aa r X i v : . [ h e p - ph ] M a y The Standard Model of Electroweak Interactions
A. Pich
IFIC, University of Val`encia – CSIC,Val`encia, Spain
Abstract
Gauge invariance is a powerful tool to determine the dynamics of the elec-troweak and strong forces. The particle content, structure and symmetriesof the Standard Model Lagrangian are discussed. Special emphasis is givento the many phenomenological tests which have established this theoreticalframework as the Standard Theory of electroweak interactions.
The Standard Model (SM) is a gauge theory, based on the symmetry group SU (3) C ⊗ SU (2) L ⊗ U (1) Y ,which describes strong, weak and electromagnetic interactions, via the exchange of the correspondingspin-1 gauge fields: eight massless gluons and one massless photon, respectively, for the strong and elec-tromagnetic interactions, and three massive bosons, W ± and Z , for the weak interaction. The fermionicmatter content is given by the known leptons and quarks, which are organized in a three-fold familystructure: (cid:20) ν e ue − d ′ (cid:21) , (cid:20) ν µ cµ − s ′ (cid:21) , (cid:20) ν τ tτ − b ′ (cid:21) , (1.1)where (each quark appears in three different colours) (cid:20) ν l q u l − q d (cid:21) ≡ (cid:18) ν l l − (cid:19) L , (cid:18) q u q d (cid:19) L , l − R , q uR , q dR , (1.2)plus the corresponding antiparticles. Thus, the left-handed fields are SU (2) L doublets, while their right-handed partners transform as SU (2) L singlets. The three fermionic families in Eq. (1.1) appear to haveidentical properties (gauge interactions); they differ only by their mass and their flavour quantum number.The gauge symmetry is broken by the vacuum, which triggers the Spontaneous Symmetry Break-ing (SSB) of the electroweak group to the electromagnetic subgroup: SU (3) C ⊗ SU (2) L ⊗ U (1) Y SSB −→ SU (3) C ⊗ U (1) QED . (1.3)The SSB mechanism generates the masses of the weak gauge bosons, and gives rise to the appearanceof a physical scalar particle in the model, the so-called Higgs. The fermion masses and mixings are alsogenerated through the SSB.The SM constitutes one of the most successful achievements in modern physics. It provides avery elegant theoretical framework, which is able to describe the known experimental facts in particlephysics with high precision. These lectures [1] provide an introduction to the electroweak sector ofthe SM, i.e., the SU (2) L ⊗ U (1) Y part [2–5]. The strong SU (3) C piece is discussed in more detailin Ref. [6]. The power of the gauge principle is shown in Section 2, where the simpler Lagrangiansof quantum electrodynamics and quantum chromodynamics are derived. The electroweak theoreticalframework is presented in Sections 3 and 4, which discuss, respectively, the gauge structure and the SSBmechanism. Section 5 summarizes the present phenomenological status and shows the main precisiontests performed at the Z peak. The flavour structure is discussed in Section 6, where knowledge of thequark mixing angles is briefly reviewed and the importance of CP violation tests is emphasized. Finally,a few comments on open questions, to be investigated at future facilities, are given in the summary.ome useful but more technical information has been collected in several appendices: a minimalamount of quantum field theory concepts are given in Appendix A; Appendix B summarizes the most im-portant algebraic properties of SU ( N ) matrices; and a short discussion on gauge anomalies is presentedin Appendix C. Let us consider the Lagrangian describing a free Dirac fermion: L = i ψ ( x ) γ µ ∂ µ ψ ( x ) − m ψ ( x ) ψ ( x ) . (2.1) L is invariant under global U (1) transformations ψ ( x ) U(1) −→ ψ ′ ( x ) ≡ exp { iQθ } ψ ( x ) , (2.2)where Qθ is an arbitrary real constant. The phase of ψ ( x ) is then a pure convention-dependent quantitywithout physical meaning. However, the free Lagrangian is no longer invariant if one allows the phasetransformation to depend on the space-time coordinate, i.e., under local phase redefinitions θ = θ ( x ) ,because ∂ µ ψ ( x ) U(1) −→ exp { iQθ } ( ∂ µ + iQ ∂ µ θ ) ψ ( x ) . (2.3)Thus, once a given phase convention has been adopted at the reference point x , the same conventionmust be taken at all space-time points. This looks very unnatural.The ‘gauge principle’ is the requirement that the U (1) phase invariance should hold locally . Thisis only possible if one adds an extra piece to the Lagrangian, transforming in such a way as to cancelthe ∂ µ θ term in Eq. (2.3). The needed modification is completely fixed by the transformation (2.3): oneintroduces a new spin-1 (since ∂ µ θ has a Lorentz index) field A µ ( x ) , transforming as A µ ( x ) U(1) −→ A ′ µ ( x ) ≡ A µ ( x ) − e ∂ µ θ , (2.4)and defines the covariant derivative D µ ψ ( x ) ≡ [ ∂ µ + ieQA µ ( x )] ψ ( x ) , (2.5)which has the required property of transforming like the field itself: D µ ψ ( x ) U(1) −→ ( D µ ψ ) ′ ( x ) ≡ exp { iQθ } D µ ψ ( x ) . (2.6)The Lagrangian L ≡ i ψ ( x ) γ µ D µ ψ ( x ) − m ψ ( x ) ψ ( x ) = L − eQA µ ( x ) ψ ( x ) γ µ ψ ( x ) (2.7)is then invariant under local U (1) transformations.The gauge principle has generated an interaction between the Dirac spinor and the gauge field A µ , which is nothing else than the familiar vertex of Quantum Electrodynamics (QED). Note that thecorresponding electromagnetic charge Q is completely arbitrary. If one wants A µ to be a true propagatingfield, one needs to add a gauge-invariant kinetic term L Kin ≡ − F µν ( x ) F µν ( x ) , (2.8)where F µν ≡ ∂ µ A ν − ∂ ν A µ is the usual electromagnetic field strength. A possible mass term for thegauge field, L m = m A µ A µ , is forbidden because it would violate gauge invariance; therefore, thephoton field is predicted to be massless. Experimentally, we know that m γ < · − eV [7].2he total Lagrangian in Eqs. (2.7) and (2.8) gives rise to the well-known Maxwell equations: ∂ µ F µν = J ν ≡ eQ ψγ ν ψ , (2.9)where J ν is the fermion electromagnetic current. From a simple gauge-symmetry requirement, we havededuced the right QED Lagrangian, which leads to a very successful quantum field theory. (a)(cid:13) (b)(cid:13) (c)(cid:13) (d)(cid:13) n(cid:13) W(cid:13) W(cid:13) g , Z(cid:13) g(cid:13) f(cid:13) f(cid:13) Fig. 1: Feynman diagrams contributing to the lepton anomalous magnetic moment.
The most stringent QED test comes from the high-precision measurements of the e [8] and µ [9]anomalous magnetic moments a l ≡ ( g γl − / , where ~µ l ≡ g γl ( e/ m l ) ~S l : a e = (1 159 652 180 . ± . · − , a µ = (11 659 208 . ± . · − . (2.10)To a measurable level, a e arises entirely from virtual electrons and photons; these contributions arefully known to O ( α ) and some O ( α ) corrections have been already computed [10–14]. The impressiveagreement achieved between theory and experiment has promoted QED to the level of the best theoryever built to describe Nature. The theoretical error is dominated by the uncertainty in the input value ofthe QED coupling α ≡ e / (4 π ) . Turning things around, a e provides the most accurate determination ofthe fine structure constant [15]: α − = 137 .
035 999 710 ± .
000 000 096 . (2.11)The anomalous magnetic moment of the muon is sensitive to small corrections from virtual heav-ier states; compared to a e , they scale with the mass ratio m µ /m e . Electroweak effects from virtual W ± and Z bosons amount to a contribution of (15 . ± . · − [10, 11], which is larger than thepresent experimental precision. Thus a µ allows one to test the entire SM. The main theoretical uncer-tainty comes from strong interactions. Since quarks have electric charge, virtual quark-antiquark pairsinduce hadronic vacuum polarization corrections to the photon propagator (Fig. 1.c). Owing to the non-perturbative character of the strong interaction at low energies, the light-quark contribution cannot bereliably calculated at present. This effect can be extracted from the measurement of the cross-section σ ( e + e − → hadrons ) and from the invariant-mass distribution of the final hadrons in τ decays, whichunfortunately provide slightly different results [16–18]: a th µ = (cid:26) (11 659 180 . ± . · − ( e + e − data) , (11 659 199 . ± . · − ( τ data) . (2.12)The quoted uncertainties include also the smaller light-by-light scattering contributions (Fig. 1.d) [19].The difference between the SM prediction and the experimental value (2.10) corresponds to . σ ( e + e − )or . σ ( τ ). New precise e + e − and τ data sets are needed to settle the true value of a th µ .3 – e + qq g , Z Fig. 2: Tree-level Feynman diagram for the e + e − annihilation into hadrons. The large number of known mesonic and baryonic states clearly signals the existence of a deeper levelof elementary constituents of matter: quarks . Assuming that mesons are M ≡ q ¯ q states, while baryonshave three quark constituents, B ≡ qqq , one can nicely classify the entire hadronic spectrum. However,in order to satisfy the Fermi–Dirac statistics one needs to assume the existence of a new quantum number, colour , such that each species of quark may have N C = 3 different colours: q α , α = 1 , , (red, green,blue). Baryons and mesons are then described by the colour-singlet combinations B = 1 √ ǫ αβγ | q α q β q γ i , M = 1 √ δ αβ | q α ¯ q β i . (2.13)In order to avoid the existence of non-observed extra states with non-zero colour, one needs to furtherpostulate that all asymptotic states are colourless, i.e., singlets under rotations in colour space. Thisassumption is known as the confinement hypothesis , because it implies the non-observability of freequarks: since quarks carry colour they are confined within colour-singlet bound states.A direct test of the colour quantum number can be obtained from the ratio R e + e − ≡ σ ( e + e − → hadrons ) σ ( e + e − → µ + µ − ) . (2.14)The hadronic production occurs through e + e − → γ ∗ , Z ∗ → q ¯ q → hadrons (Fig. 2). Since quarks areassumed to be confined, the probability to hadronize is just one; therefore, summing over all possiblequarks in the final state, we can estimate the inclusive cross-section into hadrons. The electroweakproduction factors which are common with the e + e − → γ ∗ , Z ∗ → µ + µ − process cancel in the ratio(2.14). At energies well below the Z peak, the cross-section is dominated by the γ -exchange amplitude;the ratio R e + e − is then given by the sum of the quark electric charges squared: R e + e − ≈ N C N f X f =1 Q f = N C = 2 , ( N f = 3 : u, d, s ) N C = , ( N f = 4 : u, d, s, c ) N C = , ( N f = 5 : u, d, s, c, b ) . (2.15)The measured ratio is shown in Fig. 3. Although the simple formula (2.15) cannot explain thecomplicated structure around the different quark thresholds, it gives the right average value of the cross-section (away from thresholds), provided that N C is taken to be three. The agreement is better at largerenergies. Notice that strong interactions have not been taken into account; only the confinement hypoth-esis has been used.Electromagnetic interactions are associated with the fermion electric charges, while the quarkflavours (up, down, strange, charm, bottom, top) are related to electroweak phenomena. The strongforces are flavour conserving and flavour independent. On the other side, the carriers of the electroweakinteraction ( γ , Z , W ± ) do not couple to the quark colour. Thus it seems natural to take colour as thecharge associated with the strong forces and try to build a quantum field theory based on it [20, 21].4 -1 r w f r J/ y y (2S) Z R S GeV
Fig. 3: World data on the ratio R e + e − [7]. The broken lines show the naive quark model approximation with N C = 3 . Thesolid curve is the 3-loop perturbative QCD prediction. Let us denote q αf a quark field of colour α and flavour f . To simplify the equations, let us adopt a vectornotation in colour space: q Tf ≡ ( q f , q f , q f ) . The free Lagrangian L = X f ¯ q f ( iγ µ ∂ µ − m f ) q f (2.16)is invariant under arbitrary global SU (3) C transformations in colour space, q αf −→ ( q αf ) ′ = U αβ q βf , U U † = U † U = 1 , det U = 1 . (2.17)The SU (3) C matrices can be written in the form U = exp (cid:26) i λ a θ a (cid:27) , (2.18)where λ a ( a = 1 , , . . . , ) denote the generators of the fundamental representation of the SU (3) C algebra, and θ a are arbitrary parameters. The matrices λ a are traceless and satisfy the commutationrelations (cid:20) λ a , λ b (cid:21) = i f abc λ c , (2.19)with f abc the SU (3) C structure constants, which are real and totally antisymmetric. Some useful prop-erties of SU (3) matrices are collected in Appendix B.As in the QED case, we can now require the Lagrangian to be also invariant under local SU (3) C transformations, θ a = θ a ( x ) . To satisfy this requirement, we need to change the quark derivatives bycovariant objects. Since we have now eight independent gauge parameters, eight different gauge bosons G µa ( x ) , the so-called gluons , are needed: D µ q f ≡ (cid:20) ∂ µ + ig s λ a G µa ( x ) (cid:21) q f ≡ [ ∂ µ + ig s G µ ( x )] q f . (2.20)Notice that we have introduced the compact matrix notation [ G µ ( x )] αβ ≡ (cid:18) λ a (cid:19) αβ G µa ( x ) . (2.21)5 bc f G s c G n b G m a G n c ade f abc fg s2 G b m G s d G e r q a G m a qg s g m ab a l g s b Fig. 4: Interaction vertices of the QCD Lagrangian.
We want D µ q f to transform in exactly the same way as the colour-vector q f ; this fixes the transformationproperties of the gauge fields: D µ −→ ( D µ ) ′ = U D µ U † , G µ −→ ( G µ ) ′ = U G µ U † + ig s ( ∂ µ U ) U † . (2.22)Under an infinitesimal SU (3) C transformation, q αf −→ ( q αf ) ′ = q αf + i (cid:18) λ a (cid:19) αβ δθ a q βf ,G µa −→ ( G µa ) ′ = G µa − g s ∂ µ ( δθ a ) − f abc δθ b G µc . (2.23)The gauge transformation of the gluon fields is more complicated than the one obtained in QED for thephoton. The non-commutativity of the SU (3) C matrices gives rise to an additional term involving thegluon fields themselves. For constant δθ a , the transformation rule for the gauge fields is expressed interms of the structure constants f abc ; thus, the gluon fields belong to the adjoint representation of thecolour group (see Appendix B). Note also that there is a unique SU (3) C coupling g s . In QED it waspossible to assign arbitrary electromagnetic charges to the different fermions. Since the commutationrelation (2.19) is non-linear, this freedom does not exist for SU (3) C .To build a gauge-invariant kinetic term for the gluon fields, we introduce the corresponding fieldstrengths: G µν ( x ) ≡ − ig s [ D µ , D ν ] = ∂ µ G ν − ∂ ν G µ + ig s [ G µ , G ν ] ≡ λ a G µνa ( x ) ,G µνa ( x ) = ∂ µ G νa − ∂ ν G µa − g s f abc G µb G νc . (2.24)Under a gauge transformation, G µν −→ ( G µν ) ′ = U G µν U † , (2.25)and the colour trace Tr ( G µν G µν ) = G µνa G aµν remains invariant.Taking the proper normalization for the gluon kinetic term, we finally have the SU (3) C invariantLagrangian of Quantum Chromodynamics (QCD): L QCD ≡ − G µνa G aµν + X f ¯ q f ( iγ µ D µ − m f ) q f . (2.26)It is worth while to decompose the Lagrangian into its different pieces: L QCD = −
14 ( ∂ µ G νa − ∂ ν G µa ) ( ∂ µ G aν − ∂ ν G aµ ) + X f ¯ q αf ( iγ µ ∂ µ − m f ) q αf − g s G µa X f ¯ q αf γ µ (cid:18) λ a (cid:19) αβ q βf (2.27) + g s f abc ( ∂ µ G νa − ∂ ν G µa ) G bµ G cν − g s f abc f ade G µb G νc G dµ G eν . ig. 5: Two- and three-jet events from the hadronic Z boson decays Z → q ¯ q and Z → q ¯ qG (ALEPH) [22]. The first line contains the correct kinetic terms for the different fields, which give rise to the corre-sponding propagators. The colour interaction between quarks and gluons is given by the second line; itinvolves the SU (3) C matrices λ a . Finally, owing to the non-Abelian character of the colour group, the G µνa G aµν term generates the cubic and quartic gluon self-interactions shown in the last line; the strengthof these interactions (Fig. 4) is given by the same coupling g s which appears in the fermionic piece ofthe Lagrangian.In spite of the rich physics contained in it, the Lagrangian (2.26) looks very simple because of itscolour symmetry properties. All interactions are given in terms of a single universal coupling g s , whichis called the strong coupling constant . The existence of self-interactions among the gauge fields is a newfeature that was not present in QED; it seems then reasonable to expect that these gauge self-interactionscould explain properties like asymptotic freedom (strong interactions become weaker at short distances)and confinement (the strong forces increase at large distances), which do not appear in QED [6].Without any detailed calculation, one can already extract qualitative physical consequences from L QCD . Quarks can emit gluons. At lowest order in g s , the dominant process will be the emission of asingle gauge boson; thus, the hadronic decay of the Z should result in some Z → q ¯ qG events, in additionto the dominant Z → q ¯ q decays. Figure 5 clearly shows that 3-jet events, with the required kinematics,indeed appear in the LEP data. Similar events show up in e + e − annihilation into hadrons, away from the Z peak. The ratio between 3-jet and 2-jet events provides a simple estimate of the strength of the stronginteraction at LEP energies ( s = M Z ): α s ≡ g s / (4 π ) ∼ . . Low-energy experiments have provided a large amount of information about the dynamics underlyingflavour-changing processes. The detailed analysis of the energy and angular distributions in β decays,such as µ − → e − ¯ ν e ν µ or n → p e − ¯ ν e , made clear that only the left-handed (right-handed) fermion(antifermion) chiralities participate in those weak transitions; moreover, the strength of the interactionappears to be universal. This is further corroborated through the study of other processes like π − → e − ¯ ν e or π − → µ − ¯ ν µ , which show that neutrinos have left-handed chiralities while anti-neutrinos areright-handed.From neutrino scattering data, we learnt the existence of different neutrino types ( ν e = ν µ ) and thatthere are separately conserved lepton quantum numbers which distinguish neutrinos from antineutrinos;thus we observe the transitions ¯ ν e p → e + n , ν e n → e − p , ¯ ν µ p → µ + n or ν µ n → µ − p , but we donot see processes like ν e p e + n , ¯ ν e n e − p , ¯ ν µ p e + n or ν µ n e − p .7ogether with theoretical considerations related to unitarity (a proper high-energy behaviour) andthe absence of flavour-changing neutral-current transitions ( µ − e − e − e + ), the low-energy informationwas good enough to determine the structure of the modern electroweak theory [23]. The intermediatevector bosons W ± and Z were theoretically introduced and their masses correctly estimated, before theirexperimental discovery. Nowadays, we have accumulated huge numbers of W ± and Z decay events,which bring much direct experimental evidence of their dynamical properties. W(cid:13) e(cid:13) m(cid:13) -(cid:13) n(cid:13)n(cid:13) e(cid:13) -(cid:13) m(cid:13) -(cid:13)
W(cid:13) e(cid:13) m(cid:13) +(cid:13) n(cid:13) n(cid:13) -(cid:13) m(cid:13) e(cid:13) -(cid:13)
Fig. 6: Tree-level Feynman diagrams for µ − → e − ¯ ν e ν µ and ν µ e − → µ − ν e . The interaction of quarks and leptons with the W ± bosons (Fig. 6) exhibits the following features:– Only left-handed fermions and right-handed antifermions couple to the W ± . Therefore, there isa 100% breaking of parity P (left ↔ right) and charge conjugation C (particle ↔ antiparticle).However, the combined transformation CP is still a good symmetry.– The W ± bosons couple to the fermionic doublets in Eq. (1.1), where the electric charges of thetwo fermion partners differ in one unit. The decay channels of the W − are then: W − → e − ¯ ν e , µ − ¯ ν µ , τ − ¯ ν τ , d ′ ¯ u , s ′ ¯ c . (3.1)Owing to the very high mass of the top quark [24], m t = 171 GeV > M W = 80 . , itson-shell production through W − → b ′ ¯ t is kinematically forbidden.– All fermion doublets couple to the W ± bosons with the same universal strength.– The doublet partners of the up, charm and top quarks appear to be mixtures of the three quarkswith charge − : d ′ s ′ b ′ = V dsb , V V † = V † V = 1 . (3.2)Thus, the weak eigenstates d ′ , s ′ , b ′ are different than the mass eigenstates d , s , b . They arerelated through the × unitary matrix V , which characterizes flavour-mixing phenomena.– The experimental evidence of neutrino oscillations shows that ν e , ν µ and ν τ are also mixturesof mass eigenstates. However, the neutrino masses are tiny: (cid:12)(cid:12) m ν − m ν (cid:12)(cid:12) ∼ . · − eV , m ν − m ν ∼ · − eV [7]. The neutral carriers of the electromagnetic and weak interactions have fermionic couplings (Fig. 7) withthe following properties:– All interacting vertices are flavour conserving. Both the γ and the Z couple to a fermion and itsown antifermion, i.e., γ f ¯ f and Z f ¯ f . Transitions of the type µ eγ or Z e ± µ ∓ havenever been observed. 8 – e + m – m + g , Z e – e + nn Z Fig. 7: Tree-level Feynman diagrams for e + e − → µ + µ − and e + e − → ν ¯ ν . – The interactions depend on the fermion electric charge Q f . Fermions with the same Q f haveexactly the same universal couplings. Neutrinos do not have electromagnetic interactions ( Q ν =0 ), but they have a non-zero coupling to the Z boson.– Photons have the same interaction for both fermion chiralities, but the Z couplings are different forleft-handed and right-handed fermions. The neutrino coupling to the Z involves only left-handedchiralities.– There are three different light neutrino species. SU ( ) L ⊗ U ( ) Y theory Using gauge invariance, we have been able to determine the right QED and QCD Lagrangians. Todescribe weak interactions, we need a more elaborated structure, with several fermionic flavours anddifferent properties for left- and right-handed fields; moreover, the left-handed fermions should appearin doublets, and we would like to have massive gauge bosons W ± and Z in addition to the photon.The simplest group with doublet representations is SU (2) . We want to include also the electromagneticinteractions; thus we need an additional U (1) group. The obvious symmetry group to consider is then G ≡ SU (2) L ⊗ U (1) Y , (3.3)where L refers to left-handed fields. We do not specify, for the moment, the meaning of the subindex Y since, as we will see, the naive identification with electromagnetism does not work.For simplicity, let us consider a single family of quarks, and introduce the notation ψ ( x ) = (cid:18) ud (cid:19) L , ψ ( x ) = u R , ψ ( x ) = d R . (3.4)Our discussion will also be valid for the lepton sector, with the identification ψ ( x ) = (cid:18) ν e e − (cid:19) L , ψ ( x ) = ν eR , ψ ( x ) = e − R . (3.5)As in the QED and QCD cases, let us consider the free Lagrangian L = i ¯ u ( x ) γ µ ∂ µ u ( x ) + i ¯ d ( x ) γ µ ∂ µ d ( x ) = X j =1 i ψ j ( x ) γ µ ∂ µ ψ j ( x ) . (3.6) L is invariant under global G transformations in flavour space: ψ ( x ) G −→ ψ ′ ( x ) ≡ exp { iy β } U L ψ ( x ) ,ψ ( x ) G −→ ψ ′ ( x ) ≡ exp { iy β } ψ ( x ) , (3.7) ψ ( x ) G −→ ψ ′ ( x ) ≡ exp { iy β } ψ ( x ) , SU (2) L transformation U L ≡ exp n i σ i α i o ( i = 1 , , (3.8)only acts on the doublet field ψ . The parameters y i are called hypercharges, since the U (1) Y phasetransformation is analogous to the QED one. The matrix transformation U L is non-Abelian as in QCD.Notice that we have not included a mass term in Eq. (3.6) because it would mix the left- and right-handedfields [see Eq. (A.17)], therefore spoiling our symmetry considerations.We can now require the Lagrangian to be also invariant under local SU (2) L ⊗ U (1) Y gaugetransformations, i.e., with α i = α i ( x ) and β = β ( x ) . In order to satisfy this symmetry requirement, weneed to change the fermion derivatives by covariant objects. Since we have now four gauge parameters, α i ( x ) and β ( x ) , four different gauge bosons are needed: D µ ψ ( x ) ≡ h ∂ µ + i g f W µ ( x ) + i g ′ y B µ ( x ) i ψ ( x ) ,D µ ψ ( x ) ≡ [ ∂ µ + i g ′ y B µ ( x )] ψ ( x ) , (3.9) D µ ψ ( x ) ≡ [ ∂ µ + i g ′ y B µ ( x )] ψ ( x ) , where f W µ ( x ) ≡ σ i W iµ ( x ) (3.10)denotes a SU (2) L matrix field. Thus we have the correct number of gauge fields to describe the W ± , Z and γ .We want D µ ψ j ( x ) to transform in exactly the same way as the ψ j ( x ) fields; this fixes the trans-formation properties of the gauge fields: B µ ( x ) G −→ B ′ µ ( x ) ≡ B µ ( x ) − g ′ ∂ µ β ( x ) , (3.11) f W µ G −→ f W ′ µ ≡ U L ( x ) f W µ U † L ( x ) + ig ∂ µ U L ( x ) U † L ( x ) , (3.12)where U L ( x ) ≡ exp (cid:8) i σ i α i ( x ) (cid:9) . The transformation of B µ is identical to the one obtained in QED forthe photon, while the SU (2) L W iµ fields transform in a way analogous to the gluon fields of QCD. Notethat the ψ j couplings to B µ are completely free as in QED, i.e., the hypercharges y j can be arbitraryparameters. Since the SU (2) L commutation relation is non-linear, this freedom does not exist for the W iµ : there is only a unique SU (2) L coupling g .The Lagrangian L = X j =1 i ψ j ( x ) γ µ D µ ψ j ( x ) (3.13)is invariant under local G transformations. In order to build the gauge-invariant kinetic term for the gaugefields, we introduce the corresponding field strengths: B µν ≡ ∂ µ B ν − ∂ ν B µ , (3.14) f W µν ≡ − ig h(cid:16) ∂ µ + i g f W µ (cid:17) , (cid:16) ∂ ν + i g f W ν (cid:17)i = ∂ µ f W ν − ∂ ν f W µ + ig [ W µ , W ν ] , (3.15) f W µν ≡ σ i W iµν , W iµν = ∂ µ W iν − ∂ ν W iµ − g ǫ ijk W jµ W kν . (3.16) B µν remains invariant under G transformations, while f W µν transforms covariantly: B µν G −→ B µν , f W µν G −→ U L f W µν U † L . (3.17)10herefore, the properly normalized kinetic Lagrangian is given by L Kin = − B µν B µν − Tr hf W µν f W µν i = − B µν B µν − W iµν W µνi . (3.18)Since the field strengths W iµν contain a quadratic piece, the Lagrangian L Kin gives rise to cubic andquartic self-interactions among the gauge fields. The strength of these interactions is given by the same SU (2) L coupling g which appears in the fermionic piece of the Lagrangian.The gauge symmetry forbids the writing of a mass term for the gauge bosons. Fermionic massesare also not possible, because they would communicate the left- and right-handed fields, which havedifferent transformation properties, and therefore would produce an explicit breaking of the gauge sym-metry. Thus, the SU (2) L ⊗ U (1) Y Lagrangian in Eqs. (3.13) and (3.18) only contains massless fields. W q u q d g (1- g )5 W l n l− g Fig. 8: Charged-current interaction vertices.
The Lagrangian (3.13) contains interactions of the fermion fields with the gauge bosons,
L −→ − g ψ γ µ f W µ ψ − g ′ B µ X j =1 y j ψ j γ µ ψ j . (3.19)The term containing the SU (2) L matrix f W µ = σ i W iµ = 12 W µ √ W † µ √ W µ − W µ ! (3.20)gives rise to charged-current interactions with the boson field W µ ≡ ( W µ + i W µ ) / √ and its complex-conjugate W † µ ≡ ( W µ − i W µ ) / √ (Fig. 8). For a single family of quarks and leptons, L CC = − g √ n W † µ [¯ uγ µ (1 − γ ) d + ¯ ν e γ µ (1 − γ ) e ] + h.c. o . (3.21)The universality of the quark and lepton interactions is now a direct consequence of the assumed gaugesymmetry. Note, however, that Eq. (3.21) cannot describe the observed dynamics, because the gaugebosons are massless and, therefore, give rise to long-range forces. Equation (3.19) contains also interactions with the neutral gauge fields W µ and B µ . We would like toidentify these bosons with the Z and the γ . However, since the photon has the same interaction with bothfermion chiralities, the singlet gauge boson B µ cannot be equal to the electromagnetic field. That wouldrequire y = y = y and g ′ y j = e Q j , which cannot be simultaneously true.11 f f e Q f 2 Z f f q q s ce f f (v − a ) g Fig. 9: Neutral-current interaction vertices.
Since both fields are neutral, we can try with an arbitrary combination of them: (cid:18) W µ B µ (cid:19) ≡ (cid:18) cos θ W sin θ W − sin θ W cos θ W (cid:19) (cid:18) Z µ A µ (cid:19) . (3.22)The physical Z boson has a mass different from zero, which is forbidden by the local gauge symmetry.We will see in the next section how it is possible to generate non-zero boson masses, through the SSBmechanism. For the moment, we just assume that something breaks the symmetry, generating the Z mass, and that the neutral mass eigenstates are a mixture of the triplet and singlet SU (2) L fields. Interms of the fields Z and γ , the neutral-current Lagrangian is given by L NC = − X j ψ j γ µ n A µ h g σ θ W + g ′ y j cos θ W i + Z µ h g σ θ W − g ′ y j sin θ W io ψ j . (3.23)In order to get QED from the A µ piece, one needs to impose the conditions: g sin θ W = g ′ cos θ W = e , Y = Q − T , (3.24)where T ≡ σ / and Q denotes the electromagnetic charge operator Q ≡ (cid:18) Q u/ν Q d/e (cid:19) , Q = Q u/ν , Q = Q d/e . (3.25)The first equality relates the SU (2) L and U (1) Y couplings to the electromagnetic coupling, providing thewanted unification of the electroweak interactions. The second identity fixes the fermion hyperchargesin terms of their electric charge and weak isospin quantum numbers:Quarks: y = Q u − = Q d + = , y = Q u = , y = Q d = − , Leptons: y = Q ν − = Q e + = − , y = Q ν = 0 , y = Q e = − . A hypothetical right-handed neutrino would have both electric charge and weak hypercharge equal tozero. Since it would not couple either to the W ± bosons, such a particle would not have any kind ofinteraction (sterile neutrino). For aesthetic reasons, we shall then not consider right-handed neutrinosany longer.Using the relations (3.24), the neutral-current Lagrangian can be written as L NC = L QED + L Z NC , (3.26)where L QED = − e A µ X j ψ j γ µ Q j ψ j ≡ − e A µ J µ em (3.27)12 able 1: Neutral-current couplings. u d ν e e v f − sin θ W − sin θ W − θ W a f − − is the usual QED Lagrangian and L Z NC = − e θ W cos θ W J µZ Z µ (3.28)contains the interaction of the Z boson with the neutral fermionic current J µZ ≡ X j ψ j γ µ (cid:0) σ − θ W Q j (cid:1) ψ j = J µ − θ W J µ em . (3.29)In terms of the more usual fermion fields, L Z NC has the form (Fig. 9) L Z NC = − e θ W cos θ W Z µ X f ¯ f γ µ ( v f − a f γ ) f , (3.30)where a f = T f and v f = T f (cid:0) − | Q f | sin θ W (cid:1) . Table 1 shows the neutral-current couplings of thedifferent fermions. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) W + W − g , Z g , Z (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) W − , Z g WW + + + W −− WW Fig. 10: Gauge boson self-interaction vertices.
In addition to the usual kinetic terms, the Lagrangian (3.18) generates cubic and quartic self-interactions among the gauge bosons (Fig. 10): L = ie cot θ W n ( ∂ µ W ν − ∂ ν W µ ) W † µ Z ν − (cid:16) ∂ µ W ν † − ∂ ν W µ † (cid:17) W µ Z ν + W µ W † ν ( ∂ µ Z ν − ∂ ν Z µ ) o + ie n ( ∂ µ W ν − ∂ ν W µ ) W † µ A ν − (cid:16) ∂ µ W ν † − ∂ ν W µ † (cid:17) W µ A ν + W µ W † ν ( ∂ µ A ν − ∂ ν A µ ) o ; (3.31) L = − e θ W (cid:26)(cid:16) W † µ W µ (cid:17) − W † µ W µ † W ν W ν (cid:27) − e cot θ W n W † µ W µ Z ν Z ν − W † µ Z µ W ν Z ν o − e cot θ W n W † µ W µ Z ν A ν − W † µ Z µ W ν A ν − W † µ A µ W ν Z ν o − e n W † µ W µ A ν A ν − W † µ A µ W ν A ν o . Notice that at least a pair of charged W bosons are always present. The SU (2) L algebra does not generateany neutral vertex with only photons and Z bosons.13 Spontaneous Symmetry Breaking
Fig. 11: Although Nicol´as likes the symmetric food configuration, he must break the symmetry deciding which carrot is moreappealing. In three dimensions, there is a continuous valley where Nicol´as can move from one carrot to the next without effort.
So far, we have been able to derive charged- and neutral-current interactions of the type neededto describe weak decays; we have nicely incorporated QED into the same theoretical framework and,moreover, we have got additional self-interactions of the gauge bosons, which are generated by the non-Abelian structure of the SU (2) L group. Gauge symmetry also guarantees that we have a well-definedrenormalizable Lagrangian. However, this Lagrangian has very little to do with reality. Our gauge bosonsare massless particles; while this is fine for the photon field, the physical W ± and Z bosons should bequite heavy objects.In order to generate masses, we need to break the gauge symmetry in some way; however, we alsoneed a fully symmetric Lagrangian to preserve renormalizability. This dilemma may be solved by thepossibility of getting non-symmetric results from an invariant Lagrangian.Let us consider a Lagrangian, which1. Is invariant under a group G of transformations.2. Has a degenerate set of states with minimal energy, which transform under G as the members of agiven multiplet.If one of those states is arbitrarily selected as the ground state of the system, the symmetry is said to bespontaneously broken.A well-known physical example is provided by a ferromagnet: although the Hamiltonian is in-variant under rotations, the ground state has the spins aligned into some arbitrary direction; moreover,any higher-energy state, built from the ground state by a finite number of excitations, would share thisanisotropy. In a Quantum Field Theory, the ground state is the vacuum; thus the SSB mechanism willappear when there is a symmetric Lagrangian, but a non-symmetric vacuum.The horse in Fig. 11 illustrates in a very simple way the phenomenon of SSB. Although the leftand right carrots are identical, Nicol´as must take a decision if he wants to get food. What is importantis not whether he goes left or right, which are equivalent options, but that the symmetry gets broken. Intwo dimensions (discrete left-right symmetry), after eating the first carrot Nicol´as would need to makean effort to climb the hill in order to reach the carrot on the other side; however, in three dimensions(continuous rotation symmetry) there is a marvelous flat circular valley along which Nicol´as can movefrom one carrot to the next without any effort.The existence of flat directions connecting the degenerate states of minimal energy is a generalproperty of the SSB of continuous symmetries. In a Quantum Field Theory it implies the existence ofmassless degrees of freedom. 14 .1 Goldstone theorem |f| V (f) j |f|j V (f) Fig. 12: Shape of the scalar potential for µ > (left) and µ < (right). In the second case there is a continuous set ofdegenerate vacua, corresponding to different phases θ , connected through a massless field excitation ϕ . Let us consider a complex scalar field φ ( x ) , with Lagrangian L = ∂ µ φ † ∂ µ φ − V ( φ ) , V ( φ ) = µ φ † φ + h (cid:16) φ † φ (cid:17) . (4.1) L is invariant under global phase transformations of the scalar field φ ( x ) −→ φ ′ ( x ) ≡ exp { iθ } φ ( x ) . (4.2)In order to have a ground state the potential should be bounded from below, i.e., h > . For thequadratic piece there are two possibilities, shown in Fig. 12:1. µ > : The potential has only the trivial minimum φ = 0 . It describes a massive scalar particlewith mass µ and quartic coupling h .2. µ < : The minimum is obtained for those field configurations satisfying | φ | = r − µ h ≡ v √ > , V ( φ ) = − h v . (4.3)Owing to the U (1) phase-invariance of the Lagrangian, there is an infinite number of degeneratestates of minimum energy, φ ( x ) = v √ exp { iθ } . By choosing a particular solution, θ = 0 forexample, as the ground state, the symmetry gets spontaneously broken. If we parametrize theexcitations over the ground state as φ ( x ) ≡ √ v + ϕ ( x ) + i ϕ ( x )] , (4.4)where ϕ and ϕ are real fields, the potential takes the form V ( φ ) = V ( φ ) − µ ϕ + h v ϕ (cid:0) ϕ + ϕ (cid:1) + h (cid:0) ϕ + ϕ (cid:1) . (4.5)Thus, ϕ describes a massive state of mass m ϕ = − µ , while ϕ is massless.The first possibility ( µ > ) is just the usual situation with a single ground state. The othercase, with SSB, is more interesting. The appearance of a massless particle when µ < is easy tounderstand: the field ϕ describes excitations around a flat direction in the potential, i.e., into stateswith the same energy as the chosen ground state. Since those excitations do not cost any energy, theyobviously correspond to a massless state. 15he fact that there are massless excitations associated with the SSB mechanism is a completelygeneral result, known as the Goldstone theorem [25]: if a Lagrangian is invariant under a continuoussymmetry group G , but the vacuum is only invariant under a subgroup H ⊂ G , then there must exist asmany massless spin-0 particles (Goldstone bosons) as broken generators (i.e., generators of G which donot belong to H ). At first sight, the Goldstone theorem has very little to do with our mass problem; in fact, it makes it worsesince we want massive states and not massless ones. However, something very interesting happens whenthere is a local gauge symmetry [26, 27].Let us consider [3] an SU (2) L doublet of complex scalar fields φ ( x ) ≡ (cid:18) φ (+) ( x ) φ (0) ( x ) (cid:19) . (4.6)The gauged scalar Lagrangian of the Goldstone model in Eq. (4.1), L S = ( D µ φ ) † D µ φ − µ φ † φ − h (cid:16) φ † φ (cid:17) ( h > , µ < , (4.7) D µ φ = h ∂ µ + i g f W µ + i g ′ y φ B µ i φ , y φ = Q φ − T = 12 , (4.8)is invariant under local SU (2) L ⊗ U (1) Y transformations. The value of the scalar hypercharge is fixedby the requirement of having the correct couplings between φ ( x ) and A µ ( x ) ; i.e., the photon does notcouple to φ (0) , and φ (+) has the right electric charge.The potential is very similar to the one considered before. There is a infinite set of degeneratestates with minimum energy, satisfying |h | φ (0) | i| = r − µ h ≡ v √ . (4.9)Note that we have made explicit the association of the classical ground state with the quantum vacuum.Since the electric charge is a conserved quantity, only the neutral scalar field can acquire a vacuumexpectation value. Once we choose a particular ground state, the SU (2) L ⊗ U (1) Y symmetry getsspontaneously broken to the electromagnetic subgroup U (1) QED , which by construction still remains atrue symmetry of the vacuum. According to the Goldstone theorem three massless states should thenappear.Now, let us parametrize the scalar doublet in the general form φ ( x ) = exp n i σ i θ i ( x ) o √ (cid:18) v + H ( x ) (cid:19) , (4.10)with four real fields θ i ( x ) and H ( x ) . The crucial point is that the local SU (2) L invariance of the La-grangian allows us to rotate away any dependence on θ i ( x ) . These three fields are precisely the would-bemassless Goldstone bosons associated with the SSB mechanism.The covariant derivative (4.8) couples the scalar multiplet to the SU (2) L ⊗ U (1) Y gauge bosons.If one takes the physical (unitary) gauge θ i ( x ) = 0 , the kinetic piece of the scalar Lagrangian (4.7)takes the form: ( D µ φ ) † D µ φ θ i =0 −→ ∂ µ H∂ µ H + ( v + H ) (cid:26) g W † µ W µ + g θ W Z µ Z µ (cid:27) . (4.11)16he vacuum expectation value of the neutral scalar has generated a quadratic term for the W ± and the Z , i.e., those gauge bosons have acquired masses: M Z cos θ W = M W = 12 v g . (4.12)Therefore, we have found a clever way of giving masses to the intermediate carriers of the weakforce. We just add L S to our SU (2) L ⊗ U (1) Y model. The total Lagrangian is invariant under gaugetransformations, which guarantees the renormalizability of the associated Quantum Field Theory [28].However, SSB occurs. The three broken generators give rise to three massless Goldstone bosons which,owing to the underlying local gauge symmetry, can be eliminated from the Lagrangian. Going to theunitary gauge, we discover that the W ± and the Z (but not the γ , because U (1) QED is an unbrokensymmetry) have acquired masses, which are moreover related as indicated in Eq. (4.12). Notice thatEq. (3.22) has now the meaning of writing the gauge fields in terms of the physical boson fields withdefinite mass.It is instructive to count the number of degrees of freedom (d.o.f.). Before the SSB mechanism,the Lagrangian contains massless W ± and Z bosons, i.e., × d.o.f., due to the two possiblepolarizations of a massless spin-1 field, and four real scalar fields. After SSB, the three Goldstone modesare ‘eaten’ by the weak gauge bosons, which become massive and, therefore, acquire one additionallongitudinal polarization. We have then × d.o.f. in the gauge sector, plus the remaining scalarparticle H , which is called the Higgs boson. The total number of d.o.f. remains of course the same. We have now all the needed ingredients to describe the electroweak interaction within a well-definedQuantum Field Theory. Our theoretical framework implies the existence of massive intermediate gaugebosons, W ± and Z . Moreover, the Higgs-Kibble mechanism has produced a precise prediction for the W ± and Z masses, relating them to the vacuum expectation value of the scalar field through Eq. (4.12).Thus, M Z is predicted to be bigger than M W in agreement with the measured masses [29, 30]: M Z = 91 . ± . , M W = 80 . ± .
025 GeV . (4.13)From these experimental numbers, one obtains the electroweak mixing angle sin θ W = 1 − M W M Z = 0 . . (4.14)We can easily get and independent estimate of sin θ W from the decay µ − → e − ¯ ν e ν µ . Themomentum transfer q = ( p µ − p ν µ ) = ( p e + p ν e ) . m µ is much smaller than M W . Therefore,the W propagator in Fig. 6 shrinks to a point and can be well approximated through a local four-fermioninteraction, i.e., g M W − q ≈ g M W = 4 πα sin θ W M W ≡ √ G F . (4.15)The measured muon lifetime, τ µ = (2 . ± . · − s [31], provides a very precise deter-mination of the Fermi coupling constant G F : τ µ = Γ µ = G F m µ π f ( m e /m µ ) (1 + δ RC ) , f ( x ) ≡ − x + 8 x − x − x log x . (4.16) Note, however, that the relation M Z cos θ W = M W has a more general validity. It is a direct consequence of the symmetryproperties of L S and does not depend on its detailed dynamics. δ RC , which are known to O ( α ) [32, 33], one gets [31]: G F = (1 . ± . · − GeV − . (4.17)The measured values of α − = 137 . , M W and G F imply sin θ W = 0 . , (4.18)in very good agreement with Eq. (4.14). We shall see later that the small difference between these twonumbers can be understood in terms of higher-order quantum corrections. The Fermi coupling gives alsoa direct determination of the electroweak scale, i.e., the scalar vacuum expectation value: v = (cid:16) √ G F (cid:17) − / = 246 GeV . (4.19) ZHZ HHZZ WHW − HW H M Z v W ++ − v M W v (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Fig. 13: Higgs couplings to the gauge bosons.
The scalar Lagrangian in Eq. (4.7) has introduced a new scalar particle into the model: the Higgs H . In terms of the physical fields (unitary gauge), L S takes the form L S = 14 h v + L H + L HG , (4.20)where L H = 12 ∂ µ H∂ µ H − M H H − M H v H − M H v H , (4.21) L HG = M W W † µ W µ (cid:26) v H + H v (cid:27) + 12 M Z Z µ Z µ (cid:26) v H + H v (cid:27) (4.22)and the Higgs mass is given by M H = p − µ = √ h v . (4.23)The Higgs interactions (Fig. 13) have a very characteristic form: they are always proportional to the mass(squared) of the coupled boson. All Higgs couplings are determined by M H , M W , M Z and the vacuumexpectation value v .So far the experimental searches for the Higgs have only provided a lower bound on its mass,corresponding to the exclusion of the kinematical range accessible at LEP and the Tevatron [7]: M H > . . L . ) . (4.24)18 .5 Fermion masses H ff fm v Fig. 14: Fermionic coupling of the Higgs boson.
A fermionic mass term L m = − m ψψ = − m (cid:0) ψ L ψ R + ψ R ψ L (cid:1) is not allowed, because it breaksthe gauge symmetry. However, since we have introduced an additional scalar doublet into the model, wecan write the following gauge-invariant fermion-scalar coupling: L Y = − c (cid:0) ¯ u, ¯ d (cid:1) L (cid:18) φ (+) φ (0) (cid:19) d R − c (cid:0) ¯ u, ¯ d (cid:1) L (cid:18) φ (0) ∗ − φ ( − ) (cid:19) u R − c (¯ ν e , ¯ e ) L (cid:18) φ (+) φ (0) (cid:19) e R + h.c. , (4.25)where the second term involves the C -conjugate scalar field φ c ≡ i σ φ ∗ . In the unitary gauge (afterSSB), this Yukawa-type Lagrangian takes the simpler form L Y = − √ v + H ) (cid:8) c ¯ dd + c ¯ uu + c ¯ ee (cid:9) . (4.26)Therefore, the SSB mechanism generates also fermion masses: m d = c v √ , m u = c v √ , m e = c v √ . (4.27)Since we do not know the parameters c i , the values of the fermion masses are arbitrary. Note,however, that all Yukawa couplings are fixed in terms of the masses (Fig. 14): L Y = − (cid:18) Hv (cid:19) (cid:8) m d ¯ dd + m u ¯ uu + m e ¯ ee (cid:9) . (4.28) In the gauge and scalar sectors, the SM Lagrangian contains only four parameters: g , g ′ , µ and h . Onecould trade them by α , θ W , M W and M H . Alternatively, we can choose as free parameters: G F = (1 .
166 371 ± .
000 006) · − GeV − [31] ,α − = 137 .
035 999 710 ± .
000 000 096 [15] , (5.1) M Z = (91 . ± . [29, 30]and the Higgs mass M H . This has the advantage of using the three most precise experimental determi-nations to fix the interaction. The relations sin θ W = 1 − M W M Z , M W sin θ W = πα √ G F (5.2)determine then sin θ W = 0 . and M W = 80 .
94 GeV . The predicted M W is in good agreementwith the measured value in (4.13). 19 − n l− l , d, u ij Z ff
Fig. 15: Tree-level Feynman diagrams contributing to the W ± and Z decays. At tree level (Fig. 15), the decay widths of the weak gauge bosons can be easily computed. The W partial widths, Γ (cid:0) W − → ¯ ν l l − (cid:1) = G F M W π √ , Γ (cid:0) W − → ¯ u i d j (cid:1) = N C | V ij | G F M W π √ , (5.3)are equal for all leptonic decay modes (up to small kinematical mass corrections). The quark modesinvolve also the colour quantum number N C = 3 and the mixing factor V ij relating weak and masseigenstates, d ′ i = V ij d j . The Z partial widths are different for each decay mode, since its couplingsdepend on the fermion charge: Γ (cid:0) Z → ¯ f f (cid:1) = N f G F M Z π √ (cid:0) | v f | + | a f | (cid:1) , (5.4)where N l = 1 and N q = N C . Summing over all possible final fermion pairs, one predicts the totalwidths Γ W = 2 . GeV and Γ Z = 2 . GeV, in excellent agreement with the experimental values Γ W = (2 . ± . GeV and Γ Z = (2 . ± . GeV [29, 30].The universality of the W couplings implies Br( W − → ¯ ν l l − ) = 13 + 2 N C = 11 . , (5.5)where we have taken into account that the decay into the top quark is kinematically forbidden. Similarly,the leptonic decay widths of the Z are predicted to be Γ l ≡ Γ( Z → l + l − ) = 84 .
85 MeV . As shownin Table 2, these predictions are in good agreement with the measured leptonic widths, confirming theuniversality of the W and Z leptonic couplings. There is, however, an excess of the branching ratio W → τ ¯ ν τ with respect to W → e ¯ ν e and W → µ ¯ ν µ , which represents a . σ effect [29, 30].The universality of the leptonic W couplings can also be tested indirectly, through weak decaysmediated by charged-current interactions. Comparing the measured decay widths of leptonic or semilep-tonic decays which only differ by the lepton flavour, one can test experimentally that the W interactionis indeed the same, i.e., that g e = g µ = g τ ≡ g . As shown in Table 3, the present data verify theuniversality of the leptonic charged-current couplings to the 0.2% level. Table 2: Measured values of Br ( W − → ¯ ν l l − ) and Γ( Z → l + l − ) [29,30]. The average of the three leptonic modes is shownin the last column (for a massless charged lepton l ). e µ τ l Br( W − → ¯ ν l l − ) (%) . ± .
17 10 . ± .
15 11 . ± .
22 10 . ± . Z → l + l − ) (MeV) . ± .
12 83 . ± .
18 84 . ± .
22 83 . ± . able 3: Experimental determinations of the ratios g l /g l ′ [18, 34] Γ τ → ν τ e ¯ ν e / Γ µ → ν µ e ¯ ν e Γ τ → ν τ π / Γ π → µ ¯ ν µ Γ τ → ν τ K / Γ K → µ ¯ ν µ Γ W → τ ¯ ν τ / Γ W → µ ¯ ν µ | g τ /g µ | . ± . . ± .
005 0 . ± .
017 1 . ± . τ → ν τ µ ¯ ν µ / Γ τ → ν τ e ¯ ν e Γ π → µ ¯ ν µ / Γ π → e ¯ ν e Γ K → µ ¯ ν µ / Γ K → e ¯ ν e Γ K → πµ ¯ ν µ / Γ K → πe ¯ ν e | g µ /g e | . ± . . ± . . ± .
009 1 . ± . W → µ ¯ ν µ / Γ W → e ¯ ν e Γ τ → ν τ µ ¯ ν µ / Γ µ → ν µ e ¯ ν e Γ W → τ ¯ ν τ / Γ W → e ¯ ν e | g µ /g e | . ± . | g τ /g e | . ± . . ± . Another interesting quantity is the Z decay width into invisible modes, Γ inv Γ l ≡ N ν Γ( Z → ¯ ν ν )Γ l = 2 N ν (1 − θ W ) + 1 , (5.6)which is usually normalized to the charged leptonic width. The comparison with the measured value, Γ inv / Γ l = 5 . ± . [29, 30], provides very strong experimental evidence for the existence of threedifferent light neutrinos. Z peak f - ee + f q - e + f fe g , Z Fig. 16: Tree-level contributions to e + e − → ¯ f f and kinematical configuration in the centre-of-mass system. Additional information can be obtained from the study of the process e + e − → γ, Z → ¯ f f (Fig. 16). For unpolarized e + and e − beams, the differential cross-section can be written, at lowestorder, as dσd Ω = α s N f (cid:8) A (1 + cos θ ) + B cos θ − h f (cid:2) C (1 + cos θ ) + D cos θ (cid:3)(cid:9) , (5.7)where h f = ± denotes the sign of the helicity of the produced fermion f , and θ is the scattering anglebetween e − and f in the centre-of-mass system. Here, A = 1 + 2 v e v f Re( χ ) + (cid:0) v e + a e (cid:1) (cid:0) v f + a f (cid:1) | χ | ,B = 4 a e a f Re( χ ) + 8 v e a e v f a f | χ | ,C = 2 v e a f Re( χ ) + 2 (cid:0) v e + a e (cid:1) v f a f | χ | ,D = 4 a e v f Re( χ ) + 4 v e a e (cid:0) v f + a f (cid:1) | χ | , (5.8)and χ contains the Z propagator χ = G F M Z √ πα ss − M Z + is Γ Z /M Z . (5.9)21he coefficients A , B , C and D can be experimentally determined by measuring the total cross-section, the forward–backward asymmetry, the polarization asymmetry, and the forward–backward po-larization asymmetry, respectively: σ ( s ) = 4 πα s N f A , A FB ( s ) ≡ N F − N B N F + N B = 38 BA , A Pol ( s ) ≡ σ ( h f =+1) − σ ( h f = − σ ( h f =+1) + σ ( h f = − = − CA , (5.10) A FB , Pol ( s ) ≡ N ( h f =+1) F − N ( h f = − F − N ( h f =+1) B + N ( h f = − B N ( h f =+1) F + N ( h f = − F + N ( h f =+1) B + N ( h f = − B = − DA .
Here, N F and N B denote the number of f ’s emerging in the forward and backward hemispheres, respec-tively, with respect to the electron direction. The measurement of the final fermion polarization can bedone for f = τ by measuring the distribution of the final τ decay products.For s = M Z , the real part of the Z propagator vanishes and the photon-exchange terms can beneglected in comparison with the Z -exchange contributions ( Γ Z /M Z << ). Equations (5.10) becomethen, σ ,f ≡ σ ( M Z ) = 12 πM Z Γ e Γ f Γ Z , A ,f FB ≡ A F B ( M Z ) = 34 P e P f , A ,f Pol ≡ A
Pol ( M Z ) = P f , A ,f FB , Pol ≡ A FB , Pol ( M Z ) = 34 P e , (5.11)where Γ f is the Z partial decay width into the ¯ f f final state, and P f ≡ − A f ≡ − v f a f v f + a f (5.12)is the average longitudinal polarization of the fermion f , which only depends on the ratio of the vectorand axial-vector couplings.With polarized e + e − beams, which have been available at SLC, one can also study the left–rightasymmetry between the cross-sections for initial left- and right-handed electrons, and the correspondingforward–backward left–right asymmetry: A ≡ A LR ( M Z ) = σ L ( M Z ) − σ R ( M Z ) σ L ( M Z ) + σ R ( M Z ) = −P e , A ,f FB , LR ≡ A FB , LR ( M Z ) = − P f . (5.13)At the Z peak, A measures the average initial lepton polarization, P e , without any need for finalparticle identification, while A ,f FB , LR provides a direct determination of the final fermion polarization. P f is a very sensitive function of sin θ W . Small higher-order corrections can produce largevariations on the predicted lepton polarization because | v l | = | − θ W | ≪ . Therefore, P l provides an interesting window to search for electroweak quantum effects. Before trying to analyse the relevance of higher-order electroweak contributions, it is instructive to con-sider the numerical impact of the well-known QED and QCD corrections. The photon propagator getsvacuum polarization corrections, induced by virtual fermion–antifermion pairs. This kind of QED loopcorrections can be taken into account through a redefinition of the QED coupling, which depends on the22 , Z (cid:13) g , Z (cid:13) f (cid:13) - f (cid:13) g , Z (cid:13) g , Z (cid:13) f (cid:13) - f (cid:13) g g ff- + – + – + – + – + – + – + – + – – q q Fig. 17: The photon vacuum polarization (left) generates a charge screening effect, making α ( s ) smaller at larger distances. energy scale. The resulting QED running coupling α ( s ) decreases at large distances. This can be intu-itively understood as the charge screening generated by the virtual fermion pairs (Fig. 17). The physicalQED vacuum behaves as a polarized dielectric medium. The huge difference between the electron and Z mass scales makes this quantum correction relevant at LEP energies [15, 29, 30]: α ( m e ) − = 137 .
035 999 710 (96) > α ( M Z ) − = 128 . ± . . (5.14)The running effect generates an important change in Eq. (5.2). Since G F is measured at lowenergies, while M W is a high-energy parameter, the relation between both quantities is modified byvacuum-polarization contributions. Changing α by α ( M Z ) , one gets the corrected predictions: sin θ W = 0 . , M W = 79 .
96 GeV . (5.15)The experimental value of M W is in the range between the two results obtained with either α or α ( M Z ) ,showing its sensitivity to quantum corrections. The effect is more spectacular in the leptonic asymmetriesat the Z peak. The small variation of sin θ W from 0.212 to 0.231 induces a large shift on the vector Z coupling to charged leptons from v l = − . to − . , changing the predicted average leptonpolarization P l by a factor of two.So far, we have treated quarks and leptons on an equal footing. However, quarks are strong-interacting particles. The gluonic corrections to the decays Z → ¯ qq and W − → ¯ u i d j can be directlyincorporated into the formulae given before by taking an ‘effective’ number of colours: N C = ⇒ N C n α s π + . . . o ≈ . , (5.16)where we have used the value of α s at s = M Z , α s ( M Z ) = 0 . ± . [7, 35].Note that the strong coupling also ‘runs’. However, the gluon self-interactions generate an anti-screening effect, through gluon-loop corrections to the gluon propagator, which spread out the QCDcharge [6]. Since this correction is larger than the screening of the colour charge induced by virtualquark–antiquark pairs, the net result is that the strong coupling decreases at short distances. Thus, QCDhas the required property of asymptotic freedom: quarks behave as free particles when Q → ∞ [36,37].QCD corrections increase the probabilities of the Z and the W ± to decay into hadronic modes.Therefore, their leptonic branching fractions become smaller. The effect can be easily estimated fromEq. (5.5). The probability of the decay W − → ¯ ν e e − gets reduced from 11.1% to 10.8%, improving theagreement with the measured value in Table 2. Quantum corrections offer the possibility to be sensitive to heavy particles, which cannot be kinemati-cally accessed, through their virtual loop effects. In QED and QCD the vacuum polarization contributionof a heavy fermion pair is suppressed by inverse powers of the fermion mass. At low energies, the in-formation on the heavy fermions is then lost. This ‘decoupling’ of the heavy fields happens in theories23 , Z (cid:13) g , Z (cid:13) f (cid:13) - f (cid:13) g , Z (cid:13) g , Z (cid:13) f (cid:13) - f (cid:13) W (cid:13) W (cid:13) - d (cid:13) u (cid:13) j (cid:13) i (cid:13) -W -W g , Z g , Z l , d- i n , u- l j -ff- Fig. 18: Self-energy corrections to the gauge boson propagators. with only vector couplings and an exact gauge symmetry [38], where the effects generated by the heavyparticles can always be reabsorbed into a redefinition of the low-energy parameters.The SM involves, however, a broken chiral gauge symmetry. This has the very interesting im-plication of avoiding the decoupling theorem [38]. The vacuum polarization contributions induced bya heavy top generate corrections to the W ± and Z propagators (Fig. 18), which increase quadraticallywith the top mass [39]. Therefore, a heavy top does not decouple. For instance, with m t = 171 GeV,the leading quadratic correction to the second relation in Eq. (5.2) amounts to a sizeable effect. Thequadratic mass contribution originates in the strong breaking of weak isospin generated by the top andbottom quark masses, i.e., the effect is actually proportional to m t − m b .Owing to an accidental SU (2) C symmetry of the scalar sector (the so-called custodial symmetry),the virtual production of Higgs particles does not generate any quadratic dependence on the Higgs massat one loop [39]. The dependence on M H is only logarithmic. The numerical size of the correspondingcorrection in Eq. (5.2) varies from a 0.1% to a 1% effect for M H in the range from 100 to 1000 GeV. Wb bt Z Wb bt Z
Fig. 19: One-loop corrections to the Z ¯ bb vertex, involving a virtual top. Higher-order corrections to the different electroweak couplings are non-universal and usuallysmaller than the self-energy contributions. There is one interesting exception, the Z ¯ bb vertex (Fig. 19),which is sensitive to the top quark mass [40]. The Z ¯ f f vertex gets one-loop corrections where a vir-tual W ± is exchanged between the two fermionic legs. Since the W ± coupling changes the fermionflavour, the decays Z → ¯ dd, ¯ ss, ¯ bb get contributions with a top quark in the internal fermionic lines, i.e., Z → ¯ tt → ¯ d i d i . Notice that this mechanism can also induce the flavour-changing neutral-current decays Z → ¯ d i d j with i = j . These amplitudes are suppressed by the small CKM mixing factors | V tj V ∗ ti | .However, for the Z → ¯ bb vertex, there is no suppression because | V tb | ≈ .The explicit calculation [40–43] shows the presence of hard m t corrections to the Z → ¯ bb vertex.This effect can be easily understood [40] in non-unitary gauges where the unphysical charged scalar φ ( ± ) is present. The fermionic couplings of the charged scalar are proportional to the fermion masses;therefore the exchange of a virtual φ ( ± ) gives rise to a m t factor. In the unitary gauge, the chargedscalar has been ‘eaten’ by the W ± field; thus the effect comes now from the exchange of a longitudinal W ± , with terms proportional to q µ q ν in the propagator that generate fermion masses. Since the W ± couples only to left-handed fermions, the induced correction is the same for the vector and axial-vector Z ¯ bb couplings and, for m t = 171 GeV, amounts to a 1.6% reduction of the Z → ¯ bb decay width [40].The ‘non-decoupling’ present in the Z ¯ bb vertex is quite different from the one happening in theboson self-energies. The vertex correction is not dependent on the Higgs mass. Moreover, while anykind of new heavy particle coupling to the gauge bosons would contribute to the W and Z self-energies,24he possible new physics contributions to the Z ¯ bb vertex are much more restricted and, in any case,different. Therefore, the independent experimental measurement of the two effects is very valuable inorder to disentangle possible new physics contributions from the SM corrections. In addition, since the‘non-decoupling’ vertex effect is related to W L -exchange, it is sensitive to the SSB mechanism. % CL G ll [ MeV ] s i n q l ep t e ff m t = 170.9 ± H = 114...1000 GeV m t m H Da -0.041-0.038-0.035-0.032 -0.503 -0.502 -0.501 -0.5 g Al g V l % CLl + l - e + e - m + m - t + t - m t m H m t = 172.7 ± H = 114...1000 GeV Da Fig. 20: Combined LEP and SLD measurements of sin θ lepteff and Γ l (left) and the corresponding effective vector and axial-vector couplings v l and a l (right). The shaded region shows the SM prediction. The arrows point in the direction of increasingvalues of m t and M H . The point shows the predicted values if, among the electroweak radiative corrections, only the photonvacuum polarization is included. Its arrow indicates the variation induced by the uncertainty in α ( M Z ) [29, 30]. The leptonic asymmetry measurements from LEP and SLD can all be combined to determine theratios v l /a l of the vector and axial-vector couplings of the three charged leptons, or equivalently theeffective electroweak mixing angle sin θ lepteff ≡ (cid:18) − v l a l (cid:19) . (5.17)The sum ( v l + a l ) is derived from the leptonic decay widths of the Z , i.e., from Eq. (5.4) corrected witha multiplicative factor (cid:0) απ (cid:1) to account for final-state QED corrections. The signs of v l and a l arefixed by requiring a e < .The resulting 68% probability contours are shown in Fig. 20, which provides strong evidenceof the electroweak radiative corrections. The good agreement with the SM predictions, obtained forlow values of the Higgs mass, is lost if only the QED vacuum polarization contribution is taken intoaccount, as indicated by the point with an arrow. Notice that the uncertainty induced by the input valueof α ( M Z ) − = 128 . ± . is sizeable. The measured couplings of the three charged leptons confirmlepton universality in the neutral-current sector. The solid contour combines the three measurementsassuming universality.The neutrino couplings can also be determined from the invisible Z decay width, by assumingthree identical neutrino generations with left-handed couplings, and fixing the sign from neutrino scat-tering data. Alternatively, one can use the SM prediction for Γ inv to get a determination of the numberof light neutrino flavours [29, 30]: N ν = 2 . ± . . (5.18)Figure 21 shows the measured values of A l and A b , together with the joint constraint obtainedfrom A ,b FB (diagonal band). The direct measurement of A b at SLD agrees well with the SM prediction;25 .80.91 0.14 0.145 0.15 0.155 A l A b % CL SM Fig. 21: Measurements of A l , A b (SLD) and A ,b FB . Thearrows pointing to the left (right) show the variations of theSM prediction with M H = 300 +700 − GeV ( m t = 172 . ± . ). The small arrow oriented to the left shows theadditional uncertainty from α ( M Z ) [29, 30]. R m t [ G e V ] R R Fig. 22: The SM prediction of the ratios R b and R d [ R q ≡ Γ( Z → ¯ qq ) / Γ( Z → hadrons) ], as a function ofthe top mass. The measured value of R b (vertical band)provides a determination of m t [29, 30]. however, a much lower value is obtained from the ratio A ,b FB /A l . This is the most significant discrep-ancy observed in the Z -pole data. Heavy quarks ( A ,b FB /A b ) seem to prefer a high value of the Higgsmass, while leptons ( A l ) favour a light Higgs. The combined analysis prefers low values of M H , becauseof the influence of A l .The strong sensitivity of the ratio R b ≡ Γ( Z → ¯ bb ) / Γ( Z → hadrons) to the top quark mass isshown in Fig. 22. Owing to the | V td | suppression, such a dependence is not present in the analogousratio R d . Combined with all other electroweak precision measurements at the Z peak, R b provides adetermination of m t in good agreement with the direct and most precise measurement at the Tevatron.This is shown in Fig. 23, which compares the information on M W and m t obtained at LEP1 and SLD,with the direct measurements performed at LEP2 and the Tevatron. A similar comparison for m t and M H is also shown. The lower bound on M H obtained from direct searches excludes a large portion ofthe 68% C.L. allowed domain from precision measurements. H [ GeV ]
114 300 1000 m t [ GeV ] m W [ G e V ] % CL Da LEP1 and SLDLEP2 and Tevatron (prel.) m H [ GeV ] m t [ G e V ] Excluded
High Q except m t % CL m t (Tevatron) Fig. 23: Comparison (left) of the direct measurements of M W and m t (LEP2 and Tevatron data) with the indirect determinationthrough electroweak radiative corrections (LEP1 and SLD). Also shown in the SM relationship for the masses as function of M H . The figure on the right makes the analogous comparison for m t and M H [29, 30]. m H [ GeV ] Dc Excluded
Preliminary Da had = Da (5) ± ± data Theory uncertainty m Limit = 144 GeV
Fig. 24: ∆ χ = χ − χ versus M H , from the globalfit to the electroweak data. The vertical band indicates the95% exclusion limit from direct searches [29, 30]. Measurement Fit |O meas - O fit |/ s meas Da had (m Z ) Da (5) ± Z [ GeV ] m Z [ GeV ] ± G Z [ GeV ]G Z [ GeV ] ± s had [ nb ]s ± l R l ± fb A ± l (P t )A l (P t ) 0.1465 ± b R b ± c R c ± fb A ± fb A ± b A b ± c A c ± l (SLD)A l (SLD) 0.1513 ± q eff sin q lept (Q fb ) 0.2324 ± W [ GeV ] m W [ GeV ] ± G W [ GeV ]G W [ GeV ] ± t [ GeV ] m t [ GeV ] ± Fig. 25: Comparison between the measurements includedin the combined analysis of the SM and the results fromthe global electroweak fit [29, 30].
Taking all direct and indirect data into account, one obtains the best constraints on M H . The globalelectroweak fit results in the ∆ χ = χ − χ curve shown in Fig. 24. The lower limit on M H obtainedfrom direct searches is close to the point of minimum χ . At 95% C.L., one gets [29, 30] . < M H <
144 GeV . (5.19)The fit provides also a very accurate value of the strong coupling constant, α s ( M Z ) = 0 . ± . ,in very good agreement with the world average value α s ( M Z ) = 0 . ± . [7, 35]. The largestdiscrepancy between theory and experiment occurs for A ,b FB , with the fitted value being nearly σ largerthan the measurement. As shown in Fig. 25, a good agreement is obtained for all other observables. - e - e (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) g , Ze +- W + e - W + e ZZ n e - e + e - W + W (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Fig. 26: Feynman diagrams contributing to e + e − → W + W − and e + e − → ZZ . At tree level, the W -pair production process e + e − → W + W − involves three different contribu-tions (Fig. 26), corresponding to the exchange of ν e , γ and Z . The cross-section measured at LEP2 agreesvery well with the SM predictions. As shown in Fig. 27, the ν e -exchange contribution alone would leadto an unphysical growing of the cross-section at large energies and, therefore, would imply a violation ofunitarity. Adding the γ -exchange contribution softens this behaviour, but a clear disagreement with thedata persists. The Z -exchange mechanism, which involves the ZW W vertex, appears to be crucial inorder to explain the data. 27 √ s (GeV) s WW ( pb ) YFSWW/RacoonWWno ZWW vertex (Gentle)only n e exchange (Gentle) LEP
PRELIMINARY √ s (GeV) s ZZ ( pb ) ZZTO and YFSZZ
LEP
PRELIMINARY
Fig. 27: Measured energy dependence of σ ( e + e − → W + W − ) (left) and σ ( e + e − → ZZ ) (right). The three curvesshown for the W -pair production cross-section correspond to only the ν e -exchange contribution (upper curve), ν e exchangeplus photon exchange (middle curve) and all contributions including also the ZW W vertex (lower curve). Only the e -exchangemechanism contributes to Z –pair production [29, 30]. Since the Z is electrically neutral, it does not interact with the photon. Moreover, the SM does notinclude any local ZZZ vertex. Therefore, the e + e − → ZZ cross-section only involves the contributionfrom e exchange. The agreement of the SM predictions with the experimental measurements in bothproduction channels, W + W − and ZZ , provides a test of the gauge self-interactions. There is a clearsignal of the presence of a ZW W vertex, with the predicted strength, and no evidence for any γZZ or ZZZ interactions. The gauge structure of the SU (2) L ⊗ U (1) Y theory is nicely confirmed by the data.
150 100 200 500 100010 —1 —2 —3 M H [GeV] B R ( H ) WWZZZ gggt + t — gg ttccbb
150 100 200 500 100010 —1 —2 —3
10 M H [GeV] G ( H ) [ G e V ] Fig. 28: Branching fractions of the different Higgs decay modes (left) and total decay width of the Higgs boson (right) asfunction of M H [44]. The couplings of the Higgs boson are always proportional to some mass scale. The Hf ¯ f inter-action grows linearly with the fermion mass, while the HW W and
HZZ vertices are proportional to M W and M Z , respectively. Therefore, the most probable decay mode of the Higgs will be the one intothe heaviest possible final state. This is clearly illustrated in Fig. 28. The H → b ¯ b decay channel isby far the dominant one below the W + W − production threshold. When M H is large enough to al-low the production of a pair of gauge bosons, H → W + W − and H → ZZ become dominant. For28 H > m t , the H → t ¯ t decay width is also sizeable, although smaller than the W W and ZZ onesbecause of the different dependence of the corresponding Higgs coupling with the mass scale (linearinstead of quadratic).The total decay width of the Higgs grows with increasing values of M H . The effect is very strongabove the W + W − production threshold. A heavy Higgs becomes then very broad. At M H ∼
600 GeV ,the width is around
100 GeV ; while for M H ∼ , Γ H is already of the same size as the Higgs massitself. The design of the LHC detectors has taken into account all these very characteristic properties inorder to optimize the future search for the Higgs boson. We have learnt experimentally that there are six different quark flavours u , d , s , c , b , t , three differentcharged leptons e , µ , τ and their corresponding neutrinos ν e , ν µ , ν τ . We can nicely include allthese particles into the SM framework, by organizing them into three families of quarks and leptons, asindicated in Eqs. (1.1) and (1.2). Thus, we have three nearly identical copies of the same SU (2) L ⊗ U (1) Y structure, with masses as the only difference.Let us consider the general case of N G generations of fermions, and denote ν ′ j , l ′ j , u ′ j , d ′ j themembers of the weak family j ( j = 1 , . . . , N G ), with definite transformation properties under the gaugegroup. Owing to the fermion replication, a large variety of fermion-scalar couplings are allowed by thegauge symmetry. The most general Yukawa Lagrangian has the form L Y = − X jk (cid:26)(cid:0) ¯ u ′ j , ¯ d ′ j (cid:1) L (cid:20) c ( d ) jk (cid:18) φ (+) φ (0) (cid:19) d ′ kR + c ( u ) jk (cid:18) φ (0) ∗ − φ ( − ) (cid:19) u ′ kR (cid:21) + (cid:0) ¯ ν ′ j , ¯ l ′ j (cid:1) L c ( l ) jk (cid:18) φ (+) φ (0) (cid:19) l ′ kR (cid:27) + h . c ., (6.1)where c ( d ) jk , c ( u ) jk and c ( l ) jk are arbitrary coupling constants.After SSB, the Yukawa Lagrangian can be written as L Y = − (cid:18) Hv (cid:19) (cid:8) d ′ L M ′ d d ′ R + u ′ L M ′ u u ′ R + l ′ L M ′ l l ′ R + h . c . (cid:9) . (6.2)Here, d ′ , u ′ and l ′ denote vectors in the N G -dimensional flavour space, and the corresponding massmatrices are given by ( M ′ d ) ij ≡ c ( d ) ij v √ , ( M ′ u ) ij ≡ c ( u ) ij v √ , ( M ′ l ) ij ≡ c ( l ) ij v √ . (6.3)The diagonalization of these mass matrices determines the mass eigenstates d j , u j and l j , which arelinear combinations of the corresponding weak eigenstates d ′ j , u ′ j and l ′ j , respectively.The matrix M ′ d can be decomposed as M ′ d = H d U d = S † d M d S d U d , where H d ≡ q M ′ d M ′† d is an Hermitian positive-definite matrix, while U d is unitary. H d can be diagonalized by a unitarymatrix S d ; the resulting matrix M d is diagonal, Hermitian and positive definite. Similarly, one has M ′ u = H u U u = S † u M u S u U u and M ′ l = H l U l = S † l M l S l U l . In terms of the diagonal mass The condition det M ′ f = 0 ( f = d, u, l ) guarantees that the decomposition M ′ f = H f U f is unique: U f ≡ H − f M ′ f .The matrices S f are completely determined (up to phases) only if all diagonal elements of M f are different. If there is somedegeneracy, the arbitrariness of S f reflects the freedom to define the physical fields. If det M ′ f = 0 , the matrices U f and S f are not uniquely determined, unless their unitarity is explicitly imposed. i d j i j V W u c td s b Fig. 29: Flavour-changing transitions through the charged-current couplings of the W ± bosons. matrices M d = diag( m d , m s , m b , . . . ) , M u = diag( m u , m c , m t , . . . ) , M l = diag( m e , m µ , m τ , . . . ) , (6.4)the Yukawa Lagrangian takes the simpler form L Y = − (cid:18) Hv (cid:19) (cid:8) d M d d + u M u u + l M l l (cid:9) , (6.5)where the mass eigenstates are defined by d L ≡ S d d ′ L , u L ≡ S u u ′ L , l L ≡ S l l ′ L , d R ≡ S d U d d ′ R , u R ≡ S u U u u ′ R , l R ≡ S l U l l ′ R . (6.6)Note, that the Higgs couplings are proportional to the corresponding fermions masses.Since, f ′ L f ′ L = f L f L and f ′ R f ′ R = f R f R ( f = d, u, l ), the form of the neutral-current part of the SU (2) L ⊗ U (1) Y Lagrangian does not change when expressed in terms of mass eigenstates. Therefore,there are no flavour-changing neutral currents in the SM (GIM mechanism [5]). This is a consequenceof treating all equal-charge fermions on the same footing.However, u ′ L d ′ L = u L S u S † d d L ≡ u L V d L . In general, S u = S d ; thus, if one writes the weakeigenstates in terms of mass eigenstates, a N G × N G unitary mixing matrix V , called the Cabibbo–Kobayashi–Maskawa (CKM) matrix [45, 46], appears in the quark charged-current sector: L CC = − g √ W † µ X ij ¯ u i γ µ (1 − γ ) V ij d j + X l ¯ ν l γ µ (1 − γ ) l + h . c . . (6.7)The matrix V couples any ‘up-type’ quark with all ‘down-type’ quarks (Fig. 29).If neutrinos are assumed to be massless, we can always redefine the neutrino flavours, in sucha way as to eliminate the analogous mixing in the lepton sector: ν ′ L l ′ L = ν ′ L S † l l L ≡ ν L l L . Thus,we have lepton-flavour conservation in the minimal SM without right-handed neutrinos. If sterile ν R fields are included in the model, one would have an additional Yukawa term in Eq. (6.1), giving rise toa neutrino mass matrix ( M ′ ν ) ij ≡ c ( ν ) ij v/ √ . Thus, the model could accommodate non-zero neutrinomasses and lepton-flavour violation through a lepton mixing matrix V L analogous to the one presentin the quark sector. Note, however, that the total lepton number L ≡ L e + L µ + L τ would still beconserved. We know experimentally that neutrino masses are tiny and there are strong bounds on lepton-flavour violating decays: Br( µ ± → e ± e + e − ) < . · − [47], Br( µ ± → e ± γ ) < . · − [48], Br( τ ± → µ ± γ ) < . · − [49, 50] . . . However, we do have a clear evidence of neutrino oscillationphenomena.The fermion masses and the quark mixing matrix V are all determined by the Yukawa couplingsin Eq. (6.1). However, the coefficients c ( f ) ij are not known; therefore we have a bunch of arbitraryparameters. A general N G × N G unitary matrix is characterized by N G real parameters: N G ( N G − / N G ( N G + 1) / phases. In the case of V , many of these parameters are irrelevant, becausewe can always choose arbitrary quark phases. Under the phase redefinitions u i → e iφ i u i and d j → e iθ j d j , the mixing matrix changes as V ij → V ij e i ( θ j − φ i ) ; thus, N G − phases are unobservable.The number of physical free parameters in the quark-mixing matrix then gets reduced to ( N G − : N G ( N G − / moduli and ( N G − N G − / phases.In the simpler case of two generations, V is determined by a single parameter. One then recoversthe Cabibbo rotation matrix [45] V = cos θ C sin θ C − sin θ C cos θ C ! . (6.8)With N G = 3 , the CKM matrix is described by three angles and one phase. Different (but equivalent)representations can be found in the literature. The Particle data Group [7] advocates the use of thefollowing one as the ‘standard’ CKM parametrization: V = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c . (6.9)Here c ij ≡ cos θ ij and s ij ≡ sin θ ij , with i and j being ‘generation’ labels ( i, j = 1 , , ). The realangles θ , θ and θ can all be made to lie in the first quadrant, by an appropriate redefinition of quarkfield phases; then, c ij ≥ , s ij ≥ and ≤ δ ≤ π .Notice that δ is the only complex phase in the SM Lagrangian. Therefore, it is the only possiblesource of CP -violation phenomena. In fact, it was for this reason that the third generation was assumedto exist [46], before the discovery of the b and the τ . With two generations, the SM could not explain theobserved CP violation in the K system. W(cid:13) +(cid:13)
W(cid:13) +(cid:13) c(cid:13) c(cid:13)d , s(cid:13) d , s(cid:13)e ,(cid:13) +(cid:13) m(cid:13) +(cid:13) n(cid:13) e(cid:13) n(cid:13) m(cid:13) , (cid:13) u(cid:13)d , s(cid:13) _(cid:13) _(cid:13)
Fig. 30: Determinations of V ij are done in semileptonic quark decays (left), where a single quark current is present. Hadronicdecay modes (right) involve two different quark currents and are more affected by QCD effects (gluons can couple everywhere). Our knowledge of the charged-current parameters is unfortunately not so good as in the neutral-current case. In order to measure the CKM matrix elements, one needs to study hadronic weak decaysof the type H → H ′ l − ¯ ν l or H → H ′ l + ν l , which are associated with the corresponding quarktransitions d j → u i l − ¯ ν l and u i → d j l + ν l (Fig. 30). Since quarks are confined within hadrons, thedecay amplitude T [ H → H ′ l − ¯ ν l ] ≈ G F √ V ij h H ′ | ¯ u i γ µ (1 − γ ) d j | H i (cid:2) ¯ l γ µ (1 − γ ) ν l (cid:3) (6.10)always involves an hadronic matrix element of the weak left current. The evaluation of this matrixelement is a non-perturbative QCD problem, which introduces unavoidable theoretical uncertainties.31ne usually looks for a semileptonic transition where the matrix element can be fixed at somekinematical point by a symmetry principle. This has the virtue of reducing the theoretical uncertaintiesto the level of symmetry-breaking corrections and kinematical extrapolations. The standard example is a − → − decay such as K → πlν , D → Klν or B → Dlν . Only the vector current can contributein this case: h P ′ ( k ′ ) | ¯ u i γ µ d j | P ( k ) i = C P P ′ (cid:8) ( k + k ′ ) µ f + ( t ) + ( k − k ′ ) µ f − ( t ) (cid:9) . (6.11)Here, C P P ′ is a Clebsh–Gordan factor and t = ( k − k ′ ) ≡ q . The unknown strong dynamics isfully contained in the form factors f ± ( t ) . In the limit of equal quark masses, m u i − m d j = 0 , thedivergence of the vector current is zero; thus q µ (¯ u i γ µ d j ) = 0 , which implies f − ( t ) = 0 and, moreover, f + (0) = 1 to all orders in the strong coupling because the associated flavour charge is a conservedquantity. Therefore, one only needs to estimate the corrections induced by the quark mass differences.Since q µ (cid:2) ¯ lγ µ (1 − γ ) ν l (cid:3) ∼ m l , the contribution of f − ( t ) is kinematically suppressed in theelectron and muon modes. The decay width can then be written as Γ( P → P ′ lν ) = G F M P π | V ij | C P P ′ | f + (0) | I (1 + δ RC ) , (6.12)where δ RC is an electroweak radiative correction factor and I denotes a phase-space integral, which inthe m l = 0 limit takes the form I ≈ Z ( M P − M P ′ ) dtM P λ / ( t, M P , M P ′ ) (cid:12)(cid:12)(cid:12)(cid:12) f + ( t ) f + (0) (cid:12)(cid:12)(cid:12)(cid:12) . (6.13)The usual procedure to determine | V ij | involves three steps:1. Measure the shape of the t distribution. This fixes | f + ( t ) /f + (0) | and therefore determines I .2. Measure the total decay width Γ . Since G F is already known from µ decay, one gets then anexperimental value for the product | f + (0) | | V ij | .3. Get a theoretical prediction for f + (0) .It is important to realize that theoretical input is always needed. Thus, the accuracy of the | V ij | determi-nation is limited by our ability to calculate the relevant hadronic input.The conservation of the vector and axial-vector QCD currents in the massless quark limit allowsfor accurate determinations of the light-quark mixings | V ud | and | V us | . The present values are shownin Table 4, which takes into account the recent changes in the K → πe + ν e data [7, 34] and the new | V us | determinations from Cabibbo suppressed tau decays [52] and from the ratio of decay amplitudes Γ( K + → µ + ¯ ν µ ) / Γ( π + → µ + ¯ ν µ ) [53–55]. Since | V ub | is tiny, these two light quark entries provide asensible test of the unitarity of the CKM matrix: | V ud | + | V us | + | V ub | = 0 . ± . . (6.14)It is important to notice that at the quoted level of uncertainty radiative corrections play a crucial role.In the limit of very heavy quark masses, QCD has additional symmetries [56–59] which can beused to make rather precise determinations of | V cb | , either from exclusive decays such as B → D ∗ l ¯ ν l [60, 61] or from the inclusive analysis of b → c l ¯ ν l transitions. The control of theoretical uncertaintiesis much more difficult for | V ub | , | V cd | and | V cs | , because the symmetry arguments associated with thelight and heavy quark limits get corrected by sizeable symmetry-breaking effects. This is completely analogous to the electromagnetic charge conservation in QED. The conservation of the electromagneticcurrent implies that the proton electromagnetic form factor does not get any QED or QCD correction at q = 0 and, therefore, Q ( p ) = 2 Q ( u ) + Q ( d ) = | Q ( e ) | . A detailed proof can be found in Ref. [51]. able 4: Direct determinations of the CKM matrix elements V ij . For | V tb | , 95% C.L. limits are given. CKM entry Value Source | V ud | . ± . Nuclear β decay [7] . ± . n → p e − ¯ ν e [7] . ± . π + → π e + ν e [62] . ± . average | V us | . ± . K → πl + ν l [7, 34, 63] . ± . τ decays [52] . + 0 . − . K + /π + → µ + ν µ , V ud [7, 53–55] . ± . Hyperon decays [64–66] . ± . average | V cd | . ± . D → πl ¯ ν l [7] . ± . ν d → c X [7] . ± . average | V cs | . ± . D → Kl ¯ ν l [7] . + 0 . − . W + → c ¯ s [7] . ± . W + → had . , V uj , V cd , V cb [29, 30] | V cb | . ± . B → D ∗ l ¯ ν l [7, 67] . ± . b → c l ¯ ν l [7, 67] . ± . average | V ub | . ± . B → π l ¯ ν l [7, 67] . ± . b → u l ¯ ν l [7, 67] . ± . average | V tb | / qP q | V tq | > . t → b W/q W [68, 69] | V tb | > .
68 ; ≤ p ¯ p → tb + X [70]The most precise determination of | V cd | is based on neutrino and antineutrino interactions. Thedifference of the ratio of double-muon to single-muon production by neutrino and antineutrino beams isproportional to the charm cross-section off valence d quarks and, therefore, to | V cd | . A direct determi-nation of | V cs | can be also obtained from charm-tagged W decays at LEP2. Moreover, the ratio of thetotal hadronic decay width of the W to the leptonic one provides the sum [29, 30] X i = u,cj = d,s,b | V ij | = 1 . ± . . (6.15)Although much less precise than Eq. (6.14), this result test unitarity at the 1.25% level. From Eq. (6.15)one can also obtain a tighter determination of | V cs | , using the experimental knowledge on the other CKMmatrix elements, i.e., | V ud | + | V us | + | V ub | + | V cd | + | V cb | = 1 . ± . . This gives themost accurate and final value of | V cs | quoted in Table 4.33he measured entries of the CKM matrix show a hierarchical pattern, with the diagonal elementsbeing very close to one, the ones connecting the two first generations having a size λ ≈ | V us | = 0 . ± . , (6.16)the mixing between the second and third families being of order λ , and the mixing between the firstand third quark generations having a much smaller size of about λ . It is then quite practical to use theapproximate parametrization [71]: V = − λ λ Aλ ( ρ − iη ) − λ − λ Aλ Aλ (1 − ρ − iη ) − Aλ + O (cid:0) λ (cid:1) , (6.17)where A ≈ | V cb | λ = 0 . ± . , p ρ + η ≈ (cid:12)(cid:12)(cid:12)(cid:12) V ub λ V cb (cid:12)(cid:12)(cid:12)(cid:12) = 0 . ± . . (6.18)Defining to all orders in λ [72] s ≡ λ , s ≡ Aλ and s e − iδ ≡ Aλ ( ρ − iη ) , Eq. (6.17) justcorresponds to a Taylor expansion of Eq. (6.9) in powers of λ . While parity and charge conjugation are violated by the weak interactions in a maximal way, the prod-uct of the two discrete transformations is still a good symmetry (left-handed fermions ↔ right-handedantifermions). In fact, CP appears to be a symmetry of nearly all observed phenomena. However, aslight violation of the CP symmetry at the level of . is observed in the neutral kaon system and moresizeable signals of CP violation have been recently established at the B factories. Moreover, the hugematter–antimatter asymmetry present in our Universe is a clear manifestation of CP violation and itsimportant role in the primordial baryogenesis.The CPT theorem guarantees that the product of the three discrete transformations is an exactsymmetry of any local and Lorentz-invariant quantum field theory preserving micro-causality. There-fore, a violation of CP requires a corresponding violation of time reversal. Since T is an antiunitarytransformation, this requires the presence of relative complex phases between different interfering am-plitudes.The electroweak SM Lagrangian only contains a single complex phase δ ( η ). This is the solepossible source of CP violation and, therefore, the SM predictions for CP -violating phenomena arequite constrained. The CKM mechanism requires several necessary conditions in order to generate anobservable CP -violation effect. With only two fermion generations, the quark mixing mechanism cannotgive rise to CP violation; therefore, for CP violation to occur in a particular process, all three generationsare required to play an active role. In the kaon system, for instance, CP -violation effects can only appearat the one-loop level, where the top quark is present. In addition, all CKM matrix elements must be non-zero and the quarks of a given charge must be non-degenerate in mass. If any of these conditions werenot satisfied, the CKM phase could be rotated away by a redefinition of the quark fields. CP -violationeffects are then necessarily proportional to the product of all CKM angles, and should vanish in the limitwhere any two (equal-charge) quark masses are taken to be equal. All these necessary conditions can besummarized in a very elegant way as a single requirement on the original quark mass matrices M ′ u and M ′ d [73]: CP violation ⇐⇒ Im n det h M ′ u M ′† u , M ′ d M ′† d io = 0 . (6.19)34ithout performing any detailed calculation, one can make the following general statements onthe implications of the CKM mechanism of CP violation:– Owing to unitarity, for any choice of i, j, k, l (between 1 and 3),Im (cid:2) V ij V ∗ ik V lk V ∗ lj (cid:3) = J X m,n =1 ǫ ilm ǫ jkn , (6.20) J = c c c s s s sin δ ≈ A λ η < − . (6.21)Any CP -violation observable involves the product J [73]. Thus, violations of the CP symmetryare necessarily small.– In order to have sizeable CP -violating asymmetries A ≡ (Γ − Γ) / (Γ + Γ) , one should look forvery suppressed decays, where the decay widths already involve small CKM matrix elements.– In the SM, CP violation is a low-energy phenomenon, in the sense that any effect should disappearwhen the quark mass difference m c − m u becomes negligible.– B decays are the optimal place for CP -violation signals to show up. They involve small CKMmatrix elements and are the lowest-mass processes where the three quark generations play a direct(tree-level) role.The SM mechanism of CP violation is based on the unitarity of the CKM matrix. Testing theconstraints implied by unitarity is then a way to test the source of CP violation. The unitarity tests inEqs. (6.14) and (6.15) involve only the moduli of the CKM parameters, while CP violation has to dowith their phases. More interesting are the off-diagonal unitarity conditions: V ∗ ud V us + V ∗ cd V cs + V ∗ td V ts = 0 , (6.22) V ∗ us V ub + V ∗ cs V cb + V ∗ ts V tb = 0 , (6.23) V ∗ ub V ud + V ∗ cb V cd + V ∗ tb V td = 0 . (6.24)These relations can be visualized by triangles in a complex plane which, owing to Eq. (6.20), have thesame area |J | / . In the absence of CP violation, these triangles would degenerate into segments alongthe real axis.In the first two triangles, one side is much shorter than the other two (the Cabibbo suppressionfactors of the three sides are λ , λ and λ in the first triangle, and λ , λ and λ in the second one). Thisis why CP effects are so small for K mesons (first triangle), and why certain asymmetries in B s decaysare predicted to be tiny (second triangle). The third triangle looks more interesting, since the three sideshave a similar size of about λ . They are small, which means that the relevant b -decay branching ratiosare small, but once enough B mesons have been produced, the CP -violation asymmetries are sizeable.The present experimental constraints on this triangle are shown in Fig. 31, where it has been scaled bydividing its sides by V ∗ cb V cd . This aligns one side of the triangle along the real axis and makes its lengthequal to 1; the coordinates of the 3 vertices are then (0 , , (1 , and (¯ ρ, ¯ η ) ≡ (1 − λ / ρ, η ) .One side of the unitarity triangle has been already determined in Eq. (6.18) from the ratio | V ub / V cb | .The other side can be obtained from the measured mixing between the B d and ¯ B d mesons (Fig. 32), ∆ M d = 0 . ± .
004 ps − [67], which fixes | V tb | . Additional information has been provided by therecent observation of B s – ¯ B s oscillations at CDF, implying ∆ M s = 17 . ± .
12 ps − [74]. From theexperimental ratio ∆ M d / ∆ M s = 0 . ± . , one obtains | V td | / | V ts | . A more direct constrainton the parameter η is given by the observed CP violation in K → π decays. The measured value of | ε K | = (2 . ± . · − [7] determines the parabolic region shown in Fig. 31. B decays into CP self-conjugate final states provide independent ways to determine the anglesof the unitarity triangle [75, 76]. The B (or ¯ B ) can decay directly to the given final state f , or do35 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 h a bg r -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 h BEAUTY 2006
CKM f i t t e r g ga a d m D d m D & s m D K e K e cb /V ub V b sin2 < 0 b sol. w/ cos2(excl. at CL > 0.95) e xc l uded a r ea ha s C L > . Fig. 31: Experimental constraints on the SM unitarity triangle [77]. q bu, c, t qb u, c, tW Wq bW qbu, c, t u, c, tW
Fig. 32: B – ¯ B mixing diagrams. Owing to the unitarity of the CKM matrix, the mixing vanishes for equal up-type quarkmasses (GIM mechanism). The mixing amplitude is then proportional to the mass (squared) splittings between the u , c and t quarks, and is completely dominated by the top contribution. it after the meson has been changed to its antiparticle via the mixing process. CP -violating effectscan then result from the interference of these two contributions. The time-dependent CP -violating rateasymmetries contain direct information on the CKM parameters. The gold-plated decay mode is B d → J/ψK S , which gives a clean measurement of β ≡ − arg( V cd V ∗ cb / V td V ∗ tb ) , without strong-interactionuncertainties. Including the information obtained from other b → c ¯ cs decays, one gets [67]: sin 2 β = 0 . ± . . (6.25)Many additional tests of the CKM matrix from different B decay modes are being pursued at the B factories. Determinations of the other two angles of the unitarity triangle, α ≡ − arg( V td V ∗ tb / V ud V ∗ ub ) and γ ≡ − arg( V ud V ∗ ub / V cd V ∗ cb ) , have been already obtained [67, 78], and are included in the global fitshown in Fig. 31 [77, 79]. Complementary and very valuable information could be also obtained fromthe kaon decay modes K ± → π ± ν ¯ ν , K L → π ν ¯ ν and K L → π e + e − [80]. The so-called ‘solar neutrino problem’ has been a long-standing question, since the very first chlorineexperiment at the Homestake mine [81]. The flux of solar ν e neutrinos reaching the Earth has been mea-sured by several experiments to be significantly below the standard solar model prediction [82]. More re-cently, the Sudbury Neutrino Observatory has provided strong evidence that neutrinos do change flavouras they propagate from the core of the Sun [83], independently of solar model flux predictions. SNOis able to detect neutrinos through three different reactions: the charged-current process ν e d → e − pp which is only sensitive to ν e , the neutral current transition ν x d → ν x pn which has equal probability forall active neutrino flavours, and the elastic scattering ν x e − → ν x e − which is also sensitive to ν µ and ν τ ,although the corresponding cross section is a factor . smaller than the ν e one. The measured neutrino36 -1 s -2 cm · ( e f ) - s - c m · ( tmf CCSNO f
68% C.L.
NCSNO f
68% C.L.
ESSNO f
68% C.L.
ESSK f
68% C.L.
SSMBS05 f tm NC f Fig. 33: Measured fluxes of B solar neutrinos of ν µ or ν τ type ( φ µ,τ ) versus the flux of ν e ( φ e ) [83]. fluxes, shown in Fig. 33, demonstrate the existence of a non- ν e component in the solar neutrino fluxat the . σ level. The SNO results are in good agreement with the Super-Kamiokande solar measure-ments [84] and have been further reinforced with the more recent KamLAND data, showing that ¯ ν e fromnuclear reactors disappear over distances of about 180 Km [85].Another evidence of oscillations has been obtained from atmospheric neutrinos. The known dis-crepancy between the experimental observations and the predicted ratio of muon to electron neutrinoshas become much stronger with the high precision and large statistics of Super-Kamiokande [86]. Theatmospheric anomaly appears to originate in a reduction of the ν µ flux, and the data strongly favoursthe ν µ → ν τ hypothesis. This result has been confirmed by K2K [87] and MINOS [88], observing thedisappearance of accelerator ν µ ’s at distances of 250 and 735 Km, respectively. Super-Kamiokande hasrecently reported statistical evidence of ν τ appearance at the . σ level [86]. The direct detection of theproduced ν τ is the main goal of the ongoing CERN to Gran Sasso neutrino program.Thus, we have now clear experimental evidence that neutrinos are massive particles and there ismixing in the lepton sector. Figures 34 and 35 show the present information on neutrino oscillations,from solar, atmospheric, accelerator and reactor neutrino data. A global analysis, combining the full setof data, leads to the following preferred ranges for the oscillation parameters [7]: ∆ m = (cid:0) . + 0 . − . (cid:1) · − eV , . · − < | ∆ m | / eV < . · − , (6.26) sin (2 θ ) = 0 . + 0 . − . , sin (2 θ ) > . , sin (2 θ ) < . , (6.27)where ∆ m ij ≡ m i − m j are the mass squared differences between the neutrino mass eigenstates ν i,j and θ ij the corresponding mixing angles in the standard three-flavour parametrization [7]. The rangesindicate 90% C.L. bounds. In the limit θ = 0 , solar and atmospheric neutrino oscillations decouplebecause ∆ m ⊙ ≪ ∆ m . Thus, ∆ m , θ and θ are constrained by solar data, while atmosphericexperiments constrain ∆ m , θ and θ . The angle θ is strongly constrained by the CHOOZ reactorexperiment [89]. New planned reactor experiments, T2K and NO ν A are expected to achieve sensitivitiesaround sin (2 θ ) ∼ . .Non-zero neutrino masses constitute a clear indication of new physics beyond the SM. Right-handed neutrinos are an obvious possibility to incorporate Dirac neutrino masses. However, the ν iR fieldswould be SU (3) C ⊗ SU (2) L ⊗ U (1) Y singlets, without any SM interaction. If such objects do exist, itwould seem natural to expect that they are able to communicate with the rest of the world through somestill unknown dynamics. Moreover, the SM gauge symmetry would allow for a right-handed Majorana37 tan ) e V - ( m D
68% CL95% CL99.73% CL (b)
Fig. 34: Allowed regions for ν oscillations for the com-bination of solar ( ν e ) and KamLAND ( ¯ ν e ) data, assuming CPT symmetry [83]. ) q (2 sin ) / c | ( e V m D | -3 · MINOS Best Fit MINOS 90% C.L.MINOS 68% C.L.
K2K 90% C.L. SK 90% C.L. SK (L/E) 90% C.L. ) q (2 sin ) / c | ( e V m D | -3 · Fig. 35: MINOS allowed regions for ν µ disappearanceoscillations, compared with K2K and Super-Kamiokanderesults [88]. neutrino mass term, L M = − ν ciR M ij ν jR + h . c . , (6.28)where ν ciR ≡ C ¯ ν TiR denotes the charge-conjugated field. The Majorana mass matrix M ij could havean arbitrary size, because it is not related to the ordinary Higgs mechanism. Since both fields ν iR and ν ciR absorb ν and create ¯ ν , the Majorana mass term mixes neutrinos and anti-neutrinos, violating leptonnumber by two units. Clearly, new physics is called for.Adopting a more general effective field theory language, without any assumption about the exis-tence of right-handed neutrinos or any other new particles, one can write the most general SU (3) C ⊗ SU (2) L ⊗ U (1) Y invariant Lagrangian, in terms of the known low-energy fields (left-handed neutrinosonly). The SM is the unique answer with dimension four. The first contributions from new physics ap-pear through dimension-5 operators, and have also a unique form which violates lepton number by twounits [90]: ∆ L = − c ij Λ ¯ L i ˜ φ ˜ φ t L cj + h . c . , (6.29)where L i denotes the i -flavoured SU (2) L lepton doublet, ˜ φ ≡ i τ φ ∗ and L ci ≡ C ¯ L Ti . Similar operatorswith quark fields are forbidden, due to their different hypercharges, while higher-dimension operatorswould be suppressed by higher powers of the new-physics scale Λ . After SSB, h φ (0) i = v/ √ , ∆ L generates a Majorana mass term for the left-handed neutrinos, with M ij = c ij v / Λ . Thus, Majorananeutrino masses should be expected on general symmetry grounds. Taking m ν & . eV, as suggestedby atmospheric neutrino data, one gets Λ /c ij . GeV, amazingly close to the expected scale of GranUnification.With non-zero neutrino masses, the leptonic charged-current interactions involve a flavour mix-ing matrix V L . The data on neutrino oscillations imply that all elements of V L are large, except for ( V L ) e < . ; therefore the mixing among leptons appears to be very different from the one in thequark sector. The number of relevant phases characterizing the matrix V L depends on the Dirac or Ma-jorana nature of neutrinos, because if one rotates a Majorana neutrino by a phase, this phase will appearin its mass term which will no longer be real. With only three Majorana (Dirac) neutrinos, the × matrix V L involves six (four) independent parameters: three mixing angles and three (one) phases. This relation generalizes the well-known see-saw mechanism ( m ν L ∼ m / Λ ) [91, 92]. able 5: Best published limits (90% C.L.) on lepton-flavour-violating decays [7, 49, 50]. Br( µ − → e − γ ) < . · − Br( µ − → e − γ ) < . · − Br( µ − → e − e − e + ) < . · − Br( τ − → µ − γ ) < . · − Br( τ − → e − γ ) < . · − Br( τ − → e − e − µ + ) < . · − Br( τ − → e − K S ) < . · − Br( τ − → µ − K S ) < . · − Br( τ − → µ + π − π − ) < . · − Br( τ − → Λ π − ) < . · − Br( τ − → e − π ) < . · − Br( τ − → e − π + π − ) < . · − Br( τ − → µ − π ) < . · − Br( τ − → µ − η ) < . · − Br( τ − → µ − e + µ − ) < . · − The smallness of neutrino masses implies a strong suppression of neutrinoless lepton-flavour-violating processes, which can be avoided in models with other sources of lepton-flavour violation, notrelated to m ν i . Table 5 shows the best published limits on lepton-flavour-violating decays. The B Fac-tories are pushing the experimental limits on neutrinoless τ decays beyond the − level, increasingin a drastic way the sensitivity to new physics scales. Future experiments could push further some lim-its to the − level, allowing to explore interesting and totally unknown phenomena. Complementaryinformation will be provided by the MEG experiment, which will search for µ + → e + γ events with asensitivity of − [93]. There are also ongoing projects at J-PARC aiming to study µ → e conversionsin muonic atoms, at the − level.At present, we still ignore whether neutrinos are Dirac or Majorana fermions. Another importantquestion to be addressed in the future concerns the possibility of leptonic CP violation and its relevancefor explaining the baryon asymmetry of our Universe through leptogenesis. The SM provides a beautiful theoretical framework which is able to accommodate all our present knowl-edge on electroweak and strong interactions. It is able to explain any single experimental fact and, insome cases, it has successfully passed very precise tests at the 0.1% to 1% level. In spite of this im-pressive phenomenological success, the SM leaves too many unanswered questions to be considered as acomplete description of the fundamental forces. We do not understand yet why fermions are replicated inthree (and only three) nearly identical copies. Why the pattern of masses and mixings is what it is? Arethe masses the only difference among the three families? What is the origin of the SM flavour structure?Which dynamics is responsible for the observed CP violation?In the gauge and scalar sectors, the SM Lagrangian contains only four parameters: g , g ′ , µ and h . We can trade them by α , M Z , G F and M H ; this has the advantage of using the three most preciseexperimental determinations to fix the interaction. In any case, one describes a lot of physics with onlyfour inputs. In the fermionic flavour sector, however, the situation is very different. With N G = 3 , wehave 13 additional free parameters in the minimal SM: 9 fermion masses, 3 quark mixing angles and1 phase. Taking into account non-zero neutrino masses, we have three more mass parameters plus theleptonic mixings: three angles and one phase (three phases) for Dirac (or Majorana) neutrinos.Clearly, this is not very satisfactory. The source of this proliferation of parameters is the set ofunknown Yukawa couplings in Eq. (6.1). The origin of masses and mixings, together with the reason forthe existing family replication, constitute at present the main open problem in electroweak physics. Theproblem of fermion mass generation is deeply related with the mechanism responsible for the electroweakSSB. Thus, the origin of these parameters lies in the most obscure part of the SM Lagrangian: the scalarsector. The dynamics of flavour appears to be ‘terra incognita’ which deserves a careful investigation.The SM incorporates a mechanism to generate CP violation, through the single phase naturallyoccurring in the CKM matrix. Although the present laboratory experiments are well described, thismechanism is unable to explain the matter–antimatter asymmetry of our Universe. A fundamental expla-nation of the origin of CP -violating phenomena is still lacking.39he first hints of new physics beyond the SM have emerged recently, with convincing evidenceof neutrino oscillations showing that ν e → ν µ,τ and ν µ → ν τ transitions do occur. The existence oflepton-flavour violation opens a very interesting window to unknown phenomena.The Higgs particle is the main missing block of the SM framework. The successful tests of theSM quantum corrections with precision electroweak data confirm the assumed pattern of SSB, but donot prove the validity of the minimal Higgs mechanism embedded in the SM. The present experimentalbounds (5.19) put the Higgs hunting within the reach of the new generation of detectors. The LHCshould find out whether such scalar field indeed exists, either confirming the SM Higgs mechanism ordiscovering completely new phenomena.Many interesting experimental signals are expected to be seen in the near future. New experimentswill probe the SM to a much deeper level of sensitivity and will explore the frontier of its possible exten-sions. Large surprises may well be expected, probably establishing the existence of new physics beyondthe SM and offering clues to the problems of mass generation, fermion mixing and family replication. Acknowledgements
I want to thank the organizers for the charming atmosphere of this school and all the students for theirmany interesting questions and comments. This work has been supported by the EU MRTN-CT-2006-035482 (FLAVIA net ), MEC (Spain, FPA2004-00996) and Generalitat Valenciana (GVACOMP2007-156). 40
Basic Inputs from Quantum Field Theory1.1 Wave equations
The classical Hamiltonian of a non-relativistic free particle is given by H = ~p / (2 m ) . In quantummechanics, energy and momentum correspond to operators acting on the particle wave function. Thesubstitutions H = i ~ ∂∂ t and ~p = − i ~ ~ ∇ lead then to the Schr¨odinger equation: i ~ ∂∂t ψ ( ~x, t ) = − ~ m ~ ∇ ψ ( ~x, t ) . (A.1)We can write the energy and momentum operators in a relativistic covariant way as p µ = i ∂ µ ≡ i ∂∂x µ ,where we have adopted the usual natural units convention ~ = c = 1 . The relation E = ~p + m determines the Klein–Gordon equation for a relativistic free particle: (cid:0) + m (cid:1) φ ( x ) = 0 , ≡ ∂ µ ∂ µ = ∂ ∂t − ~ ∇ . (A.2)The Klein–Gordon equation is quadratic on the time derivative because relativity puts the spaceand time coordinates on an equal footing. Let us investigate whether an equation linear in derivativescould exist. Relativistic covariance and dimensional analysis restrict its possible form to ( i γ µ ∂ µ − m ) ψ ( x ) = 0 . (A.3)Since the r.h.s. is identically zero, we can fix the coefficient of the mass term to be − ; this just deter-mines the normalization of the four coefficients γ µ . Notice that γ µ should transform as a Lorentz four-vector. The solutions of Eq. (A.3) should also satisfy the Klein–Gordon relation of Eq. (A.2). Applyingan appropriate differential operator to Eq. (A.3), one can easily obtain the wanted quadratic equation: − ( i γ ν ∂ ν + m ) ( i γ µ ∂ µ − m ) ψ ( x ) = 0 ≡ (cid:0) + m (cid:1) ψ ( x ) . (A.4)Terms linear in derivatives cancel identically, while the term with two derivatives reproduces the operator ≡ ∂ µ ∂ µ provided the coefficients γ µ satisfy the algebraic relation { γ µ , γ ν } ≡ γ µ γ ν + γ ν γ µ = 2 g µν , (A.5)which defines the so-called Dirac algebra. Eq. (A.3) is known as the Dirac equation.Obviously the components of the four-vector γ µ cannot simply be numbers. The three × Paulimatrices satisfy (cid:8) σ i , σ j (cid:9) = 2 δ ij , which is very close to the relation (A.5). The lowest-dimensionalsolution to the Dirac algebra is obtained with D = 4 matrices. An explicit representation is given by: γ = (cid:18) I − I (cid:19) , γ i = (cid:18) σ i − σ i (cid:19) . (A.6)Thus, the wave function ψ ( x ) is a column vector with four components in the Dirac space. The presenceof the Pauli matrices strongly suggests that it contains two components of spin . A proper physicalanalysis of its solutions shows that the Dirac equation describes simultaneously a fermion of spin andits own antiparticle [94].It turns useful to define the following combinations of gamma matrices: σ µν ≡ i γ µ , γ ν ] , γ ≡ γ ≡ i γ γ γ γ = − i ǫ µνρσ γ µ γ ν γ ρ γ σ . (A.7)In the explicit representation (A.6), σ ij = ǫ ijk (cid:18) σ k σ k (cid:19) , σ i = i (cid:18) σ i σ i (cid:19) , γ = (cid:18) I I (cid:19) . (A.8)41he matrix σ ij is then related to the spin operator. Some important properties are: γ γ µ γ = γ µ † , γ γ γ = − γ † = − γ , { γ , γ µ } = 0 , ( γ ) = I . (A.9)Specially relevant for weak interactions are the chirality projectors ( P L + P R = 1 ) P L ≡ − γ , P R ≡ γ , P R = P R , P L = P L , P L P R = P R P L = 0 , (A.10)which allow to decompose the Dirac spinor in its left-handed and right-handed chirality parts: ψ ( x ) = [ P L + P R ] ψ ( x ) ≡ ψ L ( x ) + ψ R ( x ) . (A.11)In the massless limit, the chiralities correspond to the fermion helicities. The Lagrangian formulation of a physical system provides a compact dynamical description and makesit easier to discuss the underlying symmetries. Like in classical mechanics, the dynamics is encoded inthe action S = Z d x L [ φ i ( x ) , ∂ µ φ i ( x )] . (A.12)The integral over the four space-time coordinates preserves relativistic invariance. The Lagrangian den-sity L is a Lorentz-invariant functional of the fields φ i ( x ) and their derivatives. The space integral L = R d x L would correspond to the usual non-relativistic Lagrangian.The principle of stationary action requires the variation δS of the action to be zero under smallfluctuations δφ i of the fields. Assuming that the variations δφ i are differentiable and vanish outside somebounded region of space-time (which allows an integration by parts), the condition δS = 0 determinesthe Euler–Lagrange equations of motion for the fields: ∂ L ∂φ i − ∂ µ (cid:18) ∂ L ∂ ( ∂ µ φ i ) (cid:19) = 0 . (A.13)One can easily find appropriate Lagrangians to generate the Klein–Gordon and Dirac equations.They should be quadratic on the fields and Lorentz invariant, which determines their possible form up toirrelevant total derivatives. The Lagrangian L = ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ (A.14)describes a complex scalar field without interactions. Both the field φ ( x ) and its complex conjugate φ ∗ ( x ) satisfy the Klein–Gordon equation; thus, φ ( x ) describes a particle of mass m without spin andits antiparticle. Particles which are their own antiparticles (i.e., with no internal charges) have onlyone degree of freedom and are described through a real scalar field. The appropriate Klein–GordonLagrangian is then L = 12 ∂ µ φ ∂ µ φ − m φ . (A.15)The Dirac equation can be derived from the Lagrangian density L = ψ ( i γ µ ∂ µ − m ) ψ . (A.16)The adjoint spinor ψ ( x ) = ψ † ( x ) γ closes the Dirac indices. The matrix γ is included to guaranteethe proper behaviour under Lorentz transformations: ψψ is a Lorentz scalar, while ψγ µ ψ transforms asa four-vector [94]. Therefore, L is Lorentz invariant as it should.Using the decomposition (A.11) of the Dirac field in its two chiral components, the fermionicLagrangian adopts the form: L = ψ L i γ µ ∂ µ ψ L + ψ R i γ µ ∂ µ ψ R − m (cid:0) ψ L ψ R + ψ R ψ L (cid:1) . (A.17)Thus, the two chiralities decouple if the fermion is massless.42 .3 Symmetries and conservation laws Let us assume that the Lagrangian of a physical system is invariant under some set of continuous trans-formations φ i ( x ) → φ ′ i ( x ) = φ i ( x ) + ǫ δ ǫ φ i ( x ) + O ( ǫ ) , (A.18)i.e., L [ φ i ( x ) , ∂ µ φ i ( x )] = L [ φ ′ i ( x ) , ∂ µ φ ′ i ( x )] . One finds then that δ ǫ L = 0 = X i (cid:26)(cid:20) ∂ L ∂φ i − ∂ µ (cid:18) ∂ L ∂ ( ∂ µ φ i ) (cid:19)(cid:21) δ ǫ φ i + ∂ µ (cid:20) ∂ L ∂ ( ∂ µ φ i ) δ ǫ φ i (cid:21)(cid:27) . (A.19)If the fields satisfy the Euler–Lagrange equations of motion (A.13), the first term is identically zero;therefore the system has a conserved current: J µ ≡ X i ∂ L ∂ ( ∂ µ φ i ) δ ǫ φ i , ∂ µ J µ = 0 . (A.20)This allows us to define a conserved charge Q ≡ Z d x J . (A.21)The condition ∂ µ J µ = 0 guarantees that d Q dt = 0 , i.e., that Q is a constant of motion.This result, known as Noether’s theorem, can be easily extended to general transformations in-volving also the space-time coordinates. For every continuous symmetry transformation which leavesthe Lagrangian invariant, there is a corresponding divergenceless Noether’s current and, therefore, a con-served charge. The selection rules observed in Nature, where there exist several conserved quantities(energy, momentum, angular momentum, electric charge, etc.), correspond to dynamical symmetries ofthe Lagrangian. The well-known Maxwell equations, ~ ∇ · ~B = 0 , ~ ∇ × ~E + ∂ ~B∂ t = 0 , (A.22) ~ ∇ · ~E = ρ , ~ ∇ × ~B − ∂ ~E∂ t = ~J , (A.23)summarize a large amount of experimental and theoretical work and provide a unified description of theelectric and magnetic forces. The first two equations in (A.22) are easily solved, writing the electromag-netic fields in terms of potentials: ~E = − ~ ∇ V − ∂ ~A∂ t , ~B = ~ ∇ × ~A . (A.24)It is very useful to rewrite these equations in a Lorentz covariant notation. The charge density ρ and the electromagnetic current ~J transform as a four-vector J µ ≡ (cid:16) ρ, ~J (cid:17) . The same is true for thepotentials which combine into A µ ≡ (cid:16) V, ~A (cid:17) . The relations (A.24) between the potentials and the fieldsthen take a very simple form, which defines the field strength tensor: F µν ≡ ∂ µ A ν − ∂ ν A µ = − E − E − E E − B B E B − B E − B B , ˜ F µν ≡ ǫ µνρσ F ρσ . (A.25)43n terms of the tensor F µν , the covariant form of the Maxwell equations turns out to be very transparent: ∂ µ ˜ F µν = 0 , ∂ µ F µν = J ν . (A.26)The electromagnetic dynamics is clearly a relativistic phenomenon, but Lorentz invariance was not veryexplicit in the original formulation of Eqs. (A.22) and (A.23). Once a covariant formulation is adopted,the equations become much simpler. The conservation of the electromagnetic current appears now as anatural compatibility condition: ∂ ν J ν = ∂ ν ∂ µ F µν = 0 . (A.27)In terms of potentials, ∂ µ ˜ F µν is identically zero while ∂ µ F µν = J ν adopts the form: A ν − ∂ ν ( ∂ µ A µ ) = J ν . (A.28)The same dynamics can be described by many different electromagnetic four-potentials, whichgive the same field strength tensor F µν . Thus, the Maxwell equations are invariant under gauge transfor-mations: A µ −→ A ′ µ = A µ + ∂ µ Λ . (A.29)Taking the Lorentz gauge ∂ µ A µ = 0 , Eq. (A.28) simplifies to A ν = J ν . (A.30)In the absence of an external current, i.e., with J µ = 0 , the four components of A µ satisfy then aKlein–Gordon equation with m = 0 . The photon is therefore a massless particle.The Lorentz condition ∂ µ A µ = 0 still allows for a residual gauge invariance under transforma-tions of the type (A.29), with the restriction Λ = 0 . Thus, we can impose a second constraint onthe electromagnetic field A µ , without changing F µν . Since A µ contains four fields ( µ = 0 , , , ) andthere are two arbitrary constraints, the number of physical degrees of freedom is just two. Therefore, thephoton has two different physical polarizations B SU(N) Algebra SU ( N ) is the group of N × N unitary matrices, U U † = U † U = 1 , with det U = 1 . Any SU ( N ) matrix can be written in the form U = exp { i T a θ a } , a = 1 , , . . . , N − , (B.1)with T a = λ a / Hermitian, traceless matrices. Their commutation relations [ T a , T b ] = i f abc T c (B.2)define the SU ( N ) algebra. The N × N matrices λ a / generate the fundamental representation of the SU ( N ) algebra. The basis of generators λ a / can be chosen so that the structure constants f abc are realand totally antisymmetric.For N = 2 , λ a are the usual Pauli matrices, σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) , (B.3)which satisfy the commutation relation [ σ i , σ j ] = 2 i ǫ ijk σ k . (B.4)Other useful properties are: { σ i , σ j } = 2 δ ij and Tr ( σ i σ j ) = 2 δ ij .44or N = 3 , the fundamental representation corresponds to the eight Gell-Mann matrices: λ = , λ = − i i , λ = − , λ = , (B.5) λ = − i i , λ = , λ = − i i , λ = 1 √ − . They satisfy the anticommutation relation n λ a , λ b o = 4 N δ ab I N + 2 d abc λ c , (B.6)where I N denotes the N -dimensional unit matrix and the constants d abc are totally symmetric in the threeindices.For SU (3) , the only non-zero (up to permutations) f abc and d abc constants are f = f = − f = f = f = f = − f = 1 √ f = 1 √ f = 12 ,d = d = − d = d = d = d = − d = − d = 12 , (B.7) d = d = d = − d = − d = − d = − d = − d = 1 √ . The adjoint representation of the SU ( N ) group is given by the ( N − × ( N − matrices ( T aA ) bc ≡ − if abc , which satisfy the commutation relations (B.2). The following equalities Tr (cid:16) λ a λ b (cid:17) = 4 T F δ ab , T F = 12 , ( λ a λ a ) αβ = 4 C F δ αβ , C F = N − N , (B.8)
Tr( T aA T bA ) = f acd f bcd = C A δ ab , C A = N , define the SU ( N ) invariants T F , C F and C A . Other useful properties are: ( λ a ) αβ ( λ a ) γδ = 2 δ αδ δ βγ − N δ αβ δ γδ , Tr (cid:16) λ a λ b λ c (cid:17) = 2 ( d abc + if abc ) , Tr( T aA T bA T cA ) = i N f abc , X b d abb = 0 , d abc d ebc = (cid:18) N − N (cid:19) δ ae , (B.9) f abe f cde + f ace f dbe + f ade f bce = 0 , f abe d cde + f ace d dbe + f ade d bce = 0 . C Anomalies
Our theoretical framework is based on the local gauge symmetry. However, so far we have only discussedthe symmetries of the classical Lagrangian. It happens sometimes that a symmetry of L gets brokenby quantum effects, i.e., it is not a symmetry of the quantized theory; one says then that there is an‘anomaly’. Anomalies appear in those symmetries involving both axial ( ψγ µ γ ψ ) and vector ( ψγ µ ψ )currents, and reflect the impossibility of regularizing the quantum theory (the divergent loops) in a waywhich preserves the chiral (left/right) symmetries. 45 p q gg Fig. 36: Triangular quark loops generating the decay π → γγ . A priori there is nothing wrong with having an anomaly. In fact, sometimes they are even wel-come. A good example is provided by the decay π → γγ . There is a chiral symmetry of the QCDLagrangian which forbids this transition; the π should then be a stable particle, in contradiction with theexperimental evidence. Fortunately, there is an anomaly generated by a triangular quark loop (Fig. 36)which couples the axial current A µ ≡ (¯ uγ µ γ u − ¯ dγ µ γ d ) to two electromagnetic currents and breaksthe conservation of the axial current at the quantum level: ∂ µ A µ = α π ǫ αβσρ F αβ F σρ + O ( m u + m d ) . (C.1)Since the π couples to A µ , h | A µ | π i = 2 i f π p µ , the π → γγ decay does finally occur, with apredicted rate Γ( π → γγ ) = (cid:18) N C (cid:19) α m π π f π = 7 . eV , (C.2)where N C = 3 denotes the number of quark colours and the so-called pion decay constant, f π =92 . MeV, is known from the π − → µ − ¯ ν µ decay rate (assuming isospin symmetry). The agreement withthe measured value, Γ = 7 . ± . eV [7], is excellent.Anomalies are, however, very dangerous in the case of local gauge symmetries, because theydestroy the renormalizability of the Quantum Field Theory. Since the SU (2) L ⊗ U (1) Y model is chiral(i.e., it distinguishes left from right), anomalies are clearly present. The gauge bosons couple to vectorand axial-vector currents; we can then draw triangular diagrams with three arbitrary gauge bosons ( W ± , Z , γ ) in the external legs. Any such diagram involving one axial and two vector currents generates abreaking of the gauge symmetry. Thus, our nice model looks meaningless at the quantum level.We have still one way out. What matters is not the value of a single Feynman diagram, but the sumof all possible contributions. The anomaly generated by the sum of all triangular diagrams connectingthe three gauge bosons G a , G b and G c is proportional to A = Tr (cid:16) { T a , T b } T c (cid:17) L − Tr (cid:16) { T a , T b } T c (cid:17) R , (C.3)where the traces sum over all possible left- and right-handed fermions, respectively, running along theinternal lines of the triangle. The matrices T a are the generators associated with the corresponding gaugebosons; in our case, T a = σ a / , Y .In order to preserve the gauge symmetry, one needs a cancellation of all anomalous contributions,i.e., A = 0 . Since Tr ( σ k ) = 0 , we have an automatic cancellation in two combinations of generators:Tr ( { σ i , σ j } σ k ) = 2 δ ij Tr ( σ k ) = 0 and Tr ( { Y, Y } σ k ) ∝ Tr ( σ k ) = 0 . However, the other twocombinations, Tr ( { σ i , σ j } Y ) and Tr ( Y ) turn out to be proportional to Tr ( Q ) , i.e., to the sum offermion electric charges: X i Q i = Q e + Q ν + N C ( Q u + Q d ) = − N C = 0 . (C.4)Equation (C.4) conveys a very important message: the gauge symmetry of the SU (2) L ⊗ U (1) Y model does not have any quantum anomaly, provided that N C = 3 . Fortunately, this is precisely the right46umber of colours to understand strong interactions. Thus, at the quantum level, the electroweak modelseems to know something about QCD. The complete SM gauge theory based on the group SU (3) C ⊗ SU (2) L ⊗ U (1) Y is free of anomalies and, therefore, renormalizable. The anomaly cancellation involvesone complete generation of leptons and quarks: ν , e , u , d . The SM would not make any sense withonly leptons or quarks. References [1] Updated version of the lectures given at the 2004 European School of High-Energy Physics (SanFeliu de Guixols, Spain): A. Pich,
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