using a new analysis method to extract excited states in ... meson sector lattice conference 2016 -...

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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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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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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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


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


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


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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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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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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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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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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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



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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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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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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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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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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Thanks

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using a new analysis method to extract excited states in ... meson sector lattice conference 2016 -...

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