The neutrino: weak interaction and properties

The neutrino: weak interaction and properties TRABAJO FIN DE GRADO Department of Atomic, Molecular and Nuclear Physics Faculty of Physics David Mu˜noz P´erez Supervisor: Juan Antonio Caballero Carretero palabra Abstract This work is divided in two parts. The first one presents a historical overview of the neutrino and a revision of some important properties of neutrinos and the weak interaction, such as parity violation and neutrino oscillations. In the second part we carry out two elaborated calculations in order to reach 1) the scattering cross section of a process where neutrino and lepton interact, and 2) the decay rate of the pion decay. In the first case, we start from the Feynman-St¨uckelberg propagator theory and present the functional form of the scattering matrix Sfi. Then, we calculate the scattering cross section of the process, where we define the invariant-scalar matrix element Mfi and make use of some useful trace theorems in order to reach the final result. For the pion decay, we start from the general expression, based on the Fermi golden rule, that connects the decay rate with the invariant-scalar matrix element Mfi. Following a similar method to the one presented in the scattering process we reach the final result for the pion decay rate and analyze some basic properties linked to the weak interaction. I Contents 1 Introduction 1 1.1 Hystoricaloverview............................... 2 1.2 Neutrinoproperties............................... 4 1.2.1 Parity violation of weak interaction . . . . . . . . . . . . . . . . . . 5 1.2.2 Neutrino oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Relativistic dispersion quantum theory 11 2.1 Quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Weakinteractions................................ 16 3 Lepton-neutrino scattering cross section 19 3.1 Differential cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Matrix element Mfi .............................. 29 4 Pion decay. Decay rate 39 4.1 Matrix element Mfi .............................. 40 5 Discussion 49 Appendix A Relativistic quantum mechanics for spin 1/2 particles. 51 A.1 Diracequation ................................. 51 A.2 Bilinearcovariants ............................... 52 A.3 Diracfreesolutions............................... 53 II Appendix B Leptonic tensor contraction Lµν Rµν. 55 B.1 Tracetheorems ................................. 55 B.2 Leptonictensor ................................. 56 B.3 Leptonic tensor contraction LµνRµν ...................... 59 Bibliography 63 III Chapter 1 Introduction The neutrino is a particle postulated by the physicist Wolfgang Pauli in 1930 and experimentally discovered by Reines and Cowan in 1956. It is usually refered as the ’elusive’ or ’invisble’ particle, since its probability of interaction with matter is very small. This is caused by the fact that they only feel the weak interaction. The study of neutrinos is paramount in current Particle Physics. The fact that they barely interact with anything provides great information about their source. An example is the study of celestial bodies where neutrinos are produced, like stars. Moreover, they are very important in Cosmology, since this property of barely interacting can lead us to achieve a deeper knowledge about the beginning of the Universe. Other topics where the study of neutrinos is of great relevance are: dark matter, study of nucleus structure, etc. In this work, we first revise the history of the neutrino and explain its essential properties. Afterwards, making use of the Feynman-St¨uckelberg propagator formalism, we study a lepton-neutrino scattering process, obtaining its scattering cross section. Finally, we study the decay of the pion, which will let us analyse and observe some of the properties mentioned at the beginning of the work. 1 Figure 1.1: βdecay energy spectrum. Image extracted from [2]. 1.1 Hystorical overview In 1914, the English physicist James Chadwick showed that βdecay energy spectrum was continuous [1] (Figure 1.1): the electron does not come out with a fixed energy equal to the difference of masses between the initial and final nucleus. After years trying to find an explanation (some risky proposals such as Niels Bohr’s non conservation of energy were made), in 1930, the well-known physicist Wolfgang Pauli proposed that in βdecay a new particle must appear that shares the energy with the electron but which cannot be detected: the neutrino [3]. Conservation of energy, charge and angular momentum requires that the neutrino must be a light and neutral particle with half-integer spin, a fermion. Although Pauli’s proposal could explain the observed spectrum, even he claimed that it was “a desperate remedy”. At a scientific congress in Rome, Pauli met Enrico Fermi, who was enthusiastic about Pauli’s idea. This enthusiasm led him to the development of the theory of βdecay, which perfectly explained the observed spectrum considering the existence of the neutrino. 2 According to Fermi theory, the process e−+ν→p+nwas possible.1Nevertheless, the calculation of two important theoretical physicists, Hans Bethe and Rudolf Peierls, showed that the process cross section would be hugely small [4]: σ≃10−44 cm2. This results made physicists very reluctant to trying to ‘capture’ the almost ‘invisible’ particle. However, in 1946, the italian physicist Bruno Pontecorvo showed the path with a very simple idea [5,6]: although the probability of interaction is very small, what if I generate a beam with a great number of neutrinos aiming at a substance with a great number of target particles? We will detect a few counts, which will allow us to prove the existence of the neutrino. Thanks to Pontecorvo’s idea, by 1951, the American physicists Clyde L. Cowan and Frederick Reines started a project that would end up with the experimental detection of the neutrino in 1956 [7]. The central process they chose to detect the particle was the inverse beta decay: ¯νe+p→n+e+.(1.1) The reactor produced antineutrinos. The detector, basically, was a big tank full of water and cadmium chloride dissolved in it. The antineutrino would collide with H2O protons (almost free protons), giving rise to a neutron and a positron, see (1.1). As we can see in Figure 1.2, the positron e+instantaneously annihilates with an electron in the medium, emitting two photons. The neutron moves through the tank losing energy until it is finally captured by a cadmium nucleus: n+108 Cd →109mCd →109 Cd + γ . (1.2) The time lapse between both events was calculated resulting a value of the order of 6µs. Thus, if these two signals were detected in accordance with the predicted time lapse, then Reines and Cowan would have one count. Fortunately, after several tries and improving significantly the detector, Cowan and Reines detected the neutrino in 1956. The final and successful detector was located in Savannah River Plant (South Car- 1Actually, the neutrino νwas an antineutrino ¯νe, but the idea of antineutrino was not developed yet. 3 Figure 1.2: The antineutrino interacts with the proton. The final particles generate the signals that would imply the presence of the antineutrino. Image extracted from [8]. olina), 12 metres under a mountain in order to eliminate the cosmic rays effect (Figure 1.3). Also, between reactor and detector there were 11 metres of concrete, so that only neutrinos could go through it and induce reactions inside the tank. 1.2 Neutrino properties Neutrinos are leptons, hence they do not feel strong interaction. They are fermions with spin 1 2and zero charge and do not feel electromagnetic interaction. There are three types of neutrinos, one of each characterised by a flavour: electron, muon and tau neutrino. Although neutrino mass is very close to zero, it is not zero. At present, different experiments on neutrino oscillations performed at several facilities are trying to improve our present knowledge on basic properties: neutrino mass hierarchy, oscillation mixing angles, etc. 4 Chapter 2 Relativistic dispersion quantum theory In this chapter, we present in detail the calculation of the scattering matrix Sfi, which can be used afterwards for the determination of the scattering cross section of a process. We will first present the results associated to processes governed by electromagnetic interaction, because these are the most commonly known. We follow closely the exhaustive analysis shown in [18]. This reference makes use of the Feynman-St¨uckelberg propagator formalism, which can be more illustrative and intuitive than the Quantum Field Theory approach. After this, we will extend the expressions obtained to the weak interaction case. 2.1 Quantum electrodynamics In Quantum Electrodynamics, the Feynman-St¨uckelberg approach considers a free particle1propagating through space-time described by an initial wavefunction φi(x).2After n interactions with an electromagnetic potential Aµ(x), the particle changes its state and is then described by ψi(x), which is not, in general, a Dirac free wavefunction. In this context, the scattering matrix or S-matrix is defined as follows, 1Let it be a spin 1/2 particle so that it can be described by Dirac free wavefunctions φ(x). 2Understand xas a four-vector. 11 Sfi = lim t→∞ hφf(x)|ψi(x)i,(2.1) where φf(x) is a free Dirac wavefunction. At first glance, (2.1) could seem confusing, but it just needs a bit of explanation. Sfi evaluates the overlap between the previously defined wavefunction ψi(x) and the final relativistic free wave function corresponding to the particle observed at the end of the scattering process φf(x). If we insert an arbitrary φf(x) and it turns out to be that the overlap is great, we will know that it is very likely that the particle that was initially described by φi(x) ends up in a state described by φf(x). If, on the contrary, Sfi is small, the probability of this to happen will be small. Moreover, the t→ ∞ appearing in (2.1) expresses the fact that we take limit for φf(t→ ∞). This limit should be intrepreted as time large enough but finite on which the particle cease to interact (Aµ≃0). We assume to be known: the initial state of the particle (previous to feeling the interaction) φi(x) characterised by the quantum numbers ~pi, Ei, si, the final relativistic free wavefunction φf(x) characterised by the quantum numbers ~pf, Ef, sfand the fourpotential Aµ(x) that characterises the interaction. Nevertheless, we do not know ψi(x). The Feynman-St¨uckelberg propagator theory allows us to calculate ψi(x) as a function of φi(x). After a laborious algebra (see [18]), the S-matrix reads Sfi =δfi |{z} Order 0 −ie Zd4xφf(x)/ A(x)φi(x) | {z } 1st order −ie2Zd4x1d4x2φf(x2)/ A(x2)SF(x2−x1)/ A(x1)φi(x1) | {z } 2nd order +... |{z} Superior orders ,(2.2) where SF(x2−x1) is the free propagator of the spin 1/2 particle3,ethe charge of the particle and / A(x) = γµAµ(x). In Figure 2.1 we show different graphs that illustrate the terms entering in (2.2). Notice that the evaluation of the scattering matrix element Sfi at nth order means considering 3If a particle propagates freely through space-time from x1to x2,SF(x2−x1) allows us to calculate the final physical state at x2from its state at x1. 12 Figure 2.1: Lines represent free propagation trajectories. Thus, in the 1st order figure, for example, we have a particle that propagates freely from (x, t) to (x1, t1). There, it interacts with Aµbeing scattered. It propagates freely from (x1, t1) to (x0, t0). Notice that in the figure, for clarity, we refer to points in space-time as (x, t), denoting xthe spatial position and tthe time. Nevertheless, in the text we always refer to xmeaning the four-vector (x0, x1, x2, x3). that the particle (projectile) in the process scatters n times with the electromagnetic potential represented by Aµ. In our context, Aµis the four-potential created by another particle (target) that collides with our projectile. Notice that the analysis of the scattering process is completely symmetric; that is, the target can also be considered as the projectile and viceversa. Looking at (2.2) the question arises authomatically: until which order do we consider when calculating Sfi? Since Aµis the electromagnetic potential, it contains an elementary charge factor e(we will see it in (2.11)). Thus, the nth order term has a factor e2n, which is proportional to the nth power of the electromagnetic coupling constant α. In the system of units we are working (natural units), α=e2 4π= 1/137 ≃7·10−3, hence the nth order term has a factor αn. Taking into account the order of magnitude of the coupling constant α, it is a good approximation considering only the first order term, since αα2. Thus, 13 Sfi =−ie Zd4x φf(x)/ A(x)φi(x).(2.3) Sfi =−ie Zd4x φf(x)γµAµ(x)φi(x) = −ie Zd4x φf(x)γµφi(x) | {z } Jµ(x) Aµ(x),(2.4) where Jµis the electromagnetic four-current.4 We now define the interaction hamiltonian as follows, HI(x) = eJµ(x)Aµ(x).(2.5) Consequently, the scattering matrix can be expressed as Sfi =−iZd4x HI(x).(2.6) Since we will consider collisions of two particles in the following chapter, we now calculate the electromagnetic potential Aµ(x) created by another particle (target) with charge ea. Maxwell equations: Fµν =∂µAν−∂νAµ,(2.7) ∂µFµν =eaJaν.(2.8) Inserting (2.7) into (2.8) and using Lorentz gauge ∂µAµ= 0, ∂µ(∂µAν−∂νAµ) = eaJaν⇒∂µ∂µAν−∂ν∂µAµ |{z} 0 =eaJaν.(2.9) Thus, Aµ(x) = eaJaµ(x),(2.10) where we have introduced the D’alembertian operator =∂µ∂µ=∂2 ∂t2−∇2. In order to solve this equation we will follow [18] and write Aµ(x) as follows, 4Notice Aµcommutes with φibecause the first acts on Minkowski space while the second acts on Dirac space. 14 Aµ(x) = eaZd4y DF(x−y)Jaµ(y),(2.11) where DF(x−y) is the Feynman photon propagator. We insert (2.11) into (2.10),5 Zd4yDF(x−y)Jaµ(y) = Jaµ(x).(2.12) Then, DF(x−y) = δ4(x−y).(2.13) We now use the Fourier transform to solve the equation above, DF(x−y) = Zd4q (2π)4e−iq(x−y)DF(q).(2.14) Also recall the Dirac delta definition given by δ4(x−y) = Zd4q (2π)4e−iq(x−y).(2.15) Inserting (2.14) and (2.15) into (2.13), we obtain Zd4q (2π)4e−iq(x−y)DF(q) = Zd4q (2π)4e−iq(x−y),(2.16) Zd4q (2π)4e−iq(x−y)(−iq)2DF(q) = Zd4q (2π)4e−iq(x−y).(2.17) It is then clear that the photon propagator in Fourier space is given by DF(q) = −1 q2,(2.18) where qis the four-momentum of the photon. With this result, equations (2.14), (2.11), (2.5) and (2.6) turn into DF(x−y) = Zd4q (2π)4e−iq(x−y) −1 q2!,(2.19) 5Notice the D’Alembertian only acts on x, not on y. 15 Aµ(x) = eaZd4yZd4q (2π)4e−iq(x−y) −1 q2!Jaµ(y),(2.20) HI(x) = eeaJµ(x)Zd4yZd4q (2π)4e−iq(x−y) −1 q2!Jaµ(y),(2.21) Sfi =−ieeaZd4xZd4yZd4q (2π)4Jµ(x)e−iq(x−y) −1 q2!Jaµ(y).(2.22) We also recall the form of the electromagnetic four-currents, Jµ(x) = φf(x)γµφi(x), Jaµ(y) = φaf (x)γµφai(x).(2.23) 2.2 Weak interactions Now we extend the previous results to weak interaction processes. As known from the electromagnetic case (2.5), the S-matrix can be written in this case as Sfi =−iZd4x HW I(x).(2.24) The weak interaction hamiltonian is given by HW I(x) = gw 2√2!Jµ(x)Wµ(x),(2.25) where Wµis the weak four-potential and Jµthe weak four-current. Notice here we have the weak charge gw 2√2!instead of the electric charge e. Like in the previous section, our goal is to evaluate the weak potential Wµ. It satisfies the following equation [19–21], hgµν(+MW2)−∂µ∂νiWν(x) = gw 2√2!Jaµ(x),(2.26) where MWis the W boson mass. For clarity, we define τµν(x)≡gµν(+MW2)−∂µ∂ν.(2.27) 16 Following a similar procedure as the one used for the electromagnetic potential, see (2.11), we can express the weak four-potential in terms of the weak charge, the W-boson propagator and the weak current,6 Wµ(x) = gw 2√2!Zd4y DFµν(x−y)Jaν(y).(2.28) Inserting (2.28) into (2.26) and using definition (2.27), we obtain τµν(x)  gw 2√2!Zd4y DFαν(x−y)Jaα(y) =   gw 2√2!Jaµ(x),(2.29) Zd4y τµν (x)DFαν(x−y)Jaα(y) = Jaµ(x).(2.30) For α=µ, we obtain τµν(x)DFµν(x−y) = δ4(x−y).(2.31) As we did in the previous section, we use the Fourier transform of the boson propagator in order to solve the equation above, DFµν(x−y) = Zd4q (2π)4e−iq(x−y)DFµν(q),(2.32) We insert (2.32) into (2.31) and recall Dirac delta definition (2.15), Zd4q (2π)4τµν(x)e−iq(x−y)DFµν(q) = Zd4q (2π)4e−iq(x−y),(2.33) Zd4q (2π)4e−iq(x−y)hgµν(−q2+MW2) + qµqνiDFµν(q) = Zd4q (2π)4e−iq(x−y).(2.34) It is then clear, hgµν(−q2+MW2) + qµqνiDFµν(q)=1.(2.35) The solution to this equation can be easily obtained and is given by 6Notice we have written a more general form of the propagator DF(x−y) with the tensor indices µν. The photon propagator is a particular case where Dµν F∝gµν , then Dµν FJν=DFgµν Jν=DFJµ 17 DFµν(q) = −gµν +qµqν MW2 q2−MW2,(2.36) where qµis the four-momentum of the W boson. It can be easily checked that DFµν (q) verifies the equation above inserting (2.36) into (2.35). In the following we will make use of the static approximation for the boson propagator (q2MW2), i.e., DFµν(q)≃gµν MW2.(2.37) This approximation makes sense since the W boson mass is of the order of 102GeV, much larger than the order of the energies we will consider in the scattering process. Thus, within the static approximation, equations (2.32), (2.28), (2.24) and (2.25) turn into DFµν(x−y) = Zd4q (2π)4e−iq(x−y) gµν MW2!,(2.38) Wµ(x) = gw 2√2!Zd4yZd4q (2π)4e−iq(x−y) gµν MW2!Jaν(y),(2.39) HI(x) = gw 2√2!2 Jµ(x)Zd4yZd4q (2π)4e−iq(x−y) gµν MW2!Jaν(y),(2.40) Sfi =−i gw 2√2!2Zd4xZd4yZd4q (2π)4Jµ(x)e−iq(x−y) gµν MW2!Jaν(y).(2.41) Notice that gµν will disappear after contracting with the four-current, gµν Jaν=Jaµ. It is important to remark that the differences with the electromagnetic results are only the charge, weak instead of electric charge, the propagator of the mediator particle, W boson instead of photon, and the functional form of the four-currents involved. For the weak interaction, we have Jµ(x) = φf(x)γµ(1 −γ5)φi(x), Jaµ(y) = φaf (x)γµ(1 −γ5)φai(x).(2.42) These currents incorporate important properties of the weak interaction like parity violation. This is analyzed in detail in the following chapters. 18 Chapter 3 Lepton-neutrino scattering cross section In this chapter, we evaluate the scattering cross section for a process in which a neutrino interacts with a lepton and analyse its properties, such as the dependence with the mass of the leptons involved or with the energy of the incoming particles. The scattering process we are interested in is presented in Figure 3.1. According to St¨uckelberg-Feynman propagator theory, the first order S-matrix element for a weak interaction process is given by (see Chapter 2) Sfi =−iZd4xHW I(x), HW I(x) = gw 2√2!Jµ l(x)Wµ(x),(3.1) where xµis the space-time four-vector1,HW Ithe weak interaction hamiltonian, Jµ lthe four-current associated with the left vertex of the first order Feynman diagram and Wµ the weak four-potential, associated with the right vertex (see Figure 3.1). The election of which vertex is associated with the current and which with the potential is arbitrary. Both possible elections give identical results for the relevant physical magnitudes. We recall (2.41) from Chapter 2, 1d4x=dx0dx1dx2dx3. All integrals performed without indicating integration limits are understood to be integrals over all posible values of the integration variables. 19 Figure 3.1: First order Feynman diagram for the scattering process studied. Sfi =−i gw 2√2!2Zd4xZd4yZd4q (2π)4Jµ l(x)e−iq(x−y) M2 W Jr µ(y),(3.2) Jµ l(x) = φ3(x)γµ(1 −γ5)φ1(x), J µ r(y) = φ4(y)γµ(1 −γ5)φ2(y),(3.3) where qand MWare, respectively, the four-momentum and the mass of the Wboson, Jr µ the leptonic four-current associated with the right vertex of the Feynman diagram, γµand γ5matrices defined in Dirac space and (φi)φithe (Dirac adjoint) solution of the free Dirac equation (see Appendix A). φ1(x) = sm1 E1Vu(~p1, s1)e−ip1x, φ2(y) = sm2 E2Vu(~p2, s2)e−ip2y,(3.4) φ3(x) = sm3 E3Vu(~p3, s3)eip3x, φ4(y) = sm4 E4Vu(~p4, s4)eip4y. 20 dσ =m1m2m3m4 (2π)2(E1+E2)|~pi| ×Z All ~p4 |Mfi|2δqm2 4+~p 2 4+qm2 3+~p 2 3−E2−E1 qm2 3+~p 2 3qm2 4+~p 2 4 δ3(~p4+~p3)d3p3d3p4,(3.27) where we have added the integral over ~p4, since we assume we are interested in detecting particle 3. Due to the presence of the delta function δ3(~p4+~p3), the integral can be evaluated explicitly and (3.27) results dσ =m1m2m3m4 (2π)2(E1+E2)|~pi||Mfi|2δqm2 4+~p 2 3+qm2 3+~p 2 3−E2−E1 qm2 3+~p 2 3qm2 4+~p 2 3 d3p3.(3.28) Writing d3p3in spherical coordinates (see Figure 3.4)d3p3=|~p3|2d|~p3|sin θ dθ dϕ = |~p3|2d|~p3|dΩ and integrating over |~p3|, we obtain the differential cross section as dσ dΩ=m1m2m3m4 (2π)2(E1+E2)|~pi|Z All |~p3||Mfi|2δqm2 4+~p 2 3+qm2 3+~p 2 3−E2−E1 qm2 3+~p 2 3qm2 4+~p 2 3|~p3|2d|~p3|. (3.29) In general, Mfi is a function of ~p1, ~p2, ~p3, ~p4. Taking into account we are working in the COM frame (~p1=−~p2,~p3=−~p4), it is clear that Mfi =Mfi (~p1, ~p3). Then, Mfi cannot be taken out from the integral, since it depends on |~p3|. In order to solve (3.29) we introduce a new set of variables, u≡qm2 3+|~p3|2+qm2 4+|~p3|2,(3.30) du d|~p3|= 2|~p3|  2qm2 3+|~p3|2+ 2|~p3|  2qm2 4+|~p3|2= u z }| { qm2 3+|~p3|2+qm2 4+|~p3|2 qm2 3+|~p3|2qm2 4+|~p3|2|~p3|. 27 x y z φ θ Figure 3.4: Definition of the coordinate system. The scattering angle in the COM frame coincides with the polar angle θ. Multiplying by |~p3|in both sides, we obtain |~p3|du u=1 qm2 3+|~p3|2qm2 4+|~p3|2|~p3|2d|~p3|.(3.31) We insert (3.30) and (3.31) into (3.29). Thus, dσ dΩ=m1m2m3m4 (2π)2(E1+E2)|~pi|Z All |~p3||Mfi(|~p3|)|2δu−(E1+E2)|~p3| udu. (3.32) 28 Notice that |~p3|can be expressed as a function of uby solving (3.30). After some algebra, we obtain |~p3|=1 2uqu4+m4 3+m4 4−2m2 3m2 4−2u2m2 3−2u2m2 4.(3.33) Making use of the delta function δu−(E2+E1)the integral in (3.32) can be solved easily. The expression for the differential cross section results dσ dΩ=m1m2m3m4 (2π)2(E1+E2)|~pi||Mfi(|~p3|)|2|~p3| E1+E2 .(3.34) Consequently, from (3.33) and (3.34), we can write dσ dΩ=m1m2m3m4 (2π)2(E1+E2)2|Mfi |2|~pf| |~pi|,(3.35) with |~pf|=1 2(E1+E2)q(E1+E2)4+m4 3+m4 4−2m2 3m2 4−2(E1+E2)2m2 3−2(E1+E2)2m2 4. (3.36) 3.2 Matrix element Mfi We proceed to calculate the matrix element for this weak interaction process (Figure 3.1). It is important to remember that the matrix element contains the relevant physical information that reveals which is the interaction responsible for the process. The magnitude we want to calculate in order to introduce it in (3.35) is |Mfi|2. We recall (3.13) and (3.15), Mfi = gw 2√2! gw 2√2!u(3)Γµu(1) 1 M2 W u(4)Γµu(2) , |Mfi|2=GF2 2|u(3)Γµu(1) u(4)Γµu(2)|2. 29 When writing the matrix element Mfi as defined above, we assume initial and final particles have fixed polarizations s1, s2, s3, s4, because the spinors are written for a defined value of s. Nevertheless, we are going to consider our detector is not sensitive to polarization of the particles, hence we have to sum over final polarizations s3, s4. Moreover, in the case we consider, incoming particles do not present a specific polarization, then we have to average over initial polarizations. We define then |Mfi|2as |Mfi|2=1 2X s1CCC 1 2X s2X s3X s4|Mfi|2.(3.37) Notice that we did average for particle 1 (lepton), because it can have both polarizations, but not for particle 2 (neutrino), since it has a defined polarization.4 Consequently, |Mfi|2=X s1X s2X s3X s4 GF2 4|u(3)Γµu(1) u(4)Γµu(2)|2.(3.38) In the following calculations we will simplify (3.38) making use of the energy projection operators, reaching an expression only dependent on matrix traces. Thereby, we will be able to obtain |Mfi|2using the trace theorems shown in Appendix B. |Mfi|2=GF2 4X s1X s2X s3X s4X µ, ν hu(3)Γµu(1) | {z } Complex number u(4)Γµu(2) | {z } Complex number i†hu(3)Γνu(1) u(4)Γνu(2)i= GF2 4X µ, ν X s1X s3hu(3)Γµu(1)i†hu(3)Γνu(1)iX s2X s4hu(4)Γµu(2)i†hu(4)Γνu(2)i= GF2 4X µ, ν X s1X s3hu†(1)(Γµ)†γ0u(3)ihu(3)Γνu(1)iX s2X s4hu†(2)(Γµ)†γ0u(4)ihu(4)Γνu(2)i= GF2 4X µ, ν X s1X s3hu†(1)γ0 | {z } u(1) γ0(Γµ)†γ0 | {z } Γµ u(3)ihu(3)Γνu(1)i ×X s2X s4hu†(2)γ0 | {z } u(2) γ0(Γµ)†γ0 | {z } Γµ u(4)ihu(4)Γνu(2)i, 4That is why we crossed out the 1 2factor associated to the neutrino. 30 where we used I=γ0γ0, u(i) = u†(i)γ0,hu(i)i†=hu†(i)γ0i†=γ0u(i), A ≡γ0A†γ0. (3.39) Thus, |Mfi|2=GF2 4X µ, ν X s1X s3 u(1)Γµu(3) u(3)Γνu(1) X s2X s4 u(2)Γµu(4) u(4)Γνu(2). (3.40) Making use of closure relations (A.27) for the energy projection operators defined in Appendix A, we can make the following substitutions, X s3 u(3) u(3) = / p3+m3 2m3 ,X s4 u(4) u(4) = / p4+m4 2m4 .(3.41) Inserting (3.41) in (3.40), we obtain |Mfi|2=GF2 4X µ, ν X s1 u(1) Γµ/ p3+m3 2m3 Γνu(1) X s2 u(2)Γµ / p4+m4 2m4 Γνu(2).(3.42) Taking into consideration Dirac spinors definitions (A.19) shown in Appendix A, we can write X s1 u(1) Γµ/ p3+m3 2m3 Γν!u(1) = 2 X r=1 wr(~p1) Γµ/ p3+m3 2m3 Γν!wr(~p1) = 4 X r=1 wr(~p1) Γµ/ p3+m3 2m3 Γν!λrΛ+(~p1)wr(~p1) = 4 X r=1 λrwr(~p1) Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!wr(~p1), where we took into account5 Λ+(~p1) = / p1+m1 2m1 , λrΛ+(~p1)wr(~p1) =        wr(~p1) for r= 1,2 0 for r= 3,4 .(3.43) Hence, X s1 u(1) Γµ/ p3+m3 2m3 Γν!u(1) = 4 X r=1 λrwr(~p1) Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!wr(~p1).(3.44) 5λr=     1 for r= 1,2 −1 for r= 3,4 31 Writing explicitly the (adjoint spinor)·(Dirac operator)·(spinor) product, (3.44) reads X s1 u(1) Γµ/ p3+m3 2m3 Γν!u(1) = X α,β 4 X r=1 λrhwr(~p1)iα Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!αβhwr(~p1)iβ= X α,β 4 X r=1 λrhwr(~p1)iβhwr(~p1)iα | {z } δαβ Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!αβ =X α Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!αα , where we made use of the spinors closure relation (A.23) shown in Appendix A. Thus, X s1 u(1) Γµ/ p3+m3 2m3 Γν!u(1) = Tr Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!.(3.45) Proceeding in a completely analogous way, we obtain for the other sum X s2 u(2) Γµ / p4+m4 2m4 Γν!u(2) = Tr Γµ / p4+m4 2m4 Γν / p2+m2 2m2!.(3.46) Inserting (3.45) and (3.46) into (3.42), we obtain |Mfi|2=GF2 2LµνRµν,(3.47) Lµν =1 2Tr Γµ/ p3+m3 2m3 Γν/ p1+m1 2m1!,Rµν = Tr Γµ / p4+m4 2m4 Γν / p2+m2 2m2!. (3.48) Lµν and Rµν are the so-called leptonic tensors, one of each associated to one fermion line6. It is interesting to recall that Lµν has a 1 2factor due to the average over initial polarizations of particle 1 (lepton) while Rµν lacks this factor because particle 2 (neutrino) has a defined polarization. The leptonic tensors contraction LµνRµν is calculated in Appendix B, given by (B.21). With this result, (3.47) becomes 6Lµν associated to the left fermion line of the Feynman diagram in Figure 3.1 and Rµν associated to the right one. 32 |Mfi|2=4GF2 m1m2m3m4 (p1·p2)(p3·p4).(3.49) Inserting (3.49) into (3.35), we obtain7 dσ dΩ=(((((( ( m1m2m3m4 ( 2π)2(E1+E2)2  4GF2 (((((( ( m1m2m3m4 (p1·p2)(p3·p4)|~pf| |~pi|, dσ dΩ=GF2 π2 (p1·p2)(p3·p4) (E1+E2)2|~pf| |~pi|.(3.50) We have achieved the differential cross section expressed as a function of magnitudes in the COM frame. Nevertheless, we want to express it as a function of the magnitudes known in a scattering experiment: the energies of the initial particles E1,E2. Taking into account we are working in the COM frame, E1and E2are directly related, so our aim will be to reach dσ dΩ(E1). In what follows, we express the magnitudes p1·p2,p3·p4,E2, |~pf|,|~pi|as functions of E1. COM frame: ~ Pi=~p1+~p2= 0 ⇒~p1=−~p2⇒ |~p1|=|~p2|=|~pi|.(3.51) Linear momentum conservation: ~ Pf=~ Pi⇒~p3+~p4= 0 ⇒~p3=−~p4⇒ |~p3|=|~p4|=|~pf|. (3.52) Since m2=mν≃0, E1=qm12+|~pi|2 E2=|~pi|    ⇒E2=qE2 1−m12.(3.53) Considering also m3=mν≃0, we obtain from (3.36) |~pf|=(E1+E2)2−m42 2(E1+E2)=E1+qE2 1−m122−m42 2E1+qE2 1−m12,(3.54) where we used (3.53) in the last step. 7We have substituted |Mfi|2→ |Mfi|2in (3.35) in order to take into account the aforementioned averaging and summing over polarizations. 33 p1·p2=p1µp2µ=E1E2−~p1·~p2=E1E2+|~pi|2=E1E2+E2 2=E2(E1+E2). Consequently, by using (3.53) we obtain p1·p2=qE2 1−m12E1+qE2 1−m12.(3.55) Similarly, p3·p4=p3µp4µ=E3E4−~p3·~p4=E3E4+|~pf|2.(3.56) As recently mentioned m3=mν≃0, what implies E3=|~pf|, E4=qm42+|~pf|2.(3.57) Thus, (3.56) reads p3·p4=|~pf|qm42+|~pf|2+|~pf|2.(3.58) The results in equations (3.53), (3.54), (3.55) and (3.58) show that the cross section (3.50) does not depend on θand ϕ. Hence, we may write σ=Z All Ω dσ dΩdΩ = dσ dΩZ All Ω dΩ = dσ dΩ·4π . (3.59) Consequently, σ=4GF2 π (p1·p2)(p3·p4) (E1+E2)2|~pf| |~pi|.(3.60) Sometimes, it is common to regard the initial lepton as an ultrarrelativisic particle (|~pi|  m1) since this approximation may be valid for a wide interesting range of energies. |~pi|>> m1⇒E1≃ |~pi| ⇒ E1≃E2. Imposing this condition, the following results emerge, |~pf|=4E12−m42 4E1 , p1·p2= 2E12, p3·p4= 2E12"1−m4 2E12#.(3.61) 34 Inserting (3.61) into (3.50), we obtain the differential cross section of the process, dσ dΩ=GF2 π2E12"1−m4 2E12#2 .(3.62) Using the Fermi constant definition (3.14), dσ dΩ=1 2 g2 wE1 4πM2 W!2"1−m4 2E12#2 .(3.63) Notice neither the general case nor the ultrarrelativistic approximation (3.63) depend on the scattering angle. Again, the total cross section is given by integrating for all solid angles, σ=1 2 g2 wE1 4πM2 W!2"1−m4 2E12#2Z All Ω dΩ | {z } 4π =1 8π g2 wE1 M2 W!2"1−m4 2E12#2 .(3.64) In what follows we evaluate the total scattering cross section σby comparing the results corresponding to the general case with the ones associated to the ultrarelativistic approximation (UA) for the initial lepton. We start by calculating the minimum value of E1required for the process to occur. Since particle 3 is massless (neutrino), using (3.53), we can write for the COM frame, Emin 1+E2=m4⇒Emin 1+q(Emin 1)2−m12=m4⇒ (Emin 1)2+ (Emin 1)2−m12+ 2Emin 1q(Emin 1)2−m12=m42⇒ h2(Emin 1)2−(m12+m42)i2= 4(Emin 1)2h(Emin 1)2−m12i⇒   4(Emin 1)4+ (m12+m42)2−4(Emin 1)2(  m12+m42) =   4(Emin 1)4−  4(Emin 1)2m12⇒ (m12+m42)2= 4m42E12. Thus, Emin 1=m12+m42 2m4 .(3.65) Notice that particles 1 and 4 refer to the initial and final leptons, respectively. 35 Figure 3.5: Total scattering cross section vs E1. Initial and final leptons are the same in each case (electron e−, muon µ−, tau τ−), hence m1=m4. Dashed lines correspond to the general case while solid ones to UA for the initial lepton. Notice that the results corresponding to electrons and muons (general and UA) are almost the same. This explains that yellow, red, green and pink lines coincide. The total cross section σis shown in Figure 3.5 in the case of the lepton being an electron, muon and tau particle. In each case we compare the full calculation (dashed lines) with the UA (solid lines). As observed, the results for the electron and muon cross sections are very similar. On the contrary, a clear difference is shown for the tau particle, for which general case and approximation differ importantly. This is consequence of the much larger value of the tau mass, which is of the order of the range of energies presented in the graph. In Figure 3.6 we present the same results as in Figure 3.5 (except we do not show τ cross section), but for a narrower range of energies. We can then appreciate that muon cross section is smaller than electron and the difference between the general case and the 36 |Mfi|2=GF2fπ2 4m3 p1µp1νX s2 v(2)hΓµ(/ p3+m3)Γνiv(2) = GF2fπ2 4m3 p1µp1ν 4 X r=3 wr(~p2)hΓµ(/ p3+m3)Γνiwr(~p2) = GF2fπ2 4m3 p1µp1ν 4 X r=1 wr(~p2)hΓµ(/ p3+m3)Γνi(−λr) Λ−(~p2) | {z } −/ p2+m2 2m2 wr(~p2),(4.13) where we used Λ−(~p2) = −/ p2+m2 2m2 ,(−λr)Λ−(~p2)wr(~p2) =        0 for r= 1,2 wr(~p2) for r= 3,4 .(4.14) We keep playing with (4.13), |Mfi|2=−GF2fπ2 8m2m3 p1µp1ν 4 X r=1 λrwr(~p2)hΓµ(/ p3+m3)Γν(−/ p2+m2)iwr(~p2) = −GF2fπ2 8m2m3 p1µp1νX α,β 4 X r=1 λrhwr(~p2)iαhΓµ(/ p3+m3)Γν(−/ p2+m2)iαβhwr(~p2)iβ= −GF2fπ2 8m2m3 p1µp1νX α,β 4 X r=1 λrhwr(~p2)iβhwr(~p2)iα | {z } δβα hΓµ(/ p3+m3)Γν(−/ p2+m2)iαβ = −GF2fπ2 8m2m3 p1µp1νX αhΓµ(/ p3+m3)Γν(−/ p2+m2)iαα ,(4.15) where we used the closure relation (A.23) given in Appendix A. It is clear from (4.15) that |Mfi|2can be expressed as |Mfi|2=−GF2fπ2 2p1µp1νTr Γµ / p3+m3 2m3 Γν−/ p2+m2 2m2!.(4.16) We now calculate the trace above, Πµν ≡Tr Γµ / p3+m3 2m3 Γν−/ p2+m2 2m2!.(4.17) 43 Notice a very similar trace was already calculated in Appendix B:Rµν, given by (B.17). We can write Πµν =Rµν(p4→p3, m4→m3, p2→ −p2).(4.18) Thus, Πµν =2 m2m3p3µ(−p2ν)+(−p2µ)p3ν−gµν p3·(−p2)−i µνσλ p4σ(−p2λ)= −2 m2m3p3µp2ν+p2µp3ν−gµν p3·p2−i µνσλ p4σp2λ,(4.19) where µνσλ is the four-dimensional Levi-Civita tensor. Inserting (4.19) into (4.16), we obtain |Mfi|2=GF2fπ2 m2m3 p1µp1νp3µp2ν+p2µp3ν−gµν p3·p2−i µνσλ p4σp2λ= GF2fπ2 m2m3 (p1·p3p1·p2+p1·p2p1·p3−p12p3·p2−i µνσλ p1µp1νp4σp2λ) = GF2fπ2 m2m3 (2 p1·p2p1·p3−p12p3·p2−i µνσλ p1µp1νp4σp2λ).(4.20) It is not difficult to see that the last term µνσλ p1µp1νp4σp2λis equal to 0. We focus on terms with fixed σλ. Let it be σ= 3, λ = 4, for example. µν34 p1µp1νp43p24= (1234 |{z} 1 p11p12+2134 |{z} −1 p12p11)p43p24= (p11p12−p11p12)p43p24= 0 . (4.21) This happens for every σλ pair of values. Thus, µνσλ p1µp1νp4σp2λ= 0 .(4.22) Inserting (4.22) into (4.20), |Mfi|2reads |Mfi|2=GF2fπ2 m2m3 (2 p1·p2p1·p3−p12p3·p2).(4.23) Inserting (4.23) into (4.4), Γ =   m2m3|~pf| 2πm12 GF2fπ2   m2m3 (2 p1·p2p1·p3−p12p3·p2),(4.24) 44 which finally results Γ = GF2fπ2 2πm12|~pf|(2 p1·p2p1·p3−p12p3·p2).(4.25) Now we proceed to calculate the scalar products appearing in the expression above and simplify (4.25) in order to give Γ as a function of known magnitudes. COM frame ⇒~ Pi=~p1= 0 .(4.26) Linear momentum conservation: ~ Pi=~ Pf⇒0 = ~p2+~p3⇒~p2=−~p3⇒ |~p2|=|~p3|=|~pf|.(4.27) Energy conservation: Eπ=El+Eν⇒Eπ2=El2+Eν2+ 2ElEν⇒ mπ2=ml2+|~pf|2+|~pf|2+ 2qml2+|~pf|2|~pf| ⇒ (mπ2−ml2−2|~pf|2)2= 4(ml2+|~pf|2)|~pf|2⇒ mπ4+ml4+  4|~pf|4−2mπ2ml2−4mπ2|~pf|2+ 4ml2|~pf|2= 4ml2|~pf|2+  4|~pf|4⇒ 4mπ2|~pf|2= (mπ2−ml2)2⇒ |~pf|=mπ2−ml2 2mπ .(4.28) It is clear,1 p1µ= (Eπ, ~pπ) = (mπ,0) , p2µ= (Eν, ~pν)=(|~pf|, ~p2),(4.29) p3µ= (El, ~pl) = (qml2+|~pf|2, ~p3).(4.30) We now calculate the scalar products p12,p1·p2,p1·p3,p2·p3. p12=p1µp1µ⇒p12=mπ2.(4.31) p1·p2=p1µp2µ=mπ|~pf| ⇒ p1·p2=mπ2−ml2 2.(4.32) 1We regard the neutrino as an ultrarrelativistic particle (mν≃0). 45 It is useful for p1·p3and p2·p3calculations to obtain previously El, El=qml2+|~pf|2=sml2+(mπ2−ml2)2 4mπ2=sml2+mπ4+ml4−2mπ2ml2 4mπ2= s4mπ2ml2+mπ4+ml4−2mπ2ml2 4mπ2=smπ4+ml4+ 2mπ2ml2 4mπ2=s(mπ4+ml2)2 4mπ2⇒ El=qml2+|~pf|2=mπ2+ml2 2mπ .(4.33) Using (4.33), p1·p3=p1µp3µ=mπqml2+|~pf|2⇒p1·p3=mπ2+ml2 2.(4.34) p2·p3=p2µp3µ=|~pf|qml2+|~pf|2+|~pf|2=|~pf|qml2+|~pf|2+|~pf|= mπ2−ml2 2mπ mπ2+ml2 2mπ +mπ2−ml2 2mπ!=mπ2−ml2 2  mπ   mπ⇒ p2·p3=mπ2−ml2 2.(4.35) Inserting (4.28), (4.31), (4.32), (4.34), (4.35) into (4.25), we obtain Γ = GF2fπ2 2πmπ2 mπ2−ml2 2mπ  2mπ2−ml2 2 mπ2+ml2  2−mπ2mπ2−ml2 2!, Γ = GF2fπ2 2πmπ2 mπ2−ml2 2mπ mπ2−ml2 2h(  mπ2+ml2)−  mπ2i, Γ = GF2fπ2 8π ml2 mπ3mπ2−ml22.(4.36) Using (3.14), we can also write Γ = fπ2 π gw 4MW!4ml2 mπ3mπ2−ml22.(4.37) 46 With this result, we now compare the decay rate for the two main paths of decay of the π−, Γ(π−→e−+νe) Γ(π−→µ−+νµ)=me2(mπ2−me2) mµ2(mπ2−mµ2)= 1,283 ·10−4,(4.38) having extracted the values of the masses from http://pdg.lbl.gov, mπ= 139,571 MeV , me= 0,511 MeV , mµ= 105,658 MeV . The experimental value of the above quotient is (1,230 ±0,004) ·10−4[2], which practically coincides with (4.38). It is interesting to calculate the density of final states for both paths of decay and compare the result with (4.38). For a two particle decay, we can write in the COM frame [24], ρf=4πV (2π)3|~pf|ElEν Eπ .(4.39) Regarding the neutrino as an ultrarrelativistic particle as we previously did, we obtain ρf=4πV (2π)3|~pf|2El mπ .(4.40) Inserting (4.33) and (4.28) into (4.40), the density of final states ρfreads ρf=πV (2π)3mπ2−ml22mπ2+ml2 2mπ4.(4.41) With this result, we now compare the density of final states for the two main paths of decay of the π−, ρf(π−→e−+νe) ρf(π−→µ−+νµ)=mπ2−me22mπ2+me2 mπ2−mµ22mπ2+mµ2= 3,489 .(4.42) We see the density of final states is greater for the electron path of decay than for the muon path. Then, intuitively, we could think the decay rate Γ will be greater for the electron path of decay. Nevertheless, (4.38) tells us the opposite. The response to this 47 Figure 4.3: Final particles for the π−decay in COM frame. In red, the spin projection of both particles. resides in the nature of neutrinos and weak interaction, and hence in the matrix element of the process Mfi. Considering neutrinos as massless particles, we can claim the antineutrino will always have positive helicity (spin projection parallel to its momentum). It is then clear from conservation laws (linear momentum and spin angular momentum) that the lepton must also have positive helicity, as we can see in Figure 4.3. If the lepton were massless, it would have negative helicity, just like a neutrino. This goes against the recently mentioned condition required by conservation laws. That is why the closer to zero the lepton mass is, the less likely the process will be. In that sense, the electron path of decay is less likely than the muon path. It is important to notice how the fact that massless particles always have negative helicity and massless antiparticles always have positive helicity was mathematically introduced in previous calculations. The key is the addition of the 1 −γ5factor (this factor does not appear in electromagnetic interaction) in the matrix element Mfi. This factor acts as a projector for massless particles, picking out the negative helicity components of the spinor for particles and the positive helicity components for antiparticles. See [25] for a clear mathematical development of these arguments. 48 Chapter 5 Discussion The general objective of this Bachelor thesis was twofold. Firstly, present a general overview on neutrino history and show some of its most important properties. Secondly, apply the general Feynman-St¨uckelberg propagator formalism and the Fermi golden rule to a scattering process and a decay, respectively. We have started from basic principles and have performed all the calculations in detail in order to reach the final expressions for σand Γ. These final results coincide with those presented (without details) in the literature [2,19]. The general topic studied in this Bachelor thesis connects directly with the contents of the courses on Relativistic Quantum Mechanics and Particle Physics. The specific evaluation of neutrino-lepton scattering as well as pion decay has needed a cumbersome analytical calculation involving gamma matrices trace properties that have been presented in detail in the work memory. Furthermore, the final results, that have been also computed using numerical codes, have been presented in several graphs and discussed at length. The pion decay exposed some of the most distinctive properties of the weak interaction: parity violation and the particular helicity of neutrinos. It is important to remark the interest of this process, since it constitutes the main method for generating neutrino beams. Moreover, (4.38) explains why most of neutrinos generated in an accelerator are muon neutrinos and only a few are electron neutrinos. It is also noteworthy the importance of the pion decay for the measurement of neutrinos 49 helicity. As already mentioned, neutrinos barely interact, what makes them very hard to detect. Consequently, it is tough to measure their helicity directly. Nevertheless, the pion decay provides a method to measure it indirectly: measure the muon helicity and apply conservation laws. As we see, the study of these processes provides a deeper knowledge on the behaviour of neutrinos and the weak interaction, and allows us to know more about its particular properties. Current experiments all around the world develop the experimental setup needed for the observation and data collection of these processes, especially neutrinonucleus interactions, what allows us to find more information about neutrino oscillations (mixing angles, CP violation phase, etc) and nucleus structure. 50 Appendix A Relativistic quantum mechanics for spin 1/2 particles. We present the main results of Dirac theory for spin 1/2 particles, all based on the wellknown equation proposed by the English physicist in 1928. For a detailed study, [26] can be consulted. A.1 Dirac equation Dirac equation for free particles reads i∂ψ ∂t = (ˆ α·ˆ p+βM)ψ="−i α1 ∂ ∂x1+α2 ∂ ∂x2+α3 ∂ ∂x3!+βM#ψ≡ˆ H ψ , (A.1) with αi, β the 4x4 Dirac matrices given by αi=   0σi σi0  , β =   I0 0I  ,(A.2) where σidenote the Pauli matrices, σ1=σx=   0 1 1 0  , σ2=σy=   0−i i0  , σ3=σz=   1 0 0−1  . (A.3) 51 Since Dirac equation describes relativistic spin 1/2 particles, it can be written in a covariant form, (i/ ∂−M)ψ= 0 ,(A.4) with / ∂=γµ∂µand γµmatrices constructed with αi, β, γ0≡β, γi≡βαi, i = 1,2,3.(A.5) Dirac matrices γµsatisfy {γµ, γν}=γµγν+γνγµ= 2gµν I.(A.6) Another relations of great utility during this work are given below, (γµ)†=γ0γµγ0,(A.7) γ5≡i γ0γ1γ2γ3,(A.8) (γ5)2=I,(γ5)†=γ5,nγµ, γ5o= 0 .(A.9) A.2 Bilinear covariants Dirac matrices belong to 4x4 matrices space. Consequently, there exists a set of 16 linearly independent matrices within the Dirac algebra. These are given by ΓS=I,ΓV µ=γµ,ΓT µν =σµν ≡i 2[γµ, γν],ΓP=γ5,ΓA µ=γ5γµ, (A.10) where S, V, T, P, A allude to scalar, vector, tensor, pseudoscalar and axial vector, respectively. These names are directly linked with the properties of these matrices under Lorentz transformations. If Λ defines the Lorentz transformation and ψ=ψγ0, we can then write1 ψ0(x0)ΓSψ0(x0) = ψ(x)ΓSψ(x)→Scalar (A.11) ψ0(x0)ΓV µψ0(x0) = Λν µψ(x)ΓV νψ(x)→Vector (A.12) 1xrefers to the space-time four-vector. 52 B.3 Leptonic tensor contraction LµνRµν Now, we are in conditions to calculate the tensor contraction LµνRµν, LµνRµν = 2 m1m2m3m4p3µp1νp4µp2ν+p3µp1νp2µp4ν−p3µp1νgµν p4·p2−ip3µp1νµνσλ p4σp2λ+ p1µp3νp4µp2ν+p1µp3νp2µp4ν−p1µp3νgµν p4·p2−ip1µp3νµνσλ p4σp2λ+ −gµν p3·p1p4µp2ν−gµν p3·p1p2µp4ν+gµν p3·p1gµν p4·p2+igµν p3·p1µνσλ p4σp2λ+ −i µναβ p3αp1βp4µp2ν−i µναβ p3αp1βp2µp4ν+i µναβ p3αp1βgµν p4·p2−µναβ p3αp1βµνσλ p4σp2λ= 2 m1m2m3m4p1·p2p3·p4+p1·p4p2·p3−(((((( ( p1·p3p2·p4−iµνσλp3µp1νp4σp2λ+ p1·p4p2·p3+p1·p2p3·p4−(((((( ( p1·p3p2·p4−iµνσλp1µp3νp4σp2λ+ −(((((( ( p1·p3p2·p4−(((((( ( p1·p3p2·p4+((((((( ( 4p1·p3p2·p4+ip1·p3gµν µνσλ |{z } 0 p4σp2λ+ −i µναβ p4µp2νp3αp1β−i µναβ p2µp4νp3αp1β+i p2·p4gµνµναβ | {z } 0 p3αp1β−µναβµνσλ p3αp1βp4σp2λ Notice gµν is only different from 0 when µ=ν, but in that case µνσλ = 0. Then, gµν µνσλ = 0. LµνRµν =2 m1m2m3m42p1·p2p3·p4+ 2 p1·p4p2·p3−i µνσλ(p1µp3ν+p3µp1ν) | {z } 0 p4σp2λ+ −i µναβ (p2µp4ν+p4µp2ν) | {z } 0 p3αp1β−µναβµνσλ p3αp1βp4σp2λ (B.18) Since the Levi-Civita tensor µνσλ is antisymmetric, it is clear those two sums above 59 are 0. To determine the last term we use a tensorial identity given in [2], µναβµνσλ =−2(δα σδβ λ−δα λδβ σ) (B.19) Inserting (B.19) into (B.18), we obtain LµνRµν =2 m1m2m3m42p1·p2p3·p4+ 2 p1·p4p2·p3+ 2 (δα σδβ λ−δα λδβ σ)p3αp1βp4σp2λ= 2 m1m2m3m42p1·p2p3·p4+((((((( ( 2p1·p4p2·p3+ 2 p1·p2p3·p4−((((((( ( 2p1·p4p2·p3 (B.20) Then, we can finally write the tensor contraction LµνRµν as follows, LµνRµν =8 m1m2m3m4 p1·p2p3·p4(B.21) 60 Bibliography [1] CHADWICK J. Intensit¨atsverteilung im magnetischen Spectrum der β-Strahlen von radium B + C. Verhandl. Dtsc. Phys. Ges., 16:383, 1914. [2] GRIFFITHS D. Introduction to elementary particles. Wiley-VCH Verlag, M¨orlenbach (Germany), 2nd edition, 2011. [3] PAULI W. 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