Current Situation in the Neutrino (and Charged-Lepton) Sector

AndrŽ de Gouva Physics & Astronomy Department, Northwestern University, Evanston, IL, USA

Abstract Neutrino masses are nonzero. Theoretically, we are still in the process of un- derstanding what these tiny masses and the pattern of lepton mixing mean, and how they Þt into a new and improved standard model. Nonetheless, the very successful parameterization of the new neutrino sector(threeneutrino masses plus a 3 × 3 unitary leptonic mixing matrix) allows one to identify what we know we donÕt know about neutrinos and to deÞne a rich experimen- tal program in neutrino physics. This experimental program must not only answer these Òneutrino questions,Ó but also test the underlying three-active- ßavors paradigm and point the way toward a deeper understanding of neutri- nos (and particle physics). Given what we learned about neutrinos, searches for charged-lepton ßavor-violating phenomena are poised toÞndnewheavy physics at the weak scale (or well above) and provide vital clues regarding the neutrino mass mystery.

1 Neutrinos: Where We Are Over the past decade, our understanding of neutrinos changeddramatically.Afterdecadesofconfusion, it is now established that neutrinos change ßavor after propagating a Þnite distance. The transition probability depends on the neutrino energy, the distance between the neutrino source and the neutrino detector, and, in some cases, the medium through which the neutrinos propagate. For a detailed summary of all data, see, for example, [1, 2]. For the most recent results and updates see [3, 4]. The only consistent explanation to all long-baseline neutrino oscillation data is to postulate that neutrinos have non-zero, distinct masses and that leptons mix. This means that the charged-current weak interactions are not ÒalignedÓ with the charged and neutral lepton mass eigenstates. In this case, the observed neutrino ßavor-change is a consequence of neutrino oscillations and the data constrain the neutrino mass-squared differences and different combinations of the elements of the lepton mixing matrix, often referred to as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. It is often the case that lepton mixing is described in the weakbasiswherethecharged-lepton mass matrix is diagonal (with eigenstates e,µ, τ)sothatthePMNSmatrixU relates the neutrino weak- eigenstates να, α = e,µ, τ and the neutrino mass eigenstates νi (with mass mi), i =1, 2, 3 (for now and until further notice, IÕll assume there are three neutrino species):

να = Uαiνi. (1)

The elements of U are often parameterized as prescribed by the Particle Data Group [4] by three mixing angles θ12, θ13, θ23 and three CP-odd phases δ, α1, α2.IftheneutrinosareDiracfermions,theconvention is such that only the phase δ is a physical observable. In order to match mixing parameters to data, it is imperative to properly deÞne the neutrino mass- eigenstates. This is usually done in the following unusual way. Given three neutrinos, one can deÞne three mass-squared differences, two of which are independent. States ν1 and ν2 deÞne the smallest (in 2 2 magnitude) mass-squared difference and we further impose m1

10 normal hierarchy inverted hierarchy

2 2 (m3) (m2) 2 (!m )sol 2 (m1)

"e 2 (!m )atm "µ 2 (!m )atm

"#

2 (m2) 2 (!m )sol 2 2 (m1) (m3)

Fig. 1: Cartoon of the relationship between neutrino mass and ßavor eigenstates, for both the normal and inverted neutrino mass hierarchy. The fractions of the different horizontal bars with a speciÞc color are representative of 2 the different |Uαi| . From [1], where one can go for more details.

2 2 is associated to ∆m13 > 0 (∆m13 < 0). Both mass hierarchies are depicted in Fig. 1. For a careful discussion of the neutrino oscillation parameters, see [5]. Current data constrain, according to [3]1

2 −5 2 2 ∆m12 =(7.65 ± 0.20) × 10 eV , sin θ12 =0.32 ± 0.02, 2 −3 2 2 +0.10 |∆m13| =(2.45 ± 0.11) × 10 eV , sin θ23 =0.45−0.05, 2 sin θ13 < 0.050 (3σ bound), δ ∈ [0, 2π], (2)

2 while the sign of ∆m13 is unconstrained. The so-called Majorana phases α1 and α2,ifphysically observable, are currently completely unconstrained. Other than masses, the current neutrino data do not require any other new neutrino properties, including anomalous magnetic moments, non-standard neutrino interactions, or new neutrino states. In- deed, many of these are currently constrained by neutrino oscillation data, sometimes rather strongly [2].

2 Neutrinos: Where WeÕd Like to Go While neutrino data have revealed new lepton properties (neutrino mass-squared differences and leptonic mixing angles), there are still several Òknown unknownsÓ that need to be uncovered by next-generation neutrino experiments. These are not new, exotic neutrino properties that may or may not be present. Here IÕll discuss questions to which, given the fact that neutrinos have mass, there are well-deÞned, always relevant, answers.

2.1 Unknown Oscillation Parameters AquickinspectionoftheresultslistedinEq.(2)revealsthat some of the oscillation parameters are either unknown or only poorly constrained. In more detail:

1The results below are simpliÞed and we refer to [3] for more precise results and comments. For example, For different 2 2 values for the sign of ∆m13, the authors of [3] Þnd slightly different best Þt values for |∆m13|.Thedifference,however,is statistically insigniÞcant.

11 2 2 1. sin θ13 is only constrained to be less than several percent. sin θ13 is also the magnitude of the Ue3 matrix element, and is equal to the probability that a ν3 is measured as what one normally refers to as an electron neutrino νe. 2. The sign of cos 2θ23 is virtually unconstrained. This means that one canÕt tell whether the ν3 state has a larger ντ or νµ component, or whether the two components (Uτ3 and Uµ3)havethesame magnitude. 3. The CP-odd phase δ is completely unconstrained. δ =0 , π indicates that CP-invariance is violated in neutrino oscillations, i.e., P (να → νβ) = P (να → νβ).TheimportanceofunderstandingCP- invariance violating phenomena in fundametal physics has been emphasized over the past several decades. So far, all observed CP-invariance violating phenomena (in kaon and B-meson mixing and decay) are parametrized by one CP-odd Lagrangian parameter: the phase in the quark mixing matrix. Neutrino oscillations provide the Þrst opportunitytoobserveaCP-invarianceviolating phenomenon that probes a guaranteed new source of CP-invariance violation. 2 4. The neutrino mass hierarchy, here parameterized by the sign of ∆m13 is unconstrained. Different neutrino mass hierarchies point to potentially very different neutrino mass theories [6] and po- tentially qualitative neutrino mass spectra. In the case of an inverted neutrino mass hierarchy, for example, two of the neutrino masses are guaranteed to be almost degenerate: m2 −m1 ≪ m2,m1. This featured is not shared by any other standard model entities, except for those which are related by approximate symmetries. Examples of composite states include the pions and nucleons, which are related by isospin symmetry. At the fundamental level, ontheotherhand,theW and Z-boson masses are related by custodial symmetry. If the neutrino mass spectrum is inverted, one will be very tempted to believe that there is some new approximate symmetry that relates at least some of the different lepton-doublet Þelds.

Obtaining the answers to these four questions is the current driving force behind the next-generation neutrino oscillation experimental program. Whether one canobtainasatisfactoryanswertoanyofthem 2 depends on the magnitude of |Ue3| .Ultimately,however,thegoalofnextandnext-next-generation experiments is not to precisely measure all neutrino oscillation parameters, but rather to test the three- massive-neutrino paradigm. While the three-massive-neutrino paradigm works very well, there is a lot of room for expansion. Only after over-constraining the formalism will it be fair to state that the neutrino sector has been properly understood. Before proceeding, IÕll schematically describe how CP-invariance violation manifests itself in neutrino oscillations. The amplitude related to the probability that, say, a neutrino produced as a νµ will be detected as a νe is given by (in vacuum)

∗ i∆12 ∗ i∆13 Aµe = U Uµ2 e − 1 + U Uµ3 e − 1 , (3) e2 e3 ∆ 2 m1iL where ∆1i = 2E , i =2, 3, L is the distance travelled by the neutrino and E its energy. The amplitude for the CP-conjugate process is

∗ i∆12 ∗ i∆13 Aµe = Ue2U e − 1 + Ue3U e − 1 . (4) µ2 µ3 Above, it was assumed that U is a unitary matrix. |A|2 = |A|2 if (a) there are non-trivial Òweak phasesÓ, ∗ i.e.,therelativephaseofthedifferentUeiUµi must be non-zero; (b) there are non-trivial Òstrong-phasesÓ, i.e., ∆12 = ∆13 =0 .Condition(a)translatesintoδ =0 , π and θ13 =0 ,whilecondition(b)translates 2 2 into L =0 and ∆m12, ∆m13 =0 .All(b)requirementscanbemetbyappropriatelychoosingthe baseline L.OurabilitytoprobeCP-violationdependsonthemagnitudeof Ue3 (or θ13).

2.2 How Light is the Lightest Neutrino? Neutrino oscillation experiments are only sensitive to neutrino mass-squared differences. Knowledge of these does not allow one to determine the neutrino masses themselves. This uncertainty can be param-

12 eterized in terms of the lightest neutrino mass, mlightest.OncethatisknownÐalongwiththeneutrino mass hierarchy and the mass-squared differences Ð all neutrino masses are determined:

2 2 2 2 2 m1 = mlightest; m1 = mlightest − ∆m13, 2 2 2 2 2 2 2 m2 = mlightest + ∆m12; m2 = mlightest − ∆m13 + ∆m12, 2 2 2 2 2 m3 = mlightest + ∆m13; m3 = mlightest. (5)

The left-hand (right-hand) expressions apply in the case of anormal(inverted)hierarchy.Different 2 2 values of mlightest lead to qualitatively different neutrino spectra. If, for example, mlightest ≫ |∆m13|, 2 all neutrino masses are quasi-degenerate: |mi − mj| ≪ mi for all i,j.Ontheotherhand,ifmlightest ≪ 2 ∆m12 and the neutrino mass hierarchy is normal the neutrino massesarehierarchical:m1 ≪ m2 ≪ m3. Current constraints on mlightest and prospects for the future were discussed in detail in this meeting [7,8].

2.3 Are Neutrinos Majorana Fermions? Unlike all other standard model fermions, neutrinos have zero electric charge. This means that, unlike all other standard model fermions, they may be Majorana fermions. Majorana fermions are their own antiparticles in the same sense that the photon or the neutralpionaretheirownantiparticles.Another way to contrast a massive Majorana fermion from a Dirac one (all charged leptons and quarks are Dirac Fermions) is that a Dirac fermion Þeld describes four degreesoffreedom,whileaMajoranafermionÞeld describes only two. The neutrino was ßagged as special because it had no electric charge. In order for it to be a massive Majorana fermion, it cannot possess any quantum numbers after electroweak symmetry is broken. In the standard model, however, the neutrinos possess a quantumnumber.Neutrinosarechargedunder aglobal,non-anomalousU(1)B−L symmetry. Hence, if the neutrinos are massive Majorana fermions, this symmetry cannot be exact and lepton number minus baryon number violation is guaranteed to occur. Searches for lepton number violation provide the deepest probes for the Majorana nature of the neutrinos. It is important to appreciate that the issue in question Ð whether neutrinos are Majorana fermions Ðcanonlybeaddressediftheneutrinoshavemass.Inthecaseof massless neutrinos, helicity is a good quantum number and only two neutrino degrees (the left-handed neutrino and the right-handed antineutrino) of freedom interact with other standard modelparticlesregardlessofwhetherright-handed chiral neutrino Þelds exist. Furthermore, nothing ÒmixesÓ the left- and right-handed neutrino states and U(1)B−L can be a conserved global symmetry. All of these imply that anyobservablethatissensitiveto the Majorana nature of the neutrino vanishes in the limit whenneutrinomassesarezero.Theamplitudes for such processes are proportional to mν/E,wheremν is some combination of neutrino masses and E is the neutrino energy associated to the process in question. Since neutrino masses are tiny, all phenomena capable of addressing the Majorana versus Dirac question arevery,veryrare. The best probe of the Majorana nature of the neutrino and of theconservationofleptonnumber is neutrinoless double-beta decay. This was discussed in detail in this meeting [7, 9]. It consists of the decay − − Z → (Z +2)e e , (6) where Z,Z +2stand for nuclei with atomic numbers Z,Z +2respectively. It is easy to see that Eq. (6) violates lepton number by two units. The massive neutrino, short-distance contribution to this amplitude is easy to compute. 2 2 mi g A ∝ g UeiUei ≡ mee, (7) 0νββ Q2 Q2 i where Q is representative of typical neutrino energies and I assumedthatthesearemuchlargerthanthe neutrino masses. mee is an effective neutrino mass. It is a weighted average of the different neutrino

13 masses, but the weights are complex. Not only can mee vanish, but it also depends, in principle, on the Majorana phases α1 and α2.Formoredetailssee[7,9].

3 Neutrinos: Why Do We Care? In the previous sections I argued that nonzero neutrino masses have been discovered, along with lepton mixing. It remains to introduce these new ingredients into the standard model of strong and electroweak interactions. Before discussing this, it is worthwhile to inspect what has been uncovered regarding neutrino masses, and how they compare with the rest of the standard model. Neutrino masses, while non-zero, are really tiny. Fig. 2 depicts the masses of all standard model fermions, including the neutrinos. Two features stand out immediately. First, neutrino masses are at least six orders of magnitude less than the electron mass. The electron mass itself is already over 100 times smaller than the muon mass and tiny compared to the weak scale,around100GeV.Second,tothebest of our knowledge, there is a ÒgapÓ between the largest allowedneutrinomassandtheelectronmass,in contrast with the fact that, in the charged-fermion part of the mass-space one encounters a new mass every order of magnitude or so. We donÕt know why neutrino masses are so small or why there is such a large gap between the neutrino and the charged fermion masses. We suspect, however, that this may be NatureÕs way of telling us that neutrino masses are Òdifferent.Ó This suspicion is only magniÞed by the fact that neutrinos may be Majorana fermions.

10 12 t 10 11

mass mass (eV) 10 10 b # 10 9 c

10 8 µ s

10 7 d u 10 6 e 10 5

10 4

10 3

10 2

10

1 -1 10 "3 -2 10 "2 -3 10 "1 -4 10 -5 10

Fig. 2: Standard model fermion masses. For the neutrino masses, the normal mass hierarchy was assumed, and a loose upper bound mi < 1 eV, for all i =1, 2, 3 was imposed. From [1].

On top of these theoretical hunches there is the fact that, in the standard model, neutrinos were

14 predicted to be exactly massless. The discovery of neutrino masses, hence, qualiÞes as the Þrst concrete instance where the standard model failed. More important is the fact that all modiÞcations to the standard model that lead to massive neutrinos change it qualitatively. For a more detailed discussion of this point see, for example, [10]. It is natural to ask what augmented, ÒnewÓ standard model (νSM) leads to non-zero neutrino masses. The answer is that we are not sure. There are many different ways to modify the standard model in order to accommodate neutrino masses. While most differ greatly from one another, all succeed Ð by design Ð in explaining small neutrino masses and are all allowed by the current particle physics exper- imental data. The most appropriate question, therefore, is not what are the candidate νSMÕs, but how can one identify the ÒcorrectÓ νSM? The answer lies in next-generation experiments, brießy summarized below. ModiÞcations to the standard model that allow neutrinos to acquire neutrino masses include aug- menting the Higgs sector with, say, SU(2)L Higgs triplets, augmenting the fermion sector with either SU(2)L singlets (right-handed neutrinos) or triplets, or adding several new Þelds and interactions that explicitly violate U(1)B−L.Inordertoexplainwhyneutrinomassesaresosmall,otheradditions are often employed including large extra dimensions, new spontaneously broken gauge symmetries, etc. For detailed examples see, for example, [6]. The key point is that different models lead to different phenomenology in the neutrino sector and elsewhere. Neutrino data also provide another perceived puzzle: the pattern of neutrino mixing. The absolute value of the entries of the CKM quark mixing matrix are, qualitatively, given by

10.20.001   |VCKM| ∼ 0.210.01 , (8)  0.001 0.01 1  while those of the entries of the PMNS matrix are given by

0.80.5 < 0.2   |UPMNS| ∼ 0.40.60.7 . (9)  0.40.60.7 

It is clear that the two matrices ÒlookÓ very different. WhiletheCKMmatrixisalmostproportionaltothe identity matrix plus hierarchically ordered off-diagonal elements, the PMNS matrix is far from diagonal 0 and, with the possible exception of the Ue3 element, all elements are O(10 ).SigniÞcantresearchefforts are concentrated on understanding what, if any, is the relationship between the quark and lepton mixing matrices and what, if any, is the Òorganizing principleÓ responsible for the observed pattern of neutrino masses and lepton mixing. There are several different theoretical ideas in the market (for summaries, overviews and references see, for example, [6, 11, 12]). Typical results, which are very relevant for next-generation experiments, include predictions for the currently unknown neutrino mass and mixing 2 parameters (sin θ13, cos 2θ23,themasshierarchy,etc)andtheestablishmentofÒsumrulesÓ involving different parameters. Progress in our understanding of the physics behind neutrinomassesandleptonmixingwillrely heavily on the availability of new experimental data. These include

Ðtheprecisedeterminationofallneutrinooscillationparameters. This is the goal of the next (and next-next) generation of neutrino oscillation experiments. The ultimate goal is to validate (or falsify) the three-neutrino-mixing hypothesis by over-constraining the parameters. Ðsearchesforleptonnumberviolation.Thisistherealmofsearches for neutrinoless double-beta decay [7, 9]; Ðsearchesforßavorviolationintheleptonsector.Thisisthe subject of the next section;

15 Ðprecisionmeasurementsofchargedandneutralleptonelectromagnetic properties, like the g − 2 of the muon [13], searches for the electron electric dipole moment or searches for neutrino electromagnetic moments [14]; Ðsearchesforthephysicsresponsibleforelectroweaksymmetry breaking and new degrees of free- dom at the TeV scale. This is the main purpose of the LHC experiments. See, for example [15]; ÐPrecisionmeasurementsofhigh[16]andlow-energy[17]neutrino scattering; Ðetc.

4 Charged Lepton Flavor Violation Neutrino oscillations have revealed that lepton-ßavor symmetry is not conserved in the standard model. This indicates that, at some order in perturbation theory, ßavor violation must also be observed in the charged-lepton sector. These charged-lepton ßavor violation (CLFV) processes include ℓ → ℓ′γ, ℓ → ℓ′ℓ′′ℓ′′′, ℓ + X → ℓ′ + X and X → ℓℓ′,whereℓ, ℓ′,... = e,µ, or τ and X are states that carry no lepton- ßavor number. Here, I concentrate on rare processes involving muons in the initial state (and electrons in the Þnal state). While lepton-ßavor-violation has been established in the neutral fermions sector, measurements of neutrino oscillation processes do not allow us to reliablyestimatetherateforthevariousCLFV processes. The reason is that while neutrino oscillating phenomena depend only on neutrino masses and lepton mixing angles, the rates for the various CLFV processes may depend on the mechanism behind neutrino masses and lepton mixing. Different neutrino mass-generating Lagrangians [6] lead to very different rates for CLFV. The massive neutrino contribution to CLFV involving only active neutrinos is absurdly small. For example [18], 2 2 3α ∗ ∆m1i −54 B(µ → eγ)= U Uei < 10 , (10) 32π µi M 2 i=2,3 W 2 where Uαi are elements of the neutrino mixing matrix, ∆mij are the neutrino mass-squared differences, and MW is the W -boson mass. The estimate above applies to some neutrino massmodels,including minimal scenarios with Dirac neutrinos. The reason for the tiny expected branching ratio is well-known. CLFV are ßavor-changing neutral current procesess and hencesubjecttotheGlashow-Illiopoulos-Maiani (GIM) mechanism [19]. In some neutrino-mass generating scenarios, the active neutrino contribution turns out to be severely subdominant. In the seesaw mechanism [20], for example, heavy neutrino contributions to CLFV are naively expected to be of order those of the light neutrinos, but there is no theorem that prevents them from being much, much larger. Current experimental constraints on the seesaw Lagrangian allow B(τ → µγ) as large as 10−9, B(µ → eγ) as large are 4 × 10−13,andnormalizedrates2 for µ → e- conversion in 48Ti that saturate the current experimental upper bound (around 4 × 10−12 [4]) [21]. Regardless of the mechanism behind neutrino masses, it is widely speculated that the rates for different CLFV processes should be just beneath current experimental upper bounds (see current bounds in [4] and references therein). The reason is the community strongly suspect that there are new degrees of freedom with masses around 1 TeV. Since lepton-ßavor numbers are not conserved, it is generically expected that virtual processes involving the new degrees offreedomwillmediateCLFV.Concreteex- amples are plentiful in the literature. See for example, [22Ð26].

2In the case of the so-called µ → e-conversion in nuclei process, one always refers to the rate for µ → e-conversion normalized to that for µ → νµ-conversion, often referred to as the capture rate. It is not uncommon to refer to this normalized rate as a branching ratio.

16 Instead of discussing speciÞc scenarios, IÕll discuss one effective Lagrangian that mediates CLFV processes involving muons. The goal is to illustrate how sensitive searches for CLFV are to new physics, and how different CLFV channels compare with one another. After electroweak symmetry breaking, CLFV is mediated by effective operators of dimension Þve and higher. Consider the following effective Lagrangian3

mµ µν L = µRσµν eLF + CLFV (κ +1)Λ2 κ µ µ µLγµeL uLγ uL + dLγ dL . (11) (1 + κ)Λ2

The subscripts L, R indicate the chirality of the different standard model fermion Þelds, F µν is the photon Þeld strength and mµ is the muon mass. The coefÞcients of the two types of operatorsareparameterized by two constants: the mass-dimension one Λ parameter, which is meant to represent the effective mass scale of the new degrees of freedom, and the dimensionless parameter κ,whichgovernstherelativesize of the two different types of operators. The magnetic-momenttypeoperatorintheÞrstlineofEq.(11) directly mediates µ → eγ and mediates µ → eee and µ → e-conversion in nuclei at order α.The four-fermion operators in the second line of Eq. (11), on the other hand, mediate µ → e-conversion at the leading order and µ → eγ at the one-loop level. For κ ≪ 1,thedipole-typeoperatordominates CLFV phenomena, while for κ ≫ 1 the four-fermion operators are dominant. The sensitivity to Λ as a function of κ for µ → eγ and µ → e-conversion efforts is depicted in Fig. 3. For κ ≪ 1,anexperimentsensitivetoBr(µ → eγ) > 10−13 will probe Λ values less than 2500 TeV, while for κ ≫ 1 an experiment sensitive to Br(µ → e conv in 48Ti) > 10−16 will probe Λ values less than 8200 TeV. AlotcanbeextractedfromFig.3.First,CLFValreadyprobesΛ values close to 1000 TeV, while next-generation experiments [28] will start to probe Λ ∼ 103 TeV and beyond. Second, a µ → e- conversion experiment is guaranteed to out-perform a µ → eγ experiment for any value of κ as long as it is a couple of orders of magnitude more sensitive. Since it is, experimentally, very hard to achieve sensitivity to branching ratios for µ → eγ smaller than 10−14 [29], µ → e-conversion searches, which are not expected to hit any ÒwallÓ before normalized rates around 10−18 (at least) [29], seem to be the most effective way of pursuing CLFV after the on-going MEG experiment is done at the Paul Scherrer Institut [28]. The discussion above also serves to illustrate another aspect of searches for CLFV violation. In the case of a positive signal, the amount of information regarding the new physics is limited. For ex- ample, a positive signal in a µ → e-conversion experiment does not allow one to measure either Λ or κ but only a function of the two. In order to learn more about the new physics, one needs to combine information involving the rate of a particular CLFV process with other observables. These include other CLFV observables, and the list of experimental efforts reported in Sec. 3. Valuable information can also be obtained by observing µ → e-conversion in different nuclei [27, 30] or by studying the kinematic distribution of the Þnal-state electrons in µ → eee (see [27] and references therein). The MEG experiment at PSI [28], currently taking data, aims atbeingsensitivetoµ+ → e+γ decays if the branching ratio exceeds 10−13.UpgradestoMEGareunderconsideration,andoneenvi- sions sensitivity to branching ratios larger than 10−14.Asdiscussedearlier,itisexpectedthatfurther sensitivity to this process is exceedingly challenge. For this reason, the community is dedicating a lot of its energy and resources in order to explore µ → e-conversion processes in nuclei. In the coming decade, experiments at Fermilab [31] and in J-PARC [32], sensitive tonormalizedconversionrateslargerthan 10−16,maybereadytostartcollectingdata.

3The most general effective Lagragian includes several other terms [27]. The subset included in Eq. (11), however, is sufÞcient to illustrate all issues discussed here.

17 (TeV) %

B(µ& e conv in 48Ti)>10-18

4 10 B(µ& e conv in 48Ti)>10-16

B(µ& e')>10-14

B(µ& e')>10-13

10 3

EXCLUDED -2 -1 2 10 10 1 10 10 $

Fig. 3: Sensitivity of a µ → e-conversion in 48Ti experiment that can probe a normalized capture rate of 10−16 and 10−18,andofaµ → eγ search that is sensitive to a branching ratio of 10−13 and 10−14, to the new physics scale Λ as a function of κ, as deÞned in Eq. (11). Also depicted is the currently excluded region of the parameter space.

5 Concluding Thoughts Physics beyond the standard model has manifested itself in one clear way Ð neutrino masses are nonzero! We can interpret the current neutrino data very well in terms of neutrino oscillations and have devised averysuccessfulparameterizationoftheneutrinosector. This parameterization allows us to identify what we know we donÕt know about neutrinos which, in turn, helps deÞne a rich experimental program in neutrino physics for the coming decade or two. This experimental program must not only answer all of the Òleft-overÓ neutrino questions, but should also test the three-neutrino paradigm and point the way toward a deeper understanding of neutrinos in particular andparticlephysicsingeneral. Theoretically, what we learned can be summarized in two phrases: neutrino masses are very, very small, and lepton mixing is very different from quark mixing.WedonÕthaveanexplanationforeither of these two observations, but feel that both of these facts are important clues towards a more satisfying understanding of fundamental physics. Right now, there are many very different candidates for the ÒnewÓ standard model and at most one of them can be correct! Amorefundamentalunderstandingofneutrinomasses,leptonmixing,andhowtheseνSM param- eters Þt into our understanding of particle physics, can onlycomewithmoreexperimentalinformation from several different sources. Besides the neutrino program highlighted above, information will come from cosmology, high-energy collider experiments, and studies of charged-lepton properties. Here, I highlighted searches for charged-lepton ßavor violating muon processes. These are expected to not only shed light on the neutrino mass puzzle, but also provide amongthemostpowerfulprobesofphysicsat the electroweak symmetry breaking scale and beyond.

18 Acknowledgements Iamhappytothanktheorganizersfortheinvitationandforputting together a very informative meeting. This work is sponsored in part by the US Department of Energy Contract DE-FG02-91ER40684.

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