Th Wak Nuclar Forc Contnts Frmi Thory and Parity Violation Higgs Mchanism Symmtry Braking - Goldston's thorm, massiv W s Hyprcharg and th Z Frmion Masss and Cabibbo Mixing Higgs Sarchs Larning Outcoms Know main xprimntal rsults on th wak forc B abl to xplain th U(1) Higgs mchanism in dtail Know Q = T 3 + Y=2 B abl to writ down W 3 ; B mass matrix and diagonaliz it B abl to xplain how th Higgs gnrats frmion masss Know th Higgs sarch mthods. Rfrnc Books Introduction to High Enrgy Physics, Prkins Elmntary Particl Physics, Knyon. 1
Th Wak Forc W now turn our attntion to th dscription of th wak nuclar forc and th associatd gaug thory. W will hav to xplor th Higgs mchanism of symmtry braking to xplain th gaug boson masss. 1 Introduction Th wak nuclar forc is rsponsibl for dcay of nucli n p +? + (1) This was th procss which lad Pauli to prdict th xistnc of th nutrino in ordr to rconcil th xprimntal data with nrgy and momntum consrvation. At th quark lvl th procss is d u +? + (2) Th forc is a wak forc in day to day lif bcaus it is vry short rang. Frmi rst succssfully dscribd th forc by postulating a Fynman rul for a four frmion intraction at a point. In modrn languag w would writ d u _ ν M = (u p u n )G F (u u ) (3) whr G F is th Frmi constant. Not it rplacs th 1=q 2 typ bit of th gaug boson propagator w hav prviously sn in this typ of xprssion. G F thrfor has dimnsion of mass?2 - xprimntally it is about 10?5 GV?2. Such a thory on its own though is not a rnormalizabl thory - at loop lvl w gt many divrgncs that may not b absorbd into paramtrs of th thory. Th story will gt considrably mor complicatd but lt's start to s th mix of idas that will ntr. 2
1.1 A Massiv Propagator Th Fynman propagator for a masslss particl controlld by th Klin Gordon quation (2 = 0) is?i=q 2. If w introduc a mass trm ((2 + m 2 ) = 0) w may trat it as a prturbation to th masslss thory. Th Fynman rul that mrgs turns out to b 00 11 im 2 Now w can work out th massiv propagator by rsumming th prturbation sris for this particl?i q 2 +?i q 2 (?im 2 )?i q 2 +?i q 2 (?im 2 )?i q 2 (?im 2 )?i q 2 + ::: =?i q 1 + m 2 2 q 2 + m4 q 4 =?i q 2 1? m 2 q 2?1 + ::: (4) =?i q 2?m 2 Not that at small momntum th propagator bcoms a constant i=m 2. This suggst that G F in th dscription of th wak forc should corrspond to th mass of a massiv xchang particl. 1.2 SU(2) Thr ar hints of an SU(2) gaug thory in th wak forc -dcay procss. SU(2) transformations act on a doublt of wav functions as ia T a (5) whr th T a ar th gnrators of SU(2) which ar just th Pauli matrics T 1 = 1 2 0 1 1 0 ; T 2 = 1 2 0?i i 0 ; T 3 = 1 2 1 0 0?1 Howvr, to plac u; d or ; in doublts w would rquir thm to b idntical particls which thy manifstly ar not in low nrgy xprimnts A gaug thory would also prdict th xistnc of thr masslss SU(2) gaug bosons which w do not obsrv. (6) 3
1.3 Parity Violation An addd confusion to th mix is providd by Mdm Wu's xprimnt that shows th wak forc violats parity. Sh studid th dcay of Cobalt 60 in a magntic ld Th spins in th xprimnt align as Bz 60 Co 60 Ni +? + (7) J=5 J=4 J=1/2 J=1/2 On would xpct th lctrons to mrg at all angls. If w tak th masslss limit and think about hlicity in this xprimnt w can s that at 0 o and 180 o thr could b procsss at 0 o ν RH LH o at 180 ν LH Exprimntally though on masurs an angular distribution of th mrging lctrons that ts th prol 1? cos. In othr words at = 0 o thr is no production. This mans that th wak forc violats parity and only coupls to lft handd particls. In th masslss limit whr lft and right handd particls dcoupl in th Dirac quation thr is no rason why this should not b th cas. Th confusion w will hav to rsolv is how this can b consistnt with th lctron having as mass that mixs ths dirnt hlicity stats. RH 4
2 Th Higgs Mchanism Th ky additional ida w nd to mak sns of th wak forc is th Higgs Mchanism. W will now xplor ths idas in simpl nvironmnts bfor rturning to th full thory of natur. 2.1 Introduction Our rst task will b to undrstand how to giv masss to gaug bosons. This sms lik a hard task givn that th much prizd gaug symmtry principl activly forbids such masss. Th Higgs mchanism introducs an xtra ingrdint - w will ll spac with mattr chargd undr th gaug symmtry. A gaug boson thn trying to propagat through this \goo" will intract continually and w will s that th nrgy of intraction will bcom th ctiv mass of th gaug boson. Mass is a scalar quantity and this tlls us th background goo must b a scalar ld (or wav function),, th Higgs. Lts do th basics of th maths in QED whr w will giv th photon a mass. W hav sn how such a chargd particl with a scalar wav function (satisfying th Klin Gordon quation) would ntr into th QED Maxwll quations 2A = J = iq( D? (D ) ) (8) th currnt is just th charg tims th numbr dnsity currnt but whr w hav st @ D = @ + iqa as minimal substitution rquirs. Now imagin that throughout spac = v - w say it has a vacuum xpctation valu (vv) oftn writtn hi = v (9) Spatial drivativs of v ar just zro and th only surviving bits in th wav quation abov ar or 2A =?2q 2 v 2 A (10) (2 + m 2 )A = 0; m 2 = 2q 2 v 2 (11) which maks th gaug ld massiv as w wantd Whn w giv a vv w ar ctivly changing th normalization of th wav function so thr ar a larg numbr of particls pr unit volum - w can think of as a classical ld in this limit. Why would hav this non-zro vacuum valu though? Wll w'll just cook things so it dos without answring that qustion Hr's an xampl potntial for that dos th trick 5
V v This looks a bit adhoc but so far so good. W will s that, sinc this background goo has lctric charg, all lctrically chargd particls will nd up coupling to it and gt a mass as a rsult. W will gnrat th lctron mass tc in th SU(2) vrsion of this story. This all looks quit simpl but to undrstand how w hav got a mass and gaug invarianc togthr w will nd to think mor dply about what w'r doing. 2.2 Symmtry Braking Th rason w can suddnly hav a mass for th gaug ld in th Higgs Mchanism is bcaus th symmtry is bing brokn by th vacuum of th thory. To bgin to undrstand this ida think about a frromagnt. Abov th critical tmpratur th atomic spins ar randomly pointing du to thrmal xcitation. Blow th critical tmpratur th thrmal nrgy is insucint to disrupt a natural alignmnt of th spins du to thir intractions. Φ T > Tc T < T c spins random spins alignd In th absnc of dfcts, th dirction in which th spins choos to nally li arrangd, blow T c, is truly random in a quantum thory. Th nal arrangmnt picks out a particular dirction in spac and thrfor braks rotational symmtry that xistd at high tmpratur or without th atoms prsnt. This is th basic ida of spontanous symmtry braking. 2.2.1 Braking an Intrnal Symmtry Considr a complx scalar ld with a global U(1) intrnal symmtry i (12) 6
W can writ a potntial for th ld V = 1 2 2 + 4 ( ) 2 (13) Whn 2 is positiv th potntial looks lik V R φ Im φ Th minimum is at = 0. Th potntial clarly xhibits th U(1) symmtry. If 2 is ngativ though th potntial looks lik V Im φ R φ Clos to th origin th jj 2 trm dominats, whilst at larg valus th jj 4 trm dominats. Thr is a ring of minima. To nd th valu of jj look on th ral axis and rquir so th minima is at dv d = 0 = 2 + 3 (14) jj 2 =? 2 Th valu of th potntial at th minimum is V min =? 4 4 (15) (16) 7
Globally th potntial again xhibits th U(1) symmtry (th potntial dos not dpnd on th phas of ). Howvr, th minima hav a symmtry braking form. thr ar many dgnrat vacua and which on w nd up in will b random. Whn on looks locally around a minimum thr is not a rotational symmtry. 2.2.2 Goldston's Thorm In th abov xampl thr is a masslss stat. W can s this by writing = r i ; = r 2 (17) Th potntial dos not dpnd on. Thus thr is no nrgy cost to a conguration whr changs in spac V vacuum 1 vacuum 2 drivativ nrgy Im φ vacuum 1 R φ vacuum 2 Th only nrgy cost is du to th drivativ from th changing ld. In trms of th Klin Gordon quation though (2 + m 2 ) = 0 (18) w know that drivativs of ar not a mass but momntum. So this stat is masslss. This is not tru for th r dgr of frdom. If w chang r thr is a dirct nrgy cost from th potntial which corrsponds to a mass. Goldston's Thorm: thr is a masslss dgr of frdom for vry brokn gnrator of a group. In th abov cas th U(1) dgr of frdom is brokn so thr is on masslss mod. If w wr to brak an SU(2) symmtry with thr gnrators w would thrfor xpct to nd 3 masslss mods in th thory. Is this a block to dscribing th wak forc as a brokn symmtry? - thr ar no physical masslss mods. In fact in a gaug thory somthing xtra happns. 8
2.3 Suprconductivity Th rst xampl of a brokn gaug symmtry to b undrstood was in th phnomna of suprconductivity. In a suprconductor intractions with th lattic of ions producs forcs that bind lctrons into Coopr pairs - a charg -2, spinlss bound stat of two lctrons. W can trat th Coopr pair as a scalar ld. Lt's look at a rlativistic vrsion of this thory. W will introduc som potntial as abov that givs th scalar a non-zro vacuum xpctation valu This will ntr into th photon's Klin Gordon quation hi = v (19) giving 2A = iq D? iq(d ) (20) (2 + m 2 )A = 0; m 2 = 2q 2 v 2 (21) Th spatial bhaviour of a constant magntic ld dscribd by a vctor potntial is thn th solution of with solution r 2 ~ A = 2q2 v 2 ~ A (22) ~A = ~A 0?p 2qvx (23) Th vctor potntial thrfor dcays away xponntially in th matrial and magntic lds ar xplld by th suprconductor. W wantd to ask whthr thr was a masslss mod associatd with th at dirction in th scalar potntial. In fact a spatial uctuation in th phas of th scalar ld can b gaugd away. If w writ a conguration thn w can mak a gaug transformation = r i(x) (24) r i(x) iq(x) (25) Clarly w can choos to cancl. In this way w can always rmov th masslss uctuations - it is not physical. W should think about th dgrs of frdom in this problm. Initially w had a masslss photon with two polarizations. Now it is massiv it bcoms possibl for thr to b uctuations in th longitudinal dirction of motion. Whr has this xtra dgr of frdom com from? - it is prcisly th dgr of frdom corrsponding to th uctuation that th gaug thory has canclld. Th gaug ld has \atn" th Goldston boson to bcom massiv 9
This modl thrfor producs a massiv photon and a massiv dgr of frdom associatd with uctuations in r. This latr ld is calld th Higgs Boson. 3 A Toy Modl of th Wak Forc Lt's try to put all of our pics togthr to mak a thory of th wak forc - it will turn out w'r still lacking on ingrdint though. SU(2) L If all th frmions wr masslss w could plac lft handd hlicity spinors in SU(2) gaug doublts L ; th right handd lds u r ; d r ::: ar singlts of SU(2). Sinc w'r in an SU(2) gaug thory w hav thr gaug lds W 1 ; W 2 ; W 3 associatd with th thr gnrators 1 2 0 1 1 0 ; 1 2 0?i i 0 u d ; L 1 2 1 0 0?1 rathr than W 1 ; W 2 it's mor physical to think about th W bosons associatd with T 1 it 2 = 0 1 0 0 or 0 0 1 0 Ths chang th top componnt of a doublt into th bottom and vic vrsa. W call this basis th W + and W? gaug bosons. Th Higgs (26) (27) (28) W nxt introduc an SU(2) doublt scalar ld = 1 + i 2 3 + i 4 (29) with a symmtry braking potntial V =? 2 2 jj2 + 4 jj4 (30) W can look at th vacuum whr = v 0 (Not you can gt to any othr vacuum by making a gaug transformation so thr is no lack of gnrality hr). (31) 10
W will hav 3 Goldston Bosons that ar atn by th 3 W gaug bosons to gt masss M 2 W ' g 2 W v 2 (32) and nally w will hav a Higgs ld whos mass dpnds on th choic of and. What's missing: w hav not yt includd QED in this discussion and th thr W particls hav ndd up dgnrat whilst in natur w s two dgnrat W and th Z. 4 Hyprcharg Including th U(1) of QED into our modl is not straightforward. Th complt st of symmtry transformations w can mak on a lft handd doublt ar whr th full st of possibl T a ar L ia T a T 1 = 1 0 1 ; T 2 1 0 2 = 1 0?i ; T 2 i 0 3 = 1 1 0 ; T 2 0?1 4 = 1 1 0 2 0 1 (34) Th rst thr ar th gnrators of SU(2). Th last though can not b QED bcaus and do not hav th sam lctric charg. Th lctric charg matrix is givn by Q = 0 0 0?1 L (33) = T 3? T 4 (35) W hav no choic but to gaug th U(1) associatd with T 4 - it's calld hyprcharg. In ordr to rmov this symmtry at low nrgis w will mak th Higgs hav hyprcharg so th associatd gaug boson, B, bcoms massiv. To obtain QED th Higgs vv will hav zro T 3? T 4 charg so a mixtur of W 3 and B will b abl to propagat and bcom th masslss photon. W will thus rquir Y Higgs =?1; T 3 Higgs = 1 2 ; Q Higgs = 0 (36) Th T 3 charg mans w ar placing th vv in th top lmnt of it's doublt - (v; 0). Not w ar using convntions whr Q = T 3 + 1 2 Y (37) 11
4.1 Th Masslss Photon Th 3 W bosons gt mass from thir intraction with th Higgs. W can draw this as from which w s that W v 2 T g 3 W T g 3 W W Similarly th B gaug boson gts a mass M 2 W = 1 4 g2 W v2 (38) B v 2 B -Y/2 g Y -Y/2 g Y Thr can also b a mass mixing btwn th W 3 and B M 2 B = 1 4 Y 2 g 2 Y v 2 = 1 4 g2 Y v 2 (39) W T g 3 W v 2 -Y/2 g Y B M 2 mix = 1 4 g W g Y Y v 2 = + 1 4 g W g Y v 2 (40) Th full W 3 ; B gaug boson mass matrix can thn b writtn (W 3 ; B ) v2 4 g 2 W + g W g Y + g W g Y g 2 Y W 3 B To nd th mass ignstats w can insrt \on" writtn as U y U, with U a unitary matrix, to th lft and right of th mass matrix, i, w group things as h (W 3 ; B )U yi " U v2 4 g 2 W + g W g Y + g W g Y g 2 Y U y # " U W 3 B Now by choosing an appropriat U w can diagonaliz th mass matrix to its ignvalus. Factoring out th v 2 =4 # (41) (42) (g 2 W? )(g 2 Y? )? g 2 W g 2 Y = 0 (43) Th ignvalus ar 2? (g 2 W? g 2 Y ) = 0 (44) 12
= 0; = v2 4 (g2 W + g 2 Y ) (45) Thr is a masslss stat - th photon - and a massiv stat - th Z boson. Th ignfunctions ar Z = g W W 3 + g Y B q gw 2 + gy 2?? g Y W 3 A = g W B q gw 2 + gy 2 W can think of this transformation as a rotation and writ (46) (47) Z = cos W W 3 + sin W B (48) with A = sin W W 3?? cos W B (49) cos W = g W q g 2 W + g 2 Y ; sin W = g Y q g 2 W + g 2 Y (50) 4.2 Frmion Chargs W hav to assign all th chiral frmions appropriat chargs to th hyprcharg gaug boson so that thir QED chargs work out right. This givs T 3 Y Q = T 3 + 1 2 Y L + 1 2?1 0 l? 1 2?1?1 R 0?2?1 (51) u L + 1 2 + 1 3 + 2 3 d L? 1 2 + 1 3? 1 3 u R 0 + 4 3 + 2 3 d R 0? 2 3? 1 3 13
So for xampl th photons coupling to th lft handd lctron is givn in trms of th W 3 and B gaug bosons by sin θ W W 3µ g /2 W L L cosθ W W want to gt? as th answr so w larn that B µ g /2 Y g W sin W = g Y cos W = (52) L L Dos th right handd lctron - photon coupling work now? Thr is no coupling to th W 3 cos θ w B µ g Y R W rquir R which is consistnt with what w hav abov.?g Y cos W =? (53) Th L should not coupl to th photon. W gt th two contributions 1 2 g W sin W? 1 2 g Y cos W = 0 (54) using th rlation abov again. It works Finally w should mntion th right handd nutrino which is not in th list abov. It has no wak intractions and QED charg zro, so it's hyprcharg is zro as wll. If this particl xists w can not tll from lctrowak intractions. Furthr th nutrinos wr thought to b masslss until vry rcntly so right handd nutrinos wr not includd in th original \Standard Modl". 4.3 Z Intractions Each chiral frmion has a contribution to its coupling to th Z on from W 3 and on from B cos θ w W 3µ B µ sinθ w g wt3 g Y/2 Y g Z = g W cos W T 3? g Y sin W Y 2 (55) 14
5 Exprimntal Conrmation of th Elctrowak Thory Th thory dscribd abov works vry wll (in fact astonishing wll) and combind with QCD is calld th \Standard Modl" of particl physics. Lt's look at a fw tstd prdictions (thr ar vry many). Firstly on must x th paramtrs of th thory which ar g W, g Y and v. Thr standard masurmnts ar G F dtrmins v G F g2 W M 2 W 1 v 2 d W u _ ν EM (q 2 = 0) dtrmins and hnc on of g W and g Y γ M Z dtrmins th othr coupling Z σ + Ths masurmnts tll us Mz s v = 246GV; g Y = 0:356; g W = 0:668; sin 2 W = 0:231 (56) Now w can mak som prdictions 5.1 W Mass W can look for W production in pp collisions d W u W nd a rsonant pak at about 81 GV. W prdictd that M W M Z = g W q g 2 W + g 2 Y _ ν = cos W (57) Currnt prcision masurmnts conrm this prdiction at around th 0.1% lvl. 15
5.2 Z Width Th dcay rat of th Z boson (which dtrmins th width of th Z rsonanc pak in +? scattring data) is proportional to th sum of th squars of th couplings to all its possibl dcay products - s abov for how to work out ths couplings. Clarly this tsts a larg numbr of paramtrs of th modl and also counts th numbr of ach sort of frmion. Again xprimnt matchs prdiction at th 0.1% lvl. 6 Frmion Masss So far w hav workd in th ultra-rlativistic limit which allowd us to trat lft and right handd frmions as sparat. Lts now s how th Higgs coupls ths particls by giving thm mass. Th masss aris through intractions with th Higgs that pictorially look lik h L y R whn th Higgs gts a vv thr is a lft right coupling m = y v (58) Thr is such an intraction linking ach familiar massiv particl's lft and right handd partnrs. Not that in this intraction th hyprcharg of th lft handd particl minus th right handd particl quals +1 for all th frmion cass. Combining in th -1 hyprcharg of th Higgs w s that th intraction consrvs hyprcharg. SU(2) charg is also consrvd bcaus th Higgs and lft handd particl ar both doublts and 2 2 = 3 1 (59) In othr words thy produc a singlt which matchs th right handd particls quantum numbrs. Th couplings y f ar calld \Yukawa couplings" and ar fr choics for ach of th frmions. So w can gnrat all th frmion masss in this way but w ar just paramtrizing thm not xplaining thm. 6.1 Cabibbo Mixing Th Higgs can gnrat masss btwn any lft handd frmion and a right handd frmion such that th intraction consrvs SU(2) L U(1) Y chargs. Howvr thr ar a numbr of particls with idntical quantum numbrs - for xampl th d and s quarks ar idntical. In fact sinc ths xtra masss ar possibl thy occur i thr ar intractions of th form 16
v v v d L d R s L s R d L s R W ar lft with an o-diagonal mass matrix again (d L ; s L ) m dd m sd m ds m ss Hr th s and d notation is for wak ignstats - i th stats that th W boson producs. Now though w should diagonaliz th matrix to obtain th mass ignstats. Th wak ignstats ar found by making a rotation on th mass ignstats d w s w = cos c sin c? sin c cos c c is calld th Cabibbo angl and sin c ' 0:23. Th upshot is that as wll as th dcay W + ud m thr is also a small chanc of sing W + us m. W talk about Cabibbo allowd and supprssd intractions dpnding on whthr thr is a larg cos c or a small sin c. g d d R s R d m s m W π Cabibbo allowd cos θ c µ (60) (61) u _ ν _ µ s K u _ sin θ c W µ ν _ µ Cabibbo supprss In th full modl thr is a 3x3 matrix calld th CKM matrix that mixs in th b quark too. (Not for rasons w will not addrss hr th charg -1/3 quark sctor is th only sctor of th Standard Modl whr thr is this distinction btwn wak and mass ignstats). 7 Higgs Sarchs W hav not to dat discovrd th Higgs boson but ar closing in on it w bliv. Th Higgs potntial involvs th paramtrs and that dtrmin th vv and m h - w can't prdict th Higgs mass thrfor. If w want th Higgs to hav prturbativ couplings though m h < 1:5 TV. Th Higgs also plays a crucial rol in WW scattring. In th Standard Modl thr ar th diagrams 17
If w only includ th rst thr diagrams w nd th probability of scattring riss with nrgy until it bcoms gratr than on Th fourth diagram damps this bhaviour and provids a snsibl answr. Again th Higgs nds to b blow a TV or so for things to mak sns. W xpct thrfor to nd somthing soon. 7.1 Dirct Sarchs Th +? machin LEP was usd to sarch for th Higgs through th procss _ Z h f _ f m f + Z quarks or lptons On looks for a pak in th cross sction indicating on has passd th Higgs mass thrshold. Th main dcay product will b th b quark sinc th higsg frmion coupling is proportional to th frmion mass. Th xprimnt is now ovr and did not nd th Higgs, only placing th limit m h 115 GV (62) W wait on th LHC in 2008 which can sarch upto and abov a TV. 7.2 Indirct Sarchs At loop lvl thr ar corrctions to th LEP procsss of th form + f _ + γ,z γ,z γ,z f _ f all particls in thory f Whn w squar th amplitud w gt a cross trm that is ordr EW smallr than th tr lvl procss. In othr words this is a 1% ct w can hop to prob with prcision data. Such computations hav bn mad and w nd a limit 70 GV m h 250 GV (63) This provids mor vidnc for optimism that LHC will nd th Higgs. 18