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The Seesaw Mechanism C. Amsler, Nuclear and Particle Physics The seesaw mechanism This section deals with a model to explain the triflingly small neutrino mass (< 2 eV), which is much smaller than that of other fermions such as the next lightest one, the elec- tron (0.5 MeV). In the standard model neutrinos and antineutrinos are different (Dirac) particles. The chirality of the neutrino is negative, that of the antineutrino positive (see *15.79*). The neutrinos and antineutrinos involved in the weak interaction with other particles are represented by the spinors 1 − γ5 1 + γ5 = and ( c) = c; (1) L 2 R 2 respectively, where and c are solution of the Dirac equation (chapter *15*). The spinors 1 + γ5 1 − γ5 = and ( c) = c (2) R 2 L 2 correspond to sterile neutrinos and antineutrinos. The following discussion deals with only one flavour of neutrinos, say νe, but is ap- plicable to νµ and ντ as well. We shall denote the neutrino spinor by and that of the charge conjugated antineutrino by c. We have shown in section *15.5* that T T c = C = iγ2γ0 : (3) Table1 lists a few useful relations satisfied by the charge conjugation C, which are easily verified by using the properties of γ matrices derived in chapter *15*. Table 1: Some properties of the charge conjugation. Cy = CT = −C C2 = −1 CCy = 1 CCT = 1 Cγ0C = γ0 γ0Cγ0 = −C We have seen e.g. in section *14.4* that the neutrino is observed to be left-handed and the corresponding antineutrino right-handed. As discussed in section *15.3* the spinors (1) and (2) are eigenstates of the chirality operator γ5, but chirality is equivalent to han- dedness in the limit of vanishing masses, hence the subscripts L and R in (1) and (2). However, a massive neutrino can be brought to rest and become right-handed by a suitable Lorentz boost, thus interacting nevertheless with other fermions. This is the mechanism at work in the hypothetical neutrinoless double β-decay in which neutrinos and antineutrinos are identical (Majorana) particles (section *14.5*). Let us now interpret the spinor as a field and write for the energy of a Dirac neutrino with mass MD in its rest frame, following the Dirac equation, y y 0 H = βMD = γ MD = MD = MD R L + MD L R; (4) 1 Seesaw mechanism because only terms with opposite chiralities contribute to H: indeed, can be decom- posed into L and R fields 1 − γ5 1 + γ5 = + = + : (5) L R 2 2 The products L L and R R vanish, 1−(γ5)2=0 1 z }| { = (1 + γ5)(1 − γ5) = 0; (6) L L 4 and likewise for R R. Furthermore, 1 + γ5 1 − γ5 1 − γ5 = yγ0 = y γ0 = [ y γ0 ]y = [ ]y; L R 2 R 2 R R 2 R L (7) so that H becomes simply H = MD R L + h:c:; (8) where h:c: stands for “hermitian conjugate”. Further terms can be added to the Hamiltonian when including charge conjugated L and R fields. The following fields c c c c L; R ( )L; ( )R and L; R; ( )L; ( )R (9) are available. The charge conjugated L and R fields satisfy the useful relations c c c c ( L) = ( )R and ( R) = ( )L: (10) The proof uses the properties of γ matrices listed in chapter *15*. One gets, e.g. for the L field, T 1 − γ5 c 1 − γ5 ( )c = = C L 2 2 " #T 1 − γ5 y 1 − γ5 T = C γ0 = C y γ0 2 2 1 − γ5 1 + γ5 = C γ0 yT = C γ0 yT 2 2 1 + γ5 1 + γ5 1 + γ5 = C ( yγ0)T = C ¯T = c = ( c) ; 2 2 2 R (11) c and similarly for ( R) . Keeping only the pairs of fields in (9) with opposite chiralities leaves the contributions to H c c c c R L; ( )R L; R( )L; ( )R( )L (12) 2 C. Amsler, Nuclear and Particle Physics and their corresponding hermitian conjugates. However, the fourth term is not indepen- dent. By using the relations listed in table1 one finds that c c c c ¯T ¯T ( )L( )R = ( R) ( L) = (C R)C L ¯T y 0 ¯T 0 ∗ y 0 ¯T = (C R) γ C L = (Cγ R) γ C L −γ0 z }| { ∗† 0 y 0 ¯T T ¯T ¯ = R γ C γ C L = − R L = L R; (13) since fermion fields anticommute. The most general expression for the Hamiltonian is therefore 1 1 H = M + M ( c) + M ( c) + h:c: (14) D R L 2 L R L 2 R R L 1 1 = M + M ( )c + M ( )c + h:c: (15) D R L 2 L L L 2 R R R where we have used the equalities (10). The first term conserves lepton number and endows the neutrino with a Dirac mass MD. The second and last terms violate lepton number conservation by ∆L = ±2 and contribute Majorana masses ML and MR, as illustrated in figure1. Figure 1: Contributions to the Hamiltonian from Dirac neutrino fields (a) and Majorana fields (b), (c). (The mass MD is due to direct coupling (×) to the Higgs field, in contrast to ML and MR, which require more complicated mass generation mechanisms, such as loops.) So far, we have listed the contributions to the Hamiltonian with the couplings MD, ML and MR, which are not the physical neutrino masses. Let us therefore identify the eigenstates of the Hamiltonian (the physically observed states) and the corresponding eigenvalues (the observable neutrino masses). Let us define the fields + ( )c + ( c) + ( )c + ( c) f ≡ L p L = L p R and F ≡ R p R = R p L : (16) 2 2 2 2 These fields are of the Majorana type because f c = f and F c = F . This can be seen by conjugating L and R twice, e.g. for L: T T c c c ¯ T 0 ∗ T 0 ∗ y 0 T ( L) = C L = CC L = C[Cγ L] = C[(Cγ L) γ ] C T z 0 }|y {0 T T = C[ L γ C γ ] C = CC L = L (17) 3 Seesaw mechanism (see table1). By keeping only terms with opposite chiralities we can rewrite the Hamil- tonian (14) as ¯ ¯ ¯ ¯ H = MLff + MRFF + MD(fF + F f) M M f = (f;¯ F¯) L D : (18) MD MR F The eigenvalues m and eigenstates φ are found by diagonalizing the matrix, that is by and solving M − m M L D φ = 0: (19) MD MR − m The secular equation 2 (ML − m)(MR − m) − MD = 0 (20) leads to r M + M (M − M )2 m = R L ∓ R L + M 2 : (21) 1;2 2 4 D Consider first the purely Dirac neutrino scenario with MR = ML = 0, hence m1 = −MD and m2 = MD. The normalized eigenfunctions in the (f; F ) basis are 1 1 f + F 1 1 F − f φ1 = p ≡ ν = p ; and φ2 = p ≡ N = p : (22) 2 1 2 2 −1 2 The Hamiltonian now reads in terms of ν and N: =1 z}|{ ¯ 5 2 ¯ 0 0 ¯ H = −MDνν¯ + MDNN = −MDν¯(γ ) ν + MDNN = MDν¯ ν + MDNN; (23) where we have redefined the field ν as ν0 = γ5ν to obtain a positive mass. Our Majorana 0 c c fields ν and N are superpositions of L, ( )R, ( )L and R with degenerate masses MD. The first two fields occur in the weak interaction where they couple to the known 80 GeV W ± boson (section *16.7*), while the last two ones correspond to sterile neutrinos. ± In another hypothetical scenario, a gauge boson WR might exist, which couples to R fermions or L antifermions. So far WR has not been observed, being too massive and therefore beyond reach of present accelerators. Let us assume that this boson decays into heavy neutrinos and set ML = 0;MR >> MD: (24) Then from (21) r 2 2 MR MR 2 MR MR MD 2 m1 = − + MD ' − 1 + 2 2 = −MD=MR: (25) 2 4 2 2 MR The corresponding eigenstate in the (f,F ) basis is found by solving the equation p −m M 1 − b2 1 D = 0; (26) MD MR − m1 b 4 C. Amsler, Nuclear and Particle Physics hence M b ' D ; (27) MR and therefore M ν ' f + D F ' f: (28) MR 2 We get a light Majorana neutrino ν with mass MD=MR. (A positive mass is again obtained by the substitution ν ! γ5ν in the Hamiltonian.) The neutrino ν is essentially the superposition of a Dirac L neutrino and Dirac R antineutrino (see (16)) which couple to the 80 GeV W ± boson. This is would be our familiar neutrino, a Majorana particle. The second solution of (21) is r 2 2 MR MR 2 MR MR MD m2 = + + MD ' + 1 + 2 2 ' MR: (29) 2 4 2 2 MR The corresponding eigenstate is orthogonal to ν: M N = F − D f ' F; (30) MR a heavy Majorana neutrino with mass MR. According to (16) N is essentially a superpo- sition of a Dirac R neutrino and Dirac L antineutrino which couple to the very massive WR boson. Figure 2: The seesaw mechanism would explain the tiny mass of the left-handed neutrino by the existence of a heavy specie with opposite chirality. From (28) and (30) one finds that 2 m1m2 = MνMR ' MD : (31) This is the essence of the seesaw mechanism originally proposed by [1]. The tiny mass of the neutrino is compensated by the very high mass of a neutrino with opposite chirality (figure2).
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