Assessing Theory Errors from Residual Cutoff Dependence H. W. Grießhammer INS Institute for Nuclear Studies Institute for Nuclear Studies The George Washington University, DC, USA The EFT Promise: Serious Theorists Have Error Bars 1 The EFT-Cookbook 2 a a Error Plots Test Power Counting & Renormalisation 3 Concluding Questions 4 Providing reliable theoretical uncertainties, testing non-perturbative EFTs. hg: Nucl. Phys. A744 (2004) 192; hg: NNPSS 2008, Saclay workshop 04.03.2013, Benasque workshop 24.07.2014; hg: Chiral Dynamics proceedings [arXiv:1511.00490 [nucl-th]] Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 0-1
1. The EFT Promise: Serious Theorists Have Error Bars Scientific Method: Quantitative results with corridor of theoretical uncertainties for falsifiable predictions . “Double-Blind” Theory Errors: Assess with pretense of no/very limited data. PHYSICAL REVIEW A 83 , 040001 (2011) Editorial: Uncertainty Estimates Editorial: Uncertainty Estimates The purpose of this Editorial is to discuss the importance of including uncertainty estimates in papers involving theoretical calculations of physical quantities. It is not unusual for manuscripts on theoretical work to be submitted without uncertainty estimates for numerical results. In is not unusual for manuscripts on theoretical work to be submitted without uncertainty estimates contrast, papers presenting the results of laboratory measurements would usually not be considered acceptable for publication ry measurements would usually not be considered acceptable in Physical Review A without a detailed discussion of the uncertainties involved in the measurements. For example, a graphical presentation of data is always accompanied by error bars for the data points. The determination of these error bars is often the most difficult part of the measurement. Without them, it is impossible to tell whether or not bumps and irregularities in the data are real physical effects, or artifacts of the measurement. Even papers reporting the observation of entirely new phenomena need to contain enough information to convince the reader that the effect being reported is real. The standards become much more rigorous for papers claiming high accuracy. The question is to what extent can the same high standards be applied to papers reporting the results of theoretical calculations. It is all too often the case that the numerical results are presented without uncertainty estimates. Authors sometimes say that it es. Authors sometimes say that it is difficult to arrive at error estimates. Should this be considered an adequate reason for omitting them? In order to answer this is difficult to arrive at error estimates. Should this be considered an adequate reason for omitting them? question, we need to consider the goals and objectives of the theoretical (or computational) work being done. Theoretical papers can be broadly classified as follows: Workshop “Predictive Capabilities of Nuclear Theories” , Krakow (Poland), 25 Aug 2012 Special Issue J. Phys. G (Feb 2015): “Enhancing the Interaction between Nuclear Experiment and Theory through Information and Statistics” Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 1-1
2. The EFT-Cookbook (a) Power-Counting Non-Perturbative EFTs −15 E [MeV] λ [fm=10 m] Correct long-range + symmetries: Chiral SSB, gauge, iso-spin,. . . 0.2 p,n (940) Short-range: ignorance into minimal parameter-set at given order. ω,ρ (770) Systematic ordering in Q = typ . momentum p typ ≪ 1 breakdown scale Λ EFT λ Controlled approximation: model-independent, error-estimate. M −M ∆ N R 1 π (140) 2 = ⇒ Chiral Effective Field Theory χ EFT ≡ low-energy QCD H 5 0 8 = ⇒ Pion-less Effective Field Theory EFT( / π ) ≡ low-energy χ EFT Shallow real/virtual QCD bound states = ⇒ Few- N non-perturbative! Λ EFT cut−off Λ T LO = V LO + V LO G T LO T NLO = ( ✶ + T † (Λ) LO ) V NLO ( ✶ + T LO ) strict perturbation about LO physical observable momenta !! unphysical = ⇒ Analytic results rare; regularisation by cut-off Λ � = Λ EFT . momenta = ⇒ saturated at Λ EFT � Λ . Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 2-1
prepared for Orsay Workshop by Grießhammer 7.3.2013 (b) NN χ EFT Power Counting Comparison based on and approved by the authors in private communications Derived with explicit & implicit assumptions; contentious issue. Proposed order Q n at which counter-term enters differs . = ⇒ Predict different accuracy, # of parameters. wave order Yang/Long Pavon Valderrama Birse PR C86 (2012) 024001 etc. PR C74 (2006) 054001 etc. PR C74 (2006) 014003 1 S 0 − 1 LO 0 NLO N 2 LO 1 2 3 S 1 − 1 LO 1 1 2 NLO 2 3 SD 1 − 1 − 1 1 LO 2 1 2 NLO 2 3 D 1 − 1 − 1 LO 2 1 2 NLO 2 3 P 0 (attr. triplet) − 1 − 1 LO 2 1 2 TPE LO # of param. at Q − 1 2 3 4 # of param. at Q 0 4 6 6 # of param. at Q 1 8 6 9 Weinberg: LO: 2 ; NLO: + 0 ; N 2 LO: + 7 = 9 – different channels; consistency questioned Beane/. . . 2002; Nogga/. . . 2005 With same χ 2 , proposal with least parameters wins : minimum information bias. Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 3-1
(c) (Some) Ways to Estimate Theoretical Uncertainties at fixed k Choose most conservative/worst-case error for final estimate! Clearly state your choice! n typ. low scale p typ c i ( Λ ) Q i complete up to order Q n ( n = 0 is LO). ∑ Expansion parameter Q = = ⇒ O = typ. high scale Λ EFT i = 0 – A priori: Q n + 1 of LO. – Convergence pattern of series: smaller corrections LO → NLO → N 2 LO → . . . | c i | captures corridor with n + 1 ⇒ Bayesian estimate: error Q n + 1 × max = n + 2 × 100% degree of belief. i Furnstahl/Klco/Phillips/Wesolowski (B UQEYE ) 2015 – Less dependence on particular low-E data taken for LECs. (e.g. Z -param. vs. ERE; fit H 0 to a 3 vs. B 3 ,. . . ) – Include selected higher-order RG- & gauge-invariant effects: does not increase accuracy. – Corridor mapped by cutoff Λ in wide range . � � � � 2.5 Should decrease order-by-order. � � � � � � � ��� � � � � �� � � � 1 � 2 � 2.0 � � � � � � c � � � � rad MeV Example: PV coefficient in nd at k = 0 . � � � � � � � 1.5 � � � � � ��� � � � � hg/Schindler/Springer 2012 � � � 1.0 � � � � 0.5 � 0.0 200 500 1000 2000 5000 � � MeV � Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 4-1
3. Error Plots Test Power Counting & Renormalisation hg 2004-; 1511.00490 (a) Using Cut-Offs to Your Advantage Observable O ( k ) at momentum k , order Q n in EFT, cut-off Λ : � k , p typ. � i � k , p typ. � n + 1 n ∑ O n ( k ; µ ) = O i + C ( Λ ; k , p typ , Λ EFT ) Λ EFT Λ EFT i � �� � � �� � renormalised, Λ -indep. residual Λ -dependence parametrically small C “of natural size” � k , p typ. � n + 1 ⇒ Difference between any two cut-offs: O n ( k ; Λ 1 ) −O n ( k ; Λ 2 ) × C ( Λ 1 ) −C ( Λ 2 ) = = O n ( k ; Λ 1 ) C ( Λ 1 ) Λ EFT Isolate breakdown scale Λ EFT , order n by double-ln plot of “derivative of observable w. r. t. cut-off”. Test consistency: Does numerics match predicted convergence pattern? � n + 1 dln C ( Λ ) � k , p typ. ⇒ Λ d O Renormalisation Group Evolution: Λ 1 → Λ 2 = d Λ = → 0 if exact RGE. O dln Λ Λ EFT Residual Λ -dependence decreases parametrically order-by-order. Complication: Several intrinsic low-energy scales in few-N EFT: scattering momentum k , m π , inverse NN scatt. lengths γ ( 3 S 1 ) ≈ 45 MeV , γ ( 1 S 0 ) ≈ 8 MeV ,. . . Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 5-1
(b) Example: nd Doublet- S Wave in EFT( / π ) Bedaque/hg/Hammer/Rupak 2002, hg 2004 Does momentum-dependent 3NI H 2 enter at N 2 LO hg/. . . 2002-4 – or higher Platter/Phillips 2006 ? k � γ , other scales = ⇒ plateau obscures slope cutoff dependence decreases with order γ , ··· ≪ k ≪ Λ / π = ⇒ extract slope � n + 1 � � � NLO N 2 LO N 2 LO without H 2 LO � 1 − k cot δ ( µ = 200 MeV ) k , p typ . � � � ∼ � � k cot δ ( µ = ∞ ) Λ / n + 1 fitted ∼ 1 . 9 2 . 9 4 . 8 3 . 1 π � �� � Qn + 1 n + 1 predicted 2 3 4 not renormalised ⇒ Fit to k ∈ [ 70;100 ... 130 ] MeV ≫ γ ,... : H 2 is N 2 LO ; re-confirmed by Ji/Phillips 2013 = Slope Confirms Power Counting; Estimates Λ / π ≈ 140 MeV ; Determines Mom.-Dep. Uncertainties. Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 6-1
(c) Comments: It’s Not The Golden Bullet, but Worth A Try � k , p typ. � n + 1 O n ( k ; Λ 1 ) −O n ( k ; Λ 2 ) × C ( Λ 1 ) −C ( Λ 2 ) = O n ( k ; Λ 1 ) C ( Λ 1 ) Λ EFT – n , Λ EFT regulator independent. – But not C : flexible regulator. . . – Non-integer powers, non-analyticities: n + 1 → n + Re [ α ] with n �∈ Z . – Fit quality also tests assumed functional dependencies. � k , p typ. � n + 1 – Estimate k -dependence of expansion parameter Q ( k ) = = ⇒ Res. theoret. error. Λ EFT – Needs “Window of Opportunity” p typ ≪ k ≪ Λ EFT . – Any two cutoffs Λ 1 , Λ 2 – Numerical leverage?! Error plots, RRTF Coulomb Darmstadt (40+X)’, 12.01.2016 Grießhammer, INS@GWU 7-1
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