Sterile neutrino as a pseudo-Goldstone fermion Stphane Lavignac - PowerPoint PPT Presentation

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 in short-baseline reactor ν e experiments) could be due to oscillations into sterile neutrinos 1.2 1.15 Krasnoyarsk − III Krasnoyarsk − II ROVNO88 − 2S Krasnoyarsk − I Goesgen − III Goesgen − II (arXiv:1101.2755) Goesgen − I ROVNO91 PaloVerde CHOOZ Bugey − 3 Bugey − 4 Bugey − 3 G. Mention et al. SRP − II SRP − I 1.1 Bugey − 3 1.05 N OBS /(N EXP ) pred,new 1 0.95 ILL 0.9 0.85 ROVNO88 − 1S ROVNO88 − 3S ROVNO88 − 2I ROVNO88 − 1I 0.8 0.75 1 2 3 10 10 10 Distance to Reactor (m) - measurement of CMB anisotropies and other cosmological data are consistent with extra light degrees of freedom [WMAP 7yr + BAO + H 0 ] N e ff = 4 . 34 +0 . 86 (68% C.L.) − 0 . 88 [WMAP 9yr + eCMB + BAO + H 0 ] N e ff = 3 . 84 ± 0 . 40 (68% C.L.)

Recent Planck data leave less room for a sterile neutrino. One usually quotes: N e ff = 3 . 30 +0 . 54 (95% C.L.) [Planck + WMAP + highL + BAO] − 0 . 51 Planck +WP+highL 1.0 However the constraint strongly +BAO arXiv:1303.5076 + H 0 0.8 depends on the set of data used: +BAO+ H 0 P / P max 0.6 N e ff = 3 . 52 +0 . 48 (95% C.L.) − 0 . 45 0.4 [Planck + WMAP + highL + BAO + H 0 ] 0.2 0.0 2.4 3.0 3.6 4.2 N e ff Assuming a fully thermalized massive sterile neutrino, the constraint becomes: [Planck + WMAP + highL] N e ff < 3 . 91 , m ν s < 0 . 59 eV (95% C.L.) [Planck + WMAP + highL + BAO] N e ff < 3 . 80 , m ν s < 0 . 42 eV (95% C.L.)

These bounds are in tension with the sterile neutrino interpretation of the neutrino anomalies, which require m ν s ∼ 1 eV [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] : sin 2 2 θ ee = 0 . 14 ± 0 . 08 SBL | > 1 . 5 eV 2 , | ∆ m 2 (95% C.L.) 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: ✓ 0 ◆ ✓ ν c − 1 ◆ − m ν L ν R − 1 m 2 M ν T ν c R 2 ( ¯ ν L ¯ L ) + h.c. R C ν R + h.c. = m M ν R the mass eigenstates and are admixtures of the active neutrino ν L 1 ν L 2 ν L and of the sterile neutrino ν c L ≡ C ν T R However, for m << M, the sterile neutrino is very heavy and has ν c L ' ν L 2 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 >> M SUSY The supersymmetric effective field theory below f involves a (pseudo-)Goldstone multiplet A = s + ia √ 2 θχ + θ 2 F + √ 2 with a shift symmetry A → A + i α 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] 1 Z from ( A + A † ) 2 ( X + X † ) , d 4 θ h X i = F θ 2 m χ ∼ m 3 / 2 M P ⇒ low-scale supersymmetry breaking needed

Assuming no R-parity (but baryon number), the most general Lagrangian compatible with the shift symmetry is ( α = 0, 1, 2, 3): A → A + i α f W = µ α H u L α + 1 k − λ u αβ k L α L β e k + λ d 2 λ e α jk L α Q j d jk H u Q j u k A + A † K = 1 2 ( A + A † ) 2 + H † u H u + L α † L α + C u H † u H u f + C αβ L α † L β A + A † C u α H u L α A + A † ✓ ◆ + + h.c. + · · · f f V soft = generic MSSM soft terms with leptonic RPV After minimization of the scalar potential, H u , L α get vevs (assume <A> = 0) ⇒ non-canonical kinetic terms for H u and L α → redefine H u and L α = (H d , L i ) such that (i) the charged fields have canonical kinetic terms H + u , H − d , e − i (ii) the sneutrino vevs vanish h ˜ ν i i (iii) real λ e 0 jk = λ e j δ jk , λ e j

Neutralino and chargino mass matrices As a consequence of the bilinear RPV terms ( µ i H u L i ), leptons mix with charginos and neutralinos. d , ν i , A H 0 u , H 0 Furthermore, since the kinetic terms of the neutral fields are not canonical, the neutralino mass matrix receives contribution from the Kähler potential and mixes χ with the standard neutrinos and neutralinos Charginos: 0 1 0 1 f W + M 2 gv u 0 1 × 3 ⇣ ⌘ @ A @ A f e e H − e − gv d µ 0 1 × 3 H + W − i d u λ e i v d δ ij e + µ i 0 3 × 1 k 2 heavy mass eigenstates (charginos) m i = λ e 3 light mass eigenstates (charged leptons) with masses i v d chargino / charged lepton mixing suppressed by or smaller µ i /µ

Neutralinos: ( f W 3 , e B, e u , e The neutralino mass matrix, written in the basis H 0 H 0 d , ν i , χ ) is a 8x8 matrix with a seesaw structure   M 4 × 4 µ 4 × 4 M N = m, µ ⌧ M   µ T m 4 × 4 4 × 4 hence the neutrino masses and mixing are given by the diagonalization of the 4x4 effective neutrino mass matrix 0 ⇣ ⌘ 1 µ j A µ i B µ i v µ + D i M ν = m − µ T M − 1 µ = µ µ f @ ⇣ ⌘ A B µ j C v 2 v µ + D j f 2 + m χ f Z cos 2 β A = µ 2 M 11 M 2 where depends only on MSSM parameters, while det M B, D i and C depend also on the Kähler parameters C ud , C u , C dd , C ı | (Rp-conserving) and (RPV). Assuming the former are of order 1 , C uı , C dı one has B, C = O ( µ )

One gets a consistent neutrino phenomenology by assuming that all RPV parameters ( ) are small, while the Rp-conserving Kähler µ i , C ui , C di parameters are of order 1 µ i v In practice, need µ . 10 − 5 , C di . 10 − 6 , f . 10 − 6 , m χ . 1 eV Can rewrite the 4x4 neutrino mass matrix in a more compact form: ✓ D ✏ α ✏ β ◆ E ⌘ α X X ✏ 2 ⌘ 2 M ν = α = α = 1 E ⌘ β F α α where we have renamed the indices i, j → α , β = e, µ, τ This structure implies: 1) the matrix has rank 3, so m 1 = 0 U α 4 ' E 2) the active-sterile mixing is given by ( m 4 ' F ) F η α 3) at order 1 in the active-sterile mixing, the active neutrino parameters are given by the matrix ( m ν ) αβ = D ✏ α ✏ β − E 2 F ⌘ α ⌘ β

3 m i U α i U β i = D ✏ α ✏ β − E 2 X ( m ν ) αβ = F ⌘ α ⌘ β i =2 Since we know the active neutrino parameters (neglecting CPV), we can ‟ reconstruct” the sterile neutrino parameters using [when the first term dominates] m 2 ' � E 2 ⇣ 2 , ~ m 3 ' D F U α 2 ' � ( ⇠ µ ⇠ τ , ⇠ e ⇠ τ , ⇠ e ⇠ µ ) /  , U α 3 ' ( ✏ e , ✏ µ , ✏ τ ) ⇣ 2 ✏ β ✏ γ ⇠ α ≡ ⇣ β ⇣ γ + ~ ~ where ( ↵ , � , � all di ff erent) ⇣ ≡ ~ ✏ × ~ ⌘ | ξ µ ξ τ | 2 + | ξ e ξ τ | 2 + | ξ e ξ µ | 2 � 1 / 4 and � κ ≡  / ~ In the reconstruction process, one obtains the as a function of , ζ 2 ⇣ 2 α which is the solution of a polynomial of degree 4. Among the solutions, ⇣ 2 ≤ 1 ~ only the ones that satisfy the constraint (if any) are acceptable E 2 Finally, one identifies ⇒ correlations F η α η β ≡ m 4 U α 4 U β 4

Numerical results Normal hierarchy, case 1 D ✏ 2 α ⌧ ( E 2 /F ) ⌘ 2 α 0.28 0.28 0.26 0.26 m 4 H eV L 0.24 m 4 H eV L 0.24 0.22 0.22 0.20 0.20 0.18 0.18 3 4 5 6 7 90 100 110 120 130 140 2 x 10 3 s 14 2 x 10 3 s 34 0.28 0.0180 0.26 0.0175 m 4 H eV L 0.24 m Β H eV L 0.22 0.0170 0.20 0.0165 0.18 60 70 80 90 100 0.18 0.20 0.22 0.24 0.26 0.28 2 x 10 3 s 24 m 4 H eV L

Not relevant to the reactor anomaly Not relevant to LSND / MiniBooNE 10 2 10 1 10 - 1 10 - 2 10 - 3 10 - 2 10 - 1 1