rpp2010-rev-kinematics - 39 Kinematics 1 39 KINEMATICS...

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39. Kinematics 1 39. KINEMATICS Revised January 2000 by J.D. Jackson (LBNL) and June 2008 by D.R. Tovey (Sheffield). 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. 39.1. Lorentz transformations The energy E and 3-momentum p of a particle of mass m form a 4-vector p = ( E, p ) whose square p 2 E 2 − | p | 2 = m 2 . The velocity of the particle is β = p /E . The energy and momentum ( E , p ) viewed from a frame moving with velocity β f are given by E p = γ f γ f β f γ f β f γ f E p , p T = p T , (39 . 1) where γ f = (1 β 2 f ) 1 / 2 and p T ( p ) are the components of p perpendicular (parallel) to β f . 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 p 1 · p 2 = E 1 E 2 p 1 · p 2 is invariant (frame independent). 39.2. Center-of-mass energy and momentum In the collision of two particles of masses m 1 and m 2 the total center-of-mass energy can be expressed in the Lorentz-invariant form E cm = ( E 1 + E 2 ) 2 ( p 1 + p 2 ) 2 1 / 2 , = m 2 1 + m 2 2 + 2 E 1 E 2 (1 β 1 β 2 cos θ ) 1 / 2 , (39 . 2) where θ is the angle between the particles. In the frame where one particle (of mass m 2 ) is at rest (lab frame), E cm = ( m 2 1 + m 2 2 + 2 E 1 lab m 2 ) 1 / 2 . (39 . 3) The velocity of the center-of-mass in the lab frame is β cm = p lab / ( E 1 lab + m 2 ) , (39 . 4) where p lab p 1 lab and γ cm = ( E 1 lab + m 2 ) /E cm . (39 . 5) The c.m. momenta of particles 1 and 2 are of magnitude p cm = p lab m 2 E cm . (39 . 6) 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 E cm dE cm = m 2 dE 1 lab = m 2 β 1 lab dp lab . (39 . 7) K. Nakamura et al. , JPG 37 , 075021 (2010) (http://pdg.lbl.gov) July 30, 2010 14:36
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2 39. Kinematics 39.3. Lorentz-invariant amplitudes The matrix elements for a scattering or decay process are written in terms of an invariant amplitude i M . As an example, the S -matrix for 2 2 scattering is related to M by p 1 p 2 | S | p 1 p 2 = I i (2 π ) 4 δ 4 ( p 1 + p 2 p 1 p 2 ) × M ( p 1 , p 2 ; p 1 , p 2 ) (2 E 1 ) 1 / 2 (2 E 2 ) 1 / 2 (2 E 1 ) 1 / 2 (2 E 2 ) 1 / 2 . (39 . 8) The state normalization is such that p | p = (2 π ) 3 δ 3 ( p p ) . (39 . 9) 39.4. 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 d Γ = (2 π ) 4 2 M | M | 2 d Φ n ( P ; p 1 , . . . , p n ) , (39 . 10) where d Φ n is an element of n -body phase space given by d Φ n ( P ; p 1 , . . . , p n ) = δ 4 ( P n i =1 p i ) n i =1 d 3 p i (2 π ) 3 2 E i . (39 . 11) This phase space can be generated recursively, viz. d Φ n ( P ; p 1 , . . . , p n ) = d Φ j ( q ; p 1 , . . ., p j ) × d Φ n j +1 ( P ; q, p j +1 , . . ., p n )(2 π ) 3 dq 2 , (39 . 12) where q 2 = ( j i =1 E i ) 2 j i =1 p i 2 . This form is particularly useful in the case where a particle decays into another particle that subsequently decays.
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