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3_The_Standard_Model_of_Electroweak_Inte

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A 2007 arXiv lecture-note review (hep-ph/0705.4264) by A. Pich of IFIC, Valencia. It derives QED and QCD from the gauge principle, then covers the SU(2)xU(1) gauge structure, spontaneous symmetry breaking and the Higgs, Z-peak precision tests, and quark mixing and CP violation. Appendices cover quantum field theory basics, SU(N) matrices and gauge anomalies. This is a downloaded book or paper by another author, filed among Phil's particle theory books.

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arXiv:0705.4264v1 [hep-ph] 29 May 2007The Standard Model ofElectroweakInteractions A.Pich IFIC,University of Val` encia –CSIC, Val` encia, Spain Abstract Gauge invariance is a powerful tool to determine the dynamic s of the elec- troweak and strong forces. The particle content, structure and symmetries of the Standard Model Lagrangian are discussed. Special emp hasis is given to the many phenomenological tests which have established t his theoretical framework asthe Standard Theory of electroweak interactio ns. 1 Introduction TheStandardModel(SM)isagaugetheory, basedonthesymmet rygroupSU(3)C⊗SU(2)L⊗U(1)Y, which describes strong, weak and electromagnetic interact ions, via the exchange of the corresponding spin-1gaugeﬁelds: eightmasslessgluonsandonemasslessp hoton,respectively, forthestrongandelec- tromagnetic interactions, and three massive bosons, W±andZ,for the weak interaction. Thefermionic matter content is given by the known leptons and quarks, whic h are organized in a three-fold family structure: /bracketleftbiggνeu e−d′/bracketrightbigg ,/bracketleftbiggνµc µ−s′/bracketrightbigg ,/bracketleftbiggντt τ−b′/bracketrightbigg , (1.1) where (each quark appears in three different colours) /bracketleftbiggνlqu l−qd/bracketrightbigg ≡/parenleftbiggνl l−/parenrightbigg L,/parenleftbiggqu qd/parenrightbigg L, l− R, quR, qdR, (1.2) plusthecorresponding antiparticles. Thus,theleft-hand ed ﬁeldsare SU(2)Ldoublets, whiletheir right- handed partners transform as SU(2)Lsinglets. The three fermionic families in Eq. (1.1) appear t o have identicalproperties(gaugeinteractions); theydifferon lybytheirmassandtheirﬂavourquantumnumber. Thegauge symmetry is broken by the vacuum, which triggers th e Spontaneous Symmetry Break- ing (SSB)of the electroweak group tothe electromagnetic su bgroup: SU(3)C⊗SU(2)L⊗U(1)YSSB−→SU(3)C⊗U(1)QED. (1.3) The SSB mechanism generates the masses of the weak gauge boso ns, and gives rise to the appearance of aphysical scalar particle in the model, the so-called Hig gs. The fermion masses and mixings are also generated through theSSB. The SM constitutes one of the most successful achievements i n modern physics. It provides a very elegant theoretical framework, which is able to descri be the known experimental facts in particle physics with high precision. These lectures [1] provide an i ntroduction to the electroweak sector of the SM, i.e., the SU(2)L⊗U(1)Ypart [2–5]. The strong SU(3)Cpiece is discussed in more detail in Ref. [6]. The power of the gauge principle is shown in Secti on 2, where the simpler Lagrangians of quantum electrodynamics and quantum chromodynamics are derived. The electroweak theoretical frameworkispresented inSections 3and4,whichdiscuss, re spectively, thegaugestructure andtheSSB mechanism. Section 5 summarizes the present phenomenologi cal status and shows the main precision tests performed at the Zpeak. The ﬂavour structure is discussed in Section 6, where k nowledge of the quark mixingangles isbrieﬂyreviewedandtheimportance of CPviolation tests isemphasized. Finally, afew comments on open questions, to beinvestigated at futur e facilities, aregiven inthe summary. Some useful but more technical information has been collect ed in several appendices: a minimal amountofquantumﬁeldtheoryconceptsaregiveninAppendix A;AppendixBsummarizesthemostim- portant algebraic properties of SU(N)matrices; andashort discussion ongauge anomalies isprese nted inAppendix C. 2 Gauge Invariance 2.1 Quantumelectrodynamics Let usconsider the Lagrangian describing afree Dirac fermi on: L0=iψ(x)γµ∂µψ(x)−mψ(x)ψ(x). (2.1) L0isinvariant under globalU(1)transformations ψ(x)U(1)−→ψ′(x)≡exp{iQθ}ψ(x), (2.2) whereQθis an arbitrary real constant. The phase of ψ(x)is then a pure convention-dependent quantity without physical meaning. However, the free Lagrangian is n o longer invariant if one allows the phase transformation to depend on the space-time coordinate, i.e ., underlocalphase redeﬁnitions θ=θ(x), because ∂µψ(x)U(1)−→ exp{iQθ}(∂µ+iQ∂ µθ)ψ(x). (2.3) Thus, once a given phase convention has been adopted at the re ference point x0, the same convention must betaken at all space-time points. Thislooks very unnat ural. The ‘gauge principle’ is the requirement that the U(1)phase invariance should hold locally. This is only possible if one adds an extra piece to the Lagrangian, transforming in such a way as to cancel the∂µθterm in Eq. (2.3). The needed modiﬁcation is completely ﬁxed by the transformation (2.3): one introduces a newspin-1 (since ∂µθhas aLorentz index) ﬁeld Aµ(x), transforming as Aµ(x)U(1)−→A′ µ(x)≡Aµ(x)−1 e∂µθ, (2.4) and deﬁnes the covariant derivative Dµψ(x)≡[∂µ+ieQA µ(x)]ψ(x), (2.5) which has the required property of transforming like the ﬁel ditself: Dµψ(x)U(1)−→ (Dµψ)′(x)≡exp{iQθ}Dµψ(x). (2.6) TheLagrangian L ≡iψ(x)γµDµψ(x)−mψ(x)ψ(x) =L0−eQA µ(x)ψ(x)γµψ(x) (2.7) isthen invariant under local U(1)transformations. The gauge principle has generated an interaction between th e Dirac spinor and the gauge ﬁeld Aµ, which is nothing else than the familiar vertex of Quantum El ectrodynamics (QED). Note that the correspondingelectromagneticcharge Qiscompletelyarbitrary. Ifonewants Aµtobeatruepropagating ﬁeld, one needs toadd agauge-invariant kinetic term LKin≡ −1 4Fµν(x)Fµν(x), (2.8) whereFµν≡∂µAν−∂νAµis the usual electromagnetic ﬁeld strength. A possible mass term for the gauge ﬁeld, Lm=1 2m2AµAµ, is forbidden because it would violate gauge invariance; th erefore, the photon ﬁeld ispredicted tobe massless. Experimentally, we know thatmγ<6·10−17eV [7]. 2 Thetotal Lagrangian in Eqs. (2.7) and (2.8) gives rise tothe well-known Maxwell equations: ∂µFµν=Jν≡eQψγνψ, (2.9) whereJνis the fermion electromagnetic current. From asimple gauge -symmetry requirement, wehave deduced theright QEDLagrangian, which leads to avery succe ssful quantum ﬁeldtheory. 2.1.1 Lepton anomalous magnetic moments (a) (b) (c) (d) n W W g , Z g f f Fig.1: Feynman diagrams contributing tothe leptonanomalo us magnetic moment. The most stringent QED test comes from the high-precision me asurements of the e[8] andµ[9] anomalous magnetic moments al≡(gγ l−2)/2, where/vector µl≡gγ l(e/2ml)/vectorSl: ae= (1 159 652 180 .85±0.76)·10−12, a µ= (11 659 208 .0±6.3)·10−10.(2.10) Toameasurablelevel, aearisesentirelyfromvirtualelectronsandphotons; thesec ontributions are fullyknownto O(α4)andsomeO(α5)correctionshavebeenalreadycomputed[10–14]. Theimpres sive agreement achieved between theory and experiment has promo ted QED to the level of the best theory ever built to describe Nature. The theoretical error is domi nated by the uncertainty in the input value of the QEDcoupling α≡e2/(4π). Turning things around, aeprovides the most accurate determination of the ﬁnestructure constant [15]: α−1= 137.035 999 710 ±0.000 000 096 . (2.11) The anomalous magnetic moment of the muon issensitive tosma ll corrections from virtual heav- ier states; compared to ae, they scale with the mass ratio m2 µ/m2 e. Electroweak effects from virtual W±andZbosons amount to a contribution of (15.4±0.2)·10−10[10,11], which is larger than the present experimental precision. Thus aµallows one to test the entire SM. The main theoretical uncer- tainty comes from strong interactions. Since quarks have el ectric charge, virtual quark-antiquark pairs inducehadronic vacuum polarization corrections tothe photon propagator (Fig. 1.c). Owingtoth e non- perturbative character of the strong interaction at low ene rgies, the light-quark contribution cannot be reliably calculated at present. This effect can be extracte d from the measurement of the cross-section σ(e+e−→hadrons )and from the invariant-mass distribution of the ﬁnal hadron s inτdecays, which unfortunately provide slightly different results [16–18] : ath µ=/braceleftbigg(11659180.2±5.6)·10−10(e+e−data), (11659199.7±6.3)·10−10(τdata).(2.12) The quoted uncertainties include also the smaller light-by-light scattering contributions (Fig. 1.d) [19]. ThedifferencebetweentheSMpredictionandtheexperiment al value(2.10)corresponds to 3.3σ(e+e−) or0.9σ(τ). Newprecise e+e−andτdata sets are needed to settle the true value of ath µ. 3 e– e+q qg, Z Fig.2: Tree-level Feynmandiagram for the e+e−annihilation intohadrons. 2.2 Quantumchromodynamics 2.2.1 Quarks and colour The large number of known mesonic and baryonic states clearl y signals the existence of a deeper level of elementary constituents of matter: quarks. Assuming that mesons are M≡q¯qstates, while baryons have three quark constituents, B≡qqq, one can nicely classify the entire hadronic spectrum. Howe ver, inordertosatisfytheFermi–Diracstatisticsoneneedstoa ssumetheexistenceofanewquantumnumber, colour, such that each species of quark may have NC= 3different colours: qα,α= 1,2,3(red, green, blue). Baryons and mesons are then described by thecolour-s inglet combinations B=1√ 6ǫαβγ|qαqβqγ∝an}b∇acket∇i}ht, M =1√ 3δαβ|qα¯qβ∝an}b∇acket∇i}ht. (2.13) In order to avoid the existence of non-observed extra states with non-zero colour, one needs to further postulate that all asymptotic states are colourless, i.e., singlets under rotations in colour space. This assumption is known as the conﬁnement hypothesis , because it implies the non-observability of free quarks: since quarks carry colour they are conﬁned within co lour-singlet bound states. A direct test of the colour quantum number can beobtained fro m theratio Re+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 are assumed to be conﬁned, the probability to hadronize is just o ne; therefore, summing over all possible quarks in the ﬁnal state, we can estimate the inclusive cross -section into hadrons. The electroweak production factors which are common with the e+e−→γ∗,Z∗→µ+µ−process cancel in the ratio (2.14). At energies well below the Zpeak, the cross-section is dominated bythe γ-exchange amplitude; the ratioRe+e−is then given by the sum of the quark electric charges squared : Re+e−≈NCNf/summationdisplay f=1Q2 f=  2 3NC= 2, (Nf= 3 :u,d,s) 10 9NC=10 3, (Nf= 4 :u,d,s,c ) 11 9NC=11 3, (Nf= 5 :u,d,s,c,b ).(2.15) The measured ratio is shown in Fig. 3. Although the simple for mula (2.15) cannot explain the complicated structure around the different quark threshol ds, it gives the right average value of thecross- section (away from thresholds), provided that NCis taken to be three. The agreement is better at larger energies. Notice that strong interactions have not been tak en intoaccount; only theconﬁnement hypoth- esis has been used. Electromagnetic interactions are associated with the ferm ion electric charges, while the quark ﬂavours (up, down, strange, charm, bottom, top) are related to electroweak phenomena. The strong forces are ﬂavour conserving and ﬂavour independent. Onthe other side, the carriers of the electroweak interaction ( γ,Z,W±) do not couple to the quark colour. Thus it seems natural to ta ke colour as the charge associated withthe strong forces and tryto build aqu antum ﬁeldtheory based onit [20,21]. 4 10-1110102103 1 10 102rwf rJ/yy(2S)Z R S GeV Fig. 3: World data on the ratio Re+e−[7]. The broken lines show the naive quark model approximati on with NC= 3. The solidcurve isthe 3-loop perturbative QCDprediction. 2.2.2 Non-Abelian gauge symmetry Let us denote qα faquark ﬁeld of colour αand ﬂavourf. Tosimplify the equations, let us adopt avector notation incolour space: qT f≡(q1 f, q2 f, q3 f). Thefree Lagrangian L0=/summationdisplay f¯qf(iγµ∂µ−mf)qf (2.16) isinvariant under arbitrary globalSU(3)Ctransformations in colour space, qα f−→(qα f)′=Uα βqβ f, UU†=U†U= 1, detU= 1.(2.17) TheSU(3)Cmatrices can bewritten inthe form U= exp/braceleftbigg iλa 2θa/bracerightbigg , (2.18) where1 2λa(a= 1,2,...,8) denote the generators of the fundamental representation o f theSU(3)C algebra, and θaare arbitrary parameters. The matrices λaare traceless and satisfy the commutation relations /bracketleftbiggλa 2,λb 2/bracketrightbigg =ifabcλc 2, (2.19) withfabctheSU(3)Cstructure constants, which are real and totally antisymmet ric. Some useful prop- erties ofSU(3)matrices arecollected in Appendix B. As in the QED case, we can now require the Lagrangian to be also invariant under localSU(3)C transformations, θa=θa(x). To satisfy this requirement, we need to change the quark der ivatives by covariant objects. Since wehave now eight independent gaug e parameters, eight different gauge bosons Gµ a(x), the so-called gluons, are needed: Dµqf≡/bracketleftbigg ∂µ+igsλa 2Gµ a(x)/bracketrightbigg qf≡[∂µ+igsGµ(x)]qf. (2.20) Notice that wehave introduced thecompact matrix notation [Gµ(x)]αβ≡/parenleftbiggλa 2/parenrightbigg αβGµ a(x). (2.21) 5 abcfGscGnbGma Gnc adefabcfgs2Gb m Gsd GerqaGma q gs2gmabal gsb Fig.4: Interactionvertices of the QCDLagrangian. WewantDµqftotransforminexactlythesamewayasthecolour-vector qf;thisﬁxesthetransformation properties of the gauge ﬁelds: Dµ−→(Dµ)′=UDµU†, Gµ−→(Gµ)′=UGµU†+i gs(∂µU)U†.(2.22) Under aninﬁnitesimal SU(3)Ctransformation, qα f−→ (qα f)′=qα f+i/parenleftbiggλa 2/parenrightbigg αβδθaqβ f, Gµ a−→ (Gµ a)′=Gµ a−1 gs∂µ(δθa)−fabcδθbGµ c. (2.23) The gauge transformation of the gluon ﬁelds is more complica ted than the one obtained in QED for the photon. The non-commutativity of the SU(3)Cmatrices gives rise to an additional term involving the gluon ﬁelds themselves. For constant δθa, the transformation rule for the gauge ﬁelds is expressed in terms of the structure constants fabc; thus, the gluon ﬁelds belong to the adjoint representation of the colour group (see Appendix B). Note also that there is a uniqu eSU(3)Ccouplinggs. In QED it was possible to assign arbitrary electromagnetic charges to th e different fermions. Since the commutation relation (2.19) isnon-linear, this freedom does not exist f orSU(3)C. To build a gauge-invariant kinetic term for the gluon ﬁelds, we introduce the corresponding ﬁeld strengths: Gµν(x)≡ −i gs[Dµ,Dν] =∂µGν−∂νGµ+igs[Gµ,Gν]≡λa 2Gµν a(x), Gµν a(x) =∂µGν a−∂νGµ a−gsfabcGµ bGν c. (2.24) Under agauge transformation, Gµν−→(Gµν)′=UGµνU†, (2.25) and the colour trace Tr (GµνGµν) =1 2Gµν aGa µνremains invariant. Taking the proper normalization for the gluon kinetic term, we ﬁnally have the SU(3)Cinvariant Lagrangian of Quantum Chromodynamics (QCD): LQCD≡ −1 4Gµν aGa µν+/summationdisplay f¯qf(iγµDµ−mf)qf. (2.26) It is worth whileto decompose the Lagrangian into its differ ent pieces: LQCD =−1 4(∂µGν a−∂νGµ a)(∂µGa ν−∂νGa µ) +/summationdisplay f¯qα f(iγµ∂µ−mf)qα f −gsGµ a/summationdisplay f¯qα fγµ/parenleftbiggλa 2/parenrightbigg αβqβ f(2.27) +gs 2fabc(∂µGν a−∂νGµ a)Gb µGc ν−g2 s 4fabcfadeGµ bGν cGd µGe ν. 6 Fig.5: Two- and three-jetevents from the hadronic Zboson decays Z→q¯qandZ→q¯qG(ALEPH)[22]. The ﬁrst line contains the correct kinetic terms for the diff erent ﬁelds, which give rise to the corre- sponding propagators. The colour interaction between quar ks and gluons is given by the second line; it involves the SU(3)Cmatricesλa. Finally, owing to the non-Abelian character of the colour g roup, the Gµν aGa µνterm generates the cubic and quartic gluon self-interactio ns shown in the last line; the strength of these interactions (Fig. 4) is given by the same coupling gswhich appears in the fermionic piece of the Lagrangian. In spite of the rich physics contained in it, the Lagrangian ( 2.26) looks very simple because of its colour symmetry properties. All interactions are given in t erms of a single universal coupling gs, which iscalled the strongcoupling constant . Theexistence ofself-interactions amongthegauge ﬁeldsi sanew feature thatwasnotpresent inQED;itseemsthenreasonable toexpect that thesegaugeself-interactions could explain properties like asymptotic freedom (strong i nteractions become weaker at short distances) and conﬁnement (the strong forces increase at large distanc es), which donot appear in QED[6]. Without any detailed calculation, one can already extract q ualitative physical consequences from LQCD. Quarks can emit gluons. At lowest order in gs, the dominant process will be the emission of a singlegaugeboson; thus,thehadronicdecayofthe Zshouldresultinsome Z→q¯qGevents,inaddition to the dominant Z→q¯qdecays. Figure 5 clearly shows that 3-jet events, with the re quired kinematics, indeedappear intheLEPdata. Similareventsshowupin e+e−annihilation intohadrons, awayfromthe Zpeak. The ratio between 3-jet and 2-jet events provides a sim ple estimate of the strength of the strong interaction at LEPenergies ( s=M2 Z):αs≡g2 s/(4π)∼0.12. 3 Electroweak Uniﬁcation 3.1 Experimental facts Low-energy experiments have provided a large amount of info rmation about the dynamics underlying ﬂavour-changing processes. The detailed analysis of the en ergy and angular distributions in βdecays, such asµ−→e−¯νeνµorn→pe−¯νe, made clear that only the left-handed (right-handed) fermi on (antifermion) chiralities participate in those weak trans itions; moreover, the strength of the interaction appears to be universal. This is further corroborated throu gh the study of other processes like π−→ e−¯νeorπ−→µ−¯νµ, which show that neutrinos have left-handed chiralities wh ile anti-neutrinos are right-handed. Fromneutrinoscatteringdata,welearnttheexistenceofdi fferentneutrinotypes( νe∝ne}ationslash=νµ)andthat there are separately conserved lepton quantum numbers whic h distinguish neutrinos from antineutrinos; thus we observe the transitions ¯νep→e+n,νen→e−p,¯νµp→µ+norνµn→µ−p, but we do not see processes like νep∝ne}ationslash→e+n,¯νen∝ne}ationslash→e−p,¯νµp∝ne}ationslash→e+norνµn∝ne}ationslash→e−p. 7 Together with theoretical considerations related to unita rity (a proper high-energy behaviour) and theabsenceofﬂavour-changing neutral-current transitio ns(µ−∝ne}ationslash→e−e−e+),thelow-energyinformation was good enough to determine the structure of the modern elec troweak theory [23]. The intermediate vectorbosons W±andZweretheoretically introduced andtheirmassescorrectlye stimated,beforetheir experimental discovery. Nowadays, we have accumulated hug e numbers of W±andZdecay events, which bring muchdirect experimental evidence of their dyna mical properties. 3.1.1 Charged currents W e m - n n e - m - W e m + n n - m e - Fig.6: Tree-level Feynmandiagrams for µ−→e−¯νeνµandνµe−→µ−νe. Theinteraction of quarks and leptons withthe W±bosons (Fig. 6) exhibits the following features: – Only left-handed fermions and right-handed antifermions couple to the W±. Therefore, there is a 100% breaking of parity P(left↔right) and charge conjugation C(particle ↔antiparticle). However, the combined transformation CPis still agood symmetry. – TheW±bosons couple to the fermionic doublets in Eq. (1.1), where t he electric charges of the twofermion partners differ inone unit. Thedecay channels o f theW−are then: W−→e−¯νe, µ−¯νµ, τ−¯ντ, d′¯u, s′¯c. (3.1) Owing to the very high mass of the top quark [24], mt= 171 GeV > M W= 80.4 GeV, its on-shell production through W−→b′¯tiskinematically forbidden. – All fermion doublets couple tothe W±bosons with the sameuniversal strength. – The doublet partners of the up, charm and top quarks appear t o be mixtures of the three quarks with charge −1 3:  d′ s′ b′ =V d s b , VV†=V†V= 1. (3.2) Thus, the weak eigenstates d′, s′, b′are different than the mass eigenstates d, s, b. They are related through the 3×3unitary matrix V, which characterizes ﬂavour-mixing phenomena. – The experimental evidence of neutrino oscillations shows thatνe,νµandντare also mixtures of mass eigenstates. However, the neutrino masses are tiny:/vextendsingle/vextendsinglem2 ν3−m2 ν2/vextendsingle/vextendsingle∼2.5·10−3eV2, m2 ν2−m2 ν1∼8·10−5eV2[7]. 3.1.2 Neutral currents Theneutral carriers of theelectromagnetic and weakintera ctions have fermionic couplings (Fig. 7) with the following properties: – All interacting vertices are ﬂavour conserving. Both the γand theZcouple to a fermion and its own antifermion, i.e., γf¯fandZf¯f. Transitions of the type µ∝ne}ationslash→eγorZ∝ne}ationslash→e±µ∓have never been observed. 8 e– e+m– m+g, Ze– e+n n Z Fig.7: Tree-level Feynman diagrams for e+e−→µ+µ−ande+e−→ν¯ν. – The interactions depend on the fermion electric charge Qf. Fermions with the same Qfhave exactly the same universal couplings. Neutrinos do not have electromagnetic interactions ( Qν= 0), but theyhave anon-zero coupling to the Zboson. – Photonshavethesameinteraction forbothfermionchirali ties, buttheZcouplingsaredifferent for left-handed and right-handed fermions. The neutrino coupl ing to theZinvolves only left-handed chiralities. – There are three different light neutrino species. 3.2 The SU(2)L⊗U(1)Ytheory Using gauge invariance, we have been able to determine the ri ght QED and QCD Lagrangians. To describe weak interactions, we need a more elaborated struc ture, with several fermionic ﬂavours and different properties for left- and right-handed ﬁelds; mor eover, the left-handed fermions should appear in doublets, and we would like to have massive gauge bosons W±andZin addition to the photon. The simplest group with doublet representations is SU(2). Wewant to include also the electromagnetic interactions; thus weneed anadditional U(1)group. Theobvious symmetry group toconsider isthen G≡SU(2)L⊗U(1)Y, (3.3) whereLrefers to left-handed ﬁelds. We do not specify, for the momen t, the meaning of the subindex Y since, as wewill see, the naive identiﬁcation withelectrom agnetism does not work. For simplicity, let usconsider asingle family of quarks, an d introduce thenotation ψ1(x) =/parenleftbiggu d/parenrightbigg L, ψ 2(x) =uR, ψ 3(x) =dR. (3.4) Our discussion will also be valid for thelepton sector, with theidentiﬁcation ψ1(x) =/parenleftbiggνe e−/parenrightbigg L, ψ 2(x) =νeR, ψ 3(x) =e− R. (3.5) Asinthe QEDand QCDcases, let us consider thefree Lagrangia n L0=i¯u(x)γµ∂µu(x) +i¯d(x)γµ∂µd(x) =3/summationdisplay j=1iψj(x)γµ∂µψj(x). (3.6) L0isinvariant under global Gtransformations inﬂavour space: ψ1(x)G−→ψ′ 1(x)≡exp{iy1β}ULψ1(x), ψ2(x)G−→ψ′ 2(x)≡exp{iy2β}ψ2(x), (3.7) ψ3(x)G−→ψ′ 3(x)≡exp{iy3β}ψ3(x), 9 where theSU(2)Ltransformation UL≡exp/braceleftig iσi 2αi/bracerightig (i= 1,2,3) (3.8) only acts on the doublet ﬁeld ψ1. The parameters yiare called hypercharges, since the U(1)Yphase transformation is analogous to the QED one. The matrix trans formationULis non-Abelian as in QCD. NoticethatwehavenotincludedamassterminEq.(3.6)becau seitwouldmixtheleft-andright-handed ﬁelds [see Eq.(A.17)], therefore spoiling our symmetry con siderations. We can now require the Lagrangian to be also invariant under l ocalSU(2)L⊗U(1)Ygauge transformations, i.e., with αi=αi(x)andβ=β(x). In order to satisfy this symmetry requirement, we need to change the fermion derivatives by covariant objects . Since we have now four gauge parameters, αi(x)andβ(x), four different gauge bosons areneeded: Dµψ1(x)≡/bracketleftig ∂µ+ig/tildewiderWµ(x) +ig′y1Bµ(x)/bracketrightig ψ1(x), Dµψ2(x)≡[∂µ+ig′y2Bµ(x)]ψ2(x), (3.9) Dµψ3(x)≡[∂µ+ig′y3Bµ(x)]ψ3(x), where /tildewiderWµ(x)≡σi 2Wi µ(x) (3.10) denotes aSU(2)Lmatrix ﬁeld. Thus wehave the correct number of gauge ﬁelds to describe the W±,Z andγ. We wantDµψj(x)to transform in exactly the same way as the ψj(x)ﬁelds; this ﬁxes the trans- formation properties of the gauge ﬁelds: Bµ(x)G−→B′ µ(x)≡Bµ(x)−1 g′∂µβ(x), (3.11) /tildewiderWµG−→/tildewiderW′ µ≡UL(x)/tildewiderWµU† L(x) +i g∂µUL(x)U† L(x), (3.12) whereUL(x)≡exp/braceleftbig iσi 2αi(x)/bracerightbig . Thetransformation of Bµisidentical totheoneobtained inQEDfor the photon, while the SU(2)LWi µﬁelds transform in awayanalogous to the gluon ﬁelds of QCD.N ote that theψjcouplings to Bµare completely free as in QED, i.e., the hypercharges yjcan be arbitrary parameters. Since the SU(2)Lcommutation relation is non-linear, this freedom does not e xist for the Wi µ: there is only aunique SU(2)Lcouplingg. TheLagrangian L=3/summationdisplay j=1iψj(x)γµDµψj(x) (3.13) isinvariantunderlocal Gtransformations. Inordertobuildthegauge-invariant kin etictermforthegauge ﬁelds, weintroduce the corresponding ﬁeldstrengths: Bµν≡∂µBν−∂νBµ, (3.14) /tildewiderWµν≡ −i g/bracketleftig/parenleftig ∂µ+ig/tildewiderWµ/parenrightig ,/parenleftig ∂ν+ig/tildewiderWν/parenrightig/bracketrightig =∂µ/tildewiderWν−∂ν/tildewiderWµ+ig[Wµ,Wν],(3.15) /tildewiderWµν≡σi 2Wi µν, Wi µν=∂µWi ν−∂νWi µ−gǫijkWj µWk ν. (3.16) Bµνremains invariant under Gtransformations, while /tildewiderWµνtransforms covariantly: BµνG−→Bµν,/tildewiderWµνG−→UL/tildewiderWµνU† L. (3.17) 10 Therefore, the properly normalized kinetic Lagrangian isg iven by LKin=−1 4BµνBµν−1 2Tr/bracketleftig /tildewiderWµν/tildewiderWµν/bracketrightig =−1 4BµνBµν−1 4Wi µνWµν i.(3.18) Since the ﬁeld strengths Wi µνcontain a quadratic piece, the Lagrangian LKingives rise to cubic and quartic self-interactions among the gauge ﬁelds. The stren gth of these interactions is given by the same SU(2)Lcouplinggwhich appears inthe fermionic piece of the Lagrangian. The gauge symmetry forbids the writing of a mass term for the g auge bosons. Fermionic masses are also not possible, because they would communicate the le ft- and right-handed ﬁelds, which have different transformation properties, and therefore would produce an explicit breaking of the gauge sym- metry. Thus, the SU(2)L⊗U(1)YLagrangian inEqs. (3.13) and (3.18) only contains massless ﬁelds. 3.3 Charged-current interaction 23/2W quqdg(1- g )5 23/2W l nl− 5(1- g )g Fig.8: Charged-current interaction vertices. TheLagrangian (3.13) contains interactions of the fermion ﬁelds withthe gauge bosons, L −→ − gψ1γµ/tildewiderWµψ1−g′Bµ3/summationdisplay j=1yjψjγµψj. (3.19) Theterm containing the SU(2)Lmatrix /tildewiderWµ=σi 2Wi µ=1 2/parenleftigg W3 µ√ 2W† µ√ 2Wµ−W3 µ/parenrightigg (3.20) givesrisetocharged-current interactions withthebosonﬁ eldWµ≡(W1 µ+iW2 µ)/√ 2anditscomplex- conjugateW† µ≡(W1 µ−iW2 µ)/√ 2(Fig. 8). Forasingle family of quarks and leptons, LCC=−g 2√ 2/braceleftig W† µ[¯uγµ(1−γ5)d+ ¯νeγµ(1−γ5)e] +h.c./bracerightig . (3.21) The universality of the quark and lepton interactions is now a direct consequence of the assumed gauge symmetry. Note, however, that Eq. (3.21) cannot describe th e observed dynamics, because the gauge bosons are massless and, therefore, give rise tolong-range forces. 3.4 Neutral-current interaction Equation (3.19) contains also interactions with the neutra l gauge ﬁelds W3 µandBµ. We would like to identifythesebosonswiththe Zandtheγ. However,sincethephotonhasthesameinteraction withbot h fermion chiralities, the singlet gauge boson Bµcannot beequal tothe electromagnetic ﬁeld. That would requirey1=y2=y3andg′yj=eQj, which cannot besimultaneously true. 11 g f f e Qf2Z f f q qs ce ff(v − a )g5 Fig.9: Neutral-current interaction vertices. Since both ﬁelds are neutral, wecan try withan arbitrary com bination of them: /parenleftbiggW3 µ Bµ/parenrightbigg ≡/parenleftbiggcosθWsinθW −sinθWcosθW/parenrightbigg/parenleftbiggZµ Aµ/parenrightbigg . (3.22) The physical Zboson has a mass different from zero, which is forbidden by th e local gauge symmetry. We will see in the next section how it is possible to generate n on-zero boson masses, through the SSB mechanism. For the moment, we just assume that something bre aks the symmetry, generating the Z mass, and that the neutral mass eigenstates are a mixture of t he triplet and singlet SU(2)Lﬁelds. In terms of the ﬁelds Zandγ, theneutral-current Lagrangian isgiven by LNC=−/summationdisplay jψjγµ/braceleftig Aµ/bracketleftig gσ3 2sinθW+g′yjcosθW/bracketrightig +Zµ/bracketleftig gσ3 2cosθW−g′yjsinθW/bracketrightig/bracerightig ψj. (3.23) Inorder toget QEDfrom the Aµpiece, one needs toimpose the conditions: gsinθW=g′cosθW=e, Y =Q−T3, (3.24) whereT3≡σ3/2andQdenotes theelectromagnetic charge operator Q1≡/parenleftbiggQu/ν 0 0Qd/e/parenrightbigg , Q 2=Qu/ν, Q 3=Qd/e.(3.25) Theﬁrstequalityrelatesthe SU(2)LandU(1)Ycouplingstotheelectromagneticcoupling,providingthe wanted uniﬁcation of the electroweak interactions. The sec ond identity ﬁxes the fermion hypercharges interms of their electric charge and weakisospin quantum nu mbers: Quarks: y1=Qu−1 2=Qd+1 2=1 6, y 2=Qu=2 3, y 3=Qd=−1 3, Leptons: y1=Qν−1 2=Qe+1 2=−1 2, y 2=Qν= 0, y 3=Qe=−1. A hypothetical right-handed neutrino would have both elect ric charge and weak hypercharge equal to zero. Since it would not couple either to the W±bosons, such a particle would not have any kind of interaction (sterile neutrino). For aesthetic reasons, we shall then not consider right-handed neutrinos any longer. Using the relations (3.24), the neutral-current Lagrangia n can be written as LNC=LQED+LZ NC, (3.26) where LQED=−eAµ/summationdisplay jψjγµQjψj≡ −eAµJµ em (3.27) 12 Table 1: Neutral-current couplings. u d ν ee 2vf 1−8 3sin2θW−1 +4 3sin2θW1−1 + 4sin2θW 2af 1 −1 1 −1 isthe usual QEDLagrangian and LZ NC=−e 2sinθWcosθWJµ ZZµ (3.28) contains the interaction of the Zboson withthe neutral fermionic current Jµ Z≡/summationdisplay jψjγµ/parenleftbig σ3−2sin2θWQj/parenrightbig ψj=Jµ 3−2sin2θWJµ em. (3.29) Interms of themore usual fermion ﬁelds, LZ NChas the form (Fig. 9) LZ NC=−e 2sinθWcosθWZµ/summationdisplay f¯fγµ(vf−afγ5)f, (3.30) whereaf=Tf 3andvf=Tf 3/parenleftbig 1−4|Qf|sin2θW/parenrightbig . Table1shows theneutral-current couplings of the different fermions. 3.5 Gaugeself-interactions /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1W+ W−g, Z g, Z /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 W−, Zg WW+ + + W− −W W Fig.10: Gauge boson self-interactionvertices. In addition to the usual kinetic terms, the Lagrangian (3.18 ) generates cubic and quartic self- interactions among the gauge bosons (Fig. 10): L3=iecotθW/braceleftig (∂µWν−∂νWµ)W† µZν−/parenleftig ∂µWν†−∂νWµ†/parenrightig WµZν+WµW† ν(∂µZν−∂νZµ)/bracerightig +ie/braceleftig (∂µWν−∂νWµ)W† µAν−/parenleftig ∂µWν†−∂νWµ†/parenrightig WµAν+WµW† ν(∂µAν−∂νAµ)/bracerightig ; (3.31) L4=−e2 2sin2θW/braceleftbigg/parenleftig W† µWµ/parenrightig2 −W† µWµ†WνWν/bracerightbigg −e2cot2θW/braceleftig W† µWµZνZν−W† µZµWνZν/bracerightig −e2cotθW/braceleftig 2W† µWµZνAν−W† µZµWνAν−W† µAµWνZν/bracerightig −e2/braceleftig W† µWµAνAν−W† µAµWνAν/bracerightig . Noticethatatleastapairofcharged Wbosonsarealwayspresent. The SU(2)Lalgebradoesnotgenerate any neutral vertex withonly photons and Zbosons. 13 4 SpontaneousSymmetry Breaking Fig. 11: Although Nicol´ as likes the symmetric food conﬁgur ation, he must break the symmetry deciding which carrot is mo re appealing. Inthree dimensions, there isa continuous valle ywhere Nicol´ ascanmove fromone carrottothe nextwithout e ffort. So far, we have been able to derive charged- and neutral-curr ent interactions of the type needed to describe weak decays; we have nicely incorporated QED int o the same theoretical framework and, moreover, wehave got additional self-interactions of theg auge bosons, which aregenerated by the non- Abelian structure of the SU(2)Lgroup. Gauge symmetry also guarantees that we have a well-de ﬁned renormalizableLagrangian. However,thisLagrangianhasv erylittletodowithreality. Ourgaugebosons are massless particles; while this is ﬁne for the photon ﬁeld , the physical W±andZbosons should be quite heavy objects. Inorder togenerate masses, weneedtobreakthegaugesymmet ryinsomeway; however,wealso need a fully symmetric Lagrangian to preserve renormalizab ility. This dilemma may be solved by the possibility of getting non-symmetric results from an invar iant Lagrangian. Let us consider aLagrangian, which 1. Is invariant under agroup Gof transformations. 2. Hasadegenerate set of states withminimal energy, whicht ransform under Gasthemembers of a given multiplet. If one of those states is arbitrarily selected as the ground s tate of the system, the symmetry is said to be spontaneously 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 alig ned into some arbitrary direction; moreover, any higher-energy state, built from the ground state by a ﬁni te number of excitations, would share this anisotropy. In a Quantum Field Theory, the ground state is th e vacuum; thus the SSB mechanism will appear when there isa symmetric Lagrangian, but anon-symme tric vacuum. The horse in Fig. 11 illustrates in a very simple way the pheno menon of SSB. Although the left and right carrots are identical, Nicol´ as must take a decisi on if he wants to get food. What is important is not whether he goes left or right, which are equivalent opt ions, but that the symmetry gets broken. In two dimensions (discrete left-right symmetry), after eati ng the ﬁrst carrot Nicol´ as would need to make an effort to climb the hill in order to reach the carrot on the o ther side; however, in three dimensions (continuous rotation symmetry) there is a marvelous ﬂat cir cular valley along which Nicol´ as can move from one carrot to thenext without any effort. The existence of ﬂat directions connecting the degenerate s tates of minimal energy is a general property of the SSB of continuous symmetries. In a Quantum Fi eld Theory it implies the existence of massless degrees of freedom. 14 4.1 Goldstonetheorem |f|V(f) 2j|f|j1V(f) Fig. 12: Shape of the scalar potential for µ2>0(left) and µ2<0(right). In the second case there is a continuous set of degenerate vacua, corresponding todifferent phases θ, connected through a massless ﬁeldexcitation ϕ2. Let us consider acomplex scalar ﬁeld φ(x), with Lagrangian L=∂µφ†∂µφ−V(φ), V (φ) =µ2φ†φ+h/parenleftig φ†φ/parenrightig2 . (4.1) Lis invariant under global phase transformations of the scal ar ﬁeld φ(x)−→φ′(x)≡exp{iθ}φ(x). (4.2) In order to have a ground state the potential should be bounde d from below, i.e., h >0. For the quadratic piece there are twopossibilities, shown inFig. 1 2: 1.µ2>0: The potential has only the trivial minimum φ= 0. It describes a massive scalar particle with massµand quartic coupling h. 2.µ2<0: Theminimum is obtained for those ﬁeldconﬁgurations satis fying |φ0|=/radicalbigg −µ2 2h≡v√ 2>0, V (φ0) =−h 4v4. (4.3) Owing to the U(1)phase-invariance of the Lagrangian, there is an inﬁnite num ber of degenerate states of minimum energy, φ0(x) =v√ 2exp{iθ}. By choosing a particular solution, θ= 0for example, as the ground state, the symmetry gets spontaneous ly broken. If we parametrize the excitations over the ground state as φ(x)≡1√ 2[v+ϕ1(x) +iϕ2(x)], (4.4) whereϕ1andϕ2are real ﬁelds, thepotential takes the form V(φ) =V(φ0)−µ2ϕ2 1+hvϕ 1/parenleftbig ϕ2 1+ϕ2 2/parenrightbig +h 4/parenleftbig ϕ2 1+ϕ2 2/parenrightbig2. (4.5) Thus,ϕ1describes amassive state of mass m2 ϕ1=−2µ2,whileϕ2ismassless. The ﬁrst possibility ( µ2>0) is just the usual situation with a single ground state. The o ther case, with SSB, is more interesting. The appearance of a mass less particle when µ2<0is easy to understand: the ﬁeld ϕ2describes excitations around a ﬂat direction in the potenti al, i.e., into states with the same energy as the chosen ground state. Since those e xcitations do not cost any energy, they obviously correspond toamassless state. 15 The fact that there are massless excitations associated wit h the SSB mechanism is a completely general result, known as the Goldstone theorem [25]: if a Lag rangian is invariant under a continuous symmetry group G, but the vacuum is only invariant under a subgroup H⊂G, then there must exist as many massless spin-0 particles (Goldstone bosons) as broke n generators (i.e., generators of Gwhich do not belong to H). 4.2 TheHiggs–Kibble mechanism Atﬁrstsight,theGoldstonetheoremhasverylittletodowit hourmassproblem; infact,itmakesitworse sincewewantmassivestatesandnotmasslessones. However, something veryinteresting happens when there is alocal gauge symmetry [26,27]. Let us consider [3] an SU(2)Ldoublet of complex scalar ﬁelds φ(x)≡/parenleftbiggφ(+)(x) φ(0)(x)/parenrightbigg . (4.6) Thegauged scalar Lagrangian of the Goldstone model in Eq.(4 .1), LS= (Dµφ)†Dµφ−µ2φ†φ−h/parenleftig φ†φ/parenrightig2 (h>0, µ2<0), (4.7) Dµφ=/bracketleftig ∂µ+ig/tildewiderWµ+ig′yφBµ/bracketrightig φ, y φ=Qφ−T3=1 2,(4.8) is invariant under local SU(2)L⊗U(1)Ytransformations. The value of the scalar hypercharge is ﬁxe d by the requirement of having the correct couplings between φ(x)andAµ(x); i.e., the photon does not couple toφ(0),andφ(+)has the right electric charge. The potential is very similar to the one considered before. T here is a inﬁnite set of degenerate states withminimum energy, satisfying |∝an}b∇acketle{t0|φ(0)|0∝an}b∇acket∇i}ht|=/radicalbigg −µ2 2h≡v√ 2. (4.9) Note that we have made explicit the association of the classi cal ground state with the quantum vacuum. Since the electric charge is a conserved quantity, only the n eutral scalar ﬁeld can acquire a vacuum expectation value. Once we choose a particular ground state , theSU(2)L⊗U(1)Ysymmetry gets spontaneously broken to the electromagnetic subgroup U(1)QED, which by construction still remains a true symmetry of the vacuum. According to the Goldstone theo rem three massless states should then appear. Now,let usparametrize the scalar doublet in thegeneral for m φ(x) = exp/braceleftig iσi 2θi(x)/bracerightig1√ 2/parenleftbigg0 v+H(x)/parenrightbigg , (4.10) with four real ﬁelds θi(x)andH(x). The crucial point is that the local SU(2)Linvariance of the La- grangianallowsustorotateawayanydependence on θi(x). Thesethreeﬁeldsarepreciselythewould-be massless Goldstone bosons associated withthe SSBmechanis m. Thecovariant derivative (4.8) couples the scalar multiple t to theSU(2)L⊗U(1)Ygauge bosons. If one takes the physical (unitary) gauge θi(x) = 0, the kinetic piece of the scalar Lagrangian (4.7) takes theform: (Dµφ)†Dµφθi=0−→1 2∂µH∂µH+ (v+H)2/braceleftbiggg2 4W† µWµ+g2 8cos2θWZµZµ/bracerightbigg .(4.11) 16 The vacuum expectation value of the neutral scalar has gener ated a quadratic term for the W±and the Z,i.e., those gauge bosons have acquired masses: MZcosθW=MW=1 2vg. (4.12) Therefore, we have found a clever way of giving masses to the i ntermediate carriers of the weak force. We just add LSto ourSU(2)L⊗U(1)Ymodel. The total Lagrangian is invariant under gauge transformations, which guarantees the renormalizability of the associated Quantum Field Theory [28]. However, SSBoccurs. The three broken generators give rise t o three massless Goldstone bosons which, owing to the underlying local gauge symmetry, can be elimina ted from the Lagrangian. Going to the unitary gauge, we discover that the W±and theZ(but not the γ, becauseU(1)QEDis an unbroken symmetry) have acquired masses, which are moreover related as indicated in Eq. (4.12). Notice that Eq. (3.22) has now the meaning of writing the gauge ﬁelds in te rms of the physical boson ﬁelds with deﬁnite mass. It is instructive to count the number of degrees of freedom (d .o.f.). Before the SSB mechanism, the Lagrangian contains massless W±andZbosons, i.e., 3×2 = 6d.o.f., due to the two possible polarizations ofamasslessspin-1ﬁeld,andfourrealscala rﬁelds. AfterSSB,thethreeGoldstonemodes are ‘eaten’ by the weak gauge bosons, which become massive an d, therefore, acquire one additional longitudinal polarization. We have then 3×3 = 9d.o.f. in the gauge sector, plus the remaining scalar particleH, which iscalled the Higgs boson. Thetotal number of d.o.f. r emains of course thesame. 4.3 Predictions We have now all the needed ingredients to describe the electr oweak interaction within a well-deﬁned Quantum Field Theory. Our theoretical framework implies th e existence of massive intermediate gauge bosons,W±andZ. Moreover, the Higgs-Kibble mechanism has produced a preci se prediction1for the W±andZmasses, relating them to the vacuum expectation value of the scalar ﬁeldthrough Eq. (4.12). Thus,MZispredicted to bebigger than MWinagreement withthe measured masses [29,30]: MZ= 91.1875±0.0021 GeV, M W= 80.398±0.025 GeV. (4.13) From these experimental numbers, one obtains the electrowe ak mixing angle sin2θW= 1−M2 W M2 Z= 0.223. (4.14) We can easily get and independent estimate of sin2θWfrom the decay µ−→e−¯νeνµ. The momentum transfer q2= (pµ−pνµ)2= (pe+pνe)2/lessorsimilarm2 µis much smaller than M2 W. Therefore, theWpropagator inFig.6shrinks toapoint andcanbewellapproxi mated through alocal four-fermion interaction, i.e., g2 M2 W−q2≈g2 M2 W=4πα sin2θWM2 W≡4√ 2GF. (4.15) The measured muon lifetime, τµ= (2.197019 ±0.000021) ·10−6s [31], provides a very precise deter- mination of the Fermi coupling constant GF: 1 τµ= Γ µ=G2 Fm5 µ 192π3f(m2 e/m2 µ) (1 +δRC), f (x)≡1−8x+ 8x3−x4−12x2logx.(4.16) 1Note,however,thattherelation MZcosθW=MWhasamoregeneralvalidity. Itisadirectconsequenceofthe symmetry properties of LSand does not depend on itsdetaileddynamics. 17 Taking into account theradiative corrections δRC,which are known to O(α2)[32,33], one gets [31]: GF= (1.166371 ±0.000006) ·10−5GeV−2. (4.17) Themeasured values of α−1= 137.035999710(96) ,MWandGFimply sin2θW= 0.215, (4.18) in very good agreement with Eq. (4.14). We shall see later tha t the small difference between these two numberscanbeunderstood intermsofhigher-order quantumc orrections. TheFermicouplinggivesalso adirect determination of the electroweak scale, i.e.,the s calar vacuum expectation value: v=/parenleftig√ 2GF/parenrightig−1/2 = 246GeV. (4.19) 4.4 TheHiggs boson Z HZ H H ZZW HW −H W H2MZ v22 Z2 M v W+ +− 2 W v 2MW v22 M/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1 Fig.13: Higgs couplings tothe gauge bosons. The scalar Lagrangian in Eq. (4.7) has introduced a new scala r particle into the model: the Higgs H. Interms of the physical ﬁelds(unitary gauge), LStakes the form LS=1 4hv4+LH+LHG2, (4.20) where LH=1 2∂µH∂µH−1 2M2 HH2−M2 H 2vH3−M2 H 8v2H4, (4.21) LHG2=M2 WW† µWµ/braceleftbigg 1 +2 vH+H2 v2/bracerightbigg +1 2M2 ZZµZµ/braceleftbigg 1 +2 vH+H2 v2/bracerightbigg (4.22) and the Higgs massis given by MH=/radicalbig −2µ2=√ 2hv. (4.23) TheHiggsinteractions (Fig.13)haveaverycharacteristic form: theyarealwaysproportional tothemass (squared) of the coupled boson. All Higgs couplings are dete rmined byMH,MW,MZand the vacuum expectation value v. So far the experimental searches for the Higgs have only prov ided a lower bound on its mass, corresponding tothe exclusion of the kinematical range acc essible at LEPand the Tevatron [7]: MH>114.4 GeV (95% C .L.). (4.24) 18 4.5 Fermionmasses Hf ffm v Fig.14: Fermionic coupling of the Higgs boson. Afermionicmassterm Lm=−mψψ=−m/parenleftbig ψLψR+ψRψL/parenrightbig isnotallowed,becauseitbreaks thegaugesymmetry. However, sincewehaveintroduced anadd itional scalar doublet intothemodel, we can writethe following gauge-invariant fermion-scalar co upling: LY=−c1/parenleftbig ¯u,¯d/parenrightbig L/parenleftbiggφ(+) φ(0)/parenrightbigg dR−c2/parenleftbig ¯u,¯d/parenrightbig L/parenleftbiggφ(0)∗ −φ(−)/parenrightbigg uR−c3(¯νe,¯e)L/parenleftbiggφ(+) φ(0)/parenrightbigg eR+h.c., (4.25) where the second term involves the C-conjugate scalar ﬁeld φc≡iσ2φ∗. In the unitary gauge (after SSB),this Yukawa-type Lagrangian takes the simpler form LY=−1√ 2(v+H)/braceleftbig c1¯dd+c2¯uu+c3¯ee/bracerightbig . (4.26) Therefore, the SSBmechanism generates also fermion masses : md=c1v√ 2, m u=c2v√ 2, m e=c3v√ 2. (4.27) Since we do not know the parameters ci, the values of the fermion masses are arbitrary. Note, however, that all Yukawacouplings are ﬁxedin termsof the ma sses (Fig. 14): LY=−/parenleftbigg 1 +H v/parenrightbigg/braceleftbig md¯dd+mu¯uu+me¯ee/bracerightbig . (4.28) 5 Electroweak Phenomenology In the gauge and scalar sectors, the SM Lagrangian contains o nly four parameters: g,g′,µ2andh. One could trade them by α,θW,MWandMH. Alternatively, wecan choose as free parameters: GF= (1.166371 ±0.000006) ·10−5GeV−2[31], α−1= 137.035999710 ±0.000000096 [15], (5.1) MZ= (91.1875±0.0021)GeV [29,30] and the Higgs mass MH. This has the advantage of using the three most precise exper imental determi- nations to ﬁxtheinteraction. Therelations sin2θW= 1−M2 W M2 Z, M2 Wsin2θW=πα√ 2GF(5.2) determine then sin2θW= 0.212andMW= 80.94 GeV. The predicted MWis in good agreement withthe measured value in(4.13). 19 W− nl−l , d , ui jZ ff Fig.15: Tree-level Feynman diagrams contributingtothe W±andZdecays. At tree level (Fig. 15), the decay widths of the weak gauge bos ons can be easily computed. The Wpartial widths, Γ/parenleftbig W−→¯νll−/parenrightbig =GFM3 W 6π√ 2, Γ/parenleftbig W−→¯uidj/parenrightbig =NC|Vij|2GFM3 W 6π√ 2,(5.3) are equal for all leptonic decay modes (up to small kinematic al mass corrections). The quark modes involve also the colour quantum number NC= 3and the mixing factor Vijrelating weak and mass eigenstates, d′ i=Vijdj. TheZpartial widths are different for each decay mode, since its c ouplings depend on the fermion charge: Γ/parenleftbig Z→¯ff/parenrightbig =NfGFM3 Z 6π√ 2/parenleftbig |vf|2+|af|2/parenrightbig , (5.4) whereNl= 1andNq=NC. Summing over all possible ﬁnal fermion pairs, one predicts the total widths ΓW= 2.09GeV and ΓZ= 2.48GeV, in excellent agreement with the experimental values ΓW= (2.147±0.060)GeV and ΓZ= (2.4952±0.0023)GeV [29,30]. Theuniversality of the Wcouplings implies Br(W−→¯νll−) =1 3 + 2NC= 11.1%, (5.5) wherewehavetakenintoaccount thatthedecayintothetopqu arkiskinematically forbidden. Similarly, the leptonic decay widths of the Zare predicted to be Γl≡Γ(Z→l+l−) = 84.85 MeV. As shown in Table 2, these predictions are in good agreement with the m easured leptonic widths, conﬁrming the universality of the WandZleptonic couplings. There is, however, an excess of the bran ching ratio W→τ¯ντwithrespect to W→e¯νeandW→µ¯νµ, which represents a 2.8σeffect [29,30]. The universality of the leptonic Wcouplings can also be tested indirectly, through weak decay s mediatedbycharged-current interactions. Comparingthem easureddecaywidthsofleptonicorsemilep- tonic decays which only differ by the lepton ﬂavour, one can t est experimentally that the Winteraction is indeed the same, i.e., that ge=gµ=gτ≡g. As shown in Table 3, the present data verify the universality of the leptonic charged-current couplings to the0.2% level. Table2: Measuredvaluesof Br (W−→¯νll−)andΓ(Z→l+l−)[29,30]. Theaverageofthethreeleptonicmodesisshown inthe lastcolumn (for a massless charged lepton l). e µ τ l Br(W−→¯νll−) (%) 10.65±0.17 10.59±0.15 11.44±0.22 10.84±0.09 Γ(Z→l+l−)(MeV) 83.92±0.12 83.99±0.18 84.08±0.22 83.985±0.086 20 Table 3: Experimental determinations of the ratios gl/gl′[18,34] Γτ→ντe¯νe/Γµ→νµe¯νeΓτ→ντπ/Γπ→µ¯νµΓτ→ντK/ΓK→µ¯νµΓW→τ¯ντ/ΓW→µ¯νµ |gτ/gµ|1.0004±0.0022 0 .996±0.005 0 .979±0.017 1 .039±0.013 Γτ→ντµ¯νµ/Γτ→ντe¯νeΓπ→µ¯νµ/Γπ→e¯νeΓK→µ¯νµ/ΓK→e¯νeΓK→πµ¯νµ/ΓK→πe¯νe |gµ/ge|1.0000±0.0020 1.0017±0.0015 1 .012±0.009 1 .0002±0.0026 ΓW→µ¯νµ/ΓW→e¯νe Γτ→ντµ¯νµ/Γµ→νµe¯νeΓW→τ¯ντ/ΓW→e¯νe |gµ/ge| 0.997±0.010 |gτ/ge| 1.0004±0.0023 1 .036±0.014 Another interesting quantity isthe Zdecay width into invisible modes, Γinv Γl≡NνΓ(Z→¯νν) Γl=2Nν (1−4 sin2θW)2+ 1, (5.6) which is usually normalized to the charged leptonic width. T he comparison with the measured value, Γinv/Γl= 5.942±0.016[29,30], provides very strong experimental evidence for th eexistence of three different light neutrinos. 5.1 Fermion-pairproduction at the Zpeak f-e e+f q-e+ f feg , Z Fig.16: Tree-level contributions to e+e−→¯ffand kinematical conﬁguration inthe centre-of-mass system . Additional information can be obtained from the study of the processe+e−→γ,Z→¯ff (Fig. 16). For unpolarized e+ande−beams, the differential cross-section can be written, at lo west order, as dσ dΩ=α2 8sNf/braceleftbig A(1 + cos2θ) +Bcosθ−hf/bracketleftbig C(1 + cos2θ) +Dcosθ/bracketrightbig/bracerightbig ,(5.7) wherehf=±1denotes thesignof thehelicity oftheproduced fermion f,andθisthescattering angle betweene−andfinthe centre-of-mass system. Here, A= 1 + 2vevfRe(χ) +/parenleftbig v2 e+a2 e/parenrightbig/parenleftbig v2 f+a2 f/parenrightbig |χ|2, B= 4aeafRe(χ) + 8veaevfaf|χ|2, C= 2veafRe(χ) + 2/parenleftbig v2 e+a2 e/parenrightbig vfaf|χ|2, D= 4aevfRe(χ) + 4veae/parenleftbig v2 f+a2 f/parenrightbig |χ|2, (5.8) andχcontains the Zpropagator χ=GFM2 Z 2√ 2παs s−M2 Z+isΓZ/MZ. (5.9) 21 The coefﬁcients A,B,CandDcan be experimentally determined by measuring the total cro ss- section, the forward–backward asymmetry, the polarizatio n asymmetry, and the forward–backward po- larization asymmetry, respectively: σ(s) =4πα2 3sNfA, AFB(s)≡NF−NB NF+NB=3 8B A, APol(s)≡σ(hf=+1)−σ(hf=−1) σ(hf=+1)+σ(hf=−1)=−C A, (5.10) AFB,Pol(s)≡N(hf=+1) F−N(hf=−1) F−N(hf=+1) B+N(hf=−1) B N(hf=+1) F+N(hf=−1) F+N(hf=+1) B+N(hf=−1) B=−3 8D A. Here,NFandNBdenotethenumberof f’semergingintheforwardandbackwardhemispheres, respec - tively, with respect to the electron direction. The measure ment of the ﬁnal fermion polarization can be done forf=τby measuring thedistribution of the ﬁnal τdecay products. Fors=M2 Z, the real part of the Zpropagator vanishes and the photon-exchange terms can be neglected in comparison with the Z-exchange contributions ( Γ2 Z/M2 Z<<1). Equations (5.10) become then, σ0,f≡σ(M2 Z) =12π M2 ZΓeΓf Γ2 Z, A0,f FB≡ A FB(M2 Z) =3 4PePf, A0,f Pol≡ A Pol(M2 Z) =Pf, A0,f FB,Pol≡ A FB,Pol(M2 Z) =3 4Pe, (5.11) where ΓfistheZpartial decay width into the ¯ffﬁnal state, and Pf≡ −Af≡−2vfaf v2 f+a2 f(5.12) is the average longitudinal polarization of the fermion f, which only depends on the ratio of the vector and axial-vector couplings. With polarized e+e−beams, which have been available at SLC, one can also study th e left–right asymmetry between the cross-sections for initial left- and right-handed electrons, and thecorresponding forward–backward left–right asymmetry: A0 LR≡ A LR(M2 Z) =σL(M2 Z)−σR(M2 Z) σL(M2 Z) +σR(M2 Z)=−Pe, A0,f FB,LR≡ A FB,LR(M2 Z) =−3 4Pf. (5.13) At theZpeak,A0 LRmeasures the average initial lepton polarization, Pe, without any need for ﬁnal particle identiﬁcation, while A0,f FB,LRprovides adirect determination of the ﬁnal fermion polariz ation. Pfis a very sensitive function of sin2θW. Small higher-order corrections can produce large variations on the predicted lepton polarization because |vl|=1 2|1−4 sin2θW| ≪1. Therefore, Pl provides an interesting window tosearch for electroweak qu antum effects. 5.2 QEDand QCDcorrections Before trying to analyse the relevance of higher-order elec troweak contributions, it is instructive to con- sider the numerical impact of the well-known QED and QCD corr ections. The photon propagator gets vacuum polarization corrections, induced by virtual fermi on–antifermion pairs. This kind of QED loop corrections can betaken into account through aredeﬁnition of the QEDcoupling, which depends onthe 22 g , Z g , Z f -f g , Z g , Z f -f g gf f-+ –+ – + –+ –+ –+ –+ – + –– q q Fig.17: The photon vacuum polarization (left)generates a c harge screening effect, making α(s)smaller atlarger 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). Thephysical QED vacuum behaves as a polarized dielectric medium. The hug e difference between the electron and Zmass scales makes this quantum correction relevant at LEPen ergies [15,29,30]: α(m2 e)−1= 137.035999710(96) > α(M2 Z)−1= 128.93±0.05. (5.14) The running effect generates an important change in Eq. (5.2 ). SinceGFis measured at low energies, while MWis a high-energy parameter, the relation between both quant ities is modiﬁed by vacuum-polarization contributions. Changing αbyα(M2 Z),one gets thecorrected predictions: sin2θW= 0.231, M W= 79.96GeV. (5.15) Theexperimental valueof MWisintherangebetweenthetworesults obtained witheither αorα(M2 Z), showingitssensitivitytoquantumcorrections. Theeffect ismorespectacularintheleptonicasymmetries at theZpeak. The small variation of sin2θWfrom 0.212 to 0.231 induces a large shift on the vector Zcoupling to charged leptons from vl=−0.076to−0.038, changing the predicted average lepton polarization Plby afactor 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 deca ysZ→¯qqandW−→¯uidjcan be directly incorporated into the formulae given before by taking an‘ef fective’ number of colours: NC=⇒NC/braceleftig 1 +αs π+.../bracerightig ≈3.115, (5.16) where wehave used the value of αsats=M2 Z,αs(M2 Z) = 0.119±0.002[7,35]. Note that the strong coupling also ‘runs’. However, the gluo n self-interactions generate an anti- screening effect, through gluon-loop corrections to the gl uon propagator, which spread out the QCD charge [6]. Since this correction is larger than the screeni ng of the colour charge induced by virtual quark–antiquark pairs, the net result is that the strong cou pling decreases at short distances. Thus, QCD hastherequiredpropertyofasymptoticfreedom: quarksbeh aveasfreeparticleswhen Q2→ ∞[36,37]. QCD corrections increase the probabilities of the Zand theW±to decay into hadronic modes. Therefore, their leptonic branching fractions become smal ler. The effect can be easily estimated from Eq. (5.5). The probability of the decay W−→¯νee−gets reduced from 11.1% to 10.8%, improving the agreement with the measured value inTable 2. 5.3 Higher-order electroweak corrections Quantum corrections offer the possibility to be sensitive t o heavy particles, which cannot be kinemati- callyaccessed, throughtheirvirtualloopeffects. InQEDa ndQCDthevacuumpolarization contribution of 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 ‘decoupli ng’ of the heavy ﬁelds happens in theories 23 g , Z g , Z f -f g , Z g , Z f -f W W -d u j i -W-W g , Z g , Zl , d- i n, u- l j -f f- Fig.18: Self-energycorrections tothe gauge boson propaga tors. with only vector couplings and an exact gauge symmetry [38], where the effects generated by the heavy particles can always be reabsorbed into aredeﬁnition of the low-energy parameters. The SM involves, however, a broken chiral gauge symmetry. Th is has the very interesting im- plication of avoiding the decoupling theorem [38]. The vacu um polarization contributions induced by a heavy top generate corrections to the W±andZpropagators (Fig. 18), which increase quadratically with the top mass [39]. Therefore, a heavy top does not decoup le. For instance, with mt= 171GeV, the leading quadratic correction to the second relation in E q. (5.2) amounts to a sizeable 3%effect. The quadratic mass contribution originates in the strong break ing of weak isospin generated by the top and bottom quark masses, i.e., theeffect isactually proportio nal tom2 t−m2 b. Owingtoanaccidental SU(2)Csymmetryofthescalarsector (theso-called custodial symm etry), the virtual production of Higgs particles does not generate any quadratic dependence on the Higgs mass at one loop [39]. The dependence on MHis only logarithmic. The numerical size of the correspondin g correction in Eq.(5.2) varies from a0.1% to a1% effect for MHinthe range from 100 to 1000 GeV. W b bt ZWb bt Z Fig.19: One-loop corrections tothe Z¯bbvertex,involving a virtual top. Higher-order corrections to the different electroweak cou plings are non-universal and usually smaller than the self-energy contributions. There is one in teresting exception, the Z¯bbvertex (Fig. 19), which is sensitive to the top quark mass [40]. The Z¯ffvertex gets one-loop corrections where a vir- tualW±is exchanged between the two fermionic legs. Since the W±coupling changes the fermion ﬂavour, thedecays Z→¯dd,¯ss,¯bbgetcontributions withatopquark intheinternal fermionic lines, i.e., Z→¯tt→¯didi. Noticethat thismechanism canalsoinducetheﬂavour-chan ging neutral-current decays Z→¯didjwithi∝ne}ationslash=j. These amplitudes are suppressed by the small CKM mixing fac tors|VtjV∗ ti|2. However, for the Z→¯bbvertex, there isno suppression because |Vtb| ≈1. Theexplicit calculation [40–43] showsthe presence of hard m2 tcorrections tothe Z→¯bbvertex. This effect can be easily understood [40] in non-unitary gau ges where the unphysical charged scalar φ(±)is present. The fermionic couplings of the charged scalar ar e proportional to the fermion masses; therefore the exchange of a virtual φ(±)gives rise to a m2 tfactor. In the unitary gauge, the charged scalar has been ‘eaten’ by the W±ﬁeld; thus the effect comes now from the exchange of a longitu dinal W±, with terms proportional to qµqνin the propagator that generate fermion masses. Since the W± couples only to left-handed fermions, the induced correcti on is the same for the vector and axial-vector Z¯bbcouplings and, for mt= 171GeV,amounts toa 1.6% reduction of the Z→¯bbdecay width [40]. The ‘non-decoupling’ present in the Z¯bbvertex is quite different from the one happening in the boson self-energies. The vertex correction is not dependen t on the Higgs mass. Moreover, while any kind of newheavy particle coupling tothegauge bosons would contribute tothe WandZself-energies, 24 the possible new physics contributions to the Z¯bbvertex are much more restricted and, in any case, different. Therefore, the independent experimental measu rement of the two effects is very valuable in order to disentangle possible new physics contributions fr om the SM corrections. In addition, since the ‘non-decoupling’ vertex effect is related to WL-exchange, it is sensitive tothe SSBmechanism. 5.4 SMelectroweak ﬁt 0.2310.2320.233 83.6 83.8 84 84.268% CL Gll [MeV]sin2qlept effmt= 170.9 ± 1.8 GeV mH= 114...1000 GeV mtmH Da -0.041-0.038-0.035-0.032 -0.503 -0.502 -0.501 -0.5 gAlgVl 68% CLl+l- e+e- m+m- t+t-mtmHmt= 172.7 ± 2.9 GeV mH= 114...1000 GeV Da Fig. 20: Combined LEP and SLD measurements of sin2θlept eﬀandΓl(left) and the corresponding effective vector and axial- vector couplings vlandal(right). The shaded region shows the SMprediction. Thearro ws point inthe directionof increasing values of mtandMH. The point shows the predicted values if, among the electrow eak radiative corrections, only the photon vacuum polarization is included. Itsarrow indicates the va riation induced by theuncertainty in α(M2 Z)[29,30]. The leptonic asymmetry measurements from LEPand SLDcan all be combined to determine the ratiosvl/alof the vector and axial-vector couplings of the three charge d leptons, or equivalently the effective electroweak mixing angle sin2θlept eﬀ≡1 4/parenleftbigg 1−vl al/parenrightbigg . (5.17) Thesum (v2 l+a2 l)isderived fromtheleptonic decaywidthsof the Z,i.e.,fromEq.(5.4)corrected with a multiplicative factor/parenleftbig 1 +3 4α π/parenrightbig to account for ﬁnal-state QED corrections. The signs of vlandalare ﬁxedby requiring ae<0. The resulting 68% probability contours are shown in Fig. 20, which provides strong evidence of the electroweak radiative corrections. The good agreeme nt with the SM predictions, obtained for low values of the Higgs mass, is lost if only the QED vacuum pol arization contribution is taken into account, as indicated by the point with an arrow. Notice that the uncertainty induced by the input value ofα(M2 Z)−1= 128.93±0.05issizeable. Themeasured couplings of thethree charged lep tons conﬁrm lepton universality in the neutral-current sector. The sol id contour combines the three measurements assuming universality. The neutrino couplings can also be determined from the invis ibleZdecay width, by assuming three identical neutrino generations with left-handed cou plings, and ﬁxing the sign from neutrino scat- tering data. Alternatively, one can use the SM prediction fo rΓinvto get a determination of the number of light neutrino ﬂavours [29,30]: Nν= 2.9840±0.0082. (5.18) Figure 21 shows the measured values of AlandAb, together with the joint constraint obtained fromA0,b FB(diagonal band). The direct measurement of Abat SLD agrees well with the SM prediction; 25 0.80.91 0.14 0.145 0.15 0.155 Al Ab 68.3 95.5 99.5 % CLSM Fig. 21: Measurements of Al,Ab(SLD) and A0,b FB. The arrowspointingtotheleft(right)showthevariationsofth e SMpredictionwith MH= 300+700 −186GeV(mt= 172 .7± 2.9 GeV). The small arrow oriented to the left shows the additional uncertaintyfrom α(M2 Z)[29,30].100175250 0.213 0.217 0.221 R0 bmt [GeV]R0 b R0 d Fig. 22: The SM prediction of the ratios RbandRd [Rq≡Γ(Z→¯qq)/Γ(Z→hadrons) ], as a function of the top mass. The measured value of Rb(vertical band) provides a determinationof mt[29,30]. however, a much lower value is obtained from the ratio4 3A0,b FB/Al. This is the most signiﬁcant discrep- ancy observed in the Z-pole data. Heavy quarks (4 3A0,b FB/Ab) seem to prefer a high value of the Higgs mass,whileleptons( Al)favouralightHiggs. Thecombinedanalysispreferslowval uesofMH,because of the inﬂuence of Al. The strong sensitivity of the ratio Rb≡Γ(Z→¯bb)/Γ(Z→hadrons) to the top quark mass is shown in Fig. 22. Owing to the |Vtd|2suppression, such a dependence is not present in the analogo us ratioRd. Combined with all other electroweak precision measuremen ts at theZpeak,Rbprovides a determination of mtin good agreement with the direct and most precise measureme nt at the Tevatron. This is shown in Fig. 23, which compares the information on MWandmtobtained at LEP1 and SLD, with the direct measurements performed at LEP2 and the Tevat ron. A similar comparison for mtand MHis also shown. The lower bound on MHobtained from direct searches excludes a large portion of the 68% C.L.allowed domain from precision measurements. 80.380.480.5 150 175 200mH [GeV] 114300 1000 mt [GeV]mW [GeV]68% CL DaLEP1 and SLD LEP2 and Tevatron (prel.) 160180200 10 102103 mH [GeV]mt [GeV] ExcludedHigh Q2 except mt 68% CL mt (Tevatron) Fig.23: Comparison(left)ofthedirectmeasurementsof MWandmt(LEP2andTevatrondata)withtheindirectdetermination through electroweak radiative corrections (LEP1 and SLD). Also shown in the SM relationship for the masses as function o f MH. The ﬁgure onthe right makes the analogous comparison for mtandMH[29,30]. 26 0123456 100 30 300 mH [GeV]Dc2 Excluded PreliminaryDahad =Da(5) 0.02758±0.00035 0.02749±0.00012 incl. low Q2 dataTheory uncertaintymLimit = 144 GeV Fig. 24: ∆χ2=χ2−χ2 minversus MH, from the global ﬁttotheelectroweakdata. Theverticalbandindicatesthe 95% exclusion limitfrom direct searches [29,30].Measurement Fit |Omeas-Ofit|/smeas 0 1 2 3 0 1 2 3Dahad(mZ) Da(5)0.02758 ± 0.00035 0.02768 mZ [GeV]mZ [GeV]91.1875 ± 0.0021 91.1875 GZ [GeV]GZ [GeV]2.4952 ± 0.0023 2.4957 shad [nb]s041.540 ± 0.037 41.477 RlRl20.767 ± 0.025 20.744 AfbA0,l0.01714 ± 0.00095 0.01645 Al(Pt)Al(Pt) 0.1465 ± 0.0032 0.1481 RbRb0.21629 ± 0.00066 0.21586 RcRc0.1721 ± 0.0030 0.1722 AfbA0,b0.0992 ± 0.0016 0.1038 AfbA0,c0.0707 ± 0.0035 0.0742 AbAb0.923 ± 0.020 0.935 AcAc0.670 ± 0.027 0.668 Al(SLD)Al(SLD) 0.1513 ± 0.0021 0.1481 sin2qeffsin2qlept(Qfb) 0.2324 ± 0.0012 0.2314 mW [GeV]mW [GeV]80.398 ± 0.025 80.374 GW [GeV] GW [GeV]2.140 ± 0.060 2.091 mt [GeV]mt [GeV]170.9 ± 1.8 171.3 Fig.25: Comparisonbetweenthemeasurementsincluded in the combined analysis of the SM and the results from the global electroweak ﬁt[29,30]. Takingalldirectandindirectdataintoaccount,oneobtain sthebestconstraintson MH. Theglobal electroweak ﬁtresults inthe ∆χ2=χ2−χ2 mincurveshowninFig.24. Thelowerlimiton MHobtained from direct searches isclose tothe point of minimum χ2. At95% C.L.,one gets [29,30] 114.4 GeV< M H<144 GeV. (5.19) Theﬁtprovides also averyaccurate valueof thestrong coupl ing constant, αs(M2 Z) = 0.1186±0.0027, in very good agreement with the world average value αs(M2 Z) = 0.119±0.002[7,35]. The largest discrepancy between theory andexperiment occurs for A0,b FB,withtheﬁttedvaluebeing nearly 3σlarger than the measurement. Asshown in Fig.25, agood agreement is obtained for all other observables. 5.5 Gaugeself-interactions -e -e /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 g , Z e+- W+e -W +eZ Zne-e +e-W +W/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 Fig.26: Feynmandiagrams contributing to e+e−→W+W−ande+e−→ZZ. Attree level, the W-pair production process e+e−→W+W−involves three different contribu- tions(Fig.26),correspondingtotheexchangeof νe,γandZ. Thecross-sectionmeasuredatLEP2agrees very well with the SM predictions. As shown in Fig. 27, the νe-exchange contribution alone would lead toanunphysical growingofthecross-section atlargeenerg ies and,therefore, wouldimplyaviolation of unitarity. Adding the γ-exchange contribution softens this behaviour, but a clear disagreement with the data persists. The Z-exchange mechanism, which involves the ZWWvertex, appears to be crucial in order toexplain the data. 27 0102030 160 180 200 √s (GeV)sWW (pb) YFSWW/RacoonWW no ZWW vertex (Gentle) only ne exchange (Gentle)LEP PRELIMINARY17/02/2005 00.51 180 190 200 √s (GeV)sZZ (pb) ZZTOandYFSZZLEPPRELIMINARY11/07/2003 Fig. 27: Measured energy dependence of σ(e+e−→W+W−)(left) and σ(e+e−→ZZ)(right). The three curves shown for the W-pair production cross-section correspond to only the νe-exchange contribution (upper curve), νeexchange plusphotonexchange(middlecurve)andallcontributionsi ncludingalsothe ZWWvertex(lowercurve). Onlythe e-exchange mechanism contributes to Z–pair production [29,30]. SincetheZiselectrically neutral, itdoesnotinteract withthephoto n. Moreover, theSMdoesnot includeanylocal ZZZvertex. Therefore, the e+e−→ZZcross-section onlyinvolvesthecontribution fromeexchange. The agreement of the SM predictions with the exper imental measurements in both production channels, W+W−andZZ, provides a test of the gauge self-interactions. There is a c lear signal of the presence of a ZWWvertex, with the predicted strength, and no evidence for any γZZor ZZZinteractions. The gauge structure of the SU(2)L⊗U(1)Ytheory isnicely conﬁrmed bythe data. 5.6 Higgs decays 1 50 100 200 500 100010Ð1 10Ð2 10Ð3 MH [GeV]BR(H) WW ZZ Zgggt+tÐ ggttccbb 1 50 100 200 500 100010Ð1 10Ð2 10Ð3102 10 MH [GeV]G(H) [GeV] Fig. 28: Branching fractions of the different Higgs decay mo des (left) and total decay width of the Higgs boson (right) as function of MH[44]. The couplings of the Higgs boson are always proportional to s ome mass scale. The Hf¯finter- action grows linearly with the fermion mass, while the HWWandHZZvertices are proportional to M2 WandM2 Z, respectively. Therefore, the most probable decay mode of t he Higgs will be the one into the heaviest possible ﬁnal state. This is clearly illustrat ed in Fig. 28. The H→b¯bdecay channel is by far the dominant one below the W+W−production threshold. When MHis large enough to al- low the production of a pair of gauge bosons, H→W+W−andH→ZZbecome dominant. For 28 MH>2mt, theH→t¯tdecay width is also sizeable, although smaller than the WWandZZones because of the different dependence of the corresponding Hi ggs coupling with the mass scale (linear instead of quadratic). Thetotal decay widthof the Higgsgrowswithincreasing valu es ofMH. Theeffect isverystrong abovetheW+W−production threshold. AheavyHiggsbecomesthenverybroad . AtMH∼600 GeV , thewidthisaround 100 GeV ;whileforMH∼1 TeV,ΓHisalready ofthesamesizeastheHiggsmass itself. The design of the LHC detectors has taken into account all the se very characteristic properties in order tooptimize the future search for the Higgs boson. 6 Flavour Dynamics Wehave learnt experimentally that there aresix different q uark ﬂavours u,d,s,c,b,t, three different charged leptons e,µ,τand their corresponding neutrinos νe,νµ,ντ. We can nicely include all these particles into the SM framework, by organizing them in to three families of quarks and leptons, as indicated in Eqs. (1.1) and (1.2). Thus, we have three nearly identical copies of the same SU(2)L⊗ U(1)Ystructure, with masses asthe only difference. Let us consider the general case of NGgenerations of fermions, and denote ν′ j,l′ j,u′ j,d′ jthe membersoftheweakfamily j(j= 1,...,N G),withdeﬁnitetransformation properties under thegauge group. Owing to the fermion replication, a large variety of f ermion-scalar couplings are allowed by the gauge symmetry. Themost general YukawaLagrangian has the f orm LY=−/summationdisplay jk/braceleftbigg/parenleftbig ¯u′ j,¯d′ j/parenrightbig L/bracketleftbigg c(d) jk/parenleftbiggφ(+) φ(0)/parenrightbigg d′ kR+c(u) jk/parenleftbiggφ(0)∗ −φ(−)/parenrightbigg u′ kR/bracketrightbigg +/parenleftbig ¯ν′ j,¯l′ j/parenrightbig Lc(l) jk/parenleftbiggφ(+) φ(0)/parenrightbigg l′ kR/bracerightbigg + h.c., (6.1) wherec(d) jk,c(u) jkandc(l) jkare arbitrary coupling constants. After SSB,the YukawaLagrangian can bewritten as LY=−/parenleftbigg 1 +H v/parenrightbigg/braceleftbig d′ LM′ dd′ R+u′ LM′ uu′ R+l′ LM′ ll′ R+ h.c./bracerightbig . (6.2) Here,d′,u′andl′denote vectors in the NG-dimensional ﬂavour space, and the corresponding mass matrices aregiven by (M′ d)ij≡c(d) ijv√ 2,(M′ u)ij≡c(u) ijv√ 2,(M′ l)ij≡c(l) ijv√ 2. (6.3) The diagonalization of these mass matrices determines the m ass eigenstates dj,ujandlj, which are linear combinations of the corresponding weak eigenstates d′ j,u′ jandl′ j, respectively. Thematrix M′ dcanbedecomposed as2M′ d=HdUd=S† dMdSdUd,where Hd≡/radicalig M′ dM′† d is an Hermitian positive-deﬁnite matrix, while Udis unitary. Hdcan be diagonalized by a unitary matrix Sd; the resulting matrix Mdis diagonal, Hermitian and positive deﬁnite. Similarly, on e has M′ u=HuUu=S† uMuSuUuandM′ l=HlUl=S† lMlSlUl. In terms of the diagonal mass 2The condition detM′ f/negationslash= 0(f=d, u, l) guarantees that the decomposition M′ f=HfUfis unique: Uf≡H−1 fM′ f. The matrices Sfare completely determined (up to phases) only if all diagona l elements of Mfare different. If there is some degeneracy, the arbitrariness of Sfreﬂects the freedom to deﬁne the physical ﬁelds. If detM′ f= 0, the matrices UfandSf are not uniquely determined, unless their unitarityisexpl icitlyimposed. 29 uidj i jV Wu c t d s b Fig.29: Flavour-changingtransitions through the charged -current couplings of the W±bosons. matrices Md= diag(md,ms,mb,...),Mu= diag(mu,mc,mt,...),Ml= diag(me,mµ,mτ,...), (6.4) the YukawaLagrangian takes the simpler form LY=−/parenleftbigg 1 +H v/parenrightbigg/braceleftbig dMdd+uMuu+lMll/bracerightbig , (6.5) where the masseigenstates are deﬁned by dL≡Sdd′ L, uL≡Suu′ L, lL≡Sll′ L, dR≡SdUdd′ R,uR≡SuUuu′ R,lR≡SlUll′ R. (6.6) Note, that the Higgs couplings are proportional to thecorre sponding fermions masses. Since, f′ Lf′ L=fLfLandf′ Rf′ R=fRfR(f=d,u,l),theformoftheneutral-current partofthe SU(2)L⊗U(1)YLagrangian does not change when expressed interms of mass ei genstates. Therefore, there are no ﬂavour-changing neutral currents in the SM (GIM mechanism [5]). This is a consequence of treating all equal-charge fermions on thesame footing. However, u′ Ld′ L=uLSuS† ddL≡uLVdL. In general, Su∝ne}ationslash=Sd; thus, if one writes the weak eigenstates in terms of mass eigenstates, a NG×NGunitary mixing matrix V, called the Cabibbo– Kobayashi–Maskawa (CKM)matrix [45,46], appears in thequa rk charged-current sector: LCC=−g 2√ 2  W† µ /summationdisplay ij¯uiγµ(1−γ5)Vijdj+/summationdisplay l¯νlγµ(1−γ5)l + h.c.  .(6.7) Thematrix Vcouples any‘up-type’ quark with all ‘down-type’ quarks (Fi g. 29). If neutrinos are assumed to be massless, we can always redeﬁn e the neutrino ﬂavours, in such a way as to eliminate the analogous mixing in the lepton secto r:ν′ Ll′ L=ν′ LS† llL≡νLlL. Thus, we have lepton-ﬂavour conservation in the minimal SM withou t right-handed neutrinos. If sterile νR ﬁelds are included in the model, one would have an additional Yukawa term in Eq. (6.1), giving rise to a neutrino mass matrix (M′ ν)ij≡c(ν) ijv/√ 2. Thus, the model could accommodate non-zero neutrino masses and lepton-ﬂavour violation through a lepton mixing matrix VLanalogous to the one present in the quark sector. Note, however, that the total lepton num berL≡Le+Lµ+Lτwould still be conserved. Weknowexperimentally thatneutrino massesare tinyandtherearestrongboundsonlepton- ﬂavour violating decays: Br(µ±→e±e+e−)<1.0·10−12[47],Br(µ±→e±γ)<1.2·10−11[48], Br(τ±→µ±γ)<4.5·10−8[49,50] ... However, we do have a clear evidence of neutrino o scillation phenomena. The fermion masses and the quark mixing matrix Vare all determined by the Yukawa couplings in Eq. (6.1). However, the coefﬁcients c(f) ijare not known; therefore we have a bunch of arbitrary parameters. Ageneral NG×NGunitarymatrixischaracterized by N2 Grealparameters: NG(NG−1)/2 30 moduli and NG(NG+ 1)/2phases. Inthecase of V,manyof these parameters areirrelevant, because we can always choose arbitrary quark phases. Under the phase redeﬁnitions ui→eiφiuianddj→ eiθjdj, the mixing matrix changes as Vij→Vijei(θj−φi); thus, 2NG−1phases are unobservable. The number of physical free parameters in the quark-mixing m atrix then gets reduced to (NG−1)2: NG(NG−1)/2moduli and (NG−1)(NG−2)/2phases. In the simpler case of two generations, Vis determined by a single parameter. Onethen recovers the Cabibbo rotation matrix [45] V=/parenleftigg cosθCsinθC −sinθCcosθC/parenrightigg . (6.8) WithNG= 3, the CKM matrix is described by three angles and one phase. Di fferent (but equivalent) representations can be found in the literature. The Particl e data Group [7] advocates the use of the following one asthe ‘standard’ CKMparametrization: V= c12c13 s12c13 s13e−iδ13 −s12c23−c12s23s13eiδ13c12c23−s12s23s13eiδ13s23c13 s12s23−c12c23s13eiδ13−c12s23−s12c23s13eiδ13c23c13 .(6.9) Herecij≡cosθijandsij≡sinθij, withiandjbeing ‘generation’ labels ( i,j= 1,2,3). The real anglesθ12,θ23andθ13canallbemadetolieintheﬁrstquadrant, byanappropriate r edeﬁnition ofquark ﬁeldphases; then, cij≥0,sij≥0and0≤δ13≤2π. Notice thatδ13istheonly complex phase inthe SMLagrangian. Therefore, it isthe only possible source of CP-violation phenomena. In fact, it was for this reason that th e third generation was assumed toexist [46], before thediscovery ofthe bandtheτ. Withtwogenerations, theSMcouldnot explainthe observed CPviolation in the Ksystem. 6.1 Quarkmixing W + W + c c d , s d , s e , + m + n e n m , u d , s _ _ Fig. 30: Determinations of Vijare done insemileptonic quark decays (left),where a single quark current is present. Hadronic decaymodes(right)involvetwodifferentquarkcurrentsan daremoreaffectedbyQCDeffects(gluonscancoupleeverywh ere). Our knowledge of the charged-current parameters is unfortu nately not so good as in the neutral- current case. In order to measure the CKM matrix elements, on e needs to study hadronic weak decays of the type H→H′l−¯νlorH→H′l+νl, which are associated with the corresponding quark transitionsdj→uil−¯νlandui→djl+νl(Fig. 30). Since quarks are conﬁned within hadrons, the decay amplitude T[H→H′l−¯νl]≈GF√ 2Vij∝an}b∇acketle{tH′|¯uiγµ(1−γ5)dj|H∝an}b∇acket∇i}ht/bracketleftbig¯lγµ(1−γ5)νl/bracketrightbig (6.10) always involves an hadronic matrix element of the weak left c urrent. The evaluation of this matrix element isanon-perturbative QCDproblem, which introduce s unavoidable theoretical uncertainties. 31 One usually looks for a semileptonic transition where the ma trix element can be ﬁxed at some kinematical point by a symmetry principle. This has the virt ue of reducing the theoretical uncertainties tothelevelofsymmetry-breaking corrections andkinemati cal extrapolations. Thestandard exampleisa 0−→0−decay such as K→πlν,D→KlνorB→Dlν. Only the vector current can contribute inthis case: ∝an}b∇acketle{tP′(k′)|¯uiγµdj|P(k)∝an}b∇acket∇i}ht=CPP′/braceleftbig (k+k′)µf+(t) + (k−k′)µf−(t)/bracerightbig .(6.11) Here,CPP′is a Clebsh–Gordan factor and t= (k−k′)2≡q2. The unknown strong dynamics is fully contained in the form factors f±(t). In the limit of equal quark masses, mui−mdj= 0, the divergenceofthevectorcurrentiszero; thus qµ(¯uiγµdj) = 0,whichimplies f−(t) = 0and,moreover, f+(0) = 1 to all orders in the strong coupling because the associated ﬂ avour charge is a conserved quantity.3Therefore, one only needs to estimate the corrections induc ed by the quark massdifferences. Sinceqµ/bracketleftbig¯lγµ(1−γ5)νl/bracketrightbig ∼ml, the contribution of f−(t)is kinematically suppressed in the electron and muonmodes. Thedecay width can then be written a s Γ(P→P′lν) =G2 FM5 P 192π3|Vij|2C2 PP′|f+(0)|2I(1 +δRC), (6.12) whereδRCis an electroweak radiative correction factor and Idenotes a phase-space integral, which in theml= 0limit takes the form I ≈/integraldisplay(MP−MP′)2 0dt M8 Pλ3/2(t,M2 P,M2 P′)/vextendsingle/vextendsingle/vextendsingle/vextendsinglef+(t) f+(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (6.13) Theusual procedure todetermine |Vij|involves three steps: 1. Measure the shape of the tdistribution. Thisﬁxes |f+(t)/f+(0)|and therefore determines I. 2. Measure the total decay width Γ. SinceGFis already known from µdecay, one gets then an experimental value for the product |f+(0)||Vij|. 3. Get atheoretical prediction for f+(0). It isimportant torealize that theoretical input isalways n eeded. Thus, the accuracy of the |Vij|determi- nation islimited byour ability to calculate therelevant ha dronic input. The conservation of the vector and axial-vector QCD current s in the massless quark limit allows for accurate determinations of the light-quark mixings |Vud|and|Vus|. The present values are shown in Table 4, which takes into account the recent changes in the K→πe+νedata [7,34] and the new |Vus|determinations from Cabibbo suppressed tau decays [52] and from the ratio of decay amplitudes Γ(K+→µ+¯νµ)/Γ(π+→µ+¯νµ)[53–55]. Since |Vub|2is tiny, these two light quark entries provide a sensible test of the unitarity of the CKMmatrix: |Vud|2+|Vus|2+|Vub|2= 0.9980±0.0012. (6.14) It is important tonotice that at the quoted level of uncertai nty radiative corrections play acrucial role. In the limit of very heavy quark masses, QCD has additional sy mmetries [56–59] which can be used to make rather precise determinations of |Vcb|, either from exclusive decays such as B→D∗l¯νl [60,61] or from the inclusive analysis of b→cl¯νltransitions. The control of theoretical uncertainties is much more difﬁcult for |Vub|,|Vcd|and|Vcs|, because the symmetry arguments associated with the light and heavy quark limits get corrected bysizeable symme try-breaking effects. 3Thisiscompletelyanalogoustotheelectromagneticcharge conservationinQED.Theconservationoftheelectromagnet ic current implies that the proton electromagnetic form facto r does not get any QED or QCD correction at q2= 0and, therefore, Q(p) = 2Q(u) +Q(d) =|Q(e)|. Adetailed proof can be found inRef.[51]. 32 Table 4: Direct determinations of the CKMmatrixelements Vij. For|Vtb|, 95% C.L.limitsare given. CKMentry Value Source |Vud| 0.97377±0.00027 Nuclearβdecay [7] 0.9746±0.0019 n→pe−¯νe[7] 0.9728±0.0030 π+→π0e+νe[62] 0.97378±0.00027 average |Vus| 0.2234±0.0024 K→πl+νl[7,34,63] 0.2220±0.0033 τdecays [52] 0.2226+ 0.0026 −0.0014K+/π+→µ+νµ,Vud[7,53–55] 0.226±0.005 Hyperon decays [64–66] 0.2230±0.0015 average |Vcd| 0.213±0.022 D→πl¯νl[7] 0.230±0.011 νd→cX[7] 0.227±0.010 average |Vcs| 0.957±0.095 D→Kl¯νl[7] 0.94+ 0.35 −0.29 W+→c¯s[7] 0.974±0.013W+→had.,Vuj,Vcd,Vcb[29,30] |Vcb| 0.0392±0.0016 B→D∗l¯νl[7,67] 0.0417±0.0007 b→cl¯νl[7,67] 0.0413±0.0006 average |Vub| 0.0039±0.0006 B→πl¯νl[7,67] 0.0045±0.0003 b→ul¯νl[7,67] 0.0044±0.0003 average |Vtb|//radicalig/summationtext q|Vtq|2>0.78 t→bW/qW [68,69] |Vtb| >0.68 ; ≤1 p¯p→tb+X[70] The most precise determination of |Vcd|is based on neutrino and antineutrino interactions. The difference of the ratio of double-muon to single-muon produ ction by neutrino and antineutrino beams is proportional to the charm cross-section off valence dquarks and, therefore, to |Vcd|. A direct determi- nation of |Vcs|can be also obtained from charm-tagged Wdecays at LEP2. Moreover, the ratio of the total hadronic decay width of the Wto theleptonic one provides thesum [29,30] /summationdisplay i=u,c j=d,s,b|Vij|2= 1.999±0.025. (6.15) Although much less precise than Eq. (6.14), this result test unitarity at the 1.25% level. From Eq. (6.15) onecanalsoobtainatighterdeterminationof |Vcs|,usingtheexperimentalknowledgeontheotherCKM matrix elements, i.e., |Vud|2+|Vus|2+|Vub|2+|Vcd|2+|Vcb|2= 1.0512±0.0058. This gives the most accurate and ﬁnal value of |Vcs|quoted in Table4. 33 The measured entries of the CKM matrix show a hierarchical pa ttern, with the diagonal elements being very close to one, the ones connecting the twoﬁrst gene rations having asize λ≈ |Vus|= 0.2230±0.0015, (6.16) the mixing between the second and third families being of ord erλ2, and the mixing between the ﬁrst and third quark generations having a much smaller size of abo utλ3. It is then quite practical to use the approximate parametrization [71]: V= 1−λ2 2λ Aλ3(ρ−iη) −λ 1−λ2 2Aλ2 Aλ3(1−ρ−iη)−Aλ21 +O/parenleftbig λ4/parenrightbig , (6.17) where A≈|Vcb| λ2= 0.831±0.014,/radicalbig ρ2+η2≈/vextendsingle/vextendsingle/vextendsingle/vextendsingleVub λVcb/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.478±0.033.(6.18) Deﬁning to all orders in λ[72]s12≡λ,s23≡Aλ2ands13e−iδ13≡Aλ3(ρ−iη), Eq. (6.17) just corresponds toa Taylor expansion of Eq.(6.9) in powers of λ. 6.2 CPViolation 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 symme try (left-handed fermions ↔right-handed antifermions). In fact, CPappears to be a symmetry of nearly all observed phenomena. Ho wever, a slight violation ofthe CPsymmetryatthelevelof 0.2%isobserved intheneutral kaonsystem andmore sizeable signals of CPviolation have been recently established at the B factories . Moreover, the huge matter–antimatter asymmetry present in our Universe is a cl ear manifestation of CPviolation and its important role in the primordial baryogenesis. TheCPTtheorem guarantees that the product of the three discrete tr ansformations is an exact symmetry of any local and Lorentz-invariant quantum ﬁeld th eory preserving micro-causality. There- fore, a violation of CPrequires a corresponding violation of time reversal. Since Tis an antiunitary transformation, this requires the presence of relative com plex phases between different interfering am- plitudes. The electroweak SM Lagrangian only contains a single comple x phaseδ13(η). This is the sole possible source of CPviolation and, therefore, the SM predictions for CP-violating phenomena are quite constrained. The CKM mechanism requires several nece ssary conditions in order to generate an observable CP-violation effect. Withonlytwofermiongenerations, theq uarkmixingmechanismcannot giveriseto CPviolation; therefore,for CPviolationtooccurinaparticularprocess,allthreegenera tions arerequired toplayanactiverole. Inthekaonsystem, forin stance, CP-violation effectscanonlyappear attheone-loop level, wherethetopquark ispresent. Inaddi tion, all CKMmatrixelements mustbenon- zero and the quarks of a given charge must be non-degenerate i n mass. If any of these conditions were not satisﬁed, the CKM phase could be rotated away by a redeﬁni tion of the quark ﬁelds. CP-violation effects arethennecessarily proportional totheproduct of allCKMangles, andshould vanish inthelimit where anytwo (equal-charge) quark masses aretaken tobe equ al. All these necessary conditions can be summarized in a very elegant way as a single requirement on th e original quark mass matrices M′ uand M′ d[73]: CPviolation ⇐⇒ Im/braceleftig det/bracketleftig M′ uM′† u,M′ dM′† d/bracketrightig/bracerightig ∝ne}ationslash= 0. (6.19) 34 Without performing any detailed calculation, one can make t he following general statements on the implications of theCKMmechanism of CPviolation: – Owingto unitarity, for any choice of i,j,k,l(between 1and 3), Im/bracketleftbig VijV∗ ikVlkV∗ lj/bracketrightbig =J3/summationdisplay m,n=1ǫilmǫjkn, (6.20) J=c12c23c2 13s12s23s13sinδ13≈A2λ6η <10−4. (6.21) AnyCP-violation observable involves the product J[73]. Thus, violations of the CPsymmetry are necessarily small. – In order to have sizeable CP-violating asymmetries A ≡(Γ−Γ)/(Γ +Γ), one should look for very suppressed decays, wherethe decay widths already invo lve small CKMmatrix elements. – IntheSM, CPviolationisalow-energyphenomenon, inthesensethatanye ffectshoulddisappear when the quark mass difference mc−mubecomes negligible. –Bdecays are the optimal place for CP-violation signals to show up. They involve small CKM matrixelementsandarethelowest-massprocesses wherethe threequark generations playadirect (tree-level) role. The SM mechanism of CPviolation is based on the unitarity of the CKM matrix. Testin g the constraints implied by unitarity is then a way to test the sou rce ofCPviolation. The unitarity tests in Eqs. (6.14) and (6.15) involve only the moduli of the CKM para meters, while CPviolation has to do withtheir phases. More interesting are the off-diagonal un itarity conditions: V∗ udVus+V∗ cdVcs+V∗ tdVts= 0, (6.22) V∗ usVub+V∗ csVcb+V∗ tsVtb= 0, (6.23) V∗ ubVud+V∗ cbVcd+V∗ tbVtd= 0. (6.24) These relations can be visualized by triangles in a complex p lane which, owing to Eq. (6.20), have the same area |J |/2. In the absence of CPviolation, these triangles would degenerate into segments along the real axis. In the ﬁrst two triangles, one side is much shorter than the ot her two (the Cabibbo suppression factors of the three sides are λ,λandλ5in the ﬁrst triangle, and λ4,λ2andλ2in the second one). This is why CPeffects are so small for Kmesons (ﬁrst triangle), and why certain asymmetries in Bsdecays are predicted to be tiny (second triangle). The third triang le looks more interesting, since the three sides have a similar size of about λ3. They are small, which means that the relevant b-decay branching ratios are small, but once enough Bmesons have been produced, the CP-violation asymmetries are sizeable. The present experimental constraints on this triangle are s hown in Fig. 31, where it has been scaled by dividing its sides by V∗ cbVcd. Thisaligns one side of the triangle along thereal axis and m akes itslength equal to1; thecoordinates of the3 vertices arethen (0,0),(1,0)and(¯ρ,¯η)≡(1−λ2/2)(ρ,η). Onesideoftheunitaritytrianglehasbeenalreadydetermin edinEq.(6.18)fromtheratio |Vub/Vcb|. The other side can be obtained from the measured mixing betwe en theB0 dand¯B0 dmesons (Fig. 32), ∆Md= 0.507±0.004 ps−1[67], which ﬁxes |Vtb|. Additional information has been provided by the recent observation of B0 s–¯B0 soscillations at CDF, implying ∆Ms= 17.77±0.12 ps−1[74]. From the experimental ratio ∆Md/∆Ms= 0.0286±0.0003, one obtains |Vtd|/|Vts|. A more direct constraint on the parameter ηis given by the observed CPviolation in K0→2πdecays. The measured value of |εK|= (2.232±0.007)·10−3[7] determines the parabolic region shown inFig. 31. B0decays into CPself-conjugate ﬁnal states provide independent ways to det ermine the angles of the unitarity triangle [75,76]. The B0(or¯B0) can decay directly to the given ﬁnal state f, or do 35 r-0.4 -0.2 0 0.2 0.4 0.6 0.8 1h 00.10.20.30.40.50.6 a bg r-0.4 -0.2 0 0.2 0.4 0.6 0.8 1h 00.10.20.30.40.50.6 BEAUTY 2006CKM f i t t e rg g aadmDdmD & smD KeKe cb/VubVbsin2 < 0b sol. w/ cos2 (excl. at CL > 0.95)excluded area has CL > 0.95 Fig.31: Experimental constraints on the SMunitaritytrian gle [77]. q bu, c, t q bu, c, tWWq b W qbu, c, t u, c, tW Fig. 32: B0–¯B0mixing diagrams. Owing to the unitarity of the CKM matrix, th e mixing vanishes for equal up-type quark masses (GIM mechanism). The mixing amplitude is then propor tional to the mass (squared) splittings between the u,candt quarks, andis completelydominated bythe topcontribution . it after the meson has been changed to its antiparticle via th e mixing process. CP-violating effects can then result from the interference of these two contribut ions. The time-dependent CP-violating rate asymmetries contain direct information on the CKM paramete rs. The gold-plated decay mode is B0 d→ J/ψK S, which gives a clean measurement of β≡ −arg(VcdV∗ cb/VtdV∗ tb), without strong-interaction uncertainties. Including theinformation obtained from ot herb→c¯csdecays, one gets [67]: sin 2β= 0.68±0.03. (6.25) Manyadditional testsoftheCKMmatrixfromdifferent Bdecaymodesarebeingpursuedatthe B factories. Determinations of the other two angles of the uni tarity triangle, α≡ −arg(VtdV∗ tb/VudV∗ ub) andγ≡ −arg(VudV∗ ub/VcdV∗ cb), have been already obtained [67,78], and are included inthe global ﬁt shown in Fig. 31 [77,79]. Complementary and very valuable in formation could be also obtained from the kaon decay modes K±→π±ν¯ν,KL→π0ν¯νandKL→π0e+e−[80]. 6.3 Leptonmixing The so-called ‘solar neutrino problem’ has been a long-stan ding question, since the very ﬁrst chlorine experiment attheHomestake mine[81]. Theﬂuxof solar νeneutrinos reaching theEarthhasbeen mea- suredbyseveralexperimentstobesigniﬁcantly belowthest andardsolarmodelprediction[82]. Morere- cently, theSudbury Neutrino Observatory hasprovided stro ng evidence that neutrinos dochange ﬂavour as they propagate from the core of the Sun [83], independentl y of solar model ﬂux predictions. SNO is able to detect neutrinos through three different reactio ns: the charged-current process νed→e−pp which is only sensitive to νe, the neutral current transition νxd→νxpnwhich has equal probability for all active neutrino ﬂavours, and the elastic scattering νxe−→νxe−which isalso sensitive to νµandντ, although thecorresponding cross section isafactor 6.48smaller thanthe νeone. Themeasured neutrino 36 )-1 s-2 cm6 10· (ef0 0.5 1 1.5 2 2.5 3 3.5)-1 s-2 cm6 10· (tmf 0123456 68% C.L.CCSNOf 68% C.L.NCSNOf 68% C.L.ESSNOf 68% C.L.ESSKf 68% C.L.SSMBS05f 68%, 95%, 99% C.L.tmNCf Fig.33: Measured ﬂuxes of8Bsolar neutrinos of νµorντtype (φµ,τ)versus the ﬂuxof νe(φe)[83]. ﬂuxes, shown in Fig. 33, demonstrate the existence of a non- νecomponent in the solar neutrino ﬂux at the 5.3σlevel. The SNO results are in good agreement with the Super-K amiokande solar measure- ments[84]andhavebeenfurther reinforced withthemorerec ent KamLANDdata,showingthat ¯νefrom nuclear reactors disappear over distances of about 180 Km[8 5]. Another evidence of oscillations has been obtained from atm ospheric neutrinos. The known dis- crepancy between the experimental observations and the pre dicted ratio of muon to electron neutrinos has become much stronger with the high precision and large st atistics of Super-Kamiokande [86]. The atmospheric anomaly appears to originate in a reduction of t heνµﬂux, and the data strongly favours theνµ→ντhypothesis. This result has been conﬁrmed by K2K [87] and MIN OS [88], observing the disappearance of accelerator νµ’s at distances of 250 and 735 Km, respectively. Super-Kamio kande has recently reported statistical evidence of ντappearance at the 2.4σlevel [86]. Thedirect detection of the producedντisthe maingoal of the ongoing CERNtoGran Sasso neutrino pro gram. Thus, we have now clear experimental evidence that neutrino s are massive particles and there is mixing in the lepton sector. Figures 34 and 35 show the presen t information on neutrino oscillations, from solar, atmospheric, accelerator and reactor neutrino data. A global analysis, combining the full set of data, leads tothe following preferred ranges for the osci llation parameters [7]: ∆m2 21=/parenleftbig 8.0+ 0.4 −0.3/parenrightbig ·10−5eV2,1.9·10−3<|∆m2 32|/eV2<3.0·10−3,(6.26) sin2(2θ12) = 0.86+ 0.03 −0.04,sin2(2θ23)>0.92,sin2(2θ13)<0.19,(6.27) where ∆m2 ij≡m2 i−m2 jare the mass squared differences between the neutrino mass e igenstatesνi,j andθijthe corresponding mixing angles in the standard three-ﬂavo ur parametrization [7]. The ranges indicate 90% C.L. bounds. In the limit θ13= 0, solar and atmospheric neutrino oscillations decouple because ∆m2 ⊙≪∆m2 atm. Thus, ∆m2 21,θ12andθ13are constrained by solar data, while atmospheric experiments constrain ∆m2 32,θ23andθ13. The angleθ13is strongly constrained by the CHOOZreactor experiment [89]. Newplanned reactorexperiments, T2KandN OνAareexpectedtoachievesensitivities around sin2(2θ13)∼0.01. Non-zero neutrino masses constitute a clear indication of n ew physics beyond the SM. Right- handedneutrinosareanobviouspossibilitytoincorporate Diracneutrinomasses. However,the νiRﬁelds would beSU(3)C⊗SU(2)L⊗U(1)Ysinglets, without any SM interaction. If such objects do exi st, it would seem natural to expect that they are able to communicat e with the rest of the world through some still unknown dynamics. Moreover, the SM gauge symmetry wou ld allow for a right-handed Majorana 37 q2tan)2 eV-5 (102 mD 5101520 0 0.2 0.4 0.6 0.8 168% CL 95% CL 99.73% CL(b) Fig.34: Allowedregions for 2νoscillations forthe com- binationofsolar( νe)andKamLAND( ¯νe)data,assuming CPTsymmetry [83].)23q(22sin0.2 0.4 0.6 0.8 1.0)4/c2| (eV322mD| 1.52.02.53.03.54.0-310· 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. )23q(22sin0.2 0.4 0.6 0.8 1.0)4/c2| (eV322mD| 1.52.02.53.03.54.0-310· Fig. 35: MINOS allowed regions for νµdisappearance oscillations,comparedwithK2KandSuper-Kamiokande results [88]. neutrino massterm, LM=−1 2νc iRMijνjR+ h.c., (6.28) whereνc iR≡ C¯νT iRdenotes the charge-conjugated ﬁeld. The Majorana mass matr ixMijcould have an arbitrary size, because it is not related to the ordinary H iggs mechanism. Since both ﬁelds νiRand νc iRabsorbνand create ¯ν, the Majorana mass term mixes neutrinos and anti-neutrinos , violating lepton number by twounits. Clearly, new physics iscalled for. Adopting a more general effective ﬁeld theory language, wit hout 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)Yinvariant Lagrangian, in terms of the known low-energy ﬁeld s (left-handed neutrinos only). The SM is the unique answer with dimension four. The ﬁr st contributions from new physics ap- pear through dimension-5 operators, and have also a unique f orm which violates lepton number by two units [90]: ∆L=−cij Λ¯Li˜φ˜φtLc j+ h.c., (6.29) whereLidenotes thei-ﬂavouredSU(2)Llepton doublet, ˜φ≡iτ2φ∗andLc i≡ C¯LT i. Similar operators with quark ﬁelds are forbidden, due to their different hyper charges, while higher-dimension operators would be suppressed by higher powers of the new-physics scal eΛ. After SSB, ∝an}b∇acketle{tφ(0)∝an}b∇acket∇i}ht=v/√ 2,∆L generates a Majorana mass term for the left-handed neutrino s, with4Mij=cijv2/Λ. Thus, Majorana neutrino masses should be expected on general symmetry grou nds. Taking mν/greaterorsimilar0.05eV, as suggested byatmospheric neutrinodata,onegets Λ/cij/lessorsimilar1015GeV,amazinglyclosetotheexpectedscaleofGran Uniﬁcation. With non-zero neutrino masses, the leptonic charged-curre nt interactions involve a ﬂavour mix- ing matrix VL. The data on neutrino oscillations imply that all elements o fVLare large, except for (VL)e3<0.18; therefore the mixing among leptons appears to be very diffe rent from the one in the quark sector. The number of relevant phases characterizing the matrix VLdepends on the Dirac or Ma- jorana nature of neutrinos, because if one rotates aMajoran a neutrino by aphase, this phase will appear in its mass term which will no longer be real. With only three M ajorana (Dirac) neutrinos, the 3×3 matrixVLinvolves six (four) independent parameters: three mixing a ngles and three (one) phases. 4This relationgeneralizes the well-knownsee-saw mechanis m (mνL∼m2/Λ) [91,92]. 38 Table5: Bestpublished limits(90% C.L.)on lepton-ﬂavour- violatingdecays [7,49,50]. Br(µ−→e−γ)<1.2·10−11Br(µ−→e−2γ)<7.2·10−11Br(µ−→e−e−e+)<1.0·10−12 Br(τ−→µ−γ)<4.5·10−8Br(τ−→e−γ)<1.1·10−7Br(τ−→e−e−µ+)<1.1·10−7 Br(τ−→e−KS)<5.6·10−8Br(τ−→µ−KS)<4.9·10−8Br(τ−→µ+π−π−)<0.7·10−7 Br(τ−→Λπ−)<7.2·10−8Br(τ−→e−π0)<1.4·10−7Br(τ−→e−π+π−)<1.2·10−7 Br(τ−→µ−π0)<1.1·10−7Br(τ−→µ−η)<1.3·10−7Br(τ−→µ−e+µ−)<1.3·10−7 The smallness of neutrino masses implies a strong suppressi on of neutrinoless lepton-ﬂavour- violating processes, which can be avoided in models with oth er sources of lepton-ﬂavour violation, not related tomνi. Table 5 shows the best published limits on lepton-ﬂavour-v iolating decays. The B Fac- tories are pushing the experimental limits on neutrinoless τdecays beyond the 10−7level, increasing in a drastic way the sensitivity to new physics scales. Futur e experiments could push further some lim- its to the 10−9level, allowing to explore interesting and totally unknown phenomena. Complementary information will be provided by the MEG experiment, which wi ll search for µ+→e+γevents with a sensitivity of 10−13[93]. Therearealsoongoing projects atJ-PARCaimingtostu dyµ→econversions inmuonic atoms, at the 10−18level. At present, we still ignore whether neutrinos are Dirac or Ma jorana fermions. Another important question to be addressed in the future concerns the possibil ity of leptonic CP violation and its relevance for explaining the baryon asymmetry of our Universe through leptogenesis. 7 Summary TheSMprovides abeautiful theoretical frameworkwhichisa bletoaccommodate allourpresent knowl- edge on electroweak and strong interactions. It is able to ex plain any single experimental fact and, in some cases, it has successfully passed very precise tests at the 0.1% to 1% level. In spite of this im- pressivephenomenological success, theSMleavestoomanyu nanswered questions tobeconsidered asa completedescriptionofthefundamental forces. Wedonotun derstandyetwhyfermionsarereplicatedin three (and only three) nearly identical copies. Why the patt ern of masses and mixings is what it is? Are themasses theonly difference among thethree families? Wha t istheorigin of theSMﬂavour structure? Which dynamics is responsible for the observed CPviolation? In the gauge and scalar sectors, the SM Lagrangian contains o nly four parameters: g,g′,µ2and h. We can trade them by α,MZ,GFandMH; this has the advantage of using the three most precise experimental determinations to ﬁx the interaction. In any c ase, one describes a lot of physics with only four inputs. In the fermionic ﬂavour sector, however, the si tuation is very different. With NG= 3, we have 13 additional free parameters in the minimal SM: 9 fermi on masses, 3 quark mixing angles and 1 phase. Taking into account non-zero neutrino masses, we ha ve three more mass parameters plus the leptonic mixings: three angles and one phase (three phases) for Dirac (or Majorana) neutrinos. Clearly, this is not very satisfactory. The source of this pr oliferation of parameters is the set of unknown Yukawacouplings inEq.(6.1). Theorigin ofmasses a ndmixings, together withthereason for the existing family replication, constitute at present the main open problem in electroweak physics. The problemoffermionmassgenerationisdeeplyrelatedwithth emechanismresponsiblefortheelectroweak SSB.Thus, the origin of these parameters lies in themost obs cure part of the SMLagrangian: the scalar sector. Thedynamics of ﬂavour appears tobe ‘terra incognit a’ which deserves acareful investigation. The SM incorporates a mechanism to generate CPviolation, through the single phase naturally occurring in the CKM matrix. Although the present laborator y experiments are well described, this mechanism isunabletoexplainthematter–antimatter asymm etryofourUniverse. Afundamental expla- nation of the origin of CP-violating phenomena isstill lacking. 39 The ﬁrst hints of new physics beyond the SM have emerged recen tly, with convincing evidence of neutrino oscillations showing that νe→νµ,τandνµ→ντtransitions do occur. The existence of lepton-ﬂavour violation opens avery interesting window to unknown phenomena. The Higgs particle is the main missing block of the SM framewo rk. The successful tests of the SM quantum corrections with precision electroweak data con ﬁrm the assumed pattern of SSB, but do not prove the validity of the minimal Higgs mechanism embedd ed in the SM. The present experimental bounds (5.19) put the Higgs hunting within the reach of the ne w generation of detectors. The LHC should ﬁnd out whether such scalar ﬁeld indeed exists, eithe r conﬁrming the SM Higgs mechanism or discovering completely new phenomena. Manyinterestingexperimental signalsareexpectedtobese eninthenearfuture. Newexperiments willprobe theSMtoamuchdeeper levelofsensitivity andwil lexplorethefrontier ofitspossible exten- sions. Large surprises maywell beexpected, probably estab lishing the existence of new physics beyond the SMand offering clues tothe problems of massgeneration, fermion mixing and family replication. Acknowledgements I want to thank the organizers for the charming atmosphere of this school and all the students for their many 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 A BasicInputsfrom QuantumField Theory 1.1 Waveequations The classical Hamiltonian of a non-relativistic free parti cle is given by H=/vector p2/(2m). In quantum mechanics, energy and momentum correspond to operators act ing on the particle wave function. The substitutions H=i/planckover2pi1∂ ∂ tand/vector p=−i/planckover2pi1/vector∇lead then to the Schr¨ odinger equation: i/planckover2pi1∂ ∂tψ(/vector x,t) =−/planckover2pi12 2m/vector∇2ψ(/vector x,t). (A.1) Wecan write the energy and momentum operators in arelativis tic covariant way as pµ=i∂µ≡i∂ ∂xµ, where we have adopted the usual natural units convention /planckover2pi1=c= 1. The relation E2=/vector p2+m2 determines theKlein–Gordon equation for arelativistic fr ee particle: /parenleftbig 2+m2/parenrightbig φ(x) = 0, 2≡∂µ∂µ=∂2 ∂t2−/vector∇2. (A.2) The Klein–Gordon equation is quadratic on the time derivati ve because relativity puts the space and time coordinates on an equal footing. Let us investigate whether an equation linear in derivatives could exist. Relativistic covariance and dimensional anal ysis restrict its possible form to (iγµ∂µ−m)ψ(x) = 0. (A.3) Since the r.h.s. is identically zero, we can ﬁx the coefﬁcien t of the mass term to be −1; this just deter- mines the normalization of the four coefﬁcients γµ. Notice that γµshould transform as a Lorentz four- vector. The solutions of Eq. (A.3) should also satisfy the Kl ein–Gordon relation of Eq. (A.2). Applying an appropriate differential operator to Eq.(A.3), one can e asily obtain the wanted quadratic equation: −(iγν∂ν+m)(iγµ∂µ−m)ψ(x) = 0 ≡/parenleftbig 2+m2/parenrightbig ψ(x). (A.4) Termslinearinderivativescancelidentically, whilethet ermwithtwoderivativesreproducestheoperator 2≡∂µ∂µprovided the coefﬁcients γµsatisfy the algebraic relation {γµ,γν} ≡γµγν+γνγµ= 2gµν, (A.5) which deﬁnes the so-called Dirac algebra. Eq.(A.3) isknown asthe Dirac equation. Obviously thecomponents of thefour-vector γµcannot simplybenumbers. Thethree 2×2Pauli matrices satisfy/braceleftbig σi,σj/bracerightbig = 2δij, which is very close to the relation (A.5). The lowest-dimen sional solution tothe Dirac algebra isobtained with D= 4matrices. Anexplicit representation isgiven by: γ0=/parenleftbiggI20 0−I2/parenrightbigg , γi=/parenleftbigg0σi −σi0/parenrightbigg . (A.6) Thus,thewavefunction ψ(x)isacolumnvector withfourcomponents intheDiracspace. Th epresence of the Pauli matrices strongly suggests that it contains two components of spin1 2. A proper physical analysis of its solutions shows that the Dirac equation desc ribes simultaneously a fermion of spin1 2and its ownantiparticle [94]. It turns useful todeﬁne the following combinations of gamma matrices: σµν≡i 2[γµ,γν], γ 5≡γ5≡iγ0γ1γ2γ3=−i 4!ǫµνρσγµγνγργσ.(A.7) Inthe explicit representation (A.6), σij=ǫijk/parenleftbiggσk0 0σk/parenrightbigg , σ0i=i/parenleftbigg0σi σi0/parenrightbigg , γ 5=/parenleftbigg0I2 I20/parenrightbigg .(A.8) 41 Thematrixσijisthen related to the spin operator. Someimportant propert ies are: γ0γµγ0=γµ†, γ0γ5γ0=−γ5†=−γ5,{γ5,γµ}= 0,(γ5)2=I4.(A.9) Specially relevant for weakinteractions are the chirality projectors (PL+PR= 1) PL≡1−γ5 2, P R≡1 +γ5 2, P2 R=PR, P2 L=PL, P LPR=PRPL= 0,(A.10) which allow to decompose theDirac spinor inits left-handed and right-handed chirality parts: ψ(x) = [PL+PR]ψ(x)≡ψL(x) +ψR(x). (A.11) Inthe massless limit, the chiralities correspond to thefer mion helicities. 1.2 Lagrangian formalism The Lagrangian formulation of a physical system provides a c ompact dynamical description and makes it easier to discuss the underlying symmetries. Like in clas sical mechanics, the dynamics is encoded in the action S=/integraldisplay d4xL[φi(x),∂µφi(x)]. (A.12) The integral over the four space-time coordinates preserve s relativistic invariance. The Lagrangian den- sityLis a Lorentz-invariant functional of the ﬁelds φi(x)and their derivatives. The space integral L=/integraltext d3xLwould correspond to the usual non-relativistic Lagrangian . The principle of stationary action requires the variation δSof the action to be zero under small ﬂuctuations δφioftheﬁelds. Assumingthatthevariations δφiaredifferentiable andvanishoutsidesome bounded region of space-time (which allows an integration b y parts), the condition δS= 0determines the Euler–Lagrange equations of motion for the ﬁelds: ∂L ∂φi−∂µ/parenleftbigg∂L ∂(∂µφi)/parenrightbigg = 0. (A.13) One can easily ﬁnd appropriate Lagrangians to generate the K lein–Gordon and Dirac equations. Theyshould bequadratic ontheﬁeldsandLorentz invariant, whichdetermines their possible form upto irrelevant total derivatives. TheLagrangian L=∂µφ∗∂µφ−m2φ∗φ (A.14) describes a complex scalar ﬁeld without interactions. Both the ﬁeldφ(x)and its complex conjugate φ∗(x)satisfy the Klein–Gordon equation; thus, φ(x)describes a particle of mass mwithout spin and its antiparticle. Particles which are their own antipartic les (i.e., with no internal charges) have only one degree of freedom and are described through a real scalar ﬁeld. The appropriate Klein–Gordon Lagrangian isthen L=1 2∂µφ∂µφ−1 2m2φ2. (A.15) TheDirac equation can bederived from the Lagrangian densit y L=ψ(iγµ∂µ−m)ψ. (A.16) The adjoint spinor ψ(x) =ψ†(x)γ0closes the Dirac indices. The matrix γ0is included to guarantee the proper behaviour under Lorentz transformations: ψψis a Lorentz scalar, while ψγµψtransforms as afour-vector [94]. Therefore, Lis Lorentz invariant asit should. Using the decomposition (A.11) of the Dirac ﬁeld in its two ch iral components, the fermionic Lagrangian adopts the form: L=ψLiγµ∂µψL+ψRiγµ∂µψR−m/parenleftbig ψLψR+ψRψL/parenrightbig . (A.17) Thus, the twochiralities decouple if the fermion ismassles s. 42 1.3 Symmetries andconservation laws Let us assume that the Lagrangian of a physical system is inva riant under some set of continuous trans- formations φi(x)→φ′ i(x) =φi(x) +ǫδǫφi(x) +O(ǫ2), (A.18) i.e.,L[φi(x),∂µφi(x)] =L[φ′ i(x),∂µφ′ i(x)]. Oneﬁndsthen that δǫL= 0 =/summationdisplay i/braceleftbigg/bracketleftbigg∂L ∂φi−∂µ/parenleftbigg∂L ∂(∂µφi)/parenrightbigg/bracketrightbigg δǫφi+∂µ/bracketleftbigg∂L ∂(∂µφi)δǫφi/bracketrightbigg/bracerightbigg . (A.19) If the ﬁelds satisfy the Euler–Lagrange equations of motion (A.13), the ﬁrst term is identically zero; therefore the system has aconserved current: Jµ≡/summationdisplay i∂L ∂(∂µφi)δǫφi, ∂µJµ= 0. (A.20) Thisallows us todeﬁne aconserved charge Q ≡/integraldisplay d3xJ0. (A.21) Thecondition ∂µJµ= 0guarantees thatdQ dt= 0, i.e., that Qisa constant of motion. This result, known as Noether’s theorem, can be easily exten ded to general transformations in- volving also the space-time coordinates. For every continu ous symmetry transformation which leaves theLagrangianinvariant, thereisacorresponding diverge nceless Noether’scurrentand,therefore, acon- served charge. The selection rules observed in Nature, wher e there exist several conserved quantities (energy, momentum, angular momentum, electric charge, etc .), correspond to dynamical symmetries of the Lagrangian. 1.4 Classical electrodynamics Thewell-known Maxwell equations, /vector∇ ·/vectorB= 0, /vector∇ ×/vectorE+∂/vectorB ∂t= 0, (A.22) /vector∇ ·/vectorE=ρ, /vector∇ ×/vectorB−∂/vectorE ∂t=/vectorJ, (A.23) summarize alarge amount of experimental and theoretical wo rk and provide a uniﬁed description of the electric andmagnetic forces. Theﬁrsttwoequations in(A.2 2)areeasily solved, writingtheelectromag- netic ﬁelds in termsof potentials: /vectorE=−/vector∇V−∂/vectorA ∂t, /vectorB=/vector∇ ×/vectorA. (A.24) It is very useful to rewrite these equations in a Lorentz cova riant notation. The charge density ρ and the electromagnetic current /vectorJtransform as a four-vector Jµ≡/parenleftig ρ,/vectorJ/parenrightig . The same is true for the potentials whichcombine into Aµ≡/parenleftig V,/vectorA/parenrightig . Therelations (A.24) betweenthe potentials andthe ﬁelds then take avery simple form, which deﬁnes the ﬁeldstrength t ensor: Fµν≡∂µAν−∂νAµ= 0−E1−E2−E3 E10−B3B2 E2B30−B1 E3−B2B10 , ˜Fµν≡1 2ǫµνρσFρσ.(A.25) 43 Intermsofthetensor Fµν,thecovariant formoftheMaxwell equations turnsout tobev erytransparent: ∂µ˜Fµν= 0, ∂ µFµν=Jν. (A.26) Theelectromagnetic dynamics isclearly arelativistic phe nomenon, but Lorentz invariance wasnot very explicit in the original formulation of Eqs. (A.22) and (A.2 3). Once a covariant formulation is adopted, the equations become much simpler. The conservation of the e lectromagnetic current appears now as a natural compatibility condition: ∂νJν=∂ν∂µFµν= 0. (A.27) Interms of potentials, ∂µ˜Fµνisidentically zero while ∂µFµν=Jνadopts the form: 2Aν−∂ν(∂µAµ) =Jν. (A.28) The same dynamics can be described by many different electro magnetic four-potentials, which givethesameﬁeldstrengthtensor Fµν. Thus,theMaxwellequations areinvariant under gaugetran sfor- mations: Aµ−→A′µ=Aµ+∂µΛ. (A.29) Taking the Lorentz gauge ∂µAµ= 0,Eq. (A.28) simpliﬁes to 2Aν=Jν. (A.30) In the absence of an external current, i.e., with Jµ= 0, the four components of Aµsatisfy then a Klein–Gordon equation with m= 0. The photon istherefore amassless particle. The Lorentz condition ∂µAµ= 0still allows for a residual gauge invariance under transfor ma- tions of the type (A.29), with the restriction 2Λ = 0. Thus, we can impose a second constraint on the electromagnetic ﬁeld Aµ, without changing Fµν. SinceAµcontains four ﬁelds ( µ= 0,1,2,3) and there aretwoarbitrary constraints, thenumber of physical degrees offreedom isjust two. Therefore, the photon has twodifferent physical polarizations B SU(N) Algebra SU(N)is the group of N×Nunitary matrices, UU†=U†U= 1, with detU= 1. AnySU(N) matrix can be written inthe form U= exp{iTaθa}, a = 1,2,...,N2−1, (B.1) withTa=λa/2Hermitian, traceless matrices. Their commutation relatio ns [Ta,Tb] =ifabcTc(B.2) deﬁne theSU(N)algebra. The N×Nmatricesλa/2generate the fundamental representation of the SU(N)algebra. Thebasis of generators λa/2can bechosen sothat the structure constants fabcarereal and totally antisymmetric. ForN= 2,λaare the usual Pauli matrices, σ1=/parenleftbigg0 1 1 0/parenrightbigg , σ 2=/parenleftbigg0−i i0/parenrightbigg , σ 3=/parenleftbigg1 0 0−1/parenrightbigg , (B.3) which satisfy the commutation relation [σi,σj] = 2iǫijkσk. (B.4) Other useful properties are: {σi,σj}= 2δijand Tr (σiσj) = 2δij. 44 ForN= 3, the fundamental representation corresponds tothe eight G ell-Mann matrices: λ1= 0 1 0 1 0 0 0 0 0 , λ2= 0−i0 i0 0 0 0 0 , λ3= 1 0 0 0−1 0 0 0 0 , λ4= 0 0 1 0 0 0 1 0 0 , (B.5) λ5= 0 0 −i 0 0 0 i0 0 , λ6= 0 0 0 0 0 1 0 1 0 , λ7= 0 0 0 0 0 −i 0i0 , λ8=1√ 3 1 0 0 0 1 0 0 0 −2 . Theysatisfy the anticommutation relation /braceleftig λa,λb/bracerightig =4 NδabIN+ 2dabcλc, (B.6) whereINdenotestheN-dimensional unitmatrixandtheconstants dabcaretotallysymmetricinthethree indices. ForSU(3), the only non-zero (up to permutations) fabcanddabcconstants are 1 2f123=f147=−f156=f246=f257=f345=−f367=1√ 3f458=1√ 3f678=1 2, d146=d157=−d247=d256=d344=d355=−d366=−d377=1 2, (B.7) d118=d228=d338=−2d448=−2d558=−2d668=−2d778=−d888=1√ 3. The adjoint representation of the SU(N)group is given by the (N2−1)×(N2−1)matrices (Ta A)bc≡ −ifabc,which satisfy the commutation relations (B.2). Thefollow ing equalities Tr/parenleftig λaλb/parenrightig = 4TFδab, T F=1 2, (λaλa)αβ= 4CFδαβ, C F=N2−1 2N, (B.8) Tr(Ta ATb A) =facdfbcd=CAδab, C A=N, deﬁne theSU(N)invariantsTF,CFandCA. Other useful properties are: (λa)αβ(λa)γδ= 2δαδδβγ−2 Nδαβδγδ, Tr/parenleftig λaλbλc/parenrightig = 2(dabc+ifabc), Tr(Ta ATb ATc A) =iN 2fabc,/summationdisplay bdabb= 0, dabcdebc=/parenleftbigg N−4 N/parenrightbigg δae,(B.9) fabefcde+facefdbe+fadefbce= 0, fabedcde+faceddbe+fadedbce= 0. C Anomalies Ourtheoreticalframeworkisbasedonthelocalgaugesymmet ry. However,sofarwehaveonlydiscussed the symmetries of the classical Lagrangian. It happens some times that a symmetry of Lgets broken by quantum effects, i.e., it is not a symmetry of the quantize d theory; one says then that there is an ‘anomaly’. Anomalies appear in those symmetries involving both axial (ψγµγ5ψ) and vector ( ψγµψ) currents, and reﬂect the impossibility of regularizing the quantum theory (the divergent loops) in a way which preserves thechiral (left/right) symmetries. 45 +p0qg g Fig.36: Triangular quarkloops generating the decay π0→γγ. A priori there is nothing wrong with having an anomaly. In fac t, sometimes they are even wel- come. A good example is provided by the decay π0→γγ. There is a chiral symmetry of the QCD Lagrangianwhichforbidsthistransition; the π0shouldthenbeastableparticle, incontradiction withthe experimental evidence. Fortunately, there is an anomaly ge nerated by a triangular quark loop (Fig. 36) which couples the axial current A3 µ≡(¯uγµγ5u−¯dγµγ5d)to two electromagnetic currents and breaks the conservation of the axial current at the quantum level: ∂µA3 µ=α 4πǫαβσρFαβFσρ+O(mu+md). (C.1) Since theπ0couples toA3 µ,∝an}b∇acketle{t0|A3 µ|π0∝an}b∇acket∇i}ht= 2ifπpµ, theπ0→γγdecay does ﬁnally occur, with a predicted rate Γ(π0→γγ) =/parenleftbiggNC 3/parenrightbigg2α2m3 π 64π3f2π= 7.73eV, (C.2) whereNC= 3denotes the number of quark colours and the so-called pion de cay constant, fπ= 92.4MeV,isknownfromthe π−→µ−¯νµdecayrate(assumingisospinsymmetry). Theagreement with the measured value, Γ = 7.7±0.6eV [7], is excellent. Anomalies are, however, very dangerous in the case of local g auge symmetries, because they destroy the renormalizability of the Quantum Field Theory. Since theSU(2)L⊗U(1)Ymodel is chiral (i.e., it distinguishes left from right), anomalies are cle arly present. The gauge bosons couple to vector andaxial-vector currents; wecanthendrawtriangular diag rams withthreearbitrary gauge bosons ( W±, Z,γ) in the external legs. Any such diagram involving one axial a nd two vector currents generates a breaking of the gauge symmetry. Thus, our nice model looks me aningless at the quantum level. Wehavestillonewayout. Whatmattersisnotthevalueofasin gleFeynmandiagram,butthesum of all possible contributions. The anomaly generated by the sum of all triangular diagrams connecting the three gauge bosons Ga,GbandGcisproportional to A=Tr/parenleftig {Ta,Tb}Tc/parenrightig L−Tr/parenleftig {Ta,Tb}Tc/parenrightig R, (C.3) where the traces sum over all possible left- and right-hande d fermions, respectively, running along the internallinesofthetriangle. Thematrices Taarethegeneratorsassociatedwiththecorresponding gauge bosons; inour case, Ta=σa/2, Y. Inorder topreserve thegauge symmetry, oneneeds acancella tion of all anomalous contributions, i.e.,A= 0. Since Tr (σk) = 0, we have an automatic cancellation in two combinations of ge nerators: Tr({σi,σj}σk) = 2δijTr(σk) = 0and Tr ({Y,Y}σk)∝Tr(σk) = 0. However, the other two combinations, Tr ({σi,σj}Y)and Tr (Y3)turn out to be proportional to Tr (Q), i.e., to the sum of fermion electric charges: /summationdisplay iQi=Qe+Qν+NC(Qu+Qd) =−1 +1 3NC= 0. (C.4) Equation (C.4) conveys a very important message: the gauge s ymmetry of the SU(2)L⊗U(1)Y modeldoesnothaveanyquantumanomaly, providedthat NC= 3. Fortunately, thisispreciselytheright 46 number of colours to understand strong interactions. Thus, at the quantum level, the electroweak model seems to know something about QCD. The complete SM gauge theo ry based on the group SU(3)C⊗ SU(2)L⊗U(1)Yisfreeofanomaliesand,therefore, renormalizable. Thean omalycancellation involves one complete generation of leptons and quarks: ν, e, u, d . The SM would not make any sense with only leptons or quarks. 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