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Analysis of LQCD scattering data Martin Ueding mu@martin-ueding.de 2015-03-20 Martin Ueding mu@martin-ueding.de Analysis of LQCD scattering data 2015-03-20 1 / 33 Contents of this presentation Data generation

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Analysis of π–π LQCD scattering data

Martin Ueding – mu@martin-ueding.de 2015-03-20

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 1 / 33

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Contents of this presentation

Data generation Analysis methods Importing data Bootstrap Correlated fit Scattering length Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 2 / 33

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Data generation

Section 1 Data generation

Data generation Analysis methods Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 3 / 33

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Data generation

Monte Carlo for quantum mechanics

Andreas Kell, Martin Efferz and Simon Blanke will introduce

◮ Feynman’s path integral formalism ◮ Markov chains for weighted generation ◮ Metropolis algorithm ◮ Energy from correlation functions

Those ideas can be used for QCD

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 4 / 33

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Data generation

Lattice QCD

◮ Use action of QCD; parametrized by quark mass, . . . ◮ Generate configurations, weighted by exp(−S) ◮ Examine observables on those configurations, meson operators in this

case

◮ Correlation functions lead to masses

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 5 / 33

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Analysis methods

Section 2 Analysis methods

Data generation Analysis methods Importing data Bootstrap Correlated fit Scattering length Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 6 / 33

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Analysis methods Importing data

Subsection 1 Importing data

Data generation Analysis methods Importing data Bootstrap Correlated fit Scattering length Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 7 / 33

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Analysis methods Importing data

Input data

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Analysis methods Importing data

Folding

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Analysis methods Bootstrap

Subsection 2 Bootstrap

Data generation Analysis methods Importing data Bootstrap Correlated fit Scattering length Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 10 / 33

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Analysis methods Bootstrap

Alternatives?

Gaussian error propagation . . .

◮ is tedious ◮ assumes small errors ◮ does not scale

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 11 / 33

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Analysis methods Bootstrap

A quick review

The bootstrap method:

◮ Analysis function f: Transforms input data X into output data Y

E.g. samples X = {xi}i to median Y

◮ f(X) is estimate for value ◮ Generate R samples from X: {˜

Xi}i

◮ Apply f to each sample: {f(˜

Xi)}i

◮ Error is standard deviation: ∆Y = σ({f(˜

Xi)}i) Value and error without computing derivatives!

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 12 / 33

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Analysis methods Bootstrap

Generation of bootstrap samples

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 13 / 33

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Analysis methods Bootstrap

Averaging for further analysis

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Analysis methods Correlated fit

Subsection 3 Correlated fit

Data generation Analysis methods Importing data Bootstrap Correlated fit Scattering length Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 15 / 33

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Analysis methods Correlated fit

Correlation in data points

◮ Correlation functions are correlated in time ◮ Regular fit will give wrong χ2 and p ≈ 1 ◮ Correlated fit is needed

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 16 / 33

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Analysis methods Correlated fit

New χ2

New χ2:

χ2 :=

T

  • i,j

[¯

xiR − f(ti,λ)] C−1

ij

  • ¯

xjR − f(tj,λ)

  • ,

¯

xiR := 1 R

R

  • r=1

xir Correlation matrix: Cij := 1 R[R − 1]

R

  • r=1

[xir − ¯

xiR][xjr − ¯ xjR]

λ Fit parameters

i, j Time slice number R Number of bootstrap samples

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 17 / 33

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Analysis methods Correlated fit

Correlation matrix

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Analysis methods Correlated fit

Correlated fit

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Analysis methods Scattering length

Subsection 4 Scattering length

Data generation Analysis methods Importing data Bootstrap Correlated fit Scattering length Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 20 / 33

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Analysis methods Scattering length

From masses to scattering length

There is a relation between mass difference to scattering length (Lüscher 1986, (1.3)): W = 2m − 4πa0 mL3

  • 1 + c1

a0 L + c2 a2 L2

  • W Mass of π-π-system

m Mass of single π a0 s-wave scattering length L Number of spatial lattice sites c1 −2,837 297 c2 6,375 183

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 21 / 33

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Analysis methods Scattering length

Motivation of Lüscher’s formula

◮ T

wo identical particles in L3 box

◮ H = H0 + V, V short ranged ◮ Probability to be in interaction range ∝ L−3 ◮ First order of V: Energy shift ∝ L−3 ◮ Born series: V(0, 0) ∝ scattering length ◮ Higher order in V gives L−4 and higher terms

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 22 / 33

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Analysis methods Scattering length

Application of Lüscher’s formula

Solve for a0 2m − W − 4πa0 mL3

  • 1 + c1

a0 L + c2 a2 L2

  • = 0

using root finding like

◮ Newton from a0 = 0 ◮ Brent (1973)

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 23 / 33

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Analysis methods Scattering length

End results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 24 / 33

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Results

Section 3 Results

Data generation Analysis methods Results

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 25 / 33

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Results Correlation functions

5 5 10 15 20 25 30 t/a 10 10

1

10

2

10

3 1 2[C(t) +C(T−t)]

Folded Correlator

0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0.05 Residual

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 26 / 33

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Results Correlation functions

5 5 10 15 20 25 30 t/a 10

1

10

2

10

3

10

4

10

5

10

6 1 2[C(t) +C(T−t)]

Folded Correlator

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Residual

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 27 / 33

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Results Effective mass

5 10 15 20 t/a 0.25 0.30 0.35 0.40 0.45 0.50 meff(t)

Effective Mass cosh−1 ([C(t−1) +C(t +1)]/[2C(t)])

10 12 14 16 18 20 22 t/a 0.222 0.223 0.224 0.225 0.226 meff(t)

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 28 / 33

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Results Effective mass

5 10 15 20 t/a 0.3 0.4 0.5 0.6 0.7 0.8 meff(t)

Effective Mass cosh−1 ([C(t−1) +C(t +1)]/[2C(t)])

10 12 14 16 18 20 22 t/a 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 meff(t)

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 29 / 33

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Results Bootstrap distribution

0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26 50 100 150 200 250

Bootstrap distribution of a_0*m_2 in A100.24

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 30 / 33

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Results Extrapolation

2.0 2.2 2.4 2.6 2.8 mπ /fπ 0.35 0.30 0.25 0.20 0.15 0.10 0.05 mπ a0

A30.32 A40.24 A40.24 A40.32 A40.20 A60.24 A60.24 B55.32 D45.32 A80.24 A100.24 A100.24 A100.24

Diamond data points from draft paper, slightly shifted. Pion decay constants from (Helmes et al. 2014, table 1).

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 31 / 33

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References

References

π–π Scattering with Nf = 2 + 1 + 1 Twisted Mass Fermions (2014). Bonn,

  • Germany. arXiv: 1412.0408 [hep-lat].

Lüscher, M. (1986). “Volume Dependence of the Energy Spectum in Massive Quantum Field Theories”. In: Commun. Math. Phys. 105,

  • pp. 153–188.

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 32 / 33

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Download

Get the paper

To get the paper and slides, go to:

◮ martin-ueding.de ◮ University ◮ Master of Science in Physics ◮ physics760

Computational Physics Made with L

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EX Beamer, SciPy, matplotlib and Inkscape. Or scan the code:

Martin Ueding – mu@martin-ueding.de Analysis of π–π LQCD scattering data 2015-03-20 33 / 33