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| 3 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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| 4 | %% Introduction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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| 5 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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| 6 | |
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| 7 | \section{Introduction} |
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| 8 | |
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| 9 | Underground water Cherenkov detectors have |
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| 10 | found unambiguous evidence for |
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| 11 | neutino oscillations and therefore beyond-the Standard Model |
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| 12 | physics. |
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| 13 | % focused much attention on neutrino physics. |
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| 14 | The atmospheric neutrino results of Super- Kamiokande(SK),IMB and Frjus, |
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| 15 | followed by the solar observations of SK, SNO and KamLAND, |
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| 16 | have confirmed that neutrinos have mass and |
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| 17 | two large mixing angles. |
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| 18 | However, there remain many questions |
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| 19 | about the parameters and properties of leptons, |
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| 20 | some of which could be addressed by a larger (megatonne) |
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| 21 | underground neutrino detector. |
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| 22 | %nonetheless there are questions |
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| 23 | %remaining. More statistics are required to increase |
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| 24 | %the sensitivity to unknown neutino parameters, |
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| 25 | If the location of such a detector was |
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| 26 | judiciously selected, it could be |
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| 27 | a suitable distance along the path of |
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| 28 | a new high intensity |
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| 29 | $\nu_\mu$ beam (superbeam), and/or or $\nu_e$ beam ($\beta$ beam). |
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| 30 | %source = beam, not astro |
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| 31 | %{\it build beam and detector so can do an accelerator expt}. |
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| 32 | |
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| 33 | The observation of neutrinos from SN1987A forshadowed |
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| 34 | the linked results on astrophysics and neutrino physics |
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| 35 | that can be obtained from a supernova. Such an exploding |
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| 36 | star is an extraordinary source, for which it |
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| 37 | would be reasonable to have a detector. |
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| 38 | A megatonne detector could perhaps even |
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| 39 | see relic neutrinos |
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| 40 | accumulated from past supernovae. |
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| 41 | |
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| 42 | Originally, large underground detectors were built |
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| 43 | to look for proton decay, a prediction of |
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| 44 | Grand Unified Theories. Nucleon decay is |
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| 45 | a ``smoking gun'' for quark lepton |
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| 46 | unification, observation of which would |
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| 47 | confirm many years of theoretical speculation. |
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| 48 | The current lower bound on the proton lifetime from SK has |
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| 49 | ruled out the simplest non-supersymmetric GUT, |
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| 50 | a megaton detector would |
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| 51 | cover a substantial area of interesting parameter |
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| 52 | space. |
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| 53 | |
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| 54 | |
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| 55 | |
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| 56 | \section{Bread and Butter: $\nu$ Physics} |
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| 57 | |
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| 58 | A megatonne detector would have improved sensitivity to |
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| 59 | currently unknown parameters of neutrino mixing. |
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| 60 | The neutrinos could be of astrophysical origin--- |
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| 61 | solar, atmospheric or from supernovae--- or $\nu$ |
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| 62 | beams of specific flavour and energy could be directed |
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| 63 | at the detector. |
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| 64 | %The solar and atmospheric neutrino fluxes would |
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| 65 | %arrive for free. |
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| 66 | A high intensity $\nu_\mu$ ``superbeam'', |
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| 67 | could be produced by increasing the intensity of the |
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| 68 | proton driver at the source, |
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| 69 | or a very pure $\nu_e$ beam could be produced |
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| 70 | in the $\beta$ decay of an ion beam. |
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| 71 | |
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| 72 | |
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| 73 | |
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| 74 | \subsection{status} |
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| 75 | |
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| 76 | A review of our current knowledge of neutrino parameters |
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| 77 | was presented by G. Fogli. |
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| 78 | |
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| 79 | Information |
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| 80 | \footnote{The numerical values are from the global fit |
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| 81 | presented by Fogli} on $\sin ^2 \theta_{23} = 0.45 |
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| 82 | \pm \stackrel{0.18}{_{0.11}}$, |
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| 83 | $\Delta m_{23}^2 = 2.4 \pm \stackrel{0.5}{_{0.6}} \times 10^{-3}$ eV$^{2}$ |
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| 84 | and $\sin ^2 \theta_{13} \leq 0.035$ |
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| 85 | is obtained from SuperKamiokande, K2K and CHOOZ. |
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| 86 | The evidence for atmospheric neutrino |
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| 87 | oscillations with large, or maximal mixing |
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| 88 | is robust, and confirmed with neutrinos |
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| 89 | from the K2K beam. |
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| 90 | SK has found evidence for a decrease |
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| 91 | in $\nu_\mu$ flux at the location |
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| 92 | expected for the first dip in the oscillation |
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| 93 | probability---this despite the smearing in |
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| 94 | energy and path length. |
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| 95 | As discussed by Fogli, the data sets can be |
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| 96 | combined in various ways to determine the parameters. |
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| 97 | The results quoted were obtained |
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| 98 | from the combined data of all three experiments, by |
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| 99 | using a three-dimensional simulation for |
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| 100 | the atmospheric neutrino fluxes, by including |
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| 101 | subleading effects due to |
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| 102 | $\Delta m_{12}^2$ and $\sin ^2 \theta_{12}$, |
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| 103 | and leaving $\sin ^2 \theta_{13}$ free. |
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| 104 | Letting |
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| 105 | $\sin ^2 \theta_{13}$ float has little effect |
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| 106 | because the data prefers it small. |
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| 107 | |
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| 108 | |
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| 109 | SNO, SK and KamLAND are sensitive to the solar mass |
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| 110 | difference $\Delta m_{12}^2 = 8.0 |
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| 111 | \pm \stackrel{0.8}{_{0.7}} \times |
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| 112 | 10^{-5} $ eV$^2$ and |
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| 113 | a large but not maximal mixing |
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| 114 | angle $\sin ^2 \theta_{23} = 0.31 \pm \stackrel{0.05}{_{ 0.04}} $. |
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| 115 | These data also prefer |
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| 116 | $\sin ^2 \theta_{13} \sim 0$ (a non-trivial |
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| 117 | consistency check with atmospheric and CHOOZ), |
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| 118 | so the allowed ranges for |
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| 119 | $\Delta m_{12}^2 $ and |
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| 120 | $\sin ^2 \theta_{23} $ are not significantly |
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| 121 | affected when $\theta_{13}$ is allowed to float. |
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| 122 | |
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| 123 | |
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| 124 | |
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| 125 | \subsection{ agenda for future experiments} |
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| 126 | |
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| 127 | The current bounds on the unknown neutrino parameters, |
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| 128 | and future prospects for measuring them were discussed by |
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| 129 | J. Ellis and G. Fogli, and T Schwetz. Some of these unknowns |
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| 130 | (items 4-7 of the following list) |
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| 131 | could be determined from more precise oscillation |
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| 132 | experiments. |
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| 133 | %---in particular from neutrino beams |
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| 134 | %directed at a megatonne detector. |
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| 135 | \begin{enumerate} |
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| 136 | \item the number of light neutrinos participating |
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| 137 | in oscillations is usually taken to be the three |
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| 138 | active neutrinos expected in the Standard Model. |
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| 139 | However, the LSND experiment found evidence |
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| 140 | for $\Delta m^2 \sim$ eV$^2$, which would require |
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| 141 | one (or more) |
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| 142 | additional light sterile neutrinos. |
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| 143 | MiniBoone is searching for oscillations |
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| 144 | in the LSND window; their results, |
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| 145 | expected in 2005, will confirm or |
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| 146 | rule out the LSND claim. |
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| 147 | \item The absolute neutrino mass scale |
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| 148 | is probed in three ways. |
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| 149 | Firstly, the endpoint spectrum of electrons in nucleon |
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| 150 | ($^3H$) $\beta$ decay is sensitive to the ``effective |
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| 151 | electron neutrino mass'' |
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| 152 | $$ m_e^2 = [c^2_{13} c_{12}^2 m_1^2 + |
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| 153 | c^2_{13} s_{12}^2 m_2^2 + |
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| 154 | s^2_{13} m_3^2 ]^2 \leq 1.8 ~{\rm eV}~~.$$ |
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| 155 | |
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| 156 | Cosmological Large Scale Structure is affected |
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| 157 | by neutrino masses, because neutrino free-streaming |
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| 158 | in the early Universe would suppress density fluctuations |
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| 159 | on small scales. Current cosmological data sets the constraint: |
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| 160 | $$ m_1 + m_2 + m_3 \leq 0.47 - 1.4 {\rm eV}$$ |
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| 161 | The range of the bound is representative of different |
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| 162 | results in the literature, which are based on |
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| 163 | inequivalent data sets. The strong bound uses |
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| 164 | Ly$\alpha$ data to probe small scale structure; |
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| 165 | this data is sometimes left out because of |
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| 166 | uncertain systematic errors. |
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| 167 | |
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| 168 | The final observable to which neutrino masses |
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| 169 | could contribute---if they are majorana--- |
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| 170 | is lepton number violating neutrino-less |
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| 171 | double $\beta$ decay ($0 \nu 2 \beta$). |
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| 172 | The amplitude can be written as a nuclear |
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| 173 | matrix element, |
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| 174 | $\times$ the coefficient of a $\Delta L = 2$ |
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| 175 | non-renormalisable operator. This coefficient |
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| 176 | can be calculated perturbatively from the |
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| 177 | new physics that permits this type of decay. |
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| 178 | When this new physics is |
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| 179 | majorana neutrino masses, the coefficient |
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| 180 | is proportional to |
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| 181 | $ m_{ee}$, where |
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| 182 | $$ |
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| 183 | m_{ee} = [c_{13}^2c_{12}^2m_1 + c_{13}^2s_{12}^2m_2e^{i \phi_2} |
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| 184 | + s_{13}^2m_3e^{i \phi_3} ] |
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| 185 | $$ |
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| 186 | The PMNS matrix has be taken |
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| 187 | $U = V P$, with $V$ CKM-like with one phase $\delta$ |
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| 188 | ($V_{13} = \sin \theta_{13}e ^{-i \delta}$), and $P = diag |
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| 189 | \{ 1, e^{ \phi_2/2}, e^{i (\phi_3/2 + \delta)} \} |
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| 190 | $ (See talk by G. Fogli.) |
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| 191 | |
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| 192 | There is a controversial claim that $0 \nu 2 \beta$ |
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| 193 | has been detected in $^{76}Ge$, with a rate corresponding |
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| 194 | to $|m_{ee}| \simeq 0.23 \pm 0.18 $ eV. A |
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| 195 | disagreement with the cosmological bound |
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| 196 | can be avoided by not using Ly$\alpha$ data. |
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| 197 | \item Are neutrinos Majorana or Dirac? |
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| 198 | Oscillation experiments are sensitive to |
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| 199 | mass$^2$ differences, so do not distinguish whether |
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| 200 | neutrinos are majorana |
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| 201 | or dirac. |
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| 202 | The majorana nature of neutrinos, which is |
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| 203 | ``natural'' in the popular seesaw mechanism, |
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| 204 | can be tested in processes that violate lepton number, |
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| 205 | such as $0 \nu 2 \beta$. |
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| 206 | \item Is the mass pattern hierarchical |
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| 207 | ($\Delta m_{13}^2 >0)$ or inverted ($\Delta m_{13}^2<0$)? |
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| 208 | Oscillation probabililities in matter, |
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| 209 | for neutrinos and antineutrinos, depend on this sign, |
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| 210 | because the matter contribution to the mass matrix |
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| 211 | changes sign between neutrinos and anti-neutrinos. |
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| 212 | Long baseline neutrino beams and the flux of |
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| 213 | neutrinos from supernovae are sensitive to this sign. |
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| 214 | \item What is the value of $\theta_{13}$? There are |
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| 215 | only upper bounds on this remaining angle of the PMNS matrix, |
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| 216 | It can be probed by looking for a $\nu_e$ |
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| 217 | contribution to $\Delta m_{13}^2$ oscillations. |
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| 218 | This angle controls ``three flavour'' effects, like |
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| 219 | CP violation. |
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| 220 | \item What is the value of $\delta$, the ``Dirac phase'' of the PMNS |
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| 221 | matrix, which contributes to CP violation in neutrino |
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| 222 | oscillations (multiplied by $\sin \theta_{13}$)? |
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| 223 | \item is $\theta_{23}$ maximal? |
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| 224 | \end{enumerate} |
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| 225 | The sensitivity of various beam and |
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| 226 | detector combinations is illustrated in figure |
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| 227 | \ref{Ellis}. |
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| 228 | |
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| 229 | \begin{figure}[ht] |
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| 230 | %\vspace{4cm} |
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| 231 | \epsfig{figure=./figures/Fig2.eps,height=7.cm} |
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| 232 | \hspace{1cm} |
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| 233 | \epsfig{figure=./figures/fig3a.ps,height=7.cm} |
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| 234 | \caption{ plots shown in the presentation of J Ellis, |
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| 235 | showing the sensitivity to $\theta_{13}$, $\Delta m_{12}^2$, |
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| 236 | and $\delta$ of various beams. } |
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| 237 | %\vspace{4cm} |
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| 238 | \protect\label{Ellis} |
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| 239 | \end{figure} |
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| 240 | |
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| 241 | |
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| 242 | \subsection{$\theta_{13}$, $\delta$ and and the sign of $\Delta m_{13}^2$ } |
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| 243 | \label{TS} |
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| 244 | |
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| 245 | |
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| 246 | %Summary of discussions by Kajita, Nakahata, elsewhere? |
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| 247 | |
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| 248 | Determining items 4-6 (of the above list) |
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| 249 | at a future megatonne detector was |
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| 250 | discussed by T. Schwetz, and |
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| 251 | J Ellis presented prospects for beams from CERN. |
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| 252 | |
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| 253 | It is known that |
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| 254 | the 3-flavour oscillation probability has degeneracies, |
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| 255 | as can be |
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| 256 | seen from |
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| 257 | %\beq |
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| 258 | \begin{equation} |
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| 259 | P_{\mu e} \simeq \sin^2 2\theta_{13} \sin^2 \theta_{23} \sin^2 \Delta_{ 31} |
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| 260 | + \alpha^2 \sin^2 \theta_{12} \cos^2 \theta_{23} |
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| 261 | \Delta^2_{31} + \alpha \sin 2\theta_{12} |
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| 262 | \sin 2\theta_{13} \sin2\theta_{23} \Delta_{ 31} \sin \Delta_{ 31} \cos( |
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| 263 | \Delta_{ 31} \pm \delta). |
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| 264 | \end{equation} |
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| 265 | %\end{equation} |
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| 266 | |
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| 267 | where $\alpha = \Delta_{21}/ \Delta_{31}$, and |
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| 268 | $ \Delta_{31} = (m_3^2 - m_1^2)L/4 E_\nu$. |
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| 269 | For instance, a measured $P_{\mu e}$ could corresponds |
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| 270 | to several solutions in the ($\delta, \theta_{13}$) plane. |
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| 271 | This is refered to as the ``intrinsic'' degeneracy. |
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| 272 | There are additional degeneracies associated with |
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| 273 | the sign of $\Delta m_{13}^2$ (``hierarchy'' degeneracy), and with |
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| 274 | the sign of $\pi/4 - \theta_{23}$ (``quadrant'' degeneracy), if |
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| 275 | $\theta_{23}$ is not maximal. |
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| 276 | |
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| 277 | The degeneracies can be resolved with |
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| 278 | spectral information, and by looking at |
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| 279 | different channels. Having a $\beta$-beam and |
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| 280 | superbeam is helpful in this second respect. |
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| 281 | Spectral information is available with |
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| 282 | an off-axis beam, so the ($\delta, \theta_{13}$) |
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| 283 | degeneracy wouuld be absent at T2K-II |
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| 284 | (T2K to HyperK). |
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| 285 | |
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| 286 | T Schwetz discussed using atmospheric neutrino data to |
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| 287 | address the degeneracies, by measuring sub-dominant |
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| 288 | effects due to three-flavour mixing. He showed that |
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| 289 | there is an enhancement in the $\nu_e$ (or $\bar{\nu}_e$) |
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| 290 | flux, for multi-GeV events, due to $\theta_{13}$. |
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| 291 | The enhancement is for neutrinos in the |
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| 292 | normal hierarchy, and anti-neutrinos in the |
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| 293 | inverted case. Since the $\nu_e$ and $\bar{\nu}_e$ |
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| 294 | detection cross-sections are different, |
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| 295 | mesuring this enhancement would give information |
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| 296 | on $\theta_{13}$ and the sign of $\Delta m_{13}^2$. |
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| 297 | Sub-GeV events could be sensitive to |
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| 298 | the octant of $\theta_{23}$ via |
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| 299 | contributions arising due to $\Delta m_{12}^2$. |
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| 300 | |
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| 301 | |
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| 302 | The hierarchy and octant degeneracies could be reduced at T2K-II |
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| 303 | by using the the atmospheric neutrino data of HyperK. |
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| 304 | This was shown by combining |
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| 305 | a numerical 3-flavour atmospheric analysis, |
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| 306 | with long baseline simulation of |
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| 307 | the beam and detector using with |
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| 308 | the GloBES software |
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| 309 | ( http://www.ph.tum.de/globes/ ). |
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| 310 | An example figure is shown on the right below |
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| 311 | (figure \ref{TSfig}). |
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| 312 | Preliminary results, assuming a superbeam and |
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| 313 | $\beta$-beam from CERN, and including atmospheric data |
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| 314 | at a 450 kt Cherenkov detector at Frejus, were also |
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| 315 | shown. |
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| 316 | |
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| 317 | In summary, the combined analysis of atmospheric and |
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| 318 | long baseline neutrino data at a megaton detector |
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| 319 | could resolve parameter degeneracies---with the advantage |
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| 320 | that atmospheric neutrinos arrive ``for free''. |
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| 321 | |
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| 322 | |
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| 323 | |
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| 324 | \begin{figure}[ht] |
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| 325 | %\vspace{4cm} |
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| 326 | \epsfig{figure=./figures/TS.eps,height=7.cm,width=12.cm} |
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| 327 | \caption{ Resolving hierarchy(H) and octant (O) |
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| 328 | degeneracies using atmospheric neutrinos. The |
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| 329 | figures compare $\beta$-beam and SPL from CERN to Fr\'ejus, |
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| 330 | (details of the experiments can be found |
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| 331 | in the NuFact05 talks of Mezzetto and Campagne), and T2K |
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| 332 | to HK |
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| 333 | The detector in all cases is 450 kt water Cherenkov. } |
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| 334 | %\vspace{4cm} |
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| 335 | \protect\label{TSfig} |
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| 336 | \end{figure} |
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| 337 | |
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| 338 | |
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| 339 | |
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| 340 | |
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| 341 | |
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| 342 | \subsection{ Theoretical interest} |
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| 343 | |
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| 344 | One of the outstanding puzzles for particle theorists |
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| 345 | is the origin of Yukawa couplings. There are many models, |
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| 346 | which fit the masses |
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| 347 | and mixing angles observed in the quark and lepton sector, |
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| 348 | %with a variety of free parameters, |
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| 349 | %However, |
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| 350 | but none are particularily compelling. Additional hints from |
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| 351 | the data --- symmetries respected by the masses, |
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| 352 | constraints on the Yukawa parameters--- would be particularily |
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| 353 | welcome. Measuring the third leptonic mixing angle $\theta_{13}$, |
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| 354 | and determining whether $\theta_{23}$ is maximal, |
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| 355 | are both important in this respect. |
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| 356 | |
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| 357 | |
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| 358 | A popular mechanism to explain the smallness of |
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| 359 | neutrino masses is the seesaw, which has 18 parameters |
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| 360 | in its simplest form (type I) with three $\nu_R$. |
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| 361 | Twelve of these parameters appear among |
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| 362 | the light leptons (although not all are realistically |
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| 363 | measurable), and some of the remaining unknowns |
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| 364 | affect $\mu$ and $\tau$ decays in SUSY. So |
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| 365 | measuring many neutrino parameters with |
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| 366 | good accuracy would reduce the parameter space of seesaw models. |
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| 367 | |
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| 368 | |
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| 369 | If $\theta_{13}$ is found to be large ($\gappeq .01$, see |
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| 370 | figure \ref{Ellis}), the phase $\delta$ of the PMNS matrix |
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| 371 | could be experimentally accessible. Observing CP violation |
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| 372 | in the leptons, for the first time, would be an exciting |
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| 373 | phenomenological novelty. |
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| 374 | %\footnote{ |
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| 375 | %The PMNS matrix contains one |
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| 376 | %unremoveable phase, so CP violation in oscillations |
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| 377 | %is phenomenologically ``expected''. But it is |
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| 378 | %important to verify expectations---we also ``expected'' |
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| 379 | %mixing angles in the lepton sector to be small.} |
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| 380 | It is also tempting to relate $\delta$ to |
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| 381 | the CP violation required in the generation of |
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| 382 | the matter excess of the Universe (baryo/lepto-genesis). |
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| 383 | Various leptogenesis mechanisms |
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| 384 | can be implemented in the seesaw model, |
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| 385 | and depend on some combination |
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| 386 | of the seesaw's complex couplings. Observing |
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| 387 | $\delta \neq 0$ would demonstrate that at least one |
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| 388 | combination of couplings is complex, thereby |
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| 389 | suggesting that the phases relevant for leptogenesis |
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| 390 | might also be present. |
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| 391 | |
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| 392 | |
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| 393 | |
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| 394 | |
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| 395 | |
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| 396 | \section{Theoretical expectations : Nucleon Decay} |
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| 397 | |
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| 398 | Nucleon decay was |
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| 399 | the original motivations for large underground detectors, |
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| 400 | ancestors of the megatonne, and |
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| 401 | attracted attention from many speakers during the |
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| 402 | workshop. |
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| 403 | The theoretical expectations for |
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| 404 | the proton's lifetime were discussed in some |
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| 405 | detail in the talks of of J. Ellis and L. Covi. |
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| 406 | |
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| 407 | Our concept of theoretical progress is |
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| 408 | that we advance by unifying apparently diverse |
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| 409 | concepts. An example of |
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| 410 | successful unification is the Standard Model, which |
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| 411 | united electromagnetism with the weak interactions. |
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| 412 | Some hints that quarks and leptons might be united |
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| 413 | in a larger theory are the curious anomaly cancellation |
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| 414 | among known fermions---where |
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| 415 | the quarks and leptons cancel each others contributions |
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| 416 | to dangerous operators which would destroy |
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| 417 | the consistency (and experimental accuracy) |
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| 418 | of the SM. Another tantalising hint is |
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| 419 | that the strong, and electroweak gauge couplings become equal |
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| 420 | at $\Lambda \sim 10^{16}$ GeV, suggesting a |
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| 421 | unique gauge interaction at this scale. |
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| 422 | |
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| 423 | Unifying the quarks and leptons into |
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| 424 | a multiplet means that there are particles |
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| 425 | in the theory that turn quarks into leptons, |
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| 426 | so baryons can decay. Observing proton decay would |
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| 427 | be a smoking gun for such theories, |
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| 428 | confirming that our theoretical preference |
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| 429 | for unified theories is reflected in nature---and |
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| 430 | it could probe higher energy scales, |
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| 431 | or shorter distances, than any previous observation. |
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| 432 | It also could give some information on mixing |
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| 433 | angles in the right-handed quark sector, about which |
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| 434 | the Standard Model says nothing. |
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| 435 | |
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| 436 | |
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| 437 | \subsection{SU(5)} |
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| 438 | |
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| 439 | |
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| 440 | The simplest GUT is SU(5), the |
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| 441 | lowest rank (``smallest'') group capable |
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| 442 | of accomodating all the SM particles. % is SU(5), |
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| 443 | %of rank 4, which was much studied at the birth of GUTS. |
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| 444 | SO(10) is the one possibility at rank 5, and it |
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| 445 | has the advantage over SU(5) of accomodating |
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| 446 | the right-handed neutrino (SM gauge singlet) |
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| 447 | in its 16-dimensional multiplets. At rank six there |
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| 448 | is a group $E_6$, which appears in some string models. |
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| 449 | |
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| 450 | In the minimal SU(5) GUT, |
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| 451 | the colour-triplet $d^c = \overline{d_R}$ are combined with |
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| 452 | the lepton SU(2) doublet $\ell_L$ into a |
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| 453 | $\bar{5}$, and the $e^c$ shares a 10 |
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| 454 | with the $q_L$ and $u^c$. |
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| 455 | The X and Y gauge bosons, |
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| 456 | which acquire masses $\sim M_{GUT}$ when |
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| 457 | SU(5) is broken, have Baryon + Lepton |
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| 458 | number violating gauge interactions because |
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| 459 | they mix different multiplet members. |
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| 460 | They mediate proton decay |
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| 461 | via dimension six operators such as |
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| 462 | \begin{equation} |
---|
| 463 | \frac{ g_5^2}{M_X^2} \epsilon_{\alpha \beta \gamma} |
---|
| 464 | (\overline{d^c}_{\alpha,k} |
---|
| 465 | \overline{u^c}_{\beta,j} q_{\gamma , j} \ell _k - |
---|
| 466 | \overline{e^c}_{k} |
---|
| 467 | \overline{u^c}_{\alpha,j} q_{\beta , j} q_{\gamma ,k} |
---|
| 468 | ) |
---|
| 469 | \end{equation} |
---|
| 470 | There are also operators induced by GUT Higgses, |
---|
| 471 | with baryon number violating Yukawa-strength couplings. |
---|
| 472 | |
---|
| 473 | Proton decay is expected |
---|
| 474 | at rates |
---|
| 475 | \begin{equation} |
---|
| 476 | \Gamma_{p} = C \frac{\alpha_{5}^2 m_p^5}{M_X^4} |
---|
| 477 | \end{equation} |
---|
| 478 | where $C$ is a constant englobing mixing angles, |
---|
| 479 | renormalisation group running, and strong |
---|
| 480 | interaction effects. The dominant decay channel in |
---|
| 481 | non-supersymmetric SU(5) is $ p \rightarrow \pi^0 e^+$. |
---|
| 482 | The experimental limit |
---|
| 483 | $\tau_{p \rightarrow \pi e} > 6.9 \times 10^{33}$ years, |
---|
| 484 | imposes $M_X \geq 7.3 \times 10^{15}$ GeV, |
---|
| 485 | so non-SUSY SU(5) is |
---|
| 486 | ruled out because this is above |
---|
| 487 | the mass scale where the gauge couplings approximately |
---|
| 488 | unify. |
---|
| 489 | |
---|
| 490 | |
---|
| 491 | |
---|
| 492 | |
---|
| 493 | Proton decay in supersymmetric SU(5) |
---|
| 494 | is different in many respects. The GUT scale |
---|
| 495 | (determined from gauge coupling unification) is |
---|
| 496 | higher, so decays mediated by |
---|
| 497 | $X$ and $Y$ are slower. However, there are new |
---|
| 498 | {\it dimension 5} operators, induced by |
---|
| 499 | the coloured triplet Higgsino |
---|
| 500 | that shares a 5 with SM-type doublet Higgsinos, and which |
---|
| 501 | has Yukawa couplings to SM fields. Schematically |
---|
| 502 | these operators can be written |
---|
| 503 | $$ |
---|
| 504 | \frac{Y^{ij}_{qq} Y^{km}_{ql}}{2 M_c } |
---|
| 505 | Q_iQ_jQ_kL_m + |
---|
| 506 | \frac{ Y^{ij}_{ue} Y^{km}_{ud} }{ M_c } U^c_i E^c_j U^c_k D^c_m |
---|
| 507 | $$ |
---|
| 508 | where $M_c$ is the triplet |
---|
| 509 | Higgsino mass $\leq M_X$, |
---|
| 510 | the capitals are superfields, two of which |
---|
| 511 | are scalars and two fermions. |
---|
| 512 | Dressing this operator with the exchange |
---|
| 513 | of a ``-ino'' gives a 4-fermion operator |
---|
| 514 | $\propto 1/(m_{SUSY} M_{c})$. This is |
---|
| 515 | enhanced with respect to the $X$-boson |
---|
| 516 | exchange, but suppressed by small Yukawa couplings. |
---|
| 517 | In addition, the SM SU(2) and SU(3) |
---|
| 518 | contractions are antisymmetric, so |
---|
| 519 | the operator is flavour non-diagonal, giving |
---|
| 520 | a dominant decay $p \rightarrow K^+ \bar{\nu}$. |
---|
| 521 | |
---|
| 522 | There are relations among the quark and lepton |
---|
| 523 | Yukawa couplings, |
---|
| 524 | which depend on the GUT Higgs content of |
---|
| 525 | the model. |
---|
| 526 | The simplest would be for all the Yukawa matrices |
---|
| 527 | to be equal at the GUT scale, but some |
---|
| 528 | differences must be included to |
---|
| 529 | fit the observed fermion masses. |
---|
| 530 | The proton lifetime in SUSY SU(5) depends |
---|
| 531 | which Yukawa matrices are equal at the GUT scale: |
---|
| 532 | setting $Y_{ql} = Y_{ud}$ equal to the down |
---|
| 533 | Yukawa matrix $Y_d$ predicts a a proton lifetime shorter |
---|
| 534 | than the current SK limit of $ |
---|
| 535 | \tau_{p \rightarrow K \bar{\nu}} >1.9 \times 10^{33}$ years. |
---|
| 536 | However, setting $Y_{ql} = Y_{ud}$ equal to the |
---|
| 537 | charged lepton Yukawa $Y_e$ changes the |
---|
| 538 | dependence of $\tau_p$ on the fermion mixing |
---|
| 539 | angles, so lifetimes |
---|
| 540 | in excess |
---|
| 541 | of the bound |
---|
| 542 | can be found. |
---|
| 543 | The proton lifetime in SUSY SU(5) |
---|
| 544 | is uncertain due to the non-unification of |
---|
| 545 | Yukawa couplings. |
---|
| 546 | |
---|
| 547 | |
---|
| 548 | |
---|
| 549 | A possible string-motivated GUT model, discussed |
---|
| 550 | by J Ellis, is |
---|
| 551 | flipped SU(5)$\times U(1)$, where |
---|
| 552 | the SU(2) doublets of the SM are inverted |
---|
| 553 | ($\nu \leftrightarrow e, u \leftrightarrow d$) |
---|
| 554 | in the GUT multiplets. This extends |
---|
| 555 | the $p \rightarrow K^+ \bar{\nu}$ lifetime |
---|
| 556 | to $\tau \gsim 10^{35} - 10^{36}$ years, |
---|
| 557 | %CITE ? %\cite{Ellis:2002vk} |
---|
| 558 | %\bibitem{Ellis:2002vk} |
---|
| 559 | %J.~R.~Ellis, D.~V.~Nanopoulos and J.~Walker, |
---|
| 560 | %%``Flipping SU(5) out of trouble,'' |
---|
| 561 | %Phys.\ Lett.\ B {\bf 550} (2002) 99 |
---|
| 562 | %[arXiv:hep-ph/0205336]. |
---|
| 563 | %%%CITATION = HEP-PH 0205336;%%, |
---|
| 564 | potentially testable at a megaton detector. |
---|
| 565 | |
---|
| 566 | \subsection{ SO(10) in six space dimensions} |
---|
| 567 | |
---|
| 568 | |
---|
| 569 | In recent years, theorists have |
---|
| 570 | constructed models in $d>4$ dimensional |
---|
| 571 | space, with the additional dimensions |
---|
| 572 | compactified at some scale $\ll m_{pl}$. |
---|
| 573 | These models offer a framework to |
---|
| 574 | study new physics possibilities not |
---|
| 575 | included in the MSSM. L Covi discussed proton |
---|
| 576 | decay in a 6-dimensional SUSY SO(10) model, where |
---|
| 577 | the extra 2 dimensions are compactified |
---|
| 578 | on a torus (that has additional discrete symmetries). |
---|
| 579 | The four fixed points of this torus correspond |
---|
| 580 | to 4-dimensional branes, where SM |
---|
| 581 | particles can reside. Each |
---|
| 582 | SM generation lives at a different fixed point, |
---|
| 583 | with a different breaking of SO(10), so the Yukawas |
---|
| 584 | in this model are different from 4-dimensional |
---|
| 585 | SO(10). The higgsino mixing |
---|
| 586 | which allowed the dimension 5 proton decay |
---|
| 587 | operators is suppressed, so |
---|
| 588 | the dimension 6 $X$-mediated diagrams |
---|
| 589 | dominate in this supersymmetric extra-dimensional |
---|
| 590 | model. The proton decay rates |
---|
| 591 | are slightly larger than 4-dimensional SU(5) due to |
---|
| 592 | the sum over the tower of Kaluza-Klein $X$ modes, |
---|
| 593 | but they differ in the flavour |
---|
| 594 | structure. This has characteristic |
---|
| 595 | signatures, such as suppressing |
---|
| 596 | $p \rightarrow K^0 \mu^+$. The |
---|
| 597 | current bound $\tau_{p \rightarrow \pi^0 e^+} |
---|
| 598 | \geq 6.9 \times 10^{33}$ years implies in |
---|
| 599 | this model |
---|
| 600 | $M_X > 9.6 \times 10^{15}$ GeV $ \sim M_{GUT}$, |
---|
| 601 | suggesting that the proton could |
---|
| 602 | be discovered to have a lifetime $\sim 10^{34}$ years. |
---|
| 603 | |
---|
| 604 | |
---|
| 605 | |
---|
| 606 | In summary, proton decay is an unmistakable |
---|
| 607 | footprint of Unification, and is just around |
---|
| 608 | the corner in many models. Looking to the |
---|
| 609 | future, once proton decay is observed, |
---|
| 610 | the branching ratios will open a new |
---|
| 611 | perspective on the structure and origin |
---|
| 612 | of the Yukawa matrices, giving new |
---|
| 613 | information on the Yukawa puzzle. |
---|
| 614 | |
---|
| 615 | |
---|
| 616 | |
---|
| 617 | \section{From the Sky: Supernova Neutrinos} |
---|
| 618 | |
---|
| 619 | Supernova neutrinos were discussed by A Dighe |
---|
| 620 | (galactic supernovae) and S Ando(relic neutrinos), |
---|
| 621 | and also by G Fogli. Astrophysical |
---|
| 622 | observation of nearby galaxies suggests |
---|
| 623 | that 1-4 supernovae should take place in our galaxy |
---|
| 624 | per century. Neutrinos carry $ 99 \%$ of the |
---|
| 625 | star's binding energy, |
---|
| 626 | so these infrequent events could |
---|
| 627 | be a fund of information about |
---|
| 628 | neutrino parameters and supernova astrophysics. |
---|
| 629 | |
---|
| 630 | |
---|
| 631 | A real-time SN within 10 kpc may determine whether the |
---|
| 632 | hierarchy is normal or inverted, and be sensitive to |
---|
| 633 | very small values of $\sin \theta_{13}$. |
---|
| 634 | A megatonne detector is probably required to see |
---|
| 635 | these effects. |
---|
| 636 | The neutrino signal could also trace |
---|
| 637 | the outward propagation of the shock which powers the optical |
---|
| 638 | explosion. |
---|
| 639 | |
---|
| 640 | |
---|
| 641 | %determine the location |
---|
| 642 | %of the SN in the sky to $\sim 10 ^o$ ( this could |
---|
| 643 | %be improved by a factor of 2 to 3 with Gadolinium). |
---|
| 644 | |
---|
| 645 | |
---|
| 646 | |
---|
| 647 | While waiting for the next galactic supernova, |
---|
| 648 | detectors could look for ``supernovae relic |
---|
| 649 | neutrinos'' (SRN), the diffuse background of neutrinos |
---|
| 650 | emitted by past supernovae. SK's present limit on |
---|
| 651 | this flux is background-limited, and |
---|
| 652 | just above predictions. Detecting these neutrinos |
---|
| 653 | could give useful information on neutrinos and the |
---|
| 654 | history of star formation. |
---|
| 655 | |
---|
| 656 | \subsection{soon in our galaxy?} |
---|
| 657 | |
---|
| 658 | |
---|
| 659 | A star of mass $\gsim 8 {\cal M}_{\odot}$ becomes |
---|
| 660 | unstable at the end of its life. It resembles |
---|
| 661 | an onion, with the different layers burning lighter |
---|
| 662 | elements into heavier, the end-products of one |
---|
| 663 | layer serving as fuel for the one underneath. |
---|
| 664 | At the centre develops an iron core, which eventually |
---|
| 665 | cannot support the outer layers, and collapses. |
---|
| 666 | Most of the binding energy is released as |
---|
| 667 | neutrinos. |
---|
| 668 | |
---|
| 669 | The SN neutrino flux has various components. |
---|
| 670 | The neutronisation burst takes place |
---|
| 671 | in the first 10 ms, as the |
---|
| 672 | heavy nuclei break up. It consists of $\nu_e$ |
---|
| 673 | from $p + e \rightarrow n + \nu_e$, and is |
---|
| 674 | emitted from the ``neutrinosphere'', that is, |
---|
| 675 | the radius from which neutrinos can free-stream |
---|
| 676 | outwards. The core density is near nuclear, above |
---|
| 677 | the $\sim 10^{10}$ g/cm$^3$ required |
---|
| 678 | to trap a 10 MeV neutrino. |
---|
| 679 | |
---|
| 680 | For the following 10 seconds, the core cools |
---|
| 681 | by emitting $\nu$ and $\bar{\nu}$ of all flavours. |
---|
| 682 | 99 $\%$ of the SN energy is emitted in |
---|
| 683 | these fluxes, refered to as ``initial'' |
---|
| 684 | fluxes $F^0$, whose |
---|
| 685 | characteristics are predicted to be flavour dependent. |
---|
| 686 | In particular, the average energies |
---|
| 687 | of $\nu_e$, $\bar{\nu}_e$ |
---|
| 688 | and $\nu_x$ are predicted to differ: |
---|
| 689 | %with the average energies |
---|
| 690 | $E_0(\nu_e) \sim 10-12$ MeV, |
---|
| 691 | $E_0(\bar{\nu}_e) \sim 13-16$ MeV, |
---|
| 692 | and $E_0({\nu}_x) \sim 15-25$ MeV. |
---|
| 693 | The more weakly interacting neutrinos are |
---|
| 694 | more energetic because they escape |
---|
| 695 | from closer to the hot centre of the star. |
---|
| 696 | |
---|
| 697 | As the neutrinos travel outwards, they pass |
---|
| 698 | through ever-decreasing density, so |
---|
| 699 | matter effects on the mixing are |
---|
| 700 | crucial. Level-crossing occurs when |
---|
| 701 | $\Delta m^2 \cos 2 \theta = \pm 2 \sqrt{2} E_\nu G_F n_e$, |
---|
| 702 | where the $+$ ($-$) refers to (anti) neutrinos. |
---|
| 703 | Flavour conversion is |
---|
| 704 | possible at two level crossings, |
---|
| 705 | corresponding to the solar and atmospheric |
---|
| 706 | mass differences, and can |
---|
| 707 | appear in the $\nu$ or the $\bar{\nu}$ |
---|
| 708 | depending on the mass hierarchy. This will mix the |
---|
| 709 | initial neutrino fluxes, which were labelled by flavour. |
---|
| 710 | |
---|
| 711 | Towards the centre of the star, $\nu_e$ is the heaviest neutrino. |
---|
| 712 | In the normal mass hierarchy, $\nu_e$ |
---|
| 713 | has a level crossing at the |
---|
| 714 | H resonance, which arises at a matter density |
---|
| 715 | $\sim 10^3$ g/cm$^3$, |
---|
| 716 | where $\nu_3$ can |
---|
| 717 | transform to $\nu_2$ via the atmospheric |
---|
| 718 | mass difference and $\theta_{13}$. % at this |
---|
| 719 | %resonance. |
---|
| 720 | The H resonance takes place in the $\bar{\nu}_e$ |
---|
| 721 | channel, for the inverted mass hierachy. |
---|
| 722 | The L resonance arises at a matter density |
---|
| 723 | $\sim 10$ g/cm$^3$. It is in the $\nu$ channel for |
---|
| 724 | both hierarchies, and crosses $\nu_2$ with |
---|
| 725 | $\nu_1$ via the solar mass difference and angle. |
---|
| 726 | The level crossing probability is adiabatic |
---|
| 727 | for the L resonance, and for the H resonance |
---|
| 728 | when $\sin^2 \theta_{13} \gappeq 10^{-3}$. |
---|
| 729 | %(refered to as ``large'' for the remainder |
---|
| 730 | %of this section.) |
---|
| 731 | It is non-adiabatic |
---|
| 732 | at the H resonance if |
---|
| 733 | $\sin^2 \theta_{13} \lappeq 10^{-3}$. |
---|
| 734 | %(``small, for the remainder of this section.) |
---|
| 735 | The fluxes arriving at the earth ($F$) depend on |
---|
| 736 | the initial fluxes ($F^0$) and the oscillation probabilities |
---|
| 737 | ($p$ and $\bar{p}$): |
---|
| 738 | $$ |
---|
| 739 | F_{\nu_e} = pF^0_{\nu_e} + (1 - p)F^0_{\nu_x} |
---|
| 740 | ~~~ |
---|
| 741 | F_{\bar{\nu}_e} = \bar{p} F^0_{\bar{\nu}_e} + (1 - \bar{p})F^0_{\nu_x} |
---|
| 742 | $$ |
---|
| 743 | (There is a related formula for $F_{{\nu}_x}$.) |
---|
| 744 | There are three interesting cases: |
---|
| 745 | \begin{itemize} |
---|
| 746 | \item Case A: normal hierarchy, $\sin^2 \theta_{13} \gappeq 10^{-3}$, |
---|
| 747 | ($p = 0$, $\bar{p} = \cos^2 \theta_{\odot}$) |
---|
| 748 | \item Case B: inverted hierarchy, $\sin^2 \theta_{13} \gappeq 10^{-3}$ |
---|
| 749 | (($p = \sin^2 \theta_{\odot}$, $\bar{p} = 0$) |
---|
| 750 | \item Case C: any hierarchy, $\sin^2 \theta_{13} \lappeq 10^{-3}$ |
---|
| 751 | ($p = \sin^2 \theta_{\odot}$, $\bar{p} = \cos^2 \theta_{\odot}$) |
---|
| 752 | \end{itemize} |
---|
| 753 | |
---|
| 754 | |
---|
| 755 | A Dighe discussed whether these cases could be distinguished |
---|
| 756 | in the observable signal, given that the initial |
---|
| 757 | spectra are poorly known, and only the final spectra for |
---|
| 758 | $\bar{\nu}_e$ are cleanly available. It is |
---|
| 759 | difficult to find observables that do not |
---|
| 760 | depend on assumptions about the initial spectra. |
---|
| 761 | A possibility, if the SN neutrino flux crosses |
---|
| 762 | the earth, is to look for oscillations in the |
---|
| 763 | signal due to matter effects in the earth. |
---|
| 764 | This would contribute high frequency |
---|
| 765 | wiggles to the spectrum, which could be |
---|
| 766 | extracted from the data at a megaton |
---|
| 767 | detector. |
---|
| 768 | For the normal hierarchy or small |
---|
| 769 | $\theta_{13}$, these earth effects would |
---|
| 770 | appear in the $\bar{\nu}_e$ channel, so |
---|
| 771 | observing such wiggles would eliminate case B. |
---|
| 772 | |
---|
| 773 | It could also be possible to identify |
---|
| 774 | earth effects if the SN is observed with two detectors, |
---|
| 775 | where one is in the earth's shadow and |
---|
| 776 | the other not. As A. Dighe discussed, IceCube could |
---|
| 777 | be the second detector, which would be complementary |
---|
| 778 | to Hyper-K. |
---|
| 779 | |
---|
| 780 | |
---|
| 781 | Neutrinos have a crucial role in the explosion of supernovae, |
---|
| 782 | for instance the energy they deposit in the shock may |
---|
| 783 | be the critical contribution that allows |
---|
| 784 | the star to explode. The interactions between |
---|
| 785 | the shock and the outgoing neutrinos may also |
---|
| 786 | provide information on the neutrino parameters. As the shock passes |
---|
| 787 | through the $H$ resonance region, it can |
---|
| 788 | make adiabatic transitions non-adiabatic, |
---|
| 789 | thereby temporarily turning scenarios A and B, |
---|
| 790 | into scenario C. One can therefore hope to |
---|
| 791 | to track the shock fronts through the |
---|
| 792 | star in the time-dependent neutrino signal. |
---|
| 793 | |
---|
| 794 | |
---|
| 795 | A nearby supernova would illuminate |
---|
| 796 | the earth with neutrinos. This flux can be |
---|
| 797 | used to simultaneously obtain information about |
---|
| 798 | the source, and about neutrino properties. |
---|
| 799 | At a megatonne detector, |
---|
| 800 | ``earth effects'' in the |
---|
| 801 | neutrino spectra could be observed, |
---|
| 802 | which would give SN-model |
---|
| 803 | independent information on the hierarchy |
---|
| 804 | (inverted vs normal) and whether $\theta_{13}$ |
---|
| 805 | is large or small. Alternatively, if |
---|
| 806 | the SN neutrinos do not cross the earth, |
---|
| 807 | information about neutrino parameters |
---|
| 808 | could be extracted from shock wave |
---|
| 809 | propagation effects in the neutrino |
---|
| 810 | spectra. |
---|
| 811 | |
---|
| 812 | |
---|
| 813 | \subsection{relics} |
---|
| 814 | |
---|
| 815 | |
---|
| 816 | Most of the energy of a supernova is released |
---|
| 817 | as neutrinos. The diffuse background of |
---|
| 818 | these neutrinos, today, depends on the |
---|
| 819 | neutrino spectrum emitted from each explosion, |
---|
| 820 | on the oscillation of those neutrinos in |
---|
| 821 | the SN and in the earth, and on the |
---|
| 822 | supernova rate over the past history of |
---|
| 823 | the Universe. |
---|
| 824 | |
---|
| 825 | As discussed in the previous section, the neutrino |
---|
| 826 | fluxes emitted from the SN core are expected to |
---|
| 827 | be flavour dependent, and to oscillate |
---|
| 828 | due to matter effects as they leave the star. For |
---|
| 829 | instance, in the normal hierarchy, a $\bar{\nu}_e$ |
---|
| 830 | emitted from the core is the lightest $\bar{\nu}$, |
---|
| 831 | due to matter effects, so it will exit |
---|
| 832 | the star as $\bar{\nu}_1$. The observed $\bar{\nu}_e$ |
---|
| 833 | flux will therefore be |
---|
| 834 | $$ F_{\bar{\nu}_e} = | U_{ei}|^2 F_{\bar{\nu}_i} |
---|
| 835 | = | U_{e1}|^2 F^0_{\bar{\nu}_e } |
---|
| 836 | + (1 - | U_{e1}|^2) F^0_{\bar{\nu}_x} |
---|
| 837 | $$ |
---|
| 838 | so $ (1 - | U_{e1}|^2) \sim 30 \% $ comes from the |
---|
| 839 | harder $\nu_x$ spectrum. The oscillations |
---|
| 840 | enhance the high-energy tail, but not dramatically |
---|
| 841 | in the detectable energy range ($< 30$ MeV). |
---|
| 842 | |
---|
| 843 | |
---|
| 844 | The SN rate is infered from the star formation rate, |
---|
| 845 | which can be extracted from other cosmological observables. |
---|
| 846 | Using the recent Galactic Evolution Explorer data, |
---|
| 847 | the event rate at SK can be calculated, and is |
---|
| 848 | found to be mostly due to SN at $z < 1$. |
---|
| 849 | A few $\bar{\nu}_e p \rightarrow n e^+$ events |
---|
| 850 | per year are predicted in the $E > 18$ MeV window |
---|
| 851 | where the flux exceeds the solar and armospheric |
---|
| 852 | neutrinos. Unfortunately, in this range there |
---|
| 853 | is a background from the decays of slowly moving muons, |
---|
| 854 | which are produced |
---|
| 855 | by atmospheric $\nu_\mu$ and are invisible at SK. |
---|
| 856 | So SK can set an upper limit on the SRN flux, |
---|
| 857 | which can then be inverted into a constraint |
---|
| 858 | on the supernova rate. The bound is just above |
---|
| 859 | theoretical predictions, so SRN might be seen |
---|
| 860 | using 5-10 years of data. |
---|
| 861 | |
---|
| 862 | The background could be reduced by |
---|
| 863 | adding Gadolinium to a water Cherenkov |
---|
| 864 | detector. This would tag the neutrons produced |
---|
| 865 | in $\bar{\nu}_e p \rightarrow n e^+$, |
---|
| 866 | and therefore distinguish the $\bar{\nu}_e$ |
---|
| 867 | from other neutrinos. Liquid Argon detectors |
---|
| 868 | are sensitive to $\nu_e$, so would be complementary |
---|
| 869 | to a water detector. |
---|
| 870 | |
---|
| 871 | S. Ando also discussed the possibility of observing, |
---|
| 872 | at a megatonne detector, a few neutrinos from SN |
---|
| 873 | in nearby galaxies ($\sim$ Mpc away). This would give |
---|
| 874 | the time of the collapse, helpful for gravitational |
---|
| 875 | wave searches. |
---|
| 876 | |
---|
| 877 | In summary, the SK limit on supernovae relic |
---|
| 878 | neutrinos is just above the theoretical prediction; |
---|
| 879 | a future megatonne detector should therefore |
---|
| 880 | have a good chance to see them. |
---|
| 881 | At a megatonne Cerenkov detector, a 5 $\sigma$ detection could |
---|
| 882 | be possible with pure water after a few years, |
---|
| 883 | ($\sim$ 300 events/yr would be expected with Gd). |
---|
| 884 | A 100 kt liquid Argon detector would expect |
---|
| 885 | $\sim 57 \pm 12 $ events after 5 years. |
---|
| 886 | |
---|
| 887 | |
---|
| 888 | |
---|
| 889 | |
---|
| 890 | |
---|
| 891 | |
---|
| 892 | |
---|
| 893 | |
---|
| 894 | |
---|
| 895 | |
---|
| 896 | |
---|
| 897 | |
---|
| 898 | |
---|
| 899 | |
---|
| 900 | |
---|
| 901 | |
---|
| 902 | |
---|
| 903 | |
---|
| 904 | |
---|
| 905 | |
---|
| 906 | |
---|
| 907 | |
---|
| 908 | |
---|
| 909 | |
---|