Functions In QED Research Articles

Spinor Field Singular Functions in QED with Strong External Backgrounds

We construct and study singular functions in strong-field $QED$ with two external electromagnetic fields that represent principally different types of external backgrounds, the first one belongs to the class of so-called $t$-potential electric steps (electric-like fields that are switched on and off at initial and final time instants), and the second one belongs to the class of so-called $x$-potential electric steps (time-independent electric-like fields of constant direction that are concentrated in a restricted spatial area). As the first background ($T$-constant electric field) is chosen an uniform electric field which acts during a finite time interval $T$ , whereas as the second background ($L$-constant electric field) is chosen a constant electric field confined between two capacitor plates separated by a large distance $L$. For the both cases we find \textrm{in}- and \textrm{out}-solutions of the Dirac equation in terms of light cone variables. With the help of these solutions, we construct Fock-Schwinger proper-time integral representations for all the singular functions that provide nonperturbative (with respect to the external backgrounds) calculations of any transition amplitudes and mean values of any physical quantities. Considering calculations in the $T$-constant field and in the $L$-constant field as different regularizations of the corresponding calculations in the constant uniform electric field, we have demonstrated their equivalence for sufficiently large $T$ and $L$.

Mellin-Barnes approach to hadronic vacuum polarization and gμ−2

It is shown that with a precise determination of a few derivatives of the hadronic vacuum polarization (HVP) self-energy function $\Pi(Q^2)$ at $Q^2=0$, from lattice QCD (LQCD) or from a dedicated low-energy experiment, one can obtain an evaluation of the lowest order HVP contribution to the anomalous magnetic moment of the muon $a_{\mu}^{\rm HVP}$ with an accuracy comparable to the one reached using the $e^+ e^-$ annihilation cross-section into hadrons. The technique of Mellin-Barnes approximants (MBa) that we propose is illustrated in detail with the example of the two loop vacuum polarization function in QED. We then apply it to the first few moments of the hadronic spectral function obtained from experiment and show that the resulting MBa evaluations of $a_{\mu}^{\rm HVP}$ converge very quickly to the full experimental determination.

Vector correlator in massless QCD at order $ \mathcal{O}\left( {\alpha_s^4} \right) $ and the QED β-function at five loop

We present a concise summary of recent results for the vector correlator in massless QCD at order ${\cal O}(\alpha_s^4)$, with all colour factors being given for a generic colour group. As a direct consequence we arrive at: (i) the full QCD contribution to the QED $\beta$--function of order $\alpha^2\, \alpha_s^4$ in the MSbar- and MOM-schemes; (ii) the full five-loop result of order $\alpha^6$ for the $\beta$-function of QED with a generic number of single-charged fermions, again for the MSbar- and MOM-schemes.

Properties of Physical Systems: Transient Singularities on Borders and Surface Transitive Zones

Certain alternative properties of physical systems are describable by supports of arguments of response functions (e.g. light cone, borders of media) and expressed by projectors; corresponding equations of restraints lead to dispersion relations, theorems of counting, etc. As supports are measurable, their absolutely strict borders contradict the spirit of quantum theory and their quantum evolution leading to appearance of subtractions or certain needed flattening would be considered. “Flattening” of projectors introduce transitive zones that can be examined as a specification of adiabatic hypothesis or the Bogoliubov regulatory function in QED. For demonstration of their possibilities the phenomena of refraction and reflection of electromagnetic wave are considered; they show, in particular, the inevitable appearing of double electromagnetic layers on all surfaces that formerly were repeatedly postulated, etc. Quantum dynamics of projectors proves the neediness of subtractions that usually are artificially adding and express transient singularities and zones in squeezed forms.

Analytic, nonperturbative, almost exact QED: The two-point functions

Based on the choice of a special gauge, in which a useful form of scaling invariance holds, a new method is suggested for the analytic, nonperturbative calculation of the n-point functions of QED. A modified functional analysis is employed in configuration space, where the dressed electron and photon propagators (in quenched approximation) are each found to be simple products of the relevant free propagator with an appropriate function of configuration space variables containing all powers of the square of the coupling constant.

Various Full Green Functions in QED

We are interested in deriving various full Green functions through general Ward–Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the various WTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields.

On the two-point Green function in QED

It is well known how multiplicative renormalizability of the fermion propagator results in a simple form of the corresponding Schwinger-Dyson equation in the momentum space in massless quenched quantum electrodynamics in 4-dimensions (QED4). We extend this idea to arbitrary dimensions d. The constraint of multiplicative renormalizability of the fermion propagator is generalized to a Landau-Khalatnikov-Fradkin transformation law in d-dimensions

Erratum: Four-point spectral functions and Ward identities in hot QED [Phys. Rev. D61, 085013 (2000)

We derive spectral representations for the different components of the 4-point function at finite temperature in the real time formalism in terms of five real spectral densities. We explicitly calculate all these functions in QED in the hard thermal loop approximation. The Ward identities obeyed by the 1-loop 3- and 4-point functions in real time and their spectral functions are derived. We compare our results with those derived previously in the imaginary-time formalism for retarded functions in hot QCD, and we discuss the generalization of our results to non-equilibrium situations.

The Relativistic Covariance of the Fermion Green Function and Minimal Quantization of Electrodynamics**The project supported in part by Institute of Theoretical Physics, the Chinese Academy of Sciences, the Third World Academy of Sciences and Vietnam National Research Programme in National Sciences

This paper is devoted to the one-loop calculation of the fermion Green function in QED within the framework of the minimal quantization method, based on an explicit solution of the constraint equations and the gauge-invariance principle. The relativistic invariant expression for the fermion Green function with correct analytical properties is obtained.

Why use a Hamilton approach in QCD?

We discuss QCD in the Hamiltonian frame work. We treat finite density QCD in the strong coupling regime. We present a parton-model inspired regularization scheme to treat the spectrum (θ-angles) and distribution functions in QED 1+1. We suggest a Monte Carlo method to construct low-dimensional effective Hamiltonians. Finally, we discuss improvement in Hamiltonian QCD.

Four-point spectral functions and Ward identities in hot QED

We derive spectral representations for the different components of the four-point function at finite temperature in the real time formalism in terms of five real spectral densities. We explicitly calculate all these functions in QED in the hard thermal loop approximation. The Ward identities obeyed by the one-loop three- and four-point functions in real time and their spectral functions are derived. We compare our results with those derived previously in the imaginary-time formalism for retarded functions in hot QCD, and we discuss the generalization of our results to non-equilibrium situations.

Non-singlet splitting functions in QED

Iterative solution of QED evolution equations for non-singlet electron structure functions is considered. Analytical expressions in the fourth and fifth orders are presented in terms of splitting functions. Relation to the existing exponentiated solution is discussed.

Nonlinear dynamics in three-dimensional QED and nontrivial infrared structure

In this work we consider a coupled system of Schwinger-Dyson equations for self-energy and vertex functions in QED_3. Using the concept of a semi-amputated vertex function, we manage to decouple the vertex equation and transform it in the infrared into a non-linear differential equation of Emden-Fowler type. Its solution suggests the following picture: in the absence of infrared cut-offs there is only a trivial infrared fixed-point structure in the theory. However, the presence of masses, for either fermions or photons, changes the situation drastically, leading to a mass-dependent non-trivial infrared fixed point. In this picture a dynamical mass for the fermions is found to be generated consistently. The non-linearity of the equations gives rise to highly non-trivial constraints among the mass and effective (`running') gauge coupling, which impose lower and upper bounds on the latter for dynamical mass generation to occur. Possible implications of this to the theory of high-temperature superconductivity are briefly discussed.

The infinite mass limit of the two-particle Green function in QED

The behaviour of the two-particle Green function in QED is analysed in the limit when one of the particles becomes infinitely massive. It is found that the dependences of the Green function on the relative times of the ingoing and outgoing particles factorize and that the bound-state spectrum is the same as that of the Dirac equation with the static potential created by the heavy particle. The Bethe - Salpeter wavefunction is also determined in terms of the Dirac wavefunction. The present result excludes the existence, in the above limit, of abnormal solutions due to relative time excitations as predicted by the Bethe - Salpeter equation in the ladder approximation.

Barton expansion and the path integral approach in thermal field theory.

It has been shown how on-shell forward scattering amplitudes (the ``Barton expansion'') and quantum mechanical path integral (QMPI) can both be used to compute temperature dependent effects in thermal field theory. We demonstrate the equivalence of these two approaches and then apply the QMPI to compute the high temperature expansion for the four-point function in QED, obtaining results consistent with those previously obtained from the Barton expansion.

Effective field theory description of Green and vertex functions in the infrared domain.

This article discusses applications of a path-integral approach to fermionic Green and vertex functions which has a systematic expansion in terms of contours (particle ``world lines''), thereby enabling the evaluation of all-order results in the coupling constant. Our approach, provides an effective field theoretical framework for studying the low-energy regime of the underlying dynamics. In particular, the infrared behavior of the full fermion Green functions in QED can be analyzed without imposing assumptions on the asymptotic states of the charged sector, and keeping the \ensuremath{\gamma}-matrix structure of spinorial fields intact. Analytic solutions are derived for one and n fermions and the renormalization constants and associated anomalous dimensions are determined nonperturbatively; they comply with previously obtained results from exactly solvable models (e.g., Bloch-Nordsieck QED). Within an Abelian infrared setup, a universal vertex function has been calculated which embeds the soft interactions into a nontrivial contour with an infinitesimal self-intersecting loop. Adjusting the involved scales to those of the heavy quark effective theory, the result emulates the Isgur-Wise form factor and complies with the recent CLEO II data.

Exact and approximate fermion Green's functions in QED and QCD.

That special variant of the Fradkin representation, previously defined for scalar Green's functions ${\mathit{G}}_{\mathit{c}}$(x,y\ensuremath{\Vert}A) in an arbitrary potential A(z), is here extended to the case of vector interactions and spinor Green's functions of QED and QCD. An exact representation is given which may again be approximated by a finite number N of quadratures, with the order of magnitude of the errors generated specified in advance, and decreasing with increasing N. A feature appears for both exact and approximate ${\mathit{G}}_{\mathit{c}}$[A]: the possibility of chaotic behavior of a function central to the representation, which in turn generates chaotic behavior in ${\mathit{G}}_{\mathit{c}}$[A] for certain A(z). An example is given to show how the general criterion specified here works for a known case of ``quantum chaos,'' in a potential theory context of first quantization. When the full, nonperturbative, radiative corrections of quantum field theory are included, such chaotic effects are removed.

Chiral symmetry breaking and the polarized photon structure function in QED

The structure function of a polarized photon g 1 γ ( x) satisfies a remarkable sum rule: ∫ g 1 γ ( x) dx=0. The zero arises as a cancellation between explicit (mass-induced) and anomalous chiral symmetry breaking. We analyse this cancellation by studying the helicity structure of the cross section γ∗(λ 1)+γ(λ 2)→l +(λ 3)+l −(λ 4) . The structure function g 1 γ ( x) is found to consist of two pieces, g 1,like and g 1,unlike, corresponding to final l + l − states with like and unlike helicities. The function g 1,like is constant in x, and integrates to +1 2 , while g 1,unlike is x-dependent and integrates to − −1 2 . When the target photon is virtual (| k 2| ⪢ m 2), the integral ∫ g 1,like dx drops to zero, while ∫ 1, unlike dx stays constant at the value −1 2 . We conclude that the “explicit” and “anomalous” parts of g 1 γ ( x) are uniquely associated with final l + l − states with equal or opposite helicities. The survival of the like-helicity component is a “quasi-paradox” analogous to the survival of helicity-flip bremsstrahlung at high energies. Some implications for the QCD process γ∗g→Q Q are noted.

Soft pair real and virtual infrared functions in QED.

We calculate the soft pair production analogues of the Yennie-Frautschi-Suura (YFS) real and virtual infrared functions [ital B][sub [gamma]] and [ital [tilde B]][sub [gamma]], where the latter describe the respective infrared singularities in QED to all orders in [alpha] via YFS exponentiation. In our work, we extend the discussion of [ital B][sub [gamma]] and [ital [tilde B]][sub [gamma]] by YFS to treat the case of soft pairs. The respective pair versions of [ital B][sub [gamma]] and [ital [tilde B]][sub [gamma]] are exhibited explicitly. We also discuss some possible applications to high precision [ital Z][sup 0] physics at SLC and/or LEP.

On the infrared structure of the one-fermion Green function in QED

We derive the infrared structure of the full one-fermion Green function in QED without invoking an asymptotic version of the theory. The basis of the proposed, nonperturbative, approach is the reformulation of the theory in terms of field-excitation path-integrals. The connection to the Bloch-Nordsieck model is discussed.