Forces. Quantum ElectroDynamics. α = = We have now:

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1 W hav now: Forcs Considrd th gnral proprtis of forcs mdiatd by xchang (Yukawa potntial); Examind consrvation laws which ar obyd by (som) forcs. W will nxt look at thr forcs in mor dtail: Elctromagntic Wak Strong 1 Quantum ElctroDynamics In th modl of intractions w hav proposd, a charg, for xampl, intracts by mitting and absorbing virtual photons. W now xamin th possibility of obsrving consquncs of this which ar not prdictd by standard quantum mchanics. Our modl of an lctron intracting with an xtrnal lctromagntic fild involvs it in absorbing a virtual photon, and thus changing its momntum. Howvr, othr intrnal intractions can occur. An lctron, of momntum p, may mit a virtual photon of momntum k, and hnc continu with a rducd momntum p k until it rabsorbs th virtual photon. Similarly a photon of momntum k may convrt into a virtual lctron positron pair, with th lctron and positron sharing th original momntum, until th virtual pair rcombins and producs th photon again. Indd, mor complicatd cass may occur, involving combinations of mission of virtual photons 2 with virtual pair productions. Each coupling of a photon to a frmion lin, known as a vrtx, involvs a factor α in th amplitud (so α in th 2 cross-sction), whr 1 α = = 4πεħc 137 Th abov intrnal intractions ar known as slf-nrgy trms. Th first contributs to th apparnt mass of th lctron, whil th scond contributs to th ffctiv charg (and is rsponsibl for th procss known as vacuum polarisation ). W can sum ovr all diagrams, and intgrat ovr all intrnal loop momnta, in an attmpt to calculat th ffctiv mass and charg. Howvr, naiv attmpts to do this rsult in valus that ar infinitly larg! A littl thought shows that it is not rasonabl to put in th masurd valus of m and as th bar paramtrs of th thory, and thn calculat som modifid, ffctiv valus. Whnvr w do an xprimnt with an lctron, it is surroundd by its cloud of virtual photons and pairs it is 3 m ffctiv that w masur in xprimnts! 0 A rigorous mathmatical procss known as rnormalisation allows us to us th physical mass and charg and (for most procsss) ignor intrnal loops in th particl lins. Extrnal loops (.g. thos coupling incoming and outgoing particls) will, howvr, b xpctd to hav a finit, obsrvabl ffct. 4

2 g 2 Thory Thr ar svral proprtis which xhibit th quantum natur of th lctromagntic intraction. On of ths is th magntic momnt of chargd lptons, in particular th muon. Th Dirac quation for a point-lik lctron or muon, which ariss from a rlativistic quantum mchanical tratmnt of th particl but without th us of a fild thortical approach to dscrib intractions, prdicts that a componnt of th lpton s intrinsic magntic momnt must b z = ± B, whr B is th Bohr magnton, B = ħ 2m or, by analogy with atomic physics, writing = g B s, with s z = ±½ thn g = 2. 6 Without QED, th prdiction is thrfor that g = 2. W can allow for th xistnc of xtra physics by writing g = 2 (1 + a), whr a is known as th anomalous magntic momnt. Th full QED calculation is vry involvd, but fild thortical considrations lad us to th xpctation that th highst ordr contribution to a will b of th ordr of 1 α, 137. W can vn produc a qualitativ xplanation of why ths highr ordr procsss might modify th obsrvd magntic momnt. 7 Whn a lpton, of spin ½, mits a photon, of spin 1, its spin, and hnc magntic momnt, must b flippd. Th avrag magntic momnt of a ral lpton might thrfor b rducd, whn compard with th xpctation for a bar particl. Th full QED calculation for th lading ordr contribution to a is 0.5 α π Anothr ffct givs a largr, positiv contribution to a. A classical pictur of th magntic momnt of a particl is that it ariss from a currnt loop around th cntr of mass of th particl. Emission and absorption of virtual photons lads to a jittr in th position of th cntr of th loop, so incrasing. Highr ordr trms dpnd on th lpton mass: a = = ( ± 0.4) 10 a = = ( ± 2.3)

3 Fynman graphs contributing to α 3 corrctions to g Rading for nxt wk from Idas of Particl Physics : Chaptr 12* Wak intraction & bta dcay Chaptr 18* Th W boson Rmindr: QED prdicts that g 2 for or is non-zro. Dfining g = 2 (1 + a), whr a is known as th anomalous magntic momnt, QED calculations giv: a = = ( ± 0.4) 10 (* in 3 rd dition! Chaptrs 13 & 19 in 2 nd dition) a = = ( ± 2.3)

4 Exprimnt A pionring xprimnt to masur g 2 for th muon was prformd by a tam including th lat Prof. Combly of this dpartmnt. A polarisd bam of muons was circulatd in a circular orbit in a uniform magntic fild. Th intraction btwn th magntic fild and th magntic momnt xrts a torqu on th muon, causing th spin dirction to prcss at a rat which dpnds on th magntic fild (Larmor and Thomas prcssions). If g = 2, th momntum and spin would turn at prcisly th sam rat, and so th polarisation would not chang. 15 Th actual polarisation of th muons was masurd through thir parity non-consrving wak dcays, + + ν ν This procss is mdiatd by a W boson (s latr), and w will show, by considring th hlicitis of th nutrino and antinutrino, that, in th rst fram of th +, th + dirction tnds to follow th dirction of th muon s spin. In th lab fram, th highst nrgy lctrons hav thir dcay momntum paralll to th muon s momntum, so thy tag tims whn th muon s spin is paralll to its momntum. Exploiting ths facts, th prcssion rat, and so th valu of a, can b masurd. 16 Tim dpndnc of th lctron counting rat from th dcay of muons in th CERN g 2 xprimnt. Not that th abscissa is foldd vry 43 s, in ordr to display th full tim rang of 360 s. Th gnral xponntial dcras corrsponds to th loss of muons by radioactiv dcay, with a man liftim dilatd by th rlativistic γ-factor of 30. (It is of intrst to not that this xprimnt providd th most prcis (0.1% accuracy) chck of Einstin's tim-dilation 17 formula.) Th ovrall dcras is modulatd by th g 2 frquncy. Exprimntally a = ( ± 9. ) QED calculation, including highr ordrs, prdicts a = ( ± 2.3) 10 9, laving a discrpancy of ( 72.3 ± 9.3) Though th abov thortical rsult is vry clos to th xprimntal valu, th diffrnc is significant somthing must b missing! This is th ffct of othr (strongly intracting) particls, which can also contribut to vacuum polarisation but ar not includd in th QED calculation. A calculation of this additional trm yilds ( 70.2 ±8.0) 10 9, nicly accounting for th abov discrpancy. 18

5 Th Standard Modl of Particl Physics Frmions Lptons Quarks Bosons ν d u γ ν s c W ±, Z 0 τ ν τ b t g What you should hav larnd Virtual procsss, highr ordrs & rnormalisation Corrctions to magntic momnt of lptons 1 st ordr ~ α Exprimntal tst: g 2 xprimnt Parity violating wak dcay of muon Enrgtic positrons idntify + spin dirction Confirmation of thortical prdiction H 0 19 (Prcis with strongly intracting particls includd) 20