The Standard Model of Particle Physics: An Introduction to the Theory

Transcription

1 Th Standard Modl of Particl Physics: An Introduction to th Thory Fawzi Boudjma LAPTH, Chmin d Bllvu, B.P. 110, F Anncy-l-Viux, Cdx, Franc Ditr Zppnfld Dpartmnt of Physics, Univrsity of isconsin, Madison, I 53706, USA Abstract: Th ky concpts of gaug invarianc and spontanous symmtry braking that hlpd build th Standard Modl of particl physics ar introducd. A short dscription of radiativ corrctions that hav mad th modl pass all prcision tsts, in particular thos from LEP, is prsntd. ky words: gaug invarianc, spontanous symmtry braking, Higgs, radiativ corrctions Résumé: Ls concpts importants d invarianc d jaug t d la brisur spontané d symétri qui ont prmis la construction du modèl standard sont xposés. Un dscription ds corrctions radiativs qui sont ssntills pour tnir compt ds msurs d précision, surtout clls ffctués au LEP, st présnté. mots clés: invarianc d jaug, brisur spontané d symétri, Higgs, corrctions radiativs URA du CNRS, associé à l Univrsité d Savoi. LAPTH-945/0 Compts Rndus d l Acadmi ds Scincs, Paris, sous prss. May 00

2 Th Standard Modl of Particl Physics: An Introduction to th Thory Fawzi Boudjma LAPTH, Chmin d Bllvu, B.P. 110, F Anncy-l-Viux, Cdx, Franc Ditr Zppnfld Dpartmnt of Physics, Univrsity of isconsin, Madison, I 53706, USA Abstract: Th ky concpts of gaug invarianc and spontanous symmtry braking that hlpd build th Standard Modl of particl physics ar introducd. A short dscription of radiativ corrctions that hav mad th modl pass all prcision tsts, in particular thos from LEP, is prsntd. ky words: gaug invarianc, spontanous symmtry braking, Higgs, radiativ corrctions Résumé: Ls concpts importants d invarianc d jaug t d la brisur spontané d symétri qui ont prmis la construction du modèl standard sont xposés. Un dscription ds corrctions radiativs qui sont ssntills pour tnir compt ds msurs d précision, surtout clls ffctués au LEP, st présnté. mots clés: invarianc d jaug, brisur spontané d symétri, Higgs, corrctions radiativs 1 Th Gaug symmtry principl Th cntral rsult of 0 th cntury particl physics is th succssful dscription of lctromagntic, wak and strong intractions as gaug fild thoris, rsulting from an undrlying symmtry calld gaug invarianc. Postulatd som thirty yars ago for th wak [1,, 3, 4] and strong intractions[5], th prturbativ prdictions of this Standard Modl SM of particl physics hav bn tstd and confirmd with amazing accuracy at LEP. 1.1 Elctromagntism as a prototyp On manifstation of th gaug invarianc principl is th fact that in lctrostatics th lctric fild, and hnc th lctrostatic forc, dpnds only on th diffrnc of potntial. Th gaug fild ida is bst known in th quantisd vrsion of Maxwll s quations. Th quantum fild crating and annihilating photons is th vctor potntial A µ x, which is rlatd to th lctromagntic fild strngth by F µν = µ A ν ν A µ. Th fild strngth, and thrby Maxwll s quations, ar invariant undr gaug transformations, A µ x A µ x + µ Λx. 1 Gaug invarianc is also familiar from th quantum mchanics dscription of a chargd particl intracting with an lctromagntic fild. Schrodingr s quation for a fr particl of mass m, 1/m i ψ = i ψ/ t, is invariant undr a global phas transformation ψ xpiλψ. If on insists that th quation rmains valid undr local phas transformation, in othr words that th transformation can b diffrnt at diffrnt points in spac-tim, λ qλx = t, x, on is thn ld to introduc a compnsating vctor fild which transforms xactly lik Eq. 1. This prscription givs th familiar Schrodingr s quation of a particl of charg q with th lctromagntic fild A µ = V, A, 1/m i + q A ψ = i / t + qv ψ. This principl is carrid ovr to th rlativistic quantum cas by rquiring that all drivativs µ b rplacd by covariant drivativs D µ = µ iqa µ, in analogy with Eq. which clarly displays th combination of th spac-tim drivativs and th vctor potntial componnts. An important consqunc of this, is that all chargd particls coupl xactly th sam way to th lctromagntic fild, th coupling is univrsal. Th ssntial point is that chargd filds and covariant drivativs of chargd filds hav idntical local transformations, ψx Uxψx, D µ ψx UxD µ ψx, Ux = iqλx 3

3 Th group of transformations, Ux, in Eq. 3, is an Ablian U1 group sinc it dos not mattr in which ordr w apply succssiv transformations of th form Ux. Elctromagntism is a long rang forc whr th mssngr of th forc, th photon is masslss. As is known also, lctromagntic filds hav only two transvrs indpndnt polarisations on also spaks of th photon as having only hlicity stats, dspit th fact that th photon is a spin-1 dscribd by a vctor A µ. In fact all ths proprtis, th drivation of Maxwll s quations that unify th lctric and magntic filds ar mbodid in th simpl Lagrangian dscribing fr photons L m = 1 4 F µνf µν 1 4 E + i B + E i B. 4 E ± i B displays th two hlicity stats of th photon, or polarisation of th fild. A mass trm for th photon would b rprsntd by m A µ A µ which braks gaug invarianc as it is not invariant undr Eq. 1. QED which is th gaug thory dscribing th intraction of lctrons with th photons has bn trmndously succssful. 1. ak intractions and non-ablian gaug thoris Considring how lgant and powrful th gaug invarianc principl is and how it dictats th form of th intraction, it was natural to xtnd it to th wak intractions. So much so, sinc procsss as diffrnt as β-dcay n p + + ν, muon dcay µ + ν + ν µ or muon captur µ + p n + ν µ smd to b of th sam natur and hav th sam strngth, pointing to univrsality. Thr ar howvr som important diffrncs with QED. Ths procsss ar chargd currnt procsss rprsnting ν or n p transitions and thus contrary to QED, thy involv a chang in th idntity of th mattr, spin-1/ fild. Morovr, th structur of th wak currnt suggstd that only lft-handd filds ar involvd. Chirality lft-handd and right-handd stats of a frmion corrsponds, at high-nrgy, to th hlicity stat of th frmion. An lctron fild can b dcomposd thn as = L + R. Elctromagntism trats ths two componnts on a qual footing, prsrving symmtry undr parity. Indd th gaug coupling of lctrons to photons writs ēγ µ A µ = ē L γ µ A µ L + ē R γ µ A µ R. γ µ ar a st of matrics which may b thought of as a rlativistic gnralisation of th Pauli spin matrics. Not in passing that th gaug intraction dos not mix ths two stats. In contrast, th chargd currnt to which on would lik to associat a chargd gaug boson, ±, is of th form, ē L γ µ µ + ν L. To mak this rsmbl th lctromagntic currnt, on can writ it as ĒLγ µ µ + τ + ν E L. Th ntity E L = should b considrd as a doublt on L spaks of an isospin doublt and τ ± ar th raising and lowring Pauli matrics. This two-lvl transition is also vry familiar to us from quantum mchanics as is th us of th Pauli matrics τ. Th smallst group of gaug transformation acting on th doublt E L and n, p gnralising Eq. 3, is th non-ablian SU L, which bsids τ ± has also a nutral gnrator τ 3. Thr will thrfor b 3 compnsating gaug filds: µ ±, µ. 3 SU L symmtry prdicts th coupling of 3 : Ē L γ µ µτ 3 3 E L = ν γ µ µν 3 ē L γ µ µ 3 L. Unfortunatly this nutral currnt dos not corrspond to th lctromagntic currnt. For a start it involvs nutral particls. On th othr hand, it dos includ a part lft-handd of th lctromagntic currnt. Thrfor w do hav a partial unification or at last a unifid dscription of th wak and lctromagntic intraction. To fully rconstruct th lctromagntic currnt from th nutral isospin currnt, on must postulat th xistnc of at last anothr nutral currnt. In th standard modl this is introducd via a U1 Y nutral currnt, associatd with th hyprcharg Y and a gaug fild B µ. Th lattr coupls to both th lft-handd doublt and right-handd.g R isospin singlt. Th photon and th Z will appar as a suprposition of th filds B µ and µ. 3 It is also appropriat to say, at this stag, that th nutron and th proton ar mad up of quarks, u and d, that form a doublt undr SU and which ar bound by th strong forc. Each quark carris a st of thr colours. Th mssngr of th colour forc, strong intraction, is th gluon. Th gaug group hr is SU3 which mans in fact that w hav ight typs of gluons, corrsponding to th ight gnrators of th fundamntal rprsntation of SU3 rprsntd by th 8 Gll-Mann matrics T a., ν, u, d form th first gnration of mattr particls of th SM modl. On has discovrd thr such familis. Ths frmions ar listd in Tabl. 1 which givs thir rspctiv charg undr th thr gaug groups SU L, U1 Y, SU3 c. Th gaug coupling constants of ths thr fundamntal intractions, which hlp dfin th gnralisation of th gaug transformations Eq. 3 and th covariant drivativ ar, rspctivly, g, g and g s. Th gaug transformations

4 Tabl 1: Quantum numbrs of th first gnration of quarks and lptons, and of th Higgs doublt fild, Φ, within th SM. Th rows giv th irrducibl rprsntations undr colour SU3 and wak SU, and th hypr-chargs for th various multiplts. Th lctric charg in units of is givn in th last column. Th assignmnts for th quarks and lptons of th scond and third gnrations c, s, ν µ, µ and t, b, ν τ, τ ar idntical to th ons for th first gnration. For th xprimntal dtrmination of th numbr of nutrinos and light gnrations s Chap. 7. SU3 SU L U1 Y Q = T 3 + Y 1 Q = u L, d L 3 6 3, 1 3 u R d R L = ν L, L 1 1 0, 1 R ν R and th covariant drivativ ar thn ψx Uxψx = iθa 3 xt a iθj τj x iθ 1 xy ψx, D µ = µ ig s T a A a µ ig τ i i µ ig Y B µ. 5 Th covariant drivativs compltly spcify th intractions of all known frmions and gaug bosons and ncod th univrsality of th gaug couplings s Chap. 8 for xprimntal tsts via th mattr Lagrangian L mattr = ψ j iγ µ D µ ψ j. 6 j=q,u R,d R,L, R,ν R An important obsrvation concrning non-ablian gaug thoris is that th gaug filds ar slf-intracting. Not only th mattr filds carry charg but also th gaug filds isospin for 1,,3, colour for th gluons. This is imposd by th non-ablian gaug transformations. For xampl th gnralisation of th fild strngth, F µν for SU is i µν = µ i ν µ i ν + gɛ ijk j µ k ν. 7 Th first two trms ntr th kintic trm as in th Ablian cas, whras th last trm dscribs th slf-intraction. Th gaug Lagrangian is constructd as in Eq. 4 using th appropriat filds strngths for SU, U1 and SU3. Th gaug Lagrangian uniquly fixs th + Z and + γ tripl gaug vrtics TGV which hav bn masurd at LEP s Chaptr 9, as wll as th quartic couplings such as + Zγ in trms of a singl paramtr, th gaug coupling g. On important proprty is that th non-ablian gaug symmtry prdicts th gyromagntic momnt of th ± to b, lik that of th chargd lmntary frmions. Spontanous symmtry braking and mass gnration Th main blow to th construct so far is that th wak intractions dscrib a short rang forc, in othr words th ± and th Z ar massiv. As pointd out abov for QED, a mass trm introducd by hand dstroys th gaug invarianc of th thory. This major hurdl was solvd by borrowing and adapting an ida that is ncountrd in many solid-stat physics phnomna. In such systms, th Hamiltonian is invariant undr a symmtry but th ground stat of th systm braks this symmtry. Such is th cas with a frromagnt blow th Curi tmpratur. In this cas, rotational symmtry is brokn by th ground stat all spins of th atoms alignd in th sam dirction dspit th fact that th dynamics th Hisnbrg spin-spin Hamiltonian dos not slct any prfrrd dirction. This spontanous symmtry braking is also at work in suprconductivity whr, with th Missnr ffct, th fact that th magntic fild ntrs th solid ovr a vry short rang could b associatd with a massiv photon.

5 .1 Th Higgs mchanism and mass gnration for th gaug bosons Figur 1: Scalar potntials. Th first panl shows a symmtric potntial, undr rotation around th vrtical axis, with a stabl minimum whr th ball is rsting. In this situation < 0 φ 0 >= 0. Th scond panl, rsmbling a mxican hat, illustrats spontanous symmtry braking. Th symmtric configuration at th top of th hat is unstabl. Th systm will pick up any stabl configuration along th brim with < 0 φ 0 > 0. Th Goldston mod thrfor rprsnts this azimuthal dirction whras th radial componnt is th Higgs fild. On usually thinks about and most oftn dfins th vacuum as a stat whr all filds hav zro xpctation valu. Howvr it may happn, lik what is dpictd in Fig., that th stat with zro nrgy is not th most stabl. Th systm will choos stability or minimum nrgy bottom of th wll rathr than th stat with maximum symmtry top of th mountain. Such potntials ar providd by scalar filds, with a potntial of th form V = λ φ v / λ > 0, whr v is th vacuum xpctation valu v..v of th scalar fild < 0 φ 0 >= v/. This scal is th origin of th mass of th gaug bosons and also th frmions. It is as if som of th chargd frmions and gaug bosons moving in such vacuua fl a rsistanc and bhav lik thy hav mass. To s how this works, it is most simpl to tak QED as an xampl[] to illustrat how a mass for th photon can b introducd in a gaug invariant way. On nds to tak a chargd scalar fild φ. Th lattr can b rprsntd by a complx fild φ = φ 1 + iφ /. For our purposs it is bst to rwrit this in polar coordinat as φ = h + v iθ/v /, whr both h and θ hav zro v..v. θ charactriss th rotational symmtry of th potntial. Th intraction of this scalar fild with th lctromagntic fild is dscribd by a fully gaug invariant Lagrangian constructd through th us of th covariant drivativ, Eq. 3. Upon xpansion of this Lagrangian w find L = 1 4 F µνf µν + 1 A v µ + 1 v µθ + 1 µh µ h λ 4 h + vh + 1 A µ + 1 v µθ h + vh. 8 On ss clarly that th gaug fild has acquird a mass, m γ = v scond trm in Eq. 8: + v A µ A µ /. Bcaus of gaug invarianc w can always mak a local phas transformation on th fild φ = h + v iθ/v / such that th phas θ/v is st to zro. This gaug whr θ 0 in Eq. 8 is th unitary gaug whr only th physical stats h, A µ ar lft. Counting th numbr of dgrs of frdom bfor and aftr symmtry braking, w find th sam numbr of cours. Bfor, on had two scalars and on masslss spin-1 which has only two transvrs stats of polarisation. Aftr symmtry braking, on of th scalars, θ, transmuts into th longitudinal polarisation of th havy photon. In fact it would b mor appropriat to say that th gaug symmtry is hiddn, onc w choos a particular gaug, sinc th gaug symmtry is prsnt in th dynamics. Th θ fild is a Goldston boson, it corrsponds, as sn in Eq. 8, to a masslss psudo-scalar. h is th Higgs scalar fild whos mass is givn by λv. Thr is also a coupling haa which proportional to th mass of th gaug boson. A vry similar approach is applid to th wak intraction. Sinc w nd thr massiv gaug bosons, th thr longitudinal stats will b providd by thr Goldston bosons. This is most simply and conomically providd

6 by a Higgs doublt, Φ, with quantum numbrs such that th vacuum is lft invariant undr lctromagntic gaug transformations, so that th photon rmains masslss. Thus Y Φ = 1/, Φ = 0 1 v + H τj iωj v L Higgs = D µ Φ D µ Φ V Φ Φ, with V Φ Φ = λ Φ Φ v. 9 whr H is th physical Higgs fild of th lctrowak thory, ω 1,,3 th Goldston bosons and v th vacuum xpctation valu.. Frmion masss alrady strssd th fact that QED tratd both lctron chiralitis on th sam footing, in particular both L and R hav th sam lctric charg. Thrfor th lctron mass trm m ē R L + ē L R is gaug invariant in QED. According to Tabl 1 th SM has no lft and right-handd multiplts with idntical SU and U1 Y charg, hnc, a frmion mass trm introducd by hand would brak th symmtry of th SM. Onc again, howvr, mass trms ar possibl via th Higgs mchanism. Lt us considr th masss for th chargd lptons. Th lft doublt L, right-handd singlt l R and th Higgs fild Φ combin so that thy form a nutral symmtric objct undr SU U1. Th masss ar introducd via Yukawa couplings y l as L l m = i=,µ,τ y i l Li Φl R,i + l R,i Φ L i unitary gaug i=,µ,τ yl i v 1 + H li l i, v y i l v = m i l. 10 This xhibits a common important fatur that applis also to quarks, namly that th couplings of th Higgs to frmions is proportional to th frmion mass. Bcaus in th quark sctor on is gnrating masss for both th up and down quark, on also inducs mixing btwn th thr-familis in th chargd currnt[6] but not in th nutral currnts[7]. This is also th sourc of CP-violation in th modl. It is important to strss that although masss and mixing ar introducd in a gaug invariant way, on nonthlss nds to introduc a larg numbr of ad-hoc Yukawa couplings, contrary to th gaug boson masss that ar xprssd through th univrsal gaug coupling. 3 Salint faturs of th modl ithin th SM th scal of all particl masss is st by th Higgs v..v., v. For th gaug bosons, th mass trm is producd by th kintic nrgy trm of th Higgs doublt fild, L Higgs in Eq. 9. Th lattr rvals svral ky prdictions of th SM which can b tstd at LEP. 1. Th Higgs Lagrangian gnrats mass trms for th chargd ± and for Z filds. 3 µ and B µ combin to giv th masslss photon and th Z, Z µ = cos θ 3 µ sin θ B µ = 1 g + g g 3 µ g B µ A µ = sin θ 3 µ + cos θ B µ. 11 This rlation dfins th wak mixing angl, or, mor prcisly, sin θ and cos θ.. Th and Z masss ar givn by M = gv and M Z = g + g v = M cos θ. 1 Th mass rlation M = ρm Z cos θ with ρ = 1 is a consqunc of th braking th lctrowak symmtry by a scalar doublt Higgs fild.

7 3. Th mass trms for th and Z ar dirctly rlatd to H and HZZ couplings, of strngth M /v and MZ /v, rspctivly. This coupling of a singl scalar to two gaug bosons rquirs th xistnc of a v..v. for th Higgs doublt fild. Normally, gaug bosons coupl to pairs of scalars only. 4. Th Higgs boson mass is givn by M H = λv. Sinc th quartic coupling λ is a fr paramtr, thr is no intrinsic prdiction for M H within th SM. Sarchs for this missing boson ar rviwd in Chap Th sam Higgs mchanism is rsponsibl for th mass of th frmions. Th coupling of th Higgs bing proportional to th mass of th frmion, an intrmdiat mass Higgs, M H < M, will dcay prdominantly to b-quarks. It is important to strss that th modl givs a vry nic unifid dscription of th wak and lctromagntic intractions. Howvr th modl dos not fully unify ths intractions, sinc w still hav two indpndnt gaug couplings g, g. Viwd anothr way th modl dos not prdict sin θ. Nonthlss th gaug principl automatically lads to th univrsality of th gaug coupling. Most importantly for LEP1, th nutral intraction Lagrangian drivd from Eq. 6, summariss th main proprtis and paramtrs of Z physics: L NC mattr = i Q i ψ i γ µ ψ i A µ + ψ sin θ cos θ i γ µ gv i gaγ i 5 ψ i Z µ, 13 i with th vctor and axial-vctor couplings givn by s = sin θ gv i = T 3 i Q i s ;, ga i = T 3 i and g =, g =. 14 sin θ cos θ 4 Th nd for radiativ corrctions Laving asid frmion masss, with th assignmnt of th quantum numbrs givn in Tabl 1, th frmion and gaug fild Lagrangians considrd abov ar compltly dtrmind in trms of just fiv fr paramtrs. Ths can b takn as th gaug couplings g s, g, g, th Higgs v..v, v, and th quartic Higgs coupling, λ. In particular th thr wak paramtrs g, g, v alon allow to dtrmin all th proprtis of th gaug wak bosons masss and slf-couplings as wll as thir intraction to mattr. This is th powr and bauty of gaug invarianc in that with an xtrmly limitd numbr of paramtrs on can dscrib a vry larg numbr of procsss and obsrvabls. For xampl at th Z pak, at LEP1 and SLC, th mixing angl s = sin θ can b xtractd from th ratio of a numbr of cross sctions that masur th coupling g f V /gf A. From th obsrvabls with lptons in th final stats, th latst LEP masurmnt of s from lptonic obsrvabls, which w will idntify as s ff,l, givs an avrag valu s = s ff,l = ± i. 0.08% prcision. 15 hav also sn that this angl also dscribs th ratio M /M Z and thrfor could b xtractd from a combination of ntirly diffrnt xprimnts LEP1 and LEP. Th latst data [8] giv M Z = ± 0.001GV M Z /M Z 10 5 and M = ± 0.033GV M /M , lading to s M = 0.16 ± i. 0.3% prcision. 16 Not only do w gt a much bttr prcision on th ffctiv mixing angl xtractd from th Z obsrvabls, but mor tlling is that this valu is about 14σ away from that xtractd from th mass ratio, M /M Z. Th Born approximation which has bn discussd so far, is insufficint to xplain th LEP masurmnts which ar dscribd in Chap. 10. Improving on th Born approximation ncssitats th inclusion of radiativ corrctions which ar contributions from th quantum fluctuations of th vacuum. Thy ar rprsntd by loops. A slction of radiativ corrctions to frmion pair production in + is shown blow. Computing som of th closd loops naivly can lad to infinit and inconsistnt rsults, whras thy ar supposd considring th small lctrowak coupling xpansion paramtr to b small corrctions! Th infinitis appar bcaus on is summing ovr all possibl stats with all possibl nrgis. Mathmatically within th loops

8 Figur : Som th diagrams rprsnting th quantum loop corrctions to f f production at +. On can indirctly accss virtual particls. Th first tr diagrams ar slf-nrgy diagrams. Diagram d is a vrtx corrction involving virtual top particls which contribut to b b production. F i F i a f f b f f Z Z H c Z f f t t d b b on is intgrating ovr all possibl momnta, k, with intgrals of th form d 4 k/k + m n with n = 1,, 3.. that divrg for small n at high momnta. This problm had occurrd in QED but was sid-stppd with th ralisation that as long as ths infinitis do not show up in masurabl quantitis xprssd in trms of physical paramtrs, ths infinitis wr not disastrous. Tak for instanc th vacuum polarisation of th photon, first figur in th slf-nrgy diagrams. This is calculatd following som simpl st of ruls drivd from th QED Lagrangian. Th divrgnc can b liminatd mathmatically by invoking that th paramtr apparing in th Lagrangian is a mathmatical quantity which is also infinit and can absorb th corrsponding divrgnc of th loop. Diffrncs in th valu of at diffrnt nrgis ar on th othr hand finit. Thus by choosing a rfrnc valu, and thrfor dfining th paramtr as a physical paramtr xtractd from som obsrvabl, all subsqunt obsrvabls will b finit. In a rnormalisabl thory, lik QED, onc th fundamntal paramtrs of th thory hav bn proprly dfind in trms of a fw physical obsrvabls, all othr obsrvabls ar prcisly dtrmind. Bfor th advnt of spontanous symmtry braking this program could not b achivd for th wak intractions. Infinitis appard all ovr th plac and rquird mor than just a rscaling of th fundamntal paramtrs of th modl. Lacking gaug invarianc in th dynamics of th systm mant that many infinitis smd to b uncorrlatd. Gaug invarianc, through th Higgs mchanism, was ssntial for proving th rnormalisability of th standard modl. In fact it was only whn this proof was achivd that th modl rcivd all its rcognition and popularity. Tchnically, for th standard modl, on also had to dvis a nw rgularisation mthod dimnsional rgularisation[4] to mathmatically handl th divrgncis and to kp th symmtris of th modl at th quantum lvl. Th modl is thn highly prdictiv and is amnabl to high prcision masurmnts. On can say that with th advnt of LEP/SLC and som othr low-nrgy prcision masurmnts th modl has bn crownd and lvatd to th status of a thory sinc th modl has bn vrifid at bttr than.1%. Rnormalisation hlps not only gt finit rsults and highly improvs th tr-lvl approximation, but th radiativ corrctions can giv accss, as w s in Figur, to nw particls lik th Higgs, th top... that ar too havy to b producd dirctly at LEP. Combining prcision masurmnts, on can wigh th ffcts of ths nw stats and litrally gt a bound or a masur on th masss of ths particls. In this light it is not difficult also to undrstand that som qualitis or natural rlations lik th diffrnt dfinitions of sin θ at th tr-lvl will b corrctd. This is th cas for th diffrnt manifstations of s in sction 3. Th most prcisly masurd quantitis of th standard modl ar th QED fin structur constant, α = /4π, th mass of th Z, M Z and G µ th Frmi constant drivd from muon dcay which at tr-lvl is G µ / = g /8M. Using ths paramtrs, in liu of th thr fundamntal paramtrs g, g and v of th original Lagrangian this is just a trad-off, on finds that s M s Z s ff,l s Z c c s c c s ˆr κ s ε 1 ; ˆr c ; κ = ε 1 + ε 3 c + c s s ε + ε 3,

9 s Z = παM Z Gµ M Z ; αm Z = α 1 αm Z. 17 Th most important corrctions for LEP physics ar ncodd in th nw quantum functions αmz, ε 1,,3[9]. αmz is a purly QED corrctions rprsnting th scrning of th lctric charg. It contains no Nw Physics paramtrs. It is givn solly in trms of known paramtrs blow th Z scal and corrspond to th ffctiv lctric charg at th Z mass. s Z constituts a QED-only improvd mixing angl which nonthlss taks into account a rathr larg corrction of about 6%. ε ar gnuin lctrowak corrctions that ar snsitiv to th top mass and th Higgs mass of th SM, thy ar also th most likly prob of a contribution byond th SM. For xampl, ε 1 = 3G µmz { } m t 8π 1 s M lnm H /M Z + + ε NP M Z 5 Conclusions Th confrontation of th prcision tsts with thory byond th tr-lvl has not only allowd to xtract th mass of th top-quark soon aftr LEP1 startd opration, but has also put strong constraints on th Higgs mass. Most modls without a light Higgs ar now strongly disfavourd. This has addd imptus to suprsymmtric modls which prdicts a light Higgs, s Chaptr 1. Anothr argumnt for suprsymmtric grand unification drivs from th vry prcis masurmnt of s ff,l togthr with α smz s Chap. 14,15: thy suggst that th thr gaug couplings g, g, g s unify at som high scal with suprsymmtric particls at or blow th TV scal. Apart from th still missing Higgs particl, th Standard Modl has passd all stringnt tsts with flying colours. It is ssntial that on discovrs th Higgs to confirm that th pattrn of symmtry braking and mass gnration is indd as dscribd by th modl or ls that th modl is in nd of mor symmtry lik suprsymmtry or nw undrlying dynamics. Rfrncs [1] S.L. Glashow, Nucl.Phys ; S. inbrg, Phys. Rv. Ltt ; A. Salam, Proc. 8th Nobl Symposium, d. N. Svartholm, Almqvist and ikslls, Stockholm 1968 p.367. [] P.. Higgs, Phys. Ltt ; Phys. Rv ; T..Kibbl, Phys. Rv S also, J. Goldston, A. Salam and S. inbrg, Phys. Rv F. Englrt and R. Brout, Phys. Rv. Ltt [3] G. t Hooft, Nucl. Phys. B , ibid B S also B.. L and J. Zinn-Justin, Phys. Rv. D , 3137, 3155; C. Bcchi, A. Rout and R. Stora, Ann. Phys [4] G. t Hooft and M. Vltman, Nucl. Phys [5] H.Fritzsch, M. Gll-Mann and H. Lutwylr, Phys. Ltt. B ; D.J. Gross and F. ilczk, Phys.Rv.Ltt ; H.D. Politzr, Phys. Rv. Ltt ; S. inbrg, Phys. Rv. Ltt [6] N. Cabibbo, Phys. Rv. Ltt ; M. Kobayashi and K. Maskawa, Prog. Tho. Phys [7] S.L. Glashow, J. I. Illiopoulos and L. Maiani, Phys. Rv. D [8] G. Myatt, Talk at Moriond 00 lctrowak sssion, [9] G. Altarlli and R. Barbiri, Phys. Ltt. B For anothr, qually popular, paramtrisation s, M.E. Pskin and T. Takuchi, Phys.Rv. D [10] A good introduction to radiativ corrctions applid for LEP physics can b found in Z Physics at LEP 1, CERN Yllow Book, CERN Vol. 1, Editd by G. Altarlli, R. Kliss and C. Vrzgnassi. S also Rports of th orking Group on Prcision Calculations for th Z Rsonanc CERN Yllow Book, CERN 95-03, Editd by D. Bardin,. Hollik and G. Passarino.