Sterile neutrino as a pseudo-Goldstone fermion
Stéphane Lavignac (IPhT Saclay)
• introduction and motivations • theoretical framework • numerical results: correlations among sterile parameters • conclusions
based on a work in progress with Enrico Bertuzzo (IPhT Saclay)
Probing the Standard Model and New Physics at Low and High Energies Portoroz, Slovenia, 17 April 2013 Introduction and motivations Renewed interest in the past few years for sterile neutrinos, mainly driven by experimental anomalies and cosmology:
- the reactor anti-neutrino anomaly (deficit of ⌫ e in short-baseline reactor experiments) could be due to oscillations into sterile neutrinos10
1.2 II I III − 1.15 − − (arXiv:1101.2755) 2S − II III I G. Mention et al. et Mention G. − − − 3 3 4 − − − II I −
1.1 − Krasnoyarsk SRP Krasnoyarsk Goesgen Krasnoyarsk Goesgen PaloVerde CHOOZ Bugey ROVNO88 Bugey Bugey Goesgen SRP ROVNO91 3 1.05 − Bugey 1 pred,new )
EXP 0.95 /(N 0.9 OBS ILL N 0.85 1S 3S 2I 1I − − − − 0.8
0.75 ROVNO88 ROVNO88 ROVNO88 ROVNO88
1 2 3 10 10 10 Distance to Reactor (m)
FIG. 5. Illustration of the short baseline reactor antineutrino anomaly. The experimental results are compared to the prediction without oscillation, taking into account the new antineutrino spectra, the corrections of the neutron mean lifetime, and the - measurementoff-equilibrium eff ects.of PublishedCMB experimental anisotropies errors and antineutrino and spectra other errors are added cosmological in quadrature. The mean data are averaged ratio including possible correlations is 0.943 ± 0.023. The red line shows a possible 3 active neutrino mixing solution, 2 2 2 with sin (2θ13)=0.06. The blue line displays a solution including a new neutrinomassstate,suchas|∆mnew,R| ! 1eV and consistent2 with extra light degrees of freedom sin (2θnew,R)=0.12 (for illustration purpose only).
N =4.34+0.86 (68% C.L.) [WMAP 7yr + BAO + H0] e↵ ting ∼ 1 MeV electron0.88 neutrinos. [57], following the 2 2 2 Experiment(s) sin (2θnew) |∆mnew| (eV ) C.L. (%) methodology developed in Ref. [56, 58]. However we ∗ decided to include possible correlations between these Reactors (no ILL-S,R ) 0.02-0.20 > 0.40 96.5 Gallium (G) > 0.06 > 0.13 96.1 four measurements in this present work. Details are Ne↵ =3.84 0.40 (68% C.L.) MiniBooNE (M)[WMAP— 9yr— + eCMB72.4 + BAO + H0] given in Appendix B. This has the effect of being ILL-S — — 68.1 slightly more conservative,± with the no-oscillation hy- R∗ +G 0.05-0.22 > 1.45 99.7 pothesis disfavored at 97.7% C.L., instead of 98% C.L. R∗ +M 0.04-0.20 > 1.45 97.6 ∗ in Ref. [56]. Gallex and Sage observed an average deficit R +ILL-S 0.02-0.21 > 0.23 95.3 of RG =0.86 ± 0.06 (1σ). Considering the hypothesis of All 0.06-0.22 > 1.5 99.8 νe disappearance caused by short baseline oscillations we 2 used Eq. (13), neglecting the ∆m31 driven oscillations TABLE III. Best fit parameter intervals or limits at 95% C.L. 2 2 because of the very short baselines of order 1 meter. Fit- for sin (2θnew)and|∆mnew| parameters, and significance of ting the data leads to |∆m2 | > 0.3eV2 (95%) and the sterile neutrino oscillation hypothesis in %, for different new,G ∗ 2 combinations of the reactor experimental rates only (R ), the sin (2θnew,G) ∼ 0.26. Combining the reactor antineu- trino anomaly with the gallium anomaly gives a good fit ILL-energy spectrum information (ILL-S), the gallium experi- ments (G), and MiniBooNE-ν (M) re-analysis of Ref. [56]. We to the data and disfavors the no-oscillation hypothesis at 2 2 2 quantify the difference between the sin (2θnew)constraints 99.7% C.L. Allowed regions in the sin (2θnew) − ∆mnew obtained from the reactor and gallium results. Following pre- plane are displayed in Figure 6 (left). The associated scription of Ref. [77], the parameter goodness-of-fit is 27.0%, 2 2 best-fit parameters are |∆mnew,R&G| > 1.5eV (95%) indicating reasonable agreement between the neutrino and an- 2 and sin (2θnew,R&G) ∼ 0.12. tineutrino data sets (see Appendix B). We then reanalyzed the MiniBooNE electron neutrino excess assuming the very short baseline neutrino os- cillation explanation of Ref. [56]. Details of our re- production of the latter analysis are provided in Ap- analysis leads to a good fit with the sterile neutrino 2 2 pendix B. The best fit values are |∆mnew,MB| =1.9eV hypothesis and disfavors the absence of oscillations at 2 and sin (2θnew,MB) ∼ 0.2, but are not significant at 98.5% C.L., dominated by the reactor experiments data. 2 2 95% C.L. The no-oscillation hypothesis is only disfa- Allowed regions in the sin (2θnew) − ∆mnew plane are vored at the level of 72.4% C.L., less significant than displayed in Figure 6 (right). The associated best- 2 2 the reactor and gallium anomalies. Combining the re- fit parameters are |∆mnew,R&MB| > 0.4eV (95%) and 2 actor antineutrino anomaly with our MiniBooNE re- sin (2θnew,R&MB) ∼ 0.1. Recent Planck data leave less room for a sterile neutrino. One usually quotes: +0.54 Ne↵ =3.30 0.51 (95% C.L.) [Planck + WMAP + highL + BAO] Planck Collaboration: Cosmological parameters
which favour higher values. Increasing the neutrino mass will only make this tension worse and drive us to artificially tight 1.0 Planck+WP+highL constraints on m⌫. If we relax spatial flatness, the CMB ge- +BAO ometric degeneracy becomes three-dimensional in models with arXiv:1303.5076 However the constraint strongly +H0 massive neutrinosP and the constraints on m⌫ weaken consider- 0.8 dependsably to on the set of data used: +BAO+H0 P
0.98 eV (95%; Planck+WP+highL) max 0.6 P m⌫ < +0.48 (73) / 8 0.32 eV (95%; Planck+WP+highL+BAO). P NeX↵ =3> .52 0.45 (95% C.L.) 0.4 < :> 6.3.2. Constraints on Ne↵ [Planck + WMAP + highL + BAO + H0] 0.2 As discussed in Sect. 2, the density of radiation in the Universe (besides photons) is usually parameterized by the e↵ective neu- 0.0 trino number Ne↵. This parameter specifies the energy density when the species are relativistic in terms of the neutrino tem- 2.4 3.0 3.6 4.2 perature assuming exactly three flavours and instantaneous de- Ne coupling. In the Standard Model, Ne↵ = 3.046, due to non- instantaneous decoupling corrections (Mangano et al. 2005). Fig. 27. Marginalized posterior distribution of Ne↵ for AssumingHowever, a fully there thermalized has been some mild massive preference forsterilePlanck +neutrino,WP+highL (black) the and constraint additionally BAO becomes: (blue), N > 3.046 from recent CMB anisotropy measurements e↵ the H0 measurement (red), and both BAO and H0 (green). (Komatsu et al. 2011; Dunkley et al. 2011; Keisler et al. 2011; Ne↵
43 These bounds are in tension with the sterile neutrino interpretation of the neutrino anomalies, which require m 1 eV ⌫s ⇠ [see e.g. Mirizzi et al., arXiv:1303.5368] e.g. a combined analysis of SBL reactor data, gallium calibration experiments and MiniBooNE neutrino data gives [G. Mention et al., arXiv:1101.2755]: m2 > 1.5eV2 , sin2 2✓ =0.14 0.08 (95% C.L.) | SBL| ee ± From a theoretical point of view, sterile neutrinos also pose a problem: since they are gauge singlets, their mass is not protected by any symmetry Sterile neutrinos are present e.g. in the seesaw mechanism: c 1 T 1 c 0 m ⌫R m ⌫L⌫R M⌫R C⌫R +h.c.= (¯⌫L ⌫¯L ) +h.c. 2 2 mM ⌫R ✓ ◆✓ ◆ the mass eigenstates ⌫ L 1 and ⌫ L 2 are admixtures of the active neutrino ⌫ L and of the sterile neutrino ⌫c C ⌫T L ⌘ R c However, for m << M, the sterile neutrino ⌫ L ⌫ L 2 is very heavy and has negligible mixing with the active neutrino ' Theoretical scenarios for naturally light sterile neutrinos
1) (very) low-energy seesaw with M < 10 eV [de Gouvêa, Huang, arXiv:1110.6122] but the seesaw explanation of small neutrino masses is lost
2) singular seesaw mechanism [Glashow ’91] det M = 0 leading to a fourth light mass eigenstate accidental or due to symmetries of the neutrino sector
3) flat extra dimensions [arkani-Hamed et al. ’98; Dienes et al. ’98] a massless bulk RH neutrino generates a tower of Kaluza-Klein sterile neutrinos with masses n/R 4) singlet fermions (modulinos) in supersymmetry/string theory [Benakli, Smirnov ’97; Dvali, Nir ’98]
5) pseudo-Goldstone fermion [Chun, Joshipura, Smirnov ’95; Chun ’99] supersymmetric partner of the Goldstone boson of a spontaneously broken global symmetry (e.g. lepton number or Peccei-Quinn) Theoretical framework
Some global symmetry spontaneously broken at a scale f >> MSUSY The supersymmetric effective field theory below f involves a (pseudo-)Goldstone multiplet s + ia A = + p2✓ + ✓2F p2 with a shift symmetry A A + i↵f ! In the supersymmetric limit and in the absence of explicit global symmetry breaking, all components s, a, χ are massless Supersymmetry breaking can give a mass to χ and s, while some explicit breaking of the global symmetry is needed to give a mass to a (or the symmetry must be anomalous) Irreducible χ mass from supersymmetry breaking [Cheung, Elor, Hall, 1104.0692] 4 1 2 2 m m 3 / 2 from d ✓ (A + A†) (X + X†) , X = F ✓ ⇠ MP h i Z ⇒ low-scale supersymmetry breaking needed Assuming no R-parity (but baryon number), the most general Lagrangian compatible with the shift symmetry A A + i ↵ f is (α = 0, 1, 2, 3): ! 1 k W = µ H L↵ + e L↵L ek + d L↵Qjd u H Qjuk ↵ u 2 ↵ k ↵jk jk u
1 2 ↵ ↵ A + A† K = (A + A†) + H†H + L †L + C H†H 2 u u u u u f
↵ A + A† ↵ A + A† + C L †L + C H L +h.c. + ↵ f u↵ u f ··· ✓ ◆ Vsoft = generic MSSM soft terms with leptonic RPV
After minimization of the scalar potential, Hu, Lα get vevs (assume = 0) ⇒ non-canonical kinetic terms for Hu and Lα → redefine Hu and Lα = (Hd, Li) such that + (i) the charged fields H u ,H d ,e i have canonical kinetic terms
(ii) the sneutrino vevs ⌫˜ i vanish e e e h i (iii) 0 jk = j jk , j real Neutralino and chargino mass matrices
As a consequence of the bilinear RPV terms (µi Hu Li), leptons mix with charginos and neutralinos. 0 0 i Furthermore, since the kinetic terms of the neutral fields Hu,Hd , ⌫ ,A are not canonical, the neutralino mass matrix receives contribution from the Kähler potential and mixes χ with the standard neutrinos and neutralinos
Charginos: + M2 gvu 01 3 W ⇥ + W Hd ei gvd µ 01 3 Hu 0 e ⇥ 1 0 + 1 03 1 µi i vd ij fe ⇣ ⌘ ⇥ k f e @ A @ e A 2 heavy mass eigenstates (charginos) e 3 light mass eigenstates (charged leptons) with masses mi = i vd chargino / charged lepton mixing suppressed by µ i /µ or smaller Neutralinos: 3 0 0 The neutralino mass matrix, written in the basis (W , B,Hu, Hd , ⌫i, ) is a 8x8 matrix with a seesaw structure f e e e M4 4 µ4 4 ⇥ ⇥ MN = m, µ M 0 T 1 ⌧ µ4 4 m4 4 ⇥ @ ⇥ A hence the neutrino masses and mixing are given by the diagonalization of the 4x4 effective neutrino mass matrix
A µi µj B µi + D v T 1 µ µ µ i f M⌫ = m µ M µ = 2 0 µj v ⇣ v ⌘ 1 B µ + Dj f C f 2 + m
2 2 2 @ ⇣ ⌘ A µ M11MZ cos where A = det M depends only on MSSM parameters, while
B, Di and C depend also on the Kähler parameters Cud,Cu,Cdd,Cı| (Rp-conserving) and C uı ,C dı (RPV). Assuming the former are of order 1, one has B,C = (µ) O One gets a consistent neutrino phenomenology by assuming that all RPV parameters ( µ i ,C ui ,C di ) are small, while the Rp-conserving Kähler parameters are of order 1
µi 5 6 v 6 In practice, need 10 ,C 10 , 10 ,m 1 eV µ . di . f . . Can rewrite the 4x4 neutrino mass matrix in a more compact form:
D✏↵✏ E⌘↵ 2 2 M⌫ = ✏↵ = ⌘↵ =1 E⌘ F ↵ ↵ ✓ ◆ X X where we have renamed the indices i, j ↵, = e, µ, ⌧ ! This structure implies:
1) the matrix has rank 3, so m1 =0 E 2) the active-sterile mixing is given by U ⌘ (m F ) ↵4 ' F ↵ 4 ' 3) at order 1 in the active-sterile mixing, the active neutrino parameters are given by the matrix E2 (m ) = D ✏ ✏ ⌘ ⌘ ⌫ ↵ ↵ F ↵ 3 E2 (m ) = m U U = D ✏ ✏ ⌘ ⌘ ⌫ ↵ i ↵i i ↵ F ↵ i=2 X Since we know the active neutrino parameters (neglecting CPV), we can ‟reconstruct” the sterile neutrino parameters using [when the first term dominates] E2 m ⇣~ 2 ,mD 2 ' F 3 ' U ( ⇠ ⇠ , ⇠ ⇠ , ⇠ ⇠ ) / ,U (✏ , ✏ , ✏ ) ↵2 ' µ ⌧ e ⌧ e µ ↵3 ' e µ ⌧ 2 where ⇣~ ~✏ ~⌘ ⇠↵ ⇣ ⇣ + ⇣~ ✏ ✏ (↵, , all di↵erent) ⌘ ⇥ ⌘ 1/4 and ⇠ ⇠ 2 + ⇠ ⇠ 2 + ⇠ ⇠ 2 ⌘ | µ ⌧ | | e ⌧ | | e µ| 2 ~ 2 In the reconstruction process, one obtains the ⇣ ↵ as a function of / ⇣ , which is the solution of a polynomial of degree 4. Among the solutions, only the ones that satisfy the constraint ⇣~ 2 1 (if any) are acceptable E2 Finally, one identifies ⌘ ⌘ m U U ⇒ correlations F ↵ ⌘ 4 ↵4 4 Numerical results
Normal hierarchy, case 1 D ✏2 (E2/F ) ⌘2 ↵ ⌧ ↵
0.28 0.28
0.26 0.26 L 0.24 L 0.24 eV eV H H
4 0.22 4 0.22 m m 0.20 0.20
0.18 0.18
3 4 5 6 7 90 100 110 120 130 140 2 3 s14 x 10 2 3 s34 x 10
0.28 0.0180 0.26
L 0.24 0.0175 L eV H eV 4 0.22 H b
m 0.0170
0.20 m
0.18 0.0165
60 70 80 90 100 0.18 0.20 0.22 0.24 0.26 0.28 s 2 x 103 24 m 4 eV
H L Not relevant to the reactor anomaly Not relevant to LSND / MiniBooNE
102
10
1
10-1
10- 2 10- 3 10- 2 10-1 1 Normal hierarchy, case 2 (E2/F ) ⌘2 D ✏2 ↵ ⌧ ↵
0.140 0.140 0.135 0.135 L L 0.130 0.130 eV eV H H 0.125 0.125 4 4 0.120 0.120 m m 0.115 0.115 0.110 0.110
0.0 0.5 1.0 1.5 300 350 400 450 2 3 2 3 s14 x 10 s34 x 10
0.140 0.0105 0.135
L 0.0100 L 0.130 eV eV H H 0.125 0.0095 b 4 0.120 m m 0.115 0.0090 0.110 0.0085 0.110 0.115 0.120 0.125 0.130 0.135 0.140 280 300 320 340 360 2 3 m 4 eV s24 x 10
H L Not relevant to the reactor anomaly Not relevant to LSND / MiniBooNE
102
10
1
10-1
10- 2 10- 3 10- 2 10-1 1 Inverted hierarchy
No numerical solution found so far Under study Conclusions
Sterile neutrino as a pseudo-Goldstone fermion within R-parity violating supersymmetry provides a surprisingly predictive scenario
Correlations between the sterile neutrino mass and the active-sterile mixing (in spite of a large number of parameters)
Does not explain the reactor neutrino anomaly nor LSND/MiniBooNE, but could be tested in future appearance experiments
Does not seem to be consistent with an inverted hierarchy in the active neutrino sector (to be confirmed)
In progress... BACK UP Active-sterile neutrino mixing Standard case (3 flavours):
Add a sterile neutrino: 4 ⌫s flavour eigenstate ⌫↵ = U↵i ⌫i [ ↵ = e, µ, ⌧ ] i=1 ⌫4 mass eigenstate (m4) U = 4x4P unitary matrix
Only ⌫ e , ⌫ µ , ⌫ ⌧ couple to electroweak gauge boson, but all four mass eigenstate are produced in a beta decay:
e
4 ⌫e = i=1 Uei ⌫i P 2-flavour oscillations: m2L P (⌫ ⌫ )=sin2 2✓ sin2 ↵ ! 4E ✓ ◆ ⌫ cos ✓ sin ✓ ⌫ ↵ = 1 2 2 2 ⌫ sin ✓ cos ✓ ⌫ m m2 m1 ✓ ◆ ✓ ◆✓ 2 ◆ ⌘
N-flavour oscillations: 2 ? ? 2 mijL P⌫↵ ⌫ (¯⌫↵ ⌫¯ ) = ↵ 4 Re U↵iU iU↵jU j sin ! ! 4E i LSND/MiniBooNE: vacuum 4 oscillations Smirnov 2 2 P ~ 4|Ue4 | |U4 | 2 m LSND mass Restricted by short baseline experiments CHOOZ, CDHS, 2 3 m atm NOMAD 2 m2 1 sun With new reactor data: 2 2 m41 = 1.78 eV Ue4 = 0.15 U4 = 0.23 We are interested in short baseline oscillations with m2 L m2 L m2 L m2 L 41 1= sin2 41 sin2 31 , sin2 21 4E . ) 4E 4E 4E ✓ ◆ ✓ ◆ ✓ ◆ 2 2 2 2 2 2 m41L P⌫ ⌫ 1 4 U↵1 + U↵2 + U↵3 U↵4 sin ↵! ↵ ' | | | | | | | | 4E ✓ ◆ m2 L 1 sin2 2✓ sin2 41 ⌘ ↵↵ 4E ✓ ◆ where sin2 2✓ 4(1 U 2) U 2 ↵↵ ⌘ | ↵4| | ↵4| 2 2 m41L P⌫↵ ⌫ 4Re U↵1U ⇤1 + U↵2U ⇤2 + U↵3U ⇤3 U↵⇤4U 4 sin ! ' 4E ✓ ◆ ⇥ m2 L ⇤ sin2 2✓ sin2 41 ⌘ ↵ 4E ✓ ◆ where sin2 2✓ 4 U U 2 ↵ ⌘ | ↵4 4| 3+2 case: Assume m m m ,m ,m 5 ⇠ 4 3 2 1 2 2 ⇒ two relevant squared mass differences m51 and m41 ⇒ CP-violating effects possible due to interference between the two oscillations frequencies 2 2 2 2 m41L 2 2 m51L P⌫ ⌫ (¯⌫ ⌫¯ ) =4U↵4U 4 sin +4U↵5U 5 sin ↵! ↵! | | 4E | | 4E ✓ ◆ ✓ ◆ m2 L m2 L m2 L +8 U U U U sin 41 sin2 51 cos 54 ⌘ | ↵4 4 ↵5 5| 4E 4E 4E ⌥ ✓ ◆ ✓ ◆ ✓ ◆ ⌘ arg U U ⇤ U ⇤ U ⌘ ↵4 4 ↵5 5 ⇥ ⇤ Table 3 90% CL limit on the neutrino oscillation probabilities from the negative searches at short baseline experiments. 2 2 Experiment Beam Channel Limit (90%) ∆mmin (eV ) Ref. CDHSW CERN ν ν P > 0.95 0.25 [98] µ → µ µµ E776 BNL ν ν P < 1.5 10 3 0.075 [99] µ → e eµ × − E734 BNL ν ν P < 1.6 10 3 0.4 [100] µ → e eµ × − KARMEN2 Rutherford ν¯ ν¯ P < 6.5 10 4 0.05 [101] µ → e eµ × − ExperimentalE531 FNAL ν situationν P < 2.5 10 3 0.9 [102] µ → τ µτ × − CCFR/ FNAL ν ν P < 8 10 4 1.6 [103, 104] µ → e µe × − NUTEV ν¯ ν¯ P < 5.5 10 4 2.4 [104] Several experimental anomalies suggest theµ → existencee µe × of− sterile neutrinos ν ν P < 4 10 3 1.6 [105] µ → τ µτ × − LSND: ⌫¯ µ ⌫¯ e oscillations ν ν P < 0.1 20.0 [106] ! e → τ eτ Chorus CERN ν ν P < 3.4 10 4 0.6 [107] Excess of ⌫¯ e events over background at 3.8µ → στ (stillµτ controversial)× − ν ν P < 2.6 10 2 7.5 [107] Not observed by KARMEN e → τ eτ × − Nomad CERN ν ν P < 1.7 10 4 0.7 [108] µ → τ µτ × − ν ν P < 7.5 10 3 5.9 [108] MiniBooNE: e → τ eτ × − 4 νµ νe Pµe < 6 10− 0.4 [108] ⌫ ⌫ data: no excess in the 475-1250→ MeV range,× but unexplained 3σ µ ! e 100 2 ⌫ e excess at low energy2 10 [eV ] 2 sin2(2θ) upper limit m Δ MiniBooNE 90% C.L.y ⌫¯ µ ⌫¯ e data: ⌫¯ e excess in the E > 475 MeVts 10 MiniBooNE 90% C.L. sensitivity ! 10 sta- BDT analysis 90% C.L. kground region consistent with LSND-like oscillations, ) KARMEN 2 dashed 4 90% C.I. CCFR /c oscilla- 2 ers but also (after the 2011 update) with a 1 1 | (eV 2 ex- m ∆ background-only hypothesis represent | ond LSND -1 Likelihood Favored regions 10 A low-energy ⌫¯ e excess0.1 is also seen LSND 90% C.L. Bugey LSND 99% C.L. -2 -3 -2 -1 with 10 10 10 10 1 10-3 10-2 10-1 1 recon-2 sin (2θ) sin2(2θ) nor- simpler Fig. 7. (Left) Allowed regions (at 90% and 99% CL) for ν ν oscillations from e → µ the LSND experiment compared with the exclusion regions (at 90% CL) from KAR- MEN2 and other experiments (from Ref. [101]). (Right) ν ν excluded regions µ → e from MiniBooNE compared to the LSND allowed region (from Ref.[110]). 39 Reactor antineutrino anomaly: New computation of the reactor antineutrino spectra ⇒ increase of the flux by about 3% ⇒ deficit of antineutrinos in SBL reactor experiments mean observed to predicted rate 0.943 ± 0.023 10 1.2 II I III G. Mention et al. et Mention G. − 1.15 − − 2S − II III I − − − 3 3 4 − − − II I − 1.1 − Krasnoyarsk SRP Krasnoyarsk Goesgen Krasnoyarsk Goesgen PaloVerde CHOOZ Bugey ROVNO88 Bugey Bugey Goesgen SRP ROVNO91 3 1.05 − Bugey 1 pred,new ) EXP 0.95 /(N 0.9 OBS ILL N 0.85 1S 3S 2I 1I − − − − 0.8 0.75 ROVNO88 ROVNO88 ROVNO88 ROVNO88 1 2 3 10 10 10 Distance to Reactor (m) FIG. 5. Illustration of the short baseline reactor antineutrino anomaly. The experimental results are compared to the prediction without oscillation, taking into account the new antineutrino spectra, the corrections of the neutron mean lifetime, and the off-equilibrium effects. Published experimental errors and antineutrino spectra errors are added in quadrature. The mean averaged ratio including possible correlations is 0.943 ± 0.023. The red line shows a possible 3 active neutrino mixing solution, 2 2 2 with sin (2θ13)=0.06. The blue line displays a solution including a new neutrinomassstate,suchas|∆mnew,R| ! 1eV and 2 sin (2θnew,R)=0.12 (for illustration purpose only). ting ∼ 1 MeV electron neutrinos. [57], following the 2 2 2 Experiment(s) sin (2θnew) |∆mnew| (eV ) C.L. (%) methodology developed in Ref. [56, 58]. However we ∗ decided to include possible correlations between these Reactors (no ILL-S,R ) 0.02-0.20 > 0.40 96.5 Gallium (G) > 0.06 > 0.13 96.1 four measurements in this present work. Details are MiniBooNE (M) — — 72.4 given in Appendix B. This has the effect of being ILL-S — — 68.1 slightly more conservative, with the no-oscillation hy- R∗ +G 0.05-0.22 > 1.45 99.7 pothesis disfavored at 97.7% C.L., instead of 98% C.L. R∗ +M 0.04-0.20 > 1.45 97.6 ∗ in Ref. [56]. Gallex and Sage observed an average deficit R + ILL-S 0.02-0.21 > 0.23 95.3 of RG =0.86 ± 0.06 (1σ). Considering the hypothesis of All 0.06-0.22 > 1.5 99.8 νe disappearance caused by short baseline oscillations we 2 used Eq. (13), neglecting the ∆m31 driven oscillations TABLE III. Best fit parameter intervals or limits at 95% C.L. 2 2 because of the very short baselines of order 1 meter. Fit- for sin (2θnew)and|∆mnew| parameters, and significance of ting the data leads to |∆m2 | > 0.3eV2 (95%) and the sterile neutrino oscillation hypothesis in %, for different new,G ∗ 2 combinations of the reactor experimental rates only (R ), the sin (2θnew,G) ∼ 0.26. Combining the reactor antineu- trino anomaly with the gallium anomaly gives a good fit ILL-energy spectrum information (ILL-S), the gallium experi- ments (G), and MiniBooNE-ν (M) re-analysis of Ref. [56]. We to the data and disfavors the no-oscillation hypothesis at 2 2 2 quantify the difference between the sin (2θnew) constraints 99.7% C.L. Allowed regions in the sin (2θnew) − ∆mnew obtained from the reactor and gallium results. Following pre- plane are displayed in Figure 6 (left). The associated scription of Ref. [77], the parameter goodness-of-fit is 27.0%, 2 2 best-fit parameters are |∆mnew,R&G| > 1.5eV (95%) indicating reasonable agreement between the neutrino and an- 2 and sin (2θnew,R&G) ∼ 0.12. tineutrino data sets (see Appendix B). We then reanalyzed the MiniBooNE electron neutrino excess assuming the very short baseline neutrino os- cillation explanation of Ref. [56]. Details of our re- production of the latter analysis are provided in Ap- analysis leads to a good fit with the sterile neutrino 2 2 pendix B. The best fit values are |∆mnew,MB| =1.9eV hypothesis and disfavors the absence of oscillations at 2 and sin (2θnew,MB) ∼ 0.2, but are not significant at 98.5% C.L., dominated by the reactor experiments data. 2 2 95% C.L. The no-oscillation hypothesis is only disfa- Allowed regions in the sin (2θnew) − ∆mnew plane are vored at the level of 72.4% C.L., less significant than displayed in Figure 6 (right). The associated best- 2 2 the reactor and gallium anomalies. Combining the re- fit parameters are |∆mnew,R&MB| > 0.4eV (95%) and 2 actor antineutrino anomaly with our MiniBooNE re- sin (2θnew,R&MB) ∼ 0.1. Gallex-SAGE calibration experiments: Calibration of the Gallex and SAGE experiments with radioactive sources ⇒ observed deficit of ⌫ e with respect to predictions R = 0.86 ± 0.05 [tension with ⌫ e - Carbon cross-section measurements at LSND and KARMEN, 1106.5552] Combined analysis of SBL reactor data, gallium calibration experiments and MiniBooNE neutrino data [G. Mention et al.]: m2 > 1.5eV2 , sin2 2✓ =0.14 0.08 (95% C.L.) | SBL| ee ± However, no coherent picture of the data with an additional (or even 2) sterile neutrinos (even if the global fit has improved with the new reactor antineutrino flux): 1) tension between appearance (LSND/MiniBooNE antineutrino data) and disappearance experiments (reactors, ⌫ µ disappearence experiments) 2 2 2 m41L Reactors: P⌫¯ ⌫¯ 1 sin 2✓ee sin e! e ' 4E ✓ ◆ require relatively small sin2 2✓ 4(1 U 2) U 2 4 U 2 ee ⌘ | e4| | e4| ' | e4| (using info from solar neutrino data) 2 2 2 m41L CDHS: P⌫ ⌫ 1 sin 2✓µµ sin µ! µ ' 4E ✓ ◆ require relatively small sin2 2✓ 4(1 U 2) U 2 4 U 2 µµ ⌘ | µ4| | µ4| ' | µ4| (using info from atm. neutrino data) Other implications of sterile neutrinos Tritium beta decay: 3 3 3 3 H H + e +¯ ⌫ E0 = m H m He ! e e The electron energy spectrum is given by: dN 2 2 = R(Ee) (E0 Ee) m⌫ Ee = E0 E⌫ dEe p Effect of the non-vanishing neutrino mass: Emax = E E m e 0 ! 0 ⌫ ⇒ distorsion of the Ee spectrum close to the endpoint Present bound (Troitsk/Mainz): m⌫e < 2.2 eV (95% C.L.) KATRIN will reach a sensitivity of about 0.3 eV In pratice, there is no electron neutrino mass, but 3 (or more) strongly mixed mass eigenstates, and dN 2 2 2 = R(Ee) Uei (E0 Ee) mi ⇥(E0 Ee mi) dEe | | i q If all mi are smaller thanX the energy resolution, this can be rewritten as: dN 2 2 2 2 2 = R(Ee) (E0 Ee) m m mi Uei dEe ⌘ | | q i If there is an eV-scale sterile neutrino (comparable to Xthe energy resolution of KATRIN), its mass may be resolved (but difficult measurement):8.3. -decay 101 1 dN 2 2 2 =(1 U ) (E E ) m 2 e4 0 e / 1 R(E ) dE | | ) e e e q dE / 2 2 2 counts + U (E E ) m ⇥(E E m ) dN 2ν2β 0ν2β | e4| 0 e 4 0 e 4 ( q H L (also: upper bound on m4 from beta decay) Q − mν Q − mν Q 0 Q Ee Total energy in electrons Figure 8.2: Fig. 8.2a: -decay spectrum close to end-point for a massless (dotted) and massive (continuous line) neutrino. Fig. 8.2b: 2⇧2 and 0⇧2 spectra. needed to plan a -decay experiment able of reaching the neutrino mass scale suggested by oscillation data. In line of principle, a -decay experiment is sensitive to neutrino masses mi and mixings Vei : | | dN e = V 2F (E )(Q E ) (Q E )2 m2. (8.3) dE | ei| e e e i e i ⇥ This is illustrated in fig. 8.2a, where we show the combined e ect of a Heavier neutrino with little e component and of a Lighter neutrino with sizable e component. Following Kurie we plotted the square root of dNe/dEe, that in absence of neutrino masses is a linear function close to the end-point, assumed to be known with negligible error. In fig. 8.3 we show the predicted reduction of the -decay rate around its end-point. The various curves are for di erent values of the lightest neutrino mass. In practice the energy resolution is limited, and only broad features can be seen. If it is not possible to resolve the di erence between neutrino masses, it is useful to approximate eq. (8.3) with (8.1) and present the experimental bound in terms of the single e ective parameter m2 V 2 m2 = cos2 ⇤ (m2 cos2 ⇤ + m2 sin2 ⇤ )+m2 sin2 ⇤ . (8.4) ⇥e ⇤ | ei| i 13 1 12 2 12 3 13 i The last equality holds in the standard three-neutrino case. The expected ranges of m⇥e are reported in table 8.1 in the limiting case where the lightest neutrino is massless. From this it is immediate to obtain the ranges corresponding to the generic case of a non vanishing lightest 2 neutrino mass mlightest: as clear from the definition m⇥e (m m†)ee or from the more explicit 2 ⇤2 · expression in eq. (8.4) one just needs to add mlightest to m⇥e . The resulting bands at 99% CL are plotted in fig. 8.5b. Searches for ⇧ and ⇧ masses have been performed by studying decays like ⌃ µ⇧¯ . The µ ⇤ ⇧ µ resulting bounds, m⇥µ,⇥ < MeV are very loose. Notice that -decay experiments probe anti- neutrinos. If one does not⌅ trust CPT and allows neutrinos and anti-neutrinos to have di erent masses, the looser bound m⇥e < 200 eV applies to neutrinos. Finally, [82] explores the futuristic possibility of studying atomic decays into ⇧⇧⇥¯ , which would be convenient since atomic energy di erences are comparable to neutrino masses, allowing Neutrinoless double beta decay: (A, Z) (A, Z + 2) + e + e ! Possible if lepton number violated (Majorana neutrinos), in nuclei where the single beta decay is forbidden Sensitive to the effective mass parameter: 106 Chapter 8. Non-oscillation experiments 2 m miU possible cancellations in the sum (phases in U) ⌘ ei i 1 X 1 3 disfavoured by 0 2⌥ disfavoured by cosmology Bound from MAINZ and TROITSK 1 10⇤1 Sensitivity of KATRIN 0.3 2 0.3 ⌃m23 ⌅ 0 eV 3-neutrino case eV eV in in ⇤2 ⇤ in 10 0.1 ee e ⌥ cosmo (Strumia, Vissani) m m 2 ⇤ m 2 ⌃m23 ⌅ 0 2 cosmology 0.1 ⌃m23 ⌅ 0 cosmology ⌃m ⇧ 0 0.03 23 by by ⇤3 2 10 ⌃m23 ⇧ 0 0.01 2 ⌃m23 ⇧ 0 ⇥ ⇥ disfavoured 99 CL 1 dof disfavoured 99 CL 1 dof 99⇥ CL 1 dof 0.03 0.003 10⇤4 10⇤3 0.01 0.1 1 10⇤3 0.01 0.1 1 10⇤4 10⇤3 10⇤2 10⇤1 1 ⇥ ⇥ lightest neutrino mass in eV lightest ⇥ neutrino mass in eV lightest neutrino mass in eV Figure 8.5: 99% CL expected ranges as function of the lightest neutrino mass for the parameters: 1/2 mcosmo = m1 + m2 + m3 probed by cosmology (fig. 8.5a), m e (m m†)ee probed by -decay 2 ⇥ · (fig. 8.5b), mee probed by 0⇤2 (fig. 8.5c). m23 > 0 corresponds to normal hierarchy (mlightest = | 2 | m1) and m23 < 0 corresponds to inverted hierarchy (mlightest = m3), see fig. 2.4. The darker regions show how the ranges would shrink if the present best-fit values of oscillation parameters were confirmed with negligible error. Many other experiments and proposals are based on (various combinations of) these concepts and other important considerations (background control, isotopic enrichment, double tag, etc.). The so called “pulse shape discrimination” is a good example of how the background can be reduced in 76Ge detectors; in the terminology above, it might be classified as a rough “electron tracking”. In 0⇤2 events the energy is deposited by two electrons in a single point. Background from ⇥ radiation deposits monochromatic energy in the crystal, producing a line in the energy spectrum, at energies that can be dangerously close to the 0⇤2 line. However, ⇥ tend to manifest as multi-site events, making a few Compton scatterings, until their energy is so low that ⇥ get photoelectrically absorbed. The electric pulse from charge collection of multi-site events has on average a di⇥erent time structure from single-site events: the HM collaboration [17] tried to exploit this di⇥erence to suppress the background by a factor (2) (IGEX also employs the same technique). O If a signal is seen, measuring the energy and/or angular distributions of the events (as say in NEMO3) and/or related modes of decay such as electron capture or double positron emission (say with a setup as in COBRA) would allow to test if 0⇤2 is due to neutrino masses or to some other speculative source, such as new gauge interactions among right-handed fermions. 2 i An additional sterile neutrino will contribute m 4 U e 4 e to the effective 2 | | mass m i m i U ei ; depending on the active neutrino parameters it may dominate⌘ or lead to cancellations P 1+3, Normal, SN 1+3, Inverted, SI 0 10 3 ν (best-fit) 3 ν (best-fit) 3 ν (2σ) 3 ν (2σ) 1+3 ν (best-fit) 1+3 ν (best-fit) 1+3 ν (2σ) 1+3 ν (2σ) -1 10 Barry, Rodejohann, > (eV) ee Zhang, 1105.3911 -3 10 0.001 0.01 0.1 0.001 0.01 0.1 mlight (eV) 3+1, Normal, NS 3+1, Inverted, IS 0 TABLE using10 I: Best-fitthe fit (from of Kopp, Ref. [7]) Maltoni and estimated and 2Schwetz:σ values of the sterile neutrino parameters. 2 2 2 2 3 ν (best-fit)parameter ∆m41 [eV] 3 ν (best-fit)Ue4 ∆m51 [eV] Ue5 3 ν (2σ) 3 ν |(2σ) | | | -1 3+1 ν (best-fit) best-fit 1.783+1 0.023ν (best-fit) 10 3+1/1+33+1 ν (2σ) 3+1 ν (2σ) 2σ 1.61–2.01 0.006–0.040 > (eV) ee best-fit 0.47 0.016 0.87 0.019 3+2/2+3 -3 10 0.001 0.01 0.1 0.001 0.01 0.1 the 3+2, 2+3 and 1+3+1 cases to have 2σ uncertainties of the same relative magnitude as mlight (eV) those in the 3+1/1+3 scenario. The data favor the presence of two sterile neutrinos, mostly FIG. 1: The allowed ranges in the m m parameter space, both in the standard three- because they allow different! ee neutrino"− light and anti-neutrino probabilities, thus alleviating the neutrino picture (unshaded regions) and with one sterile neutrino (shaded regions), for the 1+3 (top)tension and 3+1 between (bottom) the cases. LSND and MiniBooNE results. As mentioned above, 1+3+1 scenarios have a slightly better fit than 3+2/2+3 cases. holds, whereWe notem in addition= c2 c2 m that+s the2 c2 recentm eiα + resultss2 m ei fromβ is the the standard T2K [29] expression and MIN forOS three [30] experiments ! ee"3ν 12 13 1 12 13 2 13 3 activestrengthen neutrinos. the The existing upper panel hints of [31–33] Fig. 1 for displays non-zero the allowedθ , and rang thateof themee analysis(1+3)ν as in a Ref. [34] finds 13 ! " function of the lightest mass m , using data from Refs. [7, 33]. Also shown in this plot evidence for θ13 > 0 at thelight level of > 3σ. It is not yet evident whether these new data will and the following ones is the allowed range of m , i.e. the standard case. Its crucial improve or worsen the fits in the various|! sterileee"3ν| neutrino scenarios; on the other hand it can features (see [36] and references therein) are that m can vanish exactly in the normal also be argued that the T2K result is not|! ee due"3ν| to θ but is actually another signature of hierarchy case, but cannot vanish in neither the inverted hierarch13y nor the quasi-degenerate case.sterile This neutrinos standard behavior [35]. can be completely mixed up by the presence of one or more sterile neutrinos: if the lightest neutrino mass is reasonably small, e.g. mlight < 0.01 eV, the B. Neutrino-less double beta decay 7 Neutrino-less double beta decay (0νββ) is the only realistic test of lepton number vio- lation. While there are several mechanisms to mediate the process (e.g. heavy neutrinos, right-handed currents or SUSY particles), light Majorana neutrino exchange is presumably the best motivated scenario. We will work in this standard interpretation of 0νββ, and study the effect of massive sterile neutrinos in the eV range. This section is an update of the results in Refs. [12, 13]; we also use the best-fit values in Table I to report approximate numerical lower limits for the complementary mass observables m U 2m2 and m , β ≡ | ei| i i constrained by beta decay and cosmology, respectively. ! " In the presence of one sterile neutrino, the effective neutrino mass in 0νββ is given by m = c2 c2 c2 m + s2 c2 c2 m eiα + s2 c2 m eiβ + s2 m eiγ , (6) " ee#4ν 12 13 14 1 12 13 14 2 13 14 3 14 4 # # using the parameterization# in Eq. (2). If the sterile neutrino is heavier than the# active ones, the approximation m c2 m + s2 ∆m2 eiγ (7) " ee#(1+3)ν $ 14" ee#3ν 14 41 # # # $ # # # # # 6