EFT power counting and

Bayesian analysisGautam Rupak

Bayesian Inference in Sub-Atomic Physics Chalmers University, Göteborg, Sept 18, 2019

Outline

• EFT and power counting overview

• Model comparison: A toy problem

• Model comparison: Capture reaction — two or more power countings with the same particle content

What is EFT?

EFT : Effective Field Theory

Quantum Field Theory

Low-energy approximations that can be systematically improved

What is EFT?

EFT : Effective Field Theory

Quantum Field Theory

Low-energy approximations that can be systematically improved

λ ≪ R

n

p

What is EFT?

EFT : Effective Field Theory

Quantum Field Theory

Low-energy approximations that can be systematically improved

λ ≪ R

n

p

λ ∼ R

n

p

light dof

heavy dof

Q

Λ

momentum separation can be fuzzy

expand in Q

⇤⌧ 1

4

systematic self-consistent

model-independent error-estimates

Power counting: Naive Expectation

2ο 7ο

c c

Momentum separation determines (subatomic) particle content at low p ~ Q, and all parameters (couplings) of the theory given by appropriate powers of �Λ

representative Feynman diagram Chen, Rupak, Savage PLB410, 285 (1997)

Each such contributions can be assigned some factor � a priori. This is what we mean by power counting. It has to come out right at theend.

Qn/Λm

Using Brynjarsdottir’s notation:

Physical world is given by some �ζ(p)

We calculate in EFT

with � |cn | ∼ 1, | fn | ∼ 1

However, it practice its more interesting as I show in the toy model

Toy problem: n-p scatteringMeasured cross section:

We know from analyticity:

How should we expand for small momentum p?

if � , etc.|p | ≪ 1/ |a | , 1/ |r |

This works if naive expectations hold: � where � MeV/c and we look at �

1/ |a | ∼ Λ, 1/ |r | ∼ ΛΛ ∼ 140 p ∼ Q ≪ Λ

Reality is different (singlet channel): � MeV/c and � MeV/c

1/a ≈ − 8.311/r ≈ 72.3

Treat as small or large scale?

small scale

Generated “n-p” data“Real vales”: � MeV/c, � MeV/c, rest of couplings natural sized factors of 200 MeV/c

1/a = − 8.31 1/r = 72.3

Model 1: � |γ | ∼ p ∼ Q, 1/r ∼ Λ

Model 2: � |γ | ∼ 1/r ∼ p ∼ Q

� �� �� �� �� ��� ��� ����

��

��

��

��

� (���/�)

δ(���) True value

Model 1: LOModel 1: NLOModel 1: NNLOModel 2: LO

Model 1 and 2 have different accuracy and precision

Bayesian analysis

!9

P (✓|D, I) =P (D|✓, I)P (✓|I)

P (D|I)posterior PDF:

prior PDF: P (✓|I) EFT expectations/bias

likelihood PDF:

evidence: P (D|I) =Z[d✓]P (D|✓, I)P (✓|I)

— harder to calculate — not needed for parameter estimation — needed to compare models

P (M1|D, I)

P (M2|D, I)=

P (M1|I)P (M2|I)

⇥ P (D|M1, I)

P (D|M2, I)Models:

bias goes here

P (D|✓, I) / exp(�1

2�2), �2 =

NX

i=1

[Di � µi(✓)]2

�2i

Competing EFT power counting

• Some parameters might not show a clear scaling behavior

• Poorly constrained parameter space could be multi-modal

• EFTs with different particle content because energy separation fuzzy

• EFT power counting fits in nicely with priors for the parameters

• EFT power counting and prior model odd ratio

Following are some of the issues I would like to discuss

P (M1|D, I)

P (M2|D, I)=

P (M1|I)P (M2|I)

⇥ P (D|M1, I)

P (D|M2, I)Models:

I allow both models equal number of input parameters, andaccordingly assign equal prior odd to the models

! in halo EFT3He(α, γ)7Be! and ! as point-particles.

! ground and ! excited state of ! as p-wave bound states.

E1 capture from initial s-wave state, and d-wave state

3He α

32

− 12

−7Be

!11

(a1) (a2)

(a3)

(b1) (b2)

(b3)

L(ζ)E1

Higa, Rupak, Vaghani; EPJA 54, 89 (2018)Zhang, Nollett, Phillips; arXiv:1909.07287

2-body currentsinitial state interactionjust Coulombno unknown parameters

Contributions to !3He(α, γ)7BeWhich is better?

���� � ������� � ���� ����� � �������� ����� ����������� � ���� �

����� ������ �

χ� ��

������������������������

� ��(�����)

� ��� ��� ��� ���� ���� �������(���)

Two-body currentNo two-body current

Not a consistent powercounting. Can’t just droptwo-body currents!

p-wave final state asymptotic normalization constant (ANC)

C21 ∝

1# + ρ1 sits near a pole …

small change in � gives large change in cross section

ρ1

!12

Competing power countingsFrom � fit, !χ2 a0 ∼ 20-30 fm ∼ Λ2/Q3

• Initial state interaction LO

• Two-body currents LO

• LO capture: !

• NLO capture: !

• LO phase shifts: !

• NLO phase shifts: !

a0, r0, ρ(±)1 , L (±)

E1

s0

a0, r0, ρ(±)1

σ(±)1

7/9-parameters up to NLO

From � fit, !χ2 a0 ∼ 5-10 fm ∼ Λ /Q2

• Initial state interaction NLO

• Two-body currents NLO

• LO capture: !

• NLO capture: !

• NNLO capture: !

• LO s-wave phase shift: !

• NLO s-wave phase shift: !

• NNLO s-wave phase shift: !

• p-wave phase shifts unchanged

ρ(±)1

a0, L(±)E1

r0

a0

r0

s0

6/9-parameters up to NNLO!13

Premarathna, Rupak; arXiv:1906.04143

5

smaller a0 ∼ 5−10 fm ∼ Λ/Q2 [2]. This will make initialstate interaction a0(B + µJ0)/µ2 ∼ Q/Λ a NLO contri-bution. In EFT, there is always going to be two-bodycurrents unless some symmetry prevents them. Thenthe only consideration is at what order in perturbationthey contribute. Again assuming natural sized couplings

L(ζ)E1 ∼ 1, the two-body contribution a0k0L

(ζ)E1 ∼ Q/Λ is

also a NLO effect for a0 ∼ Λ/Q2.

!"#$%&

!! '%( )"*+, -!.#+/ )"*+, -!.#+/ )"*+, !

(!)

-!"-#"-$""

! !

(")

%#"

%!"

%&"

! "(-)

(#)

%#"

%!"

%&"

! "(+)

" % $ ' # (!!"(!"#)

($)

(%)

(&)

" % $ ' # (!!"(!"#)

$%& '(")%*&%+,

FIG. 2. 3He(α, γ)7Be: Model comparison for scattering phaseshifts. Notation for the curves the same as in Fig. 1. Panelson the left are fits to phase shifts, and the panels on the rightare predictions as discussed in the text.

We consider two power countings. The first one as-sumes a large a0 ∼ Λ2/Q3. This we call “Model A”though EFT is a model-independent formulation. In thispower counting [2], all the other parameters and cou-plings are natural sized determined by their naive dimen-

sions r0 ∼ 1/Λ, s0 ∼ 1/Λ3, ρ(±)1 ∼ Λ, σ(±)

1 ∼ 1/Λ, L(±)E1 ∼

1. The capture cross section depends on a0, r0, ρ(±)1 , L(±)

E1at LO. The NLO contribution gets an additional contri-bution from s0. The s-wave phase shift δ0 depends on a0and r0 at LO, and at NLO brings in s0. A fine-tuningin the linear combination r0p2/2 − 2kCH(ηp) ∼ Q3/Λ2

promotes the effective range contribution to LO [2]. The

p-wave phase shifts δ(±)1 depends on ρ(±)

1 at LO and the

NLO contribution comes from σ(±)1 . The d wave con-

tribution Y (p) is included at NLO due to a large nearcancellation at LO [2].In the second power-counting that we call “Model B”,

one assumes a smaller a0 ∼ Λ/Q2. The rest of the pa-rameters have the same scaling as Model A. However,the perturbative expansion is now different. The LOcapture cross section has no initial state strong interac-

tion. It only depends on ρ(±)1 . a0 and L(±)

E1 contributes atNLO, and r0 at next-to-next-to-leading order (NNLO).

In phase shift δ0, a0 contributes at LO, r0 at NLO and s0at NNLO. The p-wave phase shift contributions remainunchanged from Model A above. The d wave contribu-tion Y (p) is now included at NNLO.Model A and Model B are compared by calculating

the posterior odd ratio P (MA|D,H)/P (MB|D,H) fromEq. (8). The χ2 for the fit without two-body currents waslarger than the one with two-body currents when phaseshift data, especially δ0, was used. We explore the possi-bility that the uncertainty in the phase shifts are actuallylarger than estimated. We consider σ2 → K2 + σ2 to de-scribe an unaccounted noise. We draw K from the uni-form distribution U(0◦, 10◦). We also consider the pos-sibility σ2 → K2σ2 with K drawn from U(1, 10). Bothof these give similar fits so we present the analysis forK2 + σ2 only. Note that the errors σ in the phase shiftdata are estimated to be around ∼ 1◦–3◦ [24]. Alter-natively, one could also explore the possibility of an un-certainty in the overall normalization of the phase shiftmeasurements. As the s and p wave phase shifts are fromthe same measurement and analysis, we did not considerthe possibility of introducing separate measurement er-rors of the different elastic scattering channels. We usethe following uniform prior distributions for the param-eters and couplings in the EFT expressions:

a0 ∼ U(1 fm, 70 fm) ,

r0 ∼ U(−5 fm, 5 fm) ,

s0 ∼ U(−30 fm3, 30 fm3) ,

ρ(+)1 ∼ U(−300 MeV,−48 MeV) ,

ρ(−)1 ∼ U(−300 MeV,−33 MeV) ,

σ(±)1 ∼ U(−5 fm, 5 fm) ,

L(±)1 ∼ U(−10, 10) ,

K ∼ U(0◦, 10◦) . (10)

Fits without phase shifts do not depend on σ(±)1 and K.

Model B without phase shift data also does not dependon s0. The range for each of the uniform distributionsabove was guided by EFT power counting estimates. Theranges were wide enough that about 95% of the poste-rior distributions of the parameters and couplings are notpressed against the boundaries. For certain parameters

such as the p wave effective ranges ρ(±)1 physical con-

straint that the ANCs be positive determines the upperbounds. All these assumptions are considered part ofthe background information in the proposition H in theprobability distributions.In Figs. 1 and 2, we present fits in region I with

and without phase shifts. In Fig. 1, the upper pan-els (a) and (b) are fits with phase shift data, and thelower panels (c) and (d) are without. The panels onthe left in Fig. 2 are fits to phase shift data. The oneson the right are phase shift predictions from capturedata. We include the χ2 fit of Model A to data in re-gion I (with phase shift) for comparison [2]. The pa-rameter estimates are in Table V. The Bayesian curves

Priors for !3He(α, γ)7Be

ANCs must be positive

���� ���� ����

���� � ���� � ������� ����� � ����� ���� ������������ � ������� �

χ� ��� ����� ������ ����� ������ ����� �

(�)

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(�)

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δ �

(�)

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δ �(-)

(�)

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���

���

δ �(+)

� � � � � ����(���)

(�)

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��� ����������

!15

Fits to capture in region II Ecm 2000 keV We also have fits in region I Ecm 1000 keV

≲≲

Nested sampling

Parameters for !3He(α, γ)7Be11

Fits a0 (fm) r0 (fm) s0 (fm3) ⇢(+)1 (MeV) �(+)

1 (fm) ⇢(�)1 (MeV) �(�)

1 (fm) L(+)1 L(�)

1 K

�2 22± 3 1.2± 0.1 �0.9± 0.7 �55.4± 0.5 1.59± 0.03 �41.9± 0.7 1.74± 0.05 0.78± 0.06 0.83± 0.08 –

Model A I 48+2�2 1+0.09

�0.1 �1.8+1�0.9 �72+5

�8 2.1+0.2�0.2 �49+3

�6 2+0.2�0.1 1.4+0.2

�0.1 1.2+0.2�0.1 0.3+0.4

�0.2

Model B I 38+3�2 1.1+0.1

�0.1 �2+1�1 �61.6+0.6

�0.6 1.77+0.03�0.04 �48+1

�7 2+0.2�0.08 1.13+0.03

�0.02 1.2+0.3�0.08 0.3+0.3

�0.2

Model A⇤ I 20+8�5 �0.1+0.5

�0.7 �16+6�8 �89+9

�20 – �130+50�70 – 3+1

�0.9 7+2�3 –

Model B⇤ I 37+3�10 1.1+0.1

�0.9 – �61.4+1�0.8 – �47+2

�6 – 1.14+0.09�0.04 1.2+0.2

�0.1 –

Model A II 40+5�6 1.09+0.09

�0.1 �2.2+0.8�0.8 �59+1

�2 1.69+0.05�0.06 �45+2

�2 1.84+0.08�0.08 1.02+0.06

�0.06 1.07+0.08�0.09 0.3+0.3

�0.2

Model B II 7.3+0.7�0.7 1.31+0.02

�0.02 6+1�1 �53.5+0.1

�0.1 1.53+0.05�0.06 �40.1+0.2

�0.2 1.67+0.06�0.06 �0.04+0.08

�0.1 �0.01+0.09�0.1 2.2+0.6

�0.5

Model A⇤ II 46+10�4 1+0.1

�0.3 �3+5�2 �62+5

�4 – �51+4�70 – 1.1+0.1

�0.2 1.3+2�0.2 –

Model B⇤ II 5+1�2 1.24+0.04

�0.2 – �53.5+0.1�0.2 – �40.2+0.2

�0.2 – �0.5+0.2�1 �0.4+0.2

�0.9 –

TABLE V. 3He(↵, �)7Be: EFT parameters. We estimate the parameters from fits to capture data in region I (E . 1000 keV)and in region II (E . 2000 keV) as indicated. The fits that do not use elastic scattering phase shift data are indicated by theasterisk ⇤ sign. We show the errors to one significant figure.

Fits a0 (fm) r0 (fm) s0 (fm3) ⇢(+)1 (MeV) ⇢(�)

1 (MeV) L(+)1 L(�)

1 K

�2 17± 2 0.6± 0.3 2± 2 �149± 3 �129± 4 1.44± 0.07 1.5± 0.1 –

Model A 13+2�1 �0.1± 0.7 11+10

�7 �190+30�30 �230+80

�60 2.2+0.7�0.6 4+1

�2 2+2�1

Model B 21+2�2 2.7+0.1

�0.1 �24+3�2 �219+9

�8 �200+10�10 2.14+0.01

�0.01 2.4+0.1�0.1 6+2

�2

Model A⇤ 10+2�2 �2+0.8

�0.9 36+6�10 �220+30

�30 �240+50�50 3.3+0.9

�0.8 5+2�1 –

Model B⇤ 20+1�1 2.7+0.1

�0.1 – �213+6�4 �191+8

�6 2.14+0.01�0.01 2.37+0.1

�0.09 –

TABLE VI. 3H(↵, �)7Li: EFT parameters. The fits that do not use elastic scattering phase shift data are indicated by theasterisk ⇤ sign. We show the errors to one significant figure.

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Added an unaccounted noise to phase shifts from late 60s: ! Boykin et al. estimated !

σ2 → K2 + σ2

σ ∼ 5∘

!16

Elastic scattering of 3He + ↵ with SONIK

~ 500 keV to 3 MeV

Spokespersons: Connolly, Davids, Greife

Bayesian inferences for !3He(α, γ)7Be

8

have a power counting where two-body currents appearat higher order in the perturbation. This would requirea smaller s-wave scattering length. We call this powercounting Model B. If we ignore the poorly known elasticscattering data, then this second power counting can beused to describe the capture data. The relatively smallerinitial state interaction contributions in Model B can becompensated by a larger wave function normalizationconstant which requires only small variation in the re-spective p-wave e↵ective ranges as described earlier. Wediscuss the results of the analysis for 3He(↵, �)7Be and3H(↵, �)7Li separately below.

A. 3He(↵,�)7Be

We present the 8 di↵erent fits used to draw Bayesianinference for this reaction. First we start with the fits inthe smaller capture energy region I (E . 1000 keV). Theposterior odds favored Model A both with and withoutphase shift data in the fits. However, if we also look atthe overall trend then we see the data for S-factor S34

rises upward from around E ⇠ 1500 keV. The Model Afit with phase shift, we call Model A I, best describesthe capture data over the energy range E . 2000 keV.We note that the Model B fits in this region are not self-consistent in that they suggest an a0 value larger thanthe power counting, table V.

In the fits to capture energy region II (E . 2000 keV),Model A with phase shift, we call Model A II, is favoredover Model B by the posterior odd. For the fits with-out phase shifts, both Model A⇤ II and Model B⇤ II areequally favored by the posterior odd ratio. The asterisk⇤ indicates fits without phase shift data.

Fit S34(E?) (keV b) S034(E?) (10

�4 b)

�2 0.558 ± 0.008±0.056 �2.71± 0.20± 0.27

Model A I 0.541+0.012�0.014 ± 0.054 �1.34+0.64

�0.59 ± 0.13

Model A II 0.550+0.009�0.010 ± 0.055 �2.00+0.36

�0.35 ± 0.20

Model A⇤ II 0.551+0.021�0.014 ± 0.055 �1.86+0.72

�1.69 ± 0.19

Model B⇤ II 0.573+0.007�0.007 ± 0.017 �3.72+0.11

�0.10 ± 0.11

TABLE I. 3He(↵, �)7Be: S34 and S034 at threshold (defined as

E? = 60⇥ 10�3 keV). The second set of errors are estimatedfrom the EFT perturbation as detailed in the text.

Tables I and II has the S-factors S34 and branchingratios R0 for the 4 fits described above. We also includethe �2 fit of Model A for comparison [2]. We include thederivative S0

34 as well. All the numbers were evaluatedat E? = 60 ⇥ 10�3 keV. We include the estimated EFTerrors. The NLO Model A results have a 10% error, andthe NNLO Model B results have a 3% error. The di↵er-ent EFT error estimates has to do with the distinctionbetween “accuracy and precision”. The error estimatesfrom higher order corrections represent precision, and dif-

Fit R0

�2 0.395 ± 0.014±0.039

Model A I 0.387+0.018�0.015 ± 0.039

Model A II 0.389+0.008�0.007 ± 0.039

Model A⇤ II 0.379+0.010�0.005 ± 0.038

Model B⇤ II 0.401+0.005�0.005 ± 0.012

TABLE II. 3He(↵, �)7Be: Branching ratio R0 at threshold(defined as E? = 60⇥10�3 keV). The second set of errors areestimated from the EFT perturbation as detailed in the text.

ferent power countings have di↵erent accuracy and pre-cision.

FIG. 7. 3He(↵, �)7Be: S-factor S34 and branching ratio R0

at threshold (E? = 60 ⇥ 10�3 keV). The posterior distribu-tions for fits Model A I, Model A II, Model A⇤ II and ModelB⇤ II are presented as green, red, blue and purple coloredhistograms, respectively. The corresponding Gaussian proba-bility distributions are given by the dotted, dashed, solid anddot-dashed curves, respectively.

Fig. 7 shows the posterior distributions for the 4 S-factors and branching ratios from Tables I and II. Thesymmetric distributions can be described with a Gaus-sian form shown by the various smooth curves unlike theskewed distributions as expected. The spread in some ofthe quantities is related to the uncertainty in the parame-

ter estimates, especially the p-wave e↵ective ranges ⇢(±)1

for Model A fits, see Table V. Large magnitude |⇢(±)1 |

makes the wave function normalization constant smallerwhich can be compensated by a larger two-body current

L(±)1 as the parameter estimates indicate. The planned

TRIUMF 3He-↵ elastic scattering experiments and phaseshift analysis at low energies E & 500 keV would be ableto shed some light on this [26], and help establish theappropriate EFT power counting.The EFT S-factor at threshold can be compared to

other recent calculations such as – 0.593 keV b fromFMD [27], 0.59 keV b from NCSM [28]; and evalua-tions such as – [0.580 ± 0.043(stat.) ± 0.054(sys.)] keVb from Cyburt-Davids [6], (0.57 ± 0.04) keV b fromERNA [7], (0.567± 0.018± 0.004) keV b from LUNA [8],(0.554±0.020) keV b from Notre Dame [9], (0.595±0.018)

8

have a power counting where two-body currents appearat higher order in the perturbation. This would requirea smaller s-wave scattering length. We call this powercounting Model B. If we ignore the poorly known elasticscattering data, then this second power counting can beused to describe the capture data. The relatively smallerinitial state interaction contributions in Model B can becompensated by a larger wave function normalizationconstant which requires only small variation in the re-spective p-wave e↵ective ranges as described earlier. Wediscuss the results of the analysis for 3He(↵, �)7Be and3H(↵, �)7Li separately below.

A. 3He(↵,�)7Be

We present the 8 di↵erent fits used to draw Bayesianinference for this reaction. First we start with the fits inthe smaller capture energy region I (E . 1000 keV). Theposterior odds favored Model A both with and withoutphase shift data in the fits. However, if we also look atthe overall trend then we see the data for S-factor S34

rises upward from around E ⇠ 1500 keV. The Model Afit with phase shift, we call Model A I, best describesthe capture data over the energy range E . 2000 keV.We note that the Model B fits in this region are not self-consistent in that they suggest an a0 value larger thanthe power counting, table V.

In the fits to capture energy region II (E . 2000 keV),Model A with phase shift, we call Model A II, is favoredover Model B by the posterior odd. For the fits with-out phase shifts, both Model A⇤ II and Model B⇤ II areequally favored by the posterior odd ratio. The asterisk⇤ indicates fits without phase shift data.

Fit S34(E?) (keV b) S034(E?) (10

�4 b)

�2 0.558 ± 0.008±0.056 �2.71± 0.20± 0.27

Model A I 0.541+0.012�0.014 ± 0.054 �1.34+0.64

�0.59 ± 0.13

Model A II 0.550+0.009�0.010 ± 0.055 �2.00+0.36

�0.35 ± 0.20

Model A⇤ II 0.551+0.021�0.014 ± 0.055 �1.86+0.72

�1.69 ± 0.19

Model B⇤ II 0.573+0.007�0.007 ± 0.017 �3.72+0.11

�0.10 ± 0.11

TABLE I. 3He(↵, �)7Be: S34 and S034 at threshold (defined as

E? = 60⇥ 10�3 keV). The second set of errors are estimatedfrom the EFT perturbation as detailed in the text.

Tables I and II has the S-factors S34 and branchingratios R0 for the 4 fits described above. We also includethe �2 fit of Model A for comparison [2]. We include thederivative S0

34 as well. All the numbers were evaluatedat E? = 60 ⇥ 10�3 keV. We include the estimated EFTerrors. The NLO Model A results have a 10% error, andthe NNLO Model B results have a 3% error. The di↵er-ent EFT error estimates has to do with the distinctionbetween “accuracy and precision”. The error estimatesfrom higher order corrections represent precision, and dif-

Fit R0

�2 0.395 ± 0.014±0.039

Model A I 0.387+0.018�0.015 ± 0.039

Model A II 0.389+0.008�0.007 ± 0.039

Model A⇤ II 0.379+0.010�0.005 ± 0.038

Model B⇤ II 0.401+0.005�0.005 ± 0.012

TABLE II. 3He(↵, �)7Be: Branching ratio R0 at threshold(defined as E? = 60⇥10�3 keV). The second set of errors areestimated from the EFT perturbation as detailed in the text.

ferent power countings have di↵erent accuracy and pre-cision.

FIG. 7. 3He(↵, �)7Be: S-factor S34 and branching ratio R0

at threshold (E? = 60 ⇥ 10�3 keV). The posterior distribu-tions for fits Model A I, Model A II, Model A⇤ II and ModelB⇤ II are presented as green, red, blue and purple coloredhistograms, respectively. The corresponding Gaussian proba-bility distributions are given by the dotted, dashed, solid anddot-dashed curves, respectively.

Fig. 7 shows the posterior distributions for the 4 S-factors and branching ratios from Tables I and II. Thesymmetric distributions can be described with a Gaus-sian form shown by the various smooth curves unlike theskewed distributions as expected. The spread in some ofthe quantities is related to the uncertainty in the parame-

ter estimates, especially the p-wave e↵ective ranges ⇢(±)1

for Model A fits, see Table V. Large magnitude |⇢(±)1 |

makes the wave function normalization constant smallerwhich can be compensated by a larger two-body current

L(±)1 as the parameter estimates indicate. The planned

TRIUMF 3He-↵ elastic scattering experiments and phaseshift analysis at low energies E & 500 keV would be ableto shed some light on this [26], and help establish theappropriate EFT power counting.The EFT S-factor at threshold can be compared to

other recent calculations such as – 0.593 keV b fromFMD [27], 0.59 keV b from NCSM [28]; and evalua-tions such as – [0.580 ± 0.043(stat.) ± 0.054(sys.)] keVb from Cyburt-Davids [6], (0.57 ± 0.04) keV b fromERNA [7], (0.567± 0.018± 0.004) keV b from LUNA [8],(0.554±0.020) keV b from Notre Dame [9], (0.595±0.018)

We recommend A II if using phase shift information. Alternatively A* II or B* II if not using phase shift information

9

keV b from Seattle [10], and (0.53 ± 0.02 ± 0.01) keV bfrom Weizmann [11]. We also compare to a recent EFTwork using Bayesian inference [29] that finds at thresh-old S34(0) = 0.578+0.015

�0.016 keV b and R0(0) = 0.406+0.013�0.011.

Ref. [29] only used capture data to draw their inferences.Looking at tables I and II, it would seem the results ofRef. [29] are more aligned with Model B⇤ II, though theexact power counting used the article is not clear. The“best” recommended value from the review in Ref. [1] is:S34(0) = [0.56± 0.02(expt.)± 0.02(theory)] keV b.

From the various fits, we recommend the following:Model A II if we want to include phase shift information,and either Model A⇤ II or Model B⇤ II if no phase shiftinformation is used.

B. 3H(↵,�)7Li

Fit S34(E?) (keV b) S034(E?) (10

�4 b)

�2 0.098 ± 0.003±0.016 �1.13± 0.24± 0.18

Model A 0.097+0.003�0.002 ± 0.016 �1.04+0.12

�0.20 ± 0.17

Model A⇤ 0.102+0.002�0.002 ± 0.016 �1.81+0.16

�0.15 ± 0.29

TABLE III. 3H(↵, �)7Li: S34 and S034 at threshold (defined as

E? = 60⇥ 10�3 keV). The second set of errors are estimatedfrom the EFT perturbation as detailed in the text.

To draw our Bayesian inferences, we performed 4 dif-ferent fits in this system. Here the EFT power countingof Model A (with phase shift data) and Model A⇤ (with-out phase shift data) are favored. Moreover, the ModelB and Model B⇤ fits are not consistent with the powercounting estimate for the a0 values, table VI. The S-factors and branching ratios are in Tables III and IV.The corresponding posterior distributions, and the as-sociated Gaussian forms are shown in Fig. 8. The fitwithout phase shift data gives a larger S34 at thresh-old. We included a 16% EFT error estimate from NNLOcorrections. In this channel the binding energies arelarger, resulting in a larger perturbation error comparedto 3He(↵, �)7Be.

The fits, especially without the phase shift data, givea central s-wave shape parameter value s0 that is largerthan ideally expected from the power counting. Powercounting consistency would bias one towards Model A(with phase shift) over Model A⇤ (without phase shift).

The EFT S-factor calculation can be compared to re-cent theoretical results – 0.12 keV b from FMD [27], 0.13keV b from NCSM [28]; and experimental evaluation –[0.1067 ± 0.0004(stat.) ± 0.0060(sys.)] keV b from Cal-tech [14].

We leave for future a more careful study of the system-atic errors associated with the estimation of the param-eters and evidences using the various Nested Samplingmethods. In our evaluations, for example, the Multi-Nest [21] and Di↵usive Nested Sampling [30] algorithms

Fit R0

�2 0.434 ± 0.006±0.069

Model A 0.438+0.008�0.009 ± 0.070

Model A⇤ 0.439+0.008�0.009 ± 0.070

TABLE IV. 3H(↵, �)7Li: Branching ratio R0 at threshold (de-fined as E? = 60 ⇥ 10�3 keV). The second set of errors areestimated from the EFT perturbation as detailed in the text.

FIG. 8. 3H(↵, �)7Li: S-factor S34 and branching ratio R0 atthreshold (E? = 60 ⇥ 10�3 keV). The posterior distributionsfor fits Model A and Model A⇤ are presented as red and bluecolored histograms, respectively. The corresponding Gaussianprobability distributions are given by the solid and dashedcurves, respectively.

gave similar results. The EFT parameters (tables V andVI) agreed within the error bars. Similarly, the capturecross sections and branching ratios (tables I, II, III,IV) also agreed within the error bars from the fits.

ACKNOWLEDGMENTS

The authors thank Barry Davids, Renato Higa, DanielR. Phillips, Prakash Patil and Xilin Zhang for many valu-able discussions. This work was supported in part by U.S.NSF grants PHY-1615092 and PHY-1913620. The fig-ures for this article have been created using SciDraw [31].

Appendix A: s and d wave capture

The capture from initial s-wave state is given by:

|A(p)|C0(⌘p)

=

�����X(p)� 2⇡

µ2

B(p) + µJ0(p) + µ2k0L(⇣)E1

[C0(⌘p)]2p(cot �0 � i)

����� ,

(A1)

where k0 is the photon energy and L(⇣)E1 for ⇣ = 2P3/2 and

⇣ = 2P1/2 are the 2 two-body currents. We modified somedefinitions compared to Ref. [2] but they are equivalentexpressions.

!17

Adelberger et al., RMP 83, 195 (2011)

Conclusions

!18

• Bayesian analysis incorporates EFT power counting estimates naturally

• Theory error propagation in the analysis: GP, other, mean/variance?

• Bayesian analysis as a tool for constructing EFT power counting —with same or different particle content

• Need better quantification of prior odd ratio.

Thank you!