11/22/2004excited states (c. morningstar)1 extracting excited-state energies with application to the...

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


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


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)


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


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


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


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


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,


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


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


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


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


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


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 /


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


String spectrum (String spectrum (N=1,2,3N=1,2,3))

level orderings for N=1,2,3


String spectrum (String spectrum (N=4N=4))

N=4 levels


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


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


Gluon excitation gaps (Gluon excitation gaps (N=1,2N=1,2))

comparison of gaps with RN /


Gluon excitation gaps (Gluon excitation gaps (N=3N=3))



Gluon excitation gaps (Gluon excitation gaps (N=4N=4))



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)

qq


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



first excitation



gap of first excitation above ground state


3d SU(2) gaps3d SU(2) gaps


1NAu 2NAg 2 NSg

3 NAu 3NSu 3 NAu


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


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


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

)()()( )(


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!


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


Preliminary resultsPreliminary results

principal effective masses for small set of 10 operators


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

11/22/2004excited states (c. morningstar)1 extracting excited-state energies with application to the...

Documents