Die Lorentz-Transformation als Ausdruck der Superposition gegenläufiger Raum-Zeit-Beziehungen,Preisgabe des Relativitätsprinzips

The Lorentz Transformation as an Expression of Opposite Spacetime Relations. Abandonment of the Principle of Relativity Any increase of the characteristic energy of any body endowed with a clock, ΔE = E — E₀ (E₀ being the rest energy), is connected with an increase of its time lapse, t/t₀ = E/E₀ (EINSTEIN 1907). Effective observation of this accelerating influence on the speed of any clock is restricted on the increase of the potential energy only. Increase of the kinetic energy \documentclass{article}\pagestyle{empty}\begin{document}$ \left({\frac{E}{{E_0 }}\, = \,\frac{1}{{\sqrt{1 - \frac{v}{{c^2 }}} }}} \right) $\end{document} is, on the contrary, connected with a decrease of the time lapse, a decrease of exactly the same but inverse (reciprocal) amount to the increase of the energy: \documentclass{article}\pagestyle{empty}\begin{document}$ t/t_0{\rm = }E_0 /E{\rm = }\sqrt {1 - \frac{{v^2 }}{{c^2 }}.} $\end{document}. Moreover this amount is that one postulated by the Lorentz Transformation. This effect is the well-known “time dilatation” of the Special Theory of Relativity, the “transversal Doppler effect”. The Lorentztransformation is of exclusively kinematical meaning and therefore takes no account of the energy increase connected with any motion. There is no reason, why the time accelerating effect of any energy rises should be absent in the case of kinetic energy, paying regard to is seem indispensable. Therefore the actual effect \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {1 - \frac{{v^2 }}{{c^2 }}} $\end{document} has to be given as a superposition of the time accelerating energy effect \documentclass{article}\pagestyle{empty}\begin{document}$ 1/\sqrt {1 - \frac{{v^2 }}{{c^2 }}} $\end{document} and a decelerating kinematic effect of “double” (inverse square) amount: 1 – v²/c². Modified transformation equations are derived which pay regard to this subdivision of the actual relations concerning times and local scales, and whose interated form is nevertheless identical with the classical Lorentz Transformation, if kinetic energy is the sole one being present. Of course this new subdivision of the content of meaning in the transformations is in contradiction with the ?principle of relativity”?, it presumes the existence of an inertial frame absolutely at rest related to the universe, A series of arguments is asserted which let appear the existence of such an absolute frame more fascinating than the equivalence of the variety of all inertial frames.     

Einfluß des Randes auf das effektive Verhalten heterogener Körper - Anwendung auf Rayleighwellen

Effective Elastic Properties of Finite Heterogeneous Media - Application to Rayleigh-waves Rayleigh waves in a heterogeneous material (multiphase mixtures, composite materials, polycrystals) are governed by integrodifferential equations derived by the aid of known methods for infinite heterogeneous media. According to this wave equation the velocity depends on the frequency, and the waves are damped. After some simplifications (isotropy, nonrandom elastic constants) the following is obtained: if the fluctuations of the mass density are restricted to the vicinity of the boundary, the frequency dependent part of the velocity behaves like \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^3 \omega ^3}}{{{\mathop c\limits^\circ} _t^3}} $\end{document} and the damping is proportional to \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^4 \omega ^5}}{{{\mathop c\limits^\circ} _t^5}} $\end{document}, whereas \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^2 \omega ^2}}{{{\mathop c\limits^\circ} _t^2}} $\end{document} respectively \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{l^3 \omega ^4}}{{{\mathop c\limits^\circ} _t^4}} $\end{document} is found if the fluctuations are present in the whole half-space. From this it is seen, what assumptions are necessary to describe the waves by differential equations with frequenc y-dependent mass density.     

Intensity Effects in Scattering of Electromagnetic Waves by Electrons Moving in External Magnetic Field

In this paper we consider the emission processes of a relativistic electron moving in the field of a plane electromagnetic wave and in a homogeneous magnetic field. A detailed analysis of the most important characteristics of the radiation properties for arbitrary values of the magnetic field, compared with \documentclass{article}\pagestyle{empty}\begin{document}$ [H_0 = \frac{{m^2 c^3}}{{e\hbar}}]$\end{document} = 4.41.10¹³ gauss, is presented.     

Nonlinear Interaction of S-Polarized Incident Radiation and Surface Waves in Magnetoactive Plasma

The effect of an external magnetic field on the nonlinear interaction of S-polarized electromagnetic radiation incident on a S-polarized surface wave in a plasma layer was studied analytically. We have calculated the amplitudes of generated waves at combination frequencies. The generated waves are of P-polarization and can be either electromagnetic or surface waves, depending on the signal of the value=\documentclass{article}\pagestyle{empty}\begin{document}$ ^{\chi '^2 = \frac{{k'^2 }}{{\varepsilon '}} - \frac{{\omega '^2 }}{{c^2 }} + k'\frac{\partial }{{\partial x}}\frac{{\varepsilon '_2 }}{{\varepsilon '\varepsilon '_1 }}} $\end{document}.     

A Remark about the Point Interaction in one Dimension

The zero range limit of one dimensional Schrödinger operator is studied by scaling technique and new results are obtained for potentials V with \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \smallint \limits_{\rm R} $\end{document} V(x)dx = 0.     

Paketimpulse in der magnetischen Kernresonanz

Composite Pulses in Nuclear Magnetic Resonance For the compensation of spatial inhomogeneity of the radiofrequency field and a resonance offset in NMR experiments, composite pulses are used instead of the conventional single pulses. In the present work the effect of a resonance offset on composite pulses is treated quantitatively. It will be shown also experimentally that the various constructions for \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{\pi }{2} $\end{document} composite pulses (contrary to π composite pulses) lead to only two different degrees of compensation depending on the choice of the phase of the pulses or the sign of the resonance offset.     

The Scattering on the Bound State and the Two-Particle Resonances

In the given paper the scattering of a spinless particle by another spinless particle bound in the external field is considered in the three-dimensional case. The external field is represented by the rectangular well and the two-particle interaction is parametric. The influence of the single-particle basis and of the strength of the two-particle interaction on the resonance structure of the cross-section is investigated in the limit of weak coupling between channels. It is shown that the dependence of the number of resonances N_r on the number of single-particle levels N is given by the following formula: \documentclass{article}\pagestyle{empty}\begin{document}$ N_r = \frac{{N^2 + (N - 4)^2 }}{2}. $\end{document}. The scattering of a particle by another particle bound in the field of a core is considered.     

More corrections to the sixth-order anomalous magnetic moment of the muon

The contribution to the sixth-order muon anomaly from second-order electron vacuum polarization is determined analytically to orderm _e/m _μ. The result, including the contributions from graphs containing proper and improper fourth-order electron vacuum polarization subgraphs, is $$\begin{gathered} \left( {\frac{\alpha }{\pi }} \right)^3 \left\{ {\frac{2}{9}\log ^2 } \right.\frac{{m_\mu }}{{m_e }} + \left[ {\frac{{31}}{{27}}} \right. + \frac{{\pi ^2 }}{9} - \frac{2}{3}\pi ^2 \log 2 \hfill \\ \left. { + \zeta \left( 3 \right)} \right]\log \frac{{m_\mu }}{{m_e }} + \left[ {\frac{{1075}}{{216}}} \right. - \frac{{25}}{{18}}\pi ^2 + \frac{{5\pi ^2 }}{3}\log 2 \hfill \\ \left. { - 3\zeta \left( 3 \right) + \frac{{11}}{{216}}\pi ^4 - \frac{2}{9}\pi ^2 \log ^2 2 - \frac{1}{9}log^4 2 - \frac{8}{3}a_4 } \right] \hfill \\ + \left[ {\frac{{3199}}{{1080}}\pi ^2 - \frac{{16}}{9}\pi ^2 \log 2 - \frac{{13}}{8}\pi ^3 } \right]\left. {\frac{{m_e }}{{m_\mu }}} \right\} \hfill \\ \end{gathered} $$ . To obtain the total sixth-order contribution toa _μ?a _e, one must add the light-by-light contribution to the above expression.     

Electrodynamic Model of Initiation and Extinction of Emitting Sites Interacting Together in Cold Cathode Spots (Type A) in Air

From an electrodynamic and simple quantum-mechanical point of view a model is proposed which explains the phenomena of minimum arc current as well as the formation and extinction of tiny emitting sites interacting together in cold cathode spots (called type A) on the base of a specific coupling between the tunnelling “average” electrons and the metal bulk phonon field. The model seems to be especially applicable to such experimental conditions where typical trumpetlike microcraters with pronounced rims with diameters in the range 0.5—1 μm are left by microspot ensembles on the cathode surface. The model yields emitting-site lifetimes, currents, current densities and radii in the order of τ_p^s ? \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {3M/m} $\end{document} τ₀ ? 10^?11 sec, I_min = 4π ? \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {n/\mu _0 m} $\end{document}? 0.4 A, j = nev_s ? 4 · 10¹³ A/m² and r_a ? 2c/ω_Pl ? 30 nm (τ_p^s…lifetime of short wave phonons, M … atom mass, m … electron mass, τ₀ … mean free collision time of Fermi electrons at room temperature, n … conduction electron density in the metal bulk, v_s … metal bulk sound velocity, c … light velocity, ω_Pl … metal bulk plasma frequency (values for copper). The lifetime and the interaction diameter of an emitting site (event) ensemble are derived to τ_p^l ?(M/m) τ_p ? 3 nsec and Λ_p^l = ν_sτ_p^l ? 10 μ (τ_pΛ_p^l … lifetime and mean free path of long wave phonons).     

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

Study of the mechanisms of pre-equilibrium nuclear reactions in the microscopic approach

The mechanisms of pre-equilibrium nuclear reactions are investigated within the Statistical Multistep Direct Process (SMDP) + Statistical Multistep Compound Process (SMCP) formalism. It has been shown that from an analysis of linear part in such dependences as $$\ln \left[ {{{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} \mathord{\left/ {\vphantom {{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} {\varepsilon _b^{1/2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon _b^{1/2} }}} \right]upon\varepsilon _b $$ and $$\ln \left[ {{{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} \mathord{\left/ {\vphantom {{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right. \kern-\nulldelimiterspace} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right]upon{{U_B } \mathord{\left/ {\vphantom {{U_B } {\left( {E_a - B_b } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {E_a - B_b } \right)}}$$ one can extract information about the type of mechanism (SMDP, SMCP, SMDP→SMCP) and the number of stages of the multistep emission of secondary particles. In the above approach, we have discussed the experimental data for a broad class of reactions in various entrance and exit channels.     

Einsteins hermitesche Relativitätstheorie als Unifikation von Gravo- und Chromodynamik

Einstein's Hermitian Theory of Relativity as Unification of Gravo- and Chromodynamics Einstein's Hermitian unified field theory is the continuation of the Riemannian GRG to complexe values with a Hermitian fundamental tensor gμ_v = g_v*μ This complexe continuation of GRG implies the possibility of matter and anti matter with a sort of CPT theorem. — Einstein himself has interpreted his theory as a unification and generalization of the Einstein and Maxwell theory, th. i. of gravodynamics and of electrodynamics. However — according the EIH approximation —, from Einstein's equations no Coulomb-like forces between the charges are resulting (INFELD, 1950). But, the forces between two charges ?_A and ?_B have the form (Treder 1957) It is interesting that such forces are postulated in the classical models of the chromodynamics of the interactions between quarks (for the confinement of their motions. If we interprete the purely imaginary part gμ_v of the hermitian metrics gμ_v=gμ_v+gμ_v as the dual of the field of gluons then, all peculiarities of Einstein's theory become physically meaningful. — Einstein's own interpretation suggests that the both long-range fields, gravitation and electromagnetism, must be unified in a geometrical field theory. However, the potential α/r + ε/2 has a “longer range” than the Coulomb potential ～1, and such an asymptotical potential ～ ε/2 is resulting from Einstein's equations (TREDER 1957). In Einstein's theory there are no free charges with \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_A^n {\varepsilon A} $\end{document}. (Wyman 1950) because the field mass of a charged particle becomes infinite asymptotically: That means, in a chromodynamics we dont's have free quarks. The same divergence are resulting from one-particle systems with non-vanishing total charges: M～ε²r. However, if the total charges vanish because in a domain ～L³ the positive sources are compensated by negative sources, the field masses of the n-charge systems become finite. From the gravitational part of Einstein's equations we get field masses which are the masses measured by observers in distances r ? L. That means, the masses of quark systems with the colour condition \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_A^n {\varepsilon A} $\end{document} are proportional to the linear dimension L of the system.     

Die teleskopischen Prinzipien der Gravitationstheorie. (Machsches Prinzip,Mach-Einsteinsche Relativität der Trägheit und die Hertzsche Mechanik)

The Telescopical Principles in the Theory of Gravitation. (Machs Principle, Relativity of Inertia According to Mach and Einstein and Hertz' Mechanics) We give an explication and analytical formulation of Mach's principle of the “relativity of inertia” and of the Mach-Einstein doctrine on the determination of inertia by gravitation. These principles are whether “philosophical” nor “epistemological” postulates but well defined physical axioms with exactly analytical expressions. - The fundamental principle is the Galileian “reciprocity of motions”. According to this “generalized Galilei invariance” the principal functions of analytical dynamics (Lagrangian L and Hamiltonian H) are depending upon the differences ??_AB of the coordinate vectors ??_A and ??_B of the velocity differences ??_AB = ??_A-??_B, only. The Galileian reciprocity of motions means that whether the vectors ??_A and ??_A nor the accelerations ??_A of one particle have a physical significance. A mechanics obtaining this generalized Gailei-invariance cannot depend upon a kinematical Term \documentclass{article}\pagestyle{empty}\begin{document}$ T = \frac{1}{2}\mathop {\Sigma m_A \mathop r\nolimits_A^2}\limits_A $\end{document} in the Lagrangian. Therefore, the inertial masses of the particles must be homogeneous function of interaction potentials Φ_A,B. According to the Einsteinian equivalence of inertia and gravity these interactions have to be the Newtonian gravitation. In a universe with N mass points the Mach-Einsteinian Lagrangian for our “gravodynamics without inertia” is In such a Mach-Einstein universe the celestical dynamics becomes in the first approximation the Newtonian dynamics, in the second (the “post-Newtonian”) approximation the general relativistic Einstein effects are resulting.-However, our gravodynamics gives new effects for large masses (no gravitational collapses) and in cosmology (secular accelerations a.o.). Generally, the space of our gravodynamics is whether the Newtonian “absolute space” V₃ nor the relativistic Einstein-Minkowski world V₄ but the Hertzian configuration space V_3N of the N particles. According to the relativity of inertia the Hertzian metrics become Riemannian metrics which are homogenous functions of the Newtonian gravitational potentials. .     

Calibration of the electrical aerosol analyzer at subambient pressures

A commercial Electrical Aerosol Analyzer (EAA, TSI Inc. model 3030) was calibrated experimentally at three subambient pressures (i. e., 0.901, 0.878, and 0.853 atm). Each calibration resulted in a 19 × 11 response matrix and a size dependent sensitivity curve $ \left({\frac{{pA}}{{{\# \mathord{\left/ {\vphantom {\# {cm^3}}} \right. \kern-\nulldelimiterspace} {cm^3}}}}} \right) $. The results of the calibration were incorporated into a data reduction computer program for size distribution inversion. The accuracy of the calibration was tested by measuring the size distribution of a NaCl polydisperse aerosol at the three subambient pressures. All the tests gave good agreement in the inverted mean geometric diameter and geometric standard deviation of the aerosol number size distribution.     

A universal sum rule for hadronic elastic scattering with applications to the Tevatron,RHIC and LHC

Under the assumption that at high energies total absorption prevails so that the imaginary part of the scattering amplitude dominates, we present a sum rule for all hadronic elastic differential cross-sections. We find that the dimensionless quantity , at asymptotic energies. A comparison with experimental data from ISR and Tevatron confirms a trend towards its saturation and some estimates are presented for LHC. Its universality and further consequences for the nature of absorption in QCD based models for elastic and total cross-sections are explored.Received: 15 August 2004, Revised: 27 April 2005, Published online: 8 July 2005     

Adsorption,Desorption, Dissoziation und Rekombination von SO2 an einer Palladium(111)-Oberfläche

Adsorption, Desorption, Dissociation and Recombination of SO₂ on a Palladium (111) Surface The adsorption, desorption as well as decomposition- and recombination-reactions of SO₂ on Pd(1 1 1) were studied for temperatures T = 160-1200 K using LEED, AES, thermal desorption-mass-spectrometry and molecular beam techniques. At 160 K SO₂ adsorption with an initial sticking coefficient s₀ = 1 is molecular and non-ordered; it is characterized by a precursor state and leads to a saturation coverage Θ ≈ 0,3. Heating up the adlayer SO₂ is the only desorption product, namely directly from (SO₂)_ad in the α-peak (T_max = 240 K) and as the product of recombination of (SO)_ad and O_ad in the β-peak (T_max = 330-370 K). A great part of the oxygen originating from SO₂-dissociation is incorporated into the subsurface region, resulting in an atomic S-adlayer with Θ_S = 1/7 which exhibits a (\documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt 7 {\rm x}\sqrt 7 $\end{document}) R ± 19,1°-superstructure. This structure is also observed, if a 320 K-SO₂-exposure induced (2 × 2)-SO saturation layer with Θ_SO = 0,5 is heated up or if SO, is exposed at T > 500 K, where it corresponds to Θ_S, values of 3/7 and 2/7, respectively. Furthermore the poisoning effect of adsorbed sulfur on the dissociative O₂,-adsorption and the oxidation of sulfur by heating up an O? S-coadsorption layer were studied. As a result the following kinetic parameters (activation energies and frequency factors) were determined: .     

Na‐site energy of P2‐type NaxM O2 (M = Mn and Co)

P2‐type Na_xM O₂ (M = Mn and Co) is a promising cathode material for low‐cost sodium ion secondary batteries. In this structure, there are two different crystallographic Nai (i = 1 and 2) sites with different Coulomb potential $ (\varphi _i)$ provided by M^4–x and O^2–. Here, we experimentally determine a difference ${(\rm \Delta }\varepsilon \equiv \varepsilon _1 - \varepsilon _2)$ of Na‐site energies ${(}\varepsilon _i \equiv e\varphi {\kern 1pt} _i)$ based on the temperature dependence of the site occupancies. We find that ${\rm \Delta }\varepsilon \;{=}\;56\;{K}$ for Na_0.52MnO₂ is significantly smaller than 190 K for Na_0.59CoO₂. We interpret the suppressed ${\rm \Delta }\varepsilon $ in Na_0.52MnO₂ in terms of the screening effect of the Na⁺ charge. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spin- {\frac{{3}}{{2}}} beyond the Rarita-Schwinger framework

We employ the two independent Casimir operators of the Poincaré group, the squared four-momentum, p², and the squared Pauli-Lubanski vector, W², in the construction of a covariant mass m, and spin- projector in the four-vector spinor, ψ_μ. This projector provides the basis for the construction of an interacting Lagrangian that describes a causally propagating spin- particle coupled to the electromagnetic field by a gyromagnetic ratio of .     

Light quark masses in quantum chromodynamics and chiral symmetry breaking

We derive model independent lower bounds for the sums of effective quark masses \(\bar m_u + \bar m_d \) and \(\bar m_u + \bar m_s \) . The bounds follow from the combination of the spectral representation properties of the hadronic axial currents two-point functions and their behavior in the deep euclidean region (known from a perturbative QCD calculation to two loops and the leading non-perturbative contribution). The bounds incorporate PCAC in the Nambu-Goldstone version. If we define the invariant masses \(\hat m\) by $$\bar m_i = \hat m_i \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^{{{\gamma _1 } \mathord{\left/ {\vphantom {{\gamma _1 } {\beta _1 }}} \right. \kern-\nulldelimiterspace} {\beta _1 }}} $$ and <F ²> is the vacuum expectation value of $$F^2 = \Sigma _a F_{(a)}^{\mu v} F_{\mu v(a)} $$ , we find, e.g., $$\hat m_u + \hat m_d \geqq \sqrt {\frac{{2\pi }}{3} \cdot \frac{{8f_\pi m_\pi ^2 }}{{3\left\langle {\alpha _s F^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} $$ ; with the value <α _u F ²?0.04GeV⁴, recently suggested by various analysis, this gives $$\hat m_u + \hat m_d \geqq 35MeV$$ . The corresponding bounds on \(\bar m_u + \bar m_s \) are obtained replacingm _π ² f _π bym _K ² f _K . The PCAC relation can be inverted, and we get upper bounds on the spontaneous masses, \(\hat \mu \) : $$\hat \mu \leqq 170MeV$$ where \(\hat \mu \) is defined by $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .