Accounting for systematic theory errors in nuclear ...ntg/talks/MSU2015_furnstahl.pdfRef: “A...

Accounting for systematic theoryerrors in nuclear calculations

Dick Furnstahl

Department of PhysicsOhio State University

NSCL SeminarJanuary 28, 2015

LENPIC'

Collaborators: D. Phillips, N. Klco, A. Thapilaya (OU),S. Wesolowski (OSU)

Ref: “A recipe for EFT uncertainty quantification in nuclear physics,”Journal of Physics G special issue on UQ (coming soon!)

Outline

Theory errors and nuclear EFT

Bayesian methods applied to a model problem

Application to chiral EFT =⇒ building on EKM

Going forward . . .

Where are the error bars from the chiral EFT Hamiltonian?Oxygen isotopes with different methods

Present status

•  Ab-initio computations with chiral two- and

three-nucleon forces have advanced to mid

mass nuclei

•  Theory has provided predictions in exotic

nuclei, some of which have been confirmed

experimentally

•  Optimization of interactions from chiral effective

field theory with consistent currents

•  First steps towards including two-body currents

in weak decays

experiment

Groundstateenergies

ofOxygenisotopes

Two‐bodycurrents

quenchtheIkedasum

ruleinisotopesof

carbonandoxygenfor

arangeofthree‐body

couplings

Hergert&et&al.,&PRL&110,&242501&(2013)&

S2n for Calcium isotopes (MBPT, also CC)

Exciting advances for neutron-rich nuclei

3N forces key to explain 24O as heaviest oxygen isotope Otsuka, Suzuki, Holt, Schwenk, Akaishi, Phys. Rev. Lett. 105, 032501 (2010).

predicted increased binding for neutron-rich calcium

confirmed in precision Penning trap exp.

5! and 3! deviation in 51,52Ca from AME TITAN collaboration + Holt, Menendez, Schwenk, submitted.

Impact on global predictions?

Nature'498,'346'(2013)'

Theory'preceded'accurate'experiment'

22O spectrum with CCEI (also IM-SRG)

The frontier: medium-mass nuclei

38Neutron number

S2n

(M

eV)

3634323028

2

4

6

8

10

12

14

16

18

ExperimentMBPTCCCI (KB3G)CI (GXPF1A)

Ca

2

3

Theory

Experiment

3432

E(2

+)

(MeV

) 1

(a)

(b)

Microscopic valence-space Shell Model

Coupled Cluster Effective Interaction (valence cluster expansion)

In-medium SRG Effective Interaction

MBPT IM-SRG NN+3N-ind

IM-SRG NN+3N-full

Expt.0

1

2

3

4

5

6

7

8

Ener

gy (M

eV)

0+

2+

2+ 2+

0+ 0+

(2+)2+

2+

(0+)0+

0+

4+

4+ (4+)

4+

2+

0+

2+

0+

3+

3+ 3+

3+

0+

22O

CCEI

Exp.

USD

01234

5678

Ener

gy(M

eV)

0+

2+

3+

0+

2+4+

3+

0+

2+

3+0+

0+

2+

3+

2+4+

4+

22OJansen&et&al.,&PRL&113,&142504&(2014)&

Benchmarking methods: uncertainty?

Chiral EFT Hamiltonian UQ

errors in input data for fit

truncation + regulator artifacts

We seek UQ of all errors

Uncertainty Quantification (UQ) for nuclear theoryPhysical)Review)A)Editorial,)April)2011)

UNCERTAINTY QUANTIFICATION (UQ)

NUCLEI, JET, TORUS, Lattice QCD for hadrons and light nuclei: confront experiment with precision theory!

Realizing full potential of such efforts requires quantification of theory uncertainties!

These arise from: starting Hamiltonian (H), computational/many-body technique, input parameters in H!

Multiple sources of theory uncertainty that connect to and correlate with one another in complicated ways!

Goal: ability to propagate uncertainties to predictions

Physical Review A Editorial, 29 April 2011

To truly assess precision and accuracy, we need to know theory error bars.Much work to be done to establish rigorous UQ. But a lot of activity!

See J. Phys. G special issue: Enhancing the interaction between nuclear experimentand theory through information and statistics, eds. D. Ireland and W. Nazarewicz

Uncertainty Quantification (UQ) for nuclear theoryPhysical)Review)A)Editorial,)April)2011)

UNCERTAINTY QUANTIFICATION (UQ)

NUCLEI, JET, TORUS, Lattice QCD for hadrons and light nuclei: confront experiment with precision theory!

Realizing full potential of such efforts requires quantification of theory uncertainties!

These arise from: starting Hamiltonian (H), computational/many-body technique, input parameters in H!

Multiple sources of theory uncertainty that connect to and correlate with one another in complicated ways!

Goal: ability to propagate uncertainties to predictions

Physical Review A Editorial, 29 April 2011To truly assess precision and accuracy, we need to know theory error bars.Much work to be done to establish rigorous UQ. But a lot of activity!

See J. Phys. G special issue: Enhancing the interaction between nuclear experimentand theory through information and statistics, eds. D. Ireland and W. Nazarewicz

Types of systematic theory errors (not exhaustive)

Truncation of [harmonic oscillator] model space

Analog problem of finite volume and finite spacing on latticeCan understand and approximate what is missing

=⇒ extrapolation to correct for errorsMany recent developments [in JPhysG issue and elsewhere]

Truncation of [EFT] expansion but unknown higher coefficients

But not arbitrary =⇒ use theoretical constraints for statisticsExploit completeness of theory (EFT!)Test theory or alternative theories for which is better

Incomplete or possibly incorrect model

Systematics purely from theory may be uncontrolledUse feedback from data to constrainSee Error Estimates of Theoretical Models: a Guide[Dobaczewski, Nazarewicz, Reinhard, J. Phys. G 41 (2014) 074001]

Principles common to all E(F)Ts (but can be in different forms)

In coordinate space, use R toseparate short and longdistance physics

In momentum space, use Λ toseparate high and low momenta

Much freedom how this is done(e.g., different regulator forms)=⇒ different scales / schemes

Principles common to all E(F)Ts (but can be in different forms)

In coordinate space, use R toseparate short and longdistance physics

In momentum space, use Λ toseparate high and low momenta

Much freedom how this is done(e.g., different regulator forms)=⇒ different scales / schemes

Long distance solved explicitly (symmetries);short-distance captured in some LECs.Naturalness =⇒ scaled LECs are O(1)

Power counting =⇒ expansion parameter(s);e.g., ratio of scales: {p,mπ}/Λ

If Λ < Λbreakdown =⇒ regulator artifacts (use RG!)

Model independence comes from completenessof operator basis (use QFT).

LO:

NLO:renormalization of 1π-exchange renormalization of contact terms7 LECs leading 2π-exchange

2 LECs

N2LO: subleading 2π-exchangerenormalization of 1π-exchange

N3LO:

sub-subleading 2π-exchange 3π-exchange (small)

15 LECs renormalization of contact termsrenormalization of 1π-exchange

��5�*6-�2;7;826�+:.*3260�,7::.,<276;H

V2N$=$V2N$$+V2N$+$V2N$+$V2N$+$… Chiral expansion for the 2N force: (0) (2) (3) (4)

Nucleon-nucleon force up to N3LOOrdonez et al. ’94; Friar & Coon ’94; Kaiser et al. ’97; Epelbaum et al. ’98,‘03; Kaiser ’99-’01; Higa et al. ’03; …

Short4range'LECs'are'fi9ed'to'NN4data

Single4nucleon'LECs'are'fi9ed'to'πN4data

figure from H. Krebs

Hierarchy of nuclear forces in chiral EFT

figure from U.-G. Meißner

breakdown scale Λb = Λχ ~ 500-1000 MeV













New alternative approaches to EFT Hamiltonians

NN potentials unchanged for 10 years but now many parallel developments

Different philosophies, regulators, fitting protocols, . . .

If not strictly renormalizable (regulator dependence completelyremoved at each order), then not EFT =⇒ new power counting

Weinberg power counting with strict adherence to EFT principles(e.g., fix ci ’s in πN to isolate physics, then check predictions)

High-accuracy, sophisticated fitting protocol, covariance analysis

Simultaneous sophisticated fit of πN, NN, NNN LECs

Broaden range of fit beyond few-body systems to improvemany-body accuracy (e.g., energies and radii)

How do we reconcile? Different approaches for different problems?What can each approach tell about the others? UQ: Bayes for all!

[Editorial comment: best progress with competition and cooperation,avoiding “mine is better than yours” attitudes]

Validated Nuclear Interac/ons

Structure and Reac/ons: Light and Medium Nuclei

Structure and Reac/ons: Heavy Nuclei

Chiral EFT Ab-‐ini/o

Op/miza/on Model valida/on Uncertainty Quan/fica/on

Neutron Stars Neutrinos and

Fundamental Symmetries

Ab-‐ini/o RGM CI

Load balancing Eigensolvers Nonlinear solvers Model valida/on Uncertainty Quan/fica/on

DFT TDDFT

Load balancing Op/miza/on Model valida/on Uncertainty Quan/fica/on Eigensolvers Nonlinear solvers Mul/resolu/on analysis

Stellar burning

fusion

Neutron drops

EOS Correla/ons

Fission

Initiative to make N4LO forces available for calculations

LENPICLow Energy Nuclear Physics International Collaboration

Jacek Golak, Roman Skibinski, Kacper Topolniki, Henryk Witala

Sven Binder, Angelo Calci, Kai Hebeler, Joachim Langhammer, Robert Roth

Pieter Maris, Hugh Potter, James Vary

Andreas Nogga

Hiroyuki Kamada

Kai Hebeler (OSU)

Hirschegg, January 29, 2013

Hirschegg 2013:Astrophysics and Nuclear Structure

Chiral three-nucleon forces:From neutron matter to neutron stars

Richard J. Furnstahl

Evgeny Epelbaum, Hermann Krebs Ulf-G. Meißner

Veronique Bernard

1

Nuclear Physics fromLattice Simulations

Ulf-G. Meißner, Univ. Bonn & FZ Julich

Supported by DFG, SFB/TR-16 and by EU, I3HP EPOS and by BMBF 06BN9006 and by HGF VIQCD VH-VI-417

Nuclear Astrophysics Virtual Institute

NLEFT

Nuclear Physics from Lattice Simulations – Ulf-G. Meißner – Bad Honnef, March 20-21, 2012 · � C < ⇥ O > B •

LENPIC'

Limited UQ: Error bands in chiral EFT

right: chiral EFT predictions for p–dspin observables with theory errorsfrom cutoff variation

below: neutron-proton 1S0 phaseshift at NLO, N2LO, and N3LO

0 50 100 150 200 250

Elab

[MeV]

-20

0

20

40

60

Ph

ase

Sh

ift

[d

eg]

1S

0

Problems with this as UQ:

Often underestimates uncertainty

Unpleasing systematics of bands

Statistical interpretation???

What can go wrong in an EFT fit?

[from pingax.com/regulatization-implementation-r]

Overfitting (high variance) or underfitting (high bias) or misfitting?

Well-defined for statistical fits how to checkValidate with subset: if overtrained then fail on additional set(overfit); but how to avoid?If underfit, then chi-squared fails (if theory were correct, thenyou wouldn’t get that data)

What can happen in an EFT fit? What are the complications?More statistical power if larger energy range includedBut EFT is less accurate approaching breakdown scaleHow do we combine data and theory uncertainties?Is the EFT working? Are there too few or too many LECs?

Famous von Neumann quote

Attributed to John von Neumann by Enrico Fermi,as quoted by Freeman Dyson in “A meeting withEnrico Fermi” in Nature 427 (22 January 2004).

Drawing an elephant with four complex parametersJürgen MayerMax Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstr. 108, 01307 Dresden,Germany

Khaled KhairyEuropean Molecular Biology Laboratory, Meyerhofstraße. 1, 69117 Heidelberg, Germany

Jonathon HowardMax Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstr. 108, 01307 Dresden,Germany

!Received 20 August 2008; accepted 5 October 2009"

We define four complex numbers representing the parameters needed to specify an elephantineshape. The real and imaginary parts of these complex numbers are the coefficients of a Fouriercoordinate expansion, a powerful tool for reducing the data required to define shapes. © 2010American Association of Physics Teachers.

#DOI: 10.1119/1.3254017$

A turning point in Freeman Dyson’s life occurred during ameeting in the Spring of 1953 when Enrico Fermi criticizedthe complexity of Dyson’s model by quoting Johnny vonNeumann:1 “With four parameters I can fit an elephant, andwith five I can make him wiggle his trunk.” Since then it hasbecome a well-known saying among physicists, but nobodyhas successfully implemented it.

To parametrize an elephant, we note that its perimeter canbe described as a set of points !x!t" ,y!t"", where t is a pa-rameter that can be interpreted as the elapsed time whilegoing along the path of the contour. If the speed is uniform,t becomes the arc length. We expand x and y separately2 as aFourier series

x!t" = %k=0

!

!Akx cos!kt" + Bk

x sin!kt"" , !1"

y!t" = %k=0

!

!Aky cos!kt" + Bk

y sin!kt"" , !2"

where Akx, Bk

x, Aky, and Bk

y are the expansion coefficients. Thelower indices k apply to the kth term in the expansion, andthe upper indices denote the x or y expansion, respectively.

Using this expansion of the x and y coordinates, we cananalyze shapes by tracing the boundary and calculating thecoefficients in the expansions !using standard methods fromFourier analysis". By truncating the expansion, the shape issmoothed. Truncation leads to a huge reduction in the infor-mation necessary to express a certain shape compared to apixelated image, for example. Székely et al.3 used this ap-proach to segment magnetic resonance imaging data. A simi-lar approach was used to analyze the shapes of red bloodcells,4 with a spherical harmonics expansion serving as a 3Dgeneralization of the Fourier coordinate expansion.

The coefficients represent the best fit to the given shape inthe following sense. The k=0 component corresponds to thecenter of mass of the perimeter. The k=1 component corre-sponds to the best fit ellipse. The higher order components

trace out elliptical corrections analogous to Ptolemy’sepicycles.5 Visualization of the corresponding ellipses can befound at Ref. 6.

We now use this tool to fit an elephant with four param-eters. Wei7 tried this task in 1975 using a least-squares Fou-rier sine series but required about 30 terms. By analyzing thepicture in Fig. 1!a" and eliminating components with ampli-tudes less than 10% of the maximum amplitude, we obtainedan approximate spectrum. The remaining amplitudes were

Fig. 1. !a" Outline of an elephant. !b" Three snapshots of the wiggling trunk.

648 648Am. J. Phys. 78 !6", June 2010 http://aapt.org/ajp © 2010 American Association of Physics Teachers

This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:128.146.34.68 On: Fri, 23 Jan 2015 12:40:58

“Drawing an elephant with fourcomplex parameters,” J. Mayer,K. Khairy, and J. Howard,Am. J. Phys. 78, 648 (2010).

Outline




Going forward . . .

Why is a Bayesian framework well suited to EFT errors?

Frequentist approach to probabilities: long-run relative frequency

Outcomes of experiments treated as random variablesPredict probabilities of observing various outcomesWell adapted to quantities that fluctuate statisticallyBut systematic errors are problematic

Bayesian probabilities: pdf is a measure of state of our knowledge

Ideal for systematic errors (such as theory errors!)Assumptions (or expectations) about EFT encoded in prior pdfsFor EFT, makes explicit what is usually implicit, allowing assumptions tobe applied consistently, tested, and modified in light of new information

Widespread application of Bayesian approaches in theoretical physics

Interpretation of dark-matter searches; structure determination incondensed-matter physics; constrained curve-fitting in lattice QCDIs supersymmetry a “natural” approach to the hierarchy problem?Estimating uncertainties in perturbative QCD (e.g., parton distributions)

Bayesian rules of probability as principles of logicNotation: pr(x |I) is the probability (or pdf) of x being true given information I

1 Sum rule: If set {x i} is exhaustive and exclusive,∑

i

pr(x i |I) = 1 −→∫

dx pr(x |I) = 1

cf. complete and orthonormalimplies marginalization (cf. inserting complete set of states)

pr(x |I) =∑

j

pr(x , y j |I) −→ pr(x |I) =

∫dy pr(x , y |I) = 1

2 Product rule: expanding a joint probability of x and y

pr(x , y |I) = pr(x |y , I) pr(y |I) = pr(y |x , I) pr(x |I)

If x and y are mutually independent: pr(x |y , I) = pr(x |I), then

pr(x , y |I) −→ pr(x |I) pr(y |I)Rearranging the second equality yields Bayes’ theorem

pr(x |y , I) =pr(y |x , I) pr(x |I)

pr(y |I)

Applying Bayesian methods to LEC estimation

Definitions:a ≡ vector of LECs =⇒ coefficients of an expansion (a0,a1, . . .)D ≡ measured data (e.g., cross sections)I ≡ all background information (e.g., data errors, EFT details)

Bayes theorem: How knowledge of a is updated

pr(a|D, I)︸︷︷︸posterior

= pr(D|a, I)︸︷︷︸likelihood

× pr(a|I)︸︷︷︸prior

/ pr(D|I)︸︷︷︸evidence

Posterior: probability distribution for LECs given the data

Likelihood: probability to get data D given a set of LECs

Prior: What we know about the LECs a priori

Evidence: Just a normalization factor here[Note: The evidence is important in model selection]

The posterior lets us find the most probable values of parameters orthe probability they fall in a specified range (“credibility interval”)

Limiting cases in applying Bayes’ theoremSuppose we are fitting a parameter H0 to some data D given a model M1 andsome information (e.g., about the data or the parameter)

Bayes’ theorem tells us how to findthe posterior distribution of H0:

pr(H0|D,M1, I) =

pr(D|H0,M1, I)× pr(H0|M1, I)pr(D|I)

[From P. Gregory, “Bayesian Logical Data Analysis

for the Physical Sciences”]Special cases:(a) If the data is overwhelming, the prior has no effect on the posterior(b) If the likelihood is unrestrictive, the posterior returns the prior

Toy model for natural EFT [Schindler/Phillips, Ann. Phys. 324, 682 (2009)]

“Real world”: g(x) = (1/2 + tan (πx/2))2

“Model” ≈ 0.25 + 1.57x + 2.47x2 +O(x3)

atrue = {0.25,1.57,2.47,1.29, . . .}

Generate synthetic data D with noise with 5% relative error:

D : dj = gj × (1 + 0.05ηj ) where gj ≡ g(xj )

η is normally distributed random noise→ σj = 0.05 gj ηj

Pass 1: pr(a|D, I) ∝ pr(D|a, I) pr(a|I) with pr(a|I) ∝ constant

=⇒ pr(a|D, I) ∝ e−χ2/2 where χ2 =

N∑

j=1

1σ2

j

(dj −

M∑

i=0

aix i

)2

That is, if we assume no prior information about the LECs (uniformprior), the fitting procedure is the same as least squares!

Toy model Pass 1: Uniform prior

Find the maximum of the posterior distribution; this is the same asfitting the coefficients with conventional χ2 minimization.Pseudo-data: 0.03 < x < 0.32.

M χ2/dof a0 a1 a2

true 0.25 1.57 2.471 2.24 0.203±0.01 2.55±0.112 1.64 0.25±0.02 1.6±0.4 3.33±1.33 1.85 0.27±0.04 0.95±1.1 8.16±8.14 1.96 0.33±0.07 −1.9±2.7 44.7±32.65 1.39 0.57±0.3 −14.8±6.9 276±117

Pass 1 results

Results highly unstable with changing order M (e.g., see a1)

The errors become large and also unstable

But χ2/dof is not bad! Check the plot . . .

Toy model Pass 1

Would we know the results were unstable if we didn’t know theunderlying model? Maybe some unusual structure at M = 3 . . .

Toy model Pass 2: A prior for naturalnessNow, add in our knowledge of the coefficients in the form of a prior

pr(a|D) =

(M∏

i=0

1√2πR

)exp

(− a2

2R2

)

R encodes “naturalness” assumption, and M is order of expansion.Same procedure: find the maximum of the posterior . . .

Results for R = 5: Much more stable!

M a0 a1 a2

true 0.25 1.57 2.472 0.25±0.02 1.63±0.4 3.2±1.33 0.25±0.02 1.65±0.5 3±2.34 0.25±0.02 1.64±0.5 3±2.45 0.25±0.02 1.64±0.5 3±2.4

What to choose for R? =⇒ marginalize over R (integrate).

We used a Gaussian prior; where did this come from?=⇒ Maximum entropy distribution for 〈∑i a2

i 〉 = (M + 1)R2

Diagnostic tools 1: Triangle plots of posteriors from MCMCSample the posterior with an implementation of Markov Chain Monte Carlo(MCMC) [note: MCMC not actually needed for this example!]

M=3$(up$to$x3)$Uniform$prior$

M=3$(up$to$x3)$Gaussian$prior$with$R=5$

With uniform prior, parameters play off each other

With naturalness prior, much less correlation; note that a2 and a3return prior =⇒ no information from data (but marginalized)

Diagnostic tools 2: Variable xmax plots =⇒ change fit range

Plot a0 with M = 0, 1, 2, 3, 4, 5 as a function of endpoint of fit data (xmax)

Uniform prior Naturalness prior (R = 5)

For M = 0, g(x) = a0 should be fit only at lowest x (range too large)

Prior is irrelevant given a0 values; we need to account for higher orders

Very small error (sharp posterior), but wrong!




For M = 1, g(x) = a0 + a1x works with smallest xmax only

Prior is irrelevant given ai values; we need to account for higher orders

Errors (yellow band) from sampling posterior




For M = 2, entire fit range is usable

Priors on a1, a2 important for a1 stability with xmax

For this problem, higher M is same as marginalization




For M = 3, uniform prior is off the screen at lower xmax

Prior gives ai stability with xmax =⇒ accounts for higher orders not in model





For M = 4, uniform prior has lost a0 as well

Prior gives ai stability with xmax





For M = 5, g(x) = a0 uniform prior has lost a0 as well (range too large)

Prior gives ai stability with xmax


Diagnostic tools 3: How do you know what R to use?Gaussian naturalness prior but let R vary over a large range

actual&value&

actual&value&

actual&value&

Error bands from posteriors (integrating over other variables)Light dashed lines are maximum likelihood (uniform prior) resultsEach ai has a reasonable plateau from about 2 to 10 =⇒ marginalize!

Diagnostic tools 4: error plots (a la Lepage)

Plot residuals (data − predicted) from truncated expansion

5% relative data error shown by bars on selected points

Theory error dominates data error for residual over 0.05 or so

Slope increase order =⇒ reflects truncation =⇒ “EFT” works!

Intersection of different orders at breakdown scale

How the Bayes way fixes issues in the model problem

By marginalizing over higher-order terms, we are able to use allthe data, without deciding where to break; we find stability withrespect to expansion order and amount of data

Prior on naturalness suppresses overfitting by limiting how muchdifferent orders can play off each other

Statistical and systematic uncertainties are naturally combined

Diagnostic tools identify sensitivity to prior, whether the EFT isworking, breakdown scale, theory vs. data error dominance, . . .

Could we have done all this just adding a “theory error” to our χ2

likelihood function?

When everything is a gaussian, we can combine the prior andlikelihood into an “augmented χ2”. But in general, no.

Even so, it doesn’t take the form of a simple extra weighting fortheory error added in quadrature

Many other tests with model problems . . .

Alternative functions (including non-linear) to test robustness, e.g.,

gα(x) =α

(x2 + α2)2

For α = 1.1, Taylor series is

gth(x) = 0.751− 1.242x2 + 1.540x4 − 1.700x6 +O(x8)

Different kinds of error on data from gα=1.1(x)

5% relative error 1% relative error High in UV, low in IR

Alternative priors, error propagation to non-fit observables

Blind tests of fitting protocols

Nucleon mass in chiral perturbation theory [in progress]

The chiral expansion of the nucleon mass MχPT in SU(2) χPT as a function ofthe lowest-order pion mass m is (with renormalization scale µ):

MχPT(m) = M0 + k1m2 + k2m3 + k3m4 log(

mµ

)+ k4m4 + k5m5 log

(mµ

)+ k6m5

+ k7m6 log(

mµ

)2

+ k8m6 log(

mµ

)+ k9m6 +O(m7)

Goal: fit to lattice data and extract sigma term, etc.

When scaled to Λ = 0.5 GeV, phenomenological ki ’s are natural:

M0 = 1.76, k1 = 1.92, k2 = −1.41, k3 = 0.81 , k4 = 1.03,

k5 = 2.97, k6 = 4.41, k7 = 0.4, k8 = 0.31, k9 = −3.12,

If non-analytic terms are given, then this is just like our toy models!

Plan: use pseudo-data to test fitting robustness based on including anaturalness prior, fit range, lattice error, etc.

Can we fit the non-analytic terms as well?


The chiral expansion of the nucleon mass MχPT in SU(2) χPT as a function ofthe lowest-order pion mass m is (with renormalization scale µ):

MχPT(m)

Λ=

M0

Λ+

k1

Λ2 m2 +k2

Λ3 m3 +k3

Λ4 m4 log(

mµ

)+

k4

Λ4 m4 +k5

Λ5 m5 log(

mµ

)+

k6

Λ5 m5

+k7

Λ6 m6 log(

mµ

)2

+k8

Λ6 m6 log(

mµ

)+

k9

Λ6 m6 +O(m7)

Goal: fit to lattice data and extract sigma term, etc.

When scaled to Λ = 0.5 GeV, phenomenological ki ’s are natural:

M0 = 1.76, k1 = 1.92, k2 = −1.41, k3 = 0.81 , k4 = 1.03,

k5 = 2.97, k6 = 4.41, k7 = 0.4, k8 = 0.31, k9 = −3.12,

If non-analytic terms are given, then this is just like our toy models!

Plan: use pseudo-data to test fitting robustness based on including anaturalness prior, fit range, lattice error, etc.

Can we fit the non-analytic terms as well?


Fitting window is limitedby available lattice data

Proof-of-principle testswith pseudo-data

Fits at different orders inχPT expansion

with uniform prior

with naturalness prior

Lattice nf = 2 data, fits

2.0

3.0

4.0

m⇡ [MeV]500400300200

r 0M

N

2.0

3.0

4.0

r 0M

N

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

r 0M

N

(r0m⇡)2

0

0.2

0.4

0.6

400300200138

r 0�

m⇡ [MeV]

0

0.2

0.4

0.6

r 0�

0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8 1.0

r 0�

(r0m⇡)2

Figure 4: Simultaneous fits to the nucleon mass (left) and �-term data (right) for three fitting windowswith (r0m⇡)2max = 3.0, 1.6 and 1.3 (from top to bottom). These fits are labeled Soo3, Soo2 and Soo1

in Table B.4, where c2 ⌘ 3.3 GeV�1, c3 ⌘ �4.7 GeV�1 and l3 ⌘ 3.2. Lines, error bands and (full red)points are shown for the limit L ! 1 (cf. Fig. 3). A black-framed circle marks the location of thephysical point using—for each plot separately—the r0-value for which the fit is self-consistent. Openpoints did not enter any fit. In the left plots, the overlap of red points at (r0m⇡)2 = 0.436 and 0.538indicates the quality of the (fitted) finite-volume corrections.

10

Goal: Use Bayesian framework with naturalness prior plus diagnostic toolsto improve stability and robustness of fits

Outline




Going forward . . .

New NN potential and theory errors“Improved chiral nucleon-nucleon potential up to next-to-next-to-next-to-leading

[i.e., fourth] order” by E. Epelbaum, H. Krebs, and U.-G. Meißner, arXiv:1412.0142

New choices of regulators to minimize cutoff artifacts

Local regulator for long-distance parts (pion exchange):

Vlong−range(r)f (r/R) with f (x) = [1− e−x2]n (n ≥ 4)

Non-local regulator for contact interactions:

Vcontact(p,p′)e−((p2+p′2)/Λ2)m/2(m = 2 and Λ = 2/R)

Order-by-order convergence of total np cross section for R = 0.8 to 1.2 fm

22

160

170

180

190

LO NLO N2LO N3LO Exp

�tot [mb], Elab=50 MeV

60

70

80

90

100



30

40

50

60

70



20

30

40

50

60



FIG. 7: Order-by-order convergence of the chiral expansion for the np total cross section at energies of Elab = 50 MeV,Elab = 96 MeV and Elab = 143 MeV and Elab = 200 MeV. Dotted (color online: light brown), dashed (color online: green),dashed-dotted (color online: blue), solid (color online: red) and dashed-double-dotted (color online: pink) lines show the resultsbased on the cuto↵ R = 1.2 fm, R = 1.1 fm, R = 1.0 fm, R = 0.9 fm and R = 0.8 fm, respectively. The horizontal band refersto the result of the NPWA with the uncertainty estimated by means of deviations from the results based on the Nijmegen I, IIand Reid 93 potentials as explained in the text. Also shown are experimental data of Ref. [108].

expects to see in chiral EFT: one observes fast convergence at the lowest energy which becomes increasingly slower athigher energies. Notice that the large size of higher-order corrections at the energy of Elab = 200MeV relative to theleading ones is actually due to the NLO contributions being smaller than expected as will be shown below. One alsoobserves another feature which persists at all energies, namely that the size of the N2LO corrections decreases withincreasing the values of R. Given that the only new ingredient in the potential at N2LO is the subleading TPEP, thispattern simply reflects that the TPEP is stronger cut o↵ for soft cuto↵ choices.

The results shown in Fig. 7 provide a good illustration of the above mentioned issues associated with the estimationof the theoretical uncertainty by means of a cuto↵ variation. In particular, while the spread in the predictionsdoes, in general, decrease with the chiral order, it remains nearly the same at NLO and N2LO. Furthermore, atNLO, it misses (albeit barely) the result of the NPWA which is consistent with the expected underestimation of thetheoretical uncertainty at this order. On the other hand, while the spread in the predictions based on di↵erent cuto↵sis roughly consistent with the deviations between the theory and the NPWA result for the lowest energy, it appearsto significantly overestimate the uncertainty of the calculation based on lower (i.e. harder) cuto↵s R if one estimatesit via the deviation between the theory and the NPWA results. This behavior at high energy suggests that the spreadbetween the predictions for di↵erent values of R is actually governed by artefacts associated with too soft cuto↵sand does not reflect the true theoretical uncertainty of chiral EFT. We, therefore, conclude that while being a usefulconsistency check of the calculation, cuto↵ variation in the employed range does not provide a reliable approach forestimating the theoretical uncertainty. As we will show below, estimating the uncertainty via the expected size ofhigher-order corrections, as it is common e.g. in the Goldstone boson and single-baryon sectors of chiral perturbationtheory, provides a natural and more reliable approach which, in addition, has an advantage to be applicable at anyfixed value of the cuto↵ R.

For a given observable X(p), where p is the cms momentum corresponding to the considered energy, the expansionparameter in chiral EFT is given by

Q = max

✓p

⇤b,

M⇡

⇤b

◆, (7.33)

where ⇤b is the breakdown scale. Based on the results presented in sections IV and V, we will use ⇤b = 600 MeVfor the cuto↵s R = 0.8, 0.9 and 1.0 fm, ⇤b = 500 MeV for R = 1.1 fm and ⇤b = 400 MeV for R = 1.2 to account forthe increasing amount of cuto↵ artefacts which is reflected by the larger values of �2/datum in Table III. We haveverified the consistency of the choice ⇤b = 400MeV for the softest cuto↵ R = 1.2 fm by making the error plot similarto the one shown in Fig. 5. We can now confront the expected size of corrections to the np total cross section atdi↵erent orders in the chiral expansion with the result of the actual calculations. In particular, for the cuto↵ choice

R"="0.8"fm"

R"="1.2"fm"



Local regulator with cutoff Rfor long-distance parts

Non-local regulator withcutoff Λ = 2/R for contacts

24

160

165

170

175

Bands R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=50 MeV

65

70

75

80

85


� tot

[m

b]

Elab=96 MeV

30

40

50

60

70


� tot

[m

b]

Elab=143 MeV

20

30

40

50

60


� tot

[m

b]

Elab=200 MeV

FIG. 8: Predictions for the np total cross section based on the improved chiral NN potentials at NLO (filled squares, coloronline: orange), N2LO (solid diamonds, color online: green) and N3LO (filled triangles, color online: blue) at the energies ofElab = 50 MeV, Elab = 96 MeV, Elab = 143 MeV and Elab = 200 MeV for the di�erent choices of the cuto�: R1 = 0.8 fm,R2 = 0.9 fm, R3 = 1.0 fm, R4 = 1.1 fm, R5 = 1.2 fm. Vertical boxes depict the cuto� dependence of the theoretical predictionsat di�erent orders. The horizontal band refers to the result of the NPWA with the uncertainty estimated by means of deviationsfrom the results based on the Nijmegen I, II and Reid 93 potentials as explained in the text. Also shown are experimental dataof Ref. [108].

In all cases shown in Fig. 8, the predicted results calculated using di�erent values of the cuto� R agree with each otherwithin the theoretical uncertainty. It is comforting to see that our procedure for estimating the uncertainty yields thepattern which is qualitatively similar to the one found based on the �2/datum for the description of the Nijmegennp and pp phase shifts as shown in Table III. In particular, we see that the most accurate results at the lowestenergy are achieved with the cuto� R = 1.0 fm (with the uncertainty for the R = 0.9 fm case being of a comparablesize). At higher energies, the cuto� R = 0.9 fm clearly provides the most accurate choice. We also observe that atthe lowest energy, the cuto� variation does considerably underestimate the theoretical uncertainty at NLO and, toa lesser extent, at N3LO as expected based on the arguments given above. This pattern changes at higher energies.For example, at Elab = 200MeV, the cuto� bands at NLO and N3LO appear to be of the same size as the estimateduncertainty based on the optimal cuto� R = 0.9 fm. It is actually a combination of two e�ects which work againsteach other which results in a “reasonable” estimation of the NLO and N3LO uncertainties at higher energies by thecuto� bands: on the one hand, as already mentioned above, cuto� bands measure the impact of the order-Q4 andorder-Q6 contact interactions and, therefore, underestimate the uncertainty at NLO and N3LO. On the other hand,at higher energies, cuto� bands get increased due to using softer values of R as it is clearly visible from Fig. 8. Thisconclusion is further supported by the N2LO cuto� band which strongly overestimates the estimated uncertaintyin the case of R = 0.9 fm. We also learn from Fig. 8 that N2LO results for the total cross section for the cuto�sof R = 0.9 fm and R = 1.0 fm have the accuracy which is comparable to N3LO calculations with the softest cuto�

Old way: “Bands” show range of central results for R1–R5

New way: Estimate error range at each Ri by comparison to lower orders

NLO (orange), N2LO(green), N3LO (blue)

Right: error bands at fixedR2 = 0.9 fm





Local regulator with cutoff Rfor long-distance parts

Non-local regulator withcutoff Λ = 2/R for contacts

24

160

165

170

175


� tot

[m

b]

Elab=50 MeV

65

70

75

80

85


� tot

[m

b]

Elab=96 MeV

30

40

50

60

70


� tot

[m

b]

Elab=143 MeV

20

30

40

50

60


� tot

[m

b]

Elab=200 MeV

FIG. 8: Predictions for the np total cross section based on the improved chiral NN potentials at NLO (filled squares, coloronline: orange), N2LO (solid diamonds, color online: green) and N3LO (filled triangles, color online: blue) at the energies ofElab = 50 MeV, Elab = 96 MeV, Elab = 143 MeV and Elab = 200 MeV for the di�erent choices of the cuto�: R1 = 0.8 fm,R2 = 0.9 fm, R3 = 1.0 fm, R4 = 1.1 fm, R5 = 1.2 fm. Vertical boxes depict the cuto� dependence of the theoretical predictionsat di�erent orders. The horizontal band refers to the result of the NPWA with the uncertainty estimated by means of deviationsfrom the results based on the Nijmegen I, II and Reid 93 potentials as explained in the text. Also shown are experimental dataof Ref. [108].

In all cases shown in Fig. 8, the predicted results calculated using di�erent values of the cuto� R agree with each otherwithin the theoretical uncertainty. It is comforting to see that our procedure for estimating the uncertainty yields thepattern which is qualitatively similar to the one found based on the �2/datum for the description of the Nijmegennp and pp phase shifts as shown in Table III. In particular, we see that the most accurate results at the lowestenergy are achieved with the cuto� R = 1.0 fm (with the uncertainty for the R = 0.9 fm case being of a comparablesize). At higher energies, the cuto� R = 0.9 fm clearly provides the most accurate choice. We also observe that atthe lowest energy, the cuto� variation does considerably underestimate the theoretical uncertainty at NLO and, toa lesser extent, at N3LO as expected based on the arguments given above. This pattern changes at higher energies.For example, at Elab = 200MeV, the cuto� bands at NLO and N3LO appear to be of the same size as the estimateduncertainty based on the optimal cuto� R = 0.9 fm. It is actually a combination of two e�ects which work againsteach other which results in a “reasonable” estimation of the NLO and N3LO uncertainties at higher energies by thecuto� bands: on the one hand, as already mentioned above, cuto� bands measure the impact of the order-Q4 andorder-Q6 contact interactions and, therefore, underestimate the uncertainty at NLO and N3LO. On the other hand,at higher energies, cuto� bands get increased due to using softer values of R as it is clearly visible from Fig. 8. Thisconclusion is further supported by the N2LO cuto� band which strongly overestimates the estimated uncertaintyin the case of R = 0.9 fm. We also learn from Fig. 8 that N2LO results for the total cross section for the cuto�sof R = 0.9 fm and R = 1.0 fm have the accuracy which is comparable to N3LO calculations with the softest cuto�



NLO (orange), N2LO(green), N3LO (blue)

Right: error bands at fixedR2 = 0.9 fm

Phase shifts at one cutoff R = 0.9 fm

25

-20

0

20

40

60

� [d

eg]

1S0

-20

0

20

40

60

� [d

eg]

1S0

-10

0

10

20 3P0

-10

0

10

20 3P0

-40

-30

-20

-10

0

10

1P1

-40

-30

-20

-10

0

10

1P1

-30

-20

-10

03P1

-30

-20

-10

03P1

0

60

120

180

� [d

eg]

3S1

0

60

120

180

� [d

eg]

3S1

-5

0

5

10�1

-5

0

5

10�1

-30

-20

-10

0

3D1

-30

-20

-10

0

3D1

0

5

10

15

20

1D2

0

5

10

15

20

1D2

0

10

20

30

0 100 200 300

� [d

eg]

Elab [MeV]

3D2

0

10

20

30

0 100 200 300

� [d

eg]

Elab [MeV]

3D2

0

5

10

15

20

25

30

0 100 200 300

Elab [MeV]

3P2

0

5

10

15

20

25

30

0 100 200 300

Elab [MeV]

3P2

-4

-3

-2

-1

0

1

0 100 200 300

Elab [MeV]

�2

-4

-3

-2

-1

0

1

0 100 200 300

Elab [MeV]

�2

-10

-5

0

5

10

15

0 100 200 300

Elab [MeV]

3D3

-10

-5

0

5

10

15

0 100 200 300

Elab [MeV]

3D3

FIG. 9: Estimated theoretical uncertainty of the np phase shifts at NLO, N2LO and N3LO based on the cuto� of R = 0.9 fmin comparison with the NPWA [41] (solid dots) and the GWU single-energy np partial wave analysis [89] (open triangles). Thelight- (color online: yellow), medium- (color-online: green) and dark- (color-online: blue) shaded bands depict the estimatedtheoretical uncertainties at NLO, N2LO and N3LO, as explained in the text. Only those partial waves are shown which havebeen used in the fits at N3LO.

R = 1.2 fm. In summary, we find that the suggested approach for error estimation is more reliable than the standardprocedure by means of cuto� bands and, in addition, has the advantage of being applicable for a fixed value of R.This allows one to avoid the artificial increase of the theoretical uncertainty due to cuto� artefacts, the issue whichis especially relevant at high energies where the chiral expansion converges slower. The issue with using the cuto�bands is expected to become particularly important at next-to-next-to-next-to-next-to-leading order (N4LO) in thechiral expansion. In particular, we expect that the residual cuto� dependence at N4LO will be comparable to thatat N3LO, and that it will significantly overestimate the real N4LO uncertainty at higher energies in a close analogyto what is observed at N2LO. Last but not least, the ability to carry out independent calculations with quantifieduncertainties also provides a useful consistency check.

Next, we show in Fig. 9 the estimated uncertainty of the S-, P- and D-wave phase shifts and the mixing angles �1 and�2 at NLO, N2LO and N3LO based on R = 0.9 fm. The various bands result by adding/subtracting the estimatedtheoretical uncertainty, ±��(Elab) and ±��(Elab), to/from the results shown in Fig. 3. In a similar way, we alsolooked at selected neutron-proton scattering observables at di�erent energies shown in Figs. 10-13. For the lowestconsidered energy of Elab = 50MeV, we show, in addition to the results using R = 0.9 fm, also our predictions for thesoftest cuto� choice of R = 1.2 fm. While the uncertainty is clearly increased, the results actually still appear to berather accurate at this energy. Our results agree with the ones of the NPWA for all considered observables and energiesindicating that the employed way to estimate the uncertainties is quite reliable. Generally, we find that chiral EFTat N3LO allows for very accurate results at energies below Elab � 100 MeV and still provides accurate description ofthe data at energies of the order of Elab � 200 MeV. These findings are particularly promising for the ongoing studiesof the three-nucleon force whose contributions to nucleon-deuteron scattering observables are believed to increase atenergies above EN, lab � 100 MeV. It would be interesting to perform a similar analysis of nucleon-deuteron scatteringdata based on the improved chiral NN potentials in order to see whether accurate predictions are to be expected atsuch energies at N3LO. Work along these lines is in progress.

Finally, we emphasize that our results depend little on the specific choice of the regulator function. In order to

Old way: “Bands” show range of predicted observables for cutoff rangeNew way: Estimate error range at fixed Ri by comparison to lower orders

Polarization observables at one cutoff R = 0.9 fm

27

0

5

10

15

20

d�/d� [mb/sr]

0

5

10

15

20

d�/d� [mb/sr]

-0.1

0

0.1

0.2

0.3

0.4

0.5

Ay

-0.1

0

0.1

0.2

0.3

0.4

0.5

Ay

-0.2

0

0.2

0.4

0.6

0.8

1D

-0.2

0

0.2

0.4

0.6

0.8

1D

-0.8

-0.6

-0.4

-0.2

0

0.2

0 60 120 180

�CM [deg]

A

-0.8

-0.6

-0.4

-0.2

0

0.2

0 60 120 180

�CM [deg]

A

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 60 120 180

�CM [deg]

Axx

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 60 120 180

�CM [deg]

Axx

-0.5

0

0.5

1

0 60 120 180

�CM [deg]

Ayy

-0.5

0

0.5

1

0 60 120 180

�CM [deg]

Ayy

FIG. 11: Estimated theoretical uncertainty of the chiral EFT results for np di�erential cross section d�/d�, vector analyzingpower A, polarization transfer coe�cients D and A and spin correlation parameters Axx and Ayy at laboratory energy ofElab = 96 MeV calculated using on the cuto� of R = 0.9 fm. The light- (color online: yellow), medium- (color-online: green) anddark- (color-online: blue) shaded bands depict the estimated theoretical uncertainties at NLO, N2LO and N3LO, respectively.Open circles refer to the result of the NPWA. Data for the cross section are taken from [116–118]. Data for the analyzing powerare at Elab = 95 MeV and taken from [119].

in Eq. (3.27), namely n = 5 and n = 7. In Table VII, we show the resulting phase shifts in the 3S1 and pp 1S0,3P0,

3P1 and 3P2 partial waves at the energies of 10, 100 and 200 MeV as representative examples. Clearly, the observedspread in the results is negligibly small compared to the estimated accuracy of our calculations. Furthermore, asalready pointed out in section III, the employed local regularization of the pion-exchange contributions makes thespectral function regularization obsolet. In particular, phase shifts resulting from fits using di�erent values of the SFRcuto� � = 1GeV, � = 1.5 GeV and � = 2 GeV, see the last three columns in Table VII, are nearly indistinguishablefrom each other and from the DR result corresponding to � = � and shown in the third column of this Table.

VIII. SUMMARY AND CONCLUSIONS

In this paper we have presented a new generation of NN potentials derived in chiral EFT up to N3LO. The new chiralforces o�er a number of substantial improvements as compared to the widely used N3LO potentials of Refs. [1, 2]introduced a decade ago. First of all, we employ a local regularization scheme for the pion exchange contributionswhich, di�erently to the standard nonlocal regularization applied e.g. in Refs. [1, 2], does not distort the low-energyanalytic structure of the amplitude and, as a consequence, leads to a better description of phase shifts and experimentaldata. The employed regulator, by construction, removes the short-range part of the chiral two-pion exchange and thusmakes the additional spectral function regularization used in the potential of Ref. [1] obsolete. This is a particularlywelcome feature given that the expressions for the three-nucleon force at N3LO and N4LO are only available in theframework of dimensional regularization. Further, in contrast to the earlier studies of Refs. [1, 2], we have taken allpion-nucleon LECs and especially the subleading LECs ci from pion-nucleon scattering without any fine tuning. TheLECs accompanying NN contact interactions were determined by fits to the Nijmegen phase shifts and mixing anglesfor five di�erent values of the coordinate-space cuto� R chosen in the range of R = 0.8 . . . 1.2 fm and appear to be of

Old way: “Bands” show range of predicted observables for cutoff rangeNew way: Estimate error range at fixed Ri by comparison to lower orders



Fitting protocol

1 Restrict the range in energy to fit NPWA phase shifts at each order;

=⇒ use all data and marginalize over missing orders

2 Check by hand that LECs from the fit are natural;

=⇒ include a naturalness prior on LECs

3 Add a data point with assumed error for the D-state probability of thedeuteron (PD = 5%± 1%);

=⇒ add a prior on PD

4 Use an augmented χ2 to penalize deviations from Wigner SU(4)symmetry (which implies C1S0 ≈ C3S1);

=⇒ add as a prior on these LECs

5 Assumption for the error of the phase shifts from Nijmegen 1993 PWA.Uncertainty for calculating χ2/datum combining statistical plus systematicerrors in phase shifts by (“X ” is the channel):

∆X = max(

∆NPWAX , |δNijm I

X − δNPWAX |, |δNijm II

X − δNPWAX |, |δReid93

X − δNPWAX |

)

=⇒ all combined in the posterior PDF

The determination of errors from omitted higher-order is calculated separately.

=⇒What is a Bayesian alternative?

Fitting protocol with proposed Bayesian upgrades

1 Restrict the range in energy to fit NPWA phase shifts at each order;=⇒ use all data and marginalize over missing orders

2 Check by hand that LECs from the fit are natural;=⇒ include a naturalness prior on LECs

3 Add a data point with assumed error for the D-state probability of thedeuteron (PD = 5%± 1%);

=⇒ add a prior on PD

4 Use an augmented χ2 to penalize deviations from Wigner SU(4)symmetry (which implies C1S0 ≈ C3S1);

=⇒ add as a prior on these LECs

5 Assumption for the error of the phase shifts from Nijmegen 1993 PWA.Uncertainty for calculating χ2/datum combining statistical plus systematicerrors in phase shifts by (“X ” is the channel):

∆X = max(

∆NPWAX , |δNijm I

X − δNPWAX |, |δNijm II

X − δNPWAX |, |δReid93

X − δNPWAX |

)

=⇒ all combined in the posterior PDF

The determination of errors from omitted higher-order is calculated separately.=⇒What is a Bayesian alternative?

Diagnostics with proposed Bayesian upgrades18

10-6

10-4

10-2

1

50 100 200 300 400 500

|1-c

ot(�

R1)

/cot

(�R

2)|

k [MeV]

1S0

10-6

10-4

10-2

1

50 100 200 300 400 500

|1-c

ot(�

R1)

/cot

(�R

2)|

k [MeV]

3S1

10-4

10-2

1

50 100 200 300 400 500

|1-c

ot(�

R1)

/cot

(�R

2)|

k [MeV]

3P1

10-4

10-2

1

50 100 200 300 400 500

|1-c

ot(�

R1)

/cot

(�R

2)|

k [MeV]

3P2

FIG. 5: Error plots for np scattering in the 1S0,3S1,

3P1 and 3P2 partial waves as explained in the text. Dotted, dashed (coloronline: brown), dashed-dotted (color online: blue) and solid (color online: red) lines show the results at LO, NLO, N2LO andN3LO, respectively.

from LO to NLO/N2LO and from NLO/N2LO to N3LO can be viewed as a self-consistency check of the calculationand indicates that the theory is properly renormalized, see Refs. [22, 93] for more details. Finally, we read o↵ fromthe plots that the breakdown scale ⇤b at N3LO, i.e. the momenta at which the N3LO curves cross the ones of lowerorders, is about ⇠ 500 MeV for S-waves and even higher for P-waves. These observations are in line with our previousfindings and, in particular, with the size of the LECs accompanying the corresponding contact interactions which arelisted in Table II. We will use ⇤b = 400 . . . 600 MeV, depending on the cuto↵ R, in our estimation of the theoreticaluncertainties in section VII.

VI. DEUTERON PROPERTIES

We now turn to the deuteron properties. First, as already emphasized in section IV, we stress that we used thebinding energy Bd = 2.224575 MeV [94] to constrain the fit. While this choice di↵ers from our early work [1], it isactually the standard procedure for all high-precision phenomenological potentials such as the Nijmegen I, II, Reid93,CD-Bonn and AV18 one. Also the N3LO potential of Ref. [2] was tuned to reproduce the experimental value of thedeuteron binding energy. We anticipate that relaxing this condition in the fits would have little impact on few-nucleonobservables.

In Table V, we collect various deuteron properties at N3LO using di↵erent values of the cuto↵ R in comparison withthe results based on the CD-Bonn [81], N3LO Idaho (500) [2] and N3LO (550/600) [1] potentials and with empiricalnumbers. In all cases with the exception of the N3LO Idaho potential, the deuteron binding energy is calculatedbased on relativistic kinematics, see Eq. (A.4) and Refs. [1, 81] for more details. We remind the reader that theasymptotic S state normalization AS and the asymptotic D/S state ratio ⌘ are observable quantities which can beextracted from the S-matrix at the deuteron pole by means of analytic continuation, see [1] and references therein.

Is the EFT working? Use error plots to test!

Expected improvement when no LECs are added in an odd order

Make sure this is not confirmation bias

What is the breakdown scale?

Error plot not with data residuals (cf. Lepage) but with higher order

Use other tools: triangle plots, prior self-consistency checks, xmax plots

New NN potential and theory errors: N4LO

“Precision nucleon-nucleon potential at fifth order in the chiral expansion”by E. Epelbaum, H. Krebs, and U.-G. Meißner, arXiv:1412.4623

Identify the expansion parameter Q by

Q = max(

pΛb,

mπ

Λb

)which entails identifying Λb, thebreakdown scale of the EFT.

Uncertainty for observable X (p) at agiven order determined from calculationsat all lower orders. Example: uncertainty∆X N3LO(p) of N3LO prediction X N3LO(p):

∆X N3LO(p) = max(

Q5 × |X LO(p)|,

Q3 × |X LO(p)− X NLO(p)|,

Q2 × |X NLO(p)− X N2LO(p)|,

Q × |X N2LO(p)− X N3LO(p)|)

Figure below showsorder-by-order convergence oftotal cross sections

Breakdown scale when errorstops improving (R ≈ 0.9 fm)

160

165

170

175

R1 R2 R3 R4 R5 Exp� t

ot [

mb

]

Elab=50 MeV

65

70

75

80

85

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=96 MeV

30

40

50

60

70

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=143 MeV

20

30

40

50

60

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=200 MeV

Phase shifts and spin observables for R = 0.9 fm

-20

0

20

40

60�

[deg]

1S0

-20

0

20

40

60�

[deg]

1S0

-10

0

10

20 3P0

-10

0

10

20 3P0

-30-20

-10

0

10

� [d

eg]

1P1

-30-20

-10

0

10

� [d

eg]

1P1

-30

-20

-10

0 3P1

-30

-20

-10

0 3P1

0

60

120

180

� [d

eg]

3S1

0

60

120

180

� [d

eg]

3S1

-5

0

5

10 �1

-5

0

5

10 �1

-30

-20

-10

0

� [d

eg]

3D1

-30

-20

-10

0

� [d

eg]

3D1

0

5

10

15

201D2

0

5

10

15

201D2

0

10

20

30

� [d

eg]

3D2

0

10

20

30

� [d

eg]

3D2

0

10

20

303P2

0

10

20

303P2

-4

-3

-2

-1

0

0 100 200 300

� [d

eg]

Elab [MeV]

�2

-4

-3

-2

-1

0

0 100 200 300

� [d

eg]

Elab [MeV]

�2

-10

-5

0

5

10

0 100 200 300

Elab [MeV]

3D3

-10

-5

0

5

10

0 100 200 300

Elab [MeV]

3D3

0

5

10

15

20

d�/d� [mb/sr]

0

5

10

15

20

d�/d� [mb/sr]

-0.5

0

0.5 Ay

-0.5

0

0.5 Ay

-1

-0.5

0

0.5

1

1.5

D

-1

-0.5

0

0.5

1

1.5

D

-1

-0.5

0

0.5

1 A

-1

-0.5

0

0.5

1 A

-1

-0.5

0

0.5

0 60 120 180

�CM [deg]

Axx

-1

-0.5

0

0.5

0 60 120 180

�CM [deg]

Axx

-1

-0.5

0

0.5

1

1.5

0 60 120 180

�CM [deg]

Ayy

-1

-0.5

0

0.5

1

1.5

0 60 120 180

�CM [deg]

Ayy



New NN potential and theory errors: N4LO

“Precision nucleon-nucleon potential at fifth order in the chiral expansion”by E. Epelbaum, H. Krebs, and U.-G. Meißner, arXiv:1412.4623

160

165

170

175

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=50 MeV

65

70

75

80

85

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=96 MeV

30

40

50

60

70

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=143 MeV

20

30

40

50

60

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=200 MeV

Can we justify these “error bars” in a Bayesian framework?

Estimating nuclear EFT truncation errorsAdapt Bayesian technology used in pQCD [Cacciari and Houdeau (2011)]

k th order: σQCD ≈k∑

n=0

cnαks −→ σnp ≈ σ0

k∑

n=0

cn

(p

Λb

)k

where Λb ≈ 600 MeV (in the future: determine self-consistently!)

Goal: find ∆k ≡∑∞

n=k+1 cnzn where z = αs or p/Λb)Underlying assumption: all cn ’s are about the same size or have a pdfwith the same upper bound, denoted c. Possible priors for c:

set pr(cn|c) pr(c)

A 12c θ(c − |cn|) 1

2| ln ε|1c θ(c − ε)θ( 1

ε − c)

B 12c θ(c − |cn|) 1√

2πcσe−(log c)/2σ2

C 1√2πc

e−c2n/2c2 1

2| ln ε|1c θ(c − ε)θ( 1

ε − c)

For set A, apply Bayes’ theorem and marginalization repeatedly:

pr(∆k |c0, . . . , ck ) ≈(

nc

nc + 1

)1

2zk+1c(k)

1 if |∆k | ≤ zk+1c(k)(zk+1c(k)

|∆k |

)nc+1

if |∆k | > zk+1c(k)

Estimating nuclear EFT truncation errorsAdapt Bayesian technology used in pQCD [Cacciari and Houdeau (2011)]

k th order: σQCD ≈k∑

n=0

cnαks −→ σnp ≈ σ0

k∑

n=0

cn

(p

Λb

)k

where Λb ≈ 600 MeV (in the future: determine self-consistently!)Goal: find ∆k ≡

∑∞n=k+1 cnzn where z = αs or p/Λb)

Underlying assumption: all cn ’s are about the same size or have a pdfwith the same upper bound, denoted c. Possible priors for c:

set pr(cn|c) pr(c)

A 12c θ(c − |cn|) 1

2| ln ε|1c θ(c − ε)θ( 1

ε − c)

B 12c θ(c − |cn|) 1√

2πcσe−(log c)/2σ2

C 1√2πc

e−c2n/2c2 1

2| ln ε|1c θ(c − ε)θ( 1

ε − c)

For set A, apply Bayes’ theorem and marginalization repeatedly:

pr(∆k |c0, . . . , ck ) ≈(

nc

nc + 1

)1

2zk+1c(k)

1 if |∆k | ≤ zk+1c(k)(zk+1c(k)

|∆k |

)nc+1

if |∆k | > zk+1c(k)

Estimating nuclear EFT truncation errors for R2

Let’s try it! First, check whether the assumption of cn’s being the samesize works for σnp ≈ σ0(1 + c2Q2 + c3Q3 + · · · ) with Q = p/600 MeV

Yes! So apply set A (left plot blue) to σnp at Elab = 96 MeV =⇒ Q ≈ 1/3

-0.02 -0.01 0.01 0.02Dk

20

40

60

80

100

fHDk»d1...dkL

-0.02 -0.01 0.01 0.02Dk

20

40

60

80

fHDk»d1...dkLCH, l=1Log-Normal

Figure 1 – Dependence of the posterior distribution on the prior on the hidden parameter c. On the left, comparisonat LO of the default log-uniform prior with respect to a log-normal one. On the right, the same comparison butat NNLO.

expansion parameter is not uniquely defined. To address these issues, we introduce a parameter� which reflects our ignorance of the optimal expansion parameter for QCD observables. Ourexpression for a generic QCD observable is then given by

Ok =kX

n=l

⇣↵s

�

⌘n(n � 1)! �n cn

(n � 1)!⌘

kX

n=l

⇣↵s

�

⌘n(n � 1)! dn, (7)

where we have isolated a factor (n � 1)! in the expansion which can be motivated from theoryby looking at the behaviour of the coe�cients cn for large n due to renormalon chains. Weassume the same priors on the modified coe�cients dn as on the original cn. Then, we tune �by measuring the performance of the model for a fixed � value on a given set of observables.At a given order and for a given degree of belief (DoB), we calculate for each observable thecorresponding interval. Then, we compute the success rate defined as the ratio between thenumber of observables whose next-order is within the computed DoB interval over the totalnumber of observables in the set. We define the optimal � value to be such that the success rateis equal to requested DoB.

With these modifications, the analytic expression for the posterior density distribution for�k is given by

f(�k|dl, . . . , dk) '✓

nc

nc + 1

◆�k+1

2k!↵k+1s dk

8><>:

1 if |�k| k!�↵s�

�k+1dk

⇣k!↵k+1

s dk

|�k|�k+1

⌘nc+1if |�k| > k!

�↵s�

�k+1dk

(8)

where nc is the number of coe�cients available in the computation and dk = max(|d1|, . . . , |dk|).This analytic expression captures the general features of the posterior distributions for �k

produced by this class of models: a flat top with power suppressed tails.

3.1 The model’s dependence on the prior on c

The choice of the priors in a Bayesian model is arbitrary and subjective. In the original formu-lation, the prior on c was chosen as non-informative as possible, i.e. a log-uniform distribution.However, the downside of such a conservative choice is that at low perturbative orders, the tailsof the posterior distribution for �k are very long and give rise to very large intervals if a largeDoB is given as an input to the model. As a test, we use a more informative prior. If we scaleall coe�cients dn by the first coe�cient dl, the hypothesis that all coe�cients are of order unityis equivalent to the original hypothesis that all coe�cients are of the same order of magnitude.Hence, we can also consider a log-normal distribution around zero

f(c) =1p

2⇡c�e�

log2 c

2�2 , c > 0 (9)

pr(Δk|c0,*c1,*c2)*

160

165

170

175

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=50 MeV

65

70

75

80

85

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=96 MeV

30

40

50

60

70

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=143 MeV

20

30

40

50

60

R1 R2 R3 R4 R5 Exp

� tot

[m

b]

Elab=200 MeV68% credibility interval widths are 4.0, 1.2, 0.40, 0.13 mb =⇒ agrees!

Outline




Going forward . . .

Goals of theory UQ for EFT calculations

Reflect all sources of uncertainty in an EFT prediction

=⇒ likelihood or prior for each

Compare theory predictions and experimental results statistically

=⇒ error bands as credibility intervals

Distinguish uncertainties from IR (long-range) vs. UV(short-range) physics

=⇒ separate priors

Guidance on how to extract EFT parameters (LECs)

=⇒ Bayes propagates new info (e.g., will an additional or bettermeasurement or lattice calculation help and by how much?)

Test whether EFT is working as advertised— do our predictionsexhibit the anticipated systematic improvement?

=⇒ Trends of credibility interval; model selection

The Bayesian framework lets us consistently achieve our UQ goals!

Goals of theory UQ for EFT calculations

Reflect all sources of uncertainty in an EFT prediction=⇒ likelihood or prior for each

Compare theory predictions and experimental results statistically=⇒ error bands as credibility intervals

Distinguish uncertainties from IR (long-range) vs. UV(short-range) physics

=⇒ separate priors

Guidance on how to extract EFT parameters (LECs)=⇒ Bayes propagates new info (e.g., will an additional or bettermeasurement or lattice calculation help and by how much?)

Test whether EFT is working as advertised— do our predictionsexhibit the anticipated systematic improvement?

=⇒ Trends of credibility interval; model selection

The Bayesian framework lets us consistently achieve our UQ goals!

Next steps and open questions

Apply full Bayesian framework to EFT fitting

First NN, then NNNComputationally feasible?

Test propagation of EFT errors order-by-order

Try applying Bayesian model selection (what’s that?)

(Some) questions to address:

Is EFT renormalization working?Is it ok to fine-tune an observable?Can nuclei resolve pions?What are the best measurements to better constrain LECs?How well can lattice calculations constrain LECs?

Bayesian model selection

Determine the evidence for different models M1 and M2 viamarginalization by integrating over possible sets of parameters a indifferent models, same D and information I.

The evidence ratio for two different model:

pr(M1|D, I)pr(M2|D, I)

=pr(D|M1, I) pr(M1|I)pr(D|M2, I) pr(M2|I)

The Bayes Ratio (implements Occam’s Razor):

pr(D|M1, I)pr(D|M2, I)

=

∫pr(D|a1,M1, I) pr(a1|M1, I)∫pr(D|a1,M2, I) pr(a2|M2, I)

Examples of how we could use this in EFT context:

Which EFT parameters =⇒ improve the fit to data?

Which EFT power counting is more effective? (cf. more parameters)

Pionless vs. chiral EFT?

Bayesian model selection: polynomial fitting

degree%1% degree%2% degree%3%

degree%4% degree%5% degree%1%degree%6%

Likelihood%

Evidence%

degree%[adapted%from%Tom%Minka,%h?p://alumni.media.mit.edu/~tpminka/statlearn/demo/]%

The likelihood considers the single most probable curve, and always increaseswith increasing degree. The evidence is a maximum at 3, the true degree!

Is it ok to fine-tune an observable?Convergence of low-energy deuteronobservables improves if LECs fit to Ed

and ηS instead of a0[3S1] and r0[3S1]=⇒ equivalent up to higher-order terms[Phillips et al., PLB 473, 209 (2000)]

Dangers of fine-tuning

If tuned observable correlated withcalculated observable, then samecancellations, but other observables?

Correlations can lead to instabilitieswhen fitting LECs; e.g., cD and cE fitsfor NNN force to E(3H) and E(4He)

7.6 7.8 8.0 8.2 8.4 8.6 8.8

E( H) [MeV]

24

25

26

27

28

29

30

31

E(

He)

[M

eV]

Vlow k

AV18

Vlow k

CD-Bonn

Λ=3.0 fm-1

Λ=1.6 fm-1

Λ=1.3 fm-1

Λ=1.9 fm-1

Λ=1.0 fm-1

Exp.

Λ=2.1 fm-1

"bare" AV18

3

4

"bare" CD-Bonn

6 !!"!#$%&'#((((((!

Lawrence Livermore National Laboratory

7Be(p,!)8B astrophysical S-factor!!! The 7Be(p,!)8B is the final step in the

nucleosynthetic chain leading to 8B

!! ~10% error in latest S17(0): dominated by uncertainty in theoretical models

!! NCSM/RGM results with largest realistic model space

•! SRG-N3LO NN potential (" = 1.86 fm-1)

•! Nmax = 10

•! p+7Be(g.s., 1/2-, 7/2-, 5/21-, 5/22

-)

!! Sep. energy: 136 keV (Expt. 137 keV)"

!! S17(0)=19.4 eV b on the lower side of, but consistent with latest evaluation"

!! Run time: ~150,000 CPU hrs

Reactions important for solar astrophysics P. Navrátil, R. Roth and S. Quaglioni, submitted to Phys. Rev. Lett., arXiv:1105.5977

0 0.5 1 1.5 2 2.5E

kin [MeV]

0

5

10

15

20

25

30

35

S1

7 [

eV b

]

Filippone

StriederHammacheHassBaby

Junghans

SchuemannDavidsNCSM/RGM E1

7Be(p,!)

8B

(a)

Ab initio theory predicts simultaneously

both normalization and shape of S17.

Navratil et al., arXiv:1105.5977

Tune NN-only SRG to λ = 1.86 fm−1

to match separation energy. Ok?

Advertisement: INT Workshop in 2016Bayesian Methods in Nuclear Physics (INT-16-2a)

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Existing efforts using Bayesian statistics will continue to develop over the nexttwo years, but Summer 2016 will be an opportune time to bring the statisticiansand nuclear practitioners together. [See computingnuclei.org for more meetings]

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