Bayesian Fitting in Effective Field Theory

Transcription

1 Bayesian Fitting in Effective Field Theory Department of Physics Ohio State University February, 26 Collaborators: D. Phillips (Ohio U.), U. van Kolck (Arizona), R.G.E. Timmermans (Groningen, Nijmegen)

2 Outline Overview Method Request What Does The Typical Physicist Know About Bayesian Statistics? Nothing! But there are exceptions: Experimentalists searching for weak signatures of particles Some Beyond the Standard Model theorists Some lattice gauge theorists doing constrained curve fitting Even where applied, Bayesian methods are controversial Not applied as yet in EFT or nuclear structure calculations The problem described here is just one (simple) example where Bayesian methods might be useful

3 Outline Overview Method Request Constrained Curve Fitting Example (hep-lat/11175) Monte Carlo estimates of a meson correlator G(t) are generated at 24 time steps, t =, 1,, 23. Theory says the exact correlator has the form: G th (t; A n, E n ) = A n e Ent n=1 Challenge: Fit an infinite number of amplitudes A n and energies E n using only 24 G(t) s. Standard procedure: Keep only first few terms in G th Fit using only Monte Carlo data from t t min But choosing t min is ad hoc

4 Outline Overview Method Request Results From Standard Fits 2 2 Energy 1 E 2 E 1 Energy 1 E 2 E t min Number of Terms 8 Left: 2-term fit to n=1 A ne Ent for t t min Competition between large systematic errors for small t min and large statistical errors for large t max Right: Fit values for lowest two energies vs. # of terms

5 Outline Overview Method Request Unconstrained vs. Constrained Fits Goal: Fit all data using as many terms as we wish Plan: Add priors for reasonable A n s and E n s in n A ne Ent 2 2 Energy 1 E 2 E 1 Energy 1 E 2 E Number of Terms Number of Terms 8 Left: unconstrained. Right: constrained.

6 Outline Outline Overview Method Request Overview: Problem(s) to Be Solved Method: Effective Field Theory (EFT) Request: Advice on Applying Bayesian Methods

7 Outline Outline Overview Method Request Goal Overview: Problem(s) to Be Solved Method: Effective Field Theory (EFT) Request: Advice on Applying Bayesian Methods

8 Figure 1: From QCD vacuum to heavy nuclei: the intellectual Bayesian connection Fitting inbetween EFT the hadronic many-body Outline Overview Method Request Goal The Islands of Strong Interaction Physics RHIC CEBAF quarks gluons few nucleons RIA heavy nuclei vacuum quark-gluon plasma QCD nucleon QCD few-body systems free NN force many-body systems effective NN force

9 Outline Overview Method Request Goal The Big Picture (adapted from

10 Outline Overview Method Request Goal Table of the Nuclides Stable nuclei 126 Known nuclei 82 r-process Protons 2 28 rp-process Terra incognita Neutron stars Neutrons Figure The nuclear landscape, defining the territory of nuclear physics research. On this chart of the nuclides, black squares represent stable nuclei and nuclei with half-lives comparable to or longer than the age of the Earth. These nuclei define the valley of stability. Dick By Furnstahl adding either protons Bayesian or neutrons, Fitting inone EFTmoves away from the valley of

11 Outline Overview Method Request Goal Problems with Extrapolations Mass formulas and energy functionals do well where there is data, but elsewhere... two-neutron separation energies S 2n (MeV) Sn data exist Experiment HFB-SLy4 HFB-SkP HFB-D1S SkX RHB-NL3 LEDF Mass Formulae Neutron Number data do not exist Neutron Number Neutron Number exp FRDM CKZ JM MJ T+ Figure 6: Predicted two-neutron separation energies Bayesian for the even-even FittingSn inisotopes EFT using several

12 Outline Overview Method Request Goal Input to Many-Body Problem: Internucleon Force Reproduce data from scattering protons from neutrons, etc. Difficult problem with long history M.L. Goldberger, at the Midwestern Conference on Theoretical Physics, Purdue University, 196: There are few problems in nuclear theoretical physics which have attracted more attention than that of trying to determine the fundamental interaction between two nucleons. It is also true that scarcely ever has the world of physics owed so little to so many.... It is hard to believe that many of the authors are talking about the same problem or, in fact, that they know what the problem is.

13 Outline Overview Method Request Goal Successful Fits to Phase Shift Data Achieved Fit energies from to 35 MeV, with goal of χ 2 /dof 1 Account only for measurement errors Table from Argonne v 18 paper: No theoretical errors, all data treated equally

14 Outline Outline Overview Method Request EFT Chiral Analogs Overview: Problem(s) to Be Solved Method: Effective Field Theory (EFT) Request: Advice on Applying Bayesian Methods

15 Resolution and the Pointillists George Seurat painted using closely spaced small dots (.4 mm wide) of pure pigment Why do the dots blend together?

16 Resolution and the Pointillists George Seurat painted using closely spaced small dots (.4 mm wide) of pure pigment Why do the dots blend together?

17 Wavelength and Resolution

18 Wavelength and Resolution

19 Wavelength and Resolution

20 Wavelength and Resolution

21 Wavelength and Resolution

22 Wavelength and Resolution

23 Wavelength and Resolution

24 Wavelength and Resolution

25 Wavelength and Resolution

26 Principles of Effective Low-Energy Theories

27 Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved

28 Principles of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved use low-energy variables for low-energy processes short-distance structure can be replaced by something simpler without distorting low-energy observables

29 Effective Field Theory Ingredients From Crossing the Border [nucl-th/864] 1 Use the most general L with low-energy dof s consistent with the global and local symmetries of the underlying theory 2 Declaration of regularization and renormalization scheme 3 Well-defined power counting = expansion parameters

30 Effective Field Theory Ingredients: Chiral NN From Crossing the Border [nucl-th/864] 1 Use the most general L with low-energy dof s consistent with the global and local symmetries of the underlying theory L eft = L ππ + L πn + L NN chiral symmetry = systematic long-distance pion physics 2 Declaration of regularization and renormalization scheme momentum cutoff and Weinberg counting use cutoff sensitivity as measure of uncertainties! 3 Well-defined power counting = expansion parameters use the separation of scales = {p, m π} with Λ Λ χ 1 GeV χ chiral symmetry = V NN = ν=ν min c ν Q ν with ν naturalness: parameters are O(1) in appropriate units

31 Chiral Lagrangian Unified description of ππ, πn, and NN N Lowest orders: L () = 1 2 µπ µ π 1 2 M2 π 2 + N [ i + g A 2F τ σ π 1 4F 1 2 C S(N N)(N N) 1 2 C T (N σn)(n σn) +..., [ L (1) = N 4c 1 M 2 2c 1 F 2 M2 π 2 + c 2 F 2 π2 + c 3 F 2 ( µπ µ π) c ] 4 2F 2 ɛ ijk ɛ abc σ i τ a ( j π b )( k π c ) N ] τ (π π) N 2 D 4F (N N)(N στ N) π 1 2 E (N N)(N τ N) (N τ N) +... Infinite # of unknown parameters in hierarchy

32 Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N 1S 3S P 1D D3 3G

33 Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q 1S 3S P 1D D3 3G

34 Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q Q 1 1S 3S P 1D D3 3G

35 Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q Q 1 Q 2 1S 3S P 1D D3 3G

36 Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q Q 1 Q 2 Q 3 1S 3S P 1D D3 3G

37 Chiral Effective Field Theory for Two Nucleons Epelbaum, Meißner, et al. Also Entem, Machleidt L πn + match at low energy Q ν 1π 2π 4N Q Q 1 Q 2 Q 3 Q 4 many many 4 (15) 1S 3S P 1D D3 3G

38 Motivation For Applying Effective Field Theory Systematic calculations with theoretical error estimates Reliable, model independent extrapolation Analogy between EFT and basic numerical analysis naive error analysis: pick a method and reduce the mesh size (e.g., increase grid points) until the error is acceptable sophisticated error analysis: understand scaling and sources of error (e.g., algorithm vs. round-off errors) = Does it work as well as it should? representation dependence = not all are equally effective! extrapolation: completeness of an expansion basis

39 Error Plots in Numerical Analysis relative error Numerical Derivatives f (x) = [f(x+h)-f(x)]/h + O(h) mesh size h

40 Error Plots in Numerical Analysis relative error Numerical Derivatives f (x) = [f(x+h)-f(x)]/h + O(h) f (x) = [f(x+h/2)-f(x-h/2)]/h + O(h 2 ) mesh size h

41 Error Plots in Numerical Analysis relative error Numerical Derivatives f (x) = [f(x+h)-f(x)]/h + O(h) f (x) = [f(x+h/2)-f(x-h/2)]/h + O(h 2 ) Richardson extrapolation O(h 4 ) mesh size h

42 Error Plots in Numerical Analysis relative error h n to h n Numerical Integration trapezoid rule O(h 2 ) mesh size h

43 Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) mesh size h

44 Error Plots in Numerical Analysis Numerical Integration relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) mesh size h

45 The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx

46 The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration mesh size h

47 The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

48 The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

49 The Representation Can Make A Difference! E.g., elliptic integral: 1 (1 x 2 )(2 x) dx How do the numerical errors behave? After transformation: π/2 sin 2 y 2 cos y dy relative error h n to h n trapezoid rule O(h 2 ) Simpson s rule O(h 4 ) Milne s rule O(h 6 ) before Numerical Integration after mesh size h

50 Error Plots in Effective Field Theory G. P. Lepage, How to Renormalize the Schrödinger Equation! "$# % % & Errors in the 1 S phase shifts versus energy through orders Λ 2 and Λ 4. Fit using data at low energy only.

51 Error Plots in Effective Field Theory G. P. Lepage, How to Renormalize the Schrödinger Equation!#"%$ & & '(' & ) & Errors in the 1 S phase shifts versus energy for different values of the cutoff Λ.

52 Best Calculation: Theoretical Error A Posteriori Phase Shift [deg] Phase Shift [deg] Phase Shift [deg] Phase Shift [deg] ε 1 3 D2 1 S 3 P Lab. Energy [MeV] D2 3 D3 3 S1 3 P P1 3 P2 3 D Lab. Energy [MeV] Lab. Energy [MeV] ε 2

53 Naturalness of Coefficients (Epelbaum et al.) Georgi-Manohar naive dimensional analysis (NDA): ( N ) l ( ) m ( ( )N π µ ) n, m π L χ eft = c lmn fπλ 2 f χ f π Λ πλ 2 2 χ χ f π 1 MeV and Λ χ 1 MeV check NLO, NNLO constants from L NN (cutoff MeV): fπ 2 C S fπ 2 C T fπ 2 Λ 2 χ C fπ 2 Λ 2 χ C f 2 π Λ 2 χ C f 2 π Λ 2 χ C f 2 π Λ 2 χ C f 2 π Λ 2 χ C f 2 π Λ 2 χ C /3 c lmn 3 = natural! = truncation error estimates f 2 π C T unnaturally small = SU(4) spin-isospin symmetry

54 Outline Outline Overview Method Request Overview: Problem(s) to Be Solved Method: Effective Field Theory (EFT) Request: Advice on Applying Bayesian Methods

55 Outline Overview Method Request Summary of (Some of the) Questions How can we incorporate the expected behavior of the theoretical error based on naturalness and the EFT hierarchy? Frequently Asked Question: How much of the scattering data should be fit (i.e., up to what energy)? Traditionalist says: fit all data up to 35 MeV with χ 2 /dof 1 EFT practitioner says: only use the low-energy data How do we use all the data, accounting for the expected better description at low energy? How do we calculate uncertainties in our best-fit parameters? This is particularly important when using the fit to calculate elsewhere (e.g., nuclei)