Excited Hadrons: Lattice results

Christian B. LangInst. F. Physik – FB Theoretische PhysikUniversität Graz

Oberwölz, September 2006

BernGrazRegensburgQCD collaboration

PR D 73 (2006) 017502 ;[hep-lat/0511054]PR D 73 (2006) 094505 [ hep-lat/0601026]PR D 74 (2006) 014504; [hep-lat/0604019]

In collaboration with T. Burch, C. Gattringer, L.Y. Glozman, C. Hagen, D. Hierl and A. Schäfer

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Lattice simulation with Chirally Improved Dirac actions

Quenched lattice simulation results: Hadron ground state masses /K decay constants:

f=96(2)(4) MeV), fK=106(1)(8) MeV Quark masses:

mu,d=4.1(2.4) MeV, ms=101(8) MeV

Light quark condensate: -(286(4)(31) MeV)3

Pion form factor

Excited hadrons

Dynamical fermions First results on small lattices

BGR (2004)

Gattringer/Huber/CBL (2005)

Capitani/Gattringer/Lang (2005)

CBL/Majumdar/Ortner (2006)

C. B. Lang © 2006

Lattice simulation with Chirally Improved Dirac actions

Quenched lattice simulation results: Hadron ground state masses /K decay constants:

f=96(2)(4) MeV), fK=106(1)(8) MeV Quark masses:

mu,d=4.1(2.4) MeV, ms=101(8) MeV

Light quark condensate: -(286(4)(31) MeV)3

Pion form factor

Excited hadrons

Dynamical fermions First results on small lattice

BGR (2004)

Gattringer/Huber/CBL (2005)

Capitani/Gattringer/Lang (2005)

CBL/Majumdar/Ortner (2006)

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Motivation

Little understanding of excited states from lattice calculations

Non-trivial test of QCD Classification! Role of chiral symmetry? It‘s a challenge…

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Quenched Lattice QCD

QCD on Euclidean lattices:

Quark propagators

t

mte

log ( )C t

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Quenched Lattice QCD

QCD on Euclidean lattices:

Quark propagators

t

mte

log ( )C t

“quenched”approximation

C. B. Lang © 2006

Quenched Lattice QCD

QCD on Euclidean lattices:

Quark propagators

t

mte

log ( )C t

“quenched”approximation

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The lattice breaks chiral symmetry

Nogo theorem: Lattice fermions cannot have simultaneously: Locality, chiral symmetry, continuum limit of fermion propagator

Original simple Wilson Dirac operator breaks the chiral symmetry badly: Duplication of fermions, no chiral zero modes, spurious small eigenmodes (…

problems to simulate small quark masses)

But: the lattice breaks chiral symmetry only locally Ginsparg Wilson equation for lattice Dirac operators

Is related to non-linear realization of chiral symmetry (Lüscher) Leads to chiral zero modes! No problems with small quark masses

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The lattice breaks chiral symmetry locally Nogo theorem: Lattice fermions cannot have simultaneously:

Locality, chiral symmetry, continuum limit of fermion propagator

Original simple Wilson Dirac operator breaks the chiral symmetry badly: Duplication of fermions, no chiral zero modes, spurious small eigenmodes (…

problems to simulate small quark masses)

But: the lattice breaks chiral symmetry only locally Ginsparg Wilson equation for lattice Dirac operators

Is related to non-linear realization of chiral symmetry (Lüscher) Leads to chiral zero modes! No problems with small quark masses

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GW-type Dirac operators

Overlap (Neuberger) „Perfect“ (Hasenfratz et al.) Domain Wall (Kaplan,…) We use „Chirally Improved“ fermions

Gattringer PRD 63 (2001) 114501Gattringer /Hip/CBL., NP B697 (2001) 451

This is a systematic (truncated) expansion

…obey the Ginsparg-Wilson relations approximately and have similar circular shaped Dirac operator spectrum (still some fluctuation!)

+ . . .+ +=

+

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Quenched simulation environment

Lüscher-Weisz gauge action Chirally improved fermions Spatial lattice size 2.4 fm Two lattice spacings, same volume:

203x32 at a=0.12 fm 163x32 at a=0.15 fm (100 configs. each)

Two valence quark masses (mu=md varying, ms fixed) Mesons and Baryons

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Usual method: Masses from exponential decay

( )Mt M N te e

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Interpolators and propagator analysis

Propagator: sum of exponential decay terms:

Previous attempts: biased estimators (Bayesian analysis), maximum entropy,...

Significant improvement: Variational analysis

ground state (large t)

excited states (smaller t)

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Variational method

Use several interpolators Compute all cross-correlations

Solve the generalized eigenvalue problem

Analyse the eigenvalues

The eigenvectors are „fingerprints“ over t-ranges:

For t>t0 the eigenvectors allow to trace the state composition from high to low quark masses

Allows to cleanly separate ghost contributions (cf. Burch et al.)

(MichaelLüscher/Wolff)

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Interpolating fields (I)

Inspired from heavy quark theory:

Baryons: ( ) ( ) ( ) ( )1 2 2

( ) ( ) ( ) ( )1 2 2

0 8, , , ,

i i T i T iabc a b c b c

i i T i T iabc a b c b c

N u u d d u

u u s s u

(plus projection to parity)

( ) ( )1 2

5

5

4 5

1 1

2

3 1

i i

i C

i C

i i C

Mesons: *0 1 1, , , , , , ,a K K a b

i.e., different Dirac structure of interpolating hadron fields…..

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Interpolating fields (II)

(1) (2) (3), ,N N N

are not sufficient to identify the Roper state

However:

…excited states have nodes!

→ smeared quark sourcesof different widths (n,w)using combinations like:

nw nw, wwnnn, nwn, nww etc.

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*0 1 1, , , , , , ,a K K a b

Mesons

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„Effective mass“ example:mesons

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Mesons: typeud

pseudoscalar vector

4 interpolaters: n5n, n45n, n45w, w45w

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Mesons: type us

pseudoscalar vector

4 interpolaters: n5n, n45n, n45w, w45w

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Meson summary (chiral extrapolations)

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, , , , ,N

Baryons

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Nucleon (uud)

RoperLevel crossing (from + - + - to + - - +)?

Positive parity: w(ww)(1,3), w(nw)(1,3) , n(ww)(1,3) Negative parity: w(n,n)(1,2), n(nn)(1,2)

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Masses (1)

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Masses (2)

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Mass dependence of eigenvector (at t=4)

1[w(nw)]1[n(ww)]1[w(ww)]

3[w(nw)]3[n(ww)]3[w(ww)]

1 5( ) ( ) ( ) ( )Tabc a b cx u x C d x u x

3 4 5( ) ( ) ( ) ( )Tabc a b cx i u x C d x u x

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(uus)

Positive parity: w(ww)(1,3), w(nw)(1,3) , n(ww)(1,3) Negative parity: w(n,n)(1,2), n(nn)(1,2)

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(ssu)

Positive parity: w(ww)(1,3), w(nw)(1,3) , n(ww)(1,3) Negative parity: w(n,n)(1,2), n(nn)(1,2)

??

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octet (uds )

Positive parity: w(ww)(1,3), w(nw)(1,3) , n(ww)(1,3) Negative parity: w(n,n)(1,2), n(nn)(1,2)

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(uuu ), (sss)

Positive/Negative parity: n(nn), w(nn), n(wn), w(nw), n(ww), w(ww)

?

?

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Baryon summary (chiral extrapolations)

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Baryon summary (chiral extrapolations)

1st excited state, pos.parity: 2300(70) MeV ground state, neg.parity: 1970(90) MeV ground state, neg.parity: 1780(90) MeV 1st excited stated, neg.parity: 1780(110) MeV

Bold predictions:

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Summary and outlook Method works

Large set of basis operators Non-trivial spatial structure Ghosts cleanly separated Applicable for dynamical quark configurations

Physics Larger cutoff effects for excited states Positive parity excited states: too high Negative parity states quite good Chiral limit seems to affect some states strongly

Further improvements Further enlargement of basis, e.g. p-wave sources (talk by C. Hagen)

and non-fermionic interpolators (mesons) Studies at smaller quark mass

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Thank you