Hooft-Veltman Scheme Research Articles

Higher-twist contributions to the structure functions coming from 4-fermion operators

We evaluate the contribution of a class of higher-twist operators to the lowest moment of the Structure Functions, by computing appropriate matrix elements of six four-fermion operators in the quenched approximation. Their perturbative renormalization constants and mixing coefficients are calculated in the 't Hooft-Veltman scheme of dimensional regularization, using codes written in the algebraic manipulation computer language FORM.

The b→ sγ decay revisited

In this work we compute the leading logarithmic corrections to the b→ sγ decay in a dimensional scheme which does not require any definition of the γ 5 matrix. The scheme does not exhibit unconsistencies and it is therefore a vaible alternative to the 't Hooft Veltman scheme, particularly in view of the next-to-leading computation. We confirm the recent results of Ciuchini et al.

One-loop helicity amplitudes for all 2 → 2 processes in QCD and N = 1 supersymmetric Yang-Mills theory

One-loop corrections to the helicity amplitudes of all 2 → 2 subprocesses are calculated in QCD and in N = 1 supersymmetric Yang-Mills theory using two versions of dimensional regularization: the 't Hooft-Veltman scheme and dimensional reduction. Studying the structure of the soft and collinear singularities, we found universal transition rules for the squared matrix element which can be used to translate the results obtained in these schemes to the results valid in the conventional dimensional regularization scheme. With explicit calculation it is demonstrated that the one-loop helicity amplitudes of the 2 → 2 subprocesses calculated using dimensional reduction in the N = 1 supersymmetric SU( N) gauge theory respect the supersymmetry Ward identities. Our transition rules can also be used to calculate the next-to-leading order Altarelli-Parisi kernels in the dimensional reduction scheme when they satisfy supersymmetry Ward identities as well.

Two-loop anomalous dimension matrix for ΔS = 1 weak non-leptonic decays (I). O( α2s)

We calculate the two-loop 10×10 anomalous dimension matrix O( α 2 s) involving current-current operators, QCD penguin operators, and electroweak penguin operators especially relevant for ΔS = 1 weak non-leptonic decays, but also important for ΔB = 1 decays. The calculation is performed in two schemes for γ 5: the dimensional regularization scheme with anticommuting γ 5 (NDR), and in the 't Hooft-Veltman scheme. We demonstrate how a direct calculation of diagrams involving γ 5 in closed fermion loops can be avoided thus allowing a consistent calculation in the NDR scheme. The compatibility of the results obtained in the two schemes considered is verified and the properties of the resulting matrices are discussed. The two-loop corrections are found to be substantial. The two-loop anomalous dimension matrix O( αα s), required for a consistent inclusion of electroweak penguin operators, is presented in a subsequent publication.

O( αs) QCD corrections to charm quark decay in dimensional regularization with non-anticommuting γ5

Starting from the effective hamiltonian for weak decays including next-to-leading QCD corrections, the O( α s) corrections to the matrix elements of charm decay into massless s-, u-, d̄d-quarks are recalculated to obtain the complete next-to-leading order expression within the parton model for the inclusive non-leptonic c-quark decay rate. The calculation performed in the 't Hooft-Veltman scheme (non-anticommuting γ 5 in D ≠ 4 dimensions) confirms the previous result of Altarelli et al. obtained using dimensional reduction.

TRACER version 1.1: A mathematica package for γ-algebra in arbitrary dimensions

This paper describes the first MATHEMATICA implementation of γ-algebra in arbitrary space-time dimensions according to the 't Hooft-Veltman scheme. It is the only system based on a general purpose computer algebra system treating the γ 5-problem mathematically consistently in arbitrary dimensions. The TRACER package is capable of doing just purely algebraic manipulations as well as trace operations on strings of γ-algebra objects. In addition, it provides a set of utility functions for reordering, simplifying and improving the readability of the output. Optionally the output can be obtained in a form suitable to be fed into a T EX system for high quality text processing. As a whole, the TRACER package is intended as a computerized aid to a researcher working on higher order corrections in Relativistic Quantum Field Theories. A short comparison of procedural versus rule-based programming approaches is given and the discussion is supplemented by a toy implementation of the γ-algebra in rule-based style. The paper describes in detail the usage of the TRACER package and for further illustration a correlation function of two weak currents is calculated with the aid of TRACER. Finally, data on the performance of TRACER on some common platforms are given.

Effective hamiltonians for ΔS = 1 and ΔB = 1 non-leptonic decays beyond the leading logarithmic approximation

We calculate the effective hamiltonians for ΔB = 1 and ΔS = 1 decays including next-to-leading QCD corrections. In particular we present the two-loop 6 × 6 anomalous dimension matrices describing the mixing of current-current and QCD penguin operators calculated in the dimensional regularization scheme with anticommuting γ 5 and in the 't Hooft-Veltman scheme. The renormalization scheme dependences and their cancellation for physical quantities are discussed in detail. We point out that beyond the leading order, matching between coefficients and hadronic matrix elements of local operators both in the renormalization scale μ and in the renormalization scheme for the operators has to be made. The next-to-leading corrections enhance substantially the Wilson coefficients of QCD penguin operators and these effects are stronger than the ones found for current-current operators. consequently, the contributions of QCD penguins to B-decays, the ΔI = 1 2 rule and ϵ′ / ϵ are enlarged. For μ ≈ 1 GeV and Λ MS ≈ 300 MeV , these effects amount to 20–30%. In particular we emphasize that for μ < 1 GeV the next-to-leading corrections to the coefficients of penguin operators are large, reaching 40% for μ ≈ 0.8 GeV. For such low values of μ the renormalization group improved perturbation theory cannot be trusted any longer. Moreover the results are very sensitive functions of Λ MS .

QCD nonleading corrections to weak decays in dimensional regularization and 't Hooft-Veltman schemes

We calculate next-to-leading QCD corrections to the effective hamiltonian for weak decays in the limit of massless quarks using three regularization schemes with different treatment of γ5: (i) dimensional regularization (anticommuting γ5 in D ≠ 4), (ii) 't Hooft-Veltman scheme (non-anticommuting γ5 in D ≠ 4) and (iii) dimensional reduction (γ5 in D = 4). The main part of the calculation is the evaluation of the two-loop anomalous dimensions of relevant four-quark operators. Although all three schemes give rather different results at the intermediate stages of the calculation, the final result for the complete next-to-leading order corrections turns out to be the same in all schemes considered and to be equal to the one of Altarelli et al. obtained previously in the dimensional reduction scheme. This result follows after a subtle analysis of operator mixing.

A FOUR-DIMENSIONAL COUNTERPART OF DIMENSIONAL REGULARIZATION AND FERMION LOOPS

Within the framework of a four-dimensional counterpart of the dimensional cut-off scheme the number of spurious axial-vector anomalies in regularized fermion loops is shown to be reduced, in comparison with the canonical ’t Hooft-Veltman scheme. The considered cut-off procedure leads to the same result as the canonical dimensional regularization for the loops involving only vector and axial-vector vertices.