Feynman Diagrams Research Articles

Jet transport coefficients by elastic and radiative scatterings in the strongly interacting quark-gluon plasma

We extend the investigation of jet transport coefficients within the effective dynamical quasiparticle model (DQPM)—constructed to describe nonperturbative QCD phenomena of the strongly interacting quark-gluon plasma (sQGP) in line with the lattice QCD equation of state—by accounting for inelastic 2→3 reactions with gluon radiation in addition to the elastic scattering of partons. The elastic and inelastic reactions are calculated explicitly within leading-order Feynman diagrams with effective propagators and vertices from the DQPM by accounting for all channels and their interferences. We present the results for the jet transport coefficients such as the transverse momentum transfer squared q̂ per unit length as well as the energy loss dE/dx per unit length in the sQGP and investigate their dependence on the temperature T and momentum of the jet parton depending on the choice of the strong coupling constant αs in thermal, jet parton, and radiative vertices. For the latter, we consider different scenarios used in the literature and find a very strong dependence of q̂ and dE/dx on the choice of αs. Moreover, we explore the relation of q̂/T3 to the ratio of specific shear viscosity to entropy density η/s and show that the ratio T3/q̂ to η/s has a strong T dependence—especially when approaching Tc—on the choice of αs in scattering vertices. Published by the American Physical Society 2024

Nonperturbative anomalous thresholds

Feynman diagrams (notably the triangle diagram) involving heavy enough particles contain branch cuts on the physical sheet—anomalous thresholds—which, unlike normal thresholds and bound-state poles, do not correspond to any asymptotic n-particle state. “Who ordered that?” We show that anomalous thresholds arise as a consequence of established S-matrix principles and two reasonable assumptions: unitarity below the physical region and analyticity in the mass. We find explicit nonperturbative formulas for the anomalous threshold singularity and test them against the Coleman-Thun poles of the E8 integrable model. Published by the American Physical Society 2024

Thermal masses and trapped-ion quantum spin models: a self-consistent approach to Yukawa-type interactions in the λϕ4 model

The quantum simulation of magnetism in trapped-ion systems makes use of the crystal vibrations to mediate pairwise interactions between spins, which are encoded in the internal electronic states of the ions, and measured in experiments that probe the real-time dynamics. These interactions can be accounted for by a long-wavelength relativistic theory, where the phonons are described by a coarse-grained Klein-Gordon field ϕ(x) locally coupled to the spins that acts as a carrier, leading to an analogue of pion-mediated Yukawa interactions. In the vicinity of a structural transition of the ion crystal, one must go beyond the Klein-Gordon fields, and include additional λϕ4 terms responsible for phonon-phonon scattering. This leads to quantum effects that can be expressed by Feynman loop integrals that modify the range of the Yukawa-type spin interactions; an effect that could be used to probe the underlying fixed point of this quantum field theory (QFT). Unfortunately, the rigidity of the trapped-ion crystal makes it challenging to observe genuine quantum effects, such as the flow of the critical point with the quartic coupling λ. We hereby show that thermal effects, which can be controlled by laser cooling, can unveil this flow through the appearance of thermal masses in interacting QFTs. We perform self-consistent calculations that resum certain Feynman diagrams and, additionally, go beyond mean-field theory to predict how measurements on the trapped-ion spin system can probe key properties of the λϕ4 QFT.

Role of Higgs and Leggett modes for the third harmonic response in noncentrosymmetric superconductors

The amplitude and phase of the third harmonic response in superconductors provide important insights into collective Higgs and Leggett modes, respectively, in the superconducting state. In particular, for twice the incident frequency equal to the binding energy (2ω=2Δ), one finds a resonance of the amplitude and a corresponding phase jump in the third harmonic response, respectively. We generalize these concepts to superconductors without an inversion symmetry, which can be effectively described by a two-band model with an order parameter consisting of spin-singlet (even parity) and spin-triplet (odd parity) components. In our work we use an effective action approach for the derivation of the nonlinear response and assign the underlying physical processes to their respective Feynman diagrams. We calculate the third harmonic signal exemplary for the noncentrosymmetric compound CePt3Si, showing that it contains contributions from three distinguishable sources, namely the Higgs mode, the Leggett mode, and quasiparticles (broken Cooper-pairs). Only in the clean limit diamagnetic Raman-like processes contribute to the third harmonic signal, whereas the quasiparticle contributions dominate the collective modes for all triplet-singlet ratios of the gap structure. In the dirty limit, we find a significant enhancement of the Higgs mode contributions to the third harmonic response, due to the inclusion of nonvanishing paramagnetic diagrams. Finally, we argue that the phase difference between the third harmonic and the fundamental signal might reveal a jump, where its size is dependent on the light-matter coupling. Published by the American Physical Society 2024

Multiparton Cwebs at five loops

Scattering amplitudes involving multiple partons are plagued with infrared singularities. The soft singularities of the amplitude are captured by the soft function which is defined as the vacuum expectation value of Wilson line correlators. Renormalization properties of soft function allows us to write it as an exponential of the finite soft anomalous dimension. An efficient way to study the soft function is through a set of Feynman diagrams known as Cwebs (webs). We present the mixing matrices and exponentiated colour factors (ECFs) for the Cwebs at five loops that connect six Wilson lines, except those that are related by relabeling of Wilson lines. Further, we express these ECFs in terms of 29 basis colour factors. We also find that this basis can be categorized into two colour structures. Our results are the first key ingredients for the calculation of the soft anomalous dimension at five loops.

The in-out formalism for in-in correlators

Cosmological correlators, the natural observables of the primordial universe, have been extensively studied in the past two decades using the in-in formalism pioneered by Schwinger and Keldysh for the study of dissipative open systems. Ironically, most applications in cosmology have focused on non-dissipative closed systems. We show that, for non-dissipative systems, correlators can be equivalently computed using the in-out formalism with the familiar Feynman rules. In particular, the myriad of in-in propagators is reduced to a single (Feynman) time-ordered propagator and no sum over the labelling of vertices is required. In de Sitter spacetime, this requires extending the expanding Poincaré patch with a contracting patch, which prepares the bra from the future. Our results are valid for fields of any mass and spin but assuming the absence of infrared divergences.We present three applications of the in-out formalism: a representation of correlators in terms of a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation that computes Minkowski correlators in terms of lower order ones; and the derivation of cutting rules from Veltman’s largest time equation, which we explicitly develop and exemplify for two-vertex diagrams to all loop orders.The in-out formalism leads to a natural definition of a de Sitter scattering matrix, which we discuss in simple examples. Remarkably, we show that our scattering matrix satisfies the standard optical theorem and the positivity that follows from it in the forward limit.

Global sampling of Feynman's diagrams through normalizing flow

Normalizing flows (NF) are powerful generative models with increasing applications in augmenting Monte Carlo algorithms due to their high flexibility and expressiveness. In this work we explore the integration of NF in the diagrammatic Monte Carlo (DMC) method, presenting an architecture designed to sample the intricate multidimensional space of Feynman's diagrams through dimensionality reduction. By decoupling the sampling of diagram order and interaction times, the flow focuses on one interaction at a time. This enables one to construct a general diagram by employing the same unsupervised model iteratively, dressing a zero-order diagram with interactions determined by the previously sampled order. The resulting NF-augmented DMC method is tested on the widely used single-site Holstein polaron model in the entire electron-phonon coupling regime. The obtained data show that the model accurately reproduces the diagram distribution by reducing sample correlation and observables' statistical error, constituting the first example of global sampling strategy for connected Feynman's diagrams in the DMC method. Published by the American Physical Society 2024

Field theory equivalences as spans of L∞ -algebras

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying L∞ -algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of L∞ -algebras in which the quasi-isomorphic L∞ -algebras are obtained from a correspondence L∞ -algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out fields in a Feynman diagram expansion. Spans of L∞ -algebras allow for a clean definition of quasi-isomorphisms of cyclic L∞ -algebras. Furthermore, they appear naturally in many contexts within physics. As examples, we first consider scalar field theory with interaction vertices blown up in different ways. We then show that (non-Abelian) T-duality can be seen as a span of L∞ -algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of L∞ -algebras for the Penrose–Ward transform in the context of self-dual Yang–Mills theory and Bogomolny monopoles.

Nonlocal QED and lepton g-2 anomalies

Quantum electrodynamics is generally extended to a nonlocal QED by introducing the correlation functions. The gauge link is introduced to guarantee that the nonlocal QED is locally U(1) gauge invariant. The corresponding Feynman rules as well as the proof of Ward–Takahashi identity are presented. As an example, the anomalous magnetic moments of leptons are studied in nonlocal QED. At one-loop level, besides the ordinary diagrams, there are many additional Feynman diagrams which are generated from the gauge link. It shows the nonlocal QED can provide a reasonable explanation for lepton g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g-2$$\\end{document} anomalies.

Revisit the heavy quarkonium double-gluon hybrid mesons with exotic quantum numbers

We revisit the masses of heavy quarkonium double-gluon hybrid mesons with exotic quantum numbers JPC = 1−+ and 2+− in the framework of the QCD sum rules. Considering the double-gluon hybrid meson operators in the octet-octet color structure, we have constructed two independent interpolating currents with JPC = 1−+ and five independent currents with JPC = 2+−. For the interpolating currents with antisymmetric glueball operator, there exist non-local divergences in one kind of additional Feynman diagrams of the tri-gluon condensate, which will give important contributions to the sum rule stabilities and mass predictions. We use the diagrammatic renormalization to cancel out such divergences. At the leading order of αs, the two-point correlation functions and spectral densities can be expressed in the analytic form of the generalized hypergeometric functions and Meijer’s G-functions. After performing the numerical analysis, we predict the masses of the 1−+ and 2+− charmonium double-gluon hybrid mesons to be around 6.1 − 7.2 GeV and 6.3 − 6.4 GeV, respectively. For the bottomonium systems, their masses are predicted to be 13.7 − 14.3 GeV and 12.6 − 13.3 GeV for the 1−+ and 2+− channels, respectively. Besides, it is possible to hunt for these charmonium hybrids in the radiative decays of bottomonium mesons in BelleII experiment. Further investigations on these hybrid states in various theoretical and phenomenological methods are also anticipated in the future.

Алгоритмы перечисления симметричных хордовых диаграмм

Numerous applications require exhaustive lists of strings, which may be subject to various requirements, for example, such as their non-equivalence under the action of symmetry groups on them. The necklace is the equivalence class of r-ary strings under rotation. An unlabeled r-ary necklace is an equivalence class of r-ary strings under rotation and permutation of alphabetic characters. Let G be a symmetry group acting on a given set of necklaces T. A necklace x  T is called symmetric if there exists an element g  G, g  e such that gx = x. Other definitions of the symmetry of necklaces are possible, equivalent to the above. All of them, in one way or another, are involved in determining the symmetry of the figure compared to the unlabelled necklace. A flat figure is called symmetrical if it is self-aligned during movements of space R2 i. e. during its isometric transformations. A special case of figures and, accordingly, necklaces are chord diagrams. Chord diagrams represent an object of research that is interesting from different aspects (knot theory, Feynman diagrams, representations of Lie algebras), and has been studied by many authors. The article presents algorithms for enumerating symmetric chord diagrams and shows their connection with the knot theory using simple examples. The list of symmetrical chord diagrams may, in particular, be of interest in the field of mathematical poetry, which studies the dynamics of poetic stanzas horizontally (rhythm, syllabic volume of verses) and vertically (rhyme schemes, refrains, and other stylistic devices).

Strebel Differentials and String Field Theory

Abstract A closed string worldsheet of genus g with n punctures can be presented as a contact interaction in which n semi-infinite cylinders are glued together in a specific way via the Strebel differential on it, if $n\ge 1,\ 2g-2+n\gt 0$. We construct a string field theory of closed strings such that all the Feynman diagrams are represented by such contact interactions. In order to do so, we define off-shell amplitudes in the underlying string theory using the combinatorial Fenchel–Nielsen coordinates to describe the moduli space and derive a recursion relation satisfied by them. Utilizing the Fokker–Planck formalism, we construct a string field theory from which the recursion relation can be deduced through the Schwinger–Dyson equation. The Fokker–Planck Hamiltonian consists of kinetic terms and three-string interaction terms.

A complete set of 4-point amplitudes in the constructive Standard Model

We present a complete set of 4-point amplitudes in the constructive Standard Model at tree level. Any 4-point amplitude can be obtained from the results presented here by a suitable choice of masses, a permutation of the particles (by crossing symmetry), and a reversal of the momenta of the outgoing particles. We have validated all of these amplitudes by comparing with Feynman diagrams for a variety of masses, scattering energy and angles, and helicities of the photons and gluons, when they are in the initial states. The standard constructive techniques work for these amplitudes without the need for any contact terms and indeed, contact terms are not allowed. Only three 4-point vertices are used (allowed), involving the Higgs boson and the W and Z bosons. When external photons or gluons are present, the amplitude simplifies to a single spinor-product structure, present in the numerator. In a few cases, however, the propagator structure is more complex, with different terms depending on the charge or color structure. In the case of internal photons or gluons, we find that the massless limit of a massive photon or gluon diagram gives the correct result in all cases. We have additionally found that using the x factor directly gives the correct result in all cases and agrees with the massless limit calculation.

Tangleoids with quantum field theories in biosystems

Abstract The open question whether quantum field theories can be used in biological systems will be addressed in this study. Quantum effects were reported for certain biological systems like the magnetic orientation sense in migratory birds. In quantum field theories it is possible to derive effective quantum field theories with welded tangleoids including braid relations that describe composite particles based on the dynamics of microscopic particle where these composite particles are made of. We observe that the generators of the tangleoid category that will depict the Feynman graphs are X X for a scattering vertex. This arises after carrying out the integral over bosonic fields A μ {A}_{\mu } in the partition function (D) that will lead to an four-valent interaction vertex generated by a quartic term in the fermionic fields. With X + {X}_{+} and X − {X}_{-} we will depict order changes like that regarding time orderings. Ordinary propagators are depicted by ∪ \cup and ∩ \cap . Finally, the generators ! \! and ¡ \hspace{0.1em}\text{¡}\hspace{0.1em} come into play if other foreign fields are picked up.

Next-to-Next-to-Leading Order QCD Corrections to Semi-Inclusive Deep-Inelastic Scattering.

We present the first results for the next-to-next-to leading order (NNLO) corrections to the semi-inclusive deep-inelastic scattering process in perturbative quantum chromodynamics. We consider the quark initiated flavor nonsinglet process and obtain the complete contributions analytically at leading color. All relevant virtual and real emission Feynman diagrams have been computed using integration-by-parts reduction to master integrals and two approaches for their subsequent evaluation (parametric phase-space integration and method of differential equations). The numerical analysis demonstrates the significance of the NNLO corrections and their great impact on the reduction of the residual scale dependence.

A diagram-free approach to the stochastic estimates in regularity structures

In this paper, we explore the version of Hairer’s regularity structures based on a greedier index set than trees, as introduced in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039) and algebraically characterized in (Linares et al. in Comm. Am. Math. Soc. 3:1–64, 2023). More precisely, we construct and stochastically estimate the renormalized model postulated in (Otto et al. in A priori bounds for quasi-linear SPDEs in the full sub-critical regime, 2021, arXiv:2103.11039), avoiding the use of Feynman diagrams but still in a fully automated, i. e. inductive way. This is carried out for a class of quasi-linear parabolic PDEs driven by noise in the full singular but renormalizable range. We assume a spectral gap inequality on the (not necessarily Gaussian) noise ensemble. The resulting control on the variance of the model naturally complements its vanishing expectation arising from the BPHZ-choice of renormalization. We capture the gain in regularity on the level of the Malliavin derivative of the model by describing it as a modelled distribution. Symmetry is an important guiding principle and built-in on the level of the renormalization Ansatz. Our approach is analytic and top-down rather than combinatorial and bottom-up.

Perturbative unitarity and the four-point vertices in the constructive standard model

We find a complete set of four-point vertices in the constructive Standard Model (CSM). This set is smaller than in Feynman diagrams as the CSM does not need or allow any additional four-point vertices (or “contact” terms) beyond what is present in Feynman diagrams and, furthermore, it does need or allow a four-point vertex for Z,Z,W¯,W, W,W,W¯,W¯, γ,Z,W,W¯ or γ,γ,W,W¯, in addition to the already known absence of the four-gluon vertex. We show that with this set of four-point vertices, perturbative unitarity is satisfied in the CSM. Additionally, we show that many constructive diagrams are not Feynman diagrams rewritten in spinor form. In fact, we show that there is a significant rearrangement of contributions from the diagrams in constructive calculations relative to Feynman diagrams for some processes. In addition to the already known or expected rearrangement in diagrams involving external photons, we also find that diagrams involving four-vector bosons are also significantly different from their Feynman counterparts. Published by the American Physical Society 2024

Polarized ZZ pairs in gluon fusion and vector boson fusion at the LHC

Pair production of helicity-polarized weak bosons (Vλ=Wλ±,Zλ) from gluon fusion (gg→VλVλ′′) and weak boson fusion (V1V2→VλVλ′′) are powerful probes of the Standard Model, new physics, and properties of quantum systems. Measuring cross sections of polarized processes is a chief objective of the Large Hadron Collider's (LHC) Run 3 and high luminosity programs, but progress is limited by the simulation tools that are presently available. We propose a method for computing polarized cross sections that works by directly modifying Feynman rules instead of (squared) amplitudes. The method is applicable to loop-induced processes, and can capture the interference between arbitrary polarization configurations, interference with non-resonant diagrams, as well as off-shell/finite-width effects. By construction, previous results that work at the (squared) amplitude level are recoverable. As a demonstration, we report the prospect of observing and studying polarized ZλZλ′ pairs when produced via gluon fusion and electroweak processes in final-states with four charged leptons at the LHC, using the new method to simulate the gluon fusion process. Our Feynman rules are publicly available as a set of Universal FeynRules Object libraries called SM_Loop_VPolar.

Principal Landau determinants

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/. Program summaryProgram title:PLD.jlCPC Library link to program files:https://doi.org/10.17632/7h5644mm4n.1Developer's repository link:https://mathrepo.mis.mpg.de/PLD/Licensing provisions: Creative Commons by 4.0Programming language:JuliaSupplementary material: The repository includes the source code with documentation (PLD_code.zip), a jupyter notebook tutorial providing installation and usage instructions (PLD_notebook.zip), a database containing the output of our algorithm on 114 examples of Feynman integrals (PLD_database.zip).Nature of problem: A fundamental challenge in scattering amplitude is to determine the values of complexified kinematic invariants for which an amplitude can develop singularities. Bjorken, Landau, and Nakanishi wrote a system of polynomial constraints, nowadays known as the Landau equations. This project aims to rigorously revisit the Landau analysis of the singularity locus of Feynman integrals with a practical view towards explicit computations.Solution method: We define the principal Landau determinant (PLD), which is a variety inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ). We conjecture that it provides a subset of the singularity locus, and we implement effective algorithms to compute its defining equation explicitly.

Motivic Geometry of two-Loop Feynman Integrals

Abstract We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic or rational curves depending on the space-time dimension. For two-loop graphs with a small number of edges, we give more precise results. In particular, we recover a result of Bloch (Double box motive. SIGMA 2021;17,048) that in the well-known double-box example, there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the non-planar two-loop tardigrade graph is that of a K3 surface. In an appendix by Eric Pichon-Pharabod, we argue via high-precision numerical computations that the Picard number of this K3 surface is generically 11 and we compute the expected lattice polarization. Lastly, we show that generic members of the ice cream cone family of graph hypersurfaces correspond to the pairs of sunset Calabi–Yau varieties.