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Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia 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 → f 0 (600), a 0 (980) Scalar states above 1 GeV → f 0 (1370), a 0 (1450) – additional scalar states under 1 GeV required (tetraquarks?) f 0 (600), „sigma“f 0 (1370)
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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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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?)
f0(600), „sigma“ f0(1370)
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Scenario I: Scalars under 1 GeV
Nf = 2Scalars → Pseudoscalars →Vectors → Axialvectors →
(980)),600( 00 af
,)770(),782(
(1260),(1285) 11 af
scalarspseudoscalarsvectors
axialvectors
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Lagrangian of a Linear Sigma Model with Vector and Axialvector Mesons (Nf=2)
m12 parameters
photon
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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]
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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.]
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Scenario II: Scalars above 1 GeV
Nf = 2Scalars → Pseudoscalars →Vectors → Axialvectors →
(1450)),1370( 00 af
,)770(),782(
(1260),(1285) 11 af
scalarspseudoscalarsvectors
axialvectors
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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)]
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Scenario II – Scattering Lengths Scattering lengths saturated
Additional scalars: tetraquarks, quasi-molecular states
Glueball
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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]]
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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)
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Spare Slides
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Comparison: the Model with and without Vectors and Axial-Vectors
decrease values
vectors Include
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Relevant Vertices for Pion-Pion Scattering
Feynman Diagrams At Tree Level:
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Tree-Level Scattering Amplitude
s-channel isospin I components of the scattering amplitude:
scattering lengths (on the threshold):partial wave amplitude
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Results I
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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)
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
Results I: as function of
E865
m
0150047020 ..a NA48/2 Ke4
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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) ]
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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] ]
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
s-wave, isospin-two scattering length
Denis Parganlija (Frankfurt U.) Excited QCD 2010, Tatranska Lomnica/Slovakia
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