Dispersion Sum Rules Research Articles

Chiral perturbation theory of muonic-hydrogen Lamb shift: polarizability contribution

The proton polarizability effect in the muonic-hydrogen Lamb shift comes out as a prediction of baryon chiral perturbation theory at leading order and our calculation yields for it: $\Delta E^{(\mathrm{pol})} (2P-2S) = 8^{+3}_{-1}\, \mu$eV. This result is consistent with most of evaluations based on dispersive sum rules, but is about a factor of two smaller than the recent result obtained in {\em heavy-baryon} chiral perturbation theory. We also find that the effect of $\Delta(1232)$-resonance excitation on the Lamb-shift is suppressed, as is the entire contribution of the magnetic polarizability; the electric polarizability dominates. Our results reaffirm the point of view that the proton structure effects, beyond the charge radius, are too small to resolve the `proton radius puzzle'.

On renormalization group flows and the a-theorem in 6d

We study the extension of the approach to the a-theorem of Komargodski and Schwimmer to quantum field theories in d=6 spacetime dimensions. The dilaton effective action is obtained up to 6th order in derivatives. The anomaly flow a_UV - a_IR is the coefficient of the 6-derivative Euler anomaly term in this action. It then appears at order p^6 in the low energy limit of n-point scattering amplitudes of the dilaton for n > 3. The detailed structure with the correct anomaly coefficient is confirmed by direct calculation in two examples: (i) the case of explicitly broken conformal symmetry is illustrated by the free massive scalar field, and (ii) the case of spontaneously broken conformal symmetry is demonstrated by the (2,0) theory on the Coulomb branch. In the latter example, the dilaton is a dynamical field so 4-derivative terms in the action also affect n-point amplitudes at order p^6. The calculation in the (2,0) theory is done by analyzing an M5-brane probe in AdS_7 x S^4. Given the confirmation in two distinct models, we attempt to use dispersion relations to prove that the anomaly flow is positive in general. Unfortunately the 4-point matrix element of the Euler anomaly is proportional to stu and vanishes for forward scattering. Thus the optical theorem cannot be applied to show positivity. Instead the anomaly flow is given by a dispersion sum rule in which the integrand does not have definite sign. It may be possible to base a proof of the a-theorem on the analyticity and unitarity properties of the 6-point function, but our preliminary study reveals some difficulties.

Accuracy of the pion elastic form factor extracted from a local-duality sum rule

We analyze the accuracy of the pion elastic form factor predicted by a local-duality (LD) version of dispersive sum rules. To probe the precision of this theoretical approach, we adopt potential models with interactions that involve both Coulomb and confining terms. In this case, the exact form factor may be obtained from the solution of the Schrödinger equation and confronted with the LD sum rule results. We use parameter values appropriate for hadron physics and observe that, independently of the details of the confining interaction, the deviation of the LD form factor from the exact form factor culminates in the region Q2 ≈ 4–6 GeV2. For larger Q2, the accuracy of the LD description increases rather fast with Q2. A similar picture is expected for QCD. For the pion form factor, existing data suggest that the LD limit may be reached already at the relatively low values Q2 = 4–10 GeV2. Thus, large deviations of the pion form factor from the behavior predicted by LD QCD sum rules for higher values of Q2, as found by some recent analyses, appear to us quite improbable. New accurate data on the pion form factor at Q2 = 4–10 GeV2 expected soon from JLab will have important implications for the behavior of the pion form factor in a broad Q2 range up to asymptotically large values of Q2.

Extraction of bound-state parameters from dispersive sum rules

The procedure of extracting the ground-state parameters from vacuum-to-vacuum and vacuum-to-hadron correlators within the method of sum rules is considered. The emphasis is laid on the crucial ingredient of this method - the effective continuum threshold. A new algorithm to fix this quantity is proposed and tested. First, a quantum-mechanical potential model which provides the only possibility to probe the reliability and the actual accuracy of the sum-rule method is used as a study case. In this model, our algorithm is shown to lead to a remarkable improvement of the accuracy of the extracted ground-state parameters compared to the standard procedures adopted in the method and used in all previous applications of dispersive sum rules in QCD. As a next step, it is demonstrated that the procedures of extracting the ground-state decay constant in the potential model and in QCD are quantitatively very close to each other. Therefore, the application of the proposed algorithm in QCD promises a considerable increase of the accuracy of the extracted hadron parameters.

Extraction of ground-state decay constant from dispersive sum rules: QCD vs potential models

We compare the extraction of the ground-state decay constant from the two-point correlator in QCD and in potential models and show that the results obtained at each step of the extraction procedure follow a very similar pattern. We prove that allowing for a Borel-parameter-dependent effective continuum threshold yields two essential improvements compared to employing a Borel-parameter-independent quantity: (i) It reduces considerably the (unphysical) dependence of the extracted bound-state mass and the decay constant on the Borel parameter. (ii) In a potential model, where the actual value of the decay constant is known from the Schrödinger equation, a Borel-parameter-dependent threshold leads to an improvement of the accuracy of the extraction procedure. Our findings suggest that in QCD a Borel-parameter-dependent threshold leads to a more reliable and accurate determination of bound-state characteristics by the method of sum rules.

Bound-state parameters from dispersive sum rules for vacuum-to-vacuum correlators

We study the extraction of the ground-state parameters from vacuum-to-vacuum correlators. We work in a quantum-mechanical potential model which provides the only possibility to probe the reliability and the actual accuracy of this method; one obtains the bound-state parameters from the correlators by the standard procedures adopted in the method of sum rules and compares these results with the exact values calculated from the Schrödinger equation. We focus on the crucial ingredient of the method of sum rules—the effective continuum threshold—and propose a new algorithm to fix this quantity. In a quantum-mechanical model, our procedure leads to a remarkable improvement of the accuracy of the extracted ground-state parameters compared to the standard procedures adopted in the method and used in all previous applications of dispersive sum rules in QCD. The application of the proposed procedure in QCD promises a considerable increase of the accuracy of the extracted hadron parameters.

Quadrupole polarizabilities of the pion in the Nambu–Jona-Lasinio model

The electromagnetic dipole and quadrupole polarizabilities of the neutral and charged pions are calculated in the Nambu–Jona-Lasinio model. Our results agree with the recent experimental analysis of these quantities based on dispersion sum rules. Comparison is made with the results from the chiral perturbation theory.

Effective continuum threshold in dispersive sum rules

We study the accuracy of the bound-state parameters obtained with the method of dispersive sum rules, one of the most popular theoretical approaches in nonperturbative QCD and hadron physics. We make use of a quantum-mechanical potential model since it provides the only possibility to probe the reliability and the accuracy of this method: one obtains the bound-state parameters from sum rules and compares these results with the exact values calculated from the Schr\"odinger equation. We investigate various possibilities to fix the crucial ingredient of the method of sum rules---the effective continuum threshold---and propose modifications which lead to a remarkable improvement of the accuracy of the extracted ground-state parameters compared to the standard procedures adopted in the method. Although the rigorous control of systematic uncertainties in the method of sum rules remains unfeasible, the application of the proposed procedures in QCD promises a considerable increase of the actual accuracy of the extracted hadron parameters.

Dispersion theory and sum rules for the non-minimum phase problem in optical spectroscopy

Dispersion relations and sum rules for integer powers of an optical response function are given in the case of the non-minimum phase problem. These relations were obtained using the concept of the Hilbert transform and Blaschke product. The theory presented in this paper is useful both in basic and applied studies of non-minimum phase functions in optics, and also other fields of physics such as high energy physics.

Density of States and Extinction Mean Free Path of Waves in Random Media: Dispersion Relations and Sum Rules

We establish a fundamental relationship between the averaged local density of states and the extinction mean free path of waves propagating in random media. From the principle of causality and the Kramers-Kronig relations, we show that both quantities are connected by dispersion relations and are constrained by a frequency sum rule. The results should be helpful in the analysis of wave transport through complex media and in the design of materials with specific transport properties.

Evidence of Dispersion Relations for the Nonlinear Response of the Lorenz 63 System

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.

Accuracy of bound-state form factors extracted from dispersive sum rules

We discuss the extraction of form factors from three-point sum rules making use of a harmonic-oscillator model, where we derive the exact expression for the relevant correlator. We determine the form factor of the ground state by the standard procedures adopted in the method of sum rules, and compare the obtained results with the known exact values. We show that the uncontrollable uncertainty in the extracted value of the form factor is typically much larger than that for the decay constant. In the example considered, we find the uncontrolled systematic error in the extracted form factor to exceed the 10% level.

Fluctuation-dissipation theorem for the microcanonical ensemble

A derivation of the fluctuation-dissipation theorem for the microcanonical ensemble is presented using linear response theory. The theorem is stated as a relation between the frequency spectra of the symmetric correlation and response functions. When the system is not in the thermodynamic limit, this result can be viewed as an extension of the fluctuation-dissipation relations to a situation where dynamical fluctuations determine the response. Therefore, the relation presented here between equilibrium fluctuations and response can have a very different physical nature from the usual one in the canonical ensemble. These considerations imply that the fluctuation-dissipation theorem is not restricted to the context of the canonical ensemble, where it is usually derived. Dispersion relations and sum rules are also obtained and discussed in the present case. Although analogous to the Kramers-Kronig relations, they are not related to the frequency spectrum but to the energy dependence of the response function.

Determination ofπ±meson polarizabilities from theγγ→π+π−process

A fit of the experimental data to the total cross section of the process $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ in the energy region from threshold to 2500 MeV has been carried out using dispersion relations with subtractions for the invariant amplitudes, where the dipole and the quadrupole polarizabilities of the charged pion are free parameters. As a result, the sum and the difference of the electric and magnetic dipole and quadrupole polarizabilities of the charged pion have been found: $({\ensuremath{\alpha}}_{1}+{\ensuremath{\beta}}_{1}){}_{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}=(0.{18}_{\ensuremath{-}0.02}^{+0.11})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.3em}{0ex}}{\mathrm{fm}}^{3},({\ensuremath{\alpha}}_{1}\ensuremath{-}{\ensuremath{\beta}}_{1}){}_{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}=(13.{0}_{\ensuremath{-}1.9}^{+2.6})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.3em}{0ex}}{\mathrm{fm}}^{3},({\ensuremath{\alpha}}_{2}+{\ensuremath{\beta}}_{2}){}_{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}=(0.133\ifmmode\pm\else\textpm\fi{}0.015)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.3em}{0ex}}{\mathrm{fm}}^{5},({\ensuremath{\alpha}}_{2}\ensuremath{-}{\ensuremath{\beta}}_{2}){}_{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}=(25.{0}_{\ensuremath{-}0.3}^{+0.8})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.3em}{0ex}}{\mathrm{fm}}^{5}$. These values agree with the dispersion sum rule predictions. The value found for the difference of the dipole polarizabilities is consistent with the results obtained from scattering of high energy ${\ensuremath{\pi}}^{\ensuremath{-}}$ mesons off the Coulomb field of heavy nuclei [Yu. M. Antipov et al., Phys. Lett. B121, 445 (1983)] and from radiative ${\ensuremath{\pi}}^{+}$ photoproduction from the proton at MAMI [J. Ahrens et al., Eur. Phys. J. A 23, 113 (2005)], whereas it is at variance with the recent calculations in the framework of chiral perturbation theory.

Determination ofπ0meson quadrupole polarizabilities from the processγγ→π0π0

A fit of the experimental data to the total cross section of the process $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ in the energy region from 270 to 2250 MeV has been carried out using dispersion relations for the invariant amplitudes where the quadrupole polarizabilities are free parameters. As a result the sum and difference of the electric and magnetic quadrupole polarizabilities of the ${\ensuremath{\pi}}^{0}$ meson have been found for the first time: $({\ensuremath{\alpha}}_{2}+{\ensuremath{\beta}}_{2}){}_{{\ensuremath{\pi}}^{0}}=(\ensuremath{-}0.181\ifmmode\pm\else\textpm\fi{}0.004)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.3em}{0ex}}{\text{fm}}^{5}$, $({\ensuremath{\alpha}}_{2}\ensuremath{-}{\ensuremath{\beta}}_{2}){}_{{\ensuremath{\pi}}^{0}}=(39.70\ifmmode\pm\else\textpm\fi{}0.02)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\phantom{\rule{0.3em}{0ex}}{\text{fm}}^{5}$. In addition, dispersion sum rules have been constructed for this sum and difference, respectively. The values of $({\ensuremath{\alpha}}_{2}\ensuremath{-}{\ensuremath{\beta}}_{2}){}_{{\ensuremath{\pi}}^{0}}$ and $({\ensuremath{\alpha}}_{2}+{\ensuremath{\beta}}_{2}){}_{{\ensuremath{\pi}}^{0}}$ extracted from the experimental data on the process $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ are in good agreement with the result of calculations in the framework of these dispersion sum rules.

Kramers-Kronig relations and sum rules of negative refractive index media

Negative refractive index media have become a hot topic in physics due to their proposed revolutionary properties, which would have drastic consequences in design of novel optical devices. We show that Kramers-Kronig relations connecting the real and imaginary parts of the complex refractive index of absorbing media are valid even though the real refractive index may take negative value at some spectral range. In addition universal sum rules for linear optical constants of negative index media are also valid. This means that negative refractive index media are not fundamentally different from regular media. Hence, any spectrum measured from negative refractive index media can be analyzed using dispersion relations and sum rules, which have so far provided information on the optical properties of materials.

THE GERASIMOV-DRELL-HEARN SUM RULE AND THE SPIN STRUCTURE OF THE NUCLEON

▪ Abstract The Gerasimov-Drell-Hearn sum rule is one of several dispersive sum rules that connect the Compton scattering amplitudes to the inclusive photoproduction cross sections of the target under investigation. Being based on such universal principles as causality, unitarity, and gauge invariance, these sum rules provide a unique testing ground to study the internal degrees of freedom that hold the system together. The present article reviews these sum rules for the spin-dependent cross sections of the nucleon by presenting an overview of recent experiments and theoretical approaches. The generalization from real to virtual photons provides a microscope of variable resolution: At small virtuality of the photon, the data sample information about the long-range phenomena, which are described by effective degrees of freedom (Goldstone bosons and collective resonances), whereas the primary degrees of freedom (quarks and gluons) become visible at the larger virtualities. Through a rich body of new data and several theoretical developments, a unified picture of virtual Compton scattering emerges, which ranges from coherent to incoherent processes, and from the generalized spin polarizabilities on the low-energy side to higher twist effects in deep-inelastic lepton scattering.

Derivation of general dispersion relations and sum rules for meromorphic nonlinear optical spectroscopy

Dispersion relations and sum rules for nonlinear susceptibilities are derived using complex analysis and especially the concept of a meromorphic function. The dispersion relations and sum rules provide frames to investigate the consistency between the theory and experiments.

Dispersion theory and sum rules in linear and nonlinear optics

Dispersion theory and sum rules in linear and nonlinear optics

K → ππ Electroweak penguins in the chiral limit

We report on dispersive and finite energy sum rule analyses of the electroweak penguin matrix elements 〈( ππ) 2| Q 7,8| K 0〉 in the chiral limit. We accomplish the correct perturbative matching (scale and scheme dependence) at NLO in α s , and we describe two different strategies for numerical evaluation.