View
216
Download
1
Embed Size (px)
Citation preview
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
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 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)
C. B. Lang © 2006
Motivation
Little understanding of excited states from lattice calculations
Non-trivial test of QCD Classification! Role of chiral symmetry? It‘s a challenge…
C. B. Lang © 2006
Quenched Lattice QCD
QCD on Euclidean lattices:
Quark propagators
t
mte
log ( )C t
C. B. Lang © 2006
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
C. B. Lang © 2006
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
C. B. Lang © 2006
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
C. B. Lang © 2006
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!)
+ . . .+ +=
+
C. B. Lang © 2006
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
C. B. Lang © 2006
Usual method: Masses from exponential decay
( )Mt M N te e
C. B. Lang © 2006
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)
C. B. Lang © 2006
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)
C. B. Lang © 2006
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…..
C. B. Lang © 2006
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.
C. B. Lang © 2006
*0 1 1, , , , , , ,a K K a b
Mesons
C. B. Lang © 2006
„Effective mass“ example:mesons
C. B. Lang © 2006
Mesons: typeud
pseudoscalar vector
4 interpolaters: n5n, n45n, n45w, w45w
C. B. Lang © 2006
Mesons: type us
pseudoscalar vector
4 interpolaters: n5n, n45n, n45w, w45w
C. B. Lang © 2006
Meson summary (chiral extrapolations)
C. B. Lang © 2006
, , , , ,N
Baryons
C. B. Lang © 2006
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)
C. B. Lang © 2006
Masses (1)
C. B. Lang © 2006
Masses (2)
C. B. Lang © 2006
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
C. B. Lang © 2006
(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)
C. B. Lang © 2006
(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)
??
C. B. Lang © 2006
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)
C. B. Lang © 2006
(uuu ), (sss)
Positive/Negative parity: n(nn), w(nn), n(wn), w(nw), n(ww), w(ww)
?
?
C. B. Lang © 2006
Baryon summary (chiral extrapolations)
C. B. Lang © 2006
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:
C. B. Lang © 2006
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
C. B. Lang © 2006
Thank you