Self-energy Diagrams Research Articles

UV-finite \u201cold\u201d conformal bootstrap on AdS: scalar case

The double-trace from UV to IR flow subtraction of infinities used earlier for the UV-convergent calculations of the Witten tadpole diagrams being applied to the bubble self-energy diagrams gives for them the amazingly simple expressions in case of the four- dimensional boundary space. For every N = 1 … 4 in the O(N) symmetric scalar fields model with the conformal Hubbard-Stratonovich field there are three roots of the “old” conformal bootstrap spectral equations that obey unitarity bound demand.

Efficient computation of the second-Born self-energy using tensor-contraction operations.

In the nonequilibrium Green's function approach, the approximation of the correlation self-energy at the second-Born level is of particular interest, since it allows for a maximal speed-up in computational scaling when used together with the generalized Kadanoff-Baym ansatz for the Green's function. The present day numerical time-propagation algorithms for the Green's function are able to tackle first principles simulations of atoms and molecules, but they are limited to relatively small systems due to unfavorable scaling of self-energy diagrams with respect to the basis size. We propose an efficient computation of the self-energy diagrams by using tensor-contraction operations to transform the internal summations into functions of external low-level linear algebra libraries. We discuss the achieved computational speed-up in transient electron dynamics in selected molecular systems.

Loop-enhanced rate of neutrinoless double beta decay

Neutrino masses can be generated radiatively. In such scenarios their masses are calculated by evaluating a self-energy diagram with vanishing external momentum, i.e. taking only the leading order term in a momentum expansion. The difference between the full self-energy and the mass is experimentally difficult to access, since one needs off-shell neutrinos to observe it. However, massive Majorana neutrinos that mediate neutrinoless double beta decay (0νββ) are off-shell, with the virtuality of order 100 MeV. If the energy scale of the self-energy loop is of the order of this virtuality, the amplitude of double beta decay can be modified by the unsuppressed loop effect. This can have a drastic impact on the interpretation of future observations or limits of the 0νββ decay.

Electron-magnon scattering in elementary ferromagnets from first principles: Lifetime broadening and band anomalies

We study the electron-magnon scattering in bulk Fe, Co, and Ni within the framework of many-body perturbation theory implemented in the full-potential linearized augmented-plane-wave method. To this end, a $\mathbf{k}$-dependent self-energy ($GT$ self-energy) describing the scattering of electrons and magnons is constructed from the solution of a Bethe-Salpeter equation for the two-particle (electron-hole) Green function, in which single-particle Stoner and collective spin-wave excitations (magnons) are treated on the same footing. Partial self-consistency is achieved by the alignment of the chemical potentials. The resulting renormalized electronic band structures exhibit strong spin-dependent lifetime effects close to the Fermi energy, which are strongest in Fe. The renormalization can give rise to a loss of quasiparticle character close to the Fermi energy, which we attribute to electron scattering with spatially extended spin waves. This scattering is also responsible for dispersion anomalies in conduction bands of iron and for the formation of satellite bands in nickel. Furthermore, we find a band anomaly at a binding energy of 1.5~eV in iron, which results from a coupling of the quasihole with single-particle excitations that form a peak in the Stoner continuum. This band anomaly was recently observed in photoemission experiments. On the theory side, we show that the contribution of the Goldstone mode to the $GT$ self-energy is expected to (nearly) vanish in the long-wavelength limit. We also present an in-depth discussion about the possible violation of causality when an incomplete subset of self-energy diagrams is chosen.

Microscopic calculation of absorption spectra of macromolecules: An analytic approach.

We develop a cross-disciplinary approach to analytically compute optical response functions of open macromolecular systems by exploiting the mathematical formalism of quantum field theory (QFT). Indeed, the entries of the density matrix for the electronic excitations interacting with their open dissipative environment are mapped into vacuum-to-vacuum Green's functions in a fictitious relativistic closed quantum system. We show that by re-summing appropriate self-energy diagrams in this dual QFT, it is possible to obtain analytic expressions for the response functions in Mukamel's theory. This yields physical insight into the structure and dynamics of vibronic resonances, since their frequency and width is related to fundamental physical constants and microscopic model parameters. For illustration, we apply this scheme to compute the linear absorption spectrum of the Fenna-Matthews-Olson light harvesting complex, comparing analytic calculations, numerical Monte Carlo simulations, and experimental data.

Pair production of the lightest chargino at γγ-collider

Chargino pair production via photon–photon collision is investigated in the Minimal Supersymmetric Standard Model at a future linear collider. The process is computed using all the possible diagrams at the next-to-leading order, including box, triangle, and self-energy diagrams. The numerical analysis is carried out for the production rates of the lightest chargino pair in RNS, NS, mSUGRA, BB, and NUGM scenarios. These distinct benchmark models were introduced in the light of the LHC results presented at s=7–8 TeV. Among these scenarios, the RNS has the highest production rate for the γγ→χ˜1+χ˜1−. The partonic cross section reaches up to 3.94 pb at sˆ=310 GeV. The total convoluted cross section with the photon luminosity in a e+e− machine is calculated as a function of the center-of-mass energy up to 1 TeV. The convoluted cross section is ∼1.05 pb at s=350 GeV depending on the polarization of the initial electron and laser polarization. The RNS along with the BB scenarios are accessible at s=500 GeV with a γγ collision mode on e+e−-collider.

Transformed Lattice Rules for Feynman Loop Integrals on GPUs

We focus on the application of rank-1 lattice rules with tanh and sinm type transformations for the integration of functions which may have singularities on the boundaries of the integration domain. Using this technique we compute Feynman loop integrals for classes of 2-loop box and 3-loop self-energy Feynman diagrams that have up to eight internal lines with masses. The d-dimensional rule approximation with n points relies on a generator vector of length d with integer components, which is computed once and for all using the Component-by-Component method by Nuyens and Cools (2006). The generator vector is incorporated in the (CUDA) integration program that implements the lattice integration for fast execution on GPUs. Results are obtained efficiently and without special attention to specific problem characteristics.

High-speed evaluation of loop integrals using lattice rules

Lattice rules are implemented in CUDA for many-core computations on GPUs. A high-speed evaluation of Feynman loop integrals is presented, based on lattice rules and suitable transformations. The accuracy and efficiency of the method are compared for higher order sinm-transformations. Extensive results are reported for classes of diagrams including 2-loop box and 3-loop self-energy diagrams with massive internal lines. The method is further combined with an extrapolation with respect to the dimensional regularization parameter.

The MSR mass and the mathcal{O}left({Lambda}_{mathrm{QCD}}right) renormalon sum rule

We provide a detailed description and analysis of a low-scale short-distance mass scheme, called the MSR mass, that is useful for high-precision top quark mass determinations, but can be applied for any heavy quark Q. In contrast to earlier low-scale short-distance mass schemes, the MSR scheme has a direct connection to the well known overline{mathrm{MS}} mass commonly used for high-energy applications, and is determined by heavy quark on-shell self-energy Feynman diagrams. Indeed, the MSR mass scheme can be viewed as the simplest extension of the overline{mathrm{MS}} mass concept to renormalization scales ≪ mQ. The MSR mass depends on a scale R that can be chosen freely, and its renormalization group evolution has a linear dependence on R, which is known as R-evolution. Using R-evolution for the MSR mass we provide details of the derivation of an analytic expression for the normalization of the mathcal{O}left({varLambda}_{mathrm{QCD}}right) renormalon asymptotic behavior of the pole mass in perturbation theory. This is referred to as the mathcal{O}left({varLambda}_{mathrm{QCD}}right) renormalon sum rule, and can be applied to any perturbative series. The relations of the MSR mass scheme to other low-scale short-distance masses are analyzed as well.

Role of three-particle vertex within dual fermion calculations

We investigate the influence of self-energy diagrams beyond the two-particle vertex level within dual fermion theory. Specifically, we calculate the local three-particle vertex and construct from it selected dual fermion self-energy corrections to dynamical mean field theory. For the two-dimensional Hubbard model, the thus obtained self-energy corrections are small in the parameter space where dual fermion corrections based on the two-particle vertex only are small. However, in other parts of the parameter space, they are of a similar magnitude and qualitatively different from standard dual fermion theory. The high-frequency behaviour of the self-energy correction is - surprisingly - even dominated by corrections stemming from the three-particle vertex.

Regularization with numerical extrapolation for finite and UV-divergent multi-loop integrals

We give numerical integration results for Feynman loop diagrams such as those covered by Laporta (2000) and by Baikov and Chetyrkin (2010), and which may give rise to loop integrals with UV singularities. We explore automatic adaptive integration using multivariate techniques from the ParInt package for multivariate integration, as well as iterated integration with programs from the Quadpack package, and a trapezoidal method based on a double exponential transformation. ParInt is layered over MPI (Message Passing Interface), and incorporates advanced parallel/distributed techniques including load balancing among processes that may be distributed over a cluster or a network/grid of nodes. Results are included for 2-loop vertex and box diagrams and for sets of 2-, 3- and 4-loop self-energy diagrams with or without UV terms. Numerical regularization of integrals with singular terms is achieved by linear and non-linear extrapolation methods.

Numerical integration and extrapolation for finite and UV-divergent 3-loop Feynman integrals

Automatic numerical integration attempts to alleviate the detailed application of solution methods to the problem. We use automatic multivariate integration with PARINT combined with numerical extrapolation for classes of 3-loop UV-divergent self-energy diagrams with massive internal lines, and finite or UV-divergent massless vertex diagrams. The extrapolation is performed with respect to the dimensional regularization parameter. It is shown that the PARINT software handles the integrations accurately and efficiently.

One-particle many-body Green's function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms.

A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams.

Radiative corrections to the magnetic moments of the proton and the neutron

We estimate the radiative corrections of order to the magnetic moments of the proton and the neutron. The photon-loop diagram of the vertex-correction type is evaluated with phenomenological nucleon vector form factors. Infrared-finiteness and gauge-invariance require the inclusion of the wave-function renormalization factor from the self-energy diagram. Using recent empirical form factor parametrizations, the corrections amount to and . We also study the effects from photon loops with internal isobars. For two customary versions of the vertex and spin-3/2 propagator, these radiative corrections have values of and , respectively. Taking these ranges as a systematic error, our estimates read and .

Single-Particle Properties of a Strongly Interacting Bose–Fermi Mixture Above the BEC Phase Transition Temperature

We theoretically investigate the normal state properties of a Bose–Fermi mixture with a strong attractive interaction between Fermi and Bose atoms. We extend the ordinary T-matrix approximation (TMA) with respect to Bose–Fermi pairing fluctuations, to include the Hugenholtz–Pines’ relation for all Bose Green’s functions appearing in TMA self-energy diagrams. This extension is shown to be essentially important to correctly describe the physical properties of the Bose–Fermi mixture, especially near the Bose–Einstein condensation instability. Using this improved TMA, we clarify how the formation of composite fermions affects Bose and Fermi single-particle excitation spectra, over the entire interaction strength.

Complex-mass shell renormalization of the higher-derivative electrodynamics

We consider a higher-derivative extension of QED modified by the addition of a gauge-invariant dimension-6 kinetic operator in the U(1) gauge sector. The Feynman diagrams at one-loop level are then computed. The modification in the spin-1 sector leads the electron self-energy and vertex corrections diagrams finite in the ultraviolet regime. Indeed, no regularization prescription is used to calculate these diagrams because the modified propagator always occurs coupled to conserved currents. Moreover, besides the usual massless pole in the spin-1 sector, there is the emergence of a massive one, which becomes complex when computing the radiative corrections at one-loop order. This imaginary part defines the finite decay width of the massive mode. To check consistency, we also derive the decay length using the electron--positron elastic scattering and show that both results are equivalent. Because the presence of this unstable mode, the standard renormalization procedures cannot be used and is necessary adopt an appropriate framework to perform the perturbative renormalization. For this purpose, we apply the complex-mass shell scheme (CMS) to renormalize the aforementioned model. As an application of the formalism developed, we estimate a quantum bound on the massive parameter using the measurement of the electron anomalous magnetic moment and compute the Uehling potential. At the end, the renormalization group is analyzed.

Adaptive Integration and Singular Boundary Transformations

We apply and compare results of transformations used to annihilate boundary singularities for multivariate integration over hyper-rectangular and simplicial domains. While classically these transformations are applied with a product trapezoidal rule, we use adaptive methods in the PARINT software package, based on rules of higher polynomial degree for the integration over subdomains. PARINT is layered over the MPI environment (Message Passing Interface) and deploys advanced parallel computation techniques such as load balancing among processes that are distributed over a network of nodes. The message passing is performed in a non-blocking and asynchronous manner, and permits overlapping of computation and communication. Comparisons of computation times using long double vs. double precision confirm that the extended format does not considerably increase the time for long doubles. We further apply the proposed methods to problems arising from self-energy Feynman loop diagrams with massless internal lines, in particular where the corresponding integrand has singularities on the boundaries of the integration domain.

Complex-mass renormalization in hadronic EFT: Applicability at two-loop order

We discuss the application of the complex-mass scheme to multi-loop diagrams in hadronic effective field theory by considering as an example a two-loop self-energy diagram. We show that the renormalized two-loop diagram satisfies the power counting.

Automatic numerical integration methods for Feynman integrals through 3-loop

We give numerical integration results for Feynman loop diagrams through 3-loop such as those covered by Laporta [1]. The methods are based on automatic adaptive integration, using iterated integration and extrapolation with programs from the QUADPACK package, or multivariate techniques from the ParInt package. The Dqags algorithm from QuadPack accommodates boundary singularities of fairly general types. PARINT is a package for multivariate integration layered over MPI (Message Passing Interface), which runs on clusters and incorporates advanced parallel/distributed techniques such as load balancing among processes that may be distributed over a network of nodes. Results are included for 3-loop self-energy diagrams without IR (infra-red) or UV (ultra-violet) singularities. A procedure based on iterated integration and extrapolation yields a novel method of numerical regularization for integrals with UV terms, and is applied to a set of 2-loop self-energy diagrams with UV singularities.

A development of an accelerator board dedicated for multi-precision arithmetic operations and its application to Feynman loop integrals

Higher order corrections in perturbative quantum field theory are required for precise theoretical analysis to investigate new physics beyond the Standard Model. This indicates that we need to evaluate Feynman loop diagrams with multi-loop integrals which may require multi-precision calculation. We developed a dedicated accelerator system for multiprecision calculations (GRAPE9-MPX). We present performance results of our system for the case of Feynman two-loop box and three-loop selfenergy diagrams with multi-precision.