Three-point Vertex Research Articles

All-electron first-principles GWΓ simulations for accurately predicting core-electron binding energies considering first-order three-point vertex corrections.

In the conventional GW method, the three-point vertex function (Γ) is approximated to unity (Γ ∼ 1). Here, we developed an all-electron first-principles GWΓ method beyond a conventional GW method by considering a first-order three-point vertex function (Γ(1) = 1 + iGGW) in a one-electron self-energy operator. We applied the GWΓ method to simulate the binding energies (BEs) of B1s, C1s, N1s, O1s, and F1s for 19 small-sized molecules. Contrary to the one-shot GW method [or G0W0(LDA)], which underestimates the experimentally determined absolute BEs by about 3.7eV for B1s, 5.1eV for C1s, 6.9eV for N1s, 7.8eV for O1s, and 5.8eV for F1s, the GWΓ method successfully reduces these errors by approximately 1-2eV for all the elements studied here. Notably, the first-order three-point vertex corrections are more significant for heavier elements, following the order of F > O > N > C > B1s. Finally, the computational cost analysis revealed that one term in the GWΓ one-electron self-energy operator, despite being computationally intensive, contributes negligibly (<0.1eV) to the C1s, N1s, O1s, and F1s.

Geometry from integrability: multi-leg fishnet integrals in two dimensions

We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their associated geometry, are completely fixed by their Yangian and permutation symmetries. As a new feature for the hexagonal fishnets, the star-triangle identity introduces an ambiguity in the graph representation of a given Feynman integral. This translates into a map between different geometric interpretations attached to a graph. We demonstrate explicitly how these fishnet integrals can be understood as Calabi-Yau varieties, whose Picard-Fuchs ideals are generated by the Yangian over the conformal algebra. In analogy to elliptic curves, which represent the simplest examples of fishnet integrals with four-point vertices, we find that the simplest examples of three-point fishnets correspond to Picard curves with natural generalisations at higher loop orders.

Hunting for hypercharge anapole dark matter in all spin scenarios

We conduct a combined analysis to investigate dark matter (DM) with hypercharge anapole moments, focusing on scenarios where Majorana DM particles with spin 1/2, 1, 3/2, and 2 interact exclusively with Standard Model particles through U(1)Y hypercharge anapole terms for the first time. For completeness, we construct general effective U(1) gauge-invariant three-point vertices. These enable the generation of hypercharge gauge-invariant interaction vertices for both a virtual photon γ and a virtual Z boson with two identical massive Majorana particles of any nonzero spin s, after the spontaneous breaking of electroweak gauge symmetry. For complementarity, we adopt effective operators tailored to each dark matter spin allowing crossing symmetry. We calculate the relic abundance, analyze current constraints and future sensitivities from dark matter direct detection and collider experiments, and apply the conceptual naive perturbativity bound. Our estimations based on a generalized vertex calculation demonstrate that the scenario with a higher-spin DM is more stringently constrained than a lower-spin DM, primarily due to the reduced annihilation cross section and/or the enhanced rate of LHC monojet events. As a remarkable outcome, the spin-2 anapole DM scenario is almost entirely excluded, while the high-luminosity LHC exhibits high sensitivities in probing spin-1 and -3/2 scenarios, except for a tiny parameter range of DM mass around 1 TeV. A significant portion of the remaining parameter space in the spin-1/2 DM scenario can be explored through upcoming Xenon experiments, with more than 20 ton-yr exposure equivalent to approximately 5 years of running the XENONnT experiment. Published by the American Physical Society 2024

Self-Dual Gravity and Color-Kinematics Duality in AdS_{4}.

We show that self-dual gravity in Euclidean four-dimensional anti-de Sitter space (AdS_{4}) can be described by a scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalization of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS_{4} version of the so-called kinematic algebra. We also obtain the three-point interaction vertex of self-dual gravity in AdS_{4} from that of self-dual Yang-Mills by replacing the structure constants of the Lie group with the structure constants of the new kinematic algebra, implying that self-dual gravity in AdS_{4} can be derived from self-dual Yang-Mills in this background via a double copy. This provides a concrete starting point for defining the double copy for Einstein gravity in AdS_{4} by expanding around the self-dual sector. Moreover, we show that the new kinematic Lie algebra can be lifted to a deformed version of the w_{1+∞} algebra, which plays a prominent role in celestial holography.

One-loop fermion-photon vertex in arbitrary gauge and dimensions: A novel approach

We compute a one-loop electron-photon vertex with fully off-shell external momenta in an arbitrary covariant gauge and space-time dimension. There exist numerous efforts in literature where a one-loop off-shell vertex is calculated by employing the standard first-order Feynman rules in different covariant gauges and space-time dimensions of interest. The tensor structure which decomposes this three-point vertex into the components transverse and longitudinal to the photon momentum gets intertwined in this first-order formalism. The Ward-Takahashi identity is explicitly invoked to untangle the two pieces and the results are expressed in a preferred basis of 12 spin amplitudes. We propose a novel approach based upon an efficient combination of the first- and second-order formalisms of quantum electrodynamics to compute this one-loop vertex. Among some conspicuous advantages is the fact that this less known second-order formalism separates the spin and scalar degrees of freedom of an electron interacting electromagnetically. More noticeably, the longitudinal and transverse contributions naturally disentangle from the onset in our approach. Moreover, this decomposition leads to identities between one-loop scalar Feynman integrals with higher powers in the propagators and shifted space-time dimensions that can be used to prove the Ward-Takahashi identity at one-loop order without the need to evaluate any Feynman integral. Additionally, this natural decomposition allows us to establish the gauge independence of the Pauli form factor through explicit cancellations of scalar Feynman integrals that depend on the gauge parameter. These cancellations naturally lead to a compact expression for the Pauli form factor in arbitrary dimensions. Wherever necessary and insightful, we make comparisons with earlier works.

Analytic structure of three-point functions from contour deformations

We explore the analytic structure of three-point functions using contour deformations. This method allows continuing calculations analytically from the spacelike to the timelike regime. We first elucidate the case of two-point functions with explicit explanations how to deform the integration contour and the cuts in the integrand to obtain the known cut structure of the integral. This is then applied to one-loop three-point integrals. We explicate individual conditions of the corresponding Landau analysis in terms of contour deformations. In particular, the emergence and position of singular points in the complex integration plane are relevant to determine the physical thresholds. As an exploratory demonstration of this method's numerical implementation we apply it to a coupled system of functional equations for the propagator and the three-point vertex of $\phi^3$ theory. We demonstrate that under generic circumstances the three-point vertex function displays cuts which can be determined from modified Landau conditions.

Vertex-Based Diagrammatic Treatment of Light-Matter-Coupled Systems.

We propose a diagrammatic MonteCarlo approach for quantum impurity models, which can be regarded as a generalization of the strong-coupling expansion for fermionic impurity models. The algorithm is based on a self-consistently computed three-point vertex and a stochastically sampled four-point vertex, and it allows one to obtain numerically exact results in a wide parameter regime. The performance of the algorithm is demonstrated with applications to a spin-boson model representing an emitter in a waveguide. As a function of the coupling strength, the spin exhibits a delocalization-localization crossover at low temperatures, signaling a qualitative change in the real-time relaxation. In certain parameter regimes, the response functions of the emitter coupled to the electromagnetic continuum can be described by an effective Rabi model with appropriately defined parameters. We also discuss the spatial distribution of the photon density around the emitter.

Pseudoparticle vertex solver for quantum impurity models

We present a quantum impurity solver based on a pseudo-particle framework, which combines diagrammatic resummations for a three-point vertex with diagrammatic Monte Carlo sampling of a four-point vertex. This recently proposed approach [A. J. Kim et al., arXiv:2112.15549] is generalized here to fermionic impurity problems and we discuss the technical details of the implementation, including the time-stepping approach, the Monte Carlo updates, and the routines for checking the two-particle irreducibility of the four-point vertex. We also explain how the vertex information can be efficiently stored using a Dubiner basis representation. The convergence properties of the algorithm are demonstrated with applications to exactly solvable impurity models and dynamical mean field theory simulations of the single-orbital Hubbard model. It is furthermore shown that the algorithm can handle a two-orbital problem with off-diagonal hybridizations, which would cause a severe sign problem in standard hybridization-expansion Monte Carlo simulations. Since the vertex-based algorithm successfully handles sign oscillating integrals in equilibrium and samples only connected diagrams, it may be a promising approach for real-time simulations.

Constructing the covariant three-point vertices systematically

An algorithm is developed for efficiently constructing the Lorentz covariant effective three-point vertices of the decay of a particle into two daughter particles in which all the masses and spins of the three particles can be arbitrary. The closely related one-to-one correspondence between the helicity formalism and the covariant formulation is exploited for counting the number of independent terms and identifying the basic covariant three-point vertices. Assembling the basic operators according to the developed algorithm enables us to construct all the covariant three-point vertices systematically.

Weaving the covariant three-point vertices efficiently

An efficient algorithm is developed for compactly weaving all the Lorentz covariant three-point vertices in relation to the decay of a massive particle $X$ of mass $m_X$ and spin $J$ into two particles $ M_{1,2}$ with equal mass $m$ and spin $s$. The closely-related equivalence between the helicity formalism and the covariant formulation is utilized so as to identify the basic building blocks for constructing the covariant three-point vertex corresponding to each helicity combination explicitly. The massless case with $m=0$ is worked out straightforwardly and the (anti)symmetrization of the three-point vertex required by spin statistics of identical particles is made systematically. It is shown that the off-shell electromagnetic photon coupling to the states $M_1$ and $M_2$ can be accommodated in this framework. The power of the algorithm is demonstrated with a few typical examples with specific $J$ and $s$ values.

Open charm and charmonium states in strong magnetic fields

The mass modifications of the open charm ([Formula: see text] and [Formula: see text]) mesons, and their effects on the decay widths [Formula: see text] as well as of the charmonium state, [Formula: see text] to open charm mesons ([Formula: see text]), are investigated in the presence of strong magnetic fields. These are studied accounting for the mixing of the pseudoscalar ([Formula: see text]) and vector ([Formula: see text]) mesons ([Formula: see text], [Formula: see text] mixings), with the mixing parameter, [Formula: see text] of a phenomenological three-point ([Formula: see text]) vertex interaction determined from the observed radiative decay width of [Formula: see text]. For charged [Formula: see text] mixing, this parameter is dependent on the magnetic field, because of the Landau level contributions to the vacuum masses of these mesons. The masses of the charged [Formula: see text] and [Formula: see text] mesons modified due to [Formula: see text] mixing, in addition, have contributions from the lowest Landau levels in the presence of a strong magnetic field. The effects of the magnetic field on the decay widths are studied using a field theoretical model of composite hadrons with quark (and antiquark) constituents. The matrix elements for these decays are evaluated using the light quark–antiquark pair creation term of the free Dirac Hamiltonian for the constituent quark field, with explicit constructions for the charmonium state [Formula: see text], the open charm ([Formula: see text], [Formula: see text], [Formula: see text]) mesons and the pion states in terms of the constituent quark fields. The parameter for the charged [Formula: see text] mixing is observed to increase appreciably with increase in the magnetic field. This leads to dominant modifications to their masses, and hence the decay widths of charged [Formula: see text] as well as [Formula: see text] at large values of the magnetic field. The modifications of the masses and decay widths of the open and hidden charm mesons in the presence of strong magnetic fields should have observable consequences on the production of the open charm ([Formula: see text] and [Formula: see text]) mesons as well as of the charmonium states resulting from noncentral ultra-relativistic heavy ion collision experiments.

Landau gauge Yang-Mills propagators in the complex momentum plane

We calculate the dressed gluon and ghost propagators of Landau gauge Yang-Mills theory in the complex momentum plane from their Dyson-Schwinger equations. To this end, we develop techniques for a direct calculation such that no mathematically ill-posed inverse problem needs to be solved. We provide a detailed account of the employed ray technique and discuss a range of tools to monitor the stability of the numerical calculation. Within a truncation employing model Ans\"atze for the three-point vertices and neglecting effects due to four-point functions, we find a singularity in the gluon propagator in the second quadrant of the complex ${p}^{2}$ plane. Although the location of this singularity turns out to be strongly dependent on the model for the three-gluon vertex, it always occurs at complex momenta for the range of models considered.

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: II. Anomalous Dimension η for \(d \lesssim 4\) at Finite Temperatures

We study the anomalous dimension $\eta$ of homogeneous interacting single-component Bose-Einstein condensates at finite temperatures for $d\lesssim 4$ dimensions. This $\eta$ is defined in terms of the one-particle density matrix $\rho({\bf r})\equiv \langle \hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$ through its asymptotic behavior $\rho({\bf r})\rightarrow N_{\bf 0}/V+C r^{-d+2-\eta}$ for $r\rightarrow \infty$, where $N_{\bf 0}/V$ is the condensate density and $C$ is a constant. It is shown that the anomalous dimension is given by $\eta=0.181\epsilon^2$ to the leading order in $\epsilon\equiv d-4$. The change of the prefactor $0.181$ from the value $0.02$ at the transition point of the ${\rm O}(2)$ symmetric $\phi^4$ model is attributed to the emergence of three-point vertices and the anomalous Green's function when $N_{\bf 0}$ acquires a finite value.

Fermion Propagator in QED - Landau-Khalatnikov-Fradkin Transformations

Gauge theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD) describe the physical world accurately at the level of fundamental particles. They possess gauge symmetry reflected in terms of several identities and transformation laws which impose tight constraints on all conceivable Green functions which define the theory. In this article, we describe and summarize the role played by the Landau-Khalatnikov-Fradkin (LKF) transformations in this context. Within the set of covariant gauges, these transformations tell us how to construct a Green function in an arbitrary gauge, starting from its explicit expression in a particular gauge. In perturbation theory, these transformations are satisfied at every order of approximation. A non-perturbative description of QED and QCD in the continuum is provided by the Schwinger-Dyson Equations (SDEs). These are the fundamental equations of motion encoding the dynamics of Green functions. These equations provide a unified description of weak and strong coupling regimes and are thus increasingly employed to study strongly interacting theories and their transition to the perturbative limit. As these equations are an infinite set of coupled non-linear equations, a truncation is essential to reduce them to a solvable number. LKF transformations provide a stringent constraint on the acceptable truncations which preserve the original symmetries of the gauge theory involved. Most of these truncations consist in cleverly constructing an Anstaz for the electron-photon vertex in QED and the quark-gluon vertex in QCD. In this article, we review the LKF transformations for the fermion propagator. Very importantly, they imply the gauge invariance of the chiral fermion condensate and the pole mass of a fermion. We provide the first demonstration of the latter in this article. Moreover, we also describe how the LKF transformations of the fermion propagator provide gauge-symmetry constraints on a non-perturbative construction of the three-point fermion-boson vertex.

NSVZ Relation in Supersymmetric Theories Regularized by Higher Derivatives

We investigate the NSVZ equation in $$\mathcal{N} = 1$$ supersymmetric gauge theories, which relates the $$\beta $$ ‑function to the anomalous dimension of the matter superfields. In particular, we argue that it is closely connected with the non-renormalization theorem for the three-point gauge-ghost vertices (in which one external leg corresponds to the quantum gauge superfield). Using finiteness of these vertices the exact NSVZ $$\beta $$ -function can be equivalently presented in the form of a relation between the $$\beta $$ -function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This equation allows explaining how the NSVZ equation appears in the perturbation theory for theories regularized by higher covariant derivatives and constructing the NSVZ scheme in all orders in the case of using this regularization. The results are verified by an explicit three-loop calculation of the terms quartic in the Yukawa couplings.

Finite temperature QCD four-point function in the presence of a weak magnetic field within the hard thermal loop approximation

We compute the weak magnetic field correction to the four-point quark-gluon vertex at finite temperature and chemical potential within the hard thermal loop approximation. The resulting four-point quark-gluon vertex satisfies the Ward identity with the three-point quark-gluon vertex.

QCD propagators and vertices from lattice QCD (in memory of Michael Müller-Preußker)

We review lattice calculations of the elementary Greens functions of QCD with a special emphasis on the Landau gauge. These lattice results have been of interest to continuum approaches to QCD over the past 20 years. They are used as reference for Dyson-Schwinger- and functional renormalization group equation calculations as well as for hadronic bound-state equations. The lattice provides low-energy data for propagators and three-point vertices in Landau gauge at zero and finite temperature even including dynamical fermions. We summarize Michael M\"uller-Preu{\ss}ker's important contributions to this field and put them into the perspective of his other research interests.

Three-point vertex functions in Yang-Mills Theory and QCD in Landau gauge

Solutions for the three-gluon and quark-gluon vertices from Dyson-Schwinger equations and the three-particle irreducible formalism are discussed. Dynamical quarks (``unquenching'') change the three-gluon vertex via the quark-triangle diagrams which themselves include fully dressed quark-gluon vertex functions. On the other hand, the quark-swordfish diagram is, at least with the model used for the two-quark-two-gluon vertex employed here, of minor importance. For the leading tensor structure of the three-gluon vertex the "unquenching" effect can be summarized for the nonperturbative part as a shift of the related dressing function towards the infrared.

Interaction vertex for classical spinning particles

We consider a model of the classical spinning particle in which the coadjoint orbits of the Poincare group are parametrized by two pairs of canonically conjugate four vectors, one representing the standard position and momentum variables and the other which encodes the spinning degrees of freedom. This "Dual Phase Space Model" is shown to be a consistent theory of both massive and massless particles and allows for coupling to background fields such as electromagnetism. The on-shell action is derived and shown to be a sum of two terms, one associated with motion in spacetime and the other with motion in "spin space." Interactions between spinning particles are studied and a necessary and sufficient condition for consistency of a three-point vertex is established.

Subsubleading soft theorems of gravitons and dilatons in the bosonic string

Starting from the amplitude with an arbitrary number of massless closed states of the bosonic string, we compute the soft limit when one of the states becomes soft to subsubleading order in the soft momentum expansion, and we show that when the soft state is a graviton or a dilaton, the full string amplitude can be expressed as a soft theorem through subsubleading order. It turns out that there are string corrections to the field theoretical limit in the case of a soft graviton, while for a soft dilaton the string corrections vanish. We then show that the new soft theorems, including the string corrections, can be simply obtained from the exchange diagrams where the soft state is attached to the other external states through the three-point string vertex of three massless states. In the soft-limit, the propagator of the exchanged state is divergent, and at tree-level these are the only divergent contributions to the full amplitude. However, they do not form a gauge invariant subset and must be supplemented with extra non-singular terms. The requirement of gauge invariance then fixes the complete amplitude through subsubleading order in the soft expansion, reproducing exactly what one gets from the explicit calculation in string theory. From this it is seen that the string corrections at subsubleading order arise as a consequence of the three-point amplitude having string corrections in the bosonic string. When specialized to a soft dilaton, it remarkably turns out that the string corrections vanish and that the non-singular piece of the subsubleading term of the dilaton soft theorem is the generator of space-time special conformal transformation.