The Standard Model Lagrangian

Transcription

1 Th Standard Modl aranian Elmntary Particl Physics Stron Intraction Fnomnoloy Dio Bttoni Acadmic ar -

2 D. Bttoni Fnomnoloia Intrazioni Forti Dirac Formalism m i j Consrvd Currnt i i i 5 i m

3 Gau Invarianc U D D ia U D A i U U UA U Ablian D i i i ijk j k Non Ablian Gau invarianc rquirs th introduction o vctor bosons, which act as quanta o nw intractions. In au thoris th symmtris prscrib th intractions. D. Bttoni Fnomnoloia Intrazioni Forti 3

4 Th Symmtris o th Standard Modl U() invarianc. All particls appar to hav this kind o invarianc, rlatd to lctromantism. It rquirs a vctor boson, B, whos connction with th photon will b dtrmind latr. SU() invarianc. Non ablian au invarianc (lctrowak isospin). It rquirs thr vctor bosons, i, on or ach nrator o SU(). Th physical particls hav dinit lctromantic chars. 3 i i SU(3) invarianc. It rquirs iht vctor bosons, G a, th luons, th quanta o th stron intraction, dscribd by Quantum ChromoDynamics (QCD). D. Bttoni Fnomnoloia Intrazioni Forti 4

5 Th aranian In ordr to obtain th Standard Modl aranian w start rom th r particl aranian and rplac th ordinary drivativ by th convariant drivativ. It will contain two parts: au kintic nris o th au ilds rm covariant drivativ rmion kintic nris Nxt w must spciy th particls and thir transormation proprtis undr th thr intrnal symmtris. Notation D. Bttoni Fnomnoloia Intrazioni Forti 5

6 ptons P D. Bttoni Fnomnoloia Intrazioni Forti 6 P t-handd and riht-handd particls bhav dirntly undr lctrowak SU() transormations: th lctrons ar SU() sinlts, whras th lctrons ar put in doublts tothr with th nutrinos. SU() sinlt SU() doublt otations in lctrowak SU() turn lctrons in nutrinos and vv. Ordinary spin: raisin and lowrin oprators (vctors). Stron isospin: pions (vctors). ak isospin: th bosons connct th mmbrs o an lctrowak doublt is not connctd to any othr stat by lctrowak transitions. p,q,r=,..: p =, = -.

7 Quarks Q u d d, u th indx dscribs how th quark transorms undr color SU(3). Th basic rprsntation is a triplt:,, =,,3 or r,,b. Color (.. r) and anticolor (.. r ). Sinlt (rr+ + bb) All lptons ar color sinlts. All quarks ar color triplts. Th luons nrat th transitions rom on color to anothr: thy ar th quanta o th stron intraction, but unlik photons thy carry color char. Thr ar iht bi-colord luons (.. b): octt rprsntation o color SU(3). D. Bttoni Fnomnoloia Intrazioni Forti 7

8 In th Standard Modl thr ar no nutrinos: Exprimntally ony ar obsrvd. Nutrino masss ar vry small (but non-zro, oscillations). I riht-handd nutrinos xist ithr thy ar vry havy or thy intract vry wakly. and rmions wr put in dirnt lctrowak SU() multiplts: that implis a violation o parity, sinc clarly th thory is not invariant undr th rvrsal o th componnt o spin in th dirction o motion. This is th way in which parity violation is dscribd in th standard modl, but it is not xplaind in a undamntal sns. Th sam thory can b applid to th two othr rmion amilis: (,, c,s) and (,,t,b). Th univrs consists o rmions rom th irst nration. Th othr amilis ar producd in hih-nry cosmic ray collisions and in particl acclrators. No rason has bn ound or th xistnc o thr amilis o particls with idntical quantum numbrs and intractions. D. Bttoni Fnomnoloia Intrazioni Forti 8

9 Th Quark and pton aranian D i B is th spin-on ild ndd to maintain th U() au invarianc. is th couplin strnth (to b masurd xprimntally. is th nrator o U(), transormations, a constant, but in principl dirnt or th dirnt rmions. Analoous rmarks dscrib th SU() and SU(3) trms. introduc 3 and, rspctivly, 8 vctor bosons which ar ndd to maintain th local 3 3 au invarianc. i i D ivs a zro rsult whn it acts on a trm o dirnt matrix orm. For xampl i i is a matrix in SU() and it ivs zro actin on, u,d. rm D. Bttoni Fnomnoloia Intrazioni Forti 9 D i i B i D i 3 a G a,, Q, u, d

10 D. Bttoni Fnomnoloia Intrazioni Forti i i i Gauin th Global Symmtris Dirac kintic nry aranian or th irst nration: Global SU() symmtry Global U() symmtry put and in a doublt, in a sinlt. mak ths symmtris locals by introducin potntials i and B. and by rplacin with th covariant drivativ D. Thus w obtain th sam rsult. Som attmpts to xtnd th Standard Modl procd alon ths lins, by addin particls and symmtris and thn auin th symmtris.

11 Th Elctrowak aranian Sinc th trm is always prsnt w will not writ it. All th calculations in SU() will b don only or th lptons. Sinc th color labls o th quarks do not oprat in th U() or SU() spac, quarks will bhav th sam way as lptons or U() and SU() intractions. D. Bttoni Fnomnoloia Intrazioni Forti

12 Th U() Trms rm U,lptoni i i B i i B rm U,lptoni B D. Bttoni Fnomnoloia Intrazioni Forti

13 D. Bttoni Fnomnoloia Intrazioni Forti rm,lptoni i i i i SU i i Th SU() Trms

14 D. Bttoni Fnomnoloia Intrazioni Forti 4 QA EM B B A B Z B A B Z Th Nutral Currnt Elctromantic intraction o particls o char Q: Thr ar trms involvin nutrinos assum th th lctromantic ild A is th orthoonal combination:

15 D. Bttoni Fnomnoloia Intrazioni Forti 5 B B Z A B Z A Z A Trms involvin lctrons: Th trm in A must b th usual lctromantic currnt. Th trm in Z can b an additional intraction, to b chckd xprimntally.

16 D. Bttoni Fnomnoloia Intrazioni Forti 6 can choos =-, sinc any chan in can b absorbd by a rdinition o. Th thory w hav bn writin can b intrprtd to contain th usual lctromantic intraction, plus an additional nutral currnt intraction with Z or both lctrons and nutrinos.

17 Din: sin cos wak mixin anl (inbr anl) cos sin and ar writtn in trms o th known ( /4/37) and th lctrowak mixin anl, which nds to b masurd or calculatd som othr way. sin.3 D. Bttoni Fnomnoloia Intrazioni Forti 7

18 D. Bttoni Fnomnoloia Intrazioni Forti 8 Z Z cos sin cos sin cos sin cos -Z Couplin cos quantity to b associatd to ach -Z vrtx. lctrowak char o th lt-handd nutrino.

19 D. Bttoni Fnomnoloia Intrazioni Forti 9 -Z Couplin Z sin sin cos sin cos sin sin cos sin cos cos Couplin Couplin

20 cos sin T 3 Q sin This xprssion ivs th lctrowak char o any rmion, i.. th strnth o th couplin to th Z. T 3 is th invalu o T 3 or any rmion. For a sinlt (=,u,d cc) T 3 =. For th uppr mmbr o a doublt (=,u cc) T 3 = +/. For th lowr mmbr o a doublt (=,d cc) T 3 = -/. Q is th lctric char o th rmion in units o : Q =-. Q =, Q u =/3, Q d =-/3) Th lctrowak thory contains both th lctromantic intraction, mdiatd by th photon, and th wak nutral currnt, mdiatd by th Z, which coupls to any rmion with lctric char or lctrowak isospin. Th strnth o th Z intraction is not intrinsically small, but it ts rducd by th hih valu o its mass. D. Bttoni Fnomnoloia Intrazioni Forti

21 Th procss (or ) is not purly lctromantic, but it has a wak componnt, du to th xchan o a Z. G G d d d d d d d d QED ( wak) intrrnc Z s G s G Th asymmtry coms rom th intrrnc trm, th ct is o th ordr o % or s = GV. D. Bttoni Fnomnoloia Intrazioni Forti

22 Th Chard Currnt Th U() part o th aranian contains only trms diaonal in th rmions, whras th SU() part has also non diaonal trms. rm 4 4 sin 37 chard currnt 5 V-A intraction thus xpct bosons and th associatd chard currnt transitions. Th obsrvd chard currnts occur with a strnth much smallr than on would xpct: D. Bttoni Fnomnoloia Intrazioni Forti

23 Exampl o a Char Currnt: dcay n d p u s u - u - M = GV/c G - d d q << M Th intraction is practically pointlik, dscribd by a 4-rmion couplin G M As is th cas with nutral currnts, also or chard currnts th chard currnt strnhts ar rducd by th hih valu o th mass. D. Bttoni Fnomnoloia Intrazioni Forti 3

24 Th Quark Elctrowak Trms Th SU() and spin structur o quarks and lptons ar th sam, consquntly all th prvious conclusions hold without modiications or th quarks: Thy coupl to th sam au bosons, Z,. Normal lctromantic couplin to th photon. Chard currnt couplin nratin transitions u d, but no chard currnt transitions or u d. Nutral currnt transitions with a Q T 3 univrsal strnth: u +/3 +/ T Q 3 sin d -/3 -/ cos sin u +/3 d -/3 D. Bttoni Fnomnoloia Intrazioni Forti 4

25 Th Quark QCD aranian 3 q a G a q,,,3 a,...,8 It contains only quarks, sinc lptons hav no color char. In th lctrowak cas th i ar rlatd to stats o lctromantic char bcaus o th intraction with th photon. Th luons ar lctrically nutral, i.. thy hav no intraction with th lctromantic ild. Sinc th nrators a ar not all diaonal, th intraction with a luon can chan th color char o a quark. Gluons and quarks ar conind within hadrons. D. Bttoni Fnomnoloia Intrazioni Forti 5

26 D. Bttoni Fnomnoloia Intrazioni Forti 6 Th Scond and Third Familis All known phnomnoloy is consistnt with th abov rplacmnts It is unknown whthr mor amilis or additional quarks and lptons xist All rmions o th thr amilis hav bn obsrvd xprimntally. Th sam st o au bosons (,, Z,) intracts with all th amilis o rmions: lpton univrsality u- and d- univrsality b t s c d u

27 D. Bttoni Fnomnoloia Intrazioni Forti 7 Th Frmion-Gau Boson aranian d u q a a d u d u G q q d u Z Q Q T A Q, 3,,, 3,,, h.c. sin sin cos

28 Masss For rmions a mass trm would b o th orm m. m m Sinc rmions ar mmbrs o an SU() doublt, whras th rmions ar sinlts, th trms and ar not SU() sinlts and would not iv and SU() invariant aranian, For au bosons mass trms ar o th orm m B B B which is clarly not invariant undr au transormations. Th solution o this problm lis in th His mchanism. D. Bttoni Fnomnoloia Intrazioni Forti 8