Particle Physics of the Early Universe Alexey Boyarsky Spring semester 2015 Fermi theory of β-decay 1 − Continuum Neutron decay n ! p + e +ν ¯e spectrum of electrons (1927) Two papers by E. Fermi: Prediction of An attempt of a theory of beta radiation. 1. (In neutrino German) Z.Phys. 88 (1934) 161-177 (1930, 1934) DOI: 10.1007/BF01351864 Fermi theory Trends to a Theory of beta Radiation. (In (1934) Italian) Nuovo Cim. 11 (1934) 1-19 DOI: 10.1007/BF02959820 Universality of Fermi interactions Fermi 4-fermion theory: (1949) GF µ LFermi = −p [¯p(x)γµn(x)][¯e(x)γ ν(x)] (1) 2 0History of β-decay (see [hep-ph/0001283], Sec. 1,1); Cheng & Li, Chap. 11, Sec. 11.1) Alexey Boyarsky PPEU 2015 1 Neutrino-electron scattering Fermi Lagrangian includes leptonic and hardronic terms: GF h y y i h i LFermi = −p J (x) + J (x) · Jlepton(x) + Jhadron(x) (2) 2 lepton hadron where current J µ has leptonic (e±, µ±, ν, ν¯) and hadronic (p, n, π) parts Fermi theory predicts lepton-only weak interactions, such as e + νe ! e + νe scattering 4GF λ Lνe = p (¯eγλνe)(¯νeγ e) (3) 2 Matrix element for e + νe ! e + νe scattering X 2 2 4 jMj / GF Ec:m: (4) spins Alexey Boyarsky PPEU 2015 2 Massive intermediate particle Unitarity means that for any process the matrix element should be bounded from above: jMj ≤ const The matrix element for e+νe ! e+νe scattering grows with energy, 2 2 4 jMj / GF Ec:m:. Therefore, the Fermi theory wouldp predict meaningless answers for scattering at energies Ec:m: & GF ≈ 300 GeV Promote point-like 4-fermion Fermi interaction to interaction, mediated by a new massive particle : W – massive particles with MW Alexey Boyarsky PPEU 2015 3 Massive intermediate particle p For s MW – looks like a point, 4-fermion interaction p −1 For s MW – behaves as s What is the spin of this particle? It should couple to current so it is a vector field: g + − − + Lint = p J W + J W (5) 2 2 µ µ µ µ g – new coupling constant responsible for weak interactions ± The currents Jµ made of electron and νe (or muon and νµ) carries charge ±1. Alexey Boyarsky PPEU 2015 4 Theory of massive vector boson Free massive vector boson obeys Proka equation: µ ν ν µ 2 µ @ν(@ W − @ W ) = MW W (6) 2 µ Taking @µ derivative of (4) )MW @µW = 0 Eq. (6) can be rewritten as a set of Klein-Gordon equations µ 2 µ W = MW W (7) 0 Three independent plane wave solution (because now W 6= 0): µ (i) (i) ix·k k · µ = 0; i = 1; 2; 3 Wµ = µ e where µ 2 k · kµ = MW (i) µ are 3 vectors of polarizations Alexey Boyarsky PPEU 2015 5 Theory of massive vector boson Consider W boson at rest The three polarization vectors are just three unit vectors along the axes x, y, and z µ Now, make boost along z axis. W -bosons 4-momentum k = (E; 0; 0; kz) Three polarization vectors are 8 (1) > µ (k) = (0; 1; 0; 0) < (2) µ (k) = (0; 0; 1; 0) > (3) 1 : µ (k) = (kz; 0; 0;E) MW µ Consider W boson with energy E MW , then k = (E; 0; 0; kz) ≈ E(1; 0; 0; 1) when E ≈ kz MW Alexey Boyarsky PPEU 2015 6 Theory of massive vector boson (3) E E The third polarization vector: µ = (kz; 0; 0;E) ≈ (1; 0; 0; 1) MW MW kµ L ≈ MW µ There is a subtle difference here! k is time-like vector and L is a space- like vector. In the relativistic limits they both approach light-cone, but from two different sides Longitudinally polarized Wµ in the limit E MW looks like a derivative of a scalar function W µ ≈ 1 @µφ L MW Interaction with currents: − + longitudinal @µφ − g µ − gJµ Wµ −−−−−−! g Jµ = φ(@ Jµ ) MW MW . this looks like a new dimensionful coupling constant Alexey Boyarsky PPEU 2015 7 Massive vector field and St¨uckelberg field Introduce new scalar field, θ. It interacts with a gauge field W , satisfying Maxwell’s equation: µ ν ν µ @ν(@ W − @ W ) = 0 (8) Under gauge transformation Wµ ! Wµ + @µλ the field θ shifts as θ ! θ − λ Equation of motion for θ (Dµθ = @µθ + Wµ): µ µ µ @µ(D θ) = @µ(@ θ + W ) = 0 (9) The full equation for W becomes: µ ν ν µ 2 µ @ν(@ W − @ W ) = MW D θ (10) Gauge condition θ = 0 reduces these two equations for W and θ to the old Proka equation Alexey Boyarsky PPEU 2015 8 Vector boson vs. photon (i) P3 (i) (i) kµkν Show that if are 3 polarization vectors than = −ηµν + µ i=1 µ ν M2 W Define a propagator of massive Klein-Gordon equation with µ additional condition @µW = 0: 4 Z d p 0 1 X hW (x)W (x0)i = e−ip(x−x ) i (p)i (p) µ ν (2π)4 p2 − M 2 µ ν W polarizations pµpν 4 η − Z 0 µν 2 d p ip(x−x ) MW = 4 e 2 2 (2π) p − MW (11) Try to put MW ! 0. Will you recover photon-like propagator? No! Trouble with the term in the numerator Alexey Boyarsky PPEU 2015 9 Scattering eν ! eν The relevant part of the interaction Lagrangian is + 1 µ − 1 µ ∆L = gW p ν¯eγ PLe + gW p eγ¯ PLνe (12) µ 2 µ 2 1 PL = 2(1 − γ5) – projector of the 4-component spinor on the left chirality. The relevant matrix element is given by 0 rµrν 1 2 g − ig µν M2 p µ W ν iM = [¯u(k1)γ PLu(p2)] (−i) @ 2 2 A [¯u(k2)γ PLu(p1)] 2 r − MW (13) After average over polarization of the colliding electron and summation over polarizations of other particles we obtain: 4 2 ¯ 2 g s 2 jMj = 2 2 2; s = (p1 + p2) (14) 2(r − MW ) Alexey Boyarsky PPEU 2015 10 Scattering eν ! eν 2 2 At low energies, when s; r MW , but nevertheless s; r me, we ¯ 2 2 2 get the result of the Fermi theory jMj = 16GF s , provided that we identify p 2g2 GF = 2 (15) 8MW In the general case 4 ¯ 2 2g 2 2 s jMj = ; r = (p2−k1) = − (1+cos θ) (16) 2 2 2MW 2 1 + cos θ + s (j) The unitarity requirement M ≤ 1 then leads to 16π s ≤ M 2 exp − 1 (17) W g2 2 p 28 For the known value αW = g =4π ≈ 1=30, we get s . 10 GeV. Alexey Boyarsky PPEU 2015 11 Lagrangian of W boson Recall that the Lagrangian of the massive vector field Bµ would have the form (Proka Lagrangian): 1 1 L = − (@ B − @ B )2 + m2 BµB (18) Proka 4 µ ν ν µ 2 B µ However W -boson is charged! Under the gauge transformation: ± ±iα ± Wµ ! e Wµ (19) Therefore, we should change @µ ! Dµ in Eq. (18), where ± ± DµWν = (@µ ± i e Aµ)Wν We follow largely the book by J. Horejsi “Introduction to Electroweak Unification: Standard Model from Tree Unitarity”, Chapters 3 and 4. Alexey Boyarsky PPEU 2015 12 Lagrangian of W boson The kinetic term for W -boson is therefore 1 M 2 L = − (D+W −−D+W −)(D−W +−D−W +)+ W W +W − (20) W 2 µ ν µ ν µ ν µ ν 2 µ µ Show that in addition to (18) Lagrangian (20) also contains terms of the form W +W −γ and W +W −γγ. Write down explicit form of the W W γ and W W γγ interactions Alexey Boyarsky PPEU 2015 13 1 + − νeν¯e ! W W scattering The amplitude of the process: ig µ i =q ig ν ∗ ∗ iM =v ¯(p1) p γ PL p γ PL u(p2)µ(k1)ν(k2) 2 q2 2 2 Take s me (so that mass of electron was neglected in the fermion propagator). X 2 jMj¯ 2 = jMj = (21) s g2 h i k k k k = Tr =p γµ =qγν =p γρ =qγλ −g + 1µ 1λ −g + 2ν 2ρ 2 2 1 2 µλ 2 νρ 2 4q q MW MW 2 In the high-energy approximation s; q MW dominates longitudinal part, coming from k1;2=MW (i.e. one can neglect gµν term in Alexey Boyarsky PPEU 2015 14 1 + − νeν¯e ! W W scattering the numerator) 2 2 ¯ 2 g s jMj = 4 (22) MW grows as a fourth power of energy. Therefore, the considered process still violates unitarity. + − + − Similarly the process e e ! W W violates unitarity Can this be amended? Alexey Boyarsky PPEU 2015 15 More contributions to ee ! WW We saw that as W -bosons are charged, there is W W γ interaction term. Therefore the real process ee ! WW is described by the sum of two diagrams Can these two diagrams give rise to the unitary behavior of the resulting cross-section? No! (left diagram is parity violating, right diagram is parity conserved!) Alexey Boyarsky PPEU 2015 16 New particle is needed + − + − What could restore the unitarity of e e ! W W and νν¯ ! WW ? )new particle New vector boson that couples to electrons and to neutrinos in the parity violating way and that also couples to W +W −. New boson (Z-bosons) interacts with ν: 1 L = g νγ¯ µ(1 − γ )νZ (23) ννZ¯ 2 ννZ¯ 5 µ + − New boson also interacts with W W and the vertex WWZ is similar to the vertex W W γ + − As result there are two processes contributing to νν¯ ! W W scattering Alexey Boyarsky PPEU 2015 17 New particle is needed 2 2 E As discussed before Ma / g from longitudinally MW polarized final W states 2 E Mb / (gννZ¯ gZWW ) Can cancel contribution of Ma if MW 1 2 gννZ¯ gZWW = 2g Alexey Boyarsky PPEU 2015 18 Z-boson contribution to e+e− ! W +W − + − + − Similarly, for e e ! W W we would have three contributions to 2 2 the matrix element: jMj = Ma + Mb + Mc Interaction with electrons: µ µ Vint = gLe¯Lγ eLZµ + gRe¯Rγ eRZµ (24) Alexey Boyarsky PPEU 2015 19 New symmetry? Neutrino and electron – different charge.