Evidence of Dynamical Processes
from Staggered Meson
Propagators
Eric B. Gregory
University of Liverpool
Evidence of Dynamical Processes from Staggered Meson Propagators – p.1/25
MILC Collaboration:
C. Aubin (Washington U.→ Columbia)
C. Bernard (Washington U.)
T. Burch (Arizona→Regensburg)
T. DeGrand (Colorado)
C. DeTar (Utah)
S. Gottlieb (Indiana)
E. Gregory (Arizona→Liverpool)
U. Heller (APS)
J. Osbourn (Utah→Boston)
R. Sugar (UCSB)
D. Toussaint (Arizona)
Evidence of Dynamical Processes from Staggered Meson Propagators – p.2/25
Overview
Quenched vs. dynamical spectra of:
•
•
1−+ hybrids
Excited States
Asqtad staggered action.
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Gauge configurations:
Z =
Z
DU DψDψ exp − SG [U ] + ψM [U ] ψ
=
Z
DU exp {− (SG [U ])} det M [U ]
Z
Z
=
DU exp {− (SG [U ])} × 1
−→ quenched approximation
1
φ
Z =
DU Dφ Dφ exp −SG [U ] − φ
†
M [U ] M [U ]
−→ dynamical fermions
Z
†
†
Evidence of Dynamical Processes from Staggered Meson Propagators – p.4/25
Hybrid Mesons
q
q
Flux tube in an excited state, with either J P C = 1−+
or J P C = 1+− gluonic excitations.
•
•
•
The q q̄g system is a hybrid meson.
Hybrid can have exotic quantum numbers,
(e.g., ρ meson with a 1+− gluonic excitation:
J P C = 1−+ .
An exotic J P C = 1−+ state is not necessarily a
hybrid. (e.g. q̄ q̄qq state.)
Evidence of Dynamical Processes from Staggered Meson Propagators – p.5/25
1−+ hybrid meson operator
H = ρS × B
H x = ρ y Bz − ρ z By
ρS “taste-singlet” rho (1−− ): γspin ⊗ γtaste = γi ⊗ 1
ρk (x) = χ̄(x)ηk Dk χ(x),
x−
µ
x
x+
µ
Uµ (x)
U (x)
−µ
1
Dµ χ(x) = (U−µ (x)χ(x − µ̂) + U+µ (x)χ(x + µ̂))
2
ηk a the Kogut-Susskind phase (±1).
Evidence of Dynamical Processes from Staggered Meson Propagators – p.6/25
1
−+
hybrid meson operator
To get an eigenstate of charge conjugation, we use a symmetrized
combination of the B field at the site of the quark and at the site of the
antiquark.
χ̄ijk (Bj ηi Di + ηi Di Bj ) χ
x−µ
Fµν(x−µ )
x
x+ µ
Fµν (x+ µ )
+
x−µ
x
x+ µ
Fµν(x)
Evidence of Dynamical Processes from Staggered Meson Propagators – p.7/25
Simulation Parameters
10/g 2
msea a
mval a
a2 σ
Nconfigs
8.40
—
0.040
0.0499(5)
416
8.40
—
0.016
0.0499(5)
416
7.18
0.031
0.031
0.0405(7)
509
7.11
0.0124, 0.031
0.031
0.0424(9)
526
7.11
0.0124, 0.031 0.0124 0.0424(9)
All have lattice spacing a ≈ 0.09fm.
All lattices have dimensions 283 × 96
526
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Quenched 1
−+
propagator
Quenched,
Evidence of Dynamical Processes from Staggered Meson Propagators – p.9/25
10/g 2 = β = 8.40, m = 0.016
Fit propagators:
C(t) = A1 e−M1−+ t + A2 (−1)t e−m2 t + A3 (−1)t e−m3 t
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Three-flavor 10/g 2 = β = 7.11, m = 0.0124
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Three-flavor 10/g 2 = β = 7.11, m = 0.0124
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Fit Results
C(t) = A1 e−M1−+ t + A2 (−1)t e−m2 t + A3 (−1)t e−ma1 t
Here:
M = Mphys a, and t = tphys /a
msea a
mval a
Range
aM1−+
[c.l.]
8.40
—
0.040
4-15
1.062(12)(20)
[0.27]
8.40
—
0.016
4-15
0.973(26)(20)
[0.49]
7.18
0.031
0.031
5-15
0.986(30)(30)
[0.83]
7.11
0.0124, 0.031
0.031
6-15
0.911(34)(100)
[0.25]
7.11
0.0124, 0.031
0.0124
na
10/g 2
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Global Hybrid Results
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Light Quark 1−+ Hybrid Result
To estimate the mass of a light quark 1−+ hybrid meson we use the
jackknife extrapolation of the quenched results to
(mP S /mV )2 = 0.033:
√
M1−+ σ = 4.11(20).
√
Using σ = 472(11)MeV (revised with Υ splitting data),
M1−+
= 1942(105)(3%)(2%)(5%?)
= 1942(160)
Errors are:
• statistical & fitting/mixing
• lattice spacing (∼ a2 g 2 )
• “box”size
• quenching & chiral extrapolation
Evidence of Dynamical Processes from Staggered Meson Propagators – p.15/25
And about the 2+1 flavor simulations??
Illustrates that dynamical quarks introduce new and significant
processes that contribute to the 1−+ propagator.
For example b1 + π can have 1−+ , and as the sum of these masses is
less than the predicted mass of the lowest 1−+ hybrid, we expect that
dynamical quarks introduce the possibility of the hybrid coupling to
this two-meson state.
π
hybrid
b1
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Comparison with other cases
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Excited States: psuedoscalars
Goldstone 0−+ state has no oscillating parity partner (partner
would be exotic 0+− ), hence it is relatively easy to fit the
propagator:
C(t) = A0 e−M0 t + A1 e−M1 t .
Propagator and fits for psuedoscalar pion at β=7.11, amq =0.0124.
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0
−+
psuedoscalar summary
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Kaon propagators
Kaons, J P = 0− , have one light and one strange quark, so no definite
charge-congugations quantum number when mu,d 6= mS .
Their parity partner then has J P = 0+ and is non-exotic.
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Kaon fits
Must fit to three-state form to get excited states:
(1)
C(t) = A0 e−M0 t + A1 e−M1 t + A2 (−1)t e−M2 t
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Kaon summary
Oscillating state is very consistent with mass of π + K. Indication that
dynamical light quark simulations are resoving K0∗ (1430) → K + π
decay channel.
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Compare with quenched fits:
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0
++
system
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Summary & Outlook
• Full QCD simulations with light dynamical quarks probe
important physics that is missed completely by quenched
simulations!!!!
• Quenched estimate for the mass of the light quark 1−+ meson
(1942 ± 160 MeV).
• Suggestion of 1−+ hybrid decays?
• Evidence for K0∗ (1430) → K + π decay.
• Predictions should improve with better
understanding of chiral limit, better full QCD simulations.
• Need to investigate mixing.
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