Lepton number is a conserved quantity in fundamental particle physics. As would be expected, this quantity is associated with the class of elementary particles called leptons: the electron, muon, and tau particles, and their corresponding neutrinos. All of these particles have lepton number +1, and their antiparticles have lepton number -1. All other Standard Model particles have lepton number 0.

Conservation of lepton number is obeyed by all known interactions, i.e. it is conserved absolutely. This has a significant effect on the nature of particle interactions and decays. Conservation of lepton number often prescribes the production of neutrinos and antineutrinos, since they have no charge and thus do not disturb the conservation of charge in an interaction. Two illustrative examples of conservation of lepton number are the decay of the pion and the decay of the neutron.

Examples

When a positive pion, π+, decays, it generally (99.99% of the time) decays into an antimuon, μ+. However, while an antimuon has lepton number -1, a pion has zero muon number. Thus the reaction π+ -> μ+ never occurs. Rather, we must add a muon neutrino, νμ, to the final state so that the lepton number of the final state is also zero. So lepton number conservation dictates that pion decay proceeds by the reaction: π+ -> μ+ + νμ

A free neutron will decay into a proton after an average lifetime of about 900 seconds. Conservation of charge requires that an electron also be produced, so that the final state is electrically neutral. This electron provides a lepton number of +1 to the final state, and since the initial state has lepton number 0, this requires an antilepton in the final state. A positron would destroy conservation of charge, so an antineutrino, νe, is the only option. This provides a lepton number of -1, balancing the final state and giving the final reaction: n -> p + e- + νe

Both of these examples have the initial lepton number equal to 0, that is, they're hadron decays. An example with nonzero initial lepton number is electron capture by a proton in a nucleus, which is a form of inverse beta decay. Here, a proton absorbs an electron to become a neutron. To conserve lepton number, there must also be a particle in the final state with lepton number +1, i.e. a neutrino. So the reaction for electron capture is p + e- -> n + νe

True Structure of the Lepton Number

Lepton number, it turns out, is not actually a basic physical quantity but the sum of three quantities corresponding to the three 'generations' of leptons. These quantities are electron number, muon number, and tau number. Each particle in a given generation has its lepton number due to the corresponding lepton family number (e.g. the electron has +1 electron number and 0 muon and tau number). Each of these quantities is conserved separately, although neutrino oscillation violates the conservation of lepton family numbers. This explains the choices of neutrino type in the above reactions.

This separate conservation of lepton numbers is evident in muon decay. Muons decay into electrons, and if only conservation of total lepton number is considered, the decay μ- -> e- would appear to be possible, as the lepton number of both sides is +1. However, this is not the case, as the muon number of the initial state is +1 and the muon number of the final state is 0. To fully conserve lepton numbers, two neutrinos must be emitted, a muon neutrino and an electron antineutrino. Then, not only is the total lepton number conserved, but so are the electron number and muon number. The reaction is thus μ- -> e- + νe + νμ

Summary

Lepton number conservation is an important symmetry of the Standard Model of particle physics. It, along with the other conservation laws of the standard model, specifies the form of particle interactions at the fundamental level. Unlike many other conservation laws in the Standard Model, lepton number is conserved absolutely, i.e. no lepton-number violating interactions have ever been observed. Neutrino oscillation violates the conservation of the separate lepton family numbers, but at a low enough level such that the probability of lepton family violation in other reactions may be unobservably small. Searches for lepton number violating interactions are ongoing, as they would be signs of physics beyond the standard model.

Sources include my senior undergraduate particle physics course and the (very technical) Particle Data Group website at http://pdg.lbl.gov/
(CC)
This writeup is copyright 2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .

My friend Al came over for dinner one night, as he does once a month or so. It hadn't been very long since the last time, and he said he had something to tell me. Normally I like to experiment when I cook, but when Al joins me I pretty much go by the book. Al says he doesn't like it when cooks "play dice with the recipe".

Small talk lasted through the dinner, and when we adjourned to the library for some cheese and crackers, he told me the big news. There'd been a large reorganization at Aquinas Corporation where he works; he'd gotten a big promotion and there would be some hiring. He'd heard me complain before that my job at Dogmacorp was a dead end; people taking a position there tended to get stuck in it. He suggested that I look into moving.

As he said good night, he gave me the number of someone to call and said I shouldn't wait too long if I wanted to give it a try.

The manager I met with the next day told me that Al was right. Aquinas was a conservative company, rarely hiring or firing anyone but rather shifting them around, and when there was an opening it didn't tend to last long. He had a few positions we could talk about, but I was not qualified for most of them. He knew how things were at Dogma, though, and he could see the desire in my eyes. I guess something about me rubbed him the right way, because he seemed determined to get me into something.

The next Monday morning, I reported to my new boss, Mr. Pauli, in the animal husbandry department. "Call me Wolfy," he ordered me in his booming voice, with a smile that seemed like he'd pulled it out of storage for his annual practice with it. He handed me to someone passing by who turned out to be the head of the canine section and said "Here's the new guy. He's yours; train him and don't let me ever see him again." Being a dog lover, I couldn't have been happier with that accident. I started out feeding, watering, and walking the dogs; after a month, I was helping out in the veterinary office. My second day there, a beagle was brought in with a deep gash over one eye. "What happened here? This was no accident" I exclaimed as I held him down so the doc could clean and bandage the wound. "I heard he was wandering around and made a mess on the wrong rug." "But…" "Just drop it. It happens around here sometimes."

That seemed like an odd attitude for a vet to take, but I was new so I shut up. Until a week later, eating my lunch out by the paddock, I saw Wolfy slam the door of his Beemer and stomp toward his office, kicking a squirrel running across the path in front of him. I couldn't believe my eyes. I picked it up and took it to the vet, and said that I had to do something about this; I was sure it wasn't an isolated incident. Then after lunch, I was summoned to Pauli's office.

I closed the door and he said in his steely voice, "Boy, this department's not big enough for the both of us. One of us has got to go, and it's not gonna be me." He gave me a slip of paper from his desk and told me to hie myself to Personnel and don't come back.

But first, I got myself cleaned up and trekked across campus to see Al in his new office in Nikola Tower. That was a pretty rarefied atmosphere for an ex-junior stablehand. In the elevator, I got a few looks from suits that made my hair stand on end. But I found my way to Al's office and told him what happened. I was afraid my stay at Aquinas was over, but he said not to worry about it. Everybody knew some people just didn't fit in with Pauli. He was sure they would find something new for me. He said he'd phone down and put in a word for me at tech support, work that I'd done in the past.

And so it came to pass. Jacked in with my doppelgänger Barry on the phones for training, I was told that everybody accepted the principle that the customer on the phone could be fulsome with praise for your help, or Werner the supervisor could be standing behind you monitoring your call, but not both. But he told me that I'd do fine as long as I never treated a customer badly; then I would feel Werner's wrath.

Over the first few weeks, I found out why Werner was one of the most popular employees in the whole company. He had his quirks, but despite the strangeness, I recognized his charm, which had captivated most everyone in the company, top to bottom.

I passed almost a year this way, getting along with my co-workers, good reviews from Werner, no problems; until the day I lost it with a customer.

He'd made a big order of particles but insisted on the phone with me that he'd received nothing but waves: transverse, longitudinal, standing, you name it; packets and boxcars full of them. I pointed out to him that he'd be happy with the shipment if he just looked at it another way, but he raised his voice and we got into a shouting match. As if by magic, there was Werner standing behind me; he muted my phone and reminded me of the customer's always right rule, then took over the call and promised the customer a new shipment of particles would be on the way to him.

After he was done with the mollified customer, I said "Why send him a whole new shipment? You know he could see them as particles if he chose to." He turned to me and said "Copenhagen, your interpretation doesn't matter!" He started getting violet in the face and I feared catastrophe. I was sure that I would be annhilated and my short tenure there would bring down the average half-life of the department. I was right, but it wasn't so bad, because it turned into a promotion.

Once again in Personnel. Ms. Florsheim looked up from my file. "You didn't stay too long in animals, and people didn't work out in the end. Let's try numbers. I'm going to send you over to Dirac in accounting."

This was great! Accounting was easy, the hours were better. Or so I thought.

Dirac had always been thought of as a little bit crazy; he refused to deal in liabilities, insisting they were just contra-assets. He claimed he could see the company's books as a vast sea of assets and "holes". Finally upper management heaved a big sigh and ejected him from the company. Then they decided not to trust what he left behind, and ordered a complete sight inventory. The accounting department somehow took the blame for this, and we found ourselves voluntold to get into the warehouses and start counting.

So a week after my supposed good luck, here I am in warehouse number 271828 with a clipboard and a Segway. And I'm hungry! But I have to finish this section before I can call it a day. Everybody else who was in here has finished and gone home. Only the security guard is left, and he's giving me the evil eye. "Come on, Copenhagen, get a move on!"

"Dammit, you made me lose my place! Now I have to start this aisle over!" I retreated back to the start of aisle 314, section 602, floor 23: "Leptons".

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