The Quark Model. Introduction to Elementary Particle Physics. Diego Bettoni Anno Accademico

Transcription

1 The Qark Moel trocto to Eleetary Partcle Phyc Dego Betto Ao Accaeco -

2 Otle yetre a grop overvew The flavor yetry q q tate: eo Peocalar a vector eo Zweg rle q q q tate: baryo Color Haro ae a agetc oet

3 yetre a Grop lltratve exaple: the rotato grop Two cceve rotato R followe by R are eqvalet to a gle rotato R=R R. The grop cloe er ltplcato. There a etty eleet o rotato. Every rotato R ha a vere R - rotate back aga. The proct ot ecearly cotatve R R R R bt the aocatve law alway hol: R R R = R R R. t a coto grop: each rotato ca be labele by a et of cotoly varyg paraeter whch ca be regare a the copoet of a vector recte alog the ax of rotato wth agte gve by the agle of rotato. The rotato grop a Le grop: every rotato ca be expree a the proct of a cceo of fteal rotato arbtrarly cloe to the etty. The grop the copletely efe by the eghborhoo of the etty.

4 Rotato are a bet of the Loretz traforato a they for a yetry grop of a phycal yte: Phyc varat er rotato. For exaple ppoe that er a rotato R the tate of a yte trafor a R Probablte t be chage by R: U + U = U t be a tary operator. U The operator UR for a grop wth exactly the ae trctre a the orgal grop R R : they are a to for a tary repreetato of the rotato grop. The Haltoa chage by a yetry operato R of the yte a the atrx eleet are preerve U U H U HU H U HU H [ U H ]

5 The traforato U ha o explct te epeace a the eqato of oto t t H t chage by the the yetry operato. A a coeqece the expectato vale of U a cotat of the oto t U UH HU

6 All grop properte follow fro coerg fteal rotato the eghborhoo of the etty. Exaple rotato throgh aro the -ax: calle the geerator of rotato aro the -ax. + = therefore herta a hece a obervable. Coer the effect of a rotato R o the wave fcto. varace er rotato reqre: For a fteal rotato aro the -ax: copoet alog the -ax of the aglar oet. varace er rotato correpo to the coervato of aglar oet. U O U U r U r R r xp y yp x y x x y z y x z x y y x r R z y x U U

7 For a rotato throgh a fte agle : The cotator algebra of the geerator : The are a to for a Le Algebra jk = trctre cotat of the grop Nolear fcto of the geerator whch cote wth all the geerator are calle varat or Car operator. For the rotato grop the oly Car operator : t follow that we ca cotrct ltaeo egetate of a oe of the geerator e.g. : ] [ e U U l l ] [ l l j j j j j j

8 The Grop U the lowet-eo otrval repreetato of the rotato grop j=½ the geerator ay be wrtte: k k k k atrc Pal The ba for th repreetato gve by the egevector of : ecrbg a p ½ partcle of p projecto p a ow. The traforato atrce: U are tary. The et of all tary atrce U Utary Grop. However U larger tha the grop U ce the all have zero trace. For ay herta tracele atrx t ca be how that: et e e Th property preerve atrx ltplcato. The et of tracele tary atrce for a bgrop of U calle U pecal Utary. The two-eoal repreetato the faetal repreetato. e Tr

9 For a copote yte j A j B A B > the operator = A + B atfe the Le algebra a the egevale + a M of a are coerve qat ber. The proct of the two rrecble repreetato j A + a j B + ay be ecopoe to the of rrecble repreetato of eo + wth ba j A j B M> where j A -j B j A +j B M= A + B j A j B M A C B A B M j A j B A B Clebch-Gora coeffcet Exaple: fro two two-eoal repreetato j=½ we obta oe -eoal = trplet a oe -eoal = golet repreetato. 4 qarplet p / oblet p /

10 U of op l l Geerator the faetal repreetato: k k k ba p

11 op for Atpartcle p p p e p C p Cp p p orer for the atoblet to trafor the ae way a the oblet we t: Reorer the oblet troce a g p p pp p pp p For NN

12 The Grop U t the grop of tary atrce wth etu=. The geerator ay be take to be ay -=8 learly epeet tracele herta atrce. There are therefore 8 geerator of whch are agoal. Th alo the ax ber of tally cotg geerator : Rak of the grop. t ca be how that the rak of the grop eqal to the ber of Car operator. The faetal repreetato of U a trplet e.g. the three color charge of a qark. The geerator are atrce: =..8 Gell-Ma atrce. 8 R G B ltaeo egevector of a 8. G- λ 6 +- λ 7 λ8 λ +- λ B- R λ 4 +- λ 5 λ

13 op a tragee: Flavor U The trocto of a eco atve qat ber ato to gget to elarge op yetry to a larger grop a grop of rak. 96 U wa propoe. The aget of partcle to U ltplet ot traghtforwar e to the hgh a fferece betwee the varo partcle trage a o trage. For exaple the baryoc octet grop partcle wth a fferece p to 4 MeV over a average octet a of MeV. U flavor yetry ch ore approxate tha U of op. We wll ee that th e to the fact that the trage qark ch heaver tha a. U yetry for the ba of the qark oel a t tr ot to be very efl to clafy haro a to erta oe of ther properte. Color U o the other ha a exact yetry of faetal org.

14 The Qark Moel Alreay 949 Fer a Yag otce that the eo ha the ae qat ber of a NN par a tate: B= P = - = Q=. They ggete that the col be coere a a NN bo tate wth a very hgh bg eergy. Wth the covery of trage partcle akata propoe to exte the oel to cle the. Th trplet allow to cotrct oe partcle all partcle kow the 96 bt there were everal reao to prefer a fferet et of ore faetal cottet: The reglarty wth whch partcle fale occre atre ggete a teral trctre alo for p. Thee reglarte col be well explae by a U yetry. 964 Gell-Ma a epeetly Zweg propoe a et of eleetary fero ter of whch all haro col be cotrcte. Gell-Ma calle the qark Zweg calle the ace.

15 They t be fero orer to cotrct both fero a boo. aalogy to the ea of Fer a Yag eo are q q par; baryo atbaryo are q q q q q q tate. orer to for otrage partcle wth charge at leat two qark are eee. Thee t for a op oblet orer to have both = a =. orer to for trage partcle a thr qark eee to whch by coveto age =-. The al ber of cottet th. To properly accot for baryo ber qark are age B=/. The plet p-party aget P =½ +. B Fro the Gell-Ma a Nhja forla Q t follow Q Q Q Qark have a fractoal electrc charge.

16 U lagage: the faetal ltplet fro whch all other ca be cotrcte a trplet. for the faetal ltplet. B coervato ple that t poble to create or etroy a gle qark. Qark-atqark par ca be create or ahlate. Flavor coervato trog teracto ple that flavor-chagg trato ca take place oly at the weak level for exaple: Q Y B etc lepto

17 q q tate: Meo Let tart wth two qark a - - Y = x Ag a thr qark there are 9 poble cobato: octet a glet er traforato U the 8 tate trafor aog theelve bt they ever x wth the glet. - - Y A B C 8

18 The glet tate C yetrc flavor: C A the etral eber of the op trplet: A A Qark have p ½ therefore the total p of the q q par ca be = or =. The p of the eo relt a the cobato of a of the relatve aglar oet L. The party P of the eo th: P L L Proct of the trc parte of fero a atfero The vale of C obtae lke potro : C L L Fero terchage

19 Qat Nber of Meo wth the Qark 5? 4 5? 8 5 * * * D Q A f f A H Q B L L MeV M qq qq L PC oet

20 Peocalar Meo PC = -+ π - - o - - η - π - + o + π η Meoe Dec M MeV

21 Vector Meo PC = -- ρ - * * ρ * - φ * o + ρ φ The q aq are a tate L= = =. The wave fcto for the glet a for the etral eber of the octet 8 are: 8 6 The phycal tate.e. the oe oberve atre are lear cobato of a 8 : 8 co co 8 the cae of eal xg 5

22 Meo Decay a the Zweg Rle 5.% 4.4% 49.% L.% 8.5% 88.8% For the phae pace wol favor the ecay wth repect to : MeV Q MeV M M Q MeV M M M Q Φ - + ω + π π π π + π Φ - π The agra for the ecay ppree becae t cota coecte qark le Zweg Rle. The keatcal ppreo of the ecay reflecte the all total wth of the : = MeV to be copare for exaple wth = 5..6 MeV.

23 Leptoc Decay of Vector Meo Coer the ecay e l V l l V The partal wth gve by: 6 M V Q l l V ; ; V l V M a V M a Q Q have lar ae cotate M V 9 8 : : : Q l l V 9 :: : :.. e e e e e e e e Experetally Va Roye - Wekopf

24 Qark tate: Baryo Y - - = x Let ow a the thr qark Y - - Y

25 The Baryo Decplet + ++ Δ Δ Δ Δ + Σ - Σ Σ Ξ - They have p-party - Ω Ξ Δ Σ 84 Ξ 5 Ω67 The wave fcto yetrc wth repect to the exchage of ay qark par. They are wave they have parallel p therefore alo the pace a p wave fcto are yetrc. P Pal prcple? The olto le the fact that qark have a frther teral egree of freeocolor whch ca take o three vale RGB. Qark for the faetal trplet of a U color yetry. Haro are color etral.e. they belog to a glet repreetato of color U. th way the overall wave fcto atyetrc er terchage of ay qark par. qqq c.. 6 RGB RBG BRG BGR GBR GRB

26 The Baryo Octet p N99 Σ Σ Σ - Λ + Σ 9 Λ 6 P Ξ - Ξ Ξ 8 The octet tate are copletely yetrc er the ltaeo exchage of flavor a p of ay qark par.

27 + ++ Δ Δ Δ Δ + - Σ Σ Σ Ξ - Ξ Δ Σ Ξ 5 5 p N99 N99 Σ Σ Σ - Λ + Σ 9 9 Λ Ω Ω67 67 Ξ - Ξ Ξ 8 8 f a fferece were olely e to the fact that the qark heaver tha the a qark we hol have: P MeV 49 MeV 9 MeV P N MeV 49 MeV The orer of agte correct bt crepace are tll gfcat. A qattatve ertag of haro ae t take to accot the effect of the hyperfe plttg qark teracto.

28 Haro Mae f flavor U yetry were exact all eber of a gve ltplet wol have exactly the ae a. Yet t ot o..78 GeV. GeV *. 89 GeV f we coer haro ae a the of the ae of the cottet qark we obta:.9 GeV Effectve ae of qark bo haro. Cottet ae. There are frther proble:.5gev N a N cota the ae qark a o a. P

29 ce haro ae caot be explae olely ter of the ae of the cottet qark t eceary to coer the effect of qark teracto. the hyroge ato the p-p teracto lea to the hyperfe trctre of level. For two potlke fero of agetc oet a j the teracto eergy j r Drac theory gve: j e The hyperfe eparato gve by: E hf t a cotact teracto: t cota the qare of the wave fcto at zero eparato a therefore t oly apple to L= tate.

30 For qark the agetc teracto aocate to charge a p of the orer of the MeV. Bt qark teract throgh ther color charge wth a potetal of the for: V r r At all tace the ter /r oate a all eoght to ake the trog hyperfe plttg portat: 8 th chee haro ae are gve by: 4 E QQ E QQ 9 9 kr 4 q q q q q a a j j j

31 For two qark or for qark-atqark: Hece the egevale of are: ] [ 4 larly for -qark yte: ] [ 4 j j * a a a E a E N

32 Ug the experetally eare a vale t poble to ft the paraeter a a a. The relt are: 6 58 MeV MeV a a 6 MeV MeV th way the agreeet wth experetal ata of the orer of % or better.

33 Electroagetc Ma Dfferece A frther cotrbto to haro a coe fro the electroagetc teracto. Let take a a exaple the baryo the octet a let ae that the charge trbto are lar. We expect lar electroagetc cotrbto : p Let a the bare haro ae a thee eqato: p p p. MeV 8 MeV 6.4 MeV.6 MeV Colea-Glahow. Ma fferece are aocate wth op yetry breakg.

34 Electroagetc a fferece are e to three effect: Dfferece a of the a qark; ce > p we expect >. Colob eergy fferece aocate wth the electrcal eergy betwee par of qark of the orer of: e MeV R Magetc eergy fferece aocate wth the agetc oet hyperfe teracto betwee qark par: e c R MeV Fttg the exact for of thee ter to the ata t fo that: MeV The approxate op varace ca be aocate wth the ear eqalty of the a qark ae.

35 Baryo Magetc Moet Baryo agetc oet ca be calclate a the vector of the oet of the cottet qark. For a Drac potlke partcle of a a charge e: A a exaple let calclate the agetc oet of the proto. The two qark are a trplet tate. Cobg wth a frther we get: e p p 4 p p e.79

36 Coparo betwee precte a eare agetc oet for oe baryo: exp p th