Gauge_theories

Gauge Theories of the Strong and Electroweak Interactions G. Münster, G. Bergner Summer term 2011 Notes by B. Echtermeyer Is nature obeying fundamental laws? Does a comprehensive description of the laws of nature, a kind of theory of everything, exist? Gauge theories and symmetry principles provide us with a comprehensive description of the presently known fundamental particles and interactions. The Standard Model of elementary particle physics is based on gauge theo- ries, and the interactions between the elementary particles are governed by a symmetry principle, namely local gauge invariance, which represents an infinite dimensional symmetry group. These notes are not free of errors and typos. Please notify us if you find some. Contents 1 Introduction 4 1.1 Particles and Interactions . . . . . . . . . . . . . . . . . . . . 4 1.2 Relativistic Field Equations . . . . . . . . . . . . . . . . . . . 13 1.2.1 Klein-Gordon equation . . . . . . . . . . . . . . . . . . 13 1.2.2 Dirac equation . . . . . . . . . . . . . . . . . . . . . . 16 1.2.3 Maxwell’s equations . . . . . . . . . . . . . . . . . . . 19 1.2.4 Lagrangian formalism for fields . . . . . . . . . . . . . 22 1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.1 Symmetries and conservation laws . . . . . . . . . . . . 30 1.3.2 U(1) symmetry, electric charge . . . . . . . . . . . . . . 32 1.3.3 SU(2) symmetry, isospin . . . . . . . . . . . . . . . . . 34 1.3.4 SU(3) flavour symmetry . . . . . . . . . . . . . . . . . 42 1.3.5 Some comments about symmetry . . . . . . . . . . . . 44 1 2 CONTENTS 1.4 Field Quantisation . . . . . . . . . . . . . . . . . . . . . . . . 46 1.4.1 Quantisation of the real scalar field . . . . . . . . . . . 47 1.4.2 Quantisation of the complex scalar field . . . . . . . . . 52 1.4.3 Quantisation of the Dirac field . . . . . . . . . . . . . . 54 1.4.4 Quantisation of the Maxwell field . . . . . . . . . . . . 55 1.4.5 Symmetries and Noether charges . . . . . . . . . . . . 57 1.5 Interacting Fields . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.5.1 Interaction picture . . . . . . . . . . . . . . . . . . . . 58 1.5.2 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . 61 1.5.3 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . 62 1.5.4 Feynman diagrams . . . . . . . . . . . . . . . . . . . . 63 1.5.5 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.5.6 Limitations of the perturbative approach . . . . . . . . 68 2 Quantum Electrodynamics (QED) 69 2.1 Local U(1) Gauge Symmetry . . . . . . . . . . . . . . . . . . . 69 2.2 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . 71 3 Non-abelian Gauge Theory 74 3.1 Local Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . 74 3.2 Geometry of Gauge Fields . . . . . . . . . . . . . . . . . . . . 80 3.2.1 Differential geometry . . . . . . . . . . . . . . . . . . . 80 3.2.2 Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . 82 4 Quantum Chromodynamics (QCD) 87 4.1 Lagrangian Density and Symmetries . . . . . . . . . . . . . . 87 4.1.1 Local SU(3) colour symmetry . . . . . . . . . . . . . . 88 4.1.2 Global flavour symmetry . . . . . . . . . . . . . . . . . 90 4.1.3 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . 91 4.1.4 Broken chiral symmetry . . . . . . . . . . . . . . . . . 95 4.2 Running Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2.1 Quark-quark scattering . . . . . . . . . . . . . . . . . . 98 4.2.2 Renormalisation . . . . . . . . . . . . . . . . . . . . . . 100 4.2.3 Running coupling . . . . . . . . . . . . . . . . . . . . . 101 4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3 Confinement of Quarks and Gluons . . . . . . . . . . . . . . . 105 4.4 Experimental Evidence for QCD . . . . . . . . . . . . . . . . . 108 5 Electroweak Theory 111 5.1 Weak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.1.1 Fermi theory of weak interaction . . . . . . . . . . . . 111 CONTENTS 3 5.1.2 Parity violation . . . . . . . . . . . . . . . . . . . . . . 111 5.1.3 V-A theory . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2.1 Spontaneous breakdown of a global symmetry . . . . . 117 5.2.2 Higgs mechanism . . . . . . . . . . . . . . . . . . . . . 119 5.3 Glashow-Weinberg-Salam Model . . . . . . . . . . . . . . . . . 121 4 1 INTRODUCTION 1 Introduction 1.1 Particles and Interactions When reflecting on the constituents of matter, one is lead to the physics of elementary particles. A classification of elementary particles is done by re- garding their properties, which are mass, spin, (according to the representations of the inhomogeneous Lorentz group) lifetime, additional quantum numbers, (obtained from conservation laws) participation in interactions. From these properties the following classification arose. Leptons e−, νeelectron number µ−, νµmuon number τ−, ντtauon number Hadrons strongly interacting particles Mesons integer spin, baryon number=0 π+,π−,π0,K+,K−,K0,η,ρ+,ρ−,ρ0,J/ψetc. Baryons half integer spin, baryon number= ±1 n,p,Λ0,Σ,Ξ,∆,Ω−,Yetc. Quark model of hadrons (Gell-Mann, SU(3), eightfold way) The hadrons are build out of two or three quarks. Mesons q¯q (quark, antiquark) Baryons qqq There are six quarks and their antiparticles. They all have spin 1/2. The six quark types are called “flavours”, which are denoted by u, c, t d, s, b 1.1 Particles and Interactions 5 Some baryons some mesons p=uud π+=u¯d n=udd K+=u¯s Λ =sud ρ+=u¯d Ω−=sss D+=c¯d ηc=c¯c 3 Generations of constituents mass [MeV] Q B νe≈0 0 0 e−0.511−1 0 u≈4 2/3 1/3 d≈7−1/3 1/3 νµ≈0 0 0 µ−105.66−1 0 c≈1300 2 /3 1/3 s≈150−1/3 1/3 ντ<18.2 0 0 τ−1777.0−1 0 t≈174 000 2 /3 1/3 b≈4 200−1/3 1/3 Table 1: The 3 generations of elementary particles 6 1 INTRODUCTION Fig. 1 and Fig. 2 show multiplets of mesons and baryons arranged in 3- dimensional multiplets1. The coordinates are (x,y,z ) = (IsospinI,Hypercharge Y,CharmC) Figure 1: SU(4) multiplets of mesons; 16-plets of pseudoscalar (a) and vector mesons (b). In the central planes the c¯cstates have been added. – From The Particle Data Group, 2010. 1The meson multiplets form an Archimedean solid called cubooctahedron 1.1 Particles and Interactions 7 Figure 2: SU(4) multiplets of baryons. (a) The 20-plet with an SU(3) octet. (b) The 20-plet with an SU(3) decuplet. – From The Particle Data Group, 2010. Quark confinement Quarks do not exist as single free particles. There is an additional quantum number, called“colour”. E.g., Ω−=ssshasspin 3/2; thereforethewavefunc- tion has to be antisymmetric in the spin-coordinates. It is also symmetric in space coordinates, so the Pauli-principle can only be fulfilled, if the three charmed quarks are different in some additional quantum number. Allhadronsare colourless combinationsofquarks. Thisphenomenoniscalled confinement. 8 1 INTRODUCTION There is a characteristic feature for each single generation of leptons and quarks:/summationdisplay Qi= 0. e−νedr, dg, dbur, ug, ub BQ 1 0 −1 0 1 The reason, why (ν,e−)and(u,d)belong to this same generation and not, for instance (ν,e−)and (c,s)will be given later in the chapter on weak interactions. Interactions An important guiding principle in the history of understanding interactions has been unification. When Newton postulated that the gravitational force which pulls us down to earth and the force between moon and earth are es- sentially the same, this was a step towards unification of fundamental forces, as was the unification of magnetism and electricity by Faraday and Maxwell, which led to a new understanding of light, or – about a century later – the unification of electromagnetism and weak interactions. Nowadays one distinguishes four fundamental interactions: a) Electromagnetic interactions. They apply to electrically charged par- ticles only (no to neutrinos, for instance). Since the electrostatic force is proportional to 1/r2, one says that the rangeof the electromagnetic interactions is infinite. A further charac- teristic of interactions is their relative strength , when compared with the strength of other interactions. For electromagnetism it is given by Sommerfeld’s “Feinstrukturkonstante” α. range =∞ (1.1) relative strength =e2 4π/epsilon10~c≈1 137(1.2) 1.1 Particles and Interactions 9 b) Weak interactions. They are responsible for the β- decay and other processes. range≈10−18m (1.3) relative strength≈10−5(1.4) c) Strong interactions. They are responsible for the binding of quarks and for the hadronic interactions. Nuclear forces are also remnants of the strong interactions. range≈10−15m (1.5) relative strength≈1 (1.6) d) Gravitation acts on every sort of matter. E.g., it has been shown ex- perimentally that a neutron falls down through a vacuum tube just like any other object on earth. The gravitational force is always attractive. Whereas positive and negative electric charges exist, there are no neg- ative masses and thus the gravitational force cannot be screened. The range of this force is infinite like that of electromagnetism. Comparing the gravitational force between proton and electron in an H-atom with their electrostatic attraction, one finds that the gravitational force is extremely weak. range =∞ (1.7) relative strength≈10−39(1.8) Forces are mediated by the exchange of bosons. The range is given by the Compton wavelength of the exchange boson. (But there is an exception to this law in QCD due to confinement.) rangeR≈~ mc(1.9) Interaction bosons spin mass, range electromagnetic photon γ 1m= 0, R=∞ weak W+, W−, Z01mW= 80.4GeV mZ= 91.2GeV strong gluons G 1m= 0, R/negationslash= 0 gravitation graviton 2 m= 0 For gluons the spin 1 is a consequence of gauge theory, and the finite range Rarises from confinement, which holds for gluons as for quarks. The spin of the exchange boson is related to the possibility of a force being only at- tractive or both attractive and repulsive. Spin 2 implies that there is only 10 1 INTRODUCTION attraction. The existence of the graviton with zero mass is predicted theo- retically and may never be verified by experiment. Measuring gravitational waves is already very challenging, and to identify the quanta of these waves would be extremely difficult. Theories a) Quantum Electrodynamics originated in 1927, when in an appendix to thearticleofBorn, HeisenbergandJordanaboutmatrixmechanicsJor- dan quantised the free electromagnetic field. It was developed further by Dirac, Jordan, Pauli, Heisenberg and others and culminated before 1950 in the work of Tomonaga, Schwinger, Feynman and Dyson. The calculation of the Lamb shift and the exact value of the gyromagnetic ratiogof the electron are highlights of QED. Here is an example of a Feynman diagram for the scattering of two electrons by exchanging a photon. e−e−e− e− The vertex stands for a number, in QED this is α≈1/137. The propagation of electrons is affected by the emission and absorption of virtual photons, as shown in the following Feynman diagram. b) The theory of weak interactions begun in 1932 with Fermi’s theory for theβ−-decay. The Feynman graph for the decay of neutrons involves a 4–fermion coupling. 1.1 Particles and Interactions 11 n¯νee−p Improvements of the theory of β-decay in nucleons were made by the V-A theory, taking care of parity violation. Theoretical problems: while in QED perturbation theory in powers of αworks extremely well, it leads to infinities in the Fermi theory of weak interactions. The problems were overcome in 1961 – 1968 by Glashow, Weinberg, Salam and others, developing the unified theory of weak and electromagnetic interactions. The bosons mediating the electroweak interactions are Vector bosons W±,Z0and photon γ. c) Strong interactions between quarks are described by Quantum Chro- modynamics (QCD), which was formulated by Fritzsch, Gell-Mann and Leutwyler, and further developed by ’t Hooft and others. There are three “strong charges”, sources for the forces, named red, green and blue charge. The gauge bosons which mediate strong interactions are called gluons. Unlike the electrically neutral photons in QED, gluons carry colour charges themselves and interact with each other. Due to their self- interactions, gluons may form glueballs, and a “theory of pure glue” is a non-trivial theory. q q Feynman diagrams with quarks and gluons d) Gravitation is described by General Relativity (GRT), a nonlinear the- ory. A quantum theory of gravitation is not yet known. String theory, Superstring theory or Loop gravity might be candidates. 12 1 INTRODUCTION The Standard Model This means the theory of Glashow, Weinberg and Salam (G.W.S.) plus QCD. There is no mixing between the Lagrangians for electroweak and strong in- teractions, therefore, we do not speak of a unification of these interactions. The theoretical predictions of the Standard Model are so far consistent with the experimental results. Common to all parts of the Standard Model are exchange bosons, which are related to gauge fields showing local gauge symmetries. (Gravitation is also based on a local symmetry.) Gauge theories are based on gauge groups. The groups belonging to the Standard Model are SU(3)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright QCD⊗SU(2)⊗U(1)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright G.W.S.. (1.10) The principles of the Standard Model are: •local gauge symmetry, •Higgs mechanism giving masses to W±,Z0and quarks. The Higgs mechanism is due to P. Anderson, F. Englert, R. Brout, P. Higgs, G. Guralnik, C. R. Hagen and T. Kibble. It uses the Higgs field, associated with a Higgs-boson. This does not fit into a local gauge theory, so the Higgs boson might not be a fundamental particle. There is no other reason for the Higgs field than the mechanism to give the above mentioned masses. Outlook A further unification of interactions is attempted in Grand Unified Theories (GUT). The idea is to extend the semisimple2Lie group SU(3)⊗SU(2)⊗U(1) to a simple Lie group as for example SU(5),SO(10)or the exceptional Lie groupE6. GUTs predict proton decay and several Higgs particles. 2A group is called semisimple, if it is the direct product of simple groups. A group is simple, if it has no normal subgroups besides the trivial ones. 1.2 Relativistic Field Equations 13 1.2 Relativistic Field Equations In classical physics there are two distinct kinds of objects: particles – point particles or continuous distributions of mass – and secondly fields, like grav- itational or electromagnetic fields. In quantum mechanics the dichotomy be- tween particles and fields is upheld, although the wave-particle duality shows up. But in the relativistic quantum mechanics of particles one encounters contradictions. These are resolved in Quantum Field Theory (QFT). QFT deals with quantised fields, functions of space and time, where the values of the fieldsf(/vector r,t)themselves become operators. fieldf−→operator. QFT is a quantum theory of many particles. In this lecture we consider the three most prominent relativistic field equations, •Klein-Gordon equation for spin 0 particles, •Dirac equation for spin 1/2particles, •Maxwell’s equations for massless spin 1 particles. There are other relativistic equations, too (Proca, etc). In QFT the spin of fundamental fields does not exceed 2. 1.2.1 Klein-Gordon equation For a non-relativistic free particle the equation E=/vector p2 2m(1.11) together with de Broglie’s plane wave ansatz ψ=Aei(/vectork·/vector r−ωt), E =~ω, /vector p =~/vectork (1.12) leads to the non-relativistic Schrödinger equation i~∂ ∂tψ=−~2 2m∇2ψ. (1.13) For a relativistic particle with E=~ω=cp0energy and momentum are components of a four-vector and our starting point is the equation for the square of the 4-momentum E2=c2/vector p2+m2c4, (1.14) 14 1 INTRODUCTION which leads to −~2∂2 ∂t2φ=−c2~2∇2φ+m2c4φ or /parenleftBigg −∂2 ∂(ct)2+∇2−m2c2 ~2/parenrightBigg φ= 0. (1.15) ThisistheKlein-Gordonequation, inventedbySchrödinger, Fockandothers, and rediscovered by Klein and Gordon. Relativistic notations x0=ct, x1=x, x2=y, x3=z x= (x0, x1, x2, x3) = (x0, /vector x) = (xµ) gµν= 1 −1 −1 −1  x·y=x0y0−/vector x·/vector y=xµxµ, xµ=gµνxν ∂µ=∂ ∂xµ=/parenleftBigg1 c∂ ∂t,∇/parenrightBigg ∂µ=∂ ∂xµ=/parenleftBigg1 c∂ ∂t,−∇/parenrightBigg =−∂µ∂µ=−∂2 ∂(ct)2+ ∆ pµ=/parenleftbiggE c, /vector p/parenrightbigg p2=pµpµ=E2 c2−/vector p2=m2c2Note3 pµ→i~∂µde Broglie /parenleftBigg −m2c2 ~2/parenrightBigg φ(x) = 0 Klein-Gordon From now on we use natural units setting ~=c= 1.4 3The symbol p2is ambiguous. Its meaning must be determined from the context. 4To go back to SI-units in an equation one may analyse the dimension of the terms and insert ~and/orcto get the right dimension. 1.2 Relativistic Field Equations 15 Solution of the Klein-Gordon equation Letφ(x)be a complex scalar field ( φ∈C), that means, it is not quantised yet (φis not operator-valued), Spin = 0 ( φis scalar). It will turn out that complex scalar fields describe particles with positive and negative charges. Examples are the mesons π+andπ−. A general solution to the Klein-Gordon equation for free particles, being linear and of second order, is a superposition of two plane waves φ(x) =/integraldisplayd3k (2π)32ωk/braceleftBig a(k) e−ikx+b∗(k) eikx/bracerightBig . (1.16) Here we denoted ωk=k0=/radicalBig /vectork2+m2 (1.17) to be apositive frequency. The solution is verified by ∂µ∂µeikµxµ= (∂02−∂j∂j) ei(k0x0−kjxj) = i2(k0k0−kjkj)eikµxµ =−kµkµeikµxµ (−m2)eikµxµ= (kµkµ−m2)eikµxµ =/braceleftbigg (k0)2−(/vectork2+m2)/bracerightbigg eikµxµ= 0. Now letφ(x)be arealscalar field, φ∈R, being used for neutral spin 0 particles, like π0. Then the general solution is φ(x) =/integraldisplayd3k (2π)32ωk/braceleftBig a(k) e−ikx+a∗(k) eikx/bracerightBig . (1.18) The problem with negative frequencies ei(/vectork·/vector x−ωt)⇒Eφ= i~∂tφ= +~ωφ e−i(/vectork·/vector x−ωt)⇒Eφ= i~∂tφ=−~ωφ So, free particles could have arbitrarily large negative energies, which is un- physical. In the presence of interactions, e.g. with the electromagnetic field, this would lead to instabilities, because a particle would jump to lower and lower states, emitting an unbounded amount of energy. This problem will be solved by field quantisation. 16 1 INTRODUCTION 1.2.2 Dirac equation The Dirac equation was found by P. A. M. Dirac in 1928. He was searching for a covariant version of the Schrödinger equation i∂tψ=Hψ. (1.19) Tobemanifestlycovariant, ithastobeoffirstorderinthespatialderivatives, too. Hlinear in∂ ∂xk(k= 1,2,3), H=3/summationdisplay k=1αkPk+βm=−3/summationdisplay k=1αki∂k+βm. (1.20) We will now derive conditions for the constant terms αkandβ. Squaring both sides of the equation we get −(∂0)2ψ=H2ψ (1.21) =1 23/summationdisplay j,k=1(αjαk+αkαj)PjPkψ −m3/summationdisplay k=1(αkβ+βαk)Pkψ +β2m2ψ. Consider a plane wave ψ= eipx, for which one should have i∂0ψ=Eψ,/vectorPψ=/vector pψ, E2=/vector p2+m2. (1.22) This wave satisfies the Dirac equation only if αjαk+αkαj= 2δjk1 (1.23) αkβ+βαk= 0 (1.24) β2=1. (1.25) From this one concludes that αkandβcannot be numbers. The relations can be satisfied by matrices, which must at least be of size 4 by 4. They can be composed by blocks of Pauli spin matrices αk=/parenleftBigg 0σk σk0/parenrightBigg , β =/parenleftBigg 10 0−1/parenrightBigg . (1.26) 1.2 Relativistic Field Equations 17 By convention one uses the Dirac matrices γµ: γ0:=β, γk:=βαk, (k= 1,2,3) (1.27) The Hamiltonian can be written H=−γ03/summationdisplay k=1γki∂k+γ0m, and multiplying with γ0we get γ0i∂0ψ=γ0Hψ=−3/summationdisplay k=1iγk∂kψ+mψ. This is the Dirac equation, reading (iγµ∂µ−m)ψ(x) = 0. (1.28) Equivalent notations of Dirac’s equation are (γµPµ−m)ψ(x) = (p /−m)ψ(x) = 0. The algebra of the γ-symbols is γµγν+γνγµ= 2gµν1. (1.29) The matrices given above are Dirac’s representation of the γ’s. There are others representations, e.g. by Weyl or by Majorana. The Dirac matrices written in blocks of Pauli spin matrices are γ0=/parenleftBigg 10 0−1/parenrightBigg , γk=/parenleftBigg 0σk −σk0/parenrightBigg . (1.30) Solutions of Dirac’s equation will be given by spinor wavefunctions or fields ψ(x) = ψ1(x)... ψ4(x) . (1.31) These are made out of two kinds of plane waves, given by ψ(x) =u(k)e−ikx, k0=ωk>0 (1.32) with 2 independent spinors u(r)(k), r= 1,2, and ψ(x) =v(k)eikx, k0=ωk>0 (1.33) 18 1 INTRODUCTION with another 2 independent spinors v(r)(k), r= 1,2. The general solution is a superposition ψ(x) =/integraldisplayd3k (2π)32ωk2/summationdisplay r=1/braceleftBig br(/vectork)u(r)(/vectork) e−ikx+d∗ r(/vectork)v(r)(/vectork) eikx/bracerightBig .(1.34) Spin The Dirac Hamiltonian Hand the orbital angular momentum operator /vectorL= /vectorR×/vectorPdo not commute /bracketleftBig/vectorL, H/bracketrightBig /negationslash= 0. (1.35) The angular momentum of free particles should be conserved! So there must be an additional hidden contribution to the angular momentum, which is called spin. /vectorS=~ 2/vectorΣ =~ 2/parenleftBigg /vector σ0 0/vector σ/parenrightBigg . (1.36) The algebra of /vectorSis given by [Sk,Sl] = iSm (k, l, m ) = (1,2,3) +cycl. (1.37) /vectorS2=3 41=s(s+ 1)1⇒s=1 2. (1.38) The total angular momentum /vectorJ=/vectorL+/vectorSobeys [/vectorJ, H ] = 0. (1.39) Thus the total angular momentum is conserved. Notice: The last commutator can be verified with the help of Σ1=1 2[γ2, γ3]. (1.40) Covariant expressions To write down Lorentz covariant expressions with ψ ψ(x) = ψ1(x)... ψ4(x), , ψ†=/parenleftBig ψ∗ 1(x), ...,ψ∗ 4(x)/parenrightBig ,(1.41) we define the Dirac conjugate ¯ψ(x) =ψ†(x)γ0. (1.42) 1.2 Relativistic Field Equations 19 Covariant scalar and vector expressions are ¯ψ(x)ψ(x), ¯ψ(x)γµψ(x); (1.43) Objects which transform under an antisymmetric tensor representation of the Lorentz group are ¯ψσµνψ, σ µν= [γµ, γν]. (1.44) With γ5=γ5:= iγ1γ2γ3γ4=/parenleftBigg 01 10/parenrightBigg (1.45) pseudoscalars and pseudovectors are given by ¯ψγ5ψ, ¯ψγ5γµψ. (1.46) 1.2.3 Maxwell’s equations Maxwell’s equations in the MKSA system read ∇·/vectorE=ρ /epsilon10(1.47) ∇·/vectorB= 0 (1.48) ∇×/vectorE=−∂/vectorB ∂t(1.49) ∇×/vectorB=µ0/vectorj+µ0/epsilon10∂/vectorE ∂t(1.50) InQFToftentheHeaviside-Lorentzunitsystemisused. Conversionformulae are: /vectorEH=√/epsilon10/vectorE (1.51) /vectorBH=1 µ0/vectorB (1.52) ΦH=√/epsilon10Φ (1.53) /vectorAH=1√µ0/vectorA (1.54) ρH=1√/epsilon10ρ (1.55) /vectorjH=1√/epsilon10/vectorj. (1.56) 20 1 INTRODUCTION Now the Maxwell equations in Heaviside-Lorentz units read ∇·/vectorE=ρ (1.57) ∇·/vectorB= 0 (1.58) ∇×/vectorE+1 c∂/vectorB ∂t= 0 (1.59) ∇×/vectorB−1 c∂/vectorE ∂t=/vectorj (1.60) The fields can be derived from potentials /vectorB=∇×/vectorA,/vectorE=−∇Φ−1 c∂/vectorA ∂t(1.61) Equivalently the potentials are written in covariant form Aµ(x) := (Φ(x),/vectorA(x)). (1.62) From these the field strengths are derived by Fµν=∂µAν−∂νAµ, (µ, ν= 0,1,2,3) (1.63) Fµν= 0−Ex−Ey−Ez Ex0−BzBy EyBz 0−Bx Ez−ByBx 0 (1.64) or Ei=Fi0, Bi=−1 2/epsilon1ijkFjk. (i,j,k∈{1,2,3}) (1.65) For the the 4-vector current density jµ:= (ρ,/vectorj) (1.66) Maxwell’s equations give the continuity equation ∂µjµ= 0 (1.67) or ∂ ∂tρ+∇·/vectorj= 0. (1.68) Maxwell’s equations themselves may be written in covariant form also: inhomogeneous equations ∂µFµν=jν(1.69) 1.2 Relativistic Field Equations 21 and homogeneous equations ∂µFνρ+∂νFρµ+∂ρFµν= 0. (1.70) Gauge freedom, Lorenz gauge (Ludvig Lorenz, 1867; George F. FitzGerald, 1888) ∂µAµ= 0 (1.71) If one fixes the Lorenz gauge in one inertial frame, then it is fulfilled in all inertial frames. Let us consider again free fields, jµ= 0, (1.72) ∂µFµν= 0. (1.73) Together with the Lorenz gauge we get 0 =∂µFµν=∂µ(∂µAν−∂νAµ) =∂µ∂µAν−∂ν∂µAµ=∂µ∂µAν, Aν= 0. (1.74) To solve the wave equation we take the plane wave ansatz Aµ(x) =/epsilon1(λ) µeikx. (1.75) From Lorenz gauge it follows k·k= 0, k0=|/vectork|=ωk. (1.76) There are remaining superfluous degrees of freedom. The Coulomb gauge for a field free of sources fixes Φ = 0,∇·/vectorA= 0. (1.77) For the plane wave solutions this implies /epsilon10= 0, /epsilon1·k= 0. (1.78) Thusthereare2linearlyindependentsolutions,representingthe2transversal polarisations of radiation /epsilon1(1) µ(k), /epsilon1(2) µ(k)⊥[(1,0,0,0),k]. (1.79) The two transversal polarisations imply that the photon spin ( s= 1) is in the direction of propagation. 22 1 INTRODUCTION For a massive particle moving in a certain direction and having its spin paralleltoitsvelocity,adifferentinertialframecanbechosensuchthatinthis frame the particle moves in the opposite direction and its spin is antiparallel to the velocity. Therefore the projection of its spin on the velocity is not invariant under Lorentz transformation. On the other hand, for massless particles travelling with the velocity of light, the projection of the spin on the velocity is Lorentz-invariant and is called “helicity”: JS=±1. (1.80) The general solution of the electromagnetic wave equation in the Coulomb gauge is Aµ(x) =/integraldisplayd3k (2π)32ωk2/summationdisplay λ=1/epsilon1(λ) µ(k)/parenleftBig a(λ)(k) e−ikx+a(λ)∗(k) eikx/parenrightBig .(1.81) 1.2.4 Lagrangian formalism for fields Recapitulation: Classical mechanics m¨/vector r=−∇V(/vector r), (1.82) H=p2 2m+V(/vector r)with/vector p=m˙/vector r. (1.83) Hamilton’s equations give the equation of motion. Hamilton’s principle uses the actionS, build from the Lagrangian L: S=/integraldisplay dtL(/vector r(t),˙/vector r(t)) (1.84) L=m 2˙/vector r2−V(/vector r). (1.85) The realised trajectories /vector r(t)/vector r1 /vector r0/vector r(t) are such that the action Sis stationary under infinitesimal variations δ/vector r(t) provided the endpoints /vector r(t0)and/vector r(t1)are fixed: /vector r/prime(t) =/vector r(t) +δ/vector r(t) (1.86) δ/vector r(t0) =δ/vector r(t1) = 0. (1.87) δS= 0 (1.88) 1.2 Relativistic Field Equations 23 The calculus of variation leads to the equations of motion: δS=/integraldisplayt1 t0dtδL =/integraldisplayt1 t0dt/summationdisplay i/parenleftBigg∂L ∂xiδxi+∂L ∂˙xiδ˙xi/parenrightBigg (1.89) =/integraldisplayt1 t0dt/summationdisplay i/parenleftBigg∂L ∂xi−d dt∂L ∂˙xi/parenrightBigg δxi= 0. (1.90) The fundamental lemma of the calculus of variation then yields the Euler- Lagrange equations ∂L ∂xi−d dt∂L ∂˙xi= 0. (1.91) This procedure can be taken over to field theory. An advantage is that sym- metries in the action Sdirectly show up as symmetries in the field equations. The Lagrangian density Lin the field variables φaand their derivatives ∂µφa shall be denoted by L(φa(x),∂µφa(x)); (1.92) hereφa(x)and∂µφa(x)take over the rôle of infinitely many coordinates xi and velocities ˙xi, while the argument x= (xµ)ofφa(x)takes over the rôle of timetin mechanics. The action is S=/integraldisplay Gd4xL(φa(x),∂µφa(x)). (1.93) Consider now small variations of φa(x)fixed at the boundary ∂Gof the domain of integration G. Hamilton’s principle δS= 0leads to 0 =/integraldisplay Gd4x/parenleftBigg∂L ∂φa(x)δφa(x) +∂L ∂(∂µφa(x))δ∂µφa(x)/parenrightBigg (1.94) =/integraldisplay Gd4x/parenleftBigg∂L ∂φa(x)−∂µ∂L ∂(∂µφa(x))/parenrightBigg δφa(x). (1.95) Here we performed a partial integration and used the fact that the integrated part vanishes due to δφa= 0on the boundary ∂G. To see this in detail, let Bµ:=∂L ∂(∂µφa(x)), ∂µ(Bµδφa) = (∂µBµ)δφa+Bµ∂µδφa, from Leibniz’s product rule. From Stokes theorem we get /integraldisplay Gd4x∂µ(Bµδφa) =/integraldisplay ∂GdxµBµδφa= 0, 24 1 INTRODUCTION sinceδφa= 0on the boundary ∂G. Thus we can replace Bµ∂µδφaby −(∂µBµ)δφain the integral. We end up with the Lagrange field equations ∂L ∂φa(x)−∂µ∂L ∂(∂µφa(x))= 0. (1.96) Reasons for using Lagrangian densities: a) There is a single function Linstead of many field equations. b) There are advances when non-Cartesian coordinates are used – similar as in mechanics. c) Symmetries can be expressed in a simple manner. Noether theorems lead to conservation laws. d) Gauge theories can be quantised in a simpler way. Real scalar field L=1 2∂µφ∂µφ−m2 2φ2(1.97) The field equations are linear equations, therefore the Lagrangian has to be quadratic in the field and its derivatives. Here we have the simplest expres- sion for Lbeing quadratic and Lorentz invariant. We derive the Lagrange equations of motion: ∂L ∂(∂µφ)=1 2/parenleftBigg ∂µφ+∂λφ∂ ∂(∂µφ)gλν∂νφ/parenrightBigg =1 2/parenleftBigg ∂µφ+gλν∂λφ∂(∂νφ) ∂(∂µφ)/parenrightBigg =1 2(∂µφ+∂νφδνµ) =∂µφ (1.98) ∂µ∂L ∂(∂µφ)=∂µ∂µφ=−φ (1.99) ∂L ∂φ=−m2φ (1.100) This gives the Klein-Gordon-equation /parenleftBig ∂µ∂µ+m2/parenrightBig φ= 0 (1.101) 1.2 Relativistic Field Equations 25 Complex scalar field L=∂µφ∗∂µφ−m2φ∗φ (1.102) φ∗andφare not totally independent complex-valued fields – there are not 4 independent real-valued fields. The derivatives ∂φand∂φ∗also do not give further freedom, as they are connected by Cauchy-Riemann differential equations.5 Weseparate Lintoindependentpartsbymeansof φ=1√ 2(φ1+iφ2), φ∗= 1√ 2(φ1−iφ2): L=1 2∂µφ1∂µφ1+1 2∂µφ2∂µφ2 +i 2∂µφ1∂µφ2−i 2∂µφ2∂µφ1 −1 2m2φ2 1−1 2m2φ2 2 L=1 2/parenleftBig ∂µφ1∂µφ1−m2φ2 1/parenrightBig +1 2/parenleftBig ∂µφ2∂µφ2−m2φ2 2/parenrightBig . (1.103) This is a Lagrangian density for two real scalar fields φ1andφ2, each giving a Klein-Gordon-equation for the real and imaginary parts of φ. (∂µ∂µ+m2)φa= 0 (a= 1,2). (1.104) Sum and difference of both equations give two identical equations: (∂µ∂µ+m2)φ= 0 (1.105) (∂µ∂µ+m2)φ∗= 0 (1.106) Dirac field ψ(x) = (ψ1(x),..., ψ 4(x))/latticetop(1.107) ψis not a 4-vector like xorAµbut a spinor with 4 complex-valued compo- nents. Thus ψdescribes 8 real fields. L(ψ,∂ψ ) =¯ψ(x)(iγµ∂µ−m)ψ(x), (1.108) 5The Cauchy-Riemann differential equations with z=x+ iy, φ (z) =φ1(z) + iφ2(z) read∂φ1 ∂x=∂φ2 ∂y∧∂φ1 ∂y=−∂φ2 ∂x.– Furthermore a term like∂φ∗ ∂φmust not be seen as a derivative in the complex plane, for that would not exist. Instead the partialderivative ∂ ∂φshould be considered as a vector∂ ∂φ= (∂ ∂φ1,∂ ∂iφ2). 26 1 INTRODUCTION where ¯ψ:=ψ†γ0= (ψ∗ 1, ψ∗ 2,−ψ∗ 3,−ψ∗ 4). (1.109) In this form of Lthe derivatives act on ψand not on ¯ψ. A more symmetric alternative would be L=1 2/braceleftBig¯ψ/parenleftBig iγµ− →∂µ−m/parenrightBig ψ+¯ψ/parenleftBig −iγµ← −∂µ−m/parenrightBig ψ/bracerightBig , (1.110) where the arrows indicate whether the derivative acts to the right on ψor to the left on ¯ψ. The two versions for Ldiffer by a total derivative1 2∂µ(¯ψiγµψ), which does not change the field equations. The resulting field equations are the Dirac equation and its Dirac-conjugate equation (iγµ∂µ−m)ψ= 0, (1.111) i∂µ¯ψγµ+m¯ψ= 0. (1.112) Taking the first Lagrangian, Eq. (1.108), we have ∂L ∂ψ=−m¯ψ, ∂ µ∂L ∂(∂µψ)=∂µ¯ψiγµ, ∂L ∂ψ= (iγµ∂µ−m)ψ(x), ∂ µ∂L ∂(∂µψ)= 0. This leads to the given Lagrangian equations (1.111) and (1.112). Maxwell field The expression of the field strengths in terms of the vector potential Aµ(x), Fµν=∂µAν−∂νAµ, (1.113) implies that the homogeneous Maxwell equations ∂µFνρ+∂νFρµ+∂ρFµν= 0 (1.114) are automatically fulfilled. The inhomogeneous equations ∂µFµν=jν(1.115) should be derived from the Lagrangian. Let us look for a Lorentz invariant and gauge invariant Lagrangian. The gauge transformations of electrody- namics, which leave the field strengths invariant, /vectorA/prime=/vectorA+∇χand Φ/prime= Φ−1 c˙χ, 1.2 Relativistic Field Equations 27 read covariantly A/primeµ=Aµ−∂µχ. The Lagrangian cannot contain a mass term m2AµAµsince this is not gauge invariant,A/prime µA/primeµ/negationslash=AµAµ. The field strengths are gauge invariant per defini- tion, therefore the Lagrangian density L(Aµ,∂νAµ) =−1 4FµνFµν(1.116) is both Lorentz invariant and gauge invariant. It is the only such choice being quadratic in the fields. In the Lagrangian, Fµνis to be considered as a function of Aµ(x)and ∂νAµ(x), thus L=−1 4(∂µAν−∂νAµ)(∂µAν−∂νAµ) (1.117) and6 ∂L ∂Aµ= 0,∂L ∂(∂µAν)=Fνµ. (1.118) From this the field equations are ∂µFµν= 0. (1.119) The charges and currents, which enter the inhomogeneous equations, are themselves fields and should be dealt with by other field equations. On the 6 −∂L ∂(∂αAβ)=1 4∂ ∂(∂αAβ)/braceleftbig (∂µAν−∂νAµ)gµλgνρ(∂λAρ−∂ρAλ)/bracerightbig =1 4gµλgνρ/bracketleftbigg∂ ∂(∂αAβ)(∂µAν−∂νAµ)/bracketrightbigg (∂λAρ−∂ρAλ) +1 4gµλgνρ(∂µAν−∂νAµ)/bracketleftbigg∂ ∂(∂αAβ)(∂λAρ−∂ρAλ)/bracketrightbigg =1 4gµλgνρ(δαµδβν−δανδβµ)(∂λAρ−∂ρAλ) +1 4gµλgνρ(∂µAν−∂νAµ)(δαλδβρ−δαρδβλ) =1 4/braceleftbig (δαµδβν−δανδβµ)Fµν+Fλρ(δαλδβρ−δαρδβλ)/bracerightbig =1 4(Fαβ−Fβα+Fαβ−Fβα) =Fαβ 28 1 INTRODUCTION other hand, it is possible to introduce them as external sources jµin the Lagrangian. The sources obey a continuity equation ∂µjµ= 0. (1.120) With the Lagrangian L=−1 4FµνFµν−jµ(x)Aµ(x) (1.121) one gets the inhomogeneous Maxwell equations ∂νFµν+jµ= 0. (1.122) The Lagrangian with external currents is not gauge invariant. The action, however, does not change under a gauge transformation, since /integraldisplay d4xjµ∂µχ=−/integraldisplay d4xχ∂µjµ= 0. (1.123) Lagrangian for a massive vector field The field, denoted by Bµ(x), (1.124) is a Lorentz vector. In analogy to the Maxwell field we define Gµν:=∂µBν−∂νBµ, (1.125) and set L=−1 4GµνGµν+m2 2BµBµ. (1.126) With∂L ∂Bµ=m2Bµ,∂L ∂(∂µBν)=Gνµ(1.127) we get the field equations ∂νGµν−m2Bµ= 0, (1.128) and with ∂νGµν=∂ν∂µBν−∂ν∂νBµthis yields /parenleftBig ∂ν∂ν+m2/parenrightBig Bµ=∂µ∂νBν. (1.129) These are four field equations containing the Klein-Gordon operator. Taking the 4-divergence we get /parenleftBig ∂ν∂ν+m2/parenrightBig ∂µBµ=∂µ∂µ∂νBν, 1.2 Relativistic Field Equations 29 ⇒m2∂µBµ= 0. (1.130) Thus the field equations can be represented equivalently by four Klein- GordonequationsaugmentedbyanequationwhichlookslikeaLorenzgauge: /parenleftBig ∂ν∂ν+m2/parenrightBig Bµ= 0, (1.131) ∂µBµ= 0. (1.132) This field describes massive spin 1 particles, like the ρ±,0andωmesons. 30 1 INTRODUCTION 1.3 Symmetries 1.3.1 Symmetries and conservation laws What does it mean to be symmetric? Hermann Weyl described it as follows. One needs three things: •An object, which turns out to be symmetric. •A procedure, doing something with this object. •An observer, who states that after the procedure nothing has changed. In physics we may say, a symmetry is “a mapping, which does not change the physics.” The meaning of the stated invariance depends on the structure that defines what “is the same”. For instance, in Euclidean geometry a circle is symmetric, any closed loop in general not. circle closed loop On the other hand, in topology the closed loop is regarded as equivalent to the circle, the deformation of the circle to a closed loop is a symmetry transformation in topology, but not in Euclidean geometry. Symmetry transformations can be concatenated to give a symmetry transfor- mation again. As with any mapping an associative rule is valid. The inverse transformation is also a symmetry, so symmetries form groups. (With few exceptions, when a semigroup is also regarded as a symmetry.) The most prominent symmetry in everyday life, the mirror reflection symmetry, be- longs to a very small group of only two elements: reflection and identity. In this lecture we consider continuous symmetry groups and their associated conservation laws, which Emmy Noether found in classical mechanics and field theory. Most of her work can be transferred to quantum mechanics. In quantum theory also discrete symmetry groups are being considered, for instance parity. (Here the word “parity” denotes both the symmetry and the conserved quantity.) Lagrange formalism In the Lagrange formalism symmetries can be dealt with by infinitesimal symmetrytransformations, whichissimplerthanusingfinitetransformations 1.3 Symmetries 31 although this would be possible, too. Let the fields undergo an infinitesimal transformation, which is very close to the identity. φa−→φ/prime a=φa+δφa. (1.133) The change of the action is written S−→S/prime=S+δS. (1.134) If the transformation is a symmetry, the system should not change, which means that the action does not change. Thus symmetry is characterised by δS= 0,ifφa−→φa+δφa. (1.135) Here the field equations are not used. In this case one has: ifφis a solution of the field equations, thenφ/primeis also a solution of the field equations. The Noether theorem states that associated with such a symmetry there exists a conserved current jµ(x), ∂µjµ(x) = 0. (1.136) The conserved quantity is given by Q=/integraldisplay d3xj0(x), dQ dt=/integraldisplay R3d3x∂0j0(x) =−/integraldisplay R3d3x∇·/vectorj(x) =−/integraldisplay ∂R3d2/vector x·/vectorj(x) = 0. Symmetries might involve space-time transformations, where δx/negationslash= 0. For example, under translations the fields transform as φ/prime a(x) =φa(x−δx). (1.137) If the system is invariant under translations, the conserved quantity is the energy-momentum four-vector pµ. Similarly, invariance under rotations gives the conservation of angular momentum. For the application of Noether’s theorem to such space-time symmetries we refer to the textbooks. Here we restrict our discussion to internal symmetries , for which δx= 0. φa(x)−→φa(x) +δφa(x). (1.138) 32 1 INTRODUCTION Symmetry means invariance of the action S=/integraldisplay d4xL(φa(x),∂φa(x)). (1.139) We consider the stronger condition that the Lagrangian density is invariant: δL= 0. δL=∂L ∂φaδφa+∂L ∂(∂µφa)δ∂µφa =∂L ∂φaδφa−/bracketleftBigg ∂µ∂L ∂(∂µφa)/bracketrightBigg δφa+∂µ/bracketleftBigg∂L ∂(∂µφa)δφa/bracketrightBigg (1.140) = 0. Here we have used δ∂µφa=∂µδφaand the Leibniz product rule for differ- entiation. The last term in the above equation already has the form of a 4-divergence ∂µ( )µ.We write δφa=/epsilon1fa (1.141) with infinitesimal small /epsilon1and finitefaand define jµ(x) :=∂L ∂(∂µφa)fa (1.142) to get ∂µjµ=/braceleftBigg ∂µ∂L ∂(∂µφa)−∂L ∂φa/bracerightBigg fa. (1.143) Consequence: If the field equations hold, the current jµis conserved ∂µjµ= 0. (1.144) 1.3.2 U(1) symmetry, electric charge Letφbe a complex scalar7field. φ(x)−→φ/prime(x) = e−iqαφ(x), (q∈Z, α∈R). (1.145) The transformation group has the representation U(1) ={c∈C|c∗c= 1}={c∈C|c= e−iγ,−π≤γ <π}.(1.146) 7It could also be a Dirac field or a field for particles with higher spin. 1.3 Symmetries 33 U(1) is an abelian group generated by the infinitesimal transformation φ/prime(x) = (1−iq/epsilon1)φ(x) =φ(x)−iqφ(x)/epsilon1 (1.147) δφ(x) =−iqφ(x)/epsilon1 (1.148) δφ∗(x) = iqφ∗(x)/epsilon1. (1.149) We assume a symmetry under U(1), δL= 0, (1.150) as is the case with the Lagrangian for a complex scalar field, L=1 2∂µφ∗∂µφ−m2 2φ∗φ. (1.151) The Lagrangian is in fact invariant under finite U(1)-transformations. Now we have to find the Noether current. jµ=∂L ∂(∂µφa)fa, f 1=−iqφ, f 2= iqφ∗ =∂L ∂(∂µφ)(−iqφ) +∂L ∂(∂µφ∗)(iqφ∗) jµ= iq(φ∗∂µφ−φ∂µφ∗). (1.152) We note that the spatial part of this current has the same form as the prob- ability current in in Schrödinger theory, which is defined by /vectorj=~ 2mi(ψ∇ψ∗− ψ∗∇ψ). Let us check the conservation law 1 iq∂µjµ=∂µ(φ∗∂µφ−φ∂µφ∗) (1.153) =∂µφ∗∂µφ+φ∗∂µ∂µφ−∂µφ∂µφ∗−φ∂µ∂µφ∗ =φ∗∂µ∂µφ−φ∂µ∂µφ∗ =φ∗(∂µ∂µ+m2)φ−φ(∂µ∂µ+m2)φ∗. (1.154) If the field equation holds, then ∂µjµ= 0. (1.155) The conserved charge is given by Q=/integraldisplay d3xj0= iq/integraldisplay d3x(φ∗∂0φ−φ∂0φ∗), (1.156) 34 1 INTRODUCTION qmay be an integer number. For the Dirac field we have a similar U(1) symmetry. The transformation is ψ/prime(x) = e−iqαψ(x), (1.157) ¯ψ/prime(x) = eiqα¯ψ(x), (1.158) and the Lagrangian density L=¯ψ(iγµ∂µ−m)ψ (1.159) again is invariant under this transformation. For the current and charge we get jµ=∂L ∂(∂µψα)(−iqψα) =q¯ψγµψ, (1.160) Q=q/integraldisplay d3x¯ψγ0ψ=q/integraldisplay d3xψ†ψ, (1.161) ψ†ψis the charge density. 1.3.3 SU(2) symmetry, isospin Consider neutron and proton, described by two Dirac fields p(x) = (pα(x)),α= 1,..., 4, (1.162) n(x) = (nα(x)),α= 1,..., 4. (1.163) The nuclear forces are independent of the electric charge. They are the same for proton and neutron. The idea of isospin (Werner Heisenberg, Dmitri Ivanenko) is to describe this in terms of a symmetry. The situation is in analogy with the two spin states of the electron, which form a basis of a two-dimensional sub-Hilbert space |↑/angbracketright=/parenleftBigg 1 0/parenrightBigg ,|↓/angbracketright=/parenleftBigg 0 1/parenrightBigg . (1.164) In the absence of a magnetic field both states carry the same energy, the Hamiltonian is invariant under a rotation in spin space, made up of the Pauli spinors ψ(x) =/parenleftBigg ψ+(x) ψ−(x)/parenrightBigg . (1.165) The symmetry transformation shall be denoted by ψ−→U(α)ψ (1.166) 1.3 Symmetries 35 with angles αthat parameterise the rotation. In analogy to spin Heisenberg introduced isotopic spin for proton and neu- tron, sometimes called isobaric spin; today it is mostly called isospin. The nucleon is represented by N=/parenleftBigg N1(x) N2(x)/parenrightBigg , (1.167) whereN1andN2are Dirac spinors Ni,α(x), i = 1,2;α= 1,..., 4. (1.168) The pure proton or neutron states are (somewhat ambiguously) denoted as p(x) =/parenleftBigg p(x) 0/parenrightBigg , n (x) =/parenleftBigg 0 n(x)/parenrightBigg , (1.169) and the nucleon field is N(x) =/parenleftBigg p(x) n(x)/parenrightBigg . (1.170) The corresponding isospin observable is /vectorI=1 2/vector τ. (1.171) I1=1 2τ1, τ 1=/parenleftBigg 0 1 1 0/parenrightBigg I2=1 2τ2, τ 2=/parenleftBigg 0−i i 0/parenrightBigg I3=1 2τ3, τ 3=/parenleftBigg 1 0 0−1/parenrightBigg Thus I3p(x) = +1 2p(x), I 3n(x) =−1 2n(x). (1.172) Summary: Isospin Proton Neutron I1 21 2 I3 +1 2−1 2. The symmetry group SU(2) Symmetries may be approximate symmetries, but here we shall make the hy- pothesis that the Hamiltonian is invariant under a rotation in 2-dimensional 36 1 INTRODUCTION isospin space. Because of the form of the free Hamiltonian, the symmetry transformation must be unitary. We write N−→N/prime=UN. (1.173) The group of such transformations, U(2), is made from 2×2unitary matrices U. U†U=1 (1.174) det(U†U) = det(U†) det(U) =|det(U)|2= 1 (1.175) |det(U)|= 1 (1.176) The subgroup that has detU= +1is called SU(2): SU(2) ={U∈GL2(C)|U†U= 1,detU= +1}.(1.177) U=/parenleftBigg a b c d/parenrightBigg (1.178) U†U=/parenleftBigg a∗c∗ b∗d∗/parenrightBigg/parenleftBigg a b c d/parenrightBigg =/parenleftBigg a∗a+c∗c a∗b+c∗d b∗a+d∗c b∗b+d∗d/parenrightBigg =/parenleftBigg 1 0 0 1/parenrightBigg . The unitarity of Uimplies that the vector norm of the columns has to be 1, the same holds for the rows of UsinceU†is also unitary. Together we have 1 = detU=ad−bc 1 =a∗a+c∗c=b∗b+d∗d=a∗a+b∗b=c∗c+d∗d leading toa∗a=d∗d, b∗b=c∗c.From this we can restrict the general form ofUto be8 U=/parenleftBigg a b −b∗eiβa∗eiα/parenrightBigg , a∗a+b∗b= 1. The determinant gives 1 =a∗aeiα+b∗beiβ= eiα(a∗a+b∗bei(β−α)). Together with a∗a+b∗b= 1this is only possible if α−β= 0.Thenα= 0 follows, too, and we have U=/parenleftBigg a b −b∗a∗/parenrightBigg , a∗a+b∗b= 1. (1.179) 8c=b∗b c∗,|b|=|c|givesc=−b∗eiαandd=a∗a d∗,|a|=|d|givesd=a∗eiβ. 1.3 Symmetries 37 Now we may choose four real parameters, together with one condition, to characterise U9 U=/parenleftBigg a0−ia3−a2−ia1 a2−ia1a0+ ia3/parenrightBigg ,3/summationdisplay k=0a2 k= 1. (1.180) This can be written as U=a01−ia1τ1−ia2τ2−ia3τ3. Since/summationtext3 k=0a2 k= 1, one may write a0= cos(α 2), 0≤α<2π, (1.181) (a1, a2, a3) = sin(α 2)/vector n,|/vector n|= 1. (1.182) Then U= cos(α 2)1−sin(α 2)(in1τ1+ in2τ2+ in3τ3). (1.183) With the definition /vector α:=α/vector n (1.184) we finally write it in a way standard for Lie group representations: U= exp(−iα 23/summationdisplay k=1nkτk) = exp(−i/vector α·/vectorI). (1.185) That the two expressions for U, (1.183) and (1.185), are equal can easily seen to first order in α: cos(α 2)1−i sin(α 2)(/vector n·/vector τ)≈1−iα 2/vector n·/vector τ≈exp(−iα 2/vector n·/vector τ). It can be shown that they are equal in general. /vectorI=/vector τ/2is the Hermitian matrix for the isospin observable. The anti- Hermitian, traceless exponent is a common feature of Lie algebras. Once again we state that the situation is similar to quantum mechanics, whereU(/vector α)in Pauli spinor space describes a rotation by an angle αwith a rotation axis given by /vector n.The matrix elements in this representation of SU(2) are determined by 3 real parameters (α1, α2, α3) =/vector α. 9In general, the matrices representing the group SU(N) ={U∈GLn|U†U= 1,detU= 1} haveN2−1real parameters. 38 1 INTRODUCTION Lie algebra Consideraninfinitesimaltransformation,thatmeansatransformation(1.185) with infinitesimal small α: δ/vector α=δα/vector n, (1.186) U(δ/vector α) =1−iδ/vector α·/vectorI, (1.187) N/prime=N−iδ/vector α·/vectorIN, (1.188) δN=−iδ/vector α·/vectorIN. (1.189) Theisospinoperators (I1, I2, I3)arecalledgeneratorsoftheLiegroupSU(2). From unitarity it follows that they are Hermitian and traceless,10 I† k=Ik, tr(Ik) = 0. (1.190) Their commutators are the same as those of the usual spin operators: [Ik,Il] = i/epsilon1klmIm, (1.191) This is the Lie algebra of SU(2). /epsilon1klmare the structure constants of the Lie algebra spanned by I1, I2, I3. Isospin symmetry Invariant expressions Expressionswhichareinvariantunderisospinsymmetryareusefulasbuilding blocks for an invariant Lagrangian. Here are some expressions containing N=/parenleftBig p n/parenrightBig . N†N=p†p+n†n: (UN)†UN=N†U†UN=N†N (1.192) N†γµN=p†γµp+n†γµn (1.193) ¯NN (1.194) ¯N∂µN11(1.195) An invariant Lagrangian for free nucleons is L=¯N(x)(iγµ∂µ−m)N(x) (1.196) = ¯p(x)(iγµ∂µ−m)p(x) + ¯n(x)(iγµ∂µ−m)n(x). 10Every anti-Hermitian operator can be decomposed into a traceless part and an imagi- nary multiple of the identity iH= iJ+ i tr(H)1. Then exp(iH) = exp(i tr( H) ) exp(iJ)∈ U(1)⊗SU(n)withn=dimension of the Hilbert(sub)space. 11The isospin symmetry does not depend on space-time. 1.3 Symmetries 39 Isospin symmetry implies that m=mp=mn. Experimental values for proton and neutron masses are mp= 938.272 MeV (1.197) mn= 939.565 MeV. (1.198) The isospin symmetry is nearly perfect, it is broken by the electromagnetic interaction and by differences between the masses of up- and down-quarks. pn I1/21/2 I31/2−1/2Q=1 2+I3. (1.199) Other representations of SU(2) Similar to higher spin quantum numbers, belonging to operators in higher dimensional spin subspaces, there are other representations of the isospin symmetry group. The group is represented by matrices in isospin space with an arbitrary dimension. The dimension of the isospin multiplets is given by 2I+ 1,whereI= 0,1 2,1, ...denotes the isospin quantum number. ConsiderI= 1,I3=−1,0,1, giving an isospin triplet, e.g. the pion triplet π= π+ π0 π− . (1.200) Here the quantum numbers for charge and isospin components are equal Q=I3. (1.201) In thisI= 1representation the generators of SU(2) have the following form 12 I(1) 1=1 2 0 1 0 1 0 1 0 1 0 , (1.202) I(1) 2=1 2 0−i 0 i 0−i 0 i 0 , (1.203) 12Thisfollowsinthesamewayasonegetsthematricesforangularmomentumoperators: L±=Lx±iLy, L±|l, m/angbracketright=/radicalbig l(l+ 1)∓m(m±1)|l, m±1/angbracketright. From this the images Lx,y|l, m/angbracketrightas matrix-columns are found using Lx=1 2(L++L−), Ly=1 2i(L+−L−). 40 1 INTRODUCTION I(1) 3=1 2 1 0 0 0 0 0 0 0−1 . (1.204) Quarks From the composition of the nucleons or the pions one can deduce the isospin quantum numbers for the quarks. This is true under the assumption that the quantum numbers of quark content add up to the total quantum number for the composed particles. From π+=u¯d π0=u¯u−d¯d π−=d¯u p=uu¯d n=dd¯u we infer that for quarks I=1 2, I 3u=1 2u, I 3d=−1 2d. (1.205) Thus theu,dquarks belong to an isospin doublet and their electric charges Qobey Q=1 6+I3. (1.206) In general, one defines a hypercharge Yby13 Q=I3+1 2Y, Y = 2(Q−I3). (1.207) Q I I 3Y u 2/3 1/2 1/2 1/3 d−1/3 1/2−1/2 1/3 p 1 1/2 1/2 1 n 0 1/2−1/2 1 π+1 1 1 0 π00 1 0 0 π−−1 1−1 0 13HereYis equal to the baryon number B. In general Y=B+S−1 3C, withSandC expressing strangeness and charm. 1.3 Symmetries 41 Noether currents TheNoetherprocedureleadstoconservationofcurrentsintheform ∂µjµ= 0. For the SU(2) isospin symmetry the field φ(x)has components φa(x), one for each particle of the multiplet (φa) =/parenleftBigg u d/parenrightBigg or/parenleftBigg p n/parenrightBigg or π+ π0 π− . (1.208) The infinitesimal transformation is14 δφ=−iδ/vector α·/vectorI(I)φ. (1.209) The superscript (I)stands for the isospin quantum number. In components of the multiplet this reads δφa=−iδαk(I(I) k)abφb. (1.210) From this we find the current for each of the three isospin components jµ k=∂L ∂(∂µφa)δφa δαk=−i∂L ∂(∂µφa)(I(I) k)abφb (k= 1,2,3).(1.211) As an example consider a quark isospin doublet q=/parenleftBig u d/parenrightBig with Lagrangian15 L= ¯q(x)(iγµ∂µ−m)q(x). (1.212) jµ k(x) = ¯q(x)γµI(1/2) kq(x), (1.213) jµ 1=1 2/parenleftBig ¯uγµd+¯dγµu/parenrightBig , (1.214) jµ 2=−i 2/parenleftBig ¯uγµd−¯dγµu/parenrightBig , (1.215) jµ 3=1 2/parenleftBig ¯uγµu−¯dγµd/parenrightBig . (1.216) The conserved charge ˆI3is ˆI3=/integraldisplay d3xj0 3(x) =1 2/integraldisplay d3x(¯uγ0u−¯dγ0d). (1.217) 14See the procedure for the charge conservation by U(1) symmetry, Eqs. (1.145) – (1.156). 15See the Lagrangian for the Dirac field in subsection 1.2.4, Eq. (1.108), page 25. 42 1 INTRODUCTION 1.3.4 SU(3) flavour symmetry The experimentally found hadrons motivated to extend the approximate isospin SU(2) symmetry to an approximate SU(3) flavour symmetry. It should be noted that the SU(3) flavour symmetry is a global symmetry, in contrast to the SU(3) coloursymmetry, which is a local gauge symmetry. In nature the SU(3) flavouris broken by the electromagnetic interaction and through the differences between the up-, down- and strange-quark masses. This leads to considerable mass differences within the SU(3) multiplets. Be- cause the mass differences to the heavier charm-, top- and bottom-quarks are even much bigger, one cannot speak of an approximate higher SU( N), N≥4, flavour symmetry.16Arranging the quarks by their masses, we depict the symmetries by SU(2)/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright u, d, s/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright SU(3), c, b, t . (1.218) Let the flavour symmetry transformation be q−→q/prime=Uq, (1.219) U∈SU(3) =/braceleftBig U∈GL3(C)|U+U= 1,detU= 1/bracerightBig . (1.220) The group elements are characterised by n2−1 = 8independent real param- etersαk, which we write as /vector α= (α1,...,α 8).For the representation of the group by its corresponding algebra we note: Algebra−→Group by exponentiation Group−→Algebra by infinitesimal transformations U(/vector α) = exp/parenleftBigg −i8/summationdisplay k=1αkTk/parenrightBigg , (1.221) U(δ/vector α) =1−i8/summationdisplay k=1δαkTk. (1.222) T† k=Tk, tr(Tk) = 0. (1.223) 16Figs. 1 and 2 on pages 6 – 7 show SU(4) multiplets. There the SU(3) multiplets have been extended in a third, vertical dimension to incorporate charm quantum numbers C= 1,2,3. That gives 20-plets for baryons and 16-plets for mesons. 1.3 Symmetries 43 The structure constants of the algebra shall be denoted by f [Tk, Tl] =fklmTm. (1.224) Analogous to the Pauli matrices σk, the Gell-Mann matrices are defined by λk=1 2Tkfork= 1,..., 8. An additional property of the SU(3) generators is tr(TkTl) =1 2δkl. (1.225) The group SU(2) is a subgroup of SU(3), the elements of SU(2) have the representation U= 0 U/prime 0 0 0 1 . (1.226) The generators of {U/prime}form a subalgebra of the algebra of SU(3). Using the isospin operators of SU(2) one can express the corresponding generators of SU(3) as T1= 0 I10 0 0 0 , T 2= 0 I20 0 0 0 , T 3= 0 I30 0 0 0 .(1.227) A maximum of 2 generators of SU(3) can be chosen such that they commute witheachotherandcanthusbediagonalisedsimultaneously. Suchamaximal abelian subalgebra is called Cartan subalgebra. Usually one chooses T3and T8. Explicitly one writes T3=1 2 1 0 0 0−1 0 0 0 0 , T 8=1 21√ 3 1 0 0 0 1 0 0 0−2 . (1.228) We see that [T3,T8] = 0.The quantum numbers belonging to T3andT8 are isospin and the additional quantum number Y, called hypercharge. It is defined by T8=√ 3 2Y, Y = 1/3 0 0 0 1/3 0 0 0−2/3 . (1.229) In this 3-dimensional representation hypercharge Ycan have the values 1/3 and−2/3. In addition to the two up- and down-quark states, having isospin I3=±1/2and hypercharge Y= 1/3, one introduces a third state with isospinI3= 0and hypercharge Y=−2/3, called strange quark. The quan- tum number strangeness Sis defined to be zero for up- and down-quarks and 44 1 INTRODUCTION S=−1for strange quarks. Here we list the quantum numbers which are assigned to the three light quarks. qQII3YSB u2/31/21/21/301/3 d-1/31/2-1/21/301/3 s-1/300-2/3-11/3 We see that Q=I3+1 2Y, Y =B+S.17(1.231) The (partially) conserved quantities now are Quantity conserved by broken by interactions interactions B,Q all none I strong em, weak S strong, em weak The remaining Gell-Mann matrices for SU(3) are λ4= 0 0 1 0 0 0 1 0 0 λ5= 0 0−i 0 0 0 i 0 0  (1.232) λ6= 0 0 0 0 0 1 0 1 0 λ7= 0 0 1 0 0−i 0 i 0  (1.233) 1.3.5 Some comments about symmetry As we have seen, symmetries play an important rôle in the physics of ele- mentary particles. The U(1) symmetry leads to charge conservation. The SU(2) isospin symmetry is a property of the strong nuclear interactions and is associated with the equality of the masses of proton and neutron or of the pion masses. This symmetry is only approximately true, but approximate symmetries give a starting point for perturbative calculations. Symmetries help to find more or less good quantum numbers and to classify particles. 17Including the charm quark the hypercharge is Y=B+S−1 3C (1.231) 1.3 Symmetries 45 For extensions of the Standard Model, the established symmetries must been taken into account. There are two kinds of symmetries: firstly, there are the space-time symme- tries like translations, rotations, Lorentz transformations, assembled in the inhomogeneous Lorentz group, also called Poincaré group. Secondly, we have internal symmetries: U(1), SU(2) flavour, SU(3) flavour, the local gauge symme- try SU(3) colourand the chiral symmetry. The integration of these two kinds of symmetries into a unified framework requires an extension of the concept of symmetry, namely supersymmetry (SUSY). 46 1 INTRODUCTION 1.4 Field Quantisation So far we have considered relativistic classical fields , e.g. the Klein-Gordon field or Maxwell field, which are consistent with special relativity. In this chapter we develop field theory in the framework of quantum theory. In classical physics, fields are continuous systems, described by functions of space and time. Examples are elastic media, electric and magnetic force fields. In the context of quantum theory, Schrödinger’s wave function can be con- sidered as a field. Its interpretation as a probability amplitude is, however, different from classical fields. The probability density is ρ(x) =ψ∗(x)ψ(x). The attempts to reconcile quantum mechanics and special relativity by intro- ducing relativistic wave equations, i.e. the Klein-Gordon and Dirac-equation, led to problems: •An infinite number of negative energy states arouse. •When analysing scattering on quantum wells, it turned out that nega- tiveprobabilities,aswellasprobabilitieslargerthan1,occurred(Klein’s paradox). This is related to particle creation in strong fields. The solution of these problems lies in the quantisation of relativistic field equations. The fields are then not considered as wave functions, but as physical systems with an infinite number of degrees of freedom, and are subject to quantisation. It turns out that this leads to quantum theory of many particles. Quantisation in quantum mechanics and in quantum field theory Quantisation of a classical theory implies the replacement of observables by operators. The fundamental Poisson brackets are replaced by commutators. In classical mechanics the Poisson brackets are defined by {f,g}=/summationdisplay i/parenleftBigg∂f ∂qi∂g ∂pi−∂g ∂qi∂f ∂pi/parenrightBigg . (1.234) For fields the analogous definition is {F, G}=/integraldisplay d3x/summationdisplay a/braceleftBiggδF δφa(x0,/vector x)δG δπa(x0,/vector x)−δG δφa(x0,/vector x)δF δπa(x0,/vector x)/bracerightBigg , (1.235) whereFandGare functionals of the fields φa(x0,/vector x)and their conjugate momentaπa(x0,/vector x)at fixed time x0. 1.4 Field Quantisation 47 Using the functional derivatives δφa(x0,/vector x) δφb(x0,/vector x/prime)=δabδ3(/vector x−/vector x/prime) (1.236) one gets the fundamental Poisson brackets for fields {πa(x0,/vector x),φb(x0,/vector x/prime)}=−δabδ3(/vector x−/vector x/prime) (1.237) Below we summarise variables in the Hamiltonian formalism of classical physics and their quantum counterparts. Mechanical description −→quantum mechanical description variables operators qi, pi Qi, Pi pi=∂L ∂˙qi Poisson bracket commutator /i~ {qi,pk}=δik [Qi,Pk] = i~δik classical field −→ quantum field fieldφa(x) field operator φa(x) π(x) =∂L ∂˙φ(x) { , }1 i~[,] states|ψ/angbracketright∈H 1.4.1 Quantisation of the real scalar field With the expression of the Lagrangian density for a real scalar field L=1 2∂µφ∂µφ−m2 2φ2(1.238) the conjugate momentum density is π(x) =∂L ∂(∂0φ(x))=∂0φ(x) = ˙φ(x). (1.239) The fundamental Poisson brackets for classical fields are {π(x0,/vector x),φ(x0,/vector x/prime)}=−δ3(/vector x−/vector x/prime) (1.240) {φ(x0,/vector x),φ(x0,/vector x/prime)}={π(x0,/vector x),π(x0,/vector x/prime)}= 0.(1.241) 48 1 INTRODUCTION Their counterparts in quantum field theory will be /bracketleftBig π(x0,/vector x), φ(x0,/vector x/prime/bracketrightBig =~ iδ3(/vector x−/vector x/prime), (1.242) /bracketleftBig π(x0,/vector x), π(x0,/vector x/prime/bracketrightBig =/bracketleftBig φ(x0,/vector x), φ(x0,/vector x/prime/bracketrightBig = 0.(1.243) A Legendre transformation gives the Hamiltonian density H=π˙φ−L (1.244) =1 2π2+1 2(∇φ)2+m2 2φ2, (1.245) and the Hamiltonian H=1 2/integraldisplay d3x/parenleftBig π2+ (∇φ)2+m2φ2/parenrightBig . (1.246) The classical canonical equations of motion then read ˙φ(x) ={φ(x),H}=π(x), (1.247) ˙π(x) ={π(x),H}= ∆φ(x)−m2φ(x) (1.248) Combining them yields ¨φ(x) = (∆−m2)φ(x), (1.249) which is the Klein-Gordon equation, (∂µ∂µ+m2)φ= 0. As we see, the result- ing field equation is Lorentz invariant, although we introduced a distinction of space and time with the definition of the canonical momentum π,and by using only equal time Poisson brackets or – after quantisation – equal time commutators.18 The corresponding quantum canonical equations read ˙φ(x) =1 i~[φ(x),H] =π(x), (1.250) ˙π(x) =1 i~[π(x),H] = ∆φ(x)−m2φ(x). (1.251) 18This approach to field quantisation was introduced by Heisenberg and Pauli. They proceeded from classical mechanics to field theory by dividing space into small cells, to each of which was associated a pair of conjugate generalised coordinates (qi.pi). These undergo time evolution like the many conjugate variables of a many particle mechanical systems. Going to the limit of a continuum of infinitely small cells one arrives at the time evolution of fields. 1.4 Field Quantisation 49 The quantised scalar field obeys the Klein-Gordon equation. Therefore, like in the classical case it can be decomposed into plane wave solutions φ(x) =/integraldisplayd3k (2π)32ωk/parenleftBig a(/vectork) e−ik·x+a†(/vectork) eik·x/parenrightBig , (1.252) where k0=ωk=/radicalBig /vectork2+m2. (1.253) In contrast to the classical case, here the coefficients a(/vectork),a†(/vectork)are oper- ators. In the following we just denote them a(k),a†(k), where it is to be understood that k0=ωk. From the definition of the canonical conjugate momentum, Eq. (1.239), we have π(x) =/integraldisplayd3k (2π)32ωk(−iωk)/parenleftBig a(k) e−ik·x−a†(k) eik·x/parenrightBig . (1.254) We can invert the mode expansions of the field to get an expression for a(k): π(x)−iωkφ(x) =/integraldisplayd3k (2π)32ωk(−2iωk)a(k) e−ikx, a(k) = i/integraldisplay d3xeikx(π(x)−iωkφ(x))/vextendsingle/vextendsingle/vextendsingle/vextendsingle x0=0. (1.255) From these mode decompositions and the commutator rules for the fields, Eq. (1.242) and (1.243), we find that a(k),a(k†)satisfy the commutation rules [a(k),a†(k/prime)] = (2π)32ωkδ3(/vectork−/vectork/prime) (1.256) [a(k),a(k/prime)] = [a†(k),a†(k/prime)] = 0. (1.257) We can express the Hamiltonian (1.246) in these operators,too. Let us do 50 1 INTRODUCTION this in detail for the three terms. Using (1.252) and (1.254) we obtain ∇φ(x) =/integraldisplayd3k (2π)32ωk(i/vectork)/parenleftBig a(k) e−ik·x−a†(k) eik·x/parenrightBig , 1 2/integraldisplay d3xπ2=1 2/integraldisplayd3k (2π)32ωk/integraldisplayd3k/prime (2π)32ωk/prime/integraldisplay d3x(−iωk)(−iωk/prime) /parenleftBig a(k)a(k/prime) e−i(k+k/prime)·x+a†(k)a†(k/prime) ei(k+k/prime)·x −a(k)a†(k/prime) e−i(k−k/prime)·x−a†(k)a(k/prime) ei(k−k/prime)·x/parenrightBig =1 8/integraldisplayd3k (2π)3/parenleftBig −a(k)a(−k)−a†(k)a†(−k) +a(k)a†(k) +a†(k)a(k)/parenrightBig 1 2/integraldisplay d3x(∇φ)2=1 2/integraldisplayd3k (2π)32ωk/integraldisplayd3k/prime (2π)32ωk/prime/integraldisplay d3x(i/vectork)·(i/vectork/prime) /parenleftBig a(k)a(k/prime) e−i(k+k/prime)·x+a†(k)a†(k/prime) ei(k+k/prime)·x −a(k)a†(k/prime) e−i(k−k/prime)·x−a†(k)a(k/prime) ei(k−k/prime)·x/parenrightBig =1 8/integraldisplayd3k (2π)3/vectork2 ω2 k/parenleftBig a(k)a(−k) +a†(k)a†(−k) +a(k)a†(k) +a†(k)a(k)/parenrightBig m2 2/integraldisplay d3xφ2=m2 2/integraldisplayd3k (2π)32ωk/integraldisplayd3k/prime (2π)32ωk/prime/integraldisplay d3x /parenleftBig a(k)a(k/prime) e−i(k+k/prime)·x+a†(k)a†(k/prime) ei(k+k/prime)·x a(k)a†(k/prime) e−i(k−k/prime)·x+a†(k)a(k/prime) ei(k−k/prime)·x/parenrightBig =1 8/integraldisplayd3k (2π)3m2 ω2 k/parenleftBig a(k)a(−k) +a†(k)a†(−k) +a(k)a†(k) +a†(k)a(k)/parenrightBig (1.258) Usingω2 k=/vectork2+m2and adding the terms up, we finally find H=1 2/integraldisplayd3k (2π)32ωkωk/parenleftBig a†(k)a(k) +a(k)a†(k)/parenrightBig (1.259) =/integraldisplayd3k (2π)32ωkωk/parenleftbigg a†(k)a(k) +1 2[a(k),a†(k)]/parenrightbigg (1.260) =/integraldisplayd3k (2π)32ωkωka†(k)a(k) +/integraldisplay d3k1 2ωkδ3(0). (1.261) Thesecondintegralisaninfiniteconstant. Wemayinterpretitasadivergent zero point energy. The Hamiltonian can be compared with the Hamiltonian of an infinite number of harmonic oscillators with creation operators a† iand 1.4 Field Quantisation 51 annihilation operators ai H=/summationdisplay k~ωk(a† kak+1 2). (1.262) The ground state |0/angbracketrightobeysak|0/angbracketright= 0for allk, and the excited states are given by expressions like a† j|0/angbracketright, a† ja† k|0/angbracketright, a† ja† ka† l|0/angbracketright, ... The “quantum number” operator is N=/summationdisplay ka† kak. (1.263) In our field theoretical context we interpret a†(k)as particle creation operator , (1.264) a(k)as particle annihilation operator. (1.265) We deal with the zero point energy by requiring that H|0/angbracketright= 0. This is achieved by removing the infinite constant from H: H→/integraldisplayd3k (2π)32ωkωka†(k)a(k). (1.266) As this is a fixing of the absolute scale of the energy, physical meaningful energy differences are not affected by it. The subtraction of the zero point energy can be expressed in terms of normal ordering. Normal ordering means, that one has to put each creation operator to the left of every annihilation operator. Normal ordering is symbolised by colons, e.g. :a†a:=a†a, :aa†:=a†a. (1.267) Then we define the Hamiltonian to be :H:, such that :H:|0/angbracketright= 0. (1.268) In the following we leave out the colons and understand Hto be normal ordered. Applying creation operators onto the ground state yields excited states Ha†(k)|0/angbracketright=ωka†|0/angbracketright, (1.269) Ha†(k1)a†(k2)|0/angbracketright= (ωk1+ωk2)a†(k1)a†(k2)|0/angbracketright.(1.270) 52 1 INTRODUCTION From these equations we see that the energies of these states are the energies of non-interacting relativistic multi-particle states. Each creation operator creates one particle. The ground state is empty and is the vacuum state. Therepresentationofthefieldoperatorsbyparticlecreationandannihilation operators is called the Fock representation ; the states created in this way out of the vacuum state are Fock states. The Hilbert space spanned by all these multi-particle states is the direct sum of n-particle Hilbert spaces Hn, H=∞/circleplusdisplay n=0Hn (1.271) and is called Fock space. A remark about the zero-point energy Thezero-pointenergychanges, ifthereareboundaryconditions. Anexample is theCasimir effect , where two parallel conducting plates feel a force pulling them together, although they are not electrically charged. Between the con- ductors the possible “cavity modes” are restricted by boundary conditions. Their number increases, when the distance aof the plates is increased. Thus the zero-point energy in between the plates grows with distance, while out- side there is no restriction on the possible modes. The increasing zero-point energy leads to an attractive force on the plates, ∂ ∂aEnergy =attractive force∼1 a4. (1.272) 1.4.2 Quantisation of the complex scalar field The complex scalar field φ(x) =/integraldisplayd3k (2π)32ωk/parenleftBig a(k)e−ikx+b†(k)eikx/parenrightBig (1.273) can be written as a combination of two real scalar fields φ=1√ 2(φ1+ iφ2), (1.274) so that φj=/integraldisplayd3k (2π)32ωk/parenleftBig aj(k)e−ikx+a† j(k)eikx/parenrightBig , (j= 1,2.) φ=/integraldisplayd3k (2π)32ωk/parenleftbigg1√ 2(a1(k) + ia2(k)) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =a(k)e−ikx+1√ 2(a† 1(k) + ia† 2(k)) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =b†(k)eikx/parenrightbigg . 1.4 Field Quantisation 53 This leads to creation and annihilation operators a(k) =1√ 2(a1(k) + ia2(k)), (1.275) b†(k) =1√ 2(a† 1(k) + ia† 2(k)), (1.276) b†(k)/negationslash=a†(k). (1.277) The complex scalar field is equivalent to a pair of two independent real scalar fields. The commutation relations are [a(k),a†(k/prime)] = [b(k),b†(k/prime)] = (2π)32ωkδ3(/vectork−/vectork/prime), (1.278) [a,b] = [a†,b†] = [a,b†] = [a†,b] = 0. (1.279) The field describes two sorts of particles, which can be created from the vacuum state|0/angbracketright, obeying a(k)|0/angbracketright=b(k)|0/angbracketright= 0. (1.280) For example, one gets Fock states like one-particle states a†(k)|0/angbracketright, b†(k)|0/angbracketright, two-particle states a†(k)a†(k/prime)|0/angbracketright, a†(k)b†(k/prime)|0/angbracketright. The conjugate momentum and the Hamiltonian density are π=∂ ∂(∂0φ)/parenleftBig (∂µφ†)(∂µφ)−m2φ†φ)/parenrightBig =∂0φ†=˙φ†, π†=˙φ,(1.281) H=ππ†−L. (1.282) The Hamiltonian looks like the classic Hamiltonian with the fields replaced by field operators, H=/integraldisplay d3x(ππ†+ (∇φ†)(∇φ) +m2φ†φ). (1.283) In the Fock representation after normal ordering this is :H: =/integraldisplayd3k (2π)32ωkωk/parenleftBig a†(k)a(k) +b†(k)b(k)/parenrightBig .(1.284) In a similar way one finds an expression for the charge. From the U(1) symmetry and the Noether theorem a conserved charge Qfollows, which can be expressed by field operators Q= iq/integraldisplay d3x/parenleftBig φ†˙φ−φ˙φ†/parenrightBig . (1.285) 54 1 INTRODUCTION In the Fock representation this is Q=q/integraldisplayd3k (2π)32ωk/parenleftBig a†(k)a(k)−b†(k)b(k)/parenrightBig . (1.286) This immediately gives Qa†(k)|0/angbracketright=qa†(k)|0/angbracketright for a particle state of sort a,(1.287) Qb†(k)|0/angbracketright=−qb†(k)|0/angbracketright for a particle state of sort b.(1.288) We see that the state b†(k)|0/angbracketrightrepresents a particle with opposite charge than the particle represented by a†(k)|0/angbracketright, and we call it the corresponding antiparticle. We define a charge conjugation operator C, which exchanges a andb: Ca†C=b†, Cb†C=a†. (1.289) Then, since CC= 1, Ca†(k)|0/angbracketright=Ca†(k)CC|0/angbracketright=b†(k)C|0/angbracketright=b†(k)|0/angbracketright. (1.290) 1.4.3 Quantisation of the Dirac field Similar as for the complex scalar field, the expansion of the Dirac field in terms of plane waves, Eq. (1.34), contains two sorts of coefficients, labelled bandd. Quantisation turns them into creation and annihilation operators, such that b† r(k)creates particles, e.g. electrons , br(k)annihilates particles , d† r(k)creates antiparticles, e.g. positrons, dr(k)annihilates antiparticles . Starting from the Lagrangian (1.110) L=¯ψ(iγµ∂µ−m)ψ one defines the conjugate momentum field Π(x) =∂L ∂(∂0ψ)= iψ†. (1.291) Then the Hamiltonian is H=/integraldisplay d3x(Π˙ψ−L) =/integraldisplay d3x(ψ†i∂0ψ−L) (1.292) =/integraldisplayd3k (2π)32ωkωk4/summationdisplay r=1/parenleftBig b† r(k)br(k)−dr(k)d† r(k)/parenrightBig . (1.293) 1.4 Field Quantisation 55 In contrast to the scalar field, we have to impose anticommutation rules for the Dirac field, because it describes Fermions: /bracketleftBig ψα(x0,/vector x), ψ† β(x0,/vector x/prime)/bracketrightBig +=δαβδ3(/vector x−/vector x/prime). (1.294) This leads to /bracketleftBig br(k), b† r/prime(k/prime)/bracketrightBig +=/bracketleftBig dr(k), d† r/prime(k/prime)/bracketrightBig += (2π)32ωkδr,r/primeδ3(/vectork−/vectork/prime),(1.295) [br(k), br/prime(k/prime)]+=/bracketleftBig b† r(k), b† r/prime(k/prime)/bracketrightBig += 0, (1.296) [dr(k), dr/prime(k/prime)]+=/bracketleftBig d† r(k), d† r/prime(k/prime)/bracketrightBig += 0, (1.297) [br(k), dr/prime(k/prime)]+=mixed anticommutators = 0,(1.298) The definition of normal ordering contains a minus sign, when two operators are interchanged. Therefore the normal ordered Hamiltonian H=/integraldisplayd3k (2π)32ωkωk4/summationdisplay r=1/parenleftBig b† r(k)br(k) +d† r(k)dr(k)/parenrightBig (1.299) is positive for all states in the Fock space. 1.4.4 Quantisation of the Maxwell field The vector potential Aµ(x)is only determined up to gauge transformations. Therefore it contains redundant, unphysical degrees of freedom, and the quantisation procedure is non-trivial. There are two possibilities, which are considered in this context. The first possibility is to impose the Coulomb (or radiation) gauge on the fields. In this way, however, manifest Lorentz invariance is lost. To show the relativistic invariance of the physical results of the theory is a non-trivial task. The second possibility is to keep mani- fest Lorentz invariance and to quantise the theory covariantly. But then the gauge freedom in the field variables leads to unphysical states, which must be removed afterwards. This second possibility is called the Gupta-Bleuler quantisation. Coulomb or radiation gauge When external currents are absent, in the radiation gauge the field only contains transversal modes with amplitudes a(λ)(k)anda(λ)∗(k), whereλ= 1,2labelstwopolarisationstatesortwohelicitymodesofthephoton, k0=ωk and/vectork⊥/vectorA,seeEq.(1.81). Quantisationpromotestheamplitudestooperators a(λ)(k), a(λ)†(k). They are photon annihilation and creation operators, and we have Ha(λ)†(k)|0/angbracketright=k0a(λ)†(k)|0/angbracketright. (1.300) 56 1 INTRODUCTION Let us study the conjugate momenta of the field. We have ∂L ∂(∂µAν)=−Fµν, (1.301) so that Π0(x) =∂L ∂˙A0(x)= 0, (1.302) Πi(x) =∂L ∂˙Ai(x)=−F0i=Ei. (1.303) Canonical commutation relations [Ai(x0,/vector x),Πj(x0,/vector x/prime)] =−iδijδ3(/vector x−/vector x/prime) (1.304) would be in contradiction with the transversality ∂iAi(x) = 0, since 0 = [∂iAi(x0,/vector x),Πj(x0,/vector x/prime)] =−i∂jδ3(/vector x−/vector x/prime)/negationslash= 0. (1.305) Instead, the standard commutators for a(λ)(k)anda(λ)†(k)lead to [Ai(x0,/vector x),Πj(x0,/vector x/prime)] =−i˜δij(/vector x−/vector x/prime), (1.306) where the transverse δ-function is defined by ˜δij(/vector x−/vector x/prime) :=/integraldisplayd3k (2π)3ei/vectork·(/vector x−/vector x/prime)/parenleftBigg δij−kikj /vectork2/parenrightBigg . (1.307) Covariant quantisation Covariant quantisation starts with a Lagrangian L=−1 4FµνFµν−1 2(∂µAµ)2, (1.308) which is manifestly Lorentz invariant, but not gauge invariant. Covariant commutators are imposed on the fields and momenta. There are four types of annihilation and creation operators a(λ)(k), a(λ)†(k)forλ= 0,..., 3. The Fock space contains unphysical states, e.g. with negative norm. The physical states are restricted by (∂µAµ)+|physical state/angbracketright= 0, (1.309) where (∂µAµ)+is the positive frequency part of ∂µAµ. With this formalism all physical, gauge invariant results are the same as with the radiation gauge. To summarise •explicit Lorentz invariance is kept, •unphysical states in the Fock space have to be removed by constraints. 1.4 Field Quantisation 57 1.4.5 Symmetries and Noether charges According to the Noether theorem, to each continuous symmetry belongs a conserved charge. Quantisation turns it into a charge operator. Consider, for example, the isospin for up and down quarks: ˆI3:=/integraldisplay d3x¯q(x)γ0I3q(x) =1 2/integraldisplay d3x/parenleftBig ¯u(x)γ0u(x)−¯d(x)γ0d(x)/parenrightBig .(1.310) In the quantised field theory the quark fields ¯u,u,¯danddare field operators. In Fock space with creation and annihilation operators b(u)†, b(u), b(d)†, b(d) and states |u/angbracketright=b(u)†|0/angbracketright,|d/angbracketright=b(d)†|0/angbracketright, (1.311) the charge operator acts on these states as ˆI3|u/angbracketright=1 2|u/angbracketright, ˆI3|d/angbracketright=−1 2|d/angbracketright. (1.312) compare Eqs. (1.286) ff. 58 1 INTRODUCTION 1.5 Interacting Fields In the previous sections free field theories have been considered. They de- scribe particles that don’t interact with each other. Now we turn to the consideration of interactions. Consider a scattering process between parti- cles, as indicated in the picture. p1 p/prime 1 p/prime mpn|in/angbracketright | out/angbracketright interaction A number of nparticle approach each other and interact with each other. They form the ingoing state. After the interaction there are particles in an outgoing state. During the scattering process the particles are in a highly complicated state. But in the far past and in the far future they are far away from each other and can be considered as non-interacting (we neglect self-interactions here). The corresponding asymptotic states describe free particles. The transition probability from the ingoing state |in/angbracketrightto an outgoing state |out/angbracketrightwill be described by the matrix element of an unitary time evolution operatorU(t1,t0), /angbracketleftout|U(+∞,−∞)|in/angbracketright. (1.313) 1.5.1 Interaction picture Here is a short reminder about the interaction picture. In the Schrödinger picture the states are time dependent, their time evolution is given by an unitary operator: |ψS(t)/angbracketright=U(t,t0)|ψS(t0/angbracketright= e−iH(t−t0)|ψS(t0)/angbracketright. (1.314) This holds, provided the Hamiltonian has no explicit time dependence. In the Heisenberg picture the time dependence is shifted from the states to the operators: OH(t) =U†(t,t0)OSU(t,t0). (1.315) In the free field theories considered so far, the field operators and the opera- tors formed out of them are understood to be in the Heisenberg picture, e.g. 1.5 Interacting Fields 59 the Hamiltonian :H0: =1 2/integraldisplay d3x: [π2+ (∇φ)2+m2φ2] :=/integraldisplayd3k (2π)32ωkωka† kak.(1.316) For free fields the Hamiltonians in the different pictures are identical: H(H) 0=H(S) 0=H0. (1.317) In free theories there are no non-trivial transition probabilities S/angbracketleftout,t|in,t0/angbracketrightS=H/angbracketleftout|U(t,t0|in/angbracketrightH. (1.318) As for the harmonic oscillator, where the time dependence of the ladder operators in the Heisenberg picture is given by ˙aH(t) = i[H,aH] = iω[a†a,aH] =−iωaH(t), (1.319) resulting in aH(t) = exp(iωa†at)aexp(−iωa†at) =aexp(−iωt) a† H(t) = exp(iωa†at)a†exp(−iωa†at) =a† Hexp(iωt), the time dependence of annihilation and creation operators in field theory is given by ak,H(t) = exp(iH0t)ak,Sexp(−iH0t) =ak,Sexp(−iωkt) a† k,H(t) = exp(iH0t)a† k,Sexp(−iH0t) =a† k,Sexp(iωkt), and we find that in the transition probabilities only terms of the type e−iωk(t−t0)δin,out (1.320) survive. So there are no transitions in the free theory. Now we include interactions. In the Schrödinger picture we have an interac- tion Hamiltonian H(I) S, i∂t|ψ(t)/angbracketright=/parenleftBig H0+H(I) S/parenrightBig |ψ(t)/angbracketright. (1.321) If one thinks of small interactions one may define slowly varying states |φ(t)/angbracketright |φ(t)/angbracketright= eiH0t|ψ(t)/angbracketright=U−1 0|ψ(t)/angbracketright, (1.322) 60 1 INTRODUCTION and with i∂tU−1 0=−H0U−1 0one finds i∂tU−1 0|ψ(t)/angbracketright=−H0U−1 0|ψ(t)/angbracketright+U−1 0(H0+H(I) S)ψ(t)/angbracketright (1.323) =U−1 0H(I) SU0U−1 0|ψ(t)/angbracketright. (1.324) WithHI(t) :=U−1 0(t)H(I) SU0(t)we get the time dependence of states in the interaction picture i∂t|φ(t)/angbracketright=HI(t)|φ(t)/angbracketright. (1.325) Operators in the interaction picture are consistently defined to be OI(t) =U† 0(t)OSU0(t). (1.326) They evolve like Heisenberg operators in the free theory, ˙OI(t) = i[H0,OI(t)]. (1.327) The advantages of the interaction picture are •one keeps the formulations of the free theory for the operators, •the|in/angbracketright,|out/angbracketrightstates can be prepared as states of the free theory. While at times t=±∞the states are simple eigenstates of H0, the time evo- lution during the interacting becomes complicated, since it involves HI(t). Therefore, one has to find approximations. Integrating the Schrödinger equa- tion in the interaction picture (1.325) and iterating the result leads to |φ(t)/angbracketright=φ(t0)/angbracketright+ (−i)/integraldisplayt t0dt1HI(t1)|φ(t1)/angbracketright =φ(t0)/angbracketright+ (−i)/integraldisplayt t0dt1HI(t1)|φ(t0)/angbracketright + (−i)2/integraldisplayt t0dt1/integraldisplayt1 t0dt2HI(t1)HI(t2)|φ(t0)/angbracketright + (−i)3/integraldisplayt t0dt1/integraldisplayt1 t0dt2/integraldisplayt2 t0dt3HI(t1)HI(t2)HI(t3)|φ(t0)/angbracketright ......... . The limits of integration obviously underlie the restriction t>t 1>t2>t3> ...>t 0. The restriction can be implemented by defining an operator Tthat generates time ordered products from arbitrary products of operators, i.e. T[O(t1)O(t2)] =/braceleftBigg O(t1)O(t2),ift1>t2, O(t2)O(t1)else.(1.328) 1.5 Interacting Fields 61 In the same manner Torders higher operator products with respect to their time arguments. With the aid of Tthe integral over the simplex t > t 1> t2>t3>...>t 0can be transformed into an integral over a hypercube |φ(t)/angbracketright=∞/summationdisplay i=0(−i)n n!t/integraldisplay t0t/integraldisplay t0...t/integraldisplay t0dt1dt2...dtnT[HI(t1)HI(t2)...HI(tn)]|φ(t0)/angbracketright (1.329) =Texp/parenleftbigg −i/integraldisplayt t0dt/primeHI(t/prime)/parenrightbigg |φ(t0)/angbracketright. (1.330) This series is called Dyson series. It represents the unitary time evolution operator in the interaction picture, |φ(t)/angbracketright=UI(t,t0)|φ(t0)/angbracketright. (1.331) 1.5.2 The S-matrix In the limit t0→−∞,t→∞, the time evolution operator contains the description of scattering processes. The S-matrix is defined by S:=UI(+∞,−∞) =Texp/parenleftbigg −i/integraldisplay∞ −∞dtHI(t)/parenrightbigg .(1.332) It gives the transition probability /angbracketleftout|S|in/angbracketright. In a free theory the S-matrix is the identity. In general the interaction Hamiltonian contains field operators φI(x) =/integraldisplayd3k (2π)32ωkake−ikx /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright φ++/integraldisplayd3k (2π)32ωka† ke−ikx /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright φ−, (1.333) whereφ+andφ−are the positive and negative frequency parts, respectively. Let us assume that HI=−/integraltextd3xLIarises from a Lagrangian density of the formLI=λF(φ(x))with smallλ. Then one can try to expand Sin powers ofλ, S=1+S(1)+S(2)+... (1.334) S(0)=1 (1.335) S(1)= iλ/integraldisplay d4x[F(φ(x))] (1.336) S(2)=(iλ)2 2/integraldisplay/integraldisplay d4x1d4x2T[F(φ(x1))F(φ(x2))]. (1.337) 62 1 INTRODUCTION For the calculation of matrix elements /angbracketleftout|S(n)|in/angbracketright, it is convenient to bring this into normal ordered form by use of the commutation relations. The terms in the matrix element are of the form (constants, integrals ) /angbracketleft0|(creation operators for out states )† T[creation and annihilation operators from HI] (creation operators for in states )|0/angbracketright. Toevaluatethisoneusescommutationoranticommutationrelationstoarrive at (constants, integrals )functions (x1,..., xn)/angbracketleft0|0/angbracketright +(constants, integrals )functions (x1,..., xn)from [,]± ×/angbracketleft0|N[creation and annihilation operators ]|0/angbracketright. Since the vacuum expectation values of all terms, which contain normal or- deredoperatorsvanish,thistransformationoftimeorderedoperatorproducts to normal ordered products is especially useful. It can be accomplished with the help of Wick’s theorem. 1.5.3 Wick’s theorem Define contractions φ(x1)φ(x2)as the difference between time ordered prod- ucts and normal ordered products: φ(x1)φ(x2) =T[φ(x1)φ(x2)]−N[φ(x1)φ(x2)] (1.338) =/angbracketleft0|T[φ(x1)φ(x2)]|0/angbracketright (1.339) =: i∆F(x1−x2). (1.340) ∆F(x)is called Feynman propagator ∆F(x) = lim /epsilon1→0+/integraldisplayd4k (2π)41 k2−m2+ i/epsilon1e−ikx. (1.341) 1.5 Interacting Fields 63 Wick’s theorem expresses time ordered products in terms of normal ordered products and contractions. Some examples are T[φ1φ2φ3] =N[φ1φ2φ3] +φ1φ2φ3+φ1φ2φ3+ˇφ1φ2ˇφ3 T[φ1φ2φ3φ4] =N[φ1φ2φ3φ4] +φ1φ2N[φ3φ4] +φ1φ3N[φ2φ4] +φ1φ4N[φ2φ3] +φ2φ3N[φ1φ4] +φ2φ4N[φ1φ3] +φ3φ4N[φ1φ2] +φ1φ2φ3φ4+ˇφ1φ2φ3ˇφ4+φ1φ3ˇφ2ˇφ4, where we write φiforφ(xi). The general rule is T[fields ] =/summationdisplay subsets of fieldsN[subset of fields ]×(all possible contractions of other fields ). (1.342) From Wick’s theorem it follows that in /angbracketleft0|...|0/angbracketrightonly the complete contrac- tions survive. 1.5.4 Feynman diagrams The contractions resulting from Wick’s the- orem are represented by lines. The lines for bosonic fields are drawn dotted or dashed.- - - - - - - - The integral−iλ/integraltextd4xin the interaction Hamiltonian gives a vertex, with lines connected to it like the following or x x Sometimes the integration variable ( x) is denoted near the vertex. It remains to take into account the operators that generate the inandoutstates out of the vacuum. The contraction of these operators leads to diagram lines which come from outside. These lines are called external lines, whileinternal lines arise from contractions of field operators in the interaction Hamiltonian part. By means of Fourier transform the x-integrations can be evaluated in mo- mentum space. External lines are associated with solutions of the free field equations and are subject to the restriction k0=/radicalBig /vectork2+m2; (1.343) 64 1 INTRODUCTION the corresponding propagators are denoted to be on the mass shell . At each vertex the sum of incoming momenta minus the sum of outgoing momenta has to be zero. We illustrate the Feynman diagrams for the case of a real scalar field with quartic self-interaction. Let the Lagrangian be L0=1 2(∂µφ∂µφ−m2φ2), (1.344) LI=−gφ4. (1.345) Heregis for a small, real coupling constant. Other powers of φin the interaction terms are not useful, for φ1would give only a constant shift , φ2gives an extra mass , φ3would lead to instabilities. The Hamiltonian density for the interaction is HI=gφ4(x) (1.346) and for two-particle scattering to first order in gone has to evaluate a term of the type /angbracketleftout|/integraldisplay d4xgφ4(x)|in/angbracketright. (1.347) This is represented by 1 vertex for/integraldisplay d4xg, 4 external lines. The graphs for the lowest contributions to two-particle scattering are 1.5 Interacting Fields 65 ++ +S(0) S(1) S(2) Classification of graphs Let us comment on some particular types of graphs. •Graphs with no external lines, the so called “vacuum bubbles”. It can be shown that they can be neglected in S-matrix elements. •Graphs with 2 external lines are called self-energy graphs. Sunset Tadpole (Kaulquappe) They contribute tothe full particle propagator and lead to corrections to the mass of the particle. •Scattering graphs with more than 2 external lines. •Connected graphs, on which it is possible to go from each vertex to each other vertex by moving on lines, like the two previous ones. 66 1 INTRODUCTION •One-particle-irreducible graphs, which cannot be decomposed into 2 disconnected graphs by removing a single line. All graphs can be com- posed of these. 1.5.5 Fermions For fermions the calculations explained above can be performed, too. It has to be observed that commutators between the basic fields have to be replaced by anticommutators. The resulting Feynman rules are analogous to the bosonic case. Due to the anticommuting nature of fermions, closed loops of fermion lines get an extra factor of (-1). Examples for graphs including fermions are inelastic fermion scattering in a field fermion bosonfermion t e+–e−annihilation tfermion antifermionboson pair production fermion antifermionboson t 1.5 Interacting Fields 67 Yukawa coupling The Lagrangian for the simplest Yukawa model contains a real scalar field φ(x)for aπ0-meson, a Dirac field for a proton, and an interaction term: L=Lφ 0+Lψ 0+Lφψ I, (1.348) Lφ 0=1 2/parenleftBig ∂µφ∂µφ−M2φ2/parenrightBig , (1.349) Lψ 0=¯ψ(iγµ∂µ−m)ψ, (1.350) Lφψ I=−G:¯ψ(x)ψ(x)φ(x) =−HI. (1.351) A self-interaction for the pions, likeλ 4!φ4can also be included. The propagator for fermion fields is given by SF(x) := lim /epsilon1→0+i/integraldisplayd4k (2π)4γµkµ+m k2−m2+ i/epsilon1e−ikx. (1.352) InSF, the time ordering for fermions is defined by T[ψ(x),¯ψ(y)] =/braceleftBigg ψ(x)¯ψ(y),ifx0>y0, −¯ψ(y)ψ(x),ifx0<y0.(1.353) The contractions are defined by ψ(x)¯ψ(y) =T[ψ(x)¯ψ(y)]−N[ψ(x)¯ψ(y)] =SF(x−y).(1.354) Based on this we can draw Feynman graphs for the perturbative contributions. The fermion contraction is represented by a line with an arrow. The arrow gives the direction of charge flow. If charge flows in the same direction as momentum, the particle is a fermion, else it is the antiparticle. Sometimes the flow of momentum or the direction of time is indicated addi- tionally as in these graphs for an incoming antifermion. or t /vectork To distinguish fermions from bosons one draws dotted or dashed lines for bosons. 68 1 INTRODUCTION 1.5.6 Limitations of the perturbative approach •The coupling constant, e.g. λ, should be small, higher orders in the Dyson series should be less relevant. In various interesting cases, e.g. in strong interactions, this is not the case. •At higher orders the number of graphs increases rapidly. They can only be managed by computer programs. •Complicated integrals need special technical tricks. •Some integrals are divergent. Renormalisation is needed. •In general the perturbative series are not convergent, but only asymp- totic. For practical purposes they have to be truncated. 69 2 Quantum Electrodynamics (QED) 2.1 Local U(1) Gauge Symmetry QED deals with the interactions of electrons and positrons with the Maxwell field. Let us begin with the Dirac Lagrangian L=¯ψ(x)(iγµ∂µ−m)ψ(x), (2.1) which has a globalU(1) symmetry: ψ(x)−→ψ/prime(x) = e−iqαψ(x). (2.2) If instead of a constant α, one considers α(x)to depend on x, one has local transformations ψ(x)−→ψ/prime(x) = e−iqα(x)ψ(x). (2.3) But since∂µψdoes not transform like ψ, ∂µψ/prime(x) = e−iqα(x)∂µψ(x)−iq(∂µα(x)) e−iqα(x)ψ(x), the Lagrangian is not any more invariant under this transformation: L/prime=L+¯ψiγµ(−iq∂µα)ψ=L+q¯ψγµψ∂µα=L+jµ∂µα.(2.4) The invariance of the Lagrangian can be restored by coupling the Dirac field to the Maxwell field: L=¯ψ(iγµ∂µ−m)ψ−jµAµ =¯ψ(iγµ∂µ−m)ψ−q¯ψγµψAµ =¯ψ(iγµ(∂µ+ iqAµ)−m)ψ. Under gauge transformations the Maxwell field transforms as Aµ(x)−→A/prime µ(x) =Aµ(x) +∂µα(x). (2.5) Thus we find that Lis invariant: L/prime=L+q¯ψγµψ∂µα−q¯ψγµψ∂µα=L. (2.6) The Lagrangian contains the covariant derivative Dµ:=∂µ+ iqAµ. (2.7) 70 2 QUANTUM ELECTRODYNAMICS (QED) The covariant derivative of the fermion field transforms as Dµψ−→D/prime µψ/prime(x) =(∂µ+ iqA/prime µ)ψ/prime(x) =(∂µ+ iq(Aµ(x) +∂µα(x))) e−iqα(x)ψ(x) =e−iqα(x)(∂µ−iq(∂µα(x)) + iq(Aµ(x) +∂µα(x)))ψ(x) =e−iqα(x)(∂µ+ iqAµ(x))ψ(x) =e−iqα(x)(∂µ+ iqAµ(x))eiqα(x)ψ/prime(x) =e−iqα(x)Dµeiqα(x)ψ/prime(x). So we get the following transformation of the covariant derivative operator D/prime µ= e−iqα(x)Dµeiqα(x). (2.8) The case considered here is the simplest example of a more general situation, where a gauge field is coupled to matter fields. The general picture We start with a global symmetry. (U(1)) This gives a Noether current. ( jµ=q¯ψγµψ) We generalise to a local symmetry. ( α→α(x)) We restore the invariance of the action by adding a gauge field coupled to this Noether current. Thus a symmetry determines the interaction. ( −q¯ψγµψAµ) TocompletetheLagrangian, wehavetoaddthedynamicsofthefreeMaxwell field L(Aµ)=−1 4FµνFµν, (2.9) with the field strength Fµν=∂µAν−∂νAµ=−i q[Dµ,Dν]. (2.10) The complete Lagrangian density of QED thus reads L=¯ψ(iγµDµ−m)ψ−1 4FµνFµν. (2.11) 2.2 Quantum Electrodynamics 71 2.2 Quantum Electrodynamics Now we consider quantisation of the theory. The Hamiltonian contains the free parts for fermions and photons and an interaction term HI=−e0¯ψγµψAµ. (2.12) The free parts give a fermion (electron or positron) propagator and a photon propagator µ ν The photon propagator in momentum space is given by −igµν k2+i/epsilon1. (2.13) The interaction vertex is ie0γµ(/integraltextd4x).µ The external photon lines get polarisation vectors /epsilon1µ(p), /epsilon1∗ µ(p)⊥/vector p. (2.14) Some QED processes e−e−scattering This graph represents the leading order contri- bution. In the non-relativistic limit this gives the Coulomb interaction with potential19 V=−α r, α =e2 4π~c t Compton scattering There are two graphs in leading order ( O(α)), resulting in two interaction processes. γe−e− γ γe−e− γ 19In SI units α=e2/4π/epsilon10~c= 1/137.036 72 2 QUANTUM ELECTRODYNAMICS (QED) e+e−scattering The leading order graphs are e− e+e+e− + e+e−e− e+ e+e−annihilation The leading order graphs describe e++e−→2γ. +e−e− e+e+γγγ γ The graphs considered so far do not contain closed loops of lines – these are tree level graphs. In higher orders graphs with loops give corrections to the leading order, or they may correspond to new processes. For example, corrections to e+e−scattering are The following loop graph contributes to photon–photon scattering by ex- change of virtual electrons. This nonlinear breaking of the superposition principle is also called vacuum birefringence . 2.2 Quantum Electrodynamics 73 74 3 NON-ABELIAN GAUGE THEORY 3 Non-abelian Gauge Theory 3.1 Local Gauge Invariance As in the case of QED, we start by considering a global symmetry. This time, it is given by the non-abelian group SU( N). Consider a field having N complex components: φ(x) = φ1(x) ... φN(x) , φ+(x) =/parenleftBig φ∗ 1(x),... φ∗ N(x)/parenrightBig .(3.1) The scalar product between two complex n-component vectors φandφ/primeis denoted φ+·φ/prime:=N/summationdisplay a=1φ∗ aφ/prime a. (3.2) Let the Lagrangian for the free field be L=∂µφ+(x)·∂µφ(x)−m2φ+(x)·φ(x). (3.3) It is invariant under the transformation φ−→φ/prime=U·φ, (3.4) whereUis aN×Nmatrix obeying U+·U= 1, namely ∂µφ/prime+·∂µφ/prime=∂µ(U·φ)+·∂µ(U·φ) =∂µφ+·U+·U∂µφ=∂µφ+·∂µφ(3.5) and analogously φ/prime+·φ/prime=φ+·φ. (3.6) In particular, matrices Uwhich are in the group SU( N) obey the condition, so that SU( N) is a symmetry group of the Lagrangian. Let us now consider localtransformations U(x)∈SU(N) U(x) = exp −iN2−1/summationdisplay a=1αa(x)Ta , (3.7) whereTaare theN2−1generators of SU( N). The spacetime dependency of the local transformations U(x)gives an additional inhomogeneous term in the derivative of U(x)·φ(x), ∂µφ/prime(x) =U(x)·∂µφ(x) + (∂µU(x))φ(x), (3.8) 3.1 Local Gauge Invariance 75 such that the Lagrangian is no longer invariant. Again, the invariance can be restored by replacing the derivative by the covariant derivative Dµφ(x) := ∂µ−igN2−1/summationdisplay a=1Aa µ(x)Ta φ(x). (3.9) Thisintroduces N2−1gaugefields Aa(x), oneforeachgenerator TaofSU(N). The coupling constant, denoted by −g, replaces the coupling constant +qof QED. A more compact notation is Aµ:=Aa µTa:=N2−1/summationdisplay a=1Aa µTa. (3.10) ThefieldsAµ(x)areelementsoftheLiealgebraofSU( N). Thenthecovariant derivation reads Dµ=∂µ−igAµ. (3.11) In terms of the covariant derivatives the Lagrangian is written L=Dµφ+(x)·Dµφ(x)−m2φ+(x)·φ(x). (3.12) Requiring the invariance of the Lagrangian, we can determine the transfor- mation law for the gauge fields. The covariant derivative should obey D/prime µφ/prime(x)!=U(x)·Dµφ(x), (3.13) which is D/prime µφ/prime(x)!=U(x)·DµU−1(x)φ(x). (3.14) As this should hold for all φ/prime(x), we get D/prime µ!=U(x)·DµU−1(x). (3.15) With the explicit expression for Dµ, Eq. (3.14) reads (∂µ−igA/prime µ(x))·φ/prime(x)!=U(x)·(∂µ−igAµ(x))·U−1(x)φ/prime(x).(3.16) which implies −igA/prime µ(x) =U(x)·∂µU−1(x)−igU(x)Aµ(x)U−1(x). or A/prime µ(x) =U(x)Aµ(x)U−1(x) +i gU(x)∂µU−1(x). (3.17) 76 3 NON-ABELIAN GAUGE THEORY This is the transformation law for the gauge fields. It generalises the trans- formation law for the potentials Aµ(x)in QED. Let us compare the formulae for gauge groups U(1) and SU( N): U(1):U(x) = exp(−iqα(x)1) Aµ(x) =Maxwell field Dµ=∂µ+ iqAµ(x) A/prime µ(x) =Aµ(x) +∂µα(x) SU(N):U(x) = exp (−iαa(x)Ta),and [Ta,Tb]/negationslash= 0in general Aµ(x) =Aa µ(x)Ta Dµ=∂µ−igAµ(x) A/prime µ(x) =U(x)Aµ(x)U−1(x) +i gU(x)∂µU−1(x). Infinitesimal transformations It is also instructive to consider the formulae for the case of infinitesimal transformations U(x) =1−iδαa(x)Ta, (3.18) φ/prime(x) =φ(x)−iδαa(x)Taφ(x), (3.19) A/prime µ(x) =Aµ(x)−iδαa(x)[Ta,Aµ(x)]−1 g∂µδαa(x)Ta+O((δα)2).(3.20) Remembering that Aµ=Aa µTa, we see that the commutators of the genera- torsTaare involved. These commutators are given by the structure constants of the Lie algebra associated with the group SU( N) [Ta,Tb] = ifc abTc. (3.21) With this, the transformation rule for the gauge fields are A/prime µaTa=Aa µTa−iδαaAb µ[Ta,Tb]−1 g∂µδαa(x)Ta =Aa µTa+δαaAb µfc abTc−1 g∂µδαa(x)Ta = (Aa µ+δαbAc µfa bc−1 g∂µδαa(x))Ta, A/prime µa(x) =Aa µ(x) +fa bcδαb(x)Ac µ(x)−1 g∂µδαa(x). (3.22) 3.1 Local Gauge Invariance 77 Field strengths and Yang-Mills Lagrangian The missing piece is now the Lagrangian for the gauge fields. Motivated by QED we attempt to express it in terms of field strengths. We have to find appropriate expressions for the field strengths. Just defining them by Fµν= ∂µAν−∂νAµdoes not work, because this expression does not transform in suchawaythatagaugeinvariantexpressioncanbebuiltfromit. Considering the commutators of the covariant derivatives leads to the proper expression: [Dµ,Dν]φ(x) = [(∂µ−igAµ),(∂ν−igAν)]φ(x) = (−ig[Aµ,∂ν]−ig[∂µ,Aν] + (−ig)2[Aµ,Aν])φ(x) =−ig{Aµ∂ν−∂νAµ+∂µAν−Aν∂µ−ig[Aµ,Aν])}φ(x) =−ig{(∂µAν−∂νAµ)−ig[Aµ,Aν]}φ(x) In the last line, the derivatives act on the Aµonly, and dividing by φ(x)we define the field strengths through Fµν=i g[Dµ,Dν] = (∂µAν−∂νAµ)−ig[Aµ,Aν]. (3.23) Expanding the Aµin the generators yields Fµν=∂µAa νTa−∂νAa µTa+gAa µAb νfc abTc =/parenleftBig ∂µAa ν−∂νAa µ+gAb µAc νfa bc/parenrightBig Ta =:Fa µνTa. (3.24) So the components of Fµνare Fa µν=∂µAa ν−∂νAa µ+gfa bcAb µAc ν. (3.25) How does the field strength transform under gauge transformations? From Eq. (3.15) we obtain F/prime µν=i g[D/prime µ,D/prime ν] =i g[UDµU−1,UDνU−1] =i gU[Dµ,Dν]U−1=UFµνU−1. (3.26) This homogeneous transformation law allows us to form gauge invariant ex- pressions from Fµν. In particular, we can write down a gauge invariant La- grangian for the gauge fields, which parallels the Lagrangian of the Maxwell 78 3 NON-ABELIAN GAUGE THEORY theory. This is the Yang-Mills Lagrangian: LYM LYM=−1 2tr(FµνFµν) (3.27) =−1 2tr(Fa µνFa,µνTaTb) =−1 2tr(Fa µνFa,µν1 2δab) =−1 4Fa µνFa,µν. (3.28) In fact, it is invariant, as can be checked by using the cyclicity of the trace. Remarks: •A mass term like m2Aa µAa,µis forbidden by gauge invariance. •LYMcontains cubic ( Aa λAb µAc ν) and quartic ( Aa κAb λAc µAd ν) terms. They represent self-interactions of the gauge field. Field equations WecannowformulatethecompleteLagrangianforan N-componentcomplex scalar field interacting with non-abelian gauge fields, including an optional φ4-term: L= (Dµφ)+·Dµφ−m2φ+·φ−λ/parenleftBig φ+·φ/parenrightBig2−1 4Fa µνFa,µν.(3.29) From this we may derive field equations for the field φ(x)and for the gauge field strength Fa,µν(x): /parenleftBig DµDµ+m2/parenrightBig φ(x) = 0, (3.30) ∂µFµν−ig[Aµ, Fµν] =jν, (3.31) where jν:=ja,νTa (3.32) is the current formed out of the scalar fields. The first equation is a gauge covariant generalisation of the Klein-Gordon equation. The second equation is the non-abelian generalisation of the inhomogeneous Maxwell equations. The analogue of the homogeneous Maxwell equations is [Dρ, Fµν] + [Dµ, Fνρ] + [Dν, Fρµ] = 0. (3.33) It holds identically due to the definition of the field strengths. This equa- tion is called Bianchi identity , because it has the same structure as the first 3.1 Local Gauge Invariance 79 Bianchi identity, which expresses a symmetry of the curvature tensor in dif- ferential geometry. (Since Fµνis a Lie bracket, the Bianchi identity can also be considered as a Jacobi identity.) The second field equation can also be written as [Dµ, Fµν] =jν, (3.34) because [Dµ, Fµν] = [∂µ−igAµ,Fµν] =∂µFµν−Fµν∂µ−ig[Aµ, Fµν] = (∂µFµν)−ig[Aµ, Fµν]. (3.35) The current jµis not conserved, ∂µjµ/negationslash= 0, since the gauge field itself is charged and contributes to the total current. Historical remarks on gauge theory The concept of local gauge invariance has its origin in the work of Hermann Weyl in 1918. He tried to extend general relativity by allowing spacetime dependent scale changes, “Umeichungen”, of the meter stick and the time- normal. Thus the metric tensor undergoes a local gauge transformation gµν(x)−→Ω(x)gµν(x). (3.36) In Weyl’s approach 0<Ω(x) = eα(x).Einstein pointed out, however, that the length of a meter stick or the frequency of a clock could change if it is moved around a closed path and returned to its origin, which is physically unacceptable. After the introduction of Schrödinger’s wave function, London and others found that by considering purely imaginary α(x)in Weyl’s gauge transformations, andtransformingthewavefunctioncorrespondinglyinstead of the metric, the coupling to the electromagnetic field is recovered. Here are some cornerstones in the development of gauge theory. H. Weyl 1918 V. Fock 1926 Gauge covariance of the Schrödinger equation. F. London 1927 Ω(x) = eiα(x). H. Weyl 1929 Local gauge theory for the Maxwell and matter fields. O. Klein 1938 Conference in Warsaw. Non-abelian gauge field with covariant derivative, no Lagrangian. W. Pauli 1953 Non-abelian gauge theory, based on non-abelian Kaluza-Klein theory. Unpublished, because there is no mass term in the theory. C. N. Yang, and R. Mills 1954 Phys. Rev. 96(1954) 191. Gauge theory for SU(2). R. Shaw 1955 PhD thesis as student of A. Salam, Cambridge (UK). R. Utiyama 1957 Phys. Rev. 101(1957) 1597. Gauge model of gravity. 80 3 NON-ABELIAN GAUGE THEORY 3.2 Geometry of Gauge Fields 3.2.1 Differential geometry A manifold Mis, roughly spoken, a space with a curvilinear coordinate system. The coordinates are sufficiently smooth. To each pointxof the manifold is associated a tan- gent space. Its elements are tangent vectors. A vector field associates a particular vector v(x)to each point xin a smooth way. Vectors can be represented by their components vµ(x)relative to a basis in the tangent space at x. Now we would like to define how to take the derivative of a vector field along a particular curve. One is tempted to define the derivative via dvµ(x) =vµ(x+dx)−vµ(x), (3.37) wheredxis an infinitesimal line element along the curve. The problem with this is, however, that the direction of the basis vectors, which de- termine the components vµ, in general can change from point to point. So, we need an appropriate way to compare vectors at differ- ent points. This can be done by taking the vector at a point x,v(x), andfirstshift it by parallel transport to the point x+dxand thencompare it with the vector v(x+dx). In order to put this into practice, a notion of parallel transport for vectors in the surface is needed. Parallel transport does not mean that the transported vectors are parallel in the embedding space. This is illustrated in the picture, where vectors are parallel trans- ported along great circles on a sphere. The embedding space is not constitutive for the vector fields. We have introduced it here to illustrate the concepts. Parallel transport must be defined without referring to an em- bedding space. 3.2 Geometry of Gauge Fields 81 Thus we must specify a rule for parallel transport. A vector v(x)which is parallel transported from xto the point x/prime=x+dxis a vectorvp(x+dx), v(x)−→vp(x+dx). (3.38) Althoughvp(x+dx)is parallelly transported from v(x), its components change due to, say a rotation of the vector space basis, when going from xtox+dx. ⇒v(x) vp(x+dx) Therefore, associated with an infinitesimal parallel transportation is an in- finitesimal rotation. vp µ(x+dx) =/parenleftBig 1−Γλdxλ/parenrightBigν µ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright infinitesimal rotation matrixvν(x). (3.39) The elements of the infinitesimal rotation matrix are known as Christoffel symbols. With vp µ(x+dx) =vµ(x) +δvµ(x) (3.40) one writes δvµ(x) =−Γν µλ(x)vν(x)dxλ. (3.41) Differentiation Becausev(x)andv(x+dx)belong to different vector spaces, there is a priori no way to add or subtract these vectors. To define differentiation, one may instead use the difference between vµ(x+dx)and the components of the parallel transported vwhose coordinates are denoted by vp µ(x+dx). ⇒v(x)vp(x+dx)v(x+dx)Dv Inordertodistinguishitfromthenaivedifference dvµ(x) =vµ(x+dx)−vµ(x), it is denoted by a capital D: Dvµ(x) :=vµ(x+dx)−vp µ(x+dx). (3.42) 82 3 NON-ABELIAN GAUGE THEORY Usingδvµ(x), introduced above, we find Dvµ(x) =dvµ(x)−δvµ(x) =∂λvµ(x)dxλ+ Γν µλ(x)vν(x)dxλ =/bracketleftBig ∂λvµ(x) + Γν µλ(x)vν(x)/bracketrightBig dxλ =:Dλvµ(x)dxλ, (3.43) with the covariant derivative Dλvµ=∂λvµ+ Γν µλvν. (3.44) Dλvµtransformscovariantlyundercoordinatetransformations. Itisatensor, while∂λvµ,Γν µλvν, and Γν λνare not tensors. 3.2.2 Gauge Theory In contrast to differential geometry, the vector fields φ(x)of gauge theory are not vectors in the tangent space of a manifold. They belong to an “internal” vector space Vxlike isospin, flavour, and colour vector space20. φ(x) = φ1(x) ... φN(x) . (3.45) The local gauge transformation with gauge group elements U(x), denoted by φ(x)−→φ/prime(x) =U(x)φ(x), (3.46) can be considered as a spacetime dependent change of the basis in Vx. It is a passive transformation, the components φachange because the basis vectors change. Physics should not depend on the local choice of the basis. Therefore, diffe- rentiation has to be defined based on the change of φrelative to the parallel transported φp. Similar to the geometric case of the last section we write δφa(x) =φp a(x+dx)−φa(x). (3.47) In this case we do not have Christoffel symbols, however, we can write the equation, which fixes the meaning of parallel transport, in an analogous way: δφ(x) = igAµ(x)·φ(x)dxµ. (3.48) 20The mathematics of gauge theory and fiber bundles was developed by Élie Cartan, and Charles Ehresmann. 3.2 Geometry of Gauge Fields 83 HereAµ(x)is a Hermitian matrix, and igAµ(x)dxµdescribes an infinitesimal rotation in the internal vector space. Aµ(x)is an element of the Lie algebra belonging to the gauge group element U(x). We define Dφ(x) :=φ(x+dx)−(φ(x) +δφ(x)) =dφ(x)−δφ(x) =∂µφ(x)dxµ−igAµ(x)φ(x)dxµ =(∂µ−igAµ(x)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:Dµ)φ(x)dxµ. (3.49) The gauge covariant derivative Dµ=∂µ−igAµ(x) (3.50) corresponds to the covariant derivative of geometry. There is another object of geometry that can be taken over to gauge theory, the curvature. As the picture about parallel displacement of vectors on a sphere showed, curvature is related to the differing results of parallel trans- portation, when the vectors are transported along different paths. Consider two different infinitesimal paths from a point to a neighbouring point, along which vectors are parallelly transported. x x+ax+a+b x +b Path 1Path 2 A vectorv(x), which is transported from xtox+a+balong one path P1 differs from the vector, which is transported along path P2. Along path 1 the coordinates change according to vµ(x)→vµ(x)−Γν µα(x)vν(x)aα →vµ(x)−Γν µα(x)vν(x)aα−Γλ µβ(x+a){vλ(x)−Γν λα(x)vν(x)aα}bβ =vµ(x)−Γν µα(x)vν(x)aα−Γλ µβ(x)vλ(x)bβ−∂αΓλ µβ(x)aαvλ(x)bβ + Γλ µβ(x)Γν λα(x)vν(x)aαbβ+O(a2b). 84 3 NON-ABELIAN GAUGE THEORY Along path 2, after having exchanged aandb, the parallel transported vector is vµ(x)→vµ(x)−Γν µα(x)vν(x)bα−Γλ µβ(x)vλ(x)aβ−∂αΓλ µβ(x)bαvλ(x)aβ + Γλ µβ(x)Γν λα(x)vν(x)bαaβ+O(a2b). Then both displaced vectors differ by ∆vµ=δ(path 2 )−δ(path 1 ) =∂αΓλ µβ(x)aαvλ(x)bβ−∂αΓλ µβ(x)bαvλ(x)aβ + Γλ µβ(x)Γν λα(x)vν(x)bαaβ−Γλ µβ(x)Γν λα(x)vν(x)aαbβ. The result is ∆vµ=Rν µαβvνaαbβ, (3.51) where we introduced the Riemann-Christoffel curvature tensor Rν µαβ Rν µαβ=∂αΓν µβ−∂βΓν µα+ Γλ µαΓν λβ−Γλ µβΓν λα. (3.52) In a flat manifold both parallel transports must give the same result ∆v= 0, and thusRν µαβ= 0. Manifolds can be flat locally, i.e. in an infinitesimal neighbourhood of a point, as is the case with a saddle point on a two dimen- sional surface. Curvature of gauge fields Ingaugetheorywehavethesamealgebraicstructureforparalleltransporting the vector field φ δφ(x) = igAµ(x)·φ(x)dxµ. (3.53) Writing the matrix elements of AµasAd cµ, we have δφc(x) = igAd cµ(x)φd(x)dxµ. (3.54) Comparing this with δvν(x) =−Γλ νµ(x)vλ(x)dxµ, (3.55) we see that the analogue of the Christoffel symbol Γis−igA. The difference of the parallel transport along two infinitesimally paths is then ∆φc=−ig/bracketleftBig ∂αAd cβ−∂βAd cα−igAλ cαAd λβ+ igAλ cβAd λα/bracketrightBig φdaαbβ,(3.56) 3.2 Geometry of Gauge Fields 85 or using matrix notation ∆φ=−ig{∂αAβ−∂βAα−igAα·Aβ+ igAβ·Aα}·φaαbβ ={[∂α,∂β]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =0−[∂α,igAβ] + [∂β,igAα] + [igAα,igAβ]}·φaαbβ = [∂α−igAα,∂β−igAβ]·φaαbβ = [Dα,Dβ]·φaαbβ. As we have already seen, the commutator of the D’s gives the field strengths [Dµ,Dν] =−igFµν, therefore, ∆φ(x) =−igFµν(x)φ(x)aµbν /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright :=fµν, (3.57) wherefµνis a surface element. Based on the analogy with differential geom- etry, on calls Fµν=“Curvature of the gauge field” , (3.58) and ifFµν≡0, one says “the gauge field is flat.” In this case Aµcan be transformed to 0 by a special choice of gauge transformation. Let us consider parallel transport at finite distances. To see, how this works, we will look at the abelian case with symmetry group U(1). x x/primeφp(x/prime) φ(x)∈C Then an infinitesimal parallel transport along a curve is given by δφ(x) =−iqAµ(x)dxµ, (3.59) which formally may be integrated φp(x/prime) = exp/braceleftbigg −iq/integraldisplay Aµ(z)dzµ/bracerightbigg φ(x). (3.60) We check this by returning to infinitesimal steps φp(x+dx) ={1−iqAµ(x)dxµ}φ(x) =φ(x)−iqAµ(x)dxµφ(x) =φ(x) +δφ(x). In the non-abelian case the exponentiated line integral/integraltextx/prime xAµ(z)dzµhas to be path ordered in much the same way as the time ordered expressions for the S-matrix. 86 3 NON-ABELIAN GAUGE THEORY For aclosedcurveCthe change ∆φin the abelian case is given by φ(x) + ∆φ(x) = exp{−iq/contintegraldisplay CAµ(z)dzµ}φ(x). (3.61) From Stokes theorem /contintegraldisplay CAµ(z)dzµ=/integraldisplay AFµνdfµν=flux through area A21(3.62) we have φ(x) + ∆φ(x) = exp/braceleftbigg −iq/integraldisplay Fµνdfµν/bracerightbigg φ(x). (3.63) This is a gauge invariant expression and consequently the line integral of Aµ along a closed loop is gauge invariant. In electrodynamics the phase of the elec- tron wavefunction changes along a curve due to the gauge dependent vector potential. But the phase is only measurable through interference, which means that one needs two different paths to the point of measure- ment. The resulting phase difference equals the phase of ∆φfor a closed curve. The Aharonov-Bohm interference relies on this gauge invariant phase difference.thin solenoid magnetic fluxdouble slit screen B= 0 The infinitesimal version of Eq. (3.63) is ∆φ(x) =−iqFµν(x)fµνφ(x). (3.64) 21Remind e.g. the magnetic flux /integraldisplay /vectorB·df=/integraldisplay ∇×/vectorA·df=/contintegraldisplay /vectorA·d/vector r. 87 4 Quantum Chromodynamics (QCD) 4.1 Lagrangian Density and Symmetries Quarks are represented by Dirac fields, which we now denote qinstead ofψ. q= (qα) = q1 ··· q4 . (4.1) Today, six “flavours” of quarks are known: /parenleftBigg u d/parenrightBigg ,/parenleftBigg c s/parenrightBigg ,/parenleftBigg t b/parenrightBigg , and we write qf= (qα,f), f = 1,2,... Nf, Nf= 6. (4.2) Finally, there are three “colours” for each quark, which have been introduced originally to solve the problem of quark statistics: qi= (qα,i,f), i = 1,2,3,ori=red,green,blue.(4.3) Thus in total we characterise a quark state by the three quantum numbers α, i, f, which gives 4×3×Nfpossibilities. Let us begin with the Lagrangian density of free quarks L=/summationdisplay i,f¯qif(iγµ∂µ−mf)qif. (4.4) For clarity we have explicitly written out the summation over the colour and flavour indices. The spinor indices are hidden. From this Lagrangian one gets3×NfDirac equations, one for each colour and flavour. Symmetries Symmetries put very strong restrictions on the structure of physical objects or relations. For instance, symmetry around one axis of rotation restricts objects to look as being made on a turning lathe like some chess figures, and symmetry under the full rotation group allows only spherical objects. Symmetries also restrict interactions, which to a large extent are fixed by symmetry requirements. An example is the term FµνFµνin the Lagrangian. Higher powers of FµνFµνwould show the same symmetry, but they must be excluded, because they lead to non-renormalisable theories. 88 4 QUANTUM CHROMODYNAMICS (QCD) Global SU(3) colour symmetry: qif(x)−→q/prime if(x) = (Uq)if=/summationdisplay i,fUijqjf(x) (4.5) Uis an unitary 3×3 matrix. Since there is another U(1) symmetry, U can be restricted to detU= +1, so we have U∈SU(3).Ucommutes with operators on flavour and spinor subspace, therefore, the symmetry of the Lagrangian is proven by ¯q/prime if(iγµ∂µ−mf)q/prime if= ¯qif(U†)ij(iγµ∂µ−mf)Ujkqkf = ¯qif(U†)ijUjk(iγµ∂µ−mf)qkf = ¯qif(iγµ∂µ−mf)qif. U(1) baryon symmetry qif(x)−→q/prime if(x) = e−iαqjf(x) (4.6) holds for the same reason. The corresponding Noether current is jµ(x) = ¯q(x)γµq(x) =/summationdisplay i,f¯qif(x)γµqif(x). (4.7) The conserved charge is the integral of the zero component of the Noether current, which counts the number of quarks. By convention the baryon number is B=1 3/integraldisplay d3xj0(x). (4.8) Approximate flavour symmetry SU( Nf) This symmetry would be exact, if all quark masses mfwere equal. We shall give the transformation laws later, when discussing interactions. 4.1.1 Local SU(3) colour symmetry This symmetry was introduced in the early 1970’s. The idea that colour is the source of a SU(3) gauge field, was formulated by Fritzsch, Gell-Mann and Leutwyler. Local SU(3) gauge symmetry is introduced in the following way. 1. Replace the derivative ∂µ−→Dµ=∂µ−igAa µ(x)Ta, (4.9) Ta=1 2λa, (a= 1,... 8). (4.10) 4.1 Lagrangian Density and Symmetries 89 2. This introduces 8 gauge fields, called gluon fields. In the quantised theory they are associated with 8 new particles, the gluons. The field strengths are denotedGa µν, Ga µν=∂µAa ν−∂νAa µ+gfabcAb µAc ν. (4.11) 3. The Lagrangian of QCD now gets an extra Yang-Mills-action part LYM=−1 4Ga µνGµν,a, (4.12) LQCD=/summationdisplay ¯q(iγµDµ−mf)q−1 4Ga µνGµν,a. (4.13) The gluon fields lead to new Feynman graphs. The quark propagators are fermionic lines, and the gluon propagators are similar to photon lines. The term ¯qiγµDµqleads to quark-gluon ver- tices, similar to QED. The quadratic term in the gluon field strengths contains self-interaction of gluons. Ga µνGµν,a =−1 4(∂µAa ν−∂νAa µ+gfabcAb µAc ν)(∂µAν,a−∂νAµ,a+gfadeAµ,dAν,e) =−1 4(∂µAa ν−∂νAa µ)(∂µAν,a−∂νAµ,a) −1 2gfabcAb µAc ν(∂µAν,a−∂νAµ,a),3-gluon vertex −1 4g2fabcfadeAb µAc νAµ,dAν,e. 4-gluon vertex 90 4 QUANTUM CHROMODYNAMICS (QCD) Quantisation of gluon fields in perturbation theory requires gauge fixing. This introduces new fields, the ghost fields . Ghost propaga- tors are represented by dotted lines. There are also ghost-gluon vertices.22/triangleleft 4.1.2 Global flavour symmetry While the masses of up and down quarks are nearly the same, the other quarks are much more massive. As a consequence the flavour symmetry SU(Nf) is strongly broken. In spite of this, we can look not only for approx- imate symmetries but also for symmetries in subspaces of the quark states. ForU∈SU(Nf) we write qf−→q/prime f/prime(x) =Uf/primefqf(x). (4.14) Here, as elsewhere, we hide indices, which are unimportant in the context. Some examples are: for quarks q means (qαif(x)), (γµq)αifmeansγµ αβqβif, ifU∈SU(Nf),U·qmeans (U·q)αif=Uff/primeqαif/prime(x), ifU∈SU(3)colour,U·qmeans (U·q)αif=Uijqαjf(x). The term ¯q(iγµDµ−m)qin the Lagrangian LQCDtransforms under U∈ SU(Nf) as ¯q(iγµDµ−m)q−→¯q/prime(iγµDµ−m)q/prime =¯q(U†iγµDµU−U†mU)q =¯q(U†UiγµDµ−mU†U)q =¯q(iγµDµ−m)q. Since the quark masses of different flavours are not the same, the mass term in the Lagrangian is /summationdisplay fmf¯qfqf= ¯qMq, (4.15) 22See Faddeev-Popov ghosts. 4.1 Lagrangian Density and Symmetries 91 with the quark mass matrix M= mu0 0... 0 0md0... 0 0 0ms... 0 0 0 0...... 0 0 0 ... mb . (4.16) A total SU( Nf) flavour symmetry would arise, if ¯q/primeMq/prime= ¯qU†MUq = ¯qU†UMq = ¯qMq. (4.17) This would be the case, if all Ucommute with M, which means M=m1. Then all quark masses are equal. If some quark masses are approximately equal, likemu≈md, one has an approximate symmetry in their subspace, e.g. the SU(2)-isospin symmetry or the SU(3)-flavour symmetry for mu≈ md≈ms. In these cases the unitary transformation has the form U=/parenleftBigg V0 01/parenrightBigg , V∈SU(n). (4.18) 4.1.3 Chiral symmetry For processes at very high energies the masses of the quarks are negligible. This motivates another symmetry, which holds, if all quark masses vanish, mf= 0. The transformations, which act in flavour and Dirac space, are called axial transformations and read qf−→q/prime f= [exp(−iωaTaγ5)]ff/primeqf/prime. (4.19) The invariance of the kinetic term relies on the relation [γµ,γ5]+= 0. (4.20) Let us check this: γ0γ5=/parenleftBigg 10 0−1/parenrightBigg/parenleftBigg 01 10/parenrightBigg =/parenleftBigg 01 −10/parenrightBigg =−/parenleftBigg 01 10/parenrightBigg/parenleftBigg 10 0−1/parenrightBigg =−γ5γ0 γkγ5=/parenleftBigg 0σk −σk0/parenrightBigg/parenleftBigg 01 10/parenrightBigg =/parenleftBigg σk0 0−σk/parenrightBigg =−/parenleftBigg 01 10/parenrightBigg/parenleftBigg 0σk −σk0/parenrightBigg =−γ5γk The axial transformations do not form a group: e−iωaTaγ5e−iω/primebTbγ5/negationslash= e−iω/prime/primeaTaγ5. (4.21) 92 4 QUANTUM CHROMODYNAMICS (QCD) They are elements of a larger group, the chiral symmetry group, which we shall discuss now. In order to find the chiral symmetry group, we introduce the notion of chi- rality. The chiral projections are defined by PL:q(x)−→qL(x) :=1 2(1−γ5)q, (4.22) PR:q(x)−→qR(x) :=1 2(1 +γ5)q. (4.23) We speak of left-handed fieldsqLorright-handed fieldsqR. We see immediately that q(x) =qL(x) +qR(x)orPL+PR=1. (4.24) PLandPRare projection operators P2 R,L=1 4(1±γ5±γ5+γ5γ5/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =1) =1 4(2±2γ5) =PR,L.(4.25) One also has PLPR=PRPL= 0, (4.26) from (1−γ5)(1 +γ5) = 1−γ2 5= 0. (4.27) For high energetic particles described by a Dirac wavefunction, PLψ=ψor PRψ=ψmeans that the momentum of the particle is parallel (right-handed) or antiparallel (left-handed) to the direction of motion. We shall make use of the following relations. ¯qL= ¯q·1 2(1 +γ5), (4.28) ¯qR= ¯q·1 2(1−γ5), (4.29) Check: since 1−γ5is Hermitian and γ5γ0=−γ0γ5, one has ¯qL= (1 2(1−γ5)q)†γ0=q†1 2(1−γ5)γ0=q†γ01 2(1 +γ5) = ¯q1 2(1 +γ5), and similarly for ¯qR. Also ¯qLqL= ¯qRqR= 0 (4.30) 4.1 Lagrangian Density and Symmetries 93 holds due to (1 +γ5)(1−γ5) = 0. The mass term can thus be decomposed as ¯qq= (¯qL+ ¯qR)(qL+qR) = ¯qLqR+ ¯qRqL. (4.31) On the other hand, for the terms in the kinetic part of the Lagrangian we find ¯qLγµqR=1 4¯q(1 +γ5)γµ(1 +γ5)q=1 4¯q(1 +γ5)(1−γ5)γµq= 0, ¯qRγµqL=1 4¯q(1−γ5)γµ(1−γ5)q=1 4¯q(1−γ5)(1 +γ5)γµq= 0. UsingtheserelationstheLagrangiandensityforQCDcanbewritteninterms of the left-handed and right-handed fields as L=/summationdisplay f{¯qfLiγµDµqfL+ ¯qfRiγµDµqfR−mf(¯qfLqfR+ ¯qfRqfL)}−1 4Ga µνGµν,a. (4.32) The mass terms couple the left-handed to the right-handed fields. If all quark masses vanish, the left-handed and right-handed fields decouple. In this case there is a larger symmetry, which acts on the left-handed and the right-handed fields independently. The symmetry transformations are qL−→q/prime L= e−iωa LTaqL, e−iωa LTa∈SU(Nf)L, (4.33) qR−→q/prime R= e−iωa RTaqR, e−iωa RTa∈SU(Nf)R. (4.34) BothgroupsSU( Nf)LandSU(Nf)RareisomorphictoSU( Nf). Theindices L orRjust indicate that the symmetry transformations act on the left-handed and the right-handed fields, respectively. The invariance of the kinetic part of the Lagrangian holds because of ¯q/prime LiγµDµq/prime L= ¯qLiγµDµqL, (4.35) ¯q/prime RiγµDµq/prime R= ¯qRiγµDµqR, (4.36) whereas in the mass terms the exponentials do not cancel, i.e. ¯q/prime Lq/prime R= ¯qLeiωa LTae−iωa RTaqR/negationslash= ¯qLqR. So, formf= 0there is a symmetry of the Lagrangian under the group SU(Nf)L⊗SU(Nf)R, (4.37) which has 2(N2 f−1)parameters. This symmetry is called chiral symmetry. 94 4 QUANTUM CHROMODYNAMICS (QCD) The currents, belonging to this symmetry according to Noether’s theorem are jµ La= ¯qLγµTaqL, (4.38) jµ Ra= ¯qRγµTaqR. (4.39) For example, for the SU(Nf)Ltransformations we find jµ La=∂L ∂(∂µqLf)δqLf δωa L= ¯qLfiγµ(−iTa)qLf= ¯qLγµTaqL, and similarly for the right handed transformations. Using PRγµPL=1 2(1 +γ5)γµ1 2(1−γ5) =γµ1 2(1−γ5)1 2(1−γ5) =γµP2 L=γµPL, the currents can be written jµ La= ¯qγµTa1 2(1−γ5)q, (4.40) jµ Ra= ¯qγµTa1 2(1 +γ5)q. (4.41) Relation to flavour symmetry The flavour and chiral symmetries are related, flavourSU (Nf)←→chiral SU(Nf)L⊗SU(Nf)R. (4.42) To see this, consider the infinitesimal transformations q/prime L= (1L−iωa LTa)qL, q/prime R= (1R−iωa RTa)qR, q/prime= (q/prime L+q/prime R) = (1−iTa[ωa LPL+ωa RPR])q.(4.43) If we restrict the chiral transformations to ωa L=ωa R≡ωa, (4.44) we obtain the infinitesimal flavour transformations q/prime= (1−iωaTa)q. (4.45) Therefore, the flavour symmetry group is a subgroup of the chiral symmetry group SU(Nf)L⊗SU(Nf)R. It is called the diagonal subgroup , because it consists of those group elements UL⊗URwhich obey UL=UR. flavour group ={UL⊗UR|UL,UR∈SU(Nf), UL=UR} /similarequalSU(Nf)@SU(Nf)⊗SU(Nf)(4.46) 4.1 Lagrangian Density and Symmetries 95 Theflavour current belonging to the flavour symmetry is jµ a= ¯qγµTaq=jµ La+jµ Ra. (4.47) The flavour current is also called vector current, since it transforms as a Lorentz vector. Infinitesimal axial transformations are defined by ωa L=−ωa R≡−Ωa(4.48) and read (PL−PR=−γ5) q/prime= (1−iTa[ωLPL+ωRPR])q = (1−iTa(−Ωa)(PL−PR)q = (1−iΩaγ5Ta)q.(4.49) The corresponding axial current or axial vector current is given by jµ 5a:= ¯qγµγ5Taq=jµ Ra−jµ La. (4.50) It transforms as an axial vector under Lorentz transformations, i.e. if the axial vector current is looked at through a mirror, symbolised by the space reflection (parity) operator P, then left and right would be interchanged, and the current changes sign: P(jµ Ra−jµ La) =−jµ Ra+jµ La. 4.1.4 Broken chiral symmetry Symmetries of a physical theory are not just mathematical curiosities. In general they give strong restrictions on the structure and behaviour of the corresponding models. In case of a field theory, if the Lagrangian possesses a certain symmetry and if this symmetry is also realised by the physical states, then as a consequence various relations between observables emerge. Now we shall consider the question, in which way chiral symmetry of the QCD Lagrangian is realised in nature. If chiral symmetry were realised on the physical states, then e. g. the mesons and baryons should form multiplets of the symmetry group, such that the members of a multiplet have the same mass. Consider the case Nf= 2with the chiral SU(2)L⊗SU(2)Rsymmetry. The flavour symmetry subgroup, the SU(2) isospin symmetry, is an approximate symmetry in nature, as is shown by the up and down quarks, by the nucleons p,n, and by the pions π±,0. 96 4 QUANTUM CHROMODYNAMICS (QCD) On the other hand the chiral SU(2)⊗SU(2)group is not found as a symmetry group in nature. One can show that this symmetry would imply that to each isospin multiplet of particles with definite parity there should belong another isospin multiplet with the same mass but opposite parity. This is, however, not observed in nature. For instance, the pions form an isospin triplet of pseudoscalar particles, but there is no isospin triplet of scalar particles with masses near the pion masses. The same holds for other SU(N)-multiplets. If the Lagrangian Lpossesses a symmetry that is not realised in nature, one speaks of a “spontaneously broken symmetry”, in contrast to an explicitly broken symmetry, where the breaking results from symmetry breaking terms in the Lagrangian. Spontaneously broken symmetries are already familiar from everyday life and classical physics, as the following examples show. a) Swimming If one puts a rectangular piece of wood in water, it can take two positions, in which one end of the wood is immersed deeper than the other (supposed the density of the wood is not too different from the density of water). There is obviously a reflection symmetry in the Lagrangian, which is broken by the outcome of the experiment. b) Ferromagnet a1a2 A ferromagnet shows spontaneous magnetisation below the Curie point, T < TC. The directions of the molecular magnetic moments /vector m(/vector r)give up their random distribution in favour of one macroscopic direction, although the 4.2 Running Coupling 97 Hamiltonian H=−J/summationdisplay /vector r,µ=1,2,3/vector m(/vector r)·/vector m(/vector r+/vector aµ) (4.51) obeys rotation symmetry according to /vector m(/vector r)−→R/vector m(/vector r), R∈SO(3). (4.52) The average values of the molecular magnetic moments turn out to be non- zero: /angbracketleft/vector m(/vector r)/angbracketright=/vector m0/negationslash= 0. (4.53) In addition to explicitly and spontaneously broken symmetries, there is a another kind of symmetry breaking in quantum theory, which originates from a pure quantum effect. An example of such a broken symmetry is given by the axial U(1)Asymmetry q(x)−→q/prime(x) = e−iαγ5q(x). (4.54) The corresponding current density jµ 5(x) = ¯q(x)γµγ5q(x) (4.55) is an axial vector current. If mf= 0then U(1)Ais a symmetry of the Lagrangian. Classically the axial vector current is conserved. But this sym- metry is not realised in nature. (We shall not discuss the arguments for this fact here.) It turns out that in the quantum theory the conservation of the axial vector current is violated, ∂µjµ 5/negationslash= 0. (4.56) A situation, where a classically conserved current is not conserved in the quantum theory, is called an “anomaly”. So, anomalies are different from spontaneous symmetry breaking, they are quantum effects. 4.2 Running Coupling The running of the QCD coupling is very important for phenomenology and experiments. A thorough discussion would require to treat renormalisation and the Callan-Symanzik equations. Here we just give a short outline of this topic, presenting some results and experimental evidence for the varying coupling “constants” of QCD. 98 4 QUANTUM CHROMODYNAMICS (QCD) 4.2.1 Quark-quark scattering We start with a perturbative theory along the lines of QED. There is no 4-quark di- agram, so we expand the graph for quark- quark scattering in increasing order of the coupling. The simplest term corresponds to the Coulomb scattering of electrically charged particles, i. e. Rutherford scattering.p1 q q q The simplest, tree level diagram describes a one gluon exchange. The mediating gluon carries 4-momentum q=p/prime 1−p1=−(p/prime 2−p2).(4.57) qis a spacelike vector, q2<0. (4.58)p1p/prime 1 q p2p/prime 2 The scattering amplitude is determined by the gluon propagator and two vertices M=−g2¯u(p/prime 1)γµu(p1)1 q2¯u(p/prime 2)γµu(p2)1 3/braceleftBigg 1rep.6 −2rep.3∗, , (4.59) whereu(pi)are spinors, denoting solutions of the Dirac equation, and the simple form of the gluon propagator 1/q2is due to Feynman gauge. This factor is dominant for the behaviour of the scattering. Let us compare this with the e−µ−scattering in QED, where the muon mass is about 200 times the electron mass, so we are near the static limit of Rutherford scattering. Therefore, |p1|=|p/prime 1|=:kand M∼1 q2≈−1 4k2sin2(θ 2). (4.60) The differential cross-section then obeys dσ dΩ∼|M|2∼1 sin4(θ 2). (4.61) Quantum mechanics tells us that in the Born-approximation Mis propor- tional to the Fourier transformation (FT)of the potential, thus 1 q2∼FT(V(r))⇒V(r)∼1 r. (4.62) 4.2 Running Coupling 99 Thus, in lowest order of perturbation theory the quark-quark potential is a Coulomb potential. Since we only considered one-gluon exchange, we should not be surprised to get the same result as in first order of perturbative QED. In QCD, higher orders of perturbation theory add corrections to 1/q2, which will modify this result. The 1-loop corrections are the following: (a) (b) (c) (d) (e) (f) (g) The diagrams are denoted as: quark-loop (a), gluon-loop (b), gluon-tadpole (c), ghost-loop (d), quark-propagator correction (e), and vertex corrections (f), (g). Additional graphs, which are similar to (e) – (f) are not shown. The result of these calculations in the limit of large momentum transfer, Q2:=−q2large, (4.63) is essentially a modification of the one-gluon exchange case, with a corrected coupling constant g2−→g2/braceleftBigg 1−b0g2 16π2ln/parenleftBiggQ2 M2/parenrightBigg/bracerightBigg , (4.64) with b0=1 3(11Nc−2Nf) = 11−2 3Nf>0 (forNc= 3colours ).(4.65) b0is positive as long as the number of flavours is not larger than 16, and there is no experimental evidence for that. The number 11 comes from the graphs (b) to (g), while2 3Nfstems from graph (a). 100 4 QUANTUM CHROMODYNAMICS (QCD) The expression contains a mass M. It enters the result due to renormali- sation. The loops in the Feynman diagrams involve divergent integrals over the momenta of the type/integraldisplayd4p p4. (4.66) The renormalisation procedure removes the divergencies in two steps, 1. Regularisation: introduces a cutoff for the momentum integration, or analogous divergent integrals. Typical regularisations introduce a cut- off massM, which characterises the largest allowed momenta. Loop integrals are then finite, but depend on M. 2. Renormalisation. 4.2.2 Renormalisation The coupling gis a parameter in the Lagrangian and is not directly mea- surable. We call it “bare” coupling and from now on denote it by g0.A measurable, physical relevant coupling should be defined or fixed through some process. Here, for example, we fix it by choosing a fixed mass scale µ and considering quark-quark scattering at a momentum transfer Q2=µ2. The modified coupling at this momentum transfer is defined to be the renor- malised coupling g2 R(µ) =g2 0/braceleftBigg 1−b0g2 0 16π2ln/parenleftBiggµ2 M2/parenrightBigg +O(g4 0)/bracerightBigg . (4.67) Renormalisation now means •express everything in terms of gRin place ofg0, •then remove the cutoff by taking the limit M−→∞. Renormalisable theories are those which give finite physical results by this procedure. (Eventually other parameters like quark masses, etc., have to be renormalised, too.) Inourcase, expressing g0intermsofgR(µ)andexpandinginpowersof gR(µ), one gets the scattering amplitude M∼1 Q2g2 R(µ)/braceleftBigg 1−b0g2 R(µ) 16π2ln/parenleftBiggQ2 µ2/parenrightBigg +O(g4 R)/bracerightBigg .(4.68) Indeed,Mdoes not longer contain the cutoff M, and is finite. 4.2 Running Coupling 101 4.2.3 Running coupling The result for Msuggests to define an effective coupling for this process, g2 R(Q) :=g2 R(µ)/braceleftBigg 1−b0g2 R(µ) 16π2ln/parenleftBiggQ2 µ2/parenrightBigg +O(g4 R)/bracerightBigg ,(4.69) which indicates that the scattering at momentum transfer Q2is like Ruther- ford scattering with an effective coupling g2 R(Q). We see that the effective coupling is not constant. Therefore it is called “running coupling”. More- over, in the one-loop expression above the corrections get large for large Q2. (Throughout our calculation we have made the assumption of large Q2.) Therefore, we have to extend the perturbation series. Calculating higher or- ders of perturbation, the dominant contributions can be found to lead to a geometric series, so g2 R(Q)∼g2 R(µ)/bracketleftBigg 1 +b0g2 R(µ) 16π2ln/parenleftBiggQ2 µ2/parenrightBigg/bracketrightBigg−1 . (4.70) The figure shows the dependence of the running coupling on Q2. µ2Λ2g2 R(µ)g2 R(Q) Q2running coupling perturbative regime The coupling gets weaker for higher momentum transfer, supporting the suggestion that perturbation theory is reliable at large Q. A more solid consideration is based on the renormalisation group, in par- ticular the Callan-Symanzik equations, which are beyond the scope of this lecture. The renormalisation group equation for the running coupling is Qd dQgR=β(gR) (4.71) =−b0g3 R 16π2+...(odd powers ), (4.72) 102 4 QUANTUM CHROMODYNAMICS (QCD) whereβ(gR)is the famous beta-function of QCD. Thus we have Qincreases−→gRdecreases. (4.73) So, perturbation theory is reliable at large Q. Here, a remark is in order. The perturbation series of QCD or QED are not convergentinthemathematicalsense. However,theerrormadebytruncating the series at some finite order can be controlled due to Borel summability. This allows to get useful results from finite orders of perturbation theory in a suitable regime of applicability, which in QCD is the high energy regime. Let us now consider the region of small Q, or the low energy range. In QCD, b0>0, therefore, for small enough Q,ln(Q2/µ2)becomes negative, and in the considered approximation, gR(Q)gets a pole at a certain point Q= Λ. The QCD Λ-parameter is (roughly) in one-loop order defined by b0g2 R(µ) 16π2ln/parenleftBiggΛ2 µ2/parenrightBigg =−1. (4.74) Its value (in a particular scheme, the MSscheme) is about ΛMS≈200 MeV. The apparent pole at Q= Λis actually an artefact of the approximation and is not present for the full gR(Q). A more precise definition of Λ, which requires two-loop terms, can nevertheless be given. Using the Λ-parameter, for µ≈ΛandQ2/greatermuchΛ2, the high energy behaviour of the coupling constant can be written as g2 R(µ) =16π2 b0ln/parenleftBig Q2 Λ2/parenrightBig. (4.75) This decrease of the running coupling towards zero is called “asymptotic freedom” , lim Q→∞gR(Q) = 0. (4.76) Asymptotic freedom in QCD was first realised by Gerard ’t Hooft in the work on his thesis as a student of Martinus Veltman, but was not published. In 1973 asymptotic freedom in field theories has been discovered by D. J. Gross, F. Wilczek and H. D. Politzer, who were awarded the Nobel prize in 2004. 4.2.4 Discussion a)Asymptoticfreedomguarantiesagoodapplicabilityofperturbationtheory on processes at high energies or momenta. At low energies perturbation calculation fails. 4.2 Running Coupling 103 b)In QED the renormalisation group parameter is b0<0.This comes from the 1-loop calculation of e−e+scattering. Taking the electric charge for the coupling constant, one has e2 R(Q) =e2 R/bracketleftBigg 1−|b0|e2 R 16π2ln/parenleftBiggQ2 m2 e/parenrightBigg/bracketrightBigg−1 . (4.77) In QED the running fine-structure constant α=e2(Q) 4π/epsilon10~c=e2 R(Q) 4π≈1 137.036 shows a tiny increase with momentum. Q2g2 R(Q) e2 R 4π Measurements at 100 GeV have given a value of α−1≈120. The Λ-point (“Landau pole”) belongs to a momentum already beyond the Planck-scale. c)Interpretation SinceQ∼1 randb0<0in QED, the electric charge eR(1/r)increases as r→0. The decrease of the measured charge, when going away from r= 0, has been interpreted as a screening of charges due to vacuum polarisation. In QCD on the other hand, where b0>0, there are also gluon loops and the gluons have self-interactions. This leads to an anti screening effect. d)Experiments The strong “fine-structure constant” αS=g2 R(Q) 4πof QCD has been measured in several experiments: •deep inelastic e−pscattering, 104 4 QUANTUM CHROMODYNAMICS (QCD) •τ-lepton decay, •e+e−scattering, which has contributions of virtual q¯qprocesses, •Ψ,Υ(heavy quarkonia) spectroscopy, •Z0-decay. The results are in very good agreement with the prediction from asymptotic freedom. Figure 3: Summary of measurements of αs(Q),from S. Bethke, Prog. Part. Nucl. Phys. 58 (2007) 351–386, hep-ex/0606035. 4.3 Confinement of Quarks and Gluons 105 4.3 Confinement of Quarks and Gluons In the QCD Lagrangian the quarks and gluons carry colour charge, but these particles do not exist as free particles. Accessible particles are the hadrons: mesons (q¯q)as quark-antiquark states and baryons (qqq), made of three quarks. All these particles are colour neutral. Therefore, one states the Confinement Hypothesis: Physical states are colour neutral; in particular, quarks and gluons do not exist as free particles. Since perturbation theory works with quark and gluon fields, and assumes these constituents as ingoing and outgoing particles, confinement cannot be proven in the framework of perturbation theory. To demonstrate confine- ment is a non-perturbative problem. A rigorous proof of confinement is still missing. Instead, there are some phenomenological descriptions. •The bag model is based on antiscreening effects. The particles are assumed to be con- fined in a region with boundary conditions similar to boundary con- ditions of electrodynamics. In electrodynamics or optics one has the polarisability of the matter, which in the bag model is replaced by the polarisability of vacuum. /epsilon1 /epsilon1vac</epsilon1/vector r--- - - - --+++ + + + +++ •In the string picture the quark-antiquark potential at large distances grows proportional to the distance23, V(r) =kr. (4.78) 23For the static QCD potential see G.S. Ball, Phys. Rept. 343(2001) 1. 106 4 QUANTUM CHROMODYNAMICS (QCD) The force, therefore, remains constant, and when one tries to separate the quarks, energy grows until it suffices to generate a new quark- antiquark pair. The situation may depicted by lines of force, similar to electrodynamics, with the exception that the vacuum compresses the lines to flux tubes, which eventually split. Thus the chromoelectric flux tubesare squeezed by being “repelled” from the vacuum. q ¯q The bag model and the string model are based on postulated dielectric properties of the vacuum with a (relative) dielectric constant /epsilon1r<0←→antiscreening . (4.79) Lattice QCD is able to provide the spectrum of particles and many other physical quan- tities from first principles. The calculated hadron masses are in very good agreement with the experimental values. Practical limitations are the num- ber of lattice points and/or the lower limit to the lattice spacing, which typically is of the order of 0.1 fm. This also puts an upper limit to momenta. ïż£643×128 lattice points Inpure gauge theory ,fora gluon field with fixed quarks, the static quark-antiquark potential can rather well be described by V(r) =c r+k·r+const. (4.80) Herethec/rtermcomesfromperturbationtheory, whichisvalidatsmalldis- tances. Latticecalculationsatlarge rleadtothestaticconfinement V(r)∼r. 4.3 Confinement of Quarks and Gluons 107 Lattice QCD with quarks supports the string picture as it results in string breaking at larger quark distance. ¯q q q ¯q ¯q q ¯q q In this case the quark-quark potential is of the form V(r) rV(r) r∼1 r∼kr(+const.)string breaking→V=const. Lattice simulations in pure gauge theory reveal pure gluonic, colour neutral particles, the so-called glue-balls. These also exist in full QCD with quarks. Jets In perturbation theory, scattering processes in QCD are based on the scat- tering of quarks and gluons. On the other hand, all experimental information stems from the interaction of observable particles, namely hadrons. There- fore, the perturbative results for scattering processes in QCD have to be “hadronised”. Consider for example proton-antiproton scattering. 108 4 QUANTUM CHROMODYNAMICS (QCD) −→ ←− ←− ¯q q−→? Perturbation theory gives cross-sections for quark-quark scattering. Hadro- nisation turns the outgoing quarks and antiquarks into hadrons. This leads to the formation of jets of hadrons (S. Weinberg, G. Sterman). At high momenta the primary quarks and antiquarks combine with newly created quarks and antiquarks and form jets. q−→←−¯qcreation of q¯qg The formation of jets is a non-perturbative process. 4.4 Experimental Evidence for QCD In general, there is a good consistency between the theoretical and experi- mental results for QCD. In the following we list some examples, where ex- periments agree well with theoretical modelling and/or predictions. •Running coupling αS(Q). •Jet distribution. The statistical angular distribution and the momentum distributions of the particles in jets were quantitatively predicted by QCD. •Evidence for gluons. –Deep inelastic e−p+scattering at HERA. e− p+ 4.4 Experimental Evidence for QCD 109 It is found that quarks carry only about 50% of the momentum of the nucleon. The other half has to come from particles, which are neutral with respect to electroweak interactions. This is an indication for gluons. –Three-jet events. Figure 4: A three-jet event observed at PETRA. The figure is from: F. Halzen, A. D. Martin, Quarks&Leptons: An Introductory Course in Modern Particle Physics , J. Wiley & Sons (1984). They are explained by additional particles. The angular distribu- tion of the jets leads to spin 1 particles, which are identified with the gluons, in agreement with the QCD prediction. •Lattice QCD. The masses of stable hadrons have been calculated in lattice QCD. The masses mu=md, msof up, down, and strange quarks, which are needed as input parameters, have been obtained by fitting the proton and Kaon masses, respectively. The lattice simulations from the Budapest-Marseille-Wuppertal collaboration have given the baryon 110 4 QUANTUM CHROMODYNAMICS (QCD) masses for N,Λ,Σ,∆,Σ∗,Ξ,Ω,and the masses of the mesons ρandK∗.The results are within the experimental errors. There have been many other experiments that support QCD theory. Let us mention one more example: Calculating the anomalous magnetic moment of the muon in QED in the same way, as it was done for the electron, the result does not agree with the extremely precise experimental value. How- ever, taking hadronic contributions from QCD into account, the agreement is perfect. 111 5 Electroweak Theory 5.1 Weak Interactions The weak interactions of elementary particles are distinguished from other interactions by some characteristic properties like lifetimes, strength of cou- pling, cross-sections, and violation of symmetries. We refer to the introduc- tory lectures on particle physics for details. Some typical processes of weak interactions are the following. a)Leptonic processes Muon decay: µ−−→e−+ ¯νe+νµ eν-scattering: e−νµ−→µ−+νe b)Semi-leptonic processes. They involve hadrons. β-decay n−→p++e−+ ¯νe β-decay in quark picture d−→u+e−+ ¯νe Pion decay24π−−→µ−+ ¯νµ (d¯u−→µ−+ ¯νµ) c)Non-leptonic weak interaction processes Λdecay Λ0−→p++π−(uds−→uud+d¯u) Kaon decay K−−→π−+π0(s¯u−→d¯u+1√ 2(−u¯u+d¯d)) These weak interaction processes violate isospin symmetry. 5.1.1 Fermi theory of weak interaction As mentioned in the introduction, in 1932 Enrico Fermi formulated a theory for the β-decay of the neutronasafourfermionprocess. TheLagrangian consists of the free parts for n,p+,ν, ande−, plus the four-fermion interaction term Lw=G(¯e(x)γµνe(x)) (¯p(x)γµn(x)).(5.1)n¯νee−p Lwisinvariantunderspacereflections. (ParityoperatorP.)InFermi’stheory parity is conserved. 5.1.2 Parity violation In1957parityviolation, occurringinthe β-decayof60Co, wasexperimentally established by Wu. At that time, when the neutrinos were regarded to be 24There also is a dominant hadronic decay mode giving the same particles. The leptonic mode is distinguished by its timescale. 112 5 ELECTROWEAK THEORY massless particles, the νewas always considered to be left-handed, and the ¯νeright handed. Madame Wu found that the e−arising in this decay is dominantly left-handed, violating parity conservation. Parity violation in Kaon-decay was proposed by Lee and Yang in 1966. In order to describe parity violation and handedness of neutrinos, let us recall some facts about chirality from Sec. 4.1.3. The projection operators on left- and right-handed spinors are, respectively PL=1 2(1−γ5), PR=1 2(1 +γ5). (5.2) WithψL=PLψ, ψR=PRψ, the Dirac Lagrangian is L=¯ψ(iγµ∂µ−m)ψ (5.3) =¯ψLiγµ∂µψL+¯ψRiγµ∂µψR−m(¯ψLψR+¯ψRψL).(5.4) Ifm= 0,Lseparates into a left-handed and a right-handed part. Consider a left-handed Dirac spinor plane wave solution u(k) =uL(k) =1 2(1−γ5)u(k), (5.5) γ5uL(k) =γ51 2(1−γ5)uL(k) =1 2(γ5−1)uL(k) =−uL(k),(5.6) since (γ5)2= 1. With the spin-operator /vectorS= (1/2)/vectorΣwe define he- licityλto be the projection of spin onto the direc- tion of momentum. In the massless case one can show that helicity and chirality are equal: /vectorΣ·/vectork |/vectork|uL(k) =γ5uL(k) =−uL(k).(5.7) The helicity in this case is λ=−1 2./vectork /vectorΣ Similarlyγ5uR(/vectork) = +uR(/vectork), the particle is right-handed, and the helicity is λ= +1 2. Since all neutrinos were found to be left-handed and all antineutrinos to be right-handed, we have ν=νL=1 2(1−γ5)ν, (5.8) ¯ν= ¯νR=1 2(1 +γ5)¯ν. (5.9) 5.1 Weak Interactions 113 The non-existence of right-handed neutrinos implies that parity is violated. Nowadays, experimentshaveshownthatneutrinosaremassiveparticles. Due to their very small mass, they almost always move with a speed very close to the speed of light. Nevertheless, in a moving frame that moves even faster, they appear to be right-handed. So, helicity (and chirality) can take both values. 5.1.3 V-A theory The V-A theory was developed by Feynman and Gell-Mann in 1958 and independently by Marshak and Sudarshan in the same year. It assumes massless neutrinos and takes chirality and parity violation into account. The neutrino part in the Lagrangian is replaced by25 ¯e(x)γµνe(x)→¯e(x)γµ1 2(1−γ5)νe(x) (5.10) = ¯eL(x)γµνe,L(x) =1 2¯e(x)γµνe(x) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright vector current−1 2¯e(x)γµγ5νe(x) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright axial vector current(5.11) =1 2/parenleftBig V(e) µ(x)−A(e) µ(x)/parenrightBig . (5.12) Thus, the V-A theory modifies the Fermi theory, represented by the vector currentV, by subtracting the axial vector current term A. There are further contributions, which come from the muon and tau lepton. The total weak leptonic current then reads J(l) µ= 2¯eL(x)γµνe,L(x) + 2¯µL(x)γµνµ.L(x) + 2¯τL(x)γµντ,Le(x).(5.13) Hadronic Currents Hadronic currents enter in the same way as leptonic currents. For the process that a down quark is converted to an up quark, the current is J(h) µ= 2¯uL(x)γµdL(x). (5.14) (For nucleons one would write 2¯pLγµnL.) 25Compare the calculations for vector currents on page 94. 114 5 ELECTROWEAK THEORY In total we have Jµ(x) =J(l) µ(x) +Jh) µ(x). (5.15) Lw=GF√ 2Jµ(x)Jµ(x)+(5.16) =GF√ 2/parenleftBig J(l) µ(x)Jµ(l)(x)+leptonic interactions +J(l) µ(x)Jµ(h)(x)++J(h) µ(x)Jµ(l)(x)+semi-leptonic interactions +J(h) µ(x)Jµ(h)(x)+/parenrightBig . hadronic interactions The Fermi coupling constant GFis the same for all weak interaction pro- cesses. Its value is GF= 1.03·10−5m−2 p= 4.51·10−33cm2= 1.17·10−5GeV−2.(5.17) Modifications of the V-A theory a)For nucleons (and other hadrons) the hadronic current is replaced by J(h) µ=gVV(h)−gAA(h). (5.18) WhilegV= 1always, the value of the other parameter is in the case of nucleons gA≈1.24, (5.19) whereas for quarks gA= 1. b)For the u, d, s and c-quarks, forming the doublets /parenleftBigg u d/parenrightBigg ,/parenleftBigg c s/parenrightBigg , (5.20) the Lagrangian of V-A theory only allows transitions u↔dandc↔s between quarks. At times, when the charmed quark was not yet known, weak processes had been observed that change strangeness, like the weak Kaon decay K+−→µ++ ¯νµ. (5.21) Numericalrelationsbetweenstrangeness-conservingandstrangeness-changing processes motivate to incorporate strangeness-changing processes by postu- lating that mixtures of sanddstates d/prime=dcosθC+ssinθC (5.22) s/prime=scosθC−dsinθC. (5.23) 5.1 Weak Interactions 115 take part in the weak interactions. Including charmed particles, the hadronic part of the current is J(h) µ= 2¯uLγµd/prime L+ 2¯cLγµs/prime L. (5.24) The angleθCis called Cabibbo angle, its value is θC≈13◦. (5.25) The V-A theory very successfully describes many weak processes at relatively low energies. Owing to the structure of the currents Jµ, in the processes described by the V-A theory, the electric charge does always change in the leptonic sector. One speaks of “charged currents”. Problems of the V-A theory •It is not renormalisable. We have outlined the theory based on tree level diagrams. Adding loop-corrections will give non-renormalisable infinities. •It shows bad behaviour at high energies. Scattering cross-section be- have like σ∼G2 F·s, s = (p+q)2=E2 CMS (5.26) At high energies this violates the rigorous “unitary bound” σ≤4π s. (5.27) This happens at s≈1 TeV2. •The discovery of neutral currents at CERN, 1973. The following pro- cesses have been found. ( Xstands for a hadron.) ¯νµ+e−−→¯νµ+e−, νµ+N−→νµ+X, ¯νµ+N−→¯νµ+X. Intermediate vector bosons In orderto improve thehigh-energy behaviour, it wasproposed thatthe weak interactions are mediated by bosons, analogous to the photon in QED. For the charged currents two charged spin 1 particles, the vector bosons W(+) µ andW(−) µhave been postulated. If these particles were massless like photons, 116 5 ELECTROWEAK THEORY their range would be infinite in contrast to the short range nature of weak interactions. Therefore, the W-bosons are massive. The coupling to other fields are given by Lw=gWJµW(−)µ+gWJ+ µW(+)µ. (5.28) Inβ-decay, for example, the 4-fermion-vertex of the Fermi theory is replaced by a graph containing a W(+)propagator. −→u d ¯νee− u d ¯νee−W(+) For low energies the V-A theory should be reproduced. This is the case if g2 W m2 W=GF√ 2, (5.29) and consequently the mass mWhas to be large. To describe the neutral currents within this approach, there should in addi- tion exist a neutral intermediate vector boson, denoted Z(0). In fact, in 1983 the WandZbosons have been found with the Super-Proton- Synchrotron (SPS) at CERN. Nevertheless, there remains a serious problem: The theory of intermediate vector bosons is not renormalisable. This fact is related to the mass of the intermediate particles. (The renormalisability of QED is connected with the fact that photons are massless.) So we are faced with the question: Is there a theory with intermediate vector bosons, which is renormalisable ? One can think of two different approaches to answer this question. On the one hand, one could start with the most general Lagrangian, having all imag- inable kinds of intermediate particles and appropriate symmetries, and then sort out all those theories, which are non-renormalisable. The other path, which we shall follow, is to try to find a gauge theory that describes the weak interactions. The immediate problem with this attempt is that no mass term is allowed for gauge bosons, the gauge bosons are massless. The solution to this problem is the “Higgs mechanism”26 26Here, the word “mechanism” does not mean a physical process but a theoretical con- struction. 5.2 Higgs Mechanism 117 5.2 Higgs Mechanism 5.2.1 Spontaneous breakdown of a global symmetry We shall now consider a concept in field theory, which is also known in other areas of physics like phase transitions, e.g. ferromagnetism, spin waves or even laser theory. Consider a complex scalar field with Lagrangian L=∂µφ∗∂µφ−m2φ∗φ−λ(φ∗φ)2. (5.30) The coefficient m2should be thought of as a real parameter, which can be negative, too. Lhas a global U(1) symmetry φ(x)−→eiαφ(x). (5.31) Let us look at the potential part of the Lagrangian V(φ) =m2φ∗φ+λ(φ∗φ)2(5.32) as a real valued function over the complex plane with φ=1√ 2(φ1+ iφ2). (5.33) Case a)m2>0 V(φ) iφ2 φ1 The symmetry is unbroken, the minimum of the potential is not degenerate, and the mean value for the field is /angbracketleftφ/angbracketright= 0. (5.34) 118 5 ELECTROWEAK THEORY Case b)m2<0 φ1iφ2V(φ) Now the minima of the potential are lo- cated on a circle with radius |φ|=/radicalBigg −m2 2λ=:v√ 2. (5.35) In the classical theory, a groundstate corresponds to a field configuration with minimal energy, which in this case is a constant field φ0with a value in one of the potential minima. Without loss of generality we choose it to be real, φ0=v√ 2. (5.36) Let us assume that in the quantum theory the corresponding situation holds and the mean value of the field is /angbracketleftφ/angbracketright=v√ 2. (5.37) (More precisely there will be corrections to this value.) The symmetry is then spontaneously broken. We decompose φ(x)into its mean value and a complex remainder, φ(x) =1√ 2(v+ρ(x) + iϕ(x)). (5.38) iϕ√ 2 ρ√ 2iφ2 −φ1 v√ 2 5.2 Higgs Mechanism 119 The Lagrangian density, written in terms of the real fields ρ(x)andϕ(x), becomes up to a constant L=1 2(∂µρ)2+1 2(∂µϕ)2−λv2ρ2−λv(ρ3+ρϕ2)−λ 4(ρ2+ϕ2)2.(5.39) We see that ρ(x)is a massive field coupled to the massless field ϕ(x). The massmρis given by m2 ρ= 2λv2= 2|m2|. (5.40) ρ(x)describes radial excitations of the original field. The massless field ϕ(x) corresponds to tangential excitations. These are called Goldstone bosons. A general statement about the existence of Goldstone bosons is made by the Goldstone theorem: Spontaneous breakdown of a continuous, global symmetry leads to one massless spin 0 particle for each generator of the group that is spontaneously broken. In the above situation, the U(1) group has one generator, and one Goldstone boson field arises. 5.2.2 Higgs mechanism Now we turn to a seemingly similar situation, where the symmetry is, how- ever, a local gauge symmetry. Will there be massless Goldstone bosons? Let us consider the abelian Higgs model, describing a complex scalar field coupled to gauge field. L= (Dµφ)∗(Dµφ)−V(φ)−1 4FµνFµν, (5.41) Dµ=∂µ+ iqAµ. (5.42) In this “scalar QED”, Aµ(x)is the gauge field and the gauge group is U(1). As before the potential is a quartic one, V(φ) =m2φ∗φ+λ(φ∗φ)2, (5.43) and we consider the case m2<0. As in the previous section, we expand the field φ(x)near the real minimum ofV(φ), but this time we choose a different parameterisation in terms of a 120 5 ELECTROWEAK THEORY radial variable ρ(x)and an angular variable ξ(x): φ(x) =1√ 2(v+ρ(x))eiξ(x)/v(5.44) =1√ 2(v+ρ(x) +iξ(x) +...). (5.45) After a few lines of algebra the Lagrangian becomes L=1 2(∂µρ)2−λv2ρ2 +1 2(∂µξ)2+qvAµ∂µξ +1 2q2v2AµAµ−1 4FµνFµν +...(interaction terms ).(5.46) As before, the mass belonging to the radial excitations is m2 ρ= 2λv2. (5.47) There is also term quadratic in the gauge field, namely1 2q2v2AµAµ, from which we read off a mass for the field Aµ, which is mA=qv. (5.48) There is no mass term for the field ξ(x)and it looks like a massless Goldstone field. However, there is a mixing term qvAµ∂µξ, quadratic in the fields, and we have to clarify the situation. Looking at the parameterisation of φ(x), we recognise that the factor exp/parenleftbiggi vξ(x)/parenrightbigg , (5.49) has precisely the form of a local gauge transformation. Therefore the field ξ(x)can be transformed away by means of a compensating gauge transfor- mation: φ(x)−→φ/prime(x) := exp/parenleftbigg −i vξ(x)/parenrightbigg φ(x) =1√ 2(v+ρ(x)).(5.50) At the same time the gauge field Aµis transformed as Aµ−→A/prime µ(x) =Aµ(x) +1 qv∂µξ(x) =:Bµ(x). (5.51) 5.3 Glashow-Weinberg-Salam Model 121 This gauge is called “unitary gauge”. The Lagrangian now becomes L=1 2(∂µρ)2−m2 ρ 2ρ2−λvρ3−λ 4ρ4 −1 4FµνFµν+1 2q2v2BµBµ +vq2ρBµBµ+1 2q2ρ2BµBµ,(5.52) whereFµνis now formed from Bµ(x). The mixing term has disappeared. The remaining physical fields are the following. •ρ(x)is a massive scalar field with m2 ρ= 2λv2. •Bµ(x)is amassivevector field with mass mv=qv. It describes “mas- sive photons”. The usual massless photons have two transverse degrees of freedom (com- ponents), whereas the massive field Bµ(x)has three degrees of freedom. Its additional “longitudinal” degree of freedom originates from the alleged Gold- stonefieldξ(x). Thisissometimesexpressedbysaying, “theGoldstoneboson is eaten by the vector boson.” The last two terms in the Lagrangian are interaction terms between the two fields. In the unitary gauge the local gauge symmetry is hidden. In the literature one sometimes finds the formulation of a “spontaneous breaking of local gauge symmetry”. This is not quite correct. Local gauge symmetries cannot be broken spontaneously (“Elitzur’s theorem”). Instead, the fields ρ(x)and Bµ(x)are gauge invariant combinations of the original fields. 5.3 Glashow-Weinberg-Salam Model The Glashow-Weinberg-Salam model is a gauge theory describing the elec- troweakinteractions. Thegaugebosons, mediatingtheweakandelectromag- netic interactions, are the massive W+,W−,Z0, and the massless photon Aµ. The crucial piece in the formulation of a gauge theory with massive vector bosons is the Higgs mechanism. The gauge group is gauge group = SU(2)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright weak isospin⊗ U(1)Y/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright weak hypercharge(5.53) and the model uses the Higgs mechanism to create the masses mWandmZ. 122 5 ELECTROWEAK THEORY Historically, the basic structure was formulated by Sheldon Glashow (1961), but without the Higgs mechanism, and the complete form was found by Steven Weinberg (1967) and Abdus Salam (1968). Let us start by concentrating on leptons. In the weak interactions only the left handed components couple to the SU(2) gauge field. Therefore the group is denoted SU(2) L. It acts on the doublets /parenleftBigg νe e−/parenrightBigg L=1 2(1−γ5)/parenleftBigg νe e−/parenrightBigg , (5.54) and on the doublets of the remaining two families /parenleftBigg νµ µ−/parenrightBigg L,/parenleftBigg ντ τ−/parenrightBigg L. (5.55) This SU(2) symmetry is called weak isospin, its generators are denoted T1, T2,T3, and the corresponding quantum numbers of the lepton doublets are t=1 2, t 3=±1 2. (5.56) The right-handed parts of the leptons, (e−)R,(µ−)R,(τ−)R, do not have right-handed neutrino-partners, and are singlets ( t= 0) under weak isospin. The other symmetry group U(1) Yis associated with the weak hypercharge Y, which is related to the electric charge Qby Q=t3+y 2. (5.57) νe− L, µ− L, τ− Le− R, µ− R, τ− R weak isospin t3+1 2−1 20 hypercharge y−1−1−2 electric charge Q0−1−1 Gauge fields The symmetry groups are now gauged by introducing corresponding gauge fields. symmetry group generators gauge field field strength SU(2) Ta=1 2τaWa µ(x)Wa µν(x) U(1) Y=y1Bµ(x)Bµν(x) 5.3 Glashow-Weinberg-Salam Model 123 The gauge field dynamics is given by the gauge part Lagrangian Lg=−1 2Wa µνWaµν−1 4BµνBµν. (5.58) The fermionic part of the Lagrangian is Lf=(νe, e−)Liγµ/parenleftBigg ∂µ+ ig/prime 2YBµ+ ig 2τaWa µ/parenrightBigg/parenleftBigg νe e−/parenrightBigg L +e− Riγµ/parenleftBigg ∂µ+ ig/prime 2YBµ/parenrightBigg e− R +the same terms for muon and tauon fields.(5.59) Higgs field Finally, we have to add the Higgs field part of the Lagrangian. For the Higgs field we take a complex SU(2) doublet φ(x) =/parenleftBigg φ+ φ0/parenrightBigg =1√ 2/parenleftBigg φ1+ iφ2 φ3+ iφ4/parenrightBigg (5.60) The Higgs field has weak isospin t= 1/2and weak hypercharge yφ= 1. The upper component φ+hast3= +1/2,yφ= 1and therefore charge Q= +1, whereas the lower component φ0hast3=−1/2,yφ= 1and charge Q= 0. In matrix form this reads Qφ=/parenleftbigg T3+Y 2/parenrightbigg φ=/parenleftBigg1 2/parenleftBigg 1 0 0−1/parenrightBigg +1 2/parenleftBigg 1 0 0 1/parenrightBigg/parenrightBigg φ =/parenleftBigg 1 0 0 0/parenrightBigg φ.(5.61) The covariant derivative of the Higgs field is Dµφ= (∂µ+ ig/prime 2Bµ+ ig 2τaWa µ)φ. (5.62) Its Lagrangian is Lh= (Dµφ)†(Dµφ)−µ2φ†φ−λ(φ†φ)2, (µ2<0, λ> 0).(5.63) The potential has minima at φ†φ=−µ2 2λ. (5.64) 124 5 ELECTROWEAK THEORY The vacuum state is chosen to be φ0=1√ 2/parenleftBigg 0 v/parenrightBigg , v2=−µ2 λ. (5.65) It is chosen such that Qφ0= 0. This guarantees that the electromagnetic gauge group U(1), which is generated by Q, is unaffected by the Higgs mech- anism (it is “unbroken”), so that the photon remains massless. We shall see this explicitly below. Without the gauge interactions we would have a spontaneously broken global symmetry, and there would be 3 massless Goldstone bosons, 1 massive Higgs field. In the presence of the gauge fields, the fields are transformed into the unitary gauge as in Sec. 5.2.2, so that we have φ(x)−→1√ 2/parenleftBigg 0 v+ρ(x)/parenrightBigg . (5.66) The Lagrangian of the Higgs and gauge fields then becomes Lh=1 2(∂µρ)2−1 2m2 ρρ2−λvρ3−λ 4ρ4 +1 2m2 W(W1 µWµ1+W2 µWµ2) +v2 8(g/primeBµ−gW3 µ)(g/primeBµ−gWµ3) +interaction terms .(5.67) As a result one gets a massive neutral Higgs field ρwithm2 ρ= 2λv2, and massive vector fields W1andW2withmW=1 2gv. The charged bosons W± are defined as the complex linear combinations W±=1√ 2(W1 µ∓iW2 µ) (5.68) and we have W1 µWµ1+W2 µWµ2= 2W+ µWµ−. (5.69) The remaining fields BµandW3 µappear mixed in the quadratic terms. They can be diagonalised by means of Zµ:=gW3 µ−g/primeBµ√g2+g/prime2, (5.70) Aµ:=g/primeW3 µ+gBµ √g2+g/prime2. (5.71) 5.3 Glashow-Weinberg-Salam Model 125 The quadratic term then reads 1 2m2 ZZµZµ, (5.72) and we see that the masses belonging to the fields ZµandAµare mZ=1 2v/radicalBig g2+g/prime2, mA= 0. (5.73) In this way fields for the massive Z0boson and the massless photon emerge. Defining an angle θWthrough tanθW=g/prime g,ormW mZ= cosθW, (5.74) the fields can be represented as /parenleftBigg Zµ Aµ/parenrightBigg =/parenleftBigg cosθW−sinθW sinθW cosθW/parenrightBigg/parenleftBigg W3 µ Bµ/parenrightBigg . (5.75) θWis called weak mixing angle or “Weinberg angle”. We also have cosθW=g/radicalBig g2+g/prime2, sinθW=g/prime /radicalBig g2+g/prime2. (5.76) To identify the charges of the fields, we consider the covariant derivative Dµ=∂µ+ ig/prime 2YBµ+ igTaWa µ (5.77) and recall Q=T3+1 2Y. (5.78) With T±:=T1±iT2 (5.79) we have [Q,T±] = [T3,T±] =±T±, [Q,Y] = 0. (5.80) Then we get the charged field components as parts of gTaWa µ=g√ 2(T+W+ µ+T−W− µ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ±charged fields) +gT3W3 µ. (5.81) 126 5 ELECTROWEAK THEORY For the fields W3 µandBµwe find gT3W3 µ+g/primeY 2Bµ= (gsinθWT3+g/primecosθWY 2)Aµ + (gcosθWT3−g/primesinθWY 2)Zµ =gg/prime /radicalBig g2+g/prime2(T3+Y 2)Aµ +/radicalBig g2+g/prime2/parenleftbigg T3−sin2θW(T3+Y 2)/parenrightbigg Zµ =gg/prime /radicalBig g2+g/prime2QAµ +/radicalBig g2+g/prime2/parenleftBig T3−sin2θWQ/parenrightBig Zµ.(5.82) The photon field Aµcouples to the charge Q, as it should. From the expres- sion we can identify the coupling constant of the photon, the electric charge unit e0=gg/prime /radicalBig g2+g/prime2=gsinθW=g/primecosθW. (5.83) Altogether we can write the fields appearing in the covariant derivative in the form g/primeY 2Bµ+gTaWa µ=e0√ 2 sinθW(T+W+ µ+T−W− µ) +e0QAµ +e0 sinθWcosθW(T3−sin2θWQ)Zµ.(5.84) From the discussion above it is already clear that the masses of the Wand Zbosons stem from the non-vanishing vacuum value of the Higgs field. To makeitmanifest, letusneverthelessshowthisexplicitlyagain. Thecovariant derivative acts on the constant vacuum value of the Higgs field as Dµφ0=ig√ 2(W+ µT++W− µT−)φ0+ig cosθWZµT3φ0 =ig 2W+ µ/parenleftBigg v 0/parenrightBigg −ig 2√ 2 cosθwZµ/parenleftBigg 0 v/parenrightBigg .(5.85) In the vacuum state the kinetic term of the Higgs field then yields (Dµφ0)†(Dµφ0) =−g2v2 8(2W+ µWµ−+1 cos2θWZµZµ), (5.86) 5.3 Glashow-Weinberg-Salam Model 127 and we get the masses of the gauge bosons mW=1 2gv=e0v 2 cosθW, mZ=mw cosθW. (5.87) Fermion masses Under the local gauge symmetry group SU(2)L⊗U(1)Y (5.88) the right- and left-handed fields eRandeLtransform differently. Therefore, the mass term m¯ψψ=m(¯ψLψR+¯ψRψL) (5.89) cannot be invariant, and the bare fermion masses have to be zero. How can we get fermion masses? The solution is that the fermion masses come from the Higgs field, like the masses of the vector bosons. This can be achieved by coupling the Higgs field to the fermion fields through a Yukawa coupling LY=−Ge(νe,e−)L/parenleftBigg φ+ φ0/parenrightBigg e− R+Hermitian conjugate . (5.90) The same terms are added for muon and tauon fields. These Yukawa terms are invariant under SU(2) L. They are also invariant under U(1) Ybecause the hypercharges add to zero: (νe,e−)Lφ e− R/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Yφ = 1 +1−2 = 0(5.91) In the unitary gauge/parenleftBigg φ+ φ0/parenrightBigg =1√ 2/parenleftBigg 0 v+ρ(x)/parenrightBigg (5.92) the Lagrangian becomes LY=−Gev√ 2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright masse−e−−Ge√ 2ρe−e− /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright remaining Yukawa coupling(5.93) The electron mass is me=Gev√ 2. (5.94) In this way all masses (of W,Z, leptons, quarks) are generated by the Higgs mechanism. 128 5 ELECTROWEAK THEORY Reduction to the Fermi theory As discussed in Sec. 5.1.3, the reduction to the V-A theory at low energies follows the scheme κ κ2/lessmuchm2 W,Z where the vector boson propagator is replaced by igµν/m2. The resulting effective theory has a four-fermion vertex with coupling constant GF√ 2=g2 8m2 W=1 2v2. (5.95) From the known value of the Fermi coupling GFone derives v= (√ 2GF)−1 2= 246 GeV. (5.96) From independent measurements the values sin2θW= 0.231, (5.97) mW= 80.42 GeV, (5.98) mZ= 91.19 GeV (5.99) have been obtained. Recent experiments indicate the value mρ≈125 GeV (5.100) for the Higgs mass. Quarks The quarks couple to the gauge fields of SU(2)L⊗U)(1) Yin a similar way as leptons. The left-handed quarks form weak iso-doublets, i.e. t= 1/2, /parenleftBigg u d/prime/parenrightBigg L,/parenleftBigg c s/prime/parenrightBigg L,/parenleftBigg t b/prime/parenrightBigg L. (5.101) Their weak hypercharge is y= 1/3in order to yield the correct electric charges. The d/prime,s/primeandb/primecomponents have been denoted with a prime, because mixing among them will occur, generalising the Cabibbo angle. To remind you, for two generations of quarks the mixing of the dandsquarks is d/prime=dcosθC+ssinθC (5.102) s/prime=−dsinθC+scosθC. (5.103) 5.3 Glashow-Weinberg-Salam Model 129 The right-handed quarks uR, d/prime R, cR, s/prime R, tR, b/prime R (5.104) form weak iso-singlets with t= 0andy= 4/3andy=−2/3, respectively. Quark masses are generated through Yukawa couplings to the Higgs field. The most general form of the resulting mass terms for the three generations is /parenleftBig ¯uR,¯cR,¯tR/parenrightBig Mu uL cL tL +/parenleftBig¯d/prime R,¯s/prime R,¯b/prime R/parenrightBig Md d/prime L s/prime L b/prime L +h.c. (5.105) with two 3×3mass matrices MuandMd. The mass matrices cannot be diagonalised simultaneously and one ends up with a diagonal Muand Md=V md0 0 0ms0 0 0mb V+. (5.106) The relation between the mass eigenstates d,s,band the mixed states d/prime,s/prime, b/prime, entering the weak interactions, is therefore  d/prime s/prime b/prime =V d s b  (5.107) with the unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix. In a theory with only 2 generations one has the Cabibbo mixing V=/parenleftBigg cosθcsinθc −sinθccosθc/parenrightBigg . (5.108) The Glashow-Weinberg-Salam model gives a consistent theory of the weak and electromagnetic interactions. A detailed study of the interactions de- scribed by the theory shows •there are charged currents as in the V-A theory, •it predicts neutral currents coupling to left-handed and right-handed fermions: J(neutral ) µ =J3 µ−sin2θWJ(e.m.) µ, •it contains further interactions of Higgs and gauge boson fields. 130 5 ELECTROWEAK THEORY Properties of the GWS theory are •Renormalisability. Although the Lagrangian does not look renormalisable in the unitary gauge, it is! The proof (G. ’t Hooft and others) is complicated. The basic idea is: without unitary gauge the theory is renormalisable, and it has been shown that this property is independent of the gauge. •Phenomenologically the theory is extremely successful. •It provides a unified description of the electromagnetic and weak inter- actions. However, because the gauge group is the direct product of SU(2) for weak isospin and U(1) for electromagnetism, this is not really a full unification. This motivates to search for a unified theory with a sim- ple Lie group, which cannot be decomposed into the direct product of subgroups. •The theory is free of anomalies. In field theory an anomaly occurs, when the classical theory has a sym- metry and an associated conserved Noether current, ∂µjµ= 0, which is not conserved in the quantum theory: ∂µjµ/negationslash= 0. The symmetry is then broken by a quantum effect. In the GWS theory there are possible anomalies, but they cancel each other. The condition for the absence of anomalies is that the electric charges of all left-handed particles and all right-handed particles in a generation cancel against each other: /summationdisplay LQ−/summationdisplay RQ= 0. (5.109) Here the quark charges are counted with a factor of 3 due to the three colours. The condition is satisfied for leptons and quarks within each generation. As a consequence the number of colours is related to the cancellation of anomalies. •The Glashow-Iliopoulos-Maiani (GIM) mechanism explains the suppression of flavour-changing neutral currents, if there is an even number Nfof flavours. It was suggested at a time, when only three quark flavours had been known, and led to the prediction of the charm quark. Adding the strong interactions of quarks via QCD leads to the complete Standard Model. The total gauge group then is SU(3)c⊗SU(2)L⊗U(1)Y. (5.110)