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Line shape of ψ(3770) in
N.N. Achasov and G.N. Shestakov
Laboratory of Theoretical Physics
Sobolev Institute for Mathematics
Novosibirsk, Russia
MEPHI, November 12–16, 2012, Moscow – p. 1/24
Abstract
1. Parameters of the ψ(3770) resonance should be extracted
from the processing of data on the e+e− → D ¯
D reactions by
using models that satisfy the elastic unitarity requirement.
2. As the first working candidate, a model with the mixing
ψ(3770) and ψ(2S) resonances is proposed.
3. Selection of theoretical models can be toughened by
comparing their predictions with the data on the shape of the
ψ(3770) peak in the non-D ¯
D channels e+e− → γχc0,
J/ψη, φη, etc.
MEPHI, November 12–16, 2012, Moscow – p. 2/24
1. Introduction: ψ(3770) in e+e− → D ¯
Current data. Interference patterns
2. The D meson electromagnetic form factor F 0
Unitarity requirement
Description of the data on e+e− → D ¯
A simplest model for F 0
D with the mixing ψ(3770) and ψ(2S)
resonances
3. The ψ(3770) shapes in non-D ¯
D decay channels
4. A comment on ambiguity of resonance parameters
5. Conclusion
MEPHI, November 12–16, 2012, Moscow – p. 3/24
ψ(3770) = ψ . Current data
= 3773 MeV, Γtot
D = 27.2 MeV, Γψ e+e− = 0.262 keV.
The resonance ψ was investigated in e+e− → D ¯
D by MARK-I
(1977), DELCO (1978), and MARK-II (1980), and after 24 years, by
BES (2004-2010), CLEO (2006-2010), BABAR (2007, 2009), Belle
(2008) and KEDR (2010-2012).
New super-high-statistics era.
New data are expected from BESIII and CLEO-c heaving millions
D events. In this regard, we believe it is timely
to discuss some dangers which are hidden in the commonly used
schemes for the description of the ψ peak.
MEPHI, November 12–16, 2012, Moscow – p. 4/24
Current data on e+e− → D ¯
D, interference patterns
D) has the following features: (1) the right-side of the ψ peak turns
out to be more steep than its left-side, (2) there is a deep dip near 3.81 GeV. These
features are hard to describe with the help of a single ψ
resonance contribution.
MEPHI, November 12–16, 2012, Moscow – p. 5/24
Fits to 87 points with a single ψ resonance contribution
χ2 values are very bad.
MEPHI, November 12–16, 2012, Moscow – p. 6/24
ψ interferes with background
In order to qualitatively improve the data description in the ψ
resonance region, in particular, to explain a dip near 3.81 GeV, it is
necessary to take into account the interference between the
resonant and nonresonant D ¯
D production.
Note that, in this way, there unexpectedly arose the problem
with the ambiguity of the interfering ψ resonance parameters
determination [KEDR(2010-2012), PDG(2012)]. However, the

parametrizations used for the e+e− → D ¯
D reaction amplitude
have no clear dynamical justification.
MEPHI, November 12–16, 2012, Moscow – p. 7/24
The D meson electromagnetic form factor F 0
In the process e+e− → D ¯
D we investigate the D meson electromagnetic form
factor, the phase of which in the elastic region (between the D ¯
thresholds: 2mD ≈ 3.739 GeV and mD + mD∗ ≈ 3.872 GeV) is completely
fixed by the unitarity condition. Here we consider the isoscalar form factor
F 0
D is a real function of energy and δ0
1 is the phase of the P -wave D ¯
scattering amplitude T 0
1 in the channel with isospin I = 0,
δbg is the elastic background phase and δres is the phase of the resonance
amplitude
Tres.
MEPHI, November 12–16, 2012, Moscow – p. 8/24
The D meson electromagnetic form factor F 0
A similar representation of the e+e− → D ¯
D reaction amplitude used
for the data description guarantees the unitarity requirement on the model level.
The sum of the e+e− → D0 ¯
D0 and e+e− → D+D− reaction cross
sections is expressed in terms of F 0
D in the following way
where ν(s) = [p3(s) + p3 (s)]/
To understand how the form factor F 0
D and strong amplitude T 0
1 can be con-
structed satisfying the unitarity requirement, the easiest way to use, as a guide,
the field-theory model shown in the next slide.
MEPHI, November 12–16, 2012, Moscow – p. 9/24
Unitarity construction of T 0 and F 0
The graphical representation of the strong D ¯
D scattering amplitude T 0
the D meson electromagnetic form factor F 0
MEPHI, November 12–16, 2012, Moscow – p. 10/24
The model for F 0 with the mixing ψ and ψ(2S) resonances
It is clear that the main sources of the background in the ψ
region are the tails from the J/ψ, ψ(2S), ψ(4040), ψ(4160)
and other resonances. It is easy to incorporate the right number of
resonances in our scheme.
Here we present the simplest variant of the model taking into
account the background contribution from the nearest neighbor
resonance ψ(2S) and also discuss how it can be checked.
In the considered model the ψ and ψ(2S) resonances mix
via transitions ψ
D → ψ(2S).
MEPHI, November 12–16, 2012, Moscow – p. 11/24
The model for F 0 with the mixing ψ and ψ(2S) resonances
− ψ(2S) mixing amplitude caused by ψ → D ¯
D → ψ(2S) transitions
via the real D ¯
D intermediate states has the form
MEPHI, November 12–16, 2012, Moscow – p. 12/24
The model for F 0 with the mixing ψ and ψ(2S) resonances
, gψ D ¯
D , gψ γ , and gψ(2S)D ¯
D are determined by fitting;
mψ(2S) and gψ γ are fixed by the PDG data.
Note that F 0
D in the considered model is proportional to the
first-degree polynomial in s with real coefficients (see RD ¯
above ). Hence the dip observed in σ(e+e− → D ¯
D) near 3.81
GeV can be explained by the F 0
D (s) zero, caused by compensation
between the ψ and ψ(2S) contributions.
MEPHI, November 12–16, 2012, Moscow – p. 13/24
The simplest variant of the ψ − ψ(2S) mixing model for F 0
The solid curve is the fit to the data. The dashed and dot-dashed curves show the
ψ and ψ(2S) contributions, respectively. Bare parameters: = 3.794 GeV,
D = 56.8 MeV, Γψ e+e− = 0.062 keV, g2
/4π = 32.2.
MEPHI, November 12–16, 2012, Moscow – p. 14/24
The simplest variant of the ψ − ψ(2S) mixing model for T 0
From the fitting of the e+e− → D ¯
D data we all know, at the
model level, about the I = 0 P wave D ¯
D elastic scattering
amplitude T 0
MEPHI, November 12–16, 2012, Moscow – p. 15/24
Cross section and phase for D ¯
D elastic scattering in the P wave
(a) The cross section σ(D0 ¯
D0) = 3π| sin δ0(s)
and (b) the
phase δ0(s)
for the simplest variant of the ψ
− ψ(2S) mixing model.
Unfortunately, these predictions are not possible to verify. However, there are many
other reactions which can be measured experimentally.
MEPHI, November 12–16, 2012, Moscow – p. 16/24
The ψ shapes in non-D ¯
D decay channels
The solid curves show predictions of the model with the mixing ψ
and ψ(2S)
resonances for the ψ
peak shapes in the e+e− → γχc0, e+e− → J/ψη, and
e+e− → φη cross sections; the dashed and dotted curves show the contributions
from
ψ
and ψ(2S) production amplitudes proportional to gψ ab and gψ(2S)ab,
respectively (ab = γχc0, J/ψη, φη). The points with errors are the CLEO data.
MEPHI, November 12–16, 2012, Moscow – p. 17/24
The ψ shapes in non-D ¯
D decay channels
The above examples tell us that the mass spectra in the ψ
region in the non-D ¯
D channels can be very diverse. Therefore we
should expect that the future data on such spectra, together with
the high-statistics data on D ¯
D channels, will impose severe
restrictions on the constructed dynamical models.
MEPHI, November 12–16, 2012, Moscow – p. 18/24
A comment on ambiguity of resonance parameters
Here we illustrate the root of the ambiguity of the interfering resonances
parameters determination by using a simplest example. Consider the “usual”
h amplitude
At fixed M and Γ, there are two solutions for parameters Ax, ϕx, and Cx :
(I) Ax = A, ϕx = ϕ, Cx = C and (II) Ax =
A2 − 2AC sin ϕ + C2Γ2,
tan ϕx = − tan ϕ + CΓ/(A cos ϕ), Cx = C, which yield the same cross
section as a function of energy,
σ(E) = |F (E)|2, and differ in the magnitude and
phase of the resonance contribution. For example, at
M = 3.77 GeV, Γ = 0.03 GeV,
A = 0.045 nb1/2GeV, ϕ = 0, and B = 1.5 nb1/2, solution (II) gives Ax =
ϕx = π/4.
MEPHI, November 12–16, 2012, Moscow – p. 19/24
A comment on ambiguity of resonance parameters
For each energy, the two solutions also give the different overall phase of the
amplitude F (E), δ = δres + δbg. For the above numerical example, δbg for
solutions (I) and (II) is shown by the dashed and solid curves, respectively; the
phase δres is shown by the dotted curve.
MEPHI, November 12–16, 2012, Moscow – p. 20/24
A comment on ambiguity of resonance parameters
The origin of the rapid change of the phase δbg (which is
additional to δres) requires a special dynamical explanation (for
example, the presence of extra intermediate states), for which we

do not see at present any reasons.
————————–
If h and ¯h interact with each other only in the resonance way, then to resolve
ambiguity one must take into account the resonance final state interaction of

hadrons produced via the amplitude Bx and put the phase ϕx = 0 in the elastic
region according to the unitarity requirement.
MEPHI, November 12–16, 2012, Moscow – p. 21/24
Conclusion
1. We tried to show that the shape of the ψ resonance keep
important information about the production mechanism and
interference with background.
2. We considered the models satisfying the unitarity requirement
and obtained good descriptions of the current data on the
e+e− → D ¯
D reaction cross section, in particular, in the
model with the mixing ψ and ψ(2S) resonances.
(See arXiv:1208.4240 for details.)
3. The parametrization suggested in this model for the D meson
electromagnetic form factor in the ψ resonance region can be
used in the processing of new series data on the reactions
e+e− → D ¯
4. We also extracted from experiment
≈ 13 − 30.
MEPHI, November 12–16, 2012, Moscow – p. 22/24
Conclusion
5. New high-statistics data on the reactions e+e− → D ¯
should help reveal the complex mechanism of the ψ
production.
6. As we shown the measurements of mass spectra in the ψ
region in the non-D ¯
D channels, such as e+e− → γχc0,
J/ψη, φη, etc., will promote comprehensive study of the ψ
resonance physics and effective selection of theoretical
7. Additional information about the ψ in the D ¯
D mass spectra
can be extracted, for example, from weak decays B → ψ K
and photoproduction reactions at high energies γA → ψ A.
MEPHI, November 12–16, 2012, Moscow – p. 23/24
Appendix: another description of the data on e+e− → D ¯
The model with the mixing ψ
and ψ(2S) resonances and residual backgrounds.
The dashed, dot-dashed, and dotted curves show the ψ , ψ(2S), and residual
background contributions, respectively (see arXiv:1208.4240 for details).
MEPHI, November 12–16, 2012, Moscow – p. 24/24

Source: http://www.icssnp.mephi.ru/content/file/symposium/2/SQ_15_Achasov_MIPHI.pdf

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