Introduction to Particle Physics I

Transcription

1 Introuction to Particle Phyic I the Feynan calculu Rito Orava Srin 06

2 outline Lecture I: Introuction, the Stanar oel Lecture II: Particle etection Lecture III: Relativitic kineatic Lecture IV: Non-relativitic quantu echanic Lecture V: ecay rate an cro ection Lecture VI: The irac equation Lecture VII: Feynan calculu Lecture VIII: lectron-oitron annihilation

3 outline continue... Lecture IX: lectron-roton elatic catterin Lecture X: eely inelatic catterin Lecture XI: Syetrie an the quark oel Lecture XII: Quantu hrooynaic Lecture XIII: The Weak Interaction Lecture XIV: lectroweak unification Lecture XV: Tet of the Stanar oel Lecture XVI: The Hi boon

4 Lecture VII; Feynan calculu Introuction to Feynan alculu The Phae Sace ecay Rate Two oy Scatterin Procee Suary of the Rule

5 Feynan calculu... intro to the technique of calculatin article ecay an reaction cro ection Γ µ µ σ or alitue hae ace δ i f flux i f for a ecay..., the "flux" i ily, which ha to o with the noralization of the wave function ince alway want to evaluate a ecay in the article ret frae** for the cro ection of...: the flux i v where v therefore, the cro ection will have a factor / v in it. / / i the relativevelocity between an the alitue quare i the relativitic analoue of f H int i thi i failiar to u fro the tie eenent erturbation theory. 5 **For noralization an the flux factor ee Halzen & artin, Quark & Leton.88

6 ince thi i an eleent of a atrix between the et of final tate f, an intial tate i, it i calle a atrix eleent quare evaluate the alitue by alyin Feynan rule to Feynan iara. the hae ace correon to the iea that there i an a riori equal robability of a tranition to each oentu tate - the ore tate, the fater the tranition the hae ace will be / for each final tate article in the enoinator ake thi a Lorentz invariant quantity - both x an et boote by γ for Feynan calculu... [ ] Γ δ µ µ µ 6

7 ecay rate interate over the oenta to et the total ecay rate. Γ 8 norally coul not o any further without knowin, ince it i a function of an - for a two-boy ecay, in the centre-of-a yte an thi i jut eterine by the ae. δ µ µ µ therefore, can take it out of the interal, an o the interal, over : Γ 8 δ or in herical coorinate Γ δ 8 Ω chane variable: 7

8 ecay rate Γ the relative hae ace in a two-boy ecay i jut the centre-of-a oentu. Note: the total ecay rate, Γ, i the u of the rate for all the ecay channel. the rate Γ-N /t/n, lea to the exonential ecay law for the nuber of article: N tn 0e -Γt for catterin,, we have σ v with a reaonable aount of alebra excercie v can be written a an invariant: 8 µ µ µ δ interatin in a iilar way to the two-boy ecay cae exercie, obtain σ Ω 8 f i 8

9 Feynan rule in the centre-of-a where f i one of the final oenta an i i one of the initial oent. calculation of i colicate by the in of article ince have to averae over the intial in an u over the final in to ee how calculation are one without the colication of in, will firt o oe calculation in a toy oel with all inle article. let,, an be in 0 article. an are the antiarticle of an, an i it own antiarticle - the funaental vertice are an the Feynan rule are: for each vertex, ut in a factor of -i. for each internal line, ut in a factor of i/ -, where i the - oentu of the line, conervin an at each vertex. the reult i -i. i are only iortant for ore colicate cae 9

10 0 nee ore rule to eal with loo uch a: calculate the ecay : - Feynan rule 6 8 i i Γ τ τ

11 thi i one of the rawback of thi oel -it ive ifferent enery eenence than real calculation will o in all of further calculation will be ienionle. Feynan rule [ ] [ ] [ ] t

12 c..frae two boy roce i i i i i f 8 Ω σ µ µ µ, i i next, calculate a nuber of two boy catterin roble. -> the eneral two boy catterin for i in the centre-of-a yte: i

13 6 0 6 where, 6 0 co an yiel the interation over anle i trivial, In thi cae, an then 0, aue: Let u Ω Ω Ω Ω σ σ φ θ σ σ σ two boy roce...

14 [ ] oe fro roaator The 0, ae : lenth We can now check the unit : σ >> two boy roce

15 5 -> -thi i jut a rotation fro θ two boy roce... [ ] ' ' co 8, Since co in co,0 in, co 0, i f Ω θ σ θ θ θ θ θ µ µ µ µ µ i i i ' '

16 two boy roce ae : >>, σ Ω 8 σ 6, i.e. the ae a in the cae ae : 0 σ Ω 8 coθ If we try to interate over θ, we fin that the cro ection iinfinite. t all anle,- coθ θ /, coθ inθθ θθ σ θ 8 θ θ θ for allθ 6

17 Feynan alculu... ince the are initinuihable, one cannot tell fro when the total cro ection i calculate, the co interal houl only be taken over 0 to to avoi ouble countin; thee iara will how interference to ee what haen, o back to the exale. It ha oe oo hyic in it, an it i worth takin a little further. 7

18 Feynan alculu... exan the oel to four iara: where 0, 0. thi i rouhly a e -, µ -, γ, Z o i i i If thi were the only iara, then σ 6 8

19 9 Feynan calculu... 6 σ [ ] 6 σ ter ter interference ter now if both an exit, there will be two initinuihable rocee an we have to a the alitue before quarin

20 relative cro ection contribution fro, an interference ter -ter / -ter /- -ter /- 0

21 Feynan calculu... the coulin contant, or chare, ha now unit of oentu cannot avoi in a theory with all calar in all real theorie, the coulin contant will be ienionle. take the cae of ienionle coulin contant, an ee what to learn fro jut a ienional analyi - thi i articularly eay with natural unit. the cae of e e f fany quark-antiquark or leton-antileton air for enerie >> f σ [ ] l e - e γ f - f the only enery of inificance in the roble i e- or c e- σ e

22 Feynan calculu... a tyical weak ecay, the ecay of τ-leton. the Feynan iara i: τ W ν τ f - f a roaator for the W of the for / W - W - ince t << W, then W << W roaator / W W τ Γ W 5 - nee to fin oethin that behave a. - all of the ecay rouct ae, ay for τ e ν, are neliible coare to τ. τ τ 5 τ - can continue by calculatin the τ lifetie exactly, iven the µ lifetie an the τ branchin ratio. eν τ

23 Feynan alculu... µ - ν µ W- - e- τ- ν e ν τ e- W - -the only ifference between the Feynan iara above i ue to the ae of µ an τ. τ τ τ µ µ τ τ e ν ν Note that µ ha only one ecay oe wherea τ ha everal. 5 e τ - ν ε xeriental eaureent: τ.970± τ e ν ν 7.8± 0.07 τ µ τ e τ 90.6 ± µ τ ata : τ ± eV/c ± 0.9 τ ev/c 90.0 ± areeent to 0.%.

24 Feynan alculu... the total cro ection for ν catterin at >>article ae, but with c.. << W. a factor of / W fro the W roaator, but no other ae are inificant the only other enery available i ν in the c.. frae ν µ µ - W - σ νn l ν, c W σ νn ν, c n - no neutrino factorie yet - it i ore ueful to know the above relationhi in ter of ν lab we know fro kineatic c taret lab c lab, i.e. σ νn ν lab - when ealin with the toy oel, it wa not not ecifie what to o when the fouroentu of an internal line i not calculable fro the four-oenta of the external line, i.e. when there i a loo

25 5 - the anwer i iven by the interal over all oible value for the unknown four-oenta. Feynan alculu... i i i i i i i µ µ µ Ω Ω ln ln k ~ k the anular interator. i ' an the anitue, where i ', coorinate : herical to ;oin becoe the interal where, interal aear at lare roble with thi The 0 k k

26 Feynan calculu... - the interal i loarithically iverent - a ajor roble in the eveloent of relativitic quantu echanic, the roble wa eventually olve by Toonaa, Schwiner an Feynan, all workin ineenently. - the technique wa to ut a cut-off in the interal of the for:, which i for << >> - the interal can now be one, an then can be taken to infinity, et a finite iece, ineenent of fro the << art of the interation, an an infinite iece. - the infinite iece can be exree a infinite oification to ae an coulin contant. hyical o δ an hyical o δ - by uin the hyical ae an coulin contant in our calculation, can inore the infinite art of the interal - ay not be too atifyin hiloohically, but it work an 0 for 6

27 Feynan calculu... - the roce i calle renoralization - a theory in which it can be carrie out i calle renoralizable - in 97, Gerar t Hooft howe that all of our theorie for the three funaental force are renoralizable. - there are alo finite correction to - thee ive rie to the runnin coulin contant of Q that were icue in forer lecture - reently there i le concern about renoralizability - the reaon i that the reent theorie are not coniere a funaental, but ily a low enery aroxiation to ore funaental theorie uch a trin theorie... - the low enery theorie are calle effective fiel theorie. 7

28 8 n 8 S.,,where S S i n n δ δ Γ Γ uary:article ecay tranition rate Phae Sace conier the ecay of article into n article....n the ifferential ecay rate i: S i the rouct of tatitical factor: /j for each rou of j ientical article in the final tate. if there are only article in the final tate, the total ecay rate reuce to:

29 9 uary:article catterin tranition rate Phae Sace conier the catterin of article by article : 5...n the ifferential cro ection i i: i f i n n n S S 8.,,where Ω σ δ σ S i the rouct of tatitical factor: /j for each rou of j ientical article in the final tate. for article catterin in the center-of-a frae:

30 uary:cro ection i x noralization of the non-relativitic free article wave function: φ Ne robability enity, ρ, of article ecribe by φ i obtaine a follow: the robability of finin the article in a volue eleent x i: Ψ x ρ x i xit therefore, for Ψ Ne the robability enity i: ρ N φ by ultilyin the Klein-Goron equation: φ φ by iφ * an the colex t conjuate equation by t iφ an ubtractin, we et: φ φ * i φ * φ t t r j reeberin the continuity equation: j 0 we ientify the ter in the quare t bracket a the robability an flux enitie. i xit a free relativitic article, ecribe the Klein-Goron function: φ Ne ive: ρ i i N N j ii N N [ i φ * φ φ φ *] 0 ρ -nee to coenate for the Lorentz contraction of the volue eleent x n to kee the no. Of article r x unchane 0

31 uary:cro ection we now work within a volue, V, an o our noralization to article in V: aot a covariant noralization: N the tranition rate er unit volue i: where T i the tie interval of the interaction, an the tranition alitue i ee H&. 88: T in N N N δ on quarin, one elta-function reain, an tie the other ive TV. therefore, by uin the relation: N / V we obtain: V ρv V exeriental reult for catterin roce: are uually iven a a cro ection : the factor in bracket allow for the enity of the incoin an outoin tate. for a inle article, quantu theory retrict the no. of final tate in a volue, V with oenta in eleent to be V / xercie - have article in V, an: W fi T fi fi W fi δ W fi ro ection nuber of finaltate intial flux V TV V no.of finaltate/article V thu no.of available final tate V

32 uary:cro ection the initial flux we will calculate in the laboratory frae -the no. of bea article ain throuh unit area er unit tie i: v / V an the no. of taret article er unit volue i /V to obtain a noralization-ineenent eaure of the inoin enity, therefore take: Initial flux v V V by inertin the erive reult into our efinition of the cro ection, et a ifferential cro ection for catterin into : V σ δ 6 v V the arbitrary noralization volue, V, cancel, a require- V i roe, work in unit volue, i.e. noralize to article/unit volue, an the noralization factor of the wave function i: N. Thi i the oriin of the ultilicative factor N aociate with the external line of inle article ee the Feynan rule. can write the cro ection forula ro ection nuber of finaltate intial flux ybolically for the countin rate, n, a: n nbvb ntσ where n b i the nuber of bea article bea flux, v b i the the relative velocity of the bea an taret article, n t i the nuber of taret article an σ i the hyic eenent intrinic catterin robability the effective area of the bea a een by a taret article. W fi

33 uary:cro ection ay write the ifferential cro ection in the ybolic for: where Q i the Lorentz invariant hae ace factor: an the incient flux in the laboratory frae i: for a eneral collinear colliion between an : Note: F i Lorentz invariant. Q F σ Q δ v v F / withn v v F