using a new analysis method to extract excited states in ... meson sector lattice conference 2016 -...
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IntroductionAMIASResults
Using a new analysis method to extract excited states in thescalar meson sector
Lattice Conference 2016 - Southampton
A. Abdel-Rehim, C. Alexandrou, J. Berlin, M. Brida, J. Finkenrath, T. Leontiou,M. Wagner
25.07.2016
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IntroductionAMIASResults
Motivationqq
Content
* Motivation- Scalar channel qq
* AMIAS (Athens Model Independent Analysis Scheme)- Idea- Application
* AMIAS in practice- Search for a0 excluding loops- Search for a0 including loops
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IntroductionAMIASResults
Motivationqq
TARGET: extract mass states from correlation matrix
→ for large time distances lowest state can be extracted, but . . .
- how to extract if signal is only good for few time slices ?
- how to extract if signal is dominated by more than one states ?
We are using: ensemble generated with 2+1 dynamical clover fermions andIwasaki gauge action by the PACS-CS Collaboration [Aoki et.al. 2008]
Lattice 64× 323 with a ∼ 0.09 fm and∼ 500 configurations at Mπ ∼ 300 MeV
We are using a0 interpolators with JP = 0+ [Joshua Berlin’s Talk]interpolators quark content: du + ss
→ 6x6 correlation matrix with qq, diQdiQ, πηs, 2K
→ we will start with the qq correlator
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IntroductionAMIASResults
Motivationqq
The Scalar channel: qqOperator with JP = 0+: O(x) = d(x)u(x)Correlator C(t) = 〈O(t)O†(0)〉 on a periodic lattice:
0 5 10 15 20 25
10−4
10−3
10−2
10−1
100
t/a
C(t
/a)
Standard technique to extract effective masses using asymptotic behavior:
C(t/a) =∑i
Aicosh(Ei(t− T/2)) −→t�1,T�1
A0cosh(E0(t− T/2)) + . . . (1)
→ here dominated by two states4 / 18
IntroductionAMIASResults
Motivationqq
The Scalar channel: qqOperator with JP = 0+: O(x) = d(x)u(x)Effective mass on a periodic lattice:
2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t/a
a E
eff (
t/a)
Standard technique to extract effective masses using asymptotic behavior:
C(t/a)
C(t/a− 1)=
cosh(Eeff (t− T/2))
cosh(Eeff (t+ 1− T/2))(2)
→ dominated by two states5 / 18
IntroductionAMIASResults
Motivationqq
The Scalar channel: qqhere: lowest energy easy to identify (high statistics)however: we are not interesting in lowest state (unphysical)−→ lattice artefact, backwards traveling pion
2 4 6 8 10 12 140
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t/a
a E
eff (
t/a)
- higher state do not have a plateau- can’nt be extracted by a simple plateau average
→ what is with the correlator ?6 / 18
IntroductionAMIASResults
Motivationqq
The Scalar channel: qqhere: we simply fit the logarithm behavior for t ∈ [2; 4] and t ∈ [8; 12]
0 5 10 15 20 25
10−4
10−3
10−2
10−1
100
t/a
C(t
/a)
Another possiblity: fitting the correlator
f(t) =N∑n=0
An cosh(En(t− T/2))
This is a non–linear minimization
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IntroductionAMIASResults
IdeaApplication
AMIAS
- Standard least-squares algorithms:are numerically unstable,depend on the initial condition andfail for very low signal-to-noise ratio
⇒ to circumvent this we will use AMIAS
AMIAS (Athens Model Independent Analysis Scheme)
- relies on statistical concepts
- sample probability distributions of parameters
- also insensitive parameters are fully accounted and do not bias the results
- can access a large number of parameters by using Monte Carlo techniques
[Alexandrou et.al. arXiv:1411.6765],[C. Papanicolas and E. Stiliaris, arXiv:1205.6505],
[E. Stiliaris and C. Papanicolas, AIP Conf. Proc. 904, 257 (2007)]
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IntroductionAMIASResults
IdeaApplication
AMIAS: Basic Idea
- Using the χ2 of the fit function f(t) =∑Nn=0 Ai cosh(En(t− T/2))
χ2 =
Nt∑k=1
(C(tk)−∑∞n=0 Ancosh{En(tk − T/2)})2
(σ2tk/N)
.
- Using the central limit theorem⇒ each value assigned to the model parameters has a statistical weightproportional to
P (C(tn);n) = e−χ2
2
Model parametersThe probability for parameter Ai, (Ei) to have a specific value ai is given by
Π(Ai = ai) =
∫ cibidAi
∫∞−∞
∏j 6=i dAj , Aie
−χ̃2/2∫∞−∞
(∏j dAj
)Aie−χ̃
2/2.
⇒ Multi-dimensional integrals → Monte Carlo sampling with P (C(tn);n)
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IntroductionAMIASResults
IdeaApplication
Scalar channel: qq
Results from AMIAS:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
aE
0 0.5 1
A
E1
E2
A1
A2
⇒ distribution of two energies and their corresponding apmlitudes
- clean distinction
- need to set intervalls for identifying themwe use multiple tempering to explore the whole parameter region
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IntroductionAMIASResults
IdeaApplication
Scalar channel: qqResults from AMIAS:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
aE
0 0.5 1
A
E1
E2
A1
A2
lattice artefact:the π is propagating backwarts in timegenerate a lattice artefact state with→ aEart = a(mηs −mπ) ∼ 0.2
state around the a0-region ∼ 0.6:in this region is the is π + ηs and the 2 kaon channel−→ we need to couple the meson-meson interpolators to qq
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IntroductionAMIASResults
Without loopsWith loops
Using in search for a0 particleAMIAS: can be also adapted for correlation matrices (“Search for a0 interpolatorsJP = 0+”)
(j = 0) : Oqq̄ =∑
x
(dxux
)(j = 1) : OKK̄point =
∑x
(sxγ5ux
)(dxγ5sx
)(j = 2) : Oηssπpoint =
∑x
(sxγ5sx
)(dxγ5ux
)(j = 3) : OdiQdiQ̄ =
∑x εabc
(sx,bCγ5d
Tx,c
)εade
(uTx,dCγ5sx,e
)(j = 4) : OKK̄2−part =
∑x,y
(sxγ5ux
)(dyγ5sy
)(j = 5) : Oηssπ2−part =
∑x,y
(sxγ5sx
)(dyγ5uy
)⇒ 6× 6 correlation matrix:Cjk(t) = 〈Oj(t)O†k(0)〉 ∼
t�T
∑∞n=0〈0|Oj(t)|n〉〈n|O
†k(0)|0〉e−Ent.
To analyze Correlation matrix:Generalized Eigenvalue Problem (GEVP) and AMIAScross check with GEVP by solving:
[C(t)] vn(t, t0) = λn(t, t0) [C(t0)] vn(t, t0),
on the otherside: using Amias by fitting every matrix element with:
Cjk(t) =∞∑n=0
A(n)j A∗k
(n)cosh{−En(t− T/2)}.12 / 18
IntroductionAMIASResults
Without loopsWith loops
Measurements: GEVP 5x5 without loops
- Lowest states coincidence with ηsπ and 2–kaon
- Diquark-antidiquark interpolating field (j = 4) mixed with excited states
- No a0 candidate −→ expected to be around the ηsπ and the 2–kaon states
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IntroductionAMIASResults
Without loopsWith loops
Measurements: AMIAS 4x4 without loopsUsing AMIAS for correlation matrixwith 2–meson interpolators (2× (π + ηs) and 2× 2K)
0.45 0.5 0.55 0.6 0.65 0.7 0.75
aE
-1e-09 -5e-10 0 5e-10 1e-09 1.5e-09
Ain
A11
A12
A13
A14
E1E2
π+η=0.4988(15)
π+η (p)=0.6514(15)
2K=0.5456(21)
2K(p)=0.6723(21)
E3 E4
- Energies correspondes to expectations from single channels
- No a0 candidate −→ expected to be around the ηsπ and the 2–kaon states
- Clear signal for energies and corresponding amplitudes
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IntroductionAMIASResults
Without loopsWith loops
Measurements: AMIAS 4x4 without loops
Overlap of states with different interpolating fields ?→ using energies and corresponding amplitude to reconstruct correlation matrix→ extract overlap from eigenvectors of the orthogonalized GEVP
1 2 4 50
0.2
0.4
0.6
0.8
1
E1
1 2 4 50
0.2
0.4
0.6
0.8
1
E2
1 2 4 50
0.2
0.4
0.6
0.8
1
E3
1 2 4 50
0.2
0.4
0.6
0.8
1
E4
→ agree with results of pure GEVP
→ needs to identify the amplitudes→ can be difficult due to overlapping amplitudes/energies
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IntroductionAMIASResults
Without loopsWith loops
Measurements: AMIAS 6x6 with loops PRELIMINARY4 lowest states can be resolved (higher states can not be identify)
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
aE
- strange quark loops introduces large noise(higher states very difficult to resolve)
- indication for an addtional state in the a0 region→ measurements are ongoing (this are PRELIMINARY results)
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IntroductionAMIASResults
Without loopsWith loops
Conclusions
AMIAS:- sampling of fit parameters by using χ2
- can be used for single channels and correlation matrices
avoid plateau fitscan extrace states from channels which are dominated by more than one state→ like it is done in case of qq
AMIAS in the search for a0:
without loops: resolve four lowest statewith loops : a0 cannidate
Ongoing:- increasing statistics
- investigate AMIAS: for example:- excluding noisy matrix elements- identification of amplitudes and energies
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