{'fraction_non_alphanumeric': 0.05087872608991164, 'fraction_numerical': 0.0268958151276823, 'mean_word_length': 4.418200946870068, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
KamLAND results and the radiogenic terrestrial heat

We find that recent results from the KamLAND collaboration on geologically produced antineutrinos, N(U+Th) = 28+16-15 events, correspond to a radiogenic heat production from Uranium and Thorium decay chains H(U+Th) = 38+35-33 TW. The 99% confidence limit on the geo-neutrino signal translates into the upper bound H(U+Th)<162 TW, which is much weaker than that claimed by KamLAND, H(U+Th)<60 TW, based on a too narrow class of geological models. We also performed an analysis of KamLAND data including recent high precision measurements of the C13(\alpha,n)O16 cross section. The result, N(U+Th) = 31+14-13, corroborates the evidence (approx 2.5\sigma) for geo-neutrinos in KamLAND data.


I. INTRODUCTION
Geologically produced antineutrinos (geo-neutrinos) were introduced by Eder [1] in the sixties and Marx [2] soon realized their relevance. In the eighties Krauss et al. discussed their potential as probes of the Earth's interior in an extensive publication [3]. In the nineties the first paper on a geophysical journal was published by Kobayashi et al. [4]. In 1998, Raghavan et al. [5] and Rotschild et al. [6] pointed out that KamLAND and Borexino should be capable of geo-neutrino detection.
The potential of geo-neutrinos for providing information on the energetics and composition of the Earth has been discussed in Refs. [7,8] and more recently in Ref. [9]. The KamLAND collaboration has just published [10] their experimental results, claiming some 28 geo-neutrino events from Uranium and Thorium decay chains in a two-year exposure.
This important step shows that the technique for exploiting geo-neutrinos in the investigation of the Earth's interior is now available. In order to understand where to go with geo-neutrinos, we have to know where we stand, in the light of the available data. In this spirit, the aim of this letter is to discuss the implication of the KamLAND result on the contribution of Uranium and Thorium decay chains to the terrestrial heat.

II. THE GEO-NEUTRINO SIGNAL AND THE RADIOGENIC TERRESTRIAL
HEAT For a given value of, e.g., the Uranium mass in the Earth, m(U), the contributed heat production rate from the Uranium decay chain is uniquely determined, H(U) = 0.95 TW ×m(U)/(10 16 kg), whereas the flux and signal of geo-neutrinos depend on the detector location and on the Uranium distribution inside the Earth. The connection between the signal of geo-neutrinos from the Uranium decay chain, the mass of Uranium in the Earth and the heat production rate from that element was found in Ref. [9], by using global mass balance together with a detailed geochemical and geophysical study of the region surrounding Kamioka.
We remark that the mass of Uranium in the crust, m c (U), is rather well constrained by geological data, in the interval (3 ÷ 4) × 10 16 kg. The main uncertainty is the amount of Uranium (and Thorium) in the mantle. Geo-neutrinos should provide us with this informa-tion.
For a total uranium mass m(U) in the Earth, the maximal and minimal signals can be derived by using a proximity argument: the maximal (minimal) signal is obtained by placing the sources as close (as far) as possible to (from) the detector, consistently with geochemical and geophysical constraints. The maximal signal is thus obtained for an Uranium rich We have extended the same calculations to Thorium, assuming a global chondritic Uranium-to-Thorium mass ratio, m(Th)/m(U) = 3.9 ± 0.1, so that we can now connect the combined signal of geo-neutrinos from Uranium and Thorium progenies, S(U+Th), with the radiogenic heat production rate from these elements, H(U+Th), see Fig. 1.
The geo-neutrino signal is expressed in Terrestrial Neutrino Units, one TNU corresponding to 10 −32ν e captures per target proton per year. The allowed band in Fig. 1 is estimated by considering rather extreme 1 models for the distributions of radioactive elements, chosen so as to maximize or minimize the signal for a given heat production rate [9].
We also remark that, in comparison with the experimental error,the width of the band is so narrow that we can limit the discussion to the median line of the allowed band in Fig. 1, which represents our best estimate for the relationship between signal and power.
For the Bulk Silicate Earth (BSE) model, H(U+Th) = 16 TW, our prediction for Kamioka is centered at 37 TNU.
By assuming that Uranium and Potassium in the Earth are in the ratio 1/10,000 and that there is no Potassium in the core, the total radiogenic power is H(U+Th+K) = 1.18 H(U+Th). With these assumptions, a maximal and fully radiogenic heat production rate, H(U+Th+K) = 44 TW, corresponds to H(U+Th) = 37 TW, which gives a signal S(U+Th) ≈ 56 TNU.
The KamLAND collaboration has reported [10] data from an exposure of N p = (0.346 ± 0.017) × 10 32 free protons over a time T = 749 days with a detection efficiency ǫ = 69%: the effective exposure is thus E eff =N p × T × ǫ = (0.487 ± 0.025) × 10 32 protons · yr. In the energy region where geo-neutrinos are expected, there are C = 152 counts, implying a statistical fluctuation of ±12.5. Of these counts, a number R = 80.4 ± 7.2 are attributed to reactor events, based on an independent analysis of higher energy data. Fake geo-neutrino events, originating from 13 C(α, n) 16 This "rate only" study has been improved in Ref. [10] by exploiting the shape of the spec- The best fit value is close to the maximal and fully radiogenic model, however the BSE is within 1σ.
By using the median line in Fig. 1 This 162 TW upper bound is much higher than the 60 TW upper bound claimed in Ref. [10], which was obtained by using a family of geological models which are too narrow and are also incompatible with well-known geochemical and geophysical data. is assumed to scale with the total mass of U+Th, so that heat production and signal are also proportional: In this way, the 99% upper limit on the signal, 145 TNU, is translated into 60 TW [10], see point B in Fig. 1.
This scaling assumption, however, produces a too limited series of models. The points in the shaded area of Fig. 1 correspond to all models which are compatible with available geochemical and geophysical data 2 : most of these models cannot be obtained by Eq. 3 and predict, for a given signal, a larger power than Eq. 3, which therefore cannot be used to derive an upper bound on the radiogenic power production. On the other hand, what is important for deciding the potential of future experiments is the relationship between geo-neutrino signal and heat production in the physically interesting region, H(U+Th)≤ 37 TW. The basic parameter is the slope, dS/dH, which expresses how the experimental error translates into an uncertainty on the deduced heat production. For our models we find from Fig. 1 dS/dH ≃ 1 TNU/TW. Discrimination between BSE and fully radiogenic models, which requires a precision ∆H ∼ 7 TW, requires thus an experiment with an accuracy ∆S ∼ 7 TNU. The "rescaled models" of ref. [10], yielding dS/dH≃ 2.4 TNU/TW, might misleadingly suggest that the same goal can be achieved for ∆S=17 TNU.
III. THE GEO-NEUTRINO SIGNAL AND THE 13 C(α, n) 16 O CROSS SECTION.
As already remarked, a major uncertainty for extracting the geo-neutrino signal originates from the 13 C(α, n) 16 O cross section 3 . The values used in Ref. [10] are taken from the JENDL [14] compilation, which provides an R-matrix fit of relatively old data. A 20% overall uncertainty has been adopted in [10], corresponding to the accuracy claimed in the original experimental papers (see, e.g., Ref. [17]).
Recently a series of high precision measurements for this cross section has been performed [15]. In the relevant energy range (1 ÷ 5.3) MeV, the absolute normalization has been determined within a 4% accuracy. The measured values are generally in very good agreement with those recommended in JENDL, see Fig. 2; however, we find that the neutron yield per alpha particle is 5% smaller. It follows that the number of fake neutrinos is lower, F = 40 ± 5.8, and geo-neutrino events obviously increase 4 .
The "rate only" analysis gives now 27 +16 −15 geo-neutrino events, corresponding to S(U+Th) = 55 +33 −31 TNU. From the median line of Fig. 1, the radiogenic power is now: We also performed an analysis 5 of the binned spectrum reported in Fig. 3 of Ref. [10].
This analysis gives N(U+Th) = 31 +14 −13 counts, corresponding to S(U+Th) = 63 +28 −25 TNU and 3 In fact, the claim of 9 geo-neutrino events in Ref. [16] should be dismissed: more than half of these events are to be considered as fake signal, produced from 13 C(α, n) 16 O reaction. 4 Indeed Ref. [10] mentions that an alternative analysis including the time structure of the scintillation light from different particles produced a slightly larger geo-neutrino signal, which is consistent with the result presented here. 5 A complete analysis requires several details (the un-binned spectrum, the energy dependence of the detection efficiency, . . . ) which are not available to us. Just for a comparison, the binned spectrum analysis using the JENDL cross sections with 20% uncertainty gives us N(U+Th)=28.5 +15 −14 , in agreement with [10].

IV. CONCLUDING REMARKS
In summary, the new data on 13 C(α, n) 16 O corroborate the evidence for geo-neutrinos in KamLAND data, which becomes near to 2.5σ.
On the other hand, the determination of radiogenic heat power from geo-neutrino measurements is still affected by a 70% uncertainty. The best fit of H(U+Th) is close to the prediction of a maximal and fully radiogenic model, however the BSE prediction is within 1σ from it.
With more statistics KamLAND should be capable of providing a three sigma evidence of geo-neutrinos, but discrimination between BSE and fully radiogenic models definitely requires new detectors, with class and size similar to that of KamLAND, far away from nuclear power plants. The shaded area denotes the region allowed by geochemical and geophysical constraints. The dashed median line represents our best estimate for the relationship between signal and power.
The dotted line denotes the "rescaled models" of Eq. 3, used in [10]. Note that most of these models are outside the allowed area.
One TNU corresponds to 10 −32ν e captures per target proton per year.