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Home , GSI anomaly, Nicola Cabibbo, Weak interaction, Weinberg angle

People who have no weaknesses are terrible; there is no way of taking advantage of them. Anatole France (1844–1924)

27 Weak

The decay of nuclei proceeds via α-, β-, γ-emission, or by fission. The first is governed by nuclear , the third by electromagnetic interactions. The β-emission has an entirely different origin. It is caused by weak interactions. Initially, such processes posed a serious puzzle: The observed emerging particles, or , did not come out with a definite equal to the difference between initial and final nuclei. Instead, they had an energy distribution with a maximal energy at the expected value. This led Nils Bohr to speculate that energy conservation could possibly be violated in quantum . In 1939, however, Wolfgang Pauli suggested in a historic letter to colleagues [1] meeting in T¨ubingen, that the missing energy was carried off by a neutral particle, now called . Its had to be 1/2, to preserve angular momentum. The simplest nuclear particle which shows β-decay is the itself which decays, after about 900 seconds, as follows:

n p + e− +¯ν , (27.1) → e whereν ¯e denotes a right-handed version of Pauli’s particle, which is nowadays re- ferred to as an antineutrino associated with the . As time went on, many more particles were discovered that arose from similar interactions. All of them are now called . Recall the classification in Section 24.1.

27.1 Fermi Theory

In 1934, Fermi made a first theoretical attempt to describe β-decay in lowest-order perturbation theory [2]. He suggested that the is due to a local current- current interaction with a Lagrangian density of the form

= C p¯(x)γµn(x)¯e(x)γ ν(x) + c.c.. (27.2) L µ In accordance with the standard short notation for the fields, we have denoted ψe(x) by e(x) and ψp(x) by p(x), etc. This interaction looks similar to the electric

1457 1458 27 Weak Interactions interaction in Eq. (12.85), except that the Coulomb potential describing the electric long-range interaction is replaced by a short-range δ-function potential. Fermi also realized that Lorentz invariance and locality allow a more general Lagrangian density of the form

= CS p¯(x)n(x)¯e(x)ν(x) + CP p¯(x)iγ5n(x)¯e(x)iγ5ν(x) L µ µ + CV p¯(x)γ n(x)¯e(x)γµν(x) + CA p¯(x)γ γ5n(x)¯e(x)γµγ5ν(x) 1 λκ + 2 CT p¯(x)σ n(x)¯e(x)σλκν(x) + c.c.. (27.3) Fermi postulated the validity of time-reversal invariance, which to our present knowl- edge is fulfilled to a high degree of accuracy. A tiny violation of this is observable only in very few particularly sensitive processes. Hence the coefficients Ci in (27.3) have to be real, with only very small imaginary parts. If Γi denotes the five Dirac µ µ µν matrices 1, iγ5, γ , γ γ5, and σ /√2, one may write the interaction (27.3) short as = C p¯(x)Γ n(x)¯e(x)Γ ν(x) + c.c.. (27.4) L i i i i=S,P,V,A,TX

The decay of an initial nucleus at rest Ψi into a final nucleus Ψf is governed by the elements | i | i Ψ p¯(x)Γ n(x) Ψ . (27.5) h f | i | ii Due to the small energy differences involved, the final state will be almost at rest (since the decaying nucleus is usually big enough), and one may use the nonrelativis- tic limit for the matrix elements between . Taking the Dirac matrices in Dirac’s original representations, one can use the direct-product expressions (27.6),

0 3 2 5 1  γ = σ 1, ­ = iσ , γ = σ 1, (27.6) D × D × D × to rewrite the matrix elements ofp ¯(x)Γin(x) as follows:

S :p ¯(x)n(x) p†(x)n(x), ≈ 5 2 P :p ¯(x)iγ n(x) p†(x) σ 1 n(x) 0, D ≈ − × ≈ 0 V :p ¯(x)γDn(x) p†(x)n(x), i ≈ 1 i p¯(x)γ n(x) p†(x) σ σ n(x) 0, D ≈ − × ≈ 0 5 1 A :p ¯(x)γDγDn(x) p†(x) σ 1 n(x) 0, i 5 ≈ × i ≈ p¯(x)γ γ n(x) p†(x)1 σ n(x), D D ≈ × 0i 2 i T :p ¯(x)σ n(x) p†(x) σ σ n(x) 0, ij ≈ − × k ≈ p¯(x)σ n(x) ǫ p†(x)1 σ n(x). (27.7) ≈ ijk × The lines containing mixtures of upper and lower components are very small for small momenta, and can be omitted. Thus we are left with only two types of nuclear matrix elements:

Ψ p†(x)n(x) Ψ 1 ,

h f | | ii≡h i  Ψ p†(x)  n(x) Ψ , (27.8) h f | | ii≡h i 27.1 Fermi Theory 1459

where  in front of a Dirac spinor acts only upon the spin label. The S-matrix elements are then approximately given by

S =1 2πiδ(E + E + E E )T, (27.9) − f e ν¯ − i with the T -matrix [recall (9.293)]

√2m T = ν 1 C u¯(p )v(p )+ C u¯(p )γ0v(p ) V h i S e ν¯ V e ν¯

n h 1 i

 ­ +  [ C u¯(p ) v(p )+ C u¯(p ) γ v(p )] . (27.10) h i 2 T e ν¯ A e 5 ν¯ }

The spin labels s3 for the electron and antineutrino have been omitted, for brevity. We have also normalized the antineutrino states in the same way as states like 0 (3) p′ p =2p δ (p′ p ), (27.11) h ν¯| ν¯i ν¯ ν¯ − ν¯ to allow a smooth limit as mν 0. When calculating decay rates,→ we may sum over the antineutrino polarizations which are hard to measure. The polarization sum yields

√2m v¯(p )v(p )= /p m , (27.12) ν ν¯ ν¯ ν¯ − ν spinX pol and mν is set equal to zero at the end. For unpolarized nuclei, we have

i j 1 2 i

σ σ =  , σ 1 =0. (27.13) h i 3| | h i If also the electrons are unpolarized, we obtain 1 2 2 1 2 2 1 2 T = 2 CS tr(/p ν¯ /p e)+ CV tr(γ0 /p ν¯γ0 /p e) | | 2meV |h i| | | |h i| spinX pol h

1 2 2 i i 1 2 2 i i  + C  tr(γ γ /p γ γ /p )+ C tr(σ /p γ σ /p ) 3 A|h i| 5 ν¯ 5 e 12 T |h i| ν¯ 5 e 2 1 2 i

+ 2C C 1 m tr(/p γ )+ C C  m tr(σ /p γ /p ) , (27.14) S V |h i| e ν¯ 0 3 S V |h i| e ν¯ 0 e i or more explicitly 4E E p p p p 2 ν ν¯ 2 1 2 e ν¯ 2 1 2 e ν¯ T = 2 CS 1 + CV 1+ | | 2meV |h i| − EeEν¯ | | |h i| EeEν¯ spinX pol     

2 2 1 pepν¯ 1 2 2 1 pepν¯  + CA  1 + CT 1+ |h i|  − 3 EeEν¯  4 |h i|  3 EeEν¯  2 2me 2 me + 2CSCV 1 me CACT  . (27.15) |h i| Ee − |h i| Ee 

The interference terms CSCV and CT CA show a strong threshold-dependence pro- portional to 1/Ee. Since this is not observed experimentally, one concludes that

C C 0, C C 0. (27.16) S V ≈ T A ≈ 1460 27 Weak Interactions

2 One distinguishes now Gamow-Teller transitions,by for which  = 0, and Fermi transitions, for which 1 2 = 0 (both without nuclear |h change).i| 6 For tran- |h i| 6 2 2 sitions in which initial and final nuclear spins are the same, only CS and CV con- tribute. These are distinguishable by the electron-antineutrino correlation in mo- mentum space. In the first case, electron and antineutrino come out preferably in opposite direction (pepν¯ < 0), in the second case in the same direction (pepν¯ > 0). Experiments done for the decay Ne19 F19 + e +¯ν, (27.17) → in which the nuclear spin is 1/2 before and after the decay show that the latter is true. Thus we can approximate CS 0. ≈ 2 Transitions which change the nuclear spin by one unit are sensitive to CA and 2 CT , with similar momentum correlations. The decay He6 Li6 + e +¯ν (27.18) → involves a change from nuclear spin 0 to 1. The electron and the neutrino emerge antiparallel, from which one deduces that C2 0. Thus only C and C are T ≈ V A appreciable. The antineutrinos which appear in these decays have been found to interact with extremely weakly. Only many years later, in 1953, was it possible to detect them directly via the inverse reaction ν¯ + p n + e+, (27.19) e → using a prolific antineutrino source of a large nuclear reactor filled with a high density of thermal . After many precise measurements of angular correlations and electron and nu- clear polarizations, the coefficients of the general interaction (27.3) have been quite well determined and led to

GW λ gA = pγ¯ 1 γ5 n eγ¯ λ (1 γ5) ν + c.c., (27.20) L − √2 " − gV ! # − h i with the weak-interaction

5 2 G = (1.14730 0.000641) 10− GeV− , (27.21) W ± × and the ratio g /g = 1.255 0.006. (27.22) A V ± Theeν ¯ -part of the interaction violates space reflection in the maximal possible way. Under a parity transformation, the left-helicity field (1 γ0)ν(x) that possesses nonzero components only in the first two entries is multiplied− by 0 1 γ0 = . 1 0 ! 27.2 -Number Conservation 1461

This moves the nonzero entries to the lower two components producing a right- helicity field (1 + γ5)γ0ν. However, such a field is not present in the Lagrangian density (27.20). This maximal parity violation is responsible for the of the decay electrons with respect to the neutron spin as observed in Madame Wu’s experiment [3]. The ratio gA/gV can be measured in neutron decay via the up-down asymmetry by analogy with Madame Wu’s experiment. The expectation s p is proportional h n · ei to 2 C + Re C C∗ (1 g /g ). (27.23) | A| V A ∝ − A V The deviation of the ratio gA/gV from unity can be explained by strong interactions. This possibility was first suggested by the weak decay of the 1937 discovered (initially mistaken for a ). The muon decays weakly as follows

µ− e− +¯ν + ν . (27.24) → e µ 27.2 Lepton-Number Conservation

In 1962, a Brookhaven experiment showed that the antineutrino emerging in the decay had to be distinguished from that emerging in the β-decay of the muon. The two are now referred to as muon neutrinoν ¯µ and electron antineutrinoν ¯e, respectively. With this assignment, the total number of electrons plus the associated electron neutrinos, and the total number of plus the associated muon neutri- nos, are both separately conserved if antiparticles are counted negatively. These laws are called lepton-number conservation laws. The muon has no and its decay is described by the Lagrangian density

Gµ λ µ = ν¯µγ (1 γ5)µ eγ¯ λ(1 γ5)νe(x) + c.c.. (27.25) L √2 − − h ih i

The value of Gµ is given by

5 2 G =1.16637(1) 10− GeV− . (27.26) µ ×

It is 2% larger than GW of the β-decay in Eq. (27.21). The parameter Gµ is defined as the Fermi constant. There are two reasons for choosing it rather than GW . First, the experimental value of the lifetime of µ− is extremely well measured:

6 1 τ =2.197035(40) 10− s− . (27.27) µ ×

Second, this lifetime determines Gµ via a very reliable theoretical relation that follows from QED [4]:

2 5 2 2 1 Gµmµ me 3 mµ τµ− = 3 f 2 (1+R.C.) 1+ 2 + ... . (27.28) 192π mµ ! 5 mW ! 1462 27 Weak Interactions

The abbreviation R.C. denotes radiative corrections α 25 α 2 m R.C. = π2 1+ log µ 3.7 2π  4 −  π 3 me −  α 2 4 m m + log2 µ 2.0 log µ + C + ... . (27.29)  π  9 me − me  # These can be calculated with great accuracy by solving the group equation 2 α2 1 α3 (m ∂ e + β(α)∂ )R.C. =0, β(α)= + + ..., (27.30) e m α 3 π 2 π2 where β(α) is the beta-function that governs the change of the interaction strength upon a change in the scale of the theory. It was defined in (20.31) for a scalar field theory with φ4-interaction. For QED it was first introduced by St¨uckelberg and Petermann [5], and soon after that by Gell-Mann and Low [6].

27.3 Cabibbo Angle

In order to explain why GW is smaller than Gµ, , in 1964, put forward the hypothesis that the reduction is due to a simple geometric property of the weak [7]. Studying the β-decay the strange particles

Λ p + e− +¯ν , → e (27.31) Σ+ Λ+ e+ + ν , → e strange he observed that their weak coupling constant GW was considerably smaller than that of neutrons. Moreover, he was able to fit the three decays (27.1), (27.24), and (27.31), quite well by assuming that the weak coupling constant defined as in strange (27.20) for the ordinary β-decay, and a similar GW for the decays (27.31) can by be parametrized by setting

GW = cos θcGµ, (27.32) strange GW = sin θc Gµ, (27.33) where Gµ is the weak coupling constant (27.26) of the leptonic decay, if one chooses the value

θc =0.21 (27.34) for the so-called Cabibbo angle. Nowadays it is believed that weak interactions of all hadrons proceed via the decay of inside their wave functions. The non-strange hadrons decay via the elementary transitions d u + e− + ν , → e d u + µ + ν , − µ (27.35) u → d + e+ +¯ν , → c u d + µ+ +¯ν . → µ 27.4 Cabibbo Mass Matrix 1463

The strange hadrons decay by similar transitions, with the down quarks d getting replaced by a strange s. Cabibbo’s hypothesis amounts to the ansatz for the interaction Lagrangian density

Gµ λ = uγ¯ (1 γ5)(cos θc d + sin θc s) eγ¯ λ (1 γ5) ν + h.c. L −√2 − · − h i Gµ λ λ = cos θc uγ¯ (1 γ5) d +sin θc uγ¯ (1 γ5) s eγ¯ λ (1 γ5) ν +h.c..(27.36) −√2 − − · − n h i h io h i The first expression implies that the weak interactions couple only to an - rotated combination of down and strange quarks. Moreover, they appear only in a special combination of vector and axial vector currents, which were previously found to be conserved Noether currents of QCD (or partially conserved in the axial vector case). In fact we can write

λ λ λ uγ¯ d = √2j √2j + , (1+i2)/√2 ≡ π λ √ √ λ uγ¯ γ5d = 2j5 (1+i2)/√2 = 2j5 π+ , λ √ √ λ uγ¯ s = 2j(4+i5)/√2 = 2jK+ , λ d √ √ λ uγ¯ γ5 = 2j5 (4+i5)/√2 = 2j5 K+ , (27.37) where the subscripts π+,K+ indicate the SU(3)-quantum numbers which can be created by these currents. In terms of these currents, we can rewrite (27.36) as

λ λ = G cos θ j + j + + sin θ [j + j + ] eγ¯ (1 γ ) ν +h.c.. (27.38) L − µ c π − 5 π c K − 5 K λ − 5 n h i o h i 27.4 Cabibbo Mass Matrix

The Cabibbo theory can be phrased in a different way. The combination of quarks dθ cos θc d + sin θc s entering the interaction Lagrangian density (27.36) may be thought≡ of as a result of a diagonalization of a nondiagonal mass matrix of down and strange quarks:

d′ md mds d′ (d′,u′) =(d′,u′) . (27.39) M u′ ! mds ms ! u′ !

This matrix is diagonalized by the physical down and strange quarks

d V V d′ cos θ sin θ d′ = ud us = − , (27.40) s V V s′ sin θ cos θ s′ ! − us ud ! ! ! ! with a Cabibbo angle solving the equation m m m¯ tan θ = ds = d − , (27.41) m m¯ m s − ds 1464 27 Weak Interactions where 1 m¯ = (m + m )+ (m + m )2 +4m2 . (27.42) 2 d s d s ds  q  The weak interactions involve only the Cabibbo-rotated bare quark field d′ = dθ. This means that the weak quark current entering the Lagrangian density (27.36) has the form uγ¯ λ(1 γ )d , (27.43) − 5 θ so that Lagrangian reads

Gµ λ = uγ¯ (1 γ5) dθ eγ¯ λ (1 γ5) ν + h.c.. (27.44) L −√2 − · − h i 27.5 Heavy Vector

Historically, the fact that the coupling constant of weak interactions carries a di- mension of an inverse square mass was a great obstacle in calculating higher-order corrections to weak processes: Up to that point, the only successful field theory of fundamental interactions, , had only a three-particle inter- action, with the photon mediating the interactions: This led to the hypothesis that weak interactions could be due to a similar coupling of a massive charged vector called W . As we shall see later in the analysis of perturbation theory, this can generate a weak coupling of the same type as in (27.20), but with GW replaced by

e2 G (q2)= W , (27.45) W −q2 M 2 − W where q is the momentum carried off by the leptons in the decay, and eW is the coupling strength of the heavy vector meson W to the leptons and quarks. If MW is much larger than 1 BeV, the q-dependence can be ignored and one can approxiamate

2 eW GW (0) 2 . (27.46) ≈ MW

Since eW is dimensionless, it was suggestive to assume it to be equal to the coupling strength of the . Then weak interactions would carry only one additional parameter, the mass of the vector meson. Using

e2 =4πα, (27.47) this requires as mass

4πα MW = 90GeV. (27.48) s G ≈ 27.6 of Electroweak Interactions 1465

A charged meson which has the desired properties was found near this predicted mass in 1983 [8]. Its experimental value is

M = 80.423 0.039 GeV. (27.49) W ± With the usual weak decays carring off a non-zero charge via a lepton, the question has been raised whether weak interactions could also lead to a decay into νeν¯e, νµν¯µ, ee,¯ or µµ¯. The first two could have easily escaped experimental detection, the last two would be hard to find since they would hide under the similar but much larger electromagnetic interaction. Such decays would come from couplings such as

= G0 uγ¯ µ(1 γ )u eγ e +¯ν (1 γ )ν + e µ . (27.50) L − 5 µ e − 5 → h inh i h io The field combinationsuγ ¯ (1 γ )u,eγ ¯ (1 γ )e contained in (27.50) do not change µ − 5 µ − 5 the charge of the decaying particles and are called neutral currents. They appear in the same combination V µ Aµ of vector and axial vector currents as in the interaction (27.38). − Experimental evidence for such neutral currents has been observed in 1973 in a purely leptonic reaction

ν¯ + e− ν¯ + e−, (27.51) µ ← µ as well as in sinilar reactions involving quarks u u νµ + νµ + ,  d  →  d  u u ν¯µ + ν¯µ + . (27.52)  d  →  d  Also a heavy neutral vector meson has been found which, by analogy with W , mediates the neutral weak interactions. It is called Z- and its mass is given by [11]

M = 92.9 1.6GeV. (27.53) Z ± 27.6 Standard Model of Electroweak Interactions

All these interactions, plus the electromagnetic interactions, were unified in one field theory by Glashow, Weinberg, and Salam [9] in the standard model of elec- tromagnetic and weak interactions (the combination being also called electroweak interactions). We shall briefly review this theory here in a reduced version in which the only leptons involved are electrons and electron-neutrinos [10]. All terms involv- ing these leptons will have to be extended by similar terms involving the other two families of leptons discovered so far:

ν ν µ , τ . (27.54) µ !  τ  1466 27 Weak Interactions

The charged lepton are well known:

me =0.511 MeV, mµ = 105.7 MeV, mτ = 1777 MeV. (27.55)

The free lepton Lagrangian density for the electron part is

νeL = (¯νeL, e¯L) /∂ +¯eRi/∂eR. (27.56) L eL !

It is invariant under the rotation group SUL(2) of the left-handed leptons, and a trivial group UR(1) of the right-handed leptons. In the latter, the neutrino νR is absent. This Lagrangian density is made locally SU (2) U (1)-invariant by means L × R of two types of gauge fields. For the SUL(2)-symmetry of the left-handed particles one uses a 2 2-matrix gauge field × τ W = W a a , (27.57) µ µ 2 in which τa acts on the two SUL(2)-isospin indices. Since these are mathematically equivalent to isospin, one speaks of a . For the UR(1)-symmetry of the right-handed electrons one uses a single vector field Bµ. In the Lagrangian, the ordinary derivatives are replaced by the covariant derivatives

Dµ = ∂µ + igWµ,

Dµ = ∂µ + ieBµ. (27.58)

After this, one adds dynamics to the gauge fields themselves via the Lagrangian density 1 1 = tr(W Wµν) B2 , (27.59) LGF −4 µν − 4 µν where

W = ∂ W ∂ W + ig[W , W ] (27.60) µν µ ν − ν µ µ ν is the covariant curl of the SU(2)L-gauge field, and

B = ∂ B ∂ B (27.61) µν µ ν − ν µ the usual abelian curl of the U(1)R-gauge field. Instead of the 2 2 matrix notation (27.60), one can use the vector notation of × Eq. (27.57), and write the covariant curl in (27.60) as

Wa = ∂ Wa ∂ Wa g (W W )a . (27.62) µν µ ν − ν µ − µ × ν The Lagrangian density (27.59) becomes 1 1 = (Wa )2 B2 . (27.63) LGF −4 µν − 4 µν 27.6 Standard Model of Electroweak Interactions 1467

We now introduce the charged W -boson fields

1 1 2 W ± = W iW , (27.64) µ √2 µ ± µ   and rewrite the interactions as

λ a τa νeL λ int = g(¯νeL e¯L)γ Wλ g′e¯Rγ BλeR L − 2 eL ! − 3 λ Wλ /2 Wλ−/√2 νeL λ = g(¯νeLe¯L)γ + 3 g′e¯Rγ BλeR. (27.65) − W /√2 W /2 ! eL ! − λ − λ

Actually, there is some freedom in choosing the coupling to the vector meson Bν, since we can always absorb an equal coupling to (νeL, eL) into the term involving 3 λ the field W . Hence we may choose, instead of g′e¯ γ B e , the interaction − R λ R

νeL λ g′(¯νeL, e¯L, eR)γ Y  eL  , (27.66) e  R    where the 3 3-matrix Y is called the weak and has the form ×

yL Y =  yL  . (27.67) y  R    The parameter yL can be chosen at will since it can be absorbed into g, for instance

yL =1. (27.68)

Then the interaction reads

g + λ int = W ν¯eLγ eL + c.c. L −√2 λ   1 3 λ gW g′B ν¯ γ ν −2 − λ eL eL   1 3 λ g′ λ + (gW + g′B )¯e γ e y B e¯ γ e . (27.69) 2 λ λ L L − 2 R λ R R 3 Now we observe that the two fields Wλ and Bλ are initially both massless and allow for arbitrary linear mixing. One combination of them remains massless after the mixing. This may be identified as the physical photon field:

3 Aµ = cos θW Bµ + sin θW Wµ . (27.70)

The masslessness fixes the mixing angle θW to satisfy g g sin θ = ′ , cos θ = . (27.71) W 2 2 W 2 2 √g + g′ √g + g′ 1468 27 Weak Interactions

The mixing angle θW is also called weak angle or [9]. The orthogonal linear combination

Z = sin θ B cos θ W 3 (27.72) µ W µ − W µ has a mass v M = g2 + g 2. (27.73) Z 2 ′ q The exchange of the Zµ-field gives rise to another contribution to the weak interac- tions with no charge transfer, a so-called ∆Q = 0 -. If we choose the other coupling in (27.68) to be y = 1 and require that the R − photon field couples to the electron via the usual electromagnetic current density, which consists of a sum of a right-handed and left-handed part:

µ µ µ jem =e ¯Lγ eL +¯eRγ eR. (27.74) This is possible if we set gg′ e = 2 2 , (27.75) √g + g′ and the interaction density (27.69) becomes

g λ + λ int = ν¯eLγ eLW + c.c. e j Aλ (27.76) L −2√2 λ − em e   tan θ (2¯e γλe +¯e γλe +¯ν γλν ) cot θ e¯ γλe ν¯ γλν Z . − 2 W R R L L eL eL − W L L − eL eL µ h  i This can be decomposed into three terms: the desired contribution from the electro- magnetic current (27.74), a contribution from the left-handed weak currents written as vectors in weak isospace

µ µ 1 µ νeL ν¯eLγ eL jL± = (νeL eL)γ τ ± = µ , (27.77) √2 eL ! ( e¯Lγ νeL ) 3 3 µ µ τ νeL 1 µ µ jL =(νeL , e¯L)γ = (νeLγ νeL e¯Lγ eL), (27.78) 2 eL ! 2 − and a contribution from the weak :

3 2 jnc = jL + sin θW jem. (27.79) The total interaction (27.76) may then be written as

µ 1 µ + 1 µ int = e jemAµ (j+Wµ + c.c.)+ jncZλ . (27.80) L  − sin θW sin θW 

The weak angle θW parametrizes the admixture of the ordinary electric current to the neutral weak isospin current. 27.7 Masses from Meissner-Ochsenfeld-Higgs Effect 1469

Up to this point, the model had been constructed at the phenomenological level by Glashow in 1961. To explain the weakness of the interaction, he had added to the Lagrangian suitable mass terms for the W and the Z meson 2 2 MW +2 2 MZ 2 W + W − + Z , (27.81) 2 2   choosing the value for MW to achieve the correct weak coupling constant. At that time he did not know what value he should choose for MZ . In this way he obtained a model which parametrized the known weak interactions, while predicting unknown neutral currents.

27.7 Masses from Meissner-Ochsenfeld-Higgs Effect

The theory constructed so far had an important drawback: It did not permit the calculation of higher-order corrections to weak interaction processes. The masses of W ∓- and Z-bosons in (27.81) gave rise to divergencies in the higher-order Feynman diagrams which could not be removed by a redefinition of couplings, masses, and field normalization. The theory was not renormalizable. This defect was corrected by generating the masses mW and mZ by a mechanism that had been discovered in 1950 to explain the finite range of the magnetic field ob- served in a superconductor. It is known as the Meissner-Ochsenfeld effect discussed in Section 17.2. In contrast to the supercondutor, however, the theory needed more degrees of freedom. The single complex field of the Ginzburg-Landau Lagrangian (17.9) was not sufficient. An extension was needed that contained at least a doublet of complex fields, one charged and one uncharged: ϕ+ ϕ(x)= 0 . (27.82) ϕ ! Their dynamics is governed by a generalization of the Ginzburg-Landau Lagrangian (17.9), the so-called Higgs model:

2 D 1 µ m 2 g 4 1 µν se[ϕ,ϕ∗, A]= d x Dµ∗ϕ∗D ϕ ϕ ϕ Fµν F . (27.83) A Z (2 − 2 | | − 4| | − 4 )

The nongradient part involving the scalar field is now invariant under SUR(2) + 0 ×

UL(1)-rotations of the two complex fields ϕ and ϕ :  i«(x) /2 iγ(x)/2 ϕ(x) e− · e− ϕ(x). (27.84) → × The gradient term ∂ ϕ 2 is not invariant under (27.84), but picks up extra gradient | µ | terms of « (x) and γ(x). These can be compensated by gauge fields of SUL(2) µ iγ(x)/×2

UR(1)-rotations: a singlet vector field B (x) for the UR(1) transformations e− ,  µ i«(x) /2 and an SUL(2)-triplet vector field W (x) for the SUL(2) transformations e− · . For this purpose one defines a covariant derivative

g′ g

D ϕ(x)= ∂ ϕ(x)+ i B ϕ(x)+ i W  ϕ(x). (27.85) µ µ 2 µ 2 µ · 1470 27 Weak Interactions

The extra gradients can be absorbed by the gauge transformations

1 iγ(x)/2 iγ(x)/2 1 Bµ(x) Bµ(x)+ e− ∂µe = Bµ(x)+ ∂µγ(x), (27.86) → ig′ 2g′

1

 «  «  «  i« (x) /2 i (x) /2 i (x) /2 i (x) /2 W (x) e− · W e · + e− · ∂ e · . (27.87) µ → µ ig µ

µν The term Fµν F is a gauge-invariant Lagrangian density of the gauge fields them- selves. It contains the field strength of the Bµ-field and the field strength of the Wµ-field 1 1 1 F F µν = B Bµν tr(W Wµν), (27.88) −4 µν −4 µν − 4 µν

µν where Bµν B is the square of the ordinary curl of the vector potential Bµ(x):

B (x)= ∂ B (x) ∂ B (x), (27.89) µν µ ν − ν µ µν a and tr(WµνW ) the nonabelian generalization of it associated with Wµ(x) [recall (27.62)]

Wa (x)= ∂ Wa(x) ∂ Wa(x) g(W (x) W (x))a. (27.90) µν µ ν − ν µ − µ × ν Just as the finite penetration depth in a superconductor, nonzero masses of the vector bosons are now assumed to be the results of a nonzero expectation value of the ϕ-field:

1 0 ϕ = . (27.91) h i √2 v ! Allowing for a spacetime dependence of the neutral components of the ϕ-field by setting

1 0 ϕ(x)= , (27.92) √2 v + ρ(x) ! the Lagrangian density of the scalar field in (27.83) takes then the form

1 2 1 2 3 2 2 1 1µ 2 2µ = [∂ ρ(x)] + [v + ρ(x)] [g′B gW ) + g (W W + W W )] L 2 µ 8 µ − µ µ m2 g [v + ρ(x)]2 [v + ρ(x)]4. (27.93) − 2 − 8

1 2 By diagonalizing the quadratic terms in the vector fields Wµ and Wµ , we observe the generation of a charged field

1 1 2 W ± = (W W ) (27.94) √2 µ ± µ 27.8 Lepton Masses 1471 of mass vg M = . (27.95) W 2 3 The quadratic terms in the neutral field Wµ and Bµ are diagonalized by the two mixed fields

1 3 A = (gB + g′W ), (27.96) µ 2 µ µ √g + g′ 1 3 Z = (g′B gW ). (27.97) µ 2 µ µ √g + g′ − These are precisely the field combinations (27.70) and (27.72) introduced by Wein- berg. The first remains massless and can be identified as the photon field. The second is massive and heavier than the W -boson: v M = g2 + g 2 > M . (27.98) Z 2 ′ W q The mass of the ρ-field is determined by 2m2 m2 = − . (27.99) ρ 2 This is the relation anticipated in Eq. (17.37).

27.8 Lepton Masses

Thus we have found an explanaton for the masses of the vector bosons by the spontaneous breakdown of a continuous symmetry. The nonzero ground-state ex- pectation value of a scalar field v sets the mass scale of mW in Eq. (27.95) and of mZ in Eq. (27.73). It is now suggestive that also the masses are due a similar mechanism. Thus we assume the leptons to start life as massless fermi fields whose mass is generated by the simplest possible coupling to the scalar field via a Lagrangian density = G ϕ0(¯e e +¯e e )+(e µ). (27.100) L − e R L L R → Inserting the scalar field (27.92), we see that the spontaneous breakdown gives to the initially massless electrons and muons the new masses

me = Gev/√2, mµ = Gµv/√2. (27.101)

27.9 More Leptons

Apart from electrons an muons, more leptons have been discovered experimentally. They are listed in Table 27.1. Their weak interactions proceed in the sare way as those of electrons and their neutrinos by the coupling to the vector bosons W ± and Z0. 1472 27 Weak Interactions

Table 27.1 List of leptons and their properties taken from the [22]. Lepton Charge Mass (MeV) 6 ν 0 < 2 10− e · νµ 0 < 0.19 ντ 0 < 18.2 e -1 0.510998928(11) µ -1 105.6583715(35) τ -1 1776.82 0.16 ±

27.10 Weak Interaction of Hadrons

In order to accommodate also the weak interactions of hadrons, the above La- grangian density (27.80) is extended by a term involving the quarks

g + µ + µ em int = W j + c.c. + eA j L 2√2 µ h µ  τ  g2 + g 2Zµ q¯ 3 γ q sin2 θ jem , (27.102) − ′ L 2 µ L − W µh q   where

µ + µ jh =u ¯Lγ dθL, (27.103) 2 1 jem = uγ¯ u dγ¯ d +¯sγ s . (27.104) µh 3 µ − 3 µ µ   with

dθ = d cos θc + s sin θc. (27.105)

When the theory arrived at this stage it was realized that it contained an unpleasant contradiction with experiments. The neutral vector meson Z was coupled to a quark current

q¯ γµτ q =u ¯ γµu cos2 θ d¯ γµd , L 3 L L L − c L L i sin θ cos θ d¯ γµs +¯s γµd i sin2 θ q¯ γµc . (27.106) − c c L L L c − c L L   The third term changed the by one unit. However, processes with a strangeness-changing neutral current were known to be strongly suppressed in nature. Otherwise one would have observed particle decays

K± π±ν ν,¯ 0 → + K µ µ−, (27.107) → 7 8 with a much larger branching ratio than the experimentally observed 6 10− , 10− , × respectively (see Fig. 27.1). 27.10 Weak Interaction of Hadrons 1473

Figure 27.1 Quark diagrams for K+ and K0 decays involving strangeness changing neutral currents.

In order to achieve the desired suppression, Glashow, Iliopoulos, and Maiani [12] postulated the existence of a further quark, called charmed quark, denoted by c, which was supposed to form a weak isodoublet with s in the same way as u does with d. Then they considered the two doublets u c , (27.108) d s θ !L θ !L with s = u sin θ + s cos θ (27.109) θ − c c as two families of the same sort, both with 1/3, and wrote the left-handed neutral current as

¯ τ3 µ u τ2 µ c 2 µ u,¯ dθ γ +(¯c, s¯θ) γ i sin θW jem. (27.110) 2 dθ ! 2 sθ ! −   L L After this, the strangeness changing currents cancel exactly. The cancellation in 0 0 + the K decays goes as follows. Take, for instance the decay K µ µ−. The → compensating diagrams are shown in Fig. 27.2.

Figure 27.2 Diagrams for the K0 µ+µ decay with compensating strangeness- → − changing neutral currents.

In addition to the previous electromagnetic current (27.104), the charmed quark contributes an extra term 2 ∆jem = cγ¯ µc. (27.111) µh 3 1474 27 Weak Interactions

The cancellation of the strangess-chniging neutral corrents in weak interactions if names after the authors [12] as GIM mehanism. GIM mechanismThe mixig mech- anism needed in the space of all leptons to suppress the strangeness changin neutral currents is discussed in Ref. [23].

27.11 Quantum Oscillations

The standard model of weak interactions explains some interesting novel phenomena in . The nature of a fundamental particle is sometimes not fixed but can oscillate as a function of time. The first particles, where this phenomenon was observed, was the system of neuitral , where the K- K0 and K¯ 0 oscillate between each other.

27.11.1 Oscillations between Neutral Kaons In our discussion of the SU(3)-symmetry of strong interactions in Section 24.4, we mentioned that the mesons K0 and K¯ 0 have strangeness 1 and +1, respectively. It − is believed that strong interactions strictly conserve these quantum numbers. The weak interactions described by the standard model, however, conserve the symmetry CP . If we want to describe the weak decay of the strongly produced K0 and K¯ 0 mesons, we must decompose them into eigenstates of CP . These are the states 1 1 K0 K0 + K¯ 0 , K0 K0 K¯ 0 . (27.112) | 1 i ≡ √2 | i | i | 2 i ≡ √2 | i−| i     They have different decay channels. At rest, the first state is even under CP (recall Subsections 7.1.6 and 7.1.6), the second is odd. Hence the first state can decay into 0 10 two , while the second cannot. The lifetime of K1 is 0.8922 0.0020) 10− sec, + 0 0 ± × producing almost exclusively π π− and π π at a ratio 2:1. 0 The CP -odd state K2 must at least decay into three pions and has therefore a 0 0 0 0 much longer lifetime. This is why K1 and K2 are also called KS and KL. 0 Let mS, mL be their masses and ΓS, ΓL their decay rates. A beam of K produced in a strong-interaction process will then evolve in its rest frame as an oscillating superposition of KS and KL:

0 1 (imS +ΓS /2)τ (imL+ΓL/2)τ K = e− KS + e− KL , (27.113) | i √2 | i | i h i where τ is the proper time. Since the two components decay at a different rate, the beam will contain oscillating admixtures of K¯ 0. These can be detected if the beam is directed towards a strongly interacting target, say a slab of copper. In it the forward scattering amplitude for K0 is smaller than for K¯ 0. Thus, if one changes the thickness of the slab, the outcoming beam will have different admixtures of the 0 0 short-lived component KS. In this way, one is able to regenerate KS from a beam 0 + 0 0 which had turned almost completely into KL by the decay into π π− or π π . 27.11 Quantum Oscillations 1475

The process was analyzed in 1957 [14], and the experiment showed clearly the regeneration effect [15, 16]. 0 When experimentalists tried to set an upper limit to the decay of KL into two pions they found that this decay was not completely forbidden [17]. This implied that the weak decay of K0-mesons contains a small term violating CP -symmetry. Thus the correct mixtures of short- and long-lived K0 mesons should really be slightly different from (27.112):

0 1 0 ¯ 0 KS (1 + ǫ) K + (1 ǫ) K , (27.114) | i ≡ 2 | i − | i 2(1 + ǫ ) h i q 0 1 0 ¯ 0 K2 (1 ǫ) K (1 + ǫ) K , (27.115) | i ≡ 2 − | i − | i 2(1 + ǫ ) h  q 3 where ǫ is of the order of 10− . The parameter ǫ can be measured by studying a beam of K0 as a function of 0 time. Due to the decay of the KS content, the beam is expected to perform a damped oscillation from K0 = ds¯ to K¯ 0 = sd¯, and back. The content of the two components 0 0 can be studied by looking at the decay rates K π∓e±νe and K¯ π∓e±νe arising from the decay of the quarkss ¯ and s, respectively,→ in accordance→ with the ∆Q = ∆S rule. This permits measuring ǫ directly from the asymmetry

− − NK0 NK¯ 0 Nπ Nπ+ Ne+ Ne 2Re ǫ 3 − = − = − = 2 (3.320 0.074) 10− . NK0 + NK¯ 0 Nπ− + Nπ+ Ne+ + Ne− 1+ ǫ ≡ ± × | | (27.116) 0 0 Thanks to CP -violation, not only KS but also KL decays into two pions, and 0 + this makes it possible to observe oscillations in the decay of a K -beam into π π−. The ratio of the decay rates changes as a function of the proper time τ as follows:

+ 0 π π− K (τ) ΓSτ (ΓS +ΓL)τ/2 2 ΓLτ + − − − − Iπ π (τ)= |h + | 0 i| =e +2 η+ e cos(∆mτ φ+ )+ η+ e , π π− KS(τ) | −| − − | −| |h | i| (27.117) where + 0 iφ+− π π− KL(τ) η+ η+ e = |h | i|, ∆m mL mS. (27.118) − ≡| −| π+π K0 (τ) ≡ − |h −| S i| This damped oscillation was indeed observed experimentally [18], as shown in Fig. 27.3. A best fit to the data yields the parameters

8 τS = 1/ΓS = (0.8926 0.0012) 10− sec , (27.119) ± ×8 τ = 1/Γ = (5.17 0.04) 10− sec , (27.120) L L ± × ∆m = (0.5333 0.0027) 1010h/¯ sec , (27.121) × × 3 η+ = (2.269 0.023) 10− ǫ (27.122) − | | ± o × ≈ | | φ+ = (44.3 0.8) . (27.123) − ± 1476 27 Weak Interactions

N

(NK0 NK¯ 0 )/(NK0 + NK¯ 0 ) − (a)

(b)

τ[10−10 sec] τ[10−10 sec] Figure 27.3 Left-hand: Asymmetry of the number of K0 = ds¯ mesons with respect to the K¯ 0 = sd¯ mesons as a function of time [19]. The asymmetry is measured by the ratio of the decays K0 π e ν K¯ 0 π e ν . After a long time, only K0 survives which → ∓ ± e → ∓ ± e L does not have an equal content of K0 and K¯ 0 as a signal of CP -violation. Right-hand: + 0 Oscillation in the decay rate into π π− of the K -beam. Curve (a) shows the histogram of the raw data. Curve (b) shows the theoretical decay curve without oscillations, and the insert shows the best fit to the oscillations if the nonoscillating background is subtracted (figure from [18]).

1 0.8 (NB0 NB¯0 )/(NB0 + NB¯0 ) 0.6 − 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 2 4 6 8 10 12 τ[10−12 sec]

0 ¯ ¯0 ¯ Figure 27.4 Asymmetry of the number of Bd = db mesons with respect to the Bd = bd mesons as a function of time [19]. Here the long- and short-lived combinations have almost the same lifetime. 27.11 Quantum Oscillations 1477

27.11.2 Mesons containing the 0 ¯ 0 0 ¯ By analogy with the strange mesons K and K there exist neutral mesons Bd = db ¯0 ¯ and Bd = bd. The masses of their short-lived combination is ms = 5721 MeV which oscillated with the long-lived combination of mass ms = 5747 MeV. For these mesons, the asymmetry measurement yields (see Fig. 27.4) [20]:

NB0 NB¯0 N − N + N + N − 2Re ǫ d − d π π e e Bd = − = − = 2 (0.0035 0.0103 0.0015). N 0 + N ¯0 N − + N + N + + N − 1+ ǫ ≡ ± ± Bd Bd π π e e | Bd | (27.124) 0 ¯ ¯0 Similar can be observed for Bs = sb and Bs = bs¯. All such neutral particle mixings can simply be understood by assuming that the Lagrangian density for these particles contains a nondiagonal mass matrix i i M11 Γ11 M12 Γ12 =  − 2 − 2  . (27.125) M i i  M21 Γ21 M22 Γ22   − 2 − 2    This matrix must be Hermitian, implying that

M21 = M12∗ , Γ21 =Γ12∗ . (27.126) Moreover, as a consequence of CP T -invariance, the diagonal elements must satisfy

M11 = M22 = M, Γ11 =Γ22 =Γ. (27.127) The most general mass matrix is therefore i i M Γ M12 Γ12 =  − 2 − 2  . (27.128) M i i  M12∗ Γ12∗ M Γ   − 2 − 2    27.11.3 General Flavor Mixing The Cabibbo theory was set up at a time when only the three quarks u,d,s were known. After the discovery of the heavier three quarks it turned out that all quarks can be grouped into three families of charge 2/3 and 1/3, respectively, and that the weak current (27.43) has the generalization −

Vud Vus Vub d λ (u,c,t) γ (1 γ5)  Vcd Vcs Vcb   s  . (27.129) − V V V b  td ts tb        The 3 3-matrix V is called CKM-matrix, after the initials of the authors Cabibbo, Kobayashi,× and Maskawa [7, 13]. It is often expressed in terms of four angles θ12, θ13, θ23, and δ13 as

iδ13 c12 c13 s12 c13 s13 e− V = s c c s s eiδ13 c c s s s eiδ13 s c , (27.130)  − 12 23 − 12 23 13 12 23 − 12 23 13 23 13  s s c c s eiδ13 c s s c s eiδ13 c c  12 23 12 23 13 12 23 12 23 13 23 13   − − −  1478 27 Weak Interactions

where s12 sinθ12, c12 cosθ12. The angle δ13 serves to explain the small CP - ≡ ≡ 0 violation observed in the decay of KL discussed in the previous subsection. The mixing matrix is permanently updated whenever new experimental data are avail- able. The presently used numbers are [21]:

0.974 to 0.9756 0.219 to 0.226 0.0025 to 0.0048 V =  0.219 to 0.226 0.9732 to 0.9748 0.038 to 0.044  . (27.131) 0.004 to 0.014 0.037 to 0.044 0.9990 to 0.9993     The most precise constraints on the size of the elements of the CKM-matrix are extracted from the low-energy s u and d u semileptonic transitions. The → → determination is a specialized part of phenomenological particle physics, on which excellent reviews are available from the Particle Data Group [13].

27.12 Neutrino Mixing

By analogy with the flavor mixing matrix V for quarks in Eq. (27.129), there exists a mixing matrix for the three neutrinos νe, νµ, ντ . It is known as the Pontecorvo-Maki- Nakagawa-Sakata (PMNS) matrix which transforms three bare neutrinos ν1, ν2, ν2 into the physical particles:

νe ν1 Ue1 Ue2 Ue3 ν1  νµ  = U  ν2  =  Uµ1 Uµ2 Vµ3   ν2  . (27.132) ν ν U U V ν  τ   3   τ1 τ2 τ3   3          The matrix U is parametrized in terms of angles in the same way as the matrix V in (27.130), and the best determination at present yields [24]

0.82 0.01 0.54 0.02 0.15 0.03 ± ± ± U =  0.35 0.06 0.70 0.06 0.62 0.06  . (27.133) 0.44 ± 0.06 0.45 ± 0.06 0.77 ± 0.06    ± ± ±  Neutrino flavor mixing is a cause for the notorious detection problem of solar neu- trinos [25]. It is also the source of the so-called atmospheric neutrino [26]. The neutrino masses are too small to be experimentally measurable. What is measurable is the mass difference between the neutrinos associated with the electron and the muon. This is done by counting the number of times by which electron neutrinos are transformed into muon neutrinos and back on their way from the sun to us. An alternative possibility to measure small mass differences comes from recent observations of the time modulation of two-body weak decays of heavy ions. This reveals the mass content of the electron neutrinos via interference patterns in the recoiling ion wave function. From the modulation period one derives the difference 2 5 2 of the square masses ∆m 22.5 10− eV , which is about 2.8 times larger than ≈ × 27.13 Simple Theory of Two-Neutrino Mixing 1479 that derived from a combined analysis of KamLAND and solar experiments. It is, however, compatible with a data regime to which the KamLAND analysis attributes a smaller probability. At the GSI in Darmstadt, Germany, the experimental storage ring ESR permits observing completely ionized heavy I or hydrogen-like heavy ions IH over a long time [27, 28] and thus to measure the time dependence of their weak two-body decays IH I +νe or I IH +¯νe. The first decay is due to the well-known electron- capture (EC)→ process. In→ the second an electron comes out of the nucleus together with an antineutrino, and the electron remains in an orbit. The virtue of such experiments is that the properties of the neutrino or antineutrino can be deduced from the time dependence of the transition, by observing only the initial and final ions. The special efficiency of these experiments becomes clear in the Dirac sea interpretation of the second process, where the initial ion simply absorbs a negative- energy antineutrino in the vacuum. Since the vacuum has all negative-energy states filled, the vacuum is a source of negative-energy neutrinos of maximally possible current density, i.e., the best possible neutrino source in the . This is why the ESR experiments yield information on neutrino properties with great precision even if the targets and exposure times are quite small, in particular much smaller than the 2.44 1032 -yrs (2881 ton-yrs) in the famous KamLAND experiments [29], which are× only sensitive to the much less abundant positive-energy neutrinos produced by nuclear reactors.

27.13 Simple Theory of Two-Neutrino Mixing

To illustrate this we consider here the small mass difference of the two lightest neutrinos. According to Pontecorvo [30, 31], the Dirac fields of the physical electron and muon-neutrinos νf = (νe, νµ), the so-called flavor fields, are superpositions of neutrino fields νi(x)=(ν1(x), ν2(x)) of masses m1 and m2: ν (x)= ν (x) cos θ + ν (x) sin θ, ν (x) = ν (x) sin θ + ν (x) cos θ . (27.134) e 1 2 µ − 1 2 Here θ is a mixing angle. The free Dirac action has the form

= d4x ν¯ (x)(iγµ∂ ) ν (x), (27.135) A f µ −M f Xf Z where γµ are the Dirac matrices, and is a mass matrix, whose diagonal and off- M diagonal elements are mf =(me, mµ) and meµ = mµe, respectively. The eigenvalues mi =(m1, m2) are related to mf by [30, 31, 32, 33], 2 2 2 2 me = m1 cos θ + m2 sin θ, mµ = m1 sin θ + m2 cos θ, m = m =(m m ) sin θ cos θ. (27.136) eµ µe 2 − 1 The weak transition between the electron e and its neutrino νe is governed by the interaction g 4 +µ g 4 µ int = d x W −(x)J (x)+h.c. d x W −(x)¯e(x)γ (1 γ5)νe(x)+h.c., A √2 µ ≡√2 µ − Z Z (27.137) 1480 27 Weak Interactions

0 1 2 3 where γ5 is the product of Dirac matrices iγ γ γ γ . Since the interaction (27.137) involves only two flavor fields in (27.134), the states of masses mi will always be produced as coherent superpositions. The weakness of the interaction allows us to calculate the shape of the mixed wave packet from perturbation theory. Consider the decay I I +ν ¯ which is a superposition of → H e the states of masses m1 and m2. Let us write down formulas which are applicable for and, after the interchange of MH with the mass M of the bare ion, treats the case of outgoing neutrinos. In the center-of-mass (CM) frame of the initial bare ion of mass M, the final H-like ion has the same momentum as the antineutrinoν ¯ (i =1, 2), whose energy is ω ωk = k2 + m2 determined by i i ≡ i,i i i q M M + Q = ω + M 2 + k2 = ω + M 2 + ω2 m2, i =1, 2, (27.138) ≡ H i H i i H i − i q q 2 so that ωi = [(2MH + Q)Q + mi ]/2(MH + Q). Subtracting ω2 and ω1 from each other we find the energy difference

m2 m2 ∆m2 ∆ω ω ω = 2 − 1 . (27.139) ≡ 2 − 1 2M ≡ 2M

The denominator M is of the order of 100 GeV and much larger than ∆m2, so that ∆ω is extremely small. It is the difference of the recoil transferred to the outcoming ion by the antineutrinos of masses m1 and m2. Without recoil, we would have found an energy difference at the same momentum which is four orders of magnitude larger than that, namely

2 2 1/2 2 2 ∆ωk = ωk ωk = (∆m + ωk ) ωk ∆m /2ωk ∆m /2Q. (27.140) ,2 − ,1 ,1 − ,1 ≈ ,1 ≈ This is the frequency with which the incoming negative-energy neutrino current of momentum k oscillates in the vacuum. Note that although ∆ω is small, the momentum difference ∆k k k associated with the energies ω is as large as ≡ 2 − 1 1,2 ∆ωk, but has the opposite sign.

27.14 Experiments

The best experimental results are available for the EC-processes reported in Ref. [27], where an electron is captured from the K-shell and converted into an electron- neutrino which runs off to infinity. On the average, the decay is exponential with a rate expected from a standard-model calculation. In addition, however, the rate shows oscillations with a frequency ∆ω. The experimental results are [34]

140 58+ 140 58+ 1 59Pr 58Ce : ∆ω 0.890(11) sec− (Q = 3386keV), (27.141) 142 60+ → 142 60+ ≈ 1 Pm Nd : ∆ω 0.885(31) sec− (Q = 4470keV). (27.142) 61 → 60 ≈ In both cases the period of oscillations is roughly 7 sec, and it scales with M (see Fig. 27.5). 27.14 Experiments 1481

Figure 27.5 Oscillations of decay rate for the processes (27.141) and (27.142). The period is in both cases roughly 7 sec. The inserts show the frequency analyses. Plots are from Ref. [27].

We expect these oscillations to be associated with the frequency ∆ω determined by Eq. (27.139), and thus to give information on ∆m2. Inserting the experimental numbers for ∆ω into Eq. (27.139) and taking into account that the particles in the storage ring run around with 71% of the velocity with a Lorentz factor γ 1.43, we find from the two processes [35] ≈ 2 5 2 ∆m 22.5 10− eV . (27.143) ≈ × 2 +0.3 5 2 This is by a factor 2.8 larger than the result ∆m 7.58 0.3 10− eV favored by ≈ ≈ − × the KamLAND experiment [29, 37, 38], but it lies close to their less favored result [36], which the authors excluded by 2.2σ in 2005, now by 6σ [29] (see Fig. 27.6).

Figure 27.6 The upper KamLAND regime of 2006 [36] is compatibel with the result ∆m2 22.5 10 5eV2. ≈ × − 1482 27 Weak Interactions

So far, the origin of this discrepancy is unclear. One explanation has been attempted in Ref. [39] where the authors investigate the influence of the strong Coulomb field around the ion upon the process. We refer the reader to recent discussions of this question [41].

27.15 Entangled Wavefuntion

It is easy to describe theoretically the oscillations in Fig. 27.5. Consider the decay of the initial ion I into the ion IH plus an electron-antineutrinoν ¯e. Ignoring all spins and the finite size of the ions, the effective interaction for this process is

4 4 = d xI† (x)¯ν†(x)I(x)= d x cos θI† (x)¯ν†(x)I(x) + sin θI† (x)¯ν†(x)I(x) , Aint H e H 1 H 2 Z Z h (27.144)i where I(x),ν ¯e(x), and IH (x) are the field operators of the involved particles. In the CM frame, the initial ion is at rest, while the final moves nonrelativistically. The outgoing wave is spherical. The role of the antineutrinosν ¯1 andν ¯2 is simply to create a coherent superposition of two such waves with the two different radial momenta k and accociated frequencies ω whose diference was calculated in (27.143):

(+) g i(k1r ω1t) i(k2r ω2t) x ψ ; t cos θ e − + sin θ e − . (27.145) h | i ≡ −r h i The resulting wave carries a radial current density of ions IH : g2 j = cos2 θk + sin2 θk + sin θ cos θ (k + k ) cos(∆kr ∆ω t) . (27.146) r M r2 1 2 1 2 − H h i In order to find the decay rate we integrate this over a sphere of radius R surrounding the initial ion. We can choose R any size 1/∆k 104 m. From this surface we find the outgoing probability current density≪ ≈

k¯ P˙ =4πg2 [1 + sin(2θ) cos(∆ω t)] , (27.147) M where we have approximated k1 and k2 by their average k¯. This P˙ explains directly the observed oscillating decay rate of the initial ions. A comment is in place on several recent publications [42, 43, 44, 45, 46, 47]. These deny a relation between neutrino oscillations and the nonexponential decay seen in the GSI experiment for various reasons. In Ref. [42], the basic argument is that the antineutrino oscillations set in after their emission, so that they cannot be observed in the GSI experiment. The present discussion shows that this argument is true, although it fails to give the correct explanation of the data [48]. Indeed, the GSI data do not care about the neutrino oscillations while they propagate away from the decay center. The neutrinos merely serve to give the ion a coherent kick of two different momenta and energies. What is measured are the resulting oscillations of the ion wave function caused by this kick. Notes and References 1483

It is noteworthy that this analysis, in which we extract the properties of the unobserved antineutrino from the behavior of the ion, corresponds precisely to the usual entanglement analysis of decay processes, such as π0 γ + γ. There the measurement of the polarization of one photon tells us immediately→ the polarization properties of the other, unobserved photon. Let us finally remind the reader of the similarities of these neutrino oscillations with similar oscillation phenomena in associate production processes of particles, where they appear together with an oscillating partner. These have been proposed and controversially discussed before by many authors. See Subsection 27.11.1, where Λ- are produced together with neutral Kaons [49]. Or the mixing arising in the production of neutral mesons containing a bottom quark discussed in Subsection 27.11.2. Or the production of µ−-particles together with antineutrinos in the decay π− µ− +¯ν [50]. The presently-discussed data provide us with the most definite → e experimental confirmation of such a phenomenon. We refer the reader to the many interesting ideas presented in other publications [44, 45, 46, 47].

Notes and References

[1] See the www address (http://www.library.ethz.ch/exhibit/pauli/neutrino e.html).

[2] E. Fermi, Nuovo Cimento 11, 1 (1934); Z. Phys. 88, 161 (1934).

[3] C.S. Wu, E. Ambler, R.W. Hayward, D.D. Hoppes, and R.P. Hudson, Phys. Rev. 105, 1413 (1957).

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