Subtracted Dispersion Relations Research Articles

Neutron skins: A perspective from dispersive optical models

An overview of neutron skin predictions obtained using an empirical nonlocal dispersive optical model (DOM) is presented. The DOM links both scattering and bound-state experimental data through a subtracted dispersion relation which allows for fully consistent, data-informed predictions for nuclei where such data exist. Large skins were predicted for both 48Ca (Rskin48=0.25±0.023 fm in 2017) and 208Pb (Rskin208=0.25±0.05 fm in 2020). Whereas the DOM prediction in 208Pb is within 1σ of the subsequent PREX-2 measurement, the DOM prediction in 48Ca is over 2σ larger than the thin neutron skin resulting from CREX. From the moment it was revealed, the thin skin in 48Ca has puzzled the nuclear-physics community as no adequate theories simultaneously predict both a large skin in 208Pb and a small skin in 48Ca. The DOM is unique in its ability to treat both structure and reaction data on the same footing, providing a unique perspective on this Rskin puzzle. It appears vital that more neutron data be measured in both the scattering and bound-state domain for 48Ca to clarify the situation.

Flavoured axions in the tail of Bq \u2192 \u03bc+\u03bc\u2212 and B \u2192 \u03b3* form factors

We discuss how LHC di-muon data collected to study Bq → μμ can be used to constrain light particles with flavour-violating couplings to b-quarks. Focussing on the case of a flavoured QCD axion, a, we compute the decay rates for Bq → μμa and the SM background process Bq → μμγ near the kinematic endpoint. These rates depend on non-perturbative Bq → γ(*) form factors with on- or off-shell photons. The off-shell form factors — relevant for generic searches for beyond-the-SM particles — are discussed in full generality and computed with QCD sum rules for the first time. This includes an extension to the low-lying resonance region using a multiple subtracted dispersion relation. With these results, we analyse available LHCb data to obtain the sensitivity on Bq → μμa at present and future runs. We find that the full LHCb dataset alone will allow to probe axion-coupling scales of the order of 106 GeV for both b → d and b → s transitions. As a spin-off application of the off-shell form factors we further analyse the case of light, Beyond the Standard Model, vectors.

Mass corrections to the DGLAP equations

We propose a mass-dependent MOM scheme to renormalize UV divergence of unpolarized PDFs at one-loop order. This approach which is based on a once subtracted dispersion relation does not need any regulator. The overall counterterms are obtained from the imaginary part of large transverse momentum region in loop integrals. The mass-dependent characteristic of the scheme yields to mass-dependent splitting functions for the DGLAP evolution equations. While the flavor number is fixed at any renormalization scale, the decoupling theorem is automatically imposed by the mass-dependent splitting functions. The required symmetries are also automatically respected by our prescription.

Nonperturbative Mellin amplitudes: existence, properties, applications

We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories.

Comments on the dispersion relation method to vector–vector interaction

Abstract We study in detail the method proposed recently to study the vector–vector interaction using the $N/D$ method and dispersion relations, which concludes that, while, for $J=0$, one finds bound states, in the case of $J=2$, where the interaction is also attractive and much stronger, no bound state is found. In that work, approximations are done for $N$ and $D$ and a subtracted dispersion relation for $D$ is used, with subtractions made up to a polynomial of second degree in $s-s_\mathrm{th}$, matching the expression to $1-VG$ at threshold. We study this in detail for the $\rho\rho$ interaction and to see the convergence of the method we make an extra subtraction matching $1-VG$ at threshold up to $(s-s_\mathrm{th})^3$. We show that the method cannot be used to extrapolate the results down to 1270 MeV where the $f_2(1270)$ resonance appears, due to the artificial singularity stemming from the “on-shell” factorization of the $\rho$ exchange potential. In addition, we explore the same method but folding this interaction with the mass distribution of the $\rho$, and we show that the singularity disappears and the method allows one to extrapolate to low energies, where both the $(s-s_\mathrm{th})^2$ and $(s-s_\mathrm{th})^3$ expansions lead to a zero of $\mathrm{Re}\,D(s)$, at about the same energy where a realistic approach produces a bound state. Even then, the method generates a large $\mathrm{Im}\,D(s)$ that we discuss is unphysical.

Proton scalar dipole polarizabilities from real Compton scattering data using fixed-t subtracted dispersion relations and the bootstrap method

We perform a fit of the real Compton scattering (RCS) data below pion-production threshold to extract the electric (αE1) and magnetic (βM1) static scalar dipole polarizabilities of the proton, using fixed-t subtracted dispersion relations and a bootstrap-based fitting technique. The bootstrap method provides a convenient tool for including the effects of the systematic errors on the best values of and , and for propagating the statistical errors of the model parameters fixed by other measurements. We also implement various statistical tests to investigate the consistency of the available RCS data sets below pion-production threshold, and conclude that there are not strong motivations to exclude any data point from the global set. Our analysis yields and , with p-value = 12%.

Comment on “Nucleon spin-averaged forward virtual Compton tensor at large Q2 ”

In recent work, Hill and Paz apply the operator product expansion to forward doubly virtual Compton scattering. The resulting large-${Q}^{2}$ form of the amplitude ${W}_{1}(0,{Q}^{2})$ is compatible with the one we obtain by extrapolation of low-${Q}^{2}$ results from a chiral effective field theory, providing support for our approach. That paper also presents a result for the two-photon contribution to the Lamb shift in muonic hydrogen that has a much larger uncertainty than in previous work. We show that this is an overestimate arising from the inclusion of the proton pole term in the subtracted dispersion relation for ${W}_{1}$.

Nucleon form factors in dispersively improved chiral effective field theory. II. Electromagnetic form factors

We study the nucleon electromagnetic form factors (EM FFs) using a recently developed method combining Chiral Effective Field Theory ($\chi$EFT) and dispersion analysis. The spectral functions on the two-pion cut at $t > 4 M_\pi^2$ are constructed using the elastic unitarity relation and an $N/D$ representation. $\chi$EFT is used to calculate the real functions $J_\pm^1 (t) = f_\pm^1(t)/F_\pi(t)$ (ratios of the complex $\pi\pi \rightarrow N \bar N$ partial-wave amplitudes and the timelike pion FF), which are free of $\pi\pi$ rescattering. Rescattering effects are included through the empirical timelike pion FF $|F_\pi(t)|^2$. The method allows us to compute the isovector EM spectral functions up to $t \sim 1$ GeV$^2$ with controlled accuracy (LO, NLO, and partial N2LO). With the spectral functions we calculate the isovector nucleon EM FFs and their derivatives at $t = 0$ (EM radii, moments) using subtracted dispersion relations. We predict the values of higher FF derivatives with minimal uncertainties and explain their collective behavior. We estimate the individual proton and neutron FFs by adding an empirical parametrization of the isoscalar sector. Excellent agreement with the present low-$Q^2$ FF data is achieved up to $\sim$0.5 GeV$^2$ for $G_E$, and up to $\sim$0.2 GeV$^2$ for $G_M$. Our results can be used to guide the analysis of low-$Q^2$ elastic scattering data and the extraction of the proton charge radius.

Dispersive analysis of the S -, P -, D -, and F -wave ππ amplitudes

A reanalysis of $\pi\pi$ amplitudes for all important partial-waves below about 2 GeV is presented. A set of once subtracted dispersion relations with imposed crossing symmetry condition is used to modify unitary multi-channel amplitudes in the $S$, $P$, $D$, and $F$ waves. So far, these specific amplitudes constructed in our works and many other analyzes have been fitted only to experimental data and therefore do not fulfill the crossing symmetry condition. In the present analysis, the self consistent, i.e. unitary and fulfilling the crossing symmetry, amplitudes for the $S$, $P$, $D$, and $F$ waves are formed. The proposed very effective and simple method of modification of the $\pi\pi$ amplitudes does not change their previous-original mathematical structure and the method can be easily applied in various other analyzes.

Subtracted dispersion relation formalism for the two-photon exchange correction to elastic electron-proton scattering: Comparison with data

We apply a subtracted dispersion relation formalism with the aim to improve predictions for the two-photon exchange corrections to elastic electron-proton scattering observables at finite momentum transfers. We study the formalism on the elastic contribution, and make a detailed comparison with existing data for unpolarized cross sections as well as polarization transfer observables.

Dispersive estimate of the electromagnetic charge symmetry violation in the octet baryon masses

We explore the electromagnetic contribution to the charge symmetry breaking in the octet baryon masses using a subtracted dispersion relation based on the Cottingham formula. For the proton-neutron mass splitting we report a minor revision of the recent analysis of Walker-Loud, Carlson, and Miller [Phys. Rev. Lett. 108, 232301 (2012)]. For the electromagnetic structure of the hyperons we constrain our analysis, where possible, by a combination of lattice QCD and SU(3) symmetry-breaking estimates. The results for the baryon mass splittings are found to be compatible with recent lattice $\mathrm{QCD}+\mathrm{QED}$ determinations. The uncertainties in the dispersive analysis are dominated by the lack of knowledge of the hyperon inelastic structure.

Modifiedππamplitude withσpole

A set of well known once subtracted dispersion relations with imposed crossing symmetry condition is used to modify unitary multichannel $S$ ($\pi\pi$, $K \bar K$, and $\eta\eta$) and $P$ ($\pi\pi$, $\rho 2\pi$, and $\rho\sigma$) wave amplitudes mostly below 1 GeV. Before the modifications, these amplitudes significantly did not satisfy the crossing symmetry condition and did not describe the $\pi\pi$ threshold region. Moreover, the pole of the $S$ wave amplitude related with the $f_0(500)$ meson (former $f_0(600)$ or $\sigma$) had much smaller imaginary part and bigger real one in comparison with those in the newest Particle Data Group Tables. Here, these amplitudes are supplemented by near threshold expansion polynomials and refitted to the experimental data in the effective two pion mass from the threshold to 1.8 GeV and to the dispersion relations up to 1.1 GeV. In result the self consistent, i.e. unitary and fulfilling the crossing symmetry condition, $S$ and $P$ wave amplitudes are formed and the $\sigma$ pole becomes much narrower and lighter. To eliminate doubts about the uniqueness of the so obtained sigma pole position short and purely mathematical proof of the uniqueness of the results is also presented. This analysis is addressed to a wide group of physicists and aims at providing a very effective and easy method of modification of, many presently used, $\pi\pi$ amplitudes with a heavy and broad $\sigma$ meson without changing of their original mathematical structure.

Precise determination of resonance pole parameters through Padé approximants

In this work, we present a precise and model--independent method to extract resonance pole parameters from phase-shift scattering data. These parameters are defined from the associated poles in the second Riemann sheet, unfolded by the analytic continuation to the complex pole using Pad\'e approximants. Precise theoretical parameterizations of pion-pion scattering phase shifts based on once-- and twice-- subtracted dispersion relations are used as input, whose functional form allows us to show the benefit and accuracy of the method. In particular, we extract from these parametrization the pole positions of the $f_0(500)$ at $\sqrt{s}=(453\pm 15) - i(297 \pm 15)$ MeV, the $\rho(770)$ at $\sqrt{s}=(761.4\pm 1.2)-i(71.8 \pm 1.0)$ MeV, and the pole of the $f_2(1270)$, located at $\sqrt{s}=(1267.3\pm 1.7)-i(95.0\pm 2.3)$ MeV. The couplings of the resonances to two pions are also determined with high precision, obtaining respectively, $3.8\pm 0.4$ GeV, $5.92\pm 0.15$ and $4.41\pm 0.23$ GeV$^{-1}$. Special attention is dedicated to the systematic treatment of the theoretical and statistical uncertainties, together with their comparison with previous determinations.

Precise determination of the [formula omitted] and [formula omitted] parameters in dispersive analysis of the ππ data

Use of the new and precise dispersive equations with imposed crossing symmetry condition to solve the long-standing puzzle in the parameters of the f0(500), as well as the f0(980) is presented. This puzzle is finally being settled thanks to analyzes carried out during the last years [J. Beringer et al. (Particle Data Group), Phys. Rev. D86, (2012) 010001]. In this report we show how our very recent dispersive data analysis allowed for a precise and model independent determination of the amplitudes for the S, P, D and F waves [R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira and F.J. Yndurain, Phys. Rev. D83, (2011) 074004; R. Garcia-Martin, R. Kamiński, J.R. Pelaez and J. Ruiz de Elvira, Phys. Rev. Lett. 107, (2011) 072001; R. Kamiński, Phys. Rev. D83, (2011) 076008]. Especially we present that the analytic continuation of once subtracted dispersion relations for the S 0 wave to the complex energy plane leads to very precise results for the f0(500) pole: spole=457−13+14−i279−7+11 MeV and for the f0(980) pole: spole=996±7−i25−6+10MeV. We also mention on first (or one of the first) practical application of presented dispersion relations in refitting and in significant improving of the ππ S-wave amplitudes below 1000 MeV.

Precise determination of the parameters of resonancesf0(500) andf0(980) by fitting the data and dispersion relations

The long-standing puzzle in the parameters of the f0(500), as well as the f0(980), is finally being settled [1] thanks to precise dispersive analyses carried out during the last years. Here we report on our very recent dispersive data analysis which allowed for a precise and model independent determination of the amplitudes for the S, P, D and F waves [2–4]. The analytic continuation of once subtracted dispersion relations for the S0 wave to the complex energy plane leads to very precise results for the f0(500) pole: √spole = 457-13+14 - i279-7+11 MeV and for the f0(980) pole: √spole = 996 ± 7 - i25-6+10 MeV.

Dispersion relations with crossing symmetry forππD- andF-wave amplitudes

A set of once subtracted dispersion relations with imposed crossing symmetry condition for the $\ensuremath{\pi}\ensuremath{\pi}$ $D$- and $F$-wave amplitudes is derived and analyzed. An example of numerical calculations in the effective two-pion mass range from the threshold to 1.1 GeV is presented. It is shown that these new dispersion relations impose quite strong constraints on the analyzed $\ensuremath{\pi}\ensuremath{\pi}$ interactions and are very useful tools to test the $\ensuremath{\pi}\ensuremath{\pi}$ amplitudes. One of the goals of this work is to provide a complete set of equations required for easy use. Full analytical expressions are presented. Along with the well-known dispersion relations successful in testing the $\ensuremath{\pi}\ensuremath{\pi}$ $S$- and $P$-wave amplitudes, those presented here for the $D$ and $F$ waves give a complete set of tools for analyses of the $\ensuremath{\pi}\ensuremath{\pi}$ interactions.

Generalized parton distributions and their singularities

A new approach to building models of generalized parton distributions (GPDs) is discussed that is based on the factorized DD (double distribution) ansatz within the single-DD formalism. The latter was not used before, because reconstructing GPDs from the forward limit one should start in this case with a very singular function $f(\ensuremath{\beta})/\ensuremath{\beta}$ rather than with the usual parton density $f(\ensuremath{\beta})$. This results in a nonintegrable singularity at $\ensuremath{\beta}=0$ exaggerated by the fact that $f(\ensuremath{\beta})$'s, on their own, have a singular ${\ensuremath{\beta}}^{\ensuremath{-}a}$ Regge behavior for small $\ensuremath{\beta}$. It is shown that the singularity is regulated within the GPD model of Szczepaniak et al., in which the Regge behavior is implanted through a subtracted dispersion relation for the hadron-parton scattering amplitude. It is demonstrated that using proper softening of the quark-hadron vertices in the regions of large parton virtualities results in model GPDs $H(x,\ensuremath{\xi})$ that are finite and continuous at the ``border point'' $x=\ensuremath{\xi}$. Using a simple input forward distribution, we illustrate implementation of the new approach for explicit construction of model GPDs. As a further development, a more general method of regulating the $\ensuremath{\beta}=0$ singularities is proposed that is based on the separation of the initial single DD $f(\ensuremath{\beta},\ensuremath{\alpha})$ into the ``plus'' part $[f(\ensuremath{\beta},\ensuremath{\alpha}){]}_{+}$ and the $D$ term. It is demonstrated that the ``DD+D'' separation method allows one to (re)derive GPD sum rules that relate the difference between the forward distribution $f(x)=H(x,0)$ and the border function $H(x,x)$ with the $D$-term function $D(\ensuremath{\alpha})$.

ONCE SUBTRACTED DISPERSION RELATIONS IN DESCRIPTION OF THE ππ D- AND F-WAVE AMPLITUDES

Once subtracted dispersion relations with imposed crossing symmetry condition are applied in description of the ππ D- and F-wave amplitudes. We show that these equations impose strong constraints on experimental data and model amplitudes.

Crossing symmetry implemented in ππ D- and F-wave amplitudes

Once subtracted dispersion relations with imposed crossing symmetry condition are applied in description of the ππ D and F wave amplitudes. Equations similar to those with one subtraction (so called GKPY equations) and to those with two subtractions (the Roy's equations) for the S and P waves are derived. We show that they impose strong constraints on experimental data and model amplitudes. Therefore, together with already known GKPY equations these relations can be efficiently used to test the ππ amplitudes in the S , P and D wave.

NEW DISPERSION RELATIONS IN THE DESCRIPTION OF ππ SCATTERING AMPLITUDES

We present a set of once subtracted dispersion relations which implement crossing symmetry conditions for the ππ scattering amplitudes below 1 GeV. We compare and discuss the results obtained for the once and twice subtracted dispersion relations, known as Roy's equations, for three ππ partial JI waves, S0, P and S2. We also show that once subtracted dispersion relations provide a stringent test of crossing and analyticity for ππ partial wave amplitudes, remarkably precise in the 400 to 1.1 GeV region, where the resulting uncertainties are significantly smaller than those coming from standard Roy's equations, given the same input.