Neutrino Neutrino Properties Properties Boris Kayser Neutrino 2008 May 28, 2008 1
What Is the What Is the Absolute Scale Absolute Scale of Neutrino Mass? of Neutrino Mass? 2
} ν 3 ν Oscillation Δ m 2 atm (Mass) 2 ν 2 Δ m 2 ν 1 sol } β Decay, Cosmology ?? 0 How far above zero is the whole pattern? Oscillation Data ~ Mass[Heaviest ν i ] > √Δ m 2atm = 0.05 eV 3
A Cosmic Connection Cosmological Data + Cosmological Assumptions Σ m i < (0.17 – 1.0) eV . ) ( Seljak, Slosar, McDonald Mass( ν i ) Pastor If there are only 3 neutrinos, 0.05 eV < Mass[Heaviest ν i ] < (0.07 – 0.4) eV ~ √Δ m 2atm Cosmology The cosmological assumptions seem reasonable, but are not guaranteed. A laboratory determination of the absolute ν mass scale will be essential. 4
Does ν = ν ? Does ν = ν ? 5
What Is the Question? For each mass eigenstate ν i , and given helicty h, does — • ν i (h) = ν i (h) (Majorana neutrinos) or • ν i (h) ≠ ν i (h) (Dirac neutrinos) ? Equivalently, do neutrinos have Majorana masses ? If they do, then the mass eigenstates are Majorana neutrinos . 6
Majorana Masses Out of, say, a left-handed neutrino field, ν L , and its charge-conjugate, ν Lc , we can build a Left-Handed Majorana mass term — ( ν ) R ν L m L ν L ν Lc X m L Majorana masses do not conserve the Lepton Number L defined by — L( ν ) = L( l – ) = –L( ν ) = –L( l + ) = 1. 7
A Majorana mass for any fermion f causes f f. Quark and charged-lepton Majorana masses are forbidden by electric charge conservation. Neutrino Majorana masses would make the neutrinos very distinctive. Majorana ν masses cannot come from , the H SM � R � L analogue of the q and l mass terms. 8
Possible Majorana mass terms: c � L , c � L , c � R H SM H SM � L H I W = 1 � L m R � R No Higgs Not renormalizable This Higgs not in SM Majorana neutrino masses must have a different origin than the masses of quarks and charged leptons. 9
Why Majorana Masses Majorana Neutrinos The objects ν L and ν Lc in m L ν L ν Lc are not the mass eigenstates, but just the neutrinos in terms of which the model is constructed. m L ν L ν Lc induces ν L ν Lc mixing. As a result of K 0 K 0 mixing, the neutral K mass eigenstates are — K S,L ≅ (K 0 ± K 0 )/ √ 2 . K S,L = K S,L . As a result of ν L ν Lc mixing, the neutrino mass eigenstate is — ν i = ν L + ν Lc = “ ν + ν ”. ν i = ν i . 10
Why Most Theorists Expect Majorana Masses The Standard Model (SM) is defined by the fields it contains, its symmetries (notably weak isospin invariance), and its renormalizability. Leaving neutrino masses aside, anything allowed by the SM symmetries occurs in nature. Since , Right-Handed Majorana mass ( ) = 0 I W � R terms are allowed by the SM symmetries. c � R m R � R Then quite likely Majorana masses occur in nature too. 11
To Determine To Determine Whether Whether Majorana Masses Majorana Masses Occur in Nature Occur in Nature 12
The Promising Approach — Seek Neutrinoless Double Beta Decay [0 νββ ] e – e – Nucl Nucl’ We are looking for a small Majorana neutrino mass. Thus, we will need a lot of parent nuclei (say, one ton of them). 13
Whatever diagrams cause 0 νββ , its observation would imply the existence of a Majorana mass term: (Schechter and Valle) e – e – ( ν ) R ν L 0 νββ u d d u W W ( ν ) R → ν L : A (tiny) Majorana mass term ∴ 0 νββ ν i = ν i 14
We anticipate that 0 νββ is dominated by a diagram with Standard Model vertices: SM vertex e – e – ν i ν i ∑ U ei Mixing matrix U ei i W – W – Nuclear Process Nucl Nucl’ 15
But there could be other contributions to 0 νββ , which at the quark level is the process dd → uuee. An example from Supersymmetry: e e ∼ γ u u ∼ ∼ e e d d 16
If the dominant mechanism is — SM vertex e – e – ν i ν i ∑ U ei Mixing matrix U ei i W – W – Nuclear Process Nucl Nucl’ then — Mass ( ν i ) Amp[0 νββ ] ∝ ∑ m i U ei2 ≡ m ββ i 17
Why Amp[0 νββ ] Is ∝ Neutrino Mass e – e – Nucl’ Nucl — manifestly does not conserve L. But the Standard Model (SM) weak interactions do conserve L. Absent any non-SM L-violating interactions, the Δ L = 2 of 0 νββ can only come from Majorana neutrino masses , such as — ν L ( ν ) R m L ( ν Lc ν L + ν L ν Lc ) X m L 18
Treating the neutrino masses perturbatively, we have — e – e – ( ν ) R ν L X m L W – W – Nuclear Process Nucl’ Nucl A Left-Handed Majorana mass term is just what is needed to — 1) Violate L 2) Flip handedness — and allow the decay to occur. 19
How Large is m ββ ? How sensitive need an experiment be? Suppose there are only 3 neutrino mass eigenstates. (More might help.) Then the spectrum looks like — ν 2 ν 3 sol < ν 1 or atm atm ν 2 sol < ν 3 ν 1 Inverted hierarchy Normal hierarchy 20
Takes 1 ton 95% CL m ββ Smallest Takes m ββ For Each Hierarchy 100 tons 21
There is no clear theoretical preference for either hierarchy. If the hierarchy is inverted— then 0 νββ searches with sensitivity to m ββ = 0.01 eV have a very good chance to see a signal. Sensitivity in this range is a good target for the next generation of experiments. 22
Determining m ββ Determining m ββ 23
The most important goal of 0 νββ searches is to observe the process. Observation at any non-zero level would establish that — Neutrinos have Majorana masses Neutrinos are Majorana particles Lepton number is not conserved 24
What We Would Learn From Information On m ββ Suppose accelerator experiments have determined the hierarchy to be inverted . Suppose 0 νββ searches are negative, but establish convincingly that m ββ < 0.01 eV. Then, barring unlikely cancellations from exotic mechanisms, we can say that neutrinos are Dirac particles : . � � � Suppose accelerator experiments have not determined the hierarchy, but 0 νββ searches have found a convincing signal with m ββ < 0.01 eV. Then, barring exotic mechanisms, the hierarchy must be normal . Bahcall, Murayama, Pena-Garay; de Gouvêa, Jenkins 25
According to the Standard Model, the leptonic mixing matrix U is unitary. Then, if m Heaviest is the mass of the heaviest neutrino mass eigenstate, m ββ ≡ ∑ m i U ei2 ≤ m Heaviest ∑ U ei 2 = m Heaviest i i A measured value of m ββ would be a lower bound on the mass of the heaviest neutrino. 26
Majorana CP-Violating Phases Although the Cabibbo-Kobayashi-Maskawa quark mixing matrix can have only one CP phase, the Pontecorvo-Maki-Nakagawa-Sakata leptonic mixing matrix U can have three : � s 13 e � i � � � 1 0 0 � c 13 0 � c 12 s 12 0 � � � � � � � U = 0 c 23 s 23 0 1 0 � s 12 c 12 0 � � � � � � � � � s 13 e i � � � � � � � 0 � s 23 c 23 0 c 13 0 0 1 � � � � � � � e i � 1 /2 � 0 0 � � c ij ≡ cos θ ij Analogue of the e i � 2 /2 0 0 � � � quark CP phase s ij ≡ sin θ ij � � 0 0 1 � � Majorana CP phases 27
The Majorana CP phases are physical only if neutrinos are Majorana particles. They only affect processes involving violation of lepton number L, such as 0 νββ . Consider 0 νββ when the neutrino mass spectrum is inverted : sol < Average mass m 0 (From β decay exps.) atm 28
For an inverted spectrum, { Majorana CP phases α 2 – α 1 m 0 [ 1 - sin 2 2 θ 12 sin 2 ( ––––– )] ½ . m ββ ≅ 2 Solar mixing angle m 0 cos 2 θ 12 ≤ m ββ ≤ m 0 0.4 m 0 ≤ m ββ ≤ m 0 From SNO CP is violated if α 2 – α 1 ≠ 0, π . To establish CP, we must determine m ββ to within a factor of ∼ 2. Pascoli, Petcov, Rodejohann; Barger, Glashow, Langacker, Marfatia 29
Nuclear Matrix Nuclear Matrix Elements for 0 νββ Elements for 0 νββ 30
If 0 νββ is dominated by light neutrino exchange, then — Γ (0 νββ ) = (m ββ ) 2 x (Nuclear m. e.) 2 x (Phase space) The nuclear m. e. M 0 ν is calculated by the Quasi Particle Random Phase Approximation (QRPA) or the Nuclear Shell Model (NSM). 31
Vogel 32
Sources of Uncertainty in the QRPA Calculations 0 νββ is nn → pp + ee. If the two neutrons are separated by > 2 – 3 fm, there is near cancellation between the J (nn) = 0 and the J (nn) ≠ 0 contributions. As a result, there is great sensitivity to short-distance features, such as which separation distances dominate, nucleon structure, and short-range repulsion. There is also sensitivity to the strength g pp of the particle-particle neutron-proton interaction. This parameter is fixed by reference to 2 νββ decay. 33
The Bottom Line For the commonly-considered 0 νββ candidates, such as 76 Ge, the nuclear m. e. is uncertain by a factor of 2, and perhaps a factor of 3. Hopefully, this will improve, to permit cleaner interpretation of 0 νββ results. Special thanks to Petr Vogel for nuclear-physics wisdom. 34
What Are the What Are the Neutrino Neutrino Dipole Moments? Dipole Moments? 35
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