Lecture 3 summary. C4 Lecture 3 - Jim Libby 1

Transcription

1 Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb

2 Fes of efeece Cosde fe of efeece O whch geec se s descbed.e. he H wve fuco of e - he p se. If O s dffee fe of efeece coeced o O b D g whee g G whee G s goup of sfoos.e. slos oos o Loe sfoos of he coode sse The wve fuco O wll geel be wh D γ γ γ g whee γ s ohool bss s pobbles he se O d O C4 Lecue - Lbb

3 Eple: fe D well Fo he obseve O he goud se s gve b B cos π Now sle D. I O he goud se s o bu - B cos B cos π I logous fsho O whee he slo s D/ π B cos B cos B π B s π / B cos π π π [ cos s ] π D Bs Suo ove egefucos ' Dffee egefuco π π 4 0 -/ / The phscs s v C4 Lecue - Lbb

4 Soe oc phscs We wll cosde he H o oe of s eced ses The egefucos e: u whee l s l l he eeg egevlue s s he egevlue of he copoe of L We defe ew efeece fe O whch s oed bou he s of he ogl fe b gle he egevlue of E E gul oeu l l The Hlo d L e uchged s e he especve egevlues d l. I ohe wods he coue wh. Howeve he deco hs chged so s o he se s ow pojeced o ew -s L p wvefuco C4 Lecue - Lbb 4

5 oo H wvefuco The ew wve fuco u l c be epessed b supeposo of wvefucos wh he se d l bu wh dffee γ γ l u u d u l l γ whch c be coped o Now we wll use he p se l0 s eple: u 0 u d ϑ The d ϑ ϕ wheey cosϑ Y coeffces ssf : ϑ d Y ϑ ϕ l u The dl copoe does o cobue heefoe cosde ol depedece o ϑ d ϕ u Y Y l π 0 Y l 0 0 l ϑ ϕ e he sphecl hocs. Fo he p cse hese e 8π sϑ e ϕ d Y ϑ ϕ sϑ e C4 Lecue - Lbb 5 8π ϕ

6 cos cosϑ cosϑ s oe ges he elo cos s H coued The fl pece of foo we eque s descpo of he oo : cos s cos s Nog ples cos cosϑ s sϑ cosϕ b ovg o sphecl coodes whch gve he vce of he odulus ude oo gves : cosϑ cos cosϑ s sϑ cosϕ We he fd epessos fo cosϑ d sϑ cosϕ es of he sphecl hocs ϕ ϕ π ϑ d sϑ cosϕ [ sϑ e sϑ e ] [ Y ϑ ϕ Y ϑ ] 4 π Y ϕ 0 C4 Lecue - Lbb 6

7 H coued We use he pevous esuls o epess Y 0 es he d sphecl hocs he oed fe Y ϑ cosϑ 0 4π 4π cos cos cosϑ s sϑ cosϕ Y ϑ s [ Y ϑ ϕ Y ϑ ϕ ] Copg o he geel epesso Y ϑ d Y ϑ d Y ϑ ϕ d ϑ ϕ we ge he d coeffces I sl fsho ll d l Y d00 cos d0 s d d 0 coeffces c be clculed Soewh lbouous ee ehod le Wok ou he pobbl h oed se s egese ϑ ϕ ϑ ϕ ϑ ϕ ϑ ϕ s 0 '0 '0 P Y Y d Y Y d C4 Lecue - Lbb 7

8 Tbulos of d fucos fo he PDG hp://pdg.lbl.gov/ C4 Lecue - Lbb 8

9 Eple: e e - µ µ - Sps he elvsc l µ H LH e e - LH µ - H θ Il se Fl se Ol phoo echge elvsc l M µ <<CoM eeg<<m Z0 Lef-hded eleco hles wh gh-hded eleco. Theefoe fl se pcles hve oppose helc s well e H µ H LH θ LH µ Il se Fl se e Two pludes us be of equl es A dcosθ becuse of p cosevo A d cosθ A e e L L e e µ µ L µ µ µ cos θ cosθ µ C4 Lecue - Lbb 9 L dσ d cosθ A e d Le L µ A e Le µ L

10 Eple: e e - µ µ - Wek P volg effecs dso he dsbuo cos θ C4 Lecue - Lbb 0

11 C4 Lecue - Lbb Agul oeu opeos s geeos of oos [ ] becuse Tlo epdg 0 fo s cos s cos ke specfc po As he se coode d eloshp bewee We w o fd s cos s cos d vese s cos s cos ' s : oo bou he cosde Ag we wll ε ε ε ε ε ε ε s he geeo of oos bou he s. Sl esuls fo d

12 Agul oeu opeos s geeos of oos [ ] If s soluo of he Schoedge equo s ε? The Schoedge equo s d H d Opee fo lef wh U U d d ε U ε ε H d d d U HU ε ε d Theefoe s soluo f U [ U ε ] U ε HU ε [ U ε ] ε C4 Lecue - Lbb U ε HU ε H [ U ε H ] 0 U opeo eques [ H]0

13 C4 Lecue - Lbb Agul oeu opeos s geeos of oos We wll ow cosde e vo of elees of The elee s v wh e d he egevlues of e cos. Ths ples: he pojeco of gul oeu s coseved wvefuco s v ude oos H H ϕ φ ϕ φ ϕ φ ϕ φ ϕ φ ϕ φ 0 ] e d [ s depede of If d Whee we hve used * H H d d H d d φ φ ϕ ϕ φ φ

14 Fe oos A fe oo of gle c be geeed b successve pplcos of fsl oos U Theefoe U ε ecllg l ε ε 0 ep j ep j d j d j ep ep ε j j ep U α ep α. ˆ Geel cse whee α s veco he deco of he oo s wh gude equl o he gle of oo C4 Lecue - Lbb 4

15 d If d ces fo oo opeos ep Usg we c fd elo fo we c fd d Opee g wh d g j 4 [ ] 0 0 [ ] 0 C C [ ] [ ] 0 0 j C 0 j ep 0 0 coeffces 0 whee C d! j ± j 0! j j j ± C j j s eple d use he fc h c d j j C4 Lecue - Lbb 5 c

16 C4 Lecue - Lbb 6 d ces fo oo opeos cos ep 0 0 ep 0 s 0 0 d 0 s ep 0 cos - 0 d - s ep cos 0 d s ep he d coeffces c be clculed. d 0 Wh he elos 00!!!! 0 4 4!! 4 4!!!! 4 4!! 4 4!!!! d d d d

17 Eule gles Geec oos descbed b Eule gles Defe hee successve oos: gle α bou he Z s gle bou he 0 s gle γ bou he ew s C be ecs s fs γ bou ogl bou he ogl d α bou he ogl The oos of wvefucos c be epeseed b D ces Usg gul oeu opeos s geeos of he oos d j D αγ j ep α ep ep α γ d j ep γ 0 C4 Lecue - Lbb 7 j 0 0

18 Eule gles γ bou bou α bou C4 Lecue - Lbb 8

19 Eule gles -γ bou cue - bou cue -α bou cue C4 Lecue - Lbb 9

20 C4 Lecue - Lbb 0 Tslos Sl lss c be ppled o slos: Assue s fesl: ST U esul s : We c we he geel ecllg h he oeu opeo s : p p U U ST ST

21 Tslos So fo fe slos: U ST Ivce of he wve equo ude slos cosevo of oeu Sll fo e f Hlo H s e depede: U TT A τ ep A p ep τ H Ivce of he wvefuco wh espec o e slos cosevo of eeg C4 Lecue - Lbb

22 Se Pcples A vce o se pcple ess fo phscl sse S d sfoo g G f he phscl lws epessed fo S b he obseve O hs coode sse lso hold good fo he se sse S he coode sse of he obseve O wheeu g s If O he equos of Noehe s heoe: O' D g O' U g he duced u sfoo. Se pcple Ivce of heo Cosevo lw O oo e he se he he Hlos e he se : H O H O' C4 Lecue - Lbb