Gallas The chiral partner in the mirror assignmentexcitedqcd/Talks09/Gallas_ZakopaneExQCD09.pdfThe...
Transcript of Gallas The chiral partner in the mirror assignmentexcitedqcd/Talks09/Gallas_ZakopaneExQCD09.pdfThe...
The chiral partner of the nucleon
in the mirror assignment
Susanna Gallas
in collaboration with:
Francesco Giacosa and Dirk Rischke
Excited QCD8-14 February 2009- Zakopane (Poland)
])()[(2
])()[(4
1
)]([)]det()[det(
)()]([][)])([(
22
2
122
2
2
2
1
2
µµν
µ
µµν
µ λλ
RLTrm
RLTr
HTrc
TrTrTrmDDTrLmes
+++
−Φ+Φ+Φ+Φ
+ΦΦ−ΦΦ−ΦΦ−ΦΦ=
++
+++
)(ˆLRRL
RRRRLLLL
g
RgiLgi
ΨΦΨ+ΦΨΨ−
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ
+
µµ
µµ
µµ
µµ γγγγ
Talk 09.02.09 by D. Parganlija:
+ Baryons:
Motivation
Introduction of N* for chiral realization of the nucleonic Lagrangian
z chiral invariant mass term =/ m0
Correct description of the axial coupling constantgA(N)=1.26
gA(N*)=?
Generation of the nucleonic mass: how is mN generated?
Quark condensate vs. m0
Study of pion-nucleon scattering
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
The chiral partner:
what is N*?
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Mirror assignment vs naive assignment: chiral transformation
LRL
LLL
U
U
22
11
Ψ→Ψ
Ψ→Ψ
Mirror
RLR
RRR
U
U
22
11
Ψ→Ψ
Ψ→Ψ
Naive
RRR
RRR
U
U
22
11
Ψ→Ψ
Ψ→Ψ
LLL
LLL
U
U
22
11
Ψ→Ψ
Ψ→Ψ
⇓
)( 121221210 RLLRLRRLm ΨΨ+ΨΨ−ΨΨ−ΨΨ−
Nucleon mass not only
dynamically generated
No chiral invariant mass term
⇓
2 independent Linear Sigma models
RLRL SUSUUU )2()2( ×∈×
22
*
2
1
1ˆxM
N
N
=
=Ψ
=Ψ
Ψ
Ψ=
2
1
*M̂
N
N221ˆ
xM ≠
The chiral partner of the nucleonin the mirror assignment
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
The chiral partner of the nucleonin the mirror assignment
Candidates with the correct quantum numbers from PDG:
N*(1535) lowest nucleonic resonance
N*(1650) −
=2
1PJ
Other possibility (speculative):
N*(1200) (D. Zschiesche, L. Tolos, J. Schaffner-Bielich, R. Pisarski, Phys.Rev.C75:055202,2007)
Very broad resonance? Not yet measured?
A little bit of scattering theory
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
A little bit of scattering theory
cbacabba iBqqABqqAT τεγδγ µµµ
µµµ
+++
++= −−++ )(
21
)()(
21
)( )(2
1)(
2
1
General form of the Lorentz-invariant scattering amplitude: )()( 12 puTpu ba
)()(
0
)(
0
)(
0
±±±+= BmAa πηs-wave scattering lengths: )1(4
1
Nm
mππη
+
=
22 )(,)(,0 ππ mmummst NN −=+==
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Some important data
H. Schroder: Measured pion nucleon scattering lengths from pionic
hydrogen and deuterium at PSI (Eur.Phys.J.C21:473-488,2001)
14)(
0
16)(
0
10)1146.041256.6(
10)165.785783.8(
−−−
−−+
±=
±−=
MeVa
MeVa
T. Takahashi and T. Kunihiro: Axial charge of N(1535) in lattice
QCD (e-Print: arXiv:0801.4707)
3.02.0*
±=N
Ag
L. Ya. Glozman: Phys.Rept.444:1-49,2007
resultdifferent apresent shall we
1partner theis if
partner not the is 0*
*
*
*
−≈→
→≈
N
A
N
A
gN
Ng
The model
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Linear sigma model with vector and axial vector
mesons,as well as the nucleonic
fieldsin the mirror assignmentlocal symmetric case
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Spontaneus symmetry breaking
ϕσσ +→
leads to mixing terms in the Lagrangian: µ
µµ
µ ϕπηϕ 11 agfgrr
⋅∂−∂−
we need to perform the shifts of the axial fields:
η
πµµµ
µµµ
∂+→
∂+→
wff
waa
11
11
rrr
2)(1
ϕ
ϕ
gm
gw
a +=
and to renormalize the pseudoscalar fields:
ηη →Z
1ππrr
→Z
1
S. Gasiorowicz and D.A.Geffen , Rev.Mod.Phys.41:531-573,1969.
ρm
mZ
relation-KSFRby predicted as 2
a=
=Z
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Mesonic sector of the lagrangian
])()[(2
])()[(4
1
)]([)]det()[det(
)()]([][)])([(
22
2
122
2
2
2
1
2
µµν
µ
µµν
µ λλ
RLTrm
RLTr
HTrc
TrTrTrmDDTrLmes
+++
−Φ+Φ+Φ+Φ
+ΦΦ−ΦΦ−ΦΦ−ΦΦ=
++
+++
tatfR
tatfL
tiati
rrr
rrr
rrr
)()(
)()(
)()(
101
101
00
µµµµµ
µµµµµ
ρω
ρω
πησ
−+−=
+++=
+++=Φ [ ][ ]νµµννµµν
νµµννµµν
RRigRRR
LLigLLL
,
,
−∂−∂=
−∂−∂=
Fields Field strength tensor
nucleonL+)( Φ−Φ+Φ∂=Φ µµµµLRigD
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Introducing the chiral partner
)(
)(ˆ)(ˆ
121221210
2222211111
22222222
11111111
RLLRLRRL
LRRLLRRL
RRRRLLLL
RRRRLLLLnucl
m
gg
LgiRgi
RgiLgiL
ΨΨ+ΨΨ−ΨΨ−ΨΨ−
ΦΨΨ+ΨΦΨ−ΨΦΨ+ΦΨΨ−
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ+
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ=
++
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
γγγγ
γγγγ
0~ mqqmN +⟩⟨ ?
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Mixing angle and mass m0
+=
02cosh
*
m
mmar NNδMixing angle:
*
2,1
2
*
1
*
21
20
,
,
NN
NN
NN
±=Ψ⇒→
=Ψ=Ψ⇒−∞→
=Ψ=Ψ⇒∞→
δ
δ
δno mixing
maximal mixing
−±++=
2)(4)(
2
121
2
0
22
21*,
ϕϕ ggmggm NN
))))
−=
−
−
2/2/
5
2/
5
2/
cosh2
1ˆδδ
δδ
γ
γ
δ ee
eeM
Ψ
Ψ=
2
1
*M̂
N
N with
:massesnucleon ˆ ediagonaliz ⇒M
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Mixing angle and mass m0
1 2 3 4 5d
200
400
600
800
1000
1200
m0
δ
δδ
sinh4
),,(ˆ),,(ˆ ** 21
0NNNN mmgmmg
m+
=
[MeV]
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Linear sigma model with vector and axial vector
mesons,as well as the nucleonic
fieldsin the mirror assignmentlocal symmetric case:
Results
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
The s-wave scattering lengths
2=Z14)(
exp,0
16)(
exp,0
10)1146.041256.6(
10)165.785783.8(
−−−
−−+
±=
±−=
MeVa
MeVa
Axial coupling constant of N*(1535)
26.1
3.02.0
*
*
,
,
−=
±=
N
GlozmanA
N
LatticeA
g
g
][1−
MeV
][ 1−MeV
S.W.(S.G.), F. Giacosa, D.Rischke,e-Print: nucl-th/0702076
Results: locally symmetric model
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
modelour improve toneed we
consistentnot are Results
)a with compatible(not smallor very 03.02.0
1.26 is valuealexperimentbut 1)tanh(
valuesalexperimentwith agreement in not is alength Scattering
5.1for valuesalexperimentwith agreement in alength Scattering
)(
0
2
(-)
0
)(
0
*
⇒
⇒
<→±=•
<=•
•
>•
+
+
δ
δ
δ
N
A
N
A
g
Zg
Conclusions: locally symmetric model
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Global chiral symmetry
in agreement with mesonic decays
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Model with global symmetry
)(
)(ˆ)(ˆ
][2][
])()[(][2
121221210
2222211111
2222222222
1111111111
32
221
RLLRLRRL
LRRLLRRL
RRRRLLLL
RRRRLLLL
mes
m
gg
LciRci
RciLci
LRTrhRRLLTrh
RLTrTrh
LL
ΨΨ+ΨΨ−ΨΨ−ΨΨ−
ΦΨΨ+ΨΦΨ−ΨΦΨ+ΦΨΨ−
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ+
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ
+ΦΦ+ΦΦ+ΦΦ+
+ΦΦ+=
++
+++
+
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
γγγγ
γγγγ
[ ][ ]νµµννµµν
νµµννµµν
RRigRRR
LLigLLL
,
,
2
2
−∂−∂=
−∂−∂=
Changes in :
)(1 Φ−Φ+Φ∂=Φ µµµµLRigD
mesL
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Model with global symmetry
We determine the free parameters of the modelm0, c1, and c2 from
N*(1200): speculative candidate, decay width GN*zpN > 800 MeV
NN
N
A
N
A ggπ→
Γ *,, *
N*(1500) : PDG , decay width GN*zpN = 67.5 MeV
N*(1650): PDG, decay width GN*zpN =92.5+/-37.5 MeV
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Results
Global chiral symmetry
in agreement with mesonic decays
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
m0 vs. the axial coupling constant of N*(1535)
350 MeV < m0 < 600 MeV
[MeV]
Results: globally symmetric model
S.G., F. Giacosa, D.Rischke,e-Print: hep-ph/0901.4043
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Results: globally symmetric model
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Results: globally symmetric model
s-wave scattering lengths
mN* = 1535 MeV
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Results: globally symmetric model
s-wave scattering lengths
N*(1535)N*(1650)
N*(1200) is off by over a factor 1000!
N*(1200)
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Introducing a
scalar tetraquark(preliminary)
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Introducing the tetraquark
)(
)(ˆ)(ˆ
][2
][
12122121
0
0
2222211111
2222222222
1111111111
3
2
RLLRLRRL
LRRLLRRL
RRRRLLLL
RRRRLLLL
mes
m
gg
LciRci
RciLci
LRTrh
RRLLTrhLL
ΨΨ+ΨΨ−ΨΨ−ΨΨ−
ΦΨΨ+ΨΦΨ−ΨΦΨ+ΦΨΨ−
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ+
ΨΨ+Ψ∂Ψ+ΨΨ+Ψ∂Ψ
+ΦΦ+
ΦΦ+ΦΦ+=
++
+
++
χ
χ
γγγγ
γγγγ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
µµ
(almost) as before:
)()(2
1)(
2
1 2222222
µµχµχ χπσχχχ LRbgmL ++++−∂=
and add:
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Introducing the tetraquark
[ ][ ]dudu ,,2
1=χ
−=
σ
χ
ϑϑ
ϑϑ
cossin
sincos
S
H
22
4arctan
2
1
χσ
ϕϑ
mm
g
−=
)(2
1dduu +=σ
)1370(tly predominan
)600(tly predominan
0
0
fS
fH
)( 22 πσχχσ += gL
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Results: the tetraquark
b=-100
b=0
)()( 2222
µµχπσχ LRbg ++++
The isospin even scattering length as a function of g
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Results: the tetraquark
b=100
b=0
b=-100
b=-200
0 ,MeV 2500 21 === hhg
The isospin even scattering length as a function of the axial charge of the
chiral partner of the nucleon for g=2500 MeV and different values of b.
by A. Heinz et. al in finite temperature studies (hep-ph 08051134) GeVg 5.2≈
][])()[(][2
2
221 +++ ΦΦ+ΦΦ++ΦΦ µµ
µµ
µµRRLLTrhRLTrTr
h
Excited QCD 8-14 February 2009- Zakopane (Poland) Susanna Gallas
Conclusions and outlook
future. in the also included be should resonance delta The
included. israquark scalar tet a scalars on the depends length scattering The
studyour by favorednot is MeV 1200 massth partner wi broad very e,speculativ of hypothesis The
studies) densities and re temperatufiniteat also(relevant
limit chiral in the survives massnucleon theof half :MeV 500 topointsresult The
small. tooarelength scattering one and eta-nucleon intodecay al theoretichealthough t
,assignmentmirror in thenucleon theofpartner chiral thebe tocandidate good a is (1535)
)(
0
0
*
•
⇒•
•
ť
•
+a
m
N
The scalar tetraquark field shifts the isospin even scattering length in the “right direction”. It has no
influence on isospin odd scattering length.
A large value of the coupling constant g is favoured. i.e. a strong mixing between the quarkonium and
tetraquark fields.
Thank you!