Errata 18/05/2018. Chapter 1. Chapter 2

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1 Errata 8/05/08 Fundamentals of Neutrino Physics and Astrophysics C. Giunti and C.W. Kim Oxford University Press publication date: 5 March 007; 78 pages ± Lines are calculated before or after + the Anchor. If the Anchor is a page, t and b indicate, respectively, top and bottom. page iii t + 5 Universita Università Chapter page 4 b Chapter eqn.3 eqn.89 /p P mγ 0 u h p P = 0 /p mγ 0 u h p P = 0 eqn.76 + following eqn.76 following eqn.78 eqn.36 transform as transform as normal ordering is implicitly assumed eqn.378 a parity transformation a passive parity transformation eqn.385 he transformation he active transformation eqn.404 he transformation he active transformation eqn.408 transform as transform as normal ordering is implicitly assumed eqn.44 V µ A µ W µ V ba µ A ba µ W µ eqn.44 V ba µ Aba µ W µ V µ A µ W µ eqn.46 a time reversal a passive time reversal eqn.46 x µ = x0, x = x µ x µ = x0, x = x µ eqn.44 the transformation the active transformation

2 Chapter eqn.434 eqn.435 V µ = S = ξ a ξb ψ bψ a = S = ξ ξ a ξs b aξb ψ a ψ b = ba ξξ a b S ξb V ba µ eqn.436 µν = ξ a eqn.437 eqn.438 eqn.439 eqn.440 ξ a ξ b ba µν ξb ψ bγ µ ψ a = ξb ψ bσ µν ψ a = A µ = ξb ψ bγ µ γ 5 ψ a = ξb Aba µ V µ = ξ aξb ψ aγ µ ψ b = ξ aξb V µ µν = ξ a ξb ξ a ξ b µν ψ aσ µν ψ b = A µ = ξ a ξb ψ aγ µ γ 5 ψ b = ξ aξb A µ P = ξ a ξb ψ bγ 5 ψ a = P = ξ ξ a ξp b aξb ψ a γ 5 ψ b = ba ξξ a b P W µ ξ W W µ W µ ξ W ξ a ξb ξw V µ A µ W µ + ξξ a b ξ W ξ aξb ξw V Vµ ba A ba µ W µ ξξ b W V ba ξ a W µ µ A µ µ A ba µ W µ + W µ

3 Chapter eqn.454 transform as transform as normal ordering is implicitly assumed eqn.454 S CP = CP ξb CP ψ aψ b = CP ξb CP S eqn.455 V µ CP = ξcp a ξb CP ψ aγ µ ψ b = ξcp a ξb CP V µ eqn.456 µν CP = CP ξb CP ψ aσ µν ψ b = CP ξb CP µν eqn.457 A µ CP = ξ a CP ξb CP ψ aγ µ γ 5 ψ b = ξ a CP ξb CP Aµ eqn.458 P CP = CP ξb CP ψ aγ 5 ψ b = CP ξb CP P eqn Since all the covariant bilinears are left invariant by a CP transformation, apart for a possible irrelevant phase which is the same for the vector and axial currents, any possible interaction Lagrangian is invariant under CP, in agreement with the CP theorem, which says that CP is a symmetry of any relativistic local field theory. S CP = ξ a CP ξb CP ψ bψ a = ξ a CP ξb CP S ba V µ CP = ξ a CP ξb CP ψ bγ µ ψ a = ξcp a ξb CP V µ ba µν CP = ξcp a ξb CP ψ bσ µν ψ a = ξ a CP ξb CP µν ba A µ CP = ξcp a ξb ξcp a ξb CP Aµ ba P CP = ξ a CP ξb CP ψ bγ µ γ 5 ψ a = CP ψ bγ 5 ψ a = ξcp a ξb CP P ba Choosingξ a CP = ξb CP, CP transforms each covariant bilinear into its Hermitian conjugate, with a minus sign for V µ,aµ and P. Since an interaction Lagrangian containing a covariant bilinear must contain also its Hermitian conjugate the Lagrangian is Hermitian, it is invariant under CP. he minus sign in the transformation ofv µ, Aµ and P is compensated by a corresponding minus sign in the transformation of the fields to which they are coupled. he invariance under CP of any interaction Lagrangian containing a covariant bilinear is in agreement with the CP theorem, which says that CP is a symmetry of any relativistic local field theory. 3

4 Chapter eqn.487 fp,h = fp,h = EV ah p 0, EV bh p 0 fp,h = EV a h p 0, fp,h = EV b h p 0 Chapter 3 eqn 3.44 eqn 3. k his equation is wrong. Erase all the sentence. [ I k I k + ] [ I3 k v k k I k I k + ] I k 3 vk k I k 3 v k k I k 3 vk eqn ε 0 µ p p = ω ε 0 p p = ω eqn ε µ p p = ε µ p p = 0 ε p p = ε p p = 0 eqn ε 3 µ p p = ω ε 3 p p = ω Chapter 4 cosϑe eqn 4. iω sinϑe iω +η cosϑe iω sinϑe iω η cosϑe iω ω 0 e iω 0 eqn ω 0 e ω eqn 4.6 W 3 = W 3 ϑ 3,η 3 = D η 3 R 3 D η 3 sinϑe iω +η sinϑe iω η cosϑe iω W 3 = W 3 ϑ 3,η 3 η η 3 eqn 4.78 η 3 η 3 η η 3 eqn One can parameterize the mixing matrix as a product of the type in eqn 4.65 with W ϑ = π/,η on the extreme left or the extreme right. Using IfW ϑ = π/,η is on the extreme left or on the extreme right of the product in eqn 4.45 which parameterizes the mixing matrix, using eqn 4.78 δ 3 = η 3 δ 3 = η 3 +η +η 3 eqn eqn 4.5 eqn 4.6 4

5 Chapter 5 eqn 5.0 eqn 5. eqn th e g νe = g νe g νe = +g νe g νe = g νe g νe = g νe = 0.9 GeV Eth eqn 5.50 A π l ν l Aūd l ν l ν = 0.9 GeV eqn 5.50 u ν p ν γ ρ γ 5 v l p l u l p l γ ρ γ 5 v ν p ν eqn 5.5 v u p u γ ρ γ 5 u d p d m π 0 h ρ W 0 π p π eqn he factor/m π serves to keep the dimensions right the left-hand side has dimension of energy, the current h ρ W has dimension ofe 3, and the one-pion state has dimension of E, with the normalization π p π π p π = π 3 E π δ 3 p π p π as the one-fermion state in eqn.34. eqn 5.54 eqn 5.55 v u p u γ ρ γ 5 u d p d = 0 h ρ W 0 ūp u,dp d 0 h ρ W 0 π p π Note the change of dimensions in Eq. 5.5: the left-hand side has dimension of energy E, whereas the right-hand side has dimension ofe the current h ρ W has dimension ofe 3, and each one-particle state of a fermion or a boson has dimension ofe. In this way, the amplitude acquires the correct dimension ofe needed in eqn E.44 for a two-body decay rate. u ν p ν / p π γ 5 v l p l m π u l p l /p π γ 5 v ν p ν m l u ν p ν +γ 5 v l p l m π m l u l p l +γ 5 v ν p ν eqn 5.83 v ρ x v ρ W x eqn 5.84 a ρ x a ρ W x eqn 5.86 v ρ 0 v ρ W 0 eqn 5.4 Q 6 rn i G N i 0 Q 6 rn i eqn 5.47 Erase in the loratory frame. eqn 5.47 eqn 5.54 eqn 5.87 p Ni eqn Q 4m N p N G P 4 + Q G 4m P N 5

6 Chapter 5 eqn 5.60 G F V ud m N 4π G F V ud m N 4π eqn A ZN, B ZN, and C ZN A N, B N, and C N eqn 5.56 eqn 5.57 eqn 5.58 F,Q ZN F,q ZN F,Q ZN = xfzn,q F,q ZN = xf,q ZN F3,Q ZN F3,q ZN Chapter 6 eqn 6. L αl L αl eqn 6.69 iγµ ν L i / ν L page 08 t + 5 [53,79,79,408] [53,79,N,408] New Reference: [N] P. Langacker, M.-X. Luo, Phys. Rev. D44, 87, 99. eqn 6.06 ν L ML Cν L ν L ML Cν L eqn 6.0 ν L W L ML W L Cν L ν L W L ML W L Cν L eqn M l. M l = 0. page 7 t + [53,79,79,408] [53,79,N,408] page 8 b [84] [N,N3] New References: [N] G. Lazarides, Q. Shafi, C. Wetterich, Nucl. Phys. B8, 87, 98. [N3] R.N. Mohapatra and G. Senjanovic, Phys. Rev. D3 65, 98. eqn n i M n i n M eqn 6.48 l αl γ ρ U αk ν kl l αl γ ρ U αk ν kl W ρ eqn 6.48 l L γ ρ U n L l L γ ρ U n L W ρ n 6

7 Chapter 7 [ ] [ ] L/E L/E eqn 7.95 exp σl/e exp L/E L/E σl/e eqn eqn 7.70 eqn 7.93 eqn 7.3 m m kj twice eqn j NA k>n A Im... j N A Chapter 8 eqn 8.9 M P αk ormd αk M P αk and MD αk eqn σ I t σi p σ I t σi x page 309 t+3 eqn 8.5 eqn 8.4 eqn 8.44 twice m Chapter 9 m kj eqn 9. G F G F eqn 9. G F G F eqn 9.57 σ M = U M σ FU M = H M σ M = U M σ FU M Chapter 0 k>n A Im... eqn as an effectively incoherent sum as effectively incoherent sums Chapter eqn.67 R multi-gev µ/e R sub-gev µ/e Chapter eqn.3 Neglecting the small recoil energy of the neutron, the he 7

8 Chapter 3 page 476 b 9 m 3 03 ev m ev Chapter 6 eqn 6.40 = eqn χ-dec χ decoupling = χ-dec γ at the eqn because γ is the monopole moment of the temperature: γ = γ θ,φyl m θ,φ dcosθ dφ. eqn the wavelength of each Fourier mode χ-dec χ = χ-dec γ becausey0 0 = / 4π and γ = γ θ,φ dcosθ dφ. 4π the amplitude of each wavelength Chapter 7 eqn Ω 0 M. 0 4 Ω 0 M eqn He H eqn an upper limit a lower limit Appendix A δ eqn A.9 ǫ ijk ǫ lmn = il δ im δ in δ δ jl δ jm δ in δ kl δ km δ kn ǫ ijk ǫ lmn = il δ im δ in δ jl δ jm δ jn δ kl δ km δ kn σ eqn A.05 σd 0k = iαk D = i k 0 0 σ 0 σ k σd 0k = iαk D = i k σ k 0 Appendix B eqn B.7 g αρ Λ ρ µ g µν Λ ρ ν = δ α ν g αρ Λ ρ µ g µν Λ ν σ = δ α σ 8

9 Appendix C eqn C. [ψ r t, x, π s t, y] ± = iδ rs δ 3 x y eqn C. [ψ r t, x, ψ s t, y] ± = [π r t, x, π s t, y] ± = 0 [ψ r t, x, π s t, y] ± = iδ rs δ 3 x y [ψ r t, x, ψ s t, y] ± = [π r t, x, π s t, y] ± = 0 Bibliography Ref. [83] J. Phys. Conf. Ser., 53, 44-8, 006 page 693 [73] and [73] are the same Ann. Rev. Nucl. Part. Sci., 56, , 006 9