Neutral kaon mixing. K + + p+ K 0 + p+ + π+, conserves antistrangeness. Conversely, the baryon-number-conserving transition,

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1 These lectures explore how the weak force gives rise to flavour oscillations and CP violation. Flavour oscillation means a reversable transmutation from one flavour to another. This phenomenon is observed within all neutral meson systems K 0 K 0, D0 D0, B0 B0, Bs Bs ) and amongst the three neutrino species νe, νµ, ντ ). CP-violation is mathematical terminology for a difference between matter and antimatter. It is observed only in a handful of rare kaon and B-meson decays. The search for CP-violation in neutrinos is a major topic for the next generation of neutrino experiments. 1 Neutral kaon mixing Fig. 1 show early proof of neutral kaon mixing from a bubble chamber image. The strangeness of the neutral meson appears to change during the flight between creation and decay. This statement is under the [correct] assumption that the creation of the neutral kaon is via the strong force, which is flavour conserving. The transition, K + + p+ K 0 + p+ + π+, K + i = s ui, K 0 i = s di, conserves antistrangeness. Conversely, the baryon-number-conserving transition, K 0 + p+ Λ0 + π+ + π0, Λ0 i = sudi, K 0 i = sdi, in a few centimetres. For this to occur, the K 0 conserves strangeness. The neutral kaon has transmuted from s di to sdi 0 and K, cannot be the mass eigenstates of the Hamiltonian, those that propagate in time. The mass eigenstates i.e. the K-short, KS0 and K-long, KL0 as we know them today) are linear superpositions of the flavour eigenstates states of definite flavour) K 0 and K 0 and hence have access to both in interactions. C B A Figure 1: a K + beam enters the left edge of a bubble chamber and scatters off a proton at A. A pion and a neutral kaon is formed. The kaon flies to the right without leaving an trace and the recoiling proton and pion move to the bottom of the image. At B, the neutral kaon scatters off another proton to form a Λ0 baryon with an associated π+, as well as an untraced π0. At C, the Λ0 baryon decays into a proton and a π in a characteristic V decay. 1

2 1.1 Formalism The time evolution of one neutral meson, in its rest frame, may be written as, K 0 t) = e imt e Γt/ K 0 where the first exponential is a plane-wave solution, exp i p.x Et)) for a state with energy E = M in its rest frame, p = 0, with = 1. The second term describes the exponential decay for a state with proper lifetime τ i.e. width Γ = /τ, again with = 1) such that, K 0 K 0 t) e t/τ Generalise to a two state system with matrices, M and Γ encoding the time evolution, ) Kt) K 0 ) = Σ Kt) K 0 where Σ = e i Mt Γt/, 1) where the presence absence) of the bar above the Kt) shows the flavour of the state at t = 0: pure K 0 or pure K 0 ). The states K 0 and K 0 are of well defined flavour so it is in this basis that it is appropriate to discuss their interaction by the weak force. Thus they are the weak eigenstates, as opposed to the mass eigenstates of the Hamiltonian. Apply Schrödinger s equation, i dψ/dt = Hψ, identifies the hamiltonian. H = M i Γ. Any matrix can be decomposed in the form H 1 + i H where H 1 and H are hermitian. So M and Γ are hermitian matrices, which implies measurable quantities because the eigenvalues of hermitian matrices are real. Imposing CPT-invariance identical mass and lifetimes of the K 0 and K 0 ) gives, M 1 =M1 and Γ 1 =Γ1 m = M 11 =M and Γ 11 =Γ = Γ So the hamiltonian of the meson-antimeson system becomes, ) m M1 H = i ) Γ Γ1 m Γ M 1 The off-diagonal terms describe the transitions [mixing] between the meson and antimeson. If M 1 or Γ 1 have an imaginary component, CP violation in the meson mixing occurs. Γ 1 Oscilation mechanism The process described by Γ operates on-shell, driven by low-energy rescattering pion exchange). Off-shell exchange of high-mass objects a W ± box in the SM) is possible at short distances and is described by the mass matrix, M. π + K 0 π K 0 s d K 0 W W + K 0 d s

3 The eigenstates of the hamiltonian are observable with measurable mass and lifetimes. Let s call the two mass eigenstates Light/Heavy and describe them by a trivial, orthogonal, linear combination of the weak eigenstates, {p, q} R. Or equivalently, K L = p K 0 + q K 0 K H = p K 0 q K 0, ) K 0 = 1 KL + K p H ) K 0 = 1 KL K q H ). 3) Eq. 1 describes the time evolution of the weak eigenstates. A similar, but diagonal matrix describes the time evolution of the mass eigenstates, K L and K H. The two are related from Eq. &3 through the similarity transformation, ) 1 1 ) Kt) p p KL t) = Kt) 1 1 K q q H t) e i M L t Γ L t/ ) ) 0 KL = 0 e i M Ht Γ H t/ K H ) p q K 0 ) = p q K 0 which diagonalises the weak interaction matrix, Σ = PDP 1. The columns of P are eigenvectors of Σ. 1 1 p p e i M L t Γ L t/ ) ) 0 p q Σ = e i M Ht Γ H t/ 4) p q q q q g + p = g [ p q g where g g ± = 1 e i M L t Γ t/ L ± e ] i M Ht Γ H t/ 5) + Hence the time evolution of the decaying state is, Or illustratively, Kt) = g + K 0 + q p g K 0 6) Kt) = g + K 0 + p q g K 0 7) K 0 K 0 K 0 g + t) q p g t) K 0 p q g t) g + t) K 0 K 0 From which we note, as a precursor to the discussion on CP violation, that a difference in the temporal evolution of matter and antimatter can arise if q and p have different magnitudes, q p p, q q p 1. 3

4 1. Measuring K 0 -K 0 oscillations For now, we concentrate on kaon oscillations and ignore CP violation, i.e. p = q. Neutral kaons can be produced in a state of definite antistrangeness by a strong interaction of negatively-charged pions on a proton target, see Fig. a). By strangeness conservation strong interaction) and baryon number conservation, it is impossible to make a Λ 0. In the semileptonic decay, the charge of the muon must be that of the strange quark, see Fig. b) and c). Hence by counting the number of µ + π decays versus the number of µ π + decays as a function of decay time, this decay can pick out the proportion of K 0 and K 0 in the propagating neutral kaon wavefunction. p + π d u u ū d d u s s d Λ 0 K 0 s d W + ū d µ + ν µ W µ ν µ s u π K 0 π d d + a) b) c) Figure : Quark flow diagrams for a) K 0 production from pion-nucleon scattering b) semileptonic decay of a K 0 and c) semleptonic decay of a K 0. The charge of the muon unabiguously identifies the kaon flavour. As the system starts with a pure K 0, Eq. 6 with p = q is appropriate. Writing out completely, Kt) = g + K 0 + g K 0 = 1 e i M Lt Γ L t/ } {{ } + e i M Ht Γ H t/ } {{ } ) K0 + 1 e i M Lt Γ L t/ } {{ } e i M Ht Γ H t/ } {{ } ) K0 8) a b a b The K 0 K 0 ) intensity is found from the probability of finding a K 0 K 0 ) at time t from this combined wavefunction, K 0 Kt) = 1 a + b)a + b) 4 = 1 4 aa + bb ) ba + ab ) K 0 Kt) = 1 a b)a b) 4 Taking first the direct amplitude-squared term in each line, = 1 4 aa + bb ) 1 4 ba + ab ) aa + bb = e im L Γ L )t e +im L Γ L )t + e im H Γ H )t e +im H Γ H )t = e Γ Lt The interference term reveals a sinusoidal dependence, + e Γ Ht So, ba + ab = e im H Γ H )t e +im L Γ L )t + e im L Γ L )t e +im H Γ H )t = e Γ L +Γ H t e im H M L )t = e Γ L +Γ H t cos Mt) + e Γ L +Γ H t e im H M L )t P K 0t) = K 0 Kt) = 1 4 P K 0t) = K 0 Kt) = 1 4 e Γ L t + e Γ Ht ) + 1 e Γ L +Γ H t cos Mt) e Γ L t + e Γ Ht ) 1 e Γ L +Γ H t cos Mt) 9) 4

5 which depends on the K L and K H lifetimes and their mass difference 1 in the interference term. The two kaon mass eigenstates have remarkably different lifetimes and are thus always labelled with reference to that property: the K-short K 0 S and the K-long, K 0 L. The K 0 S has lifetime of 89.5 ps whereas the K 0 L lives 571 times longer. Experiment determines the K 0 S is the lighter-mass eigenstate though there is no fundamental reason why this should be the case. The distributions of P K 0t) and P K 0t) are shown over 1ns in Fig. 3. As expected for the initially well-defined K 0, P K 00) = 1 and P K 00) = 0. One nanosecond is over 11 K 0 S lifetimes so this component will have fallen to e Γ Lt 0, leaving 1 ns is about one fiftieth the K 0 L lifetime, 1 4 exp 1 50 = P K 01 ns) P K 01 ns) 1 4 e Γ Ht. Considering the oscillatory term note that P K 0 = P K 0 when cos Mt) = 0. This occurs at t = π M) 1, 3π M) 1... The sketch shown this occurring at 300 ps and 900 ps, M = π 300) 1 = ps 1. Or in ev, [s 1 ] 10 1 e = [ev]. Incredibly, this is fourteen orders of magnitude smaller than the mean kaon mass, 49 MeV, yet it readily measurable due of the sensitivity of interference phenomena. Figure 3: Probabilities of finding each kaon flavour eigenstates from from an initially K 0 source. 1 To reiterate, K L and K H are not each other s antiparticle. CPT theorem does not apply so they can, and do, have different masses and lifetimes. 5

6 CP violation in kaons Kaon oscillation occurs because the flavour eigenstates states of definite flavour) mix into orthogonal superpositions, KS 0 = p K 0 + q K 0 p d s + q s d KL 0 = p K 0 q K 0 p d s q s d. 10) If p = q = 1 when normalised), these eigenstates of the Hamiltonian are CP eigenstates, in that they map onto themselves under application of the CP operator, with eigenvalue ±1, CP d s = s d, CP s d = d s CP KS 0 = + KS 0 and CP KL 0 = KL 0 if p = q. Consider the decay of a neutral kaon to pions π + π, π 0 π 0, π + π π 0, π 0 π 0 π 0 ). An ensemble of two J P = 0 pions from a ground-state kaon decay must have parity 1) 1) L = +1 because L must be 0. The particle antiparticle transformation is symmetric so this is a CP+ final state. For the same reason, the three-pion state is CP because it has parity 1) 3 1) L 1 1) L 3 = 1. This conclusion assumes L 1 = L 3 = 0 which is verified by inspecting the Dalitz plot of the three-pion decay. Thus if CP symmetry is a conserved quantity, one expects only K 0 S ππ and only K 0 L πππ. To a good approximation, this expectation is correct and indeed, the reduced phase-space for the three-pion final state is the reason why the K 0 L has a much-longer lifetime than the KS. 0 The large difference of lifetime is experimentally useful for separating the two mass eigenstates. Imagine a neutral kaon beam of average energy 10 GeV γ 0) passed into a long tube. The average distance it will travel before decay is, γ c τ K 0 S ms 1 89ps 0.5m. After 5 metres all K 0 S will decay and K 0 L, whose lifetime is 570 longer, can be studied in isolation..1 Regeneration Evacuating the decay tube is vital because it is possible to regenerate a K 0 S component in a pure K 0 L beam by interaction with matter Pais & Piccioni, 1955, Muller, 1960). The strange neutral meson, K 0, interacts with nucleons to produce hyperons, K 0 +N Λ 0 +π; there is no such process for the anti-strange meson, K 0. Neglecting CP violation p = q = 1 ), the K 0 L wavefunction can be written in terms of proportions f and f, K 0 L = 1 f K 0 f K 0 = 1 f K 0 S + K 0 L ) f K 0 S K 0 L ) ) = f f ) K 0 S + f + f ) K 0 L ), f = f = 1. Due to the differing interaction lengths in matter, it can evolve that f f and a K 0 S component appears in the beam. This means two-pion events are observed whereas only three-pion events would be expected for a pure K 0 L CP wavefunction.. Discovery of CP violation In 1963 Blair et. al. reported and excess of K 0 S regeneration. A year later, Cronin & Fitch confirmed that observation but concluded that the excess could not be attributed to regeneration alone: the K 0 L meson was itself decaying into π + π and hence violating the supposed invariance of CP symmetry. Their experiment, shown below, was setup to count the number of π + π decays from a pure K 0 L beam passing through a volume of well controlled density. They reported an excess of decays consistent with two-pion decays along the beam line of 45 ± 10 events, with 10 attributable to K 0 S regeneration. 6

7 The 35 anomolous decays from around 7 million K 0 L particles estimated to pass through the experiment suggests the CP symmetry is violated at a rate of ) = ɛ. 11) This result implies that the K 0 L mass eigenstate is not a CP eigenstate. Relabelling the CP eigenstates K 1 and K, we write, which is Eq. 10 with p = 1 + ɛ) and q = 1 ɛ), K 1 = 1 K 0 + K 0 ) K = 1 K 0 K 0 ). K 0 S = K 1 + ɛ K K 0 L = ɛ K 1 + K, K 0 S = ɛ) K ɛ) K 0 ) KL 0 = 1 + ɛ ) ɛ) K 0 1 ɛ) K 0 ). 1) 1 + ɛ ) Note that Eq. 10, and hence Eq. 1, attributes the CP-violation to the mixing parameters p and q such that some amount of K 1 is present in the K 0 L wavefunction. Most generally, one should consider including direct CP-violation in the decay K ππ; but as we shall see later, this makes a very small contribution to the K 0 L ππ rate..3 Categories of CP violation Let us define the amplitude of each flavour eigenstates decaying to a CP eigenstate, f e.g. ππ), A f = f H weak K 0 A f = f H weak K 0. We wish to know the probability of observing f at time t from a particle that was of known strangeness K 0 or K 0 ) at t = 0. A K 0 at t=0 t) = A q f g + t) + A f p g t) A K 0 at t=0 t) = A p f g + t) + A f q g t) = A f g+ t) + λ g t) ) = A f g+ t) + λ 1 g t) ) where λ = qa f pa f. 7

8 K 0 K 0 K 0 g + t) q p g t) A f A f f 0 K p q g t) g + t) A f A f f K 0 K 0 ΓK 0 t=0 f ) = A f g + t) + λ g t) = A f [ g + t) + λ g t) + R λ g + t) g t) ) ] ΓK 0 t=0 f ) = A f g + t) + λ 1 g t) [ = A f g + t) + 1 λ g t) + λ R λ g + t) g t) ) ] Differences between Eqs. 13 and 14 give rise to CP violation. This can be of three distinct types: 13) 14) 1. A f A f : direct CP violation in the decay amplitude;. 1 : CP violation in mixing amplitudes so λ 1; q p 3. λ λ : Iλ) 0 is CP violation in the interference of mixing and decay. Extension to consider CP-conjugate decays Eqs. 13 and 14 have been formed considering a common CP-eigenstate final state e.g. π + π ) in which the K 0 or K 0 is reconstructed. More generally, one can consider the case of two final states, f and its CP conjugate f by taking the CP conjugates of Eqs. 13 and 14, ΓK 0 t=0 f ) = A f [ g + t) + λ g t) + R λ g + t) g t) )] 15) [ ΓK 0 t=0 f ) = A f g + t) + 1 λ g t) + λ R λ g + t) g t) ) ] 16) where we introduce an almost-reciprocal CP violating parameter for the CP-conjugate final state f ), λ = pa f qa f Comparing λ to λ, the p and q have swapped places, as have A and A. Notice that λ = λ 1 in the case that f is a CP eigenstate. In this case Eqs. 15 and 16 are redundant copies of Eqs. 13 and 14. Semileptonic final states are a special case of this extension. As f can only come from K 0, and f only from K 0, A f = A f = 0. Furthermore, only one type of decay amplitude is possible for semileptonic weak decays so not CP violation in the decay amplitude is possible, A f = Ā f = A. This means Eqs become, ΓK 0 t=0 f sl ) = A g + t) ΓK 0 t=0 f sl ) = A ) p q g t) ΓK 0 t=0 f sl ) = A g + t) ΓK 0 t=0 f sl ) = A ) q g t) 8 p

9 Thus semileptonic decays probe the CP violation in mixing only i.e. that which is encoded in p q)..4 Semileptonic asymmetry Unlike K ππ decays, semileptonic decay amplitudes are unambiguous about which flavour eigenstate they have decayed from: π l + ν must come from a K 0 sd, π + l ν from a K 0 s d. By comparing the number of K 0 and K 0 decays, an asymmetry can be formed: A = N K 0 N K 0)/N K 0 + N K 0). This asymmetry, plotted in Fig. 4, is dominated by the oscillation of the mixing, but the oscillation is not about zero! At t > 10 9 seconds more than ten K 0 S lifetimes) one sees the bias: there are more K 0 than K 0. If CP were conserved, i.e. if the K 0 L were a CP eigenstate, an equal amount of K 0 and K 0 should be seen. Figure 4: Charge asymmetry in semileptonic kaon decays. Gjesdal et. al., Phys. Lett. 5B ). Using the definition of the K 0 L including the CP-violating parameter, ɛ from Eq. 1, one derives the time-independent asymmetry in semileptonic K 0 L decays, δ sl = ΓK0 L π l + ν) ΓK 0 L π + l ν) ΓK 0 L π l + ν) + ΓK 0 L π + l ν) = 1 + ɛ 1 ɛ 1 + ɛ + 1 ɛ = Rɛ) 1 + ɛ 17) = p q p + q 1 q/p or 1 + q/p as p = 1 + ɛ and q = 1 ɛ. From this and other) data the semileptonic asymmetry is measured as, δ sl = 3.3 ± 0.06) Neglecting the relatively small ɛ term in the denominator of Eq. 17, one concludes that Rɛ) Comparing this with the Eq. 11 result, ɛ , one can conclude that the phase of ɛ is cos ) 1 Rɛ) ɛ 44. The similarity of the size of δ sl which is sensitive to type ) and the effect in K 0 L π + π sensitive to types 1,,3) suggests the pionic CP-violation is dominated by mixing. To fully explore this, a time-dependent measurement is needed. 9

10 .5 Time-dependent CP violation: CPLear A time dependent study of K π + π decays require the initial flavour of the neutral kaon to be tagged. The CPLear experiment CERN ) used proton-antiproton annihilation to produce neutral kaons. The charges of the associated π ± and K ± tag the neutral kaon s flavour: p p π + K K 0 or p p π K + K 0. The decay time is measured from the flight distance on the neutral kaon with its relativistic γ-factor calculated from the fourvector sum of the π + and π. Fig. 5 shows the time-dependent decay rate data of initially-tagged K 0 and K 0 mesons to two-pions. By taking the difference of these two curves, an asymmetry is plotted below. The CP-violation is expressed in terms of the observables Figure 5: Clockwise from top left: a schematic of the CPLear detector, the decay distribution of initiallytagged K 0 and K 0 mesons, the normalised asymmetry in % and an event display showing the cleanliness of the tagging. by multiplying out the time-dependent g ± factors labelling the light and heavy mass eigenstates according to their lifetime properties, K 0 S and K 0 L, g ± = 1 [ e i M St Γ S t/ } {{ } ± e i M Lt Γ L t/ } {{ } ] From Eq. 13, using the a and b labels to clarify the algebra: ΓK 0 t=0 ππ) = A ππ 1 4 a + b) + λa b) = A ππ 1 4 a1 + λ) + b1 λ) a = e Γ St a b b = e Γ Lt = A ππ λ a + ηb η = 1 λ 1 + λ = A ππ λ a + η b + R[a η b ] ) = A ππ λ e Γ St + η e Γ Lt + R [ η e i M St Γ S t/ e i M Lt Γ L t/ ]) = A ππ λ e Γ St + η e Γ Lt + η R [ e iφ e i M L M S )t ] e Γ L+Γ S )t/ ) η = η e iφ = A ππ λ e Γ St + η e Γ Lt + η cos Mt φ) e Γ L+Γ S )t/ ) M = M L M S 18) 10

11 Similarly, ΓK 0 t=0 ππ) = Ā ππ λ e ΓSt + η e ΓLt η cos Mt φ) e ) Γ L+Γ S )t/ = A ππ p q λ e ΓSt + η e ΓLt η cos Mt φ) e ) Γ L+Γ S )t/ λ = q A f 19) pa f The factor p q 1 4R[ɛ] from p = 1 + ɛ) and q = 1 ɛ) and R[ɛ] It is thus neglected = 1) here to form the asymmetry, a π + π t)=γk0 t=0 π + π ) ΓK 0 t=0 π + π ) ΓK 0 t=0 π + π ) + ΓK 0 t=0 π + π ) = 4 η + cos Mt φ + ) e Γ L+Γ S )t/ e Γ St + η + e Γ Lt ) = η + e Γ S Γ L )t/ cos Mt φ + ) 1 + η + e Γ S Γ L )t Which is the form of the time-dependent asymmetry plotted in Fig. 5. A fit to these CPLear data finds, 0) η + =.6 ± 0.03) 10 3 argη + ) = 43. ± 0.7). The similarity of this result to that seen in semileptonic decays is evidence that the CP violation is dominated by a mixing effect and there is very little, or no, direct CP violation in the decay amplitudes. In this derivation, we introduce an alternative CP-violation parameter, η that approaches zero in the limit of CP conservation λ 1) and relate it to the amplitude for K 0 L π + π compared to that of K 0 S, η + = 1 λ 1 + λ = pa + qā + pa + + qā + = π+ π K 0 L π + π K 0 S = ɛ + ɛ. The ɛ is introduced to parametrise the additional effect due to direct CP violation in the decay amplitudes. In the [very good] approximation implied so far, there is no direct CP violation in K ππ decays, η + = ɛ. Finally, we note the advantage measuring CP violation in the interference between mixing and decay: the time-integrated effect in K 0 L decays detected by Cronin&Fitch is dependent on η where the oscillatory interference term is proportional to η..6 Direct CP violation: NA48 By the 1990s, CP violation was well studied but the underlying mechanism was still debated. The CKM mechanism predicted direct CP violation but there was not experimental evidence for it. The NA48 experiment CERN ) using a simultaneous K 0 L,K 0 S beam to search for direct CP violation. The experiment reported a measurement of the double ratio of the CP violation in two different final states, i.e. with differing A f, Ā f. η 00 η + = ΓK 0 L π 0 π 0 ) / ΓK 0 L π + π ) ɛ ) ΓK 0 S π 0 π 0 ) ΓK 0 S π + π ) 1 6Re, 1) ɛ because, η + ɛ + ɛ η 00 ɛ ɛ from isospin. ) The world average from NA48 and competitor experiments is Reɛ /ɛ) = 1.65 ± 0.6) 10 3 from the analysiss of over a trillion K 0 L on target. This result established that direct CP violation was possible, thus giving strong weight to the CKM mechanism of the Standard Model, but that it is a tiny effect in kaon decays. 11

12 The derivation of Eq. follows from a isospin decomposition of the two pion state. As pions are bosons, the total final state wavefunction, Ψ = ψ space ψ f lavour, must be symmetric. Decay from the spin-0 kaon, L = 0 so the spatial wavefunction is symmetric. The isospin flavour) wavefunction must also be symmetric with total isospin I = 0 or, not 1. The total I 3 of the ππ combination must be 0 by charge conservation, so the Clebsch-Gordan coefficients are, 0, 0 =, 0 = 1 3 1, +1 1, , 0 1, , 1 1, +1 = 3 π+ π 6 1, +1 1, , 0 1, , 1 1, +1 = 1 3 π+ π + 1 We decompose η + into the two isospin amplitudes, For η +, divide by η + = π+ π K 0 L π + π K 0 S = 3 ππ I=0 K 0 L + 3 ππ I=0 K 0 S + 1 η 00 = π0 π 0 KL 0 π 0 π 0 KS = 3 ππ I=0 KL ππ I=0 KS ππ I=0 K 0 S top and bottom gives, ππ I= K 0 L 3 ππ I= K 0 S 3 ππ I= KL 0 3 ππ I= KS π0 π 0, 3 π0 π 0. η + = ππ I=0 KL ππ ππ I=0 KS 0 I= KL 0 ππ I=0 KS 0 = ππ I=0 KS ππ ππ I=0 KS 0 I= KS 0 ππ I=0 KS 0 ɛ + 1 ππ I= KL 0 ɛ + ω ππ ππ I=0 KS 0 I= KL 0 ππ 1 + ω = I= KS ω, where we now specify ɛ to be the CP violation parameter for the I = 0 final state, ɛ = ππ I=0 KL 0 ππ I=0 KS, and ω = ππ 0 I= KS 0 ππ I=0 KS 0 is the proportion of the I = to I = 0 in K 0 S decays. For η 00 it is the same logic but dividing by η 00 = ɛ ω ππ I= K 0 L ππ I= K 0 S 1 ω. 1 3 ππ I=0 K 0 S, The last step is to add and subtract terms to the numerator and identify a common expression for the difference in CP violation for the I = and I = 0 amplitudes, which is by definition direct CP violation, and labelled ɛ. η + = η 00 = ɛ + ω ππ I= K 0 L ππ I= K 0 S + ɛω ɛω 1 + ω = ɛ + ɛ ω ππ I= K 0 L ππ I= K 0 S + ɛω ɛω 1 ω ) ω ππi= KL 0 ɛ ππ I= KS ω = ɛ + ) ππi= K ω L 0 ɛ ππ I= KS 0 = ɛ 1 = ɛ ω ) ω ππi= KL 0 ππ I=0 KL 0 ππ I= KS 0 ππ I=0 KS ω = ɛ + ) ω ππi= KL 0 ππ I=0 KL 0 ππ I= KS 0 ππ I=0 KS 0 1 ω = ɛ ɛ 1 + ω. ɛ 1 ω. Empirically, it is observed that the I = 1 transition is favoured over the I = 3 transition from the I = 1 neutral kaon; ω 0.05 and so the following approximations are commonly quoted, η + ɛ + ɛ η 00 ɛ ɛ. The double ratio of Eq. follows using a simple binomial expansion of the ratio, η 00 η + = ɛ ɛ ɛ + ɛ 1

13 3 Introduction to CP violation in the Standard Model Neutral kaon mixing is a second-order weak transition which is described in the standard model by a box diagram. The vertex factors of the box diagram are proportional to the relevant CKM element. For example, s d W + V cs V ud c ū V cd V us W d s s d V cs c V cd W W + u Vud V us d s M f i V ud V csv usv cd ) u W ± c Vud V us V cd V cs ) ) d s M i f V udv cs V us V cd this is just illustrative for the moment... By comparing the K 0 K 0 box diagram with the CP-conjugate K 0 K 0 which is also the T-conjugate K 0 K 0 by the CPT ansantz) one sees that the rate differs if at least one of the contributing CKM elements is complex. Most generally, a complex N N matrix has N parameters but there are two important constraints. First, the matrix must be unitary; total probability needs to be respected, V T V = 1, so the number of parameters reduces N N. Second, one has arbitrary freedom to rotate the quark fields, q i V i j q j q i e iφ i V i j e iφ j q j : V i j e iφ j φ i ) V i j, such that only phase-differences count. N 1 parameters are absorbed in in this pairing. Thus the quark mixing matrix has N N + 1 independent physical parameters, of which NN 1)/ are rotations. N= gives 1 physical parameter, a rotation between the first and second generation: the Cabibbo angle, θ 1. N=3 gives 4 physical parameters, three of which are rotations, leaving one complex phase cos θ 13 0 sin θ 13 e iδ cos θ 1 sin θ cos θ 3 sin θ sin θ 1 cos θ sin θ 3 cos θ 3 sin θ 13 e iδ 0 cos θ With three generations, a single complex phase in V CKM generates all CP violation phenonoma. Kobayashi and Maskawa predicted three generations of quarks after the Cronin&Fitch discovery but before the discovery of the charm quark. Multiplied out, one finds the the imaginary part of V cs and V cd are indeed non-zero compared to V ud ). u cos θ 1 cos θ 13 sin θ 1 cos θ 13 sin θ 13 e iδ d c W± sin θ 1 cos θ 3 cos θ 1 sin θ 3 sin θ 13 e iδ cos θ 1 cos θ 3 sin θ 1 sin θ 3 sin θ 13 e iδ sin θ 3 cos θ 13 s t sin θ 1 cos θ 3 cos θ 1 cos θ 3 sin θ 13 e iδ cos θ 1 sin θ 3 sin θ 1 cos θ 3 sin θ 13 e iδ cos θ 3 cos θ 13 b But it is not immediately obvious from this parameterisation which CKM element has the largest imaginary part. There is a more convenient form, known as the Wolfenstein parameterisation expands the matrix in powers of the sine of the Cabibbo angle, usually labelled λ = sin θ 1, V ud V us V ub 1 λ / λ Aλ 3 ρ iη) V cd V cs V cb = λ 1 λ / Aλ + Oλ4 ) 3) V td V ts V tb Aλ 3 1 ρ iη) Aλ 1 There is a clash of notation but the CP-violating parameter λ = qa/pā and the Wolfenstein expansion parameter, λ = sin θ Cabibbo are unrelated. 13

14 which makes the hierarchy in the CKM coupling transparent. Between the first and second generations the vertex factor is suppressed by λ 0.. From generations one to three, the transition is suppressed by λ at the amplitude level! Also, the size of the imaginary parts of V cs and V cd are, in fact, small; the largest imaginary contributions are found in V td and V ub. The underlying origin of the hierarchy of the CKM elements is unknown and the four CKM parameters either θ 1, θ 3, θ 13, δ, or λ, A, ρ, η) are fundamental parameters of the SM that must be measured from decays sensitive to the CKM elements. CKM element magnitudes can be deduced from comparing branching fractions of processes that differ only by the ratio of CKM factors, for example: V cd = 0.56 ± : from comparing D 0 π e + ν decays to D 0 K e + ν decays, assuming a known V cs. V ub = ± : from comparing B 0 π e + ν to B 0 D e + ν, assuming a known V cb. Sensitivity to CKM phase information comes from CP violation. A CP-violating transition must have at least two contributing amplitudes, at least one of which has a non-zero imaginary part. 3.1 The Unitarity triangle If the CKM matrix describes all possible quark coupling via the weak force then total probability must be conserved, the matrix must be unitary. This, in turn, requires the matrix to satisfy unitarity relations, for example that the dot product of any two rows, or any two columns must equal 1 and thus the dot product of two different columns must be V usv ud +V csv cd +V tsv td = 0 first and second columns. V ubv ud +V cbv cd +V tbv td = 0 first and third columns 3. V ubv us +V cbv cs +V tbv ts = 0 second and third columns The sum of three complex numbers equalling zero are triangles in the complex plane. It is informative to notice the size of the triangles, 1. Oλ) + Oλ) + Oλ 5 ) s d triangle : K 0 decays. Oλ 3 ) + Oλ 3 ) + Oλ 3 ) b d triangle : B 0 decays 3. Oλ 4 ) + Oλ ) + Oλ ) b s triangle : B s decays The relative height of these triangles give an indication of the magnitude of the CP violation effect involved. The first triangle relates to the neutral kaon system and its modest height reflect the size of the observed CP violation, 0.) 5 1) = and ɛ = The second triangle suggests large CP asymmetries O1) seem possible. Graphically, it is, V tbv td 1 λ )ρ, η) V ubv ud V ud V ub V cbv cd α V td V tb V cbv cd V cbv cd γ β 0, 0) 1, 0) where we rotate and scale in the second diagram, i.e. choose a convention where one side is unity. The three internal angles are CP-violating quantities that can be studied in many B-decay modes. ) ) ) α = arg V tbv td VubV β = arg V cbv cd ud VtbV γ = arg V ubv ud td VcbV cd 14

15 The Wolfenstein parameterisation places the CP violation only in the off-diagonal corners of the CKM matrix: in V td and V ub to Oλ 3 )). Thus, and noticing the minus sign of V cd, these Unitarity Triangle angles simplify to, arg V td ) = β arg Vub ) = γ α = π + arg Vtd ) + arg Vub ). 4) Generally speaking, to measure phases we two amplitude to interfere. To measure β, at least one of the interfering processes must involve t d quark transitions. To measure γ, at least one of the interfering processes must involve b u quark transitions. As we shall see next, this can be achieved with B-mesons. 4 B physics B mesons come in four varieties, differing by the flavour of the quark in the q b bound state. In increasing mass, these are, B + u 579 MeV/c ) B 0 d 580 MeV/c ) B 0 s 5366 MeV/c ) B + c 676 MeV/c ). In contrast to kaons, B mesons decay to a range of final states all with considerable phase-space. This means the lifetime difference evident in kaons does not occur and we default back to labelling the mass eigenstates Heavy and Light. B-mesons decay by a weak transition of the b-quark, or perhaps the c-quark in the case of the B c meson. Due to the inaccessibility of its generational partner, the top, it must decay at tree-level to a charm, or occasionally an up quark V cb >> V ub ). This leads to a characteristically low transition probability, or to put it another way, a long lifetime, B ± u 1.64 ps) B 0 d 1.53 ps) B0 s 1.47 ps) B ± c 0.46 ps), where the shorter lifetime of the B ± c is due to the additional partial width of Cabibbo-favoured c-quark decays. Only kaons, charged pions and muons, which also decay via the weak-force but with much smaller phase-space, live longer. Over the 40 years since their discovery, B mesons have been studied by many experiments. These are of two types: e + e beams can be tuned precisely to collide at specific energies. The Υ4S ) state is just above the kinematic threshold for B + B,B 0 B 0 production. With nothing else in the event, this means a clean experimental environment although the Υ4S ) production cross-section is relatively low. A further advantage is that the two B mesons are produced in a quantum-entangled state. Experiments are ARGUS, CLEO, Babar, Belle and from 019, Belle. pp collisions have much larger production cross-sections than e + e, especially at the TeV scales of the LHC. Backgrounds are much larger so a high granularity detector and sophisticated albeit inefficient) selections are needed to cleanly identify signals. The principle handles for identifying a relatively heavy, relatively long-lived B meson is high transverse momentum of the decay products, and a large impact parameter with respect to the primary pp collision vertex. A big advantage of the LHC is that all types of b-hadrons are produced, B 0,B +,B s,b c,λ b etc..., according to their hadronisation fractions. The dedicated B-physics experiment at the LHC is called LHCb. 4.1 B-meson mixing The formalism for B mixing is almost identical to that of kaons though an important simplification is that the B mesons are heavy and there is lots of phase space available for the decay products. This means the lifetime difference between the two mass eigenstates is negligible Γ B 0 << M B 0) and we put Γ H = Γ L = Γ. This means for the probability of finding B 0 or B 0 from a B 0 initial state at t=0 is, P B 0t) = B 0 B 0 t=0t) = 1 e Γt 1 + cos M B 0t) ) P B 0t) = B 0 B 0 t=0t) = 1 e Γt 1 cos M B 0t) ), 5) which is equivalent to the kaon rates in Eq. 9 with Γ H = Γ L = Γ. 15

16 We remember M K ps 1 with τk 0 L) = 89 ps. The B 0 meson lifetime 1.5 ps) is the same order of magnitude as D 0 mesons, so the potential to observe B 0 mixing depends on M B 0 being at least as large as that of the lighter neutral mesons. Once again, let s consider the second-order W ± box diagram. V tb t b d B 0 W + W B 0 t d b V td V td V tb Figure 6: The second-order W ± box process that drives B-mixing. Note, the asterisks imply CP-conjugation for the antiquark currents The transition rate of this box process dominates the mixing probability: the larger the mass difference, the higher the transition rate. M B 0 received a contribution from u, c and t quark currents in the box. If all these quarks had identical mass, GIM suppression would be exact and there would be no mixing. However, as the top mass is much larger than the others, its contribution dominates M B 0. This is not the case for M K as the top contribution is moderated by the CKM coupling. In the SM, ) M B V tb Vtd m ) t ps m t ps 1. 6) GeV/c GeV/c For m t in the range GeV/c, M B 0 would be similar to M 0 K, in the range ps 1. The Argus experiment DESY 1987) first measured M B 0 from e + e Υ4S ) B 0 B 0 pair production by counting the number of B 0 that oscillate after the B 0 decays semileptonically. In 17% of doublesemileptonic events, the two leptons were of the same charge. Same lepton charge means the same B-meson flavour, i.e. one had mixed. µ + The event display shows this: two D µ + ν decays, D [K + π ] D 0π and D [K + π π ] D [γγ] π 0 are reconstructed hence indicating the B 0 b d ) has oscillated to a B 0 bd ). The analysis of the ARGUS data estimated M B ps 1 and thus, from Eq. 6, m t 130 GeV/c. This was the first indication of a very-high top quark mass and predated the direct discovery of the top quark by eight years. The B d mass difference is now known precisely, M B 0 = ± ps 1. µ + And what of the final neutral meson, the B s meson? The B s mixing diagram is identical to that of B 0 above except the spectator d quark is replaced with an s quark so V ts appears in the M formula instead of V td. Inspection of the CKM matrix, indicates that these two elements differ by at least a factor λ = sin θ C 0., so expect, M Bs = V ts V td M B 0 ) ps 1 11ps In the early 1980s, following the confirmation of the third quark generation by the observation of the Υ b b) resonances around 10 GeV/c, there was an expectation that the top mass would not be much higher. If fact, m t is much, much higher, 17.4 ± 0.5 GeV/c TeVatron 1995). 16

17 which is so fast that it presents an experimental problem to see such rapid oscillations. The CDF experiment TeVatron 006) were first to detect a high Bs oscillation frequency finding MBs 18ps 1. LHCb has superior vertexing capability and could spatially resolve the small distances involved, see Fig. 7. LHCb data for the slower oscillation of the B0 meson is also shown. The final state flavour is tagged from the decay, B0s) D s) π+ or B 0s) D+s) π. The initial flavour is tagged by the other B in the pp BBX production. This is the same idea as tagging the initial kaon flavour from the other K in CPLear, though the tagging at LHC is not 100% accurate. Such mistagging reduces the amplitude of the flavour oscillations but does not effect the period. Figure 7: Bs oscillations from D±s π data left). Asymmetry= NB0 ) NB0 ) NB0 )+NB0 ) 0 from D± π data right). Note the differing x-scales; the Bs mesons oscillates 35 times faster than B mesons. The initial flavour at t = 0 is know from flavour tagging. The oscillations do not seem to fall to zero go to asymmetries of ±1) only because of imperfect tagging and time resolution. The plots does not start at t = 0 because a flight distance selection is mandatory to get a clean samples of B mesons. 4. Formalism of CP violation with neutral B mesons The derivations of Eqs. 13&14 imagined kaons but they are equally applicable to B mesons decaying to final state f, q p AB0 at t=0 t) = A f g+ t) + A f g t) A B0 at t=0 t) = A f g+ t) + A f g t) p q = A f g+ t) + λ 1 g t) = A f g+ t) + λ g t) h i ΓB0t=0 f ) = A f g+ t) + λ g t) + < λ g+ t) g t) h i ΓB0t=0 f ) = A f g+ t) + λ g t) + λ < λ g+ t) g t), 7) 8) and again, use some shorthand symbol a, b) in the expansion of the time-dependent factors for the Light and Heavy mass eigenstates. An important property of B0 mesons, simplifying the algebra, is that the difference in the lifetimes of the two mass eigenstate is negligibly small due to the large available phase space: Γ = 0, ΓH = ΓL = Γ. i MH t Γt/ L t Γt/ e i M{z g± = 1 [ } ± e {z } ] a b a = e Γt ab = ei Mt e Γt b = e Γt ab ba = i sin Mt) e Γt M = MH ML Thus developing Eq. 7, + λ a + b < ab + < λ a + b)a b ) ΓB0t=0 f ) = A f 14 a + b + < ab h i h i = A f 14 e Γt + e Γt + e Γt < ei Mt + λ e Γt + e Γt e Γt < ei Mt e Γt < λ i sin Mt) = A f 1 e Γt 1 + cos Mt) + λ 1 cos Mt) ) = [λ] sin Mt) = A f 1 e Γt 1 + λ ) 1 + C cos Mt) S sin Mt)), 17 9)

18 where C = 1 λ I [λ] S = λ = qā f. 1 + λ 1 + λ pa f Similarly for Eq. 8, ΓB 0 t=0 f ) = Ā f 1 4 a + b + R [ ab ] + λ a + b R [ ab ] ) + λ R [ λa + b)a b ) ] ) = Ā f 1 4 e Γt + e Γt + e Γt R [ e i Mt] + λ e Γt + e Γt e Γt R [ e i Mt] ) e Γt λ R [λi sin Mt)] ) = Ā f ) 1 e Γt 1 + cos Mt) + λ 1 cos Mt) ) + λ I [λ] sin Mt) = Ā f 1 e Γt λ ) λ + λ cos Mt) + 1 cos Mt) + I [λ] sin Mt) = A f 1 e Γt p 1 + λ ) 1 C cos Mt) + S sin Mt)). 30) q For the derivation of the CP violation in kaons in the context of CPLear, we considered a final state f = ππ, which is manifestly self conjugate. To be more general The CP conjugate equations of Eq. 9 and Eq. 30 should also be considered, ΓB 0 t=0 f ) = Ā f 1 e Γt 1 + λ ) 1 + C cos Mt) S sin Mt) ), 31) ΓB 0 t=0 f ) = Ā f 1 e Γt q p 1 + λ ) 1 C cos Mt) + S sin Mt) ), 3) where C = 1 λ S 1 + λ f = I [ λ ] λ = pa f. 1 + λ qā If f is a self-conjugate final state e.g. π + π ), then λ = λ 1. Eqs. 31&3 become identical to Eqs. 30&9. In the limit of CP conservation, q = p and A f /Ā f = Ā f /A f. Thus λ = λ, C = C and S = S = 0. Inspection of Eqs with these substitutions gives ΓB 0 t=0 f ) = ΓB 0 t=0 f ) and ΓB 0 t=0 f ) = ΓB 0 t=0 f ), as would be expected. 4.3 CP violation in B mixing f As with kaons, CP violation in B 0 -B 0 mixing can be isolated with semileptonic decays. In this case, the charge of the muon in B 0 Dµν decays uniquely identifies the flavour of the decaying B meson: µ + identifies a B 0 decay, µ means a B 0 decay. In the formalism of the previous section with f = D µ + ν, Ā f = 0, A f = 0. Thus λ = λ = 0. B 0 b d ν W + µ+ c d D In addition, each semileptonic decays has just one contributing process so there is no direct CP violation: Ā f = A f. CP violation in mixing is parameterised by p q, or equivalently, by a non-zero semileptonic decay asymmetry, a sl = 1 q p This is measured by counting the number of B D µ + ν decays as well as the number of B D + µ ν decays where B is an untagged sample of B 0 and B 0 mesons. It is assumed the B mesons are produced in equal number. 0. A B sl = NB f ) NB f ) NB f ) + NB f ) = ΓB0 f ) + ΓB 0 f ) ΓB 0 f ) + ΓB 0 f ) ΓB 0 f ) + ΓB 0 f ) + ΓB 0 f ) + ΓB 0 f ), 18

19 then substitute in Eqs with λ = λ = 0, Ā f = A f and q/p = 1 a sl ), A B sl = 1 + cos Mt) + 1 a sl) 1 1 cos Mt)) [ 1 + cos Mt) + 1 a sl )1 cos Mt)) ] 1 + cos Mt) + 1 a sl ) 1 1 cos Mt)) cos Mt) + 1 a sl )1 cos Mt)) 1 + a = sl +... )1 cos Mt)) 1 a sl )1 cos Mt)) 1 + cos Mt)) a sl +... )1 cos Mt)) + 1 a sl )1 cos Mt)) = a sl 1 cos Mt)). This is a measurement that does not require tagging of the initial flavour. This is experimentally useful because tagging is never perfect; typical correct tagging efficiencies are 5% in hadron collider experiments and 30% at e + e B-factories. The assumption that B 0 and B 0 are produced in equal numbers is appropriate for colliders with a symmetric initial state like the e + e B-factories or the p p collisions of the TeVatron. The pp collisions at the LHC are matter-antimatter asymmetric. A 0.5% production asymmetry is expected and must be taken into account in a measurement of A B sl. 33) With kaons, the semileptonic asymmetry δ sl = 1 q/p 1+ q/p, Eq. 17. This is equivalent to A B sl in the case that 1 + q/p, and M << t such that cos Mt 0. Thus for kaons a K sl δ sl = With B 0 mesons, a smaller CP-violating effect in mixing is expected in the SM: a B slsm) = Current data does not have statistical sensitivity at this level. Results for a B sl are consistent with zero with errors around the CP violation in the interference of mixing and decay: the Unitarity Triangle angle β The decay B 0 J/ψ K 0 S offers an excellent example of CP violation in interference between mixing and decay amplitudes. This type of CP violation is that which occurs when λ has a non-zero imaginary part even though the its modulus is unity. The two paths to the same final state are mixed and unmixed B 0 mesons followed by CP-conjugate decays, g + t) B 0 A f J/ψ K 0 B 0 q p g t) B 0 Ā f J/ψ K 0 J/ψ K 0 S B J/ψ K 0 S is chosen because it is a CP eigenstate with high branching fraction, clean signature and good reconstruction efficency. 4 We remember that p and q relate the flavour to the mass eigenstates. This is also the function of the CKM matrix as it rotates the down-type mass eigenstates to connect via a weak interaction with the up-type flavour eigenstates. From the discussion of semileptonic decays above, q p 1 for B 0 mesons. This means the Γ1 M1, so, q p = M 1 i Γ 1 M 1 M 1 i Γ = V tdvtb 1 M 1 VtdV = e iβ. tb using the definition of β from Eq. 4, argv td ) = β as well as taking V tb = 1. 4 BB 0 J/ψ K 0 S ) = , BJ/ψ µµ) = 5.9%, BK 0 S π+ π ) = 69%. Total 10 5 so need 10 8 B 0 to record O1000) of these decays. 19

20 The decay amplitudes are almost identical and involve only real CKM elements, A f = A B 0 J/ψ K 0) A f A B 0 J/ψ K 0) A K 0 KS) 0 A ) = V cbvcs K 0 K 0 V VcsVcd S cbv cs VcsV cd = 1. The J/ψ K 0 S final state, is a CP eigenstate with CP eigenvalue, η = 1. The similar J/ψ K 0 L final state has η = +1, CP J/ψ = 1 J/ψ, CP K 0 S + K 0 S hence CP J/ψ K 0 S = J/ψ K 0 S. CP eigenvalue, η = 1 for f CP = J/ψ K 0 S. Similarly, CP K 0 L K 0 L hence CP J/ψ K 0 L = + J/ψ K 0 L. CP eigenvalue, η = +1 for f CP = J/ψ K 0 L. Altogether, for J/ψ K 0 S and J/ψ K 0 L, λ = q p Ā f A f = ηe iβ = cos β ± i sin β. Finally, we form a time-dependent asymmetry from the rates defined in Eq. 9&30 knowing that q/p = 1 is established, A CP t) = ΓB0 f CP ) ΓB 0 f CP ) = C cos Mt + S sin Mt. ΓB 0 f CP ) + ΓB 0 f CP ) And furthermore, as λ = 1, so C = 0 and S = Iλ) = ± sin β, A CP t) = Iλ) sin Mt = + sin β sin Mt for J/ψ K 0 S, = sin β sin Mt for J/ψ K 0 L. These data come from the B-factories, KEKB and PEP ) operated with an asymmetric e + e collision energy of s = GeV/c 9.0 vs. 3.1 GeV/c ). This is the same CoM energy as used by ARGUS, to produce B B pairs at the Υ4S ) resonance. This measurement relies on quantum coherence. Only when one B decays is the clock started. Hence the time-dependent CP study is conducted in both positive and negative time expressed as t in the plot). The sign of the time depends on when the flavour tagging associated decay of the other B ) occurred before or after the signal J/ψ K 0 S or J/ψ K 0 L) decay. The period of the time-dependent CP asymmetry is defined by M but the amplitude is sin β. The amplitude is diluted by the imperfect tagging and the final measurement of sin β requires a measurement of the tagging efficiency. This is achieved by applying the tagging algorithms to samples of charged B ± decays, as their b-flavour is known. 0

21 4.5 Direct CP violation: the Unitarity Triangle angle γ Of the three Unitarity angles, γ is the only one that is not dependent on a virtual coupling to the top quark in a box diagram. This means mixing is not needed and we can look for direct CP violation. Charged B ± DK ± decays do not mix and so can be effected by direct CP violation in the decay amplitudes only. B b ū W u c s ū D 0 K W ū s K b c B ū ū D 0 Because the the left and right processes produce a D 0 and D 0 respectively, it is necessary to reconstruct them in a decay that is accessible to both; for example D K + K. Also, the γ sensitivity aries because these two diagrams interference and one of them has a dependency on V ub the second has a negligible weak phase). We write that the amplitude of the right-hand [most abundant] process as A fav and the left-hand suppressed process as, A sup = A fav r B e iδ B e iγ CP A sup = A fav r B e iδ B e +iγ. The symbol r B is the relative magnitude, r B = A sup and δ A f av B is a CP-conserving strong phase difference between the two diagrams. The manner in which the complex amplitudes sum is shown in the cartoon for γ = 0 as well as a [large] finite value. By definition in this derivation, A fav = A fav so the partial rate equations are, ΓB [K + K ] D K )= A fav + A sup = A fav [ 1 + r B + r B cosδ B γ) ] ΓB + [K + K ] D K + )= A fav + A sup = A fav [ 1 + r B + r B cosδ B + γ) ]. As charged B mesons cannot mix, this is an example of direct CP violation where the difference between matter and antimatter originates in the decay amplitude only. The following data LHCb 011) exemplifies the large, O10 1 ), CPviolation effects in B-decays. One remembers that the direct CP-violation seen with kaons is small, Rɛ ) O10 6 ). 1

22 5 D-mixing The neutral D meson, at 1864 MeV/c is much heavier than the kaons and pions that it predominantly decays into 490 and 140 MeV/c respectively) which allows a wide range of different decay modes. This large phase-space is available to both mass eigenstates and so, unlike with kaons, the difference in the lifetime is small and we default back to labelling them Light/Heavy mass eigenstates described by a orthogonal linear combination of the weak eigenstates, D L = p D 0 + q D 0 D H = p D 0 q D 0, And following the kaon formalism above, the time evolution is described by, Dt) = g + D 0 + q p g D 0 Dt) = g + D 0 + p q g D 0 where the time-dependent factors are again, g ± t) = 1 [ e i M L t Γ L t/ ± e i M Ht Γ H t/ ]. As the phase space is large and charm-to-strange transitions are Cabibbo-favoured as opposed to strange-to-up transitions in kaon decays), the lifetimes of the two D mass eigenstates are several orders or magnitude smaller than that of the kaons s rather than s for the K 0 L). This is bad for observing mixing because essentially all D-mesons will decay before having a chance to oscillate 5. The equivalent of Fig. 3 for D mesons would show both red and blue curves dropping to zero immediately in a few picoseconds) long before one oscillation period. The shortness and similarity of the lifetimes means that both the mass difference M and lifetime difference Γ need simultaneous consideration. This makes both the algebra and the measurement more complicated. It is useful to define, M M H + M L, Γ Γ H + Γ L and rewrite the time-dependent factors, 1 g + t) = e imt Γt/ cos Mt i ) 4 Γt 1 g t) = e imt Γt/ i sin Mt i ) 4 Γt, M M H M L, Γ Γ H Γ L, x M Γ x = e imt Γt/ cos Γt iy ) Γt, y Γ Γ 34) x = e imt Γt/ i sin Γt iy ) Γt. 35) The kaon oscillation measurement is more straightforward because the the kaon flavour is tagged at both production and decay. Here, a beam of D 0 mesons is not feasible but large numbers of D ± mesons are produced in collider experiments. The flavour of neutral D mesons charm or anticharm) can be known by inspecting the charge on the pion from D ± decays: D + D 0 π + or D D 0 π. The shortness of the D 0 lifetime compared to the oscillation period means the D 0 equivalent of Fig. 3 cannot be used. Instead one uses an additional interference: that between a suppressed decay and mixing. The transition c s + W + is preferred over c d + W +. This means a K + π decay can come from a D 0, but is orders of magnitude more likely to have come from a D 0 favoured: c s + W ), K + π D 0 K + π D 0 r De iδ D, r D Thus, if the mixing effect is occurs rarely, say in 1 / 500 times, one has two paths or similar magnitude, going to an identical final state, see Fig 8. This is exactly the criteria for a system to exhibit interference and we next show this new interference

23 A r D e iδ K + π W + s u K + W ū d π D 0oscillation D 0 A 1 D 0 c ū d ū π c s D 0 K + u u Figure 8: Illustration of the two ways in which a D 0 may transition of a K + π final state: 1) directly with a suppressed transition or ) via an oscillation to a D 0 and a favoured transition. Inspection of the Feynman diagrams reveal the reason for the suppression: one diagram have small CKM vertex factors, V cd V us whilst the other s are of-order unity, V csv ud. is a larger effect than looking for the mixing directly with semileptonic decays. As the final state is accessible to both D 0 and D 0, we must consider the time-dependent rate of finding an initially-tagged D 0 in the suppressed final state K + π by either route. ΓD 0 K + π ) = K + π Dt) where we reasonably neglect CP violation, q/p = 1. = g+ t) K + π D 0 + g t) K + π D 0 = g+ t) r D e iδ D + g t) Plugging Eq. 34 and 35 into Eq. 36 and expanding, [ ΓD 0 K + π ) = x e imt Γt/ cos Γt iy ) Γt r D e iδ D x + i sin Γt iy )] Γt [ x = e Γt rd cos Γt iy ) x Γt + sin Γt iy )] Γt + e Γt [ ir D [cos x Γt + iy Γt ) sin x Γt iy Γt ) e iδ D cos Use of standard identities, remembering cosiθ) = cosh θ and siniθ) = i sinh θ gives, x Γt iy ) x Γt sin Γt + iy ) ]] Γt e iδ D [ 1 ΓD 0 K + π ) = e Γt r D coshyγt) + cosxγt)) + 1 coshyγt) cosxγt)) + r D cos δd sinhyγt) sin δ D sinxγt) )] e [r Γt D 1 + y x ) Γt) ) y + x ] + r 4 D y cos δd x sin δ D Γt + Γt) 4 Where the last step uses a small-angle approximation for the trigonometric and hyperbolic functions: cos x 1 x x ; cosh x 1 + ; sinh x sin x x. The derivation for the favoured events initially tagged D 0, for which a tiny fraction also oscillate before decay, is almost identical except one must take the complex conjugate of the decay amplitude which propagates to one sign-flip in the interference term. However in this case, the interference term is orders of magnitude smaller than the direct rate which is no longer moderated by rd) and can thus be safely neglected. [ ΓD 0 K + π ) e Γt 1 + y x ) Γt) ) + r 4 D y cos δd + x sin δ D Γt + r y + x ] D Γt) 4 e Γt 5 This statement assumes the oscillation rate is similar to that of kaons due to the mixing box-diagram being dominated by the first two generations of quarks which are somewhat similar in mass and couplings. 36) 3

24 Taking the the ratio of rates, remembering the total width is just the inverse lifetime, Γ = 1/τ and further neglecting the quadruply-suppressed rdx + y )/4 term one arrives at a quadratic in multiples of the D 0 lifetime, R = D0 K + π D 0 K + π ) t ) r D + r D y cos δd x sin δ D + y + x t ). 37) τ 4 τ This function is fitted to the data in Fig. 9. Figure 9: a) Histogram of right-sign D [K + π ] D π events. b) Histogram of wrong-sign D + [K + π ] D π + events. c) The ratio of wrong-sign-to-right-sign events in bins of D 0 lifetime with a best-fit curve of Eq. 37. One sees the dominant feature of the data is the approximately linear rise in number of wrong-sign D + [K + π ] D π + events as a function of time. The slight curving of the fit is the quadratic term - to which the semileptonic decays have sensitivity - which we see yields comparatively little information. 6 This justifies the use of D Kπ decays rather than semileptonic decays as done with kaons) for which the linear term, proportional to x&y, is not present. From the fit values of the mixing parameters and the phase can be extracted; the world average of such measurements is, x = ± y = ± , and δ D = 1 ± 13) which are small quantities. For the purpose of comparison to kaon oscillations, one can calculate the mass-difference, M = M H M L for a D 0 lifetime of 0.41 ps, M [ev] = x Γ = x τ = [ps 1 ] = x τ e = [ev]. Which is less than a factor away from the same quantity in kaons, as might be expected from the CKM matrix, and is again tiny compared to the D 0 mass, 1.86 GeV. Concluding comparison The exceptionally long K 0 L lifetime allows oscillations to be seen rather easily but the short lifetime of the D meson this is a significant challenge. Neutral B mesons also have short lifetimes but their mixing rate is much higher so mixing effects are readily seen. Small CP violation effects O ) are seen in kaons. In B mesons CP violation produces large, O0.1 1), effects in certain decays, in accordance with the prediction of the CKM mechanism. Charm CP violation is as-yet unobserved. 6 to further persuade that semileptonic decays can only access the quadratic term, look again at Eq. 9 and notice the mixing dependence on cos Mt which for small Mt = xt/γ in the case of D mesons, is 1 + x Γt)... 4

25 In 1914, Chadwick 7 showed that the energy spectrum of the electron in nuclear β decay was continuous. It had been expected the emitted electron be monochromatic, and equal to the mass-energy lost by the nucleus, as had been observed in α and γ decay. A number of confirmations followed in the 190s. Twenty years later, in 1934, Fermi published his theory of β decay describing the contact interaction of four fermions. He imagined a vector current, like electromagnetism, but with weaker strength. His conclusion from the data was that the neutrino mass was small, Two decades later, Cowan&Reines 1956) positively detected the neutrino by observing the coincidence of detection of a neutron and a positron from ν + p n + e + in a water tank placed next to a nuclear reactor. A short while after, Lederman/Schwartz/Steinberger 196) identified a second flavour of neutrino associated with muon production. Direct confirmation of a third type of neutrino waited until 000 when the DONUT experiment Fermilab) found a neutrino associated with τ lepton production. The ν τ source was D + s τ + ν τ decays; the charm mesons were produced from TeVatron protons on a fixed target. 6 Direct neutrino detection Neutrinos only feel the weak force so their detection can only proceed by the exchange of the massive weak-force boson, W ± or Z 0. W ± exchange precipitates a change of neutrino and lepton flavour. A Z 0 exchange leaves the fermion s flavour unchanged. Neutrino flavour is inferred by the production of their associated lepton in the charged current process. An electron neutrino will interact via the charged weak interaction with the electron only, and so forth. For an interaction to occur, the CoM energy must be sufficient to produce the lepton. Consider three cases: 7 Chadwick is most famous for detecting the the neutron by identifying 193) a highly penetrating neutral emission when beryllium is bombarded with α particles. The high mass of this particle precluded it being the neutral particle associated with β decay. 5

26 Scattering off an atomic electron ES): ν l + e ν e + l s = P ν + P e ) > m l m ν + P ν P e + m e > m l m ν = 0, P ν = E ν, P e = m e as p e = 0 initially at rest. E ν > m l m e m e 38) which requires energies greater than zero, 11GeV and 3.1TeV for e, µ and τ production respectively. This means that in practice, scattering off atomic electrons in neutrino experiments is sensitive to electron neutrinos only. Charged current scattering off a nucleon CC), where the nucleon changes flavour, ν l + n l + p: s = P ν + m n ) > m l + m p ) 39) m ν + E ν m N + m N > m l + m l m N + m N m n m p = m N E ν m l m N + m l 40) which give the production thresholds for electrons, muons and taus as: 511keV, 11MeV and 3.45GeV respectively. The muon threshold is still high compared to nuclear processes though reasonable for cosmic ray neutrinos. The electron threshold may differ from the above calculation due to the binding energy of specific target material. Neutral current scattering off a nucleon NC), where the nucleon does not change its flavour, ν l + N ν l + N. This process has no energy threshold and is equally sensitive to all neutrino flavours. However the absence of the lepton signature means this process is experimentally challenging to identify. 6.1 Neutrino masses A direct measure of the electron neutrino mass comes from the electron energy spectrum in beta decay. The maximum electron energy is equal to the change in mass the transmuting nuclei, E 0, minus the [at-rest] neutrino mass. It is possible 6

27 to form the Kurie variable which is proportional to the electron energy in the absence of neutrino mass, KE) [ E 0 E) [ E 0 E) mν] 1 ] 1 not derived) Deviations from linearity at the end point the Kurie spectrum see Fig. 10) would be evidence for neutrino mass. The distributions seems linear and an upper limit on the ν e mass is < ev. Upper limits on muon and tau neutrinos come from Figure 10: Kurie plot for Rhenium 187 Re. particle decay: mµ µ ) < 170keV comes from a momentum analysis of stopped-pion decay; a kinematic analysis of a few thousand τ πππν τ decays from e + e τ + τ LEP) yields the limit on mµ τ ) < 19MeV. Limits on the neutrino mass are also inferred from astronomy and cosmology. By comparing the arrival time of neutrinos from supernovae to that of light, an upper limit on the neutrino mass can be made. The neutrino velocity is, β = p E 1 E m ν = 1 m ν E E = 1 m ν E +... where the energy distribution is estimated from models of stellar collapse. SN1987N is 50kpc = 163,000 lightyears from earth and no significant difference in arrive time is measured. An upper limit of m νe < 11eV is inferred. The most stringent limits on the sum of all three neutrino masses come from cosmological models of the power spectrum of the cosmic microwave background: i=1,,3 m i < 0.3eV is claimed Planck 015). 7 Solar Neutrinos The Sun s core produces ν e s 1 radiating out in 4π. 91% of these come from proton-proton fusion, p + p 1D + e + ν e, with neutrino energy up to 0.4 MeV. These low-energy neutrinos are difficult to detect and so most experiments exploit the neutrino flux from subdominant processes. The next-most abundant process 9%) is electron capture by beryllium 7 4Be + e 7 3Li + ν e that releases a monochromatic 0.9 MeV neutrino. The largest source of high-energy solar neutrinos is the Boron-8 process by which 0.01% of the total neutrino flux are in the 1-15 MeV range. The energies of 7 4Be neutrinos are above threshold for Chlorine capture E th > 0.81 MeV). This fact is exploited by the Homestake mine neutrino experiment ). For this, chlorine atoms were placed 1478m underground in the form of 613 ton of dry-cleaning fluid C Cl 4 ). The principal experimental challenge is filtering out the few radioactive atoms of 37 Ar solar neutrino interaction rate 0.5 day 1 ). The first publication of a discrepancy between the measured neutrino flux R. Davis Jr.) and a calculation using the emerging Standard Solar Model J. Bahcall) occurred in 1968 and 7

28 Detection by radio-chemical capture: 7 νe Ga 3 Ge + e 37 νe Cl 18 Ar + e Final state isotopes must be separated out and counted. Sensitive to rate only. Or by Cherenkov detection in water: νe + e e + νe Useful for Eν > 5 MeV. Measures the direction of incoming neutrino. reported the observed flux to be.5 times lower than expectation. Following this initial work, Davis Jr. built, improved and maintained the Homestake mine experiment until The Standard Solar Model also developed over this time and in by the late 1990s a strong statement was made. R37Cl) =.56 ± 0.3 interactions per 1036 chlorine atoms per second. which is around one third that expected according to the Standard Solar Model SSM). From , the GALLEX experiment Gran Sasso, Italy) performed a similar experiment with gallium- rather than chlorine-capture, with the important change of a lower threshold of 0.33 MeV giving sensitivity to the higher flux of pp fusion neutrinos. GALLEX reported a flux 55% that expected from the SSM. Further evidence of a diminished neutrino flux came from the Kamiokande experiment in Japan which, from 1987 until 1995, collected data from ES neutrinos by detecting the Cherenkov light from the recoiling electron in 1.5kton of water. As the electron must recoil at relativistic speeds to generate Cherenkov, and to keep backgrounds from geo-radioactivity, the threshold is set higher than the calculation of Eq. 38. Over the lifetime of the experiment, techniques improved and the threshold was varied from 10 to 7 MeV. The cross section for electrons from ES neutrino interactions is, Z 1 1 σνe νe = M f i dω 64π me + Eν 1 cos θ)) which is heavily peaked towards the the direction of the incoming neutrino θ 0). The electron direction in ES interaction thus points back to the neutrino source. The flux of neutrinos in this energy region is again seen to be around a quarter that expected for the 8 B process in the SSM, though with larger error. The annual flux varied however, by about 7% from summer to winter just as expected from the known eccentricity of the Earth s orbit. Thus by the late 1990s, there was consistent evidence of a electron-neutrino deficit from the Sun, though it was not known if it was due to mis-modelling of the Sun or due to neutrino oscillations. 7.1 SNO The Sudbury Neutrino Observatory, located km underground in Ontario, Canada was built to measure the flux of all neutrino flavours and thus measure the total neutrino flux from the Sun. It held 1000 tonnes of heavy water, D O which 8

29 make a neutron-enriched target for performing neutrino astronomy. The fiducial spherical volume was contained within an outer water jacket which served as a shield, and veto counter for geo-radioactive backgrounds. SNO detect the flux of neutrinos from the Sun by all three reactions discussed above: Charged current: reactions are detected by a cone of Cherenkov light from the fast-recoiling electron. This component is sensitive to the the electron flux only as it requires the production of a lepton and only the electron is energetically accessible. Expect 30 CC reactions per day. ν e + D p + p + e charged current rate φν e ) Neutral current: reaction causes the neutron to fly off and is eventually captured by another deuterium nuclei with an associated cascade of gamma-rays, which in turn scatter electrons to give a haze of Cherenkov light. Expect 30 NC per day though the neutron capture by deuterium is only 30% efficient. This component is sensitive to all neutrino species equally. ν x + D p + n + ν x neutral current rate φν e ) + φν µ ) + φν τ ) Elastic scatter: off atomic electrons can occur via both W ± and Z 0 exchange. This scattering has enhanced sensitivity to the election flux because only the CC is sensitive only to the ν e flux whereas the NC process sees all three flavours. The neutrino cross section for electron CC scattering is a factor ten smaller than that of the nucleon CC reactions and the out-going electron is strongly peaked in the forward direction low scattering angle). elastic scattering rate φν e ) φνµ ) + φν τ ) ) In the analysis, these three components are distinguished by their visible energy, their direction of any produced light with respect to the Sun and distance from the edge of the reservoir distinguishes geo-radioactive backgrounds). Using calculable interaction cross sections and efficiencies the flux were found: φ CC = 1.76 ± 0.1) 10 6 cm s 1 φ NC = 5.09 ± 0.63) 10 6 cm s 1 φ ES =.39 ± 0.7) 10 6 cm s 1. The neutral current result compares well to the expectation from the SSM, 5.05 ± 0.95) 10 6 cm s 1. Is is trivial to deduce the individual components and so confirm the Homestake result: neutrinos change flavour en-route from the Sun. φ νe = 1.76 ± 0.1) 10 6 cm s 1 φ µ + φ τ = 3.41 ± 0.64) 10 6 cm s 1. 9

30 7. Two particle mixing With neutral mesons we saw that flavour oscillation occur if the eigenstates of the Hamiltonian the mass eigenstates that propagate in space are not the [weak] interaction basis. Here we apply a similar idea to a two-neutrino system, though unlike the meson formalism, here we work in the laboratory frame. The weak-force interaction eigenstates are labelled: ν e and ν µ and the mass eigenstates: ν 1 and ν. At the point of production in the Sun t = 0) the neutrino wavefunction is of a definite flavour, but can be written as a linear combination in the mass basis, ψ e 0) = ν e = cos θ ν 1 + sin θ ν. The mass eigenstates propagate in time as plane waves, The neutrino travels close to c in direction x. As m E, ν 1 t) = e ip 1 x E 1t) ν 1 ν t) = e ip x E t) ν p = In natural p x Et can be written x p E) i.e. c = 1), so: e ip 1 x E 1t) E m E 1 1 m +... ) = E m E E e iφ 1 where φ 1 = x E m 1 E E 1 = m 1L E. Similarly φ = m L E, for some point of detection at distance x = L, and where the tiny energy difference is indistinguishable E 1 = E = E). The two-neutrino Hamiltonian for the mass-eigenstate dispersion is, i d ) ) ) ν1 χ1 0 ν1 =, φ dt ν 0 χ ν i = χ i t = χ i L forc = 1). Use the similarity transform, H = UHU 1, to write it in the interaction basis where U is the two-particle mixing matrix. i d ) ) ) ) ) νe cos θ sin θ χ1 0 cos θ sin θ νe = dt ν µ sin θ cos θ 0 χ sin θ cos θ ν µ χ +χ 1 χ χ 1 χ cos θ χ 1 sin θ = χ χ 1 sin θ χ +χ 1 + χ χ 1 cos θ ) ) νe. ν µ Hence the amplitude of detecting either a muon neutrino, or electron neutrino at distance L is, ν µ ψ e L) = 1 sin θ e iφ e iφ 1 ) ν e ψ e L) = 1 e iφ + e iφ 1 ) 1 cos θ e iφ e iφ 1 ) 41) which we see that there is no oscillation if the neutrino masses are the same: φ 1 = φ. Eq. 41 can be derived equally well from rotating ψ e L) into the interaction basis, In the interaction basis, ψ e L) = cos θ e iφ 1 ν 1 + sin θ e iφ ν. ψ e L) = cos θ cos θ ν e sin θ ν µ ) e iφ 1 + sin θ sin θ ν e + cos θ ν µ ) e iφ = cos θ e iφ 1 + sin θ e iφ ) ν e + sin θ cos θ e iφ e iφ 1 ) ν µ. 30

31 so the probability of detecting either a muon neutrino, or electron neutrino at distance L, ν µ ψ e L) = sin θ cos θ e iφ e iφ 1 ) ν e ψ e L) = cos θ e iφ 1 + sin θ e iφ = 1 sin θ e iφ e iφ 1 ) = cos θ) e iφ cos θ) e iφ A little more trigonometry give the oscillation probability, Pν e ν µ ) = ν µ ψl) = 1 4 sin θ e iφ e iφ 1 where m 1 = m 1 m. Also: cosx) = cos x 1 = 1 sin x. = 1 e iφ + e iφ 1 ) 1 cos θ e iφ e iφ 1 ) ) = 1 e iφ 1 iφ e iφ iφ = ) e iφ 1 + e iφ 1 φ 1 = φ 1 φ = cos φ 1 = sin 1 φ )) 1 = sin θ sin 1 m 1L E m L E = sin θ sin m 1 L 4) 4 E A similar piece of trigonometry give the no-oscillation probability, Pν e ν e ) = ν e ψl) = cos θ + sin θ cos m 1 L or 1 sin θ sin m 1 4 E 4 L[ ev 1 ]. E[ ev] Last step is to note that all this is derived in natural units so need to transform to practical units of GeV and kilometres, E: [ev] [GeV] is simple factor of L: [ev 1 ] [km] is converted by remembering E = ck. [J] = c[m 1 ] e[ ev] = c[m 1 ] [ ev 1 ] = e c [m] Hence work out the conversion factor using c = 197 MeVfm: c e = , 1 L[eV 1 ] = 1 e L[m] 4 E[eV] 4 c E[eV] = 1 1 L[m] E[eV] = L[km] E[GeV] = 1000 L[km] }{{} 788 E[GeV] 1.7 Thus the widely-used survival probability in the two-flavour model is, from Eq. 4, Pν e ν e ) = 1 sin θ sin 1.7 m 1[eV ] L[km] ) E[GeV] 43) 31

32 For a monochromatic neutrino, a regular oscillation with distance is expected, see Fig. 11. For a fixed distance, the resolvability of the oscillations depends on the energy resolution of the detector. Due the large distance to the Sun, these plots demonstrate that solar neutrino oscillation could be sensitive to neutrino mass-difference ranging down to ev if the energy resolution of the detector is good enough. -10 Example: m1 = ev, E=1MeV Pν e ν e ) L [km] 6 10 Figure 11: Neutrino survival probability plotted against distance left) and neutrino energy right) imagining m 1 = ev. The amplitude of the oscillation is defined by the mixing angle which is assumed maximal 45 ) in these plots. It can be useful to know the distance to the first minimum at π/ for a sin function), for a given neutrino energy range, E [GeV] L π [km] = 1.4 m [ev ]. 7.3 MSW-effect Though each have different energy ranges, the experiments discussed so far Homestake, GALLEX, Kamiokande, SNO) have poor, or no, intrinsic energy resolution. They count the average rate with little sensitivity to the periodicity of the L/E term of Eq. 43. If the oscillations are not resolved, the average value of the sin term is 1 so the survival probability should be P e = 1 1 sin θ, which lies in the range 0.5 < Pν e ν e ) < 1. However, except for the gallium experiments, all the experimental data indicates that the survival probability for electron neutrinos is closer to a third. This discrepancy arises because the standard two-neutrino derivation is for transmission through vacuum. MSW L. Wolfenstein 1978) and S. P. Mikheyev, A. Yu Smirnov 1986)) predicted an electron neutrino perturbation by charge current weak interactions in the dense solar matter. An MSW potential is introduced into the Hamiltonian for electron neutrinos only, V = G F N e r), where G F is the Fermi coupling constant and N e is the number density of electrons which varies as a function of radius from the Sun s core. i d dt ) νe ν µ = χ +χ 1 χ χ 1 cos θ + G F N e r) χ χ 1 sin θ χ χ 1 sin θ χ +χ 1 + χ χ 1 cos θ ) ) νe. ν µ Conceptually, the additional scattering can be thought of raise the potential energy of the electron neutrino eigenstate such that he heavier mass eigenstate not only rises in mass but takes a greater proportion of that weak eigenstate. Mathematically the MSW effect modifies effective mixing from the intrinsic neutrino mixing in vacuum. To calculate the matter-mixing angle, θ m the Hamiltonian is rediagonalised, sin θ tan θ m r) = cos θ. GF N e r) χ χ 1 3

33 Diagonalisation of a matrix Take a symmetric n n matrix A. There exists a diagonal matrix D and orthonormal matrices U that are related by the similarity transform, A = UDU 1 or equivalently, AU = UD. A has n orthonormal eigenvectors v i and n eigenvalues λ i related by, Av i = v i λ i. By comparing the matrix multiplication in these two equations, one sees that U is a matrix where the columns are the eigenvectors v i and the diagonal elements of D are the eigenvalues, λ i. Continuing for the case n = where the orthonormality of U means it is a rotation matrix, parameterised by an angle x, ) ) ) ) A11 A 1 cos x sin x λ1 0 cos x sin x = A 1 A sin x cos x 0 λ sin x cos x 1 = λ + λ 1 ) 1 λ 1 λ 1 ) cos x λ ) λ 1 ) sin x 1 λ λ 1 ) sin x 1 λ + λ 1 ) + 1 λ λ 1 ) cos x The rotation angle x that is required to diagonalise A align with the eigenvectors) is thus, tan x = A 1 A 11 A Three regimes are considered: Least dense outer regions of the Sun: N e r) 0, tan θ m tan θ, θ m θ. Intrinsic mixing, no matter effect. MSW resonant density: G F N e r) = χ χ 1 ) cos θ. This gives tan θ m, θ m π/4. Maximal mixing. The dense core of the Sun: N e r), tan θ m 0, θ m π/. Mixing suppressed by matter effects. The MSW resonant density depends on the neutrino energy. The threshold for water cherenkov experiments SNO and Kamiokande) is 7 MeV. For neutrinos of this energy, and assuming θ = 33, the MSW resonant density is, N e r) = χ χ 1 ) cos θ GF = m cos θ E G F [ ev ] cos θ = [ ev] [ ev ] = ev 3 = m 3, divide by ) 3 which is the electron number density, assuming a neutral hydrogen plasma, for ρ = 30.5 gcm 3. From solar models, the core density is 160 gcm 3, far above the MSW resonant density. As the density of matter is much above the MSW resonant density then electron neutrinos produced in the fusion process) 33

34 will be mostly aligned with the heavier mass eigenstate, ) ν1 cos θ = m sin θ m sin θ m cos θ m ν ) νe ν µ ) ) νe ν µ ) as θ m π. Thus is is predominantly the heavier mass eigenstate, ν that is emitted from the Sun and propagates to Earth where is its detected as an electron neutrino with probability P e = sin θ from ν = sin θ ν e + cos θ ν µ. This is always lower than the energy-averaged survival probability when matter effects are negligible, P e = 1 1 sin θ. If the number density calculation is redone in the context of the GALLEX experiment, which has a lower neutrino energy threshold by a factor 0-30, sees that the density of the Sun s core is not large enough to suppress mixing and the MSW effect is less important. GALLEX measures P e = 0.55 in accordance with this ansatz. Understanding of the MSW effect identifies the hierarchy of the two mass eigenstates: ν is heavier. ν 1 is the lighter mass eigenstate that is normally most aligned with the ν e flavour eigenstate. 7.4 Reactor experiments: KamLand Upon inspection of Eq. 43 we notice that the probability depends on the ratio of L/E on the experiment, not their absolute values. Therefore one can probe the same phenomena at different places in the oscillation probability if one chooses the L and E correctly. The case in point is that the maximal oscillation criteria for m 10 4 can be accessed by using MeV neutrinos from a reactor over a distanct of O10 ) km. In 00 Kamland reactor experiment has made such a complementary, and precise measurement of m 1 at much shorter distance, free on the MSW effect that complicates solar-based measurements. This give m ev and sin θ Note that, in contrast to Cabibbo mixing λ = sin θ C = 0.) = 0.05), the first two generation neutrino mixing angle is close to maximal. 8 Atmospheric neutrinos A large neutrino flux in incident on Earth from the showering caused by cosmic rays incident on the upper atmosphere. These hadronic showers contain many pions which decay by π + µ + ν µ, with µ + e + ν e ν µ so expect the ratio or fluxes, φν µ + ν µ ) φν e + ν e ) =. Experiments in the 1990s, including Kamiokande, reported evidence that this ratio was less than expected. Could this atmospheric neutrino anomaly be caused by the same mechanism as the solar neutrino problem? No. The solar neutrino oscillations have m A deficit of ν µ could not be due to an oscillation to ν e because of the distances 34

35 involved. The typical energy of atmospheric neutrinos is O1 10 GeV). The two-neutrino mixing, ) Pν µ ν x ) sin 1.7 m L[km], E[GeV] shows that ν µ would need km to oscillate to ν e with the m Thus the oscillation must be predominantly to a third mass eigenstate, ν 3 perhaps dominated by ν τ flavour), driven by m Super-K To investigate the atmospheric regime, Superkamiokande, the 50 kiloton water Cherenkov detector built under Mt Ikenoyama in Japan was built. It is equiped with It observes neutrinos from charged-current interactions that produce a relativistic lepton that, in turn, produces Cherenkov light. The typical energies 1 GeV) are well above muon threshold from the charged-current interaction so two lepton species are observable. Electron and muons are distinguished by the shape of the Cherenkov ring, elections scatter more easily so the Cherenkov ring is fuzzier. Importantly, the incoming neutrinos are of large energy and the kinematics of the scattering off the fixed nuclear matter target the water) means the direction of the incoming neutrino and the scattered lepton are highly correlated. This allows the analysis to know the distance that he neutrino has passed through since creation: between 50km zenith angle = 0 ) and 1400km zenith angle = 180 ). Figure 1: Superkamiokande data. The data shows the election flux to be as expected whereas the muonic flux exhibits a clear dependence on zenith angle: where the neutrino has travelled farthest, there are fewer than expected. This implies ν µ disappearance, ν µ ν τ. Analysis of the data reveals, m ev and sin θ The m 3 m 3) is larger by a factor 6 37) than the m 1 m 1) which precipitates solar oscillations, though the mixing angle is similarly close to maximal. 35

36 8. MINOS: accelerator-based study of atmospheric oscillations Just as the KamLand experiment chose a suitable L and E for solar oscillations, the MINOS experiment was designed to target maximal atmospheric oscillations. With m we need hundreds of kilometres per GeV; GeV neutrinos from an accelerator benefits from low natural background. The distances involved and the difficulty of perfectly collimating a neutrino source means a high intensity initial beam of decaying pions is needed, π + µ + ν µ. Fermilab points the focussed pion beam towards MINOS, which first passes though a near hall detector which measures the initial muon neutrino spectrum, then a larger, but otherwise identical detector is in the far hall, 735km away in Minnesota. At the far detector, the beam is spread over kilometres so a large detector volume is needed. The MINOS experiment uses cheap scintillator layers interspersed with magnitised iron plates. The scintillator measures the energy of the recoiling nucleon whereas the bending of the resultant muon in the magnetised material measures its momentum. The energy spectra are shown below for both neutrinos and antineutrinos and are compared with the expected flux, assuming no oscillation i.e. just from solid angle). The data confirms the atmospheric disappearance. Note that the charge of the pion beam is reversed and antineutrinos can be studied. This is first step towards probing CP violation in neutrinos. Figure 13: Minos data. 8.3 Double-Chooz: probing the third generation With two distinct oscillation phenomena, visible a different L/E, one understands that there must be a neutrino mass hierarchy with the a third generation neutrino somewhat difference in mass compared to the first two: m 1 is small, m 3 is large, thus m 13 is similarly large. Because this involves squared quantities, it is not known if the m 3 > m 1, m or m 3 < m 1, m, these scenarios are known as the normal and inverted hierarchies. As m 13 m 3, one can postulate that it should be possible to detect mixing between electron and tau neutrinos at atmospheric L/E. The Double-Chooz experiment in France placed a 5 ton fiducial target of Gadolinium-loaded scintillator next to a reactor of known flux. The distance is 1km and the typical energies, 3MeV. This energy is far below the kinematics threshold for tau or muon) production so this is a electron neutrino disappearance [to ν τ ] experiment. In the first results 011), no evidence of neutrino disappearance is seen indicating a low value of the mixing between the first and the third generation, sin θ 13 < 0.1. At this stage, the full 3 generation mixing needs consideration. 9 Three generations: the PMNS matrix Inspired by the CKM mechanism we write the Pontecorvo-Maka-Nakagawa-Sakata PMNS) matrix as a starting point to 36

37 describing mixing between three neutrino species, with inverse U 1 = U = U ) T : ν e ν µ ν τ ν 1 ν ν 3 U e1 U e U e3 = U µ1 U µ U µ3 U τ1 U τ U τ3 Ue1 Uµ1 U τ1 = Ue Uµ Uτ Ue3 Uµ3 Uτ3 We know from CKM that a three-generation mixing implies three mixing angles and the presence of an irreducible complex phase that can give rise to CP violation cos θ 13 0 sin θ 13 e iδ cos θ 1 sin θ 1 0 U = 0 cos θ 3 sin θ sin θ 1 cos θ sin θ 3 cos θ 3 sin θ 13 e iδ 0 cos θ From this decomposition we can see how the regimes factorise: the first rotation describes the rotation between the nd and 3 rd generations whilst preserving the first. This dominates ν µ disappearance in Atmospheric neutrino physics. the third rotation describes the rotation between the 1 st and nd generations. This dominates ν e disappearance in traditional Solar neutrino physics. θ 13 is the smallest and least known rotation angle. ν 1 ν ν 3 ν e ν µ ν τ. The three rotation angles, the CP violating phase parameter, plus two of the three mass differences the 3rd is derivable) are the six fundamental parameters that can be measured in neutrino oscillation experiments. Currently, much interest concentrates on θ 13 because it must be non-zero to access neutrino CP-violation. The complete PMNS matrix looks like, cos θ 1 cos θ 13 sin θ 1 cos θ 13 sin θ 13 e iδ U = sin θ 1 cos θ 3 cos θ 1 sin θ 3 sin θ 13 e iδ cos θ 1 cos θ 3 sin θ 1 sin θ 3 sin θ 13 e iδ sin θ 3 cos θ 13 sin θ 1 cos θ 3 cos θ 1 cos θ 3 sin θ 13 e iδ cos θ 1 sin θ 3 sin θ 1 cos θ 3 sin θ 13 e iδ cos θ 3 cos θ 13 The magnitudes of the PMNS matrix elements can be numerically summarised: U e1 U e U e U = U µ1 U µ U µ U τ1 U τ U τ With the exception of U e3 all the elements are O)1). An interesting contrast to the approximately-diagonal CKM matrix. 9.1 Oscillation probabilities with three neutrinos To see how a measurement of θ 13 may be made we extend the two-neutrino oscillation probabilities for the PMNS matrix. Consider the production of an electron neutrino at t = 0 in the weak interaction) eigenstate, ψ0) = ν e = U e1 ν 1 + U e ν + U e3 ν 3. 37

38 The evolution over time and space is: ψt) = U e1 ν 1 e ip 1 x E 1 t) + U e ν e ip x E t) + U e3 ν 3 e ip 3 x E 3 t) with which we make the same observation/simplification as in the two-neutrino derivation, ψl) = U e1 ν 1 e iφ 1 + U e ν e iφ + U e3 ν 3 e iφ 3 where φ i = m i L E and use the inverse PMNS matrix to substitute in the definitions of the mass eigenstates in terms of the flavour eigenstates. ψl) = U e1 U e1e iφ 1 + U e U ee iφ + U e3 U e3e iφ 3 ) ν e U e1 U µ1e iφ 1 + U e U µe iφ + U e3 U µ3e iφ 3 ) ν µ U e1 U τ1e iφ 1 + U e U τe iφ + U e3 U τ3e iφ 3 ) ν τ Let s develop the probability of electron survival, ν e ν e, From the first line we have, and with the complex number identity, one can write, Pν e ν e ) = Ue1 U e1e iφ 1 + U e U ee iφ + U e3 U e3e iφ 3 z 1 + z + z 3 = z 1 + z + z 3 + Rz 1 z + z 1 z 3 + z z 3) 44) Pν e ν e ) = U e1 U e1 + U e U e + U e3 U e3 + R [ U e1 U e1u eu e e iφ φ 1 ) + U e1 U e1u e3u e3 e iφ 3 φ 1 ) + U e U eu e3u e3 e iφ 3 φ ) ] 45) Just as with the CKM matrix, the PMNS must be unitary. This requires that the dot-product of rows or columns) i and j is δ i j. Thus, a relationship between PMNS matrix elements can be formed, Using the Eq. 44 again, we can relate the PMNS elements: U e1 U e1 + U e U e + U e3 U e3 =1. U e1 U e1 + U e U e + U e3 U e3 =1 U e1 U e1 + U e U e + U e3 U e3 =1 R U e1 U e1u eu e + U e1 U e1u e3u e3 + U e U eu e3u e3 ) 46) The right hand side of Eq. 46 replaced the first three terms of Eq. 45. Unifying, one gets, Pν e ν e ) = 1 + R [ U e1 U e1u eu e e iφ 1 φ ) 1) ] + R [ U e1 U e1u e3u e3 e iφ 1 φ 3 ) 1) ] 47) + R [ U e U eu e3u e3 e iφ φ 3 ) 1) ] = 1 + U e1 U e R [ e iφ 1 φ ) 1 ] + U e1 U e3 R [ e iφ 1 φ 3 ) 1 ] 48) + U e U e3 R [ e iφ φ 3 ) 1 ] Note that the survival probability caries no sensitivity to the complex phase of the PMNS matrix and is not sensitivity to CP violation as expected, Pν e ν e ) Pν e ν e ). The exponential contains the time dependence, R [ e iφ i φ ) j 1 ] = cosφ i φ j ) 1 ) φi = sin φ j = sin i j where i j = m i m j 4 38 L E natural units) = 1.7 m i j[ev ] L[km] E[GeV]

39 in practical units. So, Pν e ν e ) = 1 4 U e1 U e sin 1 4 U e1 U e3 sin 13 4 U e U e3 sin 3 Which can be simplified a little with reasonable assumptions. Pν e ν e ) 1 4 U e1 U e sin 1 4 U e1 + U e ) U e3 sin 3 mass hierarchy: 1 is small, U e1 U e sin U e3 ) U e3 sin 3 unitarity: U e1 + U e + U e3 = 1 Next we substitute in the mixing angles and do some trigonometry), Pν e ν e ) = 1 4cos θ 1 cos θ 13 ) sin θ 1 cos θ 13 ) sin 1 41 sin θ 13 ) sin θ 13 sin 3 = 1 cos 4 θ 13 sin θ 1 sin 1 sin θ 13 sin 3 Finally, we see that the electron neutrino survival probability has two parts; a dominant part that has large mixing angle but depends on m 1 and so needs long distances to occur. The second part is subdominant and depends on m θ 13 at Daya Bay In 01, the Daya Bay experiment in southern China measured θ 13 by looking at electron neutrino disappearance in the flux of ν e from a set of nuclear power stations. The detection technique is inverse β decay, ν e + p e + + n in a gadolinium-doped liquid scintillator. The coincidence of the prompt scintillation from the e + and the haze of light from the delayed neutron capture on Gd provides a distinctive signature. A near and far detector setup was used. Two sets of detectors were placed 363m and 500m from the reactors and one far detector, located 1615m and 1985m from the two reactors measured the flux at distance. At least two identical detectors to reduced systematic errors) are bathed in a water bath to protect them from geological radioactivity backgrounds. From the formula for the oscillation wavelength we see that the oscillations involving m 11 are insignificant, λ 1 = 30km, λ 3 = 0.8km so the electron survival probability can be simplified and approximated further to, Pν e ν e ) 1 sin θ 13 sin 3 By placing the near and far detectors a little over a km apart, the deficit of ν e in the far hall compared to the near hall, should be near-maximal. The deficit is shown below and the Daya Bay collaboration s measurement is, sin θ 13 = 0.09 ± 0.016stat) ± 0.005syst). or, θ 13 9, assuming a 1 st quadrant solution. Double-Chooz was doing the right thing, just not enough sensitivity. 9.3 ν µ ν e appearance We consider the ν e appearance in an initially ν µ beam from an accelerator, Pν µ ν e ) = ν e ν µ L) = Uµ1 U e1e iφ 1 + U µ U ee iφ + U µ3 U e3e iφ 3 The derivation is the same as far as Eq. 47 though the relevant PMNS unitarity condition is not equal to one, but zero: U µ1 U e1 + U µ U e + U µ3 U e3 = 0. So, Pν µ ν e ) = R [ U µ1 U e1u µu e e iφ 1 φ ) 1 )]. + R [ U µ1 U e1u µ3u e3 e iφ 1 φ 3 ) 1 )] 49) + R [ U µ U eu µ3u e3 e iφ φ 3 ) 1 )]. 39

40 Figure 14: Daya Bay 01 result. The simplification at Eq. 48 does not apply when the creation and detection species differ and sensitivity to the imaginary part of the PMNS matrix arise. The general form of each line is R [ Ue ix 1) ] = 4R [U] sin x I [U] sin x, where we note the oscillation is twice as frequent for the imaginary part. So with C i j = C m i jl[ km] E[ GeV], Eq. 49 is rewritten, Pν µ ν e ) = 4 R [ U µ1 U e1u µu e ] sin I [ U µ1 U e1u µu e ] sin R [ U µ1 U e1u µ3u e3 ] sin I [ U µ1 U e1u µ3u e3 ] sin ) 4 R [ U µ U eu µ3u e3 ] sin I [ U µ U eu µ3u e3 ] sin CP violation in neutrinos As with quarks, the CP violation of neutrinos depends the δ parameter of the mixing metric being non-zero, in other words an imaginary component exists. The size of the CP violation depends upon the transition considered but it is clear that a transition must be studied; survival probabilities are not sensitive to the imaginary part. Let s state what the effect of CP and T transformations are, ν µ ν e CP ν µ ν e ν e ν µ T ν µ ν e ν e ν µ CPT ν µ ν e A signature of Γν µ ν e ) Γ ν µ ν e ) or Γν e ν µ ) Γν µ ν e ) would be CP or equivalently T) violation. Reactor and solar experiments have no sensitivity to the phase δ because their source is ν e only, not ν e. Cosmic rays produce equal proportions of ν µ and ν µ. These will produce, in a CC detection event, a µ and µ + respectively. However it is prohibitively expensive to build a magnet and tracking system the size of a neutrino observatory. Accelerator-based experiments, similar to MINOS, are the future of CP violation searches because they can be a source of both ν µ and ν µ by switching between decays of π and π + decays. 8 Such a search would look for CP violation described by Eq One can also imagine using the µ ± decays though here one needs to instrument the far detector with a magnet and tracker - again, expensive. 40

41 CP violation in the lepton sector would herald interesting new possibilities for explaining leptogenesis, and thus potentially the baryogenesis. In the early Universe before the electroweak symmetry-breaking, it would have been possible to change an excess of lepton number, L, to an excess of baryon number, B, via phase transitions, sphalerons, which are required to maintain B L only. This may be a more fruitful avenue for baryogenesis than the relatively small CP asymmetry effect between quarks. 9.5 θ 13 at TK However, let s neglect CP violation δ = 0, I[U] = 0) and use Pν µ ν e ) appearance to measure θ 13 by looking for electron rings in SuperKamiokande in a ν µ beam produced at J-PARC, 45km away. Developing Eq. 50, neglecting the imaginary parts, approximating 1 = 0 13 = 3 ), and using the PMNS unitarity condition, the electron appearance probability is, U µ1 U e1 + U µ U e + U µ3 U e3 = 0, Pν µ ν e ) sin θ 3 sin θ 13 sin 3 In July 013, TK release a result using an off-axis technique 9 to tune the average ν µ energy to 600 MeV to maximise the oscillation. They report, sin θ 13 = , which is compatible with the Daya Bay result. In 017 a reanalysis of their ν µ and ν µ data combined with information from Daya Bay for θ 13, they map-out the likelihood of non-zero value of δ where the full development of Eq.50 is used. This is shown in Fig The central axis of the beam is contains the most collomated, highest energy neutrinos. The average energy of the beam decreases away from the axis. Pointing the J-PARC beam 10 off axis achieves the optimal L E. 41

42 Figure 15: TK - January 017. The full expression including CP violation is shown. 4