Linking ab initio models, phenomenology, & data

Kenneth NollettSan Diego State University

The 2016 R-Matrix Workshop on Methods and ApplicationsSanta Fe

June 30, 2016

Why compute cross sections? (Probably just my answers, not yours)

For astrophysics, cross sections must be estimated and thermally averaged

We just want the most accurate estimates possible, from theory or experimentor a combination

We need error estimates on the “evaluated” cross sections

We need some test of individual data sets and of theory inputs

For the Sun and the big bang, we need percent-level accuracy

Some of us are also trying to understand the structure & dynamics of nuclei (abinitio or otherwise)

That’s a separate agenda with different needs

An example: 7Be(p, γ)8B (S17)

“Standard” rates for 7Be(p, γ)8B have mostly not come from R-matrix (paceBarker)

At low energy, this is dominantly external direct capture (have to integratebeyond 100 fm)

Going back to Tombrello (1965), data have been fitted to Woods-Saxon models

Cluster models of 8B are very poorly constrained (no scattering data until 2003)

Woods-Saxon parameters have foundation in “tradition” & in fitting π = +1

states including ground state

But capture comes from π = −1 s- & d-wave states

You assume that the model is missing spectrocopic factors, & adjust overallscale to the nonresonant capture data

An example: 7Be(p, γ)8B (S17)

In recent decades, nucleon-level models (usually RGM/GCM) have been assumedto give the best energy dependences

But: Simplifications easily screw up the overall scale by 20-30%e.g. RGM/GCM with simple forces, or kludges on variational Monte Carlo

In Solar Fusion II (Adelberger et al. 2010) we assumed a model energy dependence& adjusted scale to match data

in Solar Fusion I. Total errors, including systematic errors, areshown on each data point, to facilitate a meaningful com-parison of different data sets. All data sets exhibit a similarS17ðEÞ energy dependence, indicating that they differ mainlyin absolute normalization.

Following the discussion in Sec. IX.B, we determine ourbest estimate of S17ð0Þ by extrapolating the data using thescaled theory of Descouvemont (2004) (Minnesota calcula-tion). We performed two sets of fits, one to data below theresonance, with E # 475 keV, where we felt the resonancecontribution could be neglected. In this region, all the indi-vidual S17ð0Þ error bars overlap, except for the Bochumresult, which lies low.

We also made a fit to data with E # 1250 keV, where the1þ resonance tail contributions had to be subtracted. We didthis using the resonance parameters of Junghans et al. (2003)(Ep ¼ 720 keV, !p ¼ 35:7 keV, and !! ¼ 25:3 meV), add-ing in quadrature to data errors an error of 20% of theresonance subtraction. In order to minimize the error inducedby variations in energy averaging between experiments, weexcluded data close to the resonance, from 490 to 805 keV,

where the S factor is strongly varying and the induced error islarger than 1.0 eV b. Above the resonance, the data havesmaller errors. Only the Filippone et al. (1983) andWeizmann group error bars overlap the UW–Seattle/TRIUMF error bars.

Figure 9 shows the best-fit Descouvemont (2004)(Minnesota interaction) curve from the E # 475 keV fit [to-gether with the 1þ resonance shape determined by Junghanset al. (2003), shown here for display purposes]. Our fit resultsare shown in Table VII. The errors quoted include the in-flation factors, calculated as described in the Appendix. Themain effect of including the inflation factors is to increase theerror on the combined result by the factor 1.7 for E #475 keV, and by 2.0 for E # 1250 keV. Both the S17ð0Þcentral values and uncertainties from the combined fits forthese two energy ranges agree well, the latter because theadded statistical precision in the E # 1250 keV fit is mostlyoffset by the larger inflation factor.

We also did fits in which the low-energy cutoff was variedfrom 375 to 475 keV and the high-energy exclusion regionwas varied from 425–530 to 805–850 keV. The central valueof S17ð0Þ changed by at most 0.1 eV b. On this basis weassigned an additional systematic error of &0:1 eV b to theresults for each fit region.

To estimate the theoretical uncertainty arising from ourchoice of the nuclear model, we also performed fits using theshapes from other plausible models: Descouvemont (2004)plus and minus the theoretical uncertainty shown in Fig. 8 ofthat paper; Descouvemont and Baye (1994); the CD-Bonn2000 calculation shown in Fig. 15 of Navratil et al. (2006b);and four potential-model calculations fixed alternately toreproduce the 7Liþ n scattering lengths, the best-fit 7Beþp scattering lengths, and their upper and lower limits (Davidsand Typel, 2003). The combined-fit results for all thesecurves, including Descouvemont (2004), are shown inTable VIII.

We estimate the theoretical uncertainty on S17ð0Þ from thespread of results in Table VIII: &1:4 eV b for the E #475 keV fits, and þ1:5

'0:6 eV b from the E # 1250 keV fits

(the smaller error estimate in the latter case reflects theexclusion of the poorer potential-model fits). We note thatthe estimated uncertainties are substantially larger than thosegiven by Junghans et al. (2003) and by Descouvemont(2004).

FIG. 9 (color online). S17ðEÞ vs center-of-mass energy E, for E #1250 keV. Data points are shown with total errors, includingsystematic errors. Dashed line: scaled Descouvemont (2004) curvewith S17ð0Þ ¼ 20:8 eV b; solid line: including a fitted 1þ resonanceshape.

TABLE VII. Experimental S17ð0Þ values and (inflated) uncertainties in eV b, and "2=dof deter-mined by fitting the Descouvemont (2004) Minnesota calculation to data with E # 475 keV and withE # 1250 keV, omitting data near the resonance in the latter case.

Fit range E # 475 keV E # 1250 keVExperiment S17ð0Þ # "2=dof S17ð0Þ # "2=dof

Baby 20.2 1.4a 0:5=2 20.6 0.5a 5:2=7Filippone 19.4 2.4 4:7=6 18.0 2.2 15:8=10Hammache 19.3 1.1 4:8=6 18.2 1.0 12:5=12Hass 18.9 1.0 0=0Junghans BE3 21.6 0.5 7:4=12 21.5 0.5 12:3=17Strieder 17.2 1.7 3:5=2 17.1 1.5 5:1=6

Mean 20.8 0.7 9:1=4 20.3 0.7 18:1=5

aWe include an additional 5% target damage error on the lowest three points, consistent with thetotal error given in the text by Baby et al. (2003a) [M. Hass, 2009 (private communication)].

224 Adelberger et al.: Solar fusion cross . . .. II. The pp chain . . .

Rev. Mod. Phys., Vol. 83, No. 1, January–March 2011

S17(0) = 20.8± 0.7 (ex)± 1.4 (th) eV b

Theory error dominates

It’s estimated by normalizing 9 “plausible”models to the data

Quantitatively predictive theory

Ab initio theory can do a lot of stuff at percent precision in light (A . 12) nuclei:energies, charge radii, magnetic moments

Percent precision should be achievable now-ish for low-energy radiative capture

Ab initio definition:

1. Start from bare nucleon-nucleon and (ideally) three-nucleon forces, tuned toNN scattering, deuteron, some light-nucleus masses (∼ 30 parameters)

2. Compute e.g. A = 8 system with no particular tuning to A = 8 properties

A = 3,4 reactions have been treated successfully by hyperspherical harmonic(Pisa) & Faddeev-like (Lisbon) methods

E.g., Pisa calculation of d(p, γ)3He is probably more precise than lab data atbig bang energies (∼ 120 keV; < 5% vs. 9%)

Quantitatively predictive theory

Livermore/TRIUMF group has gotten a long way doing this with a modified no-core shell model (NCSM)

They’ve gotten way ahead of me, but quantumMonte Carlo is also usable for this

3H(α, γ)7Li, 3He(α, γ)7Be have been donewith FMD (T. Neff, maybe a one-off)

NNN forces have been “left out” in all but somenα scattering & semi-ab-initio (α, γ)

But probably NNN sensitivity is mainly inthreshold placement

Are these models purely take-it-or-leave-it?

Or can you combine them with experimentalinformation on the probed channels?

382 P. Navrátil et al. / Physics Letters B 704 (2011) 379–383

Fig. 4. (Color online.) Calculated 7Be(p,γ )8B S-factor as function of the energyin the c.m. compared to data and the microscopic cluster model calculations ofRef. [15] with the Minnesota (MN) interaction (a). Only E1 transitions were con-sidered. Initial-state partial wave contributions are shown in panel (b). Calculationas in Fig. 2.

tials with the aim to match closely the experimental s.e. in eachof the largest calculation. From these results we conclude that theuse of the Nmax = 10 space is justified and a limitation to the fivelowest 7Be eigenstates is quite reasonable (or that the Nmax = 8space is insufficient and a limitation to just 3 states is unrealis-tic). Also, based on these results we estimate the uncertainty ofour calculated S17(0) to be ±0.7 eV b.

An interesting feature of the S-factor is its flattening around1.5 MeV. As seen in Fig. 4(b), this phenomenon is due to theS-wave contribution that dominates the J i = 2− and 1− partialwaves at low energies. The increase of flattening with the num-ber of 7Be eigenstates included in the calculation, seen in Fig. 5(a),indicates that this is an effect due to the many-body correlations.This finding corroborates the observations of Ref. [15], where theflattening was attributed to the deformation of the 7Be core. Atthe same time, we find that the flattening is somewhat corre-lated with the S-wave scattering length. For example, the S-waves = 2 scattering length increases in absolute value from −13.4 fm,in the calculation with four states, to −14.5 fm, in that with theeight states [see Fig. 5(a)]. We also note that the flattening foundin the present work is slightly larger than that obtained in themicroscopic three-cluster model of Ref. [15], as seen in Fig. 4(a).Presumably, this is because in the three-cluster model the 7Bestructure was assumed to be of 3He–4He nature only, while theNCSM wave functions include in addition 6Li-p configurations, par-ticularly for the 5/2−

27Be state, as discussed earlier.

In Table 2, we summarize our calculated S17(0) S-factor, theS-wave scattering lengths and selected ANCs. We note that theANCs from our ab initio approach are smaller than those obtainedwithin the microscopic three-cluster model [15] (consistently with

Fig. 5. (Color online.) Convergence of the 7Be(p,γ )8B S-factor with the number of7Be eigenstates (a) and the size of the HO basis used to expand the 7Be eigenstatesand localized parts of the integration kernels (b). The number of eigenstates and thecalculated separation energy in each case is shown in the legend. HO frequencies ofhΩ = 19 MeV (a) and 17 MeV (b) corresponding to the respective minima of 7Beg.s. were used.

Table 2Calculated S17(0) S-factor, the p-7Be S-wave scattering lengths, a01 (s = 1), a02(s = 2) and the 8B ground-state ANCs for 7Be(g.s.)-p in the P -wave and the channelspins s = 1 (C11) and s = 2 (C12). The NCSM/RGM calculation as described in Fig. 2.The S17(0) uncertainty was obtained as discussed in the text.

S17(0) [eV b] a01 [fm] a02 [fm] C11 [fm−1/2] C12 [fm−1/2]19.4(7) −5.2 −15.3 0.294 0.650

our smaller S17(0) value as seen from Fig. 4(a)) but in fairly goodagreement with recent ab initio variational Monte Carlo calcula-tions [38]. They are, however, still larger than the experimentalones from the DWBA analysis of Ref. [39].

The present calculations improve significantly the first NCSM-based 7Be(p,γ )8B S-factor calculations [16] by a consistent treat-ment of both the bound and the scattering states. In Ref. [16],the scattering states were obtained within a potential model. Theshape of the S-factor presented in this work clearly superseeds thatfound in Ref. [16]. The presently found S17(0) S-factor value is inbetween the values found in Ref. [16] with the two different NNpotential models used there.

In conclusion, we performed ab initio many-body calculations ofthe 7Be(p,γ )8B radiative capture that predict simultaneously boththe normalization and the shape of the S-factor. Our S-factor resultat zero energy, S17(0) = 19.4(7) eV b, is on the lower side of, butconsistent with, the latest evaluation, and its shape follows closelythe Coulomb breakup data from Ref. [8]. Our calculations can befurther improved by including effects of additional higher-lying7Be resonances. This can be best done by coupling the NCSM/RGMbinary-cluster basis with the NCSM calculations for 8B as outlinedin Ref. [40]. The inclusion of three-nucleon interactions, both chiral

Navratil et al., PLB 704, 379 (2011)

Do we really want purely ab initio calculations?

That depends on what you’re doing

If you want to test the theory/computation/potential, then emphatically yes

If you want to do astrophysics or technological applications, probably not

Some things will be determined more easily by experiment, others ab initio

In fact, NCSMC 7Be(p, γ)8B does this slightly, not purely ab initio

Cross sections depend strongly on threshold placement, which is hard to buildinto NN with high precision

Sophia chooses Λ ≥ k cutoff ofNN so that induced-but-omittedNNN cancels“bare” NNN in binding energy

How to combine experiment & ab initio for applications?

Refitting NN & NNN is not out of the question, but a lot of labor& compromises ab initio as independent information

You could take advantage of “computation” parameters as opposed to potentialparameters, as with Sophia’s Λ, independence compromised

You might also decide to use pure ab initio cross sections for extrapolation/thermalaveraging

But you only do that once precision is tested

Data would then provide the precision tests of a given calculation:Test the model polarization, dσ/dΩ, elastic, etc. instead of getting total σ

“Fewer-body” methods

Probably a better way to combine ab initio & empirical data is in a simpler modelthat captures small-scale or many-body effects in parameters

This could be R-matrix, potential models, effective field theory. . .

It’s already done with simpler “microscopic” theories – e.g. spectroscopic factorsin potential models or DWBA from shell model

You could imagine producing “artificial data” to fold in with measured data

It’s more appealing if some parameters can be computed directly from ab initio (likespectroscopic factors, only better)

Fewer-body methods: weighting experiment & ab initio

Data won’t be fully independent of computed parameters, so you’ll still needsome relative weighting of theory & experiment (& attention to conflict)

Theory could restrict range for parameter search or just give the starting point

Probably the best way to use the ab initio information is as Bayesian priors

Or almost the same thing, treat some output from ab initio as data even if it isn’tactually an observable

Asymptotic normalization constants (ANCs)

Channel ANCs are natural parameters for nonresonant direct capture(R-matrix, effective field theory, even potential model)

E.g. : Φ3He(rpd →∞) =∑

l=0,2

CljφdφpYlm(rpd)W−η,l+12(2krpd)/rpd

There is a Clj for each d+ p channel in3He (Clebsch-Gordan hidden)

l = 0 (red) & l = 2 (blue) possible

j = 1/2 bound state

We saw these in R-matrix Tuesday

Points are Monte Carlo integration of awave function

But there’s a smarter way to get Clj, used for solid curves

Extracting ANCs from ab initio

Pulling tails out of variational (VMC) or relaxation (GFMC) methods is hard todo accurately

It’s easier with Sofia’s computed R-matrix approach

For quantum Monte Carlo methods, it’s best to use a sort of negative-energyLippman-Schwinger

Pinkston & Satchler (or Kawai & Yazaki):

Clj =2µ

k~2wA∫ M−η,l+1

2(2krcc)

rccΨ†A−1χ

†Y †lm(rcc) (Urel − VC) ΨAdR

First applied to ANCs from shell model by Mukhamedzhanov & Timofeyuk, Carlfinally got me to understand it

This extracts the correct tail by integration over just the wave function interior

Survey of ANCs from variational Monte Carlo

to

2.13

(full range to 2.0)

Nollett & Wiringa, PRC 83, 41001 (2011)

We computed ANCs from nearly all VMCbound state wave functions

Small error bars are VMC statistics

Large ones are “experimental”

Sensitivity to wave function constructionseems weak but hard to quantify

A ≤ 4 clearly dominated by systematics,also old

In a potential model, our 8B ANCs giveS17(0) = 20.8 eV · b, same asrecommended value

With less justification, you can estimate resonance widths similarly

The integral estimate should apply to states that are in some sense narrow

I’ve chosen low-lying states in A ≤ 9 with width mainly/all in nucleon emission

Nollett, PRC 86, 44330 (2012)

Red: overlaps inconsistent withresonance

Asterisk: uncomputed channels

Dynamic range of 0.0005 to. 1.0 MeV, not otherwisepossible for QMC

Yes, you can compute widths asANCs, but the math is fuzzy forreal-energy methods

ANCs as input for effective field theory (EFT)

About the time I finished the ANCs & widths, I arrived at Ohio University

Daniel Phillips works with the halo EFT formalism

EFT can be applied to a range of particle, nuclear, & atomic problems wheremomenta k are small compared to some high-momentum scale Λ

You write out the most general Lagrangian consistent with symmetries & sortout a hierarchy of operators in (k/Λ)n

Then you apply the machinery of quantum field theory

If k/Λ is small, you get precision near threshold & quantitative estimate oftrunction error

Halo EFT at lowest order (LO)

Xilin Zhang took this up and developed “halo EFT” for 7Li(n, γ)8Li & 7Be(p, γ)8B

Λ is 7Li or 7Be breakup momentum (not energy)

Binding momentum of a “halo” is γ ≡ √2µEsep Λ

Each has two angular-momentum channels, which Xilin paid careful attentionto (I had ANCs)

RAPID COMMUNICATIONS

XILIN ZHANG, KENNETH M. NOLLETT, AND D. R. PHILLIPS PHYSICAL REVIEW C 89, 051602(R) (2014)

TABLE I. Key physical scales in our EFT. The 8B → p + 7Bethreshold is B8B , the 7Be → 3He + 4He threshold is B7Be, and the7Be core-excitation energy is E∗. The p-7Be effective mass MR ≡MnMc/(Mn + Mc) = 7/8Mn and the 3He-4He effective mass M ′

R =12/7Mn. The scattering parameters a and r in the incoming s-wavechannels are not well determined, but generically obey the hierarchygiven here. The numbers for a1 and r1 are extracted from ab initioANCs.

Momentum scale Definition Value

kC ∼ γ QcQnαEMMR 24.02 MeV

γ√

2MRB8B 15.04 MeV

#√

2M ′RB7Be 70 MeV

γ ∗ ∼ γ√

2MR(B8B + E∗) 30.53 MeV

γ$ ∼ γ√

2MRE∗ 26.57 MeVa3S1 , a5S2

∼ 1/γ Scattering lengths Variesr0 ∼ 1/# l = 0 effective ranges Variesa1 ∼ γ −2#−1 Scattering volume 1054.1 fm3

r1 ∼ # l = 1 effective “range” −0.34 fm−1

Here nσ , ca , dδ , πα are fields of the proton (“nucleon”),7Be core ( 3

2−

), 7Be∗ ( 12

−), and 8B ground state (2+), re-

spectively; g(3S1) and g(5S2) in LS are related to “unnaturally”enhanced ∼1/γ s-wave charged-particle scattering lengths(see Refs. [14,17,18]); g(3S∗

1 ) describes 7Be + p ↔ 7Be∗ + pand is assumed to be natural, i.e., ∼1/# [23]; and hY arethe p-wave couplings. The fields’ indices are their spinprojections with a specific convention: σ,δ,σ ′,δ′ = ±1/2,a,a′ = ±3/2, ± 1/2, α,β = ±2, ± 1, 0, and i,j,k = ±1, 0;$ is πα’s bare binding energy; T ...

... s are the C-G coefficients(cf. [23]); V n;c are proton and core velocities. To implementelectromagnetic interactions, we use minimal substitution∂µ → ∂µ + ieQAµ with e ≡

√4παEM and Q the particle’s

charge.

III. p-WAVE SCATTERING AND SHALLOWBOUND STATE

We denote πα’s dressed propagator as Dβα ≡ Dδβ

α . Itsself energy receives contributions from p-7Be intermediatestates in the 3P2 and 5P2 channels, and p-7Be∗ in the 3P2channel [23]. In each case the proton and 7Be (7Be∗) interactvia the Coulomb interaction. We include these Coulomb effectsto all orders in αEM, thereby extending the calculation ofRefs. [14,17,18] to p waves. In practice this involves replacingthe plane waves of Ref. [23] by Coulomb wave functions inboth external legs and intermediate states. D(E) can then bewritten in terms of the a1 and r1 in the 2+ channel:

−6πMR

h2t D

= − 1a1

+ r1

2k2 − 2kC

(k2 + k2

C

)H (kC/k)

− 2kC

h2(3P ∗

2 )

h2t

(k2∗ + k2

C

)H (kC/k∗). (3)

with H (η) = ψ(iη) + 1/(2iη) − ln(iη) [24], h2t ≡ h2

(3P2) +h2

(5P2), and k∗ ≡√

2MR(E − E∗). a1 and r1 are then functions

of the Lagrangian parameters $, h(3P2), h(5P2), and h(3P ∗2 ). If k <

γ$ we recover the standard Coulomb-modified effective-rangeexpansion for l = 1; i.e., the right-hand side of Eq. (3) becomesC2

η,1k3(cot δ1 − i), with δ1 the phase shift relative to a Coulomb

wave, Cη,l ≡ 2l exp (−πη/2)|,(l + 1 + iη)|/,(2l + 2), andη ≡ kC/k [25,26].

We write the p-wave Coulomb-distorted T -matrix,TCS [14,26], in the p-7Be-7Be∗ Hilbert space schematicallyas V × D(E) × V . [V stands for n-c(d)-π interactions andD is given by Eq. (3).] Since TCS has a pole at B8B we haveD−1(k = iγ ) = 0. The residue then yields a relation for thewave-function renormalization factor, Z:

6π

Z= −h2

t r1 + 2kC

γ

h2

t

γ 2

[2γ 3H

(kC

γ

)

+(k3C − kCγ 2)H ′

(kC

γ

)]

+h2

(3P ∗2 )

γ ∗2

[2γ ∗3H

(kC

γ ∗

)+

(k3C − kCγ ∗2)H ′

(kC

γ ∗

)],

(4)

where, for convenience, we define H (z) ≡ H (−iz), H′(z) ≡

dH (z)/dz. The 8B → p + 7Be(7Be∗) ANCs, C(5P2), C(3P2),and C(3P ∗

2 ) are then given by (cf. Ref. [23])

C2(Y )

h2Y γ 2,2(2 + kC/γ )

=C2

(3P ∗2 )

h2(3P ∗

2 )γ∗2,2(2 + kC/γ ∗)

= Z

3π, (5)

with Y = 3P2 and 5P2.

IV. LEADING-ORDER PROTON CAPTURE

The relevant diagrams are shown in Fig. 1. Let us firstfocus on initial total spin Si = 1 and use the notation,⟨πα|LEM|χ (+)

p ,δ,a⟩ ≡ T δai T

ijα Mj , with LEM the term in the

Lagrangian that governs the interaction with the (transverse)photon, jEM · A, and |χ (+)

p ,δ,a⟩ the incoming Coulomb wavefunction for the p-7Be state, labeled by the asymptotic relativemomentum p and the individual spin projections of the twoparticles. The LO Mj is computed in coordinate spaceand decomposed into contributions from incoming s and

apc

σpn

k λ

p α

FIG. 1. Diagrams for capture. The line assignments are des-ignated by the spin indices; the shaded blobs denote the fullCoulomb-Green functions; the bold vertex in the last two diagramsmeans S-wave multiple scattering with the Coulomb interactionincluded to all orders in αEM [18,23]. Diagrams with the photoncoupled to the proton are not shown but are included in ourcalculation.

051602-2

Thin solid: “nucleon” n or p Dashed: 7Be (s = 1,2) Thick: 8B

Blob: Coulomb (all orders) Dot: s-wave coupling (all orders; a1 & a2)

Simple vertex: p-wave coupling from ANC

Halo EFT at LO

Xilin’s LO model is quite nice – ANCs from me, scattering lengths from Anguloet al.

It does well with branching ratios & partial cross sections

-3

-2

-1

0

1

2

3

-6 -5 -4 -3 -2 -1 0 1 2 3

Log[σ

(mb)

]

Log[En(KeV)]

cal1.4 cal0.6 cal

HeilBlackmon

Imhof(a)Imhof(b)

LynnNagai

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5S

(eV

b)E (MeV)

BabyHammache

StriederFilippone

Junghans

junk Zhang, Nollett, Phillips, PRC 89, 051602 (2014) & 89, 024613 (2014).

7Li(n, γ)8Li 7Be(p, γ)8B

Nothing to see here

But the absolute cross section is really only good to ∼ 40%, & reproducing1/v isn’t very interesting

Halo EFT at next-to-leading order (NLO)

Thus encouraged, we (Xilin) pressed on to next order in k/Λ, with 9 parameters:

2 ANCs: Cs (s = 1,2)

2 short-distance contributions from 7Be ground state (like R-matrix internalcapture): Ls

1 coupling to excited 7Be (off-diagonal in R-matrix internal capture): εs

2-term effective-range expansion in each s-wave channel, modeled as an unbound“dimer” analogous to bound state: (as & rs)

S(E) = f(E)∑

sC2s

[ ∣∣∣SEC (E; δs(E)) + LsSSD (E; δs(E))

+εsSCX (E; δs(E))∣∣∣2

+ |DEC(E)|2]

The S & D matrix elements are close to parts of Barker & Kajino R-matrix; δspretty much effect of elastic channel in Barker-Kajino

NLO halo EFT in pictures

In diagrams, it looks like this: (I) (II) (III) (IV)

(V) (VI) (VII)

(I)

FIG. 8: Diagrams for radiative capture to p-wave shallow bound state. The long dashed and short

dashed line are for core (c) and core excitation (d) fields. The first four diagrams ∼ V12

(VVΛ

) 12

(LO),

the last three diagrams ∼ V12

(VVΛ

) 32

(NLO). The radiative corrections would come in effectively

at N2LO.

We note that these values agree with the naive power couning, i.e., h(3P2) ∼ h(5P2) ∼ h(3P ∗2 ),

r1 ∼ Λ, and a1 ∼ 1/ (Λγ2).

III. CAPTURE REACTION AMPLITUDE AND S FACTOR

The capture reaction is studied in detail in this section. Fig. 8 shows the relevant Feynman

diagrams up to NLO. The thin solid, dashed, and dotted lines denote proton, 7Be, and 7Be∗

fields; the open and filled box are for φ(1) (and φ(2)), and π fields. According to the power

counting of interaction vertices and propagators, the first four diagrams ∼ V12

(VVΛ

) 12

are

LO, while the last three are NLO ∼ V12

(VVΛ

) 32. The 5th and 6th diagrams differ the 3rd and

4th by having 7Be∗ in the intermediate state insead of 7Be; the former coupling strength is

also suppressed by a factor of klow/Λ. The last diagram originates from the NLO contact

terms in the lagrangian,

Lc = eZeffL1π†αT ij

α Eiφ(1)j + eZeffL2π†αT iβ

α Eiφ(2)β + C.C. (40)

These terms are built based on Ref. [18] and in a similar way used in Ref. [5]. Its structure

is the same as expression (5), except the spin degrees of freedom. Its power counting can be

found in the toy model discussion (see Section I A), according to which L1,2 should scale as

1/Λ.

18

S(E) = f(E)∑

sC2s

[ ∣∣∣SEC (E; δs(E)) + LsSSD (E; δs(E))

+εsSCX (E; δs(E))∣∣∣2

+ |DEC(E)|2]

Choices of the 9 parameters seem to span space of models pretty well afterdetailed comparison

So where do we get Cs, Ls, εs, as, rs?

We fit capture data, take as from Angulo

Bayesian treatment of parameters

We have too little information to determine all 9 parameters

But if what we want is S(E < 500 keV), that’s well determined

You can look at this as finding posterior probability of S(E) from capture data,with Angulo as & data norms as Gaussian-distributed priors

We fitted atE < 500 keV, where resonant channels can be ignored, (k/Λ)2 . 4%estimates truncation error conservatively (marginalizes out to 0.2% on S(0))

We also tried experiment & theory Cs priors, but eventually left them outX. Zhang et al. / Physics Letters B 751 (2015) 535–540 537

All data are for energies above 0.1 MeV. We subtracted the M1 contribution of the 8B 1+ resonance from the data using the resonance parameters of Ref. [49] (a resonance energy of E p =0.72 MeV and a width !p ≈ 0.036 MeV). This has negligible im-pact (well below 1%) for E ≤ 0.5 MeV, so we retain only points in this region, thus eliminating the resonance’s effects. This strategy for dealing with the resonance has been applied, with a smaller upper energy for the fit, elsewhere in the literature [1,48,54].

4. Bayesian analysis

To extrapolate S(E) we must use these data to constrain the EFT parameters. We do this via Bayesian methods, which have been applied to the extraction of EFT parameters and the estima-tion of EFT errors in Refs. [55–57]. Here we compute the poste-rior probability distribution function (PDF) of the parameter vector g given data, D , our theory, T , and prior information, I . To ac-count for the common-mode errors in the data we introduce data-normalization corrections, ξi . We then employ Bayes’ theorem to write the desired PDF as:

pr (g, ξi|D; T ; I) = pr (D|g, ξi; T ; I)pr (g, ξi|I) , (3)

with the first factor proportional to the likelihood:

ln pr (D|g, ξi; T ; I) = c −N∑

j=1

[(1 − ξ j)S(g; E j) − D j

]2

2σ 2j

,

where S(g; E j) is the NLO EFT S-factor at the energy E j of the jth data point D j , whose statistical uncertainty is σ j . The constant censures pr (g, ξi|D; T ; I) is normalized. Since the CME affects all data from a particular experiment in a correlated way there are only five parameters ξi : one for each experiment.

In Eq. (3) pr (g, ξi, |I) is the prior for g and ξi. We choose independent Gaussian priors for each data set’s ξi , all centered at 0and with width equal to the assigned CMEs. We also choose Gaus-sian priors for the s-wave scattering lengths (a1,a2), with centers at the experimental values of Ref. [58], (25,−7) fm, and widths equal to their errors, (9,3) fm. All the other EFT parameters are assigned flat priors over ranges that correspond to, or exceed, val-ues that are natural when expressed in units of the theory’s short-distance scale: 0.001 ≤ C2

1,2 ≤ 1 fm−1, 0 ≤ r1,2 ≤ 10 fm [59,60], −1 ≤ ϵ1 ≤ 1, −10 ≤ L1,2 ≤ 10 fm. (For further discussion of the naturalness of these observable parameters, and of the related, but distinct, parameters in the Halo EFT Lagrangian, see Ref. [28].) We do, though, restrict the parameter space by the requirement that there is no s-wave resonance in 7Be-p scattering below 0.6 MeV.

To determine pr (g, ξi|D; T ; I), we use a Markov chain Monte Carlo algorithm [61] with Metropolis–Hastings sampling [62], gen-erating 2 × 104 uncorrelated samples in the 14-dimensional (14d) g

⊕ ξi space. Making histograms, e.g., over two parameters g1 and g2, produces the marginalized distribution, in that case: pr (g1, g2|D; T ; I) =

∫pr (g, ξi|D; T ; I) dξ1 . . .dξ5dg3 . . .dg9. Simi-

larly, to compute the PDF of a quantity F (g), e.g., S(E; g), we con-struct pr

(F |D; T ; I

)≡

∫pr (g, ξi|D; T ; I) δ( F − F (g))dξ1 . . .dξ5dg ,

and histogramming again suffices.

5. Constraints on parameters and the S-factor

The tightest parameter constraint we find is on the sum C21 +

C22 = 0.564(23) fm−1, which sets the overall scale of S(E).1 Fig. 1

1 The second moments of the MCMC sample distribution imply that C21 + 0.94C2

2is best constrained, but we consider C2

1 + C22 for simplicity.

Fig. 1. (Color online.) 2d distribution for C21 (x-axis) and C2

2 (y-axis). Shading repre-sents the 68% and 95% regions. The small and large ellipse are the 1σ contours of an ab initio calculation [63] and empirical results [64], with their best values marked as red squares. The inset is the histogram and the corresponding smoothed 1d PDF of the quantity [C2

1 + C22 ] × fm; the larger and smaller error bars show the empirical

and ab initio values.

Fig. 2. (Color online.) 2d distribution for ϵ1 (x-axis) and L1 (y-axis). The shaded area is the 68% region. The inset is the histogram and corresponding smoothed 1d PDF of the quantity 0.33 L1/fm − ϵ1.

shows contours of 68% and 95% probability for the 2d joint PDF of the ANCs. Neither ANC is strongly constrained by itself, but they are strongly anticorrelated; the 1d PDF of C2

1 + C22 is shown

in the inset. The ellipses in Fig. 1 show ANCs from an ab initiovariational Monte Carlo calculation (the smaller ellipse) [63]2 and inferred from transfer reactions by Tabacaru et al. (larger ellipse) [64]. These are also shown as error bars in the inset. The ab initioANCs shown compare well with the present results. (The ab initioANCs of Ref. [10] sum to 0.509 fm−1 and appear to be in mod-erate conflict.) Tabacaru et al. recognized that their result was 1σto 2σ below existing analyses of S-factor data; a 1.8σ conflict re-mains in our analysis. We suggest that for 8B the combination of simpler reaction mechanism, fewer assumptions, and more precise cross sections makes the capture reaction a better probe of ANCs than transfer reactions.

Fig. 2 depicts the 2d distribution of L1 and ϵ1. There is a positive correlation: in S(E) below the 7Be-p inelastic threshold, the effect of core excitation, here parameterized by ϵ1, can be traded against the short-distance contribution to the spin-1 E1

2 We recomputed the sampling errors of Ref. [63] in the basis of good s, taking more careful account of correlations between ANCs.

X. Zhang et al. / Physics Letters B 751 (2015) 535–540 537

All data are for energies above 0.1 MeV. We subtracted the M1 contribution of the 8B 1+ resonance from the data using the resonance parameters of Ref. [49] (a resonance energy of E p =0.72 MeV and a width !p ≈ 0.036 MeV). This has negligible im-pact (well below 1%) for E ≤ 0.5 MeV, so we retain only points in this region, thus eliminating the resonance’s effects. This strategy for dealing with the resonance has been applied, with a smaller upper energy for the fit, elsewhere in the literature [1,48,54].

4. Bayesian analysis

To extrapolate S(E) we must use these data to constrain the EFT parameters. We do this via Bayesian methods, which have been applied to the extraction of EFT parameters and the estima-tion of EFT errors in Refs. [55–57]. Here we compute the poste-rior probability distribution function (PDF) of the parameter vector g given data, D , our theory, T , and prior information, I . To ac-count for the common-mode errors in the data we introduce data-normalization corrections, ξi . We then employ Bayes’ theorem to write the desired PDF as:

pr (g, ξi|D; T ; I) = pr (D|g, ξi; T ; I)pr (g, ξi|I) , (3)

with the first factor proportional to the likelihood:

ln pr (D|g, ξi; T ; I) = c −N∑

j=1

[(1 − ξ j)S(g; E j) − D j

]2

2σ 2j

,

where S(g; E j) is the NLO EFT S-factor at the energy E j of the jth data point D j , whose statistical uncertainty is σ j . The constant censures pr (g, ξi|D; T ; I) is normalized. Since the CME affects all data from a particular experiment in a correlated way there are only five parameters ξi : one for each experiment.

In Eq. (3) pr (g, ξi, |I) is the prior for g and ξi. We choose independent Gaussian priors for each data set’s ξi , all centered at 0and with width equal to the assigned CMEs. We also choose Gaus-sian priors for the s-wave scattering lengths (a1,a2), with centers at the experimental values of Ref. [58], (25,−7) fm, and widths equal to their errors, (9,3) fm. All the other EFT parameters are assigned flat priors over ranges that correspond to, or exceed, val-ues that are natural when expressed in units of the theory’s short-distance scale: 0.001 ≤ C2

1,2 ≤ 1 fm−1, 0 ≤ r1,2 ≤ 10 fm [59,60], −1 ≤ ϵ1 ≤ 1, −10 ≤ L1,2 ≤ 10 fm. (For further discussion of the naturalness of these observable parameters, and of the related, but distinct, parameters in the Halo EFT Lagrangian, see Ref. [28].) We do, though, restrict the parameter space by the requirement that there is no s-wave resonance in 7Be-p scattering below 0.6 MeV.

To determine pr (g, ξi|D; T ; I), we use a Markov chain Monte Carlo algorithm [61] with Metropolis–Hastings sampling [62], gen-erating 2 × 104 uncorrelated samples in the 14-dimensional (14d) g

⊕ ξi space. Making histograms, e.g., over two parameters g1 and g2, produces the marginalized distribution, in that case: pr (g1, g2|D; T ; I) =

∫pr (g, ξi|D; T ; I) dξ1 . . .dξ5dg3 . . .dg9. Simi-

larly, to compute the PDF of a quantity F (g), e.g., S(E; g), we con-struct pr

(F |D; T ; I

)≡

∫pr (g, ξi|D; T ; I) δ( F − F (g))dξ1 . . .dξ5dg ,

and histogramming again suffices.

5. Constraints on parameters and the S-factor

The tightest parameter constraint we find is on the sum C21 +

C22 = 0.564(23) fm−1, which sets the overall scale of S(E).1 Fig. 1

1 The second moments of the MCMC sample distribution imply that C21 + 0.94C2

2is best constrained, but we consider C2

1 + C22 for simplicity.

Fig. 1. (Color online.) 2d distribution for C21 (x-axis) and C2

2 (y-axis). Shading repre-sents the 68% and 95% regions. The small and large ellipse are the 1σ contours of an ab initio calculation [63] and empirical results [64], with their best values marked as red squares. The inset is the histogram and the corresponding smoothed 1d PDF of the quantity [C2

1 + C22 ] × fm; the larger and smaller error bars show the empirical

and ab initio values.

Fig. 2. (Color online.) 2d distribution for ϵ1 (x-axis) and L1 (y-axis). The shaded area is the 68% region. The inset is the histogram and corresponding smoothed 1d PDF of the quantity 0.33 L1/fm − ϵ1.

shows contours of 68% and 95% probability for the 2d joint PDF of the ANCs. Neither ANC is strongly constrained by itself, but they are strongly anticorrelated; the 1d PDF of C2

1 + C22 is shown

in the inset. The ellipses in Fig. 1 show ANCs from an ab initiovariational Monte Carlo calculation (the smaller ellipse) [63]2 and inferred from transfer reactions by Tabacaru et al. (larger ellipse) [64]. These are also shown as error bars in the inset. The ab initioANCs shown compare well with the present results. (The ab initioANCs of Ref. [10] sum to 0.509 fm−1 and appear to be in mod-erate conflict.) Tabacaru et al. recognized that their result was 1σto 2σ below existing analyses of S-factor data; a 1.8σ conflict re-mains in our analysis. We suggest that for 8B the combination of simpler reaction mechanism, fewer assumptions, and more precise cross sections makes the capture reaction a better probe of ANCs than transfer reactions.

Fig. 2 depicts the 2d distribution of L1 and ϵ1. There is a positive correlation: in S(E) below the 7Be-p inelastic threshold, the effect of core excitation, here parameterized by ϵ1, can be traded against the short-distance contribution to the spin-1 E1

2 We recomputed the sampling errors of Ref. [63] in the basis of good s, taking more careful account of correlations between ANCs.

Zhang, Nollett, Phillips, PLB 751, 535 (2015)

What we really want for now is S(0) or S(20 keV)

After marginalizing over all parameters, we find S(0) = 21.3± 0.7 eV b

Solar Fusion II recommends S(0) = 20.8± 0.7 (ex)± 1.4 (th) eV b

Navratil et al. compute S(0) = 19.4± 0.7 eV b, error from truncation

538 X. Zhang et al. / Physics Letters B 751 (2015) 535–540

Table 1A representative EFT parameter set that gives a curve almost on the top of the median value curve (solid blue) in Fig. 3. The LO curve (dashed black) uses the LO parameters listed here, with the strictly NLO parameters set to zero. Because the parameter space is very degenerate, many such parameter sets could be given that have similar S(E)

curves but very different parameter values.

C21 (fm−1) a1 (fm) r1 (fm) ϵ1 L1 (fm) C2

2 (fm−1) a2 (fm) r2 (fm) L2 (fm)

0.2336 24.44 3.774 −0.04022 1.641 0.3269 −7.680 3.713 0.1612

Fig. 3. (Color online.) The right panel shows the NLO S-factor (y-axis) at different energies (x-axis), including the median values (solid blue curve). Shading indicates the 68% interval. The dashed line is the LO result. The data used for parameter determination together with a few above 0.5 MeV are shown, but have not been rescaled in accord with our fitted ξi. They are: Junghans et al., BE1 and BE3 [48](filled black circle and filled grey circle), Filippone et al. [49] (open circle), Baby et al. [50,51] (filled purple diamond), and Hammache et al. [52,53] (filled red box). The left panel shows 1d PDFs for S(0) (blue line and histogram) and S(20 keV)

(red-dashed line). In this case the y-axis is S(0) or S(20 keV), while the PDFs shown along the x-axis are normalized to unit total probability.

Table 2The median values of S , S ′/S , and S ′′/S at E = 0 keV [E = 20 keV], as well as the upper and lower limits of the (asymmetric) 68% interval. The sampling errors are 0.02%, 0.07%, 0.01% for median values, as estimated from ⟨X2 − ⟨X⟩2⟩1/2

/√

N with N = 2 × 104.

S (eV b) S ′/S (MeV−1) S ′′/S (MeV−2)

Median 21.33 [20.67] −1.82 [−1.34] 31.96 [22.30]+σ 0.66 [0.60] 0.12 [0.12] 0.33 [0.34]−σ 0.69 [0.63] 0.12 [0.12] 0.37 [0.38]

matrix element. The inset shows the 1d distribution of the quan-tity 0.33 L1/fm−ϵ1, for which there is a slight signal of a non-zero value. In contrast, the data prefers a positive L2: its 1d pdf [65]yields a 68% interval −0.58 fm < L2 < 7.94 fm.

We now compute the PDF of S at many energies, and extract each median value (the thin solid blue line in Fig. 3), and 68% interval (shaded region in Fig. 3). The PDFs for S at E = 0 and 20 keV are singled out and shown on the left of the figure: the blue line and histogram are for E = 0 and the red-dashed line is for E = 20 keV. We found choices of the EFT-parameter vector g(listed in Table 1) that correspond to natural coefficients, produce curves close to the median S(E) curve of Fig. 3, and have large values of the posterior probability.

6. S(20 keV) and the thermal reaction rate

Table 2 compiles median values and 68% intervals for the S-factor and its first two derivatives, S ′/S and S ′′/S , at E = 0and 20 keV. Ref. [1] recommends S(0) = 20.8 ± 1.6 eV b (quadra-ture sum of theory and experimental uncertainties). Our S(0) is consistent with this, but the uncertainty is more than a factor

Table 3The median values and 68% interval bounds for S in the energy range from 0 to 0.5 MeV. At each energy point, the histogram of S is drawn from the Monte-Carlo simulated ensemble and then is used to compute the median and the bounds.

E (MeV) Median (eV b) −σ (eV b) +σ (eV b)

0. 21.33 0.69 0.660.01 20.97 0.65 0.630.02 20.67 0.63 0.600.03 20.42 0.60 0.580.04 20.20 0.57 0.550.05 20.02 0.55 0.530.1 19.46 0.45 0.440.2 19.27 0.34 0.340.3 19.65 0.32 0.300.4 20.32 0.35 0.310.5 21.16 0.42 0.41

of two smaller. Ref. [1] also provides effective values of S ′/S =−1.5 ± 0.1 MeV−1 and S ′′/S = 11 ± 4 MeV−2. These are not literal derivatives but results of quadratic fits to several plausible models over 0 < E < 50 keV, useful for applications. Our values are consis-tent, considering the large higher derivatives (rapidly changing S ′′) left out of quadratic fits.

In Table 3, we list the median values and 68% interval bounds for S in 10 keV intervals to 50 keV and then in 100 keV intervals to 500 keV.

The important quantity for astrophysics is in fact not S(E) but the thermal reaction rate; derivatives of S(E) are used mainly in a customary approximation to the rate integral [1,2,66]. By using our S ′ and S ′′ in a Taylor series for S(E) about 20 keV, then regrouping terms and applying the approximation formula, we get

N A⟨σ v⟩ = 2.7648 × 105

T 2/39

exp

(−10.26256

T 1/39

)

× (1 + 0.0406T 1/39 − 0.5099T 2/3

9 − 0.1449T9

+ 0.9397T 4/39 + 0.6791T 5/3

9 ), (4)

in units of cm3 s−1 mol−1, where N A is Avogadro’s number and T9 ≡ T /(109 K). Up to T9 = 0.6, the lower and upper limits of the 68% interval for S(E) produce a numerically integrated rate that is 0.969(1 +0.0576T9 −0.0593T 2

9 ) and 1.030(1 −0.05T9 +0.0511T 29 )

times that of Eq. (4). At T9 ! 0.7 energies beyond the LER, and hence resonances, come into play and so these results no longer hold. We know of no astrophysical environment with such high T9where 7Be(p, γ )8B matters.

We also check this approximation against direct numerical in-tegration of our median S(E): the two differ by only 0.01% at tem-perature T9 = 0.016 (characteristic of the Sun), and 1% at T9 = 0.1(relevant for novae).

7. How accurate is NLO?

Our improved precision for S(0) is achieved because, by appro-priate choices of its nine parameters, NLO Halo EFT can represent all the models whose disagreement constitutes the 1.4 eV b uncer-tainty quoted in Ref. [1]—including the microscopic calculation of Ref. [9]. Halo EFT matches their S(E) and phase-shift curves with

Zhang, Nollett, Phillips, PLB 751, 535 (2015)

Full histogram: S(0)

Dashed histogram: S(20 keV)

Green band: Marginalized S(E)

Solid curve: Parameters matching bandmedian

Dashed curve: Keeping only LOparameters from solid curve

Some thoughts on the future

All models describe the same physics, & a lot of the “internals” (some areexternal) are very close

For example, integration of the large-separation region is pretty much the same

The question is how to get & assemble the information that isn’t trivially computable

Ab initio purism is good for testing theory & computation but probably not forapplications (when any pertinent experiment can be done)

This will probably go case-by-case depending on reaction details & desiderata

But applications will (& should) probably involve some combination of ab initio &data in “fewer-body” models likeR-matrix, EFT, potential model, CDCC, . . .

More thoughts on the future

Probably more of the EFT couplings can be related to matrix elements of abinitio wave functions (bound & unbound)

There might be an advantage over R-matrix in how directly that can be done

But halo EFT will only ever be good at low energies with shallow bound states

Bayesian methods with priors from theory & integration over nuisance parametersare probably good for many situations

It beats handing fewer-body fits ab initio information as a light suggestion or adiktat, or as fake data to be weighted against real data