Direct determination of the solar neutrino fluxes from solar neutrino data
PPreprint typeset in JHEP style - PAPER VERSION
YITP-SB-09-34IFT-UAM/CSIC-09-50EURONU-WP6-09-11
Direct Determination of the Solar Neutrino Fluxesfrom Solar Neutrino Data
M. C. Gonzalez-Garcia
C.N. Yang Institute for Theoretical PhysicsState University of New York at Stony BrookStony Brook, NY 11794-3840, USA, and:
Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),Departament d’Estructura i Constituents de la Mat`eria and Institut de Ciencies delCosmos, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, SpainE-mail: [email protected]
Michele Maltoni
Instituto de F´ısica Te´orica UAM/CSIC, Facultad de Ciencias, Universidad Aut´onomade Madrid, Cantoblanco, E-28049 Madrid, SpainE-mail: [email protected]
Jordi Salvado
Departament d’Estructura i Constituents de la Mat`eria and Institut de Ciencies delCosmos, Universitat de Barcelona, 647 Diagonal, E-08028 Barcelona, SpainE-mail: [email protected]
Abstract:
We determine the solar neutrino fluxes from a global analysis of the solarand terrestrial neutrino data in the framework of three-neutrino mixing. Using a Bayesianapproach we reconstruct the posterior probability distribution function for the eight nor-malization parameters of the solar neutrino fluxes plus the relevant masses and mixing,with and without imposing the luminosity constraint. This is done by means of a MarkovChain Monte Carlo employing the Metropolis-Hastings algorithm. We also describe howthese results can be applied to test the predictions of the Standard Solar Models. Ourresults show that, at present, both models with low and high metallicity can describe thedata with good statistical agreement.
Keywords: solar neutrinos. a r X i v : . [ h e p - ph ] J un ontents
1. Introduction 12. Data analysis 33. Results 6
4. Summary 16A. Analysis of Borexino spectra 17B. Details of the Markov Chain Monte Carlo 19
1. Introduction
The idea that the Sun generates power through nuclear fusion in its core was first suggestedin 1919 by Sir Arthur Eddington who pointed out that the nuclear energy stored in theSun could explain the apparent age of the Solar System. In 1939, Hans Bethe describedin an epochal paper [1] two nuclear fusion mechanisms by which main sequence stars likethe Sun could produce the energy necessary to power their observed luminosities. Thetwo mechanisms have become known as the pp-chain and the CNO-cycle [2]. For bothchains the basic energy source is the burning of four protons to form an alpha particle,two positrons, and two neutrinos. In the pp-chain, fusion reactions among elements lighterthan A = 8 produce a characteristic set of neutrino fluxes, whose spectral energy shapesare known but whose fluxes must be calculated with a detailed solar model. In the CNO-cycle the abundance of C plus N acts as a catalyst, while the N and O beta decaysprovide the primary source of neutrinos.In order to precisely determine the rates of the different reactions in the two chains,which are responsible for the final neutrino fluxes and their energy spectrum, a detailedknowledge of the Sun and its evolution is needed. Standard Solar Models (SSM’s) [3–9]describe the properties of the Sun and its evolution after entering the main sequence. Themodels are based on a set of observational parameters (the present surface abundances ofheavy elements and surface luminosity of the Sun, as well as its age, radius and mass) and onseveral basic assumptions: spherical symmetry, hydrostatic and thermal equilibrium, andequation of state. Over the past five decades the solar models were steadily refined as theresult of increased observational and experimental information about the input parameters– 1 –such as nuclear reaction rates and the surface abundances of different elements), moreaccurate calculations of constituent quantities (such as radiative opacity and equation ofstate), the inclusion of new physical effects (such as element diffusion) and the developmentof faster computers and more precise stellar evolution codes.Despite the progress of the theory, only neutrinos, with their extremely small interac-tion cross sections, can enable us to see into the interior of a star and thus verify directlyour understanding of the Sun [10]. Indeed from the earliest days of solar neutrino re-search this test has been a primary goal of solar neutrino experiments, but for many yearsthe task was made difficult by the increasing discrepancy between the predictions of theSSM’s and the solar neutrino observations. This so-called “solar neutrino problem” [11,12]was finally solved by the modification of the Standard Model of Particle Physics with theinclusion of neutrino masses and mixing. In this new framework leptonic flavors are nolonger symmetries of Nature, and neutrinos can change their flavor from the productionpoint in the Sun to their detection on the Earth. This flavor transition probability isenergy dependent [13–16], which explains the apparent disagreement among experimentswith different energy windows. This mechanism is known as the LMA-MSW solution tothe solar neutrino problem, and affects both the overall number of events in solar neutrinoexperiments and the relative contribution expected from the different components of thesolar neutrino spectrum. Due to these complications, at first it was necessary to assumethe SSM predictions for all the solar neutrino fluxes and their uncertainties in order toextract reasonably constrained values for neutrino masses and mixing. The upcoming ofthe real-time experiments Super-Kamiokande and SNO and the independent determinationof the flavor oscillation probability using reactor antineutrinos at KamLAND opened upthe possibility of extracting the solar neutrino fluxes and their uncertainties directly fromthe data [9, 17–22]. Nevertheless, in these works some set of simplifying assumptions hadto be imposed in order to reduce the number of free parameters to be determined.In parallel to the increased precision of the SSM-independent determination of theneutrino flavor parameters, a new puzzle has emerged in the consistency of SSM’s [23]. Tillrecently SSM’s have had notable successes in predicting other observations. In particular,quantities measured by helioseismology such as the radial distributions of sound speed anddensity [5–8] showed good agreement with the predictions of the SSM calculations andprovided accurate information on the solar interior. A key element to this agreement is theinput value of the abundances of heavy elements on the surface of the Sun [24]. However,recent determinations of these abundances point towards substantially lower values thanpreviously expected [25, 26]. A SSM which incorporates such lower metallicities fails atexplaining the helioseismological observations [23], and changes in the Sun modeling (inparticular of the less known convective zone) are not able to account for this discrepancy [27,28]. So far there has not been a successful solution of this puzzle. Thus the situation isthat, at present, there is no fully consistent SSM. This led to the construction of twodifferent sets of SSM’s, one (labeled “GS”) based on the older solar abundances [24] imply-ing high metallicity, and one (labeled “AGS”) assuming lower metallicity as inferred frommore recent determinations of the solar abundances [25, 26]. In Ref. [9] the solar fluxes– 2 –orresponding to such two models were detailed, based on updated versions of the solarmodel calculations presented in Ref. [8]. These fluxes were denoted as “BPS08(GS)” and“BPS08(AGS)”, respectively. In a very recent work [29] an update of the BPS08(AGS)solar model has been constructed using the latest determination of the compositions [26] aswell as some improvement in the equation of state. For what concerns the overall normal-ization of solar neutrino fluxes, the predictions of this new model are very close to those ofBPS08(AGS).In this work we perform a solar model independent analysis of the solar and terrestrialneutrino data in the framework of three-neutrino masses and mixing. The aim of this anal-ysis is to simultaneously determine the flavor parameters and all the solar neutrino fluxeswith a minimum set of theoretical priors. In Sec. 2 we present the method employed, thedata included in the analysis and the physical assumptions used in this study. The resultsof the analysis are given in Sec. 3, where we show the reconstructed posterior probabilitydistribution function for the eight normalization parameters of the solar neutrino fluxes.We discuss in detail the effect of the luminosity constraint [30] as well as the role of theBorexino experiment and its potential for improvement. In addition, we use the resultsof this analysis to statistically test to what degree the present solar neutrino data candiscriminate between the two SSM’s. Finally in Sec. 4 we summarize our conclusions.
2. Data analysis
In the analysis of solar neutrino experiments we include the total rates from the radio-chemical experiments Chlorine [31], Gallex/GNO [32] and SAGE [32, 33]. For real-timeexperiments in the energy range of B neutrinos we include the 44 data points of the elec-tron scattering (ES) Super-Kamiokande phase I (SK-I) energy-zenith spectrum [34], the 34data points of the day-night spectrum from SNO-I [35], the separate day and night rates forneutral current (NC) and ES events and the day-night energy-spectrum for charge current(CC) events from SNO-II (a total of 38 data points) [36], the three rates for CC, ES andNC from SNO-III [37], and the 6 points of the high-energy spectrum from the 246 live daysof Borexino [38] (which we denote as Borexino-HE). Finally, we include the main set ofthe 192 days of Borexino data (denoted as Borexino-LE) [40] in two different forms: inone analysis we use the total event rates from Be neutrinos as extracted by the Borexinocollaboration, while in the other we perform our own fit to the Borexino energy spectrum inthe region above 365 keV (corresponding to a total of 160 data points). Full details of ourBorexino data analysis are presented in Appendix A. In the framework of three neutrinomasses and mixing the expected values for these solar neutrino observables depend on theparameters ∆ m , θ , and θ as well as on the normalizations of the eight solar fluxes. We have not included here the very recent results on the low energy threshold analysis of the combinedSNO phase I and phase II [39]. These results provide information on the B and hep fluxes and showno major difference with the results from their previous analysis, hence we expect that they will have noimportant impact on the results of the global analysis here presented. In particular we notice that theirbest fit determination of the B flux as well as of the oscillation parameters are in perfect agreement withour results. – 3 –lux Φ ref i [cm − s − ] α i [MeV] β i pp 5 . × . . × − Be 5 . × . . × − pep 1 . × . . × − N 2 . × . . × − O 2 . × . . × − F 5 . × .
363 1 . × − B 5 . × . . × − hep 7 . × . . × − Table 1:
The reference neutrino flux Φ ref i used for normalization, the energy α i provided to thestar by nuclear fusion reactions associated with the i th neutrino flux (taken from Ref. [30]), and thefractional contribution β i of the i th nuclear reaction to the total solar luminosity. Besides solar experiments, we also include the latest results from the long baselinereactor experiment KamLAND [41, 42], which in the framework of three neutrino mixingalso yield information on the parameters ∆ m , θ , and θ . In addition, we include theinformation on θ obtained after marginalizing over ∆ m , θ and δ cp the results fromthe complete SK-I and SK-II atmospheric neutrino data sets (see the Appendix of Ref. [42]for full details on our analysis), the CHOOZ reactor experiment [43], K2K [44], the latestMINOS ν µ disappearance data corresponding to an exposure of 3 . × p.o.t. [45], andthe first MINOS ν µ → ν e appearance data presented in Ref. [46]. Details of the oscillationanalysis of these observables will be presented elsewhere [47].In what follows, for convenience, we will use as normalization parameters for the solarfluxes the reduced quantities: f i = Φ i Φ ref i (2.1)with i = pp, Be, pep, N, O, F, B, and hep. The numerical values of Φ ref i areconventionally set to the predictions of the BPS08(GS) solar model and are listed in Table 1.With this, the theoretical predictions for the relevant observables (after marginalizing over∆ m , θ and δ cp ) depend on eleven parameters: the three relevant oscillation parameters∆ m , θ , θ and the eight reduced solar fluxes f i . With the data from the differentdata samples (D) and the theoretical predictions for them in terms of these parameters (cid:126)ω = (∆ m , θ , θ , f pp , . . . , f hep ) we build the corresponding likelihood function L (D | (cid:126)ω ) = 1 N exp (cid:20) − χ (D | (cid:126)ω ) (cid:21) (2.2)where N is a normalization factor. In Bayesian statistics our knowledge of (cid:126)ω is summarizedby the posterior probability distribution function (p.d.f.) p ( (cid:126)ω | D , P ) = L (D | (cid:126)ω ) π ( (cid:126)ω |P ) (cid:82) L (D | (cid:126)ω (cid:48) ) π ( (cid:126)ω (cid:48) |P ) d(cid:126)ω (cid:48) (2.3)where π ( (cid:126)ω |P ) is the prior probability density for the parameters. In our model independentanalysis we assume a uniform prior probability over which we impose the following set of– 4 –onstraints to ensure consistency in the pp-chain and CNO-cycle, as well as some relationsfrom nuclear physics: • The fluxes must be positive: Φ i ≥ ⇒ f i ≥ . (2.4) • The number of nuclear reactions terminating the pp-chain should not exceed thenumber of nuclear reactions which initiate it [30, 48]:Φ Be + Φ B ≤ Φ pp + Φ pep ⇒ . × − f Be + 9 . × − f B ≤ f pp + 2 . × − f pep . (2.5) • The N( p, γ ) O reaction must be the slowest process in the main branch of theCNO-cycle [48]: Φ O ≤ Φ N ⇒ f O ≤ . f N (2.6)and the CNO-II branch must be subdominant:Φ F ≤ Φ O ⇒ f F ≤ f O . (2.7) • The ratio of the pep neutrino flux to the pp neutrino flux is fixed to high accuracybecause they have the same nuclear matrix element. We have constrained this ratioto match the average of the BPS08(GS) and BPS08(AGS) values, with 1 σ Gaussianuncertainty given by the difference between the values in the two models f pep f pp = 1 . ± . . (2.8)Following standard techniques we reconstruct the posterior p.d.f. in Eq. (2.3) using a Monte-Carlo algorithm; full details of our approach are given in Appendix B.The number of independent fluxes is reduced when imposing the so-called “luminosityconstraint”, i.e. , the requirement that the sum of the thermal energy generation ratesassociated with each of the solar neutrino fluxes coincides with the solar luminosity [49]: L (cid:12) π (A.U.) = (cid:88) i =1 α i Φ i . (2.9)Here the constant α i is the energy provided to the star by the nuclear fusion reactionsassociated with the i th neutrino flux; its numerical value is independent of details of thesolar model to an accuracy of one part in 10 or better [30]. A detailed derivation of thisequation and the numerical values of the coefficients α i , which we reproduce for convenience We have verified that assuming a flat distribution over the 1 σ uncertainty interval does not producesignificant differences in the results of our analysis. – 5 –n Table 1, is presented in Ref. [30]. In terms of the reduced fluxes Eq. (2.9) can be writtenas: 1 = (cid:88) i =1 β i f i with β i ≡ α i Φ ref i L (cid:12) (cid:14) [4 π (A.U.) ] (2.10)where β i is the fractional contribution to the total solar luminosity of the nuclear reactionsresponsible for the production of the Φ ref i neutrino flux, and L (cid:12) (cid:14) [4 π (A.U.) ] = 8 . × MeV cm − s − [30]. The analysis performed incorporating the priors in Eqs. (2.4–2.9)will be named “analysis with luminosity constraint”, P = L (cid:12) , and for this case the priorprobability distribution is: π ( (cid:126)ω (cid:48) | L (cid:12) ) = N exp (cid:34) − (cid:0) f pep (cid:14) f pp − . (cid:1) σ (cid:35) if Eqs. (2.4–2.7) and (2.9) are verified,0 otherwise, (2.11)where N is a normalization factor and σ = 0 . P = /L (cid:12) , so: π ( (cid:126)ω (cid:48) | /L (cid:12) ) = N exp (cid:34) − (cid:0) f pep (cid:14) f pp − . (cid:1) σ (cid:35) if Eqs. (2.4–2.7) are verified,0 otherwise. (2.12)Let us notice that the conditions in Eqs. (2.4–2.7) and Eq. (2.9) are constraints on somelinear combinations of the solar fluxes and they are model independent, i.e. , they do notimpose any prior bias favouring either of the SSM’s. Furthermore we have chosen to centerthe condition (2.8) at the average of the BPS08(GS) and BPS08(AGS) values, with 1 σ Gaussian uncertainty given by the difference between the values in the two models, toavoid the introduction of a bias towards one of the models. In the next sections we willcomment on how our results are affected when this prior is centered about the BPS08(GS)or the BPS08(AGS) prediction.
3. Results
Our results for the analysis with luminosity constraint are displayed in Fig. 1, where weshow the marginalized one-dimensional probability distributions p ( f i | D , L (cid:12) ) for the eightsolar neutrino fluxes as well as the 90% and 99% CL two-dimensional allowed regions.The corresponding ranges at 1 σ (and at the 99% CL in square brackets) on the oscillationparameters are: ∆ m = 7 . ± . ± . × − eV , sin θ = 0 . ± .
02 [ ± . , sin θ = 0 . ± .
012 [ +0 . − . ] , (3.1)– 6 – igure 1: Constraints from our global analysis on the solar neutrino fluxes. The curves in therightmost panels show the marginalized one-dimensional probability distributions, before and afterthe inclusion of the Borexino spectral data. The rest of the panels show the 90% and 99% CLtwo-dimensional credibility regions (see text for details). while for the solar neutrino fluxes are: f pp = 0 . +0 . − . [ +0 . − . ] , Φ pp = 5 . +0 . − . [ +0 . − . ] × cm − s − ,f Be = 1 . +0 . − . [ +0 . − . ] , Φ Be = 5 . +0 . − . [ +1 . − . ] × cm − s − ,f pep = 0 . ± .
014 [ ± . , Φ pep = 1 . +0 . − . [ +0 . − . ] × cm − s − ,f N = 2 . +1 . − . [ +5 . − . ] , Φ N = 7 . +5 . − . [ +16 − . ] × cm − s − ,f O = 1 . ± . +2 . − . ] , Φ O = 4 . +1 . − . [ +4 . − . ] × cm − s − ,f F ≤
32 [72] , Φ F ≤ . × cm − s − ,f B = 0 . ± .
03 [ ± . , Φ B = 5 . +0 . − . [ +0 . − . ] × cm − s − ,f hep = 1 . +1 . − . [ +3 . − . ] , Φ hep = 1 . ± . +3 . − . ] × cm − s − . (3.2)– 7 –s mentioned above we have checked the stability of the results under changes in theassumption of the prior in Eq. (2.8). We find that if we center this prior at the BPS08(GS)prediction ( f pep (cid:14) f pp = 1) the best fit value for pep neutrinos is changed to f pep = 0 . pep = 1 . × cm − s − ). Conversely if the Gaussian prior in Eq. (2.8) is centeredat the BPS08(AGS) prediction ( f pep (cid:14) f pp = 1 . f pep = 1 .
003 (Φ pep = 1 . × cm − s − ). All other fluxes are unaffected.For the sake of illustration we have also performed a Gaussian fit to the two-dimensionalp.d.f. for the eight fluxes. The best Gaussian approximation to the real p.d.f. is charac-terized by flux uncertainties obtained by symmetrizing the 1 σ ranges quoted in Eq. (3.2),and by the following error correlation matrix: f pp f Be f pep f N f O f F f B f hep f pp − .
81 0 . − . − . − .
26 0 .
06 0 . f Be − . − .
10 0 .
10 0 . − .
05 0 . f pep − . − . − .
20 0 .
04 0 . f N .
31 0 . − .
02 0 . f O . − .
03 0 . f F − .
02 0 . f B − . f hep Be flux is directly dictated by the luminosity constraint (see comparison withFig. 3). All these results imply the following share of the energy production between thepp-chain and the CNO-cycle L pp-chain L (cid:12) = 0 . +0 . − . [ +0 . − . ] ⇐⇒ L CNO L (cid:12) = 0 . +0 . − . [ +0 . − . ] , (3.4)in perfect agreement with the SSM’s which predict L CNO /L (cid:12) ≤
1% at the 3 σ level.The sensitivity of the various experiments is illustrated in Fig. 2, where we plot thecontribution of each flux to the total event rates at the radiochemical experiments as well asSNO and SK (for Borexino see Appendix A) together with the corresponding experimentalvalues and uncertainties. To highlight the sensitivity to the hep flux we plot separatelythe rate for the last energy bin in SK (SK-hi, E e ≥
16 MeV); similar results hold forthe highest energy bins of SNO. The rates are computed for the best fit value of theoscillations parameters, Eq. (3.1). We show the contributions as predicted by BPS08(GS)and BPS08(AGS) solar models and by our best fit values for the fluxes given in Eq. (3.2).In order to check the consistency of our results we have performed the same analy-sis without imposing the luminosity constraint, Eq. (2.9). The corresponding results for p ( f i | D , /L (cid:12) ) and the two-dimensional allowed regions are shown in Fig. 3. As expected, thepp flux is the most affected by the release of this constraint. This is so because the ppreaction gives the largest contribution to the solar energy production, as can be seen inTable 1. Imposing the luminosity constraint as an upper bound on the pp flux would imply– 8 – a Cl SNO-I SNO-II SNO-III SK-lo SK-hi R / R b e s t G S B e s t - f i t A G S G S B e s t - f i t A G S G S B e s t - f i t A G S G S B e s t - f i t A G S G S B e s t - f i t A G S G S B e s t - f i t A G S G S B e s t - f i t A G S G A LL EX G N O SA G E pp Be pep CNO B hep
Figure 2:
Contribution of each solar neutrino flux to the total event rates at different experiments.The oscillation parameters are set to their best fit value, Eq. (3.1). We show the contributions aspredicted by BPS08(GS) and BPS08(AGS) solar models as well as our best fit values given inEq. (3.2). that this flux cannot exceed its SSM prediction by more than 9%. Conversely, releasingthis constraint allows for a much larger pp flux. The pep flux is also severely affected dueto its strong correlation with the pp flux, Eq. (2.8). On a smaller scale the CNO fluxesare also affected, mainly as an indirect effect due to the modified contribution of the ppand pep fluxes to the Gallium and Chlorine experiments, which leads to a change in theallowed contribution of the CNO fluxes to these experiments. Thus in this case we get: f pp = f pep = 0 . +0 . − . [ +0 . − . ] ,f Be = 1 . +0 . − . [ +0 . − . ] ,f N = 2 . +1 . − . [ +5 . − . ] ,f O = 1 . ± +2 . − . ] ,f F ≤
34 [79] . (3.5)The determination of the B and hep fluxes (as well as the oscillation parameters) isbasically unaffected by the luminosity constraint.Interestingly, the idea that the Sun shines because of nuclear fusion reactions canbe tested accurately by comparing the observed photon luminosity of the Sun with theluminosity inferred from measurements of solar neutrino fluxes. We find that the energyproduction in the pp-chain and the CNO-cycle without imposing the luminosity constraint– 9 – igure 3:
Same as Fig. 1 but without the luminosity constraint, Eq. (2.9). The curves in therightmost panels show the marginalized one-dimensional probability distributions, before and afterthe inclusion of the Borexino spectral data. The rest of the panels show the 90% and 99% CLtwo-dimensional credibility regions (see text for details). are given by: L pp-chain L (cid:12) = 0 . +0 . − . [ ± .
40] and L CNO L (cid:12) = 0 . +0 . − . [ +0 . − . ] . (3.6)Comparing Eqs. (3.4) and (3.6) we see that the luminosity constraint has only a limitedimpact on the amount of energy produced in the CNO-cycle. However, as discussed above,the amount of energy in the pp-chain can now significantly exceed the total quantity allowedby the luminosity constraint. Altogether we find that the present value for the ratio of the– 10 –eutrino-inferred solar luminosity, L (cid:12) (neutrino-inferred), to the photon luminosity L (cid:12) is: L (cid:12) (neutrino-inferred) L (cid:12) = 1 . ± .
14 [ +0 . − . ] . (3.7)Thus we find that, at present, the neutrino-inferred luminosity perfectly agrees with themeasured one, and this agreement is known with a 1 σ uncertainty of 14%. As seen in Figs. 1 and 3 the inclusion of Borexino has a very important impact on thedetermination of the Be, pep and CNO fluxes, and indirectly on the pp flux. As mentionedabove and described in Appendix A, in our analysis we have fitted the 160 data points of theBorexino energy spectrum in the 365–2000 keV energy range, leaving as free parametersthe normalization of the the C, C, Bi and Kr backgrounds, the three relevantoscillation parameters, and the normalization of all the solar neutrino fluxes. In contrast,the Borexino collaboration fits the spectrum in the full energy range 160–2000 keV, andallows for free normalizations of the C, C, Bi+CNO and Kr backgrounds as well as C (which introduces an overwhelming background but is only relevant for events below250 keV, hence it does not contribute to our analysis). Besides the normalization of thesebackground components only the Be flux normalization is fitted to the data, and no directinformation on the normalization of the other solar fluxes is extracted. In particular, theCNO fluxes are added to the
Bi background and fitted as a unique “background”, whilethe other solar fluxes are fixed to the BPS08(GS) prediction and the oscillation parametersare fixed to the best fit point of the global pre-Borexino analysis. With this procedurethey determine the interaction rate for the 0 .
862 MeV Be neutrinos to be 49 ± stat ± sys ,which corresponds to f Be = 1 . ± .
10. Given the precision of the data and the energyspectrum of the irreducible backgrounds, their procedure is perfectly acceptable for thepurpose of extracting the Be normalization alone. However, in this work we are interestedin testing the full set of SSM fluxes, not just Be. In this case the consistent procedure isto allow for independent normalizations of all the solar fluxes.In order to illustrate the impact of these two different approaches on the determinationof the SSM fluxes, we have repeated our analysis using as unique data input from Borexino-LE the total interaction rate for Be neutrinos quoted by the collaboration. The resultsare shown in Fig. 4. As seen in this figure the constraints imposed on the Be flux afterthe inclusion of Borexino are basically the same, irrespective of whether one includes thefull Borexino energy spectrum in the range 365–2000 keV or just the total Be event rateextracted by the Borexino collaboration. However, the best fit and allowed ranges forthe CNO fluxes are not the same. This proves that, despite the unknown level of
Bicontamination, the Borexino spectral data can still provide useful information on the CNOfluxes. When using only the total Be event rate this information is lost and the constraintson CNO arise exclusively from the Gallium and Chlorine experiments.As shown in Fig. 4, the inclusion of the complete Borexino-LE spectrum leads to animprovement (albeit not very significant) of the determination of the N flux. WithoutBorexino this flux is mostly (and poorly) constrained by the Gallium experiment, and– 11 – igure 4:
Marginalized one-dimensional probability distributions for the fluxes contributing toBorexino-LE. The full line shows the determination obtained by fitting the Borexino spectrum dataas described in Appendix A. The dashed-dotted line shows what the results would be if instead onehad used the Borexino result for the extracted interaction rate of the 0 .
862 MeV Be neutrinos. Thedotted line represents the precision obtainable with the simulated “ideal” spectrum as described inSec. 3.1. – 12 –ncluding the additional information from Borexino positively adds to its knowledge. Wenotice, however, that the best fit value of the N flux in either analysis is always higherthan the prediction of any of the SSM’s (although fully compatible at better than 1 . σ ).This behavior is driven by the Gallium rate which is slightly higher than expected in anyof the SSM’s within the framework of three neutrino oscillations. A higher best fit valueof N can easily accommodate this observation without conflicting with any of the otherexperiments nor with the observed spectrum at Borexino. On the contrary, this is notthe case for the O flux. Adding the information from Borexino-LE spectrum leads to a(also small) worsening of the precision in the determination of this flux. We traced thisapparently counter-intuitive result to the existing tension between the low Chlorine rate andthe predicted rate within the framework of three neutrino oscillations. As a consequence ofthis tension, Chlorine pushes towards lower values of the O flux, whereas the spectrumof Borexino-LE prefers a higher amount of O. When only the total event rate of Be isused in the fit the extracted O flux is mostly driven by the Chlorine result. When theBorexino-LE spectrum is also included the tension results into a higher best fit for the Oflux and a worsening of the precision.Among all solar neutrino fluxes, the CNO ones are those for which the differencesbetween the BPS08(GS) and BPS08(AGS) SSM predictions are largest. It is thereforeinteresting to explore whether this discrepancy can be resolved with future Borexino data,and to what degree the CNO and pep fluxes can be better determined. Besides the accu-mulation of more statistics, in the near future Borexino aims at reducing the systematicuncertainties with the deployment of calibration sources in the detector [40]. Ideally, ifthe C background could be subtracted from the signal the pep and CNO fluxes wouldbecome directly accessible. In order to illustrate the potential of this perspective we havesimulated an “ideal” spectrum of 85 bins in the energy range 365–1238 keV according tothe expectations from the central values of the BPS08(GS) fluxes and the best fit point ofoscillations. In our simulation we have assumed that the C has been fully removed, whilethe other backgrounds are added under the same assumptions as in the present Borexinoanalysis. We have also assumed double statistics and a reduction by a factor three of thesystematic uncertainties. The results of this fit are presented in Fig. 4. As can be seen,with this ideal experiment the level of accuracy can be substantially improved for mostfluxes, with the exception of N flux whose determination becomes less precise. This isa consequence of the tension between the higher value of N preferred by the Galliumexperiments and the SSM value used for the simulated spectrum: if we had simulated datacorresponding to a higher value of N flux, the precision in the determination of this fluxwould also have improved. In any case, our results show that even with this optimisticimprovement of Borexino the precision of the CNO fluxes remains far below the presentuncertainties of the SSM’s.
In Fig. 5 we show the marginalized one-dimensional probability distributions for the solarneutrino fluxes as determined by our analysis, together with the predictions for the twoSSM’s given in Ref. [9]. In order to statistically compare our results with the SSM’s– 13 – igure 5:
Marginalized one-dimensional probability distributions for the best determined solarfluxes in our analysis as compared to the predictions for the two SSM’s in Ref. [9]. predictions we perform a significance test. We start by constructing a posterior probabilitydistribution function for the solar fluxes as well as for the SSM central values. This isdefined as the probability distribution from the data subject to the prior distribution of anarbitrary SSM: p ( (cid:126)f , (cid:126) ¯ f SSM | D , SSM) = L (D | (cid:126)f ) π ( (cid:126)f , (cid:126) ¯ f SSM | SSM) (cid:82) L (D | (cid:126)f ) π ( (cid:126)f , (cid:126) ¯ f SSM | SSM) d (cid:126)f d (cid:126) ¯ f SSM (3.8)where − (cid:104) π ( (cid:126)f , (cid:126) ¯ f SSM | SSM) (cid:105) = (cid:88) i,j ( f i − ¯ f SSM i ) V − ,ij ( f j − ¯ f SSM j ) (3.9)and V SSM is the covariance matrix for the assumed SSM model. We build the covariancematrix for arbitrary models by interpolating the covariance matrices for the BPS08(GS)– 14 – igure 6:
Probability distribution function g ( t ) (full line, see text for details). For comparison weshow the corresponding distribution for a χ p.d.f. with 8 degrees of freedom (dashed line). and BPS08(AGS) models given in Ref. [9]. Since these covariance matrices are very similarto each other, our results are not sensitive to this assumption. Furthermore, in order toimprove this approximation one would need a continuous model dependence for the fluxcovariance matrix, which is currently unavailable.The posterior probability distribution for a SSM characterized by given central valuesand covariance matrix, subject to the constraints imposed by the data is then p ( (cid:126) ¯ f SSM | D) = (cid:90) p ( (cid:126)f , (cid:126) ¯ f SSM | D , SSM) d (cid:126) ¯ f (3.10)From p ( (cid:126) ¯ f SSM | D) we define a probability distribution function for the statistics t as g ( t ) = (cid:90) p ( (cid:126) ¯ f SSM | D) δ (cid:104) t + 2 ln (cid:16) p ( (cid:126) ¯ f SSM | D) (cid:17)(cid:105) d (cid:126) ¯ f SSM (3.11)By definition g ( t ) is a function normalized to 1 in the interval t min ≤ t ≤ ∞ . With thisdefinition, t would follow a χ distribution if p ( (cid:126)f | D , L (cid:12) ) were exactly Gaussian. In Fig. 6we plot the function g ( t ). For comparison we show the corresponding distribution for a χ p.d.f. with 8 degrees of freedom (dashed line).The significance of the agreement between the data and what is expected under theassumption of a given model is quantified in terms of the probability P agr , defined as the – 15 –robability to find t in the region of equal or larger compatibility with the data than thelevel of compatibility observed within the given model: P agrGS(AGS) = (cid:90) t max t GS(AGS) g ( t ) dt (3.12)where t GS(AGS) is the value of the statistic obtained for the central value fluxes of thespecific model: t GS(AGS) = − (cid:104) p ( (cid:126) ¯ f GS(AGS) | D) (cid:105) . (3.13)We found that the GS model has a lower t , t GS = 8 .
5, while t AGS = 11 .
0. With theprobability distribution g ( t ) shown in Fig. 6 this corresponds to P agrGS = 43% and P agrAGS =20%.For comparison we have also constructed a χ function comparing the best fit valuesof the fluxes in each of the models with those obtained in the analysis without the SSMpriors and with the uncertainties given by the combined covariance matrix χ = (cid:88) ij ( ¯ f GS(AGS) i − ¯ f D i ) (cid:2) V GS(AGS) + V D (cid:3) − ij ( ¯ f GS(AGS) j − ¯ f D j ) . (3.14)Here V D is the covariance matrix obtained by the best Gaussian approximation to the p ( f i | D , L (cid:12) ) probability distribution function, Eq. (3.3), and ¯ f D j are the best fit fluxes fromthe data analysis without any SSM prior, Eq. (3.2). If the distribution p ( (cid:126)f | D , L (cid:12) ) wereexactly Gaussian, both tests would be equivalent. We found that this test still yields abetter fit for the GS model, χ = 5 . P agrGS = 74%) and χ = 5 . P agrAGS = 68%),but gives a higher probability for both models. We also find that if the prior in Eq. (2.8)is centered at the BPS08(AGS) prediction the analysis renders the same probability forboth models. Conversely if Eq. (2.8) is centered at the BPS08(GS) prediction the slightpreference for the BPS08(GS) model found above is enhanced to χ = 5 . P agrGS = 74%)versus χ = 7 . P agrAGS = 50%).From these results we conclude that, while the fit shows a slightly better agreementwith the GS model corresponding to higher metallicities, the difference between the twois not statistically significant. This is partly due to the lack of precision of present data.But we also notice that, while the measurements of SNO and SK favor a lower B flux aspredicted by the low metallicity models, the determination of the Be flux in Borexino andthe corresponding determination of the pp flux from the luminosity constraint show betteragreement with the GS predictions.
4. Summary
We have performed a solar model independent analysis of the solar and terrestrial neu-trino data in the framework of three-neutrino oscillations, following a Bayesian approachin terms of a Markov Chain Monte Carlo using the Metropolis-Hastings algorithm. Thisapproach has allowed us to reconstruct the probability distribution function in the entireeleven-dimensional parameter space, consistently incorporating the required set of theoret-ical priors. The best fit values and allowed ranges for the eight solar neutrino fluxes are– 16 –ummarized in Eq. (3.2) and Fig. 1 for the analysis with the luminosity constraint, and inEq. (3.5) and Fig. 3 for the more general case of unconstrained solar luminosity. We foundthat at present the neutrino-inferred luminosity perfectly agrees with the measured oneand it is known with a 1 σ uncertainty of 14%. We have also tested the fractional energyproduction in the pp-chain and the CNO-cycle, finding that the total amount of the solarluminosity produced in the CNO-cycle is bounded to be L CNO /L (cid:12) < .
8% at 99% CLirrespective of whether the luminosity constraint is imposed or not.We have then presented a statistical test which can be performed with these results inorder to shed some light on the so-called solar composition problem, which at present arisesin the construction of the Standard Solar Model. We found that the low value of the B fluxmeasured at SK and SNO points towards low metallicity models, whereas the measurementof Be in Borexino and the corresponding value of the pp flux implied by the luminosityconstraint show better agreement with high metallicity models. Altogether the fit shows aslight preference for models with higher metallicities, however the difference between thetwo models is not very significant at statistical level. While a realistic improvement of theBorexino data analysis in the near future can positively affect the direct determination ofmost solar neutrino fluxes, it is unlikely that enough precision will be achieved to go beyondthe the present theoretical uncertainties of the SSM’s. The largest difference between themodels lies on the CNO fluxes that give predictions which differ by about 30%. Thusideally in order to achieve a statistically meaningful discrimination between the modelsone would need a low energy solar neutrino experiment capable of measuring the neutrinoenergy spectrum for energies between 0 . (cid:46) E ν (cid:46) . O (30%) precision. Acknowledgments
We thank Stefan Schoenert, Raju Raghavan and Gianpaolo Bellini for illuminating clarifi-cations on the Borexino data and its analysis. This work is supported by Spanish MICINNgrants 2007-66665-C02-01 and FPA2006-01105 and consolider-ingenio 2010 grant CSD2008-0037, by CSIC grant 200950I111, by CUR Generalitat de Catalunya grant 2009SGR502,by Comunidad Autonoma de Madrid through the HEPHACOS project P-ESP-00346, byUSA-NSF grant PHY-0653342, and by EU grant EURONU.
A. Analysis of Borexino spectra
In our analysis of Borexino we include both the low-energy (LE) data presented in Ref. [40],which are crucial for the reconstruction of the Be line, as well as the high-energy (HE) datadiscussed in Ref. [38], which are mostly sensitive to the B flux. For the low-energy part weextracted the 180 experimental data points and the corresponding statistical uncertaintiesfrom Fig. 2 of Ref. [40], checking explicitly that the statistical error σ stat b is just the squareroot of the number of events N ex b in each bin b (except in the region where statistical α ’ssubtraction had been performed). Similarly, for the high-energy part we extracted the 6– 17 –xperimental data points and statistical uncertainties from Fig. 3 of Ref. [38]. For bothdata sets the theoretical prediction N th b for the bin b is calculated as follows: N th b ( (cid:126)ω, (cid:126)ξ ) = n el T run b (cid:88) α (cid:90) d Φ det α dE ν ( E ν | (cid:126)ω ) dσ α dT e ( E ν , T e ) R b ( T e | (cid:126)ξ ) dE ν + N bkg b ( (cid:126)ξ ) (A.1)where (cid:126)ω describes both the neutrino oscillation parameters and the eight solar flux normal-izations, and (cid:126)ξ is a set of variables parametrizing the systematic uncertainties as requiredby the “pulls” approach to χ calculation. Here n el is the number of electron targets in afiducial mass of 78 . /
20 for pseudocumene, and T run b is the total data-taking time which we set to 192 and 246 live days for LE and HEdata, respectively. In the previous formula dσ α /dT e is the elastic scattering differentialcross-section for neutrinos of type α ∈ { e, µ, τ } , and d Φ det α /dE ν is the corresponding flux ofsolar neutrinos at the detector – hence it incorporates the neutrino oscillation probabilities.The detector response function R b ( T e | (cid:126)ξ ) depends on the true electron kinetic energy T e andon the three systematic variables ξ vol , ξ scl and ξ res : R b ( T e | (cid:126)ξ ) = (1 + π vol ξ vol ) (cid:90) T max b (1+ π b scl ξ scl ) T min b (1+ π b scl ξ scl ) Gauss (cid:2) T e − T (cid:48) , σ T (1 + π res ξ res ) (cid:3) dT (cid:48) (A.2)where Gauss( x, σ ) ≡ exp (cid:2) − x / σ (cid:3) / √ πσ is the normal distribution function, while T min b and T max b are the boundaries of the reconstructed electron kinetic energy T (cid:48) in the bin b . Note that we assumed an energy resolution σ T /T e = 6% / (cid:112) T e [MeV], rather than the“official” value 5% (cid:14)(cid:112) T e [MeV] quoted by the collaboration, since our choice lead to aperfect match of the Be line shown in Fig. 2 of Ref. [40]. We verified that also the othersolar fluxes plotted in Ref. [50] are carefully reproduced. As for the effects introduced bysystematic uncertainties, we assumed π vol = 6% for the fiducial mass ratio uncertainty, π b scl = 2 .
4% (1%) for the energy scale uncertainty in LE (HE) data, and an arbitrary π res = 10% for the energy resolution uncertainty.The backgrounds N bkg b ( (cid:126)ξ ) which appear in Eq. (A.1) only affect the low-energy data,and are not included in the calculation of the high-energy event rates. The C, C, Cand Kr background shapes were taken from Fig. 2 of Ref. [40], whereas the U, Pband
Bi were extracted from slide 7 of Ref. [50]. We explicitly verified that with thenormalizations as inferred from these figures the sum of all these backgrounds with theexpected SSM fluxes precisely reproduces the “Fit” line shown in Fig. 2 of Ref. [40]. Notethat, because of the overwhelming C at energies below ∼
250 keV, our fit in this energyregion is never good. We do not know if this is due to the loss of numerical precision inour extraction of the C shape or to the fact that there are additional free parameters tobe fitted for this background. In any case, in order to avoid biasing our analysis by thelow quality description on these data points we use only the 160 points of the spectrumabove 365 keV. Hence the C background is irrelevant. Following the procedure outlinedin Ref. [40] the normalization of the
U and
Pb backgrounds are assumed to be known,whereas the normalizations of the Kr,
Bi, C and C backgrounds are introduced asfree parameters and fitted against the data – taking care to ensure their positivity. Hence: N bkg b ( (cid:126)ξ ) = N U238 b + N Pb214 b + N Kr85 b ξ Kr85 + N Bi210 b ξ Bi210 + N C11 b ξ C11 + N C10 b ξ C10 . (A.3)– 18 – igure 7: Spectrum for the best fit point of our spectral fit to the Borexino-LE data in the energyregion between 350–2000 KeV under the assumptions described in the Appendix (left) and ∆ χ asa function of the Be flux for the different test analysis of Borexino data (right).
The χ ( (cid:126)ω ) function is constructed in the usual way in the context of the pull method,by introducing standard penalties for the (cid:126)ξ variables (except for those parametrizing thefree normalizations of the backgrounds) and marginalizing over them: χ ( (cid:126)ω ) = min (cid:126)ξ (cid:88) b (cid:34) N th b ( (cid:126)ω, (cid:126)ξ ) − N ex b σ stat b (cid:35) + ξ + ξ + ξ . (A.4)As a test of our procedure we first perform a fit under the same assumptions as Ref. [40], i.e. , besides the backgrounds we only fit the Be flux normalization to the data. The othersolar fluxes are fixed to their BPS08(GS) prediction and the oscillation parameters are fixedto the best fit point of the global pre-Borexino analysis. The results of this test are shown inFig. 7. Comparing the left panel with Fig. 2 of Ref. [40] we observe a perfect agreement inthe best fit Be flux spectra. In the right panel we plot the ∆ χ for this test fit as a functionof f Be , for both Borexino-LE alone and the combination of Borexino-LE and Borexino-HEdata. As can be seen, for Borexino-LE our procedure leads to a determination of the Benormalization in very good agreement with the value f Be = 1 . ± .
10 obtained by theBorexino collaboration. The inclusion of the Borexino-HE tends to push the extractedvalue of f Be towards a slightly higher value. This is due to the assumed correlation of thesystematic uncertainties (in particular the one associated with the total fiducial volume)between LE and the HE. This small effect is diluted once both data sets are included inthe global fit, and the final results are practically independent of the degree of correlationassumed between the systematic errors. B. Details of the Markov Chain Monte Carlo
In this analysis we have used the Metropolis-Hasting algorithm including an adaptingalgorithm for the kernel function to increase the efficiency. The algorithm is defined as– 19 –ollows:1. Given a parameter set (cid:126)ω , a new value (cid:126)ω (cid:48) is generated according to a transition kernel q ( (cid:126)ω, (cid:126)ω (cid:48) ). We start with a flat kernel and, once the chain has reached a certain size,we use a kernel in terms of the covariance matrix V computed with the points in thechain. If U is the matrix diagonalizing V and d i are the eigenvalues, (cid:126)ω (cid:48) = (cid:126)ω + U (cid:126) ˜ ω with ˜ ω i generated according to a distribution | ˜ ω i | /d i × exp( − ˜ ω i /d i ). The kernel isadapted, i.e. , the covariance matrix is recalculated, every several steps.2. With (cid:126)ω and (cid:126)ω (cid:48) we compute the value: h = min (cid:18) , L (D | (cid:126)ω (cid:48) ) π ( (cid:126)ω (cid:48) |P ) L (D | (cid:126)ω ) π ( (cid:126)ω |P ) (cid:19) . (B.1)3. A random number 0 ≤ r ≤ r ≤ h , (cid:126)ω (cid:48) is accepted in the chain.4. We go back to step (1), starting with (cid:126)ω (cid:48) if it has been accepted or again with (cid:126)ω ifnot.All the points accepted in this algorithm constitute the Markov Monte-Carlo Chain { (cid:126)ω α } with α = 1 , . . . , N tot , where N tot is the total number of points in the chain. The methodensures that, once convergence has been reached, the chain takes values over the parameterspace with frequency proportional to the posterior probability distribution function.Technically, in order to reconstruct the posterior p.d.f. from the chain we discretizethe parameter space by dividing the physically relevant range of each parameter ω i into n i subdivisions ∆ k i i of length (cid:96) k i i (with 1 ≤ k i < n i ). Denoting as Ω k ...k m the cell corre-sponding to subdivisions ∆ k . . . ∆ k m m ( m = 10 or 11 for the analysis with or without theluminosity constraint, respectively), we compute the value of the posterior p.d.f. as p ( (cid:126)ω ∈ Ω k ...k m | D , P ) = 1 V k ...k m M k ...k m N tot (B.2)where M k ...k m is the number of points in the chain with parameter values within the cellΩ k ...k m , and V k ...k m = (cid:96) k × · · · × (cid:96) k m m is the volume of the cell. In order to ensure that theprocedure generates a smooth p.d.f., a sufficiently large N tot is needed.The marginalized one-dimensional p.d.f. for the parameter ω i is reconstructed as p ( ω i ∈ ∆ k i i | D , P ) = 1 (cid:96) k i i n j (cid:88) k j (cid:54) = i =1 M k ...k i ...k m N tot . (B.3)Similarly, the marginalized two-dimensional p.d.f.’s for the parameters ( ω i , ω j ) is p ( ω i ∈ ∆ k i i , ω j ∈ ∆ k j j | D , P ) = 1 (cid:96) k i i (cid:96) k j j n l (cid:88) k l (cid:54) = i,j =1 M k ...k i ...k j ...k m N tot . (B.4)From these, we obtain the two-dimensional credibility regions with a given CL as the regionwith smallest area and with CL integral posterior probability. In practice they are obtainedas the regions surrounded by a two-dimensional isoprobability contour which contains thepoint of highest posterior probability and within which the integral posterior probabilityis CL. – 20 – eferences [1] H. A. Bethe, Energy production in stars , Phys. Rev. (1939) 434–456.[2] J. N. Bahcall, Neutrino astrophysics , . Cambridge, UK: University press, 567p.[3] J. N. Bahcall and R. K. Ulrich,
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