Right-hand Cut Research Articles

Cut structures and an observable singularity in the three-body threshold dynamics: The Tcc+ case

The three-body threshold effect, the distinctive and intriguing nonperturbative dynamics in the low-energy hadron-hadron scattering, has acquired compelling significance in the wake of the recent observation of the double-charm tetraquark Tcc+. This dynamics is characterized by the emergence of singular points and branch cuts within the interaction potential, occurring when the on-shell condition of the mediated particle is satisfied. The presence of these potential singularities indicates that the system is no longer Hermitian and also poses intractable challenges in obtaining exact solutions for dynamical scattering amplitudes. In this work, we develop a complex scaled Lippmann-Schwinger equation as an operation of analytical continuation of the T matrix to resolve this problem. Through a practical application to the DD*→DD* process, we reveal complicated cut structures of the three-body threshold dynamics in the complex plane, primarily stemming from the one-pion exchange. Notably, our methodology succeeds in reproducing the Tcc+ structure, in alignment with the quasibound pole derived from the complex scaling method within the Schrödinger equation framework. More remarkably, after solving the on-shell T matrix on the positive real axis of momentum plane, we find an extra new structure in the DD* mass spectrum, which arises from a right-hand cut at a physical pion mass and should be observable in lattice QCD simulations and future high-energy experiments. Published by the American Physical Society 2024

Analytical dispersive parametrization for elastic scattering of spinless particles

In this paper, we present an improved parametrization of the elastic scattering of spin-0 particles, which is based on a dispersive representation for the inverse scattering amplitude. Besides being based on well-known general principles, the requirement that the inverse amplitude should satisfy the dispersion relation significantly constrains its possible forms and have not been incorporated in the existing parametrizations so far. While the right-hand cut of the inverse scattering amplitude is controlled by unitarity, the contribution from the left-hand cut, which comes from the crossing symmetry, is commonly ignored or incorporated improperly. The latter is parametrized using the expansion in a suitably constructed conformal variable, which accounts for its analytic structure. The correct implementations of the Adler zero and threshold factors for angular momentum $J>0$ are discussed in detail as well. The amplitudes are written in a compact analytic form and provide a useful tool to analyze current and future lattice data in the elastic region improving upon the commonly used Breit-Wigner or K-matrix approaches.

Micro-End-Milling with Small Diameter Left Hand Helical Tool for High Quality Vertical Wall Machining

The demand for micro-end-milling for products in fields such as the medical, optical, and electronics industry is increasing. However, when machining with a small diameter end-mill (micro-end-mill) with diameters such as 0.5 mm, the rigidity of the tool itself is low; hence, the cutting conditions must be set to low values to achieve stable machining. Therefore, we examined various cutting phenomena that occur during actual machining processes to achieve high machining accuracy, high finished-surface quality, and long tool life. Some studies on micromachining achieved high accuracy, high-grade machining by considering the cutting phenomena. In previous papers, we dealt with the side-cutting phenomena in micro-end-milling of hardened die steels using a high-speed air-turbine spindle with rolling bearing. Cutting experiments were carried out by measuring the cutting force and flank wear of a cutting tool to investigate the difference in cutting phenomena caused by cutting direction in high-speed micro-end-milling. Observation of the machined surface and measurement of the profile of the cutting edge and machined surface were demonstrated. It was revealed that machining quality in high-speed up-cut milling was better than that in down-cut milling. Shoulder cutting, in which both peripheral and bottom cutting edges act simultaneously on the workpiece, was also investigated. A novel small diameter end-mill with left-hand helical tool with right-hand cut was developed to avoid damaging the cutting edge in the initial cutting stage. In the present study, high-quality shoulder cutting of a vertical wall using the new tool was proposed and demonstrated.

On coupled-channel dynamics in the presence of anomalous thresholds

We explore a general framework to treat coupled-channel systems in the presence of overlapping left- and right-hand cuts as well as anomalous thresholds. Such systems are studied in terms of a generalized potential, where we exploit the known analytic structure of $t$- and $u$-channel forces as the exchange masses approach their physical values. Given an approximate generalized potential the coupled-channel reaction amplitudes are defined in terms of nonlinear systems of integral equations. For large exchange masses, where there are no anomalous thresholds present, conventional $N/D$ methods are applicable to derive numerical solutions to the latter. At a formal level a generalization to the anomalous case is readily formulated by use of suitable contour integrations with amplitudes to be evaluated at complex energies. However, it is a considerable challenge to find numerical solutions to anomalous systems set up on a set of complex contours. By suitable deformations of left-hand and right-hand cut lines we manage to establish a framework of linear integral equations defined for real energies. Explicit expressions are derived for the driving terms that hold for an arbitrary number of channels. Our approach is illustrated in terms of schematic three-channel systems. It is demonstrated that despite the presence of anomalous thresholds the scattering amplitude can be represented in terms of three phase shifts and three inelasticity parameters, as one would expect from the coupled-channel unitarity condition.

Amplitude analysis of the anomalous decay η′→π+π−γ

In this paper we perform an amplitude analysis of $\eta'\to\pi^+\pi^-\gamma$ and confront it with the latest BESIII data. Based on the final-state interaction theorem, we represent the amplitude in terms of an Omn\'es function multiplied by a form factor that corresponds to the contributions from left-hand cuts and right-hand cuts in the inelastic channels. We also take into account the isospin violation effect induced by $\rho-\omega$ mixing. Our results show that the anomaly contribution is mandatory in order to explain the data. Its contribution to the decay width of $\Gamma(\eta'\to\pi\pi\gamma)$ is larger than that induced by isospin violation. Finally we extract the pole positions of the $\rho$ and $\omega$ as well as their corresponding residues.

The N/D method with non-perturbative left-hand-cut discontinuity and the [formula omitted]NN partial wave

In this letter we introduce an integral equation that allows to calculate the exact left-hand-cut discontinuity for an uncoupled S-wave partial-wave amplitude in potential scattering for a given finite-range potential. In particular this is applied here to the S01 nucleon–nucleon (NN) partial wave. The calculation of Δ(A) is completely fixed by the potential because short-range physics (corresponding to integrated out degrees of freedom within the low-energy Effective Field Theory) does not contribute to Δ(A). The results obtained from the N/D method for a partial-wave amplitude are rigorous, since now the discontinuities along the left-hand cut and right-hand cut are exactly known. This solves in this case the open question with respect to the N/D method and the effect on the final result of the non-perturbative iterative diagrams in the evaluation of Δ(A). The solution of this problem also implies the equivalence of the N/D method and the Lippmann–Schwinger (LS) equation for the nonsingular one-pion exchange S01NN potential (Yukawa potential). The equivalence between the N/D method with one extra subtraction and the LS equation renormalized with one counterterm or with subtractive renormalization also holds for the singular attractive S01NN potentials calculated by including higher orders in Chiral Perturbation Theory (ChPT). However, the N/D method is more flexible and, rather straightforwardly, it allows to evaluate partial-wave amplitudes with a higher number of extra subtractions, that we fix in terms of shape parameters within the effective range expansion. We give results up to three extra subtractions in the N/D method, which provide a rather accurate reproduction of the S01NN phase shifts when the NNLO ChPT potential is employed. Our new method then provides a general theory to renormalize non-perturbatively singular and regular potentials in scattering that can be extended to higher partial waves as well as to coupled channel scattering.

Nucleon-nucleon scattering from dispersion relations: Next-to-next-to-leading order study

Nucleon-nucleon $(\mathit{NN})$ scattering is studied by applying an approach based on the $\mathit{N}/\mathit{D}$ method and chiral perturbation theory (ChPT), whose dynamical input per partial wave consists of the imaginary part of the $\mathit{NN}$ partial-wave amplitude along the left-hand cut. The latter is calculated in one-loop ChPT up to and including next-to-next-to-leading order (NNLO). A power counting for the subtraction constants is established, which is appropriate for those subtractions attached to both the left- and the right-hand cuts. A quite good reproduction of the Nijmegen partial-wave analysis phase shifts and mixing angles results, which implies a steady improvement in the accurateness achieved by increasing the chiral order in the calculation of the dynamical input. I discuss that it is not necessary to fine tune the chiral counterterms ${c}_{i}$ determined from pion-nucleon scattering to agree with $\mathit{NN}$ data, but instead one should perform the iteration of two-nucleon intermediate states in a well-defined way so as to keep proper unitarity and analyticity. It is also confirmed at NNLO the long-range correlations between the $\mathit{NN} S$-wave effective ranges and scattering lengths, when employing only once-subtracted dispersion relations, that hold up to around 10% when compared with experimental values.

Unified dispersive approach to real and virtual photon-photon scattering at low energy

Previous representations of pion pair production amplitudes by two real photons at low energy, which combine dispersion theoretical constraints with elastic unitariy, chiral symmetry and soft photon constraints are generalized to the case where one photon is virtual. The constructed amplitudes display explicitly the dependence on the $\pi\pi$ phase-shifts, on pion form factors and on pion polarizabilities. They apply both for space-like and time-like virtualities despite the apparent overlap of the left and right-hand cuts, by implementing a definition of resonance exchange amplitudes complying with analyticity and consistent limiting prescriptions for the energy variables. Applications are made to the pion generalized polarizabilies, to vector meson radiative decays, and to the $\sigma\gamma$ electromagnetic form factor. Finally, and evaluation of the contribution of $\gamma\pi\pi$ states in the hadronic vacuum polarization to the muon $g-2$ is given, which should be less model dependent than previous estimates.

Nucleon–nucleon interactions from effective field theory

We have established a new convergent scheme to treat analytically nucleon–nucleon interactions from a chiral effective field theory. The Kaplan–Savage–Wise (KSW) amplitudes are resummed to fulfill the unitarity or right-hand cut to all orders below pion production threshold. This is achieved by matching order by order in the KSW power counting the general expression of a partial wave with resummed unitarity cut, with the inverses of the KSW amplitudes. As a result, a new convergent and systematic KSW expansion is derived for an on-shell interacting kernel R in terms of which the partial waves are computed. The agreement with data for the S-waves is fairly good up to laboratory energies around 350 MeV and clearly improves and reestablishes the phenomenological success of the KSW amplitudes when treated within this scheme.

The Cauchy-singular integral equation appearing in the theory of chiral spontaneous symmetry breaking

It is shown how the combination of associated hypergeometric functions, in the case of equal vector and axial couplings, satisfies an analytic integral equation on the right-hand cut with Cauchy-singular kernel having the same discontinuity as the original integral equation. This is obtained by using an extension of the quadratic transformation of the Gauss function. A projective quasi-solution is found, fixing an accurate value of the ratioGA/GV=1.154, due to radiative corrections, once favored by Schwinger as twice the Euler numerical constant. Also noticed is a quasi-solution withɛ=d−4=15% andGA=GV indicating possible functioning between Minkowski and de Sitter spaced=5. Finally, evaluation of the conjugate function by the initial—discontinuous—integral operator allows one to understand both the massless confluent limit obtained by the author (J.-P. Lebrun,Nuovo Cimento A,103, 1735 (1990);104, 1071 (1991)) and the one-half the expected rate mentioned in solar-neutrino observations.

MatrixNDmethod with absorption and the unitarity problem in coupled-channel Regge theory

In a crossing-symmetric Regge theory with several coupled two-particle channels, there are many-particle absorptive effects due to crossed two-particle processes. Also, some of the partial-wave amplitudes have overlapping left-hand and right-hand cuts in the $s$ plane. To enforce unitarity in such a theory, one needs a coupled-channel $\frac{N}{D}$ method allowing arbitrary absorption parameters and overlapping cuts. These requirements lead to a nonlinear marginally singular $\frac{N}{D}$ equation, which is proposed as part of a scheme to construct a coupled-channel Regge theory with exact crossing symmetry.

An important correction to πd scattering in the resonance region

A third-order correction to πd elastic scattering involving nucleon-nucleon rescattering is calculated and found to give an important contribution to the differential cross section. Due to strong off-shell effects, we are forced to examine the controversial question of the range of the pion-nucleon (πN) interaction. The difficulties in defining a range are intimately related to the separation of left- and right-hand cuts in the πN amplitude which are not distinguished in the on-shell Chew-Low theory. Failure to draw this distinction can lead to spurious results.

Analyticity properties in q2 for moments of deep inelastic structure functions

The moments of deep inelastic structure functions introduced by Nachtmann and the familiar Cornwall-Norton moments are shown to be analytic functions of q 2 with a right-hand cut as the only physical sheet singularity. For the Nachtmann moments the analytic continuations to q 2 > 0 can again be expressed as projections of a virtual Compton amplitude. This representation is used to determine for q 2 > 0 the contribution of a single nucleon resonance as a first step towards an estimate in magnitude of these functions on the cut.

Convergent polynomial expansion, lines of zeros, and slopes of diffraction scattering

Using Mandelstam analyticity and conformal mapping new variables are constructed so as to extend the domain of applicability of the convergent polynomial expansion (CPE) to all energies. It is found that the CPE exists for all energies if the scattering amplitude possesses at least one zero in the physical region. The CPE goes over to the optimized polynomial expansion (OPE) for higher energies. The approach from CPE to OPE is faster the higher the energy is, the farther the left-hand cut is than the right-hand cut, and the closer the position of zero is to the backward direction. The variables are found to be potentially useful in describing diffraction scattering at forward angles at all energies. A universal formula has been developed that relates slope parameters to equations of boundaries of spectral functions and lines of zeros. The formula gives a good account of the world data on shrinkage, antishrinkage, and shrinkage-antishrinkage of forward peaks at all energies. Good fits to the data on shrinkage for $\mathrm{pp}$ and ${K}^{+}p$ scattering, and antishrinkage in $\overline{p}p$ scattering have been obtained with known theoretical boundaries and suitable lines of zeros. Reasonably good fits to the data for ${K}^{\ensuremath{-}}p$ and ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ scattering with oscillations at lower energies have been obtained with effective shapes of spectral functions and suitable lines of zeros. In some cases our lines of zero found from this analysis appear to be different from those of Odorico. From the present approach to the scattering problem we observe that the imaginary part of the amplitude that yields $b(s)\ensuremath{\rightarrow}\ensuremath{\infty}$ for some values of $s$ must vanish at one point at least in the backward hemisphere.

Accurate tests of analytic extrapolations

Analytic pole extrapolations are tested rigorously for a number of model amplitudes having a realistic singularity structure. Amplitude extrapolations in the energy variable using the discrepancy function method have errors of 3% in favourable cases (np scattering) and of order 30% in typical few-body reactions (nt, n alpha scattering). Cross section extrapolations in the angular variable are less reliable due to the presence of the right-hand cut. The extrapolation error is about 5% for the favourable case of nd scattering and about 50% for nt scattering. All numbers quoted are obtained using conformal mapping techniques. Results without mapping are, in general, worse. Nearby singularities, however, cannot be reliably suppressed by mapping alone. The corresponding errors in the literature are underestimated.

Dispersion calculation of the transition form factorFπωγ(t)with cut contributions

In this paper we study the $t$ dependence of the $\ensuremath{\gamma}\ensuremath{\pi}\ensuremath{\omega}$ vertex functions on the basis of partial-wave dispersion relations and unitarity. The right-hand cut is approximated by the $2\ensuremath{\pi}$ contribution and the left-hand cut by the nearest $s$- and $u$-channel poles. The electromagnetic $\ensuremath{\pi}\ensuremath{\omega}$ transition form factor is calculated as a function of $t$ in the timelike and spacelike region and is compared with predictions of the $\ensuremath{\rho}$-dominance model and with recent experimental data for ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ near threshold. The influence of the left-hand cut and the finite width of the $\ensuremath{\rho}$ resonance is explicitly shown.

The unitary equation in a fixed momentum transfer representation

An extension of the fixed momentum transfer representation for an S-matrix representation of elementary particles is developed. In this representation it is postulated that a function S U (2) on the second sheet can be found such that the matrix product S S U (2) is analytic everywhere in a complex energy plane, and S U (2) is a continuation of S ∗ in the neighborhood of the right-hand cut. The problem of π +p scattering is considered. The kinematic variables are continued from the physical region to the neighborhood of the Born pole, and the unitary equation is there applied. A function for S U (2) was chosen there with parameters determined by unitarity. Two calculations were made, each with a different choice for the form of S U (2). It is shown how unitarity determines the coupling constant together with the values of the parameters of S U (2). The calculated value of the coupling constant, in both cases, is ∼ 16, in good agreement with experiment.

Stable analytic continuation and unitarity

In theoretical physics, in inverting the operator equationAf=h (wheref is «the cause»,h is «the effect» andA is some continuous operator representing a physical law) it is not sufficient forA−1 to exist and to be unique; it is also necessary that it be continuous, in order to preserve the theory from the instabilities that might arise from the never exact knowledge of the inputh. Usually stability is obtained by restricting the set off's in which the solution is sought, to a compact set. The basic idea is then to exploit the fact that different theoretical methods which are logically equivalent (are tautological) and, thus, yield exactly the same result when fed with «absolutely correct» data, could behave in totally different ways when faced with approximate (incomplete and/or error-affected) data. It makes then sense to find among all tautological methods the one which is most insensitive to the imprecision of the knowledge of the input data. In this paper we will first treat the problem of analytic continuation of the scattering amplitude from a limited part of the cuts, first without imposing unitarity and then with imposing it, on the right-hand cut. In Sect.4 we come back to this problem by means of theN/D method, seeking those equations which are less sensitive to the errors of the left-hand data (in this case «correct» data mean that the given left-hand cut limiting values are absolutely consistent with both analyticity and unitarity). Some applications connected with the localization of singularities of the scattering amplitude are then displayed in Sect.5.

Finite-contour dispersive inequalities

Finite-contour dispersive inequalities are derived for a function f(ξ) which is analytic in the ξ plane except for a right-hand cut. Inequalities are also derived for the derivative of the function. These inequalities are rigorous and the sharpest ones that can be derived using only analyticity.

High-energy reactions seen from the s-channel: A complex pole in the impact parameter plane

An s-channel model for inelastic reactions at high energies is presented. The scattering amplitude is split up into the contribution from the right-hand cut ( T r.h.c.) and that from the left-hand cut ( T l.h.c.) in a fixed- t dispersion relation; T r.h.c. is assumed to be dominated by the peripheral s-channel resonances associated with a complex pole b 0 ( s) in the impact-parameter plane. It yields the rotating-phase part of the amplitude; T l.h.c. is assumed to be the analytic continuation of the analogous term in the u-channel. It provides a real, non-peripheral background in the s-channel. The predictions of this model for the t-dependence of high-energy amplitudes are confronted with experiment. A very good agreement is found in all cases. Moreover, we present an analytic expression for the complex pole trajectory b 0 ( s), which agrees with the experimentally determined one and with analyticity requirements. This crucially interlocks the two terms T r.h.c. (being a function of b 0 ( s)) and T l.h.c. (being a function of b 0( u)).