scalar and axial-vector mesons in a three- flavour sigma model
Post on 22-Feb-2016
50 Views
Preview:
DESCRIPTION
TRANSCRIPT
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)György Wolf and Péter Kovács
(Budapest)
Institut für Theoretische Physik Technische Universität Wien, Goethe-Universität 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++ [σ or f0(600), a0(980), a0(1450)…]
Pseudoscalar mesons: JPC = 0-+ [π, K, η, η´…]Vector mesons: JPC = 1-- [ρ, K*, ω, φ(1020)…]Axial-Vector mesons: JPC = 1++ [a1(1260), f1(1285), K1(1270), K1(1400)…]
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 ± 10 40 - 100f0(1370) 1200 - 1500 200 - 500f0(1500) 1505 ± 6 109 ± 7f0(1710) 1720 ± 6 135 ± 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 → Classification neededLook for tetraquarks, glueballs…
Walecka Model: Nucleon-nucleon interaction via σ meson
Restoration of chiral invariance and decofinement ↔ Degeneration of chiral partners π and σ → σ has to be a quarkonium
Identify the scalar quarkonia → 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 → calculations at T≠0Chiral-Partners degeneration above TC → 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
→ 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
η, η'
𝑲∗≡ 𝑲∗(𝟖𝟗𝟐) 𝝆≡ 𝝆(𝟕𝟕𝟎)
ωN ≡ ω(782) = 𝒏ഥ𝒏 ωS ≡ 𝝋(1020) = 𝒔ത𝒔
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 ≡ f1(1285) = 𝒏ഥ𝒏
𝑲𝟏 ≡ ൜𝑲𝟏(𝟏𝟐𝟕𝟎)𝑲𝟏(𝟏𝟒𝟎𝟎)
a1 ≡ a1(1260) f1S ≡ f1(1420) = 𝒔ത𝒔
൜𝝈𝑵≡ 𝒏ഥ𝒏𝝈𝑺≡ 𝒔ത𝒔 →ቊ
𝒇𝟎𝑳 ≡ 𝐩𝐫𝐞𝐝𝐨𝐦𝐢𝐧𝐚𝐧𝐭𝐥𝐲 𝒏ഥ𝒏𝒇𝟎𝑯≡ 𝐩𝐫𝐞𝐝𝐨𝐦𝐢𝐧𝐚𝐧𝐭𝐥𝐲 𝒔ത𝒔
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
𝒂𝟎 = ൜𝒂𝟎(𝟗𝟖𝟎)𝒂𝟎(𝟏𝟒𝟓𝟎)
𝑲𝑺= ൜𝑲𝟎∗ሺ𝟖𝟎𝟎ሻ / 𝜿𝑲𝟎∗(𝟏𝟒𝟑𝟎)
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 †2†† cH
)(1 LRigD
2†2
2 †1
†2 0
†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 – (Axial-)Vector Interactions
Perform Spontaneous Symmetry Breaking (SSB): σN → σN + ϕN, σS → σS + ϕS
18 parameters, 10 independent, none free → fixed via fit of masses and decay widths/amplitudes
])()[(Tr )(Tr )(Tr 2
222
22†1INT. RLhRL
hL
)(Tr † LRh 32
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Isospin 1
Isospin ½
Isospin 0 (Isoscalars)
𝒂𝟎 = ൜𝒂𝟎(𝟗𝟖𝟎)𝒂𝟎(𝟏𝟒𝟓𝟎)
𝑲𝑺= ൜𝑲𝟎∗ሺ𝟖𝟎𝟎ሻ / 𝜿𝑲𝟎∗(𝟏𝟒𝟑𝟎)
Possible Assignments
൜𝝈𝑵≡ 𝒏ഥ𝒏𝝈𝑺≡ 𝒔ത𝒔 →ቊ
𝒇𝟎𝑳 ≡ 𝐩𝐫𝐞𝐝𝐨𝐦𝐢𝐧𝐚𝐧𝐭𝐥𝐲 𝒏ഥ𝒏𝒇𝟎𝑯≡ 𝐩𝐫𝐞𝐝𝐨𝐦𝐢𝐧𝐚𝐧𝐭𝐥𝐲 𝒔ത𝒔
𝒇𝟎ሺ𝟔𝟎𝟎ሻ 𝒇𝟎ሺ𝟗𝟖𝟎ሻ 𝒇𝟎ሺ𝟏𝟑𝟕𝟎ሻ 𝒇𝟎ሺ𝟏𝟓𝟎𝟎ሻ 𝒇𝟎ሺ𝟏𝟕𝟏𝟎ሻ
Check all possibilities
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Best Fit
൜𝒇𝟎(𝟏𝟑𝟕𝟎) 𝐩𝐫𝐞𝐝𝐨𝐦𝐢𝐧𝐚𝐧𝐭𝐥𝐲 𝒏ഥ𝒏𝒇𝟎ሺ𝟏𝟕𝟏𝟎ሻ 𝐩𝐫𝐞𝐝𝐨𝐦𝐢𝐧𝐚𝐧𝐭𝐥𝐲 𝒔ത𝒔
}
𝒂𝟏ሺ𝟏𝟐𝟔𝟎ሻ/ 𝑲𝟏ሺ𝟏𝟐𝟕𝟎ሻ 𝒒ഥ𝒒 𝐬𝐭𝐚𝐭𝐞𝐬 𝒎𝝆 ↔ 𝐆𝐥𝐮𝐨𝐧 𝐂𝐨𝐧𝐝𝐞𝐧𝐬𝐚𝐭𝐞 +𝐐𝐮𝐚𝐫𝐤 𝐂𝐨𝐧𝐝𝐞𝐧𝐬𝐚𝐭𝐞; Gluon Condensate dominant 𝛈− 𝛈′ mixing angle ~ 45°
Denis Parganlija (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
No fit with 𝒇𝟎ሺ𝟔𝟎𝟎ሻ and 𝒇𝟎ሺ𝟗𝟖𝟎ሻ as 𝒒ഥ𝒒 states No fit with 𝑲𝟎∗ሺ𝟖𝟎𝟎ሻ as 𝒒ഥ𝒒 state No reasonable fit with 𝒇𝟎ሺ𝟔𝟎𝟎ሻ and 𝒇𝟎ሺ𝟏𝟑𝟕𝟎ሻ as 𝒒ഥ𝒒 states → 𝒎𝑲𝟎∗ ~ 𝟏.𝟏 GeV or 𝒎𝒂𝟎 ~ 𝟏.𝟐 GeV
What We Did Not Find
ቊ𝒎𝑲𝟎∗ሺ𝟖𝟎𝟎ሻ / 𝜿= ሺ𝟔𝟕𝟔 ± 𝟒𝟎ሻ 𝐌𝐞𝐕 𝒎𝑲𝟎∗(𝟏𝟒𝟑𝟎) = ሺ𝟏𝟒𝟐𝟓± 𝟓𝟎ሻ 𝐌𝐞𝐕
Thus: scalar 𝒒ഥ𝒒 states above 1 GeV → 𝒇𝟎ሺ𝟏𝟑𝟕𝟎ሻ predominantly 𝒏ഥ𝒏 → 𝒇𝟎ሺ𝟏𝟕𝟏𝟎ሻ predominantly 𝒔ത𝒔
ቊ𝒎𝒂𝟎ሺ𝟗𝟖𝟎ሻ= ሺ𝟗𝟖𝟎± 𝟐𝟎ሻ 𝐌𝐞𝐕 𝒎𝒂𝟎(𝟏𝟒𝟓𝟎) = ሺ𝟏𝟒𝟕𝟒± 𝟏𝟗ሻ 𝐌𝐞𝐕
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 – eLSM
Predominantly scalar states above 1 GeV:
Axial-Vectors a1 and K1 seen as states
𝒒ഥ𝒒 𝒇𝟎ሺ𝟏𝟑𝟕𝟎ሻ, 𝒇𝟎ሺ𝟏𝟕𝟏𝟎ሻ, 𝒂𝟎ሺ𝟏𝟒𝟓𝟎ሻ, 𝑲𝟎∗ሺ𝟏𝟒𝟑𝟎ሻ 𝒒ഥ𝒒
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 ± 10 40 - 100
f0(1370) 1200 - 1500 200 - 500
f0(1500) 1505 ± 6 109 ± 7
f0(1710) 1720 ± 6 135 ± 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 (𝒂𝟏,𝑲𝟏): 𝒒ഥ𝒒
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 jμμ 5j 1
current cons. a
Chiral CurrentsNoether Theorem:
Vector current Vμ = (Lμ + Rμ)/2Axial-vector current Aμ = (Lμ - Rμ)/2Vector transformation of
Axial-vector Transformation offfff
f
qqUqq
N
j
jjV t
V
12
0i
' e
quarksQCDL
ff qtqV jμμ j current cons. ρ(770)-like
quarksQCDL
ffff
f
qqUqq
N
j
jjA t
A
12
0
5i' e ff qtqA jμμ 5j current cons.
a1(1260)-like
ff qtqρ jμjμ 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
μV
μμ
μV
μμ
αραρρ
11Vector
1
Vector
aaa
μA
μμ
μA
μμ
ρααρρ
1Axial
1
1Axial
aaa
Theory: ρ and a1 should degenerate
Experiment:ρmma 2
1
Spontaneous Breaking of the Chiral Symmetry (SSB)→ Goldstone Bosons (pions, kaons…)
yclassicall ,0eSinglet Axial
quarks
050i
fm μ
μQCD AtAα
Llevel quantumat ,~
00μν
μνμ
μ 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 †2†1
† cH
)(1 LRigD
2†2
2 †1
†2 0
†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
→ Goldstone Bosons (Nf = 2 → π)Restoration of Chiral Invariance and
Deconfinement ↔ Degeneration of Chiral Partners (π/σ)
Nature of scalar mesonsScalar states under 1 GeV → f0(600),
a0(980) – not preferred by Nf = 2 resultsScalar states above 1 GeV → f0(1370),
a0(1450) – preferred by Nf = 2 results
f0(600), „sigma“ 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 ρ → ππ)
9 parameters, none free → 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 Resultsη – η’ mixing angle θη = 43.9° ↔ KLOE Collaboration: θη = 41.4° ± 0.5°Rho meson mass has two contributions:
K* → Kπ Data: 48.7 MeV Our value: 44.2 MeVφ(1020) → 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, mσ
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 01hZgZΓ 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 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 (TU Vienna / GU Frankfurt) Scalar and Axial-Vector Mesons in a Three-Flavour Sigma Model
Scenario I (Nf =2): a1→σπ Decay m1 = 0 → mρ generated from the quark condensate only;
our result: m1 = 652 MeV a1→σπ
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 (ππ scattering lengths, a0(980)→ηπ 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 → 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 (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, mσ
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
mm
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
])(det4)det [(det)]([Tr †2†† cH
],[)( 31 tieALRigD
2†2
2 †1
†2 0
†SP )(Tr )](Tr [)(Tr )]()[(Tr mDDL
pseudoscalars
scalarsExplicit Symmetry Breaking Chiral Anomaly
photon})1450(),1370({or )}980(),600({},{ 00000 afafa state? scalar the is Where qq
top related