11/22/2004 Excited states (C. Morn ingstar) 1 Extracting excited-state energies Extracting excited-state energies with application to with application to the static quark-antiquark system the static quark-antiquark system and hadrons and hadrons Colin Morningstar Carnegie Mellon University Quantum Fields in the Era of Teraflop Computing ZiF, University of Bielefeld, November 22, 2004
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Colin Morningstar Carnegie Mellon University Quantum Fields in the Era of Teraflop Computing
Extracting excited-state energies with application to the static quark-antiquark system and hadrons. Colin Morningstar Carnegie Mellon University Quantum Fields in the Era of Teraflop Computing ZiF, University of Bielefeld, November 22, 2004. Outline. - PowerPoint PPT Presentation
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11/22/2004 Excited states (C. Morningstar) 1
Extracting excited-state energies with application toExtracting excited-state energies with application tothe static quark-antiquark system and hadronsthe static quark-antiquark system and hadrons
Colin Morningstar
Carnegie Mellon University
Quantum Fields in the Era of Teraflop Computing
ZiF, University of Bielefeld, November 22, 2004
11/22/2004 Excited states (C. Morningstar) 2
OutlineOutline
spectroscopy is a powerful tool for distilling key degrees of freedom
spectrum determination requires extraction of excited-state energies
will discuss how to extract excited-state energies from Monte Carlo
estimates of correlation functions in Euclidean lattice field theory
application: gluonic excitations of the static quark-antiquark system
application: excitations in 3d SU(2) static source-antisource system
application: ongoing efforts of LHPC to determine baryon spectrum
with an eye toward future meson, tetraquark, pentaquark systems
11/22/2004 Excited states (C. Morningstar) 3
Energies from correlation functionsEnergies from correlation functions
stationary state energies can be extracted from asymptotic decay rate of
temporal correlations of the fields (in the imaginary time formalism)
consider an action depending on a real scalar field
evolution in Heisenberg picture ( Hamiltonian)
spectral representation of a simple correlation function assume transfer matrix, ignore temporal boundary conditions focus only on one time ordering
extract and as
(assuming and )
][SS ),( tx
HtHt eet )0()(
n
tEEn
tEE
n
n
HtHt
nn eAen
nneet
002
0)0(
0)0()0(00)0()(0
H
1A 01 EE t
00)0(0 00)0(1
insert complete set of energy eigenstates (discrete and continuous)
11/22/2004 Excited states (C. Morningstar) 4
Fitting procedureFitting procedure
extraction of and done by correlated- fit using single
exponential
where represents the MC estimates of the correlation function
with covariance matrix and model function is
uncertainties in fit parameters obtained by
jackknife or bootstrap
fit must be done for time range for acceptable
can fit to sum of two exponentials to reduce sensitivity to second exponential is garbage discard!
fits using large numbers of exponentials with a Bayesian prior is one
way to try to extract excited-state energies (not discussed here)
1A 01 EE 2
tttt tMtCtMtC )),()(()),()((minimize 12
)(tC
tt tetM 0
1),(
11,010 AEE
1dof/2 maxmin tt
mint
11/22/2004 Excited states (C. Morningstar) 5
Effective massEffective mass
the “effective mass” is given by
notice that (take )
the effective mass tends to the actual mass (energy) asymptotically
effective mass plot is convenient visual tool to see signal extraction seen as a plateau
plateau sets in quickly
for good operator
excited-state
contamination before
plateau
)1(
)(ln)(eff tC
tCtm
1)1(1
21eff
1
1
21
lnln)(lim EeeA
eAeAtm E
tE
tEtE
t
00 E
11/22/2004 Excited states (C. Morningstar) 6
Reducing contaminationReducing contamination
statistical noise generally increases with temporal separation
effective masses associated with correlation functions of simple local
fields do not reach a plateau before noise swamps the signal need better operators
better operators have reduced couplings with higher-lying
contaminating states
recipe for making better operators crucial to construct operators using smeared fields spatially extended operators large set of operators with variational coefficients
t
11/22/2004 Excited states (C. Morningstar) 7
Link variable smearingLink variable smearing
link variables: add staples with weight, project onto gauge group
define
common 3-d spatial smearing
APE smearing
or new analytic stout link method (hep-lat/0311018)
iterate
)ˆ()ˆ()()(
xUxUxUxC
0, 44 kkjk
)()()3(
)1( nnSU
n CUPU
UUUU n ~)()1(
)()()1( exp nnn UiQU
Tr22 Nii
Q
UC
x
11/22/2004 Excited states (C. Morningstar) 8
Quark field smearingQuark field smearing
quark fields: gauge covariant smearing
tunable parameters
three-dimensional gauge covariant Laplacian defined by
– uses the smeared links
square of smeared field is zero, like simple Grassmann field
preserves transformation properties of the quark field
3
1
)2( )(2)ˆ()ˆ(~
)ˆ()(~
)(~
kkk xOkxOkxUkxOxUxO
)(~
1)(~ )2( xxn
n,
11/22/2004 Excited states (C. Morningstar) 9
Unleashing the variational methodUnleashing the variational method
consider the correlation function of an operator given by a linear
superposition of a set of operators
choose coefficients to minimize excited-state contamination minimize effective mass at some early time separation
simply need to solve an eigenvalue problem
this is essentially a variational method!
yields the “best” operator by the above criterion
added benefit other eigenvectors yield excited states!!
)(0)0(~
)(~
0)(~
0)0()(0)()()(~
tCccOtOtC
OtOtCxOcxO
)(~
xO
)(xO
c
ctCectCtmdcd tm
)()1(0)( )(eff
eff
11/22/2004 Excited states (C. Morningstar) 10
Principal correlatorsPrincipal correlators
application of such variational techniques to extract excited-state
energies was first described in Luscher, Wolff, NPB339, 222 (1990)
for a given correlator matrix they
defined the principal correlators as the eigenvalues of
where (the time defining the “metric”) is small
L-W showed that
so the principal effective masses defined by
now tend (plateau) to the lowest-lying stationary-state
energies
0)0()(0)( OtOtC ),( 0tt
2/10
2/10 )()()( tCtCtC
),1(
),(ln)(
0
0eff
tt
tttm
0t
NNN
N
N
)1(),(lim )(0
0
EtEttt eett
11/22/2004 Excited states (C. Morningstar) 11
Principal effective masses Principal effective masses
just need to perform single-exponential fit to each principal correlator
to extract spectrum! can again use sum of two-exponentials to minimize sensitivity to
note that principal effective
masses (as functions of time)
can cross, approach asymptotic
behavior from below
final results are independent
of , but choosing larger values
of this reference time can introduce
larger errors
0t
mint
11/22/2004 Excited states (C. Morningstar) 12
Excitations of static quark potentialExcitations of static quark potential
gluon field in presence of static quark-antiquark pair can be excited
classification of states: (notation from molecular physics)
magnitude of glue spin
projected onto molecular axis
charge conjugation + parity
about midpoint
chirality (reflections in plane
containing axis)
,…doubly degenerate
( doubling)
,...,,
,...2,1,0
(odd)
(even)
u
g
Juge, Kuti, Morningstar, PRL 90, 161601 (2003)
several higher levelsnot shown
,
,...,,
,,, ,
guu
uugg
11/22/2004 Excited states (C. Morningstar) 13
Dramatis personaeDramatis personae
the gluon excitation team
Jimmy Juge Julius Kuti CM Mike Peardon
ITP, Bern UC San Diego Carnegie-Mellon,
PittsburghTrinity College,
Dublin
student: Francesca Maresca
11/22/2004 Excited states (C. Morningstar) 14
Initial remarksInitial remarks
viewpoint adopted: the nature of the confining gluons is best revealed in its
excitation spectrum
robust feature of any bosonic string description: gaps for large quark-antiquark separations
details of underlying string description encoded in the fine structure
study different gauge groups, dimensionalities
several lattice spacings, finite volume checks
very large number of fits to principal correlators web page
interface set up to facilitate scrutining/presenting the results
RN /
11/22/2004 Excited states (C. Morningstar) 15
String spectrumString spectrum
spectrum expected for a non-interacting bosonic string at large R standing waves: with circular polarization occupation numbers: energies E string quantum number N spin projection CP
flavor structure from isospin, color structure from gauge invariance
(3) group-theoretical projections onto irreps of
wrote Grassmann package in Maple to do these calculations
ugugug HHGGGG ,,,,, 2211
hO
)3,2,1,0(nt displacemelink -))(~~( )( jpxD Aa
pj
Ccp
jBbp
jAap
jabcFABC
Fi xDxDxDxB ))(~~
())(~~())(~~
()( )()()(
hO
RFiR
ORO
Fi UtBURD
gd
tBDh
Dh
)()()( )(
11/22/2004 Excited states (C. Morningstar) 31
Incorporating orbital and radial structureIncorporating orbital and radial structure
displacements of different lengths build up radial structure
displacements in different directions build up orbital structure
operator design minimizes number of sources for quark propagators
useful for mesons, tetraquarks, pentaquarks even!
11/22/2004 Excited states (C. Morningstar) 32
Spin identification and other remarksSpin identification and other remarks
spin identification possible by pattern matching
total numbers of operators is huge uncharted territory
ultimately must face two-hadron states
total numbers of operators assuming two different displacement lengths
11/22/2004 Excited states (C. Morningstar) 33
Preliminary resultsPreliminary results
principal effective masses for small set of 10 operators
11/22/2004 Excited states (C. Morningstar) 34
SummarySummary
discussed how to extract excited-state energies in lattice field theory
simulations
studied energies of 16 stationary states of gluons in presence of static
quark-antiquark pair as a function of separation R unearthed the effective QCD string for R>2 fm for the first time tantalizing fine structure revealedeffective string action clues dramatic level rearrangement between small and large separations
showed similar results in 3d SU(2)
outlined our method for extracting the baryon spectrum