The chiral partner of the nucleon in the mirror assignment

The chiral partner of the nucleon

in the mirror assignment

Hadrons@FAIR University Frankfurt

Susanna Wilms

in collaboration with: Francesco Giacosa and Dirk Rischke

Motivation

Susanna Wilms Hadrons@FAIR 25-27 June 2008 Susanna Wilms Hadrons@FAIR 25-27 June 2008

Introduction of N* for chiral realization of the nucleonic Lagrangian

chiral invariant mass term / m0

Correct description of the axial coupling constant gA(N)=1.26 gA(N*)=?

Generation of the nucleonic mass: how is mN generated?

Quark condensate vs. m0

Study of pion-nucleon scattering

The chiral partner:

what is N*?


The chiral partner of the nucleonin the mirror assignment


Mirror assignment vs naive assignment: chiral transformation

LRL

LLL

U

U

22

11

Mirror

RLR

RRR

U

U

22

11

Naive

RRR

RRR

U

U

22

11

LLL

LLL

U

U

22

11

)( 121221210 RLLRLRRLm

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

2

1

* M̂N

N221ˆ

xM

The chiral partner of the nucleonin the mirror assignment


Candidates with the correct quantum numbers from PDG:

N*(1535) lowest nucleonic resonanceN*(1650)

2

1PJOther 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

A little bit of scattering theory

cbacabba iBqqABqqAT

)(

21)()(

21)( )(

2

1)(

2

1


General form of the Lorentz-invariant scattering amplitude: )()( 12 puTpu ba

)( )(0

)(0

)(0

BmAa s-wave scattering lengths: )1(4

1

Nmm

Pion-nucleon scattering


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*

NAg

L. Ya. Glozman: Phys.Rept.444:1-49,2007

resultdifferent apresent shall we

1partner theis if

partner not the is 0*

*

*

*

NA

NA

gN

Ng


The model

The model


Linear sigma model with vector and axial vector mesons,

as well as the nucleonic fields in the mirror assignment

local symmetric case

Weinberg-Tomozawa-termlocal chiral symmetry

Global chiral symmetryin agreement with mesonic

decays


Linear sigma model with vector and axial vector

mesons, as well as the nucleonic fields

in the mirror assignmentlocal symmetric case

Mesonic sector of the lagrangian


])()[(2

])()[(4

1

)]([)]det()[det(

)()]([][)])([(

222

122

22

21

2

RLTrm

RLTr

HTrc

TrTrTrmDDTrLmes

tatfR

tatfL

tiati

)()(

)()(

)()(

101

101

00

RRigRRR

LLigLLL

,

,

Fields Field strength tensor

nucleonL)( LRigD

Spontaneus symmetry breaking


leads to mixing terms in the Lagrangian:

11 agfg

we need to perform the shifts of the axial fields:

wff

waa

11

11

2)(1

gm

gw

a

and to renormalize the pseudovector fields:

Z

1

Z

1

KSFR 2Z

Introducing the chiral partner


)(

)(ˆ)(ˆ

121221210

2222211111

22222222

11111111

RLLRLRRL

LRRLLRRL

RRRRLLLL

RRRRLLLLnucl

m

gg

LgiRgi

RgiLgiL

0~ mqqmN ?


Mixing angle and mass m0

02cosh

*

m

mmar NNMixing angle:

*

2,1

2*

1

*21

20

,

,

NN

NN

NN

no mixing

maximal mixing

2)(4)(

2

121

20

2221*,

ggmggm NN

2/2/5

2/5

2/

cosh2

1ˆ

ee

eeM

2

1

* M̂N

N with

:massesnucleon ˆ ediagonaliz M


Mixing angle and mass m0

1 2 3 4 5

200

400

600

800

1000

1200

m 0

sinh4

),,(ˆ),,(ˆ ** 210

NNNN mmgmmgm

[MeV]


Linear sigma model with vector and axial vector

mesons, as well as the nucleonic fields

in the mirror assignmentlocal symmetric case:

Results

Introducing the chiral partner


The s-wave scattering lengths

2Z14)(

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*

*

,

,

NGlozmanA

NLatticeA

g

g

][ 1MeV

][ 1MeV

S.W., F. Giacosa, D.Rischke,e-Print: nucl-th/0702076


Introducing the chiral partnerConclusions

modelour improve toneed we

consistentnot are Results

)a with compatible(not smallor very 03.02.0

1.26 is valuealexperimentbut 11

valuesalexperimentwith agreement in not is alength Scattering

5.1for valuesalexperimentwith agreement in alength Scattering

)(0

2

(-)0

)(0

*

NA

NA

g

Zg


Global chiral symmetry

in agreement with mesonic decays


Model with global symmetry

)(

)(ˆ)(ˆ

121221210

2222211111

2222222222

1111111111

RLLRLRRL

LRRLLRRL

RRRRLLLL

RRRRLLLLnucl

m

gg

LciRci

RciLciL

RRigRRR

LLigLLL

,

,

2

2

And in the mesonic sector:

)(1 LRigD


Model with global symmetry

We determine the free parameters of the model m0, c1, and c2 from

N*(1200): speculative candidate, decay width N*N > 800 MeV

NN

NA

NA gg

*,, *

N*(1500) : PDG , decay width N*N = 67.5 MeV

N*(1650): PDG, decay width N*N =92.5+/-37.5 MeV


Results

Global chiral symmetry

in agreement with mesonic decays


m0 vs. the axial coupling constant of N*(1535)

Global symmetry: Results

350 MeV < m0 < 600 MeV

[MeV]





s-wave scattering lengths



s-wave scattering lengths

N*(1535)N*(1650)

N*(1535)N*(1650)

N*(1200)

N*(1200) is off by over a factor 1000!


Conclusions

lenghts. scattering ofn descriptiobetter afor resonance, delta theinclude also shall studies Future

scalarslight of issue intricate thefrom

t independenrather are i.e. meson. sigma theof mass on the weakly depend Results

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

large. tooarelength scattering one and eta-nucleon intodecay al theoretichealthough t

,assignmentmirror in thenucleon theofpartner chiral thebe tocandidate good a is (1535)

0

*

m

N


Thank you!

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The chiral partner of the nucleon in the mirror assignment