Neutrinos Lecture 16 1 Introduction Neutrino physics is discussed in some detail for several reasons. In the first place, the physics is interesting and easily understood, yet it is representative of the present theory of fundamental particles. In addition, neutrinos not only provided the key to connecting the weak and electromagnetic forces, but impact a wide range of physics from symmetries to cosmology. As discuss below, neutrinos were introduced by Fermi in order to maintain energy conservation in beta decay (a weak interaction process). This is because neutrinos, being uncharged and weakly interacting, are not observed in the outgoing products of the decay, and thus energy changes in the decay were not observed to be conserved. Fermi proposed that to conserve energy an unobserved particle must be emitted, but apologized having felt he had postulated a particle that could never be seen. Neutrinos are now seen, and indeed the detection of neutrinos has been one of the most productive components of physics in the past 10-20 years. Neutrinos are placed in the standard model in families and the number of families can be determined by measuring the number of neutrino flavors. 1
Particle Families Quark (Hadron) Lepton 3 MeV u d c 6 MeV 95 MeV 1.24 GeV 172.5 GeV 2/3 2/3 2/3 up charm top 1/3 down s t 4.2 GeV b 1/3 1/3 strange bottom <2 ev <0.1 MeV <15 MeV υ e υ µ υ τ neutrino neutrino neutrino 0.511 MeV 106 MeV 1.78 GeV 1 e 1 µ 1 τ electron muon tau Flavor Fermion 0 0 Boson 0 γ photon 0 g gluon 90.2 GeV 0 0 Z weak force 80.4 GeV + 1 W + weak force Figure 1: A table of the generations of the standard model 2
Figure 2: A plot of the ratio R in the text that verifies only 3 generations This number can be determined by measuring the ratio of the ratios of W boson decays into electron neutrinos to that of Z bosons decaying to e + e pairs. R = σ W(W eν e ) σ W (Z e + e ) This ratio depends on the number of neutrinos as well as the top quark mass. The figure shows that only 3 generations of neutrinos are required to fit the data. 3
2 Lepton number conservation The decay of the muon into an electron and a gamma is not observed in the the 2-body process, µ eγ. The decay proceeds with neutrino emission, but the number of leptons released in the process must not only match the initial number of leptons, but these leptons must have the same flavor, µ ± e ± + ν µ (ν µ ) ν e (ν e ). Therefore, neutrinos are leptons, neutrinos carry family number, and lepton number appears to be conserved by flavor. Recall the standard model characterization of fundamental particles in terms of families and flavor (weak isospin). Of course, the characterization was developed to explain the observations. In the weak decay of baryons, baryon number is not conserved by flavor, although total baryon number remains conserved. As an example, Λ p + π. This weak decay process changes flavor and family, but preserves total baryon number. 3 Phase space Phase space is determined by calculating the number of states available for group of interacting particles after production when using Fermi statistics. Fermi statistics permits one particle per quantum bin, where a bin is defined by the quantum numbers of the state. Neglecting spin, only 2 fermions per momentum (p and -p) bin are possible. A momentum bin for an unbound particle is defined by dp x dp y dp z. In spherical coordinates this is p 2 dpdω. There are 3 4
particles so one has a particle density of; ρ = d p e d pν d pa δ(e 0 E e E ν E A )δ( p e + p ν + p A ) = 0 where relativistic momentum and energy are connected by E 2 = (pc) 2 + (mc 2 ) 2, and energy conservation is imposed by the δ functions. A common volume constant (L 3 ) has been ignored as it cancels in the transition matrix. Note that when detecting only one of 3 final state particles there are 9 - (4 + 3) = 2 degrees of freedom remaining. To simplify exposition of the density of states, consider the nonrelativistic Schrodinger equation for a free particle; (T)ψ = Eψ The solution is ψ = Ae ikx and apply periodic boundry conditions on the bin boundries 0, and L. Therefore k = 2πn/L. Thus the number of states per momentum interval is dn dp = L 2π The value of A is normalized so that it is proportional to 1/L. The relativistic formulation is identical. There is then a 3-D density function obtained by evaluation of the above integral. The result of the 3-particle density of states when one particle is detected is then; ρ(e) = p 2 ep 2 Ω e Ω ν ν (2π ) 6 c dp e 5
The transition probability in time dependent perturbation theory for a transition ψ i ψ f is written (Fermi s golden rule) ; P(i f) = 2π ψ f H int ψ i 2 ρ(e) Here ρ(e) is the density of final states (phase space). 4 Beta decay Weak decay was first observed in the radioactive decay of nuclei. For example, the decay of a neutron proceeds as n p + e + ν e In radioactive decay, the neutron is embedded in nucleus which makes the transition from a nuclear system, A(Z, N), to a nucleus of lower mass (energy), A(Z + 1, N 1). Baryon number is constant, lepton number, and generation do not change. In neutron rich nulcei, proton decay also occurs with the emission of a positron and an electron neutrino. Of course, the final nucleus can be left in an excited state if conservation of energy, and angular monentum are preserved. It is also possible in some cases for a nuclus to capture an electron (electron capture), leaving just the nuclear recoil and the neutrino in the final state. p + e n + ν e 6
Figure 3: A representative spectrum of e energies emitted in beta decay As illustrated above, beta decay has 3 particles in the final state so that the energetics of the decay are not completely determined and the energy released in the decay is distributed over the 3 particles in the final state. This distribution can be assumed to be a statistical distribution of 3 non-interacting particles (phase space or density of states). A typical electron energy distribution is shown in the figure. At the maximum energy the probability of an electron emission with that energy approaches zero. This end point is of interest as its energy value is related to the mass of the neutrino. The curve of the electron energy distribution can be linearized and extrapolated to its intersection with the energy axis (Kouri plot). This allows the end point energy to be measured, albiet with whatever error is inherent in the process. The best value of the neutrino mass is obtained from 7
the decay of the 3 H isotope, as the energy release is lower, and the nucleus less complicated that other possibilities. 3 H = 3 He + e + ν e This decay releases, 19 kev of energy. Although most of the effects on the curve shape can be theoretically included in the extrapolation, atomic effects, and measurement error including energy resolution, background, and statistics limit the measurement of the neutrino mass to < a few ev. The expectation, however is that neutrino masses are < 0.1 ev. 5 Dirac neutrino The Dirac equation has the form; [ α pc + β(mc 2 )]ψ = Wψ with the solution ψ composed of an upper and lower 2 component spinor. ψ = ( ψu ψ l ) Using this form the equation is decomposed and decoupled to give the equations; 8
Figure 4: A Kurie plot of the beta spectrum of 3 H decay showing the end point energy 9
Figure 5: The region of the end point of 3 H decay for an assumed neutrino mass of 10 ev ( σ p) 2 ψ u = (W 2 (mc 2 ) 2 )ψ u (1) ( σ p) 2 ψ l = (W 2 (mc 2 ) 2 )ψ l (2) Now suppose the mass equals 0. The above equations have the form ( σ p)ψ u = ±W 2 ψ u (3) ( σ p)ψ l = ±W 2 ψ l (4) So separation occurs naturally by choosing 10
( ) σ 0 α = 0 σ The solutions of this form are to be written as A ± where A + (A ) is the solution for positive (negative) helicity. e iφ/2 cos(θ/2) A + e = iφ/2 sin(θ/2) 0 0 0 A 0 = e iφ/2 sin(θ/2) e iφ/2 cos(θ/2) The general solution is a superposition of both of the above. However, for states of specific chirality we expect pure states of A + or A. Charge congugate states (operation by the C operation) have opposite chirality. The neutrino having right handed chirality is the antineutrino. 6 2-component neutrino The Dirac equation from the last section; Wψ = [ α pc + βmc 2 ]ψ and has a 4-component spinor solution. For the case when m = 0 11
the β matrix does not contribute and one only has the commutation relations; α i α j + α j α i = 2δ ij These may be satisfied by the Pauli matricies where α i = ±σ i. The Driac equation divides into two decoupled equations with 2- component spinor solutions. Wχ ± = (± σ p)χ ± Each equation has the classical analogue E 2 = p 2, so there is one positive and one negative energy solution. The positive energy solution is set to satisfy the equation; σ pχ + = χ + Thus χ + solution corresponds to the left handed neutrino. The other solution corresponds to the right handed neutrino. The equation is not invarient under the parity operation as this would be ν L ν R. However, the weak interaction does not conserved parity. Now the weak interaction couples by a charged current to the electron in a weak decay process; ψ e γ µ [1/2(1 γ 5 )]ψ ν This is the standard Vector/Axial-vector form for the interaction. 12
The (1 γ 5 ) operator mixes the vector and axial-vector in a maximal way. Of more importance here, it projects out either left handed or right handed neutrinos. Since the neutrino only interacts weakly we do not know if there is a right (left) handed neutrino (anti-neutrino). On the other hand if the neutrino mass were not zero then a velocity transformation could change the helicity of the neutrino. In the case when m 0 we can make the neutrino its own anti-particle to preserve the left handed coupling. The ν L and ν R are 2-component spinors. The other two components,ν R and ν L, may be Fermions of different mass. These are Majorana neutrinos as compared to the 4 component Dirac neutrinos. 7 Double beta decay The process of double beta decay is second order in the weak interaction. There are a few nuclei in which it is impossible by the weak interaction to change a single neutron into a proton, but a siultaneous decay of two neutrons is possible. As an example; 100 Mo e + e + ν e + ν e + 100 Ru The mass of the Mo atom (Z = 42,N = 58) is 93.0632 GeV while the mass of the Ru atom (Z = 44, N = 56) is 93.0602 GeV. A single weak transition would produce 100 Tc (Z + 1, N -1) which has a mass of 93.0634 GeV, so this transition is forbidden. The lifetime of 100 Mo is some 10 19 years. Note that this process releases two anti-electron 13
u e e u W ν _ ν _ W d d Figure 6: Neutrinoless double beta decay with a Majorana neutrino neutrinos. The lifetime is so long that direct observation is difficult, but double beta decay has been observed for a few nuclei. Note that charge and lepton number are conserved in the process. Suppose that the neutrino is its own anti-particle, a Majorana neutrino. In this case, double beta decay can occur without the emission of neutrinos. The figure shows a Feynman diagram of this process. In the figure d is a down quark (in a neutron) and u is an up quark (which was converted in the neutron). The charged current exchange of a virtual W boson and neutrino is illustrated. The neutrino, being its own anti-particle can annihilate with a W producing an electron, as shown. The signature of this decay is the emission of 2 electrons, essentially back to back with their summed energy equal to the energy release in the decay. An observation of this decay would confirm that the neutrino is Majorana particle and provide a measurement of the neutrino mass. A controversal observation of the double beta decay of 76 Ge gave a neutrino mass of 0.2 ev. 14
8 Neutrinos from the sun The major source of neutrinos from the sun is produced in hydrogen burning in the ppi cycle. These reactions are; p + p d + e + ν e p + e + p d + ν e d + p γ + 3 He 3 He + 3 He 4 He + p + p A much smaller fraction of neutrinos are produced in the ppii and ppiii chains which use nuclei produced in the burning process. hep and ppii hep 3 He + p 4 He + e + ν e 3 He + 4 He 7 Be + γ 7 Be + e 7 Li + ν e 7 Li + p 4 He + 4 He 15
Table 1: Integrated neutrino flux from the sun Chain Reaction Flux on Earth cm 2 s 1 pp 5.95 10 10 pep 1.4 10 8 pp hep 9.3 10 3 7 Be 4.77 10 9 8 B 5.05 10 6 13 N decay 5.48 10 8 CNO 15 O decay 4.8 10 8 17 F decay 5.63 10 6 ppiii 7 Be + p 8 B + γ 8 B 8 Be + e + + ν e 8 Be 4 He + 4 He There is also a CNO cycle but the temperature of the sun is too low for this reaction chain to significantly contribute. The figure shows the neutrino flux on the surface of the earth in units of cm 2 s 1 MeV 1 and the table gives the integrated flux over energy. Note all neutrinos are ν e 16
Figure 7: The spectrum of neutrinos from the sun 17
9 Atmospheric neutrinos Atmospheric neutrinos are produced by decays of pions and muons produced in collisions of cosmic rays with residual gas in the upper atmosphere. The reactions have the form; p + X π ± + Y π ± µ ± + ν µ (ν µ ) µ ± e ± + ν e (ν e ) + ν µ (ν µ ) To a lesser extent K ± are also produced and decay producing both electron and muon neutrinos. 10 Neutrino mass Now quark masses mix. That is the strong eigenstates are not the same as the weak eigenstates. Therefore the coupling of a d quark and a u quark is not the same as an s quark and a u quark. Listed by generations the quarks are; ( u d ) ( c s ) ( t b ) All allowed parents are listed are above and daughters below. The primed states are linear combinations of the mass eigenstates as obtained from the Cabibbo-Kobayashi-Maskawa mixing matrix equation; 18
d s b = V ud V us V ub V cd V cs V cb V td V ts V tb d s b For 3 types of neutrinos it is also possible that they mix if their masses are not zero. In this case we would have; ν e ν µ ν τ = V e1 V e2 V e3 V µ1 V µ2 V µ3 V τ1 V τ2 V τ3 ν 1 ν 2 ν 3 In the case of only 2 mixing neutrinos; ν e = cos(θ 12 )ν 1 + sin(θ 12 )ν 2 ν µ = sin(θ 12 )ν 1 + cos(θ 12 )ν 2 The two solutions mix as a function of time because they do not propagate with the same velocity for a given energy eigenvalue. In the standard model, neutrino mass is set equal to zero. A Majorana neutrino must have mass because it should preserve charility. One does not have separate conservation of lepton flavor, but total lepton number is conserved. A Majorana mass does not conserved lepton number. The lowest upperlimit on the neutrino mass, obtained from astrophysical analysis, is less than 0.3 ev. Recent experiments have essentially confirmed that neutrinos mix and thus have mass. This is the first observation that the standard model must be modified, 19
however it is not clear just how to do this. The most important recent experiments are listed below. 1. SNO - Mainly Sensitive to the Solar Neutrino Flux, ν e 2. Kamiokande, Super-Kamiokande, IMP - Sensitive to Solar Neutrinos and Atmospheric Neutrinos (anti-neutrinos), ν e and ν µ. 3. Chooz - Mainly Sensitive to Reactor Neutrinos, ν e 4. LSND, Karmen - Mainly Sensitive to ν µ SNO demonstrates that 2/3 of the highest energy ν e oscillate to either ν µ or ν τ. It also demonstrates that the standard solar model is correct. Without inclusion of the LSND data a consistent picture arises having three light neutrinos which mix in the following way; m 2 = µ 2 2 µ2 1 5.0 10 5 ev 2 m 2 at = µ 2 3 µ2 2 2.5 10 3 ev 2 In a 2-component mixing scheme this means ν e ν µ and ν µ ν τ. Finally, if the LSND result is valid then an additional, light, sterile neutrino is required. The best available theory of neutrino mass is the see-saw mechanism, which explains why the neutrino masses are so small. It implies 20
Flux of ν µ Plus ν τ ( 10 6 cm 2 s 1 ) 7 6 5 4 3 2 1 ES CC NC SSM 1 2 3 4 6 2 1 ν e Flux ( 10 cm s ) SNO Neutrino Flux 21
that the light neutrinos are obtained from the mixing of Dirac neutrinos and heavy Majorana right-handed neutrinos. The decay of a right-handed neutrino would help to explain the baryon asymmetry in the universe. Two mixing angles are large, θ 23 45 and θ 12 35 The 3 rd mixing angle is constrained so that sin(θ 13 ) < 0.17. The CP phase is unknown. If θ 13 = 0 this phase vanishes. The mixing matrix has the values; 0.84 0.54 0.0(0.17) 0.38 0.60(0.54) 0.17 0.38( 0.28) 0.60(0.66) 0.71 In this matrix the CP phase is taken equal to zero, and the value of θ 13 is chosen to be 0.0(0.17), which is the least(maximum) value for this element. Note that the last 2 rows look similar, and may suggest an unknown symmetry. Also the 2 nd column is similar and could be 1/ 3 = 0.58 which would indicate maximal mixing. 11 The MSW mechanism The sun is a significant source of neutrinos released in the thermonuclear burning of its hydrogen. The nuclear reactions are now believed to be reasonably predicted as verified by experimental measurements of the neutrino spectrum modified by their oscillations. Electron neutrino traveling from the interior of the sun can interact via charged current interactions while muon and tau neutrinos can only interact via neutral currents. An additional term arises from electron- 22
Figure 8: Effective neutrino mass in matter neutrino electron scattering. This increases the effective mass of ν e relative to the other neutrinos and changes the mixing angle, enhancing the oscillation. The figure shows the effective neutrino mass in media. The value of A is proportional to the matter density. The solid lines are the mass squared values of the eigenstates, and the dashed lines are the expectation values of the ν e and νµ states. 12 The seesaw mechanism The problem arises as to why the neutrino mass is so small. The seesaw mechanism is a way to naturally generate small mass numbers. In this case one produces a light neutrino and one heavy sterile 23
neutrino. To see how this works, assume a 2 2 matrix; A = ( 0 M M B ) Here B M. The eigenvalues of this matrix are; λ ± = B ± B 2 + 4M 2 2 The larger eigenvalue is B while the smaller eigenvalue is M2 B. Now if one of the eignevalues increases the other decreases and vice versa (a seesaw). The matrix A is the mass matrix of a sterile, right handed neutrino. The element B is the mass of a Majorana neutrino near the GUT scale, and M the mass of a Dirac neutrino at the electroweak scale. The small eigenvalue has a mass near 1 ev, which is at least near expectations. 24