Three-loop Corrections Research Articles

Three-loop radiative corrections to Lamb shift and hyperfine splitting

We consider three-loop radiative corrections to the Lamb shift and hyperfine splitting. Corrections of order α3(Zα)5m are the largest still unknown contributions to the Lamb shift in hydrogen. We calculate radiative corrections to the Lamb shift and hyperfine splitting generated by the diagrams with insertions of one radiative photon and electron polarization loops in the graphs with two external photons. We also obtain corrections generated by the gauge-invariant sets of diagrams with two reducible radiative photon insertions in the electron line and a polarization operator insertion in one of the radiative photons, and diagrams with two reducible radiative photon insertions in the electron line and a polarization operator insertion in one of the external photons. Corrections to the Lamb shift and hyperfine splitting generated by the diagrams with insertions of the three-loop one-particle reducible diagrams with radiative photons in the electron line are calculated in the Yennie gauge.PACS Nos.: 12.20.Ds, 31.30.Jv, 32.10.Fn, 36.10.Dr

Three-loop corrections to electroweak observables in the large Higgs mass limit

We review the calculation of leading three-loop radiative corrections to some electroweak precision observables in the standard model in the large Higgs mass limit.

QUARK MIXING AND CP VIOLATION — THE CKM MATRIX

I summarize the theoretical progress in the determination of the CKM elements since Lepton-Photon 2003 and present the status of the elements and parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. One finds |Vus| = 0.2227 ± 0.0017 from K and τ decays and |Vcb| = (41.6 ± 0.5) · 10-3 from inclusive semileptonic B decays. The unitarity triangle can now be determined from tree-level quantities alone and the result agrees well with the global fit including flavour-changing neutral current (FCNC) processes, which are sensitive to new physics. From the global fit one finds the three CKM angles θ12 = 12.9° ± 0.1°, θ23 = 2.38° ± 0.03° and θ13 = 0.223° ± 0.007° in the standard PDG convention. The CP phase equals [Formula: see text] at 1σ CL and [Formula: see text] at 2σ CL. A major progress are first results from fully unquenched lattice QCD computations for the hadronic quantities entering the UT fit. I further present the calculation of three-loop QCD corrections to the charm contribution in [Formula: see text] decays, which removes the last relevant theoretical uncertainty from the [Formula: see text] system. Finally I discuss mixing-induced CP asymmetries in [Formula: see text] penguin decays, whose naive average is below its Standard Model value by 3σ.

Symbolic summation and higher orders in perturbation theory

Higher orders in perturbation theory require the calculation of Feynman integrals at multiple loops. We report on an approach to systematically solve Feynman integrals by means of symbolic summation and discuss the underlying algorithms. Examples such as the non-planar vertex at two loops, or integrals from the recent calculation of the three-loop QCD corrections to structure functions in deep-inelastic scattering are given.

Three-loop electroweak corrections to the W-boson mass and [formula omitted] in the large Higgs mass limit

We present an analytical calculation of the leading three-loop radiative correction to the S-parameter in the standard model in the large Higgs mass limit. Numerically, S ( 3 ) = 1.1105 × g 4 / ( 1024 π 3 ) × m H 4 / M W 4 . When combined with the corresponding three-loop correction to the ρ-parameter, this leads to shifts of Δ ( 3 ) sin 2 θ eff lept = 4.6 × 10 −9 × m H 4 / M W 4 in the effective weak mixing angle and Δ ( 3 ) M W = − 6.3 × 10 −4 MeV × m H 4 / M W 4 in the W-boson mass. For both of these observables, the sign of the three-loop correction is equal to that of the one-loop correction.

The third-order QCD corrections to deep-inelastic scattering by photon exchange

We compute the full three-loop coefficient functions for the structure functions F 2 and F L in massless perturbative QCD. The results for F L complete the next-to-next-to-leading order description of unpolarized electromagnetic deep-inelastic scattering. The third-order coefficient functions for F 2 form, at not too small values of the Bjorken variable x, the dominant part of the next-to-next-to-next-to-leading order corrections, thus facilitating improved determinations of the strong coupling α s from scaling violations. The three-loop corrections to F L are larger than those for F 2 . Especially for the latter quantity the expansion in powers of α s is very stable, for photon virtualities Q 2 ≫ 1 GeV 2 , over the full x-range accessible to fixed-target and collider measurements.

Three-loop radiative-recoil corrections to hyperfine splitting in muonium

We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium. These corrections are enhanced by the large logarithm of the electron–muon mass ratio. The leading logarithm-cubed and logarithm-squared contributions were obtained a long time ago. We calculate the single-logarithmic and nonlogarithmic contributions of order α3(m/M)EF generated by gauge invariant sets of diagrams with one- and two-loop polarization insertions in diagrams with two exchanged photons and radiative photons, and by diagrams with one-loop radiative photon insertions both in the electron and muon lines. The results of this paper constitute a next step in the implementation of the program of reduction of the theoretical uncertainty of hyperfine splitting below 10 Hz. They improve the theory of hyperfine splitting, and affect the value of the electron–muon mass ratio extracted from experimental data on the muonium hyperfine splitting.PACS Nos.: 12.20.Ds, 31.30.Jv, 32.10.Fn, 36.10.Dr

Three-loop electroweak correction to the rho parameter in the large Higgs mass limit

We present an analytical calculation of the leading three-loop radiative correction to the ρ-parameter in the Standard Model in the large Higgs mass limit. This correction, of order g 6 m H 4 / M W 4 , is opposite in sign to the leading two-loop correction of order g 4 m H 2 / M W 2 . The two corrections cancel each other for a Higgs mass of approximately 480 GeV . The result shows that it is extremely unlikely that a strongly interacting Higgs sector could fit the data of electroweak precision measurements.

Three-loop QCD corrections andb-quark decays

We present three-loop (NNNLO) corrections to the heavy-to-heavy quark transitions in the limit of equal initial and final quark masses. In analogy with the previously found NNLO corrections, the bulk of the result is due to the beta_0^2 alpha_s^3 corrections. The remaining genuine three-loop effects for the semileptonic b --> c decays are estimated to increase the decay amplitude by 0.2(2)%. The perturbative series for the heavy-heavy axial current converges very well.

Three-loop radiative-recoil corrections to hyperfine splitting generated by one-loop fermion factors

We consider three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by diagrams with one-loop radiative photon insertions both in the electron and muon lines. An analytic result for these nonlogarithmic corrections of order $\ensuremath{\alpha}({Z}^{2}\ensuremath{\alpha})(Z\ensuremath{\alpha})(m/M){\stackrel{\texttildelow{}}{E}}_{F}$ is obtained. This result constitutes a next step in the implementation of the program of reduction of the theoretical uncertainty of hyperfine splitting below 10 Hz.

Higher-order results in the electroweak theory

The present status of higher-order results in the electroweak theory is summarised, with particular emphasis on recent two-loop results for the prediction of the W-boson mass in the Standard Model and leading three-loop corrections to the rho parameter. The remaining theoretical uncertainties in the prediction for the W-boson mass and the effective weak mixing angle are discussed. PACS: 12.15.Lk – 13.66.Jn

Three loop soft running, benchmark points and semi-perturbative unification

We consider three-loop β-function corrections to the sparticle spectrum in the MSSM, with particular emphasis on Snowmass Benchmark points. The three loop running has little effect on the weakly interacting particle spectrum, but for the squark masses the effect can be comparable to, or greater than, that of two loop running. We extend the analysis to the semi-perturbative unification scenario, where the impact of the three loop corrections becomes even more dramatic.

Three-loop radiative-recoil corrections to hyperfine splitting in muonium

We calculate three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by the diagrams with the first order electron and muon polarization loop insertions in graphs with two exchanged photons. These corrections are enhanced by the large logarithm of the electron-muon mass ratio. The leading logarithm squared contribution was obtained a long time ago. Here we calculate the single-logarithmic and nonlogarithmic contributions. We previously calculated the three-loop radiative-recoil corrections generated by two-loop polarization insertions in the exchanged photons. The current paper therefore concludes calculation of all three-loop radiative-recoil corrections to hyperfine splitting in muonium generated by diagrams with closed fermion loop insertions in the exchanged photons. The new results obtained here improve the theory of hyperfine splitting, and affect the value of the electron-muon mass ratio extracted from experimental data on the muonium hyperfine splitting.

Three-Loop -Function of N=1 Supersymmetric Electrodynamics Regularized by Higher Derivatives

We calculate three-loop corrections to the effective action for N=1 supersymmetric electrodynamics regularized by higher derivatives. Using the obtained results, we investigate the anomaly problem in the considered model.

Gluon condensate in charmonium sum rules with three-loop corrections

Charmonium sum rules are analyzed with the primary goal to obtain the restrictions on the value of dimension 4 gluon condensate. The moments M_n(Q^2) of the polarization operator of vector charm currents are calculated and compared with experimental data. The 3-loop (alpha_s^2) perturbative corrections, the gluon condensate contribution with alpha_s corrections and dimension 6 operator G^3 contribution are accounted. It is shown that the sum rules for the moments do not work at Q^2=0, where the perturbation series diverges and G^3 contribution is large. The domain in the plane (n, Q^2), where the sum rules are legitimate, is found. Strong correlation of the values of gluon condensate and MS-bar charm quark mass quark is determined. The absolute limits are found to be for gluon condensate <(alpha_s/pi)G^2>=0.009+-0.007 GeV^4 and for charm quark mass m(m)=1.275+-0.015 GeV in MS-bar scheme.

Effective method for calculating the analytic QCD coupling constant

The analytic running coupling constant αan for strong interactions is considered for approximations of standard perturbation theory up to the three-loop level. Nonperturbative contributions are singled out explicitly in αan. They are represented in the form of an expansion in a series in inverse powers of the Euclidean momentum squared. It is shown that two-and three-loop corrections lead to a partial compensation of the nonperturbative one-loop contribution of order 1/q2, which is leading in the ultraviolet region. An efficient method for calculating the analytic running coupling constant for all q>Λ is developed on the basis of the above expansion. A comparative analysis of perturbative and nonperturbative contributions is performed in the infrared region, where the latter play the most important role. A simultaneous consideration of the momentum dependence of αan and its perturbative component for one-to three-loop cases leads to the conclusion that the analytic running coupling constant is stable with respect to higher corrections and that it depends only slightly on conditions imposed in matching solutions that involve different numbers nf of active-quark flavors.

O(a) improved QCD: The 3-loop beta-function, and the critical hopping parameter

We calculate the 3-loop bare β-function of QCD, formulated on the lattice with the clover fermionic action. The dependence of our result on the number of colors N, the number of fermionic flavors N f , and the clover parameter c SW , is shown explicitly. A direct outcome of our calculation is the two-loop relation between the bare coupling constant g 0 and the one renormalized in the MS scheme. Further, we can immediately derive the three-loop correction to the relation between the lattice Λ-parameter and g 0, which turns out to be very pronounced. We also calculate the critical value of the hopping parameter, κ c , in the clover action, to two loops in perturbation theory. This quantity is an additive renormalization; as such, it exhibits a linear divergence in the lattice spacing. We compare our results to non perturbative evaluations of κ c coming from MC simulations.

FBandfBsfrom QCD sum rules

The decay constants of the pseudoscalar mesons B and B_s are evaluated from QCD sum rules for the pseudoscalar two-point function. Recently calculated perturbative three-loop QCD corrections are incorporated into the sum rule. An analysis in terms of the bottom quark pole mass turns out to be unreliable due to large higher order radiative corrections. On the contrary, in the MS_bar scheme the higher order corrections are under good theoretical control and a reliable determination of f_B and f_{B_s} becomes feasible. Including variations of all input parameters within reasonable ranges, our final results for the pseudoscalar meson decay constants are f_B = 210 +- 19 MeV and f_B_s = 244 +- 21 MeV. Employing additional information on the product sqrt(B_B) f_B from global fits to the unitarity triangle, we are in a position to also extract the B-meson B-parameter B_B = 1.26 +- 0.45. Our results are well compatible with analogous determinations of the above quantities in lattice QCD.

The three-loop β-function of QCD with the clover action

We calculate, to 3 loops in perturbation theory, the bare β-function of QCD, formulated on the lattice with the clover fermionic action. The dependence of our result on the number of colors N, the number of fermionic flavors N f , as well as the clover parameter c SW, is shown explicitly. A direct outcome of our calculation is the two-loop relation between the bare coupling constant g 0 and the one renormalized in the MS scheme. Further, we can immediately derive the three-loop correction to the relation between the lattice Λ-parameter and g 0, which is important in checks of asymptotic scaling. For typical values of c SW, this correction is found to be very pronounced.

Impact of the three-loop corrections on the QCD analysis of the deep-inelastic scattering data

We perform the analysis of the existing inclusive deep inelastic scattering (DIS) data within NNLO QCD approximation. The parton distributions functions (PDFs) and the value of strong coupling constant $\alpha_{s}(M_Z)=0.1143\pm0.0013 (exp)$ are obtained. The sensitivity of the PDFs to the uncertainty in the value of the NNLO corrections to the splitting functions is analyzed. It is shown that the PDFs errors due to this uncertainty is generally less than the experimental uncertainty in PDFs through the region of $x$ spanned by the existing DIS data.