Calculation of the one- and two-loop lamb shift for arbitrary excited hydrogenic states

General expressions for quantum electrodynamic corrections to the one-loop self-energy [of order alpha(Zalpha)6] and for the two-loop Lamb shift [of order alpha2(Zalpha)6] are derived. The latter includes all diagrams with closed fermion loops. The general results are valid for arbitrary excited non-S hydrogenic states and for the normalized Lamb shift difference of states, defined as Delta N = n3deltaE(nS) - delta E(1S). We present numerical results for one-loop and two-loop corrections for excited S, P, and D states. In particular, the normalized Lamb shift difference of states is calculated with an uncertainty of order 0.1 kHz.     

Leading logarithmic contribution to the second-order lamb shift induced by the loop-after-loop diagram

The contribution of order alpha(2)(Zalpha)(6)ln (3)(Zalpha)(-2) to the ground-state Lamb shift in hydrogen induced by the loop-after-loop diagram is evaluated analytically. An additional contribution of this order is found compared to the previous calculation by Karshenboim [Sov. Phys. JETP 76, 541 (1993)]. As a result, agreement is achieved for this correction between different numerical and analytical methods.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.``     

Heavy-quarkonium creation and annihilation with O(alpha(3)(s)ln(alpha(s)) accuracy

We calculate the O(alpha(3)(s)ln(alpha(s)) contributions to the heavy-quarkonium production and annihilation rates. Our result sheds new light on the structure of the high-order perturbative corrections and opens a new perspective for a high-precision theoretical analysis. We also determine the three-loop anomalous dimensions of the nonrelativistic vector and pseudoscalar currents.     

Anomalous Dimension in the Solution of a Nonlinear Diffusion Equation

A new method is used to obtain the anomalous dimension in the solution of the nonlinear diffusion equation.The result is the same as that in the renormalization group (RG) approach.It gives us an insight into the anomalous dimension in the solution of the nonlinear diffusion equation in the RG approach.Based on this discussion,we can see anomalous dimension appears naturally in this system.     

O(alpha(3) lnalpha) corrections to muonium and positronium hyperfine splitting

We compute O(alpha(3)lnalpha) relative corrections to the ground-state hyperfine splitting of a QED two-body bound state with different masses of constituents. The general result is then applied to muonium and positronium. In particular, a new value of the muon-to-electron mass ratio is derived from the muonium ground-state hyperfine splitting.     

General theory of radiative corrections to atomic decay rates

A general expression is derived for the radiative corrections to the one-photon decay rates of hydrogen-like ions. Our approach is based on the evaluation of the imaginary part of the fourth-order Lamb shift for excited states. We thereby avoid the ambiguities connected with electron wave function renormalization. The result may be cast in a form which is manifestly gauge invariant. As an application, we show that the formulas used by Lin and Feinberg in their study of relativistic M1 transitions, obtained by heuristic arguments, are derivable from first principles.     

Order alpha(3) ln(1/alpha) corrections to positronium decays

The logarithmically enhanced alpha(3)ln(1/alpha) corrections to the para- and orthopositronium decay widths are calculated in the framework of dimensionally regularized nonrelativistic quantum electrodynamics. In the case of parapositronium, the correction is negative, approximately doubles the effect of the leading logarithmic alpha(3)ln (2)(1/alpha) one, and is comparable to the nonlogarithmic O(alpha(2)) one. As for orthopositronium, the correction is positive and almost cancels the alpha(3)ln (2)(1/alpha) one. The uncertainty in the theoretical prediction for the parapositronium decay width is reduced to 10(-2) &mgr;s(-1).     

New value of m(mu)/m(e) from muonium hyperfine splitting

The complete contribution to the muonium hyperfine splitting of relative order alpha(3)(m(e)/m(mu))lnalpha is calculated. The result is much smaller than suggested by a previous estimate and leads to a 2sigma upward shift of the most precise value for the muon-electron mass ratio. Analogous contributions are calculated for the positronium hyperfine splitting, where a discrepancy with experiment remains.     

Mass of the eta b and alpha s from the nonrelativistic renormalization group

We sum up the next-to-leading logarithmic corrections to the heavy-quarkonium hyperfine splitting, using the nonrelativistic renormalization group. On the basis of this result, we predict the mass of the eta(b) meson to be M(eta(b))=9421+/-11(th)+9-8(delta alpha(s)) MeV. The experimental measurement of M(eta(b)) with a few MeV error would be sufficient to determine alpha(s)(M(Z)) with an accuracy of +/-0.003. For the hyperfine splitting in charmonium, the use of the nonrelativistic renormalization group brings the perturbative prediction significantly closer to the experimental figure.     

Anomalous Dimension in the Solution of the Modified Porous Medium Equation

A new method - perturbative summation to infinite order is presented to obtain the anomalous dimension in the solution of the modified porous medium equation. The result is the same as that in the renormalization group (RG) approach. It gives us an insight into the anomalous exponent in the asymptotic solution of the modified porous medium equation in the RG approach. Based on this discussion, we can see that the anomalous dimension appears naturally in the problem and the nonlinearity reflects the anomalous long-time behavior of the system.     

Relativistic and QED corrections for the beryllium atom

Complete relativistic and quantum electrodynamics corrections of order alpha(2) Ry and alpha(3) Ry are calculated for the ground state of the beryllium atom and its positive ion. A basis set of correlated Gaussian functions is used, with exponents optimized against nonrelativistic binding energies. The results for Bethe logarithms ln(k(0)(Be)=5.750 34(3) and ln(k(0)(Be+)=5.751 67(3) demonstrate the availability of high precision theoretical predictions for energy levels of the beryllium atom and light ions. Our recommended value of the ionization potential 75 192.514(80) cm(-1) agrees with equally accurate available experimental values.     

The renormalization group and the ϵ expansion

The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ? = 4?d is explained [d is the dimension of space (statistical mechanics) or space-time (quantum field theory)]. The emphasis is on principles, not particular applications. Sections 1–8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6–8 include the approximate renormalization group recursion formula and the Feyman graph method for calculating exponents. Sections 10–13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10; sections 11–13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory. An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.     

Constraints for anomalous dimensions of local light-cone operators in [ϕ 3]6 theory

Using the MS scheme, we derive in [? ³]₆ theory the collinear conformal Ward identity for the Green's functions of local light-cone operators of leading twist. The Ward identity for special collinear conformal transformations and renormalization group invariance give constraints for the off-diagonal part of the anomalous dimension matrix for the general case of β#0. We compute the anomaly of special conformal transformation in lowest loop order and obtain from the constraints the off-diagonal part of the anomalous dimension in 2-loop order.     

Experimental tests of quantum electrodynamics in weak field

This paper resumes the most precise tests of QED. We mainly describe the recent measurements of the Lamb shifts in hydrogen, deuterium and helium. The new optical frequency measurements of the 1s–2s and 2s–8s transitions in hydrogen and deuterium give now the most precise determination of the Rydberg constant and of the Lamb shifts of the 1s and 2s levels. The accurate measurement of the 2³S₁ Lamb shift in helium is also presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Unification of effective field theories

We consider the renormalization of the coupling constants in theories with extended gauge hierarchies. An effective field theory approach is used to include an interesting class of higher-order effects in the renormalization group formulas. We calculate these corrections for all possible breakdowns of O(10) to SU(3)×SU(2)×U(1).     

Improvement of Renormalization Group for Barenblatt Equation

We construct a Hartree-Fock (self-consistent)-like algorithm with renormalization group (RG) approach to calculate the anomalous dimension in a nonlinear diffusion equation. We find that our result improves the original RG work because we include the effect of Heaviside function.     

Improvement of Renormalization Group for Barenblatt Equation

We construct a Hartree-Fock (seff-consistent)-like algorithm with renormalization group (RG) approach to calculate the anomalous dimension in a nonlinear diffusion equation. We find that our result improves the original RG work because we include the effect of Heaviside function.     

Inclusion of retardation effects in the nonrelativistic treatment of the lamb shift

Summary The retardation effects are included in the nonrelativistic calculation of the Lamb shift, by using a method which exploits theO(4) symmetry of the nonrelativistic hydrogen atom. This method has been previously applied by Lieber to the treatment of the Lamb shift. However, in contrast to his calculations, no kind of dipole approximation, leading to the introduction of an inconsistent cut-off frequency, is used in our treatment. Apart from obtaining cut-off-independent results, the inclusion of retardation effects makes also possible the calculation of the spin-radiation-field Lamb-shift contribution (which vanishes in the dipole approximation). Furthermore, our unambiguous renormalization, consisting of removing the unobservable free-electron self-energy contribution from our expressions, leads to finite analytical and numerical results for the energy level (Lamb) shifts. The comparison of these results with those obtained in a nonrelativistic treatment by other authors shows that our results are in better agreement with the experiment.     

Gluino contribution to the 3-loop quark mass anomalous dimension in the minimal supersymmetric standard model

We deduce the gluino contribution to the three-loop QCD quark mass anomalous dimension function within the minimal supersymmetric Standard Model (MSSM) from its standard QCD expression. This work is a continuation of the program of computation of MSSM renormalization group functions.