In stars that lie on the main sequence in the HertzsprungRussel diagram like our sun hydrogen is fused to helium in a number of nuclear reaction chains and series such as the protonproton chain and the carbonnitrogenoxygen cycles. Precisely determined thermonuclear rates of these reactions lie at the foundation of the standard solar model. This review the third decadal evaluation of the nuclear physics of hydrogenburning stars is motivated by the great advances made in recent years by solar neutrino observatories putting experimental knowledge of the protonproton chain neutrino fluxes in the fewpercent precision range. The basis of the review is a oneweek community meeting held in July 2022 in Berkeley California and many subsequent digital meetings and exchanges. The relevant reactions of solar and stellar hydrogen burning are reviewed here from both theoretical and experimental perspectives. Recommendations for the state of the art of the astrophysical Sfactor and its uncertainty are formulated for each of them. Several other topics of paramount importance for the solar model are reviewed as well recent and future neutrino experiments electron screening radiative opacities and current and upcoming experimental facilities. In addition to reactionspecific recommendations also general recommendations are formed.
['Astrophysics - Solar and Stellar Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'Nuclear Experiment', 'Nuclear Theory']
Solar fusion III New data and theory for hydrogenburning stars
{'Solar fusion III: New data and theory for hydrogen-burning stars.': 'B. Acharya  \nOak Ridge National Laboratory, Oak Ridge, TN 37831, USA  \nM. Aliotta  \nSUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  \nA. B. Balantekin  \nDepartment of Physics, University of Wisconsin-Madison, Madison WI 53706, USA', 'D. Bemmerer': "Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  \nC. A. Bertulani  \nDepartment of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429-3011, USA  \nA. Best  \nUniversity of Napoli Federico II, 80126 Napoli, Italy Istituto Nazionale di Fisica Nucleare - Napoli, 80126 Napoli, Italy  \nC. R. Brune  \nEdwards Accelerator Laboratory, Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA  \nR. Buompane and L. Gialanella  \nDipartimento di Matematica e Fisica, Università degli Studi della Campania 'Luigi Vanvitelli', 81100 Caserta, Italy Istituto Nazionale di Fisica Nucleare - Napoli, 80126 Napoli, Italy", 'F. Cavanna': 'Istituto Nazionale di Fisica Nucleare - Torino, 10125 Torino, Italy', 'J. W. Chen': 'Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan  \nJ. Colgan  \nLos Alamos National Laboratory,  \nLos Alamos, NM 87545,  \nUSA', 'A. Czarnecki': 'Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada  \nB. Davids  \nTRIUMF, Vancouver, BC, V6T 2A3,  \nCanada  \nDepartment of Physics,  \nSimon Fraser University,  \nBurnaby, BC, V5A 1S6,  \nCanada', 'R. J. deBoer': 'Department of Physics and Astronomy and the The Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN 46556,  \nUSA', 'F. Delahaye': 'Observatoire de Paris - PSL - Sorbonne Université, 5 place Jules Janssen, 92195 Meudon cedex,  \nFrance  \nR. Depalo and A. Guglielmetti  \nUniversità degli Studi di Milano, 20133 Milano, Italy Istituto Nazionale di Fisica Nucleare - Milano, 20133 Milano, Italy', 'A. García and R. G. H. Robertson': 'Department of Physics and Center for Experimental Nuclear Physics and Astrophysics,  \nUniversity of Washington,  \nSeattle WA 98195,  \nUSA  \nM. Gatu Johnson  \nMassachusetts Institute of Technology,  \nCambridge, MA 02139,  \nUSA  \nD. Gazit  \nRacah Institute of Physics, The Hebrew University, The Edmond J. Safra Campus, Givat Ram, Jerusalem 9190401, Israel  \nU. Greife  \nColorado School of Mines, Golden CO 80401, USA', 'D. Guffanti': 'Dipartimento di Fisica, Università degli Studi e INFN Milano-Bicocca, 20126 Milano, Italy', 'K. Hambleton': 'Department of Physics, University of California, Berkeley, CA 94720, USA  \nW. C. Haxton  \nDepartment of Physics, University of California, Berkeley, CA 94720 Lawrence Berkeley National Laboratory, Berkeley, CA 94720,  \nUSA', 'Y. Herrera and A. Serenelli': "Institute of Space Sciences,  \n08193 Cerdanyola del Vallès, Barcelona,  \nSpain  \nInstitut d'Estudis Espacials de Catalunya,  \n08034 Barcelona,  \nSpain", 'M. Huang': 'Department of Physics & Astronomy,  \nIowa State University,  \nAmes, IA 50011,  \nUSA  \nLawrence Livermore National Laboratory,  \nP.O. Box 808, L-414,  \nLivermore, CA 94551,  \nUSA  \nC. Iliadis  \nDepartment of Physics & Astronomy,  \nUniversity of North Carolina at Chapel Hill, NC 27599-3255,  \nUSA  \nTriangle Universities Nuclear Laboratory (TUNL),  \nDuke University, Durham,  \nNorth Carolina 27708,  \nUSA', 'K. Kravvaris': 'Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, CA 94551, USA', 'M. La Cognata': 'Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Sud, 95123 Catania,  \nItaly', 'K. Langanke': "GSI Helmholtzzentrum fuer Schwerionenforschung, Darmstadt,  \nGermany  \nInstitut fuer Kernphysik,  \nTechnical University Darmstadt, Darmstadt, Germany  \nL. E. Marcucci  \nDepartment of Physics 'E. Fermi', University of Pisa, 56127 Pisa, Italy Istituto Nazionale di Fisica Nucleare - Pisa, 56127 Pisa, Italy", 'T. Nagayama': 'Sandia National Laboratories, Albuquerque, New Mexico 87123, USA', 'K. M. Nollett': 'Department of Physics, San Diego State University, San Diego, CA 92182,  \nUSA  \nD. Odell  \nDepartment of Physics & Astronomy, Ohio University, Athens, Ohio 45701, USA  \nG. D. Orebi Gann  \nUniversity of California Berkeley, Berkeley CA 94720, USA Lawrence Berkeley National Laboratory, Berkeley CA 94720, USA', 'D. Piatti': 'Università degli Studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy INFN, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy', 'M. Pinsonneault': 'Department of Astronomy,  \nThe Ohio State University, 140 W. 18th Avenue, Columbus, OH 43210, USA Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 W. Woodruff Avenue, Columbus, OH 43210, USA', 'L. Platter': 'Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory,  \nOak Ridge, TN 37831, USA', 'G. Rupak': 'Department of Physics & Astronomy and HPC2 Center for Computational Sciences, Mississippi State University, Mississippi State, MS 39762,  \nUSA', 'M. Sferrazza': 'Université Libre de Bruxelles', 'T. Szücs': 'HUN-REN Institute for Nuclear Research (HUN-REN ATOMKI), 4026 Debrecen, Hungary  \nX. Tang  \nIMP Chinese Academy of Science', 'A. Tumino': "Dipartimento di Ingegneria e Architettura, Università degli Studi di Enna 'Kore', Enna, Italy Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Sud, 95123 Catania, Italy", 'F. L. Villante': "Dipartimento di Scienze Fisiche e Chimiche,  \nUniversità dell'Aquila, 67100 L'Aquila,  \nItaly  \nIstituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Gran Sasso,  \n67100 Assergi (AQ),  \nItaly", 'A. Walker-Loud': 'Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA', 'X. Zhang': 'Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA', 'K. Zuber': 'Institute for Nuclear and Particle Physics, Technical University of Dresden, 01062 Dresden, Germany  \nIn stars that lie on the main sequence in the Hertzsprung Russel diagram, like our Sun, hydrogen is fused to helium in a number of nuclear reaction chains and series, such as the proton-proton chain and the carbon-nitrogen-oxygen cycles. Precisely determined thermonuclear rates of these reactions lie at the foundation of the standard solar model. This review, the third decadal evaluation of the nuclear physics of hydrogen-burning stars, is motivated by the great advances made in recent years by solar neutrino observatories, putting experimental knowledge of the pp chain neutrino fluxes in the few-percent precision range. The basis of the review is a one-week community meeting held in July 2022 in Berkeley, California, and many subsequent digital meetings and exchanges. The relevant reactions of solar and stellar hydrogen burning are reviewed here, from  \nboth theoretical and experimental perspectives. Recommendations for the state of the art of the astrophysical S-factor and its uncertainty are formulated for each of them. Several other topics of paramount importance for the solar model are reviewed, as well: recent and future neutrino experiments, electron screening, radiative opacities, and current and upcoming experimental facilities. In addition to reaction-specific recommendations, also general recommendations are formed.', 'Contents': 'D. Fitting Procedures  \n38  \n2.  \nPANDORA  \n66  \n7  \n36', 'I. Introduction': 'The present review summarizes the state of our understanding, in the third decade of the 21 st century, of the nuclear reactions and decays taking place in solar and stellar hydrogen burning. It also addresses related issues, including solar neutrino detection, radiative opacities, electron screening of nuclear reactions, and the status of the current facilities for measuring cross sections and opacities.  \nAs was the case for two previous decadal reviews (Adelberger et al. , 1998, 2011), the present review summarizes the progress made over the past decade in advancing the nuclear physics of main-sequence stars and makes recommendations for future work.', 'A. Purpose': 'Ray Davis\'s measurements of the flux of solar neutrinos (Davis, 2003; Davis et al. , 1968) showed a significant deficit with respect to the predictions of the standard solar model (SSM) (Bahcall, 1966). This discrepancy became known as the "solar neutrino problem." An intense debate ensued about possible explanations for this discrepancy, summarized in the entertaining review by Bahcall et al. (1988). Suggestions included plausible flaws in the SSM that might produce a cooler solar core (e.g. Schatzman et al. 1981; Cumming and Haxton 1996), new particle physics such as neutrino oscillations (Gribov and Pontecorvo 1969; Wolfenstein 1978), changes in the nuclear physics governing He synthesis, such as an unidentified low-energy resonance in the 3 He+ 3 He reaction affecting extrapolations of laboratory cross sections to solar energies (Fetisov and Kopysov, 1972; Fowler, 1972), or the presence of dark matter particles modifying energy transport in the solar core (Spergel and Press, 1985). By the mid-1990s, with new results from Kamioka II/III (Hirata et al. 1989; Fukuda et al. 1996) and the SAGE/GALLEX (Abazov et al. 1991; Anselmann et al. 1992) experiments revealed a pattern of neutrino fluxes inconsistent with the expected scaling of those fluxes with the solar core temperature. Attention increasingly turned to neutrino  \noscillations and other potential particle physics solutions. Concerns about the nuclear physics of the SSM also evolved, focusing more on the uncertainties that might inhibit extraction of any such \'new physics.\'  \nAt that time, during a meeting on the Solar Neutrino Problem hosted by the Institute for Nuclear Theory (INT), University of Washington, Seattle, a suggestion was made to convene the community working on solar nuclear reactions, in order to reach consensus on the best current values of cross sections and their uncertainties. The INT hosted the proposed workshop in February, 1997, drawing representatives from almost every experimental group active in this area, as well as many of the theorists who were engaged in solar neutrino physics. The working groups that formed during this meeting worked over the following year to evaluate past work, determining the needed cross sections and their uncertainties. The results of this evaluation, which came to be known as Solar Fusion I or SF I (Adelberger et al. , 1998), became the standard for use in solar modeling over the following decade.  \nAn update of SF I was launched with the January, 2009, workshop \'Solar Fusion Cross Sections II\', hosted again by the INT in Seattle. Initial results from SuperKamiokande, the Sudbury Neutrino Observatory, and Borexino were then in hand, and the conversion of approximately two thirds of solar electron neutrinos into other flavors had been firmly established (Ahmad et al. , 2002a). Consequently, the motivation for the study was to ensure that solar model predictions would be based on the most current nuclear physics, so that meaningful uncertainties could be placed on neutrino parameters derived from solar neutrino measurements, such as the mixing angle θ 12 . An additional source of uncertainty had arisen at the time, with the advent of three-dimensional (3D) radiative-hydrodynamic models of the Sun\'s atmosphere for determining element abundances from photo-absorption lines (Asplund et al. , 2006, 2009). This improved analysis, though, lowered the inferred metallicities, leading to tension with the Sun\'s interior sound speed profile determined from helioseismology (Christensen-Dalsgaard, 2002). The discrepancy between higher metallicity models that accurately reproduce the Sun\'s interior sound speed and lower metallicity models based on the most current treatment of the photosphere was termed the \'solar composition problem\'. While the neutrino fluxes were not known to the precision needed to distinguish between the competing models, it had been shown that the solar core metallicity could be extracted directly from future CN solar neutrino measurements, if the precision of associated nuclear cross sections were improved (Haxton and Serenelli, 2008). Reflecting the impact of new experimental work and improved theory, the SF II study of Adelberger et al. (2011) significantly revised the SF I cross section for the driving reaction of the CNO-I cycle, and updated key pp-chain cross sections.  \nThe present SF III review began with a four-day workshop in July, 2022, \'Solar Fusion Cross Sections III.\' The meeting was hosted by Physics Frontier Center N3AS (Network for Neutrinos, Nuclear Astrophysics, and Symmetries) 1 , a consortium of institutions involved in multimessenger astrophysics, whose central hub is the University of California, Berkeley. Sessions were held at the David Brower Center, Berkeley, and the Physics Department, UC Berkeley, with the assistance of N3AS staff. The workshop was organized by N3AS, by Solar Fusion\'s long-term sponsor, the INT, and by the European ChETEC-INFRA Starting Community for Nuclear Astrophysics 2 .  \nThe 47 workshop participants represented most of the leading experimental and theoretical research groups that are active in the field. Workshop participants were organized into nine working groups, who were charged with reviewing and evaluating work completed since SF II, to produce updated recommendations. The structure of this review largely follows that of the working groups. The timing of SF III was driven in part by planned large-scale asteroseismic surveys of hydrogen-burning stars such as ESA\'s PLATO 3 mission (Rauer et al. , 2014), and the expectation that associated data analysis will require large libraries of stellar models in which masses, ages, metallicities, and other parameters are varied. SF III will provide the best current nuclear astrophysics input for such modeling. This review summarizes that input, based on the recommendations of the nine SF III working groups.', 'B. Terminology used': 'The terminology used here follows recent practice, as summarized for example in the NACRE compilation (Angulo et al. , 1999). It is briefly described to aid the reader. Nuclear reactions are designated as  \nA ( b, c ) D, (1)  \nwhere A is the target nucleus, b is the bombarding particle, and c and D are the reaction products, with D distinguished as the heavier product. Frequently in nuclear astrophysics, the reaction cross section σ is re-expressed in terms of the S-factor S ij , with S related to σ as described below, and with i, j the mass numbers of entrance channel nuclei b and A .  \nThe relation between the nuclear cross section σ ( E ) and astrophysical S-factor is  \nS ( E ) = σ ( E ) E exp ( 2 π Z A Z b α β ) , (2)  \nwith Z A and Z b the charge numbers of the target and projectile, α ∼ 1 / 137 the fine structure constant, β ≡ v/c = √ 2 E/µc 2 the relative velocity of the interacting particles in units of c , µ the reduced mass, and E the center-ofmass energy. (In this review we use energy units of MeV unless otherwise stated.) This removes the sharp energy dependence from the cross section associated with s-wave Coulomb scattering off a point nucleus, thereby accounting for the leading penetration effects associated with the Coulomb barrier. The quantity Z A Z b α/β is known as the Sommerfeld parameter. Consequently, unless there are resonances, S ( E ) varies gently, and at low energies can generally be expressed as a low-order polynomial in E .  \nThe S-factor in Eq. (2) is defined in terms of the nuclear (or \'bare") cross section, while laboratory experiments are performed with targets consisting of neutral atoms. As electrons shield the nucleus, lowering the Coulomb barrier, their presence enhances the laboratory cross section. The S-factors derived from laboratory data must be corrected for this effect. As the energy is lowered, the screening correction becomes increasingly important. Thus particular care must be taken when extrapolating laboratory cross sections to lower energies, as Fig. 4 of SF II illustrates. Unless otherwise noted, the S-factors discussed here are those for bare nuclei.  \nThe rapid rise in the nuclear cross section with energy, reflecting the higher probability of penetrating the Coulomb barrier, competes with a rapid decrease in the probability of finding interacting particles with the requisite center-of-mass energy E in the high-energy tail of the Maxwell-Boltzmann distribution. Consequently, the thermonuclear reaction rate 〈 σv 〉 for a given stellar temperature T takes the form  \nwhere b = πZ A Z b α √ 2 µc 2 is proportional to the Sommerfeld parameter of Eq. (2) and k B is Boltzmann\'s constant. The maximum of the integrand defines the Gamow peak - the most probable energy for interactions - while the ∼ 1 σ range around the peak is frequently termed the Gamow window.  \n〈 σv 〉 = √ 8 µπ ( k B T ) -3 2 ∫ ∞ 0 S ( E ) exp [ -b √ E -E k B T ] dE, (3)  \nFor the solar reactions of interest, the energy at the Gamow peak is well below the height of the Coulomb potential. The slowly varying S ( E ) can be expanded as a power series around E = 0 , with the leading term being S (0) , and with corrections given by derivatives taken at E = 0 ,  \nS ( E ) /similarequal S (0) + S \' (0) E + S \'\' (0) 2 E 2 + · · · . (4)  \nMost of the results presented in this review will be given in terms of S (0) and its derivatives.  \nFor most of the nuclear reactions studied here, the notation S ij is adopted, where i and j are the mass numbers  \nof the projectile and target nuclei, respectively. A summary of SF III reactions and recommended astrophysical S-factors are presented in Table I.  \nFinally, we provide a short list of abbreviations that are frequently used throughout several sections:  \nANC: asymptotic normalization coefficient.  \nBBN: Big Bang nucleosynthesis.  \nCME: common-mode error.  \n- EFT: effective field theory, and variants: chiral ( χ EFT), pion-less ( / π EFT), hybrid (EFT*), etc.\n- LO, NLO, NNLO: leading order, next to (next to) leading order, etc.  \nSSM: Standard Solar Model.', 'C. Scope and structure of the review': "In the Sun, the primary mechanism for the conversion of four protons into 4 He is the pp chain of Fig. 1, during which two charge-changing weak interactions take place, each converting a proton into a neutron with the emission of a neutrino. The pp chain consists of three main branches, ppI, ppII, and ppIII. These branches are distinguished by their dependence on the solar core temperature and by the neutrinos they produce, with the pp, 7 Be, and 8 B neutrinos serving as 'tags' for the ppI, ppII, and ppIII chains, respectively. The SSM predicts that the fluxes of pp, 7 Be, and 8 B neutrinos have an approximate relative temperature scaling of ∼ 1:T 10 c :T 22 c , where T c is the solar core temperature (Haxton, 1995). By the mid-1990s, solar neutrino experiments established that the measured fluxes differed significantly from this expected pattern. This contributed to growing expectations that new neutrino physics might be the solution of the solar neutrino problem. It also was an important motivation for SF I, as errors in the nuclear physics of the pp chain could distort the relationship between the various fluxes.  \nThe pp chain accounts for 99% of solar hydrogen burning and are the main focus of this review. The remaining 1% is generated through the CNO-I cycle, see Fig. 2). Motivated in part by Borexino's recent success in measuring the flux of solar CN neutrinos, this review also considers in more detail than past Solar Fusion reviews the reactions driving hydrogen burning in stars more massive than our Sun.  \nThe review is structured as follows. Section II begins with a summary of past solar neutrino experiments, highlighting some of the open questions that remain. It discusses the second-generation experiments SuperKamiokande, the Sudbury Neutrino Observatory, and Borexino that contributed in the resolution of the solar neutrino problem. It also describes Hyper-Kamiokande, SNO+, JUNO, and DUNE, new detectors in various stages of construction.  \nFIG. 1 Nuclear reactions of the proton-proton (pp) chain. The percentage branchings are applicable for the Magg et al. (2022) solar composition.  \n<!-- image -->  \nThe cross section of the driving reaction of the pp chain, 1 H ( p, e + ν ) 2 H, is too small to be measured and thus must be taken from theory. This important reaction is reviewed in section III. With the exception of the minor hep branch, all other reactions and decays have been studied in laboratories, with measurements then combined with theory to predict rates under solar conditions. The 2 H( p, γ ) 3 He, 3 He( 3 He,2 p ) 4 He, and 3 He ( α, γ ) 7 B reactions are reviewed in Secs. IV, V, and VI, respectively. The hep reaction is reviewed in Section VII, and the electroncapture reactions on pp and 7 Be are discussed in Section VIII. The 7 Be( p , γ ) 8 B reaction responsible for the high energy neutrinos measured in Super-Kamiokande is treated in Section IX.  \nWhile a modest contributor to solar energy production, the CNO-I cycle is a potentially important probe of core metallicity as its hydrogen burning is catalyzed by the Sun's primordial C and N (Section XI).  \nA current focus of nuclear astrophysics laboratories is the higher-temperature pathways for hydrogen burning that operate in massive stars. Many of the contributing reactions have been re-measured. That progress is reviewed in Section XI.  \nCharged-particle nuclear reactions occurring at energies below the Coulomb barrier are affected by electron screening, the shielding of the nuclear charge by electrons. Because of their very different atomic environments, laboratory reactions and those occurring in the solar plasma are affected in distinct ways. Section XII describes the current status of efforts to account for these differences, when solar cross sections are extracted from laboratory measurements.  \nThe energy transport inside the radiative zone is af-  \n×  \n-1  \nTABLE I List of nuclear reactions reviewed in SF III. Denoting the astrophysical S-factor by S ij , its value at zero-energy S (0) is given along with, where applicable, derivatives parameterized in Eq. (4). See the corresponding section for uncertainties, higher precision values, and detailed discussion.  \nfected by the metals 4 that were incorporated into the Sun when it first formed. While metals comprise less than 2% of the Sun by mass, they play an outsized role in determining the opacity. Very few of the needed radiative opacities have been measured, and the conditions under which measurements are made typically are not identical to those in the Sun. Section XIII describes the current state of the art, and discusses connections between opacities and solar composition that have complicated efforts to resolve the solar metallicity problem.  \nMuch of the progress made in constraining the reactions of the pp chain and CNO-I cycles has been possible thanks to new laboratories and the facilities they host. The underground LUNA laboratory is a prominent example (Broggini et al. , 2018). Section XIV describes the capabilities of selected laboratories involved in the measurements reviewed here.", 'II. Solar neutrino observations': 'Solar neutrinos offer a unique tool box for probing both the fundamental properties of these elusive particles, and their interactions with matter, as well as understanding their source: the fusion reactions that power our Sun. The original motivation for observations of solar neutrinos was precisely the hope to probe solar fusion. The first  \nsuccessful experiment beginning in 1967 at the Homestake mine (Cleveland et al. , 1998) offered the surprising result of a neutrino flux suppressed to approximately one third of expectation. The intervening years have seen a tour de force of experimental efforts, leading to the resolution of the Solar Neutrino Problem and the confirmation of neutrino oscillations (Ahmad et al. , 2002a, 2001; Ahmed et al. , 2004; Eguchi et al. , 2003; Fukuda et al. , 1998). These high-precision flux and spectral measurements mapped out the details of solar fusion, probed the structure of the Sun, and demonstrated (together with atmospheric neutrino measurements) that neutrinos have masses and mix. Table II lists the pp-chain and CNO cycle neutrino sources, their endpoint ( β decay) or line (electron capture) energies E ν , and predicted SSM fluxes.', 'A. Open questions': "The solar neutrino spectrum has been measured for energies above 233 keV, with some fluxes determined to a precision equivalent to or better than theoretical predictions. However, improved measurements can offer further insight on several aspects. For instance, more precise determinations of the fluxes of neutrinos from the CNO cycle (Fig. 2) could determine the contemporary abundances of N and O in the solar core (Castro et al. , 2007; Kunitomo et al. , 2022; Serenelli et al. , 2011). While these abundances differ significantly from the primordial values that are input into the SSM -the primordial abundances are altered by out-of-equilibrium CNO-cycle burning that occurs in the early Sun (Roxburgh, 1985)- this burning  \nFIG. 2 Nuclear reactions of the CNO-I (green), CNO-II (red), CNO-III (orange), CNO-IV (blue), and Ne-Na (purple) cycles. Other reactions reported in this work are also present (in black). For visual aid, ( p, γ ) reactions are styled using full arrows, while ( p, α ) reactions use dashed-line arrows. Greyed isotopes are short-lived and quickly ( β + ) -decay (dotted arrows), resulting in e + + ν e emissions.  \n<!-- image -->  \nTABLE II Solar neutrino sources, energies, and SSM flux predictions. All β decay sources produce continuous spectra, while the pep and 7 Be electron-capture sources produce line spectra. The fluxes are taken from the SSM calculations of Herrera and Serenelli (2023), computed with nuclear reaction rates shown in Table I, for the compositions of GS98 (Grevesse and Sauval, 1998) (high-Z), AAG21 (Asplund et al. , 2021) (low-Z), and MB22p (Magg et al. , 2022) (high-Z) with associated uncertainties indicated.  \nNotes: (a) 90% and (b) 10% of the 7 Be neutrino flux, respectively.  \nredistributes the C, N, and O, but does not alter the net  \nnumber of such nuclei. Consequently, CNO neutrino flux measurements might allow one to experimentally address the solar composition problem discussed below.  \nSolar neutrino flux measurements can also constrain physical processes in stars such as chemical mixing. The fluxes are sensitive to solar/stellar model inputs, including the radiative opacities (see Sect. XIII). Measurements of the neutrino fluxes from 8 B and 7 Be can offer a handle on the temperature of the solar core, and other environmental factors (Bahcall and Ulmer, 1996).  \nA measurement of the pp flux with percent-level precision would provide a test of the luminosity constraint, further probing solar power generation mechanisms (Bahcall, 2002; Vescovi et al. , 2021). The one branch of the pp chain that yet remains undetected, the hep flux, has both the highest energy but also the lowest flux of neutrinos from the Sun. The spectrum extends beyond that of the 8 B neutrinos, offering a small window for unequivocal observation. Limits have been placed on this flux (Aharmim et al. , 2020), but a definitive observation will likely require a large, nextgeneration detector such as Hyper-Kamiokande (HyperK) (Hyper-Kamiokande Proto-Collaboration et al. , 2018) or the Deep Underground Neutrino Experiment  \nFinally, the observation of monochromatic neutrinos produced by electron capture reactions on 13 N, 15 O and 17 F (which we refer to as ecCNO neutrinos) (Bahcall, 1990; Stonehill et al. , 2004; Villante, 2015) could be rewarding in terms of physical implications. Indeed, they can be used as probes of the metallicity of the solar core. Moreover, they provide a measure of the electron neutrino survival probability at specific neutrino energies E ν ∼ 2 . 5 MeV within the broad transition region between vacuum and matter-enhanced oscillations, where currently we have no constraints. This would constitute a new test of the large-mixing-angle (LMA) MSW flavor oscillation paradigm. The detection of this subdominant component of the CNO-cycle is extremely difficult but it could be within reach of future very large ultra-pure liquid scintillator detectors (Villante, 2015).  \nSolar neutrinos provide an opportunity to understand the interaction of neutrinos with matter. The effect of MSW oscillations has a significant impact on the observed solar neutrino spectrum: with vacuum oscillation dominating at low energy, below approximately 1 MeV, where the survival probability is roughly one half, while matter effects further suppress the flux to a survival probability of roughly one third above approximately 5 MeV. The transition region between these two regimes offers an extremely sensitive probe of the details of the interactions of neutrinos with matter, including the potential to search for new physics such as sterile neutrinos, or nonstandard interactions, by looking for distortions to the expected spectral shape (Bhupal Dev et al. , 2019; Borexino Collaboration et al. , 2020a; Coloma et al. , 2023).  \nThe same theory of matter effects predicts a small regeneration of electron neutrinos during the night time as they propagate through the bulk of the earth. This so-called 'day/night effect' has been sought after by both Super-Kamiokande (SuperK) and the Sudbury Neutrino Observatory (SNO) (Ahmad et al. , 2002b; Renshaw et al. , 2014), but a significant observation is still limited by statistics. Future data from HyperKamiokande or DUNE may be needed to confirm our understanding of this prediction of the MSW effect.  \nThe slight tension between measurements of the mass splitting parameters, ∆ m 2 12 in solar neutrino experiments and terrestrial data sensitive to the same parameter, from the KamLAND reactor experiment has now disappeared with the addition of new data on day-night effects and spectral shapes (Capozzi et al. , 2018; Esteban et al. , 2020). Further tightening of the oscillation parameters can be expected from JUNO, which will obtain highly precise data on both the solar mass splitting and the mixing angle (An et al. , 2016).", '1. The solar composition problem': "Solar photospheric abundances, determined with spectroscopic techniques, are a fundamental input for the construction of SSMs, but also they are used as input to nearly all models in astrophysics, including stellar evolution, protoplanetary disks, and galactic chemical evolution. Over the last two decades, the development of three dimensional radiation hydrodynamic models of the solar atmosphere (Asplund et al. , 2000; Ludwig et al. , 2009) and of techniques to study line formation under non-local thermodynamic conditions (Asplund, 2005), together with improved descriptions of atomic properties (e.g. transition strengths Bergemann et al. 2021; Magg et al. 2022), have led to a significant revision of solar abundances. Initial results based on these more sophisticated methods favored a markedly lower solar metallicity (Asplund et al. , 2009), particularly in the CNO elements, than obtained in the 90s from older techniques (Grevesse and Noels, 1993; Grevesse and Sauval, 1998).  \nRecently, two groups have revisited the solar photospheric abundances using modern methods (Asplund et al. , 2021; Magg et al. , 2022). While Asplund et al. (2021) obtain results consistent with their previous findings, Magg et al. (2022) find an O abundance intermediate between that of Asplund's group and those from the 90s, in agreement with another 3D-based determination by Caffau et al. (2011). Interestingly, Magg et al. (2022) find higher C and N abundances and, indirectly, a higher Ne abundance due to the larger Ne to O ratio measured in the solar corona (Young, 2018). This leads to a combined metal-to-hydrogen ratio that is by chance comparable to those from Grevesse and Noels (1993) and Grevesse and Sauval (1998), albeit with a different mixture of elements.  \nConsidering that uncertainties in element abundances are difficult to quantify, it has become customary to consider two canonical sets of abundances, which we refer to as high metallicity (HZ) and low metallicity (LZ) solar admixtures. See, for example, Vinyoles et al. (2017) for SSM reference values. In this context, the new solar abundance determinations by Asplund et al. (2021) fall into the LZ category, while those from Magg et al. (2022) are HZ. Solar models employing the LZ abundances fail to reproduce most helioseismic probes of solar properties. This disagreement constitutes the so-called solar composition problem (Bahcall et al. , 2005a; Basu and Antia, 2004; Delahaye and Pinsonneault, 2006) that has defied a complete solution. All proposed modifications to physical processes in SSMs offer, at best, only partial improvements in some helioseismic probes, see e.g. (Basu and Antia, 2008; Castro et al. , 2007; Guzik and Mussack, 2010; Guzik et al. , 2005; Serenelli et al. , 2011). The same conclusions are obtained with SSMs computed with the newest LZ (Asplund et al. , 2021) and HZ (Magg et al. , 2022) abundances as discussed also there. An al-  \nternative possibility is to consider modifications to the physical inputs of SSMs at the level of the constitutive physics, radiative opacities in particular. This is possible because most helioseismic probes depend actually not directly on the solar composition, but on the radiative opacity profile in the solar interior, i.e. on the combination of solar composition and atomic opacities (see § XIII). The same can be said for solar neutrinos from the pp chain. Early work (Bahcall et al. , 2005b; Montalbán et al. , 2004) already suggested that a localized increase in opacities could solve or, at least, alleviate the disagreement of low-Z solar models with helioseismology, and Christensen-Dalsgaard et al. (2009) and Villante (2010) showed that a tilted increase in radiative opacities, with a few percent increase in the solar core and a larger (15-20%) increase at the base of the convective envelope could lead to LZ SSMs that would satisfy helioseismic probes equally well as HZ SSMs.  \nThe degeneracy between solar composition and opacities can be broken using CNO solar neutrinos, e.g., following the methodology developed in Haxton and Serenelli (2008). Such a study was recently carried out by the Borexino collaboration (Appel et al. , 2022; Basilico et al. , 2023; Borexino Collaboration et al. , 2020b). The Borexino measurement of the CNO-I cycle neutrino flux (the 13 N and 15 O fluxes) were used to determine the C+N core abundance. Results show a ∼ 2 σ tension with LZ metallicity determinations, while being in better agreement with HZ mixtures. While the error budget is presently dominated by the uncertainty of the Borexino CNO neutrino measurement, a significant contributor to the error ( ∼ 10% ) is nuclear, due to uncertainties in S 114 , S 34 , and S 17 . The interpretation of future improved CNO neutrino measurements will be impacted, unless these nuclear physics uncertainties are reduced.  \nVery recently, Gonzalez-Garcia et al. (2024) presented a new global determination of all solar neutrino fluxes using all available experimental data, including the latest phases of Borexino. Results from this global analysis are in line with those from Borexino, although the added 13 N and 15 O fluxes are about 10% lower than the Borexino result alone. A comparison of the solar fluxes with SSM calculations (Herrera and Serenelli, 2023) shows that HZ SSMs are in better agreement with solar neutrino fluxes than the LZ SSMs, pointing toward a C+N solar core abundance consistent with HZ abundances. The discrimination that solar neutrino fluxes can offer between solar compositions is at most, however, of the order of 2 σ . (See Table 2 in Gonzalez-Garcia et al. (2024)).  \nHelioseismology also offers the potential to determine the total solar metallicity, i.e. without disentangling individual element abundances, rather independently from opacities. This relies on using the so-called adiabatic index Γ 1 , which deviates from 5/3 in regions of partial ionization. The underlying technique has been used widely to determine the helium abundance in the solar envelope  \nand it has also been extended to determine solar metallicity. Attempting the latter is difficult because the imprint of partial ionization of metals is quite subtle. Previous work along this line (Antia and Basu, 2006) found an overall metallicity consistent with HZ abundances, but new work (Buldgen et al. , 2024) claims to favor LZ values. While this method depends very weakly on radiative opacities, the abundance determination is degenerate with the equation of state. Independent confirmation of these results would be desirable. In short, the controversy related to the solar composition is far from being resolved, currently with different indicators showing contradictory results.", '2. The Gallium anomaly': 'With the aid of very intense radioactive sources of 51 Cr and 37 Ar, tests have been made of the rate of production of 71 Ge by neutrino interactions on 71 Ga, the basis of radiochemical measurements of the low-energy solar neutrino flux. Initial experiments showed lower rates than expected, with interesting but inconclusive statistical precision. Very recently the BEST experiment (Barinov et al. , 2018, 2022a,b) has confirmed this anomaly at more than 4 σ . Many possible explanations have been explored, but for each there are contradictions (Akhmedov and Smirnov, 2022; Brdar et al. , 2023; Giunti et al. , 2023; Krueger and Schwetz, 2023). A sterile neutrino explanation is disfavored (Giunti et al. , 2022; Goldhagen et al. , 2022), particularly because of conflict with solar neutrino limits (see Fig. 3), as well as cosmological bounds (Hagstotz et al. , 2021).  \nThe sterile neutrino contribution to the solar flux is fundamentally limited by the luminosity constraint, that the Sun\'s total energy output is the result of nuclear reactions that produce neutrinos, whether active or sterile. If precise neutral-current data were available across the solar spectrum, a completely model-independent limit on a possible sterile component could be determined. The present neutral-current data at low energies (from Borexino) are imprecise but can be supplemented by much more precise charged-current data together with 3-flavor oscillation physics. Still better limits, at the cost of some model dependence, can be obtained with fits of experimental data to solar models, as shown in Fig. 3 above, as well as Fig. 1 of Goldhagen et al. (2022).  \nThe precisely known rate of electron capture on 71 Ge to 71 Ga, for which the half-life is 11.43(3) d (Hampel and Remsberg, 1985), places an important constraint on the neutrino absorption rate (Elliott et al. , 2023; Giunti et al. , 2023). The half-life has now been remeasured by two groups, Norman et al. (2024) and Collar and Yoon (2023), yielding values of 11.468(8) d and 11.46(4) d, consistent with the previous determination while increasing the precision. The allowed matrix element that governs  \n<latexit sha1\\_base64="u76Fj/F/XgmLYRG81RFiSEOAlB4=">AAACD3icbZDLSgMxFIYz9VbrrerSTbAoFaTMlKIuK7pwWcFeoFNLJj1tQ5OZIckIZSj4AG58FTcuFHHr1p1vY6atoK0/BH6+c8I55/dCzpS27S8rtbC4tLySXs2srW9sbmW3d2oqiCSFKg14IBseUcCZD1XNNIdGKIEIj0PdG1wk9fodSMUC/0YPQ2gJ0vNZl1GiDWpnD91L4JpgVxDdlyIWo9sido9x/gdAzZCjdjZnF+yx8LxxpiaHpqq0s59uJ6CRAF9TTpRqOnaoWzGRmlEOo4wbKQgJHZAeNI31iQDVisf3jPCBIR3cDaR5vsZj+vtHTIRSQ+GZzmRLNVtL4H+1ZqS7Z62Y+WGkwaeTQd2IYx3gJBzcYRKo5kNjCJXM7Ippn0hCtYkwY0JwZk+eN7ViwTkplK5LufL5/SSONNpD+yiPHHSKyugKVVAVUfSAntALerUerWfrzXqftKasaYS76I+sj2+kv5ww</latexit>  \n<latexit sha1\\_base64="u76Fj/F/XgmLYRG81RFiSEOAlB4=">AAACD3icbZDLSgMxFIYz9VbrrerSTbAoFaTMlKIuK7pwWcFeoFNLJj1tQ5OZIckIZSj4AG58FTcuFHHr1p1vY6atoK0/BH6+c8I55/dCzpS27S8rtbC4tLySXs2srW9sbmW3d2oqiCSFKg14IBseUcCZD1XNNIdGKIEIj0PdG1wk9fodSMUC/0YPQ2gJ0vNZl1GiDWpnD91L4JpgVxDdlyIWo9sido9x/gdAzZCjdjZnF+yx8LxxpiaHpqq0s59uJ6CRAF9TTpRqOnaoWzGRmlEOo4wbKQgJHZAeNI31iQDVisf3jPCBIR3cDaR5vsZj+vtHTIRSQ+GZzmRLNVtL4H+1ZqS7Z62Y+WGkwaeTQd2IYx3gJBzcYRKo5kNjCJXM7Ippn0hCtYkwY0JwZk+eN7ViwTkplK5LufL5/SSONNpD+yiPHHSKyugKVVAVUfSAntALerUerWfrzXqftKasaYS76I+sj2+kv5ww</latexit>  \n<latexit sha1\\_base64="CVn0nR4LyVoj9SztHbA59ZQShQg=">AAAB+XicbZBLSwMxFIXv+Kz1NerSTbAIrspMKeqy4sZlBfuAdiyZNG1DM5khuVMoQ8Ef4saFIm79J+78N6aPhbYeCHycc0NuTphIYdDzvp219Y3Nre3cTn53b//g0D06rps41YzXWCxj3Qyp4VIoXkOBkjcTzWkUSt4Ih7fTvDHi2ohYPeA44UFE+0r0BKNorY7rto1Qj6WsRNo44EgnHbfgFb2ZyCr4CyjAQtWO+9XuxiyNuEImqTEt30swyKhGwSSf5Nup4QllQ9rnLYuKRtwE2WzzCTm3Tpf0Ym2PQjJzf9/IaGTMOArtZERxYJazqflf1kqxdx1kQiUpcsXmD/VSSTAm0xpIV2jOUI4tUKaF3ZWwAdWUoS0rb0vwl7+8CvVS0b8slu/LhcrN07yOHJzCGVyAD1dQgTuoQg0YjOAZXuHNyZwX5935mI+uOYsKT+CPnM8fKeiTxQ==</latexit>  \n<latexit sha1\\_base64="CVn0nR4LyVoj9SztHbA59ZQShQg=">AAAB+XicbZBLSwMxFIXv+Kz1NerSTbAIrspMKeqy4sZlBfuAdiyZNG1DM5khuVMoQ8Ef4saFIm79J+78N6aPhbYeCHycc0NuTphIYdDzvp219Y3Nre3cTn53b//g0D06rps41YzXWCxj3Qyp4VIoXkOBkjcTzWkUSt4Ih7fTvDHi2ohYPeA44UFE+0r0BKNorY7rto1Qj6WsRNo44EgnHbfgFb2ZyCr4CyjAQtWO+9XuxiyNuEImqTEt30swyKhGwSSf5Nup4QllQ9rnLYuKRtwE2WzzCTm3Tpf0Ym2PQjJzf9/IaGTMOArtZERxYJazqflf1kqxdx1kQiUpcsXmD/VSSTAm0xpIV2jOUI4tUKaF3ZWwAdWUoS0rb0vwl7+8CvVS0b8slu/LhcrN07yOHJzCGVyAD1dQgTuoQg0YjOAZXuHNyZwX5935mI+uOYsKT+CPnM8fKeiTxQ==</latexit>  \n2  \nThe best-fit point is sin The best-fit point is sin  \n2  \n2  \n2  \nθ  \n=0.32,  \nΔ  \nm  \nSAGE+GALLEX+BEST oscillation analysis with 2 new cross sections.  \nSAGE+GALLEX+BEST oscillation analysis with 2 new cross sections.  \n1. Haxton- Krofcheck  -  \ns  \n[10  \n2. Haxton- Frekers  -  \ns  \n[10  \n-45  \ncm  \n2  \n/ (  \n71  \nGa atom  \n51  \nCr decay)] Ð  \n2. Haxton- Frekers -  \ns  \n[10  \n1. Haxton- Krofcheck -  \ns  \n[10  \n-45  \ncm  \n2  \n/ (  \n71  \nGa atom  \n51  \nCr decay)] Ð  \nThe best-fit point is sin 2 2 θ =0.32, Δ m 2 = 1.25. 2. Haxton- Frekers  s [10 -45 cm 2 / ( 71 Ga atom 51 Cr decay)] Ð The best-fit point is sin 2 2 θ =0.32, Δ m 2 = 1.25. 2. Haxton- Frekers s [10 -45 cm 2 / ( 71 Ga atom 51 Cr decay)] Ð FIG. 3 The contours are the sterile neutrino interpretation of the Ga anomaly, from (Elliott et al. , 2023). Limits at the 99% C.L. on the mixing angle from solar neutrinos, from (Goldhagen et al. , 2022), are indicated at left for two solar models, GS98 and AGSS09, and essentially exclude the indicated sterile-neutrino space.  \n<!-- image -->  \n.  \nneutrino capture to the ground state of 71 Ge can be extracted to a precision of ∼ 1% from the electron capture rate. The ground-state transition by itself generates a significant anomaly: inclusion of the excited-state contributions doubles the effect. In Appendix A we describe some of the details of both the 1997 extraction of the allowed (Gamow Teller) strength by Bahcall (1997) and the recent update of Elliott et al. (2024, 2023).  \nθ  \n=0.32,  \nΔ  \nm  \n-45  \ncm  \n2  \n/ (  \n71  \n2  \n71  \n= 1.25.  \n2  \n= 1.25.  \nGa atom  \n51  \nCr decay)] Ð  \n.  \n19  \n19  \n.  \n0  \n0  \n85  \n.  \n5  \n-  \n27  \n27  \n.  \n.  \n0  \n.  \n10  \n0  \n+  \n0  \n71  \n.  \n5  \n-  \n-45  \ncm  \n2  \n/ (  \nGa atom  \n51  \nCr decay)] Ð  \n85  \n.  \n5  \n-  \n10  \n.  \n0  \n71  \n.  \n5  \n-', '3. The Boron-8 neutrino spectrum': 'SAGE/GALLEX/BEST Frekers et al. SAGE/GALLEX/BEST Frekers et al. The neutrino spectrum from 8 B , extending to approximately 17 MeV, has been the most accessible part of the solar neutrino spectrum (Abe et al. , 2016; Aharmim et al. , 2020), playing an important role in disclosing the physics of neutrino oscillation and testing the SSM. Knowing it with precision, particularly at the high energy end, has renewed interest in the context of observing the hep neutrinos (Aharmim et al. , 2020, 2006), which could be accessible with future observatories (Askins et al. , 2022).  \nThe best-fit point is sin 2 2 θ =0.34, Δ m 2 = 1.25. 19 . 0 13 . 0 85 . 5 + -0.0               0.2                0.4               0.6                0.8               1.0 sin 2 2 θ The best-fit point is sin 2 2 θ =0.34, Δ m 2 = 1.25. 19 . 0 13 . 0 85 . 5 + -0.0               0.2 0.4               0.6 0.8               1.0 sin 2 2 θ In SF II it was decided to recommend the spectrum calculated by Winter et al. (Winter, 2007; Winter et al. , 2003, 2006) based on their measurement of the alpha spectrum from 8 B . This spectrum showed excellent agreement with an independent experiment (Bhattacharya et al. , 2006). Both radiative and recoil-order corrections were included by Winter et al. The radiative corrections are relatively small, due to a cancellation between the real and virtual contributions (Batkin and Sundaresan, 1995). The recoil-order corrections are dominated by the weak magnetism part, which has been deduced from measurements of the analog electromagnetic decays (de Braeckeleer et al. , 1995).  \nThe best-fit point is sin 2 2 θ =0.34, Δ m 2 = 1.25. The best-fit point is sin 2 2 θ =0.34, Δ m 2 = 1.25. Elliott et al. (2024, 2023) find the ground-state cross section for absorbing 51 Cr neutrinos is 2.5% lower than Bahcall\'s value. This work includes the contributions of weak magnetism and radiative corrections, which are shown to be sub-1% effects. An improved extraction of the ∼ 6% excited-state contribution to neutrino absorption was performed using data from forward-angle (p,n) scattering. This led to a slight increase in that contribution relative to Bahcall (1997), thereby reducing the 2.5% difference above by about half, for the total 51 Cr and 37 Ar capture cross sections.  \nThe procedures followed by Elliott et al. (2024, 2023) could be extended to the solar neutrino 71 Ga capture cross section, including the contribution from the highenergy 8 B neutrinos. That has not been done, but as noted in Appendix A, could help resolve small discrepancies noted there.  \nResults from the SAGE and GALLEX/GNO experiments remain part of the solar neutrino database used in various global fits to neutrino parameters. A resolution of the anomaly that emerged from BEST and the four gallium detector calibration experiments is important as it would increase confidence in these data.  \nSince then three additional measurements (Kirsebom et al. , 2011; Longfellow et al. , 2023; Roger et al. , 2012) of the 8 B β -decay alpha spectrum have been performed. All three find that the peak of the alpha spectrum appears about 20 keV lower than determined by Winter et al. Overall, these new measurements yield differences in the 8 B neutrino spectrum of /lessorsimilar 5% below E ν = 15 MeV, the energy above which the 8 B contribution becomes small compared to the hep component. Figure 4 shows the 8 B spectral uncertainties on the scale of the hep spectrum. With resolution effects considered, it will be difficult to extract the hep flux from solar neutrino measurements without further reduction in 8 B uncertainties.  \nIn producing a recommendation for the 8 B neutrino spectrum we considered the following:  \n- 1. Apparently the uncertainty estimations for the neutrino spectrum in Roger et al. (2012) are not quite correct. Longfellow et al. (2023) show that the uncertainty due only to the weak magnetism part, for which all authors follow similar prescriptions, is larger in the 0-12 MeV range than the overall uncertainty estimated by Roger et al. We estimated the uncertainties due to the weak magnetism term only and found agreement with those of Longfellow et al.\n- 2. All three of the recent efforts (Kirsebom et al. , 2011; Longfellow et al. , 2023; Roger et al. , 2012) used detectors (Si strip) more complicated than those used by Winter et al. (a Si surface barrier  \n13  \n.  \n0  \n.  \n13  \n+  \n<latexit sha1\\_base64="CVn0nR4LyVoj9SztHbA59ZQShQg=">AAAB+XicbZBLSwMxFIXv+Kz1NerSTbAIrspMKeqy4sZlBfuAdiyZNG1DM5khuVMoQ8Ef4saFIm79J+78N6aPhbYeCHycc0NuTphIYdDzvp219Y3Nre3cTn53b//g0D06rps41YzXWCxj3Qyp4VIoXkOBkjcTzWkUSt4Ih7fTvDHi2ohYPeA44UFE+0r0BKNorY7rto1Qj6WsRNo44EgnHbfgFb2ZyCr4CyjAQtWO+9XuxiyNuEImqTEt30swyKhGwSSf5Nup4QllQ9rnLYuKRtwE2WzzCTm3Tpf0Ym2PQjJzf9/IaGTMOArtZERxYJazqflf1kqxdx1kQiUpcsXmD/VSSTAm0xpIV2jOUI4tUKaF3ZWwAdWUoS0rb0vwl7+8CvVS0b8slu/LhcrN07yOHJzCGVyAD1dQgTuoQg0YjOAZXuHNyZwX5935mI+uOYsKT+CPnM8fKeiTxQ==</latexit>  \n<latexit sha1\\_base64="CVn0nR4LyVoj9SztHbA59ZQShQg=">AAAB+XicbZBLSwMxFIXv+Kz1NerSTbAIrspMKeqy4sZlBfuAdiyZNG1DM5khuVMoQ8Ef4saFIm79J+78N6aPhbYeCHycc0NuTphIYdDzvp219Y3Nre3cTn53b//g0D06rps41YzXWCxj3Qyp4VIoXkOBkjcTzWkUSt4Ih7fTvDHi2ohYPeA44UFE+0r0BKNorY7rto1Qj6WsRNo44EgnHbfgFb2ZyCr4CyjAQtWO+9XuxiyNuEImqTEt30swyKhGwSSf5Nup4QllQ9rnLYuKRtwE2WzzCTm3Tpf0Ym2PQjJzf9/IaGTMOArtZERxYJazqflf1kqxdx1kQiUpcsXmD/VSSTAm0xpIV2jOUI4tUKaF3ZWwAdWUoS0rb0vwl7+8CvVS0b8slu/LhcrN07yOHJzCGVyAD1dQgTuoQg0YjOAZXuHNyZwX5935mI+uOYsKT+CPnM8fKeiTxQ==</latexit>  \nFIG. 4 The hep spectrum is in black, and the uncertainties in 8 B from Longfellow et al. (Longfellow et al. , 2023) span the region between the green lines. The red line indicates the difference between Longfellow et al. and Winter et al. (Winter et al. , 2006) deduced 8 B spectra. The hep spectrum is used here as a convenient metric for the size of the uncertainties in the 8 B spectrum.  \n<!-- image -->  \ndetector). The strip detectors allow for better position resolution and reduced beta summing. However, for detection from sources external to the detectors (as opposed to detection from sources implanted deep into a detector), the complicated pattern of dead layers needs to be measured and taken into consideration for accurate calibrations. Effects of partial charge collection in-between strips are also significant. The effects are different for a calibration source than from the emission from 8 B trapped as an ion.  \nBased on these considerations, we recommend the spectrum of Longfellow et al. (2023), but suggest that conclusions sensitive to the choice of 8 B neutrino spectrum, such as evidence for hep neutrinos, should be evaluated using both the Winter et al. and the Longfellow et al. spectra.  \nGiven the importance of the weak magnetism contributions, it would be good to have a new experiment with reduced uncertainties that could be compared directly to the results of de Braeckeleer et al. (1995). Recoil-order effects in the A = 8 system were most recently measured by Sumikama et al. (2011), making use of the alignmentβ correlation. Using combinations of their measurements and β -α correlation data from McKeown et al. (1980), they were able to extract some of the recoil-order matrix elements. Calculations of these matrix elements in the symmetry-adapted no-core shell model were recently performed by Sargsyan et al. (2022). The comparison to the experimentally derived matrix elements of McKeown et al. (1980), de Braeckeleer et al. (1995), and Sumikama et al. (2011) is not straightforward because the experiments quote averages over the whole beta spectrum as  \nopposed to fits for each of the four levels used in the calculation. A more detailed comparison of theory to experiment that takes into account the averaging would be helpful. An additional point of comparison that has been used previously is the beta spectrum from 8 B (Bahcall et al. , 1996). An experiment with improved statistics and systematics that extends over the full range of beta energies would be very useful. The existing data from Napolitano et al. (1987) is in good agreement with the alpha spectrum from Winter et al. But given the apparent 20 keV difference that the newer alpha-spectrum experiments described above report, it would be good to have a modern beta spectrum measurement specifically designed to address this difference.', 'B. Experimental program on solar neutrinos': "A broad range of technology can be used to interrogate solar neutrinos. Radiochemical experiments utilizing chlorine and gallium played a crucial role in the early days of the solar neutrino problem. Large monolithic detectors have achieved great success in real-time measurements - from the water Cherenkov detectors such as Kamiokande, SNO, and SuperK, which could use the unique topology of Cherenkov light to point back to the neutrinos' origin, to liquid scintillator detectors such as KamLAND and Borexino, whose high light yield and low threshold allow for precision spectroscopy.", '1. Super-Kamiokande and Hyper-Kamiokande': "The first real-time detection of solar neutrinos was achieved by the Kamiokande experiment (Hirata et al. , 1989), which detected the neutrinos via elastic scattering of electrons, a process in which the outgoing electron's direction is highly correlated to that of the incoming neutrino. This allowed Kamiokande to directly point back to the neutrinos' origin: our Sun. The successor experiment, Super-Kamiokande, is a 50-kton volume of pure water, surrounded by over 10,000 photon detectors, which has achieved an unparalleled program of neutrino observations and other physics over its several decades of operation. This program has included the highest-precision measurement of the elastic scattering signal from solar neutrino interactions, as well as sensitive searches for the day/night distortions of the spectral shape (Abe et al. , 2016). Now filled with a gadolinium additive to enhance neutron capture, SuperK's primary current focus is the search for the Diffuse Supernova Neutrino Background (DSNB) via inverse beta decay (Harada et al. , 2023). It also continues its atmospheric neutrino program: SuperK measurements made 25 years ago demonstrated the neutrino oscillations were responsible for the puzzling zenith-angle-dependence of  \nthis flux (Fukuda et al. , 1998).  \nIn parallel to ongoing operation of SuperK, an even larger sister project is under construction. At over 250 kton in total mass, and with improved light collection relative to SuperK, Hyper-Kamiokande (HyperK) will impact statistics-limited searches the involve the higher energy 8 B solar neutrinos, including day/night effects, the search for hep neutrinos, and the shape of the 8 B neutrino spectrum. After 10 years of operations, HyperK will reach a sensitivity to day/night effects /greaterorsimilar 4 (8) σ , given the oscillation parameters deduced from reactor (solar) experiments. HyperK will serve as the far detector for JPARC's long-baseline neutrino program.", '2. Sudbury Neutrino Observatory and SNO+': "SNO's unique use of heavy water as a target medium offered the ability to detect solar neutrinos via two additional interactions besides elastic scattering. In the charged current interaction an electron neutrino interacts with the neutron in the deuteron, producing an electron and a proton. This interaction path is sensitive only to electron-flavor neutrinos at the few-MeV energy scale of solar neutrinos, providing a measurement of the pure ν e flux. SNO also had access to the neutral current interaction, in which a neutrino of any flavor interacts with the deuteron, breaking it apart into its constituent nucleons. Being equally sensitive to all active flavors, the neutral current interaction yields a measurement of the total, flavor-blind neutrino flux. It was this capability that allowed SNO to resolve the solar neutrino problem, demonstrating that most of the ν e s produced by the Sun were arriving at earth in a different flavor state (Ahmad et al. , 2002a, 2001; Ahmed et al. , 2004).  \nSNO ceased data taking in 2006. Analysis of the data set has continued, and produced a number of new results, including constraints on non-standard effects such as Lorentz violation and neutrino decay (Aharmim et al. , 2018, 2019a), limits on the fluxes of hep neutrinos and the diffuse supernova neutrino background (DSNB) (Aharmim et al. , 2020), and neutron production from cosmogenic muons and from atmospheric neutrino interactions (Aharmim et al. , 2019b,c).  \nAfter decommissioning, the detector was repurposed for the SNO+ experiment, in which the target material was replaced with a pure organic liquid scintillator: linear alkyl benzene (LAB), loaded with 2.2 g/L of the secondary fluor, PPO. The high light yield of this scintillator, coupled with the location in SNOLAB -at 6-km water equivalent, one of the deepest underground laboratories in the world- and an extremely well-understood detector, allow for a range of high-precision measurement programs. In a preliminary water phase, SNO+ demonstrated detection of neutron captures on hydrogen (Anderson et al. , 2020), an impressive technical achievement  \nin an unloaded water detector; a low background measurement of 8 B solar neutrinos (Anderson et al. , 2019b); several searches for invisible modes of nucleon decay (Allega et al. , 2022; Anderson et al. , 2019a); and detection of antineutrinos from reactors over 200 km away (Allega et al. , 2023). The low backgrounds achieved and the detector's demonstrated technical capabilities set the stage for future measurements. In a 'partial-fill' stage, in which the upper half of the detector was filled with liquid scintillator while the lower half still contained water, the first demonstration of event-level direction reconstruction was achieved for 8 B neutrinos in a scintillator detector (Allega et al. , 2024). This was facilitated by a lower loading of PPO at that time, which results in lower scintillation yield and a slower time profile, effectively enhancing the Cherenkov component, which can be leveraged for directional information.  \nNow fully filled with liquid scintillator, SNO+ is the deepest, largest operating liquid scintillator detector in the world. The future program will include measurements of several solar neutrino fluxes, as well as the 8 B neutrino energy spectrum.", '3. Borexino': "Water Cherenkov experiments such as Kamiokande, SuperK and SNO provided the first real-time measurement of solar neutrinos but their relatively high energy threshold made them sensitive only to a small fraction of the solar neutrino flux. To study in real time the bulk of solar neutrino emission a different detector technology was required. The Borexino experiment, located deep underground at the INFN Laboratori Nazionali del Gran Sasso, used an organic liquid scintillator target made of pseudocumene with 1 . 5 g / L PPO to detect the elastic scattering of solar neutrinos off electrons. The high light yield of the liquid scintillator made it possible to significantly lower the energy threshold, but given the small signal rate and the lack of a clear signature to separate it from the background (such as the direction indicated by the emission of Cherenkov radiation), the experiment required a long preparatory phase to develop the most advanced techniques to suppress the background, especially the one due to radioactive contamination of the liquid scintillator itself.  \nThe detector's extreme radiopurity was key to its success, and over its 14 years of data taking, ending in October 2021, Borexino proved itself capable of covering the entire solar neutrino spectrum. At the beginning of its data taking in 2007 the level of 238 U and 232 Th contamination in Borexino were lower than 2 × 10 -17 and 7 × 10 -18 g / g respectively (BOREXINO Collaboration et al. , 2008), paving the way for the first measurement of the 7 Be sub-MeV solar neutrinos, followed by a lowthreshold measurement of the 8 B flux and a first indica-  \non of the pep neutrinos. After a purification campaign that further reduced the liquid scintillator contamination, Borexino aimed at improving the accuracy of its results in the measurement of the pp-chain solar neutrinos. The flux of neutrinos produced in the pp fusion process was first reported in BOREXINO Collaboration et al. (2014) and later improved, along with all the other fluxes from the pp chain, apart from the hep neutrinos (Agostini et al. , 2020b, 2019; Borexino Collaboration et al. , 2018). The pp flux was determined to an uncertainty of 11%, improving the neutrino-based estimate of solar luminosity, while 7 Be neutrinos were measured with an uncertainty of 2 . 7% , half that of the SSM prediction. Furthermore, using the SSM to constrain the flux of CNO neutrinos, the pep neutrino signal was established with a significance exceeding 5 σ for the first time, and the 8 B flux was measured with a threshold as low as 3 MeV.  \nAfter completing the investigation of the pp-chain neutrinos, Borexino reported the first detection of solar neutrinos produced in the CNO-cycle (Borexino Collaboration et al. , 2020b), demonstrating directly that this mode of hydrogen burning operates in stars. Profiting from a larger exposure and a better understanding of the radioactive backgrounds enabled by the unprecedented thermal stability of the detector, Borexino further improved its measurement of the CNO neutrino flux (Appel et al. , 2022; Basilico et al. , 2023), where the new CNO result was used in combination with existing solar neutrino results to probe solar composition.  \nBorexino's physics program was not limited to solar neutrinos: its outstanding radiopurity made it a excellent detector for geoneutrinos (Agostini et al. , 2020a) and for searches for various rare processes. Borexino data were used to constrain exotic properties of neutrinos, such as their magnetic moments (Agostini et al. , 2017) and non standard interactions with matter (Borexino Collaboration et al. , 2020a).", '4. JUNO': "The Jiangmen Underground Neutrino Observatory (JUNO) is a large liquid scintillator detector currently under construction in an underground laboratory with a vertical overburden of about 650 m (roughly 1800 m water equivalent) in Jiangmen city in Southern China (Adam et al. , 2015). JUNO is located 52.5 km from two nuclear power plants, a baseline optimized for JUNO's primary goal, the determination of the neutrino mass ordering [28, 29]. To achieve this, JUNO requires a large target mass (20 kton) and excellent energy resolution, attributes also import in solar neutrino detection. The relatively shallow overburden limits the solar neutrino program, due to cosmogenic activity, but the low threshold and large detector mass may allow measurements of day/night effects, non-standard neutrino interactions af-  \nfecting the 8 B spectral shape, and 7 Be neutrinos, as well as providing a new solar determination of ∆ m 2 12 . JUNO's reactor data will constrain ∆ m 2 12 to extremely high precision, allowing comparisons between the solar and reactor determinations of this parameter. To the extent that oscillation effects can be treated with greater confidence, the connections between solar neutrino flux measurements and the solar fusion reactions generating those fluxes will be sharpened (An et al. , 2016).", '5. DUNE': 'As recommended in the 2023 P5 report (P52023, 2023), the DUNE Collaboration seeks to construct two 10-kton fiducial liquid argon time projection chambers (LArTPC), deep underground in the Homestake mine in South Dakota (Acciarri et al. , 2016). A third LArTPC will follow in Phase II, along with other upgrades. The newly developed Sanford Underground Research Facility (SURF) offers 4800-m water equivalent overburden, and will form the far site for a long baseline neutrino program utilizing a high-energy neutrino beam directed from Fermilab to SURF. The DUNE LArTPC detectors may be sensitive to measurements of the high-energy solar neutrinos via charged current interactions on argon, which offers good precision on the spectral shape and, thus, the potential to measure day/night effects (Capozzi et al. , 2019). The solar neutrino program at DUNE is limited primarily by backgrounds (the detector is optimized for GeV-scale physics) and energy threshold. Multiple technologies are under consideration for a fourth detector module, with the motivation of expanding the physics program, including improved sensitivity to solar neutrinos. Use of underground argon, or alternative technologies such as organic or water-based scintillators, could preserve the long baseline neutrino sensitivity while also opening up a rich program of low-energy physics.', '6. Future prospects': "A 'hybrid' detector that could utilize both Cherenkov and scintillation light simultaneously could achieve unprecedented levels of particle and event identification and, hence, background rejection (Alonso et al. , 2014). The Cherenkov signature offers directional information, while the high light yield scintillation offers precision energy and vertex reconstruction. A full waveform analysis of detected light offers yet more information, based on the pulse shape of the scintillation, which is subject to species-dependent quenching effects, and the impact of the Cherenkov threshold. As a result, both the shape of the waveform and the ratio of the two signals will differ for different particle types. A full-scale detector utilizing novel scintillators along with fast and spectrally-sensitive  \nphoton detectors, such as the proposed Theia experiment (Askins et al. , 2020), or the Jinping detector (Beacom et al. , 2017) could achieve percent-level precision on the CNO neutrino flux, as well as improving precision across the suite of solar neutrino measurements, such as the 8 B spectral shape, and hep neutrino flux.  \nAs experiments grow larger, and capabilities increase, there are also opportunities to apply detectors designed for other purposes to the detection of solar neutrinos. Noble liquid detectors built primarily for dark matter searches and long-baseline neutrino experiments have now reached the size and background levels to be sensitive to solar neutrinos (Aprile et al. , 2024). The detection of neutrinos from the high-energy tail of the 8 B spectrum via coherent elastic neutrino-nucleus scattering ('CE ν NS') is particularly interesting because it is a purely neutral-current process sensitive to the total flux of active neutrinos (above the experimental threshold, about 8 MeV) from the Sun. Only the SNO experiment has had such a capability. Initial results reported in (Aprile et al. , 2024) are in agreement, with limited statistical precision. Solid state detectors built purposely for observing CE ν NS have observed neutrinos from stopped pion decay at the theoretically expected rate (Akimov et al. , 2017).", 'III. The 1 H ( p, e + ν ) 2 H reaction ( S 11 )': 'The cross section for the initial reaction in the pp chain (Figure 1), p + p → d + e + + ν e , is too small to be measured in the laboratory. It must be calculated from the standard theory of weak interactions.', 'A. Introduction and terminology': "Near the Gamow peak energy E ∼ 6 keV for temperatures characteristic of the solar core, the first and second derivatives of the astrophysical S factor at zero energy, S ' 11 (0) and S '' 11 (0) , generate ∼ 7% and ∼ 0.5% corrections, respectively, in Taylor's series expansion of S 11 ( E ) around E = 0 . Higher derivative terms are neglected in this review since they only contribute at the 10 -4 level. The recommended values for S ' 11 (0) and S '' 11 (0) will be discussed in Section III.E. Here we focus on S 11 (0) .  \nAt zero relative energy, S 11 (0) can be written as (Bahcall and May, 1968, 1969),  \nS 11 (0) = 6 π 2 m p α ln 2 Λ 2 γ 3 ( G A G V ) 2 f R pp ( ft ) 0 + → 0 + , (5)  \nwhere α = 1 / 137 . 04 is the fine-structure constant, m p = 938 . 272 MeV is the proton mass, γ = (2 µB d ) 1 / 2 = 0 . 23161 fm -1 is the deuteron binding wave number, µ being the proton-neutron reduced mass and B d the  \ndeuteron binding energy, G V and G A are the Fermi vector and axial-vector weak coupling constants. Finally, f R pp is the phase-space factor for the pp reaction with radiative corrections, ( ft ) 0 + → 0 + is the ft value for superallowed 0 + → 0 + transitions, and Λ is proportional to the transition matrix element connecting the pp and deuteron states.", 'B. Adopted parameters for this review': "For the phase-space factor f R pp , we use the same value f R pp = 0 . 144(1 ± 0 . 001) as in SF II. It comes from the value without radiative corrections, f pp = 0 . 142 (Bahcall and May, 1969), increased by 1.62% to take into account radiative corrections to the cross section (Kurylov et al. , 2003). The main source of uncertainty in f R pp arises from neglected diagrams in which the lepton exchanges a weak boson and a photon with different nucleons. These diagrams are estimated to modify f R pp by ∼ 0 . 1% , based on scaling the similar nucleus-dependent correction in superallowed β decay (Kurylov et al. , 2003). Direct computations of these diagrams were recommended in Solar Fusion II. Here we urge again that this computation be carried out.  \nFor G A /G V , we use the value from the Particle Data Group (PDG) compilation (Workman et al. , 2022), 1 . 2754 ± 0 . 0013 , whose central value is larger than the 2008 PDG value, 1 . 2695 ± 0 . 0029 , used in SF II by 0 . 45% (or 1.9 σ ). Naively, this would lead to a ∼ 1% increase in the central value of S 11 (0) according to Eq. (5). Its effect will be discussed below. For ( ft ) 0 + → 0 + , we take 3072 . 24 ± 1 . 85 s from the most updated comprehensive analysis of experimental rates with the radiative and Coulomb effects corrected (Hardy and Towner, 2020). This value is consistent with 3071 . 4 ± 0 . 8 s (Hardy and Towner, 2009) used in SF II with a larger error.  \nThe dominant uncertainty in S 11 (0) comes from the normalized Gamow-Teller (GT) matrix element Λ . Reducing this uncertainty has been the main focus of theoretical work since SF I. In SF I, Λ was written as Λ = Λ + δ Λ , i.e. the sum of the one- and two-body current matrix elements, Λ and δ Λ , respectively, with their uncertainties estimated independently. In SF II, two major steps had contributed to reducing the uncertainty on Λ . The first was a much deeper understanding of the correlation between the uncertainties in Λ and δ Λ : the overall uncertainty in Λ could be described by a universal parameter that could be fixed by a single measurement. The study of Schiavilla et al. (1998) demonstrated this phenomenologically in the context of potential-model approaches, while later analysis via effective field theory (EFT) provided a more formal justification (Butler et al. , 2001; Park et al. , 2003). The second step was the use of the precisely known tritium β decay rate Γ T β to fix this universal parameter, as first proposed by Carlson et al.  \n(1991). This has been done in both potential models (Schiavilla et al. , 1998) and in the hybrid EFT approach (Park et al. , 2003) as explained below.  \nIn SF II, Λ was determined with three approaches. The first one was the potential model approach. In the most elaborate calculation for the pp fusion process, a comparison of the results for five representative modern potentials designed to accurately reproduce nucleonnucleon scattering data was carried out (Schiavilla et al. , 1998). After adjusting the unknown strength of the twobody exchange currents to reproduce Γ T β , the variation in S 11 (0) that otherwise would come from the choice of the phenomenological potential was largely removed. Predictions for five representative high-precision phenomenological potentials fell in a narrow interval 7 . 03 /lessorsimilar Λ 2 /lessorsimilar 7 . 04 . There were additional uncertainties in the threebody potentials and three-body currents in Γ T β , of the order of ∼ 0 . 8% , and a 0.5% uncertainty due to effective range parameters for nucleon-nucleon scattering. Hence, the recommended S 11 (0) value from the potential model approach was S 11 (0) = 4 . 01(1 ± 0 . 009) × 10 -25 MeV b .  \nThe second and third approaches were both based on EFT. The second one was a hybrid EFT (EFT*), which used the current operators derived from EFT in conjunction with the initial and final state wave functions generated by a potential model (Park et al. , 2003). For pp fusion, the relevant two-body current contained only one unknown low-energy constant (LEC) ˆ d R which parameterized the contact axial coupling to two nucleons (Park et al. , 2003). A weakness of this approach was the mismatch between the operators and wave functions. However, it was argued that the mismatch only happened for short distance physics which could be absorbed by the LECs. Hence, when the ultraviolet cutoff was changed over a physically reasonable range, the residual cutoff dependence of physical observables provided a measure of the model dependence of the EFT* calculation. By combining the 0 . 8% error from changing the cutoff Λ NN in the range of 500 -800 MeV and the ∼ 0.4% higher order correction, obtained by multiplying the 1 . 8% contribution of the highest calculated order with the small expansion parameter m π / Λ NN ∼ 1 / 4 , Park et al. (2003) provided the value S 11 (0) = 4 . 01(1 ± 0 . 009) × 10 -25 MeV b , in perfect agreement with the one obtained within the phenomenological approach.  \nThe third study was performed with the pionless EFT ( / π EFT) approach. It is a framework applicable to processes with the characteristic momentum p much smaller than the pion mass m π , such that the pion field can be 'integrated out' and becomes non-dynamical (Kaplan, 1996; Bedaque et al. , 1999; Chen et al. , 1999). In this approach, all nucleon-nucleon interactions and twobody currents are described by point-like contact interactions with a systematic expansion in powers of p/m π . For all the deuteron weak breakup processes (e.g. νd and ¯ νd scattering) and their inverse processes, includ-  \ne pp fusion, only one two-body current (with coupling L 1 ,A ) is needed up to next-to-next-to-leading order (N 2 LO) (Butler et al. , 2001). Therefore a single measurement will fix L 1 ,A and the rates of all such processes. This feature is shared by the other approaches discussed above. The computation of Λ in / π EFT was carried out to the second order in the p/m π expansion by Kong and Ravndal (2001) and then to the fifth order by Butler and Chen (2001). Constraints on L 1 ,A from two-nucleon systems (Butler et al. , 2002; Chen et al. , 2003) yielded S 11 (0) = 3 . 99(1 ± 0 . 030) × 10 -25 MeV b .  \nBased on the consistent results of the above three approaches, SF II recommended S 11 (0) = 4 . 01(1 ± 0 . 009) × 10 -25 MeV b . The SF I value S ' 11 (0) = S 11 (0)(11 . 2 ± 0 . 1) MeV -1 from Bahcall and May (1969) was not recomputed in SF II. However, with S 11 (0) reaching a 1% accuracy, new calculations of S ' 11 (0) and S '' 11 (0) , together with the full pionful chiral EFT ( χ EFT) computations to remove the unknown systematics of the hybrid EFT, were called for in SF II. These two challenges have been met in SF III and will be described below.", 'C. Experimental progress on muon capture of the deuteron': 'In SF II it was also recommended to carry out the experimental determination of the muon capture rate on deuteron, as proposed in the MuSun experiment (Kammel, 2003; Salvat and MuSun Collaboration, 2017). This quantity could be used to constrain S 11 (0) without the need to rely on the three-body calculation of Γ T β . The theoretical calculations for the muon capture rate on deuteron have been carried out in / π EFT (Chen et al. , 2005) (it is possible to impose neutron energy cut to isolate the low-energy neutron events so / π EFT is applicable (Kammel, 2003)), chiral hybrid EFT* (Ando et al. , 2002), phenomenological potential model (Marcucci et al. , 2011), and recently in χ EFT (Marcucci et al. , 2012, 2018; Acharya et al. , 2018; Ceccarelli et al. , 2023; Bonilla et al. , 2023; Gnech et al. , 2024). However, the MuSun result is yet to be released. On the other hand, we should notice that the muon capture processes happen at a rather large momentum transfer compared to pp fusion. The momentum transfer dependence of the single nucleon axial coupling constant g A ( q 2 ) , with g A ( q 2 = 0) ≡ g A ≡ G A /G V , has been recently studied by Hill et al. (2018), who have provided an experimental determination for the axial charge radius, given by r 2 A = 0 . 46(16) fm 2 , where r A is defined by the relation g A ( q 2 ) = g A (1 -r 2 A q 2 / 6) for small q 2 . The ∼ 30 % uncertainty on r 2 A has an impact on the ability of the MuSun experiment alone to directly constrain S 11 (0) (Acharya et al. , 2018; Bonilla et al. , 2023; Ceccarelli et al. , 2023; Gnech et al. , 2024).  \nIn the next few years, lattice Quantum Chromodynamics (QCD) calculations of g A ( q 2 ) are expected to reduce  \nthe r A uncertainty by a factor of two or more (see Meyer (2023) for the most recent review of the lattice results). Currently, lattice QCD results are consistent with Hill et al. (2018) on r A , often with comparable uncertainties. At larger q 2 ( | q 2 | /greaterorsimilar 0 . 25 GeV 2 ), there is a growing tension between lattice QCD predictions of g A ( q 2 ) and the phenomenological determination from older neutrinodeuterium bubble chamber data (Meyer et al. , 2016) with the lattice QCD results yielding a 30% larger neutrinonucleon cross section over a large range of q 2 (Meyer et al. , 2022).  \nIn any case, it is evident that accurate experimental determinations of muon capture rates on the deuteron and other light nuclei, which can be addressed theoretically ab initio approaches, represent fundamental tests for the theoretical approaches themselves, either within χ EFT or / π EFT, and might be able to provide the necessary experimental information to fix the unknown parameters of the theory.', 'D. Progress in S 11 (0) calculations since SF II': 'Below we summarize the S 11 (0) calculations performed after SF II using different approaches.', '1. χ EFT': 'The pioneering work of Marcucci et al. (2013) used the next-to-next-to-next-to-leading-order (N 3 LO) chiral twonucleon potential (Entem and Machleidt, 2003; Machleidt and Entem, 2011) augmented with higher order ( O ( α 2 ) ) two-photon and vacuum-polarization electromagnetic interactions. These O ( α 2 ) corrections reduced S 11 (0) by ∼ 0 . 8 %mainly due to the vacuum-polarizationinduced pp wave function distortion. This was consistent with the 0.84% first found in the potential model calculation of Schiavilla et al. (1998). This correction was also included in the EFT* calculation of (Park et al. , 2003). The relevant LECs were fitted to reproduce the A = 3 binding energies, magnetic moments, and GT matrix element in Γ T β to obtain S 11 (0) = 4 . 030(1 ± 0 . 006) × 10 -25 MeV b , with the p -wave initial state contributing at ∼ 1% , which was about the accuracy level of the calculation (Marcucci et al. , 2013). However, using / π EFT at next-to-leading-order (NLO), Acharya et al. (2019) later found that p -wave only contributed at the order of 10 -30 MeV b . Re-examining the computer programs, Marcucci et al. (2013) found an error in the determination of one of the p -wave reduced matrix elements (associated with the longitudinal multipole operator). Consequently, Marcucci et al. (2019) reported in the Erratum that S 11 = 4 . 008(1 ± 0 . 005) × 10 -25 MeV b .  \nIn more recent work, Acharya et al. (2016) used chiral interactions and consistent currents up to N 3 LO (called  \nnext-to-next-to-leading-order (N 2 LO) in the original literature since the second order vanished) and developed a robust procedure for the error quantification. In particular, they analyzed a family of 42 interactions (Carlsson et al. , 2016) with 7 different cutoff values from 450 to 600 MeV. The 26 LECs were fitted to 6 different pools of input data including NN and πN scatterings, as well as the binding energies and charge radii of 3 H and 3 He, the quadrupole moment of 2 H, and Γ T β . This thorough study yielded S 11 (0) = 4 . 047(1 +0 . 006 -0 . 008 ) × 10 -25 MeV b .  \nIn both of the χ EFT calculations mentioned above, a widely used relation first proposed by Gazit et al. (2009) that linked the two-body axial current LEC ˆ d R with the LEC c D from the πNN vertex was employed. However, later Marcucci et al. (2018) found that there was a factor -1 / 4 missing in this ˆ d R -c D relation which was then acknowledged in the Erratum of Gazit et al. (2019). Fortunately, this error was unimportant in muon capture on deuteron (Marcucci et al. , 2018) and it only affected S 11 (0) at the 0 . 1% level (Acharya et al. , 2023).  \nIn the most recent and comprehensive χ EFT study, Acharya et al. (2023) compared the above calculations of Marcucci et al. (2019) and Acharya et al. (2016) in detail. In addition to the 0 . 1% increase of S 11 (0) from using the correct ˆ d R -c D relation, updating the input parameters to their most recent values increased S 11 (0) by ∼ 1% in both calculations, mainly due to the 0 . 45% increase of G A /G V from its SF II value mentioned above. Furthermore, Marcucci et al. (2019) received a ∼ 1% increase from removing the truncation error of the basis functions which effectively cut off the long distance part of the wave functions, as Acharya et al. (2017) advocated. After these corrections, the ∼ 1% difference between Marcucci et al. (2019) and Acharya et al. (2016) was reconciled and the combined result was found to be (Acharya et al. , 2023)  \nS 11 (0) = 4 . 100(1 ± 0 . 007) × 10 -25 MeV b . (6)  \nAcharya et al. (2023) obtained consistent values of S 11 (0) using four different χ EFT models for the nuclear interaction. This value is also in agreement with the result obtained in / π EFT by De-Leon and Gazit (2022) (see below). In estimating the order-by-order convergence, however, Acharya et al. (2023) used one of the χ EFT models, which is able to nicely reproduce the deuteron properties already at leading order. Therefore, the χ EFT error of Eq. (6) is likely to be an underestimate, warranting the enlarged error advocated in Section III.D.4.  \nFinally, we would like to mention the work of Liu et al. (2022a), where the power counting of the χ EFT weak current operator involved in the pp reaction is revisited using renormalization group (RG) invariance as the guideline. In particular, it is argued that the contact two-body axial current proportional to the ˆ d R LEC must appear one order lower than assessed by naive dimensional analysis. Then it can be shown that RG in-  \nriance is fulfilled at N 2 LO. However, the estimate for Λ obtained by Liu et al. (2022a) does not use a value for ˆ d R obtained by fitting Γ T β , but extracted in order to match the value of Λ obtained in SF II. Therefore, we will not consider the work of Liu et al. (2022a) in the present S 11 (0) evaluation.', '2. / π EFT': 'The universal two-body current coupling L 1 ,A was determined using Γ T β in / π EFT for the first time by DeLeon et al. (2019). This calculation was carried out up to NLO using the dibaryon formulation of Beane and Savage (2001), which partially resumed higher order effective range contributions to improve the convergence. Then, this result was used by De-Leon and Gazit (2022), with updated input parameters, to obtain  \nS 11 (0) = 4 . 12(1 ± 0 . 015) × 10 -25 MeV b , (7)  \nwhere the O ( α 2 ) electromagnetic correction was not calculated but assumed to be the same as the potential model value, 0 . 84% (Schiavilla et al. , 1998). While this number was not model independent, the model dependence was believed to be well below the assigned 1 . 5% error.  \nThe small error assigned to this NLO result has been justified by drawing an analogy from the corresponding electromagnetic processes. Using the same approach, the np → dγ matrix element at threshold was predicted at NLO within 0.5% to the experimental value after the electromagnetic two-body current L 1 was fit to the magnetic moments of 3 He and 3 H. This indicates that the contribution of the three-body current at N 2 LO is small in this electromagnetic case. The weak sector is shown to follow the same operator structure and hence provides support for the calculation procedure and the uncertainty estimate.  \nIt is important to note that the reason that the threebody current is an N 2 LO effect in / π EFT is related to the non-trivial renormalization of the non-derivative threebody contact interaction, which shows up at LO to absorb the cutoff dependence of Feynman diagrams. The subleading two-derivative three-body contact interaction is expected to show up at N 2 LO. This interaction, combined with the one-body current, renormalizes the threebody current. Hence, the three-body current should also appear at the same order, N 2 LO. However, if the nonderivative three-body contact interaction were counted as higher order, such as N 3 LO, as in χ EFT (because cutoff independence is not strictly enforced order by order in χ EFT), then the three-body current would contribute at much higher order. Although the χ EFT power counting indeed yields good convergence in the expansions, it is unsatisfactory that one can not remove the cutoff dependence at each order of the expansion. In addition,  \nthe uncertainty estimate of the χ EFT still lacks a broad inspection of the specific nuclear χ EFT potential implementation. As a consequence, we consider the / π EFT 1.5% error a better estimate for the S 11 (0) theoretical uncertainty.', '3. Lattice QCD and Lattice EFT': 'Ideally lattice QCD would provide a first principles prediction of the pp fusion rate and the GT matrix element of Γ T β , for both pure QCD and with QED effects incorporated. However, such calculations are quite challenging and not yet available at the required precision. A proof-of-principle calculation was carried out by Savage et al. (2017) using a background field method to determine both the pp fusion GT matrix element and Γ T β . This exploratory calculation utilized a single pion mass at the SU(3)-flavor-symmetric point with m π ≈ 806 MeV, a single and relatively coarse lattice spacing of a ≈ 0 . 145 fm, and a single volume. The calculation was performed under the assumption that the two- and three-nucleon systems were deeply bound. Without this assumption, matrix elements computed in the finite volume can be significantly different from those in infinite volume, due to Lellouch-Lüscher factors (Lellouch and Lüscher, 2001) that lead to power-law finite volume corrections. These can range from the few-percent level to O(1) (Briceño and Davoudi, 2013; Briceño et al. , 2015).  \nMore recent lattice QCD calculations have found that two-nucleon systems at heavy pion masses are in fact not bound (Iritani et al. , 2016; Francis et al. , 2019; Hörz et al. , 2021; Amarasinghe et al. , 2021). These efforts, which employ interpolating operators more sophisticated than those of Savage et al. (2017), suggest there could be large systematic uncertainties affecting the conclusions of Savage et al. (2017) stemming from misidentification of the spectrum and inaccurate Lellouch-Lüscher factors. In addition, Green et al. (2021) found that the two-baryon spectrum may be particularly sensitive to discretization effects, which would also have important implications for the continuum extrapolation of the matrix elements. These issues are discussed in some detail in Tews et al. (2022). Finally, results with pion masses m π /lessorsimilar 300 MeV are needed for accurate extrapolation to the physical pion mass.  \nGiven these unresolved systematic issues, the result of Savage et al. (2017) is not included in the present S 11 (0) evaluation - even though the extracted value S 11 (0) = 4 . 07(1 ± 0 . 008) × 10 -25 MeV b is consistent with our recommended range quoted in Section III.D.4.  \nThe lattice EFT computation of Rupak and Ravi (2015) performed the pp fusion calculation by implementing / π EFT on a spacetime lattice. The purpose of this leading-order study was to demonstrate that lattice EFT could reproduce the infinite volume and continuum re-  \nult of / π EFT such that it could be applied to various reactions of astrophysical interest in the future. Therefore, for S 11 (0) , this result is considered as a subset of the / π EFT calculation.', '4. Final Recommendation of S 11 (0)': 'The above discussions show that determinations of S 11 (0) from χ EFT in Eq.(6), / π EFT in Eq.(7), and lattice QCD (although with unquantified systematics) are all consistent with each other. Furthermore, these values are also consistent with the recommended value of SF II, provided the central value is increased by 0 . 9% 5 to S 11 (0) = 4 . 05(1 ± 0 . 009) × 10 -25 MeV b to account for the G A /G V update. Averaging this value with the χ EFT value in Eq.(6) and the / π EFT value in Eq.(7) yields S 11 (0) = 4 . 09(1 ± 0 . 005) × 10 -25 MeV b with χ 2 per degree of freedom to be 0 . 9 . This shows that the χ EFT, / π EFT and SF II estimates of S 11 (0) are all mutually consistent.  \nIn addition, we would like to advocate adding an additional correlated error to account for any input that would tend to move all results in a coordinated way. For example, from SF II to SF III, we experienced the ∼ 1% shift due to the update of the PDG value of G A /G V . It is not inconceivable that G A /G V or other input parameters or physics could change again by similar amounts: the large PDG inflation factor of 2.7 reflects the tension that continues to exist among G A /G V measurements (Workman et al. , 2022). Therefore, assigning an additional 1% correlated error seems reasonable. This is also in line with the subtleties discussed in Secs.III.D.1 and III.D.2 which call for an enlarged error. Therefore, our final recommended value for S 11 (0) is  \nS 11 (0) = 4 . 09(1 ± 0 . 015) × 10 -25 MeV b , (8)  \nwhere we have added the correlated and uncorrelated errors linearly to be conservative.', "E. Progress in S ' 11 (0) and S '' 11 (0)": "Using / π EFT, Chen et al. (2013) computed S 11 ( E ) analytically with all partial waves included up to N 2 LO. The Fermi matrix element only contributed at the 10 -4 level and was neglected compared with the GT matrix element. The energy dependence of the phase factor f R pp of Eq. (5) was the dominant effect in S ' 11 (0) and S '' 11 (0) , the energy dependence of pp scattering was subdominant,  \nwhile the L 1 ,A contribution was much less important in these derivatives than in S 11 (0) . Therefore, these derivatives could be predicted more reliably than S 11 (0) . Furthermore, the derivatives were computed analytically and were free from errors of fitting S 11 ( E ) to a polynomial. The result was S ' 11 (0) /S 11 (0) = (11 . 3 ± 0 . 1) MeV -1 and S '' 11 (0) /S 11 (0) = (170 ± 2) MeV -2 .  \nIn χ EFT, S 11 ( E ) for E < 100 keV was fit to polynomials of E (Marcucci et al. , 2019). Depending on using a quadratic or quartic fit, S ' 11 (0) /S 11 (0) changed from 12.23 to 10.82 MeV -1 and S '' 11 (0) /S 11 (0) changed from 178.4 to 317.4 MeV -2 . Acharya et al. (2016) used cubit fit and E < 30 keV to obtain S ' 11 (0) /S 11 (0) =10.84(2) MeV -1 and S '' 11 (0) /S 11 (0) =317.8(13) MeV -2 . Recently Acharya et al. (2023) repeated the calculation of Marcucci et al. (2019) with the same energy range and cubic fit as Acharya et al. (2016), obtaining a consistent result with Acharya et al. (2016): S ' 11 (0) /S 11 (0) = 10 . 83 MeV -1 and S '' 11 (0) /S 11 (0) = 313 . 72 MeV -2 . We will take these as the recommended values from χ EFT.  \nAlthough the face values of S ' 11 (0) /S 11 (0) and S '' 11 (0) /S 11 (0) from / π EFT and χ EFT look quite different, we would like to remark that they actually agree on S 11 ( E ) /S 11 (0) better than 0.1% below the ∼ 6 keV Gamow peak. For massive stars with central temperatures ∼ 15 keV, the agreement is better than 0.8% to second order in the derivatives and 0.5% if S ''' 11 (0) /S 11 (0) = -5382 MeV -3 is included in the χ EFT result. Hence, we take the average of the / π EFT and χ EFT results as the recommended value:  \nS ' 11 (0) = S 11 (0)(11 . 0 ± 0 . 2)MeV -1 , S '' 11 (0) = S 11 (0)(242 ± 72)MeV -2 . (9)  \nFinally, we comment on the work of Gaspard et al. (2019) which performed a study of S 11 ( E ) in a wide energy range. The main focus of this work was to perform a proper energy-dependence analysis of the pp process, in order to reliably extract S ' 11 (0) /S 11 (0) and S '' 11 (0) /S 11 (0) . However, the calculation was performed within a phenomenological approach, using a quite simplified model for the nuclear currents (i.e. not including two-body currents, which are well known to be significant) and structure (i.e. neglecting the d -wave components in the deuteron wave function). They found results for S ' 11 (0) /S 11 (0) and S '' 11 (0) /S 11 (0) compatible with those of Eq. (9). However, since these values were not obtained with state of the art calculations, they have not been considered in the determination of S ' 11 (0) /S 11 (0) and S '' 11 (0) /S 11 (0) .", 'A. Introduction': 'Opacity is a measure of the photon absorption of matter and is an essential quantity for understanding radiative heat transfer in the Sun. Radiative heat transfer occurs through the absorption and emission processes taking place within the material that the radiation traverses. In local thermodynamic equilibrium (LTE) 11 , the emission and absorption are straightforwardly related through  \nthe Planck function. In this scenario, determining the absorption coefficients or opacity, denoted as κ ν , provides a complete description of how radiation is transported through the material.  \nFor example, in the equilibrium diffusion limit, the radiation heat flux F R is directly related to the Rosselandmean opacity (RMO) κ R through the following equation (Huebner and Barfield, 2014)  \nF R = -16 π 3 σT 3 κ R ∇ T , (29)  \nwhere T represents temperature, ∇ T denotes its gradient, and σ is the Stefan-Boltzmann constant. The quantity κ R is the solar Rosseland mean opacity (RMO). Thus, any error in this quantity introduces error in the radiative heat transport and the simulated solar evolution. Quantifying the error in κ R is very challenging because κ R is a weighted mean of a complex solar mixture opacity:  \n1 κ R = ∫ ∞ 0 1 κ ν w ν dν , (30)  \nwhere w ν is a weighting function related to the temperature derivative of the Planck function B ν :  \nB ν = 2 hν 3 c 2 ( e hν/kT ) -1 , (31)  \nwhere ν denotes the photon frequency, h the Planck constant, k the Boltzmann constant, and c the speed of light. Moreover, the frequency-resolved solar opacity κ ν is a sum of elemental opacities κ ν,i weighted by their mass fractions ( b i ):  \nκ ν = ∑ i b i κ ν,i (32)  \nEquation 30 reveals that the accuracy of κ ν is especially important over the spectral range where w ν is high and κ ν is low (due to the inverse weighting). Equation 32 also indicates that accurate κ ν is possible only when we have both accurate element opacities κ ν,i and accurate abundances b i .  \nFigure 11 shows the total monochromatic opacity of the Sun at three different solar radii, highlighting the variations in w ν (gray), κ ν (black), and some of the leading individual element contributions κ ν,i (red, green, blue) to the RMO determination (Eq. 30), as a function of radius. This figure was made assuming the GS98 solar abundances and using OPLIB opacities (Colgan et al. , 2016), and is included primarily for illustrative purposes. As shown in the figure, the solar opacity varies considerably in both magnitude (note the log axes) and shape across different solar radii. Moreover, each element contributes distinctively to the overall monochromatic opacity, with either a dominant continuous opacity spectrum,  \nFIG. 11 Monochromatic opacity of the Sun at three different solar radii as indicated. The temperature and electron density at each solar radius are displayed, as well as some of the major elemental opacity contributions at these conditions. Note that not all contributions are plotted for clarity. The percentages of three elements displayed indicate the leading elemental contributors to the Rosseland mean opacity (RMO) at these conditions. These are roughly quantified by calculating how much the RMO is reduced if each element is completely removed. This figure was made using the GS98 solar abundances and OPLIB opacities, and is mainly intended to show the spectral complexity of the solar opacity.  \n<!-- image -->  \nor strong lines that arise from bound-bound transitions. If there is an error in the calculated solar opacity κ ν , it would stem from inaccuracies in the calculated element opacities and/or abundances. Additionally, the error is expected to be a complex function of photon energy and would vary at each radius due to changes in temperature and density. As a result, it is not possible to provide a single factor for uncertainty or correction. It is crucial to assess the accuracy of the intricate spectral features depicted in Fig. 11 in order to better understand the nature of the errors.  \nThese complex spectral features arise from microscopic  \nphysics at the atomic level - that is, the absorption of radiation through quantum mechanical processes. If one knows the atomic energy levels and transition probabilities of all the relevant atomic states (of each atom or atomic ion of each material in the Sun), then the opacity of that material may be computed. 12 Since atomic structure and transition probabilities can be predicted (under various approximations), starting from the Schrödinger (or Dirac) equation, this seems like an achievable goal. However, complications quickly arise from a multitude of factors, such as: how to deal with the infinity of atomic states that may be populated in each atom or atomic ion; how to account for the perturbations caused by the plasma environment on the atomic states; and how to handle the large quantities of data that arise with such calculations, particularly for multi-electron atoms and ions. In solar modeling, one requires the opacity of all relevant elements (around 30) at all (plasma) conditions that are found at all radii throughout the solar evolution. Thus, the opacity must be tabulated as a function of species, plasma temperature, plasma density, and photon energy.  \nNevertheless, steady progress has been made in determining the opacity of solar mixtures over the last five decades. In the 1980s, insufficient accuracy in stellarenvelope opacity was hypothesized through multiple astrophysical puzzles (Simon, 1982), and opacity models were subsequently improved by incorporating more detailed atomic physics and a more complete set of states (Berrington et al. , 1987; Iglesias and Rogers, 1996; Magee et al. , 1995; Seaton, 1987). The improved accuracy was confirmed through a few benchmark experiments (Davidson et al. , 1988; Perry et al. , 1991; Springer et al. , 1992; Winhart et al. , 1996) and improved agreement with some astrophysical observations (Daszyńska-Daszkiewicz and Walczak, 2009, 2010; Iglesias and Rogers, 1991). However, the calculated solar opacities are not experimentally validated at most radii, and in particular more deeply into the Sun where the conditions (temperature and density) are more extreme.  \nIn the last decade, experimental methods have steadily improved, in concert with the availability of powerful xray sources, such as the Sandia Z facility. Monochromatic opacities were successfully measured at multiple conditions relevant to solar convection-base conditions (Bailey et al. , 2015, 2007). Systematic experimental studies performed on chromium, iron, and nickel helped narrow down hypotheses for the discrepancies (Nagayama et al. , 2019). All of these experimental results provided essential clues for testing various approximations for so-  \nlar interior opacity calculations; however, the reported model-data disagreements also raised significant controversy. This led to various theoretical investigations and the development of independent experimental methods (Opachich et al. , 2022; Perry et al. , 2020). Ongoing theoretical and experimental investigations are crucial for accurately interpreting the reported disagreements and to reach a consensus in this field.  \nIn this section, our aim is not to provide a consensus but rather to summarize the history and challenges associated with solar opacity calculations and experiments, as well as provide an overview of the current status of stellar opacity investigations.', 'B. Data sets used for the present review': 'The 2 H( p , γ ) 3 He reaction has a Q value of 5.5 MeV and proceeds through the radiative capture of a proton on deuterium. Different experimental approaches were followed to measure its cross section. Tišma et al. (2019) irradiated deuterated titanium targets with a proton beam and detected the γ -rays with two high-purity germanium (HPGe) detectors placed at different angles. The final S-factor is provided at four energies in the 47 - 210 keV range, with approximately 15% statistical uncertainty. More recently, the LUNA Collaboration performed a measurement in the energy range of 32 -263 keV at the underground in the Gran Sasso Laboratories, exploiting the six orders of magnitude suppression of the cosmic radiation background (Cavanna and Prati, 2018; Ferraro et al. , 2021). A windowless deuterium gas target was used and the γ -rays emitted by the 2 H( p , γ ) 3 He reaction were detected by a large HPGe detector at 90 · with respect to the beam axis. Great care was taken to minimize all sources of systematic uncertainties in the S-factor overall uncertainty at the 3% level (Mossa et al. , 2020a,b). These new results provided stringent constraints on cosmological parameters obtained by comparing the precise primordial deuterium abundance predictions of the standard BBN model with astronomical observations (Cooke et al. , 2018). A deeper discussion of the LUNA results and their implications can be found in (Moscoso et al. , 2021; Pisanti et al. , 2021; Pitrou et al. , 2021; Yeh et al. , 2021) 6  \nFinally, a new measurement was performed at the Helmholtz-Zentrum Dresden-Rossendorf in the 3001000 keV energy range using implanted deuterium targets on tantalum backings and two HPGe detectors (Turkat et al. , 2021). The resulting S-factors show ≈ 1 -2 σ tension with the analysis of the LUNA results extrapolated to E /greaterorsimilar 300 keV (Mossa et al. , 2020b). However, they are affected by large systematic uncertainties. A new measurement of the 2 H( p, γ ) 3 He reaction is planned at the Felsenkeller laboratory in Germany (Bemmerer et al. , 2019).  \nIn addition to the new accelerator-based results, two recent sets of measurements (Mohamed et al. , 2022; Zylstra et al. , 2020) have also been performed using the inertial confinement fusion plasma-based platform (Gatu Johnson et al. , 2023, 2017; Mohamed et al. , 2022), which has recently begun to be exploited for this type of work (e.g., Casey et al. , 2017; Zylstra et al. , 2016). Both measurements, which were performed at the OMEGA laser  \nfacility (see section XIV.D), used laser-driven implosions of spherical plastic-shell capsules filled with H 2 D 2 gas, and measured the emitted γ -rays using a gas-Cherenkovdetector (Herrmann et al. , 2014) that was calibrated applying the technique described in Zylstra et al. (2019). The initial experiment (Zylstra et al. , 2020) obtained good statistics at an energy of 16 keV by making several repeated measurements, with a final statistical uncertainty of 6% and a systematic one of 17% (dominated by uncertainty in the absolute calibration of the detector). The second experiment (Mohamed et al. , 2022) obtained data at three different energies in the 17-37 keV region, with comparable systematic uncertainty, but with larger statistical uncertainty due to fewer repeated measurements. The results obtained on this unique platform agree within error bars with the accelerator-based measurements.', 'C. Theoretical studies': "Nuclear reactions of astrophysical interest in general, and the 2 H( p , γ ) 3 He in particular, are of great importance in nuclear theory because the available experimental data can be used to test the adopted theoretical framework. The 2 H( p , γ ) 3 He reaction has the great advantage of involving only A ≤ 3 nuclear systems, and can be addressed with a microscopic ab initio study. This means that the nuclear systems involved in the process are viewed as made up of A nucleons, interacting among themselves and with the external electromagnetic probes. Within such an approach, the following ingredients are essential for the calculation: (i) realistic models for the nuclear interactions and currents, possibly rooted in Quantum Chromodynamics (QCD); (ii) a numerical technique able to solve the A -body bound and scattering state problem, including the Coulomb interaction without approximation. Such a technique is usually referred to as an ab initio method. The agreement (or disagreement) between ab initio theoretical predictions and experimental data represents a validation (or indicates the necessity of improvement) mostly for ingredient (i), i.e. are the models of the nuclear interactions and currents accurate within the precision specified by the ab initio method? This is why few-nucleon reactions can be used as an 'ideal' laboratory, where the ab initio framework can be stringently tested in systems under more theoretical control.  \nThe most recent ab initio calculation of the 2 H( p , γ ) 3 He reaction is that of Marcucci et al. (2016). Here the paircorrelated Hyperspherical Harmonics ab initio method was used to calculate the A = 3 initial scattering and final bound state wave functions (see Kievsky et al. (2008) and Marcucci et al. (2020) for details). The nuclear interaction model adopted to describe the A = 3 nuclear state in Marcucci et al. (2016) consists of a two-nucleon term, the Argonne v 18 (AV18) potential (Wiringa et al. , 1995),  \naugmented by a three-nucleon contribution, the Urbana IX (UIX) potential (Pudliner et al. , 1995). The AV18 potential can reproduce the large two-nucleon database with a χ 2 /datum ∼ 1 (Wiringa et al. , 1995), while the combination AV18/UIX can describe quite accurately the properties of 3 He, the spectra of light p-shell nuclei (Pieper and Wiringa, 2001), and p -d scattering observables (see for instance Wood et al. (2002)). The electromagnetic current operator used in Marcucci et al. (2016) includes, in addition to the non-relativistic one-body operator, two- and three-body terms required by gauge invariance in a system of interacting particles. These terms were constructed in Marcucci et al. (2005). The model was then tested against various electromagnetic observables, to access the quality of its predictions. As a potential model, however, there is no systematic procedure for assigning uncertainties for observables whose values are unknown.  \nThe results of Marcucci et al. (2016) have been found to be about 10% higher than the experimental data of Mossa et al. (2020b). Given the lack of a procedure for quantifying errors in calculations based on phenomenological interactions and currents, it is difficult to access the significance of this discrepancy. This leads us to make several recommendations to the theory community. First, the 2 H( p , γ ) 3 He reaction should be studied within the framework of chiral effective field theory, which has reached a degree of accuracy and predictive power comparable to potential-based phenomenology. This approach is formally ab initio and rooted in QCD, and as an operator expansion will provide an estimate of the theoretical uncertainty. Work along this line is currently underway. Second, the angular distribution of the 2 H( p , γ ) 3 He capture reaction, and possibly also polarization observables, should be both calculated and measured. This would be valuable even if measurements were limited to higher energies, where they are less difficult. Such measurements would provide a further test of the predictive power of the theory.", 'D. Phenomenological and Bayesian Analyses': 'In the energy range of astrophysical interest, the 2 H( p , γ ) 3 He reaction does not have any resonance or coupled channels that can give rise to non-trivial energy dependence, and the S-factor can be modelled by a low-order polynomial in energy (Fowler et al. , 1967). Moreover, the recent results from LUNA (Mossa et al. , 2020b), combined with previous measurements, place stringent constraints on the cross section, reducing the uncertainty over the range of interest to Solar fusion and BBN (Casella et al. , 2002; Mossa et al. , 2020b; Pisanti et al. , 2021; Pitrou et al. , 2021; Yeh et al. , 2021).  \nA significant change since SF II has been the wide adoption of Bayesian analysis methods to evaluate ther-  \nTABLE III Logarithm of the Gaussian approximation to the Bayes factor, and corresponding weight, Eq. (B4), for each model under investigation.  \nmonuclear reaction rates and provide more rigorous uncertainty estimates in a statistical sense (Iliadis et al. , 2016). This approach has been followed in the analysis presented here, which closely follow work on the 2 H( p , γ ) 3 He reaction reported in Moscoso et al. (2021), and references therein. The mathematical details of the Bayesian method will be found in Appendix B: Here we focus on the application to 2 H( p , γ ) 3 He by the authors cited above.  \nBoth the ab initio prediction from Marcucci et al. (2005) as well as a third-order polynomial in the energy were used to constrain the data, where we define an n th -order polynomial as  \nS n ( E ; λ ) = n ∑ i =0 λ i E i . (10)  \nAs noted in the literature, the energy dependence of Marcucci et al. (2005) is more reliable than the absolute normalization. Furthermore, the updated prediction in Marcucci et al. (2016), as noted above, is ≈ 10% larger than that of Marcucci et al. (2005). Therefore, Moscoso et al. (2021) modeled the prediction of Marcucci et al. (2005), denoted S nuc ( E ) , as  \nS ( E ; λ ) = aS nuc ( E ) + b , (11)  \nwhere a and b are an unknown scale factor and offset to be determined in the analysis. While S nuc ( E ) was not determined with theoretical uncertainty, model uncertainty is introduced through the parameters a and b . The resulting mean values and uncertainties of S ( E ; λ ) determined from Eq. (11) were found to be comparable to those with the third-order polynomial.', 'E. Summary and Recommendations': "In this review, we take the further step of applying Bayes Model Averaging to a number of reasonable models that describe the data, which systematically captures additional theoretical 'model selection uncertainty'. Detailes of the Bayesian analysis and model averaging procdure are presented in Appendix B. The models we explore  \ninclude Eq. (11) with both a and b , as done in Moscoso et al. (2021), as well as setting b = 0 , and then we consider polynomials in energy of order n = 3 , 4 , 5 , 6 . 7  \nWith such a Bayes model averaging, we can quantitatively compare and contrast the polynomial parameterizations of 2 H( p , γ ) 3 He along with ab initio results predicted by Marcucci et al. (2016, 2005), including the extra variance that arises from this set of reasonable models, e.g., using polynomials of different order, as well as the phenomenological model of Eq. (11). Results are listed in Table III. The first column lists the model (i.e., polynomial or ab initio ), the second column the natural logarithm of the Gaussian approximation to the BF, and the third column the corresponding weight in the model averaging. From this model averaging, the resulting prediction for the 2 H( p , γ ) 3 He S-factor at a few representative energies is provided in Table IV, where the first uncertainty arises from the first term in Eq. (B3) and the second uncertainty is from the second term, which we denote as model selection uncertainty . The resulting Bayes model average prediction of the S-factor over the entire kinematic range considered is depicted in Fig. 5, with the gray band representing the 68% coverage probability. For this particular reaction, it is interesting to note that the Bayesian analysis strongly favors the phenomenological models of Eq. (11) over the polynomial approximations. One reason for this might be that the scaled ab initio models have only one or two free parameters, and the energy dependence of S nuc ( E ) given in Marcucci et al. (2016, 2005) is sufficient to accurately describe the various data sets. On the other hand, polynomial approximations are disfavored as they require more parameters to capture the energy dependence. Also of note, in the model of Eq. (11) with b = 0 and using Marcucci et al. (2016) for S nuc ( E ) , the scale factor is given by  \na = 0 . 921(19) , (12)  \nindicating that the prediction in Marcucci et al. (2016) overestimates the 2 H( p , γ ) 3 He data by 7.9%, consistent with expectations noted above. In comparison, the third order polynomial fit predicts values of S ( E ) that are 1 σ higher at E = 0 and 2 3 σ lower at E = 91 keV as compared to those in Table IV.  \nThe analysis in Table III and Fig. 5 can be reproduced with the code linked in Table IV. The code also can provide a prediction of the mean value and uncertainty of S 12 ( E ) at any energy over the same kinematic range.  \nFIG. 5 S-factor determined via Bayes model averaging (BMA) of the 2 H( p , γ ) 3 He data sets (Casella et al. , 2002; Ma et al. , 1997; Mossa et al. , 2020b; Schmid et al. , 1997; Tišma et al. , 2019; Turkat et al. , 2021; Warren et al. , 1963) analysed as described in the text. The left panel shows the original data with statistical uncertainties only. The right panel shows the data after normalizing by BMA systematic uncertainty posteriors of f -1 D for each data set, and the uncertainties correspond to the statistical, systematic and extrinsic uncertainties all added in quadrature. In most cases, the quoted statistical uncertainty is the dominant uncertainty. The residual is defined as ∆ S/S = ( S data -S BMA fit ) /S BMA fit and is plotted for the original data with statistical uncertainty only on the left, and the adjusted data as described above, on the right.  \n<!-- image -->  \nTABLE IV The S-factor of the 2 H( p , γ ) 3 He at a few selected energies determined from data using Bayesian model averaging as described in the text. The first uncertainty is the statistical and the second is the model selection uncertainty. The value at any energy in the fitted range can be obtained by running the analysis provided at https://github.com/nrp-g/leaner .", 'V. The 3 He ( 3 He , 2 p ) 4 He reaction ( S 33 )': 'The 3 He( 3 He, 2 p ) 4 He reaction terminates the pp-I chain. The ratio of its rate to that of the 3 He( α, γ ) 7 Be reaction controls the branching to the pp-II and pp-III chains, so that in SF I increasing S 33 was discussed as a potential solution to the Solar Neutrino Problem. Subsequent experiments, notably a very low energy measurement at the LUNA 50 kV accelerator deep underground in Gran Sasso (Bonetti et al. , 1999) and a complementary experiment at somewhat higher energies (Kudomi et al. , 2004) ruled out such an increase in S 33 , as summarized in SF II.', 'A. Shape of the particle spectrum': 'Since no new absolute measurements of S 33 have been reported since SF II, we consider the same four experiments (Bonetti et al. , 1999; Junker et al. , 1998; Krauss et al. , 1987; Kudomi et al. , 2004). However, there is new information on the energy spectrum of the emitted protons. This spectrum has recently been measured using Inertial Confinement Fusion (ICF) plasmas for a Gamow peak energy of 165 keV (Zylstra et al. , 2017). The results show significant structure, indicating the presence of a sequential reaction mechanism passing through the unbound ground state of 5 Li . This spectrum is impor-  \ntant for all of the S 33 measurements since they determined cross sections by detecting only the protons above an energy threshold.  \nThe efficiency correction by which Krauss et al. (1987) account for the threshold is not well documented, but the more recent measurements (Bonetti et al. , 1999; Junker et al. , 1998; Kudomi et al. , 2004) utilized the genbod event generator (James, 1968). It employs a simple reaction model without final state interactions, the Pauli principle, or Coulomb effects that are important near the spectrum endpoint. The only angular correlations are those required by energy and momentum conservation, and together these simplifications give simple ellipses for the singles energy distributions of the emitted nuclei. Published proton spectra obtained using accelerator beams do exist (Fig. 2 of Dwarakanath (1974) and Fig. 3 of Krauss et al. (1987)), and although they are not corrected for instrumental effects, they do not appear to be well described by ellipses.  \nThe 3 He( 3 He, 2 p ) 4 He reaction may proceed via several sequential mechanisms, including p + 5 Li (with 5 Li in its ground or first excited state) and di-proton + α (where the di-proton is two correlated protons in a singlet state) (Brune et al. , 2015). More complicated threebody decay channels, sometimes called direct decays, are also possible. Only a relatively narrow intermediate state (here, the 1 MeV wide 5 Li ground state) could produce a peak in the energy spectrum, and precise classification of the reaction mechanism is in general both experimentally and theoretically ambiguous. When coincident detection of reaction products is used [as in Bonetti et al. (1999)], possible angular correlations between the reaction products also matter. A reaction through the 3 / 2 -ground state of 5 Li would emit the second proton preferentially either along or opposite the direction of the first proton, while di-proton emission would tend to send both protons in the same direction (Brune et al. , 2015).  \nThe solid curves in Fig. 6 show two different Rmatrix models of the proton energy spectrum at E c . m . = 165 keV, fitted to the spectrum measured by Zylstra et al. (2017) for E p /greaterorsimilar 6 MeV. The measured spectrum differs significantly from the elliptical spectrum of genbod . Based on their Fig. 6, the experiment of Junker et al. (1998) had a detection threshold of about 5 MeV proton energy. The measurements by Bonetti et al. (1999) required a coincidence between two detectors, with a detection threshold of 2 MeV proton energy in each detector. Finally, Fig. 19 of Kudomi et al. (2004) indicates a detection threshold of about 4 MeV proton energy. We have estimated their sensitivities to the assumed proton spectrum by integrating the curves shown in Fig. 6 above energy thresholds of 2, 4, and 6 MeV. These integrals vary from 3% below to 6% above the result from an elliptical spectrum, depending on the specific threshold and the assumed spectrum.  \nFIG. 6 Calculated proton energy spectra in the c.m. frame for the 3 He( 3 He, 2 p ) 4 He reaction at E c . m . = 165 keV. The solid red and blue curves are models 1 and 2, respectively, from Fig. 3(a) of Zylstra et al. (2017). The black dashed curve is the elliptical spectrum. The spectra are normalized to unit area.  \n<!-- image -->', 'B. Recommendation': 'Based on these investigations, our recommended S 33 fit is the one from SF II, but with an additional 4% systematic uncertainty typical of the 3% to 6% corrections to the extrapolated proton spectra estimated above to take the uncertain spectral shape into account. This gives  \nS best 33 ( E ) = 5 . 21 -4 . 90 ( E MeV ) +11 . 21 ( E MeV ) 2 MeV b δS 33 ( E ) = [ 0 . 118 -1 . 516 ( E MeV ) +14 . 037 ( E MeV ) 2 -15 . 504 ( E MeV ) 3 +71 . 640 ( E MeV ) 4 ] 1 / 2 MeV b .  \nOnly the larger constant term in the uncertainty differs from the previous evaluation, and it is due to the spectral shape discussed above.  \nNew measurements of this reaction could provide more accurate absolute cross sections by using lower energy thresholds and improve our understanding of the proton energy spectra and angular correlations. The analysis of new measurements should use Monte Carlo simulations considering a variety of plausible energy spectra and angular correlations, in order to estimate the sensitivity to these effects. We note finally that the proton energy spectrum may depend on E c . m . , as this has been found to be the case for neutrons from the mirror reaction 3 H( t, 2 n ) α (Gatu Johnson et al. , 2018).', 'VI. The 3 He ( α, γ ) 7 Be reaction ( S 34 )': 'The 3 He( α, γ ) 7 Be reaction proceeds via non-resonant capture to the ground and 429 keV first excited states of 7 Be. It has been studied experimentally using three main methods, with references given in subsections VI.A and VI.C: First, by detecting at least two of the three γ -rays from the reaction and taking the angular correlation with the alpha beam direction into account (promptγ method). Second, by detecting the induced 7 Be (Tilley et al. , 2002) radioactivity (activation method). Third, by counting the 7 Be recoils (recoil method).', 'A. Previous S 34 recommendation in SF II': 'The 3 He( α, γ ) 7 Be S-factor recommended in SF II was developed in multiple steps. First, a model was selected for the shape of S ( E ) , based on existing nucleon-level calculations (Kajino, 1986; Nollett, 2001). A rescaling of the curve was fitted to data, but only at E ≤ 1.002 MeV center-of-mass to minimize the role of short-distance physics in the models.  \nIn SF I, a possible systematic discrepancy between data from the two methods previously used was discussed. By the time of SF II, in-depth studies from two groups using both activation and prompt methods were available, from LUNA (Bemmerer et al. , 2006; Confortola et al. , 2007; Gyürky et al. , 2007) and Seattle (Brown et al. , 2007) groups. These studies did not find any discrepancy between activation and promptγ data in direct comparisons. SF II opted to limit the fitting to the data by the activation and recoil methods. The promptγ data were left out because of their somewhat larger common mode errors, concerns about how well the γ -ray angular distribution was known, and avoidance of the correlated errors between activation and prompt data from the same experiment.  \nIn SF II, only data published after 1998 were included. The SF II recommended value was S 34 (0) = 0 . 56 ± 0 . 02 expt ± 0 . 02 theor , based on the data by the Weizmann (Singh et al. , 2004), LUNA (Bemmerer et al. , 2006; Confortola et al. , 2007; Gyürky et al. , 2007), Seattle (Brown et al. , 2007), and ERNA (di Leva et al. , 2009) groups.', 'B. Theory progress on S 34': "Significant theoretical work on this reaction has occurred since SF II, but the basic understanding of its mechanism remains unchanged from the 1960s (Tombrello and Parker, 1963). It is dominated by external non-resonant capture into the two bound states, and most of the dipole strength at low energy arises beyond the range of nuclear interaction. The external  \ncapture part of the cross section is determined by the ANCs of the bound states and by scattering phase shifts; near threshold, most of the strength lies at ∼ 5 -20 fm (Neff, 2011). In models with explicit wave functions the shorter-range strength largely cancels out due to effects of nucleon-exchange antisymmetry. All models feature a shallow minimum of S ( E ) near 1.25 MeV, where capture from d -waves becomes comparable to that from s -waves.  \nThe first fully ab initio calculation (Neff, 2011) appeared just after the SF II analysis concluded; it used the fermionic molecular dynamics (FMD) method and a softened representation of the Argonne v 18 interaction (Wiringa et al. , 1995). It agrees well with both the scale and the energy dependence of the modern S-factor data, and it is very close to the energy dependence assumed in SF II. Neff's elastic-scattering phase shifts also agree well with experiment. Notably, Neff found that the dipole strength distribution departs significantly from pure external capture at 3 He4 He separations as large as 9 fm, compared with 3-4 fm in potential models.  \nAnother ab initio model (Dohet-Eraly et al. , 2016; Vorabbi et al. , 2019) has also appeared, based on the nocore shell model with continuum (NCSMC) method and the chiral interaction of Entem and Machleidt (2003), 'softened' by the similarity renormalization group (SRG) procedure. Freedom to choose the SRG stopping point allowed exact reproduction of the 7 Be breakup energy, which corrects for the main effect on the external capture of omitting the three-nucleon potential. The results agree well with the overall scale of the modern data, but their energy dependence and phase shifts depart from experiment, possibly due to the omission of explicit three-body forces.  \nAn important advance since SF II has been the application of halo effective field theory (halo EFT) methods to astrophysical capture reactions (Bertulani et al. , 2002; Higa et al. , 2018; Premarathna and Rupak, 2020; Zhang et al. , 2020). EFTs are valid for systems with a natural separation between the momentum scales Q probed in low-energy experiment (e.g., corresponding to a binding energy or a projectile energy) and the much larger momentum scale Λ where the low-energy degrees of freedom are no longer valid: the word 'halo' here refers to nuclei with only shallow bound states. In halo EFT the only degrees of freedom are the initial- and final-state nuclei treated as point particles, plus photons. For S 34 , Λ ∼ 200 MeV corresponds to the momentum needed to separate a proton from 3 He or to excite 4 He; the thresholds to separate the two 7 Be bound states into 3 He and 4 He correspond to momenta Q = 71 . 4 MeV and 60.9 MeV.  \nGiven a sufficient separation of scales, one constructs a Lagrangian that respects the system's symmetries and known qualitative features systematically, organized by powers of Q/ Λ . This series can be truncated (at leading order or LO, next-to-leading order or NLO, next-tonext-to-leading-order or NNLO, etc.), and the precision  \nof the resulting theory depends on the sizes of the omitted terms. The precision of a calculation and the energy where it breaks down can be estimated by assuming coefficients of the first omitted term to have a 'natural' size. Coupling constants of the Lagrangian must be fitted to data, and consistency of the power counting scheme (identification of powers of Q/ Λ for the main operator terms of a system) is tested by whether the fitted constants have natural sizes. Low-order coupling constants in a halo EFT can often be identified with familiar quantities like ANCs, scattering lengths, and effective ranges; the description of elastic scattering in halo EFT reproduces the (Coulomb-modified) effective-range expansion (Higa et al. , 2008; Kong and Ravndal, 2000; Poudel and Phillips, 2022). When a separation of scales exists, halo EFT is well-suited to data extrapolation because it avoids the tacit, hard-to-test, and unavoidable prior assumptions present in the short-range parts of models based on explicit wave functions.  \nTwo groups have studied S 34 in halo EFT (Higa et al. , 2018; Premarathna and Rupak, 2020; Zhang et al. , 2020). The sizes of the Coulomb interaction and the large s -wave scattering length make the correct power counting tricky to establish for this system. The work of Higa et al. (2018) and Premarathna and Rupak (2020) includes a careful examination of possible power-counting schemes and strict adherence to a power counting once established, up to terms of NLO or NNLO, respectively, in two different power-countings. These authors fitted EFTs to S-factor data, and they examined effects of including or excluding elastic scattering constraints in their fits. When they included scattering phase shifts from Boykin et al. (1972) in their fits, they concluded that the large-scattering-length power counting of the NLO theory was favored. This result promotes two-body currents (i.e., contributions not equivalent to external nonresonant capture) to leading order in the theory. The d -wave contribution that becomes important above 1 MeV first appears at NLO. In addition to data-fitting errors, these theories have errors estimated to be 10% from EFT truncation.  \nThe work of Zhang et al. (2020) followed a different approach to power counting up through NLO. At E < 2 MeV, their derived expression for S 34 ( E ) is essentially the same as that from the Premarathna and Rupak (2020) NLO theory. Zhang et al. (2020) also developed some ad hoc (i.e., not systematically developed) higher-order EFT terms, referred to as partial-N4LO, to test for their impact on the fitting. The additional terms proved not to be required by the data and did not improve the fit; this result was taken to indicate that corrections from omitted terms are not large compared with experimental errors below 2 MeV. Scattering data were not considered in this work apart from very broad priors on scattering length and effective range; the correlated errors in the S-factor data were taken into account.  \nLike halo EFT, the phenomenological R-matrix approach avoids a model of nuclear interactions and uses a systematic parameterization to fit data (Descouvemont and Baye, 2010). At the time of SF II, S 34 had been the subject of very little R-matrix fitting, apparently consisting only of the simple treatment in Descouvemont et al. (2004) that focused on BBN energies. After SF II an Rmatrix analysis using the AZURE2 code (Azuma et al. , 2010; Uberseder and deBoer, 2015) was carried out on both elastic scattering and S-factor data in conjunction with the Notre Dame experiment (Kontos et al. , 2013), and a more elaborate analysis using Monte Carlo sampling to estimate errors was later reported in deBoer et al. (2014). In the latter analysis both the fitted value and error bars were heavily influenced by the numerous scattering data of Barnard et al. (1964). Very recently, the BRICK software package has been constructed to carry out Bayesian parameter estimation for AZURE2 and applied to both capture and elastic scattering at all energies in the 7 Be system by Odell et al. (2022b). In this work it was shown that markedly different S 34 (0) values result from inclusion or not of the older scattering data (Barnard et al. , 1964) alongside the very recent SONIK scattering data of Paneru et al. (2024).  \nA small amount of additional theoretical work in more traditional frameworks has appeared since SF II. This includes fits of potential models to S-factor data (Tursunov et al. , 2018, 2021) and some calculations of the RGM type (Solovyev and Igashov, 2017, 2019).", 'C. Experimental progress on S 34': "Since SF II, five new experiments have been reported: Four from the Madrid and ATOMKI groups by the activation technique, at relatively high center of mass energies (Bordeanu et al. , 2013; Carmona-Gallardo et al. , 2012; Szücs et al. , 2019b; Tóth et al. , 2023), and one from the Notre Dame group using the promptγ method (Kontos et al. , 2013).  \nFollowing the approach adopted in SF II, we again only use the recoil and activation data for the S 34 fits below. The γ -ray angular distribution is not known experimentally (Brown et al. , 2007), and the resultant uncertainty increases the common-mode error for the promptγ studies somewhat. (An experiment to address previouslyraised concerns about angular distribution (Brown et al. , 2007) has recently concluded at Felsenkeller Dresden. The data suggest a higher than expected anisotropy but are so far only available in the form of a PhD thesis (Turkat, 2023). If confirmed, they may lead to corrections of a few percent for some of the in-beam experiments where only one angle was instrumented.) In addition, all of the 'modern' works reporting promptγ data except for Notre Dame (Kontos et al. , 2013) also include data obtained with other methods, leading to partial cor-  \nations between data sets that would complicate fitting.  \nFollowing this restriction, our data selection proceeds as follows. First, the four data sets previously used in SF II are carried over here: Weizmann (Singh et al. , 2004), LUNA (Bemmerer et al. , 2006; Confortola et al. , 2007; Gyürky et al. , 2007) (only the activation data), Seattle (Brown et al. , 2007) (only the activation data), and ERNA (di Leva et al. , 2009).  \nTwo of the archival data sets excluded by the 1998 cutoff date in SF II merit some further discussion here: The data of Osborne et al. (1982, 1984) consist of two points measured by activation in a 3 He gas cell. However, their uncertainties are not separated into statistical and systematic components. Including them would have required an uncertain guess on how to divide the error bars, and the fit result would barely change, so we left them out. The work by Robertson et al. (1983) consists of one data point that was obtained by averaging two separate activation measurements with a 3 He and a 4 He gas cell, respectively. We do not use this result because we lack details to verify the background subtraction, an issue that was raised during our meeting by one of its authors. We also choose not to use the activation study by Volk et al. (1983). In that experiment there were thick entrance foils and the 4 He beam was completely stopped inside a high-pressure 3 He gas cell. This gave an integrated measurement over a wide energy range, so that the analysis depends strongly on the assumed shape of the S 34 curve, and the result was reported only as an extrapolated S 34 (0) .  \nWe now consider the new data since SF II. An activation experiment in Madrid (Carmona-Gallardo et al. , 2012) used a 3 He beam incident on a 4 He gas cell, reporting three data points. Another activation experiment was reported by the ATOMKI group, using a 4 He beam on 3 He gas cells (Bordeanu et al. , 2013). Two higher-energy campaigns at ATOMKI were again performed using the activation method. These latter data are at E = 2.5-4.4 MeV (Szücs et al. , 2019b) and E = 4.3-8.3 MeV (Tóth et al. , 2023), respectively, above the energy range suitable for halo EFT, and therefore not included.  \nA detailed study by the Notre Dame group reported 17 data points (Kontos et al. , 2013), using the primary γ -ray from ground state capture and the secondary γ -ray from the deexcitation of the 429 keV first excited state of 7 Be. No activation data are reported in that work, and the γ -ray detector was placed at just one angle, 90 · . Since only promptγ data are reported, we did not include this data set. In order to test the effects of this decision, we repeated some of our fits (see below, Section VI.D) with the modern promptγ experiments included: the 17 points from Kontos et al. (2013), 3 promptγ points from LUNA (Confortola et al. , 2007) and 8 from Seattle (Brown et al. , 2007). The extrapolated S-factor changed by less than 1%, well within the error bars for the recommended S 34 .  \nFinally, an indirect experiment using the 6 Li( 3 He,d) 7 Be reaction and the ANC technique has recently been reported (Kiss et al. , 2020). It is left out of our fits due to the additional normalization and theory uncertainties involved in determining an ANC from a transfer experiment, which are larger than the errors in the S-factor measurements used for the present fits. All the data used in the present fits are summarized in Table V.  \nAs in SF II, we model data uncertainties as consisting of a component that is independent for each point and a common-mode component that applies to all data from a given experiment as a multiplicative factor. This separation is well-documented for all of the modern data and is also shown in Table V. Except at the lowest energies, the common-mode error typically dominates.", 'D. Data fitting': "We base our recommended S 34 on fits to the halo EFT and R-matrix parameterizations discussed above. These avoid tacit assumptions present in potential-models and conceptual difficulties involved in combining ab initio constraints with data. The fits presented here differ from the previously published fits of Premarathna and Rupak (2020), Zhang et al. (2020), and Odell et al. (2022b) mainly in the uniform use of the agreed-upon capture data and uncertainties from Section VI.C across all fits. We restricted fitting to the E < 2 MeV range of validity for the NLO halo EFT expressions. Despite uniform handling of capture data, the fits in each framework handle scattering inputs differently for reasons discussed below.  \nIn addition to the total capture cross section and scattering data, the fitted data also include branching ratios for capture into the two 7 Be bound states, taken from Brown et al. (2007), di Leva et al. (2009), Confortola et al. (2007), and Kontos et al. (2013). These are necessarily from promptγ experiments and suffer from the concerns about angular distribution discussed in Section VI.C. However, their inclusion simplifies the fitting considerably by breaking parameter degeneracies between ground- and excited-state transitions (especially in the fitted ANCs), probably without strong impact on S 34 (0) .  \nIn constructing fits we split common-mode and pointto-point errors and 'float the norms' of data sets using the cost function  \nχ 2 = N set ∑ α =1 N ∑ i =1 ( y i -µ i /s α ) 2 σ 2 α,i + N set ∑ α =1 (1 -s α ) 2 ω 2 α (13)  \nto describe goodness of fit (D'Agostini, 1994).  \nA rescaling factor s α is fitted for each data set α , with its deviation from unity penalized by the common-mode errors ω α given in Table V. The index i sums over all  \nTABLE V Experimental data used for the S 34 fit. See text for details. For each experimental data set, the rescaling factors s α are determined for the Higa et al. (2018), Zhang et al. (2020), and BRICK ((Odell et al. , 2022b)) without or with the inclusion of elastic scattering.  \npoints within a given data set; y i is a measured cross section, µ i is a predicted cross section, and σ α,i is the point-to-point error of the i th point in data set α .  \nWe carried out both frequentist fits that minimize χ 2 and Bayesian fits based on the posterior probability distribution of a likelihood function computed from exp( -χ 2 / 2) . In the Bayesian fits parameters, extrapolated S 34 (0) , and their errors were determined by Monte Carlo sampling of the parameter space. The relatively large number of model parameters and the significant fitting degeneracies between some of them make Bayesian analysis a natural choice for finding best fit and confidence intervals for the multi-parameter models applied here. Two of the groups ran into serious difficulty with the frequentist fits due to parameter degeneracies and shallow local minima; during this study they were unable to produce frequentist fits in which they had confidence.  \nWe performed multiple fits in the NLO halo EFT of Higa et al. (2018) and Premarathna and Rupak (2020). These fits were carried out both with and without scattering constraints, which mainly impact S 34 (0) by removing parameter degeneracies that would otherwise leave the s -wave scattering length poorly constrained. The code base for this version of halo EFT incorporates scattering data through phase shifts. These were taken from the partialwave analysis of Boykin et al. (1972), which at low energy are based mainly on the data of Barnard et al. (1964).  \nOne set of fits for the Higa et al. (2018) halo EFT was produced by χ 2 minimization, proceeding in two steps: first LO parameters were fitted to E ≤ 1000 keV data, and then the results were taken as initial values in the search to minimize χ 2 over all parameters in the NLO theory for the full set of data. These fits gave S 34 (0) = 0 . 566 ± 0 . 025 keV b with the scattering constraint and S 34 (0) = 0 . 588 ± 0 . 015 keV b without (uncertainties being propagated in linear approximation using covariances and partial derivatives). Formally the EFT truncation error from stopping at NLO corresponds to an additional theoretical error that can shift S-factors by 10%. However, any fitted curve is constrained by low-energy data, which in some sense become effectively renormalization conditions of the field theory; the error  \non extrapolated S 34 (0) should probably be smaller than 10% by an amount that is hard to estimate.  \nA second set of fits to the Higa et al. (2018) halo EFT was carried out using Bayesian methods. Priors for the EFT parameters were developed based on previous experience, and data rescaling factors were incorporated as additional priors. The Bayesian results (including only experimental error) are S 34 (0) = 0 . 561 +0 . 017 -0 . 018 keV b including the phase shifts by Boykin et al. (1972) and S 34 (0) = 0 . 559 +0 . 018 -0 . 019 keV b excluding them, consistent with the χ 2 -minimization.  \nSearches of the parameter space to minimize χ 2 for the NLO halo EFT of Zhang et al. (2020) ran into difficulties with parameter degeneracy and local minima in the χ 2 surface; for this formalism we report only Bayesian results. No experimental information about scattering was used, but flat priors on scattering length and effective range were chosen over a 5 σ range around recent experiment. This fit differs from Zhang et al. (2020) mainly by excluding the Notre Dame data and including the ERNA activation data, and it gives S 34 (0) = 0 . 581 ± 0 . 016 keV b. The error associated with EFT truncation at NLO is estimated in this approach by separately fitting the partialN4LO theory discussed above. The result suggests that EFT truncation at NLO affects extrapolation from the data to threshold by ∼ 2 -3%.  \nOur R-matrix fits are based mainly on sampling Bayesian posterior probabilities with the BRICK code. We also produced frequentist fits, but we were unable to estimate their errors convincingly. The R-matrix fits used capture data both alone and in combination with the SONIK scattering data (Paneru et al. , 2024) - the latter being chosen because of concerns with the Barnard et al. (1964) data that are discussed in Odell et al. (2022b). Since BRICK fits elastic differential cross sections directly, it was not feasible to use the same phase-shiftbased scattering constraints as our halo EFT fits. The R-matrix fit that includes scattering data is essentially the 'CS' fit of Odell et al. (2022b), but with a restriction to only the capture data described in Table V, only scattering data below 2 MeV center-of-mass, and only the R-matrix parameters relevant below 2 MeV (no 7 / 2 -  \nTABLE VI Main results of the Bayesian fits performed on the S 34 data. The ANC refers to the sum of the ANCs for the ground and excited state transitions. See text for details.  \nlevel or radiative widths for d -wave background poles).  \nSeveral of the fits are compared in Fig. 7. There it is apparent that they all agree within ≤ 5% in the 0-2000 keV energy range, and within ≤ 4% in the astrophysically relevant range 0-500 keV. The fit to the Higa et al. (2018) NLO EFT lies in the middle of the range, so we adopt it as a reference in the lower panel of the figure. Relative to that fit, the fits to the NLO EFT of Zhang et al. (2020) and the R-matrix fit without scattering rise up as much as 4% higher at 0-200 keV.  \nThe most striking difference among the curves is between the BRICK R-matrix fit including the SONIK scattering data (Paneru et al. , 2024) and all the other fits. While the R-matrix fit with scattering finds rescaling factors somewhat closer to unity than the other fits do, its energy dependence in the 0-1000 keV range is qualitatively different, as is especially visible in the lower panel of Fig. 7 where residuals relative to the fitted Higa et al. (2018) EFT are shown. The BRICK fit without the scattering data is much closer to all the halo EFT fits, so the recent scattering data apparently have a large impact.", 'E. Recommended S 34 value': "Because the fit to the NLO EFT of Higa et al. (2018) gives the central result among those attempted here, we adopt it for our recommended S ( E ) . For ease of adoption, it is noted that this recommended S 34 curve can be empirically parameterized using the same shape as in SF II, by the numerical equation (following a customary form in the past literature (Adelberger et al. , 2011; Kajino et al. , 1987))  \nS 34 ( E ) = (0 . 5610 keV b) exp ( -0 . 5374 E ) (14) × [ 1 -0 . 4829 E 2 +0 . 6310 E 3 -0 . 1527 E 4 ] .  \nHere E is the center of mass energy in MeV. Equation (14) reproduces our recommended curve within 0.3% and is only applicable for E = 0-1600 keV.  \nPropagation of uncertainties from measurement to extrapolated S 34 (0) is relatively straightforward and unambiguous (at least in the Bayesian fitting). However, the uncertainty due to theoretical approximation is more  \n<!-- image -->  \nEFT - NLO  \net al.  \nData and fits, relative to Higa  EFT - NLO  \net al.  \nData and fits, relative to Higa  \nFIG. 7 Top panel: 3 He( α, γ ) 7 Be astrophysical S-factors from the experiments used for the fitting of the excitation function: Weizmann (Singh et al. , 2004), LUNA (Bemmerer et al. , 2006; Confortola et al. , 2007; Gyürky et al. , 2007), Seattle (Brown et al. , 2007), Madrid (Carmona-Gallardo et al. , 2012), and ATOMKI (Bordeanu et al. , 2013). The fit curves from two different EFT based fits (approaches based on Higa et al. (2018) and Zhang et al. (2020)) and from two R-matrix fits (BRICK Odell et al. (2022b), without or with elastic scattering data) are also shown. Bottom panel: Experimental data and fits, normalized to the Higa et al. (2018) based EFT fit. See Table V and text for details.  \n<!-- image -->  \ncomplicated, and error estimation on extrapolated quantities in EFTs remains an open area of research; estimates from the groups working on S 34 are discussed above. We estimate the overall theoretical uncertainty on our recommended S-factor to be the measured by the 4% dispersion among fitted curves; this is comparable to the more formal estimates.  \nOur recommendations for the zero energy astrophysical S-factor and the low-energy slope are then  \nS 34 (0) = 0 . 561 ± 0 . 018 exp ± 0 . 022 theor keV b (15)  \nS ' 34 (0) S 34 (0) = -0 . 54 ± 0 . 07 MeV -1 . (16)  \nThe given value of S ' 34 /S 34 applies over E = 0-500 keV. Its considerable uncertainty is given by the full span of slopes among the four fits discussed above, which is dom-  \nated by the inclusion of elastic scattering in one of the R-matrix fits. The S 34 ( E ) curve is quite straight at low energy in all models; its second derivative is consistent with zero and even difficult to compute for a given fit because of numerical cancellations.  \nThe first error bar for S 34 (0) is given by the fit uncertainty, conservatively using the largest value among the various fits, and it encodes the experimental uncertainty. The second error bar reflects the 4% dispersion among theoretical formalisms. The two errors can be added in quadrature for applications. Our recommendation is essentially the same as in SF II, but derived from completely different theoretical approaches that are much more directly connected to the data and much less dependent on model assumptions. It is also based on data over a wider energy range. The dispersion of S 34 and S ' 34 among models at the solar Gamow peak is illustrated in Table VI, along with some indication of how fitted scattering lengths a 0 and squared ANCs C 2 vary among frameworks. (The latter are highly degenerate with short-range parameters in the fitting; in the Higa et al. (2018) EFT they are sensitive to small variations in the p -wave effective ranges that are fitted instead of ANCs.)", 'F. Recommendations for future work on S 34': 'Here we present recommendations for further work that would improve understanding of the 3 He( α, γ ) 7 Be reaction. Our first recommendation for experiment concerns elastic 3 He+ 4 He scattering. New experiments over a wide energy range including at least part of the astrophysical range are needed to address the marked impact of the recent SONIK scattering data (Paneru et al. , 2024) on fitting, for independent confirmation. Data below 2 MeV would be most useful for consistent fitting of halo EFTs.  \nSecond, new measurements (besides the preliminary data of Turkat (2023)) are needed for the angular distribution of the emitted γ -rays. This would relieve concerns about including promptγ data, most notably the Notre Dame (Kontos et al. , 2013) that do not include activation measurements, in future fits. These experiments should also be designed to provide new angle-corrected branching ratios for capture into the two bound states.  \nOur third recommendation is a study of the astrophysical S-factor over a wide energy range, 50-1500 keV and extending to even lower energies when technically feasible, with small point-to-point errors. Even with a large common-mode error, this would significantly constrain theoretical curve shapes near threshold, and providing an independent check on the influence of scattering data on S ( E ) .  \nOn the theoretical side, further development of ab initio reaction calculations is desirable, improving on points  \nof current difficulty like explicit inclusion of three-body potentials. Calculations with the same potential but by different methods would help to separate computational issues from the impact of potential choice. Further exploration of how ab initio inputs like ANCs and scattering lengths could serve as fitting priors would also be a useful avenue of research.  \nAuthors of future theoretical calculations should consider including γ -ray angular distributions in their results; until very recently angular distributions were only available from two potential models (Kim et al. , 1981; Tombrello and Parker, 1963). New calculations would inform analysis of experiments, and the angular distribution measurements we recommend would provide tests of the calculations (or fitting constraints).  \nA source of complication here has been the need for a good meeting point between scattering data and theory. Given the difficulties of constructing a unique phase shift analysis without an underlying theory, and the recent arrival of modern scattering data, theorists should consider including elastic differential cross sections directly as outputs and fitting constraints.', 'VII. The hep reaction ( S hep )': "The reaction p + 3 He → 4 He + e + + ν e , also known as the hep reaction, is one of the possible processes taking place after the 2 H( p, γ ) 3 He , and it produces the most energetic neutrinos, with an endpoint energy of 18.8 MeV. It is however the least probable, with its S-factor, and consequently its rate, even beyond the reach of current experiments. This is due to the fact that the hep reaction is induced by the weak interaction and further suppressed by Coulomb barrier and by the fact that the leading onebody axial (Gamow-Teller) current operator cannot connect the main s -state components of the p + 3 He and 4 He initial- and final-state wave functions. As a consequence, the one-body axial current non-zero contribution is due to the small components of the 3 He and 4 He wave functions. Furthermore, other contributions, such as weak magnetism and other one-body corrections to the vector current, and two-body vector and axial currents, and the p -waves in the p + 3 He initial state, normally suppressed at solar temperatures, are significantly enhanced. This is further complicated by cancellations between the onebody and two-body axial current contributions. As a consequence, similarly to the pp fusion, the hep reaction cross section is too small to be measured in laboratory and only theoretical predictions are available.  \nThe most recent studies of this reaction, already evaluated in SF II, are those of Marcucci et al. (2000) and Park et al. (2003). The first one was performed within a phenomenological approach, similar to the calculation of Schiavilla et al. (1998) for the pp fusion. In this particular case, the p + 3 He and 4 He initial and final nuclear  \nwave functions were obtained with the correlated Hyperspherical Harmonics (CHH) variational method (Viviani et al. , 1995, 1998), using the Argonne v 18 (AV18) twonucleon potential (Wiringa et al. , 1995) augmented by the Urbana IX (UIX) three-nucleon interaction (Pudliner et al. , 1995). In this way, the binding energies of 3 He and 4 He, and the singlet and triplet p + 3 He scattering lengths were in good agreement with experiment. The weak vector and axial transition operators were obtained within a phenomonological approach. In particular, the largest two-body contribution to the axial current arising from the excitation of intermediate ∆ -isobar degrees of freedom was included using the transition-correlation operator scheme of Carlson et al. (1991); Schiavilla et al. (1992) and fixing the nucleon-to-∆ axial coupling constant to reproduce Γ T β as in the pp case (Schiavilla et al. , 1998).  \nThe second study of Park et al. (2003) was performed within the same hybrid EFT approach used for the pp funsion (EFT ∗ ), with the same nuclear weak current operators obtained within EFT as discussed in Section III, the LEC ˆ d R being fixed to reproduce Γ T β . The initial and final wave functions were derived using the CHH method with the AV18/UIX potential, as in Marcucci et al. (2000).  \nFollowing the results of Marcucci et al. (2000) and Park et al. (2003), SF II recommended for the zero-energy Sfactor, S hep (0) , the value (8 . 6 ± 2 . 6) × 10 -20 keV b. Three recommendations were also made: (i) to perform new studies with a broad spectrum of Hamiltonian models, in order to properly access the theoretical uncertainty; (ii) to study other weak reactions for which experiments can test the predictions made by employing the same theoretical ingredients as those used for the hep reaction calculation; (iii) to further understand the relation between the hep reaction and the so-called hen process ( n + 3 He → 4 He + γ ). Of these recommended works, only the second one has been performed so far, applying the theoretical framework used for the hep study to the muon capture reactions on light nuclei (Marcucci et al. , 2011).  \nGiven the lack of further studies on the hep reaction after SF II, we decide to maintain  \nS hep (0) = (8 . 6 ± 2 . 6) × 10 -20 keV b (17)  \nas recommended value. Notice that we could have increased S hep (0) by ∼ 1 %, in analogy with S 11 (0) , due to the updated values of input parameters (mostly G A /G V ). However, we have decided not to apply such an increase, because it does not derive from a direct calculation, and, most of all, because such an increase lies well within the quoted ∼ 30 % uncertainty. However, given the advances in ab initio studies of the last decade and in the development of the EFT framework, we update the recommendations of SF II as follows: (i) the hep process should be  \nrevisited within a fully consistent χ EFT approach, similarly to what has been done in these years for the pp fusion and other processes, as, for instance, muon captures on light nuclei (Acharya et al. , 2018; Bonilla et al. , 2023; Ceccarelli et al. , 2023; Gnech et al. , 2024; Marcucci et al. , 2012, 2018); (ii) the initial and final wave functions should be calculated with the most accurate ab initio methods, as the (uncorrelated) HH method (Kievsky et al. , 2008; Marcucci et al. , 2020), which has been proven to provide more accurate A = 4 bound and scattering wave functions than the CHH method, using both local and non-local interactions; (iii) the relation between the hep and the hen process should be understood. The hen process has been studied within χ EFT and the HH method by Viviani et al. (2022), but a consistent parallel study of both A = 4 reactions is still missing.  \nA relatively recent SNO Collaboration analysis of data from all three of the detector's running phases yielded a one-sided confidence-interval bound on the hep neutrino flux of Aharmim et al. (2020)  \nφ hep < 30 × 10 3 cm -2 s -1 (90% C . I . ) , (18)  \na result in agreement with the SSM prediction. The lack of a definite measurement, however, has meant that there is no substantive experimental test of the predicted S hep (0) . Various global analyses of solar neutrino data have provided weak evidence for a nonzero flux, e.g., from Bergstrom et al. (2016)  \nφ hep ∣ ∣ ∣ global analysis = 19 +12 -9 × 10 3 cm -2 s -1 , (19)  \nwhich is consistent with the experiment bound given above.  \nAn extraction of the hep flux from experiment will likely require a detailed shape analysis to separate hep neutrinos from the high-energy tail of the 8 B spectrum and the low-energy tail of atmospheric neutrinos. Typically the hep spectrum is assumed to have an allowed shape: indeed, in the work reported here spectra are not provided, where p -waves, weak magnetism, and similar corrections have been included. Because these corrections are unusually large for hep neutrinos, future studies should provide spectra and evaluate the impact of such corrections on the shape.", 'VIII. Electron capture by p+p and 7 Be': 'Electron capture reactions are the sources of lines found in the solar neutrino spectrum. In particular, electron capture on p + p , i.e., the process p + p + e -→ d + ν e , also known as the pep reaction, competes with p + p fusion and depends on the same nuclear matrix element. Therefore, the ratio between the pp and pep rates  \nis independent of nuclear physics. Based on this consideration, in SF II the result of SF I, which is based on the work of Bahcall and May (1969), was multiplied by the radiative corrections calculated by Kurylov et al. (2003), leading to the final result given by  \nR ( pep ) = 1 . 130(1 ± 0 . 01) × 10 -4 ( ρ/µ e ) × T -1 / 2 6 [1 + 0 . 02( T 6 -16)] R ( pp ) , (20)  \nwhere ρ is the density in units of g/cm 3 , µ e is the mean molecular weight per free electron, and T 6 is the temperature in units of MK. The range of validity is 10 < T 6 < 16 . Given that no new evaluation of the pep rate has been performed since SF II, the result of Eq. (20) represents also the present recommended value.  \nCompetition between electron and proton capture on 7 Be fixes the branching ratio of the ppII and ppIII chains, and thereby the 7 Be and 8 B neutrino fluxes. Superallowed electron capture on the J π = 3 / 2 -ground state  \nof 7 Be leads to the J π = 3 / 2 -ground state or the J π = 1 / 2 -, 478 keV first excited state of 7 Li with a measured (terrestrial) branching ratio of 10.44(4)% (Tilley et al. , 2002). It is customary to calculate the solar decay rate in terms of the measured terrestrial decay rate, which has the advantage of removing the nuclear physics dependence, but it requires a proper calculation of the densities of electrons at the nucleus. This has usually been done by considering continuum and bound electrons separately, using the Debye-Hückel (DH) approximation pioneered by Salpeter (1954) for the screening of bound electrons in the solar plasma; the capture rate of continuum electrons is not significantly influenced by electron screening. Furthermore, it is necessary to calculate the atomic probability densities governing the K and L terrestrial electron capture rates. Fretwell et al. (2020) recently measured the ratio of L to K capture using superconducting tunnel junctions. The recommendation of both SF I and SF II was  \nR ( 7 Be + e -) = 5 . 60(1 ± 0 . 02) × 10 -9 ( ρ/µ e ) T -1 / 2 6 [1 + 0 . 004( T 6 -16)] s -1 , (21)  \nwhich is valid for 10 < T 6 < 16 . This expression is based on the continuum capture rate calculated by Bahcall and Moeller (1969), the average ratio of the total capture rate to the continuum rate calculated using three solar models in Bahcall (1994), and the current value of the half-life, 53.22(6) d (Tilley et al. , 2002). It agrees within 1% with a density matrix calculation of the rate that makes no assumptions regarding the nature of electronic quantum states in the solar plasma and allows for aspherical fluctuations in the spatial distributions of plasma ions (Gruzinov and Bahcall, 1997). The estimated uncertainty of 2% accounts for possible corrections to the Debye-Hückel approximation due to thermal fluctuations in the small number of ions in the Debye sphere, and breakdowns in the adiabatic approximation (Johnson et al. , 1992).  \nAlthough calculations have appeared in the literature claiming new plasma effects based on model calculations (see Simonucci et al. (2013); Vescovi et al. (2019)), we continue to regard the arguments in Bahcall et al. (2002) as definitive. That reference presented five distinct derivations demonstrating that the Salpeter formula for screening corrections is valid for pp-chain reactions and solar conditions, up to corrections on the order of a few percent.', '1. Direct Measurements': 'The thorough review of published work done in SF II investigated the influence of beam-target overlap, target stoichiometry, beam energy loss, and the backscattering of 8 B recoils on CMEs and led to the selection of a homogeneous group of well-documented data sets. Since then, no further information became available on these aspects of direct measurements. The data that formed the basis of the SF II recommendation were those of Baby et al. (2003a), Baby et al. (2003b), Hammache et al. (1998), Hammache et al. (2001), Hass (1999), Strieder et al. (2001), and the BE3 data set of Junghans et al. (2003). We considered the complete set of radiative capture measurements that were the basis of the SF II recommendation for S 17 (0) , where we adopted the revised BE3 data presented by Junghans et al. (2010), which were reanalyzed using a more sophisticated model of the target and more accurate stopping powers. In addition, we included in our analysis the only new experiment reported thereafter of Buompane et al. (2022), which used a radioactive 7 Be ion beam and the recoil mass separator ERNA to detect the 8 B recoils. While these new results contribute to the determination of S 17 with different, well controlled systematics (Buompane et al. , 2018), their large statistical uncertainty relative to the other experiments, due to the low counting statistics, limits their impact.  \nNew scattering data have been published. Elastic and inelastic scattering cross sections at relative kinetic energies between 474 keV and 2.74 MeV were measured and s wave scattering lengths were inferred by Paneru et al. (2019). At higher energies between 1.6 and 3.4 MeV, thick target elastic and inelastic scattering excitation function measurements were used to infer the existence of new resonances above 1.8 MeV (Mitchell et al. , 2013). These high lying resonances have no direct influence on S 17 (0) because of their small partial widths in the 7 Be gs + p channel and/or because they cannot be populated by s- or d-wave capture at low energy.', '2. Indirect Measurements': 'In SF II, Coulomb dissociation measurements based on the formalism developed by Baur et al. (1986) were considered (Davids et al. , 2001a,b; Iwasa et al. , 1999; Kikuchi et al. , 1998; Kikuchi et al. , 1997; Motobayashi et al. , 1994; Schümann et al. , 2003; Schümann et al. , 2006), where in  \nsome cases the most recent publications supersede the previously published ones. Finally, the results of these experiments were not included in the determination of S 17 (0) due to disagreements over whether the analyses of the various measurements properly accounted for the contributions of E 2 transitions. We disagree with these reservations and find the agreement of these measurements with radiative capture measurements significant. However, in order to include these consistent data in formulating our recommendation for S 17 (0) , we inflated the CMEs of the Coulomb breakup measurements to adequately account for the uncertainty in the E2 components assumed in each analysis.  \nThe common mode errors are obtained from a linear sum of those given in the manuscripts and an additional common mode error due to the estimated size of the E 2 component in each measurement. This additional error is applicable in one direction only, downward for the Kikuchi et al. (1998) and Schümann et al. (2006) analyses of the RIKEN and GSI measurements, respectively, and upward for the Davids and Typel (2003) analysis of the Michigan State University (MSU) measurement, according to whether the E 2 component was included or excluded. Though the Davids and Typel (2003) analysis did include a common mode error contribution of 2.5% due to the E 2 component (which represented 5% of the total cross section), we have included an additional 2.5% contribution to the MSU experiment. The size of the additional common mode errors for the RIKEN and GSI measurements was based on scaling this 2.5% according to the predicted size of the E 2 contribution to the cross section in these measurements relative to that in the MSU measurement, evaluated using first-order perturbation theory and the E 1 and E 2 S-factors calculated in a potential model of 8 B (Bertulani, 1996). Thus the additional single-sided common mode errors added to the GSI, MSU, and RIKEN measurements are -2.1%, 2.5%, and -4.3%, respectively.', 'C. Theory': "The size of the theoretical contribution to the uncertainty in the SF II estimate of S 17 (0) was twice that of the experimental contribution. Since then, significant progress has been made. A new potential model calculation was performed by Dubovichenko et al. (2019). The large-scale computational demands of calculations using high-quality nucleon-nucleon (NN) and three-nucleon (3N) interactions beyond A =5 had hindered the ab initio approach in the past. Nevertheless progress has been rapid and it has been possible to extend ab initio calculations to 7 Be( p, γ ) 8 B, albeit without 3N forces. Navrátil et al. (2011) performed such a calculation using the no-core shell model in the continuum with a nucleonnucleon interaction derived from chiral EFT at fourth  \norder. The calculation converged with respect to the model space size and reproduces experimental data for the S-factor despite having virtually no free parameters (though the similarity renormalization group evolution parameter was tuned to reproduce the binding energy of 8 B). The final value of S 17 (0) =19.4(7) eV b agrees with the SF II recommendation and, as will be shown later, with our analysis. More recently, a new approach was introduced to combine experimental measurements with ab initio predictions, resulting in an ' ab initio -informed evaluation' Kravvaris et al. (2023) that arrived at a value of S 17 (0) = 19 . 8(3) eV b.  \nAsignificant theoretical advance was the application of the formalism of halo EFT developed by Bertulani et al. (2002) to 7 Be( p, γ ) 8 B calculations by Zhang et al. (2015), Zhang et al. (2018), and Higa et al. (2022). It treats the incoming p and 7 Be as point-like particles and the 8 B final state as a bound state of the two. The calculation and the associated errors depend upon the accuracy with which one is able to describe the incoming scattering states, the final bound state and the relevant electromagnetic currents. The relative kinetic or CM energy range E /lessorsimilar 1 MeV, corresponding to a relative momentum p /lessorsimilar 40 MeV/c, is considered within the domain of applicability of halo EFT. The two cluster physical description of 7 Be( p, γ ) 8 B is expected to hold at 3 Heα relative momenta below a physical cutoff Λ ∼ 70 MeV/c set by the threshold (binding momentum) for breaking the 7 Be core into these constituents. Further, at energies above the excitation energy of the first excited state, E /star = 0 . 4291 MeV, including the contribution of the excited core 7 Be /star is imperative in the EFT.  \nZhang et al. (2015) and Zhang et al. (2018) include such an excited 7 Be contribution. Higa et al. (2022) provides a calculation without the 7 Be /star contribution below E /star and one with it above E /star that also includes an M 1 contribution to the 1 + resonance. The two groups' halo EFT calculations with the 7 Be /star contributions differ in several respects that are elaborated in Higa et al. (2022). Differences in the final bound state calculation affect the treatment of the short-range interactions and divergences in the EFTs. The interpretation of the asymptotic normalization constants (ANCs) in terms of elastic scattering parameters is affected. However, the momentum dependence of the capture cross section is not impacted by the ANCs, which are fitted to capture data. Thus there is no effect on the cross section. In the incoming channel, Higa et al. (2022) use a more general form of the s -wave short-ranged interaction involving the 7 Be /star core based on the low-energy symmetry that is included only at NNLO. In contrast, Zhang et al. (2015) and Zhang et al. (2018) include a short-ranged interaction at LO that could make a difference in the momentum dependence of the cross section. However, as shown in an order-by-order calculation (Higa et al. , 2022), the short-ranged interaction is a sub-leading effect so the two halo EFTs should have  \nsimilar low-momentum dependence. The third important difference is in the treatment of two-body currents, which affects EFT error estimates. Zhang et al. (2015) and Zhang et al. (2018) include this at NLO whereas Higa et al. (2022) estimate it to be an N 3 LO effect. Bayesian estimates of the two-body current (Zhang et al. , 2015, 2018) are not strongly constrained by capture data, consistent with this being a higher-order contribution.  \nAnother widely used approach for describing the cross sections of low-energy nuclear reactions is R-matrix theory. In R-matrix theory the 7 Be( p, γ ) 8 B cross section can be parametrized in terms of a few parameters representing either 'real' or 'background' poles. The real poles correspond to the observed states and resonances, while the background terms correspond to the 'mean-field' effects that are of the non-resonant type. In the present case for example, a background pole in the 1 -channel is needed so as to reproduce the non-resonant E 1 part of the capture. The parameters of the real pole corresponding to the bound state are determined by the binding energy and the two ANCs, while the 1 + resonance parameters are determined by its energy and partial decay widths (both proton and radiative). The R-matrix analysis of this reaction carried out by Barker (1995) arrived at a value of S 17 (0) = 17(3) eV b; however, it was based on data that were excluded in SF II and here, with the exception of those of Filippone et al. (1983).  \nWe fit data using R-matrix theory and halo EFTs, with both frequentist and Bayesian approaches, investigating the impact of different choices for the energy range in which data were fitted, and obtained consistent results. Our recommended S 17 (0) value and its uncertainty are based on a Bayesian fit of both direct and Coulomb dissociation data taken at energies E ≤ 1250 keV using the Higa et al. (2022) halo EFT /star . This theory provides a good description of capture data over a wide energy range including the M 1 contribution from the 1 + resonance. The theoretical uncertainties are well understood. Further, the contributions of the S = 1 and S = 2 spin channels to the capture cross section are self-consistently parametrized with the elastic scattering information contained in the known ANCs for this system.  \nThe Bayesian formalism naturally incorporates prior information concerning the ANCs in the fits. The upper limit on the energy in the fit was determined to be high enough to include some data from all the modern experiments, and yet low enough that the contribution of the wide 3 + resonance at E ≈ 2456 keV does not exceed 1% of the total cross section at any energy, as estimated by our R-matrix analysis. The fitting procedure is described in the next section, whereas the details, as well as the results of the fits done in different energy ranges and with different theories are reported in the supplemental material.", 'D. Fitting Procedures': "In SF II, it was noted that the different S 17 ( E ) data sets had similar energy dependences. The discrepancies among the data sets were primarily due to different absolute normalizations. We address the normalization issue by introducing scaling factors associated with the CMEs. This contrasts with the SF II approach, in which the errors were inflated by the factor √ χ 2 /χ 2 ( P = 0 . 5) . The procedure for handling discrepant data sets adopted here is mathematically rigorous and aligns naturally with the method of estimating theoretical uncertainty discussed below. We prefer this approach to the inflation factor method used in SF II, since the absolute scales of the different data sets can be used as constraints for each other, while there is no information available for a reevaluation of the CMEs. The procedure we used, described below, can be applied to both frequentist and Bayesian analyses.  \nAlthough our final recommendation is based on Bayesian analysis alone, both χ 2 minimization (a frequentist approach) and Bayesian posterior probability distribution function evaluation were performed to fit EFTs to the data; in the R-matrix analysis, only the former was used. The likelihood function, defined as the conditional probability P ( D | θ ) for the data D given theoretical parameters θ , enters both fitting procedures. It can be derived from the χ 2 expression given in D'Agostini (1994)  \nP ( D | θ , s ) = N set ∏ α =1 1 √ 2 πω 2 α exp [ -(1 -s α ) 2 2 ω 2 α ] × N α ∏ i =1 1 √ 2 πσ 2 α,i exp { -[ y α,i -µ α,i ( θ ) /s α ] 2 2 σ 2 α,i } , (22)  \nwith α = 1 , . . . , N set labeling the different experimental data sets and i = 1 , . . . , N α labeling the individual data points of each experiment. In the second term, we divide theoretical predictions µ α,i ( θ ) for the data points of the experiment α by a scaling factor s α and consider the combination as our full model for y α,i , the i th data point of experiment α with point-to-point uncertainty σ α,i .  \nIn the first term, we assign s α a Gaussian distribution centered at 1 with a width equal to the CME ω α ; for the GSI, MSU and RIKEN data shown in Table VIII, asymmetric Gaussians are used with ω α depending on the sign of 1 -s α . In our χ 2 minimization, the χ 2 definition is derived from equating the full exponents in Eq. 22 to -χ 2 / 2 . In the absence of CMEs with s α = 1 , Eq. 22 leads to the usual uncorrelated least-squares minimization formula.  \nIn the Bayesian analysis, the terms in the first exponential are considered as part of the prior distribution P ( θ , s ) that are conditioned on prior knowledge and assumptions such as the CMEs ω in the scaling factors s or estimates of theoretical parameters θ in the EFT. The  \nposterior distribution for the theoretical parameters is then derived via Bayes's theorem,  \nP ( θ , s | D ) = P Bayes ( D | θ ) P ( θ , s ) P ( D ) , (23)  \nwhere the likelihood function P Bayes ( D | θ ) is defined without the first exponential in Eq. (22). However, since the GSI, MSU and RIKEN data have asymmetric CMEs (see Table VIII), we draw priors from a uniform distribution spanning 50% to 150% of the central values for these that are then used in the asymmetric normalization exponentials in the likelihood function. This was computationally simpler than generating asymmetric Gaussian priors that span the entire ( -∞ , ∞ ) range, which is neither necessary nor physical for a normalization constant.  \nThe constant P ( D ) in Eq. (23), known as the Bayesian evidence, guarantees the correct normalization of the posterior distribution. It is useful in comparisons of different theories but does not affect the estimation of parameters.", 'E. Data Analysis and S 17 Determination': "We performed χ 2 fits using several theoretical expressions over multiple energy ranges: E ≤ 475 keV , E ≤ 1250 keV , and over a 'non-resonant' energy range E ≤ 490 keV and 805 keV ≤ E ≤ 1250 keV , in which the M 1 contribution from the 1 + resonance can be ignored. Fits with and without Coulomb dissociation data were performed that resulted in overlapping S 17 (0) determinations within the estimated fitting uncertainties. The different low energy E ≤ 475 keV fits included one using the halo EFT from Zhang et al. (2018) and one using the halo EFT /star from Higa et al. (2022), both including the excited 7 Be contribution. The simpler halo EFT gs expressions, not including an excited component 7 Be /star , were also fitted in this low energy region with comparable results. For the fits in the other two energy ranges, halo EFT expressions with excited 7 Be /star contributions from both Zhang et al. (2018) and Higa et al. (2022) were used. The Zhang et al. (2018) halo EFT expression, which does not include an M 1 contribution, was supplemented with a Breit-Wigner resonance (Rolfs and Rodney, 1988) with its energy (around the 1 + resonance energy E ≈ 630 keV ), width, and proton-decay branching ratio determined by the fits. An R-matrix model was fitted to data in the three energy regions with the AZURE2 code (Azuma et al. , 2010), again resulting in an S 17 (0) consistent with the halo EFT values. The results from the various fits are included in the supplemental material.  \nBayesian fits were performed with halo EFT expressions from both Zhang et al. (2020) and Higa et al. (2018) that are compatible with the various χ 2 fits. The Bayesian analysis had advantages over the χ 2 fits in exploring parameter space and in determining realistic uncertainties (King et al. , 2019).  \nThe 7 Be( p, γ ) 8 B capture proceeds through the S = 2 and S = 1 channels. The reaction is known to be peripheral, resulting in a S 17 (0) that has a sub-leading dependence on the strong interaction in the initial state (Baye, 2000). This behavior was shown to persist away from the threshold in halo EFT, in which for E 1 capture the strong interaction in the incoming s -wave channel only contributes at NNLO.  \nTherefore, to high order, the capture is only sensitive to the sum of the squares of the ANCs in the S = 2 and S = 1 channels. This results in an arbitrariness in the relative contributions from the two spin channels in the χ 2 fits, without affecting the final S 17 determination. R-matrix, EFT gs , and EFT /star fits confirm this behaviour. On the other hand, the ANCs have been extracted experimentally (Tabacaru et al. , 2006; Trache et al. , 2003) and also theoretically (Nollett and Wiringa, 2011; Zhang et al. , 2018) in ab initio calculations. Although the determination of S 17 (0) is insensitive to the relative contributions of the two spin channels, knowledge of the relative contributions of the ANCs is important in the EFT framework. It establishes the hierarchy of different contributions in the perturbative expansion. This is crucial in developing a self-consistent expansion that is necessary to quantify the theoretical errors at any finite order of the perturbation. Prior knowledge of the ANCs and other parameters can be accounted for naturally in the Bayesian framework in which one specifies the prior probabilities P ( θ , s ) . Similar constraints can be included in the χ 2 fit by modifying the first exponential in Eq. (22) to center the theoretical parameters around known values.  \nThe Bayesian fits involved six theoretical parameters and 11 normalization constants s α for the data sets. The priors for the normalization constants were as described earlier. The priors for the theoretical parameters are based on the underlying EFT assumptions (Higa et al. , 2022) about the expansion in a small ratio Q/ Λ where Q ∼ 40 MeV / c represents the low momentum of infrared physics and Λ ∼ 70 MeV /c the cutoff of the theory. The 5 S 2 and 3 S 1 scattering lengths a (2) 0 = -3 . 18 +0 . 55 -0 . 50 fm and a (1) 0 = 17 . 34 +1 . 11 -1 . 33 fm (Paneru et al. , 2019) are taken as inputs. For E 1 capture at LO, the only fit parameter, the 5 P 2 effective momentum ρ is assumed to scale as Q and accordingly we assume a uniform prior for ρ between -100 MeV/ c and 1.5 MeV/ c , where the upper limit is set by the physical constraint that the ANC squares have to be positive. At NLO, the 3 P 2 ANC 2 C 2 1 , 1 is a fit parameter whose prior is taken from a normal distribution determined by experiment. At NNLO, we use the ab initio calculation of (Zhang et al. , 2018) to draw the 3 P /star 2 ANC 2 from a normal distribution. A ratio of the s -wave scattering lengths in the coupled-channel 3 S 1 -3 S /star 1 that is assumed to be O (1) in the EFT is drawn from a normal distribution with a mean of 0 and a standard deviation of 10 to cover a wide range. The M 1 capture is domi-  \nthe transition 5 P 1 → 5 P 2 and it requires three parameters: a p -wave scattering volume a 1 , an effective momentum r 1 and a two-body current to regulate divergences. Higa et al. (2022) determined a 1 = -108 . 13 fm 3 and r 1 = -111 . 23 MeV from the the narrow 1 + resonance energy and width. Here, we kept a 1 fixed while drawing r 1 from a uniform distribution between -150 and -50 MeV. The Bayesian fit gave r 1 ∼ -111 . 23 MeV . The two-body coupling was drawn from a uniform distribution and the fits gave a numerical value consistent with the EFT estimate. We obtain the posterior distribution P ( θ , s | D ) using a probabilistic integration method called Nested Sampling (Skilling, 2006) implemented in Python (Feroz et al. , 2009) that calculates both the posterior and the evidence.  \nThe experimental data and EFT /star fits at LO, NLO, and NNLO are shown in Fig. 8. S 17 and its derivatives at threshold are given in Table VII. The quality of the fits, the consistency of the numerical values of the fit parameters with EFT assumptions, and the order-by-order improvement in the numerical value of the fitted capture cross section and S-factor give us confidence in the estimated theoretical uncertainty. We find the posterior probability distribution functions to be symmetric and well described by Gaussians, and as such we report the mean and standard deviation in Table VII. Table VIII gives the scaling factors for the experimental data sets determined from the NNLO fit along with their common mode errors.  \nFIG. 8 Astrophysical S-factor for 7 Be( p, γ ) 8 B . Data with E ≤ 1250 keV were included in the Bayesian fits shown here, which use the halo EFT of Higa et al. (2022) that includes an excited 7 Be component. The dot-dashed (black) curve is the LO calculation, the dashed (blue) curve is NLO, and the solid (red) curve is NNLO. The inset shows the posterior probability distribution function for S 17 (0) in eV b. The median as well as the 16th and 84th percentiles of the distribution are specified.  \n<!-- image -->  \nTABLE VII S 17 and its first two energy derivatives at E = 50 eV determined from Bayesian fits of experimental data below 1250 keV using the halo EFT of Higa et al. (2022) that includes an excited 7 Be component. The first error in each is from the fit. The second error is the estimated LO (30%), NLO (10%), and NNLO (3%) EFT error, respectively, from higher order corrections.TABLE VIII Scaling factors for the data sets at NNLO, the number of data points used in the E ≤ 1250 keV fits, and the common mode errors.", 'F. Theoretical Uncertainty': 'The theoretical uncertainties in halo EFT are estimated from the perturbative expansion in Q/ Λ (Higa et al. , 2022). Q is associated with the physics at low momenta of interest: relative momentum p /lessorsimilar 40 MeV /c , binding momentum γ ∼ 15 MeV /c , inverse Bohr radius κ C ∼ 24 MeV / /planckover2pi1 c , excitation momentum γ ∆ ∼ 27 MeV / c and resonance momentum p R ∼ 32 MeV / c . The cutoff scale of Λ ∼ 70 MeV is estimated as described above. Except for a narrow momentum range around the 1 + resonance, capture is dominated by E 1 transitions from s - and d -waves that preserve the channel spin. Thus the threshold theoretical estimates of S 17 (0) are based on E 1 capture. The ANC for the 5 P 2 final bound channel is known to be about 5 times larger than the 3 P 2 bound channel, so capture in the S = 2 channel dominates. The LO contribution in the Q/ Λ expansion comes only from s -wave Coulomb interactions in the S = 2 channel. The weakly bound 8 B state enables peripheral capture without initial state strong interactions.  \nAt NLO, d -wave Coulomb interactions in the S = 2 channel and s -wave Coulomb interactions in the S = 1 channel are added. Strong interactions in the initial s and d waves of the S = 1 channel contribute at NNLO. Perturbatively, the s -wave scattering lengths a (1) 0 , a (2) 0 contribute at this order. Though a (1) 0 is much larger than  \na (2) 0 , the S = 1 channel is one order higher in the perturbation, making these contributions comparable. We also see mixing in the coupled 3 S 1 -3 S /star 1 channels due to the excited core contribution, which is parameterized by scattering lengths. The mixing requires the 3 P /star 2 ANC in the capture cross section. The theory error is estimated as 30% at LO, 10% at NLO, and 3% at NNLO from an estimated Q/ Λ ∼ 1 / 3 . This is validated by the relative sizes of the cross section at different orders of the perturbation, and also by the sizes of the fitted parameters that are consistent with theoretical estimates.  \nThe uncertainty from the higher order contributions is estimated as follows. Two-body currents make a relative contribution ∼ k 0 a 0 L E1 with photon energy k 0 = ( p 2 + γ 2 ) / (2 µ ) ∼ Q 3 / Λ 2 . In the S = 2 channel, a (2) 0 ∼ 1 / Λ so this is a N 3 LO contribution. In the S = 1 channel, the larger a (1) 0 ∼ 1 /Q makes this a Q 2 / Λ 2 contribution; however, this spin channel is suppressed by one order in the perturbation. Thus two-body currents constitute a N 3 LO 3% uncertainty; s -wave effective range corrections would also constitute a N 3 LO uncertainty. Initial state d -wave interactions (suppressed by two powers of momentum) in the S = 2 channel also constitute a N 3 LO uncertainty. The uncertainty in the experimental determination of the s -wave scattering lengths is included in the 3% N 3 LO uncertainty.', 'G. Conclusions and Recommendation for S 17': 'R-matrix and EFT fits of direct and Coulomb dissociation data below 1250 keV, where the influence of resonances other than the 1 + can be neglected, were carried out. Many different theories and energy ranges were considered and consistent results were obtained. Our recommendation was obtained using the halo EFT of Higa et al. (2022) that includes an excited 7 Be component. This theory was selected due to the generality and rigor of its treatment of short-range interactions in the initial and final states and of two-body currents, as well as its inclusion of M 1 transitions and robust uncertainty quantification. The resulting recommendation is S 17 (0) = 20 . 5(3)(6) eV b, where the first error is due to fitting and the second due to neglected higher-order contributions; for the purposes of solar modeling, we intend that the two errors be added together in quadrature and recommend S 17 (0) = 20 . 5(7) eV b.  \nThe recommended S 17 (0) has a notably smaller uncertainty than the SF II value for two primary reasons. Significant theoretical developments in halo EFT with a NNLO calculation have reduced the theoretical uncertainty by a factor of two. The use of floating data normalization factors allows discrepant data sets to be brought into agreement for low-energy extrapolation without inflating the measurement uncertainties. In the Bayesian fits, the normalization factors are assigned priors, as is  \ndone for some of the theoretical parameters. Frequentist fits presented in the supplemental material include these in the definition of the χ 2 cost function. Normalization factors different from unity increase the cost, but the overall cost is lowered by bringing precise measurements from different data sets into agreement. This is effective because the number of scale factors is much less than the number of high precision data points. Due to the small uncertainties and large number of data points, the measurement of Junghans et al. (2010) exerts a strong influence on the fit. Planned measurements, such as those recently approved at E = 1 and 1.2 MeV using a 7 Be beam and the DRAGON recoil separator at TRIUMF, must have comparably small statistical and systematic uncertainties in order to compete effectively.', 'X. The 14 N ( p, γ ) 15 O reaction ( S 1 14 )': 'The 14 N ( p, γ ) 15 O reaction proceeds through a number of bound-state transitions, but the total capture cross section is dominated by the capture to the 6.79 MeV state (3/2 + ) in 15 O. (In this chapter the reported level energies are taken from the most recent compilation (AjzenbergSelove, 1991)). The contribution of this transition to the total reaction cross section at solar energies is 71 -78 % (Adelberger et al. , 2011; Azuma et al. , 2010). The second strongest transition with a contribution of 16 -17 % is the capture to the ground state of 15 O, while the capture to the 6.18 MeV state (3/2 -) is responsible for about 7 -8 % of the total cross section. The remaining 5 -7 % is attributed to the capture to the other bound states (i.e. 5.18 MeV, 1/2 + ; 5.24 MeV, 5/2 + ; 6.86 MeV, 5/2 + and 7.28 MeV, 7/2 + ).  \nIn SF II a lower value, with a greatly reduced uncertainty, for S 1 14 (0) of 1.66(12) keV b was recommended compared to that of SF I, 3.5 +0 . 4 -1 . 6 keV b. This was the result of both the new implementation of the more rigorous R-matrix technique (Angulo and Descouvemont, 2001) over the previously used Breit-Wigner analysis, higher precision low-energy capture data sets (Bemmerer et al. , 2006; Formicola et al. , 2004; Imbriani et al. , 2005; Lemut et al. , 2006; Marta et al. , 2008; Runkle et al. , 2005) over the yields presented in Schröder et al. (1987), transfer measurements of proton ANCs for the bound states (Bertone et al. , 2002; Mukhamedzhanov et al. , 2003), and constraints on the lifetime of the 6.79 MeV bound state (Bertone et al. , 2001; Schürmann et al. , 2008). Since SF II, several new experiments and Rmatrix calculations have been completed, but our understanding of the ground state transition still remains incomplete.', 'A. Current Status and Results': 'Since SF II, several new measurements have been reported for the capture reaction (Daigle et al. , 2016; Frentz et al. , 2022; Gyürky et al. , 2019b, 2022; Li et al. , 2016; Sharma et al. , 2020; Wagner et al. , 2018), the lifetime of the 6.79 MeV state (Frentz et al. , 2021; Galinski et al. , 2014; Sharma et al. , 2020), the proton bound state ANCs (Artemov et al. , 2012), as well as a measurement of the low energy 14 N ( p, p ) 14 N scattering cross section (deBoer et al. , 2015). These and past works now include several R-matrix fits (Angulo and Descouvemont, 2001; Formicola et al. , 2004; Frentz et al. , 2022; Imbriani et al. , 2005; Li et al. , 2016; Runkle et al. , 2005; Wagner et al. , 2018), resulting in a range of different approaches to the fitting, which gives insight into the systematic uncertainties of the model.  \nThe recomended partial S(0) are summarised in Table. X and discussed in details in sec. X.E. The 6.79 MeV transition, being thre strongest, has seen a great deal of experimental attention. Its ease of measurement and relatively well understood reaction contributions result in a small uncertainty in the low energy extrapolation of the S-factor of ≈ 3%. The ground state and 6.18 MeV transitions have uncertainties of ≈ 40%, owing mainly to the complexity of the underlying reaction contributions and the much smaller cross section.', 'B. Absolute strength of the 259 keV resonance': 'The E = 259 keV resonance in 14 N ( p, γ ) 15 O is often used as a reference for normalization in many experiments because it is a narrow resonance, has a precisely known strength, and has γ -ray transitions that span a wide energy range. The absolute strength of the resonance has been determined in three works (Daigle et al. , 2016; Gyürky et al. , 2019b; Sharma et al. , 2020) since SF II. Based largely on the work of Daigle et al. (2016), a new resonance strength of ωγ 259 = 12 . 86 ± 0 . 45 meV is recommended, consistent with the previous value but with higher precision thanks to the new data sets. For the averaging we adopted the approach followed by SF II. The common systematic uncertainty of the nitrogen stopping power of 2.9% (Ziegler et al. , 2010) was excluded from the weighted mean calculation, and summed in quadrature with the weighted mean uncertainty to obtain the final uncertainty. Additional to this, the stopping power and branching ratio corrected resonance strengths from (Daigle et al. , 2016) were used in the calculations instead of the originally quoted values. The resonance strength determination is summarised in Table IX. For the branching ratios, those of Daigle et al. (2016) are recommended.  \nTABLE IX Summary of the E =259 keV resonance strength values along with their total uncertainties. The last row gives the recommended value with its total uncertainty.', 'C. Indirect Studies': 'There are several different indirect techniques that have been used to help constrain the extrapolation of the low energy cross section. A new transfer study of proton ANCs has been reported by Artemov et al. (2012). Since the 6.79 MeV level also acts a subthreshold resonance in the ground state transition, its lifetime is also needed. New lifetime measurements have been made by Galinski et al. (2014), Sharma et al. (2020), and Frentz et al. (2021) in addition to the previously avaliable data by (Bertone et al. , 2001) and Schürmann et al. (2008). Becasue the lifetime is in a range that is very difficult to access experimentally ( t 1 / 2 < 2 fs) these works still report values either with very large uncertainties or upper limits only. Scattering cross sections can also be very helpful in constraining both bound and unbound level parameters for the R-matrix fit to the capture data. A consistent set of scattering measurements has been made by deBoer et al. (2015) superseding the unutilisable data for 14 N ( p, p ) showing a large amount of discrepancy (Gurbich, 2008). In the new work only yield ratios are reported in order to reduce the uncertainties associated with target degradation. This scattering data give added constraint to the broad, high energy, 3/2 + resonance at E ≈ 2.2 MeV.', 'D. R-matrix considerations': "The phenomenological R-matrix has been the primary model used to fit and extrapolate cross section data, following the analysis of Angulo and Descouvemont (2001). Many of the investigations have used the same multichannel R-matrix code, AZURE2 , which was introduced by Azuma et al. (2010). The model has proven to be very popular because it is very flexible in its level structure description, yet at the same time retains enough physical constraints that it can be used for extrapolations into unobserved regions of the cross section. Of course this flexibility also allows for additional complications like over  \nfitting, mistaking experimental resolution effects for inconsistencies between data sets or the need for additional reaction components (Wiescher et al. , 2023), to give just a few common examples.  \nIn the R-matrix framework, the cross section is described by including a number of reaction components, which includes resonances, subthreshold states, and background levels. For radiative capture reactions, an additional component is introduced, external capture (Angulo and Descouvemont, 2001; Barker and Kajino, 1991; Holt et al. , 1978), which is used to model the non-resonant capture process. Fig. 9 shows an example multichannel AZURE2 R-matrix fit to scattering data and the three strongest transitions of the 14 N ( p, γ ) 15 O reaction: 6.79 MeV, ground state, and 6.17 MeV.  \nUsing the alternative R-matrix parameterization of Brune (2002), the resonance components are characterized by observable energies and partial widths, while subthreshold resonances have their strength quantified by the state's asymptotic normalization coefficient (ANC). The strength of the external capture contributions are likewise given by ANCs.  \nThe strongest transition, to the 6.79 MeV state, is dominated by external capture at solar fusion energies. As pointed out in SF II, discrepancies among the data sets over the higher energy range make it unclear whether an additional contribution is needed from a background level. However, as Wagner et al. (2018) showed, even when a background level is introduced, it has a very small effect on the low energy extrapolation owing to the dominance of the external capture component.  \nThe next strongest transition is to the ground state, where the R-matrix description of the low energy cross section is rather complicated. It is thought to have significant competing and interfering contributions from the 6.79 MeV subthreshold state, the narrow E = 259 keV resonance, external capture, a background level, and the broad, higher energy, resonance at E = 2.2 MeV. Despite all of these different contributions, which gives the R-matrix fit a considerable amount of flexibility for this transition, a satisfactory fit to the ground state transition data remains elusive (Frentz et al. , 2022; Li et al. , 2016; Wagner et al. , 2018).  \nFinally, the 6.18 MeV transition is dominated by the narrow E = 259 keV resonance and external capture contributions at low energy. A contribution from the 6.79 MeV subthreshold state may also be significant.", 'E. Partial S-factors': 'In this compilation a different approach is adopted to determine the S (0) value for the two strongest transitions than in SF II. Because several are already available in recent publications, no new R-matrix calculation was performed. Instead the reported S (0) values from the  \nFIG. 9 A representative yield ratio for the 14 Np scattering cross section (deBoer et al. , 2015) (a), and low energy S-factors for the three strongest transitions in the 14 N ( p, γ ) 15 O reaction to the ground (b), 6.79 MeV (c), and 6.18 MeV (d) final states (Imbriani et al. , 2005; Li et al. , 2016; Runkle et al. , 2005; Schröder et al. , 1987; Wagner et al. , 2018). An example Rmatrix fit is shown where individual resonance contributions are indicated by dashed lines, external capture by dotted-dashed lines, and cross sections indicated by the solid lines.  \n<!-- image -->  \noriginal publications (Frentz et al. , 2022; Imbriani et al. , 2005; Li et al. , 2016; Runkle et al. , 2005; Wagner et al. , 2018) are considered. In the case of Runkle et al. (2005), Li et al. (2016) and Frentz et al. (2022), the data sets are normalized to the strength of the 259 keV resonance, while in Imbriani et al. (2005) and Wagner et al. (2018) an absolute normalization is established without the resonance strength. For the three former works, the reported S (0) values were re-normalised by the ratio of the resonance strengths used by those works and the present recommended one.', '1. Ground State Transition': 'The ground state transition cross section of the 14 N ( p, γ ) 15 O reaction is very challenging to measure accurately and to high precision because the cross section  \nis very low and most experiments perform measurements with high efficiency γ -ray detectors in very close geometry. This increases the yield for these measurements but also greatly increases the effect of γ -ray summing into the ground state transition, with the largest contribution from the 6.79 MeV transition at low energies. In addition, the cross section on the high energy side of the strong, low energy, 259 keV resonance is rather inconsistent between different measurements, likely a result of nitrogen diffusion into the thick target backing material.  \nIn SF II, the R-matrix fit included the data of Schröder et al. (1987), Imbriani et al. (2005), Runkle et al. (2005) and Marta et al. (2008), below 2 MeV. A good description of the data was obtained over most of this energy region, with the exception of a consistent over estimation of the experimental data from ≈ 300 to 500 keV. The fit there favored the higher cross sections of Marta et al.  \n(2008), consistently overestimating the data of (Runkle et al. , 2005) throughout this energy range. It should also be noted that the data of Schröder et al. (1987) were clerified to be yields that were uncorrected for experimental effects. Cross sections were deduced for these data in SF II but where never pubished.  \nSince SF II, there have been three new data sets reporting partial cross sections (Frentz et al. , 2022; Li et al. , 2016; Wagner et al. , 2018), which all used the AZURE2 R-matrix code to derive S (0) . These values are considered here as an indication of the uncertainty in the low energy extrapolation. Li et al. (2016) concentrated on differential cross section measurements between 0.5 and 3.5 MeV, while both Wagner et al. (2018) and Frentz et al. (2022) focused on bridging the gap between the measurements of Li et al. (2016) and the lower energy LUNA (Imbriani et al. , 2005) and LENA (Runkle et al. , 2005) measurements.  \nThe new data and accompanying R-matrix fits have repeatedly confirmed a discrepancy between the R-matrix fit and the experimental data for the ground state transition, especially when the high energy data of Schröder et al. (1987) and Li et al. (2016) are included in the fit. On the R-matrix side, this could point to a missing reaction component (i.e. missing resonance or subthreshold state) or an incorrect spin-parity assignement. On the data side, the close geometry setups of most experiments result in large summing corrections. Inaccurate corrections for angular distributions could also play a role. Because of this unresolved inconsistency between data and model, a larger uncertainty has been attributed to this transition compared to SF II (see Table X).', '2. 6.79 MeV Transition': 'In contrast to the ground state, repeated measurements of the 6.79 MeV transition continue to produce consistent results, especially at very low energies. The measurements of Wagner et al. (2018) do show a more rapidly increasing cross section at high energies than those of previous works, but are consistent in the low energy region. In addition, the ANC determined from the normalization of the external capture cross section used to model the non-resonant capture in the R-matrix fit yields consistent results with those determined from transfer studies (Bertone et al. , 2002; Mukhamedzhanov et al. , 2003). Because of this continued consistency across measurements, the value adopted here is the weighted average of the reported values, where the uncertainty has been further reduced from that of SF II.  \nTABLE X Total S 1 14 (0) as the sum of the partial transitions.  \n- a The uncertaity is the sum of the partial transition uncertainties.\n- b From the normalisation to the E = 259 keV resonance strength.', '3. Other Transitions': 'For the three weaker transitions reported in SF II, there were no new experimental data. Here the values of S (0) have been determined by scaling those from SF II by the ratio of the 259 keV resonance strengths adopted here and from SF II.', 'F. Total S-factor and conclusions': 'Figure 9 shows the low energy S-factors and R-matrix fits to the strongest capture transitions and the scattering yields at a representative angle and Table X summarizes the uncertainties estimated for the zero energy S-factor, S 114 (0) for each of the significant capture transitions. Their sum gives the adopted total S(0) value. The absolute uncertainty of the R-matrix sum is the quadratic sum of the absolute uncertatinties of the partial transitions, while the recomended uncertainty conservatively includes an additional 3.5% relative uncertainty from the resonance strength (see Table IX) to which most of the datasets were normalised. As described in Section X.E.1, the uncertainty in the ground state transition has increased to 37% over the 19% estimate of SF II, owing to repeated confirmation of a tension between different experimental data sets and the R-matrix description of the cross section. The uncertainty in the 6.79 MeV transition decreased from 4.0% to 2.9% due to the consistency of the various measurements for this transition.  \nIn addition, the new total capture cross section of Gyürky et al. (2022), performed using the activation technique over the energy range from 0.55 to 1.4 MeV, is larger than the sum of the partial R-matrix cross sections by ≈ 25%, a significant margin compared to the systematic uncertainties of the different data sets. However, it should be emphasized that for several of the transitions the only data available are those of Schröder et al. (1987), which are yields not cross sections and often give only upper limits. For example, the capture to the 7.56 MeV level in this energy range may be significant.  \nGiven the above considerations, a somewhat larger uncertainty of 8.4% has been estimated for S 1 14 (0) , compared to the 7% uncertainty recommended in SF II. As one of the main sources of uncertainty remains the ground state transition, far geometry angular distribution measurements are recommended extending the differential measurements of Li et al. (2016) to lower energies. These measurements will both reduce the effect of summing and provide more information on the underlying reaction mechanisms at these low energies. Further, since the data of Schröder et al. (1987) have significant uncorrected experimental effects, new measurements should be made over the wide energy range that they cover to replace them. Such measurements are planned at both the LUNA and JUNA underground facilities. At very low energies, the LUNA and LENA measurements of the ground state transition by Imbriani et al. (2005) and Runkle et al. (2005), respectively, remain the only measurements. The discrepancy between the two measurements and the tension between the R-matrix fits make new measurements in this range also of high priority. Despite several additional attempts since SF II, the lifetime of the 6.79 MeV transition still remains unmeasured. This remains a high priority, as having a well constrained value will greatly improve our understanding of the low energy behaviour of the ground state transition cross section. Finally, we recommend that the cross section to the 6.18 MeV and 7.56 MeV final states be measured at high energies in order to address the tension between the measured total cross section and the sum of the partial cross sections.', 'XI. Other CNO, Ne-Na reactions': 'In the following, we provide information on the protoninduced reactions involving the target nuclei 12 , 13 C, 15 N, 16 , 17 , 18 O, 19 F, 20 , 21 , 22 Ne, and 23 Na. Unless otherwise noted, we will focus on the low-energy region, i.e., below 150 keV in the center of mass. The reactions on lighter target nuclei up to and including 16 O are of direct relevance for solar neutrinos; the other reactions play a role in hydrogen burning at somewhat higher temperatures, including asymptotic giant branch (AGB) stars.  \nTable XI lists, where applicable, the S-factor at zero energy, the Gamow peak location, and references. Table XII provides recommended values of the S-factor and corresponding uncertainty on a grid of seven energy values. These results are displayed in Fig. 10, together with their polynomial fits. The fit coefficients are listed in Table I. Table XIII gives values of recommended strengths of narrow resonances. A detailed discussion of each reaction is provided below.', '1. 12 C ( p, γ ) 13 N': 'The low-energy reaction cross section is influenced by the tail of a broad resonance and the non-resonant capture to the ground state of 13 N. The results of six new direct measurements have been published since SF II (Csedreki et al. , 2023; Gyürky et al. , 2023; Kettner et al. , 2023; Nesaraja et al. , 2001; Skowronski et al. , 2023a,b).  \nThe first experiment by Nesaraja et al. (2001) reports the rection cross section relative to that of the 13 C( p, γ ) 14 N reactions at E lab p = 160 keV to be 0 . 24 ± 0 . 03 .  \nTwo of the newest experiments were performed at ATOMKI (Csedreki et al. , 2023; Gyürky et al. , 2023). The first employed the activation method in a wide laboratory bombarding energy range, E lab p = 300 -1900 keV, to provide a dataset with an absolute normalization having independent uncertatinties, from that of the older data sets obtained using in-beam γ -ray spectroscopy. The main focus of the second experiment was the precise determination of the energies and widths of the two low-energy resonances using in-beam γ -ray spectroscopy with detectors located at several angles.  \nThe fourth experiment, by Skowronski et al. (2023b), was performed at the Felsenkeller underground accelerator facility in Dresden, Germany, using in-beam γ -ray spectroscopy. The focus was on determining the parameters of the lowest-lying resonance employing a detector in close geometry with high solid-angle coverage. The data covered the energy range of E lab p = 345 -670 keV, and allowed for an extrapolation to solar and stellar energies using R-matrix theory. The absolute normalization of this data set is about 25% lower than the previous results. The low-energy slope of the S-factor also changed, resulting in an extrapolated value of S (25 keV) = 1 . 34 ± 0 . 09 keVb in the solar Gamow peak.  \nThe fifth experiment, by Kettner et al. (2023), was performed at Bochum and Notre Dame, and focused on the second resonance in the reaction. Differential cross sections at 0 · and 55 · , thick target yields, and angular distributions at selected energies were measured in the energy range of E lab p = 1000 -2500 keV. A comprehensive R-matrix fit of the available data was also performed, resulting in S (25 keV) = 1 . 48 ± 0 . 09 keVb.  \nThe sixth experiment, by the LUNA collaboration (Skowronski et al. , 2023a), obtained data in the range of E p = 80 -400 keV, reaching the lowest bombarding energies to date. Both the activation technique and in-beam γ -ray spectroscopy were applied, with a consistent absolute normalization of the data, despite the different systematic uncertainties introduced by the two methods. Their S-factor in the solar Gamow peak, S (25 keV) = 1 . 53 ± 0 . 06 keVb, is in good agreement with Kettner et al. (2023), and is compatible within 2 σ with', 'Skowronski et al. (2023b).': 'Apart from the direct measurements discussed above, the ground state ANC was determined in a recent particle transfer experiment (Artemov et al. , 2022). That work also reported a fit of the radiative capture data available at the time. The R-matrix fits to the data of Kettner et al. (2023); Skowronski et al. (2023a,b) give ANC values consistent with the result of the transfer work.  \nThanks to the high consistency of all recent datasets, the extrapolation of the cross section to solar energies has now improved uncertainty. Owing to the self-consistency between the used methods and the precise data coovering the lowest energies to date we adopt the S-factor recommended by Skowronski et al. (2023a) (see Tables XI and XII).', '2. 13 C ( p, γ ) 14 N': 'This reaction proceeds to the ground state and five excited states in 14 N. The low-energy cross section is influenced by the tail of a broad resonance at laboratory bombarding energy of E lab r ≈ 558 keV. Previously, only a single work reported cross sections for all transitions (King et al. , 1994), which was the basis of the S-factor recommendations in both SF I and SF II.  \nA new relative measurements was carried out by Ne-  \nsaraja et al. (2001) as mentioned in the previous section. Later a new comprehensive experiment by the LUNA collaboration (Skowronski et al. , 2023a) provided data in the energy range of E lab p = 80 -400 keV. One of their data sets was obtained with HPGe detectors to study partial cross sections. Their second data set was obtained with a γ -ray calorimeter to measure total reaction cross sections. The results from both data sets agree in the absolute cross section scale, despite major differences in the systematic uncertainties of the two methods. The new results are about 30% lower than previous results of King et al. (1994). Despite this scale difference the calculated S-factor ratio for the 12 C( p, γ ) 13 N and 13 C( p, γ ) 14 N reactions at E lab p = 160 keV is 0 . 26 ± 0 . 02 , in perfect agreement with the experimental reported one of 0 . 24 ± 0 . 03 by Nesaraja et al. (2001).  \nThe adopted S-factor given in Tables XI and XII is based on the new result of Skowronski et al. (2023a), owing to its self-consistency between the different methods for measuring the cross sections, and the higher precision of the dataset compared to King et al. (1994).', '3. 15 N ( p, α ) 12 C': 'Following SF II, the experimental and phenomenological work on the 15 N ( p, α ) 12 C reaction can be divided into three groups addressing different aspects: (i) new ( p, α 0 ) data (deBoer et al. , 2012); (ii) new ( p, αγ ) data (de-  \nBoer et al. , 2021a; Gyürky et al. , 2019a; Imbriani et al. , 2012a,b; Reinhardt et al. , 2016); and (iii) R-matrix analysis of existing data sets (Brune and deBoer, 2020; deBoer et al. , 2013).  \nOther studies have been conducted for laboratory bombarding energies above 400 keV. In Reinhardt et al. (2016), the strength of a resonance at E c.m. r ≈ 403 keV ( 2 -, E x = 12530 keV) is tabulated, and the value is in agreement with previous determinations. In Gyürky et al. (2019a), the thick-target yield for the E c.m. r = 841 keV resonance is shown, but no further details are reported. New cross section data for the ( p, α 1 γ ) reaction down to about 900 keV center-of-mass energy are displayed in deBoer et al. (2021a). They used the data for a comprehensive R-matrix fit, and reported parameters of resonances above about 1500 keV center-of-mass energy.  \nImbriani et al. (2012a,b) measured the 15 N( p, αγ ) 12 C reaction down to about 140 keV laboratory bombarding energy, and showed that the S-factor of the ( p, α 1 γ ) channel is orders of magnitude lower compared to the ( p, α 0 ) channel.  \nFinally, deBoer et al. (2013) contains a detailed analysis of all available channels populating the 16 Ocompound nucleus, expanding the results presented in deBoer et al. (2012). In this paper, a wide range of experimental data were considered leading to a single consistent R-matrix fit, accurately describing the broad level structure of 16 O below E x = 13500 keV. The resulting fit was used to extract an improved determination of the low-energy Sfactor for the 15 N( p, α ) 12 C reaction. In addition, Brune and deBoer (2020) have developed the partial-wave formalism for calculating the angular distributions of secondary γ -rays following particle emission, providing the framework needed for future studies that can directly utilize differential measurements of the 15 N ( p, α 1 γ ) 12 C reaction.  \nThe available evidence confirms that the ( p, α 0 ) channel dominates the total S-factor. The global R-matrix analysis in deBoer et al. (2013) provides results that slightly exceed the experimental data of Redder et al. (1982) and also estimates a slightly larger value, S (0) = (95 ± 6) × 10 3 keVb, than previous works (see Table VII of deBoer et al. (2013)), calling for additional low-energy measurements. The extrapolated S-factor in deBoer et al. (2013) agrees with the findings in La Cognata et al. (2009), which is the most recent indirect study, determining the S-factor down to 20 keV center-of-mass energy. The global R-matrix analysis suggests a renormalization of the La Cognata et al. (2009) S-factor by 6%, which is within the quoted uncertainty. Based on the information provided above, we adopt the S-factor of La Cognata et al. (2009), S (0) = (73 ± 5) × 10 3 keVb, which was also adopted in SF II.', '1. 15 N ( p, γ ) 16 O': "The low-energy 15 N ( p, γ ) 16 O cross section is dominated by its ground state transition (Imbriani et al. , 2012b; Leblanc et al. , 2010). This transition's cross section is enhanced by the constructively interfering tails of two broad resonances at laboratory proton energies of E lab = 338 and 1028 keV, which correspond to 1 -levels in the 16 O compound system. In addition to groundstate proton and γ -ray de-excitation, these levels also decay strongly through the α 0 and α 1 channels. Currently, the favored framework for the analysis and extrapolation of the available data is the phenomenological R-matrix. Here, the main components that are used to model the low-energy cross section of the 15 N ( p, γ ) 16 O reaction are the two broad 1 -resonances, external capture, and a 1 -background level. The strength of the external capture can be characterized by the ground state proton ANC of 16 O, which has been experimentally determined via a proton transfer reaction (Mukhamedzhanov et al. , 2008). Gamma-ray de-excitation to the ground state has also been observed through a 2 + level in this energy region in the 12 C+ α reactions, but has been shown to be negligibly weak for 15 N ( p, γ ) 16 O (deBoer et al. , 2013).  \nThe cross section measurements used in SF II were those of Hebbard (1960), Rolfs and Rodney (1974), and Bemmerer et al. (2009). While the then newly measured data of Bemmerer et al. (2009) represented a significant improvement in accuracy over Rolfs and Rodney (1974) and in precision over Hebbard (1960), they only cover a limited energy range. For SF II, the twolevel R-matrix fit of Mukhamedzhanov et al. (2008) was adopted, which simultaneously fit 15 N ( p, α ) 12 C data, giving S (0) = 36 ± 6 keVb for the 15 N ( p, γ ) 16 O reaction (see also Barker (2008)). Importantly, it showed that the extrapolated S (0) value of Rolfs and Rodney (1974), 64 ± 6 keVb, was an overestimate.  \nFollowing SF II, a comprehensive measurement campaign was carried out at LUNA and the University of Notre Dame (Leblanc et al. , 2010), covering a laboratory energy range from E lab = 130 to 1800 keV. In that work, a three-level R-matrix fit was used, but the ground state proton ANC of Mukhamedzhanov et al. (2008) was not considered, resulting in a somewhat larger extrapolated value of S (0) = 39 . 6(26) keVb than that adopted in SF II. Mukhamedzhanov et al. (2011) subsequently showed that a three-level R-matrix fit of equal quality could be obtained with the ANC, and reported an extrapolated value of S (0) that ranged from 33 to 40 keVb.  \nCaciolli et al. (2011) expanded the energy range of Bemmerer et al. (2009) by measuring the cross section up to E c.m. = 370 keV. Their experiment scanned over the top of the lowest-energy broad resonance at E c.m. = 312 keV. The data were found to be consistent with the ground  \nstate data of Leblanc et al. (2010).  \ndeBoer et al. (2013) performed a comprehensive Rmatrix fit that included not only the 15 N+ p data, but also a wide range of 12 C+ α data over a similar excitation energy range (see also Section XI.A.3). The value from this analysis, S (0) = 40 ± 3 keVb, was slightly above that of Leblanc et al. (2010) and agreed with the upper limit of Mukhamedzhanov et al. (2011). However, it is lower than the recent potential model extrapolation of S (0) = 45 +9 -7 keVb found in the NACRE II compilation (Xu et al. , 2013). The difference between the two fits is likely due to the fact that NACRE II disregarded the higher-energy part of the Leblanc et al. (2010) data, which maps the interference pattern between the two 1 -resonances more accurately than older data sets. In order to reproduce experimental data in that region, a 1 -background level was added in the R-matrix fits of Leblanc et al. (2010) and later works.  \nHere we adopt the value of S (0) = 40 ± 3 keVb from deBoer et al. (2013) as it considers the most comprehensive set of available data.  \nWhile low-energy future measurements are recommended, experiments at high energy are highly desirable to better understand the reaction mechanisms responsible for the interference pattern that is observed between the two well known 1 -resonances, which can not be completely reproduced unless some background contribution is also included.", '2. 16 O ( p, γ ) 17 F': 'Below a laboratory energy of about 2700 keV, i.e., the energy of the lowest-lying resonance (Tilley et al. , 1993), the 16 O( p, γ ) 17 F reaction is a prime example of the nonresonant direct radiative capture process. This reaction has been measured many times using a variety of techniques, including activation, prompt γ -ray detection, and inverse-kinematics experiments. The E 1 capture can proceed to the ground or first-excited state in 17 F, with the latter being the dominant transition. In SF II, a value of S (0) = 10 . 6 ± 0 . 8 keVb with a a 7 . 5 % uncertainty was adopted. This result was obtained by normalizing the microscopic-model calculations of Baye et al. (1998) to the data of Rolfs (1973) and Morlock et al. (1997).  \nArecent statistically rigorous evaluation of the S-factor data of Becker et al. (1982); Chow et al. (1975); Hester et al. (1958); Morlock et al. (1997) is presented in Iliadis et al. (2022). Their analysis, including a discussion of experimental uncertainties, used a combined fit of the transitions to the ground and first-excited states in 17 F, and their sum, and was performed using a Bayesian model. The physical model was a single-particle model employing a Woods-Saxon potential for generating the radial bound-state wave function. The fit had three adjustable parameters: the radius parameter and diffuse-  \nness of the Woods-Saxon potential, and the ANCs for scaling the theoretical non-resonant capture S-factors. It was also found that a poor fit was obtained when using spectroscopic factors instead as scaling parameters. As the 16 O( p, γ ) 17 F reaction at low bombarding energies is peripheral, the analysis of the S-factor in terms of ANCs greatly reduced the sensitivity of the fit to the single-particle potential parameters. For the ANCs of the ground and first-excited state transitions, they found values of C 2 gs = 1 . 115 fm -1 ( ± 4 . 0 %) and C 2 fes = 7063 fm -1 ( ± 4 . 0 %), respectively. These results agree with those measured in the 16 O( 3 He,d) 17 F reaction by Gagliardi et al. (1999). It is not surprising that the uncertainties ( ≈ 10 %) in the latter reference are significantly larger than for the values quoted above, because the distorted wave Born approximation (DWBA) analysis of Gagliardi et al. (1999) was subject to ambiguities in the choice of optical model potentials for the incoming and outgoing channels.  \nThe total S-factor at zero energy reported by Iliadis et al. (2022) was S tot (0) = 10 . 92 keVb ( ± 4 . 0 %). The uncertainty is significantly smaller compared to the results published by Iliadis et al. (2008), which were derived from an analysis that was not statistically rigorous. We adopt the results of Iliadis et al. (2022) in this work. Their S-factor is presented in Table XII.  \nWe endorse the two recommendations made by Iliadis et al. (2022). First, they advocated for a new low-energy measurement of the 16 O( p, γ ) 17 F reaction, specifically at center-of-mass energies below 200 keV, which have so far only been reached in the experiment of Hester et al. (1958), albeit with a relatively large systematic uncertainty of 14 %. Second, it would be interesting to compare their results to those from a future R-matrix analysis of the same data sets.', '3. 17 O ( p, γ ) 18 F': 'In SF II, only the non-resonant component of the 17 O( p, γ ) 18 F reaction cross section was discussed, which dominates the total cross section at solar temperatures. At the time, the results of Chafa et al. (2007) were recommended. Since the publication of SF II, several new S-factor measurements have been reported, both at low and high energies (Buckner et al. , 2015; Di Leva et al. , 2014; Gyürky et al. , 2017; Hager et al. , 2012; Kontos et al. , 2012; Newton et al. , 2010; Scott et al. , 2012). Despite differences in the techniques exploited and energy ranges covered, the derived S (0) values agree across different experiments, both for the non-resonant capture and the total S-factor, where the latter quantity includes the low-energy tails of two broad resonances ( E c.m. r = 557 and 677 keV). Here we recommend the total S-factor from the most recent analysis of (Gyürky et al. , 2017). The 17 O( p, γ ) 18 F reaction cross section was measured in  \nthe laboratory bombarding energy range of 500-1800 keV with a typical total uncertainty of 10% using the activation method. A comparison with other independent results, when possible, at E p lab = 500 keV resulted in good agreement with Hager et al. (2012) and Kontos et al. (2012), while a 20% difference is observed with respect to Newton et al. (2010). An R-matrix analysis was performed considering all the low energy data, resulting in a value of S (0) tot = (4.7 ± 1.0 ) keVb 8 .  \nAt temperatures near 60 MK, typical for hydrogenshell burning in AGB stars, the 17 O( p, γ ) 18 F reaction rate is dominated by the E c.m. r = 65 keV resonance, which was not considered in SF II. No direct measurements are reported for the resonance strength, ωγ , which is presently estimated from experimental values of the partial and total widths. Partial widths of Γ α = (130 ± 5) eV and Γ γ = (0 . 44 ± 0 . 02) eV were measured using the 14 N+ α reaction (Berka et al. , 1977; Mak et al. , 1980). The most uncertain quantity is the proton partial width, Γ p , which is estimated from the measured strength of the corresponding resonance in the ( p, α ) channel. Our best estimate for the ( p, α ) strength is ωγ pα = 4 . 6 +5 . 4 -1 . 2 × 10 -9 eV (see Table XIII and the following section for details), which leads to Γ p = 18 . 5 +21 . 7 -4 . 7 × 10 -9 eV. Hence, our recommended value is ωγ pγ = 15 . 6 +18 . 3 -4 . 0 × 10 -12 eV (see Table XIII). This value is consistent with the strength recommended by Palmerini et al. (2013), though we have inflated the error to account for the uncertain status of the proton width. A direct measurement of the ( p, γ ) strength of the E c.m. r = 65 keV resonance is ongoing at LUNA (Ciani et al. , 2022). Moreover, the 65 keV (1 -) resonance is expected to interfere with the -2.0 keV (1 -) subthreshold resonance. We recommend the treatment of the interference contribution described in Buckner et al. (2015). At typical temperatures of classical novae, the main contributor to the 17 O( p, γ ) 18 F reaction rate is the E c.m. r = 183.9 keV resonance, which was first observed by Fox et al. (2004). It was subsequently remeasured by Buckner et al. (2015); Chafa et al. (2007); Di Leva et al. (2014); Scott et al. (2012). Here we recommend the literaturedata weighted mean adopted in Buckner et al. (2015), ωγ = (1.77 ± 0.09) × 10 -6 eV. 9', '4. 17 O ( p, α ) 14 N': 'The 17 O( p, α ) 14 N reaction rate is determined by the contributions of narrow resonances. At solar energies,  \nthe dominant contributions originate from resonances at E c.m. r = -2 and 65 keV. At typical temperatures of AGB stars and classical novae, the main contributor to the rate is a resonance at E c.m. r = 183.9 keV. Concerning the subthreshold resonance, no new results have been published since SF II.  \nThe E c.m. r = 65-keV resonance has been studied recently both in a direct experiment ( ωγ = (10 . 0 ± 1 . 4 stat ± 0 . 7 syst ) × 10 -9 eV; Bruno et al. 2016) and with the Trojan Horse indirect Method ( ωγ = (3 . 42 ± 0 . 60) × 10 -9 eV; Sergi et al. 2015). The resonance strength reported by Bruno et al. (2016) deviates by a factor of 2.5 from that of Sergi et al. (2015) and a factor of 2 from the preSF II value of Fox et al. (2005), which includes a revised analysis of the data from Blackmon et al. (1995). Since the reason behind the discrepancies between literature strengths remains unclear, an independent investigation of this resonance would be desirable. Here we recommend a weighted average of the available literature strengths, ωγ = 4 . 6 +5 . 4 -1 . 2 × 10 -9 eV. The uncertainty was enlarged to include all literature values.  \nThe resonance at E c.m. r = 183.9 keV, on the other hand, is well known. The most recent determination of its strength is reported in Bruno et al. (2015), i.e., ωγ = (1 . 68 ± 0 . 03 stat ± 0 . 12 syst ) × 10 -3 eV, which is in excellent agreement with previous results (Chafa et al. , 2007; Moazen et al. , 2007; Newton et al. , 2007).', '1. 18 O ( p, γ ) 19 F': 'Important components for this reaction channel are a E c.m. r = 20 keV resonance, for which only indirect data exist (Champagne and Pitt, 1986; La Cognata et al. , 2008; Wiescher et al. , 1980), the resonances at E c.m. r = 90 keV and E c.m. r = 143 keV and the non-resonant capture process. Reported strengths of the E c.m. r = 20 kev resonance vary by one order of magnitude between 1 . 4 × 10 -22 (Lorenz-Wirzba et al. , 1979) and 1 . 9 × 10 -21 eV (Wiescher et al. , 1980), depending on the assumed proton widths. We recommend a value of ωγ = 5 . 7 × 10 -22 eV, based on the results of Champagne and Pitt (1986). A comprehensive measurement was done by Wiescher et al. (1980), covering the 80-2200 keV range at multiple angles and measuring branching ratios, the direct component (extrapolated from high-energy data), and strengths down to the E c.m. r = 143 keV resonance. The strength of the latter resonance, ωγ = (0 . 98 ± 0 . 03) × 10 -3 eV, has been remeasured multiple times with consistent results (Becker et al. , 1982; Best et al. , 2019; Dermigny et al. , 2016; Pantaleo et al. , 2021; Vogelaar et al. , 1990; Wiescher et al. , 1980). The strength of the E c.m. r = 90 keV resonance was recently controversial (Buckner et al. , 2012; Fortune, 2013), but a measure-  \nest et al. (2019) found it to be of insignificant strength ( ωγ = (0 . 5 ± 0 . 1) × 10 -9 eV). The non-resonant component was determined in Wiescher et al. at an energy of E c.m. = 1752 keV and extrapolated to a value of S (0) = 15 . 7 keVb. Buckner et al. normalized a nonresonant capture model calculation to the experimental value at E c.m. = 1752 keV, resulting in S (0) = 7 . 1 keVb. No uncertainties were given for these two S-factors. Best et al. (2019) extracted the non-resonant component from a cross section fit at center-of-mass energies between 85 and 150 keV, resulting in S (0) = (23 ± 3 . 8 ) keVb. Here we adopt this value, but further investigation of the lowenergy cross section is necessary to resolve these discrepancies.', '2. 18 O ( p, α ) 15 N': 'The low-energy trend of the 18 O( p, α ) 15 N cross section is determined by three resonances at 20, 90 and 143 keV (see Table XIII). The most recent determination of the strength of the 20 keV resonance comes from a THM experiment (La Cognata et al. , 2008). Their result was already recommended in SF II and we maintain the same recommendation. The resonance at 90 keV was investigated in the past with direct (Lorenz-Wirzba et al. , 1979) and indirect (La Cognata et al. , 2008) techniques, finding consistent results. Nevertheless, a new direct measurement reported a resonance strength one order of magnitude higher than the previously adopted value (Bruno et al. , 2019). Since no clear explanation can be found for such a discrepancy, here we adopt a weighted average of the three resonance strengths, but we inflate the uncertainty so that it embraces all three values. New experiments to solve this discrepancy are advised. The resonance at 143 keV is well known, and different experiments provided consistent results (Becker et al. , 1995; Bruno et al. , 2015; Lorenz-Wirzba et al. , 1979). Here we adopt a weighted average of Becker et al. (1995) and Bruno et al. (2015), which have independent systematic uncertainties (while no information on individual contributions to the total uncertainty is given in Lorenz-Wirzba et al. (1979)). The non-resonant differential cross section has recently been measured down to E c.m. = 55 keV by Bruno et al. (2019). At overlapping energies, the S-factor is consistent with the previous measurement by LorenzWirzba et al. (1979). Bruno et al. (2019) found a broad structure at E ≈ 110 keV that they could only explain assuming the existence of a so-far unobserved resonance at E c.m. r = 106 keV. Further investigations are needed to shed light on this supposed resonance.  \nThe 19 F ( p, γ ) 20 Ne reaction has been challenging to access experimentally because it has a much smaller cross section than the competing 19 F ( p, αγ ) 16 O reaction. The cross section of both reactions is dominated by a mixture of narrow and broad resonances corresponding to mostly unnatural parity states in 20 Ne (mainly J π = 1 + and 2 -). At very low energies, the cross section may be enhanced by a near threshold state at E c.m. ≈ 11 keV and subthreshold and non-resonant capture contributions may also contribute significantly, but have not yet been well characterized (deBoer et al. , 2021b), as few particle transfer studies have been published (Betts et al. , 1975; Kious, 1990).  \nThe first low-energy cross section measurement was that of Couture et al. (2008), who measured only the 19 F ( p, γ 1 ) 20 Ne transition. While this was likely a strong transition, there can certainly be other transitions with sizable contributions. Couture et al. (2008) covered the center-of-mass energy range from 200 to 760 keV, where three strong resonances were observed at E c.m. r = 323.9, 564 and 634 keV. Only an upper limit was reported for an expected resonance at E c.m. r = 213 keV. Also, no evidence of either a near-threshold state, subthreshold state, or non-resonant capture has been observed.  \nAn investigation of the γ -ray branching ratios of the E c.m. r = 323 . 9 keV resonance was undertaken by Williams et al. (2021), using the DRAGON recoil separator at TRIUMF. Recoil detection provided an effective method of discriminating the large background from the 19 F ( p, αγ ) 16 O reaction. It was found that the primary transitions to both the ground and the 4967 keV state dominate over the primary transition to the 1633 keV first-excited-state.  \nFinally, a new direct measurement of the low-energy cross section has been reported by Zhang et al. (2022), which overlaps with the previous low-energy data of Couture et al. (2008). A new resonance was discovered at E c.m. r = 225.2 keV in the 19 F ( p, γ 1 ) 20 Ne cross section, attributable to a 3 -state that corresponds to a weak resonance observed in the 19 F ( p, αγ ) 16 O data of Spyrou et al. (2000) at the same energy. This low-energy resonance increases the 19 F ( p, γ ) 20 Ne reaction rate by about an order of magnitude at T ≈ 0.1 GK. Like Williams et al. (2021), Zhang et al. (2022) also observed a sizeable ground state transition for the E c.m. r = 323 . 9 keV resonance. As in the 19 F ( p, αγ ) 16 O reaction (see Section XI.C.4), an upper limit was also estimated for the proposed E c.m. r = 11 keV resonance.  \nWe have therefore adopted the resonance energies and strengths of Zhang et al. (2022), which are in good agreement with the previous measurements of Couture et al. (2008) and Williams et al. (2021) in the overlapping energy region (see Tables XI and XIII).  \nThe 19 F( p, α ) 16 O reaction was not included in SF II. NACRE (Angulo et al. , 1999) pointed out that the total astrophysical S-factor was dominated by the nonresonant ( p, α 0 ) channel, though underscoring the large uncertainties ( ≈ 50% ) affecting the reaction rate at low temperatures. After NACRE, the situation greatly changed following the publication of a large number of papers addressing the measurement of the α 0 channel (Indelicato et al. , 2017; La Cognata et al. , 2011, 2015a; Lombardo et al. , 2013, 2015), of the αγ channel (Ding et al. , 2002; Fessler et al. , 2000; Pham et al. , 2021; Spyrou et al. , 2000; Zhang et al. , 2019, 2021b). In addition, several authors attempted to reach a consensus astrophysical factor carrying out refined analyses using, for instance, the phenomenological R-matrix (deBoer et al. , 2021b; He et al. , 2018; Lombardo, 2019; Lombardo et al. , 2019; Zhang et al. , 2021a), and carried out new theoretical calculations of the astrophysical S-factor (Sahoo and Basu, 2021).  \nRegarding the measurement of the astrophysical factor for the α 0 channel, populating the 16 O ground state, indirect (using the THM (Tribble et al. , 2014)) and direct methods were used, leading to consistent results. While the direct measurement (Lombardo et al. , 2015) could reach about 170 keV in the center-of-mass, the THM measurements (Indelicato et al. , 2017; La Cognata et al. , 2011, 2015a) could reach 5 keV, providing an extrapolated S (0) = (17 . 4 ± 2 . 7) × 10 3 keVb. The most striking feature of both direct and indirect measurements is the evidence of resonances populating the E c.m. /lessorsimilar 600 keV energy region, at odds with the NACRE extrapolation.  \nUntil Zhang et al. (2021b), direct measurements of the αγ channel stopped at energies well above the Gamow energy. Zhang et al. (2021b) could reach energies as low as ≈ 72 keV in the center of mass, still above the Gamow window for solar fusion, thanks to the use of the Jinping Underground Nuclear Astrophysics experimental facility (JUNA). Though the lowest energy point is affected by a large uncertainty, 87%, and no correction for the electron screening effect is reported (for instance, Assenbaum et al. (1987) assume a 20% enhancement at 70 keV centerof-mass energy for p -induced reactions on oxygen), the R-matrix analysis the authors carried out suggests the occurrence of a E c.m. r = 11 keV 1 + level at 12855 keV 20 Ne excitation energy, which might significantly enhance the astrophysical factor, leading to a S(0) ≈ 200 MeVb. If this result is confirmed, the αγ channel will dominate the low-energy total astrophysical S-factor (below about 50 keV in the center of mass) of the 19 F( p, α ) 16 O reaction. If the E c.m. r = 11 keV resonance is excluded from the fit, S(0) ≈ 1 MeVb would be deduced.  \nThe presence of new data sets triggered the publication of review papers aiming to provide the best fit of the  \navailable data. In particular, Lombardo et al. (2019) carried out a comprehensive R-matrix analysis spanning an energy range from about E c.m. = 200 keV to 12000 keV. The R-matrix fit predicts a S(0) ≈ 10 MeVb for the α 0 channel, with a conservative uncertainty of about 20%, lower than the value from Indelicato et al. (2017). An additional interesting result from this work is that the α 1 channel, corresponding to the population of the 0 + 16 O first excited state, is negligible with respect to the α 0 one below about E c.m. = 50 keV, yet if the 2 + state at 13095 keV is assumed to contribute, as supported by the 16 O( α, α 1 ) 16 O ∗ data, the α 1 channel may significantly contribute to the total astrophysical factor above about E c.m. = 50 keV, calling for new data at low energies.  \nThe most comprehensive R-matrix analysis, however not including the results in Zhang et al. (2021b), is the one carried out in deBoer et al. (2021b). For the α 1 channel, the results are consistent with Lombardo et al. (2019). For the αγ channel, the conclusions are that the α 2 channel is the major one and, as shown by Zhang et al. (2021b), low- and sub-threshold resonances can enhance the astrophysical factor to dominate over the α 0 channel at astrophysical energies. Finally, the result of the R-matrix analysis for the α 0 channel supports a flat S-factor devoid of resonances, in contrast with Indelicato et al. (2017); La Cognata et al. (2011, 2015a); Lombardo et al. (2013, 2015), in agreement with the analysis of Herndl et al. (1991). However, this is based on the data from Lorenz-Wirzba (1978), which were excluded from the NACRE compilation due to a likely large underestimate of reported absolute cross sections, by about a factor of 2.  \nIn summary, while the situation for the α 0 and α 1 channels is well constrained, the contribution of higher 16 O excited states and, in particular, of the α 2 channel, needs further studies to confirm the occurrence of the E c.m. r = 11 keV resonance. This represents, at present, the largest source of uncertainty. Therefore, the recommended reaction rate lower limit is conservatively set by the Indelicato et al. (2017) one, while the upper limit is given by Zhang et al. (2021a), assuming the existence of the E c.m. r = 11 keV resonance.', '1. 20 Ne ( p, γ ) 21 Na': 'The low energy S-factor of the 20 Ne ( p, γ ) 21 Na reaction is dominated by the high energy tail of a subthreshold resonance only a few keV below the reaction threshold. In addition, the non-resonant capture into two additional states is relevant and provides the largest contribution in the energy range where data are available. All three transitions have been measured over a wide range of energies by two direct experiments (Lyons et al. , 2018; Rolfs et al. ,  \nFIG. 10 Astrophysical S-factors of CNO and NeNa cycle reactions. The circles represent the values listed in Table XII. The solid lines represent polynomial (quadratic) fits of the S-factor values. The fit coefficients are listed in Table I. The S-factor of the 20 Ne ( p, γ ) 21 Na reaction cannot be accurately fit with a quadratic (or cubic) polynomial function.  \n<!-- image -->  \n1975), and the results can be used to arrive at S-factor recommendations. However, no consistent parametrizations have been published that could be used to compare the results.  \nHere, the S-factor data from these two experiments were either extracted from the published supplemental information, or digitized from published figures and normalized to the strength of the E c.m. r = 1113 keV resonance ( ωγ = 0 . 94 eV; Lyons et al. , 2018). The resonant capture into the tail of the subthreshold resonance ( E c.m. r = -7 . 9 keV; Firestone 2015) can be described by the Breit-Wigner equation, as shown in Iliadis (2015); Rolfs and Rodney (1988). To account, additionally, for the non-resonant capture to the ground state, a constant value of S ( E ) = 0 . 0463 keVb, as recommended by the indirect experiment of Mukhamedzhanov et al. (2006), was included in the fit to the data of Lyons et al. (2018); Rolfs et al. (1975). For the two direct-capture transitions, the energy dependence of the S-factor was calculated with the program DIRCAP (Iliadis, 2022), using a Woods-Saxon potential with a diffuseness of a = 0 . 65 fm to achieve Sfactor magnitudes comparable to the experimental data. These S-factors were fit individually to the experimental data and then summed to the total S-factor. The results are listed in Table XII. Uncertainties from the individual contributions were added in quadrature.  \nHowever, significant and unquantifiable uncertainties likely exist because the results of the two experiments differ systematically both in magnitude and energy dependence (Lyons et al. , 2018). This may be caused by experimental or data-analysis errors in either experiment, or in both. In addition, the experiments at low ener-  \ngies only measured in a small angular range and relied on sparse angular distribution measurements at higher energies to infer angular correlation effects. Also, the experiments did not reach sufficiently low energies to conclusively determine the potential parameters to be used in the DIRCAP calculation. Consequently, this will contribute to the S-factor uncertainty when the energy range of stellar burning, depending on the temperature, is located far below the experimental range.  \nA new measurement of the 20 Ne ( p, γ ) 21 Na directcapture cross section down to E c.m. = 250 keV has recently been published by the LUNA collaboration (Masha et al. , 2023). The potential-model results agree with the new experimental data.', '2. 21 Ne ( p, γ ) 22 Na': 'At temperatures below 0.1 GK, the reaction rate is dominated by three narrow resonances at 16.6, 94.6 and 126 keV. The 16.6 and 94.6 keV resonances have never been observed directly in the ( p, γ ) channel (Görres et al. , 1982). Upper limits on their strengths were derived from a 21 Ne(d, p ) 22 Na experiment (Iliadis et al. , 2001; Neogy et al. , 1972). Results are reported in Table XIII. The nonresonant capture S-factor at zero energy was estimated by Rolfs et al. (1975) using single-particle spectroscopic factors of the bound states in 22 Na. According to the authors, the result, S(0) ≈ 20 keV b, is uncertain by a factor of three.', '3. 22 Ne ( p, γ ) 23 Na': 'The 22 Ne ( p, γ ) 23 Na reaction used to be the most uncertain of the NeNa cycle, because of a number of poorlyknown low-lying resonances. At temperatures below 30 MK, the reaction rate is dominated by a resonance at 36 keV. Its strength has been determined from a protontransfer experiment (Hale et al. , 2001; Iliadis et al. , 2010). Two near-threshold resonances are expected at 5 and 28 keV, but the existing upper limits on their strengths are so low that they do not contribute significantly to the reaction rate (Hale et al. , 2001). Two supposed resonances at 68 and 100 keV, first reported in Powers et al. (1971), were discussed at length in the literature and searched for both with direct experiments (Cavanna et al. , 2015; Depalo et al. , 2016; Ferraro et al. , 2018), proton transfer reaction (Hale et al. , 2001) and inelastic scattering (Carrasco-Rojas et al. , 2023). None of such experiments lead to a positive detection of the two resonances. The latest direct upper limits on their strength suggest that, even if the two resonances exist, their contribution to the mean (recommended) thermonuclear reaction rate is negligible (Ferraro et al. , 2018). A resonance doublet dominates the reaction rate at temperatures around 0.1 GK.  \nThe strength of the stronger resonance in the doublet was measured both with direct (Cavanna et al. , 2018; Kelly et al. , 2017; Lennarz et al. , 2020; Williams et al. , 2020) and proton-transfer experiments (data from Hale et al. (2001); Powers et al. (1971) re-analyzed in Santra et al. (2020)). The value adopted in Table XIII is an average of all literature values mentioned above. The strength of the weaker resonance in the doublet was inferred from proton-transfer data (Santra et al. , 2020).  \nAt 0.2 GK, the reaction rate is mainly determined by a resonance at 181 keV. Its strength was measured in four independent experiments (Cavanna et al. , 2018; Ferraro et al. , 2018; Kelly et al. , 2017; Williams et al. , 2020). The strength recommended in Table XIII is a weighted average of all four values. The most intense higher-lying resonances, relevant to nova and supernova explosions, have been recently addressed in Depalo et al. (2015); Kelly et al. (2017); Longland et al. (2010); Williams et al. (2020). The non-resonant capture cross section has been measured by different groups (Ferraro et al. , 2018; Kelly et al. , 2017; Williams et al. , 2020). The low-energy trend of the ground state capture is described by a constant value of S = (50 ± 12) keVb, plus the contribution of a broad subthreshold state at -130 keV (E x = 8664 keV) (Ferraro et al. , 2018).', '4. 23 Na ( p, γ ) 24 Mg': 'The 23 Na ( p, γ ) 24 Mg reaction has seen several recent studies (Boeltzig et al. , 2019, 2022; Caciolli et al. , 2011; Marshall et al. , 2021). The cross section is dominated by three narrow resonances over the lowest experimentally measured energy range, located at E c.m. r = 133, 240.6, and 295.9 keV. Below the lowest known resonance, nonresonant components are thought to dominate.  \nSeveral indirect measurements have been made to characterize energies and proton spectroscopic factors of bound and unbound states in 24 Mg. The measurement of Hale et al. (2004) used the 23 Na ( 3 He,d) 24 Mg reaction to investigate unbound states (11690 < E x < 12520 keV) near the proton separation energy ( S p = 11693 keV). For lower-energy bound states that were not measured, the spectroscopic factors of Garrett et al. (1978) where adopted. A direct capture model was used to calculate the non-resonant component of the cross section. This was followed by an investigation of the lowest energy resonance at E c.m. r ≈ 133 keV by Cesaratto et al. (2013), where an upper limit for the resonance strength was determined. A subsequent study at the LUNA facility (Boeltzig et al. , 2019) was able to make a first measurement of the resonance strength, 1.46 +0 . 58 -0 . 53 × 10 -9 eV. A recent 23 Na ( 3 He,d) 24 Mg transfer study at the Triangle Universities Nuclear Laboratory (TUNL) revised the energy of the ≈ 133 keV resonance to E c.m. r = 133(3) keV (Marshall et al. , 2021).  \nDirect cross section measurements of the 23 Na ( p, γ ) 24 Mg reaction have been made at the University of Notre Dame at energies between E lab = 500 and 1000 keV by Boeltzig et al. (2022). Two, broad, strong, resonances were observed in several γ -ray transitions at E c.m. r = 841 and 981 keV whose low energy tails contribute in the low energy region. Note that the energy uncertainties presented here reflect the uncertainty in the beam calibration of the accelerator, which was neglected in Boeltzig et al. (2022). Further, measurements of the E x = 10740 keV primary transition in 24 Mg, predicted to have the largest non-resonant capture component based on the spectroscopic factors of Garrett et al. (1978), observed a trend in the off-resonance cross section that was highly suggestive of non-resonant capture. Given the level of consistency observed between the non-resonant capture S-factors calculated from the spectroscopic factors of Garrett et al. (1978) and the external capture S-factors from the R-matrix analysis, an uncertainty in the non-resonant S-factor of 50% was recommended.  \nWe have adopted the non-resonant component of Boeltzig et al. (2022) in this work, which, like all previous calculations, relies heavily on the spectroscopic factors of Garrett et al. (1978).', '5. 23 Na ( p, α ) 20 Ne': 'The 23 Na ( p, α ) 20 Ne reaction rate is dominated by narrow resonances, with a total of 52 resonances in the center-of-mass energy interval 6 -2328 keV.  \nAfter NACRE, the state-of-the-art tabulation of the resonance strengths is the one in Iliadis et al. (2010). It is mostly based on the ( 3 He,d) proton transfer measurement by Hale et al. (2004), supplying both the excitation energies and spectroscopic factors of the near-threshold states. In detail, below E c.m. = 968 keV, all the resonance strengths are taken from Table VI of Hale et al. (2004), with the exception of the E c.m. r = 133 keV resonance strength (Marshall et al. , 2021), for which the upper limit directly determined by Iliadis et al. (2005); Rowland et al. (2004) is adopted, equal to 1 . 5 × 10 -8 eV. Above E c.m. = 968 keV, the Hale et al. (2004) strengths are rescaled to the one of the E c.m. r = 338 keV resonance determined by Rowland et al. (2002). This result is considered as reference value since it provides the most recent directly-measured strength.  \nBesides the above-mentioned works, two more should be considered. One is Iliadis et al. (2005), which is not used in Iliadis et al. (2010) because the same upper limit for the E c.m. r = 133 keV resonance as in Rowland et al. (2004) is reported. The other work is Cesaratto et al. (2013), which used the upper limit on the ( p, γ ) resonance strength and the upper limit on the α branching ratio (equal to 0.13 at the 95% CL), to provide an up-  \nr limit on the ( p, α ) strength of the E c.m. r = 133 keV resonance equal to 0 . 88 × 10 -9 eV. This implies a negligible contribution to the reaction rate and confirms the conclusions drawn in Iliadis et al. (2010).', 'XII. Electron screening of nuclear reactions': 'Nuclear fusion reactions measured in the laboratory and those occurring in the solar core are both affected by the electronic environments in which they take place. The effects are, however, different requiring special care in deriving the appropriate solar reaction rates. As long as these rates cannot be measured under solar plasma conditions in the laboratory, the usual two-step strategy is to remove the laboratory screening effects from the data to obtain the bare nuclear cross section σ b , which is then modified to incorporate the solar plasma screening modifications. Thus, one must address the differential effects of screening for the solar cross section σ solar and the laboratory results, σ lab .', 'A. Screening in laboratory experiments': 'Screening diminishes the Coulomb barriers that retard interactions between bare nuclei, thus enhancing cross sections, with the effects becoming more pronounced with decreasing center-of-mass energy. Nuclear astrophysics has advanced rapidly over the last two decades with the deployment of underground accelerators and other lowbackground techniques (Broggini et al. , 2010) that have enabled low-energy, low-counting-rate experiments. In particular, laboratory measurements of the pp-chain reactions 2 H( p, γ ) 3 He and 3 He( 3 He,2 p ) 4 He have been made at energies corresponding to the solar Gamow peak. A treatment of the distorting effects of screening then becomes quite important in extracting the bare S-factor S b ( E ) .  \nThe bare cross section is given by  \nσ b ( E ) = 1 E S ( E ) P ( E ); P ( E ) ≡ exp( -2 πη ( E )) (24)  \nwith η ( E ) = Z 1 Z 2 α √ µ/ 2 E , where µ is the reduced mass and E the center-of-mass energy. The effects of screening can be incorporated into the Gamow penetration factor P ( E ) through an energy shift, E → E + U e , where the screening potential U e has the net effect of lowering the Coulomb barrier. At the lowest accessible energies in laboratory measurements, the separation of target and projectile during tunnelling is much smaller than the atomic radius: thus the electrons, originally bound to the target (assuming a projectile beam of bare nuclei), are attracted by the joint charge of target and projectile during the fusion. The screening potential can then be replaced by a constant screening energy U e . This picture was confirmed  \nTABLE XI Overview of CNO and NeNa reactions.  \nTABLE XII S-factors versus center-of-mass energy. Data are also plotted in Fig. 10.  \nin simulations of the electron dynamics in low-energy fusion reactions (Shoppa et al. , 1993). After expanding the argument of the exponential in Eq. 24 to first order in U e one obtains the following relation between the laboratory and bare S-factors (Assenbaum et al. , 1987)  \nS lab ( E ) = S b ( E ) exp ( πη ( E ) U e E ) (25)  \nThis result neglects a small normalization correction associated with the effects U e on the matching of the external wave function to that in the Coulomb region.  \nThe 1 /E dependence of the exponent is responsible for the growing importance of the screening correction at low energies. Although the screening potential U e in principle  \ncan be calculated from the electron charge distribution, in most applications it is treated as a free constant parameter, determined along with S b (0) and its derivatives from a fit to data. Figure 4 of SF II shows the results for S 33 ( E ) , where the fit yielded U e = 305 ± 90 eV. The exponential dependence of the screening correction leads to a rapid increase in S lab 33 ( E ) relative to S b 33 , reaching ∼ 40%, at the lowest data point measured. This example also underscores the potential for screening to complicate the extraction of S b ( E ) for energies relevant to the solar core, even in cases where measurements are restricted to higher energy than the Gamow window.  \nIn principle one can test the adequacy of Eq. 25 by doing laboratory experiments at low energies - even if  \nTABLE XIII Recommended strengths and relative kinetic energies of narrow resonances.  \nthe reactions studied are not directly relevant to nuclear astrophysics. Such studies began more than 30 years ago (Assenbaum et al. , 1987), using gaseous targets to limit energy loss effects and other systematics. Subsequent studies have varied greatly in their configurations, e.g., targets ranging from atomic and molecular gases to metals, 10 thin and thick targets, and use of direct and inverse kinematics (Aliotta et al. , 2022b). The experiments can be very challenging at the energies needed, subject to  \nbackgrounds from both intrinsic activities in detectors and external sources such as cosmic rays.  \nThe values of U e extracted from experiment using gaseous targets have often exceeded the adiabatic limit, defined in atomic physics as the difference between the electron binding energies of the separate atoms in the entrance channel and that of the composite atom (Aliotta et al. , 2001; Angulo et al. , 1993; Engstler et al. , 1988, 1992; Greife et al. , 1995; Prati et al. , 1994; Rolfs, 2001; Rolfs and Somorjai, 1995). This in turn has generated unease about the reliability of the values for S b ( E ) extracted from laboratory measurements for use in astrophysics (Balantekin et al. , 1997).  \nSeveral suggestions have been made to resolve this apparent discrepancy. For example, an error in the estimated stopping power could result in a smaller reaction effective energy, so that the experimentally deduced screening would be larger than the true value (Bang et al. , 1996; Langanke et al. , 1996). Indeed such deviations have been found between theoretical calculations of stopping powers and the tabulation which is traditionally being used in the analysis of low-energy fusion data (Bertulani, 2004; Bertulani and de Paula, 2000; Golser and Semrad, 1991). An accurate experimental determination of stopping powers for gaseous hydrogen and helium targets would be very desirable. Corrections associated with the non-uniform distribution of the electron cloud around the nucleus (Carraro et al. , 1988) and the effects of electronelectron interactions (Shoppa et al. , 1993) have also been suggested as potential generators of unidentified systematic error. For molecular targets the screening energy shows a significant dependence on the scattering angle, being smallest if the projectile passes the spectator nucleus before fusion (Shoppa et al. , 1996). Furthermore, the experimental deduction of screening energies requires assumptions about the bare nuclear cross sections, which is another source of uncertainty. This was demonstrated in Tumino et al. (2021, 2014), where after constraining fits by the addition of higher energy data (from a Trojan Horse measurement), a lower value of U e was obtained. Clusterization and polarization effects have also been proposed to effect the deduced screening energies in specific low-energy reactions (Spitaleri et al. , 2016, 2017)  \nFor resonances, both the energy and width can be affected by screening (Salpeter, 1954; Zinner, 2007). Particular care has to be taken for narrow, low-energy resonances where the assumption of a constant screening energy is not valid and the radial dependence of the screening potential has to be considered which leads to a significant reduction of screening on the resonance width (Iliadis, 2023).  \nParticularly large screening energies were observed in deuteron- and proton-induced fusion reactions if the projectile was implanted into metals (Czerski et al. , 2001; Raiola et al. , 2002a,b; Kasagi et al. , 2002; Czerski et al. , 2006; Cvetinovic et al. , 2015; Lipoglavsek and Cvetinović, 2020). While the d+d fusion data obtained with a gaseous target (Greife et al. , 1995) were compatible with the adiabatic limit ( U e = 20 eV, Bracci et al. , 1991, 1990), screening energies obtained for various metal hosts varied among themselves and could reach values of several 100 eV. As the adiabatic limit derived for the fusion on individual atoms in a gas, does not apply to reactions with targets implanted in a host material, an appropriate model has been developed (Czerski et al. , 2004; Huke et al. , 2008) which agrees nicely with recent screening energies obtained under high-vacuum conditions (Czerski, 2022; Czerski et al. , 2016). Huke et al. (2008) also identified oxygen and carbon contamination as the source for  \nsome particularly large deduced screening energies. We note that the studies of d+d low-energy fusion were motivated by the quest to enhance nuclear fusion rates by environmental effects (Czerski, 2022).  \nThe potential discrepancy between observed and theoretical screening energies would become astrophysically irrelevant if it were possible to measure reaction rates directly under solar plasma conditions. This may prove possible using high-intensity lasers. First measurements of light-ion reactions in plasma have been performed, though under plasma conditions for which screening effects are expected to be negligible (Barbui et al. , 2013; Casey et al. , 2023; Zylstra et al. , 2016, 2020). A direct measurement of bare nuclear cross section, thus avoiding screening effects, is proposed for storage ring experiments, in which a stored beam of ions can collide with a transverse beam (Glorius and Bruno, 2023).', 'B. Screening in the solar core': 'The energy-generating reactions that occur in the solar core involve nuclei that are almost completely ionized. Consequently the atomic environment differs substantially from that of the terrestrial experiments in which these same reactions are measured. Once S b ( E ) is determined from these laboratory measurements, the screening effects of the plasma must be folded in to yield S solar ( e ) . Adopting the notation of SF II, the correction factor is defined as  \nf 0 ( E ) = σ solar ( E ) σ b ( E ) (26)  \nIn the weak screening approximation (Salpeter, 1954), the ion-ion Coulomb potential is screened on a length scale given by the Debye radius R D  \nV ( r ) = αZ 1 Z 2 r exp ( -r R D ) (27)  \nyielding  \nf 0 ∼ exp ( αZ 1 Z 2 R D kT ) (28)  \nWe refer the reader to SF II for the functional dependence of R D on temperature, density, and the mass fractions X i , and for discussions of corrections for electron degeneracy and incomplete ionization.  \nSF II discusses the conditions under which the weak screening approximation is valid: under solar core conditions this requires Z 1 Z 2 < 10 , a condition satisfied by pp-chain and CNO bi-cycle reactions. Yet corrections are expected at some level, as the inter-ion potential is not completely negligible, in all collisions, compared to the relative kinetic energy of the ions (Bahcall et al. , 2002; Brown and Sawyer, 1997; Carraro et al. , 1988; Fiorentini  \net al. , 2003; Itoh et al. , 1979; Ogata, 1997; Salpeter, 1954; Salpeter and van Horn, 1969).  \nDynamic corrections -nonadiabatic effects that arise when the velocities of reacting nuclei momentarily exceed typical plasma velocities- have been a point of some controversy that both SF I and SF II addressed. However, the absence of such dynamic corrections even for large Gamow energies has been shown to be a consequence of the nearly exact thermodynamic equilibrium of the solar plasma (Brown and Sawyer, 1997; Gruzinov, 1998; Gruzinov and Bahcall, 1998). Specifically, Bahcall et al. (2002) found that corrections to the Salpeter formula under solar conditions would be at the ∼ few per cent level, and pointed to specific errors in several papers that had come to contrary conclusions.  \nThere have been a few recent papers advocating changes in solar rates due to dynamical corrections, including Vescovi et al. (2019), Wood et al. (2018), Simonucci et al. (2013), Mussack (2011), Mussack and Däppen (2010), and Mao et al. (2009). The approaches employed are based on modeling the plasma, and in general the authors do not relate their conclusions to earlier work, particularly Bahcall et al. (2002), making them difficult to evaluate. The essential point of Bahcall et al. (2002) is the conceptual simplifications that result from recognizing that screening in the solar plasma can be formulated as a problem in equilibrium statistical mechanics. This approach removes all need to classified rates as fast or slow relevant to some plasma timescale, as there is no time in equilibrium statistical mechanics. Consequently we continue to regard Bahcall et al. (2002) as the most realistic estimate of corrections to the Salpeter formula.', 'B. Opacity models': "Over the years, a wide range of approximations has been used for the various components of opacity calculations. To help navigate these approximations, it may be useful to briefly summarize the key components of a frequency-dependent opacity. Opacities are often broken down into various components. The first of these is 'bound-bound' absorption (a photoexcitation transition between two levels of an atomic ion), which manifests as strong line features in the opacity spectrum (such as the black 'spikes' in Fig. 11). The 'bound-free' absorption (photoionization of the ground or excited level of an atomic ion) is normally a continuous (wavelengthdependent) contribution (such as the smooth (red) hydrogen opacity contribution in the middle and lower panels of Fig. 11). The 'free-free' absorption (also known as inverse Bremsstrahlung) is where a free electron absorbs a photon, and finally 'scattering' (the scattering of a photon by a free electron) contributions can be important at high photon energies. Other contributions may be important in molecules or other material. Equation 30 shows that due to the harmonic mean inherent in the RMO, the most important photon energy range is when the spectral opacity is low and when the weighting function w ν is high. In the solar case, at a given solar radius, it is important to know which elements dominate the opacity at such a region, which will of course depend on the elemental abundance and its contribution at that distance, as illustrated in Fig. 11.  \nThe determination of opacities usually starts from one of two fundamental approaches. The first approach, which was used in the first opacity computations (Huebner and Barfield, 2014), starts from a 'mean ion' model, where a (fictitious) 'average' atom or ion of a given element is conceived, with fractional occupation numbers, which is consistent with the given temperature and density of the species under consideration. The fractional occupation numbers, usually determined through a selfconsistent iterative approach, inform the real (physical) ion stages of importance at such conditions. The average  \natom can then be unfolded through various procedures to the physical ion stages of relevance, and the opacity of such ions can be determined. In this model the plasma effects (in particular the perturbation of the average atom by the fields produced by the surrounding plasma ions and electrons) can be included in a natural way, and this approach is often most useful at high densities, where plasma perturbations on the atoms or ions are most important. The infinity of atomic states are naturally truncated by plasma effects that remove bound states from consideration. The old Inferno model (Liberman, 1979) and the more recent STAR tables (Krief et al. , 2016b, 2018a) are based on this method. We note that very recent work (Gill et al. , 2023) has considerably improved some of the approximations used in related approaches.  \nThe second approach is often referred to as detailed configuration (or term) accounting, where one determines all possible ion stages of the species and then determines all possible ground and (multiply) excited states of each ion, each with a corresponding population that is condition dependent. This method is often more spectroscopically accurate than the mean ion model, because it considers the structure of each ion separately, but can require enormous computational resources for complex species. The structure of each ion is determined from solution of the Schrödinger (or Dirac) equation, which can then also produce transition probabilities. Within this procedure, many computational difficulties must be overcome to produce acceptably accurate data for all ion stages of all relevant elements. A few examples are given for illustration. Within the solution of the Schrödinger equation it is desirable to include 'configuration-interaction' (CI), a two-body term in the Hamiltonian that accounts for electron-electron interactions (Cowan, 1981), which results in a quite accurate description of atomic energy levels. However, the inclusion of this term for atomic ions with large numbers of configurations can quickly result in a computationally intractable problem. Therefore, approximations in which CI is included in only a limited manner are often employed. Furthermore, careful consideration has to be given to the number of states that should be included in the calculations. A compromise is always necessary between including as many states as possible versus the computational resources available for the calculation. Convergence checks are necessary to ensure that sufficient states are included. However, the rate of convergence of the opacity with respect to the number of states included will depend on the plasma conditions of interest, so this in itself is not necessarily a straightforward matter. The computer codes that are used to generate the opacity must also be able to efficiently process large amounts of atomic data.  \nSimilarly, when considering photoexcitation and photoionization of atomic ions, it is desirable to include coupling of the bound electrons to the continuum electron(s). Often termed 'close-coupling', this may be ac-  \ncomplished through techniques such as the R-matrix approach (Burke, 2011; Seaton, 1987) or other related methods. However, such approaches are also computationally intensive, and can be severely limited in the number of terms that can be included in the resulting close-coupling expansions. It is also not clear how to include plasma perturbations within such approaches. As a result, perturbative approaches, such as distorted-wave methods (Badnell et al. , 2005; Cowan, 1981; Sampson et al. , 2009), are often employed, which are much more computationally tractable, and of sufficient accuracy for mid- and highlyionized ions.  \nIn concert with an atomic model, the thermodynamics of the system must be considered. That is, an equationof-state (EOS) description of the material is required so that the thermodynamic properties and atomic state populations can be determined for a given material temperature and density. In the mean ion approach the thermodynamic quantities arise naturally through the description of the atom/ion in a plasma. In a Detailed Configuration Accounting approach, a common approach to EOS calculations starts from a chemical picture based on a minimization of the chemical free energy (Däppen et al. , 1987; Hakel and Kilcrease, 2004; Hummer and Mihalas, 1988; Rogers and Iglesias, 1992). Such a model has to treat a wide range of physical conditions, ranging from ideal gas through high density conditions where pressure ionization may rapidly strip ions of all their electrons. The number of atomic states retained in the partition function is a key quantity in determining if the EOS is complete at the conditions of interest. This assessment of completeness will of course strongly depend on the plasma conditions.  \nThe final important aspect of opacity models that we wish to briefly discuss concerns line broadening. Photoabsorption lines in a plasma become broadened due to a number of factors, such as natural and Doppler broadening. Collisional broadening (often known as Stark broadening) is often a dominant broadening mechanism. This is broadening due to the plasma microfields caused by the motion of electrons and ions in a plasma. As before, line broadening is an important part of any opacity model, but any treatment of this effect has to be tractable for all ions and all conditions encountered when building an opacity table. While quite accurate line broadening models exist for one- and two-electron systems (Gomez et al. , 2022; Lee, 1988; Stambulchik et al. , 2019), the treatment of broadening for multi-electron systems often requires a number of approximations (Dimitrijevic and Konjevic, 1986, 1987).", 'C. Production of tables': "Some of the earliest determinations of opacity for stellar physics started by the pioneering work of Seaton and  \ncollaborators in the UK (Berrington et al. , 1987; Seaton, 1987). This development led to the 'Opacity Project' (OP), which in time produced a set of opacity tables for stellar modeling that are still in common use today (Badnell et al. , 2005) At around the same time, OPAL (OPacity Astrophysical Library) opacity tables were published by Lawrence Livermore National Laboratory (LLNL) (Iglesias and Rogers, 1996). Both of these sets of tables were widely adopted by the stellar modeling community. Some early theoretical opacity work was performed at Los Alamos National Laboratory (Huebner and Barfield, 2014), starting from pioneering work by Mayer and later by Arthur Cox in the 1960s. In the 1990s, the LANL work resulted in 'OPLIB' (OPacity LIBrary) tables produced from the LEDCOP code (Magee et al. , 1995); in the mid 2010s the modern ATOMIC code was used to produce a new generation of opacity tables that improved the computation of opacities in a number of ways (Colgan et al. , 2016). All of these tables are available electronically in various formats. Other important opacity tables have been produced in France (the OPAS and SCO-RCG opacity efforts) (Blancard et al. , 2011; Mondet et al. , 2015; Pain, 2021), Israel (using the STAR code) (Krief et al. , 2016b, 2018a), and elsewhere. The comparison of opacity models with each other and with experiment (when available; see following sections) have been explored in dedicated workshops and conferences, for example Mendoza et al. (2018).  \nMany of these tables (primarily OP and OPAL) were quickly adopted by the solar modeling community and were useful in solving the Cepheid variability puzzle (Simon, 1982), where updated opacities helped resolve a previous disagreement between observation and modeling of Cepheid variables. Detailed comparisons between these two sets of tables have been made on numerous occasions, both on comparisons of individual opacities (Badnell et al. , 2005) and on how solar models that use these tables compare (Delahaye and Pinsonneault, 2006). Overall, the agreement between the various opacity tables was deemed to be 'satisfactory' (within 5 %) for the accepted solar models of the late 1990s (Magee et al. , 1995). This paradigm was shaken when new determinations of the elemental solar abundances were reported (Asplund et al. , 2005), which implied that solar models disagreed with helioseismology observations. It was suggested (Serenelli et al. , 2009) that changes in the opacity of a few species (mainly Fe, Ne, O) could reconcile the solar models with the observations. Since then, intensive work has been performed in exploring some of the approximations made in the older opacity tables. The OP effort has focused on new and larger R-matrix calculations for important ions of Fe (Delahaye et al. , 2021; Nahar and Pradhan, 2016), primarily Fe 17+ . At Los Alamos, the modern ATOMIC code was used to create a new generation of opacity tables for H through Zn (Colgan et al. , 2016). Although significant improvements in  \nmany aspects of the calculations were incorporated, the resulting opacity tables did not significantly improve the comparison of solar modeling with helioseismology. The study of Krief et al. (2016a) showed that one possible way to produce larger opacities under solar conditions would be if line broadening effects were significantly increased - by several orders of magnitude. Opacity increases of ≈ 10% near the solar convective zone and ≈ 2% in the solar core have been attributed to ionic correlations (Krief et al. , 2018b). There is need for further investigation of such topics.", 'D. Experimental testing of calculated iron opacity': "Errors in calculated solar opacity can arise from inaccuracies in both the abundance and calculated element opacities in a complex way. Therefore, it is important to investigate the accuracy of abundance and elemental opacities separately. The accuracy of elemental opacities can be best evaluated by examining the frequencyresolved (monochromatic) opacity. Since each spectral feature of the element opacity often relies on different physics and approximations, disagreements between model predictions and experimental data on a frequencyresolved basis can help identify which approximations are invalid and guide refinements in opacity theory. Here, we provide an overview of the fundamentals as well as previous investigations into measurements of solar-interior opacity. A more general and tutorial discussion of the experimental methods and a historical overview can be found elsewhere (Bailey et al. , 2009).  \nThe material opacity κ ν is related to its transmission T ν by the following equation:  \nT ν = I ν -/epsilon1 ν B ν -/epsilon1 ν = e -κ ν ρL (33)  \nwhere B ν is the backlight spectrum, I ν is the backlight transmitted through the sample, /epsilon1 ν is a sum of the sample plasma emission and other backgrounds, and ρL is the areal density of the opacity sample. Thus, sample opacity κ ν can be experimentally determined by heating the sample and accurately measuring the following quantities: the backlight with and without the heated sample ( B ν , I ν ), the plasma self-emission and background ( /epsilon1 ν ), and the sample areal density ( ρL ).  \nReliable and useful opacity measurements require meeting several challenging criteria [Perry 1996, Bailey 2009]. First, the elements, conditions, and spectral ranges of interest must be identified. Second, a macroscopic opacity sample must be uniformly heated to the desired conditions to achieve local thermodynamic equilibrium. Third, the heated sample must be backlit with a bright and spectrally smooth radiation, B ν , to accurately determine frequency-resolved absorption. It is critical that the backlight is significantly brighter than the  \nFIG. 12 Comparison of measured opacity (red) and calculated opacity (blue, Badnell and Seaton, 2003), at a temperature of 182 eV and an electron density of 3 . 1 × 10 22 e cm -3 . Three types of measured-vs-modeled opacity disagreements are shown: (i) a lower quasi-continuum, (ii) narrower boundbound line features, and (iii) deeper opacity windows. These trends were observed in all opacity models compared in Bailey et al. (2015)  \n<!-- image -->  \nsample plasma emission or other background signals, /epsilon1 ν . Fourth, the absorption spectrum must be recorded with spectrometers that provide sufficient signals and spectral resolving power for the opacity study. Fifth, the condition of the heated sample ( T e , n e , ρL ) must be diagnosed independently of the opacity in question. Simulated conditions are not appropriate for this purpose since their accuracy depends heavily on the accuracy of the opacity in question. Finally, the opacity spectrum and its uncertainty must be accurately determined from the measured I ν , B ν , /epsilon1 ν , and ρ ν and their uncertainties.  \nAfter the revision of solar abundance in 2005 (Asplund et al. , 2005), solar models and helioseismology disagreed, and the accuracy of solar-interior opacity was called into question. This disagreement was primarily due to a significant reduction in solar opacity resulting from the reduced metallicity, and it was the greatest at the base of the solar convection zone (hereafter convection-zone base or CZB). Since then, solar abundance has been continuously revised, with the latest abundances determined by two groups being Z/X=0.0225 (Asplund et al. , 2021) and Z/X=0.0187 (Magg et al. , 2022), which still disagree with each other, leaving significant uncertainty in the solar abundance. For determining the accuracy of calculated solar opacity, the accuracy of calculated element opacities must also be experimentally scrutinized. Since Fe and O are the two dominant sources of opacity at the CZB, their opacities must be experimentally tested at CZB conditions (i.e., T e = 182 eV, n e = 9 × 10 22 e cm -3 ). Unfortunately, previous experimental approaches were not suitable for this purpose because their backlighters were not bright enough to mitigate the bright self-emission produced at these conditions.  \nOver the past two decades, experimental methods have been refined using one of the brightest x-ray sources, Zpinch. In 2007, iron opacities were successfully measured at 150 eV and 7 × 10 21 e cm -3 , providing support for  \nthe accuracy of calculated Fe opacities under these specific conditions. In 2015, iron opacities were measured at larger temperatures and densities. This study revealed a significant discrepancy between the calculated and measured opacities as the temperature and density approached the CZB conditions, as shown in Fig. 12. This is a significant concern since the experimental density was still approximately three times lower than the actual density at the CZB. Opacity models employ approximations to account for density effects such as line broadening and pressure ionization. If these approximations are incorrect, the disagreement between calculated and measured opacities could be even more pronounced at the densities found at the CZB.  \nThe disagreement observed in the iron opacity was complex, as depicted in Fig. 12. The modeled opacities were lower in the quasi-continuum region at short wavelengths ( < 10 Å), narrower in bound-bound lines, and deeper in the opacity valleys known as windows. These discrepancies are likely caused by various factors, but it is difficult to disentangle different sources of discrepancies based solely on the disagreement in iron opacity alone.  \nIn 2019, a comprehensive study was conducted to measure opacities for Cr, Fe, and Ni at solar interior temperatures (Nagayama et al. , 2019). This systematic study helped narrow down hypotheses on the sources of disagreement by observing how the discrepancies change as a function of atomic number. The measurements revealed that measured opacities for all three elements had narrower line features than the model predictions, suggesting potentially inaccurate density effects in opacity models. Disagreement in the opacity windows was observed for Cr and Fe, while Ni did not show significant disagreement. At the experimental conditions used in the study, Cr and Fe were in an open L-shell configuration, whereas Ni was closer to a closed L-shell configuration. It is known that the calculations of both population and equation of state (EOS) are more challenging in open L-shell configurations. Therefore, the observed trend in window disagreement may suggest a connection to the difficulties associated with EOS calculations in such configurations.  \nOne intriguing finding from this systematic study was the disagreement observed in the quasi-continuum region. Unlike the discrepancies observed in bound-bound linewidth and window opacities, no clear trend was observed in the quasi-continuum disagreement. The model and data agree well for Cr and Ni, but not for Fe, at high temperatures. Two possible explanations have been proposed for this unexpected behavior. The first hypothesis suggests the presence of missing physics in the opacity models that becomes significant under the conditions encountered by high-temperature Fe-opacity experiments. The second hypothesis proposes that the opacity experiments or analyses are flawed, but only for Fe at high temperatures. Until opacity experiments and theories are reconciled, neither possibility can be definitively ruled  \nout.  \nDespite numerous investigations, the discrepancies between experimental and theoretical iron opacities remain unresolved (Bailey et al. , 2015; Blancard et al. , 2016; Colgan et al. , 2016; Kilcrease et al. , 2015; Mancini, 2016; Nagayama et al. , 2016a, 2014a,b, 2016b, 2019, 2017; Nahar and Pradhan, 2016). On the experimental side, measurements of sample spatial gradients have been conducted and found to be negligible (Nagayama et al. , 2014a). Numerical tests have also shown that temporal gradients, self-emission, and the presence of tamping material do not have significant impact on the results (Nagayama et al. , 2014b, 2016b, 2017). Furthermore, the uncertainties in temperature and density diagnostics resulting from the choice of spectral model have been determined, and it has been found that these uncertainties are too small to account for the reported discrepancies (Nagayama et al. , 2016a). Additionally, background measurements have also been performed and found to be negligible (Dunham et al. , 2021).  \nThe persistent differences between experiment (Bailey et al. , 2015; Nagayama et al. , 2019) and opacity models has led to a sustained effort to reconcile these discrepancies. The experimental efforts have been previously discussed. On the theoretical side, tests of some of the approximations used in the opacity tables have been conducted (Colgan et al. , 2016, Nahar and Pradhan, 2016, Blancard et al. , 2016, Delahaye et al. , 2021), particularly in assessing the convergence of opacity models with respect to the number of configurations included in the calculations. However, no significant change in the final opacity has been reported from such studies.  \nSeveral groups have postulated new physical effects that could produce more opacity contributions than in previous works. Of particular note is the possibility of two-photon opacity contributions, where an atom or ion could simultaneously absorb two photons (possibly of different frequency) (More et al. , 2020, 2017). All current opacity tables omit such contributions, as they are believed to be a minor contribution to the overall opacity. Preliminary studies by More and co-workers (More et al. , 2020, 2017) have suggested this could be a factor in the comparison of theory with the Z measurements, but other studies (Iglesias, 2015; Kruse and Iglesias, 2019, 2021; Pain, 2018) find that such a contribution should be small.  \nAnother intriguing study has reported that transient spatial localization may result in increased opacities (Liu et al. , 2018; Zeng et al. , 2022). This is a densitydependent effect where the plasma effects localize the continuum wavefunctions near the absorbing ion, potentially resulting in increased broadening and bound-free cross sections. While the work of (Zeng et al. , 2022) suggests that this effect is significant, other studies cast doubt on this claim (Iglesias, 2023).  \nOpacity theorists and experimentalists continue to col-  \nlaborate closely to address discrepancies and uncertainties in the calculated solar RMO. They carefully examine each other's work, seeking to resolve the discrepancies and accurately quantify the uncertainty associated with calculated solar opacity.", 'E. Future opacity work': 'At the time of writing, the continuing disagreement between the Z Fe experiments (Bailey et al. , 2015; Nagayama et al. , 2019) and theoretical models is by far the biggest concern of the opacity community. New theoretical efforts, while intriguing, have not yet resolved these differences. New experimental results, particularly the opacity-on-NIF campaign (Opachich et al. , 2022; Perry et al. , 2020), are eagerly anticipated to shed some light on the current impasse. New approaches are also being developed for opacity measurements using the APOLLON laser at LULI (France) and on the GEKKO XII laser (Japan) (Fujioka et al. , 2005). There has been a suggestion for measuring heat conductivity under thermodynamic conditions resembling those found deeper in the radiative zone, using hydro-carbon foams doped to imitate the solar composition (Krief et al. , 2018b). Such progress will have implications for solar modeling, as the combination of opacity and elemental abundances is key to resolving the significant discrepancy between helioseismology and solar models. As previously noted, the abundance of solar elements is also the subject of renewed scrutiny (Asplund et al. , 2021; Magg et al. , 2022).', 'XIV. Experimental facilities for solar fusion studies': 'Improved experimental facilities have led to significant progress in nuclear astrophysics research in recent years, and future instrumental developments will likely continue to push boundaries in the next decade. Key goals driving technological developments include, among others, tests of the weak interactions and of solar properties that make use of high precision solar neutrino measurements, helioseismological mappings of the core metallicity, and detailed solar modeling. There is also a host of related open questions in different areas, such as Big Bang nucleosynthesis, red giant evolution, the evolution of supernova progenitors, and a variety of transient explosive phenomena in astrophysics where a quantitative understanding of the nuclear physics is essential. Due to the small cross sections of low-energy charged-particle reactions, experiments must be designed to achieve signal rates significantly lower than the background rates from cosmic rays, natural radioactivity in the laboratory, and induced activity from beam interactions with target impurities. In the following, we outline current experiences and future facilities-either in development or planned-that will  \npave the way for possible major breakthroughs in the coming years.', 'A. Above-ground facilities': 'Above-ground facilities always play a crucial role in investigating nuclear reactions of astrophysical interest. Various methods are applied to minimize background. A prevalent approach involves employing passive shielding around the detection area, typically utilizing a layered configuration of lead, copper, and polyethylene. This setup helps to decrease unwanted signals and neutron background in detectors with relatively small capacities. However, there are additional techniques that can be utilized to further diminish background interference, enabling measurements at energy levels closer to those pertinent to astrophysics.', '1. LENA': "The Laboratory for Experimental Nuclear Astrophysics (LENA) is part of the Triangle Universities Nuclear Laboratory, located on the campus of Duke University, in North Carolina, US. LENA features two accelerators: a 230-keV ECR accelerator that produces the world's most intense low-energy proton beams (Cooper et al. , 2018) and a 2-MV Singletron from High Voltage Engineering Europa B.V. (Shornikov et al. , 2023). These machines deliver ion beams with remarkable intensity-up to 20 mA DC proton current at lower energies and 2 mA at several MeV-along with advanced beam pulsing capabilities. Each accelerator has dedicated transport systems and control mechanisms for simultaneous operation. A key component of LENA is the γ -ray coincidence spectrometer, which allows measurements with sensitivities comparable to underground facilities (Buckner et al. , 2015). The detector setup includes a 130 % p-type HPGe detector, HPGe clover detectors, and a 16-segment NaI(Tl) annulus, along with the APEX detector featuring 24 position-sensitive NaI(Tl) bars for enhanced measurements. Recently, LENA added a cosmic-ray veto system with nine plastic scintillators to reduce cosmic-ray muon background, conveniently positioned due to its wheeled assembly.", '2. NSL': 'The Nuclear Science Laboratory (NSL) at the University of Notre Dame, hosts a number of low energy accelerators and the TriSOL radioactive beam facility to maintain a rigorous experimental program in nuclear astrophysics, nuclear structure physics,and fundamental symmetries as well as in a broad range of nuclear physics and nuclear chemistry applications. The 5MV single  \nended pelletron provides high intensity proton, helium, and A ≤ 40 heavy ion beams for nuclear astrophysics related experiments in forward kinematics using solid and gas target technologies and in inverse kinematic using the St. George recoil separator. The 10 MV FN pelletron tandem is the main machine for nuclear structure physics experiments, but also serves that AMS program for analysing long-lived radioactivities in terrestrial and meteoritic samples. The FN is also used for indirect measurements of reactions of astrophysical interest and serves as a driver for the production of light radioactive isotopes beams (A ≤ 40) for nuclear astrophysics and fundamental symmetry studies.', 'B. Underground Facilities': 'Underground facilities are essential to push the boundary of direct measurements towards the lowest energies of astrophysical interest (10s-100s keV). In particular, capture-reaction measurements leading to neutron- or γ -ray emissions are severely hampered in surface laboratories because of the overwhelming background associated with cosmic rays. This background can be suppressed by many orders of magnitude by exploiting the natural shielding provided by the rock overburden in underground sites. The improvements possible with this strategy have been demonstrated by pioneering work at the Laboratory for Underground Nuclear Astrophysics (LUNA) at INFN-LNGS, first with a 50-kV accelerator (LUNA I) and then with a 400-kV one (LUNA II) still in operation today (Aliotta et al. , 2022a). Since the last Solar Fusion review, a new accelerator has been installed at INFN-LNGS, and other underground laboratories have become operational in China (JUNA), in the US (CASPAR), and in Germany (Felsenkeller). These are briefly presented below.', '1. Bellotti Ion Beam Facility': 'The INFN-LNGS has recently expanded its accelerator capabilities with the installation of a new 3.5 MV Singletron TM machine designed and set up by High Voltage Engineering Europe (HVEE) (Sen et al. , 2019), see Fig. 13. The 3.5 MV machine is equipped with two independent beam lines which can be operated with solid and gas target systems. Acceptance tests at HVEE demonstrated that the machine can deliver intense proton, helium and carbon beams (1, 0.5 and 0.15 mA, respectively) with well defined energy resolution (0.01% of TV) and stability (0.001%h -1 of TV) (Di Leva, 2020 - private comm.). The 1.4km of rock overburden (ca. 3800 m.w.e.) at LNGS affords significant background reductions for both γ -ray, neutron-, and charged-particles detection (see (Aliotta et al. , 2022a) for a recent review). A first exper-  \nFIG. 13 Photo of the 3.5 MV Singletron TM accelerator recently installed at LNGS (credit: Matthias Junker).  \n<!-- image -->  \nmental proposal presented by the LUNA-Collaboration focuses on measurements of the reactions 14 N(p, γ ) 15 O, 12 C+ 12 C, 13 C( α ,n) 16 Oand 22 Ne( α ,n) 25 Mg, the latter being in the context of the ERC Starting Grant SHADES . The 3.5MV accelerator is now part of the Bellotti Ion Beam Facility (Junker et al. , 2023) and open to external users. 13', '2. JUNA': 'The Jinping laboratory for Underground Nuclear Astrophysics (JUNA) is part of the China Jinping Underground Laboratory (CJPL), established on the site of hydropower plants in the Jinping mountain, Sichuan, China (Liu et al. , 2022b). The laboratory, located near the middle of a traffic tunnel, is shielded by 2400 m of mainly marble overburden (6720 m.w.e.). In December 2020, the JUNA collaboration installed a 400 kV accelerator capable of delivering H + and He + beams with intensities of up to 10 particle mA, and a He 2+ beam with an intensity of up to 1 particle mA. Some important results have already been published. For example, the lowest energy study to date of the 19 F ( p, γ ) 20 Na reaction advocates a breakout from the CNO cycle as a possible route for calcium production in population III stars (Zhang et al. , 2022).', '3. CASPAR': 'The Compact Accelerator System for Performing Astrophysical Research (CASPAR) laboratory is the only US-based deep underground accelerator and is operated by a collaboration of the University of Notre Dame and the South Dakota School of Mines and Technology (Robertson et al. , 2016). The accelerator system has been  \nfully operational since 2018 and is located 1480 m below surface at the Sanford Underground Research Facility (SURF) 14 in Lead, South Dakota, formerly the Homestake gold mine. The rock overburden results in a 4300 m.w.e shielding effect, significantly decreasing cosmic-ray induced background with a muon flux level of 4 × 10 -9 cm -2 s -1 . The residual neutron flux consists of primarily low-energy ( < 10 MeV) neutrons generated by ( α, n ) reactions induced by the decay of naturally occurring uranium and thorium radionuclides in the surrounding rock, and is generally of the order of 10 -6 neutrons cm -2 s -1 (Mei and Hime, 2006; Mei et al. , 2009). The CASPAR accelerator is a 1 MV Van de Graff style JN accelerator with a 150 to 1100 kV operational range, well suited for overlap with higher energy measurements. The accelerator provides proton and α beams with up to ∼ 250 µ A on target. The scientific program continues to explore stellar neutron sources and expands the present studies into the magnesium range probing α -capture reactions on 24 Mg, 25 Mg, and 26 Mg isotopes. A new program has been initiated to explore the endpoint of nova nucleosynthesis, studying proton capture reactions in the Ar to Fe range. CASPAR is well suited for these measurements, but will be complemented by a new low energy machine, presently under development at Notre Dame.', '4. Felsenkeller': 'The Felsenkeller laboratory is hosted within a network of tunnels, excavated into hornblende monzonite rock, previously used as a cool storage place for the homonymous nearby brewery. Jointly funded and built by TU Dresden and HZDR, the laboratory operates a 5 MV Pelletron-type accelerator built by NEC. With the addition of a RF ion source in the high-voltage terminal, the accelerator can be operated in tandem mode or as a single-ended machine and is capable of delivering proton, α -particle, and carbon beams with currents up to several tens of µ A. Unlike the other sites mentioned in this section, Felsenkeller is a shallow underground laboratory, with a rock overburden of only 140 m.w.e. Such a depth is sufficient to shield all components of cosmic-ray induced radiation, except muons, for which further active shielding is generally required (Ludwig et al. , 2019). Detailed studies of muon, neutron, and γ backgrounds have been carried out (Grieger et al. , 2020; Ludwig et al. , 2019; Szücs et al. , 2019a; Turkat et al. , 2023), and the 12 C( p, γ ) 13 N reaction has been studied (Skowronski et al. , 2023b). The further scientific program of Felsenkeller foresees the study of several reactions of astrophysical interest, including 3 He( α , γ ) 7 Be and 12 C( α , γ ) 16 O.', 'C. Indirect methods': "Indirect methods are also pivotal in nuclear astrophysics due to the challenges associated with directly measuring reactions at extremely low energies. They can have systematic uncertainties that are different from those of direct measurements, and they provide supplementary information that can constrain R-matrix and other models used in the extrapolation of data from direct measurements. For the present review, it is worth further exploring here the ANC (asymptotic normalization coefficient) and THM (Trojan Horse) methods. For a detailed description of the methods, refer to (Tribble et al. , (2014; Tumino et al. , 2021).  \nThe ANC method determines the zero-energy crosssection for radiative-capture reactions by exploiting their peripheral nature. This means the reaction depends mostly on the long-distance behavior of the wave function. The ANC is extracted from transfer reactions using the DWBA, with uncertainties primarily arising from the optical model description, still much smaller than those affecting spectroscopic factors. While the method is focused to zero energy, it complements direct measurements and can be applied to loosely bound nuclei. The method has been applied to the 3 He( α , γ ) 7 Be (section VI.C).  \nThe THM is a powerful technique for indirectly measuring astrophysical S(E) factors of reactions involving charged particles. By studying a related reaction with a spectator particle, the THM allows for the extraction of the desired cross section without the need for extrapolation. The THM relies on the assumption of quasifree kinematics, where the spectator particle has a minimal effect on the reaction of interest. The method requires careful selection of beam energy and momentum transfer to ensure that the Coulomb barrier is overcome and the spectator particle remains relatively undisturbed. Since Solar Fusion II, significant advancements have been made, focused on improving the model's accuracy and assessing its systematic uncertainties (Tribble et al. , (2014; Tumino et al. , 2021). For reactions dominated by broad resonances, the modified R-matrix approach (La Cognata et al. , 2015b; Trippella and Cognata, 2017) allows for the incorporation of half-off-energy-shell and energy resolution effects within a well-established framework. This method enables multi-channel descriptions of reactions, such as in the 12 C+ 12 C fusion studies (Tumino et al. , 2018), and includes a DWBA-based normalization procedure that does not rely on direct data (La Cognata et al. , (2009). The modified R-matrix framework has been applied to the 15 N(p, α ) 12 C (section XI.A.3), 19 F(p, α ) 16 O (section XI.C.4), and 23 Na(p, α ) 20 Ne (section XI.D.5) reactions. For narrow resonance reactions, a simplified approach has been introduced (La Cognata et al. , 2022) to deduce resonance strengths. This method reduces systematic errors from normalization and the-  \nto the percent level through multi-resonance normalization and covariance in error propagation. The narrow resonance approach has been used in analyzing the 17 O(p, γ ) 18 F (section XI.B.3), 17 O(p, α ) 14 N (section XI.B.4), and 18 O(p, α ) 15 N (section XI.C.2) reactions.", '1. NIF/OMEGA': 'Several efforts have recently been devoted to studying nuclear reactions in a plasma environment reminiscent of stellar conditions (Gatu Johnson et al. , 2017) using the two large laser facilities: OMEGA at the Laboratory for Laser Energetics, University of Rochester (Boehly et al. , 1997), and the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory (Wonterghem et al. , 2016). Both facilities use high-power lasers to implode spherical capsules containing reactants of interest to high temperatures (of order 1-20 keV) and densities (up to 10 3 g cm -3 in the extreme case of implosions with a fuel ice layer (Abu-Shawareb et al. , 2022)). The OMEGA laser has the capability of delivering 30 kJ of laser energy divided between 60 laser beams to the target; the NIF can deliver up to 2.0 MJ using 192 laser beams, and can thus be used to implode larger amounts of materials to more extreme conditions compared to OMEGA. Nuclear experiments at these two facilities are enabled by an extensive suite of nuclear diagnostics, originally developed to do inertial confinement fusion experiments (Cerjan et al. , 2018). Initial results have been obtained on the 2 H( p , γ ) 3 He reaction (Mohamed et al. , 2022; Zylstra et al. , 2020), the 3 He( 3 H, γ ) 6 Li reaction (Zylstra et al. , 2016), and the 3 H( 3 H,2n) α reaction (Casey et al. , 2017). Proton spectra have also been measured for the 3 He( 3 He,2p) α reaction (Zylstra et al. , 2017). In addition to low-Z reaction nuclear S-factor studies, the NIF and OMEGA facilities are also promising for studies of plasma effects on nuclear reactions, including screening (Aliotta and Langanke, 2022; Casey et al. , 2023); for studies of charged-particle-induced reactions (Wiescher et al. , 2022); and, thanks to the high neutron fluxes achievable (up to 5 × 10 27 neutrons cm -2 s -1 (AbuShawareb et al. , 2022)), for studying reactions on excited states (Thompson, 2022). However, there are challenges remaining to be addressed in fully developing this new platform for nuclear experiments, including the impact on the results of rapid gradients in space and time (Crilly et al. , 2022).', '2. PANDORA': 'PANDORA(Fig. 14) is a device conceived for multidisciplinary studies, including many of astrophysical interest (Mascali et al. , 2022). Two of its main objectives are: (a)  \nFIG. 14 Schematic drawing of the PANDORA trap with holes to house the HPGe detectors. (credit: Domenico Santonocito).  \n<!-- image -->  \nto perform the first measurements of β decays in plasmas of astrophysical relevance, in order to verify the results obtained in storage rings with 187 Re (a lifetime reduction by 9 orders of magnitude), and (b) to measure the opacities of plasmas of astrophysical interest (kilonova ejecta). PANDORA will mainly consist of three subsystems:  \n- · an innovative superconducting magnetic plasma trap, able to produce and confine plasmas with electron-ion density up to 10 13 cm -3 and electron temperature of T e ∼ 0 . 1 -30 keV;\n- · an advanced plasma multi-diagnostic system, consisting of a set of non-invasive diagnostic tools capable of operating simultaneously, for non-intrusive monitoring of the thermodynamic plasma properties and parameters;\n- · an array of 14 HPGe (High-purity Germanium) detectors for γ -ray spectroscopy, surrounding the plasma trap.  \nThe next development steps will involve: preliminary numerical simulation studies to assess the possibility of measuring opacities in astrophysical plasmas (relevant for kilonova ejecta); study of expected abundances and constraints in AGB stars for some (key) nuclides of interest; experimental investigation of magnetic confinement and turbulence in plasma, using an existing trap at ATOMKIDebrecen; definition of the analysis algorithm for X-ray imaging and spatially resolved spectroscopy.  \nInitial physics cases to be investigated include 176 Lu (a potential cosmo-chronometer), 134 Cs (to reproduce adequately the observed abundance ratio of the two s-only  \nisotopes, 134 Ba and 136 Ba) and 94 Nb (to solve the puzzle about the exact contribution of s-processing to 94 Mo).', '3. Sandia Z': "The Sandia National Laboratories (SNL) Z machine is the world's largest pulsed power accelerator (Sinars et al. , 2020). It charges 22 MJ of electrical energy into a capacitor and then discharges it all at once (over approximately 100 ns) into targets ranging from millimeters to centimeters in size. This allows it to reach a peak electrical power of 80 terawatts, which means that, for that instance, it generates 15 times the electrical power generated by all of the world's power plants combined. By concentrating this massive power into small targets, the Z machine can convert them into a state of high energy density (HED), which refers to an extreme state of matter with a pressure exceeding 10 6 times atmospheric pressure.  \nThe Z machine has been used for a wide range of HED science, including inertial confinement fusion, dynamic material properties, and laboratory astrophysics. In particular, the Sandia Z machine can convert this electrical power into X-ray power using a scheme called the Z-pinch dynamic hohlraum (Rochau et al. , 2014). This is the most energetic X-ray source on earth and has been used by academic collaborations such as the Wooton Center for Astrophysical Plasma Properties (WCAPP) and the Center for Laboratory Astrophysics. For the stellar opacity project (see Sect. XIII.D), they heat iron or oxygen to the conditions relevant to the base of the convection zone and measure their frequency-resolved opacities to test hypotheses for the solar opacity-abundance problem (Bailey et al. , 2015, 2009; Basu and Antia, 2008; Nagayama et al. , 2019). For the accretion disk projects, they heat silicon, iron, and neon photoionized plasmas to conditions similar to those at black hole accretion disks to experimentally test the validity of calculated atomic data, kinetics, and spectral formation (Loisel et al. , 2017). This is important for accurately interpreting high-resolution data from upcoming X-ray space telescopes such as XRISM. For the white dwarf photosphere project, they reproduce its photosphere conditions and validate spectroscopic methods used for understanding the age of the universe as well as constraining its evolutionary paths and supernova remnants (Falcon et al. , 2015; Montgomery et al. , 2015; Schaeuble et al. , 2021, 2019). These experiments reach temperatures ranging from 10 4 to 2 × 10 6 K and densities ranging from 10 16 to 10 23 electrons cm -3 and are performed simultaneously by placing their samples at different locations from the Z-pinch dynamic hohlraum.", '1. CRYRING': 'A main limitation of experimental studies involving unstable nuclei arises from the difficulty of producing radioactive ion beams (RIBs) of adequate intensity and purity at the energies of interest for astrophysical applications. Storage rings (Steck and Litvinov, 2020) offer a key advantage over traditional ISOL and in-flight approaches as they allow for storing and re-circulating radioactive ions over and over again, thus allowing for multiple interactions of un-reacted beam particles with an in-ring target. Beam re-circulation significantly improves the quality of the beam as it results in orders-ofmagnitude increased intensity (typical boosting factors of ∼ 10 5 ), limited only by the duty cycle of the measurement (Bruno et al. , 2023), and in improved purity of the stored beam, as only beam particles with the right mass-to-charge ratio will survive in the beam orbit.  \nOne of the main technical challenges of storage rings comes from the requirement of ultra high vacuum (UHV) conditions ( < 10 -10 mbar) necessary to guarantee that ions can be recirculated with minimal losses. This, in turn, translates into a requirement for sufficiently thin targets (i.e. typically ≤ 10 11 -10 14 atoms cm -2 ) to minimise beam ion losses through scattering or electron capture (see (Bruno et al. , 2023) for further details).  \nA dedicated low-energy storage ring, CRYRING (Lestinsky et al. , 2016), inherited from Stockholm University, was installed at GSI in 2016, as a Swedish inkind contribution to the FAIR facility. The facility allows nuclear astrophysics studies to be done with radioactive nuclei. After its initial recommissioning (Herfurth et al. , 2018), CRYRING now serves as a low-energy extension for Experimental Storage Ring (ESR) beams, as well as a standalone machine with a local ion source (Geithner et al. , 2017). After in-flight production in the Fragment Recoil Separator (FRS) at relativistic energies, rare ions can now be cooled, post-decelerated and stored in the full range down to about 100 keV/ u . A study is underway to reach lower energy by means of a transverse low-energy beamline from the the FISIC project (Glorius and Bruno, 2023; Schury et al. , 2020). CRYRING is now equipped with an in-ring micro-droplet gas target, whose low density ( ≤ 10 14 atoms cm -2 ) ensures minimal beam energy loss and straggling through the target, a unique advantage for charged particle spectroscopy at storage rings compared to standard techniques.  \nIn addition, a new detection chamber (Fig. 15), CARME (CRYRING Array for Reaction MEasurements), was recently installed at CRYRING and fully commissioned in early 2022. Specifically designed to fulfil requirements of stored beam experiments and operating at a pressure of 10 -12 mbar, CARME can host up to eight double-sided-silicon-strip detectors with excellent  \nFIG. 15 Photo of the CARME chamber recently installed at CRYRING. Four silicon strip detectors are visible at the centre of the chamber (credits: Carlo Bruno).  \n<!-- image -->  \nangular and energy resolution.  \nFinally, the upcoming installation of the FISIC transverse beam line (Schury et al. , 2020) will allow to intercept the beam stored in CRYRING, thus opening up unprecedented opportunities for crossed-beam experiments. An exciting application may be the study of nuclear reactions between ion beams, i.e. unaffected by the electron screening, directly at the energies of astrophysical interest. Initial proposals for the study of nuclear astrophysics reactions have already been approved.', 'A. Summary': "This review summarizes the significant progress that has been made over the last decade in understanding the nuclear reactions that govern energy production in solar and stellar hydrogen burning. It also describes some of the plasma and atomic physics that effects the solar environment in which these reactions take place, as well as the diagnostics tools - solar neutrinos and helioseismolgy - by which we can probe that environment.  \nFor S 11 and S 12 , cross sections have been derived from first principles, using both potential theory and effective field theory. In the case of S 12 , new experimental data agree well with the theoretical predictions. S hep , which governs a minor branch of solar fusion, must also be taken from theory.  \nThe S-factors for other pp-chain reactions are based on laboratory measurements. In some cases, including notably S 17 , most of the studies reviewed here are relatively far from the solar Gamow peak energies, underlining the need for theoretical and experimental tools to bridge the gap.  \nThe nuclear physics of the pp chain remains a significant uncertainty in SSM predictions of the individual neutrino fluxes - pp, pep, hep, 7 Be, 8 B, and CNO. The nuclear errors are comparable to, and in the case of the  \nhep neutrinos greatly exceed, the 'environmental' errors that are generated by other uncertainties in the SSM (Orebi Gann et al. , 2021).  \nFor several reactions including S 33 and S 1 14 , the SF III recommended values are less precise than those of SF II. This fact should not be taken as a sign of regression, but instead as indicative of a more cautious treatment of uncertainties. In some of the error estimates made in this review, we have been mindful of past lessons on the potential impact of unidentified systematics.  \nThe status of our understanding of radiative opacities has been reviewed here, the first time the subject has been included in the Solar Fusion series. The debate generated by the solar composition problem has brought renewed attention to the complicated interplay between opacities and composition in the SSM. Given new experimental opportunities, especially luminous X-ray sources coupled with plasma targets, there is the expectation of rapid progress in this field over the next decade.  \nThere has been only very limited progress on electron screening since SF II. This field has important practical implications not only for the Sun but also for laboratory experiments, and may merit additional attention in the coming decade. Progress may depend on achieving a better understanding of low-energy stopping powers.", 'B. Recommended values for S-factors and their derivatives': 'The SF III recommendations for nuclear reaction Sfactors and their derivatives are presented in Table I of the Introduction. In the sections that follow the Introduction, we presented the details leading to these results and identified opportunities for future work, which are summarized directly below.', 'C. General recommendations for future work': "For all of the solar fusion reactions, obtaining highprecision experimental data in Gamow window remains an elusive goal, one that may not be reached in the foreseeable future. However, such direct measures are just one of many avenues for improving our understanding of the nuclear physics of the Sun. The recommendations below are in most cases summaries of more detailed discussions presented in this review, and we refer readers to the relevant sections for additional details.  \n- 1. For all of the experimentally accessible pp-chain reactions and for S 1 14 which determines the CNO cycle rate, there is a need for at least one experimental data set spanning the entire energy range, encompassing the more limited data sets that currently exist. This will provide an important crosscheck on the normalizations of those data sets. Experimentalists can now tackle this challenge due to the new  \ngeneration of underground accelerators that have the dynamic range needed for such studies.  \n- 2. We recommended continued effort on reaction theory relevant to pp-chain reactions. The needs go beyond rates to include both astrophysical (e.g., precise neutrino spectra) and laboratory (e.g., γ -ray angular distributions) observables.\n- 3. We urge the community to set an ambitious goal for the precision of future S-factor measurements: reduce the nuclear physics uncertainties to a factor of two below current SSM 'environmental' uncertainties, so that they are no longer a significant contributor to SSM neutrino flux uncertainties. With the exception of the hep neutrinos, this goal could be reached by a future reduction of a factor ∼ two, typically, in the individual S-factor uncertainties (Orebi Gann et al. , 2021). As solar neutrino data are an important input into global neutrino oscillation analyses, nuclear uncertainties will continue to feed into that program until this goal is achieved. Improvements in S 1 14 and other CNO cycle Sfactors are needed, as the current 8.4% uncertainty would be a limiting factor in extracting the solar core's metallicity from a future large-volume CN neutrino experiment. The progress reported here on S 17 , with the uncertainty reduced significantly from SF II to 3.4%, should be continued, given the current and future 8 B neutrino programs of SuperK and Hyper-K: these neutrinos are our best solar core thermometer. Finally, ongoing Super-K efforts to detect hep neutrinos provide strong motivation for improving estimates of S hep and the associated neutrino spectrum. (Uncertainties in the high-energy tail of the 8 B spectrum must also be reduced, as discussed in Section II.)\n- 4. New experimental methods are becoming available that could help us better understand electron screening in terrestrial and solar reactions. Plasma conditions resembling those of the Sun can now be produced in the laboratory. Reactions can be studied in rings, using ions stripped of their atomic shells. The level of theory activity in this field has declined over the past decade, so we hope experimental progress will help renew interest.\n- 5. In order to break the degeneracy between solar abundances and solar opacities, efforts should be made to improve our understanding of radiative opacities for the Sun's principal metals. A high priority is the resolution of the current discrepancy between measured and modeled iron opacities. Solar physicists need access to state-of-the-art, opensource opacity codes.  \n- 6. We recommend that resources be available to extend the use of current and next-generation dark matter detectors to solar neutrino detection. This includes designing new detectors so that they are highly capable for both applications. Some of the high-priority goals of solar neutrino physics - such as a 1% measurement of the pp neutrinos to check the luminosity constraint - might be achieved with a dual-purpose detector.\n- 7. The Solar Fusion program to periodically review the nuclear physics of the Sun and other hydrogenburning stars should be continued, with the separation between studies being no more than ten years. The format should continue to be open, giving all researchers working in the field an opportunity to contribute, and broadly international.\n- 8. Future Solar Fusion studies should continue to strengthen the connections between this community and others interested in main sequence stars. SF III has placed increased emphasis on hydrogen burning at higher temperatures, plasma physics, opacities, and asteroseimology. However, with observations now being made of solar-like stars in their formation stages, we anticipate additional connections developing, well beyond the limits of this study.", 'D. Outlook': "Six decades after the initial observations of solar neutrinos by Ray Davis (Davis, 2003), significant advances have been made in our understanding of the SSM and indeed of hydrogen-burning stars in general. The driver of this progress has been experiment: the observations made of all of the principal solar neutrino sources; our improved understanding of the flavor physics of those neutrinos; the advances made in helioseismology, a second quantitative probe of the solar interior; and our improved knowledge of the input microphysics of the SSM, including nuclear cross sections and opacities. Despite this progress, improvements are still needed. In several cases the dominant uncertainty in our solar neutrino flux predictions stems from the limited precision of the nuclear S-factors. Similarly, uncertainties remain in our understanding of radiative transport in the Sun, including the continuing debate about the Sun's primordial composition. As changes in composition often can be mimicked by changes in radiative opacities, our imperfect knowledge of the latter has slowed resolution of the solar composition problem.  \nThe prospects for substantial progress in the next decade are bright. Whereas at the time of SF II, there was only one underground ion accelerator worldwide, now there are five, opening up possibilities for independently  \ncross-checking important nuclear data. The low background that can be achieved underground has led to the first measurements of reactions in the Gamow window. Other new experimental capabilities have come on line that are advancing our understanding of solar atomic physics, including low-energy storage rings and new plasma facilities, the latter made possible in part by advances in high-power lasers. The field is approaching the point where opacity measurements can be made under conditions closely approximating those of the solar interior.  \nTheory has progressed as well, including advances in neutrino flavor physics that have made neutrinos once again a precise probe of solar physics. Techniques with a firmer footing in first principles, such as effective field theory and lattice QCD, are being employed in the extraction of pp chain S-factors. Conventional approaches like the R-matrix can now be applied to complex nuclear systems where the inclusion of multiple reaction channels is important.  \nOn the observational side, several very large neutrino detectors are under construction, including HyperKamiokande, JUNO, and DUNE. Certain directdetection dark matter experiments have reached a sensitivity where nuclear recoils due to the coherent solar neutrino scattering have become a background, leading to discussions of how such detectors might be optimized for solar neutrino detection (Essig et al. , 2018). This underscores the remarkable radiopurity of these underground detectors. Finally, the helioseismic studies that proved so important in establishing the credibility of the SSM are now being extended to other stars - a major reason SF III was undertaken at this time. Asteroseismic studies have the potential to tell us what is special and what is common about the Sun as a star.  \nGiven current excitement about the emerging field of multi-messenger astrophysics, it is perhaps fitting to remember that the Sun was the prototype for this field. An array of physical measurements of the Sun - its mass, radius, luminosity, abundances - were combined with neutrino flux and helioseismic measurements to constrain and then test the SSM. The result was the discovery of neutrino mass and flavor mixing.  \nThis success was a multi-disciplinary effort that has not yet run its course. We still have uncertainties in our characterization of the Sun's nuclear, atomic, and weak interaction physics that, if improved, would make the Sun an even more powerful laboratory. As our best known and nearest star, the Sun is the test of our understanding of the structure and evolution of other main-sequence stars. If the past is any indicator of the future, the efforts we make over the next decade to make the Sun an even more precisely calibrated reference will yield a rich return in our understanding of physics.", 'Acknowledgements': "The authors gratefully acknowledge discussions with many colleagues. The help they provided contributed significantly to this review.  \nWe thank the following institutions for their support of the SF III effort:  \n- · Physics Frontier Center N3AS (Network for Neutrinos, Nuclear Astrophysics, and Symmetries), UC Berkeley, host of the SF III workshop. N3AS is supported by the U.S. National Science Foundation under cooperative agreement 2020275.\n- · Institute for Nuclear Theory, University of Washington. The INT is supported by the US Department of Energy under grant DE-FG02-00ER41132.\n- · European Union, 'Chemical Elements as Tracers of the Evolution of the Cosmos - Infrastructures for Nuclear Astrophysics' (ChETEC-INFRA), HORIZON2020 project number 101008324.\n- · The Heising Simons Foundation, grant 2017-228.  \nThe authors of SF III acknowledge the following support:  \n- · U.S. National Science Foundation, through grants\n- -PHY-1430152 (Joint Institute for Astrophysics) and OISE-1927130 (IReNA), Michigan State University\n- -PHY-1913620 and PHY-2209184, Mississippi State University\n- -PHY-2011890 and PHY-2310059, Notre Dame University\n- -PHY-2111426, University of Tennessee\n- -PHY-2108339, University of Wisconsin\n- · U.S. Department of Energy, through contracts\n- -DE-AC02-07CH11359, Fermi National Accelerator Laboratory. Fermilab is operated by Fermi Research Alliance LLC for the Office of Science.\n- -DE-AC02-05CH11231, Lawrence Berkeley National Laboratory. LBNL is operated by the University of California for the Office of Science.\n- -DE-AC52-07NA27344, Lawrence Livermore National Laboratory. LLNL is operated by the Lawrence Livermore National Security LLC for NNSA.\n- -89233218NCA000001, Los Alamos National Laboratory. LANL is operated by Triad National Security LLC for NNSA.\n- -DE-NA0003525, Sandia National Laboratories. Sandia National Laboratories are managed and operated by NTESS for the NNSA.\n- -DE-AC05-00OR22725, Oak Ridge National Laboratory. ORNL is operated by UT-Battelle, LLC for the Office of Science.W\n- · U.S. Department of Energy, through awards\n- -DE-FG02-00ER41138 and DE-SC0004658, University of California, Berkeley\n- -DE-FG02-93ER40789, Colorado School of Mines\n- -DE-FG02-97ER41033 (TUNL), Duke University\n- -DE-SC0013617 (FRIB Theory Alliance), Michigan State University\n- -DE-FG02-97ER41041, University of North Carolina\n- -DE-FG02-88ER40387, DE-NA0003883, and DENA0004065, Ohio University\n- -DE-FG02-08ER41533, Texas A&M University, Commerce\n- -DE-FG02-97ER41020 (CENPA), University of Washington\n- -DE-SC0019465, University of Wisconsin\n- -DE-SC0023692, Iowa State University\n- -DE-SC0019257, San Diego State University\n- · Helmholtz Research Academy Hesse for FAIR, Germany\n- · Hungarian Scientific Research Fund grant FK134845, National Research, Development, and Innovation Office\n- · Israel Science Foundation grant 889/23\n- · Italian Ministero dell'Università e della Ricerca (MUR) grants 2022E2J4RK (program PRIN 2022) and 2017W4HA7S (program PRIN 2017)\n- · The Natural Sciences and Engineering Research Council of Canada (NSERC)\n- · Unidad de Excelencia María de Maeztu, grant CEX2020-001058-M, and Generalitat de Catalunya, grant 2021-SGR-1526, Spain\n- · U.K. Science and Technology Facilities Council, grant ST/V001051/1  \nThis paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the  \nviews of the U.S. Department of Energy, United States Government, or any other agency listed above.  \nFor the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.", 'A. Gallium Neutrino Source Cross Sections': "Here we describe in more detail the constraint that the electron capture rate for 71 Ge places on the 51 Cr and 37 Ar neutrino source cross sections for 71 Ga, expanding on the discussion of Section II.A.2. We discuss both the early treatment of Bahcall and the more recent analysis of Elliott et al. (2024, 2023), performed in support of the BEST experiment  \nFollowing Bahcall (Bahcall, 1978), the electron capture rate can be written in terms of a dimensionless phase space factor f EC :  \nf EC = 2 π 2 ( /planckover2pi1 mc ) 3 ( q mc 2 ) 2 | ψ ( R ) | 2 , (A1)  \nwhere q is the energy available to the neutrino and we have omitted atomic shell effects for brevity. The last factor is the electron density at the nucleus within the radius R . Multiplying this by the experimentally determined half-life t 1 / 2 yields the traditional ft value needed to calculate the neutrino cross section. Bahcall (1978) defines a characteristic scale σ 0 for neutrino cross sections on 71 Ga in the form  \nσ 0 = 1 . 2429 × 10 -47 ( ∑ i q 2 i g 2 i ) -1 (A2)  \n= 8 . 611 × 10 -46 cm 2 (A3)  \nwhere g i = ψ ( R ) . Bahcall (1997) reports the number given above and states that 'it is about 0.5% less than the value 8 . 8012 × 10 -46 I have used since 1984.' Since these numbers differ by 2.2%, there is clearly an error in one of the 3 numbers.  \nRecently Elliott et al. (2024, 2023) have recast the derivations with fundamental constants replacing numerical values and correction terms explicitly stated. They conclude that the ground-state cross section is 2.5% lower, and the excited-state contributions somewhat larger, than values used in the anomaly analysis. The adjusted net anomaly is minimally reduced.  \nThey find the following expression for the neutrino cross section on 71 Ga leading to the ground state of 71 Ge:  \nσ gs = G 2 F cos 2 θ C g 2 A π p e E e F ( Z f , E e ) B ( ν,e ) GT (gs) [1 + g v,b ( ν,e ) ] [1 + /epsilon1 q ] , (A4)  \nwhere E e = q -Q EC + m e -∆ 0 and ∆ 0 /similarequal 90 eV is a small correction introduced by Bahcall for the energy lost to  \nelectronic rearrangement in the charge-changing reaction. The Gamow-Teller matrix element B ( ν,e ) GT (gs) is determined from the measured half-life τ 1 / 2 of the electron capture decay of 71 Ge:  \nω = ln[2] τ 1 / 2 = G 2 F cos 2 θ C g 2 A 2 π | φ 1 s | 2 avg q 2 1 s [ 2(1 + /epsilon1 1 s o )(1 + P L + P M P K ) ] [2 B ( ν, e) GT (gs)] [1 + g v,b (EC )] [1 + /epsilon1 q ] . (A5)  \nIn this expression, /epsilon1 1 s o /similarequal 0 . 013 is a correction for overlap and exchange to account for the differences in the wave functions of the initial and final atomic states, g v,b is a radiative correction, and /epsilon1 q /similarequal 5 × 10 -4 is a correction for  \nthe weak magnetism contribution to the transition probability. The factor 2 with B ( ν, e) GT (gs) is a spin statistical factor. Hence,  \nσ gs = ω p e E e F ( Z f , E e ) | φ 1 s | 2 avg q 2 1 s [1 + g v,b (EC )] -1 [ 2(1 + /epsilon1 1 s o )(1 + P L + P M P K ) ] -1 (A6)  \n≡ σ 0 p e E e F ( Z f , E e ) m 2 e c 3 1 2 παZ . (A7)  \nThe last line extracts the Bahcall cross-section scale σ 0 and inserts the dimensionful parameters needed to scale the energy and momentum. The atomic factors of Elliott et al. (2023) are equivalent to those of Bahcall:  \n∑ i | φ i | 2 avg q 2 i = 0 . 01443 ( m e c 2 ) 5 4 π ( /planckover2pi1 c ) 3 (A8)  \nUnfortunately, none of the three most advanced calculations referenced in (Bahcall, 1997) of the densities g i has been published so the numerical factor 0.01443 is taken from the ratio in Eq. A3. (The computer code GRASP (Dyall et al. , 1989) may still be available.) Rather than use the theoretical electron density of each subshell at the nucleus, Elliott et al. use the experimentally measured subshell ratios, which has the advantage that only a single overlap and exchange correction and a single energy are needed, for the 1 s state. As Bahcall has stated, overlap and exchange have little effect on the total EC rate, so Elliott et al. write Eq. A8 as follows, with the inclusion of a small Q-value update,  \n∑ i | φ i | 2 avg q 2 i (1 + /epsilon1 i 0 ) = | φ 1 s | 2 avg q 2 1 s [ (1 + /epsilon1 1 s o )(1 + P L + P M P K ) ] (A9)  \nThe resulting expression is,  \nσ 0 = 8 π 2 ln (2) αZ | φ 1 s | 2 avg q 2 1 s t 1 / 2 [1 + g v,b (EC )] -1 [ 2(1 + /epsilon1 1 s o )(1 + P L + P M P K ) ] -1 . (A10) = ( /planckover2pi1 c m e c 2 ) 3 4 π 2 ln (2) αZ 0 . 01440 c t 1 / 2 ( 0 . 2221 MeV q 1 s ) 2 [1 + g v,b (EC )] -1 (A11) = 8 . 63 × 10 -46 [1 + g v,b (EC )] -1 cm 2 , (A12)  \nwhich is only 0.3% larger than the value given in Bahcall (1997).  \nThe above cross-section expressions give a groundstate cross section 2.6% lower than in the cross section table, Table II in Bahcall (1997). Whether this extends to the solar neutrino rates has not yet been evaluated, but in any scenario some reconsideration of the low-energy solar neutrino data may be required once a resolution to the Ga anomaly has emerged.", 'B. Bayesian Methods': "The treatment of uncertainties when dealing with multiple data sets was addressed in the appendix of SF II, including the Inflation Factor Method used by the Particle Data Group that was adopted by most of the SF II working groups. Readers are directed there. We now extend that discussion to include the Bayesian methods used by several of the SF III work groups. For erxample,  \nin the analyisis of the 2 H( p , γ ) 3 He reaction presented in Section IV.D, both Bayesian analysis and Bayes model averaging were utilized.  \nConsider an optimization that utilizes Gaussian distributed priors in a constrained Bayesian analysis. With such priors, maximizing the likelihood amounts to minimizing an augmented χ 2  \nχ 2 aug ≡ ∑ D,i [ y D i -f D S ( E ; λ )] 2 σ 2 D i +( σ ext D i ) 2 + ∑ p ( λ p -˜ µ p ˜ σ p ) 2 . (B1)  \nThe first term in the above equation is the standard χ 2 and the second term derives from the Bayesian constraint on the parameters ( λ p ) with prior mean ( ˜ µ p ) and width ( ˜ σ p ), which are chosen with some prior knowledge or can be optimized as described below. Of course, care must be taken when choosing values for the priors ( ˜ µ p and ˜ σ p ). In the absence of prior information, it is common to set ˜ µ p = 0 . Similarly, one should not make ˜ σ p too small, unless one has prior knowledge to do so, else this will bias the determination of λ p .  \nIn the first term in Eq. (B1), the double sum runs over the datasets ( D ) and individual results from each dataset ( D i ), with the mean value and stochastic uncertainty of each data point given by y D i and σ D i , respectively.  \nThe theoretical model describing the S-factor data, S ( E ; λ ) , is a function of the energy ( E ) and a set of parameters ( λ ) that must be determined. The quoted systematic uncertainties are parameterized by the normalization factors f D , with a prior of unit normalization and a width characterized by the quoted systematic uncertainty. It is straightforward within a Bayesian framework to utilize distribution functions for f D that are not Gaussian, such as a log-normal or other distributions. The parameters f D represent a normalization of the model function for a given dataset, which, viewed from a non-Bayesian perspective, can also be interpreted as normalization factors that must be applied to the data to match the 'true' underlying distribution.  \nFinally, σ ext D i are unknown extrinsic uncertainties (de Souza et al. , 2019). If one assumes that a smooth function of energy can accurately describe the data, then the data in a given experimental set may scatter about this presumed 'true' value by more than is reflected by the quoted statistical uncertainties. It was suggested in de Souza et al. (2019) that this extra scatter might be explained by some additional source of statistical uncertainty unbeknownst to the experimenter, or by some unknown systematic uncertainty that is different for each data point in the same data set, as opposed to a correlated systematic that affects all data points similarly. In either case, this extrinsic uncertainty can be accommodated in a Bayesian analysis framework by adding the additional uncertainty as a normal-distributed source of noise with a width that is constrained by the data. In de Souza et al. (2019) and Moscoso et al. (2021), σ ext . D i  \nwas added as an absolute uncertainty, independent of the energy. For some data, Odell et al. (2022a) suggested adopting instead a relative uncertainty, such that the scale of the extrinsic fluctuations is proportional to the mean value of S ( E ) . This strategy of adding extra extrinsic uncertainty to the data sets can be viewed as an alternative to that of inflating the quoted statistical uncertainties by √ χ 2 ν , which is often used for seemingly incompatible data sets (Workman et al. , 2022), where χ 2 ν is the χ 2 per degree of freedom. An advantage of the extrinsic uncertainty method is that it uses the observed scatter within a given data set as a measure of the possible size of unreported uncertainties, as opposed to uniformly increasing the uncertainty in all data sets by the same relative amount.  \nOne can further perform a Bayes model averaging, which we implement and describe here. For a given model, after optimizing the posterior parameter distributions, the Bayes factor (BF) is proportional to the probability of the model given the data (Hilbe et al. , 2017). Therefore, for a fixed data set, the BF can be used as a relative probability of each model, thus enabling a weighted model-averaging procedure. If we assume a uniform likelihood for each model, the expectation value and variance of a quantity Y is given by  \nE[ Y ] = ∑ k E[ Y | M k ] P [ M k | D ] (B2) Var[ Y ] = ∑ k Var[ Y | M k ] P [ M k | D ] + ∑ k E 2 [ Y | M k ] P [ M k | D ] -E 2 [ Y ] , (B3)  \nwhere E[ Y | M k ] denotes the expectation of Y given the model M k and P [ M k | D ] denotes the probability of the model M k given the data ( D ), which is given by  \nP [ M k | D ] = BF M k ∑ l BF M l , (B4)  \nwith BF M l the Bayes factor of model l . Similarly, for a given model, in the absence of prior information on the size of an unknown parameter, the optimal width of its prior can be estimated by finding the value of ˜ σ p that maximizes the BF, which typically provides a reasonable approximation to marginalizing over the prior width.", 'References': "Adelberger, E. G., A. Garc'ıa, R. G. H. Robertson, K. A. Snover, A. B. Balantekin, K. Heeger, M. J. Ramsey-Musolf, D. Bemmerer, A. Junghans, C. A. Bertulani, J. W. Chen,  \nFIG. 10 S-factor realizations obtained from different starting points in the R-matrix fit (see text for more details). The red solid line represents the mean of the evaluation with the 1 σ band denoted by dashed lines. Introducing a precise, fictitious experimental point at 0.35 MeV (in green) significantly reduces the evaluation uncertainty (blue).  \n<!-- image -->  \nH. Costantini, et al. , 2011, Reviews of Modern Physics 83 (1), 195. 1, 3  \nBaby, L. T., C. Bordeanu, G. Goldring, M. Hass, L. Weissman, V. N. Fedoseyev, U. Koster, Y. Nir-El, G. Haquin, H. W. Gaggeler, and R. Weinreich, 2003, Phys. Rev. Lett. 90 (2), 022501. 1, 3  \nBuompane, R., A. Di Leva, L. Gialanella, A. D'Onofrio, M. De Cesare, J. G. Duarte, Z. Fulop, L. R. Gasques, G. Gyurky, L. Morales-Gallegos, F. Marzaioli, G. Palumbo, et al. , 2022, Physics Letters B 824 , 136819. 1, 3, 6  \n- Davids, B., and S. Typel, 2003, Phys. Rev. C 68 (4), 045802. 3  \nFilippone, B. W., A. J. Elwyn, C. N. Davids, and D. D. Koetke, 1983, Phys. Rev. C 28 (6), 2222. 1, 3  \nHammache, F., G. Bogaert, P. Aguer, C. Angulo, S. Barhoumi, L. Brillard, J. F. Chemin, G. Claverie, A. Coc, M. Hussonnois, M. Jacotin, J. Kiener, et al. , 1998, Phys. Rev. Lett. 80 (5), 928. 1, 3  \nHammache, F., G. Bogaert, P. Aguer, C. Angulo,  \n- S. Barhoumi, L. Brillard, J. F. Chemin, G. Claverie, A. Coc, M. Hussonnois, M. Jacotin, J. Kiener, et al. , 2001, Phys. Rev. Lett. 86 (18), 3985. 2, 3  \nHass, M., 1999, Physics Letters B  \n462  \n, 237. 2, 3  \n- Higa, R., P. Premarathna, and G. Rupak, 2022, Phys. Rev. C 106 (1), 014601. 2, 3, 4, 5, 7, 8\n- Junghans, A. R., K. A. Snover, E. C. Mohrmann, E. G. Adelberger, and L. Buchmann, 2010, Phys. Rev. C 81 (1), 012801. 2, 3  \nKikuchi, T., T. Motobayashi, N. Isawa, Y. Ando, M. Kurokawa, S. Moriya, H. Murakami, T. Nishio, J. Ruan, S. Shirato, S. Shimoura, T. Uchibori, et al. , 1998, European Physical Journal A 3 (3), 213. 3  \nPaneru, S. N., C. R. Brune, R. Giri, R. J. Livesay, U. Greife, J. C. Blackmon, D. W. Bardayan, K. A. Chipps, B. Davids, D. S. Connolly, K. Y. Chae, A. E. Champagne, et al. , 2019, Phys. Rev. C 99 (4), 045807. 6  \nRolfs, C. E., and W. S. Rodney, 1988, Cauldrons in the cosmos : nuclear astrophysics (University of Chicago Press). 6  \n- Schumann, F., S. Typel, F. Hammache, K. Summerer, F. Uhlig, I. Bottcher, D. Cortina, A. Forster, M. Gai, H. Geissel, U. Greife, E. Grosse, et al. , 2006, Phys. Rev. C 73 (1), 015806. 3  \nStrieder, F., L. Gialanella, G. Gyurky, F. Schumann, R. Bonetti, C. Broggini, L. Campajola, P. Corvisiero, H. Costantini, A. D'Onofrio, A. Formicola, Z. Fulop, et al. , 2001, Nuclear Physics A 696 (1), 219. 2, 3  \n- Surer, O., F. M. Nunes, M. Plumlee, and S. M. Wild, 2022, Phys. Rev. C 106 , 024607, URL https://link.aps.org/ doi/10.1103/PhysRevC.106.024607 . 9, 10\n- Tabacaru, G., A. Azhari, J. Brinkley, V. Burjan, F. Carstoiu, C. Fu, C. A. Gagliardi, V. Kroha, A. M. Mukhamedzhanov, X. Tang, L. Trache, R. E. Tribble, et al. , 2006, Phys. Rev. C 73 (2), 025808. 2, 4  \nTrache, L., A. Azhari, F. Carstoiu, H. L. Clark, C. A. Gagliardi, Y. W. Lui, A. M. Mukhamedzhanov, X. Tang, N. Timofeyuk, and R. E. Tribble, 2003, Phys. Rev. C 67 (6), 062801. 2, 4  \nTrache, L., F. Carstoiu, C. A. Gagliardi, and R. E. Tribble, 2001, Phys. Rev. Lett. 87 , 271102, URL https://link. aps.org/doi/10.1103/PhysRevLett.87.271102 . 9, 10  \n- Zhang, X., K. M. Nollett, and D. R. Phillips, 2015, Physics Letters B 751 , 535. 6, 7, 8, 9, 10\n- Zhang, X., K. M. Nollett, and D. R. Phillips, 2018, Phys. Rev. C 98 (3), 034616. 4, 6", 'Data and analysis for the astrophysical S-factor of the Be( p, γ ) B reaction': 'We present additional information related to the Solar Fusion III section on the astrophysical S-factor of the 7 Be( p, γ ) 8 B reaction, S 17 .', 'I. Experimental Data': 'In this section we reproduce the data that formed the basis for the S 17 recommendation of the Solar Fusion III evaluation.We provide the original data in Tables I, II, III, IV, V, VI, VII, VIII, IX, X, and XI. The common mode errors associated with each data set are shown in Table XII. For the Coulomb dissociation datasets we provide the estimated asymmetric common mode errors, inflated to account for the uncertainty in the E 2 components as described in section B of the main text.  \nTABLE I S 17 data of Baby et al. (2003) a 5% added as per a private communication reported in SF IITABLE II S 17 data of Buompane et al. (2022)', 'II. Effective Field Theory EFTgs and EFT* fits': "Here we provide more details on some of the fits that were performed using EFT gs and EFT* in the S 17 (0) extrapolations. We performed fits in three different energy regions with and without Coulomb dissociation (CD) data. For notational purposes, we define fits to radiative capture data for E ≤ 475 keV region I, for E ≤ 1250 keV  \nTABLE III S 17 data of Filippone et al. (1983)TABLE IV S 17 data of Hammache et al. (1998)  \n- ∗ Data points with E > 1250 keV are not included in any fit.  \nregion II, and the non-resonant energy range comprised of E ≤ 490 keV and 805 keV ≤ E ≤ 1250 keV region III. When CD data is included in the fits in the three energy regions we denote them as IB, IIB and IIIB, respectively. Since EFT gs is only physically relevant below the 7 Be /star excitation energy E /star = 429 . 1 keV, it is used only in the I and IB fits but provides valuable checks for the EFT* II  \nTABLE V S 17 data of Hammache et al. (2001)TABLE VI S 17 data of Hass (1999) ∗  \nData points with E > 1250 keV are not included in any fit.  \nTABLE VII S 17 data of Junghans et al. (2010) ∗ Data points with E >  \nand IIB fits. Frequentist χ 2 minimization and Bayesian fits were performed at LO, NLO and NNLO. We will only report the NNLO numbers to keep the amount of information manageable.  \nFIG. 1 Contour plot of S 17 (0) in EFT gs (Higa et al. , 2022) normalized to 21 eV b. The boxed numbers identify the dashed contour lines. The × marks the experimental determinations of the ANCs (Tabacaru et al. , 2006; Trache et al. , 2003).  \n<!-- image -->  \nThe extrapolations of S 17 to 50 eV from the various fits of radiative capture data using χ 2 minimization are shown in Table XIII. We include a √ 2 / dof uncertainty estimate in the reduced χ 2 values. All the χ 2 fits included a theory prior for the ANCs in the 3 P 2 and 3 P /star 2 channels and a prior on the s -wave scattering length ratio a 22 /a 12 as described in the main text. This ensured that the fitted parameter values are consistent with the EFT power counting. Fits without the theory priors, indicated as I a , work just as well. However, the fitted parameters, shown in Table XIV, can take on values that would invalidate the EFT power counting and expansion. For the EFT gs I a fit, this is not a major concern. This theory has only two fit parameters and fits without priors give values close to expectations; see Table XIV. For the EFT /star I a  \n1250 keV are not included in any fit.  \nTABLE VIII S 17 data of Strieder et al. (2001) ∗ Data points with E > 1250 keV are not included in any fit.  \nthat has 4 fit parameters, the uncertainties in the fitted parameters are very large. Moreover, the central values result in a negligible contribution from the 3 P 2 channel with a much larger contribution from the 3 P /star 2 channel achieved through an unnaturally large ANC and a 22 /a 12 ratio. Despite the large uncertainties in the parameters, the S 17 (0) values are similar to the fits constrained by theory priors.  \nWe can understand the robustness of the S 17 (0) values and their small errors despite the large parameter uncertainties in the I a fit using Fig. 1 . The nearly horizontal contours show that the S-factor at threshold is mostly sensitive to the sum of the squares of the ANCs in EFT gs . The capture cross sections in the 5 P 2 and 3 P 2 channels have the same momentum dependence. The different s -wave scattering length in the two channels affects the momentum dependence, but only at NNLO. Thus the  \nTABLE IX S 17 Data of Schumann et al. (2006)TABLE X S 17 Data of Davids and Typel (2003)  \nTable XV shows the data set scaling factors for EFT* I a and I fits that had very different values for the fitted parameters. Again, we see that there is no difference in the quality of the fits. The difference is in how the fits without theoretical constraints enhance contributions from the 3 P /star 2 channel at the expense of the 3 P 2 channel. This imbalance is slightly less pronounced when we fit over a wider energy range where the contributions from the excited 7 Be /star become more important, better constraining the scattering length ratio a 22 /a 12 . However, sensitivity to the initial choice of parameter values in the χ 2 minimization remains since in the EFT, a 22 /a 12 is still a NNLO effect even at the higher energies.  \nE 1 transition can be equally described through capture in either of the two spin channels leading to a single parameter family of theories constrained only by the total sum of the squares of the ANCs C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) = C 2 p 3 / 2 + C 2 p 1 / 2 . In EFT*, the mixing in the S = 1 channels makes the analysis more complicated. However, we still find correlations among the 5 P 2 , 3 P 2 and 3 P /star 2 channels such that C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) is well constrained, as shown in the last column of Table XIV.  \nIn Table XVI, we show the S 17 (0) extrapolations that include the CD data in the χ 2 minimization at NNLO. All the fits were with theory priors. We find that including CD data results in a slightly lower S 17 (0) that is in agreement with the radiative capture fits in Table XIII. The scaling factors for the IIB fit are shown in Table XV.  \nWe emphasize that in the EFT framework it is not sufficient that the theory fit experimental data accurately, but must do so in a manner that is consistent with the power counting such that the perturbative expansion remains meaningful. Without a consistent expansion, theoretical uncertainty estimates are not reliable. This leads us to consider Bayesian estimations of S 17 (0).  \nThe χ 2 minimizations demonstrate that both EFT gs and EFT* are able to describe data selfconsistently in their respective domain of validity, once the theory priors are included. Incorporating EFT priors in the fits is natural in the Bayesian estimations.  \nTABLE XI S 17 Data of Kikuchi et al. (1998)TABLE XII Common mode errors of each experiment. a 2.3% common mode error plus 0.37% (the largest of the BE3L and BE3S data sets) varying error from Table I of Junghans et al. (2010)  \nMoreover, CD data is found to be compatible with the radiative capture data. The scaling factors in the fits are also reasonable. Therefore, the central consideration was whether one should fit data in only region IB, as was done in SF II without CD data, using the simpler EFT gs with only two fit parameters, or one should use data over the larger range of region IIB with the more complete EFT*.  \nThe S 17 (0) estimation from Bayesian analysis is shown in Table XVII. The EFT gs and EFT* fits in the low energy region IB are nearly identical. The Bayesian 'evidence'-the theory posterior probability distribution given some data D and hypothesis H -comparison gives ln[ P (EFT gs | D,H ) /P (EFT /star | D,H )] ≈ 0 . 32(33), ever so slightly favoring the simpler 2-parameter theory. However, this is not strong evidence given the systematic uncertainty of around 2 in the log-evidence calculations (Higa et al. , 2022). We also calculate the average  \nTABLE XIII S 17 (50 eV) and its first two energy derivatives at NNLO (Higa et al. , 2022) determined from χ 2 minimization with only radiative capture data. The theoretical uncertainty is not included.  \nTABLE XIV The mean and standard deviation of EFT (Higa et al. , 2022) parameters from χ 2 minimization. For comparison, the expected sizes of parameters from the ANCs are: C 2 ( 5 P 2 ) = 0 . 376(16) fm -1 , C 2 ( 3 P 2 ) = 0 . 088(7) fm -1 and C 2 ( 3 P /star 2 ) = 0 . 1215(36) fm -1 (Tabacaru et al. , 2006; Trache et al. , 2003; Zhang et al. , 2018). L 22 ∼ 1 is evaluated at p = p R . The last column is not a fit parameter but just lists the sum of ANC squares C 2 = C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) . Fits are only to radiative capture data.TABLE XV Scaling factors for data sets at NNLO (Higa et al. , 2022) from χ 2 minimization.  \nTABLE XVI S 17 (50 eV) and its first two energy derivatives determined from χ 2 minimization (Higa et al. , 2022) including CD data. The theoretical uncertainty is not included.  \n〈 χ 2 〉 from the posterior distribution to signify qualitatively the goodness of fit for the Bayesian estimates since the evidence cannot be used to compare the quality of fits when the data sets in IB and IIB are different. The standard deviation in the χ 2 posterior is negligible and so we only quote a √ 2 / dof uncertainty for 〈 χ 2 〉 .  \nIn Fig. 2 we compare the three Bayesian fits over a wide energy range. The uncertainties in the fits are very small (see Table XX) and are not shown. The M 1 contribution from the 1 + resonance is only included in the IIB fit that covers the resonance region. The Bayesian estimates of the EFT parameters are in Table XVIII. The parameter values are consistent with the EFT power counting. We find correlations among the 5 P 2 , 3 P 2 , and 3 P /star 2 channels  \nTABLE XVII S 17 (50 eV) and its first two energy derivatives at NNLO determined from Bayesian analysis (Higa et al. , 2022) including CD data. The theoretical uncertainty is not included.  \nFIG. 2 Astrophysical S-factor for 7 Be( p, γ ) 8 B at NNLO from Bayesian fits. The dot-dashed (black) curve is the E 1 contribution from the EFT gs IB fit, dashed (blue) curve is the E 1 contribution from the EFT* IB fit and the solid (red) curve is from the EFT* IIB fit that includes both the E 1 and M 1 contributions (Higa et al. , 2022).  \n<!-- image -->  \nsuch that only C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) is well constrained, as in the χ 2 minimization fits.  \nThe posterior from the EFT* IIB fit is shown in Fig. 3 as an example. The correlations between the ANCs support the general picture shown in Fig. 1. The interpretation in EFT gs is simpler than in EFT* where the mixing in the S = 1 channels forces a correlation with the s -wave  \nFIG. 3 Posterior distributions for the EFT* IIB fit at NNLO (Higa et al. , 2022). In the top panel, the dashed (red) curves are the priors. In the bottom panel, C 2 = C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) and ∆ C 2 = C 2 ( 5 P 2 ) -C 2 ( 3 P 2 ) . 99% of the posterior probability distribution is displayed. The vertical lines indicate the median and the regions containing 68% of the posterior probability around it.  \n<!-- image -->  \nscattering length ratio a 22 /a 12 as well. Nonetheless, the sum of ANC squares C 2 = C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) is well constrained in Fig. 3 and Table XVIII. Based on the parameter estimates in Table XVIII, one might conclude that the fits determine the individual ANCs well. However, this is not accurate.  \nIn the top panel of Fig. 3 we see that the C 2 ( 3 P 2 ) posterior is nearly identical to its prior. Something similar happens for C 2 ( 3 P /star 2 ) as well. On the other hand, we do gain  \nFIG. 4 Probability distribution functions (PDFs): Two different priors and the corresponding posteriors for C 2 ( 5 P 2 ) (Higa et al. , 2022). Details are in the text.  \n<!-- image -->  \ninformation about C 2 ( 5 P 2 ) from the fits, as seen from the corresponding prior and posterior. In EFT gs and EFT*, the 5 P 2 channel is dominant, thus, the information from the capture which we expect to be sensitive only to the sum C 2 = C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) is channeled into the posterior of C 2 ( 5 P 2 ) , while keeping the C 2 ( 3 P 2 ) and C 2 ( 3 P /star 2 ) posteriors fixed near their priors. Similar observations apply to the χ 2 fits with theory priors in Table XIV.  \nIn Fig. 4 we compare two C 2 ( 5 P 2 ) posteriors. One was obtained from a uniform prior on the p -wave effective range r ( 5 P 2 ) 1 ∼ U ( -100 MeV , 1 . 5 MeV) that covers the physically relevant regions. The C 2 ( 5 P 2 ) prior was obtained for the plot from the r ( 5 P 2 ) 1 prior using:  \nP Y ( y ) = P X [ g ( y )] g ' ( y ) , (1)  \nwhere P X ( x ), P Y ( y ) are the probability distribution functions for the random variables X and Y , respectively, and g ( y ) = f -1 ( y ) for y = f ( x ), an increasing function of x . Uniform priors on EFT parameters are a reasonable choice, as often we only estimate the sizes of the couplings and not their signs. Here, the sign of the ANC squares sets an upper bound on r ( 5 P 2 ) 1 (Higa et al. , 2022). A wide Gaussian prior r ( 5 P 2 ) 1 ∼ N ( -40 MeV , 10 MeV) centered around the experimental ANC is also shown. The Gaussian prior can result in an unphysical r ( 5 P 2 ) 1 /greaterorsimilar 1 . 5 MeV but with a insignificant cumulative probability that is less than 2 × 10 -5 . The priors result in nearly identical C 2 ( 5 P 2 ) posteriors, confirming the robustness of the fits.  \nThe data set scaling factors are shown in Table XIX. We find the three fits to be consistent with each other. The scaling factors from χ 2 minimization that includes the priors in the cost function are similar to the ones from the Bayesian estimates as seen, for example, for the IIB fits in Tables XV and XIX. Further, in Table XX we include the E 1 and M 1 contributions from the EFT* IIB Bayesian fit at NNLO at various energies. We find the  \nS 17 ( E ) posterior pdf to be close to a Gaussian, and so only list the mean and standard deviations.", 'III. A Second Bayesian Halo EFT Analysis': 'Next we provide some details about the set of fits that apply the EFT theory and Bayesian analysis tool developed in (Zhang et al. , 2015, 2018) to analyze the data sets summarized in Sec. I. This EFT (Zhang et al. , 2015) describes non-resonant capture up to NLO in terms of an EFT expansion. It has 9 parameters, including two ANCs of the final bound state: C ( 3 P 2 ) and C ( 5 P 2 ) for the 3 P 2 and 5 P 2 channels, a parameter /epsilon1 1 describing the relative strength of the radiative capture to the excited 7 Be core configuration in the final bound state, two s -wave scattering length parameters: a ( 3 S 1 ) and a ( 5 S 2 ) , two s -wave effective range parameters: r ( 3 S 1 ) and r ( 5 S 2 ) , and two parameters describing two-body-current-type contributions in the S = 1 and 2 channels: L 1 and L 2 .  \nZhang et al. (2018) demonstrates that this NLO EFT can represent previous model calculations with a few percent accuracy at energies up to 1 MeV. This few-percent discrepancy can serve as a simple estimate of the theoretical errors of this NLO EFT, assuming that (1) the previous models, when represented in the EFT framework, can populate beyond-NLO terms in the EFT expansion, and (2) neglecting them in the NLO EFT is the source of the observed discrepancies.  \nTo describe the resonant component in the capture data, we add a Breit-Wigner (BW) parameterization (Rolfs and Rodney, 1988) to the EFT-based nonresonant S-factor. The BW parameterization has three parameters: resonance energy E res , the proton decay width Γ p ( E res ), and the radiative width Γ γ ( E res ), and takes the following form:  \nS res ( E ) = 3 π 8 k 2 Γ p ( E ) Γ γ ( E ) Ee 2 πη ( E ) ( E -E res ) 2 +(Γ tot ( E ) / 2) 2 , (2)  \nΓ p ( E ) = Γ p ( E res ) e -2 πη ( E ) e -2 πη ( E res ) , (3)  \nΓ γ ( E ) = Γ γ ( E res ) E 3 γ E 3 γ res , (4)  \nwhere η ( E ) is the Sommerfeld parameter and E γ the energy of the emitted photon at energy E .  \nIn some fits, the frequentist approach of χ 2 minimization was explored. We found that some of the fitted results depended on the starting point in the model parameter space for the χ 2 -minimization algorithm; the computation of the Hessian matrix needed for error propagation in the S-factor calculations could become illconditioned, which renders numerical calculation unstable. The Bayesian analysis, on the other hand, encountered no such difficulties. Therefore, we report only on the Bayesian analysis. Three different choices of energy  \nranges for the direct capture measurements have been explored: region I with E ≤ 0 . 49 MeV , II with E ≤ 1 . 25 MeV, and III with E ≤ 0 . 49 MeV and 0 . 805 ≤ E ≤ 1 . 25 MeV. We then added the Coulomb-disassociation (CD) data sets in our analysis to the capture data sets with I, II, and III energy ranges. They are labeled as IB, IIB, and IIIB, following the convention in Sec. II. We note that in the I, III, IB, and IIIB fits, our model only includes non-resonant EFT components, while for the II and IIB fits, we also include a BW resonance.  \nIt is an essential advantage that Bayesian analysis allows the transparent incorporation of assumptions or previous knowledge about the physical model by specifying the prior probability distributions for model parameters. For this EFT, we choose wide ranges, as allowed by existing information. We assume C 2 ( 3 P 2 ) and C 2 ( 5 P 2 ) to have uniform prior distributions between 0 and 1 fm -1 ; r ( 3 S 1 ) and r ( 5 S 2 ) to have uniform prior distributions between 0 and 10 fm; /epsilon1 1 to have a uniform distribution between -1 and 1; L 1 and L 2 to have uniform distributions between -10 and 10 fm; a ( 3 S 1 ) and a ( 5 S 2 ) to have Gaussian distributions centered on the recently experimentally inferred central values of 17.34 and -3.18 fm respectively (Paneru et al. , 2019), with standard deviations equal to the experimental errors of 1.33 and 0.55 fm, respectively (Paneru et al. , 2019). We further enforce constraints on s -wave scattering parameters that preclude the existence of s -wave resonances below 0.6 MeV, following the same treatment as Zhang et al. (2015). For the resonance parameters, we assume uniform distributions between 0.6 and 0.7 MeV for E res , 0.003 and 0.063 MeV for Γ p ( E ( res ) ), and 1 × 10 -10 and 4 × 10 -8 MeV for Γ γ ( E ( res ) ). These represent 5 σ ranges (Buompane et al. , 2022).  \nAs mentioned in the main text, the other parameters to be fitted are the scaling factors for each experiment. Their prior distributions are Gaussian distributions centered at 1 with standard deviations equal to the commonmode errors (CMEs) listed in Table XII. The prior distributions of the scaling factors of the CD experiments are asymmetric Gaussians because their CMEs are asymmetric.  \nTable XXI summarizes the median and 1 σ error for the predicted S-factor and its derivatives at E = 0 MeV from six different fits. All these fits show consistency and stability. For each experiment, the fitted mean value and 1 σ errors for the scaling factor minus 1 are listed in Table XXII in percent for the I, II, and III analyses and in Table XXIII for the analysis with CD data included. A negative mean value suggests a preference for scaling the data downward, while a positive mean suggests upward scaling.  \nThe mean and error for the EFT parameters in all the fits are listed in Table XXIV. As can be seen, the results are consistent and stable against changes to the fitted energy range and inclusion of CD data. Table XXV  \nTABLE XVIII The median and the interval containing 68% of the posterior probability of the EFT (Higa et al. , 2022) parameters at NNLO from Bayesian fits. CD data are included in the fits. The last column lists C 2 = C 2 ( 5 P 2 ) + C 2 ( 3 P 2 ) .TABLE XIX Bayesian estimates of scaling factors for the data sets at NNLO (Higa et al. , 2022).  \nshows the mean and error of C 2 ( 3 P 2 ) + C 2 ( 5 P 2 ) , which is the key quantity determining S (0). As we can see, including the CD data sets reduces this quantity somewhat and thus reduces S (0), but these results are consistent with those excluding the CD data sets. It should be noted that individual ANCs are only weakly constrained, as shown in Fig. 5, but the two are strongly anti-correlated such that the sum of their squares is tightly constrained.  \nThe means and errors of the resonance parameters are provided in Table XXVI. The energy of the resonance is tightly constrained, while the widths are constrained on the percent level. Again, the two fits are consistent.  \nFig. 6 compares the IIB fit and the experimental data sets used in the analysis. The energy range shown exceeds that of the fit. Some of the errors are smaller than the size of the symbols. The theoretical 1 σ error band is only visible near E = 1 . 2 MeV. It should be pointed out that the experimental data points have not been renormalized by the scaling factors s α . The S ( E ) curves derived from the other 5 fits are similar and are not shown here.', 'IV. R-matrix Theory Fits': "We investigated the fractional contribution of the 2.2 MeV, 3 + resonance to the S-factor using R-matrix calculations. As illustrated in Fig. 7, it is vanishingly small in the energy range of interest.  \nIn the rest of this section, we attempt to quantify the uncertainty of the R-matrix fit of the low-energy  \nFIG. 5 Correlation plot of C 2 ( 3 P 2 ) and C 2 ( 5 P 2 ) from the Zhang et al. (2015) EFT + BW fit of the IIB experimental data. The contours enclose 68%, 95%, and 99.7% of the posterior probability from the inside out. In the 1-dimensional projections, the corresponding confidence intervals all start from 0 and end at one of the dashed lines. C 2 ( 3 P 2 ) and C 2 ( 5 P 2 ) are strongly anti-correlated such that their sum is tightly constrained (see Table XXV).  \n<!-- image -->  \n7 Be( p, γ ) 8 B capture data. The fit includes all experiments that have data below 425 keV and the parameters correspond to the two ANCs and four background pole partial widths. The normalization of each dataset is allowed to float following the prescription described in the main text. We randomly select N = 2300 starting points in this parameter space and for each we perform a χ 2 minimization. As the problem exhibits multiple local minima, most runs finish the fit procedure at a different one. Therefore, in what follows the usual definition for  \nTABLE XX The E 1 and M 1 contributions to S 17 from the Bayesian fit of EFT /star to data in region IIB at NNLO (Higa et al. , 2022) including CD data. The means and standard deviations are listed. An additional 3% NNLO theoretical uncertainty applies and should be added in quadrature.  \nTABLE XXI The S-factor and its 1st and 2nd derivatives at E = 0 from 6 different fits using the Zhang et al. (2015) EFT.  \nTABLE XXII Fitted values of 100 × ( s α -1), with s α the scaling factor for the data set α , from the Bayesian analysis of radiative capture data in energy ranges I, II, and III using the Zhang et al. (2015) EFT.  \nthe mean and standard deviation are replaced by  \n〈 x 〉 = N ∑ i =1 w i x i / N ∑ i =1 w i σ = √ √ √ √ N ∑ i =1 w i ( x i -〈 x 〉 ) 2 / N ∑ i =1 w i , (5)  \nwhere the weight of each sample is calculated based on the respective χ 2 value: w i = exp [ -( χ 2 /χ 2 min ) 2 ] . The value χ 2 min is the minimum χ 2 obtained from all the samples and appears there only to normalize the weights. By squaring the exponent we achieve a flatter, broader Gaussian, so as to avoid assigning an overwhelmingly large weight to the best fitting calculation. The resulting squared ANCs from all the calculations are shown in Figure 8, with the plotted lines showing the mean and standard deviation of C 2 1 + C 2 2 computed using Eq. (5). It is clear that what is constrained by the fit presented here is the sum of the squares of the ANCs and not each of them separately, but it is worth noting that in the majority of fits, C 2 > C 1 . The overall result is S (0) = 19 . 8 ± 0 . 8 eV b, consistent with our recommendation.  \nThe ANCs determine the overall normalization of the S-factor but, since we are allowing for floating normalizations of the data sets, the overall normalization we are trying to reproduce is also somewhat varying. For this set of calculations the value of χ 2 , without dividing by the number of degrees of freedom, clusters around a value of 7, with a minimum at 6.4, but calculations with val-  \nTABLE XXIII Fitted values of 100 × ( s α -1), with s α the scaling factor for the data set α , from the Bayesian analysis of radiative capture and Coulomb dissociation data in energy ranges I, II, and III using the Zhang et al. (2015) EFT.  \nFIG. 6 The Zhang et al. (2015) EFT + BW fit of the IIB experimental data, including both radiative capture and CD.  \n<!-- image -->  \nes exceeding 20 exist. We find that most calculations are concentrated around the same value of the ANCs (given by the slope in Figure 8) but lie on the rightmost edge of the uncertainty of the value extracted from 8 B breakup using both a Glauber model description (Trache et al. , 2001) and continuum-discretized coupled channel calculations (Surer et al. , 2022), as shown in the blue histogram in Figure 9. In a stark shift, the probability is concentrated at larger values of the sum of the squared ANCs and exhibits a significantly smaller spread after the ANC values obtained through the R-matrix fit are  \nFIG. 7 The contribution of the 2.2 MeV, 3 + resonance and the total of all contributions to the astrophysical S-factor S 17 ( E ) of a representative R-matrix calculation are shown in both absolute and relative terms. The 3 + contribution is negligibly small in the energy range of our fits.  \n<!-- image -->  \nFIG. 8 Squared ANCs in fm -1 of each of the N calculations after performing a χ 2 minimization to determine the R-matrix fit parameters.  \n<!-- image -->  \ndivided by the normalization of the Junghans data set, as can be seen in the red histogram of Figure 9. This result graphically depicts the tension between the normalization of the Junghans data set and the analyses of Trache et al. (2001) and Surer et al. (2022).  \nIt should be noted that a similar strong covariance exists between the ANCs and many other data sets, for example, dividing by the RIKEN data set normalization enhances the agreement with the breakup analyses and dividing by the Baby data leaves the histograms mostly unchanged. From this set of fits we can obtain the weighted mean and standard deviation for the the first and second  \nTABLE XXIV The mean values and 1 σ errors of the NLO EFT parameters from all six fits using the Zhang et al. (2015) EFT.  \nTABLE XXV The mean values and 1 σ errors of the sum of the squares of the ANCs from all six fits using the Zhang et al. (2015) EFT. The S (0) values are also listed.  \nTABLE XXVI Fitted values of the 3 BW resonance parameters in the II and IIB analyses, in which the (Zhang et al. , 2015) EFT is used to fit the direct capture component.  \nderivatives of the S-factor at zero energy S ' (0) /S (0) = -1 . 8 ± 0 . 1 MeV -1 and S '' (0) /S (0) = 39 . 1 ± 0 . 7 MeV -2 . The value of the first derivative is in agreement with that given in the main text, which is based on more data taken over a wider energy range, while the second derivative lies well outside the corresponding 3 σ range.  \nThe combined set of all the S-factor curves calculated using R-matrix fits is shown in Figure 10 as black lines with a transparency inversely proportional to their weight; the weighted mean ± 1 σ is also shown in red. From the fairly constant size of the standard deviation of the mean at various energies, we can postulate a high degree of correlation between the values of the S-factor at different energies. This is indeed the case, with correlation coefficients across all energies exceeding 0 . 9. The implication here is, in turn, that a precise measurement, even at high energies, will have a significant impact on the evaluation as a whole. This impact is also demonstrated in Figure 10, with a precise, fictitious data point at 0.35 MeV lying on top of the evaluated mean and having an uncertainty of 0.3 eV · b. Recomputing the weights associated with each fit we obtain a mean (blue) that is almost on top of the previous one but with significantly  \nFIG. 9 Posterior distribution of the sum of the squares of the computed ANCs with (blue) and without (red) renormalization of the Junghans data set in the R-matrix analysis. The solid and dashed lines give the mean and 1 σ uncertainty of the sum of the squares of the ANC values extracted from 8 B breakup by Trache et al. (2001) and Surer et al. (2022).  \n<!-- image -->  \nreduced uncertainty. This high degree of correlation is a direct byproduct of the small number of parameters that completely determine the capture cross section. Extending the analysis to the full energy spectrum (up to 1.25 MeV) would introduce more parameters (for example to describe the M 1 resonance) and thus would reduce the correlation somewhat, but the main argument presented here would still stand. It would therefore be highly desirable for future evaluations to benefit from such experiments, should they become available, and for theoretical approaches to seek an even more accurate description of the process at experimentally accessible energies."}