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
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
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
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
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
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
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
Analysis methods Importing data Input data Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 8 / 33
Analysis methods Importing data Folding Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 9 / 33
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
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
Analysis methods Bootstrap A quick review The bootstrap method: ◮ Analysis function f : Transforms input data X into output data Y E.g. samples X = { x i } i to median Y ◮ f ( X ) is estimate for value ◮ Generate R samples from X : { ˜ X i } i ◮ Apply f to each sample: { f (˜ X i ) } i ◮ Error is standard deviation: ∆ Y = σ ( { f (˜ X i ) } 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
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
Analysis methods Bootstrap Averaging for further analysis Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 14 / 33
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
Analysis methods Correlated fit Correlation in data points ◮ Correlation functions are correlated in time ◮ Regular fit will give wrong χ 2 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
Analysis methods Correlated fit New χ 2 New χ 2 : T R 1 χ 2 := � � x iR − f ( t i , λ )] C − 1 � � [¯ ¯ x jR − f ( t j , λ ) ¯ x iR := x ir ij , R r = 1 i , j Correlation matrix: R 1 � C ij := [ x ir − ¯ x iR ][ x jr − ¯ x jR ] R [ R − 1 ] r = 1 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
Analysis methods Correlated fit Correlation matrix Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 18 / 33
Analysis methods Correlated fit Correlated fit Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 19 / 33
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
Analysis methods Scattering length From masses to scattering length There is a relation between mass difference to scattering length (Lüscher 1986, (1.3)): a 2 � � 4 π a 0 a 0 0 W = 2 m − 1 + c 1 L + c 2 mL 3 L 2 W Mass of π - π -system m Mass of single π a 0 s-wave scattering length L Number of spatial lattice sites c 1 − 2 , 837 297 c 2 6 , 375 183 Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 21 / 33
Analysis methods Scattering length Motivation of Lüscher’s formula wo identical particles in L 3 box ◮ T ◮ H = H 0 + 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
Analysis methods Scattering length Application of Lüscher’s formula Solve for a 0 a 2 � � 4 π a 0 a 0 0 1 + c 1 L + c 2 = 0 2 m − W − mL 3 L 2 using root finding like ◮ Newton from a 0 = 0 ◮ Brent (1973) Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 23 / 33
Analysis methods Scattering length End results Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 24 / 33
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
Results Correlation functions Folded Correlator 3 10 0.05 0.04 0.03 2 10 0.02 2 [ C ( t ) + C ( T − t )] 0.01 Residual 0.00 1 1 10 0.01 0.02 0.03 0 10 0.04 5 0 5 10 15 20 25 30 t/a Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 26 / 33
Results Correlation functions Folded Correlator 6 10 2.0 1.5 5 10 1.0 0.5 4 10 2 [ C ( t ) + C ( T − t )] Residual 0.0 3 10 0.5 1 1.0 2 10 1.5 1 10 2.0 5 0 5 10 15 20 25 30 t/a Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 27 / 33
Results Effective mass Effective Mass cosh − 1 ([ C ( t − 1) + C ( t +1)] / [2 C ( t )]) 0.50 0.45 0.40 m eff ( t ) 0.35 0.30 0.25 0 5 10 15 20 t/a 0.226 0.225 m eff ( t ) 0.224 0.223 0.222 10 12 14 16 18 20 22 t/a Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 28 / 33
Results Effective mass Effective Mass cosh − 1 ([ C ( t − 1) + C ( t +1)] / [2 C ( t )]) 0.8 0.7 0.6 m eff ( t ) 0.5 0.4 0.3 0 5 10 15 20 t/a 0.46 0.44 0.42 0.40 m eff ( t ) 0.38 0.36 0.34 0.32 10 12 14 16 18 20 22 t/a Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 29 / 33
Results Bootstrap distribution Bootstrap distribution of a_0*m_2 in A100.24 250 200 150 100 50 0 0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26 Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 30 / 33
Results Extrapolation 0.05 A30.32 A40.24 A40.24 A40.32 0.10 A40.20 A60.24 A60.24 B55.32 0.15 D45.32 A80.24 A100.24 A100.24 A100.24 m π a 0 0.20 0.25 0.30 0.35 2.0 2.2 2.4 2.6 2.8 m π /f π 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
References References π – π Scattering with N f = 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
Download Get the paper To get the paper and slides, go to: Or scan the code: ◮ martin-ueding.de ◮ University ◮ Master of Science in Physics ◮ physics760 Computational Physics Made with L A T EX Beamer, SciPy, matplotlib and Inkscape. Martin Ueding – mu@martin-ueding.de Analysis of π – π LQCD scattering data 2015-03-20 33 / 33
Recommend
More recommend
