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08/03/2011 - 10:08 von Der Nürnberger Prozeß | Report spam
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#1 Der N
08/03/2011 - 10:09 | Warnen spam
On 8 Mrz., 10:08, Der Nürnberger Prozeß wrote:
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Um Quadrupole von Atomkernen auszurechnen brauchst du eigentlich
Lattice QCD

Keiko Murano zeigt euch hier wie Kerne eigentlich zusammenkleben :

Nucleon-Nucleon Potential and its Non-locality in Lattice QCD

Keiko Murano1, Noriyoshi Ishii2, Sinya Aoki2,3, Tetsuo Hatsuda4,5
1KEK Theory Center, Institute of Particle and Nuclear Studies,

§1. Introduction
The nucleon-nucleon (NN) potential1)–3) is a fundamental quantity to
study
various properties of atomic nuclei and nuclear matter. Recently, a
first attempt
to calculate the NN potential from QCD was reported on the basis of
the Nambu-
Bethe-Salpeter (NBS) wave function for the two nucleons on the lattice.
4), 5) Also, the
method has been extended to the baryon-baryon (BB) interactions with
strangeness,6)–8)
the three-nucleon interaction9) and meson-baryon interactions.10), 11)
Since the NN
interaction is short ranged, the NN potential extracted from lattice
QCD simulations
is exponentially insensitive to the spatial lattice extent Ls as long
as Ls ≫ 1/mπ.
Then one can calculate observables such as the scattering phase shifts
by employing
the lattice NN potential and solving the Schr¨odinger equation in the
infinite volume.
In general, the lattice NN potential obtained from the NBS wave
function is
energy-independent but non-local, U(r, r′). In practice, U is
rewritten in terms of
an infinite set of energy-independent local potentials V (LO)(r),V
(NLO)(r) • • • and so
on, by the derivative expansion. These local potentials are determined
successively
by measuring the NBS wave functions for different scattering energies
E. A possible
criterion for the validity of the derivative expansion at low energies
is the stability
of the local potentials against the variation of E.
typeset using PTPTEX.cls hVer.0.9i
2 Keiko Murano, Noriyoshi Ishii, Sinya Aoki, Tetsuo Hatsuda
The purpose of this paper is to check such stability through the
lattice data
at E ≃ 0 MeV and E ≃ 45 MeV: these two cases are realized on the
lattice by
taking the periodic and anti-periodic boundary conditions in the
spatial directions.
We carry out quenched lattice QCD simulations with Ls ≃ 4 fm and the
pion mass
mπ ≃ 530 MeV. We will show that the leading-order local potentials at
the above two
different energies show no difference within statistical error, which
validates the local
approximation up to E = 45 MeV for the central and tensor potentials.
∗) Difference
of the spin-singlet central potentials between ℓ = 0 and ℓ = 2 is also
studied, with
ℓ being the orbital angular momentum. A preliminary account of these
results is
given in Refs.13), 14).
This paper is organized as follows. In Sec.2, we make a brief review
on the
energy-independent non-local potential and its derivative expansion.
An explicit
construction of the leading order terms of the derivative expansion is
also presented.
In Sec.3, we explain a method to realize non-zero energy NN scattering
on the lattice
through the spatial boundary conditions. In particular, we introduce a
novel momentum
wall source operators which are suitable for the purpose of the
present paper. In
Sec.4, we present numerical results for the NBS wave functions and the
associated
leading order potentials for different E and ℓ. Sec.5 is devoted to
summary and
concluding remarks. In Appendix A, we give a brief summary of the
representation
of the cubic group used in this paper. In Appendix B, some details of
constructing
the ℓ = 2 source operator by using the cubic group representation is
presented.
§2. Non-local NN potential and its derivative expansion
To define the NN potential in QCD, we consider the equal-time Nambu-
Bethe-
Salpeter (NBS) wave function in the center of mass (CM) frame defined
by
φαβ(r; k) ≡ h0|pα(x)nβ(y)|B = 2; ki, (r ≡ x − y), (2.1)
where |B = 2; ki is a QCD eigenstate with baryon number two (B = 2),
and
pα(x), nβ(y) are local composite nucleon operators with spinor indices
α and β.
The asymptotic relative momentum k is related to the relativistic
total energy W as
W = 2
q
m2
N + k2 with mN being the nucleon mass. In the following, we consider
the elastic region where W < Wth ≡ 2mN + mπ is satisfied with the pion
mass mπ.
The asymptotic behavior of the NBS wave function for |r| > R (R being
the typical
interaction range) is characterized by the scattering phase shift for
hadrons.5), 15)–17)
On the other hand, from the NBS wave function for |r| < R, we can
define a kdependent
local potential Uk(r) and derive an associated k-independent non-local
potential U(r, r′) as follows:

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