ν and B eyond-the -S tandard -M odel after lunch : ( ν in the Early U) Sacha Davidson, IN2P3/CNRS, France 1. ν in the SM 2. why BSM 3. to build a ν mass model 4. how to know which model ? − 0 ν 2 β − Lepton Flavour Violation 5. Non-Standard ν Interactions 6. (new light ν s ?) ν :Standard Member of particle bestiary. Invisible. Magical property of demonstrating BSM in the lab 1 / 45
Definitions and such... I use Dirac spinors, with 4 degrees of freedom(dof) labelled by {± E , ± s } , in chiral decomposition � ψ L � �� 0 � � �� I 0 σ i , { γ α } = ψ = , ψ R I 0 − σ i 0 � 0 � � 0 � � 1 � 1 − i 0 { σ i } = , , 1 0 i 0 0 − 1 P L = ( 1 − γ 5 ) ψ L = P L ψ , ψ R = P R ψ avec 2 s · ˆ chirality is not an observable ( → helicity = ± ˆ k = ± 1 / 2 in relativistic limit), but P L , R simple to calculate with :) notation : ( ψ R ) = ( P R ψ ) † γ 0 = ψ † P R γ 0 = ψ † γ 0 P L = ( ψ ) L ( ψ c ) L = P L ( − i γ 0 γ 2 γ 0 ψ ∗ ) = − i γ 0 γ 2 γ 0 ψ ∗ R 2 / 45
Summary : leptons in the Standard Model • 3 generations of lepton doublets, and charged singlets : �� ν eL � � ν µ L � � ν τ L �� ℓ α L ∈ , , e α R ∈ { e R , µ R , τ R } µ L τ L e L in charged lepton mass basis (greek index, eg α ). 3 / 45
Summary : leptons in the Standard Model • 3 generations of lepton doublets, and charged singlets : �� ν eL � � ν µ L � � ν τ L �� ℓ α L ∈ , , e α R ∈ { e R , µ R , τ R } µ L τ L e L in charged lepton mass basis (greek index, eg α ). • No ν R in SM because 1 . data did not require m ν when SM was defined ( ν are shy in the lab...) 2 . ν R an SU(2) singlet ⇔ no gauge interactions ⇒ not need ν R for anomaly cancellation ⇒ if its there, its hard to see • most general, renormalisable, SU ( 2 ) × U ( 1 ) -invariant L for those particles gives : Charged Current ν production no lepton flavour change Universal Z cpling to 3 ν ( Γ inv says 2 . 994 ± 0 . 012) 3 / 45
Neutrinos have gravitational interactions 1. expected from equivalence principle : carry 4-momentum 2. Big Bang Nucleosynthesis ( τ U ∼ few minutes) : • T ∼ MeV, baryons in n , p , combine into light nuclei • light element abundances depend on τ U ↔ expansion rate ↔ ρ rad ↔ N ν = # light ν in equilibrium • observed abundances today confirm N ν < ∼ 4 3. Cosmic Microwave Background : (is a fit to a multi-parameter model), and U is mat-dim at recombination. But sensitivity for similar reasons to # of relativistic species present... Lesgourgues reviews 4 / 45
Why Beyond the Standard Models (of part phys+ cosmo) ? The SM (of particle phys + cosmo) does not explain : 1. Dark Matter 2. the origin of low-multipole ∆ T / T in the CMB 3. the Baryon Asymmetry of the U 4. ν masses but ’tis Pandoras box ! What about adding/looking for : ◮ new short-range interactions for neutrinos/leptons(new heavy particles) ◮ new long-range interactions for neutrinos/leptons (new light particles) ◮ more light neutrinos stay focussed : how to include m ν ? 5 / 45
To write a neutrino mass At low energy, only restriction on m ν is Lorentz invariance. Mass term for a four-component fermion ψ : m ψ ψ = m ψ L ψ R + m ψ R ψ L 6 / 45
1. Dirac mass term : introduce ≥ 2 new chiral gauge singlets ν R Construct fermion number conserving mass term like for other SM fermions : m ν L ν R + m ν R ν L � � H 0 In full SM : λ ( ν L , e L ) ν R ≡ λ ( ℓ H ) ν R → m = λ � H 0 � H − added new light particles...add more and have ν s ? 7 / 45
1. Dirac mass term : introduce ≥ 2 new chiral gauge singlets ν R Construct fermion number conserving mass term like for other SM fermions : m ν L ν R + m ν R ν L � � H 0 In full SM : λ ( ν L , e L ) ν R ≡ λ ( ℓ H ) ν R → m = λ � H 0 � H − added new light particles...add more and have ν s ? 2. Majorana mass term : ( ν L ) c is right-handed ! ⇒ write a mass term with ν L ; no new fields , but lepton number violating mass : m m 2 [ ν L ( ν L ) c + ( ν L ) c ν L ] 2 [( ν L ) † γ 0 ( ν L ) c + (( ν L ) c ) † γ 0 ν L ] = − i m L σ 2 ν L ] ≡ m 2 [ ν † L σ 2 ν ∗ L + ν T = 2 ν L ν L + h . c . (2nd line = 2 comp notn) 7 / 45
1. Dirac mass term : introduce ≥ 2 new chiral gauge singlets ν R Construct fermion number conserving mass term like for other SM fermions : m ν L ν R + m ν R ν L � � H 0 In full SM : λ ( ν L , e L ) ν R ≡ λ ( ℓ H ) ν R → m = λ � H 0 � H − added new light particles...add more and have ν s ? 2. Majorana mass term : ( ν L ) c is right-handed ! ⇒ write a mass term with ν L ; no new fields , but lepton number violating mass : m m 2 [ ν L ( ν L ) c + ( ν L ) c ν L ] 2 [( ν L ) † γ 0 ( ν L ) c + (( ν L ) c ) † γ 0 ν L ] = − i m L σ 2 ν L ] ≡ m 2 [ ν † L σ 2 ν ∗ L + ν T = 2 ν L ν L + h . c . (2nd line = 2 comp notn) Non -renormalisable in full SM : L = ... + K 2 M ( ℓ H )( ℓ H ) + h . c . → m m = K M � H 0 � 2 2 ν L ν L + h . c . , ⇒ requires New Heavy Particles 7 / 45
Mechanisms/Models to obtain small Majorana masses 1. suppress by small scale ratio m / M seesaw type 1 inverse seesaw 2. suppress by loops/small couplings leptoquark model neglect Dirac mass because phenomenologically boring, and we don’t understand Yukawas = whether they can be so small. 8 / 45
(Theory parenthesis : why replace non-renorm. operator with renormalisable model of heavy particles ?) renormalisable theories allow to calculate every observable to arbitrary precision as a function of a finite number of input parameters ⇔ predictive But : there are maany models, they have lots of parameters, and we only need to calculate observables to the accuracy at which they can be measured. expectation (Wilson) that all particles have renormalisable interactions at energies above their mass scale. 9 / 45
Tree-level Majorana mass models (*minimal*) Heavy new particles (mass M ) induce dimension 5 operator in L : 2 M [ ℓ H ][ ℓ H ] → νν K � H 0 � 2 K 2 M 10 / 45
Tree-level Majorana mass models (*minimal*) Heavy new particles (mass M ) induce dimension 5 operator in L : 2 M [ ℓ H ][ ℓ H ] → νν K � H 0 � 2 K 2 M Three possibilities at tree level : SU(2) singlet fermions triplet fermions triplet scalars Type I Type III Type II 10 / 45
Type 1 seesaw, one generation Add to SM a singlet N ( ≡ ν R ) with all renorm. interactions : � � � � H + H 0 N + M − L Yuk 2 N c N + h . c . lep = h e ( ν L , e L ) e R + λ ( ν L , e L ) H 0 ∗ H − + M 2 N c N + h . c . m e e L e R + m D ν L N ⇒ neutrino mass matrix : N c � � � � � � ν c 0 m D ( ν c L ≡ ( ν L ) c ) L ν L m D M N ⇒ eigenvectors ≃ : ν L with m ν ∼ m 2 , N with mass ∼ M D M 11 / 45
The type I seesaw, 3 generations Minkowski, Yanagida Gell-Mann Ramond Slansky • add 3 singlet N to the SM in charged lepton and N mass bases : L = L SM + λ α J N J ℓ α · H − 1 2 N J M J N c add 18 parameters : J M 1 , M 2 , M 3 18 - 3 ( ℓ phases) in λ 12 / 45
The type I seesaw, 3 generations Minkowski, Yanagida Gell-Mann Ramond Slansky • add 3 singlet N to the SM in charged lepton and N mass bases : L = L SM + λ α J N J ℓ α · H − 1 2 N J M J N c J • at low scale, for M ≫ m D = λ v , light ν mass diagram v λ α A M A v λ β A 9 parameters : ν L α x x ν L β X m 1 , m 2 , m 3 N A 6 in U MNS λ M − 1 λ T v 2 [ m ν ] = 12 / 45
The type I seesaw, 3 generations Minkowski, Yanagida Gell-Mann Ramond Slansky • add 3 singlet N to the SM in charged lepton and N mass bases : L = L SM + λ α J N J ℓ α · H − 1 2 N J M J N c J • at low scale, for M ≫ m D = λ v , light ν mass diagram v λ α A M A v λ β A 9 parameters : ν L α x x ν L β X m 1 , m 2 , m 3 N A 6 in U MNS λ M − 1 λ T v 2 [ m ν ] = M ∼ 10 15 GeV λ ∼ h t , for ∼ . 05 eV λ ∼ 10 − 6 , M ∼ TeV “natural” m ν ≪ m f , but N hard to detect ? 12 / 45
The type I seesaw + Higgs mass • add 3 singlet N to the SM in charged lepton and N mass bases : L = L SM + λ α J N J ℓ α · H − 1 2 N J M J N c J 13 / 45
The type I seesaw + Higgs mass • add 3 singlet N to the SM in charged lepton and N mass bases : L = L SM + λ α J N J ℓ α · H − 1 2 N J M J N c J • at low scale, Higgs mass contribution ν λ α A λ β A H H N A � [ λ † λ ] II ∼ m ν M 3 δ m 2 M 2 8 π 2 v 4 v 2 I ≃ − H I 8 π 2 I 13 / 45
The type I seesaw + Higgs mass • add 3 singlet N to the SM in charged lepton and N mass bases : L = L SM + λ α J N J ℓ α · H − 1 2 N J M J N c J • at low scale, Higgs mass contribution ν λ α A λ β A H H N A � [ λ † λ ] II ∼ m ν M 3 δ m 2 M 2 8 π 2 v 4 v 2 I ≃ − H I 8 π 2 I ∼ 10 7 GeV v 2 for M > > tuning problem ( ? adding particles to cancel 1 loop ? Need symmetry to cancel ≥ 2 loop ?) ∼ 10 8 GeV ? ⇒ do seesaw with M I < 13 / 45
a low-scale tree model detectable at the LHC : the inverse seesaw • add two singlets N , S per generation to the SM : Valle ... L = L SM + λ N ℓ · H − NMS − 1 2 S µ S c Dirac mass between N and S , small Majorana mass for S . 14 / 45
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