Expansion Of Amplitude Research Articles

Cutkosky rules and 1-loop κ -deformed amplitudes

In this paper, we show that the Cutkosky cutting rules are still valid term by term in the expansion in powers of κ of the κ-deformed 1-loop correction to the propagator of the κ-deformed complex scalar field. We first present a general argument which relates each term in the expansion to a nondeformed amplitude containing additional propagators with mass M>κ. We then show the same thing more pragmatically, by reducing the singularity structure of the coefficients in the expansion of the κ-deformed amplitude, to the singularity structure of nondeformed loop amplitudes, by using algebraic and analytic identities. We will explicitly show this up to second order in 1/κ, but the technique can be generalized to higher orders in 1/κ. Both the abstract and the more direct approach easily generalize to different deformed theories. We will then compute the full imaginary part of the κ-deformed 1-loop correction to the propagator in a specific model, up to second order in the expansion in 1/κ, highlighting the usefulness of the approach for the phenomenology of deformed models. This explicitly confirms previous qualitative arguments concerning the behavior of the decay width of unstable particles in the considered model. 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.

Mesoscale modeling of deformations and defects in thin crystalline sheets

We present a mesoscale description of deformations and defects in thin, flexible sheets with crystalline order, tackling the interplay between in-plane elasticity, out-of-plane deformation, as well as dislocation nucleation and motion. Our approach is based on the Phase-Field Crystal (PFC) model, which describes the microscopic atomic density in crystals at diffusive timescales, naturally encoding elasticity and plasticity effects. In its amplitude expansion (APFC), a coarse-grained description of the mechanical properties of crystals is achieved. We introduce surface PFC and surface APFC models in a convenient height-function formulation encoding deformation in the normal direction. This framework is proven consistent with classical aspects of strain-induced buckling, defect nucleation on deformed surfaces, and out-of-plane relaxation near dislocations. In particular, we benchmark and discuss the results of numerical simulations by looking at the continuum limit for buckling under uniaxial compression and at evidence from microscopic models for deformation at defects and defect arrangements, demonstrating the scale-bridging capabilities of the proposed framework. Results concerning the interplay between lattice distortion at dislocations and out-of-plane deformation are also illustrated by looking at the annihilation of dislocation dipoles and systems hosting many dislocations. With the novel formulation proposed here, and its assessment with established approaches, we envision applications to multiscale investigations of crystalline order on deformable surfaces.

Spectral representation in Klein space: simplifying celestial leaf amplitudes

In this paper, we explore the spectral representation in Klein space, which is the split (2, 2) signature flat spacetime. The Klein space can be foliated into Lorentzian AdS3/ℤ slices, and its identity resolution has continuous and discrete parts. We calculate the identity resolution and the Plancherel measure in these slices. Using the foliation of Klein space into the slices, the identity resolution, and the Plancherel measure in each slice, we compute the spectral representation of the massive bulk-to-bulk propagator in Klein space. It can be expressed as the sum of the product of two massive (or tachyonic) conformal primary wavefunctions, with both continuous and discrete parts, and sharing a common boundary coordinate. An interesting point in Klein space is that, since the identity resolution has discrete and continuous parts, a new type of conformal primary wavefunction naturally arises for the massive (or tachyonic) case. For the conformal primary wavefunctions, both the discrete and continuous parts involve integrating over the common boundary coordinate and the real (or imaginary) mass. The conformal dimension is summed in the discrete part, whereas it is integrated in the continuous part. The spectral representation in Klein space is a computational tool to derive conformal block expansions for celestial amplitudes in Klein space and its building blocks, called celestial leaf amplitudes, by integrating the particle interaction vertex over a single slice of foliation.

Computing NMHV gravity amplitudes at infinity

In this note we show how the solutions to the scattering equations in the NMHV sector fully decompose into subsectors in the z → ∞ limit of a Risager deformation. Each subsector is characterized by the punctures that coalesce in the limit. This naturally decomposes the E(n − 3, 1) solutions into sets characterized by partitions of n − 3 elements so that exactly one subset has more than one element. We present analytic expressions for the leading order of the solutions in an expansion around infinite z for any n. We also give a simple algorithm for numerically computing arbitrarily high orders in the same expansion. As a consequence, one has the ability to compute Yang-Mills and gravity amplitudes purely from this expansion around infinity. Moreover, we present a new analytic computation of the residue at infinity of the n = 12 NMHV tree-level gravity amplitude which agrees with the results of Conde and Rajabi. In fact, we present the analytic form of the leading order in 1/z of the Cachazo-Skinner-Mason/CHY formula for graviton amplitudes for each subsector and to all multiplicity. As a byproduct of the all-order algorithm, one has access to the numerical value of the residue at infinity for any n and hence to the corrected CSW (or MHV) expansion for NMHV gravity amplitudes.

Mixed NNLO QCD × electroweak corrections to single-Z production in pole approximation: differential distributions and forward-backward asymmetry

Radiative corrections in pole approximation, which are based on the leading contribution in a systematic expansion of amplitudes about resonance poles, naturally decompose into factorizable corrections attributed to the production or decay of the resonance and non-factorizable corrections induced by soft photon (or gluon) exchange between those subprocesses. In this paper we complete an earlier calculation of mixed QCD × electroweak corrections of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({\\alpha }_{{\ ext{s}}}\\alpha \\right)$$\\end{document} to the neutral-current Drell-Yan cross section in pole approximation by including the previously neglected corrections that are solely related to the Z-boson production process. We present numerical results both for differential distributions and for the forward-backward asymmetry differential in the lepton-pair invariant mass, which is the key observable in the measurement of the effective weak mixing angle at the LHC. Carefully disentangling the various types of factorizable and non-factorizable corrections, we find (as expected in our earlier work) that the by far most important contribution at \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({\\alpha }_{{\ ext{s}}}\\alpha \\right)$$\\end{document} originates from the interplay of initial-state QCD corrections and electroweak final-state corrections.

Coupling allows robust mammalian redox circadian rhythms despite heterogeneity and noise

Circadian clocks are endogenous oscillators present in almost all cells that drive daily rhythms in physiology and behavior. There are two mechanisms that have been proposed to explain how circadian rhythms are generated in mammalian cells: through a transcription-translation feedback loop (TTFL) and based on oxidation/reduction reactions, both of which are intrinsically stochastic and heterogeneous at the single cell level. In order to explore the emerging properties of stochastic and heterogeneous redox oscillators, we simplify a recently developed kinetic model of redox oscillations to an amplitude-phase oscillator with ‘twist’ (period-amplitude correlation) and subject to Gaussian noise. We show that noise and heterogeneity alone lead to fast desynchronization, and that coupling between noisy oscillators can establish robust and synchronized rhythms with amplitude expansions and tuning of the period due to twist. Coupling a network of redox oscillators to a simple model of the TTFL also contributes to synchronization, large amplitudes and fine-tuning of the period for appropriate interaction strengths. These results provide insights into how the circadian clock compensates randomness from intracellular sources and highlight the importance of noise, heterogeneity and coupling in the context of circadian oscillators.

Gravity Amplitudes from Double Bonus Relations.

In this Letter, we derive new expressions for tree-level graviton amplitudes in N=8 supergravity from Britto-Cachazo-Feng-Witten (BCFW) recursion relations combined with new types of bonus relations. These bonus relations go beyond the famous 1/z^{2} behavior under a large BCFW shift and use knowledge about certain zeros of graviton amplitudes in collinear kinematics. This extra knowledge can be used in the context of global residue theorems by writing the amplitude in a special form using canonical building blocks. In the next-to-maximally-helicity-violating case, these building blocks are dressed one-loop leading singularities, the same objects that appear in the expansion of Yang-Mills amplitudes, where each term corresponds to an R invariant. Unlike other approaches, our formula is not an expansion in terms of cyclic objects and does not manifest color-kinematics duality but rather preserves the permutational symmetry of its building blocks. We also comment on the possible connection to Grassmannian geometry and give some nontrivial evidence of such structure for graviton amplitudes.

Subleading effects in soft-gluon emission at one-loop in massless QCD

We elucidate the structure of the next-to-leading-power soft-gluon expansion of arbitrary one-loop massless-QCD amplitudes. The expansion is given in terms of universal colour-, spin- and flavour-dependent operators acting on process-dependent gauge-invariant amplitudes. The result is proven using the method of expansion-by-regions and tested numerically on non-trivial processes with up to six partons. In principle, collinear-region contributions are expressed in terms of convolutions of universal jet operators and process-dependent amplitudes with two collinear partons. However, we evaluate these convolutions exactly for arbitrary processes. This is achieved by deriving an expression for the next-to-leading power expansion of tree-level amplitudes in the collinear limit, which is a novel result as well. Compared to previous studies, our analysis, besides being more general, yields simpler formulae that avoid derivatives of process-dependent amplitudes in the collinear limit.

Effects of Stress Amplitude and Cold Expansion Strengthening on the Fatigue Property of TC4 Titanium Alloy

Effects of Stress Amplitude and Cold Expansion Strengthening on the Fatigue Property of TC4 Titanium Alloy

Finite-amplitude instability of the convection in a porous vertical slab with horizontal heterogeneity in permeability

The finite-amplitude instability of the natural convection in a vertical porous slab filled with variable permeability porous medium is investigated analytically. The side walls of the slab are kept at different temperatures, and the permeability in the horizontal direction is assumed to be exponential heterogeneous models. Two-dimensional, finite-amplitude solutions for the thermal buoyant flow are obtained for Darcy–Rayleigh numbers close to the critical values by using the amplitude expansion method. The dependence of the fundamental mode, the distortion of the mean flow, and the second harmonic upon the variable permeability constant are discussed. By calculating the first Landau coefficient, the primary bifurcations in the vicinity of the neutral stability curves are identified. The results show that only supercritical bifurcations are found to occur, rather than subcritical instabilities. In terms of the well-known Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.

An Alternative Scheme for Pionless EFT: Neutron-Deuteron Scattering in the Doublet S-Wave

Using the effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the breakdown of unitarity in the three-body amplitude. We discuss the extension of our alternative subtraction scheme for including effective range corrections in pionless effective field theory for spinless bosons to nucleons. In particular, we consider the neutron-deuteron system in the doublet S-wave channel explicitly.

All-order celestial OPE from on-shell recursion

We determine tree level, all-order celestial operator product expansions (OPEs) of gluons and gravitons in the maximally helicity violating (MHV) sector. We start by obtaining the all-order collinear expansions of MHV amplitudes using the inverse soft recursion relations that they satisfy. These collinear expansions are recast as celestial OPE expansions in bases of momentum as well as boost eigenstates. This shows that inverse soft recursion for MHV amplitudes is dual to OPE recursion in celestial conformal field theory.

Formation of trapped vacuum bubbles during inflation, and consequences for PBH scenarios

A class of inflationary scenarios for primordial black hole (PBH) formation include a small barrier in the slope of the potential. There, the inflaton slows down, generating an enhancement of primordial perturbations. Moreover, the background solution overcomes the barrier at a very low speed, and large backward quantum fluctuations can prevent certain regions from overshooting the barrier. This leads to localized bubbles where the field remains “trapped” behind the barrier. In such models, therefore, we have two distinct channels for PBH production: the standard adiabatic density perturbation channel and the bubble channel. Here, we perform numerical simulations of bubble formation, addressing the issues of initial conditions, critical amplitude and bubble expansion. Further, we explore the scaling behaviour of the co-moving size of bubbles with the initial amplitude of the field fluctuation. We find that for small to moderate non-Gaussianity f NL ≲ 2.6, the threshold for the formation of vacuum bubbles agrees with previous analytical estimates [1] to 5% accuracy or so. We also show that the mass distribution for the two channels is different, leading to a slightly broader range of PBH masses when both contributions are comparable. The bubble channel is subdominant for small f NL, and becomes dominant for f NL ≳ 2.6. We find that the mass of PBHs in the bubble channel is determined by an adiabatic overdensity surrounding the bubble at the end of inflation. Remarkably, the profile of this overdensity turns out to be of type-II. This represents a first clear example showing that overdensities of type-II can be dominant in comparison with the standard type-I. We also comment on exponential tails and on the fact that in models with local type non-Gaussianity (such as the one considered here), the occurrence of alternative channels can easily be inferred from unitarity considerations.

Single Ionization of Helium by Protons of Various Energies in the Parabolic Quasi-Sturmians Approach

The parabolic quasi-Sturmian approach, recently introduced for the calculation of ion–atom ionizing collisions, is adapted and applied here to the single ionization of helium induced by an intermediate-energy proton impact. Within the method, the ionization amplitude is represented as the sum of the products of the basis amplitudes associated with the asymptotic behavior of the continuum states of the two noninteracting hydrogenic subsystems (e−,He+) and (p+,He+). The p−e interaction is treated as a perturbation in the Lippmann–Schwinger-type (LS) equation for the three-body system (e−,He+,p+). This LS equation is solved numerically using separable expansions for the proton–electron potential. We examine the convergence behavior of the transition amplitude expansion as the number of terms in the representation of the p−e interaction is increased and find that, for some kinematic regimes, the convergence is poor. This difficulty, which is absent for a higher proton energy impact, is solved by varying the momentum of the auxiliary proton plane wave introduced into the basis function. Fully differential cross-sections are calculated and compared with experimental data for 75 keV protons and the results obtained with the 3C model.

Entropy current and fluid-gravity duality in Gauss-Bonnet theory

Working within the approximation of small amplitude expansion, recently an entropy current has been constructed on the horizons of dynamical black hole solution in any higher derivative theory of gravity. In this note, we have dualized this horizon entropy current to a boundary entropy current in an asymptotically AdS black hole metric with a dual description in terms of dynamical fluids living on the AdS boundary. This boundary entropy current is constructed using a set of mapping functions relating each point on the horizon to a point on the boundary. We have applied our construction to black holes in Einstein-Gauss-Bonnet theory. We have seen that up to the first order in derivative expansion, Gauss-Bonnet terms do not add any extra corrections to fluid entropy as expected. However, at the second order in derivative expansion, the boundary current will non-trivially depend on how we choose our horizon to boundary map, which need not be expressible entirely in terms of fluid variables. So generically, the boundary entropy current generated by dualizing the horizon current will not admit a fluid dynamical description.

Spatio-temporal instabilities in viscoelastic channel flows: The centre mode

We study in this work the spatio-temporal characteristics of the centre-mode linear instability in viscoelastic channel flows of Oldroyd-B and FENE-P fluids. The linear complex Ginzburg–Landau equation derived using an amplitude expansion method is adopted to determine whether the flow is convectively unstable or absolutely unstable. Comparison of the obtained results with those from the conventional saddle-point searching method shows a favourable agreement. This demonstrates the good predictability of the linear complex Ginzburg–Landau equation which is more suitable for a parametric study in a large space than the conventional method. We found that the centre-mode instability in the viscoelastic channel flow is of a convective nature, i.e., the trailing edge of the wavepacket travels downstream. This could be attributed to the relatively high phase speed, low spreading rate and low growth rate of the instability, collectively preventing a disturbance from travelling upstream. Our results show that stronger polymer elasticity (a greater elastic number or larger polymer concentration) marginally affects the trailing-edge velocity of the centre mode. Decreasing the maximum polymer extensibility in the FENE-P model reduces the spreading rate of the wavepacket and makes the centre-mode disturbance more convective. Theoretical derivations at asymptotically high Reynolds numbers confirm the convective nature of the instabilities from numerical observations. Our results may be conducive to a better understanding of the spatio-temporal instability in viscoelastic experiments.

Amplitude phase‐field crystal model for the hexagonal close‐packed lattice

AbstractThe phase‐field crystal model allows the study of materials on atomic length and diffusive time scales. It accounts for elastic and plastic deformation in crystal lattices, including several processes such as growth, dislocation dynamics, and microstructure evolution. The amplitude expansion of the phase‐field crystal model (APFC) describes the atomic density by a small set of Fourier modes with slowly‐varying amplitudes characterizing lattice deformations. This approach allows for tackling large, three‐dimensional systems. However, it has been used mostly for modeling basic lattice symmetries. In this work, we present a coarse‐grained description of the hexagonal closed‐packed (HCP) lattice that supports lattice deformation and defects. It builds on recent developments of the APFC model and introduces specific modeling aspects for this crystal structure. After illustrating the general modeling framework, we show that the proposed approach allows for simulating relatively large three‐dimensional HCP systems hosting complex defect networks.

Non-factorizable virtual corrections to Higgs boson production in weak boson fusion beyond the eikonal approximation

Non-factorizable virtual corrections to Higgs boson production in weak boson fusion at next-to-next-to-leading order in QCD were estimated in the eikonal approximation [1]. This approximation corresponds to the expansion of relevant amplitudes around the forward limit. In this paper we compute the leading power correction to the eikonal limit and show that it is proportional to first power of the Higgs boson transverse momentum or the Higgs boson mass over partonic center-of-mass energy. Moreover, this correction can be significantly enhanced by the rapidity of the Higgs boson. For realistic weak boson fusion cuts, the next-to-eikonal correction reduces the estimate of non-factorizable contributions to fiducial cross section by (20) percent.

Constructing the general partial wave and renormalization in effective field theory

We construct the general partial wave amplitude basis for the $N\ensuremath{\rightarrow}M$ scattering, which consists of Poincar\'e Clebsch-Gordan coefficients, in a Lorentz invariant form given in terms of the spinor-helicity variables. The inner product of the Clebsch-Gordan coefficients is defined, which converts on-shell phase space integration into an algebraic problem. We also develop the technique of partial wave expansions of arbitrary amplitudes, including those with infrared divergence. These are applied to the computation of anomalous dimension matrix for general effective operators, where unitarity cuts for the loop amplitudes, with an arbitrary number of external particles, are obtained via partial wave expansion.