determining the masses of excited hadrons in qcd by ab initio calculations

Determining the masses of excited hadrons in QCD by ab initio calculationsC. B. Lang© 2005

Nonperturbative QCD

Challenges

Dirac operator?

Pion

Interpolators andPropagators

Variational method

Interpolating field?

Smeared sources

Setting

Mesons:du, us

Baryons:uud, uus, uuu, sss

Conclusion

Determining the masses of excited hadrons in QCD by ab initio calculations

C. B. Lang, Graz

BernGrazRegensburg

QCD collaboration

Tommy BurchChristof GattringerLeonid GlozmanChristian HagenDieter HierlAndreas Schäfer

PR D69 (2004) 094513 NP B [PS] 129/130 (2004)251 PR D70 (2004)054502NPB 140 (2005) 284 (LAT04)NPA755 (2005)481 (BAR04)Lattice 2005: PoS(LAT2005)

ÖPG JahrestagungWien, 30.9.2005


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Nonperturbative QCD

QCD on Euclidean lattices:

( ) ( ) (0)C t X t X ( )

{ }{ }e ( ) (0)mS U D

dU d d N t N

( )

{ }e (det det ...)u d

S U

m mdU D D

1 1 1[( ) ....]u u dD D D

Quark propagators

“quenched”approximation

t

mte

log ( )C t


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Challenges

• Chirality

– Lattice chirality: Ginsparg-Wilson fermions– Chirally improved fermions– How close to the physical limit (or even the chiral limit) can we

come?– The pion is special (it is a Goldstone boson)

• Interpolators

– Quantum numbers– Coupling to the states: ground state dominance– Excited states?– Role of chiral symmetry breaking and quenching


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

What Dirac operator to use?

16

1 21

....p pp P

D c l l l

We use:

“chirally improved fermions”= approximate GW-fermions

Gattringer PRD 63 (2001) 114501;Gattringer et al. Nucl. Phys. B697 (2001) 451

= systematic (truncated) expansion

…obeying the Ginsparg-Wilson relations approximately.

+ . . .+ +=

+


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Hadron masses: pion

mres=0.002

M=280 MeV

GMOR

BGR, Nucl.Phys. B677 (2004)


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Interpolators and propagator analysis

Propagator: sum of exponential decay terms:

'

31 21 2 3

0 | ( ) (0) | 0 0 | | | | 0

0 | | ' ' | | 0 ...

...

n

n

m t

m t

m tm t m t

X t X X n e n X

X n e n X

a e a e a e

…such a fit is highly unstable!

Previous attempts: biased estimators (Bayesian analysis), maximum entropy,...

ground state (large t)

excited states (smaller t)


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Variational method

• Use several interpolating operators

• Compute all cross-correlations and then

• solve the (generalized) eigenvalue problem

• The spectral representation reads:

• Eigenvectors are dominated by physical states:

( ) ( ) (0) ( )i j i j i jC t X t X A A t

( ) (1 ( ))tE t Et e O e

Eigenvectors: best overlap with physical states - “wave functions”

ii iu A

Eigenvalues: masses!

(C. Michael 1985, Lüscher & Wolff 1990)


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Which interpolating fields?

Inspired from heavy quark theory, e.g. for the nucleon:

1 5( ) ( ) ( ) ( )Tabc a b cx u x C d x u x

2 5( ) ( ) ( ) ( )Tabc a b cx u x C d x u x

3 4 5( ) ( ) ( ) ( )Tabc a b cx i u x C d x u x

(plus projections to parity)

1 2 3, , are not sufficient to identify the Roper (cf. Brömmel et al. PR D69 (2004) 094513 )

But:

…excited states have nodes!


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Smeared quark sources

• Smear the quark sources (Jacobi smearing)• Allow for different smearing “widths”• Combine different quark sources in the hadron operators• Eigensystem analysis chooses “best physical states”

- =2.2 1.2


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Summarizing our setting

• Lüscher-Weisz gauge action

• Chirally improved fermions (+HYP smearing)

• Lattice size• 123x24, a=0.148 fm, L~1.8 fm, m ~ 300 MeV• 163x32, a=0.148 fm, L~2.4 fm, m~ 280 MeV• 203x32, a=0.119 fm, L~2.4 fm, m ~ 280 MeV• 100 configurations each

• Nucleons interpolators: each for 6 quark source combinations: (nnn), (nnw), (wnn), (nww), (wwn), (www)

• Other Baryons and Mesons analogously


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Mesons: typeud

pseudoscalar vector


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Mesons: type us

pseudoscalar vector


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Nucleon (uud)

?

RoperChiral approach affects different states differently

Level crossing (from + - + - to + - - +)?


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Sigma (uus)


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Delta (uuu) and Omega (sss)


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Conclusions

• Excited states are hard to identify

• Variational analysis allows to disentangle mixed (physical) states

• Interpolating fields allowing for nodal “wave function”- important for finding the excited states

• Contact with ChPT: – ghost adds complications– quenched chiral logs not (yet) clearly seen

• All these techniques can be used for dynamical fermion results


Nonperturbative QCD

Challenges

Dirac operator?

Pion


Variational method


Smeared sources

Setting

Mesons:du, us


Conclusion

Exit

determining the masses of excited hadrons in qcd by ab initio calculations

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