Overview of theory of neutrino mass and of the 0νββ nuclear matrix elements. Petr Vogel, Caltech INT workshop on neutrino mass measurements Seattle, Feb.8, 2010
The mixing angles and Δm 2 ij are quite accurately determined. We have reached the precision stage of oscillation studies. The opposite orientations of ellipses in the lower left panel suggest that sin 2 θ 13 > 0.
The status of the present knowledge of the neutrino oscillation phenomena is schematically depicted in this slide. Three quantities are unknown at present: a) The mass m 1 b) The angle θ 13 c) Whether the normal or inverted hierarchy is realized.
Here is then what we know today about the masses and mixing of the elementary fermions. The neutrino sector is really strange, i.e. very different compared to the charged fermions. CKM matrix is nearly diagonal PMNS matrix is `democratic, i.e. matrix elements are of similar magnitude, except U e3 which is evaluated here near the middle of its allowed range Neutrinos are many (at least 6) orders of magnitude lighter than the charged fermions.
In fact, the neutrino mixing matrix resembles the tri-bimaximal matrix, which can be a convenient zeroth order term of possible expansions. (compared to the empirical matrix above the last line and last column were multiplied by -1) ν 1 ν 2 ν 3 e (2/3) 1/2 (1/3) 1/2 0 U = μ -(1/6) 1/2 (1/3) 1/2 -(1/2) 1/2 τ -(1/6) 1/2 (1/3) 1/2 (1/2) 1/2 Note the apparent symmetries: the ν 2 column contains only y( (1/3) 1/2 and the 2nd and 3rd lines are identical up to the phases. The deviations could possibly be related to the expansion in small parameter Δmm 212 /Δm 2 31 ~ 1/30.
The mixing angle θ 13 is restricted by experiment to sin 2 θ 13 < 0.05 (90% CL) (figure from Chen 0706.2168(hep-ph)
Once the angle θ 13 is experimentally determined the following beautiful quote will be applicable to most of these models: The terrible tragedies of science are the horrible murders of beautiful theories by ugly facts. W. A. Fowler (after T. H. Huxley) quote borrowed from Gary Steigman
The ν masses are much smaller than the masses of other fermions; the pattern of masses is also different (neutrinos are closer together) D E S E R T Is that a possible Hint of a new mass-generating mechanism?
To solve the dilemma of `unnaturally small neutrino mass we can give up on renormalizability and add operators of dimension d > 4 that are suppressed by inverse powers of some scale Λ but are consistent with the SM symmetries. Weinberg already in 1979 (PLR 43, 1566) showed that there is only one dimension d=5 gauge-invariant operator given the particle content of the standard model: L (5) = C (5) /Λ (L c εh)(h T εl) +h.c. This operator clearly violates the lepton number by two units and represents neutrino Majorana mass L (M) = C (5) /Λ v 2 /2 (ν Lc ν L ) + h.c. If Λ is larger than v, the Higgs vacuum expectation value, the neutrinos will be `naturally lighter than the charged fermions. For C (5) ~ O(1) and neutrino mass ~ 0.1 ev, Λ ~ v 2 /m ν ~ 10 15 GeV, i.e. close to the GUT scale.
The most popular theory of why neutrinos are so light is the See-Saw Mechanism (Gell-Mann, Ramond, Slansky (1979), Yanagida(1979), Mohapatra, Senjanovic(1980)) ν Familiar light { neutrino Very heavy neutrino } N R It assumes that the very heavy neutrinos N R exist. Their mass plays an analogous role as the scale Λ of Weinberg, i.e., m ν ~ v 2 /M N. Both the light and heavy neutrinos are Majorana fermions.
The above ``classis see-saw with heavy gauge singlet N R neutrinos is called Type I see-saw. Another possibility is to assume the existence of SU(2) L triplet Higgs Y=2 with a small (compared to v) vacuum expectation ti value. Neutrinos are then again Majorana fermions and govern at low energies by the d=5 operator a la Weinberg. This is called Type II see-saw. saw Yet another possibility is Type III see-saw with a new Y=0 SU(2) L triplet fermion. Light neutrinos are obtained if the mass M T of the tilt triplet is large. These are all examples of models with the high mass scale associated with the small neutrino mass. It is also possible to construct models (typically with certain amount of fine tuning) of low (TeV) mass scale of the new particles. Some of them could be then explored at LHC.
(slide by A. Smirnov)
The best practical test of lepton number conservation is the study of the 0νββ decay.observing it would prove that neutrinos are indeed Majorana fermions. As long as the mass eigenstates ν i that are the components of the flavor neutrinos ν e, ν μ, and ν τ are really Majorana neutrinos, the 0νββ decay will occur, with the rate 1/T 1/2 = G(E tot,z) (M 0ν ) 2 <m ββ > 2, where G(E tot,z) is easily calculable phase space factor, M 0ν is the nuclear matrix element, calculable with difficulties (and discussed later), and <m ββ > = Σ i U ei 2 exp(iα i ) m i, where α i are unknown Majorana phases (only two of them are relevant) where α i are unknown Majorana phases (only two of them are relevant). Using the formula above we can relate <m ββ > to other observables related to the absolute neutrino mass.
minimum mass, not observable from observational cosmology, M = m 1 +m 2 +m 3 from β decay <m ββ > vs. the absolute mass scales blue shading: normal hierarchy, Δm 2 31 > 0. red shading: inverted hierarchy h Δm 2 31 < 0 shading:best fit parameters, lines 95% CL errors. Thanks to A. Piepke
Comparison of M 0ν from NSM, QRPA and IBM-2 (for IBM-2 see Barea and Iachello, Phys. Rev. C79, 044301(2009).) Note the smooth dependence on A,Z. Differences by a factor of ~2 between QRPA and IBM-2 versus NSM persist.
The 2ν matrix elements, unlike the 0ν ones, exhibit pronounced shell effects. They vary fast as a function of Z or A.
Why are the QRPA and NSM matrix elements different? Various possible explanations: a) Assumed occupancies of individual valence orbits might be different b) In QRPA more single particle states are included c) In NSM all configurations (seniorities) are included d) In NSM the deformation effects are included e) All of the above
Experiments by J.P.Schiffer et al, Phys.Rev.Lett. 100, 1120501(2008), used (d,p),(p,d),( 3 He,α),(α, 3 He) to derive occupancies of neutron orbits for 76 Ge and 76 Se New QRPA value with adjusted mean field so that t experimental occupancies are reproduced New NSM value with adjusted mean field (monopole) where experimental occupancies are better reproduced new NSM new NSM value
from isovector p-p and n-n pairing interaction 82 Se Why it is difficult to calculate the matrix elements accurately? Contributions of different angular momenta J of the neutron pair that is transformed in the decay into the proton pair with the same J. from isoscalar n-p interaction 130 Te Note the opposite signs, and thus tendency to cancel, between the J = 0 (pairing) and the J 0 (ground state correlations) parts. The same restricted s.p. space is used for QRPA and NSM. There is a reasonable agreement between the two methods J
C(r) C J (r) total pairing part Full matrix element The radial dependence of M 0ν for the three indicated nuclei. The contributions summed over all components is shown in the upper panel. The `pairing J = 0 and `broken pairs J 0 parts are shown separately below. Note that these two parts essentially cancel each other for r > 2-3 fm. This is a generic behavior. Hence the treatment of small values of r and large values of q are quite important. broken pairs part M 0ν = C(r)dr
The radial dependence of M 0ν for the indicated nuclei, evaluated in the nuclear shell model. (Menendes et al, arxiv:0801.3760). Note the similarity to the QRPA evaluation of the same function.
Dependence on the distance between the two transformed nucleons and the effect of different treatments of short range correlations. This causes changes of M 0ν by ~ 20%. (5.3) (4.0) (4.1) (5.0) C(r) Something like this appears to be the consensus treatment these days. Three different ways lead to this conclusion.
In QRPA more single particle states are included but it appears that their net effect is rather small after all. Contribution of initial neutron orbit pairs against the final proton pairs. The nonvalence orbits are labeled as r. Adding all parts with r-type orbits gives +2.83-3.22 = -0.39 which is only ~12% of the total matrix element 3.27 (The total matrix element is made of 9.74-6.46 = 3.27.) In the figure all entries are, however, normalized so that their sum is unity..
In QRPA one can describe the distribution of the GT strength, including the magnitude and position of the giant GT resonance reasonably well. For that one needs, however, inclusion of the nonvalence states. This is the case of 76 Ge. 76 Ge Giant GT resonance
Summary for 0νββ matrix elements: 1) All methods agree on the smooth dependence on A,Z, but significant differences on the actual values remain. 2) All methods agree on the severe cancellation between contributions ons of pairing (isovector interaction) and broken-pairs (isoscalar interaction) and on the dominant contribution of small (r < 2 fm) distances between the changed nucleons. 3) Since ab initio calculation is impossible, it is important to consider the ability to describe other observables (orbit occupancies, 2νββ halflife, GT strength, th etc.) 4) Consensus is emerging on the treatment of the short range nucleon repulsion and on the treatment of higher order weak currents (pseudoscalar, weak magnetism). 5) Proper treatment of nuclear deformation is still hard. That is particularly relevant for the otherwise attractive case of 150 Nd.
spares
Neutrino potential Φ(r) = 2/π r 0 dq sin(qr)/qr x 1/(1 + q 2 /M A2 ) 4 q/(q + <E>) Note that at r = 0 Φ(r) = 5/16 M A r 0 ; at r > 1.5 fm Φ(r) ~ 1/r