Three-body spin-orbit forces from chiral two-pion exchange

Using chiral perturbation theory, we calculate the density-dependent spin-orbit coupling generated by the two-pion exchange three-nucleon interaction involving virtual $\Delta$-isobar excitation. From the corresponding three-loop Hartree and Fock diagrams we obtain an isoscalar spin-orbit strength $F_{\rm so}(k_f)$ which amounts at nuclear matter saturation density to about half of the empirical value of $90$MeVfm$^5$. The associated isovector spin-orbit strength $G_{\rm so}(k_f)$ comes out about a factor of 20 smaller. Interestingly, this three-body spin-orbit coupling is not a relativistic effect but independent of the nucleon mass $M$. Furthermore, we calculate the three-body spin-orbit coupling generated by two-pion exchange on the basis of the most general chiral $\pi\pi NN$-contact interaction. We find similar (numerical) results for the isoscalar and isovector spin-orbit strengths $F_{\rm so}(k_f)$ and $G_{\rm so}(k_f)$ with a strong dominance of the p-wave part of the $\pi\pi NN$-contact interaction and the Hartree contribution.Comment: 8 pages, 4figure, published in : Physical Review C68, 054001 (2003

Induced pseudoscalar form factor of the nucleon at two-loop order in chiral perturbation theory

We calculate the imaginary part of the induced pseudoscalar form factor of the nucleon $G_P(t)$ in the framework of two-loop heavy baryon chiral perturbation theory. The effect of the calculated three-pion continuum on the pseudoscalar constant $g_P = (m_\mu/2M) G_P(t=-0.877m_\mu^2)$ measurable in ordinary muon capture $\mu^-p\to \nu_\mu n$ turns out to be negligibly small. Possible contributions from counterterms at two-loop order are numerically smaller than the uncertainty of the dominant pion-pole term proportional to the pion-nucleon coupling constant $g_{\pi N}= 13.2\pm 0.2$. We conclude that a sufficiently accurate representation of the induced pseudoscalar form factor of the nucleon at low momentum transfers $t$ is given by the sum of the pion-pole term and the Adler-Dothan-Wolfenstein term: $G_P(t) = 4g_{\pi N} M f_\pi/ (m_\pi^2 -t)- 2g_A M^2 /3$, with $= (0.44 \pm 0.02)$ fm$^2$ the axial mean square radius of the nucleon.Comment: 6 pages, 2 figures, accepted for publication in Physical Review

Chiral 3π\pi-exchange NN-potentials: Results for dominant next-to-leading order contributions

We calculate in (two-loop) chiral perturbation theory the local NN-potentials generated by the three-pion exchange diagrams with one insertion from the second order chiral effective pion-nucleon Lagrangian proportional to the low-energy constants $c_{1,2,3,4}$. The resulting isoscalar central potential vanishes identically. In most cases these $3\pi$-exchange potentials are larger than the ones generated by the diagrams involving only leading order vertices due to the large values of $c_{3,4}$ (which mainly represent virtual $\Delta$-excitation). A similar feature has been observed for the chiral $2\pi$-exchange. We also give suitable (double-integral) representations for the spin-spin and tensor potentials generated by the leading-order diagrams proportional to $g_A^6$ involving four nucleon propagators. In these cases the Cutkosky rule cannot be used to calculate the spectral-functions in the infinite nucleon mass limit since the corresponding mass-spectra start with a non-vanishing value at the $3\pi$-threshold. Altogether, one finds that chiral $3\pi$-exchange leads to small corrections in the region $r\geq 1.4$ fm where $1\pi$- and chiral $2\pi$-exchange alone provide a very good strong NN-force as shown in a recent analysis of the low-energy pp-scattering data-base.Comment: 11 pages, 7 figures, to be published in The Physical Review

Mean eigenvalues for simple, simply connected, compact Lie groups

We determine for each of the simple, simply connected, compact and complex Lie groups SU(n), Spin$(4n+2)$ and $E_6$ that particular region inside the unit disk in the complex plane which is filled by their mean eigenvalues. We give analytical parameterizations for the boundary curves of these so-called trace figures. The area enclosed by a trace figure turns out to be a rational multiple of $\pi$ in each case. We calculate also the length of the boundary curve and determine the radius of the largest circle that is contained in a trace figure. The discrete center of the corresponding compact complex Lie group shows up prominently in the form of cusp points of the trace figure placed symmetrically on the unit circle. For the exceptional Lie groups $G_2$, $F_4$ and $E_8$ with trivial center we determine the (negative) lower bound on their mean eigenvalues lying within the real interval $[-1,1]$. We find the rational boundary values -2/7, -3/13 and -1/31 for $G_2$, $F_4$ and $E_8$, respectively.Comment: 12 pages, 8 figure

Behavior of the collective rotor in wobbling motion

The behavior of the collective rotor in wobbling motion is investigated within the particle-rotor model for the nucleus $^{135}$Pr by transforming the wave functions from the $K$-representation to the $R$-representation. After reproducing the experimental energy spectra and wobbling frequencies, the evolution of the wobbling mode in $^{135}$Pr, from transverse at low spins to longitudinal at high spins, is illustrated by the distributions of the total angular momentum in the intrinsic reference frame (azimuthal plot). Finally, the coupling schemes of the angular momenta of the rotor and the high-$j$ particle for transverse and longitudinal wobbling are obtained from the analysis of the probability distributions of the rotor angular momentum ($R$-plots) and their projections onto the three principal axes ($K_R$-plots).Comment: 21 pages, 9 page

Effective field theory for triaxially deformed nuclei

Effective field theory (EFT) is generalized to investigate the rotational motion of triaxially deformed even-even nuclei. A Hamiltonian, called the triaxial rotor model (TRM), is obtained up to next-to-leading order (NLO) within the EFT formalism. Its applicability is examined by comparing with a five-dimensional collective Hamiltonian (5DCH) for the description of the energy spectra of the ground state and $\gamma$ band in Ru isotopes. It is found that by taking into account the NLO corrections, the ground state band in the whole spin region and the $\gamma$ band in the low spin region are well described. The results presented here indicate that it should be possible to further generalize the EFT to triaxial nuclei with odd mass number.Comment: 21 pages, 9 figure

Behavior of the collective rotor in nuclear chiral motion

The behavior of the collective rotor in the chiral motion of triaxially deformed nuclei is investigated using the particle rotor model by transforming the wave functions from the $K$-representation to the $R$-representation. After examining the energy spectra of the doublet bands and their energy differences as functions of the triaxial deformation, the angular momentum components of the rotor, proton, neutron, and the total system are investigated. Moreover, the probability distributions of the rotor angular momentum ($R$-plots) and their projections onto the three principal axes ($K_R$-plots) are analyzed. The evolution of the chiral mode from a chiral vibration at the low spins to a chiral rotation at high spins is illustrated at triaxial deformations $\gamma=20^\circ$ and $30^\circ$.Comment: 21 pages, 6 figure

On the electron-induced isotope fractionation in low temperature ³²O₂/³⁶O₂ ices—ozone as a case study

The formation of six ozone isotopomers and isotopologues, 16O16O16O, 18O18O18O, 16O16O18O, 18O18O16O, 16O18O16O, and 18O16O18O, has been studied in electron-irradiated solid oxygen 16O2 and 18O2 (1 : 1) ices at 11 K. Significant isotope effects were found to exist which involved enrichment of 18O-bearing ozone molecules. The heavy 18O18O18O species is formed with a factor of about six higher than the corresponding 16O16O16O isotopologue. Likewise, the heavy 18O18O16O species is formed with abundances of a factor of three higher than the lighter 16O16O18O counterpart. No isotope effect was observed in the production of 16O18O16O versus 18O16O18O. Such studies on the formation of distinct ozone isotopomers and isotopologues involving non-thermal, non-equilibrium chemistry by irradiation of oxygen ices with high energy electrons, as present in the magnetosphere of the giant planets Jupiter and Saturn, may suggest that similar mechanisms may contribute to the 18O enrichment on the icy satellites of Jupiter and Saturn such as Ganymede, Rhea, and Dione. In such a Solar System environment, energetic particles from the magnetospheres of the giant planets may induce non-equilibrium reactions of suprathermal and/or electronically excited atoms under conditions, which are quite distinct from isotopic enrichments found in classical, thermal gas phase reactions

General Relativistic Description of the Observed Galaxy Power Spectrum: Do We Understand What We Measure?

We extend the general relativistic description of galaxy clustering developed in Yoo, Fitzpatrick, and Zaldarriaga (2009). For the first time we provide a fully general relativistic description of the observed matter power spectrum and the observed galaxy power spectrum with the linear bias ansatz. It is significantly different from the standard Newtonian description on large scales and especially its measurements on large scales can be misinterpreted as the detection of the primordial non-Gaussianity even in the absence thereof. The key difference in the observed galaxy power spectrum arises from the real-space matter fluctuation defined as the matter fluctuation at the hypersurface of the observed redshift. As opposed to the standard description, the shape of the observed galaxy power spectrum evolves in redshift, providing additional cosmological information. While the systematic errors in the standard Newtonian description are negligible in the current galaxy surveys at low redshift, correct general relativistic description is essential for understanding the galaxy power spectrum measurements on large scales in future surveys with redshift depth z>3. We discuss ways to improve the detection significance in the current galaxy surveys and comment on applications of our general relativistic formalism in future surveys.Comment: accepted for publication in Physical Review