Hadron Optics: Diffraction Patterns in Deeply Virtual Compton Scattering

We show that the Fourier transform of the Deeply Virtual Compton Scattering (DVCS) amplitude with respect to the skewness variable $\zeta$ at fixed invariant momentum transfer squared $t$ provides a unique way to visualize the structure of the target hadron in the boost-invariant longitudinal coordinate space. The results are analogous to the diffractive scattering of a wave in optics. As a specific example, we utilize the quantum fluctuations of a fermion state at one loop in QED to obtain the behavior of the DVCS amplitude for electron-photon scattering. We then simulate the wavefunctions for a hadron by differentiating the above LFWFs with respect to $M^2$ and study the corresponding DVCS amplitudes in light-front longitudinal space. In both cases we observe that the diffractive patterns in the longitudinal variable conjugate to $\zeta$ sharpen and the positions of the first minima move in with increasing momentum transfer. For fixed $t$, higher minima appear at positions which are integral multiples of the lowest minimum. Both these observations strongly support the analogy with diffraction in optics.Comment: Some plots modified, clarifications and references adde

Scientific Argumentation as a Foundation for the Design of Inquiry-Based Science Instruction

Despite the attention that inquiry has received in science education research and policy, a coherent means for implementing inquiry in the classroom has been missing [1]. In recent research, scientific argumentation has received increasing attention for its role in science and in science education [2]. In this article, we propose that organizing a unit of instruction around building a scientific argument can bring inquiry practices together in the classroom in a coherent way. We outline a framework for argumentation, focusing on arguments that are central to science—arguments for the best explanation. We then use this framework as the basis for a set of design principles for developing a sequence of inquiry-based learning activities that support students in the construction of a scientific argument. We show that careful analysis of the argument that students are expected to build provides designers with a foundation for selecting resources and designing supports for scientific inquiry. Furthermore, we show that creating multiple opportunities for students to critique and refine their explanations through evidence-based argumentation fosters opportunities for critical thinking, while building science knowledge and knowledge of the nature of science

Extension dimensional approximation theorem

Let $L$ be a countable CW-complex and $F\colon X\to Y$ be upper semicontinuous $UV^{[L]}$-valued mapping of a paracompact space $X$ to a complete metric space $Y$. We prove that if $X$ is a C-space of extension dimension \ed X \le [L], then $F$ admits single-valued graph approximations. For $L=S^n$ our result implies well-known approximation theorem for $UV^{n-1}$-valued mappings of $n$-dimensional spaces. And for $L=\{\rm point\}$ our theorem implies a theorem of Ancel on approximations of $UV^\infty$-valued mappings of C-spaces.Comment: 7 pages, final version, minor correction

Hurewicz theorem for extension dimension

We prove a new selection theorem for multivalued mappings of C-space. Using this theorem we prove extension dimensional version of Hurewicz theorem for a closed mapping $f\colon X\to Y$ of $k$-space $X$ onto paracompact $C$-space $Y$: if for finite $CW$-complex $M$ we have \ed Y\le [M] and for every point $y\in Y$ and every compactum $Z$ with \ed Z\le [M] we have \ed(f^{-1}(y)\times Z)\le [L] for some $CW$-complex $L$, then \ed X\le [L]

Perturbative QCD and factorization of coherent pion photoproduction on the deuteron

We analyze the predictions of perturbative QCD for pion photoproduction on the deuteron, gamma D -> pi^0 D, at large momentum transfer using the reduced amplitude formalism. The cluster decomposition of the deuteron wave function at small binding only allows the nuclear coherent process to proceed if each nucleon absorbs an equal fraction of the overall momentum transfer. Furthermore, each nucleon must scatter while remaining close to its mass shell. Thus the nuclear photoproduction amplitude, M_{gamma D -> pi^0 D}(u,t), factorizes as a product of three factors: (1) the nucleon photoproduction amplitude, M_{gamma N_1 -> pi^0 N_1}(u/4,t/4), at half of the overall momentum transfer, (2) a nucleon form factor, F_{N_2}(t/4), at half the overall momentum transfer, and (3) the reduced deuteron form factor, f_d(t), which according to perturbative QCD, has the same monopole falloff as a meson form factor. A comparison with the recent JLAB data for gamma D -> pi^0 D of Meekins et al. [Phys. Rev. C 60, 052201 (1999)] and the available gamma p -> pi^0 p data shows good agreement between the perturbative QCD prediction and experiment over a large range of momentum transfers and center of mass angles. The reduced amplitude prediction is consistent with the constituent counting rule, p^11_T M_{gamma D -> pi^0 D} -> F(theta_cm), at large momentum transfer. This is found to be consistent with measurements for photon lab energies E_gamma > 3 GeV at theta_cm=90 degrees and \elab > 10 GeV at 136 degrees.Comment: RevTeX 3.1, 17 pages, 6 figures; v2: incorporates minor changes as version accepted by Phys Rev

Electromagnetic Form Factors and Charge Densities From Hadrons to Nuclei

A simple exact covariant model in which a scalar particle is modeled as a bound state of two different particles is used to elucidate relativistic aspects of electromagnetic form factors. The model form factor is computed using an exact covariant calculation of the lowest-order triangle diagram and shown to be the same as that obtained using light-front techniques. The meaning of transverse density is explained using coordinate space variables, allowing us to identify a true mean-square transverse size directly related to the form factor. We show that the rest-frame charge distribution is generally not observable because of the failure to uphold current conservation. Neutral systems of two charged constituents are shown to obey the lore that the heavier one is generally closer to the transverse origin than the lighter one. It is argued that the negative central charge density of the neutron arises, in pion-cloud models, from pions of high longitudinal momentum. The non-relativistic limit is defined precisely and the ratio of the binding energy to that of the mass of the lightest constituent is shown to govern the influence of relativistic effects. The exact relativistic formula for the form factor reduces to the familiar one of the three-dimensional Fourier transform of a square of a wave function for a very limited range of parameters. For masses that mimic the quark-di-quark model of the nucleon we find substantial relativistic corrections for any value of $Q^2$. A schematic model of the lowest s-states of nuclei is used to find that relativistic effects decrease the form factor for light nuclei but increase the form factor for heavy nuclei. Furthermore, these states are strongly influenced by relativity.Comment: 18 pages, 11 figure

Hadron Spin Dynamics

Spin effects in exclusive and inclusive reactions provide an essential new dimension for testing QCD and unraveling hadron structure. Remarkable new experiments from SLAC, HERMES (DESY), and the Jefferson Laboratory present many challenges to theory, including measurements at HERMES and SMC of the single spin asymmetries in pion electroproduction, where the proton is polarized normal to the scattering plane. This type of single spin asymmetry may be due to the effects of rescattering of the outgoing quark on the spectators of the target proton, an effect usually neglected in conventional QCD analyses. Many aspects of spin, such as single-spin asymmetries and baryon magnetic moments are sensitive to the dynamics of hadrons at the amplitude level, rather than probability distributions. I illustrate the novel features of spin dynamics for relativistic systems by examining the explicit form of the light-front wavefunctions for the two-particle Fock state of the electron in QED, thus connecting the Schwinger anomalous magnetic moment to the spin and orbital momentum carried by its Fock state constituents and providing a transparent basis for understanding the structure of relativistic composite systems and their matrix elements in hadronic physics. I also present a survey of outstanding spin puzzles in QCD, particularly the double transverse spin asymmetry A_{NN} in elastic proton-proton scattering, the J/psi to rho-pi puzzle, and J/psi polarization at the Tevatron.Comment: Concluding theory talk presented at SPIN2001, the Third Circum-Pan-Pacific Symposium on High Energy Physics, October, 2001, Beijin