Short-Baseline Electron Neutrino Disappearance, Tritium Beta Decay

arXiv:1005.4599v4 [hep-ph] 11 Oct 2010 trino eeae yalre qae-asdffrne h LSND oscillations which the neutrino difference: anomalies ν squared-mass short-baseline larger three a of by least generated signals at be are could there oscillations, aeieeeto etiodsperne[31]i the in [13–15] disappearance short- neutrino by MiniBooNE electron explained the be baseline can consider which we anomalies, paper Gallium and this low-energy In MiniBooNE [12]. the and anomaly 11], [10, anomaly iments hr ∆ where iain ftremsieneutrinos massive three of binations ntefaeoko he-etiomxn,i hc the which in neutrinos mixing, fitted flavor three-neutrino well three of are framework experiments the these in ex- of data accelerator The and periments. reactor long-baseline and mospheric oa SL n topei AM qae-asdif- squared-mass (ATM) atmospheric ferences and (SOL) solar ¯ ∗ † µ init.nni;as tDprmn fTertclPhys Theoretical of [email protected] Italy Department Torino, of at University also [email protected]; eie hs eletbihdosrain fneutrino of observations well-established these Besides etiooclain aebe bevdi oa,at- solar, in observed been have oscillations Neutrino → | ∆ ∆ ν m ν ¯ m j e 21 2 hr-aeieEeto etioDsperne Tritium Disappearance, Neutrino Electron Short-Baseline seRf.[1–8]). Refs. (see m 31 2 inl[] h alu aiatv oreexper- source radioactive Gallium the [9], signal | ≃ | ∆ = jk 2 r ihycmail,wt ls etfi auso h effec the of values best-fit close with separat compatible, The highly schemes. short-baseline are mixing of four-neutrino terms 3+1 in of framework anomaly experiments source tive xeiet.Tefi ftedt fteeeprmnslmt t limits experiments these of data the of fit The experiments. lotedt fteBgyadCozratratnurn osc antineutrino in reactor mass Chooz antineutrino and electron Bugey effective the the of on data the also rn eco n rtu aaa ttsia fluctuation, statistical a as data Tritium ∆ and reactor trino 2 sin β rdce otiuin of contributions predicted n nietioscos efida2 w a data find Tritium We tension and sectors. the reactor antineutrino antineutrino reconciling and the of and possibility data the Gallium consider also We ASnmes 46.q 46.m 14.60.St 14.60.Lm, 14.60.Pq, numbers: PACS = σ dcyae epciey ewe bu .6ad04 n be and 0.49 and 0.06 about between respectively, are, -decay m ecnie h nepeaino h iioN low-energy MiniBooNE the of interpretation the consider We ∆ m osdrn h eso ewe h etioMnBoEan MiniBooNE neutrino the between tension the Considering . 2 m .INTRODUCTION I. 2 2 m SOL 2 ϑ j 2 ≃ 32 2 h obndfi ie ∆ gives fit combined The . − Vad0 and eV 2 | ≃ m ∆ = ν k 2 8 e , iatmnod iia“.Glli,Uiestad Padova Università di Galilei”, “G. Fisica di Dipartimento × and ν m NN ein iTrn,VaP ira1 –02 oio Ita Torino, I–10125 1, Giuria P. Via Torino, di Sezione INFN, µ 10 ATM 2 , ein iPdv,VaF azl ,I311Pdv,Italy Padova, I–35131 8, Marzolo F. Via Padova, di Sezione . − m ν 01 τ 5 j eV ≤ r ntr iercom- linear unitary are ≃ stems fteneu- the of mass the is sin 2 2 etioesDul-eaDecay Double-Beta Neutrinoless , × m ν 2 1 2 4 10 , ϑ oteeetv etiomse in masses neutrino effective the to ν ≤ − 2 3 , 0 eV ν Dtd etme 6 2018) 26, September (Dated: . m 3a 2 at 13 3 2 . 2 ihthe with 6 , σ & ac Laveder Marco niaino iigageasymmetry. angle mixing a of indication al Giunti Carlo σ 0 . suigaheacyo masses of hierarchy a Assuming . eV 1 (2) (1) ics, β dcyotie nteMizadTotkTritium Troitsk and Mainz the in obtained -decay 2 n 0 and o30 e.I hspprw xedteaayi of analysis ∆ the of extend values we lower paper to this data 200 In from MiniBooNE MeV. range energy 3000 MiniBooNE to the in constant cally cee seRf.[,4 ,7) hc r h simplest four-neutrino the are in which ac- 7]), can obtained which 6, schemes mixing is 4, three-neutrino of [1, extension (3) Refs. (see Eq. schemes in probability survival see [8]). equal; the are Ref. antineutrinos that and implies neutrinos of invariance probabilities (CPT energy trino nScin IadII nteohrhn,teLSND the MiniBooNE hand, the other of the results On the ν by disfavored III. is and anomaly II explained Sections as in mixing, four-neutrino of framework effective nlsso h iioN aai ipie ytefact probability the survival by effective simplified is the data that MiniBooNE the of analysis noseienurnsgnrtdb qae-asdiffer- squared-mass ∆ a ence by generated neutrinos oscillations sterile very-short-baseline into to due of neutrinos disappearance electron the through 16] [12, anomaly low-energy [17–23]. explanation where aeie(B)eeto etioadatnurn sur- antineutrino short- and effective probability neutrino the vival electron of (SBL) dependence baseline energy resulting the µ h w-etiolk ffciesotbsln survival short-baseline effective two-neutrino-like The nRf.[3 5 epooe oepanteMiniBooNE the explain to proposed we 15] [13, Refs. In → ∗ . 11 † P L ν m epromacmie twihgives which fit combined a perform we ( ewe h etioMnBoEand MiniBooNE neutrino the between ≤ lcrnnurn iapaac nthe in disappearance neutrino electron e ν − SBL t fMnBoEadGlimdata Gallium and MiniBooNE of fits e ieoclainprmtr ∆ parameters oscillation tive e we bu .0 n .7e t2 at eV 0.07 and 0.003 about tween 2 stenurn ahlnt and length path neutrino the is ) → xeiet[2 6 n a eur another require may and 16] [12, experiment t ieetmxnsi h neutrino the in mixings different ith sin ( agrta bu 0eV 20 about than larger laineprmnsadtelimits the and experiments illation ν − nml n h alu radioac- Gallium the and anomaly evleo sin of value he e 2 ) β alu aaadteantineu- the and data Gallium d ( 2 dcyadnurnls double- neutrinoless and -decay ,E L, ϑ ≤ 1 = ) n INFN, and , 0 . 8a 2 at 48 m − 1 m , 2 eaDcyand Decay Beta sin 2 ly ϑ 2 σ 2 m , econsider We . Phys.Rev.D82:053005,2010 [hep-ph] arXiv:1005.4599 eo .0at 0.10 below 2 ϑ 3 sin ≪ 2 m nta ae the case, that In . 2 m 2 4 P the , ∆ m and ν e 4 m σ 2 → E E . considering , 2 ν L e steneu- the is spracti- is , (3) 2 commodate the two small solar and atmospheric squared- Section V we present the results of the combined anal- mass differences in Eqs. (1) and (2), and one larger ysis of MiniBooNE, Gallium, reactor and Tritium data squared-mass difference for short-baseline neutrino oscil- and in Section VI we present the corresponding predic- lations, tions for the effective masses measured in β-decay and neutrinoless double-β-decay experiments. In Section VII ∆m2 = ∆m2 & 0.1 eV2 . (4) we calculate the mixing angle asymmetry between the | 41| neutrino and antineutrino sectors which could explain The existence of a fourth massive neutrino corresponds, the tension between the neutrino and antineutrino data in the flavor basis, to the existence of a sterile neutrino under our short-baseline νe-disappearance hypothesis. In νs. Section VIII we draw the conclusions. In this paper we consider 3+1 four-neutrino schemes, since 2+2 four-neutrino schemes are disfavored by the combined constraints on active-sterile transitions in solar II. MINIBOONE and atmospheric neutrino experiments [4]. For simplicity, we consider only 3+1 four-neutrino schemes with The MiniBooNE experiment was made with the pur- pose of checking the indication ofν ¯ ν¯ oscillations m ,m ,m m , (5) µ → e 1 2 3 ≪ 4 generated by a ∆m2 & 0.1 eV2 found in the LSND ex- which give the ∆m2 in Eq. (4) and appear to be periment [9]. The MiniBooNE collaboration did not find 41 any indication of such oscillations in the ν ν chan- more natural than the other possible 3+1 four-neutrino µ → e schemes in which either three neutrinos or all four neu- nel [12, 16]. On the other hand, the MiniBooNE collab- trinos are almost degenerate at a mass scale larger than oration found an anomalous excess of low-energy νe-like 2 events in the data on the search for ν ν oscillations √∆m (see Refs. [1, 4, 6, 7]). µ → e In 3+1 four-neutrino schemes the effective mixing an- [12, 16], as shown in Fig. 1a. gle in the effective short-baseline electron neutrino sur- As in Refs. [13, 15], we consider an explanation of the vival probability in Eq. (3) is given by (see Refs. [1, 4, 6, low-energy MiniBooNE anomaly based on the possible 7]) short-baseline disappearance of electron neutrinos, taking into account a possible overall normalization factor fν 2 2 2 of the calculated ν -induced and misidentified ν -induced sin 2ϑ =4 Ue4 1 Ue4 . (6) e µ | | − | | events which contribute to the observed number of νe- 2 In this paper we assume that the value of Uµ4 is so like events. The normalization factor fν could be due small that the effective short-baseline muon neutrino| | sur- mainly to the uncertainty of the calculated neutrino flux vival probability is practically equal to unity and short- (see Ref.[28]). Since the misidentified νµ-induced and νe- (−) (−) induced events dominate, respectively, at low and high baseline ν ⇆ ν are negligible1. This assumption is jus- µ e energies (see Fig. 1a), the low-energy excess can be fit- tified by the lack of any indication of ν ν transitions µ e ted with f > 1 and the high-energy data can be fitted in the MiniBooNE experiment [12, 16] and→ the limits on ν compensating f > 1 with the disappearance of ν ’s. short-baseline muon neutrino disappearance found in the ν e In Refs. [13, 15] we considered only very-short-baseline CDHSW [24], CCFR [25] and MiniBooNE [26] experi- ν disappearance due to a ∆m2 & 20eV2, which gener- ments. We do not consider the MiniBooNE antineutrino e ates a survival probability P which is constant in data [27], which have at present statistical uncertainties νe→νe the MiniBooNE energy range, from 200 to 3000 MeV. which are too large to constraint new physics [15]. In this paper we extend the analysis to lower values of The plan of the paper is as follows. In Section II we ∆m2, considering the resulting energy dependence of the discuss the analysis of MiniBooNE data. In Section III survival probability. In this case, the theoretical number we present an update of the analysis of Gallium data of ν -like events in the jth energy bin is given by published in Ref. [14] and the combined analysis of Mini- e

BooNE and Gallium data. In Section IV we discuss the ν ,the νµ,the N the = N e + N , (7) implications of the measurements of the eﬀective electron νe,j νe,j νe,j neutrino mass in Tritium β-decay experiments and their where combination with reactor neutrino oscillation data. In νe,the (j) νe,cal Nνe,j = fν Pνe→νe Nνe,j (8)

is the number of νe-induced events and 1 In 3+1 four-neutrino schemes the effective short-baseline muon νµ,the νµ,cal neutrino survival probability has the form in Eq. (3) with Nνe,j = fν Nνe,j (9) 2 2 2 2 sin 2ϑ replaced by sin 2ϑµµ = 4|Uµ | 1 −|Uµ | . The ef- 4 4 (−) (−) is the number of misidentified ν -induced events. Here fective short-baseline νµ ⇆ νe transition probability is given µ ν ,cal νµ,cal SBL 2 2 ∆m2L 2 e by P (L,E) = sin 2ϑeµ sin , with sin 2ϑeµ = N and N are, respectively, the number of νe- (−) (−) 4E νe,j νe,j ⇆ νµ νe induced and misidentified νµ-induced events calculated 2 2 4|Ue4| |Uµ4| (see Refs. [1, 4, 6, 7]). by the MiniBooNE collaboration for the jth energy bin 3

MiniBooNE − νe MiniBooNE − νe

νe,cal νe,the (j) νe,cal N N = f ν P N νe,j νe,j νe→νe νe,j νµ,cal νµ,the = νµ,cal Nνe,j Nνe,j f ν Nνe,j ν ν cal νe,cal µ,cal the νe,the µ,the N = Nν + Nν N = Nν + Nν νe,j e,j e,j νe,j e,j e,j

(a) (b) events / MeV events / MeV events 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 200 400 600 800 1000 1200 1400 3000 200 400 600 800 1000 1200 1400 3000 E [MeV] E [MeV]

FIG. 1. Expected number of νe events compared with MiniBooNE data, represented by the black points. The energy bins are numbered with the index j. The uncertainty is represented by the vertical error bars, which represent the sum of statistical and cal uncorrelated systematic uncertainties. (a) Expected number of νe-like events Nνe,j calculated by the MiniBooNE collaboration. N cal ν N νe,cal ν N νµ,cal νe,j is given by the sum of the e-induced events ( νe,j ) and the misidentified µ-induced events ( νe,j ). (b) Best-fit the the value of the number of νe-like events Nνe,j obtained with the hypothesis of νe disappearance. Nνe,j is given by the sum of νe,the (j) νe,cal νµ,the cal 2 2 Nνe,j = fν Pνe→νe Nνe,j and Nνe,j = fν Nνµ . The best-fit values of fν , sin 2ϑ and ∆m are those in the first column of Tab. I (MBν).

(j) [29, 30]. Pνe→νe is the survival probability of electron events in 8 reconstructed neutrino energy bins. The the- neutrinos in Eq. (3) averaged in the jth energy bin. The oretical number of νµ events in the jth energy bin is given average in each bin is calculated using the ntuple-ﬁle of by 17,037 predicted muon-to-electron neutrino full transmu- the cal tation events given in Ref. [29], which contains informa- Nνµ,j = fν Nνµ,j , (11) tion on reconstructed neutrino energy, true neutrino en- cal ergy, neutrino baseline and event weight for each event. where Nνµ,j is the number of νµ events calculated by The MiniBooNE measurement of a ratio 1.21 0.24 of the MiniBooNE collaboration [29]. In order to take into ± detected and predicted charged-current quasi-elastic νµ account the correct statistical uncertainty corresponding events [31] allows a value of fν as large as about 15%. In to the rescaling of the number of νµ events due to fν in

Ref. [15] we used this estimate of the uncertainty of fν in Eq. (11), we used the covariance matrix Vνµ given by order to constrain its value in the least-squares analysis. cal cal Here we use directly the ν data given in Ref. [29] for the (Vν )jk = (V )jk + (fν 1) N δjk , (12) µ µ νµ − νµ,j construction of the MiniBooNE least-squares function cal where Vν is the 8 8 covariance matrix of νµ events 2 µ × 11 N the N exp presented by the MiniBooNE collaboration in Ref. [29]. χ2 = νe,j − νe,j MBν σ We did not use the complete 19 19 covariance matrix j=1 νe,j ! × X of νe and νµ events given in Ref. [29] because the corre- 8 lations involving νe events have been obtained without the exp −1 the exp + N N (V )jk N N . taking into account the energy-dependent disappearance νµ,j − νµ,j νµ νµ,k − νµ,k j,k=1 of electron neutrinos that we want to test. Assuming X (10) the correlations given in that 19 19 covariance matrix × (j) exp would suppress the energy dependence of Pνe→νe . There- Here N are the numbers of measured νe-like events νe,j fore, for the uncertainties σνe,j in Eq. (10) we used only in 11 reconstructed neutrino energy bins and N exp are cal νµ,j the diagonal elements of the νe covariance matrix Vνe the numbers of measured νµ charged-current quasi-elastic given in Ref. [29], corrected by the change of statistical 4 uncertainty corresponding to the variation of expected (j) events due to fν and Pνe→νe in Eqs. (7)–(9):

2 cal the cal

σνe,j = (Vνe )jj + Nνe,j Nνe,j , (13) 2

− ∆χ MiniBooNE − ν σ cal νe,cal νµ,cal 68.27% C.L. (1 ) σ with Nνe,j = Nνe,j + Nνe,j . 95.45% C.L. (2 ) 2 99.73% C.L. (3σ) The result of the minimization of χMBν is shown in

Fig. 1b, in which the solid histogram corresponds to the 0 2 4 6 8 10 2 2 best-fit values of fν , sin 2ϑ and ∆m in the first column of Tab. I. From Fig. 1b, one can see that the fit is acceptable for all the νe energy bins, including the first three bins which are out-of-fit in Fig. 1a. In Tab. I we 10 ]

2 2 give separately the contribution to χMBν of the ﬁrst three

low-energy νe bins and the sum of the contributions of [eV 2 the other νe energy bins and all the νµ energy bins. In m + ∆ 1 this way one can see that with fν = 1 and Pνe→νe = 1 (Null Hypothesis), although the global value χ2 = 19.7 is compatible with the number of degrees of freedom, NDF = 19, almost all the χ2 is due to the anomalous − 10 1 contribution 14.3 of the first three low-energy νe bins, − − − 10 3 10 2 10 1 0 2 4 6 8 10 whereas the other 16 νe and νµ energy bins are overfit- 2 2 sin 2ϑ ∆χ ted, with the excessively small χ2 contribution of 5.4. This overfitting, which is probably due to an overesti- mate of the uncertainties, remains in the fit of the data FIG. 2. Allowed regions in the sin2 2ϑ–∆m2 plane and 2 2 2 with our hypothesis of fν > 1 and νe disappearance. On marginal ∆χ ’s for sin 2ϑ and ∆m obtained from the fit the other hand, our hypothesis clearly explains the low- of MiniBooNE neutrino data. The best-fit point is indicated energy anomaly reducing the χ2 contribution of the first by a cross. three low-energy νe bins to the acceptable best-fit value of 2.0. Figure 2 shows the allowed regions in the sin2 2ϑ–∆m2 are 2 2 2 2 plane and the marginal ∆χ = χ χmin’s for sin 2ϑ 2 − GALLEX +0.11 and ∆m , from which one can infer the corresponding RCr1 =0.95−0.12 , (14) uncorrelated allowed intervals. One can see that the in- RGALLEX =0.81+0.10 , (15) dication in favor of neutrino oscillations is not strong, Cr2 −0.11 SAGE +0.12 being at the level of about 79% C.L. (1.2σ). It is inter- RCr =0.95−0.12 , (16) esting to notice that the best-fit value of ∆m2 is about SAGE +0.09 RAr =0.79−0.10 , (17) 2 eV2, which is approximately the same best-fit value obtained in Ref. [14] from the fit of the neutrino data of and the average ratio is Gallium radioactive source experiments and the antineu- Ga +0.05 trino data of the Bugey and Chooz reactor experiments R =0.86−0.05 . (18) under the hypothesis of νe andν ¯e disappearance. The results of the combined analysis of MiniBooNE neutrino Thus, the number of measured events is about 2.7σ data and the data of these other experiments is discussed smaller than the prediction. in the following Sections. The theoretical prediction of the rate is based on the calculation of the detection cross section presented in Ref. [38]. It is possible that a part of the observed III. GALLIUM RADIOACTIVE SOURCE deficit is due to an overestimation of this cross section EXPERIMENTS [10, 37, 39], because only the cross section of the transition from the ground state of 71Ga to the ground state of The GALLEX [11, 33, 34] and SAGE [10, 35–37] col- 71Ge is known with precision from the measured rate of laborations tested the respective Gallium solar neutrino electron capture decay of 71Ge to 71Ga. Electron neutri- detectors in so-called ”Gallium radioactive source exper- nos produced by 51Cr and 37Ar radioactive sources can iments” which consist in the detection of electron neutri- be absorbed also through transitions from the ground nos produced by intense artificial 51Cr and 37Ar radioac- state of 71Ga to two excited states of 71Ge, with cross tive sources placed inside the detectors. Taking into ac- sections which are inferred using a nuclear model from count the uncertainty of the cross section of the detection p + 71Ga 71Ge + n measurements [40]. This calcula- 71 71 − → process νe + Ga Ge + e estimated in Ref. [38], tion has large uncertainties [41, 42]. However, since the the ratio R of measured→ and predicted 71Ge event rates contribution of the transitions to the two excited states 5

MBν Ga MBν+Ga Re+3H (MBν+Ga)+(Re+3H) Null Hyp. χ2 14.3 + 5.4 9.4 51.5 NDF 3+16 4 58 GoF 0.41 0.051 0.71 2 Our Hyp. χmin 2.0 + 7.6 1.8 2.2 + 9.2 49.1 4.1 + 63.4 NDF 16 2 20 56 78 GoF 0.89 0.40 0.93 0.73 0.80 2 sin 2ϑbf 0.32 0.27 0.28 0.042 0.062 2 ∆mbf 1.84 2.09 1.92 1.85 1.85 bf fν 1.26 1.25 1.17 2 PG ∆χmin 0.098 0.01 6.97 NDF 22 2 GoF 0.95 0.99 0.03

TABLE I. Values of χ2, number of degrees of freedom (NDF) and goodness-of-fit (GoF) for the fit of different combinations of MiniBooNE (MBν), Gallium (Ga), and reactor (Re) data. The first three lines correspond to the case of fν = 1 and no oscillations (Null Hyp.). The following six lines correspond to the case fν > 1 and νe disappearance (Our Hyp.). The last three lines give the parameter goodness-of-fit (PG) [32]. In the MBν column, the value of χ2 in the Null. Hyp., the number 2 of degrees of freedom in the Null. Hyp. (which is equal to the number of energy bins), and the value of χmin in Our Hyp. are shown as the sum of the contributions of the first three low-energy νe bins and the other νe and νµ energy bins. In the 3 2 MBν+Ga and (MBν+Ga)+(Re+ H) columns the value of χmin in Our Hyp. is shown as the sum of the contribution of the first three MiniBooNE low-energy νe bins and the other contributions.

is only 5% [38], even the complete absence of such tran- Ga +0.05 sitions would reduce R to about 0.90−0.05, leaving an anomaly of about 1.8σ.

Here we consider the electron neutrino disappearance 2 explanation of the Gallium radioactive source experi- ∆χ Gallium 68.27% C.L. (1σ) ments anomaly [13–15, 43–45] (another interesting ex- 95.45% C.L. (2σ) planation through quantum decoherence in neutrino os- 99.73% C.L. (3σ) cillations has been proposed in Ref. [21]). 0 2 4 6 8 10 In Ref. [14] we have analyzed the data of the Gallium radioactive source experiments in terms of the eﬀective survival probability in Eq. (3). Here we update that anal- GALLEX ysis taking into account the revised value of RCr1 10 ] in Eq. (14) published recently in Ref. [11] and taking 2 into account the asymmetric uncertainties of RGALLEX,

Cr1 [eV GALLEX SAGE 2 RCr2 and RAr (which have been symmetrized for m + ∆ simplicity in the analysis presented in Ref. [14]). Fol- 1 lowing the method described in Ref. [14], we obtained the best-fit values of sin2 2ϑ and ∆m2 in the second column of Tab. I and the allowed regions in the sin2 2ϑ–∆m2 − plane shown in Fig. 3. The indication in favor of neutrino 10 1 − − − 10 3 10 2 10 1 0 2 4 6 8 10 oscillations is at the level of about 98% C.L. (2.3σ). 2 sin22ϑ ∆χ From Tab. 2 and the comparison of Figs. 2 and 3 one can see that the fits of MiniBooNE and Gallium data lead to remarkably similar results: the best-fit values of FIG. 3. Allowed regions in the sin2 2ϑ–∆m2 plane and the oscillation parameters are very close and the allowed marginal ∆χ2’s for sin2 2ϑ and ∆m2 obtained from the com- regions in the sin2 2ϑ–∆m2 plane are highly compatible. bined fit of the results of the two GALLEX 51Cr radioactive This is certainly an impressive success of our hypothesis source experiments and the SAGE 51Cr and 37Ar radioactive 2 of electron neutrino disappearance. source experiments. The best-fit point corresponding to χmin The results of the combined fit of MiniBooNE and Gal- is indicated by a cross. lium data are shown in the third column of Tab. I and in Fig. 4. The separate data sets are well fitted by the electron neutrino disappearance hypothesis: the χ2 con- consistency of the combined fit is also supported by the tribution of the first three MiniBooNE low-energy νe bins excellent value of the parameter goodness-of-fit. Com- is 2.2, that of the other 16 MiniBooNE νe and νµ energy bining the two data sets improves the indication in favor bins is 7.4, and that of the 4 Gallium data is 1.9. The of neutrino oscillations to the level of about 99.5% C.L. 6

99.73%

2 Mainz

∆χ MBν + Ga Troitsk 68.27% C.L. (1σ) Mainz + Troitsk 95.45% C.L. (2σ) 99.73% C.L. (3σ) 99% 0 2 4 6 8 10 2 ∆χ

95.45% 10 ] 2

90% [eV 2

m + ∆ 1

68.27%

− 0 2 4 6 8 10 10 1 − − − 10 3 10 2 10 1 0 2 4 6 8 10 0 1 2 3 4 2 sin22ϑ ∆χ mβ [eV]

2 2 2 FIG. 4. Allowed regions in the sin 2ϑ–∆m plane and FIG. 5. ∆χ as a function of mβ. The horizontal lines corre- marginal ∆χ2’s for sin2 2ϑ and ∆m2 obtained from the com- spond to the indicated value of confidence level. The dashed bined fit of the results of MiniBooNE neutrino data and the and dotted lines have been obtained, respectively, from the data of the gallium radioactive source experiments. The best- results in Eqs. (21) and (22) of the Mainz and Troitsk Tri- 2 fit point corresponding to χmin is indicated by a cross. tium β-decay experiments. The solid line is the result of the combined fit.

(2.8σ). have been performed in the Mainz [46] and Troitsk [47]: m2 = 0.6 2.2 2.1 eV2 (Mainz) , (21) IV. REACTOR AND TRITIUM EXPERIMENTS β − ± ± m2 = 2.3 2.5 2.0 eV2 (Troitsk) . (22) β − ± ± The indication of electron neutrino disappearance that These measurements can be interpreted and combined in we have found from the analysis of MiniBooNE and Gal- order to derive upper bounds for the effective mass mβ lium data must be confronted with the results of reactor 2 2 through a χ analysis in the physical region mβ 0. In electron antineutrino experiments. Assuming CPT in- 2 ≥ variance, the survival probabilities of neutrinos and an- Fig. 5 we plotted the corresponding ∆χ ’s as a function tineutrinos are equal (see Ref. [8]). Thus, we can combine of mβ. One can see that directly the results presented in the previous Section with mβ 2.3 eV (Mainz, 95% C.L.) , (23) the results of the analysis of the data of the Bugey and ≤ m 2.0 eV (Troitsk, 95% C.L.) , (24) Chooz reactor experiments obtained in Ref. [14]. We are β ≤ encouraged in this task by the coincidence of the best-fit in approximate agreement with the corresponding values 2 2 value of ∆m at about 2eV . in Refs. [46, 47]. The combined upper bound is In addition to reactor neutrino experiments, also Tri- tium β-decay experiments give information on the masses mβ 1.8 eV (Mainz+Troitsk, 95% C.L.) . (25) and mixing of neutrinos through the measurement of the ≤ electron energy spectrum in the process In 3+1 four-neutrino schemes, taking into account that the mass splittings among m1, m2, m3 in Eqs. (1) and 3 3 − H He + e +¯νe . (19) (2) are negligible for a measurement of mβ at the scale → of 0.1 - 1 eV, the effective mass is given by The most accurate measurements of the effective electron 2 2 2 2 2 2 2 2 neutrino mass (see Refs. [2, 8]) mβ 1 Ue4 m1 + Ue4 m4 = m1 + Ue4 ∆m41 . ≃ − | | | | | | (26) 1/2 In 3+1 schemes of the type in Eq. (5), we have 2 2 mβ = Uek mk (20) 2 | | ! mβ Ue4 √∆m . (27) Xk ≥ | | 7

The eﬀective mixing angle in short-baseline electron neutrino disappearance experiments is related to Ue4 by Eq. (6). Inverting this relation and taking into| account|

that the value of Ue4 must be small in order to ﬁt the 2 | | 3 data of solar neutrino experiments with neutrino oscilla- ∆χ Re + H tions, we have 68.27% C.L. (1σ) 95.45% C.L. (2σ) 99.73% C.L. (3σ) 2 1 2 Ue4 = 1 1 sin 2ϑ . (28) | | 2 − − 0 2 4 6 8 10 p In Fig. 6 we show the limits in the sin2 2ϑ–∆m2 plane obtained from Eqs. (27) and (28) and the results in Eqs. (21) and (22) of the Mainz and Troitsk Tritium β-decay ex- 10 ] periments. Notice that for small values of sin2 2ϑ the 2

bounds are practically linear in the log-log plot in Fig. 6, [eV 2 2 2 because in this case U sin 2ϑ/4 and the inequality m + e4 ∆ in Eq. (27) leads to | | ≃ 1

log ∆m2 . 2 log 2 + 2 log mub log sin2 2ϑ , (29) β −

ub −1 where m is the upper bound for mβ. 10 β −3 −2 −1 In the fourth column of Tab. I and in Fig. 6 we re- 10 10 10 0 2 4 6 8 10 2 ϑ ∆χ2 port the results of the analysis of the data of the Bugey sin 2 and Chooz reactor experiments presented in Ref. [14]2, with the addition in the analysis of the results of the FIG. 6. Allowed regions in the sin2 2ϑ–∆m2 plane and Mainz and Troitsk Tritium β-decay experiments, which marginal ∆χ2’s for sin2 2ϑ and ∆m2 obtained from the com- affect the high-∆m2 region. As already commented in bined fit of the results of the Bugey and Chooz reactor ex- Ref. [14], the reactor data are compatible with both the periments and the results of the Mainz and Troitsk Tritium Null Hypothesis of absence of electron antineutrino dis- β-decay experiments. The three lines in the upper-right cor- appearance and Our Hypothesis of electron antineutrino ner are the exclusion curves obtained from the results of the Mainz and Troitsk Tritium β-decay experiments alone. The disappearance, with a hint in favor of electron antineu- 2 trino oscillations due to a ∆m2 of about 2 eV. best-fit point corresponding to χmin is indicated by a cross.

V. COMBINED ANALYSIS This low compatibility of the neutrino and antineutrino data sets is illustrated in Fig. 8, where we have plotted the marginal ∆χ2’s for sin2 2ϑ obtained with the analysis The results of the combined analysis of MiniBooNE, of different data sets3. One can see that Gallium, reactor and Tritium data are presented in the last column of Tab. I and in Fig. 7. One can see that 1. The neutrino MiniBooNE and Gallium data agree the goodness-of-fit is high. The separate data sets are to indicate a value of sin2 2ϑ between about 0.11 fitted fairly well by the electron neutrino disappearance and 0.48 at 2σ. hypothesis: the χ2 contribution of the first three Mini- BooNE low-energy ν bins is 4.1, that of the other 16 2. The antineutrino reactor data indicate a value of e 2 MiniBooNE νe and νµ energy bins is 7.5, that of the 4 sin 2ϑ smaller than about 0.10 at 2σ. The Tritium Gallium data is 6.3, that of the 56 reactor degrees of free- data are practically irrelevant for the determination 2 dom is 49.0 and that of the 2 Tritium degrees of freedom of sin 2ϑ. is 0.57. On the other hand, the 3% parameter goodness- 3. The combined analysis is dominated by the reactor of-fit of the combined analysis of neutrino MiniBooNE data and indicates a value of sin2 2ϑ between about and Gallium data and antineutrino reactor and Tritium 0.01 and 0.13 at 2σ. data is rather low. The discrepancy between the neutrino and antineutrino determinations of sin2 2ϑ is about 2σ, in rough agreement with the above-mentioned 3% parameter goodness-of-fit 2 As erratum, let us notice that in the fourth column of Table III in of the combined analysis. In fact, the 2σ disagreement Ref. [14] there is a small mistake in the evaluation of the param- 2 eter goodness-of-fit. The correct values are ∆χmin = 0.52 and GoF = 0.47. We also notice that in the version of Ref. [14] pub- 2 lished in Phys. Rev. D the value of sin 2ϑbf for the Ga+Bu+Ch analysis (last column of Table III in Ref. [14]) is different from 3 We thank the anonymous referee of Phys. Rev. D for suggesting the correct one, which is 0.054 (see the arXiv version of Ref. [14]). this interesting figure. 8

99.73%

MBν 2 3 Ga ∆χ MBν + Ga + Re + H MBν + Ga 68.27% C.L. (1σ) Re + 3H 95.45% C.L. (2σ) MBν + Ga + Re + 3H 99.73% C.L. (3σ) 99% 0 2 4 6 8 10 2 ∆χ

10 95.45% ] 2 [eV 2 90% m + ∆ 1

68.27%

− 0 2 4 6 8 10 10 1 − − − −3 −2 −1 10 3 10 2 10 1 0 2 4 6 8 10 10 10 10 1 2 ∆χ2 sin 2ϑ sin22ϑ

2 ϑ m2 2 2 2 2 FIG. 7. Allowed regions in the sin 2 –∆ plane and FIG. 8. Marginal ∆χ = χ χmin for sin 2ϑ obtained from 2 2 2 marginal ∆χ ’s for sin 2ϑ and ∆m obtained from the com- the analysis of different combinations− of MiniBooNE, Gal- bined fit of the results of MiniBooNE, Gallium, reactor and lium, reactor and Tritium data. The marginal ∆χ2 obtained Tritium experiments. The best-fit point corresponding to from the analysis of reactor data alone and that obtained from 2 χmin is indicated by a cross. The three lines in the upper- the combined analysis of reactor and Tritium data are shown 2 2 right corner give the 1σ, 2σ and 3σ limits in the sin 2ϑ–∆m by the same line, since they practically coincide. plane obtained from Eqs. (20) and (28) and the results in Eqs. (21) and (22) of the Mainz and Troitsk Tritium β-decay experiments. remains better than in the case of no oscillations and between the neutrino and antineutrino data sets is en- fν = 1. tirely due to the different requirements on the value of Figure 9 shows the fit of MiniBooNE ν data corre- sin2 2ϑ, whereas they nicely agree on a best-fit value of e 2 2 sponding to the best-fit result of the combined analysis. ∆m at about 2eV . One can see that the fit of the first three low-energy bins Since the 3% parameter goodness-of-fit of the com- is not as good as that in Fig. 1b, but it is nevertheless bined analysis shows a tension between the neutrino and acceptable and much better than that in Fig. 1a. antineutrino data (under our νe-disappearance hypothesis) but is not sufficiently small to reject with confidence For the Gallium source experiments, the best-fit val- GALLEX the compatibility of the neutrino and antineutrino data ues of the oscillation parameters give RCr1 = 4 GALLEX SAGE SAGE sets , in the following part of this Section and in Sec- RCr2 = 0.97, RCr = 0.96 and RAr = 0.96. GALLEX tion VI we consider the results and implications of the Therefore, the experimental values of RCr1 and SAGE combined analysis. In Section VII we consider a possi- RCr in Eqs. (14) and (16) are fitted very well and ble difference between the effective mixing angles in the the Gallium χ2 contribution of 6.3 is almost equally due GALLEX SAGE neutrino and antineutrino sectors. to the loose fits of RCr2 and RAr in Eqs. (15) and Although the combined analysis of neutrino and an- (17). tineutrino data favors smaller values of sin2 2ϑ than those obtained from the analysis of MiniBooNE and Gallium Considering the combined fit of the results of Mini- data alone, the fit of the MiniBooNE and Gallium data BooNE, Gallium, reactor and Tritium data as a fair indication in favor of a possible short-baseline electron neutrino disappearance generated by the effective mixing parameters ∆m2 2eV and 0.01 . sin2 2ϑ . 0.13, in the ≃ 4 For example, the review on Statistics in the 2000 edition of next Section we present the corresponding predictions for the Review of Particle Physics [48] says that if the goodness-of- the effective neutrino masses in β-decay and neutrinoless fit “is larger than an agreed-upon value (0.001, 0.01, or 0.05 are double-β-decay experiments which could be measured in common choices), the data are consistent with the assumptions”. future experiments. 9

MiniBooNE − νe

νe,the (j) νe,cal N = f ν P N νe,j νe→νe νe,j ν ν µ,the = µ,cal 99.73% Nνe,j f ν Nνe,j ν the νe,the µ,the N = Nν + Nν νe,j e,j e,j MBν + Ga Re + 3H MBν + Ga + Re + 3H (b)

99% 2 ∆χ events / MeV events

95.45%

90%

68.27% 0.0 0.5 1.0 1.5 2.0 2.5

200 400 600 800 1000 1200 1400 3000 0 2 4 6 8 10 − − E [MeV] 10 2 10 1 1 2 |U e4| ∆m [eV]

FIG. 9. Expected number of MiniBooNE νe events in the 2 2 2 best-fit result of the combined analysis of MiniBooNE, Gal- FIG. 10. ∆χ = χ χmin as a function of the contribution 2 − lium, reactor and Tritium data (last column in Tab. I). The Ue4 √∆m to the effective β-decay electron-neutrino mass | | notation is the same as in Fig. 1. mβ in four-neutrino schemes obtained from the analysis of MiniBooNE and Gallium data (dashed line), from the analysis of reactor and Tritium data (dotted line), and from the VI. PREDICTIONS FOR BETA-DECAY AND combined analysis of the two sets of data (solid line). NEUTRINOLESS DOUBLE-BETA-DECAY EXPERIMENTS 2 2 if Ue4 √∆m is sufficiently large, as allowed by ∆χ in Fig.| 10.| In this Section we present predictions for the effec- If massive neutrinos are Majorana particles, neutrino- tive neutrino masses in β-decay and neutrinoless double- less double-β decay is possible, with a decay rate propor- β-decay experiments obtained as a consequence of the tional to the effective Majorana mass (see Refs. [2, 8, 50– combined fit of MiniBooNE, Gallium, reactor and Tri- 52]) tium data discussed in the previous Section. 2 2 2 Figure 10 shows the residual ∆χ = χ χmin as a 2 − 2 function of the contribution Ue4 √∆m to the effective m2β = Uekmk . (33) | | mass mβ in β-decay experiments (see Eq. (27)). Since Xk 2 from the last column of Tab. I we have sin 2ϑbf 1, we The results of the combined fit of MiniBooNE, Gallium, ≪ obtain reactor and Tritium data discussed in Section IV allow 2 us to estimate the contribution of the heaviest massive 2 sin 2ϑbf U =0.016 , (30) neutrino ν4 to m2β, which is approximately given by | e4|bf ≃ 4 U 2√∆m2, taking into account the mass hierarchy in | e4| and the best-fit value of U √∆m2 is Eq. (5). e4 2 2 2 | | Figure 11 shows ∆χ = χ χmin as a function of 2 2 − √ 2 the contribution Ue4 √∆m in four-neutrino schemes Ue4 ∆m =0.17eV , (31) | | | | bf to m2β. The best-fit value is: and 2 2 Ue4 √∆m =0.02eV , (34) | | bf 0.06 U √∆m2 0.49eV at2σ . (32) ≤ | e4| ≤ and

This prediction is relevant for the KATRIN experiment 2 2 0.003 Ue4 √∆m 0.07eV at2σ . (35) [49], which is under construction and scheduled to start ≤ | | ≤ in 2012. The expected sensitivity of about 0.2 eV at This range must be confronted with the expected con- 90% C.L. may be suﬃcient to observe a positive eﬀect tributions to m2β coming from the three light massive 10 neutrinos ν1, ν2, ν3. Assuming a hierarchy of masses, m m m m , (36) 1 ≪ 2 ≪ 3 ≪ 4 99.73% MBν + Ga which is the most natural case compatible with the hier- Re + 3H archy in Eq. (5), we have MBν + Ga + Re + 3H

m U 2 ∆m2 + U 2 ∆m2 + U 2 √∆m2 , 2β ≃ e2 SOL e3 ATM e4 99%

q q (37)

where we have neglected the contribution of the lightest 2 massive neutrino ν1. From the 3σ upper limits of the ∆χ three-neutrino mixing parameters given in Ref. [4], we obtain 95.45%

−3 2 2 −3 2 10 . Ue2 ∆mSOL . 4 10 eV , (38) × | | × 90% q U 2 ∆m2 . 3 10−3 eV . (39) | e3| ATM × 68.27% Therefore, strong cancellationsq between the contributions of ν2 and ν3 are possible (albeit not likely [53]), whereas 0 2 4 6 8 10 − − − the range in Eq. (35) disfavors strong cancellations be- 10 3 10 2 10 1 1 2 ∆ 2 tween the contributions of ν2 and ν3 and the contribu- |U e4| m [eV] 2 2 tion of ν4. In this case, m2β Ue4 √∆m leading to a possible observation of neutrinoless≃ | | double-β decay in 2 2 2 future experiments which will be sensitive to values of FIG. 11. ∆χ = χ χmin as a function of the contribu- 2√ 2 − m smaller that 10−1 eV (e.g. CUORE [54], EXO [55], tion Ue4 ∆m to the effective neutrinoless double-β decay 2β Majorana| | mass m in four-neutrino schemes obtained from SuperNEMO [56]; see the review in Ref. [52]). 2β the analysis of MiniBooNE and Gallium data (dashed line), On the other hand, if neutrinoless double-β decay ex- from the analysis of reactor and Tritium data (dotted line), periments which are sensitive to values of m2β of the and from the combined analysis of the two sets of data (solid −1 order of 10 eV (e.g. CUORICINO [57], GERDA [58], line). Majorana [59]; see the review in Ref. [52]) will see a positive signal, maybe compatible with the signal asserted in Ref. [60], the mass hierarchy in Eq. (36) will become The results for the two fits are those presented in Fig. 4 unlikely and the favorite 3+1 four-neutrino schemes will and the third column in Tab. I for neutrinos (MBν+Ga) be those in which the three light neutrinos ν1, ν2 and ν3 and Fig. 6 and the fourth column in Tab. I for antineu- 3 are almost degenerate at the mass scale of m2β. trinos (Re+ H). Since the fit of the data does not require a difference 2 2 of ∆mν and ∆mν¯, we consider only the mixing angle VII. MIXING ANGLE ASYMMETRY? asymmetry 2 2 A 2 = sin 2ϑ sin 2ϑ . (42) The tension between neutrino and antineutrino data sin 2ϑ ν − ν¯ discussed in Section V could be due to a difference of the Figure 12 shows the marginal ∆χ2 as a function of effective mixing angles in the neutrino and antineutrino Asin2 2ϑ. The best-fit value of Asin2 2ϑ is sectors. Such a difference could be due to a violation bf of the fundamental CPT symmetry or to another un- Asin2 2ϑ =0.23 , (43) known mechanism. Phenomenological analyses of differ- and the 2σ allowed range of A 2 is ent masses and mixings for neutrinos and antineutrinos sin 2ϑ have been presented in several publications [61–74]. 0.06 Asin2 2ϑ 0.45 , (44) In this section we consider the possibility that neutri- ≤ ≤ nos and antineutrinos have different effective masses and but there is no limit on the asymmetry at 3σ. The statistical significance of A 2 > 0 is 99.14% C.L. (2.6σ). mixings in short-baseline νe andν ¯e disappearance exper- sin 2ϑ iments. We fit the neutrino and antineutrino data with It is interesting to note that a difference between neu- the survival probabilities trino and antineutrino mixings can be tested in β-decay experiments by searching for different effective neutrino 2 − + SBL 2 2 ∆mν L masses in β and β decays. The prediction for the con- Pν →ν (L, E)=1 sin 2ϑν sin , (40) e e − 4E √ 2 tribution Ue4 ∆m to the effective electron an- 2 | | ν¯e ∆m L − P SBL (L, E)=1 sin2 2ϑ sin2 ν¯ . (41) tineutrino mass in β decays from the analysis of an- ν¯e→ν¯e − ν¯ 4E tineutrino reactor and Tritium data can be obtained from 11

+ (MBν + Ga) + (Re + 3H) will be interesting to study the possibility of making β - σ 99.73% 68.27% C.L. (1 ) decay experiments for the search of the eﬀective electron 90.00% C.L. 95.45% C.L. (2σ) antineutrino mass, for which the dashed line in Fig. 10 99.00% C.L. and Eq. (48) give a reachable lower limit. 99.73% C.L. (3σ) We do not consider here neutrinoless double-β decay in the case of a neutrino-antineutrino mixing diﬀerence, 99% since the Majorana nature of neutrinos requires a treat- ment which goes well beyond the purposes of this paper (see Ref. [76]). 2 ∆χ

95.45% VIII. CONCLUSIONS

90% In this paper we have discussed a neutrino oscillation interpretation of the MiniBooNE low-energy anomaly and the Gallium radioactive source experiments anomaly in the framework of 3+1 four-neutrino mixing schemes. 68.27% We have shown that the combined fit of MiniBooNE and Gallium data indicate a possible short-baseline electron 0 2 4 6 8 10 −3 −2 −1 neutrino disappearance generated by effective oscillation 10 10 10 1 parameters ∆m2 & 0.1 eV2 and 0.11 sin2 2ϑ 0.48 at A 2 ϑ sin 2 2σ, with best fit at ∆m2 2 eV2 and≤ sin2 2ϑ ≤0.3 (see Fig. 4). ≃ ≃ 2 2 2 We have also considered the data of the Bugey and FIG. 12. Marginal ∆χ = χ χmin as a function of the − Chooz reactor neutrino oscillation experiments and the mixing angle asymmetry Asin2 2ϑ. results of the Mainz and Troitsk Tritium β-decay experiments, which imply an upper bound on the effective elec- the dotted line in Fig. 10: the best fit is tron neutrino mass of about 2 eV (see Fig. 5 and the combined upper bound in Eq. (25)). As already discussed in bf 2 Ref. [14], the Bugey data give a faint indication of a pos- Ue4 √∆m =0.14eV , (45) | | ν¯e sible short-baseline electron neutrino disappearance generated by effective oscillation parameters ∆m2 2 eV2 and and sin2 2ϑ 0.04, which is compatible with Chooz≃ and Tritium data≃ (see Fig. 6). √ 2 Ue4 ∆m 1.29eV at2σ . (46) In Section V we have discussed the tension between | | ν¯e ≤ the neutrino MiniBooNE and Gallium data and the an- 2 tineutrino reactor and Tritium data. Considering such For the contribution Ue4 √∆m to the effective | | νe tension as a statistical fluctuation, we have presented + electron neutrino mass in β decays we must consider the results of the combined analysis of MiniBooNE, Gal- the dashed line in Fig. 10, which has been obtained from lium, reactor and Tritium data: ∆m2 2 eV2 and the analysis of MiniBooNE and Gallium neutrino data: 0.01 sin2 2ϑ 0.13at 2σ, with best fit at∆≃ m2 2 eV2 the best fit is and sin≤ 2 2ϑ 0≤.06 (see Fig. 7). ≃ ≃ bf In Section VI, we have presented predictions for the 2 Ue4 √∆m =0.38eV , (47) effective neutrino masses in β-decay and neutrinoless | | νe double-β-decay experiments obtained as a consequence and of the combined analysis of MiniBooNE, Gallium, reactor and Tritium data, assuming the hierarchy of masses 2 Ue4 √∆m 0.21eV at2σ . (48) in Eq. (5). The predicted interval for the contribution of | | νe ≥ m to the effective neutrino mass in β-decay is between 4 Unfortunately the existing and foreseen experiments about 0.06 and 0.49 eV at 2σ. The upper part of this in- are β−-decay experiments for which the contribution terval may be reached by the KATRIN experiment [49]. 2 For neutrinoless double-β-decay we obtained a prediction Ue4 √∆m to the effective electron antineutrino | | ν¯e for the contribution of m4 to the effective neutrino mass mass is expected to be small. The future β−-decay ex- between about 0.003 and 0.07 eV at 2σ, which may be periment will use either Tritium (KATRIN [49]) or 187Re reached in future experiments (see Ref. [52]). (MARE [75]). If the mixing difference between the neu- We also considered, in Section VII, the possibility of trino and antineutrino sectors will be confirmed with reconciling the tension between the neutrino MiniBooNE highest confidence by future neutrino oscillation data it and Gallium data and the antineutrino reactor and Tri- 12 tium data discussed in Section V with different mixings in cyclotrons under development for commercial uses [108]. the neutrino and antineutrino sectors. We found a 2.6σ These provide electron neutrino beams with energy up indication of a mixing angle asymmetry (99.14% C.L.). to 52 MeV from muon decay-at-rest. A low-energy νe We pointed out the possibility of checking the mixing (−) disappearance experiment (as well as νµ disappearance difference between the neutrino and antineutrino sectors (−) (−) and νµ νe measurements) can be performed with such by measuring different effective electron antineutrino and → − + devices due to the full efficiency of the ICARUS detec- neutrino masses in β and β decay experiments. tor at 20 MeV energies. The expected event rate is about The indication in favor of short-baseline disappearance 400 charged-current electron neutrino events per year per of electron neutrinos imply the possible existence of a ton with the ICARUS detector located at 50 m from the light sterile neutrino which could have important conse- source [109]. The number of events is calculated assum- quences in physics [18–20, 22, 77–83], astrophysics [84– 15 ing 10 νe’s per year and a fully efficient detector. Since 92] and cosmology [93–98]. at ∆m2 2 eV2 the oscillation length is about 20 m for As far as the effective number of neutrino species in a 20 MeV≈ neutrino energy, it is possible to measure the cosmology, Neff, is concerned, the analysis of 7-years full oscillation pattern along the beam direction inside WMAP data has provided the following result: Neff = +0.86 the ICARUS volume. 4.34−0.88 (68% C.L.) [99]. In 2011 the Planck experi- The disappearance of electron neutrinos can be inves- ment will measure Neff with a factor of 4 improvement tigated with high accuracy in future near-detector beta- in accuracy with respect to present data [100, 101]. In beam [110] and neutrino factory [73, 111] experiments in other words, the possibility of existence of a fourth light which the neutrino fluxes will be known with high preci- sterile neutrino could be pursued with 5σ significance. sion. Finally, we would like to encourage all experiments Furthermore, the MiniBooNE low-energy anomaly which can investigate the hypothesis of short-baseline may be clarified by the ArgoNeuT, MicroBooNE [112] electron neutrino disappearance. and BooNE experiments [113], and the magnetic off-axis Starting from 2010, at the same L/E of MiniBoone, the near detector at 280 m of the T2K experiment has the magnetic off-axis near detector at 280 m of the T2K ex- unique opportunity to measure the charge of the events periment [102] will count ν events with expected higher e of the low-energy anomaly [114]. statistics and similar νµ background contamination. A test of short-baseline oscillations may be done, although the accuracy suffers from the scarce knowledge of the NOTE ADDED neutrino flux and of the neutrino cross section at 1 GeV energies [15, 103]. A better measurement will be possible with the new After the completion of this work, two important ex- CERN-PS neutrino beam [104, 105], thanks to the pres- perimental results have been presented at the Neutrino ence of 2 detectors at 140 m (NEAR) and 885 m (FAR). 2010 conference: At ∆m2 2 eV2, the oscillation length is about 1 km for 1 GeV≈ neutrino energies. Therefore, one can reduce 1. The MiniBooNE collaboration presented updated results on the search for short-baselineν ¯ ν¯ the systematic error of the Monte Carlo predictions by µ → e normalizing the high energy part of the νe spectrum at oscillations which are compatible with the LSND the NEAR location. In addition, a better νµ background signal [115, 116]. The inclusion of these data in our rejection will be possible using the liquid Argon technol- framework will require a separate analysis in which the assumption of negligible U 2 is relaxed [117]. ogy. The interesting possibility of a νe tagging in the | µ4| CERN-PS beam was also studied in this context [106]. New measurements with a radioactive source could be 2. The MINOS collaboration presented an indication made in the SAGE experiment [10], with the Borexino of a possible difference between the effective mix- detector and with the future LENS detector [79]. At ings of neutrinos and antineutrinos in long-baseline ∆m2 2 eV2, the oscillation length is about 1 m for 1 νµ andν ¯µ disappearance [118]. This indication is MeV neutrino≈ energies. Therefore Borexino could mea- analogous to that discussed in Section VII. sure the oscillation pattern over a distance of 4 m (the Borexino radius) using the well known ν -e scattering e ACKNOWLEDGMENTS process and with a vertex resolution that at the moment is about 15 cm [107]. At the Gran Sasso laboratories a very interesting mea- We would like to thank E. Bellotti, C. Giganti, A. surement could be realized by using the ICARUS 600 Longhin, F. Pietropaolo and A. Rubbia for interesting ton detector and new low-cost and high-power proton discussions and suggestions.

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