The 4-loop QED mass and wave function renormalization constants $Z_2$ and $Z_m$ have been evaluated in the on-shell subtraction scheme with 1100 digits of precision. We also worked out the coefficients of the five color structures of the QCD renormalization constants $Z_2^{OS}$ and $Z_m^{OS}$ which can obtained from QED-like diagrams. The results agree with lower precision results available in the literature. Analytical fits were also obtained for all these quantities.
The general features of renormalization and the renormalization group in QED and in general quantum field theories in curved spacetime with additional Lorentz- and CPT-violating background fields are reviewed.
As a precursor to work on QCD, we study the dressed electron in QED nonperturbatively. The calculational scheme uses an invariant mass cutoff, discretized light-cone quantization, a Tamm--Dancoff truncation of the Fock space, and a small photon mass. Nonperturbative renormalization of the coupling and electron mass is developed.
We compute the on-shell wave function renormalization constant to four-loop order in QCD and present numerical results for all coefficients of the SU$(N_c)$ colour factors. We extract the four-loop HQET anomalous dimension of the heavy quark field and also discuss the application of our result to QED.
We describe a calculation of the on-shell renormalization factors in QCD and QED at the three loop level. Explicit results for the fermion mass renormalization factor Zm and the on-shell fermion wave function renormalization constant Z2 are given. We find that at O(alpha_s^3) the wave function renormalization constant Z2 in QCD becomes gauge dependent also in the on-shell scheme, thereby disproving the ``gauge-independence'' conjecture based on an earlier two-loop result. As a byproduct, we derive an O(alpha_s^3) contribution to the anomalous dimension of the heavy quark field in HQET.
We present the estimates of the five-loop QED corrections to the muon anomaly using the scheme-invariant approaches and demonstrate that they are in good agreement with the results of exact calculations of the corresponding tenth-order diagrams supplemented by the additional guess about the values of the non-calculated contributions.
The large N_f self-consistency programme is reviewed. As an application the QCD beta-function is computed at O(1/N_f) and the anomalous dimensions of polarized twist-2 singlet operators are determined at the same order.
This is a pedagogical introduction to NRQED, a low enery approximation of QED which can be made to reproduce QED to an arbitrary precision. It is especially useful when applied to nonrelativistic bound states. We start by explaining why QED is so difficult to apply to nr bound states, why NRQED makes it much simpler and then proceed to do an explicit calculation. We also briefly discuss what this can teach us about ``new'' physics beyond QED. This is based on a talk given at the XIV MRST meeting in Toronto.
Despite its very short lifetime positronium provides us with a number of accurate tests of the bound state QED. In this note a brief overview of QED theory and precision experiments on the spectrum and annihilation decay of the positronium atom is presented. Special attention is paid to the accuracy of theoretical predictions.
We obtain analytical five-loop results for the renormalization group $β$-function of Quantum Electrodynamics with the single lepton in different renormalization schemes. The theoretical consequences of the results obtained are discussed.
We study the physical electron in quantum electrodynamics expanded on the light-cone Fock space in order to address two problems: (1) the physics of the electron's anomalous magnetic moment $a_e$ in nonperturbative QED, and (2) the practical problems of ultraviolet regularization and renormalization in truncated nonperturbative light-cone Hamiltonian theory. We present results for $a_e$ computed in a light-cone gauge Fock space truncated to include one bare electron and at most two photons; i.e., up to two photons in flight. The calculational scheme uses an invariant mass cutoff, discretized l…
With an eye toward LHC processes in which theoretical precisions of 1 percent are desired, we introduce the theory of the simultaneous YFS resummation of QED and QCD to compute the size of the expected resummed soft radiative threshold effects in precision studies of heavy particle production at the LHC. Our results show that both QED and QCD soft threshold effects must be controlled to be on the conservative side to achieve such precision goals.
As an unstable light pure leptonic system, positronium is a very specific probe atom to test bound state QED. In contrast to ordinary QED for free leptons, the bound state QED theory is not so well understood and bound state approaches deserve highly accurate tests. We present a brief overview of precision studies of positronium paying special attention to uncertainties of theory as well as comparison of theory and experiment. We also consider in detail advantages and disadvantages of positronium tests compared to other QED experiments.
The combination of energetic electron beams, delivered from conventional accelerators at a high repetition rate, and ultraintense lasers, makes it possible to perform precision measurements of strong-field QED. The LUXE collaboration aims to perform precision measurements of nonlinear Compton scattering and Breit-Wheeler pair creation in the transition from the perturbative to nonperturbative regimes. Here we present an overview of recent developments in the modelling of strong-field QED processes, which are needed to reach the required precision of a few percent for intensity parameters $0.1 …
We study the chiral critical behaviors of QED by using Non-Perturbative Renormalization Group (NPRG). Taking account of the non-ladder contributions, our flow equations are free from the gauge parameter ($α$) dependence. We clarify the chiral phase structure, and calculate the anomalous dimension of $\barψψ$, which is enhanced compared to the ladder approximation. We find that the cutoff scheme dependence of the physical results is very small.
In this paper we calculate at high-precision the Laurent expansions in e=(4-D)/2 of the 17 master integrals which appeared in the analytical calculation of 3-loop QED contribution to the electron g-2, using difference and differential equations. The coefficients of the expansions so obtained are in perfect agreement with all the analytical expressions already known. The values of coefficients not previously known will be used in the high-precision calculation of the 4-loop QED contribution to the electron g-2.
A non-minimal coupling $η$ has been attracting growing interest particularly in the context of inflation models, though its quantum nature is not clear yet. We study the renormalization of a non-minimal coupling in the scalar quantum electrodynamics (QED). We find ${\it no}$ inhomogeneous term of the renormalization group equation (RGE) at the two-loop level. This is similar to other theories, where an inhomogeneous term of the RGE appears only at a higher-loop order: e.g., four-loop order in $λϕ^{4}$ theory.
We analyze a formulation of QED based on the Wilson renormalization group. Although the ``effective Lagrangian'' used at any given scale does not have simple gauge symmetry, we show that the resulting renormalized Green's functions correctly satisfies Ward identities to all orders in perturbation theory. The loop expansion is obtained by solving iteratively the Polchinski's renormalization group equation. We also give a new simple proof of perturbative renormalizability. The subtractions in the Feynman graphs and the corresponding counterterms are generated in the process of fixing the physica…
We extend the method of differential renormalization to massive quantum field theories treating in particular $\ph4$-theory and QED. As in the massless case, the method proves to be simple and powerful, and we are able to find, in particular, compact explicit coordinate space expressions for the finite parts of two notably complicated diagrams, namely, the 2-loop 2-point function in $\ph4$ and the 1-loop vertex in QED.
We present a discussion of the precision for the PHOTOS Monte Carlo algorithm, with improved implementation of QED interference and multiple-photon radiation. The main application of PHOTOS is the generation of QED radiative corrections in decays of any resonances, simulated by a "host" Monte Carlo generator. By careful comparisons automated with the help of the MC-TESTER tool specially tailored for that purpose, we found that the precision of the current version of PHOTOS is of 0.1% in the case of Z and W decays. In the general case, the precision of PHOTOS was also improved, but this will no…
We present results from a study of subtractive renormalization of the fermion propagator Dyson-Schwinger equation (DSE) in massive strong-coupling quenched QED$_4$. Results are compared for three different fermion-photon proper vertex {\it Ansätze\/}: bare $γ^μ$, minimal Ball-Chiu, and Curtis-Pennington. The procedure is straightforward to implement and numerically stable. This is the first study in which this technique is used and it should prove useful in future DSE studies, whenever renormalization is required in numerical work.
We study a QED extension that is unitary, CPT invariant and super-renormalizable, but violates Lorentz symmetry at high energies, and contains higher-dimension operators (LVQED). Divergent diagrams are only one- and two-loop. We compute the one-loop renormalizations at high and low energies and analyse the relation between them. It emerges that the power-like divergences of the low-energy theory are multiplied by arbitrary constants, inherited by the high-energy theory, and therefore can be set to zero at no cost, bypassing the hierarchy problem.
We consider the most accurate tests of bound state QED, precision theory of simple atoms, related to the hyperfine splitting in light hydrogen-like atoms. We discuss the HFS interval of the 1s state in muonium and positronium and of the 2s state in hydrogen, deuterium and helium-3 ion. We summarize their QED theory and pay attention to involved effects of strong interactions. We also consider recent optical measurements of the 2s HFS interval in hydrogen and deuterium.
We study dynamical fermion mass generation in (2+1)-dimensional quantum electrodynamics with a gauge field coupling to massless Dirac fermions and non-relativistic scalar bosons. We calculate the fermion velocity renormalization and then examine its influence on dynamical mass generation by using the Dyson-Schwinger equation. It is found that dynamical mass generation takes place even after including the scalar bosons as long as the bosonic compressibility parameter $ξ$ is sufficiently small. In addition, the fermion velocity renormalization enhances the dynamically generated mass.
We investigate textbook QED in the framework of the exact renormalization group. In the strong-coupling region, we study the influence of fluctuation-induced photonic and fermionic self-interactions on the nonperturbative running of the gauge coupling. Our findings confirm the triviality hypothesis of complete charge screening if the ultraviolet cutoff is sent to infinity. Though the Landau pole does not belong to the physical coupling domain owing to spontaneous chiral symmetry breaking (chiSB), the theory predicts a scale of maximal UV extension of the same order as the Landau pole scale. In…
We extend a previous Landau-gauge study of subtractive renormalization of the fermion propagator Dyson-Schwinger equation (DSE) in strong-coupling, quenched QED_4 to arbitrary covariant gauges. We use the fermion-photon proper vertex proposed by Curtis and Pennington with an additional correction term included to compensate for the small gauge-dependence induced by the ultraviolet regulator. We discuss the chiral limit and the onset of dynamical chiral symmetry breaking in the presence of nonperturbative renormalization. We extract the critical coupling in several different gauges and find evi…
An approximation algorithm is proposed to transform truncated QCD (or QED) series for observables. The approximation is a modification of the Baker-Gammel approximants, and is independent of the renormalization scale (RScl) $μ$ -- the coupling parameter $α(μ)$ in the series and in the resulting approximants can evolve according to the perturbative renormalization group equation (RGE) to any chosen loop order. The proposed algorithm is a natural generalization of the recently proposed method of diagonal Padé approximants, the latter making the result RScl-invariant in large-$β_0$ approximation …
The non-renormalization theorem of chiral vertices and the generalized non-renormalization theorem of the photon self energy are derived in SQED on the basis of algebraic renormalization. For this purpose the gauge coupling is extended to an external superfield. This extension already provides detailed insight into the divergence structure. Moreover, using the local supercoupling together with an additional external vector multiplet that couples to the axial current, the model becomes complete in the sense of multiplicative renormalization, with two important implications. First, a Slavnov--Ta…
We study the one-loop level renormalization of the electrodynamics of spin 1/2 fermions in the Poincaré projector formalism, in arbitrary covariant gauge and including fermion self-interactions, which are dimension four operators in this framework. We show that the model is renormalizable for arbitrary values of the tree level gyromagnetic factor g within the validity region of the perturbative expansion, αg^2 << 1. In the absence of tree level fermion self-interactions, we recover the pure QED of second order fermions, which is renormalizable only for |g|=2. Turning off the electromagnetic in…
In some processes at the LHC, theoretical precisions of 1 percent are desired. With an eye toward such precisions, we introduce the theory of the simultaneous YFS resummation of QED and QCD to compute the size of the expected resummed soft radiative threshold effects in precision studies of heavy particle production at the LHC. Our results, that the soft QED threshold effects are at the level of 0.3 percent whereas the soft QCD threshold effects enter at the level of 20 percent, show that both must be controlled to be on the conservative side to achieve such precision goals.
The lectures are a practical introduction to perturbative calculations in QED and QCD. I discuss methods of calculation of one- and two-loop diagrams in dimensional regularization, MSbar and on-shell renormalization schemes, decoupling of heavy-particle loops.
It is commonly asserted that the electromagnetic current is conserved and therefore is not renormalized. Within QED we show (a) that this statement is false, (b) how to obtain the renormalization of the current to all orders of perturbation theory, and (c) how to correctly define an electron number operator. The current mixes with the four-divergence of the electromagnetic field-strength tensor. The true electron number operator is the integral of the time component of the electron number density, but only when the current differs from the MSbar-renormalized current by a definite finite renorm…
We summarize the recent progress in a new approach to precision LHC physics based on the IR-improved DGLAP-CS theory as it relates to a new MC friendly exponentiated scheme for precision calculation of higher order corrections to LHC physics in which IR singularities from both QED and QCD are canceled to all orders in αand in α_s simultaneously in the presence of rigorous shower/ME matching. We present the first MC data comparing the implied new showers themselves with the standard ones using the HERWIG6.5 MC event generator as a test case at LHC energies.
We study the effective parameters of QED near decoupling temperatures and show that the QED perturbative series is convergent, at temperatures below the decoupling temperature. The renormalization constant of QED acquires different values if a system cools down from a hotter system to the electron mass temperature or heats up from a cooler system to the same temperature. At T = m, the first order contribution to the electron selfmass, δm/m is 0.0076 for a heating system and 0.0115 for a cooling system and the difference between two values is equal to 1/3 of the low temperature value and 1/2 of…
Despite its simplicity, the unitary gauge is not a popular choice for practical loop calculations in gauge theories, due to the lack of off-shell renormalizability. We study the renormalization properties of the off-shell Green functions of the elementary electron fields in the massive QED, in order to elucidate the origin and structure of the extra ultraviolet divergences which exist only in the unitary gauge. We find that all these divergences affect the Green functions in a trivial way such that in coordinate space the off-shell Green functions are in fact multiplicatively renormalizable. T…
I have evaluated up to 1100 digits of precision the contribution of the 891 4-loop Feynman diagrams contributing to the electron $g$-$2$ in QED. The total mass-independent 4-loop contribution is $ a_e = -1.912245764926445574152647167439830054060873390658725345{\ldots} \left(\fracαπ\right)^4$. I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument $e^{\frac{iπ}{3}}$, $e^{\frac{2iπ}{3}}$, $e^{\frac{iπ}{2}}$, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated …
The application of renormalization group techniques to bound states in non-relativistic QED and QCD is discussed. For QED bound states like Hydrogen and positronium, the renormalization group allows large logarithms of the velocity, ln v (or equivalently ln alpha's), to be predicted in a universal and simple way. The series of (alpha ln alpha)'s are shown to terminate after a few terms. For QCD one can systematically sum infinite series of the form [alpha_s ln alpha_s]^k$, and answer definitively the question ``alpha_s at what scale?''.
Erratum: Exact solutions to the fermion propagator Schwinger-Dyson equation in Minkowski space with on-shell renormalization for quenched QED \newline [Phys. Rev. D 96, 036021 (2017)]
A comparison of the perturbative series and the 1/N expansion for the QED renormalization group β-function in the Minimal Subtraction scheme is performed. The good agreement between two expansions is found which proves that the MS β-function is under perfect perturbative control.
We study renormalized quenched strong-coupling QED in four dimensions in arbitrary covariant gauge, in the Dyson-Schwinger equation formalism. Above the chiral critical coupling, we show that there is no finite chiral limit. This behaviour is found to be independent of the detailed choice of proper vertex, provided that the vertex is consistent with the Ward-Takahashi identity and multiplicative renormalizability. The finite solutions previously reported lie in an unphysical regime of the theory with multiple solutions and oscillating mass functions. This study is consistent with the assertion…
Because of properties of QED, the bremsstrahlung corrections to decays of particles or resonances can be calculated, with a good precision, separately from other effects. Thanks to the widespread use of event records such calculations can be embodied into a separate module of Monte Carlo simulation chains, as used in High Energy Experiments of today. The PHOTOS Monte Carlo program is used for this purpose since nearly 20 years now. In the following talk let us review the main ideas and constraints which shaped the program version of today and enabled it widespread use. We will concentrate spec…
We study the four most significant high precision observables of QED ---the anomalous electron and muon magnetic moments, the hydrogen Lamb shift and muonium hyperfine splitting--- in the context of SU(2) x U(1) gauge-invariant effective Lagrangians. The agreement between the theoretical predictions for these observables and the experimental data places bounds on the lowest dimension operators of the effective Lagrangians. We also place bounds on such effective operators using other experimental data. Comparison of the tow types of bounds allows us to discuss the potential of each one of the f…
With the introduction of a spectral representation, the Schwinger-Dyson equation (SDE) for the fermion propagator is formulated in Minkowski space in QED. After imposing the on-shell renormalization conditions, analytic solutions for the fermion propagator spectral functions are obtained in four dimensions with a renormalizable version of the Gauge Technique anzatz for the fermion-photon vertex in the quenched approximation in the Landau gauge. Despite the limitations of this model, having an explicit solution provides a guiding example of the fermion propagator with the correct analytic struc…
We investigate the dynamical generation of fermion mass in quantum electrodynamics (QED). This non-perturbative study is performed using a truncated set of Schwinger-Dyson equations for the fermion and the photon propagator. First, we study dynamical fermion mass generation in quenched QED with the Curtis-Pennington vertex, which satisfies the Ward-Takahashi identity and ensures the multiplicative renormalizability of the fermion propagator. We apply bifurcation analysis to determine the critical point for a general covariant gauge. In the second part of this work we investigate the dynamical …
Quantum electrodynamics is the first successful and still the most successful quantum field theory. Simple atoms, being essentially QED systems, allow highly accurate theoretical predictions. Because of their simple spectra, such atoms have been also efficiently studied experimentally frequently offering the most precisely measured quantities. Our review is devoted to comparison of theory and experiment in the field of precision physics of light simple atoms. In particular, we consider the Lamb shift in the hydrogen atom, the hyperfine structure in hydrogen, deuterium, helium-3 ion, muonium an…
We have evaluated with 1100 digits of precision the contribution of all the 891 mass-independent four-loop Feynman diagrams contributing to the slope of the Dirac form factor in QED. The total 4-loop contribution is $ m^2 F_1^{(4)'}(0) = 0.886545673946443145836821730610315359390424032660064745... \left(\fracαπ\right)^4$. We have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument $e^{\frac{iπ}{3}}$, $e^{\frac{2iπ}{3}}$, $e^{\frac{iπ}{2}}$, one-dimensional integrals of products of complete elliptic integrals and six finite parts o…
The velocity renormalization group is used to determine ln(alpha) contributions to QED bound state energies. The leading order anomalous dimension for the potential gives the alpha^5 ln(alpha) Bethe logarithm in the Lamb shift. The next-to-leading order anomalous dimension determines the alpha^6 ln(alpha), alpha^7 ln^2(alpha), and alpha^8 ln^3 (alpha) corrections to the energy. These are used to obtain the alpha^8 ln^3(alpha) Lamb shift and alpha^7 ln^2(alpha) hyperfine splitting for Hydrogen, muonium and positronium, as well as the alpha^2 ln(alpha) and alpha^3 ln^2(alpha) corrections to the …
A hidden generalized gauge symmetry of a cutoff QED is used to show the renormalizability of QED. In particular, it is shown that corresponding Ward identities are valid all along the renormalization group flow. The exact Renormalization Group flow equation corresponding to the effective action of a cutoff lambda phi^4 theory is also derived. Generalization to any gauge group is indicated.
The semi-analytical expression for the forth coefficient of the renormalization group $β$-function in the ${\rm{V}}$-scheme is obtained in the case of the $SU(N_c)$ gauge group. In the process of calculations we use the three-loop perturbative approximation for the QCD static potential, evaluated in the $\rm{\overline{MS}}$-scheme. The importance of getting more detailed expressions for the $n_f$-independent three-loop contribution to the static potential,obtained at present by two groups, is emphasised. The comparison of the numerical structure of the four-loop approximations for the RG $β$- …
I review recent estimates of the non-perturbative hadronic vacuum polarization contributions. Since these at present can only be evaluated in terms of experimental data of limited precision, the related uncertainties pose a serious limitation in our ability to make precise predictions. Besides e+e- - annihilation data also tau decay spectra can help to get better predictions. Here, it is important to account for all possible iso-spin violations in tau - decay spectra, from which e+e- cross sections may be obtained by an iso-spin rotation. The observed 10% discrepancy in the region above the rh…
We demonstrate our simple strategy for renormalization with QED at one-loop level, basing on an elaboration of the effective field theory philosophy. No artificial regularization or deformation of the original theory is introduced here and hence no manipulation of infinities, ambiguities arise instead of infinities. Ward identities first come to reduce the number of ambiguities, the residual ones could in principle be removed by imposing physical boundary conditions. Renormalization group equations arise as "decoupling theorems" in the underlying theory perspective. In addition, a technical th…
For supersymmetric gauge theories a consistent regularization scheme that preserves supersymmetry and gauge invariance is not known. In this article we tackle this problem for supersymmetric QED within the framework of algebraic renormalization. For practical calculations, a non-invariant regularization scheme may be used together with counterterms from all power-counting renormalizable interactions. From the Slavnov--Taylor identity, expressing gauge invariance, supersymmetry and translational invariance, simple symmetry conditions are derived that are important in a twofold respect: they est…
In this paper we briefly outline the quadrature method for estimating uncertainties in a function of several variables and apply it to estimate the numerical uncertainties in QCD-QED rescaling factors. We employ here the one-loop order in QED and three-loop order in QCD evolution equations of fermion mass renormalization. Our present calculations are found to be new and also reliable compared to the earlier values employed by various authors.
The precise knowledge of the QED initial state corrections is of instrumental importance in studying high luminosity measurements in $e^+e^-$ annihilation at facilities like LEP, the International Linear Collider ILC, CLIC, a Giga-$Z$ facility, and the planned FCC\_ee. Logarithmic corrections of up to $O(α^6 L^5)$ are necessary with various subleading terms taken into account. This applies to both the inclusive measurement of processes like $e^+ e^- \rightarrow γ^*/Z^*$ and also the forward-backward asymmetry. As has been shown recently, techniques from massive QCD, such as the computation of …
We present a brief overview of precision tests of bound state QED and mainly pay our attention to laser spectroscopy as an appropriate tool for these tests. We particularly consider different precision tests of bound state QED theory based on the laser spectroscopy of optical transitions in hydrogen, muonium and positronium and related experiments.
Renormalization scheme of QED (Quantum Electrodynamics) at high temperatures is used to calculate the effective parameters of relativistic plasma in the early universe. Renormalization constants of QED play role of effective parameters of the theory and can be used to determine the collective behavior of the medium. We explicitly show that the dielectric constant, magnetic reluctivity, Debye length and the plasma frequency depend on temperature in the early universe. Propagation speed, refractive index, plasma frequency and Debye shielding length of a QED plasma are computed at extremely high …
The expected experimental precision of the rates and asymmetries in the Future Circular Collider with electron positron beams (FCC-ee) in the centre of the mass energy range 88-365GeV considered for construction in CERN, will be better by a factor 5-200. This will be thanks to very high luminosity, factor up to $10^5$ higher than in the past LEP experiments. This poses the extraordinary challenge of improving the precision of the Standard Model predictions by a comparable factor. In particular the perturbative calculations of the trivial QED effects, which have to be removed from the experimen…
