Weak Interactions I. Susan Gardner. Department of Physics and Astronomy University of Kentucky Lexington, KY USA.

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1 Weak Interactions I Susan Gardner Department of Physics and Astronomy University of Kentucky Lexington, KY USA gardner@pa.uky.edu Lecture 1: A survey of basic principles and features with an emphasis on low-energy probes Lecture 2: The electroweak Standard Model and interpreting it as an effective field theory

2 How Weak is the Weak Interaction? We know of four fundamental interactions: electromagnetic, strong, weak, and gravitational. Let s set gravity aside and focus on the others. Particles of comparable mass can have very different lifetimes. π + µ + ν µ [99.98% of all π + decays] ; τ π s π 0 2γ [98.8% of all π 0 decays] ; τ π s. whereas Γ τ 1 = ρ 0 π + π ρ 0 µ + µ gem eff 2 geff weak = g em eff 10 4 g weak [ 100% of all ρ 0 decays] [ of all ρ 0 decays] eff = gem eff 2 geff str = g str 2 eff 10 2 geff em Conclude weak interaction is 10 6 times weaker than the strong interaction! S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 2

3 Building a Standard Model A first thought: perhaps we can describe all 3 interactions independently? What precepts should we impose? A list circa 1980: Particle Content (i.e., 3 generations of quarks and leptons) Symmetries CPT (and Lorentz) Symmetry Gauge Symmetry (SU(3)......) Unitarity Renormalizability This makes our theory predictive even as E! It is UV complete! This line of thinking ultimately yields a SU(3) c SU(2) L U(1) Y gauge theory we call the Standard Model (SM). It is predictive and successful once all its parameters are fixed. In 2015 the existence of known unknowns (e.g., dark matter) is now definite. Perhaps we can describe these new features within the context of a theory with a SM-like gauge symmetry? Let s take just baby steps beyond the SM! S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 3

4 on Effective Field Theory To explain phenomena at some fixed energy scale, we need only include the degrees of freedom operative at that energy scale. E.g., we can predict the outcome of chemical reactions without understanding how the electron gets it mass! A simple application of effective field theory: Why is the sky blue? Here we consider low-energy scattering of photons from neutral atoms E γ E α 2 m e a 1 0 m e α M atom The low-energy interactions of the atom (ψ) are fixed by symmetry: gauge and P and C invariance... L int = c 1 a 3 0 ψ ψf µν F µν +... with the a0 3 factor fixed by common sense and power counting so that c 1 O(1). (dim[ψ] = 3/2 and dim[f µν ] = 2) Thus A c 1 a0 3E γ 2 = σ a0 6E γ 4 [ = c = 1] and we conclude that blue light is scattered more strongly than red light! Thus a theory need not be UV complete to be predictive. Through identifying traces of new physics at low energy we hope to identify the nature of E physics! We return now to the path that led to the rise of the SM... S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 4

5 The Discrete Symmetries C, P, and T In particle interactions, can we tell... Left from Right? (P) Positive Charge from Negative Charge? (C) Forward in Time from Backward in Time? (T) Matter from Antimatter? (CP) If we observed a box of photons at constant temperature T m e, interacting via electromagnetic forces, the answer would be No. e e + e e+ However,... S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 5

6 On the Possibility of Parity Violation Context: Dirac the existence of a magnetic monopole can explain the quantization of electric charge! [Dirac, Proc. Roy. Soc. London A 133, 60 (1931)] E = 4πρ ; B = 0 = 4πρ M Dirac also showed that the circulation of opposite magnetic monopoles in the nucleon could give rise to a nonzero electric dipole moment. [Dirac, Phys. Rev. 74, 817 (1948).] The electric dipole moment d of a nonrelativistic particle with spin S is defined via H = d S S E But both quantities violate P and T! E. M. Purcell and N. F. Ramsey, On the Possibility of Electric Dipole Moments for Elementary Particles and Nuclei, Phys. Rev. 78, 807 (1950): The argument against electric dipoles, in another form, raises the question of parity... But there is no compelling reason for excluding this possibility... S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 6

7 Discrete Symmetries P, T, and C Parity P: Parity reverses the momentum of a particle without flipping its spin. Pa s pp = a s p, Pb s pp = b s p = Pψ(t, x)p = γ 0 ψ(t, x) Time-Reversal T : Time-reversal reverses the momentum of a particle and flips its spin. It is also antiunitary; note [x, p] = i. Ta s pt = a s p Tb s pt = b s p = T ψ(t, x)t = γ 1 γ 3 ψ( t, x) Charge-Conjugation C: Charge conjugation converts a fermion with a given spin into an antifermion with the same spin. Ca s pc = b s p, Cb s pc = a s p = Cψ(t, x)c = iγ 2 ψ (t, x) S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 7

8 The Weak Interactions Violate Parity There is a fore-aft asymmetry in the e intensity in 60 Co β-decay... [Wu, Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. 105, 1413 (1957); note also Garwin, Lederman, and Weinrich, Phys. Rev. 105, 1415 (1957); Schematically Co (J=5) Ni* (J=4) νe + + e e ν e P is violated in the weak interactions! Both P and C are violated maximally I e (θ) = 1 J p e E e Γ(π + µ + ν L ) Γ(π + µ + ν R ) = 0 Γ(π + µ + ν L ) Γ(π µ ν L ) = 0 ; P violation ; C violation S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 8

9 The Two-Component Neutrino A Dirac spinor can be formed from two 2-dimensional representations: ( ) ψl ψ = ψ R In the Weyl representation for γ µ, ( ) ( ) (iγ µ m i( µ m)ψ = 0 + σ ) ψl = 0 i( 0 σ ) m ψ R If m=0, ψ L and ψ R decouple and are of definite helicity for all p. Thus, e.g., i( 0 σ )ψ L (x) = Eψ L = σ p ψ L σ ˆp ψ L = ψ L Note ψ L ψ L γ0 transforms as a right-handed field. Experiments = No mirror image states : neither ν L nor ν R exist. Possible only if the neutrino is of zero mass. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 9

10 The Weak Interactions Can Also Violate CP CP could be a good symmetry even if P and C were violated. Schematically ν e CP e+ e ν e Γ(π + µ + ν L ) = Γ(π µ ν R ) Weak decays into hadrons, though, can violate CP. There are short-lived and long-lived K states: K S 1 2 (K 0 K 0 ) π + π ; CP invariance! (CP even) K L 1 2 (K 0 + K 0 ) π + π π 0 (CP odd) However, K L 2π as well! K S and K L do not have definite CP! [Christenson, Cronin, Fitch, Turlay, PRL 13, 138 (1964).] S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 10

11 Matter and Antimatter are Distinguishable The decay rates for K 0, K 0 π + π and B 0, B 0 J/ψK S are appreciably different. [I.I. Bigi, arxiv: v2 and references therein.] S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 11

12 All Observed Interactions Conserve CPT The CPT Theorem Any Lorentz-invariant, local quantum field theory in which the observables are represented by Hermitian operators must respect CPT. [Pauli, 1955; Lüders, 1954] Coda: CPT violation implies Lorentz violation. [Greenberg, PRL 89, (2002)] CPT = the lifetimes, masses, and the absolute values of the magnetic moments of particles and anti-particles are the same! Note, e.g., M K 0 M K0 < 6 10 CL M avg M p M p < 7 10 CL M p Thus CP T violation. Tests of CPT and Lorentz invariance are ongoing. e.g., ATRAP Collaboration, arxiv: ,... For the first time a single trapped p is used to measure the p magnetic moment µ p.... The 4.4 parts per million (ppm) uncertainty is 680 times smaller than previously realized. Comparing to the proton moment measured using the same method and trap electrodes gives µ p /µ p = ± to 5 ppm, for a proton moment µ p = µ ps/( /2), consistent with the prediction of the CPT theorem. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 12

13 Transformations of Lorentz Bilinears under P, T, and C Notation: ξ µ = 1 for µ = 0 and ξ µ = 1 for µ 0. γ 5 iγ 0 γ 1 γ 2 γ 3 ; σ µν i 2 [γµ, γ ν ] ψψ i ψγ 5 ψ ψγ µ ψ ψγ µ γ 5 ψ ψσ µν ψ µ S P V A T P +1 1 ξ µ ξ µ ξ µ ξ ν ξ µ T +1 1 ξ µ ξ µ ξ µ ξ ν ξ µ C CPT S is for Scalar P is for Pseudoscalar V is for Vector A is for Axial-Vector T is for Tensor All scalar fermion bilinears are invariant under CPT. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 13

14 Symmetries of a Dirac Theory A Lagrangian must be a Lorentz scalar to guarantee Lorentz-invariant equations of motion. E.g., applying the Euler-Lagrange eqns to yield Dirac equations for ψ and ψ. We can form two currents L Dirac = ψ(iγ µ µ m)ψ j µ (x) = ψ(x)γ µ ψ(x) ; j µ5 (x) = ψ(x)γ µ γ 5 ψ(x) j µ is always conserved if ψ(x) satisfies the Dirac equation: µ j µ = ( µ ψ)γ µ ψ + ψγ µ µ ψ = (im ψ)ψ + ψ( imψ) = 0, whereas µ j µ5 = 2im ψγ 5 ψ it is conserved only if m = 0. By Noether s theorem a conserved current follows from an invariance in L Dirac : ψ(x) e iα ψ(x) ; ψ(x) e iαγ5 ψ(x) The last is a chiral invariance; it only emerges if m = 0. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 14

15 Symmetries of a Dirac Theory To understand why it is a chiral invariance, we note in the m = 0 limit that ( ) ( ) j µ L = ψγ 1 γ µ 5 ψ, j µr 2 = ψγ 1 + γ µ 5 ψ. 2 The vector currents of left- and right-handed particles are separately conserved. Note in Weyl representation ( ) γ = 0 1 The factor (1 ± γ 5 ) acts to project out states of definite handedness. ( ) ( ) 1 γ γ 5 ψ L ψ, ψ R ψ. 2 2 so that L = ψ L iγ µ µ ψ L + ψ R iγ µ µ ψ R = L L + L R S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 15

16 Electromagnetism We assert that if we couple a Dirac field ψ to an electromagnetic field A µ j µ is the electric current density. ψ can describe a free electron. ψ(x) p, s = u(p)e ip x = (γ µ p µ m)u(p) = 0. By canonical substitution p µ p µ + ea µ (γ µ p µ m)u = γ 0 Vu ; γ 0 V = eγ µ A µ In O(e) the amplitude for an electron scattering from state i f is T fi = i u f V (x)u i(x) d 4 x = i jµa fi µ d 4 x with jµ fi = eū f γ µ u i For e p scattering, e.g., we have T fi = i jµ(x) ( e 1q ) 2 jµ(x) p d 4 x = im(2π) 4 δ (4) (p + k p k ) M e2 q 2 ( j em µ )p (jem µ ) e = (eū p (p )γ µ u p (p)) A current-current interaction. ( e2 q 2 ) ( eū e (k )γ µ u e (k)) S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 16

17 Fermi Theory Now consider n pe ν e. Fermi s crucial insight was to realize that the weak currents could be modelled after electromagnetism: M = G(ū p (p )γ µ u n (p))(ū e (k )γ µ u ν (k)) The observation of e p capture suggests L Fermi = G F 2 { ( ψp γ µ ψ n )( ψ e γ µ ψ ν ) + h.c. } An interaction with charged weak currents. A weak neutral current was discovered in G F is the Fermi constant, though G F 10 5 (GeV) 2. (N.B. not UV complete!!) Suggests the interaction is mediated by massive, spin-one particles. Fermi s interaction cannot explain the observation of parity violation. Nor can it explain the J = 1 ( Gamow-Teller ) transitions observed in nuclear β-decay. Some A A or T T interaction has to be present. Enter the V A Law... [Feynman, Gell-Mann, 1958; Sudarshan and Marshak, 1958] S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 17

18 Fermi vs. Gamow-Teller Transitions Nuclear β-decay spin-isospin selection rules are dictated by the form of the nonrelativistic transition operator. In allowed approximation (note π i π f = +1)... A τ ± (j) = T ± Fermi = J f = J i, T f = T i j=1 Here both parent and daughter are in isotopic analogue states, e.g., 10 C 10 B. A σ(j) τ ± (j) Gamow-Teller = J i = J f, J f ± 1 (J i = 0 J f = 0), j=1 T = 0, 1 N.B superallowed decays are pure Fermi transitions, whereas neutron β decay is mixed, containing both types. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 18

19 The V-A Law A universal charged, weak current: For the leptons... G F L = 1 {J λ J λ 2 + J λ J λ} with J λ = j l λ + jh λ 2 j l λ = ψ e γ λ (1 γ 5 )ψ νe + ψ µ (k )γ λ (1 γ 5 )ψ νµ + ψ τ (k )γ λ (1 γ 5 )ψ ντ which describes ν l l and l + ν l and asserts the leptons do not mix under the weak interactions. Here the V-A law is equivalent to a two-component neutrino picture. The interactions of the hadrons (quarks) are (and can be) much richer. The strong interaction is strong! The quarks mix under the weak interactions. E.g., K + µ + ν is observed. Recall K + is (u s). Let us continue to focus on neutron β-decay. Recall n is ddu and p is uud. Isospin is an approximate symmetry: M n = MeV M p = MeV (M n M p )/M n 1. n pe nu e occurs because isospin is broken = large τ n. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 19

20 Polarized Neutron β-decay in a V-A Theory d 3 1 Γ = (2π) 5 2m B ( d 3 p p d 3 p e d 3 p ν 2E p 2E e 2E ν )δ 4 (p n p p p e p v ) 1 2 spins M 2 M = G F V ud 2 p(p p ) J µ (0) n(p n, P) [ū e (p e )γ µ (1 γ 5 )u ν (p ν )] p(p p ) J µ (0) n(p n, P) = ū p (p p )(f 1 γ µ i σ µν q ν + f 3 q µ M n M n g 1 γ µ γ 5 + i g 2 M n σ µν γ 5 q ν g 3 M n γ 5 q µ )u n (p n, P) Note q = p n p p and for baryons with polarization P, u n (p n, P) ( 1+γ5/P 2 )u n (p n ) f 1 (g V ) Fermi or Vector g 1 (g A ) Gamow-Teller or Axial Vector f 2 (g M ) Weak Magnetism g 2 (g T ) Induced Tensor or Weak Electricity f 3 (g S ) Induced Scalar g 3 (g P ) Induced Pseudoscalar Since (M n M p )/M n 1, a recoil expansion is very useful. Also in nuclear decays! S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 20 f 2

21 Correlation Coefficients d 3 Γ E e p e (E max e E e ) 2 [1 + a p e p ν E e E ν + P (A p e + B p ν + D p e p ν E e E ν E e E ν A and B are P odd, T even, whereas D is (pseudo)t odd, P even. λ g 1 /f 1 g A /g V > 0 and predictions: )]de e dω e dω ν a = 1 λ2 λ(1 λ) λ(1 + λ) 1 + 3λ 2 A = λ 2 B = λ 2 [+O(R)] implying 1 + A B a = 0 and ab A A 2 = 0 [Mostovoy and Frank, 1976], testing the V-A structure of the SM to recoil order, O(R) with R Ee max /M n Currently a = ± A = ± (S = 2.4) B = ± so that the relations are satisfied. [λ = ± (S = 2.2)] With τ n = ± 1.1 sec (S = 1.9) and τ n f g2 1 more tests are possible. [Olive et al., Particle Data Group, Chin. Phys. C, 38, (2014).] S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 21

22 Symmetries of the Hadronic, Weak Current The values of the 6 couplings (assuming T invariance) are constrained by symmetry. Conserved-Vector Current ( CVC ) Hypothesis Absence of Second-Class Currents ( SCC ) Partially Conserved Axial Current ( PCAC ) Hypothesis CVC: The charged weak current and isovector electromagnetic current form an isospin triplet. [Feynman and Gell-Mann, 1958] τ 3 ( ψu 0 em,q Jµ = 2 3 ψ u γ µ ψ u 1 3 ψ d γ µ ψ d ( ) em,q Jµ = e 0 ψ q γ µ Iψ q + e 1 ψ q γ µ ψu τ 3 ψ q with ψ q = ψ d ) = ( ψu 0 ) ( ) ( 0 0 ; τ 3 = ψ d ψd ) ; e 0,1 = 1 2 (e u ± e d ) S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 22

23 Symmetries of the Hadronic, Weak Current Thus Jµ em N = ψ[f 1 S (q2 )γ µ i F 2 S(q2 ) σ µν q ν + F 3 S(q2 ) q µ ]e 0 Iψ M n M n + ψ[f 1 V (q 2 )γ µ i F 2 V (q2 ) σ µν q ν + F 3 V (q2 ) q µ ]e 1 τ 3 ψ M n ψ = ( ψp ψ n ) The CVC hypothesis implies and M n τ + ( ψp ψ n ) = ( ψp 0 f 1 (q 2 ) = F V 1 (q 2 ) and f 1 (q 2 ) 1 as q 2 0 f 2 (q 2 ) = F V 2 (q 2 ) f 3 (q 2 ) = F V 3 (q 2 ) = 0 ) (current conservation) f 1 (0) = 1 + V R V R starts in O(α)! [tested to O(0.3%) in decays] f 2 (0)/f 1 (0) = (κ p κ n )/ [tested to O(10%) in A = 8, 12 systems] The Ademollo-Gatto theorem makes the second test more interesting. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 23

24 Symmetries of the Hadronic, Weak Current SCC: Wrong G-parity interactions do not appear if isospin is an exact symmetry. [Weinberg, 1958] G C exp(iπt 2 ) where T 2 is a rotation about the 2-axis in isospin space. ( ) ψn exp(iπt 2 )ψ = iτ 2 ψ = ψ p GV (I) µ G = +V (I) µ ; GA (I) µ G = A (I) µ first class GV (II) µ G = V (II) µ ; GA (II) µ G = +A (II) µ second class no SCC: g 2 = 0 and f 3 = 0 (tested to O(10%) in A = 12 system (combined CVC/SCC test)) PCAC: g 1 /f 1 is set by strong-interaction physics: Goldberger-Treiman relation g 1(0) f 1 (0) = g πnn fπ M N Can test some of these relationships through experiments sensitive to recoil-order effects. S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 24

25 PCAC Tests in Muon Capture g 3 is also predicted by PCAC (HBChPT) and can be studied in µ capture. MuCap redresses the problem with OMC. [Andreev et al., MuCap, PRL 99, (2007).] This can also be viewed as a test of lepton-flavor universality, though π (radiative) β decay yields much more severe constraints! Also K l3, K l2! S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 25

26 on Mass Mass is key to explaining the relative strength of the weak and electromagnetic interactions. In the SM, the gauge boson masses, as well as those of the elementary fermions, arise through the spontaneously breaking of a local gauge symmetry the Higgs mechanism. The Higgs mechanism is also key to describing the mixing of the quarks under the weak interactions: it gives rise to the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The Higgs boson has finally been discovered, and measuring its couplings precisely will tell us whether it is just a SM Higgs or not! S. Gardner (Univ. of Kentucky) Weak Interactions I FNP Summer School, UTK, 6/15 26