Scalar and Axial-Vector Mesons in a Three- Flavour Sigma Model

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)]


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


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


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


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


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


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)


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


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


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


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


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°


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


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


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


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)


Spare Slides


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


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


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


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


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



Scalars and Pseudoscalars

])(det4)det [(det)]([Tr †2†1

† cH

)(1 LRigD

2†2

2 †1

†2 0


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



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


Motivation: QCD Features in an Effective Model

QCD Lagrangian

Chirality Projection Operators


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


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

qq

qq

f0(600), „sigma“ f0(1370)

[Parganlija, Giacosa, Rischke in Phys. Rev. D 82: 054024, 2010; arXiv: 1003.4934]


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


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


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]


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]


Scenario II (Nf =2): Scattering Lengths

Scattering lengths saturated

Additional scalars: tetraquarks, quasi-molecular states

Glueball


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


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


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


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]


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


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]


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


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


pseudoscalars

scalarsExplicit Symmetry Breaking Chiral Anomaly

photon})1450(),1370({or )}980(),600({},{ 00000 afafa state? scalar the is Where qq

Documents

Scalar and Axial-Vector Mesons in a Three- Flavour Sigma Model