{"id":1085,"date":"2015-03-06T19:50:02","date_gmt":"2015-03-06T19:50:02","guid":{"rendered":"http:\/\/www.particlebites.com\/?p=1085"},"modified":"2017-02-19T01:31:15","modified_gmt":"2017-02-19T01:31:15","slug":"muon-to-electron-conversion","status":"publish","type":"post","link":"https:\/\/www.particlebites.com\/?p=1085","title":{"rendered":"Muon to electron conversion"},"content":{"rendered":"
Presenting:<\/strong> Section 3.2 of “Charged Lepton Flavor Violation: An Experimenter’s Guide”
\nAuthors:\u00a0<\/strong>R. Bernstein, P. Cooper
\nReference<\/strong>:\u00a01307.5787<\/a>\u00a0(Phys. Rept. 532 (2013) 27<\/a>)<\/div>\n

Not all searches for new physics\u00a0involve colliding\u00a0protons at the\u00a0the highest\u00a0human-made energies.\u00a0An alternate approach is to look for\u00a0deviations\u00a0in ultra-rare events at low energies. These deviations\u00a0may be the quantum\u00a0footprints of new, much heavier particles.\u00a0In this bite, we’ll focus on\u00a0the decay of a muon to an electron in the presence of a heavy atom.<\/p>\n

\"Muons<\/a>
Muons conversion into an electron\u00a0in the presence of an atom, aluminum.<\/figcaption><\/figure>\n

The muon<\/strong><\/a> is a heavy version of the electron.There \u00a0are a few properties that make muons\u00a0nice systems\u00a0for precision measurements:<\/p>\n

    \n
  1. They’re easy to produce<\/strong>.\u00a0When you smash protons into a dense target, like tungsten, you get lots of light hadrons—among them, the charged pions. These charged pions decay into muons, which one can then collect by bending their trajectories with magnetic fields. (Puzzle: why don’t\u00a0pions decay into electrons? Answer below<\/a>.)<\/li>\n
  2. They can replace electrons in atoms<\/strong>. \u00a0If you point this beam of muons\u00a0into a\u00a0target, then some of the muons will replace electrons in the target’s atoms. This is very nice because\u00a0these\u00a0“muonic atoms” are described by non-relativistic quantum mechanics with the electron mass replaced with ~100 MeV.\u00a0(Muonic hydrogen\u00a0was previous mentioned in this bite<\/a> on the proton radius problem.)<\/li>\n
  3. They decay<\/strong>, and the decay products always include an electron that can be detected.\u00a0\u00a0In vacuum it will decay into an electron and two neutrinos through the weak force, analogous to beta decay<\/a>.<\/li>\n
  4. These decays are sensitive to\u00a0virtual effects.<\/strong> You don’t need to directly create a new particle in order to see its effects. Potential new particles\u00a0are constrained\u00a0to be\u00a0very heavy to explain their non-observation at the LHC. However, even these heavy particles can\u00a0leave an \u00a0imprint on muon decay\u00a0through ‘virtual effects’ according (roughly) to the\u00a0Heisenberg uncertainty\u00a0principle: you can quantum mechanically violate energy conservation, but only for very short\u00a0times.<\/li>\n<\/ol>\n
    \"Reach<\/a>
    Reach of muon conversion experiments from 1303.4097<\/a>. The y<\/em> axis is the energy scale that can be probed and the x<\/em> axis parameterizes\u00a0different ways that\u00a0lepton flavor violation can appear in a theory.<\/figcaption><\/figure>\n

    One should\u00a0be surprised that muon conversion is even possible.\u00a0The process $latex \\mu \\to e$ cannot occur\u00a0in vacuum because it\u00a0cannot simultaneously conserve energy and momentum. (Puzzle: why is this true? Answer below<\/a>.) However, this process is allowed in the presence of a heavy nucleus that can absorb the additional momentum, as shown in the\u00a0comic at the top of this post.<\/p>\n

    Muon \u00a0conversion experiments\u00a0exploit this by\u00a0forming muonic atoms in the 1s\u00a0<\/em>state and waiting for the muon\u00a0to convert into an electron which can then be detected. The upside is that\u00a0all\u00a0electrons from conversion\u00a0have a fixed energy because\u00a0they all come from the same initial state: 1s muonic\u00a0aluminum at rest in the lab frame.\u00a0This is in contrast with\u00a0more common muon decay modes which involve two neutrinos and an electron; because this\u00a0is a\u00a0multibody final state,\u00a0there is a\u00a0smooth distribution of electron energies. This\u00a0feature allows\u00a0physicists to distinguish between the $latex \\mu \\to e$\u00a0conversion versus the more frequent\u00a0muon decay $latex \\mu \\to e \\nu_\\mu \\bar \\nu_e$ in orbit or muon capture by the nucleus\u00a0(similar to electron capture<\/a>).<\/p>\n

    The\u00a0Standard Model prediction for this rate is miniscule—it’s weighted by powers of the\u00a0neutrino to the\u00a0W<\/em> boson mass ratio \u00a0(Puzzle: how does one see this?\u00a0Answer below<\/a>.). In fact, the current experimental bound on muon conversion comes from the Sindrum II experiment <\/a>\u00a0looking at muonic gold which constrains the relative rate of muon conversion to muon capture by the gold nucleus to be less than $latex 7 \\times 10^{-13}$.\u00a0This, in turn, constrains models of new physics that\u00a0predict some level of charged lepton flavor violation<\/strong>—that is, processes that change the flavor of a charged lepton, say going from muons to electrons.<\/p>\n

    The plot on the right shows the energy scales that are\u00a0indirectly<\/em> probed\u00a0by upcoming muonic aluminum experiments: the Mu2e experiment<\/a> at Fermilab and the COMET experiment<\/a> at J-PARC. The blue lines show bounds from\u00a0another rare muon decay: muons decaying into an electron and photon. The black solid lines show the reach for muon conversion in muonic aluminum. The dashed lines correspond to different\u00a0experimental sensitivities (capture rates for conversion, branching ratios for decay with a photon). Note that the\u00a0energy scales\u00a0probed\u00a0can reach\u00a01-10 PeV—that’s 1000-10,000 TeV—much higher than the energy scales direclty probed by the LHC! In this way, flavor experiments and high energy experiments are complimentary searches for new physics.<\/p>\n

    These “next generation” muon conversion experiments are currently under construction and\u00a0promise to push the intensity frontier in conjunction with the LHC’s energy frontier.<\/p>\n

     <\/p>\n

     <\/p>\n

    Solutions to exercises:<\/h3>\n
      \n
    1. Why do pions decay into muons and not electrons? <\/strong>[Note: this requires some background in undergraduate-level particle physics.] One might expect that\u00a0if\u00a0a charged pion can decay into a muon and a neutrino, then it should also go into an electron and a neutrino. In fact, the latter should\u00a0dominate since there’s much more phase space.\u00a0However, the matrix element\u00a0requires a virtual\u00a0W<\/em> boson exchange and\u00a0thus depends on an [axial] vector current. The only\u00a0vector available from the pion system is its 4-momentum.\u00a0By momentum conservation this is $p_\\pi = p_\\mu + p_\\nu$. The lepton momenta then contract with Dirac matrices on the leptonic current to give a dominant piece\u00a0proportional to the lepton mass. Thus the\u00a0amplitude for charged pion decay into a muon is much larger than the amplitude for decay into an electron.<\/li>\n
    2. Why can’t a muon decay into an electron in vacuum?<\/strong>\u00a0The process $latex \\mu \\to e$ cannot simultaneously conserve energy and momentum. This is simplest to see in the\u00a0reference frame where the muon is at rest.\u00a0Momentum conservation requires the electron to\u00a0also be at rest. However, a\u00a0particle has rest energy equal to its mass, but now there’s now way a\u00a0muon at rest can pass on all of its energy to\u00a0an electron at rest.<\/li>\n
    3. Why is muon conversion in the Standard Model suppressed by the ration of the\u00a0neutrino\u00a0to\u00a0W<\/em> masses?\u00a0<\/strong>This can be seen\u00a0by drawing the Feynman diagram (fig below from 1401.6077<\/a>).\u00a0Flavor violation in the Standard Model requires a W<\/i> boson. Because the\u00a0W<\/em> is\u00a0much heavier than the muon, this must be virtual and appear only\u00a0as an internal leg. Further,\u00a0W<\/em>‘s\u00a0couple charged leptons to neutrinos, so there must also be a virtual neutrino.\u00a0The evaluation of this diagram into an amplitude gives factors of the neutrino mass in the numerator (required for the fermion chirality flip) and the\u00a0W<\/em> mass in the denominator. For\u00a0some details, see this post<\/a>.
      \n      \"Screen<\/a><\/li>\n<\/ol>\n

      Further Reading:<\/h3>\n