Form Factors Of Bound States Research Articles

On-shell representations of two-body transition amplitudes: Single external current

This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, $2+\mathcal{J}\ensuremath{\rightarrow}2$. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.

Renormalisation group analysis of electromagnetic couplings in the pionless effective field theory

.The Wilsonian renormalisation group is applied to a system of two nonrelativistic particles interacting via short-range forces and coupled to an external EM field. By demanding that a fully off-shell one-particle-irreducible 5-point amplitude is independent of the cutoff, a renormalisation group equation is derived for the interaction current density. This is solved to obtain the fixed point corresponding to the unitary limit. The scaling behaviour of perturbations around this point is analysed. Some of these terms are related by gauge invariance to terms in the effective-range expansion; others describe short-range physics that is not included in the two-body potential. We construct observables including the bound-state form factor and show how the scaling of the terms in the interaction current is reflected in the power counting for their contributions to observables.

Bound state form factors from knockout in10Be(d,t) neutron pickup

Existing ${}^{10}$Be($d$,$t$)${}^{9}$Be neutron pickup data are analyzed using the same bound-state form factors for the ${\ensuremath{\langle}}^{10}\mathrm{Be}{|}^{9}\mathrm{Be}+n\ensuremath{\rangle}$ overlap as in a recent analysis of single-neutron knockout [Grinyer et al., Phys. Rev. Lett. 106, 162502 (2011)]. While the knockout data were well described by a bound-state form factor calculated using the variational Monte Carlo (VMC) technique, including an appropriate neutron binding potential, the ${}^{10}$Be($d$,$t$)${}^{9}$Be pickup data are significantly overpredicted using this form factor. In addition, the no-core shell model (NCSM) and VMC form factors yield the same calculated pickup cross sections whereas the knockout results using these form factors differed by 20$%$. We explore possible sources of ambiguity in the pickup calculations that affect our ability to compare the absolute magnitudes in cross sections between these two very different reactions.

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.

Hadron form factors from sum rules for vacuum-to-hadron correlators

We analyse the extraction of the bound-state form factor from vacuum-to-hadron correlator, which is the basic object for the calculation of hadron form factors in the method of light-cone sum rules in QCD. We study this correlator in quantum mechanics, calculate it exactly, and derive the corresponding OPE. We then apply the standard procedures of QCD sum rules to isolate the ground-state form factor from this correlator. We demonstrate that fixing the effective continuum threshold, one of the key ingredients of the sum-rule calculation of bound-state parameters, poses a serious problem for sum rules based on vacuum-to-hadron correlators.

Exact form factors in integrable quantum field theories: the sine-Gordon model (II)

A general model independent approach using the ‘off-shell Bethe Ansatz’ is presented to obtain an integral representation of generalized form factors. The general techniques are applied to the quantum sine-Gordon model alias the massive Thirring model. Exact expressions of all matrix elements are obtained for several local operators. In particular soliton form factors of charge-less operators as for example all higher currents are investigated. It turns out that the various local operators correspond to specific scalar functions called p-functions. The identification of the local operators is performed. In particular the exact results are checked with Feynman graph expansion and full agreement is found. Furthermore all eigenvalues of the infinitely many conserved charges are calculated and the results agree with what is expected from the classical case. Within the frame work of integrable quantum field theories a general model independent ‘crossing’ formula is derived. Furthermore the ‘bound state intertwiners’ are introduced and the bound state form factors are investigated. The general results are again applied to the sine-Gordon model. The integrations are performed and in particular for the lowest breathers a simple formula for generalized form factors is obtained.

Single-neutron transfer from 11Be[formula omitted] via the (p, d) reaction with a radioactive beam

The 11Be(p,d)10Be reaction has been performed in inverse kinematics with a radioactive 11Be beam of E/A=35.3 MeV. Angular distributions for the 0+ ground state, the 2+, 3.37 MeV state and the multiplet of states around 6 MeV in 10Be were measured at angles up to 16°cm by detecting the 10Be in a dispersion-matched spectrometer and the coincident deuterons in a silicon array. Distorted wave and coupled-channels calculations have been performed to investigate the amount of 2+ core excitation in 11Begs. The use of “realistic” 11Be wave functions is emphasised and bound-state form factors have been obtained by solving the particle-vibration coupling equations. This calculation gives a dominant 2s component in the 11Begs wave function with a 16% [2+⊗1d] core excitation admixture. Cross sections calculated with these form factors are in good agreement with the present data. The Separation Energy prescription for the bound-state wave function also gives satisfactory fits to the data, but leads to a significantly larger [2+⊗1d] component in 11Begs.

Exploring the time-like region for the elastic form factor in the light-front quantization

Even though the Drell–Yan–West formulation is the most rigorous and well-established framework to compute the exclusive processes, its utility has been limited only to the space-like region because of the intrinsic kinematic constraint q +=0 . We present an explicit example demonstrating how one may obtain the necessary information (i.e. nonvalence or so called Z-graph contribution) in the time-like region of exclusive process without encountering a formidable task of direct calculation that has hindered so far the progress in this area. In the analysis of q Q bound state form factors using an exactly solvable model of (3+1)-dimensional scalar field theory interacting with gauge fields, the results analytically continued from the space-like region coincide exactly with the direct results in the time-like region. This example verifies that the method of analytic continuation is capable of yielding the effect of complicate nonvalence contributions. The meson peaks analogous to the vector meson dominance (VMD) phenomena are also generated at the usual VMD positions.

Relativistic scattering and bound-state properties in a special Hamiltonian model.

We present numerical results for fermion-boson scattering cross sections and bound-state form factors in an elementary model of relativistic renormalized light-front Hamiltonian dynamics. The model Hamiltonian describes a fermion emitting and absorbing one scalar boson. Renormalization of the coupling constant leads to triviality and the cutoff cannot be arbitrarily large. Nevertheless, the resulting total fermion-boson scattering cross section is found to be practically independent of the cutoff within the triviality bounds. We also study cutoff dependence of the fermion-boson bound-state form factors. The cutoff dependence is negligible for boson masses considerably smaller than the fermion mass even for momentum transfers exceeding many times the fermion mass, as long as the momentum transfer is small compared to the cutoff. For heavy bosons the cutoff dependence of the fermion-boson sector contribution is stronger but the bound-state structure is dominated by the cutoff-independent bare fermion component. Thus, the model Hamiltonian leads to almost cutoff-independent results in a whole range of mass and coupling parameters within the triviality bounds.

Relativistic bound-state form factors in a solvable (1+1)-dimensional model including pair creation.

The relativistic two-body bound-state form factor, recently calculated by Mankiewicz and Sawicki from a light-front formula involving analytic solutions to the Weinberg equation in a (1+1)-dimensional model field theory, is compared with the form factor obtained from the Mandelstam formula using analytic solutions to the corresponding Bethe-Salpeter equation. The results are different, since the former formula does not include the contribution of pair-creation diagrams which are included in the latter. When the pair-creation diagrams are included in the light-front formula in 1+1 dimensions, both approaches yield a numerically identical form factor, not having the strange properties described by Mankiewicz and Sawicki.

Pion form factor in a scalar model

Using as an example the simple scalar modelρ (6) 3 ; we describe our method of analysing the bound-state form factors based on the systematic use of the alpha-representation for Feynman diagrams. The nature of «nonrenormalization group» logarithms specific for the form-factor-type problems is investigated. The summation of ladder contributions is performed in the leading-logarithm approximation.

A DWBA analysis of heavy-ion induced three-nucleon transfer reactions on 15N, 16O AND 18O

EFR-DWBA calculations incorporating all the Coulomb and nuclear interactions and using a parameterised cluster model potential for the bound-state form factors have been performed for the three-particle transfer reactions 16O( 13C, 10B) 19F, 16O( 11B, 8Li) 19Ne and 16O( 12C, 9Be) 19Ne. The results are compared with measured experimental angular distributions and spectroscopic factors are extracted for cluster transfer to mass 19. The cross sections computed using the DWBA are shown to have a similar Q-value dependence to that predicted by semi-classical theory. This feature has been used to facilitate the computation of three-nucleon transfer cross sections on 15N and 18O targets. Comparison with the experimentally measured cross sections yields spectroscopic factors for cluster transfer to masses 18 and 21.

Particle spectrum in model field theories from semiclassical solutions of the field equations

Bound-state spectra (of mesons or of solitons-antisolitons) are studied for the sine-Gordon and quartic-coupling nonlinear boson models in one-plus-one dimensions using methods developed previously by the authors for the example of the nonlinear Schr\"odinger equation. In addition to reproducing the spectra first derived by Dashen, Hasslacher, and Neveu (DHN) up to the order of the first quantum correction, we have also calculated (in the semiclassical approximation) all bound-state form factors as well as the matrix elements of the field between any bound state and any continuum state for the scattering of a meson on a bound state. For the sine-Gordon theory the results have been obtained in two ways: first, by an algorithm, derived in due course, for transcribing an exact classical solution into a quantum operator; second, by a systematic expansion about the weak-coupling limit which requires the techniques used previously for the nonlinear Schr\"odinger equation. Only this latter technique is available for the quartic model, but its (more complicated) application here leads to an explanation of why in leading order the same form of bound-state spectra are obtained for the two models. Compared to the work of DHN, aside from methodology, the main new results are the matrix elements of the field operators, but we also present a complete quantum interpretation of all their classical calculations as well as an explanation of why our methods are equivalent.

Relativistic effects in bound-state form factors

The formalism for bound-state form factors in a three-dimensional relativistic theory is used to expose the theoretical limitations of recent work on meson exchange currents. The pair excitation and recoil emission currents used in phenomenological calculations are found to have little theoretical support. The constraints imposed by the Ward identity are examined and the singularity structure of the form factor in relativistic and nonrelativistic theories is compared.

Quark confinement, rising trajectories, and asymptotic behavior of form factors

Several long-range force models of quark confinement are considered, with particular emphasis on gluon exchange analogous to linear and harmonic potentials. First, the nonrelativistic case is discussed and the connection between the rise of the Regge trajectories and the power of the potential is derived. Then semirelativistic and relativistic equations are considered. It is shown that in each case the rising trajectories result from large quark-gluon coupling constants. It is also shown that an asymptotic power decrease of bound-state form factors follows only if the interaction contains an additional Coulomb- or Yukawa-type part, and only in the spacelike or the timelike domain. The significance of infrared cutoffs is examined, violations of unitarity are pointed out, and the behavior in the static limit is discussed.

Form Factors for Proton Transfer Reactions to Unbound Isobaric Analog States

Proton form factors for isobaric analog resonances populated directly as residual states in ($d,n$) and ($^{3}\mathrm{He},d$) reactions are discussed and computed explicitly using a Lane-model Hamiltonian and the formalism of Feshbach's unified reaction theory. The resonance form factors thus obtained are compared in specific cases with equivalent parent bound-state form factors and with single-particle resonance wave functions. Numerical results and tests using data are presented for $^{92}\mathrm{Mo}(d,n)^{93}\mathrm{Tc}^{A}$, and it is shown that resonance form factors lead to a dramatic enhancement of the magnitude of the calculated cross section for some transitions, compared with calculations using conventional form factors. Such enhancements are consistent with the available data.

Off-Energy-Shell Partial-Wave Amplitudes

We present dispersion relations which give the full off-energy-shell $T$-matrix elements ${T}_{l{l}^{\ensuremath{'}}}(p,{p}^{\ensuremath{'}};s)$ for all values of the parametric energy $s$ in terms of bound-state form factors, a subtraction constant, and the half-off-shell $T$-matrix elements ${T}_{l{l}^{\ensuremath{'}}}(p,{s}^{\frac{1}{2}};s)$ in the scattering ($s>0$) region. Study of the half-off-shell $T$ matrix for $s>0$ shows that it can be written as the product of a real matrix ${H}_{l{l}^{\ensuremath{'}}}(p,s)$ and the on-shell $T$ matrix. We combine these results to obtain a representation of the full off-energy-shell $T$-matrix elements in terms of experimental on-shell $T$-matrix elements, the real half-off-shell factors ${H}_{l{l}^{\ensuremath{'}}}(p,s)$, a subtraction constant and bound-state form factors. Our results are based only on assumptions of time-reversal invariance, off-energy-shell unitarity, analyticity, and asymptotic behavior. The results are independent of any specific dynamical assumptions. We conclude with a discussion of the special case of uncoupled partial waves and the advantages of a separable representation of the half-off-shell factors.