Introduction to Particle Phyic I relativitic kineatic Rito Orava Sring 07
Lecture III_ relativitic kineatic
outline Lecture I: Introduction the Standard Model Lecture II: Particle detection Lecture III_: Relativitic kineatic Lecture III_: Non-relativitic Quantu Mechanic Lecture IV: Decay rate and cro ection Lecture V: The Dirac equation Lecture VI: Particle exchange Lecture VII: lectron-oitron annihilation
outline continued... Lecture VIII: lectron-roton elatic cattering Lecture IX: Deely inelatic cattering Lecture X: Syetrie and the quark odel
Lecture III; Relativitic kineatic Particle decay Two-article cattering Scattering angle latic cattering Angular ditribution Relative velocity Center of a and laboratory yte Croing yetry Interretation of antiarticle-tate
relativitic kineatic c c article energy oentu ret a v βc β in natural unit c article velocity γ β / Lorentz factor γ γβ γ β γ /γ
relativitic kineatic reference: Nachtann [I.] Hagedorn [II.] Byckling & Kajantie [II.] notation: roer tie Lorentz invariant tenor etric 0 0 0 0 0 0 0 0 0 0 0 0 g vector covariant four - x vector contravariant 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
notation The four - velocity: u dx dτ dx dt γ v dt dτ Since u γ v > 0 u i a tie- like vector. The four - oentu i defined a : 0 u γ v By calculating the correonding Lorentzinvariant u we find the energy oentu relation A article i aid to berelativitic if >>. For a non - relativitic article << and...... i.e. we recover the exreion for v << of Newtonian echanic.
article decay The four - oentu of a decaying article - in it ret frae - i given by M 000. xerientally : τ π ν 8.6 0 xerientally : τ π 8.6 0 ν GeV π π 0 γ The decay tie - lifetie - i : dτ dt v where dt dt γdτ > dτ. i the lifetie in laboratory frae: π 0GeV γ π v 0.9999 π t' π. tπ
contraint Contraint: i energy-oentu conervation and ii a-hell condition i i 0 M M
contraint... [ ] [ ] can bedirectly calculated. oenta the value of abolute energie and the the while reain unknown and of i.e.only the direction : we get we get By uing: Therefore: M M M M M M M M M M M M i i i i i i
Mandelta variable: t u two body cattering. A B è C D A B è C D calar roduct of -vector are invariant oible cobination: A B A C A D total -oentu i conerved > there are only two indeendent Lorentz-invariant kineatic variable on which the reaction cro ection can deend
Mandelta variable: t u Three convenient variable: A B t A C u A D for which: t u M M M M A B C D The Mandelta variable nicely relate to the roagator ae in the leading order diagra.
t u i two article cattering i i i... For elaticcattering and. Next conider the Lorentz invariant: i i and $ # " 6 invariant linearly indeendent linearly deendent The Mandelta variable t The center of a c... fraei defined by: u 0
cro ection and luinoity Cro ection σ can be defined by: or equivalently nuber of event σ L nuber of event er unit tie σ dl/dt where an event i an interaction uch a cattering i the luinoity i.e. nuber of chance of an event er unit area. For a fixed target within the bea of incident article dl/dt NJ where N i the nuber of target article and J i the flux er unit area of article in the incident bea.
cro ection and luinoity L f n n πσ x σ y σ σ reference N N reference
frae of reference ' ' and ' frae: In thec... labelled a ued and article oenta i 0 Breit yte the DIS In dee inelatic rocee variableare labelled with an L : the target" 0 "fixed In thelaboratory frae variableare often denoted by an aterix : the frae In thec... 0 defined by: a fraei The center of B i i L i lab i i c i
two article cattering ' Θ ' 0 %"$"# i no Lorentz invariant wherea i one. Wecan now exre i and ' in ter of eeexcercie no..: λ where we ue the Källén tringlefunction : λabc a b c -ab-ac-bc [ ] [ ] a b c a b c a a b c b c
two article cattering The Källén function ha the following roertie : yetric under a b c and aytotic behaviour: a >> bc: λ a b c a Thi allowoe roertie of in ax the cattering { } 0 roce to bedeterined. Fro ' > 0 it follow: i the threhold of the roce in the - channel. At the high energy liit >> i one obtain: '
cattering angle t and or and decribed by two indeendent variable : cattering i above the On the bai of co co we derive co By uing co ' ' defined by i the cattering angle frae In thec... Θ u t t function t i Θ Θ Θ Θ Θ λ λ
elatic cattering Θ Θ Θ 0 0 co - hyicallyallowed region yield: Relation to the co co giving for the cattering angle in elatic cattering : ' and e.g. and In elatic cattering t t t e e
angular ditribution ' co vectori.e. - axi defined by the the rotationally invariant with reect to angular ditribution i The dt d Θ πd d π dφ Ω Ω π λ λ π
relative velocity The relative velocity will be of relevance in defining the article flux v v v " $ # $ % fro which we get a frae indeendent quantity v The Moller flux factor. Note: The Moller flux factor i needed for noralizing the cro ection ince the claical volue eleent i not Lorentz invariant.
CMS and LAB yte For thec... and laboratoryyte : c... total energy lab L L >> L An exale: Fixed target and colliding bea ode at the Ferilab Tevatron bea 980 GeV.
CMS and LAB yte N fixed target: - econdary bea collider fixed t arget 960GeV.7GeV > < W W -channel
croing yetry t-channel the cattering roce exhibit underlying yetrie
croing yetry xale: When we exchange and xaine - channel reaction reviou age : for which the i not affected but t and u interchange their role. - oentu i conerved : The only oitive Mandelta variablefor thi T the reaction i hence the decribe thecattering dynaicof the roce and will be dicued ore later.it deend on three Mandelta variableand i redicted theoretically QCD QCD notation - channel. WSUSY... T t u T t u > 0 t 0 u 0 T can then be extended analytically to the whole range tu R. Deending on the region it can then decribe different croed reaction. For intanceuoe we exchange and we then get naively
croing yetry We now ake the interretation : n n in which n tand for the antiarticle of the article n leadingto the exreion : Sinceand are the incoing article we eak of the " t - channel" roce.we have : T t u T t u t 0 t> 0 u 0
anti-article tate The article with - oentu - are interreted a antiarticle with The reaon for that becoe clear when we look at the - current - oentu. j ρ QM & e i ϕ ϕ ϕ ϕ %""" $ """# j electron charge robability denity %""""" $ """"" # D charge denity Inerting the wave function of the freeelectron ϕ Ne i x Note: In the hae the ign of both and x can be flied without changing the wave function no lace here for article travelling backward in tie in the definition of the - current weget e e e - - with - oentu with - oentu with - 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 with the ubtitution Note: The whole -vector take a inu ign not only the atial art.
anti-article tate - A article with -oentu i a rereentation for the correonding antiarticle with -oentu. - Alternatively: iion of a oitron with energy correond to the abortion of an electron with energy figure above.
anti-article tate In the Dalitz lot the three reaction -t- and u-channel one are decribed by a ingle diagraatic rereentation. Function Ttu evaluated in the relevant kineatical region decribe all three.
exale: Moller & Bhabha cattering Moller: e - e - e - e - -croing yetry- Bhabha: e e - e e -
decay & roduction
NXT: Lecture III_: Non-relativitic Quantu Mechanic