George Rupp - IJS · ˆ resonances and P-wave ˇˇ phase shifts George Rupp CeFEMA, Instituto ......

“Exploring Hadron Resonances”, Bled, 9 July 2015

Unquenched quark-model calculation of excitedρ resonances and P-wave ππ phase shifts

George Rupp

CeFEMA, Instituto Superior Tecnico, Lisbon

Collab.: Susana Coito (Lanzhou) & Eef van Beveren (Coimbra)

I. Introduction

II. Experimental situation and lattice results

III. Resonance-Spectrum-Expansion (RSE) Model

IV. Modelling ρ recurrences

V. Preliminary results

VI. Complex masses in asymptotic states

VII. Conclusions and outlook

I. Introduction

• Experimental light-meson spectroscopy in the energy region of1–2 GeV is in a deplorable state, with many states missing andmany others poorly understood.

• Most detailed data on phase shifts and inelasticities are decadesold, with more recent experiments often providing mass and widthpredictions only.

• The PDG interpretation of resonances frequently seems ad hocand motivated by models.

• Dedicated meson-spectroscopy experiments with systematicpartial-wave analyses are sorely lacking (David Bugg dixit).

• This confusing and incomplete picture of the light mesons severlyhampers a serious check of QCD-inspired quark models and thepossible need to refine or even reconsider them.

• The radial excitations of ρ(770), including possible 3S1 and 3D1

states, represent a typical example, which will be analysed here.

Citation: K.A. Olive et al. (Particle Data Group), Chin. Phys. C38, 090001 (2014) (URL: http://pdg.lbl.gov)

ρ(1450) IG (JPC ) = 1+(1−−)

See our mini-review under the ρ(1700).

ρ(1450) MASSρ(1450) MASSρ(1450) MASSρ(1450) MASS

VALUE (MeV) DOCUMENT ID

1465±25 OUR ESTIMATE1465±25 OUR ESTIMATE1465±25 OUR ESTIMATE1465±25 OUR ESTIMATE This is only an educated guess; the error given is larger thanthe error on the average of the published values.

ηρ0 MODEηρ0 MODEηρ0 MODEηρ0 MODEVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •1497±14 1 AKHMETSHIN 01B CMD2 e+ e− → ηγ

1421±15 2 AKHMETSHIN 00D CMD2 e+ e− → ηπ+π−1470±20 ANTONELLI 88 DM2 e+ e− → ηπ+π−1446±10 FUKUI 88 SPEC 8.95 π− p → ηπ+π− n

1Using the data of AKHMETSHIN 01B on e+ e− → ηγ, AKHMETSHIN 00D and

ANTONELLI 88 on e+ e− → ηπ+π−.2 Using the data of ANTONELLI 88, DOLINSKY 91, and AKHMETSHIN 00D. The energy-independent width of the ρ(1450) and ρ(1700) mesons assumed.

ωπ MODEωπ MODEωπ MODEωπ MODEVALUE (MeV) EVTS DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •1491±19 7815 1 ACHASOV 13 SND 1.05–2.00 e+ e− →

π0π0γ1582±17±25 2382 2 AKHMETSHIN 03B CMD2 e+ e → π0π0γ

1349±25+10− 5 341 3 ALEXANDER 01B CLE2 B → D (∗)ωπ−

1523±10 4 EDWARDS 00A CLE2 τ− → ωπ− ντ1463±25 5 CLEGG 94 RVUE

1250 6 ASTON 80C OMEG 20–70 γp → ωπ0 p

1290±40 6 BARBER 80C SPEC 3–5 γp → ωπ0p

1 From a phenomenological model based on vector meson dominance with the interferingρ(1450) and ρ(1700) and their widths fixed at 400 and 250 MeV, respectively. Systematicuncertainty not estimated.

2Using the data of AKHMETSHIN 03B and BISELLO 91B assuming the ωπ0 and π+π−mass dependence of the total width. ρ(1700) mass and width fixed at 1700 MeV and240 MeV, respectively.

3Using Breit-Wigner parameterization of the ρ(1450) and assuming the ωπ− mass de-pendence for the total width.

4Mass-independent width parameterization. ρ(1700) mass and width fixed at 1700 MeVand 235 MeV respectively.

5Using data from BISELLO 91B, DOLINSKY 86 and ALBRECHT 87L.6Not separated from b1(1235), not pure JP = 1− effect.

HTTP://PDG.LBL.GOV Page 1 Created: 8/21/2014 12:55


4π MODE4π MODE4π MODE4π MODEVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •1435±40 ABELE 01B CBAR 0.0 pn → 2π− 2π0π+

1350±50 ACHASOV 97 RVUE e+ e− → 2(π+π−)

1449± 4 1 ARMSTRONG 89E OMEG 300 pp → pp2(π+π−)

1Not clear whether this observation has I=1 or 0.

ππ MODEππ MODEππ MODEππ MODEVALUE (MeV) EVTS DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •

1350 ±20 +20−30 63.5k 1 ABRAMOWICZ12 ZEUS e p → eπ+π− p

1493 ±15 2 LEES 12G BABR e+ e− → π+π− γ

1446 ± 7 ±28 5.4M 3,4 FUJIKAWA 08 BELL τ− → π−π0 ντ1328 ±15 5 SCHAEL 05C ALEP τ− → π−π0 ντ1406 ±15 87k 3,6 ANDERSON 00A CLE2 τ− → π−π0 ντ∼ 1368 7 ABELE 99C CBAR 0.0 pd → π+π−π− p

1348 ±33 BERTIN 98 OBLX 0.05–0.405 np → 2π+π−1411 ±14 8 ABELE 97 CBAR pn → π−π0π0

1370 +90−70 ACHASOV 97 RVUE e+ e− → π+π−

1359 ±40 6 BERTIN 97C OBLX 0.0 pp → π+π−π0

1282 ±37 BERTIN 97D OBLX 0.05 pp → 2π+2π−1424 ±25 BISELLO 89 DM2 e+ e− → π+π−1265.5±75.3 DUBNICKA 89 RVUE e+ e− → π+π−1292 ±17 9 KURDADZE 83 OLYA 0.64–1.4 e+ e− → π+π−

1Using the KUHN 90 parametrization of the pion form factor, neglecting ρ−ω interference.2Using the GOUNARIS 68 parametrization of the pion form factor leaving the masses andwidths of the ρ(1450), ρ(1700), and ρ(2150) resonances as free parameters of the fit.

3 From the GOUNARIS 68 parametrization of the pion form factor.4 ∣∣Fπ(0)

∣∣2 fixed to 1.5 From the combined fit of the τ− data from ANDERSON 00A and SCHAEL 05C ande+ e− data from the compilation of BARKOV 85, AKHMETSHIN 04, and ALOISIO 05.ρ(1700) mass and width fixed at 1713 MeV and 235 MeV, respectively. SupersedesBARATE 97M.

6 ρ(1700) mass and width fixed at 1700 MeV and 235 MeV, respectively.7 ρ(1700) mass and width fixed at 1780 MeV and 275 MeV respectively.8T-matrix pole.9Using for ρ(1700) mass and width 1600 ± 20 and 300 ± 10 MeV respectively.

K K MODEK K MODEK K MODEK K MODEVALUE (MeV) EVTS DOCUMENT ID TECN CHG COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •1422.8±6.5 27k 1 ABELE 99D CBAR ± 0.0 pp → K+K−π0

1K-matrix pole. Isospin not determined, could be ω(1420).



K K∗(892) + c.c. MODEK K∗(892) + c.c. MODEK K∗(892) + c.c. MODEK K∗(892) + c.c. MODEVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •1505±19±7 AUBERT 08S BABR 10.6 e+ e− → K K∗(892)γ

ρ(1450) WIDTHρ(1450) WIDTHρ(1450) WIDTHρ(1450) WIDTH

VALUE (MeV) DOCUMENT ID

400±60 OUR ESTIMATE400±60 OUR ESTIMATE400±60 OUR ESTIMATE400±60 OUR ESTIMATE This is only an educated guess; the error given is larger thanthe error on the average of the published values.


• • • We do not use the following data for averages, fits, limits, etc. • • •226±44 1 AKHMETSHIN 01B CMD2 e+ e− → ηγ

211±31 2 AKHMETSHIN 00D CMD2 e+ e− → ηπ+π−230±30 ANTONELLI 88 DM2 e+ e− → ηπ+π−60±15 FUKUI 88 SPEC 8.95 π− p → ηπ+π− n

1Using the data of AKHMETSHIN 01B on e+ e− → ηγ, AKHMETSHIN 00D and

ANTONELLI 88 on e+ e− → ηπ+π−.2 Using the data of ANTONELLI 88, DOLINSKY 91, and AKHMETSHIN 00D. The energy-independent width of the ρ(1450) and ρ(1700) mesons assumed.

ωπ MODEωπ MODEωπ MODEωπ MODEVALUE (MeV) EVTS DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •429± 42±10 2382 1 AKHMETSHIN 03B CMD2 e+ e → π0π0γ

547± 86+46−45 341 2 ALEXANDER 01B CLE2 B → D (∗)ωπ−

400± 35 3 EDWARDS 00A CLE2 τ− → ωπ− ντ311± 62 4 CLEGG 94 RVUE

300 5 ASTON 80C OMEG 20–70 γp → ωπ0 p

320±100 5 BARBER 80C SPEC 3–5 γp → ωπ0p

1Using the data of AKHMETSHIN 03B and BISELLO 91B assuming the ωπ0 and π+π−mass dependence of the total width. ρ(1700) mass and width fixed at 1700 MeV and240 MeV, respectively.

2Using Breit-Wigner parameterization of the ρ(1450) and assuming the ωπ− mass de-pendence for the total width.

3Mass-independent width parameterization. ρ(1700) mass and width fixed at 1700 MeVand 235 MeV respectively.

4Using data from BISELLO 91B, DOLINSKY 86 and ALBRECHT 87L.5Not separated from b1(1235), not pure JP = 1− effect.

4π MODE4π MODE4π MODE4π MODEVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •325±100 ABELE 01B CBAR 0.0 pn → 2π− 2π0π+



ρ(1450) DECAY MODESρ(1450) DECAY MODESρ(1450) DECAY MODESρ(1450) DECAY MODES

Mode Fraction (Γi /Γ)

Γ1 ππ seen

Γ2 4π seen

Γ3 ωπΓ4 a1(1260)πΓ5 h1(1170)πΓ6 π(1300)πΓ7 ρρΓ8 ρ(ππ)S-waveΓ9 e+ e− seen

Γ10 ηρ possibly seen

Γ11 a2(1320)π not seen

Γ12 K K not seen

Γ13 K K∗(892)+ c.c. possibly seen

Γ14 ηγ possibly seen

Γ15 f0(500)γ not seen




ρ(1450) Γ(i)Γ(e+ e−)/Γ(total)ρ(1450) Γ(i)Γ(e+ e−)/Γ(total)ρ(1450) Γ(i)Γ(e+e−)/Γ(total)ρ(1450) Γ(i)Γ(e+e−)/Γ(total)

Γ(ππ

)× Γ

(e+ e−

)/Γtotal Γ1Γ9/ΓΓ

(ππ

)× Γ

(e+ e−


(ππ

)× Γ

(e+ e−


(ππ

)× Γ

(e+ e−

)/Γtotal Γ1Γ9/Γ

VALUE (keV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •0.12 1 DIEKMAN 88 RVUE e+ e− → π+π−

0.027+0.015−0.010

2 KURDADZE 83 OLYA 0.64–1.4 e+ e− → π+π−

Γ(ηρ

)× Γ

(e+ e−


(ηρ

)× Γ

(e+ e−


(ηρ

)× Γ

(e+ e−


(ηρ

)× Γ

(e+ e−

)/Γtotal Γ10Γ9/Γ

VALUE (eV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •74±20 3 AKHMETSHIN 00D CMD2 e+ e− → ηπ+π−91±19 ANTONELLI 88 DM2 e+ e− → ηπ+π−

Γ(ηγ

)× Γ

(e+ e−


(ηγ

)× Γ

(e+ e−


(ηγ

)× Γ

(e+ e−


(ηγ

)× Γ

(e+ e−

)/Γtotal Γ14Γ9/Γ

VALUE (eV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •<16.4 4 AKHMETSHIN 05 CMD2 0.60-1.38 e+ e− → ηγ

2.2±0.5±0.3 5 AKHMETSHIN 01B CMD2 e+ e− → ηγ



ρ(1570) IG (JPC ) = 1+(1−−)

OMITTED FROM SUMMARY TABLEMay be an OZI-violating decay mode of ρ(1700). See our mini-review under the ρ(1700).


VALUE (MeV) EVTS DOCUMENT ID TECN COMMENT

1570±36±621570±36±621570±36±621570±36±62 54 1 AUBERT 08S BABR 10.6 e+ e− → φπ0 γ

• • • We do not use the following data for averages, fits, limits, etc. • • •1480±40 2 BITYUKOV 87 SPEC 32.5 π− p → φπ0 n

1 From the fit with two resonances.2 Systematic errors not estimated.



144±75±43144±75±43144±75±43144±75±43 54 3 AUBERT 08S BABR 10.6 e+ e− → φπ0 γ

• • • We do not use the following data for averages, fits, limits, etc. • • •130±60 4 BITYUKOV 87 SPEC 32.5 π− p → φπ0 n

3 From the fit with two resonances.4 Systematic errors not estimated.



Γ1 e+ e−

Γ2 φπ not seen

Γ3 ωπ

ρ(1570) Γ(i)Γ(e+ e−)/Γ(total)ρ(1570) Γ(i)Γ(e+ e−)/Γ(total)ρ(1570) Γ(i)Γ(e+e−)/Γ(total)ρ(1570) Γ(i)Γ(e+e−)/Γ(total)

Γ(φπ

)× Γ

(e+ e−


(φπ

)× Γ

(e+ e−


(φπ

)× Γ

(e+ e−


(φπ

)× Γ

(e+ e−

)/Γtotal Γ2Γ1/Γ

VALUE (eV) CL% EVTS DOCUMENT ID TECN COMMENT

3.5±0.9±0.33.5±0.9±0.33.5±0.9±0.33.5±0.9±0.3 54 5 AUBERT 08S BABR 10.6 e+ e− → φπ0 γ

• • • We do not use the following data for averages, fits, limits, etc. • • •<70 90 6 AULCHENKO 87B ND e+ e− → K0

S K0Lπ0

5 From the fit with two resonances.6Using mass and width of BITYUKOV 87.



ρ(1700) IG (JPC ) = 1+(1−−)

A REVIEW GOES HERE – Check our WWW List of Reviews


ηρ0 AND π+π− MODESηρ0 AND π+π− MODESηρ0 AND π+π− MODESηρ0 AND π+π− MODESVALUE (MeV) DOCUMENT ID

1720±20 OUR ESTIMATE1720±20 OUR ESTIMATE1720±20 OUR ESTIMATE1720±20 OUR ESTIMATE


The data in this block is included in the average printed for a previous datablock.

• • • We do not use the following data for averages, fits, limits, etc. • • •1740±20 ANTONELLI 88 DM2 e+ e− → ηπ+π−1701±15 1 FUKUI 88 SPEC 8.95 π− p → ηπ+π−n

1Assuming ρ+ f0(1370) decay mode interferes with a1(1260)+π background. From a

two Breit-Wigner fit.

ππ MODEππ MODEππ MODEππ MODEVALUE (MeV) EVTS DOCUMENT ID TECN COMMENT

The data in this block is included in the average printed for a previous datablock.


1780 ±20 +15−20 63.5k 2 ABRAMOWICZ12 ZEUS e p → eπ+π− p

1861 ±17 3 LEES 12G BABR e+ e− → π+π− γ

1728 ±17 ±89 5.4M 4,5 FUJIKAWA 08 BELL τ− → π−π0 ντ

1780 +37−29

6 ABELE 97 CBAR pn → π−π0π0

1719 ±15 6 BERTIN 97C OBLX 0.0 pp → π+π−π0

1730 ±30 CLEGG 94 RVUE e+ e− → π+π−1768 ±21 BISELLO 89 DM2 e+ e− → π+π−1745.7±91.9 DUBNICKA 89 RVUE e+ e− → π+π−1546 ±26 GESHKEN... 89 RVUE

1650 7 ERKAL 85 RVUE 20–70 γp → γπ

1550 ±70 ABE 84B HYBR 20 γp → π+π− p

1590 ±20 8 ASTON 80 OMEG 20–70 γp → p2π

1600 ±10 9 ATIYA 79B SPEC 50 γC → C2π

1598 +24−22 BECKER 79 ASPK 17 π− p polarized

1659 ±25 7 LANG 79 RVUE

1575 7 MARTIN 78C RVUE 17 π− p → π+π− n

1610 ±30 7 FROGGATT 77 RVUE 17 π− p → π+π− n

1590 ±20 10 HYAMS 73 ASPK 17 π− p → π+π− n



π+π−π0π0 MODEπ+π−π0π0 MODEπ+π−π0π0 MODEπ+π−π0π0 MODEVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •300±50 ATKINSON 85B OMEG 20–70 γp

ωπ0 MODEωπ0 MODEωπ0 MODEωπ0 MODEVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •350 to 580 38 ACHASOV 00I SND e+ e− → π0π0γ

490 to 1040 39 ACHASOV 00I SND e+ e− → π0π0γ

38Taking into account both ρ(1450) and ρ(1700) contributions. Using the data of

ACHASOV 00I on e+ e− → ωπ0 and of EDWARDS 00A on τ− → ωπ−ντ . ρ(1450)mass and width fixed at 1400 MeV and 500 MeV respectively.

39Taking into account the ρ(1700) contribution only. Using the data of ACHASOV 00I on

e+ e− → ωπ0 and of EDWARDS 00A on τ− → ωπ− ντ .

3(π+π− ) AND 2(π+π−π0 ) MODES3(π+π− ) AND 2(π+π−π0 ) MODES3(π+π− ) AND 2(π+π−π0 ) MODES3(π+π− ) AND 2(π+π−π0 ) MODESVALUE (MeV) DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •315±100 40 FRABETTI 04 E687 γp → 3π+3π− p

285± 20 CLEGG 90 RVUE e+ e− → 3(π+π−)2(π+π−π0)

40 From a fit with two resonances with the JACOB 72 continuum.



Γ1 4πΓ2 2(π+π−) large

Γ3 ρππ dominant

Γ4 ρ0π+π− large

Γ5 ρ0π0π0

Γ6 ρ±π∓π0 large

Γ7 a1(1260)π seen

Γ8 h1(1170)π seen

Γ9 π(1300)π seen

Γ10 ρρ seen

Γ11 π+π− seen

Γ12 ππ seen

Γ13 K K∗(892)+ c.c. seen

Γ14 ηρ seen

Γ15 a2(1320)π not seen

Γ16 K K seen

Γ17 e+ e− seen

Γ18 π0ω seen



ρ(1900) IG (JPC ) = 1+(1−−)

OMITTED FROM SUMMARY TABLESee our mini-review under the ρ(1700).



• • • We do not use the following data for averages, fits, limits, etc. • • •1909±17±25 54 1 AUBERT 08S BABR 10.6 e+ e− → φπ0 γ

1880±30 AUBERT 06D BABR 10.6 e+ e− → 3π+3π− γ

1860±20 AUBERT 06D BABR 10.6 e+ e− → 2(π+π−π0)γ

1910±10 2,3 FRABETTI 04 E687 γp → 3π+3π− p

1870±10 ANTONELLI 96 SPEC e+ e− → hadrons

1 From the fit with two resonances.2 From a fit with two resonances with the JACOB 72 continuum.3 Supersedes FRABETTI 01.



• • • We do not use the following data for averages, fits, limits, etc. • • •48±17±2 54 4 AUBERT 08S BABR 10.6 e+ e− → φπ0 γ

130±30 AUBERT 06D BABR 10.6 e+ e− → 3π+3π− γ

160±20 AUBERT 06D BABR 10.6 e+ e− → 2(π+π−π0)γ

37±13 5,6 FRABETTI 04 E687 γp → 3π+3π− p

10± 5 ANTONELLI 96 SPEC e+ e− → hadrons

4 From the fit with two resonances.5 From a fit with two resonances with the JACOB 72 continuum.6 Supersedes FRABETTI 01.

ρ(1900) Γ(i)Γ(e+ e−)/Γ2(total)ρ(1900) Γ(i)Γ(e+ e−)/Γ2(total)ρ(1900) Γ(i)Γ(e+ e−)/Γ2(total)ρ(1900) Γ(i)Γ(e+ e−)/Γ2(total)

Γ(φπ

)/Γtotal × Γ

(e+ e−

)/Γtotal Γ4/Γ× Γ6/ΓΓ

(φπ

)/Γtotal × Γ

(e+ e−


(φπ

)/Γtotal × Γ

(e+ e−


(φπ

)/Γtotal × Γ

(e+ e−

)/Γtotal Γ4/Γ× Γ6/Γ

VALUE (units 10−8) EVTS DOCUMENT ID TECN COMMENT

• • • We do not use the following data for averages, fits, limits, etc. • • •4.2±1.2±0.8 54 7 AUBERT 08S BABR 10.6 e+ e− → φπ0 γ

7 From the fit with two resonances.





Γ1 6π seen

Γ2 3π+3π− seen

Γ3 2π+2π−2π0

Γ4 φπΓ5 hadrons seen

Γ6 e+ e− seen

Γ7 NN not seen

ρ(1900) BRANCHING RATIOSρ(1900) BRANCHING RATIOSρ(1900) BRANCHING RATIOSρ(1900) BRANCHING RATIOS

Γ(6π

)/Γtotal Γ1/ΓΓ

(6π

)/Γtotal Γ1/ΓΓ

(6π

)/Γtotal Γ1/ΓΓ

(6π

)/Γtotal Γ1/Γ

VALUE EVTS DOCUMENT ID TECN COMMENT

seen 8k AKHMETSHIN 13 CMD3 e+ e− → 3π+3π−not seen AGNELLO 02 OBLX np → 3π+2π−π0

seen FRABETTI 01 E687 γp → 3π+3π− p

seen ANTONELLI 96 SPEC e+ e− → hadrons

ρ(1900) REFERENCESρ(1900) REFERENCESρ(1900) REFERENCESρ(1900) REFERENCES

AKHMETSHIN 13 PL B723 82 R.R. Akhmetshin et al. (CMD-3 Collab.)AUBERT 08S PR D77 092002 B. Aubert et al. (BABAR Collab.)AUBERT 06D PR D73 052003 B. Aubert et al. (BABAR Collab.)FRABETTI 04 PL B578 290 P.L. Frabetti et al. (FNAL E687 Collab.)AGNELLO 02 PL B527 39 M. Agnello et al. (OBELIX Collab.)FRABETTI 01 PL B514 240 P.L. Frabetti et al. (FNAL E687 Collab.)ANTONELLI 96 PL B365 427 A. Antonelli et al. (FENICE Collab.)JACOB 72 PR D5 1847 M. Jacob, R. Slansky



K ∗(1410) I (JP ) = 12 (1

−)

K∗(1410) MASSK∗(1410) MASSK∗(1410) MASSK∗(1410) MASS

VALUE (MeV) DOCUMENT ID TECN CHG COMMENT

1414±15 OUR AVERAGE1414±15 OUR AVERAGE1414±15 OUR AVERAGE1414±15 OUR AVERAGE Error includes scale factor of 1.3.

1380±21±19 ASTON 88 LASS 0 11 K− p → K−π+n

1420± 7±10 ASTON 87 LASS 0 11 K− p → K0π+π− n


1276+72−77

1,2 BOITO 09 RVUE τ− → K0S π− ντ

1367±54 BIRD 89 LASS − 11 K− p → K0π− p

1474±25 BAUBILLIER 82B HBC 0 8.25 K− p → K0 2πn

1500±30 ETKIN 80 MPS 0 6 K− p → K0π+π−n

1 From the pole position of the K π vector form factor in the complex s-plane and usingEPIFANOV 07 data.

2 Systematic uncertainties not estimated.

K∗(1410) WIDTHK∗(1410) WIDTHK∗(1410) WIDTHK∗(1410) WIDTH


232± 21 OUR AVERAGE232± 21 OUR AVERAGE232± 21 OUR AVERAGE232± 21 OUR AVERAGE Error includes scale factor of 1.1.

176± 52±22 ASTON 88 LASS 0 11 K− p → K−π+n



198+ 61− 87

3,4 BOITO 09 RVUE τ− → K0S π− ντ

114±101 BIRD 89 LASS − 11 K− p → K0π− p

275± 65 BAUBILLIER 82B HBC 0 8.25 K− p → K0 2πn

500±100 ETKIN 80 MPS 0 6 K− p → K0π+π−n

3 From the pole position of the K π vector form factor in the complex s-plane and usingEPIFANOV 07 data.

4 Systematic uncertainties not estimated.

K∗(1410) DECAY MODESK∗(1410) DECAY MODESK∗(1410) DECAY MODESK∗(1410) DECAY MODES

Mode Fraction (Γi /Γ) Confidence level

Γ1 K∗(892)π > 40 % 95%

Γ2 K π ( 6.6±1.3) %

Γ3 K ρ < 7 % 95%

Γ4 γK0 seen



K ∗(1680) I (JP ) = 12 (1

−)

K∗(1680) MASSK∗(1680) MASSK∗(1680) MASSK∗(1680) MASS



1677±10±32 ASTON 88 LASS 0 11 K− p → K−π+n

1735±10±20 ASTON 87 LASS 0 11 K− p → K0π+π− n

• • • We do not use the following data for averages, fits, limits, etc. • • •1678±64 BIRD 89 LASS − 11 K− p → K0π− p

1800±70 ETKIN 80 MPS 0 6 K− p → K0π+π−n

∼ 1650 ESTABROOKS 78 ASPK 0 13 K± p → K±π±n

K∗(1680) WIDTHK∗(1680) WIDTHK∗(1680) WIDTHK∗(1680) WIDTH



205± 16±34 ASTON 88 LASS 0 11 K− p → K−π+n


• • • We do not use the following data for averages, fits, limits, etc. • • •454±270 BIRD 89 LASS − 11 K− p → K0π− p

170± 30 ETKIN 80 MPS 0 6 K− p → K0π+π−n

250 to 300 ESTABROOKS 78 ASPK 0 13 K± p → K±π±n

K∗(1680) DECAY MODESK∗(1680) DECAY MODESK∗(1680) DECAY MODESK∗(1680) DECAY MODES


Γ1 K π (38.7±2.5) %

Γ2 K ρ (31.4+5.0−2.1) %

Γ3 K∗(892)π (29.9+2.2−5.0) %


Godfrey & Isgur, Phys. Rev. D 32 (1985) 189

Light isovector spectrum

Godfrey & Isgur, Phys. Rev. D 32 (1985) 189

Strange spectrum

D. Aston et al., Nucl. Phys. B Proc. Suppl. 21 (1991) 105

Yu.S. Surovtsev, P. Bydžovský / Nuclear Physics A 807 (2008) 145–157 151

Fig. 2. The phase shift of amplitude, Fig. 2a, and module of the S-matrix element, Fig. 2b, of the P -wave ππ -scattering.

The results of 3-resonance (dashed), 4-resonance (dash-dotted), and 5-resonance (solid) descriptions are shown. The data

are from Refs. [17] (circle), [18] (square), and [19] (plus).

that the resonance ρ(1900) is desired and that the ρ(1450) might be also included improving

slightly the description (at all events, its existence does not contradict to the data).

Masses and widths of the obtained ρ-states can be calculated from the pole positions on

sheet II for resonances of type (a) and on sheet IV for resonances of type (b). The calculated

values of the masses and total widths are given in Table 2 where we assume that the resonance

part of the amplitude reads as

T res=

√sΓel

m2res − s − i

√sΓtot

. (5)

3. The Breit–Wigner analysis of P -wave ππ scattering

In various works [1], it was shown that the ρ-like resonances obtained in the previous section

have also other considerable decay channels in addition to those considered explicitly above.

It was observed that the ρ(1450) and/or a possible ρ(1250) can also decay to the 4π (seen),

φπ (< 1%), and ηρ0 (< 4%) channels, where the fraction Γi/Γ [1] is given in the parenthesis.

Fit with ππ + ωπ S-matrix parametrisation

152 Yu.S. Surovtsev, P. Bydžovský / Nuclear Physics A 807 (2008) 145–157

Table 1

Pole clusters distributed on sheets II, III, and IV for the ρ-like resonances.√

sr in MeV is given

Three resonances

II III IV

ρ(770) 767.3 ± 0.6 − i(73.3 ± 0.5) 782 ± 10.9 − i(65.6 ± 4.7)

ρ(1250) 1249.9 ± 19.9 − i(152 ± 14.3) 1249 ± 16.9 − i(146.2 ± 14.4)

ρ(1600) 1585 ± 15.3 − i(130.5 ± 22.5) 1578 ± 8.8 − i(72.2 ± 12.5)

Four resonances

II III IV

ρ(770) 766.5 ± 0.6 − i(73.2 ± 0.5) 783.1 ± 10.6 − i(66.2 ± 4.9)

ρ(1250) 1251.4 ± 18.8 − i(152.1 ± 14.2) 1249 ± 16.3 − i(144.3 ± 13.9)

ρ(1600) 1585.2 ± 18.2 − i(141.8 ± 22.3) 1579.6 ± 8.1 − i(73.6 ± 10.3)

ρ(1900) 1871.5 ± 30.5 − i(97.2 ± 30.1) 1894 ± 33.6 − i(95.3 ± 32)

Five resonances

II III IV

ρ(770) 765.8 ± 0.6 − i(73.3 ± 0.4) 778.2 ± 9.1 − i(68.9 ± 3.9)

ρ(1250) 1251.4 ± 11.3 − i(130.9 ± 9.1) 1251 ± 11.1 − i(130.5 ± 9.2)

ρ(1470) 1469.4 ± 10.6 − i(91 ± 12.9) 1465.4 ± 12.1 − i(99.8 ± 15.6)

ρ(1600) 1634 ± 20.1 − i(144.7 ± 23.8) 1592.9 ± 7.9 − i(73.7 ± 11.7)

ρ(1900) 1882.8 ± 24.8 − i(112.4 ± 25.2) 1893 ± 21.9 − i(93.4 ± 19.9)

Table 2

Calculated masses and total widths of the ρ-states (all in MeV)

mres Γtot

ρ(770) 769.3 ± 0.6 146.6 ± 0.9

ρ(1250) 1257.8 ± 11.1 261 ± 18.3

ρ(1470) 1468.8 ± 12.1 199.6 ± 31.2

ρ(1600) 1594.6 ± 8 147.4 ± 23.4

ρ(1900) 1895.3 ± 21.9 186.8 ± 39.8

The ρ(1700) resonance has been observed to have the branching ratio to the 4π (large), ρ2π

(dominant), and ηρ0 (seen) channels.

To include explicitly an influence of some selected channels and to obtain information about

couplings with these channels from the analysis of the ππ -scattering data, we used five-channel

Breit–Wigner forms in constructing the Jost matrix determinant d(k1, . . . , k5). The resonance

poles and zeros in the S-matrix are generated utilizing the Le Couteur–Newton relation

Sres =d(−k1, . . . , k5)

d(k1, . . . , k5), (6)

where k1, k2, k3, k4 and k5 are the momenta of ππ , π+π−2π0, 2π+2π−, η2π , and ωπ0 chan-

nels, respectively. For the c.m. momenta of two-particle channels, we used the equation

ki =1

2

√

[s − (m1 + m2)2][s − (m1 − m2)2]s

, (7)

the multi-particle states being considered as the quasi-two-particle ones. The d-function is taken

as d = dresdbg where dres describes contributions of resonances and dbg the background.

ρ poles, masses, widhts with ππ + ωπ S-matrix parametrisation


Fig. 3. The phase shift of amplitude and module of the S-matrix element of the P -wave ππ -scattering. The curve shows

result of fitting to the data using the Breit–Wigner form with 5 resonances. The data are from Refs. [17] (circle), [18]

(square), and [19] (plus).

Table 3

The ρ-like resonance parameters for the five-resonance fit (all in MeV)

ρ(770) ρ(1250) ρ(1450) ρ(1600) ρ(1900)

M 777.69 ± 0.32 1249.8 ± 15.6 1449.9 ± 12.2 1587.3 ± 4.5 1897.8 ± 38

fr1 343.8 ± 0.73 87.7 ± 7.4 56.9 ± 5.4 248.2 ± 5.2 47.3 ± 12

fr2 24.6 ± 5.8 186.3 ± 39.9 100.1 ± 18.7 240.2 ± 8.6 73.7

fr3 34.8 ± 8.2 263.5 ± 56.5 141.6 ± 26.5 339.7 ± 12.5 104.3

fr4 231.8 ± 111 141.2 ± 98 141.8 ± 33 9

fr5 231 ± 115 150 ± 95 108.6 ± 40.4 10

Γtot ≈ 154.3 > 175 > 52 > 168 > 10

from various evaluations in the local [14] and non-local [13] Nambu–Jona-Lasinio (NJL) model

and from the analyses which use Roy’s equations [9,10] and the forward dispersion relations

[11]. In Ref. [10], the chiral perturbation theory (ChPT) was also applied to construct a precise

Fit with 5 Breit-Wigner resonances


Fig. 3. The phase shift of amplitude and module of the S-matrix element of the P -wave ππ -scattering. The curve shows

result of fitting to the data using the Breit–Wigner form with 5 resonances. The data are from Refs. [17] (circle), [18]

(square), and [19] (plus).

Table 3

The ρ-like resonance parameters for the five-resonance fit (all in MeV)

ρ(770) ρ(1250) ρ(1450) ρ(1600) ρ(1900)

M 777.69 ± 0.32 1249.8 ± 15.6 1449.9 ± 12.2 1587.3 ± 4.5 1897.8 ± 38

fr1 343.8 ± 0.73 87.7 ± 7.4 56.9 ± 5.4 248.2 ± 5.2 47.3 ± 12

fr2 24.6 ± 5.8 186.3 ± 39.9 100.1 ± 18.7 240.2 ± 8.6 73.7

fr3 34.8 ± 8.2 263.5 ± 56.5 141.6 ± 26.5 339.7 ± 12.5 104.3

fr4 231.8 ± 111 141.2 ± 98 141.8 ± 33 9

fr5 231 ± 115 150 ± 95 108.6 ± 40.4 10

Γtot ≈ 154.3 > 175 > 52 > 168 > 10

from various evaluations in the local [14] and non-local [13] Nambu–Jona-Lasinio (NJL) model

and from the analyses which use Roy’s equations [9,10] and the forward dispersion relations

[11]. In Ref. [10], the chiral perturbation theory (ChPT) was also applied to construct a precise

ρ masses, widhts, and BW parameters for 5 BW resonances

G. Engel, C. Lang, D. Mohler, A. Schafer, PoS Hadron2013 (2013)118 [arXiv:1311.6579 [hep-ph]: K ?′ level above 1.6 GeV:

0.0

0.5

1.0

1.5

2.0

2.5

3.0

mass

[G

eV

]

K

0-

K0

0+

K*

1-

K1

1+

K2

2-

T2/E

K2

*

2+

T2/E

φ

1--

f2

T2/E

2++

• However, K ?′ resonance at (1.33± 0.02) GeV (Exp. 1.41):

S. Prelovsek, L. Leskovec, C. Lang, D. Mohler, Phys. Rev. D 88(2013) 054508

0.6 0.8 1 1.2 1.4 1.6E

Kπ [GeV]

0

90

180

270

360

δ1

1/2

[d

egre

es]

exp: Estabrooks, elastic

exp: Aston, elastic

exp: Aston, inel.

exp: Aston, inel., shifted by 180o

Ethr

E0

III. Resonance-Spectrum Expansion

(EvB & GR, Annals Phys. 324 (2009) 1620)

⇒Building blocks of (non-exotic) RSE are:

V =

M

M

M

M

qq

V ΩV =

M

M

M

M

qq qq

• V is the effective two-meson potential;• Ω is the two-meson loop function;• the blobs are the 3P0 vertex functions, modelled by a spherical δ

shell in r space, i.e., a spherical Bessel function in p space;• the wiggly lines stand for s-channel exchanges of infinite towers

of qq states, i.e., a kind of Regge propagators.

⇒For N meson-meson channels and several qq channels:

V(Li ,Lj )ij (pi , p

′j ;E ) = λ2r0 j

iLi

(pi r0) j jLj (p′j r0)

Nqq∑

α=1

∞∑

n=0

g(α)i (n)g

(α)j (n)

E − E(α)n

≡ Rij(E ) j iLi (pi r0) j jLj (p′j r0) .

⇒ The closed-form off-energy-shell T -matrix then reads

T(Li ,Lj )ij (pi , p

′j ;E ) =

−2λ2r0√µipiµ′jp

′j j

iLi

(pi r0)N∑

m=1

Rim(E )

[11− ΩR]−1mj

j jLj (p′j r0) ,

Ω = −2iλ2r0 diag(jnLn(knr0)h

(1)nLn

(knr0)).

⇒The corresponding unitary and symmetric S-matrix is given by

S(Li ,Lj )ij (ki , k

′j ;E ) = δij + 2iT

(Li ,Lj )ij (ki , k

′j ;E ) .

IV. Modelling ρ recurrences with RSE model

• Both 3S1 and 3D1 qq states contribute to the 1−− excited ρ spec-trum and P-wave ππ scattering.

• Many other meson-meson channels couple, too, as seen in thePDG listings above, viz. of the types pseudoscalar-pseudoscalar(PP), vector-pseudoscalar (VP), vector-vector (VV), vector-scalar(VS), axialvector-pseudoscalar (AP), and axialvector-vector (AV).

• Included qq channels: nn = (uu − dd)/√

2 in 3S1 and 3D1.• Included meson-meson channels:

PP: ππ, KK in P-wave;VP: ωπ, ρη, ρη′, K ?K in P-wave;VV: ρρ, K ?K ? in P-wave;VS: ρf0(500), ωa0(980), K ?K ?

0 (800) in S- and D-wave;

AP: a1(1260)π, b1(1235)η, b1(1235)η′, h1(1170)π, K1(1270)K ,K1(1400)K in S-wave;

AV: a1(1260)ω, b1(1235)ρ, f1(1285)ρ, K1(1270)K ?, K1(1400)K ?

in S-wave;

P’P: π(1300)π in P-wave.

V. Very preliminary results

• For the bare spectrum E(α)n we take that of a harmonic oscillator

with constant frequency, i.e., E = (2n + `+ 3/2)ω + 2mn.

• The non-strange quark mass and the frequency are kept fixed atthe values mn = 406 MeV and ω = 190 MeV, respectively, as inall previous work.

• The more or less free parameters are the overall coupling λ, thestring-breaking distance r0, and a range parameter β which some-what suppresses kinematically closed channels via a form factor.But note that, typically, λ ∼ 3–5 and r0 ∼ 1 fm for light quarks.

• A tentative fit by hand to the old P-wave ππ phase shifts byB. Hyams et al., Nucl. Phys. B 64 (1973) 134, gives the curvesshown in the next plot, for 10, 19, and 26 channels, respectively.

• The corresponding parameters for the 26-channel calculation areλ = 5.3, r0 = 4.5 GeV−1, and β = 4 GeV−2.

• For these parameters, resonance poles are found at 754 − i67MeV, 1358 − i18 MeV, 1383 − i59 MeV, 1783 − i15 MeV, and1999− i131 MeV.

50

100

150

200

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

’../data/pontos1b.dat’’1.dat’

’1b.dat’’1c.dat’

RSE P-wave δππ calculation with 10 ( ), 19 ( ), 26 ( ) channels.

VI. Complex masses in the S-matrix

With complex masses, S ceases to be unitary. Nevertheless, S isalways symmetric, so can be decomposed via Takagi factorisation:

S = VDV T , (1)with V unitary and D a real non-negative diagonal matrix. Then

S†S = (V T )†DV †VDV T = (V T )†D2V T = U†D2U , (2)

where U ≡ V T is also unitary. So D =√US†SU†. Moreover, since

S = 11 + 2iT is manifestly non-singular, the eigenvalues of S†S areeven all nonzero and U is unique. Thus, we may define

S ′ ≡ SU†D−1U . (3)Then, using Eq. (1) and V = UT , we have

S ′ = UTDUU†D−1U = UTU , (4)which is obviously symmetric and, as

(UTU)† = U†(U†)T = U−1(U−1)T = (UTU)−1 , (5)

also unitary. So S ′ can be defined as the S-matrix for a scatteringprocess with complex masses in the asymptotic states.

T. Takagi, Japan J. Math. 1 (1924) 82;S. Coito, G. Rupp, E. van Beveren, Eur. Phys. J. C 71 (2011) 1762.

VII. Conclusions and outlook

• The ρ recurrences reported in the PDG tables collect wildly differ-ent measurements of masses and widths under the same entries,while ignoring a recent and strongly diverging detailed analysis.

• The notorious and long-time PDG obfuscation of a ρ(1250–1300)observed by several experimental groups has been very detrimentalto meson specroscopy.

• The very recent lattice confirmation of the K ?(1410) resonanceas the first radial excitation of K ?(892) lends further support tothe existence of a ρ(1250–1300).

• Preliminary results of an RSE calculation involving pairs of pseu-doscalar, vector, scalar, and axial-vector mesons are encouraging.

• Allowing for resonances in the two-meson final states to accountfor observed 4π and 6π decays would make the model more re-alistic. An algebraic procedure has been developed to reunitarisethe S-matrix for complex masses in the asymptotic states.

• It may turn out out to be necessary to allow different decay radiifor the several classes of two-meson channels, in order to repro-duce the finer details of the data.

George Rupp - IJS · ˆ resonances and P-wave ˇˇ phase shifts George Rupp CeFEMA, Instituto ......

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