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Lepton Number And Edward Wittena aSchool of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA

I should say at the outset that I do not have The fallacy is as follows. To argue, given that any real novelty to offer today. The study of neu- the neutrino only has one helicity state, that it trino masses and neutrino oscillations is an area should be exactly massless, one has to assume where experiment is definitely ahead of theory. that number is an exactly conserved quan- That may be even more true by the end of this tity. If is not conserved, one can meeting. All I can do is to review some ideas treat the left-handed neutrino and right-handed about neutrino masses that are not novel and are antineutrino as two different helicity states of one undoubtedly familiar to many of you but that re- , and combine them to make a massive main fresh and important. 1 1/2 particle. (Such a particle is called a In the 1950’s, the discovery of the “two com- massive Majorana ; this just means that ponent” of the neutrino – have it is its own and in particular car- left-handed spin around their direction of motion, ries no conserved .) If lepton number is an and antineutrinos have right-handed spin – was exact , then the neutrino and antineu- generally understood to show that the neutrino trino, because they have different lepton number, must be exactly massless. Indeed, a massive spin cannot be combined in this way as two different one-half particle has two helicity states (“spin up” states of one particle. and “spin down”), while the neutrino only has In fact, observed particle interactions conserve one. the number B and the three lepton num- There is a potential fallacy here, which was un- bers Le, Lµ,andLτ . I will write L for the total 2 derstood at an early stage by a few physicists, lepton number L = Le + Lµ + Lτ . In the 1970’s, but came to be widely appreciated only in the theorists generally became convinced that none 1970’s in the course of the development of uni- of these would be true symmetries, and (hence) fied gauge theories of the strong, weak, and elec- that neutrinos would be massive. tromagnetic interactions, which are often called I’ll try to list, in the rough historical order in GUT’s. In the course of this work, particle theo- which they had their impact (I depart from the rists came to widely suspect that neutrinos have historical presentation at the end), the principal and to guess a likely mass range that over- arguments that suggested that the baryon and laps with the range explored in present experi- lepton symmetries would be violated. (Some of ments. the arguments are more compelling for .) 1This article is based on my lecture at Neutrinos 2000, (1) All attempts to unify the and Sudbury, Ontario (June 19, 2000). Because the mate- in nature have required that physicists pos- rial is so standard, I have omitted references. I will just tulate symmetry-violation, as a result of putting mention one original source where some of the issues are of different types in the same gauge mul- treated: S. Weinberg, “Expectations For Baryon And Lep- ton Nonconservation,” First Workshop on Grand Unifica- tiplets. This was true in the Pati-Salam model, tion (Durham, NH 1980). and in the grand unified models based on groups 2In the talk after mine, John Bahcall reminded us of the such as SU(5), SO(10), E6, and it is certainly prescient early work of . 2 true in unified string theories. To be more exact, though other forces do not. The answer seems to the SU(5) model in its minimal form conserves be that QED conserves because with only only one linear combination of the four symme- the fields of QED ( and charged ), tries, namely B L, while in SO(10) and E6 it is impossible to violate parity by renormalizable (and the string theories),− it is natural to break all interactions. The same argument explains conser- of them. For example, in the case of the SU(5) vation of B,Le,Lµ,andLτ by ordinary , one combines antiquarks and leptons into model processes. a single gauge multiplet The distinguished role of renormalizable inter- actions comes because they have dimensionless q¯ coupling constants that are pure numbers. In  q¯ contrast, unrenormalizable interactions are ex- q¯ (1) pected to be suppressed by a power of 1/M ,where    ν  M is a mass scale at which the standard model    e  breaks down, perhaps a mass 1015 GeV suggested with a somewhat similar structure for the other by the Georgi-Quinn-Weinberg computation of fermions As a result, transitions inside the gauge running of coupling constants (and hence char- multiplets can violate the baryon and lepton sym- acteristic of many grand unified theories), or the 19 metries. mass, roughly 10 GeV, characteristic of B L symmetry is enough to make the neu- . trinos− exactly massless (assuming that the usual In the standard model, the lowest dimension left-handed neutrinos and right-handed antineu- operator that violates baryon number is trinos are the only relevant light helicity states). 1 This is because the neutrino and anti-neutrino QQQL (2) M 2 have different values of B L, so the naive argu- − ment for a massless neutrino is valid if B L is a (where Q and L are standard model gauge multi- − symmetry. Hence in practice the minimal SU(5) plets containing and lepton fields, respec- model has massless neutrinos, while most of its tively) while the lowest dimension operator that more elaborate cousins (such as the SO(10) and violates lepton number is E6 models in which all fermions in a “generation” 1 are unified in a single gauge multiplet) have mas- HHLL, (3) sive neutrinos. M (2)Atfirstsight,itmightseemthatif where H is a multiplet of Higgs fields. B,Le,Lµ,andLτ are not fundamental symme- If we assume that 1015 GeV M 1019 Gev, tries of nature, there is a mystery to explain: Why we get a lifetime in the≤ range≤ do ordinary particle interactions respect these symmetries? 30 45 10 years τp 10 years (4) It turns out that this question has a completely ≤ ≤ natural answer. Using the fields of the standard and a neutrino mass in the range model, it is impossible at the classical level to vi- 5 1 olate the baryon and lepton number symmetries 10− eV m 10− eV. (5) ≤ ν ≤ by renormalizable interactions.3 Asking why the standard model conserves B and L is thus anal- For the proton lifetime, there was an attrac- ogous to asking why QED conserves parity even tive model (the minimal SU(5) model without ) that predicted that the proton 3As I explain shortly, B and L violation in the standard lifetime would be quite close to the lower end of model is induced at an incredibly low level, much too low the suggested range. This prediction has been ex- to be detectable, by a quantum anomaly. The point at the moment is to explain why ordinary standard model cluded experimentally, and since then, though I couplings conserve B and L. think most particle theorists still believe that the 3 above estimate is on the right track, there has that is, the ordinary neutrino gets a nonzero L- 2 been no convincing quantitative prediction of τp. violating mass m /M . For neutrino masses, the considerations have al- If we set m equal to the scale of electroweak ways been qualitative, and, despite some interest- symmetry breaking, and use the above range for ing attempts, there has never been a convincing M, this leads to the range of neutrino masses sug- quantitative model of the neutrino masses. gested above. Actually, since quark and charged What I have explained above is a slightly ab- lepton masses vary over many orders of magni- stract way to estimate the neutrino mass. An ele- tude (from the to the ), we can gant mechanism that fits into this general scheme make the neutrino mass quite a lot less by making (and which enables one to see how to make the m smaller. Or we can make the neutrino masses neutrino mass bigger or smaller than the above bigger by making M smaller than the usual GUT range, if desired) is the “see-saw mechanism.” scale. For example, in conventional GUT theo- Here one assumes that a right-handed neutrino ries, this will occur if νR is naturally massless at νR exists but is very heavy. In fact, we assume tree level. that νR is a singlet of the standard model gauge Whether we take the abstract approach or rely group (as the V A theory of weak interactions on the see-saw mechanism, we would expect neu- suggests). One can− then have an ordinary Dirac trino oscillations, since it is hard to see why the neutrino mass neutrino mass matrix would be diagonal in the basis mν¯RνL + c.c. (6) νe that need not be small. It originates from a stan-  νµ  . (11) ν dard model coupling  τ  However, large mixing angles seem a bit sur- ν Hν¯ , (7) prising, at least to me. R e   (3) So far we have considered arguments that similar to the couplings that give bare masses to suggest that baryon and lepton number violation, and charged leptons. Hence one might and hence neutrino masses, are natural. Let us expect m to be comparable to the quark and now consider arguments that tend to show that B and L violation are inevitable. charged lepton masses. But νR can also have a very large Majorana mass The first such argument involves electroweak instantons. ’t Hooft showed in 1976 that via in- stantons of the SU(2) U(1) gauge group, the Mν¯Rν¯R + c.c. (8) standard model actually× has a mechanism that (this violates lepton number conservation, since it violates most of the baryon and lepton number couples twoν ¯’s rather than a ν and aν ¯). When symmetries. In fact, the only linear combination we combine these two terms, we get a 2 2mass of those symmetries that is conserved additively is matrix × B L; in addition, B is conserved mod three (be- cause− the standard model has three generations). 0 m (9) The ’t Hooft process will cause the deuteron to  mM decay to an plus antileptons at a fan- tastically slow rate (the lifetime is far more than ν 100 for the states L . Supposing that m<

Indeed, back in 1976, one could imagine adding endpoint of the evaporation, we may additional massive fields to the standard model in get back some , but only a small num- such a way as to cancel the B and L violation by ber since only a small mass remains in the black instantons. With the high precision tests of the hole. So in sum, the formation and evaporation standard model that we now have, this is virtually of a black hole does not conserve baryon number, impossible. and hence this cannot be an exactly conserved (4) Now I will consider black holes and quan- quantity in nature. tum gravity. This is the one point at which I What about lepton number? Here we have to will consider issues that, while certainly not at be a little more careful, because it may be that all novel, are perhaps not always considered in some of the neutrino species are massless, and if relation to neutrino masses. so neutrinos and antineutrinos make up (along Classically, a black hole absorbs and ra- with photons, , and any other massless diation and does not emit. Quantum mechan- particles that may exist in nature) a large part ically, this is impossible as it violates the her- of the black hole emission. However, because the miticity of the Hamiltonian H. Indeed, if f H i black hole emission is approximately thermal, no is a nonzero matrix element describing absorp-h | | i significant net lepton number is emitted during tion, then its complex conjugate i H f is a the bulk of the black hole history. In case there nonzero matrix element describing emission.h | | i In may exist massless neutrinos in nature, carrying fact, Hawking showed in 1974 that quantum black lepton number, we should worry that perhaps the holes do emit approximately thermal ,4 final pulse of radiation from a black hole could at a temperature that is fantastically small for carry away the original lepton number. (If so, this black holes of astronomical masses. Such a mas- “pulse” would last a very long time, since Fermi sive black hole, placed in empty space, will slowly statistics limit how many neutrinos carrying a lose mass to . As it loses mass, bounded energy can be emitted in a given space it shrinks. Ultimately, the hole shrinks down to in a given time.) This hypothesis would appar- the Planck scale. At that point, Hawking’s ap- ently contradict subtler aspects of the black hole proximations break down. By this time, the great evaporation story. Consider a black hole with a bulk of the hole’s mass has been radiated away. mass MS comparable to that of the . If the The radiation emitted after Hawking’s approxi- lepton number of a black hole is a well-defined mations break down is presumably highly non- , such a black hole might have a thermal, but can carry away only a very small lepton number of 1057 (if it formed from collapse fraction of the rest mass of the black hole, since of a sun-like ) or any larger number (if the the bulk of the rest mass has been radiated away black hole formed from collapse of a larger mass during the period when Hawking’s approxima- and then emitted Hawking radiation to reduce its tions were valid. mass to MS). So black holes with mass MS have Now, let us consider making a black hole from a infinitely many quantum states, labeled by the collapsing star, or from infalling tables and chairs lepton number. This contradicts the claim that – anything with large baryon and lepton number. the number of quantum states of a black hole of One does not get the baryons back in the outgo- given mass is the exponential of the Bekenstein- ing Hawking radiation, since for the great bulk of Hawking entropy (which is 1/4 of the area of the the black hole evaporation process, the Hawking horizon in Planck units) and in particular is finite. temperature is much too small compared to the I do not claim that arguments such as these proton mass to enable the emission of any signif- are absolutely air-tight, but they do show that icant number of baryons or antibaryons. At the it will be hard to reconcile a claim of absolute lepton number conservation in nature with what 4 The corrections to the thermal description of the outgo- we know about black hole physics. ing radiation are, for a macroscopic black hole, so subtle and small that they have never been successfully com- In contrast to global symmetries such as baryon puted. and lepton number, gauge symmetries such as 5 conservation cause no trouble for broken), but on closer examination, one finds that black hole physics. First of all, electric charge these symmetries are explicitly violated by world- definitely is a well-defined quantum number of sheet instantons. black holes. One sees it in the classical (Reissner- In summary, these arguments seem to show Nordstrom) black hole solution; it can be mea- that B and L cannot be conserved quantities in sured exterior to the black hole horizon, by a flux any theory that includes . This integral for the electric field on a large sphere is a strong hint of neutrino masses, but there is a at infinity. The electric field E of the hole con- loophole. Although L, for example, cannot be an tributes an energy d3x√gE2/8π to the black exactly conserved quantity, it might be conserved hole mass, so blackR holes of bounded mass also mod 7, or mod n for any integer n.Forn>2, this have bounded electric charge. It is global additive would suffice to prevent neutrino masses. If one symmetries that cause difficulties for black hole wishes in string theory to make a model in which physics. neutrino masses are exactly zero, one would ar- Since we have only used general properties of range to have a discrete Zn symmetry producing black holes and their Hawking emission (which such a mod n . Discrete sym- is deduced semi-classically and not on the basis metries that prevent neutrino masses could well of a detailed microscopic understanding of black prevent quark or charged lepton masses, so it ac- holes), the conclusion that baryon and lepton tually would take some care to arrange a model number are not symmetries of black hole physics in which the neutrinos are exactly massless and should apply in any consistent theory of quan- the other fermions have mass. tum gravity. Hence, we can test it out in string (5) Finally, the last hint that B and L are not theory, which is our only real candidate at the exact symmetries in nature is simply that the real moment for such a theory. We do not need to as- has a small but nonzero B and L den- sume that string theory describes nature; the ar- sity relative to the entropy . (In the case guments above indicate that global additive con- of L, we do not know this for sure, since the ob- servation laws such as baryon and lepton number served net density of might conceivably conservation cannot hold in any consistent quan- be canceled by not yet observed relic antineutri- tum gravity theory, whether it describes nature nos.) If B and L are not exact symmetries, then or not. making use of the C and CP violation in particle We indeed find that in string theory, we never interactions, and the CPT violation in get an additive global conservation law. At least that comes from the fact that the universe is ex- in known string vacua, the apparent additive panding, it is possible to spontaneously generate global symmetries turn out – sometimes in a nonzero B and L . There is not a con- tricky fashion – to be either gauged or explicitly vincing quantitative theory, but the observed B 9 broken. For example, one can add Chan-Paton and L densities (about 10− of the entropy den- factors to open strings, apparently introducing a sity) are in a natural range. If B and L are exactly global symmetry, but upon further examination conserved quantities, they must be postulated as this turns out to be a gauge symmetry. Similarly, initial conditions in the ; this is certainly in the heterotic string, the global symmetries on less attractive. the world sheet turn out to be gauge symmetries I hope that today I have at least managed, al- (not global symmetries) in space time. As an ex- beit without saying anything that is really new, ample of the opposite kind, in compactification to convey a sense of the exciting and wide-ranging of the Type II or heterotic string theories from theoretical ideas relevant to the neutrino mass ten to four dimensions on a compact manifold K, question. Hopefully, experimentalists are begin- one gets approximately massless “” associ- ning to come back with answers. I am sure we ated with the second Betti number b2(K). The will hear more in the next few days. shift symmetries of these axions appear to be con- tinuous global symmetries (albeit spontaneously 6

Acknowledgments This work was supported in part by NSF Grant PHY-9513835 and the Caltech Discovery Fund. On leave at Department of Physics, Caltech, Pasadena CA 91125.