Scattering Sum Rules Research Articles

Light-by-light scattering sum rule for radiative transitions of bottomonia

We generalize a forward light-by-light scattering sum rule to the case of heavy quarkonium radiative transitions. We apply such sum rule to the bottomonium states, and use available data on radiative transitions in its evaluation. For the transitions that are not known experimentally, we provide theoretical estimates within a potential model, and consider the spread between similar approaches in the literature as an estimate for the model error. For the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ states we observe that, due to a cancellation between transitions involving $\chi_{b0}, \chi_{b1}$, and $\chi_{b2}$ states, the sum rule is satisfied within experimental and theoretical error estimates. Having tested this sum rule for the low-lying bottomonium states, it may be used as a tool to investigate the nature of exotic states in the charmonium and bottomonium spectrum through the corresponding radiative transitions.

Light-by-light forward scattering sum rules for charmonium states

We apply three forward light-by-light scattering sum rules to charmonium states. We show that these sum rules imply a cancellation between charmonium bound state contributions, which are mostly known from the $\gamma \gamma$ decay widths of these states, and continuum contributions above $D \bar D$ threshold, for which we provide a duality estimate. We also show that two of these sum rules allow to predict the yet unmeasured $\gamma^\ast \gamma$ coupling of the $\chi_{c1}(1P)$ state, which can be tested at present high-luminosity $e^+ e^-$ colliders.

Light-by-light scattering sum rules in light of new data

We evaluate the light-quark meson contributions to three exact light-by-light scattering sum rules in light of new data by the Belle Collaboration, which recently has extracted the transition form factors of the tensor meson $f_2(1270)$ as well as of the scalar meson $f_0(980)$. We confirm a previous finding that the $\eta, \eta^\prime$ and helicity-2 $f_2(1270)$ contributions saturate one of these sum rules up to photon virtualities around 1 GeV$^2$. At larger virtualities, our sum rule analysis shows an important contribution of the $f_2(1565)$ meson and provides a first empirical extraction of its helicity-2 transition form factor. Two further sum rules allow us to predict the helicity-0 and helicity-1 transition form factors of the $f_2(1270)$ meson. Furthermore, our analysis also provides an update for the scalar and tensor meson hadronic light-by-light contributions to the muon's anomalous magnetic moment.

The generalized BLM approach to fix scale- dependence in QCD: the current status of investigations

I present a brief review of the generalized Brodsky-Lepage-McKenzie (BLM) approaches to fix the scale-dependence of the renormalization group (RG) invariant quantities in QCD. At first, these approaches are based on the expansions of the coefficients of the perturbative series for the RG-invariant quantities in the products of the coefficients βi of the QCD β-function, which are evaluated in the MS-like schemes. As a next step all βi-dependent terms are absorbed into the BLM-type scale(s) of the powers of the QCD couplings. The difference between two existing formulations of the above mentioned generalizations based on the seBLM approach and the Principle of Maximal Conformality (PMC) are clarified in the case of the Bjorken polarized deep-inelastic scattering sum rule. Using the conformal symmetry-based relations for the non-singlet coefficient functions of the Adler D-function and of Bjorken polarized deep-inelastic scattering sum rules CBjpNS (as) the βi-dependent structure of the NNLO approximation for CBjpNS (as) is predicted in QCD with ngl-multiplet of gluino degrees of freedom, which appear in SUSY extension of QCD. The importance of performing the analytical calculation of the N3LO additional contributions of ngl gluino multiplet to CBjpNS (as) for checking the presented in the report NNLO prediction and for the studies of the possibility to determine the discussed {β}-expansion pattern of this sum rule at the O(a4s)-level is emphasised.

Sum Rules for Parallel-Plate Waveguides: Experimental Results and Theory

An experimental approach to investigate the forward scattering sum rule for periodic structures is presented. This approach allows an upper bound on the total cross section integrated over a bandwidth from a simple static problem to be found. Based on energy conservation, the optical theorem is used to construct a relation between the total cross section and the forward scattering of periodic structures as well as single scatterers inside a parallel-plate waveguide. Dynamic measurements are performed using a parallel-plate waveguide and a parallel-plate capacitor is utilized to find the static polarizability. Convex optimization is introduced to identify the total cross section in the dynamic measurements and estimate an optimal lower bound on the polarizability for objects. The results show that the interactions between the electromagnetic field and an object over all wavelengths are given by the static polarizability of the object.

Single meson contributions to the muon’s anomalous magnetic moment

We develop the formalism to provide an improved estimate for the hadronic light-by-light correction to the muon's anomalous magnetic moment a_{\mu}, by considering single meson contributions beyond the leading pseudo-scalar mesons. We incorporate available experimental input as well as constraints from light-by-light scattering sum rules to estimate the effects of axial-vector, scalar, and tensor mesons. We give numerical evaluations for the hadronic light-by-light contribution of these states to a_{\mu}. The presented formalism allows to further improve on these estimates, once new data for such meson states will become available.

Conformal symmetry limit of QED and QCD and identities between perturbative contributions to deep-inelastic scattering sum rules

Conformal symmetry-based relations between concrete perturbative QED and QCD approximations for the Bjorken , the Ellis-Jaffe sum rules of polarized lepton- nucleon deep-inelastic scattering (DIS), the Gross-Llewellyn Smith sum rules of neutrino-nucleon DIS, and for the Adler functions of axial-vector and vector channels are derived. They result from the application of the operator product expansion to three triangle Green functions, constructed from the non-singlet axial-vector, and two vector currents, the singlet axial-vector and two non-singlet vector currents and the non-singlet axial-vector, vector and singlet vector currents in the limit, when the conformal symmetry of the gauge models with fermions is considered unbroken. We specify the perturbative conditions for this symmetry to be valid in the case of the U(1) and SU(N c) models. The all-order perturbative identity following from the conformal invariant limit between the concrete contributions to the Bjorken, the Ellis-Jaffe and the Gross-Llewellyn Smith sum rules is proved. The analytical and numerical O(α 4) and $ O\left( {\alpha_s^2} \right) $ conformal symmetry based approximations for these sum rules and for the Adler function of the non-singlet vector currents are summarized. Possible theoretical applications of the results presented are discussed.

Spin orbit magnetism and unconventional superconductivity

We find an exotic spin excitation in a magnetically ordered system with spin orbit magnetism in 2D, where the order parameter has a net spin current and no net magnetization. Starting from a Fermi liquid theory, similar to that for a weak ferromagnet, we show that this excitation emerges from an exotic magnetic Fermi liquid state that is protected by a generalized Pomeranchuck condition. We derive the propagating mode using the Landau kinetic equation, and find that the dispersion of the mode has a $\sqrt q$ behavior in leading order in 2D. We find an instability toward superconductivity induced by this exotic mode, and a further analysis based on the forward scattering sum rule strongly suggests that this superconductivity has p-wave pairing symmetry. We perform similar studies in the 3D case, with a slightly different magnetic system and find that the mode leads to a Lifshitz-like instability most likely toward an inhomogeneous magnetic state in one of the phases.

Optical Theorem and Forward Scattering Sum Rule for Periodic Structures

Based on energy conservation, an optical theorem is constructed for a slab having an arbitrary periodic microstructure in a plane. A sum rule for low pass structures is derived using analytic properties of Herglotz functions based on causality and passivity. The sum rule relates the total cross section to the static polarizability per unit cell, and quantifies the interaction between the slab and electromagnetic fields possible over all wavelengths. The results are illustrated with several numerical and experimental examples.

Light-by-light scattering sum rules constraining meson transition form factors

Relating the forward light-by-light scattering to energy weighted integrals of the ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}$ fusion cross sections, with one real photon ($\ensuremath{\gamma}$) and one virtual photon (${\ensuremath{\gamma}}^{*}$), we find two new exact superconvergence relations. They complement the known superconvergence relation based on the extension of the Gerasimov-Drell-Hearn sum rule to the light-light system. We also find a set of sum rules for the low-energy photon-photon interaction. All of the new relations are verified here exactly at leading order in scalar and spinor QED. The superconvergence relations, applied to the ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}$ production of mesons, lead to intricate relations between the $\ensuremath{\gamma}\ensuremath{\gamma}$ decay widths or the ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}$ transition form factors for (pseudo)scalar, axial-vector, and tensor mesons. We discuss the phenomenological implications of these results for mesons in both the light-quark sector and the charm-quark sector.

Adler Function, DIS sum rules and Crewther Relations

The current status of the Adler function and two closely related Deep Inelastic Scattering (DIS) sum rules, namely, the Bjorken sum rule for polarized DIS and the Gross-Llewellyn Smith sum rule are briefly reviewed. A new result is presented: an analytical calculation of the coefficient function of the latter sum rule in a generic gauge theory in order O(alpha_s^4). It is demonstrated that the corresponding Crewther relation allows to fix two of three colour structures in the O(alpha_s^4) contribution to the singlet part of the Adler function.

Time-domain approach to the forward scattering sum rule

The forward scattering sum rule relates the extinction cross section integrated over all wavelengths with the polarizability dyadics. It is useful for deriving bounds on the interaction between scatterers and electromagnetic fields, antenna bandwidth and directivity and energy transmission through sub-wavelength apertures. The sum rule is valid for linearly polarized plane waves impinging on linear, passive and time translational invariant scattering objects in free space. Here, a time-domain approach is used to clarify the derivation and the used assumptions. The time-domain forward scattered field defines an impulse response. Energy conservation shows that this impulse response is the kernel of a passive convolution operator, which implies that the Fourier transform of the impulse response is a Herglotz function. The forward scattering sum rule is finally constructed from integral identities for Herglotz functions.

Two-photon physics

It is reviewed how Compton scattering sum rules relate low-energy nucleon structure quantities to the nucleon excitation spectrum. In particular, the GDH sum rule and recently proposed extensions of it will be discussed. These extensions are sometimes more calculationally robust, which may be an advantage when estimating the chiral extrapolations of lattice QCD results, such as for anomalous magnetic moments. Subsequently, new developments in our description of the nucleon excitation spectrum will be discussed, in particular a recently developed chiral effective field theory framework for the Δ(1232)-resonance region. Within this framework, we discuss results on N and Δ masses, the γNΔ transition and the Δ magnetic dipole moment.

Two-photon physics

It is reviewed how Compton scattering sum rules relate low-energy nucleon structure quantities to the nucleon excitation spectrum. In particular, the GDH sum rule and recently proposed extensions of it will be discussed. These extensions are sometimes more calculationally robust, which may be an advantage when estimating the chiral extrapolations of lattice QCD results, such as for anomalous magnetic moments. Subsequently, new developments in our description of the nucleon excitation spectrum will be discussed, in particular a recently developed chiral effective field theory framework for the $\Delta(1232)$-resonance region. Within this framework, we discuss results on $N$ and $\Delta$ masses, the $\gamma N \Delta$ transition and the $\Delta$ magnetic dipole moment.

Heavy flavour contributions to the deep inelastic scattering sum rules

We have calculated the first and second order corrections to several deep inelastic sum rules which are due to heavy flavour contributions. A comparison is made with the existing perturbation series which has been computed up to third order for massless quarks only. In general it turns out that the effects of heavy quarks are very small except when Q∼ m or Q≫ m. Here Q and m denote the virtual mass of the vector boson and the mass of the heavy quark, respectively. For Q≫ m the radiative corrections reveal large logarithms of the type ln Q 2/ m 2 which have to be absorbed in the running coupling constant so that the number of light flavours n f is enhanced by one unit. However this has to happen at much larger values of Q i.e. Q∼6.5 m than one usually assumes for the flavour thresholds which appear in the running coupling constant. An alternative description for the heavy flavour dependence of the running coupling constant in the context of the MOM-scheme is discussed.

All-orders renormalon resummations for some QCD observablest

Exact large- N f results for the QCD Adler D-function and Deep Inelastic Scattering sum rules are used to resum to all orders the portion of QCD perturbative coefficients containing the highest power of b = 1 6 (11 N-2N f) , for SU ( N) QCD with N f quark flavours. These terms correspond to renormalon singularities in the Borel plane and are expected asymptotically to dominate the coefficients to all orders in the 1/ N f expansion. Remarkably, we note that this is already apparent in comparisons with the exact next-to-leading order (NLO) and next-to-NLO (NNLO) perturbative coefficients. The ultra-violet (UV) and infra-red (IR) renormalon singularities in the Borel transform are isolated and the Borel sum (principal value regulated for IR) performed. Resummed results are also obtained for the Minkowski quantities related to the D-function, the e + e − R-ratio and the analogous τ-lepton decay ratio, R τ . The renormalization scheme dependence of these partial resummations is discussed and they are compared with the results from other groups and with exact fixed-order perturbation theory at NNLO. Prospects for improving the resummation by including more exact details of the Borel transform are considered.

Relativistic corrections to the electromagnetic and axial moments of nuclei and other composite systems

We calculate the electromagnetic and axial nuclear moments of the deuteron and triton as a function of their radius using a relativistic two-nucleon and three-nucleon model formulated on the light-cone. The results also provide an estimate of the nuclear binding corrections to helicity-dependent deep inelastic scattering sum rules. At large nucleon radius, the moments are given by the usual nonrelativistic formulae modified by finite binding effects. At small radius, the moments take the canonical values given by the generalization of the Drell-Hearn-Gerasimov sum rule. In addition, as R → 0, the constituent helicities become completely disoriented, and the Gamow-Teller matrix element vanishes. Thus, in the pointlike limit MR → 0, the moments of spin-one bound states coincide with the canonical couplings of elementary spin-one bosons of the Standard Model, μ = e M , Q = −e M 2 , and g a = 0. Similarly, in the case of spin-half systems, the magnetic moment approaches the Dirac value μ = e 2M , and the axial coupling g a vanishes.

Robustness of a local Fermi liquid against ferromagnetism and phase separation.

We study the properties of Fermi liquids with the microscopic constraint of a local self-energy. In this case the forward scattering sum rule imposes strong limitations on the Fermi-liquid parameters, which rule out any Pomeranchek instabilities---both ferromagnetism and phase separation are suppressed. Superconductivity is possible in an $s$-wave channel only. We also study the approach to the metal-insulator transition, and find a Wilson ratio approaching 2. This ratio and other properties of ${\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{TiO}}_{3}$ are all consistent with the local Fermi-liquid scenario.

Estimates of the higher-order QCD corrections: theory and applications

We consider the further development of the formalism of the estimates of higher-order perturbative corrections in the Euclidean region, which is based on the application of the scheme-invariant methods, namely the principle of minimal sensitivity and the effective charges approach. We present the estimates of the order $O(\alpha^{4}_{s})$ QCD corrections to the Euclidean quantities: the $e^+e^-$-annihilation $D$-function and the deep inelastic scattering sum rules, namely the non-polarized and polarized Bjorken sum rules and to the Gross--Llewellyn Smith sum rule. The results for the $D$-function are further applied to estimate the $O(\alpha_s^4)$ QCD corrections to the Minkowskian quantities $R(s) = \sigma_{tot} (e^{+}e^{-} \to {\rm hadrons}) / \sigma (e^{+}e^{-} \to \mu^{+} \mu^{-})$ and $R_{\tau} = \Gamma (\tau \to \nu_{\tau} + {\rm hadrons}) / \Gamma (\tau \to \nu_{\tau} \overline{\nu}_{e} e)$. The problem of the fixation of the uncertainties due to the $O(\alpha_s^5)$ corrections to the considered quantities is also discussed.

ESTIMATES OF THE HIGHER ORDER QCD CORRECTIONS TO R(s), Rτ AND DEEP INELASTIC SCATTERING SUM RULES

We present the attempt to study the problem of the estimates of higher order perturbative corrections to physical quantities in the Euclidean region. Our considerations are based on the application of the scheme-invariant methods, namely the principle of minimal sensitivity and the effective charges approach. We emphasize that in order to obtain the concrete results for the physical quantities in the Minkowskian region the results of application of this formalism should be supplemented by the explicit calculations of the effects of the analytical continuation. We present the estimates of the order [Formula: see text] QCD corrections to the Euclidean quantities: the e+e−-annihilation D-function and the deep inelastic scattering sum rules, namely the nonpolarized and polarized Bjorken sum rules and to the Gross–Llewellyn Smith sum rule. The results for the D-function are further applied to estimate the [Formula: see text] QCD corrections to the Minkowskian quantities R(s) = σ tot (e+e− → hadrons )/σ(e+e− → µ+µ−) and [Formula: see text]. The problem of the fixation of the uncertainties due to the [Formula: see text] corrections to the considered quantities is also discussed.