Lepton number conservation and possible lepton violating processes. Prakash Gautam [email protected] Department of Physics, Drexel University, Philadelphia, PA March 20, 2019 1 Introduction In the early 1936, Muon were were first detected from cosmic ray experiments. They had the same gauge quantum numbers like electrons and the Muon underwent decay by the interaction µ ! eνν¯. This led to a belief that this particle was just an excited electron. So it was expected that muon would de- excite to an electron µ ! eγ. In the 1940s searches for such decay processes began but it was found that this expected de-excitation did not occur, with the upper limit of branching ratio of 10−6. To explain this phenomenon a new hypothesis called the ‘two neutrino hypothesis‘ was presented which postulated that there were two lepton ‘flavors‘ which was each accompanied by a conserved quantum number called ‘lepton-flavor number‘. For the two particles known at that time electron-number and muon-number was given to them. 1 Leptons are fundamental particle with half-integer spin 2 . Since all leptons have half integer spin, they all obey Fermi-Dirac statistics, hence fermions. These particles do not undergo strong interaction. In the Standard Model (SM) of particle physics there are six leptons. Electron (e−), Muon (µ−) and Tau τ − and the associated neutrinos. The associated neutrinos are Electron Neutrino νe, νµ and ντ . These charged letpon with corresponding neutrinos form weak isospin doublet. 1 Table 1: Lepton number conservation example − − 0 + − µ ! νµ + e +ν ¯e 2n ! 2p + 2e Electron # 0 = 0 + 1 + -1 0 =6 0 + 2 Muon # 1 = 1 + 0 + 0 0 = 0 + 0 Tau # 0 = 0 + 0 + 0 0 = 0 + 0 SM is largely developed to explain the observation in particle physics. In every interactions observed so far, lepton number is found to be conserved. So conservation of lepton number enters SM as an empirical fact. Unlike U(1) local gauge symmetry for electromagnetic interaction or the U(2) local gauge symmetry for electroweak interaction, conservation of lepton number was not originally associated with gauge symmetries. However, there are some models which associate conservation of lepton number to a new U(1) local gauge symmetry [1] different from the U(1) symmetry for electromag- netism. According to these models, the SM Lagrangian (L) for the leptonic interaction is symmetric under the transformation L! e−iLθL, where L is the lepton number. The value of L is called lepton number and its value is one for all leptons and it is −1 for all antileptons, all other particle in standard model have L = 0. In any particle interaction in standard model the lepton number is conserved in interacting particles. − − In the example show in in Table 1 the interaction µ ! νµ + e +ν ¯e is observed while the interaction 2n0 ! 2p+ + 2e− is not observed because it violates the conservation of electron lepton number (Le). 2 Lepton violating Processes With more experimental data available and more particles discovered, it is now established that lepton number might not be a good quantum number. In some processes, they may be violated in a charged-current processes in- volving weak interaction over long distance like, neutrino oscillations. The observation of neutrino oscillation implies, all charged-lepton flavor violating processes are allowed and will occur at some level. 2 e− + AZ ! µ− + AZ−2 µ− + AZ ! e+ + AZ−2 p + AZ ! µ+ + e+ + (A + 1)Z−1 e− + e− ! π− + π− − − 0 νµ + e ! π + π There are various lepton number violating processes. Both the charged Lep- ton number violating processes and neutral lepton number violating processes are under study. Examples of possible neutral lepton number violating processes are Neutrino- less Double Beta Decay (0νββ) and for charged lepton, the lepton violating − − − − − − 0 processes are τ ! µ γ, τ ! µ µµ, τ ! µ KS. In standard model the charged lepton number violating are suppressed more than for the neutral leptons. They are in general ` ! `0γ, ` ! `0`00`00, `+X ! `0 +X and X ! ``¯0 where ` is one of charged lepton fe; µ, τg and X are all other particles with no lepton number associated. 2.1 Neutrinoless double beta decay Paul Dirac trying to get the relativistic quantum mechanics of fundamental particles discovered the Dirac equation µ (iγ @µ − m) = 0 this equation has its adjoint equation which predicts negative energy state particles. Since no such particle with negative mass were observed Dirac argued that their existence was forbidden due to Pauli’s exclusion principle. The solution of adjoint of Dirac equation predicted the existence of anti particle. This discovery doubled the standard model particles because very particle with non-zero spin had its antiparticle. For charged particles, the antiparticles have opposite charge. But for neutral particles, it was not clear whether the particles themselves were the anti particle of itself. In 1937 Ettore Majorana, an Italian physicist took upon this theory and ar- gued that since the theory is symmetric [2, 3] it was possible to replace the complex wave function with a real one for neutral fermions. Thus the sym- metric Majorana equation predicts, for neutral fermions, that the particle and 3 ν¯ ν¯ e− e− e− ν¯ ν¯ e− p+ p+ p+ p+ W − W − W − W − n0 n0 n0 n0 (a) Double beta decay (b) Neutrinoless double beta decay. Figure 1: Feynman diagram for two neutrino double beta decay and neutri- noless double beta decay. anti particle are identical. The first idea of double beta decay was proposed by M. Goeppert-Mayer in the paper titled Double Beta-Disintegration[4] in the form. − (Z; A) ! (Z + 2;A) + 2e + 2¯νe This transition occurs regardless of whether the neutrinos are their own an- tiparticle or not, i.e., whether they are Dirac or Majorana particles. Neutrinoless double beta decay however would not produce neutrinos at all. (Z; A) ! (Z + 2;A) + 2e− Observation of this transition will confirm that neutrinos are Majorana fermions, because the virtual anti neutrino should exchange with each other, which is only possible if neutrinos are antiparticle of themselves[5]. Also clear from the interaction above the lepton number is not conserved in this interaction. Since two neutrons in such an interaction change to two proton giving off two electron n0 + n0 ! p+ + p+ + e− + e− (1) In the interaction (1) the total lepton number before the interaction is 0 ( since neutrons have lepton number 0) but the lepton umber on the right is 2. Thus the total change in lepton number is 2. ∆L = 2 The rate of 0νββ depends on the lepton number violating parameter; which is the effective Majorana mass for the exchange of light but massive virtual 4 (a) s-channel for kaon neutrinoless kaon (b) t-channel Feynman diagram for kaon decay decay Figure 2: Feynman diagram for Kaon neutrinoless double muon decay neutrino (Fig 1b). There are also other mechanisms that could require mas- sive neutrinos in 0νββ. Some of which are Leptoquark Mechanism[6], SUSY accompanied neutrino exchange mechanism [7] etc. If we denote the effective Majorana mass by mββ, the decay rate of 0νββ is related to this mass by 0ν −1 2 2 T1=2 = G(Q; Z)jM(Z; A)j hmββi , here G(Q; Z) is the phase space factor which depends on the nucleus and the decay endpoint of the isotope, M(Z; A) is the nuclear matrix element, which depends on the nucleus and the model used. The Majornana mass is given by 2 X 2 hmββi = miUai i where Uai are the elements of Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS) matrix. 2.2 K-meson neutrinoless double muon decay In SM extension with Majorana neutrinos, it is possible for for K meson to decay into two muons and a pion. One of the possible lepton number violating process is the K-meson neutrinoless double decay. K+ ! µ+ + µ+ + π− Figure. 2 show two lowest order Feynman diagram for this decay process. It involves weak interaction. 5 2.3 Neutrinoless Tau (τ) decays One Possible ways of lepton violating process is neutrinoless Tau decay. Neu- − − trinoless tau-lepton can decay into either three leptons τ ! `1 `2`3 or one 0 − − 0 lepton and one KS meson (τ ! ` Ks ) where ` is either an electron or muon. Experiments have been performed to search for such a decay, however no evidence for such a decay have been observed so far[8] with the upper limit of branching ratio of 1 × 10−7. 2.4 Possibility of ∆L > 2 So far the various processes we discussed violate the lepton flavor conservation by two units. There have been studies done for the existence of processes that violate the lepton number conservation by more than two units. One would assume that the next lepton violating process similar to 0νββ is 0ν3β. But such decays are forbidden as they do not respect Lorentz sym- metry. The next in series is Neutrinoless Quadrapule Beta Decay (0ν4β) [9]. Existence of such a decay would imply the interaction of the type (A; Z) ! (A; Z + 4) + 4e− This decay process is allowed if the neutrinos are Dirac particles. This process changes the lepton number by four units. ∆L = 4 The lower limit for the half life of 0ν4β is more than 1 × 1041yrs[10]. This makes this process extremely unlikely to be observed with our current tech- nology. Other processes have been proposed which violate lepton flavor con- servation by more than four units ∆L = 5; 6;::: but none have been put to experimental tests so far. 3 Conclusion Observation of neutrino oscillation confirms that neutrinos are massive.