115,517 research outputs found
Group Invariant Scattering
This paper constructs translation invariant operators on L2(R^d), which are
Lipschitz continuous to the action of diffeomorphisms. A scattering propagator
is a path ordered product of non-linear and non-commuting operators, each of
which computes the modulus of a wavelet transform. A local integration defines
a windowed scattering transform, which is proved to be Lipschitz continuous to
the action of diffeomorphisms. As the window size increases, it converges to a
wavelet scattering transform which is translation invariant. Scattering
coefficients also provide representations of stationary processes. Expected
values depend upon high order moments and can discriminate processes having the
same power spectrum. Scattering operators are extended on L2 (G), where G is a
compact Lie group, and are invariant under the action of G. Combining a
scattering on L2(R^d) and on Ld (SO(d)) defines a translation and rotation
invariant scattering on L2(R^d).Comment: 78 pages, 5 figure
The four-dimensional on-shell three-point amplitude in spinor-helicity formalism and BCFW recursion relations
Lecture notes on Poincar\'e-invariant scattering amplitudes and tree-level
recursion relations in spinor-helicity formalism. We illustrate the
non-perturbative constraints imposed over on-shell amplitudes by the Lorentz
Little Group, and review how they completely fix the three-point amplitude
involving either massless or massive particles. Then we present an introduction
to tree-level BCFW recursion relations, and some applications for massless
scattering, where the derived three-point amplitudes are employed.Comment: 41+2 pages, 4 figure
Renormalization group study of interacting electrons
The renormalization-group (RG) approach proposed earlier by Shankar for
interacting spinless fermions at is extended to the case of non-zero
temperature and spin. We study a model with -invariant short-range
effective interaction and rotationally invariant Fermi surface in two and three
dimensions. We show that the Landau interaction function of the Fermi liquid,
constructed from the bare parameters of the low-energy effective action, is RG
invariant. On the other hand, the physical forward scattering vertex is found
as a stable fixed point of the RG flow. We demonstrate that in and 3, the
RG approach to this model is equivalent to Landau's mean-field treatment of the
Fermi liquid. We discuss subtleties associated with the symmetry properties of
the scattering amplitude, the Landau function and the low-energy effective
action. Applying the RG to response functions, we find the compressibility and
the spin susceptibility as fixed points.Comment: 11 pages, RevTeX 3.0, 2 PostScript figure
A renormalized equation for the three-body system with short-range interactions
We study the three-body system with short-range interactions characterized by
an unnaturally large two-body scattering length. We show that the off-shell
scattering amplitude is cutoff independent up to power corrections. This allows
us to derive an exact renormalization group equation for the three-body force.
We also obtain a renormalized equation for the off-shell scattering amplitude.
This equation is invariant under discrete scale transformations. The
periodicity of the spectrum of bound states originally observed by Efimov is a
consequence of this symmetry. The functional dependence of the three-body
scattering length on the two-body scattering length can be obtained
analytically using the asymptotic solution to the integral equation. An
analogous formula for the three-body recombination coefficient is also
obtained.Comment: 12 pages, RevTex, 2 ps figures, included with epsf.te
An extension problem for the CR fractional Laplacian
We show that the conformally invariant fractional powers of the sub-Laplacian
on the Heisenberg group are given in terms of the scattering operator for an
extension problem to the Siegel upper halfspace. Remarkably, this extension
problem is different from the one studied, among others, by Caffarelli and
Silvestre.Comment: 33 pages. arXiv admin note: text overlap with arXiv:0709.1103 by
other author
The Three-Boson System at Next-To-Next-To-Leading Order
We discuss effective field theory treatments of the problem of three
particles interacting via short-range forces (range R >> a_2, with a_2 the
two-body scattering length). We show that forming a once-subtracted scattering
equation yields a scattering amplitude whose low-momentum part is
renormalization-group invariant up to corrections of O(R^3/a_2^3). Since
corrections of O(R/a_2) and O(R^2/a_2^2) can be straightforwardly included in
the integral equation's kernel, a unique solution for 1+2 scattering phase
shifts and three-body bound-state energies can be obtained up to this accuracy.
We use our equation to calculate the correlation between the binding energies
of Helium-4 trimers and the atom-dimer scattering length. Our results are in
excellent agreement with the recent three-dimensional Faddeev calculations of
Roudnev and collaborators that used phenomenological inter-atomic potentials.Comment: 20 pages, 3 eps figure
Nucleon-nucleon scattering within a multiple subtractive renormalization approach
A methodology to renormalize the nucleon-nucleon interaction, using a
recursive multiple subtraction approach to construct the kernel of the
scattering equation, is presented. We solve the subtracted scattering equation
with the next-leading-order (NLO) and next-to-next-leading-order (NNLO)
interactions. The results are presented for all partial waves up to ,
fitted to low-energy experimental data. In our renormalizaton group invariant
method, when introducing the NLO and NNLO interactions, the subtraction energy
emerges as a renormalization scale and the momentum associated with it comes to
be about the QCD scale (), irrespectively to the partial wave.Comment: Final versio
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