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DOE OSTI2024

Revealing the landscape of globally color-dual multi-loop integrands

Abstract We report on progress in understanding how to construct color-dual multi-loop amplitudes. First we identify a cubic theory, semi-abelian Yang-Mills, that unifies many of the color-dual theories studied in the literature, and provides a prescriptive approach for constructingD-dimensional color-dual numerators through one-loop directly from Feynman rules. By a simple weight counting argument, this approach does not further generalize to two-loops. As a first step in understanding the two-loop challenge, we use aD-dimensional color-dual bootstrap to successfully construct globally color-duallocaltwo-loop four-point nonlinear sigma model (NLSM) numerators. The double-copy of these NLSM numerators with themselves, pure Yang-Mills, and$$\mathcal{N}$$= 4 super-Yang-Mills correctly reproduce the known unitarity constructed integrands of special Galileons, Born-Infeld theory, and Dirac-Born-Infeld-Volkov-Akulov theory, respectively. Applying our bootstrap to two-loop four-point pure Yang-Mills, we exhaustively search the space of local numerators and find that it fails to satisfy global color-kinematics duality, completing a search previously initiated in the literature. We pinpoint the failure to the bowtie unitarity cut, and discuss a path forward towards non-local construction of color-dual integrands at generic loop order.

Physics↗

DOE OSTI2020

Carving out OPE space and precise O(2) model critical exponents

We develop new tools for isolating CFTs using the numerical bootstrap. A “cutting surface” algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d O(2) model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old 8σ discrepancy between theory and experiment.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Linking the singularities of cosmological correlators

Much of the structure of cosmological correlators is controlled by their singularities, which in turn are fixed in terms of flat-space scattering amplitudes. An important challenge is to interpolate between the singular limits to determine the full correlators at arbitrary kinematics. This is particularly relevant because the singularities of correlators are not directly observable, but can only be accessed by analytic continuation. In this paper, we study rational correlators — including those of gauge fields, gravitons, and the inflaton — whose only singularities at tree level are poles and whose behavior away from these poles is strongly constrained by unitarity and locality. We describe how unitarity translates into a set of cutting rules that consistent correlators must satisfy, and explain how this can be used to bootstrap correlators given information about their singularities. We also derive recursion relations that allow the iterative construction of more complicated correlators from simpler building blocks. In flat space, all energy singularities are simple poles, so that the combination of unitarity constraints and recursion relations provides an efficient way to bootstrap the full correlators. In many cases, these flat-space correlators can then be transformed into their more complex de Sitter counterparts. As an example of this procedure, we derive the correlator associated to graviton Compton scattering in de Sitter space, though the methods are much more widely applicable.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Heptagon functions and seven-gluon amplitudes in multi-Regge kinematics

We compute all 2 → 5 gluon scattering amplitudes in planar $ \mathcal{N} $ N = 4 super-Yang-Mills theory in the multi-Regge limit that is sensitive to the non-trivial (“long”) Regge cut. We provide the amplitudes through four loops and to all logarithmic accuracy at leading power, in terms of single-valued multiple polylogarithms of two variables. To obtain these results, we leverage the function-level results for the amplitudes in the Steinmann cluster bootstrap. To high powers in the series expansion in the two variables, our results agree with the recently conjectured all-order central emission vertex used in the Fourier-Mellin representation of amplitudes in multi-Regge kinematics. Our results therefore provide a resummation of the Fourier-Mellin residues into single-valued polylogarithms, and constitute an important cross-check between the bootstrap approach and the all-orders multi-Regge proposal.

1/N expansion↗

DOE OSTI2023

Goldstone bosons on celestial sphere and conformal soft theorems

In this paper, we study celestial amplitudes of Goldstone bosons and conformal soft theorems. Motivated by the success of soft bootstrap in momentum space and the important role of the soft limit behavior of tree-level amplitudes, our goal is to extend some of the methods to the celestial sphere. The crucial ingredient of the calculation is the Mellin transformation, which transforms four-dimensional scattering amplitudes to correlation functions of primary operators in the celestial CFT. The soft behavior of the amplitude is then translated to the singularities of the correlator. Only for amplitudes in “UV completed theories” (with sufficiently good high energy behavior) the Mellin integration can be properly performed. In all other cases, the celestial amplitude is only defined in a distributional sense with delta functions. We provide many examples of celestial amplitudes in UV-completed models, including linear sigma models and Z-theory, which is a certain completion of the SU(N) non-linear sigma model. We also comment on the BCFW-like and soft recursion relations for celestial amplitudes and the extension of soft bootstrap ideas.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Progress in the development and understanding of a high poloidal-beta tokamak operating scenario for an attractive fusion pilot plant

Abstract The high poloidal-beta ( $$\beta _{\textrm{P}}$$ β P ) regime was first proposed as a high bootstrap current scenario for a steady-state fusion pilot plant (FPP) in the 1990s (Kikuchi in Nucl Fusion 30:265, 1990). Since then, there have been many theoretical, modeling, and experimental research activities on this topic. A joint DIII-D/EAST research team began exploring the high- $$\beta _{\textrm{P}}$$ β P regime in 2013, focusing on addressing the needs of attractive FPP design by taking advantage of the extensive diagnostic set and sophisticated plasma control system on DIII-D and the well-developed integrated modeling capability at General Atomics. The ultimate goal is to demonstrate such a scenario on EAST with truly long pulse and metal wall compatibility. This paper summarizes the highlights of the research results on DIII-D by the joint team in the past decade. Experimental evidence and modeling analysis show the high- $$\beta _{\textrm{P}}$$ β P scenario has great advantages in addressing key needs for an attractive FPP design, such as high-energy confinement quality at low rotation, excellent core-edge integration, high line-averaged density above the Greenwald limit, low disruption risk, and high bootstrap current fraction for steady-state operation. This provides a relatively safe and economical option to base an FPP design on that will lead to commercial fusion energy.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Robust verification of stochastic simulation codes

We introduce a robust verification tool for computational codes, which we call Stochastic Robust Extrapolation based Error Quantification (StREEQ). Unlike the prevalent Grid Convergence Index (GCI) [1] method, our approach is suitable for both stochastic and deterministic computational codes and is generalizable to any number of discretization variables. Building on ideas introduced in the Robust Verification [2] approach, we estimate the converged solution and orders of convergence with uncertainty using multiple fits of a discretization error model. In contrast to Robust Verification, we perform these fits to many bootstrap samples yielding a larger set of predictions with smoother statistics. Here, bootstrap resampling is performed on the lack-of-fit errors for deterministic code responses, and directly on the noisy data set for stochastic responses. This approach lends a degree of robustness to the overall results, capable of yielding precise verification results for sufficiently resolved data sets, and appropriately expanding the uncertainty when the data set does not support a precise result. For stochastic responses, a credibility assessment is also performed to give the analyst an indication of the trustworthiness of the results. Furthermore, this approach is suitable for both code and solution verification, and is particularly useful for solution verification of high-consequence simulations..

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Historically inconsistent productivity and respiration fluxes in the global terrestrial carbon cycle

The terrestrial carbon cycle is a major source of uncertainty in climate projections. Its dominant fluxes, gross primary productivity (GPP), and respiration (in particular soil respiration, R S ), are typically estimated from independent satellite-driven models and upscaled in situ measurements, respectively. We combine carbon-cycle flux estimates and partitioning coefficients to show that historical estimates of global GPP and R S are irreconcilable. When we estimate GPP based on R S measurements and some assumptions about R S :GPP ratios, we found the resulted global GPP values (bootstrap mean ${149}^{+29}_{-23}$ Pg C yr ⁻1 ) are significantly higher than most GPP estimates reported in the literature (${113}^{+18}_{-18}$ Pg C yr ⁻1 ). Similarly, historical GPP estimates imply a soil respiration flux (Rs GPP , bootstrap mean of ${68}^{+10}_{-8}$ Pg C yr ⁻1 ) statistically inconsistent with most published R S values (${87}^{+9}_{-8}$ Pg C yr ⁻1 ), although recent, higher, GPP estimates are narrowing this gap. Furthermore, global R S :GPP ratios are inconsistent with spatial averages of this ratio calculated from individual sites as well as CMIP6 model results. This discrepancy has implications for our understanding of carbon turnover times and the terrestrial sensitivity to climate change. Future efforts should reconcile the discrepancies associated with calculations for GPP and Rs to improve estimates of the global carbon budget.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Design and performance of a superconducting neutron resonance spin flipper

Despite the challenges, neutron resonance spin echo still holds the promise to improve upon neutron spin echo for the measurement of slow dynamics in materials. We present a bootstrap, radio frequency neutron spin flipper using high temperature superconducting technology capable of flipping neutron spin with either nonadiabatic or adiabatic modes. A frequency of 2 MHz has been achieved, which would achieve an effective field integral of 0.35 T m for a meter of separation in a neutron resonance spin echo spectrometer at the current device specifications. In bootstrap mode, the self-cancellation of Larmor phase aberrations can be achieved with the appropriate selection of the polarity of the gradient coils.

47 OTHER INSTRUMENTATION↗

DOE OSTI2020

Optimization of quasi-axisymmetric stellarators with varied elongation

An optimization study of Quasi-Axisymmetric (QA) stellarators with varied elongation has been carried out using the optimization code STELLOPT. The starting point of our optimization is a previously obtained QA stellarator with 3 field periods and aspect ratio of 6. Here, a series of QA stellarators are obtained at zero plasma beta with varied elongation value ranging from 2.5 to 3.7. Good quasi-symmetry is kept when the elongation value is reduced from the original value of 3.7. The rotational transform profile and aspect ratio are kept fixed. The plasma volume is ether kept fixed or varied linearly with elongation. Furthermore, finite beta QA stellarators are considered. The corresponding bootstrap currents are calculated using the kinetic code SFINCS. A series of kink-stable QA stellarators are obtained via optimization with varied plasma beta up to 5% and self-consistent bootstrap current. This work demonstrates that good QA stellarators with finite beta and varied elongation exist that are stable to external kink modes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Role of edge-localized neoclassical tearing modes in quiescent H-mode plasmas in the DIII-D tokamak

The edge-harmonic oscillations (EHOs) in standard quiescent H-mode (QH-mode) plasmas in DIII-D are consistent with edge-localized neoclassical tearing modes (NTMs) based on nonlinear two-fluid MHD simulations. Using kinetic equilibria constrained by edge profile measurements, the MHD simulations show that the n = 1 NTM and its harmonics can be destabilized at the pedestal top of QH-mode plasma by the edge bootstrap current. The simulations further show that the unstable NTMs can saturate either at small (<2% ψ N ) or large (>4% ψ N ) island width depending on the magnitude of the edge bootstrap current, where ψ N is the normalized radius in poloidal flux. The onset of the EHO also results in a prompt decrease in the pedestal width and height, consistent with simulation results for the onset of the NTM at the top of the QH-mode pedestal. This suggests that the avoidance of edge-localized modes (ELMs) in QH-mode can be attributed to the enhanced local transport induced by the NTM that is sufficient to prevent the expansion of the pedestal to an unstable width, analogous to the mechanism explored for ELM suppression by resonant magnetic perturbations. Nonlinear MHD simulations scanning the E × B frequency and the ratio of parallel and perpendicular thermal diffusivity (χ | /χ ⊥ ) at the pedestal top show that edge-localized NTMs are destabilized for conditions of high E × B frequency, high pedestal temperature, and low pedestal density, qualitatively consistent with experimental conditions required for observing the EHO.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

A Tutorial on Bayesian analysis of linear shock compression data

Gas gun and other shock compression experiments often produce shock wave velocity measurements that are linearly associated with particle velocity. Traditionally, this empirical relationship is quantified with a single Hugoniot curve that is estimated using least squares regression. However, for downstream modeling and simulation tasks, it is often more useful to have multiple Hugoniot curves in the pressure–volume plane that are consistent with the data. We employ Bayesian uncertainty quantification methods as a framework for propagating measurement uncertainty through to model parameters and predictions. Specifically, this Tutorial shows how to sample multiple Hugoniot curves in the pressure–volume plane that are consistent with the shock wave-particle velocity measurements in a two-step Bayesian approach. First, we obtain an analytical expression for the posterior distribution of the linear model parameters using Bayesian linear regression. Second, we propagate samples from the posterior distribution through the Rankine–Hugoniot equations to yield Hugoniot curves in the pressure–volume plane. The procedure is demonstrated with publicly available data on argon, copper, and nickel, and compared against bootstrapping and linear regression. The Bayesian procedure is shown to be interpretable, computationally inexpensive, and less sensitive than an alternative bootstrapping approach to the removal of the point in the copper dataset that has the largest particle velocity. As a Tutorial on Bayesian methodology for the shock compression community, we provide several derivations and explanations that make this paper self-contained, and make all code and data available at github.com/llnl/BALSCD.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Core-Pedestal Plasma Configurations in Advanced Tokamaks

Here, several configurations for the core and pedestal plasma are examined for a predefined tokamak design by implementing multiple heating/current drive (H/CD) sources to achieve an optimum configuration of high fusion power in a noninductive operation while maintaining an ideally magnetohydrodynamic (MHD) stable core plasma using the IPS-FASTRAN framework. IPS-FASTRAN is a component-based lightweight coupled simulation framework that is used to simulate magnetically confined plasma by integrating a set of high-fidelity codes to construct the plasma equilibrium (EFIT, TOQ, and CHEASE), calculate the turbulent heat and particle transport fluxes (TGLF), model various H/CD systems (TORIC, TORAY, GENRAY, and NUBEAM), model the pedestal pressure and width (EPED), and estimate the ideal MHD stability (DCON). The TGLF core transport model and EPED pedestal model are used to self-consistently predict plasma profiles consistent with ideal MHD stability and H/CD (and bootstrap) current sources. In order to evaluate the achievable and sustainable plasma beta, varying configurations are produced ranging from the no-wall stability to with-wall stability regimes, simultaneously subject to the self-consistent TGLF, EPED, and H/CD source profile predictions that optimize configuration performance. The pedestal density, plasma current, and total injected power are scanned to explore their impact on the target plasma configuration, fusion power, and confinement quality. A set of fully noninductive scenarios are achieved by employing ion-cyclotron, neutral beam injection, helicon, and lower-hybrid H/CDs to provide a broad profile for the total current drive in the core region for a predefined tokamak design. These noninductive scenarios are characterized by high fusion gain (Q ~ 4) and power (P fus ~ 600 MW), optimum confinement quality (H 98 ~ 1.1), and high bootstrap current fraction (f BS ~ 0.7) for Greenwald fraction below unity. The broad current profile configurations identified are stable to low-n kink modes either because the normalized pressure β N is below the no-wall limit or a wall is present.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Fast ion studies in the extended high-performance high β P plasma on EAST

Comprehending and optimizing fast ion behaviors is critical for the enhancement of performance in Experimental Advanced Superconducting Tokamak (EAST). This study explores the potential benefits of several factors that can improve the fast ion confinement. First, experiments show the change in the direction of the NBI2 from counter-I p to co-I p leads to a significant reduction in fast ion losses. TRANSP/NUBEAM simulation and tomography results based on fast-ion D-alpha measurements reveal that after the neutral beam injection (NBI) upgrade, the beam ion prompt loss is reduced by approximately 50%. Second, the upgraded ion cyclotron resonant frequency (ICRF) antenna at the N-port features twice the coupling resistance of the original antennas at EAST. This improved ICRF power coupling has enhanced the synergistic heating effect of NBI + ICRF, where the ICRF wave field accelerates beam ions at the harmonics. Experiments demonstrate that NBI + ICRF synergistic not only enhances plasma neutron yield and β P , but also accelerates beam ions to hundreds of keV. Further, the electron density and the neutral beam voltage have been optimized to reduce the fast ion slowing-down time and beam ion losses. Experimental and simulation results indicate that increasing the electron density reduces beam ion losses and enhances the bootstrap current fraction. While higher beam voltage results in a slight decrease in beam power absorption, it can increase the fraction of bootstrap current. With the understanding of these optimization of fast ion confinement, experiments have demonstrated fully non-inductive operation at high density (n e /n G ∼ 0.67, β P ∼ 3.1, β N ∼ 2.1, H 98,y2 ∼ 1.2) even without the support of co-I p beam NBI2. This investigation presents a potential regime to enhance fast ion confinement and extend performance in the high β P plasma for future experiments.

EAST tokamak↗

DOE OSTI2025

Understanding the formation of a low-pressure pedestal in the presence of a strong internal transport barrier in DIII-D high β p plasmas

As a promising scenario for fusion reactors, the high poloidal-beta (β P ) scenario is characterized by a strong large radius internal transport barrier (ITB), which significantly enhances the overall confinement quality and the bootstrap current fraction for fully non-inductive operation. It is frequently observed that in the presence of a strong ITB, the pedestal height is lower and is accompanied by small edge localized modes (ELMs), which further improves the compatibility of a high performance core with an edge solution. A mechanism for the formation of the low pedestal is proposed in this paper. It is found that the strong ITB creates an off-axis bootstrap current to clamp the local safety factor q, and thus the magnetic shear in the outer core/pedestal region is increased. Gyrokinetic simulations with the CGYRO code show that the higher magnetic shear brings the experimental profiles into the range where the growth rate of drift-wave instabilities and thus transport is higher, and therefore a lower pedestal gradient is expected. Here, the combination of low pedestal and high magnetic shear further enhances the turbulent transport across the whole pedestal, consistent with power balance analysis. Such a positive feedback mechanism ultimately results in a lower pressure pedestal as observed in experiments. Under such a low pedestal, linear simulations with BOUT++ predict the growth rates of peeling–ballooning modes to be lower across the whole toroidal mode number spectra, and the nonlinear BOUT++ simulation exhibits lower saturated fluctuation intensity as well, consistent with the experimentally observed lower ELM size.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Current ramp-up studies for NSTX-U in support of the development of non-inductive current ramp-up scenarios in STs

The Spherical Tokamak (ST) is a low-aspect-ratio tokamak projected to enable simultaneous operation at high normalized toroidal beta and high bootstrap current fraction. However, a compact ST reactor cannot be built with a full-sized solenoid. The remaining current required for sustained operation must be ramped up non-inductively. Sustained 100% non-inductive current drive at the MA level has not yet been demonstrated on an ST. Non-inductive current ramp-up is therefore a more challenging objective that the NSTX-U research program aims to address. In support of this development, free-boundary TRANSP simulations have been used to develop a current ramp-up scenario in which a significant fraction of the current is driven non-inductively by a combination of neutral-beam and bootstrap current drive. The resulting discharge has properties favorable for non-inductive current drive, such as a low normalized internal plasma inductance of approximately 0.45, a minimum safety factor exceeding 2, and a slightly reversed-shear q-profile. In these discharges, the plasma current ramps to 900 kA within 250 ms in an L-mode configuration, with varying electron density and density profiles. These discharges should be suitable for next-step NSTX-U experiments aimed at generating a robust L-mode fiducial scenario with low central solenoid flux consumption, which could be progressively improved as new NSTX-U experimental discharges are developed.

Raman, Roger [Univ. of Washington, Seattle, WA (Un↗

DOE OSTI2024

UV massive resonance from IR double copy consistency

From the perspective of effective field theory (EFT), Wilson coefficients of the low energy theory are determined by integrating out modes of the full ultraviolet (UV) theory. The spectrum can be in principle resummed if one has access to all available infrared (IR) coefficients at low energies. In this work we show that there exists a general class of consistent massive resonance double-copy (CMRDC) models where UV massive residues are reconstructed through double-copy consistency conditions between the IR Wilson coefficients of the full EFT expansion. Through a color-dual bootstrap, we find surprisingly that double-copy consistency alone introduces the kinematic factors of CMRDC models that soften high energy behavior by exponentiating color-dual contacts. This bootstrap suggests that our massive resonance paradigm is an inevitable consequence of the duality between color and kinematics, thereby providing a path towards emergent UV structure directly from the IR. We then demonstrate how CMRDC models can capture a spectrum of massive modes compatible with general multiplicity, and use Padé extrapolation to solve the inverse problem of identifying massive UV resonance from a small number of IR Wilson coefficients. Published by the American Physical Society 2024

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2020

Comparing traditional and Bayesian approaches to ecological meta‐analysis

Abstract Despite the wide application of meta‐analysis in ecology, some of the traditional methods used for meta‐analysis may not perform well given the type of data characteristic of ecological meta‐analyses. We reviewed published meta‐analyses on the ecological impacts of global climate change, evaluating the number of replicates used in the primary studies ( n i ) and the number of studies or records ( k ) that were aggregated to calculate a mean effect size. We used the results of the review in a simulation experiment to assess the performance of conventional frequentist and Bayesian meta‐analysis methods for estimating a mean effect size and its uncertainty interval. Our literature review showed that n i and k were highly variable, distributions were right‐skewed and were generally small (median n i = 5, median k = 44). Our simulations show that the choice of method for calculating uncertainty intervals was critical for obtaining appropriate coverage (close to the nominal value of 0.95). When k was low (<40), 95% coverage was achieved by a confidence interval (CI) based on the t distribution that uses an adjusted standard error (the Hartung–Knapp–Sidik–Jonkman, HKSJ), or by a Bayesian credible interval, whereas bootstrap or z distribution CIs had lower coverage. Despite the importance of the method to calculate the uncertainty interval, 39% of the meta‐analyses reviewed did not report the method used, and of the 61% that did, 94% used a potentially problematic method, which may be a consequence of software defaults. In general, for a simple random‐effects meta‐analysis, the performance of the best frequentist and Bayesian methods was similar for the same combinations of factors ( k and mean replication), though the Bayesian approach had higher than nominal (>95%) coverage for the mean effect when k was very low ( k < 15). Our literature review suggests that many meta‐analyses that used z distribution or bootstrapping CIs may have overestimated the statistical significance of their results when the number of studies was low; more appropriate methods need to be adopted in ecological meta‐analyses.

Pappalardo, Paula↗