Low-energy Expansion Research Articles

Quantum radiation reaction: Analytical approximations and obtaining the spectrum from moments

We derive analytical χ≪1 approximations for spin-dependent quantum radiation reaction for locally constant and locally monochromatic fields. We show how to factor out fast spin oscillations and obtain the degree of polarization in the plane orthogonal to the magnetic field from the Frobenius norm of the Mueller matrix. We show that spin effects lead to a transseries in χ, with powers χk, logarithms (lnχ)k, and oscillating terms, cos(…/χ) and sin(…/χ). In our approach we can obtain each moment, ⟨(kP)m⟩, of the light-front longitudinal momentum independently of the other moments and without considering the spectrum. We show how to obtain a low-energy expansion of the spectrum from the moments by treating m as a continuous, complex parameter and performing an inverse Mellin transform. We also show how to obtain the spectrum, without making a low-energy approximation, from a handful of moments using the principle of maximum entropy. Published by the American Physical Society 2024

Non-holomorphic modular forms from zeta generators

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Large-N integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM: when resurgence meets modularity

Exact expressions for certain integrated correlators of four half-BPS operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory with gauge group SU(N) have been recently obtained thanks to a beautiful interplay between supersymmetric localisation and modular invariance. The large-N expansion at fixed Yang-Mills coupling of such integrated correlators produces an asymptotic series of perturbative terms, holographically related to higher derivative interactions in the low energy expansion of the type IIB effective action, as well as exponentially suppressed corrections at large N, interpreted as contributions from coincident (p, q)-string world-sheet instantons. In this work we define a manifestly modular invariant Borel resummation of the perturbative large-N expansion of these integrated correlators, from which we extract the exact non-perturbative large-N sectors via resurgence analysis. Furthermore, we show that in the ’t Hooft limit such modular invariant non-perturbative completions reduce to known resurgent genus expansions. Finally, we clarify how the same non-perturbative data is encoded in the decomposition of the integrated correlators based on SL(2, ℤ) spectral theory.

Integral of depth zero to three basis of Modular Graph Functions

Modular Graph Functions (MGFs) are SL(2,ℤ)-invariant functions that emerge in the study of the low-energy expansion of the one-loop closed string amplitude. To find the string scattering amplitude, we must integrate MGFs over the moduli space of the torus. In this paper, we use the iterated integral representation of MGFs to establish a depth-dependent basis for them, where “depth” refers to the number of iterations in the integral. This basis has a suitable Laplace equation. We integrate this basis from depth zero to depth three over the fundamental domain of SL(2,ℤ) with a cut-off.

The AdS Veneziano amplitude at small curvature

We compute the AdS Veneziano amplitude for type IIB gluon scattering in AdS5 × S3 to all orders in α′ in a small curvature expansion. This is achieved by combining a dispersion relation in the dual 4dN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT with an ansatz for the amplitude as a worldsheet integral in terms of multiple polylogarithms. The first curvature correction is fully fixed in this way and satisfies consistency checks in the high energy limit, the low energy expansion as previously fixed using supersymmetric localisation, and for the energy of massive string operators, which we independently compute using a semiclassical expansion. We also combine localisation with this first curvature correction to fix the unprotected D4F4 correction to the amplitude at finite curvature.

Quantum radiation reaction spectrum of electrons in plane waves

In previous works we derived equations for the average momentum of high-energy electrons experiencing quantum radiation reaction in strong electromagnetic plane-wave background fields. In this paper we derive similar equations for the momentum spectrum. We formulate the equations in terms of the cumulative function and study the relation between the equations for the spectrum and the equations for the moments, analyze the structure of the low-energy expansions, and finally explain how our formulation is essentially in terms of a “Green’s function” which allows us to study the dynamics without choosing a specific initial wave packet (or particle-bunch distribution). Published by the American Physical Society 2024

Flattening of the EFT-hedron: supersymmetric positivity bounds and the search for string theory

We examine universal positivity constraints on 2 → 2 scattering in 4d planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory with higher-derivative corrections. We present numerical evidence that the convex region of allowed Wilson coefficients (the “EFT-hedron”) flattens completely along about one-third of its dimensions when an increasing number of constraints on the spectral density from crossing-symmetry are included. Our analysis relies on the formulation of the positivity constraints as a linear optimization problem, which we implement using two numerical solvers, SDPB and CPLEX. Motivated by the flattening, we propose a novel partially resummed low-energy expansion of the 2 → 2 amplitude. As part of the analysis, we provide additional evidence in favor of the conjecture [1] that the Veneziano amplitude is the only amplitude compatible with both S-matrix bootstrap constraints and string monodromy.

Absorptive effects and classical black hole scattering

We describe an approach to incorporating the physical effects of the absorption of energy by the event horizon of black holes in the scattering amplitudes based post-Minkowskian, point-particle effective description. Absorptive dynamics are incorporated in a model-independent way by coupling the usual point-particle description to an invisible sector of gapless internal degrees-of-freedom. The leading order dynamics of this sector are encoded in the low-energy expansion of a spectral density function obtained by matching an absorption cross section in the ultraviolet description. This information is then recycled using the scattering amplitudes based Kosower-Maybee-O’Connell in-in formalism to calculate the leading absorptive contribution to the impulse and change in rest mass of a Schwarzschild black hole scattering with a second compact body sourcing a massless scalar, electromagnetic or gravitational field. The results obtained are in complete agreement with previous worldline Schwinger-Keldysh calculations and provide an alternative on-shell scattering amplitudes approach to incorporating horizon absorption effects in the gravitational two-body problem.

Anisotropic Josephson Diode Effect in the Topological Hybrid Junctions with the Hexagonal Warping

Recently the diode effect in superconductivity became an active area of research. In particular, the three-dimensional topological insulators may be one of the most suitable materials to implement the superconducting diodes. It is common to consider only linear and quadratic terms of the topological insulator Hamiltonian in the low energy expansion. Typically the effect of the hexagonal warping is neglected. However, the hexagonal warping can be very significant in consideration of the transport properties of the TI materials, such as Bi2Se3 or Bi2Te3. In this theoretical work we present the study of the Josephson diode effect based on the topological insulator weak link. We address the question of the hexagonal warping influence on the Josephson diode effect. We argue that the warping term leads to the anisotropy of the Josephson diode effect.

Two string theory flavours of generalised Eisenstein series

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, τ, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half τ-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a Poincaré series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp τ → i∞. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as τ → 0 and the non-trivial zeros of the Riemann zeta function.

Scattering three closed strings off a Dp-brane in pure spinor formalism

We compute the disk amplitude of three closed strings in the pure spinor formalism. Among others, this amplitude probes tree-level gravitational interactions in the presence of Dp-branes. After disentangling holomorphic and anti-holomorphic closed string coordinates on the disk by means of introducing monodromy phases we find a compact expression for the disk amplitude of three closed strings in terms of open superstring six-point amplitudes. Furthermore, we provide the low-energy expansion (in the inverse string tension) of our amplitude and discuss some relevant Dp-brane couplings associated to it. Finally, we write down an expression for the general structure of the disk amplitude of any number nc of closed strings in terms of pure open string amplitudes involving 2nc open strings.

Wireless Technologies for Accelerating High-capacity Transmission, Low Energy, and Application-area Expansion

Wireless Technologies for Accelerating High-capacity Transmission, Low Energy, and Application-area Expansion

The Analytic Wavefunction

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic structure of wavefunction coefficients in Minkowski as a function of their kinematics. We introduce an off-shell wavefunction in terms of amputated time-ordered correlation functions and show that it is analytic in the complex energy plane except for possible singularities on the negative real axis. These singularities are determined to all loop orders by a simple energy-conservation condition. We confirm this picture by developing a Landau analysis of wavefunction loop integrals and corroborate our findings with several explicit calculations in scalar field theories. This analytic structure allows us to derive new UV/IR sum rules for the wavefunction that fix the coefficients in its low-energy expansion in terms of integrals of discontinuities in the corresponding UV-completion. In contrast to the analogous sum rules for scattering amplitudes, the wavefunction sum rules can also constrain total-derivative interactions. We explicitly verify these new relations at one-loop order in simple UV models of a light and a heavy scalar. Our results, which apply to both Lorentz invariant and boost-breaking theories, pave the way towards deriving wavefunction positivity bounds in flat and cosmological spacetimes.

Unitarity cuts of the worldsheet

We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten’s i\varepsiloniε prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in \alpha'α′. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from Cutkosky rules in field theory, but bypasses the need for performing laborious sums over the intermediate states. We use this representation to study various physical aspects of string amplitudes, including their behavior in the (s,t)(s,t) plane, exponential suppression, decay widths of massive strings, total cross section, and low-energy expansions. We find that planar annulus amplitudes exhibit a version of low-spin dominance: at any finite energy, only a finite number of low partial-wave spins give an appreciable contribution to the imaginary part.

Effective Field Theory islands from perturbative and nonperturbative four-graviton amplitudes

Theoretical data obtained from physically sensible field and string theory models suggest that gravitational Effective Field Theories (EFTs) live on islands that are tiny compared to current general bounds determined from unitarity, causality, crossing symmetry, and a good high-energy behavior. In this work, we present explicit perturbative and nonperturbative 2 → 2 graviton scattering amplitudes and their associated low-energy expansion in spacetime dimensions D ≥ 4 to support this notion. Our new results include a first example of gravity weakly coupled to a nonperturbative effective action. We show that, at energies below the mass of its nonperturbative matter, the D = 4, mathcal{N} = 1 supersymmetric field theory in the confined phase lies on the same islands identified using four-dimensional perturbative models based on string theory and minimally-coupled matter circulating a loop. Furthermore, we generalize the previous four-dimensional perturbative models based on string theory and minimally-coupled massive spin-0 and spin-1 states circulating in the loop to D dimensions. Remarkably, we again find that the low-energy EFT coefficients lie on small islands. These results offer a useful guide towards constraining possible extensions of Einstein gravity.

Thermodynamics and quark condensates of three-flavor QCD at low temperature

We use three-flavor chiral perturbation theory ($\chi$PT) to calculate the pressure, light and $s$-quark condensates of QCD in the confined phase at finite temperature to ${\cal O}(p^6)$ in the low-energy expansion. We also include electromagnetic effects to order $e^2$, where the electromagnetic coupling $e$ counts as order $p$. Our results for the pressure and the condensates suggest that $\chi$PT converges very well for temperatures up to approximately 150 MeV. We combine $\chi$PT and the Hadron Resonance Gas (HRG) model by adding heavier baryons and mesons. Our results are compared with lattice simulations an d the agreement is very good for temperatures below {170} MeV, in contrast to the results from $\chi$PT which agree with the lattice only up to $T\approx120$ MeV. Our value for the chiral crossover temperature is 160.1 MeV, which compares favorably to the lattice result of $157.3$ MeV.

Modular graph forms from equivariant iterated Eisenstein integrals

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown’s alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. In this work, we provide the first validations beyond depth one of Brown’s conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Apart from a variety of examples at depth two and three, we spell out the systematics of the dictionary and make certain elements of Brown’s construction fully explicit to all orders.

Dispersion relations, knots polynomials, and the q -deformed harmonic oscillator

We show that the crossing symmetric dispersion relation (CSDR) for 2-2 scattering leads to a fascinating connection with knot polynomials and q-deformed algebras. In particular, the dispersive kernel can be identified naturally in terms of the generating function for the Alexander polynomials corresponding to the torus knot $(2,2n+1)$ arising in knot theory. Certain linear combinations of the low energy expansion coefficients of the amplitude can be bounded in terms of knot invariants. Pion S-matrix bootstrap data respects the analytic bounds so obtained. We correlate the $q$-deformed harmonic oscillator with the CSDR-knot picture. In particular, the scattering amplitude can be thought of as a $q$-averaged thermal two point function involving the $q$-deformed harmonic oscillator. The low temperature expansion coefficients are precisely the $q$-averaged Alexander knot polynomials.

Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

We detail the properties of the Veneziano, Virasoro, and Coon amplitudes. These tree-level four-point scattering amplitudes may be written as infinite products with an infinite sequence of simple poles. Our approach for the Coon amplitude uses the mathematical theory of q-analysis. We interpret the Coon amplitude as a q-deformation of the Veneziano amplitude for all q ≥ 0 and discover a new transcendental structure in its low-energy expansion. We show that there is no analogous q-deformation of the Virasoro amplitude.

Aligning a Majorana fermion’s anapole moment with an external current through photon emission mediated by the fermion’s generalized polarizabilities

The sole static electromagnetic property of a spin-$\frac{1}{2}$ Majorana fermion is its anapole moment. Though they cannot couple to single real photons, these particles can interact with electric currents through virtual photons. If a Majorana fermion is immersed in a background current, there is an energy difference between the spin states of the fermion; the higher energy state has its anapole moment antialigned with the current. In this paper, we address the ability of a system of initially unpolarized Majorana fermions to achieve some degree of polarization relative to a static background current. In considering processes that allow the Majorana fermion's spin to flip to the lower-energy state, we focus upon two irreversible processes: the spontaneous emission of two real photons and the emission of a single real photon emitted in virtual Compton scattering. Both of these processes involve coupling to photons via the fermion's polarizaibilities. We compute the spin-flip transition rates for these processes using a low-energy expansion of the Hamiltonian and construct a toy model to showcase how these rates depend upon the underlying parameters within a model. Applying these ideas to a thermal dark matter (DM) model, we find that when the DM thermally decouples from the Standard Model plasma in the early universe, two-photon emission is negligible but partial polarization for the DM medium can proceed via virtual Compton scattering if sufficient currents exist.