Related Sum Rules Research Articles

Supersymmetry and sum rules in the Goldberger-Wise model

In this work we demonstrate that the mixed gravitational and scalar sectors of the five-dimensional Goldberger-Wise (GW) model, in which the size of a warped extra dimension is dynamically determined, has a ``hidden'' dual $N=2$ supersymmetric structure. This symmetry structure, a generalization of one found in the unstabilized Randall-Sundrum model, is a result of the spontaneously broken five-dimensional diffeomorphism invariance of the underlying gravitational theory. The supersymmetries relate the properties of the spin-1 and spin-0 modes ``eaten'' by the massive spin-2 Kaluza-Klein states of the theory to the mode functions of the spin-2 modes. Because the symmetries relate the couplings and masses of the massive spin-2 states to those of the tower of physical spin-0 states of the GW model, they enable us to analytically prove the sum rule relations which ensure the tree-level scattering amplitudes of the massive spin-2 states will grow no faster than $\mathcal{O}(s)$. The analysis given here also explains the unconventional forms of the spin-0 mode equation, boundary condition(s), and normalization found in the GW model.

Implications of the zero 1-3 flavour mixing hypothesis: predictions for {theta}_{23}^{mathrm{PMNS}} and δPMNS

We revisit mixing sum rule relations in the lepton and quark sectors under the assumption that the 1-3 elements of the flavour mixing matrices ( {V}_L^u , {V}_L^d , {V}_L^e , {V}_L^{nu } ) are zero in the flavour basis. We consider the exact relations resulting from the validity of this “zero 1-3 flavour mixing hypothesis” and analyse their implications based on the current experimental data, including effects from RG running. In particular, we analyse how the existing precise measurement of {theta}_{13}^{mathrm{PMNS}} allows to derive predictions for {theta}_{23}^{mathrm{PMNS}} in models with constrained {theta}_{12}^{mathrm{e}} . As examples, we calculate the predictions for {theta}_{23}^{mathrm{PMNS}} which arise in 12 classes of Pati-Salam models and SU(5) GUTs that relate {theta}_{12}^{mathrm{e}} to {theta}_{12}^{mathrm{d}} . We also derive a novel “lepton phase sum rule”, valid under the additional assumption of small charged lepton mixing contributions. We furthermore point out that, in the context of GUT flavour models, the quark and lepton CP violating phases δCKM and δPMNS can both be predicted from a single imaginary element in the mass matrices.

Masses of hadrons in the chiral symmetry restored vacuum

We calculate the masses of the vector and axial-vector mesons as well as the nucleon and the delta resonance in the chiral symmetry restored vacuum. This is accomplished by separating the quark operators appearing in the QCD sum rules for these hadrons into the chiral symmetric and symmetry breaking parts depending on the contributions of the fermion zero modes. We then extract the vacuum expectation values of all the separated parts of the quark operators using the QCD sum rule relations for these hadrons with their vacuum masses and widths. By taking the chiral symmetry breaking parts to be zero while keeping the symmetric operators to their vacuum values, we obtain the chiral symmetric part of the hadron masses. We find that the masses of chiral partners, such as the $(\rho,a_1)$ and $(K^*,K_1)$, become degenerate to values between 500 and 600 MeV in the chiral symmetry restored vacuum, while parity partners $(\omega,f_1)$ that are chiral partners only in the limit where the disconnected diagrams are neglected remain non-degenerate with masses $(655,1060)$ MeV, respectively. The masses of the nucleon and the Delta are also found to reduce to about 500 and 600 MeV, respectively, in the chiral symmetric vacuum. This shows that while chiral symmetry breaking is responsible for the mass difference between chiral partner, both the meson and baryon retain non-trivial fraction of their masses in the chiral symmetry restored vacuum.

Beauty baryon nonleptonic decays into decuplet baryons and CP -asymmetries based on an SU(3) -flavor analysis

We consider charmless weak decays of beauty-baryons into decuplet baryons and pseudoscalar mesons in a general framework based on $SU(3)$-flavor decomposition of the decay amplitudes. The dynamical assumption independent $SU(3)$ analysis accounts for the effects of an arbitrarily broken $SU(3)$ symmetry in these decays. An alternative approach in terms of quark diagrams is also provided and compared with the $SU(3)$ decomposition in the limit of exact $SU(3)$-flavor symmetry. Furthermore, the symmetries of the effective Hamiltonian is used to relate or neglect reduced $SU(3)$ amplitudes to derive several sum rule relations between amplitudes and relations between $CP$ asymmetries in these decays and identify those that hold even if $SU(3)$ is broken.

Nonleptonic beauty baryon decays and CP asymmetries based on an SU(3) -flavor analysis

We consider hadronic weak decays of beauty-baryons into charmless baryons and pseudoscalar mesons in a general framework based on $SU(3)$ decomposition of the decay amplitudes. The advantage of the approach lies in the ability to perform an $SU(3)$ analysis of these decays without any particular set of dynamical assumptions while accounting for the effects of an arbitrarily broken $SU(3)$ flavor symmetry. Dictated by the symmetries of the effective Hamiltonian that allow us to relate or neglect reduced $SU(3)$ amplitudes, we derive several sum rule relations between amplitudes and relations between $CP$ asymmetries in these decays and identify those that hold even if $SU(3)$ is broken.

Light-cone sum rules for B \u2192 K \u03c0 form factors and applications to rare decays

We derive a set of light-cone sum rules relating the hadronic form factors relevant for B → Kπℓ+ℓ− decays to the B-meson light-cone distribution amplitudes (LCDAs). We obtain the sum rule relations for all B → K π form factors of (axial)vector and (pseudo)tensor b → s currents with a P -wave K π system. Our results reduce to the known light-cone sum rules for B → K∗ form factors in the limit of a single narrow-width resonance. We update the operator-product expansion for the underlying correlation function by including a more complete set of B-meson LCDAs with higher twists, and produce numerical results for all B → K∗ form factors in the narrow-width limit. We then use the new sum rules to estimate the effect of a non-vanishing K∗ width in B → K∗ transitions, and find that this effect is universal and increases the factorizable part of the rate of B → K∗X decays by a factor of 20%. This effect, by itself, goes in the direction of increasing the current tension in the differential B → K∗μμ branching fractions. We also discuss B → K π transitions outside the K∗ window, and explain how measurements of B → K πℓℓ observables above the K∗ region can be used to further constrain the B → K∗ form factors.

Discovery of collective states in 208Pb by complete spectroscopy

Most neutron and proton bound states in 208Pb are described in the shell model as one-particle onehole configurations. Besides precise data obtained with the Q3D magnetic spectrograph of the MLL (Garching, Germany) an important reaction is the inelastic proton scattering via isobaric analog resonances in 209Bi. It yields amplitudes of neutron one-particle one-hole configurations with relative signs in each stateof 208Pb. The orthogonality, normality, and sum rule relations allow to investigate the completeness of the transformation matrices of one-particle one-hole configurations describing the states in 208Pb with spins from 0− to 14− and 0+ to 12+. By this method amplitudes of unobservable one-particle one-hole configurations can be determined.The comparison of spin, parity, and dominant particle-hole components thus derived in up to 30 states of a certain spin to shell model calculations allows to identify states described as collective excitations of the entire nucleus.

Radiative leptonic decay B \u2192 \u03b3\u2113\u03bd\u2113 with subleading power corrections

We reconsider the QCD predictions for the radiative decay B → γℓνℓ with an energetic photon in the final state by taking into account the 1/Eγ, 1/mb power-suppressed hard-collinear and soft corrections from higher-twist B-meson light-cone distribution amplitudes (LCDAs). The soft contribution is estimated through a dispersion relation and light-cone QCD sum rules. The analysis of theoretical uncertainties and the dependence of the decay form factors on the leading-twist LCDA ϕ+(ω) shows that the latter dominates. The radiative leptonic decay is therefore well suited to constrain the parameters of ϕ+(ω), including the first inverse moment, 1/λB, from the expected high-statistics data of the BELLE II experiment.

Electromagnetic reactions from coupled-cluster theory

We review recent results for electromagnetic reactions and related sum rules in light and medium-mass nuclei obtained from coupled-cluster theory. In particular, we highlight our recent computations of the photodisintegration cross section of 40Ca and of the electric dipole polarizability for oxygen and calcium isotopes. We also provide new results for the Coulomb sum rule for 4He and 16O. For 4He we perform a thorough comparison of coupled-cluster theory with exact hyperspherical harmonics.

Sum rule distortions in fluorescence-yield x-ray magnetic circular dichroism

The quantitative analysis of $3d$ transition metal ${\mathrm{L}}_{2,3}$ edge x-ray magnetic circular dichroism (XMCD) spectra and the related sum rules are compared for measurements with electron yield and fluorescence yield detection. Multiplet calculations on divalent ions show that fluorescence yield detected sum rule derived expectation values of ${\mathrm{L}}_{z}$ and ${\mathrm{S}}_{z}$ show noticeable deviations and detection angle dependence. We show that small deviations of the polarization dependent fluorescent decay values lead to significant deviations in the ${\mathrm{L}}_{z}$ and ${\mathrm{S}}_{z}$ sum rule values. Fe and Co experimental XMCD spectra of a supported 10 nm ${\mathrm{CoFe}}_{2}{\mathrm{O}}_{4}$ thin film are measured simultaneously by both electron and fluorescence yield. The deviations shown in the experimental data are well explained by the calculations and are shown to mainly depend on the polarization dependent total decay. We conclude that fluorescence yield detected x-ray magnetic circular dichroism is unsuitable for quantitative analysis of the ${\mathrm{L}}_{z}$ and ${\mathrm{S}}_{z}$ sum rule values.

Sum rules and the role of pressure on the excitation spectrum of a confined hydrogen atom by a spherical cavity

Sum rule relations over the excitation spectrum of a quantum system contain information about both the energy spectrum and eigenfunctions of the system in a compact form, particularly regarding closure relations. In this work, the effects of pressure induced by a spherical cavity on an atomic hydrogen impurity on the dipole oscillator strength (DOS) sum rule, Sk, and its logarithmic version, Lk, are studied by means of a numerical approach based on a finite-difference solution to the Schrödinger equation. Pressure effects are accounted for by means of a spherical cavity of radius R0 immersed in a medium characterized by a penetrable potential height V0. The DOS sum rules Sk and Lk are investigated as a function of these cavity parameters and thus directly related to the impurity static pressure and surrounding material. One finds that the sum rules are fulfilled within the numerical precision for low pressure conditions. However, when the barrier height is large or infinite (a non-penetrable cavity), the sum rule, for positive k, differs from its closure relation. One finds that this occurs for a cavity radius au, corresponding to a pressure such that the first p-state that contributes to the sum rule has positive energy and it is due to the fact that the spherical confinement cavity potential dominates over the Coulombic interaction for the hydrogenic impurity. Thus, as pressure increases, the excitation spectrum approaches that of a particle confined by a spherical cavity while the ground state is slightly affected by the cavity and more closely resembles a hydrogenic atom. Therefore, the sum rule over the excitation spectrum tends to a particle confined by a spherical cavity, while the closure relation gives that of a confined hydrogen atom in the ground state. For negative k, low excitations are the most important and this behavior is not presented. As the sum rule is the static dipole polarizability, the results are compared to available data in the literature, showing excellent agreement. This behavior in the sum rule and oscillator strength in electron–impurity excitations affects optical transitions of importance in semiconductor nanostructures.

Laser spectroscopic probing of coexisting superfluid and insulating states of an atomic Bose-Hubbard system.

A system of ultracold atoms in an optical lattice has been regarded as an ideal quantum simulator for a Hubbard model with extremely high controllability of the system parameters. While making use of the controllability, a comprehensive measurement across the weakly to strongly interacting regimes in the Hubbard model to discuss the quantum many-body state is still limited. Here we observe a great change in the excitation energy spectra across the two regimes in an atomic Bose–Hubbard system by using a spectroscopic technique, which can resolve the site occupancy in the lattice. By quantitatively comparing the observed spectra and numerical simulations based on sum rule relations and a binary fluid treatment under a finite temperature Gutzwiller approximation, we show that the spectra reflect the coexistence of a delocalized superfluid state and a localized insulating state across the two regimes.

Analyticity and sum rules for photon GPDs

The photon is explored as an object to test the applications of QCD to the perturbatively calculable collinear parton distributions. We investigate analytic properties of DVCS amplitudes and related sum rules of generalized parton distributions of the photon using as an input their earlier calculations in the leading order. The relation of these GPDs to the quintessential functions in the framework of the dual parametrization approach is also found.

Neutralino and chargino masses and related sum rules beyond MSSM

We study the implications of dimension five operators involving Higgs chiral superfields for the masses of neutralinos and charginos in the minimal supersymmetric standard model (MSSM). These operators can arise from additional interactions beyond those of MSSM involving new degrees of freedom at or above the TeV scale. In addition to the masses of the neutralinos and charginos, we study the sum rules involving the masses and squared masses of these particles for different gaugino mass patterns in presence of the dimension five operators. We derive a relation for the higgsino mixing mass parameter and tanβ in the presence of the dimension five operators.

Discussion notes on “Disjoining pressure of planar adsorbed films”, by J.R. Henderson

Discussion notes on “Disjoining pressure of planar adsorbed films”, by J.R. Henderson

The orbital g-factor and related sum rules

The renormalization of the orbital g-factor in nuclei is discussed on the basis of gauge invariance. The relation of the orbital g-factor to the integrated E1 photoabsorption cross section is reviewed, and its relation to the M1 sum rule for the scissors mode of deformed nuclei is examined.

Radio-frequency response of strongly interacting Fermi gases at finite temperatures

The radio frequency spectrum of the fermions in the unitary limit at finite temperatures is characterized by the sum rule relations. We consider a simple picture where the atoms are removed by radio frequency excitations from the strongly interacting states into a state of negligible interaction. We calculate the moments of the response function in the range of temperature $0.08 \epsilon_F < T < 0.8 \epsilon_F$ using auxiliary field Monte Carlo technique (AFMC) in which continuum auxiliary fields with a density dependent shift are used. We estimate the effects of superfluid pairing from the clock shift. We find a qualitative agreement with the pairing gap - pseudogap transition behavior. We also find within the quasiparticle picture that in order for the gap to come into quantitative agreement with the previously known value at T=0, the effective mass has to be $m^* \sim 1.43 m$. Finally, we discuss implications for the adiabatic sweep of the resonant magnetic field.

Perturbative estimates of lepton mixing angles in unified models

Many unified models predict two large neutrino mixing angles, with the charged lepton mixing angles being small and quark-like, and the neutrino masses being hierarchical. Assuming this, we present simple approximate analytic formulae giving the lepton mixing angles in terms of the underlying high energy neutrino mixing angles together with small perturbations due to both charged lepton corrections and renormalisation group (RG) effects, including also the effects of third family canonical normalization (CN). We apply the perturbative formulae to the ubiquitous case of tri-bimaximal neutrino mixing at the unification scale, in order to predict the theoretical corrections to mixing angle predictions and sum rule relations, and give a general discussion of all limiting cases. We also discuss the implications for the sum rule relations of the measurement of a non-zero reactor angle, as hinted at by recent experimental measurements.

Evaluation of the axial-vector commutator sum rule for pion-pion scattering

We consider the sum rule proposed by one of us (S. L. A.), obtained by taking the expectation value of an axial-vector commutator in a state with one pion. The sum rule relates the pion decay constant to integrals of pion-pion cross sections, with one pion off the mass shell. We remark that recent data on pion-pion scattering allow a precise evaluation of the sum rule. We also discuss the related Adler-Weisberger sum rule (obtained by taking the expectation value of the same commutator in a state with one nucleon), especially in connection with the problem of extrapolation of the pion momentum off its mass shell. We find, with current data, that both the pion-pion and pion-nucleon sum rules are satisfied to better than 6%, and we give detailed estimates of the experimental and extrapolation errors in the closure discrepancies.

The Use of the Virial Theorem and Sum-rules in Atomic Structure Calculations

Dilatation transformations have been used sucessfully to obtain eigenvalues and eigenfunctions of metastable states such as resonances in electron-atom scattering.The applica- tion of dilatation theory requires in general the expansion of the eigenfunctions and energies of dilated Hamiltonians in powers of the dilatation parameter. These expansions yield a set of sum rules which supply stationary and stability conditions which exact solutions will satisfy auto- matically. For approximate wave functions these expansions provide tools for optimization in a given parameter space. The flrst member in the set of sum rules is the quantum virial theorem which is particularly valuable to obtain the correct balance of potential and kinetic energy. Ap- plied to resonances these expectation values are both complex numbers. Accurate calculations of properties of few electron systems are of interest for astrophysical plasma diagnostics. So far, most calculations have been performed on isolated atoms and molecules. In Debye-Huckel theory (4), the dilatation technique can be conveniently extended to allow for the inclusion of a model plasma environment. Correlated wave functions for two-electron systems have widely been applied to atomic structure calculations in screening environments modeled by the Debye-Huckel theory, e.g., (3,6). The cus- tomary expansion of the eigensolutions as linear combinations of a flnite number of correlated basis functions depends crucially on the size and choice of the basis set. In particular, moderately sized basis sets of Slater type orbitals require a careful, non-linear optimization of the exponents. The quantum virial theorem and a set of related sum rules provide tools not only to assess the quality of approximate calculations when the exact result is not known a priori but also to optimize the electronic wave function in a large space of nonlinear parameters. The present work studies the extension of the dilatation technique to ionic systems which are embedded in a Debye Plasma. The satisfaction of the virial theorem is warranted for physical reasons so that the potential and kinetic energy contributions are well balanced. In its complex version, the quantum virial theorem and the corresponding complex sum rules provide in addition a means to optimize resonance energies and widths without having to strictly orthogonalize the eigenvectors to all lower solutions of the same symmetry. In complex, non-Hermitian quantum mechanics this is not only not possible but also not wanted. Debye screening, originally developed for strong electrolytes is now used as a convenient model for plasmas at or near thermal equilibrium. In Debye plasmas, the interaction between the electrons and the atomic nucleus are screened as well as the electron-electron interaction. Two-electron systems ofier both relatively simple spectra and the potential for highly accurate calculations which makes them valuable tools for plasma diagnostics. Of course, electronic structure calculations of metastable states have been performed using a variety of difierent techniques: An extension into the complex plane is the hallmark of the dilatation transformation approach (1,5,7,10{13) | also called rotated coordinate methods | as well as the Siegert approach (14), while other approaches solve the same problem entirely with real numerics (6). In the present work we focus on the bound and resonance state calculations by augmenting the dilatation transformation approach with the implementation of the quantum virial theorem and the second order sum rule. The dilatation transformation consists mainly in the replacement of all radial variables ~r by iµ~r in the Hamiltonian operator of the system. By this procedure the computational method is extended onto relevant parts of the complex energy plane which allows for the calculation of decaying states (i.e., resonances) as eigenstates of the transformed Hamiltonian which is no longer Hermitean (8). The corresponding eigenvalues are complex-valued with its imaginary part related to the lifetime of the state. The departure from Hermiticity has consequences for the computational strategy. We lose, in particular, the well established rule of variational calculations: The lower the calculated energy the better is the wave function. One aims instead at a high degree of stability of the resonance eigenvalue against changes of the rotation angle µ. In order to calculate a resonance