Minimal Neutrinoless Double Beta Decay Rates from Anarchy and Flavor Symmetries

Transcription

1 Minimal Neutrinoless Double Beta Decay Rates from Anarchy and When is Γ ββ0ν unobservably small? Theoretical Division, T-2 Los Alamos National Laboratory Institute for Nuclear Theory, Seattle

2 References Introduction Based directly on... Minimally Allowed ββ0ν Rates From Approximate Flavor Symmetries, arxiv: [hep-ph] October 2008, JJ Submitted for publication in PRD Minimally Allowed ββ0ν Rates Within an Anarchical Framework, arxiv: [hep-ph] August 2008 JJ Submitted for publication in PRD Based loosley on previous work in collaboration with André de Gouvêa (Northwestern University)

3 Outline Introduction 1 Introduction Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν 2 Unbroken U(1) f Symmetry Spurion Analysis 3 Formalism Polynomial Measures Interpreting Bounds

4 Neutrinos Are Unusual... Besides, their properties reveal physics beyond the SM Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν They are unique (compared with charged cousins) Neutrinos have tiny masses Neutrinos mix No conserved quantum numbers More experiments are needed to discover other important properties!

5 Majorana Particles Introduction Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Neutrinos are neutral and can thus be their own antiparticle. The Majorana Condition ψ = ηψ c Mass Terms L D = m D (e R e L + e L e R ) L M = 1 2 m M(ν c ν + νν c ) Particle/antiparticle distinctions are meaningless for Majorana fields Example e R L D e L = m D ν c L M ν = m M

6 Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Neutrinoless Double Beta Decay The standard, high sensitivity probe of LNV and Majorana neutrino masses u d W e U ei m i Γ 1/2 i U 2 ei U ei mi q 2 m 2 i Q 2 i U 2 eim i e W u d A L = 2 process that proceeds via Majorana neutrino exchange ββ0ν is only realistic probe of LNV Some (as yet) unknown high scale process Experiments typically set upper limits on m eff ee m ee Standard picture: small m eff ee = m ee can arise in the normal hierarchy via cancellations among mixing angles and phases. With new physics, other nontrivial cancellations are possible, but induce a feedback mechanism!

7 Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Majorana Neutrinos and Lepton Number Violation The Black Box and extended Black Box theorems W e e m i Black Box U ei U ei W Black Box Theorem Schechter & Valle, 1982 LNV operators yield Majorana neutrino mass Majorana neutrino mass yields effective LNV operators m eff ee = i (O i + m O i ee) Exact cancelations are possible but require fine tuning Extended Black Box Theorem: Assuming 3 ν mixing & arbitrary LNV, m ee = 0 is not consistent with oscillation data! Hirsch, Kovalenko & Schmidt 2006

8 Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Majorana Neutrinos and Lepton Number Violation The Black Box and extended Black Box theorems ν c W e e m i Black Box U ei U ei ν W + Black Box Theorem Schechter & Valle, 1982 LNV operators yield Majorana neutrino mass Majorana neutrino mass yields effective LNV operators m eff ee = i (O i + m O i ee) Exact cancelations are possible but require fine tuning Extended Black Box Theorem: Assuming 3 ν mixing & arbitrary LNV, m ee = 0 is not consistent with oscillation data! Hirsch, Kovalenko & Schmidt 2006

9 Small ββ0ν rates How small can it be? Introduction Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Only light ν contributions m ee = mee eff Parameter dependent Hierarchy dependent Probing inverted and quasidegenerate spectra Data allows m ee = 0 New physics contributions (physics scale dependence) Extended BBT: m ee = 0 mee eff = 0 implies m ee mee eff for small m ee I study limits on m ee with anarchy and flavor symmetries

10 Goal Introduction Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν

11 Disclaimer! How far can this be pushed? Introduction Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Framework: Three light Majorana neutrinos that transform under an Abelian flavor symmetry What can go wrong? m ee = 0 at high scale Loop RG effects (investigating) Other flavor symmetry structures Non-Abelian and discrete Abelian groups Accounted for Discrete Non-Abelian flavor symmetries Not accounted for New physics alters m ee and mee eff relationship Heavy (> 5 TeV) Decouples... No effect de Gouvêa & JJ, PRD77 (2008) Intermediate (100 GeV 5 TeV) m ee mee eff ; small m ee? Light (< 100 GeV) Interpretation spoiled

12 Neutrinoless Double β-decay (ββ0ν) Black Box Theorems Vanishing Γ ββ0ν Example: ββ0ν With Light Sterile Neutrinos See Saw case: New light physics can contribute directly to ββ0ν! de Gouvea, Vasudevan & JJ, PRD75 (2007) New light degrees of freedom SeeSaw Mechanism ( ) 0 vy M flavor = vy T All m αβ = 0! m light ee m ee M R This may be observable by other means The observation or non-observation of ββ0ν reveals much about potential new physics!

13 Abelian Unbroken U(1) f Symmetry Spurion Analysis Exact m ee = 0 is stabilized by an appropriate flavor symmetry... Assume 3 light Majorana neutrino states charged under global U(1) f as n e, n µ and n τ provided to ν e, ν µ and ν τ respectively. Display these as 3-tuples (n e, n µ, n τ ) With this, the neutrino mass Lagrangian is L M 1 2 ( ) νe, c νµ, c ντ c a eeδ 2ne,0 a eµ δ ne+n µ,0 a eτ δ ne+n τ,0 a eµ δ ne+ν µ,0 a µµ δ 2nµ,0 a µτ δ nµ+n τ,0 a eτ δ ne+n τ,0 a µτ δ nµ+n τ,0 a ττ δ 2nτ,0 ν e ν µ ν τ Charged ν e forbids a ν c eν e mass term & implies vanishing Γ ββ0ν

14 Some things to consider Unbroken U(1) f Symmetry Spurion Analysis What happens to the charged leptons? Symmetry identifies flavor basis Left handed leptons l α transform like neutrinos Right handed leptons e R have different charges Important to consider these for model building...for my purposes one can assign e R charges to yield proper charged lepton mass matrix! What about extensions to non-abelian symmetries? Correlated quantum numbers defined by representations Reduced number of a αβ constants to tweak Therefore U(1) provides the most freedom for minimizing m ee

15 Texture Classes and Predictions Unbroken U(1) f Symmetry Spurion Analysis 11 discrete texture classes... None accommodate mixing data! Majorana ν + Oscillations = Γ ββ0ν 0

16 Symmetry Breaking The general idea Introduction Unbroken U(1) f Symmetry Spurion Analysis Expect that small m ee values are induced by associated small breakings of U(1) f symmetry Introduce new spurious scalar s with charge n s 1 Write all relevant Lagrangian terms to given order 2 Give s small VEV ɛ (breaks symmetry!) 3 Fit ɛ, M & a αβ parameters to oscillation data 4 Search for smallest allowed m ee 5 Extract other predictions 6 Do this for all nontrivial breaking patterns to given order Write structure as 4-tuple (n e, n µ, n τ ; n s )

17 Unbroken U(1) f Symmetry Spurion Analysis Building the perturbed mass matrix To leading order, the mass matrix is m αβ = M i=0 a (i) αβ ɛi δ (nα+n β ),in s At most one term survives Build the matrix to higher order O i is U(1) f invariant of order O(ɛ i ), then the operator O i+2 = sso i is also invariant of order O(ɛ i+2 ) In this analysis, I Work to ɛ 10 in m ee and ɛ 3 in other elements Set a ee = 1 (when present) Rescale ɛ ɛ gcd(p) This results in only 230 textures, divided into 11 classes!

18 Symmetry Breaking Some more details Introduction m αβ = M a αβ ɛ nα+n β ns Unbroken U(1) f Symmetry Spurion Analysis δ mod(nα +n β,n s ), a αβ provides parameter tuning Sets naturalness criterion Set a ee = 1 (rescaling freedom) Others fluctuate near 1 Parameterize range by C C = 0.5: order of magnitude! 10 C a αβ 10 C M provides mass scale M 1 ev ɛ yields texture structure ɛ < 2/3 smallest ɛ smallest m ee Fitting the mixing data requires a complex interplay between M, ɛ and a αβ!

19 m ee Distribution Introduction Unbroken U(1) f Symmetry Spurion Analysis 0th 1st 2nd 3rd (1, 1, 0; 2) (2, 1, 0; 2) (1, 2, 0; 2) (2, 1, 0; 1) (1, 1, 0; 1) (1, 2, 0; 1) (1, 3, 0; 2) (3, 1, 0; 2) (3, 2, 0; 2) (4, 5, 0; 1) (4, 1, 0; 1) (3, 1, 0; 1) (4, 3, 0; 1) (4, 1, 0; 1) (3, 1, 0; 1) (2, 1, 0; 1) (1, 4, 0; 1) (1, 3, 0; 1) (1, 4, 0; 1) (2, 5, 0; 1) (2, 3, 0; 2) (1, 2, 0; 1) (1, 3, 0; 1) (4, 5, 0; 1) (3, 4, 0; 1) (4, 3, 0; 1) (3, 5, 0; 2) (5, 3, 0; 2) (3, 2, 0; 1) (2, 3, 0; 1) (2, 4, 0; 1) (2, 5, 0; 1) (4, 3, 0; 2) (3, 2, 0; 1) (5, 2, 0; 1) (3, 4, 0; 2) (3, 5, 0; 1) (5, 3, 0; 1) (3, 7, 0; 2) (7, 3, 0; 2) (1, 5, 0; 2) (1, 3, 0; 2) (1, 7, 0; 2) (1, 5, 0; 2) (1, 6, 0; 2) (1, 4, 0; 2) (5, 1, 0; 2) (3, 1, 0; 2) (5, 3, 0; 2) (5, 1, 0; 2) (6, 1, 0; 2) (4, 1, 0; 2) Class C4 Example

20 Smallest m ee models Unbroken U(1) f Symmetry Spurion Analysis In each entry m ee is suppressed by ɛ 8! Class C4 example: ( ɛ 8 ) 1.42ɛ 3 0 M 1.42ɛ ɛ ɛ + O(ɛ 4 ), ɛ 0.50 where M = ev and ɛ = 0.37 Predictions: sin θ R ɛ 3, sin θ S ɛ and ms 2 /M2 ɛ 4 Taking allowed M and ms 2 implies ɛ The mass matrix resembles Class C1

21 Unbroken U(1) f Symmetry Spurion Analysis C = 0.5 Oscillation Parameter Scatter Plots

22 C = 0.5 Kinematic Scatter Plots Unbroken U(1) f Symmetry Spurion Analysis

23 C = 0.5 Kinematic Scatter Plots Unbroken U(1) f Symmetry Spurion Analysis

24 Formalism Polynomial Measures Interpreting Bounds Anarchical neutrino mass models Neutrino mass model s UV completions so complicated that the mass matrix appears random at low energies Hall, Murayama & Weiner PRL84 (2000) Neutrino parameters are well defined in theory but communicated as an ensemble of choices The flavor basis is just a random rotation from diagonal mass basis Must be treated statistically Must be basis independent Current large mixing angle data allowed (yields θ 13 bound) Haar Measure de Gouvea & Murayama PLB573 (2003) Weak mass hierarchy allowed Measure dependent Haba & Murayama PRD63 (2001) What about the m ee matrix element?

25 Conventions Introduction Formalism Polynomial Measures Interpreting Bounds Parameterize mass matrix as m αβ = ma αβ = mr αβ e iφ αβ r αβ is O(1) matrix and m constrained between 0.05 (hierarchical) and 1 ev (quasi-degenerate) m is added by hand and each a αβ is given by a dependent distribution function Given distribution function G(a αβ ) one can find probability of any matrix structure!

26 Conventions, continued Formalism Polynomial Measures Interpreting Bounds U(3) invariance implies G(a αβ ) and boundary conditions must be a function of and det(a) = ɛ ijk r 1i r 2j r 3k e i(φ 1i +φ 2j +φ 3k) tr(a a) = r r r r r r 2 23 Find marginalized probability distribution of radial magnitude a ee = r ee r, g(r) Integrate over all phases φ αβ Integrate over all other magnitudes r αβ Normalize to 1 0 g(r)rdr = 1.

27 Conventions, contunued... again Formalism Polynomial Measures Interpreting Bounds The marginalized distribution function is g(r) = l[tr(a a),det(a)] 0 [ ] G tr(a a), det(a) α β,β 1 r αβ dr αβ α β dφ αβ From this, find cummulative distribution function F(r) = r 0 g(r )r dr. C level confidence interval found by max [F(r), 1 F(r)] C = 0,

28 The analysis Introduction Formalism Polynomial Measures Interpreting Bounds Consider boundary condition l = Tr(a a) 3 2 r αβ live in 3-sphere The non-unit radius defines mass scale m All O(1) r αβ saturates upper bound. Consider G [ tr(a a), det(a) ] = p,q c pqtr(a a) p det(a) q Integration over nonzero powers of det(a) vanish G p htr(a i a), det(a) = Tr(a a) p = r 2 + r r r r r 2 p 13 " F p( r) = (6+p) r p 10 r p + 10 r p 5 r p + r # p 1 + p 120 r (6 + p)(5 + p)(4 + p)(3 + p)(2 + p)(1 + p) Good for all p 6 where r r/3

29 Analytic results Introduction Formalism Polynomial Measures Interpreting Bounds r 1, so truncate at quadratic order F p (r) = p p r 2 Leading to C confidence intervals bounded by 5 + p r 3 (1 ± C) 10(6 + p) for large p r (0.41, 1.31) at 90% confidence The n neutrino case: l = Tr(a a) n 2 2p + n r n (1 ± C) 2 + n 2 (n 1)(n + 2)(2p + n 2 + n)

30 Formalism Polynomial Measures Interpreting Bounds

31 What happenes as p 6? Formalism Polynomial Measures Interpreting Bounds lim p 6 F p ( r) r 2(p+6) 1 + (p + 6) ln r 2 g(r) δ(r), thus r e 1±C 4(6+p) The other coordinates in this limit behave as F p (r, s) r 2(p+6) (p + 6) r 2 F p (r) (1 ± C)/2, then F p (r, s) (1 ± C)/2 s 2 = F p(r) (p + 6) r 2 s 2. With one texture zero, the other matrix elements are unconstrained

32 Formalism Polynomial Measures Interpreting Bounds The smallest m ee bounds from Anarchy Naively, results suggest that anarchy allows arbitrarly small m ee This seems counter intuitive... With current data, m ee 0 forces a µ τ like symmetry which is a lot of accidental structure for a structureless matrix. Possible resolution from UV completion: The model classes select the weighting function. An anarchy guaranteed zero implies a mechanism selecting textures from the ensemble of UV completions. This is inconsistent with anarchy. It would require a corresponding symmetry mechanism and/or parameter fine tuning. With a parameter tuning tolerance of 10%, neutrino mass anarchy implies m ee ev at 90% confidence.

33 Summary Introduction Formalism Polynomial Measures Interpreting Bounds While exact Γ ββ0ν = 0 cannot be protected by flavor symmetries, small m ee values may be obtained from small breakings. I scan anarchial matrices and the U(1) f model space in search of the smallest m ee values Random fluctuations allow down to m ee ev Flavor symmetries yield m ee as small as ev Future measurements will increase these limits Neutrino mass hierarchy Oscillation parameters m ν and Σ May be evaded by new light states, unnatural couplings or discrete symmetries Combining information from ββ0ν, oscillation searches and other probes can reveal much about the nature of new physics.

34 Begin Backup Slides Introduction Formalism Polynomial Measures Interpreting Bounds Begin Backup Slides

35 Formalism Polynomial Measures Interpreting Bounds Giving Neutrinos Mass: Requires New Physics 1 Dirac Mass: M D (ν R ν L + ν L ν R ) Must add right handed neutrinos: ν R Normal Higgs mechanism: ν R ν L H 0 vν R ν L Requires very small Yukawa couplings: y Majorana Mass: M M (ν c ν + νν c ) Multiple ways to do this Violates global lepton number Only SM fields nonrenormalizable generation Couplings range over broad spectrum 3 Add both mass terms Just a subset of previous case Still yields LNV Majorana neutrinos See Saw Mechanisms

36 Implied neutrino mass textures Formalism Polynomial Measures Interpreting Bounds m ee = m 1 cos 2 θ 12 cos 2 θ 13 +m 2 sin 2 θ 12 cos 2 θ 13 e 2ıφ 2 +m 3 sin 2 θ 13 e 2ı(φ 3 δ) For the normal hierarchy, m 2 = m S, m 3 = m S + 2 A Solve for m 1 assuming small m ee, θ 13, and η = 2 S 2 A m 1 q m 2 12 = m (0) 1 q + δm(m ee, φ 2 ) / m sin 2 θ S cos 1/2 + cos 2 θs q mee θ2 13 2θ S cos 2θ S 2 η e2ı(φ 3 δ) sin 2 2θ S C + ı (2φ 2 π) 4 cos 1/2 A. 2θ S S In a similar way, expand m 2 and m 3. Now look at the magnitude of m αβ and compare with textures

37 Implied neutrino mass textures Analytic results Formalism Polynomial Measures Interpreting Bounds m 2 αβ 2 A = η ηθ 13 cos(2φ 3 δ) 1 0 A +.31η cos 2φ 3 1 A +.50θ A Zeroth order First order Second order 1 0 A +.19η(2φ 2 π) sin 2φ A