colin morningstar carnegie mellon university quantum fields in the era of teraflop computing
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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
RNEE /0
,3,2,1m
mm nn ,
1mmm nnN
CP
1mmm nn
NCP 1
11/22/2004 Excited states (C. Morningstar) 16
String spectrum (String spectrum (N=1,2,3N=1,2,3))
level orderings for N=1,2,3
11/22/2004 Excited states (C. Morningstar) 17
String spectrum (String spectrum (N=4N=4))
N=4 levels
11/22/2004 Excited states (C. Morningstar) 18
Generalized Wilson loopsGeneralized Wilson loops
gluonic terms extracted from generalized Wilson loops
large set of gluonic operators correlation matrix
link variable smearing, blocking
anisotropic lattice, improved actions
11/22/2004 Excited states (C. Morningstar) 19
Three scalesThree scales
studied the energies of 16 stationary
states of gluons in the presence of
static quark-antiquark pair
small quark-antiquark separations R excitations consistent with states
from multipole OPE
crossover region dramatic level rearrangement
large separations excitations consistent with
expectations from string modelsJuge, Kuti, Morningstar, PRL 90, 161601 (2003)
fm2fm5.0 R
fm2R
11/22/2004 Excited states (C. Morningstar) 20
Gluon excitation gaps (Gluon excitation gaps (N=1,2N=1,2))
comparison of gaps with RN /
11/22/2004 Excited states (C. Morningstar) 21
Gluon excitation gaps (Gluon excitation gaps (N=3N=3))
comparison of gaps with RN /
11/22/2004 Excited states (C. Morningstar) 22
Gluon excitation gaps (Gluon excitation gaps (N=4N=4))
comparison of gaps with RN /
11/22/2004 Excited states (C. Morningstar) 23
Possible interpretationPossible interpretation
small R strong E field of -pair repels physical
vacuum (dual Meissner effect) creating a bubble
separation of degrees of freedom– gluonic modes inside bubble (low lying)
– bubble surface modes (higher lying)
large R bubble stretches into thin tube of flux separation of degrees of freedom
– collective motion of tube (low lying)
– internal gluonic modes (higher lying) low-lying modes described by an effective string
theory (N/R gaps – Goldstone modes)
11/22/2004 Excited states (C. Morningstar) 24
3d SU(2) gauge theory3d SU(2) gauge theory
also studied 11 levels in (2+1)-dimensional SU(2) gauge theory
levels labeled by reflection symmetry (S or A) and CP (g or u)
GeV12 S
fm12
5 Sr
ground state
11/22/2004 Excited states (C. Morningstar) 25
3d SU(2) gauge theory3d SU(2) gauge theory
first excitation
11/22/2004 Excited states (C. Morningstar) 26
2d SU(2) gauge theory2d SU(2) gauge theory
gap of first excitation above ground state
11/22/2004 Excited states (C. Morningstar) 27
3d SU(2) gaps3d SU(2) gaps
comparison of gaps with RN /
1NAu 2NAg 2 NSg
3 NAu 3NSu 3 NAu
11/22/2004 Excited states (C. Morningstar) 28
3d SU(2) gaps (3d SU(2) gaps (N=4N=4))
comparison of gaps with
large R results consistent with string
spectrum without exception
fine structure less pronounced than 4d SU(3)
no dramatic level rearrangement between
small and large separations
RN /
4 NSg 4 NAg 4 NSg
4 NAg
11/22/2004 Excited states (C. Morningstar) 29
Baryon blitzBaryon blitz
charge from the late Nathan Isgur to apply these techniques to extract
the spectrum of baryons (Hall B at JLab)
formed the Lattice Hadron Physics Collaboration (LHPC) in 2000
current collaborators: Subhasish Basak, Robert Edwards, George
Fleming, David Richards, Ikuro Sato, Steve Wallace
for spectrum, need large sets of extended operators correlation
matrix techniques
since large sets of operators to be used and to facilitate spin
identification, we shunned the usual method of operator construction
which relies on continuum space-time constructions
focus on constructing operators which transform irreducibly under the
symmetries of the lattice
11/22/2004 Excited states (C. Morningstar) 30
Three stage approachThree stage approach
concentrate on baryons at rest (zero momentum)
operators classified according to the irreps of
(1) basic building blocks: smeared, covariant-displaced quark fields
(2) construct elemental operators (translationally invariant)
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
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