Dimensional regularization of mass and infrared singularities in two-loop on-shell vertex functions

We show how massless two-loop on-shell vertex functions can be calculated in a very elegant way if mass and infrared singularities are regularized by n-dimensional regularization. Using dispersion methods one is able to express the Feynman integrals in a product of gamma functions. As an application of our techniques we have calculated a two-loop on-shell form factor. Its infrared behaviour will be compared with the predictions of some resummation formulae which have been derived in the literature.     

Infrared problems in quantum electrodynamics

We present a self-contained treatment of the infrared problem in Quantum Electrodynamics. Our program includes a derivation and proof of finiteness of modified reduction formulae for scattering in Coulomb potentials and unitary extensions of the relativistic Coulomb amplitudes in the forward direction. The renormalization structure of the theory is discussed in connection with the infrared problem and the renormalization group is reconsidered and shown to be inadequate for the “improvement” of perturbation theoretic results. However, simple forms of the renormalization group equations are easily established, which allow for a simple discussion of the renormalization structure and the extraction of physical quantities out of Green functions normalized at an arbitrary mass μ < m (m is the fermion mass). As an example of such a quantity we consider the construction of a renormalized and infrared finite mass-operator in presence of external fields. Scattering theory in Quantum Electrodynamics is elaborated in the context of the coherent state formulation of the asymptotic condition. Dimensional regularization techniques are systematically used for the reduction of coherent states and the construction of S-matrix elements and the cross-section formulae. The latter are obtained in a relatively simple form, which allows for a direct comparison with the exact cross-section formulae derived in the traditional context. This establishes the equivalence of the two approaches at the cross-section level. Various applications illustrate the techniques presented here and relative topics are discussed.     

Perturbative aspects of dilepton production at finite temperature

First order corrections to the Born term production rate of dileptons in a Quark Gluon Plasma are examined in the framework of finite temperature perturbative QCD. While infrared divergences disappear, mass singularities do not cancel, which might cause large logarithmic terms to appear at order αs     

Hadron mass calculations with Susskind and Wilson fermions in the fundamental-adjoint plane

We report the results of hadron mass calculations in the valence (quenched) approximation, on an 8³ × 16 lattice. For Wilson fermions with the standard Wilson action we find good agreement with results form the hopping parameter expansion on a 16⁴ lattice at β = 5.7, with only a small finite size effect in the anticipated direction. The proton-to-rho mass ration is however too high by 60% and the delta-proton mass difference is too small. We have repeated these calculations at two points of the same string tension in the plane of the fundamental and adjoint couplings, in an attempt to avoid the unphysical critical point there; within statistical errors the meson masses remain the same, there is an improvement in the delta-proton splitting and the proton mass decreases slightly, but not enough to produce agreement with experiment. The estimates of lines of constant string tension obtained as a preliminary to the mass calculation are in good agreement with weak coupling expansions at the larger β values and cross over towards strong coupling predictions around β = 5.7. Also, crude estimates reveal the disappearance of the specific heat peak as one moves away from the unphysical singularity. For Susskind fermions we find some measure of agreement with other results form a 10³ × 16 lattice, but large, apparently finite size, effects in the rho mass at low quark mass. Meson masses in lattice units disagree with the Wilson fermion results by as much as a factor of 2. This disagreement persists in the fundamental-adjoint plane, suggesting the importance of studying improved fermion actions. At β = 6.0, Wilson fermion results show clear finite size effects on the 8³ × 16 lattice.     

Infrared and mass singularities cancellation in finite-temperature and -density radiative corrections to the MSW effect

Finite-temperature and -density radiative corrections to orderα to the charged-current MSW process are considered in the real-time version of thermal field theory. We confirm the absence of infrared and electron mass singularities in the real part of thev _e self-energy, calculated in a background of electrons at finite chemical potential. As a consequence, this indicates that at orderα the perturbation series with respect to electromagnetic interactions is adequate for a coherent process.     

Nonanalyticity of the perturbative expansion for super-renormalizable massless field theories

The complete infrared expansion of Feynman amplitudes is established at any dimension d. The so called infrared finite parts develop poles at rational d. We prove a conjecture by Parisi by constructing an infrared subtraction procedure which defines finite amplitudes in such dimensions. The corresponding counterterms are associated to nonlocal operators and are generated in a nonperturbative way for super-renormalizable theories. We determine at all orders the perturbative expansion which contains powers and logarithms of the coupling constant.     

Magnets with random uniaxial anisotropy: Large-N effective potential and infrared properties

We study the magnetic properties of systems with random uniaxial anisotropy using a large-N effective potential approach for d = 4, 3.It is found that the random interactions induce a strong infrared behavior that prevents the existence of a ferromagnetic phase and massless transverse modes.The transverse susceptibility is finite for all values of the temperature and at d = 4 it has an esential singularity in the couplings. We argue that this is indicative of a mechanism of dynamical mass generation due to the infrared instabilities of the theory.For both d = 4, 3 there is a spin-glass low-temperature phase and a paramagnetic high-temperature phase, the susceptibility having a cusp across the transition.We prove that these phases are stable and that the transverse and longitudinal susceptibilities are equal.     

Dynamical mass generation in three-dimensional Yang-Mills theory

We calculate an effective potential for three-dimensional Yang-Mills theory to two loops. The result implies that a gauge invariant part of the field strength has a non-zero vacuum expectation value, φ. The two-point function of this field, calculated to one loop, shows that two polarization states propagate with a mass proportional to √φ. In contrast to the usual Higgs mechanism the remaining degrees of freedom do not propagate. This dynamical generation of a “glueball” mass renders the already ultraviolet finite theory infrared finite and stabilizes vortices associated with the centre of the group.     

Flow equations for the relevant part of the pure Yang—Mills action

Wilson’s exact renormalization group equations are derived and integrated for the relevant part of the pure Yang-Mills action. We discuss in detail how modified Slavnov—Taylor identities control the breaking of BRST invariance in the presence of a finite infrared cutoff k through relations among different parameters in the effective action. In particular they imply a nonvanishing gluon mass term for nonvanishing k. The requirement of consistency between the renormalization group flow and the modified Slavnov—Taylor identities allows to control the self—consistency of truncations of the effective action.     

Quark-antiquark bound states within a Dyson-Schwinger Bethe-Salpeter formalism

Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is shown that an infrared finite gluon propagator can lead to quark confinement via generation of complex mass poles in quark propagators. Observables, including electromagnetic form factors, are calculated entirely in Euclidean metric for spacelike values of bound state momentum and final results are extrapolated to the physical region.     

Finite temperature radiative corrections to neutron decay and related processes

We present a detailed calculation of the radiative corrections at finite temperature to the processes nν ? pe, ne ? pν. and n ? peν. The temperature range considered is approximately 100 keV to 3 MeV, which is the range relevant for nucleosynthesis calculations. Photon absorption and emission, photonic corrections to the vertex and fermion wave-function renormalizations, as well as the correction to the electron “mass” are calculated explicitly. It is shown that infrared divergences are only cancelled when absorption and induced photon emission are included. We analyze the effect of these processes on the various reaction rates, and discuss the effects of these corrections on the calculated helium abundance in the universe.     

Perturbative QCD at high energies

A partial review of QCD at high energies is given. Factorization and the use of the renormalization group equation are emphasized. Topics discussed are the parton model, cut vertices in covariant and axial gauges, μ-pair production, jets, form factors, x → 1 limit of structure fu nctions, wide angle elastic scattering and heavy quarkonium exclusive decays. A discussion of mass and infrared singularities and a discussion of Sudakov effects are also included.     

Structure of the Gauge Invariant Quark Green’s Function in QCD2

We study, in two-dimensional QCD and in the large-N _c limit, the properties of the gauge invariant quark Green’s function, defined with a path-ordered phase factor along a straight line. The analysis is done by means of an exact integrodifferential equation. The Green’s function is found to be infrared finite, with singularities represented by an infinite number of threshold type branch points with a power ?3/2, starting at positive mass squared values. Its expression is analytically determined.     

Infrared singularities in the dimensional regularization scheme

Infrared singularities arising in some renormalized amplitudes of quantum electrodynamics are analyzed using the dimensional regularization method. We define infrared and ultraviolet convergent regions in the ν complex plane (ν is the number of dimensions of space time). It turns out that these regions do not overlap for quantum electrodynamics. Nevertheless, it is shown that there exists a unique analytic continuation from the infrared convergent region which allows us to interpret the infrared divergence in the renormalized electron self-energy amplitude as an isolated singularity at ν = 4. This statement seems to be true at all orders of perturbation theory. We also prove that the double limit μ → 0, ν → 4 (μ is the auxiliary photon mass) does not exist in quantum electrodynamics and we conjecture that this lack of uniformity provides theoretical support for the ansatz of Marciano and Sirlin.     

Infrared and mass regularization in AF field theories (I). φ63

In the framework of asymptotically free (AF) field theories we determine the correction terms in the renormalization group (RG) approach to deep inelastic (DI) scattering and to the Drell-Yan (DY) process. This leads us to an order g² analysis of the DI/DY parton cross sections. Some of their contributions reveal ultraviolet (UV), infrared (IR) and mass (M) divergences which, after regularization, are removed by renormalization, IR and (partially) M cancellation. The initial-state M divergences, persisting in both processes, are removed by M factorization. Consistent IR and M regularization is constrained by the double-cut rule: its meaning and implications are explained. The calculations are performed in φ₆³ theory which is AF, IR finite and technically simple. We use “on-shell”, “off-shell” and “massless” (n-dimensional) mass assignments and demonstrate explicitly the regularization independence of the correction term.     

Finite temperature corrections to fleid theory: Electron mass and magnetic moment,and vacuum energy

A finite temperature formalism which maintains Lorentz covariance is rederived and compared to other approaches. It is used to obtain the order-α temperature shifts in the electron mass and magnetic moment, vacuum energy. The mass and magnetic moment shifts, for kT ? m, are of order $\text{α(}\text{kT}\text{m}\text{)}{}^2$ and hence beyond the present experimental observation limits. An explicit form for the general temperature mass shift is obtained.     

Electron scattering form factors for fp-shell nuclei

Elastic and inelastic electron-scattering form factors for multipolarities up to L = 7 and some transition-strength distributions are calculated with shell-model wave functions for about ten target nuclei in the mass range A = 52–62. The results are compared with the available measured transition strengths and (e, e') form factors.     

Analysis of autoionization resonances in the Hg 6s 2-photoionization by measurements of photoelectron polarization

It is shown that nuclear target fragmentation in proton and heavy ion induced reactions, in particular the following experimental facts concerning the mass-yield distribution can be understood in terms of a semiclassical model:(i) its independence on the mass of the projectile at approximately the same incident energies,(ii) its trend of approaching a limit at higher bombarding energies,(iii) its “U-formed” shape at sufficiently high bombarding energies. Standard methods in statistical theory of chemical equilibrium are used to calculate the mass-yield distribution for medium and heavy target nuclei in high-energy nuclear reactions where the Coulomb interaction between the fragments is taken into account selfconsistently. The result shows: The fact that the decaying rest target nucleus and its fragments are bounded objects of finite size and finite charge have significant influences, especially on the form of the mass-yield distributions.     

Non-linear sigma model in 1 + 1 dimensions,supergravity and the spinning string

The non-linear σ supersymmetric model in 1 + 1 dimensions is coupled to supergravity. When we quantize the theory, the matter fields acquire mass dynamically, which leads to the breaking of the Weyl invariance. This fact implies that the two-point functions of the gravitino and the graviton, obtained from the effective action, become non-trivial. Particularly, the two-point function of the gravitino presents a pole in the infrared region. We conjecture that this pole is related to the confinement of all supersymmetric degrees of freedom of the theory. If we restrain the integration domain of x₁ to a finite length L (breaking all invariances of the theory), there appears a mass term in the two-point function of the gravitino, which decreases exponentially with L. In this context we relate this model with that of the supersymmetric string and define a stability criterion for the latter.     

Infrared instability of QCD at finite temperatures

The infrared behaviour of the transversal gluon propagator in QCD at T≠0 is investigated within the ghost-free axial gauge. The singularities found in this propagator for the momentum p ～ g²T remain for any choice of the gauge and indicate the infrared instability of QCD at finite temperatures.