Properties of Quarks ( ) Isospin. π = 1, 1

Transcription

1 Proprtis of Quarks Isospin So far, w hav discussd thr familis of lptons but principally concntratd on on doublt of quarks, th u and d. W will now introduc othr typs of quarks, along with th nw quantum numbrs which charactris thm. Many groupings of particls of similar mass and proprtis fittd in to common pattrns. On way to charactris ths is using isotopic spin or isospin, I. This quantity has nothing to do with th ral spin of th particl, but obys th sam addition laws as th quantum mchanical ruls for adding angular momntum or spin. Whn th orintation of an isospin vctor is considrd, it is in som hypothtical spac, not in trms of th x, y and z axs of normal co-ordinats. Nuclons (p, n), pi msons (π +, π, π ) and th baryons known as ( ++, +,, ) ar thr xampls of groups of similar mass particls diffring in charg by on unit. Th charg Q in ach cas can b considrd as du to th orintation of an isospin vctor in som hypothtical spac, such that Q dpnds on th third componnt I 3. Thus th nuclons blong to an isospin doublt p I, I =, n =, 3 Similarly th pions form an isospin triplt, + π =, π =, π =, Th rul for lctric charg can thn b writtn, ( ) Q = B + I whr B is th baryon numbr which is for nuclons and th and for msons such as th π. In trms of quarks, th u and d form an isospin doublt, (both with B = ). u, = 3 d =, Th forms a quadruplt with I = 3 /. 4 6

2 Thr quarks with I = / can combin to form I tot = / or 3 /. I tot = / givs th nuclons whil I tot = 3 / forms th. In strong intractions, th total isospin vctor (as wll as I 3 ) is consrvd. This is not tru in lctromagntic or wak intractions. 7 Strangnss It was obsrvd that som unstabl particls producd in strong intractions had a long liftim. This unusual stability for strongly intracting particls ld to th trm of strangnss. Such particls ar always producd in pairs (associatd production), and th quantum numbr of strangnss, S, was introducd, which is consrvd in strong intractions. Thus in th intraction π p Λ K, th Λ is assignd S = and th K S = +. Th strang particls can only dcay by th wak intraction, which dos not consrv strangnss (as w will discuss latr). (Tabl) 8 Th formula for lctric charg must now b modifid to rad ( ) ( ) Q = I + B + S = I + Y 3 3 whr Y = B + S is known as th hyprcharg. (This formula is known as th Gll-Mann Nishijima rlation.) Hadrons, quarks and quantum numbrs: th story so far Groups of particls with similar proprtis ar charactrisd by sam quantum numbrs,.g. isospin I and strangnss S Elctric charg dtrmind by 3 rd componnt of Familis of particls with similar proprtis (.g. sam spin and parity) can b plottd in trms of Y vrsus I 3, and form rgular gomtrical figurs (s plots). isospin, I 3 Up and down quark form isospin doublt, I = ½ ( ) Q = I + Y 3 whr Y = B + S is known as th hyprcharg. 9

3 K ( ds ) Y = S + K + ( us ) K * Y = S + K *+ π ( du ) η π + π + ( ud ) I 3 ρ φ ω ρ + ρ + I 3 K ( su ) K ( sd ) K * K * Th lowst-lying psudoscalar-mson stats (J P = ), with quark assignmnts indicatd. Th vctor-mson nont (J P = ). 3 n Y + p N ( 939) I = 3 Y + ++ (3) + Σ Σ Λ Σ + Σ ( 93) + I 3 Λ ( 6) I = Σ Σ Σ + + I 3 (384) I = Ξ Ξ (533) Ξ Ξ Ξ ( 38) Th baryon octt of spin-parity (J P ) ½ + (with masss in MV/c ). 4 I = Ω (67) Th baryon dcuplt of spin-parity (J P ) 3 / + (with masss in MV/c ). 5

4 In trms of quarks w can introduc a nw flavour of quark, th strang quark s. This has charg and baryon numbr (lik a d quark) but I = and S =. It is also somwhat havir than th u and d quarks. Sinc baryons consist of qqq, it is clar why no positiv baryons xist with S >, whil ngativ baryons ar found with S = or 3. 6 Quark Quantum Numbrs Flavour charg/ B I I 3 S... d / / u + / + / s. Not that th typ of quark is known as its flavour. Quarks. carry flavour and colour, and ach flavour of quark xists in thr colours.. 8 Th Standard Modl of Particl Physics Frmions Lptons Quarks Bosons d u γ µ µ s c W ±, Z τ τ b t g Furthr Quarks Othr, still havir quarks also xist. Th charm quark, c, has a charg of, lik th u, and can b considrd as a partnr to th s. In 3 dimnsions (s figur) particls containing c quarks can b plottd, and again show rgular pattrns. H 9

5 Quark Quantum Numbrs Flavour charg/ B I I 3 S c Disputd claim Discovrd 7! Not yt! d / / u + / + / s c + + Multiplts of hadrons containing up, down, strang and charm quarks. Th slics through ths figurs whr Charm = corrspond to th plan figurs alrady shown in th prvious diagrams, though containing nw particls in th cas of th msons, composd of cc. 3 Exampl dcays Not yt! Multiplts of hadrons containing up, down, strang and charm quarks. 4 W thus hav doublts or gnrations of quarks (d, u) and (s, c). Sinc thr ar 3 doublts of lptons, thr ar thortical rasons for xpcting a third doublt of quarks too. Particls containing b quarks (bottom or bauty) wr discovrd in 977. Th b is an vn havir vrsion of th d. 5

6 Quark Quantum Numbrs Flavour charg/ B I I 3 S c b d / / u + / + / s c + + b W thus hav doublts or gnrations of quarks (d, u) and (s, c). Sinc thr ar 3 doublts of lptons, thr ar thortical rasons for xpcting a third doublt of quarks too. Particls containing b quarks (bottom or bauty) wr discovrd in 977. Th b is an vn havir vrsion of th d. Its partnr, th t (top or truth) was first sn in 994, and its mass has now bn masurd at 74 GV/c. 6 7 Quark Quantum Numbrs Flavour charg/ B I I 3 S c b t d / / u + / + / s c + + b Th Standard Modl of Particl Physics Frmions Lptons Quarks Bosons d u γ µ µ s c W ±, Z τ τ b t g t H 9

7 What you should hav larnd Isospin, multiplts and charg Isospin and symmtry Strangnss and th strang quark Hyprcharg Multiplts in D (Y vrsus I 3 ) Charm multiplts in 3D b and t quarks 3 Quark Flavour and th Wak Intraction As w hav alrady sn, th strong and lctromagntic intractions consrv quark flavour, whras th wak intraction may chang it. In many wak dcays, th changs ar within a gnration,.g. in bta dcay th W coupls a u to a d quark; in th dcay D + K π + it coupls a c to an s. Howvr, this is not always th cas,.g. in th dcay K π th W coupls an s to a u quark, and it was sn that such strangnss-changing dcays wr slightly wakr than strangnss-consrving wak dcays. 3 Raction µ + + µ p n + π π K π Coupling Constant G.97 G.97 G.5 G Cabibbo xplaind this by proposing that th ignstats of th wak intraction ar diffrnt from thos of th strong intraction. Th strong intraction ignstats ar th u, d, s, c, b and t quarks, with wll-dfind isospin, strangnss tc. Th ignstats of th wak intraction, which dos not consrv I, S tc, ar said to b thos of wak isospin T. For simplicity, lt us first considr th first gnrations alon

8 Th wak ignstats ar th lptons and orthogonal linar combinations of th familiar quarks T3 = + µ u c T 3 = d µ c s c with d c = α d + β s s c = β d + α s (normalisation α + β = ) Raction µ + + µ p n + π π Coupling Constant G G.97 G G Cos θ C.97 G G Cos θ C α is usually known as cos θ c, whr θ c is th Cabibbo angl. K π.5 G G Sin θ C A valu of sin θ c =.5 is consistnt with th obsrvd apparnt variation of wak coupling constant with raction typ Th rlationship btwn wak and strong ignstats in gnrations can also b xprssd as dc cos θc sin θc d = s sin θ cos θ s c c c wak -stats mixing matrix strong -stats Th sam formalism can b usd for 3 gnrations, and th mixing matrix, known as th Cabibbo-Kabayashi- Maskawa or CKM matrix, can b paramtrisd in a numbr of ways. d s = b M d s b 4 Th magnituds of th matrix lmnts hav bn dtrmind xprimntally, and ar givn with 9% confidnc limits by M.9743 ±..54 ±.6.36 ±. =.5 ± ±..4±..89 ±.3.4 ± ±.5 Not that th valus along th lading diagonal ar quit clos to on, thos adjacnt to it ar significantly smallr, and th lmnts in th top-right and bottom-lft cornrs ar much smallr. This mans that th mixing rsults in stats which contain a small admixtur of th quark from th nxt gnration, whil mixing btwn st and 3 rd gnration quarks is xtrmly small. 4

9 Physically, this is shown in wak dcays by th rlativ probability of producing hadrons containing th rspctiv quarks. For xampl, whn a top quark dcays it producs a b quark. This is bound in a hadron by th strong intraction, so must b rvald as a strong ignstat. Th b is most likly to rsult in a particl containing a b quark, with a smallr probability of an s quark and almost ngligibl liklihood of producing a d quark. Thrfor, th nar-diagonal structur of th CKM matrix mans that wak dcays ar most likly to b within a gnration if allowd by consrvation of nrgy (a particl cannot dcay into on that is havir) or to th nxt gnration blow if this is not allowd. 43 Th most likly ovrall dcay chain of a b quark is thrfor b c s u. 44 For two gnrations, on paramtr was rquird to dscrib th mixing. This was th Cabibbo angl. With thr gnrations, 4 indpndnt paramtrs ar ndd to dfin a gnral unitary matrix, and th individual matrix lmnts may hav imaginary parts. On possibl paramtrisation of th CKM matrix is givn blow. Not that th following matrial is providd for compltnss only, and is not xaminabl! (Furthr dtails ar providd in th txt books.) [Th following paramtrisation and prvious valus ar takn from th Particl Proprtis Data Booklt, from "Rviw of Particl Physics", Chins Physics C38, July 4, by th Particl Data Group.] 45 Mixing btwn 3 gnrations: M= M c 3 s 3 c3 s3 c3 s 3 3 s c 3 c s Mixing btwn & 3 Mixing btwn & Mixing btwn & 3 s c but a gnral 3x3 (unitary) transformation rquirs 4 paramtrs c c s c s = s s c c s c s s c s c c sc3 cs 3s3 cc3 ss3s3 s3c

10 M c c s c s = s s c c s c s s c s c c sc3 cs3s3 cc3 ss3s3 s3c whr c ij = cos θ ij and s ij = sin θ ij, with i and j bing gnration labls {i,j =,,3}. In th limit θ 3 = θ 3 =, th third gnration dcoupls, and th situation rducs to th usual Cabibbo mixing of th first two gnrations, with θ idntifid with th Cabibbo angl. (Th ral angls θ, θ 3, θ 3 can all b chosn to li in th first quadrant.) c 3 is known to diffr from unity only in th sixth dcimal plac. If th paramtr δ is non-zro, thn th matrix is complx, and th small dgr of CP violation prsnt in th wak intraction can b xplaind naturally. This is still th subjct 48 of rsarch! What you should hav larnd Strangnss-changing and strangnssconsrving wak dcays Cabibbo thory. Wak & strong ignstats, quark mixing CKM matrix Prfrnc for wak dcays:- within a gnration (if allowd) othrwis to nxt gnration blow Anothr illustration of wak & strong ignstats nxt lctur! 49 Rading for nxt wk from Idas of Particl Physics (Edition 3): Chaptrs 34, 35.- R Chaptrs 5, 6, 7 Scaling Sction 3. Scaling violation. From Edition : Chaptrs 35, 36.- R Chaptrs 6, 7, 8 Scaling Sction 33. Scaling violation. 5