Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Denis Parganlija
In collaboration withFrancesco Giacosa and Dirk H. Rischke
(Frankfurt)Gyรถrgy Wolf and Pรฉter Kovรกcs
(Budapest)
Institut fรผr Theoretische Physik Technische Universitรคt Wien, Goethe-Universitรคt Frankfurt
[Based on: Phys.Rev. D82 (2010) 054024 (arXiv:1003.4934)
Int.J.Mod.Phys. A26 (2011) 607-609 (arXiv:1009.2250)
and PhD Thesis of D. Parganlija (2012)]
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Introduction:Definitions and Experimental Data
Mesons: quark-antiquark statesQuantum numbers: JPC
Scalar mesons: JPC = 0++ [ฯ or f0(600), a0(980), a0(1450)โฆ]
Pseudoscalar mesons: JPC = 0-+ [ฯ, K, ฮท, ฮทยดโฆ]Vector mesons: JPC = 1-- [ฯ, K*, ฯ, ฯ(1020)โฆ]Axial-Vector mesons: JPC = 1++ [a1(1260), f1(1285), K1(1270), K1(1400)โฆ]
Total Spin Parity Charge Conjugation
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Motivation: PDG Data on JPC = 0++ Mesons
Six states up to 1.8 GeV (isoscalars)ss
qqqq meson-mesonstate boundGlueball
State Mass [MeV] Width [MeV]f0(600) 400 - 1200 600 - 1000f0(980) 980 ยฑ 10 40 - 100f0(1370) 1200 - 1500 200 - 500f0(1500) 1505 ยฑ 6 109 ยฑ 7f0(1710) 1720 ยฑ 6 135 ยฑ 8f0(1790) 40
301790
6030270
dduunn
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Motivation: Reasons to Consider Mesons
Mesons: hadronic states with integer spinMore scalar mesons than predicted by quark-
antiquark picture โ Classification neededLook for tetraquarks, glueballsโฆ
Walecka Model: Nucleon-nucleon interaction via ฯ meson
Restoration of chiral invariance and decofinement โ Degeneration of chiral partners ฯ and ฯ โ ฯ has to be a quarkonium
Identify the scalar quarkonia โ Need a model with scalar and other states
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
An Effective Approach:Linear Sigma Model
Implements features of QCD: SU(Nf)L x SU(Nf)R Chiral SymmetryExplicit and Spontaneous Chiral Symmetry
Breaking; Chiral U(1)A AnomalyVacuum calculations โ calculations at Tโ 0Chiral-Partners degeneration above TC โ order
parameter for restoration of chiral symmetry
The model here: Nf = 3 (mesons with u, d, s quarks) in scalar, pseudoscalar, vector and axial-vector channels
โ extended Linear Sigma Model - eLSM
Vacuum spectroscopy of quark-antiquark states
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
ฮท, ฮท'
๐ฒโโก ๐ฒโ(๐๐๐) ๐โก ๐(๐๐๐)
ฯN โก ฯ(782) = ๐เดฅ๐ ฯS โก ๐(1020) = ๐เดค๐
Resonances I
S
N
N
KK
K
K
P
0
00
0
2
2
21
S
N
N
KK
K
K
V0
00
0
2
2
21
Pseudoscalars
Vectors
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
f1N โก f1(1285) = ๐เดฅ๐
๐ฒ๐ โก เต๐ฒ๐(๐๐๐๐)๐ฒ๐(๐๐๐๐)
a1 โก a1(1260) f1S โก f1(1420) = ๐เดค๐
เต๐๐ตโก ๐เดฅ๐๐๐บโก ๐เดค๐ โแ
๐๐๐ณ โก ๐ฉ๐ซ๐๐๐จ๐ฆ๐ข๐ง๐๐ง๐ญ๐ฅ๐ฒ ๐เดฅ๐๐๐๐ฏโก ๐ฉ๐ซ๐๐๐จ๐ฆ๐ข๐ง๐๐ง๐ญ๐ฅ๐ฒ ๐เดค๐
Resonances II
S
N
N
fKK
Kaf
a
Kaaf
A
1011
01
011
1
11
011
2
2
21
SSS
SN
SN
KK
Ka
a
Kaa
S
0
000
0
0
00
2
2
21
Axial-Vectors
Scalars
๐๐ = เต๐๐(๐๐๐)๐๐(๐๐๐๐)
๐ฒ๐บ= เต๐ฒ๐โแบ๐๐๐แป / ๐ฟ๐ฒ๐โ(๐๐๐๐)
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
The Lagrangian IScalars and Pseudoscalars
])(det4)det [(det)]([Tr โ 2โ โ cH
)(1 LRigD
2โ 2
2 โ 1
โ 2 0
โ SP )(Tr )](Tr [)(Tr )]()[(Tr mDDL
Explicit Symmetry Breaking Chiral Anomaly
SSS
SN
SN
KK
K
K
S
0
000
0
0
00
2
2
21 aa
aa
S
N
N
KK
K
K
P
0
00
0
2
2
21 PS i
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
The Lagrangian IIVectors and Axial-Vectors
)(
2Tr)(Tr
41 22
2 1 22
VA RLmRLL)]},[Tr{]},[{Tr (2 2
RRRLLLig
LLL
RRR
)()(
)(
2
2,
2,
ss
dun
dun
mm
m
S
N
N
KK
K
K
V0
00
0
2
2
21
S
N
N
fKK
Kaf
a
Kaaf
A
1011
01
011
1
11
011
2
2
21 AVL
AVR
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Sigma Model Lagrangian with Vector and Axial-Vector Mesons (Nf = 3)
INT. VA SP LLLL
More (Pseudo)scalar โ (Axial-)Vector Interactions
Perform Spontaneous Symmetry Breaking (SSB): ฯN โ ฯN + ฯN, ฯS โ ฯS + ฯS
18 parameters, 10 independent, none free โ fixed via fit of masses and decay widths/amplitudes
])()[(Tr )(Tr )(Tr 2
222
22โ 1INT. RLhRL
hL
)(Tr โ LRh 32
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Isospin 1
Isospin ยฝ
Isospin 0 (Isoscalars)
๐๐ = เต๐๐(๐๐๐)๐๐(๐๐๐๐)
๐ฒ๐บ= เต๐ฒ๐โแบ๐๐๐แป / ๐ฟ๐ฒ๐โ(๐๐๐๐)
Possible Assignments
เต๐๐ตโก ๐เดฅ๐๐๐บโก ๐เดค๐ โแ
๐๐๐ณ โก ๐ฉ๐ซ๐๐๐จ๐ฆ๐ข๐ง๐๐ง๐ญ๐ฅ๐ฒ ๐เดฅ๐๐๐๐ฏโก ๐ฉ๐ซ๐๐๐จ๐ฆ๐ข๐ง๐๐ง๐ญ๐ฅ๐ฒ ๐เดค๐
๐๐แบ๐๐๐แป ๐๐แบ๐๐๐แป ๐๐แบ๐๐๐๐แป ๐๐แบ๐๐๐๐แป ๐๐แบ๐๐๐๐แป
Check all possibilities
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Best Fit
เต๐๐(๐๐๐๐) ๐ฉ๐ซ๐๐๐จ๐ฆ๐ข๐ง๐๐ง๐ญ๐ฅ๐ฒ ๐เดฅ๐๐๐แบ๐๐๐๐แป ๐ฉ๐ซ๐๐๐จ๐ฆ๐ข๐ง๐๐ง๐ญ๐ฅ๐ฒ ๐เดค๐
}
๐๐แบ๐๐๐๐แป/ ๐ฒ๐แบ๐๐๐๐แป ๐เดฅ๐ ๐ฌ๐ญ๐๐ญ๐๐ฌ ๐๐ โ ๐๐ฅ๐ฎ๐จ๐ง ๐๐จ๐ง๐๐๐ง๐ฌ๐๐ญ๐ +๐๐ฎ๐๐ซ๐ค ๐๐จ๐ง๐๐๐ง๐ฌ๐๐ญ๐; Gluon Condensate dominant ๐โ ๐โฒ mixing angle ~ 45ยฐ
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
No fit with ๐๐แบ๐๐๐แป and ๐๐แบ๐๐๐แป as ๐เดฅ๐ states No fit with ๐ฒ๐โแบ๐๐๐แป as ๐เดฅ๐ state No reasonable fit with ๐๐แบ๐๐๐แป and ๐๐แบ๐๐๐๐แป as ๐เดฅ๐ states โ ๐๐ฒ๐โ ~ ๐.๐ GeV or ๐๐๐ ~ ๐.๐ GeV
What We Did Not Find
แ๐๐ฒ๐โแบ๐๐๐แป / ๐ฟ= แบ๐๐๐ ยฑ ๐๐แป ๐๐๐ ๐๐ฒ๐โ(๐๐๐๐) = แบ๐๐๐๐ยฑ ๐๐แป ๐๐๐
Thus: scalar ๐เดฅ๐ states above 1 GeV โ ๐๐แบ๐๐๐๐แป predominantly ๐เดฅ๐ โ ๐๐แบ๐๐๐๐แป predominantly ๐เดค๐
แ๐๐๐แบ๐๐๐แป= แบ๐๐๐ยฑ ๐๐แป ๐๐๐ ๐๐๐(๐๐๐๐) = แบ๐๐๐๐ยฑ ๐๐แป ๐๐๐
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Linear Sigma Model with Nf = 3 and vector and axial-vector mesons โ eLSM
Predominantly scalar states above 1 GeV:
Axial-Vectors a1 and K1 seen as states
๐เดฅ๐ ๐๐แบ๐๐๐๐แป, ๐๐แบ๐๐๐๐แป, ๐๐แบ๐๐๐๐แป, ๐ฒ๐โแบ๐๐๐๐แป ๐เดฅ๐
Summary
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Summary: Results onJPC = 0++ Mesons
State Mass [MeV] Width [MeV]
f0(600) 400 - 1200 600 - 1000
f0(980) 980 ยฑ 10 40 - 100
f0(1370) 1200 - 1500 200 - 500
f0(1500) 1505 ยฑ 6 109 ยฑ 7
f0(1710) 1720 ยฑ 6 135 ยฑ 8
nn tlypredominan
ss tlypredominan
glueball tlypredominan
[S. Janowski, D. Parganlija, F. Giacosa and D. H. Rischke, PR D 84 (2011) 054007]
?tetraquark
?tetraquark
Axial-vectors (๐๐,๐ฒ๐): ๐เดฅ๐
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
OutlookLagrangian With Three Flavours +
Glueball + TetraquarksMixing in the Scalar Sector: Quarkonia,
Tetraquarks and GlueballFour FlavoursExtension to Non-Zero Temperatures
and DensitiesInclude Tensor, Pseudotensor Mesons,
Baryons (Nucleons)
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Spare Slides
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Quantum Chromodynamics (QCD)QCD Lagrangian
Symmetries of the QCD Lagrangian
Local SU(3)c Colour Symmetry
Global Chiral U(Nf)x U(Nf) Symmetry
CPT Symmetry
Z Symmetry
Trace Symmetry
aa
fff m GGqDiqQCD 41)( L
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Chiral Symmetry of QCDLeft-handed and right-handed quarks:
Chirality Projection Operators
Transform quark fields
Quark part of the QCD Lagrangian:
fffff qqqqq RLRLRL ,, ; P
21 5
,
RLP
LRRLRRLLQCD qqqqqDqqDq quarksii ffffffffff mm L
invariantChiral Symmetry
Explicit Breaking of the Chiral Symmetry
1,...,0 ,e 2
i
'
fffff NjqqUqq Lt
LLLLjj
L
Rt
RRRR qqUqqjj
R
i
' e ffff
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
ff qtq jฮผฮผ 5j 1
current cons. a
Chiral CurrentsNoether Theorem:
Vector current Vฮผ = (Lฮผ + Rฮผ)/2Axial-vector current Aฮผ = (Lฮผ - Rฮผ)/2Vector transformation of
Axial-vector Transformation offfff
f
qqUqq
N
j
jjV t
V
12
0i
' e
quarksQCDL
ff qtqV jฮผฮผ j current cons. ฯ(770)-like
quarksQCDL
ffff
f
qqUqq
N
j
jjA t
A
12
0
5i' e ff qtqA jฮผฮผ 5j current cons.
a1(1260)-like
ff qtqฯ jฮผjฮผ current cons. }3,2,1{j
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Spontaneous Breaking of Chiral Symmetry
Transform the (axial-)vector fields
Chiral Anomaly
ฮผV
ฮผฮผ
ฮผV
ฮผฮผ
ฮฑฯฮฑฯฯ
11Vector
1
Vector
aaa
ฮผA
ฮผฮผ
ฮผA
ฮผฮผ
ฯฮฑฮฑฯฯ
1Axial
1
1Axial
aaa
Theory: ฯ and a1 should degenerate
Experiment:ฯmma 2
1
Spontaneous Breaking of the Chiral Symmetry (SSB)โ Goldstone Bosons (pions, kaonsโฆ)
yclassicall ,0eSinglet Axial
quarks
050i
fm ฮผ
ฮผQCD AtAฮฑ
Llevel quantumat ,~
00ฮผฮฝ
ฮผฮฝฮผ
ฮผ GGNA aa
fmf
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
ASAA
AAN
AAN
fKK
Kaf
a
Kaaf
A
,10,1,1
0,1
01,1
1
,11
01,1
2
2
21
BSBB
BBN
BBN
fKK
Kbf
b
Kbbf
B
,10,1,1
0,1
01,1
1
,11
01,1
2
2
21
)()(
)(
2
2,
2,
ss
dun
dun
mm
m
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Sigma Model Lagrangian with Vector and Axial-Vector Mesons (Nf = 3)
Scalars and Pseudoscalars
])(det4)det [(det)]([Tr โ 2โ 1
โ cH
)(1 LRigD
2โ 2
2 โ 1
โ 2 0
โ SP )(Tr )](Tr [)(Tr )]()[(Tr mDDL
Explicit Symmetry Breaking Chiral Anomaly
GeV? 1 AboveGeV? 1Under states? qqscalar are Where
SSS
SN
SN
KK
Ka
a
Kaa
S
0
000
0
0
00
2
2
21
S
N
N
KK
K
K
P
0
00
0
2
2
21 PS i
nn
ss
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Sigma Model Lagrangian with Vector and Axial-Vector Mesons (Nf = 3)
Vectors and Axial-Vectors
)(
2Tr)(Tr
41 22
2 1 22
VA RLmRLL)]},[Tr{]},[{Tr (2 2
RRRLLLig
LLL
RRR
)()(
)(
2
2,
2,
ss
dun
dun
mm
m
S
N
N
KK
K
K
V0
00
0
2
2
21
S
N
N
fKK
Kaf
a
Kaaf
A
1011
01
011
1
11
011
2
2
21 AVL
AVR
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Motivation: QCD Features in an Effective Model
QCD Lagrangian
Chirality Projection Operators
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Motivation: QCD Features in an Effective Model
Global Unitary Transformations
invariant not invariant
Chiral Symmetry Explicit Symmetry BreakingSpontaneously Broken in
Vacuum In addition:Chiral U(1)A Anomaly
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Motivation: Structure of Scalar MesonsSpontaneous Breaking of Chiral Symmetry
โ Goldstone Bosons (Nf = 2 โ ฯ)Restoration of Chiral Invariance and
Deconfinement โ Degeneration of Chiral Partners (ฯ/ฯ)
Nature of scalar mesonsScalar states under 1 GeV โ f0(600),
a0(980) โ not preferred by Nf = 2 resultsScalar states above 1 GeV โ f0(1370),
a0(1450) โ preferred by Nf = 2 results
f0(600), โsigmaโ f0(1370)
[Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010; arXiv: 1003.4934]
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Calculating the ParametersShift the (axial-)vector fields:
Canonically normalise pseudoscalars and KS:
Perform a fit of all parameters except g2 (fixed via ฯ โ ฯฯ)
9 parameters, none free โ fixed via masses
NfNN Nwff
111
111 awaaSfSS S
wff 111
KwKK K
111SK
KwKK
SNSN SNZ ,, ,
Z KZK K SKS KZKS
Kmm , SNmmmm ,, ' *, Kmm
)1420()1020( 11, ff mmmm
SS
)1270()1260( 1111, KKaa mmmm
)1430()1450(000
, KKaa mmmm
S
[Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024,
2010; arXiv: 1003.4934]withfit no :yPreliminar
GeV 1 GeV, 10
SKa mm
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Other Resultsฮท โ ฮทโ mixing angle ฮธฮท = 43.9ยฐ โ KLOE Collaboration: ฮธฮท = 41.4ยฐ ยฑ 0.5ยฐRho meson mass has two contributions:
K* โ Kฯ Data: 48.7 MeV Our value: 44.2 MeVฯ(1020) โ K+ K-
Data: 2.08 MeV Our value: 2.33 MeV
21321
221
2
2][
2 SN hhhhmm
~ Gluon Condensate Quark CondensatesMeV 761 obtain We 1 m
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Note: Nf = 2 LimitThe f0(600) state not preferred to be
quarkonium
[Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010;
arXiv: 1003.4934]
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Note: Nf = 2 Limit
MeV )500200(
MeV, )15001200( :PDG
)1370(
)1370(
0
0
f
fm
qqf tlypredominan as )1370( favours data alExperiment 0
[Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010;
arXiv: 1003.4934]
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario II (Nf =2): Scattering Lengths
Scattering lengths saturated
Additional scalars: tetraquarks, quasi-molecular states
Glueball
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario II (Nf =2): Parameter Determination
Masses:Pion Decay Constant Five Parameters: Z, h1, h2, g2, mฯ
10,,,, aa mmmmm
Zf
)(MeV 1.0)(149.4 22 Zgg )(MeV )13265( 22)1450(0
Zhha
ZZa MeV )246.0640.0(][1
)small ( 0 3,21 hh
free )1370(0fmm
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario I (Nf =2): Other Results
Our Result Experimental Value
MeV 640.01
a
exact ],[ ],,[ 22 01hZgZฮ af
MeV 640.01
a(NA48/2) 218.00
0 a218.000 a
0454.020 a (NA48/2) 0457.02
0 a
deg 5.041.4 :angle mixing ฮท'ฮท[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-
ฮท'ฮท
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario I (Nf =2): a1โฯฯ Decay m1 = 0 โ mฯ generated from the quark condensate only;
our result: m1 = 652 MeV a1โฯฯ
MeV )600250(:PDG (total) 1a
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Comparison: the Model with and without Vectors and Axial-Vectors (Nf=2)
decrease values
vectors Include
Note: other observables (ฯฯ scattering lengths, a0(980)โฮทฯ decay amplitude, phenomonology of a1, and others) are fine [Parganlija, Giacosa, Rischke, Phys. Rev. D 82: 054024, 2010]
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario I (Nf =2): a1 โ ฯฯ Decay
MeV )600250(:PDG
[M. Urban, M. Buballa and J. Wambach, Nucl. Phys. A 697, 338 (2002)]
MeV 500 ifMeV 160 m
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario I (Nf =2) : Parameter Determination
Three Independent Parameters: Z, m1, mฯ
MeV )246.0640.0(][1
Za
][ 020.0218.0],,[ 11
00
mmmZa
Isospin
Angular Momentum (s wave)[NA48/2 Collaboration, 2009]
MeV 652 1236521
m)]()([
2 321
221
2 ZhZhhmm
~ Gluon Condensate Quark Condensate
0.201.67Z
MeV ]477 ,288[m[S. Janowski (Frankfurt U.), Diploma Thesis, 2010]
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Lagrangian of a Linear Sigma Model with Vector and Axial-Vector Mesons (Nf =2)
vectors
axialvectors
Vectors and Axial-Vectors])()[(Tr
2])()[(Tr
41 22
2 1 22
VA RL
mRL
L
)]},[Tr{]},[{Tr (2 2
RRRLLLig }],]){[],[[(Tr {2 33
3
LL,LtieALLtieALg
]),[],[( 33 LtieALtieALLL ]),[],[( 33 RtieARtieARRR
}}],){],[],[[(Tr 33 RRRtieARRtieAR
)()(
)(
2
2,
2,
ss
dun
dun
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m
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Lagrangian of a Linear Sigma Model with Vector and Axial-Vector Mesons (Nf =2)Scalars and Pseudoscalars
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scalarsExplicit Symmetry Breaking Chiral Anomaly
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