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Perturbative Pions and the Effective Range Expansion  [PDF]
Thomas D. Cohen
Physics , 1999,
Abstract: In this talk the $Q$ counting scheme to implement effective field theory is discussed. It is pointed out that there are two small mass scales in the problem $m_\pi$ and $1/a$ with $1/a \ll m_\pi$. It is argued that while the expansion based on $1/a$ being small compared to the underlying short distance scales works well, the chiral expansion may not. The coefficients of the effective range expansion are sensitive to the chiral physics and are very poorly described in $Q$ counting at lowest nontrivial order. A ``shape function'' is introduced which again is sensitive to pionic physics and insensitive to fitting procedures. It is also poorly described in $Q$ counting.
Effective Range Expansion for the Interaction Defined on the Lattice  [PDF]
Gabriel Wlaz?owski,Piotr Magierski
Physics , 2009, DOI: 10.1142/S0218301310015229
Abstract: The relation between the interaction parameters for fermions on the spatial lattice and the two-body $T$ matrix is discussed. The presented method allows determination of the interaction parameters through the relatively simple computational scheme which include the effect of finite lattice spacing. In particular the relation between the interaction parameters and the effective range expansion parameters is derived in the limit of large lattices.
Phase shift effective range expansion from supersymmetric quantum mechanics  [PDF]
Boris F. Samsonov,Fl. Stancu
Physics , 2003, DOI: 10.1103/PhysRevC.67.054005
Abstract: Supersymmetric or Darboux transformations are used to construct local phase equivalent deep and shallow potentials for $\ell \neq 0$ partial waves. We associate the value of the orbital angular momentum with the asymptotic form of the potential at infinity which allows us to introduce adequate long-distance transformations. The approach is shown to be effective in getting the correct phase shift effective range expansion. Applications are considered for the $^1P_1$ and $^1D_2$ partial waves of the neutron-proton scattering.
Multi-channel analog of the effective-range expansion  [PDF]
S. A. Rakityansky,N. Elander
Physics , 2011, DOI: 10.1088/1751-8113/44/11/115303
Abstract: Similarly to the standard effective range expansion that is done near the threshold energy, we obtain a generalized power-series expansion of the multi-channel Jost-matrix that can be done near an arbitrary point on the Riemann surface of the energy within the domain of its analyticity. In order to do this, we analytically factorize its momentum dependencies at all the branching points on the Riemann surface. The remaining single-valued matrix functions of the energy are then expanded in the power-series near an arbitrary point in the domain of the complex energy plane where it is analytic. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This means that near an arbitrary point in the domain of physically interesting complex energies it is possible to obtain a semi-analytic expression for the Jost-matrix (and therefore for the S-matrix) and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles.
Modi ed effective range expansion for nucleon-nucleon scattering  [cached]
Minossi D.,Epelbaum E.,Pavón Valderrama M.,Nogga A.
EPJ Web of Conferences , 2010, DOI: 10.1051/epjconf/20100305018
Abstract: The standard e ective range expansion is commonly used in nucleon-nucleon scattering to encode the properties of the nuclear force model-independently in a small set of parameters. However, its applicability is limited by the longest-range part of the nuclear potential, i.e. by the one pion exchange, to the domain of momenta below half the pion mass. Therefore, it is not useful to study shorter-range parts of the interaction, e.g. two-pion exchange. To this aim a modi cation that explicitly takes a given long-range part of the interaction into account is required. This is known as modi ed e ective range expansion. Here, we apply this approach to the nucleonnucleon interaction to separate the known long-range interactions from the rest. To show the e ectiveness of this technique, we consider a toy model with a two-range potential. We study the scaling behaviour of the parameters of the standard and modi ed e ective range expansion in this two-scale problem and compare their convergence behaviour.
Connection between effective-range expansion and nuclear vertex constant or asymptotic normalization coefficient  [PDF]
R. Yarmukhamedov,D. Baye
Physics , 2011, DOI: 10.1103/PhysRevC.84.024603
Abstract: Explicit relations between the effective-range expansion and the nuclear vertex constant or asymptotic normalization coefficient (ANC) for the virtual decay $B\to A+a$ are derived for an arbitrary orbital momentum together with the corresponding location condition for the ($A+a$) bound-state energy. They are valid both for the charged case and for the neutral case. Combining these relations with the standard effective-range function up to order six makes it possible to reduce to two the number of free effective-range parameters if an ANC value is known from experiment. Values for the scattering length, effective range, and form parameter are determined in this way for the $^{16}$O+$p$, $\alpha+t$ and $\alpha+^3$He collisions in partial waves where a bound state exists by using available ANCs deduced from experiments. The resulting effective-range expansions for these collisions are valid up to energies larger 5 MeV.
Effective-Range Expansion of the Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential  [PDF]
Kenji Fukukawa,Yoshikazu Fujiwara
Physics , 2010, DOI: 10.1143/PTP.125.957
Abstract: The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.
Testing Dark Energy and Cardassian Expansion for Causality  [PDF]
J. Ponce de Leon
Physics , 2004,
Abstract: Causality principle is a powerful criterion that allows us to discriminate between what is possible or not. In this paper we study the transition from decelerated to accelerated expansion in the context of Cardassian and dark energy models. We distinguish two important events during the transition. The first one is the end of the matter-dominated phase, which occurs at some time $t_{eq}$. The second one is the actual crossover from deceleration to acceleration, which occurs at some $t_{T}$. Causality requires $t_{T} \geq t_{eq}$. We demonstrate that dark energy models, with constant $w$, and Cardassian expansion, are compatible with causality only if $(\Omega_{M} - \bar{q}) \leq 1/2$. However, observational data indicate that the most probable option is $(\Omega_{M} - \bar{q}) > 1/2$. Consequently, the transition from deceleration to acceleration in dark energy and Cardassian models occurs before the matter-dominated epoch comes to an end, i.e., $t_{eq} > t_{T}$. Which contradicts causality principle.
Effective range function below threshold  [PDF]
A. Deloff
Physics , 2000, DOI: 10.1088/0954-3899/26/12/306
Abstract: We demonstrate that the kernel of the Lippmann-Schwinger equation, associated with interactions consisting of a sum of the Coulomb plus a short range nuclear potential, below threshold becomes degenerate. Taking advantage of this fact, we present a simple method of calculating the effective range function for negative energies. This may be useful in practice since the effective range expansion extrapolated to threshold allows to extract low-energy scattering parameters: the Coulomb-modified scattering length and the effective range.
Surpassing Wigner's causality bound in relativistic scattering with zero-range interaction  [PDF]
Vladimir Pascalutsa
Physics , 2014,
Abstract: It is shown that the relativistic zero-range potential scattering surpasses Wigner's causality bound, while being consistent with causality. The relativistic theory shows in addition a richer analytic structure, such as a $K$-matrix pole necessarily accompanying the bound-state solution. Implications of these results for the effective-field theory of nuclear forces are briefly considered.
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