Hard thermal loops in the real-time formalism

We present a systematic discussion of Braaten and Pisarski's hard thermal loop (HTL) effective theory within the framework of the real-time (Schwinger-Keldysh) formalism. As is well known, the standard imaginary-time HTL amplitudes for hot gauge theory express the polarization of a medium made out of nonabelian charged point-particles; we show that the complete real-time HTL theory includes, in addition, a second set of amplitudes which account for Gaussian fluctuations in the charge distributions, but nothing else. We give a concise set of graphical rules which generate both set of functions, and discuss its relation to classical plasma physics.Comment: 14 pages, 6 figure

Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills

We derive a set of first-order differential equations obeyed by the S-matrix of planar maximally supersymmetric Yang-Mills theory. The equations, based on the Yangian symmetry of the theory, involve only finite and regulator-independent quantities and uniquely determine the all-loop S-matrix. When expanded in powers of the coupling they give derivatives of amplitudes as single integrals over lower-loop, higher-point amplitudes/Wilson loops. We outline a derivation for the equations using the Operator Product Expansion for Wilson loops. We apply them on a few examples at two- and three-loops, reproducing a recent result on the two-loop NMHV hexagon and fixing previously undermined coefficients in a recent Ansatz for the three-loop MHV hexagon. In addition, we consider amplitudes restricted to a two-dimensional subspace of Minkowski space and derive a particularly simple closed set of equations in that case

Heavy Quark Thermalization in Classical Lattice Gauge Theory: Lessons for Strongly-Coupled QCD

Thermalization of a heavy quark near rest is controlled by the correlator of two electric fields along a temporal Wilson line. We address this correlator within real-time, classical lattice Yang-Mills theory, and elaborate on the analogies that exist with the dynamics of hot QCD. In the weak-coupling limit, it can be shown analytically that the dynamics on the two sides are closely related to each other. For intermediate couplings, we carry out non-perturbative simulations within the classical theory, showing that the leading term in the weak-coupling expansion significantly underestimates the heavy quark thermalization rate. Our analytic and numerical results also yield a general understanding concerning the overall shape of the spectral function corresponding to the electric field correlator, which may be helpful in subsequent efforts to reconstruct it from Euclidean lattice Monte Carlo simulations.Comment: 22 pages. v2: a reference and clarifications added; published versio

Superpropagator and superconformal invariants

We construct a superpropagator in maximally supersymmetric Yang-Mills theory which is invariant off-shell under a chiral half of supersymmetries. Motivated by the duality with scattering amplitudes in this theory, we apply this superpropagator to supersymmeytric Wilson loop on polygonal contours. By performing explicit one-loop calculations we confirm the absence of anomalies and verify the duality between the object under study and NMHV amplitudes.Comment: 8 pages, 3 figure

A way to estimate the heavy quark thermalization rate from the lattice

The thermalization rate of a heavy quark is related to its momentum diffusion coefficient. Starting from a Kubo relation and using the framework of the heavy quark effective theory, we argue that in the large-mass limit the momentum diffusion coefficient can be defined through a certain Euclidean correlation function, involving color-electric fields along a Polyakov loop. Furthermore, carrying out a perturbative computation, we demonstrate that the spectral function corresponding to this correlator is relatively flat at small frequencies. Therefore, unlike in the case of several other transport coefficients, for which the narrowness of the transport peak makes analytic continuation from Euclidean lattice data susceptible to severe systematic uncertainties, it appears that the determination of the heavy quark thermalization rate could be relatively well under control.Comment: 17 pages. v2: clarifications and references added, published versio

Heavy quark diffusion in QCD and N=4 SYM at next-to-leading order

We present the full details of a calculation at next-to-leading order of the momentum diffusion coefficient of a heavy quark in a hot, weakly coupled, QCD plasma. Corrections arise at O(g_s); physically they represent interference between overlapping scatterings, as well as soft, electric scale (p ~ gT) gauge field physics, which we treat using the hard thermal loop (HTL) effective theory. In 3-color, 3-flavor QCD, the momentum diffusion constant of a fundamental representation heavy quark at NLO is kappa = (16\pi/3) alpha_s^2 T^3 (log(1/g) + 0.07428 + 1.9026 g). We extend the computation to a heavy fundamental representation ``probe'' quark in large N_c, N=4 Super Yang-Mills theory, where the result is kappa^{SYM}= (lambda^2 T^3)(6\pi) (log(1/\sqrt{\lambda}) + 0.4304 + 0.8010 \sqrt{lambda}) (where lambda=g_s^2 N_c is the t'Hooft coupling). In the absence of some resummation technique, the convergence of perturbation theory is poor.Comment: 40 pages, 14 figure

What can be measured asymptotically?

We consider asymptotic observables in quantum field theories in which the S-matrix makes sense. We argue that in addition to scattering amplitudes, a whole compendium of inclusive observables exists where the time-ordering is relaxed. These include expectation values of electromagnetic or gravitational radiation fields as well as out-of-time-order amplitudes. We explain how to calculate them in two ways: by relating them to amplitudes and products of amplitudes, and by using a generalization of the LSZ reduction formula. As an application, we discuss one-loop master integrals contributing to gravitational radiation in the post-Minkowski expansion, emphasizing the role of classical cut contributions and highlighting the different infrared physics of in-in observables.Comment: 67 pages, typos correcte

Crossing beyond scattering amplitudes

We find that different asymptotic measurements in quantum field theory can be related to one another through new versions of crossing symmetry. Assuming analyticity, we conjecture generalized crossing relations for multi-particle processes and the corresponding paths of analytic continuation. We prove them to all multiplicity at tree-level in quantum field theory and string theory. We illustrate how to practically perform analytic continuations on loop-level examples using different methods, including unitarity cuts and differential equations. We study the extent to which anomalous thresholds away from the usual physical region can cause an analytic obstruction to crossing when massless particles are involved. In an appendix, we review and streamline historical proofs of four-particle crossing symmetry in gapped theories.Comment: 108 page

Colour-electric spectral function at next-to-leading order

The spectral function related to the correlator of two colour-electric fields along a Polyakov loop determines the momentum diffusion coefficient of a heavy quark near rest with respect to a heat bath. We compute this spectral function at next-to-leading order, O(alpha_s^2), in the weak-coupling expansion. The high-frequency part of our result (omega >> T), which is shown to be temperature-independent, is accurately determined thanks to asymptotic freedom; the low-frequency part of our result (omega << T), in which Hard Thermal Loop resummation is needed in order to cure infrared divergences, agrees with a previously determined expression. Our result may help to calibrate the overall normalization of a lattice-extracted spectral function in a perturbative frequency domain T << omega << 1/a, paving the way for a non-perturbative estimate of the momentum diffusion coefficient at omega -> 0. We also evaluate the colour-electric Euclidean correlator, which could be directly compared with lattice simulations. As an aside we determine the Euclidean correlator in the lattice strong-coupling expansion, showing that through a limiting procedure it can in principle be defined also in the confined phase of pure Yang-Mills theory, even if a practical measurement could be very noisy there.Comment: 38 page