Lecture #4 a) Comments on effective ββ decay operators b) The role of measured orbit occupancies c) The ββ decay with heavy particle exchange d)

Transcription

1 Lecture #4 a) Comments on effective ββ decay operators b) The role of measured orbit occupancies c) The ββ decay with heavy particle exchange d) Neutrino magnetic moment and Majorana vs. Dirac neutrinos

2 So far we have used the `bare operator to describe the 0νββ decay In the following few slides I indicate a possible procedure how to obtain an effective transition operator consistently (work in progress by J. Engel, his slides)

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5 This we try to include by choosing proper g pp This is included phenomenologically by including correction for s.r.c

6 These are the corrections for the various graphs evaluated in the two-level model described previously. The corrections are substantial.

7 Yet another effect; population of the mean field orbits determined experimentally (Schiffer et al., PRL 100,112501(2008)) compared to those quantities used in QRPA. This effect is not included in the calculation thus far.

8 The discrepancy suggests that the mean field potential used in QRPA so far is not optimal for the case of 76 Ge. The potential used so far is the global fit to the Coulomb Corrected Woods-Saxon potential with a smooth dependence on Z,N as derived by Bertsch (1972) or Bohr & Mottelson (1969). Alternatively one can try to adjust the single-particle energies `by hand so that the measured populations of valence orbits (and their differences) are reproduced. (Work in progress, since so far only neutrons were measured and analyzed by Schiffer et al.

9 Here is an example. It is necessary to move the g,d,s shell closer to the p,f shell and in particular, g 9/2 very close to p 1/2

10 In order to calculate M 0ν it is more important to describe the population of individual orbits, rather then their s.p. energies. However, it is crucial to keep in mind that in both QRPA and NSM the ground state wave function is a superposition of many configurations, Lets consider for simplicity only pairing (only Cooper 0 + pairs) and orbits that are not very populated or almost comletely full. Then Ψ> ~ a 0 0> + a 2 2> + a 4 4> If the population of that orbit <n> is known and fixed we have a 02 = 1 - <n>/2 + a 4 2 ; a 22 = <n>/2-2a 4 2 And the matrix element of two creation operators is <Ψ final a + a + Ψ initial > = a 0 b 2 + a 2 b 4 Moreover, if b 4 can be neglected (orbit that is almost empty or almost full) <Ψ final a + a + Ψ initial > = [(1 - <n>/2) x (<n >/2)] 1/2 Thus, there is no contribution to M 0ν when the final orbit is completely empty or the initial orbit is completely full. However, even if <n> << 1 the contribution can be noticeable since it is proportional to the square root of the population. Also, what matters are <n> and <n > individually, not their differences.

11 In lecture 1 I showed that 0νββ decay can occur also by the exchange of heavy particles (like in LRSM or RPV SUSY), and that potentially this could lead to decay rates comparable to the light Majorana mediated 0νββ decay. The mass of these heavy particles is above the weak scale, and the dynamic occurs primarily at short distances. We can `integrate out the heavy degrees of freedom leaving an effective theory of light quarks and leptons. The effective lagrangian can be organized as an expansion in the small parameters p/λ H, p/λ ββ and Λ H /Λ ββ, where p is m π or the electron energy, Λ H ~ 1 GeV is hadronic mass scale, and Λ ββ is the heavy particle mass. ββ The hadronic vertices in the effective lagrangian will be of the type NNNNee, NNπee, and ππee and will contribute in different order in p/λ H and will be constrained by the symmetries of the theory. (see Prezau, Ransey-Musolf and P.V., Phys.Rev. D68, (2003))

12 Traditionally, the short-range effect were analyzed using nucleon form factors in the neutrino propagator (Vergados 1982) u(r,λ) = d 3 k e ikr /(2π) 3 (Λ 2 /(Λ 2 + k 2 )) 4 = e -Λr Λ 3 /64π [1 + Λr + (Λr) 2 /3] Here Λ ~ 1 GeV is the nucleon mass scale. The error introduced by this approximation (affecting the NNNNee vertex cannot be estimated, unlike in the EFT approach. Unlike NNNNee vertex, the ππee (and Nnπee) involve long (pion size) range effects. It has been shown by Faessler et al that ππee vertices would dominate in the case of RPV SUSY. Here a systematic analysis is performed.

13 Diagrams that contribute to 0νββ at tree level: Order in p: p -2 p -1 p -1 p 0 where p ~ m π. These powers of p are multiplied by the vertices K ππ, K NNπ, and K NNNN which need to be evaluated as a power series in p.

14 The most general quark-lepton Lorentz-invariant Lagrangian can contain only nine quark operators (in 0νββ only a=b=+ appear) By symmetry consideration in LRSM only O ++ 1+, and O ++ 3+can appear. With only left-handed currents just O appear. In RPV SUSY all operators appear. Note that the subscripts +,- on O denote parity.

15 Next we must consider the hadron-lepton lagrangian to different orders. (Since the NNNNee is p 2 compared to ππee we must expand ππee to LO, NLO (this one vanishes by parity) and NNLO, and NNπee to LO and NLO). The operators with X cannot contribute and those with can. Ιt turns out the NNπee graph in LO does not contribute to the transitions and therefore we need to consider only ππee to LO, and that automatically has no contributions in next order of p/λ H.

16 After the nonrelativistic limit for nucleons and Fourier transformation to coordinate space we obtain heavy scale unknown constants of the order of unity The nuclear matrix element is With ρ = r 1 - r 2 and x = m π ρ while F 1 (x) = (x - 2) e -x, F 2 (x) = (x + 1)e -x (thus both at pion range)

17 The formula at the preceding slide thus should be used for the 0νββ νββ-decay half-life and nuclear matrix element evaluation. By power counting it should contribute more compared to the pure NNNNee vertex contribution (considered usually in the literature) by (Λ H /m π ) (However, numerical calculations of Faessler et al. (1997) suggest less, but still very strong ~10 2, enhancement of the ππee contribution to decay rate in the case of RPV-SUSY. In RPV-SUSY one should replace 1/Λ ββ by (2π/63) λ (M/G F2 m q4 ) 2α s /m g in agreement with the general 1/Λ 5 scaling. squark mass gluino mass

18 Left-right symmetric model: We use λ = (M W1 /M W2 ) 2 < 10-2, and ζ for the mixing angle of W 1 and W 2. The following graphs and operators can contribute: considered in literature not considered in literature However, our analysis suggests that M (LL) (a) ~ ζ2 p 2 /Λ 2 H, M (RR) (a) ~ λ2 p 2 /Λ H2, while M (LR) (b) ~ λ ζ thus graph (b) might dominate and would give stronger constraints On the masses of M W 2 and M N R

19 Neutrino magnetic moment and the distinction between the Dirac and Majorana neutrinos. Neutrino mass and magnetic moment are intimately related. In the orthodox SM with massless neutrinos magnetic moments vanish. However, in the minimally extended SM with a Dirac neutrino of mass m ν the loops like this produce an unobservably small, but nonvanishing dipole magnetic moment µ ν = 3eG F /(2 1/2 π 2 8) m ν = 3x10-19 m ν /ev µ B (1977)

20 The interest in µ ν and its relation to m ν dates from ~1990 when it was suggested that there is an anticorrelation between the neutrino flux observed in the Cl (Davis) experiment, and the solar activity (number of sunspots that follows a 11 year cycle). A possible explanation of this was proposed by Voloshin, Vysotskij and Okun, with µ ν ~ µ B and its precession in solar magnetic field. Even though the effect does not exist, the possibility of a large µ ν and small mass was widely discussed. I like to describe a model independent constraint on the µ ν that depends on the magnitude of m ν and moreover depends on the charge conjugation properties of neutrinos, i.e. makes it possible, at least in principle, to decide between Dirac and Majorana nature of neutrinos.

21 Even though the correlation with solar activity was questionable the problem attracted 145 citations.

22 How can one measure µ ν? Magnetic moment could be observed in ν-e scattering by looking At the electron recoil spectrum; the scattered neutrino is not observed. The electromagnetic cross section has a characteristic shape σ elm = πα 2 µ ν2 /m e 2 (1-T/E ν )/T With a singularity as the electron recoil kinetic energy T 0. Nonvanishing µ ν will be recognizable only if the σ elm is comparable in magnitude with the well understood weak interaction cross section. The magnitude of µ ν which can be probed in this way is then given by Considering realistic values of T it will be difficult to go beyond µ ν ~ µ B this way.

23 Limits on µ ν can be also derived from bounds on unobserved energy loss in astrophysical objects. For sufficiently large µ ν the rate of plasmon decay into ν ν pairs would conflict such bounds. However, since plasmons can also decay weakly into ν ν pairs the sesitivity of this probe is again limited by the size of the weak rate, leading to where ω P is the plasma frequency. Since (hω P ) 2 << m e T, this bound is stronger than the limit from ν-e scattering.

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25 In ν-e scattering when the scattered neutrino is not observed, one cannot separate the effects of diagonal and transition magnetic dipole moments.

26 µ µ m ν ν ν It is difficult to reconcile large µ ν and small m ν

27 To overcome this difficulty Voloshin (88) proposed existence of a SU(2) ν symmetry in which ν L and (ν R ) c form a doublet. Under this symmetry m ν is forbidden but µ ν is allowed. For Dirac neutrinos such symmetry is broken by weak interactions, but for Majorana neutrinos it is broken only by the Yukawa couplings. Since Majorana neutrinos can have only transition in flavor magnetic moment in flavor basis the mass term for Majorana neutrinos is symmetric but the magnetic moments are antisymmetric. In the following I show that the existence of nonvanishing µ ν leads through loop effects to an addition to the neutrino mass δm ν that, naturally cannot exceed the magnitude of m ν. (See Bell et al, PRL95,151802, Davidson et al. Phys.Lett. B626, 151, and Bell et al., Phys.Lett. B642, 377)

28 Consider first the Dirac case: Assuming that µ ν is generated by some physics beyond the SM at a scale Λ, Its leading contribution to the neutrino mass, δm ν, arising from radiatiave corrections at one loop order scales as If we take Λ ~ 1 TeV and m ν < 0.3 ev we obtain the limit µ ν < µ B, which is several orders of magnitude more stringent than the current experimental limit. The limit is even more stringent if Λ >> 1 TeV. The dependence on Λ 2 arises from the quadratic divergence appearing in the renormalization of the dimension four neutrino mass operator. It is, therefore, highly unlikely that the magnetic dipole moment of a Dirac neutrino is observable.

29 The usual graph for µ ν can be expressed in a gauge invariant form: γ H H W B = C W + C B µ µ µ One can now close the loop and obtain a quadratically divergent contribution to the Dirac mass µ

30 The case of Majorana neutrinos is more subtle due to the relative flavor symmetries of m ν (symmetric) and µ ν (antisymmetric). µ ν µ ν These one loop contributions to the Majorana neutrino mass associated with the neutrino magnetic moment sum to zero. (Davidson, Gorbahn and Santamaria)

31 µ ν µ ν In order to get a nonvanishing contribution to the Majorana neutrino mass associated with the magnetic moment one has to make charged lepton mass insertions X. The resulting δm ν is smaller since it contains the differences between the (small) charged lepton Yukawa couplings (factor m 2 α - m2 β ). Hence the constraints on the µ ν of Majorana neutrinos are much weaker than for the Dirac neutrinos.

32 We have shown that the most general bound on Majorana magnetic moment is This is perfectly compatible with the experimental sensitivities, provided that Λ is not excessively large. Hence next improvement of sensitivity can indeed discover µ ν.

33 Thus if a neutrino magnetic moment is observed near its present experimental limit we would conclude that neutrinos are Majorana, and that the corresponding new scale Λ < 100 TeV. If we, further, could assume that all elements of the matrix µ αβ are of similar magnitude, than a discovery of µ ν at, say µ B would imply Λ < 10 TeV with a possible implication for the mechanism of 0νββ decay. Hence search for µ ν is in some sense complementary to the search for 0νββ decay. But, unlike the 0νββ decay, we have just an upper bound, and not a clear map where to look.

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35 How come that we can decide whether neutrinos are Dirac or Majorana without actually observing that the total lepton number is violated? That is so because the distinction between Dirac and Majorana does not actually require processes that violate lepton number, just amplitudes must violate it. An example is the electromagnetic decay of neutrinos These two graphs must be added for Majorana neutrinos then the angular distribution of the photons with respect to the direction of the neutrino beam is dn = 1/2(1 + a cos θ) d cosθ Where a = 0 for Majorana and a = -1 for Dirac neutrinos. Thus, also in this hypothetical case one can distinguish the two cases.

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