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## Icssnp.mephi.ru

**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**
mψ

**= 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**
mψ

**, **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: **mψ

**= 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 nb**1/2

**GeV, **ϕ

**= 0, and **B

**= 1.5 nb**1/2

**, 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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