Auxiliary Field Formalism Research Articles

Off-lightcone Wilson-line operators in gradient flow

Off-lightcone Wilson-line operators are constructed using local operators connected by time-like or space-like Wilson lines, which ensure gauge invariance. Off-lightcone Wilson-line operators have broad applications in various contexts. For instance, space-like Wilson-line operators play a crucial role in determining quasi-distribution functions (quasi-PDFs), while time-like Wilson-line operators are essential for understanding quarkonium decay and production within the potential non-relativistic QCD (pNRQCD) framework. In this work, we establish a systematic approach for calculating the matching from the gradient-flow scheme to the MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{MS}} $$\\end{document} scheme in the limit of small flow time for off-lightcone Wilson-line operators. By employing the one-dimensional auxiliary-field formalism, we simplify the matching procedure, reducing it to the matching of local current operators. We provide one-loop level matching coefficients for these local current operators. For the case of hadronic matrix element related to the quark quasi-PDFs, we show at one-loop level that the finite flow time effect is very small as long as the flow radius is smaller than the physical distance z, which is usually satisfied in lattice gradient flow computations. Applications include lattice gradient flow computations of quark/gluon quasi-PDFs, gluonic correlators related to quarkonium decay and production in pNRQCD, and spin-dependent potentials in terms of chromoelectric and chromomagnetic field insertions into a Wilson loop.

Weak invariants in dissipative systems: action principle and Noether charge for kinetic theory.

In non-equilibrium classical thermostatistics, the state of a system may be described by not only dynamical/thermodynamical variables but also a kinetic distribution function. This 'double structure' bears some analogy with that in quantum thermodynamics, where both dynamical variables and the Hilbert space are involved. Recently, the concept of weak invariants has repeatedly been discussed in the context of quantum thermodynamics. A weak invariant is defined in such a way that its value changes in time but its expectation value is conserved under time evolution prescribed by a kinetic equation. Here, a new aspect of a weak invariant is revealed for the classical Fokker-Planck equation as an example of classical kinetic equations. The auxiliary field formalism is applied to the construction of the action for the kinetic equation. Then, it is shown that the auxiliary field is a weak invariant and is the Noether charge. The action is invariant under the transformation generated by the weak invariant. The result may shed light on possible roles of the symmetry principle in the kinetic descriptions of non-equilibrium systems. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Ricci cubic gravity in d dimensions, gravitons and SAdS/Lifshitz black holes

A special class of higher curvature theories of gravity, Ricci cubic gravity (RCG), in general d dimensional space-time has been investigated in this paper. We have used two different approaches, the linearized equations of motion and the auxiliary field formalism to study the massive and massless graviton propagating modes of the AdS background. Using the auxiliary field formalism, we have found the renormalized boundary stress tensor to compute the mass of the Schwarzschild–AdS and Lifshitz black holes in RCG theory.

Variational principle for fractional kinetics and the Lévy Ansatz

A variational principle is developed for fractional kinetics based on the auxiliary-field formalism. It is applied to the Fokker-Planck equation with spatiotemporal fractionality, and a variational solution is obtained with the help of the Lévy Ansatz. It is shown how the whole range from subdiffusion to superdiffusion is realized by the variational solution as a competing effect between the long waiting time and the long jump. The motion of the center of the probability distribution is also analyzed in the case of a periodic drift.

Two-nucleon systems in a finite volume: Quantization conditions

The quantization condition for interacting energy eigenvalues of the two-nucleon system in a finite cubic volume is derived in connection to the nucleon-nucleon scattering amplitudes. This condition is derived using an auxiliary (dimer) field formalism that is generalized to arbitrary partial waves in the context of nonrelativistic effective field theory. The quantization condition presented gives access to the scattering parameters of the two-nucleon systems with arbitrary parity, spin, isospin, angular momentum and center-of-mass motion, from a lattice QCD calculation of the energy eigenvalues. In particular, as it includes all noncentral interactions, such as the two-nucleon tensor force, it makes explicit the dependence of the mixing parameters of nucleon-nucleon systems calculated from lattice QCD when there is a physical mixing among different partial waves, e.g. $S\mathrm{\text{\ensuremath{-}}}D$ partial wave mixing in the deuteron channel. We provide explicit relations among scattering parameters and their corresponding point group symmetry class eigenenergies with orbital angular momentum $l\ensuremath{\le}3$, and for center-of-mass boost vectors of the form $\frac{2\ensuremath{\pi}}{L}(2{n}_{1},2{n}_{2},2{n}_{3})$, $\frac{2\ensuremath{\pi}}{L}(2{n}_{1},2{n}_{2},2{n}_{3}+1)$ and $\frac{2\ensuremath{\pi}}{L}(2{n}_{1}+1,2{n}_{2}+1,2{n}_{3})$. $L$ denotes the special extent of the cubic volume and ${n}_{1}$, ${n}_{2}$, ${n}_{3}$ are integers. Our results are valid below inelastic thresholds up to exponential volume corrections that are governed by the pion mass.

AdS/BCFT correspondence for higher curvature gravity: an example

We consider the effects of higher curvature terms on a holographic dual description of boundary conformal field theory. Specifically, we consider three-dimensional gravity with a specific combination of Ricci tensor square and curvature scalar square, so called, new massive gravity. We show that a boundary entropy and an entanglement entropy are given by similar expression with those of the Einstein gravity case when we introduce an {\it effective} Newton's constant and an {\it effective} cosmological constant. We also show that the holographic g-theorem still holds in this extension, and we give some comments about the central charge dependence of boundary entropy in the holographic construction. In the same way, we consider new type black holes and comment on the boundary profile. Moreover, we reproduce these results through auxiliary field formalism in this specific higher curvature gravity.

Auxiliary field formalism for dilute fermionic atom gases with tunable interactions

We develop the auxiliary field formalism corresponding to a dilute system of spin-1/2 fermions. This theory represents the Fermi counterpart of the BEC theory developed recently by F. Cooper et al. [Phys. Rev. Lett. 105, 240402 (2010)] to describe a dilute gas of Bose particles. Assuming tunable interactions, this formalism is appropriate for the study of the crossover from the regime of Bardeen-Cooper-Schriffer (BCS) pairing to the regime of Bose-Einstein condensation (BEC) in ultracold fermionic atom gases. We show that when applied to the Fermi case at zero temperature, the leading-order auxiliary field (LOAF) approximation gives the same equations as those obtained in the standard BCS variational picture. At finite temperature, LOAF leads to the theory discussed by by Sa de Melo, Randeria, and Engelbrecht [Phys. Rev. Lett. 71, 3202(1993); Phys. Rev. B 55, 15153(1997)]. As such, LOAF provides a unified framework to study the interacting Fermi gas. The mean-field results discussed here can be systematically improved upon by calculating the one-particle irreducible (1-PI) action corrections, order by order.

Global symmetries and renormalizability of Lee-Wick theories

In this paper we discuss the global symmetries and the renormalizability of Lee-Wick (LW) scalar QED. In particular, in the ``auxiliary-field'' formalism we identify softly broken $SO(1,1)$ global symmetries of the theory. We introduce $SO(1,1)$ invariant gauge-fixing conditions that allow us to show in the auxiliary-field formalism directly that the number of superficially divergent amplitudes in a LW Abelian gauge theory is finite. To illustrate the renormalizability of the theory, we explicitly carry out the one-loop renormalization program in LW scalar QED and demonstrate how the counterterms required are constrained by the joint conditions of gauge and $SO(1,1)$ invariance. We also compute the one-loop beta functions in LW scalar QED and contrast them with those of ordinary scalar QED.

Renormalization of the O(N) model in the1/Nexpansion in the auxiliary field formalism

We study the renormalization of the O(N) model using the auxiliary field formalism (Hubbard-Stratonovich transformation) in the 1/N expansion at finite temperature. We provide the general strategy of renormalization for arbitrary order and make a calculation up to next-to-leading order. We show that renormalization is possible for any values of the condensates, prove the temperature independence of the counterterms, and determine the cutoff dependence of the first nontrivial counterterm parts.

Accuracy of auxiliary field approach for baryons

We provide a check of the accuracy of the auxiliary field formalism used to derive the Effective Hamiltonian for baryons in the Field Correlator Method. To this end we compare the solutions for the Effective Hamiltonian with those obtained from the solution of the Salpeter equation. Comparing these results gives a first estimate of the systematic uncertainty due to the use of the auxiliary field formalism for baryons.

Quantum-classical correspondence principles for locally nonequilibrium driven systems

Many of the core concepts and (especially field-theoretic) tools of statistical mechanics have developed within the context of thermodynamic equilibrium, where state variables are all taken to be charges, meaning that their values are inherently preserved under reversal of the direction of time. A principle concern of nonequilibrium statistical mechanics is to understand the emergence and stability of currents, quantities whose values change sign under time reversal. Whereas the correspondence between classical charge-valued state variables and their underlying statistical or quantum ensembles is quite well understood, the study of currents away from equilibrium has been more fragmentary, with classical descriptions relying on the asymmetric auxiliary-field formalism of Martin, Siggia, and Rose (and often restricted to the Markovian assumption of Doi and Peliti), while quantum descriptions employ a symmetric two-field formalism introduced by Schwinger and further clarified by Keldysh. In this paper we demonstrate that for quantum ensembles in which superposition is not violated by very strong conditions of decoherence, there is a large natural generalization of the principles and tools of equilibrium, which not only admits but requires the introduction of current-valued state variables. For these systems, not only do Martin-Siggia-Rose (MSR) and Schwinger-Keldysh (SK) field methods both exist, in some cases they provide inequivalent classical and quantum descriptions of identical ensembles. With these systems for examples, we can both study the correspondence between classical and quantum descriptions of currents, and also clarify the nature of the mapping between the structurally homologous but interpretationally different MSR and SK formalisms.

Auxiliary fields and hadron dynamics

The relations existing between the auxiliary field (einbein field) formalism and the spinless Salpeter equation are studied in the case of two particles with the same mass, interacting via a confining potential. The problem of nonorthogonality for radial excited states in the auxiliary field formalism is discussed and found to be noncrucial. It is shown that the classical equations of motion of the rotating string model, derived from the QCD Lagrangian, reduce exactly to the classical equations of motion of the phenomenological semirelativistic flux tube model, provided all auxiliary fields are eliminated correctly from the rotating string Hamiltonian.

On a New Formulation of the Real-Time Propagator

The development of theoretical tools for the study of dynamical phenomena of many-particle systems on the quantum level is a fundamental challenge since many decades. A lot of efforts have been invested on Feynman's path integral approach, however, no computationally tractable method for investigating realistic systems could be developed up to now. In this paper we propose an alternative representation of the real-time many-body evolution operator formulated within the framework of the auxiliary field formalism. Our goal is to derive a new auxiliary field functional integral representation, in which the large oscillations of the functional integrand are reduced, in order to render the auxiliary field approach more attractive for real-time computation. This objective is attained using a generalized version of the method of Gaussian equivalent representation of Efimov and Ganbold [Phys. Stat. Sol. 168 (1991) 165], which eliminates the low-order fluctuations of the auxiliary field from the interaction functional.

Far-from-equilibrium dynamics with broken symmetries from the1/Nexpansion of the 2PI effective action

We derive the nonequilibrium real-time evolution of an $O(N)$-invariant scalar quantum field theory in the presence of a nonvanishing expectation value of the quantum field. Using a systematic $1/N$ expansion of the 2PI effective action to next-to-leading order, we obtain nonperturbative evolution equations which include scattering and memory effects. The equivalence of the direct method, which requires the resummation of an infinite number of skeleton diagrams, with the auxiliary-field formalism, which involves only one diagram at next-to-leading order, is shown.

Schwinger-Dyson approach to nonequilibrium classical field theory

In this paper we discuss a Schwinger-Dyson (SD) approach for determining the time evolution of the unequal time correlation functions of a nonequilibrium classical field theory, where the classical system is described by an initial density matrix at time $t=0.$ We focus on $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in $1+1$ space-time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation (BVA), is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for $1/N$ expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\ensuremath{\varphi}$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.

Finite-temperature calculations of correlated electron systems

We describe a quantum Monte Carlo approach to study finite-temperature properties of correlated fermion systems. The algorithm retains advantages of the standard auxiliary-field formalism, which is widely applied in areas spanning condensed-matter, high-energy, and nuclear physics, while eliminating the sign decay that is the signature of the fermion sign problem. It allows simulations at larger system sizes and lower temperatures. The sign decay is eliminated by imposition of an approximate boundary condition in auxiliary-field space. We discuss the analytical origin of the boundary condition and illustrate its implementation by highlighting the similarities to a diffusion process with absorption.

Supersymmetry and loop space geometry

We present a conceptually new interpretation of Poincaré supersymmetric theories in terms of loop space symplectic geometry. We argue, that the fermionic part of a supersymmetric action can be viewed as a (pre) symplectic two form in the loop space of bosonic fields, and the bosonic part of the action can be constructed by contracting the pertinent loop space symplectic potential with a model independent vector field. Our approach entails a novel auxiliary field formalism. It also opens a new point of view to a large class of problems, including the mechanism of supersymmetry breaking, structure of topological field theories and even aspects of quantum integrability.

Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories

AbstractWe give a review of the renormalization and short distance properties of path ordered phase factors in nonabelian gauge field theories. It includes nonlocal gauge invariant meson, baryon and gluonium operators constructed with the help of such phase factors. Furthermore, the renormalization properties of functional derivatives of phase factors as they are needed in dynamical equations are considered. The discussion is based on an one dimensional auxiliary field formalism which enables the application of the usual language of local Green's functions.