Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia Nature of Light Scalar Mesons f 0 (600), a 0 (980), f 0 (1370) and a 0 (1450)

Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia

Nature of Light Scalar Mesons Nature of Light Scalar Mesons ff00(600), (600), aa00(980), (980), ff00(1370) and (1370) and aa00(1450)(1450)

Denis Parganlija

In collaboration withFrancesco Giacosa and Dirk H. Rischke

Institut für Theoretische Physik Goethe-Universität Frankfurt am Main


Motivation: Effective Theories of QCD and Linear Sigma Model

Description of low-energy hadrons (mesons)

Generalisation to T, μ ≠ 0 Linear Sigma Model:Treats chiral partners on the same footingVacuum calculations → calculations at T≠0Degeneration of chiral partners above TC


Motivation: Structure of Scalar Mesons Spontaneous Breaking of Chiral Symmetry →

Goldstone Bosons (π) Restoration of Chiral Invariance and

Deconfinement ↔ Degeneration of Chiral Partners (π/σ)

Nature of scalar mesons Scalar states under 1 GeV → f0(600), a0(980) Scalar states above 1 GeV → f0(1370),

a0(1450) – additional scalar states under 1 GeV required (tetraquarks?)

qq

qq

f0(600), „sigma“ f0(1370)


Scenario I: Scalars under 1 GeV

Nf = 2Scalars → Pseudoscalars →Vectors → Axialvectors →

(980)),600( 00 af

,)770(),782(

(1260),(1285) 11 af

scalarspseudoscalarsvectors

axialvectors


Lagrangian of a Linear Sigma Model with Vector and Axialvector Mesons (Nf=2)

m12 parameters

photon


Spontaneous Symmetry Breaking (SSB):

Shift: Shift (Diagonalise):

Renormalise Pseudoscalar Wave Functions:

[R. Pisarski, hep-ph/9503330 (1995)][S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. 41, 531 (1969)]

1 11

11

)(),( 31 hZZgZZ 11 parameters


Parameter Determination

Masses:Pion Decay Constant Five Parameters: Z, g2, m1, h2, mσ

Three Parameters: Z, m1, mσ

10,,,, aa mmmmm

Zf

)( precise very 22 Zgg )( precise 2201Zhhaf

MeV )246.0640.0(][

1 Za

][ 020.0218.0],,[ 11

00

mmmZa][ 0125.00457.0],,[ 1

120

mmmZaIsospin

Angular Momentum (s wave)[NA48/2 Collaboration, 2009]


Scenario I – Results

Z = 1.667 ± 0.205, mσ = (332 ± 456) MeV,

)]()([2 321

221

2 ZhZhhmm

MeV 652 14516521

m

quark condensate~ gluon condensateiesuncertaint in decrease ],0[1 mm


Scenario I – Results

Boundaries of mσ ← scattering lengths

MeV 332 1324

m MeV 332 44

145m

MeV ]477 ,288[m MeV ]356 ,319[m

020.0218.000 a


Scenario I – Sigma Decay

Γσ→ππsmall! too width decay )600(0f

MeV )272441(2

iΓim

[H. Leutwyler et al.]

MeV )255460(2

iΓim

[J. R. Peláez et al.]


Scenario I - Results

Our Result Experimental Value

MeV 640.01

a

exact ],[ ],,[ 22 01hZgZΓ af

MeV 640.01

a(NA48/2) 218.00

0 a218.000 a

0454.020 a (NA48/2) 0457.02

0 a

MeV 33730

aA)41.4 :angle mixing ( η'η

[KLOE Collaboration, hep-ex/0612029v3]

[D. V. Bugg et al., Phys. Rev. D 50, 4412 (1994)]

MeV 33300

aAMeV 3330

0aA

deg41.8 :angle mixing 0.50.2-

η'η


Scenario II: Scalars above 1 GeV

Nf = 2Scalars → Pseudoscalars →Vectors → Axialvectors →

(1450)),1370( 00 af

,)770(),782(

(1260),(1285) 11 af

scalarspseudoscalarsvectors

axialvectors


Decay Width f0(1370) → ππ

MeV )500200(

MeV, )15001200( :PDG

)1370(

)1370(

0

0

f

fm

[D. V. Bugg, arXiv: 0710.4452 [hep-ex]]MeV, 1282)1370(0

fmMeV 15207)1370(0

f

qqf as )1370( favours data alExperiment 0

MeV 200 Mixing Glueball Include )1370(0 f

[S. Janowski (Frankfurt U.) – Diploma Thesis (preliminary results)]


The Rho Mass Contributions

m

)]()([2 32

221

2 ZhZhmm

21.067.1MeV )246.0640.0(][1

ZZa 21.007.067.1

ZMeV 721],MeV 592[ 54

12911 mmm

m indominant condensate Gluon


Scenario II – Scattering Lengths Scattering lengths saturated

Additional scalars: tetraquarks, quasi-molecular states

Glueball


Including Baryons

LRLRLR

LLLRRR

UU

UU

,2,2,2,2

,1,1,1,1

)1535(**)940(

NNNN

MeV 5000 M

15(exp)0

150 MeV1012.64 MeV1060 aa

[S. Gallas, F. Giacosa and D. H. Rischke, arXiv: 0907.5084 [hep-ph]]


Summary LSM with global U(2)R x U(2)L invariance: scattering

lengths, low-energy meson decay widths General phenomenology in agreement with

experiment (ρ→ππ, a1→πγ, f1→aoπ, a0→ηπ decay, ππ scattering lengths)

The f0(600) → ππ decay width fails to match experiment ↔ quarkonium structure of f0(600), a0(980) excluded

Quarkonium structure of f0(1370), a0(1450) favoured, f0(1370) → ππ in agreement with experiment

Scalar states under 1 GeV required for correct description of pion-pion scattering lengths


Outlook Consequences of Global Invariance up to 4th

Order Mixing in the Scalar Sector Chiral Models With Three Flavours Extension to Non-Zero Temperature: Study

Chiral Symmetry Restoration Low Energy Constants of QCD p, d Wave Scattering Lengths Include Tensor, Pseudotensor Mesons,

Baryons (Nucleons)


Spare Slides


Scenario I – Sigma Decay m1 = 0 → mρ generated from the quark condensate only;

our result: m1 = 652 MeV a1→σπ

MeV )600250(:PDG (total) 1a


Scenario I: a1 → ρπ Decay

MeV )600250(:PDG

[M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)]

MeV 500 ifMeV 160 m


Scenario II: Parameter Determination

Masses:Pion Decay Constant Five Parameters: Z, h1, h2, g2, mσ

10,,,, aa mmmmm

Zf

)(MeV 1.0)(149.4 22 Zgg )(MeV )13265( 22)1450(0Zhha

ZZa MeV )246.0640.0(][1

)small ( 0 3,21 hh

free )1370(0fmm


Comparison: the Model with and without Vectors and Axial-Vectors

decrease values

vectors Include


Relevant Vertices for Pion-Pion Scattering

Feynman Diagrams At Tree Level:


Tree-Level Scattering Amplitude

s-channel isospin I components of the scattering amplitude:

scattering lengths (on the threshold):partial wave amplitude


Results I


Calculation of Scattering LengthsFeynman Diagrams At Tree Level:

Scattering Length

= 0.233 ± 0.023

J. R. Batley et al. [NA48/2 Collaboration],

Eur. Phys. J. C 54, 411 (2008)


Results I: as function of

E865

m

0150047020 ..a NA48/2 Ke4


Calculation of Pion-Pion Scattering Lengths and Decay Widths (Local Invariance)

Feynman Diagrams At Tree Level:

ππ Scattering Lengths: mσ ≈ (320 - 340) MeV[D. Parganlija, F. Giacosa and D. Rischke, AIP Conf. Proc. 1030, 160 (2008)]

(MeV)m

(MeV)

100 MeV Width at 800 MeV MassMeV )272441(

2i

Γim

[H. Leutwyler et al.] MeV5.86

MeV)0.14.149(:PDG


Results from the Locally Invariant Lagrangian: ππ Scattering Lengths and

Low-Energy Meson Decay Widths Pion-pion scattering lengths correspond to

experiment (NA48/2 data) → mσ = (315 - 345) MeV f0(600) decay width: Γσ→ππ < 100 MeV even at mσ =

800 MeV [Leutwyler: mσ – iΓσ /2 = (441-i272) MeV] Γρ→ππ = 86.5 MeV [PDG: Γρ→ππ = (149.4 ± 1.0) MeV] Decay widths for sigma and rho do not match

experiments How to improve decay widths? [D. Parganlija, F. Giacosa

and D. Rischke, AIP Conf. Proc. 1030, 160 (2008) ]


Improvements of the Lagrangian

Strategy 1: Higher-dimension terms (local invariance)

Strategy 2: Non-vanishing meson masses

becomes global QCD chiral symmetry: global LSM chiral symmetry global

[S. Gasiorowicz and D. A. Geffen (1969), U. G. Meissner (1988), P. Ko and S. Rudaz (1994)]

LR UU )2()2(

[M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)]

[D. Parganlija, F. Giacosa and D. Rischke, AIP Conf. Proc. 1030, 160 (2008) ][D. Parganlija, F. Giacosa and D. Rischke, arXiv:0812.2183 [hep-ph] ]


Parameter Determination k Three Parameters: Z, mσ and h1

Find global minimum of Look for per D.O.F. at minimum Parameter errors: diagonal elements of at global minimum for all parameters

)(gMeV )0.14.149( 22 Zg )(MeV )097.2748.8( 2201Zhhaf

2,0

2

)exp.(0

)exp.(0

)th.(0

decays all

2

)exp.(

)exp.()th.(

12 :),,(

ii

ii

a

aamhZ

22 1

1

22

ji pp

},,{ 1, hmZp ji


s-wave, isospin-two scattering length


Renormalisation Constant Z

2221

2

2

2

22

1

11

fZgmm

mm

Za

aa

22

1

2221

42

24

111

fg

mfgmmZ aaa

656.62

1max1

fm

g a

Documents

Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia Nature of Light Scalar Mesons f 0 (600), a 0 (980), f 0 (1370) and a 0 (1450)