Sterile Neutrinos Carlo Giunti INFN, Sezione di Torino and Dipartimento di Fisica Teorica, Universit` a di Torino giunti@to.infn.it Neutrino Unbound: http://www.nu.to.infn.it 52nd Winter School of Theoretical Physics Ladek Zdroj, Poland 14-21 February 2016 C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 1/94
Number of Flavor and Massive Neutrinos? 10 5 σ had [ nb ] Cross-section (pb) Z 2 ν ALEPH 3 ν 30 10 4 e + e − → hadrons DELPHI L3 4 ν OPAL 10 3 20 average measurements, error bars increased by factor 10 W + W - 10 2 CESR 10 DORIS PEP PETRA SLC TRISTAN KEKB PEP-II LEP I LEP II 10 0 86 88 90 92 94 0 20 40 60 80 100 120 140 160 180 200 220 E cm [ GeV ] Centre-of-mass energy (GeV) [LEP, Phys. Rept. 427 (2006) 257, arXiv:hep-ex/0509008] � � Γ Z = Γ Z → ℓ ¯ ℓ + Γ Z → q ¯ q + Γ inv Γ inv = N ν Γ Z → ν ¯ ν ℓ = e ,µ,τ q � = t N ν = 2 . 9840 ± 0 . 0082 C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 2/94
� e + e − → Z invisible − − − − → ν a ¯ ν a = ⇒ ν e ν µ ν τ a =active 3 light active flavor neutrinos N N ≥ 3 � mixing ⇒ ν α L = U α k ν kL α = e , µ, τ no upper limit! k =1 Mass Basis: ν 1 ν 2 ν 3 ν 4 ν 5 · · · Flavor Basis: ν e ν µ ν τ ν s 1 ν s 2 · · · ACTIVE STERILE N � ν α L = U α k ν kL α = e , µ, τ, s 1 , s 2 , . . . k =1 C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 3/94
Sterile Neutrinos [Pontecorvo, Sov. Phys. JETP 26 (1968) 984] ◮ Sterile means no standard model interactions. ◮ Obviously no electromagnetic interactions as normal active neutrinos. ◮ Thus sterile means no standard weak interactions. ◮ But sterile neutrinos are not absolutely sterile: ◮ Gravitational interactions (cosmology). ◮ New non-standard interactions of the physics beyond the Standard Model which generates the masses of sterile neutrinos. ◮ Observables in terrestrial experiments: ◮ Disappearance of active neutrinos ν e , ν µ , ν τ into sterile neutrinos ν s due to active-sterile oscillations (neutral current deficit). ◮ Oscillations (disappearance and transitions) of active neutrinos due to the new masses. ◮ Kinematical effects of the new masses (e.g. β decay). ◮ Contribution of the new masses to some process (e.g. neutrinoless double- β decay). C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 4/94
Extended Lepton Sector Q = I 3 + Y I I 3 Y 2 ν α L 1 / 2 0 1 / 2 L L = − 1 ℓ α L = α L − 1 / 2 − 1 ℓ α R = α R 0 0 − 2 − 1 ν s a R 0 0 0 0 α = e , µ, τ a = 1 , . . . , N s ◮ The right-handed sterile fields ν s a R belong to new physics beyond the Standard Model. ◮ Sterile neutrinos allow us to probe the new physics beyond the Standard Model. C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 5/94
General Dirac-Majorana Mass Lagrangian L mass = L D mass + L L mass + L R mass N S � � L D ν α L M D mass = − α a ν s a R + H.c. α = e ,µ,τ a =1 mass = 1 � L L ν T α L C † M L αβ ν β L + H.c. 2 α,β = e ,µ,τ N S mass = 1 � L R ν T s a R C † M R ab ν s b R + H.c. 2 a , b =1 ν s 1 R � � ν eL ν (a) . ν (F) ν (a) ν (s) L . = = ν µ L R = . L ν (s) c L ν τ L R ν s Ns R � M L � M D L mass = 1 2 ν (F) T C † M ν (F) + H.c. M = M D T M R L L C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 6/94
µ C − 1 = − γ µ , C † = C − 1 , C T = −C ◮ Charge conjugation matrix: C γ T C ( γ 5 ) T C − 1 = γ 5 ◮ Useful property: T ◮ Charge conjugation: ν (s) c = C ν (s) R R P L ≡ 1 − γ 5 P R ≡ 1 + γ 5 ◮ Left and right-handed chiral projectors: , 2 2 P 2 P 2 L = P L , R = P R , P L + P R = 1 , P L P R = P R P L = 0 ◮ P L ν (a) = ν (a) P R ν (a) P L ν (s) P R ν (s) R = ν (s) L , = 0, R = 0, L L R ◮ ν (s) c is left-handed: R T T P L ν (s) c = P L C ν (s) L ν (s) = C ( ν (s) = C P T R P L ) T R R R T γ 0 P L ) T = C ( ν (s) † P R γ 0 ) T = C ν (s) = C ( ν (s) † = ν (s) c R R R R P R ν (s) c = 0 R C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 7/94
◮ L mass has the structure of a Majorana mass Lagrangian � T � L mass = 1 ν (F) T C † M ν (F) − ν (F) L M † C ν (F) L L L 2 � � † ν (F) T C † M ν (F) = ν (F) † M † C ν (F) † T = ν (F) L γ 0 M † C ν (F) † T L L L L L T L M † C γ 0 T ν (F) † T = ν (F) L M † CC − 1 γ 0 C ν (F) † T = − ν (F) = − ν (F) L M † C ν (F) L L L ◮ ν (F) c ν (F) c = − ν (F) T = C ν L (F) T , C † L L L � � L mass = − 1 ν (F) c M ν (F) + ν (F) L M † ν (F) c L L L 2 C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 8/94
� M L � M D ◮ In general, M is a complex symmetric matrix: M = M D T M R � � T ν (F) T C † M ν (F) ν (F) T C † M ν (F) = L L L L = − ν (F) T M T ( C † ) T ν (F) L L C † M T ν (F) = ν (F) T M = M T = ⇒ L L ν (F) = U ν (M) ◮ M can be diagonalized with the unitary transformation L L ν eL · · · ν 1 L U e 1 U e 2 U e 3 U e 4 U eN ν µ L U µ 1 U µ 2 U µ 3 U µ 4 · · · U µ N ν 2 L ν τ L · · · ν 3 L U τ 1 U τ 2 U τ 3 U τ 4 U τ N = ν s 1 R U s 1 1 U s 1 2 U s 1 3 U s 1 4 · · · U s 1 N ν 4 L . . . . . . . ... . . . . . . . . . . . . . . ν s Ns R · · · ν NL U s Ns 1 U s Ns 2 U s Ns 3 U s Ns 4 U s Ns N ◮ U is a unitary N × N mixing matrix with N = 3 + N s . C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 9/94
U T M U = diag( m 1 , . . . , m N ) ◮ Diagonalization: with real and positive masses m 1 , . . . , m N . ◮ Mass Lagrangian in the mass basis: N L mass = 1 � kL C † ν kL − ν kL C ν kLT � � ν T m k 2 k =1 N = − 1 � � � kL ν kL + ν kL ν c ν c m k kL 2 k =1 N = − 1 � m k ν k ν k 2 k =1 ◮ Massive Majorana neutrino fields: ν k = ν kL + ν c ν k = ν c kL k ◮ In the general case of active-sterile neutrino mixing the massive neutrinos are Majorana particles. ◮ However, it is not excluded that the mixing is such that there are pairs of Majorana neutrino fields with exactly the same mass which form Dirac neutrino fields. C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 10/94
Charged-Current Weak Interactions ◮ The physical effects of neutrino mixing appear in Weak Interactions. ◮ In the flavor basis where the mass matrix of the charged leptons is diagonal L CC = − g � ℓ α L γ ρ ν α L W † √ ρ + H.c. 2 α = e ,µ,τ N = − g � � ℓ α L γ ρ U α k ν kL W † √ ρ + H.c. 2 α = e ,µ,τ k =1 ◮ In matrix form L CC = − g ρ + H.c. = − g 2 ℓ L γ ρ ν (a) 2 ℓ L γ ρ U ν (M) L W † W † √ √ ρ + H.c. L ◮ Effective rectangular 3 × N mixing matrix: ν (a) = U ν (M) U = U | 3 × N L L C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 11/94
◮ Effective rectangular 3 × N mixing matrix: · · · U e 1 U e 2 U e 3 U e 4 U eN U = · · · U µ 1 U µ 2 U µ 3 U µ 4 U µ N · · · U τ 1 U τ 2 U τ 3 U τ 4 U τ N ◮ The number of physical mixing parameters is smaller than the number necessary to parameterize the N × N unitary matrix U . ◮ This is due to the arbitrariness of the mixing in the sterile sector, which does not affect weak interactions. Any linear combination of the sterile neutrinos is equivalent. ◮ The effective rectangular 3 × N mixing matrix is not unitary: UU † = 1 3 × 3 , U † U � = 1 N × N but C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 12/94
◮ How many mixing parameters? ◮ A rectangular 3 × N matrix depends on 6 N real parameters, but UU † = 1 3 × 3 = ⇒ 9 constraints N real parameters = 6 N − 9 = 6 (3 + N s ) − 9 = 9 + 6 N s ◮ But how many mixing angles and physical CP-violating phases? ◮ For example, we know that for N s = 0 three phases can be eliminated by rephasing the charged lepton fields and we have 3 mixing angles 3 physical CP-violating phases (one Dirac and 2 Majorana) ◮ Standard parameterization of the mixing matrix in three-neutrino mixing: s 13 e − i δ 13 1 0 0 c 12 c 13 s 12 c 13 U (3 ν ) = − s 12 c 23 − c 12 s 23 s 13 e i δ 13 c 12 c 23 − s 12 s 23 s 13 e i δ 13 e i λ 2 s 23 c 13 0 0 s 12 s 23 − c 12 c 23 s 13 e i δ 13 − c 12 s 23 − s 12 c 23 s 13 e i δ 13 e i λ 3 0 0 c 23 c 13 C. Giunti − Sterile Neutrinos − 52nd Winter School of Theoretical Physics − 19-20 Feb 2016 − 13/94
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