Transcript Document
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
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Outline
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
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Energies 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 S S[ ] depending on a real scalar field ( x , t )
evolution in Heisenberg picture (t ) e Ht (0) e Ht ( H Hamiltonian)
spectral representation of a simple correlation function
assume transfer matrix, ignore temporal boundary conditions
insert complete set of
focus only on one time ordering
energy eigenstates
0 (t ) (0) 0 0 e Ht (0) e Ht n n (0) 0
(discrete and continuous)
n
n ( 0) 0
2
e En E0 t An e En E0 t
n
extract A1 and E1 E0 as t
n
(assuming 0 (0) 0 0 and 1 (0) 0 0 )
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Fitting procedure
extraction of A1 and E1 E0 done by correlated- 2 fit using single
exponential
minimize 2 (C (t ) M (t , )) tt1 (C (t ) M (t , ))
tt
where C (t ) represents the MC estimates of the correlation function
with covariance matrix tt and model function is M (t , ) 1e 0t
uncertainties in fit parameters 0 E1 E0, 1 A1 obtained by
jackknife or bootstrap
fit must be done for time range tmin tmax for acceptable 2 / dof 1
can fit to sum of two exponentials to reduce sensitivity to t min
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)
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Effective mass
C (t )
the “effective mass” is given by meff (t ) ln
C
(
t
1
)
notice that (take E0 0)
A1e E1t A2 e E2t
E1
limt meff (t ) ln
ln
e
E1
E1 ( t 1)
A1e
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
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Reducing contamination
statistical noise generally increases with temporal separation t
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
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Link variable smearing
link variables: add staples with weight, project onto gauge group
ˆ
define
C ( x) U ( x)U ( x ˆ )U ( x ˆ )
ˆ
x
jk ,
4k k 4 0
common 3-d spatial smearing
APE smearing U ( n1) PSU (3) U ( n ) C( n )
or new analytic stout link method (hep-lat/0311018)
C U
i
i
Q
Tr
2
2N
U ( n1) expiQ( n ) U ( n )
~
(1)
(n)
U
U
U
U
iterate
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Quark field smearing
quark fields: gauge covariant smearing
~ n
~( x) 1 ( 2) ( x)
tunable parameters , n
three-dimensional gauge covariant Laplacian defined by
3
~( 2)
~
~
O( x) U k ( x)O( x kˆ) U k ( x kˆ)O( x kˆ) 2O( x)
k 1
– uses the smeared links
square of smeared field is zero, like simple Grassmann field
preserves transformation properties of the quark field
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Unleashing the variational method
~
consider the correlation function of an operator O ( x) given by a linear
superposition of a set of operators O (x)
~
O ( x) c O ( x)
C (t ) 0 O (t )O (0) 0
~
~
~
C (t ) 0 O (t )O (0) 0 c c C (t )
choose coefficients c to minimize excited-state contamination
minimize effective mass at some early time separation
d
meff (t ) 0 C (t 1)c e meff (t )C (t )c
dc
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!!
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Principal correlators
application of such variational techniques to extract excited-state
energies was first described in Luscher, Wolff, NPB339, 222 (1990)
for a given N N correlator matrix C (t ) 0 O (t )O (0) 0 they
defined the N principal correlators (t , t0 ) as the eigenvalues of
C (t0 ) 1/ 2 C (t ) C (t0 ) 1/ 2
where t 0 (the time defining the “metric”) is small
L-W showed that limt (t , t0 ) e (t t0 ) E (1 e tE )
(t , t0 )
eff
so the N principal effective masses defined by m (t ) ln
(
t
1
,
t
)
0
now tend (plateau) to the N lowest-lying stationary-state energies
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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 t min
note that principal effective
masses (as functions of time)
can cross, approach asymptotic
behavior from below
final results are independent
of t 0, but choosing larger values
of this reference time can introduce
larger errors
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Excitations 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
L 0,1,2,...
, P, ,...
charge conjugation + parity
about midpoint
g (even)
u (odd)
chirality (reflections in plane
containing axis) ,
P,,…doubly degenerate
(L doubling)
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several higher levels
not shown
P g , g , u , u ,
u , P u , g ,...
Juge, Kuti, Morningstar, PRL 90, 161601 (2003)
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Dramatis personae
the gluon excitation team
Jimmy Juge
Julius Kuti
CM
Mike Peardon
ITP, Bern
UC San Diego
Carnegie-Mellon,
Pittsburgh
Trinity College,
Dublin
student: Francesca Maresca
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Initial remarks
viewpoint adopted:
the nature of the confining gluons is best revealed in its
excitation spectrum
robust feature of any bosonic string description:
N / R 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
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String spectrum
spectrum expected for a non-interacting bosonic string at large R
standing waves: m 1,2,3, with circular polarization
occupation numbers: nm , nm
energies E
E E0 N / R
string quantum number N
N nm nm
spin projection L
m1
CP CP
L nm nm
m1
CP 1N
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String spectrum (N=1,2,3)
level orderings for N=1,2,3
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String spectrum (N=4)
N=4 levels
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Generalized Wilson loops
gluonic terms extracted from generalized Wilson loops
large set of gluonic operators correlation matrix
link variable smearing, blocking
anisotropic lattice, improved actions
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Three 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 0.5fm R 2fm
dramatic level rearrangement
large separations R 2fm
excitations consistent with
expectations from string models
Juge, Kuti, Morningstar, PRL 90, 161601 (2003)
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Gluon excitation gaps (N=1,2)
comparison of gaps with N / R
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Gluon excitation gaps (N=3)
comparison of gaps with N / R
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Gluon excitation gaps (N=4)
comparison of gaps with N / R
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Possible interpretation
small R
strong E field of qq-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)
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3d 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)
rS
5
2
1 fm
S 2 1GeV
ground state
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3d SU(2) gauge theory
first excitation
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2d SU(2) gauge theory
gap of first excitation above ground state
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3d SU(2) gaps
comparison of gaps with N / R
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Au N 1
Ag N 2
S g N 2
Au N 3
Su N 3
Au N 3
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3d SU(2) gaps (N=4)
comparison of gaps with N / R
Ag N 4
S g N 4
S g N 4
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
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Ag N 4
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Baryon 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
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Three stage approach
concentrate on baryons at rest (zero momentum)
operators classified according to the irreps of Oh
G1g , G1u , G2 g , G2u , H g , H u
(1) basic building blocks: smeared, covariant-displaced quark fields
~
( D (j p )~( x)) Aa p - link displaceme nt ( j 0,1,2,3)
(2) construct elemental operators (translationally invariant)
~
~
~
B F ( x) F ( D ( p )~( x)) ( D ( p )~( x)) ( D ( p )~( x))
i
ABC abc
j
Aa
j
Bb
j
Cc
flavor structure from isospin, color structure from gauge invariance
(3) group-theoretical projections onto irreps of Oh
d
(L)
BiLF (t ) L D
( R) U R BiF (t ) U R
g O D ROhD
h
wrote Grassmann package in Maple to do these calculations
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Incorporating 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!
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Spin identification and other remarks
spin identification possible by pattern matching
total numbers of operators assuming two
different displacement lengths
total numbers of operators is huge uncharted territory
ultimately must face two-hadron states
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Preliminary results
principal effective masses for small set of 10 operators
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Summary
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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