Feynman Diagrams Research Articles

The Sommerfeld theory with the Uehling potential

The Uehling potential is the one-loop radiative correction to the photon propagator. The correction can be graphically represented by the Feynman diagram of the second order. The physical meaning of this diagram is the process Y-> (e- + e+) -> Y , where Y is denotation for photon, and e-, e+ is the electron-positron pair. It means that photon can exist in the intermediate state with e+, e- being virtual particles. Then, the Coulomb potential with radiative corrections can be inserted in the Sommerfeld quantum equation in order to find the Sommerfeld energy.

Bootstrapping conformal defect operators on a line

We study a conformal field theory with cubic anisotropic symmetry in presence of a line defect. We compute the correlators of the low lying defect operators using Feynman diagrams and derive explicit expressions for the two, three and four point defect correlators at the cubic fixed point in 4 – ϵ dimensions to O(ϵ). We also compute the defect g-function for this setup and demonstrate that this is in agreement with the g-theorem, which states that the g-function is monotonic under the renormalisation group flow along the defect. Next, we focus on conformal bootstrap techniques to determine the CFT data associated with the defect operators, which is the main objective of the paper. We utilize the framework of crossing symmetric Polyakov bootstrap and compute the averaged CFT data to O(ϵ) up to a finite number of ambiguities. We comment on unmixing the CFT data for the double trace operators at O(ϵ) and use this to compute the O(ϵ2) data. Finally, we study these defect correlators non-perturbatively using numerical methods and isolate them near the free theory limit close to four dimensions.

The Higgs mechanism with diagrams: A didactic approach

We present a paedagogical treatment of the electroweak Higgs mechanism based solely on Feynman diagrams and S-matrix elements, without recourse to (gauge) symmetry arguments. Throughout, the emphasis is on Feynman rules and the Schwinger-Dyson equations; it is pointed out that particular care is needed in the treatment of tadpole diagrams and their symmetry factors.

Torsional regularization of self-energy and bare mass of electron

Abstract In the presence of spacetime torsion, the momentum components do not commute; therefore, in quantum field theory, summation over the momentum eigenvalues will replace integration over the momentum. In the Einstein–Cartan theory of gravity, in which torsion is coupled to spin, the separation between the eigenvalues increases with the magnitude of the momentum. Consequently, this replacement regularizes divergent integrals in Feynman diagrams with loops by turning them into convergent sums. In this article, we apply torsional regularization to the self-energy of a charged lepton in quantum electrodynamics. We show that torsion eliminates the ultraviolet divergence of the standard self-energy. We also show that the infrared divergence is absent. In the end, we calculate the finite bare masses of the electron, muon, and tau lepton: 0.4329 MeV , 90.95 MeV , and 1543 MeV , respectively. These values constitute about 85% of the observed, re-normalized masses.

Soft limits of gluon and graviton correlators in Anti-de Sitter space

We derive formulae for the soft limit of tree-level gluon and graviton correlators in Anti-de Sitter space, which arise from Feynman diagrams encoding the Weinberg soft theorems in flat space. Other types of diagrams can also contribute to the soft limit at leading order in the soft momentum, but have a different pole structure. We derive these results at four points using explicit formulae recently obtained from the cosmological bootstrap and double copy, and extend them to any multiplicity using bootstrap techniques in Mellin-momentum space.

Feynman diagrams: visualization of phenomena and diagrammatic representation

AbstractI will argue that the development of Feynman diagrams came from the physicist’s capacity of visualizing phenomena and that such visualization-skill contributed to the forming of a narrative explanation in the sense of Wise (2011) and Morgan (2001). The second part of the paper explores the extent to which Feynman diagrams can be considered as weak representations of quantum phenomena. I will review some of the most common arguments in support of the instrumentalist view and I will suggest that a form of weak representation that does not imply ontological commitment can be applied to the diagrams. Such a form of weak representation will be characterized as non-denotative, intentional, and as conveying a physical interpretation through narrative explanations.

Low and moderate x gluon contribution to exclusive Compton scattering processes

Abstract We revisit the high energy semi-classical description of the exclusive processes DVCS, TCS, and Double DVCS by explicitly keeping track of the Feynman x dependence in both the hard and the hadronic matrix elements. This is achieved by a modification of the standard shock wave approximation to derive the effective Feynman rules, which leads to a generic expression on which we then perform a partial twist expansion to get rid of quantities suppressed by the proper physical scales. We obtain a compact factorized master formula that can be used to investigate the Bjorken limit at leading twist. In particular, we recover the full one-loop result in the collinear limit for pure gluon exchange with the target. Finally, we discuss the subtleties in taking the simultaneous collinear and small x limit.

Supergluon scattering in AdS: constructibility, spinning amplitudes, and new structures

We elaborate on a new recursive method proposed in [1] for computing tree-level n-point supergluon amplitudes as well as those with one gluon, i.e. spinning amplitudes, in AdS5 × S3. We present an improved proof for the so-called “constructibility” of supergluon and spinning amplitudes based on their factorizations and flat-space limit, which allows us to determine these amplitudes in Mellin space to all n. We present explicit and remarkably simple expressions for up to n = 7 supergluon amplitudes and n = 6 spinning amplitudes, which can be viewed as AdS generalizations of the scalar-scaffolded gluon amplitudes proposed recently. We then reveal a series of hidden structures of these AdS amplitudes including (1). an understanding of general pole structures especially the precise truncation on descendent poles (2). a derivation of simple “Feynman rules” for the all-n amplitudes with the simplest R-symmetry structures, and (3). certain universal behavior analogous to the soft/collinear limit of flat-space amplitudes.

Tunnelling amplitudes through localised external potentials from Feynman diagram summation

AbstractCurrently there is no general theory of quantum tunnelling of a particle through a potential barrier which is compatible with QFT. We present a complete calculation of tunnelling amplitudes for a scalar field for some simple potentials using quantum field-theoretic methods. Using the perturbative S-matrix formalism, starting with the Klein–Gordon Lagrangian, we show that an infinite summation of Feynman diagrams can recover tunnelling amplitudes consistent with relativistic quantum mechanics. While this work does not include many-particle effects arising from a fully quantised QFT, it is necessary to investigate QFT corrections to tunnelling amplitudes.

Multiway Junction Conditions for Spacetimes with Multiple Boundaries.

We explore the construction of gluing together an arbitrary number of spacetimes along a common interface. This provides a general tool to construct new gravitational backgrounds with multiple boundaries. We derive junction conditions across the interface, in both Einstein gravity and dilaton gravity, using two methods (1)by developing a geometrical technique of introducing an auxiliary reverse extension manifold and (2)by the variational principle. Our junction condition works for both spacelike and timelike interfaces. We illustrate applications of the junction conditions in Jackiw-Teitelboim gravity and analyze their constraints on the allowed dilaton profile. We further comment on the geometry of a web of spacetime in the presence of multiple interfaces and its resemblance to Feynman diagrams.

Quantum corrections to tunnelling amplitudes of neutral scalar fields

Though theoretical treatments of quantum tunnelling within single-particle quantum mechanics are well-established, at present, there is no quantum field-theoretic description (QFT) of tunnelling. Due to the single-particle nature of quantum mechanics, many-particle effects arising from quantum field theory are not accounted for. Such many-particle effects, including pair-production, have proved to be essential in resolving the Klein-paradox. This work seeks to determine how quantum corrections affect the tunnelling probability through an external field. We investigate a massive neutral scalar field, which interacts with an external field in accordance with relativistic quantum mechanics. To consider QFT corrections, we include another massive quantised neutral scalar field coupling to the original via a cubic interaction. This study formulates an all-order recursive expression for the loop-corrected scalar propagator, which contains only the class of vertex-corrected Feynman diagrams. This equation applies for general external potentials. Though there is no closed-form analytic solution, we also demonstrate how to approximate the QFT corrections if a perturbative coupling to the quantised field is assumed.

Identifying regions in wide-angle scattering via graph-theoretical approaches

The method of regions, which provides a systematic approach for computing Feynman integrals involving multiple kinematic scales, proposes that a Feynman integral can be approximated and even reproduced by summing over integrals expanded in certain regions. A modern perspective of the method of regions considers any given Feynman integral as a specific Newton polytope, defined as the convex hull of the points associated with Symanzik polynomials. The regions then correspond one-to-one with the lower facets of this polytope.As Symanzik polynomials correspond to the spanning trees and spanning 2-trees of the Feynman graph, a graph-theoretical study of these polynomials may allow us to identify the complete set of regions for a given expansion. In this work, our primary focus is on three specific expansions: the on-shell expansion of generic wide-angle scattering, the soft expansion of generic wide-angle scattering, and the mass expansion of heavy-to-light decay. For each of these expansions, we employ graph-theoretical approaches to derive the generic forms of the regions involved in the method of regions. The results, applicable to all orders, offer insights that can be leveraged to investigate various aspects of scattering amplitudes.

Staying on-shell: manifest properties and reformulations in particle physics

The empirical success of particle physics rests largely on an approximation method: perturbation theory. Yet even within perturbative quantum field theory, there are a variety of different formulations. This variety teaches us that reformulating approximation methods can provide a tremendous source of progress in science. Along with enabling the solution of otherwise intractable problems, reformulations clarify what we need to know to obtain solutions, which can in turn make previously hidden properties manifest. To develop these lessons, I compare and contrast three compatible formulations of perturbative QFT: (i) elementary perturbation theory, (ii) the method of Feynman diagrams, and (iii) a recent reformulation known as on-shell recursion. I propose and defend a novel account of what it means to ‘make a property manifest,’ based on the inferences that a formulation warrants.

W -boson pair production at lepton colliders in the Feynman-diagram gauge

We calculate helicity amplitudes for e−e+→W−W+ analytically in the Feynman-diagram (FD) gauge. We show that, unlike in the unitary gauge, there is no energy growth of the individual Feynman amplitudes for the longitudinally polarized bosons, and the contributions from the associated Goldstone bosons are manifest even without taking the high-energy limit. We also find that the physical distributions can be interpreted by the individual amplitudes in the FD gauge. Published by the American Physical Society 2024

Combinatorial summation of Feynman diagrams

Feynman’s diagrammatic series is a common language for a formally exact theoretical description of systems of infinitely-many interacting quantum particles, as well as a foundation for precision computational techniques. Here we introduce a universal framework for efficient summation of connected or skeleton Feynman diagrams for generic quantum many-body systems. It is based on an explicit combinatorial construction of the sum of the integrands by dynamic programming, at a computational cost that can be made only exponential in the diagram order on a classical computer and potentially polynomial on a quantum computer. We illustrate the technique by an unbiased diagrammatic Monte Carlo calculation of the equation of state of the 2D SU(N) Hubbard model in an experimentally relevant regime, which has remained challenging for state-of-the-art numerical methods.

Analytic NNLO QCD corrections to top quark pair production in electron-positron collisions

We present the analytic total cross section of top quark pair production in electron-positron annihilation at next-to-next-to-leading order (NNLO) in Quantum Chromodynamics (QCD). By utilizing the optical theorem, the NNLO corrections are related to the imaginary parts of three-loop self-energy Feynman diagrams, of which the master integrals are calculated with canonical differential equations. The analytic results for the NNLO corrections are expressed in terms of multiple polylogarithms as well as elliptic functions. We discuss the asymptotic expansions near the threshold and in the high energy limit in detail. Numerical results are provided for the total cross section of top quark pair production at future lepton colliders.

Classifying post-Minkowskian geometries for gravitational waves via loop-by-loop Baikov

We use the loop-by-loop Baikov representation to investigate the geometries in Feynman integrals contributing to the classical dynamics of a black-hole two-body system in the post-Minkowskian expansion of general relativity. These geometries determine the spaces of functions to which the corresponding Feynman diagrams evaluate. As a proof of principle, we provide a full classification of the geometries appearing up to three loops, i.e. fourth post-Minkowskian order, for all diagrams relevant to the conservative as well as the dissipative dynamics, finding full agreement with the literature. Moreover, we show that the non-planar top topology at four loops, which is the most complicated sector with respect to integration-by-parts identities, has an algebraic leading singularity and thus can only depend on non-trivial geometries through its subsectors.

FeynCalc 10: Do multiloop integrals dream of computer codes?

In this work we report on a new version of FeynCalc, a Mathematica package widely used in the particle physics community for manipulating quantum field theoretical expressions and calculating Feynman diagrams. Highlights of the new version include greatly improved capabilities for doing multiloop calculations, including topology identification and minimization, optimized tensor reduction, rewriting of scalar products in terms of inverse denominators, detection of equivalent or scaleless loop integrals, derivation of Symanzik polynomials, Feynman parametric as well as graph representation for master integrals and initial support for handling differential equations and iterated integrals. In addition to that, the new release also features completely rewritten routines for color algebra simplifications, inclusion of symmetry relations between arguments of Passarino–Veltman functions, tools for determining matching coefficients and quantifying the agreement between numerical results, improved export to ▪ and first steps towards a better support of calculations involving light-cone vectors.

De Sitter at all loops: the story of the Schwinger model

We consider the two-dimensional Schwinger model of a massless charged fermion coupled to an Abelian gauge field on a fixed de Sitter background. The theory admits an exact solution, first examined by Jayewardena, and can be analyzed efficiently using Euclidean methods. We calculate fully non-perturbative, gauge-invariant correlation functions of the electric field as well as the fermion and analyze these correlators in the late-time limit. We compare these results with the perturbative picture, for example by verifying that the one-loop contribution to the fermion two-point function, as predicted from the exact solution, matches the direct computation of the one-loop Feynman diagram. We demonstrate many features endemic of quantum field theory in de Sitter space, including the appearance of late-time logarithms, their resummation to de Sitter invariant expressions, and Boltzmann suppressed non-perturbative phenomena, with surprising late-time features.

Electric dipole moments of charged leptons in models with pseudo-Dirac sterile fermions

In this work, we address the impact of a small lepton number violation on charged lepton electric dipole moments — EDMs. Low-scale seesaw models protected by lepton number symmetry and leading to pseudo-Dirac pairs in the neutrino heavy spectrum provide a natural explanation for the smallness of neutrino masses with potentially testable consequences. Among which, it was thought that the small mass gap in each pair of pseudo-Dirac neutrinos may induce important contribution to the charged lepton EDMs. Recently, it has been shown that the contribution from some of the Feynman diagrams to charged lepton EDMs exactly cancel by virtue of the Ward-Takahashi identity in quantum electrodynamics. We thus consider here the Standard Model minimally extended with pairs of pseudo-Dirac sterile fermions and derive the complete analytical formula at two loops for the charged lepton EDMs. In addition, we numerically evaluate the order of the predicted EDMs consistent with the experimental bounds and constraints such as neutrino oscillation data, charged lepton flavour violating processes, sterile neutrino direct searches, meson decays, sterile neutrino decays, and cosmological and astrophysical observations. We find that, in the minimal setup accommodating neutrino data (masses and mixings) with only two pseudo-Dirac pairs, the predicted electron EDM is O10−36\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O}\\left({10}^{-36}\\right) $$\\end{document}e cm, at most, which is much smaller than the current experimental bound and even future sensitivity. Hopefully, the electron EDM might reach future sensitivity, once extra pseudo-Dirac neutrinos are taken into account. The analytical formulae we derive are generic to any model involving pseudo-Dirac pairs in the heavy neutrino spectrum.