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.

United States

Cyuan-Han Chang

Hao Chen

David Simmons-Duffin

China

Hua Zhu

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

<p>We study the QCD scaling behavior of the small-angle energy-energy correlator (EEC), focusing on the transition between its perturbative preconfinement and nonperturbative postconfinement regimes. Applying the light-ray operator product expansion (OPE), we develop a formalism that describes the scaling of the EEC with the input energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>Q</mi></mrow></math> in the transition and the postconfinement region, where the latter quantum scaling is determined by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>J</mi><mo>=</mo><mn>5</mn></mrow></math> DGLAP anomalous dimension. A key result of our Letter is a novel connection between the light-ray OPE and the dihadron fragmentation function (DFF), where we show that the nonperturbative 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mi>p</mi></math> collisions, showing excellent agreement. The potential role of the quantum scaling in the precision determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></math> is also discussed.</p>

Xiaohui Liu

Feng Yuan

Hua Xing Zhu

Quantum Scaling in Energy Correlators beyond the Confinement Transition

We explore light-ray operators in the critical O($\textit{N}$) model in the large-$\textit{N}$ 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/$\textit{N}$, 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.

Yue-Zhou Li

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

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.

Aike Liu

Italy

Ning Su

France

Balt Rees

Skydiving to bootstrap islands

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.

Yakov Landau

Spinning dispersive CFT sum rules and bulk scattering

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 and , and the stress tensor . We obtain new precise determinations of scaling dimensions (∆ , ∆ ) = (0.518148806(24), 1.41262528(29)) as well as OPE coefficients involving , , and . We also describe several improvements made along the way to algorithms and software tools for the numerical bootstrap.

Vasiliy Dommes

Rajeev Erramilli

Canada

Alexandre Homrich

United Kingdom

Petr Kravchuk

Matthew Mitchell

David Poland

Bootstrapping the 3d Ising stress tensor

Conformal Regge theory predicts the existence of analytically continued conformal field theory data for complex spin. How could this work when there are so many more operators with large spin compared to small spin? Using planar <math><mi>N</mi><mo>=</mo><mn>4</mn></math> SYM as a test ground, we find a simple physical picture. Operators do organize themselves into analytic families but the continuation of the higher families have zeros in their structure operator product expansion 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 complex spin half-plane.

Brazil

Pedro Vieira

Multitwist Trajectories and Decoupling Zeros in Conformal Field Theory

We show that thermal effective field theory controls the long-distance expansion of the partition function of a $\textit{d}$-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 . 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.

Nathan Benjamin

Jaeha Lee

Sridip Pal

Yixin Xu

Angular fractals in thermal QFT

Using integrability, we construct (to leading order in perturbation theory) the explicit form of twist-three light-ray operators in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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.

Light-ray wave functions and integrability

Abstract Recently, lattice formulations of Abelian chiral gauge theory in two dimensions have been devised on the basis of the Abelian bosonization. A salient feature of these 2D lattice formulations is that the gauge invariance is exactly preserved for anomaly-free theories and thus is completely free from the question of the gauge mode decoupling. In the present paper, we propose yet another lattice formulation sharing this desired property. A particularly unique point in our formulation is that the vertex operator of the dual scalar field, which carries the vector charge of the fermion and the “magnetic charge” in the bosonization, is represented by a “hole” excised from the lattice; this is the excision method formulated recently by Abe et al. in a somewhat different context.

Japan

Okuto Morikawa

Soma Onoda

Hiroshi Suzuki

Yet Another Lattice Formulation of 2D (1) Chiral Gauge Theory via Bosonization David Tong: lectures on gauge theory