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Relativistic Kinematics of Particle Interactions Introduction

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Relativistic Kinematics of Particle Interactions Introduction le kin rel.tex Relativistic Kinematics of Particle Interactions byW von Schlipp e, March2002 1. Notation; 4-vectors, covariant and contravariant comp onents, metric tensor, invariants. 2. Lorentz transformation; frequently used reference frames: Lab frame, centre-of-mass frame; Minkowski metric, rapidity. 3. Two-b o dy decays. 4. Three-b o dy decays. 5. Particle collisions. 6. Elastic collisions. 7. Inelastic collisions: quasi-elastic collisions, particle creation. 8. Deep inelastic scattering. 9. Phase space integrals. Intro duction These notes are intended to provide a summary of the essentials of relativistic kinematics of particle reactions. A basic familiarity with the sp ecial theory of relativity is assumed. Most derivations are omitted: it is assumed that the interested reader will b e able to verify the results, which usually requires no more than elementary algebra. Only the phase space calculations are done in some detail since we recognise that they are frequently a bit of a struggle. For a deep er study of this sub ject the reader should consult the monograph on particle kinematics byByckling and Ka jantie. Section 1 sets the scene with an intro duction of the notation used here. Although other notations and conventions are used elsewhere, I present only one version which I b elieveto b e the one most frequently encountered in the literature on particle physics, notably in such widely used textb o oks as Relativistic Quantum Mechanics by Bjorken and Drell and in the b o oks listed in the bibliography. This is followed in section 2 by a brief discussion of the Lorentz transformation. This can b e dealt with fairly brie y b ecause Lorentz transformations play no ma jor part in the kind of calculations characteristic for the rest of this b o ok: central to most relativistic calculations is the notion of Lorentz invariants. This is so b ecause all observables can b e expressed in terms of invariants. The transformations b etween di erent reference frames are usually straight forward for the invariants of the problem without involving the use of the Lorentz transformation formul. In the following sections the kinematics of particle reactions is develop ed b eginning with the simplest case, that of two-b o dy decays of unstable particles, and culminating in multiparticle kinematics which is of paramount imp ortance for the study of particle reactions at mo dern high energy accelerators and colliders. The notes conclude with a fairly detailed discussion of phase space calculations. The most exhaustive treatment of the sub ject of relativistic kinematics is given in the monograph byByckling and Ka jantie [1 ]. The notation and other conventions used here are 1 those of [2 ]. A summary of formulas of relativistic kinematics can b e found in the Particle Data Tables [3], 1. Notation and units 1 The covariantcomponents of 4-vectors are denoted by sup erscript Greek letters , etc. Thus we write x , =0; 1; 2; 3, such that 0 1 2 3 x = ct; x = x; x = y; x = z The contravariantcomponents of 4-vectors are lab eled by subscripts: 0 x = x ;x = x; x = y; x = z 0 1 2 3 The scalar pro duct of two 4-vectors is given by 0 0 1 1 2 2 3 3 x y = x y = x y x y x y x y where the notation x y implies summation over . The minus signs in the scalar pro duct signify that the 4-vector space is not Euclidian. This is also expressed byintro ducing the metric tensor g with diagonal elements (1; 1; 1; 1) and zero o -diagonal elements: g = diag(1; 1; 1; 1) The contravariantelements of the metric tensor coincide with its covariant elements, i.e. we havealsog = diag(1; 1; 1; 1) Using the summation convention one can checkby explicit calculation that x = g x and x = g x This pro cedure is referred to as the lowering and raising of the indices. Using the metric tensor one can express the scalar pro duct in the following equivalent forms: x y = g x y = g x y The invariant square of a 4-vector is given by 2 x = x x 2 2 2 If x > 0 the 4-vector is said to b e time-like,ifx =0 itis light-like, and if x < 0itis space-like. In the following we shall use units de ned by c = 1, where c is the sp eed of light. This is convenient in the kind of calculations characteristic of relativistic kinematics, b ecause all expressions must then b e homogeneous in energies, momenta and masses, which all havethe same dimension. Velo cities do not o ccur very often in these calculations, but one must remember that particle velo cities are dimensionless and do not exceed 1. Thus the relativistic factor of a particle of velo city v is written as 1=2 2 = 1 v (1) The units of energy, momentum and mass most frequently used in elementary particle 6 9 physics are the Mega-electron Volt(1MeV=10 eV), Giga-electron Volt (1 GeV = 10 eV) 1 c.f. P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edition, OUP, 1958, p. 254 2 12 and Tera-electron Volt (1 TeV = 10 eV). One should rememb er that the electron mass is approximately 0.5 MeV, the proton mass is nearly 1 GeV (a more accurate value is 0.94 GeV), and the highest energy accelerators in op eration accelerate protons to ab out 1 TeV. 2. Lorentz transformation. The Lorentz transformation b etween two inertial frames is of the form 0 x ! x = a x (2) The invariance of the scalar pro duct of two4-vectors implies that a a = (3) where is the Kronecker symb ol. 0 If the axes of the two frames coincide at time t = t = 0 and the frames move with relative velo city ~v ,then 00 0 x = (x + vx ) jj 0 0 x = (x + vx ) (4) jj jj 0 ~x = ~x ? ? 2 where v = j~v j, x is the comp onentof x in the direction of ~v , i.e. the mo dulus of ~x =(~x ~v )~v=v , jj jj 2 1=2 ~x = ~x ~x and =(1 v ) .Thus in this particular case the Lorentz transformation is ? jj de ned by three parameters, namely the three comp onents of the velo city ~v . If in addition the two frames are rotated with resp ect to each other, then the transfor- mation matrix dep ends also on the three Euler angles. Therefore the most general Lorentz transformation dep ends on six parameters. The inverse Lorentz transformation from the primed to the unprimed frame is obtained by replacing in Eq. (4) the primed by the unprimed co ordinates and vice versa, and changing the sign of v , i.e. 0 00 0 0 00 0 x = (x vx ) x = (x vx ) ~x = ~x (5) ? jj jj jj ? Energy-momentum 4-vector. Like time, so also energy is not invariant in relativity, but rather transforms under Lorentz transformation like the zeroth comp onentofa 4-vector, the other three comp onents b eing the ordinary 3-vector momentum. Let us denote the energy-momentum 4-vector of a particle of mass m byp, i.e. 0 p=(p ;~p) (6) 0 Frequently we shall denote the zeroth comp onent p , i.e. the energy,by E .Thus the invariant square of p is 2 2 2 2 p = p p = E p~ = m (7) The velo city ~v of the particle is de ned by ~p = m~v (8) From Eqs. (1), (7) and (8) we nd the useful relations = E=m (9) 3 and ~v = p=E~ (10) The energy-momentum 4-vector p transforms like the space-time 4-vector x (c.f. Eq. (4)): 00 0 p = (p + vp ) jj 0 0 p = (p + vp ) (11) jj jj 0 p~ = p~ ? ? Minkowski metric; rapidity. An alternative description of the Lorentz transformation (4) is based on the Minkowski trickofintro ducing an imaginary time x = it. There is now no need to havetwo kinds of 4 vectors, such as the contravariantandcovariantvectors considered ab ove. Instead we denote the 4-vector comp onents of space-time by x = x; x = y; x = z; x = it: 1 2 3 4 and the Lorentz invariant square b ecomes 2 2 2 2 x + x + x + x =invariant: 1 2 3 4 The transformation, which leaves this expression invariant, is a rotation. Thus, for instance, a rotation in the (x ;x ) plane, leaving x and x unchanged, is of the form 1 4 2 3 0 0 x = x cos x sin ; x = x sin + x cos 1 4 1 4 1 4 Of course, no mathematical trick gets us o the ho ok: nature demands space-time to b e non-Euclidian. Going back from the world of an imaginary time to the real world wemust write our rotation as 0 0 x = x cos it sin ; it = x sin + it cos and using the identities i sin =sinhi ; cos =coshi and setting y = i we recover a completely real form of the Lorentz transformation, viz 0 0 x = x cosh y + t sinh y; t = x sinh y + t cosh y For this to b e equivalent with Eq. (4) wemust demand that y b e real, and hence is imaginary.Thus the price to pay for a familiar Euclidian form of the rotation in the (x ;x ) 1 4 plane is an imaginary angle of rotation.
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