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47. Kinematics 1 47 47. Kinematics 1 47. Kinematics Reviewed August 2019 by D. Miller (Glasgow), D.R. Tovey (Sheffield) and January 2000 by J.D. Jackson (LBNL). Throughout this section units are used in which ~ = c = 1. The following conversions are useful: ~c = 197.3 MeV fm, (~c)2 = 0.3894 (GeV)2 mb. 47.1. Lorentz transformations The energy E and 3-momentum p of a particle of mass m form a 4-vector p =(E, p) whose square p2 E2 p 2 = m2. The velocity of the particle is β = p/E. The energy ≡ ∗ ∗−| | and momentum (E , p ) viewed from a frame moving with velocity βf are given by ∗ E γf γf βf E ∗ ∗ = − , p = pT , (47.1) p γ β γ p T k − f f f k where γ = (1 β2)−1/2 and p (p ) are the components of p perpendicular (parallel) to f − f T k βf . Other 4-vectors, such as the space-time coordinates of events, of course transform in the same way. The scalar product of two 4-momenta p1 p2 = E1E2 p1 p2 is invariant (frame independent). · − · 47.2. Center-of-mass energy and momentum In the collision of two particles of masses m1 and m2 the total center-of-mass energy can be expressed in the Lorentz-invariant form 1/2 2 2 Ecm = (E1 + E2) (p1 + p2) , h − i 1/2 2 2 = m1 + m2 +2E1E2(1 β1β2 cos θ) , (47.2) h − i where θ is the angle between the particles. In the frame where one particle (of mass m2) is at rest (lab frame), 2 2 1/2 Ecm =(m1 + m2 +2E1 lab m2) . (47.3) The velocity of the center-of-mass in the lab frame is βcm = plab/(E1 lab + m2) , (47.4) where p p and lab ≡ 1 lab γcm =(E1 lab + m2)/Ecm . (47.5) The c.m. momenta of particles 1 and 2 are of magnitude m2 pcm = plab . (47.6) Ecm For example, if a 0.80 GeV/c kaon beam is incident on a proton target, the center of mass energy is 1.699 GeV and the center of mass momentum of either particle is 0.442 GeV/c. It is also useful to note that Ecm dEcm = m2 dE1 lab = m2 β1 lab dplab . (47.7) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019 update December 6, 2019 12:03 2 47. Kinematics 47.3. Lorentz-invariant amplitudes The matrix elements for a scattering or decay process are written in terms of an invariant amplitude iM . As an example, the S-matrix for 2 2 scattering is related to M by − → p′ p′ S p p = I i(2π)4 δ4(p + p p′ p′ ) h 1 2 | | 1 2i − 1 2 − 1 − 2 M (p , p ; p′ , p′ ) 1 2 1 2 . (47.8) 1/2 1/2 ′ 1/2 ′ 1/2 × (2E1) (2E2) (2E1) (2E2) The state normalization is such that p′ p = (2π)3δ3(p p′) . (47.9) h | i − For a 2 2 scattering process producing unstable particles 1′ and 2′ decaying via 1′ 3′4′→and 2′ 5′6′ the matrix element for the complete process can be written in the narrow→ width approximation→ as: M ′ ′ M ′ ′ ′ M ′ ′ ′ M ′ ′ ′ ′ (12 1 2 ) (1 3 4 ) (2 5 6 ) (12 3 4 5 6 ) = 2 →2 → 2 2 → . (47.10) → (m ′ ′ m ′ + im1′ Γ1′ )(m ′ ′ m ′ + im2′ Γ2′ ) h1X′,h2′ 3 4 − 1 5 6 − 2 Here, mij is the invariant mass of particles i and j, mk and Γk are the mass and total width of particle k, and the sum runs over the helicities of the intermediate particles. This enables the cross section for such a process to be written as the product of the cross section for the initial 2 2 scattering process with the branching ratios (relative partial decay rates) of the subsequent→ decays. 47.4. Particle decays The partial decay rate of a particle of mass M into n bodies in its rest frame is given in terms of the Lorentz-invariant matrix element M by 4 (2π) 2 dΓ = M dΦn (P ; p ,...,pn), (47.11) 2M | | 1 where dΦn is an element of n-body phase space given by n n 3 4 d pi dΦn(P ; p ,...,pn) = δ (P p ) . (47.12) 1 − i (2π)32E Xi=1 iY=1 i This phase space can be generated recursively, viz. dΦn(P ; p1,...,pn) = dΦj(q; p1,...,pj) 3 2 dΦ (P ; q, p ,...,pn)(2π) dq , (47.13) × n−j+1 j+1 2 2 j 2 j where q =( i=1 Ei) i=1 pi . This form is particularly useful in the case where a − particle decaysP into another P particle that subsequently decays. December 6, 2019 12:03 47. Kinematics 3 47.4.1. Survival probability : If a particle of mass M has mean proper lifetime τ (= 1/Γ) and has momentum (E, p), then the probability that it lives for a time t0 or greater before decaying is given by −t0 Γ/γ −Mt0 Γ/E P (t0) = e = e , (47.14) and the probability that it travels a distance x0 or greater is −Mx0 Γ/|p| P (x0) = e . (47.15) 47.4.2. Two-body decays : p1, m1 P, M p2, m2 Figure 47.1: Definitions of variables for two-body decays. In the rest frame of a particle of mass M, decaying into 2 particles labeled 1 and 2, M 2 m2 + m2 E = − 2 1 , (47.16) 1 2M p = p | 1| | 2| 2 2 2 2 1/2 M (m1 + m2) M (m1 m2) = − − − , (47.17) 2M and 1 p dΓ = M 2 | 1| dΩ , (47.18) 32π2 | | M 2 where dΩ = dφ1d(cos θ1) is the solid angle of particle 1. The invariant mass M can be determined from the energies and momenta using Eq. (47.2) with M = Ecm. 47.4.3. Three-body decays : December 6, 2019 12:03 4 47. Kinematics p1, m1 P, M p2, m2 p3, m3 Figure 47.2: Definitions of variables for three-body decays. 2 2 2 2 2 2 2 2 2 Defining pij = pi + pj and mij = pij, then m12 + m23 + m13 = M + m1 + m2 + m3 2 2 2 2 and m12 = (P p3) = M + m3 2ME3, where E3 is the energy of particle 3 in the rest frame of M−. In that frame, the− momenta of the three decay particles lie in a plane. The relative orientation of these three momenta is fixed if their energies are known. The momenta can therefore be specified in space by giving three Euler angles (α,β,γ) that specify the orientation of the final system relative to the initial particle. The direction of any one of the particles relative to the frame in which the initial particle is described can be specified in space by two angles (α, β) while a third angle, γ, can be set as the azimuthal angle of a second particle around the first [1]. Then 1 1 dΓ = M 2 dE dE dα d(cos β) dγ . (47.19) (2π)5 16M | | 1 3 Alternatively 1 1 dΓ = M 2 p∗ p dm dΩ∗ dΩ , (47.20) (2π)5 16M 2 | | | 1|| 3| 12 1 3 ∗ ∗ where ( p1 , Ω1) is the momentum of particle 1 in the rest frame of 1 and 2, and Ω3 is the angle of| particle| 3 in the rest frame of the decaying particle. p∗ and p are given by | 1| | 3| 2 2 2 2 1/2 ∗ m12 (m1 + m2) m12 (m1 m2) p1 = − − − , (47.21a) | | 2m12 and 2 2 2 2 1/2 M (m12 + m3) M (m12 m3) p = − − − . (47.21b) | 3| 2M [Compare with Eq. (47.17).] If the decaying particle is a scalar or we average over its spin states, then integration over the angles in Eq. (47.19) gives 1 1 dΓ = M 2 dE dE (2π)3 8M | | 1 3 1 1 = M 2 dm2 dm2 . (47.22) (2π)3 32M 3 | | 12 23 This is the standard form for the Dalitz plot. December 6, 2019 12:03 47. Kinematics 5 2 2 47.4.3.1. Dalitz plot: For a given value of m12, the range of m23 is determined by its values when p2 is parallel or antiparallel to p3: 2 (m23)max = 2 ∗ ∗ 2 ∗2 2 ∗2 2 (E2 + E3) E2 m2 E3 m3 , (47.23a) − q − − q − 2 (m23)min = 2 ∗ ∗ 2 ∗2 2 ∗2 2 (E2 + E3) E2 m2 + E3 m3 . (47.23b) − q − q − ∗ 2 2 2 ∗ 2 2 2 Here E2 = (m12 m1 + m2)/2m12 and E3 = (M m12 m3)/2m12 are the energies − − − 2 2 of particles 2 and 3 in the m12 rest frame. The scatter plot in m12 and m23 is called a Dalitz plot. If M 2 is constant, the allowed region of the plot will be uniformly populated with events [see| Eq.| (47.22)]. A nonuniformity in the plot gives immediate information on M 2. For example, in the case of D Kππ, bands appear when m = m ∗ , | | → (Kπ) K (892) reflecting the appearance of the decay chain D K∗(892)π Kππ. → → 10 + 2 (m1 m2) M−m 2 8 ( 1) ) m2 2 ( ) 6 23 max V − 2 (M m3) (Ge 2 23 4 m 2 (m23)min + 2 2 (m2 m3) 0 0 1 2 3 4 5 2 2 m12 (GeV ) Figure 47.3: Dalitz plot for a three-body final state.
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