Two--Loop Anomalous Dimension Matrix for ΔS=1\Delta S=1 Weak Non-Leptonic Decays II: O(αeαs){\cal O}(\alpha_e \alpha_s)

We calculate the $10\times 10$ two--loop anomalous dimension matrix to order \ord(\alpha_e \alpha_s) in the dimensional regularization scheme with anticommuting $\gamma_5$ (NDR) which is necessary for the extension of the $\Delta S=1$ weak Hamiltonian involving electroweak penguins beyond the leading logarithmic approximation. We demonstrate, how a direct calculation of penguin diagrams involving $\gamma_5$ in closed fermion loops can be avoided thus allowing a consistent calculation of two--loop anomalous dimensions in the simplest renormalization scheme with anticommuting $\gamma_5$ in $D$ dimensions. We give the necessary one--loop finite terms which allow to obtain the corresponding two--loop anomalous dimension matrix in the HV scheme with non--anticommuting $\gamma_5$.Comment: 25 page

An Analytic Formula and an Upper Bound for epsilon'/epsilon in the Standard Model

Using the idea of the penguin box expansion we find an analytic expression for epsilon'/epsilon in the Standard Model as a function of m_t, m_s(m_c) and two non-perturbative parameters B_6 and B_8. This formula includes next-to-leading QCD/QED short distance effects calculated recently by means of the operator product expansion and renormalization group techniques. We also derive an analytic expression for the upper bound on epsilon'/epsilon as a function of |V_cb|, |V_ub/V_cb|, B_K and other relevant parameters. Numerical examples of the bound are given.Comment: 10 pages with 1 figure appended in PostScript format; MPI preprint MPI-Ph/93-60 and TU preprint TUM-T31-47/9

Direct CP Violation in K_L --> \pi^0 e^+e^- Beyond Leading Logarithms

We analyze the direct CP violation in the rare decay K_L --> Pi^0 e+e- with QCD effects taken into account consistently in the next-to-leading order. We calculate the two-loop mixing between the four-quark \Delta S=1 operators and the operator Q_7V = (sd)_(V-A)(ee)_V in the NDR and HV renormalization schemes. Using the known two-loop anomalous dimension matrix of the four-quark operators, we find that the coefficient C_7V(\mu) depends only very weakly on \mu, renormalization scheme and \Lambda_MSbar. The next-to-leading QCD corrections enhance the direct CP violating contribution over its leading order estimate so that it remains dominant in spite of the recent decrease of |V_ub/V_cb| and |V_cb|. We expect typically BR(K_L --> \pi^0 e^+ e^-)_dir ~ 6*10^(-12), although values as high as 10^(-11) are not yet excluded.Comment: 35 pages (with 9 PostScript figures available separately), Munich Technical University preprint TUM-T31-60/94, Max-Planck Institute preprint MPI-Ph/94-1

The Two-Loop Anomalous Dimension Matrix for ΔS=1\Delta S=1 Weak Non-Leptonic Decays I: O(αs2){\cal O}(\alpha_s^2)

We calculate the two-loop $10 \times 10$ anomalous dimension matrix ${\cal O}(\alpha_s^{2})$ involving current-current operators, QCD penguin operators, and electroweak penguin operators especially relevant for $\Delta S=1$ weak non-leptonic decays, but also important for $\Delta B=1$ decays. The calculation is performed in two schemes for $\gamma_{5}$: the dimensional regularization scheme with anticommuting $\gamma_{5}$ (NDR), and in the 't Hooft-Veltman scheme. We demonstrate how a direct calculation of diagrams involving $\gamma_{5}$ 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 ${\cal O}(\alpha_e\alpha_s)$, required for a consistent inclusion of electroweak penguin operators, is presented in a subsequent publication.Comment: 33 page

The Anatomy of ε′/ε\varepsilon'/\varepsilon Beyond Leading Logarithms with Improved Hadronic Matrix Elements

We use the recently calculated two--loop anomalous dimensions of current-current operators, QCD and electroweak penguin operators to construct the effective Hamiltonian for $\Delta S=1$ transitions beyond the leading logarithmic approximation. We solve the renormalization group equations and give the numerical values of Wilson coeff. functions. We propose a new semi-phenomenological approach to hadronic matrix elements which incorporates the data for $CP$-conserving $K \rightarrow \pi\pi$ amplitudes and allows to determine the matrix elements of all $(V-A)\otimes (V-A)$ operators in any renormalization scheme and do a renormalization group analysis of all hadronic matrix elements $\langle Q_i(\mu) \rangle$. We compare critically our treatment of these matrix elements with those given in the literature. We find in the NDR scheme \epe = (6.7 \pm 2.6)\times 10^{-4} in agreement with the experimental findings of E731. We point out however that the increase of $\langle Q_6 \rangle$ by only a factor of two gives \epe = (20.0 \pm 6.5)\times 10^{-4} in agreement with the result of NA31. The dependence of \epe on $\Lambda_{\bar{MS}}$, $m_t$ and $\langle Q_{6,8} \rangle$ is presented.Comment: 91 pages (A4) with 20 PostScript figures (distributed separately); TUM-T31-35/93, MPI-Ph/93-11 and CERN-TH 6821/9

Weak Decays Beyond Leading Logarithms

We review the present status of QCD corrections to weak decays beyond the leading logarithmic approximation including particle-antiparticle mixing and rare and CP violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective hamiltonians for all decays for which the next-to-leading corrections are known. Subsequently, we present the phenomenological implications of these calculations. In particular we update the values of various parameters and we incorporate new information on m_t in view of the recent top quark discovery. One of the central issues in our review are the theoretical uncertainties related to renormalization scale ambiguities which are substantially reduced by including next-to-leading order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated in various cases.Comment: 229 pages, 32 PostScript figures (included); uses RevTeX, epsf.sty, rotate.sty, rmpbib.sty (included), times.sty (included; requires LaTeX 2e); complete PostScript version available at ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-100-95.ps.gz or ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-100-95.ps2.gz (scaled down and rotated version to print two pages on one sheet of paper

Differential Effects of Painful and Non-Painful Stimulation on Tactile Processing in Fibromyalgia Syndrome and Subjects with Masochistic Behaviour

BACKGROUND: In healthy subjects repeated tactile stimulation in a conditioning test stimulation paradigm yields attenuation of primary (S1) and secondary (S2) somatosensory cortical activation, whereas a preceding painful stimulus results in facilitation. METHODOLOGY/PRINCIPAL FINDINGS: Since previous data suggest that cognitive processes might affect somatosensory processing in S1, the present study aims at investigating to what extent cortical reactivity is altered by the subjective estimation of pain. To this end, the effect of painful and tactile stimulation on processing of subsequently applied tactile stimuli was investigated in patients with fibromyalgia syndrome (FMS) and in subjects with masochistic behaviour (MB) by means of a 122-channel whole-head magnetoencephalography (MEG) system. Ten patients fulfilling the criteria for the diagnosis of FMS, 10 subjects with MB and 20 control subjects matched with respect to age, gender and handedness participated in the present study. Tactile or brief painful cutaneous laser stimuli were applied as conditioning stimulus (CS) followed by a tactile test stimulus (TS) 500 ms later. While in FMS patients significant attenuation following conditioning tactile stimulation was evident, no facilitation following painful stimulation was found. By contrast, in subjects with MB no attenuation but significant facilitation occurred. Attenuation as well as facilitation applied to cortical responses occurring at about 70 ms but not to early S1 or S2 responses. Additionally, in FMS patients the amount of attenuation was inversely correlated with catastrophizing tendency. CONCLUSION: The present results imply altered cortical reactivity of the primary somatosensory cortex in FMS patients and MB possibly reflecting differences of individual pain experience