYM. This construction allows us to directly compute analytically continued CFT data at complex spin. We derive analytically the "magic'' decoupling zeroes previously observed numerically. Using the Baxter equation, we also show that certain Regge trajectories merge together into a single unifying Riemann surface. Perhaps more surprisingly, we find that this unification of Regge trajectories is not unique. If we organize twist-three operators differently into what we call "cousin trajectories'' we find infinitely more possible continuations. We speculate about which of these remarkable features of twist-three operators might generalize to other operators, other regimes and other theories.

hep-th↗

arXiv2024

Angular fractals in thermal QFT

We show that thermal effective field theory controls the long-distance expansion of the partition function of a $d$-dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of $1/2^d$. Near certain rational angles, the partition function receives subleading contributions from "Kaluza-Klein vortex defects" in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.

hep-th↗

arXiv2024

Universal Asymptotics for High Energy CFT Data

Equilibrium finite temperature observables of a CFT can be described by a local effective action for background fields -- a "thermal effective action." This effective action determines the asymptotic density of states of a CFT as a detailed function of dimension and spin. We discuss subleading perturbative and nonperturbative corrections to the density, comparing with free and holographic examples. We furthermore show how to use the thermal effective action on more complicated geometries at special locations called "hot spots." The hot spot idea makes a prediction for a CFT partition function on a higher-dimensional version of a genus-2 Riemann surface, in a particular high temperature limit. By decomposing the partition function into a novel higher-dimensional version of genus-2 conformal blocks (which we compute at large scaling dimension), we extract the asymptotic density of heavy-heavy-heavy OPE coefficients in a higher-dimensional CFT. We also compute asymptotics of thermal 1-point functions using the same techniques.

hep-th↗

arXiv2023

Bounds on spectral gaps of Hyperbolic spin surfaces

We describe a method for constraining Laplacian and Dirac spectra of two dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key ingredient is an infinite family of identities satisfied by the spectra. These spectral identities follow from the consistency between 1) the spectral decomposition of functions on the spin bundle into irreducible representations of $\mathrm{SL}(2,\mathbb{R})$ and 2) associativity of pointwise multiplication of functions. Applying semidefinite programming methods to our identities produces rigorous upper bounds on the Laplacian spectral gap as well as on the Dirac spectral gap conditioned on the former. In several examples, our bounds are nearly sharp; a numerical algorithm based on the Selberg trace formula shows that the $[0;3,3,5]$ orbifold, a particular surface with signature $[1;3]$, and the Bolza surface nearly saturate the bounds at genus $0$, $1$ and $2$ respectively. Under additional assumptions on the number of harmonic spinors carried by the spin-surface, we obtain more restrictive bounds on the Laplacian spectral gap. In particular, these bounds apply to hyperelliptic surfaces. We also determine the set of Laplacian spectral gaps attained by all compact orientable two-dimensional hyperbolic spin orbifolds. We show that this set is upper bounded by $12.13798$; this bound is nearly saturated by the $[0;3,3,5]$ orbifold, whose first non-zero Laplacian eigenvalue is $λ^{(0)}_1\approx 12.13623$.

math.SP↗

arXiv2023

Skydiving to Bootstrap Islands

We study families of semidefinite programs (SDPs) that depend nonlinearly on a small number of "external" parameters. Such families appear universally in numerical bootstrap computations. The traditional method for finding an optimal point in parameter space works by first solving an SDP with fixed external parameters, then moving to a new point in parameter space and repeating the process. Instead, we unify solving the SDP and moving in parameter space in a single algorithm that we call "skydiving". We test skydiving on some representative problems in the conformal bootstrap, finding significant speedups compared to traditional methods.

hep-th↗

arXiv2023

The Gross-Neveu-Yukawa Archipelago

We perform a bootstrap analysis of a mixed system of four-point functions of bosonic and fermionic operators in parity-preserving 3d CFTs with O(N) global symmetry. Our results provide rigorous bounds on the scaling dimensions of the O(N)-symmetric Gross-Neveu-Yukawa (GNY) fixed points, constraining these theories to live in isolated islands in the space of CFT data. We focus on the cases N = 1, 2, 4, 8, which have applications to phase transitions in condensed matter systems, and compare our bounds to previous analytical and numerical results.

hep-th↗

arXiv2023

Three-point energy correlators and the celestial block expansion

We study the three-point energy correlator (EEEC), defined as a matrix element of a product of three energy detectors at different locations on the celestial sphere. Lorentz symmetry implies that the EEEC can be decomposed into special functions called celestial blocks. We compute three-point celestial blocks in an expansion around the collinear limit, where the three detectors approach each other on the celestial sphere. The leading term is a traditional $d{-}2$-dimensional four-point conformal block, and thus the collinear EEEC behaves like a conformally-invariant four-point function in $d{-}2$ dimensions. We obtain the coefficients of the conformal block decomposition for the collinear EEEC at leading nontrivial order in weakly-coupled $\mathcal{N}=4$ SYM and QCD. These data allow us to make certain all-orders predictions for the collinear EEEC in various kinematic limits, including the OPE limit and the double lightcone limit. We also study Ward identities satisfied by the EEEC and compute contact terms in the EEEC in weakly-coupled $\mathcal{N}=4$ SYM. Finally, we study the celestial block expansion of the EEEC in planar $\mathcal{N}=4$ SYM at strong coupling, determining celestial block coefficients to leading and first subleading order at large $λ$.

hep-th↗

arXiv2022

Complex Spin: The Missing Zeroes and Newton's Dark Magic

Conformal Regge theory predicts the existence of analytically continued CFT data for complex spin. How could this work when there are so many more operators with large spin compared to small spin? Using planar N=4 SYM as a testground we find a simple physical picture. Operators do organize themselves into analytic families but the continuation of the higher families have zeroes in their structure OPE constants for lower integer spins. They thus decouple. Newton's interpolation series technique is perfectly suited to this physical problem and will allow us to explore the right complex spin half-plane.

hep-th↗

arXiv2022

Detectors in weakly-coupled field theories

We initiate a study of asymptotic detector operators in weakly-coupled field theories. These operators describe measurements that can be performed at future null infinity in a collider experiment. In a conformal theory they can be identified with light-ray operators, and thus have a direct relation to the spectrum of the theory. After a general discussion of the underlying physical picture, we show how infrared divergences of general detector operators can be renormalized in perturbation theory, and how they give rise to detector anomalous dimensions. We discuss in detail how this renormalization can be performed at the intersections of the Regge trajectories where non-trivial mixing occurs, which is related to the poles in anomalous dimensions at special values of spin. Finally, we discuss novel horizontal trajectories in scalar theories and show how they contribute to correlation functions. Our calculations are done in the example of $ϕ^4$ theory in $d=4-ε$ dimensions, but the methods are applicable more broadly. At the Wilson-Fisher fixed point our results include an explicit expression for the Pomeron light-ray operator at two loops, as well as a prediction for the value of the Regge intercept at five loops.

hep-th↗