MKEP 1.2: Particle Physics WS 2012/13 ( )

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1 MKEP 1.: Particle Physics WS 01/13 ( ) Stephaie Hasma-Mezemer Physikalisches Istitut, INF 6, 3.101

2 Cotet for Today 4. Scatterig process ad trasitio amplitudes 4.1. Fermi s golde rule 4.. Loretz ivariat phase space ad matrix elemet 4.3. Decay Width ad Lifetime 4.4 Two ad Three Body decay rate, Dalitz plots 4.5 Cross sectio 5. Descriptio of free particle 5.1. Klei Gorda equatio 5.. Dirac equatio 5.3. Plae Wave solutio

3 Scatterig Process ad Trasitio Amplitudes i> iitial state x = (t,x) V(x) f> Igrediets to describe this process: fial state Fermi s golde rule (first order pertubatio theory): W fi = π V fi ρ f (E i ) Trasitio probability/uit of time e.g.: electro i coulomb potetial of proto matrix elemet: V fi = <φ f V φ i > phase space of fial state e - e - p γ p W fi is ot LI! Descriptio of free particle wave fuctios for fermios φ i, φ f Phase space factor Descriptio of free particle wave fuctios for fermios Descriptio of IA V fi ( M fi i LI represetatio) 3

4 Lorez trasformatio Lorez Ivariat Phase Space ad Matrix Elemet π π π V = (π) 3 ormalizatio of wave fuctio: 1 particle/volum π π π/γ Solutio: chage of ormalisatio: E particle /V V = (π) 3 /γ Not LI! Lorez ivariat! (π) 3 (π) 3 E φ φ E φ m φ m E m V fi = <φ f V φ i > Ei Ef V fi = M fi d 3 p j d 3 p j π 3 π 3 E j W fi = π 4 m l=1 E l V product of iitial states M fi δ( j=1 p j P) δ( j=1 E j E) LI matrix elemet eergy ad mometum coservatio j=1 d 3 p j π 3 E j LI phase space 4

5 Two Body Decay Rate A 1 + Γ = π 4 E A M fi δ E A E 1 E δ p A p 1 p d 3 p 1 π 3 E 1 d 3 p π 3 E W fi = Γ (trasitio probability per time decay rate) eed to defie referece frame (W fi is ot LI), here CMS: E A = m A p A = 0 p 1 = p = p* d 3 p 1 = p 1 d p 1 d Ω 1 Γ = M 8π m fi δ m A E 1 E A d 3 p 1 4E 1 E (some math exploitig properties of δ fuctio. e.g. Halze u. Marti, ch. 4) Γ = p 3π m A M fi dω p* = always true formula for ay two body decay 1 m A (m A m 1 + m )(m A m 1 m ) I case there is o depedece o Ω: M fi dω = 4π M fi (true for particles A with spi S=0) Γ = p 8π m A M fi 5

6 Three Body Decay Rate A Cosider agai ceter of mass system! Γ = π 4 E A M fi δ E A E 1 E E 3 δ p A p 1 p 3 p d 3 p 1 d 3 p d 3 p 3 π 3 E 1 π 3 E π 3 E 3 d 3 p j = p j d p j d Ω d p j = E j p j de j assumig there is o depedece o Ω (S A =0) Γ = 1 64π 3 m A M fi de 1 de d Γ(E 1,E ) de 1 de = 1 64π 3 1 m A M fi umber of evets (decay rate) i bis of E 1, E ca be measured/plotted direct access to matrix elemet squared! Dalitz-Plot 6

7 Dalitz Plot of decay A assume (for illustratio purposes first) fial state particles are massless: m 1 m m 3 0 E 1 + E + E 3 = p 1 + p p 1 + p + p 3 = 0 + p 3 = m A CMS of particle A E 1 m A / E 3 = 0 If matrix elemet M fi costat, tha uiform distributio i Dalitz plot. d Γ(E 1,E ) de 1 de = 1 64π 3 1 m A M fi m A / E

8 Sketch of Dalitz Plot of K 0 πμν ν E π π π π μ ν π μ ν μ μ π ν m π μ ν m μ E μ

9 Dalitz Plot d Γ(E 1,E ) de 1 de = 1 64π 3 1 m A M fi Istead of E 1 ad E use m 1, m 3 with ivariat mass m ij = (p i +p j ) m 1 = (E 1 +E ) (p 1 + p ) = m A E 3 p 3 = m A + m 3 m A E 3 dm 1 de 3 = m A = cost dγ (m 1,m 3 ) dm 1 dm 3 = 1 56π 3 1 m A M fi 9

10 X 0 Λ+ π - + π + X 0 Σ + + π - Λ + π + X 0 Σ - + π + Λ + π - X 0 Λ + π + + π - M fi is ot flat i phase space! 10

11 Dalitz Plot relativ agular mometum of decay particles = 1 D s + φ + π + K - + K + D + s K* + K + K - + π + D + s K + + K - + π + S(φ) = 1 S(K*)=1 p K parallel to p π S(D s+ ) = S(K) = S(π) = 0 p K atiparallel to p π m Kπ = (E K + E π ) (p K + p π )

12 Dalitz Plot p + p π 0 + π 0 + π 0 1

13 What do we lear from Dalitz Plots Are there itermediate resoaces? Ca read of their decay width ad thus lifetime If mother particle S=0 umber of dots i resoace lie spi of resoat particle Dalitz plot is sesitiv to phases (iterferece), resoaces ca be see as ehacemet or as lack of populatio i Dalitz plot Dalitz plots of symmetric fial states (e.g. pp 3π 0 ) are symmetric If S=0 ad o resoaces flat distributio ( M fi idepedet of phase space) 13

14 Cross Sectio: W fi = (π) 3 V fi δ (ormalisatio of 1 particle/uit volume) j=1 d 3 j=1 p j p i δ E j E i j=1 p j / π 3 3 Rate = (flux of 1) x (umber desity of ) x σ W fi = φ 1 x x σ = 1 x ( v 1 + v ) x x σ ormalisatio of 1 particle/uit volume: 1 = = 1 v v σ = π 4 v 1 + v V fi δ j=1 d 3 j=1 p j p i δ E j E i j=1 p j / π 3 use LI phase space/ormalizatio σ = π 4 M v 1 + v E 1 E fi δ j=1 p j p i δ E j E i j=1 d 3 j=1 p j /[ π 3 E j ] LI flux F LI matrix elemet LI phase factor (see homework) (still to be proove) σ is LI! 14

15 (see Halze & Marti ch 4) Cross Sectio: σ = π 4 j=1 d 3 M v 1 + v E 1 E fi δ j=1 p j p i δ E j E i j=1 p j /[ π 3 E j ] σ is LI! Computatio i CMS: p 1 = p = p i p 3 = p 4 = p f s = (E 1 + E ) dρ = 1 4π F = 4 p i s p f 4 s dω v v dσ dω CM = 1 p f M 64 π s p fi i ot LI defiitio of agles are ot LI 15

16 How to describe a free particle? 1. No-relativistic particles Schrödiger Equatio. Relativistic Spi=0 particles Klei-Gorda Equatio 3. Relativistic Spi=1/ particle Dirac Equatio 16

17 Schrödiger Equatio E = p m E p m = 0 o-relativistic eergy-mometum relatio E i ħ t E i t p iħ atural uits p i i ψ t + 1 m ψ = 0 ψ x, t = N e i(px Et) Schrödiger equatio Solutio: free particle wave fuctio Cotiuity equatio: Gauss theorem S t V ρ dv = j ds = S V j dv v desity flux outside the volume ρ t + j = 0 j 17

18 Schrödiger Equatio Schrödiger Equatio: m ψ = 0 [1] i ψ + 1 t [1] x (-iψ*): [1]* x (+iψ): * ρ t + j = 0 ρ j Normalisatio: ρ dv = 1 N dv = 1 N = 1 V ψ x, t = e i(px Et) work with uit volumes 18

19 Klei-Gorda-Equatio Start from relativistic eergy mometum relatio E = p + m t Φ Φ + m Φ = 0 Klei-Gorda equatio E = i t p = i E = - t p = eergy-eigevalues: E ± = ± p + m φ ± = Ne ipx ie egative eergy ±t eigevalues! +i Φ t Φ iφ Gebe Sie hier eie Formel ei. (+iφ*) x KlG Φ + i Φ m Φ = 0 (-iφ) x KlG* iφ t Φ + iφ Φ i Φ m Φ = 0 t (iφ t Φ i Φ t Φ ) + [ i (Φ Φ Φ Φ )] = 0 ρ ρ = i N e ipx+ie ±t ie ± e ipx ie ±t i N e ipx ie ±t +ie ± e ipx+ie ±t = N E ± j egative particle desities! ρ dv = 1 N = 1 EV j = i (Φ Φ Φ Φ ) = pn 19

20 Feyma-Stückelberg-Iterpretatio particle desity ρ = E ± N charge desity J 0 = qe ± N particle flux j = pn charge flux J = qpn ρ(e -, E - <0) = - e E - N = ee + N = ρ (e +, E + >0) j e, p = j (e +, p ) egative eergy solutio iterpreted as atiparticles charge desity ad flux idetically for e - with -E, +p ad e +, -E, -p 1 E - <0 p 1 - E - > 0 p t t particle travels i -t directio with eg. eergy particle travels i +t directio with pos. eergy 0

21 Applicatio to Feyma Diagrams e + with E>0 travels forward i time (p) e + with E<0 travels backward i time (p) e - with E>0 travels backward i time (-p) x e + γ arrows idicate particle (fermio) flux e - t 1

22 Discovery of Atiparticles postulated by Dirac 1933 discovered i cosmic ray i cloud chamber by Aderso 1936 Nobel Prize for Aderso lead layer: eergy loss idicates flight directio polarity of B field eed to be kow to decide o charge

23 Dirac Equatio Dirac Asatz: fid equatio which is liear i t ad Hψ = (α p + β m)ψ = i t ψ t ψ = H ψ = -(-iα x x still eed to fulfill relativistic eergy mometum relatio (solutio ψ of Dirac eq. must be solutios to Kl. Gorda) iα i α y y z + βm) (-iα z x x = (α + α x x y +α y z z β m ) + (α x α y + α y α x ) + (α x y xα z + α z α x ) x + (α x β + βα x ) + (α x yβ + βα y ) + (α y zβ + βα z ) z t Φ = Φ m Φ i α i α y y z + βm)ψ z + (α z yα z + α z α y ) y z α x = α y = α z = β =1 α i α j + α j α i = 0 if i j α i β + βα i = 0 3

24 Pauli-Dirac Represetatio α x = α y = α z = β =1 α i α j + α j α i = 0 if i j α i β + βα i = 0 Lowest order solutio: 4x4 matrices, thus ψ is 4 compoet vector (Dirac spior) ψ = oe choice of matrices: Dirac-Pauli represetatio I = ; σ x = ; σ y = 0 i i 0 ; σ z = ψ 1 ψ ψ 3 ψ β = I 0 0 I ; α j = 0 σ j σ j 0 ; 4

25 Covariat form of Dirac Equatio x -β (-iα x x (iβα x x i α i α y y z + βm)ψ = i ψ z t + iβ α + iβ α y y z + m)ψ = -i β ψ z t γ μ (β, βα μ ) μ = ( t, x, y, z ) (iγ μ μ m )ψ = 0 covariat form of Dirac equatio 5

26 Adjoit Equatio Dirac: Dirac : x γ 0 (iγ μ μ m )ψ = 0 (iγ 0 t + iγk x k - m )ψ = 0 -i t ψ γ 0 - i x k ψ γ k - m ψ = 0 -i t ψ γ 0 + i x k ψ γ k - m ψ = 0 -i t ψ γ 0 γ 0 - i x k ψ γ 0 γ k - m ψ γ 0 = 0 hermitia cojugated ( dagger : T* ) ψ = ψ γ 0 -i t ψγ0 - i x k ψγ k - m ψ = 0 i μ ψγ μ + m ψ= 0 adjoit equatio 6

27 Probability Desity ad Currets 7

Particle Physics WS 2012/13 ( )

Particle Physics WS 2012/13 ( ) Particle Physics WS /3 (3..) Stephanie Hansmann-Menzemer Physikalisches Institut, INF 6, 3. How to describe a free particle? i> initial state x (t,x) V(x) f> final state. Non-relativistic particles Schrödinger