Elementary Particle Physics Fall 2009

Transcription

1 Elmntary Particl Physics Fall 009 Lctur MW :50 PM :0 PM, Physics D Michal Rijssnbk mail: Cours Wb pag: Skyp: michal.rijssnbk@stonybrook.du mrijssnbk or via dpartmntal cours pag or BlackBoard /09/009

2 Th Wak Intraction Th dtaild study of th wak intraction ld ultimatly to th mrgnc of th Standard Modl. Th Standard Modl provids a full unification of th Wak and Elctromagntic intractions, with only a small numbr of a priori unknown paramtrs. In natur, th wak intraction is rsponsibl for th slow dcays of particls, and for intractions of nutrinos with othr lptons and quarks /09/009

3 Matrix Elmnt for Muon Dcay Historically, th thory of wak intractions startd with Frmi s four frmion point intraction of th form: (p 4 ) ν ( p p ) ν (p 3 ) m ( ν γ )( γ ) ( ) ν( ) i = ig i i fi F whr th Frmi constant G F dscribs th strngth of th intraction. Th Frmi constant has dimnsion Enrgy, a clar indication that W* this dscription can only b a low nrgy approximation. m fi is calculatd using th trac tchniqus dvlopd for QED: t (p ) ν mfi = G Tr F γ p γ p 3 ( m ) Tr γ x + p ( + m 4 ) γ ν p mass trms: Trac of 3 γ s =0 G F ν ν ν = 4p p + 4p p 4g p p 4p p + 4p p 4g p p 3 3 ( 3 ) 4,,, 4, ( 4 ) ν ν ν G F ν (p ) ( pp)( pp) ( pp)( pp) ( pp)( pp) ( pp)( pp) = GF = 3 ( pp )( pp 3 4) + ( pp 4)( pp 3) whr th factor ½ is from th avraging ovr th muon spin stats (s Lctur Nots 0). /09/009 3

4 Muon Liftim Th dcay width is now calculatd as: dγ ( ) = mfi dlips Flux 4 3 d j 4 4 = m p fi π δ p p p p 3 m j= ( π ) E j This is intgratd ovr dp and dp 3 using th δ 4 function, giving: ( ) ( ) dγ G mg F 4E 4 E F = m E 4.37 s de 6π m + Γ= = m 384π which is twic th masurd muon liftim valu of.0 s. ( ) This long liftim, i similar il to many othr particls il that dcay wakly, charactrizs th wak intraction and sts it apart from th EM and strong intractions. It was quickly rcognizd by Frmi that th amplitud blows up at high nrgis, and h postulatd th xistnc of th massiv vctor boson W as th wak mdiator. This xplains, at last qualitativly, why wak cross sctions at low nrgy and th wak dcays of particls ar supprssd, and why th q dpndnc is absnt. /09/009 4

5 Th Wak Intraction It was ralizd by T.D. L and C.N. Yang, that th wak intraction violats parity and th parity violation was subsquntly masurd to b maximal. It appars that only lft handd particls participat in th wak intraction. Bcaus nutrinos ar almost masslss and bcaus thy only intract via th wak intraction, thy ar producd and dtctd as lft handd. W can thus apply th whol Dirac formalism dvlopd prviously, providd w projct out th lft handd particl spinors and right handd anti particl spinors. W do this by oprating on th particl spinors with P L L,, on th anti particl spinors with P R! Furthrmor, w rplac th lctromagntic coupling =(4πα) ½ with th wak coupling constant g/ (th is an historical artifact). In th Glashow Winbrg Salam modl (th lctro wak sctor of th Standard Modl) g and ar rlatd by th Winbrg angl: g = sinθw 0.48 As mntiond bfor, in many low nrgy calculations, th xplicit W boson propagator rducs to a simpl constant M W. Thus, th Frmi coupling constant G F, vry prcisly masurd from th muon liftim, rlats to th combination of th wak coupling g and th M W as: G F / = g /(8M W ). /09/009 5

6 Many xampls xist of procsss mdiatd by th wak intraction: All dcays ar charactrizd by a liftims that ar long in comparison to dcays mdiatd by th lctromagntic or strong intraction:.g: τ(π 0 γγ) = 0.08 fs τ(ρ π +/0 π /0 ) = ħ / Γ ρ = MV s / 50 MV = s Exampls of Wak Intractions Procss Branching Fraction Man Lif n p + + ν 00% (fr nutron) 898 s ν + + ν 98.5 %.97 s π + ν π + ν 00.0 % K ν K ν 63.4 % K + π ν 5.0 % K + π ν 3.3 % K + π + + π % K + π + + π + + π 5.6 % K + π + + π 0 + π 0 8%.8 Λ p + π 63.9 % Λ n + π % 6.0 ns.4 ns 0.63 ns /09/009 6

7 Th Dcay of th Chargd Pion Th chargd mmbrs of th J P =0 psudoscalar nont dcay wakly. (p ) q π (m π ) W calculat l th dcay rats in a way similar il to muon dcay: W* Th matrix lmnt involvs th ν lft handd currnt and th pionic quark currnt J x ν (q): ( p ) G t F 5 ν imfi = igν u( p) γ ( γ ) uν ( p) J ( q) Fynman diagram for th chargd pion dcay via th Hr, th quark currnt is hiddn insid th pion. wak intraction. Th only variabl that is availabl to mak a Lorntz four currnt is th vctor q. For a spin zro pion, th currnt must thus b of th form J (q)= q f(q ) which, for th bcaus q =m π, simplifis to J (q) = q f(q ) = q f π. Combining, w can calculat th pion dcay matrix lmnt: GF 5 mfi = fπ qu( p) γ ( γ ) uν ( p) GF 5 GF 5 = f π ( p+ p) u( p) γ ( γ ) uν ( p) = f u ( ) π p p ( + p )( γ ) u ν ( p) GF 5 GF 5 = fπ u( p) ( m + 0)( γ ) uν ( p) = m ( ) ( ) ( ) fπ u p γ uν p /09/009 7

8 Liftim of th Chargd Pion Th squard and spin summd matrix lmnt bcoms: G F 5 5 mfi = m fπ u p uν p u p uν p ( ( ) ) * γ ( γ ) ( ) ( ) ( ) ( ) ( ) s, s G F = m fπ Tr 5 ( p m )( ) γ + p ( + γ 5 ) = 4 m ( ) fπ GF p p In th pion CMS th product p p rducs to ½(s m m ) = ½(m π m ), a simpl constant. Finally, th dcay width is calculatd: Γ( π + ) = Ω m * * ν p d f a + 3π mπ mπ m 4π m f GF ( m m ) π π π m m π π ( ) F G π F m 3 π m fπ mπm 8π mπ 8π mπ = 3 f = G m = m /09/009 8

9 Invrs Muon Dcay: ν Scattring off Elctrons Nutrino bams can b producd at high nrgy, high intnsity proton acclrators: Acclratd protons ar xtractd and dirctd d onto a mtal targt (cooling!). Th grat majority of particls producd in th targt ar pions, and th chargd pions ar allowd to dcay in a bam (tunnl) sction bhind th targt up to svral hundrd mtrs in lngth. Th chargd pions dcay in flight into muon nutrinos (with a 00% branching fraction); a π + into a muon nutrino, and a π into a muon anti nutrino. Thnutrinos arlorntz boostd in th dirction of th pions. Th accompanying muons that ar also producd ar absorbd in a long bam stop (a long dirt filld sction of th bam lin) so that only th nutrinos surviv. On may vn slct th wantd nutrino typ by slctivly bnding away ithr th positiv or th ngativ pions A vry larg dtctor,.g. a watr Crnkov dtctor, or a dtctor with liquid scintillator plans altrnating with tracking plans and magntizd iron absorbrs, follows th muon filtr. /09/009 9

10 Using Pions to mak Nutrinos at FrmiLab Mj Major Componnts: Proton Bam Pion Production Targt Focusing Systm Dcay Rgion & Monitor Hadron Absorbr Muon Shilding Muon flux masurmnt to assss th nutrino bam Most ν s from body dcays: π + + ν K + + ν ν nrgy is only function of νπ angl and π nrgy Most ν s from 3 body dcays: + + ν ν K + π 0 + ν /09/009 0

11 ν Scattring ν p = Th scattring of muon nutrinos on (lft) p p 3 p p 3 is rlatd to muon dcay (right) via a x q p 3 p p p 3 x q p 3 p topological rarrangmnt of 4 vctors. t W p p 4 W Intractions that hav W boson xchang p p 4 p 3 p t p p 4 ν ar calld Chargd Currnt vnts, p 4 p ν as opposd to vnts that rsult from th Fynman diagram for ν scattring xchang of a Z calld Nutral Currnt vnts; (lft), and its rlationship with th muon dcay diagram (right). ths lattr vnts wr fvrishly sarchd for whn th Glashow Winbrg Salam modl prdictd th Z as wll as th W. g m fi ν v = 8 ( pp )( pp 3 4 ) MW g g m fi ν 8 ( p3 p4)( p p) 8 ( p3 p4)( pp) + + ν = = M W 8 M W = 64G p p p p = 6G s m s m 6 G s (ignoring masss!) ( 3 4)( ) ( )( ) F F F ν /09/009

12 m ν Scattring ( )( ) fi ν + + ν = G F p3 p4 p p = GF s (ignoring all masss!) This is indpndnt of any angls: th dot products p p 3 and p p 4 ar only proportional to th total CMS nrgy squard. Thrfor, th angular distribution of th muon and lctron nutrino in th final stat ar isotropic i in th CMS: d ( σ ν ν) F = Gs * fi d m = Ω 64π s 4π And th total cross sction is: lab Gs F GF m Eν lab 4 σ ( ν ν) = = = s.68 0 mb GV = Eν.7 0 mb/gv π π i.. vry vry small! For xampl, a 0 GV bam of 0 0 nutrinos pr scond, ntring a 00 ton H O targt will produc only about intractions pr hour. Not that th muon mass cannot b ignord at currnt bam nrgis: s=m E lab ν is typically not larg compard to m! Not, that this vr incrasing cross sction is unphysical in th limit of ultra high nrgy: it violats Unitarity. Thus, at som nrgy, a nw procss must com in, anothr Fynman diagram, that modrats this bhavior of th cross sction /09/009

13 ν Scattring Elctron anti nutrinos ar copiously producd ν W mad ν p p in natural and man mad nuclar ractors. p p 3 q p 3 +p Whn scattring off atomic lctrons, w hav + x Z p a chargd currnt and a nutral currnt diagram. p 4 t x ν p ThChargd CurrntDiagram ν p 4 (lft) follows from t q p p 4 th ν (or ν ) CC diagram with substitutions: ν p p 4, p 4 p, p p 3 i.. th intrchang of th Lorntz invariants s and t. x q pp 3 p In th CMS of th ν systm w find: t W dσν ( ) ν p m p 4 = 6 * fi = Gt ν F d Ω 64 π s 64 π s In trms of CMS quantitis, and ignoring th (small) masss, w obtain: dσν ( ) ( ) ν Gs F Gs F = cos θ ; σν ( ) ( ) *, ν = = σν ν ν dω 6π 3π 3 Th angular cross sction paks for, i.. th outgoing lctron mittd in opposit dirction to th incoming nutrino bam. This maximal parity violation ffct can also b argud basd on lfthandd nutrinos and simpl spin argumnts. /09/009 3

14 Th Nutral Currnt Diagram or ν Scattring ν p p 3 Not that th Nutral Currnt diagram in ν Th only Fynman t is vry similar to ν or ν ν ν ν ν, diagram for ν x Z Both only procd via Z xchang in th t channl. ν scattring. p p 4 Immdiatly w ncountr problms: q p p 3 th ν ν currnt will hav th V A structur bcaus intracting nutrinos ar xclusivly lfthandd; but what about th currnt? Also, what is th coupling strngth to th Z boson? As a first guss w may assum that th Z coupls xactly lik th W to lfthandd lptons; thn σ(ν ν ) = σ(ν ν ). Exprimntally howvr, σ(ν ν ) = E lab ν (.6±0.4) 0 5 mb/gv, i.. ~0 smallr * ) Thrfor, w cannotassumanythingandmustparamtrizourignoranc: assum anything and must our ignoranc: iρgf ( im )( ) fi = iνγ 3 ( γ5) ν i4γ ( cv caγ5), with ρ= if th Z and th W coupl with th sam strngth to lptons and hav th sam mass, and with c V =c A = for pur V A coupling at th lctron vrtx. ν * ) L.H. Ahrns t al., Phys.Rv.Ltt. 5 (983) 54. /09/009 4

15 Th Matrix Elmnt for ν Scattring ν p p 3 Th ν Z tnsor is as bfor for th W: t Z ν Tr L x ( Zνν ) = γ p3γ ν p ( γ ) 5 = Tr γ p3γν p Tr γ p 3 γν p γ 5 p p ρ σ 4 = 4 p3 p ν + p p3 ν gν p p 3 4 iε p3 p νρσ q p p 3 whras th Z tnsor bcoms (ignoring m ): Th only Fynman diagram for ν ν scattring. 5 ν 5 = γ ( γ ) γ ( γ ) ν L Tr ( Z) c V ca p c 4 V ca p ν 5 ν = ( c ) Tr V + c A γ p γ p + cc Tr 4 V A γγ pγ p 4 = ( cv + c ) A p 4 p ν + p p ν 4 g ν p p 4 + cvc A iε νρσ p4ρp σ and th avragd matrix lmnt squard is: m fi ( cv+ ca) [ pp pp + pp pp ] 3 ( cc )[( pp)( pp) ( pp)( pp) ] GF ρ 3 ( )( ) ( )( ) = + 3 V A ( )( 3 4) ( 4)( 3) ρgf { = 3 ( )( ( ) } cv+ ca pp )( pp 3 4) + cv ca ( pp 4)( pp 3) /09/009 5 ν

16 Th Cross Sction for ν Scattring t ν x ν p p 3 Z Th CMSangular distribution of th outgoing lctron in th nutral currnt intraction is thn: d σν ( ν ) s p p 4 q p p 3 Th only Fynman diagram for ν ν scattring. * = m 6 * fi = ( ρgf ) ( cv + ca ) + ( cv ca ) ( cosθ4 ) d Ω 4 64π s 64π s 4 ( ρgf ) s * = ( cv + ca) + ( cv ca)( cosθ4 ) 6π 4 ( ) ρgf s σν ( ν ) = ( cv ca) ( cv ca) 4π i.. an llips in th c V, c A plan V A /09/009 6

17 Elctron Anti Nutrino Scattring off Elctrons In th calculation of th full procss for ν ν w must combin amplituds bfor squaring and thus th rlativ sign of th two contributing diagrams is crucial. It can b shown that th rlativ sign of th amplituds is ngativ,.g. by ralizing that th nutral currnt diagram has an intrchang of th two final stat frmions and thrby acquirsan an xtra sign. Th diffrnt nutrino xprimnts masur c V and c A in various combinations Th masurmnts ar in xcllnt agrmnt with th Standard Modl xpctations: c V = I 3 Qsin θ W and c A =I 3, whr I 3 is th z componnt of wak isospin, θ Wib W th Winbrg angl: sin θ W 0.3, 03 and Q th lpton charg. W ll gt back to this /09/009 7

18 Th Higgs Mchanism Th Higgs Mchanism, invntd by Ptr Higgs, was usd by M. Vltman and G. t Hooft and othrs to lnd mass to th gaug vctor bosons of th wak intractions. Th gaug filds (vctor bosons) that ar introducd to mak th local gaug invarianc of th particl fild in th Lagrangian possibl must ncssarily b masslss. Mass trms for a vctor boson, lik m A A, do not stay invariant undr th transformation of A A =A α/q. For th photon fild this is prfctly fin, but is clarly not accptabl for th massiv wak vctor bosons. Th introduction of a complx scalar Higgs fild into th Lagrangian, with a non zro xpctation valu (a non zro vacuum nrgy dnsity) turns th initially masslss gaug bosons into massiv bosons, whil on of th two Higgs fild componnts disappars. In addition to giving mass to th wak vctor bosons, W ± and Z 0, th Higgs fild also coupls to th frmion filds in th Standard Modl Lagrangian, and by its couplings givs thm mass as wll. W ll discuss th Higgs mchanism, first using a on dimnsional, and thn a twodimnsional toy modl (historically namd th sigma modl). /09/009 8

19 A On dimnsional Higgs Modl Considr th Lagrangian dnsity for a ral scalar (spin 0) Klin Gordon fild φ, s quation : 4 L= T V = ( φ)( φ) V( φ), with V( φ) = φ + λ φ 4 Th potntial V(φ) is a function of th gnralizd coordinat φ(x), and has a parabolic shap around th minimum at φ(x)=0. This Lagrangian has a simpl rflction symmtry undr φ φ = φ around th minimum. For small dviations from th minimum φ <<, V(φ) ½ φ, and: E L quation LK-G = T V = ( φ)( φ) φ ( + ) φ = 0 (K-G quation) which is th Lagrangian for a massiv Klin Gordon fild φ; not th rlativ minus sign btwn kintic and mass trms in L. Not that th mass trm is intimatly connctd to a positiv parabolic shap of th potntial V(φ ) around th origin of th fild; i.. no such parabola thn no mass trm! So far, nothing is nw... /09/009 9

20 Now, considr what happns if w chang th potntial to V(φ) = ½ φ + ¼ λ φ 4. This potntial has a bottoms up shap: point φ=0 is a local maximum, with two dgnrat minima on ithr sid at φ = ± /λ. Thr is no mass trm for small φ. L xhibits th rflction symmtry around φ=0. Th vacuum stat, i.. th lowst nrgy stat, is no longr at φ=0. Th tru vacuum stat must b at on of th two minima, say at φ = v + /λ. W now rwrit th Lagrangian in trms of small dviations φ φ with rspct to this minimum v: φ v+φ : φ : 4 4 L= ( ( )( ) φ)( φ) + φ λ φ = 4 ( φ' + v) ( φ' + v) + ( φ' + v) λ ( φ' + v) = = ( ')( ') ' v ' v ( ' v 4 ' v 4 ' v 6 ' v ) φ φ + φ + φ + φ + + φ + φ + φ = 4 v 4 3 = ( ')( ') ' φ φ + φ ( φ' + 4 φ' v) + v 4 v 4 Kintic Enrgy trm mass trm highr ordr trms constant In summary: w obtain a kintic K G trm (drivativ squard) for fild φ, a propr mass trm (th scond trm) for a particl of mass m=, and tripl and quartic vrtx coupling trms of th fild φ (th third trm in brackts). Th last trm is a constant and can b ignord (it dosn t lad to any trm in th Eulr Lagrang g quation of motion). W hav rgaind a mass trm for th fild whn xpanding th fild about its vacuum valu! Th original rflction symmtry, whn xprssd in th fild φ, is now hiddn; it will /09/009 rappar if w r writ th Lagrangian in trms of th original fild φ! 0

21 Brokn Symmtry A mchanical analogy can b mad: Considr a plastic rulr hld hld vrtical at th nds. A vrtical lforc is applid straight down at th top of th rulr. For a small nough vrtical forc, th rulr will rmain straight, and tapping th rulr sidways at th middl may induc small oscillations, s th lft figur. This situation is prfctly lft right symmtric. Howvr, for a somwhat largr vrtical forc downwards, th right figur, th rulr will find a nw quilibrium position whn it is bnt ithr to th lft or to th right (or any dirction in th plan). Th systm s symmtry is no longr apparnt, is apparntly brokn but is hiddn. Small oscillations may b inducd around th nw nrgy minimum in th bnding plan. NO oscillations ar possibl prpndicular to th bnding plan! Only much largr oscillations, causd by a substantial sidward tap, will pass through th local maximum in th cntr, and ar again symmtric around th vrtical position. F F /09/009

22 A QED lik Modl with a Complx Higgs Fild φ Lt us now considr a slightly mor complicatd Lagrangian, which will xhibit a local (and global) gaug invarianc. W will tak QED as th xampl of choic. In addition to th Elctromagntic photon fild kintic trm ¼F ν F ν, w will again add a potntial with local maximum nar th origin. This tim w will tak a complx scalar fild φ. φ As bfor, φ may b dcomposd into two indpndnt filds φ and φ of qual mass by dfining φ =(φ +iφ )/ ; in fact, this is a handy way of kping track of th two qual mass filds. W will show that: Th coupling of th fild φ with th photon A lads to a mass trm for th photon. In addition, of th two indpndnt componnts of φ, only on rmains, and it is massiv. Thothr fild disappars:hasbn atn by th photon, which thrby gaind wight! Not, that all this will b accomplishd without braking gaug invarianc! * ν 4 Lagrangian: L = T + T V( φ) = ( D φ) ( D φ) F F φ + λ φ with: QED φ QED ν 4 D + iqa, F A A ν ν ν ( ) 4 V ( φ) /09/009

23 Lagrangian: A QED lik Modl with a Complx Higgs Fild φ 4 ( 4 ) 4 L = T + T V( φ) = ( D φ) ( D φ) F F φ + λ φ QED * ν φ QED ν with φ a complx scalar fild, and D + iqa, F A A φ(x) and A (x) transform undr local phas/gaug transformations as: ν ν ν i α ( x ) φ ( x ) φ '( x ) = φ ( x ) A ( x) A '( x) = A ( x) α( x) q V ( φ) It is clar that th abov Lagrangian g is invariant undr such transformations by construction; it contains a masslss photon. /09/009 3

24 V(φ) Now obsrv what happns whn w considr small dviations around th tru minimum. Th locus of minima forms a full circl (dashd rd) in th Im(φ complx plan of th complx gnralizd ) coordinat φ(x), with a radius v = /λ. R(φ) Th minima ar dgnrat: v Oscillationsalong along th vallybottom cost no nrgy vally of minima and ar thus masslss. Im(φ) But, oscillations prpndicular to th vally floor ar xprincing a +v parabolic potntial ζ φ φ and will lad to a mass trm. As bfor, w will pick a convnint vacuum xpctation valu, φ = v (ral), v and considr η R(φ) small oscillations φ (x) = φ(x) v around this vacuum. Th complx fild φ (x) φ consists of two indpndnt ral filds, η(x) and ζ(x), so that φ (x) η(x) + iζ(x). For small φ (x) <<v, w may writ: i x v φ ( x ) v + φ'( x ) = v + η ( x ) + i ζ ( x ) v + η ( x ) ζ for η ( x ), ζ ( x ) << v ( )/ ( ) ( ) Not that indd th (phas) angl of φ is ζ/v. /09/009 4

25 ( ) ( ) ( )/ i x v φ( x) v+ φ'( x) = v+ η( x) + iζ( x) v+ η( x) ζ for η( x), ζ( x) << v Substituting this xprssion into th Lagrangian, vally of minima w xpct to s trms apparing in η(x) and ζ(x). Im(φ) Howvr, ζ(x) ntrs only as a phas factor xp(iζ(x)/v). ζ φ φ Bcaus th Lagrangian is xplicitly invariant undr local phas rotations, w may actually us th v η local gaug frdom to rotat th phas factor away by th gaug transformation: ( ) ( ) φ φ φ + η = + η A A x A x x qv iζ / v iζ / v iζ( x)/ v " v ( x) v ( x) "( ) = ( ) + ζ ( ) Thus, fild ζ(x) will b compltly absnt from th Lagrangian! R(φ) Not that th choic of minimum (vacuum) φ 0 = v = ral, is not so spcial; aftr all, any othr vacuum xpctation valu can b rachd simply by making an appropriat phas rotation in th wak isospin spac. Lt us s th othr consquncs that rsult from th chang in prspctiv looking at φ as dviations φ from th minimum v. Without loss of gnrality w mak th rplacmnt φ φ =v+η(x) in L: /09/009 5

26 Lt us s th othr consquncs rsulting from th chang in prspctiv looking at φ as dviations φ from th minimum at v. iζ / v iζ / v iζ( x)/ v φ φ" φ ( v + η ( x ) ) = ( v + η ( x ) ) A A "( x) = A ( x) + ζ ( x) qv Without loss of gnrality w may thus mak th rplacmnt φ v+η(x) in L: ( ) * 4 φ φ φ λ φ 4 4 ν L= ( D " ") ( D " ") " + " F " F " = ( ( ζ ))( η( )) ( η( )) λ ( η( )) ν λ= v ν = + iq A + qv v + x + v + x v + x F F = 3 4 η η v ν = ( + iqa" )( v + η( x) ) η + + F F 4 ν = v 4v ( ) ( )( ) η η v ν = q A " A " v η + + η η + F 4 ν F = v 4v 4 ν ( )( ) F F qva" A = η η η + " + q η A " A " + q vηa " A " ( ) 4 ν K-G quation for massiv η K.E. of Photon Photon mass Tripl and Quartic η -Photon couplings /09/009 η slf couplings 6 ν ( / v η /4v ) η + 3 4

27 Toy Modl with a Complx Higgs Fild φ In conclusion: w find that if w brak th symmtry (by considring i th fild φ with ihrspct to a chosn minimum) in a thory that xhibits local gaug invarianc, th initially masslss gaug fild (th fild A (x)) acquirs mass. Inrturn rturn, on of th scalarfildsdisappars disappars (th tangntial fildζ(x)) ζ(x)), lavingaa singl ral, massiv, scalar (Higgs) radial fild η(x): η-fild, (ral), mass: ζ -fild disapprs; is 'rotatd away' A -fild, acquirs mass: qv /09/009 7

28 Symmtris of th Standard Modl: U(), SU(), SU(3) W v now constructd a powrful toolst: w know how to dscrib fr lmntary particls: il bosons with ihth Klin Gordon Lagrangian; and frmions with th Dirac Lagrangian. W introduc intractions by th us of th gaug principl: rquiring invarianc of th Lagrangian undr local phas/gaug transformations of th particl filds, w ar ld to introduc compnsating gaug filds of th propr form. Th gaug filds hav to b masslss in ordr to prsrv gaug invarianc, which is clarly a problm whn considring wak intractions whr th gaug g bosons ar massiv. Howvr, w can ovrcom this problm by invoking th Higgs mchanism: postulating th xistnc of a boson fild with an odd shap, i.. a non zro xpctation valu, and by rxprssing th Higgs fild with rspct to a tru minimum (vacuum), w brak th manifst symmtry of th Lagrangian, but rap grat bnfits as wll: th gaug boson acquirs mass, just what w nd for th wak gaug bosons. A final bnfit from th us of a thory with local gaug invarianc is that such thoris ar inhrntly slf consistnt it twhn highr ordr h diagrams ar considrd. d Vry lgant cancllations btwn diagrams occur, which cur divrgncs that would othrwis mak th thory maninglss. /09/009 8

29 Symmtris of th Standard Modl: U(), SU(), SU(3) W labl th gaug symmtry by its group structur. Th simplst gaug transformation is th multiplication li li i of th fild by a phas factor function: xp{iα (x)} + iα + (iα) /! + (iα) 3 /3! +..., which is th (infinit) group of complx functions that hav modulus. Th group opration ismultiplication (addition of th phas angls), and th unit lmnt is xp{0} =. Th group s lmnts ar clarly unitary: [xp{iα(x)}] = xp{ iα(x)} = [xp{iα(x)}]. Th official nam of th group is unitary group of dimnsion on : U() Mor complicatd gaug g groups ar formd by bringing gin matrics:.g. th Pauli matrics xp{i½σ β(x)} + i½σ β + (i½σ β) /! + (i½σ β) 3 /3! This group is again an infinit group of complx oprator functions that hav dtrminant and ar unitary. Thy ar unitary bcaus th Pauli matrics ar Hrmitian (and so is iσ β); bcaus th Pauli matrics ar traclss w also find that Dt[xp{i½σ β}] = xp{tr[i½σ β]} = xp{0} =. This group s nam is spcial unitary group of dimnsion, SU(). ( ) Its gnrators ar th Pauli Matrics, bcaus any mmbr of th group can b formd by appropriat combinations of th thr Pauli matrics. /09/009 9

30 Th SU() L of Wak Isospin Th basic particl for th wak intraction is th lctron nutrino ν and lctron doublt Th wak intraction i procsss that w hav sn procd by coupling of th W vctor boson to th lft handd nutrino lctron or up down quark doublts: * * ν + W, W + ν * * n p+ W, W + ν * * = d u + W, W + ν * * π W, W + ν whr th W * is virtual, maning its mass is not its on shll mass of 80.4 GV. Exprimntally, th W coupls th sam to lft handd ν and lft handd : it dos not car about th diffrnc in lctric charg or in mass! Th lft handd u quark and d quark also form a wak doublt: lik th and th ν, thy turn into on anothr by th mission or absorption of a W boson. This apparnt invarianc undr rotations of th wak lft handd doublt is an xact analog of th spin and isospin formalisms considrd arlir. Thus, w will assign th lft handd nutrino a Wak Isospin z componnt of +½, and th lctron ½: = I, I =,, ν = I, I =, + (3) (3) W W W W /09/009 30

31 Wak Iso doublts W will rquir th wak Lagrangian to b invariant for gaug transformations of th SU() of wak isospin (rotations in th wak isospin spac): ν ν' iσβ ( x)/ ν i iβ3 i( β iβ) PLν xp ' = = i( β+ iβ) iβ 3 PL L L L with P L ½( γ 5 ) acting on th Dirac spinors for th nutrino and for th lctron, as bfor. Not that for all wak isospin doublts, th uppr mmbr of th doublt must hav on unit of charg mor than th lowr mmbr! For th SU() invarianc to b stablishd in th Lagrangian, w nd to rplac th rgular drivativ vrywhr by th SU() covariant drivativ D + ½igσ b, whr g is an arbitrary coupling constant. Th transformation of th thr gaug filds b (x) is closly linkd to th thr filds β(x), bcaus its transformation nds to cancl trms arising from drivativs of β(x): b i ' = U b U + ( U ) U, with U g i σ β( x) This was drivd arlir (nxt slid): /09/009 3

32 Transformation Proprtis of th Gaug Filds b (x) A gnral local phas transformation can b writtn as: φ ( x) φ( x )' = U ( x ) φ ( x ), with covariant drivativ: D = + igb with U(x) a unitary transformation. W can thn driv th transformation proprty for B that allows th Lagrangian to b invariant whn th covariant drivativ rplacs th normal drivativ vrywhr: invarianc rquirs: ( D φ)' = U( D φ) ( D φ)' = D ' φ' = ( + igb ') Uφ = ( + igb ') Uφ = U ( + igb ) φ ( + igb ') U φ = U ( + igb ) φ solv for B ': igb ' Uφ = ( Uφ) + U( φ) + igub φ = ( U) φ+ igub φ = ( ( U) + igub ) φ i i B ' U = ( U + UB ) B ' = UB U + ( U) U g g Not that w nd to kp th ordr of oprations bcaus U may b a transformation matrix, for instanc U(x)=xp{i½σ β(x)}, and B may b a vctor or tnsor. To chck, try U= ia : iα iα iα iα iα iα iα ' = + ( ) with = : ' = + ( ) = + ( α) = ( α) /09/009 3 rquir q q q q A UA U i U U U A A i A i i A

33 Similarly hr. W ll us th infinitsimal form to as th calculus, i.. w tak β(x) <<. W xpct b = b + δb, with δb small in th infinitsimal approximation. Thn, rquiring that D ψ transforms lik ψ itslf: σβ ( ) i ψ = ψ + σ b = + ( D )' ( D ), with D ig igb valuating th lft-hand and right-hand sids sparatly: ( ) i σβ LHS: ( D ψ)' = D ' ψ ' = + igσ b ' ψ + igσ b ' + i σβ ψ = = ψ + ig( ') ψ + i σ ( β) ψ + i ( σβ ) ψ g( σb ')( σβ ) ψ = σ b ( ) ( ) 4 = ψ + ig( σ b ') ψ + i σ ( β) ψ + i ( σβ ) ψ g ( σb )( σβ ) + ( σ δb )( σβ ) ψ 4 ( ) ( ) i σβ D i σβ D i σβ ig σb RHS: ( ψ ) + ψ = + + ψ = = ψ + ig( σ b ) ψ + i ( σβ ) ψ g( σβ )( σb ) ψ 4 { } nd cancl trms btwn sids, and ignor th ordr trm with δb : i σ b ' = σ b σ ( β) + ( )( ) ( )( ) g σ β σ b σ b σβ nd ordr /09/009 33

34 i σ ( b ' b ) = σ ( β) + ( )( ) ( )( ) g σ β σ b σ b σβ Th lastsquar bracktd trm can b valuatd furthr; in ordr to do so w ll xplicitly show th thr vctor indics as lowr roman indics. As bfor, w assum summation ovr qual indics in a product: ( σβ )( σb ) ( σ b )( σβ ) = ( σβ i i )( σ j b j ) ( σ j b j )( σβ i i ) = β i b j ( σσ i j σσ j i ) = β i b j [ σ i, σ j ] using th group structur of th Pauli matrics: [ σi, σ j] = iεijkσk ( σβ )( σb ) ( σ b )( σβ ) = iσε βb = iσ ( β b ) k ijk i j Combining i and, and dividing idi out th common factor σ, w find th transformation: i b ' = b + δb = b g ( β) + i( β b ) = b ( ) ( ) g β β b so that indd a simultanous transformation of th gaug filds b (x) can b constructd which kps th Lagrangian invariant undr SU() phas transformations of th particl filds. /09/009 34