Scattering and Decays from Fermi s Golden Rule including all the s and c s

Transcription

1 PHY 362L Supplemenary Noe Scaering and Decays from Fermi s Golden Rule including all he s and c s (originally by Dirac & Fermi) References: Griffins, Inroducion o Quanum Mechanics, Prenice Hall, Perkins, Inroducion o High Energy Physics 4 h Ed., Cambridge, Schiff, Quanum Mechanics 2 nd Ed., McGraw Hill, Fermi s Golden Rule: Assume he sysem is described by a Hamilonian, H: Hψ =i ψ (1) and ha H has he form: H = H 0 + H where H 0 is he unperurbed Hamilonian, for which he eigenfuncions ψ n are known, and H is he ime-dependen perurbaion. The eigenfuncions saisfy he following condiions: H 0 ψ n = E n ψ n ψ a ψ b = δ ab (2) where he bra ke noaion of Dirac implies inegraion over coninuous variables and summaion over discree variables. In his conex, δ ab represens a Kronecker dela-funcion for discree variables and a Dirac delafuncion for coninuous variables. The basic sraegy is o express he soluion o (1) as a sum over he eigensaes of H 0 wih ime-dependen coefficiens: ψ() = n a n () ψ n e ien/ (3) 1

2 Nex, subsiue (3) ino (1) and use he orhogonaliy condiions (2) o obain: i da k() = H kn d a n()e iω kn (4) n where H kn ψ k H () ψ n and ω kn E k E n. H kn is ofen called he marix-elemen or ransiion-ampliude ; i connecs he saes n k. Equaion (4) is equivalen o he Schrodinger equaion (1), bu is expressed in erms of he coefficiens a n (). In simple sysems, such as 2-level sysems, (4) can be solved explicily. In problems involving a coninuum of saes, scaering for example, (4) is generally solved approximaely by a perurbaion expansion. The order-(p + 1) approximaion is found from he order-(p) soluionby: i d d a(p+1) k () n H kn a(p) n ()eiω kn (5) wih he 0-h -order approximaion da (0) k ()/d = 0, which implies a(0) k is consan and no ransiions occur. As a firs approximaion, he sysem is assumed o be iniially in he sae m, inwhichcase,a (0) n () =δ nm and (5) can be inegraed o give: i a (1) k () = d H km( )e iω km (6) Nex, i is assumed ha he perurbing force described by H urns on a = 0 and is consan over he inerval 0. Equaion (6) can hen be inegraed o give: i a (1) k () 2H km e iω km/2 ( ) sin ωkm /2 ω km For our purposes, we shall sop he perurbaion expansion afer he firsorder erm, in which case a k () a (1) k (). The probabiliy P k () ha he sysem undergoes a ransiion from sae m o sae k is: P k () = a k () 2 4 H km 2 sin 2 ω km /2 2 ωkm 2 (7) The mean rae for he ransiion is given by w k = P k ()/. Because of srong peaking in 1 sin 2 ω/2,nearω = 0 (eviden in Figure 1), Equaion 7 requires ω 2 2

3 2 1 sin / Figure 1: Behavior of he funcion g(ω,) = 1 sin 2 ω/2 versus ω. g has ω 2 he effec of enforcing energy-conservaion because in he limi, g π 2 δ(ω); i explicily demonsraes he Heisenberg uncerainy relaion beween energy and ime hrough, for example, he half-widh of he peak ω and he lifeime of he perurbaion: ω π. ha saes o which ransiions can occur mus have ω km 0, forcing energy conservaion. In general, here will be some number of saes dn wihin an inerval dω km. The number of possible ransiion saes can be wrien: dn = ρ(k)de k where ρ(k) =dn/de k is he densiy of saes per uni energy inerval near E k ;dω km and de k are relaed by dω km =de k /. I is expeced ha ρ(k) and H km are smoohly varying funcions of momenum or energy near he sae k. The physically meaningful quaniy is he oal ransiion rae o saes near he sae k: W k = 1 P k () k near k 3

4 This summaion can be replaced by an inegral over de k : W k = 1 P k ()ρ(k )de k = de k ρ(k) 4 ( H km 2 1 sin 2 ) ω km /2 2 = 4 H km 2 ρ(k) dω 1 ω 2 km sin 2 ω/2 ω 2 As can be anicipaed from Figure 1, he las inegral has he value π/2 and we arrive a Fermi s Second Golden Rule : Decays W k = 2π H km 2 ρ(k) (8) Equaion (8) is used direcly o compue decay raes for quanum sysems. The mean lifeime τ of he sysem is relaed o W k by τ =1/W k. For sysems of very shor mean lifeimes, he widh Γ in energy of he sae is given by: Γ k = W k =2π H km 2 ρ(k) A deailed example: Fermi s heory of nuclear β-decay The prooypical example of nuclear β-decay is neuron decay n pe ν. There are many oher examples involving nuclei wih he same form: (Z, N) (Z +1,N 1) e ν. On dimensional grounds, he simples form for he marix elemen describing nuclear β-decay is given by Fermi s ansaz: H km = G F M (9) V where G F is a consan he Fermi consan. V is he normalizaion volume used for defining wave funcions and M 2 describes he overlap of he iniial/final nuclear wave funcions, a dimensionless quaniy expeced o be approximaely uniy. The energy difference beween iniial (Z, N) and final (Z + 1,N 1) nuclear saes is E 0 ; he sysem decays o a sae of definie energy, bu he iniial sae energy is uncerain o he exen of he finie lifeime, E W, where W is he decay rae. To compue he rae, he densiy of possible saes dn/de 0 = ρ(e 0 ) in he region E around E 0 is needed. 4

5 We firs examine he 3-body kinemaics of he problem. The neurino mass is assumed o be zero and, because ypical values of E 0 are in he MeV-range, recoil momena of all hree final-sae paricles will be ypically of order 1 MeV/c. The final-sae nucleus (or proon) will, hus, carry negligible kineic energy (O 10 3 MeV/c). Under hese assumpions, he decay kinemaics are described by: E 0 = E + cq 0=P + p + q where E is he elecron energy, cq is he neurino energy, P is he 3- momenum of he decay nucleus, p is he elecron 3-momenum and q is he neurino 3-momenum. The momenum of he decay proon or nucleus is compleely deermined by he elecron and neurino momena and, herefore, does no conribue o he densiy of saes. The densiy of saes is found from he produc of he elecron and neurino phase-space volumes: dn = V d3 p (2π) 3 V d 3 q (2π) 3 where V is he same normalizaion volume inroduced previously. Noice ha he normalizaion volume used in deermining he densiy of saes cancels he V 2 facor coming from H km 2 and, hus, can be dropped in subsequen formulas. (See foonoe on page 7.) The momenum-space volume elemens are given by d 3 p 4πp 2 dp and he (unobserved) neurino momenum volume elemen can be replaced by q 2 dq (E 0 E) 2 de 0 /c 3. The densiy of saes is, herefore, given by: 1 ρ(e 0 )= 4π 4 6 c 3 p2 (E 0 E) 2 dp and he Golden Rule (8) gives he differenial (in he elecron momenum p or energy E) decay rae: dw = G2 F 2π 3 7 c 3 M 2 p 2 (E 0 E) 2 dp (10) No only do we ge he decay rae (by inegraing over elecron momena p), bu (10) also gives us he shape of he decay elecron energy specrum! Inegraion of (10) is sraighforward; when he elecron can be reaed as being relaivisic (E cp), he expression is paricularly simple: p 2 (E 0 E) 2 dp Q5 0 30c 3 5

6 where Q 0 = E 0 m e c 2. The oal decay rae is hus: W = 1 τ = G2 F M 2 Q π 3 (c) 6 (11) This resul describes vas ranges 15 orders of magniude of bea-decay raes in nuclei and various elemenary paricles wih a common value of G F and M The Q 5 0-dependence is called Sargen s Law. The mos accurae deerminaion of G F comes from he purely leponic process of muon decay (µ eν ν), he rae for which in he fully relaivisic calculaion has exacly he same form as he β-decay model developed here; he muon decay rae is given by: W µ = where, in his case, Q 0 m µ c 2. Cross Secions G 2 F Q π 3 (c) 6 Consider a 2-body scaering process a + b c + d in he cener-of-mass frame as depiced in Figure 2. In general, he iniial and final momena p i,p f are no he same because paricles of differen masses may be creaed in he collision process. The oal cener-of-mass energy is given by s = E a + E b = E c + E d. iniial sae a p i b p i final sae p f d c p f d Figure 2: 2-body scaering (a + b c + d) in he cener-of-mass sysem. In scaering problems, he ransiion rae is governed by he cross secion σ for he process and he flux of iniial paricles j i according o: G F = MeV fm 3 W f =dσj i (12) 6

7 The inciden flux is: j i = ψ i v op n ψ i = v i /V where v op n represens he velociy operaor along he direcion of he collision axis, v i is he relaive speed of a and b in he cener-of-mass frame and V is he normalizaion volume for wavefuncions ψ. The relaive speed of he iniial paricles is: v i = v a + v b = c 2 p i ( 1 E a + 1 E b ) = c2 p i s E a E b The densiy of final saes is compued from he phase-space volume of one of he ougoing paricles, say paricle c; he oher paricle d is correlaed by momenum/energy conservaion. dn = ρ(f)d s = V d3 p f (2π) 3 g f (13) where g f is he saisical weigh of he final-sae spins. For spinless paricles, g f = 1; for paricles of spins S c,s d, respecively, g f =(2S c +1)(2S d +1). The momenum-space volume is: d 3 p f =dωp 2 f dp f where dω is he solid-angle elemen wihin which scaered paricle c is deeced. Thus, he densiy of saes in scaering problems has he form: and d s dp f ρ(f) = V dω p2 f g f (2π) 3 d s dp f = v c + v d v f = c2 p f s E c E d The inroducion of a normalizaion volume V here and elsewhere in his noe may seem arbirary and obscure. As a pracical maer, V is usually se o uniy and ignored he various powers of V ha accumulae in a calculaion always cancel in he end. V has been kep here as a placeholder o ensure consisen unis in all he expressions. Remember, we are rying o keep rack of all hose c s and s. The problem wih V arises from our convenion (2) for normalizing unnormalizable plane-wave eigenfuncions. 7

8 Combining he golden rule (8), he definiion of cross secion (12) and he densiy of saes (13), we find: dσ dω = V 2 H fi 2 p 2 f g f 4π 2 4 v i v f The normalizaion volume can be buried ino he definiion of he marixelemen M by: H fi M fi V The final resul is: dσ dω = 1 M fi 2 p 2 f g f 4π 2 = 1 4π 2 M fi 2 4 (14) v i v f [ ]( ) Ea E b E c E d pf (c) 4 g f s Equaion (14) is consisen wih he expression for he scaering ampliude f(q) for non-relaivisic, spinless (g f = 1) paricles in he Born approximaion, as derived in class: f(q) = m 2π 2 M fi = m 2π 2 e iq r V (r)d 3 r q = p f p i where V (r) is he scaering poenial and m is he reduced-mass of he scaered paricle. Anoher example: Inverse β-decay or ν-scaering Inverse β-decay, he scaering process ν e + p n + e +, can be described (over cerain ranges of energy) by exacly he same simplified model proposed by Fermi o describeβ-decay, given above in Equaion 9. The resul is an isoropic scaering disribuion in he cener-of-mass frame wih a oal cross secion σ = dω (dσ/dω) = 4π dσ/dω given by: p i σ = G2 F π M 2 p 2 f g f 4 v i v f (15) where M 2 is, again, dimensionless. The significance of (15) is ha raes of processes such as neuron or muon decay can be used o predic ineracion cross secions for neurinos because of he linkage hrough G F. 8

9 For high energy ani-neurinos (E m p c 2 ), σ G2 F s 16π(c) 4 M 2 g f For ypical ineracions in maer, we will assume M 2 g f 4. The arge nuclei are assumed o be a res and he ani-neurinos have energy E ν in he lab frame, in which case s 2m p E ν. Wih hese assumpions, he numerical value of he neurino scaering cross secion is approximaely: σ E ν (GeV) cm 2 which is iny! For example, he mean free pah for 10 GeV ani-neurinos in maerial having he same average densiy as he earh is abou λ cm, which is abou 60,000 imes he earh s radius. 9