Non-Standard Neutrinoless Double Beta Decay and its Implications Lukas Graf University College London ISS, 6th September 2017, Prague
Introduction and Motivation • neutrinos - neutral, left-handed, massive, light . . . • = ⇒ problem of the Standard Model (SM) • Dirac or Majorana nature? • Majorana masses ⇐ ⇒ LNV ⇐ ⇒ neutrinoless double beta decay ( 0 νββ ) • massive right-handed neutrinos (seesaw mechanism) = ⇒ leptogenesis Lukas Graf Non-Standard Neutrinoless Double Beta Decay 2 / 23
Neutrinoless Double Beta Decay > 2 . 1 × 10 25 y (GERDA) 76 Ge • current limit: T 1 / 2 > 1 . 07 × 10 26 y (KamLAND-Zen) 136 Xe T 1 / 2 • future experimental sensitivity: T 1 / 2 ∼ 6 . 6 × 10 27 y (nEXO) Lukas Graf Non-Standard Neutrinoless Double Beta Decay 3 / 23
Neutrinoless Double Beta Decay • L 0 νββ = L LR + L SR , general Lagrangian in terms of effective couplings ǫ corresponding to the pointlike vertices at the Fermi scale • F. F. Deppisch, M. Hirsch, H. P¨ as: Neutrinoless Double Beta Decay and Physics Beyond the Standard Model , J. Phys. G 39 (2012), 124007 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 4 / 23
General Lagrangian for 0 νββ • long-range part: L LR = G F � � ˜ J † V − Aµ j µ α,β ǫ β α J † , √ V − A + � α j β 2 where J † α = ¯ uO α d , j β = ¯ e O β ν and O V ± A = γ µ (1 ± γ 5 ) , O S ± P = (1 ± γ 5 ) , O T R,L = i 2 [ γ µ , γ ν ](1 ± γ 5 ) • short range part: L SR = G 2 [ ǫ 1 JJj + ǫ 2 J µν J µν j + ǫ 3 J µ J µ j + ǫ 4 J µ J µν j ν + ǫ 5 J µ Jj µ ] , F 2 m p where J = ¯ u (1 ± γ 5 ) d , J µ = ¯ uγ µ (1 ± γ 5 ) d , J µν = ¯ u i 2 [ γ µ , γ ν ](1 ± γ 5 ) d e (1 ± γ 5 ) e C j = ¯ j µ = ¯ eγ µ (1 ± γ 5 ) e C Lukas Graf Non-Standard Neutrinoless Double Beta Decay 5 / 23
General Lagrangian for 0 νββ 1 / 2 = | ǫ β • connection to the experimental half-life: T − 1 α | 2 G i | M i | 2 • = ⇒ 0 νββ half-life sets constraints on effective couplings • accurate calculation of nuclear matrix elements (NMEs) and phase-space factors (PSFs) is crucial for this estimation Lukas Graf Non-Standard Neutrinoless Double Beta Decay 6 / 23
Nuclear Matrix Elements and Phase-Space Factors for 0 νββ Decay • goal: a thorough theoretical description of non-standard 0 νββ decay mechanisms - involves NMEs and PSFs → a very complex, interdisciplinary project • understanding the nuclear and atomic parts of the process • older literature may cause a confusion (notations, mistakes, lack of explanation), but long-range part recently rigorously covered (checked) • = ⇒ similar analysis of the short-range part = ⇒ complete, consistent and cross-checked description of all contributions • application of the nuclear physics model (IBM2, maybe more), numerical calculation of NMEs • numerical computation of relevant PSFs Lukas Graf Non-Standard Neutrinoless Double Beta Decay 7 / 23
Nuclear Matrix Elements and Phase-Space Factors for 0 νββ Decay - Approximations • complicated calculation - a number of approximations used (nucleon current approximation, non-relativistic approximation, closure approximation) � P • considering nucleon isodoublet N = � , the nucleon matrix N elements of the quark currents are � N ( p ′ ) � � N ( p ) τ + � F (3) ( q 2 ) ± F (3) � ¯ ( q 2 ) γ 5 � P ( p ) | ¯ u (1 ± γ 5 ) d = N ( n ) , � S P � � N ( p ′ ) � N ( p ) τ + � F (3) ( q 2 ) γ µ − iF (3) W ( q 2 ) σ µν q ν � uγ µ (1 ± γ 5 ) d ¯ � P ( p ) | ¯ = N ( n ) � V N ( p ) τ + � F (3) ( q 2 ) γ µ γ 5 − F (3) ( q 2 ) γ 5 q µ � ± ¯ N ( n ) , A P � i � J µν ± uσ µν (1 ± γ 5 ) d � � N ( p ′ ) � N ( p ) τ + ε µνρσ J ρσ ¯ � P ( p ) | ¯ = N ( n ) , � 2 where we have defined: i 1 J µν = T (3) ( q 2 ) σ µν + T (3) ( γ µ q ν − γ ν q µ ) + T (3) ( σ µρ q ρ q ν − σ νρ q ρ q µ ) . ( q 2 ) ( q 2 ) q 2 3 m 2 m p p Lukas Graf Non-Standard Neutrinoless Double Beta Decay 8 / 23
Nuclear Matrix Elements and Phase-Space Factors for 0 νββ Decay - Approximations • non-relativistic limit then gives the resulting approximated nuclear bilinears � � 1 F (3) ± F (3) � τ a J S ± P = + δ ( x − r a ) ( σ a · q ) , S P S 2 m p a � � � �� F A F P J µ � τ a g µ 0 q 0 Q · σ a = + δ ( x − r a ) F V I a ± σ a · Q − V ± A 2 m p F A a � � � � � �� F V F W + g µi ∓ F A ( σ a ) i − Q I a − 1 − 2 m p i σ a × q , 2 m p F V i ( g µi g ν 0 − g µ 0 g νi ) T i J µν τ a + δ ( x − r a ) T (3) � a + g µj g νk ε ijk σ ai � = T ± T 5 1 a i � ε µνρσ ( g µi g ν 0 − g µ 0 g νi ) T ai + g µm g νn ε mni σ ai ± , 2 where we have defined: T (3) i T i q i I a + ( σ a × Q ) i 2 . a = 1 − 2 T (3) 2 m p 1 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 9 / 23
Nuclear Matrix Elements and Phase-Space Factors for 0 νββ Decay • reaction matrix element � G cos θ C � 2 2 n p 2 s ′ p 1 s ′ � � � R SR � ¯ 2 1 = √ d x d y ψ ( y ) O l 2 P c ψ ( x ) e e 0 ν 2 i =1 d k � (2 π ) 3 � F | J l 1 c 1 i ( y ) J l 2 c 2 i ( x ) | I � e i k · ( y − x ) , × • = ⇒ a bunch of matrix elements to be computed M GT = � H ( r 12 )( σ 1 · σ 2 ) � ( M GT ) − 1 g 2 χ F = A � H ( r 12 ) � V g 2 ( M GT ) − 1 � ˜ χ GT ˜ = H ( r 12 )( σ 1 · σ 2 ) � ( M GT ) − 1 g 2 A � ˜ χ F ˜ = V H ( r 12 ) � g 2 ( M GT ) − 1 �− r 12 H ′ ( r 12 )( σ 1 · σ 2 ) � χ ′ = GT ( M GT ) − 1 g 2 χ ′ A �− r 12 H ′ ( r 12 ) � = V F g 2 � � ( M GT ) − 1 �− r 12 H ′ ( r 12 ) χ ′ r 12 ) − 1 = ( σ 1 · � r 12 )( σ 2 · � 3 ( σ 1 · σ 2 ) � T χ ′ ( M GT ) − 1 g V 2 r +12 H ′ ( r 12 ) i ( σ 1 − σ 2 ) · ( � g A �− 1 = r 12 × � r +12 ) � P µ β M ′ g V g A � H ′′ ( r 12 )( σ 1 · σ 2 ) � = R 3 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 10 / 23
Nuclear Matrix Elements for 0 νββ Decay • T − 1 1 / 2 ≈ �R 0 ν � 2 • different nuclear physics models ... so far, quite different results ... • J. Engel, J. Men´ endez: Status and Future of Nuclear Matrix Elements for Neutrinoless Double-Beta Decay: A Review , arXiv: 1610.06548 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 11 / 23
LNV Effective Operators • alternatively: 0 νββ can be described using SM effective field theories with ∆ L = 2 • a long list of eff. operators, odd dimensions: 5 , 7 , 9 , 11 , . . . • A. de Gouvea, J. Jenkins: A Survey of Lepton Number Violation Via Effective Operators , Phys. Rev. D 77 (2008), 013008 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 12 / 23
LNV Effective Operators and L 0 νββ • correspondence between general 0 νββ decay Lagrangian and the set of ∆ L = 2 LNV effective operators Lukas Graf Non-Standard Neutrinoless Double Beta Decay 13 / 23
LNV Effective Operators & 0 νββ decay • there is a variety of operators of different dimensions contributing (directly) to 0 νββ decay (employing certain type of mechanism) • all the ∆ L = 2 LNV effective operators can be related by SM Feynman rules • = ⇒ all of them contribute to 0 νββ decay in all possible ways • if we know the relations, we can determine the dominant contribution of every operator to 0 νββ decay via each possible channel Lukas Graf Non-Standard Neutrinoless Double Beta Decay 14 / 23
LNV Effective Operators - Relations • example: a “web” of LNV effective operators of dimension 9 9 10 11,1 15 13 11,2 14,1 17 16 12,2 14,2 18 19 12,1 20 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 15 / 23
LNV Effective Operators - Example • let’s consider operator O 60 = L i d c ¯ u c H j ¯ u c ¯ e c ¯ Q j ¯ H i (dim 11) Lukas Graf Non-Standard Neutrinoless Double Beta Decay 16 / 23
LNV Effective Operators & 0 νββ • similar reduction can be done for each LNV effective operator • every operator can be related to all possible 0 νββ -decay-trigerring operators • automation - loop-closing algorithm, all possible contributions obtained, for some operators - quite a demanding computation • at the moment we are cross-checking the results, selecting the dominant ones, final results soon • resulting contributions - relations among operators’ scales and epsilons from L 0 νββ → use for further calculations Lukas Graf Non-Standard Neutrinoless Double Beta Decay 17 / 23
LNV Operators and 0 νββ - Illustration • contributions to 0 νββ decay generated by the LNV effective operators in terms of effective vertices, point-like at the nuclear Fermi level scale • if 0 νββ is observed, the scale of the underlying operator can be determined • m e ǫ o 5 = v 2 , G F ǫ o 7 v √ = 2Λ 3 Λ 5 2 7 • G 2 � � , v 2 F ǫ { o 9 ,o 11 } 1 = Λ 5 Λ 7 2 m p 9 11 Lukas Graf Non-Standard Neutrinoless Double Beta Decay 18 / 23
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