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New Vista on

Excited States in Lattice Gauge Theories

Ahmad Hosseinizadeh

Summer School on LQCD, Aug. 20, 2007

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Contents:

Monte Carlo HamiltonianSpectrum of excited statesTemperature windowsOutlook

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Lagrangian LGT (based on the Wilson action, ...):

Standard approach and very successful

Difficulties:Excited states spectrum Wave functions

Monte Carlo Hamiltonian

A critical review:

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Hamiltonian LGT (based on the Kogut-Susskind Hamiltoni):

Allows to compute the excited states spectra and wave functions.

Big problem:To find a set of basis states which are physically relevant

Monte Carlo Hamiltonian

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Monte Carlo Hamiltonian

infiinfi )|xHT/(exp|x0,,T|xx

Transition amplitude in the Lagrangian method (in 1-D):

Monte Carlo Hamiltonian

Transition amplitudes between position states via Hamiltonian:

]x[S1

exp]dx[

]x[S1

exp]U[O]dx[O

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Infinite degreeH

Finite degreeeffH

CC]C[O

N1

O

inefffi0,infi x|)/THexp(|xx|T,x

MC with imp. sampling :

Monte Carlo Hamiltonian

?

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Euclidean transition amplitude between two physical states:

inksfi infi U)T/exp(U0t,UTt,U H

In lattice gauge theories:

Tt

inU

fiU

Monte Carlo Hamiltonian

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,...23U,12UUBargmann link states:

IU> s are gauge invariant projection of the Bargmann link states.

fi

in

UUininvfi |S[U])exp([dU]0t,U|P|Tt,U

Monte Carlo Hamiltonian

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NNij (T)][MM(T)

Transfer matrix for a finite T

Transfer matrix corresponding to a finite T for the purpose of reconstruct the spectrum in some finite low energy domain.

Monte Carlo Hamiltonian

N...,2,1,ji,,U|)/ THexp(|U(T)M jksiij

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)T(MDiagonalization

Spectrum

Wave functions

Monte Carlo Hamiltonian

00 U)T(DU)T(M

M(T) is a positive and Hermitian matrix.

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|EEE|H effk

N

1k

effk

effkeff

Effective Hamiltonian:

Lesson:

an effective H can be found via MC, Such that transition amplitudes become a finite sum over N eigenstates, where N is in the order of N_c.

Monte Carlo Hamiltonian

Kröger et al, Phys. Lett. A258 (1999) 6.Huang et al, Phys.Lett. A299 (2002) 483.

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ikin

ji

kinj

ij

ij

U|)T/Hexp(|UU|)T/Hexp(|U

U|)HT/exp(|U

U|)HT/exp(|U)T(M ij

Computation of matrix elements

Monte Carlo Hamiltonian

Ratio of path integrals Kinetic T.A.

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Ratio

Kinetic amplitude

MC with importance sampling

Analytic methods using group theory

Monte Carlo Hamiltonian

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Regular configurations:

is not easy to do!

Distribution of configurations:

ix1ix 1ix

)x(P

Monte Carlo Hamiltonian

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Solution: Stochastic configurations

Guidance by physics: Let physics tell us which states are important.

Lesson: Apply Monte Carlo using action weight factor.

ix1ix 1ix

)x(P

Monte Carlo Hamiltonian

Note: stoch. states must be normalizable.

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A short story of MC Hamiltonian:

Stochastic configurations Transfer Matrix

Spectra & Wave func.

Monte Carlo Hamiltonian

Physical observables

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Example: Four plaquettes

Spectrum of excited statesSpectrum of excited states

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Temperature window

Monte Carlo Hamiltonian has a window of validity. Beyond this window, the effective Hamiltonian becomes unphysical.

Temperature window

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Temperature window

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Temperature Window: Wave Functions Temperature window

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Thermodynamical functions

Definition:

In lattice:

By MC Hamiltonian:

ZU

HTrZ

log)(

)],[exp()(

SaNa

NU

ttt

s 1

2)(

]exp[)(

1)(

,]exp[)(

1

1

neff

N

n

neff

effeff

N

n

neffeff

EEZ

U

EZ

Temperature window

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Average energy U

Temperature window

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Entropy S

Temperature window

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Application of Monte Carlo Hamiltonian:

- Idea works well in QM and scalar field theory.- Hadronic wave and structure functions in QCD

(for small x_B and Q^2)- S-matrix, scattering and decay amplitudes- Finite density QED and QCD

Outlook