Charge conjugation

Dirac’s hole theory led him to the remarkable prediction of the positron, and suggested a new kind of symmetry: to each charged spin-1/2 particle there must correspond an antiparticle with the opposite charge and the same mass.

Feynman’s interpretation of the negative energy solutions of the KG and Dirac equations assumes that this symmetry holds for both bosons and fermions.

We now explore the idea of particle-antiparticle symmetry more formally.

We begin with the KG equation for a spin-0 particle of mass m and charge q in an electromagnetic field A^μ, namely equation (4.1). Inspection of this equation shows at once that the wave function φ_C of a particle with the same mass and charge −q is related to the original wavefunction φ by

φ_C = η_Cφ ^∗ (4.84)

whereη_C is an arbitrary phase factor which we shall take to be unity. Equation (4.84) tells us how to connect the solutions of the particle (charge q) and antiparticle (charge −q) equations. When applied to free-particle solutions of the KG equation, the transformation (4.84) relates positive and negative 4­

momentum solutions, as expected in the Feynman interpretation of the latter.

We may extend the transformation (4.84) to a symmetry operation for the KG equation (4.1) if we introduce an operation which changes the sign of A^μ . Then the combined operation ‘take the complex conjugate of φ and change A^μ to −A^μ’ is a formal symmetry of (4.84), in the sense that the wavefunction φ^∗ in the field −A^μ satisfies exactly the same equation as does the wavefunction φ in the field A^μ . Of course, we have just seen that φ^∗ is the antiparticle wavefunction, so it is no surprise that the dynamics of the antiparticle in a field −A^μ is the same as that of the particle in a field A^μ . Still, this is symmetry of the KG equation, which we will call charge conjugation, denoted by C:

C: φ → φ_C = φ ^∗ , A^μ → Aμ_C = −A^μ . (4.85)

We can ask: how does the electromagnetic current behave under this trans­

formation? The expression for the KG current is found by multiplying the free-particle probability current by the charge q, and by replacing ∂^μ by the gauge-invariant operator D^μ = ∂^μ + iqA^μ. This leads to

jKG^μ em(φ, A^μ) = iq{φ ^∗ (∂^μ + iqA^μ)φ−[(∂^μ + iqA^μ)φ] ^∗ φ}

= iq[φ ^∗ ∂^μφ−(∂^μφ ^∗ )φ] −2q ²A^μφ ^∗ φ. (4.86) The current for φ_C, Aμ_C is then

μ μ μ

j_{KG em}(φ_C, A_C) = iq[φ ^∗ _C∂^μφ_C −(∂^μφ _C^∗ )φ_C] −2q ²A φ_{C C}^∗ φ_C

= iq[φ ∂^μφ ^∗ −(∂^μφ)φ ^∗ ] + 2q ²A^μφ φ ^∗

= −jμ _KGem(φ, A^μ). (4.87) As we would hope, the KG current changes sign under C.

Now consider the Dirac equation for a particle of mass m and charge q in a field A^μ, which we write in the form

∂ψ = (−α · ∇ + iqα · A −iβm−iqA⁰)ψ. (4.88)

∂t

We want to relate solutions of this equation to the solution ψ_C of the same equation with q replaced by −q. As in the KG case, we begin by writing down the complex conjugate equation,

∂ψ^∗

= (−α1∂¹+α2∂²−α3∂³

∂t

− iqα₁∂¹+ iqα₂∂²−iqα₃∂³+ iβm+ iqA⁰)ψ ^∗ (4.89) where we have used the fact that α₁, α₃ and β are real and α₂ is pure imag­

inary, which is the case in both the standard representation of the Dirac matrices, and the representation (3.40). Now imagine multiplying (4.89) from the left by a matrix c, with the properties that it commutes with α1 and α3, but anticommutes with α2 and β. Then (4.89) will become

∂ψ^∗

c = (−α ·∇ − iqα · A − iβm+ iqA⁰) cψ ^∗ (4.90)

∂t

which is just (4.88) with q replaced by −q. So we may identify the charge-conjugate Dirac wavefunction as

ψ_C = η_C cψ ^∗ (4.91)

where η_C is the usual arbitrary phase factor. The required cis

c= βα2= γ² (4.92)

as the reader may easily verify. It is customary to choose η_C = i, and so finally the connection between ψ_C and ψ is

ψ_C(x) = C₀ψ ^∗ (x), where C₀ = iγ². (4.93)

101 4.2. Discretetransformations: P,C andT

Letuslookattheeffectofthetransformation(4.93)onfree-particlesolu­

tionsoftheDiracequation. Referringto(3.73)wefindthatapositiveenergy spinoristransformedto

( )

φ^s∗

u_C(p,s) = (E+m)^1/2 iγ² σ_E+^∗ ·_m p φ^s∗

( _Eσ₊·p_m(−iσ2φ^s∗) )

= (E+m)^1/2 , (4.94)

−iσ2φ^s∗

where we have used σ₂^∗ = −σ₂, σ₂σ₁ = −σ₁σ₂ and σ₂σ₃ = −σ₃σ₂. The 4-spinor(4.94) is a negative energy solution v(p,s) as in (3.82), identifying

−iσ₂φ^s∗ withχ^s . Accordingly we have shownthat

u_C(p,s) =v(p,s). (4.95)

Similarly,asthereadermaycheck,

v_C(p,s) = iγ²v^∗ (p,s) =u(p,s). (4.96) Sofrom apositiveenergyfree-particlespinorassociatedwith4-momentump andspinsthetransformation(4.93)producesanegativeenergyfree-particle spinorassociatedwiththesame4-momentumand spin,andvice versa: that is,uandv arecharge-conjugatespinors.

Atthispointwemaywonderifitispossibletoconstructa self-conjugate 4-spinor. Such a spinor wouldbe appropriatefor a fermionic particle which is the sameas its antiparticle – that is, for a Majorana fermion, so named after EttoreMajorana whofirst raisedthis possibility (Majorana 1937). To pursuethisidea,itisconvenienttousetherepresentation(3.40)fortheDirac matricesagain,inordertokeeptrackoftheLorentztransformationproperty oftheMajorana spinor. Consider the4-spinor

( )

ωM= φ . (4.97)

iσ2φ^∗

Then ( ) ( ) ( )

0 −iσ₂ φ^∗ φ

ω_MC = iγ²ω_M^∗ = = =ω_M, (4.98)

iσ₂ 0 iσ₂φ iσ₂φ^∗

so that indeed ωM is self-conjugate. The Lorentz transformation property of ωM is consistent, since we may easily show (problem 4.4(c)) that the 2­

spinorσ2φ^∗ transformsasaχ-typespinor. Thereadercanconstructasimilar self-conjugate4-spinorusingχratherthanφ.

Aself-conjugatefermionhastocarrynodistinguishingquantumnumber, suchaselectromagneticcharge. Theonlyknownneutralfermionsaretheneu­

trinos,anduntilquite recentlyitwasassumed thattheyareDiracfermions, withdistinctantiparticles(therelevantdistinguishingquantumnumberbeing

lepton number). However, as we shall see in volume 2, owing to their very small mass, it is hard to discriminate between the two possibilities (Majorana and Dirac) for neutrinos, and a definitive answer will have to await the result of a crucial experiment, the search for neutrinoless double beta decay, which is only possible for Majorana neutrinos.

Returning to more conventional matters, we extend (as in the KG case) the transformation (4.93) to a formal symmetry of the Dirac equation by including the sign change of A^μ, so that C for the Dirac equation is

C: ψ →ψ_C = iγ²ψ ^∗ , A^μ → −A^μ . (4.99) We now examine how the electromagnetic current behaves under C in the Dirac case. The Dirac charge density is the probability density ψ^†ψ multiplied by the charge q, and the electromagnetic 3-current is the probability current ψ^†αψ multiplied by q:

μ ¯

j_{D em}= (qψ^†ψ, qψ^†αψ) = qψγ^μψ. (4.100) Consider the charge density: under the transformation (4.93) this becomes

† γ² ψ ^∗ ψ ^∗

qψ ψ_{C C} = qψ^{T †}γ² = qψ^Tα₂ββα₂ψ ^∗ = qψ^T . (4.101) In terms of the four components of ψ, the product ψ^Tψ^∗ is ψ₁ψ₁^∗ + ψ₂ψ₂^∗ + ψ3ψ₃^∗ +ψ4ψ₄^∗. These components are ordinary functions which commute with each other, so ψ^Tψ^∗ = ψ^∗Tψ = ψ^†ψ; hence

qψ_C†ψ_C = qψ^†ψ (4.102) and the charge density does not change sign under C. Similarly, one finds that the electromagnetic 3-current does not change sign either.

These results can be interpreted in the hole theory picture: the current due to a physical positive energy antiparticle of charge q and momentum pis regarded as the same as that of a missing negative energy particle of charge

−q and momentum p. Our charge conjugation operation explicitly constructs the positive energy antiparticle wavefunction from the negative energy particle one.

Yet this is not really what we want a true charge conjugation operator to do: which is, rather, to change a positive energy particle into a positive energy antiparticle. The same inadequacy was true in the KG case also. There is no way of representing such an operation in a single particle wavefunction formalism. The appropriate formalism is quantum field theory, in which ψ(x) becomes a quantum field operator (as do bosonic fields), and there is a unitary quantum field operator Cˆ with the required property. We shall see in chapter 7 that fermionic operators anticommute with each other, and that this is just what is needed to ensure that the current changes sign under Cˆ . Bosonic fields, on the other hand, obey commutation rather than anticommutation relations, and this safeguards the change in sign of the bosonic current.

103 4.2. Discretetransformations: P,C andT

We have approached charge conjugation following the historical route, which is to say via the electromagnetic interaction. But we can ask whether (true) C is a good symmetry of other interactions, for example the weak interaction. Consider applying Cto the reaction (4.82), so that it becomes

π⁻ → μ⁻ + ¯ν_μ. (4.103) If C was a good symmetry, the (parity-violating) longitudinal polarization of the μ⁻ in (4.103) should be the same as that of the μ⁺ in (4.82). But in fact it is the opposite, the μ⁻ spin being aligned along the direction of its momentum. So C, like P, is violated in weak interactions. It is a good symmetry in electromagnetic and strong interactions.

4.2.3 CP

It has probably occurred to the reader that, although C and P are each violated in the decays (4.82) and (4.103), the combined transformation CP might be a good symmetry: particles are changed to antiparticles, the sense of longitudinal polarization is reversed, and the corresponding decays occur.

Indeed, the rates for these two decays are the same, and CP is conserved.

For a while, after 1956, it was hoped that CP would prove to be always conserved, so as to avoid a ‘lopsided’ distinction between right and left, and between matter and antimatter. But before long Christenson et al. (1964) reported evidence for CPviolation in the decays of neutral K-mesons, a result soon confirmed by other experiments.

As we mentioned in section 1.2.2, it was the difficulty of incorporating CP violation into the 2-generation electroweak theory that led Kobayashi and Maskawa (1973) to propose a third generation of quarks, which allowed a CP violating parameter to be included quite naturally. CPviolation in K-decays is a small effect (of order one part in 10³), but in 1980 Carter and Sanda (1980) showed that considerably larger effects, up to 20%, could be expected in rare decays of neutral B mesons, according to the framework of Kobayashi and Maskawa (KM). Some 20 years later, the ‘B factories’ at the asymmetric e⁻e⁺ colliders PEPII and KEKB began producing B mesons by the many millions, and intensive study of CPviolation in the B⁰(db)¯ −B¯⁰(¯db) systems followed at the BaBar and Belle detectors. Remarkably, all observations to date are consistent with the original KM parametrization. We shall return to this topic when we discuss weak interactions in volume 2, specifically in chapter 21. Meanwhile we refer to Bettini (2008), chapter 8, for an introductory overview.

It is worth pausing here to note the significance of CP violation. First of all, it implies that there is an absolute distinction between matter and antimatter and, as a consequence, between left and right: these are not merely a matter of convention. For example, the rate for the process

B⁰→ K⁺π⁻ (4.104)

is some 20% greater (Nakamura et al. 2010) than the rate for the CP-conjugate process

B¯⁰ →K⁻π . + (4.105)

(Note that the B¯⁰ state is conventionally defined as the CPtransform of the B⁰ state). So the pion distinguished by being emitted in the higher-yielding reaction (4.104) defines ‘negatively charged’, and the polarization of the muon in its decay (4.103) defines what is a right-handed screw sense.

Secondly, CP(and C) violation is one of the three conditions²established by Sakharov (1967) that would enable a universe containing initially equal amounts of matter and antimatter, when created in the Big Bang, to evolve into the matter-dominated universe we see today – rather than simply having the required imbalance as an initial condition. Within the Standard Model, all known CP violating effects are attributable to the KM mechanism. But calculations show (Huet and Sather 1995) that the matter-antimatter asym­

metry generated from this source is very many orders of magnitude too small.

This is, therefore, one area of physics where the Standard Model fails.

Thirdly, CP violation is directly connected to the violation of another discrete symmetry, namely time reversal T, because very general principles of quantum field theory imply that the product CPT(in any order) is conserved – the CPTtheorem. This theorem states (L¨uders 1954, 1957, Pauli 1957) that CPT must be an exact symmetry for any Lorentz invariant quantum field theory constructed out of local fields, with a Hermitian Hamiltonian, and quantized according to the usual spin-statistics rule (integer spin particles are bosons, half-odd integer spin particles are fermions). Thus any violation of CPimplies a violation of Tif CPT is to be conserved.

We shall return to CPTpresently, but first let us deal with T.

Im Dokument GAUGE THEORIES (Seite 115-120)