Paticle Physics - Measueents and Theoy Outline Natual Units Relativistic Kineatics Paticle Physics Measueents Lifeties Resonances and Widths Scatteing Coss section Collide and Fixed Taget xeients Consevation Laws Chage, Leton and Bayon nube, Paity, Quak flavous Theoetical Concets Quantu Field Theoy Klein-Godon quation Anti-aticles Yukawa Potential Scatteing Alitude - Fei s Golden Rule Matix eleents Nuclea and Paticle Physics Fanz Muhei
Paticle Physics Units Paticle Physics is elativistic and quantu echanical c 99 79 458 /s ħ h/π.055 0-34 Js Length size of oton: f 0-5 Lifeties as shot as 0-3 s Chage e -.60 0-9 C negy Units: GeV 0 9 ev -- ev.60 0-9 J use also MeV, kev Mass in GeV/c, est ass is c Natual Units Set ħ c Mass [GeV/c ], enegy [GeV] and oentu [GeV/c] in GeV Tie [(GeV/ħ) - ], Length [(GeV/ħc) - ] in /GeV aea [(GeV/ħc) - ] Useful elations ħc 97 MeV f ħ 6.58 0 - MeV s Nuclea and Paticle Physics Fanz Muhei
Paticle Physics Measueents How do we easue aticle oeties and inteaction stengths? Static oeties Mass How do you weigh an electon? Magnetic oent coules to agnetic field Sin, Paity Paticle decays Lifeties Foce Lifeties Resonances & Widths Stong 0-3 -- 0-0 s Allowed/fobidden l.ag. 0-0 -- 0-6 s Decays Weak 0-3 -- 0 3 s Consevation laws Scatteing lastic scatteing e- e- Inelastic annihilation e+ e- + - Coss section total σ Foce Coss sections Diffeential dσ/dω Stong O(0 b) Luinosity L l.ag. O(0 - b) Paticle flux vent ate N Weak O(0 - b) Nuclea and Paticle Physics Fanz Muhei 3
Nuclea and Paticle Physics Fanz Muhei 4 Relativistic Kineatics Relativistic Kineatics Basics 4-oentu Invaiant ass Fou-vecto notation Useful Loentz boosts elations set ħ c invaiant ass γ /c / γβ c/c / γ / (- β ) β c/ / β ( -/γ ) -body decays P 0 P P wok in P 0 est fae xale: π + + ν wok in π + est fae use ν 0 ( ) /,,, c c c z y x ( ) ( ) ( ) ( ) 0 0 0 0 0,,,0 + + + ( ) ( ) ( ) 9.8 MeV/c 09.8 MeV,,,0 + π π ν π v
Lifeties Decay tie distibution Mean lifetie τ <dγ/dt> aka oe tie, eigen-tie of a aticle Lifetie easueents In laboatoy fae Decay Length L γβcτ xale: B d π + π - in LHCb exeient <L> 7 Aveage B eson enegy < B > 80 GeV τ.54 s xale: π + discovey Decay sequence π + + ν + e + ν eν ulsions exosed to Cosic ays dγ t Γ ex Γ dt τ τ π <L> + + ν + e + ν eν Nuclea and Paticle Physics Fanz Muhei 5
Resonances and Widths Stong Inteactions Poduction and decay of aticles Lifetie τ ~ 0-3 s cτ ~ O(0-5 ) uneasuable Heisenbeg s Uncetainty Pincile t h Tie and enegy easueents ae elated Natual width negy width Γ and lifetie τ of a aticle Γ ħ/τ Width Γ O(00 MeV) easuable xale - Delta(3) Resonance Poduction + π ++ + π Peak at negy.3 GeV (Cente-of-Mass) Width Γ 0 MeV Lifetie τ ħ/γ 5 0-4 s Nuclea and Paticle Physics Fanz Muhei 6
Scatteing Fixed Taget xeients a + b c + d + n a v a n b # of bea aticles velocity of bea aticles # of taget aticles e unit aea Incident flux F n a v a Coss Section effective aea of any scatteing haening noalised e unit of incident flux deends on undelying hysics What you want to study dn # of scatteed aticles in solid angle dω dσ/dω diffeential coss section in solid angle dω σ total coss section d σ dndn dn na vanbdσ Fnbdσ Ldσ L dω L Luinosity dω L dω dσ N N vent ate σ dω N σl σ Luinosity dω Incident flux ties nube of tagets Deends on you exeiental setu 4 30...34 ban b 0 c Luinosity [ L] 0 c s vent Rate Luinosity ties Coss Section vent L Rate N Nuclea and Paticle Physics Fanz Muhei 7
Scatteing Cente-of-Mass negy a + b c + d + s is invaiant quantity s CoM Collision of two aticles cente-of-ass enegy Mandelsta vaiable Total available enegy in cente-of-ass fae CoM is invaiant in any fae, e.g. laboatoy negy Theshold ( + ) ( + ) ( + ) s + + + ( cosθ ) fo aticle oduction Fixed Taget xeients + CoM s j j c, d,... ( lab ), (,0) CoM s + + lab CoM lab if lab >> i xale: 00 GeV oton onto oton at est CoM s ( ) 4 GeV Most of bea enegy goes into CoM oentu and is not available fo inteactions Nuclea and Paticle Physics Fanz Muhei 8
Scatteing Collide xeients Head-on collisions of two aticles θ 80 0 ( + ) CoM 4 if i i + + >> CoM s ( cosθ ) + + All of bea enegy available fo aticle oduction xale LP - Lage lecton Positon Collide at CRN 00 GeV e- onto 00 GeV e+ Cente-of-ass enegy CoM s 00 GeV Coss section σ(e+ e- + -). b Luinosity Ldt 400 b - Nube of ecoded events N σ Ldt 870 Nuclea and Paticle Physics Fanz Muhei 9
Consevation Laws Noethe s Theoe vey syety has associated with it a consevation law and vice-vesa negy and Moentu, Angula Moentu conseved in all inteactions Syeties tanslations in sace and tie, otations in sace Chage consevation Well established q + q e <.60 0 - e Valid fo all ocesses Syety gauge tansfoation Leton and Bayon nube (L and B) L+B consevation atte consevation Poton decay not obseved (B violation) Leton faily nubes L e, L, L τ conseved Syety ystey Quak Flavous, Isosin, Paity conseved in stong and electoagn ocesses Violated in weak inteactions Syety unknown Nuclea and Paticle Physics Fanz Muhei 0
Theoetical Concets Standad Model of Paticle Physics Standad Model of Paticle Physics Quantu Field Theoy (QFT) Descibes fundaental inteactions of leentay aticles Cobines quantu echanics and secial elativity Classical Physics Vey sall x ħc Quantu echanics Vey fast v c Secial elativity Quantu field theoy Natual exlanation fo antiaticles and fo Pauli exclusion incile Full QFT is beyond scoe of this couse Intoduction to Majo QFT concets Tansition Rate Matix eleents Feynan Diagas Foce ediated by exchange of bosons Nuclea and Paticle Physics Fanz Muhei
Klein-Godon quation Schoedinge quation Fo fee aticle non-elativistic st ode in tie deivative nd ode in sace deivatives not Loentz-invaiant Klein-Godon (K-G) quation Stat with elativistic equation + (ħ c ) ih t Aly quantu echanical oeatos ˆ ψ ˆ ψ h ψ ih ψ t ih t + ψ ψ o t + ψ 0 nd ode in sace and tie deivatives Loentz invaiant Plane wave solutions of K-G equation ψ ν ( x ) N ex( i x ) ± + ν negative enegies ( < 0) also negative obability densities ( ψ < 0) Negative negy solutions Diac quation, but ve enegies eain Antiatte Nuclea and Paticle Physics Fanz Muhei
Klein-Godon quation Inteetation K-G quation is fo sinless aticles Solutions ae wave-functions fo bosons Tie-Indeendent Solution Conside static case, i.e. no tie deivative ψ ψ Solution is sheically syetic g ψ ( ) ex 4π Inteetation - Potential analogous to Coulob otential Foce is ediated by exchange of assive bosons Yukawa Potential Intoduced to exlain nuclea foce g V ( ) ex 4π g stength of foce stong nuclea chage ass of boson R Range of foce ( ) R h c see also nuclea hysics Fo 0 and g e Coulob Potential R Nuclea and Paticle Physics Fanz Muhei 3
Antiaticles Klein-Godon & Diac quations edict negative enegy solutions Inteetation - Diac Vacuu filled with < 0 electons electons with oosite sins e enegy state - Diac Sea Hole of < 0, -ve chage in Diac sea -> antiaticle > 0, +ve chage -> ositon, e + discovey (93) Pedicts e + e - ai oduction and annihilation Moden Inteetation Feynan-Stueckelbeg < 0 solutions: Negative enegy aticle oving backwads in sace and tie coesond to ex ex Antiaticles Positive enegy, oosite chage oving fowad in sace and tie [ i( ( )( t) ( ) ( x) )] [ i( ( t x) ] Nuclea and Paticle Physics Fanz Muhei 4
Scatteing Alitude Tansition Rate W Scatteing eaction a + b c + d W σ F Inteaction ate e taget aticle elated to hysics of eaction Fei s Golden Rule W π h M fi non-elativistic st ode tie-deendent etubation theoy see e.g. Halzen&Matin,. 80, Quantu Physics Matix leent Contains all hysics of the inteaction M ψ H ) ψ fi f Hailtonian H is etubation st ode Incoing and outgoing lane waves woks if etubation is sall ρ f i Matix leent M fi scatteing alitude Density ρ f # of ossible final states hase sace Bon Aoxiation Nuclea and Paticle Physics Fanz Muhei 5
Matix leent Scatteing in Potential xale: e- e- Incoing and outgoing lane waves Matix eleent Moentu tansfe M fi fi ψ 3 V ( ) ψ d 3 N N * f ex ex ( i ) V ( )ex( i ) 3 ( iq ) V ( ) d q i f M fi (q) is Fouie tansfo of Potential V() Scatteing in Yukawa Potential V ( ) g π π ex ( ) ( ) M fi ex i q cosθ M fi 4π g i q 0 0 g 0 i 0 ( + q ) f q i f i d 3 ( ex( i q ) ex( i q ) ) ex( ) Poagato g ex 4π d sinθdθdφ d te in M fi /( +q ) ( ) Coss section dσ M dω Result still holds elativistically 4-oentu tansfe q ( + q ) dσ dω ( ), i f i f 4 q 0 Nuclea and Paticle Physics Fanz Muhei 6