Partial-wave Dispersion Relations Research Articles

Nucleon resonance parameters from Roy–Steiner equations

A reliable determination of the pole parameters and residues of nucleon resonances is notoriously challenging, given the required analytic continuation into the complex plane. We provide a comprehensive analysis of such resonance parameters accessible with Roy–Steiner equations for pion–nucleon scattering—a set of partial-wave dispersion relations that combines the constraints from analyticity, unitarity, and crossing symmetry—most prominently of the Δ(1232) resonance. Further, we study the Roper, N(1440), resonance, which lies beyond the strict domain of validity, in comparison to Padé approximants, comment on the role of subthreshold singularities in the S-wave, and determine the residues of the f0(500), ρ(770), and f0(980) resonances in the t-channel process ππ→N¯N. The latter allows us to test—for the first time fully model independently in terms of the respective residues—universality of the ρ(770) couplings and the Goldberger–Treiman relation expected if the scalars behaved as dilatons, in both cases revealing large deviations from the narrow-resonance limit.

F_{0}(1370) Controversy from Dispersive Meson-Meson Scattering Data Analyses.

We establish the existence of the long-debated f_{0}(1370) resonance in the dispersive analyses of meson-meson scattering data. For this, we present a novel approach using forward dispersion relations, valid for generic inelastic resonances. We find its pole at (1245±40)-i(300_{-70}^{+30}) MeV in ππ scattering. We also provide the couplings as well as further checks extrapolating partial-wave dispersion relations or with other continuation methods. A pole at (1380_{-60}^{+70})-i(220_{-70}^{+80}) MeV also appears in the ππ→KK[over ¯] data analysis with partial-wave dispersion relations. Despite settling its existence, our model-independent dispersive and analytic methods still show a lingering tension between pole parameters from the ππ and KK[over ¯] channels that should be attributed to data.

Data-driven dispersive analysis of the ππ and πK scattering

We present a data-driven analysis of the resonant $S$-wave $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}K\ensuremath{\rightarrow}\ensuremath{\pi}K$ reactions using the partial-wave dispersion relation. The contributions from the left-hand cuts are accounted for using the Taylor expansion in a suitably constructed conformal variable. The fits are performed to experimental and lattice data as well as Roy analyses. For the $\ensuremath{\pi}\ensuremath{\pi}$ scattering we present both a single- and a coupled-channel analysis by including additionally the $K\overline{K}$ channel. For the latter the central result is the Omn\`es matrix, which is consistent with the most recent Roy and Roy-Steiner results on $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}K\overline{K}$, respectively. By the analytic continuation to the complex plane, we found poles associated with the lightest scalar resonances $\ensuremath{\sigma}/{f}_{0}(500)$, ${f}_{0}(980)$, and $\ensuremath{\kappa}/{K}_{0}^{*}(700)$ for the physical pion mass value and in the case of $\ensuremath{\sigma}/{f}_{0}(500)$, $\ensuremath{\kappa}/{K}_{0}^{*}(700)$ also for unphysical pion mass values.

Analytic Euclidean bootstrap

We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension ∆ tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading frac{1}{varDelta } tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex ∆ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large ∆ expansion works very well already for small ∆ ∼ 1. We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts.

Dispersion relations for \u03b3\u2217\u03b3\u2217 \u2192 \u03c0\u03c0: helicity amplitudes, subtractions, and anomalous thresholds

We present a comprehensive analysis of the dispersion relations for the doubly-virtual process γ∗γ∗ → ππ. Starting from the Bardeen-Tung-Tarrach amplitudes, we first derive the kernel functions that define the system of Roy-Steiner equations for the partial-wave helicity amplitudes. We then formulate the solution of these partial-wave dispersion relations in terms of Omnès functions, with special attention paid to the role of subtraction constants as critical for the application to hadronic light-by-light scattering. In particular, we explain for the first time why for some amplitudes the standard Muskhelishvili-Omnès solution applies, while for others a modified approach based on their left-hand cut is required unless subtractions are introduced. In the doubly-virtual case, the analytic structure of the vector-resonance partial waves then gives rise to anomalous thresholds, even for space-like virtualities. We develop a strategy to account for these effects in the numerical solution, illustrated in terms of the D-waves in γ∗γ∗ → ππ, which allows us to predict the doubly-virtual responses of the f2(1270) resonance. In general, our results form the basis for the incorporation of two-meson intermediate states into hadronic light-by-light scattering beyond the S-wave contribution.

Pi pi rightarrow K {bar{K}} scattering up to 1.47 GeV with hyperbolic dispersion relations

In this work we provide a dispersive analysis of pi pi rightarrow K{bar{K}} scattering. For this purpose we present a set of partial-wave hyperbolic dispersion relations using a family of hyperbolas that maximizes the applicability range of the hyperbolic dispersive representation, which we have extended up to 1.47 GeV. We then use these equations first to test simple fits to different and often conflicting data sets, also showing that some of these data and some popular parameterizations of these waves fail to satisfy the dispersive analysis. Our main result is obtained after imposing these new relations as constraints on the data fits. We thus provide simple and precise parameterizations for the S, P and D waves that describe the experimental data from K{{bar{K}}} threshold up to 2 GeV, while being consistent with crossing symmetric partial-wave dispersion relations up to their maximum applicability range of 1.47 GeV. For the S-wave we have found that two solutions describing two conflicting data sets are possible. The dispersion relations also provide a representation for S, P and D waves in the pseudo-physical region.

On kinematical constraints in boson-boson systems

We consider the scattering of two-bosons with negative parity and spin 0 or 1. Starting from helicity partial-wave scattering amplitudes we derive transformations that eliminate all kinematical constraints. Such amplitudes are expected to satisfy partial-wave dispersion relations and therefore provide a suitable basis for data analysis and the construction of effective field theories. Our derivation relies on a decomposition of the various scattering amplitudes into suitable sets of invariant functions. A novel algebra was developed that permits the efficient computation of such functions in terms of computer algebra codes.

Kinematical constraints in fermion-antifermion systems

We consider the scattering of fermions off antifermions with spin 1/2 and 3/2. Starting from helicity partial-wave scattering amplitudes we derive transformations that eliminate all kinematical constraints. Such amplitudes are expected to satisfy partial-wave dispersion relations and therefore provide a suitable basis for data analysis and the construction of effective field theories. Our derivation relies on a decomposition of the various scattering amplitudes into suitable sets of invariant functions.

Formula omitted] extension of the large-[formula omitted] partial wave dispersion relations

Continuing our previous work [Z.H. Guo, J.J. Sanz-Cillero, H.Q. Zheng, JHEP 0706 (2007) 030], large-NC techniques and partial wave dispersion relations are used to discuss ππ scattering amplitudes. We get a set of predictions for O(p6) low-energy chiral perturbation theory couplings. They are provided in terms of the masses and decay widths of scalar and vector mesons.

Partial-wave dispersion relations: Exact left-handE-plane discontinuity computed from the Born series

We show that for a superposition of Yukawa potentials, the exact left-hand cut discontinuity in the complex-energy plane of the (S-wave) scattering amplitude is given, in an interval depending on n, by the discontinuity of the Born series stopped at order n. This establishes an inverse and unexpected correspondence of the Born series at positive high energies and negative low energies. With the discontinuity on the left-hand axis elucidated, we can construct a viable dispersion relation (DR) for the partial (S-) wave amplitude. The DR is numerically verified for the exponential potential at zero scattering energy. Generalization to higher partial waves, and extension of these ideas to field theory are discussed.

Pion-nucleon scattering in the P 33 channel in the linear sigma model

A consistent study of the NN interaction requires the investigation of the Δ resonance, as the Δ can affect the properties of the effective light σ meson originating from correlated two-pion exchange and vice versa. We study how the Δ appears as an effective degree of freedom in the linear σ model. Since the field-theoretic scattering amplitudes respect certain analytic properties, it is appropriate to use partial-wave dispersion relations for πN scattering, which produce non-linear integral equations that are solved by the N/ D method. We choose an amplitude that guarantees the correct threshold behavior and that gives a P 33 resonance with a driving term from one-nucleon exchange. Then we investigate the effect of adding the one-sigma-exchange force. With either a very heavy or a light σ meson, we can find a P 33 resonance. For intermediate σ masses, we find a πN bound state. We confirm that the calculated resonance mass depends sensitively on the short-range πN attraction. Nevertheless, the existence of a P 33 resonance is compatible with the existence of an effective light σ meson in this model.

A calculation of low-energy πN partial waves based on fixed- t analyticity

The real parts of πN partial waves from threshold to 500 MeV/ c and below threshold down to the nucleon-exchange cut are calculated from “partial-wave relations” which are obtained by a projection of fixed- t dispersion relations. The formalism proposed by earlier authors was modified such that the method can also be applied to the isospin-even amplitudes. Except for the partial waves P13, D13, D33 and D35, the results are compatible with the KH80 partial-wave solution. The agreement with our new fits to the partial-wave dispersion relation is reasonable, except for D33 and D35. Our results are useful as a starting solution and as constraints in partial-wave analyses. They show to what extent the “experimental” partial-wave amplitudes are compatible with various analyticity constraints. For further applications, we have calculated the coefficients of a threshold expansion which converges up to 320 MeV/ c, provided one considers the one-nucleon contribution separately.

Improved πN partial waves, consistent with analyticity and unitarity

Partial wave dispersion relations in a new formulation and fixed-s dispersion relations are used to calculate the long range contributions to the real parts of πN amplitudes. Recent results from a double spectral function calculation are applied for constraining the imaginary parts. We propose to use the new partial waves as starting values in future phase shift analyses and to treat our prediction for the tail of high partial waves as part of the input. — The differences between our new partial waves and the Karlsruhe-Helsinki 80 and CMU-LBL 80 solutions are discussed. Concerning measurable quantities, the new parital waves lead to significant changes in the forward logarithmic slopes of differential cross sections. — Thel-dependence of our partial waves is discussed up to large values ofl. We find a systematic even-l, odd-l splitting at all energies.

Solution of the partial wave dispersion relations and compound structures of particles

The general representation for the partial wave scattering amplitude compatible with the requirements of unitarity and analyticity is obtained on the basis of the theory of generalized R-functions. Its connection to the Jaffe-Low P-matrix formalism is found which permits one to attribute the CDD type poles arising in the solution to the eigenstate levels of the compound system (quark-gluon bag). The method is applied to (I = 0) S-wave ππ-scattering.

Two techniques in NN elastic scattering

Two methods in NN elastic scattering analysis are developed: (a) how to study the uniqueness of analysing NN → NN scattering amplitudes in terms of various meson exchanges; (b) how to write down partial-wave dispersion relations for NN → NN, and how to deduce the basic properties of ω- and ε-meson exchange forces.

A partial wave dispersion relation analysis of pion-nucleon scattering amplitudes

Partial wave dispersion relations have been evaluated, using various new πN partial wave solutions and our results for the ππN\(\bar N\) amplitudes. The Karlsruhe-Helsinki 80 and CMU-LBL-79 solution forS, P, D, F waves are well compatible with this constraint, which follows from the Mandelstam representation. The VPI 80 solution leads to serious discrepancies in some cases, for instance for the P11 wave.—In both, the KH 80 and CMU-LBL 79, solutions the higher partial waves are not yet satisfactory. They can be improved by adding constraints based on the information on the nearby part of the left hand cuts.—Properties of nucleon resonances are discussed in connection with our results for the “generalized potentials”.

Is there evidence for strong Van der Waals forces in πN scattering?

It was recently claimed that an analysis of the πN partial wave dispersion relation for the P33-amplitude leads to evidence for a new long-range interaction. Using our phase shifts we have studied in detail the partial wave dispersion relation for the P33-amplitude and also for other amplitudes. The result is that all structures are explained by contributions from the well-known nearby cuts, i.e. there is no evidence for a new interaction.

Virtual state of the three nucleon system

The existence of a virtual state of the three nucleon system is established on the basis of three different analyses. Values for its pole position and residue in the doublet, $s$-wave, $n\ensuremath{-}d$ elastic scattering amplitude, are obtained from a fit to the experimental data, from partial wave dispersion relations, and from an exact three-particle, separable potential calculation. The calculations indicate that these parameters are determined mainly by the one-nucleon exchange mechanism and the doublet scattering length ${a}_{2}$. For ${a}_{2}=0.65$ fm our best calculation gives an energy of 0.482 MeV below the elastic threshold, on the second Riemann sheet, and a residue parameter ${{C}_{\ensuremath{\nu}}}^{2}=0.0504$, where ${{C}_{\ensuremath{\nu}}}^{2}$ is defined in analogy to the triton asymptotic normalization parameter.NUCLEAR REACTIONS Three-nucleon virtual state; fits to data; dispersion relations; separable potential calculations.

Asymptotic normalization parameter of the triton

On the basis of exact separable potential calculations, we show that there is a significant variation of the asymptotic normalization parameter ${{C}_{t}}^{2}$ of the triton with its binding energy. We present a partial wave dispersion relation technique for determining this parameter from the triton energy, the doublet $n\ensuremath{-}d$ scattering length, the doublet, $s$-wave, $n\ensuremath{-}d$ inelasticities, and the two-nucleon, on-shell scattering amplitudes. We test the method and find it to be accurate and stable with only low-energy information used as input. For a doublet scattering length of 0.65 fm we obtain ${{C}_{t}}^{2}=3.3\ifmmode\pm\else\textpm\fi{}0.1$, where the error limits are determined from uncertainties in the inelasticities and the analytic continuation of the two-nucleon amplitudes to negative energies.

Dispersion relation approach to three-body systems

Partial-wave dispersion relations are presented which describe both 2 ..-->.. 2 bound-state amplitudes, 2 ..-->.. 3 breakup amplitudes, and 3 ..-->.. 3 three-particle scattering amplitudes. The dispersion relations are reduced to one-dimensional integral equations by absorbing the momentum dependence of the 2..-->.. 3 and 3 ..-->.. 3 amplitudes in the three-body phase-space factors. The momentum dependence is obtained by solving final-state unitarity equations. For S-wave nucleon-deuteron scattering the nature of this off-shell dependence is described in detail. DYnamical singularities of the amplitudes which occur in the dispersion relations are discussed,and the modifications of the dispersion relations due to the presence of anomalous thresholds are written out explicitly. The physical importace of three-body unitarity for the bound-state properties of three-body systems is reviewed and possible applications of the present theory to few-nucleon and pionic systems are briefly discussed.