Introduction to Particle Physics I relativistic kinematics. Risto Orava Spring 2015

Transcription

1 Introducton to Partcle Phyc I relatvtc kneatc Rto Orava Srng 05

2 outlne Lecture I: Orentaton Unt leentary Interacton Lecture II: Relatvtc kneatc Lecture III: Lorentz nvarant catterng cro ecton Lecture IV: Accelerator and collder exerent Lecture V: leent of Quantu lectrodynac

3 outlne contnued... Lecture VI: Tetng QD Lecture VII: Untary yetre and QCD a a gauge theory Lecture VIII: QCD n e e - annhlaton

4 Lecture II; Relatvtc kneatc Partcle decay Two-artcle catterng Scatterng angle latc catterng Angular dtrbuton Relatve velocty Center of a and laboratory yte Crong yetry Interretaton of antartcle-tate

5 relatvtc kneatc reference: Nachtann [I.] Hagedorn [II.] Bycklng & Kajante [II.] notaton: roer te Lorentz nvarant tenor etrc g vector covarant four - x vector contravarant four - 0 γ τ ν ν ν ν dt dt dx - dt dτ x x x x x g x t g x t x t x x x t x x

6 notaton The four - velocty: u dx dτ dx dt γ v dt dτ Snce u γ v > 0 u a te- lke vector. The four - oentu defned a : 0 u γ v By calculatng the correondng Lorentznvarant u we fnd the energy oentu relaton A artcle ad to berelatvtc f >>. For a non - relatvtc artcle << and e. we recover the exreon for v << of Newtonan echanc.

7 energy-oentu energy-oentu four vector 0 x y z where the energy of the artcle and x y and z are the coonent of artcle oentu. x y z length of a four-vector an nvarant t doe not change under Lorentz trnforaton

8 artcle decay The four - oentu of a decayng artcle - n t ret frae - gven by M 000. xerentally : τ π ν xerentally : τ π ν GeV π π 0 γ The decay te - lfete - : dτ dt v where dt dt γdτ > dτ. the lfete n laboratory frae: π 0GeV γ π v π t' π. tπ

9 contrant Contrant: energy-oentu conervaton and a-hell condton 0 M M

10 contrant... [ ] [ ] can bedrectly calculated. oenta the value of abolute energe and the the whle rean unknown and of.e.only the drecton : we get we get By ung: Therefore: M M M M M M M M M M M M

11 two artcle catterng varable Mandelta The u t u t 0 defned by: a c... frae center of The... " $ # deendent lnearly ndeendent lnearly 6 nvarant and Next conder the Lorentz nvarant:. and For elatccatterng

12 Mandelta varable: t u two body catterng. A B è C D A B è C D calar roduct of -vector are nvarant oble cobnaton: A B A C A D total -oentu conerved > there are only two ndeendent Lorentz-nvarant kneatc varable on whch the reacton cro ecton can deend

13 Mandelta varable: t u Three convenent varable: A B t A C u A D for whch: t u M M M M A B C D The Mandelta varable ncely relate to the roagator ae n the leadng order ddagra.

14 cro ecton and lunoty Cro ecton σ can be defned by: or equvalently nuber of event σ L nuber of event er unt te σ dl/dt where an event an nteracton uch a catterng the lunoty.e. nuber of chance of an event er unt area. For a fxed target wthn the bea of ncdent artcle dl/dt NJ where N the nuber of target artcle and J the flux er unt area of artcle n the ncdent bea.

15 cro ecton and lunoty

16 frae of reference ' ' and ' frae: In thec... labelled a ued and artcle oenta 0 Bret yte the DIS In dee nelatc rocee varableare labelled wth an L : the target" 0 "fxed In thelaboratory frae varableare often denoted by an aterx : the frae In thec... 0 defned by: a frae The center of B L lab c

17 two artcle catterng ' Θ ' 0 %"$"# no Lorentz nvarant wherea one. Wecan now exre and ' n ter of eeexcerce no..: λ where we ue the Källén trnglefuncton : λabc a b c -ab-ac-bc [ ] [ ] a b c a b c a a b c b c

18 two artcle catterng The Källén functon ha the followng roerte : yetrc under a b c and aytotc behavour: a >> bc: λ a b c a Th allowoe roerte of n ax the catterng { } 0 roce to bedeterned. Fro ' > 0 t follow: the threhold of the roce n the - channel. At the hgh energy lt >> one obtan: '

19 catterng angle t and or and decrbed by two ndeendent varable : catterng above the On the ba of co co we derve co By ung co ' ' defned by the catterng angle frae In thec... Θ u t t functon t Θ Θ Θ Θ Θ λ λ

20 elatc catterng Θ Θ Θ 0 0 co - hycallyallowed regon yeld: Relaton to the co co gvng for the catterng angle n elatc catterng : ' and e.g. and In elatc catterng t t t e e

21 angular dtrbuton ' co vector.e. - ax defned by the the rotatonally nvarant wth reect to angular dtrbuton The dt d Θ πd d π dφ Ω Ω π λ λ π

22 relatve velocty fro whch we get the artcle flux releance n defnng be of relatve velocty wll The v v v v %"$"# The Moller flux factor. a frae ndeendent quantty Note: The Moller flux factor needed for noralzng the cro ecton nce the clacal volue eleent not Lorentz nvarant.

23 CMS and LAB yte For thec... and laboratoryyte : c... total energy lab L L >> L An exale: Fxed target and colldng bea ode at the Ferlab Tevatron bea 980 GeV.

24 CMS and LAB yte N fxed target: - econdary bea collder fxed t arget 960GeV.7GeV > < W W -channel

25 crong yetry t-channel the catterng roce exhbt underlyng yetre

26 crong yetry xale: When we exchange and xane - channel reacton revou age : for whch the not affected but t and u nterchange ther role. - oentu conerved : The only otve Mandelta varablefor th T the reacton hence the decrbe thecatterng dynacof the roce and wll be dcued ore later.it deend on three Mandelta varableand redcted theoretcally QCD QCD notaton - channel. WSUSY... T t u T t u > 0 t 0 u 0 T can then be extended analytcally to the whole range tu R. Deendng on the regon t can then decrbe dfferent croed reacton. For ntanceuoe we exchange and we then get navely

27 crong yetry We now ake the nterretaton : n n n whch n tand for the antartcle of the artcle n leadngto the exreon : Snceand are the ncong artcle we eak of the " t - channel" roce.we have : T t u T t u t 0 t> 0 u 0

28 ant-artcle tate The artcle wth - oentu - are nterreted a antartcle wth The reaon for that becoe clear when we look at the - current - oentu. j ρ QM & e ϕ ϕ ϕ ϕ %""" $ """# j electron charge robablty denty %""""" $ """"" # D charge denty Inertng the wave functon of the freeelectron ϕ Ne x Note: In the hae the gn of both and x can be fled wthout changng the wave functon no lace here for artcle travellng backward n te n the defnton of the - current weget e e e - - wth - oentu wth - oentu wth - oentu - : : : j j j e e e - - e N e N e N e N e N e N And hence the rule : j e j e wth the ubttuton Note: The whole -vector take a nu gn not only the atal art.

29 ant-artcle tate - A artcle wth -oentu a rereentaton for the correondng antartcle wth -oentu. - Alternatvely: on of a otron wth energy correond to the aborton of an electron wth energy fgure above.

30 ant-artcle tate In the Daltz lot the three reacton -t- and u-channel one are decrbed by a ngle dagraatc rereentaton. Functon Ttu evaluated n the relevant kneatcal regon decrbe all three.

31 exale: Moller & Bhabha catterng Moller: e - e - e - e - -crong yetry- Bhabha: e e - e e -

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33 NXT: Lecture III; Lorentz nvarant catterng cro ecton and hae ace S-oerator Ferʼ golden rule Total decay rate Scatterng cro ecton Invarant hae ace for n f artcle Dfferental cro ecton catterng cro ecton Phae ace Dfferental cro ecton Untarty of the S-oerator