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 ,  ))  tt1 (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 ( n1)  PSU (3) U ( n )  C( n )

or new analytic stout link method (hep-lat/0311018)


   C U 
i
i
Q        
Tr      
2
2N
U ( n1)  expiQ( 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 tE )
  (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
m1

CP CP
L   nm  nm 
m1
CP   1N
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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)
BiLF (t )  L  D
( R) U R BiF (t ) U R
g O D ROhD
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 revealedeffective 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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