9 Lorentz Invariant phase-space

Transcription

1 9 Lorentz Invariant phae-space 9. Cro-ection The cattering amplitude M q,q 2,out p, p 2,in i the amplitude for a tate p, p 2 to make a tranition into the tate q,q 2. The tranition probability i the quare modulu of thi quantity. But here we have a problem. Let u write M M 2π 4 δ 4 p + p 2 q q 2. The quare of the energy-momentum conerving delta-function i not defined. The problem arie becaue we do not have incoming tate which are perfect eigentate of momentum, but rather a wave-packet, which i a weighted uperpoition of uch tate, o that in in -tate i really in > d 3 p 2π 3 2E d 3 p 2 2π 3 2E 2 f p f 2 p 2 p, p 2, where f and f 2 are the Fourier tranform of the wavefunction of the incident particle. The tranition probability, W, i now given by W d 3 p d 3 p 2 2π 3 2E 2π 3 2E 2 d 3 p d 3 p 2π 3 2E 2 2π 3 2E 2 f p f 2 p 2 f p f 2 p 2 2π 8 δ 4 q + q 2 p p 2 δ 4 p + p 2 p p 2 M 2 We can write the econd delta-function a 2π 4 δ 4 q + q 2 p p 2 d 4 xe ip +p 2 p p 2 x, and perform the integration over p,p 2 the invere Fourier tranform to get an expreion in term of the wave-function, ψ x, ψ 2 x of the incoming particle. For incoming wavepacket which are harply peaked at p and p 2, thi integration approximate to W d 4 x ψ x 2 2E ψ 2 x 2 2E 2 2π 4 δ 4 q + q 2 p p 2 M 2 The tranition rate per unit volume i dw d 3 x 2π4 δ 4 q + q 2 p p 2 4E E 2 M 2 ψ x 2 ψ 2 x 2 flux 34

2 i the differential cro-ection for the initial tate to go into the tate q,q 2, and the flux factor, F, i the probability to find particle per unit volume multiplied by the probability to find particle 2 per unit volume multiplied by their relative velocity, v. F ψ x 2 ψ 2 x 2 v. In the ret-frame of one of the particle 2 the relative velocity i given by v p E. Remembering that the tate are relativitically normalied, the quare-wavefunction for an almot momentum eigentate can be replaced by 2E and o we have, in the ret frame of particle 2 F 4E E 2 v 4 p E 2 4 p m 2. Thi can be written in manifet Lorentz invariant form a F 4m 2 E 2 m2 4 p p 2 2 m 2 m2 2 Since thi latter expreion i in term of mae and Lorentz-invariant calar product of 4- momenta, it i a Lorentz invariant expreion. We can write with λ a before given by F 2λ /2,m 2,m2 2, λx,y,z x 2 + y 2 + z 2 2xy 2xz 2yz. Thu finally we end up with an expreion for thew differential cro-ection 2π4 δ 4 q + q 2 p p 2 M 2. F 9.2 Lorentz-invariant phae-pace LIPS integration i the cro-ection for a tranition into the tate q,q 2. The total cro-ection i obtained by integrating over all poible final tate momenta uing the Lorentz invariant meaure. DLIPS d4 q 2π 3 d 4 q 2 2π 3 δq2 m 2 3θq 0 δq2 2 m 2 4θq 0 2, where we have taken the mae of the outgoing particle to be m 3 and m 4. In general, if we have n final-tate particle the Lorentz-invariant phae-pace i given by DLIPS n i d 4 q 2π 3 δq2 i m2 i θq0 i. It will be convenient to write ome of thee factor in the non-manifetly Lorenz invariant form d 3 q 2π 3 2E q, 35

3 and chooe a uitable frame in which to perform the integration. Thu the rule for calculating the total cro-ection are. Calculate the matrix element M from the Feynman rule, omitting the energy-momentum delta-function 2π 4 δ 4 i q i p p The cro-ection for n-particle in the final tate i σ n i d 4 q i 2π 3 δq2 i m2 i θq0 i 2π4 δ 4 i q i p p 2 M 2 F. Returning to the cae of two final-tate particle, we may not want the total cro ection but a quantity uch a dθ, where θ i the cattering angle. Since thi i frame-dependent it would be better to calculate a quantity uch a, and then tranform the reult into the differential croection w.r.t cattering angle in a choen frame. Since we are then calculating a Lorentz invariant quantity, we are at liberty to conider the ytem in a convenient frame of reference. For the two final-tate cae the eaiet frame i the centre-ofma frame for which the incoming momenta p, p 2 are given by p µ p µ 2 p 2 + m 2,0,0, p p 2 + m 22,0,0, p Uing the definition of the Mandeltam variable and λ thi can be written a p µ p µ 2 + m 2 m2 2 2,0,0, λ/2,m 2,m m 2 2 m2 2,0,0, λ/2,m 2,m2 2 2 Likewie the outgoing momenta may be written a q µ q 2 + m 2 3,qinθcoφ,qinθinφ,qcoθ q µ 2 q 2 + m 2 4, qinθcoφ, qinθinφ, qcoθ, 36

4 where θ,φ are the polar angle of the outgoing particle with momentum q. Since may alo be written q + q 2 2 we can perform the ame manipulation to obtain q µ + m 2 3 m2 4 2, λ/2,m 2 3,m2 4 2 n q µ with the unit 3-vector n given by and the Mandeltam variable t i + m 2 4 m2 3 2, λ/2,m 2 3,m2 4 2 n n inθcoφ,inθinφ,coθ t p q 2 m 2 + m 2 3 2E p E q + 2 p q coθ Now write the expreion for the cro-ection a σ d 3 q d 4 q 2 2π 3 2E q 2π 3 δq2 2 m2 4 2π4 δ 4 M 2 p + p 2 q q 2 F, where we have written the phae-pace meaure for q in non-relativitic form. We can now ue the integral over d 4 q 2 to aborb the energy-momentum conerving delta-function, but remember that q 2 mut be replaced by p + p 2 q inide the delta-function δq 2 2 m2 4, o that we are now left with σ d 3 q 2π 2 δp + p 2 q 2 m M 2E q F d 3 q d coθdφ q 2 d q The integration over φ introduce a factor of 2π. We want to replace the integral over coθ by an integral over t. From the expreion for t we have d coθ 2 p q In the centre-of-ma frame, p + p 2 µ,0,0,0, o that the argument of the remaining delta-function i 2 E q + m 2 3 m2 4 Furthermore ince Eq 2 m q 2, we have q d q E q de q, 37

5 o that leaving after integration over φ d 3 q 2E q dφ 2 p q q E q 2E q de q, deq 2π 4 p δ 2 E q + m 2 3 m2 2 4 M F Performing the integration over E q to aborb the remaining delta-function an inerting the expreion for the flux, F, we have 6π p M 2 2λ /2,m 2,m2 2 But and o we finally end up with p λ/2,m 2,m2 2 2, 6πλ,m 2,m2 2 2 M Note that λ /2 i only real if > m + m 2 2, which i the phyical threhold for the cattering to occur. In the φ 3 cae with equal mae that we have been conidering we therefore have Note that we have ued u 4m 2 t. g 4 6π 4m 2 m 2 + t m m 2. t The integration over t needed to calculate the total cro-ection i often very mey. The limit on t are obtained in term of coθ ± giving t min m 2 + m 2 3 2E p E q 2 p q t max m 2 + m2 3 2E p E q + 2 p q In thi cae where all the mae are equal, the energie of the particle are equal and o are the magnitude of their three-momenta in the centre-of-ma frame and thi implifie to t min 4m 2 t max 0 38

6 Furthermore, we can obtain the differential cro-ection with repect to the centre-of-ma cattering angle, θ by d coθ 2 p q Again, if all the mae are equal thi implifie to d coθ 4m2 2 Sometime differential cro-ection are quoted in term of dω where Ω i the olid angle. Thi i what i meaured directly a a detector will ubtend a given element of olid angle dω. Thi i imply obtained by not performing the integration over the azimuthal angle φ, i.e. dω 2π d coθ. again thi quantity i frame dependent and different in a collider experiment from a fixed-target experiment. 39